LMFDB - Embedded newform 7600.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3800)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.91223$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.656620 q^{3} +0.656620 q^{7} -2.56885 q^{9} +O(q^{10})$	$q-0.656620 q^{3} +0.656620 q^{7} -2.56885 q^{9} +0.343380 q^{11} +1.91223 q^{13} -4.48108 q^{17} +1.00000 q^{19} -0.431150 q^{21} -3.56885 q^{23} +3.65662 q^{27} +7.99230 q^{29} -5.73669 q^{31} -0.225470 q^{33} +4.16784 q^{37} -1.25561 q^{39} -9.08007 q^{41} +3.51122 q^{43} +3.40101 q^{47} -6.56885 q^{49} +2.94237 q^{51} +1.68676 q^{53} -0.656620 q^{57} +7.82446 q^{59} -12.4234 q^{61} -1.68676 q^{63} +9.48108 q^{67} +2.34338 q^{69} +9.96216 q^{71} -7.53101 q^{73} +0.225470 q^{77} +14.9923 q^{79} +5.30554 q^{81} -10.9045 q^{83} -5.24791 q^{87} -4.19533 q^{89} +1.25561 q^{91} +3.76683 q^{93} -1.73669 q^{97} -0.882090 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{9}+O(q^{10}) $ 3 * q + q^9	$ 3 q + q^{9} + 3 q^{11} - q^{13} + 2 q^{17} + 3 q^{19} - 10 q^{21} - 2 q^{23} + 9 q^{27} - q^{29} + 3 q^{31} + 10 q^{33} + q^{37} + q^{39} - 9 q^{41} + q^{43} + 13 q^{47} - 11 q^{49} + 8 q^{51} + 9 q^{53} + 10 q^{59} - 21 q^{61} - 9 q^{63} + 13 q^{67} + 9 q^{69} - q^{71} + 17 q^{73} - 10 q^{77} + 20 q^{79} - 13 q^{81} - q^{83} + 14 q^{87} + 4 q^{89} - q^{91} - 3 q^{93} + 15 q^{97} + 10 q^{99}+O(q^{100}) $ 3 * q + q^9 + 3 * q^11 - q^13 + 2 * q^17 + 3 * q^19 - 10 * q^21 - 2 * q^23 + 9 * q^27 - q^29 + 3 * q^31 + 10 * q^33 + q^37 + q^39 - 9 * q^41 + q^43 + 13 * q^47 - 11 * q^49 + 8 * q^51 + 9 * q^53 + 10 * q^59 - 21 * q^61 - 9 * q^63 + 13 * q^67 + 9 * q^69 - q^71 + 17 * q^73 - 10 * q^77 + 20 * q^79 - 13 * q^81 - q^83 + 14 * q^87 + 4 * q^89 - q^91 - 3 * q^93 + 15 * q^97 + 10 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−0.656620	−0.379100	−0.189550	−	0.981871i	$-0.560703\pi$
$3$	−0.656620	−0.379100	−0.189550	+	0.981871i	$0.560703\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.656620	0.248179	0.124090	−	0.992271i	$-0.460399\pi$
$7$	0.656620	0.248179	0.124090	+	0.992271i	$0.460399\pi$
$8$	0	0
$9$	−2.56885	−0.856283
$10$	0	0
$11$	0.343380	0.103533	0.0517664	−	0.998659i	$-0.483515\pi$
$11$	0.343380	0.103533	0.0517664	+	0.998659i	$0.483515\pi$
$12$	0	0
$13$	1.91223	0.530357	0.265178	−	0.964199i	$-0.414569\pi$
$13$	1.91223	0.530357	0.265178	+	0.964199i	$0.414569\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.48108	−1.08682	−0.543411	−	0.839467i	$-0.682867\pi$
$17$	−4.48108	−1.08682	−0.543411	+	0.839467i	$0.682867\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.431150	−0.0940847
$22$	0	0
$23$	−3.56885	−0.744157	−0.372078	−	0.928201i	$-0.621355\pi$
$23$	−3.56885	−0.744157	−0.372078	+	0.928201i	$0.621355\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.65662	0.703717
$28$	0	0
$29$	7.99230	1.48413	0.742066	−	0.670327i	$-0.233846\pi$
$29$	7.99230	1.48413	0.742066	+	0.670327i	$0.233846\pi$
$30$	0	0
$31$	−5.73669	−1.03034	−0.515170	−	0.857088i	$-0.672271\pi$
$31$	−5.73669	−1.03034	−0.515170	+	0.857088i	$0.672271\pi$
$32$	0	0
$33$	−0.225470	−0.0392493
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.16784	0.685188	0.342594	−	0.939483i	$-0.388694\pi$
$37$	4.16784	0.685188	0.342594	+	0.939483i	$0.388694\pi$
$38$	0	0
$39$	−1.25561	−0.201058
$40$	0	0
$41$	−9.08007	−1.41807	−0.709034	−	0.705174i	$-0.750869\pi$
$41$	−9.08007	−1.41807	−0.709034	+	0.705174i	$0.750869\pi$
$42$	0	0
$43$	3.51122	0.535456	0.267728	−	0.963495i	$-0.413727\pi$
$43$	3.51122	0.535456	0.267728	+	0.963495i	$0.413727\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.40101	0.496089	0.248044	−	0.968749i	$-0.420212\pi$
$47$	3.40101	0.496089	0.248044	+	0.968749i	$0.420212\pi$
$48$	0	0
$49$	−6.56885	−0.938407
$50$	0	0
$51$	2.94237	0.412014
$52$	0	0
$53$	1.68676	0.231694	0.115847	−	0.993267i	$-0.463042\pi$
$53$	1.68676	0.231694	0.115847	+	0.993267i	$0.463042\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.656620	−0.0869715
$58$	0	0
$59$	7.82446	1.01866	0.509329	−	0.860572i	$-0.329894\pi$
$59$	7.82446	1.01866	0.509329	+	0.860572i	$0.329894\pi$
$60$	0	0
$61$	−12.4234	−1.59066	−0.795330	−	0.606177i	$-0.792702\pi$
$61$	−12.4234	−1.59066	−0.795330	+	0.606177i	$0.792702\pi$
$62$	0	0
$63$	−1.68676	−0.212512
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.48108	1.15830	0.579149	−	0.815222i	$-0.303385\pi$
$67$	9.48108	1.15830	0.579149	+	0.815222i	$0.303385\pi$
$68$	0	0
$69$	2.34338	0.282110
$70$	0	0
$71$	9.96216	1.18229	0.591145	−	0.806565i	$-0.298676\pi$
$71$	9.96216	1.18229	0.591145	+	0.806565i	$0.298676\pi$
$72$	0	0
$73$	−7.53101	−0.881438	−0.440719	−	0.897645i	$-0.645276\pi$
