LMFDB - Embedded newform 2312.4.a.j.1.4

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			1.4
Root			$1.81785$ 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)

$f(q)$	$=$	$q+2.81785 q^{3} -4.22994 q^{5} -16.4587 q^{7} -19.0597 q^{9} +O(q^{10})$	$q+2.81785 q^{3} -4.22994 q^{5} -16.4587 q^{7} -19.0597 q^{9} -46.4193 q^{11} -40.4963 q^{13} -11.9194 q^{15} -40.5756 q^{19} -46.3781 q^{21} -161.693 q^{23} -107.108 q^{25} -129.789 q^{27} +35.9239 q^{29} +242.334 q^{31} -130.803 q^{33} +69.6193 q^{35} -237.719 q^{37} -114.113 q^{39} +220.797 q^{41} +157.486 q^{43} +80.6215 q^{45} -311.331 q^{47} -72.1121 q^{49} -387.213 q^{53} +196.351 q^{55} -114.336 q^{57} +217.629 q^{59} +56.5390 q^{61} +313.697 q^{63} +171.297 q^{65} +399.651 q^{67} -455.626 q^{69} +175.230 q^{71} +1073.51 q^{73} -301.813 q^{75} +763.999 q^{77} -661.392 q^{79} +148.884 q^{81} -395.143 q^{83} +101.228 q^{87} -1000.50 q^{89} +666.515 q^{91} +682.861 q^{93} +171.632 q^{95} -1818.84 q^{97} +884.737 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.81785	0.542296	0.271148	−	0.962538i	$-0.412597\pi$
$3$	2.81785	0.542296	0.271148	+	0.962538i	$0.412597\pi$
$4$	0	0
$5$	−4.22994	−0.378338	−0.189169	−	0.981945i	$-0.560579\pi$
$5$	−4.22994	−0.378338	−0.189169	+	0.981945i	$0.560579\pi$
$6$	0	0
$7$	−16.4587	−0.888685	−0.444342	−	0.895857i	$-0.646563\pi$
$7$	−16.4587	−0.888685	−0.444342	+	0.895857i	$0.646563\pi$
$8$	0	0
$9$	−19.0597	−0.705915
$10$	0	0
$11$	−46.4193	−1.27236	−0.636179	−	0.771542i	$-0.719486\pi$
$11$	−46.4193	−1.27236	−0.636179	+	0.771542i	$0.719486\pi$
$12$	0	0
$13$	−40.4963	−0.863973	−0.431986	−	0.901880i	$-0.642187\pi$
$13$	−40.4963	−0.863973	−0.431986	+	0.901880i	$0.642187\pi$
$14$	0	0
$15$	−11.9194	−0.205171
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−40.5756	−0.489930	−0.244965	−	0.969532i	$-0.578776\pi$
$19$	−40.5756	−0.489930	−0.244965	+	0.969532i	$0.578776\pi$
$20$	0	0
$21$	−46.3781	−0.481930
$22$	0	0
$23$	−161.693	−1.46588	−0.732940	−	0.680293i	$-0.761853\pi$
$23$	−161.693	−1.46588	−0.732940	+	0.680293i	$0.761853\pi$
$24$	0	0
$25$	−107.108	−0.856861
$26$	0	0
$27$	−129.789	−0.925111
$28$	0	0
$29$	35.9239	0.230031	0.115016	−	0.993364i	$-0.463308\pi$
$29$	35.9239	0.230031	0.115016	+	0.993364i	$0.463308\pi$
$30$	0	0
$31$	242.334	1.40401	0.702007	−	0.712170i	$-0.252287\pi$
$31$	242.334	1.40401	0.702007	+	0.712170i	$0.252287\pi$
$32$	0	0
$33$	−130.803	−0.689994
$34$	0	0
$35$	69.6193	0.336223
$36$	0	0
$37$	−237.719	−1.05624	−0.528119	−	0.849170i	$-0.677103\pi$
$37$	−237.719	−1.05624	−0.528119	+	0.849170i	$0.677103\pi$
$38$	0	0
$39$	−114.113	−0.468529
$40$	0	0
$41$	220.797	0.841040	0.420520	−	0.907283i	$-0.361848\pi$
$41$	220.797	0.841040	0.420520	+	0.907283i	$0.361848\pi$
$42$	0	0
$43$	157.486	0.558521	0.279261	−	0.960215i	$-0.409911\pi$
$43$	157.486	0.558521	0.279261	+	0.960215i	$0.409911\pi$
$44$	0	0
$45$	80.6215	0.267074
$46$	0	0
$47$	−311.331	−0.966218	−0.483109	−	0.875560i	$-0.660493\pi$
$47$	−311.331	−0.966218	−0.483109	+	0.875560i	$0.660493\pi$
$48$	0	0
$49$	−72.1121	−0.210239
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−387.213	−1.00354	−0.501771	−	0.865000i	$-0.667318\pi$
$53$	−387.213	−1.00354	−0.501771	+	0.865000i	$0.667318\pi$
$54$	0	0
$55$	196.351	0.481381
$56$	0	0
$57$	−114.336	−0.265687
$58$	0	0
$59$	217.629	0.480219	0.240109	−	0.970746i	$-0.422817\pi$
$59$	217.629	0.480219	0.240109	+	0.970746i	$0.422817\pi$
$60$	0	0
$61$	56.5390	0.118673	0.0593366	−	0.998238i	$-0.481101\pi$
$61$	56.5390	0.118673	0.0593366	+	0.998238i	$0.481101\pi$
$62$	0	0
$63$	313.697	0.627336
$64$	0	0
$65$	171.297	0.326874
$66$	0	0
$67$	399.651	0.728733	0.364367	−	0.931256i	$-0.381285\pi$
$67$	399.651	0.728733	0.364367	+	0.931256i	$0.381285\pi$
$68$	0	0
$69$	−455.626	−0.794941
$70$	0	0
$71$	175.230	0.292900	0.146450	−	0.989218i	$-0.453215\pi$
$71$	175.230	0.292900	0.146450	+	0.989218i	$0.453215\pi$
$72$	0	0
$73$	1073.51	1.72117	0.860584	−	0.509309i	$-0.170099\pi$
$73$	1073.51	1.72117	0.860584	+	0.509309i	$0.170099\pi$
$74$	0	0
$75$	−301.813	−0.464672
$76$	0	0
$77$	763.999	1.13072
$78$	0	0
$79$	−661.392	−0.941929	−0.470965	−	0.882152i	$-0.656094\pi$
$79$	−661.392	−0.941929	−0.470965	+	0.882152i	$0.656094\pi$
$80$	0	0
$81$	148.884	0.204231
$82$	0	0
$83$	−395.143	−0.522561	−0.261280	−	0.965263i	$-0.584145\pi$
$83$	−395.143	−0.522561	−0.261280	+	0.965263i	$0.584145\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	101.228	0.124745
