LMFDB - Embedded newform 476.4.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [476,4,Mod(1,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, 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("476.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 476 = 2^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	476.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:	$28.0849091627$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 87x^{4} + 184x^{3} + 2031x^{2} - 4232x - 7516 $ x^6 - 2x^5 - 87x^4 + 184x^3 + 2031x^2 - 4232*x - 7516
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$5.08046$ of defining polynomial
Character	$\chi$	$=$	476.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.08046 q^{3} +5.06198 q^{5} -7.00000 q^{7} -1.18888 q^{9} +O(q^{10})$	$q+5.08046 q^{3} +5.06198 q^{5} -7.00000 q^{7} -1.18888 q^{9} -22.7042 q^{11} -77.2366 q^{13} +25.7172 q^{15} -17.0000 q^{17} -43.9551 q^{19} -35.5632 q^{21} -69.9126 q^{23} -99.3763 q^{25} -143.213 q^{27} -122.651 q^{29} +247.920 q^{31} -115.348 q^{33} -35.4339 q^{35} +246.229 q^{37} -392.398 q^{39} +194.092 q^{41} -149.773 q^{43} -6.01811 q^{45} +50.6882 q^{47} +49.0000 q^{49} -86.3679 q^{51} -112.398 q^{53} -114.928 q^{55} -223.312 q^{57} -495.317 q^{59} -507.437 q^{61} +8.32219 q^{63} -390.970 q^{65} +993.485 q^{67} -355.188 q^{69} +717.227 q^{71} +1098.36 q^{73} -504.878 q^{75} +158.929 q^{77} -269.441 q^{79} -695.487 q^{81} -1490.94 q^{83} -86.0537 q^{85} -623.126 q^{87} +1121.49 q^{89} +540.656 q^{91} +1259.55 q^{93} -222.500 q^{95} -938.210 q^{97} +26.9927 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 10 q^{5} - 42 q^{7} + 16 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 + 10 * q^5 - 42 * q^7 + 16 * q^9	$ 6 q + 2 q^{3} + 10 q^{5} - 42 q^{7} + 16 q^{9} - 20 q^{11} + 42 q^{13} - 136 q^{15} - 102 q^{17} - 104 q^{19} - 14 q^{21} - 230 q^{23} + 108 q^{25} - 130 q^{27} - 52 q^{29} - 564 q^{31} - 346 q^{33} - 70 q^{35} - 564 q^{37} - 626 q^{39} - 548 q^{41} - 648 q^{43} - 174 q^{45} - 366 q^{47} + 294 q^{49} - 34 q^{51} - 74 q^{53} - 1460 q^{55} - 316 q^{57} - 558 q^{59} - 620 q^{61} - 112 q^{63} - 1378 q^{65} - 164 q^{67} - 540 q^{69} - 822 q^{71} + 940 q^{73} - 2698 q^{75} + 140 q^{77} - 1838 q^{79} - 3094 q^{81} - 1118 q^{83} - 170 q^{85} - 1354 q^{87} - 1634 q^{89} - 294 q^{91} + 268 q^{93} - 2642 q^{95} + 182 q^{97} + 12 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 + 10 * q^5 - 42 * q^7 + 16 * q^9 - 20 * q^11 + 42 * q^13 - 136 * q^15 - 102 * q^17 - 104 * q^19 - 14 * q^21 - 230 * q^23 + 108 * q^25 - 130 * q^27 - 52 * q^29 - 564 * q^31 - 346 * q^33 - 70 * q^35 - 564 * q^37 - 626 * q^39 - 548 * q^41 - 648 * q^43 - 174 * q^45 - 366 * q^47 + 294 * q^49 - 34 * q^51 - 74 * q^53 - 1460 * q^55 - 316 * q^57 - 558 * q^59 - 620 * q^61 - 112 * q^63 - 1378 * q^65 - 164 * q^67 - 540 * q^69 - 822 * q^71 + 940 * q^73 - 2698 * q^75 + 140 * q^77 - 1838 * q^79 - 3094 * q^81 - 1118 * q^83 - 170 * q^85 - 1354 * q^87 - 1634 * q^89 - 294 * q^91 + 268 * q^93 - 2642 * q^95 + 182 * q^97 + 12 * 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
$3$	5.08046	0.977736	0.488868	−	0.872358i	$-0.337410\pi$
$3$	5.08046	0.977736	0.488868	+	0.872358i	$0.337410\pi$
$4$	0	0
$5$	5.06198	0.452757	0.226379	−	0.974039i	$-0.427311\pi$
$5$	5.06198	0.452757	0.226379	+	0.974039i	$0.427311\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−1.18888	−0.0440328
$10$	0	0
$11$	−22.7042	−0.622325	−0.311162	−	0.950357i	$-0.600718\pi$
$11$	−22.7042	−0.622325	−0.311162	+	0.950357i	$0.600718\pi$
$12$	0	0
$13$	−77.2366	−1.64781	−0.823907	−	0.566725i	$-0.808210\pi$
$13$	−77.2366	−1.64781	−0.823907	+	0.566725i	$0.808210\pi$
$14$	0	0
$15$	25.7172	0.442677
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−43.9551	−0.530736	−0.265368	−	0.964147i	$-0.585493\pi$
$19$	−43.9551	−0.530736	−0.265368	+	0.964147i	$0.585493\pi$
$20$	0	0
$21$	−35.5632	−0.369549
$22$	0	0
$23$	−69.9126	−0.633817	−0.316908	−	0.948456i	$-0.602645\pi$
$23$	−69.9126	−0.633817	−0.316908	+	0.948456i	$0.602645\pi$
$24$	0	0
$25$	−99.3763	−0.795011
$26$	0	0
$27$	−143.213	−1.02079
$28$	0	0
$29$	−122.651	−0.785372	−0.392686	−	0.919673i	$-0.628454\pi$
$29$	−122.651	−0.785372	−0.392686	+	0.919673i	$0.628454\pi$
$30$	0	0
$31$	247.920	1.43638	0.718189	−	0.695848i	$-0.244971\pi$
$31$	247.920	1.43638	0.718189	+	0.695848i	$0.244971\pi$
$32$	0	0
$33$	−115.348	−0.608469
$34$	0	0
$35$	−35.4339	−0.171126
$36$	0	0
$37$	246.229	1.09405	0.547025	−	0.837116i	$-0.315760\pi$
$37$	246.229	1.09405	0.547025	+	0.837116i	$0.315760\pi$
$38$	0	0
$39$	−392.398	−1.61113
$40$	0	0
$41$	194.092	0.739318	0.369659	−	0.929167i	$-0.379474\pi$
$41$	194.092	0.739318	0.369659	+	0.929167i	$0.379474\pi$
$42$	0	0
$43$	−149.773	−0.531167	−0.265584	−	0.964088i	$-0.585565\pi$
$43$	−149.773	−0.531167	−0.265584	+	0.964088i	$0.585565\pi$
$44$	0	0
