LMFDB - Embedded newform 336.6.h.b.239.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,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, 6, names="a")

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

chi := DirichletCharacter("336.239");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
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:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.19
Character	$\chi$	$=$	336.239
Dual form			336.6.h.b.239.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.989377 - 15.5570i) q^{3} +62.7716i q^{5} -49.0000i q^{7} +(-241.042 + 30.7835i) q^{9} +O(q^{10})$	\(q+(-0.989377 - 15.5570i) q^{3} +62.7716i q^{5} -49.0000i q^{7} +(-241.042 + 30.7835i) q^{9} +304.782 q^{11} -316.809 q^{13} +(976.539 - 62.1047i) q^{15} -2196.56i q^{17} +1299.24i q^{19} +(-762.294 + 48.4795i) q^{21} -817.500 q^{23} -815.271 q^{25} +(717.382 + 3719.44i) q^{27} +3653.97i q^{29} +3099.50i q^{31} +(-301.544 - 4741.49i) q^{33} +3075.81 q^{35} -6738.18 q^{37} +(313.443 + 4928.60i) q^{39} +9263.00i q^{41} -249.564i q^{43} +(-1932.33 - 15130.6i) q^{45} -5600.97 q^{47} -2401.00 q^{49} +(-34171.9 + 2173.22i) q^{51} +31010.2i q^{53} +19131.6i q^{55} +(20212.3 - 1285.44i) q^{57} +6698.53 q^{59} +1689.15 q^{61} +(1508.39 + 11811.1i) q^{63} -19886.6i q^{65} +31569.5i q^{67} +(808.815 + 12717.9i) q^{69} +10440.3 q^{71} +49449.0 q^{73} +(806.610 + 12683.2i) q^{75} -14934.3i q^{77} +59796.1i q^{79} +(57153.7 - 14840.3i) q^{81} +68276.9 q^{83} +137881. q^{85} +(56845.0 - 3615.16i) q^{87} -15967.7i q^{89} +15523.6i q^{91} +(48218.9 - 3066.57i) q^{93} -81555.3 q^{95} +2710.14 q^{97} +(-73465.2 + 9382.25i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 8 q^{9}+O(q^{10}) $ 40 * q + 8 * q^9	$ 40 q + 8 q^{9} - 1048 q^{13} + 980 q^{21} - 43416 q^{25} + 20296 q^{33} - 16192 q^{37} + 56488 q^{45} - 96040 q^{49} + 31088 q^{57} + 173112 q^{61} - 114176 q^{69} - 267488 q^{73} + 64888 q^{81} + 508112 q^{85} - 224544 q^{93} - 276400 q^{97}+O(q^{100}) $ 40 * q + 8 * q^9 - 1048 * q^13 + 980 * q^21 - 43416 * q^25 + 20296 * q^33 - 16192 * q^37 + 56488 * q^45 - 96040 * q^49 + 31088 * q^57 + 173112 * q^61 - 114176 * q^69 - 267488 * q^73 + 64888 * q^81 + 508112 * q^85 - 224544 * q^93 - 276400 * 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
$3$	−0.989377	−	15.5570i	−0.0634686	−	0.997984i
$3$	−0.989377	−	15.5570i	−0.0634686	−	0.997984i
$4$		0			0
$5$			62.7716i			1.12289i	0.827513	+	0.561446i	$0.189755\pi$
$5$			62.7716i			1.12289i	−0.827513	+	0.561446i	$0.810245\pi$
$6$		0			0
$7$		−	49.0000i		−	0.377964i
$7$		−	49.0000i		−	0.377964i
$8$		0			0
$9$	−241.042	+	30.7835i	−0.991943	+	0.126681i
$10$		0			0
$11$	304.782			0.759464			0.379732	−	0.925097i	$-0.376016\pi$
$11$	304.782			0.759464			0.379732	+	0.925097i	$0.376016\pi$
$12$		0			0
$13$	−316.809			−0.519923			−0.259961	−	0.965619i	$-0.583710\pi$
$13$	−316.809			−0.519923			−0.259961	+	0.965619i	$0.583710\pi$
$14$		0			0
$15$	976.539	−	62.1047i	1.12063	−	0.0712683i
$16$		0			0
$17$		−	2196.56i		−	1.84340i	−0.387900	−	0.921702i	$-0.626799\pi$
$17$		−	2196.56i		−	1.84340i	0.387900	−	0.921702i	$-0.373201\pi$
$18$		0			0
$19$			1299.24i			0.825667i	0.910806	+	0.412834i	$0.135461\pi$
$19$			1299.24i			0.825667i	−0.910806	+	0.412834i	$0.864539\pi$
$20$		0			0
$21$	−762.294	+	48.4795i	−0.377202	+	0.0239889i
$22$		0			0
$23$	−817.500			−0.322232			−0.161116	−	0.986936i	$-0.551509\pi$
$23$	−817.500			−0.322232			−0.161116	+	0.986936i	$0.551509\pi$
$24$		0			0
$25$	−815.271			−0.260887
$26$		0			0
$27$	717.382	+	3719.44i	0.189383	+	0.981903i
$28$		0			0
$29$			3653.97i			0.806809i	0.915022	+	0.403404i	$0.132173\pi$
$29$			3653.97i			0.806809i	−0.915022	+	0.403404i	$0.867827\pi$
$30$		0			0
$31$			3099.50i			0.579278i	0.957136	+	0.289639i	$0.0935352\pi$
$31$			3099.50i			0.579278i	−0.957136	+	0.289639i	$0.906465\pi$
$32$		0			0
$33$	−301.544	−	4741.49i	−0.0482021	−	0.757932i
$34$		0			0
$35$	3075.81			0.424413
$36$		0			0
$37$	−6738.18			−0.809168			−0.404584	−	0.914501i	$-0.632584\pi$
$37$	−6738.18			−0.809168			−0.404584	+	0.914501i	$0.632584\pi$
$38$		0			0
$39$	313.443	+	4928.60i	0.0329987	+	0.518874i
$40$		0			0
$41$			9263.00i			0.860581i	0.902690	+	0.430291i	$0.141589\pi$
$41$			9263.00i			0.860581i	−0.902690	+	0.430291i	$0.858411\pi$
$42$		0			0
$43$		−	249.564i		−	0.0205831i	−0.999947	−	0.0102915i	$-0.996724\pi$
$43$		−	249.564i		−	0.0205831i	0.999947	−	0.0102915i	$-0.00327596\pi$
$44$		0			0
$45$	−1932.33	−	15130.6i	−0.142249	−	1.11385i
$46$		0			0
$47$	−5600.97			−0.369844			−0.184922	−	0.982753i	$-0.559203\pi$
$47$	−5600.97			−0.369844			−0.184922	+	0.982753i	$0.559203\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	−34171.9	+	2173.22i	−1.83969	+	0.116998i
$52$		0			0
$53$			31010.2i			1.51641i	0.652019	+	0.758203i	$0.273922\pi$
$53$			31010.2i			1.51641i	−0.652019	+	0.758203i	$0.726078\pi$
$54$		0			0
$55$			19131.6i			0.852796i
$56$		0			0
$57$	20212.3	−	1285.44i	0.824002	−	0.0524039i
$58$		0			0
$59$	6698.53			0.250524			0.125262	−	0.992124i	$-0.460023\pi$
$59$	6698.53			0.250524			0.125262	+	0.992124i	$0.460023\pi$
$60$		0			0
$61$	1689.15			0.0581224			0.0290612	−	0.999578i	$-0.490748\pi$
$61$	1689.15			0.0581224			0.0290612	+	0.999578i	$0.490748\pi$
$62$		0			0
$63$	1508.39	+	11811.1i	0.0478810	+	0.374919i
$64$		0			0
$65$		−	19886.6i		−	0.583817i
$66$		0			0
$67$			31569.5i			0.859172i	0.903026	+	0.429586i	$0.141340\pi$
$67$			31569.5i			0.859172i	−0.903026	+	0.429586i	$0.858660\pi$
$68$		0			0
$69$	808.815	+	12717.9i	0.0204516	+	0.321582i
$70$		0			0
$71$	10440.3			0.245792			0.122896	−	0.992420i	$-0.460782\pi$
$71$	10440.3			0.245792			0.122896	+	0.992420i	$0.460782\pi$
$72$		0			0
$73$	49449.0			1.08605			0.543026	−	0.839716i	$-0.317278\pi$
