LMFDB - Embedded newform 6008.2.a.e.1.23

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:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
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)

$f(q)$	$=$	$q-0.217342 q^{3} +1.76324 q^{5} +1.63306 q^{7} -2.95276 q^{9} +O(q^{10})$	$q-0.217342 q^{3} +1.76324 q^{5} +1.63306 q^{7} -2.95276 q^{9} -4.73515 q^{11} +4.29715 q^{13} -0.383227 q^{15} -6.17157 q^{17} +3.23590 q^{19} -0.354933 q^{21} -5.20023 q^{23} -1.89097 q^{25} +1.29379 q^{27} -5.37319 q^{29} +3.65068 q^{31} +1.02915 q^{33} +2.87948 q^{35} +10.2663 q^{37} -0.933952 q^{39} +12.3293 q^{41} -4.50192 q^{43} -5.20644 q^{45} +11.0393 q^{47} -4.33312 q^{49} +1.34134 q^{51} -2.84055 q^{53} -8.34922 q^{55} -0.703298 q^{57} -0.223774 q^{59} +11.2299 q^{61} -4.82204 q^{63} +7.57691 q^{65} +7.71113 q^{67} +1.13023 q^{69} -5.66749 q^{71} +6.65373 q^{73} +0.410989 q^{75} -7.73278 q^{77} +8.73124 q^{79} +8.57709 q^{81} -1.13562 q^{83} -10.8820 q^{85} +1.16782 q^{87} +17.7119 q^{89} +7.01749 q^{91} -0.793447 q^{93} +5.70568 q^{95} +14.4828 q^{97} +13.9818 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * 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
$3$	−0.217342	−0.125483	−0.0627413	−	0.998030i	$-0.519984\pi$
$3$	−0.217342	−0.125483	−0.0627413	+	0.998030i	$0.519984\pi$
$4$	0	0
$5$	1.76324	0.788546	0.394273	−	0.918993i	$-0.370996\pi$
$5$	1.76324	0.788546	0.394273	+	0.918993i	$0.370996\pi$
$6$	0	0
$7$	1.63306	0.617238	0.308619	−	0.951186i	$-0.400133\pi$
$7$	1.63306	0.617238	0.308619	+	0.951186i	$0.400133\pi$
$8$	0	0
$9$	−2.95276	−0.984254
$10$	0	0
$11$	−4.73515	−1.42770	−0.713851	−	0.700298i	$-0.753051\pi$
$11$	−4.73515	−1.42770	−0.713851	+	0.700298i	$0.753051\pi$
$12$	0	0
$13$	4.29715	1.19181	0.595907	−	0.803053i	$-0.296793\pi$
$13$	4.29715	1.19181	0.595907	+	0.803053i	$0.296793\pi$
$14$	0	0
$15$	−0.383227	−0.0989489
$16$	0	0
$17$	−6.17157	−1.49683	−0.748413	−	0.663233i	$-0.769184\pi$
$17$	−6.17157	−1.49683	−0.748413	+	0.663233i	$0.769184\pi$
$18$	0	0
$19$	3.23590	0.742366	0.371183	−	0.928560i	$-0.378952\pi$
$19$	3.23590	0.742366	0.371183	+	0.928560i	$0.378952\pi$
$20$	0	0
$21$	−0.354933	−0.0774527
$22$	0	0
$23$	−5.20023	−1.08432	−0.542162	−	0.840274i	$-0.682394\pi$
$23$	−5.20023	−1.08432	−0.542162	+	0.840274i	$0.682394\pi$
$24$	0	0
$25$	−1.89097	−0.378195
$26$	0	0
$27$	1.29379	0.248989
$28$	0	0
$29$	−5.37319	−0.997776	−0.498888	−	0.866666i	$-0.666258\pi$
$29$	−5.37319	−0.997776	−0.498888	+	0.866666i	$0.666258\pi$
$30$	0	0
$31$	3.65068	0.655681	0.327840	−	0.944733i	$-0.393679\pi$
$31$	3.65068	0.655681	0.327840	+	0.944733i	$0.393679\pi$
$32$	0	0
$33$	1.02915	0.179152
$34$	0	0
$35$	2.87948	0.486721
$36$	0	0
$37$	10.2663	1.68777	0.843884	−	0.536525i	$-0.180263\pi$
$37$	10.2663	1.68777	0.843884	+	0.536525i	$0.180263\pi$
$38$	0	0
$39$	−0.933952	−0.149552
$40$	0	0
$41$	12.3293	1.92552	0.962760	−	0.270359i	$-0.0871423\pi$
$41$	12.3293	1.92552	0.962760	+	0.270359i	$0.0871423\pi$
$42$	0	0
$43$	−4.50192	−0.686536	−0.343268	−	0.939238i	$-0.611534\pi$
$43$	−4.50192	−0.686536	−0.343268	+	0.939238i	$0.611534\pi$
$44$	0	0
$45$	−5.20644	−0.776130
$46$	0	0
$47$	11.0393	1.61024	0.805122	−	0.593109i	$-0.202100\pi$
$47$	11.0393	1.61024	0.805122	+	0.593109i	$0.202100\pi$
$48$	0	0
$49$	−4.33312	−0.619017
$50$	0	0
$51$	1.34134	0.187826
$52$	0	0
$53$	−2.84055	−0.390179	−0.195090	−	0.980785i	$-0.562500\pi$
$53$	−2.84055	−0.390179	−0.195090	+	0.980785i	$0.562500\pi$
$54$	0	0
$55$	−8.34922	−1.12581
$56$	0	0
$57$	−0.703298	−0.0931541
$58$	0	0
$59$	−0.223774	−0.0291329	−0.0145665	−	0.999894i	$-0.504637\pi$
$59$	−0.223774	−0.0291329	−0.0145665	+	0.999894i	$0.504637\pi$
$60$	0	0
$61$	11.2299	1.43784	0.718921	−	0.695092i	$-0.244636\pi$
$61$	11.2299	1.43784	0.718921	+	0.695092i	$0.244636\pi$
$62$	0	0
$63$	−4.82204	−0.607519
$64$	0	0
$65$	7.57691	0.939800
$66$	0	0
$67$	7.71113	0.942064	0.471032	−	0.882116i	$-0.343882\pi$
$67$	7.71113	0.942064	0.471032	+	0.882116i	$0.343882\pi$
$68$	0	0
$69$	1.13023	0.136064
$70$	0	0
$71$	−5.66749	−0.672607	−0.336304	−	0.941754i	$-0.609177\pi$
$71$	−5.66749	−0.672607	−0.336304	+	0.941754i	$0.609177\pi$
$72$	0	0
$73$	6.65373	0.778760	0.389380	−	0.921077i	$-0.372689\pi$
$73$	6.65373	0.778760	0.389380	+	0.921077i	$0.372689\pi$
$74$	0	0
$75$	0.410989	0.0474569
$76$	0	0
$77$	−7.73278	−0.881232
$78$	0	0
$79$	8.73124	0.982341	0.491171	−	0.871063i	$-0.336569\pi$
$79$	8.73124	0.982341	0.491171	+	0.871063i	$0.336569\pi$
