LMFDB - Embedded newform 7600.2.a.ch.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$1$
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:	no (minimal twist has level 3800)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.77008$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.77008 q^{3} +2.31077 q^{7} +4.67334 q^{9} +O(q^{10})$	$q+2.77008 q^{3} +2.31077 q^{7} +4.67334 q^{9} -2.16026 q^{11} -6.25643 q^{13} -7.10756 q^{17} +1.00000 q^{19} +6.40100 q^{21} -8.99784 q^{23} +4.63527 q^{27} +3.16424 q^{29} +9.95490 q^{31} -5.98410 q^{33} -9.43207 q^{37} -17.3308 q^{39} -10.8193 q^{41} +1.05217 q^{43} -6.98410 q^{47} -1.66036 q^{49} -19.6885 q^{51} +2.69517 q^{53} +2.77008 q^{57} -11.2021 q^{59} -4.36457 q^{61} +10.7990 q^{63} +6.25408 q^{67} -24.9247 q^{69} +13.9390 q^{71} +8.64016 q^{73} -4.99186 q^{77} +9.93758 q^{79} -1.17994 q^{81} -9.82384 q^{83} +8.76520 q^{87} -1.63686 q^{89} -14.4572 q^{91} +27.5759 q^{93} -9.27994 q^{97} -10.0956 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9	$ 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}+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

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
$3$	2.77008	1.59931	0.799653	−	0.600463i	$-0.205017\pi$
$3$	2.77008	1.59931	0.799653	+	0.600463i	$0.205017\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.31077	0.873387	0.436694	−	0.899610i	$-0.356149\pi$
$7$	2.31077	0.873387	0.436694	+	0.899610i	$0.356149\pi$
$8$	0	0
$9$	4.67334	1.55778
$10$	0	0
$11$	−2.16026	−0.651344	−0.325672	−	0.945483i	$-0.605591\pi$
$11$	−2.16026	−0.651344	−0.325672	+	0.945483i	$0.605591\pi$
$12$	0	0
$13$	−6.25643	−1.73522	−0.867611	−	0.497243i	$-0.834346\pi$
$13$	−6.25643	−1.73522	−0.867611	+	0.497243i	$0.834346\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.10756	−1.72384	−0.861918	−	0.507047i	$-0.830737\pi$
$17$	−7.10756	−1.72384	−0.861918	+	0.507047i	$0.830737\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	6.40100	1.39681
$22$	0	0
$23$	−8.99784	−1.87618	−0.938090	−	0.346392i	$-0.887407\pi$
$23$	−8.99784	−1.87618	−0.938090	+	0.346392i	$0.887407\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.63527	0.892059
$28$	0	0
$29$	3.16424	0.587585	0.293793	−	0.955869i	$-0.405083\pi$
$29$	3.16424	0.587585	0.293793	+	0.955869i	$0.405083\pi$
$30$	0	0
$31$	9.95490	1.78795	0.893977	−	0.448114i	$-0.147904\pi$
$31$	9.95490	1.78795	0.893977	+	0.448114i	$0.147904\pi$
$32$	0	0
$33$	−5.98410	−1.04170
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.43207	−1.55062	−0.775311	−	0.631579i	$-0.782407\pi$
$37$	−9.43207	−1.55062	−0.775311	+	0.631579i	$0.782407\pi$
$38$	0	0
$39$	−17.3308	−2.77515
$40$	0	0
$41$	−10.8193	−1.68970	−0.844848	−	0.535007i	$-0.820309\pi$
$41$	−10.8193	−1.68970	−0.844848	+	0.535007i	$0.820309\pi$
$42$	0	0
$43$	1.05217	0.160455	0.0802275	−	0.996777i	$-0.474435\pi$
$43$	1.05217	0.160455	0.0802275	+	0.996777i	$0.474435\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.98410	−1.01874	−0.509368	−	0.860549i	$-0.670121\pi$
$47$	−6.98410	−1.01874	−0.509368	+	0.860549i	$0.670121\pi$
$48$	0	0
$49$	−1.66036	−0.237194
$50$	0	0
$51$	−19.6885	−2.75694
$52$	0	0
$53$	2.69517	0.370210	0.185105	−	0.982719i	$-0.440738\pi$
$53$	2.69517	0.370210	0.185105	+	0.982719i	$0.440738\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.77008	0.366906
$58$	0	0
$59$	−11.2021	−1.45840	−0.729198	−	0.684303i	$-0.760106\pi$
$59$	−11.2021	−1.45840	−0.729198	+	0.684303i	$0.760106\pi$
$60$	0	0
$61$	−4.36457	−0.558826	−0.279413	−	0.960171i	$-0.590140\pi$
$61$	−4.36457	−0.558826	−0.279413	+	0.960171i	$0.590140\pi$
$62$	0	0
$63$	10.7990	1.36054
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.25408	0.764058	0.382029	−	0.924150i	$-0.375225\pi$
$67$	6.25408	0.764058	0.382029	+	0.924150i	$0.375225\pi$
$68$	0	0
$69$	−24.9247	−3.00059
$70$	0	0
$71$	13.9390	1.65426	0.827128	−	0.562014i	$-0.189973\pi$
$71$	13.9390	1.65426	0.827128	+	0.562014i	$0.189973\pi$
$72$	0	0
$73$	8.64016	1.01125	0.505627	−	0.862752i	$-0.331261\pi$
$73$	8.64016	1.01125	0.505627	+	0.862752i	$0.331261\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.99186	−0.568876
$78$	0	0
$79$	9.93758	1.11806	0.559032	−	0.829146i	$-0.311173\pi$
$79$	9.93758	1.11806	0.559032	+	0.829146i	$0.311173\pi$
$80$	0	0
$81$	−1.17994	−0.131104
$82$	0	0
$83$	−9.82384	−1.07831	−0.539153	−	0.842208i	$-0.681256\pi$
$83$	−9.82384	−1.07831	−0.539153	+	0.842208i	$0.681256\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.76520	0.939728
