LMFDB - Embedded newform 7400.2.a.be.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7400,2,Mod(1,7400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7400 = 2^{3} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7400.a (trivial)

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:	yes
Analytic conductor:	$59.0892974957$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 33 x^{14} + 66 x^{13} + 404 x^{12} - 796 x^{11} - 2273 x^{10} + 4284 x^{9} + \cdots + 16 $ x^16 - 2x^15 - 33x^14 + 66x^13 + 404x^12 - 796x^11 - 2273x^10 + 4284x^9 + 6039x^8 - 10158x^7 - 7660x^6 + 10040x^5 + 3914x^4 - 3314x^3 - 232x^2 + 172*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$2.53062$ of defining polynomial
Character	$\chi$	$=$	7400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53062 q^{3} -0.665840 q^{7} +3.40405 q^{9} +0.212137 q^{11} +4.76390 q^{13} +4.45920 q^{17} +1.83455 q^{19} -1.68499 q^{21} +3.65776 q^{23} +1.02251 q^{27} +3.11206 q^{29} -2.62327 q^{31} +0.536840 q^{33} +1.00000 q^{37} +12.0556 q^{39} -4.61554 q^{41} +12.4107 q^{43} -3.87039 q^{47} -6.55666 q^{49} +11.2846 q^{51} -9.63817 q^{53} +4.64256 q^{57} -6.55947 q^{59} +10.5841 q^{61} -2.26655 q^{63} +2.61116 q^{67} +9.25641 q^{69} -5.49658 q^{71} -13.2091 q^{73} -0.141249 q^{77} +8.46684 q^{79} -7.62458 q^{81} +5.30556 q^{83} +7.87544 q^{87} +1.40902 q^{89} -3.17199 q^{91} -6.63851 q^{93} +1.96770 q^{97} +0.722127 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{3} + 9 q^{7} + 22 q^{9} + 3 q^{11} - 4 q^{13} - 9 q^{17} + 14 q^{19} + 8 q^{21} + 18 q^{23} - 4 q^{27} + q^{29} + 23 q^{31} - 10 q^{33} + 16 q^{37} + 23 q^{41} - 11 q^{43} + 32 q^{47} + 45 q^{49}+ \cdots - 15 q^{99}+O(q^{100}) $ 16 * q + 2 * q^3 + 9 * q^7 + 22 * q^9 + 3 * q^11 - 4 * q^13 - 9 * q^17 + 14 * q^19 + 8 * q^21 + 18 * q^23 - 4 * q^27 + q^29 + 23 * q^31 - 10 * q^33 + 16 * q^37 + 23 * q^41 - 11 * q^43 + 32 * q^47 + 45 * q^49 + 28 * q^51 + q^53 - 32 * q^57 + 28 * q^59 + 31 * q^61 + 51 * q^63 + 14 * q^67 + 14 * q^69 + 18 * q^71 - 6 * q^73 - 31 * q^77 + 8 * q^79 + 92 * q^81 + 8 * q^83 + 70 * q^87 + 38 * q^91 - 34 * q^93 - 15 * q^97 - 15 * q^99

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
$3$	2.53062	1.46106	0.730528	−	0.682883i	$-0.239274\pi$
$3$	2.53062	1.46106	0.730528	+	0.682883i	$0.239274\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.665840	−0.251664	−0.125832	−	0.992052i	$-0.540160\pi$
$7$	−0.665840	−0.251664	−0.125832	+	0.992052i	$0.540160\pi$
$8$	0	0
$9$	3.40405	1.13468
$10$	0	0
$11$	0.212137	0.0639618	0.0319809	−	0.999488i	$-0.489818\pi$
$11$	0.212137	0.0639618	0.0319809	+	0.999488i	$0.489818\pi$
$12$	0	0
$13$	4.76390	1.32127	0.660634	−	0.750708i	$-0.270287\pi$
$13$	4.76390	1.32127	0.660634	+	0.750708i	$0.270287\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.45920	1.08151	0.540757	−	0.841179i	$-0.318138\pi$
$17$	4.45920	1.08151	0.540757	+	0.841179i	$0.318138\pi$
$18$	0	0
$19$	1.83455	0.420875	0.210437	−	0.977607i	$-0.432511\pi$
$19$	1.83455	0.420875	0.210437	+	0.977607i	$0.432511\pi$
$20$	0	0
$21$	−1.68499	−0.367695
$22$	0	0
$23$	3.65776	0.762696	0.381348	−	0.924432i	$-0.375460\pi$
$23$	3.65776	0.762696	0.381348	+	0.924432i	$0.375460\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.02251	0.196781
$28$	0	0
$29$	3.11206	0.577894	0.288947	−	0.957345i	$-0.406695\pi$
$29$	3.11206	0.577894	0.288947	+	0.957345i	$0.406695\pi$
$30$	0	0
$31$	−2.62327	−0.471154	−0.235577	−	0.971856i	$-0.575698\pi$
$31$	−2.62327	−0.471154	−0.235577	+	0.971856i	$0.575698\pi$
$32$	0	0
$33$	0.536840	0.0934518
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	12.0556	1.93045
$40$	0	0
$41$	−4.61554	−0.720827	−0.360414	−	0.932793i	$-0.617364\pi$
$41$	−4.61554	−0.720827	−0.360414	+	0.932793i	$0.617364\pi$
$42$	0	0
$43$	12.4107	1.89262	0.946308	−	0.323268i	$-0.104781\pi$
$43$	12.4107	1.89262	0.946308	+	0.323268i	$0.104781\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.87039	−0.564555	−0.282277	−	0.959333i	$-0.591090\pi$
$47$	−3.87039	−0.564555	−0.282277	+	0.959333i	$0.591090\pi$
$48$	0	0
$49$	−6.55666	−0.936665
$50$	0	0
$51$	11.2846	1.58015
$52$	0	0
$53$	−9.63817	−1.32390	−0.661952	−	0.749546i	$-0.730272\pi$
$53$	−9.63817	−1.32390	−0.661952	+	0.749546i	$0.730272\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.64256	0.614922
$58$	0	0
$59$	−6.55947	−0.853970	−0.426985	−	0.904259i	$-0.640424\pi$
$59$	−6.55947	−0.853970	−0.426985	+	0.904259i	$0.640424\pi$
$60$	0	0
$61$	10.5841	1.35515	0.677576	−	0.735453i	$-0.263031\pi$
$61$	10.5841	1.35515	0.677576	+	0.735453i	$0.263031\pi$
$62$	0	0
$63$	−2.26655	−0.285559
