LMFDB - Embedded newform 2312.4.a.o.1.12

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:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 345 x^{16} - 182 x^{15} + 48165 x^{14} + 48078 x^{13} - 3485278 x^{12} - 4881882 x^{11} + \cdots - 119632152329 $ x^18 - 345x^16 - 182x^15 + 48165x^14 + 48078x^13 - 3485278x^12 - 4881882x^11 + 139131876x^10 + 239770606x^9 - 3013674039x^8 - 5867731896x^7 + 32610490343x^6 + 66112081788x^5 - 151840532421x^4 - 282437582582x^3 + 230444683878x^2 + 237530449410x - 119632152329
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{2}\cdot 17^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$2.69228$ 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)

$f(q)$	$=$	$q+2.69228 q^{3} -16.6683 q^{5} +18.8441 q^{7} -19.7516 q^{9} +O(q^{10})$	$q+2.69228 q^{3} -16.6683 q^{5} +18.8441 q^{7} -19.7516 q^{9} +4.51771 q^{11} -62.5555 q^{13} -44.8758 q^{15} +64.6289 q^{19} +50.7336 q^{21} +87.5140 q^{23} +152.833 q^{25} -125.868 q^{27} +97.1463 q^{29} +256.677 q^{31} +12.1629 q^{33} -314.100 q^{35} +209.151 q^{37} -168.417 q^{39} -0.0409033 q^{41} +366.294 q^{43} +329.227 q^{45} -185.878 q^{47} +12.1004 q^{49} +610.984 q^{53} -75.3027 q^{55} +173.999 q^{57} -842.820 q^{59} -923.090 q^{61} -372.202 q^{63} +1042.70 q^{65} -113.617 q^{67} +235.612 q^{69} -254.291 q^{71} -649.475 q^{73} +411.469 q^{75} +85.1323 q^{77} -820.245 q^{79} +194.422 q^{81} +178.939 q^{83} +261.545 q^{87} -202.424 q^{89} -1178.80 q^{91} +691.046 q^{93} -1077.26 q^{95} -1277.52 q^{97} -89.2323 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 30 q^{5} - 33 q^{7} + 204 q^{9}+O(q^{10}) $ 18 * q - 30 * q^5 - 33 * q^7 + 204 * q^9	$ 18 q - 30 q^{5} - 33 q^{7} + 204 q^{9} + 66 q^{11} - 30 q^{13} - 102 q^{15} - 168 q^{19} - 510 q^{21} - 153 q^{23} + 594 q^{25} + 546 q^{27} - 447 q^{29} - 303 q^{31} + 153 q^{33} - 117 q^{35} - 939 q^{37} - 516 q^{39} - 1257 q^{41} + 306 q^{43} - 672 q^{45} + 633 q^{47} + 1239 q^{49} - 489 q^{53} + 1089 q^{55} - 1494 q^{57} + 696 q^{59} - 1686 q^{61} - 1908 q^{63} - 855 q^{65} + 513 q^{67} - 1329 q^{69} - 324 q^{71} - 1863 q^{73} + 3054 q^{75} + 1833 q^{77} - 3699 q^{79} + 2622 q^{81} + 1188 q^{83} - 3927 q^{87} + 1713 q^{89} - 252 q^{91} - 1470 q^{93} - 2109 q^{95} - 4611 q^{97} + 3918 q^{99}+O(q^{100}) $ 18 * q - 30 * q^5 - 33 * q^7 + 204 * q^9 + 66 * q^11 - 30 * q^13 - 102 * q^15 - 168 * q^19 - 510 * q^21 - 153 * q^23 + 594 * q^25 + 546 * q^27 - 447 * q^29 - 303 * q^31 + 153 * q^33 - 117 * q^35 - 939 * q^37 - 516 * q^39 - 1257 * q^41 + 306 * q^43 - 672 * q^45 + 633 * q^47 + 1239 * q^49 - 489 * q^53 + 1089 * q^55 - 1494 * q^57 + 696 * q^59 - 1686 * q^61 - 1908 * q^63 - 855 * q^65 + 513 * q^67 - 1329 * q^69 - 324 * q^71 - 1863 * q^73 + 3054 * q^75 + 1833 * q^77 - 3699 * q^79 + 2622 * q^81 + 1188 * q^83 - 3927 * q^87 + 1713 * q^89 - 252 * q^91 - 1470 * q^93 - 2109 * q^95 - 4611 * q^97 + 3918 * 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
$3$	2.69228	0.518129	0.259065	−	0.965860i	$-0.416586\pi$
$3$	2.69228	0.518129	0.259065	+	0.965860i	$0.416586\pi$
$4$	0	0
$5$	−16.6683	−1.49086	−0.745430	−	0.666584i	$-0.767756\pi$
$5$	−16.6683	−1.49086	−0.745430	+	0.666584i	$0.767756\pi$
$6$	0	0
$7$	18.8441	1.01749	0.508743	−	0.860918i	$-0.330110\pi$
$7$	18.8441	1.01749	0.508743	+	0.860918i	$0.330110\pi$
$8$	0	0
$9$	−19.7516	−0.731542
$10$	0	0
$11$	4.51771	0.123831	0.0619155	−	0.998081i	$-0.480279\pi$
$11$	4.51771	0.123831	0.0619155	+	0.998081i	$0.480279\pi$
$12$	0	0
$13$	−62.5555	−1.33460	−0.667299	−	0.744790i	$-0.732550\pi$
$13$	−62.5555	−1.33460	−0.667299	+	0.744790i	$0.732550\pi$
$14$	0	0
$15$	−44.8758	−0.772458
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	64.6289	0.780362	0.390181	−	0.920738i	$-0.372412\pi$
$19$	64.6289	0.780362	0.390181	+	0.920738i	$0.372412\pi$
$20$	0	0
$21$	50.7336	0.527189
$22$	0	0
$23$	87.5140	0.793389	0.396694	−	0.917951i	$-0.370157\pi$
$23$	87.5140	0.793389	0.396694	+	0.917951i	$0.370157\pi$
$24$	0	0
$25$	152.833	1.22266
$26$	0	0
$27$	−125.868	−0.897162
$28$	0	0
$29$	97.1463	0.622056	0.311028	−	0.950401i	$-0.399327\pi$
$29$	97.1463	0.622056	0.311028	+	0.950401i	$0.399327\pi$
$30$	0	0
$31$	256.677	1.48712	0.743558	−	0.668672i	$-0.233137\pi$
$31$	256.677	1.48712	0.743558	+	0.668672i	$0.233137\pi$
$32$	0	0
$33$	12.1629	0.0641605
$34$	0	0
$35$	−314.100	−1.51693
$36$	0	0
$37$	209.151	0.929303	0.464651	−	0.885494i	$-0.346180\pi$
$37$	209.151	0.929303	0.464651	+	0.885494i	$0.346180\pi$
$38$	0	0
$39$	−168.417	−0.691494
$40$	0	0
$41$	−0.0409033	−0.000155805 0	−7.79027e−5	−	1.00000i	$-0.500025\pi$
$41$	−0.0409033	−0.000155805 0	−7.79027e−5		1.00000i	$0.500025\pi$
$42$	0	0
$43$	366.294	1.29905	0.649527	−	0.760339i	$-0.274967\pi$
$43$	366.294	1.29905	0.649527	+	0.760339i	$0.274967\pi$
$44$	0	0
$45$	329.227	1.09063
$46$	0	0
