LMFDB - Embedded newform 3800.2.d.p.3649.5

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 28x^{10} + 274x^{8} + 1078x^{6} + 1385x^{4} + 478x^{2} + 49 $ x^12 + 28x^10 + 274x^8 + 1078x^6 + 1385x^4 + 478*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.5
Root			$-0.486697i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.p.3649.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.486697i q^{3} +3.63249i q^{7} +2.76313 q^{9} -2.79460 q^{11} -2.86457i q^{13} +1.17400i q^{17} -1.00000 q^{19} +1.76793 q^{21} +0.617328i q^{23} -2.80490i q^{27} +4.96700 q^{29} +0.745377 q^{31} +1.36012i q^{33} +8.23679i q^{37} -1.39418 q^{39} +9.98217 q^{41} -10.4955i q^{43} +5.07742i q^{47} -6.19502 q^{49} +0.571381 q^{51} +7.45370i q^{53} +0.486697i q^{57} -3.83310 q^{59} +11.2450 q^{61} +10.0370i q^{63} +6.10730i q^{67} +0.300452 q^{69} -9.40599 q^{71} +9.52367i q^{73} -10.1514i q^{77} +3.70094 q^{79} +6.92424 q^{81} +4.66397i q^{83} -2.41743i q^{87} -10.6888 q^{89} +10.4055 q^{91} -0.362773i q^{93} +0.629221i q^{97} -7.72183 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{9} + 6 q^{11} - 12 q^{19} + 22 q^{21} - 14 q^{29} + 10 q^{31} - 16 q^{39} + 22 q^{41} + 4 q^{49} + 26 q^{51} + 8 q^{59} + 26 q^{61} - 14 q^{69} + 58 q^{71} - 56 q^{79} + 76 q^{81} + 24 q^{89}+ \cdots - 20 q^{99}+O(q^{100}) $ 12 * q - 20 * q^9 + 6 * q^11 - 12 * q^19 + 22 * q^21 - 14 * q^29 + 10 * q^31 - 16 * q^39 + 22 * q^41 + 4 * q^49 + 26 * q^51 + 8 * q^59 + 26 * q^61 - 14 * q^69 + 58 * q^71 - 56 * q^79 + 76 * q^81 + 24 * q^89 - 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
$3$	− 0.486697i	− 0.280995i	−0.990081	−	0.140497i	$-0.955130\pi$
$3$	− 0.486697i	− 0.280995i	0.990081	−	0.140497i	$-0.0448702\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.63249i	1.37295i	0.727152	+	0.686477i	$0.240844\pi$
$7$	3.63249i	1.37295i	−0.727152	+	0.686477i	$0.759156\pi$
$8$	0	0
$9$	2.76313	0.921042
$10$	0	0
$11$	−2.79460	−0.842603	−0.421302	−	0.906921i	$-0.638427\pi$
$11$	−2.79460	−0.842603	−0.421302	+	0.906921i	$0.638427\pi$
$12$	0	0
$13$	− 2.86457i	− 0.794489i	−0.917713	−	0.397244i	$-0.869967\pi$
$13$	− 2.86457i	− 0.794489i	0.917713	−	0.397244i	$-0.130033\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.17400i	0.284736i	0.989814	+	0.142368i	$0.0454716\pi$
$17$	1.17400i	0.284736i	−0.989814	+	0.142368i	$0.954528\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.76793	0.385793
$22$	0	0
$23$	0.617328i	0.128722i	0.997927	+	0.0643609i	$0.0205009\pi$
$23$	0.617328i	0.128722i	−0.997927	+	0.0643609i	$0.979499\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.80490i	− 0.539803i
$28$	0	0
$29$	4.96700	0.922349	0.461175	−	0.887309i	$-0.347428\pi$
$29$	4.96700	0.922349	0.461175	+	0.887309i	$0.347428\pi$
$30$	0	0
$31$	0.745377	0.133874	0.0669369	−	0.997757i	$-0.478677\pi$
$31$	0.745377	0.133874	0.0669369	+	0.997757i	$0.478677\pi$
$32$	0	0
$33$	1.36012i	0.236767i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.23679i	1.35412i	0.735928	+	0.677060i	$0.236746\pi$
$37$	8.23679i	1.35412i	−0.735928	+	0.677060i	$0.763254\pi$
$38$	0	0
$39$	−1.39418	−0.223247
$40$	0	0
$41$	9.98217	1.55895	0.779476	−	0.626432i	$-0.215485\pi$
$41$	9.98217	1.55895	0.779476	+	0.626432i	$0.215485\pi$
$42$	0	0
$43$	− 10.4955i	− 1.60055i	−0.599630	−	0.800277i	$-0.704686\pi$
$43$	− 10.4955i	− 1.60055i	0.599630	−	0.800277i	$-0.295314\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.07742i	0.740618i	0.928909	+	0.370309i	$0.120748\pi$
$47$	5.07742i	0.740618i	−0.928909	+	0.370309i	$0.879252\pi$
$48$	0	0
$49$	−6.19502	−0.885003
$50$	0	0
$51$	0.571381	0.0800093
$52$	0	0
$53$	7.45370i	1.02384i	0.859032	+	0.511922i	$0.171066\pi$
$53$	7.45370i	1.02384i	−0.859032	+	0.511922i	$0.828934\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.486697i	0.0644646i
$58$	0	0
$59$	−3.83310	−0.499027	−0.249513	−	0.968371i	$-0.580271\pi$
$59$	−3.83310	−0.499027	−0.249513	+	0.968371i	$0.580271\pi$
$60$	0	0
$61$	11.2450	1.43978	0.719889	−	0.694089i	$-0.244193\pi$
$61$	11.2450	1.43978	0.719889	+	0.694089i	$0.244193\pi$
$62$	0	0
$63$	10.0370i	1.26455i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.10730i	0.746125i	0.927806	+	0.373063i	$0.121692\pi$
$67$	6.10730i	0.746125i	−0.927806	+	0.373063i	$0.878308\pi$
$68$	0	0
$69$	0.300452	0.0361702
$70$	0	0
$71$	−9.40599	−1.11629	−0.558143	−	0.829745i	$-0.688486\pi$
$71$	−9.40599	−1.11629	−0.558143	+	0.829745i	$0.688486\pi$
$72$	0	0
$73$	9.52367i	1.11466i	0.830291	+	0.557331i	$0.188174\pi$
$73$	9.52367i	1.11466i	−0.830291	+	0.557331i	$0.811826\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.1514i	− 1.15686i
$78$	0	0
$79$	3.70094	0.416388	0.208194	−	0.978088i	$-0.433241\pi$
