LMFDB - Embedded newform 3800.2.a.z.1.1

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.253565184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 22x^{2} - 32x + 8 $ x^6 - 2x^5 - 11x^4 + 20x^3 + 22x^2 - 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.97875$ 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)

$f(q)$	$=$	$q-2.97875 q^{3} -4.30732 q^{7} +5.87292 q^{9} +1.96218 q^{11} -5.83724 q^{13} +6.46615 q^{17} +1.00000 q^{19} +12.8304 q^{21} -4.45534 q^{23} -8.55770 q^{27} -5.31383 q^{29} -0.713620 q^{31} -5.84482 q^{33} +8.56529 q^{37} +17.3876 q^{39} +2.71362 q^{41} +7.62116 q^{43} +4.25038 q^{47} +11.5530 q^{49} -19.2610 q^{51} +7.90861 q^{53} -2.97875 q^{57} -6.04542 q^{59} +1.00899 q^{61} -25.2966 q^{63} -12.1099 q^{67} +13.2713 q^{69} +0.272709 q^{71} +15.0808 q^{73} -8.45172 q^{77} +1.95749 q^{79} +7.87245 q^{81} +5.52960 q^{83} +15.8286 q^{87} -0.632880 q^{89} +25.1429 q^{91} +2.12569 q^{93} +7.10856 q^{97} +11.5237 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47}+ \cdots - 22 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^7 + 8 * q^9 - 2 * q^11 - 14 * q^13 - 10 * q^17 + 6 * q^19 + 18 * q^21 - 2 * q^23 - 2 * q^27 - 2 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^37 - 18 * q^39 + 4 * q^41 - 4 * q^43 - 4 * q^47 + 2 * q^49 - 34 * q^51 + 14 * q^53 - 2 * q^57 - 2 * q^59 + 10 * q^61 - 44 * q^63 - 42 * q^67 + 18 * q^69 - 8 * q^71 + 2 * q^73 - 24 * q^77 - 20 * q^79 + 6 * q^81 - 16 * q^83 - 30 * q^87 + 32 * q^89 + 10 * q^91 - 12 * q^93 - 40 * q^97 - 22 * 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
$3$	−2.97875	−1.71978	−0.859890	−	0.510480i	$-0.829468\pi$
$3$	−2.97875	−1.71978	−0.859890	+	0.510480i	$0.829468\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.30732	−1.62801	−0.814007	−	0.580855i	$-0.802718\pi$
$7$	−4.30732	−1.62801	−0.814007	+	0.580855i	$0.802718\pi$
$8$	0	0
$9$	5.87292	1.95764
$10$	0	0
$11$	1.96218	0.591618	0.295809	−	0.955247i	$-0.404411\pi$
$11$	1.96218	0.591618	0.295809	+	0.955247i	$0.404411\pi$
$12$	0	0
$13$	−5.83724	−1.61896	−0.809479	−	0.587148i	$-0.800250\pi$
$13$	−5.83724	−1.61896	−0.809479	+	0.587148i	$0.800250\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.46615	1.56827	0.784136	−	0.620589i	$-0.213106\pi$
$17$	6.46615	1.56827	0.784136	+	0.620589i	$0.213106\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	12.8304	2.79983
$22$	0	0
$23$	−4.45534	−0.929003	−0.464501	−	0.885572i	$-0.653766\pi$
$23$	−4.45534	−0.929003	−0.464501	+	0.885572i	$0.653766\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−8.55770	−1.64693
$28$	0	0
$29$	−5.31383	−0.986754	−0.493377	−	0.869816i	$-0.664238\pi$
$29$	−5.31383	−0.986754	−0.493377	+	0.869816i	$0.664238\pi$
$30$	0	0
$31$	−0.713620	−0.128170	−0.0640850	−	0.997944i	$-0.520413\pi$
$31$	−0.713620	−0.128170	−0.0640850	+	0.997944i	$0.520413\pi$
$32$	0	0
$33$	−5.84482	−1.01745
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.56529	1.40812	0.704062	−	0.710138i	$-0.251368\pi$
$37$	8.56529	1.40812	0.704062	+	0.710138i	$0.251368\pi$
$38$	0	0
$39$	17.3876	2.78425
$40$	0	0
$41$	2.71362	0.423796	0.211898	−	0.977292i	$-0.432035\pi$
$41$	2.71362	0.423796	0.211898	+	0.977292i	$0.432035\pi$
$42$	0	0
$43$	7.62116	1.16222	0.581108	−	0.813827i	$-0.302620\pi$
$43$	7.62116	1.16222	0.581108	+	0.813827i	$0.302620\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.25038	0.619982	0.309991	−	0.950739i	$-0.399674\pi$
$47$	4.25038	0.619982	0.309991	+	0.950739i	$0.399674\pi$
$48$	0	0
$49$	11.5530	1.65043
$50$	0	0
$51$	−19.2610	−2.69708
$52$	0	0
$53$	7.90861	1.08633	0.543165	−	0.839626i	$-0.317226\pi$
$53$	7.90861	1.08633	0.543165	+	0.839626i	$0.317226\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.97875	−0.394544
$58$	0	0
$59$	−6.04542	−0.787047	−0.393524	−	0.919315i	$-0.628744\pi$
$59$	−6.04542	−0.787047	−0.393524	+	0.919315i	$0.628744\pi$
$60$	0	0
$61$	1.00899	0.129187	0.0645937	−	0.997912i	$-0.479425\pi$
$61$	1.00899	0.129187	0.0645937	+	0.997912i	$0.479425\pi$
$62$	0	0
$63$	−25.2966	−3.18707
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12.1099	−1.47947	−0.739733	−	0.672901i	$-0.765048\pi$
$67$	−12.1099	−1.47947	−0.739733	+	0.672901i	$0.765048\pi$
$68$	0	0
$69$	13.2713	1.59768
$70$	0	0
$71$	0.272709	0.0323646	0.0161823	−	0.999869i	$-0.494849\pi$
$71$	0.272709	0.0323646	0.0161823	+	0.999869i	$0.494849\pi$
$72$	0	0
$73$	15.0808	1.76507	0.882537	−	0.470243i	$-0.155834\pi$
$73$	15.0808	1.76507	0.882537	+	0.470243i	$0.155834\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.45172	−0.963163
