LMFDB - Embedded newform 6008.2.a.d.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, 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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.52312 q^{3} +3.48921 q^{5} -1.44076 q^{7} -0.680107 q^{9} +O(q^{10})$	$q-1.52312 q^{3} +3.48921 q^{5} -1.44076 q^{7} -0.680107 q^{9} +2.95674 q^{11} -4.96257 q^{13} -5.31448 q^{15} -1.40385 q^{17} -0.462641 q^{19} +2.19445 q^{21} -2.06485 q^{23} +7.17456 q^{25} +5.60524 q^{27} -7.55113 q^{29} +3.39545 q^{31} -4.50347 q^{33} -5.02711 q^{35} +4.87368 q^{37} +7.55859 q^{39} -7.84681 q^{41} -0.828254 q^{43} -2.37303 q^{45} +8.33285 q^{47} -4.92421 q^{49} +2.13822 q^{51} -2.22945 q^{53} +10.3167 q^{55} +0.704657 q^{57} +6.19427 q^{59} +11.5576 q^{61} +0.979870 q^{63} -17.3154 q^{65} +7.86771 q^{67} +3.14502 q^{69} +14.4594 q^{71} +11.9761 q^{73} -10.9277 q^{75} -4.25995 q^{77} -13.6591 q^{79} -6.49713 q^{81} +10.1688 q^{83} -4.89831 q^{85} +11.5013 q^{87} +4.52922 q^{89} +7.14988 q^{91} -5.17168 q^{93} -1.61425 q^{95} +5.83835 q^{97} -2.01090 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.52312	−0.879374	−0.439687	−	0.898151i	$-0.644911\pi$
$3$	−1.52312	−0.879374	−0.439687	+	0.898151i	$0.644911\pi$
$4$	0	0
$5$	3.48921	1.56042	0.780210	−	0.625517i	$-0.215112\pi$
$5$	3.48921	1.56042	0.780210	+	0.625517i	$0.215112\pi$
$6$	0	0
$7$	−1.44076	−0.544556	−0.272278	−	0.962219i	$-0.587777\pi$
$7$	−1.44076	−0.544556	−0.272278	+	0.962219i	$0.587777\pi$
$8$	0	0
$9$	−0.680107	−0.226702
$10$	0	0
$11$	2.95674	0.891491	0.445745	−	0.895160i	$-0.352939\pi$
$11$	2.95674	0.891491	0.445745	+	0.895160i	$0.352939\pi$
$12$	0	0
$13$	−4.96257	−1.37637	−0.688185	−	0.725535i	$-0.741592\pi$
$13$	−4.96257	−1.37637	−0.688185	+	0.725535i	$0.741592\pi$
$14$	0	0
$15$	−5.31448	−1.37219
$16$	0	0
$17$	−1.40385	−0.340483	−0.170241	−	0.985402i	$-0.554455\pi$
$17$	−1.40385	−0.340483	−0.170241	+	0.985402i	$0.554455\pi$
$18$	0	0
$19$	−0.462641	−0.106137	−0.0530685	−	0.998591i	$-0.516900\pi$
$19$	−0.462641	−0.106137	−0.0530685	+	0.998591i	$0.516900\pi$
$20$	0	0
$21$	2.19445	0.478868
$22$	0	0
$23$	−2.06485	−0.430551	−0.215276	−	0.976553i	$-0.569065\pi$
$23$	−2.06485	−0.430551	−0.215276	+	0.976553i	$0.569065\pi$
$24$	0	0
$25$	7.17456	1.43491
$26$	0	0
$27$	5.60524	1.07873
$28$	0	0
$29$	−7.55113	−1.40221	−0.701105	−	0.713058i	$-0.747310\pi$
$29$	−7.55113	−1.40221	−0.701105	+	0.713058i	$0.747310\pi$
$30$	0	0
$31$	3.39545	0.609841	0.304920	−	0.952378i	$-0.401370\pi$
$31$	3.39545	0.609841	0.304920	+	0.952378i	$0.401370\pi$
$32$	0	0
$33$	−4.50347	−0.783953
$34$	0	0
$35$	−5.02711	−0.849736
$36$	0	0
$37$	4.87368	0.801228	0.400614	−	0.916247i	$-0.368797\pi$
$37$	4.87368	0.801228	0.400614	+	0.916247i	$0.368797\pi$
$38$	0	0
$39$	7.55859	1.21034
$40$	0	0
$41$	−7.84681	−1.22547	−0.612733	−	0.790290i	$-0.709930\pi$
$41$	−7.84681	−1.22547	−0.612733	+	0.790290i	$0.709930\pi$
$42$	0	0
$43$	−0.828254	−0.126307	−0.0631537	−	0.998004i	$-0.520116\pi$
$43$	−0.828254	−0.126307	−0.0631537	+	0.998004i	$0.520116\pi$
$44$	0	0
$45$	−2.37303	−0.353751
$46$	0	0
$47$	8.33285	1.21547	0.607736	−	0.794139i	$-0.292078\pi$
$47$	8.33285	1.21547	0.607736	+	0.794139i	$0.292078\pi$
$48$	0	0
$49$	−4.92421	−0.703459
$50$	0	0
$51$	2.13822	0.299411
$52$	0	0
$53$	−2.22945	−0.306238	−0.153119	−	0.988208i	$-0.548932\pi$
$53$	−2.22945	−0.306238	−0.153119	+	0.988208i	$0.548932\pi$
$54$	0	0
$55$	10.3167	1.39110
$56$	0	0
$57$	0.704657	0.0933341
$58$	0	0
$59$	6.19427	0.806425	0.403213	−	0.915106i	$-0.367894\pi$
$59$	6.19427	0.806425	0.403213	+	0.915106i	$0.367894\pi$
$60$	0	0
$61$	11.5576	1.47980	0.739898	−	0.672719i	$-0.234874\pi$
$61$	11.5576	1.47980	0.739898	+	0.672719i	$0.234874\pi$
$62$	0	0
$63$	0.979870	0.123452
$64$	0	0
$65$	−17.3154	−2.14772
$66$	0	0
$67$	7.86771	0.961194	0.480597	−	0.876942i	$-0.340420\pi$
$67$	7.86771	0.961194	0.480597	+	0.876942i	$0.340420\pi$
$68$	0	0
$69$	3.14502	0.378616
$70$	0	0
$71$	14.4594	1.71602	0.858008	−	0.513636i	$-0.171702\pi$
$71$	14.4594	1.71602	0.858008	+	0.513636i	$0.171702\pi$
$72$	0	0
$73$	11.9761	1.40170	0.700851	−	0.713308i	$-0.252804\pi$
$73$	11.9761	1.40170	0.700851	+	0.713308i	$0.252804\pi$
$74$	0	0
$75$	−10.9277	−1.26182
$76$	0	0
$77$	−4.25995	−0.485467
$78$	0	0
$79$	−13.6591	−1.53677	−0.768386	−	0.639987i	$-0.778940\pi$
$79$	−13.6591	−1.53677	−0.768386	+	0.639987i	$0.778940\pi$
$80$	0	0
$81$	−6.49713	−0.721904
$82$	0	0