$79$	3.70094	0.416388	0.208194	+	0.978088i	$0.433241\pi$
$80$	0	0
$81$	6.92424	0.769360
$82$	0	0
$83$	4.66397i	0.511937i	0.966685	+	0.255968i	$0.0823943\pi$
$83$	4.66397i	0.511937i	−0.966685	+	0.255968i	$0.917606\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.41743i	− 0.259175i
$88$	0	0
$89$	−10.6888	−1.13301	−0.566506	−	0.824058i	$-0.691705\pi$
$89$	−10.6888	−1.13301	−0.566506	+	0.824058i	$0.691705\pi$
$90$	0	0
$91$	10.4055	1.09080
$92$	0	0
$93$	− 0.362773i	− 0.0376178i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.629221i	0.0638877i	0.999490	+	0.0319439i	$0.0101698\pi$
$97$	0.629221i	0.0638877i	−0.999490	+	0.0319439i	$0.989830\pi$
$98$	0	0
$99$	−7.72183	−0.776073
$100$	0	0
$101$	3.82531	0.380633	0.190316	−	0.981723i	$-0.439049\pi$
$101$	3.82531	0.380633	0.190316	+	0.981723i	$0.439049\pi$
$102$	0	0
$103$	10.0702i	0.992251i	0.868251	+	0.496125i	$0.165244\pi$
$103$	10.0702i	0.992251i	−0.868251	+	0.496125i	$0.834756\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.53412i	0.438330i	0.975688	+	0.219165i	$0.0703333\pi$
$107$	4.53412i	0.438330i	−0.975688	+	0.219165i	$0.929667\pi$
$108$	0	0
$109$	−15.9974	−1.53227	−0.766137	−	0.642678i	$-0.777823\pi$
$109$	−15.9974	−1.53227	−0.766137	+	0.642678i	$0.777823\pi$
$110$	0	0
$111$	4.00882	0.380501
$112$	0	0
$113$	2.34828i	0.220908i	0.993881	+	0.110454i	$0.0352304\pi$
$113$	2.34828i	0.220908i	−0.993881	+	0.110454i	$0.964770\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 7.91517i	− 0.731757i
$118$	0	0
$119$	−4.26454	−0.390929
$120$	0	0
$121$	−3.19022	−0.290020
$122$	0	0
$123$	− 4.85829i	− 0.438058i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.4542i	1.54881i	0.632693	+	0.774403i	$0.281950\pi$
$127$	17.4542i	1.54881i	−0.632693	+	0.774403i	$0.718050\pi$
$128$	0	0
$129$	−5.10815	−0.449748
$130$	0	0
$131$	9.00877	0.787100	0.393550	−	0.919303i	$-0.371247\pi$
$131$	9.00877	0.787100	0.393550	+	0.919303i	$0.371247\pi$
$132$	0	0
$133$	− 3.63249i	− 0.314977i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.4369i	1.31887i	0.751764	+	0.659433i	$0.229203\pi$
$137$	15.4369i	1.31887i	−0.751764	+	0.659433i	$0.770797\pi$
$138$	0	0
$139$	−15.1144	−1.28199	−0.640993	−	0.767547i	$-0.721477\pi$
$139$	−15.1144	−1.28199	−0.640993	+	0.767547i	$0.721477\pi$
$140$	0	0
$141$	2.47117	0.208110
$142$	0	0
$143$	8.00532i	0.669439i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.01510i	0.248681i
$148$	0	0
$149$	11.8422	0.970153	0.485076	−	0.874472i	$-0.338792\pi$
$149$	11.8422	0.970153	0.485076	+	0.874472i	$0.338792\pi$
$150$	0	0
$151$	−8.31262	−0.676471	−0.338236	−	0.941061i	$-0.609830\pi$
$151$	−8.31262	−0.676471	−0.338236	+	0.941061i	$0.609830\pi$
$152$	0	0
$153$	3.24390i	0.262254i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 10.6650i	− 0.851162i	−0.904920	−	0.425581i	$-0.860070\pi$
$157$	− 10.6650i	− 0.851162i	0.904920	−	0.425581i	$-0.139930\pi$
$158$	0	0
$159$	3.62770	0.287695
$160$	0	0
$161$	−2.24244	−0.176729
$162$	0	0
$163$	1.65583i	0.129694i	0.997895	+	0.0648472i	$0.0206560\pi$
$163$	1.65583i	0.129694i	−0.997895	+	0.0648472i	$0.979344\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.83568i	0.761108i	0.924759	+	0.380554i	$0.124267\pi$
$167$	9.83568i	0.761108i	−0.924759	+	0.380554i	$0.875733\pi$
$168$	0	0
$169$	4.79424	0.368788
$170$	0	0
$171$	−2.76313	−0.211302
$172$	0	0
$173$	8.05586i	0.612476i	0.951955	+	0.306238i	$0.0990703\pi$
$173$	8.05586i	0.612476i	−0.951955	+	0.306238i	$0.900930\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.86556i	0.140224i
$178$	0	0
$179$	14.0058	1.04685	0.523423	−	0.852073i	$-0.324655\pi$
$179$	14.0058	1.04685	0.523423	+	0.852073i	$0.324655\pi$
$180$	0	0
$181$	2.43741	0.181171	0.0905856	−	0.995889i	$-0.471126\pi$
$181$	2.43741	0.181171	0.0905856	+	0.995889i	$0.471126\pi$
$182$	0	0
$183$	− 5.47292i	− 0.404570i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.28085i	− 0.239919i
$188$	0	0
$189$	10.1888	0.741124
$190$	0	0
$191$	14.3568	1.03882	0.519412	−	0.854524i	$-0.326151\pi$
$191$	14.3568	1.03882	0.519412	+	0.854524i	$0.326151\pi$
$192$	0	0
$193$	10.3628i	0.745929i	0.927846	+	0.372964i	$0.121659\pi$
$193$	10.3628i	0.745929i	−0.927846	+	0.372964i	$0.878341\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.9168i	− 0.920286i	−0.887845	−	0.460143i	$-0.847798\pi$
$197$	− 12.9168i	− 0.920286i	0.887845	−	0.460143i	$-0.152202\pi$
$198$	0	0
$199$	−14.9400	−1.05907	−0.529536	−	0.848288i	$-0.677634\pi$
$199$	−14.9400	−1.05907	−0.529536	+	0.848288i	$0.677634\pi$
$200$	0	0
$201$	2.97241	0.209657
$202$	0	0
$203$	18.0426i	1.26634i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.70576i	0.118558i
$208$	0	0
