LMFDB - Embedded newform 2312.4.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, 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("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.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:	$136.412415933$
Analytic rank:	$0$
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} - x^{5} - 92x^{4} + 123x^{3} + 2120x^{2} - 3573x - 261 $ x^6 - x^5 - 92x^4 + 123x^3 + 2120x^2 - 3573x - 261
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$6.86357$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.86357 q^{3} +12.8315 q^{5} -9.16275 q^{7} +34.8357 q^{9} +O(q^{10})$	$q+7.86357 q^{3} +12.8315 q^{5} -9.16275 q^{7} +34.8357 q^{9} +20.9973 q^{11} +90.1467 q^{13} +100.902 q^{15} -150.102 q^{19} -72.0519 q^{21} +135.052 q^{23} +39.6482 q^{25} +61.6168 q^{27} +17.3415 q^{29} +92.9706 q^{31} +165.114 q^{33} -117.572 q^{35} +217.891 q^{37} +708.875 q^{39} -211.722 q^{41} -64.9457 q^{43} +446.996 q^{45} +588.306 q^{47} -259.044 q^{49} +493.259 q^{53} +269.427 q^{55} -1180.34 q^{57} +605.620 q^{59} -528.657 q^{61} -319.191 q^{63} +1156.72 q^{65} +362.208 q^{67} +1061.99 q^{69} +230.555 q^{71} -394.114 q^{73} +311.776 q^{75} -192.393 q^{77} -414.500 q^{79} -456.037 q^{81} -265.968 q^{83} +136.366 q^{87} -56.6809 q^{89} -825.991 q^{91} +731.081 q^{93} -1926.04 q^{95} -1535.62 q^{97} +731.456 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 7 q^{3} + 3 q^{5} + 7 q^{7} + 31 q^{9}+O(q^{10}) $ 6 * q + 7 * q^3 + 3 * q^5 + 7 * q^7 + 31 * q^9	$ 6 q + 7 q^{3} + 3 q^{5} + 7 q^{7} + 31 q^{9} + 72 q^{11} + 15 q^{13} - 60 q^{15} + 83 q^{19} + 96 q^{21} + 141 q^{23} + 263 q^{25} + 94 q^{27} - 249 q^{29} + 106 q^{31} + 289 q^{33} - 267 q^{35} + 170 q^{37} + 329 q^{39} - 100 q^{41} - 90 q^{43} - 286 q^{45} + 372 q^{47} + 323 q^{49} - 23 q^{53} - 457 q^{55} - 193 q^{57} + 784 q^{59} + 92 q^{61} - 722 q^{63} + 1412 q^{65} + 238 q^{67} + 1178 q^{69} + 940 q^{71} + 692 q^{73} + 1814 q^{75} - 45 q^{77} + 84 q^{79} - 2182 q^{81} + 1393 q^{83} - 1247 q^{87} + 976 q^{89} - 384 q^{91} + 1437 q^{93} - 1022 q^{95} - 325 q^{97} + 3981 q^{99}+O(q^{100}) $ 6 * q + 7 * q^3 + 3 * q^5 + 7 * q^7 + 31 * q^9 + 72 * q^11 + 15 * q^13 - 60 * q^15 + 83 * q^19 + 96 * q^21 + 141 * q^23 + 263 * q^25 + 94 * q^27 - 249 * q^29 + 106 * q^31 + 289 * q^33 - 267 * q^35 + 170 * q^37 + 329 * q^39 - 100 * q^41 - 90 * q^43 - 286 * q^45 + 372 * q^47 + 323 * q^49 - 23 * q^53 - 457 * q^55 - 193 * q^57 + 784 * q^59 + 92 * q^61 - 722 * q^63 + 1412 * q^65 + 238 * q^67 + 1178 * q^69 + 940 * q^71 + 692 * q^73 + 1814 * q^75 - 45 * q^77 + 84 * q^79 - 2182 * q^81 + 1393 * q^83 - 1247 * q^87 + 976 * q^89 - 384 * q^91 + 1437 * q^93 - 1022 * q^95 - 325 * q^97 + 3981 * 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
$3$	7.86357	1.51334	0.756672	−	0.653794i	$-0.226824\pi$
$3$	7.86357	1.51334	0.756672	+	0.653794i	$0.226824\pi$
$4$	0	0
$5$	12.8315	1.14769	0.573843	−	0.818965i	$-0.305452\pi$
$5$	12.8315	1.14769	0.573843	+	0.818965i	$0.305452\pi$
$6$	0	0
$7$	−9.16275	−0.494742	−0.247371	−	0.968921i	$-0.579567\pi$
$7$	−9.16275	−0.494742	−0.247371	+	0.968921i	$0.579567\pi$
$8$	0	0
$9$	34.8357	1.29021
$10$	0	0
$11$	20.9973	0.575538	0.287769	−	0.957700i	$-0.407086\pi$
$11$	20.9973	0.575538	0.287769	+	0.957700i	$0.407086\pi$
$12$	0	0
$13$	90.1467	1.92324	0.961622	−	0.274376i	$-0.0884713\pi$
$13$	90.1467	1.92324	0.961622	+	0.274376i	$0.0884713\pi$
$14$	0	0
$15$	100.902	1.73685
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−150.102	−1.81241	−0.906203	−	0.422843i	$-0.861032\pi$
$19$	−150.102	−1.81241	−0.906203	+	0.422843i	$0.861032\pi$
$20$	0	0
$21$	−72.0519	−0.748715
$22$	0	0
$23$	135.052	1.22436	0.612182	−	0.790717i	$-0.290292\pi$
$23$	135.052	1.22436	0.612182	+	0.790717i	$0.290292\pi$
$24$	0	0
$25$	39.6482	0.317185
$26$	0	0
$27$	61.6168	0.439191
$28$	0	0
$29$	17.3415	0.111042	0.0555212	−	0.998458i	$-0.482318\pi$
$29$	17.3415	0.111042	0.0555212	+	0.998458i	$0.482318\pi$
$30$	0	0
$31$	92.9706	0.538646	0.269323	−	0.963050i	$-0.413200\pi$
$31$	92.9706	0.538646	0.269323	+	0.963050i	$0.413200\pi$
$32$	0	0
$33$	165.114	0.870988
$34$	0	0
$35$	−117.572	−0.567809
$36$	0	0
$37$	217.891	0.968138	0.484069	−	0.875030i	$-0.339158\pi$
$37$	217.891	0.968138	0.484069	+	0.875030i	$0.339158\pi$
$38$	0	0
$39$	708.875	2.91053
$40$	0	0
$41$	−211.722	−0.806473	−0.403237	−	0.915096i	$-0.632115\pi$
$41$	−211.722	−0.806473	−0.403237	+	0.915096i	$0.632115\pi$
$42$	0	0
$43$	−64.9457	−0.230329	−0.115164	−	0.993346i	$-0.536739\pi$
$43$	−64.9457	−0.230329	−0.115164	+	0.993346i	$0.536739\pi$
$44$	0	0
$45$	446.996	1.48076
$46$	0	0
$47$	588.306	1.82581	0.912907	−	0.408168i	$-0.133832\pi$
$47$	588.306	1.82581	0.912907	+	0.408168i	$0.133832\pi$
$48$	0	0
$49$	−259.044	−0.755230
$50$	0	0