$73$	49449.0			1.08605			0.543026	+	0.839716i	$0.317278\pi$
$74$		0			0
$75$	806.610	+	12683.2i	0.0165581	+	0.260361i
$76$		0			0
$77$		−	14934.3i		−	0.287050i
$78$		0			0
$79$			59796.1i			1.07797i	0.842316	+	0.538983i	$0.181191\pi$
$79$			59796.1i			1.07797i	−0.842316	+	0.538983i	$0.818809\pi$
$80$		0			0
$81$	57153.7	−	14840.3i	0.967904	−	0.251321i
$82$		0			0
$83$	68276.9			1.08787			0.543937	−	0.839126i	$-0.316933\pi$
$83$	68276.9			1.08787			0.543937	+	0.839126i	$0.316933\pi$
$84$		0			0
$85$	137881.			2.06994
$86$		0			0
$87$	56845.0	−	3615.16i	0.805182	−	0.0512070i
$88$		0			0
$89$		−	15967.7i		−	0.213682i	−0.994276	−	0.106841i	$-0.965926\pi$
$89$		−	15967.7i		−	0.213682i	0.994276	−	0.106841i	$-0.0340736\pi$
$90$		0			0
$91$			15523.6i			0.196512i
$92$		0			0
$93$	48218.9	−	3066.57i	0.578110	−	0.0367659i
$94$		0			0
$95$	−81555.3			−0.927135
$96$		0			0
$97$	2710.14			0.0292457			0.0146229	−	0.999893i	$-0.495345\pi$
$97$	2710.14			0.0292457			0.0146229	+	0.999893i	$0.495345\pi$
$98$		0			0
$99$	−73465.2	+	9382.25i	−0.753345	+	0.0962097i
$100$		0			0
$101$			179932.i			1.75512i	0.479471	+	0.877558i	$0.340829\pi$
$101$			179932.i			1.75512i	−0.479471	+	0.877558i	$0.659171\pi$
$102$		0			0
$103$			126173.i			1.17186i	0.810363	+	0.585928i	$0.199270\pi$
$103$			126173.i			1.17186i	−0.810363	+	0.585928i	$0.800730\pi$
$104$		0			0
$105$	−3043.13	−	47850.4i	−0.0269369	−	0.423558i
$106$		0			0
$107$	153839.			1.29899			0.649497	−	0.760364i	$-0.274980\pi$
$107$	153839.			1.29899			0.649497	+	0.760364i	$0.274980\pi$
$108$		0			0
$109$	41287.3			0.332851			0.166426	−	0.986054i	$-0.446777\pi$
$109$	41287.3			0.332851			0.166426	+	0.986054i	$0.446777\pi$
$110$		0			0
$111$	6666.60	+	104826.i	0.0513567	+	0.807536i
$112$		0			0
$113$		−	235321.i		−	1.73367i	−0.498599	−	0.866833i	$-0.666152\pi$
$113$		−	235321.i		−	1.73367i	0.498599	−	0.866833i	$-0.333848\pi$
$114$		0			0
$115$		−	51315.7i		−	0.361831i
$116$		0			0
$117$	76364.3	−	9752.49i	0.515734	−	0.0658644i
$118$		0			0
$119$	−107631.			−0.696741
$120$		0			0
$121$	−68159.2			−0.423215
$122$		0			0
$123$	144105.	−	9164.60i	0.858846	−	0.0546199i
$124$		0			0
$125$			144985.i			0.829945i
$126$		0			0
$127$			335608.i			1.84639i	0.384332	+	0.923195i	$0.374432\pi$
$127$			335608.i			1.84639i	−0.384332	+	0.923195i	$0.625568\pi$
$128$		0			0
$129$	−3882.47	+	246.913i	−0.0205416	+	0.00130638i
$130$		0			0
$131$	−337011.			−1.71579			−0.857897	−	0.513821i	$-0.828229\pi$
$131$	−337011.			−1.71579			−0.857897	+	0.513821i	$0.828229\pi$
$132$		0			0
$133$	63662.7			0.312073
$134$		0			0
$135$	−233475.	+	45031.2i	−1.10257	+	0.212657i
$136$		0			0
$137$			212732.i			0.968347i	0.874972	+	0.484173i	$0.160880\pi$
$137$			212732.i			0.968347i	−0.874972	+	0.484173i	$0.839120\pi$
$138$		0			0
$139$			266889.i			1.17164i	0.810441	+	0.585820i	$0.199227\pi$
$139$			266889.i			1.17164i	−0.810441	+	0.585820i	$0.800773\pi$
$140$		0			0
$141$	5541.47	+	87134.4i	0.0234735	+	0.369098i
$142$		0			0
$143$	−96557.4			−0.394862
$144$		0			0
$145$	−229366.			−0.905959
$146$		0			0
$147$	2375.49	+	37352.4i	0.00906694	+	0.142569i
$148$		0			0
$149$		−	494950.i		−	1.82640i	−0.407514	−	0.913199i	$-0.633604\pi$
$149$		−	494950.i		−	1.82640i	0.407514	−	0.913199i	$-0.366396\pi$
$150$		0			0
$151$			530999.i			1.89518i	0.319484	+	0.947592i	$0.396490\pi$
$151$			530999.i			1.89518i	−0.319484	+	0.947592i	$0.603510\pi$
$152$		0			0
$153$	67617.8	+	529463.i	0.233524	+	1.82855i
$154$		0			0
$155$	−194560.			−0.650466
$156$		0			0
$157$	−330668.			−1.07064			−0.535319	−	0.844650i	$-0.679809\pi$
$157$	−330668.			−1.07064			−0.535319	+	0.844650i	$0.679809\pi$
$158$		0			0
$159$	482427.	−	30680.8i	1.51335	−	0.0962441i
$160$		0			0
$161$			40057.5i			0.121792i
$162$		0			0
$163$		−	268000.i		−	0.790071i	−0.918666	−	0.395035i	$-0.870732\pi$
$163$		−	268000.i		−	0.790071i	0.918666	−	0.395035i	$-0.129268\pi$
$164$		0			0
$165$	297631.	−	18928.4i	0.851076	−	0.0541257i
$166$		0			0
$167$	380811.			1.05662			0.528309	−	0.849052i	$-0.322826\pi$
$167$	380811.			1.05662			0.528309	+	0.849052i	$0.322826\pi$
$168$		0			0
$169$	−270925.			−0.729681
$170$		0			0
$171$	−39995.2	−	313171.i	−0.104596	−	0.819015i
$172$		0			0
$173$		−	425651.i		−	1.08128i	−0.841254	−	0.540640i	$-0.818182\pi$
$173$		−	425651.i		−	1.08128i	0.841254	−	0.540640i	$-0.181818\pi$
$174$		0			0
$175$			39948.3i			0.0986059i
$176$		0			0
$177$	−6627.37	−	104209.i	−0.0159004	−	0.250019i
$178$		0			0
$179$	428199.			0.998878			0.499439	−	0.866349i	$-0.333539\pi$
$179$	428199.			0.998878			0.499439	+	0.866349i	$0.333539\pi$
$180$		0			0
$181$	111059.			0.251976			0.125988	−	0.992032i	$-0.459790\pi$
$181$	111059.			0.251976			0.125988	+	0.992032i	$0.459790\pi$
$182$		0			0
$183$	−1671.21	−	26278.2i	−0.00368895	−	0.0580052i
$184$		0			0
$185$		−	422966.i		−	0.908608i
$186$		0			0
$187$		−	669470.i		−	1.40000i
$188$		0			0
$189$	182253.	−	35151.7i	0.371125	−	0.0715800i
$190$		0			0
$191$	−273510.			−0.542489			−0.271244	−	0.962511i	$-0.587435\pi$
$191$	−273510.			−0.542489			−0.271244	+	0.962511i	$0.587435\pi$
$192$		0			0
$193$	−201023.			−0.388465			−0.194232	−	0.980956i	$-0.562222\pi$
$193$	−201023.			−0.388465			−0.194232	+	0.980956i	$0.562222\pi$
$194$		0			0
$195$	−309376.	+	19675.3i	−0.582640	+	0.0370540i
$196$		0			0
$197$		−	28067.0i		−	0.0515265i	−0.999668	−	0.0257633i	$-0.991798\pi$
$197$		−	28067.0i		−	0.0515265i	0.999668	−	0.0257633i	$-0.00820161\pi$
$198$		0			0
$199$		−	759424.i		−	1.35941i	−0.733484	−	0.679707i	$-0.762107\pi$
$199$		−	759424.i		−	1.35941i	0.733484	−	0.679707i	$-0.237893\pi$
$200$		0			0
$201$	491127.	−	31234.1i	0.857440	−	0.0545304i
$202$		0			0
$203$	179045.			0.304945
$204$		0			0
$205$	−581453.			−0.966340
$206$		0			0
