LMFDB - Embedded newform 6004.2.a.g.1.27

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.27
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.16967 q^{3} -2.87894 q^{5} -4.05578 q^{7} +7.04683 q^{9} +O(q^{10})$	$q+3.16967 q^{3} -2.87894 q^{5} -4.05578 q^{7} +7.04683 q^{9} -0.0722189 q^{11} -0.263471 q^{13} -9.12532 q^{15} +3.42694 q^{17} +1.00000 q^{19} -12.8555 q^{21} +2.37948 q^{23} +3.28832 q^{25} +12.8271 q^{27} +1.44989 q^{29} -9.00254 q^{31} -0.228910 q^{33} +11.6764 q^{35} -7.21477 q^{37} -0.835116 q^{39} +6.92976 q^{41} -8.85392 q^{43} -20.2874 q^{45} -1.76721 q^{47} +9.44932 q^{49} +10.8623 q^{51} +3.29917 q^{53} +0.207914 q^{55} +3.16967 q^{57} +0.351290 q^{59} -10.3711 q^{61} -28.5804 q^{63} +0.758517 q^{65} -7.05499 q^{67} +7.54217 q^{69} +1.30367 q^{71} -13.6132 q^{73} +10.4229 q^{75} +0.292904 q^{77} -1.00000 q^{79} +19.5174 q^{81} -0.269951 q^{83} -9.86598 q^{85} +4.59567 q^{87} -7.38755 q^{89} +1.06858 q^{91} -28.5351 q^{93} -2.87894 q^{95} -2.87132 q^{97} -0.508915 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * 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
$3$	3.16967	1.83001	0.915006	−	0.403440i	$-0.132186\pi$
$3$	3.16967	1.83001	0.915006	+	0.403440i	$0.132186\pi$
$4$	0	0
$5$	−2.87894	−1.28750	−0.643752	−	0.765235i	$-0.722623\pi$
$5$	−2.87894	−1.28750	−0.643752	+	0.765235i	$0.722623\pi$
$6$	0	0
$7$	−4.05578	−1.53294	−0.766470	−	0.642280i	$-0.777988\pi$
$7$	−4.05578	−1.53294	−0.766470	+	0.642280i	$0.777988\pi$
$8$	0	0
$9$	7.04683	2.34894
$10$	0	0
$11$	−0.0722189	−0.0217748	−0.0108874	−	0.999941i	$-0.503466\pi$
$11$	−0.0722189	−0.0217748	−0.0108874	+	0.999941i	$0.503466\pi$
$12$	0	0
$13$	−0.263471	−0.0730736	−0.0365368	−	0.999332i	$-0.511633\pi$
$13$	−0.263471	−0.0730736	−0.0365368	+	0.999332i	$0.511633\pi$
$14$	0	0
$15$	−9.12532	−2.35615
$16$	0	0
$17$	3.42694	0.831156	0.415578	−	0.909558i	$-0.363579\pi$
$17$	3.42694	0.831156	0.415578	+	0.909558i	$0.363579\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−12.8555	−2.80530
$22$	0	0
$23$	2.37948	0.496156	0.248078	−	0.968740i	$-0.420201\pi$
$23$	2.37948	0.496156	0.248078	+	0.968740i	$0.420201\pi$
$24$	0	0
$25$	3.28832	0.657664
$26$	0	0
$27$	12.8271	2.46858
$28$	0	0
$29$	1.44989	0.269238	0.134619	−	0.990897i	$-0.457019\pi$
$29$	1.44989	0.269238	0.134619	+	0.990897i	$0.457019\pi$
$30$	0	0
$31$	−9.00254	−1.61690	−0.808452	−	0.588563i	$-0.799694\pi$
$31$	−9.00254	−1.61690	−0.808452	+	0.588563i	$0.799694\pi$
$32$	0	0
$33$	−0.228910	−0.0398482
$34$	0	0
$35$	11.6764	1.97366
$36$	0	0
$37$	−7.21477	−1.18610	−0.593050	−	0.805165i	$-0.702077\pi$
$37$	−7.21477	−1.18610	−0.593050	+	0.805165i	$0.702077\pi$
$38$	0	0
$39$	−0.835116	−0.133726
$40$	0	0
$41$	6.92976	1.08225	0.541123	−	0.840943i	$-0.317999\pi$
$41$	6.92976	1.08225	0.541123	+	0.840943i	$0.317999\pi$
$42$	0	0
$43$	−8.85392	−1.35021	−0.675105	−	0.737722i	$-0.735902\pi$
$43$	−8.85392	−1.35021	−0.675105	+	0.737722i	$0.735902\pi$
$44$	0	0
$45$	−20.2874	−3.02427
$46$	0	0
$47$	−1.76721	−0.257773	−0.128887	−	0.991659i	$-0.541140\pi$
$47$	−1.76721	−0.257773	−0.128887	+	0.991659i	$0.541140\pi$
$48$	0	0
$49$	9.44932	1.34990
$50$	0	0
$51$	10.8623	1.52103
$52$	0	0
$53$	3.29917	0.453175	0.226588	−	0.973991i	$-0.427243\pi$
$53$	3.29917	0.453175	0.226588	+	0.973991i	$0.427243\pi$
$54$	0	0
$55$	0.207914	0.0280352
$56$	0	0
$57$	3.16967	0.419834
$58$	0	0
$59$	0.351290	0.0457340	0.0228670	−	0.999739i	$-0.492721\pi$
$59$	0.351290	0.0457340	0.0228670	+	0.999739i	$0.492721\pi$
$60$	0	0
$61$	−10.3711	−1.32788	−0.663941	−	0.747785i	$-0.731118\pi$
$61$	−10.3711	−1.32788	−0.663941	+	0.747785i	$0.731118\pi$
$62$	0	0
$63$	−28.5804	−3.60079
$64$	0	0
$65$	0.758517	0.0940825
$66$	0	0
$67$	−7.05499	−0.861905	−0.430952	−	0.902375i	$-0.641822\pi$
$67$	−7.05499	−0.861905	−0.430952	+	0.902375i	$0.641822\pi$
$68$	0	0
$69$	7.54217	0.907971
$70$	0	0
$71$	1.30367	0.154717	0.0773585	−	0.997003i	$-0.475351\pi$
$71$	1.30367	0.154717	0.0773585	+	0.997003i	$0.475351\pi$
$72$	0	0
$73$	−13.6132	−1.59330	−0.796650	−	0.604441i	$-0.793397\pi$
$73$	−13.6132	−1.59330	−0.796650	+	0.604441i	$0.793397\pi$
$74$	0	0
$75$	10.4229	1.20353
$76$	0	0
$77$	0.292904	0.0333795
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	19.5174	2.16860
$82$	0	0
$83$	−0.269951	−0.0296310	−0.0148155	−	0.999890i	$-0.504716\pi$
$83$	−0.269951	−0.0296310	−0.0148155	+	0.999890i	$0.504716\pi$
$84$	0	0
$85$	−9.86598	−1.07012