$209$	2.79460	0.193306
$210$	0	0
$211$	5.28130	0.363579	0.181790	−	0.983337i	$-0.441811\pi$
$211$	5.28130	0.363579	0.181790	+	0.983337i	$0.441811\pi$
$212$	0	0
$213$	4.57787i	0.313670i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.70758i	0.183802i
$218$	0	0
$219$	4.63514	0.313214
$220$	0	0
$221$	3.36299	0.226219
$222$	0	0
$223$	− 11.5273i	− 0.771926i	−0.922514	−	0.385963i	$-0.873869\pi$
$223$	− 11.5273i	− 0.771926i	0.922514	−	0.385963i	$-0.126131\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	29.1127i	1.93228i	0.258021	+	0.966139i	$0.416930\pi$
$227$	29.1127i	1.93228i	−0.258021	+	0.966139i	$0.583070\pi$
$228$	0	0
$229$	−2.60133	−0.171901	−0.0859503	−	0.996299i	$-0.527393\pi$
$229$	−2.60133	−0.171901	−0.0859503	+	0.996299i	$0.527393\pi$
$230$	0	0
$231$	−4.94064	−0.325070
$232$	0	0
$233$	11.2634i	0.737890i	0.929451	+	0.368945i	$0.120281\pi$
$233$	11.2634i	0.737890i	−0.929451	+	0.368945i	$0.879719\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.80124i	− 0.117003i
$238$	0	0
$239$	22.6347	1.46412	0.732059	−	0.681242i	$-0.238560\pi$
$239$	22.6347	1.46412	0.732059	+	0.681242i	$0.238560\pi$
$240$	0	0
$241$	−13.2940	−0.856339	−0.428169	−	0.903698i	$-0.640841\pi$
$241$	−13.2940	−0.856339	−0.428169	+	0.903698i	$0.640841\pi$
$242$	0	0
$243$	− 11.7847i	− 0.755989i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.86457i	0.182268i
$248$	0	0
$249$	2.26994	0.143852
$250$	0	0
$251$	16.2433	1.02527	0.512633	−	0.858608i	$-0.328670\pi$
$251$	16.2433	1.02527	0.512633	+	0.858608i	$0.328670\pi$
$252$	0	0
$253$	− 1.72518i	− 0.108461i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 27.4124i	− 1.70994i	−0.518680	−	0.854968i	$-0.673577\pi$
$257$	− 27.4124i	− 1.70994i	0.518680	−	0.854968i	$-0.326423\pi$
$258$	0	0
$259$	−29.9201	−1.85914
$260$	0	0
$261$	13.7244	0.849522
$262$	0	0
$263$	24.0460i	1.48274i	0.671097	+	0.741370i	$0.265824\pi$
$263$	24.0460i	1.48274i	−0.671097	+	0.741370i	$0.734176\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.20221i	0.318370i
$268$	0	0
$269$	12.5398	0.764565	0.382283	−	0.924045i	$-0.375138\pi$
$269$	12.5398	0.764565	0.382283	+	0.924045i	$0.375138\pi$
$270$	0	0
$271$	17.5270	1.06469	0.532344	−	0.846528i	$-0.321311\pi$
$271$	17.5270	1.06469	0.532344	+	0.846528i	$0.321311\pi$
$272$	0	0
$273$	− 5.06435i	− 0.306508i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.79301i	0.348068i	0.984740	+	0.174034i	$0.0556803\pi$
$277$	5.79301i	0.348068i	−0.984740	+	0.174034i	$0.944320\pi$
$278$	0	0
$279$	2.05957	0.123303
$280$	0	0
$281$	19.2228	1.14673	0.573367	−	0.819298i	$-0.305637\pi$
$281$	19.2228	1.14673	0.573367	+	0.819298i	$0.305637\pi$
$282$	0	0
$283$	− 28.3864i	− 1.68740i	−0.536818	−	0.843698i	$-0.680374\pi$
$283$	− 28.3864i	− 1.68740i	0.536818	−	0.843698i	$-0.319626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	36.2602i	2.14037i
$288$	0	0
$289$	15.6217	0.918925
$290$	0	0
$291$	0.306240	0.0179521
$292$	0	0
$293$	− 19.8463i	− 1.15943i	−0.814818	−	0.579717i	$-0.803163\pi$
$293$	− 19.8463i	− 1.15943i	0.814818	−	0.579717i	$-0.196837\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.83856i	0.454840i
$298$	0	0
$299$	1.76838	0.102268
$300$	0	0
$301$	38.1250	2.19749
$302$	0	0
$303$	− 1.86177i	− 0.106956i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6.35995i	− 0.362982i	−0.983393	−	0.181491i	$-0.941908\pi$
$307$	− 6.35995i	− 0.362982i	0.983393	−	0.181491i	$-0.0580923\pi$
$308$	0	0
$309$	4.90116	0.278817
$310$	0	0
$311$	23.4371	1.32899	0.664497	−	0.747291i	$-0.268646\pi$
$311$	23.4371	1.32899	0.664497	+	0.747291i	$0.268646\pi$
$312$	0	0
$313$	3.38952i	0.191587i	0.995401	+	0.0957934i	$0.0305388\pi$
$313$	3.38952i	0.191587i	−0.995401	+	0.0957934i	$0.969461\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 5.37856i	− 0.302090i	−0.988527	−	0.151045i	$-0.951736\pi$
$317$	− 5.37856i	− 0.302090i	0.988527	−	0.151045i	$-0.0482639\pi$
$318$	0	0
$319$	−13.8808	−0.777174
$320$	0	0
$321$	2.20674	0.123168
$322$	0	0
$323$	− 1.17400i	− 0.0653229i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.78589i	0.430561i
$328$	0	0
$329$	−18.4437	−1.01683
$330$	0	0
$331$	−3.80034	−0.208885	−0.104443	−	0.994531i	$-0.533306\pi$
$331$	−3.80034	−0.208885	−0.104443	+	0.994531i	$0.533306\pi$
$332$	0	0
$333$	22.7593i	1.24720i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 28.2975i	− 1.54146i	−0.637159	−	0.770732i	$-0.719891\pi$
$337$	− 28.2975i	− 1.54146i	0.637159	−	0.770732i	$-0.280109\pi$
$338$	0	0
$339$	1.14290	0.0620739
$340$	0	0
$341$	−2.08303	−0.112802
$342$	0	0
$343$	2.92409i	0.157886i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 14.0367i	− 0.753530i	−0.926309	−	0.376765i	$-0.877036\pi$