$207$	197052.	−	25165.5i	0.319635	−	0.0408207i
$208$		0			0
$209$			395984.i			0.627064i
$210$		0			0
$211$			740964.i			1.14575i	0.819642	+	0.572876i	$0.194172\pi$
$211$			740964.i			1.14575i	−0.819642	+	0.572876i	$0.805828\pi$
$212$		0			0
$213$	−10329.4	−	162421.i	−0.0156001	−	0.245297i
$214$		0			0
$215$	15665.5			0.0231126
$216$		0			0
$217$	151875.			0.218946
$218$		0			0
$219$	−48923.7	−	769280.i	−0.0689302	−	1.08386i
$220$		0			0
$221$			695888.i			0.958427i
$222$		0			0
$223$		−	512545.i		−	0.690191i	−0.938568	−	0.345096i	$-0.887846\pi$
$223$		−	512545.i		−	0.690191i	0.938568	−	0.345096i	$-0.112154\pi$
$224$		0			0
$225$	196515.	−	25096.9i	0.258785	−	0.0330494i
$226$		0			0
$227$	710528.			0.915201			0.457600	−	0.889158i	$-0.348709\pi$
$227$	710528.			0.915201			0.457600	+	0.889158i	$0.348709\pi$
$228$		0			0
$229$	391979.			0.493940			0.246970	−	0.969023i	$-0.420565\pi$
$229$	391979.			0.493940			0.246970	+	0.969023i	$0.420565\pi$
$230$		0			0
$231$	−232333.	+	14775.6i	−0.286471	+	0.0182187i
$232$		0			0
$233$			1.13075e6i			1.36450i	0.731117	+	0.682252i	$0.238999\pi$
$233$			1.13075e6i			1.36450i	−0.731117	+	0.682252i	$0.761001\pi$
$234$		0			0
$235$		−	351582.i		−	0.415295i
$236$		0			0
$237$	930250.	−	59160.9i	1.07579	−	0.0684170i
$238$		0			0
$239$	−267970.			−0.303453			−0.151727	−	0.988423i	$-0.548483\pi$
$239$	−267970.			−0.303453			−0.151727	+	0.988423i	$0.548483\pi$
$240$		0			0
$241$	−1.51200e6			−1.67691			−0.838454	−	0.544972i	$-0.816540\pi$
$241$	−1.51200e6			−1.67691			−0.838454	+	0.544972i	$0.816540\pi$
$242$		0			0
$243$	−287417.	−	874460.i	−0.312246	−	0.950001i
$244$		0			0
$245$		−	150715.i		−	0.160413i
$246$		0			0
$247$		−	411610.i		−	0.429283i
$248$		0			0
$249$	−67551.6	−	1.06219e6i	−0.0690458	−	1.08568i
$250$		0			0
$251$	211650.			0.212048			0.106024	−	0.994364i	$-0.466188\pi$
$251$	211650.			0.212048			0.106024	+	0.994364i	$0.466188\pi$
$252$		0			0
$253$	−249159.			−0.244723
$254$		0			0
$255$	−136417.	−	2.14502e6i	−0.131376	−	2.06577i
$256$		0			0
$257$			275906.i			0.260572i	0.991476	+	0.130286i	$0.0415895\pi$
$257$			275906.i			0.260572i	−0.991476	+	0.130286i	$0.958410\pi$
$258$		0			0
$259$			330171.i			0.305837i
$260$		0			0
$261$	−112482.	−	880762.i	−0.102207	−	0.800309i
$262$		0			0
$263$	−2.02994e6			−1.80965			−0.904824	−	0.425787i	$-0.859997\pi$
$263$	−2.02994e6			−1.80965			−0.904824	+	0.425787i	$0.859997\pi$
$264$		0			0
$265$	−1.94656e6			−1.70276
$266$		0			0
$267$	−248410.	+	15798.1i	−0.213251	+	0.0135621i
$268$		0			0
$269$			826554.i			0.696451i	0.937411	+	0.348225i	$0.113216\pi$
$269$			826554.i			0.696451i	−0.937411	+	0.348225i	$0.886784\pi$
$270$		0			0
$271$		−	577284.i		−	0.477492i	−0.971082	−	0.238746i	$-0.923264\pi$
$271$		−	577284.i		−	0.477492i	0.971082	−	0.238746i	$-0.0767363\pi$
$272$		0			0
$273$	241501.	−	15358.7i	0.196116	−	0.0124723i
$274$		0			0
$275$	−248479.			−0.198134
$276$		0			0
$277$	550847.			0.431352			0.215676	−	0.976465i	$-0.430805\pi$
$277$	550847.			0.431352			0.215676	+	0.976465i	$0.430805\pi$
$278$		0			0
$279$	−95413.4	−	747109.i	−0.0733836	−	0.574611i
$280$		0			0
$281$		−	435111.i		−	0.328726i	−0.986400	−	0.164363i	$-0.947443\pi$
$281$		−	435111.i		−	0.328726i	0.986400	−	0.164363i	$-0.0525568\pi$
$282$		0			0
$283$		−	1.81477e6i		−	1.34696i	−0.739206	−	0.673480i	$-0.764799\pi$
$283$		−	1.81477e6i		−	1.34696i	0.739206	−	0.673480i	$-0.235201\pi$
$284$		0			0
$285$	80688.9	+	1.26876e6i	0.0588439	+	0.925266i
$286$		0			0
$287$	453887.			0.325269
$288$		0			0
$289$	−3.40501e6			−2.39813
$290$		0			0
$291$	−2681.35	−	42161.7i	−0.00185618	−	0.0291868i
$292$		0			0
$293$		−	2.40035e6i		−	1.63345i	−0.577029	−	0.816723i	$-0.695788\pi$
$293$		−	2.40035e6i		−	1.63345i	0.577029	−	0.816723i	$-0.304212\pi$
$294$		0			0
$295$			420477.i			0.281312i
$296$		0			0
$297$	218645.	+	1.13362e6i	0.143829	+	0.745720i
$298$		0			0
$299$	258991.			0.167535
$300$		0			0
$301$	−12228.6			−0.00777968
$302$		0			0
$303$	2.79921e6	−	178021.i	1.75158	−	0.111395i
$304$		0			0
$305$			106031.i			0.0652652i
$306$		0			0
$307$			2.96289e6i			1.79420i	0.441831	+	0.897098i	$0.354329\pi$
$307$			2.96289e6i			1.79420i	−0.441831	+	0.897098i	$0.645671\pi$
$308$		0			0
$309$	1.96288e6	−	124833.i	1.16949	−	0.0743760i
$310$		0			0
$311$	2.85407e6			1.67326			0.836631	−	0.547767i	$-0.184522\pi$
$311$	2.85407e6			1.67326			0.836631	+	0.547767i	$0.184522\pi$
$312$		0			0
$313$	−2.74326e6			−1.58273			−0.791363	−	0.611346i	$-0.790628\pi$
$313$	−2.74326e6			−1.58273			−0.791363	+	0.611346i	$0.790628\pi$
$314$		0			0
$315$	−741400.	+	94684.2i	−0.420994	+	0.0537652i
$316$		0			0
$317$			1.10180e6i			0.615823i	0.951415	+	0.307912i	$0.0996301\pi$
$317$			1.10180e6i			0.615823i	−0.951415	+	0.307912i	$0.900370\pi$
$318$		0			0
$319$			1.11366e6i			0.612742i
$320$		0			0
$321$	−152205.	−	2.39328e6i	−0.0824453	−	1.29637i
$322$		0			0
$323$	2.85385e6			1.52204
$324$		0			0
$325$	258285.			0.135641
$326$		0			0
$327$	−40848.7	−	642307.i	−0.0211256	−	0.332180i
$328$		0			0
$329$			274447.i			0.139788i
$330$		0			0
$331$			1.65859e6i			0.832086i	0.909345	+	0.416043i	$0.136583\pi$
$331$			1.65859e6i			0.832086i	−0.909345	+	0.416043i	$0.863417\pi$
$332$		0			0
$333$	1.62419e6	−	207425.i	0.802649	−	0.102506i
$334$		0			0
$335$	−1.98166e6			−0.964757
$336$		0			0
$337$	−1.41769e6			−0.679998			−0.339999	−	0.940426i	$-0.610427\pi$
$337$	−1.41769e6			−0.679998			−0.339999	+	0.940426i	$0.610427\pi$
$338$		0			0
$339$	−3.66090e6	+	232822.i	−1.73017	+	0.110033i
$340$		0			0
$341$			944669.i			0.439940i
$342$		0			0
$343$			117649.i			0.0539949i
$344$		0			0
$345$	−798321.	+	50770.6i	−0.361102	+	0.0229649i
$346$		0			0
$347$	−1.22664e6			−0.546881			−0.273441	−	0.961889i	$-0.588162\pi$
