LMFDB - Embedded newform 208.4.f.e.129.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(129,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("208.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.f (of order $2$, degree $1$, not 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:	$12.2723972812$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 170x^{8} + 8945x^{6} + 145432x^{4} + 614160x^{2} + 20736 $ x^10 + 170x^8 + 8945x^6 + 145432x^4 + 614160x^2 + 20736
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.5
Root			$-0.184491i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.4.f.e.129.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.18449 q^{3} -18.9843i q^{5} -9.32401i q^{7} -25.5970 q^{9} +O(q^{10})$	$q-1.18449 q^{3} -18.9843i q^{5} -9.32401i q^{7} -25.5970 q^{9} +39.8107i q^{11} +(-45.2865 - 12.0885i) q^{13} +22.4867i q^{15} +92.6362 q^{17} +128.604i q^{19} +11.0442i q^{21} -158.981 q^{23} -235.403 q^{25} +62.3007 q^{27} -126.583 q^{29} -189.472i q^{31} -47.1555i q^{33} -177.010 q^{35} -53.7542i q^{37} +(53.6415 + 14.3187i) q^{39} +136.223i q^{41} -518.457 q^{43} +485.940i q^{45} -309.742i q^{47} +256.063 q^{49} -109.727 q^{51} -149.390 q^{53} +755.778 q^{55} -152.331i q^{57} -74.3413i q^{59} +99.0118 q^{61} +238.667i q^{63} +(-229.491 + 859.732i) q^{65} -435.290i q^{67} +188.312 q^{69} +827.267i q^{71} -981.880i q^{73} +278.833 q^{75} +371.196 q^{77} -299.012 q^{79} +617.324 q^{81} -169.626i q^{83} -1758.63i q^{85} +149.937 q^{87} -265.124i q^{89} +(-112.713 + 422.252i) q^{91} +224.428i q^{93} +2441.46 q^{95} -88.9929i q^{97} -1019.03i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{3} + 72 q^{9}+O(q^{10}) $ 10 * q - 6 * q^3 + 72 * q^9	$ 10 q - 6 q^{3} + 72 q^{9} - 24 q^{13} + 58 q^{17} - 180 q^{23} + 28 q^{25} - 354 q^{27} - 392 q^{29} - 154 q^{35} + 532 q^{39} + 234 q^{43} - 128 q^{49} + 510 q^{51} - 1244 q^{53} + 576 q^{55} - 56 q^{61} + 566 q^{65} + 1748 q^{69} - 472 q^{75} - 304 q^{77} - 1908 q^{79} + 2282 q^{81} + 64 q^{87} - 582 q^{91} + 2340 q^{95}+O(q^{100}) $ 10 * q - 6 * q^3 + 72 * q^9 - 24 * q^13 + 58 * q^17 - 180 * q^23 + 28 * q^25 - 354 * q^27 - 392 * q^29 - 154 * q^35 + 532 * q^39 + 234 * q^43 - 128 * q^49 + 510 * q^51 - 1244 * q^53 + 576 * q^55 - 56 * q^61 + 566 * q^65 + 1748 * q^69 - 472 * q^75 - 304 * q^77 - 1908 * q^79 + 2282 * q^81 + 64 * q^87 - 582 * q^91 + 2340 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−1.18449			−0.227955			−0.113978	−	0.993483i	$-0.536359\pi$
$3$	−1.18449			−0.227955			−0.113978	+	0.993483i	$0.536359\pi$
$4$		0			0
$5$		−	18.9843i		−	1.69801i	−0.528388	−	0.849003i	$-0.677203\pi$
$5$		−	18.9843i		−	1.69801i	0.528388	−	0.849003i	$-0.322797\pi$
$6$		0			0
$7$		−	9.32401i		−	0.503449i	−0.967799	−	0.251725i	$-0.919002\pi$
$7$		−	9.32401i		−	0.503449i	0.967799	−	0.251725i	$-0.0809977\pi$
$8$		0			0
$9$	−25.5970			−0.948036
$10$		0			0
$11$			39.8107i			1.09122i	0.838040	+	0.545609i	$0.183701\pi$
$11$			39.8107i			1.09122i	−0.838040	+	0.545609i	$0.816299\pi$
$12$		0			0
$13$	−45.2865	−	12.0885i	−0.966171	−	0.257903i
$13$	−45.2865	−	12.0885i	−0.966171	−	0.257903i
$14$		0			0
$15$			22.4867i			0.387070i
$16$		0			0
$17$	92.6362			1.32162			0.660811	−	0.750552i	$-0.270212\pi$
$17$	92.6362			1.32162			0.660811	+	0.750552i	$0.270212\pi$
$18$		0			0
$19$			128.604i			1.55283i	0.630219	+	0.776417i	$0.282965\pi$
$19$			128.604i			1.55283i	−0.630219	+	0.776417i	$0.717035\pi$
$20$		0			0
$21$			11.0442i			0.114764i
$22$		0			0
$23$	−158.981			−1.44130			−0.720650	−	0.693299i	$-0.756156\pi$
$23$	−158.981			−1.44130			−0.720650	+	0.693299i	$0.756156\pi$
$24$		0			0
$25$	−235.403			−1.88323
$26$		0			0
$27$	62.3007			0.444065
$28$		0			0
$29$	−126.583			−0.810550			−0.405275	−	0.914195i	$-0.632824\pi$
$29$	−126.583			−0.810550			−0.405275	+	0.914195i	$0.632824\pi$
$30$		0			0
$31$		−	189.472i		−	1.09775i	−0.835906	−	0.548873i	$-0.815057\pi$
$31$		−	189.472i		−	1.09775i	0.835906	−	0.548873i	$-0.184943\pi$
$32$		0			0
$33$		−	47.1555i		−	0.248749i
$34$		0			0
$35$	−177.010			−0.854860
$36$		0			0
$37$		−	53.7542i		−	0.238842i	−0.992844	−	0.119421i	$-0.961896\pi$
$37$		−	53.7542i		−	0.238842i	0.992844	−	0.119421i	$-0.0381038\pi$
$38$		0			0
$39$	53.6415	+	14.3187i	0.220244	+	0.0587904i
$40$		0			0
$41$			136.223i			0.518889i	0.965758	+	0.259445i	$0.0835395\pi$
$41$			136.223i			0.518889i	−0.965758	+	0.259445i	$0.916460\pi$
$42$		0			0
$43$	−518.457			−1.83870			−0.919349	−	0.393443i	$-0.871284\pi$
$43$	−518.457			−1.83870			−0.919349	+	0.393443i	$0.871284\pi$
$44$		0			0
$45$			485.940i			1.60977i
$46$		0			0
$47$		−	309.742i		−	0.961288i	−0.876916	−	0.480644i	$-0.840403\pi$
$47$		−	309.742i		−	0.961288i	0.876916	−	0.480644i	$-0.159597\pi$
$48$		0			0
$49$	256.063			0.746539
$50$		0			0
$51$	−109.727			−0.301271
$52$		0			0
$53$	−149.390			−0.387176			−0.193588	−	0.981083i	$-0.562012\pi$
$53$	−149.390			−0.387176			−0.193588	+	0.981083i	$0.562012\pi$
$54$		0			0
$55$	755.778			1.85289
$56$		0			0
$57$		−	152.331i		−	0.353977i
$58$		0			0
$59$		−	74.3413i		−	0.164041i	−0.996631	−	0.0820204i	$-0.973863\pi$
$59$		−	74.3413i		−	0.164041i	0.996631	−	0.0820204i	$-0.0261373\pi$
$60$		0			0
$61$	99.0118			0.207822			0.103911	−	0.994587i	$-0.466864\pi$
$61$	99.0118			0.207822			0.103911	+	0.994587i	$0.466864\pi$
$62$		0			0
$63$			238.667i			0.477288i
$64$		0			0
$65$	−229.491	+	859.732i	−0.437921	+	1.64056i
$66$		0			0
