LMFDB - Embedded newform 3800.2.a.bb.1.2

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			1.2
Root			$2.70452$ 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)

$f(q)$	$=$	$q-2.70452 q^{3} -3.77934 q^{7} +4.31442 q^{9} +O(q^{10})$	$q-2.70452 q^{3} -3.77934 q^{7} +4.31442 q^{9} +4.71724 q^{11} +5.44195 q^{13} +2.40944 q^{17} +1.00000 q^{19} +10.2213 q^{21} -4.38924 q^{23} -3.55488 q^{27} +9.05146 q^{29} -9.92581 q^{31} -12.7579 q^{33} -10.0429 q^{37} -14.7179 q^{39} +1.11712 q^{41} +12.1169 q^{43} -6.72432 q^{47} +7.28339 q^{49} -6.51637 q^{51} -4.34694 q^{53} -2.70452 q^{57} -4.03914 q^{59} -9.14562 q^{61} -16.3057 q^{63} -4.39672 q^{67} +11.8708 q^{69} +7.90222 q^{71} -3.07165 q^{73} -17.8280 q^{77} +11.3996 q^{79} -3.32902 q^{81} +4.37652 q^{83} -24.4798 q^{87} +2.53933 q^{89} -20.5670 q^{91} +26.8445 q^{93} +0.302105 q^{97} +20.3522 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

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
$3$	−2.70452	−1.56145	−0.780727	−	0.624872i	$-0.785151\pi$
$3$	−2.70452	−1.56145	−0.780727	+	0.624872i	$0.785151\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.77934	−1.42846	−0.714228	−	0.699914i	$-0.753222\pi$
$7$	−3.77934	−1.42846	−0.714228	+	0.699914i	$0.753222\pi$
$8$	0	0
$9$	4.31442	1.43814
$10$	0	0
$11$	4.71724	1.42230	0.711150	−	0.703040i	$-0.248175\pi$
$11$	4.71724	1.42230	0.711150	+	0.703040i	$0.248175\pi$
$12$	0	0
$13$	5.44195	1.50933	0.754663	−	0.656113i	$-0.227800\pi$
$13$	5.44195	1.50933	0.754663	+	0.656113i	$0.227800\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.40944	0.584375	0.292187	−	0.956361i	$-0.405617\pi$
$17$	2.40944	0.584375	0.292187	+	0.956361i	$0.405617\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	10.2213	2.23047
$22$	0	0
$23$	−4.38924	−0.915220	−0.457610	−	0.889153i	$-0.651294\pi$
$23$	−4.38924	−0.915220	−0.457610	+	0.889153i	$0.651294\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.55488	−0.684137
$28$	0	0
$29$	9.05146	1.68081	0.840407	−	0.541956i	$-0.182316\pi$
$29$	9.05146	1.68081	0.840407	+	0.541956i	$0.182316\pi$
$30$	0	0
$31$	−9.92581	−1.78273	−0.891364	−	0.453288i	$-0.850251\pi$
$31$	−9.92581	−1.78273	−0.891364	+	0.453288i	$0.850251\pi$
$32$	0	0
$33$	−12.7579	−2.22086
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.0429	−1.65105	−0.825524	−	0.564367i	$-0.809120\pi$
$37$	−10.0429	−1.65105	−0.825524	+	0.564367i	$0.809120\pi$
$38$	0	0
$39$	−14.7179	−2.35674
$40$	0	0
$41$	1.11712	0.174465	0.0872326	−	0.996188i	$-0.472198\pi$
$41$	1.11712	0.174465	0.0872326	+	0.996188i	$0.472198\pi$
$42$	0	0
$43$	12.1169	1.84781	0.923904	−	0.382625i	$-0.124980\pi$
$43$	12.1169	1.84781	0.923904	+	0.382625i	$0.124980\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.72432	−0.980842	−0.490421	−	0.871486i	$-0.663157\pi$
$47$	−6.72432	−0.980842	−0.490421	+	0.871486i	$0.663157\pi$
$48$	0	0
$49$	7.28339	1.04048
$50$	0	0
$51$	−6.51637	−0.912474
$52$	0	0
$53$	−4.34694	−0.597098	−0.298549	−	0.954394i	$-0.596503\pi$
$53$	−4.34694	−0.597098	−0.298549	+	0.954394i	$0.596503\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.70452	−0.358222
$58$	0	0
$59$	−4.03914	−0.525851	−0.262926	−	0.964816i	$-0.584687\pi$
$59$	−4.03914	−0.525851	−0.262926	+	0.964816i	$0.584687\pi$
$60$	0	0
$61$	−9.14562	−1.17098	−0.585488	−	0.810681i	$-0.699097\pi$
$61$	−9.14562	−1.17098	−0.585488	+	0.810681i	$0.699097\pi$
$62$	0	0
$63$	−16.3057	−2.05432
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.39672	−0.537145	−0.268572	−	0.963259i	$-0.586552\pi$
$67$	−4.39672	−0.537145	−0.268572	+	0.963259i	$0.586552\pi$
$68$	0	0
$69$	11.8708	1.42907
$70$	0	0
$71$	7.90222	0.937821	0.468910	−	0.883246i	$-0.344647\pi$
$71$	7.90222	0.937821	0.468910	+	0.883246i	$0.344647\pi$
$72$	0	0
$73$	−3.07165	−0.359510	−0.179755	−	0.983711i	$-0.557530\pi$
$73$	−3.07165	−0.359510	−0.179755	+	0.983711i	$0.557530\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−17.8280	−2.03169
$78$	0	0
$79$	11.3996	1.28256	0.641280	−	0.767307i	$-0.278404\pi$
$79$	11.3996	1.28256	0.641280	+	0.767307i	$0.278404\pi$
$80$	0	0
$81$	−3.32902	−0.369891
$82$	0	0
$83$	4.37652	0.480386	0.240193	−	0.970725i	$-0.422789\pi$
$83$	4.37652	0.480386	0.240193	+	0.970725i	$0.422789\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−24.4798	−2.62451
$88$	0	0
$89$	2.53933	0.269169	0.134584	−	0.990902i	$-0.457030\pi$
$89$	2.53933	0.269169	0.134584	+	0.990902i	$0.457030\pi$
$90$	0	0
$91$	−20.5670	−2.15600
