LMFDB - Embedded newform 3800.2.d.c.3649.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.c.3649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-2.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +6.00000i q^{17} +1.00000 q^{19} +8.00000 q^{21} -8.00000i q^{23} -4.00000i q^{27} +6.00000 q^{29} -8.00000 q^{31} +8.00000i q^{33} -8.00000i q^{37} -2.00000 q^{41} +12.0000i q^{47} -9.00000 q^{49} +12.0000 q^{51} -4.00000i q^{53} -2.00000i q^{57} -8.00000 q^{59} -14.0000 q^{61} -4.00000i q^{63} -2.00000i q^{67} -16.0000 q^{69} -8.00000 q^{71} +2.00000i q^{73} -16.0000i q^{77} -4.00000 q^{79} -11.0000 q^{81} -12.0000i q^{83} -12.0000i q^{87} -6.00000 q^{89} +16.0000i q^{93} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 8 q^{11} + 2 q^{19} + 16 q^{21} + 12 q^{29} - 16 q^{31} - 4 q^{41} - 18 q^{49} + 24 q^{51} - 16 q^{59} - 28 q^{61} - 32 q^{69} - 16 q^{71} - 8 q^{79} - 22 q^{81} - 12 q^{89} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 8 * q^11 + 2 * q^19 + 16 * q^21 + 12 * q^29 - 16 * q^31 - 4 * q^41 - 18 * q^49 + 24 * q^51 - 16 * q^59 - 28 * q^61 - 32 * q^69 - 16 * q^71 - 8 * q^79 - 22 * q^81 - 12 * q^89 + 8 * q^99

Display 100 coefficients 10 coefficients

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
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	8.00000	1.74574
$22$	0	0
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	8.00000i	1.39262i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	12.0000	1.68034
$52$	0	0
$53$	− 4.00000i	− 0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	− 4.00000i	− 0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.00000i	− 0.264906i
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	− 4.00000i	− 0.503953i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	0	0
$69$	−16.0000	−1.92617
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 16.0000i	− 1.82337i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 12.0000i	− 1.28654i
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	16.0000i	1.65912i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	6.00000i	0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$	6.00000i	0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.0000i	1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$	18.0000i	1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−16.0000	−1.51865
$112$	0	0
$113$	16.0000i	1.50515i	0.658505	+	0.752577i	$0.271189\pi$
$113$	16.0000i	1.50515i	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−24.0000	−2.20008
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	4.00000i	0.360668i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 18.0000i	− 1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$127$	− 18.0000i	− 1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	4.00000i	0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	24.0000	2.02116
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	18.0000i	1.48461i
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	32.0000	2.52195
$162$	0	0
$163$	− 24.0000i	− 1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$	− 24.0000i	− 1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 6.00000i	− 0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$	− 6.00000i	− 0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	20.0000i	1.52057i	0.649589	+	0.760286i	$0.274941\pi$
$173$	20.0000i	1.52057i	−0.649589	+	0.760286i	$0.725059\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.0000i	1.20263i
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	28.0000i	2.06982i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 24.0000i	− 1.75505i
$188$	0	0
$189$	16.0000	1.16383
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	− 16.0000i	− 1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$	− 16.0000i	− 1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	24.0000i	1.68447i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.00000i	0.556038i
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	16.0000i	1.09630i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 32.0000i	− 2.17230i
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6.00000i	− 0.398234i	−0.979976	−	0.199117i	$-0.936193\pi$
$227$	− 6.00000i	− 0.398234i	0.979976	−	0.199117i	$-0.0638074\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	−32.0000	−2.10545
$232$	0	0
$233$	− 10.0000i	− 0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$	− 10.0000i	− 0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000i	0.519656i
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	10.0000i	0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−24.0000	−1.52094
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	32.0000i	2.01182i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.0000i	0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$	12.0000i	0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$	0	0
$259$	32.0000	1.98838
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	12.0000i	0.734388i
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 8.00000i	− 0.472225i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 4.00000i	− 0.233682i	−0.993151	−	0.116841i	$-0.962723\pi$