$86$	0	0
$87$	4.59567	0.492708
$88$	0	0
$89$	−7.38755	−0.783078	−0.391539	−	0.920161i	$-0.628057\pi$
$89$	−7.38755	−0.783078	−0.391539	+	0.920161i	$0.628057\pi$
$90$	0	0
$91$	1.06858	0.112017
$92$	0	0
$93$	−28.5351	−2.95895
$94$	0	0
$95$	−2.87894	−0.295373
$96$	0	0
$97$	−2.87132	−0.291538	−0.145769	−	0.989319i	$-0.546566\pi$
$97$	−2.87132	−0.291538	−0.145769	+	0.989319i	$0.546566\pi$
$98$	0	0
$99$	−0.508915	−0.0511479
$100$	0	0
$101$	−15.4464	−1.53697	−0.768486	−	0.639867i	$-0.778990\pi$
$101$	−15.4464	−1.53697	−0.768486	+	0.639867i	$0.778990\pi$
$102$	0	0
$103$	−6.91399	−0.681256	−0.340628	−	0.940198i	$-0.610640\pi$
$103$	−6.91399	−0.681256	−0.340628	+	0.940198i	$0.610640\pi$
$104$	0	0
$105$	37.0102	3.61183
$106$	0	0
$107$	6.66449	0.644281	0.322140	−	0.946692i	$-0.395598\pi$
$107$	6.66449	0.644281	0.322140	+	0.946692i	$0.395598\pi$
$108$	0	0
$109$	−7.72585	−0.740002	−0.370001	−	0.929031i	$-0.620643\pi$
$109$	−7.72585	−0.740002	−0.370001	+	0.929031i	$0.620643\pi$
$110$	0	0
$111$	−22.8685	−2.17058
$112$	0	0
$113$	−12.3685	−1.16353	−0.581763	−	0.813358i	$-0.697637\pi$
$113$	−12.3685	−1.16353	−0.581763	+	0.813358i	$0.697637\pi$
$114$	0	0
$115$	−6.85039	−0.638802
$116$	0	0
$117$	−1.85663	−0.171646
$118$	0	0
$119$	−13.8989	−1.27411
$120$	0	0
$121$	−10.9948	−0.999526
$122$	0	0
$123$	21.9651	1.98052
$124$	0	0
$125$	4.92783	0.440759
$126$	0	0
$127$	17.8391	1.58297	0.791484	−	0.611190i	$-0.209309\pi$
$127$	17.8391	1.58297	0.791484	+	0.611190i	$0.209309\pi$
$128$	0	0
$129$	−28.0640	−2.47090
$130$	0	0
$131$	−9.17054	−0.801234	−0.400617	−	0.916246i	$-0.631204\pi$
$131$	−9.17054	−0.801234	−0.400617	+	0.916246i	$0.631204\pi$
$132$	0	0
$133$	−4.05578	−0.351680
$134$	0	0
$135$	−36.9286	−3.17831
$136$	0	0
$137$	−7.33244	−0.626453	−0.313226	−	0.949679i	$-0.601410\pi$
$137$	−7.33244	−0.626453	−0.313226	+	0.949679i	$0.601410\pi$
$138$	0	0
$139$	15.6965	1.33136	0.665678	−	0.746239i	$-0.268142\pi$
$139$	15.6965	1.33136	0.665678	+	0.746239i	$0.268142\pi$
$140$	0	0
$141$	−5.60146	−0.471728
$142$	0	0
$143$	0.0190276	0.00159116
$144$	0	0
$145$	−4.17415	−0.346644
$146$	0	0
$147$	29.9513	2.47034
$148$	0	0
$149$	4.52585	0.370772	0.185386	−	0.982666i	$-0.440646\pi$
$149$	4.52585	0.370772	0.185386	+	0.982666i	$0.440646\pi$
$150$	0	0
$151$	−11.7300	−0.954572	−0.477286	−	0.878748i	$-0.658379\pi$
$151$	−11.7300	−0.954572	−0.477286	+	0.878748i	$0.658379\pi$
$152$	0	0
$153$	24.1491	1.95234
$154$	0	0
$155$	25.9178	2.08177
$156$	0	0
$157$	17.4585	1.39334	0.696669	−	0.717393i	$-0.254665\pi$
$157$	17.4585	1.39334	0.696669	+	0.717393i	$0.254665\pi$
$158$	0	0
$159$	10.4573	0.829316
$160$	0	0
$161$	−9.65064	−0.760577
$162$	0	0
$163$	15.3777	1.20447	0.602235	−	0.798318i	$-0.294277\pi$
$163$	15.3777	1.20447	0.602235	+	0.798318i	$0.294277\pi$
$164$	0	0
$165$	0.659020	0.0513047
$166$	0	0
$167$	−2.34625	−0.181558	−0.0907790	−	0.995871i	$-0.528936\pi$
$167$	−2.34625	−0.181558	−0.0907790	+	0.995871i	$0.528936\pi$
$168$	0	0
$169$	−12.9306	−0.994660
$170$	0	0
$171$	7.04683	0.538885
$172$	0	0
$173$	−10.1774	−0.773771	−0.386885	−	0.922128i	$-0.626449\pi$
$173$	−10.1774	−0.773771	−0.386885	+	0.922128i	$0.626449\pi$
$174$	0	0
$175$	−13.3367	−1.00816
$176$	0	0
$177$	1.11347	0.0836938
$178$	0	0
$179$	−16.2076	−1.21141	−0.605707	−	0.795688i	$-0.707110\pi$
$179$	−16.2076	−1.21141	−0.605707	+	0.795688i	$0.707110\pi$
$180$	0	0
$181$	−0.278393	−0.0206928	−0.0103464	−	0.999946i	$-0.503293\pi$
$181$	−0.278393	−0.0206928	−0.0103464	+	0.999946i	$0.503293\pi$
$182$	0	0
$183$	−32.8730	−2.43004
$184$	0	0
$185$	20.7709	1.52711
$186$	0	0
$187$	−0.247490	−0.0180983
$188$	0	0
$189$	−52.0240	−3.78419
$190$	0	0
$191$	14.3934	1.04147	0.520735	−	0.853718i	$-0.325658\pi$
$191$	14.3934	1.04147	0.520735	+	0.853718i	$0.325658\pi$
$192$	0	0
$193$	13.9120	1.00141	0.500705	−	0.865618i	$-0.333074\pi$
$193$	13.9120	1.00141	0.500705	+	0.865618i	$0.333074\pi$
$194$	0	0
$195$	2.40425	0.172172
$196$	0	0
$197$	−10.9664	−0.781327	−0.390663	−	0.920534i	$-0.627754\pi$
$197$	−10.9664	−0.781327	−0.390663	+	0.920534i	$0.627754\pi$
$198$	0	0
$199$	1.31103	0.0929367	0.0464684	−	0.998920i	$-0.485203\pi$
$199$	1.31103	0.0929367	0.0464684	+	0.998920i	$0.485203\pi$
$200$	0	0
$201$	−22.3620	−1.57730
$202$	0	0
$203$	−5.88042	−0.412725
$204$	0	0
$205$	−19.9504	−1.39340
$206$	0	0
$207$	16.7678	1.16544
$208$	0	0
$209$	−0.0722189	−0.00499549
$210$	0	0
$211$	−3.17734	−0.218737	−0.109368	−	0.994001i	$-0.534883\pi$
