LMFDB - Embedded newform 3800.2.a.bd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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, 0, 0]))

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

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

chi := DirichletCharacter("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 16x^{3} + 33x^{2} - 4x - 7 $ x^6 - 2x^5 - 12x^4 + 16x^3 + 33x^2 - 4*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.486697$ of defining polynomial
Character	$\chi$	$=$	3800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.486697 q^{3} +3.63249 q^{7} -2.76313 q^{9} -2.79460 q^{11} +2.86457 q^{13} +1.17400 q^{17} +1.00000 q^{19} +1.76793 q^{21} -0.617328 q^{23} -2.80490 q^{27} -4.96700 q^{29} +0.745377 q^{31} -1.36012 q^{33} +8.23679 q^{37} +1.39418 q^{39} +9.98217 q^{41} +10.4955 q^{43} +5.07742 q^{47} +6.19502 q^{49} +0.571381 q^{51} -7.45370 q^{53} +0.486697 q^{57} +3.83310 q^{59} +11.2450 q^{61} -10.0370 q^{63} +6.10730 q^{67} -0.300452 q^{69} -9.40599 q^{71} -9.52367 q^{73} -10.1514 q^{77} -3.70094 q^{79} +6.92424 q^{81} -4.66397 q^{83} -2.41743 q^{87} +10.6888 q^{89} +10.4055 q^{91} +0.362773 q^{93} +0.629221 q^{97} +7.72183 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 2 q^{7} + 10 q^{9} + 3 q^{11} - 3 q^{13} - 2 q^{17} + 6 q^{19} + 11 q^{21} + 4 q^{23} + 20 q^{27} + 7 q^{29} + 5 q^{31} - 16 q^{33} + 8 q^{39} + 11 q^{41} - 7 q^{43} + 20 q^{47} - 2 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 + 2 * q^7 + 10 * q^9 + 3 * q^11 - 3 * q^13 - 2 * q^17 + 6 * q^19 + 11 * q^21 + 4 * q^23 + 20 * q^27 + 7 * q^29 + 5 * q^31 - 16 * q^33 + 8 * q^39 + 11 * q^41 - 7 * q^43 + 20 * q^47 - 2 * q^49 + 13 * q^51 - 7 * q^53 + 2 * q^57 - 4 * q^59 + 13 * q^61 - q^63 + 25 * q^67 + 7 * q^69 + 29 * q^71 - 19 * q^73 + 24 * q^77 + 28 * q^79 + 38 * q^81 - 15 * q^83 + 57 * q^87 - 12 * q^89 - 27 * q^93 - 13 * q^97 + 10 * q^99

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
$3$	0.486697	0.280995	0.140497	−	0.990081i	$-0.455130\pi$
$3$	0.486697	0.280995	0.140497	+	0.990081i	$0.455130\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.63249	1.37295	0.686477	−	0.727152i	$-0.259156\pi$
$7$	3.63249	1.37295	0.686477	+	0.727152i	$0.259156\pi$
$8$	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
$13$	2.86457	0.794489	0.397244	−	0.917713i	$-0.369967\pi$
$13$	2.86457	0.794489	0.397244	+	0.917713i	$0.369967\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.17400	0.284736	0.142368	−	0.989814i	$-0.454528\pi$
$17$	1.17400	0.284736	0.142368	+	0.989814i	$0.454528\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.76793	0.385793
$22$	0	0
$23$	−0.617328	−0.128722	−0.0643609	−	0.997927i	$-0.520501\pi$
$23$	−0.617328	−0.128722	−0.0643609	+	0.997927i	$0.520501\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.80490	−0.539803
$28$	0	0
$29$	−4.96700	−0.922349	−0.461175	−	0.887309i	$-0.652572\pi$
$29$	−4.96700	−0.922349	−0.461175	+	0.887309i	$0.652572\pi$
$30$	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
$33$	−1.36012	−0.236767
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.23679	1.35412	0.677060	−	0.735928i	$-0.263254\pi$
$37$	8.23679	1.35412	0.677060	+	0.735928i	$0.263254\pi$
$38$	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
$43$	10.4955	1.60055	0.800277	−	0.599630i	$-0.204686\pi$
$43$	10.4955	1.60055	0.800277	+	0.599630i	$0.204686\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.07742	0.740618	0.370309	−	0.928909i	$-0.379252\pi$
$47$	5.07742	0.740618	0.370309	+	0.928909i	$0.379252\pi$
$48$	0	0
$49$	6.19502	0.885003
$50$	0	0
$51$	0.571381	0.0800093
$52$	0	0
$53$	−7.45370	−1.02384	−0.511922	−	0.859032i	$-0.671066\pi$
$53$	−7.45370	−1.02384	−0.511922	+	0.859032i	$0.671066\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.486697	0.0644646
$58$	0	0
$59$	3.83310	0.499027	0.249513	−	0.968371i	$-0.419729\pi$
$59$	3.83310	0.499027	0.249513	+	0.968371i	$0.419729\pi$
$60$	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
$63$	−10.0370	−1.26455
