LMFDB - Embedded newform 9675.2.a.ck.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9675,2,Mod(1,9675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9675 = 3^{2} \cdot 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9675.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:	$77.2552639556$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.282109865.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 11x^{4} + 8x^{3} + 34x^{2} - 12x - 27 $ x^6 - x^5 - 11x^4 + 8x^3 + 34x^2 - 12x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1075)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.920974$ of defining polynomial
Character	$\chi$	$=$	9675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.920974 q^{2} -1.15181 q^{4} -2.23083 q^{7} +2.90273 q^{8} +O(q^{10})$	$q-0.920974 q^{2} -1.15181 q^{4} -2.23083 q^{7} +2.90273 q^{8} +6.02758 q^{11} -0.827279 q^{13} +2.05454 q^{14} -0.369730 q^{16} +3.19701 q^{17} +1.19701 q^{19} -5.55124 q^{22} +5.21678 q^{23} +0.761903 q^{26} +2.56949 q^{28} -9.80965 q^{29} -6.20030 q^{31} -5.46495 q^{32} -2.94436 q^{34} +2.53033 q^{37} -1.10241 q^{38} -8.55689 q^{41} +1.00000 q^{43} -6.94260 q^{44} -4.80452 q^{46} -11.7549 q^{47} -2.02339 q^{49} +0.952865 q^{52} +8.72019 q^{53} -6.47551 q^{56} +9.03444 q^{58} -3.63808 q^{59} +4.61943 q^{61} +5.71032 q^{62} +5.77254 q^{64} +7.63693 q^{67} -3.68233 q^{68} -0.418505 q^{71} -9.88154 q^{73} -2.33037 q^{74} -1.37872 q^{76} -13.4465 q^{77} -10.3498 q^{79} +7.88067 q^{82} -7.74888 q^{83} -0.920974 q^{86} +17.4964 q^{88} +7.86050 q^{89} +1.84552 q^{91} -6.00872 q^{92} +10.8260 q^{94} -5.93400 q^{97} +1.86349 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + 11 q^{4} - 2 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + q^2 + 11 * q^4 - 2 * q^7 + 6 * q^8	$ 6 q + q^{2} + 11 q^{4} - 2 q^{7} + 6 q^{8} + 6 q^{11} - 17 q^{14} + 5 q^{16} + 7 q^{17} - 5 q^{19} - 34 q^{22} - 3 q^{23} - 14 q^{26} + 21 q^{28} - 18 q^{29} - 12 q^{31} + 3 q^{32} + q^{34} - 7 q^{37} - q^{38} + 6 q^{43} - 9 q^{44} - 14 q^{46} - 6 q^{47} - 3 q^{52} - 8 q^{53} - 20 q^{56} + 24 q^{58} - 21 q^{59} - 8 q^{61} + 47 q^{62} + 6 q^{64} - 35 q^{68} - 14 q^{71} + q^{73} - 2 q^{74} - 57 q^{76} + 13 q^{77} - 31 q^{79} - 32 q^{82} - 13 q^{83} + q^{86} - 39 q^{88} - 40 q^{89} + 11 q^{91} - 32 q^{92} + 35 q^{94} - 11 q^{97} - 44 q^{98}+O(q^{100}) $ 6 * q + q^2 + 11 * q^4 - 2 * q^7 + 6 * q^8 + 6 * q^11 - 17 * q^14 + 5 * q^16 + 7 * q^17 - 5 * q^19 - 34 * q^22 - 3 * q^23 - 14 * q^26 + 21 * q^28 - 18 * q^29 - 12 * q^31 + 3 * q^32 + q^34 - 7 * q^37 - q^38 + 6 * q^43 - 9 * q^44 - 14 * q^46 - 6 * q^47 - 3 * q^52 - 8 * q^53 - 20 * q^56 + 24 * q^58 - 21 * q^59 - 8 * q^61 + 47 * q^62 + 6 * q^64 - 35 * q^68 - 14 * q^71 + q^73 - 2 * q^74 - 57 * q^76 + 13 * q^77 - 31 * q^79 - 32 * q^82 - 13 * q^83 + q^86 - 39 * q^88 - 40 * q^89 + 11 * q^91 - 32 * q^92 + 35 * q^94 - 11 * q^97 - 44 * q^98

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 $

$2$	−0.920974	−0.651227	−0.325614	−	0.945503i	$-0.605571\pi$
$2$	−0.920974	−0.651227	−0.325614	+	0.945503i	$0.605571\pi$
$3$	0	0
$3$	0	0
$4$	−1.15181	−0.575903
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.23083	−0.843175	−0.421588	−	0.906788i	$-0.638527\pi$
$7$	−2.23083	−0.843175	−0.421588	+	0.906788i	$0.638527\pi$
$8$	2.90273	1.02627
$9$	0	0
$10$	0	0
$11$	6.02758	1.81738	0.908692	−	0.417468i	$-0.137082\pi$
$11$	6.02758	1.81738	0.908692	+	0.417468i	$0.137082\pi$
$12$	0	0
$13$	−0.827279	−0.229446	−0.114723	−	0.993398i	$-0.536598\pi$
$13$	−0.827279	−0.229446	−0.114723	+	0.993398i	$0.536598\pi$
$14$	2.05454	0.549099
$15$	0	0
$16$	−0.369730	−0.0924324
$17$	3.19701	0.775389	0.387694	−	0.921788i	$-0.373272\pi$
$17$	3.19701	0.775389	0.387694	+	0.921788i	$0.373272\pi$
$18$	0	0
$19$	1.19701	0.274613	0.137306	−	0.990529i	$-0.456156\pi$
$19$	1.19701	0.274613	0.137306	+	0.990529i	$0.456156\pi$
$20$	0	0
$21$	0	0
$22$	−5.55124	−1.18353
$23$	5.21678	1.08777	0.543887	−	0.839159i	$-0.316952\pi$
$23$	5.21678	1.08777	0.543887	+	0.839159i	$0.316952\pi$
$24$	0	0
$25$	0	0
$26$	0.761903	0.149421
$27$	0	0
$28$	2.56949	0.485587
$29$	−9.80965	−1.82161	−0.910804	−	0.412840i	$-0.864537\pi$
$29$	−9.80965	−1.82161	−0.910804	+	0.412840i	$0.864537\pi$
$30$	0	0
$31$	−6.20030	−1.11361	−0.556803	−	0.830644i	$-0.687972\pi$
$31$	−6.20030	−1.11361	−0.556803	+	0.830644i	$0.687972\pi$
$32$	−5.46495	−0.966076
$33$	0	0
$34$	−2.94436	−0.504954
$35$	0	0
$36$	0	0
$37$	2.53033	0.415984	0.207992	−	0.978131i	$-0.433307\pi$
$37$	2.53033	0.415984	0.207992	+	0.978131i	$0.433307\pi$
$38$	−1.10241	−0.178835
$39$	0	0
$40$	0	0
$41$	−8.55689	−1.33636	−0.668181	−	0.743999i	$-0.732927\pi$
$41$	−8.55689	−1.33636	−0.668181	+	0.743999i	$0.732927\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−6.94260	−1.04664
$45$	0	0
$46$	−4.80452	−0.708388
$47$	−11.7549	−1.71463	−0.857315	−	0.514791i	$-0.827869\pi$
$47$	−11.7549	−1.71463	−0.857315	+	0.514791i	$0.827869\pi$