$211$	−3.17734	−0.218737	−0.109368	+	0.994001i	$0.534883\pi$
$212$	0	0
$213$	4.13221	0.283134
$214$	0	0
$215$	25.4899	1.73840
$216$	0	0
$217$	36.5123	2.47861
$218$	0	0
$219$	−43.1493	−2.91576
$220$	0	0
$221$	−0.902899	−0.0607355
$222$	0	0
$223$	6.51962	0.436586	0.218293	−	0.975883i	$-0.429951\pi$
$223$	6.51962	0.436586	0.218293	+	0.975883i	$0.429951\pi$
$224$	0	0
$225$	23.1722	1.54482
$226$	0	0
$227$	−20.4346	−1.35629	−0.678146	−	0.734927i	$-0.737217\pi$
$227$	−20.4346	−1.35629	−0.678146	+	0.734927i	$0.737217\pi$
$228$	0	0
$229$	19.4075	1.28248	0.641241	−	0.767339i	$-0.278420\pi$
$229$	19.4075	1.28248	0.641241	+	0.767339i	$0.278420\pi$
$230$	0	0
$231$	0.928410	0.0610849
$232$	0	0
$233$	−20.8895	−1.36852	−0.684260	−	0.729239i	$-0.739874\pi$
$233$	−20.8895	−1.36852	−0.684260	+	0.729239i	$0.739874\pi$
$234$	0	0
$235$	5.08768	0.331884
$236$	0	0
$237$	−3.16967	−0.205892
$238$	0	0
$239$	17.2485	1.11571	0.557856	−	0.829938i	$-0.311624\pi$
$239$	17.2485	1.11571	0.557856	+	0.829938i	$0.311624\pi$
$240$	0	0
$241$	−13.1053	−0.844187	−0.422094	−	0.906552i	$-0.638705\pi$
$241$	−13.1053	−0.844187	−0.422094	+	0.906552i	$0.638705\pi$
$242$	0	0
$243$	23.3823	1.49997
$244$	0	0
$245$	−27.2041	−1.73800
$246$	0	0
$247$	−0.263471	−0.0167642
$248$	0	0
$249$	−0.855656	−0.0542250
$250$	0	0
$251$	16.8078	1.06090	0.530450	−	0.847716i	$-0.322023\pi$
$251$	16.8078	1.06090	0.530450	+	0.847716i	$0.322023\pi$
$252$	0	0
$253$	−0.171843	−0.0108037
$254$	0	0
$255$	−31.2719	−1.95832
$256$	0	0
$257$	3.76837	0.235064	0.117532	−	0.993069i	$-0.462502\pi$
$257$	3.76837	0.235064	0.117532	+	0.993069i	$0.462502\pi$
$258$	0	0
$259$	29.2615	1.81822
$260$	0	0
$261$	10.2171	0.632424
$262$	0	0
$263$	−10.5467	−0.650338	−0.325169	−	0.945656i	$-0.605421\pi$
$263$	−10.5467	−0.650338	−0.325169	+	0.945656i	$0.605421\pi$
$264$	0	0
$265$	−9.49812	−0.583465
$266$	0	0
$267$	−23.4161	−1.43304
$268$	0	0
$269$	−8.61940	−0.525534	−0.262767	−	0.964859i	$-0.584635\pi$
$269$	−8.61940	−0.525534	−0.262767	+	0.964859i	$0.584635\pi$
$270$	0	0
$271$	15.2640	0.927223	0.463611	−	0.886039i	$-0.346553\pi$
$271$	15.2640	0.927223	0.463611	+	0.886039i	$0.346553\pi$
$272$	0	0
$273$	3.38704	0.204993
$274$	0	0
$275$	−0.237479	−0.0143205
$276$	0	0
$277$	13.0188	0.782224	0.391112	−	0.920343i	$-0.372090\pi$
$277$	13.0188	0.782224	0.391112	+	0.920343i	$0.372090\pi$
$278$	0	0
$279$	−63.4394	−3.79802
$280$	0	0
$281$	−26.8262	−1.60032	−0.800158	−	0.599789i	$-0.795251\pi$
$281$	−26.8262	−1.60032	−0.800158	+	0.599789i	$0.795251\pi$
$282$	0	0
$283$	−23.8068	−1.41517	−0.707584	−	0.706629i	$-0.750215\pi$
$283$	−23.8068	−1.41517	−0.707584	+	0.706629i	$0.750215\pi$
$284$	0	0
$285$	−9.12532	−0.540537
$286$	0	0
$287$	−28.1056	−1.65902
$288$	0	0
$289$	−5.25606	−0.309180
$290$	0	0
$291$	−9.10113	−0.533518
$292$	0	0
$293$	9.32254	0.544629	0.272314	−	0.962208i	$-0.412211\pi$
$293$	9.32254	0.544629	0.272314	+	0.962208i	$0.412211\pi$
$294$	0	0
$295$	−1.01134	−0.0588827
$296$	0	0
$297$	−0.926362	−0.0537530
$298$	0	0
$299$	−0.626923	−0.0362559
$300$	0	0
$301$	35.9095	2.06979
$302$	0	0
$303$	−48.9600	−2.81268
$304$	0	0
$305$	29.8578	1.70965
$306$	0	0
$307$	−12.7288	−0.726469	−0.363234	−	0.931698i	$-0.618328\pi$
$307$	−12.7288	−0.726469	−0.363234	+	0.931698i	$0.618328\pi$
$308$	0	0
$309$	−21.9151	−1.24671
$310$	0	0
$311$	−15.3508	−0.870465	−0.435233	−	0.900318i	$-0.643334\pi$
$311$	−15.3508	−0.870465	−0.435233	+	0.900318i	$0.643334\pi$
$312$	0	0
$313$	11.8578	0.670240	0.335120	−	0.942175i	$-0.391223\pi$
$313$	11.8578	0.670240	0.335120	+	0.942175i	$0.391223\pi$
$314$	0	0
$315$	82.2813	4.63603
$316$	0	0
$317$	−30.2373	−1.69830	−0.849148	−	0.528155i	$-0.822884\pi$
$317$	−30.2373	−1.69830	−0.849148	+	0.528155i	$0.822884\pi$
$318$	0	0
$319$	−0.104709	−0.00586260
$320$	0	0
$321$	21.1243	1.17904
$322$	0	0
$323$	3.42694	0.190680
$324$	0	0
$325$	−0.866376	−0.0480579
$326$	0	0
$327$	−24.4884	−1.35421
$328$	0	0
$329$	7.16739	0.395151
$330$	0	0
$331$	14.5943	0.802176	0.401088	−	0.916040i	$-0.368632\pi$
$331$	14.5943	0.802176	0.401088	+	0.916040i	$0.368632\pi$
$332$	0	0
$333$	−50.8413	−2.78609
$334$	0	0
$335$	20.3109	1.10970
$336$	0	0
$337$	15.1756	0.826667	0.413334	−	0.910580i	$-0.364364\pi$
$337$	15.1756	0.826667	0.413334	+	0.910580i	$0.364364\pi$
$338$	0	0
$339$	−39.2040	−2.12927
$340$	0	0
$341$	0.650153	0.0352078
$342$	0	0
$343$	−9.93390	−0.536380
$344$	0	0