$88$	0	0
$89$	−1.63686	−0.173507	−0.0867533	−	0.996230i	$-0.527649\pi$
$89$	−1.63686	−0.173507	−0.0867533	+	0.996230i	$0.527649\pi$
$90$	0	0
$91$	−14.4572	−1.51552
$92$	0	0
$93$	27.5759	2.85948
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.27994	−0.942235	−0.471117	−	0.882070i	$-0.656149\pi$
$97$	−9.27994	−0.942235	−0.471117	+	0.882070i	$0.656149\pi$
$98$	0	0
$99$	−10.0956	−1.01465
$100$	0	0
$101$	9.86462	0.981567	0.490783	−	0.871282i	$-0.336711\pi$
$101$	9.86462	0.981567	0.490783	+	0.871282i	$0.336711\pi$
$102$	0	0
$103$	9.99492	0.984829	0.492414	−	0.870361i	$-0.336114\pi$
$103$	9.99492	0.984829	0.492414	+	0.870361i	$0.336114\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.929973	0.0899039	0.0449520	−	0.998989i	$-0.485687\pi$
$107$	0.929973	0.0899039	0.0449520	+	0.998989i	$0.485687\pi$
$108$	0	0
$109$	8.00037	0.766296	0.383148	−	0.923687i	$-0.374840\pi$
$109$	8.00037	0.766296	0.383148	+	0.923687i	$0.374840\pi$
$110$	0	0
$111$	−26.1276	−2.47992
$112$	0	0
$113$	3.06548	0.288376	0.144188	−	0.989550i	$-0.453943\pi$
$113$	3.06548	0.288376	0.144188	+	0.989550i	$0.453943\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−29.2384	−2.70309
$118$	0	0
$119$	−16.4239	−1.50558
$120$	0	0
$121$	−6.33326	−0.575751
$122$	0	0
$123$	−29.9704	−2.70234
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−13.1998	−1.17129	−0.585646	−	0.810567i	$-0.699159\pi$
$127$	−13.1998	−1.17129	−0.585646	+	0.810567i	$0.699159\pi$
$128$	0	0
$129$	2.91460	0.256617
$130$	0	0
$131$	1.04021	0.0908837	0.0454419	−	0.998967i	$-0.485530\pi$
$131$	1.04021	0.0908837	0.0454419	+	0.998967i	$0.485530\pi$
$132$	0	0
$133$	2.31077	0.200369
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.8603	−1.35504	−0.677519	−	0.735505i	$-0.736945\pi$
$137$	−15.8603	−1.35504	−0.677519	+	0.735505i	$0.736945\pi$
$138$	0	0
$139$	15.2366	1.29235	0.646174	−	0.763190i	$-0.276368\pi$
$139$	15.2366	1.29235	0.646174	+	0.763190i	$0.276368\pi$
$140$	0	0
$141$	−19.3465	−1.62927
$142$	0	0
$143$	13.5156	1.13023
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.59933	−0.379346
$148$	0	0
$149$	4.82039	0.394902	0.197451	−	0.980313i	$-0.436734\pi$
$149$	4.82039	0.394902	0.197451	+	0.980313i	$0.436734\pi$
$150$	0	0
$151$	−19.2233	−1.56437	−0.782183	−	0.623049i	$-0.785894\pi$
$151$	−19.2233	−1.56437	−0.782183	+	0.623049i	$0.785894\pi$
$152$	0	0
$153$	−33.2160	−2.68536
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.62844	0.209772	0.104886	−	0.994484i	$-0.466552\pi$
$157$	2.62844	0.209772	0.104886	+	0.994484i	$0.466552\pi$
$158$	0	0
$159$	7.46583	0.592078
$160$	0	0
$161$	−20.7919	−1.63863
$162$	0	0
$163$	13.3875	1.04859	0.524293	−	0.851538i	$-0.324330\pi$
$163$	13.3875	1.04859	0.524293	+	0.851538i	$0.324330\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.6800	−1.44550	−0.722752	−	0.691108i	$-0.757123\pi$
$167$	−18.6800	−1.44550	−0.722752	+	0.691108i	$0.757123\pi$
$168$	0	0
$169$	26.1430	2.01100
$170$	0	0
$171$	4.67334	0.357379
$172$	0	0
$173$	8.72637	0.663454	0.331727	−	0.943375i	$-0.392369\pi$
$173$	8.72637	0.663454	0.331727	+	0.943375i	$0.392369\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−31.0308	−2.33242
$178$	0	0
$179$	−17.1065	−1.27860	−0.639298	−	0.768959i	$-0.720775\pi$
$179$	−17.1065	−1.27860	−0.639298	+	0.768959i	$0.720775\pi$
$180$	0	0
$181$	4.16620	0.309671	0.154836	−	0.987940i	$-0.450515\pi$
$181$	4.16620	0.309671	0.154836	+	0.987940i	$0.450515\pi$
$182$	0	0
$183$	−12.0902	−0.893733
$184$	0	0
$185$	0	0
$186$	0	0
$187$	15.3542	1.12281
$188$	0	0
$189$	10.7110	0.779113
$190$	0	0
$191$	−6.27884	−0.454321	−0.227160	−	0.973857i	$-0.572944\pi$
$191$	−6.27884	−0.454321	−0.227160	+	0.973857i	$0.572944\pi$
$192$	0	0
$193$	7.25648	0.522333	0.261166	−	0.965294i	$-0.415893\pi$
$193$	7.25648	0.522333	0.261166	+	0.965294i	$0.415893\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.78451	0.269635	0.134817	−	0.990870i	$-0.456955\pi$
$197$	3.78451	0.269635	0.134817	+	0.990870i	$0.456955\pi$
$198$	0	0
$199$	−4.17722	−0.296115	−0.148058	−	0.988979i	$-0.547302\pi$
$199$	−4.17722	−0.296115	−0.148058	+	0.988979i	$0.547302\pi$
$200$	0	0
$201$	17.3243	1.22196
$202$	0	0
$203$	7.31182	0.513189
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−42.0499	−2.92267
$208$	0	0
$209$	−2.16026	−0.149429
$210$	0	0
$211$	−22.2575	−1.53227	−0.766134	−	0.642680i	$-0.777822\pi$
$211$	−22.2575	−1.53227	−0.766134	+	0.642680i	$0.777822\pi$
$212$	0	0
$213$	38.6121	2.64566
$214$	0	0
$215$	0	0
