LMFDB - Embedded newform 448.2.m.a.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,2,Mod(113,448)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("448.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$3.57729801055$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	448.337
Dual form			448.2.m.a.113.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 - 2.00000i) q^{5} -1.00000i q^{7} -3.00000i q^{9} +O(q^{10})$	$q+(2.00000 - 2.00000i) q^{5} -1.00000i q^{7} -3.00000i q^{9} +(-1.00000 + 1.00000i) q^{11} -2.00000 q^{17} +(-2.00000 - 2.00000i) q^{19} -6.00000i q^{23} -3.00000i q^{25} +(7.00000 + 7.00000i) q^{29} +8.00000 q^{31} +(-2.00000 - 2.00000i) q^{35} +(-5.00000 + 5.00000i) q^{37} -10.0000i q^{41} +(1.00000 - 1.00000i) q^{43} +(-6.00000 - 6.00000i) q^{45} +12.0000 q^{47} -1.00000 q^{49} +(-1.00000 + 1.00000i) q^{53} +4.00000i q^{55} +(-8.00000 + 8.00000i) q^{59} +(6.00000 + 6.00000i) q^{61} -3.00000 q^{63} +(-3.00000 - 3.00000i) q^{67} +6.00000i q^{73} +(1.00000 + 1.00000i) q^{77} -10.0000 q^{79} -9.00000 q^{81} +(10.0000 + 10.0000i) q^{83} +(-4.00000 + 4.00000i) q^{85} +14.0000i q^{89} -8.00000 q^{95} -2.00000 q^{97} +(3.00000 + 3.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5}+O(q^{10}) $ 2 * q + 4 * q^5	$ 2 q + 4 q^{5} - 2 q^{11} - 4 q^{17} - 4 q^{19} + 14 q^{29} + 16 q^{31} - 4 q^{35} - 10 q^{37} + 2 q^{43} - 12 q^{45} + 24 q^{47} - 2 q^{49} - 2 q^{53} - 16 q^{59} + 12 q^{61} - 6 q^{63} - 6 q^{67} + 2 q^{77} - 20 q^{79} - 18 q^{81} + 20 q^{83} - 8 q^{85} - 16 q^{95} - 4 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q + 4 * q^5 - 2 * q^11 - 4 * q^17 - 4 * q^19 + 14 * q^29 + 16 * q^31 - 4 * q^35 - 10 * q^37 + 2 * q^43 - 12 * q^45 + 24 * q^47 - 2 * q^49 - 2 * q^53 - 16 * q^59 + 12 * q^61 - 6 * q^63 - 6 * q^67 + 2 * q^77 - 20 * q^79 - 18 * q^81 + 20 * q^83 - 8 * q^85 - 16 * q^95 - 4 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

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			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$	2.00000	−	2.00000i	0.894427	−	0.894427i	−0.100509	−	0.994936i	$-0.532047\pi$
$5$	2.00000	−	2.00000i	0.894427	−	0.894427i	0.994936	+	0.100509i	$0.0320471\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$	−1.00000	+	1.00000i	−0.301511	+	0.301511i	−0.841605	−	0.540094i	$-0.818389\pi$
$11$	−1.00000	+	1.00000i	−0.301511	+	0.301511i	0.540094	+	0.841605i	$0.318389\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$	−2.00000	−	2.00000i	−0.458831	−	0.458831i	0.439440	−	0.898272i	$-0.355177\pi$
$19$	−2.00000	−	2.00000i	−0.458831	−	0.458831i	−0.898272	+	0.439440i	$0.855177\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$		−	3.00000i		−	0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.00000	+	7.00000i	1.29987	+	1.29987i	0.928477	+	0.371391i	$0.121119\pi$
$29$	7.00000	+	7.00000i	1.29987	+	1.29987i	0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.00000	−	2.00000i	−0.338062	−	0.338062i
$36$		0			0
$37$	−5.00000	+	5.00000i	−0.821995	+	0.821995i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−5.00000	+	5.00000i	−0.821995	+	0.821995i	0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$		0			0
$43$	1.00000	−	1.00000i	0.152499	−	0.152499i	−0.626734	−	0.779233i	$-0.715609\pi$
$43$	1.00000	−	1.00000i	0.152499	−	0.152499i	0.779233	+	0.626734i	$0.215609\pi$
$44$		0			0
$45$	−6.00000	−	6.00000i	−0.894427	−	0.894427i
$46$		0			0
$47$	12.0000			1.75038			0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000			1.75038			0.875190	+	0.483779i	$0.160736\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.00000	+	1.00000i	−0.137361	+	0.137361i	−0.772444	−	0.635083i	$-0.780966\pi$
$53$	−1.00000	+	1.00000i	−0.137361	+	0.137361i	0.635083	+	0.772444i	$0.280966\pi$
$54$		0			0
$55$			4.00000i			0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.00000	+	8.00000i	−1.04151	+	1.04151i	−0.0424110	+	0.999100i	$0.513504\pi$
$59$	−8.00000	+	8.00000i	−1.04151	+	1.04151i	−0.999100	+	0.0424110i	$0.986496\pi$
$60$		0			0
$61$	6.00000	+	6.00000i	0.768221	+	0.768221i	0.977793	−	0.209572i	$-0.0672070\pi$
$61$	6.00000	+	6.00000i	0.768221	+	0.768221i	−0.209572	+	0.977793i	$0.567207\pi$
$62$		0			0
$63$	−3.00000			−0.377964
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.00000	−	3.00000i	−0.366508	−	0.366508i	0.499694	−	0.866202i	$-0.333446\pi$
$67$	−3.00000	−	3.00000i	−0.366508	−	0.366508i	−0.866202	+	0.499694i	$0.833446\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.00000	+	1.00000i	0.113961	+	0.113961i
$78$		0			0
$79$	−10.0000			−1.12509			−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000			−1.12509			−0.562544	+	0.826767i	$0.690177\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	10.0000	+	10.0000i	1.09764	+	1.09764i	0.994686	+	0.102957i	$0.0328303\pi$
$83$	10.0000	+	10.0000i	1.09764	+	1.09764i	0.102957	+	0.994686i	$0.467170\pi$
$84$		0			0
$85$	−4.00000	+	4.00000i	−0.433861	+	0.433861i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			14.0000i			1.48400i	0.670402	+	0.741999i	$0.266122\pi$
$89$			14.0000i			1.48400i	−0.670402	+	0.741999i	$0.733878\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.00000			−0.820783
