LMFDB - Embedded newform 448.2.ba.b.401.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,2,Mod(81,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 9, 8])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.81"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.ba (of order $12$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.57729801055$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			401.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	448.401
Dual form			448.2.ba.b.305.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.133975 - 0.500000i) q^{3} +(0.232051 + 0.866025i) q^{5} +(1.73205 - 2.00000i) q^{7} +(2.36603 + 1.36603i) q^{9} +(-2.86603 - 0.767949i) q^{11} +(3.73205 + 3.73205i) q^{13} +0.464102 q^{15} +(-3.23205 - 5.59808i) q^{17} +(2.86603 - 0.767949i) q^{19} +(-0.767949 - 1.13397i) q^{21} +(3.86603 + 2.23205i) q^{23} +(3.63397 - 2.09808i) q^{25} +(2.09808 - 2.09808i) q^{27} +(-0.267949 - 0.267949i) q^{29} +(1.86603 + 3.23205i) q^{31} +(-0.767949 + 1.33013i) q^{33} +(2.13397 + 1.03590i) q^{35} +(0.303848 + 1.13397i) q^{37} +(2.36603 - 1.36603i) q^{39} -4.92820i q^{41} +(-6.46410 + 6.46410i) q^{43} +(-0.633975 + 2.36603i) q^{45} +(2.13397 - 3.69615i) q^{47} +(-1.00000 - 6.92820i) q^{49} +(-3.23205 + 0.866025i) q^{51} +(3.96410 + 1.06218i) q^{53} -2.66025i q^{55} -1.53590i q^{57} +(-11.3301 - 3.03590i) q^{59} +(-6.96410 + 1.86603i) q^{61} +(6.83013 - 2.36603i) q^{63} +(-2.36603 + 4.09808i) q^{65} +(-1.33013 + 4.96410i) q^{67} +(1.63397 - 1.63397i) q^{69} -0.535898i q^{71} +(-6.23205 + 3.59808i) q^{73} +(-0.562178 - 2.09808i) q^{75} +(-6.50000 + 4.40192i) q^{77} +(-8.33013 + 14.4282i) q^{79} +(3.33013 + 5.76795i) q^{81} +(-1.53590 - 1.53590i) q^{83} +(4.09808 - 4.09808i) q^{85} +(-0.169873 + 0.0980762i) q^{87} +(4.50000 + 2.59808i) q^{89} +(13.9282 - 1.00000i) q^{91} +(1.86603 - 0.500000i) q^{93} +(1.33013 + 2.30385i) q^{95} -2.92820 q^{97} +(-5.73205 - 5.73205i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 6 q^{5} + 6 q^{9} - 8 q^{11} + 8 q^{13} - 12 q^{15} - 6 q^{17} + 8 q^{19} - 10 q^{21} + 12 q^{23} + 18 q^{25} - 2 q^{27} - 8 q^{29} + 4 q^{31} - 10 q^{33} + 12 q^{35} + 22 q^{37} + 6 q^{39}+ \cdots - 16 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 6 * q^5 + 6 * q^9 - 8 * q^11 + 8 * q^13 - 12 * q^15 - 6 * q^17 + 8 * q^19 - 10 * q^21 + 12 * q^23 + 18 * q^25 - 2 * q^27 - 8 * q^29 + 4 * q^31 - 10 * q^33 + 12 * q^35 + 22 * q^37 + 6 * q^39 - 12 * q^43 - 6 * q^45 + 12 * q^47 - 4 * q^49 - 6 * q^51 + 2 * q^53 - 28 * q^59 - 14 * q^61 + 10 * q^63 - 6 * q^65 + 12 * q^67 + 10 * q^69 - 18 * q^73 + 22 * q^75 - 26 * q^77 - 16 * q^79 - 4 * q^81 - 20 * q^83 + 6 * q^85 - 18 * q^87 + 18 * q^89 + 28 * q^91 + 4 * q^93 - 12 * q^95 + 16 * q^97 - 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	0.133975	−	0.500000i	0.0773503	−	0.288675i	−0.916406	−	0.400251i	$-0.868923\pi$
$3$	0.133975	−	0.500000i	0.0773503	−	0.288675i	0.993756	+	0.111576i	$0.0355897\pi$
$4$		0			0
$5$	0.232051	+	0.866025i	0.103776	+	0.387298i	0.998203	−	0.0599153i	$-0.0190830\pi$
$5$	0.232051	+	0.866025i	0.103776	+	0.387298i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$8$		0			0
$9$	2.36603	+	1.36603i	0.788675	+	0.455342i
$10$		0			0
$11$	−2.86603	−	0.767949i	−0.864139	−	0.231545i	−0.200587	−	0.979676i	$-0.564285\pi$
$11$	−2.86603	−	0.767949i	−0.864139	−	0.231545i	−0.663552	+	0.748130i	$0.730952\pi$
$12$		0			0
$13$	3.73205	+	3.73205i	1.03508	+	1.03508i	0.999362	+	0.0357229i	$0.0113734\pi$
$13$	3.73205	+	3.73205i	1.03508	+	1.03508i	0.0357229	+	0.999362i	$0.488627\pi$
$14$		0			0
$15$	0.464102			0.119831
$16$		0			0
$17$	−3.23205	−	5.59808i	−0.783887	−	1.35773i	−0.929661	−	0.368415i	$-0.879901\pi$
$17$	−3.23205	−	5.59808i	−0.783887	−	1.35773i	0.145774	−	0.989318i	$-0.453433\pi$
$18$		0			0
$19$	2.86603	−	0.767949i	0.657511	−	0.176180i	0.0853887	−	0.996348i	$-0.472787\pi$
$19$	2.86603	−	0.767949i	0.657511	−	0.176180i	0.572123	+	0.820168i	$0.306120\pi$
$20$		0			0
$21$	−0.767949	−	1.13397i	−0.167580	−	0.247454i
$22$		0			0
$23$	3.86603	+	2.23205i	0.806122	+	0.465415i	0.845607	−	0.533805i	$-0.179239\pi$
$23$	3.86603	+	2.23205i	0.806122	+	0.465415i	−0.0394853	+	0.999220i	$0.512572\pi$
$24$		0			0
$25$	3.63397	−	2.09808i	0.726795	−	0.419615i
$26$		0			0
$27$	2.09808	−	2.09808i	0.403775	−	0.403775i
$28$		0			0
$29$	−0.267949	−	0.267949i	−0.0497569	−	0.0497569i	0.681791	−	0.731547i	$-0.261202\pi$
$29$	−0.267949	−	0.267949i	−0.0497569	−	0.0497569i	−0.731547	+	0.681791i	$0.761202\pi$
$30$		0			0
$31$	1.86603	+	3.23205i	0.335148	+	0.580493i	0.983513	−	0.180836i	$-0.0578803\pi$
$31$	1.86603	+	3.23205i	0.335148	+	0.580493i	−0.648365	+	0.761329i	$0.724547\pi$
$32$		0			0
$33$	−0.767949	+	1.33013i	−0.133683	+	0.231545i
$34$		0			0
$35$	2.13397	+	1.03590i	0.360708	+	0.175099i
$36$		0			0
$37$	0.303848	+	1.13397i	0.0499522	+	0.186424i	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	0.303848	+	1.13397i	0.0499522	+	0.186424i	−0.936442	+	0.350823i	$0.885902\pi$
$38$		0			0
$39$	2.36603	−	1.36603i	0.378867	−	0.218739i
$40$		0			0
$41$		−	4.92820i		−	0.769656i	−0.922988	−	0.384828i	$-0.874261\pi$
$41$		−	4.92820i		−	0.769656i	0.922988	−	0.384828i	$-0.125739\pi$
$42$		0			0
$43$	−6.46410	+	6.46410i	−0.985766	+	0.985766i	−0.999900	−	0.0141339i	$-0.995501\pi$
$43$	−6.46410	+	6.46410i	−0.985766	+	0.985766i	0.0141339	+	0.999900i	$0.495501\pi$
$44$		0			0
$45$	−0.633975	+	2.36603i	−0.0945074	+	0.352706i
$46$		0			0
$47$	2.13397	−	3.69615i	0.311272	−	0.539139i	−0.667366	−	0.744730i	$-0.732578\pi$
$47$	2.13397	−	3.69615i	0.311272	−	0.539139i	0.978638	+	0.205591i	$0.0659116\pi$
$48$		0			0
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$		0			0
$51$	−3.23205	+	0.866025i	−0.452578	+	0.121268i
$52$		0			0
$53$	3.96410	+	1.06218i	0.544511	+	0.145901i	0.520581	−	0.853812i	$-0.325715\pi$
$53$	3.96410	+	1.06218i	0.544511	+	0.145901i	0.0239302	+	0.999714i	$0.492382\pi$
$54$		0			0
$55$		−	2.66025i		−	0.358709i
$56$		0			0
$57$		−	1.53590i		−	0.203435i
$58$		0			0
$59$	−11.3301	−	3.03590i	−1.47506	−	0.395240i	−0.570395	−	0.821370i	$-0.693210\pi$
$59$	−11.3301	−	3.03590i	−1.47506	−	0.395240i	−0.904662	+	0.426130i	$0.859877\pi$
$60$		0			0
$61$	−6.96410	+	1.86603i	−0.891662	+	0.238920i	−0.675432	−	0.737422i	$-0.736043\pi$
$61$	−6.96410	+	1.86603i	−0.891662	+	0.238920i	−0.216230	+	0.976342i	$0.569376\pi$
$62$		0			0
$63$	6.83013	−	2.36603i	0.860515	−	0.298091i
$64$		0			0
$65$	−2.36603	+	4.09808i	−0.293469	+	0.508304i
$66$		0			0
$67$	−1.33013	+	4.96410i	−0.162501	+	0.606462i	0.835845	+	0.548966i	$0.184978\pi$
$67$	−1.33013	+	4.96410i	−0.162501	+	0.606462i	−0.998346	+	0.0574958i	$0.981688\pi$
$68$		0			0
$69$	1.63397	−	1.63397i	0.196707	−	0.196707i
$70$		0			0
$71$		−	0.535898i		−	0.0635994i	−0.999494	−	0.0317997i	$-0.989876\pi$
$71$		−	0.535898i		−	0.0635994i	0.999494	−	0.0317997i	$-0.0101239\pi$
$72$		0			0
$73$	−6.23205	+	3.59808i	−0.729406	+	0.421123i	−0.818205	−	0.574927i	$-0.805031\pi$