$83$	10.1688	1.11617	0.558083	−	0.829785i	$-0.311537\pi$
$83$	10.1688	1.11617	0.558083	+	0.829785i	$0.311537\pi$
$84$	0	0
$85$	−4.89831	−0.531296
$86$	0	0
$87$	11.5013	1.23307
$88$	0	0
$89$	4.52922	0.480096	0.240048	−	0.970761i	$-0.422837\pi$
$89$	4.52922	0.480096	0.240048	+	0.970761i	$0.422837\pi$
$90$	0	0
$91$	7.14988	0.749511
$92$	0	0
$93$	−5.17168	−0.536278
$94$	0	0
$95$	−1.61425	−0.165618
$96$	0	0
$97$	5.83835	0.592795	0.296397	−	0.955065i	$-0.404215\pi$
$97$	5.83835	0.592795	0.296397	+	0.955065i	$0.404215\pi$
$98$	0	0
$99$	−2.01090	−0.202103
$100$	0	0
$101$	−12.6428	−1.25801	−0.629004	−	0.777402i	$-0.716537\pi$
$101$	−12.6428	−1.25801	−0.629004	+	0.777402i	$0.716537\pi$
$102$	0	0
$103$	−7.63859	−0.752653	−0.376326	−	0.926487i	$-0.622813\pi$
$103$	−7.63859	−0.752653	−0.376326	+	0.926487i	$0.622813\pi$
$104$	0	0
$105$	7.65689	0.747236
$106$	0	0
$107$	14.5320	1.40486	0.702430	−	0.711752i	$-0.252098\pi$
$107$	14.5320	1.40486	0.702430	+	0.711752i	$0.252098\pi$
$108$	0	0
$109$	11.4202	1.09385	0.546927	−	0.837180i	$-0.315798\pi$
$109$	11.4202	1.09385	0.546927	+	0.837180i	$0.315798\pi$
$110$	0	0
$111$	−7.42319	−0.704578
$112$	0	0
$113$	5.52935	0.520158	0.260079	−	0.965587i	$-0.416251\pi$
$113$	5.52935	0.520158	0.260079	+	0.965587i	$0.416251\pi$
$114$	0	0
$115$	−7.20470	−0.671841
$116$	0	0
$117$	3.37508	0.312026
$118$	0	0
$119$	2.02260	0.185412
$120$	0	0
$121$	−2.25769	−0.205245
$122$	0	0
$123$	11.9516	1.07764
$124$	0	0
$125$	7.58750	0.678647
$126$	0	0
$127$	−7.02888	−0.623712	−0.311856	−	0.950129i	$-0.600951\pi$
$127$	−7.02888	−0.623712	−0.311856	+	0.950129i	$0.600951\pi$
$128$	0	0
$129$	1.26153	0.111071
$130$	0	0
$131$	−2.63093	−0.229865	−0.114933	−	0.993373i	$-0.536665\pi$
$131$	−2.63093	−0.229865	−0.114933	+	0.993373i	$0.536665\pi$
$132$	0	0
$133$	0.666554	0.0577976
$134$	0	0
$135$	19.5578	1.68327
$136$	0	0
$137$	9.10539	0.777926	0.388963	−	0.921253i	$-0.372833\pi$
$137$	9.10539	0.777926	0.388963	+	0.921253i	$0.372833\pi$
$138$	0	0
$139$	5.88668	0.499302	0.249651	−	0.968336i	$-0.419684\pi$
$139$	5.88668	0.499302	0.249651	+	0.968336i	$0.419684\pi$
$140$	0	0
$141$	−12.6919	−1.06885
$142$	0	0
$143$	−14.6730	−1.22702
$144$	0	0
$145$	−26.3475	−2.18804
$146$	0	0
$147$	7.50016	0.618603
$148$	0	0
$149$	15.4920	1.26916	0.634579	−	0.772858i	$-0.281174\pi$
$149$	15.4920	1.26916	0.634579	+	0.772858i	$0.281174\pi$
$150$	0	0
$151$	17.8073	1.44914	0.724568	−	0.689204i	$-0.242040\pi$
$151$	17.8073	1.44914	0.724568	+	0.689204i	$0.242040\pi$
$152$	0	0
$153$	0.954765	0.0771882
$154$	0	0
$155$	11.8474	0.951608
$156$	0	0
$157$	−5.95200	−0.475022	−0.237511	−	0.971385i	$-0.576331\pi$
$157$	−5.95200	−0.475022	−0.237511	+	0.971385i	$0.576331\pi$
$158$	0	0
$159$	3.39572	0.269298
$160$	0	0
$161$	2.97496	0.234459
$162$	0	0
$163$	5.45696	0.427422	0.213711	−	0.976897i	$-0.431445\pi$
$163$	5.45696	0.427422	0.213711	+	0.976897i	$0.431445\pi$
$164$	0	0
$165$	−15.7135	−1.22330
$166$	0	0
$167$	2.45300	0.189819	0.0949093	−	0.995486i	$-0.469744\pi$
$167$	2.45300	0.189819	0.0949093	+	0.995486i	$0.469744\pi$
$168$	0	0
$169$	11.6271	0.894395
$170$	0	0
$171$	0.314645	0.0240615
$172$	0	0
$173$	−19.4048	−1.47532	−0.737658	−	0.675174i	$-0.764068\pi$
$173$	−19.4048	−1.47532	−0.737658	+	0.675174i	$0.764068\pi$
$174$	0	0
$175$	−10.3368	−0.781390
$176$	0	0
$177$	−9.43461	−0.709149
$178$	0	0
$179$	20.7839	1.55346	0.776731	−	0.629833i	$-0.216877\pi$
$179$	20.7839	1.55346	0.776731	+	0.629833i	$0.216877\pi$
$180$	0	0
$181$	9.73001	0.723226	0.361613	−	0.932328i	$-0.382226\pi$
$181$	9.73001	0.723226	0.361613	+	0.932328i	$0.382226\pi$
$182$	0	0
$183$	−17.6036	−1.30129
$184$	0	0
$185$	17.0053	1.25025
$186$	0	0
$187$	−4.15081	−0.303537
$188$	0	0
$189$	−8.07581	−0.587429
$190$	0	0
$191$	−12.4922	−0.903906	−0.451953	−	0.892042i	$-0.649273\pi$
$191$	−12.4922	−0.903906	−0.451953	+	0.892042i	$0.649273\pi$
$192$	0	0
$193$	−2.27515	−0.163769	−0.0818846	−	0.996642i	$-0.526094\pi$
$193$	−2.27515	−0.163769	−0.0818846	+	0.996642i	$0.526094\pi$
$194$	0	0
$195$	26.3735	1.88865
$196$	0	0
$197$	0.0108839	0.000775449 0	0.000387725	−	1.00000i	$-0.499877\pi$
$197$	0.0108839	0.000775449 0	0.000387725		1.00000i	$0.499877\pi$
$198$	0	0
$199$	−5.72341	−0.405722	−0.202861	−	0.979208i	$-0.565024\pi$
$199$	−5.72341	−0.405722	−0.202861	+	0.979208i	$0.565024\pi$
$200$	0	0
$201$	−11.9835	−0.845249
$202$	0	0
$203$	10.8794	0.763582
$204$	0	0
$205$	−27.3791	−1.91224
$206$	0	0
