LMFDB - Embedded newform 4680.2.l.e.2809.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4680,2,Mod(2809,4680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4680.2809"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4680.l (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2809.5
Root			$-1.75233 + 1.75233i$ of defining polynomial
Character	$\chi$	$=$	4680.2809
Dual form			4680.2.l.e.2809.6

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.75233 - 1.38900i) q^{5} -1.50466 q^{11} -1.00000i q^{13} +2.72666i q^{17} -0.726656 q^{19} -4.72666i q^{23} +(1.14134 - 4.86799i) q^{25} -7.55602 q^{29} -3.00933 q^{31} -5.00933i q^{37} -5.78734 q^{41} -2.72666i q^{43} +10.2313i q^{47} +7.00000 q^{49} -7.55602i q^{53} +(-2.63667 + 2.08998i) q^{55} -12.5140 q^{59} +6.28267 q^{61} +(-1.38900 - 1.75233i) q^{65} -12.5653i q^{67} -4.77801 q^{71} -12.0187i q^{73} -5.27334 q^{79} +7.78734i q^{83} +(3.78734 + 4.77801i) q^{85} +1.78734 q^{89} +(-1.27334 + 1.00933i) q^{95} -6.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 12 q^{11} + 4 q^{19} - 10 q^{25} - 20 q^{29} + 24 q^{31} + 20 q^{41} + 42 q^{49} - 20 q^{55} - 12 q^{59} + 4 q^{61} - 2 q^{65} - 16 q^{71} - 40 q^{79} - 32 q^{85} - 44 q^{89} - 16 q^{95}+O(q^{100})$ 6 * q + 12 * q^11 + 4 * q^19 - 10 * q^25 - 20 * q^29 + 24 * q^31 + 20 * q^41 + 42 * q^49 - 20 * q^55 - 12 * q^59 + 4 * q^61 - 2 * q^65 - 16 * q^71 - 40 * q^79 - 32 * q^85 - 44 * q^89 - 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$ .

$n$	$937$	$1081$	$2081$	$2341$	$3511$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$		0			0
$4$		0			0
$5$	1.75233	−	1.38900i	0.783667	−	0.621181i
$5$	1.75233	−	1.38900i	0.783667	−	0.621181i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.50466			−0.453673			−0.226837	−	0.973933i	$-0.572838\pi$
$11$	−1.50466			−0.453673			−0.226837	+	0.973933i	$0.572838\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.72666i			0.661311i	0.943751	+	0.330656i	$0.107270\pi$
$17$			2.72666i			0.661311i	−0.943751	+	0.330656i	$0.892730\pi$
$18$		0			0
$19$	−0.726656			−0.166706			−0.0833532	−	0.996520i	$-0.526563\pi$
$19$	−0.726656			−0.166706			−0.0833532	+	0.996520i	$0.526563\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	4.72666i		−	0.985576i	−0.870149	−	0.492788i	$-0.835978\pi$
$23$		−	4.72666i		−	0.985576i	0.870149	−	0.492788i	$-0.164022\pi$
$24$		0			0
$25$	1.14134	−	4.86799i	0.228267	−	0.973599i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.55602			−1.40312			−0.701558	−	0.712612i	$-0.747512\pi$
$29$	−7.55602			−1.40312			−0.701558	+	0.712612i	$0.747512\pi$
$30$		0			0
$31$	−3.00933			−0.540491			−0.270246	−	0.962791i	$-0.587105\pi$
$31$	−3.00933			−0.540491			−0.270246	+	0.962791i	$0.587105\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	5.00933i		−	0.823529i	−0.911290	−	0.411764i	$-0.864913\pi$
$37$		−	5.00933i		−	0.823529i	0.911290	−	0.411764i	$-0.135087\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.78734			−0.903830			−0.451915	−	0.892061i	$-0.649259\pi$
$41$	−5.78734			−0.903830			−0.451915	+	0.892061i	$0.649259\pi$
$42$		0			0
$43$		−	2.72666i		−	0.415811i	−0.978149	−	0.207906i	$-0.933335\pi$
$43$		−	2.72666i		−	0.415811i	0.978149	−	0.207906i	$-0.0666647\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.2313i			1.49239i	0.665727	+	0.746196i	$0.268122\pi$
$47$			10.2313i			1.49239i	−0.665727	+	0.746196i	$0.731878\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	7.55602i		−	1.03790i	−0.854805	−	0.518949i	$-0.826323\pi$
$53$		−	7.55602i		−	1.03790i	0.854805	−	0.518949i	$-0.173677\pi$
$54$		0			0
$55$	−2.63667	+	2.08998i	−0.355529	+	0.281813i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.5140			−1.62918			−0.814592	−	0.580035i	$-0.803039\pi$
$59$	−12.5140			−1.62918			−0.814592	+	0.580035i	$0.803039\pi$
$60$		0			0
$61$	6.28267			0.804414			0.402207	−	0.915549i	$-0.368243\pi$
$61$	6.28267			0.804414			0.402207	+	0.915549i	$0.368243\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.38900	−	1.75233i	−0.172285	−	0.217350i
$66$		0			0
$67$		−	12.5653i		−	1.53510i	−0.640988	−	0.767551i	$-0.721475\pi$
$67$		−	12.5653i		−	1.53510i	0.640988	−	0.767551i	$-0.278525\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.77801			−0.567045			−0.283523	−	0.958966i	$-0.591503\pi$
$71$	−4.77801			−0.567045			−0.283523	+	0.958966i	$0.591503\pi$
$72$		0			0
$73$		−	12.0187i		−	1.40668i	−0.710855	−	0.703339i	$-0.751692\pi$
$73$		−	12.0187i		−	1.40668i	0.710855	−	0.703339i	$-0.248308\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.27334			−0.593297			−0.296649	−	0.954987i	$-0.595869\pi$
