LMFDB - Embedded newform 336.4.h.b.239.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,4,Mod(239,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("336.239");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	336.h (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.8
Character	$\chi$	$=$	336.239
Dual form			336.4.h.b.239.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.33643 + 4.64124i) q^{3} +12.0055i q^{5} -7.00000i q^{7} +(-16.0822 - 21.6878i) q^{9} +O(q^{10})$	\(q+(-2.33643 + 4.64124i) q^{3} +12.0055i q^{5} -7.00000i q^{7} +(-16.0822 - 21.6878i) q^{9} -33.9879 q^{11} -13.8712 q^{13} +(-55.7206 - 28.0501i) q^{15} +28.7374i q^{17} -75.7520i q^{19} +(32.4887 + 16.3550i) q^{21} -104.296 q^{23} -19.1331 q^{25} +(138.233 - 23.9692i) q^{27} +242.660i q^{29} -316.604i q^{31} +(79.4103 - 157.746i) q^{33} +84.0388 q^{35} +303.265 q^{37} +(32.4091 - 64.3795i) q^{39} -382.946i q^{41} -12.0228i q^{43} +(260.374 - 193.076i) q^{45} -314.619 q^{47} -49.0000 q^{49} +(-133.377 - 67.1430i) q^{51} -418.518i q^{53} -408.043i q^{55} +(351.583 + 176.989i) q^{57} -477.106 q^{59} +112.267 q^{61} +(-151.815 + 112.575i) q^{63} -166.531i q^{65} -180.543i q^{67} +(243.680 - 484.062i) q^{69} -25.4197 q^{71} -349.113 q^{73} +(44.7032 - 88.8015i) q^{75} +237.915i q^{77} -452.651i q^{79} +(-211.725 + 697.577i) q^{81} +1081.66 q^{83} -345.009 q^{85} +(-1126.24 - 566.957i) q^{87} +816.177i q^{89} +97.0984i q^{91} +(1469.44 + 739.724i) q^{93} +909.444 q^{95} -1070.07 q^{97} +(546.600 + 737.124i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 136 q^{9}+O(q^{10}) $ 24 * q - 136 * q^9	$ 24 q - 136 q^{9} + 96 q^{13} - 112 q^{21} + 168 q^{25} - 320 q^{33} - 864 q^{37} + 592 q^{45} - 1176 q^{49} - 1120 q^{57} - 2304 q^{61} + 1168 q^{69} + 3312 q^{73} - 968 q^{81} - 5808 q^{85} + 1776 q^{93} + 6480 q^{97}+O(q^{100}) $ 24 * q - 136 * q^9 + 96 * q^13 - 112 * q^21 + 168 * q^25 - 320 * q^33 - 864 * q^37 + 592 * q^45 - 1176 * q^49 - 1120 * q^57 - 2304 * q^61 + 1168 * q^69 + 3312 * q^73 - 968 * q^81 - 5808 * q^85 + 1776 * q^93 + 6480 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−2.33643	+	4.64124i	−0.449646	+	0.893207i
$3$	−2.33643	+	4.64124i	−0.449646	+	0.893207i
$4$		0			0
$5$			12.0055i			1.07381i	0.843643	+	0.536904i	$0.180406\pi$
$5$			12.0055i			1.07381i	−0.843643	+	0.536904i	$0.819594\pi$
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$		0			0
$9$	−16.0822	−	21.6878i	−0.595637	−	0.803254i
$10$		0			0
$11$	−33.9879			−0.931612			−0.465806	−	0.884887i	$-0.654236\pi$
$11$	−33.9879			−0.931612			−0.465806	+	0.884887i	$0.654236\pi$
$12$		0			0
$13$	−13.8712			−0.295937			−0.147968	−	0.988992i	$-0.547273\pi$
$13$	−13.8712			−0.295937			−0.147968	+	0.988992i	$0.547273\pi$
$14$		0			0
$15$	−55.7206	−	28.0501i	−0.959133	−	0.482834i
$16$		0			0
$17$			28.7374i			0.409991i	0.978763	+	0.204996i	$0.0657180\pi$
$17$			28.7374i			0.409991i	−0.978763	+	0.204996i	$0.934282\pi$
$18$		0			0
$19$		−	75.7520i		−	0.914668i	−0.889295	−	0.457334i	$-0.848804\pi$
$19$		−	75.7520i		−	0.914668i	0.889295	−	0.457334i	$-0.151196\pi$
$20$		0			0
$21$	32.4887	+	16.3550i	0.337600	+	0.169950i
$22$		0			0
$23$	−104.296			−0.945530			−0.472765	−	0.881189i	$-0.656744\pi$
$23$	−104.296			−0.945530			−0.472765	+	0.881189i	$0.656744\pi$
$24$		0			0
$25$	−19.1331			−0.153065
$26$		0			0
$27$	138.233	−	23.9692i	0.985297	−	0.170848i
$28$		0			0
$29$			242.660i			1.55382i	0.629612	+	0.776910i	$0.283214\pi$
$29$			242.660i			1.55382i	−0.629612	+	0.776910i	$0.716786\pi$
$30$		0			0
$31$		−	316.604i		−	1.83432i	−0.398523	−	0.917158i	$-0.630477\pi$
$31$		−	316.604i		−	1.83432i	0.398523	−	0.917158i	$-0.369523\pi$
$32$		0			0
$33$	79.4103	−	157.746i	0.418896	−	0.832122i
$34$		0			0
$35$	84.0388			0.405862
$36$		0			0
$37$	303.265			1.34747			0.673737	−	0.738971i	$-0.264688\pi$
$37$	303.265			1.34747			0.673737	+	0.738971i	$0.264688\pi$
$38$		0			0
$39$	32.4091	−	64.3795i	0.133067	−	0.264333i
$40$		0			0
$41$		−	382.946i		−	1.45869i	−0.684148	−	0.729343i	$-0.739826\pi$
$41$		−	382.946i		−	1.45869i	0.684148	−	0.729343i	$-0.260174\pi$
$42$		0			0
$43$		−	12.0228i		−	0.0426385i	−0.999773	−	0.0213193i	$-0.993213\pi$
$43$		−	12.0228i		−	0.0426385i	0.999773	−	0.0213193i	$-0.00678664\pi$
$44$		0			0
$45$	260.374	−	193.076i	0.862541	−	0.639600i
$46$		0			0
$47$	−314.619			−0.976424			−0.488212	−	0.872725i	$-0.662351\pi$
$47$	−314.619			−0.976424			−0.488212	+	0.872725i	$0.662351\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	−133.377	−	67.1430i	−0.366207	−	0.184351i
$52$		0			0
$53$		−	418.518i		−	1.08468i	−0.840160	−	0.542338i	$-0.817539\pi$
$53$		−	418.518i		−	1.08468i	0.840160	−	0.542338i	$-0.182461\pi$
$54$		0			0
$55$		−	408.043i		−	1.00037i
$56$		0			0
$57$	351.583	+	176.989i	0.816988	+	0.411277i
$58$		0			0
$59$	−477.106			−1.05278			−0.526389	−	0.850244i	$-0.676454\pi$
$59$	−477.106			−1.05278			−0.526389	+	0.850244i	$0.676454\pi$
$60$		0			0
$61$	112.267			0.235644			0.117822	−	0.993035i	$-0.462409\pi$
$61$	112.267			0.235644			0.117822	+	0.993035i	$0.462409\pi$
$62$		0			0
$63$	−151.815	+	112.575i	−0.303601	+	0.225130i
$64$		0			0
$65$		−	166.531i		−	0.317779i
$66$		0			0
$67$		−	180.543i		−	0.329207i	−0.986360	−	0.164604i	$-0.947366\pi$
$67$		−	180.543i		−	0.329207i	0.986360	−	0.164604i	$-0.0526345\pi$
$68$		0			0
$69$	243.680	−	484.062i	0.425154	−	0.844554i
$70$		0			0
$71$	−25.4197			−0.0424896			−0.0212448	−	0.999774i	$-0.506763\pi$
$71$	−25.4197			−0.0424896			−0.0212448	+	0.999774i	$0.506763\pi$
$72$		0			0
$73$	−349.113			−0.559734			−0.279867	−	0.960039i	$-0.590290\pi$
$73$	−349.113			−0.559734			−0.279867	+	0.960039i	$0.590290\pi$