$51$	0	0
$52$	0	0
$53$	493.259	1.27838	0.639192	−	0.769047i	$-0.279269\pi$
$53$	493.259	1.27838	0.639192	+	0.769047i	$0.279269\pi$
$54$	0	0
$55$	269.427	0.660538
$56$	0	0
$57$	−1180.34	−2.74280
$58$	0	0
$59$	605.620	1.33636	0.668178	−	0.744001i	$-0.267074\pi$
$59$	605.620	1.33636	0.668178	+	0.744001i	$0.267074\pi$
$60$	0	0
$61$	−528.657	−1.10963	−0.554816	−	0.831973i	$-0.687211\pi$
$61$	−528.657	−1.10963	−0.554816	+	0.831973i	$0.687211\pi$
$62$	0	0
$63$	−319.191	−0.638322
$64$	0	0
$65$	1156.72	2.20728
$66$	0	0
$67$	362.208	0.660458	0.330229	−	0.943901i	$-0.392874\pi$
$67$	362.208	0.660458	0.330229	+	0.943901i	$0.392874\pi$
$68$	0	0
$69$	1061.99	1.85289
$70$	0	0
$71$	230.555	0.385378	0.192689	−	0.981260i	$-0.438279\pi$
$71$	230.555	0.385378	0.192689	+	0.981260i	$0.438279\pi$
$72$	0	0
$73$	−394.114	−0.631884	−0.315942	−	0.948778i	$-0.602321\pi$
$73$	−394.114	−0.631884	−0.315942	+	0.948778i	$0.602321\pi$
$74$	0	0
$75$	311.776	0.480011
$76$	0	0
$77$	−192.393	−0.284743
$78$	0	0
$79$	−414.500	−0.590315	−0.295157	−	0.955449i	$-0.595372\pi$
$79$	−414.500	−0.590315	−0.295157	+	0.955449i	$0.595372\pi$
$80$	0	0
$81$	−456.037	−0.625564
$82$	0	0
$83$	−265.968	−0.351733	−0.175866	−	0.984414i	$-0.556273\pi$
$83$	−265.968	−0.351733	−0.175866	+	0.984414i	$0.556273\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	136.366	0.168046
$88$	0	0
$89$	−56.6809	−0.0675074	−0.0337537	−	0.999430i	$-0.510746\pi$
$89$	−56.6809	−0.0675074	−0.0337537	+	0.999430i	$0.510746\pi$
$90$	0	0
$91$	−825.991	−0.951510
$92$	0	0
$93$	731.081	0.815157
$94$	0	0
$95$	−1926.04	−2.08007
$96$	0	0
$97$	−1535.62	−1.60741	−0.803705	−	0.595028i	$-0.797141\pi$
$97$	−1535.62	−1.60741	−0.803705	+	0.595028i	$0.797141\pi$
$98$	0	0
$99$	731.456	0.742566
$100$	0	0
$101$	−147.523	−0.145337	−0.0726687	−	0.997356i	$-0.523152\pi$
$101$	−147.523	−0.145337	−0.0726687	+	0.997356i	$0.523152\pi$
$102$	0	0
$103$	525.420	0.502633	0.251316	−	0.967905i	$-0.419137\pi$
$103$	525.420	0.502633	0.251316	+	0.967905i	$0.419137\pi$
$104$	0	0
$105$	−924.536	−0.859290
$106$	0	0
$107$	1688.98	1.52598	0.762989	−	0.646411i	$-0.223731\pi$
$107$	1688.98	1.52598	0.762989	+	0.646411i	$0.223731\pi$
$108$	0	0
$109$	1311.71	1.15265	0.576326	−	0.817220i	$-0.304486\pi$
$109$	1311.71	1.15265	0.576326	+	0.817220i	$0.304486\pi$
$110$	0	0
$111$	1713.40	1.46513
$112$	0	0
$113$	−1659.20	−1.38128	−0.690638	−	0.723200i	$-0.742670\pi$
$113$	−1659.20	−1.38128	−0.690638	+	0.723200i	$0.742670\pi$
$114$	0	0
$115$	1732.93	1.40519
$116$	0	0
$117$	3140.32	2.48139
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−890.114	−0.668756
$122$	0	0
$123$	−1664.89	−1.22047
$124$	0	0
$125$	−1095.19	−0.783658
$126$	0	0
$127$	1761.98	1.23111	0.615553	−	0.788096i	$-0.288933\pi$
$127$	1761.98	1.23111	0.615553	+	0.788096i	$0.288933\pi$
$128$	0	0
$129$	−510.705	−0.348567
$130$	0	0
$131$	425.185	0.283577	0.141788	−	0.989897i	$-0.454715\pi$
$131$	425.185	0.283577	0.141788	+	0.989897i	$0.454715\pi$
$132$	0	0
$133$	1375.35	0.896673
$134$	0	0
$135$	790.638	0.504054
$136$	0	0
$137$	2623.11	1.63582	0.817910	−	0.575346i	$-0.195132\pi$
$137$	2623.11	1.63582	0.817910	+	0.575346i	$0.195132\pi$
$138$	0	0
$139$	2388.19	1.45729	0.728645	−	0.684891i	$-0.240150\pi$
$139$	2388.19	1.45729	0.728645	+	0.684891i	$0.240150\pi$
$140$	0	0
$141$	4626.19	2.76309
$142$	0	0
$143$	1892.83	1.10690
$144$	0	0
$145$	222.518	0.127442
$146$	0	0
$147$	−2037.01	−1.14292
$148$	0	0
$149$	−2762.13	−1.51867	−0.759336	−	0.650699i	$-0.774476\pi$
$149$	−2762.13	−1.51867	−0.759336	+	0.650699i	$0.774476\pi$
$150$	0	0
$151$	−2734.96	−1.47396	−0.736981	−	0.675913i	$-0.763749\pi$
$151$	−2734.96	−1.47396	−0.736981	+	0.675913i	$0.763749\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1192.96	0.618197
$156$	0	0
$157$	−2316.23	−1.17742	−0.588712	−	0.808343i	$-0.700365\pi$
$157$	−2316.23	−1.17742	−0.588712	+	0.808343i	$0.700365\pi$
$158$	0	0
$159$	3878.78	1.93464
$160$	0	0
$161$	−1237.45	−0.605744
$162$	0	0
$163$	1372.21	0.659387	0.329693	−	0.944088i	$-0.393055\pi$
$163$	1372.21	0.659387	0.329693	+	0.944088i	$0.393055\pi$
$164$	0	0
$165$	2118.66	0.999621
$166$	0	0
$167$	668.677	0.309843	0.154921	−	0.987927i	$-0.450488\pi$
$167$	668.677	0.309843	0.154921	+	0.987927i	$0.450488\pi$
$168$	0	0
$169$	5929.42	2.69887
$170$	0	0
$171$	−5228.91	−2.33839
$172$	0	0
$173$	−3216.21	−1.41343	−0.706717	−	0.707496i	$-0.749825\pi$
$173$	−3216.21	−1.41343	−0.706717	+	0.707496i	$0.749825\pi$
$174$	0	0
$175$	−363.286	−0.156925
$176$	0	0
$177$	4762.33	2.02237
$178$	0	0
$179$	717.219	0.299483	0.149742	−	0.988725i	$-0.452156\pi$
$179$	717.219	0.299483	0.149742	+	0.988725i	$0.452156\pi$
$180$	0	0
$181$	3939.88	1.61795	0.808975	−	0.587843i	$-0.200023\pi$
$181$	3939.88	1.61795	0.808975	+	0.587843i	$0.200023\pi$
$182$	0	0
