LMFDB - Embedded newform 448.2.ba.c.177.6

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 = [48,0,0] 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:	$48$
Relative dimension:	$12$ over $\Q(\zeta_{12})$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			177.6
Character	$\chi$	$=$	448.177
Dual form			448.2.ba.c.81.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.827840 - 0.221819i) q^{3} +(4.10878 - 1.10095i) q^{5} +(2.50325 - 0.856573i) q^{7} +(-1.96196 - 1.13274i) q^{9} +(0.318576 - 1.18894i) q^{11} +(-1.73092 + 1.73092i) q^{13} -3.64563 q^{15} +(-0.931914 - 1.61412i) q^{17} +(0.989804 + 3.69400i) q^{19} +(-2.26230 + 0.153836i) q^{21} +(-1.23305 - 0.711901i) q^{23} +(11.3399 - 6.54709i) q^{25} +(3.19099 + 3.19099i) q^{27} +(-0.181826 + 0.181826i) q^{29} +(-3.23200 - 5.59799i) q^{31} +(-0.527459 + 0.913587i) q^{33} +(9.34229 - 6.27542i) q^{35} +(0.238000 - 0.0637719i) q^{37} +(1.81687 - 1.04897i) q^{39} +0.440182i q^{41} +(5.54503 + 5.54503i) q^{43} +(-9.30835 - 2.49417i) q^{45} +(3.61427 - 6.26009i) q^{47} +(5.53257 - 4.28844i) q^{49} +(0.413433 + 1.54295i) q^{51} +(-2.59716 + 9.69274i) q^{53} -5.23583i q^{55} -3.27760i q^{57} +(0.438601 - 1.63688i) q^{59} +(2.41596 + 9.01647i) q^{61} +(-5.88156 - 1.15497i) q^{63} +(-5.20632 + 9.01761i) q^{65} +(-9.59422 - 2.57076i) q^{67} +(0.862854 + 0.862854i) q^{69} +11.2912i q^{71} +(-8.13092 + 4.69439i) q^{73} +(-10.8399 + 2.90454i) q^{75} +(-0.220938 - 3.24910i) q^{77} +(-6.52948 + 11.3094i) q^{79} +(1.46441 + 2.53643i) q^{81} +(3.06926 - 3.06926i) q^{83} +(-5.60609 - 5.60609i) q^{85} +(0.190855 - 0.110190i) q^{87} +(-5.66084 - 3.26829i) q^{89} +(-2.85027 + 5.81558i) q^{91} +(1.43384 + 5.35117i) q^{93} +(8.13378 + 14.0881i) q^{95} +1.70409 q^{97} +(-1.97179 + 1.97179i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 4 q^{5} + 4 q^{11} - 24 q^{13} + 40 q^{15} + 8 q^{17} + 4 q^{19} - 8 q^{21} + 24 q^{27} + 24 q^{29} - 28 q^{31} + 16 q^{33} - 28 q^{35} - 24 q^{37} + 40 q^{43} - 28 q^{45} + 20 q^{47} - 24 q^{51}+ \cdots + 24 q^{99}+O(q^{100}) $ 48 * q + 4 * q^5 + 4 * q^11 - 24 * q^13 + 40 * q^15 + 8 * q^17 + 4 * q^19 - 8 * q^21 + 24 * q^27 + 24 * q^29 - 28 * q^31 + 16 * q^33 - 28 * q^35 - 24 * q^37 + 40 * q^43 - 28 * q^45 + 20 * q^47 - 24 * q^51 - 16 * q^53 + 20 * q^59 + 8 * q^61 + 16 * q^63 + 8 * q^65 - 48 * q^67 - 40 * q^69 + 4 * q^75 - 20 * q^77 + 36 * q^79 + 8 * q^83 - 64 * q^91 + 8 * q^93 + 4 * q^95 - 48 * q^97 + 24 * q^99

Character values

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

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

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
$3$	−0.827840	−	0.221819i	−0.477954	−	0.128067i	0.0117959	−	0.999930i	$-0.496245\pi$
$3$	−0.827840	−	0.221819i	−0.477954	−	0.128067i	−0.489750	+	0.871863i	$0.662912\pi$
$4$		0			0
$5$	4.10878	−	1.10095i	1.83750	−	0.492358i	0.838857	−	0.544352i	$-0.183225\pi$
$5$	4.10878	−	1.10095i	1.83750	−	0.492358i	0.998647	+	0.0519945i	$0.0165578\pi$
$6$		0			0
$7$	2.50325	−	0.856573i	0.946141	−	0.323754i
$7$	2.50325	−	0.856573i	0.946141	−	0.323754i
$8$		0			0
$9$	−1.96196	−	1.13274i	−0.653987	−	0.377579i
$10$		0			0
$11$	0.318576	−	1.18894i	0.0960542	−	0.358479i	−0.901123	−	0.433564i	$-0.857256\pi$
$11$	0.318576	−	1.18894i	0.0960542	−	0.358479i	0.997177	+	0.0750845i	$0.0239226\pi$
$12$		0			0
$13$	−1.73092	+	1.73092i	−0.480070	+	0.480070i	−0.905154	−	0.425084i	$-0.860245\pi$
$13$	−1.73092	+	1.73092i	−0.480070	+	0.480070i	0.425084	+	0.905154i	$0.360245\pi$
$14$		0			0
$15$	−3.64563			−0.941297
$16$		0			0
$17$	−0.931914	−	1.61412i	−0.226022	−	0.391482i	0.730603	−	0.682802i	$-0.239239\pi$
$17$	−0.931914	−	1.61412i	−0.226022	−	0.391482i	−0.956626	+	0.291320i	$0.905906\pi$
$18$		0			0
$19$	0.989804	+	3.69400i	0.227077	+	0.847462i	0.981562	+	0.191145i	$0.0612199\pi$
$19$	0.989804	+	3.69400i	0.227077	+	0.847462i	−0.754485	+	0.656317i	$0.772113\pi$
$20$		0			0
$21$	−2.26230	+	0.153836i	−0.493674	+	0.0335697i
$22$		0			0
$23$	−1.23305	−	0.711901i	−0.257108	−	0.148442i	0.365906	−	0.930652i	$-0.380759\pi$
$23$	−1.23305	−	0.711901i	−0.257108	−	0.148442i	−0.623015	+	0.782210i	$0.714092\pi$
$24$		0			0
$25$	11.3399	−	6.54709i	2.26798	−	1.30942i
$26$		0			0
$27$	3.19099	+	3.19099i	0.614106	+	0.614106i
$28$		0			0
$29$	−0.181826	+	0.181826i	−0.0337642	+	0.0337642i	−0.723787	−	0.690023i	$-0.757600\pi$
$29$	−0.181826	+	0.181826i	−0.0337642	+	0.0337642i	0.690023	+	0.723787i	$0.257600\pi$
$30$		0			0
$31$	−3.23200	−	5.59799i	−0.580485	−	1.00543i	−0.995422	−	0.0955792i	$-0.969530\pi$
$31$	−3.23200	−	5.59799i	−0.580485	−	1.00543i	0.414937	−	0.909850i	$-0.363804\pi$
$32$		0			0
$33$	−0.527459	+	0.913587i	−0.0918189	+	0.159035i
$34$		0			0
$35$	9.34229	−	6.27542i	1.57914	−	1.06074i
$36$		0			0
$37$	0.238000	−	0.0637719i	0.0391269	−	0.0104840i	−0.239202	−	0.970970i	$-0.576886\pi$
$37$	0.238000	−	0.0637719i	0.0391269	−	0.0104840i	0.278329	+	0.960486i	$0.410219\pi$
$38$		0			0
$39$	1.81687	−	1.04897i	0.290933	−	0.167970i
$40$		0			0
$41$			0.440182i			0.0687448i	0.999409	+	0.0343724i	$0.0109432\pi$
$41$			0.440182i			0.0687448i	−0.999409	+	0.0343724i	$0.989057\pi$
$42$		0			0
$43$	5.54503	+	5.54503i	0.845609	+	0.845609i	0.989582	−	0.143972i	$-0.0459877\pi$
$43$	5.54503	+	5.54503i	0.845609	+	0.845609i	−0.143972	+	0.989582i	$0.545988\pi$
$44$		0			0
$45$	−9.30835	−	2.49417i	−1.38761	−	0.371808i
$46$		0			0
$47$	3.61427	−	6.26009i	0.527195	−	0.913128i	−0.472303	−	0.881436i	$-0.656577\pi$
$47$	3.61427	−	6.26009i	0.527195	−	0.913128i	0.999498	−	0.0316918i	$-0.0100895\pi$
$48$		0			0
$49$	5.53257	−	4.28844i	0.790366	−	0.612634i
$50$		0			0
$51$	0.413433	+	1.54295i	0.0578921	+	0.216056i
$52$		0			0
$53$	−2.59716	+	9.69274i	−0.356747	+	1.33140i	0.521524	+	0.853237i	$0.325364\pi$
$53$	−2.59716	+	9.69274i	−0.356747	+	1.33140i	−0.878271	+	0.478163i	$0.841303\pi$
$54$		0			0
$55$		−	5.23583i		−	0.706000i
$56$		0			0
$57$		−	3.27760i		−	0.434129i
$58$		0			0
$59$	0.438601	−	1.63688i	0.0571010	−	0.213104i	−0.931480	−	0.363791i	$-0.881482\pi$
$59$	0.438601	−	1.63688i	0.0571010	−	0.213104i	0.988581	+	0.150688i	$0.0481488\pi$
$60$		0			0
$61$	2.41596	+	9.01647i	0.309331	+	1.15444i	0.929152	+	0.369697i	$0.120539\pi$
$61$	2.41596	+	9.01647i	0.309331	+	1.15444i	−0.619821	+	0.784743i	$0.712795\pi$
$62$		0			0
$63$	−5.88156	−	1.15497i	−0.741007	−	0.145513i
$64$		0			0
$65$	−5.20632	+	9.01761i	−0.645765	+	1.11850i
$66$		0			0
$67$	−9.59422	−	2.57076i	−1.17212	−	0.314069i	−0.380324	−	0.924853i	$-0.624187\pi$
$67$	−9.59422	−	2.57076i	−1.17212	−	0.314069i	−0.791797	+	0.610785i	$0.790854\pi$
$68$		0			0
$69$	0.862854	+	0.862854i	0.103875	+	0.103875i
$70$		0			0
$71$			11.2912i			1.34002i	0.742352	+	0.670010i	$0.233710\pi$
$71$			11.2912i			1.34002i	−0.742352	+	0.670010i	$0.766290\pi$
$72$		0			0
$73$	−8.13092	+	4.69439i	−0.951652	+	0.549437i	−0.893594	−	0.448877i	$-0.851824\pi$