$88$	0	0
$89$	−1000.50	−1.19160	−0.595802	−	0.803131i	$-0.703166\pi$
$89$	−1000.50	−1.19160	−0.595802	+	0.803131i	$0.703166\pi$
$90$	0	0
$91$	666.515	0.767799
$92$	0	0
$93$	682.861	0.761391
$94$	0	0
$95$	171.632	0.185359
$96$	0	0
$97$	−1818.84	−1.90386	−0.951932	−	0.306309i	$-0.900906\pi$
$97$	−1818.84	−1.90386	−0.951932	+	0.306309i	$0.900906\pi$
$98$	0	0
$99$	884.737	0.898176
$100$	0	0
$101$	947.783	0.933742	0.466871	−	0.884325i	$-0.345381\pi$
$101$	947.783	0.933742	0.466871	+	0.884325i	$0.345381\pi$
$102$	0	0
$103$	−440.330	−0.421233	−0.210617	−	0.977569i	$-0.567547\pi$
$103$	−440.330	−0.421233	−0.210617	+	0.977569i	$0.567547\pi$
$104$	0	0
$105$	196.177	0.182332
$106$	0	0
$107$	9.33573	0.00843475	0.00421738	−	0.999991i	$-0.498658\pi$
$107$	9.33573	0.00843475	0.00421738	+	0.999991i	$0.498658\pi$
$108$	0	0
$109$	761.211	0.668907	0.334453	−	0.942412i	$-0.391448\pi$
$109$	761.211	0.668907	0.334453	+	0.942412i	$0.391448\pi$
$110$	0	0
$111$	−669.858	−0.572794
$112$	0	0
$113$	842.369	0.701269	0.350635	−	0.936512i	$-0.385966\pi$
$113$	842.369	0.701269	0.350635	+	0.936512i	$0.385966\pi$
$114$	0	0
$115$	683.951	0.554598
$116$	0	0
$117$	771.847	0.609891
$118$	0	0
$119$	0	0
$120$	0	0
$121$	823.747	0.618893
$122$	0	0
$123$	622.172	0.456093
$124$	0	0
$125$	981.802	0.702520
$126$	0	0
$127$	1509.09	1.05441	0.527207	−	0.849737i	$-0.323239\pi$
$127$	1509.09	1.05441	0.527207	+	0.849737i	$0.323239\pi$
$128$	0	0
$129$	443.773	0.302884
$130$	0	0
$131$	−286.684	−0.191204	−0.0956018	−	0.995420i	$-0.530478\pi$
$131$	−286.684	−0.191204	−0.0956018	+	0.995420i	$0.530478\pi$
$132$	0	0
$133$	667.820	0.435393
$134$	0	0
$135$	549.002	0.350004
$136$	0	0
$137$	1891.92	1.17984	0.589919	−	0.807462i	$-0.299160\pi$
$137$	1891.92	1.17984	0.589919	+	0.807462i	$0.299160\pi$
$138$	0	0
$139$	901.877	0.550332	0.275166	−	0.961397i	$-0.411267\pi$
$139$	901.877	0.550332	0.275166	+	0.961397i	$0.411267\pi$
$140$	0	0
$141$	−877.284	−0.523976
$142$	0	0
$143$	1879.81	1.09928
$144$	0	0
$145$	−151.956	−0.0870294
$146$	0	0
$147$	−203.201	−0.114012
$148$	0	0
$149$	−2731.46	−1.50181	−0.750905	−	0.660411i	$-0.770382\pi$
$149$	−2731.46	−1.50181	−0.750905	+	0.660411i	$0.770382\pi$
$150$	0	0
$151$	1621.93	0.874109	0.437054	−	0.899435i	$-0.356022\pi$
$151$	1621.93	0.874109	0.437054	+	0.899435i	$0.356022\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1025.06	−0.531192
$156$	0	0
$157$	1709.12	0.868806	0.434403	−	0.900719i	$-0.356959\pi$
$157$	1709.12	0.868806	0.434403	+	0.900719i	$0.356959\pi$
$158$	0	0
$159$	−1091.11	−0.544217
$160$	0	0
$161$	2661.25	1.30271
$162$	0	0
$163$	−1952.37	−0.938166	−0.469083	−	0.883154i	$-0.655416\pi$
$163$	−1952.37	−0.938166	−0.469083	+	0.883154i	$0.655416\pi$
$164$	0	0
$165$	553.288	0.261051
$166$	0	0
$167$	2551.39	1.18223	0.591115	−	0.806587i	$-0.298688\pi$
$167$	2551.39	1.18223	0.591115	+	0.806587i	$0.298688\pi$
$168$	0	0
$169$	−557.051	−0.253551
$170$	0	0
$171$	773.358	0.345849
$172$	0	0
$173$	1138.76	0.500454	0.250227	−	0.968187i	$-0.419495\pi$
$173$	1138.76	0.500454	0.250227	+	0.968187i	$0.419495\pi$
$174$	0	0
$175$	1762.85	0.761479
$176$	0	0
$177$	613.247	0.260421
$178$	0	0
$179$	−4192.93	−1.75081	−0.875403	−	0.483394i	$-0.839404\pi$
$179$	−4192.93	−1.75081	−0.875403	+	0.483394i	$0.839404\pi$
$180$	0	0
$181$	−2645.51	−1.08640	−0.543202	−	0.839602i	$-0.682788\pi$
$181$	−2645.51	−1.08640	−0.543202	+	0.839602i	$0.682788\pi$
$182$	0	0
$183$	159.318	0.0643561
$184$	0	0
$185$	1005.54	0.399615
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2136.16	0.822132
$190$	0	0
$191$	−2865.26	−1.08546	−0.542730	−	0.839907i	$-0.682609\pi$
$191$	−2865.26	−1.08546	−0.542730	+	0.839907i	$0.682609\pi$
$192$	0	0
$193$	−4176.24	−1.55758	−0.778789	−	0.627286i	$-0.784166\pi$
$193$	−4176.24	−1.55758	−0.778789	+	0.627286i	$0.784166\pi$
$194$	0	0
$195$	482.690	0.177262
$196$	0	0
$197$	1330.14	0.481057	0.240529	−	0.970642i	$-0.422679\pi$
$197$	1330.14	0.481057	0.240529	+	0.970642i	$0.422679\pi$
$198$	0	0
$199$	−1306.79	−0.465506	−0.232753	−	0.972536i	$-0.574773\pi$
$199$	−1306.79	−0.465506	−0.232753	+	0.972536i	$0.574773\pi$
$200$	0	0
$201$	1126.16	0.395189
$202$	0	0
$203$	−591.260	−0.204425
$204$	0	0
$205$	−933.958	−0.318197
$206$	0	0
$207$	3081.82	1.03479
$208$	0	0
$209$	1883.49	0.623366
$210$	0	0
$211$	3946.14	1.28751	0.643753	−	0.765234i	$-0.277377\pi$
$211$	3946.14	1.28751	0.643753	+	0.765234i	$0.277377\pi$
