LMFDB - Embedded newform 207.6.c.a.206.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(206,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("207.206");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.c (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:	$33.1994507013$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			206.13
Character	$\chi$	$=$	207.206
Dual form			207.6.c.a.206.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.94564i q^{2} -31.1332 q^{4} -36.1472 q^{5} -129.914i q^{7} -6.88755i q^{8} +O(q^{10})$	\(q-7.94564i q^{2} -31.1332 q^{4} -36.1472 q^{5} -129.914i q^{7} -6.88755i q^{8} +287.213i q^{10} +161.441 q^{11} -436.884 q^{13} -1032.25 q^{14} -1050.99 q^{16} -707.940 q^{17} -2179.32i q^{19} +1125.38 q^{20} -1282.75i q^{22} +(833.061 + 2396.32i) q^{23} -1818.38 q^{25} +3471.33i q^{26} +4044.62i q^{28} +3683.19i q^{29} +2407.23 q^{31} +8130.36i q^{32} +5625.03i q^{34} +4696.02i q^{35} -10233.0i q^{37} -17316.1 q^{38} +248.966i q^{40} +7242.22i q^{41} +23136.8i q^{43} -5026.18 q^{44} +(19040.3 - 6619.20i) q^{46} +14630.9i q^{47} -70.5680 q^{49} +14448.2i q^{50} +13601.6 q^{52} +6085.92 q^{53} -5835.66 q^{55} -894.787 q^{56} +29265.3 q^{58} -1746.76i q^{59} -33889.2i q^{61} -19127.0i q^{62} +30969.3 q^{64} +15792.2 q^{65} +16243.7i q^{67} +22040.4 q^{68} +37312.9 q^{70} -33252.5i q^{71} -841.330 q^{73} -81307.4 q^{74} +67849.1i q^{76} -20973.4i q^{77} +68291.7i q^{79} +37990.3 q^{80} +57544.1 q^{82} -37877.9 q^{83} +25590.1 q^{85} +183837. q^{86} -1111.94i q^{88} -69995.4 q^{89} +56757.3i q^{91} +(-25935.8 - 74605.0i) q^{92} +116252. q^{94} +78776.4i q^{95} +2494.38i q^{97} +560.708i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 600 q^{4}+O(q^{10}) $ 40 * q - 600 * q^4	$ 40 q - 600 q^{4} - 1048 q^{13} + 9728 q^{16} + 14704 q^{25} + 4640 q^{31} - 91864 q^{46} - 8192 q^{49} + 150360 q^{52} + 134592 q^{55} - 195704 q^{58} - 183416 q^{64} - 257448 q^{70} + 31088 q^{73} - 77096 q^{82} - 368760 q^{85} - 123512 q^{94}+O(q^{100}) $ 40 * q - 600 * q^4 - 1048 * q^13 + 9728 * q^16 + 14704 * q^25 + 4640 * q^31 - 91864 * q^46 - 8192 * q^49 + 150360 * q^52 + 134592 * q^55 - 195704 * q^58 - 183416 * q^64 - 257448 * q^70 + 31088 * q^73 - 77096 * q^82 - 368760 * q^85 - 123512 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/207\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	7.94564i		−	1.40460i	−0.711879	−	0.702302i	$-0.752156\pi$
$2$		−	7.94564i		−	1.40460i	0.711879	−	0.702302i	$-0.247844\pi$
$3$		0			0
$3$		0			0
$4$	−31.1332			−0.972911
$5$	−36.1472			−0.646622			−0.323311	−	0.946293i	$-0.604796\pi$
$5$	−36.1472			−0.646622			−0.323311	+	0.946293i	$0.604796\pi$
$6$		0			0
$7$		−	129.914i		−	1.00210i	−0.865419	−	0.501049i	$-0.832948\pi$
$7$		−	129.914i		−	1.00210i	0.865419	−	0.501049i	$-0.167052\pi$
$8$		−	6.88755i		−	0.0380487i
$9$		0			0
$10$			287.213i			0.908247i
$11$	161.441			0.402284			0.201142	−	0.979562i	$-0.435535\pi$
$11$	161.441			0.402284			0.201142	+	0.979562i	$0.435535\pi$
$12$		0			0
$13$	−436.884			−0.716982			−0.358491	−	0.933533i	$-0.616709\pi$
$13$	−436.884			−0.716982			−0.358491	+	0.933533i	$0.616709\pi$
$14$	−1032.25			−1.40755
$15$		0			0
$16$	−1050.99			−1.02635
$17$	−707.940			−0.594120			−0.297060	−	0.954859i	$-0.596006\pi$
$17$	−707.940			−0.594120			−0.297060	+	0.954859i	$0.596006\pi$
$18$		0			0
$19$		−	2179.32i		−	1.38496i	−0.721438	−	0.692479i	$-0.756518\pi$
$19$		−	2179.32i		−	1.38496i	0.721438	−	0.692479i	$-0.243482\pi$
$20$	1125.38			0.629105
$21$		0			0
$22$		−	1282.75i		−	0.565050i
$23$	833.061	+	2396.32i	0.328365	+	0.944551i
$23$	833.061	+	2396.32i	0.328365	+	0.944551i
$24$		0			0
$25$	−1818.38			−0.581881
$26$			3471.33i			1.00708i
$27$		0			0
$28$			4044.62i			0.974952i
$29$			3683.19i			0.813260i	0.913593	+	0.406630i	$0.133296\pi$
$29$			3683.19i			0.813260i	−0.913593	+	0.406630i	$0.866704\pi$
$30$		0			0
$31$	2407.23			0.449898			0.224949	−	0.974371i	$-0.427778\pi$
$31$	2407.23			0.449898			0.224949	+	0.974371i	$0.427778\pi$
$32$			8130.36i			1.40357i
$33$		0			0
$34$			5625.03i			0.834503i
$35$			4696.02i			0.647978i
$36$		0			0
$37$		−	10233.0i		−	1.22885i	−0.788977	−	0.614423i	$-0.789389\pi$
$37$		−	10233.0i		−	1.22885i	0.788977	−	0.614423i	$-0.210611\pi$
$38$	−17316.1			−1.94532
$39$		0			0
$40$			248.966i			0.0246031i
$41$			7242.22i			0.672841i	0.941712	+	0.336420i	$0.109216\pi$
$41$			7242.22i			0.672841i	−0.941712	+	0.336420i	$0.890784\pi$
$42$		0			0
$43$			23136.8i			1.90824i	0.299427	+	0.954119i	$0.403205\pi$
$43$			23136.8i			1.90824i	−0.299427	+	0.954119i	$0.596795\pi$
$44$	−5026.18			−0.391387
$45$		0			0
$46$	19040.3	−	6619.20i	1.32672	−	0.461223i
$47$			14630.9i			0.966112i	0.875589	+	0.483056i	$0.160473\pi$
$47$			14630.9i			0.966112i	−0.875589	+	0.483056i	$0.839527\pi$
$48$		0			0
$49$	−70.5680			−0.00419873
$50$			14448.2i			0.817312i
$51$		0			0
$52$	13601.6			0.697560
$53$	6085.92			0.297603			0.148801	−	0.988867i	$-0.452459\pi$
$53$	6085.92			0.297603			0.148801	+	0.988867i	$0.452459\pi$
$54$		0			0
$55$	−5835.66			−0.260126
$56$	−894.787			−0.0381285
$57$		0			0
$58$	29265.3			1.14231
$59$		−	1746.76i		−	0.0653287i	−0.999466	−	0.0326644i	$-0.989601\pi$
$59$		−	1746.76i		−	0.0653287i	0.999466	−	0.0326644i	$-0.0103992\pi$
$60$		0			0
$61$		−	33889.2i		−	1.16610i	−0.812435	−	0.583051i	$-0.801859\pi$
$61$		−	33889.2i		−	1.16610i	0.812435	−	0.583051i	$-0.198141\pi$
$62$		−	19127.0i		−	0.631928i
$63$		0			0
$64$	30969.3			0.945109
$65$	15792.2			0.463616
$66$		0			0
$67$			16243.7i			0.442077i	0.975265	+	0.221039i	$0.0709447\pi$
$67$			16243.7i			0.442077i	−0.975265	+	0.221039i	$0.929055\pi$
$68$	22040.4			0.578026
$69$		0			0
$70$	37312.9			0.910152
$71$		−	33252.5i		−	0.782849i	−0.920210	−	0.391425i	$-0.871982\pi$