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.10730	0.746125	0.373063	−	0.927806i	$-0.378308\pi$
$67$	6.10730	0.746125	0.373063	+	0.927806i	$0.378308\pi$
$68$	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
$73$	−9.52367	−1.11466	−0.557331	−	0.830291i	$-0.688174\pi$
$73$	−9.52367	−1.11466	−0.557331	+	0.830291i	$0.688174\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10.1514	−1.15686
$78$	0	0
$79$	−3.70094	−0.416388	−0.208194	−	0.978088i	$-0.566759\pi$
$79$	−3.70094	−0.416388	−0.208194	+	0.978088i	$0.566759\pi$
$80$	0	0
$81$	6.92424	0.769360
$82$	0	0
$83$	−4.66397	−0.511937	−0.255968	−	0.966685i	$-0.582394\pi$
$83$	−4.66397	−0.511937	−0.255968	+	0.966685i	$0.582394\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.41743	−0.259175
$88$	0	0
$89$	10.6888	1.13301	0.566506	−	0.824058i	$-0.308295\pi$
$89$	10.6888	1.13301	0.566506	+	0.824058i	$0.308295\pi$
$90$	0	0
$91$	10.4055	1.09080
$92$	0	0
$93$	0.362773	0.0376178
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.629221	0.0638877	0.0319439	−	0.999490i	$-0.489830\pi$
$97$	0.629221	0.0638877	0.0319439	+	0.999490i	$0.489830\pi$
$98$	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
$103$	−10.0702	−0.992251	−0.496125	−	0.868251i	$-0.665244\pi$
$103$	−10.0702	−0.992251	−0.496125	+	0.868251i	$0.665244\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.53412	0.438330	0.219165	−	0.975688i	$-0.429667\pi$
$107$	4.53412	0.438330	0.219165	+	0.975688i	$0.429667\pi$
$108$	0	0
$109$	15.9974	1.53227	0.766137	−	0.642678i	$-0.222177\pi$
$109$	15.9974	1.53227	0.766137	+	0.642678i	$0.222177\pi$
$110$	0	0
$111$	4.00882	0.380501
$112$	0	0
$113$	−2.34828	−0.220908	−0.110454	−	0.993881i	$-0.535230\pi$
$113$	−2.34828	−0.220908	−0.110454	+	0.993881i	$0.535230\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−7.91517	−0.731757
$118$	0	0
$119$	4.26454	0.390929
$120$	0	0
$121$	−3.19022	−0.290020
$122$	0	0
$123$	4.85829	0.438058
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.4542	1.54881	0.774403	−	0.632693i	$-0.218050\pi$
$127$	17.4542	1.54881	0.774403	+	0.632693i	$0.218050\pi$
$128$	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
$133$	3.63249	0.314977
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.4369	1.31887	0.659433	−	0.751764i	$-0.270797\pi$
$137$	15.4369	1.31887	0.659433	+	0.751764i	$0.270797\pi$
$138$	0	0
$139$	15.1144	1.28199	0.640993	−	0.767547i	$-0.278523\pi$
$139$	15.1144	1.28199	0.640993	+	0.767547i	$0.278523\pi$
$140$	0	0
$141$	2.47117	0.208110
$142$	0	0
$143$	−8.00532	−0.669439
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.01510	0.248681
$148$	0	0
$149$	−11.8422	−0.970153	−0.485076	−	0.874472i	$-0.661208\pi$
$149$	−11.8422	−0.970153	−0.485076	+	0.874472i	$0.661208\pi$
$150$	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
$153$	−3.24390	−0.262254
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.6650	−0.851162	−0.425581	−	0.904920i	$-0.639930\pi$
$157$	−10.6650	−0.851162	−0.425581	+	0.904920i	$0.639930\pi$
$158$	0	0
$159$	−3.62770	−0.287695
$160$	0	0
$161$	−2.24244	−0.176729
$162$	0	0
$163$	−1.65583	−0.129694	−0.0648472	−	0.997895i	$-0.520656\pi$
$163$	−1.65583	−0.129694	−0.0648472	+	0.997895i	$0.520656\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.83568	0.761108	0.380554	−	0.924759i	$-0.375733\pi$
$167$	9.83568	0.761108	0.380554	+	0.924759i	$0.375733\pi$
$168$	0	0
$169$	−4.79424	−0.368788
$170$	0	0
$171$	−2.76313	−0.211302
$172$	0	0
$173$	−8.05586	−0.612476	−0.306238	−	0.951955i	$-0.599070\pi$
$173$	−8.05586	−0.612476	−0.306238	+	0.951955i	$0.599070\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.86556	0.140224
$178$	0	0
$179$	−14.0058	−1.04685	−0.523423	−	0.852073i	$-0.675345\pi$
$179$	−14.0058	−1.04685	−0.523423	+	0.852073i	$0.675345\pi$
$180$	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
$183$	5.47292	0.404570
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.28085	−0.239919
$188$	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
$193$	−10.3628	−0.745929	−0.372964	−	0.927846i	$-0.621659\pi$