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.61116	0.319004	0.159502	−	0.987198i	$-0.449011\pi$
$67$	2.61116	0.319004	0.159502	+	0.987198i	$0.449011\pi$
$68$	0	0
$69$	9.25641	1.11434
$70$	0	0
$71$	−5.49658	−0.652324	−0.326162	−	0.945314i	$-0.605756\pi$
$71$	−5.49658	−0.652324	−0.326162	+	0.945314i	$0.605756\pi$
$72$	0	0
$73$	−13.2091	−1.54601	−0.773003	−	0.634402i	$-0.781246\pi$
$73$	−13.2091	−1.54601	−0.773003	+	0.634402i	$0.781246\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.141249	−0.0160969
$78$	0	0
$79$	8.46684	0.952593	0.476297	−	0.879285i	$-0.341979\pi$
$79$	8.46684	0.952593	0.476297	+	0.879285i	$0.341979\pi$
$80$	0	0
$81$	−7.62458	−0.847176
$82$	0	0
$83$	5.30556	0.582361	0.291181	−	0.956668i	$-0.405952\pi$
$83$	5.30556	0.582361	0.291181	+	0.956668i	$0.405952\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.87544	0.844336
$88$	0	0
$89$	1.40902	0.149356	0.0746778	−	0.997208i	$-0.476207\pi$
$89$	1.40902	0.149356	0.0746778	+	0.997208i	$0.476207\pi$
$90$	0	0
$91$	−3.17199	−0.332515
$92$	0	0
$93$	−6.63851	−0.688382
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.96770	0.199790	0.0998950	−	0.994998i	$-0.468149\pi$
$97$	1.96770	0.199790	0.0998950	+	0.994998i	$0.468149\pi$
$98$	0	0
$99$	0.722127	0.0725765
$100$	0	0
$101$	15.1334	1.50583	0.752913	−	0.658120i	$-0.228648\pi$
$101$	15.1334	1.50583	0.752913	+	0.658120i	$0.228648\pi$
$102$	0	0
$103$	12.3638	1.21824	0.609120	−	0.793078i	$-0.291523\pi$
$103$	12.3638	1.21824	0.609120	+	0.793078i	$0.291523\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.00842	0.870877	0.435438	−	0.900219i	$-0.356593\pi$
$107$	9.00842	0.870877	0.435438	+	0.900219i	$0.356593\pi$
$108$	0	0
$109$	1.33330	0.127707	0.0638535	−	0.997959i	$-0.479661\pi$
$109$	1.33330	0.127707	0.0638535	+	0.997959i	$0.479661\pi$
$110$	0	0
$111$	2.53062	0.240196
$112$	0	0
$113$	−5.83790	−0.549184	−0.274592	−	0.961561i	$-0.588543\pi$
$113$	−5.83790	−0.549184	−0.274592	+	0.961561i	$0.588543\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16.2166	1.49922
$118$	0	0
$119$	−2.96911	−0.272178
$120$	0	0
$121$	−10.9550	−0.995909
$122$	0	0
$123$	−11.6802	−1.05317
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.4120	1.19012	0.595060	−	0.803682i	$-0.297128\pi$
$127$	13.4120	1.19012	0.595060	+	0.803682i	$0.297128\pi$
$128$	0	0
$129$	31.4068	2.76522
$130$	0	0
$131$	−4.67076	−0.408086	−0.204043	−	0.978962i	$-0.565408\pi$
$131$	−4.67076	−0.408086	−0.204043	+	0.978962i	$0.565408\pi$
$132$	0	0
$133$	−1.22152	−0.105919
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.20240	−0.444471	−0.222236	−	0.974993i	$-0.571335\pi$
$137$	−5.20240	−0.444471	−0.222236	+	0.974993i	$0.571335\pi$
$138$	0	0
$139$	−4.69871	−0.398539	−0.199270	−	0.979945i	$-0.563857\pi$
$139$	−4.69871	−0.398539	−0.199270	+	0.979945i	$0.563857\pi$
$140$	0	0
$141$	−9.79450	−0.824846
$142$	0	0
$143$	1.01060	0.0845107
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−16.5924	−1.36852
$148$	0	0
$149$	−4.97695	−0.407727	−0.203864	−	0.978999i	$-0.565350\pi$
$149$	−4.97695	−0.407727	−0.203864	+	0.978999i	$0.565350\pi$
$150$	0	0
$151$	−0.121787	−0.00991092	−0.00495546	−	0.999988i	$-0.501577\pi$
$151$	−0.121787	−0.00991092	−0.00495546	+	0.999988i	$0.501577\pi$
$152$	0	0
$153$	15.1794	1.22718
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−16.3676	−1.30628	−0.653139	−	0.757238i	$-0.726548\pi$
$157$	−16.3676	−1.30628	−0.653139	+	0.757238i	$0.726548\pi$
$158$	0	0
$159$	−24.3906	−1.93430
$160$	0	0
$161$	−2.43548	−0.191943
$162$	0	0
$163$	−5.86638	−0.459490	−0.229745	−	0.973251i	$-0.573789\pi$
$163$	−5.86638	−0.459490	−0.229745	+	0.973251i	$0.573789\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.24159	0.328224	0.164112	−	0.986442i	$-0.447524\pi$
$167$	4.24159	0.328224	0.164112	+	0.986442i	$0.447524\pi$
$168$	0	0
$169$	9.69475	0.745750
$170$	0	0
$171$	6.24491	0.477560
$172$	0	0
$173$	13.5657	1.03138	0.515690	−	0.856775i	$-0.327536\pi$
$173$	13.5657	1.03138	0.515690	+	0.856775i	$0.327536\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−16.5995	−1.24770
$178$	0	0
$179$	8.65357	0.646798	0.323399	−	0.946263i	$-0.395174\pi$
$179$	8.65357	0.646798	0.323399	+	0.946263i	$0.395174\pi$
$180$	0	0
$181$	19.5784	1.45525	0.727625	−	0.685975i	$-0.240624\pi$
$181$	19.5784	1.45525	0.727625	+	0.685975i	$0.240624\pi$
$182$	0	0
$183$	26.7843	1.97995
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.945963	0.0691756
$188$	0	0
$189$	−0.680825	−0.0495227
$190$	0	0
$191$	18.6616	1.35030	0.675152	−	0.737678i	$-0.264078\pi$
$191$	18.6616	1.35030	0.675152	+	0.737678i	$0.264078\pi$
$192$	0	0
$193$	11.9126	0.857490	0.428745	−	0.903426i	$-0.358956\pi$