$345$	−21.7135	−1.16902
$346$	0	0
$347$	7.39441	0.396953	0.198476	−	0.980106i	$-0.436401\pi$
$347$	7.39441	0.396953	0.198476	+	0.980106i	$0.436401\pi$
$348$	0	0
$349$	−7.83343	−0.419314	−0.209657	−	0.977775i	$-0.567235\pi$
$349$	−7.83343	−0.419314	−0.209657	+	0.977775i	$0.567235\pi$
$350$	0	0
$351$	−3.37957	−0.180388
$352$	0	0
$353$	2.67972	0.142627	0.0713135	−	0.997454i	$-0.477281\pi$
$353$	2.67972	0.142627	0.0713135	+	0.997454i	$0.477281\pi$
$354$	0	0
$355$	−3.75319	−0.199199
$356$	0	0
$357$	−44.0550	−2.33164
$358$	0	0
$359$	−37.1345	−1.95988	−0.979942	−	0.199281i	$-0.936139\pi$
$359$	−37.1345	−1.95988	−0.979942	+	0.199281i	$0.936139\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−34.8499	−1.82914
$364$	0	0
$365$	39.1915	2.05138
$366$	0	0
$367$	−4.09056	−0.213525	−0.106763	−	0.994285i	$-0.534049\pi$
$367$	−4.09056	−0.213525	−0.106763	+	0.994285i	$0.534049\pi$
$368$	0	0
$369$	48.8329	2.54214
$370$	0	0
$371$	−13.3807	−0.694690
$372$	0	0
$373$	32.5937	1.68764	0.843818	−	0.536629i	$-0.180302\pi$
$373$	32.5937	1.68764	0.843818	+	0.536629i	$0.180302\pi$
$374$	0	0
$375$	15.6196	0.806594
$376$	0	0
$377$	−0.382003	−0.0196742
$378$	0	0
$379$	−26.2772	−1.34977	−0.674885	−	0.737923i	$-0.735807\pi$
$379$	−26.2772	−1.34977	−0.674885	+	0.737923i	$0.735807\pi$
$380$	0	0
$381$	56.5443	2.89685
$382$	0	0
$383$	16.3002	0.832900	0.416450	−	0.909159i	$-0.363274\pi$
$383$	16.3002	0.832900	0.416450	+	0.909159i	$0.363274\pi$
$384$	0	0
$385$	−0.843254	−0.0429762
$386$	0	0
$387$	−62.3921	−3.17157
$388$	0	0
$389$	5.78641	0.293383	0.146691	−	0.989182i	$-0.453138\pi$
$389$	5.78641	0.293383	0.146691	+	0.989182i	$0.453138\pi$
$390$	0	0
$391$	8.15434	0.412383
$392$	0	0
$393$	−29.0676	−1.46627
$394$	0	0
$395$	2.87894	0.144855
$396$	0	0
$397$	3.80345	0.190890	0.0954449	−	0.995435i	$-0.469573\pi$
$397$	3.80345	0.190890	0.0954449	+	0.995435i	$0.469573\pi$
$398$	0	0
$399$	−12.8555	−0.643579
$400$	0	0
$401$	−9.65652	−0.482224	−0.241112	−	0.970497i	$-0.577512\pi$
$401$	−9.65652	−0.482224	−0.241112	+	0.970497i	$0.577512\pi$
$402$	0	0
$403$	2.37190	0.118153
$404$	0	0
$405$	−56.1894	−2.79207
$406$	0	0
$407$	0.521043	0.0258271
$408$	0	0
$409$	15.2449	0.753812	0.376906	−	0.926252i	$-0.376988\pi$
$409$	15.2449	0.753812	0.376906	+	0.926252i	$0.376988\pi$
$410$	0	0
$411$	−23.2414	−1.14642
$412$	0	0
$413$	−1.42475	−0.0701074
$414$	0	0
$415$	0.777174	0.0381499
$416$	0	0
$417$	49.7527	2.43640
$418$	0	0
$419$	5.81011	0.283842	0.141921	−	0.989878i	$-0.454672\pi$
$419$	5.81011	0.283842	0.141921	+	0.989878i	$0.454672\pi$
$420$	0	0
$421$	−3.37672	−0.164571	−0.0822857	−	0.996609i	$-0.526222\pi$
$421$	−3.37672	−0.164571	−0.0822857	+	0.996609i	$0.526222\pi$
$422$	0	0
$423$	−12.4532	−0.605495
$424$	0	0
$425$	11.2689	0.546621
$426$	0	0
$427$	42.0628	2.03556
$428$	0	0
$429$	0.0603112	0.00291185
$430$	0	0
$431$	21.9934	1.05939	0.529693	−	0.848190i	$-0.322307\pi$
$431$	21.9934	1.05939	0.529693	+	0.848190i	$0.322307\pi$
$432$	0	0
$433$	0.306165	0.0147134	0.00735668	−	0.999973i	$-0.497658\pi$
$433$	0.306165	0.0147134	0.00735668	+	0.999973i	$0.497658\pi$
$434$	0	0
$435$	−13.2307	−0.634363
$436$	0	0
$437$	2.37948	0.113826
$438$	0	0
$439$	1.57346	0.0750974	0.0375487	−	0.999295i	$-0.488045\pi$
$439$	1.57346	0.0750974	0.0375487	+	0.999295i	$0.488045\pi$
$440$	0	0
$441$	66.5878	3.17085
$442$	0	0
$443$	27.1032	1.28771	0.643857	−	0.765146i	$-0.277333\pi$
$443$	27.1032	1.28771	0.643857	+	0.765146i	$0.277333\pi$
$444$	0	0
$445$	21.2683	1.00822
$446$	0	0
$447$	14.3455	0.678518
$448$	0	0
$449$	−6.20036	−0.292613	−0.146307	−	0.989239i	$-0.546739\pi$
$449$	−6.20036	−0.292613	−0.146307	+	0.989239i	$0.546739\pi$
$450$	0	0
$451$	−0.500460	−0.0235657
$452$	0	0
$453$	−37.1802	−1.74688
$454$	0	0
$455$	−3.07638	−0.144223
$456$	0	0
$457$	34.7792	1.62690	0.813450	−	0.581635i	$-0.197587\pi$
$457$	34.7792	1.62690	0.813450	+	0.581635i	$0.197587\pi$
$458$	0	0
$459$	43.9579	2.05178
$460$	0	0
$461$	17.5163	0.815814	0.407907	−	0.913023i	$-0.366259\pi$
$461$	17.5163	0.815814	0.407907	+	0.913023i	$0.366259\pi$
$462$	0	0
$463$	−20.0968	−0.933977	−0.466988	−	0.884263i	$-0.654661\pi$
$463$	−20.0968	−0.933977	−0.466988	+	0.884263i	$0.654661\pi$
$464$	0	0
$465$	82.1510	3.80966
$466$	0	0
$467$	24.9978	1.15676	0.578380	−	0.815767i	$-0.303685\pi$
$467$	24.9978	1.15676	0.578380	+	0.815767i	$0.303685\pi$
$468$	0	0
$469$	28.6135	1.32125
$470$	0	0
$471$	55.3376	2.54982
$472$	0	0
$473$	0.639420	0.0294006