$183$	−4157.13	−1.67926
$184$	0	0
$185$	2795.88	1.11112
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−564.579	−0.217286
$190$	0	0
$191$	3523.31	1.33475	0.667376	−	0.744721i	$-0.267418\pi$
$191$	3523.31	1.33475	0.667376	+	0.744721i	$0.267418\pi$
$192$	0	0
$193$	1455.89	0.542990	0.271495	−	0.962440i	$-0.412482\pi$
$193$	1455.89	0.542990	0.271495	+	0.962440i	$0.412482\pi$
$194$	0	0
$195$	9095.94	3.34038
$196$	0	0
$197$	1654.71	0.598441	0.299221	−	0.954184i	$-0.403273\pi$
$197$	1654.71	0.598441	0.299221	+	0.954184i	$0.403273\pi$
$198$	0	0
$199$	−5033.93	−1.79320	−0.896598	−	0.442845i	$-0.853969\pi$
$199$	−5033.93	−1.79320	−0.896598	+	0.442845i	$0.853969\pi$
$200$	0	0
$201$	2848.24	0.999501
$202$	0	0
$203$	−158.896	−0.0549374
$204$	0	0
$205$	−2716.72	−0.925579
$206$	0	0
$207$	4704.65	1.57969
$208$	0	0
$209$	−3151.73	−1.04311
$210$	0	0
$211$	−2131.31	−0.695380	−0.347690	−	0.937609i	$-0.613034\pi$
$211$	−2131.31	−0.695380	−0.347690	+	0.937609i	$0.613034\pi$
$212$	0	0
$213$	1812.98	0.583210
$214$	0	0
$215$	−833.353	−0.264345
$216$	0	0
$217$	−851.866	−0.266491
$218$	0	0
$219$	−3099.14	−0.956259
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−4929.73	−1.48036	−0.740178	−	0.672411i	$-0.765259\pi$
$223$	−4929.73	−1.48036	−0.740178	+	0.672411i	$0.765259\pi$
$224$	0	0
$225$	1381.17	0.409236
$226$	0	0
$227$	−1126.17	−0.329280	−0.164640	−	0.986354i	$-0.552646\pi$
$227$	−1126.17	−0.329280	−0.164640	+	0.986354i	$0.552646\pi$
$228$	0	0
$229$	−1783.75	−0.514733	−0.257366	−	0.966314i	$-0.582855\pi$
$229$	−1783.75	−0.514733	−0.257366	+	0.966314i	$0.582855\pi$
$230$	0	0
$231$	−1512.89	−0.430914
$232$	0	0
$233$	−5184.16	−1.45762	−0.728810	−	0.684716i	$-0.759926\pi$
$233$	−5184.16	−1.45762	−0.728810	+	0.684716i	$0.759926\pi$
$234$	0	0
$235$	7548.87	2.09546
$236$	0	0
$237$	−3259.45	−0.893350
$238$	0	0
$239$	2688.87	0.727733	0.363867	−	0.931451i	$-0.381456\pi$
$239$	2688.87	0.727733	0.363867	+	0.931451i	$0.381456\pi$
$240$	0	0
$241$	32.6672	0.00873145	0.00436573	−	0.999990i	$-0.498610\pi$
$241$	32.6672	0.00873145	0.00436573	+	0.999990i	$0.498610\pi$
$242$	0	0
$243$	−5249.73	−1.38589
$244$	0	0
$245$	−3323.93	−0.866768
$246$	0	0
$247$	−13531.2	−3.48570
$248$	0	0
$249$	−2091.46	−0.532293
$250$	0	0
$251$	−1592.00	−0.400343	−0.200172	−	0.979761i	$-0.564150\pi$
$251$	−1592.00	−0.400343	−0.200172	+	0.979761i	$0.564150\pi$
$252$	0	0
$253$	2835.73	0.704668
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5507.87	1.33685	0.668427	−	0.743777i	$-0.266968\pi$
$257$	5507.87	1.33685	0.668427	+	0.743777i	$0.266968\pi$
$258$	0	0
$259$	−1996.48	−0.478978
$260$	0	0
$261$	604.103	0.143268
$262$	0	0
$263$	981.423	0.230103	0.115052	−	0.993360i	$-0.463297\pi$
$263$	981.423	0.230103	0.115052	+	0.993360i	$0.463297\pi$
$264$	0	0
$265$	6329.27	1.46718
$266$	0	0
$267$	−445.714	−0.102162
$268$	0	0
$269$	4366.75	0.989760	0.494880	−	0.868961i	$-0.335212\pi$
$269$	4366.75	0.989760	0.494880	+	0.868961i	$0.335212\pi$
$270$	0	0
$271$	−2670.53	−0.598608	−0.299304	−	0.954158i	$-0.596755\pi$
$271$	−2670.53	−0.598608	−0.299304	+	0.954158i	$0.596755\pi$
$272$	0	0
$273$	−6495.24	−1.43996
$274$	0	0
$275$	832.504	0.182552
$276$	0	0
$277$	−5659.56	−1.22762	−0.613809	−	0.789455i	$-0.710364\pi$
$277$	−5659.56	−1.22762	−0.613809	+	0.789455i	$0.710364\pi$
$278$	0	0
$279$	3238.70	0.694967
$280$	0	0
$281$	5674.96	1.20477	0.602384	−	0.798207i	$-0.294218\pi$
$281$	5674.96	1.20477	0.602384	+	0.798207i	$0.294218\pi$
$282$	0	0
$283$	−5088.25	−1.06878	−0.534390	−	0.845238i	$-0.679459\pi$
$283$	−5088.25	−1.06878	−0.534390	+	0.845238i	$0.679459\pi$
$284$	0	0
$285$	−15145.5	−3.14787
$286$	0	0
$287$	1939.95	0.398996
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−12075.5	−2.43257
$292$	0	0
$293$	3122.93	0.622674	0.311337	−	0.950300i	$-0.399223\pi$
$293$	3122.93	0.622674	0.311337	+	0.950300i	$0.399223\pi$
$294$	0	0
$295$	7771.03	1.53372
$296$	0	0
$297$	1293.79	0.252771
$298$	0	0
$299$	12174.5	2.35475
$300$	0	0
$301$	595.081	0.113953
$302$	0	0
$303$	−1160.06	−0.219946
$304$	0	0
$305$	−6783.47	−1.27351
$306$	0	0
$307$	−3894.95	−0.724093	−0.362046	−	0.932160i	$-0.617922\pi$
$307$	−3894.95	−0.724093	−0.362046	+	0.932160i	$0.617922\pi$
$308$	0	0
$309$	4131.68	0.760657
$310$	0	0
$311$	8451.98	1.54105	0.770527	−	0.637407i	$-0.219993\pi$
$311$	8451.98	1.54105	0.770527	+	0.637407i	$0.219993\pi$
$312$	0	0
$313$	1167.49	0.210832	0.105416	−	0.994428i	$-0.466383\pi$
$313$	1167.49	0.210832	0.105416	+	0.994428i	$0.466383\pi$
$314$	0	0
$315$	−4095.71	−0.732594
$316$	0	0
$317$	−1679.31	−0.297538	−0.148769	−	0.988872i	$-0.547531\pi$
$317$	−1679.31	−0.297538	−0.148769	+	0.988872i	$0.547531\pi$
$318$	0	0
$319$	364.124	0.0639092
$320$	0	0
$321$	13281.4	2.30933
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3574.15	0.610025