$96$		0			0
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$		0			0
$99$	3.00000	+	3.00000i	0.301511	+	0.301511i
$100$		0			0
$101$	6.00000	−	6.00000i	0.597022	−	0.597022i	−0.342497	−	0.939519i	$-0.611273\pi$
$101$	6.00000	−	6.00000i	0.597022	−	0.597022i	0.939519	+	0.342497i	$0.111273\pi$
$102$		0			0
$103$			4.00000i			0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$			4.00000i			0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.00000	+	5.00000i	−0.483368	+	0.483368i	−0.906206	−	0.422837i	$-0.861034\pi$
$107$	−5.00000	+	5.00000i	−0.483368	+	0.483368i	0.422837	+	0.906206i	$0.361034\pi$
$108$		0			0
$109$	−3.00000	−	3.00000i	−0.287348	−	0.287348i	0.548683	−	0.836031i	$-0.315129\pi$
$109$	−3.00000	−	3.00000i	−0.287348	−	0.287348i	−0.836031	+	0.548683i	$0.815129\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$	−12.0000	−	12.0000i	−1.11901	−	1.11901i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2.00000i			0.183340i
$120$		0			0
$121$			9.00000i			0.818182i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.00000	+	4.00000i	0.357771	+	0.357771i
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	14.0000	+	14.0000i	1.22319	+	1.22319i	0.966493	+	0.256693i	$0.0826328\pi$
$131$	14.0000	+	14.0000i	1.22319	+	1.22319i	0.256693	+	0.966493i	$0.417367\pi$
$132$		0			0
$133$	−2.00000	+	2.00000i	−0.173422	+	0.173422i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$		0			0
$139$	2.00000	−	2.00000i	0.169638	−	0.169638i	−0.617182	−	0.786820i	$-0.711726\pi$
$139$	2.00000	−	2.00000i	0.169638	−	0.169638i	0.786820	+	0.617182i	$0.211726\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	28.0000			2.32527
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	−	3.00000i	0.245770	−	0.245770i	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	3.00000	−	3.00000i	0.245770	−	0.245770i	0.819232	+	0.573462i	$0.194400\pi$
$150$		0			0
$151$			10.0000i			0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$		0			0
$153$			6.00000i			0.485071i
$154$		0			0
$155$	16.0000	−	16.0000i	1.28515	−	1.28515i
$156$		0			0
$157$	−12.0000	−	12.0000i	−0.957704	−	0.957704i	0.0414369	−	0.999141i	$-0.486806\pi$
$157$	−12.0000	−	12.0000i	−0.957704	−	0.957704i	−0.999141	+	0.0414369i	$0.986806\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.00000			−0.472866
$162$		0			0
$163$	−15.0000	−	15.0000i	−1.17489	−	1.17489i	−0.981029	−	0.193862i	$-0.937899\pi$
$163$	−15.0000	−	15.0000i	−1.17489	−	1.17489i	−0.193862	−	0.981029i	$-0.562101\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$	−6.00000	+	6.00000i	−0.458831	+	0.458831i
$172$		0			0
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$	−3.00000			−0.226779
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.00000	+	3.00000i	0.224231	+	0.224231i	0.810277	−	0.586047i	$-0.199317\pi$
$179$	3.00000	+	3.00000i	0.224231	+	0.224231i	−0.586047	+	0.810277i	$0.699317\pi$
$180$		0			0
$181$	−4.00000	+	4.00000i	−0.297318	+	0.297318i	−0.839962	−	0.542645i	$-0.817423\pi$
$181$	−4.00000	+	4.00000i	−0.297318	+	0.297318i	0.542645	+	0.839962i	$0.317423\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			20.0000i			1.47043i
$186$		0			0
$187$	2.00000	−	2.00000i	0.146254	−	0.146254i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	18.0000			1.30243			0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000			1.30243			0.651217	+	0.758891i	$0.274259\pi$
$192$		0			0
$193$	−16.0000			−1.15171			−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000			−1.15171			−0.575853	+	0.817554i	$0.695330\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	−0.506188	−	0.862423i	$-0.668946\pi$
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	0.862423	+	0.506188i	$0.168946\pi$
$198$		0			0
$199$		−	24.0000i		−	1.70131i	−0.525720	−	0.850657i	$-0.676204\pi$
$199$		−	24.0000i		−	1.70131i	0.525720	−	0.850657i	$-0.323796\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.00000	−	7.00000i	0.491304	−	0.491304i
$204$		0			0
$205$	−20.0000	−	20.0000i	−1.39686	−	1.39686i
$206$		0			0
$207$	−18.0000			−1.25109
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	9.00000	+	9.00000i	0.619586	+	0.619586i	0.945425	−	0.325840i	$-0.105647\pi$
$211$	9.00000	+	9.00000i	0.619586	+	0.619586i	−0.325840	+	0.945425i	$0.605647\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	4.00000i		−	0.272798i
$216$		0			0
$217$		−	8.00000i		−	0.543075i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4.00000			−0.267860			−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000			−0.267860			−0.133930	+	0.990991i	$0.542760\pi$
$224$		0			0
$225$	−9.00000			−0.600000
$226$		0			0
$227$	2.00000	+	2.00000i	0.132745	+	0.132745i	0.770357	−	0.637613i	$-0.220078\pi$
$227$	2.00000	+	2.00000i	0.132745	+	0.132745i	−0.637613	+	0.770357i	$0.720078\pi$
$228$		0			0
$229$	8.00000	−	8.00000i	0.528655	−	0.528655i	−0.391516	−	0.920171i	$-0.628049\pi$
$229$	8.00000	−	8.00000i	0.528655	−	0.528655i	0.920171	+	0.391516i	$0.128049\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.0000i			1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$			16.0000i			1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		0			0