$80$	0	0
$81$	8.57709	0.953010
$82$	0	0
$83$	−1.13562	−0.124650	−0.0623252	−	0.998056i	$-0.519852\pi$
$83$	−1.13562	−0.124650	−0.0623252	+	0.998056i	$0.519852\pi$
$84$	0	0
$85$	−10.8820	−1.18032
$86$	0	0
$87$	1.16782	0.125204
$88$	0	0
$89$	17.7119	1.87745	0.938727	−	0.344660i	$-0.112006\pi$
$89$	17.7119	1.87745	0.938727	+	0.344660i	$0.112006\pi$
$90$	0	0
$91$	7.01749	0.735633
$92$	0	0
$93$	−0.793447	−0.0822766
$94$	0	0
$95$	5.70568	0.585390
$96$	0	0
$97$	14.4828	1.47051	0.735254	−	0.677791i	$-0.237063\pi$
$97$	14.4828	1.47051	0.735254	+	0.677791i	$0.237063\pi$
$98$	0	0
$99$	13.9818	1.40522
$100$	0	0
$101$	−6.51307	−0.648075	−0.324037	−	0.946044i	$-0.605040\pi$
$101$	−6.51307	−0.648075	−0.324037	+	0.946044i	$0.605040\pi$
$102$	0	0
$103$	13.7546	1.35529	0.677643	−	0.735391i	$-0.263002\pi$
$103$	13.7546	1.35529	0.677643	+	0.735391i	$0.263002\pi$
$104$	0	0
$105$	−0.625833	−0.0610750
$106$	0	0
$107$	−5.72856	−0.553801	−0.276901	−	0.960899i	$-0.589307\pi$
$107$	−5.72856	−0.553801	−0.276901	+	0.960899i	$0.589307\pi$
$108$	0	0
$109$	16.8740	1.61624	0.808118	−	0.589021i	$-0.200487\pi$
$109$	16.8740	1.61624	0.808118	+	0.589021i	$0.200487\pi$
$110$	0	0
$111$	−2.23130	−0.211786
$112$	0	0
$113$	14.9079	1.40241	0.701207	−	0.712957i	$-0.252645\pi$
$113$	14.9079	1.40241	0.701207	+	0.712957i	$0.252645\pi$
$114$	0	0
$115$	−9.16928	−0.855040
$116$	0	0
$117$	−12.6885	−1.17305
$118$	0	0
$119$	−10.0785	−0.923899
$120$	0	0
$121$	11.4217	1.03833
$122$	0	0
$123$	−2.67969	−0.241619
$124$	0	0
$125$	−12.1505	−1.08677
$126$	0	0
$127$	−10.6094	−0.941435	−0.470718	−	0.882284i	$-0.656005\pi$
$127$	−10.6094	−0.941435	−0.470718	+	0.882284i	$0.656005\pi$
$128$	0	0
$129$	0.978457	0.0861483
$130$	0	0
$131$	−22.3149	−1.94966	−0.974831	−	0.222943i	$-0.928434\pi$
$131$	−22.3149	−1.94966	−0.974831	+	0.222943i	$0.928434\pi$
$132$	0	0
$133$	5.28442	0.458217
$134$	0	0
$135$	2.28126	0.196340
$136$	0	0
$137$	10.0300	0.856919	0.428460	−	0.903561i	$-0.359056\pi$
$137$	10.0300	0.856919	0.428460	+	0.903561i	$0.359056\pi$
$138$	0	0
$139$	2.27084	0.192611	0.0963053	−	0.995352i	$-0.469298\pi$
$139$	2.27084	0.192611	0.0963053	+	0.995352i	$0.469298\pi$
$140$	0	0
$141$	−2.39930	−0.202058
$142$	0	0
$143$	−20.3476	−1.70156
$144$	0	0
$145$	−9.47424	−0.786793
$146$	0	0
$147$	0.941770	0.0776759
$148$	0	0
$149$	−7.92170	−0.648971	−0.324485	−	0.945891i	$-0.605191\pi$
$149$	−7.92170	−0.648971	−0.324485	+	0.945891i	$0.605191\pi$
$150$	0	0
$151$	−5.29063	−0.430546	−0.215273	−	0.976554i	$-0.569064\pi$
$151$	−5.29063	−0.430546	−0.215273	+	0.976554i	$0.569064\pi$
$152$	0	0
$153$	18.2232	1.47326
$154$	0	0
$155$	6.43703	0.517035
$156$	0	0
$157$	−7.69056	−0.613773	−0.306887	−	0.951746i	$-0.599287\pi$
$157$	−7.69056	−0.613773	−0.306887	+	0.951746i	$0.599287\pi$
$158$	0	0
$159$	0.617371	0.0489607
$160$	0	0
$161$	−8.49229	−0.669286
$162$	0	0
$163$	14.2163	1.11350	0.556752	−	0.830679i	$-0.312047\pi$
$163$	14.2163	1.11350	0.556752	+	0.830679i	$0.312047\pi$
$164$	0	0
$165$	1.81464	0.141269
$166$	0	0
$167$	25.5366	1.97608	0.988040	−	0.154199i	$-0.0492798\pi$
$167$	25.5366	1.97608	0.988040	+	0.154199i	$0.0492798\pi$
$168$	0	0
$169$	5.46547	0.420421
$170$	0	0
$171$	−9.55484	−0.730677
$172$	0	0
$173$	−7.30483	−0.555376	−0.277688	−	0.960671i	$-0.589568\pi$
$173$	−7.30483	−0.555376	−0.277688	+	0.960671i	$0.589568\pi$
$174$	0	0
$175$	−3.08807	−0.233436
$176$	0	0
$177$	0.0486356	0.00365567
$178$	0	0
$179$	−6.19399	−0.462960	−0.231480	−	0.972840i	$-0.574357\pi$
$179$	−6.19399	−0.462960	−0.231480	+	0.972840i	$0.574357\pi$
$180$	0	0
$181$	8.51132	0.632641	0.316321	−	0.948652i	$-0.397552\pi$
$181$	8.51132	0.632641	0.316321	+	0.948652i	$0.397552\pi$
$182$	0	0
$183$	−2.44073	−0.180424
$184$	0	0
$185$	18.1020	1.33088
$186$	0	0
$187$	29.2233	2.13702
$188$	0	0
$189$	2.11283	0.153686
$190$	0	0
$191$	−10.9988	−0.795842	−0.397921	−	0.917420i	$-0.630268\pi$
$191$	−10.9988	−0.795842	−0.397921	+	0.917420i	$0.630268\pi$
$192$	0	0
$193$	−12.9670	−0.933382	−0.466691	−	0.884421i	$-0.654554\pi$
$193$	−12.9670	−0.933382	−0.466691	+	0.884421i	$0.654554\pi$
$194$	0	0
$195$	−1.64678	−0.117929
$196$	0	0
$197$	7.60271	0.541671	0.270835	−	0.962626i	$-0.412700\pi$
$197$	7.60271	0.541671	0.270835	+	0.962626i	$0.412700\pi$
$198$	0	0
$199$	−7.96777	−0.564820	−0.282410	−	0.959294i	$-0.591134\pi$
$199$	−7.96777	−0.564820	−0.282410	+	0.959294i	$0.591134\pi$
$200$	0	0
$201$	−1.67595	−0.118213