$67$		−	435.290i		−	0.793718i	−0.917880	−	0.396859i	$-0.870100\pi$
$67$		−	435.290i		−	0.793718i	0.917880	−	0.396859i	$-0.129900\pi$
$68$		0			0
$69$	188.312			0.328552
$70$		0			0
$71$			827.267i			1.38280i	0.722474	+	0.691398i	$0.243005\pi$
$71$			827.267i			1.38280i	−0.722474	+	0.691398i	$0.756995\pi$
$72$		0			0
$73$		−	981.880i		−	1.57425i	−0.616793	−	0.787126i	$-0.711568\pi$
$73$		−	981.880i		−	1.57425i	0.616793	−	0.787126i	$-0.288432\pi$
$74$		0			0
$75$	278.833			0.429291
$76$		0			0
$77$	371.196			0.549372
$78$		0			0
$79$	−299.012			−0.425841			−0.212920	−	0.977070i	$-0.568298\pi$
$79$	−299.012			−0.425841			−0.212920	+	0.977070i	$0.568298\pi$
$80$		0			0
$81$	617.324			0.846809
$82$		0			0
$83$		−	169.626i		−	0.224324i	−0.993690	−	0.112162i	$-0.964222\pi$
$83$		−	169.626i		−	0.224324i	0.993690	−	0.112162i	$-0.0357776\pi$
$84$		0			0
$85$		−	1758.63i		−	2.24412i
$86$		0			0
$87$	149.937			0.184769
$88$		0			0
$89$		−	265.124i		−	0.315765i	−0.987458	−	0.157883i	$-0.949533\pi$
$89$		−	265.124i		−	0.315765i	0.987458	−	0.157883i	$-0.0504668\pi$
$90$		0			0
$91$	−112.713	+	422.252i	−0.129841	+	0.486418i
$92$		0			0
$93$			224.428i			0.250237i
$94$		0			0
$95$	2441.46			2.63672
$96$		0			0
$97$		−	88.9929i		−	0.0931532i	−0.998915	−	0.0465766i	$-0.985169\pi$
$97$		−	88.9929i		−	0.0931532i	0.998915	−	0.0465766i	$-0.0148312\pi$
$98$		0			0
$99$		−	1019.03i		−	1.03451i
$100$		0			0
$101$	−456.498			−0.449735			−0.224868	−	0.974389i	$-0.572195\pi$
$101$	−456.498			−0.449735			−0.224868	+	0.974389i	$0.572195\pi$
$102$		0			0
$103$	663.161			0.634400			0.317200	−	0.948359i	$-0.397257\pi$
$103$	663.161			0.634400			0.317200	+	0.948359i	$0.397257\pi$
$104$		0			0
$105$	209.666			0.194870
$106$		0			0
$107$	−1567.99			−1.41667			−0.708334	−	0.705877i	$-0.750553\pi$
$107$	−1567.99			−1.41667			−0.708334	+	0.705877i	$0.750553\pi$
$108$		0			0
$109$		−	108.767i		−	0.0955777i	−0.998857	−	0.0477889i	$-0.984783\pi$
$109$		−	108.767i		−	0.0955777i	0.998857	−	0.0477889i	$-0.0152175\pi$
$110$		0			0
$111$			63.6714i			0.0544453i
$112$		0			0
$113$	−200.018			−0.166514			−0.0832569	−	0.996528i	$-0.526532\pi$
$113$	−200.018			−0.166514			−0.0832569	+	0.996528i	$0.526532\pi$
$114$		0			0
$115$			3018.15i			2.44734i
$116$		0			0
$117$	1159.20	+	309.429i	0.915965	+	0.244502i
$118$		0			0
$119$		−	863.741i		−	0.665370i
$120$		0			0
$121$	−253.894			−0.190755
$122$		0			0
$123$		−	161.355i		−	0.118284i
$124$		0			0
$125$			2095.93i			1.49972i
$126$		0			0
$127$	797.353			0.557115			0.278558	−	0.960419i	$-0.410144\pi$
$127$	797.353			0.557115			0.278558	+	0.960419i	$0.410144\pi$
$128$		0			0
$129$	614.108			0.419141
$130$		0			0
$131$	−1438.36			−0.959312			−0.479656	−	0.877457i	$-0.659239\pi$
$131$	−1438.36			−0.959312			−0.479656	+	0.877457i	$0.659239\pi$
$132$		0			0
$133$	1199.11			0.781773
$134$		0			0
$135$		−	1182.73i		−	0.754026i
$136$		0			0
$137$		−	1078.50i		−	0.672575i	−0.941759	−	0.336287i	$-0.890829\pi$
$137$		−	1078.50i		−	0.672575i	0.941759	−	0.336287i	$-0.109171\pi$
$138$		0			0
$139$	1390.39			0.848427			0.424213	−	0.905562i	$-0.360551\pi$
$139$	1390.39			0.848427			0.424213	+	0.905562i	$0.360551\pi$
$140$		0			0
$141$			366.887i			0.219131i
$142$		0			0
$143$	481.251	−	1802.89i	0.281428	−	1.05430i
$144$		0			0
$145$			2403.10i			1.37632i
$146$		0			0
$147$	−303.304			−0.170178
$148$		0			0
$149$		−	1858.02i		−	1.02157i	−0.859707	−	0.510787i	$-0.829354\pi$
$149$		−	1858.02i		−	1.02157i	0.859707	−	0.510787i	$-0.170646\pi$
$150$		0			0
$151$		−	2188.59i		−	1.17951i	−0.807584	−	0.589753i	$-0.799225\pi$
$151$		−	2188.59i		−	1.17951i	0.807584	−	0.589753i	$-0.200775\pi$
$152$		0			0
$153$	−2371.21			−1.25295
$154$		0			0
$155$	−3596.99			−1.86398
$156$		0			0
$157$	−2290.52			−1.16435			−0.582176	−	0.813063i	$-0.697798\pi$
$157$	−2290.52			−1.16435			−0.582176	+	0.813063i	$0.697798\pi$
$158$		0			0
$159$	176.951			0.0882588
$160$		0			0
$161$			1482.34i			0.725621i
$162$		0			0
$163$			2257.67i			1.08487i	0.840097	+	0.542436i	$0.182498\pi$
$163$			2257.67i			1.08487i	−0.840097	+	0.542436i	$0.817502\pi$
$164$		0			0
$165$	−895.213			−0.422377
$166$		0			0
$167$		−	1181.49i		−	0.547465i	−0.961806	−	0.273732i	$-0.911742\pi$
$167$		−	1181.49i		−	0.547465i	0.961806	−	0.273732i	$-0.0882583\pi$
$168$		0			0
$169$	1904.74	+	1094.89i	0.866972	+	0.498357i
$170$		0			0
$171$		−	3291.88i		−	1.47214i
$172$		0			0
$173$	428.168			0.188168			0.0940839	−	0.995564i	$-0.470008\pi$
$173$	428.168			0.188168			0.0940839	+	0.995564i	$0.470008\pi$
$174$		0			0
$175$			2194.90i			0.948109i
$176$		0			0
$177$			88.0566i			0.0373940i
$178$		0			0
$179$	−2575.03			−1.07523			−0.537616	−	0.843190i	$-0.680675\pi$
$179$	−2575.03			−1.07523			−0.537616	+	0.843190i	$0.680675\pi$
$180$		0			0
$181$	−1724.04			−0.707993			−0.353997	−	0.935247i	$-0.615178\pi$
$181$	−1724.04			−0.707993			−0.353997	+	0.935247i	$0.615178\pi$
$182$		0			0
$183$	−117.279			−0.0473742
$184$		0			0
$185$	−1020.49			−0.405555
$186$		0			0
$187$			3687.92i			1.44218i
$188$		0			0
$189$		−	580.892i		−	0.223564i
$190$		0			0
$191$	−4437.81			−1.68120			−0.840599	−	0.541659i	$-0.817797\pi$
$191$	−4437.81			−1.68120			−0.840599	+	0.541659i	$0.817797\pi$
$192$		0			0
$193$		−	5112.76i		−	1.90686i	−0.301608	−	0.953432i	$-0.597523\pi$
$193$		−	5112.76i		−	1.90686i	0.301608	−	0.953432i	$-0.402477\pi$
$194$		0			0
$195$	271.830	−	1018.35i	0.0998265	−	0.373975i
$196$		0			0