$73$	−8.13092	+	4.69439i	−0.951652	+	0.549437i	−0.0580583	+	0.998313i	$0.518491\pi$
$74$		0			0
$75$	−10.8399	+	2.90454i	−1.25168	+	0.335388i
$76$		0			0
$77$	−0.220938	−	3.24910i	−0.0251783	−	0.370270i
$78$		0			0
$79$	−6.52948	+	11.3094i	−0.734624	+	1.27241i	0.220264	+	0.975440i	$0.429308\pi$
$79$	−6.52948	+	11.3094i	−0.734624	+	1.27241i	−0.954888	+	0.296966i	$0.904025\pi$
$80$		0			0
$81$	1.46441	+	2.53643i	0.162712	+	0.281825i
$82$		0			0
$83$	3.06926	−	3.06926i	0.336895	−	0.336895i	−0.518302	−	0.855198i	$-0.673436\pi$
$83$	3.06926	−	3.06926i	0.336895	−	0.336895i	0.855198	+	0.518302i	$0.173436\pi$
$84$		0			0
$85$	−5.60609	−	5.60609i	−0.608066	−	0.608066i
$86$		0			0
$87$	0.190855	−	0.110190i	0.0204618	−	0.0118137i
$88$		0			0
$89$	−5.66084	−	3.26829i	−0.600047	−	0.346438i	0.169013	−	0.985614i	$-0.445942\pi$
$89$	−5.66084	−	3.26829i	−0.600047	−	0.346438i	−0.769060	+	0.639176i	$0.779275\pi$
$90$		0			0
$91$	−2.85027	+	5.81558i	−0.298789	+	0.609639i
$92$		0			0
$93$	1.43384	+	5.35117i	0.148682	+	0.554890i
$94$		0			0
$95$	8.13378	+	14.0881i	0.834508	+	1.44541i
$96$		0			0
$97$	1.70409			0.173024			0.0865122	−	0.996251i	$-0.472428\pi$
$97$	1.70409			0.173024			0.0865122	+	0.996251i	$0.472428\pi$
$98$		0			0
$99$	−1.97179	+	1.97179i	−0.198172	+	0.198172i
$100$		0			0
$101$	0.632838	−	2.36178i	0.0629697	−	0.235006i	−0.927267	−	0.374400i	$-0.877849\pi$
$101$	0.632838	−	2.36178i	0.0629697	−	0.235006i	0.990237	+	0.139394i	$0.0445154\pi$
$102$		0			0
$103$	−12.6328	−	7.29353i	−1.24474	−	0.718653i	−0.274687	−	0.961534i	$-0.588574\pi$
$103$	−12.6328	−	7.29353i	−1.24474	−	0.718653i	−0.970056	+	0.242881i	$0.921908\pi$
$104$		0			0
$105$	−9.12593	+	3.12275i	−0.890600	+	0.304749i
$106$		0			0
$107$	4.80256	−	1.28684i	0.464281	−	0.124404i	−0.0190927	−	0.999818i	$-0.506078\pi$
$107$	4.80256	−	1.28684i	0.464281	−	0.124404i	0.483374	+	0.875414i	$0.339411\pi$
$108$		0			0
$109$	−9.76613	−	2.61683i	−0.935426	−	0.250647i	−0.241259	−	0.970461i	$-0.577560\pi$
$109$	−9.76613	−	2.61683i	−0.935426	−	0.250647i	−0.694167	+	0.719814i	$0.744227\pi$
$110$		0			0
$111$	−0.211172			−0.0200435
$112$		0			0
$113$	5.22780			0.491791			0.245895	−	0.969296i	$-0.420918\pi$
$113$	5.22780			0.491791			0.245895	+	0.969296i	$0.420918\pi$
$114$		0			0
$115$	−5.85009	−	1.56753i	−0.545524	−	0.146173i
$116$		0			0
$117$	5.35667	−	1.43532i	0.495224	−	0.132695i
$118$		0			0
$119$	−3.71543	−	3.24230i	−0.340593	−	0.297222i
$120$		0			0
$121$	8.21419	+	4.74247i	0.746745	+	0.431133i
$122$		0			0
$123$	0.0976407	−	0.364400i	0.00880396	−	0.0328568i
$124$		0			0
$125$	24.3460	−	24.3460i	2.17757	−	2.17757i
$126$		0			0
$127$	1.20463			0.106893			0.0534467	−	0.998571i	$-0.482979\pi$
$127$	1.20463			0.106893			0.0534467	+	0.998571i	$0.482979\pi$
$128$		0			0
$129$	−3.36041	−	5.82039i	−0.295867	−	0.512457i
$130$		0			0
$131$	0.809509	+	3.02113i	0.0707272	+	0.263957i	0.992231	−	0.124413i	$-0.0397048\pi$
$131$	0.809509	+	3.02113i	0.0707272	+	0.263957i	−0.921503	+	0.388370i	$0.873038\pi$
$132$		0			0
$133$	5.64191	+	8.39918i	0.489216	+	0.728301i
$134$		0			0
$135$	16.6242	+	9.59798i	1.43078	+	0.826063i
$136$		0			0
$137$	0.286357	−	0.165329i	0.0244652	−	0.0141250i	−0.487718	−	0.873001i	$-0.662170\pi$
$137$	0.286357	−	0.165329i	0.0244652	−	0.0141250i	0.512183	+	0.858877i	$0.328837\pi$
$138$		0			0
$139$	2.41295	+	2.41295i	0.204663	+	0.204663i	0.801995	−	0.597331i	$-0.203772\pi$
$139$	2.41295	+	2.41295i	0.204663	+	0.204663i	−0.597331	+	0.801995i	$0.703772\pi$
$140$		0			0
$141$	−4.38064	+	4.38064i	−0.368917	+	0.368917i
$142$		0			0
$143$	1.50653	+	2.60939i	0.125982	+	0.218208i
$144$		0			0
$145$	−0.546903	+	0.947264i	−0.0454179	+	0.0786660i
$146$		0			0
$147$	−5.53134	+	2.32291i	−0.456217	+	0.191591i
$148$		0			0
$149$	−2.36381	+	0.633381i	−0.193651	+	0.0518886i	−0.354341	−	0.935116i	$-0.615295\pi$
$149$	−2.36381	+	0.633381i	−0.193651	+	0.0518886i	0.160690	+	0.987005i	$0.448628\pi$
$150$		0			0
$151$	−11.4320	+	6.60029i	−0.930326	+	0.537124i	−0.886915	−	0.461934i	$-0.847156\pi$
$151$	−11.4320	+	6.60029i	−0.930326	+	0.537124i	−0.0434110	+	0.999057i	$0.513823\pi$
$152$		0			0
$153$			4.22246i			0.341365i
$154$		0			0
$155$	−19.4427	−	19.4427i	−1.56167	−	1.56167i
$156$		0			0
$157$	−3.45433	−	0.925585i	−0.275685	−	0.0738697i	0.118327	−	0.992975i	$-0.462247\pi$
$157$	−3.45433	−	0.925585i	−0.275685	−	0.0738697i	−0.394013	+	0.919105i	$0.628913\pi$
$158$		0			0
$159$	4.30007	−	7.44794i	0.341018	−	0.590660i
$160$		0			0
$161$	−3.69643	−	0.725873i	−0.291319	−	0.0572068i
$162$		0			0
$163$	2.90781	+	10.8521i	0.227757	+	0.850001i	0.981281	+	0.192581i	$0.0616857\pi$
$163$	2.90781	+	10.8521i	0.227757	+	0.850001i	−0.753524	+	0.657420i	$0.771648\pi$
$164$		0			0
$165$	−1.16141	+	4.33443i	−0.0904155	+	0.337435i
$166$		0			0
$167$		−	11.7171i		−	0.906693i	−0.891334	−	0.453346i	$-0.850230\pi$
$167$		−	11.7171i		−	0.906693i	0.891334	−	0.453346i	$-0.149770\pi$
$168$		0			0
$169$			7.00785i			0.539065i
$170$		0			0
$171$	2.24238	−	8.36867i	0.171479	−	0.639968i
$172$		0			0
$173$	2.83415	+	10.5772i	0.215476	+	0.804169i	0.985998	+	0.166755i	$0.0533290\pi$
$173$	2.83415	+	10.5772i	0.215476	+	0.804169i	−0.770522	+	0.637413i	$0.780004\pi$
$174$		0			0
$175$	22.7786	−	26.1025i	1.72190	−	1.97316i
$176$		0			0
$177$	−0.726183	+	1.25779i	−0.0545833	+	0.0945410i
$178$		0			0
$179$	16.1464	+	4.32641i	1.20684	+	0.323371i	0.805520	−	0.592568i	$-0.201886\pi$
$179$	16.1464	+	4.32641i	1.20684	+	0.323371i	0.401317	+	0.915939i	$0.368553\pi$
$180$		0			0
$181$	−15.9791	−	15.9791i	−1.18772	−	1.18772i	−0.977696	−	0.210025i	$-0.932646\pi$
$181$	−15.9791	−	15.9791i	−1.18772	−	1.18772i	−0.210025	−	0.977696i	$-0.567354\pi$
$182$		0			0
$183$		−	8.00010i		−	0.591385i
$184$		0			0
$185$	0.907680	−	0.524050i	0.0667340	−	0.0385289i
$186$		0			0
$187$	−2.21598	+	0.593770i	−0.162048	+	0.0434208i
$188$		0			0
$189$	10.7212	+	5.25454i	0.779851	+	0.382212i
$190$		0			0
$191$	−4.97963	+	8.62497i	−0.360313	+	0.624081i	−0.988012	−	0.154375i	$-0.950664\pi$
$191$	−4.97963	+	8.62497i	−0.360313	+	0.624081i	0.627699	+	0.778456i	$0.283997\pi$
$192$		0			0
$193$	−4.02312	−	6.96824i	−0.289590	−	0.501585i	0.684122	−	0.729368i	$-0.260186\pi$
$193$	−4.02312	−	6.96824i	−0.289590	−	0.501585i	−0.973712	+	0.227783i	$0.926852\pi$
$194$		0			0
$195$	6.31028	−	6.31028i	0.451889	−	0.451889i
$196$		0			0
$197$	6.01951	+	6.01951i	0.428873	+	0.428873i	0.888244	−	0.459372i	$-0.151925\pi$
$197$	6.01951	+	6.01951i	0.428873	+	0.428873i	−0.459372	+	0.888244i	$0.651925\pi$
$198$		0			0
$199$	−12.7812	+	7.37922i	−0.906035	+	0.523099i	−0.879153	−	0.476539i	$-0.841891\pi$
$199$	−12.7812	+	7.37922i	−0.906035	+	0.523099i	−0.0268814	+	0.999639i	$0.508558\pi$
$200$		0			0
$201$	7.37224	+	4.25636i	0.519998	+	0.300221i
$202$		0			0
$203$	−0.299410	+	0.610904i	−0.0210144	+	0.0428771i
$204$		0			0
$205$	0.484616	+	1.80861i	0.0338470	+	0.126319i
$206$		0			0
$207$	1.61279	+	2.79344i	0.112097	+	0.194158i