$74$		0			0
$75$	44.7032	−	88.8015i	0.0688251	−	0.136719i
$76$		0			0
$77$			237.915i			0.352116i
$78$		0			0
$79$		−	452.651i		−	0.644648i	−0.946629	−	0.322324i	$-0.895536\pi$
$79$		−	452.651i		−	0.644648i	0.946629	−	0.322324i	$-0.104464\pi$
$80$		0			0
$81$	−211.725	+	697.577i	−0.290433	+	0.956895i
$82$		0			0
$83$	1081.66			1.43046			0.715230	−	0.698889i	$-0.246322\pi$
$83$	1081.66			1.43046			0.715230	+	0.698889i	$0.246322\pi$
$84$		0			0
$85$	−345.009			−0.440252
$86$		0			0
$87$	−1126.24	−	566.957i	−1.38788	−	0.698669i
$88$		0			0
$89$			816.177i			0.972075i	0.873938	+	0.486037i	$0.161558\pi$
$89$			816.177i			0.972075i	−0.873938	+	0.486037i	$0.838442\pi$
$90$		0			0
$91$			97.0984i			0.111854i
$92$		0			0
$93$	1469.44	+	739.724i	1.63842	+	0.824793i
$94$		0			0
$95$	909.444			0.982178
$96$		0			0
$97$	−1070.07			−1.12009			−0.560046	−	0.828462i	$-0.689216\pi$
$97$	−1070.07			−1.12009			−0.560046	+	0.828462i	$0.689216\pi$
$98$		0			0
$99$	546.600	+	737.124i	0.554903	+	0.748321i
$100$		0			0
$101$		−	1356.15i		−	1.33606i	−0.744134	−	0.668031i	$-0.767137\pi$
$101$		−	1356.15i		−	1.33606i	0.744134	−	0.668031i	$-0.232863\pi$
$102$		0			0
$103$		−	757.734i		−	0.724872i	−0.932009	−	0.362436i	$-0.881945\pi$
$103$		−	757.734i		−	0.724872i	0.932009	−	0.362436i	$-0.118055\pi$
$104$		0			0
$105$	−196.351	+	390.044i	−0.182494	+	0.362518i
$106$		0			0
$107$	−987.341			−0.892055			−0.446027	−	0.895019i	$-0.647162\pi$
$107$	−987.341			−0.892055			−0.446027	+	0.895019i	$0.647162\pi$
$108$		0			0
$109$	−1257.53			−1.10504			−0.552522	−	0.833498i	$-0.686334\pi$
$109$	−1257.53			−1.10504			−0.552522	+	0.833498i	$0.686334\pi$
$110$		0			0
$111$	−708.558	+	1407.53i	−0.605886	+	1.20357i
$112$		0			0
$113$			449.026i			0.373812i	0.982378	+	0.186906i	$0.0598460\pi$
$113$			449.026i			0.373812i	−0.982378	+	0.186906i	$0.940154\pi$
$114$		0			0
$115$		−	1252.13i		−	1.01532i
$116$		0			0
$117$	223.079	+	300.836i	0.176271	+	0.237712i
$118$		0			0
$119$	201.162			0.154962
$120$		0			0
$121$	−175.823			−0.132099
$122$		0			0
$123$	1777.35	+	894.727i	1.30291	+	0.655892i
$124$		0			0
$125$			1270.99i			0.909446i
$126$		0			0
$127$			1557.87i			1.08849i	0.838925	+	0.544247i	$0.183184\pi$
$127$			1557.87i			1.08849i	−0.838925	+	0.544247i	$0.816816\pi$
$128$		0			0
$129$	55.8006	+	28.0904i	0.0380850	+	0.0191722i
$130$		0			0
$131$	−1864.44			−1.24349			−0.621743	−	0.783221i	$-0.713575\pi$
$131$	−1864.44			−1.24349			−0.621743	+	0.783221i	$0.713575\pi$
$132$		0			0
$133$	−530.264			−0.345712
$134$		0			0
$135$	287.764	+	1659.57i	0.183458	+	1.05802i
$136$		0			0
$137$			579.299i			0.361261i	0.983551	+	0.180631i	$0.0578139\pi$
$137$			579.299i			0.361261i	−0.983551	+	0.180631i	$0.942186\pi$
$138$		0			0
$139$			1410.44i			0.860661i	0.902671	+	0.430330i	$0.141603\pi$
$139$			1410.44i			0.860661i	−0.902671	+	0.430330i	$0.858397\pi$
$140$		0			0
$141$	735.086	−	1460.22i	0.439045	−	0.872149i
$142$		0			0
$143$	471.453			0.275698
$144$		0			0
$145$	−2913.26			−1.66851
$146$		0			0
$147$	114.485	−	227.421i	0.0642351	−	0.127601i
$148$		0			0
$149$		−	2848.23i		−	1.56601i	−0.622013	−	0.783007i	$-0.713685\pi$
$149$		−	2848.23i		−	1.56601i	0.622013	−	0.783007i	$-0.286315\pi$
$150$		0			0
$151$			556.913i			0.300138i	0.988675	+	0.150069i	$0.0479497\pi$
$151$			556.913i			0.300138i	−0.988675	+	0.150069i	$0.952050\pi$
$152$		0			0
$153$	623.253	−	462.161i	0.329327	−	0.244206i
$154$		0			0
$155$	3801.01			1.96971
$156$		0			0
$157$	−1054.63			−0.536104			−0.268052	−	0.963405i	$-0.586380\pi$
$157$	−1054.63			−0.536104			−0.268052	+	0.963405i	$0.586380\pi$
$158$		0			0
$159$	1942.44	+	977.837i	0.968840	+	0.487720i
$160$		0			0
$161$			730.071i			0.357377i
$162$		0			0
$163$			2521.74i			1.21177i	0.795553	+	0.605884i	$0.207180\pi$
$163$			2521.74i			1.21177i	−0.795553	+	0.605884i	$0.792820\pi$
$164$		0			0
$165$	1893.83	+	953.364i	0.893540	+	0.449814i
$166$		0			0
$167$	1267.63			0.587379			0.293689	−	0.955901i	$-0.405117\pi$
$167$	1267.63			0.587379			0.293689	+	0.955901i	$0.405117\pi$
$168$		0			0
$169$	−2004.59			−0.912421
$170$		0			0
$171$	−1642.90	+	1218.26i	−0.734710	+	0.544810i
$172$		0			0
$173$		−	356.100i		−	0.156496i	−0.996934	−	0.0782479i	$-0.975067\pi$
$173$		−	356.100i		−	0.156496i	0.996934	−	0.0782479i	$-0.0249326\pi$
$174$		0			0
$175$			133.932i			0.0578532i
$176$		0			0
$177$	1114.72	−	2214.36i	0.473377	−	0.940349i
$178$		0			0
$179$	−4277.21			−1.78600			−0.893000	−	0.450057i	$-0.851404\pi$
$179$	−4277.21			−1.78600			−0.893000	+	0.450057i	$0.851404\pi$
$180$		0			0
$181$	3200.35			1.31426			0.657128	−	0.753779i	$-0.271771\pi$
$181$	3200.35			1.31426			0.657128	+	0.753779i	$0.271771\pi$
$182$		0			0
$183$	−262.303	+	521.056i	−0.105956	+	0.210479i
$184$		0			0
$185$			3640.87i			1.44693i
$186$		0			0
$187$		−	976.725i		−	0.381953i
$188$		0			0
$189$	−167.785	−	967.634i	−0.0645743	−	0.372407i
$190$		0			0
$191$	3437.92			1.30240			0.651201	−	0.758905i	$-0.274265\pi$
$191$	3437.92			1.30240			0.651201	+	0.758905i	$0.274265\pi$
$192$		0			0
$193$	−4846.78			−1.80766			−0.903831	−	0.427889i	$-0.859258\pi$
$193$	−4846.78			−1.80766			−0.903831	+	0.427889i	$0.859258\pi$
$194$		0			0
$195$	772.912	+	389.088i	0.283843	+	0.142888i
$196$		0			0
$197$			1330.88i			0.481326i	0.970609	+	0.240663i	$0.0773648\pi$
$197$			1330.88i			0.481326i	−0.970609	+	0.240663i	$0.922635\pi$
$198$		0			0
$199$		−	3197.38i		−	1.13898i	−0.821999	−	0.569489i	$-0.807141\pi$
$199$		−	3197.38i		−	1.13898i	0.821999	−	0.569489i	$-0.192859\pi$
$200$		0			0
$201$	837.945	+	421.827i	0.294050	+	0.148027i
$202$		0			0
$203$	1698.62			0.587289