$326$	0	0
$327$	10314.7	1.74436
$328$	0	0
$329$	−5390.50	−0.903306
$330$	0	0
$331$	−6909.94	−1.14745	−0.573723	−	0.819049i	$-0.694501\pi$
$331$	−6909.94	−1.14745	−0.573723	+	0.819049i	$0.694501\pi$
$332$	0	0
$333$	7590.40	1.24910
$334$	0	0
$335$	4647.68	0.757999
$336$	0	0
$337$	−2336.07	−0.377608	−0.188804	−	0.982015i	$-0.560461\pi$
$337$	−2336.07	−0.377608	−0.188804	+	0.982015i	$0.560461\pi$
$338$	0	0
$339$	−13047.2	−2.09035
$340$	0	0
$341$	1952.13	0.310011
$342$	0	0
$343$	5516.38	0.868386
$344$	0	0
$345$	13627.0	2.12653
$346$	0	0
$347$	4545.32	0.703185	0.351593	−	0.936153i	$-0.385640\pi$
$347$	4545.32	0.703185	0.351593	+	0.936153i	$0.385640\pi$
$348$	0	0
$349$	−3483.92	−0.534355	−0.267177	−	0.963647i	$-0.586091\pi$
$349$	−3483.92	−0.534355	−0.267177	+	0.963647i	$0.586091\pi$
$350$	0	0
$351$	5554.55	0.844672
$352$	0	0
$353$	−5057.80	−0.762604	−0.381302	−	0.924450i	$-0.624524\pi$
$353$	−5057.80	−0.762604	−0.381302	+	0.924450i	$0.624524\pi$
$354$	0	0
$355$	2958.37	0.442293
$356$	0	0
$357$	0	0
$358$	0	0
$359$	514.466	0.0756336	0.0378168	−	0.999285i	$-0.487960\pi$
$359$	514.466	0.0756336	0.0378168	+	0.999285i	$0.487960\pi$
$360$	0	0
$361$	15671.6	2.28482
$362$	0	0
$363$	−6999.47	−1.01206
$364$	0	0
$365$	−5057.09	−0.725205
$366$	0	0
$367$	−938.814	−0.133531	−0.0667653	−	0.997769i	$-0.521268\pi$
$367$	−938.814	−0.133531	−0.0667653	+	0.997769i	$0.521268\pi$
$368$	0	0
$369$	−7375.49	−1.04052
$370$	0	0
$371$	−4519.61	−0.632470
$372$	0	0
$373$	−8562.32	−1.18858	−0.594290	−	0.804251i	$-0.702567\pi$
$373$	−8562.32	−1.18858	−0.594290	+	0.804251i	$0.702567\pi$
$374$	0	0
$375$	−8612.14	−1.18594
$376$	0	0
$377$	1563.28	0.213562
$378$	0	0
$379$	5788.04	0.784464	0.392232	−	0.919866i	$-0.371703\pi$
$379$	5788.04	0.784464	0.392232	+	0.919866i	$0.371703\pi$
$380$	0	0
$381$	13855.4	1.86309
$382$	0	0
$383$	−4949.94	−0.660392	−0.330196	−	0.943912i	$-0.607115\pi$
$383$	−4949.94	−0.660392	−0.330196	+	0.943912i	$0.607115\pi$
$384$	0	0
$385$	−2468.69	−0.326796
$386$	0	0
$387$	−2262.43	−0.297173
$388$	0	0
$389$	10592.0	1.38056	0.690279	−	0.723543i	$-0.257488\pi$
$389$	10592.0	1.38056	0.690279	+	0.723543i	$0.257488\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	3343.47	0.429149
$394$	0	0
$395$	−5318.67	−0.677496
$396$	0	0
$397$	−2882.19	−0.364366	−0.182183	−	0.983265i	$-0.558316\pi$
$397$	−2882.19	−0.364366	−0.182183	+	0.983265i	$0.558316\pi$
$398$	0	0
$399$	10815.1	1.35698
$400$	0	0
$401$	3565.37	0.444005	0.222002	−	0.975046i	$-0.428741\pi$
$401$	3565.37	0.444005	0.222002	+	0.975046i	$0.428741\pi$
$402$	0	0
$403$	8380.99	1.03595
$404$	0	0
$405$	−5851.65	−0.717952
$406$	0	0
$407$	4575.12	0.557200
$408$	0	0
$409$	−9096.65	−1.09976	−0.549878	−	0.835245i	$-0.685326\pi$
$409$	−9096.65	−1.09976	−0.549878	+	0.835245i	$0.685326\pi$
$410$	0	0
$411$	20627.0	2.47556
$412$	0	0
$413$	−5549.14	−0.661151
$414$	0	0
$415$	−3412.78	−0.403679
$416$	0	0
$417$	18779.7	2.20538
$418$	0	0
$419$	12893.6	1.50333	0.751663	−	0.659547i	$-0.229252\pi$
$419$	12893.6	1.50333	0.751663	+	0.659547i	$0.229252\pi$
$420$	0	0
$421$	−7200.75	−0.833594	−0.416797	−	0.909000i	$-0.636847\pi$
$421$	−7200.75	−0.833594	−0.416797	+	0.909000i	$0.636847\pi$
$422$	0	0
$423$	20494.1	2.35569
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4843.95	0.548981
$428$	0	0
$429$	14884.4	1.67512
$430$	0	0
$431$	5642.91	0.630648	0.315324	−	0.948984i	$-0.397887\pi$
$431$	5642.91	0.630648	0.315324	+	0.948984i	$0.397887\pi$
$432$	0	0
$433$	14820.6	1.64488	0.822439	−	0.568853i	$-0.192613\pi$
$433$	14820.6	1.64488	0.822439	+	0.568853i	$0.192613\pi$
$434$	0	0
$435$	1749.78	0.192864
$436$	0	0
$437$	−20271.6	−2.21905
$438$	0	0
$439$	515.053	0.0559957	0.0279979	−	0.999608i	$-0.491087\pi$
$439$	515.053	0.0559957	0.0279979	+	0.999608i	$0.491087\pi$
$440$	0	0
$441$	−9023.99	−0.974408
$442$	0	0
$443$	−11633.8	−1.24772	−0.623859	−	0.781537i	$-0.714436\pi$
$443$	−11633.8	−1.24772	−0.623859	+	0.781537i	$0.714436\pi$
$444$	0	0
$445$	−727.302	−0.0774774
$446$	0	0
$447$	−21720.2	−2.29827
$448$	0	0
$449$	−14782.9	−1.55378	−0.776891	−	0.629635i	$-0.783204\pi$
$449$	−14782.9	−1.55378	−0.776891	+	0.629635i	$0.783204\pi$
$450$	0	0
$451$	−4445.58	−0.464156
$452$	0	0
$453$	−21506.6	−2.23061
$454$	0	0
$455$	−10598.7	−1.09204
$456$	0	0
$457$	−18883.4	−1.93289	−0.966444	−	0.256876i	$-0.917307\pi$
$457$	−18883.4	−1.93289	−0.966444	+	0.256876i	$0.917307\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	281.824	0.0284726	0.0142363	−	0.999899i	$-0.495468\pi$
$461$	281.824	0.0284726	0.0142363	+	0.999899i	$0.495468\pi$
$462$	0	0
$463$	−9400.90	−0.943622	−0.471811	−	0.881700i	$-0.656400\pi$
$463$	−9400.90	−0.943622	−0.471811	+	0.881700i	$0.656400\pi$
$464$	0	0
$465$	9380.89	0.935545
$466$	0	0
$467$	−9450.27	−0.936416	−0.468208	−	0.883618i	$-0.655100\pi$