$48$	0	0
$49$	−2.02339	−0.289056
$50$	0	0
$51$	0	0
$52$	0.952865	0.132139
$53$	8.72019	1.19781	0.598905	−	0.800820i	$-0.295603\pi$
$53$	8.72019	1.19781	0.598905	+	0.800820i	$0.295603\pi$
$54$	0	0
$55$	0	0
$56$	−6.47551	−0.865326
$57$	0	0
$58$	9.03444	1.18628
$59$	−3.63808	−0.473637	−0.236819	−	0.971554i	$-0.576105\pi$
$59$	−3.63808	−0.473637	−0.236819	+	0.971554i	$0.576105\pi$
$60$	0	0
$61$	4.61943	0.591458	0.295729	−	0.955272i	$-0.404438\pi$
$61$	4.61943	0.591458	0.295729	+	0.955272i	$0.404438\pi$
$62$	5.71032	0.725211
$63$	0	0
$64$	5.77254	0.721568
$65$	0	0
$66$	0	0
$67$	7.63693	0.933000	0.466500	−	0.884521i	$-0.345515\pi$
$67$	7.63693	0.933000	0.466500	+	0.884521i	$0.345515\pi$
$68$	−3.68233	−0.446549
$69$	0	0
$70$	0	0
$71$	−0.418505	−0.0496674	−0.0248337	−	0.999692i	$-0.507906\pi$
$71$	−0.418505	−0.0496674	−0.0248337	+	0.999692i	$0.507906\pi$
$72$	0	0
$73$	−9.88154	−1.15655	−0.578273	−	0.815843i	$-0.696273\pi$
$73$	−9.88154	−1.15655	−0.578273	+	0.815843i	$0.696273\pi$
$74$	−2.33037	−0.270900
$75$	0	0
$76$	−1.37872	−0.158150
$77$	−13.4465	−1.53237
$78$	0	0
$79$	−10.3498	−1.16445	−0.582224	−	0.813029i	$-0.697817\pi$
$79$	−10.3498	−1.16445	−0.582224	+	0.813029i	$0.697817\pi$
$80$	0	0
$81$	0	0
$82$	7.88067	0.870275
$83$	−7.74888	−0.850550	−0.425275	−	0.905064i	$-0.639823\pi$
$83$	−7.74888	−0.850550	−0.425275	+	0.905064i	$0.639823\pi$
$84$	0	0
$85$	0	0
$86$	−0.920974	−0.0993112
$87$	0	0
$88$	17.4964	1.86513
$89$	7.86050	0.833211	0.416606	−	0.909087i	$-0.363220\pi$
$89$	7.86050	0.833211	0.416606	+	0.909087i	$0.363220\pi$
$90$	0	0
$91$	1.84552	0.193463
$92$	−6.00872	−0.626452
$93$	0	0
$94$	10.8260	1.11661
$95$	0	0
$96$	0	0
$97$	−5.93400	−0.602507	−0.301253	−	0.953544i	$-0.597405\pi$
$97$	−5.93400	−0.602507	−0.301253	+	0.953544i	$0.597405\pi$
$98$	1.86349	0.188241
$99$	0	0
$100$	0	0
$101$	−4.16338	−0.414272	−0.207136	−	0.978312i	$-0.566414\pi$
$101$	−4.16338	−0.414272	−0.207136	+	0.978312i	$0.566414\pi$
$102$	0	0
$103$	6.70757	0.660916	0.330458	−	0.943821i	$-0.392797\pi$
$103$	6.70757	0.660916	0.330458	+	0.943821i	$0.392797\pi$
$104$	−2.40137	−0.235474
$105$	0	0
$106$	−8.03108	−0.780047
$107$	2.42160	0.234105	0.117052	−	0.993126i	$-0.462656\pi$
$107$	2.42160	0.234105	0.117052	+	0.993126i	$0.462656\pi$
$108$	0	0
$109$	−5.23781	−0.501692	−0.250846	−	0.968027i	$-0.580709\pi$
$109$	−5.23781	−0.501692	−0.250846	+	0.968027i	$0.580709\pi$
$110$	0	0
$111$	0	0
$112$	0.824805	0.0779367
$113$	16.9968	1.59893	0.799464	−	0.600713i	$-0.205117\pi$
$113$	16.9968	1.59893	0.799464	+	0.600713i	$0.205117\pi$
$114$	0	0
$115$	0	0
$116$	11.2988	1.04907
$117$	0	0
$118$	3.35058	0.308446
$119$	−7.13199	−0.653788
$120$	0	0
$121$	25.3317	2.30288
$122$	−4.25438	−0.385173
$123$	0	0
$124$	7.14154	0.641330
$125$	0	0
$126$	0	0
$127$	5.00267	0.443915	0.221958	−	0.975056i	$-0.428755\pi$
$127$	5.00267	0.443915	0.221958	+	0.975056i	$0.428755\pi$
$128$	5.61354	0.496172
$129$	0	0
$130$	0	0
$131$	3.58651	0.313355	0.156677	−	0.987650i	$-0.449922\pi$
$131$	3.58651	0.313355	0.156677	+	0.987650i	$0.449922\pi$
$132$	0	0
$133$	−2.67033	−0.231547
$134$	−7.03342	−0.607595
$135$	0	0
$136$	9.28006	0.795759
$137$	−5.88193	−0.502527	−0.251264	−	0.967919i	$-0.580846\pi$
$137$	−5.88193	−0.502527	−0.251264	+	0.967919i	$0.580846\pi$
$138$	0	0
$139$	3.84348	0.326000	0.163000	−	0.986626i	$-0.447883\pi$
$139$	3.84348	0.326000	0.163000	+	0.986626i	$0.447883\pi$
$140$	0	0
$141$	0	0
$142$	0.385432	0.0323447
$143$	−4.98649	−0.416991
$144$	0	0
$145$	0	0
$146$	9.10064	0.753174
$147$	0	0
$148$	−2.91445	−0.239566
$149$	17.7268	1.45224	0.726119	−	0.687569i	$-0.241322\pi$
$149$	17.7268	1.45224	0.726119	+	0.687569i	$0.241322\pi$
$150$	0	0
$151$	0.490789	0.0399398	0.0199699	−	0.999801i	$-0.493643\pi$
$151$	0.490789	0.0399398	0.0199699	+	0.999801i	$0.493643\pi$
$152$	3.47460	0.281827
$153$	0	0
$154$	12.3839	0.997923
$155$	0	0
$156$	0	0
$157$	2.49569	0.199177	0.0995887	−	0.995029i	$-0.468247\pi$
$157$	2.49569	0.199177	0.0995887	+	0.995029i	$0.468247\pi$
$158$	9.53193	0.758320
$159$	0	0
$160$	0	0
$161$	−11.6378	−0.917184
$162$	0	0
$163$	22.1544	1.73527	0.867634	−	0.497204i	$-0.165640\pi$
$163$	22.1544	1.73527	0.867634	+	0.497204i	$0.165640\pi$
$164$	9.85588	0.769615
$165$	0	0
$166$	7.13652	0.553902
$167$	−13.6238	−1.05424	−0.527120	−	0.849791i	$-0.676728\pi$
$167$	−13.6238	−1.05424	−0.527120	+	0.849791i	$0.676728\pi$
$168$	0	0
$169$	−12.3156	−0.947355
$170$	0	0
$171$	0	0
$172$	−1.15181	−0.0878244
$173$	−9.58487	−0.728724	−0.364362	−	0.931257i	$-0.618713\pi$
$173$	−9.58487	−0.728724	−0.364362	+	0.931257i	$0.618713\pi$
$174$	0	0
$175$	0	0
$176$	−2.22857	−0.167985
$177$	0	0
$178$	−7.23932	−0.542610
$179$	−22.7649	−1.70153	−0.850764	−	0.525549i	$-0.823860\pi$
$179$	−22.7649	−1.70153	−0.850764	+	0.525549i	$0.823860\pi$
$180$	0	0
$181$	1.04780	0.0778822	0.0389411	−	0.999242i	$-0.487602\pi$
$181$	1.04780	0.0778822	0.0389411	+	0.999242i	$0.487602\pi$
