LMFDB - Embedded newform 1216.4.a.y.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.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:	$71.7463225670$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 28x^{3} - 8x^{2} + 73x - 12 $ x^5 - 2x^4 - 28x^3 - 8x^2 + 73x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$6.33779$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.31962 q^{3} -9.41363 q^{5} +16.2508 q^{7} +1.29831 q^{9} -8.71440 q^{11} +43.4140 q^{13} -50.0769 q^{15} -124.442 q^{17} +19.0000 q^{19} +86.4480 q^{21} +71.6374 q^{23} -36.3835 q^{25} -136.723 q^{27} -279.955 q^{29} +252.949 q^{31} -46.3573 q^{33} -152.979 q^{35} -17.9550 q^{37} +230.946 q^{39} -466.221 q^{41} +340.723 q^{43} -12.2218 q^{45} -305.434 q^{47} -78.9116 q^{49} -661.985 q^{51} +293.326 q^{53} +82.0342 q^{55} +101.073 q^{57} +408.311 q^{59} -334.310 q^{61} +21.0985 q^{63} -408.683 q^{65} +754.117 q^{67} +381.084 q^{69} -758.951 q^{71} -393.936 q^{73} -193.546 q^{75} -141.616 q^{77} +91.7062 q^{79} -762.369 q^{81} -1396.71 q^{83} +1171.45 q^{85} -1489.25 q^{87} +931.233 q^{89} +705.512 q^{91} +1345.59 q^{93} -178.859 q^{95} -282.166 q^{97} -11.3140 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 6 q^{3} - 5 q^{5} - 7 q^{7} - 5 q^{9} - 13 q^{11} + 72 q^{13} - 72 q^{15} - 59 q^{17} + 95 q^{19} + 224 q^{21} - 52 q^{23} - 86 q^{25} - 54 q^{27} + 128 q^{29} + 110 q^{31} - 68 q^{33} + 45 q^{35}+ \cdots - 601 q^{99}+O(q^{100}) $ 5 * q - 6 * q^3 - 5 * q^5 - 7 * q^7 - 5 * q^9 - 13 * q^11 + 72 * q^13 - 72 * q^15 - 59 * q^17 + 95 * q^19 + 224 * q^21 - 52 * q^23 - 86 * q^25 - 54 * q^27 + 128 * q^29 + 110 * q^31 - 68 * q^33 + 45 * q^35 + 436 * q^37 - 356 * q^39 - 804 * q^41 + 143 * q^43 + 579 * q^45 - 661 * q^47 - 406 * q^49 - 570 * q^51 + 898 * q^53 - 17 * q^55 - 114 * q^57 - 196 * q^59 + 1079 * q^61 - 143 * q^63 - 1632 * q^65 - 832 * q^67 + 1644 * q^69 - 834 * q^71 - 1375 * q^73 - 1054 * q^75 + 1473 * q^77 - 154 * q^79 - 1859 * q^81 - 2224 * q^83 + 1807 * q^85 - 2004 * q^87 - 542 * q^89 - 1890 * q^91 + 2788 * q^93 - 95 * q^95 - 1528 * q^97 - 601 * 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
$3$	5.31962	1.02376	0.511880	−	0.859057i	$-0.328949\pi$
$3$	5.31962	1.02376	0.511880	+	0.859057i	$0.328949\pi$
$4$	0	0
$5$	−9.41363	−0.841981	−0.420990	−	0.907065i	$-0.638317\pi$
$5$	−9.41363	−0.841981	−0.420990	+	0.907065i	$0.638317\pi$
$6$	0	0
$7$	16.2508	0.877461	0.438730	−	0.898619i	$-0.355428\pi$
$7$	16.2508	0.877461	0.438730	+	0.898619i	$0.355428\pi$
$8$	0	0
$9$	1.29831	0.0480854
$10$	0	0
$11$	−8.71440	−0.238863	−0.119431	−	0.992842i	$-0.538107\pi$
$11$	−8.71440	−0.238863	−0.119431	+	0.992842i	$0.538107\pi$
$12$	0	0
$13$	43.4140	0.926221	0.463110	−	0.886301i	$-0.346733\pi$
$13$	43.4140	0.926221	0.463110	+	0.886301i	$0.346733\pi$
$14$	0	0
$15$	−50.0769	−0.861987
$16$	0	0
$17$	−124.442	−1.77539	−0.887697	−	0.460428i	$-0.847696\pi$
$17$	−124.442	−1.77539	−0.887697	+	0.460428i	$0.847696\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	86.4480	0.898309
$22$	0	0
$23$	71.6374	0.649454	0.324727	−	0.945808i	$-0.394728\pi$
$23$	71.6374	0.649454	0.324727	+	0.945808i	$0.394728\pi$
$24$	0	0
$25$	−36.3835	−0.291068
$26$	0	0
$27$	−136.723	−0.974532
$28$	0	0
$29$	−279.955	−1.79263	−0.896315	−	0.443418i	$-0.853766\pi$
$29$	−279.955	−1.79263	−0.896315	+	0.443418i	$0.853766\pi$
$30$	0	0
$31$	252.949	1.46551	0.732757	−	0.680490i	$-0.238233\pi$
$31$	252.949	1.46551	0.732757	+	0.680490i	$0.238233\pi$
$32$	0	0
$33$	−46.3573	−0.244538
$34$	0	0
$35$	−152.979	−0.738805
$36$	0	0
$37$	−17.9550	−0.0797778	−0.0398889	−	0.999204i	$-0.512700\pi$
$37$	−17.9550	−0.0797778	−0.0398889	+	0.999204i	$0.512700\pi$
$38$	0	0
$39$	230.946	0.948228
$40$	0	0
$41$	−466.221	−1.77589	−0.887944	−	0.459951i	$-0.847867\pi$
$41$	−466.221	−1.77589	−0.887944	+	0.459951i	$0.847867\pi$
$42$	0	0
$43$	340.723	1.20837	0.604184	−	0.796845i	$-0.293499\pi$
$43$	340.723	1.20837	0.604184	+	0.796845i	$0.293499\pi$
$44$	0	0
$45$	−12.2218	−0.0404870
$46$	0	0
$47$	−305.434	−0.947916	−0.473958	−	0.880547i	$-0.657175\pi$
$47$	−305.434	−0.947916	−0.473958	+	0.880547i	$0.657175\pi$
$48$	0	0
$49$	−78.9116	−0.230063
$50$	0	0
$51$	−661.985	−1.81758
$52$	0	0
$53$	293.326	0.760216	0.380108	−	0.924942i	$-0.375887\pi$
$53$	293.326	0.760216	0.380108	+	0.924942i	$0.375887\pi$
$54$	0	0
$55$	82.0342	0.201118
$56$	0	0
$57$	101.073	0.234867
$58$	0	0
$59$	408.311	0.900976	0.450488	−	0.892782i	$-0.351250\pi$
$59$	408.311	0.900976	0.450488	+	0.892782i	$0.351250\pi$
$60$	0	0
$61$	−334.310	−0.701704	−0.350852	−	0.936431i	$-0.614108\pi$
$61$	−334.310	−0.701704	−0.350852	+	0.936431i	$0.614108\pi$
$62$	0	0
$63$	21.0985	0.0421930
