LMFDB - Embedded newform 6003.2.a.t.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6003,2,Mod(1,6003)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6003, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6003.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.a (trivial)

Newform invariants

comment:select newform

sage:traces = [22,-3,0,17,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	6003.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.424349 q^{2} -1.81993 q^{4} -2.43761 q^{5} +3.41293 q^{7} +1.62098 q^{8} +1.03440 q^{10} +5.84264 q^{11} -6.73637 q^{13} -1.44828 q^{14} +2.95199 q^{16} -4.57994 q^{17} +3.36654 q^{19} +4.43628 q^{20} -2.47932 q^{22} +1.00000 q^{23} +0.941962 q^{25} +2.85857 q^{26} -6.21129 q^{28} +1.00000 q^{29} -2.25875 q^{31} -4.49464 q^{32} +1.94349 q^{34} -8.31942 q^{35} -4.84124 q^{37} -1.42859 q^{38} -3.95133 q^{40} -4.80077 q^{41} +7.09307 q^{43} -10.6332 q^{44} -0.424349 q^{46} -3.50228 q^{47} +4.64812 q^{49} -0.399721 q^{50} +12.2597 q^{52} +0.803109 q^{53} -14.2421 q^{55} +5.53231 q^{56} -0.424349 q^{58} +8.37512 q^{59} -12.3218 q^{61} +0.958499 q^{62} -3.99669 q^{64} +16.4207 q^{65} +7.03184 q^{67} +8.33516 q^{68} +3.53034 q^{70} +6.87668 q^{71} -3.29707 q^{73} +2.05438 q^{74} -6.12686 q^{76} +19.9406 q^{77} -12.0450 q^{79} -7.19582 q^{80} +2.03720 q^{82} +15.2301 q^{83} +11.1641 q^{85} -3.00994 q^{86} +9.47082 q^{88} +7.50144 q^{89} -22.9908 q^{91} -1.81993 q^{92} +1.48619 q^{94} -8.20633 q^{95} -9.65226 q^{97} -1.97243 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

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 $

$2$	−0.424349	−0.300060	−0.150030	−	0.988681i	$-0.547937\pi$
$2$	−0.424349	−0.300060	−0.150030	+	0.988681i	$0.547937\pi$
$3$	0	0
$3$	0	0
$4$	−1.81993	−0.909964
$5$	−2.43761	−1.09013	−0.545067	−	0.838392i	$-0.683496\pi$
$5$	−2.43761	−1.09013	−0.545067	+	0.838392i	$0.683496\pi$
$6$	0	0
$7$	3.41293	1.28997	0.644984	−	0.764196i	$-0.276864\pi$
$7$	3.41293	1.28997	0.644984	+	0.764196i	$0.276864\pi$
$8$	1.62098	0.573104
$9$	0	0
$10$	1.03440	0.327106
$11$	5.84264	1.76162	0.880811	−	0.473468i	$-0.156998\pi$
$11$	5.84264	1.76162	0.880811	+	0.473468i	$0.156998\pi$
$12$	0	0
$13$	−6.73637	−1.86833	−0.934167	−	0.356837i	$-0.883855\pi$
$13$	−6.73637	−1.86833	−0.934167	+	0.356837i	$0.883855\pi$
$14$	−1.44828	−0.387068
$15$	0	0
$16$	2.95199	0.737998
$17$	−4.57994	−1.11080	−0.555399	−	0.831584i	$-0.687435\pi$
$17$	−4.57994	−1.11080	−0.555399	+	0.831584i	$0.687435\pi$
$18$	0	0
$19$	3.36654	0.772337	0.386169	−	0.922428i	$-0.373798\pi$
$19$	3.36654	0.772337	0.386169	+	0.922428i	$0.373798\pi$
$20$	4.43628	0.991983
$21$	0	0
$22$	−2.47932	−0.528593
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0.941962	0.188392
$26$	2.85857	0.560612
$27$	0	0
$28$	−6.21129	−1.17382
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−2.25875	−0.405684	−0.202842	−	0.979212i	$-0.565018\pi$
$31$	−2.25875	−0.405684	−0.202842	+	0.979212i	$0.565018\pi$
$32$	−4.49464	−0.794548
$33$	0	0
$34$	1.94349	0.333306
$35$	−8.31942	−1.40624
$36$	0	0
$37$	−4.84124	−0.795895	−0.397948	−	0.917408i	$-0.630277\pi$
$37$	−4.84124	−0.795895	−0.397948	+	0.917408i	$0.630277\pi$
$38$	−1.42859	−0.231748
$39$	0	0
$40$	−3.95133	−0.624760
$41$	−4.80077	−0.749754	−0.374877	−	0.927075i	$-0.622315\pi$
$41$	−4.80077	−0.749754	−0.374877	+	0.927075i	$0.622315\pi$
$42$	0	0
$43$	7.09307	1.08168	0.540841	−	0.841125i	$-0.318106\pi$
$43$	7.09307	1.08168	0.540841	+	0.841125i	$0.318106\pi$
$44$	−10.6332	−1.60301
$45$	0	0
$46$	−0.424349	−0.0625669
$47$	−3.50228	−0.510860	−0.255430	−	0.966828i	$-0.582217\pi$
$47$	−3.50228	−0.510860	−0.255430	+	0.966828i	$0.582217\pi$
$48$	0	0
$49$	4.64812	0.664017
$50$	−0.399721	−0.0565291
$51$	0	0
$52$	12.2597	1.70012
$53$	0.803109	0.110316	0.0551578	−	0.998478i	$-0.482434\pi$
$53$	0.803109	0.110316	0.0551578	+	0.998478i	$0.482434\pi$
$54$	0	0
$55$	−14.2421	−1.92040
$56$	5.53231	0.739286
$57$	0	0
$58$	−0.424349	−0.0557198
$59$	8.37512	1.09035	0.545174	−	0.838323i	$-0.316464\pi$
$59$	8.37512	1.09035	0.545174	+	0.838323i	$0.316464\pi$
$60$	0	0
$61$	−12.3218	−1.57764	−0.788821	−	0.614623i	$-0.789308\pi$
$61$	−12.3218	−1.57764	−0.788821	+	0.614623i	$0.789308\pi$
$62$	0.958499	0.121729
$63$	0	0
$64$	−3.99669	−0.499586
$65$	16.4207	2.03673
$66$	0	0
$67$	7.03184	0.859076	0.429538	−	0.903049i	$-0.358677\pi$
$67$	7.03184	0.859076	0.429538	+	0.903049i	$0.358677\pi$
$68$	8.33516	1.01079
$69$	0	0
$70$	3.53034	0.421956
$71$	6.87668	0.816112	0.408056	−	0.912957i	$-0.366207\pi$
$71$	6.87668	0.816112	0.408056	+	0.912957i	$0.366207\pi$
$72$	0	0
$73$	−3.29707	−0.385893	−0.192946	−	0.981209i	$-0.561804\pi$
$73$	−3.29707	−0.385893	−0.192946	+	0.981209i	$0.561804\pi$
$74$	2.05438	0.238816
$75$	0	0
$76$	−6.12686	−0.702799
$77$	19.9406	2.27244
