LMFDB - Embedded newform 3800.2.d.m.3649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.4227136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9x^4 + 22x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.4
Root			$1.91223i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.m.3649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.656620i q^{3} +0.656620i q^{7} +2.56885 q^{9} -0.343380 q^{11} +1.91223i q^{13} +4.48108i q^{17} +1.00000 q^{19} -0.431150 q^{21} +3.56885i q^{23} +3.65662i q^{27} -7.99230 q^{29} +5.73669 q^{31} -0.225470i q^{33} -4.16784i q^{37} -1.25561 q^{39} -9.08007 q^{41} -3.51122i q^{43} +3.40101i q^{47} +6.56885 q^{49} -2.94237 q^{51} +1.68676i q^{53} +0.656620i q^{57} +7.82446 q^{59} -12.4234 q^{61} +1.68676i q^{63} +9.48108i q^{67} -2.34338 q^{69} -9.96216 q^{71} -7.53101i q^{73} -0.225470i q^{77} +14.9923 q^{79} +5.30554 q^{81} +10.9045i q^{83} -5.24791i q^{87} +4.19533 q^{89} -1.25561 q^{91} +3.76683i q^{93} +1.73669i q^{97} -0.882090 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{9} - 6 q^{11} + 6 q^{19} - 20 q^{21} + 2 q^{29} - 6 q^{31} + 2 q^{39} - 18 q^{41} + 22 q^{49} - 16 q^{51} + 20 q^{59} - 42 q^{61} - 18 q^{69} + 2 q^{71} + 40 q^{79} - 26 q^{81} - 8 q^{89} + 2 q^{91}+ \cdots + 20 q^{99}+O(q^{100}) $ 6 * q - 2 * q^9 - 6 * q^11 + 6 * q^19 - 20 * q^21 + 2 * q^29 - 6 * q^31 + 2 * q^39 - 18 * q^41 + 22 * q^49 - 16 * q^51 + 20 * q^59 - 42 * q^61 - 18 * q^69 + 2 * q^71 + 40 * q^79 - 26 * q^81 - 8 * q^89 + 2 * q^91 + 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.656620i	0.379100i	0.981871	+	0.189550i	$0.0607029\pi$
$3$	0.656620i	0.379100i	−0.981871	+	0.189550i	$0.939297\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.656620i	0.248179i	0.992271	+	0.124090i	$0.0396010\pi$
$7$	0.656620i	0.248179i	−0.992271	+	0.124090i	$0.960399\pi$
$8$	0	0
$9$	2.56885	0.856283
$10$	0	0
$11$	−0.343380	−0.103533	−0.0517664	−	0.998659i	$-0.516485\pi$
$11$	−0.343380	−0.103533	−0.0517664	+	0.998659i	$0.516485\pi$
$12$	0	0
$13$	1.91223i	0.530357i	0.964199	+	0.265178i	$0.0854309\pi$
$13$	1.91223i	0.530357i	−0.964199	+	0.265178i	$0.914569\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.48108i	1.08682i	0.839467	+	0.543411i	$0.182867\pi$
$17$	4.48108i	1.08682i	−0.839467	+	0.543411i	$0.817133\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.431150	−0.0940847
$22$	0	0
$23$	3.56885i	0.744157i	0.928201	+	0.372078i	$0.121355\pi$
$23$	3.56885i	0.744157i	−0.928201	+	0.372078i	$0.878645\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.65662i	0.703717i
$28$	0	0
$29$	−7.99230	−1.48413	−0.742066	−	0.670327i	$-0.766154\pi$
$29$	−7.99230	−1.48413	−0.742066	+	0.670327i	$0.766154\pi$
$30$	0	0
$31$	5.73669	1.03034	0.515170	−	0.857088i	$-0.327729\pi$
$31$	5.73669	1.03034	0.515170	+	0.857088i	$0.327729\pi$
$32$	0	0
$33$	− 0.225470i	− 0.0392493i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 4.16784i	− 0.685188i	−0.939483	−	0.342594i	$-0.888694\pi$
$37$	− 4.16784i	− 0.685188i	0.939483	−	0.342594i	$-0.111306\pi$
$38$	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
$43$	− 3.51122i	− 0.535456i	−0.963495	−	0.267728i	$-0.913727\pi$
$43$	− 3.51122i	− 0.535456i	0.963495	−	0.267728i	$-0.0862728\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.40101i	0.496089i	0.968749	+	0.248044i	$0.0797878\pi$
$47$	3.40101i	0.496089i	−0.968749	+	0.248044i	$0.920212\pi$
$48$	0	0
$49$	6.56885	0.938407
$50$	0	0
$51$	−2.94237	−0.412014
$52$	0	0
$53$	1.68676i	0.231694i	0.993267	+	0.115847i	$0.0369583\pi$
$53$	1.68676i	0.231694i	−0.993267	+	0.115847i	$0.963042\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.656620i	0.0869715i
$58$	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
$63$	1.68676i	0.212512i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.48108i	1.15830i	0.815222	+	0.579149i	$0.196615\pi$
$67$	9.48108i	1.15830i	−0.815222	+	0.579149i	$0.803385\pi$
$68$	0	0
$69$	−2.34338	−0.282110
$70$	0	0
$71$	−9.96216	−1.18229	−0.591145	−	0.806565i	$-0.701324\pi$
$71$	−9.96216	−1.18229	−0.591145	+	0.806565i	$0.701324\pi$
$72$	0	0