$71$		−	33252.5i		−	0.782849i	0.920210	−	0.391425i	$-0.128018\pi$
$72$		0			0
$73$	−841.330			−0.0184782			−0.00923909	−	0.999957i	$-0.502941\pi$
$73$	−841.330			−0.0184782			−0.00923909	+	0.999957i	$0.502941\pi$
$74$	−81307.4			−1.72604
$75$		0			0
$76$			67849.1i			1.34744i
$77$		−	20973.4i		−	0.403128i
$78$		0			0
$79$			68291.7i			1.23112i	0.788090	+	0.615560i	$0.211070\pi$
$79$			68291.7i			1.23112i	−0.788090	+	0.615560i	$0.788930\pi$
$80$	37990.3			0.663663
$81$		0			0
$82$	57544.1			0.945074
$83$	−37877.9			−0.603519			−0.301760	−	0.953384i	$-0.597574\pi$
$83$	−37877.9			−0.603519			−0.301760	+	0.953384i	$0.597574\pi$
$84$		0			0
$85$	25590.1			0.384171
$86$	183837.			2.68032
$87$		0			0
$88$		−	1111.94i		−	0.0153064i
$89$	−69995.4			−0.936687			−0.468344	−	0.883546i	$-0.655149\pi$
$89$	−69995.4			−0.936687			−0.468344	+	0.883546i	$0.655149\pi$
$90$		0			0
$91$			56757.3i			0.718486i
$92$	−25935.8	−	74605.0i	−0.319470	−	0.918964i
$93$		0			0
$94$	116252.			1.35701
$95$			78776.4i			0.895544i
$96$		0			0
$97$			2494.38i			0.0269174i	0.999909	+	0.0134587i	$0.00428416\pi$
$97$			2494.38i			0.0269174i	−0.999909	+	0.0134587i	$0.995716\pi$
$98$			560.708i			0.00589755i
$99$		0			0
$100$	56611.8			0.566118
$101$			85360.2i			0.832630i	0.909221	+	0.416315i	$0.136679\pi$
$101$			85360.2i			0.832630i	−0.909221	+	0.416315i	$0.863321\pi$
$102$		0			0
$103$		−	141370.i		−	1.31300i	−0.754328	−	0.656498i	$-0.772037\pi$
$103$		−	141370.i		−	1.31300i	0.754328	−	0.656498i	$-0.227963\pi$
$104$			3009.06i			0.0272802i
$105$		0			0
$106$		−	48356.5i		−	0.418014i
$107$	46706.3			0.394381			0.197191	−	0.980365i	$-0.436818\pi$
$107$	46706.3			0.394381			0.197191	+	0.980365i	$0.436818\pi$
$108$		0			0
$109$		−	63506.1i		−	0.511975i	−0.966680	−	0.255988i	$-0.917599\pi$
$109$		−	63506.1i		−	0.511975i	0.966680	−	0.255988i	$-0.0824007\pi$
$110$			46368.0i			0.365373i
$111$		0			0
$112$			136538.i			1.02851i
$113$	−163947.			−1.20783			−0.603916	−	0.797048i	$-0.706394\pi$
$113$	−163947.			−1.20783			−0.603916	+	0.797048i	$0.706394\pi$
$114$		0			0
$115$	−30112.8	−	86620.4i	−0.212328	−	0.610767i
$116$		−	114669.i		−	0.791230i
$117$		0			0
$118$	−13879.2			−0.0917610
$119$			91971.1i			0.595366i
$120$		0			0
$121$	−134988.			−0.838167
$122$	−269271.			−1.63791
$123$		0			0
$124$	−74944.8			−0.437711
$125$	178689.			1.02288
$126$		0			0
$127$	−205801.			−1.13224			−0.566120	−	0.824323i	$-0.691556\pi$
$127$	−205801.			−1.13224			−0.566120	+	0.824323i	$0.691556\pi$
$128$			14100.5i			0.0760695i
$129$		0			0
$130$		−	125479.i		−	0.651197i
$131$			149239.i			0.759806i	0.925026	+	0.379903i	$0.124043\pi$
$131$			149239.i			0.759806i	−0.925026	+	0.379903i	$0.875957\pi$
$132$		0			0
$133$	−283123.			−1.38786
$134$	129067.			0.620943
$135$		0			0
$136$			4875.97i			0.0226055i
$137$	206345.			0.939276			0.469638	−	0.882859i	$-0.344385\pi$
$137$	206345.			0.939276			0.469638	+	0.882859i	$0.344385\pi$
$138$		0			0
$139$	−196099.			−0.860871			−0.430435	−	0.902621i	$-0.641640\pi$
$139$	−196099.			−0.860871			−0.430435	+	0.902621i	$0.641640\pi$
$140$		−	146202.i		−	0.630425i
$141$		0			0
$142$	−264212.			−1.09959
$143$	−70531.2			−0.288431
$144$		0			0
$145$		−	133137.i		−	0.525872i
$146$			6684.91i			0.0259545i
$147$		0			0
$148$			318585.i			1.19556i
$149$	−496593.			−1.83246			−0.916231	−	0.400649i	$-0.868785\pi$
$149$	−496593.			−1.83246			−0.916231	+	0.400649i	$0.868785\pi$
$150$		0			0
$151$	−359543.			−1.28324			−0.641622	−	0.767021i	$-0.721738\pi$
$151$	−359543.			−1.28324			−0.641622	+	0.767021i	$0.721738\pi$
$152$	−15010.2			−0.0526959
$153$		0			0
$154$	−166647.			−0.566235
$155$	−87014.8			−0.290914
$156$		0			0
$157$		−	481507.i		−	1.55903i	−0.626385	−	0.779514i	$-0.715466\pi$
$157$		−	481507.i		−	1.55903i	0.626385	−	0.779514i	$-0.284534\pi$
$158$	542621.			1.72924
$159$		0			0
$160$		−	293890.i		−	0.907581i
$161$	311315.	−	108226.i	0.946532	−	0.329054i
$162$		0			0
$163$	7949.51			0.0234353			0.0117177	−	0.999931i	$-0.496270\pi$
$163$	7949.51			0.0234353			0.0117177	+	0.999931i	$0.496270\pi$
$164$		−	225473.i		−	0.654614i
$165$		0			0
$166$			300964.i			0.847705i
$167$		−	41126.5i		−	0.114112i	−0.998371	−	0.0570560i	$-0.981829\pi$
$167$		−	41126.5i		−	0.114112i	0.998371	−	0.0570560i	$-0.0181713\pi$
$168$		0			0
$169$	−180425.			−0.485937
$170$		−	203329.i		−	0.539608i
$171$		0			0
$172$		−	720323.i		−	1.85655i
$173$			251461.i			0.638786i	0.947622	+	0.319393i	$0.103479\pi$
$173$			251461.i			0.638786i	−0.947622	+	0.319393i	$0.896521\pi$
$174$		0			0
$175$			236232.i			0.583101i
$176$	−169673.			−0.412886
$177$		0			0
$178$			556158.i			1.31567i
$179$		−	485811.i		−	1.13327i	−0.823968	−	0.566637i	$-0.808244\pi$
$179$		−	485811.i		−	1.13327i	0.823968	−	0.566637i	$-0.191756\pi$
$180$		0			0
$181$			28952.8i			0.0656893i	0.999460	+	0.0328446i	$0.0104567\pi$
$181$			28952.8i			0.0656893i	−0.999460	+	0.0328446i	$0.989543\pi$
$182$	450973.			1.00919
$183$		0			0
$184$	16504.8	−	5737.75i	0.0359389	−	0.0124939i
$185$			369893.i			0.794598i
$186$		0			0
$187$	−114291.			−0.239005
$188$		−	455508.i		−	0.939942i
$189$		0			0
$190$	625929.			1.25788
$191$	−220082.			−0.436518			−0.218259	−	0.975891i	$-0.570038\pi$
$191$	−220082.			−0.436518			−0.218259	+	0.975891i	$0.570038\pi$
$192$		0			0
$193$	332552.			0.642638			0.321319	−	0.946971i	$-0.395874\pi$
$193$	332552.			0.642638			0.321319	+	0.946971i	$0.395874\pi$
$194$	19819.4			0.0378082
$195$		0			0
$196$	2197.01			0.00408499
$197$		−	413036.i		−	0.758267i	−0.925342	−	0.379134i	$-0.876222\pi$
$197$		−	413036.i		−	0.758267i	0.925342	−	0.379134i	$-0.123778\pi$
$198$		0			0
$199$			454666.i			0.813880i	0.913455	+	0.406940i	$0.133404\pi$
$199$			454666.i			0.813880i	−0.913455	+	0.406940i	$0.866596\pi$
$200$			12524.2i			0.0221398i
$201$		0			0