$197$			3586.24i			1.29700i	0.761214	+	0.648500i	$0.224604\pi$
$197$			3586.24i			1.29700i	−0.761214	+	0.648500i	$0.775396\pi$
$198$		0			0
$199$	−647.142			−0.230526			−0.115263	−	0.993335i	$-0.536771\pi$
$199$	−647.142			−0.230526			−0.115263	+	0.993335i	$0.536771\pi$
$200$		0			0
$201$			515.597i			0.180932i
$202$		0			0
$203$			1180.27i			0.408071i
$204$		0			0
$205$	2586.10			0.881078
$206$		0			0
$207$	4069.44			1.36640
$208$		0			0
$209$	−5119.83			−1.69448
$210$		0			0
$211$	4851.47			1.58289			0.791443	−	0.611243i	$-0.209330\pi$
$211$	4851.47			1.58289			0.791443	+	0.611243i	$0.209330\pi$
$212$		0			0
$213$		−	979.891i		−	0.315216i
$214$		0			0
$215$			9842.54i			3.12212i
$216$		0			0
$217$	−1766.64			−0.552659
$218$		0			0
$219$			1163.03i			0.358859i
$220$		0			0
$221$	−4195.17	−	1119.83i	−1.27691	−	0.340851i
$222$		0			0
$223$			2629.48i			0.789609i	0.918765	+	0.394804i	$0.129188\pi$
$223$			2629.48i			0.789609i	−0.918765	+	0.394804i	$0.870812\pi$
$224$		0			0
$225$	6025.61			1.78537
$226$		0			0
$227$		−	5158.70i		−	1.50835i	−0.656676	−	0.754173i	$-0.728038\pi$
$227$		−	5158.70i		−	1.50835i	0.656676	−	0.754173i	$-0.271962\pi$
$228$		0			0
$229$			4191.85i			1.20963i	0.796366	+	0.604815i	$0.206753\pi$
$229$			4191.85i			1.20963i	−0.796366	+	0.604815i	$0.793247\pi$
$230$		0			0
$231$	−439.678			−0.125232
$232$		0			0
$233$	−2358.40			−0.663108			−0.331554	−	0.943436i	$-0.607573\pi$
$233$	−2358.40			−0.663108			−0.331554	+	0.943436i	$0.607573\pi$
$234$		0			0
$235$	−5880.23			−1.63227
$236$		0			0
$237$	354.176			0.0970727
$238$		0			0
$239$		−	1647.39i		−	0.445862i	−0.974834	−	0.222931i	$-0.928438\pi$
$239$		−	1647.39i		−	0.445862i	0.974834	−	0.222931i	$-0.0715625\pi$
$240$		0			0
$241$		−	4164.11i		−	1.11300i	−0.830846	−	0.556502i	$-0.812143\pi$
$241$		−	4164.11i		−	1.11300i	0.830846	−	0.556502i	$-0.187857\pi$
$242$		0			0
$243$	−2413.33			−0.637100
$244$		0			0
$245$		−	4861.17i		−	1.26763i
$246$		0			0
$247$	1554.63	−	5824.04i	0.400481	−	1.50030i
$248$		0			0
$249$			200.921i			0.0511359i
$250$		0			0
$251$	3117.04			0.783848			0.391924	−	0.919998i	$-0.371810\pi$
$251$	3117.04			0.783848			0.391924	+	0.919998i	$0.371810\pi$
$252$		0			0
$253$		−	6329.16i		−	1.57277i
$254$		0			0
$255$			2083.08i			0.511560i
$256$		0			0
$257$	6224.92			1.51089			0.755447	−	0.655210i	$-0.227420\pi$
$257$	6224.92			1.51089			0.755447	+	0.655210i	$0.227420\pi$
$258$		0			0
$259$	−501.205			−0.120245
$260$		0			0
$261$	3240.16			0.768431
$262$		0			0
$263$	3214.69			0.753713			0.376857	−	0.926272i	$-0.377005\pi$
$263$	3214.69			0.753713			0.376857	+	0.926272i	$0.377005\pi$
$264$		0			0
$265$			2836.06i			0.657427i
$266$		0			0
$267$			314.037i			0.0719805i
$268$		0			0
$269$	−3098.88			−0.702387			−0.351193	−	0.936303i	$-0.614224\pi$
$269$	−3098.88			−0.702387			−0.351193	+	0.936303i	$0.614224\pi$
$270$		0			0
$271$		−	4688.31i		−	1.05090i	−0.850824	−	0.525451i	$-0.823897\pi$
$271$		−	4688.31i		−	1.05090i	0.850824	−	0.525451i	$-0.176103\pi$
$272$		0			0
$273$	133.508	−	500.154i	0.0295980	−	0.110882i
$274$		0			0
$275$		−	9371.57i		−	2.05501i
$276$		0			0
$277$	5680.02			1.23206			0.616028	−	0.787724i	$-0.288741\pi$
$277$	5680.02			1.23206			0.616028	+	0.787724i	$0.288741\pi$
$278$		0			0
$279$			4849.90i			1.04070i
$280$		0			0
$281$			7674.08i			1.62917i	0.580044	+	0.814585i	$0.303036\pi$
$281$			7674.08i			1.62917i	−0.580044	+	0.814585i	$0.696964\pi$
$282$		0			0
$283$	6647.25			1.39625			0.698124	−	0.715977i	$-0.254019\pi$
$283$	6647.25			1.39625			0.698124	+	0.715977i	$0.254019\pi$
$284$		0			0
$285$	−2891.89			−0.601055
$286$		0			0
$287$	1270.15			0.261235
$288$		0			0
$289$	3668.47			0.746686
$290$		0			0
$291$			105.411i			0.0212348i
$292$		0			0
$293$			4732.11i			0.943526i	0.881725	+	0.471763i	$0.156382\pi$
$293$			4732.11i			0.943526i	−0.881725	+	0.471763i	$0.843618\pi$
$294$		0			0
$295$	−1411.32			−0.278542
$296$		0			0
$297$			2480.23i			0.484572i
$298$		0			0
$299$	7199.71	+	1921.84i	1.39254	+	0.371716i
$300$		0			0
$301$			4834.10i			0.925691i
$302$		0			0
$303$	540.718			0.102520
$304$		0			0
$305$		−	1879.67i		−	0.352884i
$306$		0			0
$307$		−	58.9492i		−	0.0109590i	−0.999985	−	0.00547949i	$-0.998256\pi$
$307$		−	58.9492i		−	0.0109590i	0.999985	−	0.00547949i	$-0.00174419\pi$
$308$		0			0
$309$	−785.509			−0.144615
$310$		0			0
$311$	−7906.34			−1.44157			−0.720784	−	0.693160i	$-0.756218\pi$
$311$	−7906.34			−1.44157			−0.720784	+	0.693160i	$0.756218\pi$
$312$		0			0
$313$	2090.15			0.377452			0.188726	−	0.982030i	$-0.439564\pi$
$313$	2090.15			0.377452			0.188726	+	0.982030i	$0.439564\pi$
$314$		0			0
$315$	4530.91			0.810438
$316$		0			0
$317$		−	1439.70i		−	0.255084i	−0.991833	−	0.127542i	$-0.959291\pi$
$317$		−	1439.70i		−	0.255084i	0.991833	−	0.127542i	$-0.0407088\pi$
$318$		0			0
$319$		−	5039.38i		−	0.884486i
$320$		0			0
$321$	1857.27			0.322937
$322$		0			0
$323$			11913.4i			2.05226i
$324$		0			0
$325$	10660.6	+	2845.67i	1.81952	+	0.485690i
$326$		0			0
$327$			128.833i			0.0217875i
$328$		0			0
$329$	−2888.04			−0.483960
$330$		0			0
$331$		−	8721.66i		−	1.44829i	−0.689645	−	0.724147i	$-0.742234\pi$
$331$		−	8721.66i		−	1.44829i	0.689645	−	0.724147i	$-0.257766\pi$
$332$		0			0
$333$			1375.95i			0.226431i
$334$		0			0
$335$	−8263.67			−1.34774
$336$		0			0
$337$	−1270.70			−0.205398			−0.102699	−	0.994712i	$-0.532748\pi$
$337$	−1270.70			−0.205398			−0.102699	+	0.994712i	$0.532748\pi$