$193$	11.9126	0.857490	0.428745	+	0.903426i	$0.358956\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.7536	1.40739	0.703693	−	0.710504i	$-0.251533\pi$
$197$	19.7536	1.40739	0.703693	+	0.710504i	$0.251533\pi$
$198$	0	0
$199$	−12.5990	−0.893118	−0.446559	−	0.894754i	$-0.647351\pi$
$199$	−12.5990	−0.893118	−0.446559	+	0.894754i	$0.647351\pi$
$200$	0	0
$201$	6.60787	0.466083
$202$	0	0
$203$	−2.07213	−0.145435
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.4512	0.865419
$208$	0	0
$209$	0.389177	0.0269199
$210$	0	0
$211$	0.549320	0.0378167	0.0189084	−	0.999821i	$-0.493981\pi$
$211$	0.549320	0.0378167	0.0189084	+	0.999821i	$0.493981\pi$
$212$	0	0
$213$	−13.9098	−0.953082
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.74668	0.118572
$218$	0	0
$219$	−33.4272	−2.25880
$220$	0	0
$221$	21.2432	1.42897
$222$	0	0
$223$	−6.36784	−0.426422	−0.213211	−	0.977006i	$-0.568392\pi$
$223$	−6.36784	−0.426422	−0.213211	+	0.977006i	$0.568392\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.56839	0.303215	0.151607	−	0.988441i	$-0.451555\pi$
$227$	4.56839	0.303215	0.151607	+	0.988441i	$0.451555\pi$
$228$	0	0
$229$	14.4092	0.952186	0.476093	−	0.879395i	$-0.342053\pi$
$229$	14.4092	0.952186	0.476093	+	0.879395i	$0.342053\pi$
$230$	0	0
$231$	−0.357449	−0.0235184
$232$	0	0
$233$	6.86619	0.449819	0.224909	−	0.974380i	$-0.427791\pi$
$233$	6.86619	0.449819	0.224909	+	0.974380i	$0.427791\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	21.4264	1.39179
$238$	0	0
$239$	−27.2778	−1.76445	−0.882226	−	0.470826i	$-0.843956\pi$
$239$	−27.2778	−1.76445	−0.882226	+	0.470826i	$0.843956\pi$
$240$	0	0
$241$	11.2625	0.725482	0.362741	−	0.931890i	$-0.381841\pi$
$241$	11.2625	0.725482	0.362741	+	0.931890i	$0.381841\pi$
$242$	0	0
$243$	−22.3625	−1.43455
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.73962	0.556089
$248$	0	0
$249$	13.4264	0.850862
$250$	0	0
$251$	14.4759	0.913710	0.456855	−	0.889541i	$-0.348976\pi$
$251$	14.4759	0.913710	0.456855	+	0.889541i	$0.348976\pi$
$252$	0	0
$253$	0.775947	0.0487834
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.87425	−0.366426	−0.183213	−	0.983073i	$-0.558650\pi$
$257$	−5.87425	−0.366426	−0.183213	+	0.983073i	$0.558650\pi$
$258$	0	0
$259$	−0.665840	−0.0413733
$260$	0	0
$261$	10.5936	0.655728
$262$	0	0
$263$	29.0895	1.79374	0.896869	−	0.442297i	$-0.145836\pi$
$263$	29.0895	1.79374	0.896869	+	0.442297i	$0.145836\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.56569	0.218217
$268$	0	0
$269$	−16.9280	−1.03212	−0.516060	−	0.856552i	$-0.672602\pi$
$269$	−16.9280	−1.03212	−0.516060	+	0.856552i	$0.672602\pi$
$270$	0	0
$271$	−1.97302	−0.119852	−0.0599262	−	0.998203i	$-0.519087\pi$
$271$	−1.97302	−0.119852	−0.0599262	+	0.998203i	$0.519087\pi$
$272$	0	0
$273$	−8.02712	−0.485824
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.88133	0.173123	0.0865613	−	0.996247i	$-0.472412\pi$
$277$	2.88133	0.173123	0.0865613	+	0.996247i	$0.472412\pi$
$278$	0	0
$279$	−8.92976	−0.534611
$280$	0	0
$281$	17.8396	1.06422	0.532111	−	0.846674i	$-0.321399\pi$
$281$	17.8396	1.06422	0.532111	+	0.846674i	$0.321399\pi$
$282$	0	0
$283$	−10.8916	−0.647441	−0.323720	−	0.946153i	$-0.604934\pi$
$283$	−10.8916	−0.647441	−0.323720	+	0.946153i	$0.604934\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.07321	0.181406
$288$	0	0
$289$	2.88446	0.169674
$290$	0	0
$291$	4.97952	0.291904
$292$	0	0
$293$	−17.1425	−1.00148	−0.500738	−	0.865599i	$-0.666938\pi$
$293$	−17.1425	−1.00148	−0.500738	+	0.865599i	$0.666938\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.216912	0.0125865
$298$	0	0
$299$	17.4252	1.00773
$300$	0	0
$301$	−8.26354	−0.476303
$302$	0	0
$303$	38.2968	2.20010
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.66495	−0.0950234	−0.0475117	−	0.998871i	$-0.515129\pi$
$307$	−1.66495	−0.0950234	−0.0475117	+	0.998871i	$0.515129\pi$
$308$	0	0
$309$	31.2881	1.77992
$310$	0	0
$311$	20.7686	1.17768	0.588840	−	0.808250i	$-0.299585\pi$
$311$	20.7686	1.17768	0.588840	+	0.808250i	$0.299585\pi$
$312$	0	0
$313$	11.9556	0.675772	0.337886	−	0.941187i	$-0.390288\pi$
$313$	11.9556	0.675772	0.337886	+	0.941187i	$0.390288\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.8309	−1.73164	−0.865818	−	0.500360i	$-0.833201\pi$
$317$	−30.8309	−1.73164	−0.865818	+	0.500360i	$0.833201\pi$
$318$	0	0
$319$	0.660183	0.0369632
$320$	0	0
$321$	22.7969	1.27240
$322$	0	0
$323$	8.18063	0.455182
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.37408	0.186587
$328$	0	0
$329$	2.57706	0.142078
$330$	0	0
$331$	26.3314	1.44730	0.723652	−	0.690165i	$-0.242462\pi$
$331$	26.3314	1.44730	0.723652	+	0.690165i	$0.242462\pi$
$332$	0	0