$73$	−6.23205	+	3.59808i	−0.729406	+	0.421123i	0.0887986	+	0.996050i	$0.471697\pi$
$74$		0			0
$75$	−0.562178	−	2.09808i	−0.0649147	−	0.242265i
$76$		0			0
$77$	−6.50000	+	4.40192i	−0.740744	+	0.501646i
$78$		0			0
$79$	−8.33013	+	14.4282i	−0.937213	+	1.62330i	−0.166572	+	0.986029i	$0.553270\pi$
$79$	−8.33013	+	14.4282i	−0.937213	+	1.62330i	−0.770640	+	0.637270i	$0.780063\pi$
$80$		0			0
$81$	3.33013	+	5.76795i	0.370014	+	0.640883i
$82$		0			0
$83$	−1.53590	−	1.53590i	−0.168587	−	0.168587i	0.617771	−	0.786358i	$-0.288036\pi$
$83$	−1.53590	−	1.53590i	−0.168587	−	0.168587i	−0.786358	+	0.617771i	$0.788036\pi$
$84$		0			0
$85$	4.09808	−	4.09808i	0.444499	−	0.444499i
$86$		0			0
$87$	−0.169873	+	0.0980762i	−0.0182123	+	0.0105149i
$88$		0			0
$89$	4.50000	+	2.59808i	0.476999	+	0.275396i	0.719165	−	0.694839i	$-0.244525\pi$
$89$	4.50000	+	2.59808i	0.476999	+	0.275396i	−0.242166	+	0.970235i	$0.577858\pi$
$90$		0			0
$91$	13.9282	−	1.00000i	1.46007	−	0.104828i
$92$		0			0
$93$	1.86603	−	0.500000i	0.193498	−	0.0518476i
$94$		0			0
$95$	1.33013	+	2.30385i	0.136468	+	0.236370i
$96$		0			0
$97$	−2.92820			−0.297314			−0.148657	−	0.988889i	$-0.547495\pi$
$97$	−2.92820			−0.297314			−0.148657	+	0.988889i	$0.547495\pi$
$98$		0			0
$99$	−5.73205	−	5.73205i	−0.576093	−	0.576093i
$100$		0			0
$101$	−18.8923	−	5.06218i	−1.87985	−	0.503706i	−0.999572	−	0.0292559i	$-0.990686\pi$
$101$	−18.8923	−	5.06218i	−1.87985	−	0.503706i	−0.880283	−	0.474450i	$-0.842647\pi$
$102$		0			0
$103$	−5.59808	−	3.23205i	−0.551595	−	0.318463i	0.198170	−	0.980168i	$-0.436500\pi$
$103$	−5.59808	−	3.23205i	−0.551595	−	0.318463i	−0.749765	+	0.661704i	$0.769833\pi$
$104$		0			0
$105$	0.803848	−	0.928203i	0.0784475	−	0.0905834i
$106$		0			0
$107$	0.866025	+	3.23205i	0.0837218	+	0.312454i	0.995069	−	0.0991843i	$-0.0316233\pi$
$107$	0.866025	+	3.23205i	0.0837218	+	0.312454i	−0.911347	+	0.411638i	$0.864957\pi$
$108$		0			0
$109$	0.571797	−	2.13397i	0.0547682	−	0.204398i	−0.933120	−	0.359565i	$-0.882925\pi$
$109$	0.571797	−	2.13397i	0.0547682	−	0.204398i	0.987888	+	0.155167i	$0.0495917\pi$
$110$		0			0
$111$	0.607695			0.0576799
$112$		0			0
$113$	−1.46410			−0.137731			−0.0688655	−	0.997626i	$-0.521938\pi$
$113$	−1.46410			−0.137731			−0.0688655	+	0.997626i	$0.521938\pi$
$114$		0			0
$115$	−1.03590	+	3.86603i	−0.0965980	+	0.360509i
$116$		0			0
$117$	3.73205	+	13.9282i	0.345028	+	1.28766i
$118$		0			0
$119$	−16.7942	−	3.23205i	−1.53952	−	0.296282i
$120$		0			0
$121$	−1.90192	−	1.09808i	−0.172902	−	0.0998251i
$122$		0			0
$123$	−2.46410	−	0.660254i	−0.222181	−	0.0595331i
$124$		0			0
$125$	5.83013	+	5.83013i	0.521462	+	0.521462i
$126$		0			0
$127$	−9.46410			−0.839803			−0.419902	−	0.907570i	$-0.637935\pi$
$127$	−9.46410			−0.839803			−0.419902	+	0.907570i	$0.637935\pi$
$128$		0			0
$129$	2.36603	+	4.09808i	0.208317	+	0.360815i
$130$		0			0
$131$	−7.59808	+	2.03590i	−0.663847	+	0.177877i	−0.574982	−	0.818166i	$-0.694991\pi$
$131$	−7.59808	+	2.03590i	−0.663847	+	0.177877i	−0.0888654	+	0.996044i	$0.528324\pi$
$132$		0			0
$133$	3.42820	−	7.06218i	0.297263	−	0.612368i
$134$		0			0
$135$	2.30385	+	1.33013i	0.198284	+	0.114479i
$136$		0			0
$137$	−15.2321	+	8.79423i	−1.30136	+	0.751342i	−0.980638	−	0.195831i	$-0.937260\pi$
$137$	−15.2321	+	8.79423i	−1.30136	+	0.751342i	−0.320724	+	0.947173i	$0.603926\pi$
$138$		0			0
$139$	11.9282	−	11.9282i	1.01174	−	1.01174i	0.0118067	−	0.999930i	$-0.496242\pi$
$139$	11.9282	−	11.9282i	1.01174	−	1.01174i	0.999930	−	0.0118067i	$-0.00375827\pi$
$140$		0			0
$141$	−1.56218	−	1.56218i	−0.131559	−	0.131559i
$142$		0			0
$143$	−7.83013	−	13.5622i	−0.654788	−	1.13413i
$144$		0			0
$145$	0.169873	−	0.294229i	0.0141072	−	0.0244344i
$146$		0			0
$147$	−3.59808	−	0.428203i	−0.296764	−	0.0353176i
$148$		0			0
$149$	2.03590	+	7.59808i	0.166787	+	0.622459i	0.997805	+	0.0662134i	$0.0210918\pi$
$149$	2.03590	+	7.59808i	0.166787	+	0.622459i	−0.831018	+	0.556245i	$0.812242\pi$
$150$		0			0
$151$	9.86603	−	5.69615i	0.802886	−	0.463546i	−0.0415935	−	0.999135i	$-0.513243\pi$
$151$	9.86603	−	5.69615i	0.802886	−	0.463546i	0.844479	+	0.535588i	$0.179910\pi$
$152$		0			0
$153$		−	17.6603i		−	1.42775i
$154$		0			0
$155$	−2.36603	+	2.36603i	−0.190044	+	0.190044i
$156$		0			0
$157$	4.89230	−	18.2583i	0.390448	−	1.45717i	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	4.89230	−	18.2583i	0.390448	−	1.45717i	0.829396	−	0.558661i	$-0.188685\pi$
$158$		0			0
$159$	1.06218	−	1.83975i	0.0842362	−	0.145901i
$160$		0			0
$161$	11.1603	−	3.86603i	0.879551	−	0.304685i
$162$		0			0
$163$	12.0622	−	3.23205i	0.944783	−	0.253154i	0.246636	−	0.969108i	$-0.420675\pi$
$163$	12.0622	−	3.23205i	0.944783	−	0.253154i	0.698147	+	0.715954i	$0.254008\pi$
$164$		0			0
$165$	−1.33013	−	0.356406i	−0.103550	−	0.0277462i
$166$		0			0
$167$			21.8564i			1.69130i	0.533738	+	0.845650i	$0.320787\pi$
$167$			21.8564i			1.69130i	−0.533738	+	0.845650i	$0.679213\pi$
$168$		0			0
$169$			14.8564i			1.14280i
$170$		0			0
$171$	7.83013	+	2.09808i	0.598785	+	0.160444i
$172$		0			0
$173$	−2.23205	+	0.598076i	−0.169700	+	0.0454709i	−0.342668	−	0.939456i	$-0.611331\pi$
$173$	−2.23205	+	0.598076i	−0.169700	+	0.0454709i	0.172969	+	0.984927i	$0.444664\pi$
$174$		0			0
$175$	2.09808	−	10.9019i	0.158600	−	0.824108i
$176$		0			0
$177$	−3.03590	+	5.25833i	−0.228192	+	0.395240i
$178$		0			0
$179$	2.40192	−	8.96410i	0.179528	−	0.670008i	−0.816208	−	0.577759i	$-0.803928\pi$
$179$	2.40192	−	8.96410i	0.179528	−	0.670008i	0.995736	−	0.0922498i	$-0.0294058\pi$
$180$		0			0
$181$	13.3923	−	13.3923i	0.995442	−	0.995442i	−0.00454748	−	0.999990i	$-0.501448\pi$
$181$	13.3923	−	13.3923i	0.995442	−	0.995442i	0.999990	+	0.00454748i	$0.00144751\pi$
$182$		0			0
$183$			3.73205i			0.275881i
$184$		0			0
$185$	−0.911543	+	0.526279i	−0.0670180	+	0.0386928i
$186$		0			0
$187$	4.96410	+	18.5263i	0.363011	+	1.35478i
$188$		0			0
$189$	−0.562178	−	7.83013i	−0.0408924	−	0.569558i
$190$		0			0
$191$	4.33013	−	7.50000i	0.313317	−	0.542681i	−0.665761	−	0.746165i	$-0.731893\pi$
$191$	4.33013	−	7.50000i	0.313317	−	0.542681i	0.979078	+	0.203484i	$0.0652264\pi$
$192$		0			0
$193$	11.5000	+	19.9186i	0.827788	+	1.43377i	0.899770	+	0.436365i	$0.143734\pi$
$193$	11.5000	+	19.9186i	0.827788	+	1.43377i	−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$	1.73205	+	1.73205i	0.124035	+	0.124035i
$196$		0			0
$197$	−16.6603	+	16.6603i	−1.18699	+	1.18699i	−0.209100	+	0.977894i	$0.567053\pi$
$197$	−16.6603	+	16.6603i	−1.18699	+	1.18699i	−0.977894	+	0.209100i	$0.932947\pi$
$198$		0			0
$199$	−10.3301	+	5.96410i	−0.732283	+	0.422784i	−0.819257	−	0.573427i	$-0.805614\pi$
$199$	−10.3301	+	5.96410i	−0.732283	+	0.422784i	0.0869736	+	0.996211i	$0.472280\pi$
$200$		0			0
$201$	2.30385	+	1.33013i	0.162501	+	0.0938199i
$202$		0			0
$203$	−1.00000	+	0.0717968i	−0.0701862	+	0.00503915i
$204$		0			0
$205$	4.26795	−	1.14359i	0.298087	−	0.0798720i