$73$	−7.53101	−0.881438	−0.440719	+	0.897645i	$0.645276\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.225470	0.0256947
$78$	0	0
$79$	14.9923	1.68677	0.843383	−	0.537314i	$-0.180561\pi$
$79$	14.9923	1.68677	0.843383	+	0.537314i	$0.180561\pi$
$80$	0	0
$81$	5.30554	0.589504
$82$	0	0
$83$	−10.9045	−1.19693	−0.598464	−	0.801150i	$-0.704222\pi$
$83$	−10.9045	−1.19693	−0.598464	+	0.801150i	$0.704222\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.24791	−0.562634
$88$	0	0
$89$	−4.19533	−0.444704	−0.222352	−	0.974966i	$-0.571373\pi$
$89$	−4.19533	−0.444704	−0.222352	+	0.974966i	$0.571373\pi$
$90$	0	0
$91$	1.25561	0.131624
$92$	0	0
$93$	3.76683	0.390602
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.73669	−0.176334	−0.0881670	−	0.996106i	$-0.528101\pi$
$97$	−1.73669	−0.176334	−0.0881670	+	0.996106i	$0.528101\pi$
$98$	0	0
$99$	−0.882090	−0.0886534
$100$	0	0
$101$	2.99230	0.297745	0.148872	−	0.988856i	$-0.452436\pi$
$101$	2.99230	0.297745	0.148872	+	0.988856i	$0.452436\pi$
$102$	0	0
$103$	−3.96986	−0.391162	−0.195581	−	0.980688i	$-0.562659\pi$
$103$	−3.96986	−0.391162	−0.195581	+	0.980688i	$0.562659\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.05763	0.682287	0.341144	−	0.940011i	$-0.389186\pi$
$107$	7.05763	0.682287	0.341144	+	0.940011i	$0.389186\pi$
$108$	0	0
$109$	−1.88209	−0.180272	−0.0901358	−	0.995929i	$-0.528730\pi$
$109$	−1.88209	−0.180272	−0.0901358	+	0.995929i	$0.528730\pi$
$110$	0	0
$111$	−2.73669	−0.259755
$112$	0	0
$113$	7.59899	0.714853	0.357426	−	0.933941i	$-0.383654\pi$
$113$	7.59899	0.714853	0.357426	+	0.933941i	$0.383654\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.91223	−0.454136
$118$	0	0
$119$	−2.94237	−0.269726
$120$	0	0
$121$	−10.8821	−0.989281
$122$	0	0
$123$	5.96216	0.537590
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.3555	1.80626	0.903128	−	0.429372i	$-0.141265\pi$
$127$	20.3555	1.80626	0.903128	+	0.429372i	$0.141265\pi$
$128$	0	0
$129$	−2.30554	−0.202991
$130$	0	0
$131$	7.25561	0.633925	0.316963	−	0.948438i	$-0.397337\pi$
$131$	7.25561	0.633925	0.316963	+	0.948438i	$0.397337\pi$
$132$	0	0
$133$	0.656620	0.0569362
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.5310	1.15603	0.578016	−	0.816025i	$-0.303827\pi$
$137$	13.5310	1.15603	0.578016	+	0.816025i	$0.303827\pi$
$138$	0	0
$139$	−12.9243	−1.09623	−0.548113	−	0.836404i	$-0.684654\pi$
$139$	−12.9243	−1.09623	−0.548113	+	0.836404i	$0.684654\pi$
$140$	0	0
$141$	−2.23317	−0.188067
$142$	0	0
$143$	0.656620	0.0549094
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.31324	0.355750
$148$	0	0
$149$	−5.94237	−0.486818	−0.243409	−	0.969924i	$-0.578266\pi$
$149$	−5.94237	−0.486818	−0.243409	+	0.969924i	$0.578266\pi$
$150$	0	0
$151$	8.91223	0.725267	0.362633	−	0.931932i	$-0.381878\pi$
$151$	8.91223	0.725267	0.362633	+	0.931932i	$0.381878\pi$
$152$	0	0
$153$	11.5112	0.930627
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.9819	1.51492	0.757462	−	0.652879i	$-0.226439\pi$
$157$	18.9819	1.51492	0.757462	+	0.652879i	$0.226439\pi$
$158$	0	0
$159$	−1.10756	−0.0878353
$160$	0	0
$161$	−2.34338	−0.184684
$162$	0	0
$163$	−2.36317	−0.185098	−0.0925489	−	0.995708i	$-0.529501\pi$
$163$	−2.36317	−0.185098	−0.0925489	+	0.995708i	$0.529501\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.0801	1.32170	0.660848	−	0.750520i	$-0.270197\pi$
$167$	17.0801	1.32170	0.660848	+	0.750520i	$0.270197\pi$
$168$	0	0
$169$	−9.34338	−0.718722
$170$	0	0
$171$	−2.56885	−0.196445
$172$	0	0
$173$	−2.36581	−0.179870	−0.0899348	−	0.995948i	$-0.528666\pi$
$173$	−2.36581	−0.179870	−0.0899348	+	0.995948i	$0.528666\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.13770	−0.386173
$178$	0	0
$179$	8.47338	0.633330	0.316665	−	0.948537i	$-0.397437\pi$
$179$	8.47338	0.633330	0.316665	+	0.948537i	$0.397437\pi$
$180$	0	0
$181$	−22.5886	−1.67900	−0.839500	−	0.543359i	$-0.817152\pi$
$181$	−22.5886	−1.67900	−0.839500	+	0.543359i	$0.817152\pi$
$182$	0	0
$183$	8.15749	0.603019
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.53871	−0.112522
$188$	0	0
$189$	2.40101	0.174648
$190$	0	0
$191$	11.9320	0.863371	0.431685	−	0.902024i	$-0.357919\pi$
$191$	11.9320	0.863371	0.431685	+	0.902024i	$0.357919\pi$
$192$	0	0
$193$	20.6687	1.48777	0.743883	−	0.668310i	$-0.232982\pi$
$193$	20.6687	1.48777	0.743883	+	0.668310i	$0.232982\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.81675	−0.699415	−0.349707	−	0.936859i	$-0.613719\pi$
$197$	−9.81675	−0.699415	−0.349707	+	0.936859i	$0.613719\pi$
$198$	0	0
$199$	0.518921	0.0367853	0.0183927	−	0.999831i	$-0.494145\pi$
$199$	0.518921	0.0367853	0.0183927	+	0.999831i	$0.494145\pi$