$202$	0	0
$203$	−8.77474	−0.615866
$204$	0	0
$205$	21.7396	1.51836
$206$	0	0
$207$	15.3551	1.06725
$208$	0	0
$209$	−15.3225	−1.05988
$210$	0	0
$211$	12.7030	0.874513	0.437256	−	0.899337i	$-0.355950\pi$
$211$	12.7030	0.874513	0.437256	+	0.899337i	$0.355950\pi$
$212$	0	0
$213$	1.23178	0.0844005
$214$	0	0
$215$	−7.93797	−0.541365
$216$	0	0
$217$	5.96177	0.404711
$218$	0	0
$219$	−1.44614	−0.0977209
$220$	0	0
$221$	−26.5202	−1.78394
$222$	0	0
$223$	−11.7706	−0.788219	−0.394109	−	0.919064i	$-0.628947\pi$
$223$	−11.7706	−0.788219	−0.394109	+	0.919064i	$0.628947\pi$
$224$	0	0
$225$	5.58360	0.372240
$226$	0	0
$227$	−4.58907	−0.304587	−0.152294	−	0.988335i	$-0.548666\pi$
$227$	−4.58907	−0.304587	−0.152294	+	0.988335i	$0.548666\pi$
$228$	0	0
$229$	6.76699	0.447175	0.223588	−	0.974684i	$-0.428223\pi$
$229$	6.76699	0.447175	0.223588	+	0.974684i	$0.428223\pi$
$230$	0	0
$231$	1.68066	0.110579
$232$	0	0
$233$	6.23664	0.408576	0.204288	−	0.978911i	$-0.434512\pi$
$233$	6.23664	0.408576	0.204288	+	0.978911i	$0.434512\pi$
$234$	0	0
$235$	19.4649	1.26975
$236$	0	0
$237$	−1.89767	−0.123267
$238$	0	0
$239$	−0.462154	−0.0298943	−0.0149471	−	0.999888i	$-0.504758\pi$
$239$	−0.462154	−0.0298943	−0.0149471	+	0.999888i	$0.504758\pi$
$240$	0	0
$241$	−14.7355	−0.949194	−0.474597	−	0.880203i	$-0.657406\pi$
$241$	−14.7355	−0.949194	−0.474597	+	0.880203i	$0.657406\pi$
$242$	0	0
$243$	−5.74553	−0.368576
$244$	0	0
$245$	−7.64034	−0.488123
$246$	0	0
$247$	13.9051	0.884763
$248$	0	0
$249$	0.246818	0.0156415
$250$	0	0
$251$	29.5708	1.86649	0.933247	−	0.359235i	$-0.116962\pi$
$251$	29.5708	1.86649	0.933247	+	0.359235i	$0.116962\pi$
$252$	0	0
$253$	24.6239	1.54809
$254$	0	0
$255$	2.36512	0.148109
$256$	0	0
$257$	−20.5357	−1.28098	−0.640490	−	0.767967i	$-0.721269\pi$
$257$	−20.5357	−1.28098	−0.640490	+	0.767967i	$0.721269\pi$
$258$	0	0
$259$	16.7655	1.04176
$260$	0	0
$261$	15.8658	0.982065
$262$	0	0
$263$	−20.3586	−1.25537	−0.627683	−	0.778469i	$-0.715997\pi$
$263$	−20.3586	−1.25537	−0.627683	+	0.778469i	$0.715997\pi$
$264$	0	0
$265$	−5.00858	−0.307674
$266$	0	0
$267$	−3.84954	−0.235588
$268$	0	0
$269$	14.5190	0.885237	0.442619	−	0.896710i	$-0.354050\pi$
$269$	14.5190	0.885237	0.442619	+	0.896710i	$0.354050\pi$
$270$	0	0
$271$	−14.2851	−0.867759	−0.433880	−	0.900971i	$-0.642856\pi$
$271$	−14.2851	−0.867759	−0.433880	+	0.900971i	$0.642856\pi$
$272$	0	0
$273$	−1.52520	−0.0923092
$274$	0	0
$275$	8.95405	0.539949
$276$	0	0
$277$	17.8056	1.06983	0.534916	−	0.844905i	$-0.320343\pi$
$277$	17.8056	1.06983	0.534916	+	0.844905i	$0.320343\pi$
$278$	0	0
$279$	−10.7796	−0.645357
$280$	0	0
$281$	18.7804	1.12035	0.560173	−	0.828376i	$-0.310735\pi$
$281$	18.7804	1.12035	0.560173	+	0.828376i	$0.310735\pi$
$282$	0	0
$283$	−27.5587	−1.63819	−0.819097	−	0.573655i	$-0.805525\pi$
$283$	−27.5587	−1.63819	−0.819097	+	0.573655i	$0.805525\pi$
$284$	0	0
$285$	−1.24009	−0.0734563
$286$	0	0
$287$	20.1345	1.18850
$288$	0	0
$289$	21.0883	1.24049
$290$	0	0
$291$	−3.14773	−0.184523
$292$	0	0
$293$	−7.80088	−0.455733	−0.227866	−	0.973692i	$-0.573175\pi$
$293$	−7.80088	−0.455733	−0.227866	+	0.973692i	$0.573175\pi$
$294$	0	0
$295$	−0.394568	−0.0229726
$296$	0	0
$297$	−6.12628	−0.355483
$298$	0	0
$299$	−22.3462	−1.29231
$300$	0	0
$301$	−7.35190	−0.423756
$302$	0	0
$303$	1.41557	0.0813221
$304$	0	0
$305$	19.8011	1.13381
$306$	0	0
$307$	−6.75243	−0.385382	−0.192691	−	0.981260i	$-0.561721\pi$
$307$	−6.75243	−0.385382	−0.192691	+	0.981260i	$0.561721\pi$
$308$	0	0
$309$	−2.98947	−0.170065
$310$	0	0
$311$	−32.2693	−1.82982	−0.914912	−	0.403654i	$-0.867740\pi$
$311$	−32.2693	−1.82982	−0.914912	+	0.403654i	$0.867740\pi$
$312$	0	0
$313$	32.0508	1.81162	0.905810	−	0.423685i	$-0.139264\pi$
$313$	32.0508	1.81162	0.905810	+	0.423685i	$0.139264\pi$
$314$	0	0
$315$	−8.50242	−0.479057
$316$	0	0
$317$	−17.4027	−0.977431	−0.488716	−	0.872443i	$-0.662534\pi$
$317$	−17.4027	−0.977431	−0.488716	+	0.872443i	$0.662534\pi$
$318$	0	0
$319$	25.4429	1.42453
$320$	0	0
$321$	1.24506	0.0694924
$322$	0	0
$323$	−19.9706	−1.11119
$324$	0	0
$325$	−8.12579	−0.450738
$326$	0	0
$327$	−3.66743	−0.202809
$328$	0	0
$329$	18.0278	0.993904
$330$	0	0
$331$	29.5846	1.62612	0.813059	−	0.582182i	$-0.197801\pi$
$331$	29.5846	1.62612	0.813059	+	0.582182i	$0.197801\pi$
$332$	0	0
$333$	−30.3139	−1.66119
$334$	0	0
$335$	13.5966	0.742861
$336$	0	0
$337$	−6.02154	−0.328014	−0.164007	−	0.986459i	$-0.552442\pi$