$47$	−185.878	−0.576874	−0.288437	−	0.957499i	$-0.593136\pi$
$47$	−185.878	−0.576874	−0.288437	+	0.957499i	$0.593136\pi$
$48$	0	0
$49$	12.1004	0.0352782
$50$	0	0
$51$	0	0
$52$	0	0
$53$	610.984	1.58349	0.791747	−	0.610850i	$-0.209172\pi$
$53$	610.984	1.58349	0.791747	+	0.610850i	$0.209172\pi$
$54$	0	0
$55$	−75.3027	−0.184615
$56$	0	0
$57$	173.999	0.404328
$58$	0	0
$59$	−842.820	−1.85976	−0.929880	−	0.367864i	$-0.880089\pi$
$59$	−842.820	−1.85976	−0.929880	+	0.367864i	$0.880089\pi$
$60$	0	0
$61$	−923.090	−1.93753	−0.968767	−	0.247971i	$-0.920236\pi$
$61$	−923.090	−1.93753	−0.968767	+	0.247971i	$0.920236\pi$
$62$	0	0
$63$	−372.202	−0.744334
$64$	0	0
$65$	1042.70	1.98970
$66$	0	0
$67$	−113.617	−0.207172	−0.103586	−	0.994621i	$-0.533032\pi$
$67$	−113.617	−0.207172	−0.103586	+	0.994621i	$0.533032\pi$
$68$	0	0
$69$	235.612	0.411078
$70$	0	0
$71$	−254.291	−0.425054	−0.212527	−	0.977155i	$-0.568169\pi$
$71$	−254.291	−0.425054	−0.212527	+	0.977155i	$0.568169\pi$
$72$	0	0
$73$	−649.475	−1.04131	−0.520653	−	0.853769i	$-0.674311\pi$
$73$	−649.475	−1.04131	−0.520653	+	0.853769i	$0.674311\pi$
$74$	0	0
$75$	411.469	0.633498
$76$	0	0
$77$	85.1323	0.125996
$78$	0	0
$79$	−820.245	−1.16816	−0.584081	−	0.811696i	$-0.698545\pi$
$79$	−820.245	−1.16816	−0.584081	+	0.811696i	$0.698545\pi$
$80$	0	0
$81$	194.422	0.266696
$82$	0	0
$83$	178.939	0.236639	0.118320	−	0.992976i	$-0.462249\pi$
$83$	178.939	0.236639	0.118320	+	0.992976i	$0.462249\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	261.545	0.322305
$88$	0	0
$89$	−202.424	−0.241089	−0.120545	−	0.992708i	$-0.538464\pi$
$89$	−202.424	−0.241089	−0.120545	+	0.992708i	$0.538464\pi$
$90$	0	0
$91$	−1178.80	−1.35794
$92$	0	0
$93$	691.046	0.770518
$94$	0	0
$95$	−1077.26	−1.16341
$96$	0	0
$97$	−1277.52	−1.33724	−0.668620	−	0.743604i	$-0.733115\pi$
$97$	−1277.52	−1.33724	−0.668620	+	0.743604i	$0.733115\pi$
$98$	0	0
$99$	−89.2323	−0.0905877
$100$	0	0
$101$	1660.29	1.63570	0.817848	−	0.575435i	$-0.195167\pi$
$101$	1660.29	1.63570	0.817848	+	0.575435i	$0.195167\pi$
$102$	0	0
$103$	−1435.67	−1.37341	−0.686703	−	0.726938i	$-0.740943\pi$
$103$	−1435.67	−1.37341	−0.686703	+	0.726938i	$0.740943\pi$
$104$	0	0
$105$	−845.644	−0.785965
$106$	0	0
$107$	15.9401	0.0144018	0.00720090	−	0.999974i	$-0.497708\pi$
$107$	15.9401	0.0144018	0.00720090	+	0.999974i	$0.497708\pi$
$108$	0	0
$109$	−1084.71	−0.953178	−0.476589	−	0.879126i	$-0.658127\pi$
$109$	−1084.71	−0.953178	−0.476589	+	0.879126i	$0.658127\pi$
$110$	0	0
$111$	563.092	0.481499
$112$	0	0
$113$	1312.32	1.09250	0.546249	−	0.837623i	$-0.316055\pi$
$113$	1312.32	1.09250	0.546249	+	0.837623i	$0.316055\pi$
$114$	0	0
$115$	−1458.71	−1.18283
$116$	0	0
$117$	1235.57	0.976315
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1310.59	−0.984666
$122$	0	0
$123$	−0.110123	−8.07273e−5 0
$124$	0	0
$125$	−463.930	−0.331961
$126$	0	0
$127$	−499.725	−0.349161	−0.174580	−	0.984643i	$-0.555857\pi$
$127$	−499.725	−0.349161	−0.174580	+	0.984643i	$0.555857\pi$
$128$	0	0
$129$	986.165	0.673078
$130$	0	0
$131$	−60.2897	−0.0402102	−0.0201051	−	0.999798i	$-0.506400\pi$
$131$	−60.2897	−0.0402102	−0.0201051	+	0.999798i	$0.506400\pi$
$132$	0	0
$133$	1217.87	0.794008
$134$	0	0
$135$	2098.02	1.33754
$136$	0	0
$137$	1451.26	0.905035	0.452517	−	0.891756i	$-0.350526\pi$
$137$	1451.26	0.905035	0.452517	+	0.891756i	$0.350526\pi$
$138$	0	0
$139$	2783.12	1.69828	0.849142	−	0.528165i	$-0.177120\pi$
$139$	2783.12	1.69828	0.849142	+	0.528165i	$0.177120\pi$
$140$	0	0
$141$	−500.435	−0.298895
$142$	0	0
$143$	−282.608	−0.165265
$144$	0	0
$145$	−1619.27	−0.927398
$146$	0	0
$147$	32.5777	0.0182787
$148$	0	0
$149$	2284.55	1.25609	0.628045	−	0.778177i	$-0.283856\pi$
$149$	2284.55	1.25609	0.628045	+	0.778177i	$0.283856\pi$
$150$	0	0
$151$	−2789.04	−1.50311	−0.751553	−	0.659673i	$-0.770695\pi$
$151$	−2789.04	−1.50311	−0.751553	+	0.659673i	$0.770695\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4278.38	−2.21708
$156$	0	0
$157$	1317.95	0.669962	0.334981	−	0.942225i	$-0.391270\pi$
$157$	1317.95	0.669962	0.334981	+	0.942225i	$0.391270\pi$
$158$	0	0
$159$	1644.94	0.820454
$160$	0	0
$161$	1649.12	0.807262
$162$	0	0
$163$	1876.96	0.901929	0.450965	−	0.892542i	$-0.351080\pi$
$163$	1876.96	0.901929	0.450965	+	0.892542i	$0.351080\pi$
$164$	0	0
$165$	−202.736	−0.0956543
$166$	0	0
$167$	−354.379	−0.164207	−0.0821037	−	0.996624i	$-0.526164\pi$
$167$	−354.379	−0.164207	−0.0821037	+	0.996624i	$0.526164\pi$
$168$	0	0
$169$	1716.19	0.781152
$170$	0	0
$171$	−1276.53	−0.570868
$172$	0	0
$173$	−4052.21	−1.78083	−0.890417	−	0.455146i	$-0.849587\pi$
$173$	−4052.21	−1.78083	−0.890417	+	0.455146i	$0.849587\pi$
$174$	0	0
$175$	2880.00	1.24404
$176$	0	0
$177$	−2269.11	−0.963595
$178$	0	0
$179$	708.949	0.296030	0.148015	−	0.988985i	$-0.452712\pi$
$179$	708.949	0.296030	0.148015	+	0.988985i	$0.452712\pi$
