LMFDB - Embedded newform 6240.2.w.h.3121.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(3121,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6240.3121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.w (of order $2$, degree $1$, 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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3121.4
Character	$\chi$	$=$	6240.3121
Dual form			6240.2.w.h.3121.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} -2.68192 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} -2.68192 q^{7} -1.00000 q^{9} -3.65883i q^{11} -1.00000i q^{13} -1.00000 q^{15} -2.06298 q^{17} -5.41299i q^{19} -2.68192i q^{21} +0.461619 q^{23} -1.00000 q^{25} -1.00000i q^{27} +3.76763i q^{29} -1.63167 q^{31} +3.65883 q^{33} -2.68192i q^{35} +7.44611i q^{37} +1.00000 q^{39} -4.68586 q^{41} +3.33338i q^{43} -1.00000i q^{45} +3.52987 q^{47} +0.192701 q^{49} -2.06298i q^{51} +5.62294i q^{53} +3.65883 q^{55} +5.41299 q^{57} -3.65036i q^{59} -4.27777i q^{61} +2.68192 q^{63} +1.00000 q^{65} +0.885885i q^{67} +0.461619i q^{69} +4.89129 q^{71} +4.30767 q^{73} -1.00000i q^{75} +9.81270i q^{77} +15.7644 q^{79} +1.00000 q^{81} -3.08546i q^{83} -2.06298i q^{85} -3.76763 q^{87} +1.90866 q^{89} +2.68192i q^{91} -1.63167i q^{93} +5.41299 q^{95} +9.32678 q^{97} +3.65883i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q + 4 q^{7} - 26 q^{9}+O(q^{10}) $ 26 * q + 4 * q^7 - 26 * q^9	$ 26 q + 4 q^{7} - 26 q^{9} - 26 q^{15} + 28 q^{17} + 12 q^{23} - 26 q^{25} + 20 q^{31} + 26 q^{39} - 36 q^{47} + 42 q^{49} - 12 q^{57} - 4 q^{63} + 26 q^{65} + 52 q^{71} - 32 q^{73} - 28 q^{79} + 26 q^{81} + 20 q^{87} - 12 q^{95} + 56 q^{97}+O(q^{100}) $ 26 * q + 4 * q^7 - 26 * q^9 - 26 * q^15 + 28 * q^17 + 12 * q^23 - 26 * q^25 + 20 * q^31 + 26 * q^39 - 36 * q^47 + 42 * q^49 - 12 * q^57 - 4 * q^63 + 26 * q^65 + 52 * q^71 - 32 * q^73 - 28 * q^79 + 26 * q^81 + 20 * q^87 - 12 * q^95 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6240\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$5761$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−2.68192	−1.01367	−0.506835	−	0.862043i	$-0.669185\pi$
$7$	−2.68192	−1.01367	−0.506835	+	0.862043i	$0.669185\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 3.65883i	− 1.10318i	−0.834115	−	0.551590i	$-0.814021\pi$
$11$	− 3.65883i	− 1.10318i	0.834115	−	0.551590i	$-0.185979\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−2.06298	−0.500347	−0.250174	−	0.968201i	$-0.580488\pi$
$17$	−2.06298	−0.500347	−0.250174	+	0.968201i	$0.580488\pi$
$18$	0	0
$19$	− 5.41299i	− 1.24183i	−0.783879	−	0.620913i	$-0.786762\pi$
$19$	− 5.41299i	− 1.24183i	0.783879	−	0.620913i	$-0.213238\pi$
$20$	0	0
$21$	− 2.68192i	− 0.585243i
$22$	0	0
$23$	0.461619	0.0962543	0.0481271	−	0.998841i	$-0.484675\pi$
$23$	0.461619	0.0962543	0.0481271	+	0.998841i	$0.484675\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	3.76763i	0.699631i	0.936819	+	0.349815i	$0.113756\pi$
$29$	3.76763i	0.699631i	−0.936819	+	0.349815i	$0.886244\pi$
$30$	0	0
$31$	−1.63167	−0.293056	−0.146528	−	0.989207i	$-0.546810\pi$
$31$	−1.63167	−0.293056	−0.146528	+	0.989207i	$0.546810\pi$
$32$	0	0
$33$	3.65883	0.636921
$34$	0	0
$35$	− 2.68192i	− 0.453327i
$36$	0	0
$37$	7.44611i	1.22413i	0.790807	+	0.612066i	$0.209661\pi$
$37$	7.44611i	1.22413i	−0.790807	+	0.612066i	$0.790339\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−4.68586	−0.731808	−0.365904	−	0.930653i	$-0.619240\pi$
$41$	−4.68586	−0.731808	−0.365904	+	0.930653i	$0.619240\pi$
$42$	0	0
$43$	3.33338i	0.508335i	0.967160	+	0.254168i	$0.0818015\pi$
$43$	3.33338i	0.508335i	−0.967160	+	0.254168i	$0.918199\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	3.52987	0.514884	0.257442	−	0.966294i	$-0.417120\pi$
$47$	3.52987	0.514884	0.257442	+	0.966294i	$0.417120\pi$
$48$	0	0
$49$	0.192701	0.0275287
$50$	0	0
$51$	− 2.06298i	− 0.288876i
$52$	0	0
$53$	5.62294i	0.772370i	0.922421	+	0.386185i	$0.126207\pi$
$53$	5.62294i	0.772370i	−0.922421	+	0.386185i	$0.873793\pi$
$54$	0	0
$55$	3.65883	0.493357
$56$	0	0
$57$	5.41299	0.716969
$58$	0	0
$59$	− 3.65036i	− 0.475237i	−0.971359	−	0.237618i	$-0.923633\pi$
$59$	− 3.65036i	− 0.475237i	0.971359	−	0.237618i	$-0.0763668\pi$
$60$	0	0
$61$	− 4.27777i	− 0.547712i	−0.961771	−	0.273856i	$-0.911701\pi$
$61$	− 4.27777i	− 0.547712i	0.961771	−	0.273856i	$-0.0882992\pi$
$62$	0	0
$63$	2.68192	0.337890
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	0.885885i	0.108228i	0.998535	+	0.0541140i	$0.0172335\pi$
$67$	0.885885i	0.108228i	−0.998535	+	0.0541140i	$0.982767\pi$
$68$	0	0
$69$	0.461619i	0.0555724i
$70$	0	0
$71$	4.89129	0.580489	0.290245	−	0.956952i	$-0.406263\pi$
$71$	4.89129	0.580489	0.290245	+	0.956952i	$0.406263\pi$
$72$	0	0