$200$	0	0
$201$	−6.22547	−0.439111
$202$	0	0
$203$	5.24791	0.368331
$204$	0	0
$205$	0	0
$206$	0	0
$207$	9.16784	0.637209
$208$	0	0
$209$	0.343380	0.0237521
$210$	0	0
$211$	−10.4131	−0.716867	−0.358434	−	0.933555i	$-0.616689\pi$
$211$	−10.4131	−0.716867	−0.358434	+	0.933555i	$0.616689\pi$
$212$	0	0
$213$	−6.54136	−0.448206
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.76683	−0.255709
$218$	0	0
$219$	4.94501	0.334153
$220$	0	0
$221$	−8.56885	−0.576403
$222$	0	0
$223$	−8.25296	−0.552659	−0.276330	−	0.961063i	$-0.589118\pi$
$223$	−8.25296	−0.552659	−0.276330	+	0.961063i	$0.589118\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.393308	−0.0261048	−0.0130524	−	0.999915i	$-0.504155\pi$
$227$	−0.393308	−0.0261048	−0.0130524	+	0.999915i	$0.504155\pi$
$228$	0	0
$229$	15.2479	1.00761	0.503805	−	0.863817i	$-0.331933\pi$
$229$	15.2479	1.00761	0.503805	+	0.863817i	$0.331933\pi$
$230$	0	0
$231$	−0.148048	−0.00974086
$232$	0	0
$233$	23.3055	1.52680	0.763398	−	0.645928i	$-0.223529\pi$
$233$	23.3055	1.52680	0.763398	+	0.645928i	$0.223529\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−9.84425	−0.639453
$238$	0	0
$239$	−0.934664	−0.0604584	−0.0302292	−	0.999543i	$-0.509624\pi$
$239$	−0.934664	−0.0604584	−0.0302292	+	0.999543i	$0.509624\pi$
$240$	0	0
$241$	−14.2178	−0.915847	−0.457923	−	0.888992i	$-0.651407\pi$
$241$	−14.2178	−0.915847	−0.457923	+	0.888992i	$0.651407\pi$
$242$	0	0
$243$	−14.4536	−0.927198
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.91223	0.121672
$248$	0	0
$249$	7.16013	0.453755
$250$	0	0
$251$	−24.9518	−1.57494	−0.787472	−	0.616350i	$-0.788611\pi$
$251$	−24.9518	−1.57494	−0.787472	+	0.616350i	$0.788611\pi$
$252$	0	0
$253$	−1.22547	−0.0770446
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.8168	1.17376	0.586878	−	0.809675i	$-0.300357\pi$
$257$	18.8168	1.17376	0.586878	+	0.809675i	$0.300357\pi$
$258$	0	0
$259$	2.73669	0.170049
$260$	0	0
$261$	−20.5310	−1.27084
$262$	0	0
$263$	9.97251	0.614931	0.307466	−	0.951559i	$-0.400519\pi$
$263$	9.97251	0.614931	0.307466	+	0.951559i	$0.400519\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.75474	0.168587
$268$	0	0
$269$	17.3632	1.05865	0.529326	−	0.848419i	$-0.322445\pi$
$269$	17.3632	1.05865	0.529326	+	0.848419i	$0.322445\pi$
$270$	0	0
$271$	−0.228115	−0.0138570	−0.00692851	−	0.999976i	$-0.502205\pi$
$271$	−0.228115	−0.0138570	−0.00692851	+	0.999976i	$0.502205\pi$
$272$	0	0
$273$	−0.824458	−0.0498985
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.2530	0.616041	0.308020	−	0.951380i	$-0.400333\pi$
$277$	10.2530	0.616041	0.308020	+	0.951380i	$0.400333\pi$
$278$	0	0
$279$	14.7367	0.882262
$280$	0	0
$281$	13.2376	0.789686	0.394843	−	0.918749i	$-0.370799\pi$
$281$	13.2376	0.789686	0.394843	+	0.918749i	$0.370799\pi$
$282$	0	0
$283$	8.74439	0.519800	0.259900	−	0.965636i	$-0.416310\pi$
$283$	8.74439	0.519800	0.259900	+	0.965636i	$0.416310\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.96216	−0.351935
$288$	0	0
$289$	3.08007	0.181180
$290$	0	0
$291$	1.14034	0.0668482
$292$	0	0
$293$	10.8218	0.632217	0.316109	−	0.948723i	$-0.397624\pi$
$293$	10.8218	0.632217	0.316109	+	0.948723i	$0.397624\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.25561	0.0728578
$298$	0	0
$299$	−6.82446	−0.394669
$300$	0	0
$301$	2.30554	0.132889
$302$	0	0
$303$	−1.96480	−0.112875
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−3.68411	−0.210263	−0.105132	−	0.994458i	$-0.533526\pi$
$307$	−3.68411	−0.210263	−0.105132	+	0.994458i	$0.533526\pi$
$308$	0	0
$309$	2.60669	0.148290
$310$	0	0
$311$	4.65398	0.263903	0.131951	−	0.991256i	$-0.457876\pi$
$311$	4.65398	0.263903	0.131951	+	0.991256i	$0.457876\pi$
$312$	0	0
$313$	9.42080	0.532495	0.266248	−	0.963905i	$-0.414216\pi$
$313$	9.42080	0.532495	0.266248	+	0.963905i	$0.414216\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.25561	0.295184	0.147592	−	0.989048i	$-0.452848\pi$
$317$	5.25561	0.295184	0.147592	+	0.989048i	$0.452848\pi$
$318$	0	0
$319$	2.74439	0.153656
$320$	0	0
$321$	−4.63419	−0.258655
$322$	0	0
$323$	−4.48108	−0.249334
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.23582	0.0683409
$328$	0	0
$329$	2.23317	0.123119
$330$	0	0
$331$	17.4080	0.956832	0.478416	−	0.878133i	$-0.341211\pi$
$331$	17.4080	0.956832	0.478416	+	0.878133i	$0.341211\pi$
$332$	0	0
$333$	−10.7065	−0.586715
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.5836	−0.685471	−0.342736	−	0.939432i	$-0.611354\pi$
$337$	−12.5836	−0.685471	−0.342736	+	0.939432i	$0.611354\pi$
$338$	0	0
$339$	−4.98965	−0.271001
$340$	0	0
$341$	−1.96986	−0.106674
$342$	0	0
$343$	−8.90958	−0.481072