$216$	0	0
$217$	23.0034	1.56158
$218$	0	0
$219$	23.9339	1.61730
$220$	0	0
$221$	44.4680	2.99124
$222$	0	0
$223$	−4.75443	−0.318380	−0.159190	−	0.987248i	$-0.550888\pi$
$223$	−4.75443	−0.318380	−0.159190	+	0.987248i	$0.550888\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.95011	−0.129433	−0.0647166	−	0.997904i	$-0.520614\pi$
$227$	−1.95011	−0.129433	−0.0647166	+	0.997904i	$0.520614\pi$
$228$	0	0
$229$	2.00794	0.132689	0.0663443	−	0.997797i	$-0.478866\pi$
$229$	2.00794	0.132689	0.0663443	+	0.997797i	$0.478866\pi$
$230$	0	0
$231$	−13.8279	−0.909806
$232$	0	0
$233$	18.0832	1.18467	0.592335	−	0.805692i	$-0.298206\pi$
$233$	18.0832	1.18467	0.592335	+	0.805692i	$0.298206\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	27.5279	1.78813
$238$	0	0
$239$	−0.946241	−0.0612072	−0.0306036	−	0.999532i	$-0.509743\pi$
$239$	−0.946241	−0.0612072	−0.0306036	+	0.999532i	$0.509743\pi$
$240$	0	0
$241$	16.2017	1.04364	0.521822	−	0.853054i	$-0.325253\pi$
$241$	16.2017	1.04364	0.521822	+	0.853054i	$0.325253\pi$
$242$	0	0
$243$	−17.1743	−1.10173
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.25643	−0.398087
$248$	0	0
$249$	−27.2128	−1.72454
$250$	0	0
$251$	−1.57875	−0.0996499	−0.0498249	−	0.998758i	$-0.515866\pi$
$251$	−1.57875	−0.0996499	−0.0498249	+	0.998758i	$0.515866\pi$
$252$	0	0
$253$	19.4377	1.22204
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0581	−1.12643	−0.563217	−	0.826309i	$-0.690436\pi$
$257$	−18.0581	−1.12643	−0.563217	+	0.826309i	$0.690436\pi$
$258$	0	0
$259$	−21.7953	−1.35429
$260$	0	0
$261$	14.7876	0.915328
$262$	0	0
$263$	−3.77369	−0.232696	−0.116348	−	0.993209i	$-0.537119\pi$
$263$	−3.77369	−0.232696	−0.116348	+	0.993209i	$0.537119\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.53422	−0.277490
$268$	0	0
$269$	−9.49161	−0.578714	−0.289357	−	0.957221i	$-0.593441\pi$
$269$	−9.49161	−0.578714	−0.289357	+	0.957221i	$0.593441\pi$
$270$	0	0
$271$	20.7162	1.25842	0.629211	−	0.777234i	$-0.283378\pi$
$271$	20.7162	1.25842	0.629211	+	0.777234i	$0.283378\pi$
$272$	0	0
$273$	−40.0475	−2.42378
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.3988	0.624806	0.312403	−	0.949950i	$-0.398866\pi$
$277$	10.3988	0.624806	0.312403	+	0.949950i	$0.398866\pi$
$278$	0	0
$279$	46.5226	2.78524
$280$	0	0
$281$	25.9898	1.55042	0.775212	−	0.631701i	$-0.217643\pi$
$281$	25.9898	1.55042	0.775212	+	0.631701i	$0.217643\pi$
$282$	0	0
$283$	17.6225	1.04755	0.523774	−	0.851857i	$-0.324524\pi$
$283$	17.6225	1.04755	0.523774	+	0.851857i	$0.324524\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−25.0009	−1.47576
$288$	0	0
$289$	33.5174	1.97161
$290$	0	0
$291$	−25.7062	−1.50692
$292$	0	0
$293$	17.6057	1.02853	0.514267	−	0.857630i	$-0.328064\pi$
$293$	17.6057	1.02853	0.514267	+	0.857630i	$0.328064\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−10.0134	−0.581037
$298$	0	0
$299$	56.2944	3.25559
$300$	0	0
$301$	2.43133	0.140139
$302$	0	0
$303$	27.3258	1.56982
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−19.1166	−1.09104	−0.545521	−	0.838097i	$-0.683668\pi$
$307$	−19.1166	−1.09104	−0.545521	+	0.838097i	$0.683668\pi$
$308$	0	0
$309$	27.6867	1.57504
$310$	0	0
$311$	−5.46289	−0.309772	−0.154886	−	0.987932i	$-0.549501\pi$
$311$	−5.46289	−0.309772	−0.154886	+	0.987932i	$0.549501\pi$
$312$	0	0
$313$	1.02818	0.0581164	0.0290582	−	0.999578i	$-0.490749\pi$
$313$	1.02818	0.0581164	0.0290582	+	0.999578i	$0.490749\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.9622	1.90751	0.953754	−	0.300590i	$-0.0971835\pi$
$317$	33.9622	1.90751	0.953754	+	0.300590i	$0.0971835\pi$
$318$	0	0
$319$	−6.83560	−0.382720
$320$	0	0
$321$	2.57610	0.143784
$322$	0	0
$323$	−7.10756	−0.395475
$324$	0	0
$325$	0	0
$326$	0	0
$327$	22.1617	1.22554
$328$	0	0
$329$	−16.1386	−0.889751
$330$	0	0
$331$	−0.992479	−0.0545516	−0.0272758	−	0.999628i	$-0.508683\pi$
$331$	−0.992479	−0.0545516	−0.0272758	+	0.999628i	$0.508683\pi$
$332$	0	0
$333$	−44.0792	−2.41553
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−15.1204	−0.823658	−0.411829	−	0.911261i	$-0.635110\pi$
$337$	−15.1204	−0.823658	−0.411829	+	0.911261i	$0.635110\pi$
$338$	0	0
$339$	8.49161	0.461201
$340$	0	0
$341$	−21.5052	−1.16457
$342$	0	0
$343$	−20.0121	−1.08055
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.14208	−0.0613099	−0.0306550	−	0.999530i	$-0.509759\pi$
$347$	−1.14208	−0.0613099	−0.0306550	+	0.999530i	$0.509759\pi$
$348$	0	0
$349$	16.4472	0.880398	0.440199	−	0.897900i	$-0.354908\pi$