$78$	0	0
$79$	1.95749	0.220235	0.110117	−	0.993919i	$-0.464877\pi$
$79$	1.95749	0.220235	0.110117	+	0.993919i	$0.464877\pi$
$80$	0	0
$81$	7.87245	0.874717
$82$	0	0
$83$	5.52960	0.606953	0.303476	−	0.952839i	$-0.401853\pi$
$83$	5.52960	0.606953	0.303476	+	0.952839i	$0.401853\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	15.8286	1.69700
$88$	0	0
$89$	−0.632880	−0.0670851	−0.0335426	−	0.999437i	$-0.510679\pi$
$89$	−0.632880	−0.0670851	−0.0335426	+	0.999437i	$0.510679\pi$
$90$	0	0
$91$	25.1429	2.63569
$92$	0	0
$93$	2.12569	0.220424
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.10856	0.721765	0.360883	−	0.932611i	$-0.382476\pi$
$97$	7.10856	0.721765	0.360883	+	0.932611i	$0.382476\pi$
$98$	0	0
$99$	11.5237	1.15818
$100$	0	0
$101$	−10.5199	−1.04677	−0.523387	−	0.852095i	$-0.675332\pi$
$101$	−10.5199	−1.04677	−0.523387	+	0.852095i	$0.675332\pi$
$102$	0	0
$103$	−0.546734	−0.0538713	−0.0269356	−	0.999637i	$-0.508575\pi$
$103$	−0.546734	−0.0538713	−0.0269356	+	0.999637i	$0.508575\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.62193	−0.736840	−0.368420	−	0.929659i	$-0.620101\pi$
$107$	−7.62193	−0.736840	−0.368420	+	0.929659i	$0.620101\pi$
$108$	0	0
$109$	−16.0308	−1.53548	−0.767738	−	0.640764i	$-0.778618\pi$
$109$	−16.0308	−1.53548	−0.767738	+	0.640764i	$0.778618\pi$
$110$	0	0
$111$	−25.5138	−2.42166
$112$	0	0
$113$	14.3257	1.34765	0.673825	−	0.738891i	$-0.264650\pi$
$113$	14.3257	1.34765	0.673825	+	0.738891i	$0.264650\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−34.2816	−3.16934
$118$	0	0
$119$	−27.8518	−2.55317
$120$	0	0
$121$	−7.14987	−0.649988
$122$	0	0
$123$	−8.08318	−0.728836
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−9.04871	−0.802943	−0.401472	−	0.915872i	$-0.631501\pi$
$127$	−9.04871	−0.802943	−0.401472	+	0.915872i	$0.631501\pi$
$128$	0	0
$129$	−22.7015	−1.99875
$130$	0	0
$131$	−2.95881	−0.258512	−0.129256	−	0.991611i	$-0.541259\pi$
$131$	−2.95881	−0.258512	−0.129256	+	0.991611i	$0.541259\pi$
$132$	0	0
$133$	−4.30732	−0.373492
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.5171	−1.15484	−0.577422	−	0.816446i	$-0.695941\pi$
$137$	−13.5171	−1.15484	−0.577422	+	0.816446i	$0.695941\pi$
$138$	0	0
$139$	19.6552	1.66713	0.833566	−	0.552419i	$-0.186295\pi$
$139$	19.6552	1.66713	0.833566	+	0.552419i	$0.186295\pi$
$140$	0	0
$141$	−12.6608	−1.06623
$142$	0	0
$143$	−11.4537	−0.957806
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−34.4135	−2.83838
$148$	0	0
$149$	−1.18393	−0.0969911	−0.0484955	−	0.998823i	$-0.515443\pi$
$149$	−1.18393	−0.0969911	−0.0484955	+	0.998823i	$0.515443\pi$
$150$	0	0
$151$	6.70332	0.545508	0.272754	−	0.962084i	$-0.412065\pi$
$151$	6.70332	0.545508	0.272754	+	0.962084i	$0.412065\pi$
$152$	0	0
$153$	37.9752	3.07011
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.6414	0.849273	0.424636	−	0.905364i	$-0.360402\pi$
$157$	10.6414	0.849273	0.424636	+	0.905364i	$0.360402\pi$
$158$	0	0
$159$	−23.5577	−1.86825
$160$	0	0
$161$	19.1906	1.51243
$162$	0	0
$163$	−18.6748	−1.46272	−0.731360	−	0.681991i	$-0.761114\pi$
$163$	−18.6748	−1.46272	−0.731360	+	0.681991i	$0.761114\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.41076	0.109168	0.0545838	−	0.998509i	$-0.482617\pi$
$167$	1.41076	0.109168	0.0545838	+	0.998509i	$0.482617\pi$
$168$	0	0
$169$	21.0733	1.62103
$170$	0	0
$171$	5.87292	0.449114
$172$	0	0
$173$	−1.54582	−0.117527	−0.0587633	−	0.998272i	$-0.518716\pi$
$173$	−1.54582	−0.117527	−0.0587633	+	0.998272i	$0.518716\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	18.0078	1.35355
$178$	0	0
$179$	−1.08504	−0.0810997	−0.0405498	−	0.999178i	$-0.512911\pi$
$179$	−1.08504	−0.0810997	−0.0405498	+	0.999178i	$0.512911\pi$
$180$	0	0
$181$	0.148526	0.0110398	0.00551991	−	0.999985i	$-0.498243\pi$
$181$	0.148526	0.0110398	0.00551991	+	0.999985i	$0.498243\pi$
$182$	0	0
$183$	−3.00551	−0.222174
$184$	0	0
$185$	0	0
$186$	0	0
$187$	12.6877	0.927819
$188$	0	0
$189$	36.8608	2.68123
$190$	0	0
$191$	−21.5605	−1.56007	−0.780033	−	0.625739i	$-0.784798\pi$
$191$	−21.5605	−1.56007	−0.780033	+	0.625739i	$0.784798\pi$
$192$	0	0
$193$	−20.2318	−1.45632	−0.728160	−	0.685408i	$-0.759624\pi$
$193$	−20.2318	−1.45632	−0.728160	+	0.685408i	$0.759624\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.8206	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$197$	−20.8206	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$198$	0	0
$199$	5.31009	0.376422	0.188211	−	0.982129i	$-0.439731\pi$
$199$	5.31009	0.376422	0.188211	+	0.982129i	$0.439731\pi$
$200$	0	0
$201$	36.0724	2.54435