$293$	− 4.00000i	− 0.233682i	0.993151	−	0.116841i	$-0.0372769\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.0000i	0.928414i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	12.0000i	0.689382i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 22.0000i	− 1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$307$	− 22.0000i	− 1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	14.0000i	0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$	14.0000i	0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.0000i	1.79730i	0.438667	+	0.898650i	$0.355451\pi$
$317$	32.0000i	1.79730i	−0.438667	+	0.898650i	$0.644549\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	36.0000	2.00932
$322$	0	0
$323$	6.00000i	0.333849i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 4.00000i	− 0.221201i
$328$	0	0
$329$	−48.0000	−2.64633
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	0	0
$333$	8.00000i	0.438397i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.00000i	0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$	8.00000i	0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$	0	0
$339$	32.0000	1.73800
$340$	0	0
$341$	32.0000	1.73290
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	48.0000i	2.54043i
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 10.0000i	− 0.524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	36.0000i	1.87918i	0.342296	+	0.939592i	$0.388796\pi$
$367$	36.0000i	1.87918i	−0.342296	+	0.939592i	$0.611204\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	16.0000	0.830679
$372$	0	0
$373$	12.0000i	0.621336i	0.950518	+	0.310668i	$0.100553\pi$
$373$	12.0000i	0.621336i	−0.950518	+	0.310668i	$0.899447\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	−36.0000	−1.84434
$382$	0	0
$383$	− 22.0000i	− 1.12415i	−0.827087	−	0.562074i	$-0.810004\pi$
$383$	− 22.0000i	− 1.12415i	0.827087	−	0.562074i	$-0.189996\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	0	0
$393$	40.0000i	2.01773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	32.0000i	1.58618i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	−20.0000	−0.986527
$412$	0	0
$413$	− 32.0000i	− 1.57462i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.00000i	0.391762i
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	− 12.0000i	− 0.583460i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 56.0000i	− 2.71003i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.00000	0.192673	0.0963366	−	0.995349i	$-0.469287\pi$
$431$	4.00000	0.192673	0.0963366	+	0.995349i	$0.469287\pi$
$432$	0	0
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.00000i	− 0.382692i
$438$	0	0
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	− 36.0000i	− 1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$	− 36.0000i	− 1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 12.0000i	− 0.567581i
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	− 24.0000i	− 1.12762i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	0	0
$459$	24.0000	1.12022
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4.00000i	− 0.185098i	−0.995708	−	0.0925490i	$-0.970499\pi$
$467$	− 4.00000i	− 0.185098i	0.995708	−	0.0925490i	$-0.0295015\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.00000i	0.183147i
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 64.0000i	− 2.91210i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.0000i	1.17817i	0.808070	+	0.589086i	$0.200512\pi$
$487$	26.0000i	1.17817i	−0.808070	+	0.589086i	$0.799488\pi$
$488$	0	0
$489$	−48.0000	−2.17064
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	36.0000i	1.62136i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 32.0000i	− 1.43540i
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	− 24.0000i	− 1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$	− 24.0000i	− 1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 26.0000i	− 1.15470i
$508$	0	0
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 48.0000i	− 2.11104i
$518$	0	0
$519$	40.0000	1.75581
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	− 22.0000i	− 0.961993i	−0.876723	−	0.480996i	$-0.840275\pi$
$523$	− 22.0000i	− 0.961993i	0.876723	−	0.480996i	$-0.159725\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 48.0000i	− 2.09091i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	36.0000	1.55063
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	20.0000i	0.858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	26.0000i	1.11168i	0.831289	+	0.555840i	$0.187603\pi$
$547$	26.0000i	1.11168i	−0.831289	+	0.555840i	$0.812397\pi$
$548$	0	0
$549$	14.0000	0.597505
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−48.0000	−2.02656
$562$	0	0
$563$	18.0000i	0.758610i	0.925272	+	0.379305i	$0.123837\pi$
$563$	18.0000i	0.758610i	−0.925272	+	0.379305i	$0.876163\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 44.0000i	− 1.84783i
$568$	0	0
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	48.0000i	2.00523i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.0000i	0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$	18.0000i	0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	0	0