$45$	−6.01811	−0.0199362
$46$	0	0
$47$	50.6882	0.157311	0.0786557	−	0.996902i	$-0.474937\pi$
$47$	50.6882	0.157311	0.0786557	+	0.996902i	$0.474937\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−86.3679	−0.237136
$52$	0	0
$53$	−112.398	−0.291303	−0.145651	−	0.989336i	$-0.546528\pi$
$53$	−112.398	−0.291303	−0.145651	+	0.989336i	$0.546528\pi$
$54$	0	0
$55$	−114.928	−0.281762
$56$	0	0
$57$	−223.312	−0.518920
$58$	0	0
$59$	−495.317	−1.09296	−0.546482	−	0.837471i	$-0.684033\pi$
$59$	−495.317	−1.09296	−0.546482	+	0.837471i	$0.684033\pi$
$60$	0	0
$61$	−507.437	−1.06509	−0.532546	−	0.846401i	$-0.678765\pi$
$61$	−507.437	−1.06509	−0.532546	+	0.846401i	$0.678765\pi$
$62$	0	0
$63$	8.32219	0.0166428
$64$	0	0
$65$	−390.970	−0.746060
$66$	0	0
$67$	993.485	1.81154	0.905772	−	0.423765i	$-0.139292\pi$
$67$	993.485	1.81154	0.905772	+	0.423765i	$0.139292\pi$
$68$	0	0
$69$	−355.188	−0.619705
$70$	0	0
$71$	717.227	1.19886	0.599431	−	0.800427i	$-0.295394\pi$
$71$	717.227	1.19886	0.599431	+	0.800427i	$0.295394\pi$
$72$	0	0
$73$	1098.36	1.76101	0.880506	−	0.474035i	$-0.157203\pi$
$73$	1098.36	1.76101	0.880506	+	0.474035i	$0.157203\pi$
$74$	0	0
$75$	−504.878	−0.777310
$76$	0	0
$77$	158.929	0.235217
$78$	0	0
$79$	−269.441	−0.383727	−0.191864	−	0.981422i	$-0.561453\pi$
$79$	−269.441	−0.383727	−0.191864	+	0.981422i	$0.561453\pi$
$80$	0	0
$81$	−695.487	−0.954028
$82$	0	0
$83$	−1490.94	−1.97171	−0.985855	−	0.167601i	$-0.946398\pi$
$83$	−1490.94	−1.97171	−0.985855	+	0.167601i	$0.946398\pi$
$84$	0	0
$85$	−86.0537	−0.109810
$86$	0	0
$87$	−623.126	−0.767886
$88$	0	0
$89$	1121.49	1.33570	0.667851	−	0.744295i	$-0.267215\pi$
$89$	1121.49	1.33570	0.667851	+	0.744295i	$0.267215\pi$
$90$	0	0
$91$	540.656	0.622815
$92$	0	0
$93$	1259.55	1.40440
$94$	0	0
$95$	−222.500	−0.240295
$96$	0	0
$97$	−938.210	−0.982070	−0.491035	−	0.871140i	$-0.663381\pi$
$97$	−938.210	−0.982070	−0.491035	+	0.871140i	$0.663381\pi$
$98$	0	0
$99$	26.9927	0.0274027
$100$	0	0
$101$	−1883.96	−1.85605	−0.928024	−	0.372520i	$-0.878494\pi$
$101$	−1883.96	−1.85605	−0.928024	+	0.372520i	$0.878494\pi$
$102$	0	0
$103$	−900.180	−0.861140	−0.430570	−	0.902557i	$-0.641687\pi$
$103$	−900.180	−0.861140	−0.430570	+	0.902557i	$0.641687\pi$
$104$	0	0
$105$	−180.021	−0.167316
$106$	0	0
$107$	690.617	0.623967	0.311983	−	0.950088i	$-0.399007\pi$
$107$	690.617	0.623967	0.311983	+	0.950088i	$0.399007\pi$
$108$	0	0
$109$	−1931.09	−1.69693	−0.848463	−	0.529255i	$-0.822472\pi$
$109$	−1931.09	−1.69693	−0.848463	+	0.529255i	$0.822472\pi$
$110$	0	0
$111$	1250.96	1.06969
$112$	0	0
$113$	453.310	0.377379	0.188689	−	0.982037i	$-0.439576\pi$
$113$	453.310	0.377379	0.188689	+	0.982037i	$0.439576\pi$
$114$	0	0
$115$	−353.896	−0.286965
$116$	0	0
$117$	91.8254	0.0725578
$118$	0	0
$119$	119.000	0.0916698
$120$	0	0
$121$	−815.519	−0.612712
$122$	0	0
$123$	986.076	0.722858
$124$	0	0
$125$	−1135.79	−0.812704
$126$	0	0
$127$	−972.878	−0.679755	−0.339878	−	0.940470i	$-0.610386\pi$
$127$	−972.878	−0.679755	−0.339878	+	0.940470i	$0.610386\pi$
$128$	0	0
$129$	−760.917	−0.519341
$130$	0	0
$131$	−513.598	−0.342544	−0.171272	−	0.985224i	$-0.554788\pi$
$131$	−513.598	−0.342544	−0.171272	+	0.985224i	$0.554788\pi$
$132$	0	0
$133$	307.685	0.200599
$134$	0	0
$135$	−724.940	−0.462169
$136$	0	0
$137$	816.228	0.509015	0.254508	−	0.967071i	$-0.418087\pi$
$137$	816.228	0.509015	0.254508	+	0.967071i	$0.418087\pi$
$138$	0	0
$139$	1551.97	0.947024	0.473512	−	0.880787i	$-0.342986\pi$
$139$	1551.97	0.947024	0.473512	+	0.880787i	$0.342986\pi$
$140$	0	0
$141$	257.520	0.153809
$142$	0	0
$143$	1753.59	1.02548
$144$	0	0
$145$	−620.859	−0.355583
$146$	0	0
$147$	248.943	0.139677
$148$	0	0
$149$	2363.17	1.29932	0.649659	−	0.760226i	$-0.274912\pi$
$149$	2363.17	1.29932	0.649659	+	0.760226i	$0.274912\pi$
$150$	0	0
$151$	−839.184	−0.452264	−0.226132	−	0.974097i	$-0.572608\pi$
$151$	−839.184	−0.452264	−0.226132	+	0.974097i	$0.572608\pi$
$152$	0	0
$153$	20.2110	0.0106795
$154$	0	0
$155$	1254.97	0.650331
$156$	0	0
$157$	−980.766	−0.498558	−0.249279	−	0.968432i	$-0.580194\pi$
$157$	−980.766	−0.498558	−0.249279	+	0.968432i	$0.580194\pi$
$158$	0	0
$159$	−571.033	−0.284817
$160$	0	0
$161$	489.388	0.239560
$162$	0	0
$163$	1699.92	0.816858	0.408429	−	0.912790i	$-0.366077\pi$
$163$	1699.92	0.816858	0.408429	+	0.912790i	$0.366077\pi$
$164$	0	0
$165$	−583.889	−0.275489
$166$	0	0
$167$	−2047.19	−0.948600	−0.474300	−	0.880363i	$-0.657299\pi$
$167$	−2047.19	−0.948600	−0.474300	+	0.880363i	$0.657299\pi$
$168$	0	0
$169$	3768.49	1.71529
$170$	0	0
$171$	52.2575	0.0233698
$172$	0	0
$173$	894.235	0.392991	0.196495	−	0.980505i	$-0.437044\pi$
$173$	894.235	0.392991	0.196495	+	0.980505i	$0.437044\pi$
$174$	0	0
$175$	695.634	0.300486
$176$	0	0
$177$	−2516.44	−1.06863
$178$	0	0
$179$	−3524.38	−1.47164	−0.735822	−	0.677175i	$-0.763204\pi$
$179$	−3524.38	−1.47164	−0.735822	+	0.677175i	$0.763204\pi$