$235$	24.0000	−	24.0000i	1.56559	−	1.56559i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	22.0000			1.41714			0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000			1.41714			0.708572	+	0.705638i	$0.249340\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.00000	+	2.00000i	−0.127775	+	0.127775i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	14.0000	−	14.0000i	0.883672	−	0.883672i	−0.110234	−	0.993906i	$-0.535160\pi$
$251$	14.0000	−	14.0000i	0.883672	−	0.883672i	0.993906	+	0.110234i	$0.0351599\pi$
$252$		0			0
$253$	6.00000	+	6.00000i	0.377217	+	0.377217i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.00000			−0.124757			−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000			−0.124757			−0.0623783	+	0.998053i	$0.519869\pi$
$258$		0			0
$259$	5.00000	+	5.00000i	0.310685	+	0.310685i
$260$		0			0
$261$	21.0000	−	21.0000i	1.29987	−	1.29987i
$262$		0			0
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$		0			0
$265$			4.00000i			0.245718i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.00000	−	8.00000i	−0.487769	−	0.487769i	0.419833	−	0.907601i	$-0.362089\pi$
$269$	−8.00000	−	8.00000i	−0.487769	−	0.487769i	−0.907601	+	0.419833i	$0.862089\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.00000	+	3.00000i	0.180907	+	0.180907i
$276$		0			0
$277$	−15.0000	+	15.0000i	−0.901263	+	0.901263i	−0.995545	−	0.0942828i	$-0.969944\pi$
$277$	−15.0000	+	15.0000i	−0.901263	+	0.901263i	0.0942828	+	0.995545i	$0.469944\pi$
$278$		0			0
$279$		−	24.0000i		−	1.43684i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−4.00000	+	4.00000i	−0.237775	+	0.237775i	−0.815928	−	0.578153i	$-0.803774\pi$
$283$	−4.00000	+	4.00000i	−0.237775	+	0.237775i	0.578153	+	0.815928i	$0.303774\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.0000			−0.590281
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.0000	−	14.0000i	0.817889	−	0.817889i	−0.167913	−	0.985802i	$-0.553703\pi$
$293$	14.0000	−	14.0000i	0.817889	−	0.817889i	0.985802	+	0.167913i	$0.0537028\pi$
$294$		0			0
$295$			32.0000i			1.86311i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−1.00000	−	1.00000i	−0.0576390	−	0.0576390i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	24.0000			1.37424
$306$		0			0
$307$	−18.0000	−	18.0000i	−1.02731	−	1.02731i	−0.999616	−	0.0276979i	$-0.991182\pi$
$307$	−18.0000	−	18.0000i	−1.02731	−	1.02731i	−0.0276979	−	0.999616i	$-0.508818\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			20.0000i			1.13410i	0.823685	+	0.567048i	$0.191915\pi$
$311$			20.0000i			1.13410i	−0.823685	+	0.567048i	$0.808085\pi$
$312$		0			0
$313$		−	14.0000i		−	0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$		−	14.0000i		−	0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$		0			0
$315$	−6.00000	+	6.00000i	−0.338062	+	0.338062i
$316$		0			0
$317$	−7.00000	−	7.00000i	−0.393159	−	0.393159i	0.482653	−	0.875812i	$-0.339673\pi$
$317$	−7.00000	−	7.00000i	−0.393159	−	0.393159i	−0.875812	+	0.482653i	$0.839673\pi$
$318$		0			0
$319$	−14.0000			−0.783850
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.00000	+	4.00000i	0.222566	+	0.222566i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	12.0000i		−	0.661581i
$330$		0			0
$331$	−21.0000	+	21.0000i	−1.15426	+	1.15426i	−0.168576	+	0.985689i	$0.553917\pi$
$331$	−21.0000	+	21.0000i	−1.15426	+	1.15426i	−0.985689	+	0.168576i	$0.946083\pi$
$332$		0			0
$333$	15.0000	+	15.0000i	0.821995	+	0.821995i
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$	8.00000			0.435788			0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000			0.435788			0.217894	+	0.975972i	$0.430081\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.00000	+	8.00000i	−0.433224	+	0.433224i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.0000	+	15.0000i	−0.805242	+	0.805242i	−0.983910	−	0.178667i	$-0.942821\pi$
$347$	−15.0000	+	15.0000i	−0.805242	+	0.805242i	0.178667	+	0.983910i	$0.442821\pi$
$348$		0			0
$349$	12.0000	+	12.0000i	0.642345	+	0.642345i	0.951131	−	0.308786i	$-0.0999228\pi$
$349$	12.0000	+	12.0000i	0.642345	+	0.642345i	−0.308786	+	0.951131i	$0.599923\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6.00000i			0.316668i	0.987386	+	0.158334i	$0.0506123\pi$
$359$			6.00000i			0.316668i	−0.987386	+	0.158334i	$0.949388\pi$
$360$		0			0
$361$		−	11.0000i		−	0.578947i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.0000	+	12.0000i	0.628109	+	0.628109i
$366$		0			0
$367$	32.0000			1.67039			0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000			1.67039			0.835193	+	0.549957i	$0.185356\pi$
$368$		0			0
$369$	−30.0000			−1.56174
$370$		0			0
$371$	1.00000	+	1.00000i	0.0519174	+	0.0519174i
$372$		0			0
$373$	−11.0000	+	11.0000i	−0.569558	+	0.569558i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−11.0000	+	11.0000i	−0.569558	+	0.569558i	0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−13.0000	+	13.0000i	−0.667765	+	0.667765i	−0.957198	−	0.289433i	$-0.906533\pi$
$379$	−13.0000	+	13.0000i	−0.667765	+	0.667765i	0.289433	+	0.957198i	$0.406533\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.00000			−0.204390			−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000			−0.204390			−0.102195	+	0.994764i	$0.532587\pi$
