LMFDB - Embedded newform 624.6.c.f.337.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,6,Mod(337,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("624.337");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	624.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$100.079503563$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 4385x^{6} + 4890448x^{4} + 396656640x^{2} + 3049690176 $ x^8 + 4385x^6 + 4890448x^4 + 396656640*x^2 + 3049690176
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.7
Root			$42.7585i$ of defining polynomial
Character	$\chi$	$=$	624.337
Dual form			624.6.c.f.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +83.5169i q^{5} +187.367i q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +83.5169i q^{5} +187.367i q^{7} +81.0000 q^{9} -786.334i q^{11} +(259.909 + 551.127i) q^{13} -751.652i q^{15} +923.787 q^{17} -434.886i q^{19} -1686.30i q^{21} -3790.69 q^{23} -3850.08 q^{25} -729.000 q^{27} -3735.14 q^{29} -2690.43i q^{31} +7077.00i q^{33} -15648.3 q^{35} +1734.37i q^{37} +(-2339.18 - 4960.14i) q^{39} -12144.6i q^{41} -1808.26 q^{43} +6764.87i q^{45} -5647.45i q^{47} -18299.4 q^{49} -8314.09 q^{51} -30508.2 q^{53} +65672.2 q^{55} +3913.97i q^{57} -38919.0i q^{59} +9010.71 q^{61} +15176.7i q^{63} +(-46028.4 + 21706.8i) q^{65} +28172.0i q^{67} +34116.2 q^{69} +80313.2i q^{71} +5864.17i q^{73} +34650.7 q^{75} +147333. q^{77} +17250.7 q^{79} +6561.00 q^{81} -74769.4i q^{83} +77151.9i q^{85} +33616.3 q^{87} -45780.6i q^{89} +(-103263. + 48698.3i) q^{91} +24213.9i q^{93} +36320.4 q^{95} -77912.8i q^{97} -63693.0i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 72 q^{3} + 648 q^{9}+O(q^{10}) $ 8 * q - 72 * q^3 + 648 * q^9	$ 8 q - 72 q^{3} + 648 q^{9} - 52 q^{13} + 600 q^{17} + 1656 q^{23} - 10128 q^{25} - 5832 q^{27} - 3360 q^{29} - 27144 q^{35} + 468 q^{39} - 16944 q^{43} - 59480 q^{49} - 5400 q^{51} + 9576 q^{53} - 5424 q^{55} + 58344 q^{61} - 192888 q^{65} - 14904 q^{69} + 91152 q^{75} + 243960 q^{77} - 155968 q^{79} + 52488 q^{81} + 30240 q^{87} + 159280 q^{91} + 368232 q^{95}+O(q^{100}) $ 8 * q - 72 * q^3 + 648 * q^9 - 52 * q^13 + 600 * q^17 + 1656 * q^23 - 10128 * q^25 - 5832 * q^27 - 3360 * q^29 - 27144 * q^35 + 468 * q^39 - 16944 * q^43 - 59480 * q^49 - 5400 * q^51 + 9576 * q^53 - 5424 * q^55 + 58344 * q^61 - 192888 * q^65 - 14904 * q^69 + 91152 * q^75 + 243960 * q^77 - 155968 * q^79 + 52488 * q^81 + 30240 * q^87 + 159280 * q^91 + 368232 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	−9.00000			−0.577350
$3$	−9.00000			−0.577350
$4$		0			0
$5$			83.5169i			1.49400i	0.664826	+	0.746998i	$0.268506\pi$
$5$			83.5169i			1.49400i	−0.664826	+	0.746998i	$0.731494\pi$
$6$		0			0
$7$			187.367i			1.44527i	0.691232	+	0.722633i	$0.257068\pi$
$7$			187.367i			1.44527i	−0.691232	+	0.722633i	$0.742932\pi$
$8$		0			0
$9$	81.0000			0.333333
$10$		0			0
$11$		−	786.334i		−	1.95941i	−0.200447	−	0.979705i	$-0.564239\pi$
$11$		−	786.334i		−	1.95941i	0.200447	−	0.979705i	$-0.435761\pi$
$12$		0			0
$13$	259.909	+	551.127i	0.426543	+	0.904467i
$13$	259.909	+	551.127i	0.426543	+	0.904467i
$14$		0			0
$15$		−	751.652i		−	0.862559i
$16$		0			0
$17$	923.787			0.775264			0.387632	−	0.921814i	$-0.373293\pi$
$17$	923.787			0.775264			0.387632	+	0.921814i	$0.373293\pi$
$18$		0			0
$19$		−	434.886i		−	0.276370i	−0.990406	−	0.138185i	$-0.955873\pi$