$202$	0	0
$203$	22.8884	1.60645
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−26.1659	−1.81865
$208$	0	0
$209$	1.96218	0.135727
$210$	0	0
$211$	−4.26809	−0.293827	−0.146914	−	0.989149i	$-0.546934\pi$
$211$	−4.26809	−0.293827	−0.146914	+	0.989149i	$0.546934\pi$
$212$	0	0
$213$	−0.812331	−0.0556600
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.07379	0.208663
$218$	0	0
$219$	−44.9218	−3.03554
$220$	0	0
$221$	−37.7445	−2.53897
$222$	0	0
$223$	−9.21370	−0.616995	−0.308497	−	0.951225i	$-0.599826\pi$
$223$	−9.21370	−0.616995	−0.308497	+	0.951225i	$0.599826\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.74453	−0.182161	−0.0910805	−	0.995844i	$-0.529032\pi$
$227$	−2.74453	−0.182161	−0.0910805	+	0.995844i	$0.529032\pi$
$228$	0	0
$229$	8.56192	0.565787	0.282894	−	0.959151i	$-0.408706\pi$
$229$	8.56192	0.565787	0.282894	+	0.959151i	$0.408706\pi$
$230$	0	0
$231$	25.1755	1.65643
$232$	0	0
$233$	−6.64463	−0.435305	−0.217652	−	0.976026i	$-0.569840\pi$
$233$	−6.64463	−0.435305	−0.217652	+	0.976026i	$0.569840\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5.83086	−0.378755
$238$	0	0
$239$	−7.23627	−0.468075	−0.234038	−	0.972228i	$-0.575194\pi$
$239$	−7.23627	−0.468075	−0.234038	+	0.972228i	$0.575194\pi$
$240$	0	0
$241$	9.61249	0.619195	0.309597	−	0.950868i	$-0.399806\pi$
$241$	9.61249	0.619195	0.309597	+	0.950868i	$0.399806\pi$
$242$	0	0
$243$	2.22309	0.142611
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.83724	−0.371415
$248$	0	0
$249$	−16.4713	−1.04382
$250$	0	0
$251$	16.1727	1.02081	0.510406	−	0.859934i	$-0.329495\pi$
$251$	16.1727	1.02081	0.510406	+	0.859934i	$0.329495\pi$
$252$	0	0
$253$	−8.74216	−0.549615
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.5792	−1.53321	−0.766606	−	0.642118i	$-0.778056\pi$
$257$	−24.5792	−1.53321	−0.766606	+	0.642118i	$0.778056\pi$
$258$	0	0
$259$	−36.8934	−2.29245
$260$	0	0
$261$	−31.2077	−1.93171
$262$	0	0
$263$	18.6106	1.14758	0.573788	−	0.819004i	$-0.305473\pi$
$263$	18.6106	1.14758	0.573788	+	0.819004i	$0.305473\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.88519	0.115372
$268$	0	0
$269$	−8.74586	−0.533245	−0.266622	−	0.963801i	$-0.585908\pi$
$269$	−8.74586	−0.533245	−0.266622	+	0.963801i	$0.585908\pi$
$270$	0	0
$271$	30.5881	1.85809	0.929047	−	0.369963i	$-0.120630\pi$
$271$	30.5881	1.85809	0.929047	+	0.369963i	$0.120630\pi$
$272$	0	0
$273$	−74.8942	−4.53280
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−27.6239	−1.65976	−0.829880	−	0.557942i	$-0.811591\pi$
$277$	−27.6239	−1.65976	−0.829880	+	0.557942i	$0.811591\pi$
$278$	0	0
$279$	−4.19104	−0.250911
$280$	0	0
$281$	−15.8579	−0.946002	−0.473001	−	0.881062i	$-0.656829\pi$
$281$	−15.8579	−0.946002	−0.473001	+	0.881062i	$0.656829\pi$
$282$	0	0
$283$	−17.1223	−1.01782	−0.508909	−	0.860821i	$-0.669951\pi$
$283$	−17.1223	−1.01782	−0.508909	+	0.860821i	$0.669951\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.6884	−0.689946
$288$	0	0
$289$	24.8111	1.45948
$290$	0	0
$291$	−21.1746	−1.24128
$292$	0	0
$293$	−19.7436	−1.15343	−0.576717	−	0.816944i	$-0.695667\pi$
$293$	−19.7436	−1.15343	−0.576717	+	0.816944i	$0.695667\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−16.7917	−0.974354
$298$	0	0
$299$	26.0069	1.50402
$300$	0	0
$301$	−32.8268	−1.89210
$302$	0	0
$303$	31.3362	1.80022
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.66335	0.0949325	0.0474662	−	0.998873i	$-0.484885\pi$
$307$	1.66335	0.0949325	0.0474662	+	0.998873i	$0.484885\pi$
$308$	0	0
$309$	1.62858	0.0926467
$310$	0	0
$311$	−21.3711	−1.21185	−0.605923	−	0.795523i	$-0.707196\pi$
$311$	−21.3711	−1.21185	−0.605923	+	0.795523i	$0.707196\pi$
$312$	0	0
$313$	−0.140748	−0.00795557	−0.00397779	−	0.999992i	$-0.501266\pi$
$313$	−0.140748	−0.00795557	−0.00397779	+	0.999992i	$0.501266\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.226895	−0.0127437	−0.00637184	−	0.999980i	$-0.502028\pi$
$317$	−0.226895	−0.0127437	−0.00637184	+	0.999980i	$0.502028\pi$
$318$	0	0
$319$	−10.4267	−0.583782
$320$	0	0
$321$	22.7038	1.26720
$322$	0	0
$323$	6.46615	0.359786
$324$	0	0
$325$	0	0
$326$	0	0
$327$	47.7518	2.64068
$328$	0	0
$329$	−18.3078	−1.00934
$330$	0	0
$331$	−15.1854	−0.834663	−0.417332	−	0.908754i	$-0.637035\pi$
$331$	−15.1854	−0.834663	−0.417332	+	0.908754i	$0.637035\pi$
$332$	0	0
$333$	50.3033	2.75660
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.70005	0.201555	0.100777	−	0.994909i	$-0.467867\pi$
$337$	3.70005	0.201555	0.100777	+	0.994909i	$0.467867\pi$
$338$	0	0
$339$	−42.6727	−2.31766
$340$	0	0
$341$	−1.40025	−0.0758277