$180$	0	0
$181$	−3986.08	−1.63692	−0.818462	−	0.574561i	$-0.805173\pi$
$181$	−3986.08	−1.63692	−0.818462	+	0.574561i	$0.805173\pi$
$182$	0	0
$183$	−2485.22	−1.00389
$184$	0	0
$185$	−3486.19	−1.38546
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2371.88	−0.912850
$190$	0	0
$191$	−2723.71	−1.03184	−0.515918	−	0.856638i	$-0.672549\pi$
$191$	−2723.71	−1.03184	−0.515918	+	0.856638i	$0.672549\pi$
$192$	0	0
$193$	676.876	0.252449	0.126224	−	0.992002i	$-0.459714\pi$
$193$	676.876	0.252449	0.126224	+	0.992002i	$0.459714\pi$
$194$	0	0
$195$	2807.23	1.03092
$196$	0	0
$197$	43.2465	0.0156405	0.00782027	−	0.999969i	$-0.497511\pi$
$197$	43.2465	0.0156405	0.00782027	+	0.999969i	$0.497511\pi$
$198$	0	0
$199$	−1474.68	−0.525314	−0.262657	−	0.964889i	$-0.584599\pi$
$199$	−1474.68	−0.525314	−0.262657	+	0.964889i	$0.584599\pi$
$200$	0	0
$201$	−305.888	−0.107342
$202$	0	0
$203$	1830.64	0.632933
$204$	0	0
$205$	0.681789	0.000232284 0
$206$	0	0
$207$	−1728.55	−0.580397
$208$	0	0
$209$	291.975	0.0966331
$210$	0	0
$211$	−5298.35	−1.72869	−0.864345	−	0.502899i	$-0.832267\pi$
$211$	−5298.35	−1.72869	−0.864345	+	0.502899i	$0.832267\pi$
$212$	0	0
$213$	−684.623	−0.220233
$214$	0	0
$215$	−6105.51	−1.93671
$216$	0	0
$217$	4836.85	1.51312
$218$	0	0
$219$	−1748.57	−0.539530
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−420.020	−0.126128	−0.0630642	−	0.998009i	$-0.520087\pi$
$223$	−420.020	−0.126128	−0.0630642	+	0.998009i	$0.520087\pi$
$224$	0	0
$225$	−3018.70	−0.894430
$226$	0	0
$227$	−1495.98	−0.437409	−0.218705	−	0.975791i	$-0.570183\pi$
$227$	−1495.98	−0.437409	−0.218705	+	0.975791i	$0.570183\pi$
$228$	0	0
$229$	−428.828	−0.123746	−0.0618729	−	0.998084i	$-0.519707\pi$
$229$	−428.828	−0.123746	−0.0618729	+	0.998084i	$0.519707\pi$
$230$	0	0
$231$	229.200	0.0652824
$232$	0	0
$233$	138.947	0.0390675	0.0195338	−	0.999809i	$-0.493782\pi$
$233$	138.947	0.0390675	0.0195338	+	0.999809i	$0.493782\pi$
$234$	0	0
$235$	3098.27	0.860038
$236$	0	0
$237$	−2208.33	−0.605258
$238$	0	0
$239$	−6410.82	−1.73507	−0.867535	−	0.497376i	$-0.834297\pi$
$239$	−6410.82	−1.73507	−0.867535	+	0.497376i	$0.834297\pi$
$240$	0	0
$241$	−888.741	−0.237547	−0.118773	−	0.992921i	$-0.537896\pi$
$241$	−888.741	−0.237547	−0.118773	+	0.992921i	$0.537896\pi$
$242$	0	0
$243$	3921.88	1.03535
$244$	0	0
$245$	−201.694	−0.0525949
$246$	0	0
$247$	−4042.89	−1.04147
$248$	0	0
$249$	481.752	0.122610
$250$	0	0
$251$	6891.40	1.73299	0.866497	−	0.499183i	$-0.166366\pi$
$251$	6891.40	1.73299	0.866497	+	0.499183i	$0.166366\pi$
$252$	0	0
$253$	395.363	0.0982462
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3819.92	−0.927160	−0.463580	−	0.886055i	$-0.653435\pi$
$257$	−3819.92	−0.927160	−0.463580	+	0.886055i	$0.653435\pi$
$258$	0	0
$259$	3941.26	0.945552
$260$	0	0
$261$	−1918.80	−0.455060
$262$	0	0
$263$	4045.73	0.948558	0.474279	−	0.880375i	$-0.342709\pi$
$263$	4045.73	0.948558	0.474279	+	0.880375i	$0.342709\pi$
$264$	0	0
$265$	−10184.1	−2.36077
$266$	0	0
$267$	−544.983	−0.124915
$268$	0	0
$269$	−3229.89	−0.732082	−0.366041	−	0.930599i	$-0.619287\pi$
$269$	−3229.89	−0.732082	−0.366041	+	0.930599i	$0.619287\pi$
$270$	0	0
$271$	−6387.30	−1.43174	−0.715868	−	0.698235i	$-0.753969\pi$
$271$	−6387.30	−1.43174	−0.715868	+	0.698235i	$0.753969\pi$
$272$	0	0
$273$	−3173.66	−0.703586
$274$	0	0
$275$	690.456	0.151404
$276$	0	0
$277$	−443.088	−0.0961103	−0.0480552	−	0.998845i	$-0.515302\pi$
$277$	−443.088	−0.0961103	−0.0480552	+	0.998845i	$0.515302\pi$
$278$	0	0
$279$	−5069.80	−1.08789
$280$	0	0
$281$	1758.37	0.373293	0.186646	−	0.982427i	$-0.440238\pi$
$281$	1758.37	0.373293	0.186646	+	0.982427i	$0.440238\pi$
$282$	0	0
$283$	−7199.46	−1.51224	−0.756119	−	0.654434i	$-0.772907\pi$
$283$	−7199.46	−1.51224	−0.756119	+	0.654434i	$0.772907\pi$
$284$	0	0
$285$	−2900.27	−0.602797
$286$	0	0
$287$	−0.770786	−0.000158530 0
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−3439.43	−0.692863
$292$	0	0
$293$	−7523.76	−1.50015	−0.750073	−	0.661355i	$-0.769982\pi$
$293$	−7523.76	−1.50015	−0.750073	+	0.661355i	$0.769982\pi$
$294$	0	0
$295$	14048.4	2.77264
$296$	0	0
$297$	−568.637	−0.111097
$298$	0	0
$299$	−5474.48	−1.05885
$300$	0	0
$301$	6902.48	1.32177
$302$	0	0
$303$	4469.97	0.847501
$304$	0	0
$305$	15386.4	2.88859
$306$	0	0
$307$	246.578	0.0458402	0.0229201	−	0.999737i	$-0.492704\pi$
$307$	246.578	0.0458402	0.0229201	+	0.999737i	$0.492704\pi$
$308$	0	0
$309$	−3865.23	−0.711602
$310$	0	0
$311$	−5275.89	−0.961957	−0.480978	−	0.876733i	$-0.659718\pi$
$311$	−5275.89	−0.961957	−0.480978	+	0.876733i	$0.659718\pi$
$312$	0	0
$313$	−535.313	−0.0966699	−0.0483349	−	0.998831i	$-0.515391\pi$
$313$	−535.313	−0.0966699	−0.0483349	+	0.998831i	$0.515391\pi$
$314$	0	0
$315$	6203.98	1.10970
$316$	0	0
$317$	−8902.44	−1.57732	−0.788661	−	0.614828i	$-0.789225\pi$