$64$	0	0
$65$	−408.683	−0.779860
$66$	0	0
$67$	754.117	1.37507	0.687537	−	0.726149i	$-0.258692\pi$
$67$	754.117	1.37507	0.687537	+	0.726149i	$0.258692\pi$
$68$	0	0
$69$	381.084	0.664885
$70$	0	0
$71$	−758.951	−1.26861	−0.634303	−	0.773085i	$-0.718713\pi$
$71$	−758.951	−1.26861	−0.634303	+	0.773085i	$0.718713\pi$
$72$	0	0
$73$	−393.936	−0.631599	−0.315800	−	0.948826i	$-0.602273\pi$
$73$	−393.936	−0.631599	−0.315800	+	0.948826i	$0.602273\pi$
$74$	0	0
$75$	−193.546	−0.297984
$76$	0	0
$77$	−141.616	−0.209593
$78$	0	0
$79$	91.7062	0.130605	0.0653023	−	0.997866i	$-0.479199\pi$
$79$	91.7062	0.130605	0.0653023	+	0.997866i	$0.479199\pi$
$80$	0	0
$81$	−762.369	−1.04577
$82$	0	0
$83$	−1396.71	−1.84710	−0.923550	−	0.383478i	$-0.874726\pi$
$83$	−1396.71	−1.84710	−0.923550	+	0.383478i	$0.874726\pi$
$84$	0	0
$85$	1171.45	1.49485
$86$	0	0
$87$	−1489.25	−1.83522
$88$	0	0
$89$	931.233	1.10911	0.554553	−	0.832148i	$-0.312889\pi$
$89$	931.233	1.10911	0.554553	+	0.832148i	$0.312889\pi$
$90$	0	0
$91$	705.512	0.812722
$92$	0	0
$93$	1345.59	1.50034
$94$	0	0
$95$	−178.859	−0.193164
$96$	0	0
$97$	−282.166	−0.295357	−0.147679	−	0.989035i	$-0.547180\pi$
$97$	−282.166	−0.295357	−0.147679	+	0.989035i	$0.547180\pi$
$98$	0	0
$99$	−11.3140	−0.0114858
$100$	0	0
$101$	−57.3942	−0.0565440	−0.0282720	−	0.999600i	$-0.509000\pi$
$101$	−57.3942	−0.0565440	−0.0282720	+	0.999600i	$0.509000\pi$
$102$	0	0
$103$	−1320.03	−1.26278	−0.631390	−	0.775466i	$-0.717515\pi$
$103$	−1320.03	−1.26278	−0.631390	+	0.775466i	$0.717515\pi$
$104$	0	0
$105$	−813.789	−0.756359
$106$	0	0
$107$	−749.733	−0.677377	−0.338689	−	0.940898i	$-0.609983\pi$
$107$	−749.733	−0.677377	−0.338689	+	0.940898i	$0.609983\pi$
$108$	0	0
$109$	−645.208	−0.566970	−0.283485	−	0.958977i	$-0.591491\pi$
$109$	−645.208	−0.566970	−0.283485	+	0.958977i	$0.591491\pi$
$110$	0	0
$111$	−95.5136	−0.0816734
$112$	0	0
$113$	−2308.79	−1.92206	−0.961029	−	0.276448i	$-0.910843\pi$
$113$	−2308.79	−1.92206	−0.961029	+	0.276448i	$0.910843\pi$
$114$	0	0
$115$	−674.369	−0.546828
$116$	0	0
$117$	56.3646	0.0445377
$118$	0	0
$119$	−2022.29	−1.55784
$120$	0	0
$121$	−1255.06	−0.942945
$122$	0	0
$123$	−2480.11	−1.81808
$124$	0	0
$125$	1519.21	1.08705
$126$	0	0
$127$	311.645	0.217749	0.108874	−	0.994056i	$-0.465275\pi$
$127$	311.645	0.217749	0.108874	+	0.994056i	$0.465275\pi$
$128$	0	0
$129$	1812.52	1.23708
$130$	0	0
$131$	514.521	0.343160	0.171580	−	0.985170i	$-0.445113\pi$
$131$	514.521	0.343160	0.171580	+	0.985170i	$0.445113\pi$
$132$	0	0
$133$	308.765	0.201303
$134$	0	0
$135$	1287.06	0.820538
$136$	0	0
$137$	−2201.49	−1.37289	−0.686446	−	0.727181i	$-0.740830\pi$
$137$	−2201.49	−1.37289	−0.686446	+	0.727181i	$0.740830\pi$
$138$	0	0
$139$	188.406	0.114967	0.0574835	−	0.998346i	$-0.481692\pi$
$139$	188.406	0.114967	0.0574835	+	0.998346i	$0.481692\pi$
$140$	0	0
$141$	−1624.79	−0.970439
$142$	0	0
$143$	−378.327	−0.221240
$144$	0	0
$145$	2635.39	1.50936
$146$	0	0
$147$	−419.779	−0.235529
$148$	0	0
$149$	2002.07	1.10078	0.550390	−	0.834908i	$-0.314479\pi$
$149$	2002.07	1.10078	0.550390	+	0.834908i	$0.314479\pi$
$150$	0	0
$151$	−1324.14	−0.713621	−0.356810	−	0.934177i	$-0.616136\pi$
$151$	−1324.14	−0.713621	−0.356810	+	0.934177i	$0.616136\pi$
$152$	0	0
$153$	−161.564	−0.0853705
$154$	0	0
$155$	−2381.17	−1.23394
$156$	0	0
$157$	−360.790	−0.183403	−0.0917013	−	0.995787i	$-0.529230\pi$
$157$	−360.790	−0.183403	−0.0917013	+	0.995787i	$0.529230\pi$
$158$	0	0
$159$	1560.38	0.778279
$160$	0	0
$161$	1164.17	0.569870
$162$	0	0
$163$	634.639	0.304962	0.152481	−	0.988306i	$-0.451274\pi$
$163$	634.639	0.304962	0.152481	+	0.988306i	$0.451274\pi$
$164$	0	0
$165$	436.390	0.205897
$166$	0	0
$167$	3740.13	1.73305	0.866526	−	0.499131i	$-0.166348\pi$
$167$	3740.13	1.73305	0.866526	+	0.499131i	$0.166348\pi$
$168$	0	0
$169$	−312.227	−0.142115
$170$	0	0
$171$	24.6678	0.0110315
$172$	0	0
$173$	−978.817	−0.430162	−0.215081	−	0.976596i	$-0.569002\pi$
$173$	−978.817	−0.430162	−0.215081	+	0.976596i	$0.569002\pi$
$174$	0	0
$175$	−591.262	−0.255401
$176$	0	0
$177$	2172.06	0.922384
$178$	0	0
$179$	259.331	0.108287	0.0541434	−	0.998533i	$-0.482757\pi$
$179$	259.331	0.108287	0.0541434	+	0.998533i	$0.482757\pi$
$180$	0	0
$181$	−892.412	−0.366478	−0.183239	−	0.983068i	$-0.558658\pi$
$181$	−892.412	−0.366478	−0.183239	+	0.983068i	$0.558658\pi$
$182$	0	0
$183$	−1778.40	−0.718377
$184$	0	0
$185$	169.022	0.0671714
$186$	0	0
$187$	1084.44	0.424076
$188$	0	0
$189$	−2221.86	−0.855114
$190$	0	0
$191$	−4020.76	−1.52320	−0.761601	−	0.648046i	$-0.775587\pi$
$191$	−4020.76	−1.52320	−0.761601	+	0.648046i	$0.775587\pi$
$192$	0	0
$193$	407.021	0.151803	0.0759015	−	0.997115i	$-0.475817\pi$