$342$	0	0
$343$	−19.6113	−1.05891
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−23.1560	−1.24308	−0.621539	−	0.783383i	$-0.713492\pi$
$347$	−23.1560	−1.24308	−0.621539	+	0.783383i	$0.713492\pi$
$348$	0	0
$349$	−31.7444	−1.69924	−0.849620	−	0.527396i	$-0.823169\pi$
$349$	−31.7444	−1.69924	−0.849620	+	0.527396i	$0.823169\pi$
$350$	0	0
$351$	49.9534	2.66631
$352$	0	0
$353$	0.522123	0.0277898	0.0138949	−	0.999903i	$-0.495577\pi$
$353$	0.522123	0.0277898	0.0138949	+	0.999903i	$0.495577\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	82.9634	4.39089
$358$	0	0
$359$	1.23778	0.0653275	0.0326637	−	0.999466i	$-0.489601\pi$
$359$	1.23778	0.0653275	0.0326637	+	0.999466i	$0.489601\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	21.2976	1.11784
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.8293	1.71368	0.856838	−	0.515586i	$-0.172426\pi$
$367$	32.8293	1.71368	0.856838	+	0.515586i	$0.172426\pi$
$368$	0	0
$369$	15.9369	0.829641
$370$	0	0
$371$	−34.0649	−1.76856
$372$	0	0
$373$	−2.45943	−0.127344	−0.0636721	−	0.997971i	$-0.520281\pi$
$373$	−2.45943	−0.127344	−0.0636721	+	0.997971i	$0.520281\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.0181	1.59751
$378$	0	0
$379$	14.1150	0.725037	0.362519	−	0.931977i	$-0.381917\pi$
$379$	14.1150	0.725037	0.362519	+	0.931977i	$0.381917\pi$
$380$	0	0
$381$	26.9538	1.38088
$382$	0	0
$383$	−9.67102	−0.494166	−0.247083	−	0.968994i	$-0.579472\pi$
$383$	−9.67102	−0.494166	−0.247083	+	0.968994i	$0.579472\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	44.7585	2.27520
$388$	0	0
$389$	−34.5556	−1.75204	−0.876019	−	0.482277i	$-0.839810\pi$
$389$	−34.5556	−1.75204	−0.876019	+	0.482277i	$0.839810\pi$
$390$	0	0
$391$	−28.8089	−1.45693
$392$	0	0
$393$	8.81354	0.444584
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−13.8088	−0.693046	−0.346523	−	0.938041i	$-0.612638\pi$
$397$	−13.8088	−0.693046	−0.346523	+	0.938041i	$0.612638\pi$
$398$	0	0
$399$	12.8304	0.642324
$400$	0	0
$401$	31.6660	1.58133	0.790663	−	0.612251i	$-0.209736\pi$
$401$	31.6660	1.58133	0.790663	+	0.612251i	$0.209736\pi$
$402$	0	0
$403$	4.16557	0.207502
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.8066	0.833072
$408$	0	0
$409$	−6.07381	−0.300331	−0.150165	−	0.988661i	$-0.547981\pi$
$409$	−6.07381	−0.300331	−0.150165	+	0.988661i	$0.547981\pi$
$410$	0	0
$411$	40.2640	1.98608
$412$	0	0
$413$	26.0396	1.28132
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−58.5478	−2.86710
$418$	0	0
$419$	−8.09929	−0.395676	−0.197838	−	0.980235i	$-0.563392\pi$
$419$	−8.09929	−0.395676	−0.197838	+	0.980235i	$0.563392\pi$
$420$	0	0
$421$	−23.8540	−1.16257	−0.581287	−	0.813699i	$-0.697451\pi$
$421$	−23.8540	−1.16257	−0.581287	+	0.813699i	$0.697451\pi$
$422$	0	0
$423$	24.9622	1.21370
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.34602	−0.210319
$428$	0	0
$429$	34.1176	1.64721
$430$	0	0
$431$	2.24782	0.108274	0.0541368	−	0.998534i	$-0.482759\pi$
$431$	2.24782	0.108274	0.0541368	+	0.998534i	$0.482759\pi$
$432$	0	0
$433$	−30.4523	−1.46345	−0.731723	−	0.681602i	$-0.761284\pi$
$433$	−30.4523	−1.46345	−0.731723	+	0.681602i	$0.761284\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.45534	−0.213128
$438$	0	0
$439$	−15.0133	−0.716546	−0.358273	−	0.933617i	$-0.616634\pi$
$439$	−15.0133	−0.716546	−0.358273	+	0.933617i	$0.616634\pi$
$440$	0	0
$441$	67.8500	3.23095
$442$	0	0
$443$	8.40078	0.399133	0.199567	−	0.979884i	$-0.436047\pi$
$443$	8.40078	0.399133	0.199567	+	0.979884i	$0.436047\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.52662	0.166803
$448$	0	0
$449$	34.8655	1.64541	0.822703	−	0.568471i	$-0.192465\pi$
$449$	34.8655	1.64541	0.822703	+	0.568471i	$0.192465\pi$
$450$	0	0
$451$	5.32460	0.250726
$452$	0	0
$453$	−19.9675	−0.938153
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.1075	−0.519586	−0.259793	−	0.965664i	$-0.583654\pi$
$457$	−11.1075	−0.519586	−0.259793	+	0.965664i	$0.583654\pi$
$458$	0	0
$459$	−55.3354	−2.58284
$460$	0	0
$461$	20.3212	0.946454	0.473227	−	0.880941i	$-0.343089\pi$
$461$	20.3212	0.946454	0.473227	+	0.880941i	$0.343089\pi$
$462$	0	0
$463$	21.0142	0.976614	0.488307	−	0.872672i	$-0.337615\pi$
$463$	21.0142	0.976614	0.488307	+	0.872672i	$0.337615\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.54731	0.210425	0.105212	−	0.994450i	$-0.466448\pi$
$467$	4.54731	0.210425	0.105212	+	0.994450i	$0.466448\pi$
$468$	0	0
$469$	52.1614	2.40859
$470$	0	0
$471$	−31.6979	−1.46056
$472$	0	0
$473$	14.9540	0.687588
$474$	0	0
$475$	0	0
$476$	0	0
$477$	46.4466	2.12665
$478$	0	0