$202$	678241.			1.16951
$203$	478497.			0.814966
$204$		0			0
$205$		−	261786.i		−	0.435073i
$206$	−1.12327e6			−1.84424
$207$		0			0
$208$	459160.			0.735878
$209$		−	351832.i		−	0.557147i
$210$		0			0
$211$	491123.			0.759424			0.379712	−	0.925105i	$-0.376023\pi$
$211$	491123.			0.759424			0.379712	+	0.925105i	$0.376023\pi$
$212$	−189474.			−0.289541
$213$		0			0
$214$		−	371112.i		−	0.553950i
$215$		−	836333.i		−	1.23391i
$216$		0			0
$217$		−	312732.i		−	0.450841i
$218$	−504596.			−0.719123
$219$		0			0
$220$	181683.			0.253079
$221$	309288.			0.425973
$222$		0			0
$223$	−1.14702e6			−1.54458			−0.772289	−	0.635272i	$-0.780888\pi$
$223$	−1.14702e6			−1.54458			−0.772289	+	0.635272i	$0.780888\pi$
$224$	1.05625e6			1.40652
$225$		0			0
$226$			1.30266e6i			1.69652i
$227$	1.28677e6			1.65744			0.828719	−	0.559665i	$-0.189070\pi$
$227$	1.28677e6			1.65744			0.828719	+	0.559665i	$0.189070\pi$
$228$		0			0
$229$			518304.i			0.653124i	0.945176	+	0.326562i	$0.105890\pi$
$229$			518304.i			0.653124i	−0.945176	+	0.326562i	$0.894110\pi$
$230$	−688254.	+	239266.i	−0.857885	+	0.298237i
$231$		0			0
$232$	25368.2			0.0309435
$233$		−	170515.i		−	0.205766i	−0.994693	−	0.102883i	$-0.967193\pi$
$233$		−	170515.i		−	0.205766i	0.994693	−	0.102883i	$-0.0328067\pi$
$234$		0			0
$235$		−	528868.i		−	0.624709i
$236$			54382.3i			0.0635591i
$237$		0			0
$238$	730769.			0.836253
$239$		−	240329.i		−	0.272152i	−0.990698	−	0.136076i	$-0.956551\pi$
$239$		−	240329.i		−	0.272152i	0.990698	−	0.136076i	$-0.0434492\pi$
$240$		0			0
$241$		−	298553.i		−	0.331114i	−0.986200	−	0.165557i	$-0.947058\pi$
$241$		−	298553.i		−	0.331114i	0.986200	−	0.165557i	$-0.0529423\pi$
$242$			1.07256e6i			1.17729i
$243$		0			0
$244$			1.05508e6i			1.13451i
$245$	2550.84			0.00271499
$246$		0			0
$247$			952111.i			0.992990i
$248$		−	16579.9i		−	0.0171180i
$249$		0			0
$250$		−	1.41980e6i		−	1.43674i
$251$	−313290.			−0.313879			−0.156940	−	0.987608i	$-0.550163\pi$
$251$	−313290.			−0.313879			−0.156940	+	0.987608i	$0.550163\pi$
$252$		0			0
$253$	134490.	+	386865.i	0.132096	+	0.379978i
$254$			1.63522e6i			1.59035i
$255$		0			0
$256$	1.10306e6			1.05196
$257$		−	1.18110e6i		−	1.11546i	−0.830022	−	0.557730i	$-0.811672\pi$
$257$		−	1.18110e6i		−	1.11546i	0.830022	−	0.557730i	$-0.188328\pi$
$258$		0			0
$259$	−1.32940e6			−1.23142
$260$	−491660.			−0.451057
$261$		0			0
$262$	1.18580e6			1.06723
$263$	−543901.			−0.484876			−0.242438	−	0.970167i	$-0.577947\pi$
$263$	−543901.			−0.484876			−0.242438	+	0.970167i	$0.577947\pi$
$264$		0			0
$265$	−219989.			−0.192436
$266$			2.24960e6i			1.94940i
$267$		0			0
$268$		−	505718.i		−	0.430102i
$269$			641388.i			0.540431i	0.962800	+	0.270215i	$0.0870949\pi$
$269$			641388.i			0.540431i	−0.962800	+	0.270215i	$0.912905\pi$
$270$		0			0
$271$	735270.			0.608168			0.304084	−	0.952645i	$-0.401650\pi$
$271$	735270.			0.608168			0.304084	+	0.952645i	$0.401650\pi$
$272$	744036.			0.609778
$273$		0			0
$274$		−	1.63954e6i		−	1.31931i
$275$	−293561.			−0.234081
$276$		0			0
$277$	−915728.			−0.717079			−0.358539	−	0.933515i	$-0.616725\pi$
$277$	−915728.			−0.717079			−0.358539	+	0.933515i	$0.616725\pi$
$278$			1.55813e6i			1.20918i
$279$		0			0
$280$	32344.1			0.0246547
$281$	−511915.			−0.386751			−0.193376	−	0.981125i	$-0.561944\pi$
$281$	−511915.			−0.386751			−0.193376	+	0.981125i	$0.561944\pi$
$282$		0			0
$283$		−	1.27834e6i		−	0.948814i	−0.880306	−	0.474407i	$-0.842663\pi$
$283$		−	1.27834e6i		−	0.948814i	0.880306	−	0.474407i	$-0.157337\pi$
$284$			1.03525e6i			0.761643i
$285$		0			0
$286$			560415.i			0.405131i
$287$	940864.			0.674252
$288$		0			0
$289$	−918678.			−0.647022
$290$	−1.05786e6			−0.738641
$291$		0			0
$292$	26193.3			0.0179776
$293$	−1.42386e6			−0.968943			−0.484471	−	0.874807i	$-0.660988\pi$
$293$	−1.42386e6			−0.968943			−0.484471	+	0.874807i	$0.660988\pi$
$294$		0			0
$295$			63140.7i			0.0422430i
$296$	−70480.1			−0.0467560
$297$		0			0
$298$			3.94575e6i			2.57388i
$299$	−363951.	−	1.04692e6i	−0.235432	−	0.677226i
$300$		0			0
$301$	3.00579e6			1.91224
$302$			2.85680e6i			1.80245i
$303$		0			0
$304$			2.29044e6i			1.42146i
$305$			1.22500e6i			0.754027i
$306$		0			0
$307$	199416.			0.120758			0.0603789	−	0.998176i	$-0.480769\pi$
$307$	199416.			0.120758			0.0603789	+	0.998176i	$0.480769\pi$
$308$			652970.i			0.392208i
$309$		0			0
$310$			691388.i			0.408618i
$311$			1.83325e6i			1.07478i	0.843333	+	0.537391i	$0.180590\pi$
$311$			1.83325e6i			1.07478i	−0.843333	+	0.537391i	$0.819410\pi$
$312$		0			0
$313$		−	1.72157e6i		−	0.993260i	−0.867962	−	0.496630i	$-0.834571\pi$
$313$		−	1.72157e6i		−	0.993260i	0.867962	−	0.496630i	$-0.165429\pi$
$314$	−3.82588e6			−2.18982
$315$		0			0
$316$		−	2.12614e6i		−	1.19777i
$317$		−	2.78107e6i		−	1.55440i	−0.629251	−	0.777202i	$-0.716638\pi$
$317$		−	2.78107e6i		−	1.55440i	0.629251	−	0.777202i	$-0.283362\pi$
$318$		0			0
$319$			594620.i			0.327162i
$320$	−1.11946e6			−0.611128
$321$		0			0
$322$	−859925.	−	2.47360e6i	−0.462190	−	1.32950i
$323$			1.54283e6i			0.822831i
$324$		0			0
$325$	794421.			0.417198
$326$		−	63163.9i		−	0.0329174i
$327$		0			0
$328$	49881.2			0.0256007
$329$	1.90076e6			0.968139
$330$		0			0
$331$	1.52534e6			0.765240			0.382620	−	0.923906i	$-0.375022\pi$
$331$	1.52534e6			0.765240			0.382620	+	0.923906i	$0.375022\pi$
$332$	1.17926e6			0.587171
$333$		0			0
$334$	−326777.			−0.160282
$335$		−	587165.i		−	0.285857i
$336$		0			0
$337$			3.26172e6i			1.56449i	0.622971	+	0.782245i	$0.285925\pi$
$337$			3.26172e6i			1.56449i	−0.622971	+	0.782245i	$0.714075\pi$
$338$			1.43359e6i			0.682549i
$339$		0			0
$340$	−796700.			−0.373764
$341$	388627.			0.180987
$342$		0			0
$343$		−	2.17429e6i		−	0.997890i
$344$	159356.			0.0726060
$345$		0			0
$346$	1.99802e6			0.897241