$204$		0			0
$205$	4597.48			1.56635
$206$		0			0
$207$	1677.31	+	2261.95i	0.563193	+	0.759500i
$208$		0			0
$209$			2574.65i			0.852116i
$210$		0			0
$211$		−	85.7881i		−	0.0279900i	−0.999902	−	0.0139950i	$-0.995545\pi$
$211$		−	85.7881i		−	0.0279900i	0.999902	−	0.0139950i	$-0.00445490\pi$
$212$		0			0
$213$	59.3912	−	117.979i	0.0191053	−	0.0379520i
$214$		0			0
$215$	144.340			0.0457856
$216$		0			0
$217$	−2216.23			−0.693307
$218$		0			0
$219$	815.678	−	1620.32i	0.251682	−	0.499958i
$220$		0			0
$221$		−	398.623i		−	0.121332i
$222$		0			0
$223$			335.779i			0.100832i	0.998728	+	0.0504158i	$0.0160546\pi$
$223$			335.779i			0.100832i	−0.998728	+	0.0504158i	$0.983945\pi$
$224$		0			0
$225$	307.703	+	414.957i	0.0911713	+	0.122950i
$226$		0			0
$227$	−3532.32			−1.03281			−0.516405	−	0.856344i	$-0.672730\pi$
$227$	−3532.32			−1.03281			−0.516405	+	0.856344i	$0.672730\pi$
$228$		0			0
$229$	−5711.26			−1.64808			−0.824040	−	0.566531i	$-0.808285\pi$
$229$	−5711.26			−1.64808			−0.824040	+	0.566531i	$0.808285\pi$
$230$		0			0
$231$	−1104.22	−	555.872i	−0.314513	−	0.158328i
$232$		0			0
$233$		−	1026.46i		−	0.288607i	−0.989534	−	0.144303i	$-0.953906\pi$
$233$		−	1026.46i		−	0.288607i	0.989534	−	0.144303i	$-0.0460941\pi$
$234$		0			0
$235$		−	3777.18i		−	1.04849i
$236$		0			0
$237$	2100.86	+	1057.59i	0.575804	+	0.289863i
$238$		0			0
$239$	−2404.61			−0.650800			−0.325400	−	0.945576i	$-0.605499\pi$
$239$	−2404.61			−0.650800			−0.325400	+	0.945576i	$0.605499\pi$
$240$		0			0
$241$	−6537.60			−1.74740			−0.873700	−	0.486464i	$-0.838286\pi$
$241$	−6537.60			−1.74740			−0.873700	+	0.486464i	$0.838286\pi$
$242$		0			0
$243$	−2742.94	−	2612.51i	−0.724114	−	0.689681i
$244$		0			0
$245$		−	588.272i		−	0.153401i
$246$		0			0
$247$			1050.77i			0.270684i
$248$		0			0
$249$	−2527.23	+	5020.27i	−0.643200	+	1.27770i
$250$		0			0
$251$	76.2619			0.0191777			0.00958886	−	0.999954i	$-0.496948\pi$
$251$	76.2619			0.0191777			0.00958886	+	0.999954i	$0.496948\pi$
$252$		0			0
$253$	3544.80			0.880867
$254$		0			0
$255$	806.088	−	1601.27i	0.197958	−	0.393236i
$256$		0			0
$257$		−	5213.42i		−	1.26539i	−0.774403	−	0.632693i	$-0.781949\pi$
$257$		−	5213.42i		−	1.26539i	0.774403	−	0.632693i	$-0.218051\pi$
$258$		0			0
$259$		−	2122.86i		−	0.509297i
$260$		0			0
$261$	5262.77	−	3902.50i	1.24811	−	0.925513i
$262$		0			0
$263$	−6662.59			−1.56210			−0.781051	−	0.624468i	$-0.785316\pi$
$263$	−6662.59			−1.56210			−0.781051	+	0.624468i	$0.785316\pi$
$264$		0			0
$265$	5024.54			1.16473
$266$		0			0
$267$	−3788.07	−	1906.94i	−0.868264	−	0.437089i
$268$		0			0
$269$			5591.40i			1.26734i	0.773605	+	0.633669i	$0.218452\pi$
$269$			5591.40i			1.26734i	−0.773605	+	0.633669i	$0.781548\pi$
$270$		0			0
$271$		−	2023.00i		−	0.453462i	−0.973957	−	0.226731i	$-0.927196\pi$
$271$		−	2023.00i		−	0.453462i	0.973957	−	0.226731i	$-0.0728039\pi$
$272$		0			0
$273$	−450.657	−	226.863i	−0.0999084	−	0.0502945i
$274$		0			0
$275$	650.295			0.142597
$276$		0			0
$277$	6383.29			1.38460			0.692300	−	0.721609i	$-0.256597\pi$
$277$	6383.29			1.38460			0.692300	+	0.721609i	$0.256597\pi$
$278$		0			0
$279$	−6866.47	+	5091.70i	−1.47342	+	1.09259i
$280$		0			0
$281$		−	7466.36i		−	1.58507i	−0.609824	−	0.792537i	$-0.708760\pi$
$281$		−	7466.36i		−	1.58507i	0.609824	−	0.792537i	$-0.291240\pi$
$282$		0			0
$283$			1593.17i			0.334644i	0.985902	+	0.167322i	$0.0535120\pi$
$283$			1593.17i			0.334644i	−0.985902	+	0.167322i	$0.946488\pi$
$284$		0			0
$285$	−2124.85	+	4220.95i	−0.441633	+	0.877289i
$286$		0			0
$287$	−2680.62			−0.551332
$288$		0			0
$289$	4087.16			0.831907
$290$		0			0
$291$	2500.14	−	4966.44i	0.503645	−	1.00047i
$292$		0			0
$293$		−	7812.67i		−	1.55775i	−0.627179	−	0.778875i	$-0.715790\pi$
$293$		−	7812.67i		−	1.55775i	0.627179	−	0.778875i	$-0.284210\pi$
$294$		0			0
$295$		−	5727.92i		−	1.13048i
$296$		0			0
$297$	−4698.26	+	814.664i	−0.917915	+	0.159164i
$298$		0			0
$299$	1446.71			0.279817
$300$		0			0
$301$	−84.1594			−0.0161158
$302$		0			0
$303$	6294.23	+	3168.55i	1.19338	+	0.600754i
$304$		0			0
$305$			1347.82i			0.253036i
$306$		0			0
$307$		−	5120.06i		−	0.951848i	−0.879486	−	0.475924i	$-0.842114\pi$
$307$		−	5120.06i		−	0.951848i	0.879486	−	0.475924i	$-0.157886\pi$
$308$		0			0
$309$	3516.83	+	1770.39i	0.647460	+	0.325936i
$310$		0			0
$311$	7740.01			1.41124			0.705620	−	0.708590i	$-0.250668\pi$
$311$	7740.01			1.41124			0.705620	+	0.708590i	$0.250668\pi$
$312$		0			0
$313$	4208.89			0.760066			0.380033	−	0.924973i	$-0.375913\pi$
$313$	4208.89			0.760066			0.380033	+	0.924973i	$0.375913\pi$
$314$		0			0
$315$	−1351.53	−	1822.62i	−0.241746	−	0.326010i
$316$		0			0
$317$		−	8363.38i		−	1.48181i	−0.671609	−	0.740905i	$-0.734397\pi$
$317$		−	8363.38i		−	1.48181i	0.671609	−	0.740905i	$-0.265603\pi$
$318$		0			0
$319$		−	8247.49i		−	1.44756i
$320$		0			0
$321$	2306.85	−	4582.48i	0.401109	−	0.796789i
$322$		0			0
$323$	2176.92			0.375006
$324$		0			0
$325$	265.400			0.0452976
$326$		0			0
$327$	2938.14	−	5836.51i	0.496879	−	0.987033i
$328$		0			0
$329$			2202.34i			0.369054i
$330$		0			0
$331$			11782.9i			1.95663i	0.207111	+	0.978317i	$0.433594\pi$
$331$			11782.9i			1.95663i	−0.207111	+	0.978317i	$0.566406\pi$
$332$		0			0
$333$	−4877.18	−	6577.17i	−0.802605	−	1.08236i
$334$		0			0
$335$	2167.52			0.353505
$336$		0			0
$337$	4424.50			0.715187			0.357593	−	0.933877i	$-0.383597\pi$
$337$	4424.50			0.715187			0.357593	+	0.933877i	$0.383597\pi$
$338$		0			0
$339$	−2084.04	−	1049.12i	−0.333892	−	0.168083i
$340$		0			0
$341$			10760.7i			1.70887i
$342$		0			0
$343$			343.000i			0.0539949i
$344$		0			0