$92$	0	0
$93$	26.8445	2.78365
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.302105	0.0306741	0.0153371	−	0.999882i	$-0.495118\pi$
$97$	0.302105	0.0306741	0.0153371	+	0.999882i	$0.495118\pi$
$98$	0	0
$99$	20.3522	2.04547
$100$	0	0
$101$	4.77080	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$101$	4.77080	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$102$	0	0
$103$	17.9014	1.76388	0.881939	−	0.471364i	$-0.156238\pi$
$103$	17.9014	1.76388	0.881939	+	0.471364i	$0.156238\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.1673	1.27293	0.636465	−	0.771305i	$-0.280396\pi$
$107$	13.1673	1.27293	0.636465	+	0.771305i	$0.280396\pi$
$108$	0	0
$109$	0.756212	0.0724320	0.0362160	−	0.999344i	$-0.488470\pi$
$109$	0.756212	0.0724320	0.0362160	+	0.999344i	$0.488470\pi$
$110$	0	0
$111$	27.1613	2.57804
$112$	0	0
$113$	20.6082	1.93866	0.969329	−	0.245766i	$-0.0790395\pi$
$113$	20.6082	1.93866	0.969329	+	0.245766i	$0.0790395\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	23.4789	2.17062
$118$	0	0
$119$	−9.10608	−0.834753
$120$	0	0
$121$	11.2523	1.02294
$122$	0	0
$123$	−3.02128	−0.272419
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−19.3178	−1.71418	−0.857089	−	0.515168i	$-0.827730\pi$
$127$	−19.3178	−1.71418	−0.857089	+	0.515168i	$0.827730\pi$
$128$	0	0
$129$	−32.7703	−2.88527
$130$	0	0
$131$	−4.29192	−0.374986	−0.187493	−	0.982266i	$-0.560036\pi$
$131$	−4.29192	−0.374986	−0.187493	+	0.982266i	$0.560036\pi$
$132$	0	0
$133$	−3.77934	−0.327710
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.2413	1.81476	0.907382	−	0.420306i	$-0.138077\pi$
$137$	21.2413	1.81476	0.907382	+	0.420306i	$0.138077\pi$
$138$	0	0
$139$	1.94814	0.165239	0.0826197	−	0.996581i	$-0.473671\pi$
$139$	1.94814	0.165239	0.0826197	+	0.996581i	$0.473671\pi$
$140$	0	0
$141$	18.1860	1.53154
$142$	0	0
$143$	25.6710	2.14671
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−19.6981	−1.62467
$148$	0	0
$149$	−4.56655	−0.374107	−0.187053	−	0.982350i	$-0.559894\pi$
$149$	−4.56655	−0.374107	−0.187053	+	0.982350i	$0.559894\pi$
$150$	0	0
$151$	−8.84246	−0.719590	−0.359795	−	0.933031i	$-0.617153\pi$
$151$	−8.84246	−0.719590	−0.359795	+	0.933031i	$0.617153\pi$
$152$	0	0
$153$	10.3953	0.840413
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.63494	−0.449717	−0.224859	−	0.974391i	$-0.572192\pi$
$157$	−5.63494	−0.449717	−0.224859	+	0.974391i	$0.572192\pi$
$158$	0	0
$159$	11.7564	0.932341
$160$	0	0
$161$	16.5884	1.30735
$162$	0	0
$163$	−3.71114	−0.290679	−0.145340	−	0.989382i	$-0.546427\pi$
$163$	−3.71114	−0.290679	−0.145340	+	0.989382i	$0.546427\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.403668	−0.0312368	−0.0156184	−	0.999878i	$-0.504972\pi$
$167$	−0.403668	−0.0312368	−0.0156184	+	0.999878i	$0.504972\pi$
$168$	0	0
$169$	16.6148	1.27806
$170$	0	0
$171$	4.31442	0.329932
$172$	0	0
$173$	−13.9048	−1.05716	−0.528580	−	0.848884i	$-0.677275\pi$
$173$	−13.9048	−1.05716	−0.528580	+	0.848884i	$0.677275\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.9239	0.821093
$178$	0	0
$179$	3.27673	0.244914	0.122457	−	0.992474i	$-0.460923\pi$
$179$	3.27673	0.244914	0.122457	+	0.992474i	$0.460923\pi$
$180$	0	0
$181$	13.0254	0.968173	0.484086	−	0.875020i	$-0.339152\pi$
$181$	13.0254	0.968173	0.484086	+	0.875020i	$0.339152\pi$
$182$	0	0
$183$	24.7345	1.82843
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.3659	0.831156
$188$	0	0
$189$	13.4351	0.977260
$190$	0	0
$191$	−17.6403	−1.27641	−0.638203	−	0.769868i	$-0.720322\pi$
$191$	−17.6403	−1.27641	−0.638203	+	0.769868i	$0.720322\pi$
$192$	0	0
$193$	−16.8445	−1.21250	−0.606248	−	0.795276i	$-0.707326\pi$
$193$	−16.8445	−1.21250	−0.606248	+	0.795276i	$0.707326\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.8742	1.41598	0.707988	−	0.706224i	$-0.249603\pi$
$197$	19.8742	1.41598	0.707988	+	0.706224i	$0.249603\pi$
$198$	0	0
$199$	−8.54003	−0.605386	−0.302693	−	0.953088i	$-0.597886\pi$
$199$	−8.54003	−0.605386	−0.302693	+	0.953088i	$0.597886\pi$
$200$	0	0
$201$	11.8910	0.838727
$202$	0	0
$203$	−34.2085	−2.40097
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−18.9370	−1.31622
$208$	0	0
$209$	4.71724	0.326298
$210$	0	0
$211$	−0.0127177	−0.000875525 0	−0.000437762	−	1.00000i	$-0.500139\pi$
$211$	−0.0127177	−0.000875525 0	−0.000437762		1.00000i	$0.500139\pi$
$212$	0	0
$213$	−21.3717	−1.46437
$214$	0	0
$215$	0	0
$216$	0	0
$217$	37.5130	2.54655
$218$	0	0
$219$	8.30734	0.561358
$220$	0	0