$193$	−10.3628	−0.745929	−0.372964	+	0.927846i	$0.621659\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.9168	−0.920286	−0.460143	−	0.887845i	$-0.652202\pi$
$197$	−12.9168	−0.920286	−0.460143	+	0.887845i	$0.652202\pi$
$198$	0	0
$199$	14.9400	1.05907	0.529536	−	0.848288i	$-0.322366\pi$
$199$	14.9400	1.05907	0.529536	+	0.848288i	$0.322366\pi$
$200$	0	0
$201$	2.97241	0.209657
$202$	0	0
$203$	−18.0426	−1.26634
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.70576	0.118558
$208$	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
$213$	−4.57787	−0.313670
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.70758	0.183802
$218$	0	0
$219$	−4.63514	−0.313214
$220$	0	0
$221$	3.36299	0.226219
$222$	0	0
$223$	11.5273	0.771926	0.385963	−	0.922514i	$-0.373869\pi$
$223$	11.5273	0.771926	0.385963	+	0.922514i	$0.373869\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	29.1127	1.93228	0.966139	−	0.258021i	$-0.0830703\pi$
$227$	29.1127	1.93228	0.966139	+	0.258021i	$0.0830703\pi$
$228$	0	0
$229$	2.60133	0.171901	0.0859503	−	0.996299i	$-0.472607\pi$
$229$	2.60133	0.171901	0.0859503	+	0.996299i	$0.472607\pi$
$230$	0	0
$231$	−4.94064	−0.325070
$232$	0	0
$233$	−11.2634	−0.737890	−0.368945	−	0.929451i	$-0.620281\pi$
$233$	−11.2634	−0.737890	−0.368945	+	0.929451i	$0.620281\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1.80124	−0.117003
$238$	0	0
$239$	−22.6347	−1.46412	−0.732059	−	0.681242i	$-0.761440\pi$
$239$	−22.6347	−1.46412	−0.732059	+	0.681242i	$0.761440\pi$
$240$	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
$243$	11.7847	0.755989
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.86457	0.182268
$248$	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
$253$	1.72518	0.108461
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−27.4124	−1.70994	−0.854968	−	0.518680i	$-0.826423\pi$
$257$	−27.4124	−1.70994	−0.854968	+	0.518680i	$0.826423\pi$
$258$	0	0
$259$	29.9201	1.85914
$260$	0	0
$261$	13.7244	0.849522
$262$	0	0
$263$	−24.0460	−1.48274	−0.741370	−	0.671097i	$-0.765824\pi$
$263$	−24.0460	−1.48274	−0.741370	+	0.671097i	$0.765824\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.20221	0.318370
$268$	0	0
$269$	−12.5398	−0.764565	−0.382283	−	0.924045i	$-0.624862\pi$
$269$	−12.5398	−0.764565	−0.382283	+	0.924045i	$0.624862\pi$
$270$	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
$273$	5.06435	0.306508
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.79301	0.348068	0.174034	−	0.984740i	$-0.444320\pi$
$277$	5.79301	0.348068	0.174034	+	0.984740i	$0.444320\pi$
$278$	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
$283$	28.3864	1.68740	0.843698	−	0.536818i	$-0.180374\pi$
$283$	28.3864	1.68740	0.843698	+	0.536818i	$0.180374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	36.2602	2.14037
$288$	0	0
$289$	−15.6217	−0.918925
$290$	0	0
$291$	0.306240	0.0179521
$292$	0	0
$293$	19.8463	1.15943	0.579717	−	0.814818i	$-0.303163\pi$
$293$	19.8463	1.15943	0.579717	+	0.814818i	$0.303163\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.83856	0.454840
$298$	0	0
$299$	−1.76838	−0.102268
$300$	0	0
$301$	38.1250	2.19749
$302$	0	0
$303$	1.86177	0.106956
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6.35995	−0.362982	−0.181491	−	0.983393i	$-0.558092\pi$
$307$	−6.35995	−0.362982	−0.181491	+	0.983393i	$0.558092\pi$
$308$	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
$313$	−3.38952	−0.191587	−0.0957934	−	0.995401i	$-0.530539\pi$
$313$	−3.38952	−0.191587	−0.0957934	+	0.995401i	$0.530539\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.37856	−0.302090	−0.151045	−	0.988527i	$-0.548264\pi$
$317$	−5.37856	−0.302090	−0.151045	+	0.988527i	$0.548264\pi$
$318$	0	0
$319$	13.8808	0.777174
$320$	0	0
$321$	2.20674	0.123168
$322$	0	0
$323$	1.17400	0.0653229
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.78589	0.430561
$328$	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
$333$	−22.7593	−1.24720
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−28.2975	−1.54146	−0.770732	−	0.637159i	$-0.780109\pi$
$337$	−28.2975	−1.54146	−0.770732	+	0.637159i	$0.780109\pi$
$338$	0	0
$339$	−1.14290	−0.0620739