$345$	5811.43	+	2925.51i	0.906889	+	0.456534i
$346$		0			0
$347$	3752.27			0.580497			0.290248	−	0.956951i	$-0.406262\pi$
$347$	3752.27			0.580497			0.290248	+	0.956951i	$0.406262\pi$
$348$		0			0
$349$	3908.60			0.599492			0.299746	−	0.954019i	$-0.403098\pi$
$349$	3908.60			0.599492			0.299746	+	0.954019i	$0.403098\pi$
$350$		0			0
$351$	−1917.46	+	332.482i	−0.291586	+	0.0505601i
$352$		0			0
$353$			2100.72i			0.316742i	0.987380	+	0.158371i	$0.0506242\pi$
$353$			2100.72i			0.316742i	−0.987380	+	0.158371i	$0.949376\pi$
$354$		0			0
$355$		−	305.177i		−	0.0456257i
$356$		0			0
$357$	−470.001	+	933.642i	−0.0696781	+	0.138413i
$358$		0			0
$359$	−3016.76			−0.443505			−0.221752	−	0.975103i	$-0.571178\pi$
$359$	−3016.76			−0.443505			−0.221752	+	0.975103i	$0.571178\pi$
$360$		0			0
$361$	1120.64			0.163382
$362$		0			0
$363$	410.799	−	816.038i	0.0593976	−	0.117991i
$364$		0			0
$365$		−	4191.29i		−	0.601047i
$366$		0			0
$367$		−	6833.27i		−	0.971918i	−0.873982	−	0.485959i	$-0.838470\pi$
$367$		−	6833.27i		−	0.971918i	0.873982	−	0.485959i	$-0.161530\pi$
$368$		0			0
$369$	−8305.28	+	6158.62i	−1.17170	+	0.868848i
$370$		0			0
$371$	−2929.62			−0.409969
$372$		0			0
$373$	11866.5			1.64725			0.823627	−	0.567132i	$-0.191947\pi$
$373$	11866.5			1.64725			0.823627	+	0.567132i	$0.191947\pi$
$374$		0			0
$375$	−5898.97	−	2969.58i	−0.812323	−	0.408929i
$376$		0			0
$377$		−	3365.98i		−	0.459832i
$378$		0			0
$379$			13985.9i			1.89554i	0.318956	+	0.947769i	$0.396668\pi$
$379$			13985.9i			1.89554i	−0.318956	+	0.947769i	$0.603332\pi$
$380$		0			0
$381$	−7230.45	−	3639.85i	−0.972250	−	0.489436i
$382$		0			0
$383$	−7569.70			−1.00990			−0.504952	−	0.863147i	$-0.668490\pi$
$383$	−7569.70			−1.00990			−0.504952	+	0.863147i	$0.668490\pi$
$384$		0			0
$385$	−2856.30			−0.378106
$386$		0			0
$387$	−260.748	+	193.353i	−0.0342495	+	0.0253971i
$388$		0			0
$389$		−	2949.39i		−	0.384422i	−0.981354	−	0.192211i	$-0.938434\pi$
$389$		−	2949.39i		−	0.384422i	0.981354	−	0.192211i	$-0.0615657\pi$
$390$		0			0
$391$		−	2997.20i		−	0.387659i
$392$		0			0
$393$	4356.13	−	8653.31i	0.559129	−	1.11069i
$394$		0			0
$395$	5434.32			0.692229
$396$		0			0
$397$	−9934.24			−1.25588			−0.627941	−	0.778261i	$-0.716102\pi$
$397$	−9934.24			−1.25588			−0.627941	+	0.778261i	$0.716102\pi$
$398$		0			0
$399$	1238.92	−	2461.08i	0.155448	−	0.308792i
$400$		0			0
$401$			12531.7i			1.56060i	0.625403	+	0.780302i	$0.284934\pi$
$401$			12531.7i			1.56060i	−0.625403	+	0.780302i	$0.715066\pi$
$402$		0			0
$403$			4391.68i			0.542842i
$404$		0			0
$405$	−8374.79	−	2541.88i	−1.02752	−	0.311869i
$406$		0			0
$407$	−10307.4			−1.25532
$408$		0			0
$409$	−9543.29			−1.15375			−0.576877	−	0.816831i	$-0.695729\pi$
$409$	−9543.29			−1.15375			−0.576877	+	0.816831i	$0.695729\pi$
$410$		0			0
$411$	−2688.66	−	1353.49i	−0.322681	−	0.162440i
$412$		0			0
$413$			3339.74i			0.397913i
$414$		0			0
$415$			12986.0i			1.53604i
$416$		0			0
$417$	−6546.18	−	3295.39i	−0.768748	−	0.386993i
$418$		0			0
$419$	−889.368			−0.103696			−0.0518478	−	0.998655i	$-0.516511\pi$
$419$	−889.368			−0.103696			−0.0518478	+	0.998655i	$0.516511\pi$
$420$		0			0
$421$	293.443			0.0339704			0.0169852	−	0.999856i	$-0.494593\pi$
$421$	293.443			0.0339704			0.0169852	+	0.999856i	$0.494593\pi$
$422$		0			0
$423$	5059.77	+	6823.42i	0.581595	+	0.784316i
$424$		0			0
$425$		−	549.838i		−	0.0627554i
$426$		0			0
$427$		−	785.866i		−	0.0890650i
$428$		0			0
$429$	−1101.52	+	2188.12i	−0.123967	+	0.246256i
$430$		0			0
$431$	−7891.67			−0.881968			−0.440984	−	0.897515i	$-0.645370\pi$
$431$	−7891.67			−0.881968			−0.440984	+	0.897515i	$0.645370\pi$
$432$		0			0
$433$	10888.9			1.20851			0.604255	−	0.796791i	$-0.293471\pi$
$433$	10888.9			1.20851			0.604255	+	0.796791i	$0.293471\pi$
$434$		0			0
$435$	6806.63	−	13521.1i	0.750236	−	1.49032i
$436$		0			0
$437$			7900.61i			0.864846i
$438$		0			0
$439$			7101.91i			0.772109i	0.922476	+	0.386054i	$0.126162\pi$
$439$			7101.91i			0.772109i	−0.922476	+	0.386054i	$0.873838\pi$
$440$		0			0
$441$	788.028	+	1062.70i	0.0850910	+	0.114751i
$442$		0			0
$443$	10843.3			1.16294			0.581468	−	0.813569i	$-0.302478\pi$
$443$	10843.3			1.16294			0.581468	+	0.813569i	$0.302478\pi$
$444$		0			0
$445$	−9798.66			−1.04382
$446$		0			0
$447$	13219.3	+	6654.68i	1.39877	+	0.704151i
$448$		0			0
$449$			11771.1i			1.23722i	0.785698	+	0.618611i	$0.212304\pi$
$449$			11771.1i			1.23722i	−0.785698	+	0.618611i	$0.787696\pi$
$450$		0			0
$451$			13015.5i			1.35893i
$452$		0			0
$453$	−2584.76	−	1301.19i	−0.268086	−	0.134956i
$454$		0			0
$455$	−1165.72			−0.120109
$456$		0			0
$457$	15242.9			1.56025			0.780125	−	0.625624i	$-0.215155\pi$
$457$	15242.9			1.56025			0.780125	+	0.625624i	$0.215155\pi$
$458$		0			0
$459$	688.815	+	3972.48i	0.0700460	+	0.403964i
$460$		0			0
$461$		−	537.066i		−	0.0542595i	−0.999632	−	0.0271298i	$-0.991363\pi$
$461$		−	537.066i		−	0.0542595i	0.999632	−	0.0271298i	$-0.00863673\pi$
$462$		0			0
$463$			1043.87i			0.104780i	0.998627	+	0.0523898i	$0.0166838\pi$
$463$			1043.87i			0.104780i	−0.998627	+	0.0523898i	$0.983316\pi$
$464$		0			0
$465$	−8880.78	+	17641.4i	−0.885670	+	1.75935i
$466$		0			0
$467$	1482.78			0.146927			0.0734633	−	0.997298i	$-0.476595\pi$
$467$	1482.78			0.146927			0.0734633	+	0.997298i	$0.476595\pi$
$468$		0			0
$469$	−1263.80			−0.124429
$470$		0			0
$471$	2464.06	−	4894.77i	0.241057	−	0.478851i
$472$		0			0
$473$			408.629i			0.0397226i
$474$		0			0
$475$			1449.37i			0.140004i
$476$		0			0
$477$	−9076.75	+	6730.69i	−0.871270	+	0.646074i
$478$		0			0
$479$	−722.425			−0.0689111			−0.0344556	−	0.999406i	$-0.510970\pi$