$73$	4.30767	0.504175	0.252087	−	0.967704i	$-0.418883\pi$
$73$	4.30767	0.504175	0.252087	+	0.967704i	$0.418883\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	9.81270i	1.11826i
$78$	0	0
$79$	15.7644	1.77364	0.886818	−	0.462119i	$-0.152911\pi$
$79$	15.7644	1.77364	0.886818	+	0.462119i	$0.152911\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 3.08546i	− 0.338673i	−0.985558	−	0.169336i	$-0.945838\pi$
$83$	− 3.08546i	− 0.338673i	0.985558	−	0.169336i	$-0.0541624\pi$
$84$	0	0
$85$	− 2.06298i	− 0.223762i
$86$	0	0
$87$	−3.76763	−0.403932
$88$	0	0
$89$	1.90866	0.202318	0.101159	−	0.994870i	$-0.467745\pi$
$89$	1.90866	0.202318	0.101159	+	0.994870i	$0.467745\pi$
$90$	0	0
$91$	2.68192i	0.281142i
$92$	0	0
$93$	− 1.63167i	− 0.169196i
$94$	0	0
$95$	5.41299	0.555362
$96$	0	0
$97$	9.32678	0.946991	0.473496	−	0.880796i	$-0.342992\pi$
$97$	9.32678	0.946991	0.473496	+	0.880796i	$0.342992\pi$
$98$	0	0
$99$	3.65883i	0.367727i
$100$	0	0
$101$	10.7969i	1.07433i	0.843478	+	0.537164i	$0.180504\pi$
$101$	10.7969i	1.07433i	−0.843478	+	0.537164i	$0.819496\pi$
$102$	0	0
$103$	10.1628	1.00137	0.500687	−	0.865628i	$-0.333081\pi$
$103$	10.1628	1.00137	0.500687	+	0.865628i	$0.333081\pi$
$104$	0	0
$105$	2.68192	0.261729
$106$	0	0
$107$	9.13091i	0.882718i	0.897331	+	0.441359i	$0.145504\pi$
$107$	9.13091i	0.882718i	−0.897331	+	0.441359i	$0.854496\pi$
$108$	0	0
$109$	7.57707i	0.725751i	0.931838	+	0.362876i	$0.118205\pi$
$109$	7.57707i	0.725751i	−0.931838	+	0.362876i	$0.881795\pi$
$110$	0	0
$111$	−7.44611	−0.706753
$112$	0	0
$113$	10.4655	0.984515	0.492258	−	0.870450i	$-0.336172\pi$
$113$	10.4655	0.984515	0.492258	+	0.870450i	$0.336172\pi$
$114$	0	0
$115$	0.461619i	0.0430462i
$116$	0	0
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	5.53276	0.507187
$120$	0	0
$121$	−2.38706	−0.217006
$122$	0	0
$123$	− 4.68586i	− 0.422510i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−15.2502	−1.35323	−0.676617	−	0.736335i	$-0.736555\pi$
$127$	−15.2502	−1.35323	−0.676617	+	0.736335i	$0.736555\pi$
$128$	0	0
$129$	−3.33338	−0.293487
$130$	0	0
$131$	− 1.58278i	− 0.138288i	−0.997607	−	0.0691440i	$-0.977973\pi$
$131$	− 1.58278i	− 0.138288i	0.997607	−	0.0691440i	$-0.0220268\pi$
$132$	0	0
$133$	14.5172i	1.25880i
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	17.0293	1.45492	0.727458	−	0.686152i	$-0.240702\pi$
$137$	17.0293	1.45492	0.727458	+	0.686152i	$0.240702\pi$
$138$	0	0
$139$	− 4.60208i	− 0.390344i	−0.980769	−	0.195172i	$-0.937474\pi$
$139$	− 4.60208i	− 0.390344i	0.980769	−	0.195172i	$-0.0625264\pi$
$140$	0	0
$141$	3.52987i	0.297268i
$142$	0	0
$143$	−3.65883	−0.305967
$144$	0	0
$145$	−3.76763	−0.312884
$146$	0	0
$147$	0.192701i	0.0158937i
$148$	0	0
$149$	15.9180i	1.30406i	0.758195	+	0.652028i	$0.226082\pi$
$149$	15.9180i	1.30406i	−0.758195	+	0.652028i	$0.773918\pi$
$150$	0	0
$151$	−21.2193	−1.72680	−0.863402	−	0.504516i	$-0.831671\pi$
$151$	−21.2193	−1.72680	−0.863402	+	0.504516i	$0.831671\pi$
$152$	0	0
$153$	2.06298	0.166782
$154$	0	0
$155$	− 1.63167i	− 0.131059i
$156$	0	0
$157$	9.74439i	0.777687i	0.921304	+	0.388843i	$0.127125\pi$
$157$	9.74439i	0.777687i	−0.921304	+	0.388843i	$0.872875\pi$
$158$	0	0
$159$	−5.62294	−0.445928
$160$	0	0
$161$	−1.23803	−0.0975701
$162$	0	0
$163$	3.07725i	0.241029i	0.992712	+	0.120514i	$0.0384544\pi$
$163$	3.07725i	0.241029i	−0.992712	+	0.120514i	$0.961546\pi$
$164$	0	0
$165$	3.65883i	0.284840i
$166$	0	0
$167$	−3.08440	−0.238678	−0.119339	−	0.992854i	$-0.538078\pi$
$167$	−3.08440	−0.238678	−0.119339	+	0.992854i	$0.538078\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	5.41299i	0.413942i
$172$	0	0
$173$	15.6744i	1.19170i	0.803096	+	0.595850i	$0.203185\pi$
$173$	15.6744i	1.19170i	−0.803096	+	0.595850i	$0.796815\pi$
$174$	0	0
$175$	2.68192	0.202734
$176$	0	0
$177$	3.65036	0.274378
$178$	0	0
$179$	− 1.98266i	− 0.148191i	−0.997251	−	0.0740954i	$-0.976393\pi$
$179$	− 1.98266i	− 0.148191i	0.997251	−	0.0740954i	$-0.0236069\pi$
$180$	0	0
$181$	12.4177i	0.923004i	0.887139	+	0.461502i	$0.152689\pi$
$181$	12.4177i	0.923004i	−0.887139	+	0.461502i	$0.847311\pi$
$182$	0	0
$183$	4.27777	0.316222
$184$	0	0
$185$	−7.44611	−0.547449
$186$	0	0
$187$	7.54811i	0.551973i
$188$	0	0
$189$	2.68192i	0.195081i
$190$	0	0
$191$	−12.2350	−0.885295	−0.442647	−	0.896696i	$-0.645961\pi$
$191$	−12.2350	−0.885295	−0.442647	+	0.896696i	$0.645961\pi$
$192$	0	0
$193$	15.8183	1.13862	0.569311	−	0.822122i	$-0.307210\pi$
$193$	15.8183	1.13862	0.569311	+	0.822122i	$0.307210\pi$
$194$	0	0
$195$	1.00000i	0.0716115i
$196$	0	0
$197$	− 14.8146i	− 1.05550i	−0.849400	−	0.527749i	$-0.823036\pi$
$197$	− 14.8146i	− 1.05550i	0.849400	−	0.527749i	$-0.176964\pi$
$198$	0	0
$199$	19.3825	1.37399	0.686994	−	0.726663i	$-0.258930\pi$
$199$	19.3825	1.37399	0.686994	+	0.726663i	$0.258930\pi$