$337$	−6.02154	−0.328014	−0.164007	+	0.986459i	$0.552442\pi$
$338$	0	0
$339$	−3.24011	−0.175979
$340$	0	0
$341$	−17.2865	−0.936117
$342$	0	0
$343$	−18.5077	−0.999319
$344$	0	0
$345$	1.99287	0.107293
$346$	0	0
$347$	1.10663	0.0594068	0.0297034	−	0.999559i	$-0.490544\pi$
$347$	1.10663	0.0594068	0.0297034	+	0.999559i	$0.490544\pi$
$348$	0	0
$349$	−18.5033	−0.990458	−0.495229	−	0.868763i	$-0.664916\pi$
$349$	−18.5033	−0.990458	−0.495229	+	0.868763i	$0.664916\pi$
$350$	0	0
$351$	5.55959	0.296749
$352$	0	0
$353$	−6.31975	−0.336366	−0.168183	−	0.985756i	$-0.553790\pi$
$353$	−6.31975	−0.336366	−0.168183	+	0.985756i	$0.553790\pi$
$354$	0	0
$355$	−9.99316	−0.530382
$356$	0	0
$357$	2.19049	0.115933
$358$	0	0
$359$	27.3704	1.44456	0.722278	−	0.691602i	$-0.243095\pi$
$359$	27.3704	1.44456	0.722278	+	0.691602i	$0.243095\pi$
$360$	0	0
$361$	−8.52895	−0.448892
$362$	0	0
$363$	−2.48241	−0.130293
$364$	0	0
$365$	11.7321	0.614088
$366$	0	0
$367$	−21.6094	−1.12800	−0.564001	−	0.825774i	$-0.690738\pi$
$367$	−21.6094	−1.12800	−0.564001	+	0.825774i	$0.690738\pi$
$368$	0	0
$369$	−36.4056	−1.89520
$370$	0	0
$371$	−4.63878	−0.240834
$372$	0	0
$373$	−22.6293	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$373$	−22.6293	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$374$	0	0
$375$	2.64081	0.136371
$376$	0	0
$377$	−23.0894	−1.18916
$378$	0	0
$379$	−22.8768	−1.17510	−0.587550	−	0.809188i	$-0.699908\pi$
$379$	−22.8768	−1.17510	−0.587550	+	0.809188i	$0.699908\pi$
$380$	0	0
$381$	2.30588	0.118134
$382$	0	0
$383$	−13.6820	−0.699119	−0.349559	−	0.936914i	$-0.613669\pi$
$383$	−13.6820	−0.699119	−0.349559	+	0.936914i	$0.613669\pi$
$384$	0	0
$385$	−13.6348	−0.694892
$386$	0	0
$387$	13.2931	0.675726
$388$	0	0
$389$	12.1384	0.615439	0.307719	−	0.951477i	$-0.400434\pi$
$389$	12.1384	0.615439	0.307719	+	0.951477i	$0.400434\pi$
$390$	0	0
$391$	32.0936	1.62304
$392$	0	0
$393$	4.84997	0.244649
$394$	0	0
$395$	15.3953	0.774621
$396$	0	0
$397$	30.6967	1.54063	0.770313	−	0.637666i	$-0.220100\pi$
$397$	30.6967	1.54063	0.770313	+	0.637666i	$0.220100\pi$
$398$	0	0
$399$	−1.14853	−0.0574983
$400$	0	0
$401$	6.65165	0.332167	0.166084	−	0.986112i	$-0.446888\pi$
$401$	6.65165	0.332167	0.166084	+	0.986112i	$0.446888\pi$
$402$	0	0
$403$	15.6875	0.781450
$404$	0	0
$405$	15.1235	0.751493
$406$	0	0
$407$	−48.6125	−2.40963
$408$	0	0
$409$	38.2720	1.89243	0.946214	−	0.323541i	$-0.104873\pi$
$409$	38.2720	1.89243	0.946214	+	0.323541i	$0.104873\pi$
$410$	0	0
$411$	−2.17994	−0.107528
$412$	0	0
$413$	−0.365436	−0.0179819
$414$	0	0
$415$	−2.00237	−0.0982927
$416$	0	0
$417$	−0.493551	−0.0241693
$418$	0	0
$419$	31.5974	1.54364	0.771818	−	0.635844i	$-0.219348\pi$
$419$	31.5974	1.54364	0.771818	+	0.635844i	$0.219348\pi$
$420$	0	0
$421$	23.8718	1.16344	0.581720	−	0.813389i	$-0.302380\pi$
$421$	23.8718	1.16344	0.581720	+	0.813389i	$0.302380\pi$
$422$	0	0
$423$	−32.5964	−1.58489
$424$	0	0
$425$	11.6703	0.566092
$426$	0	0
$427$	18.3391	0.887491
$428$	0	0
$429$	4.42240	0.213516
$430$	0	0
$431$	1.75101	0.0843431	0.0421715	−	0.999110i	$-0.486572\pi$
$431$	1.75101	0.0843431	0.0421715	+	0.999110i	$0.486572\pi$
$432$	0	0
$433$	−6.36595	−0.305928	−0.152964	−	0.988232i	$-0.548882\pi$
$433$	−6.36595	−0.305928	−0.152964	+	0.988232i	$0.548882\pi$
$434$	0	0
$435$	2.05915	0.0987288
$436$	0	0
$437$	−16.8274	−0.804966
$438$	0	0
$439$	11.6213	0.554655	0.277328	−	0.960775i	$-0.410551\pi$
$439$	11.6213	0.554655	0.277328	+	0.960775i	$0.410551\pi$
$440$	0	0
$441$	12.7947	0.609270
$442$	0	0
$443$	26.8654	1.27641	0.638206	−	0.769865i	$-0.279677\pi$
$443$	26.8654	1.27641	0.638206	+	0.769865i	$0.279677\pi$
$444$	0	0
$445$	31.2303	1.48046
$446$	0	0
$447$	1.72172	0.0814345
$448$	0	0
$449$	14.8136	0.699099	0.349549	−	0.936918i	$-0.386335\pi$
$449$	14.8136	0.699099	0.349549	+	0.936918i	$0.386335\pi$
$450$	0	0
$451$	−58.3813	−2.74907
$452$	0	0
$453$	1.14988	0.0540260
$454$	0	0
$455$	12.3735	0.580081
$456$	0	0
$457$	−11.2920	−0.528216	−0.264108	−	0.964493i	$-0.585078\pi$
$457$	−11.2920	−0.528216	−0.264108	+	0.964493i	$0.585078\pi$
$458$	0	0
$459$	−7.98470	−0.372694
$460$	0	0
$461$	37.1345	1.72953	0.864763	−	0.502180i	$-0.167469\pi$
$461$	37.1345	1.72953	0.864763	+	0.502180i	$0.167469\pi$
$462$	0	0
$463$	−18.9567	−0.880991	−0.440495	−	0.897755i	$-0.645197\pi$
$463$	−18.9567	−0.880991	−0.440495	+	0.897755i	$0.645197\pi$
$464$	0	0
$465$	−1.39904	−0.0648789
$466$	0	0
$467$	3.79272	0.175506	0.0877531	−	0.996142i	$-0.472031\pi$