$340$	0	0
$341$	−2.08303	−0.112802
$342$	0	0
$343$	−2.92409	−0.157886
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.0367	−0.753530	−0.376765	−	0.926309i	$-0.622964\pi$
$347$	−14.0367	−0.753530	−0.376765	+	0.926309i	$0.622964\pi$
$348$	0	0
$349$	−31.9225	−1.70877	−0.854387	−	0.519637i	$-0.826067\pi$
$349$	−31.9225	−1.70877	−0.854387	+	0.519637i	$0.826067\pi$
$350$	0	0
$351$	−8.03482	−0.428867
$352$	0	0
$353$	−2.65209	−0.141157	−0.0705783	−	0.997506i	$-0.522484\pi$
$353$	−2.65209	−0.141157	−0.0705783	+	0.997506i	$0.522484\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.07554	0.109849
$358$	0	0
$359$	23.7750	1.25480	0.627398	−	0.778699i	$-0.284120\pi$
$359$	23.7750	1.25480	0.627398	+	0.778699i	$0.284120\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.55267	−0.0814941
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.3923	0.907869	0.453934	−	0.891035i	$-0.350020\pi$
$367$	17.3923	0.907869	0.453934	+	0.891035i	$0.350020\pi$
$368$	0	0
$369$	−27.5820	−1.43586
$370$	0	0
$371$	−27.0755	−1.40569
$372$	0	0
$373$	17.8099	0.922164	0.461082	−	0.887358i	$-0.347461\pi$
$373$	17.8099	0.922164	0.461082	+	0.887358i	$0.347461\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14.2283	−0.732796
$378$	0	0
$379$	25.7998	1.32525	0.662624	−	0.748952i	$-0.269443\pi$
$379$	25.7998	1.32525	0.662624	+	0.748952i	$0.269443\pi$
$380$	0	0
$381$	8.49489	0.435206
$382$	0	0
$383$	2.08964	0.106776	0.0533879	−	0.998574i	$-0.482998\pi$
$383$	2.08964	0.106776	0.0533879	+	0.998574i	$0.482998\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−29.0005	−1.47418
$388$	0	0
$389$	−17.8701	−0.906052	−0.453026	−	0.891497i	$-0.649656\pi$
$389$	−17.8701	−0.906052	−0.453026	+	0.891497i	$0.649656\pi$
$390$	0	0
$391$	−0.724741	−0.0366517
$392$	0	0
$393$	4.38455	0.221171
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.4105	1.42588	0.712941	−	0.701224i	$-0.247363\pi$
$397$	28.4105	1.42588	0.712941	+	0.701224i	$0.247363\pi$
$398$	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
$403$	2.13518	0.106361
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−23.0185	−1.14099
$408$	0	0
$409$	29.1270	1.44024	0.720119	−	0.693850i	$-0.244087\pi$
$409$	29.1270	1.44024	0.720119	+	0.693850i	$0.244087\pi$
$410$	0	0
$411$	7.51311	0.370594
$412$	0	0
$413$	13.9237	0.685141
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.35613	0.360231
$418$	0	0
$419$	16.3711	0.799782	0.399891	−	0.916563i	$-0.369048\pi$
$419$	16.3711	0.799782	0.399891	+	0.916563i	$0.369048\pi$
$420$	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
$423$	−14.0296	−0.682140
$424$	0	0
$425$	0	0
$426$	0	0
$427$	40.8475	1.97675
$428$	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
$433$	−18.9901	−0.912608	−0.456304	−	0.889824i	$-0.650827\pi$
$433$	−18.9901	−0.912608	−0.456304	+	0.889824i	$0.650827\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.617328	−0.0295308
$438$	0	0
$439$	−13.3540	−0.637352	−0.318676	−	0.947864i	$-0.603238\pi$
$439$	−13.3540	−0.637352	−0.318676	+	0.947864i	$0.603238\pi$
$440$	0	0
$441$	−17.1176	−0.815125
$442$	0	0
$443$	8.67208	0.412023	0.206012	−	0.978550i	$-0.433952\pi$
$443$	8.67208	0.412023	0.206012	+	0.978550i	$0.433952\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5.76358	−0.272608
$448$	0	0
$449$	40.5277	1.91262	0.956310	−	0.292354i	$-0.0944386\pi$
$449$	40.5277	1.91262	0.956310	+	0.292354i	$0.0944386\pi$
$450$	0	0
$451$	−27.8962	−1.31358
$452$	0	0
$453$	−4.04573	−0.190085
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.9859	−0.794566	−0.397283	−	0.917696i	$-0.630047\pi$
$457$	−16.9859	−0.794566	−0.397283	+	0.917696i	$0.630047\pi$
$458$	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
$463$	8.11961	0.377350	0.188675	−	0.982040i	$-0.439581\pi$
$463$	8.11961	0.377350	0.188675	+	0.982040i	$0.439581\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.1221	0.746042	0.373021	−	0.927823i	$-0.378322\pi$
$467$	16.1221	0.746042	0.373021	+	0.927823i	$0.378322\pi$
$468$	0	0
$469$	22.1847	1.02440
$470$	0	0
$471$	−5.19064	−0.239172
$472$	0	0
$473$	−29.3308	−1.34863
$474$	0	0
