LMFDB - Embedded newform 608.4.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,4,Mod(1,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("608.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	608.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:	$35.8731612835$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664 $ x^7 - 3x^6 - 121x^5 + 402x^4 + 4234x^3 - 14542x^2 - 40996x + 141664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$6.59304$ of defining polynomial
Character	$\chi$	$=$	608.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.59727 q^{3} -20.9584 q^{5} +13.9085 q^{7} +4.32941 q^{9} -58.3230 q^{11} +65.0857 q^{13} -117.310 q^{15} +31.7022 q^{17} -19.0000 q^{19} +77.8498 q^{21} +151.699 q^{23} +314.256 q^{25} -126.893 q^{27} +110.944 q^{29} +94.0614 q^{31} -326.450 q^{33} -291.501 q^{35} -291.678 q^{37} +364.302 q^{39} +64.4169 q^{41} +449.123 q^{43} -90.7375 q^{45} +530.261 q^{47} -149.553 q^{49} +177.446 q^{51} +621.195 q^{53} +1222.36 q^{55} -106.348 q^{57} -244.887 q^{59} +801.180 q^{61} +60.2157 q^{63} -1364.10 q^{65} -7.34129 q^{67} +849.098 q^{69} -1027.68 q^{71} +592.854 q^{73} +1758.97 q^{75} -811.188 q^{77} +120.481 q^{79} -827.150 q^{81} -502.722 q^{83} -664.428 q^{85} +620.983 q^{87} +253.862 q^{89} +905.247 q^{91} +526.487 q^{93} +398.210 q^{95} +511.232 q^{97} -252.504 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} - 17 q^{5} + 42 q^{7} + 86 q^{9} - 33 q^{11} - 35 q^{13} + 120 q^{15} + 66 q^{17} - 133 q^{19} + 33 q^{21} + 389 q^{23} + 44 q^{25} - 39 q^{27} + 233 q^{29} + 158 q^{31} - 206 q^{33} + 123 q^{35}+ \cdots - 1591 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 - 17 * q^5 + 42 * q^7 + 86 * q^9 - 33 * q^11 - 35 * q^13 + 120 * q^15 + 66 * q^17 - 133 * q^19 + 33 * q^21 + 389 * q^23 + 44 * q^25 - 39 * q^27 + 233 * q^29 + 158 * q^31 - 206 * q^33 + 123 * q^35 - 436 * q^37 + 807 * q^39 - 94 * q^41 + 645 * q^43 + 103 * q^45 + 1451 * q^47 + 93 * q^49 + 1741 * q^51 + 3 * q^53 + 1971 * q^55 - 57 * q^57 + 297 * q^59 + 93 * q^61 + 2999 * q^63 - 788 * q^65 + 1641 * q^67 + 945 * q^69 + 2392 * q^71 + 324 * q^73 + 1909 * q^75 - 711 * q^77 + 2492 * q^79 + 143 * q^81 + 310 * q^83 + 2353 * q^85 + 4795 * q^87 - 440 * q^89 - 107 * q^91 + 900 * q^93 + 323 * q^95 - 532 * q^97 - 1591 * q^99

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
$3$	5.59727	1.07719	0.538597	−	0.842563i	$-0.318954\pi$
$3$	5.59727	1.07719	0.538597	+	0.842563i	$0.318954\pi$
$4$	0	0
$5$	−20.9584	−1.87458	−0.937290	−	0.348552i	$-0.886674\pi$
$5$	−20.9584	−1.87458	−0.937290	+	0.348552i	$0.886674\pi$
$6$	0	0
$7$	13.9085	0.750990	0.375495	−	0.926824i	$-0.377473\pi$
$7$	13.9085	0.750990	0.375495	+	0.926824i	$0.377473\pi$
$8$	0	0
$9$	4.32941	0.160348
$10$	0	0
$11$	−58.3230	−1.59864	−0.799321	−	0.600904i	$-0.794807\pi$
$11$	−58.3230	−1.59864	−0.799321	+	0.600904i	$0.794807\pi$
$12$	0	0
$13$	65.0857	1.38858	0.694290	−	0.719695i	$-0.255719\pi$
$13$	65.0857	1.38858	0.694290	+	0.719695i	$0.255719\pi$
$14$	0	0
$15$	−117.310	−2.01929
$16$	0	0
$17$	31.7022	0.452289	0.226144	−	0.974094i	$-0.427388\pi$
$17$	31.7022	0.452289	0.226144	+	0.974094i	$0.427388\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	77.8498	0.808963
$22$	0	0
$23$	151.699	1.37528	0.687639	−	0.726053i	$-0.258647\pi$
$23$	151.699	1.37528	0.687639	+	0.726053i	$0.258647\pi$
$24$	0	0
$25$	314.256	2.51405
$26$	0	0
$27$	−126.893	−0.904468
$28$	0	0
$29$	110.944	0.710406	0.355203	−	0.934789i	$-0.384412\pi$
$29$	110.944	0.710406	0.355203	+	0.934789i	$0.384412\pi$
$30$	0	0
$31$	94.0614	0.544965	0.272483	−	0.962161i	$-0.412155\pi$
$31$	94.0614	0.544965	0.272483	+	0.962161i	$0.412155\pi$
$32$	0	0
$33$	−326.450	−1.72205
$34$	0	0
$35$	−291.501	−1.40779
$36$	0	0
$37$	−291.678	−1.29599	−0.647994	−	0.761645i	$-0.724392\pi$
$37$	−291.678	−1.29599	−0.647994	+	0.761645i	$0.724392\pi$
$38$	0	0
$39$	364.302	1.49577
$40$	0	0
$41$	64.4169	0.245371	0.122686	−	0.992446i	$-0.460849\pi$
$41$	64.4169	0.245371	0.122686	+	0.992446i	$0.460849\pi$
$42$	0	0
$43$	449.123	1.59281	0.796403	−	0.604766i	$-0.206733\pi$
$43$	449.123	1.59281	0.796403	+	0.604766i	$0.206733\pi$
$44$	0	0
$45$	−90.7375	−0.300586
$46$	0	0
$47$	530.261	1.64567	0.822835	−	0.568281i	$-0.192391\pi$
$47$	530.261	1.64567	0.822835	+	0.568281i	$0.192391\pi$
$48$	0	0
$49$	−149.553	−0.436013
$50$	0	0
$51$	177.446	0.487203
$52$	0	0
$53$	621.195	1.60996	0.804978	−	0.593304i	$-0.202177\pi$
$53$	621.195	1.60996	0.804978	+	0.593304i	$0.202177\pi$
$54$	0	0
$55$	1222.36	2.99678
$56$	0	0
$57$	−106.348	−0.247125
$58$	0	0
$59$	−244.887	−0.540367	−0.270183	−	0.962809i	$-0.587084\pi$
$59$	−244.887	−0.540367	−0.270183	+	0.962809i	$0.587084\pi$
$60$	0	0