$221$	13.1120	0.882012
$222$	0	0
$223$	18.6403	1.24825	0.624124	−	0.781326i	$-0.285456\pi$
$223$	18.6403	1.24825	0.624124	+	0.781326i	$0.285456\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.75187	−0.514510	−0.257255	−	0.966344i	$-0.582818\pi$
$227$	−7.75187	−0.514510	−0.257255	+	0.966344i	$0.582818\pi$
$228$	0	0
$229$	−19.3735	−1.28023	−0.640117	−	0.768277i	$-0.721114\pi$
$229$	−19.3735	−1.28023	−0.640117	+	0.768277i	$0.721114\pi$
$230$	0	0
$231$	48.2162	3.17240
$232$	0	0
$233$	8.63389	0.565625	0.282813	−	0.959175i	$-0.408733\pi$
$233$	8.63389	0.565625	0.282813	+	0.959175i	$0.408733\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−30.8306	−2.00266
$238$	0	0
$239$	9.47644	0.612980	0.306490	−	0.951874i	$-0.400845\pi$
$239$	9.47644	0.612980	0.306490	+	0.951874i	$0.400845\pi$
$240$	0	0
$241$	29.3641	1.89151	0.945755	−	0.324880i	$-0.105324\pi$
$241$	29.3641	1.89151	0.945755	+	0.324880i	$0.105324\pi$
$242$	0	0
$243$	19.6681	1.26171
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.44195	0.346263
$248$	0	0
$249$	−11.8364	−0.750101
$250$	0	0
$251$	−7.57377	−0.478052	−0.239026	−	0.971013i	$-0.576828\pi$
$251$	−7.57377	−0.478052	−0.239026	+	0.971013i	$0.576828\pi$
$252$	0	0
$253$	−20.7051	−1.30172
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.7573	0.733400	0.366700	−	0.930339i	$-0.380488\pi$
$257$	11.7573	0.733400	0.366700	+	0.930339i	$0.380488\pi$
$258$	0	0
$259$	37.9556	2.35845
$260$	0	0
$261$	39.0518	2.41725
$262$	0	0
$263$	−2.20334	−0.135864	−0.0679319	−	0.997690i	$-0.521640\pi$
$263$	−2.20334	−0.135864	−0.0679319	+	0.997690i	$0.521640\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.86767	−0.420295
$268$	0	0
$269$	6.14642	0.374754	0.187377	−	0.982288i	$-0.440001\pi$
$269$	6.14642	0.374754	0.187377	+	0.982288i	$0.440001\pi$
$270$	0	0
$271$	25.6620	1.55886	0.779428	−	0.626491i	$-0.215510\pi$
$271$	25.6620	1.55886	0.779428	+	0.626491i	$0.215510\pi$
$272$	0	0
$273$	55.6238	3.36650
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4.64202	0.278912	0.139456	−	0.990228i	$-0.455465\pi$
$277$	4.64202	0.278912	0.139456	+	0.990228i	$0.455465\pi$
$278$	0	0
$279$	−42.8241	−2.56381
$280$	0	0
$281$	−0.369967	−0.0220704	−0.0110352	−	0.999939i	$-0.503513\pi$
$281$	−0.369967	−0.0220704	−0.0110352	+	0.999939i	$0.503513\pi$
$282$	0	0
$283$	1.21325	0.0721203	0.0360602	−	0.999350i	$-0.488519\pi$
$283$	1.21325	0.0721203	0.0360602	+	0.999350i	$0.488519\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.22198	−0.249216
$288$	0	0
$289$	−11.1946	−0.658506
$290$	0	0
$291$	−0.817050	−0.0478963
$292$	0	0
$293$	−8.66265	−0.506077	−0.253039	−	0.967456i	$-0.581430\pi$
$293$	−8.66265	−0.506077	−0.253039	+	0.967456i	$0.581430\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−16.7692	−0.973049
$298$	0	0
$299$	−23.8860	−1.38137
$300$	0	0
$301$	−45.7938	−2.63951
$302$	0	0
$303$	−12.9027	−0.741242
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−8.26123	−0.471493	−0.235747	−	0.971815i	$-0.575754\pi$
$307$	−8.26123	−0.471493	−0.235747	+	0.971815i	$0.575754\pi$
$308$	0	0
$309$	−48.4147	−2.75421
$310$	0	0
$311$	7.54408	0.427786	0.213893	−	0.976857i	$-0.431386\pi$
$311$	7.54408	0.427786	0.213893	+	0.976857i	$0.431386\pi$
$312$	0	0
$313$	3.43152	0.193961	0.0969804	−	0.995286i	$-0.469082\pi$
$313$	3.43152	0.193961	0.0969804	+	0.995286i	$0.469082\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.50784	−0.140854	−0.0704270	−	0.997517i	$-0.522436\pi$
$317$	−2.50784	−0.140854	−0.0704270	+	0.997517i	$0.522436\pi$
$318$	0	0
$319$	42.6979	2.39062
$320$	0	0
$321$	−35.6112	−1.98762
$322$	0	0
$323$	2.40944	0.134065
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.04519	−0.113099
$328$	0	0
$329$	25.4135	1.40109
$330$	0	0
$331$	29.1167	1.60040	0.800200	−	0.599734i	$-0.204727\pi$
$331$	29.1167	1.60040	0.800200	+	0.599734i	$0.204727\pi$
$332$	0	0
$333$	−43.3295	−2.37444
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−20.4033	−1.11144	−0.555720	−	0.831370i	$-0.687557\pi$
$337$	−20.4033	−1.11144	−0.555720	+	0.831370i	$0.687557\pi$
$338$	0	0
$339$	−55.7353	−3.02713
$340$	0	0
$341$	−46.8224	−2.53557
$342$	0	0
$343$	−1.07103	−0.0578300
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.9967	1.39557	0.697787	−	0.716305i	$-0.254168\pi$
$347$	25.9967	1.39557	0.697787	+	0.716305i	$0.254168\pi$
$348$	0	0
$349$	3.32628	0.178052	0.0890258	−	0.996029i	$-0.471625\pi$
$349$	3.32628	0.178052	0.0890258	+	0.996029i	$0.471625\pi$