$338$		0			0
$339$	236.919			0.0379577
$340$		0			0
$341$	7543.01			1.19788
$342$		0			0
$343$		−	5585.67i		−	0.879294i
$344$		0			0
$345$		−	3574.97i		−	0.557883i
$346$		0			0
$347$	−4211.68			−0.651570			−0.325785	−	0.945444i	$-0.605628\pi$
$347$	−4211.68			−0.651570			−0.325785	+	0.945444i	$0.605628\pi$
$348$		0			0
$349$		−	3314.57i		−	0.508380i	−0.967154	−	0.254190i	$-0.918191\pi$
$349$		−	3314.57i		−	0.508380i	0.967154	−	0.254190i	$-0.0818089\pi$
$350$		0			0
$351$	−2821.38	−	753.120i	−0.429043	−	0.114526i
$352$		0			0
$353$		−	6919.86i		−	1.04336i	−0.853140	−	0.521681i	$-0.825305\pi$
$353$		−	6919.86i		−	1.04336i	0.853140	−	0.521681i	$-0.174695\pi$
$354$		0			0
$355$	15705.1			2.34800
$356$		0			0
$357$			1023.09i			0.151675i
$358$		0			0
$359$		−	2835.92i		−	0.416920i	−0.978031	−	0.208460i	$-0.933155\pi$
$359$		−	2835.92i		−	0.416920i	0.978031	−	0.208460i	$-0.0668451\pi$
$360$		0			0
$361$	−9680.06			−1.41129
$362$		0			0
$363$	300.736			0.0434835
$364$		0			0
$365$	−18640.3			−2.67309
$366$		0			0
$367$	−1961.20			−0.278947			−0.139474	−	0.990226i	$-0.544541\pi$
$367$	−1961.20			−0.278947			−0.139474	+	0.990226i	$0.544541\pi$
$368$		0			0
$369$		−	3486.90i		−	0.491926i
$370$		0			0
$371$			1392.91i			0.194923i
$372$		0			0
$373$	−6879.92			−0.955037			−0.477518	−	0.878622i	$-0.658464\pi$
$373$	−6879.92			−0.955037			−0.477518	+	0.878622i	$0.658464\pi$
$374$		0			0
$375$		−	2482.61i		−	0.341870i
$376$		0			0
$377$	5732.53	+	1530.20i	0.783130	+	0.209044i
$378$		0			0
$379$		−	84.6536i		−	0.0114732i	−0.999984	−	0.00573662i	$-0.998174\pi$
$379$		−	84.6536i		−	0.0114732i	0.999984	−	0.00573662i	$-0.00182603\pi$
$380$		0			0
$381$	−944.458			−0.126997
$382$		0			0
$383$			11882.7i			1.58533i	0.609660	+	0.792663i	$0.291306\pi$
$383$			11882.7i			1.58533i	−0.609660	+	0.792663i	$0.708694\pi$
$384$		0			0
$385$		−	7046.89i		−	0.932838i
$386$		0			0
$387$	13270.9			1.74315
$388$		0			0
$389$	11618.1			1.51429			0.757146	−	0.653246i	$-0.226593\pi$
$389$	11618.1			1.51429			0.757146	+	0.653246i	$0.226593\pi$
$390$		0			0
$391$	−14727.4			−1.90485
$392$		0			0
$393$	1703.72			0.218680
$394$		0			0
$395$			5676.52i			0.723080i
$396$		0			0
$397$		−	4257.81i		−	0.538271i	−0.963102	−	0.269135i	$-0.913262\pi$
$397$		−	4257.81i		−	0.538271i	0.963102	−	0.269135i	$-0.0867379\pi$
$398$		0			0
$399$	−1420.33			−0.178209
$400$		0			0
$401$			6695.23i			0.833776i	0.908958	+	0.416888i	$0.136879\pi$
$401$			6695.23i			0.833776i	−0.908958	+	0.416888i	$0.863121\pi$
$402$		0			0
$403$	−2290.42	+	8580.51i	−0.283112	+	1.06061i
$404$		0			0
$405$		−	11719.5i		−	1.43789i
$406$		0			0
$407$	2140.00			0.260628
$408$		0			0
$409$			3900.96i			0.471614i	0.971800	+	0.235807i	$0.0757734\pi$
$409$			3900.96i			0.471614i	−0.971800	+	0.235807i	$0.924227\pi$
$410$		0			0
$411$			1277.48i			0.153317i
$412$		0			0
$413$	−693.159			−0.0825863
$414$		0			0
$415$	−3220.24			−0.380904
$416$		0			0
$417$	−1646.90			−0.193403
$418$		0			0
$419$	−14987.6			−1.74748			−0.873739	−	0.486396i	$-0.838311\pi$
$419$	−14987.6			−1.74748			−0.873739	+	0.486396i	$0.838311\pi$
$420$		0			0
$421$			14239.7i			1.64846i	0.566258	+	0.824228i	$0.308391\pi$
$421$			14239.7i			1.64846i	−0.566258	+	0.824228i	$0.691609\pi$
$422$		0			0
$423$			7928.46i			0.911336i
$424$		0			0
$425$	−21806.9			−2.48891
$426$		0			0
$427$		−	923.188i		−	0.104628i
$428$		0			0
$429$	−570.038	+	2135.51i	−0.0641531	+	0.240334i
$430$		0			0
$431$			8463.65i			0.945893i	0.881091	+	0.472946i	$0.156810\pi$
$431$			8463.65i			0.945893i	−0.881091	+	0.472946i	$0.843190\pi$
$432$		0			0
$433$	11335.2			1.25804			0.629022	−	0.777387i	$-0.283455\pi$
$433$	11335.2			1.25804			0.629022	+	0.777387i	$0.283455\pi$
$434$		0			0
$435$		−	2846.45i		−	0.313740i
$436$		0			0
$437$		−	20445.7i		−	2.23810i
$438$		0			0
$439$	1306.84			0.142077			0.0710386	−	0.997474i	$-0.477369\pi$
$439$	1306.84			0.142077			0.0710386	+	0.997474i	$0.477369\pi$
$440$		0			0
$441$	−6554.43			−0.707746
$442$		0			0
$443$	−2044.77			−0.219300			−0.109650	−	0.993970i	$-0.534973\pi$
$443$	−2044.77			−0.219300			−0.109650	+	0.993970i	$0.534973\pi$
$444$		0			0
$445$	−5033.20			−0.536172
$446$		0			0
$447$			2200.80i			0.232873i
$448$		0			0
$449$			7906.06i			0.830980i	0.909598	+	0.415490i	$0.136390\pi$
$449$			7906.06i			0.830980i	−0.909598	+	0.415490i	$0.863610\pi$
$450$		0			0
$451$	−5423.14			−0.566221
$452$		0			0
$453$			2592.37i			0.268875i
$454$		0			0
$455$	8016.15	+	2139.78i	0.825941	+	0.220471i
$456$		0			0
$457$			14420.3i			1.47605i	0.674773	+	0.738025i	$0.264241\pi$
$457$			14420.3i			1.47605i	−0.674773	+	0.738025i	$0.735759\pi$
$458$		0			0
$459$	5771.30			0.586887
$460$		0			0
$461$			17257.0i			1.74347i	0.489978	+	0.871735i	$0.337005\pi$
$461$			17257.0i			1.74347i	−0.489978	+	0.871735i	$0.662995\pi$
$462$		0			0
$463$			2186.87i			0.219508i	0.993959	+	0.109754i	$0.0350063\pi$
$463$			2186.87i			0.219508i	−0.993959	+	0.109754i	$0.964994\pi$
$464$		0			0
$465$	4260.60			0.424904
$466$		0			0
$467$	2802.47			0.277694			0.138847	−	0.990314i	$-0.455660\pi$
$467$	2802.47			0.277694			0.138847	+	0.990314i	$0.455660\pi$
$468$		0			0
$469$	−4058.65			−0.399597
$470$		0			0
$471$	2713.10			0.265420
$472$		0			0
$473$		−	20640.2i		−	2.00642i
$474$		0			0
$475$		−	30273.9i		−	2.92434i
$476$		0			0
$477$	3823.94			0.367057
$478$		0			0
$479$		−	7337.58i		−	0.699921i	−0.936764	−	0.349961i	$-0.886195\pi$