$79$	−5.27334			−0.593297			−0.296649	+	0.954987i	$0.595869\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			7.78734i			0.854771i	0.904069	+	0.427386i	$0.140565\pi$
$83$			7.78734i			0.854771i	−0.904069	+	0.427386i	$0.859435\pi$
$84$		0			0
$85$	3.78734	+	4.77801i	0.410794	+	0.518248i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.78734			0.189457			0.0947286	−	0.995503i	$-0.469802\pi$
$89$	1.78734			0.189457			0.0947286	+	0.995503i	$0.469802\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.27334	+	1.00933i	−0.130642	+	0.103555i
$96$		0			0
$97$		−	6.00000i		−	0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$		−	6.00000i		−	0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.99067			0.297583			0.148791	−	0.988869i	$-0.452462\pi$
$101$	2.99067			0.297583			0.148791	+	0.988869i	$0.452462\pi$
$102$		0			0
$103$			0.443984i			0.0437471i	0.999761	+	0.0218735i	$0.00696312\pi$
$103$			0.443984i			0.0437471i	−0.999761	+	0.0218735i	$0.993037\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	7.00933i		−	0.677617i	−0.940855	−	0.338809i	$-0.889976\pi$
$107$		−	7.00933i		−	0.677617i	0.940855	−	0.338809i	$-0.110024\pi$
$108$		0			0
$109$	13.8387			1.32551			0.662753	−	0.748838i	$-0.269388\pi$
$109$	13.8387			1.32551			0.662753	+	0.748838i	$0.269388\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	4.28267i		−	0.402880i	−0.979501	−	0.201440i	$-0.935438\pi$
$113$		−	4.28267i		−	0.402880i	0.979501	−	0.201440i	$-0.0645621\pi$
$114$		0			0
$115$	−6.56534	−	8.28267i	−0.612222	−	0.772363i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−8.73599			−0.794180
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.76166	−	10.1157i	−0.425896	−	0.904772i
$126$		0			0
$127$		−	5.71733i		−	0.507331i	−0.967292	−	0.253665i	$-0.918364\pi$
$127$		−	5.71733i		−	0.507331i	0.967292	−	0.253665i	$-0.0816362\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.55602			−0.485431			−0.242716	−	0.970097i	$-0.578038\pi$
$131$	−5.55602			−0.485431			−0.242716	+	0.970097i	$0.578038\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.67531i			0.570310i	0.958481	+	0.285155i	$0.0920450\pi$
$137$			6.67531i			0.570310i	−0.958481	+	0.285155i	$0.907955\pi$
$138$		0			0
$139$	−19.4720			−1.65159			−0.825795	−	0.563970i	$-0.809273\pi$
$139$	−19.4720			−1.65159			−0.825795	+	0.563970i	$0.809273\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1.50466i			0.125826i
$144$		0			0
$145$	−13.2406	+	10.4953i	−1.09958	+	0.871590i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−14.5140			−1.18903			−0.594516	−	0.804084i	$-0.702656\pi$
$149$	−14.5140			−1.18903			−0.594516	+	0.804084i	$0.702656\pi$
$150$		0			0
$151$	−4.46264			−0.363165			−0.181582	−	0.983376i	$-0.558122\pi$
$151$	−4.46264			−0.363165			−0.181582	+	0.983376i	$0.558122\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.27334	+	4.17997i	−0.423565	+	0.335743i
$156$		0			0
$157$			8.30133i			0.662518i	0.943540	+	0.331259i	$0.107473\pi$
$157$			8.30133i			0.662518i	−0.943540	+	0.331259i	$0.892527\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			11.0093i			0.862317i	0.902276	+	0.431159i	$0.141895\pi$
$163$			11.0093i			0.862317i	−0.902276	+	0.431159i	$0.858105\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			5.76868i			0.446394i	0.974773	+	0.223197i	$0.0716493\pi$
$167$			5.76868i			0.446394i	−0.974773	+	0.223197i	$0.928351\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$			4.90663i			0.373044i	0.982451	+	0.186522i	$0.0597215\pi$
$173$			4.90663i			0.373044i	−0.982451	+	0.186522i	$0.940278\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.45331			0.706574			0.353287	−	0.935515i	$-0.385064\pi$
$179$	9.45331			0.706574			0.353287	+	0.935515i	$0.385064\pi$
$180$		0			0
$181$	17.4720			1.29868			0.649341	−	0.760498i	$-0.275045\pi$
$181$	17.4720			1.29868			0.649341	+	0.760498i	$0.275045\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.95798	−	8.77801i	−0.511561	−	0.645372i
$186$		0			0
$187$		−	4.10270i		−	0.300019i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	11.1120			0.804038			0.402019	−	0.915631i	$-0.368309\pi$
$191$	11.1120			0.804038			0.402019	+	0.915631i	$0.368309\pi$
$192$		0			0
$193$		−	6.10270i		−	0.439282i	−0.975581	−	0.219641i	$-0.929511\pi$
$193$		−	6.10270i		−	0.439282i	0.975581	−	0.219641i	$-0.0704886\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	21.3620i		−	1.52198i	−0.648764	−	0.760990i	$-0.724714\pi$
$197$		−	21.3620i		−	1.52198i	0.648764	−	0.760990i	$-0.275286\pi$
$198$		0			0
$199$	−8.38538			−0.594423			−0.297212	−	0.954812i	$-0.596057\pi$
