LMFDB - Embedded newform 3800.2.d.l.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:	$0$
Dimension:	$6$
Coefficient field:	6.0.3356224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 8x^{4} + 16x^{2} + 1 $ x^6 + 8x^4 + 16x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.1
Root			$0.254102i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.l.3649.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68133i q^{3} +3.18953i q^{7} -4.18953 q^{9} +O(q^{10})$	$q-2.68133i q^{3} +3.18953i q^{7} -4.18953 q^{9} -0.681331i q^{13} +1.18953i q^{17} -1.00000 q^{19} +8.55220 q^{21} -2.17313i q^{23} +3.18953i q^{27} -2.81047 q^{29} +6.37907 q^{31} -7.87086i q^{37} -1.82687 q^{39} +0.983593 q^{41} +1.36266i q^{43} -11.7417i q^{47} -3.17313 q^{49} +3.18953 q^{51} -1.69774i q^{53} +2.68133i q^{57} -11.5358 q^{59} +7.36266 q^{61} -13.3627i q^{63} -7.02759i q^{67} -5.82687 q^{69} -12.7581 q^{71} -5.53579i q^{73} -5.36266 q^{79} -4.01641 q^{81} +2.37907i q^{83} +7.53579i q^{87} -3.01641 q^{89} +2.17313 q^{91} -17.1044i q^{93} -4.88727i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{9}+O(q^{10}) $ 6 * q - 8 * q^9	$ 6 q - 8 q^{9} - 6 q^{19} + 6 q^{21} - 34 q^{29} + 4 q^{31} - 22 q^{39} + 12 q^{41} - 8 q^{49} + 2 q^{51} - 30 q^{59} + 16 q^{61} - 46 q^{69} - 8 q^{71} - 4 q^{79} - 18 q^{81} - 12 q^{89} + 2 q^{91}+O(q^{100}) $ 6 * q - 8 * q^9 - 6 * q^19 + 6 * q^21 - 34 * q^29 + 4 * q^31 - 22 * q^39 + 12 * q^41 - 8 * q^49 + 2 * q^51 - 30 * q^59 + 16 * q^61 - 46 * q^69 - 8 * q^71 - 4 * q^79 - 18 * q^81 - 12 * q^89 + 2 * q^91

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
$3$	− 2.68133i	− 1.54807i	−0.633145	−	0.774033i	$-0.718236\pi$
$3$	− 2.68133i	− 1.54807i	0.633145	−	0.774033i	$-0.281764\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.18953i	1.20553i	0.797919	+	0.602765i	$0.205934\pi$
$7$	3.18953i	1.20553i	−0.797919	+	0.602765i	$0.794066\pi$
$8$	0	0
$9$	−4.18953	−1.39651
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	− 0.681331i	− 0.188967i	−0.995526	−	0.0944836i	$-0.969880\pi$
$13$	− 0.681331i	− 0.188967i	0.995526	−	0.0944836i	$-0.0301200\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.18953i	0.288504i	0.989541	+	0.144252i	$0.0460777\pi$
$17$	1.18953i	0.288504i	−0.989541	+	0.144252i	$0.953922\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	8.55220	1.86624
$22$	0	0
$23$	− 2.17313i	− 0.453128i	−0.973996	−	0.226564i	$-0.927251\pi$
$23$	− 2.17313i	− 0.453128i	0.973996	−	0.226564i	$-0.0727493\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.18953i	0.613826i
$28$	0	0
$29$	−2.81047	−0.521890	−0.260945	−	0.965354i	$-0.584034\pi$
$29$	−2.81047	−0.521890	−0.260945	+	0.965354i	$0.584034\pi$
$30$	0	0
$31$	6.37907	1.14571	0.572857	−	0.819655i	$-0.305835\pi$
$31$	6.37907	1.14571	0.572857	+	0.819655i	$0.305835\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.87086i	− 1.29396i	−0.762506	−	0.646981i	$-0.776031\pi$
$37$	− 7.87086i	− 1.29396i	0.762506	−	0.646981i	$-0.223969\pi$
$38$	0	0
$39$	−1.82687	−0.292534
$40$	0	0
$41$	0.983593	0.153611	0.0768057	−	0.997046i	$-0.475528\pi$
$41$	0.983593	0.153611	0.0768057	+	0.997046i	$0.475528\pi$
$42$	0	0
$43$	1.36266i	0.207804i	0.994588	+	0.103902i	$0.0331328\pi$
$43$	1.36266i	0.207804i	−0.994588	+	0.103902i	$0.966867\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 11.7417i	− 1.71271i	−0.516390	−	0.856354i	$-0.672724\pi$
$47$	− 11.7417i	− 1.71271i	0.516390	−	0.856354i	$-0.327276\pi$
$48$	0	0
$49$	−3.17313	−0.453304
$50$	0	0
$51$	3.18953	0.446624
$52$	0	0
$53$	− 1.69774i	− 0.233202i	−0.993179	−	0.116601i	$-0.962800\pi$
$53$	− 1.69774i	− 0.233202i	0.993179	−	0.116601i	$-0.0371999\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.68133i	0.355151i
$58$	0	0
$59$	−11.5358	−1.50183	−0.750916	−	0.660398i	$-0.770388\pi$
$59$	−11.5358	−1.50183	−0.750916	+	0.660398i	$0.770388\pi$
$60$	0	0
$61$	7.36266	0.942692	0.471346	−	0.881948i	$-0.343768\pi$
$61$	7.36266	0.942692	0.471346	+	0.881948i	$0.343768\pi$
$62$	0	0
$63$	− 13.3627i	− 1.68354i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.02759i	− 0.858556i	−0.903172	−	0.429278i	$-0.858768\pi$
$67$	− 7.02759i	− 0.858556i	0.903172	−	0.429278i	$-0.141232\pi$
$68$	0	0
$69$	−5.82687	−0.701473
$70$	0	0
$71$	−12.7581	−1.51411	−0.757056	−	0.653350i	$-0.773363\pi$
$71$	−12.7581	−1.51411	−0.757056	+	0.653350i	$0.773363\pi$
$72$	0	0
$73$	− 5.53579i	− 0.647915i	−0.946072	−	0.323958i	$-0.894987\pi$
$73$	− 5.53579i	− 0.647915i	0.946072	−	0.323958i	$-0.105013\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.36266	−0.603347	−0.301673	−	0.953411i	$-0.597545\pi$