$212$	0	0
$213$	493.771	0.158839
$214$	0	0
$215$	−666.158	−0.211310
$216$	0	0
$217$	−3988.49	−1.24773
$218$	0	0
$219$	3025.00	0.933382
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1029.67	−0.309200	−0.154600	−	0.987977i	$-0.549409\pi$
$223$	−1029.67	−0.309200	−0.154600	+	0.987977i	$0.549409\pi$
$224$	0	0
$225$	2041.44	0.604871
$226$	0	0
$227$	3179.96	0.929787	0.464893	−	0.885367i	$-0.346093\pi$
$227$	3179.96	0.929787	0.464893	+	0.885367i	$0.346093\pi$
$228$	0	0
$229$	−3681.58	−1.06238	−0.531191	−	0.847252i	$-0.678255\pi$
$229$	−3681.58	−1.06238	−0.531191	+	0.847252i	$0.678255\pi$
$230$	0	0
$231$	2152.84	0.613187
$232$	0	0
$233$	−3234.20	−0.909354	−0.454677	−	0.890656i	$-0.650245\pi$
$233$	−3234.20	−0.909354	−0.454677	+	0.890656i	$0.650245\pi$
$234$	0	0
$235$	1316.91	0.365557
$236$	0	0
$237$	−1863.71	−0.510804
$238$	0	0
$239$	−6310.98	−1.70805	−0.854024	−	0.520233i	$-0.825845\pi$
$239$	−6310.98	−1.70805	−0.854024	+	0.520233i	$0.825845\pi$
$240$	0	0
$241$	−679.187	−0.181536	−0.0907682	−	0.995872i	$-0.528932\pi$
$241$	−679.187	−0.181536	−0.0907682	+	0.995872i	$0.528932\pi$
$242$	0	0
$243$	3923.85	1.03586
$244$	0	0
$245$	305.030	0.0795415
$246$	0	0
$247$	1643.16	0.423286
$248$	0	0
$249$	−1113.45	−0.283383
$250$	0	0
$251$	−5887.62	−1.48057	−0.740285	−	0.672293i	$-0.765310\pi$
$251$	−5887.62	−1.48057	−0.740285	+	0.672293i	$0.765310\pi$
$252$	0	0
$253$	7505.65	1.86512
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2973.63	−0.721752	−0.360876	−	0.932614i	$-0.617522\pi$
$257$	−2973.63	−0.721752	−0.360876	+	0.932614i	$0.617522\pi$
$258$	0	0
$259$	3912.54	0.938663
$260$	0	0
$261$	−684.699	−0.162382
$262$	0	0
$263$	5345.48	1.25330	0.626648	−	0.779303i	$-0.284427\pi$
$263$	5345.48	1.25330	0.626648	+	0.779303i	$0.284427\pi$
$264$	0	0
$265$	1637.89	0.379678
$266$	0	0
$267$	−2819.26	−0.646203
$268$	0	0
$269$	5245.36	1.18890	0.594452	−	0.804131i	$-0.297369\pi$
$269$	5245.36	1.18890	0.594452	+	0.804131i	$0.297369\pi$
$270$	0	0
$271$	−5271.60	−1.18165	−0.590825	−	0.806800i	$-0.701197\pi$
$271$	−5271.60	−1.18165	−0.590825	+	0.806800i	$0.701197\pi$
$272$	0	0
$273$	1878.14	0.416375
$274$	0	0
$275$	4971.85	1.09023
$276$	0	0
$277$	2830.00	0.613856	0.306928	−	0.951733i	$-0.400699\pi$
$277$	2830.00	0.613856	0.306928	+	0.951733i	$0.400699\pi$
$278$	0	0
$279$	−4618.81	−0.991115
$280$	0	0
$281$	7739.35	1.64303	0.821514	−	0.570189i	$-0.193130\pi$
$281$	7739.35	1.64303	0.821514	+	0.570189i	$0.193130\pi$
$282$	0	0
$283$	5015.76	1.05355	0.526777	−	0.850003i	$-0.323400\pi$
$283$	5015.76	1.05355	0.526777	+	0.850003i	$0.323400\pi$
$284$	0	0
$285$	483.635	0.100519
$286$	0	0
$287$	−3634.02	−0.747419
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−5125.21	−1.03246
$292$	0	0
$293$	8361.00	1.66708	0.833540	−	0.552458i	$-0.186310\pi$
$293$	8361.00	1.66708	0.833540	+	0.552458i	$0.186310\pi$
$294$	0	0
$295$	−920.560	−0.181685
$296$	0	0
$297$	6024.73	1.17707
$298$	0	0
$299$	6547.95	1.26648
$300$	0	0
$301$	−2592.01	−0.496349
$302$	0	0
$303$	2670.71	0.506365
$304$	0	0
$305$	−239.157	−0.0448986
$306$	0	0
$307$	−5090.44	−0.946341	−0.473171	−	0.880971i	$-0.656891\pi$
$307$	−5090.44	−0.946341	−0.473171	+	0.880971i	$0.656891\pi$
$308$	0	0
$309$	−1240.79	−0.228433
$310$	0	0
$311$	−2901.47	−0.529027	−0.264513	−	0.964382i	$-0.585211\pi$
$311$	−2901.47	−0.529027	−0.264513	+	0.964382i	$0.585211\pi$
$312$	0	0
$313$	8672.88	1.56620	0.783099	−	0.621897i	$-0.213638\pi$
$313$	8672.88	1.56620	0.783099	+	0.621897i	$0.213638\pi$
$314$	0	0
$315$	−1326.92	−0.237345
$316$	0	0
$317$	−8160.58	−1.44588	−0.722940	−	0.690911i	$-0.757210\pi$
$317$	−8160.58	−1.44588	−0.722940	+	0.690911i	$0.757210\pi$
$318$	0	0
$319$	−1667.56	−0.292682
$320$	0	0
$321$	26.3067	0.00457413
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4337.46	0.740304
$326$	0	0
$327$	2144.98	0.362745
$328$	0	0
$329$	5124.09	0.858663
$330$	0	0
$331$	7197.33	1.19517	0.597584	−	0.801806i	$-0.296127\pi$
$331$	7197.33	1.19517	0.597584	+	0.801806i	$0.296127\pi$
$332$	0	0
$333$	4530.86	0.745615
$334$	0	0
$335$	−1690.50	−0.275707
$336$	0	0
$337$	6370.04	1.02967	0.514834	−	0.857290i	$-0.327854\pi$
$337$	6370.04	1.02967	0.514834	+	0.857290i	$0.327854\pi$
$338$	0	0
$339$	2373.67	0.380295
$340$	0	0
$341$	−11249.0	−1.78641
$342$	0	0
$343$	6832.19	1.07552
$344$	0	0
$345$	1927.27	0.300756
$346$	0	0
$347$	−3247.99	−0.502482	−0.251241	−	0.967925i	$-0.580839\pi$
$347$	−3247.99	−0.502482	−0.251241	+	0.967925i	$0.580839\pi$