$200$	0	0
$201$	−0.885885	−0.0624855
$202$	0	0
$203$	− 10.1045i	− 0.709195i
$204$	0	0
$205$	− 4.68586i	− 0.327275i
$206$	0	0
$207$	−0.461619	−0.0320848
$208$	0	0
$209$	−19.8052	−1.36996
$210$	0	0
$211$	− 2.80637i	− 0.193198i	−0.995323	−	0.0965992i	$-0.969203\pi$
$211$	− 2.80637i	− 0.193198i	0.995323	−	0.0965992i	$-0.0307965\pi$
$212$	0	0
$213$	4.89129i	0.335146i
$214$	0	0
$215$	−3.33338	−0.227334
$216$	0	0
$217$	4.37600	0.297062
$218$	0	0
$219$	4.30767i	0.291085i
$220$	0	0
$221$	2.06298i	0.138771i
$222$	0	0
$223$	−5.03056	−0.336871	−0.168436	−	0.985713i	$-0.553872\pi$
$223$	−5.03056	−0.336871	−0.168436	+	0.985713i	$0.553872\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	15.1822i	1.00768i	0.863797	+	0.503840i	$0.168080\pi$
$227$	15.1822i	1.00768i	−0.863797	+	0.503840i	$0.831920\pi$
$228$	0	0
$229$	18.8791i	1.24757i	0.781597	+	0.623784i	$0.214406\pi$
$229$	18.8791i	1.24757i	−0.781597	+	0.623784i	$0.785594\pi$
$230$	0	0
$231$	−9.81270	−0.645628
$232$	0	0
$233$	−15.7620	−1.03260	−0.516301	−	0.856407i	$-0.672691\pi$
$233$	−15.7620	−1.03260	−0.516301	+	0.856407i	$0.672691\pi$
$234$	0	0
$235$	3.52987i	0.230263i
$236$	0	0
$237$	15.7644i	1.02401i
$238$	0	0
$239$	−0.698989	−0.0452138	−0.0226069	−	0.999744i	$-0.507197\pi$
$239$	−0.698989	−0.0452138	−0.0226069	+	0.999744i	$0.507197\pi$
$240$	0	0
$241$	−11.4199	−0.735621	−0.367810	−	0.929901i	$-0.619892\pi$
$241$	−11.4199	−0.735621	−0.367810	+	0.929901i	$0.619892\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0.192701i	0.0123112i
$246$	0	0
$247$	−5.41299	−0.344421
$248$	0	0
$249$	3.08546	0.195533
$250$	0	0
$251$	9.47493i	0.598053i	0.954245	+	0.299026i	$0.0966618\pi$
$251$	9.47493i	0.598053i	−0.954245	+	0.299026i	$0.903338\pi$
$252$	0	0
$253$	− 1.68899i	− 0.106186i
$254$	0	0
$255$	2.06298	0.129189
$256$	0	0
$257$	25.6643	1.60090	0.800449	−	0.599401i	$-0.204594\pi$
$257$	25.6643	1.60090	0.800449	+	0.599401i	$0.204594\pi$
$258$	0	0
$259$	− 19.9699i	− 1.24087i
$260$	0	0
$261$	− 3.76763i	− 0.233210i
$262$	0	0
$263$	20.9294	1.29056	0.645282	−	0.763944i	$-0.276740\pi$
$263$	20.9294	1.29056	0.645282	+	0.763944i	$0.276740\pi$
$264$	0	0
$265$	−5.62294	−0.345414
$266$	0	0
$267$	1.90866i	0.116808i
$268$	0	0
$269$	− 7.41190i	− 0.451911i	−0.974138	−	0.225956i	$-0.927450\pi$
$269$	− 7.41190i	− 0.451911i	0.974138	−	0.225956i	$-0.0725504\pi$
$270$	0	0
$271$	−22.2288	−1.35030	−0.675152	−	0.737679i	$-0.735922\pi$
$271$	−22.2288	−1.35030	−0.675152	+	0.737679i	$0.735922\pi$
$272$	0	0
$273$	−2.68192	−0.162317
$274$	0	0
$275$	3.65883i	0.220636i
$276$	0	0
$277$	21.9465i	1.31864i	0.751863	+	0.659319i	$0.229155\pi$
$277$	21.9465i	1.31864i	−0.751863	+	0.659319i	$0.770845\pi$
$278$	0	0
$279$	1.63167	0.0976853
$280$	0	0
$281$	22.7782	1.35883	0.679417	−	0.733752i	$-0.262233\pi$
$281$	22.7782	1.35883	0.679417	+	0.733752i	$0.262233\pi$
$282$	0	0
$283$	24.0010i	1.42671i	0.700801	+	0.713356i	$0.252826\pi$
$283$	24.0010i	1.42671i	−0.700801	+	0.713356i	$0.747174\pi$
$284$	0	0
$285$	5.41299i	0.320638i
$286$	0	0
$287$	12.5671	0.741813
$288$	0	0
$289$	−12.7441	−0.749653
$290$	0	0
$291$	9.32678i	0.546746i
$292$	0	0
$293$	− 23.4482i	− 1.36986i	−0.728610	−	0.684928i	$-0.759833\pi$
$293$	− 23.4482i	− 1.36986i	0.728610	−	0.684928i	$-0.240167\pi$
$294$	0	0
$295$	3.65036	0.212532
$296$	0	0
$297$	−3.65883	−0.212307
$298$	0	0
$299$	− 0.461619i	− 0.0266961i
$300$	0	0
$301$	− 8.93985i	− 0.515284i
$302$	0	0
$303$	−10.7969	−0.620264
$304$	0	0
$305$	4.27777	0.244944
$306$	0	0
$307$	− 11.0199i	− 0.628939i	−0.949268	−	0.314470i	$-0.898173\pi$
$307$	− 11.0199i	− 0.628939i	0.949268	−	0.314470i	$-0.101827\pi$
$308$	0	0
$309$	10.1628i	0.578144i
$310$	0	0
$311$	−4.43687	−0.251592	−0.125796	−	0.992056i	$-0.540148\pi$
$311$	−4.43687	−0.251592	−0.125796	+	0.992056i	$0.540148\pi$
$312$	0	0
$313$	−19.0588	−1.07727	−0.538633	−	0.842540i	$-0.681059\pi$
$313$	−19.0588	−1.07727	−0.538633	+	0.842540i	$0.681059\pi$
$314$	0	0
$315$	2.68192i	0.151109i
$316$	0	0
$317$	15.2599i	0.857083i	0.903522	+	0.428542i	$0.140972\pi$
$317$	15.2599i	0.857083i	−0.903522	+	0.428542i	$0.859028\pi$
$318$	0	0
$319$	13.7851	0.771818
$320$	0	0
$321$	−9.13091	−0.509638
$322$	0	0
$323$	11.1669i	0.621344i
$324$	0	0
$325$	1.00000i	0.0554700i
$326$	0	0
$327$	−7.57707	−0.419013
$328$	0	0
$329$	−9.46682	−0.521923
$330$	0	0
$331$	17.8967i	0.983691i	0.870683	+	0.491845i	$0.163677\pi$
$331$	17.8967i	0.983691i	−0.870683	+	0.491845i	$0.836323\pi$
$332$	0	0
$333$	− 7.44611i	− 0.408044i
$334$	0	0
$335$	−0.885885	−0.0484011
$336$	0	0
$337$	−8.47933	−0.461898	−0.230949	−	0.972966i	$-0.574183\pi$
$337$	−8.47933	−0.461898	−0.230949	+	0.972966i	$0.574183\pi$
$338$	0	0
$339$	10.4655i	0.568410i
$340$	0	0
$341$	5.96999i	0.323293i
$342$	0	0