$347$	− 14.0367i	− 0.753530i	0.926309	−	0.376765i	$-0.122964\pi$
$348$	0	0
$349$	31.9225	1.70877	0.854387	−	0.519637i	$-0.173933\pi$
$349$	31.9225	1.70877	0.854387	+	0.519637i	$0.173933\pi$
$350$	0	0
$351$	−8.03482	−0.428867
$352$	0	0
$353$	2.65209i	0.141157i	0.997506	+	0.0705783i	$0.0224845\pi$
$353$	2.65209i	0.141157i	−0.997506	+	0.0705783i	$0.977516\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.07554i	0.109849i
$358$	0	0
$359$	−23.7750	−1.25480	−0.627398	−	0.778699i	$-0.715880\pi$
$359$	−23.7750	−1.25480	−0.627398	+	0.778699i	$0.715880\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.55267i	0.0814941i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.3923i	0.907869i	0.891035	+	0.453934i	$0.149980\pi$
$367$	17.3923i	0.907869i	−0.891035	+	0.453934i	$0.850020\pi$
$368$	0	0
$369$	27.5820	1.43586
$370$	0	0
$371$	−27.0755	−1.40569
$372$	0	0
$373$	− 17.8099i	− 0.922164i	−0.887358	−	0.461082i	$-0.847461\pi$
$373$	− 17.8099i	− 0.922164i	0.887358	−	0.461082i	$-0.152539\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 14.2283i	− 0.732796i
$378$	0	0
$379$	−25.7998	−1.32525	−0.662624	−	0.748952i	$-0.730557\pi$
$379$	−25.7998	−1.32525	−0.662624	+	0.748952i	$0.730557\pi$
$380$	0	0
$381$	8.49489	0.435206
$382$	0	0
$383$	− 2.08964i	− 0.106776i	−0.998574	−	0.0533879i	$-0.982998\pi$
$383$	− 2.08964i	− 0.106776i	0.998574	−	0.0533879i	$-0.0170020\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 29.0005i	− 1.47418i
$388$	0	0
$389$	17.8701	0.906052	0.453026	−	0.891497i	$-0.350344\pi$
$389$	17.8701	0.906052	0.453026	+	0.891497i	$0.350344\pi$
$390$	0	0
$391$	−0.724741	−0.0366517
$392$	0	0
$393$	− 4.38455i	− 0.221171i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.4105i	1.42588i	0.701224	+	0.712941i	$0.252637\pi$
$397$	28.4105i	1.42588i	−0.701224	+	0.712941i	$0.747363\pi$
$398$	0	0
$399$	−1.76793	−0.0885070
$400$	0	0
$401$	−20.7047	−1.03394	−0.516971	−	0.856003i	$-0.672940\pi$
$401$	−20.7047	−1.03394	−0.516971	+	0.856003i	$0.672940\pi$
$402$	0	0
$403$	− 2.13518i	− 0.106361i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 23.0185i	− 1.14099i
$408$	0	0
$409$	−29.1270	−1.44024	−0.720119	−	0.693850i	$-0.755913\pi$
$409$	−29.1270	−1.44024	−0.720119	+	0.693850i	$0.755913\pi$
$410$	0	0
$411$	7.51311	0.370594
$412$	0	0
$413$	− 13.9237i	− 0.685141i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.35613i	0.360231i
$418$	0	0
$419$	−16.3711	−0.799782	−0.399891	−	0.916563i	$-0.630952\pi$
$419$	−16.3711	−0.799782	−0.399891	+	0.916563i	$0.630952\pi$
$420$	0	0
$421$	−8.82857	−0.430278	−0.215139	−	0.976583i	$-0.569020\pi$
$421$	−8.82857	−0.430278	−0.215139	+	0.976583i	$0.569020\pi$
$422$	0	0
$423$	14.0296i	0.682140i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	40.8475i	1.97675i
$428$	0	0
$429$	3.89617	0.188109
$430$	0	0
$431$	−24.8989	−1.19934	−0.599668	−	0.800249i	$-0.704701\pi$
$431$	−24.8989	−1.19934	−0.599668	+	0.800249i	$0.704701\pi$
$432$	0	0
$433$	18.9901i	0.912608i	0.889824	+	0.456304i	$0.150827\pi$
$433$	18.9901i	0.912608i	−0.889824	+	0.456304i	$0.849173\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 0.617328i	− 0.0295308i
$438$	0	0
$439$	13.3540	0.637352	0.318676	−	0.947864i	$-0.396762\pi$
$439$	13.3540	0.637352	0.318676	+	0.947864i	$0.396762\pi$
$440$	0	0
$441$	−17.1176	−0.815125
$442$	0	0
$443$	− 8.67208i	− 0.412023i	−0.978550	−	0.206012i	$-0.933952\pi$
$443$	− 8.67208i	− 0.412023i	0.978550	−	0.206012i	$-0.0660484\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 5.76358i	− 0.272608i
$448$	0	0
$449$	−40.5277	−1.91262	−0.956310	−	0.292354i	$-0.905561\pi$
$449$	−40.5277	−1.91262	−0.956310	+	0.292354i	$0.905561\pi$
$450$	0	0
$451$	−27.8962	−1.31358
$452$	0	0
$453$	4.04573i	0.190085i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 16.9859i	− 0.794566i	−0.917696	−	0.397283i	$-0.869953\pi$
$457$	− 16.9859i	− 0.794566i	0.917696	−	0.397283i	$-0.130047\pi$
$458$	0	0
$459$	3.29294	0.153701
$460$	0	0
$461$	33.1374	1.54336	0.771681	−	0.636010i	$-0.219416\pi$
$461$	33.1374	1.54336	0.771681	+	0.636010i	$0.219416\pi$
$462$	0	0
$463$	− 8.11961i	− 0.377350i	−0.982040	−	0.188675i	$-0.939581\pi$
$463$	− 8.11961i	− 0.377350i	0.982040	−	0.188675i	$-0.0604193\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.1221i	0.746042i	0.927823	+	0.373021i	$0.121678\pi$
$467$	16.1221i	0.746042i	−0.927823	+	0.373021i	$0.878322\pi$
$468$	0	0
$469$	−22.1847	−1.02440
$470$	0	0
$471$	−5.19064	−0.239172
$472$	0	0
$473$	29.3308i	1.34863i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	20.5955i	0.943003i
$478$	0	0
$479$	5.30225	0.242266	0.121133	−	0.992636i	$-0.461347\pi$
$479$	5.30225	0.242266	0.121133	+	0.992636i	$0.461347\pi$