$199$	−8.38538			−0.594423			−0.297212	+	0.954812i	$0.596057\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−10.1413	+	8.03863i	−0.708302	+	0.561443i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.09337			0.0756303
$210$		0			0
$211$	−1.27334			−0.0876606			−0.0438303	−	0.999039i	$-0.513956\pi$
$211$	−1.27334			−0.0876606			−0.0438303	+	0.999039i	$0.513956\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.78734	−	4.77801i	−0.258294	−	0.325857i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.72666			0.183415
$222$		0			0
$223$			16.4626i			1.10242i	0.834367	+	0.551210i	$0.185834\pi$
$223$			16.4626i			1.10242i	−0.834367	+	0.551210i	$0.814166\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			12.2500i			0.813060i	0.913638	+	0.406530i	$0.133261\pi$
$227$			12.2500i			0.813060i	−0.913638	+	0.406530i	$0.866739\pi$
$228$		0			0
$229$	1.27334			0.0841449			0.0420725	−	0.999115i	$-0.486604\pi$
$229$	1.27334			0.0841449			0.0420725	+	0.999115i	$0.486604\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	18.3013i		−	1.19896i	−0.800390	−	0.599480i	$-0.795374\pi$
$233$		−	18.3013i		−	1.19896i	0.800390	−	0.599480i	$-0.204626\pi$
$234$		0			0
$235$	14.2113	+	17.9287i	0.927046	+	1.16954i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.23132			0.403071			0.201535	−	0.979481i	$-0.435407\pi$
$239$	6.23132			0.403071			0.201535	+	0.979481i	$0.435407\pi$
$240$		0			0
$241$	19.5560			1.25971			0.629857	−	0.776711i	$-0.283114\pi$
$241$	19.5560			1.25971			0.629857	+	0.776711i	$0.283114\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.2663	−	9.72303i	0.783667	−	0.621181i
$246$		0			0
$247$			0.726656i			0.0462360i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−26.4813			−1.67148			−0.835742	−	0.549122i	$-0.814962\pi$
$251$	−26.4813			−1.67148			−0.835742	+	0.549122i	$0.814962\pi$
$252$		0			0
$253$			7.11203i			0.447130i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			14.8294i			0.925030i	0.886611	+	0.462515i	$0.153053\pi$
$257$			14.8294i			0.925030i	−0.886611	+	0.462515i	$0.846947\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	22.9507i		−	1.41520i	−0.706612	−	0.707601i	$-0.749777\pi$
$263$		−	22.9507i		−	1.41520i	0.706612	−	0.707601i	$-0.250223\pi$
$264$		0			0
$265$	−10.4953	−	13.2406i	−0.644723	−	0.813367i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−27.9160			−1.70207			−0.851033	−	0.525112i	$-0.824023\pi$
$269$	−27.9160			−1.70207			−0.851033	+	0.525112i	$0.824023\pi$
$270$		0			0
$271$	−5.65872			−0.343743			−0.171871	−	0.985119i	$-0.554981\pi$
$271$	−5.65872			−0.343743			−0.171871	+	0.985119i	$0.554981\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.71733	+	7.32469i	−0.103559	+	0.441696i
$276$		0			0
$277$		−	7.73599i		−	0.464810i	−0.972619	−	0.232405i	$-0.925340\pi$
$277$		−	7.73599i		−	0.464810i	0.972619	−	0.232405i	$-0.0746595\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.11929			−0.305391			−0.152696	−	0.988273i	$-0.548795\pi$
$281$	−5.11929			−0.305391			−0.152696	+	0.988273i	$0.548795\pi$
$282$		0			0
$283$			6.90663i			0.410556i	0.978704	+	0.205278i	$0.0658099\pi$
$283$			6.90663i			0.410556i	−0.978704	+	0.205278i	$0.934190\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	9.56534			0.562667
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	26.3527i		−	1.53954i	−0.638321	−	0.769770i	$-0.720371\pi$
$293$		−	26.3527i		−	1.53954i	0.638321	−	0.769770i	$-0.279629\pi$
$294$		0			0
$295$	−21.9287	+	17.3820i	−1.27674	+	1.01202i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.72666			−0.273350
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	11.0093	−	8.72666i	0.630392	−	0.499687i
$306$		0			0
$307$		−	20.0000i		−	1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$		−	20.0000i		−	1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.8973			0.674634			0.337317	−	0.941391i	$-0.390481\pi$
$311$	11.8973			0.674634			0.337317	+	0.941391i	$0.390481\pi$
$312$		0			0
$313$			17.7360i			1.00250i	0.865303	+	0.501249i	$0.167126\pi$
$313$			17.7360i			1.00250i	−0.865303	+	0.501249i	$0.832874\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	1.32469i		−	0.0744023i	−0.999308	−	0.0372011i	$-0.988156\pi$
$317$		−	1.32469i		−	0.0744023i	0.999308	−	0.0372011i	$-0.0118442\pi$
$318$		0			0
$319$	11.3693			0.636557
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1.98134i		−	0.110245i
$324$		0			0
$325$	−4.86799	−	1.14134i	−0.270028	−	0.0633099i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−3.73599			−0.205348			−0.102674	−	0.994715i	$-0.532740\pi$
$331$	−3.73599			−0.205348			−0.102674	+	0.994715i	$0.532740\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17.4533	−	22.0187i	−0.953576	−	1.20301i
$336$		0			0
