LMFDB - Embedded newform 1400.2.x.c.657.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,2,Mod(657,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

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

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

chi := DirichletCharacter("1400.657");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1400.x (of order $4$, degree $2$, 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:	$11.1790562830$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			657.13
Character	$\chi$	$=$	1400.657
Dual form			1400.2.x.c.993.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.26512 + 1.26512i) q^{3} +(-1.98283 - 1.75168i) q^{7} +0.201074i q^{9} +O(q^{10})$	$q+(1.26512 + 1.26512i) q^{3} +(-1.98283 - 1.75168i) q^{7} +0.201074i q^{9} +4.28140 q^{11} +(-4.40575 - 4.40575i) q^{13} +(3.77213 - 3.77213i) q^{17} -8.01983 q^{19} +(-0.292427 - 4.72462i) q^{21} +(2.07403 - 2.07403i) q^{23} +(3.54099 - 3.54099i) q^{27} +0.383780i q^{29} -1.01573i q^{31} +(5.41649 + 5.41649i) q^{33} +(-5.30041 - 5.30041i) q^{37} -11.1476i q^{39} +3.42580i q^{41} +(4.67869 - 4.67869i) q^{43} +(-6.51241 + 6.51241i) q^{47} +(0.863214 + 6.94657i) q^{49} +9.54442 q^{51} +(6.18400 - 6.18400i) q^{53} +(-10.1461 - 10.1461i) q^{57} +9.39231 q^{59} +1.99639i q^{61} +(0.352218 - 0.398696i) q^{63} +(-0.224086 - 0.224086i) q^{67} +5.24781 q^{69} +7.88462 q^{71} +(-9.62980 - 9.62980i) q^{73} +(-8.48927 - 7.49965i) q^{77} -8.59338i q^{79} +9.56279 q^{81} +(6.32562 + 6.32562i) q^{83} +(-0.485529 + 0.485529i) q^{87} -1.04559 q^{89} +(1.01837 + 16.4533i) q^{91} +(1.28502 - 1.28502i) q^{93} +(-4.17461 + 4.17461i) q^{97} +0.860879i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 16 q^{11} - 40 q^{21} + 32 q^{51} + 128 q^{71}+O(q^{100}) $ 32 * q - 16 * q^11 - 40 * q^21 + 32 * q^51 + 128 * q^71

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$3$	1.26512	+	1.26512i	0.730419	+	0.730419i	0.970703	−	0.240283i	$-0.0772404\pi$
$3$	1.26512	+	1.26512i	0.730419	+	0.730419i	−0.240283	+	0.970703i	$0.577240\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.98283	−	1.75168i	−0.749439	−	0.662074i
$7$	−1.98283	−	1.75168i	−0.749439	−	0.662074i
$8$		0			0
$9$			0.201074i			0.0670248i
$10$		0			0
$11$	4.28140			1.29089			0.645445	−	0.763807i	$-0.276672\pi$
$11$	4.28140			1.29089			0.645445	+	0.763807i	$0.276672\pi$
$12$		0			0
$13$	−4.40575	−	4.40575i	−1.22194	−	1.22194i	−0.966943	−	0.254994i	$-0.917927\pi$
$13$	−4.40575	−	4.40575i	−1.22194	−	1.22194i	−0.254994	−	0.966943i	$-0.582073\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.77213	−	3.77213i	0.914876	−	0.914876i	−0.0817746	−	0.996651i	$-0.526059\pi$
$17$	3.77213	−	3.77213i	0.914876	−	0.914876i	0.996651	+	0.0817746i	$0.0260588\pi$
$18$		0			0
$19$	−8.01983			−1.83988			−0.919938	−	0.392064i	$-0.871761\pi$
$19$	−8.01983			−1.83988			−0.919938	+	0.392064i	$0.871761\pi$
$20$		0			0
$21$	−0.292427	−	4.72462i	−0.0638128	−	1.03100i
$22$		0			0
$23$	2.07403	−	2.07403i	0.432466	−	0.432466i	−0.457001	−	0.889466i	$-0.651076\pi$
$23$	2.07403	−	2.07403i	0.432466	−	0.432466i	0.889466	+	0.457001i	$0.151076\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.54099	−	3.54099i	0.681463	−	0.681463i
$28$		0			0
$29$			0.383780i			0.0712661i	0.999365	+	0.0356331i	$0.0113448\pi$
$29$			0.383780i			0.0712661i	−0.999365	+	0.0356331i	$0.988655\pi$
$30$		0			0
$31$		−	1.01573i		−	0.182430i	−0.995831	−	0.0912151i	$-0.970925\pi$
$31$		−	1.01573i		−	0.182430i	0.995831	−	0.0912151i	$-0.0290751\pi$
$32$		0			0
$33$	5.41649	+	5.41649i	0.942891	+	0.942891i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.30041	−	5.30041i	−0.871383	−	0.871383i	0.121240	−	0.992623i	$-0.461313\pi$
$37$	−5.30041	−	5.30041i	−0.871383	−	0.871383i	−0.992623	+	0.121240i	$0.961313\pi$
$38$		0			0
$39$		−	11.1476i		−	1.78505i
$40$		0			0
$41$			3.42580i			0.535019i	0.963555	+	0.267510i	$0.0862007\pi$
$41$			3.42580i			0.535019i	−0.963555	+	0.267510i	$0.913799\pi$
$42$		0			0
$43$	4.67869	−	4.67869i	0.713494	−	0.713494i	−0.253771	−	0.967264i	$-0.581671\pi$
$43$	4.67869	−	4.67869i	0.713494	−	0.713494i	0.967264	+	0.253771i	$0.0816708\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.51241	+	6.51241i	−0.949932	+	0.949932i	−0.998805	−	0.0488727i	$-0.984437\pi$
$47$	−6.51241	+	6.51241i	−0.949932	+	0.949932i	0.0488727	+	0.998805i	$0.484437\pi$
$48$		0			0
$49$	0.863214	+	6.94657i	0.123316	+	0.992367i
$50$		0			0
$51$	9.54442			1.33649
$52$		0			0
$53$	6.18400	−	6.18400i	0.849438	−	0.849438i	−0.140625	−	0.990063i	$-0.544911\pi$
$53$	6.18400	−	6.18400i	0.849438	−	0.849438i	0.990063	+	0.140625i	$0.0449112\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−10.1461	−	10.1461i	−1.34388	−	1.34388i
$58$		0			0
$59$	9.39231			1.22277			0.611387	−	0.791331i	$-0.290612\pi$
$59$	9.39231			1.22277			0.611387	+	0.791331i	$0.290612\pi$
$60$		0			0
$61$			1.99639i			0.255612i	0.991799	+	0.127806i	$0.0407935\pi$
$61$			1.99639i			0.255612i	−0.991799	+	0.127806i	$0.959207\pi$
$62$		0			0
$63$	0.352218	−	0.398696i	0.0443753	−	0.0502309i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−0.224086	−	0.224086i	−0.0273764	−	0.0273764i	0.693286	−	0.720662i	$-0.256162\pi$
$67$	−0.224086	−	0.224086i	−0.0273764	−	0.0273764i	−0.720662	+	0.693286i	$0.756162\pi$
$68$		0			0
$69$	5.24781			0.631763
$70$		0			0
$71$	7.88462			0.935732			0.467866	−	0.883799i	$-0.345023\pi$
$71$	7.88462			0.935732			0.467866	+	0.883799i	$0.345023\pi$
$72$		0			0
$73$	−9.62980	−	9.62980i	−1.12708	−	1.12708i	−0.990649	−	0.136434i	$-0.956436\pi$
$73$	−9.62980	−	9.62980i	−1.12708	−	1.12708i	−0.136434	−	0.990649i	$-0.543564\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.48927	−	7.49965i	−0.967442	−	0.854664i
$78$		0			0
$79$		−	8.59338i		−	0.966831i	−0.875391	−	0.483415i	$-0.839396\pi$