$208$		0			0
$209$	4.70727			0.325609
$210$		0			0
$211$	−14.5016	+	14.5016i	−0.998329	+	0.998329i	−0.999999	−	0.00166940i	$-0.999469\pi$
$211$	−14.5016	+	14.5016i	−0.998329	+	0.998329i	0.00166940	+	0.999999i	$0.499469\pi$
$212$		0			0
$213$	2.50461	−	9.34732i	0.171613	−	0.640468i
$214$		0			0
$215$	28.8881	+	16.6786i	1.97015	+	1.13747i
$216$		0			0
$217$	−12.8856	−	11.2448i	−0.874733	−	0.763344i
$218$		0			0
$219$	7.77241	−	2.08261i	0.525211	−	0.140730i
$220$		0			0
$221$	4.40698	+	1.18085i	0.296445	+	0.0794323i
$222$		0			0
$223$	−4.72461			−0.316383			−0.158192	−	0.987408i	$-0.550566\pi$
$223$	−4.72461			−0.316383			−0.158192	+	0.987408i	$0.550566\pi$
$224$		0			0
$225$	−29.6646			−1.97764
$226$		0			0
$227$	−15.9081	−	4.26257i	−1.05586	−	0.282917i	−0.311189	−	0.950348i	$-0.600727\pi$
$227$	−15.9081	−	4.26257i	−1.05586	−	0.282917i	−0.744671	+	0.667431i	$0.767394\pi$
$228$		0			0
$229$	−13.3085	+	3.56601i	−0.879453	+	0.235649i	−0.670171	−	0.742207i	$-0.733779\pi$
$229$	−13.3085	+	3.56601i	−0.879453	+	0.235649i	−0.209282	+	0.977855i	$0.567113\pi$
$230$		0			0
$231$	−0.537812	+	2.73875i	−0.0353854	+	0.180196i
$232$		0			0
$233$	12.0806	+	6.97475i	0.791428	+	0.456931i	0.840465	−	0.541866i	$-0.182282\pi$
$233$	12.0806	+	6.97475i	0.791428	+	0.456931i	−0.0490370	+	0.998797i	$0.515615\pi$
$234$		0			0
$235$	7.95822	−	29.7005i	0.519137	−	1.93745i
$236$		0			0
$237$	7.91401	−	7.91401i	0.514070	−	0.514070i
$238$		0			0
$239$	23.1794			1.49935			0.749674	−	0.661807i	$-0.230210\pi$
$239$	23.1794			1.49935			0.749674	+	0.661807i	$0.230210\pi$
$240$		0			0
$241$	2.09690	+	3.63195i	0.135073	+	0.233954i	0.925626	−	0.378441i	$-0.123540\pi$
$241$	2.09690	+	3.63195i	0.135073	+	0.233954i	−0.790552	+	0.612395i	$0.790206\pi$
$242$		0			0
$243$	−4.15362	−	15.5015i	−0.266455	−	0.994422i
$244$		0			0
$245$	18.0108	−	23.7113i	1.15067	−	1.51486i
$246$		0			0
$247$	−8.10728	−	4.68074i	−0.515854	−	0.297828i
$248$		0			0
$249$	−3.22168	+	1.86004i	−0.204166	+	0.117875i
$250$		0			0
$251$	16.5937	+	16.5937i	1.04739	+	1.04739i	0.998820	+	0.0485679i	$0.0154657\pi$
$251$	16.5937	+	16.5937i	1.04739	+	1.04739i	0.0485679	+	0.998820i	$0.484534\pi$
$252$		0			0
$253$	−1.23923	+	1.23923i	−0.0779095	+	0.0779095i
$254$		0			0
$255$	3.39741	+	5.88449i	0.212754	+	0.368501i
$256$		0			0
$257$	5.52727	−	9.57351i	0.344782	−	0.597179i	−0.640532	−	0.767931i	$-0.721286\pi$
$257$	5.52727	−	9.57351i	0.344782	−	0.597179i	0.985314	+	0.170752i	$0.0546196\pi$
$258$		0			0
$259$	0.541149	−	0.363501i	0.0336254	−	0.0225869i
$260$		0			0
$261$	0.562697	−	0.150774i	0.0348301	−	0.00933269i
$262$		0			0
$263$	14.1936	−	8.19466i	0.875213	−	0.505304i	0.00613562	−	0.999981i	$-0.498047\pi$
$263$	14.1936	−	8.19466i	0.875213	−	0.505304i	0.869077	+	0.494677i	$0.164714\pi$
$264$		0			0
$265$			42.6847i			2.62210i
$266$		0			0
$267$	3.96130	+	3.96130i	0.242428	+	0.242428i
$268$		0			0
$269$	−1.24494	−	0.333581i	−0.0759055	−	0.0203388i	0.220666	−	0.975349i	$-0.429177\pi$
$269$	−1.24494	−	0.333581i	−0.0759055	−	0.0203388i	−0.296572	+	0.955011i	$0.595843\pi$
$270$		0			0
$271$	−10.0688	+	17.4397i	−0.611639	+	1.05939i	0.379326	+	0.925263i	$0.376156\pi$
$271$	−10.0688	+	17.4397i	−0.611639	+	1.05939i	−0.990964	+	0.134126i	$0.957177\pi$
$272$		0			0
$273$	3.64958	−	4.18213i	0.220882	−	0.253114i
$274$		0			0
$275$	−4.17149	−	15.5682i	−0.251550	−	0.938798i
$276$		0			0
$277$	8.35382	−	31.1769i	0.501932	−	1.87324i	0.0148319	−	0.999890i	$-0.495279\pi$
$277$	8.35382	−	31.1769i	0.501932	−	1.87324i	0.487100	−	0.873346i	$-0.338055\pi$
$278$		0			0
$279$			14.6441i			0.876717i
$280$		0			0
$281$		−	13.6466i		−	0.814089i	−0.913408	−	0.407044i	$-0.866559\pi$
$281$		−	13.6466i		−	0.814089i	0.913408	−	0.407044i	$-0.133441\pi$
$282$		0			0
$283$	4.86306	−	18.1492i	0.289079	−	1.07886i	−0.656728	−	0.754128i	$-0.728060\pi$
$283$	4.86306	−	18.1492i	0.289079	−	1.07886i	0.945807	−	0.324730i	$-0.105273\pi$
$284$		0			0
$285$	−3.60846	−	13.4669i	−0.213747	−	0.797713i
$286$		0			0
$287$	0.377048	+	1.10189i	0.0222564	+	0.0650423i
$288$		0			0
$289$	6.76307	−	11.7140i	0.397828	−	0.689058i
$290$		0			0
$291$	−1.41072	−	0.378000i	−0.0826977	−	0.0221588i
$292$		0			0
$293$	2.19433	+	2.19433i	0.128194	+	0.128194i	0.768293	−	0.640099i	$-0.221107\pi$
$293$	2.19433	+	2.19433i	0.128194	+	0.128194i	−0.640099	+	0.768293i	$0.721107\pi$
$294$		0			0
$295$		−	7.20847i		−	0.419693i
$296$		0			0
$297$	4.81047	−	2.77733i	0.279132	−	0.161157i
$298$		0			0
$299$	3.36655	−	0.902063i	0.194692	−	0.0521677i
$300$		0			0
$301$	18.6303	+	9.13090i	1.07384	+	0.526296i
$302$		0			0
$303$	−1.04778	+	1.81480i	−0.0601932	+	0.104258i
$304$		0			0
$305$	19.8533	+	34.3869i	1.13680	+	1.96899i
$306$		0			0
$307$	8.28029	−	8.28029i	0.472581	−	0.472581i	−0.430168	−	0.902749i	$-0.641546\pi$
$307$	8.28029	−	8.28029i	0.472581	−	0.472581i	0.902749	+	0.430168i	$0.141546\pi$
$308$		0			0
$309$	8.84007	+	8.84007i	0.502894	+	0.502894i
$310$		0			0
$311$	7.36500	−	4.25218i	0.417631	−	0.241119i	−0.276432	−	0.961033i	$-0.589152\pi$
$311$	7.36500	−	4.25218i	0.417631	−	0.241119i	0.694063	+	0.719914i	$0.255819\pi$
$312$		0			0
$313$	−14.1001	−	8.14069i	−0.796984	−	0.460139i	0.0454317	−	0.998967i	$-0.485534\pi$
$313$	−14.1001	−	8.14069i	−0.796984	−	0.460139i	−0.842415	+	0.538829i	$0.818867\pi$
$314$		0			0
$315$	−25.4376	+	1.72975i	−1.43325	+	0.0974605i
$316$		0			0
$317$	−5.85666	−	21.8574i	−0.328943	−	1.22763i	−0.910288	−	0.413975i	$-0.864140\pi$
$317$	−5.85666	−	21.8574i	−0.328943	−	1.22763i	0.581345	−	0.813657i	$-0.302526\pi$
$318$		0			0
$319$	0.158255	+	0.274106i	0.00886058	+	0.0153470i
$320$		0			0
$321$	−4.26120			−0.237837
$322$		0			0
$323$	5.04015	−	5.04015i	0.280442	−	0.280442i
$324$		0			0
$325$	−8.29595	+	30.9609i	−0.460177	+	1.71740i
$326$		0			0
$327$	7.50434	+	4.33263i	0.414991	+	0.239595i
$328$		0			0
$329$	3.68520	−	18.7665i	0.203172	−	1.03463i
$330$		0			0
$331$	−11.2314	+	3.00945i	−0.617334	+	0.165414i	−0.553915	−	0.832573i	$-0.686867\pi$
$331$	−11.2314	+	3.00945i	−0.617334	+	0.165414i	−0.0634183	+	0.997987i	$0.520200\pi$
$332$		0			0
$333$	−0.539183	−	0.144474i	−0.0295470	−	0.00791711i
$334$		0			0
$335$	−42.2509			−2.30841
$336$		0			0
$337$	15.7590			0.858445			0.429223	−	0.903199i	$-0.358788\pi$
$337$	15.7590			0.858445			0.429223	+	0.903199i	$0.358788\pi$
$338$		0			0
$339$	−4.32779	−	1.15963i	−0.235053	−	0.0629823i
$340$		0			0
$341$	−7.68532	+	2.05928i	−0.416183	+	0.111516i
$342$		0			0
$343$	10.1761	−	15.4741i	0.549455	−	0.835523i
$344$		0			0
$345$	4.49523	+	2.59533i	0.242015	+	0.139728i
$346$		0			0
$347$	−3.95167	+	14.7478i	−0.212137	+	0.791706i	0.775018	+	0.631939i	$0.217741\pi$
$347$	−3.95167	+	14.7478i	−0.212137	+	0.791706i	−0.987155	+	0.159766i	$0.948926\pi$
$348$		0			0
$349$	−6.62124	+	6.62124i	−0.354427	+	0.354427i	−0.861754	−	0.507327i	$-0.830634\pi$
$349$	−6.62124	+	6.62124i	−0.354427	+	0.354427i	0.507327	+	0.861754i	$0.330634\pi$