$207$	1.40432	0.0976070
$208$	0	0
$209$	−1.36791	−0.0946202
$210$	0	0
$211$	−8.07736	−0.556068	−0.278034	−	0.960571i	$-0.589683\pi$
$211$	−8.07736	−0.556068	−0.278034	+	0.960571i	$0.589683\pi$
$212$	0	0
$213$	−22.0234	−1.50902
$214$	0	0
$215$	−2.88995	−0.197093
$216$	0	0
$217$	−4.89203	−0.332092
$218$	0	0
$219$	−18.2411	−1.23262
$220$	0	0
$221$	6.96669	0.468630
$222$	0	0
$223$	−20.3252	−1.36108	−0.680538	−	0.732713i	$-0.738254\pi$
$223$	−20.3252	−1.36108	−0.680538	+	0.732713i	$0.738254\pi$
$224$	0	0
$225$	−4.87947	−0.325298
$226$	0	0
$227$	12.3352	0.818717	0.409358	−	0.912374i	$-0.365753\pi$
$227$	12.3352	0.818717	0.409358	+	0.912374i	$0.365753\pi$
$228$	0	0
$229$	−8.54529	−0.564689	−0.282344	−	0.959313i	$-0.591112\pi$
$229$	−8.54529	−0.564689	−0.282344	+	0.959313i	$0.591112\pi$
$230$	0	0
$231$	6.48842	0.426906
$232$	0	0
$233$	−5.07249	−0.332310	−0.166155	−	0.986100i	$-0.553135\pi$
$233$	−5.07249	−0.332310	−0.166155	+	0.986100i	$0.553135\pi$
$234$	0	0
$235$	29.0750	1.89665
$236$	0	0
$237$	20.8045	1.35140
$238$	0	0
$239$	16.1923	1.04739	0.523697	−	0.851905i	$-0.324552\pi$
$239$	16.1923	1.04739	0.523697	+	0.851905i	$0.324552\pi$
$240$	0	0
$241$	−5.39548	−0.347554	−0.173777	−	0.984785i	$-0.555597\pi$
$241$	−5.39548	−0.347554	−0.173777	+	0.984785i	$0.555597\pi$
$242$	0	0
$243$	−6.91981	−0.443906
$244$	0	0
$245$	−17.1816	−1.09769
$246$	0	0
$247$	2.29589	0.146084
$248$	0	0
$249$	−15.4882	−0.981527
$250$	0	0
$251$	−21.0933	−1.33140	−0.665700	−	0.746220i	$-0.731867\pi$
$251$	−21.0933	−1.33140	−0.665700	+	0.746220i	$0.731867\pi$
$252$	0	0
$253$	−6.10523	−0.383833
$254$	0	0
$255$	7.46071	0.467208
$256$	0	0
$257$	17.9371	1.11888	0.559442	−	0.828870i	$-0.311016\pi$
$257$	17.9371	1.11888	0.559442	+	0.828870i	$0.311016\pi$
$258$	0	0
$259$	−7.02180	−0.436313
$260$	0	0
$261$	5.13558	0.317884
$262$	0	0
$263$	9.30652	0.573865	0.286932	−	0.957951i	$-0.407364\pi$
$263$	9.30652	0.573865	0.286932	+	0.957951i	$0.407364\pi$
$264$	0	0
$265$	−7.77901	−0.477861
$266$	0	0
$267$	−6.89854	−0.422184
$268$	0	0
$269$	30.8891	1.88334	0.941670	−	0.336537i	$-0.109256\pi$
$269$	30.8891	1.88334	0.941670	+	0.336537i	$0.109256\pi$
$270$	0	0
$271$	1.55908	0.0947071	0.0473536	−	0.998878i	$-0.484921\pi$
$271$	1.55908	0.0947071	0.0473536	+	0.998878i	$0.484921\pi$
$272$	0	0
$273$	−10.8901	−0.659100
$274$	0	0
$275$	21.2133	1.27921
$276$	0	0
$277$	−22.3747	−1.34436	−0.672182	−	0.740386i	$-0.734643\pi$
$277$	−22.3747	−1.34436	−0.672182	+	0.740386i	$0.734643\pi$
$278$	0	0
$279$	−2.30927	−0.138252
$280$	0	0
$281$	−20.7553	−1.23816	−0.619079	−	0.785329i	$-0.712494\pi$
$281$	−20.7553	−1.23816	−0.619079	+	0.785329i	$0.712494\pi$
$282$	0	0
$283$	−0.957740	−0.0569317	−0.0284659	−	0.999595i	$-0.509062\pi$
$283$	−0.957740	−0.0569317	−0.0284659	+	0.999595i	$0.509062\pi$
$284$	0	0
$285$	2.45869	0.145640
$286$	0	0
$287$	11.3054	0.667335
$288$	0	0
$289$	−15.0292	−0.884072
$290$	0	0
$291$	−8.89251	−0.521288
$292$	0	0
$293$	8.77805	0.512819	0.256410	−	0.966568i	$-0.417460\pi$
$293$	8.77805	0.512819	0.256410	+	0.966568i	$0.417460\pi$
$294$	0	0
$295$	21.6131	1.25836
$296$	0	0
$297$	16.5732	0.961677
$298$	0	0
$299$	10.2470	0.592598
$300$	0	0
$301$	1.19331	0.0687815
$302$	0	0
$303$	19.2565	1.10626
$304$	0	0
$305$	40.3268	2.30910
$306$	0	0
$307$	19.1667	1.09390	0.546952	−	0.837164i	$-0.315788\pi$
$307$	19.1667	1.09390	0.546952	+	0.837164i	$0.315788\pi$
$308$	0	0
$309$	11.6345	0.661863
$310$	0	0
$311$	7.46435	0.423264	0.211632	−	0.977349i	$-0.432122\pi$
$311$	7.46435	0.423264	0.211632	+	0.977349i	$0.432122\pi$
$312$	0	0
$313$	−16.2182	−0.916705	−0.458353	−	0.888770i	$-0.651560\pi$
$313$	−16.2182	−0.916705	−0.458353	+	0.888770i	$0.651560\pi$
$314$	0	0
$315$	3.41897	0.192637
$316$	0	0
$317$	−1.27225	−0.0714567	−0.0357284	−	0.999362i	$-0.511375\pi$
$317$	−1.27225	−0.0714567	−0.0357284	+	0.999362i	$0.511375\pi$
$318$	0	0
$319$	−22.3267	−1.25006
$320$	0	0
$321$	−22.1340	−1.23540
$322$	0	0
$323$	0.649476	0.0361378
$324$	0	0
$325$	−35.6043	−1.97497
$326$	0	0
$327$	−17.3943	−0.961906
$328$	0	0
$329$	−12.0056	−0.661892
$330$	0	0
$331$	18.6668	1.02602	0.513010	−	0.858382i	$-0.328530\pi$
$331$	18.6668	1.02602	0.513010	+	0.858382i	$0.328530\pi$
$332$	0	0
$333$	−3.31462	−0.181640
$334$	0	0
$335$	27.4521	1.49987
$336$	0	0
$337$	24.1271	1.31429	0.657145	−	0.753764i	$-0.271764\pi$
$337$	24.1271	1.31429	0.657145	+	0.753764i	$0.271764\pi$
$338$	0	0
$339$	−8.42187	−0.457413