$79$		−	8.59338i		−	0.966831i	0.875391	−	0.483415i	$-0.160604\pi$
$80$		0			0
$81$	9.56279			1.06253
$82$		0			0
$83$	6.32562	+	6.32562i	0.694327	+	0.694327i	0.963181	−	0.268854i	$-0.0866450\pi$
$83$	6.32562	+	6.32562i	0.694327	+	0.694327i	−0.268854	+	0.963181i	$0.586645\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.485529	+	0.485529i	−0.0520542	+	0.0520542i
$88$		0			0
$89$	−1.04559			−0.110832			−0.0554162	−	0.998463i	$-0.517649\pi$
$89$	−1.04559			−0.110832			−0.0554162	+	0.998463i	$0.517649\pi$
$90$		0			0
$91$	1.01837	+	16.4533i	0.106754	+	1.72478i
$92$		0			0
$93$	1.28502	−	1.28502i	0.133250	−	0.133250i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.17461	+	4.17461i	−0.423867	+	0.423867i	−0.886533	−	0.462666i	$-0.846893\pi$
$97$	−4.17461	+	4.17461i	−0.423867	+	0.423867i	0.462666	+	0.886533i	$0.346893\pi$
$98$		0			0
$99$			0.860879i			0.0865216i
$100$		0			0
$101$			0.776429i			0.0772576i	0.999254	+	0.0386288i	$0.0122990\pi$
$101$			0.776429i			0.0772576i	−0.999254	+	0.0386288i	$0.987701\pi$
$102$		0			0
$103$	5.01402	+	5.01402i	0.494046	+	0.494046i	0.909578	−	0.415533i	$-0.136405\pi$
$103$	5.01402	+	5.01402i	0.494046	+	0.494046i	−0.415533	+	0.909578i	$0.636405\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.54274	+	9.54274i	0.922531	+	0.922531i	0.997208	−	0.0746766i	$-0.0237925\pi$
$107$	9.54274	+	9.54274i	0.922531	+	0.922531i	−0.0746766	+	0.997208i	$0.523792\pi$
$108$		0			0
$109$			14.7639i			1.41412i	0.707153	+	0.707061i	$0.249979\pi$
$109$			14.7639i			1.41412i	−0.707153	+	0.707061i	$0.750021\pi$
$110$		0			0
$111$		−	13.4114i		−	1.27295i
$112$		0			0
$113$	−6.24428	+	6.24428i	−0.587412	+	0.587412i	−0.936930	−	0.349518i	$-0.886345\pi$
$113$	−6.24428	+	6.24428i	−0.587412	+	0.587412i	0.349518	+	0.936930i	$0.386345\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.885884	−	0.885884i	0.0819000	−	0.0819000i
$118$		0			0
$119$	−14.0871	+	0.871910i	−1.29136	+	0.0799278i
$120$		0			0
$121$	7.33035			0.666396
$122$		0			0
$123$	−4.33405	+	4.33405i	−0.390788	+	0.390788i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2.29780	−	2.29780i	−0.203897	−	0.203897i	0.597770	−	0.801667i	$-0.296053\pi$
$127$	−2.29780	−	2.29780i	−0.203897	−	0.203897i	−0.801667	+	0.597770i	$0.796053\pi$
$128$		0			0
$129$	11.8382			1.04230
$130$		0			0
$131$			4.47139i			0.390667i	0.980737	+	0.195333i	$0.0625789\pi$
$131$			4.47139i			0.390667i	−0.980737	+	0.195333i	$0.937421\pi$
$132$		0			0
$133$	15.9019	+	14.0482i	1.37887	+	1.21813i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.36069	−	5.36069i	−0.457995	−	0.457995i	0.440002	−	0.897997i	$-0.354978\pi$
$137$	−5.36069	−	5.36069i	−0.457995	−	0.457995i	−0.897997	+	0.440002i	$0.854978\pi$
$138$		0			0
$139$	13.1143			1.11234			0.556171	−	0.831068i	$-0.312270\pi$
$139$	13.1143			1.11234			0.556171	+	0.831068i	$0.312270\pi$
$140$		0			0
$141$	−16.4780			−1.38770
$142$		0			0
$143$	−18.8628	−	18.8628i	−1.57738	−	1.57738i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−7.69620	+	9.88034i	−0.634772	+	0.814917i
$148$		0			0
$149$		−	20.0575i		−	1.64317i	−0.570084	−	0.821586i	$-0.693089\pi$
$149$		−	20.0575i		−	1.64317i	0.570084	−	0.821586i	$-0.306911\pi$
$150$		0			0
$151$	1.84788			0.150378			0.0751892	−	0.997169i	$-0.476044\pi$
$151$	1.84788			0.150378			0.0751892	+	0.997169i	$0.476044\pi$
$152$		0			0
$153$	0.758479	+	0.758479i	0.0613194	+	0.0613194i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.72270	+	2.72270i	−0.217295	+	0.217295i	−0.807357	−	0.590063i	$-0.799103\pi$
$157$	−2.72270	+	2.72270i	−0.217295	+	0.217295i	0.590063	+	0.807357i	$0.299103\pi$
$158$		0			0
$159$	15.6470			1.24089
$160$		0			0
$161$	−7.74550	+	0.479403i	−0.610431	+	0.0377822i
$162$		0			0
$163$	12.4404	−	12.4404i	0.974407	−	0.974407i	−0.0252734	−	0.999681i	$-0.508046\pi$
$163$	12.4404	−	12.4404i	0.974407	−	0.974407i	0.999681	+	0.0252734i	$0.00804564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.73815	+	2.73815i	−0.211885	+	0.211885i	−0.805068	−	0.593183i	$-0.797871\pi$
$167$	−2.73815	+	2.73815i	−0.211885	+	0.211885i	0.593183	+	0.805068i	$0.297871\pi$
$168$		0			0
$169$			25.8213i			1.98626i
$170$		0			0
$171$		−	1.61258i		−	0.123317i
$172$		0			0
$173$	−4.36706	−	4.36706i	−0.332021	−	0.332021i	0.521332	−	0.853354i	$-0.325435\pi$
$173$	−4.36706	−	4.36706i	−0.332021	−	0.332021i	−0.853354	+	0.521332i	$0.825435\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	11.8824	+	11.8824i	0.893138	+	0.893138i
$178$		0			0
$179$		−	15.1423i		−	1.13179i	−0.824478	−	0.565893i	$-0.808531\pi$
$179$		−	15.1423i		−	1.13179i	0.824478	−	0.565893i	$-0.191469\pi$
$180$		0			0
$181$			9.26686i			0.688800i	0.938823	+	0.344400i	$0.111918\pi$
$181$			9.26686i			0.688800i	−0.938823	+	0.344400i	$0.888082\pi$
$182$		0			0
$183$	−2.52568	+	2.52568i	−0.186704	+	0.186704i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	16.1500	−	16.1500i	1.18100	−	1.18100i
$188$		0			0
$189$	−13.2239	+	0.818482i	−0.961894	+	0.0595358i
$190$		0			0
$191$	12.2586			0.886998			0.443499	−	0.896275i	$-0.353737\pi$
$191$	12.2586			0.886998			0.443499	+	0.896275i	$0.353737\pi$
$192$		0			0
$193$	1.67292	−	1.67292i	0.120419	−	0.120419i	−0.644329	−	0.764748i	$-0.722863\pi$
$193$	1.67292	−	1.67292i	0.120419	−	0.120419i	0.764748	+	0.644329i	$0.222863\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−7.41191	−	7.41191i	−0.528077	−	0.528077i	0.391922	−	0.919998i	$-0.371810\pi$
$197$	−7.41191	−	7.41191i	−0.528077	−	0.528077i	−0.919998	+	0.391922i	$0.871810\pi$
$198$		0			0
$199$	−1.36448			−0.0967252			−0.0483626	−	0.998830i	$-0.515400\pi$
$199$	−1.36448			−0.0967252			−0.0483626	+	0.998830i	$0.515400\pi$
$200$		0			0
$201$		−	0.566992i		−	0.0399925i
$202$		0			0
$203$	0.672261	−	0.760970i	0.0471835	−	0.0534096i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.417035	+	0.417035i	0.0289859	+	0.0289859i