$206$		0			0
$207$	6.09808	+	10.5622i	0.423846	+	0.734122i
$208$		0			0
$209$	−8.80385			−0.608975
$210$		0			0
$211$	15.9282	+	15.9282i	1.09654	+	1.09654i	0.994812	+	0.101731i	$0.0324380\pi$
$211$	15.9282	+	15.9282i	1.09654	+	1.09654i	0.101731	+	0.994812i	$0.467562\pi$
$212$		0			0
$213$	−0.267949	−	0.0717968i	−0.0183596	−	0.00491943i
$214$		0			0
$215$	−7.09808	−	4.09808i	−0.484085	−	0.279486i
$216$		0			0
$217$	9.69615	+	1.86603i	0.658218	+	0.126674i
$218$		0			0
$219$	0.964102	+	3.59808i	0.0651479	+	0.243135i
$220$		0			0
$221$	8.83013	−	32.9545i	0.593979	−	2.21676i
$222$		0			0
$223$	−9.85641			−0.660034			−0.330017	−	0.943975i	$-0.607054\pi$
$223$	−9.85641			−0.660034			−0.330017	+	0.943975i	$0.607054\pi$
$224$		0			0
$225$	11.4641			0.764273
$226$		0			0
$227$	2.99038	−	11.1603i	0.198479	−	0.740732i	−0.792860	−	0.609403i	$-0.791409\pi$
$227$	2.99038	−	11.1603i	0.198479	−	0.740732i	0.991339	−	0.131329i	$-0.0419243\pi$
$228$		0			0
$229$	−3.57180	−	13.3301i	−0.236031	−	0.880880i	−0.977681	−	0.210093i	$-0.932623\pi$
$229$	−3.57180	−	13.3301i	−0.236031	−	0.880880i	0.741650	−	0.670787i	$-0.234043\pi$
$230$		0			0
$231$	1.33013	+	3.83975i	0.0875159	+	0.252637i
$232$		0			0
$233$	−0.696152	−	0.401924i	−0.0456065	−	0.0263309i	0.477023	−	0.878891i	$-0.341716\pi$
$233$	−0.696152	−	0.401924i	−0.0456065	−	0.0263309i	−0.522630	+	0.852560i	$0.675049\pi$
$234$		0			0
$235$	3.69615	+	0.990381i	0.241110	+	0.0646053i
$236$		0			0
$237$	6.09808	+	6.09808i	0.396113	+	0.396113i
$238$		0			0
$239$	8.53590			0.552141			0.276071	−	0.961137i	$-0.410968\pi$
$239$	8.53590			0.552141			0.276071	+	0.961137i	$0.410968\pi$
$240$		0			0
$241$	−6.96410	−	12.0622i	−0.448597	−	0.776993i	0.549698	−	0.835364i	$-0.314743\pi$
$241$	−6.96410	−	12.0622i	−0.448597	−	0.776993i	−0.998295	+	0.0583704i	$0.981410\pi$
$242$		0			0
$243$	11.9282	−	3.19615i	0.765195	−	0.205033i
$244$		0			0
$245$	5.76795	−	2.47372i	0.368501	−	0.158040i
$246$		0			0
$247$	13.5622	+	7.83013i	0.862941	+	0.498219i
$248$		0			0
$249$	−0.973721	+	0.562178i	−0.0617070	+	0.0356266i
$250$		0			0
$251$	17.5885	−	17.5885i	1.11017	−	1.11017i	0.117047	−	0.993126i	$-0.462657\pi$
$251$	17.5885	−	17.5885i	1.11017	−	1.11017i	0.993126	−	0.117047i	$-0.0373429\pi$
$252$		0			0
$253$	−9.36603	−	9.36603i	−0.588837	−	0.588837i
$254$		0			0
$255$	−1.50000	−	2.59808i	−0.0939336	−	0.162698i
$256$		0			0
$257$	9.69615	−	16.7942i	0.604829	−	1.04760i	−0.387249	−	0.921975i	$-0.626575\pi$
$257$	9.69615	−	16.7942i	0.604829	−	1.04760i	0.992078	−	0.125620i	$-0.0400920\pi$
$258$		0			0
$259$	2.79423	+	1.35641i	0.173625	+	0.0842830i
$260$		0			0
$261$	−0.267949	−	1.00000i	−0.0165856	−	0.0618984i
$262$		0			0
$263$	3.99038	−	2.30385i	0.246057	−	0.142061i	−0.371900	−	0.928273i	$-0.621294\pi$
$263$	3.99038	−	2.30385i	0.246057	−	0.142061i	0.617958	+	0.786211i	$0.287960\pi$
$264$		0			0
$265$			3.67949i			0.226029i
$266$		0			0
$267$	1.90192	−	1.90192i	0.116396	−	0.116396i
$268$		0			0
$269$	−2.62436	+	9.79423i	−0.160010	+	0.597165i	0.838614	+	0.544726i	$0.183366\pi$
$269$	−2.62436	+	9.79423i	−0.160010	+	0.597165i	−0.998624	+	0.0524390i	$0.983300\pi$
$270$		0			0
$271$	6.06218	−	10.5000i	0.368251	−	0.637830i	−0.621041	−	0.783778i	$-0.713290\pi$
$271$	6.06218	−	10.5000i	0.368251	−	0.637830i	0.989292	+	0.145948i	$0.0466233\pi$
$272$		0			0
$273$	1.36603	−	7.09808i	0.0826756	−	0.429595i
$274$		0			0
$275$	−12.0263	+	3.22243i	−0.725212	+	0.194320i
$276$		0			0
$277$	4.50000	+	1.20577i	0.270379	+	0.0724478i	0.391461	−	0.920195i	$-0.371970\pi$
$277$	4.50000	+	1.20577i	0.270379	+	0.0724478i	−0.121082	+	0.992642i	$0.538636\pi$
$278$		0			0
$279$			10.1962i			0.610428i
$280$		0			0
$281$		−	0.928203i		−	0.0553720i	−0.999617	−	0.0276860i	$-0.991186\pi$
$281$		−	0.928203i		−	0.0553720i	0.999617	−	0.0276860i	$-0.00881385\pi$
$282$		0			0
$283$	−14.5263	−	3.89230i	−0.863498	−	0.231374i	−0.200224	−	0.979750i	$-0.564167\pi$
$283$	−14.5263	−	3.89230i	−0.863498	−	0.231374i	−0.663274	+	0.748377i	$0.730834\pi$
$284$		0			0
$285$	1.33013	−	0.356406i	0.0787899	−	0.0211117i
$286$		0			0
$287$	−9.85641	−	8.53590i	−0.581805	−	0.503858i
$288$		0			0
$289$	−12.3923	+	21.4641i	−0.728959	+	1.26259i
$290$		0			0
$291$	−0.392305	+	1.46410i	−0.0229973	+	0.0858272i
$292$		0			0
$293$	−7.92820	+	7.92820i	−0.463171	+	0.463171i	−0.899693	−	0.436523i	$-0.856210\pi$
$293$	−7.92820	+	7.92820i	−0.463171	+	0.463171i	0.436523	+	0.899693i	$0.356210\pi$
$294$		0			0
$295$		−	10.5167i		−	0.612304i
$296$		0			0
$297$	−7.62436	+	4.40192i	−0.442410	+	0.255426i
$298$		0			0
$299$	6.09808	+	22.7583i	0.352661	+	1.31615i
$300$		0			0
$301$	1.73205	+	24.1244i	0.0998337	+	1.39050i
$302$		0			0
$303$	−5.06218	+	8.76795i	−0.290815	+	0.503706i
$304$		0			0
$305$	−3.23205	−	5.59808i	−0.185067	−	0.320545i
$306$		0			0
$307$	−9.00000	−	9.00000i	−0.513657	−	0.513657i	0.401988	−	0.915645i	$-0.368319\pi$
$307$	−9.00000	−	9.00000i	−0.513657	−	0.513657i	−0.915645	+	0.401988i	$0.868319\pi$
$308$		0			0
$309$	−2.36603	+	2.36603i	−0.134598	+	0.134598i
$310$		0			0
$311$	5.59808	−	3.23205i	0.317438	−	0.183273i	−0.332812	−	0.942993i	$-0.607998\pi$
$311$	5.59808	−	3.23205i	0.317438	−	0.183273i	0.650250	+	0.759720i	$0.274664\pi$
$312$		0			0
$313$	18.6962	+	10.7942i	1.05677	+	0.610126i	0.924537	−	0.381093i	$-0.124452\pi$
$313$	18.6962	+	10.7942i	1.05677	+	0.610126i	0.132232	+	0.991219i	$0.457786\pi$
$314$		0			0
$315$	3.63397	+	5.36603i	0.204751	+	0.302341i
$316$		0			0
$317$	14.1603	−	3.79423i	0.795319	−	0.213105i	0.161791	−	0.986825i	$-0.448273\pi$
$317$	14.1603	−	3.79423i	0.795319	−	0.213105i	0.633528	+	0.773720i	$0.281606\pi$
$318$		0			0
$319$	0.562178	+	0.973721i	0.0314759	+	0.0545179i
$320$		0			0
$321$	1.73205			0.0966736
$322$		0			0
$323$	−13.5622	−	13.5622i	−0.754620	−	0.754620i
$324$		0			0
$325$	21.3923	+	5.73205i	1.18663	+	0.317957i
$326$		0			0
$327$	−0.990381	−	0.571797i	−0.0547682	−	0.0316204i
$328$		0			0
$329$	−3.69615	−	10.6699i	−0.203775	−	0.588249i
$330$		0			0
$331$	−5.25833	−	19.6244i	−0.289024	−	1.07865i	−0.945848	−	0.324609i	$-0.894767\pi$
$331$	−5.25833	−	19.6244i	−0.289024	−	1.07865i	0.656824	−	0.754044i	$-0.271899\pi$
$332$		0			0
$333$	−0.830127	+	3.09808i	−0.0454907	+	0.169774i
$334$		0			0
$335$	−4.60770			−0.251745
$336$		0			0
$337$	−6.14359			−0.334663			−0.167331	−	0.985901i	$-0.553515\pi$
$337$	−6.14359			−0.334663			−0.167331	+	0.985901i	$0.553515\pi$
$338$		0			0
$339$	−0.196152	+	0.732051i	−0.0106535	+	0.0397595i
$340$		0			0
$341$	−2.86603	−	10.6962i	−0.155204	−	0.579229i
$342$		0			0
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$		0			0
$345$	1.79423	+	1.03590i	0.0965980	+	0.0557709i
$346$		0			0
$347$	−23.5263	−	6.30385i	−1.26296	−	0.338408i	−0.435628	−	0.900127i	$-0.643474\pi$
$347$	−23.5263	−	6.30385i	−1.26296	−	0.338408i	−0.827329	+	0.561718i	$0.810140\pi$
$348$		0			0
$349$	6.12436	+	6.12436i	0.327829	+	0.327829i	0.851761	−	0.523931i	$-0.175535\pi$