$19$		−	434.886i		−	0.276370i	0.990406	−	0.138185i	$-0.0441269\pi$
$20$		0			0
$21$		−	1686.30i		−	0.834425i
$22$		0			0
$23$	−3790.69			−1.49416			−0.747082	−	0.664732i	$-0.768546\pi$
$23$	−3790.69			−1.49416			−0.747082	+	0.664732i	$0.768546\pi$
$24$		0			0
$25$	−3850.08			−1.23203
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	−3735.14			−0.824731			−0.412365	−	0.911019i	$-0.635297\pi$
$29$	−3735.14			−0.824731			−0.412365	+	0.911019i	$0.635297\pi$
$30$		0			0
$31$		−	2690.43i		−	0.502826i	−0.967880	−	0.251413i	$-0.919105\pi$
$31$		−	2690.43i		−	0.502826i	0.967880	−	0.251413i	$-0.0808952\pi$
$32$		0			0
$33$			7077.00i			1.13127i
$34$		0			0
$35$	−15648.3			−2.15922
$36$		0			0
$37$			1734.37i			0.208275i	0.994563	+	0.104138i	$0.0332082\pi$
$37$			1734.37i			0.208275i	−0.994563	+	0.104138i	$0.966792\pi$
$38$		0			0
$39$	−2339.18	−	4960.14i	−0.246265	−	0.522195i
$40$		0			0
$41$		−	12144.6i		−	1.12830i	−0.825672	−	0.564150i	$-0.809204\pi$
$41$		−	12144.6i		−	1.12830i	0.825672	−	0.564150i	$-0.190796\pi$
$42$		0			0
$43$	−1808.26			−0.149138			−0.0745690	−	0.997216i	$-0.523758\pi$
$43$	−1808.26			−0.149138			−0.0745690	+	0.997216i	$0.523758\pi$
$44$		0			0
$45$			6764.87i			0.497999i
$46$		0			0
$47$		−	5647.45i		−	0.372913i	−0.982463	−	0.186457i	$-0.940300\pi$
$47$		−	5647.45i		−	0.372913i	0.982463	−	0.186457i	$-0.0597004\pi$
$48$		0			0
$49$	−18299.4			−1.08879
$50$		0			0
$51$	−8314.09			−0.447599
$52$		0			0
$53$	−30508.2			−1.49186			−0.745929	−	0.666026i	$-0.767994\pi$
$53$	−30508.2			−1.49186			−0.745929	+	0.666026i	$0.767994\pi$
$54$		0			0
$55$	65672.2			2.92735
$56$		0			0
$57$			3913.97i			0.159563i
$58$		0			0
$59$		−	38919.0i		−	1.45556i	−0.685808	−	0.727782i	$-0.740551\pi$
$59$		−	38919.0i		−	1.45556i	0.685808	−	0.727782i	$-0.259449\pi$
$60$		0			0
$61$	9010.71			0.310052			0.155026	−	0.987910i	$-0.450454\pi$
$61$	9010.71			0.310052			0.155026	+	0.987910i	$0.450454\pi$
$62$		0			0
$63$			15176.7i			0.481755i
$64$		0			0
$65$	−46028.4	+	21706.8i	−1.35127	+	0.637253i
$66$		0			0
$67$			28172.0i			0.766709i	0.923601	+	0.383355i	$0.125231\pi$
$67$			28172.0i			0.766709i	−0.923601	+	0.383355i	$0.874769\pi$
$68$		0			0
$69$	34116.2			0.862656
$70$		0			0
$71$			80313.2i			1.89078i	0.325941	+	0.945390i	$0.394319\pi$
$71$			80313.2i			1.89078i	−0.325941	+	0.945390i	$0.605681\pi$
$72$		0			0
$73$			5864.17i			0.128795i	0.997924	+	0.0643975i	$0.0205126\pi$
$73$			5864.17i			0.128795i	−0.997924	+	0.0643975i	$0.979487\pi$
$74$		0			0
$75$	34650.7			0.711310
$76$		0			0
$77$	147333.			2.83187
$78$		0			0
$79$	17250.7			0.310985			0.155492	−	0.987837i	$-0.450304\pi$
$79$	17250.7			0.310985			0.155492	+	0.987837i	$0.450304\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$		−	74769.4i		−	1.19132i	−0.803236	−	0.595661i	$-0.796890\pi$
$83$		−	74769.4i		−	1.19132i	0.803236	−	0.595661i	$-0.203110\pi$
$84$		0			0
$85$			77151.9i			1.15824i
$86$		0			0
$87$	33616.3			0.476158
$88$		0			0
$89$		−	45780.6i		−	0.612641i	−0.951928	−	0.306320i	$-0.900902\pi$
$89$		−	45780.6i		−	0.612641i	0.951928	−	0.306320i	$-0.0990979\pi$
$90$		0			0
$91$	−103263.	+	48698.3i	−1.30720	+	0.616468i
$92$		0			0
$93$			24213.9i			0.290307i
$94$		0			0
$95$	36320.4			0.412896
$96$		0			0
$97$		−	77912.8i		−	0.840774i	−0.907345	−	0.420387i	$-0.861894\pi$
$97$		−	77912.8i		−	0.840774i	0.907345	−	0.420387i	$-0.138106\pi$
$98$		0			0