$347$	−1.22664e6			−0.546881			−0.273441	+	0.961889i	$0.588162\pi$
$348$		0			0
$349$	−1.19709e6			−0.526096			−0.263048	−	0.964783i	$-0.584728\pi$
$349$	−1.19709e6			−0.526096			−0.263048	+	0.964783i	$0.584728\pi$
$350$		0			0
$351$	−227273.	−	1.17835e6i	−0.0984645	−	0.510514i
$352$		0			0
$353$		−	4.20354e6i		−	1.79547i	−0.440534	−	0.897736i	$-0.645211\pi$
$353$		−	4.20354e6i		−	1.79547i	0.440534	−	0.897736i	$-0.354789\pi$
$354$		0			0
$355$			655356.i			0.275998i
$356$		0			0
$357$	106488.	+	1.67442e6i	0.0442211	+	0.695336i
$358$		0			0
$359$	−472238.			−0.193386			−0.0966930	−	0.995314i	$-0.530826\pi$
$359$	−472238.			−0.193386			−0.0966930	+	0.995314i	$0.530826\pi$
$360$		0			0
$361$	788078.			0.318274
$362$		0			0
$363$	67435.2	+	1.06036e6i	0.0268609	+	0.422362i
$364$		0			0
$365$			3.10399e6i			1.21952i
$366$		0			0
$367$		−	1.75523e6i		−	0.680252i	−0.940380	−	0.340126i	$-0.889530\pi$
$367$		−	1.75523e6i		−	0.680252i	0.940380	−	0.340126i	$-0.110470\pi$
$368$		0			0
$369$	−285148.	−	2.23277e6i	−0.109019	−	0.853648i
$370$		0			0
$371$	1.51950e6			0.573148
$372$		0			0
$373$	−2.27878e6			−0.848066			−0.424033	−	0.905647i	$-0.639386\pi$
$373$	−2.27878e6			−0.848066			−0.424033	+	0.905647i	$0.639386\pi$
$374$		0			0
$375$	2.25554e6	−	143445.i	0.828271	−	0.0526754i
$376$		0			0
$377$		−	1.15761e6i		−	0.419478i
$378$		0			0
$379$			1.89684e6i			0.678317i	0.940729	+	0.339159i	$0.110142\pi$
$379$			1.89684e6i			0.678317i	−0.940729	+	0.339159i	$0.889858\pi$
$380$		0			0
$381$	5.22107e6	−	332043.i	1.84267	−	0.117188i
$382$		0			0
$383$	−1.10776e6			−0.385876			−0.192938	−	0.981211i	$-0.561802\pi$
$383$	−1.10776e6			−0.385876			−0.192938	+	0.981211i	$0.561802\pi$
$384$		0			0
$385$	937449.			0.322326
$386$		0			0
$387$	7682.46	+	60155.5i	0.00260749	+	0.0204173i
$388$		0			0
$389$		−	919043.i		−	0.307937i	−0.988076	−	0.153968i	$-0.950795\pi$
$389$		−	919043.i		−	0.307937i	0.988076	−	0.153968i	$-0.0492054\pi$
$390$		0			0
$391$			1.79569e6i			0.594003i
$392$		0			0
$393$	333430.	+	5.24288e6i	0.108899	+	1.71234i
$394$		0			0
$395$	−3.75350e6			−1.21044
$396$		0			0
$397$	4.94051e6			1.57324			0.786622	−	0.617435i	$-0.211828\pi$
$397$	4.94051e6			1.57324			0.786622	+	0.617435i	$0.211828\pi$
$398$		0			0
$399$	−62986.4	−	990402.i	−0.0198068	−	0.311444i
$400$		0			0
$401$			297451.i			0.0923751i	0.998933	+	0.0461876i	$0.0147072\pi$
$401$			297451.i			0.0923751i	−0.998933	+	0.0461876i	$0.985293\pi$
$402$		0			0
$403$		−	981947.i		−	0.301179i
$404$		0			0
$405$	931547.	+	3.58763e6i	0.282207	+	1.08685i
$406$		0			0
$407$	−2.05367e6			−0.614533
$408$		0			0
$409$	2.25616e6			0.666902			0.333451	−	0.942767i	$-0.391787\pi$
$409$	2.25616e6			0.666902			0.333451	+	0.942767i	$0.391787\pi$
$410$		0			0
$411$	3.30947e6	−	210472.i	0.966395	−	0.0614596i
$412$		0			0
$413$		−	328228.i		−	0.0946892i
$414$		0			0
$415$			4.28585e6i			1.22157i
$416$		0			0
$417$	4.15201e6	−	264054.i	1.16928	−	0.0743623i
$418$		0			0
$419$	183668.			0.0511093			0.0255546	−	0.999673i	$-0.491865\pi$
$419$	183668.			0.0511093			0.0255546	+	0.999673i	$0.491865\pi$
$420$		0			0
$421$	2.33587e6			0.642309			0.321155	−	0.947027i	$-0.395929\pi$
$421$	2.33587e6			0.642309			0.321155	+	0.947027i	$0.395929\pi$
$422$		0			0
$423$	1.35007e6	−	172418.i	0.366864	−	0.0468523i
$424$		0			0
$425$			1.79079e6i			0.480919i
$426$		0			0
$427$		−	82768.3i		−	0.0219682i
$428$		0			0
$429$	95531.7	+	1.50215e6i	0.0250613	+	0.394066i
$430$		0			0
$431$	−1.75311e6			−0.454585			−0.227292	−	0.973827i	$-0.572987\pi$
$431$	−1.75311e6			−0.454585			−0.227292	+	0.973827i	$0.572987\pi$
$432$		0			0
$433$	−1.28744e6			−0.329996			−0.164998	−	0.986294i	$-0.552762\pi$
$433$	−1.28744e6			−0.329996			−0.164998	+	0.986294i	$0.552762\pi$
$434$		0			0
$435$	226929.	+	3.56825e6i	0.0574999	+	0.904132i
$436$		0			0
$437$		−	1.06213e6i		−	0.266056i
$438$		0			0
$439$			4.72854e6i			1.17102i	0.810664	+	0.585512i	$0.199106\pi$
$439$			4.72854e6i			1.17102i	−0.810664	+	0.585512i	$0.800894\pi$
$440$		0			0
$441$	578742.	−	73911.2i	0.141706	−	0.0180973i
$442$		0			0
$443$	−4.10860e6			−0.994683			−0.497342	−	0.867555i	$-0.665690\pi$
$443$	−4.10860e6			−0.994683			−0.497342	+	0.867555i	$0.665690\pi$
$444$		0			0
$445$	1.00232e6			0.239942
$446$		0			0
$447$	−7.69995e6	+	489692.i	−1.82272	+	0.115919i
$448$		0			0
$449$		−	2.09833e6i		−	0.491200i	−0.969371	−	0.245600i	$-0.921015\pi$
$449$		−	2.09833e6i		−	0.491200i	0.969371	−	0.245600i	$-0.0789850\pi$
$450$		0			0
$451$			2.82319e6i			0.653580i
$452$		0			0
$453$	8.26076e6	−	525358.i	1.89136	−	0.120285i
$454$		0			0
$455$	−974442.			−0.220662
$456$		0			0
$457$	−4.14572e6			−0.928558			−0.464279	−	0.885689i	$-0.653687\pi$
$457$	−4.14572e6			−0.928558			−0.464279	+	0.885689i	$0.653687\pi$
$458$		0			0
$459$	8.16998e6	−	1.57577e6i	1.81004	−	0.349109i
$460$		0			0
$461$		−	3.83342e6i		−	0.840105i	−0.907500	−	0.420053i	$-0.862012\pi$
$461$		−	3.83342e6i		−	0.840105i	0.907500	−	0.420053i	$-0.137988\pi$
$462$		0			0
$463$			388435.i			0.0842105i	0.999113	+	0.0421052i	$0.0134065\pi$
$463$			388435.i			0.0842105i	−0.999113	+	0.0421052i	$0.986594\pi$
$464$		0			0
$465$	192493.	+	3.02678e6i	0.0412842	+	0.649155i
$466$		0			0
$467$	2.03969e6			0.432784			0.216392	−	0.976307i	$-0.430571\pi$
$467$	2.03969e6			0.432784			0.216392	+	0.976307i	$0.430571\pi$
$468$		0			0
$469$	1.54690e6			0.324736
$470$		0			0
$471$	327155.	+	5.14421e6i	0.0679519	+	1.06848i
$472$		0			0
$473$		−	76062.5i		−	0.0156321i
$474$		0			0
$475$		−	1.05923e6i		−	0.215406i
$476$		0			0
$477$	−954605.	−	7.47478e6i	−0.192100	−	1.50419i
$478$		0			0
$479$	7.06773e6			1.40748			0.703739	−	0.710459i	$-0.251513\pi$
$479$	7.06773e6			1.40748			0.703739	+	0.710459i	$0.251513\pi$
$480$		0			0
$481$	2.13471e6			0.420704
$482$		0			0
