LMFDB - Embedded newform 8208.2.a.ca.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8208,2,Mod(1,8208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8208.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8208 = 2^{4} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,-3,0,-1,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$65.5412099791$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.7986588.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 16x^{3} + 23x^{2} + 24x - 36 $ x^5 - x^4 - 16x^3 + 23x^2 + 24*x - 36
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.29730$ of defining polynomial
Character	$\chi$	$=$	8208.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.32769 q^{5} -2.77621 q^{7} -2.53271 q^{11} +0.745822 q^{13} -4.83810 q^{17} +1.00000 q^{19} +1.29730 q^{23} +6.07351 q^{25} +2.01162 q^{29} -1.93811 q^{31} -9.23836 q^{35} -1.83001 q^{37} +11.7908 q^{41} -5.64376 q^{43} -4.39768 q^{47} +0.707340 q^{49} +1.00809 q^{53} -8.42806 q^{55} +0.602695 q^{59} +4.18161 q^{61} +2.48186 q^{65} -1.92649 q^{67} +0.548896 q^{71} -5.91971 q^{73} +7.03133 q^{77} -8.96335 q^{79} +11.0065 q^{83} -16.0997 q^{85} -10.4316 q^{89} -2.07056 q^{91} +3.32769 q^{95} +16.0146 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{5} - q^{7} - 4 q^{11} + 3 q^{13} - 5 q^{17} + 5 q^{19} - q^{23} + 10 q^{25} - 4 q^{29} - 16 q^{31} - 6 q^{35} + 7 q^{37} - 7 q^{41} - 3 q^{43} + 7 q^{47} + 18 q^{49} + 2 q^{53} - q^{55} - 15 q^{59}+ \cdots + 27 q^{97}+O(q^{100}) $ 5 * q - 3 * q^5 - q^7 - 4 * q^11 + 3 * q^13 - 5 * q^17 + 5 * q^19 - q^23 + 10 * q^25 - 4 * q^29 - 16 * q^31 - 6 * q^35 + 7 * q^37 - 7 * q^41 - 3 * q^43 + 7 * q^47 + 18 * q^49 + 2 * q^53 - q^55 - 15 * q^59 + 23 * q^61 - 32 * q^65 - 30 * q^67 - 12 * q^71 + 13 * q^73 + 4 * q^77 - 3 * q^79 + 5 * q^83 + 7 * q^85 - 29 * q^91 - 3 * q^95 + 27 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.32769	1.48819	0.744094	−	0.668075i	$-0.232882\pi$
$5$	3.32769	1.48819	0.744094	+	0.668075i	$0.232882\pi$
$6$	0	0
$7$	−2.77621	−1.04931	−0.524654	−	0.851315i	$-0.675805\pi$
$7$	−2.77621	−1.04931	−0.524654	+	0.851315i	$0.675805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.53271	−0.763640	−0.381820	−	0.924237i	$-0.624703\pi$
$11$	−2.53271	−0.763640	−0.381820	+	0.924237i	$0.624703\pi$
$12$	0	0
$13$	0.745822	0.206854	0.103427	−	0.994637i	$-0.467019\pi$
$13$	0.745822	0.206854	0.103427	+	0.994637i	$0.467019\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.83810	−1.17341	−0.586706	−	0.809800i	$-0.699576\pi$
$17$	−4.83810	−1.17341	−0.586706	+	0.809800i	$0.699576\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.29730	0.270506	0.135253	−	0.990811i	$-0.456815\pi$
$23$	1.29730	0.270506	0.135253	+	0.990811i	$0.456815\pi$
$24$	0	0
$25$	6.07351	1.21470
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.01162	0.373548	0.186774	−	0.982403i	$-0.440197\pi$
$29$	2.01162	0.373548	0.186774	+	0.982403i	$0.440197\pi$
$30$	0	0
$31$	−1.93811	−0.348094	−0.174047	−	0.984737i	$-0.555685\pi$
$31$	−1.93811	−0.348094	−0.174047	+	0.984737i	$0.555685\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−9.23836	−1.56157
$36$	0	0
$37$	−1.83001	−0.300852	−0.150426	−	0.988621i	$-0.548064\pi$
$37$	−1.83001	−0.300852	−0.150426	+	0.988621i	$0.548064\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.7908	1.84141	0.920705	−	0.390259i	$-0.127614\pi$
$41$	11.7908	1.84141	0.920705	+	0.390259i	$0.127614\pi$
$42$	0	0
$43$	−5.64376	−0.860665	−0.430333	−	0.902670i	$-0.641604\pi$
$43$	−5.64376	−0.860665	−0.430333	+	0.902670i	$0.641604\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.39768	−0.641467	−0.320733	−	0.947170i	$-0.603929\pi$
$47$	−4.39768	−0.641467	−0.320733	+	0.947170i	$0.603929\pi$
$48$	0	0
$49$	0.707340	0.101049
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.00809	0.138472	0.0692362	−	0.997600i	$-0.477944\pi$
$53$	1.00809	0.138472	0.0692362	+	0.997600i	$0.477944\pi$
$54$	0	0
$55$	−8.42806	−1.13644
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.602695	0.0784642	0.0392321	−	0.999230i	$-0.487509\pi$
$59$	0.602695	0.0784642	0.0392321	+	0.999230i	$0.487509\pi$
$60$	0	0
$61$	4.18161	0.535400	0.267700	−	0.963502i	$-0.413736\pi$
$61$	4.18161	0.535400	0.267700	+	0.963502i	$0.413736\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.48186	0.307837
$66$	0	0
$67$	−1.92649	−0.235358	−0.117679	−	0.993052i	$-0.537545\pi$
$67$	−1.92649	−0.235358	−0.117679	+	0.993052i	$0.537545\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.548896	0.0651420	0.0325710	−	0.999469i	$-0.489631\pi$