$317$	−8902.44	−1.57732	−0.788661	+	0.614828i	$0.789225\pi$
$318$	0	0
$319$	438.879	0.0770299
$320$	0	0
$321$	42.9153	0.00746199
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−9560.55	−1.63177
$326$	0	0
$327$	−2920.34	−0.493869
$328$	0	0
$329$	−3502.70	−0.586961
$330$	0	0
$331$	7616.32	1.26475	0.632373	−	0.774664i	$-0.282081\pi$
$331$	7616.32	1.26475	0.632373	+	0.774664i	$0.282081\pi$
$332$	0	0
$333$	−4131.07	−0.679824
$334$	0	0
$335$	1893.80	0.308864
$336$	0	0
$337$	5633.45	0.910605	0.455302	−	0.890337i	$-0.349531\pi$
$337$	5633.45	0.910605	0.455302	+	0.890337i	$0.349531\pi$
$338$	0	0
$339$	3533.12	0.566055
$340$	0	0
$341$	1159.59	0.184151
$342$	0	0
$343$	−6235.51	−0.981591
$344$	0	0
$345$	−3927.26	−0.612859
$346$	0	0
$347$	−1820.54	−0.281647	−0.140823	−	0.990035i	$-0.544975\pi$
$347$	−1820.54	−0.281647	−0.140823	+	0.990035i	$0.544975\pi$
$348$	0	0
$349$	8969.67	1.37575	0.687873	−	0.725831i	$-0.258545\pi$
$349$	8969.67	1.37575	0.687873	+	0.725831i	$0.258545\pi$
$350$	0	0
$351$	7873.76	1.19735
$352$	0	0
$353$	−8014.44	−1.20840	−0.604201	−	0.796832i	$-0.706507\pi$
$353$	−8014.44	−1.20840	−0.604201	+	0.796832i	$0.706507\pi$
$354$	0	0
$355$	4238.61	0.633696
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10969.1	1.61262	0.806308	−	0.591495i	$-0.201462\pi$
$359$	10969.1	1.61262	0.806308	+	0.591495i	$0.201462\pi$
$360$	0	0
$361$	−2682.10	−0.391034
$362$	0	0
$363$	−3528.47	−0.510184
$364$	0	0
$365$	10825.7	1.55244
$366$	0	0
$367$	−10752.3	−1.52933	−0.764667	−	0.644425i	$-0.777097\pi$
$367$	−10752.3	−1.52933	−0.764667	+	0.644425i	$0.777097\pi$
$368$	0	0
$369$	0.807907	0.000113978 0
$370$	0	0
$371$	11513.5	1.61118
$372$	0	0
$373$	6620.53	0.919029	0.459514	−	0.888170i	$-0.348023\pi$
$373$	6620.53	0.919029	0.459514	+	0.888170i	$0.348023\pi$
$374$	0	0
$375$	−1249.03	−0.171999
$376$	0	0
$377$	−6077.04	−0.830195
$378$	0	0
$379$	4623.18	0.626588	0.313294	−	0.949656i	$-0.398568\pi$
$379$	4623.18	0.626588	0.313294	+	0.949656i	$0.398568\pi$
$380$	0	0
$381$	−1345.40	−0.180910
$382$	0	0
$383$	5699.56	0.760401	0.380201	−	0.924904i	$-0.375855\pi$
$383$	5699.56	0.760401	0.380201	+	0.924904i	$0.375855\pi$
$384$	0	0
$385$	−1419.01	−0.187843
$386$	0	0
$387$	−7234.91	−0.950313
$388$	0	0
$389$	4318.11	0.562820	0.281410	−	0.959588i	$-0.409198\pi$
$389$	4318.11	0.562820	0.281410	+	0.959588i	$0.409198\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−162.317	−0.0208341
$394$	0	0
$395$	13672.1	1.74156
$396$	0	0
$397$	8201.80	1.03687	0.518434	−	0.855118i	$-0.326515\pi$
$397$	8201.80	1.03687	0.518434	+	0.855118i	$0.326515\pi$
$398$	0	0
$399$	3278.86	0.411399
$400$	0	0
$401$	5459.77	0.679919	0.339960	−	0.940440i	$-0.389587\pi$
$401$	5459.77	0.679919	0.339960	+	0.940440i	$0.389587\pi$
$402$	0	0
$403$	−16056.6	−1.98470
$404$	0	0
$405$	−3240.68	−0.397607
$406$	0	0
$407$	944.884	0.115077
$408$	0	0
$409$	−10306.3	−1.24600	−0.622999	−	0.782223i	$-0.714086\pi$
$409$	−10306.3	−1.24600	−0.622999	+	0.782223i	$0.714086\pi$
$410$	0	0
$411$	3907.20	0.468925
$412$	0	0
$413$	−15882.2	−1.89228
$414$	0	0
$415$	−2982.61	−0.352796
$416$	0	0
$417$	7492.94	0.879930
$418$	0	0
$419$	−10642.2	−1.24082	−0.620411	−	0.784277i	$-0.713034\pi$
$419$	−10642.2	−1.24082	−0.620411	+	0.784277i	$0.713034\pi$
$420$	0	0
$421$	−13604.2	−1.57489	−0.787443	−	0.616387i	$-0.788596\pi$
$421$	−13604.2	−1.57489	−0.787443	+	0.616387i	$0.788596\pi$
$422$	0	0
$423$	3671.39	0.422008
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−17394.8	−1.97141
$428$	0	0
$429$	−760.859	−0.0856285
$430$	0	0
$431$	−1869.77	−0.208965	−0.104482	−	0.994527i	$-0.533319\pi$
$431$	−1869.77	−0.208965	−0.104482	+	0.994527i	$0.533319\pi$
$432$	0	0
$433$	−7532.13	−0.835961	−0.417981	−	0.908456i	$-0.637262\pi$
$433$	−7532.13	−0.835961	−0.417981	+	0.908456i	$0.637262\pi$
$434$	0	0
$435$	−4359.52	−0.480512
$436$	0	0
$437$	5655.94	0.619131
$438$	0	0
$439$	14230.7	1.54714	0.773569	−	0.633713i	$-0.218470\pi$
$439$	14230.7	1.54714	0.773569	+	0.633713i	$0.218470\pi$
$440$	0	0
$441$	−239.003	−0.0258075
$442$	0	0
$443$	14587.7	1.56452	0.782260	−	0.622952i	$-0.214067\pi$
$443$	14587.7	1.56452	0.782260	+	0.622952i	$0.214067\pi$
$444$	0	0
$445$	3374.08	0.359430
$446$	0	0
$447$	6150.63	0.650816
$448$	0	0
$449$	−3381.10	−0.355377	−0.177688	−	0.984087i	$-0.556862\pi$
$449$	−3381.10	−0.355377	−0.177688	+	0.984087i	$0.556862\pi$
$450$	0	0
$451$	−0.184789	−1.92935e−5 0
$452$	0	0
$453$	−7508.87	−0.778802
$454$	0	0
$455$	19648.7	2.02449
$456$	0	0
$457$	16456.3	1.68445	0.842225	−	0.539126i	$-0.181245\pi$
$457$	16456.3	1.68445	0.842225	+	0.539126i	$0.181245\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7368.48	−0.744434	−0.372217	−	0.928146i	$-0.621402\pi$
$461$	−7368.48	−0.744434	−0.372217	+	0.928146i	$0.621402\pi$
$462$	0	0
$463$	8487.61	0.851950	0.425975	−	0.904735i	$-0.359931\pi$