$483$	623175.	−	39631.9i	0.121547	−	0.00772997i
$484$		0			0
$485$			170120.i			0.0328398i
$486$		0			0
$487$			2.45138e6i			0.468369i	0.972192	+	0.234185i	$0.0752420\pi$
$487$			2.45138e6i			0.468369i	−0.972192	+	0.234185i	$0.924758\pi$
$488$		0			0
$489$	−4.16929e6	+	265153.i	−0.788478	+	0.0501447i
$490$		0			0
$491$	3.43920e6			0.643804			0.321902	−	0.946773i	$-0.395678\pi$
$491$	3.43920e6			0.643804			0.321902	+	0.946773i	$0.395678\pi$
$492$		0			0
$493$	8.02617e6			1.48727
$494$		0			0
$495$	−588939.	−	4.61153e6i	−0.108033	−	0.845925i
$496$		0			0
$497$		−	511576.i		−	0.0929008i
$498$		0			0
$499$			1.06278e7i			1.91069i	0.295488	+	0.955346i	$0.404518\pi$
$499$			1.06278e7i			1.91069i	−0.295488	+	0.955346i	$0.595482\pi$
$500$		0			0
$501$	−376765.	−	5.92428e6i	−0.0670620	−	1.05449i
$502$		0			0
$503$	8.43009e6			1.48564			0.742818	−	0.669494i	$-0.233489\pi$
$503$	8.43009e6			1.48564			0.742818	+	0.669494i	$0.233489\pi$
$504$		0			0
$505$	−1.12946e7			−1.97081
$506$		0			0
$507$	268047.	+	4.21479e6i	0.0463118	+	0.728209i
$508$		0			0
$509$			4.42506e6i			0.757050i	0.925591	+	0.378525i	$0.123569\pi$
$509$			4.42506e6i			0.757050i	−0.925591	+	0.378525i	$0.876431\pi$
$510$		0			0
$511$		−	2.42300e6i		−	0.410489i
$512$		0			0
$513$	−4.83245e6	+	932050.i	−0.810725	+	0.156367i
$514$		0			0
$515$	−7.92009e6			−1.31587
$516$		0			0
$517$	−1.70707e6			−0.280883
$518$		0			0
$519$	−6.62187e6	+	421129.i	−1.07910	+	0.0686273i
$520$		0			0
$521$			9.63345e6i			1.55485i	0.628978	+	0.777423i	$0.283474\pi$
$521$			9.63345e6i			1.55485i	−0.628978	+	0.777423i	$0.716526\pi$
$522$		0			0
$523$		−	6.53432e6i		−	1.04459i	−0.852764	−	0.522296i	$-0.825076\pi$
$523$		−	6.53432e6i		−	1.04459i	0.852764	−	0.522296i	$-0.174924\pi$
$524$		0			0
$525$	621476.	−	39523.9i	0.0984071	−	0.00625837i
$526$		0			0
$527$	6.80822e6			1.06784
$528$		0			0
$529$	−5.76804e6			−0.896167
$530$		0			0
$531$	−1.61463e6	+	206204.i	−0.248506	+	0.0317367i
$532$		0			0
$533$		−	2.93460e6i		−	0.447436i
$534$		0			0
$535$			9.65672e6i			1.45863i
$536$		0			0
$537$	−423650.	−	6.66150e6i	−0.0633974	−	0.996864i
$538$		0			0
$539$	−731780.			−0.108495
$540$		0			0
$541$	−5.92904e6			−0.870946			−0.435473	−	0.900202i	$-0.643419\pi$
$541$	−5.92904e6			−0.870946			−0.435473	+	0.900202i	$0.643419\pi$
$542$		0			0
$543$	−109880.	−	1.72775e6i	−0.0159925	−	0.251468i
$544$		0			0
$545$			2.59167e6i			0.373756i
$546$		0			0
$547$		−	1.87571e6i		−	0.268038i	−0.990979	−	0.134019i	$-0.957212\pi$
$547$		−	1.87571e6i		−	0.268038i	0.990979	−	0.134019i	$-0.0427883\pi$
$548$		0			0
$549$	−407157.	+	51998.0i	−0.0576541	+	0.00736302i
$550$		0			0
$551$	−4.74739e6			−0.666155
$552$		0			0
$553$	2.93001e6			0.407433
$554$		0			0
$555$	−6.58010e6	+	418473.i	−0.906776	+	0.0576680i
$556$		0			0
$557$			7.81173e6i			1.06686i	0.845843	+	0.533432i	$0.179098\pi$
$557$			7.81173e6i			1.06686i	−0.845843	+	0.533432i	$0.820902\pi$
$558$		0			0
$559$			79064.0i			0.0107016i
$560$		0			0
$561$	−1.04150e7	+	662358.i	−1.39717	+	0.0888558i
$562$		0			0
$563$	−8.11831e6			−1.07943			−0.539715	−	0.841848i	$-0.681468\pi$
$563$	−8.11831e6			−1.07943			−0.539715	+	0.841848i	$0.681468\pi$
$564$		0			0
$565$	1.47715e7			1.94672
$566$		0			0
$567$	−727173.	−	2.80053e6i	−0.0949905	−	0.365833i
$568$		0			0
$569$		−	1.04412e7i		−	1.35198i	−0.736911	−	0.675989i	$-0.763717\pi$
$569$		−	1.04412e7i		−	1.35198i	0.736911	−	0.675989i	$-0.236283\pi$
$570$		0			0
$571$		−	1.19439e7i		−	1.53305i	−0.642213	−	0.766526i	$-0.721984\pi$
$571$		−	1.19439e7i		−	1.53305i	0.642213	−	0.766526i	$-0.278016\pi$
$572$		0			0
$573$	270605.	+	4.25501e6i	0.0344310	+	0.541395i
$574$		0			0
$575$	666484.			0.0840659
$576$		0			0
$577$	1.13940e7			1.42474			0.712370	−	0.701804i	$-0.247622\pi$
$577$	1.13940e7			1.42474			0.712370	+	0.701804i	$0.247622\pi$
$578$		0			0
$579$	198887.	+	3.12731e6i	0.0246553	+	0.387682i
$580$		0			0
$581$		−	3.34557e6i		−	0.411178i
$582$		0			0
$583$			9.45135e6i			1.15165i
$584$		0			0
$585$	612179.	+	4.79351e6i	0.0739586	+	0.579113i
$586$		0			0
$587$	1.43604e7			1.72017			0.860083	−	0.510154i	$-0.170411\pi$
$587$	1.43604e7			1.72017			0.860083	+	0.510154i	$0.170411\pi$
$588$		0			0
$589$	−4.02698e6			−0.478290
$590$		0			0
$591$	−436640.	+	27768.9i	−0.0514227	+	0.00327032i
$592$		0			0
$593$			8.47508e6i			0.989708i	0.868976	+	0.494854i	$0.164778\pi$
$593$			8.47508e6i			0.989708i	−0.868976	+	0.494854i	$0.835222\pi$
$594$		0			0
$595$		−	6.75619e6i		−	0.782365i
$596$		0			0
$597$	−1.18144e7	+	751356.i	−1.35667	+	0.0862800i
$598$		0			0
$599$	−7.14427e6			−0.813563			−0.406781	−	0.913526i	$-0.633349\pi$
$599$	−7.14427e6			−0.813563			−0.406781	+	0.913526i	$0.633349\pi$
$600$		0			0
$601$	−1.11832e7			−1.26293			−0.631466	−	0.775403i	$-0.717547\pi$
$601$	−1.11832e7			−1.26293			−0.631466	+	0.775403i	$0.717547\pi$
$602$		0			0
$603$	−971819.	−	7.60957e6i	−0.108841	−	0.852250i
$604$		0			0
$605$		−	4.27846e6i		−	0.475225i
$606$		0			0
$607$		−	5.76663e6i		−	0.635258i	−0.948215	−	0.317629i	$-0.897113\pi$
$607$		−	5.76663e6i		−	0.635258i	0.948215	−	0.317629i	$-0.102887\pi$
$608$		0			0
$609$	−177143.	−	2.78540e6i	−0.0193544	−	0.304330i
$610$		0			0
$611$	1.77443e6			0.192290
$612$		0			0
$613$	7.72627e6			0.830460			0.415230	−	0.909716i	$-0.363701\pi$
$613$	7.72627e6			0.830460			0.415230	+	0.909716i	$0.363701\pi$
$614$		0			0
$615$	575276.	+	9.04568e6i	0.0613322	+	0.964392i
$616$		0			0
$617$		−	9.84493e6i		−	1.04112i	−0.853826	−	0.520559i	$-0.825724\pi$
$617$		−	9.84493e6i		−	1.04112i	0.853826	−	0.520559i	$-0.174276\pi$
$618$		0			0
$619$		−	1.16942e7i		−	1.22672i	−0.789805	−	0.613358i	$-0.789818\pi$
$619$		−	1.16942e7i		−	1.22672i	0.789805	−	0.613358i	$-0.210182\pi$
$620$		0			0
$621$	−586459.	−	3.04065e6i	−0.0610252	−	0.316400i