$71$	0.548896	0.0651420	0.0325710	+	0.999469i	$0.489631\pi$
$72$	0	0
$73$	−5.91971	−0.692849	−0.346425	−	0.938078i	$-0.612604\pi$
$73$	−5.91971	−0.692849	−0.346425	+	0.938078i	$0.612604\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.03133	0.801294
$78$	0	0
$79$	−8.96335	−1.00846	−0.504228	−	0.863571i	$-0.668223\pi$
$79$	−8.96335	−1.00846	−0.504228	+	0.863571i	$0.668223\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.0065	1.20812	0.604059	−	0.796940i	$-0.293549\pi$
$83$	11.0065	1.20812	0.604059	+	0.796940i	$0.293549\pi$
$84$	0	0
$85$	−16.0997	−1.74626
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.4316	−1.10575	−0.552873	−	0.833266i	$-0.686469\pi$
$89$	−10.4316	−1.10575	−0.552873	+	0.833266i	$0.686469\pi$
$90$	0	0
$91$	−2.07056	−0.217053
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.32769	0.341414
$96$	0	0
$97$	16.0146	1.62603	0.813017	−	0.582240i	$-0.197824\pi$
$97$	16.0146	1.62603	0.813017	+	0.582240i	$0.197824\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.4596	−1.04077	−0.520383	−	0.853933i	$-0.674211\pi$
$101$	−10.4596	−1.04077	−0.520383	+	0.853933i	$0.674211\pi$
$102$	0	0
$103$	−12.2896	−1.21093	−0.605464	−	0.795873i	$-0.707012\pi$
$103$	−12.2896	−1.21093	−0.605464	+	0.795873i	$0.707012\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.74024	−0.361583	−0.180791	−	0.983521i	$-0.557866\pi$
$107$	−3.74024	−0.361583	−0.180791	+	0.983521i	$0.557866\pi$
$108$	0	0
$109$	0.796097	0.0762523	0.0381261	−	0.999273i	$-0.487861\pi$
$109$	0.796097	0.0762523	0.0381261	+	0.999273i	$0.487861\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.5716	−1.84114	−0.920572	−	0.390573i	$-0.872277\pi$
$113$	−19.5716	−1.84114	−0.920572	+	0.390573i	$0.872277\pi$
$114$	0	0
$115$	4.31701	0.402563
$116$	0	0
$117$	0	0
$118$	0	0
$119$	13.4316	1.23127
$120$	0	0
$121$	−4.58539	−0.416854
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.57231	0.319517
$126$	0	0
$127$	−14.4047	−1.27821	−0.639106	−	0.769119i	$-0.720696\pi$
$127$	−14.4047	−1.27821	−0.639106	+	0.769119i	$0.720696\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.00809	−0.437559	−0.218780	−	0.975774i	$-0.570208\pi$
$131$	−5.00809	−0.437559	−0.218780	+	0.975774i	$0.570208\pi$
$132$	0	0
$133$	−2.77621	−0.240728
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.8745	−0.929074	−0.464537	−	0.885554i	$-0.653779\pi$
$137$	−10.8745	−0.929074	−0.464537	+	0.885554i	$0.653779\pi$
$138$	0	0
$139$	13.3924	1.13592	0.567962	−	0.823054i	$-0.307732\pi$
$139$	13.3924	1.13592	0.567962	+	0.823054i	$0.307732\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.88895	−0.157962
$144$	0	0
$145$	6.69404	0.555909
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.86245	0.562194	0.281097	−	0.959679i	$-0.409302\pi$
$149$	6.86245	0.562194	0.281097	+	0.959679i	$0.409302\pi$
$150$	0	0
$151$	−8.96335	−0.729427	−0.364714	−	0.931120i	$-0.618833\pi$
$151$	−8.96335	−0.729427	−0.364714	+	0.931120i	$0.618833\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.44942	−0.518030
$156$	0	0
$157$	−13.6405	−1.08863	−0.544315	−	0.838881i	$-0.683210\pi$
$157$	−13.6405	−1.08863	−0.544315	+	0.838881i	$0.683210\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.60158	−0.283844
$162$	0	0
$163$	−0.723528	−0.0566711	−0.0283355	−	0.999598i	$-0.509021\pi$
$163$	−0.723528	−0.0566711	−0.0283355	+	0.999598i	$0.509021\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.29842	−0.100475	−0.0502373	−	0.998737i	$-0.515998\pi$
$167$	−1.29842	−0.100475	−0.0502373	+	0.998737i	$0.515998\pi$
$168$	0	0
$169$	−12.4438	−0.957212
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.66163	−0.734560	−0.367280	−	0.930110i	$-0.619711\pi$
$173$	−9.66163	−0.734560	−0.367280	+	0.930110i	$0.619711\pi$
$174$	0	0
$175$	−16.8613	−1.27460
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−22.1459	−1.65526	−0.827631	−	0.561272i	$-0.810312\pi$
$179$	−22.1459	−1.65526	−0.827631	+	0.561272i	$0.810312\pi$
$180$	0	0
$181$	8.35714	0.621181	0.310591	−	0.950544i	$-0.399473\pi$
$181$	8.35714	0.621181	0.310591	+	0.950544i	$0.399473\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.08970	−0.447723
$186$	0	0
$187$	12.2535	0.896065
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.67973	−0.338613	−0.169307	−	0.985563i	$-0.554153\pi$
$191$	−4.67973	−0.338613	−0.169307	+	0.985563i	$0.554153\pi$
$192$	0	0
$193$	4.16026	0.299462	0.149731	−	0.988727i	$-0.452159\pi$
$193$	4.16026	0.299462	0.149731	+	0.988727i	$0.452159\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.6825	−1.68731	−0.843654	−	0.536887i	$-0.819600\pi$