$78$	0	0
$79$	−12.0450	−1.35517	−0.677584	−	0.735445i	$-0.736973\pi$
$79$	−12.0450	−1.35517	−0.677584	+	0.735445i	$0.736973\pi$
$80$	−7.19582	−0.804517
$81$	0	0
$82$	2.03720	0.224971
$83$	15.2301	1.67172	0.835858	−	0.548945i	$-0.184970\pi$
$83$	15.2301	1.67172	0.835858	+	0.548945i	$0.184970\pi$
$84$	0	0
$85$	11.1641	1.21092
$86$	−3.00994	−0.324570
$87$	0	0
$88$	9.47082	1.00959
$89$	7.50144	0.795151	0.397575	−	0.917570i	$-0.369852\pi$
$89$	7.50144	0.795151	0.397575	+	0.917570i	$0.369852\pi$
$90$	0	0
$91$	−22.9908	−2.41009
$92$	−1.81993	−0.189741
$93$	0	0
$94$	1.48619	0.153289
$95$	−8.20633	−0.841951
$96$	0	0
$97$	−9.65226	−0.980038	−0.490019	−	0.871712i	$-0.663010\pi$
$97$	−9.65226	−0.980038	−0.490019	+	0.871712i	$0.663010\pi$
$98$	−1.97243	−0.199245
$99$	0	0
$100$	−1.71430	−0.171430
$101$	6.89515	0.686094	0.343047	−	0.939318i	$-0.388541\pi$
$101$	6.89515	0.686094	0.343047	+	0.939318i	$0.388541\pi$
$102$	0	0
$103$	15.4211	1.51948	0.759742	−	0.650224i	$-0.225325\pi$
$103$	15.4211	1.51948	0.759742	+	0.650224i	$0.225325\pi$
$104$	−10.9195	−1.07075
$105$	0	0
$106$	−0.340799	−0.0331013
$107$	14.2228	1.37497	0.687486	−	0.726198i	$-0.258714\pi$
$107$	14.2228	1.37497	0.687486	+	0.726198i	$0.258714\pi$
$108$	0	0
$109$	6.67649	0.639492	0.319746	−	0.947503i	$-0.396403\pi$
$109$	6.67649	0.639492	0.319746	+	0.947503i	$0.396403\pi$
$110$	6.04362	0.576237
$111$	0	0
$112$	10.0750	0.951994
$113$	−11.2542	−1.05870	−0.529352	−	0.848402i	$-0.677565\pi$
$113$	−11.2542	−1.05870	−0.529352	+	0.848402i	$0.677565\pi$
$114$	0	0
$115$	−2.43761	−0.227309
$116$	−1.81993	−0.168976
$117$	0	0
$118$	−3.55397	−0.327170
$119$	−15.6310	−1.43289
$120$	0	0
$121$	23.1365	2.10331
$122$	5.22874	0.473388
$123$	0	0
$124$	4.11076	0.369157
$125$	9.89193	0.884761
$126$	0	0
$127$	−17.8522	−1.58413	−0.792063	−	0.610440i	$-0.790993\pi$
$127$	−17.8522	−1.58413	−0.792063	+	0.610440i	$0.790993\pi$
$128$	10.6853	0.944454
$129$	0	0
$130$	−6.96810	−0.611143
$131$	−6.10711	−0.533581	−0.266790	−	0.963755i	$-0.585963\pi$
$131$	−6.10711	−0.533581	−0.266790	+	0.963755i	$0.585963\pi$
$132$	0	0
$133$	11.4898	0.996291
$134$	−2.98395	−0.257774
$135$	0	0
$136$	−7.42401	−0.636603
$137$	−16.6372	−1.42141	−0.710707	−	0.703488i	$-0.751625\pi$
$137$	−16.6372	−1.42141	−0.710707	+	0.703488i	$0.751625\pi$
$138$	0	0
$139$	−17.3748	−1.47371	−0.736855	−	0.676051i	$-0.763690\pi$
$139$	−17.3748	−1.47371	−0.736855	+	0.676051i	$0.763690\pi$
$140$	15.1407	1.27963
$141$	0	0
$142$	−2.91811	−0.244883
$143$	−39.3582	−3.29130
$144$	0	0
$145$	−2.43761	−0.202433
$146$	1.39911	0.115791
$147$	0	0
$148$	8.81071	0.724236
$149$	−15.5254	−1.27189	−0.635947	−	0.771733i	$-0.719390\pi$
$149$	−15.5254	−1.27189	−0.635947	+	0.771733i	$0.719390\pi$
$150$	0	0
$151$	−2.45313	−0.199633	−0.0998165	−	0.995006i	$-0.531826\pi$
$151$	−2.45313	−0.199633	−0.0998165	+	0.995006i	$0.531826\pi$
$152$	5.45711	0.442630
$153$	0	0
$154$	−8.46176	−0.681868
$155$	5.50596	0.442250
$156$	0	0
$157$	−2.35821	−0.188205	−0.0941027	−	0.995562i	$-0.529998\pi$
$157$	−2.35821	−0.188205	−0.0941027	+	0.995562i	$0.529998\pi$
$158$	5.11129	0.406632
$159$	0	0
$160$	10.9562	0.866164
$161$	3.41293	0.268977
$162$	0	0
$163$	−16.3414	−1.27996	−0.639978	−	0.768393i	$-0.721057\pi$
$163$	−16.3414	−1.27996	−0.639978	+	0.768393i	$0.721057\pi$
$164$	8.73705	0.682249
$165$	0	0
$166$	−6.46287	−0.501616
$167$	−5.89276	−0.455995	−0.227998	−	0.973662i	$-0.573218\pi$
$167$	−5.89276	−0.455995	−0.227998	+	0.973662i	$0.573218\pi$
$168$	0	0
$169$	32.3787	2.49067
$170$	−4.73749	−0.363349
$171$	0	0
$172$	−12.9089	−0.984292
$173$	−1.20279	−0.0914467	−0.0457234	−	0.998954i	$-0.514559\pi$
$173$	−1.20279	−0.0914467	−0.0457234	+	0.998954i	$0.514559\pi$
$174$	0	0
$175$	3.21485	0.243020
$176$	17.2474	1.30007
$177$	0	0
$178$	−3.18323	−0.238593
$179$	−15.5040	−1.15883	−0.579413	−	0.815034i	$-0.696718\pi$
$179$	−15.5040	−1.15883	−0.579413	+	0.815034i	$0.696718\pi$
$180$	0	0
$181$	−23.2463	−1.72789	−0.863943	−	0.503590i	$-0.832012\pi$
$181$	−23.2463	−1.72789	−0.863943	+	0.503590i	$0.832012\pi$
$182$	9.75612	0.723172
$183$	0	0
$184$	1.62098	0.119500
$185$	11.8011	0.867633
$186$	0	0
$187$	−26.7589	−1.95681
$188$	6.37389	0.464864
$189$	0	0
$190$	3.48235	0.252636
$191$	18.0430	1.30555	0.652773	−	0.757553i	$-0.273605\pi$
$191$	18.0430	1.30555	0.652773	+	0.757553i	$0.273605\pi$
$192$	0	0
$193$	6.24298	0.449380	0.224690	−	0.974430i	$-0.427863\pi$
$193$	6.24298	0.449380	0.224690	+	0.974430i	$0.427863\pi$
$194$	4.09593	0.294070
$195$	0	0
$196$	−8.45925	−0.604232
$197$	14.9870	1.06778	0.533891	−	0.845553i	$-0.320729\pi$
$197$	14.9870	1.06778	0.533891	+	0.845553i	$0.320729\pi$
$198$	0	0
$199$	−4.83185	−0.342521	−0.171260	−	0.985226i	$-0.554784\pi$
$199$	−4.83185	−0.342521	−0.171260	+	0.985226i	$0.554784\pi$
$200$	1.52690	0.107968