$463$	8487.61	0.851950	0.425975	+	0.904735i	$0.359931\pi$
$464$	0	0
$465$	−11518.6	−1.14873
$466$	0	0
$467$	4132.05	0.409440	0.204720	−	0.978821i	$-0.434372\pi$
$467$	4132.05	0.409440	0.204720	+	0.978821i	$0.434372\pi$
$468$	0	0
$469$	−2141.01	−0.210794
$470$	0	0
$471$	3548.29	0.347127
$472$	0	0
$473$	1654.81	0.160863
$474$	0	0
$475$	9877.43	0.954121
$476$	0	0
$477$	−12067.9	−1.15839
$478$	0	0
$479$	−2156.64	−0.205719	−0.102860	−	0.994696i	$-0.532799\pi$
$479$	−2156.64	−0.205719	−0.102860	+	0.994696i	$0.532799\pi$
$480$	0	0
$481$	−13083.5	−1.24025
$482$	0	0
$483$	4439.90	0.418266
$484$	0	0
$485$	21294.1	1.99364
$486$	0	0
$487$	−14132.3	−1.31498	−0.657488	−	0.753465i	$-0.728381\pi$
$487$	−14132.3	−1.31498	−0.657488	+	0.753465i	$0.728381\pi$
$488$	0	0
$489$	5053.29	0.467316
$490$	0	0
$491$	−14487.8	−1.33162	−0.665811	−	0.746121i	$-0.731914\pi$
$491$	−14487.8	−1.33162	−0.665811	+	0.746121i	$0.731914\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	1487.35	0.135054
$496$	0	0
$497$	−4791.90	−0.432487
$498$	0	0
$499$	693.499	0.0622150	0.0311075	−	0.999516i	$-0.490097\pi$
$499$	693.499	0.0622150	0.0311075	+	0.999516i	$0.490097\pi$
$500$	0	0
$501$	−954.086	−0.0850806
$502$	0	0
$503$	−21101.4	−1.87051	−0.935255	−	0.353976i	$-0.884829\pi$
$503$	−21101.4	−1.87051	−0.935255	+	0.353976i	$0.884829\pi$
$504$	0	0
$505$	−27674.3	−2.43859
$506$	0	0
$507$	4620.46	0.404738
$508$	0	0
$509$	19008.4	1.65527	0.827637	−	0.561264i	$-0.189685\pi$
$509$	19008.4	1.65527	0.827637	+	0.561264i	$0.189685\pi$
$510$	0	0
$511$	−12238.8	−1.05951
$512$	0	0
$513$	−8134.74	−0.700112
$514$	0	0
$515$	23930.2	2.04756
$516$	0	0
$517$	−839.743	−0.0714349
$518$	0	0
$519$	−10909.7	−0.922702
$520$	0	0
$521$	1052.96	0.0885433	0.0442716	−	0.999020i	$-0.485903\pi$
$521$	1052.96	0.0885433	0.0442716	+	0.999020i	$0.485903\pi$
$522$	0	0
$523$	−4013.16	−0.335532	−0.167766	−	0.985827i	$-0.553655\pi$
$523$	−4013.16	−0.335532	−0.167766	+	0.985827i	$0.553655\pi$
$524$	0	0
$525$	7753.76	0.644575
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−4508.29	−0.370535
$530$	0	0
$531$	16647.1	1.36049
$532$	0	0
$533$	2.55873	0.000207938 0
$534$	0	0
$535$	−265.695	−0.0214711
$536$	0	0
$537$	1908.69	0.153382
$538$	0	0
$539$	54.6662	0.00436854
$540$	0	0
$541$	−14321.6	−1.13814	−0.569071	−	0.822288i	$-0.692697\pi$
$541$	−14321.6	−1.13814	−0.569071	+	0.822288i	$0.692697\pi$
$542$	0	0
$543$	−10731.6	−0.848138
$544$	0	0
$545$	18080.3	1.42105
$546$	0	0
$547$	8955.41	0.700010	0.350005	−	0.936748i	$-0.386180\pi$
$547$	8955.41	0.700010	0.350005	+	0.936748i	$0.386180\pi$
$548$	0	0
$549$	18232.6	1.41739
$550$	0	0
$551$	6278.46	0.485429
$552$	0	0
$553$	−15456.8	−1.18859
$554$	0	0
$555$	−9385.80	−0.717847
$556$	0	0
$557$	11871.0	0.903038	0.451519	−	0.892261i	$-0.350882\pi$
$557$	11871.0	0.903038	0.451519	+	0.892261i	$0.350882\pi$
$558$	0	0
$559$	−22913.7	−1.73371
$560$	0	0
$561$	0	0
$562$	0	0
$563$	21886.3	1.63836	0.819179	−	0.573537i	$-0.194429\pi$
$563$	21886.3	1.63836	0.819179	+	0.573537i	$0.194429\pi$
$564$	0	0
$565$	−21874.1	−1.62876
$566$	0	0
$567$	3663.70	0.271360
$568$	0	0
$569$	1381.14	0.101758	0.0508791	−	0.998705i	$-0.483798\pi$
$569$	1381.14	0.101758	0.0508791	+	0.998705i	$0.483798\pi$
$570$	0	0
$571$	10946.8	0.802297	0.401148	−	0.916013i	$-0.368611\pi$
$571$	10946.8	0.802297	0.401148	+	0.916013i	$0.368611\pi$
$572$	0	0
$573$	−7332.99	−0.534625
$574$	0	0
$575$	13375.0	0.970048
$576$	0	0
$577$	−5030.40	−0.362943	−0.181472	−	0.983396i	$-0.558086\pi$
$577$	−5030.40	−0.362943	−0.181472	+	0.983396i	$0.558086\pi$
$578$	0	0
$579$	1822.34	0.130801
$580$	0	0
$581$	3371.94	0.240777
$582$	0	0
$583$	2760.25	0.196086
$584$	0	0
$585$	−20594.9	−1.45555
$586$	0	0
$587$	−11988.7	−0.842975	−0.421487	−	0.906834i	$-0.638492\pi$
$587$	−11988.7	−0.842975	−0.421487	+	0.906834i	$0.638492\pi$
$588$	0	0
$589$	16588.8	1.16049
$590$	0	0
$591$	116.432	0.00810382
$592$	0	0
$593$	−18092.5	−1.25290	−0.626451	−	0.779461i	$-0.715493\pi$
$593$	−18092.5	−1.25290	−0.626451	+	0.779461i	$0.715493\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3970.25	−0.272180
$598$	0	0
$599$	−12012.1	−0.819370	−0.409685	−	0.912227i	$-0.634361\pi$
$599$	−12012.1	−0.819370	−0.409685	+	0.912227i	$0.634361\pi$
$600$	0	0
$601$	−3108.90	−0.211006	−0.105503	−	0.994419i	$-0.533645\pi$
$601$	−3108.90	−0.211006	−0.105503	+	0.994419i	$0.533645\pi$
$602$	0	0
$603$	2244.12	0.151555
$604$	0	0
$605$	21845.3	1.46800
$606$	0	0
$607$	24245.6	1.62125	0.810626	−	0.585564i	$-0.199127\pi$
$607$	24245.6	1.62125	0.810626	+	0.585564i	$0.199127\pi$
$608$	0	0
$609$	4928.58	0.327941
$610$	0	0
$611$	11627.7	0.769895
$612$	0	0
$613$	2379.68	0.156793	0.0783967	−	0.996922i	$-0.475020\pi$
$613$	2379.68	0.156793	0.0783967	+	0.996922i	$0.475020\pi$
$614$	0	0
$615$	1.83557	0.000120353 0
$616$	0	0
$617$	−8084.41	−0.527498	−0.263749	−	0.964591i	$-0.584959\pi$