$480$	0	0
$481$	23.5949	1.07583
$482$	0	0
$483$	1.09139i	0.0496600i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 20.9188i	− 0.947922i	−0.880546	−	0.473961i	$-0.842824\pi$
$487$	− 20.9188i	− 0.947922i	0.880546	−	0.473961i	$-0.157176\pi$
$488$	0	0
$489$	0.805886	0.0364434
$490$	0	0
$491$	−22.1370	−0.999029	−0.499515	−	0.866305i	$-0.666488\pi$
$491$	−22.1370	−0.999029	−0.499515	+	0.866305i	$0.666488\pi$
$492$	0	0
$493$	5.83124i	0.262626i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 34.1672i	− 1.53261i
$498$	0	0
$499$	−5.44182	−0.243609	−0.121805	−	0.992554i	$-0.538868\pi$
$499$	−5.44182	−0.243609	−0.121805	+	0.992554i	$0.538868\pi$
$500$	0	0
$501$	4.78700	0.213867
$502$	0	0
$503$	− 17.9666i	− 0.801092i	−0.916277	−	0.400546i	$-0.868820\pi$
$503$	− 17.9666i	− 0.801092i	0.916277	−	0.400546i	$-0.131180\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.33334i	− 0.103627i
$508$	0	0
$509$	−0.433419	−0.0192110	−0.00960548	−	0.999954i	$-0.503058\pi$
$509$	−0.433419	−0.0192110	−0.00960548	+	0.999954i	$0.503058\pi$
$510$	0	0
$511$	−34.5947	−1.53038
$512$	0	0
$513$	2.80490i	0.123839i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 14.1894i	− 0.624047i
$518$	0	0
$519$	3.92077	0.172103
$520$	0	0
$521$	−1.29105	−0.0565621	−0.0282811	−	0.999600i	$-0.509003\pi$
$521$	−1.29105	−0.0565621	−0.0282811	+	0.999600i	$0.509003\pi$
$522$	0	0
$523$	1.17153i	0.0512275i	0.999672	+	0.0256138i	$0.00815401\pi$
$523$	1.17153i	0.0512275i	−0.999672	+	0.0256138i	$0.991846\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.875070i	0.0381187i
$528$	0	0
$529$	22.6189	0.983431
$530$	0	0
$531$	−10.5913	−0.459624
$532$	0	0
$533$	− 28.5946i	− 1.23857i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 6.81660i	− 0.294158i
$538$	0	0
$539$	17.3126	0.745706
$540$	0	0
$541$	−21.1522	−0.909402	−0.454701	−	0.890644i	$-0.650254\pi$
$541$	−21.1522	−0.909402	−0.454701	+	0.890644i	$0.650254\pi$
$542$	0	0
$543$	− 1.18628i	− 0.0509082i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 39.5096i	− 1.68931i	−0.535311	−	0.844655i	$-0.679806\pi$
$547$	− 39.5096i	− 1.68931i	0.535311	−	0.844655i	$-0.320194\pi$
$548$	0	0
$549$	31.0714	1.32610
$550$	0	0
$551$	−4.96700	−0.211601
$552$	0	0
$553$	13.4436i	0.571682i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.90886i	− 0.335109i	−0.985863	−	0.167554i	$-0.946413\pi$
$557$	− 7.90886i	− 0.335109i	0.985863	−	0.167554i	$-0.0535870\pi$
$558$	0	0
$559$	−30.0652	−1.27162
$560$	0	0
$561$	−1.59678	−0.0674161
$562$	0	0
$563$	5.40347i	0.227729i	0.993496	+	0.113865i	$0.0363230\pi$
$563$	5.40347i	0.227729i	−0.993496	+	0.113865i	$0.963677\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	25.1523i	1.05630i
$568$	0	0
$569$	4.60607	0.193096	0.0965482	−	0.995328i	$-0.469220\pi$
$569$	4.60607	0.193096	0.0965482	+	0.995328i	$0.469220\pi$
$570$	0	0
$571$	−38.6010	−1.61540	−0.807701	−	0.589593i	$-0.799288\pi$
$571$	−38.6010	−1.61540	−0.807701	+	0.589593i	$0.799288\pi$
$572$	0	0
$573$	− 6.98743i	− 0.291904i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 30.0011i	− 1.24896i	−0.781040	−	0.624480i	$-0.785311\pi$
$577$	− 30.0011i	− 1.24896i	0.781040	−	0.624480i	$-0.214689\pi$
$578$	0	0
$579$	5.04353	0.209602
$580$	0	0
$581$	−16.9418	−0.702866
$582$	0	0
$583$	− 20.8301i	− 0.862694i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.90807i	− 0.0787545i	−0.999224	−	0.0393772i	$-0.987463\pi$
$587$	− 1.90807i	− 0.0787545i	0.999224	−	0.0393772i	$-0.0125374\pi$
$588$	0	0
$589$	−0.745377	−0.0307127
$590$	0	0
$591$	−6.28658	−0.258596
$592$	0	0
$593$	11.5819i	0.475610i	0.971313	+	0.237805i	$0.0764279\pi$
$593$	11.5819i	0.475610i	−0.971313	+	0.237805i	$0.923572\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.27128i	0.297594i
$598$	0	0
$599$	35.9808	1.47014	0.735068	−	0.677993i	$-0.237150\pi$
$599$	35.9808	1.47014	0.735068	+	0.677993i	$0.237150\pi$
$600$	0	0
$601$	−6.25635	−0.255202	−0.127601	−	0.991826i	$-0.540728\pi$
$601$	−6.25635	−0.255202	−0.127601	+	0.991826i	$0.540728\pi$
$602$	0	0
$603$	16.8752i	0.687213i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 3.98858i	− 0.161891i	−0.996719	−	0.0809457i	$-0.974206\pi$
$607$	− 3.98858i	− 0.161891i	0.996719	−	0.0809457i	$-0.0257941\pi$
$608$	0	0
$609$	8.78129	0.355836
$610$	0	0
$611$	14.5446	0.588412
$612$	0	0
$613$	− 26.1306i	− 1.05541i	−0.849429	−	0.527703i	$-0.823053\pi$
$613$	− 26.1306i	− 1.05541i	0.849429	−	0.527703i	$-0.176947\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.30081i	0.253661i	0.991924	+	0.126831i	$0.0404805\pi$
$617$	6.30081i	0.253661i	−0.991924	+	0.126831i	$0.959520\pi$
$618$	0	0
$619$	−12.7231	−0.511387	−0.255693	−	0.966758i	$-0.582304\pi$