$579$	−32.0000	−1.32987
$580$	0	0
$581$	48.0000	1.99138
$582$	0	0
$583$	16.0000i	0.662652i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.0000i	0.654836i
$598$	0	0
$599$	44.0000	1.79779	0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000	1.79779	0.898896	+	0.438163i	$0.144371\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	2.00000i	0.0814463i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.0000i	0.730597i	0.930890	+	0.365299i	$0.119033\pi$
$607$	18.0000i	0.730597i	−0.930890	+	0.365299i	$0.880967\pi$
$608$	0	0
$609$	48.0000	1.94506
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 34.0000i	− 1.37325i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	− 34.0000i	− 1.37325i	0.727013	−	0.686624i	$-0.240908\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	−32.0000	−1.28412
$622$	0	0
$623$	− 24.0000i	− 0.961540i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.00000i	0.319489i
$628$	0	0
$629$	48.0000	1.91389
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	− 32.0000i	− 1.27189i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	− 28.0000i	− 1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$	− 28.0000i	− 1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.0000i	0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$	24.0000i	0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	−64.0000	−2.50836
$652$	0	0
$653$	− 26.0000i	− 1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$	− 26.0000i	− 1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 48.0000i	− 1.85857i
$668$	0	0
$669$	−4.00000	−0.154649
$670$	0	0
$671$	56.0000	2.16186
$672$	0	0
$673$	− 44.0000i	− 1.69608i	−0.529936	−	0.848038i	$-0.677784\pi$
$673$	− 44.0000i	− 1.69608i	0.529936	−	0.848038i	$-0.322216\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.0000i	0.768662i	0.923195	+	0.384331i	$0.125568\pi$
$677$	20.0000i	0.768662i	−0.923195	+	0.384331i	$0.874432\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	34.0000i	1.30097i	0.759517	+	0.650487i	$0.225435\pi$
$683$	34.0000i	1.30097i	−0.759517	+	0.650487i	$0.774565\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 12.0000i	− 0.457829i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	0	0
$693$	16.0000i	0.607790i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 12.0000i	− 0.454532i
$698$	0	0
$699$	−20.0000	−0.756469
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	− 8.00000i	− 0.301726i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 24.0000i	− 0.902613i
$708$	0	0
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	64.0000i	2.39682i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.0000i	1.79259i
$718$	0	0
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	0	0
$723$	4.00000i	0.148762i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 20.0000i	− 0.741759i	−0.928681	−	0.370879i	$-0.879056\pi$
$727$	− 20.0000i	− 0.741759i	0.928681	−	0.370879i	$-0.120944\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.00000i	0.294684i
$738$	0	0
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 30.0000i	− 1.10059i	−0.834969	−	0.550297i	$-0.814515\pi$
$743$	− 30.0000i	− 1.10059i	0.834969	−	0.550297i	$-0.185485\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	−72.0000	−2.63082
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	− 24.0000i	− 0.874609i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 34.0000i	− 1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$	− 34.0000i	− 1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$	0	0
$759$	64.0000	2.32305
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	0	0
$773$	− 48.0000i	− 1.72644i	−0.504828	−	0.863220i	$-0.668444\pi$
$773$	− 48.0000i	− 1.72644i	0.504828	−	0.863220i	$-0.331556\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 64.0000i	− 2.29599i
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	− 24.0000i	− 0.857690i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.0000i	1.21197i	0.795476	+	0.605985i	$0.207221\pi$
$787$	34.0000i	1.21197i	−0.795476	+	0.605985i	$0.792779\pi$
$788$	0	0
$789$	48.0000	1.70885
$790$	0	0
$791$	−64.0000	−2.27558
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	44.0000i	1.55856i	0.626676	+	0.779280i	$0.284415\pi$
$797$	44.0000i	1.55856i	−0.626676	+	0.779280i	$0.715585\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	− 8.00000i	− 0.282314i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.0000i	0.422420i
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	32.0000i	1.12229i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	52.0000i	1.81261i	0.422628	+	0.906303i	$0.361108\pi$
$823$	52.0000i	1.81261i	−0.422628	+	0.906303i	$0.638892\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.00000i	0.208640i	0.994544	+	0.104320i	$0.0332667\pi$
$827$	6.00000i	0.208640i	−0.994544	+	0.104320i	$0.966733\pi$
$828$	0	0
$829$	42.0000	1.45872	0.729360	−	0.684130i	$-0.239818\pi$
$829$	42.0000	1.45872	0.729360	+	0.684130i	$0.239818\pi$
$830$	0	0
$831$	52.0000	1.80386
$832$	0	0
$833$	− 54.0000i	− 1.87099i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	32.0000i	1.10608i
$838$	0	0