$197$	−23.6825	−1.68731	−0.843654	+	0.536887i	$0.819600\pi$
$198$	0	0
$199$	−26.0175	−1.84433	−0.922167	−	0.386793i	$-0.873583\pi$
$199$	−26.0175	−1.84433	−0.922167	+	0.386793i	$0.873583\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.58467	−0.391967
$204$	0	0
$205$	39.2360	2.74036
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.53271	−0.175191
$210$	0	0
$211$	8.40509	0.578630	0.289315	−	0.957234i	$-0.406572\pi$
$211$	8.40509	0.578630	0.289315	+	0.957234i	$0.406572\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−18.7807	−1.28083
$216$	0	0
$217$	5.38059	0.365258
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.60836	−0.242725
$222$	0	0
$223$	11.0568	0.740415	0.370207	−	0.928949i	$-0.379287\pi$
$223$	11.0568	0.740415	0.370207	+	0.928949i	$0.379287\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.1991	1.00880	0.504401	−	0.863470i	$-0.331713\pi$
$227$	15.1991	1.00880	0.504401	+	0.863470i	$0.331713\pi$
$228$	0	0
$229$	12.1372	0.802051	0.401026	−	0.916067i	$-0.368654\pi$
$229$	12.1372	0.802051	0.401026	+	0.916067i	$0.368654\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.0327	−1.63994	−0.819972	−	0.572403i	$-0.806011\pi$
$233$	−25.0327	−1.63994	−0.819972	+	0.572403i	$0.806011\pi$
$234$	0	0
$235$	−14.6341	−0.954623
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.2653	−0.664005	−0.332003	−	0.943278i	$-0.607724\pi$
$239$	−10.2653	−0.664005	−0.332003	+	0.943278i	$0.607724\pi$
$240$	0	0
$241$	15.1915	0.978573	0.489287	−	0.872123i	$-0.337257\pi$
$241$	15.1915	0.978573	0.489287	+	0.872123i	$0.337257\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.35381	0.150379
$246$	0	0
$247$	0.745822	0.0474555
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.15837	−0.136235	−0.0681177	−	0.997677i	$-0.521699\pi$
$251$	−2.15837	−0.136235	−0.0681177	+	0.997677i	$0.521699\pi$
$252$	0	0
$253$	−3.28568	−0.206569
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.47891	−0.279387	−0.139693	−	0.990195i	$-0.544612\pi$
$257$	−4.47891	−0.279387	−0.139693	+	0.990195i	$0.544612\pi$
$258$	0	0
$259$	5.08049	0.315686
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.0693	0.867551	0.433775	−	0.901021i	$-0.357181\pi$
$263$	14.0693	0.867551	0.433775	+	0.901021i	$0.357181\pi$
$264$	0	0
$265$	3.35462	0.206073
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.2321	1.11163	0.555817	−	0.831305i	$-0.312406\pi$
$269$	18.2321	1.11163	0.555817	+	0.831305i	$0.312406\pi$
$270$	0	0
$271$	10.3102	0.626302	0.313151	−	0.949703i	$-0.398615\pi$
$271$	10.3102	0.626302	0.313151	+	0.949703i	$0.398615\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−15.3824	−0.927595
$276$	0	0
$277$	19.0354	1.14373	0.571863	−	0.820349i	$-0.306221\pi$
$277$	19.0354	1.14373	0.571863	+	0.820349i	$0.306221\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−21.4559	−1.27995	−0.639977	−	0.768394i	$-0.721056\pi$
$281$	−21.4559	−1.27995	−0.639977	+	0.768394i	$0.721056\pi$
$282$	0	0
$283$	−18.1089	−1.07646	−0.538231	−	0.842797i	$-0.680907\pi$
$283$	−18.1089	−1.07646	−0.538231	+	0.842797i	$0.680907\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−32.7337	−1.93221
$288$	0	0
$289$	6.40724	0.376896
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.8295	−0.632665	−0.316333	−	0.948648i	$-0.602452\pi$
$293$	−10.8295	−0.632665	−0.316333	+	0.948648i	$0.602452\pi$
$294$	0	0
$295$	2.00558	0.116769
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.967555	0.0559551
$300$	0	0
$301$	15.6683	0.903103
$302$	0	0
$303$	0	0
$304$	0	0
$305$	13.9151	0.796776
$306$	0	0
$307$	−2.13590	−0.121902	−0.0609512	−	0.998141i	$-0.519413\pi$
$307$	−2.13590	−0.121902	−0.0609512	+	0.998141i	$0.519413\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	20.9895	1.19021	0.595104	−	0.803649i	$-0.297111\pi$
$311$	20.9895	1.19021	0.595104	+	0.803649i	$0.297111\pi$
$312$	0	0
$313$	−3.71839	−0.210176	−0.105088	−	0.994463i	$-0.533512\pi$
$313$	−3.71839	−0.210176	−0.105088	+	0.994463i	$0.533512\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.968935	0.0544208	0.0272104	−	0.999630i	$-0.491338\pi$
$317$	0.968935	0.0544208	0.0272104	+	0.999630i	$0.491338\pi$
$318$	0	0
$319$	−5.09484	−0.285256
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.83810	−0.269199
$324$	0	0
$325$	4.52976	0.251266
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.2089	0.673097
$330$	0	0
$331$	19.9099	1.09435	0.547174	−	0.837019i	$-0.315704\pi$
$331$	19.9099	1.09435	0.547174	+	0.837019i	$0.315704\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.41076	−0.350257