$467$	3.79272	0.175506	0.0877531	+	0.996142i	$0.472031\pi$
$468$	0	0
$469$	12.5927	0.581478
$470$	0	0
$471$	1.67148	0.0770179
$472$	0	0
$473$	21.3173	0.980169
$474$	0	0
$475$	−6.11900	−0.280759
$476$	0	0
$477$	8.38746	0.384036
$478$	0	0
$479$	27.5983	1.26100	0.630499	−	0.776190i	$-0.282850\pi$
$479$	27.5983	1.26100	0.630499	+	0.776190i	$0.282850\pi$
$480$	0	0
$481$	44.1158	2.01151
$482$	0	0
$483$	1.84573	0.0839838
$484$	0	0
$485$	25.5368	1.15956
$486$	0	0
$487$	−32.7583	−1.48442	−0.742210	−	0.670168i	$-0.766222\pi$
$487$	−32.7583	−1.48442	−0.742210	+	0.670168i	$0.766222\pi$
$488$	0	0
$489$	−3.08980	−0.139725
$490$	0	0
$491$	38.4034	1.73312	0.866561	−	0.499072i	$-0.166326\pi$
$491$	38.4034	1.73312	0.866561	+	0.499072i	$0.166326\pi$
$492$	0	0
$493$	33.1610	1.49350
$494$	0	0
$495$	24.6533	1.10808
$496$	0	0
$497$	−9.25534	−0.415159
$498$	0	0
$499$	−39.5752	−1.77163	−0.885816	−	0.464037i	$-0.846400\pi$
$499$	−39.5752	−1.77163	−0.885816	+	0.464037i	$0.846400\pi$
$500$	0	0
$501$	−5.55018	−0.247964
$502$	0	0
$503$	−16.2094	−0.722741	−0.361371	−	0.932422i	$-0.617691\pi$
$503$	−16.2094	−0.722741	−0.361371	+	0.932422i	$0.617691\pi$
$504$	0	0
$505$	−11.4841	−0.511037
$506$	0	0
$507$	−1.18788	−0.0527555
$508$	0	0
$509$	3.90023	0.172875	0.0864374	−	0.996257i	$-0.472452\pi$
$509$	3.90023	0.172875	0.0864374	+	0.996257i	$0.472452\pi$
$510$	0	0
$511$	10.8659	0.480681
$512$	0	0
$513$	4.18657	0.184841
$514$	0	0
$515$	24.2528	1.06870
$516$	0	0
$517$	−52.2726	−2.29895
$518$	0	0
$519$	1.58765	0.0696900
$520$	0	0
$521$	−11.0831	−0.485558	−0.242779	−	0.970082i	$-0.578059\pi$
$521$	−11.0831	−0.485558	−0.242779	+	0.970082i	$0.578059\pi$
$522$	0	0
$523$	3.54404	0.154970	0.0774850	−	0.996994i	$-0.475311\pi$
$523$	3.54404	0.154970	0.0774850	+	0.996994i	$0.475311\pi$
$524$	0	0
$525$	0.671169	0.0292922
$526$	0	0
$527$	−22.5304	−0.981441
$528$	0	0
$529$	4.04244	0.175758
$530$	0	0
$531$	0.660752	0.0286742
$532$	0	0
$533$	52.9810	2.29486
$534$	0	0
$535$	−10.1008	−0.436698
$536$	0	0
$537$	1.34621	0.0580935
$538$	0	0
$539$	20.5180	0.883772
$540$	0	0
$541$	−17.6029	−0.756808	−0.378404	−	0.925640i	$-0.623527\pi$
$541$	−17.6029	−0.756808	−0.378404	+	0.925640i	$0.623527\pi$
$542$	0	0
$543$	−1.84987	−0.0793855
$544$	0	0
$545$	29.7529	1.27448
$546$	0	0
$547$	11.9882	0.512578	0.256289	−	0.966600i	$-0.417500\pi$
$547$	11.9882	0.512578	0.256289	+	0.966600i	$0.417500\pi$
$548$	0	0
$549$	−33.1593	−1.41520
$550$	0	0
$551$	−17.3871	−0.740716
$552$	0	0
$553$	14.2586	0.606339
$554$	0	0
$555$	−3.93432	−0.167003
$556$	0	0
$557$	10.7837	0.456921	0.228461	−	0.973553i	$-0.426631\pi$
$557$	10.7837	0.456921	0.228461	+	0.973553i	$0.426631\pi$
$558$	0	0
$559$	−19.3454	−0.818223
$560$	0	0
$561$	−6.35147	−0.268159
$562$	0	0
$563$	−21.6639	−0.913025	−0.456513	−	0.889717i	$-0.650902\pi$
$563$	−21.6639	−0.913025	−0.456513	+	0.889717i	$0.650902\pi$
$564$	0	0
$565$	26.2862	1.10587
$566$	0	0
$567$	14.0069	0.588234
$568$	0	0
$569$	3.95093	0.165632	0.0828158	−	0.996565i	$-0.473609\pi$
$569$	3.95093	0.165632	0.0828158	+	0.996565i	$0.473609\pi$
$570$	0	0
$571$	−24.3652	−1.01965	−0.509826	−	0.860278i	$-0.670290\pi$
$571$	−24.3652	−1.01965	−0.509826	+	0.860278i	$0.670290\pi$
$572$	0	0
$573$	2.39050	0.0998644
$574$	0	0
$575$	9.83351	0.410086
$576$	0	0
$577$	27.6591	1.15146	0.575731	−	0.817639i	$-0.304718\pi$
$577$	27.6591	1.15146	0.575731	+	0.817639i	$0.304718\pi$
$578$	0	0
$579$	2.81827	0.117123
$580$	0	0
$581$	−1.85453	−0.0769390
$582$	0	0
$583$	13.4504	0.557060
$584$	0	0
$585$	−22.3728	−0.925002
$586$	0	0
$587$	5.03413	0.207781	0.103890	−	0.994589i	$-0.466871\pi$
$587$	5.03413	0.207781	0.103890	+	0.994589i	$0.466871\pi$
$588$	0	0
$589$	11.8132	0.486755
$590$	0	0
$591$	−1.65239	−0.0679703
$592$	0	0
$593$	24.0095	0.985954	0.492977	−	0.870042i	$-0.335909\pi$
$593$	24.0095	0.985954	0.492977	+	0.870042i	$0.335909\pi$
$594$	0	0
$595$	−17.7709	−0.728537
$596$	0	0
$597$	1.73173	0.0708751
$598$	0	0
$599$	3.61773	0.147816	0.0739082	−	0.997265i	$-0.476453\pi$
$599$	3.61773	0.147816	0.0739082	+	0.997265i	$0.476453\pi$
$600$	0	0
$601$	−40.3471	−1.64579	−0.822897	−	0.568190i	$-0.807644\pi$
$601$	−40.3471	−1.64579	−0.822897	+	0.568190i	$0.807644\pi$
$602$	0	0
$603$	−22.7691	−0.927231
$604$	0	0
$605$	20.1392	0.818773
$606$	0	0
$607$	−19.8461	−0.805528	−0.402764	−	0.915304i	$-0.631950\pi$
$607$	−19.8461	−0.805528	−0.402764	+	0.915304i	$0.631950\pi$
$608$	0	0