$479$	2.16486	0.0989148	0.0494574	−	0.998776i	$-0.484251\pi$
$479$	2.16486	0.0989148	0.0494574	+	0.998776i	$0.484251\pi$
$480$	0	0
$481$	−49.9976	−2.27970
$482$	0	0
$483$	−57.1639	−2.60105
$484$	0	0
$485$	0	0
$486$	0	0
$487$	36.3926	1.64911	0.824553	−	0.565784i	$-0.191427\pi$
$487$	36.3926	1.64911	0.824553	+	0.565784i	$0.191427\pi$
$488$	0	0
$489$	55.6274	2.51556
$490$	0	0
$491$	−11.1512	−0.503246	−0.251623	−	0.967825i	$-0.580964\pi$
$491$	−11.1512	−0.503246	−0.251623	+	0.967825i	$0.580964\pi$
$492$	0	0
$493$	−34.3601	−1.54750
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.17465	−0.0526900
$498$	0	0
$499$	26.6730	1.19405	0.597024	−	0.802223i	$-0.296350\pi$
$499$	26.6730	1.19405	0.597024	+	0.802223i	$0.296350\pi$
$500$	0	0
$501$	−4.20228	−0.187744
$502$	0	0
$503$	6.48268	0.289049	0.144524	−	0.989501i	$-0.453835\pi$
$503$	6.48268	0.289049	0.144524	+	0.989501i	$0.453835\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−62.7721	−2.78781
$508$	0	0
$509$	13.0210	0.577145	0.288572	−	0.957458i	$-0.406819\pi$
$509$	13.0210	0.577145	0.288572	+	0.957458i	$0.406819\pi$
$510$	0	0
$511$	−64.9578	−2.87357
$512$	0	0
$513$	−8.55770	−0.377832
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.34000	0.366793
$518$	0	0
$519$	4.60460	0.202120
$520$	0	0
$521$	27.7831	1.21720	0.608601	−	0.793477i	$-0.291731\pi$
$521$	27.7831	1.21720	0.608601	+	0.793477i	$0.291731\pi$
$522$	0	0
$523$	−30.1925	−1.32023	−0.660113	−	0.751166i	$-0.729492\pi$
$523$	−30.1925	−1.32023	−0.660113	+	0.751166i	$0.729492\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.61438	−0.201005
$528$	0	0
$529$	−3.14994	−0.136954
$530$	0	0
$531$	−35.5043	−1.54076
$532$	0	0
$533$	−15.8400	−0.686109
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.23206	0.139474
$538$	0	0
$539$	22.6691	0.976425
$540$	0	0
$541$	−22.7426	−0.977783	−0.488891	−	0.872345i	$-0.662599\pi$
$541$	−22.7426	−0.977783	−0.488891	+	0.872345i	$0.662599\pi$
$542$	0	0
$543$	−0.442420	−0.0189861
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−15.2404	−0.651633	−0.325816	−	0.945433i	$-0.605639\pi$
$547$	−15.2404	−0.651633	−0.325816	+	0.945433i	$0.605639\pi$
$548$	0	0
$549$	5.92569	0.252902
$550$	0	0
$551$	−5.31383	−0.226377
$552$	0	0
$553$	−8.43154	−0.358546
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23.0467	0.976520	0.488260	−	0.872698i	$-0.337632\pi$
$557$	23.0467	0.976520	0.488260	+	0.872698i	$0.337632\pi$
$558$	0	0
$559$	−44.4865	−1.88158
$560$	0	0
$561$	−37.7935	−1.59564
$562$	0	0
$563$	21.8744	0.921895	0.460947	−	0.887427i	$-0.347510\pi$
$563$	21.8744	0.921895	0.460947	+	0.887427i	$0.347510\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−33.9092	−1.42405
$568$	0	0
$569$	−2.06779	−0.0866861	−0.0433430	−	0.999060i	$-0.513801\pi$
$569$	−2.06779	−0.0866861	−0.0433430	+	0.999060i	$0.513801\pi$
$570$	0	0
$571$	13.8494	0.579580	0.289790	−	0.957090i	$-0.406415\pi$
$571$	13.8494	0.579580	0.289790	+	0.957090i	$0.406415\pi$
$572$	0	0
$573$	64.2233	2.68297
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−21.1085	−0.878756	−0.439378	−	0.898302i	$-0.644801\pi$
$577$	−21.1085	−0.878756	−0.439378	+	0.898302i	$0.644801\pi$
$578$	0	0
$579$	60.2655	2.50455
$580$	0	0
$581$	−23.8178	−0.988128
$582$	0	0
$583$	15.5181	0.642693
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.7089	−0.442004	−0.221002	−	0.975273i	$-0.570933\pi$
$587$	−10.7089	−0.442004	−0.221002	+	0.975273i	$0.570933\pi$
$588$	0	0
$589$	−0.713620	−0.0294042
$590$	0	0
$591$	62.0192	2.55113
$592$	0	0
$593$	32.6623	1.34128	0.670640	−	0.741783i	$-0.266019\pi$
$593$	32.6623	1.34128	0.670640	+	0.741783i	$0.266019\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.8174	−0.647363
$598$	0	0
$599$	38.2222	1.56172	0.780858	−	0.624708i	$-0.214782\pi$
$599$	38.2222	1.56172	0.780858	+	0.624708i	$0.214782\pi$
$600$	0	0
$601$	−19.8871	−0.811210	−0.405605	−	0.914048i	$-0.632939\pi$
$601$	−19.8871	−0.811210	−0.405605	+	0.914048i	$0.632939\pi$
$602$	0	0
$603$	−71.1208	−2.89626
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.5604	1.64630	0.823148	−	0.567828i	$-0.192216\pi$
$607$	40.5604	1.64630	0.823148	+	0.567828i	$0.192216\pi$
$608$	0	0
$609$	−68.1787	−2.76274
$610$	0	0
$611$	−24.8105	−1.00373
$612$	0	0
$613$	12.0992	0.488682	0.244341	−	0.969689i	$-0.421428\pi$
$613$	12.0992	0.488682	0.244341	+	0.969689i	$0.421428\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25.5344	−1.02798	−0.513989	−	0.857797i	$-0.671833\pi$
$617$	−25.5344	−1.02798	−0.513989	+	0.857797i	$0.671833\pi$
$618$	0	0
$619$	−4.43998	−0.178458	−0.0892289	−	0.996011i	$-0.528440\pi$