$336$	0	0
$337$	20.5821	1.12118	0.560590	−	0.828093i	$-0.310574\pi$
$337$	20.5821	1.12118	0.560590	+	0.828093i	$0.310574\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.90866	0.265819
$342$	0	0
$343$	17.4697	0.943278
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.1438	0.598228	0.299114	−	0.954217i	$-0.403309\pi$
$347$	11.1438	0.598228	0.299114	+	0.954217i	$0.403309\pi$
$348$	0	0
$349$	−11.3843	−0.609386	−0.304693	−	0.952451i	$-0.598554\pi$
$349$	−11.3843	−0.609386	−0.304693	+	0.952451i	$0.598554\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.7867	−1.21281	−0.606407	−	0.795154i	$-0.707390\pi$
$353$	−22.7867	−1.21281	−0.606407	+	0.795154i	$0.707390\pi$
$354$	0	0
$355$	1.82655	0.0969435
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.4273	−0.972557	−0.486278	−	0.873804i	$-0.661646\pi$
$359$	−18.4273	−0.972557	−0.486278	+	0.873804i	$0.661646\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−19.6989	−1.03109
$366$	0	0
$367$	−14.5173	−0.757795	−0.378897	−	0.925439i	$-0.623697\pi$
$367$	−14.5173	−0.757795	−0.378897	+	0.925439i	$0.623697\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.79868	−0.145300
$372$	0	0
$373$	−9.06395	−0.469314	−0.234657	−	0.972078i	$-0.575397\pi$
$373$	−9.06395	−0.469314	−0.234657	+	0.972078i	$0.575397\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.50031	0.0772698
$378$	0	0
$379$	32.9824	1.69419	0.847096	−	0.531440i	$-0.178349\pi$
$379$	32.9824	1.69419	0.847096	+	0.531440i	$0.178349\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.6808	0.647961	0.323980	−	0.946064i	$-0.394979\pi$
$383$	12.6808	0.647961	0.323980	+	0.946064i	$0.394979\pi$
$384$	0	0
$385$	23.3981	1.19248
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−15.2609	−0.773757	−0.386879	−	0.922131i	$-0.626447\pi$
$389$	−15.2609	−0.773757	−0.386879	+	0.922131i	$0.626447\pi$
$390$	0	0
$391$	−6.27647	−0.317415
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−29.8272	−1.50077
$396$	0	0
$397$	16.3641	0.821290	0.410645	−	0.911795i	$-0.365303\pi$
$397$	16.3641	0.821290	0.410645	+	0.911795i	$0.365303\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.3769	1.46701	0.733506	−	0.679683i	$-0.237883\pi$
$401$	29.3769	1.46701	0.733506	+	0.679683i	$0.237883\pi$
$402$	0	0
$403$	−1.44548	−0.0720046
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.63488	0.229742
$408$	0	0
$409$	−9.66647	−0.477976	−0.238988	−	0.971023i	$-0.576816\pi$
$409$	−9.66647	−0.477976	−0.238988	+	0.971023i	$0.576816\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.67321	−0.0823331
$414$	0	0
$415$	36.6261	1.79791
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.9032	−1.80284	−0.901420	−	0.432946i	$-0.857474\pi$
$419$	−36.9032	−1.80284	−0.901420	+	0.432946i	$0.857474\pi$
$420$	0	0
$421$	30.0798	1.46600	0.733000	−	0.680229i	$-0.238119\pi$
$421$	30.0798	1.46600	0.733000	+	0.680229i	$0.238119\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−29.3843	−1.42535
$426$	0	0
$427$	−11.6090	−0.561800
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.4136	−0.549772	−0.274886	−	0.961477i	$-0.588640\pi$
$431$	−11.4136	−0.549772	−0.274886	+	0.961477i	$0.588640\pi$
$432$	0	0
$433$	22.7978	1.09559	0.547797	−	0.836611i	$-0.315467\pi$
$433$	22.7978	1.09559	0.547797	+	0.836611i	$0.315467\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.29730	0.0620583
$438$	0	0
$439$	17.1584	0.818927	0.409463	−	0.912327i	$-0.365716\pi$
$439$	17.1584	0.818927	0.409463	+	0.912327i	$0.365716\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.3080	−1.29744	−0.648722	−	0.761025i	$-0.724696\pi$
$443$	−27.3080	−1.29744	−0.648722	+	0.761025i	$0.724696\pi$
$444$	0	0
$445$	−34.7131	−1.64556
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.31003	−0.297789	−0.148894	−	0.988853i	$-0.547571\pi$
$449$	−6.31003	−0.297789	−0.148894	+	0.988853i	$0.547571\pi$
$450$	0	0
$451$	−29.8626	−1.40618
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.89017	−0.323016
$456$	0	0
$457$	−15.3027	−0.715828	−0.357914	−	0.933755i	$-0.616512\pi$
$457$	−15.3027	−0.715828	−0.357914	+	0.933755i	$0.616512\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.25298	0.244655	0.122328	−	0.992490i	$-0.460964\pi$
$461$	5.25298	0.244655	0.122328	+	0.992490i	$0.460964\pi$
$462$	0	0
$463$	−39.9432	−1.85632	−0.928158	−	0.372186i	$-0.878608\pi$
$463$	−39.9432	−1.85632	−0.928158	+	0.372186i	$0.878608\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.86164	−0.178696	−0.0893478	−	0.996000i	$-0.528478\pi$