$79$	−5.36266	−0.603347	−0.301673	+	0.953411i	$0.597545\pi$
$80$	0	0
$81$	−4.01641	−0.446267
$82$	0	0
$83$	2.37907i	0.261137i	0.991439	+	0.130568i	$0.0416802\pi$
$83$	2.37907i	0.261137i	−0.991439	+	0.130568i	$0.958320\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.53579i	0.807921i
$88$	0	0
$89$	−3.01641	−0.319738	−0.159869	−	0.987138i	$-0.551107\pi$
$89$	−3.01641	−0.319738	−0.159869	+	0.987138i	$0.551107\pi$
$90$	0	0
$91$	2.17313	0.227806
$92$	0	0
$93$	− 17.1044i	− 1.77364i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.88727i	− 0.496227i	−0.968731	−	0.248114i	$-0.920189\pi$
$97$	− 4.88727i	− 0.496227i	0.968731	−	0.248114i	$-0.0798106\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.1208	1.20606	0.603032	−	0.797717i	$-0.293959\pi$
$101$	12.1208	1.20606	0.603032	+	0.797717i	$0.293959\pi$
$102$	0	0
$103$	− 17.8709i	− 1.76087i	−0.474168	−	0.880434i	$-0.657251\pi$
$103$	− 17.8709i	− 1.76087i	0.474168	−	0.880434i	$-0.342749\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.4231i	− 1.39433i	−0.716911	−	0.697165i	$-0.754445\pi$
$107$	− 14.4231i	− 1.39433i	0.716911	−	0.697165i	$-0.245555\pi$
$108$	0	0
$109$	−10.2059	−0.977552	−0.488776	−	0.872409i	$-0.662556\pi$
$109$	−10.2059	−0.977552	−0.488776	+	0.872409i	$0.662556\pi$
$110$	0	0
$111$	−21.1044	−2.00314
$112$	0	0
$113$	− 1.49180i	− 0.140336i	−0.997535	−	0.0701682i	$-0.977646\pi$
$113$	− 1.49180i	− 0.140336i	0.997535	−	0.0701682i	$-0.0223536\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.85446i	0.263895i
$118$	0	0
$119$	−3.79406	−0.347801
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	− 2.63734i	− 0.237801i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.24993i	− 0.732063i	−0.930602	−	0.366032i	$-0.880716\pi$
$127$	− 8.24993i	− 0.732063i	0.930602	−	0.366032i	$-0.119284\pi$
$128$	0	0
$129$	3.65375	0.321694
$130$	0	0
$131$	8.75814	0.765202	0.382601	−	0.923914i	$-0.375028\pi$
$131$	8.75814	0.765202	0.382601	+	0.923914i	$0.375028\pi$
$132$	0	0
$133$	− 3.18953i	− 0.276568i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.9149i	1.01795i	0.860780	+	0.508977i	$0.169976\pi$
$137$	11.9149i	1.01795i	−0.860780	+	0.508977i	$0.830024\pi$
$138$	0	0
$139$	23.4835	1.99184	0.995920	−	0.0902352i	$-0.0287619\pi$
$139$	23.4835	1.99184	0.995920	+	0.0902352i	$0.0287619\pi$
$140$	0	0
$141$	−31.4835	−2.65139
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	8.50820i	0.701745i
$148$	0	0
$149$	18.8461	1.54393	0.771967	−	0.635662i	$-0.219273\pi$
$149$	18.8461	1.54393	0.771967	+	0.635662i	$0.219273\pi$
$150$	0	0
$151$	−16.0880	−1.30922	−0.654611	−	0.755966i	$-0.727167\pi$
$151$	−16.0880	−1.30922	−0.654611	+	0.755966i	$0.727167\pi$
$152$	0	0
$153$	− 4.98359i	− 0.402900i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.7417i	1.09671i	0.836246	+	0.548355i	$0.184746\pi$
$157$	13.7417i	1.09671i	−0.836246	+	0.548355i	$0.815254\pi$
$158$	0	0
$159$	−4.55220	−0.361013
$160$	0	0
$161$	6.93126	0.546260
$162$	0	0
$163$	− 14.6373i	− 1.14648i	−0.819386	−	0.573242i	$-0.805685\pi$
$163$	− 14.6373i	− 1.14648i	0.819386	−	0.573242i	$-0.194315\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 17.9588i	− 1.38970i	−0.719156	−	0.694849i	$-0.755471\pi$
$167$	− 17.9588i	− 1.38970i	0.719156	−	0.694849i	$-0.244529\pi$
$168$	0	0
$169$	12.5358	0.964291
$170$	0	0
$171$	4.18953	0.320382
$172$	0	0
$173$	6.85446i	0.521135i	0.965456	+	0.260567i	$0.0839096\pi$
$173$	6.85446i	0.521135i	−0.965456	+	0.260567i	$0.916090\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	30.9313i	2.32494i
$178$	0	0
$179$	−5.01641	−0.374944	−0.187472	−	0.982270i	$-0.560029\pi$
$179$	−5.01641	−0.374944	−0.187472	+	0.982270i	$0.560029\pi$
$180$	0	0
$181$	16.7253	1.24318	0.621592	−	0.783341i	$-0.286486\pi$
$181$	16.7253	1.24318	0.621592	+	0.783341i	$0.286486\pi$
$182$	0	0
$183$	− 19.7417i	− 1.45935i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−10.1731	−0.739986
$190$	0	0
$191$	−11.2775	−0.816013	−0.408006	−	0.912979i	$-0.633776\pi$
$191$	−11.2775	−0.816013	−0.408006	+	0.912979i	$0.633776\pi$
$192$	0	0
$193$	13.5798i	0.977494i	0.872426	+	0.488747i	$0.162546\pi$
$193$	13.5798i	0.977494i	−0.872426	+	0.488747i	$0.837454\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	−25.6566	−1.81875	−0.909374	−	0.415980i	$-0.863438\pi$
$199$	−25.6566	−1.81875	−0.909374	+	0.415980i	$0.863438\pi$
$200$	0	0
$201$	−18.8433	−1.32910
$202$	0	0
$203$	− 8.96408i	− 0.629155i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	9.10439i	0.632799i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	16.2939	1.12172	0.560860	−	0.827911i	$-0.310471\pi$