$340$	0	0
$341$	10.0395	0.543667
$342$	0	0
$343$	17.1799	0.927629
$344$	0	0
$345$	10.9736	0.590799
$346$	0	0
$347$	19.6865	1.05682	0.528412	−	0.848988i	$-0.322788\pi$
$347$	19.6865	1.05682	0.528412	+	0.848988i	$0.322788\pi$
$348$	0	0
$349$	−8.86088	−0.474312	−0.237156	−	0.971472i	$-0.576215\pi$
$349$	−8.86088	−0.474312	−0.237156	+	0.971472i	$0.576215\pi$
$350$	0	0
$351$	−27.8164	−1.48473
$352$	0	0
$353$	14.9331	0.794807	0.397403	−	0.917644i	$-0.369911\pi$
$353$	14.9331	0.794807	0.397403	+	0.917644i	$0.369911\pi$
$354$	0	0
$355$	50.4519	2.67771
$356$	0	0
$357$	−3.08067	−0.163046
$358$	0	0
$359$	−3.88406	−0.204993	−0.102496	−	0.994733i	$-0.532683\pi$
$359$	−3.88406	−0.204993	−0.102496	+	0.994733i	$0.532683\pi$
$360$	0	0
$361$	−18.7860	−0.988735
$362$	0	0
$363$	3.43873	0.180487
$364$	0	0
$365$	41.7872	2.18724
$366$	0	0
$367$	15.8789	0.828872	0.414436	−	0.910078i	$-0.363979\pi$
$367$	15.8789	0.828872	0.414436	+	0.910078i	$0.363979\pi$
$368$	0	0
$369$	5.33667	0.277816
$370$	0	0
$371$	3.21210	0.166764
$372$	0	0
$373$	34.1320	1.76729	0.883643	−	0.468161i	$-0.155083\pi$
$373$	34.1320	1.76729	0.883643	+	0.468161i	$0.155083\pi$
$374$	0	0
$375$	−11.5567	−0.596784
$376$	0	0
$377$	37.4731	1.92996
$378$	0	0
$379$	−33.5106	−1.72132	−0.860661	−	0.509178i	$-0.829949\pi$
$379$	−33.5106	−1.72132	−0.860661	+	0.509178i	$0.829949\pi$
$380$	0	0
$381$	10.7058	0.548476
$382$	0	0
$383$	−3.12136	−0.159494	−0.0797471	−	0.996815i	$-0.525411\pi$
$383$	−3.12136	−0.159494	−0.0797471	+	0.996815i	$0.525411\pi$
$384$	0	0
$385$	−14.8639	−0.757532
$386$	0	0
$387$	0.563301	0.0286342
$388$	0	0
$389$	−7.77412	−0.394164	−0.197082	−	0.980387i	$-0.563146\pi$
$389$	−7.77412	−0.394164	−0.197082	+	0.980387i	$0.563146\pi$
$390$	0	0
$391$	2.89873	0.146595
$392$	0	0
$393$	4.00722	0.202138
$394$	0	0
$395$	−47.6595	−2.39801
$396$	0	0
$397$	28.3179	1.42124	0.710618	−	0.703578i	$-0.248415\pi$
$397$	28.3179	1.42124	0.710618	+	0.703578i	$0.248415\pi$
$398$	0	0
$399$	−1.01524	−0.0508257
$400$	0	0
$401$	−9.13451	−0.456155	−0.228078	−	0.973643i	$-0.573244\pi$
$401$	−9.13451	−0.456155	−0.228078	+	0.973643i	$0.573244\pi$
$402$	0	0
$403$	−16.8502	−0.839367
$404$	0	0
$405$	−22.6698	−1.12647
$406$	0	0
$407$	14.4102	0.714287
$408$	0	0
$409$	−7.31660	−0.361782	−0.180891	−	0.983503i	$-0.557898\pi$
$409$	−7.31660	−0.361782	−0.180891	+	0.983503i	$0.557898\pi$
$410$	0	0
$411$	−13.8686	−0.684088
$412$	0	0
$413$	−8.92445	−0.439144
$414$	0	0
$415$	35.4809	1.74169
$416$	0	0
$417$	−8.96612	−0.439073
$418$	0	0
$419$	35.1014	1.71482	0.857408	−	0.514638i	$-0.172074\pi$
$419$	35.1014	1.71482	0.857408	+	0.514638i	$0.172074\pi$
$420$	0	0
$421$	33.8419	1.64935	0.824677	−	0.565603i	$-0.191357\pi$
$421$	33.8419	1.64935	0.824677	+	0.565603i	$0.191357\pi$
$422$	0	0
$423$	−5.66723	−0.275550
$424$	0	0
$425$	−10.0720	−0.488563
$426$	0	0
$427$	−16.6517	−0.805832
$428$	0	0
$429$	22.3488	1.07901
$430$	0	0
$431$	−0.249504	−0.0120182	−0.00600910	−	0.999982i	$-0.501913\pi$
$431$	−0.249504	−0.0120182	−0.00600910	+	0.999982i	$0.501913\pi$
$432$	0	0
$433$	16.5117	0.793500	0.396750	−	0.917927i	$-0.370138\pi$
$433$	16.5117	0.793500	0.396750	+	0.917927i	$0.370138\pi$
$434$	0	0
$435$	40.1303	1.92410
$436$	0	0
$437$	0.955285	0.0456975
$438$	0	0
$439$	−14.4893	−0.691538	−0.345769	−	0.938320i	$-0.612382\pi$
$439$	−14.4893	−0.691538	−0.345769	+	0.938320i	$0.612382\pi$
$440$	0	0
$441$	3.34899	0.159476
$442$	0	0
$443$	5.72376	0.271944	0.135972	−	0.990713i	$-0.456584\pi$
$443$	5.72376	0.271944	0.135972	+	0.990713i	$0.456584\pi$
$444$	0	0
$445$	15.8034	0.749152
$446$	0	0
$447$	−23.5962	−1.11606
$448$	0	0
$449$	15.8010	0.745694	0.372847	−	0.927893i	$-0.378382\pi$
$449$	15.8010	0.745694	0.372847	+	0.927893i	$0.378382\pi$
$450$	0	0
$451$	−23.2010	−1.09249
$452$	0	0
$453$	−27.1226	−1.27433
$454$	0	0
$455$	24.9474	1.16955
$456$	0	0
$457$	32.8207	1.53529	0.767644	−	0.640877i	$-0.221429\pi$
$457$	32.8207	1.53529	0.767644	+	0.640877i	$0.221429\pi$
$458$	0	0
$459$	−7.86889	−0.367289
$460$	0	0
$461$	39.2177	1.82655	0.913275	−	0.407343i	$-0.133545\pi$
$461$	39.2177	1.82655	0.913275	+	0.407343i	$0.133545\pi$
$462$	0	0
$463$	19.8609	0.923016	0.461508	−	0.887136i	$-0.347309\pi$
$463$	19.8609	0.923016	0.461508	+	0.887136i	$0.347309\pi$
$464$	0	0
$465$	−18.0450	−0.836819
$466$	0	0
$467$	23.0684	1.06748	0.533738	−	0.845650i	$-0.320787\pi$
$467$	23.0684	1.06748	0.533738	+	0.845650i	$0.320787\pi$
$468$	0	0