$193$	407.021	0.151803	0.0759015	+	0.997115i	$0.475817\pi$
$194$	0	0
$195$	−2174.04	−0.798390
$196$	0	0
$197$	3699.86	1.33809	0.669046	−	0.743221i	$-0.266703\pi$
$197$	3699.86	1.33809	0.669046	+	0.743221i	$0.266703\pi$
$198$	0	0
$199$	−1802.62	−0.642131	−0.321066	−	0.947057i	$-0.604041\pi$
$199$	−1802.62	−0.642131	−0.321066	+	0.947057i	$0.604041\pi$
$200$	0	0
$201$	4011.61	1.40775
$202$	0	0
$203$	−4549.49	−1.57296
$204$	0	0
$205$	4388.83	1.49526
$206$	0	0
$207$	93.0073	0.0312292
$208$	0	0
$209$	−165.574	−0.0547989
$210$	0	0
$211$	−5032.87	−1.64207	−0.821035	−	0.570878i	$-0.806603\pi$
$211$	−5032.87	−1.64207	−0.821035	+	0.570878i	$0.806603\pi$
$212$	0	0
$213$	−4037.33	−1.29875
$214$	0	0
$215$	−3207.45	−1.01742
$216$	0	0
$217$	4110.62	1.28593
$218$	0	0
$219$	−2095.59	−0.646606
$220$	0	0
$221$	−5402.54	−1.64441
$222$	0	0
$223$	417.179	0.125275	0.0626376	−	0.998036i	$-0.480049\pi$
$223$	417.179	0.125275	0.0626376	+	0.998036i	$0.480049\pi$
$224$	0	0
$225$	−47.2370	−0.0139961
$226$	0	0
$227$	3166.92	0.925974	0.462987	−	0.886365i	$-0.346778\pi$
$227$	3166.92	0.925974	0.462987	+	0.886365i	$0.346778\pi$
$228$	0	0
$229$	2198.00	0.634271	0.317135	−	0.948380i	$-0.397279\pi$
$229$	2198.00	0.634271	0.317135	+	0.948380i	$0.397279\pi$
$230$	0	0
$231$	−753.343	−0.214573
$232$	0	0
$233$	1956.70	0.550161	0.275081	−	0.961421i	$-0.411295\pi$
$233$	1956.70	0.550161	0.275081	+	0.961421i	$0.411295\pi$
$234$	0	0
$235$	2875.24	0.798127
$236$	0	0
$237$	487.842	0.133708
$238$	0	0
$239$	358.707	0.0970829	0.0485415	−	0.998821i	$-0.484543\pi$
$239$	358.707	0.0970829	0.0485415	+	0.998821i	$0.484543\pi$
$240$	0	0
$241$	51.0894	0.0136554	0.00682771	−	0.999977i	$-0.497827\pi$
$241$	51.0894	0.0136554	0.00682771	+	0.999977i	$0.497827\pi$
$242$	0	0
$243$	−363.984	−0.0960887
$244$	0	0
$245$	742.845	0.193709
$246$	0	0
$247$	824.865	0.212490
$248$	0	0
$249$	−7429.98	−1.89099
$250$	0	0
$251$	460.611	0.115831	0.0579153	−	0.998322i	$-0.481555\pi$
$251$	460.611	0.115831	0.0579153	+	0.998322i	$0.481555\pi$
$252$	0	0
$253$	−624.278	−0.155130
$254$	0	0
$255$	6231.69	1.53037
$256$	0	0
$257$	1349.26	0.327489	0.163744	−	0.986503i	$-0.447643\pi$
$257$	1349.26	0.327489	0.163744	+	0.986503i	$0.447643\pi$
$258$	0	0
$259$	−291.783	−0.0700019
$260$	0	0
$261$	−363.467	−0.0861993
$262$	0	0
$263$	312.198	0.0731977	0.0365988	−	0.999330i	$-0.488348\pi$
$263$	312.198	0.0731977	0.0365988	+	0.999330i	$0.488348\pi$
$264$	0	0
$265$	−2761.26	−0.640087
$266$	0	0
$267$	4953.80	1.13546
$268$	0	0
$269$	−4178.51	−0.947093	−0.473546	−	0.880769i	$-0.657026\pi$
$269$	−4178.51	−0.947093	−0.473546	+	0.880769i	$0.657026\pi$
$270$	0	0
$271$	2140.56	0.479815	0.239907	−	0.970796i	$-0.422883\pi$
$271$	2140.56	0.479815	0.239907	+	0.970796i	$0.422883\pi$
$272$	0	0
$273$	3753.05	0.832033
$274$	0	0
$275$	317.061	0.0695254
$276$	0	0
$277$	6290.18	1.36440	0.682202	−	0.731164i	$-0.261022\pi$
$277$	6290.18	1.36440	0.682202	+	0.731164i	$0.261022\pi$
$278$	0	0
$279$	328.405	0.0704698
$280$	0	0
$281$	5960.18	1.26532	0.632659	−	0.774430i	$-0.281963\pi$
$281$	5960.18	1.26532	0.632659	+	0.774430i	$0.281963\pi$
$282$	0	0
$283$	−2151.83	−0.451990	−0.225995	−	0.974128i	$-0.572563\pi$
$283$	−2151.83	−0.451990	−0.225995	+	0.974128i	$0.572563\pi$
$284$	0	0
$285$	−951.461	−0.197753
$286$	0	0
$287$	−7576.46	−1.55827
$288$	0	0
$289$	10572.9	2.15203
$290$	0	0
$291$	−1501.02	−0.302375
$292$	0	0
$293$	−6159.82	−1.22819	−0.614096	−	0.789231i	$-0.710479\pi$
$293$	−6159.82	−1.22819	−0.614096	+	0.789231i	$0.710479\pi$
$294$	0	0
$295$	−3843.69	−0.758605
$296$	0	0
$297$	1191.46	0.232780
$298$	0	0
$299$	3110.07	0.601538
$300$	0	0
$301$	5537.03	1.06030
$302$	0	0
$303$	−305.315	−0.0578875
$304$	0	0
$305$	3147.07	0.590821
$306$	0	0
$307$	9672.80	1.79823	0.899114	−	0.437714i	$-0.144212\pi$
$307$	9672.80	1.79823	0.899114	+	0.437714i	$0.144212\pi$
$308$	0	0
$309$	−7022.04	−1.29278
$310$	0	0
$311$	−6369.85	−1.16142	−0.580709	−	0.814111i	$-0.697224\pi$
$311$	−6369.85	−1.16142	−0.580709	+	0.814111i	$0.697224\pi$
$312$	0	0
$313$	−3346.98	−0.604417	−0.302209	−	0.953242i	$-0.597724\pi$
$313$	−3346.98	−0.604417	−0.302209	+	0.953242i	$0.597724\pi$
$314$	0	0
$315$	−198.614	−0.0355257
$316$	0	0
$317$	−10720.3	−1.89941	−0.949706	−	0.313142i	$-0.898618\pi$
$317$	−10720.3	−1.89941	−0.949706	+	0.313142i	$0.898618\pi$
$318$	0	0
$319$	2439.64	0.428193
$320$	0	0
$321$	−3988.29	−0.693472
$322$	0	0
$323$	−2364.40	−0.407303
$324$	0	0
$325$	−1579.55	−0.269594
$326$	0	0
$327$	−3432.26	−0.580442
$328$	0	0
$329$	−4963.54	−0.831759
$330$	0	0
$331$	−6992.48	−1.16115	−0.580576	−	0.814206i	$-0.697173\pi$
$331$	−6992.48	−1.16115	−0.580576	+	0.814206i	$0.697173\pi$
$332$	0	0
$333$	−23.3110	−0.00383615