$211$	16.2939	1.12172	0.560860	+	0.827911i	$0.310471\pi$
$212$	0	0
$213$	34.2088i	2.34395i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	20.3463i	1.38119i
$218$	0	0
$219$	−14.8433	−1.00302
$220$	0	0
$221$	0.810466	0.0545178
$222$	0	0
$223$	− 24.2499i	− 1.62390i	−0.583730	−	0.811948i	$-0.698407\pi$
$223$	− 24.2499i	− 1.62390i	0.583730	−	0.811948i	$-0.301593\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5.31867i	− 0.353012i	−0.984300	−	0.176506i	$-0.943520\pi$
$227$	− 5.31867i	− 0.353012i	0.984300	−	0.176506i	$-0.0564796\pi$
$228$	0	0
$229$	−11.7089	−0.773747	−0.386873	−	0.922133i	$-0.626445\pi$
$229$	−11.7089	−0.773747	−0.386873	+	0.922133i	$0.626445\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	27.5163i	1.80265i	0.433142	+	0.901325i	$0.357405\pi$
$233$	27.5163i	1.80265i	−0.433142	+	0.901325i	$0.642595\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	14.3791i	0.934021i
$238$	0	0
$239$	−14.9313	−0.965823	−0.482912	−	0.875669i	$-0.660421\pi$
$239$	−14.9313	−0.965823	−0.482912	+	0.875669i	$0.660421\pi$
$240$	0	0
$241$	−9.32985	−0.600988	−0.300494	−	0.953784i	$-0.597152\pi$
$241$	−9.32985	−0.600988	−0.300494	+	0.953784i	$0.597152\pi$
$242$	0	0
$243$	20.3379i	1.30468i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.681331i	0.0433520i
$248$	0	0
$249$	6.37907	0.404257
$250$	0	0
$251$	1.27468	0.0804569	0.0402285	−	0.999191i	$-0.487191\pi$
$251$	1.27468	0.0804569	0.0402285	+	0.999191i	$0.487191\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 31.2887i	− 1.95174i	−0.218363	−	0.975868i	$-0.570072\pi$
$257$	− 31.2887i	− 1.95174i	0.218363	−	0.975868i	$-0.429928\pi$
$258$	0	0
$259$	25.1044	1.55991
$260$	0	0
$261$	11.7745	0.728826
$262$	0	0
$263$	25.5163i	1.57340i	0.617335	+	0.786700i	$0.288212\pi$
$263$	25.5163i	1.57340i	−0.617335	+	0.786700i	$0.711788\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.08798i	0.494977i
$268$	0	0
$269$	−26.7581	−1.63147	−0.815736	−	0.578424i	$-0.803668\pi$
$269$	−26.7581	−1.63147	−0.815736	+	0.578424i	$0.803668\pi$
$270$	0	0
$271$	14.1731	0.860956	0.430478	−	0.902601i	$-0.358345\pi$
$271$	14.1731	0.860956	0.430478	+	0.902601i	$0.358345\pi$
$272$	0	0
$273$	− 5.82687i	− 0.352658i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 6.08798i	− 0.365791i	−0.983132	−	0.182896i	$-0.941453\pi$
$277$	− 6.08798i	− 0.365791i	0.983132	−	0.182896i	$-0.0585471\pi$
$278$	0	0
$279$	−26.7253	−1.60000
$280$	0	0
$281$	−15.7089	−0.937115	−0.468558	−	0.883433i	$-0.655226\pi$
$281$	−15.7089	−0.937115	−0.468558	+	0.883433i	$0.655226\pi$
$282$	0	0
$283$	0.346255i	0.0205827i	0.999947	+	0.0102913i	$0.00327590\pi$
$283$	0.346255i	0.0205827i	−0.999947	+	0.0102913i	$0.996724\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.13720i	0.185183i
$288$	0	0
$289$	15.5850	0.916765
$290$	0	0
$291$	−13.1044	−0.768193
$292$	0	0
$293$	6.71414i	0.392244i	0.980579	+	0.196122i	$0.0628349\pi$
$293$	6.71414i	0.392244i	−0.980579	+	0.196122i	$0.937165\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.48062	−0.0856264
$300$	0	0
$301$	−4.34625	−0.250514
$302$	0	0
$303$	− 32.4999i	− 1.86707i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.6126i	1.00520i	0.864518	+	0.502602i	$0.167624\pi$
$307$	17.6126i	1.00520i	−0.864518	+	0.502602i	$0.832376\pi$
$308$	0	0
$309$	−47.9177	−2.72594
$310$	0	0
$311$	−6.58501	−0.373402	−0.186701	−	0.982417i	$-0.559779\pi$
$311$	−6.58501	−0.373402	−0.186701	+	0.982417i	$0.559779\pi$
$312$	0	0
$313$	− 18.6178i	− 1.05234i	−0.850379	−	0.526171i	$-0.823627\pi$
$313$	− 18.6178i	− 1.05234i	0.850379	−	0.526171i	$-0.176373\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.7857i	0.998946i	0.866330	+	0.499473i	$0.166473\pi$
$317$	17.7857i	0.998946i	−0.866330	+	0.499473i	$0.833527\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−38.6730	−2.15852
$322$	0	0
$323$	− 1.18953i	− 0.0661874i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	27.3655i	1.51332i
$328$	0	0
$329$	37.4506	2.06472
$330$	0	0
$331$	−3.53579	−0.194345	−0.0971723	−	0.995268i	$-0.530980\pi$
$331$	−3.53579	−0.194345	−0.0971723	+	0.995268i	$0.530980\pi$
$332$	0	0
$333$	32.9753i	1.80703i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.87086i	0.428753i	0.976751	+	0.214377i	$0.0687720\pi$
$337$	7.87086i	0.428753i	−0.976751	+	0.214377i	$0.931228\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	0	0
$342$	0	0
$343$	12.2059i	0.659059i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.6537i	− 0.840337i	−0.907446	−	0.420169i	$-0.861971\pi$
$347$	− 15.6537i	− 0.840337i	0.907446	−	0.420169i	$-0.138029\pi$
$348$	0	0