$73$	− 7.53101i	− 0.881438i	−0.897645	−	0.440719i	$-0.854724\pi$
$73$	− 7.53101i	− 0.881438i	0.897645	−	0.440719i	$-0.145276\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.225470i	− 0.0256947i
$78$	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
$83$	10.9045i	1.19693i	0.801150	+	0.598464i	$0.204222\pi$
$83$	10.9045i	1.19693i	−0.801150	+	0.598464i	$0.795778\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 5.24791i	− 0.562634i
$88$	0	0
$89$	4.19533	0.444704	0.222352	−	0.974966i	$-0.428627\pi$
$89$	4.19533	0.444704	0.222352	+	0.974966i	$0.428627\pi$
$90$	0	0
$91$	−1.25561	−0.131624
$92$	0	0
$93$	3.76683i	0.390602i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.73669i	0.176334i	0.996106	+	0.0881670i	$0.0281009\pi$
$97$	1.73669i	0.176334i	−0.996106	+	0.0881670i	$0.971899\pi$
$98$	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
$103$	3.96986i	0.391162i	0.980688	+	0.195581i	$0.0626593\pi$
$103$	3.96986i	0.391162i	−0.980688	+	0.195581i	$0.937341\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.05763i	0.682287i	0.940011	+	0.341144i	$0.110814\pi$
$107$	7.05763i	0.682287i	−0.940011	+	0.341144i	$0.889186\pi$
$108$	0	0
$109$	1.88209	0.180272	0.0901358	−	0.995929i	$-0.471270\pi$
$109$	1.88209	0.180272	0.0901358	+	0.995929i	$0.471270\pi$
$110$	0	0
$111$	2.73669	0.259755
$112$	0	0
$113$	7.59899i	0.714853i	0.933941	+	0.357426i	$0.116346\pi$
$113$	7.59899i	0.714853i	−0.933941	+	0.357426i	$0.883654\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.91223i	0.454136i
$118$	0	0
$119$	−2.94237	−0.269726
$120$	0	0
$121$	−10.8821	−0.989281
$122$	0	0
$123$	− 5.96216i	− 0.537590i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.3555i	1.80626i	0.429372	+	0.903128i	$0.358735\pi$
$127$	20.3555i	1.80626i	−0.429372	+	0.903128i	$0.641265\pi$
$128$	0	0
$129$	2.30554	0.202991
$130$	0	0
$131$	−7.25561	−0.633925	−0.316963	−	0.948438i	$-0.602663\pi$
$131$	−7.25561	−0.633925	−0.316963	+	0.948438i	$0.602663\pi$
$132$	0	0
$133$	0.656620i	0.0569362i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13.5310i	− 1.15603i	−0.816025	−	0.578016i	$-0.803827\pi$
$137$	− 13.5310i	− 1.15603i	0.816025	−	0.578016i	$-0.196173\pi$
$138$	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
$143$	− 0.656620i	− 0.0549094i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.31324i	0.355750i
$148$	0	0
$149$	5.94237	0.486818	0.243409	−	0.969924i	$-0.421734\pi$
$149$	5.94237	0.486818	0.243409	+	0.969924i	$0.421734\pi$
$150$	0	0
$151$	−8.91223	−0.725267	−0.362633	−	0.931932i	$-0.618122\pi$
$151$	−8.91223	−0.725267	−0.362633	+	0.931932i	$0.618122\pi$
$152$	0	0
$153$	11.5112i	0.930627i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 18.9819i	− 1.51492i	−0.652879	−	0.757462i	$-0.726439\pi$
$157$	− 18.9819i	− 1.51492i	0.652879	−	0.757462i	$-0.273561\pi$
$158$	0	0
$159$	−1.10756	−0.0878353
$160$	0	0
$161$	−2.34338	−0.184684
$162$	0	0
$163$	2.36317i	0.185098i	0.995708	+	0.0925489i	$0.0295014\pi$
$163$	2.36317i	0.185098i	−0.995708	+	0.0925489i	$0.970499\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.0801i	1.32170i	0.750520	+	0.660848i	$0.229803\pi$
$167$	17.0801i	1.32170i	−0.750520	+	0.660848i	$0.770197\pi$
$168$	0	0
$169$	9.34338	0.718722
$170$	0	0
$171$	2.56885	0.196445
$172$	0	0
$173$	− 2.36581i	− 0.179870i	−0.995948	−	0.0899348i	$-0.971334\pi$
$173$	− 2.36581i	− 0.179870i	0.995948	−	0.0899348i	$-0.0286659\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.13770i	0.386173i
$178$	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
$183$	− 8.15749i	− 0.603019i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.53871i	− 0.112522i
$188$	0	0
$189$	−2.40101	−0.174648
$190$	0	0
$191$	−11.9320	−0.863371	−0.431685	−	0.902024i	$-0.642081\pi$
$191$	−11.9320	−0.863371	−0.431685	+	0.902024i	$0.642081\pi$
$192$	0	0
$193$	20.6687i	1.48777i	0.668310	+	0.743883i	$0.267018\pi$
$193$	20.6687i	1.48777i	−0.668310	+	0.743883i	$0.732982\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.81675i	0.699415i	0.936859	+	0.349707i	$0.113719\pi$
$197$	9.81675i	0.699415i	−0.936859	+	0.349707i	$0.886281\pi$