$469$	−11.3355	−0.523424
$470$	0	0
$471$	9.06561	0.417721
$472$	0	0
$473$	−2.44893	−0.112602
$474$	0	0
$475$	−3.31925	−0.152297
$476$	0	0
$477$	1.51626	0.0694249
$478$	0	0
$479$	−7.18973	−0.328507	−0.164254	−	0.986418i	$-0.552522\pi$
$479$	−7.18973	−0.328507	−0.164254	+	0.986418i	$0.552522\pi$
$480$	0	0
$481$	−24.1860	−1.10279
$482$	0	0
$483$	−4.53121	−0.206177
$484$	0	0
$485$	20.3712	0.925009
$486$	0	0
$487$	35.6927	1.61739	0.808695	−	0.588228i	$-0.200174\pi$
$487$	35.6927	1.61739	0.808695	+	0.588228i	$0.200174\pi$
$488$	0	0
$489$	−8.31160	−0.375863
$490$	0	0
$491$	−34.9803	−1.57864	−0.789320	−	0.613982i	$-0.789567\pi$
$491$	−34.9803	−1.57864	−0.789320	+	0.613982i	$0.789567\pi$
$492$	0	0
$493$	10.6006	0.477428
$494$	0	0
$495$	−7.01644	−0.315366
$496$	0	0
$497$	−20.8325	−0.934467
$498$	0	0
$499$	−21.8984	−0.980307	−0.490153	−	0.871636i	$-0.663059\pi$
$499$	−21.8984	−0.980307	−0.490153	+	0.871636i	$0.663059\pi$
$500$	0	0
$501$	−3.73621	−0.166921
$502$	0	0
$503$	26.8133	1.19555	0.597774	−	0.801665i	$-0.296052\pi$
$503$	26.8133	1.19555	0.597774	+	0.801665i	$0.296052\pi$
$504$	0	0
$505$	−44.1134	−1.96302
$506$	0	0
$507$	−17.7095	−0.786508
$508$	0	0
$509$	−12.7459	−0.564953	−0.282477	−	0.959274i	$-0.591156\pi$
$509$	−12.7459	−0.564953	−0.282477	+	0.959274i	$0.591156\pi$
$510$	0	0
$511$	−17.2547	−0.763305
$512$	0	0
$513$	−2.59321	−0.114493
$514$	0	0
$515$	−26.6526	−1.17446
$516$	0	0
$517$	24.6381	1.08358
$518$	0	0
$519$	29.5558	1.29735
$520$	0	0
$521$	27.8620	1.22066	0.610328	−	0.792149i	$-0.291038\pi$
$521$	27.8620	1.22066	0.610328	+	0.792149i	$0.291038\pi$
$522$	0	0
$523$	13.6083	0.595050	0.297525	−	0.954714i	$-0.403839\pi$
$523$	13.6083	0.595050	0.297525	+	0.954714i	$0.403839\pi$
$524$	0	0
$525$	15.7442	0.687134
$526$	0	0
$527$	−4.76669	−0.207640
$528$	0	0
$529$	−18.7364	−0.814625
$530$	0	0
$531$	−4.21276	−0.182818
$532$	0	0
$533$	38.9404	1.68669
$534$	0	0
$535$	50.7051	2.19217
$536$	0	0
$537$	−31.6564	−1.36607
$538$	0	0
$539$	−14.5596	−0.627127
$540$	0	0
$541$	−15.2623	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$541$	−15.2623	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$542$	0	0
$543$	−14.8200	−0.635986
$544$	0	0
$545$	39.8473	1.70687
$546$	0	0
$547$	−2.19487	−0.0938460	−0.0469230	−	0.998899i	$-0.514942\pi$
$547$	−2.19487	−0.0938460	−0.0469230	+	0.998899i	$0.514942\pi$
$548$	0	0
$549$	−7.86038	−0.335473
$550$	0	0
$551$	3.49346	0.148826
$552$	0	0
$553$	19.6795	0.836858
$554$	0	0
$555$	−25.9011	−1.09944
$556$	0	0
$557$	24.2825	1.02888	0.514442	−	0.857525i	$-0.327999\pi$
$557$	24.2825	1.02888	0.514442	+	0.857525i	$0.327999\pi$
$558$	0	0
$559$	4.11027	0.173846
$560$	0	0
$561$	6.32217	0.266922
$562$	0	0
$563$	−2.80513	−0.118222	−0.0591110	−	0.998251i	$-0.518827\pi$
$563$	−2.80513	−0.118222	−0.0591110	+	0.998251i	$0.518827\pi$
$564$	0	0
$565$	19.2931	0.811665
$566$	0	0
$567$	9.36081	0.393117
$568$	0	0
$569$	−8.63450	−0.361977	−0.180989	−	0.983485i	$-0.557930\pi$
$569$	−8.63450	−0.361977	−0.180989	+	0.983485i	$0.557930\pi$
$570$	0	0
$571$	−31.2612	−1.30824	−0.654120	−	0.756390i	$-0.726961\pi$
$571$	−31.2612	−1.30824	−0.654120	+	0.756390i	$0.726961\pi$
$572$	0	0
$573$	19.0272	0.794871
$574$	0	0
$575$	−14.8144	−0.617804
$576$	0	0
$577$	−27.1321	−1.12953	−0.564763	−	0.825253i	$-0.691032\pi$
$577$	−27.1321	−1.12953	−0.564763	+	0.825253i	$0.691032\pi$
$578$	0	0
$579$	3.46533	0.144014
$580$	0	0
$581$	−14.6507	−0.607815
$582$	0	0
$583$	−6.59190	−0.273009
$584$	0	0
$585$	11.7764	0.486892
$586$	0	0
$587$	−28.5630	−1.17892	−0.589461	−	0.807797i	$-0.700660\pi$
$587$	−28.5630	−1.17892	−0.589461	+	0.807797i	$0.700660\pi$
$588$	0	0
$589$	−1.57087	−0.0647267
$590$	0	0
$591$	−0.0165776	−0.000681909 0
$592$	0	0
$593$	−45.1735	−1.85505	−0.927526	−	0.373759i	$-0.878069\pi$
$593$	−45.1735	−1.85505	−0.927526	+	0.373759i	$0.878069\pi$
$594$	0	0
$595$	7.05728	0.289320
$596$	0	0
$597$	8.71744	0.356781
$598$	0	0
$599$	4.09209	0.167198	0.0835991	−	0.996499i	$-0.473359\pi$
$599$	4.09209	0.167198	0.0835991	+	0.996499i	$0.473359\pi$
$600$	0	0
$601$	1.66970	0.0681084	0.0340542	−	0.999420i	$-0.489158\pi$
$601$	1.66970	0.0681084	0.0340542	+	0.999420i	$0.489158\pi$
$602$	0	0
$603$	−5.35088	−0.217905
$604$	0	0
$605$	−7.87755	−0.320268
$606$	0	0
$607$	−13.0792	−0.530868	−0.265434	−	0.964129i	$-0.585515\pi$
$607$	−13.0792	−0.530868	−0.265434	+	0.964129i	$0.585515\pi$
$608$	0	0
$609$	−16.5706	−0.671474
$610$	0	0