$334$	0	0
$335$	−7098.98	−1.15779
$336$	0	0
$337$	−7780.91	−1.25772	−0.628862	−	0.777517i	$-0.716479\pi$
$337$	−7780.91	−1.25772	−0.628862	+	0.777517i	$0.716479\pi$
$338$	0	0
$339$	−12281.9	−1.96773
$340$	0	0
$341$	−2204.30	−0.350057
$342$	0	0
$343$	−6856.40	−1.07933
$344$	0	0
$345$	−3587.38	−0.559821
$346$	0	0
$347$	2574.64	0.398311	0.199156	−	0.979968i	$-0.436180\pi$
$347$	2574.64	0.398311	0.199156	+	0.979968i	$0.436180\pi$
$348$	0	0
$349$	−4736.45	−0.726465	−0.363232	−	0.931699i	$-0.618327\pi$
$349$	−4736.45	−0.726465	−0.363232	+	0.931699i	$0.618327\pi$
$350$	0	0
$351$	−5935.69	−0.902632
$352$	0	0
$353$	8513.83	1.28370	0.641849	−	0.766831i	$-0.278167\pi$
$353$	8513.83	1.28370	0.641849	+	0.766831i	$0.278167\pi$
$354$	0	0
$355$	7144.49	1.06814
$356$	0	0
$357$	−10757.8	−1.59485
$358$	0	0
$359$	7961.99	1.17052	0.585261	−	0.810845i	$-0.300992\pi$
$359$	7961.99	1.17052	0.585261	+	0.810845i	$0.300992\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−6676.43	−0.965349
$364$	0	0
$365$	3708.37	0.531794
$366$	0	0
$367$	2283.26	0.324755	0.162377	−	0.986729i	$-0.448084\pi$
$367$	2283.26	0.324755	0.162377	+	0.986729i	$0.448084\pi$
$368$	0	0
$369$	−605.297	−0.0853943
$370$	0	0
$371$	4766.78	0.667060
$372$	0	0
$373$	−7740.69	−1.07452	−0.537262	−	0.843415i	$-0.680542\pi$
$373$	−7740.69	−1.07452	−0.537262	+	0.843415i	$0.680542\pi$
$374$	0	0
$375$	8081.59	1.11288
$376$	0	0
$377$	−12153.9	−1.66037
$378$	0	0
$379$	−8461.28	−1.14677	−0.573386	−	0.819285i	$-0.694371\pi$
$379$	−8461.28	−1.14677	−0.573386	+	0.819285i	$0.694371\pi$
$380$	0	0
$381$	1657.83	0.222922
$382$	0	0
$383$	11065.4	1.47628	0.738139	−	0.674649i	$-0.235705\pi$
$383$	11065.4	1.47628	0.738139	+	0.674649i	$0.235705\pi$
$384$	0	0
$385$	1333.12	0.176473
$386$	0	0
$387$	442.363	0.0581049
$388$	0	0
$389$	9051.42	1.17976	0.589878	−	0.807492i	$-0.299176\pi$
$389$	9051.42	1.17976	0.589878	+	0.807492i	$0.299176\pi$
$390$	0	0
$391$	−8914.73	−1.15304
$392$	0	0
$393$	2737.06	0.351313
$394$	0	0
$395$	−863.289	−0.109966
$396$	0	0
$397$	−10777.0	−1.36243	−0.681213	−	0.732085i	$-0.738547\pi$
$397$	−10777.0	−1.36243	−0.681213	+	0.732085i	$0.738547\pi$
$398$	0	0
$399$	1642.51	0.206086
$400$	0	0
$401$	610.869	0.0760732	0.0380366	−	0.999276i	$-0.487890\pi$
$401$	610.869	0.0760732	0.0380366	+	0.999276i	$0.487890\pi$
$402$	0	0
$403$	10981.5	1.35739
$404$	0	0
$405$	7176.66	0.880521
$406$	0	0
$407$	156.467	0.0190560
$408$	0	0
$409$	−218.808	−0.0264532	−0.0132266	−	0.999913i	$-0.504210\pi$
$409$	−218.808	−0.0264532	−0.0132266	+	0.999913i	$0.504210\pi$
$410$	0	0
$411$	−11711.1	−1.40551
$412$	0	0
$413$	6635.38	0.790571
$414$	0	0
$415$	13148.2	1.55522
$416$	0	0
$417$	1002.25	0.117699
$418$	0	0
$419$	4826.13	0.562702	0.281351	−	0.959605i	$-0.409218\pi$
$419$	4826.13	0.562702	0.281351	+	0.959605i	$0.409218\pi$
$420$	0	0
$421$	9386.61	1.08664	0.543320	−	0.839526i	$-0.317167\pi$
$421$	9386.61	1.08664	0.543320	+	0.839526i	$0.317167\pi$
$422$	0	0
$423$	−396.546	−0.0455809
$424$	0	0
$425$	4527.65	0.516761
$426$	0	0
$427$	−5432.80	−0.615718
$428$	0	0
$429$	−2012.55	−0.226496
$430$	0	0
$431$	8452.72	0.944671	0.472336	−	0.881419i	$-0.343411\pi$
$431$	8452.72	0.944671	0.472336	+	0.881419i	$0.343411\pi$
$432$	0	0
$433$	−10404.1	−1.15471	−0.577355	−	0.816494i	$-0.695915\pi$
$433$	−10404.1	−1.15471	−0.577355	+	0.816494i	$0.695915\pi$
$434$	0	0
$435$	14019.3	1.54522
$436$	0	0
$437$	1361.11	0.148995
$438$	0	0
$439$	9961.31	1.08298	0.541489	−	0.840708i	$-0.317861\pi$
$439$	9961.31	1.08298	0.541489	+	0.840708i	$0.317861\pi$
$440$	0	0
$441$	−102.451	−0.0110627
$442$	0	0
$443$	−8962.09	−0.961178	−0.480589	−	0.876946i	$-0.659577\pi$
$443$	−8962.09	−0.961178	−0.480589	+	0.876946i	$0.659577\pi$
$444$	0	0
$445$	−8766.28	−0.933846
$446$	0	0
$447$	10650.3	1.12694
$448$	0	0
$449$	8431.24	0.886180	0.443090	−	0.896477i	$-0.353882\pi$
$449$	8431.24	0.886180	0.443090	+	0.896477i	$0.353882\pi$
$450$	0	0
$451$	4062.83	0.424194
$452$	0	0
$453$	−7043.90	−0.730577
$454$	0	0
$455$	−6641.43	−0.684296
$456$	0	0
$457$	−13669.0	−1.39915	−0.699574	−	0.714560i	$-0.746627\pi$
$457$	−13669.0	−1.39915	−0.699574	+	0.714560i	$0.746627\pi$
$458$	0	0
$459$	17014.1	1.73018
$460$	0	0
$461$	−270.681	−0.0273468	−0.0136734	−	0.999907i	$-0.504353\pi$
$461$	−270.681	−0.0273468	−0.0136734	+	0.999907i	$0.504353\pi$
$462$	0	0
$463$	17732.8	1.77995	0.889973	−	0.456013i	$-0.150723\pi$
$463$	17732.8	1.77995	0.889973	+	0.456013i	$0.150723\pi$
$464$	0	0
$465$	−12666.9	−1.26325
$466$	0	0
$467$	−55.8896	−0.00553803	−0.00276902	−	0.999996i	$-0.500881\pi$
$467$	−55.8896	−0.00553803	−0.00276902	+	0.999996i	$0.500881\pi$
$468$	0	0
$469$	12255.0	1.20657
$470$	0	0
$471$	−1919.27	−0.187760