$475$	0	0
$476$	0	0
$477$	20.5955	0.943003
$478$	0	0
$479$	−5.30225	−0.242266	−0.121133	−	0.992636i	$-0.538653\pi$
$479$	−5.30225	−0.242266	−0.121133	+	0.992636i	$0.538653\pi$
$480$	0	0
$481$	23.5949	1.07583
$482$	0	0
$483$	−1.09139	−0.0496600
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−20.9188	−0.947922	−0.473961	−	0.880546i	$-0.657176\pi$
$487$	−20.9188	−0.947922	−0.473961	+	0.880546i	$0.657176\pi$
$488$	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
$493$	−5.83124	−0.262626
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−34.1672	−1.53261
$498$	0	0
$499$	5.44182	0.243609	0.121805	−	0.992554i	$-0.461132\pi$
$499$	5.44182	0.243609	0.121805	+	0.992554i	$0.461132\pi$
$500$	0	0
$501$	4.78700	0.213867
$502$	0	0
$503$	17.9666	0.801092	0.400546	−	0.916277i	$-0.368820\pi$
$503$	17.9666	0.801092	0.400546	+	0.916277i	$0.368820\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.33334	−0.103627
$508$	0	0
$509$	0.433419	0.0192110	0.00960548	−	0.999954i	$-0.496942\pi$
$509$	0.433419	0.0192110	0.00960548	+	0.999954i	$0.496942\pi$
$510$	0	0
$511$	−34.5947	−1.53038
$512$	0	0
$513$	−2.80490	−0.123839
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−14.1894	−0.624047
$518$	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
$523$	−1.17153	−0.0512275	−0.0256138	−	0.999672i	$-0.508154\pi$
$523$	−1.17153	−0.0512275	−0.0256138	+	0.999672i	$0.508154\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.875070	0.0381187
$528$	0	0
$529$	−22.6189	−0.983431
$530$	0	0
$531$	−10.5913	−0.459624
$532$	0	0
$533$	28.5946	1.23857
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−6.81660	−0.294158
$538$	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
$543$	1.18628	0.0509082
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−39.5096	−1.68931	−0.844655	−	0.535311i	$-0.820194\pi$
$547$	−39.5096	−1.68931	−0.844655	+	0.535311i	$0.820194\pi$
$548$	0	0
$549$	−31.0714	−1.32610
$550$	0	0
$551$	−4.96700	−0.211601
$552$	0	0
$553$	−13.4436	−0.571682
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.90886	−0.335109	−0.167554	−	0.985863i	$-0.553587\pi$
$557$	−7.90886	−0.335109	−0.167554	+	0.985863i	$0.553587\pi$
$558$	0	0
$559$	30.0652	1.27162
$560$	0	0
$561$	−1.59678	−0.0674161
$562$	0	0
$563$	−5.40347	−0.227729	−0.113865	−	0.993496i	$-0.536323\pi$
$563$	−5.40347	−0.227729	−0.113865	+	0.993496i	$0.536323\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	25.1523	1.05630
$568$	0	0
$569$	−4.60607	−0.193096	−0.0965482	−	0.995328i	$-0.530780\pi$
$569$	−4.60607	−0.193096	−0.0965482	+	0.995328i	$0.530780\pi$
$570$	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
$573$	6.98743	0.291904
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−30.0011	−1.24896	−0.624480	−	0.781040i	$-0.714689\pi$
$577$	−30.0011	−1.24896	−0.624480	+	0.781040i	$0.714689\pi$
$578$	0	0
$579$	−5.04353	−0.209602
$580$	0	0
$581$	−16.9418	−0.702866
$582$	0	0
$583$	20.8301	0.862694
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.90807	−0.0787545	−0.0393772	−	0.999224i	$-0.512537\pi$
$587$	−1.90807	−0.0787545	−0.0393772	+	0.999224i	$0.512537\pi$
$588$	0	0
$589$	0.745377	0.0307127
$590$	0	0
$591$	−6.28658	−0.258596
$592$	0	0
$593$	−11.5819	−0.475610	−0.237805	−	0.971313i	$-0.576428\pi$
$593$	−11.5819	−0.475610	−0.237805	+	0.971313i	$0.576428\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.27128	0.297594
$598$	0	0
$599$	−35.9808	−1.47014	−0.735068	−	0.677993i	$-0.762850\pi$
$599$	−35.9808	−1.47014	−0.735068	+	0.677993i	$0.762850\pi$
$600$	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
$603$	−16.8752	−0.687213
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.98858	−0.161891	−0.0809457	−	0.996719i	$-0.525794\pi$
$607$	−3.98858	−0.161891	−0.0809457	+	0.996719i	$0.525794\pi$
$608$	0	0
$609$	−8.78129	−0.355836
$610$	0	0
$611$	14.5446	0.588412
$612$	0	0
$613$	26.1306	1.05541	0.527703	−	0.849429i	$-0.323053\pi$
$613$	26.1306	1.05541	0.527703	+	0.849429i	$0.323053\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.30081	0.253661	0.126831	−	0.991924i	$-0.459520\pi$