$617$	−8084.41	−0.527498	−0.263749	+	0.964591i	$0.584959\pi$
$618$	0	0
$619$	207.944	0.0135024	0.00675121	−	0.999977i	$-0.497851\pi$
$619$	207.944	0.0135024	0.00675121	+	0.999977i	$0.497851\pi$
$620$	0	0
$621$	−11015.3	−0.711798
$622$	0	0
$623$	−3814.51	−0.245305
$624$	0	0
$625$	−11371.2	−0.727757
$626$	0	0
$627$	786.077	0.0500684
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−24394.2	−1.53902	−0.769508	−	0.638637i	$-0.779498\pi$
$631$	−24394.2	−1.53902	−0.769508	+	0.638637i	$0.779498\pi$
$632$	0	0
$633$	−14264.6	−0.895685
$634$	0	0
$635$	8329.58	0.520550
$636$	0	0
$637$	−756.948	−0.0470822
$638$	0	0
$639$	5022.67	0.310945
$640$	0	0
$641$	27048.5	1.66669	0.833346	−	0.552752i	$-0.186422\pi$
$641$	27048.5	1.66669	0.833346	+	0.552752i	$0.186422\pi$
$642$	0	0
$643$	−24605.3	−1.50908	−0.754540	−	0.656254i	$-0.772140\pi$
$643$	−24605.3	−1.50908	−0.754540	+	0.656254i	$0.772140\pi$
$644$	0	0
$645$	−16437.7	−1.00346
$646$	0	0
$647$	−27882.0	−1.69421	−0.847106	−	0.531424i	$-0.821657\pi$
$647$	−27882.0	−1.69421	−0.847106	+	0.531424i	$0.821657\pi$
$648$	0	0
$649$	−3807.62	−0.230296
$650$	0	0
$651$	13022.2	0.783991
$652$	0	0
$653$	−14051.8	−0.842099	−0.421049	−	0.907038i	$-0.638338\pi$
$653$	−14051.8	−0.842099	−0.421049	+	0.907038i	$0.638338\pi$
$654$	0	0
$655$	1004.93	0.0599478
$656$	0	0
$657$	12828.2	0.761759
$658$	0	0
$659$	5581.86	0.329952	0.164976	−	0.986298i	$-0.447245\pi$
$659$	5581.86	0.329952	0.164976	+	0.986298i	$0.447245\pi$
$660$	0	0
$661$	13130.6	0.772652	0.386326	−	0.922362i	$-0.373744\pi$
$661$	13130.6	0.772652	0.386326	+	0.922362i	$0.373744\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−20299.9	−1.18375
$666$	0	0
$667$	8501.67	0.493532
$668$	0	0
$669$	−1130.81	−0.0653508
$670$	0	0
$671$	−4170.26	−0.239927
$672$	0	0
$673$	−19017.9	−1.08928	−0.544640	−	0.838670i	$-0.683334\pi$
$673$	−19017.9	−1.08928	−0.544640	+	0.838670i	$0.683334\pi$
$674$	0	0
$675$	−19236.8	−1.09693
$676$	0	0
$677$	−7142.63	−0.405486	−0.202743	−	0.979232i	$-0.564986\pi$
$677$	−7142.63	−0.405486	−0.202743	+	0.979232i	$0.564986\pi$
$678$	0	0
$679$	−24073.7	−1.36062
$680$	0	0
$681$	−4027.60	−0.226634
$682$	0	0
$683$	−4064.11	−0.227685	−0.113842	−	0.993499i	$-0.536316\pi$
$683$	−4064.11	−0.227685	−0.113842	+	0.993499i	$0.536316\pi$
$684$	0	0
$685$	−24190.1	−1.34928
$686$	0	0
$687$	−1154.52	−0.0641163
$688$	0	0
$689$	−38220.4	−2.11333
$690$	0	0
$691$	11881.7	0.654127	0.327063	−	0.945002i	$-0.393941\pi$
$691$	11881.7	0.654127	0.327063	+	0.945002i	$0.393941\pi$
$692$	0	0
$693$	−1681.50	−0.0921717
$694$	0	0
$695$	−46390.0	−2.53190
$696$	0	0
$697$	0	0
$698$	0	0
$699$	374.084	0.0202420
$700$	0	0
$701$	−2657.92	−0.143207	−0.0716037	−	0.997433i	$-0.522812\pi$
$701$	−2657.92	−0.143207	−0.0716037	+	0.997433i	$0.522812\pi$
$702$	0	0
$703$	13517.2	0.725193
$704$	0	0
$705$	8341.41	0.445611
$706$	0	0
$707$	31286.7	1.66430
$708$	0	0
$709$	2229.08	0.118075	0.0590373	−	0.998256i	$-0.481197\pi$
$709$	2229.08	0.118075	0.0590373	+	0.998256i	$0.481197\pi$
$710$	0	0
$711$	16201.2	0.854559
$712$	0	0
$713$	22462.9	1.17986
$714$	0	0
$715$	4710.60	0.246387
$716$	0	0
$717$	−17259.7	−0.898990
$718$	0	0
$719$	−37414.5	−1.94065	−0.970323	−	0.241811i	$-0.922259\pi$
$719$	−37414.5	−1.94065	−0.970323	+	0.241811i	$0.922259\pi$
$720$	0	0
$721$	−27053.9	−1.39742
$722$	0	0
$723$	−2392.74	−0.123080
$724$	0	0
$725$	14847.2	0.760565
$726$	0	0
$727$	21672.0	1.10560	0.552800	−	0.833314i	$-0.313559\pi$
$727$	21672.0	1.10560	0.552800	+	0.833314i	$0.313559\pi$
$728$	0	0
$729$	5309.42	0.269746
$730$	0	0
$731$	0	0
$732$	0	0
$733$	10285.3	0.518277	0.259139	−	0.965840i	$-0.416561\pi$
$733$	10285.3	0.518277	0.259139	+	0.965840i	$0.416561\pi$
$734$	0	0
$735$	−543.016	−0.0272509
$736$	0	0
$737$	−513.289	−0.0256543
$738$	0	0
$739$	−10718.5	−0.533540	−0.266770	−	0.963760i	$-0.585956\pi$
$739$	−10718.5	−0.533540	−0.266770	+	0.963760i	$0.585956\pi$
$740$	0	0
$741$	−10884.6	−0.539616
$742$	0	0
$743$	11194.4	0.552735	0.276367	−	0.961052i	$-0.410869\pi$
$743$	11194.4	0.552735	0.276367	+	0.961052i	$0.410869\pi$
$744$	0	0
$745$	−38079.6	−1.87265
$746$	0	0
$747$	−3534.33	−0.173112
$748$	0	0
$749$	300.378	0.0146536
$750$	0	0
$751$	25605.4	1.24415	0.622074	−	0.782958i	$-0.286290\pi$
$751$	25605.4	1.24415	0.622074	+	0.782958i	$0.286290\pi$
$752$	0	0
$753$	18553.6	0.897914
$754$	0	0
$755$	46488.6	2.24092
$756$	0	0
$757$	3699.54	0.177625	0.0888124	−	0.996048i	$-0.471693\pi$
$757$	3699.54	0.177625	0.0888124	+	0.996048i	$0.471693\pi$
$758$	0	0
$759$	1064.43	0.0509042
$760$	0	0
$761$	29329.3	1.39709	0.698545	−	0.715566i	$-0.253831\pi$
$761$	29329.3	1.39709	0.698545	+	0.715566i	$0.253831\pi$
$762$	0	0
$763$	−20440.4	−0.969845
$764$	0	0
$765$	0	0
$766$	0	0
$767$	52723.0	2.48203
$768$	0	0
$769$	−6124.06	−0.287177	−0.143589	−	0.989637i	$-0.545864\pi$