$349$	6.12436	+	6.12436i	0.327829	+	0.327829i	−0.523931	+	0.851761i	$0.675535\pi$
$350$		0			0
$351$	15.6603			0.835883
$352$		0			0
$353$	13.8923	+	24.0622i	0.739413	+	1.28070i	0.952760	+	0.303724i	$0.0982301\pi$
$353$	13.8923	+	24.0622i	0.739413	+	1.28070i	−0.213347	+	0.976976i	$0.568437\pi$
$354$		0			0
$355$	0.464102	−	0.124356i	0.0246320	−	0.00660011i
$356$		0			0
$357$	−3.86603	+	7.96410i	−0.204612	+	0.421505i
$358$		0			0
$359$	−3.86603	−	2.23205i	−0.204041	−	0.117803i	0.394498	−	0.918897i	$-0.370919\pi$
$359$	−3.86603	−	2.23205i	−0.204041	−	0.117803i	−0.598539	+	0.801094i	$0.704252\pi$
$360$		0			0
$361$	−8.83013	+	5.09808i	−0.464744	+	0.268320i
$362$		0			0
$363$	−0.803848	+	0.803848i	−0.0421911	+	0.0421911i
$364$		0			0
$365$	−4.56218	−	4.56218i	−0.238795	−	0.238795i
$366$		0			0
$367$	3.06218	+	5.30385i	0.159844	+	0.276859i	0.934812	−	0.355142i	$-0.115567\pi$
$367$	3.06218	+	5.30385i	0.159844	+	0.276859i	−0.774968	+	0.632000i	$0.782234\pi$
$368$		0			0
$369$	6.73205	−	11.6603i	0.350457	−	0.607009i
$370$		0			0
$371$	8.99038	−	6.08846i	0.466757	−	0.316097i
$372$		0			0
$373$	−0.232051	−	0.866025i	−0.0120151	−	0.0448411i	0.959658	−	0.281170i	$-0.0907224\pi$
$373$	−0.232051	−	0.866025i	−0.0120151	−	0.0448411i	−0.971673	+	0.236329i	$0.924056\pi$
$374$		0			0
$375$	3.69615	−	2.13397i	0.190868	−	0.110198i
$376$		0			0
$377$		−	2.00000i		−	0.103005i
$378$		0			0
$379$	−0.124356	+	0.124356i	−0.00638772	+	0.00638772i	−0.710293	−	0.703906i	$-0.751438\pi$
$379$	−0.124356	+	0.124356i	−0.00638772	+	0.00638772i	0.703906	+	0.710293i	$0.251438\pi$
$380$		0			0
$381$	−1.26795	+	4.73205i	−0.0649590	+	0.242430i
$382$		0			0
$383$	−12.5981	+	21.8205i	−0.643732	+	1.11498i	0.340861	+	0.940114i	$0.389281\pi$
$383$	−12.5981	+	21.8205i	−0.643732	+	1.11498i	−0.984593	+	0.174862i	$0.944052\pi$
$384$		0			0
$385$	−5.32051	−	4.60770i	−0.271158	−	0.234830i
$386$		0			0
$387$	−24.1244	+	6.46410i	−1.22631	+	0.328589i
$388$		0			0
$389$	−22.3564	−	5.99038i	−1.13351	−	0.303724i	−0.357174	−	0.934038i	$-0.616260\pi$
$389$	−22.3564	−	5.99038i	−1.13351	−	0.303724i	−0.776341	+	0.630313i	$0.782926\pi$
$390$		0			0
$391$		−	28.8564i		−	1.45933i
$392$		0			0
$393$			4.07180i			0.205395i
$394$		0			0
$395$	−14.4282	−	3.86603i	−0.725962	−	0.194521i
$396$		0			0
$397$	−25.6244	+	6.86603i	−1.28605	+	0.344596i	−0.836159	−	0.548488i	$-0.815204\pi$
$397$	−25.6244	+	6.86603i	−1.28605	+	0.344596i	−0.449891	+	0.893084i	$0.648537\pi$
$398$		0			0
$399$	−3.07180	−	2.66025i	−0.153782	−	0.133179i
$400$		0			0
$401$	7.50000	−	12.9904i	0.374532	−	0.648709i	−0.615725	−	0.787961i	$-0.711137\pi$
$401$	7.50000	−	12.9904i	0.374532	−	0.648709i	0.990257	+	0.139253i	$0.0444700\pi$
$402$		0			0
$403$	−5.09808	+	19.0263i	−0.253953	+	0.947766i
$404$		0			0
$405$	−4.22243	+	4.22243i	−0.209814	+	0.209814i
$406$		0			0
$407$		−	3.48334i		−	0.172663i
$408$		0			0
$409$	7.50000	−	4.33013i	0.370851	−	0.214111i	−0.302979	−	0.952997i	$-0.597981\pi$
$409$	7.50000	−	4.33013i	0.370851	−	0.214111i	0.673830	+	0.738886i	$0.264648\pi$
$410$		0			0
$411$	2.35641	+	8.79423i	0.116233	+	0.433787i
$412$		0			0
$413$	−25.6962	+	17.4019i	−1.26442	+	0.856293i
$414$		0			0
$415$	0.973721	−	1.68653i	0.0477981	−	0.0827887i
$416$		0			0
$417$	−4.36603	−	7.56218i	−0.213805	−	0.370321i
$418$		0			0
$419$	19.0000	+	19.0000i	0.928211	+	0.928211i	0.997590	−	0.0693796i	$-0.0221020\pi$
$419$	19.0000	+	19.0000i	0.928211	+	0.928211i	−0.0693796	+	0.997590i	$0.522102\pi$
$420$		0			0
$421$	8.66025	−	8.66025i	0.422075	−	0.422075i	−0.463843	−	0.885918i	$-0.653530\pi$
$421$	8.66025	−	8.66025i	0.422075	−	0.422075i	0.885918	+	0.463843i	$0.153530\pi$
$422$		0			0
$423$	10.0981	−	5.83013i	0.490985	−	0.283470i
$424$		0			0
$425$	−23.4904	−	13.5622i	−1.13945	−	0.657862i
$426$		0			0
$427$	−8.33013	+	17.1603i	−0.403123	+	0.830443i
$428$		0			0
$429$	−7.83013	+	2.09808i	−0.378042	+	0.101296i
$430$		0			0
$431$	15.3301	+	26.5526i	0.738426	+	1.27899i	0.953204	+	0.302329i	$0.0977640\pi$
$431$	15.3301	+	26.5526i	0.738426	+	1.27899i	−0.214777	+	0.976663i	$0.568903\pi$
$432$		0			0
$433$	−33.1769			−1.59438			−0.797190	−	0.603728i	$-0.793681\pi$
$433$	−33.1769			−1.59438			−0.797190	+	0.603728i	$0.793681\pi$
$434$		0			0
$435$	−0.124356	−	0.124356i	−0.00596240	−	0.00596240i
$436$		0			0
$437$	12.7942	+	3.42820i	0.612031	+	0.163993i
$438$		0			0
$439$	6.52628	+	3.76795i	0.311482	+	0.179834i	0.647590	−	0.761989i	$-0.275777\pi$
$439$	6.52628	+	3.76795i	0.311482	+	0.179834i	−0.336107	+	0.941824i	$0.609110\pi$
$440$		0			0
$441$	7.09808	−	17.7583i	0.338004	−	0.845635i
$442$		0			0
$443$	−2.79423	−	10.4282i	−0.132758	−	0.495459i	0.867239	−	0.497892i	$-0.165892\pi$
$443$	−2.79423	−	10.4282i	−0.132758	−	0.495459i	−0.999997	+	0.00243278i	$0.999226\pi$
$444$		0			0
$445$	−1.20577	+	4.50000i	−0.0571590	+	0.213320i
$446$		0			0
$447$	4.07180			0.192589
$448$		0			0
$449$	−23.3205			−1.10056			−0.550281	−	0.834979i	$-0.685480\pi$
$449$	−23.3205			−1.10056			−0.550281	+	0.834979i	$0.685480\pi$
$450$		0			0
$451$	−3.78461	+	14.1244i	−0.178210	+	0.665090i
$452$		0			0
$453$	−1.52628	−	5.69615i	−0.0717109	−	0.267629i
$454$		0			0
$455$	4.09808	+	11.8301i	0.192121	+	0.554605i
$456$		0			0
$457$	−16.2846	−	9.40192i	−0.761762	−	0.439803i	0.0681661	−	0.997674i	$-0.478285\pi$
$457$	−16.2846	−	9.40192i	−0.761762	−	0.439803i	−0.829928	+	0.557871i	$0.811619\pi$
$458$		0			0
$459$	−18.5263	−	4.96410i	−0.864733	−	0.231704i
$460$		0			0
$461$	18.6603	+	18.6603i	0.869095	+	0.869095i	0.992372	−	0.123278i	$-0.0393405\pi$
$461$	18.6603	+	18.6603i	0.869095	+	0.869095i	−0.123278	+	0.992372i	$0.539341\pi$
$462$		0			0
$463$	−2.14359			−0.0996212			−0.0498106	−	0.998759i	$-0.515862\pi$
$463$	−2.14359			−0.0996212			−0.0498106	+	0.998759i	$0.515862\pi$
$464$		0			0
$465$	0.866025	+	1.50000i	0.0401610	+	0.0695608i
$466$		0			0
$467$	−25.9904	+	6.96410i	−1.20269	+	0.322260i	−0.803890	−	0.594777i	$-0.797240\pi$
$467$	−25.9904	+	6.96410i	−1.20269	+	0.322260i	−0.398801	+	0.917038i	$0.630574\pi$
$468$		0			0
$469$	7.62436	+	11.2583i	0.352060	+	0.519861i
$470$		0			0
$471$	−8.47372	−	4.89230i	−0.390448	−	0.225426i
$472$		0			0
$473$	23.4904	−	13.5622i	1.08009	−	0.623590i
$474$		0			0
$475$	8.80385	−	8.80385i	0.403948	−	0.403948i
$476$		0			0
$477$	7.92820	+	7.92820i	0.363007	+	0.363007i
$478$		0			0
$479$	7.79423	+	13.5000i	0.356127	+	0.616831i	0.987310	−	0.158803i	$-0.0507636\pi$
$479$	7.79423	+	13.5000i	0.356127	+	0.616831i	−0.631183	+	0.775634i	$0.717430\pi$
$480$		0			0
$481$	−3.09808	+	5.36603i	−0.141260	+	0.244670i
$482$		0			0
$483$	−0.437822	−	6.09808i	−0.0199216	−	0.277472i
$484$		0			0
$485$	−0.679492	−	2.53590i	−0.0308541	−	0.115149i
$486$		0			0
$487$	10.6699	−	6.16025i	0.483498	−	0.279148i	−0.238375	−	0.971173i	$-0.576615\pi$
$487$	10.6699	−	6.16025i	0.483498	−	0.279148i	0.721873	+	0.692025i	$0.243281\pi$
$488$		0			0
$489$		−	6.46410i		−	0.292317i
$490$		0			0