$467$	−3.86164	−0.178696	−0.0893478	+	0.996000i	$0.528478\pi$
$468$	0	0
$469$	5.34834	0.246963
$470$	0	0
$471$	0	0
$472$	0	0
$473$	14.2940	0.657239
$474$	0	0
$475$	6.07351	0.278672
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.3644	1.61584	0.807920	−	0.589292i	$-0.200593\pi$
$479$	35.3644	1.61584	0.807920	+	0.589292i	$0.200593\pi$
$480$	0	0
$481$	−1.36486	−0.0622323
$482$	0	0
$483$	0	0
$484$	0	0
$485$	53.2915	2.41984
$486$	0	0
$487$	25.6252	1.16119	0.580594	−	0.814193i	$-0.302820\pi$
$487$	25.6252	1.16119	0.580594	+	0.814193i	$0.302820\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.1502	1.13501	0.567506	−	0.823370i	$-0.307909\pi$
$491$	25.1502	1.13501	0.567506	+	0.823370i	$0.307909\pi$
$492$	0	0
$493$	−9.73241	−0.438326
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.52385	−0.0683540
$498$	0	0
$499$	25.8605	1.15767	0.578837	−	0.815443i	$-0.303507\pi$
$499$	25.8605	1.15767	0.578837	+	0.815443i	$0.303507\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.3342	−0.817480	−0.408740	−	0.912651i	$-0.634032\pi$
$503$	−18.3342	−0.817480	−0.408740	+	0.912651i	$0.634032\pi$
$504$	0	0
$505$	−34.8062	−1.54885
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−32.0888	−1.42231	−0.711156	−	0.703035i	$-0.751828\pi$
$509$	−32.0888	−1.42231	−0.711156	+	0.703035i	$0.751828\pi$
$510$	0	0
$511$	16.4343	0.727013
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−40.8959	−1.80209
$516$	0	0
$517$	11.1380	0.489850
$518$	0	0
$519$	0	0
$520$	0	0
$521$	28.9870	1.26994	0.634971	−	0.772536i	$-0.281012\pi$
$521$	28.9870	1.26994	0.634971	+	0.772536i	$0.281012\pi$
$522$	0	0
$523$	−26.8215	−1.17282	−0.586412	−	0.810013i	$-0.699460\pi$
$523$	−26.8215	−1.17282	−0.586412	+	0.810013i	$0.699460\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.37676	0.408458
$528$	0	0
$529$	−21.3170	−0.926827
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.79382	0.380903
$534$	0	0
$535$	−12.4464	−0.538103
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.79149	−0.0771647
$540$	0	0
$541$	−12.0367	−0.517498	−0.258749	−	0.965945i	$-0.583310\pi$
$541$	−12.0367	−0.517498	−0.258749	+	0.965945i	$0.583310\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.64916	0.113478
$546$	0	0
$547$	11.5681	0.494616	0.247308	−	0.968937i	$-0.420454\pi$
$547$	11.5681	0.494616	0.247308	+	0.968937i	$0.420454\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.01162	0.0856978
$552$	0	0
$553$	24.8841	1.05818
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.2520	0.688619	0.344309	−	0.938856i	$-0.388113\pi$
$557$	16.2520	0.688619	0.344309	+	0.938856i	$0.388113\pi$
$558$	0	0
$559$	−4.20924	−0.178032
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−38.8534	−1.63748	−0.818739	−	0.574167i	$-0.805326\pi$
$563$	−38.8534	−1.63748	−0.818739	+	0.574167i	$0.805326\pi$
$564$	0	0
$565$	−65.1283	−2.73997
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.91728	−0.164221	−0.0821104	−	0.996623i	$-0.526166\pi$
$569$	−3.91728	−0.164221	−0.0821104	+	0.996623i	$0.526166\pi$
$570$	0	0
$571$	−37.2665	−1.55955	−0.779776	−	0.626058i	$-0.784667\pi$
$571$	−37.2665	−1.55955	−0.779776	+	0.626058i	$0.784667\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.87917	0.328584
$576$	0	0
$577$	8.16835	0.340053	0.170026	−	0.985440i	$-0.445615\pi$
$577$	8.16835	0.340053	0.170026	+	0.985440i	$0.445615\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−30.5563	−1.26769
$582$	0	0
$583$	−2.55321	−0.105743
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.3668	−0.592980	−0.296490	−	0.955036i	$-0.595816\pi$
$587$	−14.3668	−0.592980	−0.296490	+	0.955036i	$0.595816\pi$
$588$	0	0
$589$	−1.93811	−0.0798583
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.56900	0.0644313	0.0322157	−	0.999481i	$-0.489744\pi$
$593$	1.56900	0.0644313	0.0322157	+	0.999481i	$0.489744\pi$
$594$	0	0
$595$	44.6961	1.83236
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21.2853	0.869695	0.434848	−	0.900504i	$-0.356802\pi$
$599$	21.2853	0.869695	0.434848	+	0.900504i	$0.356802\pi$
$600$	0	0
$601$	−23.6232	−0.963610	−0.481805	−	0.876278i	$-0.660019\pi$
$601$	−23.6232	−0.963610	−0.481805	+	0.876278i	$0.660019\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−15.2587	−0.620356
$606$	0	0
$607$	4.50269	0.182759	0.0913793	−	0.995816i	$-0.470872\pi$
$607$	4.50269	0.182759	0.0913793	+	0.995816i	$0.470872\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.27988	−0.132690
$612$	0	0
$613$	−3.32327	−0.134226	−0.0671128	−	0.997745i	$-0.521379\pi$