$617$	6.30081	0.253661	0.126831	+	0.991924i	$0.459520\pi$
$618$	0	0
$619$	12.7231	0.511387	0.255693	−	0.966758i	$-0.417696\pi$
$619$	12.7231	0.511387	0.255693	+	0.966758i	$0.417696\pi$
$620$	0	0
$621$	1.73154	0.0694844
$622$	0	0
$623$	38.8270	1.55557
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.36012	−0.0543181
$628$	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
$633$	2.57039	0.102164
$634$	0	0
$635$	0	0
$636$	0	0
$637$	17.7461	0.703125
$638$	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
$643$	−42.3146	−1.66872	−0.834362	−	0.551217i	$-0.814164\pi$
$643$	−42.3146	−1.66872	−0.834362	+	0.551217i	$0.814164\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.57122	−0.219027	−0.109514	−	0.993985i	$-0.534929\pi$
$647$	−5.57122	−0.219027	−0.109514	+	0.993985i	$0.534929\pi$
$648$	0	0
$649$	−10.7120	−0.420481
$650$	0	0
$651$	1.31777	0.0516475
$652$	0	0
$653$	−44.6953	−1.74906	−0.874532	−	0.484968i	$-0.838831\pi$
$653$	−44.6953	−1.74906	−0.874532	+	0.484968i	$0.838831\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	26.3151	1.02665
$658$	0	0
$659$	−37.0288	−1.44244	−0.721218	−	0.692708i	$-0.756417\pi$
$659$	−37.0288	−1.44244	−0.721218	+	0.692708i	$0.756417\pi$
$660$	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
$663$	1.63676	0.0635665
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.06627	0.118726
$668$	0	0
$669$	5.61031	0.216907
$670$	0	0
$671$	−31.4253	−1.21316
$672$	0	0
$673$	−13.5879	−0.523773	−0.261887	−	0.965099i	$-0.584345\pi$
$673$	−13.5879	−0.523773	−0.261887	+	0.965099i	$0.584345\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.0051	0.653559	0.326779	−	0.945101i	$-0.394037\pi$
$677$	17.0051	0.653559	0.326779	+	0.945101i	$0.394037\pi$
$678$	0	0
$679$	2.28564	0.0877149
$680$	0	0
$681$	14.1691	0.542960
$682$	0	0
$683$	10.9418	0.418677	0.209338	−	0.977843i	$-0.432869\pi$
$683$	10.9418	0.418677	0.209338	+	0.977843i	$0.432869\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.26606	0.0483032
$688$	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
$693$	28.0495	1.06551
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.7190	0.443890
$698$	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
$703$	8.23679	0.310656
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.8954	0.522591
$708$	0	0
$709$	25.9811	0.975740	0.487870	−	0.872916i	$-0.337774\pi$
$709$	25.9811	0.975740	0.487870	+	0.872916i	$0.337774\pi$
$710$	0	0
$711$	10.2262	0.383511
$712$	0	0
$713$	−0.460142	−0.0172325
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.0162	−0.411409
$718$	0	0
$719$	−32.4059	−1.20854	−0.604268	−	0.796781i	$-0.706535\pi$
$719$	−32.4059	−1.20854	−0.604268	+	0.796781i	$0.706535\pi$
$720$	0	0
$721$	−36.5801	−1.36231
$722$	0	0
$723$	−6.47013	−0.240627
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−47.1313	−1.74800	−0.874001	−	0.485924i	$-0.838483\pi$
$727$	−47.1313	−1.74800	−0.874001	+	0.485924i	$0.838483\pi$
$728$	0	0
$729$	−15.0371	−0.556931
$730$	0	0
$731$	12.3217	0.455735
$732$	0	0
$733$	12.7925	0.472500	0.236250	−	0.971692i	$-0.424081\pi$
$733$	12.7925	0.472500	0.236250	+	0.971692i	$0.424081\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.0675	−0.628688
$738$	0	0
$739$	−28.2097	−1.03771	−0.518856	−	0.854862i	$-0.673642\pi$
$739$	−28.2097	−1.03771	−0.518856	+	0.854862i	$0.673642\pi$
$740$	0	0
$741$	1.39418	0.0512164
$742$	0	0
$743$	16.0366	0.588325	0.294162	−	0.955755i	$-0.404959\pi$
$743$	16.0366	0.588325	0.294162	+	0.955755i	$0.404959\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.8871	0.471515
$748$	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
$753$	7.90555	0.288094
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−39.3944	−1.43181	−0.715906	−	0.698196i	$-0.753986\pi$
$757$	−39.3944	−1.43181	−0.715906	+	0.698196i	$0.753986\pi$
$758$	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
$763$	58.1105	2.10374
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.9802	0.396471
$768$	0	0
$769$	15.7229	0.566983	0.283491	−	0.958975i	$-0.408507\pi$
$769$	15.7229	0.566983	0.283491	+	0.958975i	$0.408507\pi$
