LMFDB - Embedded newform 3800.2.a.ba.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3800,2,Mod(1,3800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [6,0,-2,0,0,0,-2,0,6,0,3,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 $ x^6 - 2x^5 - 10x^4 + 16x^3 + 15x^2 - 14*x - 3
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			1.2
Root			$1.93590$ 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)

$f(q)$	$=$	$q-1.93590 q^{3} +1.24708 q^{7} +0.747704 q^{9} -0.513860 q^{11} +6.15670 q^{13} -4.51986 q^{17} -1.00000 q^{19} -2.41421 q^{21} -5.86084 q^{23} +4.36022 q^{27} +6.62700 q^{29} +6.41995 q^{31} +0.994780 q^{33} -1.40671 q^{37} -11.9187 q^{39} +10.6870 q^{41} -3.04878 q^{43} -1.99478 q^{47} -5.44480 q^{49} +8.74998 q^{51} -14.0848 q^{53} +1.93590 q^{57} -4.34261 q^{59} +10.7173 q^{61} +0.932444 q^{63} -9.89978 q^{67} +11.3460 q^{69} +7.42517 q^{71} +12.8079 q^{73} -0.640822 q^{77} +2.56138 q^{79} -10.6841 q^{81} -7.50864 q^{83} -12.8292 q^{87} -7.85353 q^{89} +7.67787 q^{91} -12.4284 q^{93} +6.74797 q^{97} -0.384215 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} + 3 q^{11} + q^{13} + 14 q^{17} - 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} + 5 q^{31} - 2 q^{33} + 8 q^{37} + 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47}+ \cdots - 6 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 + 3 * q^11 + q^13 + 14 * q^17 - 6 * q^19 + 15 * q^21 - 12 * q^23 - 8 * q^27 + 9 * q^29 + 5 * q^31 - 2 * q^33 + 8 * q^37 + 12 * q^39 + 3 * q^41 - 15 * q^43 - 4 * q^47 + 22 * q^49 + 33 * q^51 - 13 * q^53 + 2 * q^57 + 9 * q^61 - 21 * q^63 + 3 * q^67 - 11 * q^69 + 19 * q^71 - 3 * q^73 + 36 * q^77 - 16 * q^79 + 26 * q^81 - 31 * q^83 + 25 * q^87 + 14 * q^89 + 42 * q^91 - 39 * q^93 + 11 * q^97 - 6 * 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
$3$	−1.93590	−1.11769	−0.558846	−	0.829272i	$-0.688756\pi$
$3$	−1.93590	−1.11769	−0.558846	+	0.829272i	$0.688756\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.24708	0.471350	0.235675	−	0.971832i	$-0.424270\pi$
$7$	1.24708	0.471350	0.235675	+	0.971832i	$0.424270\pi$
$8$	0	0
$9$	0.747704	0.249235
$10$	0	0
$11$	−0.513860	−0.154935	−0.0774673	−	0.996995i	$-0.524683\pi$
$11$	−0.513860	−0.154935	−0.0774673	+	0.996995i	$0.524683\pi$
$12$	0	0
$13$	6.15670	1.70756	0.853780	−	0.520633i	$-0.174304\pi$
$13$	6.15670	1.70756	0.853780	+	0.520633i	$0.174304\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.51986	−1.09623	−0.548113	−	0.836404i	$-0.684654\pi$
$17$	−4.51986	−1.09623	−0.548113	+	0.836404i	$0.684654\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.41421	−0.526824
$22$	0	0
$23$	−5.86084	−1.22207	−0.611035	−	0.791603i	$-0.709247\pi$
$23$	−5.86084	−1.22207	−0.611035	+	0.791603i	$0.709247\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.36022	0.839124
$28$	0	0
$29$	6.62700	1.23060	0.615302	−	0.788292i	$-0.289034\pi$
$29$	6.62700	1.23060	0.615302	+	0.788292i	$0.289034\pi$
$30$	0	0
$31$	6.41995	1.15306	0.576528	−	0.817077i	$-0.304407\pi$
$31$	6.41995	1.15306	0.576528	+	0.817077i	$0.304407\pi$
$32$	0	0
$33$	0.994780	0.173169
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.40671	−0.231263	−0.115631	−	0.993292i	$-0.536889\pi$
$37$	−1.40671	−0.231263	−0.115631	+	0.993292i	$0.536889\pi$
$38$	0	0
$39$	−11.9187	−1.90853
$40$	0	0
$41$	10.6870	1.66903	0.834514	−	0.550987i	$-0.185749\pi$
$41$	10.6870	1.66903	0.834514	+	0.550987i	$0.185749\pi$
$42$	0	0
$43$	−3.04878	−0.464934	−0.232467	−	0.972604i	$-0.574680\pi$
$43$	−3.04878	−0.464934	−0.232467	+	0.972604i	$0.574680\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.99478	−0.290969	−0.145484	−	0.989361i	$-0.546474\pi$
$47$	−1.99478	−0.290969	−0.145484	+	0.989361i	$0.546474\pi$
$48$	0	0
$49$	−5.44480	−0.777829
$50$	0	0
$51$	8.74998	1.22524
$52$	0	0
$53$	−14.0848	−1.93469	−0.967346	−	0.253459i	$-0.918432\pi$
$53$	−14.0848	−1.93469	−0.967346	+	0.253459i	$0.918432\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.93590	0.256416
$58$	0	0
$59$	−4.34261	−0.565360	−0.282680	−	0.959214i	$-0.591223\pi$
$59$	−4.34261	−0.565360	−0.282680	+	0.959214i	$0.591223\pi$
$60$	0	0
$61$	10.7173	1.37221	0.686106	−	0.727502i	$-0.259319\pi$
$61$	10.7173	1.37221	0.686106	+	0.727502i	$0.259319\pi$
$62$	0	0
$63$	0.932444	0.117477
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.89978	−1.20945	−0.604725	−	0.796434i	$-0.706717\pi$
$67$	−9.89978	−1.20945	−0.604725	+	0.796434i	$0.706717\pi$
$68$	0	0
$69$	11.3460	1.36590
$70$	0	0
$71$	7.42517	0.881205	0.440603	−	0.897702i	$-0.354765\pi$
$71$	7.42517	0.881205	0.440603	+	0.897702i	$0.354765\pi$