$613$	−3.32327	−0.134226	−0.0671128	+	0.997745i	$0.521379\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.4475	1.22577	0.612885	−	0.790172i	$-0.290009\pi$
$617$	30.4475	1.22577	0.612885	+	0.790172i	$0.290009\pi$
$618$	0	0
$619$	40.8559	1.64214	0.821068	−	0.570831i	$-0.193379\pi$
$619$	40.8559	1.64214	0.821068	+	0.570831i	$0.193379\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	28.9603	1.16027
$624$	0	0
$625$	−18.4800	−0.739201
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.85377	0.353023
$630$	0	0
$631$	38.9565	1.55083	0.775416	−	0.631451i	$-0.217540\pi$
$631$	38.9565	1.55083	0.775416	+	0.631451i	$0.217540\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−47.9344	−1.90222
$636$	0	0
$637$	0.527549	0.0209023
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−38.6625	−1.52708	−0.763539	−	0.645762i	$-0.776540\pi$
$641$	−38.6625	−1.52708	−0.763539	+	0.645762i	$0.776540\pi$
$642$	0	0
$643$	28.1296	1.10932	0.554662	−	0.832075i	$-0.312847\pi$
$643$	28.1296	1.10932	0.554662	+	0.832075i	$0.312847\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	40.0471	1.57441	0.787207	−	0.616689i	$-0.211526\pi$
$647$	40.0471	1.57441	0.787207	+	0.616689i	$0.211526\pi$
$648$	0	0
$649$	−1.52645	−0.0599184
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−33.5455	−1.31274	−0.656368	−	0.754441i	$-0.727908\pi$
$653$	−33.5455	−1.31274	−0.656368	+	0.754441i	$0.727908\pi$
$654$	0	0
$655$	−16.6654	−0.651170
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27.0745	1.05467	0.527336	−	0.849657i	$-0.323191\pi$
$659$	27.0745	1.05467	0.527336	+	0.849657i	$0.323191\pi$
$660$	0	0
$661$	28.7930	1.11992	0.559959	−	0.828520i	$-0.310817\pi$
$661$	28.7930	1.11992	0.559959	+	0.828520i	$0.310817\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.23836	−0.358248
$666$	0	0
$667$	2.60967	0.101047
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.5908	−0.408853
$672$	0	0
$673$	−46.2242	−1.78181	−0.890907	−	0.454186i	$-0.849930\pi$
$673$	−46.2242	−1.78181	−0.890907	+	0.454186i	$0.849930\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.59160	−0.330202	−0.165101	−	0.986277i	$-0.552795\pi$
$677$	−8.59160	−0.330202	−0.165101	+	0.986277i	$0.552795\pi$
$678$	0	0
$679$	−44.4598	−1.70621
$680$	0	0
$681$	0	0
$682$	0	0
$683$	11.7018	0.447757	0.223878	−	0.974617i	$-0.428128\pi$
$683$	11.7018	0.447757	0.223878	+	0.974617i	$0.428128\pi$
$684$	0	0
$685$	−36.1870	−1.38264
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.751858	0.0286435
$690$	0	0
$691$	4.71655	0.179426	0.0897130	−	0.995968i	$-0.471405\pi$
$691$	4.71655	0.179426	0.0897130	+	0.995968i	$0.471405\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	44.5656	1.69047
$696$	0	0
$697$	−57.0450	−2.16073
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.0483	0.983831	0.491915	−	0.870643i	$-0.336297\pi$
$701$	26.0483	0.983831	0.491915	+	0.870643i	$0.336297\pi$
$702$	0	0
$703$	−1.83001	−0.0690201
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29.0380	1.09208
$708$	0	0
$709$	28.4740	1.06936	0.534681	−	0.845054i	$-0.320432\pi$
$709$	28.4740	1.06936	0.534681	+	0.845054i	$0.320432\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.51431	−0.0941615
$714$	0	0
$715$	−6.28583	−0.235077
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.82832	0.180066	0.0900330	−	0.995939i	$-0.471303\pi$
$719$	4.82832	0.180066	0.0900330	+	0.995939i	$0.471303\pi$
$720$	0	0
$721$	34.1184	1.27064
$722$	0	0
$723$	0	0
$724$	0	0
$725$	12.2176	0.453749
$726$	0	0
$727$	26.0141	0.964808	0.482404	−	0.875949i	$-0.339764\pi$
$727$	26.0141	0.964808	0.482404	+	0.875949i	$0.339764\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	27.3051	1.00992
$732$	0	0
$733$	2.93251	0.108315	0.0541574	−	0.998532i	$-0.482753\pi$
$733$	2.93251	0.108315	0.0541574	+	0.998532i	$0.482753\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.87924	0.179729
$738$	0	0
$739$	13.2892	0.488852	0.244426	−	0.969668i	$-0.421400\pi$
$739$	13.2892	0.488852	0.244426	+	0.969668i	$0.421400\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.75074	−0.174288	−0.0871439	−	0.996196i	$-0.527774\pi$
$743$	−4.75074	−0.174288	−0.0871439	+	0.996196i	$0.527774\pi$
$744$	0	0
$745$	22.8361	0.836650
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.3837	0.379412
$750$	0	0
$751$	−49.3812	−1.80194	−0.900972	−	0.433876i	$-0.857145\pi$
$751$	−49.3812	−1.80194	−0.900972	+	0.433876i	$0.857145\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−29.8272	−1.08552
$756$	0	0
$757$	13.5342	0.491910	0.245955	−	0.969281i	$-0.420898\pi$