$349$	16.4472	0.880398	0.440199	+	0.897900i	$0.354908\pi$
$350$	0	0
$351$	−29.0003	−1.54792
$352$	0	0
$353$	14.9388	0.795112	0.397556	−	0.917578i	$-0.369858\pi$
$353$	14.9388	0.795112	0.397556	+	0.917578i	$0.369858\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−45.4955	−2.40788
$358$	0	0
$359$	−2.50832	−0.132384	−0.0661920	−	0.997807i	$-0.521085\pi$
$359$	−2.50832	−0.132384	−0.0661920	+	0.997807i	$0.521085\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−17.5436	−0.920801
$364$	0	0
$365$	0	0
$366$	0	0
$367$	16.6031	0.866677	0.433339	−	0.901231i	$-0.357335\pi$
$367$	16.6031	0.866677	0.433339	+	0.901231i	$0.357335\pi$
$368$	0	0
$369$	−50.5624	−2.63217
$370$	0	0
$371$	6.22790	0.323336
$372$	0	0
$373$	−23.2873	−1.20577	−0.602887	−	0.797827i	$-0.705983\pi$
$373$	−23.2873	−1.20577	−0.602887	+	0.797827i	$0.705983\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.7969	−1.01959
$378$	0	0
$379$	−1.06225	−0.0545639	−0.0272819	−	0.999628i	$-0.508685\pi$
$379$	−1.06225	−0.0545639	−0.0272819	+	0.999628i	$0.508685\pi$
$380$	0	0
$381$	−36.5645	−1.87325
$382$	0	0
$383$	3.82460	0.195428	0.0977141	−	0.995215i	$-0.468847\pi$
$383$	3.82460	0.195428	0.0977141	+	0.995215i	$0.468847\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.91716	0.249953
$388$	0	0
$389$	−2.32225	−0.117743	−0.0588713	−	0.998266i	$-0.518750\pi$
$389$	−2.32225	−0.117743	−0.0588713	+	0.998266i	$0.518750\pi$
$390$	0	0
$391$	63.9527	3.23423
$392$	0	0
$393$	2.88147	0.145351
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.5438	1.18163	0.590814	−	0.806808i	$-0.298807\pi$
$397$	23.5438	1.18163	0.590814	+	0.806808i	$0.298807\pi$
$398$	0	0
$399$	6.40100	0.320451
$400$	0	0
$401$	−9.45351	−0.472086	−0.236043	−	0.971743i	$-0.575851\pi$
$401$	−9.45351	−0.472086	−0.236043	+	0.971743i	$0.575851\pi$
$402$	0	0
$403$	−62.2822	−3.10250
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.3758	1.00999
$408$	0	0
$409$	−21.8633	−1.08107	−0.540535	−	0.841322i	$-0.681778\pi$
$409$	−21.8633	−1.08107	−0.540535	+	0.841322i	$0.681778\pi$
$410$	0	0
$411$	−43.9343	−2.16712
$412$	0	0
$413$	−25.8855	−1.27374
$414$	0	0
$415$	0	0
$416$	0	0
$417$	42.2065	2.06686
$418$	0	0
$419$	15.2767	0.746315	0.373158	−	0.927768i	$-0.378275\pi$
$419$	15.2767	0.746315	0.373158	+	0.927768i	$0.378275\pi$
$420$	0	0
$421$	−36.2730	−1.76784	−0.883919	−	0.467641i	$-0.845104\pi$
$421$	−36.2730	−1.76784	−0.883919	+	0.467641i	$0.845104\pi$
$422$	0	0
$423$	−32.6391	−1.58697
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10.0855	−0.488071
$428$	0	0
$429$	37.4391	1.80758
$430$	0	0
$431$	23.8603	1.14931	0.574655	−	0.818395i	$-0.305136\pi$
$431$	23.8603	1.14931	0.574655	+	0.818395i	$0.305136\pi$
$432$	0	0
$433$	−12.0076	−0.577051	−0.288525	−	0.957472i	$-0.593165\pi$
$433$	−12.0076	−0.577051	−0.288525	+	0.957472i	$0.593165\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.99784	−0.430425
$438$	0	0
$439$	−5.78565	−0.276134	−0.138067	−	0.990423i	$-0.544089\pi$
$439$	−5.78565	−0.276134	−0.138067	+	0.990423i	$0.544089\pi$
$440$	0	0
$441$	−7.75943	−0.369496
$442$	0	0
$443$	1.36400	0.0648054	0.0324027	−	0.999475i	$-0.489684\pi$
$443$	1.36400	0.0648054	0.0324027	+	0.999475i	$0.489684\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.3529	0.631568
$448$	0	0
$449$	−10.6471	−0.502468	−0.251234	−	0.967926i	$-0.580836\pi$
$449$	−10.6471	−0.502468	−0.251234	+	0.967926i	$0.580836\pi$
$450$	0	0
$451$	23.3726	1.10057
$452$	0	0
$453$	−53.2499	−2.50190
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31.9955	−1.49669	−0.748344	−	0.663311i	$-0.769151\pi$
$457$	−31.9955	−1.49669	−0.748344	+	0.663311i	$0.769151\pi$
$458$	0	0
$459$	−32.9455	−1.53776
$460$	0	0
$461$	31.0876	1.44789	0.723947	−	0.689856i	$-0.242326\pi$
$461$	31.0876	1.44789	0.723947	+	0.689856i	$0.242326\pi$
$462$	0	0
$463$	13.3371	0.619827	0.309913	−	0.950765i	$-0.399700\pi$
$463$	13.3371	0.619827	0.309913	+	0.950765i	$0.399700\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.8114	−1.05558	−0.527792	−	0.849374i	$-0.676980\pi$
$467$	−22.8114	−1.05558	−0.527792	+	0.849374i	$0.676980\pi$
$468$	0	0
$469$	14.4517	0.667319
$470$	0	0
$471$	7.28099	0.335490
$472$	0	0
$473$	−2.27297	−0.104511
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.5954	0.576705
$478$	0	0
$479$	0.737298	0.0336880	0.0168440	−	0.999858i	$-0.494638\pi$
$479$	0.737298	0.0336880	0.0168440	+	0.999858i	$0.494638\pi$
$480$	0	0
$481$	59.0111	2.69068
$482$	0	0
$483$	−57.5952	−2.62067