$350$		0			0
$351$	−11.0467			−0.589628
$352$		0			0
$353$	−10.9032	−	18.8849i	−0.580319	−	1.00514i	−0.995441	−	0.0953759i	$-0.969595\pi$
$353$	−10.9032	−	18.8849i	−0.580319	−	1.00514i	0.415123	−	0.909765i	$-0.363739\pi$
$354$		0			0
$355$	12.4310	+	46.3931i	0.659769	+	2.46229i
$356$		0			0
$357$	2.35658	+	3.50826i	0.124723	+	0.185677i
$358$		0			0
$359$	−29.8216	−	17.2175i	−1.57392	−	0.908704i	−0.995681	−	0.0928386i	$-0.970406\pi$
$359$	−29.8216	−	17.2175i	−1.57392	−	0.908704i	−0.578241	−	0.815866i	$-0.696261\pi$
$360$		0			0
$361$	3.78856	−	2.18733i	0.199398	−	0.115123i
$362$		0			0
$363$	−5.74807	−	5.74807i	−0.301695	−	0.301695i
$364$		0			0
$365$	−28.2399	+	28.2399i	−1.47815	+	1.47815i
$366$		0			0
$367$	1.65812	+	2.87195i	0.0865533	+	0.149915i	0.906052	−	0.423166i	$-0.139081\pi$
$367$	1.65812	+	2.87195i	0.0865533	+	0.149915i	−0.819499	+	0.573081i	$0.805748\pi$
$368$		0			0
$369$	0.498611	−	0.863619i	0.0259566	−	0.0449582i
$370$		0			0
$371$	1.80118	+	26.4880i	0.0935127	+	1.37519i
$372$		0			0
$373$	−22.4918	+	6.02667i	−1.16458	+	0.312049i	−0.788795	−	0.614656i	$-0.789295\pi$
$373$	−22.4918	+	6.02667i	−1.16458	+	0.312049i	−0.375789	+	0.926705i	$0.622628\pi$
$374$		0			0
$375$	−25.5550	+	14.7542i	−1.31966	+	0.761903i
$376$		0			0
$377$		−	0.629452i		−	0.0324184i
$378$		0			0
$379$	−2.77271	−	2.77271i	−0.142425	−	0.142425i	0.632299	−	0.774724i	$-0.282111\pi$
$379$	−2.77271	−	2.77271i	−0.142425	−	0.142425i	−0.774724	+	0.632299i	$0.782111\pi$
$380$		0			0
$381$	−0.997239	−	0.267209i	−0.0510901	−	0.0136895i
$382$		0			0
$383$	10.6953	−	18.5248i	0.546504	−	0.946572i	−0.452007	−	0.892014i	$-0.649292\pi$
$383$	10.6953	−	18.5248i	0.546504	−	0.946572i	0.998511	−	0.0545578i	$-0.0173749\pi$
$384$		0			0
$385$	−4.48487	−	13.1066i	−0.228570	−	0.667975i
$386$		0			0
$387$	−4.59806	−	17.1602i	−0.233733	−	0.872302i
$388$		0			0
$389$	−6.51295	+	24.3067i	−0.330220	+	1.23240i	0.578740	+	0.815512i	$0.303545\pi$
$389$	−6.51295	+	24.3067i	−0.330220	+	1.23240i	−0.908959	+	0.416885i	$0.863122\pi$
$390$		0			0
$391$			2.65372i			0.134204i
$392$		0			0
$393$		−	2.68058i		−	0.135217i
$394$		0			0
$395$	−14.3772	+	53.6564i	−0.723395	+	2.69975i
$396$		0			0
$397$	0.101204	+	0.377699i	0.00507929	+	0.0189562i	0.968419	−	0.249328i	$-0.0802099\pi$
$397$	0.101204	+	0.377699i	0.00507929	+	0.0189562i	−0.963340	+	0.268285i	$0.913543\pi$
$398$		0			0
$399$	−2.80750	−	8.20466i	−0.140551	−	0.410747i
$400$		0			0
$401$	−9.92701	+	17.1941i	−0.495731	+	0.858631i	−0.999988	−	0.00492242i	$-0.998433\pi$
$401$	−9.92701	+	17.1941i	−0.495731	+	0.858631i	0.504257	+	0.863554i	$0.331766\pi$
$402$		0			0
$403$	15.2840	+	4.09534i	0.761350	+	0.204003i
$404$		0			0
$405$	8.80940	+	8.80940i	0.437743	+	0.437743i
$406$		0			0
$407$		−	0.303284i		−	0.0150332i
$408$		0			0
$409$	30.4501	−	17.5804i	1.50566	−	0.869295i	0.505685	−	0.862718i	$-0.331240\pi$
$409$	30.4501	−	17.5804i	1.50566	−	0.869295i	0.999978	−	0.00657643i	$-0.00209336\pi$
$410$		0			0
$411$	−0.273731	+	0.0733461i	−0.0135022	+	0.00361789i
$412$		0			0
$413$	−0.304178	−	4.47322i	−0.0149676	−	0.220113i
$414$		0			0
$415$	9.23185	−	15.9900i	0.453174	−	0.784920i
$416$		0			0
$417$	−1.46230	−	2.53277i	−0.0716089	−	0.124030i
$418$		0			0
$419$	17.1877	−	17.1877i	0.839673	−	0.839673i	−0.149142	−	0.988816i	$-0.547651\pi$
$419$	17.1877	−	17.1877i	0.839673	−	0.839673i	0.988816	+	0.149142i	$0.0476513\pi$
$420$		0			0
$421$	12.9543	+	12.9543i	0.631352	+	0.631352i	0.948407	−	0.317055i	$-0.102694\pi$
$421$	12.9543	+	12.9543i	0.631352	+	0.631352i	−0.317055	+	0.948407i	$0.602694\pi$
$422$		0			0
$423$	−14.1821	+	8.18803i	−0.689557	+	0.398116i
$424$		0			0
$425$	−21.1356	−	12.2026i	−1.02523	−	0.591915i
$426$		0			0
$427$	13.7710	+	20.5011i	0.666426	+	0.992117i
$428$		0			0
$429$	−0.668354	−	2.49433i	−0.0322685	−	0.120427i
$430$		0			0
$431$	−2.85225	−	4.94025i	−0.137388	−	0.237963i	0.789119	−	0.614240i	$-0.210537\pi$
$431$	−2.85225	−	4.94025i	−0.137388	−	0.237963i	−0.926507	+	0.376277i	$0.877204\pi$
$432$		0			0
$433$	−8.09187			−0.388870			−0.194435	−	0.980915i	$-0.562287\pi$
$433$	−8.09187			−0.388870			−0.194435	+	0.980915i	$0.562287\pi$
$434$		0			0
$435$	0.662870	−	0.662870i	0.0317822	−	0.0317822i
$436$		0			0
$437$	1.40928	−	5.25952i	0.0674152	−	0.251597i
$438$		0			0
$439$	−6.11379	−	3.52980i	−0.291795	−	0.168468i	0.346956	−	0.937881i	$-0.387215\pi$
$439$	−6.11379	−	3.52980i	−0.291795	−	0.168468i	−0.638751	+	0.769413i	$0.720549\pi$
$440$		0			0
$441$	−15.7124	+	2.14680i	−0.748207	+	0.102229i
$442$		0			0
$443$	14.7590	−	3.95466i	0.701221	−	0.187891i	0.109443	−	0.993993i	$-0.465093\pi$
$443$	14.7590	−	3.95466i	0.701221	−	0.187891i	0.591777	+	0.806102i	$0.298427\pi$
$444$		0			0
$445$	−26.8574	−	7.19641i	−1.27316	−	0.341142i
$446$		0			0
$447$	2.09735			0.0992014
$448$		0			0
$449$	19.5020			0.920356			0.460178	−	0.887827i	$-0.347786\pi$
$449$	19.5020			0.920356			0.460178	+	0.887827i	$0.347786\pi$
$450$		0			0
$451$	0.523350	+	0.140231i	0.0246436	+	0.00660322i
$452$		0			0
$453$	10.9280	−	2.92814i	0.513441	−	0.137576i
$454$		0			0
$455$	−5.30850	+	27.0330i	−0.248866	+	1.26733i
$456$		0			0
$457$	−8.72578	−	5.03783i	−0.408175	−	0.235660i	0.281830	−	0.959464i	$-0.409058\pi$
$457$	−8.72578	−	5.03783i	−0.408175	−	0.235660i	−0.690005	+	0.723804i	$0.742392\pi$
$458$		0			0
$459$	2.17692	−	8.12437i	0.101610	−	0.379213i
$460$		0			0
$461$	−15.9419	+	15.9419i	−0.742487	+	0.742487i	−0.973056	−	0.230569i	$-0.925941\pi$
$461$	−15.9419	+	15.9419i	−0.742487	+	0.742487i	0.230569	+	0.973056i	$0.425941\pi$
$462$		0			0
$463$	8.23452			0.382690			0.191345	−	0.981523i	$-0.438715\pi$
$463$	8.23452			0.382690			0.191345	+	0.981523i	$0.438715\pi$
$464$		0			0
$465$	11.7827	+	20.4082i	0.546409	+	0.946408i
$466$		0			0
$467$	−8.94417	−	33.3801i	−0.413887	−	1.54465i	−0.787055	−	0.616882i	$-0.788395\pi$
$467$	−8.94417	−	33.3801i	−0.413887	−	1.54465i	0.373169	−	0.927764i	$-0.378271\pi$
$468$		0			0
$469$	−26.2188	+	1.78288i	−1.21067	+	0.0823255i
$470$		0			0
$471$	2.65432	+	1.53247i	0.122305	+	0.0706126i
$472$		0			0
$473$	8.35922	−	4.82620i	0.384357	−	0.221909i
$474$		0			0
$475$	35.4092	+	35.4092i	1.62469	+	1.62469i
$476$		0			0
$477$	16.0749	−	16.0749i	0.736017	−	0.736017i
$478$		0			0
$479$	−3.70229	−	6.41255i	−0.169162	−	0.292997i	0.768964	−	0.639293i	$-0.220773\pi$
$479$	−3.70229	−	6.41255i	−0.169162	−	0.292997i	−0.938125	+	0.346296i	$0.887439\pi$
$480$		0			0
$481$	−0.301574	+	0.522342i	−0.0137506	+	0.0238167i
$482$		0			0
$483$	2.89904	+	1.42085i	0.131911	+	0.0646507i
$484$		0			0
$485$	7.00175	−	1.87611i	0.317933	−	0.0851899i
$486$		0			0
$487$	29.4458	−	17.0005i	1.33432	−	0.770368i	0.348359	−	0.937361i	$-0.386739\pi$
$487$	29.4458	−	17.0005i	1.33432	−	0.770368i	0.985958	+	0.166993i	$0.0534056\pi$
$488$		0			0
$489$		−	9.62880i		−	0.435429i
$490$		0			0
$491$	−2.03871	−	2.03871i	−0.0920055	−	0.0920055i	0.659606	−	0.751612i	$-0.270723\pi$