$208$		0			0
$209$	−34.3361			−2.37508
$210$		0			0
$211$	−14.7141			−1.01296			−0.506481	−	0.862251i	$-0.669054\pi$
$211$	−14.7141			−1.01296			−0.506481	+	0.862251i	$0.669054\pi$
$212$		0			0
$213$	9.97502	+	9.97502i	0.683477	+	0.683477i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.77923	+	2.01401i	−0.120782	+	0.136720i
$218$		0			0
$219$		−	24.3658i		−	1.64649i
$220$		0			0
$221$	−33.2382			−2.23584
$222$		0			0
$223$	−3.72778	−	3.72778i	−0.249630	−	0.249630i	0.571189	−	0.820819i	$-0.306483\pi$
$223$	−3.72778	−	3.72778i	−0.249630	−	0.249630i	−0.820819	+	0.571189i	$0.806483\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	20.1068	−	20.1068i	1.33453	−	1.33453i	0.433270	−	0.901264i	$-0.357360\pi$
$227$	20.1068	−	20.1068i	1.33453	−	1.33453i	0.901264	−	0.433270i	$-0.142640\pi$
$228$		0			0
$229$	−23.4574			−1.55011			−0.775056	−	0.631893i	$-0.782278\pi$
$229$	−23.4574			−1.55011			−0.775056	+	0.631893i	$0.782278\pi$
$230$		0			0
$231$	−1.25200	−	20.2280i	−0.0823753	−	1.33090i
$232$		0			0
$233$	−18.0820	+	18.0820i	−1.18459	+	1.18459i	−0.206047	+	0.978542i	$0.566060\pi$
$233$	−18.0820	+	18.0820i	−1.18459	+	1.18459i	−0.978542	+	0.206047i	$0.933940\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	10.8717	−	10.8717i	0.706192	−	0.706192i
$238$		0			0
$239$			10.6607i			0.689584i	0.938679	+	0.344792i	$0.112050\pi$
$239$			10.6607i			0.689584i	−0.938679	+	0.344792i	$0.887950\pi$
$240$		0			0
$241$		−	28.7323i		−	1.85081i	−0.378978	−	0.925406i	$-0.623724\pi$
$241$		−	28.7323i		−	1.85081i	0.378978	−	0.925406i	$-0.376276\pi$
$242$		0			0
$243$	1.47515	+	1.47515i	0.0946311	+	0.0946311i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	35.3334	+	35.3334i	2.24821	+	2.24821i
$248$		0			0
$249$			16.0054i			1.01430i
$250$		0			0
$251$		−	2.17077i		−	0.137018i	−0.997651	−	0.0685088i	$-0.978176\pi$
$251$		−	2.17077i		−	0.137018i	0.997651	−	0.0685088i	$-0.0218241\pi$
$252$		0			0
$253$	8.87976	−	8.87976i	0.558265	−	0.558265i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.7794	+	16.7794i	−1.04667	+	1.04667i	−0.0478117	+	0.998856i	$0.515225\pi$
$257$	−16.7794	+	16.7794i	−1.04667	+	1.04667i	−0.998856	+	0.0478117i	$0.984775\pi$
$258$		0			0
$259$	1.22516	+	19.7945i	0.0761280	+	1.22997i
$260$		0			0
$261$	−0.0771683			−0.00477660
$262$		0			0
$263$	−11.0795	+	11.0795i	−0.683191	+	0.683191i	−0.960718	−	0.277527i	$-0.910485\pi$
$263$	−11.0795	+	11.0795i	−0.683191	+	0.683191i	0.277527	+	0.960718i	$0.410485\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.32280	−	1.32280i	−0.0809541	−	0.0809541i
$268$		0			0
$269$	24.9736			1.52267			0.761335	−	0.648359i	$-0.224544\pi$
$269$	24.9736			1.52267			0.761335	+	0.648359i	$0.224544\pi$
$270$		0			0
$271$		−	6.40205i		−	0.388897i	−0.980913	−	0.194448i	$-0.937708\pi$
$271$		−	6.40205i		−	0.388897i	0.980913	−	0.194448i	$-0.0622917\pi$
$272$		0			0
$273$	−19.5271	+	22.1039i	−1.18184	+	1.33779i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	21.0777	+	21.0777i	1.26644	+	1.26644i	0.947916	+	0.318519i	$0.103185\pi$
$277$	21.0777	+	21.0777i	1.26644	+	1.26644i	0.318519	+	0.947916i	$0.396815\pi$
$278$		0			0
$279$	0.204237			0.0122273
$280$		0			0
$281$	−1.76219			−0.105123			−0.0525617	−	0.998618i	$-0.516739\pi$
$281$	−1.76219			−0.105123			−0.0525617	+	0.998618i	$0.516739\pi$
$282$		0			0
$283$	−7.33848	−	7.33848i	−0.436227	−	0.436227i	0.454513	−	0.890740i	$-0.349813\pi$
$283$	−7.33848	−	7.33848i	−0.436227	−	0.436227i	−0.890740	+	0.454513i	$0.849813\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.00091	−	6.79276i	0.354222	−	0.400964i
$288$		0			0
$289$		−	11.4580i		−	0.673997i
$290$		0			0
$291$	−10.5628			−0.619202
$292$		0			0
$293$	−15.8548	−	15.8548i	−0.926246	−	0.926246i	0.0712146	−	0.997461i	$-0.477312\pi$
$293$	−15.8548	−	15.8548i	−0.926246	−	0.926246i	−0.997461	+	0.0712146i	$0.977312\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	15.1604	−	15.1604i	0.879694	−	0.879694i
$298$		0			0
$299$	−18.2754			−1.05689
$300$		0			0
$301$	−17.4726	+	1.08146i	−1.00711	+	0.0623341i
$302$		0			0
$303$	−0.982278	+	0.982278i	−0.0564304	+	0.0564304i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−8.09271	+	8.09271i	−0.461875	+	0.461875i	−0.899270	−	0.437394i	$-0.855901\pi$
$307$	−8.09271	+	8.09271i	−0.461875	+	0.461875i	0.437394	+	0.899270i	$0.355901\pi$
$308$		0			0
$309$			12.6867i			0.721721i
$310$		0			0
$311$			32.2529i			1.82889i	0.404705	+	0.914447i	$0.367374\pi$
$311$			32.2529i			1.82889i	−0.404705	+	0.914447i	$0.632626\pi$
$312$		0			0
$313$	−3.45483	−	3.45483i	−0.195278	−	0.195278i	0.602694	−	0.797972i	$-0.294094\pi$
$313$	−3.45483	−	3.45483i	−0.195278	−	0.195278i	−0.797972	+	0.602694i	$0.794094\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	11.9389	+	11.9389i	0.670556	+	0.670556i	0.957844	−	0.287288i	$-0.0927537\pi$
$317$	11.9389	+	11.9389i	0.670556	+	0.670556i	−0.287288	+	0.957844i	$0.592754\pi$
$318$		0			0
$319$			1.64311i			0.0919967i
$320$		0			0
$321$			24.1455i			1.34767i
$322$		0			0
$323$	−30.2519	+	30.2519i	−1.68326	+	1.68326i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−18.6781	+	18.6781i	−1.03290	+	1.03290i
$328$		0			0
$329$	24.3207	−	1.50531i	1.34084	−	0.0829905i
$330$		0			0
$331$	2.08402			0.114548			0.0572739	−	0.998359i	$-0.481759\pi$
$331$	2.08402			0.114548			0.0572739	+	0.998359i	$0.481759\pi$
$332$		0			0
$333$	1.06578	−	1.06578i	0.0584042	−	0.0584042i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	11.7691	+	11.7691i	0.641103	+	0.641103i	0.950827	−	0.309723i	$-0.100236\pi$
$337$	11.7691	+	11.7691i	0.641103	+	0.641103i	−0.309723	+	0.950827i	$0.600236\pi$
$338$		0			0
$339$	−15.7996			−0.858114
$340$		0			0
$341$		−	4.34873i		−	0.235497i
$342$		0			0