$61$	801.180	1.68165	0.840825	−	0.541308i	$-0.182071\pi$
$61$	801.180	1.68165	0.840825	+	0.541308i	$0.182071\pi$
$62$	0	0
$63$	60.2157	0.120420
$64$	0	0
$65$	−1364.10	−2.60300
$66$	0	0
$67$	−7.34129	−0.0133863	−0.00669315	−	0.999978i	$-0.502131\pi$
$67$	−7.34129	−0.0133863	−0.00669315	+	0.999978i	$0.502131\pi$
$68$	0	0
$69$	849.098	1.48144
$70$	0	0
$71$	−1027.68	−1.71778	−0.858892	−	0.512156i	$-0.828847\pi$
$71$	−1027.68	−1.71778	−0.858892	+	0.512156i	$0.828847\pi$
$72$	0	0
$73$	592.854	0.950526	0.475263	−	0.879844i	$-0.342353\pi$
$73$	592.854	0.950526	0.475263	+	0.879844i	$0.342353\pi$
$74$	0	0
$75$	1758.97	2.70812
$76$	0	0
$77$	−811.188	−1.20056
$78$	0	0
$79$	120.481	0.171584	0.0857919	−	0.996313i	$-0.472658\pi$
$79$	120.481	0.171584	0.0857919	+	0.996313i	$0.472658\pi$
$80$	0	0
$81$	−827.150	−1.13464
$82$	0	0
$83$	−502.722	−0.664830	−0.332415	−	0.943133i	$-0.607863\pi$
$83$	−502.722	−0.664830	−0.332415	+	0.943133i	$0.607863\pi$
$84$	0	0
$85$	−664.428	−0.847851
$86$	0	0
$87$	620.983	0.765246
$88$	0	0
$89$	253.862	0.302352	0.151176	−	0.988507i	$-0.451694\pi$
$89$	253.862	0.302352	0.151176	+	0.988507i	$0.451694\pi$
$90$	0	0
$91$	905.247	1.04281
$92$	0	0
$93$	526.487	0.587034
$94$	0	0
$95$	398.210	0.430058
$96$	0	0
$97$	511.232	0.535131	0.267566	−	0.963540i	$-0.413781\pi$
$97$	511.232	0.535131	0.267566	+	0.963540i	$0.413781\pi$
$98$	0	0
$99$	−252.504	−0.256340
$100$	0	0
$101$	−647.433	−0.637842	−0.318921	−	0.947781i	$-0.603320\pi$
$101$	−647.433	−0.637842	−0.318921	+	0.947781i	$0.603320\pi$
$102$	0	0
$103$	825.539	0.789736	0.394868	−	0.918738i	$-0.370790\pi$
$103$	825.539	0.789736	0.394868	+	0.918738i	$0.370790\pi$
$104$	0	0
$105$	−1631.61	−1.51646
$106$	0	0
$107$	244.650	0.221040	0.110520	−	0.993874i	$-0.464748\pi$
$107$	244.650	0.221040	0.110520	+	0.993874i	$0.464748\pi$
$108$	0	0
$109$	−1011.60	−0.888937	−0.444469	−	0.895794i	$-0.646607\pi$
$109$	−1011.60	−0.888937	−0.444469	+	0.895794i	$0.646607\pi$
$110$	0	0
$111$	−1632.60	−1.39603
$112$	0	0
$113$	70.2342	0.0584697	0.0292348	−	0.999573i	$-0.490693\pi$
$113$	70.2342	0.0584697	0.0292348	+	0.999573i	$0.490693\pi$
$114$	0	0
$115$	−3179.37	−2.57807
$116$	0	0
$117$	281.783	0.222656
$118$	0	0
$119$	440.931	0.339664
$120$	0	0
$121$	2070.58	1.55566
$122$	0	0
$123$	360.558	0.264313
$124$	0	0
$125$	−3966.51	−2.83820
$126$	0	0
$127$	484.115	0.338254	0.169127	−	0.985594i	$-0.445905\pi$
$127$	484.115	0.338254	0.169127	+	0.985594i	$0.445905\pi$
$128$	0	0
$129$	2513.86	1.71576
$130$	0	0
$131$	2460.64	1.64113	0.820563	−	0.571557i	$-0.193660\pi$
$131$	2460.64	1.64113	0.820563	+	0.571557i	$0.193660\pi$
$132$	0	0
$133$	−264.262	−0.172289
$134$	0	0
$135$	2659.49	1.69550
$136$	0	0
$137$	−1872.74	−1.16788	−0.583939	−	0.811798i	$-0.698489\pi$
$137$	−1872.74	−1.16788	−0.583939	+	0.811798i	$0.698489\pi$
$138$	0	0
$139$	168.319	0.102710	0.0513548	−	0.998680i	$-0.483646\pi$
$139$	168.319	0.102710	0.0513548	+	0.998680i	$0.483646\pi$
$140$	0	0
$141$	2968.01	1.77271
$142$	0	0
$143$	−3796.00	−2.21984
$144$	0	0
$145$	−2325.21	−1.33171
$146$	0	0
$147$	−837.086	−0.469671
$148$	0	0
$149$	−719.682	−0.395696	−0.197848	−	0.980233i	$-0.563395\pi$
$149$	−719.682	−0.395696	−0.197848	+	0.980233i	$0.563395\pi$
$150$	0	0
$151$	3487.49	1.87952	0.939762	−	0.341829i	$-0.111046\pi$
$151$	3487.49	1.87952	0.939762	+	0.341829i	$0.111046\pi$
$152$	0	0
$153$	137.252	0.0725237
$154$	0	0
$155$	−1971.38	−1.02158
$156$	0	0
$157$	−416.547	−0.211746	−0.105873	−	0.994380i	$-0.533764\pi$
$157$	−416.547	−0.211746	−0.105873	+	0.994380i	$0.533764\pi$
$158$	0	0
$159$	3476.99	1.73424
$160$	0	0
$161$	2109.91	1.03282
$162$	0	0
$163$	2999.31	1.44125	0.720626	−	0.693324i	$-0.243854\pi$
$163$	2999.31	1.44125	0.720626	+	0.693324i	$0.243854\pi$
$164$	0	0
$165$	6841.87	3.22812
$166$	0	0
$167$	−2279.85	−1.05641	−0.528204	−	0.849117i	$-0.677134\pi$
$167$	−2279.85	−1.05641	−0.528204	+	0.849117i	$0.677134\pi$
$168$	0	0
$169$	2039.15	0.928154
$170$	0	0
$171$	−82.2587	−0.0367864
$172$	0	0
$173$	710.537	0.312261	0.156130	−	0.987736i	$-0.450098\pi$
$173$	710.537	0.312261	0.156130	+	0.987736i	$0.450098\pi$
$174$	0	0
$175$	4370.84	1.88802
$176$	0	0
$177$	−1370.70	−0.582080
$178$	0	0
$179$	−1830.61	−0.764392	−0.382196	−	0.924081i	$-0.624832\pi$
$179$	−1830.61	−0.764392	−0.382196	+	0.924081i	$0.624832\pi$
$180$	0	0
$181$	−225.382	−0.0925554	−0.0462777	−	0.998929i	$-0.514736\pi$
$181$	−225.382	−0.0925554	−0.0462777	+	0.998929i	$0.514736\pi$
$182$	0	0
$183$	4484.42	1.81146
$184$	0	0
$185$	6113.11	2.42943
$186$	0	0
$187$	−1848.97	−0.723048
$188$	0	0
$189$	−1764.90	−0.679247
$190$	0	0
$191$	1179.95	0.447005	0.223502	−	0.974703i	$-0.428251\pi$
$191$	1179.95	0.447005	0.223502	+	0.974703i	$0.428251\pi$