$619$	−12.7231	−0.511387	−0.255693	+	0.966758i	$0.582304\pi$
$620$	0	0
$621$	1.73154	0.0694844
$622$	0	0
$623$	− 38.8270i	− 1.55557i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.36012i	− 0.0543181i
$628$	0	0
$629$	−9.66996	−0.385567
$630$	0	0
$631$	27.1455	1.08065	0.540323	−	0.841458i	$-0.318302\pi$
$631$	27.1455	1.08065	0.540323	+	0.841458i	$0.318302\pi$
$632$	0	0
$633$	− 2.57039i	− 0.102164i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	17.7461i	0.703125i
$638$	0	0
$639$	−25.9899	−1.02815
$640$	0	0
$641$	−20.8345	−0.822914	−0.411457	−	0.911429i	$-0.634980\pi$
$641$	−20.8345	−0.822914	−0.411457	+	0.911429i	$0.634980\pi$
$642$	0	0
$643$	42.3146i	1.66872i	0.551217	+	0.834362i	$0.314164\pi$
$643$	42.3146i	1.66872i	−0.551217	+	0.834362i	$0.685836\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 5.57122i	− 0.219027i	−0.993985	−	0.109514i	$-0.965071\pi$
$647$	− 5.57122i	− 0.219027i	0.993985	−	0.109514i	$-0.0349293\pi$
$648$	0	0
$649$	10.7120	0.420481
$650$	0	0
$651$	1.31777	0.0516475
$652$	0	0
$653$	44.6953i	1.74906i	0.484968	+	0.874532i	$0.338831\pi$
$653$	44.6953i	1.74906i	−0.484968	+	0.874532i	$0.661169\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	26.3151i	1.02665i
$658$	0	0
$659$	37.0288	1.44244	0.721218	−	0.692708i	$-0.243583\pi$
$659$	37.0288	1.44244	0.721218	+	0.692708i	$0.243583\pi$
$660$	0	0
$661$	−6.16081	−0.239628	−0.119814	−	0.992796i	$-0.538230\pi$
$661$	−6.16081	−0.239628	−0.119814	+	0.992796i	$0.538230\pi$
$662$	0	0
$663$	− 1.63676i	− 0.0635665i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.06627i	0.118726i
$668$	0	0
$669$	−5.61031	−0.216907
$670$	0	0
$671$	−31.4253	−1.21316
$672$	0	0
$673$	13.5879i	0.523773i	0.965099	+	0.261887i	$0.0843447\pi$
$673$	13.5879i	0.523773i	−0.965099	+	0.261887i	$0.915655\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.0051i	0.653559i	0.945101	+	0.326779i	$0.105963\pi$
$677$	17.0051i	0.653559i	−0.945101	+	0.326779i	$0.894037\pi$
$678$	0	0
$679$	−2.28564	−0.0877149
$680$	0	0
$681$	14.1691	0.542960
$682$	0	0
$683$	− 10.9418i	− 0.418677i	−0.977843	−	0.209338i	$-0.932869\pi$
$683$	− 10.9418i	− 0.418677i	0.977843	−	0.209338i	$-0.0671310\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.26606i	0.0483032i
$688$	0	0
$689$	21.3516	0.813433
$690$	0	0
$691$	−21.4168	−0.814733	−0.407366	−	0.913265i	$-0.633553\pi$
$691$	−21.4168	−0.814733	−0.407366	+	0.913265i	$0.633553\pi$
$692$	0	0
$693$	− 28.0495i	− 1.06551i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.7190i	0.443890i
$698$	0	0
$699$	5.48186	0.207343
$700$	0	0
$701$	−27.2640	−1.02975	−0.514874	−	0.857266i	$-0.672161\pi$
$701$	−27.2640	−1.02975	−0.514874	+	0.857266i	$0.672161\pi$
$702$	0	0
$703$	− 8.23679i	− 0.310656i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.8954i	0.522591i
$708$	0	0
$709$	−25.9811	−0.975740	−0.487870	−	0.872916i	$-0.662226\pi$
$709$	−25.9811	−0.975740	−0.487870	+	0.872916i	$0.662226\pi$
$710$	0	0
$711$	10.2262	0.383511
$712$	0	0
$713$	0.460142i	0.0172325i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 11.0162i	− 0.411409i
$718$	0	0
$719$	32.4059	1.20854	0.604268	−	0.796781i	$-0.293465\pi$
$719$	32.4059	1.20854	0.604268	+	0.796781i	$0.293465\pi$
$720$	0	0
$721$	−36.5801	−1.36231
$722$	0	0
$723$	6.47013i	0.240627i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 47.1313i	− 1.74800i	−0.485924	−	0.874001i	$-0.661517\pi$
$727$	− 47.1313i	− 1.74800i	0.485924	−	0.874001i	$-0.338483\pi$
$728$	0	0
$729$	15.0371	0.556931
$730$	0	0
$731$	12.3217	0.455735
$732$	0	0
$733$	− 12.7925i	− 0.472500i	−0.971692	−	0.236250i	$-0.924081\pi$
$733$	− 12.7925i	− 0.472500i	0.971692	−	0.236250i	$-0.0759185\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 17.0675i	− 0.628688i
$738$	0	0
$739$	28.2097	1.03771	0.518856	−	0.854862i	$-0.326358\pi$
$739$	28.2097	1.03771	0.518856	+	0.854862i	$0.326358\pi$
$740$	0	0
$741$	1.39418	0.0512164
$742$	0	0
$743$	− 16.0366i	− 0.588325i	−0.955755	−	0.294162i	$-0.904959\pi$
$743$	− 16.0366i	− 0.588325i	0.955755	−	0.294162i	$-0.0950406\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.8871i	0.471515i
$748$	0	0
$749$	−16.4702	−0.601807
$750$	0	0
$751$	28.8165	1.05153	0.525765	−	0.850630i	$-0.323779\pi$
$751$	28.8165	1.05153	0.525765	+	0.850630i	$0.323779\pi$
$752$	0	0
$753$	− 7.90555i	− 0.288094i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 39.3944i	− 1.43181i	−0.698196	−	0.715906i	$-0.746014\pi$
$757$	− 39.3944i	− 1.43181i	0.698196	−	0.715906i	$-0.253986\pi$
$758$	0	0
$759$	−0.839643	−0.0304771
$760$	0	0
$761$	21.7891	0.789856	0.394928	−	0.918712i	$-0.370770\pi$
$761$	21.7891	0.789856	0.394928	+	0.918712i	$0.370770\pi$
$762$	0	0