$333$	3.40405	0.186541
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−22.5264	−1.22709	−0.613545	−	0.789660i	$-0.710257\pi$
$337$	−22.5264	−1.22709	−0.613545	+	0.789660i	$0.710257\pi$
$338$	0	0
$339$	−14.7735	−0.802388
$340$	0	0
$341$	−0.556494	−0.0301358
$342$	0	0
$343$	9.02656	0.487389
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.05241	0.163862	0.0819311	−	0.996638i	$-0.473891\pi$
$347$	3.05241	0.163862	0.0819311	+	0.996638i	$0.473891\pi$
$348$	0	0
$349$	−7.15671	−0.383090	−0.191545	−	0.981484i	$-0.561350\pi$
$349$	−7.15671	−0.383090	−0.191545	+	0.981484i	$0.561350\pi$
$350$	0	0
$351$	4.87111	0.260001
$352$	0	0
$353$	−29.3932	−1.56444	−0.782222	−	0.622999i	$-0.785914\pi$
$353$	−29.3932	−1.56444	−0.782222	+	0.622999i	$0.785914\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7.51371	−0.397667
$358$	0	0
$359$	−12.6182	−0.665962	−0.332981	−	0.942934i	$-0.608054\pi$
$359$	−12.6182	−0.665962	−0.332981	+	0.942934i	$0.608054\pi$
$360$	0	0
$361$	−15.6344	−0.822864
$362$	0	0
$363$	−27.7230	−1.45508
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−31.0497	−1.62078	−0.810391	−	0.585890i	$-0.800745\pi$
$367$	−31.0497	−1.62078	−0.810391	+	0.585890i	$0.800745\pi$
$368$	0	0
$369$	−15.7116	−0.817911
$370$	0	0
$371$	6.41748	0.333179
$372$	0	0
$373$	−6.90718	−0.357640	−0.178820	−	0.983882i	$-0.557228\pi$
$373$	−6.90718	−0.357640	−0.178820	+	0.983882i	$0.557228\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.8255	0.763554
$378$	0	0
$379$	−2.35124	−0.120775	−0.0603875	−	0.998175i	$-0.519234\pi$
$379$	−2.35124	−0.120775	−0.0603875	+	0.998175i	$0.519234\pi$
$380$	0	0
$381$	33.9406	1.73883
$382$	0	0
$383$	−20.3865	−1.04170	−0.520850	−	0.853648i	$-0.674385\pi$
$383$	−20.3865	−1.04170	−0.520850	+	0.853648i	$0.674385\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	42.2467	2.14752
$388$	0	0
$389$	21.3954	1.08479	0.542395	−	0.840124i	$-0.317518\pi$
$389$	21.3954	1.08479	0.542395	+	0.840124i	$0.317518\pi$
$390$	0	0
$391$	16.3107	0.824867
$392$	0	0
$393$	−11.8199	−0.596237
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−10.3406	−0.518981	−0.259490	−	0.965746i	$-0.583555\pi$
$397$	−10.3406	−0.518981	−0.259490	+	0.965746i	$0.583555\pi$
$398$	0	0
$399$	−3.09120	−0.154754
$400$	0	0
$401$	−24.0997	−1.20348	−0.601740	−	0.798692i	$-0.705526\pi$
$401$	−24.0997	−1.20348	−0.601740	+	0.798692i	$0.705526\pi$
$402$	0	0
$403$	−12.4970	−0.622520
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.212137	0.0105153
$408$	0	0
$409$	34.5089	1.70635	0.853176	−	0.521622i	$-0.174673\pi$
$409$	34.5089	1.70635	0.853176	+	0.521622i	$0.174673\pi$
$410$	0	0
$411$	−13.1653	−0.649397
$412$	0	0
$413$	4.36756	0.214913
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−11.8907	−0.582288
$418$	0	0
$419$	31.4018	1.53408	0.767039	−	0.641601i	$-0.221729\pi$
$419$	31.4018	1.53408	0.767039	+	0.641601i	$0.221729\pi$
$420$	0	0
$421$	33.2638	1.62118	0.810589	−	0.585615i	$-0.199147\pi$
$421$	33.2638	1.62118	0.810589	+	0.585615i	$0.199147\pi$
$422$	0	0
$423$	−13.1750	−0.640591
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−7.04730	−0.341043
$428$	0	0
$429$	2.55745	0.123475
$430$	0	0
$431$	−7.53778	−0.363082	−0.181541	−	0.983383i	$-0.558108\pi$
$431$	−7.53778	−0.363082	−0.181541	+	0.983383i	$0.558108\pi$
$432$	0	0
$433$	11.8703	0.570451	0.285225	−	0.958460i	$-0.407932\pi$
$433$	11.8703	0.570451	0.285225	+	0.958460i	$0.407932\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.71035	0.320999
$438$	0	0
$439$	40.2687	1.92192	0.960959	−	0.276690i	$-0.0892374\pi$
$439$	40.2687	1.92192	0.960959	+	0.276690i	$0.0892374\pi$
$440$	0	0
$441$	−22.3192	−1.06282
$442$	0	0
$443$	−25.3001	−1.20204	−0.601021	−	0.799233i	$-0.705239\pi$
$443$	−25.3001	−1.20204	−0.601021	+	0.799233i	$0.705239\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−12.5948	−0.595713
$448$	0	0
$449$	−38.8068	−1.83141	−0.915704	−	0.401854i	$-0.868366\pi$
$449$	−38.8068	−1.83141	−0.915704	+	0.401854i	$0.868366\pi$
$450$	0	0
$451$	−0.979129	−0.0461054
$452$	0	0
$453$	−0.308198	−0.0144804
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.0818	−0.986165	−0.493083	−	0.869982i	$-0.664130\pi$
$457$	−21.0818	−0.986165	−0.493083	+	0.869982i	$0.664130\pi$
$458$	0	0
$459$	4.55955	0.212822
$460$	0	0
$461$	19.3748	0.902376	0.451188	−	0.892429i	$-0.351000\pi$
$461$	19.3748	0.902376	0.451188	+	0.892429i	$0.351000\pi$
$462$	0	0
$463$	−35.9717	−1.67175	−0.835873	−	0.548923i	$-0.815038\pi$
$463$	−35.9717	−1.67175	−0.835873	+	0.548923i	$0.815038\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.23518	0.195981	0.0979904	−	0.995187i	$-0.468759\pi$
