LMFDB - Embedded newform 464.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,4,Mod(1,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, 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("464.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	464.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:	$27.3768862427$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.92622 q^{3} +14.3315 q^{5} -16.8804 q^{7} -18.4372 q^{9} +O(q^{10})$	$q-2.92622 q^{3} +14.3315 q^{5} -16.8804 q^{7} -18.4372 q^{9} -16.5613 q^{11} +55.3675 q^{13} -41.9372 q^{15} +6.28120 q^{17} +119.008 q^{19} +49.3957 q^{21} -22.5433 q^{23} +80.3924 q^{25} +132.959 q^{27} -29.0000 q^{29} -228.987 q^{31} +48.4621 q^{33} -241.921 q^{35} +257.186 q^{37} -162.017 q^{39} +382.208 q^{41} +170.123 q^{43} -264.234 q^{45} +172.889 q^{47} -58.0532 q^{49} -18.3802 q^{51} -69.2381 q^{53} -237.349 q^{55} -348.244 q^{57} +43.6860 q^{59} +684.892 q^{61} +311.227 q^{63} +793.500 q^{65} +528.585 q^{67} +65.9667 q^{69} +488.054 q^{71} +80.3863 q^{73} -235.246 q^{75} +279.561 q^{77} +741.898 q^{79} +108.737 q^{81} +1360.29 q^{83} +90.0191 q^{85} +84.8604 q^{87} +957.563 q^{89} -934.623 q^{91} +670.068 q^{93} +1705.57 q^{95} -1111.59 q^{97} +305.345 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{3} + 4 q^{5} - 16 q^{7} + 45 q^{9}+O(q^{10}) $ 3 * q - 6 * q^3 + 4 * q^5 - 16 * q^7 + 45 * q^9	$ 3 q - 6 q^{3} + 4 q^{5} - 16 q^{7} + 45 q^{9} + 2 q^{11} + 28 q^{13} - 136 q^{15} - 66 q^{17} + 66 q^{19} + 472 q^{21} - 176 q^{23} - 9 q^{25} - 228 q^{27} - 87 q^{29} + 190 q^{31} + 154 q^{33} - 660 q^{35} + 442 q^{37} + 656 q^{39} + 1162 q^{41} - 30 q^{43} - 254 q^{45} + 738 q^{47} + 851 q^{49} + 576 q^{51} + 312 q^{53} - 464 q^{55} + 684 q^{57} - 44 q^{59} + 54 q^{61} - 964 q^{63} + 178 q^{65} + 116 q^{67} + 812 q^{69} + 1200 q^{71} - 1118 q^{73} + 1038 q^{75} + 792 q^{77} + 2262 q^{79} + 15 q^{81} + 1804 q^{83} - 8 q^{85} + 174 q^{87} + 1578 q^{89} + 1972 q^{91} - 706 q^{93} + 1052 q^{95} + 1450 q^{97} + 482 q^{99}+O(q^{100}) $ 3 * q - 6 * q^3 + 4 * q^5 - 16 * q^7 + 45 * q^9 + 2 * q^11 + 28 * q^13 - 136 * q^15 - 66 * q^17 + 66 * q^19 + 472 * q^21 - 176 * q^23 - 9 * q^25 - 228 * q^27 - 87 * q^29 + 190 * q^31 + 154 * q^33 - 660 * q^35 + 442 * q^37 + 656 * q^39 + 1162 * q^41 - 30 * q^43 - 254 * q^45 + 738 * q^47 + 851 * q^49 + 576 * q^51 + 312 * q^53 - 464 * q^55 + 684 * q^57 - 44 * q^59 + 54 * q^61 - 964 * q^63 + 178 * q^65 + 116 * q^67 + 812 * q^69 + 1200 * q^71 - 1118 * q^73 + 1038 * q^75 + 792 * q^77 + 2262 * q^79 + 15 * q^81 + 1804 * q^83 - 8 * q^85 + 174 * q^87 + 1578 * q^89 + 1972 * q^91 - 706 * q^93 + 1052 * q^95 + 1450 * q^97 + 482 * 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
$3$	−2.92622	−0.563151	−0.281576	−	0.959539i	$-0.590857\pi$
$3$	−2.92622	−0.563151	−0.281576	+	0.959539i	$0.590857\pi$
$4$	0	0
$5$	14.3315	1.28185	0.640925	−	0.767603i	$-0.278551\pi$
$5$	14.3315	1.28185	0.640925	+	0.767603i	$0.278551\pi$
$6$	0	0
$7$	−16.8804	−0.911454	−0.455727	−	0.890120i	$-0.650621\pi$
$7$	−16.8804	−0.911454	−0.455727	+	0.890120i	$0.650621\pi$
$8$	0	0
$9$	−18.4372	−0.682860
$10$	0	0
$11$	−16.5613	−0.453948	−0.226974	−	0.973901i	$-0.572883\pi$
$11$	−16.5613	−0.453948	−0.226974	+	0.973901i	$0.572883\pi$
$12$	0	0
$13$	55.3675	1.18124	0.590622	−	0.806948i	$-0.298882\pi$
$13$	55.3675	1.18124	0.590622	+	0.806948i	$0.298882\pi$
$14$	0	0
$15$	−41.9372	−0.721876
$16$	0	0
$17$	6.28120	0.0896126	0.0448063	−	0.998996i	$-0.485733\pi$
$17$	6.28120	0.0896126	0.0448063	+	0.998996i	$0.485733\pi$
$18$	0	0
$19$	119.008	1.43696	0.718482	−	0.695545i	$-0.244837\pi$
$19$	119.008	1.43696	0.718482	+	0.695545i	$0.244837\pi$
$20$	0	0
$21$	49.3957	0.513287
$22$	0	0
$23$	−22.5433	−0.204374	−0.102187	−	0.994765i	$-0.532584\pi$
$23$	−22.5433	−0.204374	−0.102187	+	0.994765i	$0.532584\pi$
$24$	0	0
$25$	80.3924	0.643139
$26$	0	0
$27$	132.959	0.947705
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−228.987	−1.32669	−0.663344	−	0.748314i	$-0.730864\pi$
$31$	−228.987	−1.32669	−0.663344	+	0.748314i	$0.730864\pi$
$32$	0	0
$33$	48.4621	0.255642
$34$	0	0
$35$	−241.921	−1.16835
$36$	0	0
$37$	257.186	1.14273	0.571366	−	0.820696i	$-0.306414\pi$
$37$	257.186	1.14273	0.571366	+	0.820696i	$0.306414\pi$
$38$	0	0
$39$	−162.017	−0.665219
$40$	0	0
$41$	382.208	1.45588	0.727938	−	0.685643i	$-0.240479\pi$
$41$	382.208	1.45588	0.727938	+	0.685643i	$0.240479\pi$
$42$	0	0
$43$	170.123	0.603337	0.301669	−	0.953413i	$-0.402456\pi$
$43$	170.123	0.603337	0.301669	+	0.953413i	$0.402456\pi$
$44$	0	0
$45$	−264.234	−0.875325
$46$	0	0
$47$	172.889	0.536564	0.268282	−	0.963340i	$-0.413544\pi$
$47$	172.889	0.536564	0.268282	+	0.963340i	$0.413544\pi$
$48$	0	0
$49$	−58.0532	−0.169251
$50$	0	0
$51$	−18.3802	−0.0504654
$52$	0	0
$53$	−69.2381	−0.179445	−0.0897224	−	0.995967i	$-0.528598\pi$