$770$	0	0
$771$	−13.3415	−0.480483
$772$	0	0
$773$	26.6810	0.959649	0.479825	−	0.877364i	$-0.340700\pi$
$773$	26.6810	0.959649	0.479825	+	0.877364i	$0.340700\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	14.5620	0.522410
$778$	0	0
$779$	9.98217	0.357648
$780$	0	0
$781$	26.2860	0.940586
$782$	0	0
$783$	13.9319	0.497887
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−38.7243	−1.38037	−0.690187	−	0.723631i	$-0.742472\pi$
$787$	−38.7243	−1.38037	−0.690187	+	0.723631i	$0.742472\pi$
$788$	0	0
$789$	−11.7031	−0.416642
$790$	0	0
$791$	−8.53012	−0.303296
$792$	0	0
$793$	32.2121	1.14389
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.5802	1.18947	0.594736	−	0.803921i	$-0.297256\pi$
$797$	33.5802	1.18947	0.594736	+	0.803921i	$0.297256\pi$
$798$	0	0
$799$	5.96087	0.210881
$800$	0	0
$801$	−29.5345	−1.04355
$802$	0	0
$803$	26.6148	0.939217
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.10309	−0.214839
$808$	0	0
$809$	−25.5182	−0.897172	−0.448586	−	0.893740i	$-0.648072\pi$
$809$	−25.5182	−0.897172	−0.448586	+	0.893740i	$0.648072\pi$
$810$	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
$813$	8.53033	0.299172
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.4955	0.367192
$818$	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
$823$	−18.4568	−0.643363	−0.321681	−	0.946848i	$-0.604248\pi$
$823$	−18.4568	−0.643363	−0.321681	+	0.946848i	$0.604248\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.3135	0.358637	0.179318	−	0.983791i	$-0.442611\pi$
$827$	10.3135	0.358637	0.179318	+	0.983791i	$0.442611\pi$
$828$	0	0
$829$	−28.5805	−0.992640	−0.496320	−	0.868140i	$-0.665316\pi$
$829$	−28.5805	−0.992640	−0.496320	+	0.868140i	$0.665316\pi$
$830$	0	0
$831$	2.81944	0.0978053
$832$	0	0
$833$	7.27293	0.251992
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.09071	−0.0722654
$838$	0	0
$839$	5.06942	0.175016	0.0875079	−	0.996164i	$-0.472110\pi$
$839$	5.06942	0.175016	0.0875079	+	0.996164i	$0.472110\pi$
$840$	0	0
$841$	−4.32890	−0.149272
$842$	0	0
$843$	9.35567	0.322227
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−11.5885	−0.398184
$848$	0	0
$849$	13.8156	0.474149
$850$	0	0
$851$	−5.08480	−0.174305
$852$	0	0
$853$	−17.1101	−0.585839	−0.292920	−	0.956137i	$-0.594627\pi$
$853$	−17.1101	−0.585839	−0.292920	+	0.956137i	$0.594627\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.4488	−1.07427	−0.537135	−	0.843496i	$-0.680494\pi$
$857$	−31.4488	−1.07427	−0.537135	+	0.843496i	$0.680494\pi$
$858$	0	0
$859$	5.98445	0.204187	0.102093	−	0.994775i	$-0.467446\pi$
$859$	5.98445	0.204187	0.102093	+	0.994775i	$0.467446\pi$
$860$	0	0
$861$	17.6477	0.601433
$862$	0	0
$863$	20.8530	0.709844	0.354922	−	0.934896i	$-0.384507\pi$
$863$	20.8530	0.709844	0.354922	+	0.934896i	$0.384507\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−7.60306	−0.258213
$868$	0	0
$869$	10.3426	0.350850
$870$	0	0
$871$	17.4948	0.592788
$872$	0	0
$873$	−1.73862	−0.0588432
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7.11382	−0.240217	−0.120108	−	0.992761i	$-0.538324\pi$
$877$	−7.11382	−0.240217	−0.120108	+	0.992761i	$0.538324\pi$
$878$	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
$883$	9.15841	0.308205	0.154103	−	0.988055i	$-0.450751\pi$
$883$	9.15841	0.308205	0.154103	+	0.988055i	$0.450751\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.2435	0.713286	0.356643	−	0.934241i	$-0.383921\pi$
$887$	21.2435	0.713286	0.356643	+	0.934241i	$0.383921\pi$
$888$	0	0
$889$	63.4021	2.12644
$890$	0	0
$891$	−19.3505	−0.648265
$892$	0	0
$893$	5.07742	0.169909
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.860666	−0.0287368
$898$	0	0
$899$	−3.70229	−0.123478
$900$	0	0
$901$	−8.75062	−0.291525
$902$	0	0
$903$	18.5553	0.617483
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19.1139	0.634667	0.317333	−	0.948314i	$-0.397213\pi$
$907$	19.1139	0.634667	0.317333	+	0.948314i	$0.397213\pi$
$908$	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
$913$	13.0339	0.431360
$914$	0	0
$915$	0	0
$916$	0	0
$917$	32.7243	1.08065
$918$	0	0