$467$	4.23518	0.195981	0.0979904	+	0.995187i	$0.468759\pi$
$468$	0	0
$469$	−1.73862	−0.0802819
$470$	0	0
$471$	−41.4203	−1.90855
$472$	0	0
$473$	2.63277	0.121055
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−32.8088	−1.50221
$478$	0	0
$479$	−16.9389	−0.773957	−0.386979	−	0.922089i	$-0.626481\pi$
$479$	−16.9389	−0.773957	−0.386979	+	0.922089i	$0.626481\pi$
$480$	0	0
$481$	4.76390	0.217215
$482$	0	0
$483$	−6.16329	−0.280439
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4.79971	−0.217496	−0.108748	−	0.994069i	$-0.534684\pi$
$487$	−4.79971	−0.217496	−0.108748	+	0.994069i	$0.534684\pi$
$488$	0	0
$489$	−14.8456	−0.671341
$490$	0	0
$491$	25.8467	1.16645	0.583223	−	0.812312i	$-0.301791\pi$
$491$	25.8467	1.16645	0.583223	+	0.812312i	$0.301791\pi$
$492$	0	0
$493$	13.8773	0.625001
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.65984	0.164166
$498$	0	0
$499$	9.32675	0.417523	0.208761	−	0.977967i	$-0.433057\pi$
$499$	9.32675	0.417523	0.208761	+	0.977967i	$0.433057\pi$
$500$	0	0
$501$	10.7339	0.479554
$502$	0	0
$503$	−6.75244	−0.301077	−0.150538	−	0.988604i	$-0.548101\pi$
$503$	−6.75244	−0.301077	−0.150538	+	0.988604i	$0.548101\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	24.5338	1.08958
$508$	0	0
$509$	−24.7844	−1.09855	−0.549274	−	0.835642i	$-0.685096\pi$
$509$	−24.7844	−1.09855	−0.549274	+	0.835642i	$0.685096\pi$
$510$	0	0
$511$	8.79514	0.389074
$512$	0	0
$513$	1.87584	0.0828203
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.821054	−0.0361099
$518$	0	0
$519$	34.3296	1.50690
$520$	0	0
$521$	24.4210	1.06990	0.534952	−	0.844882i	$-0.320330\pi$
$521$	24.4210	1.06990	0.534952	+	0.844882i	$0.320330\pi$
$522$	0	0
$523$	−27.7585	−1.21379	−0.606896	−	0.794781i	$-0.707586\pi$
$523$	−27.7585	−1.21379	−0.606896	+	0.794781i	$0.707586\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.6977	−0.509560
$528$	0	0
$529$	−9.62079	−0.418295
$530$	0	0
$531$	−22.3288	−0.968986
$532$	0	0
$533$	−21.9880	−0.952406
$534$	0	0
$535$	0	0
$536$	0	0
$537$	21.8989	0.945009
$538$	0	0
$539$	−1.39091	−0.0599108
$540$	0	0
$541$	21.9556	0.943944	0.471972	−	0.881613i	$-0.343542\pi$
$541$	21.9556	0.943944	0.471972	+	0.881613i	$0.343542\pi$
$542$	0	0
$543$	49.5455	2.12620
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−31.7311	−1.35673	−0.678363	−	0.734727i	$-0.737310\pi$
$547$	−31.7311	−1.35673	−0.678363	+	0.734727i	$0.737310\pi$
$548$	0	0
$549$	36.0287	1.53767
$550$	0	0
$551$	5.70923	0.243221
$552$	0	0
$553$	−5.63756	−0.239733
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.83522	−0.247246	−0.123623	−	0.992329i	$-0.539451\pi$
$557$	−5.83522	−0.247246	−0.123623	+	0.992329i	$0.539451\pi$
$558$	0	0
$559$	59.1234	2.50065
$560$	0	0
$561$	2.39388	0.101069
$562$	0	0
$563$	−43.9123	−1.85068	−0.925341	−	0.379135i	$-0.876222\pi$
$563$	−43.9123	−1.85068	−0.925341	+	0.379135i	$0.876222\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.07675	0.213204
$568$	0	0
$569$	−29.3895	−1.23207	−0.616035	−	0.787719i	$-0.711262\pi$
$569$	−29.3895	−1.23207	−0.616035	+	0.787719i	$0.711262\pi$
$570$	0	0
$571$	−43.3521	−1.81423	−0.907114	−	0.420886i	$-0.861719\pi$
$571$	−43.3521	−1.81423	−0.907114	+	0.420886i	$0.861719\pi$
$572$	0	0
$573$	47.2254	1.97287
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−12.6707	−0.527487	−0.263744	−	0.964593i	$-0.584957\pi$
$577$	−12.6707	−0.527487	−0.263744	+	0.964593i	$0.584957\pi$
$578$	0	0
$579$	30.1464	1.25284
$580$	0	0
$581$	−3.53266	−0.146559
$582$	0	0
$583$	−2.04462	−0.0846793
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.95031	−0.369419	−0.184709	−	0.982793i	$-0.559134\pi$
$587$	−8.95031	−0.369419	−0.184709	+	0.982793i	$0.559134\pi$
$588$	0	0
$589$	−4.81253	−0.198297
$590$	0	0
$591$	49.9889	2.05627
$592$	0	0
$593$	−16.5839	−0.681019	−0.340510	−	0.940241i	$-0.610600\pi$
$593$	−16.5839	−0.681019	−0.340510	+	0.940241i	$0.610600\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−31.8833	−1.30490
$598$	0	0
$599$	−16.4241	−0.671071	−0.335535	−	0.942028i	$-0.608917\pi$
$599$	−16.4241	−0.671071	−0.335535	+	0.942028i	$0.608917\pi$
$600$	0	0
$601$	−9.47752	−0.386596	−0.193298	−	0.981140i	$-0.561918\pi$
$601$	−9.47752	−0.386596	−0.193298	+	0.981140i	$0.561918\pi$
$602$	0	0
$603$	8.88854	0.361969
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.4880	0.831583	0.415792	−	0.909460i	$-0.363505\pi$
$607$	20.4880	0.831583	0.415792	+	0.909460i	$0.363505\pi$
$608$	0	0
$609$	−5.24378	−0.212489
$610$	0	0
$611$	−18.4382	−0.745928
$612$	0	0
$613$	8.51856	0.344061	0.172031	−	0.985092i	$-0.444967\pi$
$613$	8.51856	0.344061	0.172031	+	0.985092i	$0.444967\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−37.8757	−1.52482	−0.762409	−	0.647096i	$-0.775983\pi$