$479$	−722.425			−0.0689111			−0.0344556	+	0.999406i	$0.510970\pi$
$480$		0			0
$481$	−4206.65			−0.398767
$482$		0			0
$483$	−3388.43	−	1705.76i	−0.319211	−	0.160693i
$484$		0			0
$485$		−	12846.7i		−	1.20276i
$486$		0			0
$487$		−	3750.20i		−	0.348948i	−0.984662	−	0.174474i	$-0.944177\pi$
$487$		−	3750.20i		−	0.348948i	0.984662	−	0.174474i	$-0.0558226\pi$
$488$		0			0
$489$	−11704.0	−	5891.87i	−1.08236	−	0.544866i
$490$		0			0
$491$	5139.62			0.472399			0.236199	−	0.971705i	$-0.424098\pi$
$491$	5139.62			0.472399			0.236199	+	0.971705i	$0.424098\pi$
$492$		0			0
$493$	−6973.42			−0.637053
$494$		0			0
$495$	−8849.58	+	6562.23i	−0.803553	+	0.595860i
$496$		0			0
$497$			177.938i			0.0160595i
$498$		0			0
$499$			13413.4i			1.20334i	0.798746	+	0.601668i	$0.205497\pi$
$499$			13413.4i			1.20334i	−0.798746	+	0.601668i	$0.794503\pi$
$500$		0			0
$501$	−2961.73	+	5883.38i	−0.264113	+	0.524651i
$502$		0			0
$503$	9602.30			0.851184			0.425592	−	0.904915i	$-0.360066\pi$
$503$	9602.30			0.851184			0.425592	+	0.904915i	$0.360066\pi$
$504$		0			0
$505$	16281.3			1.43467
$506$		0			0
$507$	4683.58	−	9303.78i	0.410267	−	0.814981i
$508$		0			0
$509$			20047.2i			1.74573i	0.487962	+	0.872865i	$0.337740\pi$
$509$			20047.2i			1.74573i	−0.487962	+	0.872865i	$0.662260\pi$
$510$		0			0
$511$			2443.79i			0.211560i
$512$		0			0
$513$	−1815.72	−	10471.5i	−0.156269	−	0.901220i
$514$		0			0
$515$	9097.02			0.778374
$516$		0			0
$517$	10693.2			0.909649
$518$		0			0
$519$	1652.75	+	832.002i	0.139783	+	0.0703677i
$520$		0			0
$521$		−	14178.2i		−	1.19224i	−0.802894	−	0.596121i	$-0.796708\pi$
$521$		−	14178.2i		−	1.19224i	0.802894	−	0.596121i	$-0.203292\pi$
$522$		0			0
$523$			3326.40i			0.278114i	0.990284	+	0.139057i	$0.0444071\pi$
$523$			3326.40i			0.278114i	−0.990284	+	0.139057i	$0.955593\pi$
$524$		0			0
$525$	−621.610	−	312.922i	−0.0516749	−	0.0260134i
$526$		0			0
$527$	9098.40			0.752054
$528$		0			0
$529$	−1289.38			−0.105974
$530$		0			0
$531$	7672.92	+	10347.4i	0.627074	+	0.845648i
$532$		0			0
$533$			5311.92i			0.431679i
$534$		0			0
$535$		−	11853.6i		−	0.957896i
$536$		0			0
$537$	9993.40	−	19851.6i	0.803067	−	1.59527i
$538$		0			0
$539$	1665.41			0.133087
$540$		0			0
$541$	1153.72			0.0916867			0.0458434	−	0.998949i	$-0.485402\pi$
$541$	1153.72			0.0916867			0.0458434	+	0.998949i	$0.485402\pi$
$542$		0			0
$543$	−7477.40	+	14853.6i	−0.590950	+	1.17390i
$544$		0			0
$545$		−	15097.4i		−	1.18661i
$546$		0			0
$547$			25250.9i			1.97377i	0.161436	+	0.986883i	$0.448387\pi$
$547$			25250.9i			1.97377i	−0.161436	+	0.986883i	$0.551613\pi$
$548$		0			0
$549$	−1805.49	−	2434.82i	−0.140358	−	0.189282i
$550$		0			0
$551$	18381.9			1.42123
$552$		0			0
$553$	−3168.56			−0.243654
$554$		0			0
$555$	−16898.1	−	8506.62i	−1.29241	−	0.650606i
$556$		0			0
$557$			6469.49i			0.492138i	0.969252	+	0.246069i	$0.0791391\pi$
$557$			6469.49i			0.492138i	−0.969252	+	0.246069i	$0.920861\pi$
$558$		0			0
$559$			166.770i			0.0126183i
$560$		0			0
$561$	4533.22	+	2282.05i	0.341163	+	0.171744i
$562$		0			0
$563$	−19640.7			−1.47026			−0.735132	−	0.677924i	$-0.762880\pi$
$563$	−19640.7			−1.47026			−0.735132	+	0.677924i	$0.762880\pi$
$564$		0			0
$565$	−5390.80			−0.401403
$566$		0			0
$567$	4883.04	+	1482.08i	0.361672	+	0.109773i
$568$		0			0
$569$		−	8396.74i		−	0.618646i	−0.950957	−	0.309323i	$-0.899898\pi$
$569$		−	8396.74i		−	0.618646i	0.950957	−	0.309323i	$-0.100102\pi$
$570$		0			0
$571$		−	4362.50i		−	0.319728i	−0.987139	−	0.159864i	$-0.948894\pi$
$571$		−	4362.50i		−	0.319728i	0.987139	−	0.159864i	$-0.0511056\pi$
$572$		0			0
$573$	−8032.45	+	15956.2i	−0.585620	+	1.16332i
$574$		0			0
$575$	1995.51			0.144728
$576$		0			0
$577$	19329.8			1.39465			0.697324	−	0.716756i	$-0.254374\pi$
$577$	19329.8			1.39465			0.697324	+	0.716756i	$0.254374\pi$
$578$		0			0
$579$	11324.2	−	22495.1i	0.812808	−	1.61462i
$580$		0			0
$581$		−	7571.65i		−	0.540663i
$582$		0			0
$583$			14224.5i			1.01050i
$584$		0			0
$585$	−3611.71	+	2678.19i	−0.255257	+	0.189281i
$586$		0			0
$587$	−21737.9			−1.52848			−0.764241	−	0.644931i	$-0.776886\pi$
$587$	−21737.9			−1.52848			−0.764241	+	0.644931i	$0.776886\pi$
$588$		0			0
$589$	−23983.4			−1.67779
$590$		0			0
$591$	−6176.93	−	3109.50i	−0.429924	−	0.216426i
$592$		0			0
$593$		−	20354.5i		−	1.40954i	−0.709435	−	0.704771i	$-0.751050\pi$
$593$		−	20354.5i		−	1.40954i	0.709435	−	0.704771i	$-0.248950\pi$
$594$		0			0
$595$			2415.06i			0.166400i
$596$		0			0
$597$	14839.8	+	7470.46i	1.01734	+	0.512137i
$598$		0			0
$599$	24269.5			1.65547			0.827733	−	0.561122i	$-0.189630\pi$
$599$	24269.5			1.65547			0.827733	+	0.561122i	$0.189630\pi$
$600$		0			0
$601$	20775.4			1.41006			0.705029	−	0.709178i	$-0.250934\pi$
$601$	20775.4			1.41006			0.705029	+	0.709178i	$0.250934\pi$
$602$		0			0
$603$	−3915.60	+	2903.53i	−0.264437	+	0.196088i
$604$		0			0
$605$		−	2110.85i		−	0.141849i
$606$		0			0
$607$		−	3498.28i		−	0.233922i	−0.993136	−	0.116961i	$-0.962685\pi$
$607$		−	3498.28i		−	0.233922i	0.993136	−	0.116961i	$-0.0373153\pi$
$608$		0			0
$609$	−3968.70	+	7883.69i	−0.264072	+	0.524570i
$610$		0			0
$611$	4364.15			0.288960
$612$		0			0
$613$	14449.4			0.952047			0.476024	−	0.879433i	$-0.342078\pi$
$613$	14449.4			0.952047			0.476024	+	0.879433i	$0.342078\pi$
$614$		0			0
$615$	−10741.7	+	21338.0i	−0.704303	+	1.39908i
$616$		0			0
$617$		−	14078.2i		−	0.918585i	−0.888285	−	0.459293i	$-0.848103\pi$
$617$		−	14078.2i		−	0.918585i	0.888285	−	0.459293i	$-0.151897\pi$
$618$		0			0
$619$		−	7166.47i		−	0.465339i	−0.972556	−	0.232670i	$-0.925254\pi$
$619$		−	7166.47i		−	0.465339i	0.972556	−	0.232670i	$-0.0747461\pi$