$479$		−	7337.58i		−	0.699921i	0.936764	−	0.349961i	$-0.113805\pi$
$480$		0			0
$481$	−649.807	+	2434.34i	−0.0615980	+	0.230762i
$482$		0			0
$483$		−	1755.82i		−	0.165409i
$484$		0			0
$485$	−1689.47			−0.158175
$486$		0			0
$487$			2948.76i			0.274376i	0.990545	+	0.137188i	$0.0438064\pi$
$487$			2948.76i			0.274376i	−0.990545	+	0.137188i	$0.956194\pi$
$488$		0			0
$489$		−	2674.19i		−	0.247302i
$490$		0			0
$491$	−7307.22			−0.671630			−0.335815	−	0.941928i	$-0.609012\pi$
$491$	−7307.22			−0.671630			−0.335815	+	0.941928i	$0.609012\pi$
$492$		0			0
$493$	−11726.2			−1.07124
$494$		0			0
$495$	−19345.6			−1.75661
$496$		0			0
$497$	7713.45			0.696168
$498$		0			0
$499$		−	6102.36i		−	0.547453i	−0.961808	−	0.273727i	$-0.911744\pi$
$499$		−	6102.36i		−	0.547453i	0.961808	−	0.273727i	$-0.0882563\pi$
$500$		0			0
$501$			1399.47i			0.124798i
$502$		0			0
$503$	−8460.72			−0.749990			−0.374995	−	0.927027i	$-0.622356\pi$
$503$	−8460.72			−0.749990			−0.374995	+	0.927027i	$0.622356\pi$
$504$		0			0
$505$			8666.29i			0.763653i
$506$		0			0
$507$	−2256.14	−	1296.89i	−0.197631	−	0.113603i
$508$		0			0
$509$		−	4715.21i		−	0.410605i	−0.978699	−	0.205302i	$-0.934182\pi$
$509$		−	4715.21i		−	0.410605i	0.978699	−	0.205302i	$-0.0658178\pi$
$510$		0			0
$511$	−9155.06			−0.792556
$512$		0			0
$513$			8012.13i			0.689560i
$514$		0			0
$515$		−	12589.6i		−	1.07722i
$516$		0			0
$517$	12331.1			1.04897
$518$		0			0
$519$	−507.161			−0.0428939
$520$		0			0
$521$	−13951.0			−1.17313			−0.586567	−	0.809901i	$-0.699521\pi$
$521$	−13951.0			−1.17313			−0.586567	+	0.809901i	$0.699521\pi$
$522$		0			0
$523$	10809.0			0.903720			0.451860	−	0.892089i	$-0.350761\pi$
$523$	10809.0			0.903720			0.451860	+	0.892089i	$0.350761\pi$
$524$		0			0
$525$		−	2599.84i		−	0.216126i
$526$		0			0
$527$		−	17551.9i		−	1.45081i
$528$		0			0
$529$	13108.0			1.07734
$530$		0			0
$531$			1902.91i			0.155517i
$532$		0			0
$533$	1646.73	−	6169.07i	0.133823	−	0.501336i
$534$		0			0
$535$			29767.2i			2.40551i
$536$		0			0
$537$	3050.10			0.245105
$538$		0			0
$539$			10194.0i			0.814636i
$540$		0			0
$541$			3077.38i			0.244560i	0.992496	+	0.122280i	$0.0390206\pi$
$541$			3077.38i			0.244560i	−0.992496	+	0.122280i	$0.960979\pi$
$542$		0			0
$543$	2042.11			0.161391
$544$		0			0
$545$	−2064.86			−0.162292
$546$		0			0
$547$	2866.88			0.224093			0.112046	−	0.993703i	$-0.464259\pi$
$547$	2866.88			0.224093			0.112046	+	0.993703i	$0.464259\pi$
$548$		0			0
$549$	−2534.40			−0.197023
$550$		0			0
$551$		−	16279.2i		−	1.25865i
$552$		0			0
$553$			2787.99i			0.214389i
$554$		0			0
$555$	1208.76			0.0924484
$556$		0			0
$557$			21840.2i			1.66140i	0.556721	+	0.830700i	$0.312059\pi$
$557$			21840.2i			1.66140i	−0.556721	+	0.830700i	$0.687941\pi$
$558$		0			0
$559$	23479.1	+	6267.36i	1.77650	+	0.474206i
$560$		0			0
$561$		−	4368.30i		−	0.328752i
$562$		0			0
$563$	4009.24			0.300123			0.150061	−	0.988677i	$-0.452053\pi$
$563$	4009.24			0.300123			0.150061	+	0.988677i	$0.452053\pi$
$564$		0			0
$565$			3797.19i			0.282742i
$566$		0			0
$567$		−	5755.94i		−	0.426325i
$568$		0			0
$569$	9968.97			0.734483			0.367242	−	0.930126i	$-0.380302\pi$
$569$	9968.97			0.734483			0.367242	+	0.930126i	$0.380302\pi$
$570$		0			0
$571$	1100.67			0.0806681			0.0403340	−	0.999186i	$-0.487158\pi$
$571$	1100.67			0.0806681			0.0403340	+	0.999186i	$0.487158\pi$
$572$		0			0
$573$	5256.55			0.383238
$574$		0			0
$575$	37424.7			2.71429
$576$		0			0
$577$			7330.22i			0.528875i	0.964403	+	0.264438i	$0.0851863\pi$
$577$			7330.22i			0.528875i	−0.964403	+	0.264438i	$0.914814\pi$
$578$		0			0
$579$			6056.02i			0.434680i
$580$		0			0
$581$	−1581.60			−0.112936
$582$		0			0
$583$		−	5947.33i		−	0.422493i
$584$		0			0
$585$	5874.28	−	22006.6i	0.415165	−	1.55531i
$586$		0			0
$587$			13209.0i			0.928776i	0.885632	+	0.464388i	$0.153726\pi$
$587$			13209.0i			0.928776i	−0.885632	+	0.464388i	$0.846274\pi$
$588$		0			0
$589$	24366.9			1.70462
$590$		0			0
$591$		−	4247.87i		−	0.295658i
$592$		0			0
$593$		−	11375.3i		−	0.787739i	−0.919166	−	0.393870i	$-0.871136\pi$
$593$		−	11375.3i		−	0.787739i	0.919166	−	0.393870i	$-0.128864\pi$
$594$		0			0
$595$	−16397.5			−1.12980
$596$		0			0
$597$	766.534			0.0525497
$598$		0			0
$599$	17080.5			1.16509			0.582546	−	0.812798i	$-0.302057\pi$
$599$	17080.5			1.16509			0.582546	+	0.812798i	$0.302057\pi$
$600$		0			0
$601$	−9710.91			−0.659095			−0.329548	−	0.944139i	$-0.606896\pi$
$601$	−9710.91			−0.659095			−0.329548	+	0.944139i	$0.606896\pi$
$602$		0			0
$603$			11142.1i			0.752474i
$604$		0			0
$605$			4820.00i			0.323902i
$606$		0			0
$607$	−17146.1			−1.14652			−0.573260	−	0.819374i	$-0.694321\pi$
$607$	−17146.1			−1.14652			−0.573260	+	0.819374i	$0.694321\pi$
$608$		0			0
$609$		−	1398.01i		−	0.0930220i
$610$		0			0
$611$	−3744.31	+	14027.1i	−0.247919	+	0.928768i
$612$		0			0
$613$		−	13936.4i		−	0.918247i	−0.888372	−	0.459124i	$-0.848164\pi$
$613$		−	13936.4i		−	0.918247i	0.888372	−	0.459124i	$-0.151836\pi$
$614$		0			0
$615$	−3063.21			−0.200846
$616$		0			0
$617$		−	16924.9i		−	1.10433i	−0.833735	−	0.552164i	$-0.813802\pi$
$617$		−	16924.9i		−	1.10433i	0.833735	−	0.552164i	$-0.186198\pi$
$618$		0			0
$619$			25880.4i			1.68048i	0.542211	+	0.840242i	$0.317587\pi$
$619$			25880.4i			1.68048i	−0.542211	+	0.840242i	$0.682413\pi$
$620$		0			0
$621$	−9904.64			−0.640031
$622$		0			0
$623$	−2472.02			−0.158972
$624$		0			0
$625$	10364.3			0.663313
$626$		0			0