$343$	10.4566	−	15.2859i	0.564603	−	0.825363i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.49402	+	6.49402i	0.348617	+	0.348617i	0.859594	−	0.510977i	$-0.170716\pi$
$347$	6.49402	+	6.49402i	0.348617	+	0.348617i	−0.510977	+	0.859594i	$0.670716\pi$
$348$		0			0
$349$	15.2983			0.818899			0.409450	−	0.912333i	$-0.365721\pi$
$349$	15.2983			0.818899			0.409450	+	0.912333i	$0.365721\pi$
$350$		0			0
$351$	−31.2014			−1.66541
$352$		0			0
$353$	19.2674	+	19.2674i	1.02550	+	1.02550i	0.999666	+	0.0258329i	$0.00822380\pi$
$353$	19.2674	+	19.2674i	1.02550	+	1.02550i	0.0258329	+	0.999666i	$0.491776\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−18.9250	−	16.7188i	−1.00161	−	0.884853i
$358$		0			0
$359$			3.44336i			0.181733i	0.995863	+	0.0908667i	$0.0289637\pi$
$359$			3.44336i			0.181733i	−0.995863	+	0.0908667i	$0.971036\pi$
$360$		0			0
$361$	45.3177			2.38514
$362$		0			0
$363$	9.27380	+	9.27380i	0.486748	+	0.486748i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−20.2373	+	20.2373i	−1.05638	+	1.05638i	−0.0580658	+	0.998313i	$0.518493\pi$
$367$	−20.2373	+	20.2373i	−1.05638	+	1.05638i	−0.998313	+	0.0580658i	$0.981507\pi$
$368$		0			0
$369$	−0.688839			−0.0358595
$370$		0			0
$371$	−23.0942	+	1.42940i	−1.19899	+	0.0742108i
$372$		0			0
$373$	19.7529	−	19.7529i	1.02277	−	1.02277i	0.0230311	−	0.999735i	$-0.492668\pi$
$373$	19.7529	−	19.7529i	1.02277	−	1.02277i	0.999735	−	0.0230311i	$-0.00733166\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.69084	−	1.69084i	0.0870827	−	0.0870827i
$378$		0			0
$379$			8.77071i			0.450521i	0.974299	+	0.225261i	$0.0723234\pi$
$379$			8.77071i			0.450521i	−0.974299	+	0.225261i	$0.927677\pi$
$380$		0			0
$381$		−	5.81400i		−	0.297860i
$382$		0			0
$383$	17.4151	+	17.4151i	0.889871	+	0.889871i	0.994510	−	0.104640i	$-0.0333689\pi$
$383$	17.4151	+	17.4151i	0.889871	+	0.889871i	−0.104640	+	0.994510i	$0.533369\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.940765	+	0.940765i	0.0478218	+	0.0478218i
$388$		0			0
$389$		−	29.5372i		−	1.49759i	−0.662800	−	0.748797i	$-0.730632\pi$
$389$		−	29.5372i		−	1.49759i	0.662800	−	0.748797i	$-0.269368\pi$
$390$		0			0
$391$		−	15.6470i		−	0.791305i
$392$		0			0
$393$	−5.65685	+	5.65685i	−0.285351	+	0.285351i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	4.02864	−	4.02864i	0.202192	−	0.202192i	−0.598747	−	0.800938i	$-0.704334\pi$
$397$	4.02864	−	4.02864i	0.202192	−	0.202192i	0.800938	+	0.598747i	$0.204334\pi$
$398$		0			0
$399$	2.34522	+	37.8906i	0.117408	+	1.89690i
$400$		0			0
$401$	1.06117			0.0529925			0.0264963	−	0.999649i	$-0.491565\pi$
$401$	1.06117			0.0529925			0.0264963	+	0.999649i	$0.491565\pi$
$402$		0			0
$403$	−4.47505	+	4.47505i	−0.222918	+	0.222918i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−22.6932	−	22.6932i	−1.12486	−	1.12486i
$408$		0			0
$409$	−16.0914			−0.795668			−0.397834	−	0.917457i	$-0.630238\pi$
$409$	−16.0914			−0.795668			−0.397834	+	0.917457i	$0.630238\pi$
$410$		0			0
$411$		−	13.5639i		−	0.669056i
$412$		0			0
$413$	−18.6233	−	16.4524i	−0.916395	−	0.809567i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	16.5912	+	16.5912i	0.812477	+	0.812477i
$418$		0			0
$419$	33.0871			1.61641			0.808204	−	0.588902i	$-0.200440\pi$
$419$	33.0871			1.61641			0.808204	+	0.588902i	$0.200440\pi$
$420$		0			0
$421$	−21.8527			−1.06504			−0.532518	−	0.846419i	$-0.678754\pi$
$421$	−21.8527			−1.06504			−0.532518	+	0.846419i	$0.678754\pi$
$422$		0			0
$423$	−1.30948	−	1.30948i	−0.0636690	−	0.0636690i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	3.49705	−	3.95851i	0.169234	−	0.191566i
$428$		0			0
$429$		−	47.7275i		−	2.30430i
$430$		0			0
$431$	−10.1827			−0.490484			−0.245242	−	0.969462i	$-0.578867\pi$
$431$	−10.1827			−0.490484			−0.245242	+	0.969462i	$0.578867\pi$
$432$		0			0
$433$	−11.0245	−	11.0245i	−0.529801	−	0.529801i	0.390712	−	0.920513i	$-0.372229\pi$
$433$	−11.0245	−	11.0245i	−0.529801	−	0.529801i	−0.920513	+	0.390712i	$0.872229\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−16.6334	+	16.6334i	−0.795683	+	0.795683i
$438$		0			0
$439$	11.4238			0.545227			0.272613	−	0.962124i	$-0.412112\pi$
$439$	11.4238			0.545227			0.272613	+	0.962124i	$0.412112\pi$
$440$		0			0
$441$	−1.39678	+	0.173570i	−0.0665132	+	0.00826525i
$442$		0			0
$443$	−2.12567	+	2.12567i	−0.100994	+	0.100994i	−0.755798	−	0.654805i	$-0.772751\pi$
$443$	−2.12567	+	2.12567i	−0.100994	+	0.100994i	0.654805	+	0.755798i	$0.272751\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	25.3752	−	25.3752i	1.20021	−	1.20021i
$448$		0			0
$449$			22.8564i			1.07866i	0.842094	+	0.539330i	$0.181323\pi$
$449$			22.8564i			1.07866i	−0.842094	+	0.539330i	$0.818677\pi$
$450$		0			0
$451$			14.6672i			0.690651i
$452$		0			0
$453$	2.33780	+	2.33780i	0.109839	+	0.109839i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	18.9220	+	18.9220i	0.885133	+	0.885133i	0.994051	−	0.108918i	$-0.0347385\pi$
$457$	18.9220	+	18.9220i	0.885133	+	0.885133i	−0.108918	+	0.994051i	$0.534739\pi$
$458$		0			0
$459$		−	26.7141i		−	1.24691i
$460$		0			0
$461$		−	21.1123i		−	0.983298i	−0.870794	−	0.491649i	$-0.836394\pi$
$461$		−	21.1123i		−	0.983298i	0.870794	−	0.491649i	$-0.163606\pi$
$462$		0			0
$463$	8.34112	−	8.34112i	0.387645	−	0.387645i	−0.486202	−	0.873847i	$-0.661618\pi$
$463$	8.34112	−	8.34112i	0.387645	−	0.387645i	0.873847	+	0.486202i	$0.161618\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.82644	+	2.82644i	−0.130792	+	0.130792i	−0.769472	−	0.638680i	$-0.779481\pi$
$467$	−2.82644	+	2.82644i	−0.130792	+	0.130792i	0.638680	+	0.769472i	$0.279481\pi$
$468$		0			0
$469$	0.0517963	+	0.836850i	0.00239173	+	0.0386422i
$470$		0			0
$471$	−6.88909			−0.317433
$472$		0			0
$473$	20.0313	−	20.0313i	0.921042	−	0.921042i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.24344	+	1.24344i	0.0569334	+	0.0569334i
$478$		0			0
$479$	−15.4149			−0.704327			−0.352163	−	0.935939i	$-0.614554\pi$