$350$	0	0
$351$	−19.3455	−1.03259
$352$	0	0
$353$	15.0904	0.803179	0.401590	−	0.915820i	$-0.368458\pi$
$353$	15.0904	0.803179	0.401590	+	0.915820i	$0.368458\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	24.6276	1.30343
$358$	0	0
$359$	−12.8292	−0.677102	−0.338551	−	0.940948i	$-0.609937\pi$
$359$	−12.8292	−0.677102	−0.338551	+	0.940948i	$0.609937\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−30.4321	−1.59727
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.11198	−0.214644	−0.107322	−	0.994224i	$-0.534228\pi$
$367$	−4.11198	−0.214644	−0.107322	+	0.994224i	$0.534228\pi$
$368$	0	0
$369$	4.81974	0.250905
$370$	0	0
$371$	16.4285	0.852927
$372$	0	0
$373$	16.5930	0.859153	0.429576	−	0.903031i	$-0.358663\pi$
$373$	16.5930	0.859153	0.429576	+	0.903031i	$0.358663\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	49.2576	2.53690
$378$	0	0
$379$	30.6822	1.57604	0.788021	−	0.615649i	$-0.211106\pi$
$379$	30.6822	1.57604	0.788021	+	0.615649i	$0.211106\pi$
$380$	0	0
$381$	52.2454	2.67661
$382$	0	0
$383$	30.4909	1.55801	0.779006	−	0.627017i	$-0.215724\pi$
$383$	30.4909	1.55801	0.779006	+	0.627017i	$0.215724\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	52.2774	2.65741
$388$	0	0
$389$	16.3448	0.828714	0.414357	−	0.910114i	$-0.364006\pi$
$389$	16.3448	0.828714	0.414357	+	0.910114i	$0.364006\pi$
$390$	0	0
$391$	−10.5756	−0.534831
$392$	0	0
$393$	11.6076	0.585524
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.5946	0.782670	0.391335	−	0.920248i	$-0.372013\pi$
$397$	15.5946	0.782670	0.391335	+	0.920248i	$0.372013\pi$
$398$	0	0
$399$	10.2213	0.511705
$400$	0	0
$401$	−20.3465	−1.01606	−0.508029	−	0.861340i	$-0.669626\pi$
$401$	−20.3465	−1.01606	−0.508029	+	0.861340i	$0.669626\pi$
$402$	0	0
$403$	−54.0158	−2.69072
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−47.3749	−2.34829
$408$	0	0
$409$	−5.65655	−0.279699	−0.139849	−	0.990173i	$-0.544662\pi$
$409$	−5.65655	−0.279699	−0.139849	+	0.990173i	$0.544662\pi$
$410$	0	0
$411$	−57.4474	−2.83367
$412$	0	0
$413$	15.2653	0.751155
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5.26879	−0.258014
$418$	0	0
$419$	−18.1490	−0.886635	−0.443318	−	0.896365i	$-0.646199\pi$
$419$	−18.1490	−0.886635	−0.443318	+	0.896365i	$0.646199\pi$
$420$	0	0
$421$	5.94874	0.289924	0.144962	−	0.989437i	$-0.453694\pi$
$421$	5.94874	0.289924	0.144962	+	0.989437i	$0.453694\pi$
$422$	0	0
$423$	−29.0115	−1.41059
$424$	0	0
$425$	0	0
$426$	0	0
$427$	34.5644	1.67269
$428$	0	0
$429$	−69.4276	−3.35200
$430$	0	0
$431$	1.93242	0.0930816	0.0465408	−	0.998916i	$-0.485180\pi$
$431$	1.93242	0.0930816	0.0465408	+	0.998916i	$0.485180\pi$
$432$	0	0
$433$	14.7014	0.706505	0.353253	−	0.935528i	$-0.385076\pi$
$433$	14.7014	0.706505	0.353253	+	0.935528i	$0.385076\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.38924	−0.209966
$438$	0	0
$439$	10.5922	0.505540	0.252770	−	0.967526i	$-0.418658\pi$
$439$	10.5922	0.505540	0.252770	+	0.967526i	$0.418658\pi$
$440$	0	0
$441$	31.4236	1.49636
$442$	0	0
$443$	38.7357	1.84039	0.920195	−	0.391459i	$-0.128030\pi$
$443$	38.7357	1.84039	0.920195	+	0.391459i	$0.128030\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12.3503	0.584151
$448$	0	0
$449$	−29.2298	−1.37944	−0.689719	−	0.724078i	$-0.742266\pi$
$449$	−29.2298	−1.37944	−0.689719	+	0.724078i	$0.742266\pi$
$450$	0	0
$451$	5.26973	0.248142
$452$	0	0
$453$	23.9146	1.12361
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.18004	−0.195534	−0.0977671	−	0.995209i	$-0.531170\pi$
$457$	−4.18004	−0.195534	−0.0977671	+	0.995209i	$0.531170\pi$
$458$	0	0
$459$	−8.56527	−0.399793
$460$	0	0
$461$	35.9590	1.67478	0.837389	−	0.546607i	$-0.184081\pi$
$461$	35.9590	1.67478	0.837389	+	0.546607i	$0.184081\pi$
$462$	0	0
$463$	32.5866	1.51443	0.757214	−	0.653167i	$-0.226560\pi$
$463$	32.5866	1.51443	0.757214	+	0.653167i	$0.226560\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.66769	−0.262269	−0.131135	−	0.991365i	$-0.541862\pi$
$467$	−5.66769	−0.262269	−0.131135	+	0.991365i	$0.541862\pi$
$468$	0	0
$469$	16.6167	0.767287
$470$	0	0
$471$	15.2398	0.702213
$472$	0	0
$473$	57.1582	2.62814
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−18.7545	−0.858711
$478$	0	0
$479$	−16.2096	−0.740635	−0.370318	−	0.928905i	$-0.620751\pi$
$479$	−16.2096	−0.740635	−0.370318	+	0.928905i	$0.620751\pi$
$480$	0	0
$481$	−54.6531	−2.49197
$482$	0	0
$483$	−44.8637	−2.04137
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−19.0329	−0.862461	−0.431231	−	0.902242i	$-0.641920\pi$