$72$	0	0
$73$	12.8079	1.49905	0.749525	−	0.661976i	$-0.230282\pi$
$73$	12.8079	1.49905	0.749525	+	0.661976i	$0.230282\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.640822	−0.0730285
$78$	0	0
$79$	2.56138	0.288178	0.144089	−	0.989565i	$-0.453975\pi$
$79$	2.56138	0.288178	0.144089	+	0.989565i	$0.453975\pi$
$80$	0	0
$81$	−10.6841	−1.18712
$82$	0	0
$83$	−7.50864	−0.824180	−0.412090	−	0.911143i	$-0.635201\pi$
$83$	−7.50864	−0.824180	−0.412090	+	0.911143i	$0.635201\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−12.8292	−1.37543
$88$	0	0
$89$	−7.85353	−0.832473	−0.416236	−	0.909256i	$-0.636651\pi$
$89$	−7.85353	−0.832473	−0.416236	+	0.909256i	$0.636651\pi$
$90$	0	0
$91$	7.67787	0.804859
$92$	0	0
$93$	−12.4284	−1.28876
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.74797	0.685152	0.342576	−	0.939490i	$-0.388701\pi$
$97$	6.74797	0.685152	0.342576	+	0.939490i	$0.388701\pi$
$98$	0	0
$99$	−0.384215	−0.0386151
$100$	0	0
$101$	17.6503	1.75627	0.878134	−	0.478415i	$-0.158788\pi$
$101$	17.6503	1.75627	0.878134	+	0.478415i	$0.158788\pi$
$102$	0	0
$103$	−5.84147	−0.575577	−0.287789	−	0.957694i	$-0.592920\pi$
$103$	−5.84147	−0.575577	−0.287789	+	0.957694i	$0.592920\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.14743	−0.400947	−0.200474	−	0.979699i	$-0.564248\pi$
$107$	−4.14743	−0.400947	−0.200474	+	0.979699i	$0.564248\pi$
$108$	0	0
$109$	5.69767	0.545738	0.272869	−	0.962051i	$-0.412027\pi$
$109$	5.69767	0.545738	0.272869	+	0.962051i	$0.412027\pi$
$110$	0	0
$111$	2.72326	0.258480
$112$	0	0
$113$	−5.36073	−0.504295	−0.252148	−	0.967689i	$-0.581137\pi$
$113$	−5.36073	−0.504295	−0.252148	+	0.967689i	$0.581137\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.60339	0.425584
$118$	0	0
$119$	−5.63660	−0.516707
$120$	0	0
$121$	−10.7359	−0.975995
$122$	0	0
$123$	−20.6889	−1.86546
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18.3991	1.63266	0.816328	−	0.577589i	$-0.196006\pi$
$127$	18.3991	1.63266	0.816328	+	0.577589i	$0.196006\pi$
$128$	0	0
$129$	5.90212	0.519653
$130$	0	0
$131$	−14.4958	−1.26650	−0.633252	−	0.773946i	$-0.718280\pi$
$131$	−14.4958	−1.26650	−0.633252	+	0.773946i	$0.718280\pi$
$132$	0	0
$133$	−1.24708	−0.108135
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.40281	0.290722	0.145361	−	0.989379i	$-0.453566\pi$
$137$	3.40281	0.290722	0.145361	+	0.989379i	$0.453566\pi$
$138$	0	0
$139$	−4.01062	−0.340176	−0.170088	−	0.985429i	$-0.554405\pi$
$139$	−4.01062	−0.340176	−0.170088	+	0.985429i	$0.554405\pi$
$140$	0	0
$141$	3.86169	0.325213
$142$	0	0
$143$	−3.16368	−0.264560
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.5406	0.869373
$148$	0	0
$149$	9.42255	0.771926	0.385963	−	0.922514i	$-0.373869\pi$
$149$	9.42255	0.771926	0.385963	+	0.922514i	$0.373869\pi$
$150$	0	0
$151$	1.60111	0.130297	0.0651483	−	0.997876i	$-0.479248\pi$
$151$	1.60111	0.130297	0.0651483	+	0.997876i	$0.479248\pi$
$152$	0	0
$153$	−3.37952	−0.273218
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.7241	1.49434	0.747172	−	0.664631i	$-0.231411\pi$
$157$	18.7241	1.49434	0.747172	+	0.664631i	$0.231411\pi$
$158$	0	0
$159$	27.2667	2.16239
$160$	0	0
$161$	−7.30892	−0.576024
$162$	0	0
$163$	15.6370	1.22479	0.612394	−	0.790553i	$-0.290207\pi$
$163$	15.6370	1.22479	0.612394	+	0.790553i	$0.290207\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.5865	1.36088	0.680442	−	0.732801i	$-0.261788\pi$
$167$	17.5865	1.36088	0.680442	+	0.732801i	$0.261788\pi$
$168$	0	0
$169$	24.9049	1.91576
$170$	0	0
$171$	−0.747704	−0.0571784
$172$	0	0
$173$	18.7799	1.42781	0.713903	−	0.700245i	$-0.246926\pi$
$173$	18.7799	1.42781	0.713903	+	0.700245i	$0.246926\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.40686	0.631898
$178$	0	0
$179$	−4.22047	−0.315453	−0.157726	−	0.987483i	$-0.550416\pi$
$179$	−4.22047	−0.315453	−0.157726	+	0.987483i	$0.550416\pi$
$180$	0	0
$181$	11.8180	0.878424	0.439212	−	0.898383i	$-0.355258\pi$
$181$	11.8180	0.878424	0.439212	+	0.898383i	$0.355258\pi$
$182$	0	0
$183$	−20.7476	−1.53371
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.32257	0.169843
$188$	0	0
$189$	5.43752	0.395521
$190$	0	0
$191$	26.7184	1.93327	0.966636	−	0.256153i	$-0.0824551\pi$
$191$	26.7184	1.93327	0.966636	+	0.256153i	$0.0824551\pi$
$192$	0	0
$193$	9.25224	0.665990	0.332995	−	0.942929i	$-0.391941\pi$
$193$	9.25224	0.665990	0.332995	+	0.942929i	$0.391941\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.32896	0.237179	0.118589	−	0.992943i	$-0.462163\pi$
$197$	3.32896	0.237179	0.118589	+	0.992943i	$0.462163\pi$
$198$	0	0