$474$	0	0
$475$	3.28832	0.150878
$476$	0	0
$477$	23.2487	1.06448
$478$	0	0
$479$	18.0570	0.825045	0.412522	−	0.910948i	$-0.364648\pi$
$479$	18.0570	0.825045	0.412522	+	0.910948i	$0.364648\pi$
$480$	0	0
$481$	1.90088	0.0866726
$482$	0	0
$483$	−30.5894	−1.39186
$484$	0	0
$485$	8.26636	0.375356
$486$	0	0
$487$	−11.4547	−0.519060	−0.259530	−	0.965735i	$-0.583568\pi$
$487$	−11.4547	−0.519060	−0.259530	+	0.965735i	$0.583568\pi$
$488$	0	0
$489$	48.7422	2.20420
$490$	0	0
$491$	37.5161	1.69308	0.846538	−	0.532329i	$-0.178683\pi$
$491$	37.5161	1.69308	0.846538	+	0.532329i	$0.178683\pi$
$492$	0	0
$493$	4.96869	0.223778
$494$	0	0
$495$	1.46514	0.0658530
$496$	0	0
$497$	−5.28739	−0.237172
$498$	0	0
$499$	−31.5209	−1.41107	−0.705535	−	0.708675i	$-0.749293\pi$
$499$	−31.5209	−1.41107	−0.705535	+	0.708675i	$0.749293\pi$
$500$	0	0
$501$	−7.43684	−0.332253
$502$	0	0
$503$	−17.2991	−0.771331	−0.385665	−	0.922639i	$-0.626028\pi$
$503$	−17.2991	−0.771331	−0.385665	+	0.922639i	$0.626028\pi$
$504$	0	0
$505$	44.4693	1.97886
$506$	0	0
$507$	−40.9857	−1.82024
$508$	0	0
$509$	28.4219	1.25978	0.629889	−	0.776685i	$-0.283100\pi$
$509$	28.4219	1.25978	0.629889	+	0.776685i	$0.283100\pi$
$510$	0	0
$511$	55.2120	2.44243
$512$	0	0
$513$	12.8271	0.566332
$514$	0	0
$515$	19.9050	0.877119
$516$	0	0
$517$	0.127626	0.00561297
$518$	0	0
$519$	−32.2589	−1.41601
$520$	0	0
$521$	−18.2741	−0.800605	−0.400302	−	0.916383i	$-0.631095\pi$
$521$	−18.2741	−0.800605	−0.400302	+	0.916383i	$0.631095\pi$
$522$	0	0
$523$	−30.7736	−1.34564	−0.672818	−	0.739808i	$-0.734916\pi$
$523$	−30.7736	−1.34564	−0.672818	+	0.739808i	$0.734916\pi$
$524$	0	0
$525$	−42.2730	−1.84494
$526$	0	0
$527$	−30.8512	−1.34390
$528$	0	0
$529$	−17.3381	−0.753829
$530$	0	0
$531$	2.47548	0.107427
$532$	0	0
$533$	−1.82579	−0.0790837
$534$	0	0
$535$	−19.1867	−0.829514
$536$	0	0
$537$	−51.3729	−2.21690
$538$	0	0
$539$	−0.682420	−0.0293939
$540$	0	0
$541$	25.4739	1.09521	0.547603	−	0.836738i	$-0.315540\pi$
$541$	25.4739	1.09521	0.547603	+	0.836738i	$0.315540\pi$
$542$	0	0
$543$	−0.882415	−0.0378681
$544$	0	0
$545$	22.2423	0.952755
$546$	0	0
$547$	8.57681	0.366718	0.183359	−	0.983046i	$-0.441303\pi$
$547$	8.57681	0.366718	0.183359	+	0.983046i	$0.441303\pi$
$548$	0	0
$549$	−73.0834	−3.11912
$550$	0	0
$551$	1.44989	0.0617673
$552$	0	0
$553$	4.05578	0.172469
$554$	0	0
$555$	65.8371	2.79463
$556$	0	0
$557$	−10.3323	−0.437793	−0.218896	−	0.975748i	$-0.570246\pi$
$557$	−10.3323	−0.437793	−0.218896	+	0.975748i	$0.570246\pi$
$558$	0	0
$559$	2.33275	0.0986647
$560$	0	0
$561$	−0.784463	−0.0331201
$562$	0	0
$563$	−9.82986	−0.414279	−0.207140	−	0.978311i	$-0.566415\pi$
$563$	−9.82986	−0.414279	−0.207140	+	0.978311i	$0.566415\pi$
$564$	0	0
$565$	35.6081	1.49804
$566$	0	0
$567$	−79.1581	−3.32433
$568$	0	0
$569$	26.1068	1.09445	0.547227	−	0.836984i	$-0.315683\pi$
$569$	26.1068	1.09445	0.547227	+	0.836984i	$0.315683\pi$
$570$	0	0
$571$	−10.0611	−0.421043	−0.210521	−	0.977589i	$-0.567516\pi$
$571$	−10.0611	−0.421043	−0.210521	+	0.977589i	$0.567516\pi$
$572$	0	0
$573$	45.6224	1.90590
$574$	0	0
$575$	7.82449	0.326304
$576$	0	0
$577$	33.4678	1.39328	0.696641	−	0.717420i	$-0.254677\pi$
$577$	33.4678	1.39328	0.696641	+	0.717420i	$0.254677\pi$
$578$	0	0
$579$	44.0966	1.83259
$580$	0	0
$581$	1.09486	0.0454225
$582$	0	0
$583$	−0.238262	−0.00986781
$584$	0	0
$585$	5.34514	0.220994
$586$	0	0
$587$	39.7684	1.64142	0.820709	−	0.571347i	$-0.193579\pi$
$587$	39.7684	1.64142	0.820709	+	0.571347i	$0.193579\pi$
$588$	0	0
$589$	−9.00254	−0.370943
$590$	0	0
$591$	−34.7600	−1.42984
$592$	0	0
$593$	−36.2119	−1.48704	−0.743522	−	0.668711i	$-0.766846\pi$
$593$	−36.2119	−1.48704	−0.743522	+	0.668711i	$0.766846\pi$
$594$	0	0
$595$	40.0142	1.64042
$596$	0	0
$597$	4.15555	0.170075
$598$	0	0
$599$	24.7400	1.01085	0.505424	−	0.862871i	$-0.331336\pi$
$599$	24.7400	1.01085	0.505424	+	0.862871i	$0.331336\pi$
$600$	0	0
$601$	−3.80667	−0.155277	−0.0776387	−	0.996982i	$-0.524738\pi$
$601$	−3.80667	−0.155277	−0.0776387	+	0.996982i	$0.524738\pi$
$602$	0	0
$603$	−49.7154	−2.02457
$604$	0	0
$605$	31.6534	1.28689
$606$	0	0
$607$	47.5058	1.92820	0.964100	−	0.265539i	$-0.0855498\pi$
$607$	47.5058	1.92820	0.964100	+	0.265539i	$0.0855498\pi$
$608$	0	0
$609$	−18.6390	−0.755291
$610$	0	0
$611$	0.465606	0.0188364
$612$	0	0
$613$	−6.08754	−0.245873	−0.122937	−	0.992415i	$-0.539231\pi$
$613$	−6.08754	−0.245873	−0.122937	+	0.992415i	$0.539231\pi$