$99$		−	63693.0i		−	0.653136i
$100$		0			0
$101$	17140.1			0.167190			0.0835949	−	0.996500i	$-0.473360\pi$
$101$	17140.1			0.167190			0.0835949	+	0.996500i	$0.473360\pi$
$102$		0			0
$103$	102916.			0.955848			0.477924	−	0.878401i	$-0.341390\pi$
$103$	102916.			0.955848			0.477924	+	0.878401i	$0.341390\pi$
$104$		0			0
$105$	140835.			1.24663
$106$		0			0
$107$	−133735.			−1.12924			−0.564619	−	0.825352i	$-0.690977\pi$
$107$	−133735.			−1.12924			−0.564619	+	0.825352i	$0.690977\pi$
$108$		0			0
$109$		−	237067.i		−	1.91120i	−0.294674	−	0.955598i	$-0.595211\pi$
$109$		−	237067.i		−	1.91120i	0.294674	−	0.955598i	$-0.404789\pi$
$110$		0			0
$111$		−	15609.3i		−	0.120248i
$112$		0			0
$113$	138273.			1.01869			0.509343	−	0.860564i	$-0.329889\pi$
$113$	138273.			1.01869			0.509343	+	0.860564i	$0.329889\pi$
$114$		0			0
$115$		−	316587.i		−	2.23228i
$116$		0			0
$117$	21052.6	+	44641.2i	0.142181	+	0.301489i
$118$		0			0
$119$			173087.i			1.12046i
$120$		0			0
$121$	−457270.			−2.83928
$122$		0			0
$123$			109302.i			0.651425i
$124$		0			0
$125$		−	60556.4i		−	0.346645i
$126$		0			0
$127$	−283497.			−1.55969			−0.779847	−	0.625970i	$-0.784703\pi$
$127$	−283497.			−1.55969			−0.779847	+	0.625970i	$0.784703\pi$
$128$		0			0
$129$	16274.3			0.0861049
$130$		0			0
$131$	−101624.			−0.517391			−0.258695	−	0.965959i	$-0.583293\pi$
$131$	−101624.			−0.517391			−0.258695	+	0.965959i	$0.583293\pi$
$132$		0			0
$133$	81483.3			0.399429
$134$		0			0
$135$		−	60883.8i		−	0.287520i
$136$		0			0
$137$		−	136816.i		−	0.622781i	−0.950282	−	0.311390i	$-0.899205\pi$
$137$		−	136816.i		−	0.622781i	0.950282	−	0.311390i	$-0.100795\pi$
$138$		0			0
$139$	−122457.			−0.537585			−0.268793	−	0.963198i	$-0.586625\pi$
$139$	−122457.			−0.537585			−0.268793	+	0.963198i	$0.586625\pi$
$140$		0			0
$141$			50827.1i			0.215302i
$142$		0			0
$143$	433369.	−	204375.i	1.77222	−	0.835772i
$144$		0			0
$145$		−	311948.i		−	1.23214i
$146$		0			0
$147$	164694.			0.628616
$148$		0			0
$149$		−	247858.i		−	0.914613i	−0.889309	−	0.457307i	$-0.848814\pi$
$149$		−	247858.i		−	0.914613i	0.889309	−	0.457307i	$-0.151186\pi$
$150$		0			0
$151$		−	324290.i		−	1.15742i	−0.815533	−	0.578711i	$-0.803556\pi$
$151$		−	324290.i		−	1.15742i	0.815533	−	0.578711i	$-0.196444\pi$
$152$		0			0
$153$	74826.8			0.258421
$154$		0			0
$155$	224696.			0.751220
$156$		0			0
$157$	407075.			1.31803			0.659014	−	0.752130i	$-0.270974\pi$
$157$	407075.			1.31803			0.659014	+	0.752130i	$0.270974\pi$
$158$		0			0
$159$	274574.			0.861324
$160$		0			0
$161$		−	710250.i		−	2.15947i
$162$		0			0
$163$			192942.i			0.568799i	0.958706	+	0.284399i	$0.0917942\pi$
$163$			192942.i			0.568799i	−0.958706	+	0.284399i	$0.908206\pi$
$164$		0			0
$165$	−591050.			−1.69011
$166$		0			0
$167$			358847.i			0.995675i	0.867270	+	0.497838i	$0.165872\pi$
$167$			358847.i			0.995675i	−0.867270	+	0.497838i	$0.834128\pi$
$168$		0			0
$169$	−236188.	+	286485.i	−0.636123	+	0.771588i
$170$		0			0
$171$		−	35225.8i		−	0.0921235i
$172$		0			0
$173$	575252.			1.46131			0.730656	−	0.682746i	$-0.239214\pi$
$173$	575252.			1.46131			0.730656	+	0.682746i	$0.239214\pi$
$174$		0			0
$175$		−	721378.i		−	1.78060i
$176$		0			0
$177$			350271.i			0.840371i
$178$		0			0
$179$	730150.			1.70326			0.851628	−	0.524147i	$-0.175616\pi$
$179$	730150.			1.70326			0.851628	+	0.524147i	$0.175616\pi$
$180$		0			0