$491$	−2.03871	−	2.03871i	−0.0920055	−	0.0920055i	−0.751612	+	0.659606i	$0.770723\pi$
$492$		0			0
$493$	0.462935	+	0.124043i	0.0208496	+	0.00558662i
$494$		0			0
$495$	−5.93083	+	10.2725i	−0.266571	+	0.461714i
$496$		0			0
$497$	9.67175	+	28.2648i	0.433837	+	1.26785i
$498$		0			0
$499$	3.72332	+	13.8956i	0.166679	+	0.622053i	0.997820	+	0.0659922i	$0.0210212\pi$
$499$	3.72332	+	13.8956i	0.166679	+	0.622053i	−0.831141	+	0.556061i	$0.812312\pi$
$500$		0			0
$501$	−2.59907	+	9.69985i	−0.116118	+	0.433357i
$502$		0			0
$503$		−	27.2980i		−	1.21716i	−0.793494	−	0.608579i	$-0.791740\pi$
$503$		−	27.2980i		−	1.21716i	0.793494	−	0.608579i	$-0.208260\pi$
$504$		0			0
$505$		−	10.4008i		−	0.462828i
$506$		0			0
$507$	1.55447	−	5.80138i	0.0690366	−	0.257648i
$508$		0			0
$509$	5.34620	+	19.9523i	0.236966	+	0.884370i	0.977253	+	0.212078i	$0.0680232\pi$
$509$	5.34620	+	19.9523i	0.236966	+	0.884370i	−0.740286	+	0.672292i	$0.765310\pi$
$510$		0			0
$511$	−16.3327	+	18.7160i	−0.722515	+	0.827946i
$512$		0			0
$513$	−8.62906	+	14.9460i	−0.380982	+	0.659881i
$514$		0			0
$515$	−59.9351	−	16.0596i	−2.64106	−	0.707669i
$516$		0			0
$517$	−6.29146	−	6.29146i	−0.276698	−	0.276698i
$518$		0			0
$519$		−	9.38489i		−	0.411951i
$520$		0			0
$521$	−9.46400	+	5.46404i	−0.414625	+	0.239384i	−0.692775	−	0.721154i	$-0.743612\pi$
$521$	−9.46400	+	5.46404i	−0.414625	+	0.239384i	0.278150	+	0.960538i	$0.410279\pi$
$522$		0			0
$523$	−11.8976	+	3.18796i	−0.520247	+	0.139400i	−0.509381	−	0.860541i	$-0.670125\pi$
$523$	−11.8976	+	3.18796i	−0.520247	+	0.139400i	−0.0108660	+	0.999941i	$0.503459\pi$
$524$		0			0
$525$	−24.6471	+	16.5560i	−1.07569	+	0.722562i
$526$		0			0
$527$	−6.02390	+	10.4337i	−0.262405	+	0.454499i
$528$		0			0
$529$	−10.4864	−	18.1630i	−0.455930	−	0.789694i
$530$		0			0
$531$	−2.71468	+	2.71468i	−0.117807	+	0.117807i
$532$		0			0
$533$	−0.761918	−	0.761918i	−0.0330023	−	0.0330023i
$534$		0			0
$535$	18.3160	−	10.5747i	0.791868	−	0.457185i
$536$		0			0
$537$	−12.4069	−	7.16316i	−0.535399	−	0.309113i
$538$		0			0
$539$	−3.33616	−	7.94408i	−0.143699	−	0.342176i
$540$		0			0
$541$	4.34231	+	16.2057i	0.186691	+	0.696739i	0.994262	+	0.106969i	$0.0341146\pi$
$541$	4.34231	+	16.2057i	0.186691	+	0.696739i	−0.807572	+	0.589769i	$0.799219\pi$
$542$		0			0
$543$	9.68370	+	16.7727i	0.415567	+	0.719784i
$544$		0			0
$545$	−43.0079			−1.84226
$546$		0			0
$547$	23.2429	−	23.2429i	0.993797	−	0.993797i	−0.00618416	−	0.999981i	$-0.501968\pi$
$547$	23.2429	−	23.2429i	0.993797	−	0.993797i	0.999981	+	0.00618416i	$0.00196849\pi$
$548$		0			0
$549$	5.47329	−	20.4266i	0.233594	−	0.871786i
$550$		0			0
$551$	−0.851637	−	0.491693i	−0.0362810	−	0.0209468i
$552$		0			0
$553$	−6.65763	+	33.9033i	−0.283111	+	1.44171i
$554$		0			0
$555$	−0.867659	+	0.232488i	−0.0368301	+	0.00986858i
$556$		0			0
$557$	−22.7871	−	6.10578i	−0.965519	−	0.258710i	−0.258584	−	0.965989i	$-0.583256\pi$
$557$	−22.7871	−	6.10578i	−0.965519	−	0.258710i	−0.706935	+	0.707279i	$0.749923\pi$
$558$		0			0
$559$	−19.1960			−0.811904
$560$		0			0
$561$	1.96619			0.0830125
$562$		0			0
$563$	37.6366	+	10.0847i	1.58620	+	0.425020i	0.940837	−	0.338860i	$-0.110041\pi$
$563$	37.6366	+	10.0847i	1.58620	+	0.425020i	0.645358	+	0.763880i	$0.276708\pi$
$564$		0			0
$565$	21.4799	−	5.75553i	0.903667	−	0.242137i
$566$		0			0
$567$	5.83842	+	5.09495i	0.245191	+	0.213968i
$568$		0			0
$569$	10.2086	+	5.89395i	0.427968	+	0.247087i	0.698481	−	0.715629i	$-0.253860\pi$
$569$	10.2086	+	5.89395i	0.427968	+	0.247087i	−0.270513	+	0.962716i	$0.587193\pi$
$570$		0			0
$571$	0.732074	−	2.73214i	0.0306363	−	0.114336i	−0.948914	−	0.315534i	$-0.897817\pi$
$571$	0.732074	−	2.73214i	0.0306363	−	0.114336i	0.979551	+	0.201197i	$0.0644832\pi$
$572$		0			0
$573$	6.03552	−	6.03552i	0.252138	−	0.252138i
$574$		0			0
$575$	−18.6435			−0.777489
$576$		0			0
$577$	21.6020	+	37.4157i	0.899302	+	1.55764i	0.828389	+	0.560154i	$0.189258\pi$
$577$	21.6020	+	37.4157i	0.899302	+	1.55764i	0.0709133	+	0.997482i	$0.477409\pi$
$578$		0			0
$579$	1.78481	+	6.66100i	0.0741741	+	0.276822i
$580$		0			0
$581$	5.05410	−	10.3122i	0.209679	−	0.427822i
$582$		0			0
$583$	10.6967	+	6.17574i	0.443012	+	0.255773i
$584$		0			0
$585$	20.4292	−	11.7948i	0.844643	−	0.487655i
$586$		0			0
$587$	−28.1127	−	28.1127i	−1.16034	−	1.16034i	−0.984402	−	0.175935i	$-0.943705\pi$
$587$	−28.1127	−	28.1127i	−1.16034	−	1.16034i	−0.175935	−	0.984402i	$-0.556295\pi$
$588$		0			0
$589$	17.4799	−	17.4799i	0.720248	−	0.720248i
$590$		0			0
$591$	−3.64795	−	6.31844i	−0.150057	−	0.259906i
$592$		0			0
$593$	8.44544	−	14.6279i	0.346812	−	0.600697i	−0.638869	−	0.769316i	$-0.720597\pi$
$593$	8.44544	−	14.6279i	0.346812	−	0.600697i	0.985681	+	0.168619i	$0.0539307\pi$
$594$		0			0
$595$	−18.8355	−	9.23145i	−0.772180	−	0.378452i
$596$		0			0
$597$	12.2176	−	3.27371i	0.500035	−	0.133984i
$598$		0			0
$599$	−25.7194	+	14.8491i	−1.05087	+	0.606718i	−0.922892	−	0.385059i	$-0.874181\pi$
$599$	−25.7194	+	14.8491i	−1.05087	+	0.606718i	−0.127975	+	0.991777i	$0.540848\pi$
$600$		0			0
$601$		−	30.4663i		−	1.24275i	−0.783515	−	0.621373i	$-0.786575\pi$
$601$		−	30.4663i		−	1.24275i	0.783515	−	0.621373i	$-0.213425\pi$
$602$		0			0
$603$	15.9115	+	15.9115i	0.647966	+	0.647966i
$604$		0			0
$605$	38.9715	+	10.4424i	1.58442	+	0.424544i
$606$		0			0
$607$	16.6272	−	28.7992i	0.674879	−	1.16892i	−0.301626	−	0.953426i	$-0.597529\pi$
$607$	16.6272	−	28.7992i	0.674879	−	1.16892i	0.976504	−	0.215498i	$-0.0691373\pi$
$608$		0			0
$609$	0.383374	−	0.439316i	0.0155351	−	0.0178020i
$610$		0			0
$611$	4.57971	+	17.0917i	0.185275	+	0.691456i
$612$		0			0
$613$	−5.29950	+	19.7780i	−0.214045	+	0.798827i	0.772456	+	0.635069i	$0.219028\pi$
$613$	−5.29950	+	19.7780i	−0.214045	+	0.798827i	−0.986501	+	0.163758i	$0.947638\pi$
$614$		0			0
$615$		−	1.60474i		−	0.0647093i
$616$		0			0
$617$			2.64884i			0.106638i	0.998578	+	0.0533192i	$0.0169801\pi$
$617$			2.64884i			0.106638i	−0.998578	+	0.0533192i	$0.983020\pi$
$618$		0			0
$619$	9.69816	−	36.1940i	0.389802	−	1.45476i	−0.440654	−	0.897677i	$-0.645253\pi$
$619$	9.69816	−	36.1940i	0.389802	−	1.45476i	0.830456	−	0.557084i	$-0.188080\pi$
$620$		0			0
$621$	−1.66298	−	6.20631i	−0.0667330	−	0.249051i
$622$		0			0
$623$	−16.9700	−	3.33243i	−0.679890	−	0.133511i
$624$		0			0
$625$	40.4934	−	70.1366i	1.61974	−	2.80546i
$626$		0			0
$627$	−3.89687	−	1.04416i	−0.155626	−	0.0416999i
$628$		0			0
$629$	−0.324731	−	0.324731i	−0.0129479	−	0.0129479i
$630$		0			0
$631$			10.0919i			0.401753i	0.979617	+	0.200877i	$0.0643791\pi$
$631$			10.0919i			0.401753i	−0.979617	+	0.200877i	$0.935621\pi$
$632$		0			0
$633$	15.2217	−	8.78826i	0.605009	−	0.349302i
$634$		0			0
$635$	4.94955	−	1.32623i	0.196417	−	0.0526298i
$636$		0			0
$637$	−2.15348	+	16.9994i	−0.0853239	+	0.673539i
$638$		0			0
$639$	12.7900	−	22.1529i	0.505964	−	0.876355i
$640$		0			0
$641$	−15.3814	−	26.6413i	−0.607528	−	1.05227i	−0.991646	−	0.128986i	$-0.958828\pi$