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−8.57681	−0.388652	−0.194326	−	0.980937i	$-0.562252\pi$
$487$	−8.57681	−0.388652	−0.194326	+	0.980937i	$0.562252\pi$
$488$	0	0
$489$	37.0843	1.67701
$490$	0	0
$491$	−1.62319	−0.0732535	−0.0366267	−	0.999329i	$-0.511661\pi$
$491$	−1.62319	−0.0732535	−0.0366267	+	0.999329i	$0.511661\pi$
$492$	0	0
$493$	−22.4900	−1.01290
$494$	0	0
$495$	0	0
$496$	0	0
$497$	32.2098	1.44481
$498$	0	0
$499$	−19.8881	−0.890312	−0.445156	−	0.895453i	$-0.646852\pi$
$499$	−19.8881	−0.890312	−0.445156	+	0.895453i	$0.646852\pi$
$500$	0	0
$501$	−51.7451	−2.31180
$502$	0	0
$503$	−1.50927	−0.0672950	−0.0336475	−	0.999434i	$-0.510712\pi$
$503$	−1.50927	−0.0672950	−0.0336475	+	0.999434i	$0.510712\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	72.4181	3.21620
$508$	0	0
$509$	8.26509	0.366344	0.183172	−	0.983081i	$-0.441364\pi$
$509$	8.26509	0.366344	0.183172	+	0.983081i	$0.441364\pi$
$510$	0	0
$511$	19.9654	0.883216
$512$	0	0
$513$	4.63527	0.204652
$514$	0	0
$515$	0	0
$516$	0	0
$517$	15.0875	0.663548
$518$	0	0
$519$	24.1727	1.06107
$520$	0	0
$521$	−20.7526	−0.909187	−0.454593	−	0.890699i	$-0.650215\pi$
$521$	−20.7526	−0.909187	−0.454593	+	0.890699i	$0.650215\pi$
$522$	0	0
$523$	1.68418	0.0736441	0.0368220	−	0.999322i	$-0.488277\pi$
$523$	1.68418	0.0736441	0.0368220	+	0.999322i	$0.488277\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−70.7551	−3.08214
$528$	0	0
$529$	57.9612	2.52005
$530$	0	0
$531$	−52.3514	−2.27186
$532$	0	0
$533$	67.6904	2.93200
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−47.3862	−2.04487
$538$	0	0
$539$	3.58682	0.154495
$540$	0	0
$541$	−35.8869	−1.54290	−0.771450	−	0.636291i	$-0.780468\pi$
$541$	−35.8869	−1.54290	−0.771450	+	0.636291i	$0.780468\pi$
$542$	0	0
$543$	11.5407	0.495259
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.3264	0.869092	0.434546	−	0.900650i	$-0.356909\pi$
$547$	20.3264	0.869092	0.434546	+	0.900650i	$0.356909\pi$
$548$	0	0
$549$	−20.3971	−0.870527
$550$	0	0
$551$	3.16424	0.134801
$552$	0	0
$553$	22.9634	0.976504
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.2922	−0.817437	−0.408719	−	0.912660i	$-0.634024\pi$
$557$	−19.2922	−0.817437	−0.408719	+	0.912660i	$0.634024\pi$
$558$	0	0
$559$	−6.58286	−0.278425
$560$	0	0
$561$	42.5324	1.79572
$562$	0	0
$563$	16.7275	0.704980	0.352490	−	0.935816i	$-0.385335\pi$
$563$	16.7275	0.704980	0.352490	+	0.935816i	$0.385335\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.72656	−0.114505
$568$	0	0
$569$	−35.3730	−1.48291	−0.741456	−	0.671002i	$-0.765864\pi$
$569$	−35.3730	−1.48291	−0.741456	+	0.671002i	$0.765864\pi$
$570$	0	0
$571$	−31.5790	−1.32154	−0.660771	−	0.750588i	$-0.729770\pi$
$571$	−31.5790	−1.32154	−0.660771	+	0.750588i	$0.729770\pi$
$572$	0	0
$573$	−17.3929	−0.726598
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.52376	0.0634350	0.0317175	−	0.999497i	$-0.489902\pi$
$577$	1.52376	0.0634350	0.0317175	+	0.999497i	$0.489902\pi$
$578$	0	0
$579$	20.1010	0.835370
$580$	0	0
$581$	−22.7006	−0.941779
$582$	0	0
$583$	−5.82227	−0.241134
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.57118	−0.147399	−0.0736993	−	0.997281i	$-0.523480\pi$
$587$	−3.57118	−0.147399	−0.0736993	+	0.997281i	$0.523480\pi$
$588$	0	0
$589$	9.95490	0.410185
$590$	0	0
$591$	10.4834	0.431229
$592$	0	0
$593$	−5.91174	−0.242766	−0.121383	−	0.992606i	$-0.538733\pi$
$593$	−5.91174	−0.242766	−0.121383	+	0.992606i	$0.538733\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.5712	−0.473579
$598$	0	0
$599$	14.1545	0.578337	0.289169	−	0.957278i	$-0.406621\pi$
$599$	14.1545	0.578337	0.289169	+	0.957278i	$0.406621\pi$
$600$	0	0
$601$	−37.4106	−1.52601	−0.763005	−	0.646393i	$-0.776277\pi$
$601$	−37.4106	−1.52601	−0.763005	+	0.646393i	$0.776277\pi$
$602$	0	0
$603$	29.2274	1.19023
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−0.745322	−0.0302517	−0.0151258	−	0.999886i	$-0.504815\pi$
$607$	−0.745322	−0.0302517	−0.0151258	+	0.999886i	$0.504815\pi$
$608$	0	0
$609$	20.2543	0.820747
$610$	0	0
$611$	43.6956	1.76773
$612$	0	0
$613$	22.7467	0.918732	0.459366	−	0.888247i	$-0.348077\pi$
$613$	22.7467	0.918732	0.459366	+	0.888247i	$0.348077\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−48.3900	−1.94811	−0.974055	−	0.226314i	$-0.927333\pi$
$617$	−48.3900	−1.94811	−0.974055	+	0.226314i	$0.927333\pi$
$618$	0	0
$619$	−26.1264	−1.05011	−0.525054	−	0.851069i	$-0.675955\pi$
$619$	−26.1264	−1.05011	−0.525054	+	0.851069i	$0.675955\pi$
$620$	0	0
$621$	−41.7075	−1.67366
$622$	0	0
$623$	−3.78239	−0.151538