$192$	0	0
$193$	−1007.61	−0.375799	−0.187899	−	0.982188i	$-0.560168\pi$
$193$	−1007.61	−0.375799	−0.187899	+	0.982188i	$0.560168\pi$
$194$	0	0
$195$	−7635.20	−2.80394
$196$	0	0
$197$	−740.845	−0.267934	−0.133967	−	0.990986i	$-0.542772\pi$
$197$	−740.845	−0.267934	−0.133967	+	0.990986i	$0.542772\pi$
$198$	0	0
$199$	4116.86	1.46652	0.733258	−	0.679951i	$-0.237999\pi$
$199$	4116.86	1.46652	0.733258	+	0.679951i	$0.237999\pi$
$200$	0	0
$201$	−41.0912	−0.0144196
$202$	0	0
$203$	1543.07	0.533508
$204$	0	0
$205$	−1350.08	−0.459968
$206$	0	0
$207$	656.765	0.220523
$208$	0	0
$209$	1108.14	0.366754
$210$	0	0
$211$	−926.033	−0.302136	−0.151068	−	0.988523i	$-0.548271\pi$
$211$	−926.033	−0.302136	−0.151068	+	0.988523i	$0.548271\pi$
$212$	0	0
$213$	−5752.18	−1.85039
$214$	0	0
$215$	−9412.92	−2.98584
$216$	0	0
$217$	1308.26	0.409264
$218$	0	0
$219$	3318.37	1.02390
$220$	0	0
$221$	2063.36	0.628039
$222$	0	0
$223$	−1588.10	−0.476893	−0.238446	−	0.971156i	$-0.576638\pi$
$223$	−1588.10	−0.476893	−0.238446	+	0.971156i	$0.576638\pi$
$224$	0	0
$225$	1360.54	0.403123
$226$	0	0
$227$	−1728.22	−0.505314	−0.252657	−	0.967556i	$-0.581304\pi$
$227$	−1728.22	−0.505314	−0.252657	+	0.967556i	$0.581304\pi$
$228$	0	0
$229$	−3648.90	−1.05295	−0.526477	−	0.850190i	$-0.676487\pi$
$229$	−3648.90	−1.05295	−0.526477	+	0.850190i	$0.676487\pi$
$230$	0	0
$231$	−4540.44	−1.29324
$232$	0	0
$233$	−732.776	−0.206033	−0.103017	−	0.994680i	$-0.532850\pi$
$233$	−732.776	−0.206033	−0.103017	+	0.994680i	$0.532850\pi$
$234$	0	0
$235$	−11113.4	−3.08494
$236$	0	0
$237$	674.362	0.184829
$238$	0	0
$239$	4003.14	1.08344	0.541719	−	0.840560i	$-0.317774\pi$
$239$	4003.14	1.08344	0.541719	+	0.840560i	$0.317774\pi$
$240$	0	0
$241$	4947.02	1.32226	0.661132	−	0.750270i	$-0.270076\pi$
$241$	4947.02	1.32226	0.661132	+	0.750270i	$0.270076\pi$
$242$	0	0
$243$	−1203.66	−0.317756
$244$	0	0
$245$	3134.39	0.817342
$246$	0	0
$247$	−1236.63	−0.318562
$248$	0	0
$249$	−2813.87	−0.716151
$250$	0	0
$251$	−3693.27	−0.928753	−0.464376	−	0.885638i	$-0.653722\pi$
$251$	−3693.27	−0.928753	−0.464376	+	0.885638i	$0.653722\pi$
$252$	0	0
$253$	−8847.53	−2.19858
$254$	0	0
$255$	−3718.98	−0.913300
$256$	0	0
$257$	−5937.42	−1.44111	−0.720556	−	0.693397i	$-0.756113\pi$
$257$	−5937.42	−1.44111	−0.720556	+	0.693397i	$0.756113\pi$
$258$	0	0
$259$	−4056.81	−0.973275
$260$	0	0
$261$	480.321	0.113912
$262$	0	0
$263$	484.571	0.113612	0.0568059	−	0.998385i	$-0.481908\pi$
$263$	484.571	0.113612	0.0568059	+	0.998385i	$0.481908\pi$
$264$	0	0
$265$	−13019.3	−3.01799
$266$	0	0
$267$	1420.93	0.325692
$268$	0	0
$269$	4691.52	1.06337	0.531685	−	0.846942i	$-0.321559\pi$
$269$	4691.52	1.06337	0.531685	+	0.846942i	$0.321559\pi$
$270$	0	0
$271$	3754.24	0.841526	0.420763	−	0.907170i	$-0.361762\pi$
$271$	3754.24	0.841526	0.420763	+	0.907170i	$0.361762\pi$
$272$	0	0
$273$	5066.91	1.12331
$274$	0	0
$275$	−18328.4	−4.01906
$276$	0	0
$277$	−2432.06	−0.527540	−0.263770	−	0.964586i	$-0.584966\pi$
$277$	−2432.06	−0.527540	−0.263770	+	0.964586i	$0.584966\pi$
$278$	0	0
$279$	407.230	0.0873843
$280$	0	0
$281$	5992.05	1.27209	0.636043	−	0.771654i	$-0.280570\pi$
$281$	5992.05	1.27209	0.636043	+	0.771654i	$0.280570\pi$
$282$	0	0
$283$	−181.186	−0.0380579	−0.0190290	−	0.999819i	$-0.506057\pi$
$283$	−181.186	−0.0380579	−0.0190290	+	0.999819i	$0.506057\pi$
$284$	0	0
$285$	2228.89	0.463256
$286$	0	0
$287$	895.944	0.184271
$288$	0	0
$289$	−3907.97	−0.795435
$290$	0	0
$291$	2861.50	0.576440
$292$	0	0
$293$	−1909.02	−0.380634	−0.190317	−	0.981723i	$-0.560952\pi$
$293$	−1909.02	−0.380634	−0.190317	+	0.981723i	$0.560952\pi$
$294$	0	0
$295$	5132.46	1.01296
$296$	0	0
$297$	7400.81	1.44592
$298$	0	0
$299$	9873.43	1.90968
$300$	0	0
$301$	6246.65	1.19618
$302$	0	0
$303$	−3623.86	−0.687080
$304$	0	0
$305$	−16791.5	−3.15238
$306$	0	0
$307$	−3890.48	−0.723261	−0.361631	−	0.932321i	$-0.617780\pi$
$307$	−3890.48	−0.723261	−0.361631	+	0.932321i	$0.617780\pi$
$308$	0	0
$309$	4620.76	0.850699
$310$	0	0
$311$	−978.939	−0.178491	−0.0892453	−	0.996010i	$-0.528446\pi$
$311$	−978.939	−0.178491	−0.0892453	+	0.996010i	$0.528446\pi$
$312$	0	0
$313$	1049.35	0.189498	0.0947492	−	0.995501i	$-0.469795\pi$
$313$	1049.35	0.189498	0.0947492	+	0.995501i	$0.469795\pi$
$314$	0	0
$315$	−1262.03	−0.225737
$316$	0	0
$317$	−4862.01	−0.861443	−0.430722	−	0.902485i	$-0.641741\pi$
$317$	−4862.01	−0.861443	−0.430722	+	0.902485i	$0.641741\pi$
$318$	0	0
$319$	−6470.59	−1.13569
$320$	0	0
$321$	1369.37	0.238103
$322$	0	0
$323$	−602.341	−0.103762
$324$	0	0
$325$	20453.6	3.49095
$326$	0	0
$327$	−5662.22	−0.957558
$328$	0	0
$329$	7375.15	1.23588
$330$	0	0
$331$	6962.05	1.15610	0.578049	−	0.816002i	$-0.303814\pi$