$53$	−69.2381	−0.179445	−0.0897224	+	0.995967i	$0.528598\pi$
$54$	0	0
$55$	−237.349	−0.581893
$56$	0	0
$57$	−348.244	−0.809229
$58$	0	0
$59$	43.6860	0.0963971	0.0481986	−	0.998838i	$-0.484652\pi$
$59$	43.6860	0.0963971	0.0481986	+	0.998838i	$0.484652\pi$
$60$	0	0
$61$	684.892	1.43756	0.718782	−	0.695235i	$-0.244700\pi$
$61$	684.892	1.43756	0.718782	+	0.695235i	$0.244700\pi$
$62$	0	0
$63$	311.227	0.622396
$64$	0	0
$65$	793.500	1.51418
$66$	0	0
$67$	528.585	0.963835	0.481917	−	0.876217i	$-0.339941\pi$
$67$	528.585	0.963835	0.481917	+	0.876217i	$0.339941\pi$
$68$	0	0
$69$	65.9667	0.115094
$70$	0	0
$71$	488.054	0.815794	0.407897	−	0.913028i	$-0.366262\pi$
$71$	488.054	0.815794	0.407897	+	0.913028i	$0.366262\pi$
$72$	0	0
$73$	80.3863	0.128884	0.0644418	−	0.997921i	$-0.479473\pi$
$73$	80.3863	0.128884	0.0644418	+	0.997921i	$0.479473\pi$
$74$	0	0
$75$	−235.246	−0.362185
$76$	0	0
$77$	279.561	0.413753
$78$	0	0
$79$	741.898	1.05658	0.528292	−	0.849063i	$-0.322833\pi$
$79$	741.898	1.05658	0.528292	+	0.849063i	$0.322833\pi$
$80$	0	0
$81$	108.737	0.149159
$82$	0	0
$83$	1360.29	1.79893	0.899466	−	0.436991i	$-0.143956\pi$
$83$	1360.29	1.79893	0.899466	+	0.436991i	$0.143956\pi$
$84$	0	0
$85$	90.0191	0.114870
$86$	0	0
$87$	84.8604	0.104575
$88$	0	0
$89$	957.563	1.14047	0.570233	−	0.821483i	$-0.306853\pi$
$89$	957.563	1.14047	0.570233	+	0.821483i	$0.306853\pi$
$90$	0	0
$91$	−934.623	−1.07665
$92$	0	0
$93$	670.068	0.747127
$94$	0	0
$95$	1705.57	1.84197
$96$	0	0
$97$	−1111.59	−1.16355	−0.581776	−	0.813349i	$-0.697642\pi$
$97$	−1111.59	−1.16355	−0.581776	+	0.813349i	$0.697642\pi$
$98$	0	0
$99$	305.345	0.309983
$100$	0	0
$101$	−1691.56	−1.66650	−0.833249	−	0.552898i	$-0.813522\pi$
$101$	−1691.56	−1.66650	−0.833249	+	0.552898i	$0.813522\pi$
$102$	0	0
$103$	439.312	0.420259	0.210130	−	0.977674i	$-0.432611\pi$
$103$	439.312	0.420259	0.210130	+	0.977674i	$0.432611\pi$
$104$	0	0
$105$	707.915	0.657957
$106$	0	0
$107$	861.531	0.778387	0.389193	−	0.921156i	$-0.372754\pi$
$107$	861.531	0.778387	0.389193	+	0.921156i	$0.372754\pi$
$108$	0	0
$109$	−459.780	−0.404027	−0.202013	−	0.979383i	$-0.564748\pi$
$109$	−459.780	−0.404027	−0.202013	+	0.979383i	$0.564748\pi$
$110$	0	0
$111$	−752.582	−0.643531
$112$	0	0
$113$	−9.56230	−0.00796058	−0.00398029	−	0.999992i	$-0.501267\pi$
$113$	−9.56230	−0.00796058	−0.00398029	+	0.999992i	$0.501267\pi$
$114$	0	0
$115$	−323.080	−0.261977
$116$	0	0
$117$	−1020.82	−0.806625
$118$	0	0
$119$	−106.029	−0.0816777
$120$	0	0
$121$	−1056.72	−0.793931
$122$	0	0
$123$	−1118.43	−0.819879
$124$	0	0
$125$	−639.294	−0.457442
$126$	0	0
$127$	−905.522	−0.632694	−0.316347	−	0.948644i	$-0.602456\pi$
$127$	−905.522	−0.632694	−0.316347	+	0.948644i	$0.602456\pi$
$128$	0	0
$129$	−497.817	−0.339770
$130$	0	0
$131$	−1039.43	−0.693250	−0.346625	−	0.938004i	$-0.612672\pi$
$131$	−1039.43	−0.693250	−0.346625	+	0.938004i	$0.612672\pi$
$132$	0	0
$133$	−2008.90	−1.30973
$134$	0	0
$135$	1905.51	1.21482
$136$	0	0
$137$	1146.58	0.715028	0.357514	−	0.933908i	$-0.383624\pi$
$137$	1146.58	0.715028	0.357514	+	0.933908i	$0.383624\pi$
$138$	0	0
$139$	573.043	0.349675	0.174838	−	0.984597i	$-0.444060\pi$
$139$	573.043	0.349675	0.174838	+	0.984597i	$0.444060\pi$
$140$	0	0
$141$	−505.912	−0.302167
$142$	0	0
$143$	−916.959	−0.536224
$144$	0	0
$145$	−415.614	−0.238034
$146$	0	0
$147$	169.877	0.0953142
$148$	0	0
$149$	2324.32	1.27796	0.638979	−	0.769224i	$-0.279357\pi$
$149$	2324.32	1.27796	0.638979	+	0.769224i	$0.279357\pi$
$150$	0	0
$151$	3188.68	1.71848	0.859242	−	0.511570i	$-0.170936\pi$
$151$	3188.68	1.71848	0.859242	+	0.511570i	$0.170936\pi$
$152$	0	0
$153$	−115.808	−0.0611929
$154$	0	0
$155$	−3281.74	−1.70062
$156$	0	0
$157$	−2960.71	−1.50503	−0.752516	−	0.658574i	$-0.771160\pi$
$157$	−2960.71	−1.50503	−0.752516	+	0.658574i	$0.771160\pi$
$158$	0	0
$159$	202.606	0.101055
$160$	0	0
$161$	380.540	0.186278
$162$	0	0
$163$	−3417.14	−1.64203	−0.821015	−	0.570907i	$-0.806592\pi$
$163$	−3417.14	−1.64203	−0.821015	+	0.570907i	$0.806592\pi$
$164$	0	0
$165$	694.536	0.327694
$166$	0	0
$167$	677.988	0.314158	0.157079	−	0.987586i	$-0.449792\pi$
$167$	677.988	0.314158	0.157079	+	0.987586i	$0.449792\pi$
$168$	0	0
$169$	868.556	0.395337
$170$	0	0
$171$	−2194.18	−0.981246
$172$	0	0
$173$	−773.271	−0.339831	−0.169915	−	0.985459i	$-0.554349\pi$
$173$	−773.271	−0.339831	−0.169915	+	0.985459i	$0.554349\pi$
$174$	0	0
$175$	−1357.05	−0.586192
$176$	0	0
$177$	−127.835	−0.0542862
$178$	0	0
$179$	−1413.92	−0.590399	−0.295199	−	0.955436i	$-0.595386\pi$
$179$	−1413.92	−0.590399	−0.295199	+	0.955436i	$0.595386\pi$
$180$	0	0
$181$	2160.51	0.887234	0.443617	−	0.896216i	$-0.353695\pi$
$181$	2160.51	0.887234	0.443617	+	0.896216i	$0.353695\pi$
$182$	0	0
$183$	−2004.15	−0.809567
$184$	0	0
$185$	3685.86	1.46481
$186$	0	0
$187$	−104.025	−0.0406795
$188$	0	0