$349$	21.4178	1.14647	0.573235	−	0.819391i	$-0.305688\pi$
$349$	21.4178	1.14647	0.573235	+	0.819391i	$0.305688\pi$
$350$	0	0
$351$	2.17313	0.115993
$352$	0	0
$353$	3.15672i	0.168015i	0.996465	+	0.0840076i	$0.0267720\pi$
$353$	3.15672i	0.168015i	−0.996465	+	0.0840076i	$0.973228\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10.1731i	0.538419i
$358$	0	0
$359$	−17.8269	−0.940866	−0.470433	−	0.882436i	$-0.655902\pi$
$359$	−17.8269	−0.940866	−0.470433	+	0.882436i	$0.655902\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	29.4946i	1.54807i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 19.2252i	− 1.00355i	−0.864999	−	0.501773i	$-0.832681\pi$
$367$	− 19.2252i	− 1.00355i	0.864999	−	0.501773i	$-0.167319\pi$
$368$	0	0
$369$	−4.12080	−0.214520
$370$	0	0
$371$	5.41499	0.281132
$372$	0	0
$373$	9.95601i	0.515503i	0.966211	+	0.257751i	$0.0829815\pi$
$373$	9.95601i	0.515503i	−0.966211	+	0.257751i	$0.917018\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.91486i	0.0986201i
$378$	0	0
$379$	19.6238	1.00801	0.504003	−	0.863702i	$-0.331860\pi$
$379$	19.6238	1.00801	0.504003	+	0.863702i	$0.331860\pi$
$380$	0	0
$381$	−22.1208	−1.13328
$382$	0	0
$383$	11.9037i	0.608250i	0.952632	+	0.304125i	$0.0983640\pi$
$383$	11.9037i	0.608250i	−0.952632	+	0.304125i	$0.901636\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 5.70892i	− 0.290201i
$388$	0	0
$389$	−10.3463	−0.524576	−0.262288	−	0.964990i	$-0.584477\pi$
$389$	−10.3463	−0.524576	−0.262288	+	0.964990i	$0.584477\pi$
$390$	0	0
$391$	2.58501	0.130730
$392$	0	0
$393$	− 23.4835i	− 1.18458i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 26.1536i	− 1.31261i	−0.754495	−	0.656306i	$-0.772118\pi$
$397$	− 26.1536i	− 1.31261i	0.754495	−	0.656306i	$-0.227882\pi$
$398$	0	0
$399$	−8.55220	−0.428145
$400$	0	0
$401$	−32.7253	−1.63422	−0.817112	−	0.576479i	$-0.804426\pi$
$401$	−32.7253	−1.63422	−0.817112	+	0.576479i	$0.804426\pi$
$402$	0	0
$403$	− 4.34625i	− 0.216502i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−4.03281	−0.199410	−0.0997049	−	0.995017i	$-0.531790\pi$
$409$	−4.03281	−0.199410	−0.0997049	+	0.995017i	$0.531790\pi$
$410$	0	0
$411$	31.9477	1.57586
$412$	0	0
$413$	− 36.7938i	− 1.81050i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 62.9669i	− 3.08350i
$418$	0	0
$419$	15.8297	0.773332	0.386666	−	0.922220i	$-0.373627\pi$
$419$	15.8297	0.773332	0.386666	+	0.922220i	$0.373627\pi$
$420$	0	0
$421$	6.05233	0.294973	0.147486	−	0.989064i	$-0.452882\pi$
$421$	6.05233	0.294973	0.147486	+	0.989064i	$0.452882\pi$
$422$	0	0
$423$	49.1924i	2.39182i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	23.4835i	1.13644i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.7089	1.23835	0.619177	−	0.785251i	$-0.287466\pi$
$431$	25.7089	1.23835	0.619177	+	0.785251i	$0.287466\pi$
$432$	0	0
$433$	2.57383i	0.123690i	0.998086	+	0.0618452i	$0.0196985\pi$
$433$	2.57383i	0.123690i	−0.998086	+	0.0618452i	$0.980301\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.17313i	0.103955i
$438$	0	0
$439$	−26.8133	−1.27973	−0.639865	−	0.768488i	$-0.721010\pi$
$439$	−26.8133	−1.27973	−0.639865	+	0.768488i	$0.721010\pi$
$440$	0	0
$441$	13.2939	0.633044
$442$	0	0
$443$	10.7909i	0.512693i	0.966585	+	0.256347i	$0.0825189\pi$
$443$	10.7909i	0.512693i	−0.966585	+	0.256347i	$0.917481\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 50.5327i	− 2.39011i
$448$	0	0
$449$	29.4835	1.39141	0.695705	−	0.718327i	$-0.255092\pi$
$449$	29.4835	1.39141	0.695705	+	0.718327i	$0.255092\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	43.1372i	2.02676i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.8269i	0.553238i	0.960980	+	0.276619i	$0.0892140\pi$
$457$	11.8269i	0.553238i	−0.960980	+	0.276619i	$0.910786\pi$
$458$	0	0
$459$	−3.79406	−0.177092
$460$	0	0
$461$	−7.45065	−0.347011	−0.173506	−	0.984833i	$-0.555509\pi$
$461$	−7.45065	−0.347011	−0.173506	+	0.984833i	$0.555509\pi$
$462$	0	0
$463$	− 37.4506i	− 1.74048i	−0.492629	−	0.870240i	$-0.663964\pi$
$463$	− 37.4506i	− 1.74048i	0.492629	−	0.870240i	$-0.336036\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.4835i	1.27178i	0.771779	+	0.635891i	$0.219367\pi$
$467$	27.4835i	1.27178i	−0.771779	+	0.635891i	$0.780633\pi$
$468$	0	0
$469$	22.4147	1.03502
$470$	0	0
$471$	36.8461	1.69778
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.11273i	0.325669i
$478$	0	0
$479$	−10.6597	−0.487054	−0.243527	−	0.969894i	$-0.578304\pi$
$479$	−10.6597	−0.487054	−0.243527	+	0.969894i	$0.578304\pi$
$480$	0	0
$481$	−5.36266	−0.244516
$482$	0	0