$180$	0	0
$181$	3042.82	1.24956	0.624782	−	0.780800i	$-0.285188\pi$
$181$	3042.82	1.24956	0.624782	+	0.780800i	$0.285188\pi$
$182$	0	0
$183$	−2578.01	−1.04138
$184$	0	0
$185$	1246.41	0.495339
$186$	0	0
$187$	385.971	0.150936
$188$	0	0
$189$	1002.49	0.385822
$190$	0	0
$191$	−2601.05	−0.985369	−0.492685	−	0.870208i	$-0.663984\pi$
$191$	−2601.05	−0.985369	−0.492685	+	0.870208i	$0.663984\pi$
$192$	0	0
$193$	−2284.66	−0.852089	−0.426045	−	0.904702i	$-0.640093\pi$
$193$	−2284.66	−0.852089	−0.426045	+	0.904702i	$0.640093\pi$
$194$	0	0
$195$	−1986.31	−0.729449
$196$	0	0
$197$	4435.90	1.60429	0.802143	−	0.597131i	$-0.203693\pi$
$197$	4435.90	1.60429	0.802143	+	0.597131i	$0.203693\pi$
$198$	0	0
$199$	1011.91	0.360465	0.180233	−	0.983624i	$-0.442315\pi$
$199$	1011.91	0.360465	0.180233	+	0.983624i	$0.442315\pi$
$200$	0	0
$201$	5047.36	1.77121
$202$	0	0
$203$	858.560	0.296843
$204$	0	0
$205$	982.489	0.334732
$206$	0	0
$207$	83.1180	0.0279087
$208$	0	0
$209$	997.964	0.330290
$210$	0	0
$211$	−2148.05	−0.700844	−0.350422	−	0.936592i	$-0.613962\pi$
$211$	−2148.05	−0.700844	−0.350422	+	0.936592i	$0.613962\pi$
$212$	0	0
$213$	3643.85	1.17217
$214$	0	0
$215$	−758.149	−0.240490
$216$	0	0
$217$	−1735.44	−0.542900
$218$	0	0
$219$	5580.20	1.72180
$220$	0	0
$221$	1313.02	0.399653
$222$	0	0
$223$	882.580	0.265031	0.132515	−	0.991181i	$-0.457695\pi$
$223$	882.580	0.265031	0.132515	+	0.991181i	$0.457695\pi$
$224$	0	0
$225$	118.147	0.0350065
$226$	0	0
$227$	2975.63	0.870042	0.435021	−	0.900420i	$-0.356741\pi$
$227$	2975.63	0.870042	0.435021	+	0.900420i	$0.356741\pi$
$228$	0	0
$229$	2257.57	0.651459	0.325730	−	0.945463i	$-0.394390\pi$
$229$	2257.57	0.651459	0.325730	+	0.945463i	$0.394390\pi$
$230$	0	0
$231$	807.435	0.229980
$232$	0	0
$233$	2089.32	0.587449	0.293725	−	0.955890i	$-0.405105\pi$
$233$	2089.32	0.587449	0.293725	+	0.955890i	$0.405105\pi$
$234$	0	0
$235$	256.583	0.0712239
$236$	0	0
$237$	−1368.88	−0.375184
$238$	0	0
$239$	1092.15	0.295586	0.147793	−	0.989018i	$-0.452783\pi$
$239$	1092.15	0.295586	0.147793	+	0.989018i	$0.452783\pi$
$240$	0	0
$241$	7130.61	1.90591	0.952953	−	0.303119i	$-0.0980279\pi$
$241$	7130.61	1.90591	0.952953	+	0.303119i	$0.0980279\pi$
$242$	0	0
$243$	333.346	0.0880005
$244$	0	0
$245$	248.037	0.0646796
$246$	0	0
$247$	3394.94	0.874554
$248$	0	0
$249$	−7574.67	−1.92781
$250$	0	0
$251$	−2293.62	−0.576780	−0.288390	−	0.957513i	$-0.593120\pi$
$251$	−2293.62	−0.576780	−0.288390	+	0.957513i	$0.593120\pi$
$252$	0	0
$253$	1587.31	0.394440
$254$	0	0
$255$	−437.193	−0.107365
$256$	0	0
$257$	1649.01	0.400243	0.200121	−	0.979771i	$-0.435866\pi$
$257$	1649.01	0.400243	0.200121	+	0.979771i	$0.435866\pi$
$258$	0	0
$259$	−1723.61	−0.413512
$260$	0	0
$261$	145.818	0.0345821
$262$	0	0
$263$	147.883	0.0346725	0.0173363	−	0.999850i	$-0.494481\pi$
$263$	147.883	0.0346725	0.0173363	+	0.999850i	$0.494481\pi$
$264$	0	0
$265$	−568.956	−0.131889
$266$	0	0
$267$	5697.68	1.30596
$268$	0	0
$269$	4313.72	0.977741	0.488871	−	0.872356i	$-0.337409\pi$
$269$	4313.72	0.977741	0.488871	+	0.872356i	$0.337409\pi$
$270$	0	0
$271$	−7036.97	−1.57736	−0.788682	−	0.614801i	$-0.789236\pi$
$271$	−7036.97	−1.57736	−0.788682	+	0.614801i	$0.789236\pi$
$272$	0	0
$273$	2746.78	0.608948
$274$	0	0
$275$	2256.26	0.494755
$276$	0	0
$277$	−513.326	−0.111346	−0.0556728	−	0.998449i	$-0.517730\pi$
$277$	−513.326	−0.111346	−0.0556728	+	0.998449i	$0.517730\pi$
$278$	0	0
$279$	−294.748	−0.0632477
$280$	0	0
$281$	6184.08	1.31285	0.656425	−	0.754391i	$-0.272068\pi$
$281$	6184.08	1.31285	0.656425	+	0.754391i	$0.272068\pi$
$282$	0	0
$283$	−7548.24	−1.58550	−0.792749	−	0.609548i	$-0.791351\pi$
$283$	−7548.24	−1.58550	−0.792749	+	0.609548i	$0.791351\pi$
$284$	0	0
$285$	−1130.40	−0.234945
$286$	0	0
$287$	−1358.64	−0.279436
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−4766.54	−0.960205
$292$	0	0
$293$	4816.27	0.960305	0.480152	−	0.877185i	$-0.340581\pi$
$293$	4816.27	0.960305	0.480152	+	0.877185i	$0.340581\pi$
$294$	0	0
$295$	−2507.29	−0.494847
$296$	0	0
$297$	3251.53	0.635262
$298$	0	0
$299$	5399.81	1.04441
$300$	0	0
$301$	1048.41	0.200762
$302$	0	0
$303$	−9571.38	−1.81472
$304$	0	0
$305$	−2568.64	−0.482228
$306$	0	0
$307$	4065.59	0.755816	0.377908	−	0.925843i	$-0.376644\pi$
$307$	4065.59	0.755816	0.377908	+	0.925843i	$0.376644\pi$
$308$	0	0
$309$	−4573.33	−0.841967
$310$	0	0
$311$	−7751.11	−1.41326	−0.706632	−	0.707582i	$-0.749786\pi$
$311$	−7751.11	−1.41326	−0.706632	+	0.707582i	$0.749786\pi$
$312$	0	0
$313$	−3853.18	−0.695829	−0.347914	−	0.937526i	$-0.613110\pi$
$313$	−3853.18	−0.695829	−0.347914	+	0.937526i	$0.613110\pi$
$314$	0	0
$315$	42.1268	0.00753516
$316$	0	0
$317$	2384.12	0.422415	0.211208	−	0.977441i	$-0.432260\pi$
$317$	2384.12	0.422415	0.211208	+	0.977441i	$0.432260\pi$
$318$	0	0
$319$	2784.70	0.488756
$320$	0	0
$321$	3508.65	0.610075
$322$	0	0