$343$	18.2566	0.985766
$344$	0	0
$345$	−0.461619	−0.0248527
$346$	0	0
$347$	25.8732i	1.38895i	0.719519	+	0.694473i	$0.244362\pi$
$347$	25.8732i	1.38895i	−0.719519	+	0.694473i	$0.755638\pi$
$348$	0	0
$349$	− 24.1506i	− 1.29275i	−0.763019	−	0.646376i	$-0.776284\pi$
$349$	− 24.1506i	− 1.29275i	0.763019	−	0.646376i	$-0.223716\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−28.4447	−1.51396	−0.756978	−	0.653440i	$-0.773325\pi$
$353$	−28.4447	−1.51396	−0.756978	+	0.653440i	$0.773325\pi$
$354$	0	0
$355$	4.89129i	0.259603i
$356$	0	0
$357$	5.53276i	0.292825i
$358$	0	0
$359$	17.6342	0.930699	0.465349	−	0.885127i	$-0.345929\pi$
$359$	17.6342	0.930699	0.465349	+	0.885127i	$0.345929\pi$
$360$	0	0
$361$	−10.3005	−0.542132
$362$	0	0
$363$	− 2.38706i	− 0.125288i
$364$	0	0
$365$	4.30767i	0.225474i
$366$	0	0
$367$	−8.57136	−0.447422	−0.223711	−	0.974656i	$-0.571817\pi$
$367$	−8.57136	−0.447422	−0.223711	+	0.974656i	$0.571817\pi$
$368$	0	0
$369$	4.68586	0.243936
$370$	0	0
$371$	− 15.0803i	− 0.782929i
$372$	0	0
$373$	− 11.2923i	− 0.584695i	−0.956312	−	0.292348i	$-0.905564\pi$
$373$	− 11.2923i	− 0.584695i	0.956312	−	0.292348i	$-0.0944364\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	3.76763	0.194043
$378$	0	0
$379$	0.879340i	0.0451687i	0.999745	+	0.0225843i	$0.00718943\pi$
$379$	0.879340i	0.0451687i	−0.999745	+	0.0225843i	$0.992811\pi$
$380$	0	0
$381$	− 15.2502i	− 0.781291i
$382$	0	0
$383$	−16.3487	−0.835382	−0.417691	−	0.908589i	$-0.637161\pi$
$383$	−16.3487	−0.835382	−0.417691	+	0.908589i	$0.637161\pi$
$384$	0	0
$385$	−9.81270	−0.500102
$386$	0	0
$387$	− 3.33338i	− 0.169445i
$388$	0	0
$389$	− 7.34928i	− 0.372623i	−0.982491	−	0.186312i	$-0.940347\pi$
$389$	− 7.34928i	− 0.372623i	0.982491	−	0.186312i	$-0.0596534\pi$
$390$	0	0
$391$	−0.952313	−0.0481605
$392$	0	0
$393$	1.58278	0.0798406
$394$	0	0
$395$	15.7644i	0.793194i
$396$	0	0
$397$	− 24.2976i	− 1.21946i	−0.792609	−	0.609731i	$-0.791278\pi$
$397$	− 24.2976i	− 1.21946i	0.792609	−	0.609731i	$-0.208722\pi$
$398$	0	0
$399$	−14.5172	−0.726770
$400$	0	0
$401$	23.4720	1.17213	0.586067	−	0.810263i	$-0.300676\pi$
$401$	23.4720	1.17213	0.586067	+	0.810263i	$0.300676\pi$
$402$	0	0
$403$	1.63167i	0.0812791i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	27.2441	1.35044
$408$	0	0
$409$	0.0588074	0.00290784	0.00145392	−	0.999999i	$-0.499537\pi$
$409$	0.0588074	0.00290784	0.00145392	+	0.999999i	$0.499537\pi$
$410$	0	0
$411$	17.0293i	0.839996i
$412$	0	0
$413$	9.78999i	0.481734i
$414$	0	0
$415$	3.08546	0.151459
$416$	0	0
$417$	4.60208	0.225365
$418$	0	0
$419$	30.7636i	1.50290i	0.659791	+	0.751449i	$0.270645\pi$
$419$	30.7636i	1.50290i	−0.659791	+	0.751449i	$0.729355\pi$
$420$	0	0
$421$	− 19.7742i	− 0.963735i	−0.876244	−	0.481868i	$-0.839959\pi$
$421$	− 19.7742i	− 0.963735i	0.876244	−	0.481868i	$-0.160041\pi$
$422$	0	0
$423$	−3.52987	−0.171628
$424$	0	0
$425$	2.06298	0.100069
$426$	0	0
$427$	11.4726i	0.555200i
$428$	0	0
$429$	− 3.65883i	− 0.176650i
$430$	0	0
$431$	−27.4222	−1.32088	−0.660440	−	0.750878i	$-0.729630\pi$
$431$	−27.4222	−1.32088	−0.660440	+	0.750878i	$0.729630\pi$
$432$	0	0
$433$	8.99002	0.432033	0.216016	−	0.976390i	$-0.430694\pi$
$433$	8.99002	0.432033	0.216016	+	0.976390i	$0.430694\pi$
$434$	0	0
$435$	− 3.76763i	− 0.180644i
$436$	0	0
$437$	− 2.49874i	− 0.119531i
$438$	0	0
$439$	−36.0850	−1.72224	−0.861122	−	0.508398i	$-0.830238\pi$
$439$	−36.0850	−1.72224	−0.861122	+	0.508398i	$0.830238\pi$
$440$	0	0
$441$	−0.192701	−0.00917623
$442$	0	0
$443$	− 8.91100i	− 0.423375i	−0.977337	−	0.211687i	$-0.932104\pi$
$443$	− 8.91100i	− 0.423375i	0.977337	−	0.211687i	$-0.0678958\pi$
$444$	0	0
$445$	1.90866i	0.0904793i
$446$	0	0
$447$	−15.9180	−0.752897
$448$	0	0
$449$	33.5911	1.58526	0.792632	−	0.609700i	$-0.208710\pi$
$449$	33.5911	1.58526	0.792632	+	0.609700i	$0.208710\pi$
$450$	0	0
$451$	17.1448i	0.807316i
$452$	0	0
$453$	− 21.2193i	− 0.996971i
$454$	0	0
$455$	−2.68192	−0.125730
$456$	0	0
$457$	−38.7184	−1.81117	−0.905586	−	0.424164i	$-0.860568\pi$
$457$	−38.7184	−1.81117	−0.905586	+	0.424164i	$0.860568\pi$
$458$	0	0
$459$	2.06298i	0.0962918i
$460$	0	0
$461$	4.05415i	0.188821i	0.995533	+	0.0944103i	$0.0300965\pi$
$461$	4.05415i	0.188821i	−0.995533	+	0.0944103i	$0.969903\pi$
$462$	0	0
$463$	14.8658	0.690875	0.345437	−	0.938442i	$-0.387731\pi$
$463$	14.8658	0.690875	0.345437	+	0.938442i	$0.387731\pi$
$464$	0	0
$465$	1.63167	0.0756667
$466$	0	0
$467$	− 33.8931i	− 1.56839i	−0.620516	−	0.784194i	$-0.713077\pi$
$467$	− 33.8931i	− 1.56839i	0.620516	−	0.784194i	$-0.286923\pi$
$468$	0	0
$469$	− 2.37587i	− 0.109708i
$470$	0	0
$471$	−9.74439	−0.448998
$472$	0	0
$473$	12.1963	0.560785
$474$	0	0
$475$	5.41299i	0.248365i
$476$	0	0
$477$	− 5.62294i	− 0.257457i
$478$	0	0