$348$	0	0
$349$	−10870.4	−1.66727	−0.833636	−	0.552314i	$-0.813745\pi$
$349$	−10870.4	−1.66727	−0.833636	+	0.552314i	$0.813745\pi$
$350$	0	0
$351$	5255.99	0.799271
$352$	0	0
$353$	−10126.9	−1.52691	−0.763455	−	0.645861i	$-0.776499\pi$
$353$	−10126.9	−1.52691	−0.763455	+	0.645861i	$0.776499\pi$
$354$	0	0
$355$	−741.211	−0.110815
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2409.84	−0.354279	−0.177140	−	0.984186i	$-0.556684\pi$
$359$	−2409.84	−0.354279	−0.177140	+	0.984186i	$0.556684\pi$
$360$	0	0
$361$	−5212.62	−0.759968
$362$	0	0
$363$	2321.20	0.335623
$364$	0	0
$365$	−4540.90	−0.651183
$366$	0	0
$367$	10200.2	1.45081	0.725407	−	0.688320i	$-0.241652\pi$
$367$	10200.2	1.45081	0.725407	+	0.688320i	$0.241652\pi$
$368$	0	0
$369$	−4208.32	−0.593703
$370$	0	0
$371$	6373.01	0.891833
$372$	0	0
$373$	−12299.7	−1.70738	−0.853692	−	0.520777i	$-0.825642\pi$
$373$	−12299.7	−1.70738	−0.853692	+	0.520777i	$0.825642\pi$
$374$	0	0
$375$	2766.57	0.380974
$376$	0	0
$377$	−1454.78	−0.198741
$378$	0	0
$379$	13.3372	0.00180761	0.000903805	−	1.00000i	$-0.499712\pi$
$379$	13.3372	0.00180761	0.000903805		1.00000i	$0.499712\pi$
$380$	0	0
$381$	4252.41	0.571804
$382$	0	0
$383$	−10866.4	−1.44973	−0.724866	−	0.688890i	$-0.758098\pi$
$383$	−10866.4	−1.44973	−0.724866	+	0.688890i	$0.758098\pi$
$384$	0	0
$385$	−3231.67	−0.427796
$386$	0	0
$387$	−3001.64	−0.394268
$388$	0	0
$389$	8781.25	1.14454	0.572271	−	0.820064i	$-0.306062\pi$
$389$	8781.25	1.14454	0.572271	+	0.820064i	$0.306062\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−807.833	−0.103689
$394$	0	0
$395$	2797.65	0.356367
$396$	0	0
$397$	−4917.12	−0.621620	−0.310810	−	0.950472i	$-0.600600\pi$
$397$	−4917.12	−0.621620	−0.310810	+	0.950472i	$0.600600\pi$
$398$	0	0
$399$	1881.82	0.236112
$400$	0	0
$401$	814.952	0.101488	0.0507441	−	0.998712i	$-0.483841\pi$
$401$	814.952	0.101488	0.0507441	+	0.998712i	$0.483841\pi$
$402$	0	0
$403$	−9813.62	−1.21303
$404$	0	0
$405$	−629.773	−0.0772683
$406$	0	0
$407$	11034.8	1.34391
$408$	0	0
$409$	−3661.82	−0.442703	−0.221352	−	0.975194i	$-0.571047\pi$
$409$	−3661.82	−0.442703	−0.221352	+	0.975194i	$0.571047\pi$
$410$	0	0
$411$	5331.16	0.639822
$412$	0	0
$413$	−3581.89	−0.426763
$414$	0	0
$415$	1671.43	0.197704
$416$	0	0
$417$	2541.36	0.298443
$418$	0	0
$419$	−9116.74	−1.06296	−0.531482	−	0.847070i	$-0.678365\pi$
$419$	−9116.74	−1.06296	−0.531482	+	0.847070i	$0.678365\pi$
$420$	0	0
$421$	−15801.6	−1.82928	−0.914638	−	0.404274i	$-0.867524\pi$
$421$	−15801.6	−1.82928	−0.914638	+	0.404274i	$0.867524\pi$
$422$	0	0
$423$	5933.87	0.682068
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−930.556	−0.105463
$428$	0	0
$429$	5297.02	0.596136
$430$	0	0
$431$	6646.15	0.742770	0.371385	−	0.928479i	$-0.378883\pi$
$431$	6646.15	0.742770	0.371385	+	0.928479i	$0.378883\pi$
$432$	0	0
$433$	−3455.21	−0.383480	−0.191740	−	0.981446i	$-0.561413\pi$
$433$	−3455.21	−0.383480	−0.191740	+	0.981446i	$0.561413\pi$
$434$	0	0
$435$	−428.190	−0.0471957
$436$	0	0
$437$	6560.77	0.718179
$438$	0	0
$439$	14022.8	1.52454	0.762271	−	0.647258i	$-0.224084\pi$
$439$	14022.8	1.52454	0.762271	+	0.647258i	$0.224084\pi$
$440$	0	0
$441$	1374.44	0.148411
$442$	0	0
$443$	10078.8	1.08095	0.540473	−	0.841361i	$-0.318245\pi$
$443$	10078.8	1.08095	0.540473	+	0.841361i	$0.318245\pi$
$444$	0	0
$445$	4232.06	0.450829
$446$	0	0
$447$	−7696.84	−0.814425
$448$	0	0
$449$	8621.06	0.906132	0.453066	−	0.891477i	$-0.350330\pi$
$449$	8621.06	0.906132	0.453066	+	0.891477i	$0.350330\pi$
$450$	0	0
$451$	−10249.2	−1.07010
$452$	0	0
$453$	4570.35	0.474026
$454$	0	0
$455$	−2819.32	−0.290488
$456$	0	0
$457$	8812.85	0.902074	0.451037	−	0.892505i	$-0.351054\pi$
$457$	8812.85	0.902074	0.451037	+	0.892505i	$0.351054\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4016.03	−0.405738	−0.202869	−	0.979206i	$-0.565027\pi$
$461$	−4016.03	−0.405738	−0.202869	+	0.979206i	$0.565027\pi$
$462$	0	0
$463$	−1316.90	−0.132185	−0.0660923	−	0.997814i	$-0.521053\pi$
$463$	−1316.90	−0.132185	−0.0660923	+	0.997814i	$0.521053\pi$
$464$	0	0
$465$	−2888.46	−0.288063
$466$	0	0
$467$	13164.3	1.30443	0.652216	−	0.758033i	$-0.273840\pi$
$467$	13164.3	1.30443	0.652216	+	0.758033i	$0.273840\pi$
$468$	0	0
$469$	−6577.72	−0.647614
$470$	0	0
$471$	4816.05	0.471150
$472$	0	0
$473$	−7310.39	−0.710638
$474$	0	0
$475$	4345.95	0.419802
$476$	0	0
$477$	7380.16	0.708416
$478$	0	0