$627$	6064.39			0.386266
$628$		0			0
$629$		−	4979.59i		−	0.315659i
$630$		0			0
$631$		−	17122.1i		−	1.08022i	−0.841593	−	0.540112i	$-0.818382\pi$
$631$		−	17122.1i		−	1.08022i	0.841593	−	0.540112i	$-0.181618\pi$
$632$		0			0
$633$	−5746.52			−0.360827
$634$		0			0
$635$		−	15137.2i		−	0.945986i
$636$		0			0
$637$	−11596.2	−	3095.41i	−0.721284	−	0.192535i
$638$		0			0
$639$		−	21175.5i		−	1.31094i
$640$		0			0
$641$	20087.4			1.23776			0.618882	−	0.785484i	$-0.287586\pi$
$641$	20087.4			1.23776			0.618882	+	0.785484i	$0.287586\pi$
$642$		0			0
$643$		−	16479.2i		−	1.01070i	−0.862916	−	0.505348i	$-0.831364\pi$
$643$		−	16479.2i		−	1.01070i	0.862916	−	0.505348i	$-0.168636\pi$
$644$		0			0
$645$		−	11658.4i		−	0.711704i
$646$		0			0
$647$	12913.8			0.784689			0.392345	−	0.919818i	$-0.371664\pi$
$647$	12913.8			0.784689			0.392345	+	0.919818i	$0.371664\pi$
$648$		0			0
$649$	2959.58			0.179004
$650$		0			0
$651$	2092.57			0.125982
$652$		0			0
$653$	13342.2			0.799570			0.399785	−	0.916609i	$-0.369085\pi$
$653$	13342.2			0.799570			0.399785	+	0.916609i	$0.369085\pi$
$654$		0			0
$655$			27306.2i			1.62892i
$656$		0			0
$657$			25133.2i			1.49245i
$658$		0			0
$659$	4506.69			0.266397			0.133199	−	0.991089i	$-0.457475\pi$
$659$	4506.69			0.266397			0.133199	+	0.991089i	$0.457475\pi$
$660$		0			0
$661$		−	24472.1i		−	1.44002i	−0.693963	−	0.720010i	$-0.744137\pi$
$661$		−	24472.1i		−	1.44002i	0.693963	−	0.720010i	$-0.255863\pi$
$662$		0			0
$663$	4969.14	+	1326.43i	0.291079	+	0.0776987i
$664$		0			0
$665$		−	22764.2i		−	1.32746i
$666$		0			0
$667$	20124.4			1.16825
$668$		0			0
$669$		−	3114.59i		−	0.179996i
$670$		0			0
$671$			3941.73i			0.226779i
$672$		0			0
$673$	−19231.8			−1.10153			−0.550767	−	0.834659i	$-0.685665\pi$
$673$	−19231.8			−1.10153			−0.550767	+	0.834659i	$0.685665\pi$
$674$		0			0
$675$	−14665.8			−0.836275
$676$		0			0
$677$	8345.13			0.473751			0.236875	−	0.971540i	$-0.423877\pi$
$677$	8345.13			0.473751			0.236875	+	0.971540i	$0.423877\pi$
$678$		0			0
$679$	−829.771			−0.0468979
$680$		0			0
$681$			6110.43i			0.343836i
$682$		0			0
$683$			14617.2i			0.818906i	0.912331	+	0.409453i	$0.134281\pi$
$683$			14617.2i			0.818906i	−0.912331	+	0.409453i	$0.865719\pi$
$684$		0			0
$685$	−20474.6			−1.14204
$686$		0			0
$687$		−	4965.21i		−	0.275742i
$688$		0			0
$689$	6765.36	+	1805.90i	0.374078	+	0.0998538i
$690$		0			0
$691$			24769.0i			1.36361i	0.731532	+	0.681807i	$0.238805\pi$
$691$			24769.0i			1.36361i	−0.731532	+	0.681807i	$0.761195\pi$
$692$		0			0
$693$	−9501.49			−0.520825
$694$		0			0
$695$		−	26395.6i		−	1.44063i
$696$		0			0
$697$			12619.2i			0.685776i
$698$		0			0
$699$	2793.51			0.151159
$700$		0			0
$701$	−15303.9			−0.824567			−0.412283	−	0.911056i	$-0.635269\pi$
$701$	−15303.9			−0.824567			−0.412283	+	0.911056i	$0.635269\pi$
$702$		0			0
$703$	6913.03			0.370882
$704$		0			0
$705$	6965.08			0.372085
$706$		0			0
$707$			4256.39i			0.226419i
$708$		0			0
$709$			15870.9i			0.840684i	0.907366	+	0.420342i	$0.138090\pi$
$709$			15870.9i			0.840684i	−0.907366	+	0.420342i	$0.861910\pi$
$710$		0			0
$711$	7653.79			0.403713
$712$		0			0
$713$			30122.5i			1.58218i
$714$		0			0
$715$	−34226.6	−	9136.21i	−1.79021	−	0.477867i
$716$		0			0
$717$			1951.32i			0.101637i
$718$		0			0
$719$	4941.77			0.256324			0.128162	−	0.991753i	$-0.459092\pi$
$719$	4941.77			0.256324			0.128162	+	0.991753i	$0.459092\pi$
$720$		0			0
$721$		−	6183.32i		−	0.319388i
$722$		0			0
$723$			4932.35i			0.253715i
$724$		0			0
$725$	29798.2			1.52645
$726$		0			0
$727$	22126.1			1.12876			0.564381	−	0.825514i	$-0.309115\pi$
$727$	22126.1			1.12876			0.564381	+	0.825514i	$0.309115\pi$
$728$		0			0
$729$	−13809.2			−0.701579
$730$		0			0
$731$	−48027.9			−2.43006
$732$		0			0
$733$		−	15333.1i		−	0.772634i	−0.922366	−	0.386317i	$-0.873747\pi$
$733$		−	15333.1i		−	0.772634i	0.922366	−	0.386317i	$-0.126253\pi$
$734$		0			0
$735$			5758.01i			0.288963i
$736$		0			0
$737$	17329.2			0.866119
$738$		0			0
$739$		−	28866.8i		−	1.43692i	−0.695568	−	0.718460i	$-0.744847\pi$
$739$		−	28866.8i		−	1.43692i	0.695568	−	0.718460i	$-0.255153\pi$
$740$		0			0
$741$	−1841.45	+	6898.52i	−0.0912918	+	0.342002i
$742$		0			0
$743$		−	21733.3i		−	1.07311i	−0.843867	−	0.536553i	$-0.819726\pi$
$743$		−	21733.3i		−	1.07311i	0.843867	−	0.536553i	$-0.180274\pi$
$744$		0			0
$745$	−35273.1			−1.73464
$746$		0			0
$747$			4341.92i			0.212668i
$748$		0			0
$749$			14620.0i			0.713220i
$750$		0			0
$751$	9.93637			0.000482801			0.000241400	−	1.00000i	$-0.499923\pi$
$751$	9.93637			0.000482801			0.000241400		1.00000i	$0.499923\pi$
$752$		0			0
$753$	−3692.11			−0.178682
$754$		0			0
$755$	−41548.9			−2.00281
$756$		0			0
$757$	26846.7			1.28898			0.644492	−	0.764611i	$-0.277069\pi$
$757$	26846.7			1.28898			0.644492	+	0.764611i	$0.277069\pi$
$758$		0			0
$759$			7496.83i			0.358521i
$760$		0			0
$761$			7716.02i			0.367550i	0.982968	+	0.183775i	$0.0588318\pi$
$761$			7716.02i			0.367550i	−0.982968	+	0.183775i	$0.941168\pi$
$762$		0			0
$763$	−1014.14			−0.0481185
$764$		0			0
$765$			45015.7i			2.12751i
$766$		0			0
$767$	−898.673	+	3366.66i	−0.0423067	+	0.158491i
$768$		0			0
$769$		−	19702.0i		−	0.923893i	−0.886908	−	0.461946i	$-0.847151\pi$
$769$		−	19702.0i		−	0.923893i	0.886908	−	0.461946i	$-0.152849\pi$
$770$		0			0
$771$	−7373.36			−0.344416
$772$		0			0
$773$			5302.95i			0.246745i	0.992360	+	0.123372i	$0.0393710\pi$