$487$	−19.0329	−0.862461	−0.431231	+	0.902242i	$0.641920\pi$
$488$	0	0
$489$	10.0369	0.453882
$490$	0	0
$491$	−16.5658	−0.747602	−0.373801	−	0.927509i	$-0.621946\pi$
$491$	−16.5658	−0.747602	−0.373801	+	0.927509i	$0.621946\pi$
$492$	0	0
$493$	21.8089	0.982224
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−29.8652	−1.33964
$498$	0	0
$499$	6.36691	0.285022	0.142511	−	0.989793i	$-0.454482\pi$
$499$	6.36691	0.285022	0.142511	+	0.989793i	$0.454482\pi$
$500$	0	0
$501$	1.09173	0.0487748
$502$	0	0
$503$	−5.79766	−0.258505	−0.129252	−	0.991612i	$-0.541258\pi$
$503$	−5.79766	−0.258505	−0.129252	+	0.991612i	$0.541258\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−44.9352	−1.99564
$508$	0	0
$509$	23.0521	1.02177	0.510883	−	0.859650i	$-0.329319\pi$
$509$	23.0521	1.02177	0.510883	+	0.859650i	$0.329319\pi$
$510$	0	0
$511$	11.6088	0.513544
$512$	0	0
$513$	−3.55488	−0.156952
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−31.7202	−1.39505
$518$	0	0
$519$	37.6057	1.65071
$520$	0	0
$521$	11.7470	0.514645	0.257323	−	0.966326i	$-0.417160\pi$
$521$	11.7470	0.514645	0.257323	+	0.966326i	$0.417160\pi$
$522$	0	0
$523$	24.0715	1.05257	0.526286	−	0.850308i	$-0.323584\pi$
$523$	24.0715	1.05257	0.526286	+	0.850308i	$0.323584\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−23.9156	−1.04178
$528$	0	0
$529$	−3.73456	−0.162372
$530$	0	0
$531$	−17.4266	−0.756248
$532$	0	0
$533$	6.07932	0.263325
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−8.86198	−0.382423
$538$	0	0
$539$	34.3575	1.47988
$540$	0	0
$541$	−3.93335	−0.169108	−0.0845540	−	0.996419i	$-0.526947\pi$
$541$	−3.93335	−0.169108	−0.0845540	+	0.996419i	$0.526947\pi$
$542$	0	0
$543$	−35.2275	−1.51176
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21.2622	0.909105	0.454553	−	0.890720i	$-0.349799\pi$
$547$	21.2622	0.909105	0.454553	+	0.890720i	$0.349799\pi$
$548$	0	0
$549$	−39.4581	−1.68403
$550$	0	0
$551$	9.05146	0.385605
$552$	0	0
$553$	−43.0831	−1.83208
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.2899	0.435996	0.217998	−	0.975949i	$-0.430047\pi$
$557$	10.2899	0.435996	0.217998	+	0.975949i	$0.430047\pi$
$558$	0	0
$559$	65.9395	2.78894
$560$	0	0
$561$	−30.7393	−1.29781
$562$	0	0
$563$	−21.8745	−0.921901	−0.460950	−	0.887426i	$-0.652491\pi$
$563$	−21.8745	−0.921901	−0.460950	+	0.887426i	$0.652491\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12.5815	0.528373
$568$	0	0
$569$	22.8560	0.958172	0.479086	−	0.877768i	$-0.340968\pi$
$569$	22.8560	0.958172	0.479086	+	0.877768i	$0.340968\pi$
$570$	0	0
$571$	−5.58019	−0.233524	−0.116762	−	0.993160i	$-0.537251\pi$
$571$	−5.58019	−0.233524	−0.116762	+	0.993160i	$0.537251\pi$
$572$	0	0
$573$	47.7085	1.99305
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.48419	0.353201	0.176601	−	0.984283i	$-0.443490\pi$
$577$	8.48419	0.353201	0.176601	+	0.984283i	$0.443490\pi$
$578$	0	0
$579$	45.5564	1.89326
$580$	0	0
$581$	−16.5404	−0.686210
$582$	0	0
$583$	−20.5055	−0.849252
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.8166	−1.06557	−0.532783	−	0.846252i	$-0.678854\pi$
$587$	−25.8166	−1.06557	−0.532783	+	0.846252i	$0.678854\pi$
$588$	0	0
$589$	−9.92581	−0.408986
$590$	0	0
$591$	−53.7501	−2.21098
$592$	0	0
$593$	−26.3984	−1.08405	−0.542025	−	0.840362i	$-0.682342\pi$
$593$	−26.3984	−1.08405	−0.542025	+	0.840362i	$0.682342\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	23.0967	0.945284
$598$	0	0
$599$	36.1825	1.47838	0.739188	−	0.673499i	$-0.235210\pi$
$599$	36.1825	1.47838	0.739188	+	0.673499i	$0.235210\pi$
$600$	0	0
$601$	3.20839	0.130873	0.0654364	−	0.997857i	$-0.479156\pi$
$601$	3.20839	0.130873	0.0654364	+	0.997857i	$0.479156\pi$
$602$	0	0
$603$	−18.9693	−0.772490
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.3390	0.825534	0.412767	−	0.910837i	$-0.364562\pi$
$607$	20.3390	0.825534	0.412767	+	0.910837i	$0.364562\pi$
$608$	0	0
$609$	92.5176	3.74900
$610$	0	0
$611$	−36.5934	−1.48041
$612$	0	0
$613$	−18.9062	−0.763616	−0.381808	−	0.924242i	$-0.624698\pi$
$613$	−18.9062	−0.763616	−0.381808	+	0.924242i	$0.624698\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.0702	−1.41187	−0.705936	−	0.708275i	$-0.749473\pi$
$617$	−35.0702	−1.41187	−0.705936	+	0.708275i	$0.749473\pi$
$618$	0	0
$619$	−45.1329	−1.81404	−0.907021	−	0.421085i	$-0.861650\pi$
$619$	−45.1329	−1.81404	−0.907021	+	0.421085i	$0.861650\pi$
$620$	0	0
$621$	15.6032	0.626136
$622$	0	0