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−5.98410	−0.238982
$628$	0	0
$629$	67.0390	2.67302
$630$	0	0
$631$	13.7707	0.548202	0.274101	−	0.961701i	$-0.411620\pi$
$631$	13.7707	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$632$	0	0
$633$	−61.6550	−2.45057
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.3879	0.411585
$638$	0	0
$639$	65.1416	2.57696
$640$	0	0
$641$	−12.6880	−0.501145	−0.250573	−	0.968098i	$-0.580619\pi$
$641$	−12.6880	−0.501145	−0.250573	+	0.968098i	$0.580619\pi$
$642$	0	0
$643$	28.4054	1.12020	0.560099	−	0.828426i	$-0.310763\pi$
$643$	28.4054	1.12020	0.560099	+	0.828426i	$0.310763\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.53899	0.0605040	0.0302520	−	0.999542i	$-0.490369\pi$
$647$	1.53899	0.0605040	0.0302520	+	0.999542i	$0.490369\pi$
$648$	0	0
$649$	24.1996	0.949917
$650$	0	0
$651$	63.7214	2.49744
$652$	0	0
$653$	−12.5269	−0.490217	−0.245109	−	0.969496i	$-0.578824\pi$
$653$	−12.5269	−0.490217	−0.245109	+	0.969496i	$0.578824\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	40.3784	1.57531
$658$	0	0
$659$	−44.8511	−1.74715	−0.873576	−	0.486688i	$-0.838205\pi$
$659$	−44.8511	−1.74715	−0.873576	+	0.486688i	$0.838205\pi$
$660$	0	0
$661$	2.81697	0.109567	0.0547837	−	0.998498i	$-0.482553\pi$
$661$	2.81697	0.109567	0.0547837	+	0.998498i	$0.482553\pi$
$662$	0	0
$663$	123.180	4.78391
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−28.4714	−1.10242
$668$	0	0
$669$	−13.1701	−0.509187
$670$	0	0
$671$	9.42862	0.363988
$672$	0	0
$673$	−20.9107	−0.806047	−0.403023	−	0.915190i	$-0.632041\pi$
$673$	−20.9107	−0.806047	−0.403023	+	0.915190i	$0.632041\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.458701	0.0176293	0.00881465	−	0.999961i	$-0.497194\pi$
$677$	0.458701	0.0176293	0.00881465	+	0.999961i	$0.497194\pi$
$678$	0	0
$679$	−21.4438	−0.822936
$680$	0	0
$681$	−5.40195	−0.207003
$682$	0	0
$683$	−44.6820	−1.70971	−0.854855	−	0.518866i	$-0.826354\pi$
$683$	−44.6820	−1.70971	−0.854855	+	0.518866i	$0.826354\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.56216	0.212209
$688$	0	0
$689$	−16.8621	−0.642396
$690$	0	0
$691$	−25.6748	−0.976714	−0.488357	−	0.872644i	$-0.662404\pi$
$691$	−25.6748	−0.976714	−0.488357	+	0.872644i	$0.662404\pi$
$692$	0	0
$693$	−23.3287	−0.886183
$694$	0	0
$695$	0	0
$696$	0	0
$697$	76.8990	2.91276
$698$	0	0
$699$	50.0919	1.89465
$700$	0	0
$701$	44.5710	1.68342	0.841711	−	0.539928i	$-0.181548\pi$
$701$	44.5710	1.68342	0.841711	+	0.539928i	$0.181548\pi$
$702$	0	0
$703$	−9.43207	−0.355737
$704$	0	0
$705$	0	0
$706$	0	0
$707$	22.7948	0.857288
$708$	0	0
$709$	29.9199	1.12366	0.561832	−	0.827251i	$-0.310097\pi$
$709$	29.9199	1.12366	0.561832	+	0.827251i	$0.310097\pi$
$710$	0	0
$711$	46.4416	1.74170
$712$	0	0
$713$	−89.5726	−3.35452
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.62116	−0.0978891
$718$	0	0
$719$	−10.4095	−0.388210	−0.194105	−	0.980981i	$-0.562180\pi$
$719$	−10.4095	−0.388210	−0.194105	+	0.980981i	$0.562180\pi$
$720$	0	0
$721$	23.0959	0.860137
$722$	0	0
$723$	44.8800	1.66911
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−39.5086	−1.46529	−0.732646	−	0.680610i	$-0.761715\pi$
$727$	−39.5086	−1.46529	−0.732646	+	0.680610i	$0.761715\pi$
$728$	0	0
$729$	−44.0345	−1.63091
$730$	0	0
$731$	−7.47839	−0.276598
$732$	0	0
$733$	−23.2658	−0.859343	−0.429672	−	0.902985i	$-0.641371\pi$
$733$	−23.2658	−0.859343	−0.429672	+	0.902985i	$0.641371\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.5105	−0.497665
$738$	0	0
$739$	−7.40919	−0.272551	−0.136276	−	0.990671i	$-0.543513\pi$
$739$	−7.40919	−0.272551	−0.136276	+	0.990671i	$0.543513\pi$
$740$	0	0
$741$	−17.3308	−0.636663
$742$	0	0
$743$	−45.7971	−1.68013	−0.840066	−	0.542484i	$-0.817484\pi$
$743$	−45.7971	−1.68013	−0.840066	+	0.542484i	$0.817484\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−45.9101	−1.67976
$748$	0	0
$749$	2.14895	0.0785209
$750$	0	0
$751$	−5.82537	−0.212571	−0.106285	−	0.994336i	$-0.533896\pi$
$751$	−5.82537	−0.212571	−0.106285	+	0.994336i	$0.533896\pi$
$752$	0	0
$753$	−4.37326	−0.159371
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−19.0463	−0.692251	−0.346125	−	0.938188i	$-0.612503\pi$
$757$	−19.0463	−0.692251	−0.346125	+	0.938188i	$0.612503\pi$
$758$	0	0
$759$	53.8440	1.95441
$760$	0	0
$761$	−5.41145	−0.196165	−0.0980824	−	0.995178i	$-0.531271\pi$
$761$	−5.41145	−0.196165	−0.0980824	+	0.995178i	$0.531271\pi$
$762$	0	0
$763$	18.4870	0.669274
$764$	0	0
$765$	0	0
$766$	0	0