$199$	−0.0700969	−0.00496903	−0.00248452	−	0.999997i	$-0.500791\pi$
$199$	−0.0700969	−0.00496903	−0.00248452	+	0.999997i	$0.500791\pi$
$200$	0	0
$201$	19.1650	1.35179
$202$	0	0
$203$	8.26437	0.580045
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.38218	−0.304583
$208$	0	0
$209$	0.513860	0.0355444
$210$	0	0
$211$	15.5952	1.07362	0.536810	−	0.843703i	$-0.319629\pi$
$211$	15.5952	1.07362	0.536810	+	0.843703i	$0.319629\pi$
$212$	0	0
$213$	−14.3744	−0.984916
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.00616	0.543494
$218$	0	0
$219$	−24.7948	−1.67547
$220$	0	0
$221$	−27.8274	−1.87187
$222$	0	0
$223$	22.4019	1.50014	0.750071	−	0.661358i	$-0.230020\pi$
$223$	22.4019	1.50014	0.750071	+	0.661358i	$0.230020\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.3464	−1.61593	−0.807963	−	0.589234i	$-0.799430\pi$
$227$	−24.3464	−1.61593	−0.807963	+	0.589234i	$0.799430\pi$
$228$	0	0
$229$	−5.27650	−0.348681	−0.174340	−	0.984685i	$-0.555779\pi$
$229$	−5.27650	−0.348681	−0.174340	+	0.984685i	$0.555779\pi$
$230$	0	0
$231$	1.24057	0.0816233
$232$	0	0
$233$	4.30335	0.281922	0.140961	−	0.990015i	$-0.454981\pi$
$233$	4.30335	0.281922	0.140961	+	0.990015i	$0.454981\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.95857	−0.322094
$238$	0	0
$239$	−1.44454	−0.0934395	−0.0467197	−	0.998908i	$-0.514877\pi$
$239$	−1.44454	−0.0934395	−0.0467197	+	0.998908i	$0.514877\pi$
$240$	0	0
$241$	5.20083	0.335015	0.167507	−	0.985871i	$-0.446428\pi$
$241$	5.20083	0.335015	0.167507	+	0.985871i	$0.446428\pi$
$242$	0	0
$243$	7.60259	0.487707
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.15670	−0.391741
$248$	0	0
$249$	14.5360	0.921180
$250$	0	0
$251$	1.57046	0.0991267	0.0495634	−	0.998771i	$-0.484217\pi$
$251$	1.57046	0.0991267	0.0495634	+	0.998771i	$0.484217\pi$
$252$	0	0
$253$	3.01165	0.189341
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.2831	1.51474	0.757368	−	0.652988i	$-0.226485\pi$
$257$	24.2831	1.51474	0.757368	+	0.652988i	$0.226485\pi$
$258$	0	0
$259$	−1.75428	−0.109006
$260$	0	0
$261$	4.95504	0.306709
$262$	0	0
$263$	−7.50729	−0.462919	−0.231460	−	0.972844i	$-0.574350\pi$
$263$	−7.50729	−0.462919	−0.231460	+	0.972844i	$0.574350\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	15.2036	0.930448
$268$	0	0
$269$	9.37783	0.571776	0.285888	−	0.958263i	$-0.407711\pi$
$269$	9.37783	0.571776	0.285888	+	0.958263i	$0.407711\pi$
$270$	0	0
$271$	16.8103	1.02115	0.510577	−	0.859832i	$-0.329432\pi$
$271$	16.8103	1.02115	0.510577	+	0.859832i	$0.329432\pi$
$272$	0	0
$273$	−14.8636	−0.899585
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.55337	0.0933328	0.0466664	−	0.998911i	$-0.485140\pi$
$277$	1.55337	0.0933328	0.0466664	+	0.998911i	$0.485140\pi$
$278$	0	0
$279$	4.80022	0.287382
$280$	0	0
$281$	−5.68294	−0.339016	−0.169508	−	0.985529i	$-0.554218\pi$
$281$	−5.68294	−0.339016	−0.169508	+	0.985529i	$0.554218\pi$
$282$	0	0
$283$	−28.2237	−1.67773	−0.838864	−	0.544342i	$-0.816780\pi$
$283$	−28.2237	−1.67773	−0.838864	+	0.544342i	$0.816780\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.3275	0.786697
$288$	0	0
$289$	3.42909	0.201711
$290$	0	0
$291$	−13.0634	−0.765789
$292$	0	0
$293$	5.92000	0.345850	0.172925	−	0.984935i	$-0.444678\pi$
$293$	5.92000	0.345850	0.172925	+	0.984935i	$0.444678\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.24054	−0.130009
$298$	0	0
$299$	−36.0835	−2.08676
$300$	0	0
$301$	−3.80206	−0.219147
$302$	0	0
$303$	−34.1691	−1.96297
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5.37023	−0.306495	−0.153248	−	0.988188i	$-0.548973\pi$
$307$	−5.37023	−0.306495	−0.153248	+	0.988188i	$0.548973\pi$
$308$	0	0
$309$	11.3085	0.643318
$310$	0	0
$311$	−3.90761	−0.221580	−0.110790	−	0.993844i	$-0.535338\pi$
$311$	−3.90761	−0.221580	−0.110790	+	0.993844i	$0.535338\pi$
$312$	0	0
$313$	30.1038	1.70156	0.850782	−	0.525518i	$-0.176129\pi$
$313$	30.1038	1.70156	0.850782	+	0.525518i	$0.176129\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.7722	0.942019	0.471010	−	0.882128i	$-0.343890\pi$
$317$	16.7722	0.942019	0.471010	+	0.882128i	$0.343890\pi$
$318$	0	0
$319$	−3.40535	−0.190663
$320$	0	0
$321$	8.02900	0.448135
$322$	0	0
$323$	4.51986	0.251491
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−11.0301	−0.609967
$328$	0	0
$329$	−2.48764	−0.137148
$330$	0	0
$331$	−24.4039	−1.34136	−0.670680	−	0.741747i	$-0.733998\pi$
$331$	−24.4039	−1.34136	−0.670680	+	0.741747i	$0.733998\pi$
$332$	0	0
$333$	−1.05181	−0.0576387
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.0027	0.599355	0.299678	−	0.954041i	$-0.403121\pi$
$337$	11.0027	0.599355	0.299678	+	0.954041i	$0.403121\pi$