$323$	747.236	0.128722
$324$	0	0
$325$	7675.49	1.31003
$326$	0	0
$327$	−9810.83	−1.65914
$328$	0	0
$329$	−354.817	−0.0594581
$330$	0	0
$331$	−11291.2	−1.87499	−0.937495	−	0.348000i	$-0.886861\pi$
$331$	−11291.2	−1.87499	−0.937495	+	0.348000i	$0.886861\pi$
$332$	0	0
$333$	−292.738	−0.0481741
$334$	0	0
$335$	5029.00	0.820190
$336$	0	0
$337$	−3319.53	−0.536577	−0.268288	−	0.963339i	$-0.586458\pi$
$337$	−3319.53	−0.536577	−0.268288	+	0.963339i	$0.586458\pi$
$338$	0	0
$339$	2303.03	0.368977
$340$	0	0
$341$	−5628.82	−0.893894
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−1797.96	−0.280576
$346$	0	0
$347$	−1159.49	−0.179380	−0.0896901	−	0.995970i	$-0.528588\pi$
$347$	−1159.49	−0.179380	−0.0896901	+	0.995970i	$0.528588\pi$
$348$	0	0
$349$	−1004.55	−0.154075	−0.0770374	−	0.997028i	$-0.524546\pi$
$349$	−1004.55	−0.154075	−0.0770374	+	0.997028i	$0.524546\pi$
$350$	0	0
$351$	11061.3	1.68207
$352$	0	0
$353$	−9112.60	−1.37398	−0.686989	−	0.726667i	$-0.741068\pi$
$353$	−9112.60	−1.37398	−0.686989	+	0.726667i	$0.741068\pi$
$354$	0	0
$355$	3630.59	0.542794
$356$	0	0
$357$	604.575	0.0896289
$358$	0	0
$359$	−10434.4	−1.53400	−0.767001	−	0.641646i	$-0.778252\pi$
$359$	−10434.4	−1.53400	−0.767001	+	0.641646i	$0.778252\pi$
$360$	0	0
$361$	−4926.95	−0.718319
$362$	0	0
$363$	−4143.22	−0.599070
$364$	0	0
$365$	5559.90	0.797311
$366$	0	0
$367$	−7994.13	−1.13703	−0.568515	−	0.822673i	$-0.692482\pi$
$367$	−7994.13	−1.13703	−0.568515	+	0.822673i	$0.692482\pi$
$368$	0	0
$369$	−230.753	−0.0325542
$370$	0	0
$371$	786.785	0.110102
$372$	0	0
$373$	−12655.8	−1.75681	−0.878407	−	0.477913i	$-0.841393\pi$
$373$	−12655.8	−1.75681	−0.878407	+	0.477913i	$0.841393\pi$
$374$	0	0
$375$	−5770.33	−0.794610
$376$	0	0
$377$	9473.17	1.29415
$378$	0	0
$379$	−2440.75	−0.330799	−0.165400	−	0.986227i	$-0.552891\pi$
$379$	−2440.75	−0.330799	−0.165400	+	0.986227i	$0.552891\pi$
$380$	0	0
$381$	−4942.67	−0.664621
$382$	0	0
$383$	−6954.63	−0.927846	−0.463923	−	0.885876i	$-0.653558\pi$
$383$	−6954.63	−0.927846	−0.463923	+	0.885876i	$0.653558\pi$
$384$	0	0
$385$	804.498	0.106496
$386$	0	0
$387$	178.063	0.0233888
$388$	0	0
$389$	−4428.95	−0.577267	−0.288633	−	0.957440i	$-0.593201\pi$
$389$	−4428.95	−0.577267	−0.288633	+	0.957440i	$0.593201\pi$
$390$	0	0
$391$	1188.51	0.153723
$392$	0	0
$393$	−2609.32	−0.334918
$394$	0	0
$395$	−1363.90	−0.173735
$396$	0	0
$397$	3053.79	0.386058	0.193029	−	0.981193i	$-0.438169\pi$
$397$	3053.79	0.386058	0.193029	+	0.981193i	$0.438169\pi$
$398$	0	0
$399$	1563.18	0.196133
$400$	0	0
$401$	9873.42	1.22956	0.614782	−	0.788697i	$-0.289244\pi$
$401$	9873.42	1.22956	0.614782	+	0.788697i	$0.289244\pi$
$402$	0	0
$403$	−19148.5	−2.36688
$404$	0	0
$405$	−3520.54	−0.431943
$406$	0	0
$407$	−5590.44	−0.680855
$408$	0	0
$409$	11207.3	1.35493	0.677464	−	0.735556i	$-0.263079\pi$
$409$	11207.3	1.35493	0.677464	+	0.735556i	$0.263079\pi$
$410$	0	0
$411$	4146.82	0.497682
$412$	0	0
$413$	3467.22	0.413101
$414$	0	0
$415$	−7547.11	−0.892706
$416$	0	0
$417$	7884.73	0.925940
$418$	0	0
$419$	16518.4	1.92596	0.962981	−	0.269568i	$-0.0868810\pi$
$419$	16518.4	1.92596	0.962981	+	0.269568i	$0.0868810\pi$
$420$	0	0
$421$	−8916.89	−1.03226	−0.516131	−	0.856510i	$-0.672628\pi$
$421$	−8916.89	−1.03226	−0.516131	+	0.856510i	$0.672628\pi$
$422$	0	0
$423$	−60.2624	−0.00692686
$424$	0	0
$425$	1689.40	0.192818
$426$	0	0
$427$	3552.06	0.402567
$428$	0	0
$429$	8909.08	1.00264
$430$	0	0
$431$	−12979.5	−1.45058	−0.725291	−	0.688442i	$-0.758295\pi$
$431$	−12979.5	−1.45058	−0.725291	+	0.688442i	$0.758295\pi$
$432$	0	0
$433$	5430.17	0.602673	0.301337	−	0.953518i	$-0.402567\pi$
$433$	5430.17	0.602673	0.301337	+	0.953518i	$0.402567\pi$
$434$	0	0
$435$	−3154.25	−0.347666
$436$	0	0
$437$	3073.01	0.336389
$438$	0	0
$439$	2291.73	0.249153	0.124577	−	0.992210i	$-0.460243\pi$
$439$	2291.73	0.249153	0.124577	+	0.992210i	$0.460243\pi$
$440$	0	0
$441$	−58.2553	−0.00629039
$442$	0	0
$443$	8938.66	0.958665	0.479332	−	0.877633i	$-0.340879\pi$
$443$	8938.66	0.958665	0.479332	+	0.877633i	$0.340879\pi$
$444$	0	0
$445$	5676.95	0.604749
$446$	0	0
$447$	12006.0	1.27039
$448$	0	0
$449$	−2381.67	−0.250330	−0.125165	−	0.992136i	$-0.539946\pi$
$449$	−2381.67	−0.250330	−0.125165	+	0.992136i	$0.539946\pi$
$450$	0	0
$451$	−4406.70	−0.460096
$452$	0	0
$453$	−4263.44	−0.442194
$454$	0	0
$455$	2736.79	0.281984
$456$	0	0
$457$	−7426.51	−0.760169	−0.380085	−	0.924952i	$-0.624105\pi$
$457$	−7426.51	−0.760169	−0.380085	+	0.924952i	$0.624105\pi$
$458$	0	0
$459$	2434.61	0.247577
$460$	0	0
$461$	−11646.2	−1.17661	−0.588307	−	0.808637i	$-0.700205\pi$
$461$	−11646.2	−1.17661	−0.588307	+	0.808637i	$0.700205\pi$
$462$	0	0
$463$	−4135.72	−0.415126	−0.207563	−	0.978222i	$-0.566553\pi$
$463$	−4135.72	−0.415126	−0.207563	+	0.978222i	$0.566553\pi$
$464$	0	0
$465$	6375.81	0.635852
$466$	0	0
$467$	−14728.6	−1.45944	−0.729721	−	0.683745i	$-0.760350\pi$