$763$	− 58.1105i	− 2.10374i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.9802i	0.396471i
$768$	0	0
$769$	−15.7229	−0.566983	−0.283491	−	0.958975i	$-0.591493\pi$
$769$	−15.7229	−0.566983	−0.283491	+	0.958975i	$0.591493\pi$
$770$	0	0
$771$	−13.3415	−0.480483
$772$	0	0
$773$	− 26.6810i	− 0.959649i	−0.877364	−	0.479825i	$-0.840700\pi$
$773$	− 26.6810i	− 0.959649i	0.877364	−	0.479825i	$-0.159300\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	14.5620i	0.522410i
$778$	0	0
$779$	−9.98217	−0.357648
$780$	0	0
$781$	26.2860	0.940586
$782$	0	0
$783$	− 13.9319i	− 0.497887i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 38.7243i	− 1.38037i	−0.723631	−	0.690187i	$-0.757528\pi$
$787$	− 38.7243i	− 1.38037i	0.723631	−	0.690187i	$-0.242472\pi$
$788$	0	0
$789$	11.7031	0.416642
$790$	0	0
$791$	−8.53012	−0.303296
$792$	0	0
$793$	− 32.2121i	− 1.14389i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.5802i	1.18947i	0.803921	+	0.594736i	$0.202744\pi$
$797$	33.5802i	1.18947i	−0.803921	+	0.594736i	$0.797256\pi$
$798$	0	0
$799$	−5.96087	−0.210881
$800$	0	0
$801$	−29.5345	−1.04355
$802$	0	0
$803$	− 26.6148i	− 0.939217i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6.10309i	− 0.214839i
$808$	0	0
$809$	25.5182	0.897172	0.448586	−	0.893740i	$-0.351928\pi$
$809$	25.5182	0.897172	0.448586	+	0.893740i	$0.351928\pi$
$810$	0	0
$811$	9.88905	0.347252	0.173626	−	0.984812i	$-0.444452\pi$
$811$	9.88905	0.347252	0.173626	+	0.984812i	$0.444452\pi$
$812$	0	0
$813$	− 8.53033i	− 0.299172i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.4955i	0.367192i
$818$	0	0
$819$	28.7518	1.00467
$820$	0	0
$821$	−8.01417	−0.279696	−0.139848	−	0.990173i	$-0.544661\pi$
$821$	−8.01417	−0.279696	−0.139848	+	0.990173i	$0.544661\pi$
$822$	0	0
$823$	18.4568i	0.643363i	0.946848	+	0.321681i	$0.104248\pi$
$823$	18.4568i	0.643363i	−0.946848	+	0.321681i	$0.895752\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.3135i	0.358637i	0.983791	+	0.179318i	$0.0573893\pi$
$827$	10.3135i	0.358637i	−0.983791	+	0.179318i	$0.942611\pi$
$828$	0	0
$829$	28.5805	0.992640	0.496320	−	0.868140i	$-0.334684\pi$
$829$	28.5805	0.992640	0.496320	+	0.868140i	$0.334684\pi$
$830$	0	0
$831$	2.81944	0.0978053
$832$	0	0
$833$	− 7.27293i	− 0.251992i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.09071i	− 0.0722654i
$838$	0	0
$839$	−5.06942	−0.175016	−0.0875079	−	0.996164i	$-0.527890\pi$
$839$	−5.06942	−0.175016	−0.0875079	+	0.996164i	$0.527890\pi$
$840$	0	0
$841$	−4.32890	−0.149272
$842$	0	0
$843$	− 9.35567i	− 0.322227i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 11.5885i	− 0.398184i
$848$	0	0
$849$	−13.8156	−0.474149
$850$	0	0
$851$	−5.08480	−0.174305
$852$	0	0
$853$	17.1101i	0.585839i	0.956137	+	0.292920i	$0.0946268\pi$
$853$	17.1101i	0.585839i	−0.956137	+	0.292920i	$0.905373\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 31.4488i	− 1.07427i	−0.843496	−	0.537135i	$-0.819506\pi$
$857$	− 31.4488i	− 1.07427i	0.843496	−	0.537135i	$-0.180494\pi$
$858$	0	0
$859$	−5.98445	−0.204187	−0.102093	−	0.994775i	$-0.532554\pi$
$859$	−5.98445	−0.204187	−0.102093	+	0.994775i	$0.532554\pi$
$860$	0	0
$861$	17.6477	0.601433
$862$	0	0
$863$	− 20.8530i	− 0.709844i	−0.934896	−	0.354922i	$-0.884507\pi$
$863$	− 20.8530i	− 0.709844i	0.934896	−	0.354922i	$-0.115493\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 7.60306i	− 0.258213i
$868$	0	0
$869$	−10.3426	−0.350850
$870$	0	0
$871$	17.4948	0.592788
$872$	0	0
$873$	1.73862i	0.0588432i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7.11382i	− 0.240217i	−0.992761	−	0.120108i	$-0.961676\pi$
$877$	− 7.11382i	− 0.240217i	0.992761	−	0.120108i	$-0.0383242\pi$
$878$	0	0
$879$	−9.65915	−0.325795
$880$	0	0
$881$	−42.4272	−1.42941	−0.714704	−	0.699427i	$-0.753438\pi$
$881$	−42.4272	−1.42941	−0.714704	+	0.699427i	$0.753438\pi$
$882$	0	0
$883$	− 9.15841i	− 0.308205i	−0.988055	−	0.154103i	$-0.950751\pi$
$883$	− 9.15841i	− 0.308205i	0.988055	−	0.154103i	$-0.0492486\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.2435i	0.713286i	0.934241	+	0.356643i	$0.116079\pi$
$887$	21.2435i	0.713286i	−0.934241	+	0.356643i	$0.883921\pi$
$888$	0	0
$889$	−63.4021	−2.12644
$890$	0	0
$891$	−19.3505	−0.648265
$892$	0	0
$893$	− 5.07742i	− 0.169909i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 0.860666i	− 0.0287368i
$898$	0	0
$899$	3.70229	0.123478
$900$	0	0
$901$	−8.75062	−0.291525
$902$	0	0
$903$	− 18.5553i	− 0.617483i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19.1139i	0.634667i	0.948314	+	0.317333i	$0.102787\pi$
$907$	19.1139i	0.634667i	−0.948314	+	0.317333i	$0.897213\pi$
$908$	0	0
$909$	10.5698	0.350579
$910$	0	0
$911$	−19.9557	−0.661161	−0.330580	−	0.943778i	$-0.607244\pi$