$467$	−9450.27	−0.936416	−0.468208	+	0.883618i	$0.655100\pi$
$468$	0	0
$469$	−3318.82	−0.326756
$470$	0	0
$471$	−18213.9	−1.78185
$472$	0	0
$473$	−1363.68	−0.132563
$474$	0	0
$475$	−5951.26	−0.574869
$476$	0	0
$477$	17183.0	1.64939
$478$	0	0
$479$	9990.90	0.953019	0.476509	−	0.879169i	$-0.341902\pi$
$479$	9990.90	0.953019	0.476509	+	0.879169i	$0.341902\pi$
$480$	0	0
$481$	19642.2	1.86197
$482$	0	0
$483$	−9730.79	−0.916700
$484$	0	0
$485$	−19704.4	−1.84480
$486$	0	0
$487$	4191.84	0.390042	0.195021	−	0.980799i	$-0.437522\pi$
$487$	4191.84	0.390042	0.195021	+	0.980799i	$0.437522\pi$
$488$	0	0
$489$	10790.5	0.997880
$490$	0	0
$491$	11574.2	1.06382	0.531910	−	0.846801i	$-0.321475\pi$
$491$	11574.2	1.06382	0.531910	+	0.846801i	$0.321475\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	9385.70	0.852234
$496$	0	0
$497$	−2112.52	−0.190663
$498$	0	0
$499$	−2265.64	−0.203254	−0.101627	−	0.994823i	$-0.532405\pi$
$499$	−2265.64	−0.203254	−0.101627	+	0.994823i	$0.532405\pi$
$500$	0	0
$501$	5258.19	0.468899
$502$	0	0
$503$	6668.62	0.591131	0.295566	−	0.955322i	$-0.404492\pi$
$503$	6668.62	0.591131	0.295566	+	0.955322i	$0.404492\pi$
$504$	0	0
$505$	−1892.94	−0.166802
$506$	0	0
$507$	46626.4	4.08432
$508$	0	0
$509$	−1758.29	−0.153114	−0.0765569	−	0.997065i	$-0.524393\pi$
$509$	−1758.29	−0.153114	−0.0765569	+	0.997065i	$0.524393\pi$
$510$	0	0
$511$	3611.17	0.312620
$512$	0	0
$513$	−9248.80	−0.795993
$514$	0	0
$515$	6741.94	0.576865
$516$	0	0
$517$	12352.8	1.05083
$518$	0	0
$519$	−25290.9	−2.13901
$520$	0	0
$521$	−5675.16	−0.477223	−0.238612	−	0.971115i	$-0.576692\pi$
$521$	−5675.16	−0.477223	−0.238612	+	0.971115i	$0.576692\pi$
$522$	0	0
$523$	2515.62	0.210326	0.105163	−	0.994455i	$-0.466464\pi$
$523$	2515.62	0.210326	0.105163	+	0.994455i	$0.466464\pi$
$524$	0	0
$525$	−2856.73	−0.237481
$526$	0	0
$527$	0	0
$528$	0	0
$529$	6072.17	0.499069
$530$	0	0
$531$	21097.2	1.72418
$532$	0	0
$533$	−19086.0	−1.55105
$534$	0	0
$535$	21672.2	1.75135
$536$	0	0
$537$	5639.90	0.453221
$538$	0	0
$539$	−5439.22	−0.434664
$540$	0	0
$541$	−2502.81	−0.198898	−0.0994492	−	0.995043i	$-0.531708\pi$
$541$	−2502.81	−0.198898	−0.0994492	+	0.995043i	$0.531708\pi$
$542$	0	0
$543$	30981.5	2.44852
$544$	0	0
$545$	16831.2	1.32288
$546$	0	0
$547$	−4548.84	−0.355566	−0.177783	−	0.984070i	$-0.556892\pi$
$547$	−4548.84	−0.355566	−0.177783	+	0.984070i	$0.556892\pi$
$548$	0	0
$549$	−18416.1	−1.43166
$550$	0	0
$551$	−2602.99	−0.201254
$552$	0	0
$553$	3797.96	0.292053
$554$	0	0
$555$	21985.6	1.68151
$556$	0	0
$557$	2839.28	0.215986	0.107993	−	0.994152i	$-0.465558\pi$
$557$	2839.28	0.215986	0.107993	+	0.994152i	$0.465558\pi$
$558$	0	0
$559$	−5854.64	−0.442978
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13378.2	−1.00147	−0.500733	−	0.865602i	$-0.666936\pi$
$563$	−13378.2	−1.00147	−0.500733	+	0.865602i	$0.666936\pi$
$564$	0	0
$565$	−21290.1	−1.58527
$566$	0	0
$567$	4178.55	0.309493
$568$	0	0
$569$	−379.258	−0.0279426	−0.0139713	−	0.999902i	$-0.504447\pi$
$569$	−379.258	−0.0279426	−0.0139713	+	0.999902i	$0.504447\pi$
$570$	0	0
$571$	−2944.67	−0.215815	−0.107908	−	0.994161i	$-0.534415\pi$
$571$	−2944.67	−0.215815	−0.107908	+	0.994161i	$0.534415\pi$
$572$	0	0
$573$	27705.8	2.01994
$574$	0	0
$575$	5354.58	0.388350
$576$	0	0
$577$	−9592.10	−0.692070	−0.346035	−	0.938222i	$-0.612472\pi$
$577$	−9592.10	−0.692070	−0.346035	+	0.938222i	$0.612472\pi$
$578$	0	0
$579$	11448.5	0.821732
$580$	0	0
$581$	2437.00	0.174017
$582$	0	0
$583$	10357.1	0.735759
$584$	0	0
$585$	40295.2	2.84786
$586$	0	0
$587$	18492.2	1.30027	0.650133	−	0.759820i	$-0.274713\pi$
$587$	18492.2	1.30027	0.650133	+	0.759820i	$0.274713\pi$
$588$	0	0
$589$	−13955.1	−0.976245
$590$	0	0
$591$	13011.9	0.905648
$592$	0	0
$593$	−10310.2	−0.713975	−0.356988	−	0.934109i	$-0.616196\pi$
$593$	−10310.2	−0.713975	−0.356988	+	0.934109i	$0.616196\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−39584.7	−2.71372
$598$	0	0
$599$	−18475.7	−1.26026	−0.630132	−	0.776488i	$-0.716999\pi$
$599$	−18475.7	−1.26026	−0.630132	+	0.776488i	$0.716999\pi$
$600$	0	0
$601$	24908.0	1.69055	0.845274	−	0.534333i	$-0.179437\pi$
$601$	24908.0	1.69055	0.845274	+	0.534333i	$0.179437\pi$
$602$	0	0
$603$	12617.8	0.852131
$604$	0	0
$605$	−11421.5	−0.767522
$606$	0	0
$607$	12102.2	0.809250	0.404625	−	0.914483i	$-0.367402\pi$
$607$	12102.2	0.809250	0.404625	+	0.914483i	$0.367402\pi$
$608$	0	0
$609$	−1249.49	−0.0831392
$610$	0	0
$611$	53033.8	3.51149
$612$	0	0
$613$	−26730.2	−1.76121	−0.880606	−	0.473849i	$-0.842864\pi$
$613$	−26730.2	−1.76121	−0.880606	+	0.473849i	$0.842864\pi$
$614$	0	0
$615$	−21363.1	−1.40072
$616$	0	0
$617$	14440.9	0.942251	0.471125	−	0.882066i	$-0.343848\pi$
$617$	14440.9	0.942251	0.471125	+	0.882066i	$0.343848\pi$
$618$	0	0