$384$		0			0
$385$	4.00000			0.203859
$386$		0			0
$387$	−3.00000	−	3.00000i	−0.152499	−	0.152499i
$388$		0			0
$389$	3.00000	−	3.00000i	0.152106	−	0.152106i	−0.626952	−	0.779058i	$-0.715698\pi$
$389$	3.00000	−	3.00000i	0.152106	−	0.152106i	0.779058	+	0.626952i	$0.215698\pi$
$390$		0			0
$391$			12.0000i			0.606866i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−20.0000	+	20.0000i	−1.00631	+	1.00631i
$396$		0			0
$397$	−12.0000	−	12.0000i	−0.602263	−	0.602263i	0.338650	−	0.940913i	$-0.390030\pi$
$397$	−12.0000	−	12.0000i	−0.602263	−	0.602263i	−0.940913	+	0.338650i	$0.890030\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.0000			0.599251			0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000			0.599251			0.299626	+	0.954057i	$0.403138\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−18.0000	+	18.0000i	−0.894427	+	0.894427i
$406$		0			0
$407$		−	10.0000i		−	0.495682i
$408$		0			0
$409$			14.0000i			0.692255i	0.938187	+	0.346128i	$0.112504\pi$
$409$			14.0000i			0.692255i	−0.938187	+	0.346128i	$0.887496\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.00000	+	8.00000i	0.393654	+	0.393654i
$414$		0			0
$415$	40.0000			1.96352
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.0000	−	12.0000i	−0.586238	−	0.586238i	0.350372	−	0.936611i	$-0.386055\pi$
$419$	−12.0000	−	12.0000i	−0.586238	−	0.586238i	−0.936611	+	0.350372i	$0.886055\pi$
$420$		0			0
$421$	−9.00000	+	9.00000i	−0.438633	+	0.438633i	−0.891552	−	0.452919i	$-0.850383\pi$
$421$	−9.00000	+	9.00000i	−0.438633	+	0.438633i	0.452919	+	0.891552i	$0.350383\pi$
$422$		0			0
$423$		−	36.0000i		−	1.75038i
$424$		0			0
$425$			6.00000i			0.291043i
$426$		0			0
$427$	6.00000	−	6.00000i	0.290360	−	0.290360i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.00000			0.385346			0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000			0.385346			0.192673	+	0.981263i	$0.438284\pi$
$432$		0			0
$433$	−6.00000			−0.288342			−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000			−0.288342			−0.144171	+	0.989553i	$0.546051\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.0000	+	12.0000i	−0.574038	+	0.574038i
$438$		0			0
$439$			16.0000i			0.763638i	0.924237	+	0.381819i	$0.124702\pi$
$439$			16.0000i			0.763638i	−0.924237	+	0.381819i	$0.875298\pi$
$440$		0			0
$441$			3.00000i			0.142857i
$442$		0			0
$443$	1.00000	−	1.00000i	0.0475114	−	0.0475114i	−0.682952	−	0.730463i	$-0.739304\pi$
$443$	1.00000	−	1.00000i	0.0475114	−	0.0475114i	0.730463	+	0.682952i	$0.239304\pi$
$444$		0			0
$445$	28.0000	+	28.0000i	1.32733	+	1.32733i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$	10.0000	+	10.0000i	0.470882	+	0.470882i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	22.0000i		−	1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$		−	22.0000i		−	1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.0000	+	16.0000i	0.745194	+	0.745194i	0.973572	−	0.228378i	$-0.0733423\pi$
$461$	16.0000	+	16.0000i	0.745194	+	0.745194i	−0.228378	+	0.973572i	$0.573342\pi$
$462$		0			0
$463$	−14.0000			−0.650635			−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000			−0.650635			−0.325318	+	0.945605i	$0.605471\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.0000	−	18.0000i	−0.832941	−	0.832941i	0.154977	−	0.987918i	$-0.450470\pi$
$467$	−18.0000	−	18.0000i	−0.832941	−	0.832941i	−0.987918	+	0.154977i	$0.950470\pi$
$468$		0			0
$469$	−3.00000	+	3.00000i	−0.138527	+	0.138527i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			2.00000i			0.0919601i
$474$		0			0
$475$	−6.00000	+	6.00000i	−0.275299	+	0.275299i
$476$		0			0
$477$	3.00000	+	3.00000i	0.137361	+	0.137361i
$478$		0			0
$479$	20.0000			0.913823			0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000			0.913823			0.456912	+	0.889512i	$0.348956\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.00000	+	4.00000i	−0.181631	+	0.181631i
$486$		0			0
$487$			22.0000i			0.996915i	0.866914	+	0.498458i	$0.166100\pi$
$487$			22.0000i			0.996915i	−0.866914	+	0.498458i	$0.833900\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	19.0000	−	19.0000i	0.857458	−	0.857458i	−0.133580	−	0.991038i	$-0.542647\pi$
$491$	19.0000	−	19.0000i	0.857458	−	0.857458i	0.991038	+	0.133580i	$0.0426473\pi$
$492$		0			0
$493$	−14.0000	−	14.0000i	−0.630528	−	0.630528i
$494$		0			0
$495$	12.0000			0.539360
$496$		0			0
$497$		0			0
$498$		0			0
$499$	23.0000	+	23.0000i	1.02962	+	1.02962i	0.999548	+	0.0300737i	$0.00957421\pi$
$499$	23.0000	+	23.0000i	1.02962	+	1.02962i	0.0300737	+	0.999548i	$0.490426\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	16.0000i		−	0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$		−	16.0000i		−	0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$		0			0
$505$		−	24.0000i		−	1.06799i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.00000	−	8.00000i	−0.354594	−	0.354594i	0.507222	−	0.861816i	$-0.330672\pi$
$509$	−8.00000	−	8.00000i	−0.354594	−	0.354594i	−0.861816	+	0.507222i	$0.830672\pi$
$510$		0			0
$511$	6.00000			0.265424
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.00000	+	8.00000i	0.352522	+	0.352522i
$516$		0			0