$611$	−41.3524	−1.67294
$612$	0	0
$613$	−3.24156	−0.130925	−0.0654626	−	0.997855i	$-0.520852\pi$
$613$	−3.24156	−0.130925	−0.0654626	+	0.997855i	$0.520852\pi$
$614$	0	0
$615$	41.7017	1.68157
$616$	0	0
$617$	36.5486	1.47139	0.735695	−	0.677313i	$-0.236856\pi$
$617$	36.5486	1.47139	0.735695	+	0.677313i	$0.236856\pi$
$618$	0	0
$619$	−8.53616	−0.343097	−0.171549	−	0.985176i	$-0.554877\pi$
$619$	−8.53616	−0.343097	−0.171549	+	0.985176i	$0.554877\pi$
$620$	0	0
$621$	−11.5740	−0.464449
$622$	0	0
$623$	−6.52551	−0.261439
$624$	0	0
$625$	−9.39846	−0.375938
$626$	0	0
$627$	2.08349	0.0832065
$628$	0	0
$629$	−6.84189	−0.272804
$630$	0	0
$631$	−24.8557	−0.989489	−0.494745	−	0.869038i	$-0.664738\pi$
$631$	−24.8557	−0.989489	−0.494745	+	0.869038i	$0.664738\pi$
$632$	0	0
$633$	12.3028	0.488992
$634$	0	0
$635$	−24.5252	−0.973254
$636$	0	0
$637$	24.4368	0.968220
$638$	0	0
$639$	−9.83394	−0.389025
$640$	0	0
$641$	47.8356	1.88939	0.944697	−	0.327945i	$-0.106356\pi$
$641$	47.8356	1.88939	0.944697	+	0.327945i	$0.106356\pi$
$642$	0	0
$643$	−22.0880	−0.871066	−0.435533	−	0.900173i	$-0.643440\pi$
$643$	−22.0880	−0.871066	−0.435533	+	0.900173i	$0.643440\pi$
$644$	0	0
$645$	4.40174	0.173318
$646$	0	0
$647$	−23.7796	−0.934873	−0.467437	−	0.884027i	$-0.654822\pi$
$647$	−23.7796	−0.934873	−0.467437	+	0.884027i	$0.654822\pi$
$648$	0	0
$649$	18.3148	0.718920
$650$	0	0
$651$	7.45114	0.292033
$652$	0	0
$653$	−16.6656	−0.652174	−0.326087	−	0.945340i	$-0.605730\pi$
$653$	−16.6656	−0.652174	−0.326087	+	0.945340i	$0.605730\pi$
$654$	0	0
$655$	−9.17986	−0.358687
$656$	0	0
$657$	−8.14505	−0.317769
$658$	0	0
$659$	−4.63528	−0.180565	−0.0902824	−	0.995916i	$-0.528777\pi$
$659$	−4.63528	−0.180565	−0.0902824	+	0.995916i	$0.528777\pi$
$660$	0	0
$661$	33.7771	1.31378	0.656889	−	0.753988i	$-0.271872\pi$
$661$	33.7771	1.31378	0.656889	+	0.753988i	$0.271872\pi$
$662$	0	0
$663$	−10.6111	−0.412101
$664$	0	0
$665$	2.32575	0.0901885
$666$	0	0
$667$	15.5920	0.603724
$668$	0	0
$669$	30.9577	1.19689
$670$	0	0
$671$	34.1727	1.31922
$672$	0	0
$673$	34.1787	1.31749	0.658746	−	0.752365i	$-0.271087\pi$
$673$	34.1787	1.31749	0.658746	+	0.752365i	$0.271087\pi$
$674$	0	0
$675$	40.2152	1.54788
$676$	0	0
$677$	−17.5876	−0.675946	−0.337973	−	0.941156i	$-0.609741\pi$
$677$	−17.5876	−0.675946	−0.337973	+	0.941156i	$0.609741\pi$
$678$	0	0
$679$	−8.41166	−0.322810
$680$	0	0
$681$	−18.7880	−0.719958
$682$	0	0
$683$	32.6815	1.25052	0.625262	−	0.780415i	$-0.284992\pi$
$683$	32.6815	1.25052	0.625262	+	0.780415i	$0.284992\pi$
$684$	0	0
$685$	31.7706	1.21389
$686$	0	0
$687$	13.0155	0.496572
$688$	0	0
$689$	11.0638	0.421497
$690$	0	0
$691$	−4.42414	−0.168302	−0.0841511	−	0.996453i	$-0.526818\pi$
$691$	−4.42414	−0.168302	−0.0841511	+	0.996453i	$0.526818\pi$
$692$	0	0
$693$	2.89722	0.110056
$694$	0	0
$695$	20.5398	0.779121
$696$	0	0
$697$	11.0157	0.417250
$698$	0	0
$699$	7.72600	0.292224
$700$	0	0
$701$	−29.6814	−1.12105	−0.560525	−	0.828137i	$-0.689401\pi$
$701$	−29.6814	−1.12105	−0.560525	+	0.828137i	$0.689401\pi$
$702$	0	0
$703$	−2.25476	−0.0850399
$704$	0	0
$705$	−44.2848	−1.66786
$706$	0	0
$707$	18.2153	0.685056
$708$	0	0
$709$	−21.2720	−0.798888	−0.399444	−	0.916758i	$-0.630797\pi$
$709$	−21.2720	−0.798888	−0.399444	+	0.916758i	$0.630797\pi$
$710$	0	0
$711$	9.28966	0.348390
$712$	0	0
$713$	−7.01110	−0.262568
$714$	0	0
$715$	−51.1973	−1.91467
$716$	0	0
$717$	−24.6628	−0.921051
$718$	0	0
$719$	−4.87863	−0.181942	−0.0909711	−	0.995854i	$-0.528997\pi$
$719$	−4.87863	−0.181942	−0.0909711	+	0.995854i	$0.528997\pi$
$720$	0	0
$721$	11.0054	0.409862
$722$	0	0
$723$	8.21796	0.305629
$724$	0	0
$725$	−54.1761	−2.01205
$726$	0	0
$727$	8.95156	0.331995	0.165998	−	0.986126i	$-0.446916\pi$
$727$	8.95156	0.331995	0.165998	+	0.986126i	$0.446916\pi$
$728$	0	0
$729$	30.0311	1.11226
$730$	0	0
$731$	1.16274	0.0430055
$732$	0	0
$733$	−8.75230	−0.323274	−0.161637	−	0.986850i	$-0.551677\pi$
$733$	−8.75230	−0.323274	−0.161637	+	0.986850i	$0.551677\pi$
$734$	0	0
$735$	26.1696	0.965281
$736$	0	0
$737$	23.2628	0.856895
$738$	0	0
$739$	31.0993	1.14401	0.572004	−	0.820251i	$-0.306166\pi$
$739$	31.0993	1.14401	0.572004	+	0.820251i	$0.306166\pi$
$740$	0	0
$741$	−3.49691	−0.128462
$742$	0	0
$743$	6.54028	0.239940	0.119970	−	0.992778i	$-0.461720\pi$
$743$	6.54028	0.239940	0.119970	+	0.992778i	$0.461720\pi$
$744$	0	0
$745$	54.0549	1.98042
$746$	0	0
$747$	−6.91584	−0.253037
$748$	0	0