$622$		0			0
$623$	−782418.			−0.0807642
$624$		0			0
$625$	−1.16487e7			−1.19282
$626$		0			0
$627$	6.16033e6	−	391777.i	0.625800	−	0.0397988i
$628$		0			0
$629$			1.48008e7i			1.49162i
$630$		0			0
$631$		−	7.21029e6i		−	0.720908i	−0.932777	−	0.360454i	$-0.882622\pi$
$631$		−	7.21029e6i		−	0.720908i	0.932777	−	0.360454i	$-0.117378\pi$
$632$		0			0
$633$	1.15272e7	−	733093.i	1.14344	−	0.0727193i
$634$		0			0
$635$	−2.10667e7			−2.07330
$636$		0			0
$637$	760658.			0.0742746
$638$		0			0
$639$	−2.51656e6	+	321390.i	−0.243812	+	0.0311373i
$640$		0			0
$641$			1.29890e7i			1.24863i	0.781174	+	0.624313i	$0.214621\pi$
$641$			1.29890e7i			1.24863i	−0.781174	+	0.624313i	$0.785379\pi$
$642$		0			0
$643$		−	1.32931e7i		−	1.26794i	−0.773357	−	0.633970i	$-0.781424\pi$
$643$		−	1.32931e7i		−	1.26794i	0.773357	−	0.633970i	$-0.218576\pi$
$644$		0			0
$645$	−15499.1	−	243709.i	−0.00146692	−	0.0230660i
$646$		0			0
$647$	5.01118e6			0.470629			0.235315	−	0.971919i	$-0.424388\pi$
$647$	5.01118e6			0.470629			0.235315	+	0.971919i	$0.424388\pi$
$648$		0			0
$649$	2.04159e6			0.190264
$650$		0			0
$651$	−150262.	−	2.36273e6i	−0.0138962	−	0.218505i
$652$		0			0
$653$		−	1.08506e7i		−	0.995794i	−0.867236	−	0.497897i	$-0.834106\pi$
$653$		−	1.08506e7i		−	0.995794i	0.867236	−	0.497897i	$-0.165894\pi$
$654$		0			0
$655$		−	2.11547e7i		−	1.92665i
$656$		0			0
$657$	−1.19193e7	+	1.52222e6i	−1.07730	+	0.137582i
$658$		0			0
$659$	9.13505e6			0.819403			0.409701	−	0.912220i	$-0.365633\pi$
$659$	9.13505e6			0.819403			0.409701	+	0.912220i	$0.365633\pi$
$660$		0			0
$661$	1.40942e7			1.25469			0.627344	−	0.778742i	$-0.284142\pi$
$661$	1.40942e7			1.25469			0.627344	+	0.778742i	$0.284142\pi$
$662$		0			0
$663$	1.08260e7	−	688496.i	0.956494	−	0.0608300i
$664$		0			0
$665$			3.99621e6i			0.350424i
$666$		0			0
$667$		−	2.98712e6i		−	0.259979i
$668$		0			0
$669$	−7.97367e6	+	507100.i	−0.688800	+	0.0438054i
$670$		0			0
$671$	514822.			0.0441418
$672$		0			0
$673$	4.58926e6			0.390575			0.195288	−	0.980746i	$-0.437436\pi$
$673$	4.58926e6			0.390575			0.195288	+	0.980746i	$0.437436\pi$
$674$		0			0
$675$	−584861.	−	3.03236e6i	−0.0494075	−	0.256165i
$676$		0			0
$677$		−	1.75529e7i		−	1.47190i	−0.677037	−	0.735949i	$-0.736737\pi$
$677$		−	1.75529e7i		−	1.47190i	0.677037	−	0.735949i	$-0.263263\pi$
$678$		0			0
$679$		−	132797.i		−	0.0110538i
$680$		0			0
$681$	−702980.	−	1.10537e7i	−0.0580865	−	0.913355i
$682$		0			0
$683$	2.00842e7			1.64741			0.823707	−	0.567016i	$-0.191902\pi$
$683$	2.00842e7			1.64741			0.823707	+	0.567016i	$0.191902\pi$
$684$		0			0
$685$	−1.33535e7			−1.08735
$686$		0			0
$687$	−387815.	−	6.09803e6i	−0.0313497	−	0.492944i
$688$		0			0
$689$		−	9.82431e6i		−	0.788414i
$690$		0			0
$691$		−	1.28308e7i		−	1.02226i	−0.859505	−	0.511128i	$-0.829228\pi$
$691$		−	1.28308e7i		−	1.02226i	0.859505	−	0.511128i	$-0.170772\pi$
$692$		0			0
$693$	459730.	+	3.59980e6i	0.0363639	+	0.284738i
$694$		0			0
$695$	−1.67531e7			−1.31563
$696$		0			0
$697$	2.03467e7			1.58640
$698$		0			0
$699$	1.75910e7	−	1.11873e6i	1.36175	−	0.0866031i
$700$		0			0
$701$			1.25119e7i			0.961675i	0.876810	+	0.480838i	$0.159667\pi$
$701$			1.25119e7i			0.961675i	−0.876810	+	0.480838i	$0.840333\pi$
$702$		0			0
$703$		−	8.75451e6i		−	0.668103i
$704$		0			0
$705$	−5.46956e6	+	347847.i	−0.414457	+	0.0263582i
$706$		0			0
$707$	8.81668e6			0.663371
$708$		0			0
$709$	−2.29997e7			−1.71833			−0.859164	−	0.511700i	$-0.829016\pi$
$709$	−2.29997e7			−1.71833			−0.859164	+	0.511700i	$0.829016\pi$
$710$		0			0
$711$	−1.84074e6	−	1.44134e7i	−0.136558	−	1.06928i
$712$		0			0
$713$		−	2.53384e6i		−	0.186662i
$714$		0			0
$715$		−	6.06106e6i		−	0.443388i
$716$		0			0
$717$	265123.	+	4.16882e6i	0.0192597	+	0.302841i
$718$		0			0
$719$	2.52852e7			1.82408			0.912041	−	0.410099i	$-0.134506\pi$
$719$	2.52852e7			1.82408			0.912041	+	0.410099i	$0.134506\pi$
$720$		0			0
$721$	6.18249e6			0.442920
$722$		0			0
$723$	1.49594e6	+	2.35222e7i	0.106431	+	1.67353i
$724$		0			0
$725$		−	2.97898e6i		−	0.210486i
$726$		0			0
$727$			4.12561e6i			0.289502i	0.989468	+	0.144751i	$0.0462382\pi$
$727$			4.12561e6i			0.289502i	−0.989468	+	0.144751i	$0.953762\pi$
$728$		0			0
$729$	−1.33196e7	+	5.33652e6i	−0.928268	+	0.371912i
$730$		0			0
$731$	−548182.			−0.0379429
$732$		0			0
$733$	1.70125e7			1.16952			0.584762	−	0.811205i	$-0.301188\pi$
$733$	1.70125e7			1.16952			0.584762	+	0.811205i	$0.301188\pi$
$734$		0			0
$735$	−2.34467e6	+	149113.i	−0.160090	+	0.0101812i
$736$		0			0
$737$			9.62179e6i			0.652510i
$738$		0			0
$739$		−	9.21118e6i		−	0.620446i	−0.950664	−	0.310223i	$-0.899596\pi$
$739$		−	9.21118e6i		−	0.620446i	0.950664	−	0.310223i	$-0.100404\pi$
$740$		0			0
$741$	−6.40343e6	+	407237.i	−0.428417	+	0.0272460i
$742$		0			0
$743$	−1.18462e7			−0.787241			−0.393620	−	0.919273i	$-0.628778\pi$
$743$	−1.18462e7			−0.787241			−0.393620	+	0.919273i	$0.628778\pi$
$744$		0			0
$745$	3.10688e7			2.05085
$746$		0			0
$747$	−1.64576e7	+	2.10181e6i	−1.07911	+	0.137813i
$748$		0			0
$749$		−	7.53811e6i		−	0.490974i
$750$		0			0
$751$			1.51545e7i			0.980488i	0.871585	+	0.490244i	$0.163092\pi$
$751$			1.51545e7i			0.980488i	−0.871585	+	0.490244i	$0.836908\pi$
$752$		0			0
$753$	−209402.	−	3.29264e6i	−0.0134584	−	0.211620i
$754$		0			0
$755$	−3.33316e7			−2.12809
$756$		0			0
$757$	6.45208e6			0.409223			0.204611	−	0.978843i	$-0.434407\pi$
$757$	6.45208e6			0.409223			0.204611	+	0.978843i	$0.434407\pi$
$758$		0			0
$759$	246512.	+	3.87617e6i	0.0155322	+	0.244230i
$760$		0			0
$761$		−	5.84103e6i		−	0.365618i	−0.983148	−	0.182809i	$-0.941481\pi$
$761$		−	5.84103e6i		−	0.365618i	0.983148	−	0.182809i	$-0.0585190\pi$
$762$		0			0
$763$		−	2.02308e6i		−	0.125806i
$764$		0			0
$765$	−3.32352e7	+	4.24448e6i	−2.05327	+	0.262223i