$331$	6962.05	1.15610	0.578049	+	0.816002i	$0.303814\pi$
$332$	0	0
$333$	−1262.79	−0.207810
$334$	0	0
$335$	153.862	0.0250937
$336$	0	0
$337$	−12140.3	−1.96239	−0.981197	−	0.193011i	$-0.938175\pi$
$337$	−12140.3	−1.96239	−0.981197	+	0.193011i	$0.938175\pi$
$338$	0	0
$339$	393.119	0.0629832
$340$	0	0
$341$	−5485.95	−0.871204
$342$	0	0
$343$	−6850.69	−1.07843
$344$	0	0
$345$	−17795.8	−2.77708
$346$	0	0
$347$	1658.87	0.256636	0.128318	−	0.991733i	$-0.459042\pi$
$347$	1658.87	0.256636	0.128318	+	0.991733i	$0.459042\pi$
$348$	0	0
$349$	−9251.45	−1.41896	−0.709482	−	0.704723i	$-0.751071\pi$
$349$	−9251.45	−1.41896	−0.709482	+	0.704723i	$0.751071\pi$
$350$	0	0
$351$	−8258.95	−1.25593
$352$	0	0
$353$	−1168.48	−0.176180	−0.0880902	−	0.996112i	$-0.528076\pi$
$353$	−1168.48	−0.176180	−0.0880902	+	0.996112i	$0.528076\pi$
$354$	0	0
$355$	21538.5	3.22012
$356$	0	0
$357$	2468.01	0.365885
$358$	0	0
$359$	−3821.35	−0.561791	−0.280896	−	0.959738i	$-0.590631\pi$
$359$	−3821.35	−0.561791	−0.280896	+	0.959738i	$0.590631\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	11589.6	1.67574
$364$	0	0
$365$	−12425.3	−1.78184
$366$	0	0
$367$	5672.77	0.806856	0.403428	−	0.915011i	$-0.367819\pi$
$367$	5672.77	0.806856	0.403428	+	0.915011i	$0.367819\pi$
$368$	0	0
$369$	278.887	0.0393449
$370$	0	0
$371$	8639.91	1.20906
$372$	0	0
$373$	−1253.21	−0.173964	−0.0869821	−	0.996210i	$-0.527722\pi$
$373$	−1253.21	−0.173964	−0.0869821	+	0.996210i	$0.527722\pi$
$374$	0	0
$375$	−22201.6	−3.05729
$376$	0	0
$377$	7220.87	0.986456
$378$	0	0
$379$	−8245.91	−1.11758	−0.558792	−	0.829308i	$-0.688735\pi$
$379$	−8245.91	−1.11758	−0.558792	+	0.829308i	$0.688735\pi$
$380$	0	0
$381$	2709.72	0.364365
$382$	0	0
$383$	10182.5	1.35849	0.679247	−	0.733910i	$-0.262307\pi$
$383$	10182.5	1.35849	0.679247	+	0.733910i	$0.262307\pi$
$384$	0	0
$385$	17001.2	2.25055
$386$	0	0
$387$	1944.44	0.255404
$388$	0	0
$389$	4232.31	0.551637	0.275819	−	0.961210i	$-0.411051\pi$
$389$	4232.31	0.551637	0.275819	+	0.961210i	$0.411051\pi$
$390$	0	0
$391$	4809.18	0.622022
$392$	0	0
$393$	13772.9	1.76781
$394$	0	0
$395$	−2525.08	−0.321647
$396$	0	0
$397$	13862.9	1.75254	0.876269	−	0.481823i	$-0.160025\pi$
$397$	13862.9	1.75254	0.876269	+	0.481823i	$0.160025\pi$
$398$	0	0
$399$	−1479.15	−0.185589
$400$	0	0
$401$	622.725	0.0775496	0.0387748	−	0.999248i	$-0.487655\pi$
$401$	622.725	0.0775496	0.0387748	+	0.999248i	$0.487655\pi$
$402$	0	0
$403$	6122.06	0.756728
$404$	0	0
$405$	17335.8	2.12697
$406$	0	0
$407$	17011.5	2.07182
$408$	0	0
$409$	−2415.98	−0.292085	−0.146042	−	0.989278i	$-0.546654\pi$
$409$	−2415.98	−0.292085	−0.146042	+	0.989278i	$0.546654\pi$
$410$	0	0
$411$	−10482.2	−1.25803
$412$	0	0
$413$	−3406.03	−0.405810
$414$	0	0
$415$	10536.3	1.24628
$416$	0	0
$417$	942.127	0.110638
$418$	0	0
$419$	538.527	0.0627894	0.0313947	−	0.999507i	$-0.490005\pi$
$419$	538.527	0.0627894	0.0313947	+	0.999507i	$0.490005\pi$
$420$	0	0
$421$	−5804.87	−0.672000	−0.336000	−	0.941862i	$-0.609074\pi$
$421$	−5804.87	−0.672000	−0.336000	+	0.941862i	$0.609074\pi$
$422$	0	0
$423$	2295.71	0.263880
$424$	0	0
$425$	9962.59	1.13707
$426$	0	0
$427$	11143.2	1.26290
$428$	0	0
$429$	−21247.2	−2.39120
$430$	0	0
$431$	3562.84	0.398181	0.199091	−	0.979981i	$-0.436201\pi$
$431$	3562.84	0.398181	0.199091	+	0.979981i	$0.436201\pi$
$432$	0	0
$433$	3848.93	0.427177	0.213589	−	0.976924i	$-0.431485\pi$
$433$	3848.93	0.427177	0.213589	+	0.976924i	$0.431485\pi$
$434$	0	0
$435$	−13014.8	−1.43451
$436$	0	0
$437$	−2882.28	−0.315510
$438$	0	0
$439$	−14995.7	−1.63031	−0.815154	−	0.579245i	$-0.803348\pi$
$439$	−14995.7	−1.63031	−0.815154	+	0.579245i	$0.803348\pi$
$440$	0	0
$441$	−647.474	−0.0699140
$442$	0	0
$443$	−7647.68	−0.820208	−0.410104	−	0.912039i	$-0.634508\pi$
$443$	−7647.68	−0.820208	−0.410104	+	0.912039i	$0.634508\pi$
$444$	0	0
$445$	−5320.55	−0.566783
$446$	0	0
$447$	−4028.26	−0.426241
$448$	0	0
$449$	16945.8	1.78111	0.890557	−	0.454871i	$-0.150315\pi$
$449$	16945.8	1.78111	0.890557	+	0.454871i	$0.150315\pi$
$450$	0	0
$451$	−3756.99	−0.392261
$452$	0	0
$453$	19520.4	2.02461
$454$	0	0
$455$	−18972.6	−1.95483
$456$	0	0
$457$	−9759.32	−0.998953	−0.499477	−	0.866327i	$-0.666474\pi$
$457$	−9759.32	−0.998953	−0.499477	+	0.866327i	$0.666474\pi$
$458$	0	0
$459$	−4022.80	−0.409081
$460$	0	0
$461$	12152.0	1.22771	0.613855	−	0.789419i	$-0.289618\pi$
$461$	12152.0	1.22771	0.613855	+	0.789419i	$0.289618\pi$
$462$	0	0
$463$	15144.7	1.52016	0.760079	−	0.649831i	$-0.225160\pi$
$463$	15144.7	1.52016	0.760079	+	0.649831i	$0.225160\pi$
$464$	0	0
$465$	−11034.3	−1.10044
$466$	0	0
$467$	11903.1	1.17947	0.589734	−	0.807598i	$-0.299233\pi$
$467$	11903.1	1.17947	0.589734	+	0.807598i	$0.299233\pi$
$468$	0	0