$619$	−4.43998	−0.178458	−0.0892289	+	0.996011i	$0.528440\pi$
$620$	0	0
$621$	38.1275	1.53000
$622$	0	0
$623$	2.72602	0.109216
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−5.84482	−0.233420
$628$	0	0
$629$	55.3845	2.20832
$630$	0	0
$631$	20.7751	0.827043	0.413521	−	0.910494i	$-0.364299\pi$
$631$	20.7751	0.827043	0.413521	+	0.910494i	$0.364299\pi$
$632$	0	0
$633$	12.7135	0.505318
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−67.4377	−2.67198
$638$	0	0
$639$	1.60160	0.0633583
$640$	0	0
$641$	40.9268	1.61651	0.808256	−	0.588831i	$-0.200412\pi$
$641$	40.9268	1.61651	0.808256	+	0.588831i	$0.200412\pi$
$642$	0	0
$643$	25.4006	1.00170	0.500851	−	0.865533i	$-0.333020\pi$
$643$	25.4006	1.00170	0.500851	+	0.865533i	$0.333020\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.974550	−0.0383135	−0.0191568	−	0.999816i	$-0.506098\pi$
$647$	−0.974550	−0.0383135	−0.0191568	+	0.999816i	$0.506098\pi$
$648$	0	0
$649$	−11.8622	−0.465632
$650$	0	0
$651$	−9.15604	−0.358854
$652$	0	0
$653$	6.64113	0.259887	0.129944	−	0.991521i	$-0.458520\pi$
$653$	6.64113	0.259887	0.129944	+	0.991521i	$0.458520\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	88.5683	3.45538
$658$	0	0
$659$	−44.0807	−1.71714	−0.858570	−	0.512696i	$-0.828647\pi$
$659$	−44.0807	−1.71714	−0.858570	+	0.512696i	$0.828647\pi$
$660$	0	0
$661$	−13.1380	−0.511009	−0.255504	−	0.966808i	$-0.582242\pi$
$661$	−13.1380	−0.511009	−0.255504	+	0.966808i	$0.582242\pi$
$662$	0	0
$663$	112.431	4.36646
$664$	0	0
$665$	0	0
$666$	0	0
$667$	23.6749	0.916697
$668$	0	0
$669$	27.4453	1.06109
$670$	0	0
$671$	1.97981	0.0764296
$672$	0	0
$673$	−33.7387	−1.30053	−0.650266	−	0.759707i	$-0.725343\pi$
$673$	−33.7387	−1.30053	−0.650266	+	0.759707i	$0.725343\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.7458	1.48912	0.744561	−	0.667554i	$-0.232659\pi$
$677$	38.7458	1.48912	0.744561	+	0.667554i	$0.232659\pi$
$678$	0	0
$679$	−30.6189	−1.17504
$680$	0	0
$681$	8.17527	0.313277
$682$	0	0
$683$	−7.85443	−0.300541	−0.150271	−	0.988645i	$-0.548015\pi$
$683$	−7.85443	−0.300541	−0.150271	+	0.988645i	$0.548015\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−25.5038	−0.973030
$688$	0	0
$689$	−46.1644	−1.75872
$690$	0	0
$691$	−11.3478	−0.431692	−0.215846	−	0.976427i	$-0.569251\pi$
$691$	−11.3478	−0.431692	−0.215846	+	0.976427i	$0.569251\pi$
$692$	0	0
$693$	−49.6363	−1.88553
$694$	0	0
$695$	0	0
$696$	0	0
$697$	17.5467	0.664628
$698$	0	0
$699$	19.7927	0.748628
$700$	0	0
$701$	49.3273	1.86307	0.931533	−	0.363656i	$-0.118472\pi$
$701$	49.3273	1.86307	0.931533	+	0.363656i	$0.118472\pi$
$702$	0	0
$703$	8.56529	0.323046
$704$	0	0
$705$	0	0
$706$	0	0
$707$	45.3128	1.70416
$708$	0	0
$709$	0.400854	0.0150544	0.00752719	−	0.999972i	$-0.497604\pi$
$709$	0.400854	0.0150544	0.00752719	+	0.999972i	$0.497604\pi$
$710$	0	0
$711$	11.4962	0.431141
$712$	0	0
$713$	3.17942	0.119070
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.5550	0.804986
$718$	0	0
$719$	13.4165	0.500349	0.250175	−	0.968201i	$-0.419512\pi$
$719$	13.4165	0.500349	0.250175	+	0.968201i	$0.419512\pi$
$720$	0	0
$721$	2.35496	0.0877032
$722$	0	0
$723$	−28.6331	−1.06488
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−51.1955	−1.89874	−0.949369	−	0.314164i	$-0.898276\pi$
$727$	−51.1955	−1.89874	−0.949369	+	0.314164i	$0.898276\pi$
$728$	0	0
$729$	−30.2394	−1.11998
$730$	0	0
$731$	49.2795	1.82267
$732$	0	0
$733$	−18.6844	−0.690126	−0.345063	−	0.938580i	$-0.612142\pi$
$733$	−18.6844	−0.690126	−0.345063	+	0.938580i	$0.612142\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−23.7618	−0.875279
$738$	0	0
$739$	−12.2935	−0.452225	−0.226112	−	0.974101i	$-0.572602\pi$
$739$	−12.2935	−0.452225	−0.226112	+	0.974101i	$0.572602\pi$
$740$	0	0
$741$	17.3876	0.638751
$742$	0	0
$743$	−11.2902	−0.414199	−0.207099	−	0.978320i	$-0.566402\pi$
$743$	−11.2902	−0.414199	−0.207099	+	0.978320i	$0.566402\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	32.4749	1.18820
$748$	0	0
$749$	32.8301	1.19959
$750$	0	0
$751$	−7.41112	−0.270436	−0.135218	−	0.990816i	$-0.543173\pi$
$751$	−7.41112	−0.270436	−0.135218	+	0.990816i	$0.543173\pi$
$752$	0	0
$753$	−48.1743	−1.75557
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.8735	0.395202	0.197601	−	0.980282i	$-0.436685\pi$
$757$	10.8735	0.395202	0.197601	+	0.980282i	$0.436685\pi$
$758$	0	0
$759$	26.0407	0.945217
$760$	0	0
$761$	−27.7149	−1.00466	−0.502332	−	0.864675i	$-0.667525\pi$
$761$	−27.7149	−1.00466	−0.502332	+	0.864675i	$0.667525\pi$
$762$	0	0
$763$	69.0500	2.49978