$483$	− 18.5850i	− 0.845647i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 7.92604i	− 0.359163i	−0.983743	−	0.179581i	$-0.942526\pi$
$487$	− 7.92604i	− 0.359163i	0.983743	−	0.179581i	$-0.0574743\pi$
$488$	0	0
$489$	−39.2475	−1.77484
$490$	0	0
$491$	−1.68656	−0.0761133	−0.0380567	−	0.999276i	$-0.512117\pi$
$491$	−1.68656	−0.0761133	−0.0380567	+	0.999276i	$0.512117\pi$
$492$	0	0
$493$	− 3.34314i	− 0.150568i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 40.6925i	− 1.82531i
$498$	0	0
$499$	20.3463	0.910823	0.455412	−	0.890281i	$-0.349492\pi$
$499$	20.3463	0.910823	0.455412	+	0.890281i	$0.349492\pi$
$500$	0	0
$501$	−48.1536	−2.15134
$502$	0	0
$503$	1.06874i	0.0476526i	0.999716	+	0.0238263i	$0.00758487\pi$
$503$	1.06874i	0.0476526i	−0.999716	+	0.0238263i	$0.992415\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 33.6126i	− 1.49279i
$508$	0	0
$509$	19.0164	0.842887	0.421444	−	0.906855i	$-0.361524\pi$
$509$	19.0164	0.842887	0.421444	+	0.906855i	$0.361524\pi$
$510$	0	0
$511$	17.6566	0.781081
$512$	0	0
$513$	− 3.18953i	− 0.140821i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	18.3791	0.806752
$520$	0	0
$521$	9.07158	0.397433	0.198717	−	0.980057i	$-0.436323\pi$
$521$	9.07158	0.397433	0.198717	+	0.980057i	$0.436323\pi$
$522$	0	0
$523$	− 13.4946i	− 0.590079i	−0.955485	−	0.295040i	$-0.904667\pi$
$523$	− 13.4946i	− 0.590079i	0.955485	−	0.295040i	$-0.0953329\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.58812i	0.330544i
$528$	0	0
$529$	18.2775	0.794675
$530$	0	0
$531$	48.3296	2.09733
$532$	0	0
$533$	− 0.670152i	− 0.0290275i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	13.4506i	0.580438i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	4.63734	0.199375	0.0996874	−	0.995019i	$-0.468216\pi$
$541$	4.63734	0.199375	0.0996874	+	0.995019i	$0.468216\pi$
$542$	0	0
$543$	− 44.8461i	− 1.92453i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18.1291i	0.775146i	0.921839	+	0.387573i	$0.126686\pi$
$547$	18.1291i	0.775146i	−0.921839	+	0.387573i	$0.873314\pi$
$548$	0	0
$549$	−30.8461	−1.31648
$550$	0	0
$551$	2.81047	0.119730
$552$	0	0
$553$	− 17.1044i	− 0.727353i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 32.2968i	− 1.36846i	−0.729267	−	0.684229i	$-0.760139\pi$
$557$	− 32.2968i	− 1.36846i	0.729267	−	0.684229i	$-0.239861\pi$
$558$	0	0
$559$	0.928423	0.0392681
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 31.3871i	− 1.32281i	−0.750029	−	0.661405i	$-0.769960\pi$
$563$	− 31.3871i	− 1.32281i	0.750029	−	0.661405i	$-0.230040\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 12.8105i	− 0.537989i
$568$	0	0
$569$	31.7969	1.33300	0.666498	−	0.745507i	$-0.267793\pi$
$569$	31.7969	1.33300	0.666498	+	0.745507i	$0.267793\pi$
$570$	0	0
$571$	18.3134	0.766394	0.383197	−	0.923667i	$-0.374823\pi$
$571$	18.3134	0.766394	0.383197	+	0.923667i	$0.374823\pi$
$572$	0	0
$573$	30.2388i	1.26324i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 26.7282i	− 1.11271i	−0.830945	−	0.556354i	$-0.812200\pi$
$577$	− 26.7282i	− 1.11271i	0.830945	−	0.556354i	$-0.187800\pi$
$578$	0	0
$579$	36.4119	1.51323
$580$	0	0
$581$	−7.58812	−0.314808
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 24.3463i	− 1.00488i	−0.864613	−	0.502439i	$-0.832436\pi$
$587$	− 24.3463i	− 1.00488i	0.864613	−	0.502439i	$-0.167564\pi$
$588$	0	0
$589$	−6.37907	−0.262845
$590$	0	0
$591$	5.36266	0.220590
$592$	0	0
$593$	12.7253i	0.522566i	0.965262	+	0.261283i	$0.0841456\pi$
$593$	12.7253i	0.522566i	−0.965262	+	0.261283i	$0.915854\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	68.7938i	2.81554i
$598$	0	0
$599$	−10.6373	−0.434630	−0.217315	−	0.976102i	$-0.569730\pi$
$599$	−10.6373	−0.434630	−0.217315	+	0.976102i	$0.569730\pi$
$600$	0	0
$601$	11.5329	0.470439	0.235219	−	0.971942i	$-0.424419\pi$
$601$	11.5329	0.470439	0.235219	+	0.971942i	$0.424419\pi$
$602$	0	0
$603$	29.4423i	1.19898i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.2827i	0.742074i	0.928618	+	0.371037i	$0.120998\pi$
$607$	18.2827i	0.742074i	−0.928618	+	0.371037i	$0.879002\pi$
$608$	0	0
$609$	−24.0357	−0.973974
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	− 6.84612i	− 0.276512i	−0.990397	−	0.138256i	$-0.955850\pi$
$613$	− 6.84612i	− 0.276512i	0.990397	−	0.138256i	$-0.0441497\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 15.0820i	− 0.607180i	−0.952803	−	0.303590i	$-0.901815\pi$
$617$	− 15.0820i	− 0.607180i	0.952803	−	0.303590i	$-0.0981853\pi$
$618$	0	0
$619$	42.7253	1.71728	0.858638	−	0.512583i	$-0.171311\pi$
$619$	42.7253	1.71728	0.858638	+	0.512583i	$0.171311\pi$
$620$	0	0
$621$	6.93126	0.278142
$622$	0	0