$469$	−102.107	−0.0100530
$470$	0	0
$471$	−2331.52	−0.228091
$472$	0	0
$473$	−26194.2	−2.54633
$474$	0	0
$475$	−5970.86	−0.576762
$476$	0	0
$477$	2689.40	0.258154
$478$	0	0
$479$	−2125.99	−0.202795	−0.101398	−	0.994846i	$-0.532331\pi$
$479$	−2125.99	−0.202795	−0.101398	+	0.994846i	$0.532331\pi$
$480$	0	0
$481$	−18984.1	−1.79958
$482$	0	0
$483$	11809.7	1.11255
$484$	0	0
$485$	−10714.6	−1.00315
$486$	0	0
$487$	9302.38	0.865567	0.432783	−	0.901498i	$-0.357531\pi$
$487$	9302.38	0.865567	0.432783	+	0.901498i	$0.357531\pi$
$488$	0	0
$489$	16788.0	1.55251
$490$	0	0
$491$	−1284.29	−0.118044	−0.0590218	−	0.998257i	$-0.518798\pi$
$491$	−1284.29	−0.118044	−0.0590218	+	0.998257i	$0.518798\pi$
$492$	0	0
$493$	3517.17	0.321309
$494$	0	0
$495$	5292.09	0.480529
$496$	0	0
$497$	−14293.5	−1.29004
$498$	0	0
$499$	270.467	0.0242640	0.0121320	−	0.999926i	$-0.496138\pi$
$499$	270.467	0.0242640	0.0121320	+	0.999926i	$0.496138\pi$
$500$	0	0
$501$	−12760.9	−1.13796
$502$	0	0
$503$	−16179.9	−1.43424	−0.717122	−	0.696947i	$-0.754541\pi$
$503$	−16179.9	−1.43424	−0.717122	+	0.696947i	$0.754541\pi$
$504$	0	0
$505$	13569.2	1.19568
$506$	0	0
$507$	11413.7	0.999802
$508$	0	0
$509$	12004.0	1.04532	0.522658	−	0.852542i	$-0.324940\pi$
$509$	12004.0	1.04532	0.522658	+	0.852542i	$0.324940\pi$
$510$	0	0
$511$	8245.74	0.713836
$512$	0	0
$513$	2410.97	0.207499
$514$	0	0
$515$	−17302.0	−1.48042
$516$	0	0
$517$	−30926.4	−2.63084
$518$	0	0
$519$	3977.07	0.336366
$520$	0	0
$521$	−7092.90	−0.596441	−0.298220	−	0.954497i	$-0.596393\pi$
$521$	−7092.90	−0.596441	−0.298220	+	0.954497i	$0.596393\pi$
$522$	0	0
$523$	1447.50	0.121023	0.0605114	−	0.998168i	$-0.480727\pi$
$523$	1447.50	0.121023	0.0605114	+	0.998168i	$0.480727\pi$
$524$	0	0
$525$	24464.8	2.03377
$526$	0	0
$527$	2981.95	0.246482
$528$	0	0
$529$	10845.5	0.891387
$530$	0	0
$531$	−1060.22	−0.0866469
$532$	0	0
$533$	4192.62	0.340718
$534$	0	0
$535$	−5127.49	−0.414356
$536$	0	0
$537$	−10246.4	−0.823398
$538$	0	0
$539$	8722.36	0.697029
$540$	0	0
$541$	14803.7	1.17645	0.588226	−	0.808697i	$-0.299827\pi$
$541$	14803.7	1.17645	0.588226	+	0.808697i	$0.299827\pi$
$542$	0	0
$543$	−1261.52	−0.0997002
$544$	0	0
$545$	21201.6	1.66638
$546$	0	0
$547$	15330.9	1.19836	0.599179	−	0.800615i	$-0.295494\pi$
$547$	15330.9	1.19836	0.599179	+	0.800615i	$0.295494\pi$
$548$	0	0
$549$	3468.63	0.269650
$550$	0	0
$551$	−2107.94	−0.162978
$552$	0	0
$553$	1675.71	0.128858
$554$	0	0
$555$	34216.7	2.61697
$556$	0	0
$557$	−12069.8	−0.918159	−0.459080	−	0.888395i	$-0.651821\pi$
$557$	−12069.8	−0.918159	−0.459080	+	0.888395i	$0.651821\pi$
$558$	0	0
$559$	29231.5	2.21174
$560$	0	0
$561$	−10349.2	−0.778863
$562$	0	0
$563$	26232.5	1.96371	0.981856	−	0.189629i	$-0.0607284\pi$
$563$	26232.5	1.96371	0.981856	+	0.189629i	$0.0607284\pi$
$564$	0	0
$565$	−1472.00	−0.109606
$566$	0	0
$567$	−11504.4	−0.852101
$568$	0	0
$569$	7982.36	0.588116	0.294058	−	0.955788i	$-0.404994\pi$
$569$	7982.36	0.588116	0.294058	+	0.955788i	$0.404994\pi$
$570$	0	0
$571$	−8020.59	−0.587831	−0.293915	−	0.955831i	$-0.594958\pi$
$571$	−8020.59	−0.587831	−0.293915	+	0.955831i	$0.594958\pi$
$572$	0	0
$573$	6604.48	0.481511
$574$	0	0
$575$	47672.2	3.45751
$576$	0	0
$577$	13707.5	0.988999	0.494499	−	0.869178i	$-0.335351\pi$
$577$	13707.5	0.988999	0.494499	+	0.869178i	$0.335351\pi$
$578$	0	0
$579$	−5639.85	−0.404809
$580$	0	0
$581$	−6992.12	−0.499281
$582$	0	0
$583$	−36230.0	−2.57374
$584$	0	0
$585$	−5905.72	−0.417387
$586$	0	0
$587$	17767.6	1.24931	0.624656	−	0.780900i	$-0.285239\pi$
$587$	17767.6	1.24931	0.624656	+	0.780900i	$0.285239\pi$
$588$	0	0
$589$	−1787.17	−0.125024
$590$	0	0
$591$	−4146.71	−0.288617
$592$	0	0
$593$	−321.358	−0.0222539	−0.0111270	−	0.999938i	$-0.503542\pi$
$593$	−321.358	−0.0222539	−0.0111270	+	0.999938i	$0.503542\pi$
$594$	0	0
$595$	−9241.22	−0.636728
$596$	0	0
$597$	23043.2	1.57972
$598$	0	0
$599$	25291.8	1.72520	0.862600	−	0.505887i	$-0.168835\pi$
$599$	25291.8	1.72520	0.862600	+	0.505887i	$0.168835\pi$
$600$	0	0
$601$	−24813.2	−1.68412	−0.842058	−	0.539387i	$-0.818656\pi$
$601$	−24813.2	−1.68412	−0.842058	+	0.539387i	$0.818656\pi$
$602$	0	0
$603$	−31.7834	−0.00214647
$604$	0	0
$605$	−43396.0	−2.91620
$606$	0	0
$607$	20853.2	1.39441	0.697203	−	0.716874i	$-0.254428\pi$
$607$	20853.2	1.39441	0.697203	+	0.716874i	$0.254428\pi$
$608$	0	0
$609$	8636.97	0.574692
$610$	0	0
$611$	34512.4	2.28514
$612$	0	0
$613$	−21342.3	−1.40621	−0.703105	−	0.711086i	$-0.748203\pi$
$613$	−21342.3	−1.40621	−0.703105	+	0.711086i	$0.748203\pi$
$614$	0	0
$615$	−7556.74	−0.495475
$616$	0	0
$617$	−14363.6	−0.937208	−0.468604	−	0.883408i	$-0.655243\pi$
$617$	−14363.6	−0.937208	−0.468604	+	0.883408i	$0.655243\pi$
$618$	0	0