$201$	0	0
$202$	−2.92595	−0.205869
$203$	3.41293	0.239541
$204$	0	0
$205$	11.7024	0.817332
$206$	−6.54393	−0.455937
$207$	0	0
$208$	−19.8857	−1.37883
$209$	19.6695	1.36057
$210$	0	0
$211$	9.98291	0.687252	0.343626	−	0.939107i	$-0.388345\pi$
$211$	9.98291	0.687252	0.343626	+	0.939107i	$0.388345\pi$
$212$	−1.46160	−0.100383
$213$	0	0
$214$	−6.03544	−0.412574
$215$	−17.2902	−1.17918
$216$	0	0
$217$	−7.70897	−0.523319
$218$	−2.83316	−0.191886
$219$	0	0
$220$	25.9196	1.74750
$221$	30.8522	2.07534
$222$	0	0
$223$	−3.51605	−0.235452	−0.117726	−	0.993046i	$-0.537561\pi$
$223$	−3.51605	−0.235452	−0.117726	+	0.993046i	$0.537561\pi$
$224$	−15.3399	−1.02494
$225$	0	0
$226$	4.77570	0.317675
$227$	−9.22552	−0.612319	−0.306160	−	0.951980i	$-0.599044\pi$
$227$	−9.22552	−0.612319	−0.306160	+	0.951980i	$0.599044\pi$
$228$	0	0
$229$	−8.97788	−0.593275	−0.296638	−	0.954990i	$-0.595865\pi$
$229$	−8.97788	−0.593275	−0.296638	+	0.954990i	$0.595865\pi$
$230$	1.03440	0.0682063
$231$	0	0
$232$	1.62098	0.106423
$233$	−10.5134	−0.688753	−0.344376	−	0.938832i	$-0.611910\pi$
$233$	−10.5134	−0.688753	−0.344376	+	0.938832i	$0.611910\pi$
$234$	0	0
$235$	8.53720	0.556905
$236$	−15.2421	−0.992177
$237$	0	0
$238$	6.63302	0.429955
$239$	14.4794	0.936596	0.468298	−	0.883571i	$-0.344867\pi$
$239$	14.4794	0.936596	0.468298	+	0.883571i	$0.344867\pi$
$240$	0	0
$241$	1.76018	0.113383	0.0566917	−	0.998392i	$-0.481945\pi$
$241$	1.76018	0.113383	0.0566917	+	0.998392i	$0.481945\pi$
$242$	−9.81793	−0.631121
$243$	0	0
$244$	22.4248	1.43560
$245$	−11.3303	−0.723868
$246$	0	0
$247$	−22.6783	−1.44298
$248$	−3.66140	−0.232499
$249$	0	0
$250$	−4.19763	−0.265482
$251$	11.6685	0.736506	0.368253	−	0.929726i	$-0.379956\pi$
$251$	11.6685	0.736506	0.368253	+	0.929726i	$0.379956\pi$
$252$	0	0
$253$	5.84264	0.367324
$254$	7.57556	0.475333
$255$	0	0
$256$	3.45909	0.216193
$257$	9.01106	0.562094	0.281047	−	0.959694i	$-0.409318\pi$
$257$	9.01106	0.562094	0.281047	+	0.959694i	$0.409318\pi$
$258$	0	0
$259$	−16.5228	−1.02668
$260$	−29.8844	−1.85335
$261$	0	0
$262$	2.59155	0.160106
$263$	−21.3050	−1.31372	−0.656861	−	0.754012i	$-0.728116\pi$
$263$	−21.3050	−1.31372	−0.656861	+	0.754012i	$0.728116\pi$
$264$	0	0
$265$	−1.95767	−0.120259
$266$	−4.87568	−0.298947
$267$	0	0
$268$	−12.7974	−0.781728
$269$	6.35901	0.387716	0.193858	−	0.981030i	$-0.437900\pi$
$269$	6.35901	0.387716	0.193858	+	0.981030i	$0.437900\pi$
$270$	0	0
$271$	−27.4931	−1.67009	−0.835044	−	0.550183i	$-0.814558\pi$
$271$	−27.4931	−1.67009	−0.835044	+	0.550183i	$0.814558\pi$
$272$	−13.5200	−0.819767
$273$	0	0
$274$	7.05999	0.426510
$275$	5.50355	0.331876
$276$	0	0
$277$	−29.7357	−1.78664	−0.893322	−	0.449417i	$-0.851632\pi$
$277$	−29.7357	−1.78664	−0.893322	+	0.449417i	$0.851632\pi$
$278$	7.37298	0.442202
$279$	0	0
$280$	−13.4856	−0.805921
$281$	−11.1902	−0.667550	−0.333775	−	0.942653i	$-0.608323\pi$
$281$	−11.1902	−0.667550	−0.333775	+	0.942653i	$0.608323\pi$
$282$	0	0
$283$	17.7812	1.05698	0.528492	−	0.848938i	$-0.322758\pi$
$283$	17.7812	1.05698	0.528492	+	0.848938i	$0.322758\pi$
$284$	−12.5151	−0.742632
$285$	0	0
$286$	16.7016	0.987587
$287$	−16.3847	−0.967158
$288$	0	0
$289$	3.97586	0.233874
$290$	1.03440	0.0607420
$291$	0	0
$292$	6.00043	0.351148
$293$	−15.5413	−0.907931	−0.453966	−	0.891019i	$-0.649991\pi$
$293$	−15.5413	−0.907931	−0.453966	+	0.891019i	$0.649991\pi$
$294$	0	0
$295$	−20.4153	−1.18863
$296$	−7.84757	−0.456131
$297$	0	0
$298$	6.58821	0.381645
$299$	−6.73637	−0.389574
$300$	0	0
$301$	24.2082	1.39534
$302$	1.04098	0.0599019
$303$	0	0
$304$	9.93800	0.569984
$305$	30.0357	1.71984
$306$	0	0
$307$	14.1411	0.807075	0.403538	−	0.914963i	$-0.367780\pi$
$307$	14.1411	0.807075	0.403538	+	0.914963i	$0.367780\pi$
$308$	−36.2904	−2.06784
$309$	0	0
$310$	−2.33645	−0.132701
$311$	17.5960	0.997780	0.498890	−	0.866665i	$-0.333741\pi$
$311$	17.5960	0.997780	0.498890	+	0.866665i	$0.333741\pi$
$312$	0	0
$313$	0.544433	0.0307732	0.0153866	−	0.999882i	$-0.495102\pi$
$313$	0.544433	0.0307732	0.0153866	+	0.999882i	$0.495102\pi$
$314$	1.00070	0.0564729
$315$	0	0
$316$	21.9210	1.23315
$317$	14.2193	0.798635	0.399318	−	0.916813i	$-0.369247\pi$
$317$	14.2193	0.798635	0.399318	+	0.916813i	$0.369247\pi$
$318$	0	0
$319$	5.84264	0.327125
$320$	9.74238	0.544616
$321$	0	0
$322$	−1.44828	−0.0807093
$323$	−15.4186	−0.857912
$324$	0	0
$325$	−6.34540	−0.351980
$326$	6.93445	0.384064
$327$	0	0
$328$	−7.78196	−0.429687
$329$	−11.9530	−0.658992
$330$	0	0
$331$	10.3007	0.566176	0.283088	−	0.959094i	$-0.408641\pi$
$331$	10.3007	0.566176	0.283088	+	0.959094i	$0.408641\pi$
$332$	−27.7176	−1.52120
$333$	0	0
$334$	2.50059	0.136826
$335$	−17.1409	−0.936508
$336$	0	0
$337$	−25.8924	−1.41045	−0.705224	−	0.708984i	$-0.749154\pi$
$337$	−25.8924	−1.41045	−0.705224	+	0.708984i	$0.749154\pi$