$769$	−6124.06	−0.287177	−0.143589	+	0.989637i	$0.545864\pi$
$770$	0	0
$771$	−10284.3	−0.480389
$772$	0	0
$773$	−17938.7	−0.834683	−0.417342	−	0.908750i	$-0.637038\pi$
$773$	−17938.7	−0.834683	−0.417342	+	0.908750i	$0.637038\pi$
$774$	0	0
$775$	39228.7	1.81824
$776$	0	0
$777$	10611.0	0.489918
$778$	0	0
$779$	−2.64353	−0.000121585 0
$780$	0	0
$781$	−1148.82	−0.0526349
$782$	0	0
$783$	−12227.7	−0.558085
$784$	0	0
$785$	−21968.0	−0.998819
$786$	0	0
$787$	5305.65	0.240313	0.120156	−	0.992755i	$-0.461660\pi$
$787$	5305.65	0.240313	0.120156	+	0.992755i	$0.461660\pi$
$788$	0	0
$789$	10892.2	0.491475
$790$	0	0
$791$	24729.4	1.11160
$792$	0	0
$793$	57744.4	2.58583
$794$	0	0
$795$	−27418.4	−1.22318
$796$	0	0
$797$	20429.0	0.907947	0.453974	−	0.891015i	$-0.350006\pi$
$797$	20429.0	0.907947	0.453974	+	0.891015i	$0.350006\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	3998.21	0.176367
$802$	0	0
$803$	−2934.14	−0.128946
$804$	0	0
$805$	−27488.1	−1.20351
$806$	0	0
$807$	−8695.77	−0.379313
$808$	0	0
$809$	−3913.24	−0.170065	−0.0850324	−	0.996378i	$-0.527099\pi$
$809$	−3913.24	−0.170065	−0.0850324	+	0.996378i	$0.527099\pi$
$810$	0	0
$811$	−12122.3	−0.524874	−0.262437	−	0.964949i	$-0.584526\pi$
$811$	−12122.3	−0.524874	−0.262437	+	0.964949i	$0.584526\pi$
$812$	0	0
$813$	−17196.4	−0.741824
$814$	0	0
$815$	−31285.7	−1.34465
$816$	0	0
$817$	23673.2	1.01373
$818$	0	0
$819$	23283.3	0.993387
$820$	0	0
$821$	−3753.45	−0.159557	−0.0797786	−	0.996813i	$-0.525421\pi$
$821$	−3753.45	−0.159557	−0.0797786	+	0.996813i	$0.525421\pi$
$822$	0	0
$823$	−23886.6	−1.01171	−0.505854	−	0.862619i	$-0.668822\pi$
$823$	−23886.6	−1.01171	−0.505854	+	0.862619i	$0.668822\pi$
$824$	0	0
$825$	1858.90	0.0784467
$826$	0	0
$827$	−33419.1	−1.40519	−0.702597	−	0.711588i	$-0.747976\pi$
$827$	−33419.1	−1.40519	−0.702597	+	0.711588i	$0.747976\pi$
$828$	0	0
$829$	−17956.2	−0.752287	−0.376144	−	0.926561i	$-0.622750\pi$
$829$	−17956.2	−0.752287	−0.376144	+	0.926561i	$0.622750\pi$
$830$	0	0
$831$	−1192.92	−0.0497976
$832$	0	0
$833$	0	0
$834$	0	0
$835$	5906.90	0.244810
$836$	0	0
$837$	−32307.5	−1.33418
$838$	0	0
$839$	−4403.82	−0.181212	−0.0906059	−	0.995887i	$-0.528880\pi$
$839$	−4403.82	−0.181212	−0.0906059	+	0.995887i	$0.528880\pi$
$840$	0	0
$841$	−14951.6	−0.613046
$842$	0	0
$843$	4734.01	0.193414
$844$	0	0
$845$	−28606.0	−1.16459
$846$	0	0
$847$	−24696.9	−1.00188
$848$	0	0
$849$	−19382.9	−0.783535
$850$	0	0
$851$	18303.6	0.737298
$852$	0	0
$853$	−14043.6	−0.563710	−0.281855	−	0.959457i	$-0.590950\pi$
$853$	−14043.6	−0.563710	−0.281855	+	0.959457i	$0.590950\pi$
$854$	0	0
$855$	21277.6	0.851085
$856$	0	0
$857$	49347.1	1.96694	0.983468	−	0.181082i	$-0.0579599\pi$
$857$	49347.1	1.96694	0.983468	+	0.181082i	$0.0579599\pi$
$858$	0	0
$859$	−26429.1	−1.04977	−0.524883	−	0.851174i	$-0.675891\pi$
$859$	−26429.1	−1.04977	−0.524883	+	0.851174i	$0.675891\pi$
$860$	0	0
$861$	−2.07517	−8.21389e−5 0
$862$	0	0
$863$	−3912.05	−0.154308	−0.0771539	−	0.997019i	$-0.524583\pi$
$863$	−3912.05	−0.154308	−0.0771539	+	0.997019i	$0.524583\pi$
$864$	0	0
$865$	67543.6	2.65497
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3705.63	−0.144655
$870$	0	0
$871$	7107.36	0.276491
$872$	0	0
$873$	25233.1	0.978248
$874$	0	0
$875$	−8742.34	−0.337766
$876$	0	0
$877$	−26253.9	−1.01087	−0.505434	−	0.862865i	$-0.668668\pi$
$877$	−26253.9	−1.01087	−0.505434	+	0.862865i	$0.668668\pi$
$878$	0	0
$879$	−20256.1	−0.777269
$880$	0	0
$881$	8894.17	0.340127	0.170064	−	0.985433i	$-0.445603\pi$
$881$	8894.17	0.340127	0.170064	+	0.985433i	$0.445603\pi$
$882$	0	0
$883$	10392.7	0.396084	0.198042	−	0.980194i	$-0.436542\pi$
$883$	10392.7	0.396084	0.198042	+	0.980194i	$0.436542\pi$
$884$	0	0
$885$	37822.2	1.43659
$886$	0	0
$887$	38393.2	1.45335	0.726673	−	0.686984i	$-0.241066\pi$
$887$	38393.2	1.45335	0.726673	+	0.686984i	$0.241066\pi$
$888$	0	0
$889$	−9416.88	−0.355266
$890$	0	0
$891$	878.341	0.0330253
$892$	0	0
$893$	−12013.1	−0.450171
$894$	0	0
$895$	−11817.0	−0.441339
$896$	0	0
$897$	−14738.8	−0.548624
$898$	0	0
$899$	24935.3	0.925069
$900$	0	0
$901$	0	0
$902$	0	0
$903$	18583.4	0.684847
$904$	0	0
$905$	66441.3	2.44042
$906$	0	0
$907$	−4236.92	−0.155110	−0.0775549	−	0.996988i	$-0.524711\pi$
$907$	−4236.92	−0.155110	−0.0775549	+	0.996988i	$0.524711\pi$
$908$	0	0
$909$	−32793.5	−1.19658
$910$	0	0
$911$	−31689.1	−1.15248	−0.576238	−	0.817282i	$-0.695480\pi$
$911$	−31689.1	−1.15248	−0.576238	+	0.817282i	$0.695480\pi$
$912$	0	0
$913$	808.393	0.0293033
$914$	0	0
$915$	41424.4	1.49666
$916$	0	0
$917$	−1136.11	−0.0409133
$918$	0	0
$919$	−1920.56	−0.0689373	−0.0344687	−	0.999406i	$-0.510974\pi$
$919$	−1920.56	−0.0689373	−0.0344687	+	0.999406i	$0.510974\pi$
$920$	0	0
$921$	663.856	0.0237511
$922$	0	0
$923$	15907.3	0.567277
$924$	0	0
$925$	31965.2	1.13622
$926$	0	0
$927$	28356.9	1.00471
$928$	0	0