$614$	0	0
$615$	−63.2363	−2.54993
$616$	0	0
$617$	−36.1983	−1.45729	−0.728644	−	0.684892i	$-0.759849\pi$
$617$	−36.1983	−1.45729	−0.728644	+	0.684892i	$0.759849\pi$
$618$	0	0
$619$	22.6631	0.910908	0.455454	−	0.890259i	$-0.349477\pi$
$619$	22.6631	0.910908	0.455454	+	0.890259i	$0.349477\pi$
$620$	0	0
$621$	30.5219	1.22480
$622$	0	0
$623$	29.9622	1.20041
$624$	0	0
$625$	−30.6286	−1.22514
$626$	0	0
$627$	−0.228910	−0.00914180
$628$	0	0
$629$	−24.7246	−0.985835
$630$	0	0
$631$	7.22078	0.287455	0.143727	−	0.989617i	$-0.454091\pi$
$631$	7.22078	0.287455	0.143727	+	0.989617i	$0.454091\pi$
$632$	0	0
$633$	−10.0711	−0.400291
$634$	0	0
$635$	−51.3579	−2.03808
$636$	0	0
$637$	−2.48962	−0.0986423
$638$	0	0
$639$	9.18674	0.363422
$640$	0	0
$641$	0.183407	0.00724413	0.00362206	−	0.999993i	$-0.498847\pi$
$641$	0.183407	0.00724413	0.00362206	+	0.999993i	$0.498847\pi$
$642$	0	0
$643$	−2.92339	−0.115287	−0.0576437	−	0.998337i	$-0.518359\pi$
$643$	−2.92339	−0.115287	−0.0576437	+	0.998337i	$0.518359\pi$
$644$	0	0
$645$	80.7948	3.18129
$646$	0	0
$647$	20.4003	0.802020	0.401010	−	0.916074i	$-0.368659\pi$
$647$	20.4003	0.802020	0.401010	+	0.916074i	$0.368659\pi$
$648$	0	0
$649$	−0.0253697	−0.000995850 0
$650$	0	0
$651$	115.732	4.53589
$652$	0	0
$653$	35.5990	1.39310	0.696548	−	0.717510i	$-0.254718\pi$
$653$	35.5990	1.39310	0.696548	+	0.717510i	$0.254718\pi$
$654$	0	0
$655$	26.4015	1.03159
$656$	0	0
$657$	−95.9297	−3.74257
$658$	0	0
$659$	−32.2051	−1.25453	−0.627266	−	0.778805i	$-0.715826\pi$
$659$	−32.2051	−1.25453	−0.627266	+	0.778805i	$0.715826\pi$
$660$	0	0
$661$	22.2873	0.866876	0.433438	−	0.901183i	$-0.357300\pi$
$661$	22.2873	0.866876	0.433438	+	0.901183i	$0.357300\pi$
$662$	0	0
$663$	−2.86189	−0.111147
$664$	0	0
$665$	11.6764	0.452790
$666$	0	0
$667$	3.44998	0.133584
$668$	0	0
$669$	20.6651	0.798958
$670$	0	0
$671$	0.748989	0.0289144
$672$	0	0
$673$	22.4339	0.864765	0.432383	−	0.901690i	$-0.357673\pi$
$673$	22.4339	0.864765	0.432383	+	0.901690i	$0.357673\pi$
$674$	0	0
$675$	42.1798	1.62350
$676$	0	0
$677$	−3.13786	−0.120598	−0.0602988	−	0.998180i	$-0.519205\pi$
$677$	−3.13786	−0.120598	−0.0602988	+	0.998180i	$0.519205\pi$
$678$	0	0
$679$	11.6454	0.446910
$680$	0	0
$681$	−64.7711	−2.48203
$682$	0	0
$683$	45.6673	1.74741	0.873705	−	0.486456i	$-0.161711\pi$
$683$	45.6673	1.74741	0.873705	+	0.486456i	$0.161711\pi$
$684$	0	0
$685$	21.1097	0.806560
$686$	0	0
$687$	61.5154	2.34696
$688$	0	0
$689$	−0.869233	−0.0331151
$690$	0	0
$691$	41.8620	1.59251	0.796253	−	0.604964i	$-0.206812\pi$
$691$	41.8620	1.59251	0.796253	+	0.604964i	$0.206812\pi$
$692$	0	0
$693$	2.06404	0.0784066
$694$	0	0
$695$	−45.1892	−1.71413
$696$	0	0
$697$	23.7479	0.899516
$698$	0	0
$699$	−66.2130	−2.50441
$700$	0	0
$701$	−4.08691	−0.154360	−0.0771802	−	0.997017i	$-0.524592\pi$
$701$	−4.08691	−0.154360	−0.0771802	+	0.997017i	$0.524592\pi$
$702$	0	0
$703$	−7.21477	−0.272110
$704$	0	0
$705$	16.1263	0.607352
$706$	0	0
$707$	62.6471	2.35609
$708$	0	0
$709$	−1.33922	−0.0502954	−0.0251477	−	0.999684i	$-0.508006\pi$
$709$	−1.33922	−0.0502954	−0.0251477	+	0.999684i	$0.508006\pi$
$710$	0	0
$711$	−7.04683	−0.264277
$712$	0	0
$713$	−21.4213	−0.802236
$714$	0	0
$715$	−0.0547793	−0.00204863
$716$	0	0
$717$	54.6721	2.04177
$718$	0	0
$719$	9.34311	0.348439	0.174220	−	0.984707i	$-0.444260\pi$
$719$	9.34311	0.348439	0.174220	+	0.984707i	$0.444260\pi$
$720$	0	0
$721$	28.0416	1.04432
$722$	0	0
$723$	−41.5396	−1.54487
$724$	0	0
$725$	4.76770	0.177068
$726$	0	0
$727$	−42.6807	−1.58294	−0.791470	−	0.611208i	$-0.790684\pi$
$727$	−42.6807	−1.58294	−0.791470	+	0.611208i	$0.790684\pi$
$728$	0	0
$729$	15.5620	0.576371
$730$	0	0
$731$	−30.3419	−1.12223
$732$	0	0
$733$	−27.4088	−1.01237	−0.506184	−	0.862425i	$-0.668944\pi$
$733$	−27.4088	−1.01237	−0.506184	+	0.862425i	$0.668944\pi$
$734$	0	0
$735$	−86.2280	−3.18057
$736$	0	0
$737$	0.509504	0.0187678
$738$	0	0
$739$	−17.9410	−0.659971	−0.329986	−	0.943986i	$-0.607044\pi$
$739$	−17.9410	−0.659971	−0.329986	+	0.943986i	$0.607044\pi$
$740$	0	0
$741$	−0.835116	−0.0306787
$742$	0	0
$743$	19.8236	0.727258	0.363629	−	0.931544i	$-0.381538\pi$
$743$	19.8236	0.727258	0.363629	+	0.931544i	$0.381538\pi$
$744$	0	0
$745$	−13.0297	−0.477370
$746$	0	0
$747$	−1.90230	−0.0696015
$748$	0	0
$749$	−27.0297	−0.987644
$750$	0	0
$751$	4.48804	0.163771	0.0818855	−	0.996642i	$-0.473906\pi$