$479$	28.5515	1.30455	0.652276	−	0.757981i	$-0.273814\pi$
$479$	28.5515	1.30455	0.652276	+	0.757981i	$0.273814\pi$
$480$	0	0
$481$	7.44611	0.339513
$482$	0	0
$483$	− 1.23803i	− 0.0563322i
$484$	0	0
$485$	9.32678i	0.423507i
$486$	0	0
$487$	35.2074	1.59540	0.797701	−	0.603054i	$-0.206049\pi$
$487$	35.2074	1.59540	0.797701	+	0.603054i	$0.206049\pi$
$488$	0	0
$489$	−3.07725	−0.139158
$490$	0	0
$491$	34.9061i	1.57529i	0.616130	+	0.787644i	$0.288699\pi$
$491$	34.9061i	1.57529i	−0.616130	+	0.787644i	$0.711301\pi$
$492$	0	0
$493$	− 7.77255i	− 0.350058i
$494$	0	0
$495$	−3.65883	−0.164452
$496$	0	0
$497$	−13.1181	−0.588425
$498$	0	0
$499$	− 8.09657i	− 0.362452i	−0.983441	−	0.181226i	$-0.941993\pi$
$499$	− 8.09657i	− 0.362452i	0.983441	−	0.181226i	$-0.0580066\pi$
$500$	0	0
$501$	− 3.08440i	− 0.137801i
$502$	0	0
$503$	22.2160	0.990561	0.495281	−	0.868733i	$-0.335065\pi$
$503$	22.2160	0.990561	0.495281	+	0.868733i	$0.335065\pi$
$504$	0	0
$505$	−10.7969	−0.480454
$506$	0	0
$507$	− 1.00000i	− 0.0444116i
$508$	0	0
$509$	− 21.2238i	− 0.940729i	−0.882472	−	0.470365i	$-0.844122\pi$
$509$	− 21.2238i	− 0.940729i	0.882472	−	0.470365i	$-0.155878\pi$
$510$	0	0
$511$	−11.5528	−0.511067
$512$	0	0
$513$	−5.41299	−0.238990
$514$	0	0
$515$	10.1628i	0.447828i
$516$	0	0
$517$	− 12.9152i	− 0.568009i
$518$	0	0
$519$	−15.6744	−0.688028
$520$	0	0
$521$	−14.9661	−0.655677	−0.327839	−	0.944734i	$-0.606320\pi$
$521$	−14.9661	−0.655677	−0.327839	+	0.944734i	$0.606320\pi$
$522$	0	0
$523$	− 6.37281i	− 0.278663i	−0.990246	−	0.139332i	$-0.955505\pi$
$523$	− 6.37281i	− 0.278663i	0.990246	−	0.139332i	$-0.0444954\pi$
$524$	0	0
$525$	2.68192i	0.117049i
$526$	0	0
$527$	3.36610	0.146630
$528$	0	0
$529$	−22.7869	−0.990735
$530$	0	0
$531$	3.65036i	0.158412i
$532$	0	0
$533$	4.68586i	0.202967i
$534$	0	0
$535$	−9.13091	−0.394764
$536$	0	0
$537$	1.98266	0.0855580
$538$	0	0
$539$	− 0.705061i	− 0.0303691i
$540$	0	0
$541$	− 5.20872i	− 0.223940i	−0.993712	−	0.111970i	$-0.964284\pi$
$541$	− 5.20872i	− 0.223940i	0.993712	−	0.111970i	$-0.0357161\pi$
$542$	0	0
$543$	−12.4177	−0.532896
$544$	0	0
$545$	−7.57707	−0.324566
$546$	0	0
$547$	1.14923i	0.0491374i	0.999698	+	0.0245687i	$0.00782124\pi$
$547$	1.14923i	0.0491374i	−0.999698	+	0.0245687i	$0.992179\pi$
$548$	0	0
$549$	4.27777i	0.182571i
$550$	0	0
$551$	20.3941	0.868820
$552$	0	0
$553$	−42.2789	−1.79788
$554$	0	0
$555$	− 7.44611i	− 0.316070i
$556$	0	0
$557$	− 17.5537i	− 0.743775i	−0.928278	−	0.371887i	$-0.878711\pi$
$557$	− 17.5537i	− 0.743775i	0.928278	−	0.371887i	$-0.121289\pi$
$558$	0	0
$559$	3.33338	0.140987
$560$	0	0
$561$	−7.54811	−0.318682
$562$	0	0
$563$	2.46277i	0.103794i	0.998652	+	0.0518968i	$0.0165267\pi$
$563$	2.46277i	0.103794i	−0.998652	+	0.0518968i	$0.983473\pi$
$564$	0	0
$565$	10.4655i	0.440289i
$566$	0	0
$567$	−2.68192	−0.112630
$568$	0	0
$569$	28.3229	1.18736	0.593678	−	0.804702i	$-0.297675\pi$
$569$	28.3229	1.18736	0.593678	+	0.804702i	$0.297675\pi$
$570$	0	0
$571$	− 19.6004i	− 0.820250i	−0.912029	−	0.410125i	$-0.865485\pi$
$571$	− 19.6004i	− 0.820250i	0.912029	−	0.410125i	$-0.134515\pi$
$572$	0	0
$573$	− 12.2350i	− 0.511125i
$574$	0	0
$575$	−0.461619	−0.0192509
$576$	0	0
$577$	19.5513	0.813932	0.406966	−	0.913443i	$-0.366587\pi$
$577$	19.5513	0.813932	0.406966	+	0.913443i	$0.366587\pi$
$578$	0	0
$579$	15.8183i	0.657384i
$580$	0	0
$581$	8.27495i	0.343303i
$582$	0	0
$583$	20.5734	0.852062
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	28.7078i	1.18490i	0.805609	+	0.592448i	$0.201838\pi$
$587$	28.7078i	1.18490i	−0.805609	+	0.592448i	$0.798162\pi$
$588$	0	0
$589$	8.83220i	0.363924i
$590$	0	0
$591$	14.8146	0.609392
$592$	0	0
$593$	44.9129	1.84435	0.922175	−	0.386772i	$-0.126410\pi$
$593$	44.9129	1.84435	0.922175	+	0.386772i	$0.126410\pi$
$594$	0	0
$595$	5.53276i	0.226821i
$596$	0	0
$597$	19.3825i	0.793272i
$598$	0	0
$599$	12.4455	0.508511	0.254255	−	0.967137i	$-0.418170\pi$
$599$	12.4455	0.508511	0.254255	+	0.967137i	$0.418170\pi$
$600$	0	0
$601$	−35.2686	−1.43864	−0.719319	−	0.694680i	$-0.755546\pi$
$601$	−35.2686	−1.43864	−0.719319	+	0.694680i	$0.755546\pi$
$602$	0	0
$603$	− 0.885885i	− 0.0360760i
$604$	0	0
$605$	− 2.38706i	− 0.0970479i
$606$	0	0
$607$	32.5475	1.32106	0.660531	−	0.750799i	$-0.270331\pi$
$607$	32.5475	1.32106	0.660531	+	0.750799i	$0.270331\pi$
$608$	0	0
$609$	10.1045	0.409454
$610$	0	0
$611$	− 3.52987i	− 0.142803i
$612$	0	0
$613$	39.2979i	1.58723i	0.608421	+	0.793614i	$0.291803\pi$
$613$	39.2979i	1.58723i	−0.608421	+	0.793614i	$0.708197\pi$
$614$	0	0
$615$	4.68586	0.188952
$616$	0	0
$617$	38.1569	1.53614	0.768070	−	0.640366i	$-0.221217\pi$
$617$	38.1569	1.53614	0.768070	+	0.640366i	$0.221217\pi$
$618$	0	0
$619$	− 38.6267i	− 1.55254i	−0.630402	−	0.776269i	$-0.717110\pi$