$491$	3.58846	−	3.58846i	0.161945	−	0.161945i	−0.621483	−	0.783428i	$-0.713469\pi$
$491$	3.58846	−	3.58846i	0.161945	−	0.161945i	0.783428	+	0.621483i	$0.213469\pi$
$492$		0			0
$493$	−0.633975	+	2.36603i	−0.0285528	+	0.106560i
$494$		0			0
$495$	3.63397	−	6.29423i	0.163335	−	0.282905i
$496$		0			0
$497$	−1.07180	−	0.928203i	−0.0480767	−	0.0416356i
$498$		0			0
$499$	24.9904	−	6.69615i	1.11872	−	0.299761i	0.348356	−	0.937362i	$-0.386740\pi$
$499$	24.9904	−	6.69615i	1.11872	−	0.299761i	0.770367	+	0.637601i	$0.220073\pi$
$500$		0			0
$501$	10.9282	+	2.92820i	0.488236	+	0.130822i
$502$		0			0
$503$			31.8564i			1.42041i	0.703996	+	0.710203i	$0.251397\pi$
$503$			31.8564i			1.42041i	−0.703996	+	0.710203i	$0.748603\pi$
$504$		0			0
$505$		−	17.5359i		−	0.780337i
$506$		0			0
$507$	7.42820	+	1.99038i	0.329898	+	0.0883959i
$508$		0			0
$509$	−9.42820	+	2.52628i	−0.417898	+	0.111975i	−0.461640	−	0.887067i	$-0.652739\pi$
$509$	−9.42820	+	2.52628i	−0.417898	+	0.111975i	0.0437420	+	0.999043i	$0.486072\pi$
$510$		0			0
$511$	−3.59808	+	18.6962i	−0.159170	+	0.827069i
$512$		0			0
$513$	4.40192	−	7.62436i	0.194350	−	0.336624i
$514$		0			0
$515$	1.50000	−	5.59808i	0.0660979	−	0.246681i
$516$		0			0
$517$	−8.95448	+	8.95448i	−0.393818	+	0.393818i
$518$		0			0
$519$			1.19615i			0.0525053i
$520$		0			0
$521$	13.3756	−	7.72243i	0.585998	−	0.338326i	−0.177516	−	0.984118i	$-0.556806\pi$
$521$	13.3756	−	7.72243i	0.585998	−	0.338326i	0.763513	+	0.645792i	$0.223473\pi$
$522$		0			0
$523$	−8.65064	−	32.2846i	−0.378266	−	1.41171i	−0.848514	−	0.529173i	$-0.822502\pi$
$523$	−8.65064	−	32.2846i	−0.378266	−	1.41171i	0.470248	−	0.882534i	$-0.344164\pi$
$524$		0			0
$525$	−5.16987	−	2.50962i	−0.225632	−	0.109529i
$526$		0			0
$527$	12.0622	−	20.8923i	0.525437	−	0.910083i
$528$		0			0
$529$	−1.53590	−	2.66025i	−0.0667782	−	0.115663i
$530$		0			0
$531$	−22.6603	−	22.6603i	−0.983371	−	0.983371i
$532$		0			0
$533$	18.3923	−	18.3923i	0.796659	−	0.796659i
$534$		0			0
$535$	−2.59808	+	1.50000i	−0.112325	+	0.0648507i
$536$		0			0
$537$	−4.16025	−	2.40192i	−0.179528	−	0.103651i
$538$		0			0
$539$	−2.45448	+	20.6244i	−0.105722	+	0.888354i
$540$		0			0
$541$	−9.23205	+	2.47372i	−0.396917	+	0.106354i	−0.451756	−	0.892142i	$-0.649202\pi$
$541$	−9.23205	+	2.47372i	−0.396917	+	0.106354i	0.0548389	+	0.998495i	$0.482535\pi$
$542$		0			0
$543$	−4.90192	−	8.49038i	−0.210362	−	0.364357i
$544$		0			0
$545$	1.98076			0.0848465
$546$		0			0
$547$	23.0526	+	23.0526i	0.985656	+	0.985656i	0.999899	−	0.0142423i	$-0.00453363\pi$
$547$	23.0526	+	23.0526i	0.985656	+	0.985656i	−0.0142423	+	0.999899i	$0.504534\pi$
$548$		0			0
$549$	−19.0263	−	5.09808i	−0.812022	−	0.217581i
$550$		0			0
$551$	−0.973721	−	0.562178i	−0.0414819	−	0.0239496i
$552$		0			0
$553$	14.4282	+	41.6506i	0.613550	+	1.77117i
$554$		0			0
$555$	0.141016	+	0.526279i	0.00598580	+	0.0223393i
$556$		0			0
$557$	−1.76795	+	6.59808i	−0.0749104	+	0.279569i	−0.993213	−	0.116310i	$-0.962893\pi$
$557$	−1.76795	+	6.59808i	−0.0749104	+	0.279569i	0.918303	+	0.395879i	$0.129560\pi$
$558$		0			0
$559$	−48.2487			−2.04070
$560$		0			0
$561$	9.92820			0.419169
$562$		0			0
$563$	−0.650635	+	2.42820i	−0.0274210	+	0.102337i	−0.978280	−	0.207287i	$-0.933536\pi$
$563$	−0.650635	+	2.42820i	−0.0274210	+	0.102337i	0.950859	+	0.309624i	$0.100203\pi$
$564$		0			0
$565$	−0.339746	−	1.26795i	−0.0142932	−	0.0533430i
$566$		0			0
$567$	17.3038	+	3.33013i	0.726693	+	0.139852i
$568$		0			0
$569$	5.08846	+	2.93782i	0.213319	+	0.123160i	0.602853	−	0.797852i	$-0.294031\pi$
$569$	5.08846	+	2.93782i	0.213319	+	0.123160i	−0.389534	+	0.921012i	$0.627364\pi$
$570$		0			0
$571$	−26.7224	−	7.16025i	−1.11830	−	0.299647i	−0.348104	−	0.937456i	$-0.613174\pi$
$571$	−26.7224	−	7.16025i	−1.11830	−	0.299647i	−0.770195	+	0.637809i	$0.779841\pi$
$572$		0			0
$573$	−3.16987	−	3.16987i	−0.132423	−	0.132423i
$574$		0			0
$575$	18.7321			0.781181
$576$		0			0
$577$	−6.62436	−	11.4737i	−0.275776	−	0.477657i	0.694555	−	0.719440i	$-0.255601\pi$
$577$	−6.62436	−	11.4737i	−0.275776	−	0.477657i	−0.970330	+	0.241782i	$0.922268\pi$
$578$		0			0
$579$	11.5000	−	3.08142i	0.477924	−	0.128059i
$580$		0			0
$581$	−5.73205	+	0.411543i	−0.237806	+	0.0170737i
$582$		0			0
$583$	−10.5455	−	6.08846i	−0.436751	−	0.252158i
$584$		0			0
$585$	−11.1962	+	6.46410i	−0.462904	+	0.267258i
$586$		0			0
$587$	−21.9282	+	21.9282i	−0.905074	+	0.905074i	−0.995870	−	0.0907957i	$-0.971059\pi$
$587$	−21.9282	+	21.9282i	−0.905074	+	0.905074i	0.0907957	+	0.995870i	$0.471059\pi$
$588$		0			0
$589$	7.83013	+	7.83013i	0.322635	+	0.322635i
$590$		0			0
$591$	6.09808	+	10.5622i	0.250841	+	0.434470i
$592$		0			0
$593$	5.69615	−	9.86603i	0.233913	−	0.405149i	−0.725043	−	0.688703i	$-0.758180\pi$
$593$	5.69615	−	9.86603i	0.233913	−	0.405149i	0.958956	+	0.283554i	$0.0915136\pi$
$594$		0			0
$595$	−1.09808	−	15.2942i	−0.0450167	−	0.627002i
$596$		0			0
$597$	1.59808	+	5.96410i	0.0654049	+	0.244094i
$598$		0			0
$599$	16.6699	−	9.62436i	0.681113	−	0.393241i	−0.119162	−	0.992875i	$-0.538021\pi$
$599$	16.6699	−	9.62436i	0.681113	−	0.393241i	0.800274	+	0.599634i	$0.204687\pi$
$600$		0			0
$601$			10.0000i			0.407909i	0.978980	+	0.203954i	$0.0653794\pi$
$601$			10.0000i			0.407909i	−0.978980	+	0.203954i	$0.934621\pi$
$602$		0			0
$603$	−9.92820	+	9.92820i	−0.404308	+	0.404308i
$604$		0			0
$605$	0.509619	−	1.90192i	0.0207190	−	0.0773242i
$606$		0			0
$607$	8.52628	−	14.7679i	0.346071	−	0.599413i	−0.639477	−	0.768810i	$-0.720849\pi$
$607$	8.52628	−	14.7679i	0.346071	−	0.599413i	0.985548	+	0.169398i	$0.0541823\pi$
$608$		0			0
$609$	−0.0980762	+	0.509619i	−0.00397425	+	0.0206508i
$610$		0			0
$611$	21.7583	−	5.83013i	0.880248	−	0.235862i
$612$		0			0
$613$	−9.23205	−	2.47372i	−0.372879	−	0.0999126i	0.0675126	−	0.997718i	$-0.478494\pi$
$613$	−9.23205	−	2.47372i	−0.372879	−	0.0999126i	−0.440392	+	0.897806i	$0.645160\pi$
$614$		0			0
$615$		−	2.28719i		−	0.0922283i
$616$		0			0
$617$			0.535898i			0.0215745i	0.999942	+	0.0107872i	$0.00343375\pi$
$617$			0.535898i			0.0215745i	−0.999942	+	0.0107872i	$0.996566\pi$
$618$		0			0
$619$	19.0622	+	5.10770i	0.766174	+	0.205296i	0.620680	−	0.784064i	$-0.286856\pi$
$619$	19.0622	+	5.10770i	0.766174	+	0.205296i	0.145493	+	0.989359i	$0.453523\pi$
$620$		0			0
$621$	12.7942	−	3.42820i	0.513415	−	0.137569i
$622$		0			0
$623$	12.9904	−	4.50000i	0.520449	−	0.180289i
$624$		0			0
$625$	6.79423	−	11.7679i	0.271769	−	0.470718i
$626$		0			0
$627$	−1.17949	+	4.40192i	−0.0471044	+	0.175796i
$628$		0			0
$629$	5.36603	−	5.36603i	0.213957	−	0.213957i
$630$		0			0
$631$		−	32.2487i		−	1.28380i	−0.766788	−	0.641900i	$-0.778146\pi$
$631$		−	32.2487i		−	1.28380i	0.766788	−	0.641900i	$-0.221854\pi$
$632$		0			0
$633$	10.0981	−	5.83013i	0.401362	−	0.231727i
$634$		0			0
$635$	−2.19615	−	8.19615i	−0.0871517	−	0.325254i
$636$		0			0
$637$	22.1244	−	29.5885i	0.876599	−	1.17234i
$638$		0			0
$639$	0.732051	−	1.26795i	0.0289595	−	0.0501593i