$609$	1.90712	0.0772805
$610$	0	0
$611$	47.4374	1.91911
$612$	0	0
$613$	18.7510	0.757345	0.378672	−	0.925531i	$-0.376381\pi$
$613$	18.7510	0.757345	0.378672	+	0.925531i	$0.376381\pi$
$614$	0	0
$615$	−4.72494	−0.190528
$616$	0	0
$617$	−17.7500	−0.714590	−0.357295	−	0.933992i	$-0.616301\pi$
$617$	−17.7500	−0.714590	−0.357295	+	0.933992i	$0.616301\pi$
$618$	0	0
$619$	−8.69003	−0.349282	−0.174641	−	0.984632i	$-0.555876\pi$
$619$	−8.69003	−0.349282	−0.174641	+	0.984632i	$0.555876\pi$
$620$	0	0
$621$	−6.72800	−0.269985
$622$	0	0
$623$	28.9245	1.15884
$624$	0	0
$625$	−11.9693	−0.478774
$626$	0	0
$627$	3.33022	0.132996
$628$	0	0
$629$	−63.3592	−2.52630
$630$	0	0
$631$	23.3145	0.928136	0.464068	−	0.885800i	$-0.346389\pi$
$631$	23.3145	0.928136	0.464068	+	0.885800i	$0.346389\pi$
$632$	0	0
$633$	−2.76091	−0.109736
$634$	0	0
$635$	−18.7070	−0.742365
$636$	0	0
$637$	−18.6200	−0.737753
$638$	0	0
$639$	16.7347	0.662016
$640$	0	0
$641$	33.3034	1.31540	0.657702	−	0.753279i	$-0.271529\pi$
$641$	33.3034	1.31540	0.657702	+	0.753279i	$0.271529\pi$
$642$	0	0
$643$	42.5841	1.67935	0.839677	−	0.543087i	$-0.182745\pi$
$643$	42.5841	1.67935	0.839677	+	0.543087i	$0.182745\pi$
$644$	0	0
$645$	1.72526	0.0679319
$646$	0	0
$647$	5.77584	0.227072	0.113536	−	0.993534i	$-0.463782\pi$
$647$	5.77584	0.227072	0.113536	+	0.993534i	$0.463782\pi$
$648$	0	0
$649$	1.05960	0.0415931
$650$	0	0
$651$	−1.29575	−0.0507843
$652$	0	0
$653$	27.6630	1.08254	0.541269	−	0.840850i	$-0.317944\pi$
$653$	27.6630	1.08254	0.541269	+	0.840850i	$0.317944\pi$
$654$	0	0
$655$	−39.3466	−1.53740
$656$	0	0
$657$	−19.6469	−0.766498
$658$	0	0
$659$	22.9222	0.892923	0.446462	−	0.894803i	$-0.352684\pi$
$659$	22.9222	0.892923	0.446462	+	0.894803i	$0.352684\pi$
$660$	0	0
$661$	34.7645	1.35218	0.676091	−	0.736818i	$-0.263673\pi$
$661$	34.7645	1.35218	0.676091	+	0.736818i	$0.263673\pi$
$662$	0	0
$663$	5.76395	0.223853
$664$	0	0
$665$	9.31771	0.361325
$666$	0	0
$667$	27.9418	1.08191
$668$	0	0
$669$	2.55825	0.0989078
$670$	0	0
$671$	−53.1753	−2.05281
$672$	0	0
$673$	1.57761	0.0608125	0.0304062	−	0.999538i	$-0.490320\pi$
$673$	1.57761	0.0608125	0.0304062	+	0.999538i	$0.490320\pi$
$674$	0	0
$675$	−2.44652	−0.0941665
$676$	0	0
$677$	17.7350	0.681612	0.340806	−	0.940134i	$-0.389300\pi$
$677$	17.7350	0.681612	0.340806	+	0.940134i	$0.389300\pi$
$678$	0	0
$679$	23.6513	0.907654
$680$	0	0
$681$	0.997399	0.0382204
$682$	0	0
$683$	−12.3120	−0.471106	−0.235553	−	0.971862i	$-0.575690\pi$
$683$	−12.3120	−0.471106	−0.235553	+	0.971862i	$0.575690\pi$
$684$	0	0
$685$	17.6853	0.675720
$686$	0	0
$687$	−1.47075	−0.0561127
$688$	0	0
$689$	−12.2063	−0.465021
$690$	0	0
$691$	16.3776	0.623035	0.311518	−	0.950240i	$-0.399163\pi$
$691$	16.3776	0.623035	0.311518	+	0.950240i	$0.399163\pi$
$692$	0	0
$693$	22.8331	0.867357
$694$	0	0
$695$	4.00405	0.151882
$696$	0	0
$697$	−76.0914	−2.88217
$698$	0	0
$699$	−1.35549	−0.0512692
$700$	0	0
$701$	17.1563	0.647984	0.323992	−	0.946060i	$-0.394975\pi$
$701$	17.1563	0.647984	0.323992	+	0.946060i	$0.394975\pi$
$702$	0	0
$703$	33.2207	1.25294
$704$	0	0
$705$	−4.23055	−0.159332
$706$	0	0
$707$	−10.6362	−0.400016
$708$	0	0
$709$	5.79811	0.217753	0.108876	−	0.994055i	$-0.465275\pi$
$709$	5.79811	0.217753	0.108876	+	0.994055i	$0.465275\pi$
$710$	0	0
$711$	−25.7813	−0.966873
$712$	0	0
$713$	−18.9844	−0.710971
$714$	0	0
$715$	−35.8778	−1.34175
$716$	0	0
$717$	0.100446	0.00375121
$718$	0	0
$719$	28.0666	1.04671	0.523354	−	0.852115i	$-0.324680\pi$
$719$	28.0666	1.04671	0.523354	+	0.852115i	$0.324680\pi$
$720$	0	0
$721$	22.4621	0.836534
$722$	0	0
$723$	3.20264	0.119107
$724$	0	0
$725$	10.1606	0.377354
$726$	0	0
$727$	−42.7540	−1.58566	−0.792830	−	0.609443i	$-0.791393\pi$
$727$	−42.7540	−1.58566	−0.792830	+	0.609443i	$0.791393\pi$
$728$	0	0
$729$	−24.4825	−0.906760
$730$	0	0
$731$	27.7839	1.02763
$732$	0	0
$733$	−10.0366	−0.370710	−0.185355	−	0.982672i	$-0.559343\pi$
$733$	−10.0366	−0.370710	−0.185355	+	0.982672i	$0.559343\pi$
$734$	0	0
$735$	1.66057	0.0612510
$736$	0	0
$737$	−36.5134	−1.34499
$738$	0	0
$739$	23.4548	0.862798	0.431399	−	0.902161i	$-0.358020\pi$
$739$	23.4548	0.862798	0.431399	+	0.902161i	$0.358020\pi$
$740$	0	0
$741$	−3.02217	−0.111022
$742$	0	0
$743$	22.6455	0.830783	0.415392	−	0.909643i	$-0.363645\pi$
$743$	22.6455	0.830783	0.415392	+	0.909643i	$0.363645\pi$
$744$	0	0
$745$	−13.9679	−0.511743
$746$	0	0
$747$	3.35322	0.122688
$748$	0	0
$749$	−9.35508	−0.341827