$338$	−13.7399	−0.747350
$339$	0	0
$340$	−20.3179	−1.10189
$341$	−13.1971	−0.714661
$342$	0	0
$343$	−8.02681	−0.433407
$344$	11.4977	0.619917
$345$	0	0
$346$	0.510404	0.0274395
$347$	−13.4609	−0.722621	−0.361310	−	0.932446i	$-0.617670\pi$
$347$	−13.4609	−0.722621	−0.361310	+	0.932446i	$0.617670\pi$
$348$	0	0
$349$	−28.3225	−1.51607	−0.758035	−	0.652213i	$-0.773841\pi$
$349$	−28.3225	−1.51607	−0.758035	+	0.652213i	$0.773841\pi$
$350$	−1.36422	−0.0729207
$351$	0	0
$352$	−26.2606	−1.39969
$353$	−10.2989	−0.548156	−0.274078	−	0.961707i	$-0.588373\pi$
$353$	−10.2989	−0.548156	−0.274078	+	0.961707i	$0.588373\pi$
$354$	0	0
$355$	−16.7627	−0.889671
$356$	−13.6521	−0.723559
$357$	0	0
$358$	6.57912	0.347717
$359$	−23.4510	−1.23770	−0.618849	−	0.785510i	$-0.712401\pi$
$359$	−23.4510	−1.23770	−0.618849	+	0.785510i	$0.712401\pi$
$360$	0	0
$361$	−7.66640	−0.403495
$362$	9.86455	0.518469
$363$	0	0
$364$	41.8416	2.19309
$365$	8.03698	0.420675
$366$	0	0
$367$	−19.1389	−0.999040	−0.499520	−	0.866302i	$-0.666490\pi$
$367$	−19.1389	−0.999040	−0.499520	+	0.866302i	$0.666490\pi$
$368$	2.95199	0.153883
$369$	0	0
$370$	−5.00778	−0.260342
$371$	2.74096	0.142304
$372$	0	0
$373$	−36.6987	−1.90019	−0.950094	−	0.311964i	$-0.899013\pi$
$373$	−36.6987	−1.90019	−0.950094	+	0.311964i	$0.899013\pi$
$374$	11.3551	0.587160
$375$	0	0
$376$	−5.67713	−0.292776
$377$	−6.73637	−0.346941
$378$	0	0
$379$	−27.5791	−1.41664	−0.708322	−	0.705889i	$-0.750548\pi$
$379$	−27.5791	−1.41664	−0.708322	+	0.705889i	$0.750548\pi$
$380$	14.9349	0.766145
$381$	0	0
$382$	−7.65654	−0.391743
$383$	−22.2252	−1.13566	−0.567829	−	0.823147i	$-0.692216\pi$
$383$	−22.2252	−1.13566	−0.567829	+	0.823147i	$0.692216\pi$
$384$	0	0
$385$	−48.6074	−2.47726
$386$	−2.64920	−0.134841
$387$	0	0
$388$	17.5664	0.891800
$389$	−8.46057	−0.428968	−0.214484	−	0.976728i	$-0.568807\pi$
$389$	−8.46057	−0.428968	−0.214484	+	0.976728i	$0.568807\pi$
$390$	0	0
$391$	−4.57994	−0.231618
$392$	7.53453	0.380551
$393$	0	0
$394$	−6.35973	−0.320399
$395$	29.3611	1.47732
$396$	0	0
$397$	−27.9140	−1.40096	−0.700481	−	0.713671i	$-0.747031\pi$
$397$	−27.9140	−1.40096	−0.700481	+	0.713671i	$0.747031\pi$
$398$	2.05039	0.102777
$399$	0	0
$400$	2.78066	0.139033
$401$	5.92439	0.295850	0.147925	−	0.988999i	$-0.452741\pi$
$401$	5.92439	0.295850	0.147925	+	0.988999i	$0.452741\pi$
$402$	0	0
$403$	15.2158	0.757952
$404$	−12.5487	−0.624320
$405$	0	0
$406$	−1.44828	−0.0718767
$407$	−28.2856	−1.40207
$408$	0	0
$409$	14.3485	0.709487	0.354744	−	0.934964i	$-0.384568\pi$
$409$	14.3485	0.709487	0.354744	+	0.934964i	$0.384568\pi$
$410$	−4.96591	−0.245249
$411$	0	0
$412$	−28.0653	−1.38268
$413$	28.5837	1.40651
$414$	0	0
$415$	−37.1250	−1.82240
$416$	30.2776	1.48448
$417$	0	0
$418$	−8.34673	−0.408252
$419$	−36.2582	−1.77133	−0.885665	−	0.464325i	$-0.846297\pi$
$419$	−36.2582	−1.77133	−0.885665	+	0.464325i	$0.846297\pi$
$420$	0	0
$421$	17.7411	0.864650	0.432325	−	0.901718i	$-0.357693\pi$
$421$	17.7411	0.864650	0.432325	+	0.901718i	$0.357693\pi$
$422$	−4.23624	−0.206217
$423$	0	0
$424$	1.30183	0.0632223
$425$	−4.31413	−0.209266
$426$	0	0
$427$	−42.0534	−2.03511
$428$	−25.8845	−1.25118
$429$	0	0
$430$	7.33707	0.353825
$431$	18.7348	0.902422	0.451211	−	0.892417i	$-0.350992\pi$
$431$	18.7348	0.902422	0.451211	+	0.892417i	$0.350992\pi$
$432$	0	0
$433$	4.71993	0.226825	0.113413	−	0.993548i	$-0.463822\pi$
$433$	4.71993	0.226825	0.113413	+	0.993548i	$0.463822\pi$
$434$	3.27129	0.157027
$435$	0	0
$436$	−12.1507	−0.581914
$437$	3.36654	0.161043
$438$	0	0
$439$	24.6405	1.17603	0.588014	−	0.808851i	$-0.299910\pi$
$439$	24.6405	1.17603	0.588014	+	0.808851i	$0.299910\pi$
$440$	−23.0862	−1.10059
$441$	0	0
$442$	−13.0921	−0.622727
$443$	−11.2204	−0.533097	−0.266549	−	0.963821i	$-0.585883\pi$
$443$	−11.2204	−0.533097	−0.266549	+	0.963821i	$0.585883\pi$
$444$	0	0
$445$	−18.2856	−0.866821
$446$	1.49203	0.0706499
$447$	0	0
$448$	−13.6404	−0.644450
$449$	−8.68202	−0.409730	−0.204865	−	0.978790i	$-0.565676\pi$
$449$	−8.68202	−0.409730	−0.204865	+	0.978790i	$0.565676\pi$
$450$	0	0
$451$	−28.0492	−1.32078
$452$	20.4818	0.963382
$453$	0	0
$454$	3.91484	0.183733
$455$	56.0427	2.62732
$456$	0	0
$457$	37.9401	1.77476	0.887382	−	0.461035i	$-0.152522\pi$
$457$	37.9401	1.77476	0.887382	+	0.461035i	$0.152522\pi$
$458$	3.80976	0.178018
$459$	0	0
$460$	4.43628	0.206843
$461$	40.1649	1.87066	0.935332	−	0.353770i	$-0.115100\pi$
$461$	40.1649	1.87066	0.935332	+	0.353770i	$0.115100\pi$
$462$	0	0
$463$	−29.2381	−1.35881	−0.679404	−	0.733764i	$-0.737762\pi$
$463$	−29.2381	−1.35881	−0.679404	+	0.733764i	$0.737762\pi$
$464$	2.95199	0.137043
$465$	0	0
$466$	4.46133	0.206667
$467$	−32.6854	−1.51250	−0.756250	−	0.654283i	$-0.772971\pi$
$467$	−32.6854	−1.51250	−0.756250	+	0.654283i	$0.772971\pi$
$468$	0	0
$469$	23.9992	1.10818