$189$	−2244.40	−0.863790
$190$	0	0
$191$	−1586.39	−0.600981	−0.300490	−	0.953785i	$-0.597150\pi$
$191$	−1586.39	−0.600981	−0.300490	+	0.953785i	$0.597150\pi$
$192$	0	0
$193$	−1970.84	−0.735047	−0.367523	−	0.930014i	$-0.619794\pi$
$193$	−1970.84	−0.735047	−0.367523	+	0.930014i	$0.619794\pi$
$194$	0	0
$195$	−2321.96	−0.852711
$196$	0	0
$197$	4803.57	1.73726	0.868629	−	0.495463i	$-0.165002\pi$
$197$	4803.57	1.73726	0.868629	+	0.495463i	$0.165002\pi$
$198$	0	0
$199$	2692.85	0.959252	0.479626	−	0.877473i	$-0.340772\pi$
$199$	2692.85	0.959252	0.479626	+	0.877473i	$0.340772\pi$
$200$	0	0
$201$	−1546.76	−0.542785
$202$	0	0
$203$	489.531	0.169253
$204$	0	0
$205$	5477.63	1.86621
$206$	0	0
$207$	415.636	0.139559
$208$	0	0
$209$	−1970.93	−0.652308
$210$	0	0
$211$	703.260	0.229452	0.114726	−	0.993397i	$-0.463401\pi$
$211$	703.260	0.229452	0.114726	+	0.993397i	$0.463401\pi$
$212$	0	0
$213$	−1428.15	−0.459415
$214$	0	0
$215$	2438.12	0.773388
$216$	0	0
$217$	3865.39	1.20922
$218$	0	0
$219$	−235.228	−0.0725810
$220$	0	0
$221$	347.774	0.105854
$222$	0	0
$223$	3965.95	1.19094	0.595469	−	0.803378i	$-0.296966\pi$
$223$	3965.95	1.19094	0.595469	+	0.803378i	$0.296966\pi$
$224$	0	0
$225$	−1482.21	−0.439174
$226$	0	0
$227$	−3976.09	−1.16256	−0.581282	−	0.813702i	$-0.697449\pi$
$227$	−3976.09	−1.16256	−0.581282	+	0.813702i	$0.697449\pi$
$228$	0	0
$229$	−611.004	−0.176316	−0.0881578	−	0.996107i	$-0.528098\pi$
$229$	−611.004	−0.176316	−0.0881578	+	0.996107i	$0.528098\pi$
$230$	0	0
$231$	−818.058	−0.233006
$232$	0	0
$233$	−1051.90	−0.295761	−0.147880	−	0.989005i	$-0.547245\pi$
$233$	−1051.90	−0.295761	−0.147880	+	0.989005i	$0.547245\pi$
$234$	0	0
$235$	2477.77	0.687794
$236$	0	0
$237$	−2170.96	−0.595017
$238$	0	0
$239$	−2085.97	−0.564561	−0.282280	−	0.959332i	$-0.591091\pi$
$239$	−2085.97	−0.564561	−0.282280	+	0.959332i	$0.591091\pi$
$240$	0	0
$241$	−6064.55	−1.62096	−0.810481	−	0.585765i	$-0.800794\pi$
$241$	−6064.55	−1.62096	−0.810481	+	0.585765i	$0.800794\pi$
$242$	0	0
$243$	−3908.09	−1.03170
$244$	0	0
$245$	−831.991	−0.216955
$246$	0	0
$247$	6589.18	1.69741
$248$	0	0
$249$	−3980.51	−1.01307
$250$	0	0
$251$	−5169.26	−1.29992	−0.649962	−	0.759967i	$-0.725215\pi$
$251$	−5169.26	−1.29992	−0.649962	+	0.759967i	$0.725215\pi$
$252$	0	0
$253$	373.347	0.0927753
$254$	0	0
$255$	−263.416	−0.0646891
$256$	0	0
$257$	−7238.84	−1.75699	−0.878495	−	0.477751i	$-0.841452\pi$
$257$	−7238.84	−1.75699	−0.878495	+	0.477751i	$0.841452\pi$
$258$	0	0
$259$	−4341.39	−1.04155
$260$	0	0
$261$	534.680	0.126804
$262$	0	0
$263$	−4855.39	−1.13839	−0.569195	−	0.822203i	$-0.692745\pi$
$263$	−4855.39	−1.13839	−0.569195	+	0.822203i	$0.692745\pi$
$264$	0	0
$265$	−992.286	−0.230021
$266$	0	0
$267$	−2802.04	−0.642255
$268$	0	0
$269$	5559.70	1.26015	0.630076	−	0.776533i	$-0.283024\pi$
$269$	5559.70	1.26015	0.630076	+	0.776533i	$0.283024\pi$
$270$	0	0
$271$	−2534.36	−0.568087	−0.284044	−	0.958811i	$-0.591676\pi$
$271$	−2534.36	−0.568087	−0.284044	+	0.958811i	$0.591676\pi$
$272$	0	0
$273$	2734.91	0.606317
$274$	0	0
$275$	−1331.41	−0.291952
$276$	0	0
$277$	7540.72	1.63566	0.817831	−	0.575459i	$-0.195177\pi$
$277$	7540.72	1.63566	0.817831	+	0.575459i	$0.195177\pi$
$278$	0	0
$279$	4221.89	0.905943
$280$	0	0
$281$	5426.97	1.15212	0.576060	−	0.817407i	$-0.304589\pi$
$281$	5426.97	1.15212	0.576060	+	0.817407i	$0.304589\pi$
$282$	0	0
$283$	−1297.75	−0.272590	−0.136295	−	0.990668i	$-0.543520\pi$
$283$	−1297.75	−0.272590	−0.136295	+	0.990668i	$0.543520\pi$
$284$	0	0
$285$	−4990.87	−1.03731
$286$	0	0
$287$	−6451.82	−1.32696
$288$	0	0
$289$	−4873.55	−0.991970
$290$	0	0
$291$	3252.75	0.655256
$292$	0	0
$293$	−6201.84	−1.23657	−0.618285	−	0.785954i	$-0.712172\pi$
$293$	−6201.84	−1.23657	−0.618285	+	0.785954i	$0.712172\pi$
$294$	0	0
$295$	626.086	0.123567
$296$	0	0
$297$	−2201.98	−0.430209
$298$	0	0
$299$	−1248.17	−0.241416
$300$	0	0
$301$	−2871.74	−0.549914
$302$	0	0
$303$	4949.87	0.938491
$304$	0	0
$305$	9815.55	1.84274
$306$	0	0
$307$	9181.06	1.70681	0.853406	−	0.521247i	$-0.174533\pi$
$307$	9181.06	1.70681	0.853406	+	0.521247i	$0.174533\pi$
$308$	0	0
$309$	−1285.52	−0.236670
$310$	0	0
$311$	−3753.29	−0.684339	−0.342170	−	0.939638i	$-0.611162\pi$
$311$	−3753.29	−0.684339	−0.342170	+	0.939638i	$0.611162\pi$
$312$	0	0
$313$	2115.29	0.381991	0.190996	−	0.981591i	$-0.438828\pi$
$313$	2115.29	0.381991	0.190996	+	0.981591i	$0.438828\pi$
$314$	0	0
$315$	4460.36	0.797818
$316$	0	0
$317$	−4739.89	−0.839807	−0.419903	−	0.907569i	$-0.637936\pi$
$317$	−4739.89	−0.839807	−0.419903	+	0.907569i	$0.637936\pi$
$318$	0	0
$319$	480.279	0.0842961
$320$	0	0
$321$	−2521.03	−0.438350
$322$	0	0
$323$	747.513	0.128770
$324$	0	0
$325$	4451.12	0.759705
$326$	0	0
$327$	1345.42	0.227528
$328$	0	0
$329$	−2918.43	−0.489053
$330$	0	0
$331$	6958.68	1.15554	0.577770	−	0.816200i	$-0.303923\pi$