$479$	−15.4149			−0.704327			−0.352163	+	0.935939i	$0.614554\pi$
$480$		0			0
$481$			46.7046i			2.12955i
$482$		0			0
$483$	−10.4055	−	9.19251i	−0.473467	−	0.418274i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	5.43657	+	5.43657i	0.246354	+	0.246354i	0.819473	−	0.573118i	$-0.194266\pi$
$487$	5.43657	+	5.43657i	0.246354	+	0.246354i	−0.573118	+	0.819473i	$0.694266\pi$
$488$		0			0
$489$	31.4773			1.42345
$490$		0			0
$491$	−2.86088			−0.129110			−0.0645548	−	0.997914i	$-0.520563\pi$
$491$	−2.86088			−0.129110			−0.0645548	+	0.997914i	$0.520563\pi$
$492$		0			0
$493$	1.44767	+	1.44767i	0.0651997	+	0.0651997i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−15.6338	−	13.8114i	−0.701274	−	0.619524i
$498$		0			0
$499$			25.4666i			1.14004i	0.821631	+	0.570020i	$0.193065\pi$
$499$			25.4666i			1.14004i	−0.821631	+	0.570020i	$0.806935\pi$
$500$		0			0
$501$	−6.92820			−0.309529
$502$		0			0
$503$	16.1711	+	16.1711i	0.721034	+	0.721034i	0.968816	−	0.247782i	$-0.0797016\pi$
$503$	16.1711	+	16.1711i	0.721034	+	0.721034i	−0.247782	+	0.968816i	$0.579702\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−32.6672	+	32.6672i	−1.45080	+	1.45080i
$508$		0			0
$509$	25.7310			1.14051			0.570253	−	0.821469i	$-0.306845\pi$
$509$	25.7310			1.14051			0.570253	+	0.821469i	$0.306845\pi$
$510$		0			0
$511$	2.22588	+	35.9626i	0.0984672	+	1.59089i
$512$		0			0
$513$	−28.3981	+	28.3981i	−1.25381	+	1.25381i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−27.8822	+	27.8822i	−1.22626	+	1.22626i
$518$		0			0
$519$		−	11.0497i		−	0.485030i
$520$		0			0
$521$			5.37847i			0.235635i	0.993035	+	0.117817i	$0.0375898\pi$
$521$			5.37847i			0.235635i	−0.993035	+	0.117817i	$0.962410\pi$
$522$		0			0
$523$	1.09945	+	1.09945i	0.0480755	+	0.0480755i	0.730736	−	0.682660i	$-0.239177\pi$
$523$	1.09945	+	1.09945i	0.0480755	+	0.0480755i	−0.682660	+	0.730736i	$0.739177\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.83146	−	3.83146i	−0.166901	−	0.166901i
$528$		0			0
$529$			14.3968i			0.625947i
$530$		0			0
$531$			1.88855i			0.0819562i
$532$		0			0
$533$	15.0932	−	15.0932i	0.653760	−	0.653760i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	19.1568	−	19.1568i	0.826679	−	0.826679i
$538$		0			0
$539$	3.69576	+	29.7410i	0.159188	+	1.28104i
$540$		0			0
$541$	17.1072			0.735497			0.367748	−	0.929925i	$-0.380129\pi$
$541$	17.1072			0.735497			0.367748	+	0.929925i	$0.380129\pi$
$542$		0			0
$543$	−11.7237	+	11.7237i	−0.503113	+	0.503113i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−21.3185	−	21.3185i	−0.911512	−	0.911512i	0.0848791	−	0.996391i	$-0.472950\pi$
$547$	−21.3185	−	21.3185i	−0.911512	−	0.911512i	−0.996391	+	0.0848791i	$0.972950\pi$
$548$		0			0
$549$	−0.401423			−0.0171323
$550$		0			0
$551$		−	3.07785i		−	0.131121i
$552$		0			0
$553$	−15.0529	+	17.0392i	−0.640113	+	0.724580i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12.9932	+	12.9932i	0.550540	+	0.550540i	0.926597	−	0.376057i	$-0.122720\pi$
$557$	12.9932	+	12.9932i	0.550540	+	0.550540i	−0.376057	+	0.926597i	$0.622720\pi$
$558$		0			0
$559$	−41.2263			−1.74369
$560$		0			0
$561$	40.8635			1.72526
$562$		0			0
$563$	9.52351	+	9.52351i	0.401368	+	0.401368i	0.878715	−	0.477347i	$-0.158401\pi$
$563$	9.52351	+	9.52351i	0.401368	+	0.401368i	−0.477347	+	0.878715i	$0.658401\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−18.9614	−	16.7510i	−0.796303	−	0.703475i
$568$		0			0
$569$		−	39.0170i		−	1.63568i	−0.575446	−	0.817840i	$-0.695171\pi$
$569$		−	39.0170i		−	1.63568i	0.575446	−	0.817840i	$-0.304829\pi$
$570$		0			0
$571$	18.7540			0.784831			0.392416	−	0.919788i	$-0.371639\pi$
$571$	18.7540			0.784831			0.392416	+	0.919788i	$0.371639\pi$
$572$		0			0
$573$	15.5086	+	15.5086i	0.647880	+	0.647880i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	7.95452	−	7.95452i	0.331151	−	0.331151i	−0.521872	−	0.853024i	$-0.674766\pi$
$577$	7.95452	−	7.95452i	0.331151	−	0.331151i	0.853024	+	0.521872i	$0.174766\pi$
$578$		0			0
$579$	4.23289			0.175913
$580$		0			0
$581$	−1.46214	−	23.6231i	−0.0606596	−	0.980051i
$582$		0			0
$583$	26.4762	−	26.4762i	1.09653	−	1.09653i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15.5572	+	15.5572i	−0.642114	+	0.642114i	−0.951075	−	0.308961i	$-0.900019\pi$
$587$	−15.5572	+	15.5572i	−0.642114	+	0.642114i	0.308961	+	0.951075i	$0.400019\pi$
$588$		0			0
$589$			8.14597i			0.335649i
$590$		0			0
$591$		−	18.7540i		−	0.771435i
$592$		0			0
$593$	−13.2956	−	13.2956i	−0.545987	−	0.545987i	0.379291	−	0.925277i	$-0.376168\pi$
$593$	−13.2956	−	13.2956i	−0.545987	−	0.545987i	−0.925277	+	0.379291i	$0.876168\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.72623	−	1.72623i	−0.0706500	−	0.0706500i
$598$		0			0
$599$		−	21.8479i		−	0.892680i	−0.894863	−	0.446340i	$-0.852727\pi$
$599$		−	21.8479i		−	0.892680i	0.894863	−	0.446340i	$-0.147273\pi$
$600$		0			0
$601$		−	27.9050i		−	1.13827i	−0.822245	−	0.569134i	$-0.807279\pi$
$601$		−	27.9050i		−	1.13827i	0.822245	−	0.569134i	$-0.192721\pi$
$602$		0			0
$603$	0.0450579	−	0.0450579i	0.00183490	−	0.00183490i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	22.0865	−	22.0865i	0.896462	−	0.896462i	−0.0986594	−	0.995121i	$-0.531455\pi$
$607$	22.0865	−	22.0865i	0.896462	−	0.896462i	0.995121	+	0.0986594i	$0.0314554\pi$
$608$		0			0
$609$	1.81321	−	0.112228i	0.0734751	−	0.00454769i
$610$		0			0
$611$	57.3841			2.32151
$612$		0			0
$613$	26.3711	−	26.3711i	1.06512	−	1.06512i	0.0673938	−	0.997726i	$-0.478532\pi$
$613$	26.3711	−	26.3711i	1.06512	−	1.06512i	0.997726	−	0.0673938i	$-0.0214684\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.34199	−	5.34199i	−0.215060	−	0.215060i	0.591353	−	0.806413i	$-0.298594\pi$
$617$	−5.34199	−	5.34199i	−0.215060	−	0.215060i	−0.806413	+	0.591353i	$0.798594\pi$
$618$		0			0