$619$	9264.35	0.601560	0.300780	−	0.953694i	$-0.402753\pi$
$619$	9264.35	0.601560	0.300780	+	0.953694i	$0.402753\pi$
$620$	0	0
$621$	8321.51	0.537730
$622$	0	0
$623$	519.352	0.0333987
$624$	0	0
$625$	−19009.0	−1.21658
$626$	0	0
$627$	−24783.9	−1.57858
$628$	0	0
$629$	0	0
$630$	0	0
$631$	6675.13	0.421129	0.210565	−	0.977580i	$-0.432470\pi$
$631$	6675.13	0.421129	0.210565	+	0.977580i	$0.432470\pi$
$632$	0	0
$633$	−16759.7	−1.05235
$634$	0	0
$635$	22608.9	1.41292
$636$	0	0
$637$	−23352.0	−1.45249
$638$	0	0
$639$	8031.55	0.497219
$640$	0	0
$641$	−5606.59	−0.345471	−0.172735	−	0.984968i	$-0.555261\pi$
$641$	−5606.59	−0.345471	−0.172735	+	0.984968i	$0.555261\pi$
$642$	0	0
$643$	8669.26	0.531698	0.265849	−	0.964015i	$-0.414348\pi$
$643$	8669.26	0.531698	0.265849	+	0.964015i	$0.414348\pi$
$644$	0	0
$645$	−6553.13	−0.400045
$646$	0	0
$647$	2022.71	0.122907	0.0614537	−	0.998110i	$-0.480426\pi$
$647$	2022.71	0.122907	0.0614537	+	0.998110i	$0.480426\pi$
$648$	0	0
$649$	12716.4	0.769124
$650$	0	0
$651$	−6698.71	−0.403292
$652$	0	0
$653$	2948.22	0.176681	0.0883407	−	0.996090i	$-0.471844\pi$
$653$	2948.22	0.176681	0.0883407	+	0.996090i	$0.471844\pi$
$654$	0	0
$655$	5455.77	0.325457
$656$	0	0
$657$	−13729.2	−0.815265
$658$	0	0
$659$	−1524.53	−0.0901174	−0.0450587	−	0.998984i	$-0.514347\pi$
$659$	−1524.53	−0.0901174	−0.0450587	+	0.998984i	$0.514347\pi$
$660$	0	0
$661$	8496.15	0.499943	0.249971	−	0.968253i	$-0.419579\pi$
$661$	8496.15	0.499943	0.249971	+	0.968253i	$0.419579\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	17647.8	1.02910
$666$	0	0
$667$	2342.01	0.135956
$668$	0	0
$669$	−38765.3	−2.24029
$670$	0	0
$671$	−11100.4	−0.638635
$672$	0	0
$673$	−26705.8	−1.52962	−0.764809	−	0.644257i	$-0.777167\pi$
$673$	−26705.8	−1.52962	−0.764809	+	0.644257i	$0.777167\pi$
$674$	0	0
$675$	2442.99	0.139305
$676$	0	0
$677$	21673.6	1.23040	0.615201	−	0.788370i	$-0.289075\pi$
$677$	21673.6	1.23040	0.615201	+	0.788370i	$0.289075\pi$
$678$	0	0
$679$	14070.5	0.795253
$680$	0	0
$681$	−8855.73	−0.498315
$682$	0	0
$683$	−33409.2	−1.87170	−0.935849	−	0.352402i	$-0.885365\pi$
$683$	−33409.2	−1.87170	−0.935849	+	0.352402i	$0.885365\pi$
$684$	0	0
$685$	33658.5	1.87741
$686$	0	0
$687$	−14026.7	−0.778968
$688$	0	0
$689$	44465.7	2.45865
$690$	0	0
$691$	−2036.82	−0.112134	−0.0560668	−	0.998427i	$-0.517856\pi$
$691$	−2036.82	−0.112134	−0.0560668	+	0.998427i	$0.517856\pi$
$692$	0	0
$693$	−6702.14	−0.367379
$694$	0	0
$695$	30644.1	1.67251
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−40766.0	−2.20588
$700$	0	0
$701$	−11339.7	−0.610978	−0.305489	−	0.952196i	$-0.598820\pi$
$701$	−11339.7	−0.610978	−0.305489	+	0.952196i	$0.598820\pi$
$702$	0	0
$703$	−32705.9	−1.75466
$704$	0	0
$705$	59361.0	3.17116
$706$	0	0
$707$	1351.72	0.0719045
$708$	0	0
$709$	7810.35	0.413715	0.206857	−	0.978371i	$-0.433676\pi$
$709$	7810.35	0.413715	0.206857	+	0.978371i	$0.433676\pi$
$710$	0	0
$711$	−14439.4	−0.761631
$712$	0	0
$713$	12555.9	0.659499
$714$	0	0
$715$	24288.0	1.27038
$716$	0	0
$717$	21144.1	1.10131
$718$	0	0
$719$	−4852.32	−0.251684	−0.125842	−	0.992050i	$-0.540163\pi$
$719$	−4852.32	−0.251684	−0.125842	+	0.992050i	$0.540163\pi$
$720$	0	0
$721$	−4814.29	−0.248673
$722$	0	0
$723$	256.881	0.0132137
$724$	0	0
$725$	687.558	0.0352210
$726$	0	0
$727$	10206.6	0.520688	0.260344	−	0.965516i	$-0.416164\pi$
$727$	10206.6	0.520688	0.260344	+	0.965516i	$0.416164\pi$
$728$	0	0
$729$	−28968.6	−1.47176
$730$	0	0
$731$	0	0
$732$	0	0
$733$	14858.2	0.748704	0.374352	−	0.927287i	$-0.377865\pi$
$733$	14858.2	0.748704	0.374352	+	0.927287i	$0.377865\pi$
$734$	0	0
$735$	−26138.0	−1.31172
$736$	0	0
$737$	7605.37	0.380119
$738$	0	0
$739$	−39804.9	−1.98139	−0.990694	−	0.136106i	$-0.956541\pi$
$739$	−39804.9	−1.98139	−0.990694	+	0.136106i	$0.956541\pi$
$740$	0	0
$741$	−106403.	−5.27507
$742$	0	0
$743$	−11865.8	−0.585888	−0.292944	−	0.956130i	$-0.594635\pi$
$743$	−11865.8	−0.585888	−0.292944	+	0.956130i	$0.594635\pi$
$744$	0	0
$745$	−35442.3	−1.74296
$746$	0	0
$747$	−9265.20	−0.453810
$748$	0	0
$749$	−15475.7	−0.754965
$750$	0	0
$751$	−13377.0	−0.649979	−0.324990	−	0.945718i	$-0.605361\pi$
$751$	−13377.0	−0.649979	−0.324990	+	0.945718i	$0.605361\pi$
$752$	0	0
$753$	−12518.8	−0.605857
$754$	0	0
$755$	−35093.8	−1.69165
$756$	0	0
$757$	26499.3	1.27230	0.636152	−	0.771563i	$-0.280525\pi$
$757$	26499.3	1.27230	0.636152	+	0.771563i	$0.280525\pi$
$758$	0	0
$759$	22299.0	1.06641
$760$	0	0
$761$	−18510.1	−0.881722	−0.440861	−	0.897575i	$-0.645327\pi$
$761$	−18510.1	−0.881722	−0.440861	+	0.897575i	$0.645327\pi$
$762$	0	0
$763$	−12018.9	−0.570265
$764$	0	0
$765$	0	0
$766$	0	0
$767$	54594.6	2.57014
$768$	0	0
$769$	−12059.0	−0.565484	−0.282742	−	0.959196i	$-0.591244\pi$
$769$	−12059.0	−0.565484	−0.282742	+	0.959196i	$0.591244\pi$
$770$	0	0