$182$	−1.69968	−0.125988
$183$	0	0
$184$	15.1429	1.11635
$185$	0	0
$186$	0	0
$187$	19.2702	1.40918
$188$	13.5394	0.987461
$189$	0	0
$190$	0	0
$191$	4.76445	0.344744	0.172372	−	0.985032i	$-0.444857\pi$
$191$	4.76445	0.344744	0.172372	+	0.985032i	$0.444857\pi$
$192$	0	0
$193$	8.50247	0.612021	0.306011	−	0.952028i	$-0.401006\pi$
$193$	8.50247	0.612021	0.306011	+	0.952028i	$0.401006\pi$
$194$	5.46507	0.392369
$195$	0	0
$196$	2.33055	0.166468
$197$	19.2227	1.36956	0.684779	−	0.728751i	$-0.259899\pi$
$197$	19.2227	1.36956	0.684779	+	0.728751i	$0.259899\pi$
$198$	0	0
$199$	−16.6617	−1.18112	−0.590559	−	0.806995i	$-0.701093\pi$
$199$	−16.6617	−1.18112	−0.590559	+	0.806995i	$0.701093\pi$
$200$	0	0
$201$	0	0
$202$	3.83437	0.269785
$203$	21.8837	1.53593
$204$	0	0
$205$	0	0
$206$	−6.17750	−0.430407
$207$	0	0
$208$	0.305870	0.0212082
$209$	7.21506	0.499076
$210$	0	0
$211$	−19.7154	−1.35726	−0.678631	−	0.734479i	$-0.737426\pi$
$211$	−19.7154	−1.35726	−0.678631	+	0.734479i	$0.737426\pi$
$212$	−10.0440	−0.689823
$213$	0	0
$214$	−2.23023	−0.152455
$215$	0	0
$216$	0	0
$217$	13.8318	0.938965
$218$	4.82389	0.326715
$219$	0	0
$220$	0	0
$221$	−2.64482	−0.177910
$222$	0	0
$223$	−23.6343	−1.58267	−0.791334	−	0.611385i	$-0.790613\pi$
$223$	−23.6343	−1.58267	−0.791334	+	0.611385i	$0.790613\pi$
$224$	12.1914	0.814572
$225$	0	0
$226$	−15.6537	−1.04127
$227$	−5.33292	−0.353958	−0.176979	−	0.984215i	$-0.556633\pi$
$227$	−5.33292	−0.353958	−0.176979	+	0.984215i	$0.556633\pi$
$228$	0	0
$229$	8.01915	0.529921	0.264960	−	0.964259i	$-0.414641\pi$
$229$	8.01915	0.529921	0.264960	+	0.964259i	$0.414641\pi$
$230$	0	0
$231$	0	0
$232$	−28.4748	−1.86946
$233$	−23.6181	−1.54727	−0.773637	−	0.633629i	$-0.781565\pi$
$233$	−23.6181	−1.54727	−0.773637	+	0.633629i	$0.781565\pi$
$234$	0	0
$235$	0	0
$236$	4.19036	0.272769
$237$	0	0
$238$	6.56838	0.425765
$239$	−8.41088	−0.544055	−0.272027	−	0.962290i	$-0.587694\pi$
$239$	−8.41088	−0.544055	−0.272027	+	0.962290i	$0.587694\pi$
$240$	0	0
$241$	−25.2640	−1.62740	−0.813698	−	0.581288i	$-0.802549\pi$
$241$	−25.2640	−1.62740	−0.813698	+	0.581288i	$0.802549\pi$
$242$	−23.3298	−1.49970
$243$	0	0
$244$	−5.32069	−0.340622
$245$	0	0
$246$	0	0
$247$	−0.990261	−0.0630088
$248$	−17.9978	−1.14286
$249$	0	0
$250$	0	0
$251$	4.07895	0.257461	0.128730	−	0.991680i	$-0.458910\pi$
$251$	4.07895	0.257461	0.128730	+	0.991680i	$0.458910\pi$
$252$	0	0
$253$	31.4445	1.97690
$254$	−4.60733	−0.289090
$255$	0	0
$256$	−16.7150	−1.04469
$257$	−17.7131	−1.10491	−0.552455	−	0.833543i	$-0.686309\pi$
$257$	−17.7131	−1.10491	−0.552455	+	0.833543i	$0.686309\pi$
$258$	0	0
$259$	−5.64474	−0.350747
$260$	0	0
$261$	0	0
$262$	−3.30308	−0.204065
$263$	7.12925	0.439608	0.219804	−	0.975544i	$-0.429458\pi$
$263$	7.12925	0.439608	0.219804	+	0.975544i	$0.429458\pi$
$264$	0	0
$265$	0	0
$266$	2.45930	0.150789
$267$	0	0
$268$	−8.79627	−0.537318
$269$	−28.4501	−1.73463	−0.867317	−	0.497756i	$-0.834158\pi$
$269$	−28.4501	−1.73463	−0.867317	+	0.497756i	$0.834158\pi$
$270$	0	0
$271$	5.50403	0.334346	0.167173	−	0.985928i	$-0.446536\pi$
$271$	5.50403	0.334346	0.167173	+	0.985928i	$0.446536\pi$
$272$	−1.18203	−0.0716711
$273$	0	0
$274$	5.41711	0.327259
$275$	0	0
$276$	0	0
$277$	2.58762	0.155475	0.0777374	−	0.996974i	$-0.475230\pi$
$277$	2.58762	0.155475	0.0777374	+	0.996974i	$0.475230\pi$
$278$	−3.53974	−0.212300
$279$	0	0
$280$	0	0
$281$	20.6194	1.23005	0.615026	−	0.788506i	$-0.289145\pi$
$281$	20.6194	1.23005	0.615026	+	0.788506i	$0.289145\pi$
$282$	0	0
$283$	10.3853	0.617342	0.308671	−	0.951169i	$-0.400116\pi$
$283$	10.3853	0.617342	0.308671	+	0.951169i	$0.400116\pi$
$284$	0.482036	0.0286036
$285$	0	0
$286$	4.59243	0.271556
$287$	19.0890	1.12679
$288$	0	0
$289$	−6.77913	−0.398773
$290$	0	0
$291$	0	0
$292$	11.3816	0.666059
$293$	−15.2912	−0.893322	−0.446661	−	0.894703i	$-0.647387\pi$
$293$	−15.2912	−0.893322	−0.446661	+	0.894703i	$0.647387\pi$
$294$	0	0
$295$	0	0
$296$	7.34487	0.426912
$297$	0	0
$298$	−16.3260	−0.945737
$299$	−4.31573	−0.249585
$300$	0	0
$301$	−2.23083	−0.128583
$302$	−0.452004	−0.0260099
$303$	0	0
$304$	−0.442570	−0.0253831
$305$	0	0
$306$	0	0
$307$	21.6017	1.23287	0.616437	−	0.787404i	$-0.288576\pi$
$307$	21.6017	1.23287	0.616437	+	0.787404i	$0.288576\pi$
$308$	15.4878	0.882498
$309$	0	0
$310$	0	0
$311$	−29.3890	−1.66650	−0.833249	−	0.552898i	$-0.813522\pi$
$311$	−29.3890	−1.66650	−0.833249	+	0.552898i	$0.813522\pi$
$312$	0	0
$313$	7.88896	0.445911	0.222955	−	0.974829i	$-0.428430\pi$
$313$	7.88896	0.445911	0.222955	+	0.974829i	$0.428430\pi$
$314$	−2.29846	−0.129710
$315$	0	0
$316$	11.9210	0.670609
$317$	1.10994	0.0623404	0.0311702	−	0.999514i	$-0.490077\pi$
$317$	1.10994	0.0623404	0.0311702	+	0.999514i	$0.490077\pi$
$318$	0	0
$319$	−59.1285	−3.31056
$320$	0	0
$321$	0	0
$322$	10.7181	0.597295
$323$	3.82685	0.212932
$324$	0	0
$325$	0	0
$326$	−20.4036	−1.13005
$327$	0	0
$328$	−24.8384	−1.37147