$623$	− 9.62093i	− 0.385454i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.36266	0.373314
$630$	0	0
$631$	26.7909	1.06653	0.533265	−	0.845948i	$-0.320965\pi$
$631$	26.7909	1.06653	0.533265	+	0.845948i	$0.320965\pi$
$632$	0	0
$633$	− 43.6894i	− 1.73650i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.16195i	0.0856595i
$638$	0	0
$639$	53.4506	2.11447
$640$	0	0
$641$	28.1208	1.11070	0.555352	−	0.831615i	$-0.312583\pi$
$641$	28.1208	1.11070	0.555352	+	0.831615i	$0.312583\pi$
$642$	0	0
$643$	15.1372i	0.596953i	0.954417	+	0.298477i	$0.0964785\pi$
$643$	15.1372i	0.596953i	−0.954417	+	0.298477i	$0.903522\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.06874i	− 0.0420164i	−0.999779	−	0.0210082i	$-0.993312\pi$
$647$	− 1.06874i	− 0.0420164i	0.999779	−	0.0210082i	$-0.00668761\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	54.5550	2.13818
$652$	0	0
$653$	− 21.3298i	− 0.834701i	−0.908745	−	0.417351i	$-0.862959\pi$
$653$	− 21.3298i	− 0.834701i	0.908745	−	0.417351i	$-0.137041\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	23.1924i	0.904821i
$658$	0	0
$659$	−45.4639	−1.77102	−0.885512	−	0.464617i	$-0.846192\pi$
$659$	−45.4639	−1.77102	−0.885512	+	0.464617i	$0.846192\pi$
$660$	0	0
$661$	28.6074	1.11270	0.556349	−	0.830949i	$-0.312202\pi$
$661$	28.6074	1.11270	0.556349	+	0.830949i	$0.312202\pi$
$662$	0	0
$663$	− 2.17313i	− 0.0843973i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.10750i	0.236483i
$668$	0	0
$669$	−65.0221	−2.51390
$670$	0	0
$671$	0	0
$672$	0	0
$673$	47.1840i	1.81881i	0.415911	+	0.909405i	$0.363463\pi$
$673$	47.1840i	1.81881i	−0.415911	+	0.909405i	$0.636537\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 20.8573i	− 0.801611i	−0.916163	−	0.400806i	$-0.868730\pi$
$677$	− 20.8573i	− 0.801611i	0.916163	−	0.400806i	$-0.131270\pi$
$678$	0	0
$679$	15.5881	0.598217
$680$	0	0
$681$	−14.2611	−0.546487
$682$	0	0
$683$	− 17.2887i	− 0.661534i	−0.943713	−	0.330767i	$-0.892693\pi$
$683$	− 17.2887i	− 0.661534i	0.943713	−	0.330767i	$-0.107307\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	31.3955i	1.19781i
$688$	0	0
$689$	−1.15672	−0.0440675
$690$	0	0
$691$	−35.8297	−1.36303	−0.681513	−	0.731806i	$-0.738678\pi$
$691$	−35.8297	−1.36303	−0.681513	+	0.731806i	$0.738678\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.17002i	0.0443176i
$698$	0	0
$699$	73.7802	2.79062
$700$	0	0
$701$	−38.0880	−1.43856	−0.719282	−	0.694719i	$-0.755529\pi$
$701$	−38.0880	−1.43856	−0.719282	+	0.694719i	$0.755529\pi$
$702$	0	0
$703$	7.87086i	0.296855i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	38.6597i	1.45395i
$708$	0	0
$709$	15.2747	0.573653	0.286826	−	0.957983i	$-0.407400\pi$
$709$	15.2747	0.573653	0.286826	+	0.957983i	$0.407400\pi$
$710$	0	0
$711$	22.4671	0.842580
$712$	0	0
$713$	− 13.8625i	− 0.519156i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	40.0357i	1.49516i
$718$	0	0
$719$	27.8597	1.03899	0.519495	−	0.854473i	$-0.326120\pi$
$719$	27.8597	1.03899	0.519495	+	0.854473i	$0.326120\pi$
$720$	0	0
$721$	56.9997	2.12278
$722$	0	0
$723$	25.0164i	0.930370i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	39.4311i	1.46242i	0.682153	+	0.731210i	$0.261044\pi$
$727$	39.4311i	1.46242i	−0.682153	+	0.731210i	$0.738956\pi$
$728$	0	0
$729$	42.4835	1.57346
$730$	0	0
$731$	−1.62093	−0.0599523
$732$	0	0
$733$	33.1372i	1.22395i	0.790877	+	0.611975i	$0.209625\pi$
$733$	33.1372i	1.22395i	−0.790877	+	0.611975i	$0.790375\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	25.8625	0.951368	0.475684	−	0.879616i	$-0.342201\pi$
$739$	25.8625	0.951368	0.475684	+	0.879616i	$0.342201\pi$
$740$	0	0
$741$	1.82687	0.0671118
$742$	0	0
$743$	− 28.6842i	− 1.05232i	−0.850386	−	0.526160i	$-0.823631\pi$
$743$	− 28.6842i	− 1.05232i	0.850386	−	0.526160i	$-0.176369\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 9.96719i	− 0.364680i
$748$	0	0
$749$	46.0028	1.68091
$750$	0	0
$751$	−34.2968	−1.25151	−0.625753	−	0.780021i	$-0.715208\pi$
$751$	−34.2968	−1.25151	−0.625753	+	0.780021i	$0.715208\pi$
$752$	0	0
$753$	− 3.41783i	− 0.124553i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 20.6597i	− 0.750889i	−0.926845	−	0.375445i	$-0.877490\pi$
$757$	− 20.6597i	− 0.750889i	0.926845	−	0.375445i	$-0.122510\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.1236	1.38198	0.690990	−	0.722864i	$-0.257175\pi$
$761$	38.1236	1.38198	0.690990	+	0.722864i	$0.257175\pi$
$762$	0	0
$763$	− 32.5522i	− 1.17847i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.85969i	0.283797i
$768$	0	0
$769$	−20.3267	−0.733001	−0.366500	−	0.930418i	$-0.619444\pi$
$769$	−20.3267	−0.733001	−0.366500	+	0.930418i	$0.619444\pi$