$472$	0	0
$473$	−2969.20	−0.288634
$474$	0	0
$475$	−691.287	−0.0667757
$476$	0	0
$477$	380.827	0.0365553
$478$	0	0
$479$	2012.67	0.191986	0.0959929	−	0.995382i	$-0.469397\pi$
$479$	2012.67	0.191986	0.0959929	+	0.995382i	$0.469397\pi$
$480$	0	0
$481$	−779.497	−0.0738919
$482$	0	0
$483$	6192.91	0.583411
$484$	0	0
$485$	2656.21	0.248685
$486$	0	0
$487$	4313.69	0.401380	0.200690	−	0.979655i	$-0.435682\pi$
$487$	4313.69	0.401380	0.200690	+	0.979655i	$0.435682\pi$
$488$	0	0
$489$	3376.03	0.312208
$490$	0	0
$491$	3762.75	0.345846	0.172923	−	0.984935i	$-0.444679\pi$
$491$	3762.75	0.345846	0.172923	+	0.984935i	$0.444679\pi$
$492$	0	0
$493$	34838.2	3.18263
$494$	0	0
$495$	106.505	0.00967083
$496$	0	0
$497$	−12333.6	−1.11315
$498$	0	0
$499$	4203.71	0.377122	0.188561	−	0.982061i	$-0.439618\pi$
$499$	4203.71	0.377122	0.188561	+	0.982061i	$0.439618\pi$
$500$	0	0
$501$	19896.0	1.77423
$502$	0	0
$503$	19614.2	1.73868	0.869340	−	0.494215i	$-0.164545\pi$
$503$	19614.2	1.73868	0.869340	+	0.494215i	$0.164545\pi$
$504$	0	0
$505$	540.288	0.0476089
$506$	0	0
$507$	−1660.93	−0.145492
$508$	0	0
$509$	18720.4	1.63019	0.815093	−	0.579330i	$-0.196686\pi$
$509$	18720.4	1.63019	0.815093	+	0.579330i	$0.196686\pi$
$510$	0	0
$511$	−6401.78	−0.554203
$512$	0	0
$513$	−2597.74	−0.223573
$514$	0	0
$515$	12426.3	1.06324
$516$	0	0
$517$	2661.67	0.226422
$518$	0	0
$519$	−5206.93	−0.440383
$520$	0	0
$521$	20279.4	1.70529	0.852646	−	0.522489i	$-0.174996\pi$
$521$	20279.4	1.70529	0.852646	+	0.522489i	$0.174996\pi$
$522$	0	0
$523$	−12501.4	−1.04522	−0.522608	−	0.852573i	$-0.675041\pi$
$523$	−12501.4	−1.04522	−0.522608	+	0.852573i	$0.675041\pi$
$524$	0	0
$525$	−3145.28	−0.261469
$526$	0	0
$527$	−31477.5	−2.60187
$528$	0	0
$529$	−7035.08	−0.578210
$530$	0	0
$531$	530.113	0.0433238
$532$	0	0
$533$	−20240.5	−1.64486
$534$	0	0
$535$	7057.71	0.570339
$536$	0	0
$537$	1379.54	0.110860
$538$	0	0
$539$	687.667	0.0549535
$540$	0	0
$541$	11028.5	0.876437	0.438219	−	0.898868i	$-0.355610\pi$
$541$	11028.5	0.876437	0.438219	+	0.898868i	$0.355610\pi$
$542$	0	0
$543$	−4747.29	−0.375185
$544$	0	0
$545$	6073.75	0.477378
$546$	0	0
$547$	20679.4	1.61643	0.808215	−	0.588887i	$-0.200434\pi$
$547$	20679.4	1.61643	0.808215	+	0.588887i	$0.200434\pi$
$548$	0	0
$549$	−434.036	−0.0337417
$550$	0	0
$551$	−5319.14	−0.411258
$552$	0	0
$553$	1490.30	0.114600
$554$	0	0
$555$	899.129	0.0687674
$556$	0	0
$557$	6393.51	0.486359	0.243179	−	0.969981i	$-0.421810\pi$
$557$	6393.51	0.486359	0.243179	+	0.969981i	$0.421810\pi$
$558$	0	0
$559$	14792.2	1.11922
$560$	0	0
$561$	5768.81	0.434152
$562$	0	0
$563$	−2530.58	−0.189434	−0.0947170	−	0.995504i	$-0.530195\pi$
$563$	−2530.58	−0.189434	−0.0947170	+	0.995504i	$0.530195\pi$
$564$	0	0
$565$	21734.1	1.61834
$566$	0	0
$567$	−12389.1	−0.917625
$568$	0	0
$569$	18965.6	1.39733	0.698664	−	0.715450i	$-0.253778\pi$
$569$	18965.6	1.39733	0.698664	+	0.715450i	$0.253778\pi$
$570$	0	0
$571$	19660.1	1.44089	0.720445	−	0.693512i	$-0.243938\pi$
$571$	19660.1	1.44089	0.720445	+	0.693512i	$0.243938\pi$
$572$	0	0
$573$	−21388.9	−1.55939
$574$	0	0
$575$	−2606.42	−0.189035
$576$	0	0
$577$	−5446.15	−0.392940	−0.196470	−	0.980510i	$-0.562948\pi$
$577$	−5446.15	−0.392940	−0.196470	+	0.980510i	$0.562948\pi$
$578$	0	0
$579$	2165.19	0.155410
$580$	0	0
$581$	−22697.7	−1.62076
$582$	0	0
$583$	−2556.16	−0.181587
$584$	0	0
$585$	−530.596	−0.0374999
$586$	0	0
$587$	3622.30	0.254699	0.127349	−	0.991858i	$-0.459353\pi$
$587$	3622.30	0.254699	0.127349	+	0.991858i	$0.459353\pi$
$588$	0	0
$589$	4806.03	0.336212
$590$	0	0
$591$	19681.8	1.36989
$592$	0	0
$593$	19137.7	1.32528	0.662641	−	0.748938i	$-0.269436\pi$
$593$	19137.7	1.32528	0.662641	+	0.748938i	$0.269436\pi$
$594$	0	0
$595$	19037.1	1.31167
$596$	0	0
$597$	−9589.23	−0.657388
$598$	0	0
$599$	−4350.42	−0.296750	−0.148375	−	0.988931i	$-0.547404\pi$
$599$	−4350.42	−0.296750	−0.148375	+	0.988931i	$0.547404\pi$
$600$	0	0
$601$	−10657.8	−0.723362	−0.361681	−	0.932302i	$-0.617797\pi$
$601$	−10657.8	−0.723362	−0.361681	+	0.932302i	$0.617797\pi$
$602$	0	0
$603$	979.074	0.0661210
$604$	0	0
$605$	11814.7	0.793941
$606$	0	0
$607$	−5515.67	−0.368821	−0.184410	−	0.982849i	$-0.559038\pi$
$607$	−5515.67	−0.368821	−0.184410	+	0.982849i	$0.559038\pi$
$608$	0	0
$609$	−24201.5	−1.61034
$610$	0	0
$611$	−13260.1	−0.877980
$612$	0	0
$613$	26144.0	1.72259	0.861293	−	0.508109i	$-0.169655\pi$
$613$	26144.0	1.72259	0.861293	+	0.508109i	$0.169655\pi$
$614$	0	0
$615$	23346.9	1.53079
$616$	0	0
$617$	−4104.01	−0.267781	−0.133891	−	0.990996i	$-0.542747\pi$
$617$	−4104.01	−0.267781	−0.133891	+	0.990996i	$0.542747\pi$
$618$	0	0
$619$	−265.000	−0.0172072	−0.00860360	−	0.999963i	$-0.502739\pi$