$338$	0	0
$339$	10.3778	0.563646
$340$	0	0
$341$	−3.29895	−0.178648
$342$	0	0
$343$	−15.5196	−0.837980
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.2447	−0.711014	−0.355507	−	0.934674i	$-0.615692\pi$
$347$	−13.2447	−0.711014	−0.355507	+	0.934674i	$0.615692\pi$
$348$	0	0
$349$	1.03534	0.0554206	0.0277103	−	0.999616i	$-0.491178\pi$
$349$	1.03534	0.0554206	0.0277103	+	0.999616i	$0.491178\pi$
$350$	0	0
$351$	26.8445	1.43286
$352$	0	0
$353$	5.37333	0.285994	0.142997	−	0.989723i	$-0.454326\pi$
$353$	5.37333	0.285994	0.142997	+	0.989723i	$0.454326\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10.9119	0.577519
$358$	0	0
$359$	23.6189	1.24656	0.623280	−	0.781999i	$-0.285800\pi$
$359$	23.6189	1.24656	0.623280	+	0.781999i	$0.285800\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	20.7837	1.09086
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7.75683	−0.404903	−0.202452	−	0.979292i	$-0.564891\pi$
$367$	−7.75683	−0.404903	−0.202452	+	0.979292i	$0.564891\pi$
$368$	0	0
$369$	7.99071	0.415980
$370$	0	0
$371$	−17.5648	−0.911918
$372$	0	0
$373$	−32.2543	−1.67006	−0.835032	−	0.550201i	$-0.814551\pi$
$373$	−32.2543	−1.67006	−0.835032	+	0.550201i	$0.814551\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	40.8004	2.10133
$378$	0	0
$379$	28.7810	1.47838	0.739191	−	0.673496i	$-0.235208\pi$
$379$	28.7810	1.47838	0.739191	+	0.673496i	$0.235208\pi$
$380$	0	0
$381$	−35.6188	−1.82481
$382$	0	0
$383$	11.0718	0.565743	0.282871	−	0.959158i	$-0.408713\pi$
$383$	11.0718	0.565743	0.282871	+	0.959158i	$0.408713\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.27958	−0.115878
$388$	0	0
$389$	−4.65488	−0.236012	−0.118006	−	0.993013i	$-0.537650\pi$
$389$	−4.65488	−0.236012	−0.118006	+	0.993013i	$0.537650\pi$
$390$	0	0
$391$	26.4902	1.33967
$392$	0	0
$393$	28.0624	1.41556
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−0.573392	−0.0287777	−0.0143889	−	0.999896i	$-0.504580\pi$
$397$	−0.573392	−0.0287777	−0.0143889	+	0.999896i	$0.504580\pi$
$398$	0	0
$399$	2.41421	0.120862
$400$	0	0
$401$	35.9965	1.79758	0.898791	−	0.438378i	$-0.144447\pi$
$401$	35.9965	1.79758	0.898791	+	0.438378i	$0.144447\pi$
$402$	0	0
$403$	39.5257	1.96891
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.722854	0.0358305
$408$	0	0
$409$	18.3961	0.909626	0.454813	−	0.890587i	$-0.349706\pi$
$409$	18.3961	0.909626	0.454813	+	0.890587i	$0.349706\pi$
$410$	0	0
$411$	−6.58750	−0.324937
$412$	0	0
$413$	−5.41557	−0.266483
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.76415	0.380212
$418$	0	0
$419$	14.0610	0.686923	0.343461	−	0.939167i	$-0.388401\pi$
$419$	14.0610	0.686923	0.343461	+	0.939167i	$0.388401\pi$
$420$	0	0
$421$	−36.7041	−1.78885	−0.894425	−	0.447218i	$-0.852415\pi$
$421$	−36.7041	−1.78885	−0.894425	+	0.447218i	$0.852415\pi$
$422$	0	0
$423$	−1.49151	−0.0725195
$424$	0	0
$425$	0	0
$426$	0	0
$427$	13.3653	0.646793
$428$	0	0
$429$	6.12456	0.295697
$430$	0	0
$431$	21.2682	1.02445	0.512226	−	0.858851i	$-0.328821\pi$
$431$	21.2682	1.02445	0.512226	+	0.858851i	$0.328821\pi$
$432$	0	0
$433$	−11.8229	−0.568172	−0.284086	−	0.958799i	$-0.591690\pi$
$433$	−11.8229	−0.568172	−0.284086	+	0.958799i	$0.591690\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.86084	0.280362
$438$	0	0
$439$	−8.03728	−0.383599	−0.191799	−	0.981434i	$-0.561432\pi$
$439$	−8.03728	−0.383599	−0.191799	+	0.981434i	$0.561432\pi$
$440$	0	0
$441$	−4.07110	−0.193862
$442$	0	0
$443$	−10.3632	−0.492368	−0.246184	−	0.969223i	$-0.579177\pi$
$443$	−10.3632	−0.492368	−0.246184	+	0.969223i	$0.579177\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−18.2411	−0.862775
$448$	0	0
$449$	−33.4637	−1.57925	−0.789625	−	0.613590i	$-0.789725\pi$
$449$	−33.4637	−1.57925	−0.789625	+	0.613590i	$0.789725\pi$
$450$	0	0
$451$	−5.49161	−0.258590
$452$	0	0
$453$	−3.09959	−0.145631
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.6968	−1.29560	−0.647801	−	0.761809i	$-0.724311\pi$
$457$	−27.6968	−1.29560	−0.647801	+	0.761809i	$0.724311\pi$
$458$	0	0
$459$	−19.7075	−0.919870
$460$	0	0
$461$	4.70500	0.219134	0.109567	−	0.993979i	$-0.465054\pi$
$461$	4.70500	0.219134	0.109567	+	0.993979i	$0.465054\pi$
$462$	0	0
$463$	−5.44945	−0.253257	−0.126629	−	0.991950i	$-0.540416\pi$
$463$	−5.44945	−0.253257	−0.126629	+	0.991950i	$0.540416\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.9687	0.507571	0.253786	−	0.967260i	$-0.418324\pi$
$467$	10.9687	0.507571	0.253786	+	0.967260i	$0.418324\pi$
$468$	0	0
$469$	−12.3458	−0.570075
$470$	0	0
$471$	−36.2479	−1.67022
$472$	0	0
$473$	1.56664	0.0720343