$181$	18004.4			0.0408490			0.0204245	−	0.999791i	$-0.493498\pi$
$181$	18004.4			0.0408490			0.0204245	+	0.999791i	$0.493498\pi$
$182$		0			0
$183$	−81096.4			−0.179009
$184$		0			0
$185$	−144849.			−0.311162
$186$		0			0
$187$		−	726405.i		−	1.51906i
$188$		0			0
$189$		−	136591.i		−	0.278142i
$190$		0			0
$191$	−718887.			−1.42586			−0.712930	−	0.701235i	$-0.752632\pi$
$191$	−718887.			−1.42586			−0.712930	+	0.701235i	$0.752632\pi$
$192$		0			0
$193$			163524.i			0.316001i	0.987439	+	0.158000i	$0.0505047\pi$
$193$			163524.i			0.316001i	−0.987439	+	0.158000i	$0.949495\pi$
$194$		0			0
$195$	414256.	−	195361.i	0.780157	−	0.367918i
$196$		0			0
$197$			41747.3i			0.0766412i	0.999265	+	0.0383206i	$0.0122008\pi$
$197$			41747.3i			0.0766412i	−0.999265	+	0.0383206i	$0.987799\pi$
$198$		0			0
$199$	−159023.			−0.284661			−0.142331	−	0.989819i	$-0.545460\pi$
$199$	−159023.			−0.284661			−0.142331	+	0.989819i	$0.545460\pi$
$200$		0			0
$201$		−	253548.i		−	0.442660i
$202$		0			0
$203$		−	699842.i		−	1.19196i
$204$		0			0
$205$	1.01428e6			1.68568
$206$		0			0
$207$	−307046.			−0.498055
$208$		0			0
$209$	−341966.			−0.541523
$210$		0			0
$211$	323039.			0.499516			0.249758	−	0.968308i	$-0.419649\pi$
$211$	323039.			0.499516			0.249758	+	0.968308i	$0.419649\pi$
$212$		0			0
$213$		−	722819.i		−	1.09164i
$214$		0			0
$215$		−	151020.i		−	0.222812i
$216$		0			0
$217$	504098.			0.726717
$218$		0			0
$219$		−	52777.5i		−	0.0743598i
$220$		0			0
$221$	240100.	+	509124.i	0.330683	+	0.701201i
$222$		0			0
$223$			1.20727e6i			1.62570i	0.582470	+	0.812852i	$0.302086\pi$
$223$			1.20727e6i			1.62570i	−0.582470	+	0.812852i	$0.697914\pi$
$224$		0			0
$225$	−311856.			−0.410675
$226$		0			0
$227$		−	885255.i		−	1.14026i	−0.821555	−	0.570130i	$-0.806893\pi$
$227$		−	885255.i		−	1.14026i	0.821555	−	0.570130i	$-0.193107\pi$
$228$		0			0
$229$		−	473761.i		−	0.596995i	−0.954410	−	0.298498i	$-0.903515\pi$
$229$		−	473761.i		−	0.596995i	0.954410	−	0.298498i	$-0.0964855\pi$
$230$		0			0
$231$	−1.32600e6			−1.63498
$232$		0			0
$233$	169230.			0.204215			0.102107	−	0.994773i	$-0.467441\pi$
$233$	169230.			0.204215			0.102107	+	0.994773i	$0.467441\pi$
$234$		0			0
$235$	471658.			0.557131
$236$		0			0
$237$	−155256.			−0.179547
$238$		0			0
$239$		−	1.46535e6i		−	1.65939i	−0.558219	−	0.829694i	$-0.688515\pi$
$239$		−	1.46535e6i		−	1.65939i	0.558219	−	0.829694i	$-0.311485\pi$
$240$		0			0
$241$			34983.5i			0.0387990i	0.999812	+	0.0193995i	$0.00617543\pi$
$241$			34983.5i			0.0387990i	−0.999812	+	0.0193995i	$0.993825\pi$
$242$		0			0
$243$	−59049.0			−0.0641500
$244$		0			0
$245$		−	1.52831e6i		−	1.62666i
$246$		0			0
$247$	239677.	−	113031.i	0.249968	−	0.117884i
$248$		0			0
$249$			672925.i			0.687810i
$250$		0			0
$251$	629330.			0.630513			0.315256	−	0.949007i	$-0.397909\pi$
$251$	629330.			0.630513			0.315256	+	0.949007i	$0.397909\pi$
$252$		0			0
$253$			2.98075e6i			2.92768i
$254$		0			0
$255$		−	694367.i		−	0.668711i
$256$		0			0
$257$	−829748.			−0.783634			−0.391817	−	0.920043i	$-0.628153\pi$
$257$	−829748.			−0.783634			−0.391817	+	0.920043i	$0.628153\pi$
$258$		0			0
$259$	−324964.			−0.301013
$260$		0			0
$261$	−302546.			−0.274910
$262$		0			0
$263$	209606.			0.186859			0.0934294	−	0.995626i	$-0.470217\pi$
$263$	209606.			0.186859			0.0934294	+	0.995626i	$0.470217\pi$
$264$		0			0
$265$		−	2.54795e6i		−	2.22883i