$329$	26.2232	1.44573
$330$	0	0
$331$	−10.9028	−0.599270	−0.299635	−	0.954054i	$-0.596865\pi$
$331$	−10.9028	−0.599270	−0.299635	+	0.954054i	$0.596865\pi$
$332$	8.92521	0.489835
$333$	0	0
$334$	12.5472	0.686550
$335$	0	0
$336$	0	0
$337$	32.6494	1.77853	0.889263	−	0.457397i	$-0.151218\pi$
$337$	32.6494	1.77853	0.889263	+	0.457397i	$0.151218\pi$
$338$	11.3424	0.616943
$339$	0	0
$340$	0	0
$341$	−37.3728	−2.02385
$342$	0	0
$343$	20.1297	1.08690
$344$	2.90273	0.156505
$345$	0	0
$346$	8.82742	0.474565
$347$	−6.75915	−0.362850	−0.181425	−	0.983405i	$-0.558071\pi$
$347$	−6.75915	−0.362850	−0.181425	+	0.983405i	$0.558071\pi$
$348$	0	0
$349$	33.5638	1.79663	0.898313	−	0.439355i	$-0.144793\pi$
$349$	33.5638	1.79663	0.898313	+	0.439355i	$0.144793\pi$
$350$	0	0
$351$	0	0
$352$	−32.9404	−1.75573
$353$	−33.8726	−1.80286	−0.901429	−	0.432928i	$-0.857481\pi$
$353$	−33.8726	−1.80286	−0.901429	+	0.432928i	$0.857481\pi$
$354$	0	0
$355$	0	0
$356$	−9.05377	−0.479849
$357$	0	0
$358$	20.9659	1.10808
$359$	−25.6258	−1.35248	−0.676238	−	0.736683i	$-0.736391\pi$
$359$	−25.6258	−1.35248	−0.676238	+	0.736683i	$0.736391\pi$
$360$	0	0
$361$	−17.5672	−0.924588
$362$	−0.964995	−0.0507190
$363$	0	0
$364$	−2.12568	−0.111416
$365$	0	0
$366$	0	0
$367$	−3.64783	−0.190415	−0.0952077	−	0.995457i	$-0.530352\pi$
$367$	−3.64783	−0.190415	−0.0952077	+	0.995457i	$0.530352\pi$
$368$	−1.92880	−0.100546
$369$	0	0
$370$	0	0
$371$	−19.4533	−1.00996
$372$	0	0
$373$	17.3176	0.896669	0.448335	−	0.893866i	$-0.352017\pi$
$373$	17.3176	0.896669	0.448335	+	0.893866i	$0.352017\pi$
$374$	−17.7474	−0.917695
$375$	0	0
$376$	−34.1214	−1.75968
$377$	8.11532	0.417960
$378$	0	0
$379$	35.4068	1.81873	0.909363	−	0.416003i	$-0.136569\pi$
$379$	35.4068	1.81873	0.909363	+	0.416003i	$0.136569\pi$
$380$	0	0
$381$	0	0
$382$	−4.38794	−0.224506
$383$	15.7587	0.805234	0.402617	−	0.915369i	$-0.368101\pi$
$383$	15.7587	0.805234	0.402617	+	0.915369i	$0.368101\pi$
$384$	0	0
$385$	0	0
$386$	−7.83056	−0.398565
$387$	0	0
$388$	6.83482	0.346986
$389$	28.3444	1.43712	0.718559	−	0.695466i	$-0.244802\pi$
$389$	28.3444	1.43712	0.718559	+	0.695466i	$0.244802\pi$
$390$	0	0
$391$	16.6781	0.843447
$392$	−5.87336	−0.296649
$393$	0	0
$394$	−17.7036	−0.891894
$395$	0	0
$396$	0	0
$397$	−22.3142	−1.11992	−0.559959	−	0.828521i	$-0.689183\pi$
$397$	−22.3142	−1.11992	−0.559959	+	0.828521i	$0.689183\pi$
$398$	15.3450	0.769176
$399$	0	0
$400$	0	0
$401$	−21.2383	−1.06059	−0.530295	−	0.847813i	$-0.677919\pi$
$401$	−21.2383	−1.06059	−0.530295	+	0.847813i	$0.677919\pi$
$402$	0	0
$403$	5.12938	0.255513
$404$	4.79541	0.238581
$405$	0	0
$406$	−20.1543	−1.00024
$407$	15.2518	0.756002
$408$	0	0
$409$	−1.54546	−0.0764180	−0.0382090	−	0.999270i	$-0.512165\pi$
$409$	−1.54546	−0.0764180	−0.0382090	+	0.999270i	$0.512165\pi$
$410$	0	0
$411$	0	0
$412$	−7.72582	−0.380624
$413$	8.11594	0.399359
$414$	0	0
$415$	0	0
$416$	4.52104	0.221662
$417$	0	0
$418$	−6.64489	−0.325012
$419$	9.37059	0.457783	0.228892	−	0.973452i	$-0.426490\pi$
$419$	9.37059	0.457783	0.228892	+	0.973452i	$0.426490\pi$
$420$	0	0
$421$	−9.15495	−0.446185	−0.223092	−	0.974797i	$-0.571615\pi$
$421$	−9.15495	−0.446185	−0.223092	+	0.974797i	$0.571615\pi$
$422$	18.1573	0.883886
$423$	0	0
$424$	25.3124	1.22928
$425$	0	0
$426$	0	0
$427$	−10.3052	−0.498703
$428$	−2.78921	−0.134822
$429$	0	0
$430$	0	0
$431$	−3.00769	−0.144875	−0.0724376	−	0.997373i	$-0.523078\pi$
$431$	−3.00769	−0.144875	−0.0724376	+	0.997373i	$0.523078\pi$
$432$	0	0
$433$	0.799530	0.0384230	0.0192115	−	0.999815i	$-0.493884\pi$
$433$	0.799530	0.0384230	0.0192115	+	0.999815i	$0.493884\pi$
$434$	−12.7388	−0.611480
$435$	0	0
$436$	6.03295	0.288926
$437$	6.24453	0.298716
$438$	0	0
$439$	−36.1043	−1.72317	−0.861583	−	0.507617i	$-0.830527\pi$
$439$	−36.1043	−1.72317	−0.861583	+	0.507617i	$0.830527\pi$
$440$	0	0
$441$	0	0
$442$	2.43581	0.115860
$443$	9.06007	0.430457	0.215228	−	0.976564i	$-0.430950\pi$
$443$	9.06007	0.430457	0.215228	+	0.976564i	$0.430950\pi$
$444$	0	0
$445$	0	0
$446$	21.7665	1.03068
$447$	0	0
$448$	−12.8776	−0.608408
$449$	6.21426	0.293269	0.146635	−	0.989191i	$-0.453156\pi$
$449$	6.21426	0.293269	0.146635	+	0.989191i	$0.453156\pi$
$450$	0	0
$451$	−51.5773	−2.42868
$452$	−19.5771	−0.920828
$453$	0	0
$454$	4.91148	0.230507
$455$	0	0
$456$	0	0
$457$	−12.8856	−0.602763	−0.301381	−	0.953504i	$-0.597448\pi$
$457$	−12.8856	−0.602763	−0.301381	+	0.953504i	$0.597448\pi$
$458$	−7.38543	−0.345099
$459$	0	0
$460$	0	0
$461$	−31.2002	−1.45314	−0.726568	−	0.687095i	$-0.758886\pi$
$461$	−31.2002	−1.45314	−0.726568	+	0.687095i	$0.758886\pi$
$462$	0	0
$463$	−12.7993	−0.594832	−0.297416	−	0.954748i	$-0.596125\pi$
$463$	−12.7993	−0.594832	−0.297416	+	0.954748i	$0.596125\pi$
$464$	3.62692	0.168376
$465$	0	0
$466$	21.7517	1.00763
$467$	−16.9429	−0.784024	−0.392012	−	0.919960i	$-0.628221\pi$
$467$	−16.9429	−0.784024	−0.392012	+	0.919960i	$0.628221\pi$
$468$	0	0
$469$	−17.0367	−0.786682