$467$	−14728.6	−1.45944	−0.729721	+	0.683745i	$0.760350\pi$
$468$	0	0
$469$	−6954.39	−0.684699
$470$	0	0
$471$	−4982.74	−0.487458
$472$	0	0
$473$	3400.48	0.330559
$474$	0	0
$475$	4368.09	0.421941
$476$	0	0
$477$	133.628	0.0128269
$478$	0	0
$479$	12824.0	1.22327	0.611634	−	0.791141i	$-0.290513\pi$
$479$	12824.0	1.22327	0.611634	+	0.791141i	$0.290513\pi$
$480$	0	0
$481$	−19017.9	−1.80279
$482$	0	0
$483$	2486.32	0.234227
$484$	0	0
$485$	−4749.20	−0.444640
$486$	0	0
$487$	−1680.72	−0.156387	−0.0781935	−	0.996938i	$-0.524915\pi$
$487$	−1680.72	−0.156387	−0.0781935	+	0.996938i	$0.524915\pi$
$488$	0	0
$489$	8636.37	0.798671
$490$	0	0
$491$	12592.5	1.15741	0.578707	−	0.815536i	$-0.303558\pi$
$491$	12592.5	1.15741	0.578707	+	0.815536i	$0.303558\pi$
$492$	0	0
$493$	2085.07	0.190481
$494$	0	0
$495$	136.636	0.0124068
$496$	0	0
$497$	−5020.59	−0.453127
$498$	0	0
$499$	−11618.6	−1.04232	−0.521162	−	0.853458i	$-0.674501\pi$
$499$	−11618.6	−1.04232	−0.521162	+	0.853458i	$0.674501\pi$
$500$	0	0
$501$	−10400.7	−0.927480
$502$	0	0
$503$	16145.9	1.43124	0.715618	−	0.698492i	$-0.246145\pi$
$503$	16145.9	1.43124	0.715618	+	0.698492i	$0.246145\pi$
$504$	0	0
$505$	−9536.56	−0.840340
$506$	0	0
$507$	19145.7	1.67710
$508$	0	0
$509$	6634.28	0.577720	0.288860	−	0.957371i	$-0.406724\pi$
$509$	6634.28	0.577720	0.288860	+	0.957371i	$0.406724\pi$
$510$	0	0
$511$	−7688.55	−0.665600
$512$	0	0
$513$	6294.92	0.541769
$514$	0	0
$515$	−4556.70	−0.389887
$516$	0	0
$517$	−1150.84	−0.0978988
$518$	0	0
$519$	4543.13	0.384241
$520$	0	0
$521$	21010.2	1.76674	0.883370	−	0.468676i	$-0.155269\pi$
$521$	21010.2	1.76674	0.883370	+	0.468676i	$0.155269\pi$
$522$	0	0
$523$	−11349.0	−0.948868	−0.474434	−	0.880291i	$-0.657347\pi$
$523$	−11349.0	−0.948868	−0.474434	+	0.880291i	$0.657347\pi$
$524$	0	0
$525$	3534.15	0.293796
$526$	0	0
$527$	−4214.64	−0.348373
$528$	0	0
$529$	−7279.23	−0.598276
$530$	0	0
$531$	588.875	0.0481262
$532$	0	0
$533$	−14991.0	−1.21826
$534$	0	0
$535$	3495.89	0.282506
$536$	0	0
$537$	−17905.5	−1.43888
$538$	0	0
$539$	−1112.51	−0.0889035
$540$	0	0
$541$	−20629.2	−1.63941	−0.819703	−	0.572788i	$-0.805862\pi$
$541$	−20629.2	−1.63941	−0.819703	+	0.572788i	$0.805862\pi$
$542$	0	0
$543$	15458.9	1.22174
$544$	0	0
$545$	−9775.14	−0.768296
$546$	0	0
$547$	12407.6	0.969858	0.484929	−	0.874553i	$-0.338845\pi$
$547$	12407.6	0.969858	0.484929	+	0.874553i	$0.338845\pi$
$548$	0	0
$549$	603.284	0.0468990
$550$	0	0
$551$	5391.15	0.416825
$552$	0	0
$553$	1886.08	0.145035
$554$	0	0
$555$	6332.33	0.484311
$556$	0	0
$557$	−19488.4	−1.48249	−0.741247	−	0.671232i	$-0.765766\pi$
$557$	−19488.4	−1.48249	−0.741247	+	0.671232i	$0.765766\pi$
$558$	0	0
$559$	11568.0	0.875265
$560$	0	0
$561$	1960.91	0.147575
$562$	0	0
$563$	−11193.2	−0.837899	−0.418949	−	0.908010i	$-0.637602\pi$
$563$	−11193.2	−0.837899	−0.418949	+	0.908010i	$0.637602\pi$
$564$	0	0
$565$	2294.65	0.170861
$566$	0	0
$567$	4868.41	0.360589
$568$	0	0
$569$	13860.2	1.02118	0.510589	−	0.859825i	$-0.329427\pi$
$569$	13860.2	1.02118	0.510589	+	0.859825i	$0.329427\pi$
$570$	0	0
$571$	−10769.6	−0.789305	−0.394652	−	0.918830i	$-0.629135\pi$
$571$	−10769.6	−0.789305	−0.394652	+	0.918830i	$0.629135\pi$
$572$	0	0
$573$	−13214.6	−0.963431
$574$	0	0
$575$	6947.66	0.503891
$576$	0	0
$577$	19796.9	1.42835	0.714174	−	0.699968i	$-0.246803\pi$
$577$	19796.9	1.42835	0.714174	+	0.699968i	$0.246803\pi$
$578$	0	0
$579$	−11607.1	−0.833118
$580$	0	0
$581$	10436.6	0.745236
$582$	0	0
$583$	2551.90	0.181285
$584$	0	0
$585$	464.818	0.0328511
$586$	0	0
$587$	18947.6	1.33229	0.666143	−	0.745824i	$-0.267944\pi$
$587$	18947.6	1.33229	0.666143	+	0.745824i	$0.267944\pi$
$588$	0	0
$589$	−10897.3	−0.762338
$590$	0	0
$591$	22536.4	1.56857
$592$	0	0
$593$	−8431.86	−0.583904	−0.291952	−	0.956433i	$-0.594305\pi$
$593$	−8431.86	−0.583904	−0.291952	+	0.956433i	$0.594305\pi$
$594$	0	0
$595$	602.376	0.0415042
$596$	0	0
$597$	5140.99	0.352440
$598$	0	0
$599$	−11004.8	−0.750661	−0.375330	−	0.926891i	$-0.622471\pi$
$599$	−11004.8	−0.750661	−0.375330	+	0.926891i	$0.622471\pi$
$600$	0	0
$601$	20009.1	1.35805	0.679025	−	0.734115i	$-0.262403\pi$
$601$	20009.1	1.35805	0.679025	+	0.734115i	$0.262403\pi$
$602$	0	0
$603$	−1181.14	−0.0797673
$604$	0	0
$605$	−4128.14	−0.277410
$606$	0	0
$607$	−4691.09	−0.313683	−0.156841	−	0.987624i	$-0.550131\pi$
$607$	−4691.09	−0.313683	−0.156841	+	0.987624i	$0.550131\pi$
$608$	0	0
$609$	4361.88	0.290234
$610$	0	0
$611$	−3914.98	−0.259220
$612$	0	0
$613$	−8727.38	−0.575034	−0.287517	−	0.957776i	$-0.592830\pi$
$613$	−8727.38	−0.575034	−0.287517	+	0.957776i	$0.592830\pi$
$614$	0	0
$615$	4991.50	0.327279
$616$	0	0
$617$	4452.92	0.290547	0.145274	−	0.989392i	$-0.453594\pi$
$617$	4452.92	0.290547	0.145274	+	0.989392i	$0.453594\pi$
$618$	0	0
$619$	−14962.7	−0.971567	−0.485784	−	0.874079i	$-0.661466\pi$