$619$	− 38.6267i	− 1.55254i	0.630402	−	0.776269i	$-0.282890\pi$
$620$	0	0
$621$	− 0.461619i	− 0.0185241i
$622$	0	0
$623$	−5.11888	−0.205084
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 19.8052i	− 0.790945i
$628$	0	0
$629$	− 15.3612i	− 0.612491i
$630$	0	0
$631$	−1.36308	−0.0542633	−0.0271317	−	0.999632i	$-0.508637\pi$
$631$	−1.36308	−0.0542633	−0.0271317	+	0.999632i	$0.508637\pi$
$632$	0	0
$633$	2.80637	0.111543
$634$	0	0
$635$	− 15.2502i	− 0.605185i
$636$	0	0
$637$	− 0.192701i	− 0.00763509i
$638$	0	0
$639$	−4.89129	−0.193496
$640$	0	0
$641$	36.3057	1.43399	0.716995	−	0.697078i	$-0.245517\pi$
$641$	36.3057	1.43399	0.716995	+	0.697078i	$0.245517\pi$
$642$	0	0
$643$	10.6297i	0.419195i	0.977788	+	0.209598i	$0.0672154\pi$
$643$	10.6297i	0.419195i	−0.977788	+	0.209598i	$0.932785\pi$
$644$	0	0
$645$	− 3.33338i	− 0.131252i
$646$	0	0
$647$	25.7943	1.01408	0.507040	−	0.861923i	$-0.330740\pi$
$647$	25.7943	1.01408	0.507040	+	0.861923i	$0.330740\pi$
$648$	0	0
$649$	−13.3561	−0.524272
$650$	0	0
$651$	4.37600i	0.171509i
$652$	0	0
$653$	9.66428i	0.378192i	0.981959	+	0.189096i	$0.0605558\pi$
$653$	9.66428i	0.378192i	−0.981959	+	0.189096i	$0.939444\pi$
$654$	0	0
$655$	1.58278	0.0618443
$656$	0	0
$657$	−4.30767	−0.168058
$658$	0	0
$659$	19.8849i	0.774604i	0.921953	+	0.387302i	$0.126593\pi$
$659$	19.8849i	0.774604i	−0.921953	+	0.387302i	$0.873407\pi$
$660$	0	0
$661$	43.8601i	1.70596i	0.521944	+	0.852980i	$0.325207\pi$
$661$	43.8601i	1.70596i	−0.521944	+	0.852980i	$0.674793\pi$
$662$	0	0
$663$	−2.06298	−0.0801197
$664$	0	0
$665$	−14.5172	−0.562954
$666$	0	0
$667$	1.73921i	0.0673424i
$668$	0	0
$669$	− 5.03056i	− 0.194493i
$670$	0	0
$671$	−15.6517	−0.604225
$672$	0	0
$673$	−15.5596	−0.599778	−0.299889	−	0.953974i	$-0.596950\pi$
$673$	−15.5596	−0.599778	−0.299889	+	0.953974i	$0.596950\pi$
$674$	0	0
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	11.0104i	0.423165i	0.977360	+	0.211583i	$0.0678617\pi$
$677$	11.0104i	0.423165i	−0.977360	+	0.211583i	$0.932138\pi$
$678$	0	0
$679$	−25.0137	−0.959938
$680$	0	0
$681$	−15.1822	−0.581784
$682$	0	0
$683$	− 43.8155i	− 1.67656i	−0.545244	−	0.838278i	$-0.683563\pi$
$683$	− 43.8155i	− 1.67656i	0.545244	−	0.838278i	$-0.316437\pi$
$684$	0	0
$685$	17.0293i	0.650658i
$686$	0	0
$687$	−18.8791	−0.720284
$688$	0	0
$689$	5.62294	0.214217
$690$	0	0
$691$	9.19588i	0.349828i	0.984584	+	0.174914i	$0.0559647\pi$
$691$	9.19588i	0.349828i	−0.984584	+	0.174914i	$0.944035\pi$
$692$	0	0
$693$	− 9.81270i	− 0.372754i
$694$	0	0
$695$	4.60208	0.174567
$696$	0	0
$697$	9.66685	0.366158
$698$	0	0
$699$	− 15.7620i	− 0.596173i
$700$	0	0
$701$	− 42.3439i	− 1.59931i	−0.600461	−	0.799654i	$-0.705016\pi$
$701$	− 42.3439i	− 1.59931i	0.600461	−	0.799654i	$-0.294984\pi$
$702$	0	0
$703$	40.3057	1.52016
$704$	0	0
$705$	−3.52987	−0.132942
$706$	0	0
$707$	− 28.9563i	− 1.08901i
$708$	0	0
$709$	35.4195i	1.33021i	0.746752	+	0.665103i	$0.231612\pi$
$709$	35.4195i	1.33021i	−0.746752	+	0.665103i	$0.768388\pi$
$710$	0	0
$711$	−15.7644	−0.591212
$712$	0	0
$713$	−0.753208	−0.0282079
$714$	0	0
$715$	− 3.65883i	− 0.136833i
$716$	0	0
$717$	− 0.698989i	− 0.0261042i
$718$	0	0
$719$	26.0594	0.971852	0.485926	−	0.874000i	$-0.338483\pi$
$719$	26.0594	0.971852	0.485926	+	0.874000i	$0.338483\pi$
$720$	0	0
$721$	−27.2559	−1.01506
$722$	0	0
$723$	− 11.4199i	− 0.424711i
$724$	0	0
$725$	− 3.76763i	− 0.139926i
$726$	0	0
$727$	21.4753	0.796474	0.398237	−	0.917283i	$-0.369622\pi$
$727$	21.4753	0.796474	0.398237	+	0.917283i	$0.369622\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 6.87670i	− 0.254344i
$732$	0	0
$733$	32.1006i	1.18566i	0.805327	+	0.592831i	$0.201990\pi$
$733$	32.1006i	1.18566i	−0.805327	+	0.592831i	$0.798010\pi$
$734$	0	0
$735$	−0.192701	−0.00710788
$736$	0	0
$737$	3.24131	0.119395
$738$	0	0
$739$	− 42.2219i	− 1.55316i	−0.630021	−	0.776578i	$-0.716954\pi$
$739$	− 42.2219i	− 1.55316i	0.630021	−	0.776578i	$-0.283046\pi$
$740$	0	0
$741$	− 5.41299i	− 0.198851i
$742$	0	0
$743$	−16.3941	−0.601442	−0.300721	−	0.953712i	$-0.597227\pi$
$743$	−16.3941	−0.601442	−0.300721	+	0.953712i	$0.597227\pi$
$744$	0	0
$745$	−15.9180	−0.583192
$746$	0	0
$747$	3.08546i	0.112891i
$748$	0	0
$749$	− 24.4884i	− 0.894786i
$750$	0	0
$751$	−34.5359	−1.26023	−0.630117	−	0.776500i	$-0.716993\pi$
$751$	−34.5359	−1.26023	−0.630117	+	0.776500i	$0.716993\pi$
$752$	0	0
$753$	−9.47493	−0.345286
$754$	0	0
$755$	− 21.2193i	− 0.772250i
$756$	0	0
$757$	22.5707i	0.820345i	0.912008	+	0.410172i	$0.134531\pi$
$757$	22.5707i	0.820345i	−0.912008	+	0.410172i	$0.865469\pi$
$758$	0	0
$759$	1.68899	0.0613064
$760$	0	0
$761$	42.1665	1.52854	0.764268	−	0.644899i	$-0.223101\pi$
$761$	42.1665	1.52854	0.764268	+	0.644899i	$0.223101\pi$
$762$	0	0
$763$	− 20.3211i	− 0.735673i
$764$	0	0
$765$	2.06298i	0.0745873i