$517$	−12.0000	+	12.0000i	−0.527759	+	0.527759i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			10.0000i			0.438108i	0.975713	+	0.219054i	$0.0702971\pi$
$521$			10.0000i			0.438108i	−0.975713	+	0.219054i	$0.929703\pi$
$522$		0			0
$523$	−24.0000	+	24.0000i	−1.04945	+	1.04945i	−0.0507346	+	0.998712i	$0.516156\pi$
$523$	−24.0000	+	24.0000i	−1.04945	+	1.04945i	−0.998712	+	0.0507346i	$0.983844\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−16.0000			−0.696971
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	24.0000	+	24.0000i	1.04151	+	1.04151i
$532$		0			0
$533$		0			0
$534$		0			0
$535$			20.0000i			0.864675i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.00000	−	1.00000i	0.0430730	−	0.0430730i
$540$		0			0
$541$	11.0000	+	11.0000i	0.472927	+	0.472927i	0.902861	−	0.429934i	$-0.141463\pi$
$541$	11.0000	+	11.0000i	0.472927	+	0.472927i	−0.429934	+	0.902861i	$0.641463\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0000			−0.514024
$546$		0			0
$547$	−23.0000	−	23.0000i	−0.983409	−	0.983409i	0.0164556	−	0.999865i	$-0.494762\pi$
$547$	−23.0000	−	23.0000i	−0.983409	−	0.983409i	−0.999865	+	0.0164556i	$0.994762\pi$
$548$		0			0
$549$	18.0000	−	18.0000i	0.768221	−	0.768221i
$550$		0			0
$551$		−	28.0000i		−	1.19284i
$552$		0			0
$553$			10.0000i			0.425243i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17.0000	−	17.0000i	−0.720313	−	0.720313i	0.248356	−	0.968669i	$-0.420110\pi$
$557$	−17.0000	−	17.0000i	−0.720313	−	0.720313i	−0.968669	+	0.248356i	$0.920110\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.0000	−	20.0000i	−0.842900	−	0.842900i	0.146336	−	0.989235i	$-0.453252\pi$
$563$	−20.0000	−	20.0000i	−0.842900	−	0.842900i	−0.989235	+	0.146336i	$0.953252\pi$
$564$		0			0
$565$	8.00000	−	8.00000i	0.336563	−	0.336563i
$566$		0			0
$567$			9.00000i			0.377964i
$568$		0			0
$569$			14.0000i			0.586911i	0.955973	+	0.293455i	$0.0948052\pi$
$569$			14.0000i			0.586911i	−0.955973	+	0.293455i	$0.905195\pi$
$570$		0			0
$571$	9.00000	−	9.00000i	0.376638	−	0.376638i	−0.493250	−	0.869888i	$-0.664191\pi$
$571$	9.00000	−	9.00000i	0.376638	−	0.376638i	0.869888	+	0.493250i	$0.164191\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−18.0000			−0.750652
$576$		0			0
$577$	−2.00000			−0.0832611			−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000			−0.0832611			−0.0416305	+	0.999133i	$0.513255\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10.0000	−	10.0000i	0.414870	−	0.414870i
$582$		0			0
$583$		−	2.00000i		−	0.0828315i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.0000	+	10.0000i	−0.412744	+	0.412744i	−0.882693	−	0.469949i	$-0.844272\pi$
$587$	−10.0000	+	10.0000i	−0.412744	+	0.412744i	0.469949	+	0.882693i	$0.344272\pi$
$588$		0			0
$589$	−16.0000	−	16.0000i	−0.659269	−	0.659269i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		0			0
$595$	4.00000	+	4.00000i	0.163984	+	0.163984i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	24.0000i		−	0.980613i	−0.871550	−	0.490307i	$-0.836885\pi$
$599$		−	24.0000i		−	0.980613i	0.871550	−	0.490307i	$-0.163115\pi$
$600$		0			0
$601$		−	10.0000i		−	0.407909i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$		−	10.0000i		−	0.407909i	0.978980	−	0.203954i	$-0.0653794\pi$
$602$		0			0
$603$	−9.00000	+	9.00000i	−0.366508	+	0.366508i
$604$		0			0
$605$	18.0000	+	18.0000i	0.731804	+	0.731804i
$606$		0			0
$607$	−28.0000			−1.13648			−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000			−1.13648			−0.568242	+	0.822861i	$0.692376\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	9.00000	−	9.00000i	0.363507	−	0.363507i	−0.501596	−	0.865102i	$-0.667253\pi$
$613$	9.00000	−	9.00000i	0.363507	−	0.363507i	0.865102	+	0.501596i	$0.167253\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	42.0000i		−	1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$		−	42.0000i		−	1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$		0			0
$619$	12.0000	−	12.0000i	0.482321	−	0.482321i	−0.423551	−	0.905872i	$-0.639217\pi$
$619$	12.0000	−	12.0000i	0.482321	−	0.482321i	0.905872	+	0.423551i	$0.139217\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	14.0000			0.560898
$624$		0			0
$625$	31.0000			1.24000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	10.0000	−	10.0000i	0.398726	−	0.398726i
$630$		0			0
$631$		−	40.0000i		−	1.59237i	−0.605050	−	0.796187i	$-0.706847\pi$
$631$		−	40.0000i		−	1.59237i	0.605050	−	0.796187i	$-0.293153\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−16.0000	+	16.0000i	−0.634941	+	0.634941i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12.0000			0.473972			0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000			0.473972			0.236986	+	0.971513i	$0.423841\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			12.0000i			0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$			12.0000i			0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$		0			0
$649$		−	16.0000i		−	0.628055i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.0000	+	25.0000i	0.978326	+	0.978326i	0.999770	−	0.0214444i	$-0.00682650\pi$
$653$	25.0000	+	25.0000i	0.978326	+	0.978326i	−0.0214444	+	0.999770i	$0.506827\pi$
$654$		0			0
$655$	56.0000			2.18810