$641$	−15.3814	−	26.6413i	−0.607528	−	1.05227i	0.384118	−	0.923284i	$-0.374505\pi$
$642$		0			0
$643$	−22.2297	+	22.2297i	−0.876653	+	0.876653i	−0.993187	−	0.116533i	$-0.962822\pi$
$643$	−22.2297	+	22.2297i	−0.876653	+	0.876653i	0.116533	+	0.993187i	$0.462822\pi$
$644$		0			0
$645$	−20.2151	−	20.2151i	−0.795970	−	0.795970i
$646$		0			0
$647$	14.3410	−	8.27978i	0.563803	−	0.325512i	−0.190867	−	0.981616i	$-0.561130\pi$
$647$	14.3410	−	8.27978i	0.563803	−	0.325512i	0.754670	+	0.656104i	$0.227797\pi$
$648$		0			0
$649$	−1.80643	−	1.04294i	−0.0709085	−	0.0409390i
$650$		0			0
$651$	8.17293	+	12.1671i	0.320322	+	0.476868i
$652$		0			0
$653$	−9.67497	−	36.1075i	−0.378611	−	1.41300i	−0.847996	−	0.530002i	$-0.822191\pi$
$653$	−9.67497	−	36.1075i	−0.378611	−	1.41300i	0.469385	−	0.882993i	$-0.344476\pi$
$654$		0			0
$655$	6.65220	+	11.5219i	0.259923	+	0.450200i
$656$		0			0
$657$	21.2701			0.829824
$658$		0			0
$659$	14.8808	−	14.8808i	0.579674	−	0.579674i	−0.355139	−	0.934813i	$-0.615567\pi$
$659$	14.8808	−	14.8808i	0.579674	−	0.579674i	0.934813	+	0.355139i	$0.115567\pi$
$660$		0			0
$661$	−2.96488	+	11.0651i	−0.115321	+	0.430382i	−0.999311	−	0.0371232i	$-0.988181\pi$
$661$	−2.96488	+	11.0651i	−0.115321	+	0.430382i	0.883990	+	0.467506i	$0.154847\pi$
$662$		0			0
$663$	−3.38634	−	1.95510i	−0.131515	−	0.0759299i
$664$		0			0
$665$	32.4284	+	28.2990i	1.25752	+	1.09739i
$666$		0			0
$667$	0.353642	−	0.0947582i	0.0136931	−	0.00366905i
$668$		0			0
$669$	3.91122	+	1.04801i	0.151217	+	0.0405183i
$670$		0			0
$671$	11.4897			0.443555
$672$		0			0
$673$	20.7704			0.800639			0.400319	−	0.916376i	$-0.368899\pi$
$673$	20.7704			0.800639			0.400319	+	0.916376i	$0.368899\pi$
$674$		0			0
$675$	57.0772	+	15.2938i	2.19690	+	0.588658i
$676$		0			0
$677$	26.5605	−	7.11687i	1.02080	−	0.273524i	0.290667	−	0.956824i	$-0.406123\pi$
$677$	26.5605	−	7.11687i	1.02080	−	0.273524i	0.730137	+	0.683301i	$0.239456\pi$
$678$		0			0
$679$	4.26578	−	1.45968i	0.163706	−	0.0560174i
$680$		0			0
$681$	12.2239	+	7.05746i	0.468420	+	0.270442i
$682$		0			0
$683$	0.558762	−	2.08533i	0.0213804	−	0.0797928i	−0.954411	−	0.298495i	$-0.903516\pi$
$683$	0.558762	−	2.08533i	0.0213804	−	0.0797928i	0.975792	+	0.218702i	$0.0701822\pi$
$684$		0			0
$685$	0.994563	−	0.994563i	0.0380003	−	0.0380003i
$686$		0			0
$687$	11.8083			0.450517
$688$		0			0
$689$	−12.2819	−	21.2728i	−0.467902	−	0.810429i
$690$		0			0
$691$	0.631272	+	2.35594i	0.0240147	+	0.0896242i	0.976893	−	0.213728i	$-0.0685607\pi$
$691$	0.631272	+	2.35594i	0.0240147	+	0.0896242i	−0.952878	+	0.303352i	$0.901894\pi$
$692$		0			0
$693$	−3.24691	+	6.62488i	−0.123340	+	0.251658i
$694$		0			0
$695$	12.5708	+	7.25775i	0.476837	+	0.275302i
$696$		0			0
$697$	0.710506	−	0.410211i	0.0269124	−	0.0155379i
$698$		0			0
$699$	−8.45369	−	8.45369i	−0.319748	−	0.319748i
$700$		0			0
$701$	6.81795	−	6.81795i	0.257510	−	0.257510i	−0.566530	−	0.824041i	$-0.691715\pi$
$701$	6.81795	−	6.81795i	0.257510	−	0.257510i	0.824041	+	0.566530i	$0.191715\pi$
$702$		0			0
$703$	0.471146	+	0.816049i	0.0177696	+	0.0307779i
$704$		0			0
$705$	−13.1763	+	22.8220i	−0.496247	+	0.859525i
$706$		0			0
$707$	−0.438885	−	6.45421i	−0.0165060	−	0.242736i
$708$		0			0
$709$	11.3630	−	3.04470i	0.426745	−	0.114346i	−0.0390527	−	0.999237i	$-0.512434\pi$
$709$	11.3630	−	3.04470i	0.426745	−	0.114346i	0.465798	+	0.884891i	$0.345767\pi$
$710$		0			0
$711$	25.6212	−	14.7924i	0.960869	−	0.554758i
$712$		0			0
$713$			9.20346i			0.344672i
$714$		0			0
$715$	9.06280	+	9.06280i	0.338929	+	0.338929i
$716$		0			0
$717$	−19.1888	−	5.14163i	−0.716619	−	0.192018i
$718$		0			0
$719$	16.4827	−	28.5490i	0.614703	−	1.06470i	−0.375734	−	0.926728i	$-0.622609\pi$
$719$	16.4827	−	28.5490i	0.614703	−	1.06470i	0.990437	−	0.137969i	$-0.0440573\pi$
$720$		0			0
$721$	−37.8705	−	7.43668i	−1.41037	−	0.276956i
$722$		0			0
$723$	−0.930267	−	3.47180i	−0.0345970	−	0.129118i
$724$		0			0
$725$	−0.871457	+	3.25232i	−0.0323651	+	0.120788i
$726$		0			0
$727$			13.8871i			0.515045i	0.966272	+	0.257523i	$0.0829062\pi$
$727$			13.8871i			0.515045i	−0.966272	+	0.257523i	$0.917094\pi$
$728$		0			0
$729$			4.96768i			0.183988i
$730$		0			0
$731$	3.78286	−	14.1178i	0.139914	−	0.522167i
$732$		0			0
$733$	−2.86505	−	10.6925i	−0.105823	−	0.394936i	0.892614	−	0.450821i	$-0.148869\pi$
$733$	−2.86505	−	10.6925i	−0.105823	−	0.394936i	−0.998437	+	0.0558846i	$0.982202\pi$
$734$		0			0
$735$	−20.1697	+	15.6341i	−0.743970	+	0.576671i
$736$		0			0
$737$	−6.11297	+	10.5880i	−0.225174	+	0.390013i
$738$		0			0
$739$	−10.7856	−	2.88999i	−0.396754	−	0.106310i	0.0549249	−	0.998490i	$-0.482508\pi$
$739$	−10.7856	−	2.88999i	−0.396754	−	0.106310i	−0.451679	+	0.892181i	$0.649175\pi$
$740$		0			0
$741$	5.67325	+	5.67325i	0.208412	+	0.208412i
$742$		0			0
$743$			49.5292i			1.81705i	0.417832	+	0.908524i	$0.362790\pi$
$743$			49.5292i			1.81705i	−0.417832	+	0.908524i	$0.637210\pi$
$744$		0			0
$745$	−9.01507	+	5.20485i	−0.330287	+	0.190691i
$746$		0			0
$747$	−9.49845	+	2.54510i	−0.347530	+	0.0931204i
$748$		0			0
$749$	10.9198	−	7.33504i	0.398999	−	0.268017i
$750$		0			0
$751$	−0.523146	+	0.906116i	−0.0190899	+	0.0330646i	−0.875413	−	0.483377i	$-0.839410\pi$
$751$	−0.523146	+	0.906116i	−0.0190899	+	0.0330646i	0.856323	+	0.516441i	$0.172744\pi$
$752$		0			0
$753$	−10.0562	−	17.4178i	−0.366467	−	0.634739i
$754$		0			0
$755$	−39.7052	+	39.7052i	−1.44502	+	1.44502i
$756$		0			0
$757$	−29.5274	−	29.5274i	−1.07319	−	1.07319i	−0.997101	−	0.0760892i	$-0.975757\pi$
$757$	−29.5274	−	29.5274i	−1.07319	−	1.07319i	−0.0760892	−	0.997101i	$-0.524243\pi$
$758$		0			0
$759$	1.30077	−	0.750998i	0.0472148	−	0.0272595i
$760$		0			0
$761$	33.9144	+	19.5805i	1.22940	+	0.709793i	0.966904	−	0.255141i	$-0.0821220\pi$
$761$	33.9144	+	19.5805i	1.22940	+	0.709793i	0.262493	+	0.964934i	$0.415455\pi$
$762$		0			0
$763$	−26.6886	+	1.81482i	−0.966193	+	0.0657009i
$764$		0			0
$765$	4.64869	+	17.3492i	0.168074	+	0.627260i
$766$		0			0
$767$	2.07412	+	3.59249i	0.0748923	+	0.129717i
$768$		0			0
$769$	28.7025			1.03504			0.517519	−	0.855672i	$-0.326856\pi$
$769$	28.7025			1.03504			0.517519	+	0.855672i	$0.326856\pi$
$770$		0			0
$771$	−6.69929	+	6.69929i	−0.241269	+	0.241269i
$772$		0			0
$773$	3.90180	−	14.5617i	0.140338	−	0.523748i	−0.859581	−	0.511000i	$-0.829275\pi$
$773$	3.90180	−	14.5617i	0.140338	−	0.523748i	0.999919	−	0.0127485i	$-0.00405807\pi$
$774$		0			0
$775$	−73.3012	−	42.3205i	−2.63306	−	1.52020i
$776$		0			0
$777$	−0.528616	+	0.180884i	−0.0189640	+	0.00648917i
$778$		0			0
$779$	−1.62603	+	0.435693i	−0.0582586	+	0.0156103i
$780$		0			0
$781$	13.4246	+	3.59710i	0.480369	+	0.128715i
$782$		0			0
$783$	−1.16041			−0.0414697
$784$		0			0
$785$	−15.2121			−0.542944
$786$		0			0
$787$	−22.6434	−	6.06728i	−0.807150	−	0.216275i	−0.168429	−	0.985714i	$-0.553869\pi$
$787$	−22.6434	−	6.06728i	−0.807150	−	0.216275i	−0.638721	+	0.769438i	$0.720536\pi$