$770$	0	0
$771$	−83.8953	−3.02142
$772$	0	0
$773$	26.4559i	0.951552i	0.879567	+	0.475776i	$0.157833\pi$
$773$	26.4559i	0.951552i	−0.879567	+	0.475776i	$0.842167\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 67.3132i	− 2.41485i
$778$	0	0
$779$	−0.983593	−0.0352409
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 8.96408i	− 0.320350i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	27.3738i	0.975772i	0.872907	+	0.487886i	$0.162232\pi$
$787$	27.3738i	0.975772i	−0.872907	+	0.487886i	$0.837768\pi$
$788$	0	0
$789$	68.4176	2.43573
$790$	0	0
$791$	4.75814	0.169180
$792$	0	0
$793$	− 5.01641i	− 0.178138i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 36.4454i	− 1.29096i	−0.763776	−	0.645481i	$-0.776657\pi$
$797$	− 36.4454i	− 1.29096i	0.763776	−	0.645481i	$-0.223343\pi$
$798$	0	0
$799$	13.9672	0.494124
$800$	0	0
$801$	12.6373	0.446518
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	71.7474i	2.52563i
$808$	0	0
$809$	−52.2444	−1.83682	−0.918408	−	0.395634i	$-0.870525\pi$
$809$	−52.2444	−1.83682	−0.918408	+	0.395634i	$0.870525\pi$
$810$	0	0
$811$	32.8984	1.15522	0.577610	−	0.816313i	$-0.303985\pi$
$811$	32.8984	1.15522	0.577610	+	0.816313i	$0.303985\pi$
$812$	0	0
$813$	− 38.0028i	− 1.33282i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.36266i	− 0.0476735i
$818$	0	0
$819$	−9.10439	−0.318133
$820$	0	0
$821$	−4.44470	−0.155121	−0.0775605	−	0.996988i	$-0.524713\pi$
$821$	−4.44470	−0.155121	−0.0775605	+	0.996988i	$0.524713\pi$
$822$	0	0
$823$	− 27.7774i	− 0.968259i	−0.874996	−	0.484129i	$-0.839136\pi$
$823$	− 27.7774i	− 0.968259i	0.874996	−	0.484129i	$-0.160864\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 46.5110i	− 1.61735i	−0.588257	−	0.808674i	$-0.700186\pi$
$827$	− 46.5110i	− 1.61735i	0.588257	−	0.808674i	$-0.299814\pi$
$828$	0	0
$829$	−28.2388	−0.980772	−0.490386	−	0.871505i	$-0.663144\pi$
$829$	−28.2388	−0.980772	−0.490386	+	0.871505i	$0.663144\pi$
$830$	0	0
$831$	−16.3239	−0.566270
$832$	0	0
$833$	− 3.77454i	− 0.130780i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	20.3463i	0.703269i
$838$	0	0
$839$	−9.01641	−0.311281	−0.155640	−	0.987814i	$-0.549744\pi$
$839$	−9.01641	−0.311281	−0.155640	+	0.987814i	$0.549744\pi$
$840$	0	0
$841$	−21.1013	−0.727630
$842$	0	0
$843$	42.1208i	1.45072i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 35.0849i	− 1.20553i
$848$	0	0
$849$	0.928423	0.0318634
$850$	0	0
$851$	−17.1044	−0.586331
$852$	0	0
$853$	5.30749i	0.181725i	0.995863	+	0.0908625i	$0.0289624\pi$
$853$	5.30749i	0.181725i	−0.995863	+	0.0908625i	$0.971038\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.9258i	1.32968i	0.746986	+	0.664839i	$0.231500\pi$
$857$	38.9258i	1.32968i	−0.746986	+	0.664839i	$0.768500\pi$
$858$	0	0
$859$	−20.1760	−0.688395	−0.344198	−	0.938897i	$-0.611849\pi$
$859$	−20.1760	−0.688395	−0.344198	+	0.938897i	$0.611849\pi$
$860$	0	0
$861$	8.41188	0.286676
$862$	0	0
$863$	− 30.3051i	− 1.03160i	−0.856710	−	0.515799i	$-0.827495\pi$
$863$	− 30.3051i	− 1.03160i	0.856710	−	0.515799i	$-0.172505\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 41.7886i	− 1.41921i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−4.78811	−0.162239
$872$	0	0
$873$	20.4754i	0.692987i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 9.43947i	− 0.318748i	−0.987218	−	0.159374i	$-0.949052\pi$
$877$	− 9.43947i	− 0.318748i	0.987218	−	0.159374i	$-0.0509476\pi$
$878$	0	0
$879$	18.0028	0.607221
$880$	0	0
$881$	−24.4671	−0.824316	−0.412158	−	0.911112i	$-0.635225\pi$
$881$	−24.4671	−0.824316	−0.412158	+	0.911112i	$0.635225\pi$
$882$	0	0
$883$	6.12080i	0.205981i	0.994682	+	0.102991i	$0.0328412\pi$
$883$	6.12080i	0.205981i	−0.994682	+	0.102991i	$0.967159\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 45.4200i	− 1.52505i	−0.646957	−	0.762526i	$-0.723959\pi$
$887$	− 45.4200i	− 1.52505i	0.646957	−	0.762526i	$-0.276041\pi$
$888$	0	0
$889$	26.3134	0.882524
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11.7417i	0.392922i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.97003i	0.132555i
$898$	0	0
$899$	−17.9282	−0.597937
$900$	0	0
$901$	2.01952	0.0672798
$902$	0	0
$903$	11.6537i	0.387812i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	10.2694i	0.340991i	0.985358	+	0.170496i	$0.0545369\pi$
$907$	10.2694i	0.340991i	−0.985358	+	0.170496i	$0.945463\pi$
$908$	0	0
$909$	−50.7805	−1.68428
$910$	0	0
$911$	−10.9180	−0.361728	−0.180864	−	0.983508i	$-0.557889\pi$
$911$	−10.9180	−0.361728	−0.180864	+	0.983508i	$0.557889\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.9344i	0.922474i
$918$	0	0