$470$	−3.62275	−0.167105
$471$	0	0
$472$	13.5759	0.624883
$473$	41.4423	1.90552
$474$	0	0
$475$	3.17115	0.145503
$476$	28.4474	1.30388
$477$	0	0
$478$	−6.14433	−0.281035
$479$	2.99873	0.137016	0.0685078	−	0.997651i	$-0.478176\pi$
$479$	2.99873	0.137016	0.0685078	+	0.997651i	$0.478176\pi$
$480$	0	0
$481$	32.6124	1.48700
$482$	−0.746932	−0.0340218
$483$	0	0
$484$	−42.1067	−1.91394
$485$	23.5285	1.06837
$486$	0	0
$487$	5.57967	0.252839	0.126420	−	0.991977i	$-0.459651\pi$
$487$	5.57967	0.252839	0.126420	+	0.991977i	$0.459651\pi$
$488$	−19.9734	−0.904153
$489$	0	0
$490$	4.80801	0.217204
$491$	−15.3909	−0.694583	−0.347292	−	0.937757i	$-0.612899\pi$
$491$	−15.3909	−0.694583	−0.347292	+	0.937757i	$0.612899\pi$
$492$	0	0
$493$	−4.57994	−0.206270
$494$	9.62350	0.432982
$495$	0	0
$496$	−6.66782	−0.299394
$497$	23.4697	1.05276
$498$	0	0
$499$	−0.711042	−0.0318306	−0.0159153	−	0.999873i	$-0.505066\pi$
$499$	−0.711042	−0.0318306	−0.0159153	+	0.999873i	$0.505066\pi$
$500$	−18.0026	−0.805101
$501$	0	0
$502$	−4.95150	−0.220996
$503$	7.73183	0.344745	0.172373	−	0.985032i	$-0.444857\pi$
$503$	7.73183	0.344745	0.172373	+	0.985032i	$0.444857\pi$
$504$	0	0
$505$	−16.8077	−0.747934
$506$	−2.47932	−0.110219
$507$	0	0
$508$	32.4897	1.44150
$509$	−29.1832	−1.29352	−0.646760	−	0.762693i	$-0.723877\pi$
$509$	−29.1832	−1.29352	−0.646760	+	0.762693i	$0.723877\pi$
$510$	0	0
$511$	−11.2527	−0.497789
$512$	−22.8384	−1.00932
$513$	0	0
$514$	−3.82383	−0.168662
$515$	−37.5907	−1.65644
$516$	0	0
$517$	−20.4625	−0.899942
$518$	7.01145	0.308066
$519$	0	0
$520$	26.6176	1.16726
$521$	31.2501	1.36909	0.684547	−	0.728969i	$-0.260000\pi$
$521$	31.2501	1.36909	0.684547	+	0.728969i	$0.260000\pi$
$522$	0	0
$523$	8.76052	0.383071	0.191535	−	0.981486i	$-0.438653\pi$
$523$	8.76052	0.383071	0.191535	+	0.981486i	$0.438653\pi$
$524$	11.1145	0.485539
$525$	0	0
$526$	9.04075	0.394196
$527$	10.3449	0.450633
$528$	0	0
$529$	1.00000	0.0434783
$530$	0.830736	0.0360849
$531$	0	0
$532$	−20.9106	−0.906588
$533$	32.3397	1.40079
$534$	0	0
$535$	−34.6698	−1.49890
$536$	11.3985	0.492340
$537$	0	0
$538$	−2.69844	−0.116338
$539$	27.1573	1.16975
$540$	0	0
$541$	−27.5448	−1.18424	−0.592121	−	0.805849i	$-0.701709\pi$
$541$	−27.5448	−1.18424	−0.592121	+	0.805849i	$0.701709\pi$
$542$	11.6667	0.501127
$543$	0	0
$544$	20.5852	0.882583
$545$	−16.2747	−0.697132
$546$	0	0
$547$	10.6989	0.457451	0.228725	−	0.973491i	$-0.426544\pi$
$547$	10.6989	0.457451	0.228725	+	0.973491i	$0.426544\pi$
$548$	30.2785	1.29343
$549$	0	0
$550$	−2.33542	−0.0995829
$551$	3.36654	0.143419
$552$	0	0
$553$	−41.1088	−1.74812
$554$	12.6183	0.536101
$555$	0	0
$556$	31.6209	1.34102
$557$	−14.8501	−0.629217	−0.314608	−	0.949222i	$-0.601873\pi$
$557$	−14.8501	−0.629217	−0.314608	+	0.949222i	$0.601873\pi$
$558$	0	0
$559$	−47.7815	−2.02094
$560$	−24.5589	−1.03780
$561$	0	0
$562$	4.74855	0.200305
$563$	−32.9448	−1.38846	−0.694229	−	0.719754i	$-0.744255\pi$
$563$	−32.9448	−1.38846	−0.694229	+	0.719754i	$0.744255\pi$
$564$	0	0
$565$	27.4333	1.15413
$566$	−7.54545	−0.317159
$567$	0	0
$568$	11.1470	0.467717
$569$	−0.826143	−0.0346337	−0.0173169	−	0.999850i	$-0.505512\pi$
$569$	−0.826143	−0.0346337	−0.0173169	+	0.999850i	$0.505512\pi$
$570$	0	0
$571$	26.0145	1.08867	0.544337	−	0.838867i	$-0.316781\pi$
$571$	26.0145	1.08867	0.544337	+	0.838867i	$0.316781\pi$
$572$	71.6291	2.99496
$573$	0	0
$574$	6.95283	0.290206
$575$	0.941962	0.0392825
$576$	0	0
$577$	−17.0333	−0.709104	−0.354552	−	0.935036i	$-0.615367\pi$
$577$	−17.0333	−0.709104	−0.354552	+	0.935036i	$0.615367\pi$
$578$	−1.68715	−0.0701762
$579$	0	0
$580$	4.43628	0.184207
$581$	51.9792	2.15646
$582$	0	0
$583$	4.69228	0.194334
$584$	−5.34449	−0.221157
$585$	0	0
$586$	6.59493	0.272434
$587$	−4.52833	−0.186904	−0.0934522	−	0.995624i	$-0.529790\pi$
$587$	−4.52833	−0.186904	−0.0934522	+	0.995624i	$0.529790\pi$
$588$	0	0
$589$	−7.60418	−0.313325
$590$	8.66322	0.356659
$591$	0	0
$592$	−14.2913	−0.587369
$593$	32.9549	1.35330	0.676648	−	0.736307i	$-0.263432\pi$
$593$	32.9549	1.35330	0.676648	+	0.736307i	$0.263432\pi$
$594$	0	0
$595$	38.1024	1.56205
$596$	28.2552	1.15738
$597$	0	0
$598$	2.85857	0.116896
$599$	19.3328	0.789918	0.394959	−	0.918699i	$-0.370759\pi$
$599$	19.3328	0.789918	0.394959	+	0.918699i	$0.370759\pi$
$600$	0	0
$601$	20.3794	0.831294	0.415647	−	0.909526i	$-0.363555\pi$
$601$	20.3794	0.831294	0.415647	+	0.909526i	$0.363555\pi$
$602$	−10.2727	−0.418685
$603$	0	0
$604$	4.46452	0.181659
$605$	−56.3977	−2.29289
$606$	0	0
$607$	21.5119	0.873143	0.436571	−	0.899670i	$-0.356193\pi$
$607$	21.5119	0.873143	0.436571	+	0.899670i	$0.356193\pi$
$608$	−15.1314	−0.613659
$609$	0	0
$610$	−12.7456	−0.516056
$611$	23.5926	0.954456
$612$	0	0
$613$	8.08436	0.326524	0.163262	−	0.986583i	$-0.447798\pi$