$656$		0			0
$657$	18.0000			0.702247
$658$		0			0
$659$	3.00000	+	3.00000i	0.116863	+	0.116863i	0.763120	−	0.646257i	$-0.223666\pi$
$659$	3.00000	+	3.00000i	0.116863	+	0.116863i	−0.646257	+	0.763120i	$0.723666\pi$
$660$		0			0
$661$	26.0000	−	26.0000i	1.01128	−	1.01128i	0.0113472	−	0.999936i	$-0.496388\pi$
$661$	26.0000	−	26.0000i	1.01128	−	1.01128i	0.999936	−	0.0113472i	$-0.00361200\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			8.00000i			0.310227i
$666$		0			0
$667$	42.0000	−	42.0000i	1.62625	−	1.62625i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.0000			−0.463255
$672$		0			0
$673$	34.0000			1.31060			0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000			1.31060			0.655302	+	0.755367i	$0.272541\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−20.0000	+	20.0000i	−0.768662	+	0.768662i	−0.977871	−	0.209209i	$-0.932911\pi$
$677$	−20.0000	+	20.0000i	−0.768662	+	0.768662i	0.209209	+	0.977871i	$0.432911\pi$
$678$		0			0
$679$			2.00000i			0.0767530i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.0000	−	11.0000i	0.420903	−	0.420903i	−0.464611	−	0.885515i	$-0.653806\pi$
$683$	11.0000	−	11.0000i	0.420903	−	0.420903i	0.885515	+	0.464611i	$0.153806\pi$
$684$		0			0
$685$	−24.0000	−	24.0000i	−0.916993	−	0.916993i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−26.0000	−	26.0000i	−0.989087	−	0.989087i	0.0108545	−	0.999941i	$-0.496545\pi$
$691$	−26.0000	−	26.0000i	−0.989087	−	0.989087i	−0.999941	+	0.0108545i	$0.996545\pi$
$692$		0			0
$693$	3.00000	−	3.00000i	0.113961	−	0.113961i
$694$		0			0
$695$		−	8.00000i		−	0.303457i
$696$		0			0
$697$			20.0000i			0.757554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−19.0000	−	19.0000i	−0.717620	−	0.717620i	0.250497	−	0.968117i	$-0.419406\pi$
$701$	−19.0000	−	19.0000i	−0.717620	−	0.717620i	−0.968117	+	0.250497i	$0.919406\pi$
$702$		0			0
$703$	20.0000			0.754314
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.00000	−	6.00000i	−0.225653	−	0.225653i
$708$		0			0
$709$	−7.00000	+	7.00000i	−0.262891	+	0.262891i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	−7.00000	+	7.00000i	−0.262891	+	0.262891i	0.563337	+	0.826227i	$0.309517\pi$
$710$		0			0
$711$			30.0000i			1.12509i
$712$		0			0
$713$		−	48.0000i		−	1.79761i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$	4.00000			0.148968
$722$		0			0
$723$		0			0
$724$		0			0
$725$	21.0000	−	21.0000i	0.779920	−	0.779920i
$726$		0			0
$727$			32.0000i			1.18681i	0.804902	+	0.593407i	$0.202218\pi$
$727$			32.0000i			1.18681i	−0.804902	+	0.593407i	$0.797782\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$	−2.00000	+	2.00000i	−0.0739727	+	0.0739727i
$732$		0			0
$733$	−10.0000	−	10.0000i	−0.369358	−	0.369358i	0.497885	−	0.867243i	$-0.334110\pi$
$733$	−10.0000	−	10.0000i	−0.369358	−	0.369358i	−0.867243	+	0.497885i	$0.834110\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.00000			0.221013
$738$		0			0
$739$	−7.00000	−	7.00000i	−0.257499	−	0.257499i	0.566537	−	0.824036i	$-0.308283\pi$
$739$	−7.00000	−	7.00000i	−0.257499	−	0.257499i	−0.824036	+	0.566537i	$0.808283\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	6.00000i		−	0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$		−	6.00000i		−	0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$		0			0
$745$		−	12.0000i		−	0.439646i
$746$		0			0
$747$	30.0000	−	30.0000i	1.09764	−	1.09764i
$748$		0			0
$749$	5.00000	+	5.00000i	0.182696	+	0.182696i
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	20.0000	+	20.0000i	0.727875	+	0.727875i
$756$		0			0
$757$	−15.0000	+	15.0000i	−0.545184	+	0.545184i	−0.925044	−	0.379860i	$-0.875972\pi$
$757$	−15.0000	+	15.0000i	−0.545184	+	0.545184i	0.379860	+	0.925044i	$0.375972\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			30.0000i			1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		0			0
$763$	−3.00000	+	3.00000i	−0.108607	+	0.108607i
$764$		0			0
$765$	12.0000	+	12.0000i	0.433861	+	0.433861i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.00000	+	6.00000i	−0.215805	+	0.215805i	−0.806728	−	0.590923i	$-0.798764\pi$
$773$	−6.00000	+	6.00000i	−0.215805	+	0.215805i	0.590923	+	0.806728i	$0.298764\pi$
$774$		0			0
$775$		−	24.0000i		−	0.862105i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−20.0000	+	20.0000i	−0.716574	+	0.716574i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−48.0000			−1.71319
$786$		0			0
$787$	−18.0000	−	18.0000i	−0.641631	−	0.641631i	0.309326	−	0.950956i	$-0.399897\pi$
$787$	−18.0000	−	18.0000i	−0.641631	−	0.641631i	−0.950956	+	0.309326i	$0.899897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	4.00000i		−	0.142224i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−32.0000	−	32.0000i	−1.13350	−	1.13350i	−0.989591	−	0.143907i	$-0.954033\pi$
$797$	−32.0000	−	32.0000i	−1.13350	−	1.13350i	−0.143907	−	0.989591i	$-0.545967\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$	42.0000			1.48400
$802$		0			0
$803$	−6.00000	−	6.00000i	−0.211735	−	0.211735i
$804$		0			0
$805$	−12.0000	+	12.0000i	−0.422944	+	0.422944i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	6.00000i		−	0.210949i	−0.994422	−	0.105474i	$-0.966364\pi$