$479$	7950.73	0.758409	0.379205	−	0.925313i	$-0.376198\pi$
$479$	7950.73	0.758409	0.379205	+	0.925313i	$0.376198\pi$
$480$	0	0
$481$	9626.75	0.912561
$482$	0	0
$483$	7499.00	0.706452
$484$	0	0
$485$	7693.58	0.720304
$486$	0	0
$487$	6612.82	0.615309	0.307655	−	0.951498i	$-0.400456\pi$
$487$	6612.82	0.615309	0.307655	+	0.951498i	$0.400456\pi$
$488$	0	0
$489$	−5501.48	−0.508764
$490$	0	0
$491$	12140.9	1.11591	0.557957	−	0.829870i	$-0.311586\pi$
$491$	12140.9	1.11591	0.557957	+	0.829870i	$0.311586\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−3742.39	−0.339814
$496$	0	0
$497$	−2884.04	−0.260296
$498$	0	0
$499$	15308.6	1.37336	0.686682	−	0.726958i	$-0.259067\pi$
$499$	15308.6	1.37336	0.686682	+	0.726958i	$0.259067\pi$
$500$	0	0
$501$	7189.44	0.641118
$502$	0	0
$503$	−1121.96	−0.0994543	−0.0497271	−	0.998763i	$-0.515835\pi$
$503$	−1121.96	−0.0994543	−0.0497271	+	0.998763i	$0.515835\pi$
$504$	0	0
$505$	−4009.07	−0.353270
$506$	0	0
$507$	−1569.69	−0.137500
$508$	0	0
$509$	−11674.7	−1.01664	−0.508321	−	0.861168i	$-0.669734\pi$
$509$	−11674.7	−1.01664	−0.508321	+	0.861168i	$0.669734\pi$
$510$	0	0
$511$	−17668.6	−1.52958
$512$	0	0
$513$	5266.28	0.453240
$514$	0	0
$515$	1862.57	0.159368
$516$	0	0
$517$	14451.7	1.22937
$518$	0	0
$519$	3208.87	0.271394
$520$	0	0
$521$	−15831.1	−1.33124	−0.665618	−	0.746292i	$-0.731832\pi$
$521$	−15831.1	−1.33124	−0.665618	+	0.746292i	$0.731832\pi$
$522$	0	0
$523$	10376.0	0.867517	0.433759	−	0.901029i	$-0.357187\pi$
$523$	10376.0	0.867517	0.433759	+	0.901029i	$0.357187\pi$
$524$	0	0
$525$	4967.45	0.412947
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13977.5	1.14881
$530$	0	0
$531$	−4147.95	−0.338994
$532$	0	0
$533$	−8941.44	−0.726636
$534$	0	0
$535$	−39.4896	−0.00319119
$536$	0	0
$537$	−11815.1	−0.949455
$538$	0	0
$539$	3347.39	0.267500
$540$	0	0
$541$	−13065.1	−1.03829	−0.519143	−	0.854688i	$-0.673749\pi$
$541$	−13065.1	−1.03829	−0.519143	+	0.854688i	$0.673749\pi$
$542$	0	0
$543$	−7454.65	−0.589152
$544$	0	0
$545$	−3219.88	−0.253073
$546$	0	0
$547$	−15315.8	−1.19717	−0.598587	−	0.801058i	$-0.704271\pi$
$547$	−15315.8	−1.19717	−0.598587	+	0.801058i	$0.704271\pi$
$548$	0	0
$549$	−1077.62	−0.0837733
$550$	0	0
$551$	−1457.63	−0.112699
$552$	0	0
$553$	10885.6	0.837078
$554$	0	0
$555$	2833.46	0.216710
$556$	0	0
$557$	−16191.3	−1.23168	−0.615842	−	0.787870i	$-0.711184\pi$
$557$	−16191.3	−1.23168	−0.615842	+	0.787870i	$0.711184\pi$
$558$	0	0
$559$	−6377.60	−0.482547
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6105.00	0.457007	0.228504	−	0.973543i	$-0.426617\pi$
$563$	6105.00	0.457007	0.228504	+	0.973543i	$0.426617\pi$
$564$	0	0
$565$	−3563.17	−0.265317
$566$	0	0
$567$	−2450.44	−0.181497
$568$	0	0
$569$	8158.61	0.601102	0.300551	−	0.953766i	$-0.402830\pi$
$569$	8158.61	0.601102	0.300551	+	0.953766i	$0.402830\pi$
$570$	0	0
$571$	24042.6	1.76209	0.881044	−	0.473035i	$-0.156842\pi$
$571$	24042.6	1.76209	0.881044	+	0.473035i	$0.156842\pi$
$572$	0	0
$573$	−8073.88	−0.588641
$574$	0	0
$575$	17318.5	1.25606
$576$	0	0
$577$	5897.08	0.425474	0.212737	−	0.977110i	$-0.431762\pi$
$577$	5897.08	0.425474	0.212737	+	0.977110i	$0.431762\pi$
$578$	0	0
$579$	−11768.0	−0.844668
$580$	0	0
$581$	6503.52	0.464392
$582$	0	0
$583$	17974.1	1.27686
$584$	0	0
$585$	−3264.87	−0.230745
$586$	0	0
$587$	−13739.6	−0.966090	−0.483045	−	0.875596i	$-0.660469\pi$
$587$	−13739.6	−0.966090	−0.483045	+	0.875596i	$0.660469\pi$
$588$	0	0
$589$	−9832.83	−0.687869
$590$	0	0
$591$	3748.13	0.260875
$592$	0	0
$593$	−17643.9	−1.22183	−0.610916	−	0.791695i	$-0.709199\pi$
$593$	−17643.9	−1.22183	−0.610916	+	0.791695i	$0.709199\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3682.34	−0.252442
$598$	0	0
$599$	14486.6	0.988157	0.494078	−	0.869417i	$-0.335506\pi$
$599$	14486.6	0.988157	0.494078	+	0.869417i	$0.335506\pi$
$600$	0	0
$601$	−8858.88	−0.601266	−0.300633	−	0.953740i	$-0.597198\pi$
$601$	−8858.88	−0.601266	−0.300633	+	0.953740i	$0.597198\pi$
$602$	0	0
$603$	−7617.23	−0.514424
$604$	0	0
$605$	−3484.40	−0.234151
$606$	0	0
$607$	−19959.5	−1.33465	−0.667323	−	0.744769i	$-0.732560\pi$
$607$	−19959.5	−1.33465	−0.667323	+	0.744769i	$0.732560\pi$
$608$	0	0
$609$	−1666.08	−0.110859
$610$	0	0
$611$	12607.7	0.834786
$612$	0	0
$613$	20390.2	1.34348	0.671741	−	0.740786i	$-0.265547\pi$
$613$	20390.2	1.34348	0.671741	+	0.740786i	$0.265547\pi$
$614$	0	0
$615$	−2631.75	−0.172557
$616$	0	0
$617$	27071.3	1.76637	0.883184	−	0.469028i	$-0.155395\pi$