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.5259	1.79977	0.899884	−	0.436130i	$-0.143651\pi$
$347$	33.5259	1.79977	0.899884	+	0.436130i	$0.143651\pi$
$348$	0	0
$349$	−18.0948	−0.968592	−0.484296	−	0.874904i	$-0.660924\pi$
$349$	−18.0948	−0.968592	−0.484296	+	0.874904i	$0.660924\pi$
$350$	0	0
$351$	6.99230	0.373221
$352$	0	0
$353$	9.88979	0.526381	0.263190	−	0.964744i	$-0.415225\pi$
$353$	9.88979	0.526381	0.263190	+	0.964744i	$0.415225\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.93202	0.102253
$358$	0	0
$359$	31.0895	1.64084	0.820421	−	0.571760i	$-0.193739\pi$
$359$	31.0895	1.64084	0.820421	+	0.571760i	$0.193739\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.14540	0.375036
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.6911	−0.714672	−0.357336	−	0.933976i	$-0.616315\pi$
$367$	−13.6911	−0.714672	−0.357336	+	0.933976i	$0.616315\pi$
$368$	0	0
$369$	23.3253	1.21427
$370$	0	0
$371$	1.10756	0.0575017
$372$	0	0
$373$	13.5062	0.699322	0.349661	−	0.936876i	$-0.386297\pi$
$373$	13.5062	0.699322	0.349661	+	0.936876i	$0.386297\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.2831	0.787120
$378$	0	0
$379$	17.9122	0.920089	0.460045	−	0.887896i	$-0.347833\pi$
$379$	17.9122	0.920089	0.460045	+	0.887896i	$0.347833\pi$
$380$	0	0
$381$	−13.3658	−0.684751
$382$	0	0
$383$	1.73669	0.0887406	0.0443703	−	0.999015i	$-0.485872\pi$
$383$	1.73669	0.0887406	0.0443703	+	0.999015i	$0.485872\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.01979	−0.458502
$388$	0	0
$389$	−11.0396	−0.559729	−0.279864	−	0.960040i	$-0.590290\pi$
$389$	−11.0396	−0.559729	−0.279864	+	0.960040i	$0.590290\pi$
$390$	0	0
$391$	15.9923	0.808765
$392$	0	0
$393$	−4.76418	−0.240321
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.6214	1.33609	0.668045	−	0.744120i	$-0.267131\pi$
$397$	26.6214	1.33609	0.668045	+	0.744120i	$0.267131\pi$
$398$	0	0
$399$	−0.431150	−0.0215845
$400$	0	0
$401$	0.771885	0.0385461	0.0192730	−	0.999814i	$-0.493865\pi$
$401$	0.771885	0.0385461	0.0192730	+	0.999814i	$0.493865\pi$
$402$	0	0
$403$	−10.9699	−0.546448
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.43115	0.0709395
$408$	0	0
$409$	−12.1403	−0.600301	−0.300151	−	0.953892i	$-0.597037\pi$
$409$	−12.1403	−0.600301	−0.300151	+	0.953892i	$0.597037\pi$
$410$	0	0
$411$	−8.88474	−0.438252
$412$	0	0
$413$	5.13770	0.252810
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.48637	0.415579
$418$	0	0
$419$	14.3253	0.699838	0.349919	−	0.936780i	$-0.386209\pi$
$419$	14.3253	0.699838	0.349919	+	0.936780i	$0.386209\pi$
$420$	0	0
$421$	0.759123	0.0369974	0.0184987	−	0.999829i	$-0.494111\pi$
$421$	0.759123	0.0369974	0.0184987	+	0.999829i	$0.494111\pi$
$422$	0	0
$423$	−8.73669	−0.424792
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−8.15749	−0.394769
$428$	0	0
$429$	−0.431150	−0.0208161
$430$	0	0
$431$	−35.7411	−1.72159	−0.860793	−	0.508955i	$-0.830032\pi$
$431$	−35.7411	−1.72159	−0.860793	+	0.508955i	$0.830032\pi$
$432$	0	0
$433$	−35.2024	−1.69172	−0.845859	−	0.533407i	$-0.820911\pi$
$433$	−35.2024	−1.69172	−0.845859	+	0.533407i	$0.820911\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.56885	−0.170721
$438$	0	0
$439$	21.4208	1.02236	0.511180	−	0.859474i	$-0.329209\pi$
$439$	21.4208	1.02236	0.511180	+	0.859474i	$0.329209\pi$
$440$	0	0
$441$	16.8744	0.803542
$442$	0	0
$443$	36.1421	1.71716	0.858581	−	0.512678i	$-0.171346\pi$
$443$	36.1421	1.71716	0.858581	+	0.512678i	$0.171346\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.90188	0.184553
$448$	0	0
$449$	21.7917	1.02841	0.514206	−	0.857667i	$-0.328087\pi$
$449$	21.7917	1.02841	0.514206	+	0.857667i	$0.328087\pi$
$450$	0	0
$451$	−3.11791	−0.146817
$452$	0	0
$453$	−5.85195	−0.274949
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.7143	0.501192	0.250596	−	0.968092i	$-0.419373\pi$
$457$	10.7143	0.501192	0.250596	+	0.968092i	$0.419373\pi$
$458$	0	0
$459$	−16.3856	−0.764815
$460$	0	0
$461$	−3.81940	−0.177887	−0.0889436	−	0.996037i	$-0.528349\pi$
$461$	−3.81940	−0.177887	−0.0889436	+	0.996037i	$0.528349\pi$
$462$	0	0
$463$	26.5611	1.23440	0.617201	−	0.786806i	$-0.288267\pi$
$463$	26.5611	1.23440	0.617201	+	0.786806i	$0.288267\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.6533	1.14082	0.570409	−	0.821361i	$-0.306785\pi$
$467$	24.6533	1.14082	0.570409	+	0.821361i	$0.306785\pi$
$468$	0	0
$469$	6.22547	0.287465
$470$	0	0
$471$	−12.4639	−0.574308
$472$	0	0
$473$	1.20568	0.0554372
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.33303	−0.198396
$478$	0	0
$479$	−19.8693	−0.907853	−0.453926	−	0.891039i	$-0.649977\pi$
$479$	−19.8693	−0.907853	−0.453926	+	0.891039i	$0.649977\pi$
$480$	0	0
$481$	7.96986	0.363394