$757$	13.5342	0.491910	0.245955	+	0.969281i	$0.420898\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.4451	0.994884	0.497442	−	0.867497i	$-0.334273\pi$
$761$	27.4451	0.994884	0.497442	+	0.867497i	$0.334273\pi$
$762$	0	0
$763$	−2.21013	−0.0800122
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.449503	0.0162306
$768$	0	0
$769$	−10.2830	−0.370814	−0.185407	−	0.982662i	$-0.559360\pi$
$769$	−10.2830	−0.370814	−0.185407	+	0.982662i	$0.559360\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−49.0937	−1.76578	−0.882889	−	0.469582i	$-0.844405\pi$
$773$	−49.0937	−1.76578	−0.882889	+	0.469582i	$0.844405\pi$
$774$	0	0
$775$	−11.7711	−0.422831
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.7908	0.422449
$780$	0	0
$781$	−1.39019	−0.0497450
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−45.3913	−1.62009
$786$	0	0
$787$	7.31410	0.260720	0.130360	−	0.991467i	$-0.458387\pi$
$787$	7.31410	0.260720	0.130360	+	0.991467i	$0.458387\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	54.3349	1.93193
$792$	0	0
$793$	3.11873	0.110750
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13.4698	−0.477126	−0.238563	−	0.971127i	$-0.576676\pi$
$797$	−13.4698	−0.477126	−0.238563	+	0.971127i	$0.576676\pi$
$798$	0	0
$799$	21.2764	0.752705
$800$	0	0
$801$	0	0
$802$	0	0
$803$	14.9929	0.529088
$804$	0	0
$805$	−11.9849	−0.422413
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23.2678	−0.818051	−0.409025	−	0.912523i	$-0.634131\pi$
$809$	−23.2678	−0.818051	−0.409025	+	0.912523i	$0.634131\pi$
$810$	0	0
$811$	−10.2519	−0.359994	−0.179997	−	0.983667i	$-0.557609\pi$
$811$	−10.2519	−0.359994	−0.179997	+	0.983667i	$0.557609\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.40767	−0.0843372
$816$	0	0
$817$	−5.64376	−0.197450
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.6122	0.544869	0.272435	−	0.962174i	$-0.412171\pi$
$821$	15.6122	0.544869	0.272435	+	0.962174i	$0.412171\pi$
$822$	0	0
$823$	49.7015	1.73249	0.866243	−	0.499624i	$-0.166528\pi$
$823$	49.7015	1.73249	0.866243	+	0.499624i	$0.166528\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.5764	1.16756	0.583782	−	0.811910i	$-0.301572\pi$
$827$	33.5764	1.16756	0.583782	+	0.811910i	$0.301572\pi$
$828$	0	0
$829$	−43.6713	−1.51677	−0.758383	−	0.651810i	$-0.774010\pi$
$829$	−43.6713	−1.51677	−0.758383	+	0.651810i	$0.774010\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.42218	−0.118572
$834$	0	0
$835$	−4.32073	−0.149525
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.1340	−0.833198	−0.416599	−	0.909090i	$-0.636778\pi$
$839$	−24.1340	−0.833198	−0.416599	+	0.909090i	$0.636778\pi$
$840$	0	0
$841$	−24.9534	−0.860462
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−41.4089	−1.42451
$846$	0	0
$847$	12.7300	0.437408
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.37407	−0.0813821
$852$	0	0
$853$	13.5896	0.465298	0.232649	−	0.972561i	$-0.425261\pi$
$853$	13.5896	0.465298	0.232649	+	0.972561i	$0.425261\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.95504	0.305898	0.152949	−	0.988234i	$-0.451123\pi$
$857$	8.95504	0.305898	0.152949	+	0.988234i	$0.451123\pi$
$858$	0	0
$859$	10.0099	0.341534	0.170767	−	0.985311i	$-0.445375\pi$
$859$	10.0099	0.341534	0.170767	+	0.985311i	$0.445375\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−49.1539	−1.67322	−0.836609	−	0.547801i	$-0.815465\pi$
$863$	−49.1539	−1.67322	−0.836609	+	0.547801i	$0.815465\pi$
$864$	0	0
$865$	−32.1509	−1.09316
$866$	0	0
$867$	0	0
$868$	0	0
$869$	22.7016	0.770098
$870$	0	0
$871$	−1.43682	−0.0486847
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.91747	−0.335272
$876$	0	0
$877$	−19.5430	−0.659920	−0.329960	−	0.943995i	$-0.607035\pi$
$877$	−19.5430	−0.659920	−0.329960	+	0.943995i	$0.607035\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−11.0905	−0.373649	−0.186825	−	0.982393i	$-0.559820\pi$
$881$	−11.0905	−0.373649	−0.186825	+	0.982393i	$0.559820\pi$
$882$	0	0
$883$	−32.6991	−1.10041	−0.550206	−	0.835029i	$-0.685451\pi$
$883$	−32.6991	−1.10041	−0.550206	+	0.835029i	$0.685451\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.29213	−0.244846	−0.122423	−	0.992478i	$-0.539066\pi$
$887$	−7.29213	−0.244846	−0.122423	+	0.992478i	$0.539066\pi$
$888$	0	0
$889$	39.9905	1.34124
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.39768	−0.147163
$894$	0	0
$895$	−73.6947	−2.46334
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.89873	−0.130030
$900$	0	0
$901$	−4.87726	−0.162485
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27.8099	0.924434
$906$	0	0
$907$	−39.6456	−1.31641	−0.658204	−	0.752839i	$-0.728684\pi$