$331$	6958.68	1.15554	0.577770	+	0.816200i	$0.303923\pi$
$332$	0	0
$333$	−4741.79	−0.780326
$334$	0	0
$335$	7575.42	1.23549
$336$	0	0
$337$	2997.55	0.484530	0.242265	−	0.970210i	$-0.422110\pi$
$337$	2997.55	0.484530	0.242265	+	0.970210i	$0.422110\pi$
$338$	0	0
$339$	27.9814	0.00448301
$340$	0	0
$341$	3792.34	0.602248
$342$	0	0
$343$	6769.93	1.06572
$344$	0	0
$345$	945.404	0.147533
$346$	0	0
$347$	−7788.46	−1.20492	−0.602459	−	0.798150i	$-0.705812\pi$
$347$	−7788.46	−1.20492	−0.602459	+	0.798150i	$0.705812\pi$
$348$	0	0
$349$	−6404.16	−0.982254	−0.491127	−	0.871088i	$-0.663415\pi$
$349$	−6404.16	−0.982254	−0.491127	+	0.871088i	$0.663415\pi$
$350$	0	0
$351$	7361.62	1.11947
$352$	0	0
$353$	−5869.46	−0.884986	−0.442493	−	0.896772i	$-0.645906\pi$
$353$	−5869.46	−0.884986	−0.442493	+	0.896772i	$0.645906\pi$
$354$	0	0
$355$	6994.55	1.04572
$356$	0	0
$357$	310.264	0.0459969
$358$	0	0
$359$	2761.48	0.405975	0.202988	−	0.979181i	$-0.434935\pi$
$359$	2761.48	0.405975	0.202988	+	0.979181i	$0.434935\pi$
$360$	0	0
$361$	7303.93	1.06487
$362$	0	0
$363$	3092.20	0.447103
$364$	0	0
$365$	1152.06	0.165209
$366$	0	0
$367$	−3039.36	−0.432298	−0.216149	−	0.976360i	$-0.569350\pi$
$367$	−3039.36	−0.432298	−0.216149	+	0.976360i	$0.569350\pi$
$368$	0	0
$369$	−7046.86	−0.994160
$370$	0	0
$371$	1168.76	0.163556
$372$	0	0
$373$	2051.57	0.284789	0.142394	−	0.989810i	$-0.454520\pi$
$373$	2051.57	0.284789	0.142394	+	0.989810i	$0.454520\pi$
$374$	0	0
$375$	1870.72	0.257609
$376$	0	0
$377$	−1605.66	−0.219351
$378$	0	0
$379$	1852.42	0.251062	0.125531	−	0.992090i	$-0.459937\pi$
$379$	1852.42	0.251062	0.125531	+	0.992090i	$0.459937\pi$
$380$	0	0
$381$	2649.76	0.356302
$382$	0	0
$383$	−5723.43	−0.763587	−0.381793	−	0.924248i	$-0.624693\pi$
$383$	−5723.43	−0.763587	−0.381793	+	0.924248i	$0.624693\pi$
$384$	0	0
$385$	4006.54	0.530369
$386$	0	0
$387$	−3136.59	−0.411995
$388$	0	0
$389$	−4946.70	−0.644750	−0.322375	−	0.946612i	$-0.604481\pi$
$389$	−4946.70	−0.644750	−0.322375	+	0.946612i	$0.604481\pi$
$390$	0	0
$391$	−141.599	−0.0183145
$392$	0	0
$393$	3041.61	0.390405
$394$	0	0
$395$	10632.5	1.35438
$396$	0	0
$397$	9275.78	1.17264	0.586320	−	0.810080i	$-0.300576\pi$
$397$	9275.78	1.17264	0.586320	+	0.810080i	$0.300576\pi$
$398$	0	0
$399$	5878.49	0.737575
$400$	0	0
$401$	5051.01	0.629016	0.314508	−	0.949255i	$-0.398161\pi$
$401$	5051.01	0.629016	0.314508	+	0.949255i	$0.398161\pi$
$402$	0	0
$403$	−12678.4	−1.56714
$404$	0	0
$405$	1558.36	0.191199
$406$	0	0
$407$	−4259.34	−0.518741
$408$	0	0
$409$	5049.42	0.610459	0.305229	−	0.952279i	$-0.401267\pi$
$409$	5049.42	0.610459	0.305229	+	0.952279i	$0.401267\pi$
$410$	0	0
$411$	−3355.14	−0.402669
$412$	0	0
$413$	−737.435	−0.0878616
$414$	0	0
$415$	19495.0	2.30596
$416$	0	0
$417$	−1676.85	−0.196920
$418$	0	0
$419$	−5332.54	−0.621746	−0.310873	−	0.950451i	$-0.600621\pi$
$419$	−5332.54	−0.621746	−0.310873	+	0.950451i	$0.600621\pi$
$420$	0	0
$421$	6195.89	0.717266	0.358633	−	0.933479i	$-0.383243\pi$
$421$	6195.89	0.717266	0.358633	+	0.933479i	$0.383243\pi$
$422$	0	0
$423$	−3187.60	−0.366398
$424$	0	0
$425$	504.961	0.0576334
$426$	0	0
$427$	−11561.2	−1.31027
$428$	0	0
$429$	2683.22	0.301975
$430$	0	0
$431$	−1399.50	−0.156408	−0.0782038	−	0.996937i	$-0.524918\pi$
$431$	−1399.50	−0.156408	−0.0782038	+	0.996937i	$0.524918\pi$
$432$	0	0
$433$	1689.45	0.187505	0.0937526	−	0.995596i	$-0.470114\pi$
$433$	1689.45	0.187505	0.0937526	+	0.995596i	$0.470114\pi$
$434$	0	0
$435$	1216.18	0.134049
$436$	0	0
$437$	−2682.84	−0.293679
$438$	0	0
$439$	12352.8	1.34298	0.671491	−	0.741013i	$-0.265654\pi$
$439$	12352.8	1.34298	0.671491	+	0.741013i	$0.265654\pi$
$440$	0	0
$441$	1070.34	0.115575
$442$	0	0
$443$	13427.1	1.44005	0.720023	−	0.693950i	$-0.244131\pi$
$443$	13427.1	1.44005	0.720023	+	0.693950i	$0.244131\pi$
$444$	0	0
$445$	13723.3	1.46191
$446$	0	0
$447$	−6801.47	−0.719684
$448$	0	0
$449$	2912.34	0.306107	0.153053	−	0.988218i	$-0.451089\pi$
$449$	2912.34	0.306107	0.153053	+	0.988218i	$0.451089\pi$
$450$	0	0
$451$	−6329.88	−0.660892
$452$	0	0
$453$	−9330.78	−0.967767
$454$	0	0
$455$	−13394.6	−1.38010
$456$	0	0
$457$	5906.26	0.604558	0.302279	−	0.953219i	$-0.402253\pi$
$457$	5906.26	0.604558	0.302279	+	0.953219i	$0.402253\pi$
$458$	0	0
$459$	835.144	0.0849263
$460$	0	0
$461$	2847.28	0.287659	0.143830	−	0.989602i	$-0.454058\pi$
$461$	2847.28	0.287659	0.143830	+	0.989602i	$0.454058\pi$
$462$	0	0
$463$	−8561.02	−0.859319	−0.429659	−	0.902991i	$-0.641366\pi$
$463$	−8561.02	−0.859319	−0.429659	+	0.902991i	$0.641366\pi$
$464$	0	0
$465$	9603.09	0.957704
$466$	0	0
$467$	−14998.9	−1.48622	−0.743112	−	0.669167i	$-0.766651\pi$
$467$	−14998.9	−1.48622	−0.743112	+	0.669167i	$0.766651\pi$
$468$	0	0
$469$	−8922.71	−0.878491
$470$	0	0
$471$	8663.68	0.847561
$472$	0	0
$473$	−2817.46	−0.273884
$474$	0	0
$475$	9567.35	0.924169