$767$	70.0855	2.53064
$768$	0	0
$769$	−18.6113	−0.671141	−0.335571	−	0.942015i	$-0.608929\pi$
$769$	−18.6113	−0.671141	−0.335571	+	0.942015i	$0.608929\pi$
$770$	0	0
$771$	−50.0224	−1.80151
$772$	0	0
$773$	22.5113	0.809676	0.404838	−	0.914389i	$-0.367328\pi$
$773$	22.5113	0.809676	0.404838	+	0.914389i	$0.367328\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−60.3747	−2.16593
$778$	0	0
$779$	−10.8193	−0.387643
$780$	0	0
$781$	−30.1119	−1.07749
$782$	0	0
$783$	14.6671	0.524160
$784$	0	0
$785$	0	0
$786$	0	0
$787$	17.8799	0.637349	0.318675	−	0.947864i	$-0.396762\pi$
$787$	17.8799	0.637349	0.318675	+	0.947864i	$0.396762\pi$
$788$	0	0
$789$	−10.4534	−0.372151
$790$	0	0
$791$	7.08360	0.251864
$792$	0	0
$793$	27.3066	0.969687
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−46.6304	−1.65173	−0.825866	−	0.563866i	$-0.809313\pi$
$797$	−46.6304	−1.65173	−0.825866	+	0.563866i	$0.809313\pi$
$798$	0	0
$799$	49.6399	1.75613
$800$	0	0
$801$	−7.64959	−0.270285
$802$	0	0
$803$	−18.6650	−0.658674
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−26.2925	−0.925540
$808$	0	0
$809$	25.3264	0.890430	0.445215	−	0.895424i	$-0.353127\pi$
$809$	25.3264	0.890430	0.445215	+	0.895424i	$0.353127\pi$
$810$	0	0
$811$	10.6696	0.374660	0.187330	−	0.982297i	$-0.440017\pi$
$811$	10.6696	0.374660	0.187330	+	0.982297i	$0.440017\pi$
$812$	0	0
$813$	57.3856	2.01260
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.05217	0.0368109
$818$	0	0
$819$	−67.5631	−2.36085
$820$	0	0
$821$	−9.61676	−0.335627	−0.167814	−	0.985819i	$-0.553671\pi$
$821$	−9.61676	−0.335627	−0.167814	+	0.985819i	$0.553671\pi$
$822$	0	0
$823$	−43.3970	−1.51272	−0.756362	−	0.654154i	$-0.773025\pi$
$823$	−43.3970	−1.51272	−0.756362	+	0.654154i	$0.773025\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.1443	1.22209	0.611044	−	0.791597i	$-0.290750\pi$
$827$	35.1443	1.22209	0.611044	+	0.791597i	$0.290750\pi$
$828$	0	0
$829$	−20.0594	−0.696691	−0.348346	−	0.937366i	$-0.613256\pi$
$829$	−20.0594	−0.696691	−0.348346	+	0.937366i	$0.613256\pi$
$830$	0	0
$831$	28.8056	0.999256
$832$	0	0
$833$	11.8011	0.408884
$834$	0	0
$835$	0	0
$836$	0	0
$837$	46.1437	1.59496
$838$	0	0
$839$	−3.08979	−0.106671	−0.0533357	−	0.998577i	$-0.516985\pi$
$839$	−3.08979	−0.106671	−0.0533357	+	0.998577i	$0.516985\pi$
$840$	0	0
$841$	−18.9876	−0.654744
$842$	0	0
$843$	71.9939	2.47960
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−14.6347	−0.502853
$848$	0	0
$849$	48.8157	1.67535
$850$	0	0
$851$	84.8683	2.90925
$852$	0	0
$853$	26.3357	0.901718	0.450859	−	0.892595i	$-0.351118\pi$
$853$	26.3357	0.901718	0.450859	+	0.892595i	$0.351118\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.4774	0.836134	0.418067	−	0.908416i	$-0.362708\pi$
$857$	24.4774	0.836134	0.418067	+	0.908416i	$0.362708\pi$
$858$	0	0
$859$	−12.7601	−0.435370	−0.217685	−	0.976019i	$-0.569851\pi$
$859$	−12.7601	−0.435370	−0.217685	+	0.976019i	$0.569851\pi$
$860$	0	0
$861$	−69.2546	−2.36019
$862$	0	0
$863$	41.6878	1.41907	0.709534	−	0.704671i	$-0.248905\pi$
$863$	41.6878	1.41907	0.709534	+	0.704671i	$0.248905\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	92.8459	3.15321
$868$	0	0
$869$	−21.4678	−0.728245
$870$	0	0
$871$	−39.1283	−1.32581
$872$	0	0
$873$	−43.3683	−1.46779
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−24.3120	−0.820957	−0.410478	−	0.911870i	$-0.634638\pi$
$877$	−24.3120	−0.820957	−0.410478	+	0.911870i	$0.634638\pi$
$878$	0	0
$879$	48.7691	1.64494
$880$	0	0
$881$	14.0162	0.472216	0.236108	−	0.971727i	$-0.424128\pi$
$881$	14.0162	0.472216	0.236108	+	0.971727i	$0.424128\pi$
$882$	0	0
$883$	−46.0960	−1.55125	−0.775627	−	0.631191i	$-0.782566\pi$
$883$	−46.0960	−1.55125	−0.775627	+	0.631191i	$0.782566\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11.0211	−0.370054	−0.185027	−	0.982733i	$-0.559237\pi$
$887$	−11.0211	−0.370054	−0.185027	+	0.982733i	$0.559237\pi$
$888$	0	0
$889$	−30.5016	−1.02299
$890$	0	0
$891$	2.54898	0.0853940
$892$	0	0
$893$	−6.98410	−0.233714
$894$	0	0
$895$	0	0
$896$	0	0
$897$	155.940	5.20668
$898$	0	0
$899$	31.4997	1.05057
$900$	0	0
$901$	−19.1561	−0.638181
$902$	0	0
$903$	6.73497	0.224126
$904$	0	0
$905$	0	0
$906$	0	0
$907$	13.1795	0.437618	0.218809	−	0.975768i	$-0.429783\pi$
$907$	13.1795	0.437618	0.218809	+	0.975768i	$0.429783\pi$
$908$	0	0
$909$	46.1007	1.52906
$910$	0	0
$911$	−4.87590	−0.161546	−0.0807730	−	0.996733i	$-0.525739\pi$
$911$	−4.87590	−0.161546	−0.0807730	+	0.996733i	$0.525739\pi$
$912$	0	0
$913$	21.2221	0.702349