$198$	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
$203$	− 5.24791i	− 0.368331i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	9.16784i	0.637209i
$208$	0	0
$209$	−0.343380	−0.0237521
$210$	0	0
$211$	10.4131	0.716867	0.358434	−	0.933555i	$-0.383311\pi$
$211$	10.4131	0.716867	0.358434	+	0.933555i	$0.383311\pi$
$212$	0	0
$213$	− 6.54136i	− 0.448206i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.76683i	0.255709i
$218$	0	0
$219$	4.94501	0.334153
$220$	0	0
$221$	−8.56885	−0.576403
$222$	0	0
$223$	8.25296i	0.552659i	0.961063	+	0.276330i	$0.0891182\pi$
$223$	8.25296i	0.552659i	−0.961063	+	0.276330i	$0.910882\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 0.393308i	− 0.0261048i	−0.999915	−	0.0130524i	$-0.995845\pi$
$227$	− 0.393308i	− 0.0261048i	0.999915	−	0.0130524i	$-0.00415482\pi$
$228$	0	0
$229$	−15.2479	−1.00761	−0.503805	−	0.863817i	$-0.668067\pi$
$229$	−15.2479	−1.00761	−0.503805	+	0.863817i	$0.668067\pi$
$230$	0	0
$231$	0.148048	0.00974086
$232$	0	0
$233$	23.3055i	1.52680i	0.645928	+	0.763398i	$0.276471\pi$
$233$	23.3055i	1.52680i	−0.645928	+	0.763398i	$0.723529\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.84425i	0.639453i
$238$	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
$243$	14.4536i	0.927198i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.91223i	0.121672i
$248$	0	0
$249$	−7.16013	−0.453755
$250$	0	0
$251$	24.9518	1.57494	0.787472	−	0.616350i	$-0.211389\pi$
$251$	24.9518	1.57494	0.787472	+	0.616350i	$0.211389\pi$
$252$	0	0
$253$	− 1.22547i	− 0.0770446i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.8168i	− 1.17376i	−0.809675	−	0.586878i	$-0.800357\pi$
$257$	− 18.8168i	− 1.17376i	0.809675	−	0.586878i	$-0.199643\pi$
$258$	0	0
$259$	2.73669	0.170049
$260$	0	0
$261$	−20.5310	−1.27084
$262$	0	0
$263$	− 9.97251i	− 0.614931i	−0.951559	−	0.307466i	$-0.900519\pi$
$263$	− 9.97251i	− 0.614931i	0.951559	−	0.307466i	$-0.0994809\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.75474i	0.168587i
$268$	0	0
$269$	−17.3632	−1.05865	−0.529326	−	0.848419i	$-0.677555\pi$
$269$	−17.3632	−1.05865	−0.529326	+	0.848419i	$0.677555\pi$
$270$	0	0
$271$	0.228115	0.0138570	0.00692851	−	0.999976i	$-0.497795\pi$
$271$	0.228115	0.0138570	0.00692851	+	0.999976i	$0.497795\pi$
$272$	0	0
$273$	− 0.824458i	− 0.0498985i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 10.2530i	− 0.616041i	−0.951380	−	0.308020i	$-0.900333\pi$
$277$	− 10.2530i	− 0.616041i	0.951380	−	0.308020i	$-0.0996665\pi$
$278$	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
$283$	− 8.74439i	− 0.519800i	−0.965636	−	0.259900i	$-0.916310\pi$
$283$	− 8.74439i	− 0.519800i	0.965636	−	0.259900i	$-0.0836896\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 5.96216i	− 0.351935i
$288$	0	0
$289$	−3.08007	−0.181180
$290$	0	0
$291$	−1.14034	−0.0668482
$292$	0	0
$293$	10.8218i	0.632217i	0.948723	+	0.316109i	$0.102376\pi$
$293$	10.8218i	0.632217i	−0.948723	+	0.316109i	$0.897624\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.25561i	− 0.0728578i
$298$	0	0
$299$	−6.82446	−0.394669
$300$	0	0
$301$	2.30554	0.132889
$302$	0	0
$303$	1.96480i	0.112875i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 3.68411i	− 0.210263i	−0.994458	−	0.105132i	$-0.966474\pi$
$307$	− 3.68411i	− 0.210263i	0.994458	−	0.105132i	$-0.0335264\pi$
$308$	0	0
$309$	−2.60669	−0.148290
$310$	0	0
$311$	−4.65398	−0.263903	−0.131951	−	0.991256i	$-0.542124\pi$
$311$	−4.65398	−0.263903	−0.131951	+	0.991256i	$0.542124\pi$
$312$	0	0
$313$	9.42080i	0.532495i	0.963905	+	0.266248i	$0.0857839\pi$
$313$	9.42080i	0.532495i	−0.963905	+	0.266248i	$0.914216\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 5.25561i	− 0.295184i	−0.989048	−	0.147592i	$-0.952848\pi$
$317$	− 5.25561i	− 0.295184i	0.989048	−	0.147592i	$-0.0471523\pi$
$318$	0	0
$319$	2.74439	0.153656
$320$	0	0
$321$	−4.63419	−0.258655
$322$	0	0
$323$	4.48108i	0.249334i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.23582i	0.0683409i
$328$	0	0
$329$	−2.23317	−0.123119
$330$	0	0
$331$	−17.4080	−0.956832	−0.478416	−	0.878133i	$-0.658789\pi$