$347$		−	4.04494e6i		−	1.80339i	−0.432376	−	0.901693i	$-0.642325\pi$
$347$		−	4.04494e6i		−	1.80339i	0.432376	−	0.901693i	$-0.357675\pi$
$348$		0			0
$349$	−2.33889e6			−1.02789			−0.513945	−	0.857823i	$-0.671817\pi$
$349$	−2.33889e6			−1.02789			−0.513945	+	0.857823i	$0.671817\pi$
$350$	1.87701e6			0.819026
$351$		0			0
$352$			1.31258e6i			0.564635i
$353$		−	3.77363e6i		−	1.61184i	−0.592022	−	0.805922i	$-0.701670\pi$
$353$		−	3.77363e6i		−	1.61184i	0.592022	−	0.805922i	$-0.298330\pi$
$354$		0			0
$355$			1.20199e6i			0.506207i
$356$	2.17918e6			0.911314
$357$		0			0
$358$	−3.86008e6			−1.59180
$359$	4.18340e6			1.71314			0.856571	−	0.516030i	$-0.172591\pi$
$359$	4.18340e6			1.71314			0.856571	+	0.516030i	$0.172591\pi$
$360$		0			0
$361$	−2.27333e6			−0.918110
$362$	230049.			0.0922674
$363$		0			0
$364$		−	1.76703e6i		−	0.699023i
$365$	30411.8			0.0119484
$366$		0			0
$367$		−	4.93886e6i		−	1.91409i	−0.289947	−	0.957043i	$-0.593638\pi$
$367$		−	4.93886e6i		−	1.91409i	0.289947	−	0.957043i	$-0.406362\pi$
$368$	−875536.	−	2.51850e6i	−0.337019	−	0.969444i
$369$		0			0
$370$	2.93904e6			1.11609
$371$		−	790645.i		−	0.298227i
$372$		0			0
$373$			1.15318e6i			0.429167i	0.976706	+	0.214584i	$0.0688394\pi$
$373$			1.15318e6i			0.429167i	−0.976706	+	0.214584i	$0.931161\pi$
$374$			908113.i			0.335707i
$375$		0			0
$376$	100771.			0.0367593
$377$		−	1.60913e6i		−	0.583093i
$378$		0			0
$379$			865668.i			0.309566i	0.987948	+	0.154783i	$0.0494679\pi$
$379$			865668.i			0.309566i	−0.987948	+	0.154783i	$0.950532\pi$
$380$		−	2.45256e6i		−	0.871285i
$381$		0			0
$382$			1.74869e6i			0.613134i
$383$	2.19858e6			0.765854			0.382927	−	0.923779i	$-0.374916\pi$
$383$	2.19858e6			0.765854			0.382927	+	0.923779i	$0.374916\pi$
$384$		0			0
$385$			758132.i			0.260671i
$386$		−	2.64234e6i		−	0.902652i
$387$		0			0
$388$		−	77657.8i		−	0.0261882i
$389$	−1.22759e6			−0.411320			−0.205660	−	0.978623i	$-0.565934\pi$
$389$	−1.22759e6			−0.411320			−0.205660	+	0.978623i	$0.565934\pi$
$390$		0			0
$391$	−589757.	−	1.69645e6i	−0.195088	−	0.561176i
$392$			486.041i			0.000159756i
$393$		0			0
$394$	−3.28184e6			−1.06507
$395$		−	2.46856e6i		−	0.796068i
$396$		0			0
$397$	1.33341e6			0.424607			0.212303	−	0.977204i	$-0.431903\pi$
$397$	1.33341e6			0.424607			0.212303	+	0.977204i	$0.431903\pi$
$398$	3.61261e6			1.14318
$399$		0			0
$400$	1.91109e6			0.597216
$401$	−4.72666e6			−1.46789			−0.733946	−	0.679208i	$-0.762323\pi$
$401$	−4.72666e6			−1.46789			−0.733946	+	0.679208i	$0.762323\pi$
$402$		0			0
$403$	−1.05168e6			−0.322569
$404$		−	2.65753e6i		−	0.810075i
$405$		0			0
$406$		−	3.80197e6i		−	1.14470i
$407$		−	1.65202e6i		−	0.494345i
$408$		0			0
$409$	−516188.			−0.152581			−0.0762903	−	0.997086i	$-0.524308\pi$
$409$	−516188.			−0.152581			−0.0762903	+	0.997086i	$0.524308\pi$
$410$	−2.08006e6			−0.611105
$411$		0			0
$412$			4.40129e6i			1.27743i
$413$	−226929.			−0.0654657
$414$		0			0
$415$	1.36918e6			0.390249
$416$		−	3.55203e6i		−	1.00634i
$417$		0			0
$418$	−2.79553e6			−0.782571
$419$	−2.89621e6			−0.805924			−0.402962	−	0.915217i	$-0.632019\pi$
$419$	−2.89621e6			−0.805924			−0.402962	+	0.915217i	$0.632019\pi$
$420$		0			0
$421$			4.95453e6i			1.36238i	0.732108	+	0.681189i	$0.238537\pi$
$421$			4.95453e6i			1.36238i	−0.732108	+	0.681189i	$0.761463\pi$
$422$		−	3.90229e6i		−	1.06669i
$423$		0			0
$424$		−	41917.1i		−	0.0113234i
$425$	1.28730e6			0.345707
$426$		0			0
$427$	−4.40267e6			−1.16855
$428$	−1.45412e6			−0.383698
$429$		0			0
$430$	−6.64520e6			−1.73315
$431$	2.08579e6			0.540850			0.270425	−	0.962741i	$-0.412836\pi$
$431$	2.08579e6			0.540850			0.270425	+	0.962741i	$0.412836\pi$
$432$		0			0
$433$		−	2.27978e6i		−	0.584351i	−0.956365	−	0.292176i	$-0.905621\pi$
$433$		−	2.27978e6i		−	0.584351i	0.956365	−	0.292176i	$-0.0943792\pi$
$434$	−2.48486e6			−0.633253
$435$		0			0
$436$			1.97715e6i			0.498107i
$437$	5.22235e6	−	1.81551e6i	1.30816	−	0.454772i
$438$		0			0
$439$	4.32178e6			1.07029			0.535144	−	0.844761i	$-0.320257\pi$
$439$	4.32178e6			1.07029			0.535144	+	0.844761i	$0.320257\pi$
$440$			40193.4i			0.00989744i
$441$		0			0
$442$		−	2.45749e6i		−	0.598324i
$443$			754886.i			0.182756i	0.995816	+	0.0913781i	$0.0291272\pi$
$443$			754886.i			0.182756i	−0.995816	+	0.0913781i	$0.970873\pi$
$444$		0			0
$445$	2.53014e6			0.605682
$446$			9.11383e6i			2.16952i
$447$		0			0
$448$		−	4.02334e6i		−	0.947091i
$449$			6.17861e6i			1.44636i	0.690662	+	0.723178i	$0.257319\pi$
$449$			6.17861e6i			1.44636i	−0.690662	+	0.723178i	$0.742681\pi$
$450$		0			0
$451$			1.16919e6i			0.270673i
$452$	5.10418e6			1.17511
$453$		0			0
$454$		−	1.02242e7i		−	2.32804i
$455$		−	2.05162e6i		−	0.464588i
$456$		0			0
$457$		−	4.15982e6i		−	0.931717i	−0.884859	−	0.465858i	$-0.845746\pi$
$457$		−	4.15982e6i		−	0.931717i	0.884859	−	0.465858i	$-0.154254\pi$
$458$	4.11825e6			0.917381
$459$		0			0
$460$	937508.	+	2.69677e6i	0.206576	+	0.594222i
$461$			8.85376e6i			1.94033i	0.242447	+	0.970165i	$0.422050\pi$
$461$			8.85376e6i			1.94033i	−0.242447	+	0.970165i	$0.577950\pi$
$462$		0			0
$463$	2.04602e6			0.443565			0.221782	−	0.975096i	$-0.428813\pi$
$463$	2.04602e6			0.443565			0.221782	+	0.975096i	$0.428813\pi$
$464$		−	3.87099e6i		−	0.834694i
$465$		0			0
$466$	−1.35485e6			−0.289020
$467$	−8.79797e6			−1.86677			−0.933384	−	0.358879i	$-0.883159\pi$
$467$	−8.79797e6			−1.86677			−0.933384	+	0.358879i	$0.883159\pi$
$468$		0			0
$469$	2.11028e6			0.443004
$470$	−4.20220e6			−0.877469
$471$		0			0
$472$	−12030.9			−0.00248567
$473$			3.73524e6i			0.767654i
$474$		0			0
$475$			3.96282e6i			0.805881i
$476$		−	2.86335e6i		−	0.579238i
$477$		0			0
$478$	−1.90957e6			−0.382266
$479$	3.10322e6			0.617979			0.308989	−	0.951065i	$-0.400009\pi$
$479$	3.10322e6			0.617979			0.308989	+	0.951065i	$0.400009\pi$