$919$	38.9313	1.28422	0.642112	−	0.766611i	$-0.278058\pi$
$919$	38.9313	1.28422	0.642112	+	0.766611i	$0.278058\pi$
$920$	0	0
$921$	47.2252	1.55612
$922$	0	0
$923$	8.69251i	0.286117i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	74.8706i	2.45907i
$928$	0	0
$929$	−21.9700	−0.720813	−0.360407	−	0.932795i	$-0.617362\pi$
$929$	−21.9700	−0.720813	−0.360407	+	0.932795i	$0.617362\pi$
$930$	0	0
$931$	3.17313	0.103995
$932$	0	0
$933$	17.6566i	0.578051i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 10.6402i	− 0.347600i	−0.984781	−	0.173800i	$-0.944395\pi$
$937$	− 10.6402i	− 0.347600i	0.984781	−	0.173800i	$-0.0556045\pi$
$938$	0	0
$939$	−49.9205	−1.62910
$940$	0	0
$941$	−20.8656	−0.680200	−0.340100	−	0.940389i	$-0.610461\pi$
$941$	−20.8656	−0.680200	−0.340100	+	0.940389i	$0.610461\pi$
$942$	0	0
$943$	− 2.13747i	− 0.0696057i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 12.6925i	− 0.412451i	−0.978504	−	0.206226i	$-0.933882\pi$
$947$	− 12.6925i	− 0.412451i	0.978504	−	0.206226i	$-0.0661181\pi$
$948$	0	0
$949$	−3.77170	−0.122435
$950$	0	0
$951$	47.6894	1.54643
$952$	0	0
$953$	16.7337i	0.542056i	0.962571	+	0.271028i	$0.0873637\pi$
$953$	16.7337i	0.542056i	−0.962571	+	0.271028i	$0.912636\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−38.0028	−1.22718
$960$	0	0
$961$	9.69251	0.312662
$962$	0	0
$963$	60.4259i	1.94720i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.62688i	− 0.148790i	−0.997229	−	0.0743952i	$-0.976297\pi$
$967$	− 4.62688i	− 0.148790i	0.997229	−	0.0743952i	$-0.0237026\pi$
$968$	0	0
$969$	−3.18953	−0.102463
$970$	0	0
$971$	−49.4506	−1.58695	−0.793473	−	0.608605i	$-0.791729\pi$
$971$	−49.4506	−1.58695	−0.793473	+	0.608605i	$0.791729\pi$
$972$	0	0
$973$	74.9013i	2.40123i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 22.5962i	− 0.722916i	−0.932388	−	0.361458i	$-0.882279\pi$
$977$	− 22.5962i	− 0.722916i	0.932388	−	0.361458i	$-0.117721\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	42.7581	1.36516
$982$	0	0
$983$	32.9424i	1.05070i	0.850886	+	0.525350i	$0.176066\pi$
$983$	32.9424i	1.05070i	−0.850886	+	0.525350i	$0.823934\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 100.418i	− 3.19633i
$988$	0	0
$989$	2.96124	0.0941618
$990$	0	0
$991$	−49.7802	−1.58132	−0.790660	−	0.612255i	$-0.790263\pi$
$991$	−49.7802	−1.58132	−0.790660	+	0.612255i	$0.790263\pi$
$992$	0	0
$993$	9.48062i	0.300858i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	36.4014i	1.15284i	0.817152	+	0.576422i	$0.195552\pi$
$997$	36.4014i	1.15284i	−0.817152	+	0.576422i	$0.804448\pi$
$998$	0	0
$999$	25.1044	0.794268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.l.3649.1		6
5.2	odd	4		760.2.a.j.1.1	✓	3
5.3	odd	4		3800.2.a.x.1.3		3
5.4	even	2	inner	3800.2.d.l.3649.6		6
15.2	even	4		6840.2.a.bg.1.1		3
20.3	even	4		7600.2.a.bq.1.1		3
20.7	even	4		1520.2.a.s.1.3		3
40.27	even	4		6080.2.a.bq.1.1		3
40.37	odd	4		6080.2.a.bv.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.j.1.1	✓	3	5.2	odd	4
1520.2.a.s.1.3		3	20.7	even	4
3800.2.a.x.1.3		3	5.3	odd	4
3800.2.d.l.3649.1		6	1.1	even	1	trivial
3800.2.d.l.3649.6		6	5.4	even	2	inner
6080.2.a.bq.1.1		3	40.27	even	4
6080.2.a.bv.1.3		3	40.37	odd	4
6840.2.a.bg.1.1		3	15.2	even	4
7600.2.a.bq.1.1		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)

\(f(q)\)	\(=\)	\(q-2.68133i q^{3} +3.18953i q^{7} -4.18953 q^{9} +O(q^{10})\)	\(q-2.68133i q^{3} +3.18953i q^{7} -4.18953 q^{9} -0.681331i q^{13} +1.18953i q^{17} -1.00000 q^{19} +8.55220 q^{21} -2.17313i q^{23} +3.18953i q^{27} -2.81047 q^{29} +6.37907 q^{31} -7.87086i q^{37} -1.82687 q^{39} +0.983593 q^{41} +1.36266i q^{43} -11.7417i q^{47} -3.17313 q^{49} +3.18953 q^{51} -1.69774i q^{53} +2.68133i q^{57} -11.5358 q^{59} +7.36266 q^{61} -13.3627i q^{63} -7.02759i q^{67} -5.82687 q^{69} -12.7581 q^{71} -5.53579i q^{73} -5.36266 q^{79} -4.01641 q^{81} +2.37907i q^{83} +7.53579i q^{87} -3.01641 q^{89} +2.17313 q^{91} -17.1044i q^{93} -4.88727i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{9}+O(q^{10}) \) 6 * q - 8 * q^9	\( 6 q - 8 q^{9} - 6 q^{19} + 6 q^{21} - 34 q^{29} + 4 q^{31} - 22 q^{39} + 12 q^{41} - 8 q^{49} + 2 q^{51} - 30 q^{59} + 16 q^{61} - 46 q^{69} - 8 q^{71} - 4 q^{79} - 18 q^{81} - 12 q^{89} + 2 q^{91}+O(q^{100}) \) 6 * q - 8 * q^9 - 6 * q^19 + 6 * q^21 - 34 * q^29 + 4 * q^31 - 22 * q^39 + 12 * q^41 - 8 * q^49 + 2 * q^51 - 30 * q^59 + 16 * q^61 - 46 * q^69 - 8 * q^71 - 4 * q^79 - 18 * q^81 - 12 * q^89 + 2 * q^91

Introduction

L-functions

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Database

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