$620$		0			0
$621$	−14417.2	+	2499.89i	−0.931628	+	0.161541i
$622$		0			0
$623$	5713.24			0.367410
$624$		0			0
$625$	−17650.6			−1.12964
$626$		0			0
$627$	−11949.6	−	6015.48i	−0.761116	−	0.383150i
$628$		0			0
$629$			8715.07i			0.552453i
$630$		0			0
$631$		−	8130.21i		−	0.512929i	−0.966554	−	0.256465i	$-0.917442\pi$
$631$		−	8130.21i		−	0.512929i	0.966554	−	0.256465i	$-0.0825577\pi$
$632$		0			0
$633$	398.163	+	200.438i	0.0250009	+	0.0125856i
$634$		0			0
$635$	−18703.1			−1.16883
$636$		0			0
$637$	679.689			0.0422767
$638$		0			0
$639$	408.804	+	551.298i	0.0253084	+	0.0341299i
$640$		0			0
$641$			21601.6i			1.33106i	0.746370	+	0.665531i	$0.231795\pi$
$641$			21601.6i			1.33106i	−0.746370	+	0.665531i	$0.768205\pi$
$642$		0			0
$643$		−	21818.4i		−	1.33816i	−0.743192	−	0.669078i	$-0.766689\pi$
$643$		−	21818.4i		−	1.33816i	0.743192	−	0.669078i	$-0.233311\pi$
$644$		0			0
$645$	−337.240	+	669.916i	−0.0205873	+	0.0408960i
$646$		0			0
$647$	−26057.5			−1.58335			−0.791673	−	0.610945i	$-0.790790\pi$
$647$	−26057.5			−1.58335			−0.791673	+	0.610945i	$0.790790\pi$
$648$		0			0
$649$	16215.8			0.980781
$650$		0			0
$651$	5178.06	−	10286.1i	0.311742	−	0.619266i
$652$		0			0
$653$		−	8250.21i		−	0.494419i	−0.968962	−	0.247209i	$-0.920486\pi$
$653$		−	8250.21i		−	0.494419i	0.968962	−	0.247209i	$-0.0795136\pi$
$654$		0			0
$655$		−	22383.6i		−	1.33527i
$656$		0			0
$657$	5614.51	+	7571.51i	0.333398	+	0.449608i
$658$		0			0
$659$	−7630.25			−0.451035			−0.225518	−	0.974239i	$-0.572407\pi$
$659$	−7630.25			−0.451035			−0.225518	+	0.974239i	$0.572407\pi$
$660$		0			0
$661$	1683.25			0.0990480			0.0495240	−	0.998773i	$-0.484230\pi$
$661$	1683.25			0.0990480			0.0495240	+	0.998773i	$0.484230\pi$
$662$		0			0
$663$	1850.10	+	931.354i	0.108374	+	0.0545562i
$664$		0			0
$665$		−	6366.11i		−	0.371229i
$666$		0			0
$667$		−	25308.4i		−	1.46918i
$668$		0			0
$669$	−1558.43	−	784.524i	−0.0900634	−	0.0453385i
$670$		0			0
$671$	−3815.71			−0.219529
$672$		0			0
$673$	−16361.7			−0.937143			−0.468571	−	0.883426i	$-0.655231\pi$
$673$	−16361.7			−0.937143			−0.468571	+	0.883426i	$0.655231\pi$
$674$		0			0
$675$	−2644.84	+	458.607i	−0.150815	+	0.0261508i
$676$		0			0
$677$			13383.2i			0.759763i	0.925035	+	0.379882i	$0.124035\pi$
$677$			13383.2i			0.759763i	−0.925035	+	0.379882i	$0.875965\pi$
$678$		0			0
$679$			7490.47i			0.423355i
$680$		0			0
$681$	8253.00	−	16394.3i	0.464399	−	0.922514i
$682$		0			0
$683$	15026.1			0.841810			0.420905	−	0.907105i	$-0.361713\pi$
$683$	15026.1			0.841810			0.420905	+	0.907105i	$0.361713\pi$
$684$		0			0
$685$	−6954.80			−0.387926
$686$		0			0
$687$	13343.9	−	26507.3i	0.741053	−	1.47208i
$688$		0			0
$689$			5805.34i			0.320996i
$690$		0			0
$691$		−	1932.62i		−	0.106397i	−0.998584	−	0.0531986i	$-0.983058\pi$
$691$		−	1932.62i		−	0.106397i	0.998584	−	0.0531986i	$-0.0169416\pi$
$692$		0			0
$693$	5159.87	−	3826.20i	0.282839	−	0.209734i
$694$		0			0
$695$	−16933.1			−0.924185
$696$		0			0
$697$	11004.9			0.598049
$698$		0			0
$699$	4764.03	+	2398.24i	0.257785	+	0.129771i
$700$		0			0
$701$		−	3730.63i		−	0.201004i	−0.994937	−	0.100502i	$-0.967955\pi$
$701$		−	3730.63i		−	0.201004i	0.994937	−	0.100502i	$-0.0320449\pi$
$702$		0			0
$703$		−	22972.9i		−	1.23249i
$704$		0			0
$705$	17530.8	+	8825.10i	0.936521	+	0.471451i
$706$		0			0
$707$	−9493.07			−0.504984
$708$		0			0
$709$	−19110.7			−1.01230			−0.506149	−	0.862446i	$-0.668931\pi$
$709$	−19110.7			−1.01230			−0.506149	+	0.862446i	$0.668931\pi$
$710$		0			0
$711$	−9817.02	+	7279.62i	−0.517816	+	0.383976i
$712$		0			0
$713$			33020.5i			1.73440i
$714$		0			0
$715$			5660.05i			0.296047i
$716$		0			0
$717$	5618.20	−	11160.4i	0.292630	−	0.581299i
$718$		0			0
$719$	−9718.45			−0.504085			−0.252043	−	0.967716i	$-0.581102\pi$
$719$	−9718.45			−0.504085			−0.252043	+	0.967716i	$0.581102\pi$
$720$		0			0
$721$	−5304.14			−0.273976
$722$		0			0
$723$	15274.6	−	30342.5i	0.785712	−	1.56079i
$724$		0			0
$725$		−	4642.84i		−	0.237836i
$726$		0			0
$727$		−	20385.8i		−	1.03998i	−0.854172	−	0.519991i	$-0.825935\pi$
$727$		−	20385.8i		−	1.03998i	0.854172	−	0.519991i	$-0.174065\pi$
$728$		0			0
$729$	18534.0	−	6626.70i	0.941622	−	0.336671i
$730$		0			0
$731$	345.504			0.0174814
$732$		0			0
$733$	−14957.5			−0.753709			−0.376855	−	0.926272i	$-0.622994\pi$
$733$	−14957.5			−0.753709			−0.376855	+	0.926272i	$0.622994\pi$
$734$		0			0
$735$	2730.31	+	1374.45i	0.137019	+	0.0689762i
$736$		0			0
$737$			6136.29i			0.306693i
$738$		0			0
$739$			31710.5i			1.57847i	0.614090	+	0.789236i	$0.289523\pi$
$739$			31710.5i			1.57847i	−0.614090	+	0.789236i	$0.710477\pi$
$740$		0			0
$741$	−4876.88	−	2455.05i	−0.241777	−	0.121712i
$742$		0			0
$743$	26608.9			1.31384			0.656922	−	0.753958i	$-0.271858\pi$
$743$	26608.9			1.31384			0.656922	+	0.753958i	$0.271858\pi$
$744$		0			0
$745$	34194.5			1.68160
$746$		0			0
$747$	−17395.6	−	23459.0i	−0.852035	−	1.14902i
$748$		0			0
$749$			6911.39i			0.337165i
$750$		0			0
$751$		−	20486.9i		−	0.995445i	−0.867336	−	0.497723i	$-0.834170\pi$
$751$		−	20486.9i		−	0.995445i	0.867336	−	0.497723i	$-0.165830\pi$
$752$		0			0
$753$	−178.180	+	353.950i	−0.00862318	+	0.0171297i
$754$		0			0
$755$	−6686.04			−0.322291
$756$		0			0
$757$	7557.64			0.362863			0.181431	−	0.983404i	$-0.441927\pi$
$757$	7557.64			0.362863			0.181431	+	0.983404i	$0.441927\pi$
$758$		0			0
$759$	−8282.16	+	16452.2i	−0.396078	+	0.786797i
$760$		0			0
$761$		−	7134.66i		−	0.339857i	−0.985456	−	0.169928i	$-0.945646\pi$
$761$		−	7134.66i		−	0.339857i	0.985456	−	0.169928i	$-0.0543537\pi$
$762$		0			0
$763$			8802.73i			0.417667i
$764$		0			0