$480$		0			0
$481$			4.47062e6i			0.881060i
$482$	−2.37219e6			−0.465085
$483$		0			0
$484$	4.20259e6			0.815463
$485$		−	90164.8i		−	0.0174054i
$486$		0			0
$487$	−9.66482e6			−1.84659			−0.923297	−	0.384087i	$-0.874516\pi$
$487$	−9.66482e6			−1.84659			−0.923297	+	0.384087i	$0.874516\pi$
$488$	−233414.			−0.0443687
$489$		0			0
$490$		−	20268.1i		−	0.00381348i
$491$			5.98560e6i			1.12048i	0.828331	+	0.560240i	$0.189291\pi$
$491$			5.98560e6i			1.12048i	−0.828331	+	0.560240i	$0.810709\pi$
$492$		0			0
$493$		−	2.60748e6i		−	0.483174i
$494$	7.56513e6			1.39476
$495$		0			0
$496$	−2.52997e6			−0.461755
$497$	−4.31995e6			−0.784491
$498$		0			0
$499$	−1.10754e7			−1.99117			−0.995585	−	0.0938598i	$-0.970079\pi$
$499$	−1.10754e7			−1.99117			−0.995585	+	0.0938598i	$0.970079\pi$
$500$	−5.56317e6			−0.995170
$501$		0			0
$502$			2.48929e6i			0.440876i
$503$	−4.10773e6			−0.723905			−0.361953	−	0.932196i	$-0.617890\pi$
$503$	−4.10773e6			−0.723905			−0.361953	+	0.932196i	$0.617890\pi$
$504$		0			0
$505$		−	3.08554e6i		−	0.538396i
$506$	3.07389e6	−	1.06861e6i	0.533718	−	0.185543i
$507$		0			0
$508$	6.40724e6			1.10157
$509$			9.84051e6i			1.68354i	0.539838	+	0.841769i	$0.318486\pi$
$509$			9.84051e6i			1.68354i	−0.539838	+	0.841769i	$0.681514\pi$
$510$		0			0
$511$			109300.i			0.0185169i
$512$		−	8.31327e6i		−	1.40151i
$513$		0			0
$514$	−9.38460e6			−1.56678
$515$			5.11013e6i			0.849012i
$516$		0			0
$517$			2.36204e6i			0.388652i
$518$			1.05629e7i			1.72966i
$519$		0			0
$520$		−	108769.i		−	0.0176400i
$521$	4.29853e6			0.693787			0.346893	−	0.937905i	$-0.387237\pi$
$521$	4.29853e6			0.693787			0.346893	+	0.937905i	$0.387237\pi$
$522$		0			0
$523$			4.44773e6i			0.711025i	0.934672	+	0.355512i	$0.115694\pi$
$523$			4.44773e6i			0.711025i	−0.934672	+	0.355512i	$0.884306\pi$
$524$		−	4.64627e6i		−	0.739224i
$525$		0			0
$526$			4.32164e6i			0.681058i
$527$	−1.70418e6			−0.267293
$528$		0			0
$529$	−5.04836e6	+	3.99256e6i	−0.784353	+	0.620315i
$530$			1.74796e6i			0.270297i
$531$		0			0
$532$	8.81453e6			1.35027
$533$		−	3.16401e6i		−	0.482415i
$534$		0			0
$535$	−1.68831e6			−0.255016
$536$	111879.			0.0168205
$537$		0			0
$538$	5.09623e6			0.759091
$539$	−11392.6			−0.00168908
$540$		0			0
$541$	−633658.			−0.0930812			−0.0465406	−	0.998916i	$-0.514820\pi$
$541$	−633658.			−0.0930812			−0.0465406	+	0.998916i	$0.514820\pi$
$542$		−	5.84219e6i		−	0.854236i
$543$		0			0
$544$		−	5.75581e6i		−	0.833891i
$545$			2.29557e6i			0.331054i
$546$		0			0
$547$	2.58765e6			0.369774			0.184887	−	0.982760i	$-0.440808\pi$
$547$	2.58765e6			0.369774			0.184887	+	0.982760i	$0.440808\pi$
$548$	−6.42418e6			−0.913832
$549$		0			0
$550$			2.33253e6i			0.328792i
$551$	8.02685e6			1.12633
$552$		0			0
$553$	8.87203e6			1.23370
$554$			7.27604e6i			1.00721i
$555$		0			0
$556$	6.10518e6			0.837551
$557$	6.86170e6			0.937117			0.468558	−	0.883433i	$-0.344774\pi$
$557$	6.86170e6			0.937117			0.468558	+	0.883433i	$0.344774\pi$
$558$		0			0
$559$		−	1.01081e7i		−	1.36817i
$560$		−	4.93546e6i		−	0.665055i
$561$		0			0
$562$			4.06749e6i			0.543232i
$563$	−1.35168e7			−1.79722			−0.898611	−	0.438747i	$-0.855422\pi$
$563$	−1.35168e7			−1.79722			−0.898611	+	0.438747i	$0.855422\pi$
$564$		0			0
$565$	5.92622e6			0.781010
$566$	−1.01572e7			−1.33271
$567$		0			0
$568$	−229028.			−0.0297864
$569$	−6.73995e6			−0.872722			−0.436361	−	0.899772i	$-0.643733\pi$
$569$	−6.73995e6			−0.872722			−0.436361	+	0.899772i	$0.643733\pi$
$570$		0			0
$571$		−	9.64699e6i		−	1.23823i	−0.785300	−	0.619115i	$-0.787491\pi$
$571$		−	9.64699e6i		−	1.23823i	0.785300	−	0.619115i	$-0.212509\pi$
$572$	2.19586e6			0.280617
$573$		0			0
$574$		−	7.47576e6i		−	0.947056i
$575$	−1.51482e6	−	4.35741e6i	−0.191069	−	0.549616i
$576$		0			0
$577$	1.32536e7			1.65727			0.828634	−	0.559791i	$-0.189118\pi$
$577$	1.32536e7			1.65727			0.828634	+	0.559791i	$0.189118\pi$
$578$			7.29948e6i			0.908809i
$579$		0			0
$580$			4.14499e6i			0.511626i
$581$			4.92086e6i			0.604785i
$582$		0			0
$583$	982519.			0.119721
$584$			5794.71i			0.000703071i
$585$		0			0
$586$			1.13135e7i			1.36098i
$587$			1.76386e6i			0.211285i	0.994404	+	0.105643i	$0.0336899\pi$
$587$			1.76386e6i			0.211285i	−0.994404	+	0.105643i	$0.966310\pi$
$588$		0			0
$589$		−	5.24613e6i		−	0.623090i
$590$	501693.			0.0593346
$591$		0			0
$592$			1.07547e7i			1.26123i
$593$		−	2.28992e6i		−	0.267413i	−0.991021	−	0.133707i	$-0.957312\pi$
$593$		−	2.28992e6i		−	0.267413i	0.991021	−	0.133707i	$-0.0426880\pi$
$594$		0			0
$595$		−	3.32450e6i		−	0.384976i
$596$	1.54605e7			1.78282
$597$		0			0
$598$	−8.31841e6	+	2.89183e6i	−0.951234	+	0.330689i
$599$			9.91751e6i			1.12937i	0.825307	+	0.564684i	$0.191002\pi$
$599$			9.91751e6i			1.12937i	−0.825307	+	0.564684i	$0.808998\pi$
$600$		0			0
$601$	1.39185e7			1.57183			0.785914	−	0.618336i	$-0.212193\pi$
$601$	1.39185e7			1.57183			0.785914	+	0.618336i	$0.212193\pi$
$602$		−	2.38829e7i		−	2.68594i
$603$		0			0
$604$	1.11937e7			1.24848
$605$	4.87943e6			0.541977
$606$		0			0
$607$	1.37959e7			1.51977			0.759887	−	0.650055i	$-0.225254\pi$
$607$	1.37959e7			1.51977			0.759887	+	0.650055i	$0.225254\pi$
$608$	1.77187e7			1.94389
$609$		0			0
$610$	9.73342e6			1.05911
$611$		−	6.39203e6i		−	0.692685i
$612$		0			0
$613$			6.04865e6i			0.650140i	0.945690	+	0.325070i	$0.105388\pi$
$613$			6.04865e6i			0.650140i	−0.945690	+	0.325070i	$0.894612\pi$
$614$		−	1.58449e6i		−	0.169617i
$615$		0			0
$616$	−144456.			−0.0153385
$617$	−3.25234e6			−0.343941			−0.171970	−	0.985102i	$-0.555013\pi$
$617$	−3.25234e6			−0.343941			−0.171970	+	0.985102i	$0.555013\pi$
$618$		0			0
$619$			7.41563e6i			0.777896i	0.921260	+	0.388948i	$0.127161\pi$
$619$			7.41563e6i			0.777896i	−0.921260	+	0.388948i	$0.872839\pi$
$620$	2.70905e6			0.283033
$621$		0			0