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.5312	−0.482193
$478$	0	0
$479$	−14.4663	−0.660980	−0.330490	−	0.943809i	$-0.607214\pi$
$479$	−14.4663	−0.660980	−0.330490	+	0.943809i	$0.607214\pi$
$480$	0	0
$481$	−8.66072	−0.394895
$482$	0	0
$483$	14.1493	0.643817
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.1973	1.18711	0.593556	−	0.804793i	$-0.297724\pi$
$487$	26.1973	1.18711	0.593556	+	0.804793i	$0.297724\pi$
$488$	0	0
$489$	−30.2717	−1.36894
$490$	0	0
$491$	12.3251	0.556224	0.278112	−	0.960549i	$-0.410291\pi$
$491$	12.3251	0.556224	0.278112	+	0.960549i	$0.410291\pi$
$492$	0	0
$493$	−29.9531	−1.34902
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.25975	0.415356
$498$	0	0
$499$	27.4827	1.23029	0.615146	−	0.788413i	$-0.289097\pi$
$499$	27.4827	1.23029	0.615146	+	0.788413i	$0.289097\pi$
$500$	0	0
$501$	−34.0457	−1.52105
$502$	0	0
$503$	21.0989	0.940756	0.470378	−	0.882465i	$-0.344118\pi$
$503$	21.0989	0.940756	0.470378	+	0.882465i	$0.344118\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−48.2134	−2.14123
$508$	0	0
$509$	−14.3975	−0.638156	−0.319078	−	0.947728i	$-0.603373\pi$
$509$	−14.3975	−0.638156	−0.319078	+	0.947728i	$0.603373\pi$
$510$	0	0
$511$	15.9724	0.706577
$512$	0	0
$513$	−4.36022	−0.192508
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.02504	0.0450811
$518$	0	0
$519$	−36.3559	−1.59585
$520$	0	0
$521$	9.60225	0.420682	0.210341	−	0.977628i	$-0.432543\pi$
$521$	9.60225	0.420682	0.210341	+	0.977628i	$0.432543\pi$
$522$	0	0
$523$	1.07927	0.0471933	0.0235966	−	0.999722i	$-0.492488\pi$
$523$	1.07927	0.0471933	0.0235966	+	0.999722i	$0.492488\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.0172	−1.26401
$528$	0	0
$529$	11.3495	0.493457
$530$	0	0
$531$	−3.24699	−0.140907
$532$	0	0
$533$	65.7966	2.84997
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.17040	0.352579
$538$	0	0
$539$	2.79786	0.120513
$540$	0	0
$541$	−3.55082	−0.152662	−0.0763309	−	0.997083i	$-0.524321\pi$
$541$	−3.55082	−0.152662	−0.0763309	+	0.997083i	$0.524321\pi$
$542$	0	0
$543$	−22.8784	−0.981807
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−44.5233	−1.90368	−0.951840	−	0.306596i	$-0.900810\pi$
$547$	−44.5233	−1.90368	−0.951840	+	0.306596i	$0.900810\pi$
$548$	0	0
$549$	8.01339	0.342003
$550$	0	0
$551$	−6.62700	−0.282320
$552$	0	0
$553$	3.19424	0.135833
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.2157	0.475227	0.237613	−	0.971360i	$-0.423635\pi$
$557$	11.2157	0.475227	0.237613	+	0.971360i	$0.423635\pi$
$558$	0	0
$559$	−18.7704	−0.793903
$560$	0	0
$561$	−4.49626	−0.189832
$562$	0	0
$563$	−36.1900	−1.52523	−0.762614	−	0.646854i	$-0.776084\pi$
$563$	−36.1900	−1.52523	−0.762614	+	0.646854i	$0.776084\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−13.3238	−0.559548
$568$	0	0
$569$	−9.60022	−0.402462	−0.201231	−	0.979544i	$-0.564494\pi$
$569$	−9.60022	−0.402462	−0.201231	+	0.979544i	$0.564494\pi$
$570$	0	0
$571$	29.4150	1.23098	0.615490	−	0.788145i	$-0.288958\pi$
$571$	29.4150	1.23098	0.615490	+	0.788145i	$0.288958\pi$
$572$	0	0
$573$	−51.7240	−2.16080
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−20.9070	−0.870371	−0.435186	−	0.900341i	$-0.643317\pi$
$577$	−20.9070	−0.870371	−0.435186	+	0.900341i	$0.643317\pi$
$578$	0	0
$579$	−17.9114	−0.744372
$580$	0	0
$581$	−9.36384	−0.388478
$582$	0	0
$583$	7.23760	0.299751
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−41.2156	−1.70115	−0.850576	−	0.525853i	$-0.823746\pi$
$587$	−41.2156	−1.70115	−0.850576	+	0.525853i	$0.823746\pi$
$588$	0	0
$589$	−6.41995	−0.264529
$590$	0	0
$591$	−6.44453	−0.265093
$592$	0	0
$593$	32.0201	1.31491	0.657454	−	0.753495i	$-0.271633\pi$
$593$	32.0201	1.31491	0.657454	+	0.753495i	$0.271633\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.135700	0.00555385
$598$	0	0
$599$	46.7364	1.90960	0.954798	−	0.297256i	$-0.0960713\pi$
$599$	46.7364	1.90960	0.954798	+	0.297256i	$0.0960713\pi$
$600$	0	0
$601$	−22.9230	−0.935050	−0.467525	−	0.883980i	$-0.654854\pi$
$601$	−22.9230	−0.935050	−0.467525	+	0.883980i	$0.654854\pi$
$602$	0	0
$603$	−7.40211	−0.301437
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−38.7571	−1.57310	−0.786552	−	0.617525i	$-0.788136\pi$
$607$	−38.7571	−1.57310	−0.786552	+	0.617525i	$0.788136\pi$
$608$	0	0
$609$	−15.9990	−0.648312
$610$	0	0
$611$	−12.2813	−0.496847
$612$	0	0
$613$	13.4487	0.543187	0.271593	−	0.962412i	$-0.412449\pi$
$613$	13.4487	0.543187	0.271593	+	0.962412i	$0.412449\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.9727	0.441744	0.220872	−	0.975303i	$-0.429110\pi$
$617$	10.9727	0.441744	0.220872	+	0.975303i	$0.429110\pi$