$929$	28550.9	1.00832	0.504158	−	0.863612i	$-0.331803\pi$
$929$	28550.9	1.00832	0.504158	+	0.863612i	$0.331803\pi$
$930$	0	0
$931$	782.037	0.0275298
$932$	0	0
$933$	−14204.2	−0.498418
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24477.0	0.853391	0.426696	−	0.904395i	$-0.359678\pi$
$937$	24477.0	0.853391	0.426696	+	0.904395i	$0.359678\pi$
$938$	0	0
$939$	−1441.21	−0.0500875
$940$	0	0
$941$	−38973.1	−1.35015	−0.675073	−	0.737751i	$-0.735888\pi$
$941$	−38973.1	−1.35015	−0.675073	+	0.737751i	$0.735888\pi$
$942$	0	0
$943$	−3.57961	−0.000123614 0
$944$	0	0
$945$	39535.2	1.36093
$946$	0	0
$947$	51993.4	1.78412	0.892059	−	0.451920i	$-0.149261\pi$
$947$	51993.4	1.78412	0.892059	+	0.451920i	$0.149261\pi$
$948$	0	0
$949$	40628.2	1.38972
$950$	0	0
$951$	−23967.9	−0.817256
$952$	0	0
$953$	31327.1	1.06483	0.532415	−	0.846483i	$-0.321284\pi$
$953$	31327.1	1.06483	0.532415	+	0.846483i	$0.321284\pi$
$954$	0	0
$955$	45399.7	1.53832
$956$	0	0
$957$	1181.59	0.0399114
$958$	0	0
$959$	27347.8	0.920861
$960$	0	0
$961$	36092.2	1.21151
$962$	0	0
$963$	−314.844	−0.0105355
$964$	0	0
$965$	−11282.4	−0.376366
$966$	0	0
$967$	28228.2	0.938735	0.469367	−	0.883003i	$-0.344482\pi$
$967$	28228.2	0.938735	0.469367	+	0.883003i	$0.344482\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32628.1	1.07836	0.539179	−	0.842191i	$-0.318735\pi$
$971$	32628.1	1.07836	0.539179	+	0.842191i	$0.318735\pi$
$972$	0	0
$973$	52445.5	1.72798
$974$	0	0
$975$	−25739.6	−0.845465
$976$	0	0
$977$	−3618.45	−0.118490	−0.0592449	−	0.998243i	$-0.518869\pi$
$977$	−3618.45	−0.118490	−0.0592449	+	0.998243i	$0.518869\pi$
$978$	0	0
$979$	−914.496	−0.0298543
$980$	0	0
$981$	21424.8	0.697290
$982$	0	0
$983$	−2729.64	−0.0885676	−0.0442838	−	0.999019i	$-0.514101\pi$
$983$	−2729.64	−0.0885676	−0.0442838	+	0.999019i	$0.514101\pi$
$984$	0	0
$985$	−720.847	−0.0233179
$986$	0	0
$987$	−9430.25	−0.304122
$988$	0	0
$989$	32055.9	1.03065
$990$	0	0
$991$	8411.57	0.269629	0.134814	−	0.990871i	$-0.456956\pi$
$991$	8411.57	0.269629	0.134814	+	0.990871i	$0.456956\pi$
$992$	0	0
$993$	20505.3	0.655302
$994$	0	0
$995$	24580.5	0.783170
$996$	0	0
$997$	30799.7	0.978371	0.489185	−	0.872180i	$-0.337294\pi$
$997$	30799.7	0.978371	0.489185	+	0.872180i	$0.337294\pi$
$998$	0	0
$999$	−26325.5	−0.833735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.o.1.12	✓	18
17.16	even	2		2312.4.a.p.1.7	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.o.1.12	✓	18	1.1	even	1	trivial
2312.4.a.p.1.7	yes	18	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.69228 q^{3} -16.6683 q^{5} +18.8441 q^{7} -19.7516 q^{9} +O(q^{10})\)	\(q+2.69228 q^{3} -16.6683 q^{5} +18.8441 q^{7} -19.7516 q^{9} +4.51771 q^{11} -62.5555 q^{13} -44.8758 q^{15} +64.6289 q^{19} +50.7336 q^{21} +87.5140 q^{23} +152.833 q^{25} -125.868 q^{27} +97.1463 q^{29} +256.677 q^{31} +12.1629 q^{33} -314.100 q^{35} +209.151 q^{37} -168.417 q^{39} -0.0409033 q^{41} +366.294 q^{43} +329.227 q^{45} -185.878 q^{47} +12.1004 q^{49} +610.984 q^{53} -75.3027 q^{55} +173.999 q^{57} -842.820 q^{59} -923.090 q^{61} -372.202 q^{63} +1042.70 q^{65} -113.617 q^{67} +235.612 q^{69} -254.291 q^{71} -649.475 q^{73} +411.469 q^{75} +85.1323 q^{77} -820.245 q^{79} +194.422 q^{81} +178.939 q^{83} +261.545 q^{87} -202.424 q^{89} -1178.80 q^{91} +691.046 q^{93} -1077.26 q^{95} -1277.52 q^{97} -89.2323 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 30 q^{5} - 33 q^{7} + 204 q^{9}+O(q^{10}) \) 18 * q - 30 * q^5 - 33 * q^7 + 204 * q^9	\( 18 q - 30 q^{5} - 33 q^{7} + 204 q^{9} + 66 q^{11} - 30 q^{13} - 102 q^{15} - 168 q^{19} - 510 q^{21} - 153 q^{23} + 594 q^{25} + 546 q^{27} - 447 q^{29} - 303 q^{31} + 153 q^{33} - 117 q^{35} - 939 q^{37} - 516 q^{39} - 1257 q^{41} + 306 q^{43} - 672 q^{45} + 633 q^{47} + 1239 q^{49} - 489 q^{53} + 1089 q^{55} - 1494 q^{57} + 696 q^{59} - 1686 q^{61} - 1908 q^{63} - 855 q^{65} + 513 q^{67} - 1329 q^{69} - 324 q^{71} - 1863 q^{73} + 3054 q^{75} + 1833 q^{77} - 3699 q^{79} + 2622 q^{81} + 1188 q^{83} - 3927 q^{87} + 1713 q^{89} - 252 q^{91} - 1470 q^{93} - 2109 q^{95} - 4611 q^{97} + 3918 q^{99}+O(q^{100}) \) 18 * q - 30 * q^5 - 33 * q^7 + 204 * q^9 + 66 * q^11 - 30 * q^13 - 102 * q^15 - 168 * q^19 - 510 * q^21 - 153 * q^23 + 594 * q^25 + 546 * q^27 - 447 * q^29 - 303 * q^31 + 153 * q^33 - 117 * q^35 - 939 * q^37 - 516 * q^39 - 1257 * q^41 + 306 * q^43 - 672 * q^45 + 633 * q^47 + 1239 * q^49 - 489 * q^53 + 1089 * q^55 - 1494 * q^57 + 696 * q^59 - 1686 * q^61 - 1908 * q^63 - 855 * q^65 + 513 * q^67 - 1329 * q^69 - 324 * q^71 - 1863 * q^73 + 3054 * q^75 + 1833 * q^77 - 3699 * q^79 + 2622 * q^81 + 1188 * q^83 - 3927 * q^87 + 1713 * q^89 - 252 * q^91 - 1470 * q^93 - 2109 * q^95 - 4611 * q^97 + 3918 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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