$266$		0			0
$267$			412025.i			0.353708i
$268$		0			0
$269$	1.68780e6			1.42213			0.711066	−	0.703125i	$-0.248213\pi$
$269$	1.68780e6			1.42213			0.711066	+	0.703125i	$0.248213\pi$
$270$		0			0
$271$			1.42123e6i			1.17555i	0.809025	+	0.587774i	$0.199996\pi$
$271$			1.42123e6i			1.17555i	−0.809025	+	0.587774i	$0.800004\pi$
$272$		0			0
$273$	929366.	−	438285.i	0.754710	−	0.355918i
$274$		0			0
$275$			3.02745e6i			2.41404i
$276$		0			0
$277$	−428628.			−0.335646			−0.167823	−	0.985817i	$-0.553674\pi$
$277$	−428628.			−0.335646			−0.167823	+	0.985817i	$0.553674\pi$
$278$		0			0
$279$		−	217925.i		−	0.167609i
$280$		0			0
$281$			243338.i			0.183842i	0.995766	+	0.0919208i	$0.0293006\pi$
$281$			243338.i			0.183842i	−0.995766	+	0.0919208i	$0.970699\pi$
$282$		0			0
$283$	1.53845e6			1.14187			0.570937	−	0.820994i	$-0.306580\pi$
$283$	1.53845e6			1.14187			0.570937	+	0.820994i	$0.306580\pi$
$284$		0			0
$285$	−326883.			−0.238386
$286$		0			0
$287$	2.27550e6			1.63070
$288$		0			0
$289$	−566474.			−0.398965
$290$		0			0
$291$			701215.i			0.485421i
$292$		0			0
$293$			1.02136e6i			0.695038i	0.937673	+	0.347519i	$0.112976\pi$
$293$			1.02136e6i			0.695038i	−0.937673	+	0.347519i	$0.887024\pi$
$294$		0			0
$295$	3.25040e6			2.17461
$296$		0			0
$297$			573237.i			0.377088i
$298$		0			0
$299$	−985233.	−	2.08915e6i	−0.637325	−	1.35142i
$300$		0			0
$301$		−	338807.i		−	0.215544i
$302$		0			0
$303$	−154261.			−0.0965271
$304$		0			0
$305$			752547.i			0.463217i
$306$		0			0
$307$		−	535631.i		−	0.324354i	−0.986762	−	0.162177i	$-0.948148\pi$
$307$		−	535631.i		−	0.324354i	0.986762	−	0.162177i	$-0.0518516\pi$
$308$		0			0
$309$	−926241.			−0.551859
$310$		0			0
$311$	−2.97584e6			−1.74465			−0.872325	−	0.488926i	$-0.837389\pi$
$311$	−2.97584e6			−1.74465			−0.872325	+	0.488926i	$0.837389\pi$
$312$		0			0
$313$	16412.0			0.00946893			0.00473447	−	0.999989i	$-0.498493\pi$
$313$	16412.0			0.00946893			0.00473447	+	0.999989i	$0.498493\pi$
$314$		0			0
$315$	−1.26751e6			−0.719741
$316$		0			0
$317$		−	2.98387e6i		−	1.66775i	−0.551951	−	0.833876i	$-0.686116\pi$
$317$		−	2.98387e6i		−	1.66775i	0.551951	−	0.833876i	$-0.313884\pi$
$318$		0			0
$319$			2.93707e6i			1.61598i
$320$		0			0
$321$	1.20361e6			0.651966
$322$		0			0
$323$		−	401742.i		−	0.214260i
$324$		0			0
$325$	−1.00067e6	−	2.12188e6i	−0.525511	−	1.11433i
$326$		0			0
$327$			2.13360e6i			1.10343i
$328$		0			0
$329$	1.05815e6			0.538959
$330$		0			0
$331$		−	848266.i		−	0.425561i	−0.977100	−	0.212781i	$-0.931748\pi$
$331$		−	848266.i		−	0.425561i	0.977100	−	0.212781i	$-0.0682520\pi$
$332$		0			0
$333$			140484.i			0.0694251i
$334$		0			0
$335$	−2.35284e6			−1.14546
$336$		0			0
$337$	403049.			0.193323			0.0966615	−	0.995317i	$-0.469184\pi$
$337$	403049.			0.193323			0.0966615	+	0.995317i	$0.469184\pi$
$338$		0			0
$339$	−1.24445e6			−0.588138
$340$		0			0
$341$	−2.11558e6			−0.985241
$342$		0			0
$343$		−	279621.i		−	0.128332i
$344$		0			0
$345$			2.84928e6i			1.28881i
$346$		0			0
$347$	455097.			0.202899			0.101450	−	0.994841i	$-0.467652\pi$
$347$	455097.			0.202899			0.101450	+	0.994841i	$0.467652\pi$
$348$		0			0
$349$		−	2.55559e6i		−	1.12312i	−0.827435	−	0.561562i	$-0.810201\pi$
$349$		−	2.55559e6i		−	1.12312i	0.827435	−	0.561562i	$-0.189799\pi$
$350$		0			0
$351$	−189473.	−	401771.i	−0.0820882	−	0.174065i
$352$		0			0