$623$	−9.59699	−0.384495
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−12.7579	−0.509500
$628$	0	0
$629$	−24.1978	−0.964830
$630$	0	0
$631$	35.0467	1.39519	0.697593	−	0.716494i	$-0.254254\pi$
$631$	35.0467	1.39519	0.697593	+	0.716494i	$0.254254\pi$
$632$	0	0
$633$	0.0343954	0.00136709
$634$	0	0
$635$	0	0
$636$	0	0
$637$	39.6359	1.57043
$638$	0	0
$639$	34.0935	1.34872
$640$	0	0
$641$	−10.8470	−0.428431	−0.214215	−	0.976786i	$-0.568719\pi$
$641$	−10.8470	−0.428431	−0.214215	+	0.976786i	$0.568719\pi$
$642$	0	0
$643$	−37.6665	−1.48542	−0.742712	−	0.669611i	$-0.766461\pi$
$643$	−37.6665	−1.48542	−0.742712	+	0.669611i	$0.766461\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.2893	1.46599	0.732997	−	0.680231i	$-0.238121\pi$
$647$	37.2893	1.46599	0.732997	+	0.680231i	$0.238121\pi$
$648$	0	0
$649$	−19.0536	−0.747918
$650$	0	0
$651$	−101.455	−3.97632
$652$	0	0
$653$	5.10026	0.199588	0.0997942	−	0.995008i	$-0.468182\pi$
$653$	5.10026	0.199588	0.0997942	+	0.995008i	$0.468182\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−13.2524	−0.517026
$658$	0	0
$659$	36.1180	1.40696	0.703479	−	0.710716i	$-0.251629\pi$
$659$	36.1180	1.40696	0.703479	+	0.710716i	$0.251629\pi$
$660$	0	0
$661$	−15.5014	−0.602934	−0.301467	−	0.953477i	$-0.597476\pi$
$661$	−15.5014	−0.602934	−0.301467	+	0.953477i	$0.597476\pi$
$662$	0	0
$663$	−35.4618	−1.37722
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−39.7290	−1.53831
$668$	0	0
$669$	−50.4131	−1.94908
$670$	0	0
$671$	−43.1420	−1.66548
$672$	0	0
$673$	−20.8897	−0.805240	−0.402620	−	0.915367i	$-0.631900\pi$
$673$	−20.8897	−0.805240	−0.402620	+	0.915367i	$0.631900\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	39.3686	1.51306	0.756529	−	0.653960i	$-0.226894\pi$
$677$	39.3686	1.51306	0.756529	+	0.653960i	$0.226894\pi$
$678$	0	0
$679$	−1.14176	−0.0438166
$680$	0	0
$681$	20.9651	0.803383
$682$	0	0
$683$	41.6596	1.59406	0.797030	−	0.603940i	$-0.206403\pi$
$683$	41.6596	1.59406	0.797030	+	0.603940i	$0.206403\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	52.3959	1.99903
$688$	0	0
$689$	−23.6558	−0.901215
$690$	0	0
$691$	3.90123	0.148410	0.0742050	−	0.997243i	$-0.476358\pi$
$691$	3.90123	0.148410	0.0742050	+	0.997243i	$0.476358\pi$
$692$	0	0
$693$	−76.9177	−2.92186
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.69164	0.101953
$698$	0	0
$699$	−23.3505	−0.883198
$700$	0	0
$701$	−24.6495	−0.931000	−0.465500	−	0.885048i	$-0.654125\pi$
$701$	−24.6495	−0.931000	−0.465500	+	0.885048i	$0.654125\pi$
$702$	0	0
$703$	−10.0429	−0.378776
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18.0305	−0.678106
$708$	0	0
$709$	9.56624	0.359268	0.179634	−	0.983734i	$-0.442509\pi$
$709$	9.56624	0.359268	0.179634	+	0.983734i	$0.442509\pi$
$710$	0	0
$711$	49.1829	1.84450
$712$	0	0
$713$	43.5668	1.63159
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−25.6292	−0.957141
$718$	0	0
$719$	−26.6273	−0.993030	−0.496515	−	0.868028i	$-0.665387\pi$
$719$	−26.6273	−0.993030	−0.496515	+	0.868028i	$0.665387\pi$
$720$	0	0
$721$	−67.6554	−2.51962
$722$	0	0
$723$	−79.4159	−2.95351
$724$	0	0
$725$	0	0
$726$	0	0
$727$	24.3779	0.904127	0.452063	−	0.891986i	$-0.350688\pi$
$727$	24.3779	0.904127	0.452063	+	0.891986i	$0.350688\pi$
$728$	0	0
$729$	−43.2056	−1.60021
$730$	0	0
$731$	29.1949	1.07981
$732$	0	0
$733$	6.53865	0.241511	0.120755	−	0.992682i	$-0.461468\pi$
$733$	6.53865	0.241511	0.120755	+	0.992682i	$0.461468\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.7404	−0.763981
$738$	0	0
$739$	11.7605	0.432617	0.216308	−	0.976325i	$-0.430598\pi$
$739$	11.7605	0.432617	0.216308	+	0.976325i	$0.430598\pi$
$740$	0	0
$741$	−14.7179	−0.540674
$742$	0	0
$743$	40.7714	1.49576	0.747879	−	0.663835i	$-0.231072\pi$
$743$	40.7714	1.49576	0.747879	+	0.663835i	$0.231072\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	18.8822	0.690863
$748$	0	0
$749$	−49.7636	−1.81832
$750$	0	0
$751$	−0.578858	−0.0211228	−0.0105614	−	0.999944i	$-0.503362\pi$
$751$	−0.578858	−0.0211228	−0.0105614	+	0.999944i	$0.503362\pi$
$752$	0	0
$753$	20.4834	0.746457
$754$	0	0
$755$	0	0
$756$	0	0
$757$	14.3710	0.522325	0.261162	−	0.965295i	$-0.415894\pi$
$757$	14.3710	0.522325	0.261162	+	0.965295i	$0.415894\pi$
$758$	0	0
$759$	55.9973	2.03257
$760$	0	0
$761$	−16.4753	−0.597230	−0.298615	−	0.954374i	$-0.596525\pi$
$761$	−16.4753	−0.597230	−0.298615	+	0.954374i	$0.596525\pi$
$762$	0	0
$763$	−2.85798	−0.103466