$618$	0	0
$619$	46.1568	1.85520	0.927600	−	0.373575i	$-0.121868\pi$
$619$	46.1568	1.85520	0.927600	+	0.373575i	$0.121868\pi$
$620$	0	0
$621$	−25.5546	−1.02547
$622$	0	0
$623$	−9.79395	−0.392386
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.994780	−0.0397277
$628$	0	0
$629$	6.35815	0.253516
$630$	0	0
$631$	43.3700	1.72653	0.863266	−	0.504749i	$-0.168415\pi$
$631$	43.3700	1.72653	0.863266	+	0.504749i	$0.168415\pi$
$632$	0	0
$633$	−30.1908	−1.19998
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−33.5220	−1.32819
$638$	0	0
$639$	5.55183	0.219627
$640$	0	0
$641$	16.6952	0.659420	0.329710	−	0.944082i	$-0.393049\pi$
$641$	16.6952	0.659420	0.329710	+	0.944082i	$0.393049\pi$
$642$	0	0
$643$	23.4243	0.923765	0.461882	−	0.886941i	$-0.347174\pi$
$643$	23.4243	0.923765	0.461882	+	0.886941i	$0.347174\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.3397	0.681694	0.340847	−	0.940119i	$-0.389286\pi$
$647$	17.3397	0.681694	0.340847	+	0.940119i	$0.389286\pi$
$648$	0	0
$649$	2.23149	0.0875938
$650$	0	0
$651$	−15.4991	−0.607458
$652$	0	0
$653$	−18.3541	−0.718251	−0.359126	−	0.933289i	$-0.616925\pi$
$653$	−18.3541	−0.718251	−0.359126	+	0.933289i	$0.616925\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.57651	0.373615
$658$	0	0
$659$	−23.4398	−0.913085	−0.456543	−	0.889702i	$-0.650912\pi$
$659$	−23.4398	−0.913085	−0.456543	+	0.889702i	$0.650912\pi$
$660$	0	0
$661$	−20.7743	−0.808027	−0.404014	−	0.914753i	$-0.632385\pi$
$661$	−20.7743	−0.808027	−0.404014	+	0.914753i	$0.632385\pi$
$662$	0	0
$663$	53.8710	2.09218
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−38.8398	−1.50388
$668$	0	0
$669$	−43.3678	−1.67670
$670$	0	0
$671$	−5.50720	−0.212603
$672$	0	0
$673$	6.74273	0.259913	0.129957	−	0.991520i	$-0.458516\pi$
$673$	6.74273	0.259913	0.129957	+	0.991520i	$0.458516\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25.1589	−0.966934	−0.483467	−	0.875363i	$-0.660623\pi$
$677$	−25.1589	−0.966934	−0.483467	+	0.875363i	$0.660623\pi$
$678$	0	0
$679$	8.41523	0.322947
$680$	0	0
$681$	47.1321	1.80611
$682$	0	0
$683$	13.7289	0.525321	0.262661	−	0.964888i	$-0.415400\pi$
$683$	13.7289	0.525321	0.262661	+	0.964888i	$0.415400\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.2148	0.389718
$688$	0	0
$689$	−86.7157	−3.30360
$690$	0	0
$691$	26.8531	1.02154	0.510770	−	0.859718i	$-0.329361\pi$
$691$	26.8531	1.02154	0.510770	+	0.859718i	$0.329361\pi$
$692$	0	0
$693$	−0.479145	−0.0182012
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−48.3037	−1.82963
$698$	0	0
$699$	−8.33085	−0.315102
$700$	0	0
$701$	−20.1359	−0.760524	−0.380262	−	0.924879i	$-0.624166\pi$
$701$	−20.1359	−0.760524	−0.380262	+	0.924879i	$0.624166\pi$
$702$	0	0
$703$	1.40671	0.0530553
$704$	0	0
$705$	0	0
$706$	0	0
$707$	22.0112	0.827818
$708$	0	0
$709$	23.4008	0.878837	0.439419	−	0.898282i	$-0.355184\pi$
$709$	23.4008	0.878837	0.439419	+	0.898282i	$0.355184\pi$
$710$	0	0
$711$	1.91515	0.0718239
$712$	0	0
$713$	−37.6263	−1.40912
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.79648	0.104437
$718$	0	0
$719$	10.5133	0.392082	0.196041	−	0.980596i	$-0.437191\pi$
$719$	10.5133	0.392082	0.196041	+	0.980596i	$0.437191\pi$
$720$	0	0
$721$	−7.28476	−0.271299
$722$	0	0
$723$	−10.0683	−0.374443
$724$	0	0
$725$	0	0
$726$	0	0
$727$	35.6031	1.32044	0.660222	−	0.751070i	$-0.270462\pi$
$727$	35.6031	1.32044	0.660222	+	0.751070i	$0.270462\pi$
$728$	0	0
$729$	17.3343	0.642011
$730$	0	0
$731$	13.7800	0.509673
$732$	0	0
$733$	37.2476	1.37577	0.687886	−	0.725818i	$-0.258539\pi$
$733$	37.2476	1.37577	0.687886	+	0.725818i	$0.258539\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.08710	0.187386
$738$	0	0
$739$	−26.1515	−0.961998	−0.480999	−	0.876721i	$-0.659726\pi$
$739$	−26.1515	−0.961998	−0.480999	+	0.876721i	$0.659726\pi$
$740$	0	0
$741$	11.9187	0.437846
$742$	0	0
$743$	−16.4700	−0.604226	−0.302113	−	0.953272i	$-0.597692\pi$
$743$	−16.4700	−0.604226	−0.302113	+	0.953272i	$0.597692\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5.61424	−0.205414
$748$	0	0
$749$	−5.17216	−0.188987
$750$	0	0
$751$	−24.0052	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$751$	−24.0052	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$752$	0	0
$753$	−3.04026	−0.110793
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−34.5457	−1.25558	−0.627792	−	0.778381i	$-0.716041\pi$
$757$	−34.5457	−1.25558	−0.627792	+	0.778381i	$0.716041\pi$
$758$	0	0
$759$	−5.83025	−0.211625
$760$	0	0
$761$	−34.6920	−1.25758	−0.628792	−	0.777573i	$-0.716450\pi$
$761$	−34.6920	−1.25758	−0.628792	+	0.777573i	$0.716450\pi$