$337$		−	23.3947i		−	1.27439i	−0.770702	−	0.637195i	$-0.780094\pi$
$337$		−	23.3947i		−	1.27439i	0.770702	−	0.637195i	$-0.219906\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.52803			0.245207
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	30.5840i		−	1.64184i	−0.571047	−	0.820918i	$-0.693462\pi$
$347$		−	30.5840i		−	1.64184i	0.571047	−	0.820918i	$-0.306538\pi$
$348$		0			0
$349$	−1.37605			−0.0736581			−0.0368290	−	0.999322i	$-0.511726\pi$
$349$	−1.37605			−0.0736581			−0.0368290	+	0.999322i	$0.511726\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			19.9087i			1.05963i	0.848112	+	0.529817i	$0.177739\pi$
$353$			19.9087i			1.05963i	−0.848112	+	0.529817i	$0.822261\pi$
$354$		0			0
$355$	−8.37266	+	6.63667i	−0.444374	+	0.352238i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	26.6940			1.40885			0.704427	−	0.709777i	$-0.251204\pi$
$359$	26.6940			1.40885			0.704427	+	0.709777i	$0.251204\pi$
$360$		0			0
$361$	−18.4720			−0.972209
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−16.6940	−	21.0607i	−0.873802	−	1.10237i
$366$		0			0
$367$			8.12136i			0.423932i	0.977277	+	0.211966i	$0.0679865\pi$
$367$			8.12136i			0.423932i	−0.977277	+	0.211966i	$0.932013\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	9.47197i		−	0.490440i	−0.969467	−	0.245220i	$-0.921140\pi$
$373$		−	9.47197i		−	0.490440i	0.969467	−	0.245220i	$-0.0788602\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			7.55602i			0.389155i
$378$		0			0
$379$	−16.7267			−0.859191			−0.429595	−	0.903022i	$-0.641344\pi$
$379$	−16.7267			−0.859191			−0.429595	+	0.903022i	$0.641344\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			19.6846i			1.00584i	0.864334	+	0.502919i	$0.167741\pi$
$383$			19.6846i			1.00584i	−0.864334	+	0.502919i	$0.832259\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.0000			−0.709828			−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000			−0.709828			−0.354914	+	0.934899i	$0.615490\pi$
$390$		0			0
$391$	12.8880			0.651773
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−9.24065	+	7.32469i	−0.464948	+	0.368545i
$396$		0			0
$397$		−	3.35061i		−	0.168162i	−0.996459	−	0.0840812i	$-0.973205\pi$
$397$		−	3.35061i		−	0.168162i	0.996459	−	0.0840812i	$-0.0267955\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	36.3713			1.81630			0.908149	−	0.418647i	$-0.137496\pi$
$401$	36.3713			1.81630			0.908149	+	0.418647i	$0.137496\pi$
$402$		0			0
$403$			3.00933i			0.149905i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			7.53736i			0.373613i
$408$		0			0
$409$	−35.5933			−1.75998			−0.879988	−	0.474995i	$-0.842450\pi$
$409$	−35.5933			−1.75998			−0.879988	+	0.474995i	$0.842450\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	10.8166	+	13.6460i	0.530968	+	0.669856i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.9160			−0.679839			−0.339919	−	0.940455i	$-0.610400\pi$
$419$	−13.9160			−0.679839			−0.339919	+	0.940455i	$0.610400\pi$
$420$		0			0
$421$	4.70800			0.229454			0.114727	−	0.993397i	$-0.463401\pi$
$421$	4.70800			0.229454			0.114727	+	0.993397i	$0.463401\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	13.2733	+	3.11203i	0.643852	+	0.150956i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.89004			−0.380050			−0.190025	−	0.981779i	$-0.560857\pi$
$431$	−7.89004			−0.380050			−0.190025	+	0.981779i	$0.560857\pi$
$432$		0			0
$433$			2.30133i			0.110595i	0.998470	+	0.0552974i	$0.0176107\pi$
$433$			2.30133i			0.110595i	−0.998470	+	0.0552974i	$0.982389\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			3.43466i			0.164302i
$438$		0			0
$439$	−41.1307			−1.96306			−0.981530	−	0.191307i	$-0.938728\pi$
$439$	−41.1307			−1.96306			−0.981530	+	0.191307i	$0.938728\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	29.5933i		−	1.40602i	−0.711179	−	0.703011i	$-0.751839\pi$
$443$		−	29.5933i		−	1.40602i	0.711179	−	0.703011i	$-0.248161\pi$
$444$		0			0
$445$	3.13201	−	2.48262i	0.148471	−	0.117687i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−11.4461			−0.540173			−0.270086	−	0.962836i	$-0.587052\pi$
$449$	−11.4461			−0.540173			−0.270086	+	0.962836i	$0.587052\pi$
$450$		0			0
$451$	8.70800			0.410044
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	16.5467i		−	0.774021i	−0.922075	−	0.387011i	$-0.873508\pi$
$457$		−	16.5467i		−	0.774021i	0.922075	−	0.387011i	$-0.126492\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.50466			−0.163228			−0.0816142	−	0.996664i	$-0.526008\pi$
$461$	−3.50466			−0.163228			−0.0816142	+	0.996664i	$0.526008\pi$
$462$		0			0
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			6.44398i			0.298192i	0.988823	+	0.149096i	$0.0476363\pi$