$764$	0	0
$765$	0	0
$766$	0	0
$767$	35.2886	1.27420
$768$	0	0
$769$	24.4585	0.881994	0.440997	−	0.897508i	$-0.354625\pi$
$769$	24.4585	0.881994	0.440997	+	0.897508i	$0.354625\pi$
$770$	0	0
$771$	73.2153	2.63678
$772$	0	0
$773$	24.1402	0.868264	0.434132	−	0.900849i	$-0.357055\pi$
$773$	24.1402	0.868264	0.434132	+	0.900849i	$0.357055\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	109.896	3.94250
$778$	0	0
$779$	2.71362	0.0972255
$780$	0	0
$781$	0.535103	0.0191475
$782$	0	0
$783$	45.4742	1.62512
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−34.5844	−1.23280	−0.616400	−	0.787433i	$-0.711410\pi$
$787$	−34.5844	−1.23280	−0.616400	+	0.787433i	$0.711410\pi$
$788$	0	0
$789$	−55.4361	−1.97358
$790$	0	0
$791$	−61.7055	−2.19399
$792$	0	0
$793$	−5.88969	−0.209149
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7.78290	−0.275685	−0.137842	−	0.990454i	$-0.544017\pi$
$797$	−7.78290	−0.275685	−0.137842	+	0.990454i	$0.544017\pi$
$798$	0	0
$799$	27.4836	0.972301
$800$	0	0
$801$	−3.71686	−0.131329
$802$	0	0
$803$	29.5912	1.04425
$804$	0	0
$805$	0	0
$806$	0	0
$807$	26.0517	0.917063
$808$	0	0
$809$	7.35072	0.258438	0.129219	−	0.991616i	$-0.458753\pi$
$809$	7.35072	0.258438	0.129219	+	0.991616i	$0.458753\pi$
$810$	0	0
$811$	4.38106	0.153840	0.0769198	−	0.997037i	$-0.475491\pi$
$811$	4.38106	0.153840	0.0769198	+	0.997037i	$0.475491\pi$
$812$	0	0
$813$	−91.1141	−3.19551
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.62116	0.266630
$818$	0	0
$819$	147.662	5.15973
$820$	0	0
$821$	36.0375	1.25772	0.628859	−	0.777520i	$-0.283522\pi$
$821$	36.0375	1.25772	0.628859	+	0.777520i	$0.283522\pi$
$822$	0	0
$823$	−36.2617	−1.26400	−0.632002	−	0.774967i	$-0.717767\pi$
$823$	−36.2617	−1.26400	−0.632002	+	0.774967i	$0.717767\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21.1896	−0.736834	−0.368417	−	0.929661i	$-0.620100\pi$
$827$	−21.1896	−0.736834	−0.368417	+	0.929661i	$0.620100\pi$
$828$	0	0
$829$	−8.49155	−0.294924	−0.147462	−	0.989068i	$-0.547110\pi$
$829$	−8.49155	−0.294924	−0.147462	+	0.989068i	$0.547110\pi$
$830$	0	0
$831$	82.2846	2.85442
$832$	0	0
$833$	74.7036	2.58833
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.10695	0.211087
$838$	0	0
$839$	5.33979	0.184350	0.0921750	−	0.995743i	$-0.470618\pi$
$839$	5.33979	0.184350	0.0921750	+	0.995743i	$0.470618\pi$
$840$	0	0
$841$	−0.763173	−0.0263163
$842$	0	0
$843$	47.2366	1.62691
$844$	0	0
$845$	0	0
$846$	0	0
$847$	30.7968	1.05819
$848$	0	0
$849$	51.0031	1.75042
$850$	0	0
$851$	−38.1613	−1.30815
$852$	0	0
$853$	37.8867	1.29722	0.648608	−	0.761123i	$-0.275352\pi$
$853$	37.8867	1.29722	0.648608	+	0.761123i	$0.275352\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.2686	−0.692363	−0.346181	−	0.938168i	$-0.612522\pi$
$857$	−20.2686	−0.692363	−0.346181	+	0.938168i	$0.612522\pi$
$858$	0	0
$859$	−32.4147	−1.10598	−0.552988	−	0.833189i	$-0.686512\pi$
$859$	−32.4147	−1.10598	−0.552988	+	0.833189i	$0.686512\pi$
$860$	0	0
$861$	34.8169	1.18656
$862$	0	0
$863$	−5.60777	−0.190891	−0.0954453	−	0.995435i	$-0.530428\pi$
$863$	−5.60777	−0.190891	−0.0954453	+	0.995435i	$0.530428\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−73.9060	−2.50998
$868$	0	0
$869$	3.84094	0.130295
$870$	0	0
$871$	70.6886	2.39519
$872$	0	0
$873$	41.7480	1.41296
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.0065	−1.28339	−0.641694	−	0.766961i	$-0.721768\pi$
$877$	−38.0065	−1.28339	−0.641694	+	0.766961i	$0.721768\pi$
$878$	0	0
$879$	58.8112	1.98365
$880$	0	0
$881$	−35.7230	−1.20354	−0.601770	−	0.798670i	$-0.705538\pi$
$881$	−35.7230	−1.20354	−0.601770	+	0.798670i	$0.705538\pi$
$882$	0	0
$883$	9.57681	0.322285	0.161143	−	0.986931i	$-0.448482\pi$
$883$	9.57681	0.322285	0.161143	+	0.986931i	$0.448482\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.5801	1.42970	0.714850	−	0.699277i	$-0.246495\pi$
$887$	42.5801	1.42970	0.714850	+	0.699277i	$0.246495\pi$
$888$	0	0
$889$	38.9757	1.30720
$890$	0	0
$891$	15.4471	0.517498
$892$	0	0
$893$	4.25038	0.142234
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−77.4679	−2.58658
$898$	0	0
$899$	3.79206	0.126472
$900$	0	0
$901$	51.1383	1.70366
$902$	0	0
$903$	97.7826	3.25400
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−7.71419	−0.256145	−0.128073	−	0.991765i	$-0.540879\pi$
$907$	−7.71419	−0.256145	−0.128073	+	0.991765i	$0.540879\pi$
$908$	0	0
$909$	−61.7828	−2.04921
$910$	0	0
$911$	−48.1383	−1.59489	−0.797446	−	0.603391i	$-0.793816\pi$
$911$	−48.1383	−1.59489	−0.797446	+	0.603391i	$0.793816\pi$
$912$	0	0
$913$	10.8501	0.359084