$788$		0			0
$789$	−13.5677	+	3.63546i	−0.483024	+	0.129426i
$790$		0			0
$791$	13.0865	−	4.47800i	0.465303	−	0.159219i
$792$		0			0
$793$	−19.7886	−	11.4249i	−0.702713	−	0.405712i
$794$		0			0
$795$	9.46828	−	35.3361i	0.335805	−	1.25324i
$796$		0			0
$797$	3.81531	−	3.81531i	0.135145	−	0.135145i	−0.636298	−	0.771443i	$-0.719535\pi$
$797$	3.81531	−	3.81531i	0.135145	−	0.135145i	0.771443	+	0.636298i	$0.219535\pi$
$798$		0			0
$799$	−13.4727			−0.476631
$800$		0			0
$801$	7.40422	+	12.8245i	0.261615	+	0.453131i
$802$		0			0
$803$	2.99104	+	11.1627i	0.105551	+	0.393923i
$804$		0			0
$805$	−15.9870	+	1.08711i	−0.563467	+	0.0383156i
$806$		0			0
$807$	0.956619	+	0.552304i	0.0336746	+	0.0194420i
$808$		0			0
$809$	−24.7881	+	14.3114i	−0.871505	+	0.503164i	−0.867848	−	0.496830i	$-0.834497\pi$
$809$	−24.7881	+	14.3114i	−0.871505	+	0.503164i	−0.00365682	+	0.999993i	$0.501164\pi$
$810$		0			0
$811$	−9.30808	−	9.30808i	−0.326851	−	0.326851i	0.524537	−	0.851388i	$-0.324238\pi$
$811$	−9.30808	−	9.30808i	−0.326851	−	0.326851i	−0.851388	+	0.524537i	$0.824238\pi$
$812$		0			0
$813$	12.2039	−	12.2039i	0.428008	−	0.428008i
$814$		0			0
$815$	23.8951	+	41.3875i	0.837009	+	1.44974i
$816$		0			0
$817$	−14.9948	+	25.9718i	−0.524603	+	0.908639i
$818$		0			0
$819$	12.1797	−	8.18134i	0.425592	−	0.285879i
$820$		0			0
$821$	−16.1169	+	4.31850i	−0.562482	+	0.150717i	−0.528846	−	0.848718i	$-0.677375\pi$
$821$	−16.1169	+	4.31850i	−0.562482	+	0.150717i	−0.0336357	+	0.999434i	$0.510709\pi$
$822$		0			0
$823$	−9.41497	+	5.43573i	−0.328185	+	0.189478i	−0.655035	−	0.755598i	$-0.727346\pi$
$823$	−9.41497	+	5.43573i	−0.328185	+	0.189478i	0.326850	+	0.945076i	$0.394013\pi$
$824$		0			0
$825$			13.8133i			0.480918i
$826$		0			0
$827$	−9.97483	−	9.97483i	−0.346859	−	0.346859i	0.512079	−	0.858938i	$-0.328875\pi$
$827$	−9.97483	−	9.97483i	−0.346859	−	0.346859i	−0.858938	+	0.512079i	$0.828875\pi$
$828$		0			0
$829$	24.9645	+	6.68921i	0.867052	+	0.232326i	0.664813	−	0.747010i	$-0.268511\pi$
$829$	24.9645	+	6.68921i	0.867052	+	0.232326i	0.202240	+	0.979336i	$0.435178\pi$
$830$		0			0
$831$	−13.8312	+	23.9564i	−0.479801	+	0.831039i
$832$		0			0
$833$	−12.0779	−	4.93378i	−0.418476	−	0.170945i
$834$		0			0
$835$	−12.8998	−	48.1428i	−0.446417	−	1.66605i
$836$		0			0
$837$	7.54985	−	28.1764i	0.260961	−	0.973920i
$838$		0			0
$839$			35.9931i			1.24262i	0.783565	+	0.621310i	$0.213399\pi$
$839$			35.9931i			1.24262i	−0.783565	+	0.621310i	$0.786601\pi$
$840$		0			0
$841$			28.9339i			0.997720i
$842$		0			0
$843$	−3.02708	+	11.2972i	−0.104258	+	0.389097i
$844$		0			0
$845$	7.71526	+	28.7937i	0.265413	+	0.990535i
$846$		0			0
$847$	24.6245	+	4.83554i	0.846107	+	0.166151i
$848$		0			0
$849$	−8.05167	+	13.9459i	−0.276333	+	0.478622i
$850$		0			0
$851$	−0.338864	−	0.0907985i	−0.0116161	−	0.00311253i
$852$		0			0
$853$	11.5045	+	11.5045i	0.393906	+	0.393906i	0.876077	−	0.482171i	$-0.160152\pi$
$853$	11.5045	+	11.5045i	0.393906	+	0.393906i	−0.482171	+	0.876077i	$0.660152\pi$
$854$		0			0
$855$		−	36.8538i		−	1.26037i
$856$		0			0
$857$	−46.8795	+	27.0659i	−1.60137	+	0.924554i	−0.610161	+	0.792277i	$0.708895\pi$
$857$	−46.8795	+	27.0659i	−1.60137	+	0.924554i	−0.991213	+	0.132276i	$0.957771\pi$
$858$		0			0
$859$	6.87557	−	1.84230i	0.234591	−	0.0628586i	−0.139608	−	0.990207i	$-0.544584\pi$
$859$	6.87557	−	1.84230i	0.234591	−	0.0628586i	0.374199	+	0.927348i	$0.377918\pi$
$860$		0			0
$861$	−0.0677157	−	0.995822i	−0.00230775	−	0.0339375i
$862$		0			0
$863$	8.29596	−	14.3690i	0.282398	−	0.489127i	−0.689577	−	0.724212i	$-0.742204\pi$
$863$	8.29596	−	14.3690i	0.282398	−	0.489127i	0.971975	+	0.235085i	$0.0755369\pi$
$864$		0			0
$865$	23.2898	+	40.3391i	0.791877	+	1.37157i
$866$		0			0
$867$	−8.19713	+	8.19713i	−0.278389	+	0.278389i
$868$		0			0
$869$	11.3661	+	11.3661i	0.385567	+	0.385567i
$870$		0			0
$871$	21.0566	−	12.1570i	0.713475	−	0.411925i
$872$		0			0
$873$	−3.34336	−	1.93029i	−0.113156	−	0.0653305i
$874$		0			0
$875$	40.0901	−	81.7984i	1.35529	−	2.76529i
$876$		0			0
$877$	−7.85634	−	29.3202i	−0.265290	−	0.990074i	−0.962073	−	0.272793i	$-0.912053\pi$
$877$	−7.85634	−	29.3202i	−0.265290	−	0.990074i	0.696783	−	0.717282i	$-0.254614\pi$
$878$		0			0
$879$	−1.32981	−	2.30330i	−0.0448534	−	0.0776883i
$880$		0			0
$881$	−48.7213			−1.64146			−0.820731	−	0.571314i	$-0.806434\pi$
$881$	−48.7213			−1.64146			−0.820731	+	0.571314i	$0.806434\pi$
$882$		0			0
$883$	−2.32744	+	2.32744i	−0.0783247	+	0.0783247i	−0.745184	−	0.666859i	$-0.767638\pi$
$883$	−2.32744	+	2.32744i	−0.0783247	+	0.0783247i	0.666859	+	0.745184i	$0.267638\pi$
$884$		0			0
$885$	−1.59898	+	5.96746i	−0.0537490	+	0.200594i
$886$		0			0
$887$	45.9577	+	26.5337i	1.54311	+	0.890914i	0.998640	+	0.0521315i	$0.0166015\pi$
$887$	45.9577	+	26.5337i	1.54311	+	0.890914i	0.544467	+	0.838782i	$0.316732\pi$
$888$		0			0
$889$	3.01549	−	1.03185i	0.101136	−	0.0346072i
$890$		0			0
$891$	3.48219	−	0.933049i	0.116658	−	0.0312583i
$892$		0			0
$893$	26.7022	+	7.15483i	0.893555	+	0.239427i
$894$		0			0
$895$	71.1052			2.37678
$896$		0			0
$897$	−2.98706			−0.0997350
$898$		0			0
$899$	1.60552	+	0.430199i	0.0535472	+	0.0143479i
$900$		0			0
$901$	18.0656	−	4.84066i	0.601852	−	0.161266i
$902$		0			0
$903$	−13.3975	−	11.6915i	−0.445842	−	0.389069i
$904$		0			0
$905$	−83.2470	−	48.0627i	−2.76723	−	1.59766i
$906$		0			0
$907$	−14.4466	+	53.9154i	−0.479691	+	1.79023i	0.123172	+	0.992385i	$0.460693\pi$
$907$	−14.4466	+	53.9154i	−0.479691	+	1.79023i	−0.602863	+	0.797845i	$0.705973\pi$
$908$		0			0
$909$	−3.91688	+	3.91688i	−0.129915	+	0.129915i
$910$		0			0
$911$	44.7554			1.48281			0.741406	−	0.671057i	$-0.234159\pi$
$911$	44.7554			1.48281			0.741406	+	0.671057i	$0.234159\pi$
$912$		0			0
$913$	−2.67138	−	4.62696i	−0.0884097	−	0.153130i
$914$		0			0
$915$	−8.80768	−	32.8707i	−0.291173	−	1.08667i
$916$		0			0
$917$	4.61423	+	6.86925i	0.152375	+	0.226843i
$918$		0			0
$919$	−19.8098	−	11.4372i	−0.653464	−	0.377277i	0.136318	−	0.990665i	$-0.456473\pi$
$919$	−19.8098	−	11.4372i	−0.653464	−	0.377277i	−0.789782	+	0.613388i	$0.789806\pi$
$920$		0			0
$921$	−8.69148	+	5.01803i	−0.286394	+	0.165350i
$922$		0			0
$923$	−19.5442	−	19.5442i	−0.643304	−	0.643304i
$924$		0			0
$925$	2.28137	−	2.28137i	0.0750111	−	0.0750111i
$926$		0			0
$927$	16.5233	+	28.6192i	0.542697	+	0.939979i
$928$		0			0
$929$	13.2871	−	23.0139i	0.435935	−	0.755062i	−0.561436	−	0.827520i	$-0.689751\pi$
$929$	13.2871	−	23.0139i	0.435935	−	0.755062i	0.997371	+	0.0724577i	$0.0230842\pi$
$930$		0			0
$931$	21.3177	+	16.1926i	0.698658	+	0.530690i
$932$		0			0
$933$	−7.04026	+	1.88643i	−0.230488	+	0.0617590i
$934$		0			0
$935$	−8.45127	+	4.87934i	−0.276386	+	0.159572i
$936$		0			0
$937$		−	11.8582i		−	0.387391i	−0.981062	−	0.193696i	$-0.937953\pi$
$937$		−	11.8582i		−	0.387391i	0.981062	−	0.193696i	$-0.0620474\pi$
$938$		0			0
$939$	9.86686	+	9.86686i	0.321993	+	0.321993i