$619$	−265.000	−0.0172072	−0.00860360	+	0.999963i	$0.502739\pi$
$620$	0	0
$621$	−9794.50	−0.632914
$622$	0	0
$623$	15133.3	0.973197
$624$	0	0
$625$	−9753.29	−0.624211
$626$	0	0
$627$	−880.788	−0.0561009
$628$	0	0
$629$	2234.36	0.141637
$630$	0	0
$631$	23088.2	1.45662	0.728309	−	0.685249i	$-0.240307\pi$
$631$	23088.2	1.45662	0.728309	+	0.685249i	$0.240307\pi$
$632$	0	0
$633$	−26772.9	−1.68109
$634$	0	0
$635$	−2933.72	−0.183340
$636$	0	0
$637$	−3425.87	−0.213089
$638$	0	0
$639$	−985.351	−0.0610014
$640$	0	0
$641$	3574.05	0.220229	0.110114	−	0.993919i	$-0.464878\pi$
$641$	3574.05	0.220229	0.110114	+	0.993919i	$0.464878\pi$
$642$	0	0
$643$	25705.2	1.57654	0.788270	−	0.615329i	$-0.210977\pi$
$643$	25705.2	1.57654	0.788270	+	0.615329i	$0.210977\pi$
$644$	0	0
$645$	−17062.4	−1.04160
$646$	0	0
$647$	1822.68	0.110752	0.0553762	−	0.998466i	$-0.482364\pi$
$647$	1822.68	0.110752	0.0553762	+	0.998466i	$0.482364\pi$
$648$	0	0
$649$	−3558.19	−0.215210
$650$	0	0
$651$	21866.9	1.31649
$652$	0	0
$653$	15678.9	0.939609	0.469804	−	0.882771i	$-0.344324\pi$
$653$	15678.9	0.939609	0.469804	+	0.882771i	$0.344324\pi$
$654$	0	0
$655$	−4843.51	−0.288934
$656$	0	0
$657$	−511.450	−0.0303707
$658$	0	0
$659$	−18817.0	−1.11230	−0.556151	−	0.831081i	$-0.687723\pi$
$659$	−18817.0	−1.11230	−0.556151	+	0.831081i	$0.687723\pi$
$660$	0	0
$661$	2463.08	0.144936	0.0724680	−	0.997371i	$-0.476912\pi$
$661$	2463.08	0.144936	0.0724680	+	0.997371i	$0.476912\pi$
$662$	0	0
$663$	−28739.4	−1.68348
$664$	0	0
$665$	−2906.60	−0.169493
$666$	0	0
$667$	−20055.2	−1.16423
$668$	0	0
$669$	2219.23	0.128252
$670$	0	0
$671$	2913.31	0.167611
$672$	0	0
$673$	−27177.3	−1.55662	−0.778312	−	0.627877i	$-0.783924\pi$
$673$	−27177.3	−1.55662	−0.778312	+	0.627877i	$0.783924\pi$
$674$	0	0
$675$	4974.47	0.283656
$676$	0	0
$677$	3111.27	0.176626	0.0883131	−	0.996093i	$-0.471852\pi$
$677$	3111.27	0.176626	0.0883131	+	0.996093i	$0.471852\pi$
$678$	0	0
$679$	−4585.43	−0.259164
$680$	0	0
$681$	16846.8	0.947976
$682$	0	0
$683$	−16068.7	−0.900221	−0.450110	−	0.892973i	$-0.648615\pi$
$683$	−16068.7	−0.900221	−0.450110	+	0.892973i	$0.648615\pi$
$684$	0	0
$685$	20724.0	1.15595
$686$	0	0
$687$	11692.5	0.649341
$688$	0	0
$689$	12734.5	0.704128
$690$	0	0
$691$	11854.9	0.652653	0.326327	−	0.945257i	$-0.394189\pi$
$691$	11854.9	0.652653	0.326327	+	0.945257i	$0.394189\pi$
$692$	0	0
$693$	−183.861	−0.0100783
$694$	0	0
$695$	−1773.59	−0.0968000
$696$	0	0
$697$	58017.6	3.15290
$698$	0	0
$699$	10408.9	0.563233
$700$	0	0
$701$	−19270.9	−1.03830	−0.519151	−	0.854682i	$-0.673752\pi$
$701$	−19270.9	−1.03830	−0.519151	+	0.854682i	$0.673752\pi$
$702$	0	0
$703$	−341.145	−0.0183023
$704$	0	0
$705$	15295.2	0.817091
$706$	0	0
$707$	−932.702	−0.0496151
$708$	0	0
$709$	952.832	0.0504716	0.0252358	−	0.999682i	$-0.491966\pi$
$709$	952.832	0.0504716	0.0252358	+	0.999682i	$0.491966\pi$
$710$	0	0
$711$	119.063	0.00628017
$712$	0	0
$713$	18120.6	0.951784
$714$	0	0
$715$	3561.43	0.186280
$716$	0	0
$717$	1908.18	0.0993897
$718$	0	0
$719$	26859.5	1.39317	0.696585	−	0.717474i	$-0.254702\pi$
$719$	26859.5	1.39317	0.696585	+	0.717474i	$0.254702\pi$
$720$	0	0
$721$	−21451.5	−1.10804
$722$	0	0
$723$	271.776	0.0139799
$724$	0	0
$725$	10185.7	0.521778
$726$	0	0
$727$	−26684.9	−1.36133	−0.680667	−	0.732593i	$-0.738310\pi$
$727$	−26684.9	−1.36133	−0.680667	+	0.732593i	$0.738310\pi$
$728$	0	0
$729$	18647.7	0.947401
$730$	0	0
$731$	−42400.4	−2.14533
$732$	0	0
$733$	−30013.3	−1.51237	−0.756183	−	0.654360i	$-0.772938\pi$
$733$	−30013.3	−1.51237	−0.756183	+	0.654360i	$0.772938\pi$
$734$	0	0
$735$	3951.65	0.198311
$736$	0	0
$737$	−6571.68	−0.328454
$738$	0	0
$739$	−12404.1	−0.617444	−0.308722	−	0.951152i	$-0.599901\pi$
$739$	−12404.1	−0.617444	−0.308722	+	0.951152i	$0.599901\pi$
$740$	0	0
$741$	4387.97	0.217538
$742$	0	0
$743$	−35368.0	−1.74634	−0.873168	−	0.487420i	$-0.837938\pi$
$743$	−35368.0	−1.74634	−0.873168	+	0.487420i	$0.837938\pi$
$744$	0	0
$745$	−18846.8	−0.926836
$746$	0	0
$747$	−1813.36	−0.0888185
$748$	0	0
$749$	−12183.8	−0.594372
$750$	0	0
$751$	−1974.41	−0.0959348	−0.0479674	−	0.998849i	$-0.515274\pi$
$751$	−1974.41	−0.0959348	−0.0479674	+	0.998849i	$0.515274\pi$
$752$	0	0
$753$	2450.27	0.118583
$754$	0	0
$755$	12464.9	0.600855
$756$	0	0
$757$	−5328.85	−0.255852	−0.127926	−	0.991784i	$-0.540832\pi$
$757$	−5328.85	−0.255852	−0.127926	+	0.991784i	$0.540832\pi$
$758$	0	0
$759$	−3320.92	−0.158816
$760$	0	0
$761$	−13374.6	−0.637092	−0.318546	−	0.947907i	$-0.603195\pi$
$761$	−13374.6	−0.637092	−0.318546	+	0.947907i	$0.603195\pi$
$762$	0	0
$763$	−10485.2	−0.497494
$764$	0	0
$765$	1520.91	0.0718804
$766$	0	0
$767$	17726.4	0.834503
$768$	0	0