$470$	0	0
$471$	0	0
$472$	−10.5604	−0.486080
$473$	6.02758	0.277148
$474$	0	0
$475$	0	0
$476$	8.21467	0.376519
$477$	0	0
$478$	7.74620	0.354303
$479$	−16.8246	−0.768735	−0.384368	−	0.923180i	$-0.625580\pi$
$479$	−16.8246	−0.768735	−0.384368	+	0.923180i	$0.625580\pi$
$480$	0	0
$481$	−2.09329	−0.0954458
$482$	23.2675	1.05980
$483$	0	0
$484$	−29.1772	−1.32624
$485$	0	0
$486$	0	0
$487$	−20.9673	−0.950119	−0.475060	−	0.879954i	$-0.657573\pi$
$487$	−20.9673	−0.950119	−0.475060	+	0.879954i	$0.657573\pi$
$488$	13.4090	0.606996
$489$	0	0
$490$	0	0
$491$	1.84207	0.0831316	0.0415658	−	0.999136i	$-0.486765\pi$
$491$	1.84207	0.0831316	0.0415658	+	0.999136i	$0.486765\pi$
$492$	0	0
$493$	−31.3616	−1.41245
$494$	0.912005	0.0410330
$495$	0	0
$496$	2.29243	0.102933
$497$	0.933613	0.0418783
$498$	0	0
$499$	−20.2655	−0.907207	−0.453604	−	0.891204i	$-0.649862\pi$
$499$	−20.2655	−0.907207	−0.453604	+	0.891204i	$0.649862\pi$
$500$	0	0
$501$	0	0
$502$	−3.75661	−0.167666
$503$	0.0128212	0.000571667 0	0.000285834	−	1.00000i	$-0.499909\pi$
$503$	0.0128212	0.000571667 0	0.000285834		1.00000i	$0.499909\pi$
$504$	0	0
$505$	0	0
$506$	−28.9596	−1.28741
$507$	0	0
$508$	−5.76211	−0.255652
$509$	−34.4372	−1.52640	−0.763201	−	0.646162i	$-0.776373\pi$
$509$	−34.4372	−1.52640	−0.763201	+	0.646162i	$0.776373\pi$
$510$	0	0
$511$	22.0440	0.975171
$512$	4.16701	0.184158
$513$	0	0
$514$	16.3133	0.719547
$515$	0	0
$516$	0	0
$517$	−70.8537	−3.11614
$518$	5.19866	0.228416
$519$	0	0
$520$	0	0
$521$	−11.5077	−0.504159	−0.252080	−	0.967706i	$-0.581114\pi$
$521$	−11.5077	−0.504159	−0.252080	+	0.967706i	$0.581114\pi$
$522$	0	0
$523$	−2.17138	−0.0949477	−0.0474739	−	0.998872i	$-0.515117\pi$
$523$	−2.17138	−0.0949477	−0.0474739	+	0.998872i	$0.515117\pi$
$524$	−4.13096	−0.180462
$525$	0	0
$526$	−6.56586	−0.286285
$527$	−19.8224	−0.863478
$528$	0	0
$529$	4.21479	0.183252
$530$	0	0
$531$	0	0
$532$	3.07570	0.133348
$533$	7.07894	0.306623
$534$	0	0
$535$	0	0
$536$	22.1680	0.957511
$537$	0	0
$538$	26.2018	1.12964
$539$	−12.1961	−0.525325
$540$	0	0
$541$	20.3867	0.876493	0.438246	−	0.898855i	$-0.355600\pi$
$541$	20.3867	0.876493	0.438246	+	0.898855i	$0.355600\pi$
$542$	−5.06907	−0.217735
$543$	0	0
$544$	−17.4715	−0.749085
$545$	0	0
$546$	0	0
$547$	−8.61617	−0.368401	−0.184200	−	0.982889i	$-0.558970\pi$
$547$	−8.61617	−0.368401	−0.184200	+	0.982889i	$0.558970\pi$
$548$	6.77484	0.289407
$549$	0	0
$550$	0	0
$551$	−11.7422	−0.500236
$552$	0	0
$553$	23.0887	0.981833
$554$	−2.38313	−0.101249
$555$	0	0
$556$	−4.42694	−0.187744
$557$	0.175038	0.00741659	0.00370829	−	0.999993i	$-0.498820\pi$
$557$	0.175038	0.00741659	0.00370829	+	0.999993i	$0.498820\pi$
$558$	0	0
$559$	−0.827279	−0.0349902
$560$	0	0
$561$	0	0
$562$	−18.9900	−0.801044
$563$	27.6675	1.16605	0.583024	−	0.812455i	$-0.301869\pi$
$563$	27.6675	1.16605	0.583024	+	0.812455i	$0.301869\pi$
$564$	0	0
$565$	0	0
$566$	−9.56460	−0.402030
$567$	0	0
$568$	−1.21481	−0.0509722
$569$	25.5683	1.07188	0.535939	−	0.844257i	$-0.319958\pi$
$569$	25.5683	1.07188	0.535939	+	0.844257i	$0.319958\pi$
$570$	0	0
$571$	21.9812	0.919883	0.459941	−	0.887949i	$-0.347870\pi$
$571$	21.9812	0.919883	0.459941	+	0.887949i	$0.347870\pi$
$572$	5.74347	0.240147
$573$	0	0
$574$	−17.5805	−0.733794
$575$	0	0
$576$	0	0
$577$	17.0335	0.709112	0.354556	−	0.935035i	$-0.384632\pi$
$577$	17.0335	0.709112	0.354556	+	0.935035i	$0.384632\pi$
$578$	6.24341	0.259692
$579$	0	0
$580$	0	0
$581$	17.2865	0.717163
$582$	0	0
$583$	52.5617	2.17688
$584$	−28.6835	−1.18693
$585$	0	0
$586$	14.0828	0.581756
$587$	−37.8869	−1.56376	−0.781879	−	0.623430i	$-0.785739\pi$
$587$	−37.8869	−1.56376	−0.781879	+	0.623430i	$0.785739\pi$
$588$	0	0
$589$	−7.42181	−0.305810
$590$	0	0
$591$	0	0
$592$	−0.935538	−0.0384504
$593$	16.8345	0.691310	0.345655	−	0.938362i	$-0.387657\pi$
$593$	16.8345	0.691310	0.345655	+	0.938362i	$0.387657\pi$
$594$	0	0
$595$	0	0
$596$	−20.4179	−0.836348
$597$	0	0
$598$	3.97468	0.162537
$599$	−13.2862	−0.542860	−0.271430	−	0.962458i	$-0.587497\pi$
$599$	−13.2862	−0.542860	−0.271430	+	0.962458i	$0.587497\pi$
$600$	0	0
$601$	19.0981	0.779028	0.389514	−	0.921021i	$-0.372643\pi$
$601$	19.0981	0.779028	0.389514	+	0.921021i	$0.372643\pi$
$602$	2.05454	0.0837368
$603$	0	0
$604$	−0.565294	−0.0230015
$605$	0	0
$606$	0	0
$607$	12.6302	0.512643	0.256322	−	0.966592i	$-0.417489\pi$
$607$	12.6302	0.512643	0.256322	+	0.966592i	$0.417489\pi$
$608$	−6.54160	−0.265297
$609$	0	0
$610$	0	0
$611$	9.72460	0.393415
$612$	0	0
$613$	0.918115	0.0370823	0.0185412	−	0.999828i	$-0.494098\pi$
$613$	0.918115	0.0370823	0.0185412	+	0.999828i	$0.494098\pi$
$614$	−19.8946	−0.802881
$615$	0	0
$616$	−39.0316	−1.57263
$617$	10.2963	0.414513	0.207256	−	0.978287i	$-0.433547\pi$
$617$	10.2963	0.414513	0.207256	+	0.978287i	$0.433547\pi$
$618$	0	0
$619$	19.5465	0.785641	0.392820	−	0.919615i	$-0.371499\pi$
$619$	19.5465	0.785641	0.392820	+	0.919615i	$0.371499\pi$
$620$	0	0