$331$	−17.4080	−0.956832	−0.478416	+	0.878133i	$0.658789\pi$
$332$	0	0
$333$	− 10.7065i	− 0.586715i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.5836i	0.685471i	0.939432	+	0.342736i	$0.111354\pi$
$337$	12.5836i	0.685471i	−0.939432	+	0.342736i	$0.888646\pi$
$338$	0	0
$339$	−4.98965	−0.271001
$340$	0	0
$341$	−1.96986	−0.106674
$342$	0	0
$343$	8.90958i	0.481072i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.5259i	1.79977i	0.436130	+	0.899884i	$0.356349\pi$
$347$	33.5259i	1.79977i	−0.436130	+	0.899884i	$0.643651\pi$
$348$	0	0
$349$	18.0948	0.968592	0.484296	−	0.874904i	$-0.339076\pi$
$349$	18.0948	0.968592	0.484296	+	0.874904i	$0.339076\pi$
$350$	0	0
$351$	−6.99230	−0.373221
$352$	0	0
$353$	9.88979i	0.526381i	0.964744	+	0.263190i	$0.0847747\pi$
$353$	9.88979i	0.526381i	−0.964744	+	0.263190i	$0.915225\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.93202i	− 0.102253i
$358$	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
$363$	− 7.14540i	− 0.375036i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 13.6911i	− 0.714672i	−0.933976	−	0.357336i	$-0.883685\pi$
$367$	− 13.6911i	− 0.714672i	0.933976	−	0.357336i	$-0.116315\pi$
$368$	0	0
$369$	−23.3253	−1.21427
$370$	0	0
$371$	−1.10756	−0.0575017
$372$	0	0
$373$	13.5062i	0.699322i	0.936876	+	0.349661i	$0.113703\pi$
$373$	13.5062i	0.699322i	−0.936876	+	0.349661i	$0.886297\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 15.2831i	− 0.787120i
$378$	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
$383$	− 1.73669i	− 0.0887406i	−0.999015	−	0.0443703i	$-0.985872\pi$
$383$	− 1.73669i	− 0.0887406i	0.999015	−	0.0443703i	$-0.0141281\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 9.01979i	− 0.458502i
$388$	0	0
$389$	11.0396	0.559729	0.279864	−	0.960040i	$-0.409710\pi$
$389$	11.0396	0.559729	0.279864	+	0.960040i	$0.409710\pi$
$390$	0	0
$391$	−15.9923	−0.808765
$392$	0	0
$393$	− 4.76418i	− 0.240321i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 26.6214i	− 1.33609i	−0.744120	−	0.668045i	$-0.767131\pi$
$397$	− 26.6214i	− 1.33609i	0.744120	−	0.668045i	$-0.232869\pi$
$398$	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
$403$	10.9699i	0.546448i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.43115i	0.0709395i
$408$	0	0
$409$	12.1403	0.600301	0.300151	−	0.953892i	$-0.402963\pi$
$409$	12.1403	0.600301	0.300151	+	0.953892i	$0.402963\pi$
$410$	0	0
$411$	8.88474	0.438252
$412$	0	0
$413$	5.13770i	0.252810i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 8.48637i	− 0.415579i
$418$	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
$423$	8.73669i	0.424792i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8.15749i	− 0.394769i
$428$	0	0
$429$	0.431150	0.0208161
$430$	0	0
$431$	35.7411	1.72159	0.860793	−	0.508955i	$-0.169968\pi$
$431$	35.7411	1.72159	0.860793	+	0.508955i	$0.169968\pi$
$432$	0	0
$433$	− 35.2024i	− 1.69172i	−0.533407	−	0.845859i	$-0.679089\pi$
$433$	− 35.2024i	− 1.69172i	0.533407	−	0.845859i	$-0.320911\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.56885i	0.170721i
$438$	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
$443$	− 36.1421i	− 1.71716i	−0.512678	−	0.858581i	$-0.671346\pi$
$443$	− 36.1421i	− 1.71716i	0.512678	−	0.858581i	$-0.328654\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.90188i	0.184553i
$448$	0	0
$449$	−21.7917	−1.02841	−0.514206	−	0.857667i	$-0.671913\pi$
$449$	−21.7917	−1.02841	−0.514206	+	0.857667i	$0.671913\pi$
$450$	0	0
$451$	3.11791	0.146817
$452$	0	0
$453$	− 5.85195i	− 0.274949i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 10.7143i	− 0.501192i	−0.968092	−	0.250596i	$-0.919373\pi$
$457$	− 10.7143i	− 0.501192i	0.968092	−	0.250596i	$-0.0806265\pi$
$458$	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
$463$	− 26.5611i	− 1.23440i	−0.786806	−	0.617201i	$-0.788267\pi$
$463$	− 26.5611i	− 1.23440i	0.786806	−	0.617201i	$-0.211733\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.6533i	1.14082i	0.821361	+	0.570409i	$0.193215\pi$
$467$	24.6533i	1.14082i	−0.821361	+	0.570409i	$0.806785\pi$
$468$	0	0
$469$	−6.22547	−0.287465