$773$			5302.95i			0.246745i	−0.992360	+	0.123372i	$0.960629\pi$
$774$		0			0
$775$			44602.3i			2.06730i
$776$		0			0
$777$	593.673			0.0274104
$778$		0			0
$779$	−17518.9			−0.805749
$780$		0			0
$781$	−32934.1			−1.50893
$782$		0			0
$783$	−7886.23			−0.359937
$784$		0			0
$785$			43483.8i			1.97708i
$786$		0			0
$787$		−	11129.4i		−	0.504092i	−0.967715	−	0.252046i	$-0.918897\pi$
$787$		−	11129.4i		−	0.504092i	0.967715	−	0.252046i	$-0.0811034\pi$
$788$		0			0
$789$	−3807.78			−0.171813
$790$		0			0
$791$			1864.97i			0.0838313i
$792$		0			0
$793$	−4483.90	−	1196.90i	−0.200792	−	0.0535981i
$794$		0			0
$795$		−	3359.29i		−	0.149864i
$796$		0			0
$797$	−23405.0			−1.04021			−0.520105	−	0.854102i	$-0.674107\pi$
$797$	−23405.0			−1.04021			−0.520105	+	0.854102i	$0.674107\pi$
$798$		0			0
$799$		−	28693.3i		−	1.27046i
$800$		0			0
$801$			6786.38i			0.299357i
$802$		0			0
$803$	39089.4			1.71785
$804$		0			0
$805$	28141.2			1.23211
$806$		0			0
$807$	3670.59			0.160113
$808$		0			0
$809$	−17062.0			−0.741493			−0.370747	−	0.928734i	$-0.620898\pi$
$809$	−17062.0			−0.741493			−0.370747	+	0.928734i	$0.620898\pi$
$810$		0			0
$811$		−	31704.6i		−	1.37275i	−0.727248	−	0.686375i	$-0.759201\pi$
$811$		−	31704.6i		−	1.37275i	0.727248	−	0.686375i	$-0.240799\pi$
$812$		0			0
$813$			5553.26i			0.239559i
$814$		0			0
$815$	42860.2			1.84212
$816$		0			0
$817$		−	66675.8i		−	2.85519i
$818$		0			0
$819$	2885.12	−	10808.4i	0.123094	−	0.461142i
$820$		0			0
$821$			11943.5i			0.507713i	0.967242	+	0.253857i	$0.0816991\pi$
$821$			11943.5i			0.507713i	−0.967242	+	0.253857i	$0.918301\pi$
$822$		0			0
$823$	−21105.8			−0.893928			−0.446964	−	0.894552i	$-0.647495\pi$
$823$	−21105.8			−0.893928			−0.446964	+	0.894552i	$0.647495\pi$
$824$		0			0
$825$			11100.5i			0.468450i
$826$		0			0
$827$		−	2380.24i		−	0.100083i	−0.998747	−	0.0500417i	$-0.984065\pi$
$827$		−	2380.24i		−	0.100083i	0.998747	−	0.0500417i	$-0.0159354\pi$
$828$		0			0
$829$	−47415.1			−1.98648			−0.993242	−	0.116064i	$-0.962972\pi$
$829$	−47415.1			−1.98648			−0.993242	+	0.116064i	$0.962972\pi$
$830$		0			0
$831$	−6727.93			−0.280854
$832$		0			0
$833$	23720.7			0.986643
$834$		0			0
$835$	−22429.8			−0.929599
$836$		0			0
$837$		−	11804.2i		−	0.487471i
$838$		0			0
$839$			10078.7i			0.414725i	0.978264	+	0.207362i	$0.0664880\pi$
$839$			10078.7i			0.414725i	−0.978264	+	0.207362i	$0.933512\pi$
$840$		0			0
$841$	−8365.62			−0.343008
$842$		0			0
$843$		−	9089.88i		−	0.371378i
$844$		0			0
$845$	20785.7	−	36160.1i	0.846213	−	1.47212i
$846$		0			0
$847$			2367.31i			0.0960353i
$848$		0			0
$849$	−7873.61			−0.318282
$850$		0			0
$851$			8545.92i			0.344242i
$852$		0			0
$853$			12683.5i			0.509115i	0.967058	+	0.254558i	$0.0819298\pi$
$853$			12683.5i			0.509115i	−0.967058	+	0.254558i	$0.918070\pi$
$854$		0			0
$855$	−62494.0			−2.49971
$856$		0			0
$857$	16182.3			0.645013			0.322507	−	0.946567i	$-0.395475\pi$
$857$	16182.3			0.645013			0.322507	+	0.946567i	$0.395475\pi$
$858$		0			0
$859$	−3402.48			−0.135147			−0.0675735	−	0.997714i	$-0.521526\pi$
$859$	−3402.48			−0.135147			−0.0675735	+	0.997714i	$0.521526\pi$
$860$		0			0
$861$	−1504.48			−0.0595498
$862$		0			0
$863$			8753.92i			0.345292i	0.984984	+	0.172646i	$0.0552316\pi$
$863$			8753.92i			0.345292i	−0.984984	+	0.172646i	$0.944768\pi$
$864$		0			0
$865$		−	8128.47i		−	0.319510i
$866$		0			0
$867$	−4345.27			−0.170211
$868$		0			0
$869$		−	11903.9i		−	0.464685i
$870$		0			0
$871$	−5261.99	+	19712.8i	−0.204702	+	0.766867i
$872$		0			0
$873$			2277.95i			0.0883127i
$874$		0			0
$875$	19542.4			0.755034
$876$		0			0
$877$		−	35118.0i		−	1.35217i	−0.736825	−	0.676083i	$-0.763676\pi$
$877$		−	35118.0i		−	1.35217i	0.736825	−	0.676083i	$-0.236324\pi$
$878$		0			0
$879$		−	5605.15i		−	0.215082i
$880$		0			0
$881$	−36752.5			−1.40548			−0.702738	−	0.711449i	$-0.748039\pi$
$881$	−36752.5			−1.40548			−0.702738	+	0.711449i	$0.748039\pi$
$882$		0			0
$883$	−25732.6			−0.980716			−0.490358	−	0.871521i	$-0.663134\pi$
$883$	−25732.6			−0.980716			−0.490358	+	0.871521i	$0.663134\pi$
$884$		0			0
$885$	1671.69			0.0634953
$886$		0			0
$887$	36526.1			1.38267			0.691334	−	0.722535i	$-0.257023\pi$
$887$	36526.1			1.38267			0.691334	+	0.722535i	$0.257023\pi$
$888$		0			0
$889$		−	7434.53i		−	0.280479i
$890$		0			0
$891$			24576.1i			0.924053i
$892$		0			0
$893$	39834.2			1.49272
$894$		0			0
$895$			48885.1i			1.82575i
$896$		0			0
$897$	−8527.99	−	2276.40i	−0.317437	−	0.0847346i
$898$		0			0
$899$			23984.0i			0.889779i
$900$		0			0
$901$	−13838.9			−0.511700
$902$		0			0
$903$		−	5725.95i		−	0.211016i
$904$		0			0
$905$			32729.7i			1.20218i
$906$		0			0
$907$	19361.5			0.708805			0.354403	−	0.935093i	$-0.384684\pi$
$907$	19361.5			0.708805			0.354403	+	0.935093i	$0.384684\pi$
$908$		0			0
$909$	11685.0			0.426365
$910$		0			0
$911$	25170.3			0.915398			0.457699	−	0.889107i	$-0.348674\pi$
$911$	25170.3			0.915398			0.457699	+	0.889107i	$0.348674\pi$
$912$		0			0
$913$	6752.95			0.244786
$914$		0			0
$915$			2226.45i			0.0804418i
$916$		0			0
$917$			13411.3i			0.482965i
$918$		0			0
$919$	−8108.11			−0.291036			−0.145518	−	0.989356i	$-0.546485\pi$
$919$	−8108.11			−0.291036			−0.145518	+	0.989356i	$0.546485\pi$
$920$		0			0
$921$			69.8248i			0.00249816i
$922$		0			0
$923$	10000.4	−	37464.0i	0.356628	−	1.33602i
$924$		0			0
$925$			12653.9i			0.449793i
$926$		0			0
$927$	−16974.9			−0.601435
$928$		0			0