$613$	8.08436	0.326524	0.163262	+	0.986583i	$0.447798\pi$
$614$	−6.00076	−0.242171
$615$	0	0
$616$	32.3233	1.30234
$617$	−5.79105	−0.233139	−0.116570	−	0.993183i	$-0.537190\pi$
$617$	−5.79105	−0.233139	−0.116570	+	0.993183i	$0.537190\pi$
$618$	0	0
$619$	−22.5549	−0.906559	−0.453279	−	0.891368i	$-0.649746\pi$
$619$	−22.5549	−0.906559	−0.453279	+	0.891368i	$0.649746\pi$
$620$	−10.0205	−0.402431
$621$	0	0
$622$	−7.46687	−0.299394
$623$	25.6019	1.02572
$624$	0	0
$625$	−28.8225	−1.15290
$626$	−0.231030	−0.00923380
$627$	0	0
$628$	4.29177	0.171260
$629$	22.1726	0.884080
$630$	0	0
$631$	45.2346	1.80076	0.900381	−	0.435101i	$-0.143287\pi$
$631$	45.2346	1.80076	0.900381	+	0.435101i	$0.143287\pi$
$632$	−19.5247	−0.776653
$633$	0	0
$634$	−6.03395	−0.239639
$635$	43.5167	1.72691
$636$	0	0
$637$	−31.3115	−1.24061
$638$	−2.47932	−0.0981572
$639$	0	0
$640$	−26.0466	−1.02958
$641$	−23.2117	−0.916809	−0.458404	−	0.888744i	$-0.651579\pi$
$641$	−23.2117	−0.916809	−0.458404	+	0.888744i	$0.651579\pi$
$642$	0	0
$643$	7.33923	0.289431	0.144715	−	0.989473i	$-0.453773\pi$
$643$	7.33923	0.289431	0.144715	+	0.989473i	$0.453773\pi$
$644$	−6.21129	−0.244759
$645$	0	0
$646$	6.54285	0.257425
$647$	22.3876	0.880148	0.440074	−	0.897962i	$-0.354952\pi$
$647$	22.3876	0.880148	0.440074	+	0.897962i	$0.354952\pi$
$648$	0	0
$649$	48.9328	1.92078
$650$	2.69267	0.105615
$651$	0	0
$652$	29.7401	1.16471
$653$	19.8442	0.776564	0.388282	−	0.921541i	$-0.373069\pi$
$653$	19.8442	0.776564	0.388282	+	0.921541i	$0.373069\pi$
$654$	0	0
$655$	14.8868	0.581675
$656$	−14.1718	−0.553317
$657$	0	0
$658$	5.07226	0.197737
$659$	−10.2160	−0.397958	−0.198979	−	0.980004i	$-0.563763\pi$
$659$	−10.2160	−0.397958	−0.198979	+	0.980004i	$0.563763\pi$
$660$	0	0
$661$	−12.0111	−0.467179	−0.233589	−	0.972335i	$-0.575047\pi$
$661$	−12.0111	−0.467179	−0.233589	+	0.972335i	$0.575047\pi$
$662$	−4.37108	−0.169887
$663$	0	0
$664$	24.6877	0.958068
$665$	−28.0077	−1.08609
$666$	0	0
$667$	1.00000	0.0387202
$668$	10.7244	0.414939
$669$	0	0
$670$	7.27373	0.281009
$671$	−71.9917	−2.77921
$672$	0	0
$673$	24.1569	0.931179	0.465590	−	0.885001i	$-0.345842\pi$
$673$	24.1569	0.931179	0.465590	+	0.885001i	$0.345842\pi$
$674$	10.9874	0.423220
$675$	0	0
$676$	−58.9269	−2.26642
$677$	−45.3916	−1.74454	−0.872271	−	0.489023i	$-0.837353\pi$
$677$	−45.3916	−1.74454	−0.872271	+	0.489023i	$0.837353\pi$
$678$	0	0
$679$	−32.9425	−1.26422
$680$	18.0969	0.693983
$681$	0	0
$682$	5.60016	0.214441
$683$	29.6148	1.13318	0.566589	−	0.824000i	$-0.308263\pi$
$683$	29.6148	1.13318	0.566589	+	0.824000i	$0.308263\pi$
$684$	0	0
$685$	40.5551	1.54953
$686$	3.40617	0.130048
$687$	0	0
$688$	20.9387	0.798280
$689$	−5.41004	−0.206106
$690$	0	0
$691$	25.0680	0.953633	0.476817	−	0.879003i	$-0.341791\pi$
$691$	25.0680	0.953633	0.476817	+	0.879003i	$0.341791\pi$
$692$	2.18900	0.0832132
$693$	0	0
$694$	5.71213	0.216830
$695$	42.3530	1.60654
$696$	0	0
$697$	21.9872	0.832826
$698$	12.0186	0.454912
$699$	0	0
$700$	−5.85080	−0.221140
$701$	−6.49421	−0.245283	−0.122641	−	0.992451i	$-0.539136\pi$
$701$	−6.49421	−0.245283	−0.122641	+	0.992451i	$0.539136\pi$
$702$	0	0
$703$	−16.2982	−0.614700
$704$	−23.3512	−0.880082
$705$	0	0
$706$	4.37034	0.164480
$707$	23.5327	0.885039
$708$	0	0
$709$	2.52860	0.0949635	0.0474817	−	0.998872i	$-0.484880\pi$
$709$	2.52860	0.0949635	0.0474817	+	0.998872i	$0.484880\pi$
$710$	7.11323	0.266955
$711$	0	0
$712$	12.1597	0.455704
$713$	−2.25875	−0.0845909
$714$	0	0
$715$	95.9401	3.58796
$716$	28.2162	1.05449
$717$	0	0
$718$	9.95142	0.371384
$719$	−50.7768	−1.89366	−0.946828	−	0.321739i	$-0.895733\pi$
$719$	−50.7768	−1.89366	−0.946828	+	0.321739i	$0.895733\pi$
$720$	0	0
$721$	52.6312	1.96009
$722$	3.25323	0.121073
$723$	0	0
$724$	42.3066	1.57231
$725$	0.941962	0.0349836
$726$	0	0
$727$	−40.8691	−1.51575	−0.757876	−	0.652399i	$-0.773763\pi$
$727$	−40.8691	−1.51575	−0.757876	+	0.652399i	$0.773763\pi$
$728$	−37.2677	−1.38123
$729$	0	0
$730$	−3.41049	−0.126228
$731$	−32.4858	−1.20153
$732$	0	0
$733$	−25.3863	−0.937665	−0.468832	−	0.883287i	$-0.655325\pi$
$733$	−25.3863	−0.937665	−0.468832	+	0.883287i	$0.655325\pi$
$734$	8.12156	0.299772
$735$	0	0
$736$	−4.49464	−0.165675
$737$	41.0845	1.51337
$738$	0	0
$739$	−21.1947	−0.779658	−0.389829	−	0.920887i	$-0.627466\pi$
$739$	−21.1947	−0.779658	−0.389829	+	0.920887i	$0.627466\pi$
$740$	−21.4771	−0.789514
$741$	0	0
$742$	−1.16312	−0.0426996
$743$	−33.9818	−1.24667	−0.623336	−	0.781954i	$-0.714223\pi$
$743$	−33.9818	−1.24667	−0.623336	+	0.781954i	$0.714223\pi$
$744$	0	0
$745$	37.8450	1.38653
$746$	15.5731	0.570171
$747$	0	0
$748$	48.6994	1.78062
$749$	48.5416	1.77367
$750$	0	0
$751$	8.40956	0.306869	0.153435	−	0.988159i	$-0.450967\pi$
$751$	8.40956	0.306869	0.153435	+	0.988159i	$0.450967\pi$
$752$	−10.3387	−0.377013
$753$	0	0