$619$	−19.8879			−0.799364			−0.399682	−	0.916654i	$-0.630879\pi$
$619$	−19.8879			−0.799364			−0.399682	+	0.916654i	$0.630879\pi$
$620$		0			0
$621$		−	14.6882i		−	0.589419i
$622$		0			0
$623$	2.07323	+	1.83154i	0.0830620	+	0.0733792i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−43.4394	−	43.4394i	−1.73480	−	1.73480i
$628$		0			0
$629$	−39.9877			−1.59441
$630$		0			0
$631$	15.9106			0.633392			0.316696	−	0.948527i	$-0.397427\pi$
$631$	15.9106			0.633392			0.316696	+	0.948527i	$0.397427\pi$
$632$		0			0
$633$	−18.6152	−	18.6152i	−0.739887	−	0.739887i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	26.8018	−	34.4080i	1.06193	−	1.36329i
$638$		0			0
$639$			1.58539i			0.0627172i
$640$		0			0
$641$	26.1569			1.03314			0.516569	−	0.856246i	$-0.327209\pi$
$641$	26.1569			1.03314			0.516569	+	0.856246i	$0.327209\pi$
$642$		0			0
$643$	2.86867	+	2.86867i	0.113129	+	0.113129i	0.761405	−	0.648276i	$-0.224510\pi$
$643$	2.86867	+	2.86867i	0.113129	+	0.113129i	−0.648276	+	0.761405i	$0.724510\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	23.0219	−	23.0219i	0.905086	−	0.905086i	−0.0907850	−	0.995871i	$-0.528938\pi$
$647$	23.0219	−	23.0219i	0.905086	−	0.905086i	0.995871	+	0.0907850i	$0.0289376\pi$
$648$		0			0
$649$	40.2122			1.57847
$650$		0			0
$651$	−4.79893	+	0.297026i	−0.188085	+	0.0116414i
$652$		0			0
$653$	10.4970	−	10.4970i	0.410778	−	0.410778i	−0.471231	−	0.882010i	$-0.656190\pi$
$653$	10.4970	−	10.4970i	0.410778	−	0.410778i	0.882010	+	0.471231i	$0.156190\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.93631	−	1.93631i	0.0755425	−	0.0755425i
$658$		0			0
$659$			34.3398i			1.33769i	0.743403	+	0.668844i	$0.233211\pi$
$659$			34.3398i			1.33769i	−0.743403	+	0.668844i	$0.766789\pi$
$660$		0			0
$661$			17.6084i			0.684887i	0.939539	+	0.342443i	$0.111254\pi$
$661$			17.6084i			0.684887i	−0.939539	+	0.342443i	$0.888746\pi$
$662$		0			0
$663$	−42.0504	−	42.0504i	−1.63310	−	1.63310i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	0.795972	+	0.795972i	0.0308202	+	0.0308202i
$668$		0			0
$669$		−	9.43220i		−	0.364670i
$670$		0			0
$671$			8.54735i			0.329967i
$672$		0			0
$673$	6.66679	−	6.66679i	0.256986	−	0.256986i	−0.566841	−	0.823827i	$-0.691835\pi$
$673$	6.66679	−	6.66679i	0.256986	−	0.256986i	0.823827	+	0.566841i	$0.191835\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	32.5877	−	32.5877i	1.25245	−	1.25245i	0.297826	−	0.954620i	$-0.403739\pi$
$677$	32.5877	−	32.5877i	1.25245	−	1.25245i	0.954620	−	0.297826i	$-0.0962615\pi$
$678$		0			0
$679$	15.5901	−	0.964941i	0.598294	−	0.0370310i
$680$		0			0
$681$	50.8751			1.94954
$682$		0			0
$683$	−17.1883	+	17.1883i	−0.657693	+	0.657693i	−0.954834	−	0.297141i	$-0.903967\pi$
$683$	−17.1883	+	17.1883i	−0.657693	+	0.657693i	0.297141	+	0.954834i	$0.403967\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−29.6766	−	29.6766i	−1.13223	−	1.13223i
$688$		0			0
$689$	−54.4904			−2.07592
$690$		0			0
$691$		−	22.3621i		−	0.850695i	−0.905030	−	0.425348i	$-0.860152\pi$
$691$		−	22.3621i		−	0.850695i	0.905030	−	0.425348i	$-0.139848\pi$
$692$		0			0
$693$	1.50799	−	1.70697i	0.0572837	−	0.0648426i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	12.9226	+	12.9226i	0.489476	+	0.489476i
$698$		0			0
$699$	−45.7518			−1.73049
$700$		0			0
$701$	−18.2586			−0.689616			−0.344808	−	0.938673i	$-0.612056\pi$
$701$	−18.2586			−0.689616			−0.344808	+	0.938673i	$0.612056\pi$
$702$		0			0
$703$	42.5084	+	42.5084i	1.60324	+	1.60324i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.36006	−	1.53953i	0.0511502	−	0.0578998i
$708$		0			0
$709$		−	34.3565i		−	1.29028i	−0.764063	−	0.645142i	$-0.776798\pi$
$709$		−	34.3565i		−	1.29028i	0.764063	−	0.645142i	$-0.223202\pi$
$710$		0			0
$711$	1.72791			0.0648016
$712$		0			0
$713$	−2.10665	−	2.10665i	−0.0788948	−	0.0788948i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−13.4871	+	13.4871i	−0.503685	+	0.503685i
$718$		0			0
$719$	3.87614			0.144556			0.0722778	−	0.997385i	$-0.476973\pi$
$719$	3.87614			0.144556			0.0722778	+	0.997385i	$0.476973\pi$
$720$		0			0
$721$	−1.15897	−	18.7249i	−0.0431621	−	0.697352i
$722$		0			0
$723$	36.3499	−	36.3499i	1.35187	−	1.35187i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	7.82189	−	7.82189i	0.290098	−	0.290098i	−0.547021	−	0.837119i	$-0.684238\pi$
$727$	7.82189	−	7.82189i	0.290098	−	0.290098i	0.837119	+	0.547021i	$0.184238\pi$
$728$		0			0
$729$		−	24.9559i		−	0.924292i
$730$		0			0
$731$		−	35.2973i		−	1.30552i
$732$		0			0
$733$	−7.96432	−	7.96432i	−0.294169	−	0.294169i	0.544556	−	0.838725i	$-0.316698\pi$
$733$	−7.96432	−	7.96432i	−0.294169	−	0.294169i	−0.838725	+	0.544556i	$0.816698\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.959400	−	0.959400i	−0.0353399	−	0.0353399i
$738$		0			0
$739$			20.6579i			0.759913i	0.925004	+	0.379957i	$0.124061\pi$
$739$			20.6579i			0.759913i	−0.925004	+	0.379957i	$0.875939\pi$
$740$		0			0
$741$			89.4023i			3.28427i
$742$		0			0
$743$	−11.9614	+	11.9614i	−0.438821	+	0.438821i	−0.891615	−	0.452794i	$-0.850427\pi$
$743$	−11.9614	+	11.9614i	−0.438821	+	0.438821i	0.452794	+	0.891615i	$0.350427\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.27192	+	1.27192i	−0.0465371	+	0.0465371i
$748$		0			0
$749$	−2.20576	−	35.6375i	−0.0805966	−	1.30216i
$750$		0			0
$751$	−37.8083			−1.37965			−0.689823	−	0.723978i	$-0.742312\pi$
$751$	−37.8083			−1.37965			−0.689823	+	0.723978i	$0.742312\pi$
$752$		0			0
$753$	2.74629	−	2.74629i	0.100080	−	0.100080i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−15.6665	−	15.6665i	−0.569407	−	0.569407i	0.362555	−	0.931962i	$-0.381904\pi$
$757$	−15.6665	−	15.6665i	−0.569407	−	0.569407i	−0.931962	+	0.362555i	$0.881904\pi$
$758$		0			0
$759$	22.4680			0.815536
$760$		0			0
$761$		−	40.3166i		−	1.46148i	−0.682658	−	0.730738i	$-0.739176\pi$