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.7445	0.420862
$918$	0	0
$919$	−34.8125	−1.14836	−0.574179	−	0.818730i	$-0.694679\pi$
$919$	−34.8125	−1.14836	−0.574179	+	0.818730i	$0.694679\pi$
$920$	0	0
$921$	−4.95470	−0.163263
$922$	0	0
$923$	−1.59187	−0.0523969
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3.21093	−0.105461
$928$	0	0
$929$	−22.0732	−0.724199	−0.362099	−	0.932139i	$-0.617940\pi$
$929$	−22.0732	−0.724199	−0.362099	+	0.932139i	$0.617940\pi$
$930$	0	0
$931$	11.5530	0.378635
$932$	0	0
$933$	63.6592	2.08411
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−14.6709	−0.479277	−0.239638	−	0.970862i	$-0.577029\pi$
$937$	−14.6709	−0.479277	−0.239638	+	0.970862i	$0.577029\pi$
$938$	0	0
$939$	0.419254	0.0136818
$940$	0	0
$941$	−36.7887	−1.19928	−0.599639	−	0.800271i	$-0.704689\pi$
$941$	−36.7887	−1.19928	−0.599639	+	0.800271i	$0.704689\pi$
$942$	0	0
$943$	−12.0901	−0.393708
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.3062	1.24478	0.622392	−	0.782706i	$-0.286161\pi$
$947$	38.3062	1.24478	0.622392	+	0.782706i	$0.286161\pi$
$948$	0	0
$949$	−88.0302	−2.85758
$950$	0	0
$951$	0.675862	0.0219163
$952$	0	0
$953$	−12.5970	−0.408058	−0.204029	−	0.978965i	$-0.565404\pi$
$953$	−12.5970	−0.408058	−0.204029	+	0.978965i	$0.565404\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	31.0584	1.00398
$958$	0	0
$959$	58.2225	1.88010
$960$	0	0
$961$	−30.4907	−0.983572
$962$	0	0
$963$	−44.7630	−1.44247
$964$	0	0
$965$	0	0
$966$	0	0
$967$	51.7595	1.66447	0.832236	−	0.554421i	$-0.187060\pi$
$967$	51.7595	1.66447	0.832236	+	0.554421i	$0.187060\pi$
$968$	0	0
$969$	−19.2610	−0.618753
$970$	0	0
$971$	−52.8251	−1.69524	−0.847619	−	0.530605i	$-0.821965\pi$
$971$	−52.8251	−1.69524	−0.847619	+	0.530605i	$0.821965\pi$
$972$	0	0
$973$	−84.6613	−2.71412
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.4195	−0.813243	−0.406622	−	0.913597i	$-0.633293\pi$
$977$	−25.4195	−0.813243	−0.406622	+	0.913597i	$0.633293\pi$
$978$	0	0
$979$	−1.24182	−0.0396888
$980$	0	0
$981$	−94.1479	−3.00591
$982$	0	0
$983$	−49.9262	−1.59240	−0.796200	−	0.605034i	$-0.793160\pi$
$983$	−49.9262	−1.59240	−0.796200	+	0.605034i	$0.793160\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	54.5342	1.73584
$988$	0	0
$989$	−33.9548	−1.07970
$990$	0	0
$991$	7.17830	0.228026	0.114013	−	0.993479i	$-0.463629\pi$
$991$	7.17830	0.228026	0.114013	+	0.993479i	$0.463629\pi$
$992$	0	0
$993$	45.2333	1.43544
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−25.5592	−0.809468	−0.404734	−	0.914434i	$-0.632636\pi$
$997$	−25.5592	−0.809468	−0.404734	+	0.914434i	$0.632636\pi$
$998$	0	0
$999$	−73.2992	−2.31908

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.z.1.1		6
4.3	odd	2		7600.2.a.cn.1.6		6
5.2	odd	4		760.2.d.e.609.12	yes	12
5.3	odd	4		760.2.d.e.609.1	✓	12
5.4	even	2		3800.2.a.be.1.6		6
20.3	even	4		1520.2.d.k.609.12		12
20.7	even	4		1520.2.d.k.609.1		12
20.19	odd	2		7600.2.a.cg.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.e.609.1	✓	12	5.3	odd	4
760.2.d.e.609.12	yes	12	5.2	odd	4
1520.2.d.k.609.1		12	20.7	even	4
1520.2.d.k.609.12		12	20.3	even	4
3800.2.a.z.1.1		6	1.1	even	1	trivial
3800.2.a.be.1.6		6	5.4	even	2
7600.2.a.cg.1.1		6	20.19	odd	2
7600.2.a.cn.1.6		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)

\(f(q)\)	\(=\)	\(q-2.97875 q^{3} -4.30732 q^{7} +5.87292 q^{9} +1.96218 q^{11} -5.83724 q^{13} +6.46615 q^{17} +1.00000 q^{19} +12.8304 q^{21} -4.45534 q^{23} -8.55770 q^{27} -5.31383 q^{29} -0.713620 q^{31} -5.84482 q^{33} +8.56529 q^{37} +17.3876 q^{39} +2.71362 q^{41} +7.62116 q^{43} +4.25038 q^{47} +11.5530 q^{49} -19.2610 q^{51} +7.90861 q^{53} -2.97875 q^{57} -6.04542 q^{59} +1.00899 q^{61} -25.2966 q^{63} -12.1099 q^{67} +13.2713 q^{69} +0.272709 q^{71} +15.0808 q^{73} -8.45172 q^{77} +1.95749 q^{79} +7.87245 q^{81} +5.52960 q^{83} +15.8286 q^{87} -0.632880 q^{89} +25.1429 q^{91} +2.12569 q^{93} +7.10856 q^{97} +11.5237 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47}+ \cdots - 22 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^7 + 8 * q^9 - 2 * q^11 - 14 * q^13 - 10 * q^17 + 6 * q^19 + 18 * q^21 - 2 * q^23 - 2 * q^27 - 2 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^37 - 18 * q^39 + 4 * q^41 - 4 * q^43 - 4 * q^47 + 2 * q^49 - 34 * q^51 + 14 * q^53 - 2 * q^57 - 2 * q^59 + 10 * q^61 - 44 * q^63 - 42 * q^67 + 18 * q^69 - 8 * q^71 + 2 * q^73 - 24 * q^77 - 20 * q^79 + 6 * q^81 - 16 * q^83 - 30 * q^87 + 32 * q^89 + 10 * q^91 - 12 * q^93 - 40 * q^97 - 22 * q^99

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