$476$	0	0
$477$	1276.56	0.122536
$478$	0	0
$479$	8422.27	0.803389	0.401695	−	0.915774i	$-0.368421\pi$
$479$	8422.27	0.803389	0.401695	+	0.915774i	$0.368421\pi$
$480$	0	0
$481$	14239.7	1.34984
$482$	0	0
$483$	−1113.54	−0.104903
$484$	0	0
$485$	−15930.7	−1.49150
$486$	0	0
$487$	1773.91	0.165058	0.0825291	−	0.996589i	$-0.473700\pi$
$487$	1773.91	0.165058	0.0825291	+	0.996589i	$0.473700\pi$
$488$	0	0
$489$	9999.30	0.924711
$490$	0	0
$491$	2157.20	0.198275	0.0991375	−	0.995074i	$-0.468392\pi$
$491$	2157.20	0.198275	0.0991375	+	0.995074i	$0.468392\pi$
$492$	0	0
$493$	−182.155	−0.0166406
$494$	0	0
$495$	4376.06	0.397352
$496$	0	0
$497$	−8238.53	−0.743558
$498$	0	0
$499$	−4303.31	−0.386058	−0.193029	−	0.981193i	$-0.561831\pi$
$499$	−4303.31	−0.386058	−0.193029	+	0.981193i	$0.561831\pi$
$500$	0	0
$501$	−1983.94	−0.176918
$502$	0	0
$503$	−16020.5	−1.42011	−0.710057	−	0.704144i	$-0.751331\pi$
$503$	−16020.5	−1.42011	−0.710057	+	0.704144i	$0.751331\pi$
$504$	0	0
$505$	−24242.6	−2.13620
$506$	0	0
$507$	−2541.59	−0.222635
$508$	0	0
$509$	18269.6	1.59094	0.795469	−	0.605994i	$-0.207224\pi$
$509$	18269.6	1.59094	0.795469	+	0.605994i	$0.207224\pi$
$510$	0	0
$511$	−1356.95	−0.117472
$512$	0	0
$513$	15823.2	1.36182
$514$	0	0
$515$	6296.01	0.538709
$516$	0	0
$517$	−2863.28	−0.243572
$518$	0	0
$519$	2262.76	0.191376
$520$	0	0
$521$	13128.7	1.10399	0.551996	−	0.833847i	$-0.313866\pi$
$521$	13128.7	1.10399	0.551996	+	0.833847i	$0.313866\pi$
$522$	0	0
$523$	−8295.67	−0.693583	−0.346792	−	0.937942i	$-0.612729\pi$
$523$	−8295.67	−0.693583	−0.346792	+	0.937942i	$0.612729\pi$
$524$	0	0
$525$	3971.04	0.330115
$526$	0	0
$527$	−1438.31	−0.118888
$528$	0	0
$529$	−11658.8	−0.958231
$530$	0	0
$531$	−805.449	−0.0658258
$532$	0	0
$533$	21161.9	1.71974
$534$	0	0
$535$	12347.1	0.997775
$536$	0	0
$537$	4137.44	0.332484
$538$	0	0
$539$	961.439	0.0768314
$540$	0	0
$541$	23563.6	1.87261	0.936303	−	0.351194i	$-0.114224\pi$
$541$	23563.6	1.87261	0.936303	+	0.351194i	$0.114224\pi$
$542$	0	0
$543$	−6322.13	−0.499647
$544$	0	0
$545$	−6589.34	−0.517902
$546$	0	0
$547$	−17151.4	−1.34066	−0.670330	−	0.742063i	$-0.733848\pi$
$547$	−17151.4	−1.34066	−0.670330	+	0.742063i	$0.733848\pi$
$548$	0	0
$549$	−12627.5	−0.981656
$550$	0	0
$551$	−3451.23	−0.266838
$552$	0	0
$553$	−12523.5	−0.963027
$554$	0	0
$555$	−10785.6	−0.824910
$556$	0	0
$557$	18572.6	1.41283	0.706415	−	0.707798i	$-0.250311\pi$
$557$	18572.6	1.41283	0.706415	+	0.707798i	$0.250311\pi$
$558$	0	0
$559$	9419.27	0.712688
$560$	0	0
$561$	304.400	0.0229087
$562$	0	0
$563$	−7835.40	−0.586541	−0.293270	−	0.956030i	$-0.594744\pi$
$563$	−7835.40	−0.586541	−0.293270	+	0.956030i	$0.594744\pi$
$564$	0	0
$565$	−137.042	−0.0102043
$566$	0	0
$567$	−1835.52	−0.135951
$568$	0	0
$569$	5891.09	0.434038	0.217019	−	0.976167i	$-0.430367\pi$
$569$	5891.09	0.434038	0.217019	+	0.976167i	$0.430367\pi$
$570$	0	0
$571$	20083.8	1.47195	0.735973	−	0.677010i	$-0.236725\pi$
$571$	20083.8	1.47195	0.735973	+	0.677010i	$0.236725\pi$
$572$	0	0
$573$	4642.13	0.338443
$574$	0	0
$575$	−1812.31	−0.131441
$576$	0	0
$577$	−21008.9	−1.51579	−0.757896	−	0.652375i	$-0.773773\pi$
$577$	−21008.9	−1.51579	−0.757896	+	0.652375i	$0.773773\pi$
$578$	0	0
$579$	5767.11	0.413943
$580$	0	0
$581$	−22962.2	−1.63964
$582$	0	0
$583$	1146.67	0.0814587
$584$	0	0
$585$	−14629.9	−1.03397
$586$	0	0
$587$	20222.3	1.42192	0.710958	−	0.703235i	$-0.248262\pi$
$587$	20222.3	1.42192	0.710958	+	0.703235i	$0.248262\pi$
$588$	0	0
$589$	−27251.4	−1.90640
$590$	0	0
$591$	−14056.3	−0.978339
$592$	0	0
$593$	18803.0	1.30210	0.651051	−	0.759034i	$-0.274328\pi$
$593$	18803.0	1.30210	0.651051	+	0.759034i	$0.274328\pi$
$594$	0	0
$595$	−1519.55	−0.104699
$596$	0	0
$597$	−7879.87	−0.540204
$598$	0	0
$599$	−10612.1	−0.723872	−0.361936	−	0.932203i	$-0.617884\pi$
$599$	−10612.1	−0.723872	−0.361936	+	0.932203i	$0.617884\pi$
$600$	0	0
$601$	27709.6	1.88070	0.940350	−	0.340210i	$-0.110498\pi$
$601$	27709.6	1.88070	0.940350	+	0.340210i	$0.110498\pi$
$602$	0	0
$603$	−9745.64	−0.658164
$604$	0	0
$605$	−15144.4	−1.01770
$606$	0	0
$607$	21945.0	1.46741	0.733707	−	0.679466i	$-0.237788\pi$
$607$	21945.0	1.46741	0.733707	+	0.679466i	$0.237788\pi$
$608$	0	0
$609$	−1432.47	−0.0953150
$610$	0	0
$611$	9572.44	0.633813
$612$	0	0
$613$	−11777.0	−0.775971	−0.387985	−	0.921665i	$-0.626829\pi$
$613$	−11777.0	−0.775971	−0.387985	+	0.921665i	$0.626829\pi$
$614$	0	0
$615$	−16028.7	−1.05096
$616$	0	0
$617$	−27161.1	−1.77223	−0.886115	−	0.463465i	$-0.846606\pi$
$617$	−27161.1	−1.77223	−0.886115	+	0.463465i	$0.846606\pi$
$618$	0	0
$619$	10366.3	0.673116	0.336558	−	0.941663i	$-0.390737\pi$
$619$	10366.3	0.673116	0.336558	+	0.941663i	$0.390737\pi$
$620$	0	0
$621$	−2997.35	−0.193687
$622$	0	0
$623$	−16164.0	−1.03948
$624$	0	0
$625$	−19211.1	−1.22951
$626$	0	0
$627$	5767.38	0.367348