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−6.42699	−0.234213
$754$	0	0
$755$	−9.32867	−0.339505
$756$	0	0
$757$	−31.1705	−1.13291	−0.566455	−	0.824093i	$-0.691686\pi$
$757$	−31.1705	−1.13291	−0.566455	+	0.824093i	$0.691686\pi$
$758$	0	0
$759$	−5.35181	−0.194259
$760$	0	0
$761$	−21.3629	−0.774403	−0.387202	−	0.921995i	$-0.626558\pi$
$761$	−21.3629	−0.774403	−0.387202	+	0.921995i	$0.626558\pi$
$762$	0	0
$763$	27.5562	0.997602
$764$	0	0
$765$	32.1319	1.16173
$766$	0	0
$767$	−0.961590	−0.0347210
$768$	0	0
$769$	−28.9756	−1.04489	−0.522444	−	0.852674i	$-0.674980\pi$
$769$	−28.9756	−1.04489	−0.522444	+	0.852674i	$0.674980\pi$
$770$	0	0
$771$	4.46327	0.160741
$772$	0	0
$773$	17.2980	0.622168	0.311084	−	0.950382i	$-0.399308\pi$
$773$	17.2980	0.622168	0.311084	+	0.950382i	$0.399308\pi$
$774$	0	0
$775$	−6.90334	−0.247975
$776$	0	0
$777$	−3.64385	−0.130722
$778$	0	0
$779$	39.8965	1.42944
$780$	0	0
$781$	26.8364	0.960282
$782$	0	0
$783$	−6.95176	−0.248436
$784$	0	0
$785$	−13.5603	−0.483989
$786$	0	0
$787$	−20.8084	−0.741740	−0.370870	−	0.928685i	$-0.620940\pi$
$787$	−20.8084	−0.741740	−0.370870	+	0.928685i	$0.620940\pi$
$788$	0	0
$789$	4.42479	0.157527
$790$	0	0
$791$	24.3454	0.865624
$792$	0	0
$793$	48.2566	1.71364
$794$	0	0
$795$	1.08858	0.0386078
$796$	0	0
$797$	25.2111	0.893025	0.446512	−	0.894778i	$-0.352666\pi$
$797$	25.2111	0.893025	0.446512	+	0.894778i	$0.352666\pi$
$798$	0	0
$799$	−68.1297	−2.41026
$800$	0	0
$801$	−52.2990	−1.84789
$802$	0	0
$803$	−31.5064	−1.11184
$804$	0	0
$805$	−14.9740	−0.527763
$806$	0	0
$807$	−3.15559	−0.111082
$808$	0	0
$809$	−27.1052	−0.952968	−0.476484	−	0.879183i	$-0.658089\pi$
$809$	−27.1052	−0.952968	−0.476484	+	0.879183i	$0.658089\pi$
$810$	0	0
$811$	−15.8230	−0.555619	−0.277810	−	0.960636i	$-0.589608\pi$
$811$	−15.8230	−0.555619	−0.277810	+	0.960636i	$0.589608\pi$
$812$	0	0
$813$	3.10476	0.108889
$814$	0	0
$815$	25.0667	0.878050
$816$	0	0
$817$	−14.5678	−0.509661
$818$	0	0
$819$	−20.7210	−0.724050
$820$	0	0
$821$	1.44493	0.0504285	0.0252143	−	0.999682i	$-0.491973\pi$
$821$	1.44493	0.0504285	0.0252143	+	0.999682i	$0.491973\pi$
$822$	0	0
$823$	9.09322	0.316970	0.158485	−	0.987361i	$-0.449339\pi$
$823$	9.09322	0.316970	0.158485	+	0.987361i	$0.449339\pi$
$824$	0	0
$825$	−1.94609	−0.0677543
$826$	0	0
$827$	3.95363	0.137481	0.0687405	−	0.997635i	$-0.478102\pi$
$827$	3.95363	0.137481	0.0687405	+	0.997635i	$0.478102\pi$
$828$	0	0
$829$	23.8653	0.828877	0.414438	−	0.910077i	$-0.363978\pi$
$829$	23.8653	0.828877	0.414438	+	0.910077i	$0.363978\pi$
$830$	0	0
$831$	−3.86990	−0.134245
$832$	0	0
$833$	26.7422	0.926561
$834$	0	0
$835$	45.0272	1.55823
$836$	0	0
$837$	4.72320	0.163258
$838$	0	0
$839$	−11.0805	−0.382541	−0.191270	−	0.981537i	$-0.561261\pi$
$839$	−11.0805	−0.382541	−0.191270	+	0.981537i	$0.561261\pi$
$840$	0	0
$841$	−0.128833	−0.00444250
$842$	0	0
$843$	−4.08178	−0.140584
$844$	0	0
$845$	9.63695	0.331521
$846$	0	0
$847$	18.6522	0.640899
$848$	0	0
$849$	5.98967	0.205565
$850$	0	0
$851$	−53.3872	−1.83009
$852$	0	0
$853$	−27.8275	−0.952795	−0.476398	−	0.879230i	$-0.658058\pi$
$853$	−27.8275	−0.952795	−0.476398	+	0.879230i	$0.658058\pi$
$854$	0	0
$855$	−16.8475	−0.576173
$856$	0	0
$857$	−30.4486	−1.04010	−0.520052	−	0.854134i	$-0.674088\pi$
$857$	−30.4486	−1.04010	−0.520052	+	0.854134i	$0.674088\pi$
$858$	0	0
$859$	−19.3728	−0.660992	−0.330496	−	0.943807i	$-0.607216\pi$
$859$	−19.3728	−0.660992	−0.330496	+	0.943807i	$0.607216\pi$
$860$	0	0
$861$	−4.37609	−0.149137
$862$	0	0
$863$	22.7950	0.775951	0.387975	−	0.921670i	$-0.373175\pi$
$863$	22.7950	0.775951	0.387975	+	0.921670i	$0.373175\pi$
$864$	0	0
$865$	−12.8802	−0.437939
$866$	0	0
$867$	−4.58338	−0.155660
$868$	0	0
$869$	−41.3437	−1.40249
$870$	0	0
$871$	33.1359	1.12277
$872$	0	0
$873$	−42.7644	−1.44735
$874$	0	0
$875$	−19.8424	−0.670796
$876$	0	0
$877$	−7.50812	−0.253531	−0.126766	−	0.991933i	$-0.540460\pi$
$877$	−7.50812	−0.253531	−0.126766	+	0.991933i	$0.540460\pi$
$878$	0	0
$879$	1.69546	0.0571865
$880$	0	0
$881$	14.1458	0.476584	0.238292	−	0.971194i	$-0.423413\pi$
$881$	14.1458	0.476584	0.238292	+	0.971194i	$0.423413\pi$
$882$	0	0
$883$	−41.6328	−1.40106	−0.700528	−	0.713625i	$-0.747052\pi$
$883$	−41.6328	−1.40106	−0.700528	+	0.713625i	$0.747052\pi$
$884$	0	0
$885$	0.0857564	0.00288267
$886$	0	0
$887$	15.6978	0.527082	0.263541	−	0.964648i	$-0.415110\pi$
$887$	15.6978	0.527082	0.263541	+	0.964648i	$0.415110\pi$
$888$	0	0