$467$			6.44398i			0.298192i	−0.988823	+	0.149096i	$0.952364\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			4.10270i			0.188642i
$474$		0			0
$475$	−0.829359	+	3.53736i	−0.0380536	+	0.162305i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	22.2313			1.01577			0.507887	−	0.861423i	$-0.330427\pi$
$479$	22.2313			1.01577			0.507887	+	0.861423i	$0.330427\pi$
$480$		0			0
$481$	−5.00933			−0.228406
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.33402	−	10.5140i	−0.378429	−	0.477416i
$486$		0			0
$487$			13.9160i			0.630592i	0.948993	+	0.315296i	$0.102104\pi$
$487$			13.9160i			0.630592i	−0.948993	+	0.315296i	$0.897896\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	28.1400			1.26994			0.634971	−	0.772536i	$-0.281012\pi$
$491$	28.1400			1.26994			0.634971	+	0.772536i	$0.281012\pi$
$492$		0			0
$493$		−	20.6027i		−	0.927897i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−38.8480			−1.73908			−0.869538	−	0.493866i	$-0.835583\pi$
$499$	−38.8480			−1.73908			−0.869538	+	0.493866i	$0.835583\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	8.19863i		−	0.365559i	−0.983154	−	0.182779i	$-0.941491\pi$
$503$		−	8.19863i		−	0.365559i	0.983154	−	0.182779i	$-0.0585094\pi$
$504$		0			0
$505$	5.24065	−	4.15405i	0.233206	−	0.184853i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.0700			0.712291			0.356145	−	0.934431i	$-0.384091\pi$
$509$	16.0700			0.712291			0.356145	+	0.934431i	$0.384091\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.616696	+	0.778008i	0.0271749	+	0.0342831i
$516$		0			0
$517$		−	15.3947i		−	0.677058i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	23.0280			1.00887			0.504437	−	0.863448i	$-0.331700\pi$
$521$	23.0280			1.00887			0.504437	+	0.863448i	$0.331700\pi$
$522$		0			0
$523$			5.47875i			0.239569i	0.992800	+	0.119784i	$0.0382204\pi$
$523$			5.47875i			0.239569i	−0.992800	+	0.119784i	$0.961780\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	8.20541i		−	0.357433i
$528$		0			0
$529$	0.658719			0.0286399
$530$		0			0
$531$		0			0
$532$		0			0
$533$			5.78734i			0.250677i
$534$		0			0
$535$	−9.73599	−	12.2827i	−0.420923	−	0.531026i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−10.5327			−0.453673
$540$		0			0
$541$	17.8387			0.766945			0.383473	−	0.923552i	$-0.374728\pi$
$541$	17.8387			0.766945			0.383473	+	0.923552i	$0.374728\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	24.2500	−	19.2220i	1.03875	−	0.823380i
$546$		0			0
$547$		−	29.1053i		−	1.24445i	−0.782838	−	0.622225i	$-0.786229\pi$
$547$		−	29.1053i		−	1.24445i	0.782838	−	0.622225i	$-0.213771\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.49063			0.233909
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$			44.0114i			1.86482i	0.361399	+	0.932411i	$0.382299\pi$
$557$			44.0114i			1.86482i	−0.361399	+	0.932411i	$0.617701\pi$
$558$		0			0
$559$	−2.72666			−0.115325
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	7.43466i		−	0.313333i	−0.987652	−	0.156667i	$-0.949925\pi$
$563$		−	7.43466i		−	0.313333i	0.987652	−	0.156667i	$-0.0500748\pi$
$564$		0			0
$565$	−5.94865	−	7.50466i	−0.250262	−	0.315724i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.48130			0.355555			0.177777	−	0.984071i	$-0.443109\pi$
$569$	8.48130			0.355555			0.177777	+	0.984071i	$0.443109\pi$
$570$		0			0
$571$	40.7826			1.70670			0.853350	−	0.521339i	$-0.174567\pi$
$571$	40.7826			1.70670			0.853350	+	0.521339i	$0.174567\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−23.0093	−	5.39470i	−0.959555	−	0.224975i
$576$		0			0
$577$		−	1.57467i		−	0.0655545i	−0.999463	−	0.0327773i	$-0.989565\pi$
$577$		−	1.57467i		−	0.0655545i	0.999463	−	0.0327773i	$-0.0104352\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			11.3693i			0.470867i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			27.8247i			1.14845i	0.818699	+	0.574223i	$0.194696\pi$
$587$			27.8247i			1.14845i	−0.818699	+	0.574223i	$0.805304\pi$
$588$		0			0
$589$	2.18675			0.0901034
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	32.6940i		−	1.34258i	−0.741195	−	0.671290i	$-0.765740\pi$
$593$		−	32.6940i		−	1.34258i	0.741195	−	0.671290i	$-0.234260\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−32.1400			−1.31321			−0.656603	−	0.754237i	$-0.728007\pi$
$599$	−32.1400			−1.31321			−0.656603	+	0.754237i	$0.728007\pi$
$600$		0			0
$601$	40.8667			1.66699			0.833493	−	0.552530i	$-0.186337\pi$
$601$	40.8667			1.66699			0.833493	+	0.552530i	$0.186337\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−15.3083	+	12.1343i	−0.622373	+	0.493330i
$606$		0			0
$607$		−	14.9907i		−	0.608453i	−0.952600	−	0.304226i	$-0.901602\pi$