$482$	0	0
$483$	1.53871	0.0700138
$484$	0	0
$485$	0	0
$486$	0	0
$487$	27.2952	1.23686	0.618432	−	0.785839i	$-0.287768\pi$
$487$	27.2952	1.23686	0.618432	+	0.785839i	$0.287768\pi$
$488$	0	0
$489$	1.55171	0.0701705
$490$	0	0
$491$	2.62913	0.118651	0.0593254	−	0.998239i	$-0.481105\pi$
$491$	2.62913	0.118651	0.0593254	+	0.998239i	$0.481105\pi$
$492$	0	0
$493$	−35.8141	−1.61299
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.54136	0.293420
$498$	0	0
$499$	7.28310	0.326036	0.163018	−	0.986623i	$-0.447877\pi$
$499$	7.28310	0.326036	0.163018	+	0.986623i	$0.447877\pi$
$500$	0	0
$501$	−11.2151	−0.501055
$502$	0	0
$503$	−35.8185	−1.59707	−0.798534	−	0.601950i	$-0.794391\pi$
$503$	−35.8185	−1.59707	−0.798534	+	0.601950i	$0.794391\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.13505	0.272467
$508$	0	0
$509$	41.9819	1.86082	0.930409	−	0.366524i	$-0.119452\pi$
$509$	41.9819	1.86082	0.930409	+	0.366524i	$0.119452\pi$
$510$	0	0
$511$	−4.94501	−0.218755
$512$	0	0
$513$	3.65662	0.161444
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.16784	0.0513615
$518$	0	0
$519$	1.55344	0.0681885
$520$	0	0
$521$	−2.72634	−0.119443	−0.0597215	−	0.998215i	$-0.519021\pi$
$521$	−2.72634	−0.119443	−0.0597215	+	0.998215i	$0.519021\pi$
$522$	0	0
$523$	13.8493	0.605588	0.302794	−	0.953056i	$-0.402081\pi$
$523$	13.8493	0.605588	0.302794	+	0.953056i	$0.402081\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.7065	1.11979
$528$	0	0
$529$	−10.2633	−0.446231
$530$	0	0
$531$	−20.0999	−0.872259
$532$	0	0
$533$	−17.3632	−0.752082
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.56379	−0.240095
$538$	0	0
$539$	−2.25561	−0.0971559
$540$	0	0
$541$	12.4982	0.537341	0.268670	−	0.963232i	$-0.413416\pi$
$541$	12.4982	0.537341	0.268670	+	0.963232i	$0.413416\pi$
$542$	0	0
$543$	14.8322	0.636509
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−37.5836	−1.60696	−0.803479	−	0.595333i	$-0.797020\pi$
$547$	−37.5836	−1.60696	−0.803479	+	0.595333i	$0.797020\pi$
$548$	0	0
$549$	31.9140	1.36205
$550$	0	0
$551$	7.99230	0.340483
$552$	0	0
$553$	9.84425	0.418620
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.7943	−0.499741	−0.249871	−	0.968279i	$-0.580388\pi$
$557$	−11.7943	−0.499741	−0.249871	+	0.968279i	$0.580388\pi$
$558$	0	0
$559$	6.71425	0.283983
$560$	0	0
$561$	1.01035	0.0426570
$562$	0	0
$563$	−4.62407	−0.194881	−0.0974406	−	0.995241i	$-0.531066\pi$
$563$	−4.62407	−0.194881	−0.0974406	+	0.995241i	$0.531066\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.48372	0.146303
$568$	0	0
$569$	−8.92258	−0.374054	−0.187027	−	0.982355i	$-0.559885\pi$
$569$	−8.92258	−0.374054	−0.187027	+	0.982355i	$0.559885\pi$
$570$	0	0
$571$	−20.1025	−0.841264	−0.420632	−	0.907231i	$-0.638192\pi$
$571$	−20.1025	−0.841264	−0.420632	+	0.907231i	$0.638192\pi$
$572$	0	0
$573$	−7.83481	−0.327304
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.8770	0.619339	0.309669	−	0.950844i	$-0.399782\pi$
$577$	14.8770	0.619339	0.309669	+	0.950844i	$0.399782\pi$
$578$	0	0
$579$	−13.5715	−0.564012
$580$	0	0
$581$	−7.16013	−0.297052
$582$	0	0
$583$	0.579199	0.0239880
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.3280	−0.839025	−0.419513	−	0.907750i	$-0.637799\pi$
$587$	−20.3280	−0.839025	−0.419513	+	0.907750i	$0.637799\pi$
$588$	0	0
$589$	−5.73669	−0.236376
$590$	0	0
$591$	6.44588	0.265148
$592$	0	0
$593$	26.9294	1.10586	0.552928	−	0.833229i	$-0.313510\pi$
$593$	26.9294	1.10586	0.552928	+	0.833229i	$0.313510\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.340734	−0.0139453
$598$	0	0
$599$	−14.4586	−0.590764	−0.295382	−	0.955379i	$-0.595447\pi$
$599$	−14.4586	−0.590764	−0.295382	+	0.955379i	$0.595447\pi$
$600$	0	0
$601$	11.0955	0.452594	0.226297	−	0.974058i	$-0.427338\pi$
$601$	11.0955	0.452594	0.226297	+	0.974058i	$0.427338\pi$
$602$	0	0
$603$	−24.3555	−0.991831
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.78397	0.356530	0.178265	−	0.983982i	$-0.442952\pi$
$607$	8.78397	0.356530	0.178265	+	0.983982i	$0.442952\pi$
$608$	0	0
$609$	−3.44588	−0.139634
$610$	0	0
$611$	6.50351	0.263104
$612$	0	0
$613$	−12.1705	−0.491561	−0.245781	−	0.969325i	$-0.579044\pi$
$613$	−12.1705	−0.491561	−0.245781	+	0.969325i	$0.579044\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.09986	−0.326088	−0.163044	−	0.986619i	$-0.552131\pi$
$617$	−8.09986	−0.326088	−0.163044	+	0.986619i	$0.552131\pi$
$618$	0	0
$619$	36.4932	1.46678	0.733392	−	0.679806i	$-0.237936\pi$
$619$	36.4932	1.46678	0.733392	+	0.679806i	$0.237936\pi$
$620$	0	0
$621$	−13.0499	−0.523676
$622$	0	0
$623$	−2.75474	−0.110366
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.225470	−0.00900441