$621$	0	0
$622$	27.0665	1.08527
$623$	−17.5355	−0.702543
$624$	0	0
$625$	0	0
$626$	−7.26553	−0.290389
$627$	0	0
$628$	−2.87455	−0.114707
$629$	8.08949	0.322549
$630$	0	0
$631$	−4.32943	−0.172352	−0.0861759	−	0.996280i	$-0.527465\pi$
$631$	−4.32943	−0.172352	−0.0861759	+	0.996280i	$0.527465\pi$
$632$	−30.0428	−1.19504
$633$	0	0
$634$	−1.02223	−0.0405978
$635$	0	0
$636$	0	0
$637$	1.67391	0.0663226
$638$	54.4558	2.15593
$639$	0	0
$640$	0	0
$641$	32.5697	1.28642	0.643212	−	0.765688i	$-0.277601\pi$
$641$	32.5697	1.28642	0.643212	+	0.765688i	$0.277601\pi$
$642$	0	0
$643$	−42.9291	−1.69296	−0.846479	−	0.532421i	$-0.821282\pi$
$643$	−42.9291	−1.69296	−0.846479	+	0.532421i	$0.821282\pi$
$644$	13.4044	0.528209
$645$	0	0
$646$	−3.52443	−0.138667
$647$	1.74432	0.0685765	0.0342882	−	0.999412i	$-0.489084\pi$
$647$	1.74432	0.0685765	0.0342882	+	0.999412i	$0.489084\pi$
$648$	0	0
$649$	−21.9288	−0.860781
$650$	0	0
$651$	0	0
$652$	−25.5176	−0.999346
$653$	5.07029	0.198416	0.0992078	−	0.995067i	$-0.468369\pi$
$653$	5.07029	0.198416	0.0992078	+	0.995067i	$0.468369\pi$
$654$	0	0
$655$	0	0
$656$	3.16374	0.123523
$657$	0	0
$658$	−24.1509	−0.941501
$659$	15.7173	0.612260	0.306130	−	0.951990i	$-0.400966\pi$
$659$	15.7173	0.612260	0.306130	+	0.951990i	$0.400966\pi$
$660$	0	0
$661$	26.5033	1.03086	0.515430	−	0.856932i	$-0.327632\pi$
$661$	26.5033	1.03086	0.515430	+	0.856932i	$0.327632\pi$
$662$	10.0412	0.390261
$663$	0	0
$664$	−22.4929	−0.872895
$665$	0	0
$666$	0	0
$667$	−51.1748	−1.98150
$668$	15.6920	0.607140
$669$	0	0
$670$	0	0
$671$	27.8440	1.07491
$672$	0	0
$673$	−34.4298	−1.32717	−0.663586	−	0.748100i	$-0.730966\pi$
$673$	−34.4298	−1.32717	−0.663586	+	0.748100i	$0.730966\pi$
$674$	−30.0692	−1.15822
$675$	0	0
$676$	14.1852	0.545584
$677$	25.5650	0.982542	0.491271	−	0.871007i	$-0.336532\pi$
$677$	25.5650	0.982542	0.491271	+	0.871007i	$0.336532\pi$
$678$	0	0
$679$	13.2378	0.508019
$680$	0	0
$681$	0	0
$682$	34.4194	1.31799
$683$	1.47320	0.0563705	0.0281852	−	0.999603i	$-0.491027\pi$
$683$	1.47320	0.0563705	0.0281852	+	0.999603i	$0.491027\pi$
$684$	0	0
$685$	0	0
$686$	−18.5389	−0.707819
$687$	0	0
$688$	−0.369730	−0.0140958
$689$	−7.21404	−0.274833
$690$	0	0
$691$	−20.1606	−0.766945	−0.383472	−	0.923552i	$-0.625272\pi$
$691$	−20.1606	−0.766945	−0.383472	+	0.923552i	$0.625272\pi$
$692$	11.0399	0.419674
$693$	0	0
$694$	6.22500	0.236298
$695$	0	0
$696$	0	0
$697$	−27.3564	−1.03620
$698$	−30.9114	−1.17001
$699$	0	0
$700$	0	0
$701$	−50.6488	−1.91298	−0.956489	−	0.291767i	$-0.905757\pi$
$701$	−50.6488	−1.91298	−0.956489	+	0.291767i	$0.905757\pi$
$702$	0	0
$703$	3.02883	0.114234
$704$	34.7944	1.31136
$705$	0	0
$706$	31.1958	1.17407
$707$	9.28781	0.349304
$708$	0	0
$709$	−1.50121	−0.0563793	−0.0281896	−	0.999603i	$-0.508974\pi$
$709$	−1.50121	−0.0563793	−0.0281896	+	0.999603i	$0.508974\pi$
$710$	0	0
$711$	0	0
$712$	22.8169	0.855101
$713$	−32.3456	−1.21135
$714$	0	0
$715$	0	0
$716$	26.2207	0.979915
$717$	0	0
$718$	23.6007	0.880770
$719$	−10.0122	−0.373393	−0.186696	−	0.982418i	$-0.559778\pi$
$719$	−10.0122	−0.373393	−0.186696	+	0.982418i	$0.559778\pi$
$720$	0	0
$721$	−14.9635	−0.557268
$722$	16.1789	0.602117
$723$	0	0
$724$	−1.20686	−0.0448526
$725$	0	0
$726$	0	0
$727$	8.61092	0.319361	0.159681	−	0.987169i	$-0.448954\pi$
$727$	8.61092	0.319361	0.159681	+	0.987169i	$0.448954\pi$
$728$	5.35705	0.198546
$729$	0	0
$730$	0	0
$731$	3.19701	0.118246
$732$	0	0
$733$	−2.35205	−0.0868749	−0.0434375	−	0.999056i	$-0.513831\pi$
$733$	−2.35205	−0.0868749	−0.0434375	+	0.999056i	$0.513831\pi$
$734$	3.35956	0.124004
$735$	0	0
$736$	−28.5095	−1.05087
$737$	46.0322	1.69562
$738$	0	0
$739$	19.5242	0.718208	0.359104	−	0.933297i	$-0.383082\pi$
$739$	19.5242	0.718208	0.359104	+	0.933297i	$0.383082\pi$
$740$	0	0
$741$	0	0
$742$	17.9160	0.657716
$743$	10.4332	0.382758	0.191379	−	0.981516i	$-0.438704\pi$
$743$	10.4332	0.382758	0.191379	+	0.981516i	$0.438704\pi$
$744$	0	0
$745$	0	0
$746$	−15.9490	−0.583936
$747$	0	0
$748$	−22.1956	−0.811550
$749$	−5.40217	−0.197391
$750$	0	0
$751$	24.3185	0.887395	0.443698	−	0.896177i	$-0.353666\pi$
$751$	24.3185	0.887395	0.443698	+	0.896177i	$0.353666\pi$
$752$	4.34614	0.158488
$753$	0	0
$754$	−7.47400	−0.272187
$755$	0	0
$756$	0	0
$757$	−21.2779	−0.773360	−0.386680	−	0.922214i	$-0.626378\pi$
$757$	−21.2779	−0.773360	−0.386680	+	0.922214i	$0.626378\pi$
$758$	−32.6088	−1.18440
$759$	0	0
$760$	0	0
$761$	−34.5183	−1.25129	−0.625644	−	0.780109i	$-0.715164\pi$
$761$	−34.5183	−1.25129	−0.625644	+	0.780109i	$0.715164\pi$
$762$	0	0
$763$	11.6847	0.423014
$764$	−5.48773	−0.198539
$765$	0	0
$766$	−14.5134	−0.524390
$767$	3.00971	0.108674
$768$	0	0
$769$	−14.9892	−0.540524	−0.270262	−	0.962787i	$-0.587110\pi$
$769$	−14.9892	−0.540524	−0.270262	+	0.962787i	$0.587110\pi$
$770$	0	0
$771$	0	0
$772$	−9.79320	−0.352465
$773$	37.7719	1.35856	0.679281	−	0.733878i	$-0.262292\pi$
$773$	37.7719	1.35856	0.679281	+	0.733878i	$0.262292\pi$