$622$	1.45663e7			1.50964
$623$			9.09336e6i			0.938652i
$624$		0			0
$625$	−776703.			−0.0795344
$626$	−1.36790e7			−1.39514
$627$		0			0
$628$			1.49909e7i			1.51680i
$629$			7.24432e6i			0.730081i
$630$		0			0
$631$		−	4.43907e6i		−	0.443832i	−0.975066	−	0.221916i	$-0.928769\pi$
$631$		−	4.43907e6i		−	0.443832i	0.975066	−	0.221916i	$-0.0712310\pi$
$632$	470363.			0.0468425
$633$		0			0
$634$	−2.20974e7			−2.18332
$635$	7.43914e6			0.732130
$636$		0			0
$637$	30830.1			0.00301041
$638$	4.72463e6			0.459533
$639$		0			0
$640$		−	509695.i		−	0.0491882i
$641$	−1.48643e7			−1.42890			−0.714448	−	0.699689i	$-0.753322\pi$
$641$	−1.48643e7			−1.42890			−0.714448	+	0.699689i	$0.753322\pi$
$642$		0			0
$643$			1.91155e7i			1.82330i	0.410966	+	0.911651i	$0.365191\pi$
$643$			1.91155e7i			1.82330i	−0.410966	+	0.911651i	$0.634809\pi$
$644$	−9.69222e6	+	3.36942e6i	−0.920892	+	0.320140i
$645$		0			0
$646$	1.22587e7			1.15575
$647$		−	7.70494e6i		−	0.723616i	−0.932253	−	0.361808i	$-0.882160\pi$
$647$		−	7.70494e6i		−	0.723616i	0.932253	−	0.361808i	$-0.117840\pi$
$648$		0			0
$649$		−	282000.i		−	0.0262807i
$650$		−	6.31218e6i		−	0.585998i
$651$		0			0
$652$	−247493.			−0.0228005
$653$			1.66199e7i			1.52526i	0.646832	+	0.762632i	$0.276093\pi$
$653$			1.66199e7i			1.52526i	−0.646832	+	0.762632i	$0.723907\pi$
$654$		0			0
$655$		−	5.39456e6i		−	0.491307i
$656$		−	7.61148e6i		−	0.690573i
$657$		0			0
$658$		−	1.51027e7i		−	1.35985i
$659$	−1.42803e7			−1.28093			−0.640464	−	0.767989i	$-0.721258\pi$
$659$	−1.42803e7			−1.28093			−0.640464	+	0.767989i	$0.721258\pi$
$660$		0			0
$661$			6.86018e6i			0.610705i	0.952239	+	0.305353i	$0.0987743\pi$
$661$			6.86018e6i			0.610705i	−0.952239	+	0.305353i	$0.901226\pi$
$662$		−	1.21198e7i		−	1.07486i
$663$		0			0
$664$			260886.i			0.0229631i
$665$	1.02341e7			0.897422
$666$		0			0
$667$	−8.82611e6	+	3.06832e6i	−0.768166	+	0.267046i
$668$			1.28040e6i			0.111021i
$669$		0			0
$670$	−4.66540e6			−0.401515
$671$		−	5.47112e6i		−	0.469105i
$672$		0			0
$673$	9.22552e6			0.785151			0.392575	−	0.919720i	$-0.371584\pi$
$673$	9.22552e6			0.785151			0.392575	+	0.919720i	$0.371584\pi$
$674$	2.59165e7			2.19749
$675$		0			0
$676$	5.61720e6			0.472774
$677$	4.21670e6			0.353591			0.176795	−	0.984248i	$-0.443427\pi$
$677$	4.21670e6			0.353591			0.176795	+	0.984248i	$0.443427\pi$
$678$		0			0
$679$	324054.			0.0269738
$680$		−	176253.i		−	0.0146172i
$681$		0			0
$682$		−	3.08789e6i		−	0.254215i
$683$		−	1.55822e7i		−	1.27814i	−0.769149	−	0.639069i	$-0.779320\pi$
$683$		−	1.55822e7i		−	1.27814i	0.769149	−	0.639069i	$-0.220680\pi$
$684$		0			0
$685$	−7.45881e6			−0.607356
$686$	−1.72761e7			−1.40164
$687$		0			0
$688$		−	2.43165e7i		−	1.95853i
$689$	−2.65884e6			−0.213376
$690$		0			0
$691$	2.24359e7			1.78751			0.893754	−	0.448558i	$-0.148062\pi$
$691$	2.24359e7			1.78751			0.893754	+	0.448558i	$0.148062\pi$
$692$		−	7.82878e6i		−	0.621482i
$693$		0			0
$694$	−3.21397e7			−2.53304
$695$	7.08843e6			0.556657
$696$		0			0
$697$		−	5.12706e6i		−	0.399748i
$698$			1.85840e7i			1.44378i
$699$		0			0
$700$		−	7.35465e6i		−	0.567306i
$701$	−6.73657e6			−0.517778			−0.258889	−	0.965907i	$-0.583356\pi$
$701$	−6.73657e6			−0.517778			−0.258889	+	0.965907i	$0.583356\pi$
$702$		0			0
$703$	−2.23009e7			−1.70190
$704$	4.99973e6			0.380202
$705$		0			0
$706$	−2.99839e7			−2.26400
$707$	1.10895e7			0.834376
$708$		0			0
$709$			1.59195e7i			1.18936i	0.803961	+	0.594682i	$0.202722\pi$
$709$			1.59195e7i			1.18936i	−0.803961	+	0.594682i	$0.797278\pi$
$710$	9.55054e6			0.711021
$711$		0			0
$712$			482097.i			0.0356397i
$713$	2.00537e6	+	5.76850e6i	0.147731	+	0.424951i
$714$		0			0
$715$	2.54951e6			0.186505
$716$			1.51248e7i			1.10257i
$717$		0			0
$718$		−	3.32398e7i		−	2.40628i
$719$		−	788435.i		−	0.0568779i	−0.999596	−	0.0284390i	$-0.990946\pi$
$719$		−	788435.i		−	0.0568779i	0.999596	−	0.0284390i	$-0.00905362\pi$
$720$		0			0
$721$	−1.83659e7			−1.31575
$722$			1.80631e7i			1.28958i
$723$		0			0
$724$		−	901393.i		−	0.0639098i
$725$		−	6.69743e6i		−	0.473220i
$726$		0			0
$727$			1.58869e7i			1.11481i	0.830239	+	0.557407i	$0.188204\pi$
$727$			1.58869e7i			1.11481i	−0.830239	+	0.557407i	$0.811796\pi$
$728$	390919.			0.0273374
$729$		0			0
$730$		−	241641.i		−	0.0167828i
$731$		−	1.63795e7i		−	1.13372i
$732$		0			0
$733$		−	2.51739e6i		−	0.173058i	−0.996249	−	0.0865288i	$-0.972423\pi$
$733$		−	2.51739e6i		−	0.173058i	0.996249	−	0.0865288i	$-0.0275775\pi$
$734$	−3.92424e7			−2.68853
$735$		0			0
$736$	−1.94830e7	+	6.77309e6i	−1.32575	+	0.460884i
$737$			2.62240e6i			0.177841i
$738$		0			0
$739$	−1.12188e7			−0.755678			−0.377839	−	0.925871i	$-0.623333\pi$
$739$	−1.12188e7			−0.755678			−0.377839	+	0.925871i	$0.623333\pi$
$740$		−	1.15160e7i		−	0.773073i
$741$		0			0
$742$	−6.28218e6			−0.418890
$743$	−2.12261e7			−1.41058			−0.705290	−	0.708919i	$-0.749183\pi$
$743$	−2.12261e7			−1.41058			−0.705290	+	0.708919i	$0.749183\pi$
$744$		0			0
$745$	1.79505e7			1.18491
$746$	9.16278e6			0.602810
$747$		0			0
$748$	3.55823e6			0.232531
$749$		−	6.06779e6i		−	0.395208i
$750$		0			0
$751$		−	1.69377e7i		−	1.09586i	−0.836524	−	0.547930i	$-0.815416\pi$
$751$		−	1.69377e7i		−	1.09586i	0.836524	−	0.547930i	$-0.184584\pi$
$752$		−	1.53769e7i		−	0.991574i
$753$		0			0
$754$	−1.27856e7			−0.819014
$755$	1.29965e7			0.829773
$756$		0			0
$757$			518622.i			0.0328936i	0.999865	+	0.0164468i	$0.00523541\pi$
$757$			518622.i			0.0328936i	−0.999865	+	0.0164468i	$0.994765\pi$
$758$	6.87829e6			0.434818
$759$		0			0
$760$	542576.			0.0340743
$761$			1.93740e7i			1.21271i	0.795194	+	0.606355i	$0.207369\pi$
$761$			1.93740e7i			1.21271i	−0.795194	+	0.606355i	$0.792631\pi$
$762$		0			0
$763$	−8.25031e6			−0.513049
$764$	6.85186e6			0.424693
$765$		0			0