$919$	54.3957	1.79435	0.897174	−	0.441677i	$-0.145616\pi$
$919$	54.3957	1.79435	0.897174	+	0.441677i	$0.145616\pi$
$920$	0	0
$921$	−3.09537	−0.101996
$922$	0	0
$923$	−26.9441	−0.886876
$924$	0	0
$925$	0	0
$926$	0	0
$927$	27.8253	0.913904
$928$	0	0
$929$	−50.0187	−1.64106	−0.820530	−	0.571603i	$-0.806322\pi$
$929$	−50.0187	−1.64106	−0.820530	+	0.571603i	$0.806322\pi$
$930$	0	0
$931$	6.19502	0.203034
$932$	0	0
$933$	11.4068	0.373441
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−50.3138	−1.64368	−0.821840	−	0.569719i	$-0.807052\pi$
$937$	−50.3138	−1.64368	−0.821840	+	0.569719i	$0.807052\pi$
$938$	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
$943$	−6.16227	−0.200671
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−0.628142	−0.0204119	−0.0102059	−	0.999948i	$-0.503249\pi$
$947$	−0.628142	−0.0204119	−0.0102059	+	0.999948i	$0.503249\pi$
$948$	0	0
$949$	−27.2812	−0.885586
$950$	0	0
$951$	−2.61773	−0.0848858
$952$	0	0
$953$	−40.6251	−1.31598	−0.657988	−	0.753028i	$-0.728592\pi$
$953$	−40.6251	−1.31598	−0.657988	+	0.753028i	$0.728592\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.75574	0.218382
$958$	0	0
$959$	56.0745	1.81074
$960$	0	0
$961$	−30.4444	−0.982078
$962$	0	0
$963$	−12.5283	−0.403720
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−25.2241	−0.811153	−0.405576	−	0.914061i	$-0.632929\pi$
$967$	−25.2241	−0.811153	−0.405576	+	0.914061i	$0.632929\pi$
$968$	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
$973$	54.9030	1.76011
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.1606	−1.09289	−0.546447	−	0.837494i	$-0.684020\pi$
$977$	−34.1606	−1.09289	−0.546447	+	0.837494i	$0.684020\pi$
$978$	0	0
$979$	−29.8709	−0.954679
$980$	0	0
$981$	−44.2028	−1.41129
$982$	0	0
$983$	46.0699	1.46940	0.734701	−	0.678391i	$-0.237323\pi$
$983$	46.0699	1.46940	0.734701	+	0.678391i	$0.237323\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.97650	0.285725
$988$	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
$993$	−1.84961	−0.0586957
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−50.3518	−1.59466	−0.797328	−	0.603546i	$-0.793754\pi$
$997$	−50.3518	−1.59466	−0.797328	+	0.603546i	$0.793754\pi$
$998$	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.a.bd.1.4	yes	6
4.3	odd	2		7600.2.a.ci.1.3		6
5.2	odd	4		3800.2.d.p.3649.5		12
5.3	odd	4		3800.2.d.p.3649.8		12
5.4	even	2		3800.2.a.bb.1.3	✓	6
20.19	odd	2		7600.2.a.cm.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.bb.1.3	✓	6	5.4	even	2
3800.2.a.bd.1.4	yes	6	1.1	even	1	trivial
3800.2.d.p.3649.5		12	5.2	odd	4
3800.2.d.p.3649.8		12	5.3	odd	4
7600.2.a.ci.1.3		6	4.3	odd	2
7600.2.a.cm.1.4		6	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q+0.486697 q^{3} +3.63249 q^{7} -2.76313 q^{9} -2.79460 q^{11} +2.86457 q^{13} +1.17400 q^{17} +1.00000 q^{19} +1.76793 q^{21} -0.617328 q^{23} -2.80490 q^{27} -4.96700 q^{29} +0.745377 q^{31} -1.36012 q^{33} +8.23679 q^{37} +1.39418 q^{39} +9.98217 q^{41} +10.4955 q^{43} +5.07742 q^{47} +6.19502 q^{49} +0.571381 q^{51} -7.45370 q^{53} +0.486697 q^{57} +3.83310 q^{59} +11.2450 q^{61} -10.0370 q^{63} +6.10730 q^{67} -0.300452 q^{69} -9.40599 q^{71} -9.52367 q^{73} -10.1514 q^{77} -3.70094 q^{79} +6.92424 q^{81} -4.66397 q^{83} -2.41743 q^{87} +10.6888 q^{89} +10.4055 q^{91} +0.362773 q^{93} +0.629221 q^{97} +7.72183 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 2 q^{7} + 10 q^{9} + 3 q^{11} - 3 q^{13} - 2 q^{17} + 6 q^{19} + 11 q^{21} + 4 q^{23} + 20 q^{27} + 7 q^{29} + 5 q^{31} - 16 q^{33} + 8 q^{39} + 11 q^{41} - 7 q^{43} + 20 q^{47} - 2 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 + 2 * q^7 + 10 * q^9 + 3 * q^11 - 3 * q^13 - 2 * q^17 + 6 * q^19 + 11 * q^21 + 4 * q^23 + 20 * q^27 + 7 * q^29 + 5 * q^31 - 16 * q^33 + 8 * q^39 + 11 * q^41 - 7 * q^43 + 20 * q^47 - 2 * q^49 + 13 * q^51 - 7 * q^53 + 2 * q^57 - 4 * q^59 + 13 * q^61 - q^63 + 25 * q^67 + 7 * q^69 + 29 * q^71 - 19 * q^73 + 24 * q^77 + 28 * q^79 + 38 * q^81 - 15 * q^83 + 57 * q^87 - 12 * q^89 - 27 * q^93 - 13 * q^97 + 10 * q^99

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