$628$	0	0
$629$	1615.43	0.102403
$630$	0	0
$631$	11208.4	0.707132	0.353566	−	0.935410i	$-0.384969\pi$
$631$	11208.4	0.707132	0.353566	+	0.935410i	$0.384969\pi$
$632$	0	0
$633$	−2057.89	−0.129216
$634$	0	0
$635$	−12977.5	−0.811018
$636$	0	0
$637$	−3214.26	−0.199927
$638$	0	0
$639$	−8998.36	−0.557073
$640$	0	0
$641$	351.957	0.0216872	0.0108436	−	0.999941i	$-0.496548\pi$
$641$	351.957	0.0216872	0.0108436	+	0.999941i	$0.496548\pi$
$642$	0	0
$643$	−27628.0	−1.69447	−0.847234	−	0.531220i	$-0.821734\pi$
$643$	−27628.0	−1.69447	−0.847234	+	0.531220i	$0.821734\pi$
$644$	0	0
$645$	−7134.48	−0.435534
$646$	0	0
$647$	10858.0	0.659771	0.329886	−	0.944021i	$-0.392990\pi$
$647$	10858.0	0.659771	0.329886	+	0.944021i	$0.392990\pi$
$648$	0	0
$649$	−723.498	−0.0437593
$650$	0	0
$651$	−11311.0	−0.680972
$652$	0	0
$653$	−9714.54	−0.582174	−0.291087	−	0.956697i	$-0.594017\pi$
$653$	−9714.54	−0.582174	−0.291087	+	0.956697i	$0.594017\pi$
$654$	0	0
$655$	−14896.7	−0.888643
$656$	0	0
$657$	−1482.10	−0.0880095
$658$	0	0
$659$	−1918.79	−0.113422	−0.0567111	−	0.998391i	$-0.518061\pi$
$659$	−1918.79	−0.113422	−0.0567111	+	0.998391i	$0.518061\pi$
$660$	0	0
$661$	−12106.3	−0.712375	−0.356188	−	0.934414i	$-0.615924\pi$
$661$	−12106.3	−0.712375	−0.356188	+	0.934414i	$0.615924\pi$
$662$	0	0
$663$	−1017.66	−0.0596120
$664$	0	0
$665$	−28790.6	−1.67887
$666$	0	0
$667$	653.756	0.0379513
$668$	0	0
$669$	−11605.2	−0.670679
$670$	0	0
$671$	−11342.7	−0.652580
$672$	0	0
$673$	16348.0	0.936355	0.468178	−	0.883634i	$-0.344911\pi$
$673$	16348.0	0.936355	0.468178	+	0.883634i	$0.344911\pi$
$674$	0	0
$675$	10688.9	0.609507
$676$	0	0
$677$	−19298.2	−1.09555	−0.547776	−	0.836625i	$-0.684525\pi$
$677$	−19298.2	−1.09555	−0.547776	+	0.836625i	$0.684525\pi$
$678$	0	0
$679$	18764.0	1.06052
$680$	0	0
$681$	11634.9	0.654700
$682$	0	0
$683$	−9938.38	−0.556781	−0.278390	−	0.960468i	$-0.589801\pi$
$683$	−9938.38	−0.556781	−0.278390	+	0.960468i	$0.589801\pi$
$684$	0	0
$685$	16432.2	0.916558
$686$	0	0
$687$	1787.93	0.0992924
$688$	0	0
$689$	−3833.54	−0.211968
$690$	0	0
$691$	18610.4	1.02456	0.512282	−	0.858817i	$-0.328800\pi$
$691$	18610.4	1.02456	0.512282	+	0.858817i	$0.328800\pi$
$692$	0	0
$693$	−5154.34	−0.282536
$694$	0	0
$695$	8212.58	0.448231
$696$	0	0
$697$	2400.73	0.130465
$698$	0	0
$699$	3078.09	0.166558
$700$	0	0
$701$	26877.8	1.44816	0.724080	−	0.689716i	$-0.242265\pi$
$701$	26877.8	1.44816	0.724080	+	0.689716i	$0.242265\pi$
$702$	0	0
$703$	30607.2	1.64206
$704$	0	0
$705$	−7250.49	−0.387332
$706$	0	0
$707$	28554.1	1.51894
$708$	0	0
$709$	−3141.28	−0.166394	−0.0831970	−	0.996533i	$-0.526513\pi$
$709$	−3141.28	−0.166394	−0.0831970	+	0.996533i	$0.526513\pi$
$710$	0	0
$711$	−13678.6	−0.721499
$712$	0	0
$713$	5162.14	0.271141
$714$	0	0
$715$	−13141.4	−0.687358
$716$	0	0
$717$	6104.00	0.317933
$718$	0	0
$719$	−27198.6	−1.41076	−0.705379	−	0.708830i	$-0.749223\pi$
$719$	−27198.6	−1.41076	−0.705379	+	0.708830i	$0.749223\pi$
$720$	0	0
$721$	−7415.75	−0.383047
$722$	0	0
$723$	17746.2	0.912847
$724$	0	0
$725$	−2331.38	−0.119428
$726$	0	0
$727$	−15713.1	−0.801603	−0.400802	−	0.916165i	$-0.631268\pi$
$727$	−15713.1	−0.801603	−0.400802	+	0.916165i	$0.631268\pi$
$728$	0	0
$729$	8500.05	0.431847
$730$	0	0
$731$	1068.58	0.0540666
$732$	0	0
$733$	10977.4	0.553153	0.276576	−	0.960992i	$-0.410800\pi$
$733$	10977.4	0.553153	0.276576	+	0.960992i	$0.410800\pi$
$734$	0	0
$735$	2434.59	0.122178
$736$	0	0
$737$	−8754.07	−0.437531
$738$	0	0
$739$	−12586.3	−0.626515	−0.313258	−	0.949668i	$-0.601420\pi$
$739$	−12586.3	−0.626515	−0.313258	+	0.949668i	$0.601420\pi$
$740$	0	0
$741$	−19281.4	−0.955897
$742$	0	0
$743$	−21237.6	−1.04863	−0.524315	−	0.851524i	$-0.675679\pi$
$743$	−21237.6	−1.04863	−0.524315	+	0.851524i	$0.675679\pi$
$744$	0	0
$745$	33311.0	1.63815
$746$	0	0
$747$	−25080.0	−1.22842
$748$	0	0
$749$	−14543.0	−0.709464
$750$	0	0
$751$	−31114.2	−1.51181	−0.755907	−	0.654679i	$-0.772804\pi$
$751$	−31114.2	−1.51181	−0.755907	+	0.654679i	$0.772804\pi$
$752$	0	0
$753$	15126.4	0.732054
$754$	0	0
$755$	45698.6	2.20284
$756$	0	0
$757$	12838.7	0.616423	0.308211	−	0.951318i	$-0.400270\pi$
$757$	12838.7	0.616423	0.308211	+	0.951318i	$0.400270\pi$
$758$	0	0
$759$	−1092.50	−0.0522466
$760$	0	0
$761$	−11665.6	−0.555689	−0.277844	−	0.960626i	$-0.589620\pi$
$761$	−11665.6	−0.555689	−0.277844	+	0.960626i	$0.589620\pi$
$762$	0	0
$763$	7761.25	0.368252
$764$	0	0
$765$	−1659.70	−0.0784401
$766$	0	0
$767$	2418.78	0.113869
$768$	0	0
$769$	−33339.3	−1.56339	−0.781693	−	0.623663i	$-0.785644\pi$
$769$	−33339.3	−1.56339	−0.781693	+	0.623663i	$0.785644\pi$
$770$	0	0
$771$	21182.4	0.989452
$772$	0	0
$773$	−17895.2	−0.832660	−0.416330	−	0.909214i	$-0.636684\pi$
$773$	−17895.2	−0.832660	−0.416330	+	0.909214i	$0.636684\pi$
$774$	0	0
$775$	−18408.9	−0.853246
$776$	0	0
$777$	12703.9	0.586549