$617$	−37.8757	−1.52482	−0.762409	+	0.647096i	$0.775983\pi$
$618$	0	0
$619$	20.3246	0.816915	0.408457	−	0.912777i	$-0.366067\pi$
$619$	20.3246	0.816915	0.408457	+	0.912777i	$0.366067\pi$
$620$	0	0
$621$	3.74008	0.150084
$622$	0	0
$623$	−0.938180	−0.0375874
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.984860	0.0393315
$628$	0	0
$629$	4.45920	0.177800
$630$	0	0
$631$	−16.1110	−0.641371	−0.320685	−	0.947186i	$-0.603913\pi$
$631$	−16.1110	−0.641371	−0.320685	+	0.947186i	$0.603913\pi$
$632$	0	0
$633$	1.39012	0.0552523
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−31.2353	−1.23759
$638$	0	0
$639$	−18.7107	−0.740182
$640$	0	0
$641$	−41.9600	−1.65732	−0.828660	−	0.559752i	$-0.810896\pi$
$641$	−41.9600	−1.65732	−0.828660	+	0.559752i	$0.810896\pi$
$642$	0	0
$643$	41.9665	1.65500	0.827499	−	0.561467i	$-0.189763\pi$
$643$	41.9665	1.65500	0.827499	+	0.561467i	$0.189763\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.45946	−0.293262	−0.146631	−	0.989191i	$-0.546843\pi$
$647$	−7.45946	−0.293262	−0.146631	+	0.989191i	$0.546843\pi$
$648$	0	0
$649$	−1.39151	−0.0546215
$650$	0	0
$651$	4.42019	0.173241
$652$	0	0
$653$	−12.0216	−0.470441	−0.235221	−	0.971942i	$-0.575581\pi$
$653$	−12.0216	−0.470441	−0.235221	+	0.971942i	$0.575581\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−44.9644	−1.75423
$658$	0	0
$659$	27.3611	1.06584	0.532919	−	0.846166i	$-0.321095\pi$
$659$	27.3611	1.06584	0.532919	+	0.846166i	$0.321095\pi$
$660$	0	0
$661$	−2.47942	−0.0964384	−0.0482192	−	0.998837i	$-0.515355\pi$
$661$	−2.47942	−0.0964384	−0.0482192	+	0.998837i	$0.515355\pi$
$662$	0	0
$663$	53.7585	2.08781
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.3832	0.440758
$668$	0	0
$669$	−16.1146	−0.623027
$670$	0	0
$671$	2.24528	0.0866779
$672$	0	0
$673$	−33.4903	−1.29096	−0.645478	−	0.763779i	$-0.723342\pi$
$673$	−33.4903	−1.29096	−0.645478	+	0.763779i	$0.723342\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.23160	0.239500	0.119750	−	0.992804i	$-0.461791\pi$
$677$	6.23160	0.239500	0.119750	+	0.992804i	$0.461791\pi$
$678$	0	0
$679$	−1.31018	−0.0502799
$680$	0	0
$681$	11.5609	0.443013
$682$	0	0
$683$	−28.3438	−1.08455	−0.542274	−	0.840202i	$-0.682436\pi$
$683$	−28.3438	−1.08455	−0.542274	+	0.840202i	$0.682436\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	36.4642	1.39120
$688$	0	0
$689$	−45.9153	−1.74923
$690$	0	0
$691$	−47.8097	−1.81876	−0.909382	−	0.415961i	$-0.863445\pi$
$691$	−47.8097	−1.81876	−0.909382	+	0.415961i	$0.863445\pi$
$692$	0	0
$693$	−0.480821	−0.0182649
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−20.5816	−0.779585
$698$	0	0
$699$	17.3757	0.657211
$700$	0	0
$701$	−2.20755	−0.0833779	−0.0416890	−	0.999131i	$-0.513274\pi$
$701$	−2.20755	−0.0833779	−0.0416890	+	0.999131i	$0.513274\pi$
$702$	0	0
$703$	1.83455	0.0691914
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.0764	−0.378962
$708$	0	0
$709$	−4.52774	−0.170043	−0.0850214	−	0.996379i	$-0.527096\pi$
$709$	−4.52774	−0.170043	−0.0850214	+	0.996379i	$0.527096\pi$
$710$	0	0
$711$	28.8216	1.08089
$712$	0	0
$713$	−9.59530	−0.359347
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−69.0298	−2.57796
$718$	0	0
$719$	−38.5228	−1.43666	−0.718329	−	0.695703i	$-0.755093\pi$
$719$	−38.5228	−1.43666	−0.718329	+	0.695703i	$0.755093\pi$
$720$	0	0
$721$	−8.23230	−0.306587
$722$	0	0
$723$	28.5012	1.05997
$724$	0	0
$725$	0	0
$726$	0	0
$727$	31.0552	1.15177	0.575887	−	0.817529i	$-0.304657\pi$
$727$	31.0552	1.15177	0.575887	+	0.817529i	$0.304657\pi$
$728$	0	0
$729$	−33.7172	−1.24879
$730$	0	0
$731$	55.3418	2.04689
$732$	0	0
$733$	−22.4131	−0.827849	−0.413924	−	0.910311i	$-0.635842\pi$
$733$	−22.4131	−0.827849	−0.413924	+	0.910311i	$0.635842\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.553925	0.0204041
$738$	0	0
$739$	17.8141	0.655301	0.327651	−	0.944799i	$-0.393743\pi$
$739$	17.8141	0.655301	0.327651	+	0.944799i	$0.393743\pi$
$740$	0	0
$741$	22.1167	0.812476
$742$	0	0
$743$	11.0836	0.406617	0.203309	−	0.979115i	$-0.434831\pi$
$743$	11.0836	0.406617	0.203309	+	0.979115i	$0.434831\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	18.0604	0.660796
$748$	0	0
$749$	−5.99816	−0.219168
$750$	0	0
$751$	12.6522	0.461684	0.230842	−	0.972991i	$-0.425852\pi$
$751$	12.6522	0.461684	0.230842	+	0.972991i	$0.425852\pi$
$752$	0	0
$753$	36.6330	1.33498
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−27.5485	−1.00127	−0.500634	−	0.865659i	$-0.666900\pi$
$757$	−27.5485	−1.00127	−0.500634	+	0.865659i	$0.666900\pi$
$758$	0	0
$759$	1.96363	0.0712753
$760$	0	0
$761$	−46.5395	−1.68705	−0.843527	−	0.537086i	$-0.819525\pi$