$761$		−	40.3166i		−	1.46148i	0.682658	−	0.730738i	$-0.260824\pi$
$762$		0			0
$763$	25.8616	−	29.2742i	0.936253	−	1.05980i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−41.3802	−	41.3802i	−1.49415	−	1.49415i
$768$		0			0
$769$	−19.6390			−0.708202			−0.354101	−	0.935207i	$-0.615213\pi$
$769$	−19.6390			−0.708202			−0.354101	+	0.935207i	$0.615213\pi$
$770$		0			0
$771$	−42.4559			−1.52901
$772$		0			0
$773$	24.8633	+	24.8633i	0.894271	+	0.894271i	0.994922	−	0.100651i	$-0.0320924\pi$
$773$	24.8633	+	24.8633i	0.894271	+	0.894271i	−0.100651	+	0.994922i	$0.532092\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−23.4924	+	26.5924i	−0.842787	+	0.953997i
$778$		0			0
$779$		−	27.4743i		−	0.984369i
$780$		0			0
$781$	33.7572			1.20793
$782$		0			0
$783$	1.35896	+	1.35896i	0.0485652	+	0.0485652i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	10.2824	−	10.2824i	0.366529	−	0.366529i	−0.499681	−	0.866210i	$-0.666549\pi$
$787$	10.2824	−	10.2824i	0.366529	−	0.366529i	0.866210	+	0.499681i	$0.166549\pi$
$788$		0			0
$789$	−28.0339			−0.998032
$790$		0			0
$791$	23.3193	−	1.44333i	0.829140	−	0.0513191i
$792$		0			0
$793$	8.79562	−	8.79562i	0.312342	−	0.312342i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.97178	+	4.97178i	−0.176110	+	0.176110i	−0.789658	−	0.613548i	$-0.789742\pi$
$797$	−4.97178	+	4.97178i	−0.176110	+	0.176110i	0.613548	+	0.789658i	$0.289742\pi$
$798$		0			0
$799$			49.1313i			1.73814i
$800$		0			0
$801$		−	0.210241i		−	0.00742851i
$802$		0			0
$803$	−41.2290	−	41.2290i	−1.45494	−	1.45494i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	31.5947	+	31.5947i	1.11219	+	1.11219i
$808$		0			0
$809$			31.9963i			1.12493i	0.826821	+	0.562465i	$0.190147\pi$
$809$			31.9963i			1.12493i	−0.826821	+	0.562465i	$0.809853\pi$
$810$		0			0
$811$		−	11.5246i		−	0.404682i	−0.979315	−	0.202341i	$-0.935145\pi$
$811$		−	11.5246i		−	0.404682i	0.979315	−	0.202341i	$-0.0648549\pi$
$812$		0			0
$813$	8.09938	−	8.09938i	0.284058	−	0.284058i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−37.5223	+	37.5223i	−1.31274	+	1.31274i
$818$		0			0
$819$	−3.30834	+	0.204768i	−0.115603	+	0.00715517i
$820$		0			0
$821$	−15.1844			−0.529939			−0.264969	−	0.964257i	$-0.585362\pi$
$821$	−15.1844			−0.529939			−0.264969	+	0.964257i	$0.585362\pi$
$822$		0			0
$823$	−16.8465	+	16.8465i	−0.587233	+	0.587233i	−0.936881	−	0.349648i	$-0.886301\pi$
$823$	−16.8465	+	16.8465i	−0.587233	+	0.587233i	0.349648	+	0.936881i	$0.386301\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	13.9187	+	13.9187i	0.484000	+	0.484000i	0.906407	−	0.422406i	$-0.138814\pi$
$827$	13.9187	+	13.9187i	0.484000	+	0.484000i	−0.422406	+	0.906407i	$0.638814\pi$
$828$		0			0
$829$	25.4450			0.883742			0.441871	−	0.897078i	$-0.354315\pi$
$829$	25.4450			0.883742			0.441871	+	0.897078i	$0.354315\pi$
$830$		0			0
$831$			53.3317i			1.85006i
$832$		0			0
$833$	29.4595	+	22.9472i	1.02071	+	0.795074i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−3.59668	−	3.59668i	−0.124319	−	0.124319i
$838$		0			0
$839$	46.9718			1.62165			0.810823	−	0.585291i	$-0.199020\pi$
$839$	46.9718			1.62165			0.810823	+	0.585291i	$0.199020\pi$
$840$		0			0
$841$	28.8527			0.994921
$842$		0			0
$843$	−2.22939	−	2.22939i	−0.0767841	−	0.0767841i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−14.5348	−	12.8405i	−0.499423	−	0.441203i
$848$		0			0
$849$		−	18.5682i		−	0.637258i
$850$		0			0
$851$	−21.9865			−0.753686
$852$		0			0
$853$	18.4660	+	18.4660i	0.632262	+	0.632262i	0.948635	−	0.316373i	$-0.102465\pi$
$853$	18.4660	+	18.4660i	0.632262	+	0.632262i	−0.316373	+	0.948635i	$0.602465\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.59503	+	6.59503i	−0.225282	+	0.225282i	−0.810718	−	0.585436i	$-0.800923\pi$
$857$	−6.59503	+	6.59503i	−0.225282	+	0.225282i	0.585436	+	0.810718i	$0.300923\pi$
$858$		0			0
$859$	48.8668			1.66731			0.833656	−	0.552284i	$-0.186244\pi$
$859$	48.8668			1.66731			0.833656	+	0.552284i	$0.186244\pi$
$860$		0			0
$861$	16.1856	−	1.00180i	0.551603	−	0.0341411i
$862$		0			0
$863$	−24.6363	+	24.6363i	−0.838628	+	0.838628i	−0.988678	−	0.150050i	$-0.952057\pi$
$863$	−24.6363	+	24.6363i	−0.838628	+	0.838628i	0.150050	+	0.988678i	$0.452057\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	14.4957	−	14.4957i	0.492301	−	0.492301i
$868$		0			0
$869$		−	36.7917i		−	1.24807i
$870$		0			0
$871$			1.97453i			0.0669045i
$872$		0			0
$873$	−0.839407	−	0.839407i	−0.0284096	−	0.0284096i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14.7489	−	14.7489i	−0.498035	−	0.498035i	0.412791	−	0.910826i	$-0.364554\pi$
$877$	−14.7489	−	14.7489i	−0.498035	−	0.498035i	−0.910826	+	0.412791i	$0.864554\pi$
$878$		0			0
$879$		−	40.1165i		−	1.35310i
$880$		0			0
$881$		−	13.7980i		−	0.464865i	−0.972612	−	0.232433i	$-0.925331\pi$
$881$		−	13.7980i		−	0.464865i	0.972612	−	0.232433i	$-0.0746685\pi$
$882$		0			0
$883$	16.6467	−	16.6467i	0.560205	−	0.560205i	−0.369160	−	0.929366i	$-0.620355\pi$
$883$	16.6467	−	16.6467i	0.560205	−	0.560205i	0.929366	+	0.369160i	$0.120355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21.7822	−	21.7822i	0.731374	−	0.731374i	−0.239518	−	0.970892i	$-0.576989\pi$
$887$	21.7822	−	21.7822i	0.731374	−	0.731374i	0.970892	+	0.239518i	$0.0769895\pi$
$888$		0			0
$889$	0.531125	+	8.58116i	0.0178134	+	0.287803i
$890$		0			0
$891$	40.9421			1.37161
$892$		0			0
$893$	52.2284	−	52.2284i	1.74776	−	1.74776i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−23.1206	−	23.1206i	−0.771974	−	0.771974i
$898$		0			0
$899$	0.389816			0.0130011
$900$		0			0
$901$		−	46.6537i		−	1.55426i
$902$		0			0
$903$	−23.4732	−	20.7369i	−0.781139	−	0.690079i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−5.71913	−	5.71913i	−0.189901	−	0.189901i	0.605753	−	0.795653i	$-0.292872\pi$
$907$	−5.71913	−	5.71913i	−0.189901	−	0.189901i	−0.795653	+	0.605753i	$0.792872\pi$