$619$	19126.3	1.24192	0.620961	−	0.783842i	$-0.286743\pi$
$619$	19126.3	1.24192	0.620961	+	0.783842i	$0.286743\pi$
$620$	0	0
$621$	−19249.6	−1.24389
$622$	0	0
$623$	3530.85	0.227064
$624$	0	0
$625$	43849.8	2.80638
$626$	0	0
$627$	6202.54	0.395065
$628$	0	0
$629$	−9246.83	−0.586161
$630$	0	0
$631$	10575.0	0.667168	0.333584	−	0.942720i	$-0.391742\pi$
$631$	10575.0	0.667168	0.333584	+	0.942720i	$0.391742\pi$
$632$	0	0
$633$	−5183.26	−0.325460
$634$	0	0
$635$	−10146.3	−0.634083
$636$	0	0
$637$	−9733.74	−0.605439
$638$	0	0
$639$	−4449.23	−0.275444
$640$	0	0
$641$	−3470.84	−0.213869	−0.106934	−	0.994266i	$-0.534103\pi$
$641$	−3470.84	−0.213869	−0.106934	+	0.994266i	$0.534103\pi$
$642$	0	0
$643$	−20693.0	−1.26913	−0.634566	−	0.772868i	$-0.718821\pi$
$643$	−20693.0	−1.26913	−0.634566	+	0.772868i	$0.718821\pi$
$644$	0	0
$645$	−52686.6	−3.21633
$646$	0	0
$647$	−23520.2	−1.42917	−0.714587	−	0.699547i	$-0.753385\pi$
$647$	−23520.2	−1.42917	−0.714587	+	0.699547i	$0.753385\pi$
$648$	0	0
$649$	14282.6	0.863853
$650$	0	0
$651$	7322.66	0.440857
$652$	0	0
$653$	11374.6	0.681660	0.340830	−	0.940125i	$-0.389292\pi$
$653$	11374.6	0.681660	0.340830	+	0.940125i	$0.389292\pi$
$654$	0	0
$655$	−51571.2	−3.07642
$656$	0	0
$657$	2566.71	0.152415
$658$	0	0
$659$	191.638	0.0113280	0.00566399	−	0.999984i	$-0.498197\pi$
$659$	191.638	0.0113280	0.00566399	+	0.999984i	$0.498197\pi$
$660$	0	0
$661$	−18403.2	−1.08291	−0.541455	−	0.840730i	$-0.682126\pi$
$661$	−18403.2	−1.08291	−0.541455	+	0.840730i	$0.682126\pi$
$662$	0	0
$663$	11549.2	0.676520
$664$	0	0
$665$	5538.52	0.322969
$666$	0	0
$667$	16830.1	0.977005
$668$	0	0
$669$	−8889.03	−0.513706
$670$	0	0
$671$	−46727.3	−2.68835
$672$	0	0
$673$	−31346.8	−1.79544	−0.897719	−	0.440569i	$-0.854777\pi$
$673$	−31346.8	−1.79544	−0.897719	+	0.440569i	$0.854777\pi$
$674$	0	0
$675$	−39877.0	−2.27388
$676$	0	0
$677$	3187.80	0.180971	0.0904853	−	0.995898i	$-0.471158\pi$
$677$	3187.80	0.180971	0.0904853	+	0.995898i	$0.471158\pi$
$678$	0	0
$679$	7110.49	0.401878
$680$	0	0
$681$	−9673.33	−0.544321
$682$	0	0
$683$	5358.92	0.300225	0.150112	−	0.988669i	$-0.452036\pi$
$683$	5358.92	0.300225	0.150112	+	0.988669i	$0.452036\pi$
$684$	0	0
$685$	39249.8	2.18928
$686$	0	0
$687$	−20423.9	−1.13424
$688$	0	0
$689$	40430.9	2.23555
$690$	0	0
$691$	−12057.1	−0.663784	−0.331892	−	0.943317i	$-0.607687\pi$
$691$	−12057.1	−0.663784	−0.331892	+	0.943317i	$0.607687\pi$
$692$	0	0
$693$	−3511.96	−0.192509
$694$	0	0
$695$	−3527.71	−0.192537
$696$	0	0
$697$	2042.15	0.110979
$698$	0	0
$699$	−4101.54	−0.221938
$700$	0	0
$701$	21103.5	1.13704	0.568522	−	0.822668i	$-0.307515\pi$
$701$	21103.5	1.13704	0.568522	+	0.822668i	$0.307515\pi$
$702$	0	0
$703$	5541.88	0.297320
$704$	0	0
$705$	−62204.8	−3.32308
$706$	0	0
$707$	−9004.85	−0.479013
$708$	0	0
$709$	5643.93	0.298959	0.149480	−	0.988765i	$-0.452240\pi$
$709$	5643.93	0.298959	0.149480	+	0.988765i	$0.452240\pi$
$710$	0	0
$711$	521.609	0.0275132
$712$	0	0
$713$	14269.0	0.749478
$714$	0	0
$715$	79558.2	4.16127
$716$	0	0
$717$	22406.7	1.16707
$718$	0	0
$719$	8547.76	0.443363	0.221681	−	0.975119i	$-0.428846\pi$
$719$	8547.76	0.443363	0.221681	+	0.975119i	$0.428846\pi$
$720$	0	0
$721$	11482.0	0.593084
$722$	0	0
$723$	27689.8	1.42434
$724$	0	0
$725$	34864.8	1.78599
$726$	0	0
$727$	−15122.0	−0.771451	−0.385726	−	0.922614i	$-0.626049\pi$
$727$	−15122.0	−0.771451	−0.385726	+	0.922614i	$0.626049\pi$
$728$	0	0
$729$	15595.9	0.792351
$730$	0	0
$731$	14238.2	0.720408
$732$	0	0
$733$	34376.8	1.73225	0.866123	−	0.499832i	$-0.166605\pi$
$733$	34376.8	1.73225	0.866123	+	0.499832i	$0.166605\pi$
$734$	0	0
$735$	17544.0	0.880436
$736$	0	0
$737$	428.167	0.0213999
$738$	0	0
$739$	−461.978	−0.0229961	−0.0114981	−	0.999934i	$-0.503660\pi$
$739$	−461.978	−0.0229961	−0.0114981	+	0.999934i	$0.503660\pi$
$740$	0	0
$741$	−6921.74	−0.343153
$742$	0	0
$743$	−12671.9	−0.625690	−0.312845	−	0.949804i	$-0.601282\pi$
$743$	−12671.9	−0.625690	−0.312845	+	0.949804i	$0.601282\pi$
$744$	0	0
$745$	15083.4	0.741763
$746$	0	0
$747$	−2176.49	−0.106604
$748$	0	0
$749$	3402.73	0.165999
$750$	0	0
$751$	−14780.8	−0.718189	−0.359095	−	0.933301i	$-0.616914\pi$
$751$	−14780.8	−0.718189	−0.359095	+	0.933301i	$0.616914\pi$
$752$	0	0
$753$	−20672.2	−1.00045
$754$	0	0
$755$	−73092.4	−3.52332
$756$	0	0
$757$	−37080.6	−1.78034	−0.890169	−	0.455630i	$-0.849414\pi$
$757$	−37080.6	−1.78034	−0.890169	+	0.455630i	$0.849414\pi$
$758$	0	0
$759$	−49522.0	−2.36829
$760$	0	0
$761$	28305.3	1.34831	0.674157	−	0.738588i	$-0.264507\pi$
$761$	28305.3	1.34831	0.674157	+	0.738588i	$0.264507\pi$
$762$	0	0
$763$	−14069.9	−0.667583
$764$	0	0
$765$	−2876.58	−0.135951
$766$	0	0
$767$	−15938.7	−0.750342
$768$	0	0