$751$	4.48804	0.163771	0.0818855	+	0.996642i	$0.473906\pi$
$752$	0	0
$753$	53.2753	1.94146
$754$	0	0
$755$	33.7700	1.22901
$756$	0	0
$757$	−24.6713	−0.896693	−0.448346	−	0.893860i	$-0.647987\pi$
$757$	−24.6713	−0.896693	−0.448346	+	0.893860i	$0.647987\pi$
$758$	0	0
$759$	−0.544688	−0.0197709
$760$	0	0
$761$	11.9705	0.433931	0.216966	−	0.976179i	$-0.430384\pi$
$761$	11.9705	0.433931	0.216966	+	0.976179i	$0.430384\pi$
$762$	0	0
$763$	31.3343	1.13438
$764$	0	0
$765$	−69.5239	−2.51364
$766$	0	0
$767$	−0.0925544	−0.00334195
$768$	0	0
$769$	−10.7158	−0.386422	−0.193211	−	0.981157i	$-0.561890\pi$
$769$	−10.7158	−0.386422	−0.193211	+	0.981157i	$0.561890\pi$
$770$	0	0
$771$	11.9445	0.430171
$772$	0	0
$773$	41.2708	1.48441	0.742203	−	0.670175i	$-0.233781\pi$
$773$	41.2708	1.48441	0.742203	+	0.670175i	$0.233781\pi$
$774$	0	0
$775$	−29.6032	−1.06338
$776$	0	0
$777$	92.7494	3.32737
$778$	0	0
$779$	6.92976	0.248284
$780$	0	0
$781$	−0.0941496	−0.00336894
$782$	0	0
$783$	18.5979	0.664636
$784$	0	0
$785$	−50.2620	−1.79393
$786$	0	0
$787$	−54.5783	−1.94551	−0.972754	−	0.231841i	$-0.925525\pi$
$787$	−54.5783	−1.94551	−0.972754	+	0.231841i	$0.925525\pi$
$788$	0	0
$789$	−33.4296	−1.19013
$790$	0	0
$791$	50.1637	1.78362
$792$	0	0
$793$	2.73248	0.0970331
$794$	0	0
$795$	−30.1059	−1.06775
$796$	0	0
$797$	−39.6699	−1.40518	−0.702590	−	0.711595i	$-0.747973\pi$
$797$	−39.6699	−1.40518	−0.702590	+	0.711595i	$0.747973\pi$
$798$	0	0
$799$	−6.05611	−0.214250
$800$	0	0
$801$	−52.0588	−1.83941
$802$	0	0
$803$	0.983128	0.0346938
$804$	0	0
$805$	27.7836	0.979245
$806$	0	0
$807$	−27.3207	−0.961734
$808$	0	0
$809$	−46.8819	−1.64828	−0.824140	−	0.566386i	$-0.808341\pi$
$809$	−46.8819	−1.64828	−0.824140	+	0.566386i	$0.808341\pi$
$810$	0	0
$811$	−27.3392	−0.960008	−0.480004	−	0.877266i	$-0.659365\pi$
$811$	−27.3392	−0.960008	−0.480004	+	0.877266i	$0.659365\pi$
$812$	0	0
$813$	48.3820	1.69683
$814$	0	0
$815$	−44.2714	−1.55076
$816$	0	0
$817$	−8.85392	−0.309759
$818$	0	0
$819$	7.53009	0.263123
$820$	0	0
$821$	−7.08097	−0.247128	−0.123564	−	0.992337i	$-0.539432\pi$
$821$	−7.08097	−0.247128	−0.123564	+	0.992337i	$0.539432\pi$
$822$	0	0
$823$	−36.5081	−1.27259	−0.636296	−	0.771445i	$-0.719534\pi$
$823$	−36.5081	−1.27259	−0.636296	+	0.771445i	$0.719534\pi$
$824$	0	0
$825$	−0.752731	−0.0262067
$826$	0	0
$827$	39.9114	1.38785	0.693927	−	0.720045i	$-0.255879\pi$
$827$	39.9114	1.38785	0.693927	+	0.720045i	$0.255879\pi$
$828$	0	0
$829$	30.0815	1.04477	0.522387	−	0.852708i	$-0.325041\pi$
$829$	30.0815	1.04477	0.522387	+	0.852708i	$0.325041\pi$
$830$	0	0
$831$	41.2654	1.43148
$832$	0	0
$833$	32.3823	1.12198
$834$	0	0
$835$	6.75471	0.233756
$836$	0	0
$837$	−115.477	−3.99146
$838$	0	0
$839$	−42.7941	−1.47741	−0.738707	−	0.674026i	$-0.764563\pi$
$839$	−42.7941	−1.47741	−0.738707	+	0.674026i	$0.764563\pi$
$840$	0	0
$841$	−26.8978	−0.927511
$842$	0	0
$843$	−85.0303	−2.92860
$844$	0	0
$845$	37.2264	1.28063
$846$	0	0
$847$	44.5924	1.53221
$848$	0	0
$849$	−75.4599	−2.58978
$850$	0	0
$851$	−17.1674	−0.588491
$852$	0	0
$853$	7.76751	0.265954	0.132977	−	0.991119i	$-0.457546\pi$
$853$	7.76751	0.265954	0.132977	+	0.991119i	$0.457546\pi$
$854$	0	0
$855$	−20.2874	−0.693816
$856$	0	0
$857$	6.07587	0.207548	0.103774	−	0.994601i	$-0.466908\pi$
$857$	6.07587	0.207548	0.103774	+	0.994601i	$0.466908\pi$
$858$	0	0
$859$	11.6755	0.398364	0.199182	−	0.979963i	$-0.436172\pi$
$859$	11.6755	0.398364	0.199182	+	0.979963i	$0.436172\pi$
$860$	0	0
$861$	−89.0855	−3.03602
$862$	0	0
$863$	−46.0962	−1.56913	−0.784566	−	0.620045i	$-0.787114\pi$
$863$	−46.0962	−1.56913	−0.784566	+	0.620045i	$0.787114\pi$
$864$	0	0
$865$	29.3001	0.996232
$866$	0	0
$867$	−16.6600	−0.565803
$868$	0	0
$869$	0.0722189	0.00244986
$870$	0	0
$871$	1.85878	0.0629825
$872$	0	0
$873$	−20.2337	−0.684806
$874$	0	0
$875$	−19.9862	−0.675656
$876$	0	0
$877$	−9.72574	−0.328415	−0.164208	−	0.986426i	$-0.552507\pi$
$877$	−9.72574	−0.328415	−0.164208	+	0.986426i	$0.552507\pi$
$878$	0	0
$879$	29.5494	0.996677
$880$	0	0
$881$	−19.2462	−0.648420	−0.324210	−	0.945985i	$-0.605098\pi$
$881$	−19.2462	−0.648420	−0.324210	+	0.945985i	$0.605098\pi$
$882$	0	0
$883$	−5.38468	−0.181209	−0.0906045	−	0.995887i	$-0.528880\pi$
$883$	−5.38468	−0.181209	−0.0906045	+	0.995887i	$0.528880\pi$
$884$	0	0
$885$	−3.20563	−0.107756
$886$	0	0
$887$	49.2197	1.65263	0.826317	−	0.563205i	$-0.190432\pi$
$887$	49.2197	1.65263	0.826317	+	0.563205i	$0.190432\pi$
$888$	0	0