$762$	0	0
$763$	7.10543	0.257234
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−26.7362	−0.965387
$768$	0	0
$769$	−30.1152	−1.08598	−0.542991	−	0.839738i	$-0.682708\pi$
$769$	−30.1152	−1.08598	−0.542991	+	0.839738i	$0.682708\pi$
$770$	0	0
$771$	−47.0096	−1.69301
$772$	0	0
$773$	−29.9879	−1.07859	−0.539295	−	0.842117i	$-0.681309\pi$
$773$	−29.9879	−1.07859	−0.539295	+	0.842117i	$0.681309\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.39611	0.121835
$778$	0	0
$779$	−10.6870	−0.382901
$780$	0	0
$781$	−3.81549	−0.136529
$782$	0	0
$783$	28.8952	1.03263
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−10.3894	−0.370342	−0.185171	−	0.982706i	$-0.559284\pi$
$787$	−10.3894	−0.370342	−0.185171	+	0.982706i	$0.559284\pi$
$788$	0	0
$789$	14.5334	0.517401
$790$	0	0
$791$	−6.68524	−0.237700
$792$	0	0
$793$	65.9833	2.34314
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32.8854	−1.16486	−0.582430	−	0.812881i	$-0.697898\pi$
$797$	−32.8854	−1.16486	−0.582430	+	0.812881i	$0.697898\pi$
$798$	0	0
$799$	9.01612	0.318967
$800$	0	0
$801$	−5.87212	−0.207481
$802$	0	0
$803$	−6.58145	−0.232254
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−18.1545	−0.639070
$808$	0	0
$809$	−16.5347	−0.581329	−0.290664	−	0.956825i	$-0.593876\pi$
$809$	−16.5347	−0.581329	−0.290664	+	0.956825i	$0.593876\pi$
$810$	0	0
$811$	2.88477	0.101298	0.0506489	−	0.998717i	$-0.483871\pi$
$811$	2.88477	0.101298	0.0506489	+	0.998717i	$0.483871\pi$
$812$	0	0
$813$	−32.5431	−1.14134
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.04878	0.106663
$818$	0	0
$819$	5.74078	0.200599
$820$	0	0
$821$	−10.0122	−0.349428	−0.174714	−	0.984619i	$-0.555900\pi$
$821$	−10.0122	−0.349428	−0.174714	+	0.984619i	$0.555900\pi$
$822$	0	0
$823$	−26.1520	−0.911602	−0.455801	−	0.890082i	$-0.650647\pi$
$823$	−26.1520	−0.911602	−0.455801	+	0.890082i	$0.650647\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.1880	0.875873	0.437936	−	0.899006i	$-0.355710\pi$
$827$	25.1880	0.875873	0.437936	+	0.899006i	$0.355710\pi$
$828$	0	0
$829$	3.32822	0.115594	0.0577969	−	0.998328i	$-0.481592\pi$
$829$	3.32822	0.115594	0.0577969	+	0.998328i	$0.481592\pi$
$830$	0	0
$831$	−3.00716	−0.104317
$832$	0	0
$833$	24.6097	0.852676
$834$	0	0
$835$	0	0
$836$	0	0
$837$	27.9924	0.967557
$838$	0	0
$839$	−50.5339	−1.74462	−0.872312	−	0.488949i	$-0.837380\pi$
$839$	−50.5339	−1.74462	−0.872312	+	0.488949i	$0.837380\pi$
$840$	0	0
$841$	14.9171	0.514384
$842$	0	0
$843$	11.0016	0.378915
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−13.3885	−0.460036
$848$	0	0
$849$	54.6383	1.87518
$850$	0	0
$851$	8.24454	0.282619
$852$	0	0
$853$	54.6636	1.87165	0.935823	−	0.352470i	$-0.114658\pi$
$853$	54.6636	1.87165	0.935823	+	0.352470i	$0.114658\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.8578	−0.951603	−0.475802	−	0.879553i	$-0.657842\pi$
$857$	−27.8578	−0.951603	−0.475802	+	0.879553i	$0.657842\pi$
$858$	0	0
$859$	−55.2474	−1.88502	−0.942509	−	0.334181i	$-0.891540\pi$
$859$	−55.2474	−1.88502	−0.942509	+	0.334181i	$0.891540\pi$
$860$	0	0
$861$	−25.8007	−0.879285
$862$	0	0
$863$	27.3258	0.930181	0.465090	−	0.885263i	$-0.346022\pi$
$863$	27.3258	0.930181	0.465090	+	0.885263i	$0.346022\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−6.63838	−0.225451
$868$	0	0
$869$	−1.31619	−0.0446487
$870$	0	0
$871$	−60.9500	−2.06521
$872$	0	0
$873$	5.04548	0.170764
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.3409	0.450489	0.225245	−	0.974302i	$-0.427682\pi$
$877$	13.3409	0.450489	0.225245	+	0.974302i	$0.427682\pi$
$878$	0	0
$879$	−11.4605	−0.386554
$880$	0	0
$881$	−44.5458	−1.50079	−0.750393	−	0.660992i	$-0.770136\pi$
$881$	−44.5458	−1.50079	−0.750393	+	0.660992i	$0.770136\pi$
$882$	0	0
$883$	23.4498	0.789148	0.394574	−	0.918864i	$-0.370892\pi$
$883$	23.4498	0.789148	0.394574	+	0.918864i	$0.370892\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40.1362	1.34764	0.673821	−	0.738895i	$-0.264652\pi$
$887$	40.1362	1.34764	0.673821	+	0.738895i	$0.264652\pi$
$888$	0	0
$889$	22.9451	0.769553
$890$	0	0
$891$	5.49010	0.183925
$892$	0	0
$893$	1.99478	0.0667528
$894$	0	0
$895$	0	0
$896$	0	0
$897$	69.8539	2.33235
$898$	0	0
$899$	42.5450	1.41895
$900$	0	0
$901$	63.6611	2.12086
$902$	0	0
$903$	7.36040	0.244939
$904$	0	0
$905$	0	0
$906$	0	0
$907$	35.8062	1.18893	0.594463	−	0.804123i	$-0.297364\pi$
$907$	35.8062	1.18893	0.594463	+	0.804123i	$0.297364\pi$
$908$	0	0
$909$	13.1972	0.437723
$910$	0	0
$911$	−29.5302	−0.978381	−0.489190	−	0.872177i	$-0.662708\pi$
$911$	−29.5302	−0.978381	−0.489190	+	0.872177i	$0.662708\pi$