$766$		0			0
$767$	−2.12215e6			−0.130253
$768$		0			0
$769$	−2.59575e7			−1.58288			−0.791439	−	0.611249i	$-0.790667\pi$
$769$	−2.59575e7			−1.58288			−0.791439	+	0.611249i	$0.790667\pi$
$770$		0			0
$771$	4.29227e6	−	272975.i	0.260047	−	0.0165381i
$772$		0			0
$773$			1.23153e7i			0.741304i	0.928772	+	0.370652i	$0.120866\pi$
$773$			1.23153e7i			0.741304i	−0.928772	+	0.370652i	$0.879134\pi$
$774$		0			0
$775$		−	2.52693e6i		−	0.151126i
$776$		0			0
$777$	5.13648e6	−	326664.i	0.305220	−	0.0194110i
$778$		0			0
$779$	−1.20348e7			−0.710554
$780$		0			0
$781$	3.18202e6			0.186670
$782$		0			0
$783$	−1.35908e7	+	2.62129e6i	−0.792208	+	0.152796i
$784$		0			0
$785$		−	2.07565e7i		−	1.20221i
$786$		0			0
$787$		−	1.27182e7i		−	0.731965i	−0.930622	−	0.365983i	$-0.880733\pi$
$787$		−	1.27182e7i		−	0.731965i	0.930622	−	0.365983i	$-0.119267\pi$
$788$		0			0
$789$	2.00838e6	+	3.15798e7i	0.114856	+	1.80600i
$790$		0			0
$791$	−1.15307e7			−0.655264
$792$		0			0
$793$	−535137.			−0.0302191
$794$		0			0
$795$	1.92588e6	+	3.02827e7i	0.108072	+	1.69933i
$796$		0			0
$797$		−	1.13864e7i		−	0.634951i	−0.948266	−	0.317476i	$-0.897165\pi$
$797$		−	1.13864e7i		−	0.634951i	0.948266	−	0.317476i	$-0.102835\pi$
$798$		0			0
$799$			1.23028e7i			0.681771i
$800$		0			0
$801$	491543.	+	3.84890e6i	0.0270695	+	0.211960i
$802$		0			0
$803$	1.50712e7			0.824817
$804$		0			0
$805$	−2.51447e6			−0.136759
$806$		0			0
$807$	1.28587e7	−	817773.i	0.695047	−	0.0442027i
$808$		0			0
$809$		−	3.09552e7i		−	1.66289i	−0.555611	−	0.831443i	$-0.687515\pi$
$809$		−	3.09552e7i		−	1.66289i	0.555611	−	0.831443i	$-0.312485\pi$
$810$		0			0
$811$		−	8.94919e6i		−	0.477784i	−0.971046	−	0.238892i	$-0.923216\pi$
$811$		−	8.94919e6i		−	0.477784i	0.971046	−	0.238892i	$-0.0767841\pi$
$812$		0			0
$813$	−8.98082e6	+	571151.i	−0.476529	+	0.0303057i
$814$		0			0
$815$	1.68228e7			0.887164
$816$		0			0
$817$	324243.			0.0169948
$818$		0			0
$819$	−477872.	−	3.74185e6i	−0.0248944	−	0.194929i
$820$		0			0
$821$			2.59222e7i			1.34219i	0.741372	+	0.671094i	$0.234175\pi$
$821$			2.59222e7i			1.34219i	−0.741372	+	0.671094i	$0.765825\pi$
$822$		0			0
$823$		−	8.57580e6i		−	0.441342i	−0.975348	−	0.220671i	$-0.929175\pi$
$823$		−	8.57580e6i		−	0.441342i	0.975348	−	0.220671i	$-0.0708247\pi$
$824$		0			0
$825$	245840.	+	3.86560e6i	0.0125753	+	0.197734i
$826$		0			0
$827$	−2.15654e7			−1.09646			−0.548232	−	0.836326i	$-0.684699\pi$
$827$	−2.15654e7			−1.09646			−0.548232	+	0.836326i	$0.684699\pi$
$828$		0			0
$829$	−1.12730e7			−0.569711			−0.284856	−	0.958570i	$-0.591946\pi$
$829$	−1.12730e7			−0.569711			−0.284856	+	0.958570i	$0.591946\pi$
$830$		0			0
$831$	−544995.	−	8.56954e6i	−0.0273773	−	0.430482i
$832$		0			0
$833$			5.27393e6i			0.263343i
$834$		0			0
$835$			2.39041e7i			1.18647i
$836$		0			0
$837$	−1.15284e7	+	2.22352e6i	−0.568795	+	0.109705i
$838$		0			0
$839$	−1.92151e6			−0.0942407			−0.0471204	−	0.998889i	$-0.515004\pi$
$839$	−1.92151e6			−0.0942407			−0.0471204	+	0.998889i	$0.515004\pi$
$840$		0			0
$841$	7.15962e6			0.349060
$842$		0			0
$843$	−6.76903e6	+	430489.i	−0.328063	+	0.0208638i
$844$		0			0
$845$		−	1.70064e7i		−	0.819353i
$846$		0			0
$847$			3.33980e6i			0.159960i
$848$		0			0
$849$	−2.82324e7	+	1.79549e6i	−1.34424	+	0.0854896i
$850$		0			0
$851$	5.50846e6			0.260739
$852$		0			0
$853$	6.88905e6			0.324180			0.162090	−	0.986776i	$-0.448176\pi$
$853$	6.88905e6			0.324180			0.162090	+	0.986776i	$0.448176\pi$
$854$		0			0
$855$	1.96583e7	−	2.51056e6i	0.919666	−	0.117451i
$856$		0			0
$857$			2.33579e7i			1.08638i	0.839609	+	0.543191i	$0.182784\pi$
$857$			2.33579e7i			1.08638i	−0.839609	+	0.543191i	$0.817216\pi$
$858$		0			0
$859$			523874.i			0.0242239i	0.999927	+	0.0121119i	$0.00385544\pi$
$859$			523874.i			0.0242239i	−0.999927	+	0.0121119i	$0.996145\pi$
$860$		0			0
$861$	−449065.	−	7.06113e6i	−0.0206444	−	0.324613i
$862$		0			0
$863$	1.77753e7			0.812436			0.406218	−	0.913776i	$-0.366847\pi$
$863$	1.77753e7			0.812436			0.406218	+	0.913776i	$0.366847\pi$
$864$		0			0
$865$	2.67188e7			1.21416
$866$		0			0
$867$	3.36884e6	+	5.29718e7i	0.152206	+	2.39330i
$868$		0			0
$869$			1.82248e7i			0.818676i
$870$		0			0
$871$		−	1.00015e7i		−	0.446703i
$872$		0			0
$873$	−653258.	+	83427.7i	−0.0290101	+	0.00370488i
$874$		0			0
$875$	7.10428e6			0.313690
$876$		0			0
$877$	−3.88984e7			−1.70778			−0.853891	−	0.520452i	$-0.825763\pi$
$877$	−3.88984e7			−1.70778			−0.853891	+	0.520452i	$0.825763\pi$
$878$		0			0
$879$	−3.73423e7	+	2.37485e6i	−1.63015	+	0.103673i
$880$		0			0
$881$		−	1.25981e7i		−	0.546846i	−0.961894	−	0.273423i	$-0.911844\pi$
$881$		−	1.25981e7i		−	0.546846i	0.961894	−	0.273423i	$-0.0881558\pi$
$882$		0			0
$883$			2.94612e7i			1.27159i	0.771857	+	0.635796i	$0.219328\pi$
$883$			2.94612e7i			1.27159i	−0.771857	+	0.635796i	$0.780672\pi$
$884$		0			0
$885$	6.54138e6	−	416010.i	0.280744	−	0.0178544i
$886$		0			0
$887$	−1.10514e7			−0.471637			−0.235819	−	0.971797i	$-0.575777\pi$
$887$	−1.10514e7			−0.471637			−0.235819	+	0.971797i	$0.575777\pi$
$888$		0			0
$889$	1.64448e7			0.697870
$890$		0			0
$891$	1.74194e7	−	4.52304e6i	0.735088	−	0.190869i
$892$		0			0
$893$		−	7.27699e6i		−	0.305368i
$894$		0			0
$895$			2.68787e7i			1.12163i
$896$		0			0
$897$	−256240.	−	4.02913e6i	−0.0106332	−	0.167198i
$898$		0			0
$899$	−1.13255e7			−0.467366
$900$		0			0
$901$	6.81158e7			2.79535
$902$		0			0
$903$	12098.7	+	190241.i	0.000493765	+	0.00776399i
$904$		0			0
$905$			6.97137e6i			0.282942i
$906$		0			0
$907$			2.73768e7i			1.10501i	0.833511	+	0.552504i	$0.186327\pi$
$907$			2.73768e7i			1.10501i	−0.833511	+	0.552504i	$0.813673\pi$
$908$		0			0
$909$	−5.53895e6	−	4.33713e7i	−0.222340	−	1.74098i
$910$		0			0
$911$	−6.57624e6			−0.262532			−0.131266	−	0.991347i	$-0.541904\pi$