$769$	15549.6	0.729172	0.364586	−	0.931170i	$-0.381211\pi$
$769$	15549.6	0.729172	0.364586	+	0.931170i	$0.381211\pi$
$770$	0	0
$771$	−33233.3	−1.55236
$772$	0	0
$773$	32107.4	1.49395	0.746975	−	0.664852i	$-0.231505\pi$
$773$	32107.4	1.49395	0.746975	+	0.664852i	$0.231505\pi$
$774$	0	0
$775$	29559.3	1.37007
$776$	0	0
$777$	−22707.1	−1.04841
$778$	0	0
$779$	−1223.92	−0.0562920
$780$	0	0
$781$	59937.2	2.74612
$782$	0	0
$783$	−14078.1	−0.642540
$784$	0	0
$785$	8730.16	0.396934
$786$	0	0
$787$	12075.6	0.546950	0.273475	−	0.961879i	$-0.411827\pi$
$787$	12075.6	0.546950	0.273475	+	0.961879i	$0.411827\pi$
$788$	0	0
$789$	2712.27	0.122382
$790$	0	0
$791$	976.855	0.0439102
$792$	0	0
$793$	52145.4	2.33510
$794$	0	0
$795$	−72872.4	−3.25096
$796$	0	0
$797$	1271.50	0.0565107	0.0282553	−	0.999601i	$-0.491005\pi$
$797$	1271.50	0.0565107	0.0282553	+	0.999601i	$0.491005\pi$
$798$	0	0
$799$	16810.4	0.744318
$800$	0	0
$801$	1099.07	0.0484817
$802$	0	0
$803$	−34577.1	−1.51955
$804$	0	0
$805$	−44220.4	−1.93610
$806$	0	0
$807$	26259.7	1.14546
$808$	0	0
$809$	−1837.55	−0.0798575	−0.0399288	−	0.999203i	$-0.512713\pi$
$809$	−1837.55	−0.0798575	−0.0399288	+	0.999203i	$0.512713\pi$
$810$	0	0
$811$	15642.0	0.677270	0.338635	−	0.940918i	$-0.390035\pi$
$811$	15642.0	0.677270	0.338635	+	0.940918i	$0.390035\pi$
$812$	0	0
$813$	21013.5	0.906488
$814$	0	0
$815$	−62860.9	−2.70174
$816$	0	0
$817$	−8533.34	−0.365415
$818$	0	0
$819$	3919.18	0.167213
$820$	0	0
$821$	31083.1	1.32133	0.660663	−	0.750683i	$-0.270275\pi$
$821$	31083.1	1.32133	0.660663	+	0.750683i	$0.270275\pi$
$822$	0	0
$823$	−14774.0	−0.625747	−0.312874	−	0.949795i	$-0.601292\pi$
$823$	−14774.0	−0.625747	−0.312874	+	0.949795i	$0.601292\pi$
$824$	0	0
$825$	−102589.	−4.32931
$826$	0	0
$827$	−5622.05	−0.236394	−0.118197	−	0.992990i	$-0.537711\pi$
$827$	−5622.05	−0.236394	−0.118197	+	0.992990i	$0.537711\pi$
$828$	0	0
$829$	−5438.89	−0.227865	−0.113933	−	0.993488i	$-0.536345\pi$
$829$	−5438.89	−0.227865	−0.113933	+	0.993488i	$0.536345\pi$
$830$	0	0
$831$	−13612.9	−0.568263
$832$	0	0
$833$	−4741.14	−0.197204
$834$	0	0
$835$	47782.1	1.98032
$836$	0	0
$837$	−11935.8	−0.492904
$838$	0	0
$839$	13224.4	0.544166	0.272083	−	0.962274i	$-0.412287\pi$
$839$	13224.4	0.544166	0.272083	+	0.962274i	$0.412287\pi$
$840$	0	0
$841$	−12080.4	−0.495323
$842$	0	0
$843$	33539.1	1.37028
$844$	0	0
$845$	−42737.5	−1.73990
$846$	0	0
$847$	28798.7	1.16828
$848$	0	0
$849$	−1014.15	−0.0409958
$850$	0	0
$851$	−44247.2	−1.78234
$852$	0	0
$853$	26559.3	1.06609	0.533045	−	0.846087i	$-0.321048\pi$
$853$	26559.3	1.06609	0.533045	+	0.846087i	$0.321048\pi$
$854$	0	0
$855$	1724.01	0.0689591
$856$	0	0
$857$	−23397.6	−0.932611	−0.466306	−	0.884624i	$-0.654415\pi$
$857$	−23397.6	−0.932611	−0.466306	+	0.884624i	$0.654415\pi$
$858$	0	0
$859$	37137.6	1.47511	0.737554	−	0.675288i	$-0.235981\pi$
$859$	37137.6	1.47511	0.737554	+	0.675288i	$0.235981\pi$
$860$	0	0
$861$	5014.84	0.198496
$862$	0	0
$863$	33399.2	1.31741	0.658703	−	0.752403i	$-0.271105\pi$
$863$	33399.2	1.31741	0.658703	+	0.752403i	$0.271105\pi$
$864$	0	0
$865$	−14891.7	−0.585358
$866$	0	0
$867$	−21874.0	−0.856838
$868$	0	0
$869$	−7026.79	−0.274301
$870$	0	0
$871$	−477.814	−0.0185879
$872$	0	0
$873$	2213.33	0.0858074
$874$	0	0
$875$	−55168.3	−2.13146
$876$	0	0
$877$	596.730	0.0229762	0.0114881	−	0.999934i	$-0.496343\pi$
$877$	596.730	0.0229762	0.0114881	+	0.999934i	$0.496343\pi$
$878$	0	0
$879$	−10685.3	−0.410017
$880$	0	0
$881$	−11250.9	−0.430253	−0.215126	−	0.976586i	$-0.569016\pi$
$881$	−11250.9	−0.430253	−0.215126	+	0.976586i	$0.569016\pi$
$882$	0	0
$883$	10586.0	0.403449	0.201725	−	0.979442i	$-0.435345\pi$
$883$	10586.0	0.403449	0.201725	+	0.979442i	$0.435345\pi$
$884$	0	0
$885$	28727.7	1.09116
$886$	0	0
$887$	15954.3	0.603939	0.301970	−	0.953318i	$-0.402356\pi$
$887$	15954.3	0.603939	0.301970	+	0.953318i	$0.402356\pi$
$888$	0	0
$889$	6733.33	0.254025
$890$	0	0
$891$	48241.9	1.81388
$892$	0	0
$893$	−10074.9	−0.377542
$894$	0	0
$895$	38366.7	1.43291
$896$	0	0
$897$	55264.2	2.05710
$898$	0	0
$899$	10435.5	0.387147
$900$	0	0
$901$	19693.2	0.728165
$902$	0	0
$903$	34964.2	1.28852
$904$	0	0
$905$	4723.66	0.173502
$906$	0	0
$907$	45468.5	1.66456	0.832280	−	0.554356i	$-0.187035\pi$
$907$	45468.5	1.66456	0.832280	+	0.554356i	$0.187035\pi$
$908$	0	0
$909$	−2803.00	−0.102277
$910$	0	0
$911$	−8802.26	−0.320123	−0.160061	−	0.987107i	$-0.551169\pi$
$911$	−8802.26	−0.320123	−0.160061	+	0.987107i	$0.551169\pi$
$912$	0	0
$913$	29320.3	1.06282
$914$	0	0
$915$	−93986.4	−3.39573
$916$	0	0
$917$	34223.9	1.23247
$918$	0	0
$919$	−28718.3	−1.03083	−0.515414	−	0.856941i	$-0.672362\pi$
$919$	−28718.3	−1.03083	−0.515414	+	0.856941i	$0.672362\pi$