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−21.9808	−0.793681
$768$	0	0
$769$	−18.8410	−0.679425	−0.339712	−	0.940529i	$-0.610330\pi$
$769$	−18.8410	−0.679425	−0.339712	+	0.940529i	$0.610330\pi$
$770$	0	0
$771$	−31.7978	−1.14517
$772$	0	0
$773$	−37.5917	−1.35208	−0.676039	−	0.736866i	$-0.736305\pi$
$773$	−37.5917	−1.35208	−0.676039	+	0.736866i	$0.736305\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−102.652	−3.68261
$778$	0	0
$779$	1.11712	0.0400251
$780$	0	0
$781$	37.2766	1.33386
$782$	0	0
$783$	−32.1769	−1.14991
$784$	0	0
$785$	0	0
$786$	0	0
$787$	44.5235	1.58709	0.793545	−	0.608511i	$-0.208233\pi$
$787$	44.5235	1.58709	0.793545	+	0.608511i	$0.208233\pi$
$788$	0	0
$789$	5.95897	0.212145
$790$	0	0
$791$	−77.8854	−2.76929
$792$	0	0
$793$	−49.7700	−1.76738
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.84212	0.136095	0.0680475	−	0.997682i	$-0.478323\pi$
$797$	3.84212	0.136095	0.0680475	+	0.997682i	$0.478323\pi$
$798$	0	0
$799$	−16.2018	−0.573179
$800$	0	0
$801$	10.9558	0.387103
$802$	0	0
$803$	−14.4897	−0.511331
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−16.6231	−0.585161
$808$	0	0
$809$	−1.51537	−0.0532774	−0.0266387	−	0.999645i	$-0.508480\pi$
$809$	−1.51537	−0.0532774	−0.0266387	+	0.999645i	$0.508480\pi$
$810$	0	0
$811$	−20.1768	−0.708502	−0.354251	−	0.935150i	$-0.615264\pi$
$811$	−20.1768	−0.708502	−0.354251	+	0.935150i	$0.615264\pi$
$812$	0	0
$813$	−69.4034	−2.43408
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.1169	0.423916
$818$	0	0
$819$	−88.7346	−3.10064
$820$	0	0
$821$	−33.0855	−1.15469	−0.577347	−	0.816499i	$-0.695912\pi$
$821$	−33.0855	−1.15469	−0.577347	+	0.816499i	$0.695912\pi$
$822$	0	0
$823$	22.9779	0.800958	0.400479	−	0.916306i	$-0.368844\pi$
$823$	22.9779	0.800958	0.400479	+	0.916306i	$0.368844\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−54.9730	−1.91160	−0.955798	−	0.294023i	$-0.905006\pi$
$827$	−54.9730	−1.91160	−0.955798	+	0.294023i	$0.905006\pi$
$828$	0	0
$829$	36.6692	1.27357	0.636787	−	0.771039i	$-0.280263\pi$
$829$	36.6692	1.27357	0.636787	+	0.771039i	$0.280263\pi$
$830$	0	0
$831$	−12.5544	−0.435508
$832$	0	0
$833$	17.5489	0.608033
$834$	0	0
$835$	0	0
$836$	0	0
$837$	35.2851	1.21963
$838$	0	0
$839$	−31.4619	−1.08619	−0.543093	−	0.839672i	$-0.682747\pi$
$839$	−31.4619	−1.08619	−0.543093	+	0.839672i	$0.682747\pi$
$840$	0	0
$841$	52.9289	1.82513
$842$	0	0
$843$	1.00058	0.0344619
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−42.5263	−1.46122
$848$	0	0
$849$	−3.28126	−0.112613
$850$	0	0
$851$	44.0808	1.51107
$852$	0	0
$853$	−9.45230	−0.323641	−0.161820	−	0.986820i	$-0.551737\pi$
$853$	−9.45230	−0.323641	−0.161820	+	0.986820i	$0.551737\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.9551	−0.613335	−0.306668	−	0.951817i	$-0.599214\pi$
$857$	−17.9551	−0.613335	−0.306668	+	0.951817i	$0.599214\pi$
$858$	0	0
$859$	−24.3628	−0.831249	−0.415624	−	0.909536i	$-0.636437\pi$
$859$	−24.3628	−0.831249	−0.415624	+	0.909536i	$0.636437\pi$
$860$	0	0
$861$	11.4184	0.389139
$862$	0	0
$863$	−45.1946	−1.53844	−0.769222	−	0.638982i	$-0.779356\pi$
$863$	−45.1946	−1.53844	−0.769222	+	0.638982i	$0.779356\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	30.2760	1.02823
$868$	0	0
$869$	53.7748	1.82419
$870$	0	0
$871$	−23.9267	−0.810727
$872$	0	0
$873$	1.30341	0.0441138
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7.55736	−0.255194	−0.127597	−	0.991826i	$-0.540726\pi$
$877$	−7.55736	−0.255194	−0.127597	+	0.991826i	$0.540726\pi$
$878$	0	0
$879$	23.4283	0.790217
$880$	0	0
$881$	−50.5760	−1.70395	−0.851974	−	0.523584i	$-0.824595\pi$
$881$	−50.5760	−1.70395	−0.851974	+	0.523584i	$0.824595\pi$
$882$	0	0
$883$	−6.40946	−0.215696	−0.107848	−	0.994167i	$-0.534396\pi$
$883$	−6.40946	−0.215696	−0.107848	+	0.994167i	$0.534396\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.14684	−0.0720838	−0.0360419	−	0.999350i	$-0.511475\pi$
$887$	−2.14684	−0.0720838	−0.0360419	+	0.999350i	$0.511475\pi$
$888$	0	0
$889$	73.0085	2.44863
$890$	0	0
$891$	−15.7038	−0.526097
$892$	0	0
$893$	−6.72432	−0.225021
$894$	0	0
$895$	0	0
$896$	0	0
$897$	64.6003	2.15694
$898$	0	0
$899$	−89.8430	−2.99643
$900$	0	0
$901$	−10.4737	−0.348929
$902$	0	0
$903$	123.850	4.12148
$904$	0	0
$905$	0	0
$906$	0	0
$907$	41.3186	1.37196	0.685980	−	0.727620i	$-0.259374\pi$
$907$	41.3186	1.37196	0.685980	+	0.727620i	$0.259374\pi$
$908$	0	0
$909$	20.5833	0.682704
$910$	0	0
$911$	59.7328	1.97903	0.989517	−	0.144415i	$-0.0461300\pi$