$940$		0			0
$941$	−2.04535	−	0.548049i	−0.0666764	−	0.0178659i	0.225327	−	0.974283i	$-0.427655\pi$
$941$	−2.04535	−	0.548049i	−0.0666764	−	0.0178659i	−0.292003	+	0.956417i	$0.594322\pi$
$942$		0			0
$943$	0.313366	−	0.542765i	0.0102046	−	0.0176749i
$944$		0			0
$945$	49.8360	+	9.78636i	1.62116	+	0.318350i
$946$		0			0
$947$	−1.38876	−	5.18291i	−0.0451285	−	0.168422i	0.939684	−	0.342045i	$-0.111119\pi$
$947$	−1.38876	−	5.18291i	−0.0451285	−	0.168422i	−0.984812	+	0.173623i	$0.944453\pi$
$948$		0			0
$949$	5.94835	−	22.1995i	0.193092	−	0.720628i
$950$		0			0
$951$			19.3935i			0.628878i
$952$		0			0
$953$			3.43130i			0.111151i	0.998454	+	0.0555753i	$0.0176993\pi$
$953$			3.43130i			0.111151i	−0.998454	+	0.0555753i	$0.982301\pi$
$954$		0			0
$955$	−10.9646	+	40.9205i	−0.354806	+	1.32415i
$956$		0			0
$957$	−0.0702080	−	0.262020i	−0.00226950	−	0.00846989i
$958$		0			0
$959$	0.575210	−	0.659145i	0.0185745	−	0.0212849i
$960$		0			0
$961$	−5.39169	+	9.33869i	−0.173926	+	0.301248i
$962$		0			0
$963$	−10.8801	−	2.91531i	−0.350606	−	0.0939446i
$964$		0			0
$965$	−24.2018	−	24.2018i	−0.779083	−	0.779083i
$966$		0			0
$967$		−	19.4110i		−	0.624217i	−0.950046	−	0.312108i	$-0.898965\pi$
$967$		−	19.4110i		−	0.624217i	0.950046	−	0.312108i	$-0.101035\pi$
$968$		0			0
$969$	−5.29044	+	3.05444i	−0.169954	+	0.0981227i
$970$		0			0
$971$	−19.4567	+	5.21341i	−0.624396	+	0.167306i	−0.557125	−	0.830428i	$-0.688096\pi$
$971$	−19.4567	+	5.21341i	−0.624396	+	0.167306i	−0.0672705	+	0.997735i	$0.521429\pi$
$972$		0			0
$973$	8.10708	+	3.97335i	0.259901	+	0.127380i
$974$		0			0
$975$	13.7354	−	23.7905i	0.439886	−	0.761905i
$976$		0			0
$977$	−8.78731	−	15.2201i	−0.281131	−	0.486933i	0.690533	−	0.723301i	$-0.257376\pi$
$977$	−8.78731	−	15.2201i	−0.281131	−	0.486933i	−0.971664	+	0.236368i	$0.924043\pi$
$978$		0			0
$979$	−5.68920	+	5.68920i	−0.181828	+	0.181828i
$980$		0			0
$981$	16.1966	+	16.1966i	0.517117	+	0.517117i
$982$		0			0
$983$	−47.7497	+	27.5683i	−1.52298	+	0.879292i	−0.523347	+	0.852119i	$0.675317\pi$
$983$	−47.7497	+	27.5683i	−1.52298	+	0.879292i	−0.999631	+	0.0271723i	$0.991350\pi$
$984$		0			0
$985$	31.3600	+	18.1057i	0.999214	+	0.576896i
$986$		0			0
$987$	−7.21352	+	14.7182i	−0.229609	+	0.468486i
$988$		0			0
$989$	−2.88978	−	10.7848i	−0.0918896	−	0.342937i
$990$		0			0
$991$	−18.4719	−	31.9943i	−0.586781	−	1.01633i	−0.994651	−	0.103294i	$-0.967062\pi$
$991$	−18.4719	−	31.9943i	−0.586781	−	1.01633i	0.407870	−	0.913040i	$-0.366272\pi$
$992$		0			0
$993$	9.96536			0.316241
$994$		0			0
$995$	−44.3910	+	44.3910i	−1.40729	+	1.40729i
$996$		0			0
$997$	−0.959891	+	3.58236i	−0.0304001	+	0.113455i	−0.979459	−	0.201645i	$-0.935371\pi$
$997$	−0.959891	+	3.58236i	−0.0304001	+	0.113455i	0.949059	+	0.315100i	$0.102038\pi$
$998$		0			0
$999$	0.962950	+	0.555960i	0.0304664	+	0.0175898i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.ba.c.177.6		48
4.3	odd	2		112.2.w.c.37.7	✓	48
7.4	even	3	inner	448.2.ba.c.305.7		48
8.3	odd	2		896.2.ba.f.737.6		48
8.5	even	2		896.2.ba.e.737.7		48
16.3	odd	4		112.2.w.c.93.2	yes	48
16.5	even	4		896.2.ba.e.289.6		48
16.11	odd	4		896.2.ba.f.289.7		48
16.13	even	4	inner	448.2.ba.c.401.7		48
28.3	even	6		784.2.x.o.165.2		48
28.11	odd	6		112.2.w.c.53.2	yes	48
28.19	even	6		784.2.m.k.197.9		24
28.23	odd	6		784.2.m.j.197.9		24
28.27	even	2		784.2.x.o.373.7		48
56.11	odd	6		896.2.ba.f.865.7		48
56.53	even	6		896.2.ba.e.865.6		48
112.3	even	12		784.2.x.o.557.7		48
112.11	odd	12		896.2.ba.f.417.6		48
112.19	even	12		784.2.m.k.589.9		24
112.51	odd	12		784.2.m.j.589.9		24
112.53	even	12		896.2.ba.e.417.7		48
112.67	odd	12		112.2.w.c.109.7	yes	48
112.83	even	4		784.2.x.o.765.2		48
112.109	even	12	inner	448.2.ba.c.81.6		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.w.c.37.7	✓	48	4.3	odd	2
112.2.w.c.53.2	yes	48	28.11	odd	6
112.2.w.c.93.2	yes	48	16.3	odd	4
112.2.w.c.109.7	yes	48	112.67	odd	12
448.2.ba.c.81.6		48	112.109	even	12	inner
448.2.ba.c.177.6		48	1.1	even	1	trivial
448.2.ba.c.305.7		48	7.4	even	3	inner
448.2.ba.c.401.7		48	16.13	even	4	inner
784.2.m.j.197.9		24	28.23	odd	6
784.2.m.j.589.9		24	112.51	odd	12
784.2.m.k.197.9		24	28.19	even	6
784.2.m.k.589.9		24	112.19	even	12
784.2.x.o.165.2		48	28.3	even	6
784.2.x.o.373.7		48	28.27	even	2
784.2.x.o.557.7		48	112.3	even	12
784.2.x.o.765.2		48	112.83	even	4
896.2.ba.e.289.6		48	16.5	even	4
896.2.ba.e.417.7		48	112.53	even	12
896.2.ba.e.737.7		48	8.5	even	2
896.2.ba.e.865.6		48	56.53	even	6
896.2.ba.f.289.7		48	16.11	odd	4
896.2.ba.f.417.6		48	112.11	odd	12
896.2.ba.f.737.6		48	8.3	odd	2
896.2.ba.f.865.7		48	56.11	odd	6

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)

\(f(q)\)	\(=\)	\(q+(-0.827840 - 0.221819i) q^{3} +(4.10878 - 1.10095i) q^{5} +(2.50325 - 0.856573i) q^{7} +(-1.96196 - 1.13274i) q^{9} +(0.318576 - 1.18894i) q^{11} +(-1.73092 + 1.73092i) q^{13} -3.64563 q^{15} +(-0.931914 - 1.61412i) q^{17} +(0.989804 + 3.69400i) q^{19} +(-2.26230 + 0.153836i) q^{21} +(-1.23305 - 0.711901i) q^{23} +(11.3399 - 6.54709i) q^{25} +(3.19099 + 3.19099i) q^{27} +(-0.181826 + 0.181826i) q^{29} +(-3.23200 - 5.59799i) q^{31} +(-0.527459 + 0.913587i) q^{33} +(9.34229 - 6.27542i) q^{35} +(0.238000 - 0.0637719i) q^{37} +(1.81687 - 1.04897i) q^{39} +0.440182i q^{41} +(5.54503 + 5.54503i) q^{43} +(-9.30835 - 2.49417i) q^{45} +(3.61427 - 6.26009i) q^{47} +(5.53257 - 4.28844i) q^{49} +(0.413433 + 1.54295i) q^{51} +(-2.59716 + 9.69274i) q^{53} -5.23583i q^{55} -3.27760i q^{57} +(0.438601 - 1.63688i) q^{59} +(2.41596 + 9.01647i) q^{61} +(-5.88156 - 1.15497i) q^{63} +(-5.20632 + 9.01761i) q^{65} +(-9.59422 - 2.57076i) q^{67} +(0.862854 + 0.862854i) q^{69} +11.2912i q^{71} +(-8.13092 + 4.69439i) q^{73} +(-10.8399 + 2.90454i) q^{75} +(-0.220938 - 3.24910i) q^{77} +(-6.52948 + 11.3094i) q^{79} +(1.46441 + 2.53643i) q^{81} +(3.06926 - 3.06926i) q^{83} +(-5.60609 - 5.60609i) q^{85} +(0.190855 - 0.110190i) q^{87} +(-5.66084 - 3.26829i) q^{89} +(-2.85027 + 5.81558i) q^{91} +(1.43384 + 5.35117i) q^{93} +(8.13378 + 14.0881i) q^{95} +1.70409 q^{97} +(-1.97179 + 1.97179i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{5} + 4 q^{11} - 24 q^{13} + 40 q^{15} + 8 q^{17} + 4 q^{19} - 8 q^{21} + 24 q^{27} + 24 q^{29} - 28 q^{31} + 16 q^{33} - 28 q^{35} - 24 q^{37} + 40 q^{43} - 28 q^{45} + 20 q^{47} - 24 q^{51}+ \cdots + 24 q^{99}+O(q^{100}) \) 48 * q + 4 * q^5 + 4 * q^11 - 24 * q^13 + 40 * q^15 + 8 * q^17 + 4 * q^19 - 8 * q^21 + 24 * q^27 + 24 * q^29 - 28 * q^31 + 16 * q^33 - 28 * q^35 - 24 * q^37 + 40 * q^43 - 28 * q^45 + 20 * q^47 - 24 * q^51 - 16 * q^53 + 20 * q^59 + 8 * q^61 + 16 * q^63 + 8 * q^65 - 48 * q^67 - 40 * q^69 + 4 * q^75 - 20 * q^77 + 36 * q^79 + 8 * q^83 - 64 * q^91 + 8 * q^93 + 4 * q^95 - 48 * q^97 + 24 * q^99

Introduction

L-functions

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Database

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