$766$	0	0
$767$	−3.65036	−0.131807
$768$	0	0
$769$	19.3667	0.698381	0.349190	−	0.937052i	$-0.386457\pi$
$769$	19.3667	0.698381	0.349190	+	0.937052i	$0.386457\pi$
$770$	0	0
$771$	25.6643i	0.924279i
$772$	0	0
$773$	7.55570i	0.271760i	0.990725	+	0.135880i	$0.0433861\pi$
$773$	7.55570i	0.271760i	−0.990725	+	0.135880i	$0.956614\pi$
$774$	0	0
$775$	1.63167	0.0586112
$776$	0	0
$777$	19.9699	0.716415
$778$	0	0
$779$	25.3645i	0.908778i
$780$	0	0
$781$	− 17.8964i	− 0.640384i
$782$	0	0
$783$	3.76763	0.134644
$784$	0	0
$785$	−9.74439	−0.347792
$786$	0	0
$787$	49.1811i	1.75312i	0.481296	+	0.876558i	$0.340166\pi$
$787$	49.1811i	1.75312i	−0.481296	+	0.876558i	$0.659834\pi$
$788$	0	0
$789$	20.9294i	0.745108i
$790$	0	0
$791$	−28.0678	−0.997974
$792$	0	0
$793$	−4.27777	−0.151908
$794$	0	0
$795$	− 5.62294i	− 0.199425i
$796$	0	0
$797$	− 7.14045i	− 0.252928i	−0.991971	−	0.126464i	$-0.959637\pi$
$797$	− 7.14045i	− 0.252928i	0.991971	−	0.126464i	$-0.0403628\pi$
$798$	0	0
$799$	−7.28206	−0.257621
$800$	0	0
$801$	−1.90866	−0.0674393
$802$	0	0
$803$	− 15.7610i	− 0.556195i
$804$	0	0
$805$	− 1.23803i	− 0.0436347i
$806$	0	0
$807$	7.41190	0.260911
$808$	0	0
$809$	25.5132	0.896995	0.448498	−	0.893784i	$-0.351959\pi$
$809$	25.5132	0.896995	0.448498	+	0.893784i	$0.351959\pi$
$810$	0	0
$811$	39.8927i	1.40082i	0.713739	+	0.700411i	$0.247000\pi$
$811$	39.8927i	1.40082i	−0.713739	+	0.700411i	$0.753000\pi$
$812$	0	0
$813$	− 22.2288i	− 0.779598i
$814$	0	0
$815$	−3.07725	−0.107791
$816$	0	0
$817$	18.0435	0.631264
$818$	0	0
$819$	− 2.68192i	− 0.0937139i
$820$	0	0
$821$	42.3623i	1.47845i	0.673456	+	0.739227i	$0.264809\pi$
$821$	42.3623i	1.47845i	−0.673456	+	0.739227i	$0.735191\pi$
$822$	0	0
$823$	−45.2288	−1.57658	−0.788289	−	0.615305i	$-0.789033\pi$
$823$	−45.2288	−1.57658	−0.788289	+	0.615305i	$0.789033\pi$
$824$	0	0
$825$	−3.65883	−0.127384
$826$	0	0
$827$	35.6657i	1.24022i	0.784515	+	0.620109i	$0.212912\pi$
$827$	35.6657i	1.24022i	−0.784515	+	0.620109i	$0.787088\pi$
$828$	0	0
$829$	− 28.6512i	− 0.995097i	−0.867436	−	0.497548i	$-0.834234\pi$
$829$	− 28.6512i	− 0.995097i	0.867436	−	0.497548i	$-0.165766\pi$
$830$	0	0
$831$	−21.9465	−0.761316
$832$	0	0
$833$	−0.397539	−0.0137739
$834$	0	0
$835$	− 3.08440i	− 0.106740i
$836$	0	0
$837$	1.63167i	0.0563986i
$838$	0	0
$839$	25.3537	0.875307	0.437654	−	0.899144i	$-0.355810\pi$
$839$	25.3537	0.875307	0.437654	+	0.899144i	$0.355810\pi$
$840$	0	0
$841$	14.8050	0.510517
$842$	0	0
$843$	22.7782i	0.784524i
$844$	0	0
$845$	− 1.00000i	− 0.0344010i
$846$	0	0
$847$	6.40191	0.219972
$848$	0	0
$849$	−24.0010	−0.823713
$850$	0	0
$851$	3.43727i	0.117828i
$852$	0	0
$853$	25.6225i	0.877299i	0.898658	+	0.438650i	$0.144543\pi$
$853$	25.6225i	0.877299i	−0.898658	+	0.438650i	$0.855457\pi$
$854$	0	0
$855$	−5.41299	−0.185121
$856$	0	0
$857$	20.7763	0.709706	0.354853	−	0.934922i	$-0.384531\pi$
$857$	20.7763	0.709706	0.354853	+	0.934922i	$0.384531\pi$
$858$	0	0
$859$	0.646354i	0.0220533i	0.999939	+	0.0110267i	$0.00350997\pi$
$859$	0.646354i	0.0220533i	−0.999939	+	0.0110267i	$0.996490\pi$
$860$	0	0
$861$	12.5671i	0.428286i
$862$	0	0
$863$	1.29317	0.0440201	0.0220101	−	0.999758i	$-0.492993\pi$
$863$	1.29317	0.0440201	0.0220101	+	0.999758i	$0.492993\pi$
$864$	0	0
$865$	−15.6744	−0.532944
$866$	0	0
$867$	− 12.7441i	− 0.432812i
$868$	0	0
$869$	− 57.6794i	− 1.95664i
$870$	0	0
$871$	0.885885	0.0300171
$872$	0	0
$873$	−9.32678	−0.315664
$874$	0	0
$875$	2.68192i	0.0906655i
$876$	0	0
$877$	− 51.3563i	− 1.73418i	−0.498151	−	0.867090i	$-0.665987\pi$
$877$	− 51.3563i	− 1.73418i	0.498151	−	0.867090i	$-0.334013\pi$
$878$	0	0
$879$	23.4482	0.790887
$880$	0	0
$881$	8.99343	0.302996	0.151498	−	0.988458i	$-0.451590\pi$
$881$	8.99343	0.302996	0.151498	+	0.988458i	$0.451590\pi$
$882$	0	0
$883$	59.2011i	1.99227i	0.0878087	+	0.996137i	$0.472014\pi$
$883$	59.2011i	1.99227i	−0.0878087	+	0.996137i	$0.527986\pi$
$884$	0	0
$885$	3.65036i	0.122706i
$886$	0	0
$887$	−17.6505	−0.592646	−0.296323	−	0.955088i	$-0.595760\pi$
$887$	−17.6505	−0.592646	−0.296323	+	0.955088i	$0.595760\pi$
$888$	0	0
$889$	40.8998	1.37173
$890$	0	0
$891$	− 3.65883i	− 0.122576i
$892$	0	0
$893$	− 19.1071i	− 0.639396i
$894$	0	0
$895$	1.98266	0.0662729
$896$	0	0
$897$	0.461619	0.0154130
$898$	0	0
$899$	− 6.14751i	− 0.205031i
$900$	0	0
$901$	− 11.6000i	− 0.386453i
$902$	0	0
$903$	8.93985	0.297500
$904$	0	0
$905$	−12.4177	−0.412780
$906$	0	0
$907$	49.1464i	1.63188i	0.578136	+	0.815940i	$0.303780\pi$
$907$	49.1464i	1.63188i	−0.578136	+	0.815940i	$0.696220\pi$
$908$	0	0
$909$	− 10.7969i	− 0.358109i
$910$	0	0
$911$	4.55660	0.150967	0.0754835	−	0.997147i	$-0.475950\pi$
$911$	4.55660	0.150967	0.0754835	+	0.997147i	$0.475950\pi$
$912$	0	0
$913$	−11.2892	−0.373617
$914$	0	0
$915$	4.27777i	0.141419i