$778$	0	0
$779$	45485.9	2.09204
$780$	0	0
$781$	−8082.82	−0.370328
$782$	0	0
$783$	−3855.82	−0.175984
$784$	0	0
$785$	−42431.4	−1.92923
$786$	0	0
$787$	15816.4	0.716384	0.358192	−	0.933648i	$-0.383393\pi$
$787$	15816.4	0.716384	0.358192	+	0.933648i	$0.383393\pi$
$788$	0	0
$789$	14208.0	0.641086
$790$	0	0
$791$	161.415	0.00725570
$792$	0	0
$793$	37920.7	1.69811
$794$	0	0
$795$	2903.65	0.129537
$796$	0	0
$797$	21146.2	0.939822	0.469911	−	0.882714i	$-0.344286\pi$
$797$	21146.2	0.939822	0.469911	+	0.882714i	$0.344286\pi$
$798$	0	0
$799$	1085.95	0.0480828
$800$	0	0
$801$	−17654.8	−0.778779
$802$	0	0
$803$	−1331.30	−0.0585065
$804$	0	0
$805$	5453.71	0.238780
$806$	0	0
$807$	−16268.9	−0.709657
$808$	0	0
$809$	−19190.2	−0.833982	−0.416991	−	0.908911i	$-0.636915\pi$
$809$	−19190.2	−0.833982	−0.416991	+	0.908911i	$0.636915\pi$
$810$	0	0
$811$	37890.9	1.64060	0.820302	−	0.571931i	$-0.193806\pi$
$811$	37890.9	1.64060	0.820302	+	0.571931i	$0.193806\pi$
$812$	0	0
$813$	7416.11	0.319919
$814$	0	0
$815$	−48972.8	−2.10484
$816$	0	0
$817$	20246.0	0.866974
$818$	0	0
$819$	17231.9	0.735201
$820$	0	0
$821$	−11006.2	−0.467868	−0.233934	−	0.972252i	$-0.575160\pi$
$821$	−11006.2	−0.467868	−0.233934	+	0.972252i	$0.575160\pi$
$822$	0	0
$823$	−42361.2	−1.79419	−0.897094	−	0.441839i	$-0.854326\pi$
$823$	−42361.2	−1.79419	−0.897094	+	0.441839i	$0.854326\pi$
$824$	0	0
$825$	3895.99	0.164413
$826$	0	0
$827$	1786.13	0.0751028	0.0375514	−	0.999295i	$-0.488044\pi$
$827$	1786.13	0.0751028	0.0375514	+	0.999295i	$0.488044\pi$
$828$	0	0
$829$	12070.8	0.505712	0.252856	−	0.967504i	$-0.418630\pi$
$829$	12070.8	0.505712	0.252856	+	0.967504i	$0.418630\pi$
$830$	0	0
$831$	−22065.8	−0.921125
$832$	0	0
$833$	−364.644	−0.0151671
$834$	0	0
$835$	9716.60	0.402703
$836$	0	0
$837$	−30446.0	−1.25731
$838$	0	0
$839$	39687.9	1.63311	0.816555	−	0.577267i	$-0.195881\pi$
$839$	39687.9	1.63311	0.816555	+	0.577267i	$0.195881\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−15880.5	−0.648818
$844$	0	0
$845$	12447.7	0.506763
$846$	0	0
$847$	17837.9	0.723632
$848$	0	0
$849$	3797.50	0.153510
$850$	0	0
$851$	−5797.82	−0.233545
$852$	0	0
$853$	36146.8	1.45093	0.725466	−	0.688258i	$-0.241624\pi$
$853$	36146.8	1.45093	0.725466	+	0.688258i	$0.241624\pi$
$854$	0	0
$855$	−31445.9	−1.25781
$856$	0	0
$857$	1778.90	0.0709054	0.0354527	−	0.999371i	$-0.488713\pi$
$857$	1778.90	0.0709054	0.0354527	+	0.999371i	$0.488713\pi$
$858$	0	0
$859$	−37890.5	−1.50501	−0.752507	−	0.658584i	$-0.771156\pi$
$859$	−37890.5	−1.50501	−0.752507	+	0.658584i	$0.771156\pi$
$860$	0	0
$861$	18879.4	0.747282
$862$	0	0
$863$	22754.4	0.897529	0.448764	−	0.893650i	$-0.351864\pi$
$863$	22754.4	0.897529	0.448764	+	0.893650i	$0.351864\pi$
$864$	0	0
$865$	−11082.1	−0.435612
$866$	0	0
$867$	14261.1	0.558629
$868$	0	0
$869$	−12286.8	−0.479634
$870$	0	0
$871$	29266.4	1.13852
$872$	0	0
$873$	20494.6	0.794543
$874$	0	0
$875$	10791.5	0.416937
$876$	0	0
$877$	28543.9	1.09904	0.549521	−	0.835480i	$-0.314810\pi$
$877$	28543.9	1.09904	0.549521	+	0.835480i	$0.314810\pi$
$878$	0	0
$879$	18147.9	0.696376
$880$	0	0
$881$	46848.4	1.79156	0.895779	−	0.444499i	$-0.146618\pi$
$881$	46848.4	1.79156	0.895779	+	0.444499i	$0.146618\pi$
$882$	0	0
$883$	−7020.06	−0.267547	−0.133773	−	0.991012i	$-0.542709\pi$
$883$	−7020.06	−0.267547	−0.133773	+	0.991012i	$0.542709\pi$
$884$	0	0
$885$	−1832.07	−0.0695867
$886$	0	0
$887$	−19200.5	−0.726820	−0.363410	−	0.931629i	$-0.618388\pi$
$887$	−19200.5	−0.726820	−0.363410	+	0.931629i	$0.618388\pi$
$888$	0	0
$889$	15285.5	0.576671
$890$	0	0
$891$	−1800.83	−0.0677104
$892$	0	0
$893$	20575.2	0.771023
$894$	0	0
$895$	−20263.6	−0.756803
$896$	0	0
$897$	3652.41	0.135954
$898$	0	0
$899$	6640.63	0.246360
$900$	0	0
$901$	−434.898	−0.0160805
$902$	0	0
$903$	8403.34	0.309685
$904$	0	0
$905$	30963.4	1.13730
$906$	0	0
$907$	36098.6	1.32154	0.660768	−	0.750590i	$-0.270231\pi$
$907$	36098.6	1.32154	0.660768	+	0.750590i	$0.270231\pi$
$908$	0	0
$909$	31187.6	1.13799
$910$	0	0
$911$	19193.8	0.698047	0.349023	−	0.937114i	$-0.386513\pi$
$911$	19193.8	0.698047	0.349023	+	0.937114i	$0.386513\pi$
$912$	0	0
$913$	−22528.2	−0.816622
$914$	0	0
$915$	−28722.5	−1.03774
$916$	0	0
$917$	17546.0	0.631866
$918$	0	0
$919$	−49242.2	−1.76752	−0.883760	−	0.467941i	$-0.844996\pi$
$919$	−49242.2	−1.76752	−0.883760	+	0.467941i	$0.844996\pi$
$920$	0	0
$921$	−26865.8	−0.961193
$922$	0	0
$923$	27022.3	0.963651
$924$	0	0
$925$	20675.8	0.734936
$926$	0	0
$927$	−8099.70	−0.286978
$928$	0	0
$929$	−43350.9	−1.53100	−0.765498	−	0.643438i	$-0.777507\pi$
$929$	−43350.9	−1.53100	−0.765498	+	0.643438i	$0.777507\pi$
$930$	0	0
$931$	−6908.80	−0.243208
$932$	0	0
$933$	10983.0	0.385387
$934$	0	0
$935$	−1490.84	−0.0521450
$936$	0	0
$937$	39167.8	1.36559	0.682794	−	0.730611i	$-0.260765\pi$
$937$	39167.8	1.36559	0.682794	+	0.730611i	$0.260765\pi$