$889$	−72.3516	−2.42659
$890$	0	0
$891$	−1.40952	−0.0472208
$892$	0	0
$893$	−1.76721	−0.0591373
$894$	0	0
$895$	46.6608	1.55970
$896$	0	0
$897$	−1.98714	−0.0663487
$898$	0	0
$899$	−13.0527	−0.435331
$900$	0	0
$901$	11.3061	0.376659
$902$	0	0
$903$	113.821	3.78774
$904$	0	0
$905$	0.801478	0.0266420
$906$	0	0
$907$	−19.3743	−0.643312	−0.321656	−	0.946857i	$-0.604239\pi$
$907$	−19.3743	−0.643312	−0.321656	+	0.946857i	$0.604239\pi$
$908$	0	0
$909$	−108.848	−3.61026
$910$	0	0
$911$	−30.2687	−1.00285	−0.501424	−	0.865202i	$-0.667190\pi$
$911$	−30.2687	−1.00285	−0.501424	+	0.865202i	$0.667190\pi$
$912$	0	0
$913$	0.0194956	0.000645209 0
$914$	0	0
$915$	94.6395	3.12869
$916$	0	0
$917$	37.1936	1.22824
$918$	0	0
$919$	14.0107	0.462170	0.231085	−	0.972934i	$-0.425772\pi$
$919$	14.0107	0.462170	0.231085	+	0.972934i	$0.425772\pi$
$920$	0	0
$921$	−40.3460	−1.32945
$922$	0	0
$923$	−0.343478	−0.0113057
$924$	0	0
$925$	−23.7245	−0.780056
$926$	0	0
$927$	−48.7218	−1.60023
$928$	0	0
$929$	28.7184	0.942220	0.471110	−	0.882074i	$-0.343853\pi$
$929$	28.7184	0.942220	0.471110	+	0.882074i	$0.343853\pi$
$930$	0	0
$931$	9.44932	0.309689
$932$	0	0
$933$	−48.6571	−1.59296
$934$	0	0
$935$	0.712510	0.0233016
$936$	0	0
$937$	27.0367	0.883250	0.441625	−	0.897200i	$-0.354402\pi$
$937$	27.0367	0.883250	0.441625	+	0.897200i	$0.354402\pi$
$938$	0	0
$939$	37.5852	1.22655
$940$	0	0
$941$	−45.9444	−1.49774	−0.748872	−	0.662715i	$-0.769404\pi$
$941$	−45.9444	−1.49774	−0.748872	+	0.662715i	$0.769404\pi$
$942$	0	0
$943$	16.4892	0.536963
$944$	0	0
$945$	149.774	4.87216
$946$	0	0
$947$	58.2620	1.89326	0.946630	−	0.322323i	$-0.104464\pi$
$947$	58.2620	1.89326	0.946630	+	0.322323i	$0.104464\pi$
$948$	0	0
$949$	3.58667	0.116428
$950$	0	0
$951$	−95.8424	−3.10790
$952$	0	0
$953$	44.9365	1.45564	0.727818	−	0.685770i	$-0.240535\pi$
$953$	44.9365	1.45564	0.727818	+	0.685770i	$0.240535\pi$
$954$	0	0
$955$	−41.4378	−1.34090
$956$	0	0
$957$	−0.331895	−0.0107286
$958$	0	0
$959$	29.7387	0.960314
$960$	0	0
$961$	50.0456	1.61438
$962$	0	0
$963$	46.9636	1.51338
$964$	0	0
$965$	−40.0519	−1.28932
$966$	0	0
$967$	21.4529	0.689879	0.344939	−	0.938625i	$-0.387899\pi$
$967$	21.4529	0.689879	0.344939	+	0.938625i	$0.387899\pi$
$968$	0	0
$969$	10.8623	0.348947
$970$	0	0
$971$	−52.9068	−1.69786	−0.848931	−	0.528504i	$-0.822753\pi$
$971$	−52.9068	−1.69786	−0.848931	+	0.528504i	$0.822753\pi$
$972$	0	0
$973$	−63.6613	−2.04089
$974$	0	0
$975$	−2.74613	−0.0879465
$976$	0	0
$977$	38.6349	1.23604	0.618020	−	0.786162i	$-0.287935\pi$
$977$	38.6349	1.23604	0.618020	+	0.786162i	$0.287935\pi$
$978$	0	0
$979$	0.533521	0.0170514
$980$	0	0
$981$	−54.4428	−1.73822
$982$	0	0
$983$	−13.2590	−0.422896	−0.211448	−	0.977389i	$-0.567818\pi$
$983$	−13.2590	−0.422896	−0.211448	+	0.977389i	$0.567818\pi$
$984$	0	0
$985$	31.5718	1.00596
$986$	0	0
$987$	22.7183	0.723131
$988$	0	0
$989$	−21.0677	−0.669914
$990$	0	0
$991$	28.7796	0.914214	0.457107	−	0.889412i	$-0.348886\pi$
$991$	28.7796	0.914214	0.457107	+	0.889412i	$0.348886\pi$
$992$	0	0
$993$	46.2592	1.46799
$994$	0	0
$995$	−3.77439	−0.119656
$996$	0	0
$997$	−42.0528	−1.33183	−0.665913	−	0.746029i	$-0.731958\pi$
$997$	−42.0528	−1.33183	−0.665913	+	0.746029i	$0.731958\pi$
$998$	0	0
$999$	−92.5449	−2.92799

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.27	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.27	✓	27	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q+3.16967 q^{3} -2.87894 q^{5} -4.05578 q^{7} +7.04683 q^{9} +O(q^{10})\)	\(q+3.16967 q^{3} -2.87894 q^{5} -4.05578 q^{7} +7.04683 q^{9} -0.0722189 q^{11} -0.263471 q^{13} -9.12532 q^{15} +3.42694 q^{17} +1.00000 q^{19} -12.8555 q^{21} +2.37948 q^{23} +3.28832 q^{25} +12.8271 q^{27} +1.44989 q^{29} -9.00254 q^{31} -0.228910 q^{33} +11.6764 q^{35} -7.21477 q^{37} -0.835116 q^{39} +6.92976 q^{41} -8.85392 q^{43} -20.2874 q^{45} -1.76721 q^{47} +9.44932 q^{49} +10.8623 q^{51} +3.29917 q^{53} +0.207914 q^{55} +3.16967 q^{57} +0.351290 q^{59} -10.3711 q^{61} -28.5804 q^{63} +0.758517 q^{65} -7.05499 q^{67} +7.54217 q^{69} +1.30367 q^{71} -13.6132 q^{73} +10.4229 q^{75} +0.292904 q^{77} -1.00000 q^{79} +19.5174 q^{81} -0.269951 q^{83} -9.86598 q^{85} +4.59567 q^{87} -7.38755 q^{89} +1.06858 q^{91} -28.5351 q^{93} -2.87894 q^{95} -2.87132 q^{97} -0.508915 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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