$908$		0			0
$909$	−0.156120			−0.00517817
$910$		0			0
$911$	43.6578			1.44645			0.723224	−	0.690614i	$-0.242660\pi$
$911$	43.6578			1.44645			0.723224	+	0.690614i	$0.242660\pi$
$912$		0			0
$913$	27.0825	+	27.0825i	0.896299	+	0.896299i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.83245	−	8.86599i	0.258650	−	0.292781i
$918$		0			0
$919$			5.83476i			0.192471i	0.995359	+	0.0962355i	$0.0306802\pi$
$919$			5.83476i			0.192471i	−0.995359	+	0.0962355i	$0.969320\pi$
$920$		0			0
$921$	−20.4766			−0.674726
$922$		0			0
$923$	−34.7377	−	34.7377i	−1.14341	−	1.14341i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1.00819	+	1.00819i	−0.0331133	+	0.0331133i
$928$		0			0
$929$	26.3688			0.865131			0.432566	−	0.901602i	$-0.357608\pi$
$929$	26.3688			0.865131			0.432566	+	0.901602i	$0.357608\pi$
$930$		0			0
$931$	−6.92283	−	55.7103i	−0.226887	−	1.82583i
$932$		0			0
$933$	−40.8039	+	40.8039i	−1.33586	+	1.33586i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	21.8652	−	21.8652i	0.714306	−	0.714306i	−0.253127	−	0.967433i	$-0.581459\pi$
$937$	21.8652	−	21.8652i	0.714306	−	0.714306i	0.967433	+	0.253127i	$0.0814592\pi$
$938$		0			0
$939$		−	8.74157i		−	0.285270i
$940$		0			0
$941$			57.2544i			1.86644i	0.359306	+	0.933220i	$0.383014\pi$
$941$			57.2544i			1.86644i	−0.359306	+	0.933220i	$0.616986\pi$
$942$		0			0
$943$	7.10521	+	7.10521i	0.231378	+	0.231378i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−10.6937	−	10.6937i	−0.347497	−	0.347497i	0.511679	−	0.859177i	$-0.329024\pi$
$947$	−10.6937	−	10.6937i	−0.347497	−	0.347497i	−0.859177	+	0.511679i	$0.829024\pi$
$948$		0			0
$949$			84.8531i			2.75445i
$950$		0			0
$951$			30.2084i			0.979574i
$952$		0			0
$953$	−18.6129	+	18.6129i	−0.602931	+	0.602931i	−0.941089	−	0.338158i	$-0.890196\pi$
$953$	−18.6129	+	18.6129i	−0.602931	+	0.602931i	0.338158	+	0.941089i	$0.390196\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−2.07874	+	2.07874i	−0.0671962	+	0.0671962i
$958$		0			0
$959$	1.23910	+	20.0196i	0.0400125	+	0.646465i
$960$		0			0
$961$	29.9683			0.966719
$962$		0			0
$963$	−1.91880	+	1.91880i	−0.0618324	+	0.0618324i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.723258	+	0.723258i	0.0232584	+	0.0232584i	0.718640	−	0.695382i	$-0.244765\pi$
$967$	0.723258	+	0.723258i	0.0232584	+	0.0232584i	−0.695382	+	0.718640i	$0.744765\pi$
$968$		0			0
$969$	−76.5447			−2.45897
$970$		0			0
$971$			20.6768i			0.663549i	0.943359	+	0.331775i	$0.107647\pi$
$971$			20.6768i			0.663549i	−0.943359	+	0.331775i	$0.892353\pi$
$972$		0			0
$973$	−26.0035	−	22.9721i	−0.833633	−	0.736453i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.12309	−	6.12309i	−0.195895	−	0.195895i	0.602343	−	0.798238i	$-0.294234\pi$
$977$	−6.12309	−	6.12309i	−0.195895	−	0.195895i	−0.798238	+	0.602343i	$0.794234\pi$
$978$		0			0
$979$	−4.47659			−0.143072
$980$		0			0
$981$	−2.96863			−0.0947812
$982$		0			0
$983$	26.4070	+	26.4070i	0.842254	+	0.842254i	0.989152	−	0.146898i	$-0.0469288\pi$
$983$	26.4070	+	26.4070i	0.842254	+	0.842254i	−0.146898	+	0.989152i	$0.546929\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	32.6730	+	28.8642i	1.03999	+	0.918758i
$988$		0			0
$989$		−	19.4075i		−	0.617123i
$990$		0			0
$991$	−24.1187			−0.766157			−0.383078	−	0.923716i	$-0.625136\pi$
$991$	−24.1187			−0.766157			−0.383078	+	0.923716i	$0.625136\pi$
$992$		0			0
$993$	2.63654	+	2.63654i	0.0836680	+	0.0836680i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−12.0465	+	12.0465i	−0.381517	+	0.381517i	−0.871648	−	0.490132i	$-0.836949\pi$
$997$	−12.0465	+	12.0465i	−0.381517	+	0.381517i	0.490132	+	0.871648i	$0.336949\pi$
$998$		0			0
$999$	−37.5374			−1.18763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.2.x.c.657.13	yes	32
5.2	odd	4	inner	1400.2.x.c.993.14	yes	32
5.3	odd	4	inner	1400.2.x.c.993.3	yes	32
5.4	even	2	inner	1400.2.x.c.657.4	yes	32
7.6	odd	2	inner	1400.2.x.c.657.3	✓	32
35.13	even	4	inner	1400.2.x.c.993.13	yes	32
35.27	even	4	inner	1400.2.x.c.993.4	yes	32
35.34	odd	2	inner	1400.2.x.c.657.14	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.x.c.657.3	✓	32	7.6	odd	2	inner
1400.2.x.c.657.4	yes	32	5.4	even	2	inner
1400.2.x.c.657.13	yes	32	1.1	even	1	trivial
1400.2.x.c.657.14	yes	32	35.34	odd	2	inner
1400.2.x.c.993.3	yes	32	5.3	odd	4	inner
1400.2.x.c.993.4	yes	32	35.27	even	4	inner
1400.2.x.c.993.13	yes	32	35.13	even	4	inner
1400.2.x.c.993.14	yes	32	5.2	odd	4	inner

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:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.26512 + 1.26512i) q^{3} +(-1.98283 - 1.75168i) q^{7} +0.201074i q^{9} +O(q^{10})\)	\(q+(1.26512 + 1.26512i) q^{3} +(-1.98283 - 1.75168i) q^{7} +0.201074i q^{9} +4.28140 q^{11} +(-4.40575 - 4.40575i) q^{13} +(3.77213 - 3.77213i) q^{17} -8.01983 q^{19} +(-0.292427 - 4.72462i) q^{21} +(2.07403 - 2.07403i) q^{23} +(3.54099 - 3.54099i) q^{27} +0.383780i q^{29} -1.01573i q^{31} +(5.41649 + 5.41649i) q^{33} +(-5.30041 - 5.30041i) q^{37} -11.1476i q^{39} +3.42580i q^{41} +(4.67869 - 4.67869i) q^{43} +(-6.51241 + 6.51241i) q^{47} +(0.863214 + 6.94657i) q^{49} +9.54442 q^{51} +(6.18400 - 6.18400i) q^{53} +(-10.1461 - 10.1461i) q^{57} +9.39231 q^{59} +1.99639i q^{61} +(0.352218 - 0.398696i) q^{63} +(-0.224086 - 0.224086i) q^{67} +5.24781 q^{69} +7.88462 q^{71} +(-9.62980 - 9.62980i) q^{73} +(-8.48927 - 7.49965i) q^{77} -8.59338i q^{79} +9.56279 q^{81} +(6.32562 + 6.32562i) q^{83} +(-0.485529 + 0.485529i) q^{87} -1.04559 q^{89} +(1.01837 + 16.4533i) q^{91} +(1.28502 - 1.28502i) q^{93} +(-4.17461 + 4.17461i) q^{97} +0.860879i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 16 q^{11} - 40 q^{21} + 32 q^{51} + 128 q^{71}+O(q^{100}) \) 32 * q - 16 * q^11 - 40 * q^21 + 32 * q^51 + 128 * q^71

Introduction

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