$619$	−14962.7	−0.971567	−0.485784	+	0.874079i	$0.661466\pi$
$620$	0	0
$621$	10012.4	0.646993
$622$	0	0
$623$	−7850.41	−0.504848
$624$	0	0
$625$	6672.70	0.427053
$626$	0	0
$627$	5070.12	0.322936
$628$	0	0
$629$	−4185.90	−0.265346
$630$	0	0
$631$	−10549.6	−0.665564	−0.332782	−	0.943004i	$-0.607987\pi$
$631$	−10549.6	−0.665564	−0.332782	+	0.943004i	$0.607987\pi$
$632$	0	0
$633$	−10913.1	−0.685240
$634$	0	0
$635$	−4924.69	−0.307764
$636$	0	0
$637$	−3784.59	−0.235402
$638$	0	0
$639$	−852.700	−0.0527892
$640$	0	0
$641$	7139.83	0.439948	0.219974	−	0.975506i	$-0.429403\pi$
$641$	7139.83	0.439948	0.219974	+	0.975506i	$0.429403\pi$
$642$	0	0
$643$	−119.599	−0.00733517	−0.00366759	−	0.999993i	$-0.501167\pi$
$643$	−119.599	−0.00733517	−0.00366759	+	0.999993i	$0.501167\pi$
$644$	0	0
$645$	−3851.75	−0.235136
$646$	0	0
$647$	−15030.9	−0.913334	−0.456667	−	0.889638i	$-0.650957\pi$
$647$	−15030.9	−0.913334	−0.456667	+	0.889638i	$0.650957\pi$
$648$	0	0
$649$	11245.8	0.680178
$650$	0	0
$651$	−8816.84	−0.530813
$652$	0	0
$653$	−26641.6	−1.59658	−0.798291	−	0.602272i	$-0.794262\pi$
$653$	−26641.6	−1.59658	−0.798291	+	0.602272i	$0.794262\pi$
$654$	0	0
$655$	−2599.82	−0.155089
$656$	0	0
$657$	−1305.83	−0.0775422
$658$	0	0
$659$	10107.4	0.597465	0.298732	−	0.954337i	$-0.403436\pi$
$659$	10107.4	0.597465	0.298732	+	0.954337i	$0.403436\pi$
$660$	0	0
$661$	407.369	0.0239710	0.0119855	−	0.999928i	$-0.496185\pi$
$661$	407.369	0.0239710	0.0119855	+	0.999928i	$0.496185\pi$
$662$	0	0
$663$	6670.76	0.390756
$664$	0	0
$665$	1557.50	0.0908228
$666$	0	0
$667$	8574.87	0.497782
$668$	0	0
$669$	4483.91	0.259130
$670$	0	0
$671$	11520.9	0.662833
$672$	0	0
$673$	10000.4	0.572791	0.286395	−	0.958112i	$-0.407543\pi$
$673$	10000.4	0.572791	0.286395	+	0.958112i	$0.407543\pi$
$674$	0	0
$675$	14231.9	0.811538
$676$	0	0
$677$	671.561	0.0381244	0.0190622	−	0.999818i	$-0.493932\pi$
$677$	671.561	0.0381244	0.0190622	+	0.999818i	$0.493932\pi$
$678$	0	0
$679$	6567.47	0.371188
$680$	0	0
$681$	15117.6	0.850671
$682$	0	0
$683$	11991.6	0.671807	0.335903	−	0.941896i	$-0.390958\pi$
$683$	11991.6	0.671807	0.335903	+	0.941896i	$0.390958\pi$
$684$	0	0
$685$	4131.73	0.230460
$686$	0	0
$687$	11469.5	0.636955
$688$	0	0
$689$	8681.23	0.480012
$690$	0	0
$691$	23066.5	1.26989	0.634943	−	0.772559i	$-0.281024\pi$
$691$	23066.5	1.26989	0.634943	+	0.772559i	$0.281024\pi$
$692$	0	0
$693$	−188.949	−0.0103572
$694$	0	0
$695$	7856.04	0.428772
$696$	0	0
$697$	−3299.56	−0.179311
$698$	0	0
$699$	10614.7	0.574370
$700$	0	0
$701$	22390.3	1.20638	0.603188	−	0.797599i	$-0.293897\pi$
$701$	22390.3	1.20638	0.603188	+	0.797599i	$0.293897\pi$
$702$	0	0
$703$	−10823.0	−0.580652
$704$	0	0
$705$	1303.56	0.0696382
$706$	0	0
$707$	13187.7	0.701520
$708$	0	0
$709$	−21625.9	−1.14552	−0.572762	−	0.819722i	$-0.694128\pi$
$709$	−21625.9	−1.14552	−0.572762	+	0.819722i	$0.694128\pi$
$710$	0	0
$711$	320.334	0.0168966
$712$	0	0
$713$	−17332.7	−0.910401
$714$	0	0
$715$	8876.66	0.464292
$716$	0	0
$717$	5548.62	0.289005
$718$	0	0
$719$	−23657.2	−1.22707	−0.613536	−	0.789667i	$-0.710253\pi$
$719$	−23657.2	−1.22707	−0.613536	+	0.789667i	$0.710253\pi$
$720$	0	0
$721$	6301.26	0.325480
$722$	0	0
$723$	36226.8	1.86347
$724$	0	0
$725$	12188.6	0.624379
$726$	0	0
$727$	−149.999	−0.00765218	−0.00382609	−	0.999993i	$-0.501218\pi$
$727$	−149.999	−0.00765218	−0.00382609	+	0.999993i	$0.501218\pi$
$728$	0	0
$729$	20471.7	1.04007
$730$	0	0
$731$	2546.14	0.128827
$732$	0	0
$733$	−17908.4	−0.902401	−0.451201	−	0.892422i	$-0.649004\pi$
$733$	−17908.4	−0.902401	−0.451201	+	0.892422i	$0.649004\pi$
$734$	0	0
$735$	1260.14	0.0632396
$736$	0	0
$737$	−22556.3	−1.12737
$738$	0	0
$739$	13822.8	0.688064	0.344032	−	0.938958i	$-0.388207\pi$
$739$	13822.8	0.688064	0.344032	+	0.938958i	$0.388207\pi$
$740$	0	0
$741$	17247.9	0.855083
$742$	0	0
$743$	286.767	0.0141595	0.00707973	−	0.999975i	$-0.497746\pi$
$743$	286.767	0.0141595	0.00707973	+	0.999975i	$0.497746\pi$
$744$	0	0
$745$	11962.3	0.588276
$746$	0	0
$747$	1772.56	0.0868198
$748$	0	0
$749$	−4834.32	−0.235837
$750$	0	0
$751$	−29096.2	−1.41376	−0.706881	−	0.707332i	$-0.749898\pi$
$751$	−29096.2	−1.41376	−0.706881	+	0.707332i	$0.749898\pi$
$752$	0	0
$753$	−11652.6	−0.563939
$754$	0	0
$755$	−4247.93	−0.204766
$756$	0	0
$757$	7745.34	0.371875	0.185937	−	0.982562i	$-0.440468\pi$
$757$	7745.34	0.371875	0.185937	+	0.982562i	$0.440468\pi$
$758$	0	0
$759$	8064.27	0.385658
$760$	0	0
$761$	31459.1	1.49854	0.749270	−	0.662264i	$-0.230404\pi$
$761$	31459.1	1.49854	0.749270	+	0.662264i	$0.230404\pi$
$762$	0	0
$763$	13517.6	0.641378
$764$	0	0
$765$	102.308	0.00483523
$766$	0	0
$767$	38256.6	1.80100
$768$	0	0
$769$	−15502.7	−0.726973	−0.363487	−	0.931599i	$-0.618414\pi$
$769$	−15502.7	−0.726973	−0.363487	+	0.931599i	$0.618414\pi$
$770$	0	0
$771$	8377.73	0.391332
$772$	0	0
$773$	21718.6	1.01056	0.505281	−	0.862955i	$-0.331389\pi$