$911$	−6.57624e6			−0.262532			−0.131266	+	0.991347i	$0.541904\pi$
$912$		0			0
$913$	2.08096e7			0.826201
$914$		0			0
$915$	1.64952e6	−	104904.i	0.0651336	−	0.00414229i
$916$		0			0
$917$			1.65135e7i			0.648509i
$918$		0			0
$919$			1.62091e7i			0.633096i	0.948577	+	0.316548i	$0.102524\pi$
$919$			1.62091e7i			0.633096i	−0.948577	+	0.316548i	$0.897476\pi$
$920$		0			0
$921$	4.60938e7	−	2.93142e6i	1.79058	−	0.113875i
$922$		0			0
$923$	−3.30759e6			−0.127793
$924$		0			0
$925$	5.49344e6			0.211101
$926$		0			0
$927$	−3.88406e6	−	3.04131e7i	−0.148452	−	1.16241i
$928$		0			0
$929$			2.67909e7i			1.01847i	0.860627	+	0.509235i	$0.170072\pi$
$929$			2.67909e7i			1.01847i	−0.860627	+	0.509235i	$0.829928\pi$
$930$		0			0
$931$		−	3.11947e6i		−	0.117952i
$932$		0			0
$933$	−2.82375e6	−	4.44009e7i	−0.106200	−	1.66989i
$934$		0			0
$935$	4.20237e7			1.57205
$936$		0			0
$937$	4.54913e7			1.69270			0.846349	−	0.532629i	$-0.178796\pi$
$937$	4.54913e7			1.69270			0.846349	+	0.532629i	$0.178796\pi$
$938$		0			0
$939$	2.71412e6	+	4.26769e7i	0.100453	+	1.57954i
$940$		0			0
$941$		−	2.90128e7i		−	1.06811i	−0.845450	−	0.534055i	$-0.820668\pi$
$941$		−	2.90128e7i		−	1.06811i	0.845450	−	0.534055i	$-0.179332\pi$
$942$		0			0
$943$		−	7.57250e6i		−	0.277306i
$944$		0			0
$945$	2.20653e6	+	1.14403e7i	0.0803767	+	0.416733i
$946$		0			0
$947$	1.99365e7			0.722395			0.361198	−	0.932489i	$-0.382368\pi$
$947$	1.99365e7			0.722395			0.361198	+	0.932489i	$0.382368\pi$
$948$		0			0
$949$	−1.56659e7			−0.564663
$950$		0			0
$951$	1.71408e7	−	1.09010e6i	0.614582	−	0.0390854i
$952$		0			0
$953$			3.47256e7i			1.23856i	0.785170	+	0.619280i	$0.212575\pi$
$953$			3.47256e7i			1.23856i	−0.785170	+	0.619280i	$0.787425\pi$
$954$		0			0
$955$		−	1.71687e7i		−	0.609156i
$956$		0			0
$957$	1.73253e7	−	1.10183e6i	0.611506	−	0.0388898i
$958$		0			0
$959$	1.04239e7			0.366001
$960$		0			0
$961$	1.90223e7			0.664437
$962$		0			0
$963$	−3.70817e7	+	4.73571e6i	−1.28853	+	0.164558i
$964$		0			0
$965$		−	1.26185e7i		−	0.436204i
$966$		0			0
$967$			3.19106e7i			1.09741i	0.836016	+	0.548705i	$0.184879\pi$
$967$			3.19106e7i			1.09741i	−0.836016	+	0.548705i	$0.815121\pi$
$968$		0			0
$969$	−2.82354e6	−	4.43975e7i	−0.0966015	−	1.51897i
$970$		0			0
$971$	2.13637e7			0.727157			0.363578	−	0.931564i	$-0.381555\pi$
$971$	2.13637e7			0.727157			0.363578	+	0.931564i	$0.381555\pi$
$972$		0			0
$973$	1.30776e7			0.442838
$974$		0			0
$975$	−255541.	−	4.01814e6i	−0.00860893	−	0.135367i
$976$		0			0
$977$			3.15697e7i			1.05812i	0.848585	+	0.529059i	$0.177455\pi$
$977$			3.15697e7i			1.05812i	−0.848585	+	0.529059i	$0.822545\pi$
$978$		0			0
$979$		−	4.86667e6i		−	0.162284i
$980$		0			0
$981$	−9.95198e6	+	1.27097e6i	−0.330170	+	0.0421660i
$982$		0			0
$983$	−1.86037e7			−0.614065			−0.307033	−	0.951699i	$-0.599336\pi$
$983$	−1.86037e7			−0.614065			−0.307033	+	0.951699i	$0.599336\pi$
$984$		0			0
$985$	1.76181e6			0.0578587
$986$		0			0
$987$	4.26959e6	−	271532.i	0.139506	−	0.00887213i
$988$		0			0
$989$			204018.i			0.00663252i
$990$		0			0
$991$			6.00220e6i			0.194145i	0.995277	+	0.0970725i	$0.0309479\pi$
$991$			6.00220e6i			0.194145i	−0.995277	+	0.0970725i	$0.969052\pi$
$992$		0			0
$993$	2.58027e7	−	1.64097e6i	0.830408	−	0.0528113i
$994$		0			0
$995$	4.76702e7			1.52647
$996$		0			0
$997$	−3.45422e6			−0.110056			−0.0550278	−	0.998485i	$-0.517525\pi$
$997$	−3.45422e6			−0.110056			−0.0550278	+	0.998485i	$0.517525\pi$
$998$		0			0
$999$	−4.83385e6	−	2.50623e7i	−0.153243	−	0.794524i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.h.b.239.19	✓	40
3.2	odd	2	inner	336.6.h.b.239.21	yes	40
4.3	odd	2	inner	336.6.h.b.239.22	yes	40
12.11	even	2	inner	336.6.h.b.239.20	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.6.h.b.239.19	✓	40	1.1	even	1	trivial
336.6.h.b.239.20	yes	40	12.11	even	2	inner
336.6.h.b.239.21	yes	40	3.2	odd	2	inner
336.6.h.b.239.22	yes	40	4.3	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.989377 - 15.5570i) q^{3} +62.7716i q^{5} -49.0000i q^{7} +(-241.042 + 30.7835i) q^{9} +O(q^{10})\)	\(q+(-0.989377 - 15.5570i) q^{3} +62.7716i q^{5} -49.0000i q^{7} +(-241.042 + 30.7835i) q^{9} +304.782 q^{11} -316.809 q^{13} +(976.539 - 62.1047i) q^{15} -2196.56i q^{17} +1299.24i q^{19} +(-762.294 + 48.4795i) q^{21} -817.500 q^{23} -815.271 q^{25} +(717.382 + 3719.44i) q^{27} +3653.97i q^{29} +3099.50i q^{31} +(-301.544 - 4741.49i) q^{33} +3075.81 q^{35} -6738.18 q^{37} +(313.443 + 4928.60i) q^{39} +9263.00i q^{41} -249.564i q^{43} +(-1932.33 - 15130.6i) q^{45} -5600.97 q^{47} -2401.00 q^{49} +(-34171.9 + 2173.22i) q^{51} +31010.2i q^{53} +19131.6i q^{55} +(20212.3 - 1285.44i) q^{57} +6698.53 q^{59} +1689.15 q^{61} +(1508.39 + 11811.1i) q^{63} -19886.6i q^{65} +31569.5i q^{67} +(808.815 + 12717.9i) q^{69} +10440.3 q^{71} +49449.0 q^{73} +(806.610 + 12683.2i) q^{75} -14934.3i q^{77} +59796.1i q^{79} +(57153.7 - 14840.3i) q^{81} +68276.9 q^{83} +137881. q^{85} +(56845.0 - 3615.16i) q^{87} -15967.7i q^{89} +15523.6i q^{91} +(48218.9 - 3066.57i) q^{93} -81555.3 q^{95} +2710.14 q^{97} +(-73465.2 + 9382.25i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{9}+O(q^{10}) \) 40 * q + 8 * q^9	\( 40 q + 8 q^{9} - 1048 q^{13} + 980 q^{21} - 43416 q^{25} + 20296 q^{33} - 16192 q^{37} + 56488 q^{45} - 96040 q^{49} + 31088 q^{57} + 173112 q^{61} - 114176 q^{69} - 267488 q^{73} + 64888 q^{81} + 508112 q^{85} - 224544 q^{93} - 276400 q^{97}+O(q^{100}) \) 40 * q + 8 * q^9 - 1048 * q^13 + 980 * q^21 - 43416 * q^25 + 20296 * q^33 - 16192 * q^37 + 56488 * q^45 - 96040 * q^49 + 31088 * q^57 + 173112 * q^61 - 114176 * q^69 - 267488 * q^73 + 64888 * q^81 + 508112 * q^85 - 224544 * q^93 - 276400 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more