$765$	5548.50	+	7482.50i	0.262231	+	0.353634i
$766$		0			0
$767$	6618.03			0.311556
$768$		0			0
$769$	−15189.2			−0.712270			−0.356135	−	0.934434i	$-0.615906\pi$
$769$	−15189.2			−0.712270			−0.356135	+	0.934434i	$0.615906\pi$
$770$		0			0
$771$	24196.7	+	12180.8i	1.13025	+	0.568976i
$772$		0			0
$773$		−	30376.4i		−	1.41340i	−0.707511	−	0.706702i	$-0.750182\pi$
$773$		−	30376.4i		−	1.41340i	0.707511	−	0.706702i	$-0.249818\pi$
$774$		0			0
$775$			6057.64i			0.280770i
$776$		0			0
$777$	9852.69	+	4959.91i	0.454908	+	0.229003i
$778$		0			0
$779$	−29008.9			−1.33421
$780$		0			0
$781$	863.961			0.0395838
$782$		0			0
$783$	5816.37	+	33543.7i	0.265466	+	1.53097i
$784$		0			0
$785$		−	12661.4i		−	0.575673i
$786$		0			0
$787$		−	19109.2i		−	0.865528i	−0.901507	−	0.432764i	$-0.857538\pi$
$787$		−	19109.2i		−	0.865528i	0.901507	−	0.432764i	$-0.142462\pi$
$788$		0			0
$789$	15566.7	−	30922.7i	0.702393	−	1.39528i
$790$		0			0
$791$	3143.18			0.141288
$792$		0			0
$793$	−1557.27			−0.0697356
$794$		0			0
$795$	−11739.5	+	23320.1i	−0.523718	+	1.04035i
$796$		0			0
$797$			15055.1i			0.669109i	0.942376	+	0.334554i	$0.108586\pi$
$797$			15055.1i			0.669109i	−0.942376	+	0.334554i	$0.891414\pi$
$798$		0			0
$799$		−	9041.36i		−	0.400326i
$800$		0			0
$801$	17701.1	−	13125.9i	0.780822	−	0.579004i
$802$		0			0
$803$	11865.6			0.521455
$804$		0			0
$805$	−8764.90			−0.383754
$806$		0			0
$807$	−25951.0	−	13063.9i	−1.13199	−	0.569853i
$808$		0			0
$809$		−	7380.46i		−	0.320746i	−0.987057	−	0.160373i	$-0.948730\pi$
$809$		−	7380.46i		−	0.320746i	0.987057	−	0.160373i	$-0.0512696\pi$
$810$		0			0
$811$		−	25037.2i		−	1.08406i	−0.840358	−	0.542032i	$-0.817655\pi$
$811$		−	25037.2i		−	1.08406i	0.840358	−	0.542032i	$-0.182345\pi$
$812$		0			0
$813$	9389.21	+	4726.59i	0.405036	+	0.203897i
$814$		0			0
$815$	−30274.9			−1.30121
$816$		0			0
$817$	−910.749			−0.0390001
$818$		0			0
$819$	2105.85	−	1561.56i	0.0898468	−	0.0666241i
$820$		0			0
$821$			13748.5i			0.584441i	0.956351	+	0.292220i	$0.0943941\pi$
$821$			13748.5i			0.584441i	−0.956351	+	0.292220i	$0.905606\pi$
$822$		0			0
$823$		−	14854.7i		−	0.629165i	−0.949230	−	0.314582i	$-0.898136\pi$
$823$		−	14854.7i		−	0.629165i	0.949230	−	0.314582i	$-0.101864\pi$
$824$		0			0
$825$	−1519.37	+	3018.17i	−0.0641183	+	0.127369i
$826$		0			0
$827$	2071.34			0.0870950			0.0435475	−	0.999051i	$-0.486134\pi$
$827$	2071.34			0.0870950			0.0435475	+	0.999051i	$0.486134\pi$
$828$		0			0
$829$	−6720.58			−0.281563			−0.140781	−	0.990041i	$-0.544961\pi$
$829$	−6720.58			−0.281563			−0.140781	+	0.990041i	$0.544961\pi$
$830$		0			0
$831$	−14914.1	+	29626.4i	−0.622580	+	1.23674i
$832$		0			0
$833$		−	1408.13i		−	0.0585702i
$834$		0			0
$835$			15218.6i			0.630733i
$836$		0			0
$837$	−7588.77	−	43765.3i	−0.313389	−	1.80735i
$838$		0			0
$839$	14950.6			0.615201			0.307600	−	0.951516i	$-0.400474\pi$
$839$	14950.6			0.615201			0.307600	+	0.951516i	$0.400474\pi$
$840$		0			0
$841$	−34494.7			−1.41436
$842$		0			0
$843$	34653.2	+	17444.6i	1.41580	+	0.712722i
$844$		0			0
$845$		−	24066.2i		−	0.979766i
$846$		0			0
$847$			1230.76i			0.0499286i
$848$		0			0
$849$	−7394.30	−	3722.34i	−0.298907	−	0.150471i
$850$		0			0
$851$	−31629.3			−1.27408
$852$		0			0
$853$	18485.1			0.741989			0.370995	−	0.928635i	$-0.379017\pi$
$853$	18485.1			0.741989			0.370995	+	0.928635i	$0.379017\pi$
$854$		0			0
$855$	−14625.9	−	19723.9i	−0.585022	−	0.788938i
$856$		0			0
$857$		−	21046.5i		−	0.838897i	−0.907779	−	0.419448i	$-0.862224\pi$
$857$		−	21046.5i		−	0.838897i	0.907779	−	0.419448i	$-0.137776\pi$
$858$		0			0
$859$			15412.2i			0.612172i	0.952004	+	0.306086i	$0.0990195\pi$
$859$			15412.2i			0.612172i	−0.952004	+	0.306086i	$0.900980\pi$
$860$		0			0
$861$	6263.09	−	12441.4i	0.247904	−	0.492453i
$862$		0			0
$863$	19843.9			0.782726			0.391363	−	0.920236i	$-0.372004\pi$
$863$	19843.9			0.782726			0.391363	+	0.920236i	$0.372004\pi$
$864$		0			0
$865$	4275.18			0.168047
$866$		0			0
$867$	−9549.36	+	18969.5i	−0.374064	+	0.743065i
$868$		0			0
$869$			15384.6i			0.600562i
$870$		0			0
$871$			2504.35i			0.0974245i
$872$		0			0
$873$	17209.0	+	23207.5i	0.667168	+	0.899718i
$874$		0			0
$875$	8896.93			0.343738
$876$		0			0
$877$	−49681.4			−1.91291			−0.956455	−	0.291881i	$-0.905719\pi$
$877$	−49681.4			−1.91291			−0.956455	+	0.291881i	$0.905719\pi$
$878$		0			0
$879$	36260.5	+	18253.7i	1.39139	+	0.700436i
$880$		0			0
$881$		−	25877.1i		−	0.989582i	−0.869012	−	0.494791i	$-0.835245\pi$
$881$		−	25877.1i		−	0.989582i	0.869012	−	0.494791i	$-0.164755\pi$
$882$		0			0
$883$			39739.5i			1.51454i	0.653101	+	0.757271i	$0.273468\pi$
$883$			39739.5i			1.51454i	−0.653101	+	0.757271i	$0.726532\pi$
$884$		0			0
$885$	26584.6	+	13382.9i	1.00975	+	0.508317i
$886$		0			0
$887$	37385.6			1.41521			0.707603	−	0.706611i	$-0.249777\pi$
$887$	37385.6			1.41521			0.707603	+	0.706611i	$0.249777\pi$
$888$		0			0
$889$	10905.1			0.411412
$890$		0			0
$891$	7196.10	−	23709.2i	0.270571	−	0.891455i
$892$		0			0
$893$			23833.0i			0.893104i
$894$		0			0
$895$		−	51350.3i		−	1.91782i
$896$		0			0
$897$	−3380.13	+	6714.52i	−0.125819	+	0.249934i
$898$		0			0
$899$	76827.1			2.85020
$900$		0			0
$901$	12027.1			0.444708
$902$		0			0
$903$	196.632	−	390.604i	0.00724642	−	0.0143948i
$904$		0			0
$905$			38422.0i			1.41126i
$906$		0			0
$907$		−	30147.4i		−	1.10367i	−0.833953	−	0.551835i	$-0.813928\pi$
$907$		−	30147.4i		−	1.10367i	0.833953	−	0.551835i	$-0.186072\pi$
$908$		0			0
$909$	−29412.0	+	21809.9i	−1.07320	+	0.795808i
$910$		0			0
$911$	−52314.2			−1.90258			−0.951288	−	0.308303i	$-0.900239\pi$
$911$	−52314.2			−1.90258			−0.951288	+	0.308303i	$0.900239\pi$