$749$	−20.9371	−0.765025
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	32.1277	1.17080
$754$	0	0
$755$	62.1333	2.26126
$756$	0	0
$757$	6.63500	0.241153	0.120577	−	0.992704i	$-0.461526\pi$
$757$	6.63500	0.241153	0.120577	+	0.992704i	$0.461526\pi$
$758$	0	0
$759$	9.29900	0.337532
$760$	0	0
$761$	−6.58533	−0.238718	−0.119359	−	0.992851i	$-0.538084\pi$
$761$	−6.58533	−0.238718	−0.119359	+	0.992851i	$0.538084\pi$
$762$	0	0
$763$	−16.4537	−0.595665
$764$	0	0
$765$	3.33137	0.120446
$766$	0	0
$767$	−30.7395	−1.10994
$768$	0	0
$769$	38.6628	1.39422	0.697109	−	0.716965i	$-0.254469\pi$
$769$	38.6628	1.39422	0.697109	+	0.716965i	$0.254469\pi$
$770$	0	0
$771$	−27.3203	−0.983917
$772$	0	0
$773$	−47.9988	−1.72640	−0.863198	−	0.504865i	$-0.831542\pi$
$773$	−47.9988	−1.72640	−0.863198	+	0.504865i	$0.831542\pi$
$774$	0	0
$775$	24.3609	0.875068
$776$	0	0
$777$	10.6950	0.383682
$778$	0	0
$779$	3.63025	0.130067
$780$	0	0
$781$	42.7527	1.52981
$782$	0	0
$783$	−42.3259	−1.51261
$784$	0	0
$785$	−20.7678	−0.741233
$786$	0	0
$787$	−19.9864	−0.712439	−0.356219	−	0.934402i	$-0.615934\pi$
$787$	−19.9864	−0.712439	−0.356219	+	0.934402i	$0.615934\pi$
$788$	0	0
$789$	−14.1749	−0.504642
$790$	0	0
$791$	−7.96647	−0.283255
$792$	0	0
$793$	−57.3553	−2.03675
$794$	0	0
$795$	11.8484	0.420218
$796$	0	0
$797$	−28.9099	−1.02404	−0.512020	−	0.858974i	$-0.671103\pi$
$797$	−28.9099	−1.02404	−0.512020	+	0.858974i	$0.671103\pi$
$798$	0	0
$799$	−11.6980	−0.413847
$800$	0	0
$801$	−3.08035	−0.108839
$802$	0	0
$803$	35.4103	1.24960
$804$	0	0
$805$	10.3802	0.365855
$806$	0	0
$807$	−47.0478	−1.65616
$808$	0	0
$809$	18.3638	0.645637	0.322819	−	0.946461i	$-0.395370\pi$
$809$	18.3638	0.645637	0.322819	+	0.946461i	$0.395370\pi$
$810$	0	0
$811$	6.80272	0.238876	0.119438	−	0.992842i	$-0.461891\pi$
$811$	6.80272	0.238876	0.119438	+	0.992842i	$0.461891\pi$
$812$	0	0
$813$	−2.37466	−0.0832829
$814$	0	0
$815$	19.0404	0.666958
$816$	0	0
$817$	0.383184	0.0134059
$818$	0	0
$819$	−4.86268	−0.169916
$820$	0	0
$821$	−22.5315	−0.786353	−0.393177	−	0.919463i	$-0.628624\pi$
$821$	−22.5315	−0.786353	−0.393177	+	0.919463i	$0.628624\pi$
$822$	0	0
$823$	36.2037	1.26198	0.630990	−	0.775791i	$-0.282649\pi$
$823$	36.2037	1.26198	0.630990	+	0.775791i	$0.282649\pi$
$824$	0	0
$825$	−32.3104	−1.12490
$826$	0	0
$827$	−41.4606	−1.44173	−0.720863	−	0.693077i	$-0.756254\pi$
$827$	−41.4606	−1.44173	−0.720863	+	0.693077i	$0.756254\pi$
$828$	0	0
$829$	−33.2836	−1.15599	−0.577994	−	0.816041i	$-0.696164\pi$
$829$	−33.2836	−1.15599	−0.577994	+	0.816041i	$0.696164\pi$
$830$	0	0
$831$	34.0793	1.18220
$832$	0	0
$833$	6.91283	0.239515
$834$	0	0
$835$	8.55901	0.296197
$836$	0	0
$837$	19.0323	0.657853
$838$	0	0
$839$	36.9460	1.27552	0.637759	−	0.770236i	$-0.279861\pi$
$839$	36.9460	1.27552	0.637759	+	0.770236i	$0.279861\pi$
$840$	0	0
$841$	28.0196	0.966194
$842$	0	0
$843$	31.6128	1.08880
$844$	0	0
$845$	40.5695	1.39563
$846$	0	0
$847$	3.25279	0.111767
$848$	0	0
$849$	1.45875	0.0500643
$850$	0	0
$851$	−10.0634	−0.344970
$852$	0	0
$853$	3.02484	0.103569	0.0517844	−	0.998658i	$-0.483509\pi$
$853$	3.02484	0.103569	0.0517844	+	0.998658i	$0.483509\pi$
$854$	0	0
$855$	1.09786	0.0375461
$856$	0	0
$857$	33.1556	1.13257	0.566287	−	0.824208i	$-0.308379\pi$
$857$	33.1556	1.13257	0.566287	+	0.824208i	$0.308379\pi$
$858$	0	0
$859$	6.91803	0.236040	0.118020	−	0.993011i	$-0.462345\pi$
$859$	6.91803	0.236040	0.118020	+	0.993011i	$0.462345\pi$
$860$	0	0
$861$	−17.2194	−0.586836
$862$	0	0
$863$	51.0722	1.73852	0.869259	−	0.494357i	$-0.164597\pi$
$863$	51.0722	1.73852	0.869259	+	0.494357i	$0.164597\pi$
$864$	0	0
$865$	−67.7072	−2.30211
$866$	0	0
$867$	22.8913	0.777429
$868$	0	0
$869$	−40.3865	−1.37002
$870$	0	0
$871$	−39.0441	−1.32296
$872$	0	0
$873$	−3.97070	−0.134388
$874$	0	0
$875$	−10.9318	−0.369561
$876$	0	0
$877$	−35.4660	−1.19760	−0.598802	−	0.800897i	$-0.704356\pi$
$877$	−35.4660	−1.19760	−0.598802	+	0.800897i	$0.704356\pi$
$878$	0	0
$879$	−13.3700	−0.450960
$880$	0	0
$881$	−16.2735	−0.548269	−0.274134	−	0.961691i	$-0.588391\pi$
$881$	−16.2735	−0.548269	−0.274134	+	0.961691i	$0.588391\pi$
$882$	0	0
$883$	40.4970	1.36283	0.681416	−	0.731896i	$-0.261364\pi$
$883$	40.4970	1.36283	0.681416	+	0.731896i	$0.261364\pi$
$884$	0	0
$885$	−32.9193	−1.10657
$886$	0	0
$887$	27.2986	0.916596	0.458298	−	0.888799i	$-0.348459\pi$
$887$	27.2986	0.916596	0.458298	+	0.888799i	$0.348459\pi$
$888$	0	0
$889$	10.1269	0.339646