$470$	0	0
$471$	12.4639	0.574308
$472$	0	0
$473$	1.20568i	0.0554372i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.33303i	0.198396i
$478$	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
$483$	− 1.53871i	− 0.0700138i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	27.2952i	1.23686i	0.785839	+	0.618432i	$0.212232\pi$
$487$	27.2952i	1.23686i	−0.785839	+	0.618432i	$0.787768\pi$
$488$	0	0
$489$	−1.55171	−0.0701705
$490$	0	0
$491$	−2.62913	−0.118651	−0.0593254	−	0.998239i	$-0.518895\pi$
$491$	−2.62913	−0.118651	−0.0593254	+	0.998239i	$0.518895\pi$
$492$	0	0
$493$	− 35.8141i	− 1.61299i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.54136i	− 0.293420i
$498$	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
$503$	35.8185i	1.59707i	0.601950	+	0.798534i	$0.294391\pi$
$503$	35.8185i	1.59707i	−0.601950	+	0.798534i	$0.705609\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.13505i	0.272467i
$508$	0	0
$509$	−41.9819	−1.86082	−0.930409	−	0.366524i	$-0.880548\pi$
$509$	−41.9819	−1.86082	−0.930409	+	0.366524i	$0.880548\pi$
$510$	0	0
$511$	4.94501	0.218755
$512$	0	0
$513$	3.65662i	0.161444i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.16784i	− 0.0513615i
$518$	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
$523$	− 13.8493i	− 0.605588i	−0.953056	−	0.302794i	$-0.902081\pi$
$523$	− 13.8493i	− 0.605588i	0.953056	−	0.302794i	$-0.0979194\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.7065i	1.11979i
$528$	0	0
$529$	10.2633	0.446231
$530$	0	0
$531$	20.0999	0.872259
$532$	0	0
$533$	− 17.3632i	− 0.752082i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.56379i	0.240095i
$538$	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
$543$	− 14.8322i	− 0.636509i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 37.5836i	− 1.60696i	−0.595333	−	0.803479i	$-0.702980\pi$
$547$	− 37.5836i	− 1.60696i	0.595333	−	0.803479i	$-0.297020\pi$
$548$	0	0
$549$	−31.9140	−1.36205
$550$	0	0
$551$	−7.99230	−0.340483
$552$	0	0
$553$	9.84425i	0.418620i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.7943i	0.499741i	0.968279	+	0.249871i	$0.0803881\pi$
$557$	11.7943i	0.499741i	−0.968279	+	0.249871i	$0.919612\pi$
$558$	0	0
$559$	6.71425	0.283983
$560$	0	0
$561$	1.01035	0.0426570
$562$	0	0
$563$	4.62407i	0.194881i	0.995241	+	0.0974406i	$0.0310656\pi$
$563$	4.62407i	0.194881i	−0.995241	+	0.0974406i	$0.968934\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.48372i	0.146303i
$568$	0	0
$569$	8.92258	0.374054	0.187027	−	0.982355i	$-0.440115\pi$
$569$	8.92258	0.374054	0.187027	+	0.982355i	$0.440115\pi$
$570$	0	0
$571$	20.1025	0.841264	0.420632	−	0.907231i	$-0.361808\pi$
$571$	20.1025	0.841264	0.420632	+	0.907231i	$0.361808\pi$
$572$	0	0
$573$	− 7.83481i	− 0.327304i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 14.8770i	− 0.619339i	−0.950844	−	0.309669i	$-0.899782\pi$
$577$	− 14.8770i	− 0.619339i	0.950844	−	0.309669i	$-0.100218\pi$
$578$	0	0
$579$	−13.5715	−0.564012
$580$	0	0
$581$	−7.16013	−0.297052
$582$	0	0
$583$	− 0.579199i	− 0.0239880i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 20.3280i	− 0.839025i	−0.907750	−	0.419513i	$-0.862201\pi$
$587$	− 20.3280i	− 0.839025i	0.907750	−	0.419513i	$-0.137799\pi$
$588$	0	0
$589$	5.73669	0.236376
$590$	0	0
$591$	−6.44588	−0.265148
$592$	0	0
$593$	26.9294i	1.10586i	0.833229	+	0.552928i	$0.186490\pi$
$593$	26.9294i	1.10586i	−0.833229	+	0.552928i	$0.813510\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.340734i	0.0139453i
$598$	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
$603$	24.3555i	0.991831i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.78397i	0.356530i	0.983982	+	0.178265i	$0.0570485\pi$
$607$	8.78397i	0.356530i	−0.983982	+	0.178265i	$0.942952\pi$
$608$	0	0
$609$	3.44588	0.139634
$610$	0	0
$611$	−6.50351	−0.263104
$612$	0	0
$613$	− 12.1705i	− 0.491561i	−0.969325	−	0.245781i	$-0.920956\pi$
$613$	− 12.1705i	− 0.491561i	0.969325	−	0.245781i	$-0.0790443\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.09986i	0.326088i	0.986619	+	0.163044i	$0.0521312\pi$