$809$		−	6.00000i		−	0.210949i	0.994422	−	0.105474i	$-0.0336361\pi$
$810$		0			0
$811$	24.0000	−	24.0000i	0.842754	−	0.842754i	−0.146462	−	0.989216i	$-0.546789\pi$
$811$	24.0000	−	24.0000i	0.842754	−	0.842754i	0.989216	+	0.146462i	$0.0467887\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−60.0000			−2.10171
$816$		0			0
$817$	−4.00000			−0.139942
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.00000	−	1.00000i	0.0349002	−	0.0349002i	−0.689441	−	0.724342i	$-0.742144\pi$
$821$	1.00000	−	1.00000i	0.0349002	−	0.0349002i	0.724342	+	0.689441i	$0.242144\pi$
$822$		0			0
$823$			24.0000i			0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$			24.0000i			0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5.00000	−	5.00000i	0.173867	−	0.173867i	−0.614809	−	0.788676i	$-0.710767\pi$
$827$	5.00000	−	5.00000i	0.173867	−	0.173867i	0.788676	+	0.614809i	$0.210767\pi$
$828$		0			0
$829$	2.00000	+	2.00000i	0.0694629	+	0.0694629i	0.740985	−	0.671522i	$-0.234359\pi$
$829$	2.00000	+	2.00000i	0.0694629	+	0.0694629i	−0.671522	+	0.740985i	$0.734359\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.00000			0.0692959
$834$		0			0
$835$	24.0000	+	24.0000i	0.830554	+	0.830554i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	4.00000i		−	0.138095i	−0.997613	−	0.0690477i	$-0.978004\pi$
$839$		−	4.00000i		−	0.138095i	0.997613	−	0.0690477i	$-0.0219961\pi$
$840$		0			0
$841$			69.0000i			2.37931i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−26.0000	−	26.0000i	−0.894427	−	0.894427i
$846$		0			0
$847$	9.00000			0.309244
$848$		0			0
$849$		0			0
$850$		0			0
$851$	30.0000	+	30.0000i	1.02839	+	1.02839i
$852$		0			0
$853$	24.0000	−	24.0000i	0.821744	−	0.821744i	−0.164614	−	0.986358i	$-0.552638\pi$
$853$	24.0000	−	24.0000i	0.821744	−	0.821744i	0.986358	+	0.164614i	$0.0526378\pi$
$854$		0			0
$855$			24.0000i			0.820783i
$856$		0			0
$857$			18.0000i			0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$			18.0000i			0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$		0			0
$859$	2.00000	−	2.00000i	0.0682391	−	0.0682391i	−0.672164	−	0.740403i	$-0.734635\pi$
$859$	2.00000	−	2.00000i	0.0682391	−	0.0682391i	0.740403	+	0.672164i	$0.234635\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14.0000			−0.476566			−0.238283	−	0.971196i	$-0.576585\pi$
$863$	−14.0000			−0.476566			−0.238283	+	0.971196i	$0.576585\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	10.0000	−	10.0000i	0.339227	−	0.339227i
$870$		0			0
$871$		0			0
$872$		0			0
$873$			6.00000i			0.203069i
$874$		0			0
$875$	4.00000	−	4.00000i	0.135225	−	0.135225i
$876$		0			0
$877$	3.00000	+	3.00000i	0.101303	+	0.101303i	0.755942	−	0.654639i	$-0.227179\pi$
$877$	3.00000	+	3.00000i	0.101303	+	0.101303i	−0.654639	+	0.755942i	$0.727179\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	25.0000	+	25.0000i	0.841317	+	0.841317i	0.989030	−	0.147713i	$-0.0471913\pi$
$883$	25.0000	+	25.0000i	0.841317	+	0.841317i	−0.147713	+	0.989030i	$0.547191\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			12.0000i			0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$			12.0000i			0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$		0			0
$889$			8.00000i			0.268311i
$890$		0			0
$891$	9.00000	−	9.00000i	0.301511	−	0.301511i
$892$		0			0
$893$	−24.0000	−	24.0000i	−0.803129	−	0.803129i
$894$		0			0
$895$	12.0000			0.401116
$896$		0			0
$897$		0			0
$898$		0			0
$899$	56.0000	+	56.0000i	1.86770	+	1.86770i
$900$		0			0
$901$	2.00000	−	2.00000i	0.0666297	−	0.0666297i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			16.0000i			0.531858i
$906$		0			0
$907$	−15.0000	+	15.0000i	−0.498067	+	0.498067i	−0.910836	−	0.412769i	$-0.864562\pi$
$907$	−15.0000	+	15.0000i	−0.498067	+	0.498067i	0.412769	+	0.910836i	$0.364562\pi$
$908$		0			0
$909$	−18.0000	−	18.0000i	−0.597022	−	0.597022i
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$	−20.0000			−0.661903
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14.0000	−	14.0000i	0.462321	−	0.462321i
$918$		0			0
$919$		−	24.0000i		−	0.791687i	−0.918318	−	0.395843i	$-0.870452\pi$
$919$		−	24.0000i		−	0.791687i	0.918318	−	0.395843i	$-0.129548\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	15.0000	+	15.0000i	0.493197	+	0.493197i
$926$		0			0
$927$	12.0000			0.394132
$928$		0			0
$929$	−50.0000			−1.64045			−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000			−1.64045			−0.820223	+	0.572043i	$0.806151\pi$
$930$		0			0
$931$	2.00000	+	2.00000i	0.0655474	+	0.0655474i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	8.00000i		−	0.261628i
$936$		0			0
$937$			58.0000i			1.89478i	0.320085	+	0.947389i	$0.396288\pi$
$937$			58.0000i			1.89478i	−0.320085	+	0.947389i	$0.603712\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	26.0000	+	26.0000i	0.847576	+	0.847576i	0.989830	−	0.142254i	$-0.0454351\pi$
$941$	26.0000	+	26.0000i	0.847576	+	0.847576i	−0.142254	+	0.989830i	$0.545435\pi$
$942$		0			0
$943$	−60.0000			−1.95387
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−33.0000	−	33.0000i	−1.07236	−	1.07236i	−0.997169	−	0.0751864i	$-0.976045\pi$