$771$	43311.5	2.02312
$772$	0	0
$773$	−27873.5	−1.29694	−0.648472	−	0.761238i	$-0.724592\pi$
$773$	−27873.5	−1.29694	−0.648472	+	0.761238i	$0.724592\pi$
$774$	0	0
$775$	3686.11	0.170850
$776$	0	0
$777$	−15699.5	−0.724859
$778$	0	0
$779$	31779.8	1.46166
$780$	0	0
$781$	4841.03	0.221800
$782$	0	0
$783$	1068.53	0.0487689
$784$	0	0
$785$	−29720.8	−1.35131
$786$	0	0
$787$	20235.2	0.916527	0.458263	−	0.888816i	$-0.348472\pi$
$787$	20235.2	0.916527	0.458263	+	0.888816i	$0.348472\pi$
$788$	0	0
$789$	7717.49	0.348226
$790$	0	0
$791$	15202.8	0.683375
$792$	0	0
$793$	−47656.6	−2.13409
$794$	0	0
$795$	49770.7	2.22036
$796$	0	0
$797$	−5719.18	−0.254183	−0.127091	−	0.991891i	$-0.540564\pi$
$797$	−5719.18	−0.254183	−0.127091	+	0.991891i	$0.540564\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−1974.52	−0.0870989
$802$	0	0
$803$	−8275.32	−0.363673
$804$	0	0
$805$	−15878.4	−0.695205
$806$	0	0
$807$	34338.3	1.49785
$808$	0	0
$809$	18892.2	0.821033	0.410517	−	0.911853i	$-0.365348\pi$
$809$	18892.2	0.821033	0.410517	+	0.911853i	$0.365348\pi$
$810$	0	0
$811$	−40569.7	−1.75659	−0.878295	−	0.478119i	$-0.841319\pi$
$811$	−40569.7	−1.75659	−0.878295	+	0.478119i	$0.841319\pi$
$812$	0	0
$813$	−20999.9	−0.905901
$814$	0	0
$815$	17607.6	0.756770
$816$	0	0
$817$	9748.47	0.417449
$818$	0	0
$819$	−28774.0	−1.22765
$820$	0	0
$821$	−1331.27	−0.0565916	−0.0282958	−	0.999600i	$-0.509008\pi$
$821$	−1331.27	−0.0565916	−0.0282958	+	0.999600i	$0.509008\pi$
$822$	0	0
$823$	17376.5	0.735973	0.367987	−	0.929831i	$-0.380047\pi$
$823$	17376.5	0.735973	0.367987	+	0.929831i	$0.380047\pi$
$824$	0	0
$825$	6546.45	0.276264
$826$	0	0
$827$	8704.79	0.366016	0.183008	−	0.983111i	$-0.441417\pi$
$827$	8704.79	0.366016	0.183008	+	0.983111i	$0.441417\pi$
$828$	0	0
$829$	−39707.6	−1.66357	−0.831786	−	0.555097i	$-0.812681\pi$
$829$	−39707.6	−1.66357	−0.831786	+	0.555097i	$0.812681\pi$
$830$	0	0
$831$	−44504.4	−1.85781
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8580.14	0.355603
$836$	0	0
$837$	5728.56	0.236568
$838$	0	0
$839$	15341.0	0.631263	0.315632	−	0.948882i	$-0.397784\pi$
$839$	15341.0	0.631263	0.315632	+	0.948882i	$0.397784\pi$
$840$	0	0
$841$	−24088.3	−0.987670
$842$	0	0
$843$	44625.4	1.82323
$844$	0	0
$845$	76083.5	3.09746
$846$	0	0
$847$	8155.89	0.330862
$848$	0	0
$849$	−40011.8	−1.61743
$850$	0	0
$851$	29426.8	1.18535
$852$	0	0
$853$	−1332.43	−0.0534834	−0.0267417	−	0.999642i	$-0.508513\pi$
$853$	−1332.43	−0.0534834	−0.0267417	+	0.999642i	$0.508513\pi$
$854$	0	0
$855$	−67094.9	−2.68374
$856$	0	0
$857$	−5345.78	−0.213079	−0.106539	−	0.994308i	$-0.533977\pi$
$857$	−5345.78	−0.213079	−0.106539	+	0.994308i	$0.533977\pi$
$858$	0	0
$859$	−8460.43	−0.336049	−0.168024	−	0.985783i	$-0.553739\pi$
$859$	−8460.43	−0.336049	−0.168024	+	0.985783i	$0.553739\pi$
$860$	0	0
$861$	15255.0	0.603819
$862$	0	0
$863$	24917.4	0.982848	0.491424	−	0.870921i	$-0.336477\pi$
$863$	24917.4	0.982848	0.491424	+	0.870921i	$0.336477\pi$
$864$	0	0
$865$	−41268.9	−1.62218
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8703.37	−0.339749
$870$	0	0
$871$	32651.8	1.27022
$872$	0	0
$873$	−53494.5	−2.07390
$874$	0	0
$875$	10035.0	0.387708
$876$	0	0
$877$	−1053.92	−0.0405798	−0.0202899	−	0.999794i	$-0.506459\pi$
$877$	−1053.92	−0.0405798	−0.0202899	+	0.999794i	$0.506459\pi$
$878$	0	0
$879$	24557.4	0.942321
$880$	0	0
$881$	−19244.6	−0.735943	−0.367972	−	0.929837i	$-0.619948\pi$
$881$	−19244.6	−0.735943	−0.367972	+	0.929837i	$0.619948\pi$
$882$	0	0
$883$	−1460.49	−0.0556617	−0.0278308	−	0.999613i	$-0.508860\pi$
$883$	−1460.49	−0.0556617	−0.0278308	+	0.999613i	$0.508860\pi$
$884$	0	0
$885$	61108.0	2.32104
$886$	0	0
$887$	35836.8	1.35657	0.678287	−	0.734797i	$-0.262723\pi$
$887$	35836.8	1.35657	0.678287	+	0.734797i	$0.262723\pi$
$888$	0	0
$889$	−16144.6	−0.609079
$890$	0	0
$891$	−9575.53	−0.360036
$892$	0	0
$893$	−88305.8	−3.30912
$894$	0	0
$895$	9203.02	0.343713
$896$	0	0
$897$	95735.3	3.56355
$898$	0	0
$899$	1612.25	0.0598125
$900$	0	0
$901$	0	0
$902$	0	0
$903$	4679.46	0.172450
$904$	0	0
$905$	50554.7	1.85690
$906$	0	0
$907$	−50682.2	−1.85543	−0.927715	−	0.373290i	$-0.878230\pi$
$907$	−50682.2	−1.85543	−0.927715	+	0.373290i	$0.878230\pi$
$908$	0	0
$909$	−5139.07	−0.187516
$910$	0	0
$911$	18662.5	0.678722	0.339361	−	0.940656i	$-0.389789\pi$
$911$	18662.5	0.678722	0.339361	+	0.940656i	$0.389789\pi$
$912$	0	0
$913$	−5584.61	−0.202435
$914$	0	0
$915$	−53342.3	−1.92726
$916$	0	0
$917$	−3895.86	−0.140297
$918$	0	0
$919$	−899.936	−0.0323027	−0.0161513	−	0.999870i	$-0.505141\pi$
$919$	−899.936	−0.0323027	−0.0161513	+	0.999870i	$0.505141\pi$
$920$	0	0
$921$	−30628.2	−1.09580
$922$	0	0
$923$	20783.8	0.741176
$924$	0	0
$925$	8638.99	0.307079
$926$	0	0
$927$	18303.4	0.648503
$928$	0	0
$929$	6429.44	0.227065	0.113532	−	0.993534i	$-0.463783\pi$