$911$	59.7328	1.97903	0.989517	+	0.144415i	$0.0461300\pi$
$912$	0	0
$913$	20.6451	0.683253
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.2206	0.535651
$918$	0	0
$919$	−34.3526	−1.13319	−0.566594	−	0.823997i	$-0.691739\pi$
$919$	−34.3526	−1.13319	−0.566594	+	0.823997i	$0.691739\pi$
$920$	0	0
$921$	22.3426	0.736215
$922$	0	0
$923$	43.0035	1.41548
$924$	0	0
$925$	0	0
$926$	0	0
$927$	77.2342	2.53670
$928$	0	0
$929$	−48.0752	−1.57729	−0.788647	−	0.614846i	$-0.789218\pi$
$929$	−48.0752	−1.57729	−0.788647	+	0.614846i	$0.789218\pi$
$930$	0	0
$931$	7.28339	0.238703
$932$	0	0
$933$	−20.4031	−0.667968
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.9537	1.40324	0.701619	−	0.712552i	$-0.252461\pi$
$937$	42.9537	1.40324	0.701619	+	0.712552i	$0.252461\pi$
$938$	0	0
$939$	−9.28060	−0.302861
$940$	0	0
$941$	−42.8745	−1.39767	−0.698835	−	0.715283i	$-0.746298\pi$
$941$	−42.8745	−1.39767	−0.698835	+	0.715283i	$0.746298\pi$
$942$	0	0
$943$	−4.90332	−0.159674
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.1350	−0.914264	−0.457132	−	0.889399i	$-0.651123\pi$
$947$	−28.1350	−0.914264	−0.457132	+	0.889399i	$0.651123\pi$
$948$	0	0
$949$	−16.7158	−0.542617
$950$	0	0
$951$	6.78249	0.219937
$952$	0	0
$953$	16.7612	0.542949	0.271474	−	0.962446i	$-0.412489\pi$
$953$	16.7612	0.542949	0.271474	+	0.962446i	$0.412489\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−115.477	−3.73285
$958$	0	0
$959$	−80.2780	−2.59231
$960$	0	0
$961$	67.5217	2.17812
$962$	0	0
$963$	56.8093	1.83065
$964$	0	0
$965$	0	0
$966$	0	0
$967$	35.3277	1.13606	0.568030	−	0.823008i	$-0.307706\pi$
$967$	35.3277	1.13606	0.568030	+	0.823008i	$0.307706\pi$
$968$	0	0
$969$	−6.51637	−0.209336
$970$	0	0
$971$	54.3917	1.74551	0.872756	−	0.488156i	$-0.162330\pi$
$971$	54.3917	1.74551	0.872756	+	0.488156i	$0.162330\pi$
$972$	0	0
$973$	−7.36269	−0.236037
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.4481	1.51800	0.759000	−	0.651091i	$-0.225688\pi$
$977$	47.4481	1.51800	0.759000	+	0.651091i	$0.225688\pi$
$978$	0	0
$979$	11.9786	0.382839
$980$	0	0
$981$	3.26262	0.104167
$982$	0	0
$983$	−42.8550	−1.36686	−0.683432	−	0.730014i	$-0.739513\pi$
$983$	−42.8550	−1.36686	−0.683432	+	0.730014i	$0.739513\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−68.7312	−2.18774
$988$	0	0
$989$	−53.1839	−1.69115
$990$	0	0
$991$	23.4118	0.743700	0.371850	−	0.928293i	$-0.378724\pi$
$991$	23.4118	0.743700	0.371850	+	0.928293i	$0.378724\pi$
$992$	0	0
$993$	−78.7467	−2.49895
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−12.9905	−0.411412	−0.205706	−	0.978614i	$-0.565949\pi$
$997$	−12.9905	−0.411412	−0.205706	+	0.978614i	$0.565949\pi$
$998$	0	0
$999$	35.7014	1.12954

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.bb.1.2	✓	6
4.3	odd	2		7600.2.a.cm.1.5		6
5.2	odd	4		3800.2.d.p.3649.10		12
5.3	odd	4		3800.2.d.p.3649.3		12
5.4	even	2		3800.2.a.bd.1.5	yes	6
20.19	odd	2		7600.2.a.ci.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.bb.1.2	✓	6	1.1	even	1	trivial
3800.2.a.bd.1.5	yes	6	5.4	even	2
3800.2.d.p.3649.3		12	5.3	odd	4
3800.2.d.p.3649.10		12	5.2	odd	4
7600.2.a.ci.1.2		6	20.19	odd	2
7600.2.a.cm.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.70452 q^{3} -3.77934 q^{7} +4.31442 q^{9} +O(q^{10})\)	\(q-2.70452 q^{3} -3.77934 q^{7} +4.31442 q^{9} +4.71724 q^{11} +5.44195 q^{13} +2.40944 q^{17} +1.00000 q^{19} +10.2213 q^{21} -4.38924 q^{23} -3.55488 q^{27} +9.05146 q^{29} -9.92581 q^{31} -12.7579 q^{33} -10.0429 q^{37} -14.7179 q^{39} +1.11712 q^{41} +12.1169 q^{43} -6.72432 q^{47} +7.28339 q^{49} -6.51637 q^{51} -4.34694 q^{53} -2.70452 q^{57} -4.03914 q^{59} -9.14562 q^{61} -16.3057 q^{63} -4.39672 q^{67} +11.8708 q^{69} +7.90222 q^{71} -3.07165 q^{73} -17.8280 q^{77} +11.3996 q^{79} -3.32902 q^{81} +4.37652 q^{83} -24.4798 q^{87} +2.53933 q^{89} -20.5670 q^{91} +26.8445 q^{93} +0.302105 q^{97} +20.3522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9	\( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} + 3 q^{11} + 3 q^{13} + 2 q^{17} + 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} + 5 q^{31} + 16 q^{33} + 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49} + 13 q^{51} + 7 q^{53} - 2 q^{57} - 4 q^{59} + 13 q^{61} + q^{63} - 25 q^{67} + 7 q^{69} + 29 q^{71} + 19 q^{73} - 24 q^{77} + 28 q^{79} + 38 q^{81} + 15 q^{83} - 57 q^{87} - 12 q^{89} + 27 q^{93} + 13 q^{97} + 10 q^{99}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more