$754$	2.85857	0.104103
$755$	5.97979	0.217627
$756$	0	0
$757$	25.3310	0.920670	0.460335	−	0.887745i	$-0.347729\pi$
$757$	25.3310	0.920670	0.460335	+	0.887745i	$0.347729\pi$
$758$	11.7032	0.425079
$759$	0	0
$760$	−13.3023	−0.482526
$761$	20.1122	0.729067	0.364534	−	0.931190i	$-0.381228\pi$
$761$	20.1122	0.729067	0.364534	+	0.931190i	$0.381228\pi$
$762$	0	0
$763$	22.7864	0.824924
$764$	−32.8370	−1.18800
$765$	0	0
$766$	9.43126	0.340765
$767$	−56.4179	−2.03713
$768$	0	0
$769$	−28.2509	−1.01875	−0.509377	−	0.860543i	$-0.670124\pi$
$769$	−28.2509	−1.01875	−0.509377	+	0.860543i	$0.670124\pi$
$770$	20.6265	0.743327
$771$	0	0
$772$	−11.3618	−0.408919
$773$	−21.5840	−0.776323	−0.388161	−	0.921591i	$-0.626890\pi$
$773$	−21.5840	−0.776323	−0.388161	+	0.921591i	$0.626890\pi$
$774$	0	0
$775$	−2.12766	−0.0764277
$776$	−15.6461	−0.561664
$777$	0	0
$778$	3.59024	0.128716
$779$	−16.1620	−0.579063
$780$	0	0
$781$	40.1780	1.43768
$782$	1.94349	0.0694992
$783$	0	0
$784$	13.7212	0.490044
$785$	5.74840	0.205169
$786$	0	0
$787$	−35.3258	−1.25923	−0.629614	−	0.776908i	$-0.716787\pi$
$787$	−35.3258	−1.25923	−0.629614	+	0.776908i	$0.716787\pi$
$788$	−27.2753	−0.971643
$789$	0	0
$790$	−12.4593	−0.443283
$791$	−38.4098	−1.36569
$792$	0	0
$793$	83.0041	2.94756
$794$	11.8453	0.420373
$795$	0	0
$796$	8.79362	0.311682
$797$	16.5841	0.587439	0.293719	−	0.955892i	$-0.405107\pi$
$797$	16.5841	0.587439	0.293719	+	0.955892i	$0.405107\pi$
$798$	0	0
$799$	16.0402	0.567462
$800$	−4.23378	−0.149687
$801$	0	0
$802$	−2.51401	−0.0887728
$803$	−19.2636	−0.679797
$804$	0	0
$805$	−8.31942	−0.293221
$806$	−6.45680	−0.227431
$807$	0	0
$808$	11.1769	0.393203
$809$	46.3267	1.62876	0.814379	−	0.580333i	$-0.197078\pi$
$809$	46.3267	1.62876	0.814379	+	0.580333i	$0.197078\pi$
$810$	0	0
$811$	−6.92923	−0.243318	−0.121659	−	0.992572i	$-0.538821\pi$
$811$	−6.92923	−0.243318	−0.121659	+	0.992572i	$0.538821\pi$
$812$	−6.21129	−0.217974
$813$	0	0
$814$	12.0030	0.420704
$815$	39.8340	1.39532
$816$	0	0
$817$	23.8791	0.835424
$818$	−6.08877	−0.212889
$819$	0	0
$820$	−21.2976	−0.743743
$821$	−12.4948	−0.436070	−0.218035	−	0.975941i	$-0.569965\pi$
$821$	−12.4948	−0.436070	−0.218035	+	0.975941i	$0.569965\pi$
$822$	0	0
$823$	20.6832	0.720971	0.360485	−	0.932765i	$-0.382611\pi$
$823$	20.6832	0.720971	0.360485	+	0.932765i	$0.382611\pi$
$824$	24.9973	0.870823
$825$	0	0
$826$	−12.1295	−0.422039
$827$	−37.9507	−1.31968	−0.659838	−	0.751408i	$-0.729375\pi$
$827$	−37.9507	−1.31968	−0.659838	+	0.751408i	$0.729375\pi$
$828$	0	0
$829$	−23.1196	−0.802977	−0.401489	−	0.915864i	$-0.631507\pi$
$829$	−23.1196	−0.802977	−0.401489	+	0.915864i	$0.631507\pi$
$830$	15.7540	0.546828
$831$	0	0
$832$	26.9232	0.933393
$833$	−21.2881	−0.737590
$834$	0	0
$835$	14.3643	0.497096
$836$	−35.7970	−1.23807
$837$	0	0
$838$	15.3861	0.531505
$839$	−35.3974	−1.22205	−0.611027	−	0.791609i	$-0.709243\pi$
$839$	−35.3974	−1.22205	−0.611027	+	0.791609i	$0.709243\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−7.52844	−0.259447
$843$	0	0
$844$	−18.1682	−0.625374
$845$	−78.9267	−2.71516
$846$	0	0
$847$	78.9632	2.71321
$848$	2.37077	0.0814127
$849$	0	0
$850$	1.83070	0.0627924
$851$	−4.84124	−0.165956
$852$	0	0
$853$	25.4871	0.872661	0.436331	−	0.899786i	$-0.356278\pi$
$853$	25.4871	0.872661	0.436331	+	0.899786i	$0.356278\pi$
$854$	17.8453	0.610655
$855$	0	0
$856$	23.0550	0.788002
$857$	−26.7263	−0.912954	−0.456477	−	0.889735i	$-0.650889\pi$
$857$	−26.7263	−0.912954	−0.456477	+	0.889735i	$0.650889\pi$
$858$	0	0
$859$	−44.6325	−1.52284	−0.761421	−	0.648257i	$-0.775498\pi$
$859$	−44.6325	−1.52284	−0.761421	+	0.648257i	$0.775498\pi$
$860$	31.4669	1.07301
$861$	0	0
$862$	−7.95009	−0.270781
$863$	38.5205	1.31125	0.655626	−	0.755085i	$-0.272405\pi$
$863$	38.5205	1.31125	0.655626	+	0.755085i	$0.272405\pi$
$864$	0	0
$865$	2.93195	0.0996892
$866$	−2.00290	−0.0680612
$867$	0	0
$868$	14.0298	0.476201
$869$	−70.3746	−2.38730
$870$	0	0
$871$	−47.3691	−1.60504
$872$	10.8225	0.366495
$873$	0	0
$874$	−1.42859	−0.0483227
$875$	33.7605	1.14131
$876$	0	0
$877$	−45.2033	−1.52641	−0.763203	−	0.646158i	$-0.776375\pi$
$877$	−45.2033	−1.52641	−0.763203	+	0.646158i	$0.776375\pi$
$878$	−10.4562	−0.352879
$879$	0	0
$880$	−42.0426	−1.41726
$881$	−34.1314	−1.14992	−0.574959	−	0.818182i	$-0.694982\pi$
$881$	−34.1314	−1.14992	−0.574959	+	0.818182i	$0.694982\pi$
$882$	0	0
$883$	24.9724	0.840387	0.420193	−	0.907435i	$-0.361962\pi$
$883$	24.9724	0.840387	0.420193	+	0.907435i	$0.361962\pi$
$884$	−56.1487	−1.88849
$885$	0	0
$886$	4.76137	0.159961
$887$	−1.69868	−0.0570362	−0.0285181	−	0.999593i	$-0.509079\pi$
$887$	−1.69868	−0.0570362	−0.0285181	+	0.999593i	$0.509079\pi$
$888$	0	0
$889$	−60.9283	−2.04347
$890$	7.75948	0.260098
$891$	0	0
$892$	6.39896	0.214253
$893$	−11.7906	−0.394556
$894$	0	0
$895$	37.7928	1.26328