$761$	−46.5395	−1.68705	−0.843527	+	0.537086i	$0.819525\pi$
$762$	0	0
$763$	−0.887764	−0.0321392
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−31.2487	−1.12832
$768$	0	0
$769$	26.4928	0.955355	0.477678	−	0.878535i	$-0.341479\pi$
$769$	26.4928	0.955355	0.477678	+	0.878535i	$0.341479\pi$
$770$	0	0
$771$	−14.8655	−0.535368
$772$	0	0
$773$	−18.3791	−0.661049	−0.330524	−	0.943797i	$-0.607226\pi$
$773$	−18.3791	−0.661049	−0.330524	+	0.943797i	$0.607226\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.68499	−0.0604487
$778$	0	0
$779$	−8.46745	−0.303378
$780$	0	0
$781$	−1.16603	−0.0417238
$782$	0	0
$783$	3.18209	0.113719
$784$	0	0
$785$	0	0
$786$	0	0
$787$	37.6988	1.34382	0.671909	−	0.740634i	$-0.265475\pi$
$787$	37.6988	1.34382	0.671909	+	0.740634i	$0.265475\pi$
$788$	0	0
$789$	73.6146	2.62075
$790$	0	0
$791$	3.88711	0.138210
$792$	0	0
$793$	50.4215	1.79052
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.7869	−0.948840	−0.474420	−	0.880299i	$-0.657342\pi$
$797$	−26.7869	−0.948840	−0.474420	+	0.880299i	$0.657342\pi$
$798$	0	0
$799$	−17.2588	−0.610574
$800$	0	0
$801$	4.79637	0.169471
$802$	0	0
$803$	−2.80214	−0.0988854
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−42.8385	−1.50799
$808$	0	0
$809$	−25.1772	−0.885182	−0.442591	−	0.896724i	$-0.645941\pi$
$809$	−25.1772	−0.885182	−0.442591	+	0.896724i	$0.645941\pi$
$810$	0	0
$811$	−50.4209	−1.77052	−0.885259	−	0.465099i	$-0.846019\pi$
$811$	−50.4209	−1.77052	−0.885259	+	0.465099i	$0.846019\pi$
$812$	0	0
$813$	−4.99297	−0.175111
$814$	0	0
$815$	0	0
$816$	0	0
$817$	22.7681	0.796554
$818$	0	0
$819$	−10.7976	−0.377300
$820$	0	0
$821$	−33.7559	−1.17809	−0.589044	−	0.808101i	$-0.700496\pi$
$821$	−33.7559	−1.17809	−0.589044	+	0.808101i	$0.700496\pi$
$822$	0	0
$823$	28.1656	0.981790	0.490895	−	0.871219i	$-0.336670\pi$
$823$	28.1656	0.981790	0.490895	+	0.871219i	$0.336670\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.8109	0.723667	0.361834	−	0.932243i	$-0.382151\pi$
$827$	20.8109	0.723667	0.361834	+	0.932243i	$0.382151\pi$
$828$	0	0
$829$	−10.1593	−0.352849	−0.176424	−	0.984314i	$-0.556453\pi$
$829$	−10.1593	−0.352849	−0.176424	+	0.984314i	$0.556453\pi$
$830$	0	0
$831$	7.29157	0.252942
$832$	0	0
$833$	−29.2374	−1.01302
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.68231	−0.0927142
$838$	0	0
$839$	4.94779	0.170817	0.0854083	−	0.996346i	$-0.472781\pi$
$839$	4.94779	0.170817	0.0854083	+	0.996346i	$0.472781\pi$
$840$	0	0
$841$	−19.3151	−0.666038
$842$	0	0
$843$	45.1454	1.55489
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.29427	0.250634
$848$	0	0
$849$	−27.5626	−0.945947
$850$	0	0
$851$	3.65776	0.125386
$852$	0	0
$853$	1.98923	0.0681099	0.0340550	−	0.999420i	$-0.489158\pi$
$853$	1.98923	0.0681099	0.0340550	+	0.999420i	$0.489158\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.281525	0.00961670	0.00480835	−	0.999988i	$-0.498469\pi$
$857$	0.281525	0.00961670	0.00480835	+	0.999988i	$0.498469\pi$
$858$	0	0
$859$	5.41937	0.184906	0.0924532	−	0.995717i	$-0.470529\pi$
$859$	5.41937	0.184906	0.0924532	+	0.995717i	$0.470529\pi$
$860$	0	0
$861$	7.77715	0.265044
$862$	0	0
$863$	38.2292	1.30134	0.650669	−	0.759362i	$-0.274489\pi$
$863$	38.2292	1.30134	0.650669	+	0.759362i	$0.274489\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.29949	0.247904
$868$	0	0
$869$	1.79613	0.0609296
$870$	0	0
$871$	12.4393	0.421491
$872$	0	0
$873$	6.69817	0.226699
$874$	0	0
$875$	0	0
$876$	0	0
$877$	55.3971	1.87063	0.935314	−	0.353818i	$-0.115117\pi$
$877$	55.3971	1.87063	0.935314	+	0.353818i	$0.115117\pi$
$878$	0	0
$879$	−43.3812	−1.46321
$880$	0	0
$881$	−0.387591	−0.0130583	−0.00652914	−	0.999979i	$-0.502078\pi$
$881$	−0.387591	−0.0130583	−0.00652914	+	0.999979i	$0.502078\pi$
$882$	0	0
$883$	−13.7648	−0.463222	−0.231611	−	0.972808i	$-0.574400\pi$
$883$	−13.7648	−0.463222	−0.231611	+	0.972808i	$0.574400\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	43.5025	1.46067	0.730336	−	0.683088i	$-0.239364\pi$
$887$	43.5025	1.46067	0.730336	+	0.683088i	$0.239364\pi$
$888$	0	0
$889$	−8.93022	−0.299510
$890$	0	0
$891$	−1.61746	−0.0541869
$892$	0	0
$893$	−7.10043	−0.237607
$894$	0	0
$895$	0	0
$896$	0	0
$897$	44.0966	1.47234
$898$	0	0
$899$	−8.16377	−0.272277
$900$	0	0
$901$	−42.9785	−1.43182
$902$	0	0
$903$	−20.9119	−0.695905
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−19.1761	−0.636731	−0.318366	−	0.947968i	$-0.603134\pi$
$907$	−19.1761	−0.636731	−0.318366	+	0.947968i	$0.603134\pi$
$908$	0	0
$909$	51.5148	1.70864
$910$	0	0
$911$	−32.8000	−1.08671	−0.543356	−	0.839502i	$-0.682847\pi$
$911$	−32.8000	−1.08671	−0.543356	+	0.839502i	$0.682847\pi$