$617$	8.09986i	0.326088i	−0.986619	+	0.163044i	$0.947869\pi$
$618$	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
$623$	2.75474i	0.110366i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 0.225470i	− 0.00900441i
$628$	0	0
$629$	18.6764	0.744677
$630$	0	0
$631$	31.8614	1.26838	0.634191	−	0.773176i	$-0.281333\pi$
$631$	31.8614	1.26838	0.634191	+	0.773176i	$0.281333\pi$
$632$	0	0
$633$	6.83745i	0.271764i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.5611i	0.497691i
$638$	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
$643$	− 14.3511i	− 0.565951i	−0.959127	−	0.282976i	$-0.908678\pi$
$643$	− 14.3511i	− 0.565951i	0.959127	−	0.282976i	$-0.0913216\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 17.5139i	− 0.688541i	−0.938871	−	0.344270i	$-0.888126\pi$
$647$	− 17.5139i	− 0.688541i	0.938871	−	0.344270i	$-0.111874\pi$
$648$	0	0
$649$	−2.68676	−0.105465
$650$	0	0
$651$	−2.47338	−0.0969392
$652$	0	0
$653$	− 2.72196i	− 0.106518i	−0.998581	−	0.0532592i	$-0.983039\pi$
$653$	− 2.72196i	− 0.106518i	0.998581	−	0.0532592i	$-0.0169610\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 19.3460i	− 0.754760i
$658$	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
$663$	− 5.62648i	− 0.218514i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 28.5233i	− 1.10443i
$668$	0	0
$669$	−5.41906	−0.209513
$670$	0	0
$671$	4.26596	0.164685
$672$	0	0
$673$	40.5706i	1.56388i	0.623353	+	0.781941i	$0.285770\pi$
$673$	40.5706i	1.56388i	−0.623353	+	0.781941i	$0.714230\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.4586i	0.555691i	0.960626	+	0.277845i	$0.0896203\pi$
$677$	14.4586i	0.555691i	−0.960626	+	0.277845i	$0.910380\pi$
$678$	0	0
$679$	−1.14034	−0.0437624
$680$	0	0
$681$	0.258254	0.00989632
$682$	0	0
$683$	− 34.3605i	− 1.31477i	−0.753555	−	0.657384i	$-0.771663\pi$
$683$	− 34.3605i	− 1.31477i	0.753555	−	0.657384i	$-0.228337\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 10.0121i	− 0.381985i
$688$	0	0
$689$	−3.22547	−0.122881
$690$	0	0
$691$	13.6489	0.519229	0.259615	−	0.965712i	$-0.416404\pi$
$691$	13.6489	0.519229	0.259615	+	0.965712i	$0.416404\pi$
$692$	0	0
$693$	− 0.579199i	− 0.0220019i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 40.6885i	− 1.54119i
$698$	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
$703$	− 4.16784i	− 0.157193i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.96480i	0.0738940i
$708$	0	0
$709$	−13.3460	−0.501220	−0.250610	−	0.968088i	$-0.580631\pi$
$709$	−13.3460	−0.501220	−0.250610	+	0.968088i	$0.580631\pi$
$710$	0	0
$711$	38.5130	1.44435
$712$	0	0
$713$	20.4734i	0.766734i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 0.613720i	− 0.0229198i
$718$	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
$723$	− 9.33568i	− 0.347198i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 27.8392i	− 1.03250i	−0.856438	−	0.516249i	$-0.827328\pi$
$727$	− 27.8392i	− 1.03250i	0.856438	−	0.516249i	$-0.172672\pi$
$728$	0	0
$729$	6.42609	0.238003
$730$	0	0
$731$	15.7340	0.581945
$732$	0	0
$733$	− 17.1755i	− 0.634393i	−0.948360	−	0.317197i	$-0.897259\pi$
$733$	− 17.1755i	− 0.634393i	0.948360	−	0.317197i	$-0.102741\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3.25561i	− 0.119922i
$738$	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
$743$	− 44.1095i	− 1.61822i	−0.587656	−	0.809111i	$-0.699949\pi$
$743$	− 44.1095i	− 1.61822i	0.587656	−	0.809111i	$-0.300051\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	28.0121i	1.02491i
$748$	0	0
$749$	−4.63419	−0.169329
$750$	0	0
$751$	−13.5688	−0.495134	−0.247567	−	0.968871i	$-0.579631\pi$
$751$	−13.5688	−0.495134	−0.247567	+	0.968871i	$0.579631\pi$
$752$	0	0
$753$	16.3839i	0.597061i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 29.7444i	− 1.08108i	−0.841319	−	0.540539i	$-0.818220\pi$
$757$	− 29.7444i	− 1.08108i	0.841319	−	0.540539i	$-0.181780\pi$
$758$	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
$763$	1.23582i	0.0447397i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.9622i	0.540252i
$768$	0	0
$769$	6.46635	0.233182	0.116591	−	0.993180i	$-0.462803\pi$