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	− 60.0000i	− 2.06651i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.0000i	0.687208i
$848$	0	0
$849$	8.00000	0.274559
$850$	0	0
$851$	−64.0000	−2.19389
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 28.0000i	− 0.956462i	−0.878234	−	0.478231i	$-0.841278\pi$
$857$	− 28.0000i	− 0.956462i	0.878234	−	0.478231i	$-0.158722\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	−16.0000	−0.545279
$862$	0	0
$863$	34.0000i	1.15737i	0.815550	+	0.578687i	$0.196435\pi$
$863$	34.0000i	1.15737i	−0.815550	+	0.578687i	$0.803565\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	38.0000i	1.29055i
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	52.0000i	1.75592i	0.478738	+	0.877958i	$0.341094\pi$
$877$	52.0000i	1.75592i	−0.478738	+	0.877958i	$0.658906\pi$
$878$	0	0
$879$	−8.00000	−0.269833
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	8.00000i	0.269221i	0.990899	+	0.134611i	$0.0429784\pi$
$883$	8.00000i	0.269221i	−0.990899	+	0.134611i	$0.957022\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 18.0000i	− 0.604381i	−0.953248	−	0.302190i	$-0.902282\pi$
$887$	− 18.0000i	− 0.604381i	0.953248	−	0.302190i	$-0.0977178\pi$
$888$	0	0
$889$	72.0000	2.41480
$890$	0	0
$891$	44.0000	1.47406
$892$	0	0
$893$	12.0000i	0.401565i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−48.0000	−1.60089
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 46.0000i	− 1.52740i	−0.645568	−	0.763702i	$-0.723379\pi$
$907$	− 46.0000i	− 1.52740i	0.645568	−	0.763702i	$-0.276621\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	−44.0000	−1.45779	−0.728893	−	0.684628i	$-0.759965\pi$
$911$	−44.0000	−1.45779	−0.728893	+	0.684628i	$0.759965\pi$
$912$	0	0
$913$	48.0000i	1.58857i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 80.0000i	− 2.64183i
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−44.0000	−1.44985
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6.00000i	− 0.197066i
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	0	0
$933$	16.0000i	0.523816i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	0	0
$939$	28.0000	0.913745
$940$	0	0
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	0	0
$943$	16.0000i	0.521032i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	64.0000	2.07534
$952$	0	0
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	48.0000i	1.55162i
$958$	0	0
$959$	40.0000	1.29167
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	− 18.0000i	− 0.580042i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 40.0000i	− 1.28631i	−0.765735	−	0.643157i	$-0.777624\pi$
$967$	− 40.0000i	− 1.28631i	0.765735	−	0.643157i	$-0.222376\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	− 16.0000i	− 0.512936i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 28.0000i	− 0.895799i	−0.894084	−	0.447900i	$-0.852172\pi$
$977$	− 28.0000i	− 0.895799i	0.894084	−	0.447900i	$-0.147828\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	34.0000i	1.08443i	0.840239	+	0.542216i	$0.182414\pi$
$983$	34.0000i	1.08443i	−0.840239	+	0.542216i	$0.817586\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	96.0000i	3.05571i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	32.0000i	1.01549i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 22.0000i	− 0.696747i	−0.937356	−	0.348373i	$-0.886734\pi$
$997$	− 22.0000i	− 0.696747i	0.937356	−	0.348373i	$-0.113266\pi$
$998$	0	0
$999$	−32.0000	−1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.c.3649.1		2
5.2	odd	4		3800.2.a.b.1.1		1
5.3	odd	4		760.2.a.d.1.1	✓	1
5.4	even	2	inner	3800.2.d.c.3649.2		2
15.8	even	4		6840.2.a.o.1.1		1
20.3	even	4		1520.2.a.c.1.1		1
20.7	even	4		7600.2.a.s.1.1		1
40.3	even	4		6080.2.a.s.1.1		1
40.13	odd	4		6080.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.d.1.1	✓	1	5.3	odd	4
1520.2.a.c.1.1		1	20.3	even	4
3800.2.a.b.1.1		1	5.2	odd	4
3800.2.d.c.3649.1		2	1.1	even	1	trivial
3800.2.d.c.3649.2		2	5.4	even	2	inner
6080.2.a.e.1.1		1	40.13	odd	4
6080.2.a.s.1.1		1	40.3	even	4
6840.2.a.o.1.1		1	15.8	even	4
7600.2.a.s.1.1		1	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)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +6.00000i q^{17} +1.00000 q^{19} +8.00000 q^{21} -8.00000i q^{23} -4.00000i q^{27} +6.00000 q^{29} -8.00000 q^{31} +8.00000i q^{33} -8.00000i q^{37} -2.00000 q^{41} +12.0000i q^{47} -9.00000 q^{49} +12.0000 q^{51} -4.00000i q^{53} -2.00000i q^{57} -8.00000 q^{59} -14.0000 q^{61} -4.00000i q^{63} -2.00000i q^{67} -16.0000 q^{69} -8.00000 q^{71} +2.00000i q^{73} -16.0000i q^{77} -4.00000 q^{79} -11.0000 q^{81} -12.0000i q^{83} -12.0000i q^{87} -6.00000 q^{89} +16.0000i q^{93} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 8 q^{11} + 2 q^{19} + 16 q^{21} + 12 q^{29} - 16 q^{31} - 4 q^{41} - 18 q^{49} + 24 q^{51} - 16 q^{59} - 28 q^{61} - 32 q^{69} - 16 q^{71} - 8 q^{79} - 22 q^{81} - 12 q^{89} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 8 * q^11 + 2 * q^19 + 16 * q^21 + 12 * q^29 - 16 * q^31 - 4 * q^41 - 18 * q^49 + 24 * q^51 - 16 * q^59 - 28 * q^61 - 32 * q^69 - 16 * q^71 - 8 * q^79 - 22 * q^81 - 12 * q^89 + 8 * q^99

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