$938$	0	0
$939$	−6189.81	−0.215119
$940$	0	0
$941$	−13329.9	−0.461789	−0.230895	−	0.972979i	$-0.574165\pi$
$941$	−13329.9	−0.461789	−0.230895	+	0.972979i	$0.574165\pi$
$942$	0	0
$943$	−8616.25	−0.297544
$944$	0	0
$945$	−32165.7	−1.10725
$946$	0	0
$947$	19115.3	0.655926	0.327963	−	0.944690i	$-0.393638\pi$
$947$	19115.3	0.655926	0.327963	+	0.944690i	$0.393638\pi$
$948$	0	0
$949$	4450.78	0.152243
$950$	0	0
$951$	13870.0	0.472938
$952$	0	0
$953$	−29008.4	−0.986018	−0.493009	−	0.870024i	$-0.664103\pi$
$953$	−29008.4	−0.986018	−0.493009	+	0.870024i	$0.664103\pi$
$954$	0	0
$955$	−22735.4	−0.770367
$956$	0	0
$957$	−1405.40	−0.0474715
$958$	0	0
$959$	−19354.7	−0.651715
$960$	0	0
$961$	22644.2	0.760103
$962$	0	0
$963$	−15884.3	−0.531529
$964$	0	0
$965$	−28245.1	−0.942220
$966$	0	0
$967$	−55491.5	−1.84538	−0.922691	−	0.385539i	$-0.874015\pi$
$967$	−55491.5	−1.84538	−0.922691	+	0.385539i	$0.874015\pi$
$968$	0	0
$969$	−2187.39	−0.0725171
$970$	0	0
$971$	−39721.0	−1.31278	−0.656389	−	0.754422i	$-0.727917\pi$
$971$	−39721.0	−1.31278	−0.656389	+	0.754422i	$0.727917\pi$
$972$	0	0
$973$	−9673.18	−0.318713
$974$	0	0
$975$	−13025.0	−0.427829
$976$	0	0
$977$	4352.89	0.142540	0.0712699	−	0.997457i	$-0.477295\pi$
$977$	4352.89	0.142540	0.0712699	+	0.997457i	$0.477295\pi$
$978$	0	0
$979$	−15858.5	−0.517712
$980$	0	0
$981$	8477.07	0.275894
$982$	0	0
$983$	10317.1	0.334754	0.167377	−	0.985893i	$-0.446470\pi$
$983$	10317.1	0.334754	0.167377	+	0.985893i	$0.446470\pi$
$984$	0	0
$985$	68842.4	2.22690
$986$	0	0
$987$	8539.98	0.275411
$988$	0	0
$989$	−3835.13	−0.123307
$990$	0	0
$991$	45439.0	1.45653	0.728264	−	0.685297i	$-0.240328\pi$
$991$	45439.0	1.45653	0.728264	+	0.685297i	$0.240328\pi$
$992$	0	0
$993$	−20362.6	−0.650744
$994$	0	0
$995$	38592.6	1.22962
$996$	0	0
$997$	31888.0	1.01294	0.506471	−	0.862257i	$-0.330950\pi$
$997$	31888.0	1.01294	0.506471	+	0.862257i	$0.330950\pi$
$998$	0	0
$999$	34195.2	1.08297

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.4.a.g.1.2		3
4.3	odd	2		232.4.a.b.1.2	✓	3
8.3	odd	2		1856.4.a.p.1.2		3
8.5	even	2		1856.4.a.u.1.2		3
12.11	even	2		2088.4.a.b.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.b.1.2	✓	3	4.3	odd	2
464.4.a.g.1.2		3	1.1	even	1	trivial
1856.4.a.p.1.2		3	8.3	odd	2
1856.4.a.u.1.2		3	8.5	even	2
2088.4.a.b.1.1		3	12.11	even	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:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	464.a (trivial)

\(f(q)\)	\(=\)	\(q-2.92622 q^{3} +14.3315 q^{5} -16.8804 q^{7} -18.4372 q^{9} +O(q^{10})\)	\(q-2.92622 q^{3} +14.3315 q^{5} -16.8804 q^{7} -18.4372 q^{9} -16.5613 q^{11} +55.3675 q^{13} -41.9372 q^{15} +6.28120 q^{17} +119.008 q^{19} +49.3957 q^{21} -22.5433 q^{23} +80.3924 q^{25} +132.959 q^{27} -29.0000 q^{29} -228.987 q^{31} +48.4621 q^{33} -241.921 q^{35} +257.186 q^{37} -162.017 q^{39} +382.208 q^{41} +170.123 q^{43} -264.234 q^{45} +172.889 q^{47} -58.0532 q^{49} -18.3802 q^{51} -69.2381 q^{53} -237.349 q^{55} -348.244 q^{57} +43.6860 q^{59} +684.892 q^{61} +311.227 q^{63} +793.500 q^{65} +528.585 q^{67} +65.9667 q^{69} +488.054 q^{71} +80.3863 q^{73} -235.246 q^{75} +279.561 q^{77} +741.898 q^{79} +108.737 q^{81} +1360.29 q^{83} +90.0191 q^{85} +84.8604 q^{87} +957.563 q^{89} -934.623 q^{91} +670.068 q^{93} +1705.57 q^{95} -1111.59 q^{97} +305.345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{3} + 4 q^{5} - 16 q^{7} + 45 q^{9}+O(q^{10}) \) 3 * q - 6 * q^3 + 4 * q^5 - 16 * q^7 + 45 * q^9	\( 3 q - 6 q^{3} + 4 q^{5} - 16 q^{7} + 45 q^{9} + 2 q^{11} + 28 q^{13} - 136 q^{15} - 66 q^{17} + 66 q^{19} + 472 q^{21} - 176 q^{23} - 9 q^{25} - 228 q^{27} - 87 q^{29} + 190 q^{31} + 154 q^{33} - 660 q^{35} + 442 q^{37} + 656 q^{39} + 1162 q^{41} - 30 q^{43} - 254 q^{45} + 738 q^{47} + 851 q^{49} + 576 q^{51} + 312 q^{53} - 464 q^{55} + 684 q^{57} - 44 q^{59} + 54 q^{61} - 964 q^{63} + 178 q^{65} + 116 q^{67} + 812 q^{69} + 1200 q^{71} - 1118 q^{73} + 1038 q^{75} + 792 q^{77} + 2262 q^{79} + 15 q^{81} + 1804 q^{83} - 8 q^{85} + 174 q^{87} + 1578 q^{89} + 1972 q^{91} - 706 q^{93} + 1052 q^{95} + 1450 q^{97} + 482 q^{99}+O(q^{100}) \) 3 * q - 6 * q^3 + 4 * q^5 - 16 * q^7 + 45 * q^9 + 2 * q^11 + 28 * q^13 - 136 * q^15 - 66 * q^17 + 66 * q^19 + 472 * q^21 - 176 * q^23 - 9 * q^25 - 228 * q^27 - 87 * q^29 + 190 * q^31 + 154 * q^33 - 660 * q^35 + 442 * q^37 + 656 * q^39 + 1162 * q^41 - 30 * q^43 - 254 * q^45 + 738 * q^47 + 851 * q^49 + 576 * q^51 + 312 * q^53 - 464 * q^55 + 684 * q^57 - 44 * q^59 + 54 * q^61 - 964 * q^63 + 178 * q^65 + 116 * q^67 + 812 * q^69 + 1200 * q^71 - 1118 * q^73 + 1038 * q^75 + 792 * q^77 + 2262 * q^79 + 15 * q^81 + 1804 * q^83 - 8 * q^85 + 174 * q^87 + 1578 * q^89 + 1972 * q^91 - 706 * q^93 + 1052 * q^95 + 1450 * q^97 + 482 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more