$617$	27071.3	1.76637	0.883184	+	0.469028i	$0.155395\pi$
$618$	0	0
$619$	−12673.5	−0.822927	−0.411463	−	0.911426i	$-0.634982\pi$
$619$	−12673.5	−0.822927	−0.411463	+	0.911426i	$0.634982\pi$
$620$	0	0
$621$	20986.0	1.35610
$622$	0	0
$623$	16466.9	1.05896
$624$	0	0
$625$	9235.48	0.591071
$626$	0	0
$627$	5307.39	0.338049
$628$	0	0
$629$	0	0
$630$	0	0
$631$	7955.03	0.501877	0.250939	−	0.968003i	$-0.419261\pi$
$631$	7955.03	0.501877	0.250939	+	0.968003i	$0.419261\pi$
$632$	0	0
$633$	11119.6	0.698209
$634$	0	0
$635$	−6383.39	−0.398924
$636$	0	0
$637$	2920.27	0.181641
$638$	0	0
$639$	−3339.82	−0.206763
$640$	0	0
$641$	−12534.3	−0.772349	−0.386175	−	0.922426i	$-0.626204\pi$
$641$	−12534.3	−0.772349	−0.386175	+	0.922426i	$0.626204\pi$
$642$	0	0
$643$	13784.3	0.845412	0.422706	−	0.906267i	$-0.361080\pi$
$643$	13784.3	0.845412	0.422706	+	0.906267i	$0.361080\pi$
$644$	0	0
$645$	−1877.13	−0.114592
$646$	0	0
$647$	12392.1	0.752986	0.376493	−	0.926419i	$-0.377130\pi$
$647$	12392.1	0.752986	0.376493	+	0.926419i	$0.377130\pi$
$648$	0	0
$649$	−10102.2	−0.611010
$650$	0	0
$651$	−11239.0	−0.676637
$652$	0	0
$653$	−8634.02	−0.517420	−0.258710	−	0.965955i	$-0.583297\pi$
$653$	−8634.02	−0.517420	−0.258710	+	0.965955i	$0.583297\pi$
$654$	0	0
$655$	1212.66	0.0723396
$656$	0	0
$657$	−20460.8	−1.21500
$658$	0	0
$659$	30592.7	1.80838	0.904190	−	0.427130i	$-0.140475\pi$
$659$	30592.7	1.80838	0.904190	+	0.427130i	$0.140475\pi$
$660$	0	0
$661$	−19706.3	−1.15959	−0.579793	−	0.814764i	$-0.696867\pi$
$661$	−19706.3	−1.15959	−0.579793	+	0.814764i	$0.696867\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2824.84	−0.164726
$666$	0	0
$667$	−5808.63	−0.337198
$668$	0	0
$669$	−2901.45	−0.167678
$670$	0	0
$671$	−2624.50	−0.150995
$672$	0	0
$673$	−2223.13	−0.127333	−0.0636665	−	0.997971i	$-0.520279\pi$
$673$	−2223.13	−0.127333	−0.0636665	+	0.997971i	$0.520279\pi$
$674$	0	0
$675$	13901.4	0.792691
$676$	0	0
$677$	−3620.34	−0.205526	−0.102763	−	0.994706i	$-0.532768\pi$
$677$	−3620.34	−0.205526	−0.102763	+	0.994706i	$0.532768\pi$
$678$	0	0
$679$	29935.6	1.69194
$680$	0	0
$681$	8960.67	0.504220
$682$	0	0
$683$	20160.4	1.12945	0.564727	−	0.825278i	$-0.308981\pi$
$683$	20160.4	1.12945	0.564727	+	0.825278i	$0.308981\pi$
$684$	0	0
$685$	−8002.73	−0.446377
$686$	0	0
$687$	−10374.2	−0.576126
$688$	0	0
$689$	15680.7	0.867033
$690$	0	0
$691$	21589.0	1.18854	0.594271	−	0.804265i	$-0.297441\pi$
$691$	21589.0	1.18854	0.594271	+	0.804265i	$0.297441\pi$
$692$	0	0
$693$	−14561.6	−0.798195
$694$	0	0
$695$	−3814.89	−0.208211
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−9113.50	−0.493139
$700$	0	0
$701$	−20599.6	−1.10989	−0.554947	−	0.831886i	$-0.687262\pi$
$701$	−20599.6	−1.10989	−0.554947	+	0.831886i	$0.687262\pi$
$702$	0	0
$703$	9645.60	0.517483
$704$	0	0
$705$	3710.86	0.198240
$706$	0	0
$707$	−15599.3	−0.829802
$708$	0	0
$709$	18558.7	0.983057	0.491528	−	0.870862i	$-0.336438\pi$
$709$	18558.7	0.983057	0.491528	+	0.870862i	$0.336438\pi$
$710$	0	0
$711$	12605.9	0.664922
$712$	0	0
$713$	−39183.6	−2.05812
$714$	0	0
$715$	−7951.48	−0.415900
$716$	0	0
$717$	−17783.4	−0.926268
$718$	0	0
$719$	−14709.0	−0.762937	−0.381469	−	0.924382i	$-0.624582\pi$
$719$	−14709.0	−0.762937	−0.381469	+	0.924382i	$0.624582\pi$
$720$	0	0
$721$	7247.25	0.374344
$722$	0	0
$723$	−1913.85	−0.0984464
$724$	0	0
$725$	−3847.72	−0.197104
$726$	0	0
$727$	15816.6	0.806884	0.403442	−	0.915005i	$-0.367814\pi$
$727$	15816.6	0.806884	0.403442	+	0.915005i	$0.367814\pi$
$728$	0	0
$729$	7036.95	0.357514
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−26143.0	−1.31734	−0.658671	−	0.752431i	$-0.728881\pi$
$733$	−26143.0	−1.31734	−0.658671	+	0.752431i	$0.728881\pi$
$734$	0	0
$735$	859.530	0.0431350
$736$	0	0
$737$	−18551.5	−0.927209
$738$	0	0
$739$	−1355.94	−0.0674953	−0.0337476	−	0.999430i	$-0.510744\pi$
$739$	−1355.94	−0.0674953	−0.0337476	+	0.999430i	$0.510744\pi$
$740$	0	0
$741$	4630.18	0.229547
$742$	0	0
$743$	−5596.90	−0.276353	−0.138177	−	0.990408i	$-0.544124\pi$
$743$	−5596.90	−0.276353	−0.138177	+	0.990408i	$0.544124\pi$
$744$	0	0
$745$	11553.9	0.568191
$746$	0	0
$747$	7531.30	0.368884
$748$	0	0
$749$	−153.654	−0.00749584
$750$	0	0
$751$	−5900.36	−0.286694	−0.143347	−	0.989672i	$-0.545786\pi$
$751$	−5900.36	−0.286694	−0.143347	+	0.989672i	$0.545786\pi$
$752$	0	0
$753$	−16590.4	−0.802908
$754$	0	0
$755$	−6860.66	−0.330708
$756$	0	0
$757$	−25378.6	−1.21850	−0.609249	−	0.792979i	$-0.708529\pi$