$947$	−33.0000	−	33.0000i	−1.07236	−	1.07236i	−0.0751864	−	0.997169i	$-0.523955\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			16.0000i			0.518291i	0.965838	+	0.259145i	$0.0834409\pi$
$953$			16.0000i			0.518291i	−0.965838	+	0.259145i	$0.916559\pi$
$954$		0			0
$955$	36.0000	−	36.0000i	1.16493	−	1.16493i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12.0000			−0.387500
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	15.0000	+	15.0000i	0.483368	+	0.483368i
$964$		0			0
$965$	−32.0000	+	32.0000i	−1.03012	+	1.03012i
$966$		0			0
$967$			2.00000i			0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$			2.00000i			0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.00000	−	4.00000i	0.128366	−	0.128366i	−0.640005	−	0.768371i	$-0.721068\pi$
$971$	4.00000	−	4.00000i	0.128366	−	0.128366i	0.768371	+	0.640005i	$0.221068\pi$
$972$		0			0
$973$	−2.00000	−	2.00000i	−0.0641171	−	0.0641171i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	38.0000			1.21573			0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000			1.21573			0.607864	+	0.794041i	$0.292027\pi$
$978$		0			0
$979$	−14.0000	−	14.0000i	−0.447442	−	0.447442i
$980$		0			0
$981$	−9.00000	+	9.00000i	−0.287348	+	0.287348i
$982$		0			0
$983$			44.0000i			1.40338i	0.712481	+	0.701691i	$0.247571\pi$
$983$			44.0000i			1.40338i	−0.712481	+	0.701691i	$0.752429\pi$
$984$		0			0
$985$		−	20.0000i		−	0.637253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.00000	−	6.00000i	−0.190789	−	0.190789i
$990$		0			0
$991$	−22.0000			−0.698853			−0.349427	−	0.936964i	$-0.613624\pi$
$991$	−22.0000			−0.698853			−0.349427	+	0.936964i	$0.613624\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−48.0000	−	48.0000i	−1.52170	−	1.52170i
$996$		0			0
$997$	−30.0000	+	30.0000i	−0.950110	+	0.950110i	−0.998813	−	0.0487037i	$-0.984491\pi$
$997$	−30.0000	+	30.0000i	−0.950110	+	0.950110i	0.0487037	+	0.998813i	$0.484491\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.m.a.337.1		2
4.3	odd	2		112.2.m.b.29.1	✓	2
8.3	odd	2		896.2.m.b.673.1		2
8.5	even	2		896.2.m.c.673.1		2
16.3	odd	4		896.2.m.b.225.1		2
16.5	even	4	inner	448.2.m.a.113.1		2
16.11	odd	4		112.2.m.b.85.1	yes	2
16.13	even	4		896.2.m.c.225.1		2
28.3	even	6		784.2.x.e.765.1		4
28.11	odd	6		784.2.x.d.765.1		4
28.19	even	6		784.2.x.e.557.1		4
28.23	odd	6		784.2.x.d.557.1		4
28.27	even	2		784.2.m.a.589.1		2
32.5	even	8		7168.2.a.k.1.2		2
32.11	odd	8		7168.2.a.b.1.1		2
32.21	even	8		7168.2.a.k.1.1		2
32.27	odd	8		7168.2.a.b.1.2		2
112.11	odd	12		784.2.x.d.373.1		4
112.27	even	4		784.2.m.a.197.1		2
112.59	even	12		784.2.x.e.373.1		4
112.75	even	12		784.2.x.e.165.1		4
112.107	odd	12		784.2.x.d.165.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.b.29.1	✓	2	4.3	odd	2
112.2.m.b.85.1	yes	2	16.11	odd	4
448.2.m.a.113.1		2	16.5	even	4	inner
448.2.m.a.337.1		2	1.1	even	1	trivial
784.2.m.a.197.1		2	112.27	even	4
784.2.m.a.589.1		2	28.27	even	2
784.2.x.d.165.1		4	112.107	odd	12
784.2.x.d.373.1		4	112.11	odd	12
784.2.x.d.557.1		4	28.23	odd	6
784.2.x.d.765.1		4	28.11	odd	6
784.2.x.e.165.1		4	112.75	even	12
784.2.x.e.373.1		4	112.59	even	12
784.2.x.e.557.1		4	28.19	even	6
784.2.x.e.765.1		4	28.3	even	6
896.2.m.b.225.1		2	16.3	odd	4
896.2.m.b.673.1		2	8.3	odd	2
896.2.m.c.225.1		2	16.13	even	4
896.2.m.c.673.1		2	8.5	even	2
7168.2.a.b.1.1		2	32.11	odd	8
7168.2.a.b.1.2		2	32.27	odd	8
7168.2.a.k.1.1		2	32.21	even	8
7168.2.a.k.1.2		2	32.5	even	8

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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(2.00000 - 2.00000i) q^{5} -1.00000i q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(2.00000 - 2.00000i) q^{5} -1.00000i q^{7} -3.00000i q^{9} +(-1.00000 + 1.00000i) q^{11} -2.00000 q^{17} +(-2.00000 - 2.00000i) q^{19} -6.00000i q^{23} -3.00000i q^{25} +(7.00000 + 7.00000i) q^{29} +8.00000 q^{31} +(-2.00000 - 2.00000i) q^{35} +(-5.00000 + 5.00000i) q^{37} -10.0000i q^{41} +(1.00000 - 1.00000i) q^{43} +(-6.00000 - 6.00000i) q^{45} +12.0000 q^{47} -1.00000 q^{49} +(-1.00000 + 1.00000i) q^{53} +4.00000i q^{55} +(-8.00000 + 8.00000i) q^{59} +(6.00000 + 6.00000i) q^{61} -3.00000 q^{63} +(-3.00000 - 3.00000i) q^{67} +6.00000i q^{73} +(1.00000 + 1.00000i) q^{77} -10.0000 q^{79} -9.00000 q^{81} +(10.0000 + 10.0000i) q^{83} +(-4.00000 + 4.00000i) q^{85} +14.0000i q^{89} -8.00000 q^{95} -2.00000 q^{97} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5}+O(q^{10}) \) 2 * q + 4 * q^5	\( 2 q + 4 q^{5} - 2 q^{11} - 4 q^{17} - 4 q^{19} + 14 q^{29} + 16 q^{31} - 4 q^{35} - 10 q^{37} + 2 q^{43} - 12 q^{45} + 24 q^{47} - 2 q^{49} - 2 q^{53} - 16 q^{59} + 12 q^{61} - 6 q^{63} - 6 q^{67} + 2 q^{77} - 20 q^{79} - 18 q^{81} + 20 q^{83} - 8 q^{85} - 16 q^{95} - 4 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q + 4 * q^5 - 2 * q^11 - 4 * q^17 - 4 * q^19 + 14 * q^29 + 16 * q^31 - 4 * q^35 - 10 * q^37 + 2 * q^43 - 12 * q^45 + 24 * q^47 - 2 * q^49 - 2 * q^53 - 16 * q^59 + 12 * q^61 - 6 * q^63 - 6 * q^67 + 2 * q^77 - 20 * q^79 - 18 * q^81 + 20 * q^83 - 8 * q^85 - 16 * q^95 - 4 * q^97 + 6 * q^99

Introduction

L-functions

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Fields

Representations

Groups

Database

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