$769$	6.46635	0.233182	0.116591	+	0.993180i	$0.462803\pi$
$770$	0	0
$771$	12.3555	0.444971
$772$	0	0
$773$	− 27.2780i	− 0.981123i	−0.871407	−	0.490562i	$-0.836792\pi$
$773$	− 27.2780i	− 0.981123i	0.871407	−	0.490562i	$-0.163208\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.79696i	0.0644658i
$778$	0	0
$779$	−9.08007	−0.325327
$780$	0	0
$781$	3.42080	0.122406
$782$	0	0
$783$	− 29.2248i	− 1.04441i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.3926i	1.22596i	0.790097	+	0.612982i	$0.210030\pi$
$787$	34.3926i	1.22596i	−0.790097	+	0.612982i	$0.789970\pi$
$788$	0	0
$789$	6.54815	0.233120
$790$	0	0
$791$	−4.98965	−0.177412
$792$	0	0
$793$	− 23.7565i	− 0.843617i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.7307i	1.05312i	0.850139	+	0.526558i	$0.176518\pi$
$797$	29.7307i	1.05312i	−0.850139	+	0.526558i	$0.823482\pi$
$798$	0	0
$799$	−15.2402	−0.539160
$800$	0	0
$801$	10.7772	0.380793
$802$	0	0
$803$	2.58599i	0.0912577i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 11.4010i	− 0.401335i
$808$	0	0
$809$	11.0653	0.389036	0.194518	−	0.980899i	$-0.437686\pi$
$809$	11.0653	0.389036	0.194518	+	0.980899i	$0.437686\pi$
$810$	0	0
$811$	−15.5079	−0.544556	−0.272278	−	0.962219i	$-0.587777\pi$
$811$	−15.5079	−0.544556	−0.272278	+	0.962219i	$0.587777\pi$
$812$	0	0
$813$	0.149785i	0.00525320i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3.51122i	− 0.122842i
$818$	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
$823$	51.1568i	1.78321i	0.452810	+	0.891607i	$0.350422\pi$
$823$	51.1568i	1.78321i	−0.452810	+	0.891607i	$0.649578\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.4613i	0.363775i	0.983319	+	0.181887i	$0.0582206\pi$
$827$	10.4613i	0.363775i	−0.983319	+	0.181887i	$0.941779\pi$
$828$	0	0
$829$	−46.6661	−1.62078	−0.810390	−	0.585891i	$-0.800745\pi$
$829$	−46.6661	−1.62078	−0.810390	+	0.585891i	$0.800745\pi$
$830$	0	0
$831$	6.73231	0.233541
$832$	0	0
$833$	29.4355i	1.01988i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	20.9769i	0.725067i
$838$	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
$843$	8.69205i	0.299370i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 7.14540i	− 0.245519i
$848$	0	0
$849$	5.74175	0.197056
$850$	0	0
$851$	14.8744	0.509887
$852$	0	0
$853$	− 50.1076i	− 1.71565i	−0.513942	−	0.857825i	$-0.671815\pi$
$853$	− 50.1076i	− 1.71565i	0.513942	−	0.857825i	$-0.328185\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 30.8915i	− 1.05523i	−0.849482	−	0.527617i	$-0.823086\pi$
$857$	− 30.8915i	− 1.05523i	0.849482	−	0.527617i	$-0.176914\pi$
$858$	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
$863$	− 23.9622i	− 0.815681i	−0.913053	−	0.407841i	$-0.866282\pi$
$863$	− 23.9622i	− 0.815681i	0.913053	−	0.407841i	$-0.133718\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 2.02243i	− 0.0686855i
$868$	0	0
$869$	−5.14805	−0.174636
$870$	0	0
$871$	−18.1300	−0.614311
$872$	0	0
$873$	4.46129i	0.150992i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.2221i	1.39197i	0.718055	+	0.695987i	$0.245033\pi$
$877$	41.2221i	1.39197i	−0.718055	+	0.695987i	$0.754967\pi$
$878$	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
$883$	− 44.2747i	− 1.48996i	−0.667085	−	0.744982i	$-0.732458\pi$
$883$	− 44.2747i	− 1.48996i	0.667085	−	0.744982i	$-0.267542\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 33.8392i	− 1.13621i	−0.822956	−	0.568104i	$-0.807677\pi$
$887$	− 33.8392i	− 1.13621i	0.822956	−	0.568104i	$-0.192323\pi$
$888$	0	0
$889$	−13.3658	−0.448275
$890$	0	0
$891$	−1.82181	−0.0610330
$892$	0	0
$893$	3.40101i	0.113811i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4.48108i	− 0.149619i
$898$	0	0
$899$	−45.8493	−1.52916
$900$	0	0
$901$	−7.55850	−0.251810
$902$	0	0
$903$	1.51386i	0.0503782i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 23.4536i	− 0.778764i	−0.921076	−	0.389382i	$-0.872689\pi$
$907$	− 23.4536i	− 0.778764i	0.921076	−	0.389382i	$-0.127311\pi$
$908$	0	0
$909$	7.68676	0.254954
$910$	0	0
$911$	−31.3099	−1.03734	−0.518672	−	0.854973i	$-0.673573\pi$
$911$	−31.3099	−1.03734	−0.518672	+	0.854973i	$0.673573\pi$