$774$	0	0
$775$	0	0
$776$	−17.2248	−0.618335
$777$	0	0
$778$	−26.1045	−0.935891
$779$	−10.2427	−0.366982
$780$	0	0
$781$	−2.52257	−0.0902646
$782$	−15.3601	−0.549276
$783$	0	0
$784$	0.748107	0.0267181
$785$	0	0
$786$	0	0
$787$	27.7105	0.987771	0.493886	−	0.869527i	$-0.335576\pi$
$787$	27.7105	0.987771	0.493886	+	0.869527i	$0.335576\pi$
$788$	−22.1408	−0.788733
$789$	0	0
$790$	0	0
$791$	−37.9171	−1.34818
$792$	0	0
$793$	−3.82156	−0.135708
$794$	20.5508	0.729321
$795$	0	0
$796$	19.1911	0.680209
$797$	−43.2267	−1.53117	−0.765584	−	0.643336i	$-0.777550\pi$
$797$	−43.2267	−1.53117	−0.765584	+	0.643336i	$0.777550\pi$
$798$	0	0
$799$	−37.5806	−1.32951
$800$	0	0
$801$	0	0
$802$	19.5599	0.690685
$803$	−59.5617	−2.10189
$804$	0	0
$805$	0	0
$806$	−4.72403	−0.166397
$807$	0	0
$808$	−12.0852	−0.425156
$809$	−4.09860	−0.144099	−0.0720496	−	0.997401i	$-0.522954\pi$
$809$	−4.09860	−0.144099	−0.0720496	+	0.997401i	$0.522954\pi$
$810$	0	0
$811$	40.7497	1.43092	0.715459	−	0.698655i	$-0.246218\pi$
$811$	40.7497	1.43092	0.715459	+	0.698655i	$0.246218\pi$
$812$	−25.2058	−0.884549
$813$	0	0
$814$	−14.0465	−0.492329
$815$	0	0
$816$	0	0
$817$	1.19701	0.0418780
$818$	1.42333	0.0497655
$819$	0	0
$820$	0	0
$821$	16.7905	0.585994	0.292997	−	0.956113i	$-0.405347\pi$
$821$	16.7905	0.585994	0.292997	+	0.956113i	$0.405347\pi$
$822$	0	0
$823$	−15.8303	−0.551810	−0.275905	−	0.961185i	$-0.588978\pi$
$823$	−15.8303	−0.551810	−0.275905	+	0.961185i	$0.588978\pi$
$824$	19.4703	0.678279
$825$	0	0
$826$	−7.47457	−0.260074
$827$	13.5789	0.472185	0.236092	−	0.971731i	$-0.424133\pi$
$827$	13.5789	0.472185	0.236092	+	0.971731i	$0.424133\pi$
$828$	0	0
$829$	−7.01805	−0.243747	−0.121874	−	0.992546i	$-0.538890\pi$
$829$	−7.01805	−0.243747	−0.121874	+	0.992546i	$0.538890\pi$
$830$	0	0
$831$	0	0
$832$	−4.77550	−0.165561
$833$	−6.46879	−0.224130
$834$	0	0
$835$	0	0
$836$	−8.31036	−0.287420
$837$	0	0
$838$	−8.63007	−0.298121
$839$	17.2572	0.595786	0.297893	−	0.954599i	$-0.403716\pi$
$839$	17.2572	0.595786	0.297893	+	0.954599i	$0.403716\pi$
$840$	0	0
$841$	67.2293	2.31825
$842$	8.43147	0.290568
$843$	0	0
$844$	22.7083	0.781651
$845$	0	0
$846$	0	0
$847$	−56.5108	−1.94173
$848$	−3.22412	−0.110717
$849$	0	0
$850$	0	0
$851$	13.2002	0.452496
$852$	0	0
$853$	23.9573	0.820283	0.410142	−	0.912022i	$-0.365479\pi$
$853$	23.9573	0.820283	0.410142	+	0.912022i	$0.365479\pi$
$854$	9.49080	0.324769
$855$	0	0
$856$	7.02925	0.240255
$857$	−24.8654	−0.849387	−0.424694	−	0.905337i	$-0.639618\pi$
$857$	−24.8654	−0.849387	−0.424694	+	0.905337i	$0.639618\pi$
$858$	0	0
$859$	36.9039	1.25914	0.629572	−	0.776942i	$-0.283230\pi$
$859$	36.9039	1.25914	0.629572	+	0.776942i	$0.283230\pi$
$860$	0	0
$861$	0	0
$862$	2.77000	0.0943466
$863$	−21.3201	−0.725743	−0.362872	−	0.931839i	$-0.618204\pi$
$863$	−21.3201	−0.725743	−0.362872	+	0.931839i	$0.618204\pi$
$864$	0	0
$865$	0	0
$866$	−0.736347	−0.0250221
$867$	0	0
$868$	−15.9316	−0.540753
$869$	−62.3844	−2.11625
$870$	0	0
$871$	−6.31788	−0.214073
$872$	−15.2040	−0.514872
$873$	0	0
$874$	−5.75105	−0.194532
$875$	0	0
$876$	0	0
$877$	−30.8431	−1.04150	−0.520749	−	0.853710i	$-0.674347\pi$
$877$	−30.8431	−1.04150	−0.520749	+	0.853710i	$0.674347\pi$
$878$	33.2512	1.12217
$879$	0	0
$880$	0	0
$881$	−26.5915	−0.895889	−0.447944	−	0.894061i	$-0.647844\pi$
$881$	−26.5915	−0.895889	−0.447944	+	0.894061i	$0.647844\pi$
$882$	0	0
$883$	−5.13923	−0.172949	−0.0864745	−	0.996254i	$-0.527560\pi$
$883$	−5.13923	−0.172949	−0.0864745	+	0.996254i	$0.527560\pi$
$884$	3.04632	0.102459
$885$	0	0
$886$	−8.34409	−0.280325
$887$	−20.4753	−0.687494	−0.343747	−	0.939062i	$-0.611696\pi$
$887$	−20.4753	−0.687494	−0.343747	+	0.939062i	$0.611696\pi$
$888$	0	0
$889$	−11.1601	−0.374298
$890$	0	0
$891$	0	0
$892$	27.2221	0.911463
$893$	−14.0707	−0.470859
$894$	0	0
$895$	0	0
$896$	−12.5229	−0.418360
$897$	0	0
$898$	−5.72318	−0.190985
$899$	60.8228	2.02855
$900$	0	0
$901$	27.8785	0.928769
$902$	47.5014	1.58162
$903$	0	0
$904$	49.3373	1.64093
$905$	0	0
$906$	0	0
$907$	−13.2349	−0.439457	−0.219728	−	0.975561i	$-0.570517\pi$
$907$	−13.2349	−0.439457	−0.219728	+	0.975561i	$0.570517\pi$
$908$	6.14249	0.203846
$909$	0	0
$910$	0	0
$911$	−47.3412	−1.56848	−0.784242	−	0.620455i	$-0.786948\pi$
$911$	−47.3412	−1.56848	−0.784242	+	0.620455i	$0.786948\pi$
$912$	0	0
$913$	−46.7070	−1.54578
$914$	11.8673	0.392535
$915$	0	0
$916$	−9.23651	−0.305183
$917$	−8.00090	−0.264213
$918$	0	0
$919$	−30.3350	−1.00066	−0.500330	−	0.865835i	$-0.666788\pi$
$919$	−30.3350	−1.00066	−0.500330	+	0.865835i	$0.666788\pi$
$920$	0	0
$921$	0	0
$922$	28.7345	0.946322
$923$	0.346220	0.0113960
$924$	0	0
$925$	0	0
$926$	11.7878	0.387371
$927$	0	0
$928$	53.6093	1.75981
$929$	25.2857	0.829598	0.414799	−	0.909913i	$-0.363852\pi$
$929$	25.2857	0.829598	0.414799	+	0.909913i	$0.363852\pi$
$930$	0	0
$931$	−2.42201	−0.0793783
$932$	27.2035	0.891080
$933$	0	0
$934$	15.6040	0.510578
$935$	0	0