$769$	−41658.8	−1.95352	−0.976760	−	0.214337i	$-0.931241\pi$
$769$	−41658.8	−1.95352	−0.976760	+	0.214337i	$0.931241\pi$
$770$	0	0
$771$	7177.55	0.335270
$772$	0	0
$773$	−24505.2	−1.14022	−0.570110	−	0.821569i	$-0.693099\pi$
$773$	−24505.2	−1.14022	−0.570110	+	0.821569i	$0.693099\pi$
$774$	0	0
$775$	−9203.17	−0.426565
$776$	0	0
$777$	−1552.17	−0.0716652
$778$	0	0
$779$	−8858.19	−0.407417
$780$	0	0
$781$	6613.81	0.303023
$782$	0	0
$783$	38276.3	1.74698
$784$	0	0
$785$	3396.35	0.154421
$786$	0	0
$787$	−24681.7	−1.11793	−0.558963	−	0.829192i	$-0.688801\pi$
$787$	−24681.7	−1.11793	−0.558963	+	0.829192i	$0.688801\pi$
$788$	0	0
$789$	1660.78	0.0749369
$790$	0	0
$791$	−37519.7	−1.68653
$792$	0	0
$793$	−14513.7	−0.649933
$794$	0	0
$795$	−14688.9	−0.655296
$796$	0	0
$797$	−7727.34	−0.343434	−0.171717	−	0.985146i	$-0.554931\pi$
$797$	−7727.34	−0.343434	−0.171717	+	0.985146i	$0.554931\pi$
$798$	0	0
$799$	38008.9	1.68293
$800$	0	0
$801$	1209.02	0.0533318
$802$	0	0
$803$	3432.92	0.150866
$804$	0	0
$805$	−10959.0	−0.479820
$806$	0	0
$807$	−22228.0	−0.969596
$808$	0	0
$809$	4220.96	0.183438	0.0917188	−	0.995785i	$-0.470764\pi$
$809$	4220.96	0.183438	0.0917188	+	0.995785i	$0.470764\pi$
$810$	0	0
$811$	26026.3	1.12689	0.563445	−	0.826154i	$-0.309476\pi$
$811$	26026.3	1.12689	0.563445	+	0.826154i	$0.309476\pi$
$812$	0	0
$813$	11387.0	0.491215
$814$	0	0
$815$	−5974.25	−0.256772
$816$	0	0
$817$	6473.75	0.277219
$818$	0	0
$819$	915.970	0.0390801
$820$	0	0
$821$	−13319.5	−0.566204	−0.283102	−	0.959090i	$-0.591364\pi$
$821$	−13319.5	−0.566204	−0.283102	+	0.959090i	$0.591364\pi$
$822$	0	0
$823$	−10436.6	−0.442038	−0.221019	−	0.975270i	$-0.570938\pi$
$823$	−10436.6	−0.442038	−0.221019	+	0.975270i	$0.570938\pi$
$824$	0	0
$825$	1686.64	0.0711774
$826$	0	0
$827$	−8612.06	−0.362117	−0.181058	−	0.983472i	$-0.557952\pi$
$827$	−8612.06	−0.362117	−0.181058	+	0.983472i	$0.557952\pi$
$828$	0	0
$829$	−10298.1	−0.431444	−0.215722	−	0.976455i	$-0.569211\pi$
$829$	−10298.1	−0.431444	−0.215722	+	0.976455i	$0.569211\pi$
$830$	0	0
$831$	33461.3	1.39682
$832$	0	0
$833$	9819.94	0.408452
$834$	0	0
$835$	−35208.2	−1.45920
$836$	0	0
$837$	−34584.0	−1.42819
$838$	0	0
$839$	14169.2	0.583045	0.291522	−	0.956564i	$-0.405838\pi$
$839$	14169.2	0.583045	0.291522	+	0.956564i	$0.405838\pi$
$840$	0	0
$841$	53985.6	2.21352
$842$	0	0
$843$	31705.9	1.29538
$844$	0	0
$845$	2939.19	0.119658
$846$	0	0
$847$	−20395.7	−0.827397
$848$	0	0
$849$	−11446.9	−0.462730
$850$	0	0
$851$	−1286.25	−0.0518120
$852$	0	0
$853$	−9822.77	−0.394285	−0.197143	−	0.980375i	$-0.563166\pi$
$853$	−9822.77	−0.394285	−0.197143	+	0.980375i	$0.563166\pi$
$854$	0	0
$855$	−232.214	−0.00928835
$856$	0	0
$857$	−11841.3	−0.471983	−0.235992	−	0.971755i	$-0.575834\pi$
$857$	−11841.3	−0.471983	−0.235992	+	0.971755i	$0.575834\pi$
$858$	0	0
$859$	−44204.6	−1.75581	−0.877905	−	0.478835i	$-0.841059\pi$
$859$	−44204.6	−1.75581	−0.877905	+	0.478835i	$0.841059\pi$
$860$	0	0
$861$	−40303.8	−1.59530
$862$	0	0
$863$	16756.6	0.660952	0.330476	−	0.943814i	$-0.392791\pi$
$863$	16756.6	0.660952	0.330476	+	0.943814i	$0.392791\pi$
$864$	0	0
$865$	9214.22	0.362188
$866$	0	0
$867$	56243.8	2.20316
$868$	0	0
$869$	−799.165	−0.0311966
$870$	0	0
$871$	32739.2	1.27362
$872$	0	0
$873$	−366.338	−0.0142024
$874$	0	0
$875$	24688.3	0.953848
$876$	0	0
$877$	40606.0	1.56347	0.781737	−	0.623609i	$-0.214334\pi$
$877$	40606.0	1.56347	0.781737	+	0.623609i	$0.214334\pi$
$878$	0	0
$879$	−32767.9	−1.25737
$880$	0	0
$881$	18464.8	0.706124	0.353062	−	0.935600i	$-0.385140\pi$
$881$	18464.8	0.706124	0.353062	+	0.935600i	$0.385140\pi$
$882$	0	0
$883$	20592.3	0.784809	0.392404	−	0.919793i	$-0.371643\pi$
$883$	20592.3	0.784809	0.392404	+	0.919793i	$0.371643\pi$
$884$	0	0
$885$	−20447.0	−0.776630
$886$	0	0
$887$	−17682.8	−0.669370	−0.334685	−	0.942330i	$-0.608630\pi$
$887$	−17682.8	−0.669370	−0.334685	+	0.942330i	$0.608630\pi$
$888$	0	0
$889$	5064.49	0.191066
$890$	0	0
$891$	6643.59	0.249796
$892$	0	0
$893$	−5803.24	−0.217467
$894$	0	0
$895$	−2441.25	−0.0911753
$896$	0	0
$897$	16544.4	0.615830
$898$	0	0
$899$	−70814.2	−2.62713
$900$	0	0
$901$	−36502.2	−1.34968
$902$	0	0
$903$	29454.9	1.08549
$904$	0	0
$905$	8400.84	0.308567
$906$	0	0
$907$	−19256.9	−0.704978	−0.352489	−	0.935816i	$-0.614665\pi$
$907$	−19256.9	−0.704978	−0.352489	+	0.935816i	$0.614665\pi$
$908$	0	0
$909$	−74.5153	−0.00271894
$910$	0	0
$911$	24122.8	0.877306	0.438653	−	0.898657i	$-0.355456\pi$
$911$	24122.8	0.877306	0.438653	+	0.898657i	$0.355456\pi$
$912$	0	0
$913$	12171.5	0.441204
$914$	0	0
$915$	16741.2	0.604860
$916$	0	0
$917$	8361.38	0.301109
$918$	0	0
$919$	−16780.0	−0.602309	−0.301154	−	0.953575i	$-0.597372\pi$