$757$	−25378.6	−1.21850	−0.609249	+	0.792979i	$0.708529\pi$
$758$	0	0
$759$	21149.8	1.01145
$760$	0	0
$761$	33288.5	1.58569	0.792844	−	0.609425i	$-0.208600\pi$
$761$	33288.5	1.58569	0.792844	+	0.609425i	$0.208600\pi$
$762$	0	0
$763$	−12528.5	−0.594447
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8813.17	−0.414896
$768$	0	0
$769$	35261.4	1.65352	0.826761	−	0.562553i	$-0.190181\pi$
$769$	35261.4	1.65352	0.826761	+	0.562553i	$0.190181\pi$
$770$	0	0
$771$	−8379.26	−0.391403
$772$	0	0
$773$	−23873.3	−1.11082	−0.555409	−	0.831577i	$-0.687438\pi$
$773$	−23873.3	−1.11082	−0.555409	+	0.831577i	$0.687438\pi$
$774$	0	0
$775$	−25955.8	−1.20304
$776$	0	0
$777$	11025.0	0.509033
$778$	0	0
$779$	−8958.95	−0.412051
$780$	0	0
$781$	−8134.02	−0.372674
$782$	0	0
$783$	−4662.54	−0.212804
$784$	0	0
$785$	−7229.48	−0.328702
$786$	0	0
$787$	3747.51	0.169739	0.0848693	−	0.996392i	$-0.472953\pi$
$787$	3747.51	0.169739	0.0848693	+	0.996392i	$0.472953\pi$
$788$	0	0
$789$	15062.8	0.679657
$790$	0	0
$791$	−13864.3	−0.623207
$792$	0	0
$793$	−2289.62	−0.102531
$794$	0	0
$795$	4615.33	0.205898
$796$	0	0
$797$	−8047.11	−0.357645	−0.178823	−	0.983881i	$-0.557229\pi$
$797$	−8047.11	−0.357645	−0.178823	+	0.983881i	$0.557229\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	19069.2	0.841172
$802$	0	0
$803$	−49831.7	−2.18994
$804$	0	0
$805$	−11256.9	−0.492863
$806$	0	0
$807$	14780.7	0.644738
$808$	0	0
$809$	−12863.4	−0.559027	−0.279513	−	0.960142i	$-0.590173\pi$
$809$	−12863.4	−0.559027	−0.279513	+	0.960142i	$0.590173\pi$
$810$	0	0
$811$	33917.8	1.46857	0.734287	−	0.678839i	$-0.237517\pi$
$811$	33917.8	1.46857	0.734287	+	0.678839i	$0.237517\pi$
$812$	0	0
$813$	−14854.6	−0.640804
$814$	0	0
$815$	8258.40	0.354944
$816$	0	0
$817$	−6390.09	−0.273636
$818$	0	0
$819$	−12703.6	−0.542001
$820$	0	0
$821$	27873.7	1.18489	0.592447	−	0.805609i	$-0.298162\pi$
$821$	27873.7	1.18489	0.592447	+	0.805609i	$0.298162\pi$
$822$	0	0
$823$	−34876.0	−1.47716	−0.738580	−	0.674166i	$-0.764503\pi$
$823$	−34876.0	−1.47716	−0.738580	+	0.674166i	$0.764503\pi$
$824$	0	0
$825$	14010.0	0.591229
$826$	0	0
$827$	18251.0	0.767410	0.383705	−	0.923456i	$-0.374648\pi$
$827$	18251.0	0.767410	0.383705	+	0.923456i	$0.374648\pi$
$828$	0	0
$829$	−3257.55	−0.136477	−0.0682386	−	0.997669i	$-0.521738\pi$
$829$	−3257.55	−0.136477	−0.0682386	+	0.997669i	$0.521738\pi$
$830$	0	0
$831$	7974.52	0.332892
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−10792.2	−0.447282
$836$	0	0
$837$	−31452.4	−1.29887
$838$	0	0
$839$	−1338.75	−0.0550879	−0.0275439	−	0.999621i	$-0.508769\pi$
$839$	−1338.75	−0.0550879	−0.0275439	+	0.999621i	$0.508769\pi$
$840$	0	0
$841$	−23098.5	−0.947086
$842$	0	0
$843$	21808.3	0.891007
$844$	0	0
$845$	2356.30	0.0959279
$846$	0	0
$847$	−13557.8	−0.550001
$848$	0	0
$849$	14133.7	0.571338
$850$	0	0
$851$	38437.5	1.54832
$852$	0	0
$853$	−4408.40	−0.176953	−0.0884764	−	0.996078i	$-0.528200\pi$
$853$	−4408.40	−0.176953	−0.0884764	+	0.996078i	$0.528200\pi$
$854$	0	0
$855$	−3271.26	−0.130848
$856$	0	0
$857$	−13581.7	−0.541356	−0.270678	−	0.962670i	$-0.587248\pi$
$857$	−13581.7	−0.541356	−0.270678	+	0.962670i	$0.587248\pi$
$858$	0	0
$859$	−11599.8	−0.460746	−0.230373	−	0.973102i	$-0.573995\pi$
$859$	−11599.8	−0.460746	−0.230373	+	0.973102i	$0.573995\pi$
$860$	0	0
$861$	−10240.1	−0.405323
$862$	0	0
$863$	−26230.6	−1.03464	−0.517322	−	0.855791i	$-0.673071\pi$
$863$	−26230.6	−1.03464	−0.517322	+	0.855791i	$0.673071\pi$
$864$	0	0
$865$	−4816.91	−0.189341
$866$	0	0
$867$	0	0
$868$	0	0
$869$	30701.3	1.19847
$870$	0	0
$871$	−16184.4	−0.629606
$872$	0	0
$873$	34666.5	1.34397
$874$	0	0
$875$	−16159.2	−0.624319
$876$	0	0
$877$	−41491.6	−1.59757	−0.798787	−	0.601614i	$-0.794525\pi$
$877$	−41491.6	−1.59757	−0.798787	+	0.601614i	$0.794525\pi$
$878$	0	0
$879$	23560.1	0.904051
$880$	0	0
$881$	−50336.0	−1.92493	−0.962465	−	0.271405i	$-0.912512\pi$
$881$	−50336.0	−1.92493	−0.962465	+	0.271405i	$0.912512\pi$
$882$	0	0
$883$	51163.6	1.94994	0.974968	−	0.222344i	$-0.0713709\pi$
$883$	51163.6	1.94994	0.974968	+	0.222344i	$0.0713709\pi$
$884$	0	0
$885$	−2594.00	−0.0985270
$886$	0	0
$887$	7151.46	0.270713	0.135357	−	0.990797i	$-0.456782\pi$
$887$	7151.46	0.270713	0.135357	+	0.990797i	$0.456782\pi$
$888$	0	0
$889$	−24837.7	−0.937041
$890$	0	0
$891$	−6911.10	−0.259855
$892$	0	0
$893$	12632.4	0.473379
$894$	0	0
$895$	17735.9	0.662396
$896$	0	0
$897$	18451.2	0.686808