$353$			492691.i			0.210445i	0.994449	+	0.105222i	$0.0335554\pi$
$353$			492691.i			0.210445i	−0.994449	+	0.105222i	$0.966445\pi$
$354$		0			0
$355$	−6.70751e6			−2.82482
$356$		0			0
$357$		−	1.55779e6i		−	0.646900i
$358$		0			0
$359$		−	2.17485e6i		−	0.890623i	−0.895376	−	0.445311i	$-0.853093\pi$
$359$		−	2.17485e6i		−	0.890623i	0.895376	−	0.445311i	$-0.146907\pi$
$360$		0			0
$361$	2.28697e6			0.923619
$362$		0			0
$363$	4.11543e6			1.63926
$364$		0			0
$365$	−489757.			−0.192419
$366$		0			0
$367$	−741107.			−0.287221			−0.143610	−	0.989634i	$-0.545871\pi$
$367$	−741107.			−0.287221			−0.143610	+	0.989634i	$0.545871\pi$
$368$		0			0
$369$		−	983716.i		−	0.376100i
$370$		0			0
$371$		−	5.71623e6i		−	2.15613i
$372$		0			0
$373$	−3.42043e6			−1.27294			−0.636471	−	0.771300i	$-0.719607\pi$
$373$	−3.42043e6			−1.27294			−0.636471	+	0.771300i	$0.719607\pi$
$374$		0			0
$375$			545008.i			0.200136i
$376$		0			0
$377$	−970796.	−	2.05854e6i	−0.351783	−	0.745942i
$378$		0			0
$379$			2.40039e6i			0.858387i	0.903213	+	0.429193i	$0.141202\pi$
$379$			2.40039e6i			0.858387i	−0.903213	+	0.429193i	$0.858798\pi$
$380$		0			0
$381$	2.55148e6			0.900490
$382$		0			0
$383$			70722.4i			0.0246354i	0.999924	+	0.0123177i	$0.00392095\pi$
$383$			70722.4i			0.0246354i	−0.999924	+	0.0123177i	$0.996079\pi$
$384$		0			0
$385$			1.23048e7i			4.23080i
$386$		0			0
$387$	−146469.			−0.0497127
$388$		0			0
$389$	−3.95890e6			−1.32648			−0.663240	−	0.748406i	$-0.730819\pi$
$389$	−3.95890e6			−1.32648			−0.663240	+	0.748406i	$0.730819\pi$
$390$		0			0
$391$	−3.50179e6			−1.15837
$392$		0			0
$393$	914617.			0.298716
$394$		0			0
$395$			1.44073e6i			0.464610i
$396$		0			0
$397$		−	559732.i		−	0.178240i	−0.996021	−	0.0891198i	$-0.971595\pi$
$397$		−	559732.i		−	0.178240i	0.996021	−	0.0891198i	$-0.0284054\pi$
$398$		0			0
$399$	−733349.			−0.230610
$400$		0			0
$401$			731268.i			0.227099i	0.993532	+	0.113550i	$0.0362221\pi$
$401$			731268.i			0.227099i	−0.993532	+	0.113550i	$0.963778\pi$
$402$		0			0
$403$	1.48277e6	−	699266.i	0.454789	−	0.214477i
$404$		0			0
$405$			547955.i			0.166000i
$406$		0			0
$407$	1.36379e6			0.408096
$408$		0			0
$409$			4.10605e6i			1.21371i	0.794812	+	0.606856i	$0.207569\pi$
$409$			4.10605e6i			1.21371i	−0.794812	+	0.606856i	$0.792431\pi$
$410$		0			0
$411$			1.23134e6i			0.359563i
$412$		0			0
$413$	7.29213e6			2.10368
$414$		0			0
$415$	6.24451e6			1.77983
$416$		0			0
$417$	1.10211e6			0.310375
$418$		0			0
$419$	2.16620e6			0.602787			0.301393	−	0.953500i	$-0.402548\pi$
$419$	2.16620e6			0.602787			0.301393	+	0.953500i	$0.402548\pi$
$420$		0			0
$421$		−	5.73337e6i		−	1.57654i	−0.615330	−	0.788270i	$-0.710977\pi$
$421$		−	5.73337e6i		−	1.57654i	0.615330	−	0.788270i	$-0.289023\pi$
$422$		0			0
$423$		−	457444.i		−	0.124304i
$424$		0			0
$425$	−3.55665e6			−0.955145
$426$		0			0
$427$			1.68831e6i			0.448108i
$428$		0			0
$429$	−3.90032e6	+	1.83937e6i	−1.02319	+	0.482533i
$430$		0			0
$431$			943230.i			0.244582i	0.992494	+	0.122291i	$0.0390241\pi$
$431$			943230.i			0.244582i	−0.992494	+	0.122291i	$0.960976\pi$
$432$		0			0
$433$	907853.			0.232700			0.116350	−	0.993208i	$-0.462881\pi$
$433$	907853.			0.232700			0.116350	+	0.993208i	$0.462881\pi$
$434$		0			0
$435$			2.80753e6i			0.711379i
$436$		0			0
$437$			1.64852e6i			0.412943i
$438$		0			0
$439$	801162.			0.198408			0.0992039	−	0.995067i	$-0.468370\pi$