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.40369	0.0793767
$918$	0	0
$919$	16.4646	0.543119	0.271559	−	0.962422i	$-0.412461\pi$
$919$	16.4646	0.543119	0.271559	+	0.962422i	$0.412461\pi$
$920$	0	0
$921$	−52.9545	−1.74491
$922$	0	0
$923$	−87.2085	−2.87050
$924$	0	0
$925$	0	0
$926$	0	0
$927$	46.7096	1.53415
$928$	0	0
$929$	−5.45900	−0.179104	−0.0895520	−	0.995982i	$-0.528544\pi$
$929$	−5.45900	−0.179104	−0.0895520	+	0.995982i	$0.528544\pi$
$930$	0	0
$931$	−1.66036	−0.0544161
$932$	0	0
$933$	−15.1326	−0.495421
$934$	0	0
$935$	0	0
$936$	0	0
$937$	32.0799	1.04800	0.524002	−	0.851717i	$-0.324439\pi$
$937$	32.0799	1.04800	0.524002	+	0.851717i	$0.324439\pi$
$938$	0	0
$939$	2.84815	0.0929458
$940$	0	0
$941$	−21.2304	−0.692092	−0.346046	−	0.938218i	$-0.612476\pi$
$941$	−21.2304	−0.692092	−0.346046	+	0.938218i	$0.612476\pi$
$942$	0	0
$943$	97.3506	3.17017
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.96640	−0.0638995	−0.0319497	−	0.999489i	$-0.510172\pi$
$947$	−1.96640	−0.0638995	−0.0319497	+	0.999489i	$0.510172\pi$
$948$	0	0
$949$	−54.0566	−1.75475
$950$	0	0
$951$	94.0780	3.05069
$952$	0	0
$953$	41.9245	1.35807	0.679034	−	0.734106i	$-0.262399\pi$
$953$	41.9245	1.35807	0.679034	+	0.734106i	$0.262399\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−18.9352	−0.612086
$958$	0	0
$959$	−36.6495	−1.18347
$960$	0	0
$961$	68.1001	2.19678
$962$	0	0
$963$	4.34608	0.140050
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−0.270634	−0.00870299	−0.00435150	−	0.999991i	$-0.501385\pi$
$967$	−0.270634	−0.00870299	−0.00435150	+	0.999991i	$0.501385\pi$
$968$	0	0
$969$	−19.6885	−0.632486
$970$	0	0
$971$	10.9907	0.352709	0.176354	−	0.984327i	$-0.443570\pi$
$971$	10.9907	0.352709	0.176354	+	0.984327i	$0.443570\pi$
$972$	0	0
$973$	35.2081	1.12872
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28.9562	−0.926389	−0.463195	−	0.886257i	$-0.653297\pi$
$977$	−28.9562	−0.926389	−0.463195	+	0.886257i	$0.653297\pi$
$978$	0	0
$979$	3.53604	0.113012
$980$	0	0
$981$	37.3884	1.19372
$982$	0	0
$983$	−27.4563	−0.875720	−0.437860	−	0.899043i	$-0.644263\pi$
$983$	−27.4563	−0.875720	−0.437860	+	0.899043i	$0.644263\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−44.7053	−1.42298
$988$	0	0
$989$	−9.46730	−0.301043
$990$	0	0
$991$	−25.4904	−0.809729	−0.404864	−	0.914377i	$-0.632681\pi$
$991$	−25.4904	−0.809729	−0.404864	+	0.914377i	$0.632681\pi$
$992$	0	0
$993$	−2.74925	−0.0872447
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−10.5382	−0.333747	−0.166873	−	0.985978i	$-0.553367\pi$
$997$	−10.5382	−0.333747	−0.166873	+	0.985978i	$0.553367\pi$
$998$	0	0
$999$	−43.7202	−1.38325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.ch.1.6		6
4.3	odd	2		3800.2.a.bc.1.1	yes	6
5.4	even	2		7600.2.a.cl.1.1		6
20.3	even	4		3800.2.d.q.3649.2		12
20.7	even	4		3800.2.d.q.3649.11		12
20.19	odd	2		3800.2.a.ba.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.ba.1.6	✓	6	20.19	odd	2
3800.2.a.bc.1.1	yes	6	4.3	odd	2
3800.2.d.q.3649.2		12	20.3	even	4
3800.2.d.q.3649.11		12	20.7	even	4
7600.2.a.ch.1.6		6	1.1	even	1	trivial
7600.2.a.cl.1.1		6	5.4	even	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+2.77008 q^{3} +2.31077 q^{7} +4.67334 q^{9} +O(q^{10})\)	\(q+2.77008 q^{3} +2.31077 q^{7} +4.67334 q^{9} -2.16026 q^{11} -6.25643 q^{13} -7.10756 q^{17} +1.00000 q^{19} +6.40100 q^{21} -8.99784 q^{23} +4.63527 q^{27} +3.16424 q^{29} +9.95490 q^{31} -5.98410 q^{33} -9.43207 q^{37} -17.3308 q^{39} -10.8193 q^{41} +1.05217 q^{43} -6.98410 q^{47} -1.66036 q^{49} -19.6885 q^{51} +2.69517 q^{53} +2.77008 q^{57} -11.2021 q^{59} -4.36457 q^{61} +10.7990 q^{63} +6.25408 q^{67} -24.9247 q^{69} +13.9390 q^{71} +8.64016 q^{73} -4.99186 q^{77} +9.93758 q^{79} -1.17994 q^{81} -9.82384 q^{83} +8.76520 q^{87} -1.63686 q^{89} -14.4572 q^{91} +27.5759 q^{93} -9.27994 q^{97} -10.0956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9	\( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} - 3 q^{11} - q^{13} - 14 q^{17} + 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} - 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} - 33 q^{51} + 13 q^{53} - 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} - 19 q^{71} + 3 q^{73} - 36 q^{77} + 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} - 42 q^{91} + 39 q^{93} - 11 q^{97} + 6 q^{99}+O(q^{100}) \) 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

Introduction

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