$628$	0	0
$629$	−18.6764	−0.744677
$630$	0	0
$631$	−31.8614	−1.26838	−0.634191	−	0.773176i	$-0.718667\pi$
$631$	−31.8614	−1.26838	−0.634191	+	0.773176i	$0.718667\pi$
$632$	0	0
$633$	6.83745	0.271764
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−12.5611	−0.497691
$638$	0	0
$639$	−25.5913	−1.01238
$640$	0	0
$641$	45.4905	1.79677	0.898384	−	0.439211i	$-0.144742\pi$
$641$	45.4905	1.79677	0.898384	+	0.439211i	$0.144742\pi$
$642$	0	0
$643$	14.3511	0.565951	0.282976	−	0.959127i	$-0.408678\pi$
$643$	14.3511	0.565951	0.282976	+	0.959127i	$0.408678\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.5139	−0.688541	−0.344270	−	0.938871i	$-0.611874\pi$
$647$	−17.5139	−0.688541	−0.344270	+	0.938871i	$0.611874\pi$
$648$	0	0
$649$	2.68676	0.105465
$650$	0	0
$651$	2.47338	0.0969392
$652$	0	0
$653$	−2.72196	−0.106518	−0.0532592	−	0.998581i	$-0.516961\pi$
$653$	−2.72196	−0.106518	−0.0532592	+	0.998581i	$0.516961\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	19.3460	0.754760
$658$	0	0
$659$	−16.6060	−0.646879	−0.323439	−	0.946249i	$-0.604839\pi$
$659$	−16.6060	−0.646879	−0.323439	+	0.946249i	$0.604839\pi$
$660$	0	0
$661$	13.4630	0.523651	0.261826	−	0.965115i	$-0.415676\pi$
$661$	13.4630	0.523651	0.261826	+	0.965115i	$0.415676\pi$
$662$	0	0
$663$	5.62648	0.218514
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−28.5233	−1.10443
$668$	0	0
$669$	5.41906	0.209513
$670$	0	0
$671$	−4.26596	−0.164685
$672$	0	0
$673$	40.5706	1.56388	0.781941	−	0.623353i	$-0.214230\pi$
$673$	40.5706	1.56388	0.781941	+	0.623353i	$0.214230\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.4586	−0.555691	−0.277845	−	0.960626i	$-0.589620\pi$
$677$	−14.4586	−0.555691	−0.277845	+	0.960626i	$0.589620\pi$
$678$	0	0
$679$	−1.14034	−0.0437624
$680$	0	0
$681$	0.258254	0.00989632
$682$	0	0
$683$	34.3605	1.31477	0.657384	−	0.753555i	$-0.271663\pi$
$683$	34.3605	1.31477	0.657384	+	0.753555i	$0.271663\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10.0121	−0.381985
$688$	0	0
$689$	3.22547	0.122881
$690$	0	0
$691$	−13.6489	−0.519229	−0.259615	−	0.965712i	$-0.583596\pi$
$691$	−13.6489	−0.519229	−0.259615	+	0.965712i	$0.583596\pi$
$692$	0	0
$693$	−0.579199	−0.0220019
$694$	0	0
$695$	0	0
$696$	0	0
$697$	40.6885	1.54119
$698$	0	0
$699$	−15.3029	−0.578809
$700$	0	0
$701$	3.88450	0.146716	0.0733578	−	0.997306i	$-0.476628\pi$
$701$	3.88450	0.146716	0.0733578	+	0.997306i	$0.476628\pi$
$702$	0	0
$703$	4.16784	0.157193
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.96480	0.0738940
$708$	0	0
$709$	13.3460	0.501220	0.250610	−	0.968088i	$-0.419369\pi$
$709$	13.3460	0.501220	0.250610	+	0.968088i	$0.419369\pi$
$710$	0	0
$711$	−38.5130	−1.44435
$712$	0	0
$713$	20.4734	0.766734
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.613720	0.0229198
$718$	0	0
$719$	−0.538711	−0.0200905	−0.0100453	−	0.999950i	$-0.503198\pi$
$719$	−0.538711	−0.0200905	−0.0100453	+	0.999950i	$0.503198\pi$
$720$	0	0
$721$	−2.60669	−0.0970783
$722$	0	0
$723$	9.33568	0.347198
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−27.8392	−1.03250	−0.516249	−	0.856438i	$-0.672672\pi$
$727$	−27.8392	−1.03250	−0.516249	+	0.856438i	$0.672672\pi$
$728$	0	0
$729$	−6.42609	−0.238003
$730$	0	0
$731$	−15.7340	−0.581945
$732$	0	0
$733$	−17.1755	−0.634393	−0.317197	−	0.948360i	$-0.602741\pi$
$733$	−17.1755	−0.634393	−0.317197	+	0.948360i	$0.602741\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.25561	0.119922
$738$	0	0
$739$	32.7917	1.20626	0.603131	−	0.797642i	$-0.293920\pi$
$739$	32.7917	1.20626	0.603131	+	0.797642i	$0.293920\pi$
$740$	0	0
$741$	−1.25561	−0.0461259
$742$	0	0
$743$	44.1095	1.61822	0.809111	−	0.587656i	$-0.199949\pi$
$743$	44.1095	1.61822	0.809111	+	0.587656i	$0.199949\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	28.0121	1.02491
$748$	0	0
$749$	4.63419	0.169329
$750$	0	0
$751$	13.5688	0.495134	0.247567	−	0.968871i	$-0.420369\pi$
$751$	13.5688	0.495134	0.247567	+	0.968871i	$0.420369\pi$
$752$	0	0
$753$	16.3839	0.597061
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29.7444	1.08108	0.540539	−	0.841319i	$-0.318220\pi$
$757$	29.7444	1.08108	0.540539	+	0.841319i	$0.318220\pi$
$758$	0	0
$759$	0.804669	0.0292076
$760$	0	0
$761$	36.1575	1.31071	0.655354	−	0.755322i	$-0.272519\pi$
$761$	36.1575	1.31071	0.655354	+	0.755322i	$0.272519\pi$
$762$	0	0
$763$	−1.23582	−0.0447397
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.9622	0.540252
$768$	0	0
$769$	−6.46635	−0.233182	−0.116591	−	0.993180i	$-0.537197\pi$
$769$	−6.46635	−0.233182	−0.116591	+	0.993180i	$0.537197\pi$
$770$	0	0
$771$	−12.3555	−0.444971
$772$	0	0