$898$	0	0
$899$	8705.57	0.322967
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−7303.91	−0.269168
$904$	0	0
$905$	11190.4	0.411028
$906$	0	0
$907$	16028.6	0.586794	0.293397	−	0.955991i	$-0.405214\pi$
$907$	16028.6	0.586794	0.293397	+	0.955991i	$0.405214\pi$
$908$	0	0
$909$	−18064.5	−0.659143
$910$	0	0
$911$	3536.66	0.128622	0.0643111	−	0.997930i	$-0.479515\pi$
$911$	3536.66	0.128622	0.0643111	+	0.997930i	$0.479515\pi$
$912$	0	0
$913$	18342.2	0.664884
$914$	0	0
$915$	−673.908	−0.0243483
$916$	0	0
$917$	4718.43	0.169920
$918$	0	0
$919$	6249.55	0.224324	0.112162	−	0.993690i	$-0.464222\pi$
$919$	6249.55	0.224324	0.112162	+	0.993690i	$0.464222\pi$
$920$	0	0
$921$	−14344.1	−0.513197
$922$	0	0
$923$	−7096.14	−0.253058
$924$	0	0
$925$	25461.5	0.905049
$926$	0	0
$927$	8392.57	0.297355
$928$	0	0
$929$	−29474.8	−1.04094	−0.520472	−	0.853878i	$-0.674244\pi$
$929$	−29474.8	−1.04094	−0.520472	+	0.853878i	$0.674244\pi$
$930$	0	0
$931$	2925.99	0.103003
$932$	0	0
$933$	−8175.92	−0.286889
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38857.0	1.35475	0.677376	−	0.735637i	$-0.263117\pi$
$937$	38857.0	1.35475	0.677376	+	0.735637i	$0.263117\pi$
$938$	0	0
$939$	24438.9	0.849343
$940$	0	0
$941$	−16450.8	−0.569905	−0.284952	−	0.958542i	$-0.591978\pi$
$941$	−16450.8	−0.569905	−0.284952	+	0.958542i	$0.591978\pi$
$942$	0	0
$943$	−35701.2	−1.23286
$944$	0	0
$945$	−9035.85	−0.311044
$946$	0	0
$947$	−13909.2	−0.477284	−0.238642	−	0.971108i	$-0.576702\pi$
$947$	−13909.2	−0.477284	−0.238642	+	0.971108i	$0.576702\pi$
$948$	0	0
$949$	−43473.3	−1.48704
$950$	0	0
$951$	−22995.3	−0.784095
$952$	0	0
$953$	−2620.36	−0.0890681	−0.0445340	−	0.999008i	$-0.514180\pi$
$953$	−2620.36	−0.0890681	−0.0445340	+	0.999008i	$0.514180\pi$
$954$	0	0
$955$	12119.9	0.410671
$956$	0	0
$957$	−4698.94	−0.158720
$958$	0	0
$959$	−31138.5	−1.04850
$960$	0	0
$961$	28934.7	0.971256
$962$	0	0
$963$	−177.936	−0.00595422
$964$	0	0
$965$	17665.3	0.589291
$966$	0	0
$967$	−49790.8	−1.65581	−0.827903	−	0.560872i	$-0.810466\pi$
$967$	−49790.8	−1.65581	−0.827903	+	0.560872i	$0.810466\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	11962.2	0.395351	0.197675	−	0.980268i	$-0.436661\pi$
$971$	11962.2	0.395351	0.197675	+	0.980268i	$0.436661\pi$
$972$	0	0
$973$	−14843.7	−0.489072
$974$	0	0
$975$	12222.3	0.401464
$976$	0	0
$977$	−16551.9	−0.542008	−0.271004	−	0.962578i	$-0.587356\pi$
$977$	−16551.9	−0.542008	−0.271004	+	0.962578i	$0.587356\pi$
$978$	0	0
$979$	46442.5	1.51615
$980$	0	0
$981$	−14508.5	−0.472191
$982$	0	0
$983$	−56940.3	−1.84752	−0.923761	−	0.382969i	$-0.874902\pi$
$983$	−56940.3	−1.84752	−0.923761	+	0.382969i	$0.874902\pi$
$984$	0	0
$985$	−5626.40	−0.182002
$986$	0	0
$987$	14438.9	0.465650
$988$	0	0
$989$	−25464.4	−0.818725
$990$	0	0
$991$	−49959.3	−1.60142	−0.800712	−	0.599050i	$-0.795545\pi$
$991$	−49959.3	−1.60142	−0.800712	+	0.599050i	$0.795545\pi$
$992$	0	0
$993$	20281.0	0.648135
$994$	0	0
$995$	5527.64	0.176119
$996$	0	0
$997$	158.971	0.00504981	0.00252490	−	0.999997i	$-0.499196\pi$
$997$	158.971	0.00504981	0.00252490	+	0.999997i	$0.499196\pi$
$998$	0	0
$999$	30853.5	0.977138

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.j.1.4	yes	6
17.16	even	2		2312.4.a.f.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.f.1.3	✓	6	17.16	even	2
2312.4.a.j.1.4	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)

\(f(q)\)	\(=\)	\(q+2.81785 q^{3} -4.22994 q^{5} -16.4587 q^{7} -19.0597 q^{9} +O(q^{10})\)	\(q+2.81785 q^{3} -4.22994 q^{5} -16.4587 q^{7} -19.0597 q^{9} -46.4193 q^{11} -40.4963 q^{13} -11.9194 q^{15} -40.5756 q^{19} -46.3781 q^{21} -161.693 q^{23} -107.108 q^{25} -129.789 q^{27} +35.9239 q^{29} +242.334 q^{31} -130.803 q^{33} +69.6193 q^{35} -237.719 q^{37} -114.113 q^{39} +220.797 q^{41} +157.486 q^{43} +80.6215 q^{45} -311.331 q^{47} -72.1121 q^{49} -387.213 q^{53} +196.351 q^{55} -114.336 q^{57} +217.629 q^{59} +56.5390 q^{61} +313.697 q^{63} +171.297 q^{65} +399.651 q^{67} -455.626 q^{69} +175.230 q^{71} +1073.51 q^{73} -301.813 q^{75} +763.999 q^{77} -661.392 q^{79} +148.884 q^{81} -395.143 q^{83} +101.228 q^{87} -1000.50 q^{89} +666.515 q^{91} +682.861 q^{93} +171.632 q^{95} -1818.84 q^{97} +884.737 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 7 q^{3} + 3 q^{5} + 7 q^{7} + 31 q^{9}+O(q^{10}) \) 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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