$929$		−	11382.7i		−	0.401997i	−0.979592	−	0.200999i	$-0.935581\pi$
$929$		−	11382.7i		−	0.401997i	0.979592	−	0.200999i	$-0.0644187\pi$
$930$		0			0
$931$			32930.8i			1.15925i
$932$		0			0
$933$	9364.99			0.328613
$934$		0			0
$935$	70012.5			2.44883
$936$		0			0
$937$	−676.773			−0.0235957			−0.0117979	−	0.999930i	$-0.503755\pi$
$937$	−676.773			−0.0235957			−0.0117979	+	0.999930i	$0.503755\pi$
$938$		0			0
$939$	−2475.77			−0.0860422
$940$		0			0
$941$		−	49902.5i		−	1.72877i	−0.502827	−	0.864387i	$-0.667707\pi$
$941$		−	49902.5i		−	1.72877i	0.502827	−	0.864387i	$-0.332293\pi$
$942$		0			0
$943$		−	21656.9i		−	0.747875i
$944$		0			0
$945$	−11027.8			−0.379614
$946$		0			0
$947$		−	12526.7i		−	0.429843i	−0.976631	−	0.214922i	$-0.931050\pi$
$947$		−	12526.7i		−	0.429843i	0.976631	−	0.214922i	$-0.0689496\pi$
$948$		0			0
$949$	−11869.4	+	44465.9i	−0.406004	+	1.52100i
$950$		0			0
$951$			1705.31i			0.0581478i
$952$		0			0
$953$	40757.3			1.38537			0.692686	−	0.721239i	$-0.256427\pi$
$953$	40757.3			1.38537			0.692686	+	0.721239i	$0.256427\pi$
$954$		0			0
$955$			84248.7i			2.85468i
$956$		0			0
$957$			5969.10i			0.201623i
$958$		0			0
$959$	−10056.0			−0.338607
$960$		0			0
$961$	−6108.53			−0.205046
$962$		0			0
$963$	40135.9			1.34305
$964$		0			0
$965$	−97062.2			−3.23787
$966$		0			0
$967$		−	20704.8i		−	0.688545i	−0.938870	−	0.344272i	$-0.888126\pi$
$967$		−	20704.8i		−	0.688545i	0.938870	−	0.344272i	$-0.111874\pi$
$968$		0			0
$969$		−	14111.3i		−	0.467824i
$970$		0			0
$971$	−17178.3			−0.567743			−0.283871	−	0.958862i	$-0.591619\pi$
$971$	−17178.3			−0.567743			−0.283871	+	0.958862i	$0.591619\pi$
$972$		0			0
$973$		−	12964.0i		−	0.427140i
$974$		0			0
$975$	−12627.4	−	3370.67i	−0.414769	−	0.110716i
$976$		0			0
$977$			13112.7i			0.429390i	0.976681	+	0.214695i	$0.0688757\pi$
$977$			13112.7i			0.429390i	−0.976681	+	0.214695i	$0.931124\pi$
$978$		0			0
$979$	10554.8			0.344569
$980$		0			0
$981$			2784.10i			0.0906111i
$982$		0			0
$983$		−	44280.0i		−	1.43674i	−0.695663	−	0.718368i	$-0.744889\pi$
$983$		−	44280.0i		−	1.43674i	0.695663	−	0.718368i	$-0.255111\pi$
$984$		0			0
$985$	68082.3			2.20232
$986$		0			0
$987$	3420.86			0.110321
$988$		0			0
$989$	82425.0			2.65011
$990$		0			0
$991$	1420.03			0.0455184			0.0227592	−	0.999741i	$-0.492755\pi$
$991$	1420.03			0.0455184			0.0227592	+	0.999741i	$0.492755\pi$
$992$		0			0
$993$			10330.7i			0.330147i
$994$		0			0
$995$			12285.5i			0.391435i
$996$		0			0
$997$	42998.9			1.36589			0.682944	−	0.730471i	$-0.260699\pi$
$997$	42998.9			1.36589			0.682944	+	0.730471i	$0.260699\pi$
$998$		0			0
$999$		−	3348.92i		−	0.106061i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.f.e.129.5		10
4.3	odd	2		104.4.f.a.25.5	✓	10
8.3	odd	2		832.4.f.k.129.6		10
8.5	even	2		832.4.f.l.129.6		10
12.11	even	2		936.4.c.a.649.10		10
13.12	even	2	inner	208.4.f.e.129.6		10
52.31	even	4		1352.4.a.k.1.3		5
52.47	even	4		1352.4.a.l.1.3		5
52.51	odd	2		104.4.f.a.25.6	yes	10
104.51	odd	2		832.4.f.k.129.5		10
104.77	even	2		832.4.f.l.129.5		10
156.155	even	2		936.4.c.a.649.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.4.f.a.25.5	✓	10	4.3	odd	2
104.4.f.a.25.6	yes	10	52.51	odd	2
208.4.f.e.129.5		10	1.1	even	1	trivial
208.4.f.e.129.6		10	13.12	even	2	inner
832.4.f.k.129.5		10	104.51	odd	2
832.4.f.k.129.6		10	8.3	odd	2
832.4.f.l.129.5		10	104.77	even	2
832.4.f.l.129.6		10	8.5	even	2
936.4.c.a.649.1		10	156.155	even	2
936.4.c.a.649.10		10	12.11	even	2
1352.4.a.k.1.3		5	52.31	even	4
1352.4.a.l.1.3		5	52.47	even	4

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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.18449 q^{3} -18.9843i q^{5} -9.32401i q^{7} -25.5970 q^{9} +O(q^{10})\)	\(q-1.18449 q^{3} -18.9843i q^{5} -9.32401i q^{7} -25.5970 q^{9} +39.8107i q^{11} +(-45.2865 - 12.0885i) q^{13} +22.4867i q^{15} +92.6362 q^{17} +128.604i q^{19} +11.0442i q^{21} -158.981 q^{23} -235.403 q^{25} +62.3007 q^{27} -126.583 q^{29} -189.472i q^{31} -47.1555i q^{33} -177.010 q^{35} -53.7542i q^{37} +(53.6415 + 14.3187i) q^{39} +136.223i q^{41} -518.457 q^{43} +485.940i q^{45} -309.742i q^{47} +256.063 q^{49} -109.727 q^{51} -149.390 q^{53} +755.778 q^{55} -152.331i q^{57} -74.3413i q^{59} +99.0118 q^{61} +238.667i q^{63} +(-229.491 + 859.732i) q^{65} -435.290i q^{67} +188.312 q^{69} +827.267i q^{71} -981.880i q^{73} +278.833 q^{75} +371.196 q^{77} -299.012 q^{79} +617.324 q^{81} -169.626i q^{83} -1758.63i q^{85} +149.937 q^{87} -265.124i q^{89} +(-112.713 + 422.252i) q^{91} +224.428i q^{93} +2441.46 q^{95} -88.9929i q^{97} -1019.03i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{3} + 72 q^{9}+O(q^{10}) \) 10 * q - 6 * q^3 + 72 * q^9	\( 10 q - 6 q^{3} + 72 q^{9} - 24 q^{13} + 58 q^{17} - 180 q^{23} + 28 q^{25} - 354 q^{27} - 392 q^{29} - 154 q^{35} + 532 q^{39} + 234 q^{43} - 128 q^{49} + 510 q^{51} - 1244 q^{53} + 576 q^{55} - 56 q^{61} + 566 q^{65} + 1748 q^{69} - 472 q^{75} - 304 q^{77} - 1908 q^{79} + 2282 q^{81} + 64 q^{87} - 582 q^{91} + 2340 q^{95}+O(q^{100}) \) 10 * q - 6 * q^3 + 72 * q^9 - 24 * q^13 + 58 * q^17 - 180 * q^23 + 28 * q^25 - 354 * q^27 - 392 * q^29 - 154 * q^35 + 532 * q^39 + 234 * q^43 - 128 * q^49 + 510 * q^51 - 1244 * q^53 + 576 * q^55 - 56 * q^61 + 566 * q^65 + 1748 * q^69 - 472 * q^75 - 304 * q^77 - 1908 * q^79 + 2282 * q^81 + 64 * q^87 - 582 * q^91 + 2340 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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