$439$	801162.			0.198408			0.0992039	+	0.995067i	$0.468370\pi$
$440$		0			0
$441$	−1.48225e6			−0.362932
$442$		0			0
$443$	5.98329e6			1.44854			0.724270	−	0.689516i	$-0.242177\pi$
$443$	5.98329e6			1.44854			0.724270	+	0.689516i	$0.242177\pi$
$444$		0			0
$445$	3.82345e6			0.915283
$446$		0			0
$447$			2.23072e6i			0.528052i
$448$		0			0
$449$		−	674540.i		−	0.157904i	−0.996878	−	0.0789518i	$-0.974843\pi$
$449$		−	674540.i		−	0.157904i	0.996878	−	0.0789518i	$-0.0251573\pi$
$450$		0			0
$451$	−9.54974e6			−2.21080
$452$		0			0
$453$			2.91861e6i			0.668237i
$454$		0			0
$455$	−4.06713e6	−	8.62420e6i	−0.921001	−	1.95295i
$456$		0			0
$457$			2.44421e6i			0.547455i	0.961807	+	0.273728i	$0.0882567\pi$
$457$			2.44421e6i			0.547455i	−0.961807	+	0.273728i	$0.911743\pi$
$458$		0			0
$459$	−673441.			−0.149200
$460$		0			0
$461$			858805.i			0.188210i	0.995562	+	0.0941049i	$0.0299989\pi$
$461$			858805.i			0.188210i	−0.995562	+	0.0941049i	$0.970001\pi$
$462$		0			0
$463$			7.81981e6i			1.69529i	0.530564	+	0.847645i	$0.321980\pi$
$463$			7.81981e6i			1.69529i	−0.530564	+	0.847645i	$0.678020\pi$
$464$		0			0
$465$	−2.02227e6			−0.433717
$466$		0			0
$467$	6.27959e6			1.33241			0.666207	−	0.745767i	$-0.267917\pi$
$467$	6.27959e6			1.33241			0.666207	+	0.745767i	$0.267917\pi$
$468$		0			0
$469$	−5.27850e6			−1.10810
$470$		0			0
$471$	−3.66367e6			−0.760964
$472$		0			0
$473$			1.42189e6i			0.292223i
$474$		0			0
$475$			1.67435e6i			0.340495i
$476$		0			0
$477$	−2.47117e6			−0.497286
$478$		0			0
$479$			2.29110e6i			0.456252i	0.973632	+	0.228126i	$0.0732598\pi$
$479$			2.29110e6i			0.456252i	−0.973632	+	0.228126i	$0.926740\pi$
$480$		0			0
$481$	−955858.	+	450778.i	−0.188378	+	0.0888383i
$482$		0			0
$483$			6.39225e6i			1.24677i
$484$		0			0
$485$	6.50704e6			1.25611
$486$		0			0
$487$			4.05799e6i			0.775333i	0.921800	+	0.387666i	$0.126719\pi$
$487$			4.05799e6i			0.775333i	−0.921800	+	0.387666i	$0.873281\pi$
$488$		0			0
$489$		−	1.73648e6i		−	0.328396i
$490$		0			0
$491$	609711.			0.114135			0.0570677	−	0.998370i	$-0.481825\pi$
$491$	609711.			0.114135			0.0570677	+	0.998370i	$0.481825\pi$
$492$		0			0
$493$	−3.45048e6			−0.639384
$494$		0			0
$495$	5.31945e6			0.975783
$496$		0			0
$497$	−1.50480e7			−2.73268
$498$		0			0
$499$		−	9.43555e6i		−	1.69635i	−0.529715	−	0.848176i	$-0.677701\pi$
$499$		−	9.43555e6i		−	1.69635i	0.529715	−	0.848176i	$-0.322299\pi$
$500$		0			0
$501$		−	3.22962e6i		−	0.574853i
$502$		0			0
$503$	4.48965e6			0.791211			0.395606	−	0.918421i	$-0.370535\pi$
$503$	4.48965e6			0.791211			0.395606	+	0.918421i	$0.370535\pi$
$504$		0			0
$505$			1.43149e6i			0.249781i
$506$		0			0
$507$	2.12569e6	−	2.57837e6i	0.367266	−	0.445477i
$508$		0			0
$509$		−	4.48242e6i		−	0.766863i	−0.923569	−	0.383432i	$-0.874742\pi$
$509$		−	4.48242e6i		−	0.766863i	0.923569	−	0.383432i	$-0.125258\pi$
$510$		0			0
$511$	−1.09875e6			−0.186143
$512$		0			0
$513$			317032.i			0.0531875i
$514$		0			0
$515$			8.59521e6i			1.42803i
$516$		0			0
$517$	−4.44078e6			−0.730690
$518$		0			0
$519$	−5.17727e6			−0.843689
$520$		0			0
$521$	7.38286e6			1.19160			0.595799	−	0.803133i	$-0.296835\pi$
$521$	7.38286e6			1.19160			0.595799	+	0.803133i	$0.296835\pi$
$522$		0			0
$523$	−2.71336e6			−0.433764			−0.216882	−	0.976198i	$-0.569589\pi$
$523$	−2.71336e6			−0.433764			−0.216882	+	0.976198i	$0.569589\pi$