$911$	−19.9557	−0.661161	−0.330580	+	0.943778i	$0.607244\pi$
$912$	0	0
$913$	− 13.0339i	− 0.431360i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	32.7243i	1.08065i
$918$	0	0
$919$	−54.3957	−1.79435	−0.897174	−	0.441677i	$-0.854384\pi$
$919$	−54.3957	−1.79435	−0.897174	+	0.441677i	$0.854384\pi$
$920$	0	0
$921$	−3.09537	−0.101996
$922$	0	0
$923$	26.9441i	0.886876i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	27.8253i	0.913904i
$928$	0	0
$929$	50.0187	1.64106	0.820530	−	0.571603i	$-0.193678\pi$
$929$	50.0187	1.64106	0.820530	+	0.571603i	$0.193678\pi$
$930$	0	0
$931$	6.19502	0.203034
$932$	0	0
$933$	− 11.4068i	− 0.373441i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 50.3138i	− 1.64368i	−0.569719	−	0.821840i	$-0.692948\pi$
$937$	− 50.3138i	− 1.64368i	0.569719	−	0.821840i	$-0.307052\pi$
$938$	0	0
$939$	1.64967	0.0538349
$940$	0	0
$941$	−24.6083	−0.802207	−0.401103	−	0.916033i	$-0.631373\pi$
$941$	−24.6083	−0.802207	−0.401103	+	0.916033i	$0.631373\pi$
$942$	0	0
$943$	6.16227i	0.200671i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 0.628142i	− 0.0204119i	−0.999948	−	0.0102059i	$-0.996751\pi$
$947$	− 0.628142i	− 0.0204119i	0.999948	−	0.0102059i	$-0.00324871\pi$
$948$	0	0
$949$	27.2812	0.885586
$950$	0	0
$951$	−2.61773	−0.0848858
$952$	0	0
$953$	40.6251i	1.31598i	0.753028	+	0.657988i	$0.228592\pi$
$953$	40.6251i	1.31598i	−0.753028	+	0.657988i	$0.771408\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.75574i	0.218382i
$958$	0	0
$959$	−56.0745	−1.81074
$960$	0	0
$961$	−30.4444	−0.982078
$962$	0	0
$963$	12.5283i	0.403720i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 25.2241i	− 0.811153i	−0.914061	−	0.405576i	$-0.867071\pi$
$967$	− 25.2241i	− 0.811153i	0.914061	−	0.405576i	$-0.132929\pi$
$968$	0	0
$969$	−0.571381	−0.0183554
$970$	0	0
$971$	2.82319	0.0906006	0.0453003	−	0.998973i	$-0.485576\pi$
$971$	2.82319	0.0906006	0.0453003	+	0.998973i	$0.485576\pi$
$972$	0	0
$973$	− 54.9030i	− 1.76011i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 34.1606i	− 1.09289i	−0.837494	−	0.546447i	$-0.815980\pi$
$977$	− 34.1606i	− 1.09289i	0.837494	−	0.546447i	$-0.184020\pi$
$978$	0	0
$979$	29.8709	0.954679
$980$	0	0
$981$	−44.2028	−1.41129
$982$	0	0
$983$	− 46.0699i	− 1.46940i	−0.678391	−	0.734701i	$-0.737323\pi$
$983$	− 46.0699i	− 1.46940i	0.678391	−	0.734701i	$-0.262677\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.97650i	0.285725i
$988$	0	0
$989$	6.47919	0.206026
$990$	0	0
$991$	56.9827	1.81011	0.905057	−	0.425290i	$-0.139828\pi$
$991$	56.9827	1.81011	0.905057	+	0.425290i	$0.139828\pi$
$992$	0	0
$993$	1.84961i	0.0586957i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 50.3518i	− 1.59466i	−0.603546	−	0.797328i	$-0.706246\pi$
$997$	− 50.3518i	− 1.59466i	0.603546	−	0.797328i	$-0.293754\pi$
$998$	0	0
$999$	23.1034	0.730958

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.p.3649.5		12
5.2	odd	4		3800.2.a.bb.1.3	✓	6
5.3	odd	4		3800.2.a.bd.1.4	yes	6
5.4	even	2	inner	3800.2.d.p.3649.8		12
20.3	even	4		7600.2.a.ci.1.3		6
20.7	even	4		7600.2.a.cm.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.bb.1.3	✓	6	5.2	odd	4
3800.2.a.bd.1.4	yes	6	5.3	odd	4
3800.2.d.p.3649.5		12	1.1	even	1	trivial
3800.2.d.p.3649.8		12	5.4	even	2	inner
7600.2.a.ci.1.3		6	20.3	even	4
7600.2.a.cm.1.4		6	20.7	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)

\(f(q)\)	\(=\)	\(q-0.486697i q^{3} +3.63249i q^{7} +2.76313 q^{9} -2.79460 q^{11} -2.86457i q^{13} +1.17400i q^{17} -1.00000 q^{19} +1.76793 q^{21} +0.617328i q^{23} -2.80490i q^{27} +4.96700 q^{29} +0.745377 q^{31} +1.36012i q^{33} +8.23679i q^{37} -1.39418 q^{39} +9.98217 q^{41} -10.4955i q^{43} +5.07742i q^{47} -6.19502 q^{49} +0.571381 q^{51} +7.45370i q^{53} +0.486697i q^{57} -3.83310 q^{59} +11.2450 q^{61} +10.0370i q^{63} +6.10730i q^{67} +0.300452 q^{69} -9.40599 q^{71} +9.52367i q^{73} -10.1514i q^{77} +3.70094 q^{79} +6.92424 q^{81} +4.66397i q^{83} -2.41743i q^{87} -10.6888 q^{89} +10.4055 q^{91} -0.362773i q^{93} +0.629221i q^{97} -7.72183 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{9} + 6 q^{11} - 12 q^{19} + 22 q^{21} - 14 q^{29} + 10 q^{31} - 16 q^{39} + 22 q^{41} + 4 q^{49} + 26 q^{51} + 8 q^{59} + 26 q^{61} - 14 q^{69} + 58 q^{71} - 56 q^{79} + 76 q^{81} + 24 q^{89}+ \cdots - 20 q^{99}+O(q^{100}) \) 12 * q - 20 * q^9 + 6 * q^11 - 12 * q^19 + 22 * q^21 - 14 * q^29 + 10 * q^31 - 16 * q^39 + 22 * q^41 + 4 * q^49 + 26 * q^51 + 8 * q^59 + 26 * q^61 - 14 * q^69 + 58 * q^71 - 56 * q^79 + 76 * q^81 + 24 * q^89 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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