$607$		−	14.9907i		−	0.608453i	0.952600	−	0.304226i	$-0.0983979\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10.2313			0.413915
$612$		0			0
$613$			13.5747i			0.548276i	0.961690	+	0.274138i	$0.0883925\pi$
$613$			13.5747i			0.548276i	−0.961690	+	0.274138i	$0.911608\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	4.69396i		−	0.188972i	−0.995526	−	0.0944859i	$-0.969879\pi$
$617$		−	4.69396i		−	0.188972i	0.995526	−	0.0944859i	$-0.0301207\pi$
$618$		0			0
$619$	20.8667			0.838702			0.419351	−	0.907824i	$-0.362258\pi$
$619$	20.8667			0.838702			0.419351	+	0.907824i	$0.362258\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−22.3947	−	11.1120i	−0.895788	−	0.444481i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13.6587			0.544609
$630$		0			0
$631$	45.9533			1.82937			0.914685	−	0.404167i	$-0.132438\pi$
$631$	45.9533			1.82937			0.914685	+	0.404167i	$0.132438\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.94139	−	10.0187i	−0.315144	−	0.397578i
$636$		0			0
$637$		−	7.00000i		−	0.277350i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20.0187			−0.790689			−0.395345	−	0.918533i	$-0.629375\pi$
$641$	−20.0187			−0.790689			−0.395345	+	0.918533i	$0.629375\pi$
$642$		0			0
$643$		−	26.5840i		−	1.04837i	−0.851604	−	0.524185i	$-0.824370\pi$
$643$		−	26.5840i		−	1.04837i	0.851604	−	0.524185i	$-0.175630\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			9.39470i			0.369344i	0.982800	+	0.184672i	$0.0591223\pi$
$647$			9.39470i			0.369344i	−0.982800	+	0.184672i	$0.940878\pi$
$648$		0			0
$649$	18.8294			0.739117
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	2.00000i		−	0.0782660i	−0.999234	−	0.0391330i	$-0.987540\pi$
$653$		−	2.00000i		−	0.0782660i	0.999234	−	0.0391330i	$-0.0124596\pi$
$654$		0			0
$655$	−9.73599	+	7.71733i	−0.380416	+	0.301541i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	47.6774			1.85725			0.928623	−	0.371024i	$-0.120993\pi$
$659$	47.6774			1.85725			0.928623	+	0.371024i	$0.120993\pi$
$660$		0			0
$661$	−22.0959			−0.859432			−0.429716	−	0.902964i	$-0.641386\pi$
$661$	−22.0959			−0.859432			−0.429716	+	0.902964i	$0.641386\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			35.7147i			1.38288i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.45331			−0.364941
$672$		0			0
$673$		−	48.6027i		−	1.87349i	−0.350006	−	0.936747i	$-0.613820\pi$
$673$		−	48.6027i		−	1.87349i	0.350006	−	0.936747i	$-0.386180\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			10.4253i			0.400678i	0.979727	+	0.200339i	$0.0642043\pi$
$677$			10.4253i			0.400678i	−0.979727	+	0.200339i	$0.935796\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			9.13795i			0.349654i	0.984599	+	0.174827i	$0.0559366\pi$
$683$			9.13795i			0.349654i	−0.984599	+	0.174827i	$0.944063\pi$
$684$		0			0
$685$	9.27203	+	11.6974i	0.354266	+	0.446933i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.55602			−0.287861
$690$		0			0
$691$	33.7173			1.28267			0.641334	−	0.767262i	$-0.278381\pi$
$691$	33.7173			1.28267			0.641334	+	0.767262i	$0.278381\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−34.1214	+	27.0466i	−1.29430	+	1.02594i
$696$		0			0
$697$		−	15.7801i		−	0.597713i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	19.9813			0.754685			0.377342	−	0.926074i	$-0.376838\pi$
$701$	19.9813			0.754685			0.377342	+	0.926074i	$0.376838\pi$
$702$		0			0
$703$			3.64006i			0.137288i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	43.3293			1.62727			0.813633	−	0.581378i	$-0.197486\pi$
$709$	43.3293			1.62727			0.813633	+	0.581378i	$0.197486\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			14.2241i			0.532695i
$714$		0			0
$715$	2.08998	+	2.63667i	0.0781610	+	0.0986059i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−34.6867			−1.29360			−0.646798	−	0.762661i	$-0.723892\pi$
$719$	−34.6867			−1.29360			−0.646798	+	0.762661i	$0.723892\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.62395	+	36.7826i	−0.320286	+	1.36607i
$726$		0			0
$727$			5.39470i			0.200078i	0.994983	+	0.100039i	$0.0318968\pi$
$727$			5.39470i			0.200078i	−0.994983	+	0.100039i	$0.968103\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7.43466			0.274981
$732$		0			0
$733$		−	9.11203i		−	0.336561i	−0.985739	−	0.168280i	$-0.946179\pi$
$733$		−	9.11203i		−	0.336561i	0.985739	−	0.168280i	$-0.0538214\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			18.9066i			0.696435i
$738$		0			0
$739$	4.82936			0.177651			0.0888254	−	0.996047i	$-0.471689\pi$
$739$	4.82936			0.177651			0.0888254	+	0.996047i	$0.471689\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			50.4087i			1.84931i	0.380801	+	0.924657i	$0.375648\pi$