$912$	0	0
$913$	− 3.74439i	− 0.123921i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.76418i	− 0.157327i
$918$	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
$923$	− 19.0499i	− 0.627036i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.1980i	0.334945i
$928$	0	0
$929$	45.4606	1.49151	0.745757	−	0.666218i	$-0.232088\pi$
$929$	45.4606	1.49151	0.745757	+	0.666218i	$0.232088\pi$
$930$	0	0
$931$	6.56885	0.215285
$932$	0	0
$933$	− 3.05590i	− 0.100046i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 7.16784i	− 0.234163i	−0.993122	−	0.117082i	$-0.962646\pi$
$937$	− 7.16784i	− 0.234163i	0.993122	−	0.117082i	$-0.0373539\pi$
$938$	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
$943$	− 32.4054i	− 1.05526i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 58.2195i	− 1.89188i	−0.324342	−	0.945940i	$-0.605143\pi$
$947$	− 58.2195i	− 1.89188i	0.324342	−	0.945940i	$-0.394857\pi$
$948$	0	0
$949$	14.4010	0.467477
$950$	0	0
$951$	3.45094	0.111904
$952$	0	0
$953$	− 49.3247i	− 1.59778i	−0.601476	−	0.798891i	$-0.705420\pi$
$953$	− 49.3247i	− 1.59778i	0.601476	−	0.798891i	$-0.294580\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.80202i	0.0582511i
$958$	0	0
$959$	8.88474	0.286903
$960$	0	0
$961$	1.90958	0.0615995
$962$	0	0
$963$	18.1300i	0.584231i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 28.6368i	− 0.920898i	−0.887686	−	0.460449i	$-0.847688\pi$
$967$	− 28.6368i	− 0.920898i	0.887686	−	0.460449i	$-0.152312\pi$
$968$	0	0
$969$	−2.94237	−0.0945225
$970$	0	0
$971$	−11.4131	−0.366264	−0.183132	−	0.983088i	$-0.558624\pi$
$971$	−11.4131	−0.366264	−0.183132	+	0.983088i	$0.558624\pi$
$972$	0	0
$973$	− 8.48637i	− 0.272061i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 52.2567i	− 1.67184i	−0.548852	−	0.835919i	$-0.684935\pi$
$977$	− 52.2567i	− 1.67184i	0.548852	−	0.835919i	$-0.315065\pi$
$978$	0	0
$979$	−1.44059	−0.0460415
$980$	0	0
$981$	4.83481	0.154364
$982$	0	0
$983$	35.5904i	1.13516i	0.823319	+	0.567578i	$0.192120\pi$
$983$	35.5904i	1.13516i	−0.823319	+	0.567578i	$0.807880\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.46635i	− 0.0466744i
$988$	0	0
$989$	12.5310	0.398463
$990$	0	0
$991$	35.2952	1.12119	0.560594	−	0.828091i	$-0.310573\pi$
$991$	35.2952	1.12119	0.560594	+	0.828091i	$0.310573\pi$
$992$	0	0
$993$	− 11.4305i	− 0.362735i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 7.45359i	− 0.236057i	−0.993010	−	0.118029i	$-0.962343\pi$
$997$	− 7.45359i	− 0.236057i	0.993010	−	0.118029i	$-0.0376575\pi$
$998$	0	0
$999$	15.2402	0.482179

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.u.1.2	✓	3	5.2	odd	4
3800.2.a.v.1.2	yes	3	5.3	odd	4
3800.2.d.m.3649.3		6	5.4	even	2	inner
3800.2.d.m.3649.4		6	1.1	even	1	trivial
7600.2.a.bt.1.2		3	20.7	even	4
7600.2.a.bu.1.2		3	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.656620i q^{3} +0.656620i q^{7} +2.56885 q^{9} -0.343380 q^{11} +1.91223i q^{13} +4.48108i q^{17} +1.00000 q^{19} -0.431150 q^{21} +3.56885i q^{23} +3.65662i q^{27} -7.99230 q^{29} +5.73669 q^{31} -0.225470i q^{33} -4.16784i q^{37} -1.25561 q^{39} -9.08007 q^{41} -3.51122i q^{43} +3.40101i q^{47} +6.56885 q^{49} -2.94237 q^{51} +1.68676i q^{53} +0.656620i q^{57} +7.82446 q^{59} -12.4234 q^{61} +1.68676i q^{63} +9.48108i q^{67} -2.34338 q^{69} -9.96216 q^{71} -7.53101i q^{73} -0.225470i q^{77} +14.9923 q^{79} +5.30554 q^{81} +10.9045i q^{83} -5.24791i q^{87} +4.19533 q^{89} -1.25561 q^{91} +3.76683i q^{93} +1.73669i q^{97} -0.882090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{9} - 6 q^{11} + 6 q^{19} - 20 q^{21} + 2 q^{29} - 6 q^{31} + 2 q^{39} - 18 q^{41} + 22 q^{49} - 16 q^{51} + 20 q^{59} - 42 q^{61} - 18 q^{69} + 2 q^{71} + 40 q^{79} - 26 q^{81} - 8 q^{89} + 2 q^{91}+ \cdots + 20 q^{99}+O(q^{100}) \) 6 * q - 2 * q^9 - 6 * q^11 + 6 * q^19 - 20 * q^21 + 2 * q^29 - 6 * q^31 + 2 * q^39 - 18 * q^41 + 22 * q^49 - 16 * q^51 + 20 * q^59 - 42 * q^61 - 18 * q^69 + 2 * q^71 + 40 * q^79 - 26 * q^81 - 8 * q^89 + 2 * q^91 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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