$912$		0			0
$913$	−36763.5			−1.33263
$914$		0			0
$915$	−6255.56	−	3149.09i	−0.226014	−	0.113777i
$916$		0			0
$917$			13051.1i			0.469994i
$918$		0			0
$919$			48228.1i			1.73112i	0.500807	+	0.865559i	$0.333037\pi$
$919$			48228.1i			1.73112i	−0.500807	+	0.865559i	$0.666963\pi$
$920$		0			0
$921$	23763.4	+	11962.7i	0.850198	+	0.427995i
$922$		0			0
$923$	352.601			0.0125742
$924$		0			0
$925$	−5802.42			−0.206251
$926$		0			0
$927$	−16433.6	+	12186.0i	−0.582256	+	0.431761i
$928$		0			0
$929$			6538.92i			0.230931i	0.993311	+	0.115466i	$0.0368360\pi$
$929$			6538.92i			0.230931i	−0.993311	+	0.115466i	$0.963164\pi$
$930$		0			0
$931$			3711.85i			0.130667i
$932$		0			0
$933$	−18084.0	+	35923.2i	−0.634559	+	1.26053i
$934$		0			0
$935$	11726.1			0.410145
$936$		0			0
$937$	−12346.7			−0.430470			−0.215235	−	0.976562i	$-0.569052\pi$
$937$	−12346.7			−0.430470			−0.215235	+	0.976562i	$0.569052\pi$
$938$		0			0
$939$	−9833.77	+	19534.5i	−0.341760	+	0.678896i
$940$		0			0
$941$			29721.4i			1.02964i	0.857298	+	0.514820i	$0.172141\pi$
$941$			29721.4i			1.02964i	−0.857298	+	0.514820i	$0.827859\pi$
$942$		0			0
$943$			39939.7i			1.37923i
$944$		0			0
$945$	11617.0	−	2014.35i	0.399894	−	0.0693404i
$946$		0			0
$947$	−2536.03			−0.0870222			−0.0435111	−	0.999053i	$-0.513854\pi$
$947$	−2536.03			−0.0870222			−0.0435111	+	0.999053i	$0.513854\pi$
$948$		0			0
$949$	4842.62			0.165646
$950$		0			0
$951$	38816.4	+	19540.4i	1.32356	+	0.666290i
$952$		0			0
$953$			31662.7i			1.07624i	0.842869	+	0.538119i	$0.180865\pi$
$953$			31662.7i			1.07624i	−0.842869	+	0.538119i	$0.819135\pi$
$954$		0			0
$955$			41274.1i			1.39853i
$956$		0			0
$957$	38278.6	+	19269.7i	1.29297	+	0.650888i
$958$		0			0
$959$	4055.09			0.136544
$960$		0			0
$961$	−70447.3			−2.36472
$962$		0			0
$963$	15878.6	+	21413.3i	0.531341	+	0.716546i
$964$		0			0
$965$		−	58188.3i		−	1.94108i
$966$		0			0
$967$		−	32923.2i		−	1.09487i	−0.836848	−	0.547435i	$-0.815604\pi$
$967$		−	32923.2i		−	1.09487i	0.836848	−	0.547435i	$-0.184396\pi$
$968$		0			0
$969$	−5086.21	+	10103.6i	−0.168620	+	0.334958i
$970$		0			0
$971$	2357.50			0.0779153			0.0389576	−	0.999241i	$-0.487596\pi$
$971$	2357.50			0.0779153			0.0389576	+	0.999241i	$0.487596\pi$
$972$		0			0
$973$	9873.07			0.325299
$974$		0			0
$975$	−620.087	+	1231.78i	−0.0203679	+	0.0404601i
$976$		0			0
$977$		−	37980.1i		−	1.24370i	−0.783138	−	0.621848i	$-0.786382\pi$
$977$		−	37980.1i		−	1.24370i	0.783138	−	0.621848i	$-0.213618\pi$
$978$		0			0
$979$		−	27740.1i		−	0.905597i
$980$		0			0
$981$	20223.9	+	27273.2i	0.658205	+	0.887631i
$982$		0			0
$983$	−8307.82			−0.269561			−0.134780	−	0.990875i	$-0.543033\pi$
$983$	−8307.82			−0.269561			−0.134780	+	0.990875i	$0.543033\pi$
$984$		0			0
$985$	−15977.9			−0.516852
$986$		0			0
$987$	−10221.6	−	5145.60i	−0.329641	−	0.165944i
$988$		0			0
$989$			1253.93i			0.0403160i
$990$		0			0
$991$		−	15692.8i		−	0.503026i	−0.967854	−	0.251513i	$-0.919072\pi$
$991$		−	15692.8i		−	0.503026i	0.967854	−	0.251513i	$-0.0809282\pi$
$992$		0			0
$993$	−54687.2	−	27529.9i	−1.74768	−	0.879793i
$994$		0			0
$995$	38386.3			1.22304
$996$		0			0
$997$	37479.3			1.19055			0.595276	−	0.803521i	$-0.297043\pi$
$997$	37479.3			1.19055			0.595276	+	0.803521i	$0.297043\pi$
$998$		0			0
$999$	41921.4	−	7269.04i	1.32766	−	0.230213i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.h.b.239.8	yes	24
3.2	odd	2	inner	336.4.h.b.239.18	yes	24
4.3	odd	2	inner	336.4.h.b.239.17	yes	24
12.11	even	2	inner	336.4.h.b.239.7	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.h.b.239.7	✓	24	12.11	even	2	inner
336.4.h.b.239.8	yes	24	1.1	even	1	trivial
336.4.h.b.239.17	yes	24	4.3	odd	2	inner
336.4.h.b.239.18	yes	24	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	336.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.33643 + 4.64124i) q^{3} +12.0055i q^{5} -7.00000i q^{7} +(-16.0822 - 21.6878i) q^{9} +O(q^{10})\)	\(q+(-2.33643 + 4.64124i) q^{3} +12.0055i q^{5} -7.00000i q^{7} +(-16.0822 - 21.6878i) q^{9} -33.9879 q^{11} -13.8712 q^{13} +(-55.7206 - 28.0501i) q^{15} +28.7374i q^{17} -75.7520i q^{19} +(32.4887 + 16.3550i) q^{21} -104.296 q^{23} -19.1331 q^{25} +(138.233 - 23.9692i) q^{27} +242.660i q^{29} -316.604i q^{31} +(79.4103 - 157.746i) q^{33} +84.0388 q^{35} +303.265 q^{37} +(32.4091 - 64.3795i) q^{39} -382.946i q^{41} -12.0228i q^{43} +(260.374 - 193.076i) q^{45} -314.619 q^{47} -49.0000 q^{49} +(-133.377 - 67.1430i) q^{51} -418.518i q^{53} -408.043i q^{55} +(351.583 + 176.989i) q^{57} -477.106 q^{59} +112.267 q^{61} +(-151.815 + 112.575i) q^{63} -166.531i q^{65} -180.543i q^{67} +(243.680 - 484.062i) q^{69} -25.4197 q^{71} -349.113 q^{73} +(44.7032 - 88.8015i) q^{75} +237.915i q^{77} -452.651i q^{79} +(-211.725 + 697.577i) q^{81} +1081.66 q^{83} -345.009 q^{85} +(-1126.24 - 566.957i) q^{87} +816.177i q^{89} +97.0984i q^{91} +(1469.44 + 739.724i) q^{93} +909.444 q^{95} -1070.07 q^{97} +(546.600 + 737.124i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 136 q^{9}+O(q^{10}) \) 24 * q - 136 * q^9	\( 24 q - 136 q^{9} + 96 q^{13} - 112 q^{21} + 168 q^{25} - 320 q^{33} - 864 q^{37} + 592 q^{45} - 1176 q^{49} - 1120 q^{57} - 2304 q^{61} + 1168 q^{69} + 3312 q^{73} - 968 q^{81} - 5808 q^{85} + 1776 q^{93} + 6480 q^{97}+O(q^{100}) \) 24 * q - 136 * q^9 + 96 * q^13 - 112 * q^21 + 168 * q^25 - 320 * q^33 - 864 * q^37 + 592 * q^45 - 1176 * q^49 - 1120 * q^57 - 2304 * q^61 + 1168 * q^69 + 3312 * q^73 - 968 * q^81 - 5808 * q^85 + 1776 * q^93 + 6480 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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