$912$	0	0
$913$	1.12551	0.0372489
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.10998	0.102701
$918$	0	0
$919$	56.8859	1.87649	0.938246	−	0.345970i	$-0.112450\pi$
$919$	56.8859	1.87649	0.938246	+	0.345970i	$0.112450\pi$
$920$	0	0
$921$	−4.21335	−0.138835
$922$	0	0
$923$	−26.1852	−0.861896
$924$	0	0
$925$	0	0
$926$	0	0
$927$	42.0870	1.38232
$928$	0	0
$929$	34.5195	1.13255	0.566274	−	0.824217i	$-0.308384\pi$
$929$	34.5195	1.13255	0.566274	+	0.824217i	$0.308384\pi$
$930$	0	0
$931$	−12.0285	−0.394219
$932$	0	0
$933$	52.5575	1.72066
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.3434	0.337905	0.168953	−	0.985624i	$-0.445961\pi$
$937$	10.3434	0.337905	0.168953	+	0.985624i	$0.445961\pi$
$938$	0	0
$939$	30.2552	0.987340
$940$	0	0
$941$	27.8705	0.908552	0.454276	−	0.890861i	$-0.349898\pi$
$941$	27.8705	0.908552	0.454276	+	0.890861i	$0.349898\pi$
$942$	0	0
$943$	−16.8826	−0.549772
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.78656	0.0580553	0.0290276	−	0.999579i	$-0.490759\pi$
$947$	1.78656	0.0580553	0.0290276	+	0.999579i	$0.490759\pi$
$948$	0	0
$949$	−62.9268	−2.04269
$950$	0	0
$951$	−78.0214	−2.53002
$952$	0	0
$953$	−54.9360	−1.77955	−0.889776	−	0.456397i	$-0.849140\pi$
$953$	−54.9360	−1.77955	−0.889776	+	0.456397i	$0.849140\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.67068	0.0540053
$958$	0	0
$959$	3.46397	0.111857
$960$	0	0
$961$	−24.1184	−0.778014
$962$	0	0
$963$	30.6651	0.988170
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−10.8335	−0.348383	−0.174192	−	0.984712i	$-0.555731\pi$
$967$	−10.8335	−0.348383	−0.174192	+	0.984712i	$0.555731\pi$
$968$	0	0
$969$	20.7021	0.665047
$970$	0	0
$971$	36.7131	1.17818	0.589090	−	0.808067i	$-0.299486\pi$
$971$	36.7131	1.17818	0.589090	+	0.808067i	$0.299486\pi$
$972$	0	0
$973$	3.12859	0.100298
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.0633	0.321954	0.160977	−	0.986958i	$-0.448535\pi$
$977$	10.0633	0.321954	0.160977	+	0.986958i	$0.448535\pi$
$978$	0	0
$979$	0.298905	0.00955305
$980$	0	0
$981$	4.53862	0.144907
$982$	0	0
$983$	−7.49574	−0.239077	−0.119538	−	0.992830i	$-0.538141\pi$
$983$	−7.49574	−0.239077	−0.119538	+	0.992830i	$0.538141\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.52157	0.207584
$988$	0	0
$989$	45.3954	1.44349
$990$	0	0
$991$	28.6355	0.909635	0.454818	−	0.890585i	$-0.349704\pi$
$991$	28.6355	0.909635	0.454818	+	0.890585i	$0.349704\pi$
$992$	0	0
$993$	66.6348	2.11459
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−21.9922	−0.696501	−0.348250	−	0.937402i	$-0.613224\pi$
$997$	−21.9922	−0.696501	−0.348250	+	0.937402i	$0.613224\pi$
$998$	0	0
$999$	1.02251	0.0323506

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7400.2.a.be.1.14		16
5.2	odd	4		1480.2.d.c.889.6	✓	32
5.3	odd	4		1480.2.d.c.889.27	yes	32
5.4	even	2		7400.2.a.bd.1.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.d.c.889.6	✓	32	5.2	odd	4
1480.2.d.c.889.27	yes	32	5.3	odd	4
7400.2.a.bd.1.3		16	5.4	even	2
7400.2.a.be.1.14		16	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:	\( N \)	\(=\)	\( 7400 = 2^{3} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7400.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53062 q^{3} -0.665840 q^{7} +3.40405 q^{9} +0.212137 q^{11} +4.76390 q^{13} +4.45920 q^{17} +1.83455 q^{19} -1.68499 q^{21} +3.65776 q^{23} +1.02251 q^{27} +3.11206 q^{29} -2.62327 q^{31} +0.536840 q^{33} +1.00000 q^{37} +12.0556 q^{39} -4.61554 q^{41} +12.4107 q^{43} -3.87039 q^{47} -6.55666 q^{49} +11.2846 q^{51} -9.63817 q^{53} +4.64256 q^{57} -6.55947 q^{59} +10.5841 q^{61} -2.26655 q^{63} +2.61116 q^{67} +9.25641 q^{69} -5.49658 q^{71} -13.2091 q^{73} -0.141249 q^{77} +8.46684 q^{79} -7.62458 q^{81} +5.30556 q^{83} +7.87544 q^{87} +1.40902 q^{89} -3.17199 q^{91} -6.63851 q^{93} +1.96770 q^{97} +0.722127 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{3} + 9 q^{7} + 22 q^{9} + 3 q^{11} - 4 q^{13} - 9 q^{17} + 14 q^{19} + 8 q^{21} + 18 q^{23} - 4 q^{27} + q^{29} + 23 q^{31} - 10 q^{33} + 16 q^{37} + 23 q^{41} - 11 q^{43} + 32 q^{47} + 45 q^{49}+ \cdots - 15 q^{99}+O(q^{100}) \) 16 * q + 2 * q^3 + 9 * q^7 + 22 * q^9 + 3 * q^11 - 4 * q^13 - 9 * q^17 + 14 * q^19 + 8 * q^21 + 18 * q^23 - 4 * q^27 + q^29 + 23 * q^31 - 10 * q^33 + 16 * q^37 + 23 * q^41 - 11 * q^43 + 32 * q^47 + 45 * q^49 + 28 * q^51 + q^53 - 32 * q^57 + 28 * q^59 + 31 * q^61 + 51 * q^63 + 14 * q^67 + 14 * q^69 + 18 * q^71 - 6 * q^73 - 31 * q^77 + 8 * q^79 + 92 * q^81 + 8 * q^83 + 70 * q^87 + 38 * q^91 - 34 * q^93 - 15 * q^97 - 15 * q^99

Introduction

L-functions

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