$936$	0	0
$937$	44.6718	1.45936	0.729681	−	0.683787i	$-0.239668\pi$
$937$	44.6718	1.45936	0.729681	+	0.683787i	$0.239668\pi$
$938$	15.6904	0.512309
$939$	0	0
$940$	0	0
$941$	39.2393	1.27917	0.639583	−	0.768722i	$-0.279107\pi$
$941$	39.2393	1.27917	0.639583	+	0.768722i	$0.279107\pi$
$942$	0	0
$943$	−44.6394	−1.45366
$944$	1.34511	0.0437795
$945$	0	0
$946$	−5.55124	−0.180487
$947$	33.4001	1.08536	0.542678	−	0.839941i	$-0.317410\pi$
$947$	33.4001	1.08536	0.542678	+	0.839941i	$0.317410\pi$
$948$	0	0
$949$	8.17479	0.265365
$950$	0	0
$951$	0	0
$952$	−20.7023	−0.670964
$953$	3.27030	0.105936	0.0529678	−	0.998596i	$-0.483132\pi$
$953$	3.27030	0.105936	0.0529678	+	0.998596i	$0.483132\pi$
$954$	0	0
$955$	0	0
$956$	9.68770	0.313323
$957$	0	0
$958$	15.4950	0.500621
$959$	13.1216	0.423718
$960$	0	0
$961$	7.44371	0.240120
$962$	1.92787	0.0621569
$963$	0	0
$964$	29.0992	0.937222
$965$	0	0
$966$	0	0
$967$	10.8952	0.350366	0.175183	−	0.984536i	$-0.443948\pi$
$967$	10.8952	0.350366	0.175183	+	0.984536i	$0.443948\pi$
$968$	73.5311	2.36338
$969$	0	0
$970$	0	0
$971$	−22.8205	−0.732346	−0.366173	−	0.930547i	$-0.619332\pi$
$971$	−22.8205	−0.732346	−0.366173	+	0.930547i	$0.619332\pi$
$972$	0	0
$973$	−8.57415	−0.274875
$974$	19.3103	0.618743
$975$	0	0
$976$	−1.70794	−0.0546699
$977$	−14.0590	−0.449787	−0.224894	−	0.974383i	$-0.572203\pi$
$977$	−14.0590	−0.449787	−0.224894	+	0.974383i	$0.572203\pi$
$978$	0	0
$979$	47.3798	1.51426
$980$	0	0
$981$	0	0
$982$	−1.69650	−0.0541376
$983$	18.9691	0.605021	0.302511	−	0.953146i	$-0.402175\pi$
$983$	18.9691	0.605021	0.302511	+	0.953146i	$0.402175\pi$
$984$	0	0
$985$	0	0
$986$	28.8832	0.919828
$987$	0	0
$988$	1.14059	0.0362869
$989$	5.21678	0.165884
$990$	0	0
$991$	1.99010	0.0632177	0.0316088	−	0.999500i	$-0.489937\pi$
$991$	1.99010	0.0632177	0.0316088	+	0.999500i	$0.489937\pi$
$992$	33.8843	1.07583
$993$	0	0
$994$	−0.859834	−0.0272723
$995$	0	0
$996$	0	0
$997$	33.2899	1.05430	0.527151	−	0.849772i	$-0.323260\pi$
$997$	33.2899	1.05430	0.527151	+	0.849772i	$0.323260\pi$
$998$	18.6640	0.590798
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9675.2.a.ck.1.3		6
3.2	odd	2		1075.2.a.q.1.4	✓	6
5.4	even	2		9675.2.a.cj.1.4		6
15.2	even	4		1075.2.b.j.474.7		12
15.8	even	4		1075.2.b.j.474.6		12
15.14	odd	2		1075.2.a.r.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1075.2.a.q.1.4	✓	6	3.2	odd	2
1075.2.a.r.1.3	yes	6	15.14	odd	2
1075.2.b.j.474.6		12	15.8	even	4
1075.2.b.j.474.7		12	15.2	even	4
9675.2.a.cj.1.4		6	5.4	even	2
9675.2.a.ck.1.3		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9675 = 3^{2} \cdot 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9675.a (trivial)

\(f(q)\)	\(=\)	\(q-0.920974 q^{2} -1.15181 q^{4} -2.23083 q^{7} +2.90273 q^{8} +O(q^{10})\)	\(q-0.920974 q^{2} -1.15181 q^{4} -2.23083 q^{7} +2.90273 q^{8} +6.02758 q^{11} -0.827279 q^{13} +2.05454 q^{14} -0.369730 q^{16} +3.19701 q^{17} +1.19701 q^{19} -5.55124 q^{22} +5.21678 q^{23} +0.761903 q^{26} +2.56949 q^{28} -9.80965 q^{29} -6.20030 q^{31} -5.46495 q^{32} -2.94436 q^{34} +2.53033 q^{37} -1.10241 q^{38} -8.55689 q^{41} +1.00000 q^{43} -6.94260 q^{44} -4.80452 q^{46} -11.7549 q^{47} -2.02339 q^{49} +0.952865 q^{52} +8.72019 q^{53} -6.47551 q^{56} +9.03444 q^{58} -3.63808 q^{59} +4.61943 q^{61} +5.71032 q^{62} +5.77254 q^{64} +7.63693 q^{67} -3.68233 q^{68} -0.418505 q^{71} -9.88154 q^{73} -2.33037 q^{74} -1.37872 q^{76} -13.4465 q^{77} -10.3498 q^{79} +7.88067 q^{82} -7.74888 q^{83} -0.920974 q^{86} +17.4964 q^{88} +7.86050 q^{89} +1.84552 q^{91} -6.00872 q^{92} +10.8260 q^{94} -5.93400 q^{97} +1.86349 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 11 q^{4} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + q^2 + 11 * q^4 - 2 * q^7 + 6 * q^8	\( 6 q + q^{2} + 11 q^{4} - 2 q^{7} + 6 q^{8} + 6 q^{11} - 17 q^{14} + 5 q^{16} + 7 q^{17} - 5 q^{19} - 34 q^{22} - 3 q^{23} - 14 q^{26} + 21 q^{28} - 18 q^{29} - 12 q^{31} + 3 q^{32} + q^{34} - 7 q^{37} - q^{38} + 6 q^{43} - 9 q^{44} - 14 q^{46} - 6 q^{47} - 3 q^{52} - 8 q^{53} - 20 q^{56} + 24 q^{58} - 21 q^{59} - 8 q^{61} + 47 q^{62} + 6 q^{64} - 35 q^{68} - 14 q^{71} + q^{73} - 2 q^{74} - 57 q^{76} + 13 q^{77} - 31 q^{79} - 32 q^{82} - 13 q^{83} + q^{86} - 39 q^{88} - 40 q^{89} + 11 q^{91} - 32 q^{92} + 35 q^{94} - 11 q^{97} - 44 q^{98}+O(q^{100}) \) 6 * q + q^2 + 11 * q^4 - 2 * q^7 + 6 * q^8 + 6 * q^11 - 17 * q^14 + 5 * q^16 + 7 * q^17 - 5 * q^19 - 34 * q^22 - 3 * q^23 - 14 * q^26 + 21 * q^28 - 18 * q^29 - 12 * q^31 + 3 * q^32 + q^34 - 7 * q^37 - q^38 + 6 * q^43 - 9 * q^44 - 14 * q^46 - 6 * q^47 - 3 * q^52 - 8 * q^53 - 20 * q^56 + 24 * q^58 - 21 * q^59 - 8 * q^61 + 47 * q^62 + 6 * q^64 - 35 * q^68 - 14 * q^71 + q^73 - 2 * q^74 - 57 * q^76 + 13 * q^77 - 31 * q^79 - 32 * q^82 - 13 * q^83 + q^86 - 39 * q^88 - 40 * q^89 + 11 * q^91 - 32 * q^92 + 35 * q^94 - 11 * q^97 - 44 * q^98

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