$640$		0			0
$641$	5.57180	+	9.65064i	0.220073	+	0.381177i	0.954830	−	0.297153i	$-0.0960372\pi$
$641$	5.57180	+	9.65064i	0.220073	+	0.381177i	−0.734757	+	0.678330i	$0.762704\pi$
$642$		0			0
$643$	5.39230	+	5.39230i	0.212652	+	0.212652i	0.805393	−	0.592741i	$-0.201954\pi$
$643$	5.39230	+	5.39230i	0.212652	+	0.212652i	−0.592741	+	0.805393i	$0.701954\pi$
$644$		0			0
$645$	−3.00000	+	3.00000i	−0.118125	+	0.118125i
$646$		0			0
$647$	30.8660	−	17.8205i	1.21347	−	0.700596i	0.249955	−	0.968257i	$-0.419584\pi$
$647$	30.8660	−	17.8205i	1.21347	−	0.700596i	0.963513	+	0.267661i	$0.0862507\pi$
$648$		0			0
$649$	30.1410	+	17.4019i	1.18314	+	0.683085i
$650$		0			0
$651$	2.23205	−	4.59808i	0.0874810	−	0.180213i
$652$		0			0
$653$	32.5526	−	8.72243i	1.27388	−	0.341335i	0.442364	−	0.896836i	$-0.354140\pi$
$653$	32.5526	−	8.72243i	1.27388	−	0.341335i	0.831516	+	0.555501i	$0.187473\pi$
$654$		0			0
$655$	−3.52628	−	6.10770i	−0.137783	−	0.238647i
$656$		0			0
$657$	−19.6603			−0.767020
$658$		0			0
$659$	−18.8564	−	18.8564i	−0.734541	−	0.734541i	0.236975	−	0.971516i	$-0.423844\pi$
$659$	−18.8564	−	18.8564i	−0.734541	−	0.734541i	−0.971516	+	0.236975i	$0.923844\pi$
$660$		0			0
$661$	22.1603	+	5.93782i	0.861934	+	0.230955i	0.662597	−	0.748976i	$-0.269454\pi$
$661$	22.1603	+	5.93782i	0.861934	+	0.230955i	0.199337	+	0.979931i	$0.436121\pi$
$662$		0			0
$663$	−15.2942	−	8.83013i	−0.593979	−	0.342934i
$664$		0			0
$665$	6.91154	+	1.33013i	0.268018	+	0.0515801i
$666$		0			0
$667$	−0.437822	−	1.63397i	−0.0169525	−	0.0632677i
$668$		0			0
$669$	−1.32051	+	4.92820i	−0.0510538	+	0.190535i
$670$		0			0
$671$	21.3923			0.825841
$672$		0			0
$673$	40.7846			1.57213			0.786066	−	0.618143i	$-0.212115\pi$
$673$	40.7846			1.57213			0.786066	+	0.618143i	$0.212115\pi$
$674$		0			0
$675$	3.22243	−	12.0263i	0.124031	−	0.462892i
$676$		0			0
$677$	11.4474	+	42.7224i	0.439961	+	1.64196i	0.728908	+	0.684611i	$0.240028\pi$
$677$	11.4474	+	42.7224i	0.439961	+	1.64196i	−0.288947	+	0.957345i	$0.593305\pi$
$678$		0			0
$679$	−5.07180	+	5.85641i	−0.194638	+	0.224748i
$680$		0			0
$681$	−5.17949	−	2.99038i	−0.198479	−	0.114592i
$682$		0			0
$683$	45.5788	+	12.2128i	1.74403	+	0.467310i	0.983335	−	0.181804i	$-0.0581936\pi$
$683$	45.5788	+	12.2128i	1.74403	+	0.467310i	0.760691	+	0.649114i	$0.224860\pi$
$684$		0			0
$685$	−11.1506	−	11.1506i	−0.426044	−	0.426044i
$686$		0			0
$687$	−7.14359			−0.272545
$688$		0			0
$689$	10.8301	+	18.7583i	0.412595	+	0.714635i
$690$		0			0
$691$	22.9904	−	6.16025i	0.874595	−	0.234347i	0.206521	−	0.978442i	$-0.433786\pi$
$691$	22.9904	−	6.16025i	0.874595	−	0.234347i	0.668074	+	0.744095i	$0.267119\pi$
$692$		0			0
$693$	−21.3923	+	1.53590i	−0.812626	+	0.0583440i
$694$		0			0
$695$	13.0981	+	7.56218i	0.496838	+	0.286850i
$696$		0			0
$697$	−27.5885	+	15.9282i	−1.04499	+	0.603324i
$698$		0			0
$699$	−0.294229	+	0.294229i	−0.0111287	+	0.0111287i
$700$		0			0
$701$	−13.3923	−	13.3923i	−0.505820	−	0.505820i	0.407420	−	0.913241i	$-0.366428\pi$
$701$	−13.3923	−	13.3923i	−0.505820	−	0.505820i	−0.913241	+	0.407420i	$0.866428\pi$
$702$		0			0
$703$	1.74167	+	3.01666i	0.0656883	+	0.113776i
$704$		0			0
$705$	0.990381	−	1.71539i	0.0372999	−	0.0646053i
$706$		0			0
$707$	−42.8468	+	29.0167i	−1.61142	+	1.09128i
$708$		0			0
$709$	−12.2321	−	45.6506i	−0.459384	−	1.71445i	−0.674868	−	0.737938i	$-0.735800\pi$
$709$	−12.2321	−	45.6506i	−0.459384	−	1.71445i	0.215484	−	0.976507i	$-0.430867\pi$
$710$		0			0
$711$	−39.4186	+	22.7583i	−1.47831	+	0.853504i
$712$		0			0
$713$			16.6603i			0.623931i
$714$		0			0
$715$	9.92820	−	9.92820i	0.371294	−	0.371294i
$716$		0			0
$717$	1.14359	−	4.26795i	0.0427083	−	0.159389i
$718$		0			0
$719$	0.205771	−	0.356406i	0.00767398	−	0.0132917i	−0.862163	−	0.506631i	$-0.830891\pi$
$719$	0.205771	−	0.356406i	0.00767398	−	0.0132917i	0.869837	+	0.493339i	$0.164224\pi$
$720$		0			0
$721$	−16.1603	+	5.59808i	−0.601839	+	0.208483i
$722$		0			0
$723$	−6.96410	+	1.86603i	−0.258998	+	0.0693982i
$724$		0			0
$725$	−1.53590	−	0.411543i	−0.0570418	−	0.0152843i
$726$		0			0
$727$			41.3205i			1.53249i	0.642547	+	0.766246i	$0.277878\pi$
$727$			41.3205i			1.53249i	−0.642547	+	0.766246i	$0.722122\pi$
$728$		0			0
$729$			13.5885i			0.503276i
$730$		0			0
$731$	57.0788	+	15.2942i	2.11114	+	0.565677i
$732$		0			0
$733$	32.8923	−	8.81347i	1.21490	−	0.325533i	0.406220	−	0.913775i	$-0.366847\pi$
$733$	32.8923	−	8.81347i	1.21490	−	0.325533i	0.808685	+	0.588242i	$0.200180\pi$
$734$		0			0
$735$	−0.464102	−	3.21539i	−0.0171186	−	0.118601i
$736$		0			0
$737$	7.62436	−	13.2058i	0.280847	−	0.486441i
$738$		0			0
$739$	10.3109	−	38.4808i	0.379292	−	1.41554i	−0.467679	−	0.883899i	$-0.654910\pi$
$739$	10.3109	−	38.4808i	0.379292	−	1.41554i	0.846971	−	0.531639i	$-0.178424\pi$
$740$		0			0
$741$	5.73205	−	5.73205i	0.210572	−	0.210572i
$742$		0			0
$743$			11.0718i			0.406185i	0.979160	+	0.203092i	$0.0650992\pi$
$743$			11.0718i			0.406185i	−0.979160	+	0.203092i	$0.934901\pi$
$744$		0			0
$745$	−6.10770	+	3.52628i	−0.223769	+	0.129193i
$746$		0			0
$747$	−1.53590	−	5.73205i	−0.0561956	−	0.209725i
$748$		0			0
$749$	7.96410	+	3.86603i	0.291002	+	0.141261i
$750$		0			0
$751$	6.52628	−	11.3038i	0.238147	−	0.412483i	−0.722035	−	0.691856i	$-0.756793\pi$
$751$	6.52628	−	11.3038i	0.238147	−	0.412483i	0.960183	+	0.279373i	$0.0901266\pi$
$752$		0			0
$753$	−6.43782	−	11.1506i	−0.234607	−	0.406352i
$754$		0			0
$755$	7.22243	+	7.22243i	0.262851	+	0.262851i
$756$		0			0
$757$	−18.6603	+	18.6603i	−0.678218	+	0.678218i	−0.959597	−	0.281378i	$-0.909208\pi$
$757$	−18.6603	+	18.6603i	−0.678218	+	0.678218i	0.281378	+	0.959597i	$0.409208\pi$
$758$		0			0
$759$	−5.93782	+	3.42820i	−0.215529	+	0.124436i
$760$		0			0
$761$	−11.7679	−	6.79423i	−0.426588	−	0.246291i	0.271304	−	0.962494i	$-0.412545\pi$
$761$	−11.7679	−	6.79423i	−0.426588	−	0.246291i	−0.697892	+	0.716203i	$0.745878\pi$
$762$		0			0
$763$	−3.27757	−	4.83975i	−0.118656	−	0.175211i
$764$		0			0
$765$	15.2942	−	4.09808i	0.552964	−	0.148166i
$766$		0			0
$767$	−30.9545	−	53.6147i	−1.11770	−	1.93592i
$768$		0			0
$769$	9.85641			0.355431			0.177716	−	0.984082i	$-0.443129\pi$
$769$	9.85641			0.355431			0.177716	+	0.984082i	$0.443129\pi$
$770$		0			0
$771$	−7.09808	−	7.09808i	−0.255631	−	0.255631i
$772$		0			0
$773$	10.9641	+	2.93782i	0.394351	+	0.105666i	0.450545	−	0.892754i	$-0.351230\pi$
$773$	10.9641	+	2.93782i	0.394351	+	0.105666i	−0.0561936	+	0.998420i	$0.517896\pi$
$774$		0			0
$775$	13.5622	+	7.83013i	0.487168	+	0.281266i
$776$		0			0
$777$	1.05256	−	1.21539i	0.0377603	−	0.0436019i
$778$		0			0
$779$	−3.78461	−	14.1244i	−0.135598	−	0.506058i
$780$		0			0
$781$	−0.411543	+	1.53590i	−0.0147262	+	0.0549588i
$782$		0			0
$783$	−1.12436			−0.0401812
$784$		0			0
$785$	16.9474			0.604880
$786$		0			0
$787$	−3.52628	+	13.1603i	−0.125698	+	0.469112i	−0.999864	−	0.0165161i	$-0.994743\pi$
$787$	−3.52628	+	13.1603i	−0.125698	+	0.469112i	0.874165	+	0.485628i	$0.161409\pi$