$919$	−16780.0	−0.602309	−0.301154	+	0.953575i	$0.597372\pi$
$920$	0	0
$921$	51455.6	1.84095
$922$	0	0
$923$	−32949.1	−1.17501
$924$	0	0
$925$	653.266	0.0232208
$926$	0	0
$927$	−1713.80	−0.0607212
$928$	0	0
$929$	32219.1	1.13786	0.568931	−	0.822386i	$-0.307357\pi$
$929$	32219.1	1.13786	0.568931	+	0.822386i	$0.307357\pi$
$930$	0	0
$931$	−1499.32	−0.0527801
$932$	0	0
$933$	−33885.1	−1.18901
$934$	0	0
$935$	−10208.5	−0.357064
$936$	0	0
$937$	−36861.2	−1.28517	−0.642584	−	0.766215i	$-0.722138\pi$
$937$	−36861.2	−1.28517	−0.642584	+	0.766215i	$0.722138\pi$
$938$	0	0
$939$	−17804.7	−0.618779
$940$	0	0
$941$	−1000.08	−0.0346460	−0.0173230	−	0.999850i	$-0.505514\pi$
$941$	−1000.08	−0.0346460	−0.0173230	+	0.999850i	$0.505514\pi$
$942$	0	0
$943$	−33398.8	−1.15336
$944$	0	0
$945$	20915.8	0.719989
$946$	0	0
$947$	50954.9	1.74848	0.874240	−	0.485494i	$-0.161360\pi$
$947$	50954.9	1.74848	0.874240	+	0.485494i	$0.161360\pi$
$948$	0	0
$949$	−17102.3	−0.585000
$950$	0	0
$951$	−57028.0	−1.94454
$952$	0	0
$953$	−34671.8	−1.17852	−0.589260	−	0.807943i	$-0.700581\pi$
$953$	−34671.8	−1.17852	−0.589260	+	0.807943i	$0.700581\pi$
$954$	0	0
$955$	37849.9	1.28251
$956$	0	0
$957$	12977.9	0.438367
$958$	0	0
$959$	−35776.0	−1.20466
$960$	0	0
$961$	34192.1	1.14773
$962$	0	0
$963$	−973.382	−0.0325720
$964$	0	0
$965$	−3831.54	−0.127815
$966$	0	0
$967$	−1487.32	−0.0494610	−0.0247305	−	0.999694i	$-0.507873\pi$
$967$	−1487.32	−0.0494610	−0.0247305	+	0.999694i	$0.507873\pi$
$968$	0	0
$969$	−12577.7	−0.416981
$970$	0	0
$971$	19161.1	0.633273	0.316636	−	0.948547i	$-0.397447\pi$
$971$	19161.1	0.633273	0.316636	+	0.948547i	$0.397447\pi$
$972$	0	0
$973$	3061.75	0.100879
$974$	0	0
$975$	−8402.62	−0.275999
$976$	0	0
$977$	−29747.5	−0.974110	−0.487055	−	0.873371i	$-0.661929\pi$
$977$	−29747.5	−0.974110	−0.487055	+	0.873371i	$0.661929\pi$
$978$	0	0
$979$	−8115.14	−0.264924
$980$	0	0
$981$	−837.678	−0.0272630
$982$	0	0
$983$	−58843.1	−1.90926	−0.954631	−	0.297791i	$-0.903750\pi$
$983$	−58843.1	−1.90926	−0.954631	+	0.297791i	$0.903750\pi$
$984$	0	0
$985$	−34829.1	−1.12665
$986$	0	0
$987$	−26404.1	−0.851522
$988$	0	0
$989$	24408.6	0.784780
$990$	0	0
$991$	48869.6	1.56649	0.783247	−	0.621711i	$-0.213562\pi$
$991$	48869.6	1.56649	0.783247	+	0.621711i	$0.213562\pi$
$992$	0	0
$993$	−37197.3	−1.18874
$994$	0	0
$995$	16969.2	0.540662
$996$	0	0
$997$	2207.55	0.0701243	0.0350621	−	0.999385i	$-0.488837\pi$
$997$	2207.55	0.0701243	0.0350621	+	0.999385i	$0.488837\pi$
$998$	0	0
$999$	2454.86	0.0777461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.y.1.5		5
4.3	odd	2		1216.4.a.bd.1.1		5
8.3	odd	2		608.4.a.e.1.5	✓	5
8.5	even	2		608.4.a.h.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.e.1.5	✓	5	8.3	odd	2
608.4.a.h.1.1	yes	5	8.5	even	2
1216.4.a.y.1.5		5	1.1	even	1	trivial
1216.4.a.bd.1.1		5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q+5.31962 q^{3} -9.41363 q^{5} +16.2508 q^{7} +1.29831 q^{9} -8.71440 q^{11} +43.4140 q^{13} -50.0769 q^{15} -124.442 q^{17} +19.0000 q^{19} +86.4480 q^{21} +71.6374 q^{23} -36.3835 q^{25} -136.723 q^{27} -279.955 q^{29} +252.949 q^{31} -46.3573 q^{33} -152.979 q^{35} -17.9550 q^{37} +230.946 q^{39} -466.221 q^{41} +340.723 q^{43} -12.2218 q^{45} -305.434 q^{47} -78.9116 q^{49} -661.985 q^{51} +293.326 q^{53} +82.0342 q^{55} +101.073 q^{57} +408.311 q^{59} -334.310 q^{61} +21.0985 q^{63} -408.683 q^{65} +754.117 q^{67} +381.084 q^{69} -758.951 q^{71} -393.936 q^{73} -193.546 q^{75} -141.616 q^{77} +91.7062 q^{79} -762.369 q^{81} -1396.71 q^{83} +1171.45 q^{85} -1489.25 q^{87} +931.233 q^{89} +705.512 q^{91} +1345.59 q^{93} -178.859 q^{95} -282.166 q^{97} -11.3140 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 6 q^{3} - 5 q^{5} - 7 q^{7} - 5 q^{9} - 13 q^{11} + 72 q^{13} - 72 q^{15} - 59 q^{17} + 95 q^{19} + 224 q^{21} - 52 q^{23} - 86 q^{25} - 54 q^{27} + 128 q^{29} + 110 q^{31} - 68 q^{33} + 45 q^{35}+ \cdots - 601 q^{99}+O(q^{100}) \) 5 * q - 6 * q^3 - 5 * q^5 - 7 * q^7 - 5 * q^9 - 13 * q^11 + 72 * q^13 - 72 * q^15 - 59 * q^17 + 95 * q^19 + 224 * q^21 - 52 * q^23 - 86 * q^25 - 54 * q^27 + 128 * q^29 + 110 * q^31 - 68 * q^33 + 45 * q^35 + 436 * q^37 - 356 * q^39 - 804 * q^41 + 143 * q^43 + 579 * q^45 - 661 * q^47 - 406 * q^49 - 570 * q^51 + 898 * q^53 - 17 * q^55 - 114 * q^57 - 196 * q^59 + 1079 * q^61 - 143 * q^63 - 1632 * q^65 - 832 * q^67 + 1644 * q^69 - 834 * q^71 - 1375 * q^73 - 1054 * q^75 + 1473 * q^77 - 154 * q^79 - 1859 * q^81 - 2224 * q^83 + 1807 * q^85 - 2004 * q^87 - 542 * q^89 - 1890 * q^91 + 2788 * q^93 - 95 * q^95 - 1528 * q^97 - 601 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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