$773$	−27.2780	−0.981123	−0.490562	−	0.871407i	$-0.663208\pi$
$773$	−27.2780	−0.981123	−0.490562	+	0.871407i	$0.663208\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.79696	−0.0644658
$778$	0	0
$779$	−9.08007	−0.325327
$780$	0	0
$781$	3.42080	0.122406
$782$	0	0
$783$	29.2248	1.04441
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.3926	1.22596	0.612982	−	0.790097i	$-0.289970\pi$
$787$	34.3926	1.22596	0.612982	+	0.790097i	$0.289970\pi$
$788$	0	0
$789$	−6.54815	−0.233120
$790$	0	0
$791$	4.98965	0.177412
$792$	0	0
$793$	−23.7565	−0.843617
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.7307	−1.05312	−0.526558	−	0.850139i	$-0.676518\pi$
$797$	−29.7307	−1.05312	−0.526558	+	0.850139i	$0.676518\pi$
$798$	0	0
$799$	−15.2402	−0.539160
$800$	0	0
$801$	10.7772	0.380793
$802$	0	0
$803$	−2.58599	−0.0912577
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11.4010	−0.401335
$808$	0	0
$809$	−11.0653	−0.389036	−0.194518	−	0.980899i	$-0.562314\pi$
$809$	−11.0653	−0.389036	−0.194518	+	0.980899i	$0.562314\pi$
$810$	0	0
$811$	15.5079	0.544556	0.272278	−	0.962219i	$-0.412223\pi$
$811$	15.5079	0.544556	0.272278	+	0.962219i	$0.412223\pi$
$812$	0	0
$813$	0.149785	0.00525320
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.51122	0.122842
$818$	0	0
$819$	−3.22547	−0.112707
$820$	0	0
$821$	24.5534	0.856921	0.428461	−	0.903561i	$-0.359056\pi$
$821$	24.5534	0.856921	0.428461	+	0.903561i	$0.359056\pi$
$822$	0	0
$823$	−51.1568	−1.78321	−0.891607	−	0.452810i	$-0.850422\pi$
$823$	−51.1568	−1.78321	−0.891607	+	0.452810i	$0.850422\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.4613	0.363775	0.181887	−	0.983319i	$-0.441779\pi$
$827$	10.4613	0.363775	0.181887	+	0.983319i	$0.441779\pi$
$828$	0	0
$829$	46.6661	1.62078	0.810390	−	0.585891i	$-0.199255\pi$
$829$	46.6661	1.62078	0.810390	+	0.585891i	$0.199255\pi$
$830$	0	0
$831$	−6.73231	−0.233541
$832$	0	0
$833$	29.4355	1.01988
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−20.9769	−0.725067
$838$	0	0
$839$	1.38560	0.0478364	0.0239182	−	0.999714i	$-0.492386\pi$
$839$	1.38560	0.0478364	0.0239182	+	0.999714i	$0.492386\pi$
$840$	0	0
$841$	34.8768	1.20265
$842$	0	0
$843$	−8.69205	−0.299370
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7.14540	−0.245519
$848$	0	0
$849$	−5.74175	−0.197056
$850$	0	0
$851$	−14.8744	−0.509887
$852$	0	0
$853$	−50.1076	−1.71565	−0.857825	−	0.513942i	$-0.828185\pi$
$853$	−50.1076	−1.71565	−0.857825	+	0.513942i	$0.828185\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.8915	1.05523	0.527617	−	0.849482i	$-0.323086\pi$
$857$	30.8915	1.05523	0.527617	+	0.849482i	$0.323086\pi$
$858$	0	0
$859$	55.4949	1.89346	0.946731	−	0.322026i	$-0.104364\pi$
$859$	55.4949	1.89346	0.946731	+	0.322026i	$0.104364\pi$
$860$	0	0
$861$	3.91487	0.133419
$862$	0	0
$863$	23.9622	0.815681	0.407841	−	0.913053i	$-0.366282\pi$
$863$	23.9622	0.815681	0.407841	+	0.913053i	$0.366282\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.02243	−0.0686855
$868$	0	0
$869$	5.14805	0.174636
$870$	0	0
$871$	18.1300	0.614311
$872$	0	0
$873$	4.46129	0.150992
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−41.2221	−1.39197	−0.695987	−	0.718055i	$-0.745033\pi$
$877$	−41.2221	−1.39197	−0.695987	+	0.718055i	$0.745033\pi$
$878$	0	0
$879$	−7.10582	−0.239673
$880$	0	0
$881$	−14.8064	−0.498840	−0.249420	−	0.968395i	$-0.580240\pi$
$881$	−14.8064	−0.498840	−0.249420	+	0.968395i	$0.580240\pi$
$882$	0	0
$883$	44.2747	1.48996	0.744982	−	0.667085i	$-0.232458\pi$
$883$	44.2747	1.48996	0.744982	+	0.667085i	$0.232458\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−33.8392	−1.13621	−0.568104	−	0.822956i	$-0.692323\pi$
$887$	−33.8392	−1.13621	−0.568104	+	0.822956i	$0.692323\pi$
$888$	0	0
$889$	13.3658	0.448275
$890$	0	0
$891$	1.82181	0.0610330
$892$	0	0
$893$	3.40101	0.113811
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.48108	0.149619
$898$	0	0
$899$	−45.8493	−1.52916
$900$	0	0
$901$	−7.55850	−0.251810
$902$	0	0
$903$	−1.51386	−0.0503782
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−23.4536	−0.778764	−0.389382	−	0.921076i	$-0.627311\pi$
$907$	−23.4536	−0.778764	−0.389382	+	0.921076i	$0.627311\pi$
$908$	0	0
$909$	−7.68676	−0.254954
$910$	0	0
$911$	31.3099	1.03734	0.518672	−	0.854973i	$-0.326427\pi$
$911$	31.3099	1.03734	0.518672	+	0.854973i	$0.326427\pi$
$912$	0	0
$913$	−3.74439	−0.123921
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.76418	0.157327
$918$	0	0
$919$	−22.4252	−0.739739	−0.369869	−	0.929084i	$-0.620598\pi$
$919$	−22.4252	−0.739739	−0.369869	+	0.929084i	$0.620598\pi$
$920$	0	0
$921$	2.41906	0.0797109
$922$	0	0
$923$	19.0499	0.627036