$788$		0			0
$789$	−0.617314	−	2.30385i	−0.0219770	−	0.0820191i
$790$		0			0
$791$	−2.53590	+	2.92820i	−0.0901662	+	0.104115i
$792$		0			0
$793$	−32.9545	−	19.0263i	−1.17025	−	0.675643i
$794$		0			0
$795$	1.83975	+	0.492958i	0.0652491	+	0.0174834i
$796$		0			0
$797$	−13.3397	−	13.3397i	−0.472518	−	0.472518i	0.430211	−	0.902729i	$-0.358439\pi$
$797$	−13.3397	−	13.3397i	−0.472518	−	0.472518i	−0.902729	+	0.430211i	$0.858439\pi$
$798$		0			0
$799$	−27.5885			−0.976009
$800$		0			0
$801$	7.09808	+	12.2942i	0.250798	+	0.434395i
$802$		0			0
$803$	20.6244	−	5.52628i	0.727818	−	0.195018i
$804$		0			0
$805$	5.93782	+	8.76795i	0.209281	+	0.309030i
$806$		0			0
$807$	4.54552	+	2.62436i	0.160010	+	0.0923817i
$808$		0			0
$809$	29.4282	−	16.9904i	1.03464	−	0.597350i	0.116330	−	0.993211i	$-0.462887\pi$
$809$	29.4282	−	16.9904i	1.03464	−	0.597350i	0.918311	+	0.395861i	$0.129554\pi$
$810$		0			0
$811$	−24.3205	+	24.3205i	−0.854009	+	0.854009i	−0.990624	−	0.136616i	$-0.956377\pi$
$811$	−24.3205	+	24.3205i	−0.854009	+	0.854009i	0.136616	+	0.990624i	$0.456377\pi$
$812$		0			0
$813$	−4.43782	−	4.43782i	−0.155641	−	0.155641i
$814$		0			0
$815$	5.59808	+	9.69615i	0.196092	+	0.339641i
$816$		0			0
$817$	−13.5622	+	23.4904i	−0.474481	+	0.821824i
$818$		0			0
$819$	34.3205	+	16.6603i	1.19926	+	0.582156i
$820$		0			0
$821$	−0.160254	−	0.598076i	−0.00559290	−	0.0208730i	0.963073	−	0.269240i	$-0.0867726\pi$
$821$	−0.160254	−	0.598076i	−0.00559290	−	0.0208730i	−0.968666	+	0.248367i	$0.920106\pi$
$822$		0			0
$823$	−14.9378	+	8.62436i	−0.520700	+	0.300626i	−0.737221	−	0.675652i	$-0.763862\pi$
$823$	−14.9378	+	8.62436i	−0.520700	+	0.300626i	0.216521	+	0.976278i	$0.430529\pi$
$824$		0			0
$825$			6.44486i			0.224381i
$826$		0			0
$827$	3.78461	−	3.78461i	0.131604	−	0.131604i	−0.638237	−	0.769840i	$-0.720336\pi$
$827$	3.78461	−	3.78461i	0.131604	−	0.131604i	0.769840	+	0.638237i	$0.220336\pi$
$828$		0			0
$829$	0.820508	−	3.06218i	0.0284974	−	0.106354i	−0.950212	−	0.311603i	$-0.899134\pi$
$829$	0.820508	−	3.06218i	0.0284974	−	0.106354i	0.978710	+	0.205250i	$0.0658006\pi$
$830$		0			0
$831$	1.20577	−	2.08846i	0.0418277	−	0.0724478i
$832$		0			0
$833$	−35.5526	+	27.9904i	−1.23182	+	0.969809i
$834$		0			0
$835$	−18.9282	+	5.07180i	−0.655037	+	0.175517i
$836$		0			0
$837$	10.6962	+	2.86603i	0.369713	+	0.0990643i
$838$		0			0
$839$		−	17.7128i		−	0.611514i	−0.952110	−	0.305757i	$-0.901090\pi$
$839$		−	17.7128i		−	0.611514i	0.952110	−	0.305757i	$-0.0989096\pi$
$840$		0			0
$841$		−	28.8564i		−	0.995048i
$842$		0			0
$843$	−0.464102	−	0.124356i	−0.0159845	−	0.00428304i
$844$		0			0
$845$	−12.8660	+	3.44744i	−0.442605	+	0.118596i
$846$		0			0
$847$	−5.49038	+	1.90192i	−0.188652	+	0.0653509i
$848$		0			0
$849$	−3.89230	+	6.74167i	−0.133584	+	0.231374i
$850$		0			0
$851$	−1.35641	+	5.06218i	−0.0464970	+	0.173529i
$852$		0			0
$853$	−11.8756	+	11.8756i	−0.406614	+	0.406614i	−0.880556	−	0.473942i	$-0.842831\pi$
$853$	−11.8756	+	11.8756i	−0.406614	+	0.406614i	0.473942	+	0.880556i	$0.342831\pi$
$854$		0			0
$855$			7.26795i			0.248559i
$856$		0			0
$857$	−11.8923	+	6.86603i	−0.406233	+	0.234539i	−0.689170	−	0.724600i	$-0.742025\pi$
$857$	−11.8923	+	6.86603i	−0.406233	+	0.234539i	0.282937	+	0.959139i	$0.408691\pi$
$858$		0			0
$859$	3.65064	+	13.6244i	0.124558	+	0.464857i	0.999824	−	0.0187858i	$-0.00598005\pi$
$859$	3.65064	+	13.6244i	0.124558	+	0.464857i	−0.875265	+	0.483643i	$0.839313\pi$
$860$		0			0
$861$	−5.58846	+	3.78461i	−0.190454	+	0.128979i
$862$		0			0
$863$	−16.3301	+	28.2846i	−0.555884	+	0.962819i	0.441950	+	0.897040i	$0.354287\pi$
$863$	−16.3301	+	28.2846i	−0.555884	+	0.962819i	−0.997834	+	0.0657797i	$0.979047\pi$
$864$		0			0
$865$	−1.03590	−	1.79423i	−0.0352216	−	0.0610056i
$866$		0			0
$867$	9.07180	+	9.07180i	0.308094	+	0.308094i
$868$		0			0
$869$	34.9545	−	34.9545i	1.18575	−	1.18575i
$870$		0			0
$871$	−23.4904	+	13.5622i	−0.795941	+	0.459537i
$872$		0			0
$873$	−6.92820	−	4.00000i	−0.234484	−	0.135379i
$874$		0			0
$875$	21.7583	−	1.56218i	0.735566	−	0.0528112i
$876$		0			0
$877$	−30.5526	+	8.18653i	−1.03169	+	0.276440i	−0.734664	−	0.678431i	$-0.762660\pi$
$877$	−30.5526	+	8.18653i	−1.03169	+	0.276440i	−0.297022	+	0.954871i	$0.595994\pi$
$878$		0			0
$879$	2.90192	+	5.02628i	0.0978795	+	0.169532i
$880$		0			0
$881$	50.0000			1.68454			0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000			1.68454			0.842271	+	0.539054i	$0.181218\pi$
$882$		0			0
$883$	5.00000	+	5.00000i	0.168263	+	0.168263i	0.786216	−	0.617952i	$-0.212037\pi$
$883$	5.00000	+	5.00000i	0.168263	+	0.168263i	−0.617952	+	0.786216i	$0.712037\pi$
$884$		0			0
$885$	−5.25833	−	1.40897i	−0.176757	−	0.0473619i
$886$		0			0
$887$	−39.7750	−	22.9641i	−1.33551	−	0.771059i	−0.349375	−	0.936983i	$-0.613606\pi$
$887$	−39.7750	−	22.9641i	−1.33551	−	0.771059i	−0.986139	+	0.165924i	$0.946939\pi$
$888$		0			0
$889$	−16.3923	+	18.9282i	−0.549780	+	0.634832i
$890$		0			0
$891$	−5.11474	−	19.0885i	−0.171350	−	0.639487i
$892$		0			0
$893$	3.27757	−	12.2321i	0.109680	−	0.409330i
$894$		0			0
$895$	8.32051			0.278124
$896$		0			0
$897$	12.1962			0.407218
$898$		0			0
$899$	0.366025	−	1.36603i	0.0122076	−	0.0455595i
$900$		0			0
$901$	−6.86603	−	25.6244i	−0.228740	−	0.853671i
$902$		0			0
$903$	12.2942	+	2.36603i	0.409126	+	0.0787364i
$904$		0			0
$905$	14.7058	+	8.49038i	0.488836	+	0.282230i
$906$		0			0
$907$	26.7224	+	7.16025i	0.887304	+	0.237752i	0.673556	−	0.739136i	$-0.264766\pi$
$907$	26.7224	+	7.16025i	0.887304	+	0.237752i	0.213748	+	0.976889i	$0.431433\pi$
$908$		0			0
$909$	−37.7846	−	37.7846i	−1.25324	−	1.25324i
$910$		0			0
$911$	−7.32051			−0.242539			−0.121270	−	0.992620i	$-0.538697\pi$
$911$	−7.32051			−0.242539			−0.121270	+	0.992620i	$0.538697\pi$
$912$		0			0
$913$	3.22243	+	5.58142i	0.106647	+	0.184718i
$914$		0			0
$915$	−3.23205	+	0.866025i	−0.106848	+	0.0286299i
$916$		0			0
$917$	−9.08846	+	18.7224i	−0.300127	+	0.618269i
$918$		0			0
$919$	−15.8660	−	9.16025i	−0.523372	−	0.302169i	0.214941	−	0.976627i	$-0.431044\pi$
$919$	−15.8660	−	9.16025i	−0.523372	−	0.302169i	−0.738313	+	0.674458i	$0.764377\pi$
$920$		0			0
$921$	−5.70577	+	3.29423i	−0.188012	+	0.108549i
$922$		0			0
$923$	2.00000	−	2.00000i	0.0658308	−	0.0658308i
$924$		0			0
$925$	3.48334	+	3.48334i	0.114531	+	0.114531i
$926$		0			0
$927$	−8.83013	−	15.2942i	−0.290019	−	0.502328i
$928$		0			0
$929$	−25.0167	+	43.3301i	−0.820770	+	1.42162i	0.0843396	+	0.996437i	$0.473122\pi$
$929$	−25.0167	+	43.3301i	−0.820770	+	1.42162i	−0.905110	+	0.425178i	$0.860211\pi$
$930$		0			0
$931$	−8.18653	−	19.0885i	−0.268303	−	0.625599i
$932$		0			0
$933$	−0.866025	−	3.23205i	−0.0283524	−	0.105813i
$934$		0			0
$935$	−14.8923	+	8.59808i	−0.487030	+	0.281187i
$936$		0			0
$937$		−	29.0718i		−	0.949734i	−0.880058	−	0.474867i	$-0.842496\pi$
$937$		−	29.0718i		−	0.949734i	0.880058	−	0.474867i	$-0.157504\pi$
$938$		0			0