$766$		−	1.74691e7i		−	1.07572i
$767$			763134.i			0.0468395i
$768$		0			0
$769$		−	2.02181e7i		−	1.23289i	−0.787397	−	0.616446i	$-0.788572\pi$
$769$		−	2.02181e7i		−	1.23289i	0.787397	−	0.616446i	$-0.211428\pi$
$770$	6.02384e6			0.366140
$771$		0			0
$772$	−1.03534e7			−0.625230
$773$	−2.37082e7			−1.42708			−0.713542	−	0.700613i	$-0.752910\pi$
$773$	−2.37082e7			−1.42708			−0.713542	+	0.700613i	$0.752910\pi$
$774$		0			0
$775$	−4.37726e6			−0.261787
$776$	17180.1			0.00102417
$777$		0			0
$778$			9.75401e6i			0.577742i
$779$	1.57831e7			0.931856
$780$		0			0
$781$		−	5.36832e6i		−	0.314928i
$782$	−1.34794e7	+	4.68600e6i	−0.788230	+	0.274022i
$783$		0			0
$784$	74166.1			0.00430939
$785$			1.74052e7i			1.00810i
$786$		0			0
$787$			370166.i			0.0213039i	0.999943	+	0.0106520i	$0.00339069\pi$
$787$			370166.i			0.0213039i	−0.999943	+	0.0106520i	$0.996609\pi$
$788$			1.28591e7i			0.737727i
$789$		0			0
$790$	−1.96143e7			−1.11816
$791$			2.12989e7i			1.21036i
$792$		0			0
$793$			1.48057e7i			0.836075i
$794$		−	1.05948e7i		−	0.596405i
$795$		0			0
$796$		−	1.41552e7i		−	0.791833i
$797$	1.53144e7			0.853993			0.426996	−	0.904253i	$-0.359572\pi$
$797$	1.53144e7			0.853993			0.426996	+	0.904253i	$0.359572\pi$
$798$		0			0
$799$		−	1.03578e7i		−	0.573987i
$800$		−	1.47841e7i		−	0.816712i
$801$		0			0
$802$			3.75564e7i			2.06181i
$803$	−135825.			−0.00743348
$804$		0			0
$805$	−1.12532e7	+	3.91207e6i	−0.612048	+	0.212773i
$806$			8.35629e6i			0.453081i
$807$		0			0
$808$	587923.			0.0316805
$809$		−	1.18351e7i		−	0.635770i	−0.948129	−	0.317885i	$-0.897027\pi$
$809$		−	1.18351e7i		−	0.635770i	0.948129	−	0.317885i	$-0.102973\pi$
$810$		0			0
$811$	−2.62417e6			−0.140101			−0.0700503	−	0.997543i	$-0.522316\pi$
$811$	−2.62417e6			−0.140101			−0.0700503	+	0.997543i	$0.522316\pi$
$812$	−1.48971e7			−0.792890
$813$		0			0
$814$	−1.31264e7			−0.694359
$815$	−287353.			−0.0151538
$816$		0			0
$817$	5.04225e7			2.64283
$818$			4.10144e6i			0.214315i
$819$		0			0
$820$			8.15024e6i			0.423288i
$821$		−	3.38177e7i		−	1.75100i	−0.483219	−	0.875499i	$-0.660533\pi$
$821$		−	3.38177e7i		−	1.75100i	0.483219	−	0.875499i	$-0.339467\pi$
$822$		0			0
$823$	−1.38353e7			−0.712015			−0.356008	−	0.934483i	$-0.615862\pi$
$823$	−1.38353e7			−0.712015			−0.356008	+	0.934483i	$0.615862\pi$
$824$	−973691.			−0.0499578
$825$		0			0
$826$			1.80309e6i			0.0919534i
$827$	−8.59551e6			−0.437027			−0.218513	−	0.975834i	$-0.570121\pi$
$827$	−8.59551e6			−0.437027			−0.218513	+	0.975834i	$0.570121\pi$
$828$		0			0
$829$	2.71726e7			1.37323			0.686617	−	0.727019i	$-0.259095\pi$
$829$	2.71726e7			1.37323			0.686617	+	0.727019i	$0.259095\pi$
$830$		−	1.08790e7i		−	0.548145i
$831$		0			0
$832$	−1.35300e7			−0.677626
$833$	49957.9			0.00249455
$834$		0			0
$835$			1.48661e6i			0.0737872i
$836$			1.09536e7i			0.542055i
$837$		0			0
$838$			2.30122e7i			1.13200i
$839$	1.18418e7			0.580782			0.290391	−	0.956908i	$-0.406215\pi$
$839$	1.18418e7			0.580782			0.290391	+	0.956908i	$0.406215\pi$
$840$		0			0
$841$	6.94523e6			0.338608
$842$	3.93669e7			1.91360
$843$		0			0
$844$	−1.52902e7			−0.738852
$845$	6.52186e6			0.314217
$846$		0			0
$847$			1.75368e7i			0.839925i
$848$	−6.39623e6			−0.305446
$849$		0			0
$850$		−	1.02284e7i		−	0.485581i
$851$	2.45215e7	−	8.52468e6i	1.16071	−	0.403510i
$852$		0			0
$853$	2.89628e7			1.36291			0.681457	−	0.731858i	$-0.261347\pi$
$853$	2.89628e7			1.36291			0.681457	+	0.731858i	$0.261347\pi$
$854$			3.49820e7i			1.64135i
$855$		0			0
$856$		−	321692.i		−	0.0150057i
$857$		−	3.64428e7i		−	1.69496i	−0.530828	−	0.847479i	$-0.678119\pi$
$857$		−	3.64428e7i		−	1.69496i	0.530828	−	0.847479i	$-0.321881\pi$
$858$		0			0
$859$	−2.43719e7			−1.12695			−0.563476	−	0.826132i	$-0.690536\pi$
$859$	−2.43719e7			−1.12695			−0.563476	+	0.826132i	$0.690536\pi$
$860$			2.60377e7i			1.20048i
$861$		0			0
$862$		−	1.65729e7i		−	0.759680i
$863$		−	1.96409e7i		−	0.897705i	−0.893606	−	0.448853i	$-0.851833\pi$
$863$		−	1.96409e7i		−	0.897705i	0.893606	−	0.448853i	$-0.148167\pi$
$864$		0			0
$865$		−	9.08962e6i		−	0.413053i
$866$	−1.81143e7			−0.820782
$867$		0			0
$868$			9.73635e6i			0.438629i
$869$			1.10251e7i			0.495260i
$870$		0			0
$871$		−	7.09662e6i		−	0.316961i
$872$	−437401.			−0.0194800
$873$		0			0
$874$	−1.44253e7	−	4.14949e7i	−0.638775	−	1.83745i
$875$		−	2.32142e7i		−	1.02502i
$876$		0			0
$877$	1.79644e7			0.788705			0.394352	−	0.918959i	$-0.370969\pi$
$877$	1.79644e7			0.788705			0.394352	+	0.918959i	$0.370969\pi$
$878$		−	3.43393e7i		−	1.50333i
$879$		0			0
$880$	6.13320e6			0.266981
$881$	−2.08526e7			−0.905150			−0.452575	−	0.891726i	$-0.649494\pi$
$881$	−2.08526e7			−0.905150			−0.452575	+	0.891726i	$0.649494\pi$
$882$		0			0
$883$	−5.66888e6			−0.244679			−0.122339	−	0.992488i	$-0.539040\pi$
$883$	−5.66888e6			−0.244679			−0.122339	+	0.992488i	$0.539040\pi$
$884$	−9.62911e6			−0.414434
$885$		0			0
$886$	5.99805e6			0.256700
$887$		−	2.96694e7i		−	1.26619i	−0.774073	−	0.633096i	$-0.781784\pi$
$887$		−	2.96694e7i		−	1.26619i	0.774073	−	0.633096i	$-0.218216\pi$
$888$		0			0
$889$			2.67364e7i			1.13461i
$890$		−	2.01036e7i		−	0.850743i
$891$		0			0
$892$	3.57104e7			1.50274
$893$	3.18855e7			1.33803
$894$		0			0
$895$			1.75607e7i			0.732799i
$896$	1.83185e6			0.0762290
$897$		0			0
$898$	4.90930e7			2.03156
$899$			8.86630e6i			0.365884i
$900$		0			0
$901$	−4.30847e6			−0.176812
$902$	9.28999e6			0.380189
$903$		0			0
$904$			1.12919e6i			0.0459564i
$905$		−	1.04656e6i		−	0.0424761i
$906$		0			0
$907$		−	7.21341e6i		−	0.291154i	−0.989347	−	0.145577i	$-0.953496\pi$
$907$		−	7.21341e6i		−	0.291154i	0.989347	−	0.145577i	$-0.0465038\pi$
$908$	−4.00613e7			−1.61254
$909$		0			0
$910$	−1.63014e7			−0.652562
$911$	3.30286e7			1.31854			0.659271	−	0.751906i	$-0.270865\pi$