$773$	21718.6	1.01056	0.505281	+	0.862955i	$0.331389\pi$
$774$	0	0
$775$	−24637.4	−1.14194
$776$	0	0
$777$	−8756.72	−0.404306
$778$	0	0
$779$	−8531.32	−0.392383
$780$	0	0
$781$	−16284.1	−0.746081
$782$	0	0
$783$	17565.2	0.801698
$784$	0	0
$785$	−4964.62	−0.225726
$786$	0	0
$787$	−18467.9	−0.836481	−0.418240	−	0.908336i	$-0.637353\pi$
$787$	−18467.9	−0.836481	−0.418240	+	0.908336i	$0.637353\pi$
$788$	0	0
$789$	751.315	0.0339005
$790$	0	0
$791$	−3173.17	−0.142636
$792$	0	0
$793$	39192.7	1.75507
$794$	0	0
$795$	−2890.56	−0.128953
$796$	0	0
$797$	42363.0	1.88278	0.941389	−	0.337324i	$-0.109522\pi$
$797$	42363.0	1.88278	0.941389	+	0.337324i	$0.109522\pi$
$798$	0	0
$799$	−861.700	−0.0381536
$800$	0	0
$801$	−1333.32	−0.0588146
$802$	0	0
$803$	−24937.5	−1.09592
$804$	0	0
$805$	2477.27	0.108463
$806$	0	0
$807$	21915.7	0.955972
$808$	0	0
$809$	−37849.4	−1.64489	−0.822444	−	0.568846i	$-0.807390\pi$
$809$	−37849.4	−1.64489	−0.822444	+	0.568846i	$0.807390\pi$
$810$	0	0
$811$	−33151.9	−1.43541	−0.717707	−	0.696346i	$-0.754808\pi$
$811$	−33151.9	−1.43541	−0.717707	+	0.696346i	$0.754808\pi$
$812$	0	0
$813$	−35751.1	−1.54225
$814$	0	0
$815$	8604.95	0.369838
$816$	0	0
$817$	6583.29	0.281910
$818$	0	0
$819$	−642.778	−0.0274243
$820$	0	0
$821$	−15860.8	−0.674233	−0.337116	−	0.941463i	$-0.609452\pi$
$821$	−15860.8	−0.674233	−0.337116	+	0.941463i	$0.609452\pi$
$822$	0	0
$823$	−3835.11	−0.162435	−0.0812173	−	0.996696i	$-0.525881\pi$
$823$	−3835.11	−0.162435	−0.0812173	+	0.996696i	$0.525881\pi$
$824$	0	0
$825$	11462.8	0.483740
$826$	0	0
$827$	19478.2	0.819014	0.409507	−	0.912307i	$-0.365701\pi$
$827$	19478.2	0.819014	0.409507	+	0.912307i	$0.365701\pi$
$828$	0	0
$829$	6423.66	0.269123	0.134561	−	0.990905i	$-0.457037\pi$
$829$	6423.66	0.269123	0.134561	+	0.990905i	$0.457037\pi$
$830$	0	0
$831$	−2607.93	−0.108867
$832$	0	0
$833$	−833.000	−0.0346479
$834$	0	0
$835$	−10362.8	−0.429486
$836$	0	0
$837$	−35505.3	−1.46624
$838$	0	0
$839$	23071.9	0.949381	0.474691	−	0.880153i	$-0.342560\pi$
$839$	23071.9	0.949381	0.474691	+	0.880153i	$0.342560\pi$
$840$	0	0
$841$	−9345.64	−0.383191
$842$	0	0
$843$	31418.0	1.28362
$844$	0	0
$845$	19076.0	0.776610
$846$	0	0
$847$	5708.64	0.231583
$848$	0	0
$849$	−38348.5	−1.55020
$850$	0	0
$851$	−17214.5	−0.693427
$852$	0	0
$853$	−16436.2	−0.659746	−0.329873	−	0.944025i	$-0.607006\pi$
$853$	−16436.2	−0.659746	−0.329873	+	0.944025i	$0.607006\pi$
$854$	0	0
$855$	264.526	0.0105808
$856$	0	0
$857$	−15649.0	−0.623756	−0.311878	−	0.950122i	$-0.600958\pi$
$857$	−15649.0	−0.623756	−0.311878	+	0.950122i	$0.600958\pi$
$858$	0	0
$859$	9334.29	0.370759	0.185379	−	0.982667i	$-0.440649\pi$
$859$	9334.29	0.370759	0.185379	+	0.982667i	$0.440649\pi$
$860$	0	0
$861$	−6902.53	−0.273215
$862$	0	0
$863$	28285.6	1.11570	0.557852	−	0.829941i	$-0.311626\pi$
$863$	28285.6	1.11570	0.557852	+	0.829941i	$0.311626\pi$
$864$	0	0
$865$	4526.60	0.177929
$866$	0	0
$867$	1468.25	0.0575139
$868$	0	0
$869$	6117.43	0.238803
$870$	0	0
$871$	−76733.4	−2.98509
$872$	0	0
$873$	1115.42	0.0432433
$874$	0	0
$875$	7950.52	0.307173
$876$	0	0
$877$	7783.86	0.299706	0.149853	−	0.988708i	$-0.452120\pi$
$877$	7783.86	0.299706	0.149853	+	0.988708i	$0.452120\pi$
$878$	0	0
$879$	24468.9	0.938924
$880$	0	0
$881$	−13092.6	−0.500684	−0.250342	−	0.968157i	$-0.580543\pi$
$881$	−13092.6	−0.500684	−0.250342	+	0.968157i	$0.580543\pi$
$882$	0	0
$883$	35728.3	1.36167	0.680835	−	0.732437i	$-0.261617\pi$
$883$	35728.3	1.36167	0.680835	+	0.732437i	$0.261617\pi$
$884$	0	0
$885$	−12738.2	−0.483830
$886$	0	0
$887$	39531.7	1.49644	0.748221	−	0.663449i	$-0.230908\pi$
$887$	39531.7	1.49644	0.748221	+	0.663449i	$0.230908\pi$
$888$	0	0
$889$	6810.14	0.256923
$890$	0	0
$891$	15790.5	0.593716
$892$	0	0
$893$	−2228.00	−0.0834908
$894$	0	0
$895$	−17840.3	−0.666298
$896$	0	0
$897$	27433.5	1.02116
$898$	0	0
$899$	−30407.7	−1.12809
$900$	0	0
$901$	1910.76	0.0706513
$902$	0	0
$903$	5326.42	0.196293
$904$	0	0
$905$	15402.7	0.565749
$906$	0	0
$907$	−41167.6	−1.50711	−0.753554	−	0.657386i	$-0.771662\pi$
$907$	−41167.6	−1.50711	−0.753554	+	0.657386i	$0.771662\pi$
$908$	0	0
$909$	2239.81	0.0817269
$910$	0	0
$911$	−9646.33	−0.350820	−0.175410	−	0.984495i	$-0.556125\pi$
$911$	−9646.33	−0.350820	−0.175410	+	0.984495i	$0.556125\pi$
$912$	0	0
$913$	33850.6	1.22704
$914$	0	0
$915$	−13049.9	−0.471492
$916$	0	0
$917$	3595.19	0.129469
$918$	0	0
$919$	16691.5	0.599132	0.299566	−	0.954076i	$-0.403158\pi$
$919$	16691.5	0.599132	0.299566	+	0.954076i	$0.403158\pi$
$920$	0	0
$921$	20655.1	0.738988
$922$	0	0
$923$	−55396.2	−1.97550
$924$	0	0
$925$	−24469.4	−0.869782
$926$	0	0
$927$	1070.21	0.0379184
$928$	0	0
$929$	−30224.8	−1.06743	−0.533715	−	0.845664i	$-0.679205\pi$
$929$	−30224.8	−1.06743	−0.533715	+	0.845664i	$0.679205\pi$
$930$	0	0
$931$	−2153.80	−0.0758194
$932$	0	0
$933$	−39379.2	−1.38180
$934$	0	0