$939$	7.90192	−	7.90192i	0.257870	−	0.257870i
$940$		0			0
$941$	6.30385	−	23.5263i	0.205500	−	0.766935i	−0.783797	−	0.621017i	$-0.786720\pi$
$941$	6.30385	−	23.5263i	0.205500	−	0.766935i	0.989297	−	0.145918i	$-0.0466135\pi$
$942$		0			0
$943$	11.0000	−	19.0526i	0.358209	−	0.620437i
$944$		0			0
$945$	6.65064	−	2.30385i	0.216345	−	0.0749442i
$946$		0			0
$947$	−5.40192	+	1.44744i	−0.175539	+	0.0470355i	−0.345518	−	0.938412i	$-0.612297\pi$
$947$	−5.40192	+	1.44744i	−0.175539	+	0.0470355i	0.169979	+	0.985448i	$0.445630\pi$
$948$		0			0
$949$	−36.6865	−	9.83013i	−1.19090	−	0.319099i
$950$		0			0
$951$		−	7.58846i		−	0.246073i
$952$		0			0
$953$			16.5359i			0.535650i	0.963468	+	0.267825i	$0.0863050\pi$
$953$			16.5359i			0.535650i	−0.963468	+	0.267825i	$0.913695\pi$
$954$		0			0
$955$	7.50000	+	2.00962i	0.242694	+	0.0650297i
$956$		0			0
$957$	0.562178	−	0.150635i	0.0181726	−	0.00486934i
$958$		0			0
$959$	−8.79423	+	45.6962i	−0.283980	+	1.47561i
$960$		0			0
$961$	8.53590	−	14.7846i	0.275352	−	0.476923i
$962$		0			0
$963$	−2.36603	+	8.83013i	−0.0762441	+	0.284547i
$964$		0			0
$965$	−14.5814	+	14.5814i	−0.469392	+	0.469392i
$966$		0			0
$967$			60.2487i			1.93747i	0.248102	+	0.968734i	$0.420193\pi$
$967$			60.2487i			1.93747i	−0.248102	+	0.968734i	$0.579807\pi$
$968$		0			0
$969$	−8.59808	+	4.96410i	−0.276210	+	0.159470i
$970$		0			0
$971$	−14.0429	−	52.4090i	−0.450659	−	1.68188i	−0.700545	−	0.713608i	$-0.747060\pi$
$971$	−14.0429	−	52.4090i	−0.450659	−	1.68188i	0.249885	−	0.968275i	$-0.419607\pi$
$972$		0			0
$973$	−3.19615	−	44.5167i	−0.102464	−	1.42714i
$974$		0			0
$975$	5.73205	−	9.92820i	0.183573	−	0.317957i
$976$		0			0
$977$	8.57180	+	14.8468i	0.274236	+	0.474991i	0.969942	−	0.243336i	$-0.0782417\pi$
$977$	8.57180	+	14.8468i	0.274236	+	0.474991i	−0.695706	+	0.718327i	$0.744908\pi$
$978$		0			0
$979$	−10.9019	−	10.9019i	−0.348427	−	0.348427i
$980$		0			0
$981$	4.26795	−	4.26795i	0.136265	−	0.136265i
$982$		0			0
$983$	17.1340	−	9.89230i	0.546489	−	0.315516i	−0.201216	−	0.979547i	$-0.564489\pi$
$983$	17.1340	−	9.89230i	0.546489	−	0.315516i	0.747705	+	0.664031i	$0.231156\pi$
$984$		0			0
$985$	−18.2942	−	10.5622i	−0.582903	−	0.336539i
$986$		0			0
$987$	−5.83013	+	0.418584i	−0.185575	+	0.0133237i
$988$		0			0
$989$	−39.4186	+	10.5622i	−1.25344	+	0.335858i
$990$		0			0
$991$	−4.20577	−	7.28461i	−0.133601	−	0.231403i	0.791461	−	0.611219i	$-0.209321\pi$
$991$	−4.20577	−	7.28461i	−0.133601	−	0.231403i	−0.925062	+	0.379816i	$0.875987\pi$
$992$		0			0
$993$	−10.5167			−0.333736
$994$		0			0
$995$	−7.56218	−	7.56218i	−0.239737	−	0.239737i
$996$		0			0
$997$	−1.50000	−	0.401924i	−0.0475055	−	0.0127291i	0.234988	−	0.971998i	$-0.424495\pi$
$997$	−1.50000	−	0.401924i	−0.0475055	−	0.0127291i	−0.282494	+	0.959269i	$0.591162\pi$
$998$		0			0
$999$	3.01666	+	1.74167i	0.0954429	+	0.0551040i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.ba.b.401.1		4
4.3	odd	2		112.2.w.a.93.1	yes	4
7.4	even	3		448.2.ba.a.81.1		4
8.3	odd	2		896.2.ba.d.289.1		4
8.5	even	2		896.2.ba.a.289.1		4
16.3	odd	4		896.2.ba.b.737.1		4
16.5	even	4		448.2.ba.a.177.1		4
16.11	odd	4		112.2.w.b.37.1	yes	4
16.13	even	4		896.2.ba.c.737.1		4
28.3	even	6		784.2.x.h.557.1		4
28.11	odd	6		112.2.w.b.109.1	yes	4
28.19	even	6		784.2.m.d.589.2		4
28.23	odd	6		784.2.m.e.589.2		4
28.27	even	2		784.2.x.a.765.1		4
56.11	odd	6		896.2.ba.b.417.1		4
56.53	even	6		896.2.ba.c.417.1		4
112.11	odd	12		112.2.w.a.53.1	✓	4
112.27	even	4		784.2.x.h.373.1		4
112.53	even	12	inner	448.2.ba.b.305.1		4
112.59	even	12		784.2.x.a.165.1		4
112.67	odd	12		896.2.ba.d.865.1		4
112.75	even	12		784.2.m.d.197.2		4
112.107	odd	12		784.2.m.e.197.2		4
112.109	even	12		896.2.ba.a.865.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.w.a.53.1	✓	4	112.11	odd	12
112.2.w.a.93.1	yes	4	4.3	odd	2
112.2.w.b.37.1	yes	4	16.11	odd	4
112.2.w.b.109.1	yes	4	28.11	odd	6
448.2.ba.a.81.1		4	7.4	even	3
448.2.ba.a.177.1		4	16.5	even	4
448.2.ba.b.305.1		4	112.53	even	12	inner
448.2.ba.b.401.1		4	1.1	even	1	trivial
784.2.m.d.197.2		4	112.75	even	12
784.2.m.d.589.2		4	28.19	even	6
784.2.m.e.197.2		4	112.107	odd	12
784.2.m.e.589.2		4	28.23	odd	6
784.2.x.a.165.1		4	112.59	even	12
784.2.x.a.765.1		4	28.27	even	2
784.2.x.h.373.1		4	112.27	even	4
784.2.x.h.557.1		4	28.3	even	6
896.2.ba.a.289.1		4	8.5	even	2
896.2.ba.a.865.1		4	112.109	even	12
896.2.ba.b.417.1		4	56.11	odd	6
896.2.ba.b.737.1		4	16.3	odd	4
896.2.ba.c.417.1		4	56.53	even	6
896.2.ba.c.737.1		4	16.13	even	4
896.2.ba.d.289.1		4	8.3	odd	2
896.2.ba.d.865.1		4	112.67	odd	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.ba (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.133975 - 0.500000i) q^{3} +(0.232051 + 0.866025i) q^{5} +(1.73205 - 2.00000i) q^{7} +(2.36603 + 1.36603i) q^{9} +(-2.86603 - 0.767949i) q^{11} +(3.73205 + 3.73205i) q^{13} +0.464102 q^{15} +(-3.23205 - 5.59808i) q^{17} +(2.86603 - 0.767949i) q^{19} +(-0.767949 - 1.13397i) q^{21} +(3.86603 + 2.23205i) q^{23} +(3.63397 - 2.09808i) q^{25} +(2.09808 - 2.09808i) q^{27} +(-0.267949 - 0.267949i) q^{29} +(1.86603 + 3.23205i) q^{31} +(-0.767949 + 1.33013i) q^{33} +(2.13397 + 1.03590i) q^{35} +(0.303848 + 1.13397i) q^{37} +(2.36603 - 1.36603i) q^{39} -4.92820i q^{41} +(-6.46410 + 6.46410i) q^{43} +(-0.633975 + 2.36603i) q^{45} +(2.13397 - 3.69615i) q^{47} +(-1.00000 - 6.92820i) q^{49} +(-3.23205 + 0.866025i) q^{51} +(3.96410 + 1.06218i) q^{53} -2.66025i q^{55} -1.53590i q^{57} +(-11.3301 - 3.03590i) q^{59} +(-6.96410 + 1.86603i) q^{61} +(6.83013 - 2.36603i) q^{63} +(-2.36603 + 4.09808i) q^{65} +(-1.33013 + 4.96410i) q^{67} +(1.63397 - 1.63397i) q^{69} -0.535898i q^{71} +(-6.23205 + 3.59808i) q^{73} +(-0.562178 - 2.09808i) q^{75} +(-6.50000 + 4.40192i) q^{77} +(-8.33013 + 14.4282i) q^{79} +(3.33013 + 5.76795i) q^{81} +(-1.53590 - 1.53590i) q^{83} +(4.09808 - 4.09808i) q^{85} +(-0.169873 + 0.0980762i) q^{87} +(4.50000 + 2.59808i) q^{89} +(13.9282 - 1.00000i) q^{91} +(1.86603 - 0.500000i) q^{93} +(1.33013 + 2.30385i) q^{95} -2.92820 q^{97} +(-5.73205 - 5.73205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 6 q^{5} + 6 q^{9} - 8 q^{11} + 8 q^{13} - 12 q^{15} - 6 q^{17} + 8 q^{19} - 10 q^{21} + 12 q^{23} + 18 q^{25} - 2 q^{27} - 8 q^{29} + 4 q^{31} - 10 q^{33} + 12 q^{35} + 22 q^{37} + 6 q^{39}+ \cdots - 16 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 6 * q^5 + 6 * q^9 - 8 * q^11 + 8 * q^13 - 12 * q^15 - 6 * q^17 + 8 * q^19 - 10 * q^21 + 12 * q^23 + 18 * q^25 - 2 * q^27 - 8 * q^29 + 4 * q^31 - 10 * q^33 + 12 * q^35 + 22 * q^37 + 6 * q^39 - 12 * q^43 - 6 * q^45 + 12 * q^47 - 4 * q^49 - 6 * q^51 + 2 * q^53 - 28 * q^59 - 14 * q^61 + 10 * q^63 - 6 * q^65 + 12 * q^67 + 10 * q^69 - 18 * q^73 + 22 * q^75 - 26 * q^77 - 16 * q^79 - 4 * q^81 - 20 * q^83 + 6 * q^85 - 18 * q^87 + 18 * q^89 + 28 * q^91 + 4 * q^93 - 12 * q^95 + 16 * q^97 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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