$929$	6429.44	0.227065	0.113532	+	0.993534i	$0.463783\pi$
$930$	0	0
$931$	38883.0	1.36878
$932$	0	0
$933$	66462.8	2.33215
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−14387.3	−0.501615	−0.250807	−	0.968037i	$-0.580696\pi$
$937$	−14387.3	−0.501615	−0.250807	+	0.968037i	$0.580696\pi$
$938$	0	0
$939$	9180.62	0.319061
$940$	0	0
$941$	24911.4	0.863004	0.431502	−	0.902112i	$-0.357984\pi$
$941$	24911.4	0.863004	0.431502	+	0.902112i	$0.357984\pi$
$942$	0	0
$943$	−28593.6	−0.987417
$944$	0	0
$945$	−7244.42	−0.249377
$946$	0	0
$947$	−11698.0	−0.401410	−0.200705	−	0.979652i	$-0.564323\pi$
$947$	−11698.0	−0.401410	−0.200705	+	0.979652i	$0.564323\pi$
$948$	0	0
$949$	−35528.1	−1.21527
$950$	0	0
$951$	−13205.4	−0.450277
$952$	0	0
$953$	7688.43	0.261336	0.130668	−	0.991426i	$-0.458288\pi$
$953$	7688.43	0.261336	0.130668	+	0.991426i	$0.458288\pi$
$954$	0	0
$955$	45209.4	1.53188
$956$	0	0
$957$	2863.31	0.0967166
$958$	0	0
$959$	−24034.9	−0.809309
$960$	0	0
$961$	−21147.5	−0.709861
$962$	0	0
$963$	58836.8	1.96884
$964$	0	0
$965$	18681.3	0.623183
$966$	0	0
$967$	−3679.54	−0.122364	−0.0611819	−	0.998127i	$-0.519487\pi$
$967$	−3679.54	−0.122364	−0.0611819	+	0.998127i	$0.519487\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20724.1	−0.684930	−0.342465	−	0.939531i	$-0.611262\pi$
$971$	−20724.1	−0.684930	−0.342465	+	0.939531i	$0.611262\pi$
$972$	0	0
$973$	−21882.4	−0.720983
$974$	0	0
$975$	28105.6	0.923178
$976$	0	0
$977$	−54936.4	−1.79895	−0.899474	−	0.436975i	$-0.856050\pi$
$977$	−54936.4	−1.79895	−0.899474	+	0.436975i	$0.856050\pi$
$978$	0	0
$979$	−1190.14	−0.0388531
$980$	0	0
$981$	45694.4	1.48716
$982$	0	0
$983$	16782.0	0.544519	0.272259	−	0.962224i	$-0.412229\pi$
$983$	16782.0	0.544519	0.272259	+	0.962224i	$0.412229\pi$
$984$	0	0
$985$	21232.4	0.686823
$986$	0	0
$987$	−42388.6	−1.36701
$988$	0	0
$989$	−8771.08	−0.282006
$990$	0	0
$991$	20887.2	0.669530	0.334765	−	0.942302i	$-0.391343\pi$
$991$	20887.2	0.669530	0.334765	+	0.942302i	$0.391343\pi$
$992$	0	0
$993$	−54336.8	−1.73648
$994$	0	0
$995$	−64593.1	−2.05803
$996$	0	0
$997$	−26458.1	−0.840457	−0.420228	−	0.907418i	$-0.638050\pi$
$997$	−26458.1	−0.840457	−0.420228	+	0.907418i	$0.638050\pi$
$998$	0	0
$999$	13425.8	0.425198

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.j.1.6	yes	6
17.16	even	2		2312.4.a.f.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.f.1.1	✓	6	17.16	even	2
2312.4.a.j.1.6	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q+7.86357 q^{3} +12.8315 q^{5} -9.16275 q^{7} +34.8357 q^{9} +O(q^{10})\)	\(q+7.86357 q^{3} +12.8315 q^{5} -9.16275 q^{7} +34.8357 q^{9} +20.9973 q^{11} +90.1467 q^{13} +100.902 q^{15} -150.102 q^{19} -72.0519 q^{21} +135.052 q^{23} +39.6482 q^{25} +61.6168 q^{27} +17.3415 q^{29} +92.9706 q^{31} +165.114 q^{33} -117.572 q^{35} +217.891 q^{37} +708.875 q^{39} -211.722 q^{41} -64.9457 q^{43} +446.996 q^{45} +588.306 q^{47} -259.044 q^{49} +493.259 q^{53} +269.427 q^{55} -1180.34 q^{57} +605.620 q^{59} -528.657 q^{61} -319.191 q^{63} +1156.72 q^{65} +362.208 q^{67} +1061.99 q^{69} +230.555 q^{71} -394.114 q^{73} +311.776 q^{75} -192.393 q^{77} -414.500 q^{79} -456.037 q^{81} -265.968 q^{83} +136.366 q^{87} -56.6809 q^{89} -825.991 q^{91} +731.081 q^{93} -1926.04 q^{95} -1535.62 q^{97} +731.456 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 7 q^{3} + 3 q^{5} + 7 q^{7} + 31 q^{9}+O(q^{10}) \) 6 * q + 7 * q^3 + 3 * q^5 + 7 * q^7 + 31 * q^9	\( 6 q + 7 q^{3} + 3 q^{5} + 7 q^{7} + 31 q^{9} + 72 q^{11} + 15 q^{13} - 60 q^{15} + 83 q^{19} + 96 q^{21} + 141 q^{23} + 263 q^{25} + 94 q^{27} - 249 q^{29} + 106 q^{31} + 289 q^{33} - 267 q^{35} + 170 q^{37} + 329 q^{39} - 100 q^{41} - 90 q^{43} - 286 q^{45} + 372 q^{47} + 323 q^{49} - 23 q^{53} - 457 q^{55} - 193 q^{57} + 784 q^{59} + 92 q^{61} - 722 q^{63} + 1412 q^{65} + 238 q^{67} + 1178 q^{69} + 940 q^{71} + 692 q^{73} + 1814 q^{75} - 45 q^{77} + 84 q^{79} - 2182 q^{81} + 1393 q^{83} - 1247 q^{87} + 976 q^{89} - 384 q^{91} + 1437 q^{93} - 1022 q^{95} - 325 q^{97} + 3981 q^{99}+O(q^{100}) \) 6 * q + 7 * q^3 + 3 * q^5 + 7 * q^7 + 31 * q^9 + 72 * q^11 + 15 * q^13 - 60 * q^15 + 83 * q^19 + 96 * q^21 + 141 * q^23 + 263 * q^25 + 94 * q^27 - 249 * q^29 + 106 * q^31 + 289 * q^33 - 267 * q^35 + 170 * q^37 + 329 * q^39 - 100 * q^41 - 90 * q^43 - 286 * q^45 + 372 * q^47 + 323 * q^49 - 23 * q^53 - 457 * q^55 - 193 * q^57 + 784 * q^59 + 92 * q^61 - 722 * q^63 + 1412 * q^65 + 238 * q^67 + 1178 * q^69 + 940 * q^71 + 692 * q^73 + 1814 * q^75 - 45 * q^77 + 84 * q^79 - 2182 * q^81 + 1393 * q^83 - 1247 * q^87 + 976 * q^89 - 384 * q^91 + 1437 * q^93 - 1022 * q^95 - 325 * q^97 + 3981 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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