$889$	−17.3258	−0.581090
$890$	0	0
$891$	−40.6138	−1.36061
$892$	0	0
$893$	35.7220	1.19539
$894$	0	0
$895$	−10.9215	−0.365066
$896$	0	0
$897$	4.85677	0.162163
$898$	0	0
$899$	−19.6158	−0.654223
$900$	0	0
$901$	17.5306	0.584031
$902$	0	0
$903$	1.59788	0.0531741
$904$	0	0
$905$	15.0075	0.498867
$906$	0	0
$907$	13.3670	0.443844	0.221922	−	0.975064i	$-0.428767\pi$
$907$	13.3670	0.443844	0.221922	+	0.975064i	$0.428767\pi$
$908$	0	0
$909$	19.2315	0.637870
$910$	0	0
$911$	−23.3969	−0.775174	−0.387587	−	0.921833i	$-0.626691\pi$
$911$	−23.3969	−0.775174	−0.387587	+	0.921833i	$0.626691\pi$
$912$	0	0
$913$	5.37733	0.177964
$914$	0	0
$915$	−4.30361	−0.142273
$916$	0	0
$917$	−36.4416	−1.20341
$918$	0	0
$919$	58.7649	1.93847	0.969237	−	0.246129i	$-0.0791588\pi$
$919$	58.7649	1.93847	0.969237	+	0.246129i	$0.0791588\pi$
$920$	0	0
$921$	1.46759	0.0483587
$922$	0	0
$923$	−24.3540	−0.801623
$924$	0	0
$925$	−19.4133	−0.638305
$926$	0	0
$927$	−40.6142	−1.33394
$928$	0	0
$929$	−55.9544	−1.83580	−0.917902	−	0.396807i	$-0.870118\pi$
$929$	−55.9544	−1.83580	−0.917902	+	0.396807i	$0.870118\pi$
$930$	0	0
$931$	−14.0215	−0.459537
$932$	0	0
$933$	7.01348	0.229611
$934$	0	0
$935$	51.5278	1.68514
$936$	0	0
$937$	3.14070	0.102602	0.0513011	−	0.998683i	$-0.483663\pi$
$937$	3.14070	0.102602	0.0513011	+	0.998683i	$0.483663\pi$
$938$	0	0
$939$	−6.96600	−0.227327
$940$	0	0
$941$	9.73923	0.317490	0.158745	−	0.987320i	$-0.449255\pi$
$941$	9.73923	0.317490	0.158745	+	0.987320i	$0.449255\pi$
$942$	0	0
$943$	−64.1155	−2.08789
$944$	0	0
$945$	3.72543	0.121188
$946$	0	0
$947$	−44.8318	−1.45684	−0.728419	−	0.685132i	$-0.759745\pi$
$947$	−44.8318	−1.45684	−0.728419	+	0.685132i	$0.759745\pi$
$948$	0	0
$949$	28.5921	0.928137
$950$	0	0
$951$	3.78234	0.122651
$952$	0	0
$953$	31.2386	1.01192	0.505958	−	0.862558i	$-0.331139\pi$
$953$	31.2386	1.01192	0.505958	+	0.862558i	$0.331139\pi$
$954$	0	0
$955$	−19.3935	−0.627558
$956$	0	0
$957$	−5.52981	−0.178753
$958$	0	0
$959$	16.3795	0.528923
$960$	0	0
$961$	−17.6726	−0.570082
$962$	0	0
$963$	16.9151	0.545081
$964$	0	0
$965$	−22.8639	−0.736015
$966$	0	0
$967$	23.3706	0.751547	0.375774	−	0.926712i	$-0.377377\pi$
$967$	23.3706	0.751547	0.375774	+	0.926712i	$0.377377\pi$
$968$	0	0
$969$	4.34045	0.139435
$970$	0	0
$971$	24.0844	0.772904	0.386452	−	0.922310i	$-0.373701\pi$
$971$	24.0844	0.772904	0.386452	+	0.922310i	$0.373701\pi$
$972$	0	0
$973$	3.70842	0.118887
$974$	0	0
$975$	1.76608	0.0565598
$976$	0	0
$977$	38.5225	1.23244	0.616222	−	0.787573i	$-0.288663\pi$
$977$	38.5225	1.23244	0.616222	+	0.787573i	$0.288663\pi$
$978$	0	0
$979$	−83.8684	−2.68045
$980$	0	0
$981$	−49.8249	−1.59079
$982$	0	0
$983$	−2.11774	−0.0675453	−0.0337726	−	0.999430i	$-0.510752\pi$
$983$	−2.11774	−0.0675453	−0.0337726	+	0.999430i	$0.510752\pi$
$984$	0	0
$985$	13.4054	0.427133
$986$	0	0
$987$	−3.91820	−0.124718
$988$	0	0
$989$	23.4110	0.744427
$990$	0	0
$991$	−26.7322	−0.849175	−0.424587	−	0.905387i	$-0.639581\pi$
$991$	−26.7322	−0.849175	−0.424587	+	0.905387i	$0.639581\pi$
$992$	0	0
$993$	−6.42999	−0.204049
$994$	0	0
$995$	−14.0491	−0.445387
$996$	0	0
$997$	45.1222	1.42903	0.714517	−	0.699618i	$-0.246646\pi$
$997$	45.1222	1.42903	0.714517	+	0.699618i	$0.246646\pi$
$998$	0	0
$999$	13.2824	0.420237

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.23	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.23	✓	50	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)

\(f(q)\)	\(=\)	\(q-0.217342 q^{3} +1.76324 q^{5} +1.63306 q^{7} -2.95276 q^{9} +O(q^{10})\)	\(q-0.217342 q^{3} +1.76324 q^{5} +1.63306 q^{7} -2.95276 q^{9} -4.73515 q^{11} +4.29715 q^{13} -0.383227 q^{15} -6.17157 q^{17} +3.23590 q^{19} -0.354933 q^{21} -5.20023 q^{23} -1.89097 q^{25} +1.29379 q^{27} -5.37319 q^{29} +3.65068 q^{31} +1.02915 q^{33} +2.87948 q^{35} +10.2663 q^{37} -0.933952 q^{39} +12.3293 q^{41} -4.50192 q^{43} -5.20644 q^{45} +11.0393 q^{47} -4.33312 q^{49} +1.34134 q^{51} -2.84055 q^{53} -8.34922 q^{55} -0.703298 q^{57} -0.223774 q^{59} +11.2299 q^{61} -4.82204 q^{63} +7.57691 q^{65} +7.71113 q^{67} +1.13023 q^{69} -5.66749 q^{71} +6.65373 q^{73} +0.410989 q^{75} -7.73278 q^{77} +8.73124 q^{79} +8.57709 q^{81} -1.13562 q^{83} -10.8820 q^{85} +1.16782 q^{87} +17.7119 q^{89} +7.01749 q^{91} -0.793447 q^{93} +5.70568 q^{95} +14.4828 q^{97} +13.9818 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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