$743$			50.4087i			1.84931i	−0.380801	+	0.924657i	$0.624352\pi$
$744$		0			0
$745$	−25.4333	+	20.1600i	−0.931805	+	0.738605i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4.88797			0.178365			0.0891823	−	0.996015i	$-0.471575\pi$
$751$	4.88797			0.178365			0.0891823	+	0.996015i	$0.471575\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−7.82003	+	6.19863i	−0.284600	+	0.225591i
$756$		0			0
$757$			32.2241i			1.17120i	0.810599	+	0.585602i	$0.199142\pi$
$757$			32.2241i			1.17120i	−0.810599	+	0.585602i	$0.800858\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−36.4740			−1.32218			−0.661091	−	0.750305i	$-0.729906\pi$
$761$	−36.4740			−1.32218			−0.661091	+	0.750305i	$0.729906\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			12.5140i			0.451854i
$768$		0			0
$769$	−13.1120			−0.472832			−0.236416	−	0.971652i	$-0.575973\pi$
$769$	−13.1120			−0.472832			−0.236416	+	0.971652i	$0.575973\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	12.5913i		−	0.452876i	−0.974026	−	0.226438i	$-0.927292\pi$
$773$		−	12.5913i		−	0.452876i	0.974026	−	0.226438i	$-0.0727081\pi$
$774$		0			0
$775$	−3.43466	+	14.6494i	−0.123376	+	0.526222i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.20541			0.150674
$780$		0			0
$781$	7.18930			0.257253
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.5306	+	14.5467i	0.411544	+	0.519194i
$786$		0			0
$787$			36.9253i			1.31624i	0.752911	+	0.658122i	$0.228649\pi$
$787$			36.9253i			1.31624i	−0.752911	+	0.658122i	$0.771351\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	6.28267i		−	0.223104i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			25.0466i			0.887198i	0.896225	+	0.443599i	$0.146298\pi$
$797$			25.0466i			0.887198i	−0.896225	+	0.443599i	$0.853702\pi$
$798$		0			0
$799$	−27.8973			−0.986935
$800$		0			0
$801$		0			0
$802$		0			0
$803$			18.0840i			0.638172i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	11.2334			0.394945			0.197473	−	0.980308i	$-0.436727\pi$
$809$	11.2334			0.394945			0.197473	+	0.980308i	$0.436727\pi$
$810$		0			0
$811$	31.8387			1.11801			0.559004	−	0.829165i	$-0.311184\pi$
$811$	31.8387			1.11801			0.559004	+	0.829165i	$0.311184\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	15.2920	+	19.2920i	0.535655	+	0.675769i
$816$		0			0
$817$			1.98134i			0.0693184i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	32.1073			1.12055			0.560277	−	0.828306i	$-0.310695\pi$
$821$	32.1073			1.12055			0.560277	+	0.828306i	$0.310695\pi$
$822$		0			0
$823$			46.7054i			1.62805i	0.580832	+	0.814023i	$0.302727\pi$
$823$			46.7054i			1.62805i	−0.580832	+	0.814023i	$0.697273\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	26.6940i		−	0.928240i	−0.885772	−	0.464120i	$-0.846371\pi$
$827$		−	26.6940i		−	0.928240i	0.885772	−	0.464120i	$-0.153629\pi$
$828$		0			0
$829$	28.8294			1.00129			0.500643	−	0.865654i	$-0.333097\pi$
$829$	28.8294			1.00129			0.500643	+	0.865654i	$0.333097\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			19.0866i			0.661311i
$834$		0			0
$835$	8.01272	+	10.1086i	0.277292	+	0.349824i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	7.68463			0.265303			0.132652	−	0.991163i	$-0.457651\pi$
$839$	7.68463			0.265303			0.132652	+	0.991163i	$0.457651\pi$
$840$		0			0
$841$	28.0934			0.968737
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.75233	+	1.38900i	−0.0602821	+	0.0477832i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−23.6774			−0.811650
$852$		0			0
$853$		−	18.4626i		−	0.632149i	−0.948734	−	0.316074i	$-0.897635\pi$
$853$		−	18.4626i		−	0.632149i	0.948734	−	0.316074i	$-0.102365\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			32.2827i			1.10276i	0.834256	+	0.551378i	$0.185898\pi$
$857$			32.2827i			1.10276i	−0.834256	+	0.551378i	$0.814102\pi$
$858$		0			0
$859$	18.9694			0.647227			0.323613	−	0.946189i	$-0.395102\pi$
$859$	18.9694			0.647227			0.323613	+	0.946189i	$0.395102\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			16.8807i			0.574626i	0.957837	+	0.287313i	$0.0927620\pi$
$863$			16.8807i			0.574626i	−0.957837	+	0.287313i	$0.907238\pi$
$864$		0			0
$865$	6.81532	+	8.59804i	0.231728	+	0.292342i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.93461			0.269163
$870$		0			0
$871$	−12.5653			−0.425760
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			28.1214i			0.949591i	0.880096	+	0.474795i	$0.157478\pi$
$877$			28.1214i			0.949591i	−0.880096	+	0.474795i	$0.842522\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	49.4066			1.66455			0.832275	−	0.554363i	$-0.187038\pi$
$881$	49.4066			1.66455			0.832275	+	0.554363i	$0.187038\pi$
$882$		0			0