$911$	3.30286e7			1.31854			0.659271	+	0.751906i	$0.270865\pi$
$912$		0			0
$913$	−6.11506e6			−0.242786
$914$	−3.30524e7			−1.30869
$915$		0			0
$916$		−	1.61364e7i		−	0.635432i
$917$	1.93881e7			0.761400
$918$		0			0
$919$		−	5.08344e7i		−	1.98550i	−0.120209	−	0.992749i	$-0.538357\pi$
$919$		−	5.08344e7i		−	1.98550i	0.120209	−	0.992749i	$-0.461643\pi$
$920$	−596602.	+	207404.i	−0.0232389	+	0.00807880i
$921$		0			0
$922$	7.03488e7			2.72539
$923$			1.45275e7i			0.561289i
$924$		0			0
$925$			1.86074e7i			0.715041i
$926$		−	1.62569e7i		−	0.623032i
$927$		0			0
$928$	−2.99457e7			−1.14147
$929$			2.97570e7i			1.13123i	0.824670	+	0.565614i	$0.191361\pi$
$929$			2.97570e7i			1.13123i	−0.824670	+	0.565614i	$0.808639\pi$
$930$		0			0
$931$			153790.i			0.00581507i
$932$			5.30869e6i			0.200192i
$933$		0			0
$934$			6.99055e7i			2.62207i
$935$	4.13130e6			0.154546
$936$		0			0
$937$		−	2.95228e7i		−	1.09852i	−0.835651	−	0.549260i	$-0.814910\pi$
$937$		−	2.95228e7i		−	1.09852i	0.835651	−	0.549260i	$-0.185090\pi$
$938$		−	1.67675e7i		−	0.622245i
$939$		0			0
$940$			1.64653e7i			0.607787i
$941$	−1.26981e7			−0.467480			−0.233740	−	0.972299i	$-0.575096\pi$
$941$	−1.26981e7			−0.467480			−0.233740	+	0.972299i	$0.575096\pi$
$942$		0			0
$943$	−1.73547e7	+	6.03321e6i	−0.635532	+	0.220937i
$944$			1.83583e6i			0.0670504i
$945$		0			0
$946$	2.96789e7			1.07825
$947$		−	3.37074e7i		−	1.22138i	−0.791870	−	0.610689i	$-0.790892\pi$
$947$		−	3.37074e7i		−	1.22138i	0.791870	−	0.610689i	$-0.209108\pi$
$948$		0			0
$949$	367564.			0.0132485
$950$	3.14872e7			1.13194
$951$		0			0
$952$	633456.			0.0226529
$953$	−4.66711e7			−1.66462			−0.832312	−	0.554308i	$-0.812983\pi$
$953$	−4.66711e7			−1.66462			−0.832312	+	0.554308i	$0.812983\pi$
$954$		0			0
$955$	7.95537e6			0.282262
$956$			7.48221e6i			0.264780i
$957$		0			0
$958$		−	2.46571e7i		−	0.868016i
$959$		−	2.68071e7i		−	0.941245i
$960$		0			0
$961$	−2.28344e7			−0.797592
$962$	3.55220e7			1.23754
$963$		0			0
$964$			9.29488e6i			0.322145i
$965$	−1.20208e7			−0.415544
$966$		0			0
$967$	2.72035e7			0.935531			0.467766	−	0.883853i	$-0.345059\pi$
$967$	2.72035e7			0.935531			0.467766	+	0.883853i	$0.345059\pi$
$968$			929735.i			0.0318912i
$969$		0			0
$970$	−716417.			−0.0244476
$971$	3.63123e7			1.23596			0.617981	−	0.786193i	$-0.287951\pi$
$971$	3.63123e7			1.23596			0.617981	+	0.786193i	$0.287951\pi$
$972$		0			0
$973$			2.54759e7i			0.862676i
$974$			7.67932e7i			2.59373i
$975$		0			0
$976$			3.56171e7i			1.19684i
$977$	−4.14754e7			−1.39013			−0.695063	−	0.718949i	$-0.744624\pi$
$977$	−4.14754e7			−1.39013			−0.695063	+	0.718949i	$0.744624\pi$
$978$		0			0
$979$	−1.13002e7			−0.376814
$980$	−79415.7			−0.00264144
$981$		0			0
$982$	4.75594e7			1.57383
$983$	3.88724e7			1.28309			0.641546	−	0.767085i	$-0.278293\pi$
$983$	3.88724e7			1.28309			0.641546	+	0.767085i	$0.278293\pi$
$984$		0			0
$985$			1.49301e7i			0.490312i
$986$	−2.07181e7			−0.678668
$987$		0			0
$988$		−	2.96422e7i		−	0.966092i
$989$	−5.54433e7	+	1.92744e7i	−1.80243	+	0.626599i
$990$		0			0
$991$	−2.70747e7			−0.875750			−0.437875	−	0.899036i	$-0.644269\pi$
$991$	−2.70747e7			−0.875750			−0.437875	+	0.899036i	$0.644269\pi$
$992$			1.95717e7i			0.631464i
$993$		0			0
$994$			3.43248e7i			1.10190i
$995$		−	1.64349e7i		−	0.526272i
$996$		0			0
$997$	1.67970e7			0.535171			0.267585	−	0.963534i	$-0.413774\pi$
$997$	1.67970e7			0.535171			0.267585	+	0.963534i	$0.413774\pi$
$998$			8.80012e7i			2.79681i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.c.a.206.13	✓	40
3.2	odd	2	inner	207.6.c.a.206.28	yes	40
23.22	odd	2	inner	207.6.c.a.206.27	yes	40
69.68	even	2	inner	207.6.c.a.206.14	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.c.a.206.13	✓	40	1.1	even	1	trivial
207.6.c.a.206.14	yes	40	69.68	even	2	inner
207.6.c.a.206.27	yes	40	23.22	odd	2	inner
207.6.c.a.206.28	yes	40	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.94564i q^{2} -31.1332 q^{4} -36.1472 q^{5} -129.914i q^{7} -6.88755i q^{8} +O(q^{10})\)	\(q-7.94564i q^{2} -31.1332 q^{4} -36.1472 q^{5} -129.914i q^{7} -6.88755i q^{8} +287.213i q^{10} +161.441 q^{11} -436.884 q^{13} -1032.25 q^{14} -1050.99 q^{16} -707.940 q^{17} -2179.32i q^{19} +1125.38 q^{20} -1282.75i q^{22} +(833.061 + 2396.32i) q^{23} -1818.38 q^{25} +3471.33i q^{26} +4044.62i q^{28} +3683.19i q^{29} +2407.23 q^{31} +8130.36i q^{32} +5625.03i q^{34} +4696.02i q^{35} -10233.0i q^{37} -17316.1 q^{38} +248.966i q^{40} +7242.22i q^{41} +23136.8i q^{43} -5026.18 q^{44} +(19040.3 - 6619.20i) q^{46} +14630.9i q^{47} -70.5680 q^{49} +14448.2i q^{50} +13601.6 q^{52} +6085.92 q^{53} -5835.66 q^{55} -894.787 q^{56} +29265.3 q^{58} -1746.76i q^{59} -33889.2i q^{61} -19127.0i q^{62} +30969.3 q^{64} +15792.2 q^{65} +16243.7i q^{67} +22040.4 q^{68} +37312.9 q^{70} -33252.5i q^{71} -841.330 q^{73} -81307.4 q^{74} +67849.1i q^{76} -20973.4i q^{77} +68291.7i q^{79} +37990.3 q^{80} +57544.1 q^{82} -37877.9 q^{83} +25590.1 q^{85} +183837. q^{86} -1111.94i q^{88} -69995.4 q^{89} +56757.3i q^{91} +(-25935.8 - 74605.0i) q^{92} +116252. q^{94} +78776.4i q^{95} +2494.38i q^{97} +560.708i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 600 q^{4}+O(q^{10}) \) 40 * q - 600 * q^4	\( 40 q - 600 q^{4} - 1048 q^{13} + 9728 q^{16} + 14704 q^{25} + 4640 q^{31} - 91864 q^{46} - 8192 q^{49} + 150360 q^{52} + 134592 q^{55} - 195704 q^{58} - 183416 q^{64} - 257448 q^{70} + 31088 q^{73} - 77096 q^{82} - 368760 q^{85} - 123512 q^{94}+O(q^{100}) \) 40 * q - 600 * q^4 - 1048 * q^13 + 9728 * q^16 + 14704 * q^25 + 4640 * q^31 - 91864 * q^46 - 8192 * q^49 + 150360 * q^52 + 134592 * q^55 - 195704 * q^58 - 183416 * q^64 - 257448 * q^70 + 31088 * q^73 - 77096 * q^82 - 368760 * q^85 - 123512 * q^94

Introduction

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