$920$	0	0
$921$	−21776.0	−0.779093
$922$	0	0
$923$	−66887.1	−2.38528
$924$	0	0
$925$	−91661.5	−3.25817
$926$	0	0
$927$	3574.09	0.126633
$928$	0	0
$929$	−39530.2	−1.39607	−0.698033	−	0.716066i	$-0.745941\pi$
$929$	−39530.2	−1.39607	−0.698033	+	0.716066i	$0.745941\pi$
$930$	0	0
$931$	2841.50	0.100028
$932$	0	0
$933$	−5479.39	−0.192269
$934$	0	0
$935$	38751.5	1.35541
$936$	0	0
$937$	−17009.2	−0.593028	−0.296514	−	0.955028i	$-0.595824\pi$
$937$	−17009.2	−0.593028	−0.296514	+	0.955028i	$0.595824\pi$
$938$	0	0
$939$	5873.52	0.204127
$940$	0	0
$941$	−27153.7	−0.940684	−0.470342	−	0.882484i	$-0.655870\pi$
$941$	−27153.7	−0.940684	−0.470342	+	0.882484i	$0.655870\pi$
$942$	0	0
$943$	9771.96	0.337454
$944$	0	0
$945$	36989.6	1.27330
$946$	0	0
$947$	−31678.3	−1.08702	−0.543510	−	0.839403i	$-0.682905\pi$
$947$	−31678.3	−1.08702	−0.543510	+	0.839403i	$0.682905\pi$
$948$	0	0
$949$	38586.4	1.31988
$950$	0	0
$951$	−27214.0	−0.927942
$952$	0	0
$953$	−39352.1	−1.33761	−0.668803	−	0.743440i	$-0.733193\pi$
$953$	−39352.1	−1.33761	−0.668803	+	0.743440i	$0.733193\pi$
$954$	0	0
$955$	−24729.8	−0.837946
$956$	0	0
$957$	−36217.6	−1.22335
$958$	0	0
$959$	−26047.1	−0.877065
$960$	0	0
$961$	−20943.5	−0.703013
$962$	0	0
$963$	1059.19	0.0354433
$964$	0	0
$965$	21117.9	0.704465
$966$	0	0
$967$	−4101.55	−0.136398	−0.0681991	−	0.997672i	$-0.521725\pi$
$967$	−4101.55	−0.136398	−0.0681991	+	0.997672i	$0.521725\pi$
$968$	0	0
$969$	−3371.47	−0.111772
$970$	0	0
$971$	−39275.2	−1.29804	−0.649022	−	0.760770i	$-0.724822\pi$
$971$	−39275.2	−1.29804	−0.649022	+	0.760770i	$0.724822\pi$
$972$	0	0
$973$	2341.07	0.0771340
$974$	0	0
$975$	114484.	3.76044
$976$	0	0
$977$	−1259.70	−0.0412501	−0.0206251	−	0.999787i	$-0.506566\pi$
$977$	−1259.70	−0.0412501	−0.0206251	+	0.999787i	$0.506566\pi$
$978$	0	0
$979$	−14806.0	−0.483353
$980$	0	0
$981$	−4379.65	−0.142540
$982$	0	0
$983$	45455.1	1.47487	0.737433	−	0.675420i	$-0.236038\pi$
$983$	45455.1	1.47487	0.737433	+	0.675420i	$0.236038\pi$
$984$	0	0
$985$	15526.9	0.502264
$986$	0	0
$987$	41280.7	1.33129
$988$	0	0
$989$	68131.4	2.19055
$990$	0	0
$991$	22713.1	0.728058	0.364029	−	0.931388i	$-0.381401\pi$
$991$	22713.1	0.728058	0.364029	+	0.931388i	$0.381401\pi$
$992$	0	0
$993$	38968.4	1.24534
$994$	0	0
$995$	−86282.9	−2.74910
$996$	0	0
$997$	−19731.0	−0.626767	−0.313384	−	0.949627i	$-0.601463\pi$
$997$	−19731.0	−0.626767	−0.313384	+	0.949627i	$0.601463\pi$
$998$	0	0
$999$	37012.0	1.17218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.4.a.k.1.5	yes	7
4.3	odd	2		608.4.a.j.1.3	✓	7
8.3	odd	2		1216.4.a.bg.1.5		7
8.5	even	2		1216.4.a.bf.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.j.1.3	✓	7	4.3	odd	2
608.4.a.k.1.5	yes	7	1.1	even	1	trivial
1216.4.a.bf.1.3		7	8.5	even	2
1216.4.a.bg.1.5		7	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	608.a (trivial)

\(f(q)\)	\(=\)	\(q+5.59727 q^{3} -20.9584 q^{5} +13.9085 q^{7} +4.32941 q^{9} -58.3230 q^{11} +65.0857 q^{13} -117.310 q^{15} +31.7022 q^{17} -19.0000 q^{19} +77.8498 q^{21} +151.699 q^{23} +314.256 q^{25} -126.893 q^{27} +110.944 q^{29} +94.0614 q^{31} -326.450 q^{33} -291.501 q^{35} -291.678 q^{37} +364.302 q^{39} +64.4169 q^{41} +449.123 q^{43} -90.7375 q^{45} +530.261 q^{47} -149.553 q^{49} +177.446 q^{51} +621.195 q^{53} +1222.36 q^{55} -106.348 q^{57} -244.887 q^{59} +801.180 q^{61} +60.2157 q^{63} -1364.10 q^{65} -7.34129 q^{67} +849.098 q^{69} -1027.68 q^{71} +592.854 q^{73} +1758.97 q^{75} -811.188 q^{77} +120.481 q^{79} -827.150 q^{81} -502.722 q^{83} -664.428 q^{85} +620.983 q^{87} +253.862 q^{89} +905.247 q^{91} +526.487 q^{93} +398.210 q^{95} +511.232 q^{97} -252.504 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 17 q^{5} + 42 q^{7} + 86 q^{9} - 33 q^{11} - 35 q^{13} + 120 q^{15} + 66 q^{17} - 133 q^{19} + 33 q^{21} + 389 q^{23} + 44 q^{25} - 39 q^{27} + 233 q^{29} + 158 q^{31} - 206 q^{33} + 123 q^{35}+ \cdots - 1591 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 - 17 * q^5 + 42 * q^7 + 86 * q^9 - 33 * q^11 - 35 * q^13 + 120 * q^15 + 66 * q^17 - 133 * q^19 + 33 * q^21 + 389 * q^23 + 44 * q^25 - 39 * q^27 + 233 * q^29 + 158 * q^31 - 206 * q^33 + 123 * q^35 - 436 * q^37 + 807 * q^39 - 94 * q^41 + 645 * q^43 + 103 * q^45 + 1451 * q^47 + 93 * q^49 + 1741 * q^51 + 3 * q^53 + 1971 * q^55 - 57 * q^57 + 297 * q^59 + 93 * q^61 + 2999 * q^63 - 788 * q^65 + 1641 * q^67 + 945 * q^69 + 2392 * q^71 + 324 * q^73 + 1909 * q^75 - 711 * q^77 + 2492 * q^79 + 143 * q^81 + 310 * q^83 + 2353 * q^85 + 4795 * q^87 - 440 * q^89 - 107 * q^91 + 900 * q^93 + 323 * q^95 - 532 * q^97 - 1591 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

Properties

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