$896$	36.4681	1.21832
$897$	0	0
$898$	3.68421	0.122944
$899$	−2.25875	−0.0753335
$900$	0	0
$901$	−3.67819	−0.122538
$902$	11.9026	0.396314
$903$	0	0
$904$	−18.2428	−0.606748
$905$	56.6655	1.88363
$906$	0	0
$907$	20.6791	0.686639	0.343320	−	0.939219i	$-0.388449\pi$
$907$	20.6791	0.686639	0.343320	+	0.939219i	$0.388449\pi$
$908$	16.7898	0.557188
$909$	0	0
$910$	−23.7817	−0.788354
$911$	56.7976	1.88179	0.940894	−	0.338700i	$-0.109987\pi$
$911$	56.7976	1.88179	0.940894	+	0.338700i	$0.109987\pi$
$912$	0	0
$913$	88.9838	2.94493
$914$	−16.0999	−0.532536
$915$	0	0
$916$	16.3391	0.539859
$917$	−20.8432	−0.688302
$918$	0	0
$919$	−38.6718	−1.27567	−0.637833	−	0.770175i	$-0.720169\pi$
$919$	−38.6718	−1.27567	−0.637833	+	0.770175i	$0.720169\pi$
$920$	−3.95133	−0.130272
$921$	0	0
$922$	−17.0439	−0.561312
$923$	−46.3239	−1.52477
$924$	0	0
$925$	−4.56027	−0.149941
$926$	12.4072	0.407724
$927$	0	0
$928$	−4.49464	−0.147544
$929$	39.0509	1.28122	0.640609	−	0.767867i	$-0.278682\pi$
$929$	39.0509	1.28122	0.640609	+	0.767867i	$0.278682\pi$
$930$	0	0
$931$	15.6481	0.512846
$932$	19.1335	0.626740
$933$	0	0
$934$	13.8700	0.453841
$935$	65.2280	2.13318
$936$	0	0
$937$	−38.2425	−1.24933	−0.624664	−	0.780894i	$-0.714764\pi$
$937$	−38.2425	−1.24933	−0.624664	+	0.780894i	$0.714764\pi$
$938$	−10.1840	−0.332521
$939$	0	0
$940$	−15.5371	−0.506764
$941$	26.9386	0.878172	0.439086	−	0.898445i	$-0.355302\pi$
$941$	26.9386	0.878172	0.439086	+	0.898445i	$0.355302\pi$
$942$	0	0
$943$	−4.80077	−0.156334
$944$	24.7233	0.804675
$945$	0	0
$946$	−17.5860	−0.571770
$947$	−16.4580	−0.534814	−0.267407	−	0.963584i	$-0.586167\pi$
$947$	−16.4580	−0.534814	−0.267407	+	0.963584i	$0.586167\pi$
$948$	0	0
$949$	22.2103	0.720976
$950$	−1.34568	−0.0436595
$951$	0	0
$952$	−25.3376	−0.821198
$953$	−17.2466	−0.558672	−0.279336	−	0.960193i	$-0.590114\pi$
$953$	−17.2466	−0.558672	−0.279336	+	0.960193i	$0.590114\pi$
$954$	0	0
$955$	−43.9819	−1.42322
$956$	−26.3515	−0.852268
$957$	0	0
$958$	−1.27251	−0.0411129
$959$	−56.7817	−1.83358
$960$	0	0
$961$	−25.8980	−0.835421
$962$	−13.8390	−0.446189
$963$	0	0
$964$	−3.20341	−0.103175
$965$	−15.2180	−0.489884
$966$	0	0
$967$	47.6981	1.53387	0.766934	−	0.641726i	$-0.221781\pi$
$967$	47.6981	1.53387	0.766934	+	0.641726i	$0.221781\pi$
$968$	37.5038	1.20542
$969$	0	0
$970$	−9.98429	−0.320576
$971$	26.3554	0.845784	0.422892	−	0.906180i	$-0.361015\pi$
$971$	26.3554	0.845784	0.422892	+	0.906180i	$0.361015\pi$
$972$	0	0
$973$	−59.2990	−1.90104
$974$	−2.36773	−0.0758670
$975$	0	0
$976$	−36.3738	−1.16430
$977$	−28.7208	−0.918858	−0.459429	−	0.888214i	$-0.651946\pi$
$977$	−28.7208	−0.918858	−0.459429	+	0.888214i	$0.651946\pi$
$978$	0	0
$979$	43.8282	1.40076
$980$	20.6204	0.658694
$981$	0	0
$982$	6.53113	0.208417
$983$	−0.767884	−0.0244917	−0.0122458	−	0.999925i	$-0.503898\pi$
$983$	−0.767884	−0.0244917	−0.0122458	+	0.999925i	$0.503898\pi$
$984$	0	0
$985$	−36.5326	−1.16403
$986$	1.94349	0.0618935
$987$	0	0
$988$	41.2728	1.31306
$989$	7.09307	0.225546
$990$	0	0
$991$	−29.3337	−0.931815	−0.465908	−	0.884833i	$-0.654272\pi$
$991$	−29.3337	−0.931815	−0.465908	+	0.884833i	$0.654272\pi$
$992$	10.1523	0.322335
$993$	0	0
$994$	−9.95933	−0.315891
$995$	11.7782	0.373394
$996$	0	0
$997$	−13.7965	−0.436940	−0.218470	−	0.975844i	$-0.570107\pi$
$997$	−13.7965	−0.436940	−0.218470	+	0.975844i	$0.570107\pi$
$998$	0.301730	0.00955110
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.10	✓	22
3.2	odd	2		6003.2.a.u.1.13	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.10	✓	22	1.1	even	1	trivial
6003.2.a.u.1.13	yes	22	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-0.424349 q^{2} -1.81993 q^{4} -2.43761 q^{5} +3.41293 q^{7} +1.62098 q^{8} +1.03440 q^{10} +5.84264 q^{11} -6.73637 q^{13} -1.44828 q^{14} +2.95199 q^{16} -4.57994 q^{17} +3.36654 q^{19} +4.43628 q^{20} -2.47932 q^{22} +1.00000 q^{23} +0.941962 q^{25} +2.85857 q^{26} -6.21129 q^{28} +1.00000 q^{29} -2.25875 q^{31} -4.49464 q^{32} +1.94349 q^{34} -8.31942 q^{35} -4.84124 q^{37} -1.42859 q^{38} -3.95133 q^{40} -4.80077 q^{41} +7.09307 q^{43} -10.6332 q^{44} -0.424349 q^{46} -3.50228 q^{47} +4.64812 q^{49} -0.399721 q^{50} +12.2597 q^{52} +0.803109 q^{53} -14.2421 q^{55} +5.53231 q^{56} -0.424349 q^{58} +8.37512 q^{59} -12.3218 q^{61} +0.958499 q^{62} -3.99669 q^{64} +16.4207 q^{65} +7.03184 q^{67} +8.33516 q^{68} +3.53034 q^{70} +6.87668 q^{71} -3.29707 q^{73} +2.05438 q^{74} -6.12686 q^{76} +19.9406 q^{77} -12.0450 q^{79} -7.19582 q^{80} +2.03720 q^{82} +15.2301 q^{83} +11.1641 q^{85} -3.00994 q^{86} +9.47082 q^{88} +7.50144 q^{89} -22.9908 q^{91} -1.81993 q^{92} +1.48619 q^{94} -8.20633 q^{95} -9.65226 q^{97} -1.97243 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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