$907$	−39.6456	−1.31641	−0.658204	+	0.752839i	$0.728684\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10.9341	0.362262	0.181131	−	0.983459i	$-0.442024\pi$
$911$	10.9341	0.362262	0.181131	+	0.983459i	$0.442024\pi$
$912$	0	0
$913$	−27.8762	−0.922567
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.9035	0.459135
$918$	0	0
$919$	0.603265	0.0198999	0.00994994	−	0.999950i	$-0.496833\pi$
$919$	0.603265	0.0198999	0.00994994	+	0.999950i	$0.496833\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.409378	0.0134749
$924$	0	0
$925$	−11.1146	−0.365445
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−34.3751	−1.12781	−0.563905	−	0.825840i	$-0.690702\pi$
$929$	−34.3751	−1.12781	−0.563905	+	0.825840i	$0.690702\pi$
$930$	0	0
$931$	0.707340	0.0231821
$932$	0	0
$933$	0	0
$934$	0	0
$935$	40.7758	1.33351
$936$	0	0
$937$	35.0958	1.14653	0.573265	−	0.819370i	$-0.305677\pi$
$937$	35.0958	1.14653	0.573265	+	0.819370i	$0.305677\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−31.0709	−1.01288	−0.506442	−	0.862274i	$-0.669039\pi$
$941$	−31.0709	−1.01288	−0.506442	+	0.862274i	$0.669039\pi$
$942$	0	0
$943$	15.2962	0.498112
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−49.2023	−1.59886	−0.799430	−	0.600760i	$-0.794865\pi$
$947$	−49.2023	−1.59886	−0.799430	+	0.600760i	$0.794865\pi$
$948$	0	0
$949$	−4.41505	−0.143318
$950$	0	0
$951$	0	0
$952$	0	0
$953$	29.7875	0.964913	0.482457	−	0.875920i	$-0.339745\pi$
$953$	29.7875	0.964913	0.482457	+	0.875920i	$0.339745\pi$
$954$	0	0
$955$	−15.5727	−0.503920
$956$	0	0
$957$	0	0
$958$	0	0
$959$	30.1900	0.974885
$960$	0	0
$961$	−27.2437	−0.878830
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.8440	0.445655
$966$	0	0
$967$	55.1653	1.77400	0.886999	−	0.461772i	$-0.152786\pi$
$967$	55.1653	1.77400	0.886999	+	0.461772i	$0.152786\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−23.5605	−0.756093	−0.378046	−	0.925787i	$-0.623404\pi$
$971$	−23.5605	−0.756093	−0.378046	+	0.925787i	$0.623404\pi$
$972$	0	0
$973$	−37.1800	−1.19194
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.8486	−0.986935	−0.493467	−	0.869764i	$-0.664271\pi$
$977$	−30.8486	−0.986935	−0.493467	+	0.869764i	$0.664271\pi$
$978$	0	0
$979$	26.4202	0.844392
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0.964602	0.0307660	0.0153830	−	0.999882i	$-0.495103\pi$
$983$	0.964602	0.0307660	0.0153830	+	0.999882i	$0.495103\pi$
$984$	0	0
$985$	−78.8080	−2.51103
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.32165	−0.232815
$990$	0	0
$991$	−59.2666	−1.88266	−0.941332	−	0.337481i	$-0.890425\pi$
$991$	−59.2666	−1.88266	−0.941332	+	0.337481i	$0.890425\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−86.5782	−2.74471
$996$	0	0
$997$	47.8885	1.51664	0.758322	−	0.651880i	$-0.226019\pi$
$997$	47.8885	1.51664	0.758322	+	0.651880i	$0.226019\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.ca.1.5		5
3.2	odd	2		8208.2.a.cd.1.1		5
4.3	odd	2		4104.2.a.p.1.5	✓	5
12.11	even	2		4104.2.a.q.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4104.2.a.p.1.5	✓	5	4.3	odd	2
4104.2.a.q.1.1	yes	5	12.11	even	2
8208.2.a.ca.1.5		5	1.1	even	1	trivial
8208.2.a.cd.1.1		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8208 = 2^{4} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8208.a (trivial)

\(f(q)\)	\(=\)	\(q+3.32769 q^{5} -2.77621 q^{7} -2.53271 q^{11} +0.745822 q^{13} -4.83810 q^{17} +1.00000 q^{19} +1.29730 q^{23} +6.07351 q^{25} +2.01162 q^{29} -1.93811 q^{31} -9.23836 q^{35} -1.83001 q^{37} +11.7908 q^{41} -5.64376 q^{43} -4.39768 q^{47} +0.707340 q^{49} +1.00809 q^{53} -8.42806 q^{55} +0.602695 q^{59} +4.18161 q^{61} +2.48186 q^{65} -1.92649 q^{67} +0.548896 q^{71} -5.91971 q^{73} +7.03133 q^{77} -8.96335 q^{79} +11.0065 q^{83} -16.0997 q^{85} -10.4316 q^{89} -2.07056 q^{91} +3.32769 q^{95} +16.0146 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{5} - q^{7} - 4 q^{11} + 3 q^{13} - 5 q^{17} + 5 q^{19} - q^{23} + 10 q^{25} - 4 q^{29} - 16 q^{31} - 6 q^{35} + 7 q^{37} - 7 q^{41} - 3 q^{43} + 7 q^{47} + 18 q^{49} + 2 q^{53} - q^{55} - 15 q^{59}+ \cdots + 27 q^{97}+O(q^{100}) \) 5 * q - 3 * q^5 - q^7 - 4 * q^11 + 3 * q^13 - 5 * q^17 + 5 * q^19 - q^23 + 10 * q^25 - 4 * q^29 - 16 * q^31 - 6 * q^35 + 7 * q^37 - 7 * q^41 - 3 * q^43 + 7 * q^47 + 18 * q^49 + 2 * q^53 - q^55 - 15 * q^59 + 23 * q^61 - 32 * q^65 - 30 * q^67 - 12 * q^71 + 13 * q^73 + 4 * q^77 - 3 * q^79 + 5 * q^83 + 7 * q^85 - 29 * q^91 - 3 * q^95 + 27 * q^97

Introduction

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