$883$			5.65872i			0.190431i	0.995457	+	0.0952155i	$0.0303540\pi$
$883$			5.65872i			0.190431i	−0.995457	+	0.0952155i	$0.969646\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	28.2640i		−	0.949013i	−0.880252	−	0.474506i	$-0.842627\pi$
$887$		−	28.2640i		−	0.949013i	0.880252	−	0.474506i	$-0.157373\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	7.43466i		−	0.248791i
$894$		0			0
$895$	16.5653	−	13.1307i	0.553718	−	0.438911i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	22.7385			0.758373
$900$		0			0
$901$	20.6027			0.686374
$902$		0			0
$903$		0			0
$904$		0			0
$905$	30.6167	−	24.2686i	1.01773	−	0.806717i
$906$		0			0
$907$		−	14.5840i		−	0.484254i	−0.970245	−	0.242127i	$-0.922155\pi$
$907$		−	14.5840i		−	0.484254i	0.970245	−	0.242127i	$-0.0778450\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−29.1680			−0.966379			−0.483190	−	0.875516i	$-0.660522\pi$
$911$	−29.1680			−0.966379			−0.483190	+	0.875516i	$0.660522\pi$
$912$		0			0
$913$		−	11.7173i		−	0.387787i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−2.36672			−0.0780708			−0.0390354	−	0.999238i	$-0.512429\pi$
$919$	−2.36672			−0.0780708			−0.0390354	+	0.999238i	$0.512429\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			4.77801i			0.157270i
$924$		0			0
$925$	−24.3854	−	5.71733i	−0.801786	−	0.187985i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	31.1753			1.02283			0.511414	−	0.859335i	$-0.329122\pi$
$929$	31.1753			1.02283			0.511414	+	0.859335i	$0.329122\pi$
$930$		0			0
$931$	−5.08660			−0.166706
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.69867	−	7.18930i	−0.186366	−	0.235115i
$936$		0			0
$937$			6.46942i			0.211347i	0.994401	+	0.105673i	$0.0336998\pi$
$937$			6.46942i			0.211347i	−0.994401	+	0.105673i	$0.966300\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.4020			0.632486			0.316243	−	0.948678i	$-0.397579\pi$
$941$	19.4020			0.632486			0.316243	+	0.948678i	$0.397579\pi$
$942$		0			0
$943$			27.3548i			0.890793i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	50.0632i		−	1.62684i	−0.581679	−	0.813418i	$-0.697604\pi$
$947$		−	50.0632i		−	1.62684i	0.581679	−	0.813418i	$-0.302396\pi$
$948$		0			0
$949$	−12.0187			−0.390142
$950$		0			0
$951$		0			0
$952$		0			0
$953$			6.30133i			0.204120i	0.994778	+	0.102060i	$0.0325434\pi$
$953$			6.30133i			0.204120i	−0.994778	+	0.102060i	$0.967457\pi$
$954$		0			0
$955$	19.4720	−	15.4347i	0.630098	−	0.499454i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−21.9439			−0.707869
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.47668	−	10.6940i	−0.272874	−	0.344251i
$966$		0			0
$967$		−	37.2334i		−	1.19735i	−0.800994	−	0.598673i	$-0.795695\pi$
$967$		−	37.2334i		−	1.19735i	0.800994	−	0.598673i	$-0.204305\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.4533			−0.431737			−0.215869	−	0.976422i	$-0.569258\pi$
$971$	−13.4533			−0.431737			−0.215869	+	0.976422i	$0.569258\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			37.4647i			1.19860i	0.800524	+	0.599301i	$0.204555\pi$
$977$			37.4647i			1.19860i	−0.800524	+	0.599301i	$0.795445\pi$
$978$		0			0
$979$	−2.68934			−0.0859517
$980$		0			0
$981$		0			0
$982$		0			0
$983$			29.2406i			0.932632i	0.884618	+	0.466316i	$0.154419\pi$
$983$			29.2406i			0.932632i	−0.884618	+	0.466316i	$0.845581\pi$
$984$		0			0
$985$	−29.6719	−	37.4333i	−0.945426	−	1.19273i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.8880			−0.409814
$990$		0			0
$991$	3.11203			0.0988569			0.0494285	−	0.998778i	$-0.484260\pi$
$991$	3.11203			0.0988569			0.0494285	+	0.998778i	$0.484260\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−14.6940	+	11.6473i	−0.465830	+	0.369245i
$996$		0			0
$997$		−	42.3200i		−	1.34029i	−0.742231	−	0.670144i	$-0.766232\pi$
$997$		−	42.3200i		−	1.34029i	0.742231	−	0.670144i	$-0.233768\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.l.e.2809.5		6
3.2	odd	2		1560.2.l.c.1249.1	✓	6
5.4	even	2	inner	4680.2.l.e.2809.6		6
12.11	even	2		3120.2.l.m.1249.4		6
15.2	even	4		7800.2.a.bj.1.3		3
15.8	even	4		7800.2.a.bp.1.3		3
15.14	odd	2		1560.2.l.c.1249.4	yes	6
60.59	even	2		3120.2.l.m.1249.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.c.1249.1	✓	6	3.2	odd	2
1560.2.l.c.1249.4	yes	6	15.14	odd	2
3120.2.l.m.1249.1		6	60.59	even	2
3120.2.l.m.1249.4		6	12.11	even	2
4680.2.l.e.2809.5		6	1.1	even	1	trivial
4680.2.l.e.2809.6		6	5.4	even	2	inner
7800.2.a.bj.1.3		3	15.2	even	4
7800.2.a.bp.1.3		3	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4680.2.l.e.2809.5

Level	$4680$
Weight	$2$
Character	4680.2809
Analytic conductor	$37.370$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$