$890$	0	0
$891$	−19.2103	−0.643570
$892$	0	0
$893$	−3.85512	−0.129007
$894$	0	0
$895$	72.5193	2.42405
$896$	0	0
$897$	−15.6074	−0.521115
$898$	0	0
$899$	−25.6395	−0.855125
$900$	0	0
$901$	3.12980	0.104269
$902$	0	0
$903$	−1.81756	−0.0604846
$904$	0	0
$905$	33.9500	1.12854
$906$	0	0
$907$	2.34855	0.0779823	0.0389912	−	0.999240i	$-0.487586\pi$
$907$	2.34855	0.0779823	0.0389912	+	0.999240i	$0.487586\pi$
$908$	0	0
$909$	8.59847	0.285193
$910$	0	0
$911$	−46.4255	−1.53815	−0.769073	−	0.639161i	$-0.779282\pi$
$911$	−46.4255	−1.53815	−0.769073	+	0.639161i	$0.779282\pi$
$912$	0	0
$913$	30.0664	0.995051
$914$	0	0
$915$	−61.4225	−2.03056
$916$	0	0
$917$	3.79054	0.125175
$918$	0	0
$919$	39.6791	1.30889	0.654447	−	0.756108i	$-0.272902\pi$
$919$	39.6791	1.30889	0.654447	+	0.756108i	$0.272902\pi$
$920$	0	0
$921$	−29.1932	−0.961950
$922$	0	0
$923$	−71.7559	−2.36187
$924$	0	0
$925$	34.9665	1.14969
$926$	0	0
$927$	5.19506	0.170628
$928$	0	0
$929$	−11.5081	−0.377567	−0.188784	−	0.982019i	$-0.560454\pi$
$929$	−11.5081	−0.377567	−0.188784	+	0.982019i	$0.560454\pi$
$930$	0	0
$931$	2.27814	0.0746631
$932$	0	0
$933$	−11.3691	−0.372207
$934$	0	0
$935$	−14.4830	−0.473645
$936$	0	0
$937$	0.609119	0.0198991	0.00994953	−	0.999951i	$-0.496833\pi$
$937$	0.609119	0.0198991	0.00994953	+	0.999951i	$0.496833\pi$
$938$	0	0
$939$	24.7022	0.806126
$940$	0	0
$941$	−47.9190	−1.56211	−0.781057	−	0.624460i	$-0.785319\pi$
$941$	−47.9190	−1.56211	−0.781057	+	0.624460i	$0.785319\pi$
$942$	0	0
$943$	16.2025	0.527626
$944$	0	0
$945$	−28.1782	−0.916636
$946$	0	0
$947$	−31.1984	−1.01381	−0.506906	−	0.862001i	$-0.669211\pi$
$947$	−31.1984	−1.01381	−0.506906	+	0.862001i	$0.669211\pi$
$948$	0	0
$949$	−59.4325	−1.92926
$950$	0	0
$951$	1.93779	0.0628371
$952$	0	0
$953$	−18.6387	−0.603765	−0.301883	−	0.953345i	$-0.597615\pi$
$953$	−18.6387	−0.603765	−0.301883	+	0.953345i	$0.597615\pi$
$954$	0	0
$955$	−43.5880	−1.41047
$956$	0	0
$957$	34.0063	1.09927
$958$	0	0
$959$	−13.1187	−0.423624
$960$	0	0
$961$	−19.4709	−0.628094
$962$	0	0
$963$	−9.88331	−0.318485
$964$	0	0
$965$	−7.93848	−0.255549
$966$	0	0
$967$	−24.5129	−0.788283	−0.394141	−	0.919050i	$-0.628958\pi$
$967$	−24.5129	−0.788283	−0.394141	+	0.919050i	$0.628958\pi$
$968$	0	0
$969$	−0.989230	−0.0317786
$970$	0	0
$971$	22.1035	0.709334	0.354667	−	0.934993i	$-0.384594\pi$
$971$	22.1035	0.709334	0.354667	+	0.934993i	$0.384594\pi$
$972$	0	0
$973$	−8.48129	−0.271898
$974$	0	0
$975$	54.2296	1.73674
$976$	0	0
$977$	−32.7711	−1.04844	−0.524220	−	0.851583i	$-0.675643\pi$
$977$	−32.7711	−1.04844	−0.524220	+	0.851583i	$0.675643\pi$
$978$	0	0
$979$	13.3917	0.428001
$980$	0	0
$981$	−7.76693	−0.247979
$982$	0	0
$983$	−48.7214	−1.55397	−0.776985	−	0.629519i	$-0.783252\pi$
$983$	−48.7214	−1.55397	−0.776985	+	0.629519i	$0.783252\pi$
$984$	0	0
$985$	0.0379763	0.00121003
$986$	0	0
$987$	18.2860	0.582051
$988$	0	0
$989$	1.71022	0.0543819
$990$	0	0
$991$	−9.77429	−0.310491	−0.155245	−	0.987876i	$-0.549617\pi$
$991$	−9.77429	−0.310491	−0.155245	+	0.987876i	$0.549617\pi$
$992$	0	0
$993$	−28.4318	−0.902255
$994$	0	0
$995$	−19.9702	−0.633097
$996$	0	0
$997$	9.43021	0.298658	0.149329	−	0.988788i	$-0.452289\pi$
$997$	9.43021	0.298658	0.149329	+	0.988788i	$0.452289\pi$
$998$	0	0
$999$	27.3181	0.864308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.12	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.12	✓	49	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.52312 q^{3} +3.48921 q^{5} -1.44076 q^{7} -0.680107 q^{9} +O(q^{10})\)	\(q-1.52312 q^{3} +3.48921 q^{5} -1.44076 q^{7} -0.680107 q^{9} +2.95674 q^{11} -4.96257 q^{13} -5.31448 q^{15} -1.40385 q^{17} -0.462641 q^{19} +2.19445 q^{21} -2.06485 q^{23} +7.17456 q^{25} +5.60524 q^{27} -7.55113 q^{29} +3.39545 q^{31} -4.50347 q^{33} -5.02711 q^{35} +4.87368 q^{37} +7.55859 q^{39} -7.84681 q^{41} -0.828254 q^{43} -2.37303 q^{45} +8.33285 q^{47} -4.92421 q^{49} +2.13822 q^{51} -2.22945 q^{53} +10.3167 q^{55} +0.704657 q^{57} +6.19427 q^{59} +11.5576 q^{61} +0.979870 q^{63} -17.3154 q^{65} +7.86771 q^{67} +3.14502 q^{69} +14.4594 q^{71} +11.9761 q^{73} -10.9277 q^{75} -4.25995 q^{77} -13.6591 q^{79} -6.49713 q^{81} +10.1688 q^{83} -4.89831 q^{85} +11.5013 q^{87} +4.52922 q^{89} +7.14988 q^{91} -5.17168 q^{93} -1.61425 q^{95} +5.83835 q^{97} -2.01090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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