$916$	0	0
$917$	4.24489i	0.140178i
$918$	0	0
$919$	−26.7052	−0.880925	−0.440462	−	0.897771i	$-0.645185\pi$
$919$	−26.7052	−0.880925	−0.440462	+	0.897771i	$0.645185\pi$
$920$	0	0
$921$	11.0199	0.363118
$922$	0	0
$923$	− 4.89129i	− 0.160999i
$924$	0	0
$925$	− 7.44611i	− 0.244826i
$926$	0	0
$927$	−10.1628	−0.333792
$928$	0	0
$929$	−32.3021	−1.05980	−0.529899	−	0.848061i	$-0.677770\pi$
$929$	−32.3021	−1.05980	−0.529899	+	0.848061i	$0.677770\pi$
$930$	0	0
$931$	− 1.04309i	− 0.0341859i
$932$	0	0
$933$	− 4.43687i	− 0.145257i
$934$	0	0
$935$	−7.54811	−0.246850
$936$	0	0
$937$	−22.9706	−0.750415	−0.375208	−	0.926941i	$-0.622429\pi$
$937$	−22.9706	−0.750415	−0.375208	+	0.926941i	$0.622429\pi$
$938$	0	0
$939$	− 19.0588i	− 0.621960i
$940$	0	0
$941$	15.7011i	0.511841i	0.966698	+	0.255920i	$0.0823785\pi$
$941$	15.7011i	0.511841i	−0.966698	+	0.255920i	$0.917622\pi$
$942$	0	0
$943$	−2.16308	−0.0704396
$944$	0	0
$945$	−2.68192	−0.0872429
$946$	0	0
$947$	− 24.9752i	− 0.811585i	−0.913965	−	0.405792i	$-0.866996\pi$
$947$	− 24.9752i	− 0.811585i	0.913965	−	0.405792i	$-0.133004\pi$
$948$	0	0
$949$	− 4.30767i	− 0.139833i
$950$	0	0
$951$	−15.2599	−0.494837
$952$	0	0
$953$	−8.31371	−0.269308	−0.134654	−	0.990893i	$-0.542992\pi$
$953$	−8.31371	−0.269308	−0.134654	+	0.990893i	$0.542992\pi$
$954$	0	0
$955$	− 12.2350i	− 0.395916i
$956$	0	0
$957$	13.7851i	0.445610i
$958$	0	0
$959$	−45.6714	−1.47481
$960$	0	0
$961$	−28.3377	−0.914118
$962$	0	0
$963$	− 9.13091i	− 0.294239i
$964$	0	0
$965$	15.8183i	0.509208i
$966$	0	0
$967$	10.6898	0.343761	0.171881	−	0.985118i	$-0.445016\pi$
$967$	10.6898	0.343761	0.171881	+	0.985118i	$0.445016\pi$
$968$	0	0
$969$	−11.1669	−0.358733
$970$	0	0
$971$	− 41.5427i	− 1.33317i	−0.745430	−	0.666584i	$-0.767756\pi$
$971$	− 41.5427i	− 1.33317i	0.745430	−	0.666584i	$-0.232244\pi$
$972$	0	0
$973$	12.3424i	0.395680i
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	24.5483	0.785371	0.392685	−	0.919673i	$-0.371546\pi$
$977$	24.5483	0.785371	0.392685	+	0.919673i	$0.371546\pi$
$978$	0	0
$979$	− 6.98348i	− 0.223193i
$980$	0	0
$981$	− 7.57707i	− 0.241917i
$982$	0	0
$983$	17.6254	0.562162	0.281081	−	0.959684i	$-0.409307\pi$
$983$	17.6254	0.562162	0.281081	+	0.959684i	$0.409307\pi$
$984$	0	0
$985$	14.8146	0.472033
$986$	0	0
$987$	− 9.46682i	− 0.301332i
$988$	0	0
$989$	1.53875i	0.0489294i
$990$	0	0
$991$	16.6848	0.530009	0.265005	−	0.964247i	$-0.414627\pi$
$991$	16.6848	0.530009	0.265005	+	0.964247i	$0.414627\pi$
$992$	0	0
$993$	−17.8967	−0.567934
$994$	0	0
$995$	19.3825i	0.614466i
$996$	0	0
$997$	− 50.1226i	− 1.58740i	−0.608311	−	0.793699i	$-0.708152\pi$
$997$	− 50.1226i	− 1.58740i	0.608311	−	0.793699i	$-0.291848\pi$
$998$	0	0
$999$	7.44611	0.235584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.w.h.3121.4		26
4.3	odd	2		1560.2.w.h.781.1	✓	26
8.3	odd	2		1560.2.w.h.781.2	yes	26
8.5	even	2	inner	6240.2.w.h.3121.3		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.w.h.781.1	✓	26	4.3	odd	2
1560.2.w.h.781.2	yes	26	8.3	odd	2
6240.2.w.h.3121.3		26	8.5	even	2	inner
6240.2.w.h.3121.4		26	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.w (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} -2.68192 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} -2.68192 q^{7} -1.00000 q^{9} -3.65883i q^{11} -1.00000i q^{13} -1.00000 q^{15} -2.06298 q^{17} -5.41299i q^{19} -2.68192i q^{21} +0.461619 q^{23} -1.00000 q^{25} -1.00000i q^{27} +3.76763i q^{29} -1.63167 q^{31} +3.65883 q^{33} -2.68192i q^{35} +7.44611i q^{37} +1.00000 q^{39} -4.68586 q^{41} +3.33338i q^{43} -1.00000i q^{45} +3.52987 q^{47} +0.192701 q^{49} -2.06298i q^{51} +5.62294i q^{53} +3.65883 q^{55} +5.41299 q^{57} -3.65036i q^{59} -4.27777i q^{61} +2.68192 q^{63} +1.00000 q^{65} +0.885885i q^{67} +0.461619i q^{69} +4.89129 q^{71} +4.30767 q^{73} -1.00000i q^{75} +9.81270i q^{77} +15.7644 q^{79} +1.00000 q^{81} -3.08546i q^{83} -2.06298i q^{85} -3.76763 q^{87} +1.90866 q^{89} +2.68192i q^{91} -1.63167i q^{93} +5.41299 q^{95} +9.32678 q^{97} +3.65883i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q + 4 q^{7} - 26 q^{9}+O(q^{10}) \) 26 * q + 4 * q^7 - 26 * q^9	\( 26 q + 4 q^{7} - 26 q^{9} - 26 q^{15} + 28 q^{17} + 12 q^{23} - 26 q^{25} + 20 q^{31} + 26 q^{39} - 36 q^{47} + 42 q^{49} - 12 q^{57} - 4 q^{63} + 26 q^{65} + 52 q^{71} - 32 q^{73} - 28 q^{79} + 26 q^{81} + 20 q^{87} - 12 q^{95} + 56 q^{97}+O(q^{100}) \) 26 * q + 4 * q^7 - 26 * q^9 - 26 * q^15 + 28 * q^17 + 12 * q^23 - 26 * q^25 + 20 * q^31 + 26 * q^39 - 36 * q^47 + 42 * q^49 - 12 * q^57 - 4 * q^63 + 26 * q^65 + 52 * q^71 - 32 * q^73 - 28 * q^79 + 26 * q^81 + 20 * q^87 - 12 * q^95 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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