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.1980	0.334945
$928$	0	0
$929$	−45.4606	−1.49151	−0.745757	−	0.666218i	$-0.767912\pi$
$929$	−45.4606	−1.49151	−0.745757	+	0.666218i	$0.767912\pi$
$930$	0	0
$931$	−6.56885	−0.215285
$932$	0	0
$933$	−3.05590	−0.100046
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.16784	0.234163	0.117082	−	0.993122i	$-0.462646\pi$
$937$	7.16784	0.234163	0.117082	+	0.993122i	$0.462646\pi$
$938$	0	0
$939$	−6.18589	−0.201869
$940$	0	0
$941$	30.5983	0.997476	0.498738	−	0.866753i	$-0.333797\pi$
$941$	30.5983	0.997476	0.498738	+	0.866753i	$0.333797\pi$
$942$	0	0
$943$	32.4054	1.05526
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−58.2195	−1.89188	−0.945940	−	0.324342i	$-0.894857\pi$
$947$	−58.2195	−1.89188	−0.945940	+	0.324342i	$0.894857\pi$
$948$	0	0
$949$	−14.4010	−0.467477
$950$	0	0
$951$	−3.45094	−0.111904
$952$	0	0
$953$	−49.3247	−1.59778	−0.798891	−	0.601476i	$-0.794580\pi$
$953$	−49.3247	−1.59778	−0.798891	+	0.601476i	$0.794580\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.80202	−0.0582511
$958$	0	0
$959$	8.88474	0.286903
$960$	0	0
$961$	1.90958	0.0615995
$962$	0	0
$963$	−18.1300	−0.584231
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−28.6368	−0.920898	−0.460449	−	0.887686i	$-0.652312\pi$
$967$	−28.6368	−0.920898	−0.460449	+	0.887686i	$0.652312\pi$
$968$	0	0
$969$	2.94237	0.0945225
$970$	0	0
$971$	11.4131	0.366264	0.183132	−	0.983088i	$-0.441376\pi$
$971$	11.4131	0.366264	0.183132	+	0.983088i	$0.441376\pi$
$972$	0	0
$973$	−8.48637	−0.272061
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.2567	1.67184	0.835919	−	0.548852i	$-0.184935\pi$
$977$	52.2567	1.67184	0.835919	+	0.548852i	$0.184935\pi$
$978$	0	0
$979$	−1.44059	−0.0460415
$980$	0	0
$981$	4.83481	0.154364
$982$	0	0
$983$	−35.5904	−1.13516	−0.567578	−	0.823319i	$-0.692120\pi$
$983$	−35.5904	−1.13516	−0.567578	+	0.823319i	$0.692120\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.46635	−0.0466744
$988$	0	0
$989$	−12.5310	−0.398463
$990$	0	0
$991$	−35.2952	−1.12119	−0.560594	−	0.828091i	$-0.689427\pi$
$991$	−35.2952	−1.12119	−0.560594	+	0.828091i	$0.689427\pi$
$992$	0	0
$993$	−11.4305	−0.362735
$994$	0	0
$995$	0	0
$996$	0	0
$997$	7.45359	0.236057	0.118029	−	0.993010i	$-0.462343\pi$
$997$	7.45359	0.236057	0.118029	+	0.993010i	$0.462343\pi$
$998$	0	0
$999$	15.2402	0.482179

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bt.1.2		3
4.3	odd	2		3800.2.a.u.1.2	✓	3
5.4	even	2		7600.2.a.bu.1.2		3
20.3	even	4		3800.2.d.m.3649.4		6
20.7	even	4		3800.2.d.m.3649.3		6
20.19	odd	2		3800.2.a.v.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.u.1.2	✓	3	4.3	odd	2
3800.2.a.v.1.2	yes	3	20.19	odd	2
3800.2.d.m.3649.3		6	20.7	even	4
3800.2.d.m.3649.4		6	20.3	even	4
7600.2.a.bt.1.2		3	1.1	even	1	trivial
7600.2.a.bu.1.2		3	5.4	even	2

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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-0.656620 q^{3} +0.656620 q^{7} -2.56885 q^{9} +O(q^{10})\)	\(q-0.656620 q^{3} +0.656620 q^{7} -2.56885 q^{9} +0.343380 q^{11} +1.91223 q^{13} -4.48108 q^{17} +1.00000 q^{19} -0.431150 q^{21} -3.56885 q^{23} +3.65662 q^{27} +7.99230 q^{29} -5.73669 q^{31} -0.225470 q^{33} +4.16784 q^{37} -1.25561 q^{39} -9.08007 q^{41} +3.51122 q^{43} +3.40101 q^{47} -6.56885 q^{49} +2.94237 q^{51} +1.68676 q^{53} -0.656620 q^{57} +7.82446 q^{59} -12.4234 q^{61} -1.68676 q^{63} +9.48108 q^{67} +2.34338 q^{69} +9.96216 q^{71} -7.53101 q^{73} +0.225470 q^{77} +14.9923 q^{79} +5.30554 q^{81} -10.9045 q^{83} -5.24791 q^{87} -4.19533 q^{89} +1.25561 q^{91} +3.76683 q^{93} -1.73669 q^{97} -0.882090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{9}+O(q^{10}) \) 3 * q + q^9	\( 3 q + q^{9} + 3 q^{11} - q^{13} + 2 q^{17} + 3 q^{19} - 10 q^{21} - 2 q^{23} + 9 q^{27} - q^{29} + 3 q^{31} + 10 q^{33} + q^{37} + q^{39} - 9 q^{41} + q^{43} + 13 q^{47} - 11 q^{49} + 8 q^{51} + 9 q^{53} + 10 q^{59} - 21 q^{61} - 9 q^{63} + 13 q^{67} + 9 q^{69} - q^{71} + 17 q^{73} - 10 q^{77} + 20 q^{79} - 13 q^{81} - q^{83} + 14 q^{87} + 4 q^{89} - q^{91} - 3 q^{93} + 15 q^{97} + 10 q^{99}+O(q^{100}) \) 3 * q + q^9 + 3 * q^11 - q^13 + 2 * q^17 + 3 * q^19 - 10 * q^21 - 2 * q^23 + 9 * q^27 - q^29 + 3 * q^31 + 10 * q^33 + q^37 + q^39 - 9 * q^41 + q^43 + 13 * q^47 - 11 * q^49 + 8 * q^51 + 9 * q^53 + 10 * q^59 - 21 * q^61 - 9 * q^63 + 13 * q^67 + 9 * q^69 - q^71 + 17 * q^73 - 10 * q^77 + 20 * q^79 - 13 * q^81 - q^83 + 14 * q^87 + 4 * q^89 - q^91 - 3 * q^93 + 15 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

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Database

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