$935$	1953.78	0.0683374
$936$	0	0
$937$	−27596.4	−0.962152	−0.481076	−	0.876679i	$-0.659754\pi$
$937$	−27596.4	−0.962152	−0.481076	+	0.876679i	$0.659754\pi$
$938$	0	0
$939$	−19575.9	−0.680337
$940$	0	0
$941$	−10199.5	−0.353342	−0.176671	−	0.984270i	$-0.556533\pi$
$941$	−10199.5	−0.353342	−0.176671	+	0.984270i	$0.556533\pi$
$942$	0	0
$943$	−13569.5	−0.468592
$944$	0	0
$945$	5074.58	0.174684
$946$	0	0
$947$	−14572.5	−0.500045	−0.250023	−	0.968240i	$-0.580438\pi$
$947$	−14572.5	−0.500045	−0.250023	+	0.968240i	$0.580438\pi$
$948$	0	0
$949$	−84833.9	−2.90182
$950$	0	0
$951$	12112.4	0.413010
$952$	0	0
$953$	−44373.7	−1.50829	−0.754147	−	0.656705i	$-0.771949\pi$
$953$	−44373.7	−1.50829	−0.754147	+	0.656705i	$0.771949\pi$
$954$	0	0
$955$	−13166.5	−0.446133
$956$	0	0
$957$	14147.6	0.477875
$958$	0	0
$959$	−5713.60	−0.192390
$960$	0	0
$961$	31673.3	1.06318
$962$	0	0
$963$	−821.064	−0.0274750
$964$	0	0
$965$	−11564.9	−0.385790
$966$	0	0
$967$	−25217.6	−0.838619	−0.419309	−	0.907843i	$-0.637728\pi$
$967$	−25217.6	−0.838619	−0.419309	+	0.907843i	$0.637728\pi$
$968$	0	0
$969$	3796.31	0.125856
$970$	0	0
$971$	51329.7	1.69644	0.848222	−	0.529640i	$-0.177673\pi$
$971$	51329.7	1.69644	0.848222	+	0.529640i	$0.177673\pi$
$972$	0	0
$973$	−10863.8	−0.357942
$974$	0	0
$975$	38995.1	1.28086
$976$	0	0
$977$	−1373.80	−0.0449866	−0.0224933	−	0.999747i	$-0.507160\pi$
$977$	−1373.80	−0.0449866	−0.0224933	+	0.999747i	$0.507160\pi$
$978$	0	0
$979$	−25462.5	−0.831240
$980$	0	0
$981$	2295.84	0.0747203
$982$	0	0
$983$	7261.69	0.235617	0.117809	−	0.993036i	$-0.462413\pi$
$983$	7261.69	0.235617	0.117809	+	0.993036i	$0.462413\pi$
$984$	0	0
$985$	22454.4	0.726353
$986$	0	0
$987$	−1802.64	−0.0581343
$988$	0	0
$989$	10471.0	0.336663
$990$	0	0
$991$	−40773.1	−1.30696	−0.653481	−	0.756943i	$-0.726692\pi$
$991$	−40773.1	−1.30696	−0.653481	+	0.756943i	$0.726692\pi$
$992$	0	0
$993$	−57364.6	−1.83324
$994$	0	0
$995$	5122.28	0.163203
$996$	0	0
$997$	−31499.7	−1.00061	−0.500304	−	0.865850i	$-0.666778\pi$
$997$	−31499.7	−1.00061	−0.500304	+	0.865850i	$0.666778\pi$
$998$	0	0
$999$	−35263.1	−1.11679

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	476.4.a.c.1.5	✓	6
4.3	odd	2		1904.4.a.n.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.4.a.c.1.5	✓	6	1.1	even	1	trivial
1904.4.a.n.1.2		6	4.3	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:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	476.a (trivial)

\(f(q)\)	\(=\)	\(q+5.08046 q^{3} +5.06198 q^{5} -7.00000 q^{7} -1.18888 q^{9} +O(q^{10})\)	\(q+5.08046 q^{3} +5.06198 q^{5} -7.00000 q^{7} -1.18888 q^{9} -22.7042 q^{11} -77.2366 q^{13} +25.7172 q^{15} -17.0000 q^{17} -43.9551 q^{19} -35.5632 q^{21} -69.9126 q^{23} -99.3763 q^{25} -143.213 q^{27} -122.651 q^{29} +247.920 q^{31} -115.348 q^{33} -35.4339 q^{35} +246.229 q^{37} -392.398 q^{39} +194.092 q^{41} -149.773 q^{43} -6.01811 q^{45} +50.6882 q^{47} +49.0000 q^{49} -86.3679 q^{51} -112.398 q^{53} -114.928 q^{55} -223.312 q^{57} -495.317 q^{59} -507.437 q^{61} +8.32219 q^{63} -390.970 q^{65} +993.485 q^{67} -355.188 q^{69} +717.227 q^{71} +1098.36 q^{73} -504.878 q^{75} +158.929 q^{77} -269.441 q^{79} -695.487 q^{81} -1490.94 q^{83} -86.0537 q^{85} -623.126 q^{87} +1121.49 q^{89} +540.656 q^{91} +1259.55 q^{93} -222.500 q^{95} -938.210 q^{97} +26.9927 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 10 q^{5} - 42 q^{7} + 16 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 + 10 * q^5 - 42 * q^7 + 16 * q^9	\( 6 q + 2 q^{3} + 10 q^{5} - 42 q^{7} + 16 q^{9} - 20 q^{11} + 42 q^{13} - 136 q^{15} - 102 q^{17} - 104 q^{19} - 14 q^{21} - 230 q^{23} + 108 q^{25} - 130 q^{27} - 52 q^{29} - 564 q^{31} - 346 q^{33} - 70 q^{35} - 564 q^{37} - 626 q^{39} - 548 q^{41} - 648 q^{43} - 174 q^{45} - 366 q^{47} + 294 q^{49} - 34 q^{51} - 74 q^{53} - 1460 q^{55} - 316 q^{57} - 558 q^{59} - 620 q^{61} - 112 q^{63} - 1378 q^{65} - 164 q^{67} - 540 q^{69} - 822 q^{71} + 940 q^{73} - 2698 q^{75} + 140 q^{77} - 1838 q^{79} - 3094 q^{81} - 1118 q^{83} - 170 q^{85} - 1354 q^{87} - 1634 q^{89} - 294 q^{91} + 268 q^{93} - 2642 q^{95} + 182 q^{97} + 12 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 + 10 * q^5 - 42 * q^7 + 16 * q^9 - 20 * q^11 + 42 * q^13 - 136 * q^15 - 102 * q^17 - 104 * q^19 - 14 * q^21 - 230 * q^23 + 108 * q^25 - 130 * q^27 - 52 * q^29 - 564 * q^31 - 346 * q^33 - 70 * q^35 - 564 * q^37 - 626 * q^39 - 548 * q^41 - 648 * q^43 - 174 * q^45 - 366 * q^47 + 294 * q^49 - 34 * q^51 - 74 * q^53 - 1460 * q^55 - 316 * q^57 - 558 * q^59 - 620 * q^61 - 112 * q^63 - 1378 * q^65 - 164 * q^67 - 540 * q^69 - 822 * q^71 + 940 * q^73 - 2698 * q^75 + 140 * q^77 - 1838 * q^79 - 3094 * q^81 - 1118 * q^83 - 170 * q^85 - 1354 * q^87 - 1634 * q^89 - 294 * q^91 + 268 * q^93 - 2642 * q^95 + 182 * q^97 + 12 * q^99

Introduction

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