$912$	0	0
$913$	3.85839	0.127694
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−18.0774	−0.596967
$918$	0	0
$919$	11.7570	0.387829	0.193914	−	0.981018i	$-0.437882\pi$
$919$	11.7570	0.387829	0.193914	+	0.981018i	$0.437882\pi$
$920$	0	0
$921$	10.3962	0.342567
$922$	0	0
$923$	45.7145	1.50471
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.36769	−0.143454
$928$	0	0
$929$	−21.5189	−0.706012	−0.353006	−	0.935621i	$-0.614841\pi$
$929$	−21.5189	−0.706012	−0.353006	+	0.935621i	$0.614841\pi$
$930$	0	0
$931$	5.44480	0.178446
$932$	0	0
$933$	7.56473	0.247658
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−31.7690	−1.03785	−0.518924	−	0.854820i	$-0.673667\pi$
$937$	−31.7690	−1.03785	−0.518924	+	0.854820i	$0.673667\pi$
$938$	0	0
$939$	−58.2778	−1.90182
$940$	0	0
$941$	−0.730603	−0.0238170	−0.0119085	−	0.999929i	$-0.503791\pi$
$941$	−0.730603	−0.0238170	−0.0119085	+	0.999929i	$0.503791\pi$
$942$	0	0
$943$	−62.6348	−2.03967
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.2171	−1.33938	−0.669688	−	0.742643i	$-0.733572\pi$
$947$	−41.2171	−1.33938	−0.669688	+	0.742643i	$0.733572\pi$
$948$	0	0
$949$	78.8543	2.55972
$950$	0	0
$951$	−32.4692	−1.05289
$952$	0	0
$953$	39.9343	1.29360	0.646800	−	0.762660i	$-0.276107\pi$
$953$	39.9343	1.29360	0.646800	+	0.762660i	$0.276107\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.59241	0.213102
$958$	0	0
$959$	4.24357	0.137032
$960$	0	0
$961$	10.2157	0.329539
$962$	0	0
$963$	−3.10105	−0.0999300
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−8.73989	−0.281056	−0.140528	−	0.990077i	$-0.544880\pi$
$967$	−8.73989	−0.281056	−0.140528	+	0.990077i	$0.544880\pi$
$968$	0	0
$969$	−8.74998	−0.281090
$970$	0	0
$971$	7.49072	0.240389	0.120194	−	0.992750i	$-0.461648\pi$
$971$	7.49072	0.240389	0.120194	+	0.992750i	$0.461648\pi$
$972$	0	0
$973$	−5.00155	−0.160342
$974$	0	0
$975$	0	0
$976$	0	0
$977$	51.6069	1.65105	0.825526	−	0.564364i	$-0.190879\pi$
$977$	51.6069	1.65105	0.825526	+	0.564364i	$0.190879\pi$
$978$	0	0
$979$	4.03561	0.128979
$980$	0	0
$981$	4.26017	0.136017
$982$	0	0
$983$	−57.9017	−1.84678	−0.923389	−	0.383865i	$-0.874593\pi$
$983$	−57.9017	−1.84678	−0.923389	+	0.383865i	$0.874593\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.81582	0.153289
$988$	0	0
$989$	17.8684	0.568182
$990$	0	0
$991$	−18.2369	−0.579314	−0.289657	−	0.957131i	$-0.593541\pi$
$991$	−18.2369	−0.579314	−0.289657	+	0.957131i	$0.593541\pi$
$992$	0	0
$993$	47.2435	1.49923
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−10.7461	−0.340332	−0.170166	−	0.985415i	$-0.554430\pi$
$997$	−10.7461	−0.340332	−0.170166	+	0.985415i	$0.554430\pi$
$998$	0	0
$999$	−6.13358	−0.194058

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.ba.1.2	✓	6
4.3	odd	2		7600.2.a.cl.1.5		6
5.2	odd	4		3800.2.d.q.3649.10		12
5.3	odd	4		3800.2.d.q.3649.3		12
5.4	even	2		3800.2.a.bc.1.5	yes	6
20.19	odd	2		7600.2.a.ch.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.ba.1.2	✓	6	1.1	even	1	trivial
3800.2.a.bc.1.5	yes	6	5.4	even	2
3800.2.d.q.3649.3		12	5.3	odd	4
3800.2.d.q.3649.10		12	5.2	odd	4
7600.2.a.ch.1.2		6	20.19	odd	2
7600.2.a.cl.1.5		6	4.3	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.93590 q^{3} +1.24708 q^{7} +0.747704 q^{9} -0.513860 q^{11} +6.15670 q^{13} -4.51986 q^{17} -1.00000 q^{19} -2.41421 q^{21} -5.86084 q^{23} +4.36022 q^{27} +6.62700 q^{29} +6.41995 q^{31} +0.994780 q^{33} -1.40671 q^{37} -11.9187 q^{39} +10.6870 q^{41} -3.04878 q^{43} -1.99478 q^{47} -5.44480 q^{49} +8.74998 q^{51} -14.0848 q^{53} +1.93590 q^{57} -4.34261 q^{59} +10.7173 q^{61} +0.932444 q^{63} -9.89978 q^{67} +11.3460 q^{69} +7.42517 q^{71} +12.8079 q^{73} -0.640822 q^{77} +2.56138 q^{79} -10.6841 q^{81} -7.50864 q^{83} -12.8292 q^{87} -7.85353 q^{89} +7.67787 q^{91} -12.4284 q^{93} +6.74797 q^{97} -0.384215 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} + 3 q^{11} + q^{13} + 14 q^{17} - 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} + 5 q^{31} - 2 q^{33} + 8 q^{37} + 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47}+ \cdots - 6 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 + 3 * q^11 + q^13 + 14 * q^17 - 6 * q^19 + 15 * q^21 - 12 * q^23 - 8 * q^27 + 9 * q^29 + 5 * q^31 - 2 * q^33 + 8 * q^37 + 12 * q^39 + 3 * q^41 - 15 * q^43 - 4 * q^47 + 22 * q^49 + 33 * q^51 - 13 * q^53 + 2 * q^57 + 9 * q^61 - 21 * q^63 + 3 * q^67 - 11 * q^69 + 19 * q^71 - 3 * q^73 + 36 * q^77 - 16 * q^79 + 26 * q^81 - 31 * q^83 + 25 * q^87 + 14 * q^89 + 42 * q^91 - 39 * q^93 + 11 * q^97 - 6 * q^99

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