$524$		0			0
$525$			6.49240e6i			1.02803i
$526$		0			0
$527$		−	2.48539e6i		−	0.389823i
$528$		0			0
$529$	7.93297e6			1.23253
$530$		0			0
$531$		−	3.15244e6i		−	0.485188i
$532$		0			0
$533$	6.69323e6	−	3.15650e6i	1.02051	−	0.481268i
$534$		0			0
$535$		−	1.11691e7i		−	1.68708i
$536$		0			0
$537$	−6.57135e6			−0.983375
$538$		0			0
$539$			1.43894e7i			2.13339i
$540$		0			0
$541$			4.37688e6i			0.642942i	0.946920	+	0.321471i	$0.104177\pi$
$541$			4.37688e6i			0.642942i	−0.946920	+	0.321471i	$0.895823\pi$
$542$		0			0
$543$	−162039.			−0.0235842
$544$		0			0
$545$	1.97991e7			2.85532
$546$		0			0
$547$	−2.72786e6			−0.389811			−0.194905	−	0.980822i	$-0.562440\pi$
$547$	−2.72786e6			−0.389811			−0.194905	+	0.980822i	$0.562440\pi$
$548$		0			0
$549$	729868.			0.103351
$550$		0			0
$551$			1.62436e6i			0.227931i
$552$		0			0
$553$			3.23221e6i			0.449456i
$554$		0			0
$555$	1.30364e6			0.179650
$556$		0			0
$557$			1.11073e7i			1.51694i	0.651706	+	0.758472i	$0.274054\pi$
$557$			1.11073e7i			1.51694i	−0.651706	+	0.758472i	$0.725946\pi$
$558$		0			0
$559$	−469981.	−	996577.i	−0.0636138	−	0.134891i
$560$		0			0
$561$			6.53765e6i			0.877030i
$562$		0			0
$563$	−1.20957e7			−1.60827			−0.804137	−	0.594444i	$-0.797372\pi$
$563$	−1.20957e7			−1.60827			−0.804137	+	0.594444i	$0.797372\pi$
$564$		0			0
$565$			1.15481e7i			1.52191i
$566$		0			0
$567$			1.22931e6i			0.160585i
$568$		0			0
$569$	−1.25640e7			−1.62685			−0.813426	−	0.581669i	$-0.802400\pi$
$569$	−1.25640e7			−1.62685			−0.813426	+	0.581669i	$0.802400\pi$
$570$		0			0
$571$	−1.01907e7			−1.30802			−0.654009	−	0.756487i	$-0.726914\pi$
$571$	−1.01907e7			−1.30802			−0.654009	+	0.756487i	$0.726914\pi$
$572$		0			0
$573$	6.46998e6			0.823221
$574$		0			0
$575$	1.45944e7			1.84085
$576$		0			0
$577$			396134.i			0.0495340i	0.999693	+	0.0247670i	$0.00788438\pi$
$577$			396134.i			0.0495340i	−0.999693	+	0.0247670i	$0.992116\pi$
$578$		0			0
$579$		−	1.47172e6i		−	0.182443i
$580$		0			0
$581$	1.40093e7			1.72178
$582$		0			0
$583$			2.39896e7i			2.92316i
$584$		0			0
$585$	−3.72830e6	+	1.75825e6i	−0.450424	+	0.212418i
$586$		0			0
$587$			4.04427e6i			0.484446i	0.970221	+	0.242223i	$0.0778765\pi$
$587$			4.04427e6i			0.484446i	−0.970221	+	0.242223i	$0.922124\pi$
$588$		0			0
$589$	−1.17003e6			−0.138966
$590$		0			0
$591$		−	375725.i		−	0.0442488i
$592$		0			0
$593$			3.61298e6i			0.421918i	0.977495	+	0.210959i	$0.0676587\pi$
$593$			3.61298e6i			0.421918i	−0.977495	+	0.210959i	$0.932341\pi$
$594$		0			0
$595$	−1.44557e7			−1.67397
$596$		0			0
$597$	1.43121e6			0.164349
$598$		0			0
$599$	−1.13849e7			−1.29647			−0.648235	−	0.761441i	$-0.724492\pi$
$599$	−1.13849e7			−1.29647			−0.648235	+	0.761441i	$0.724492\pi$
$600$		0			0
$601$	−1.38080e7			−1.55935			−0.779677	−	0.626181i	$-0.784617\pi$
$601$	−1.38080e7			−1.55935			−0.779677	+	0.626181i	$0.784617\pi$
$602$		0			0
$603$			2.28193e6i			0.255570i
$604$		0			0
$605$		−	3.81898e7i		−	4.24188i
$606$		0			0
$607$	4.80008e6			0.528782			0.264391	−	0.964416i	$-0.414829\pi$
$607$	4.80008e6			0.528782			0.264391	+	0.964416i	$0.414829\pi$
$608$		0			0
$609$			6.29858e6i			0.688176i
$610$		0			0
$611$	3.11246e6	−	1.46782e6i	0.337288	−	0.159064i
$612$		0			0
$613$		−	8.26464e6i		−	0.888327i	−0.895946	−	0.444163i	$-0.853501\pi$
$613$		−	8.26464e6i		−	0.888327i	0.895946	−	0.444163i	$-0.146499\pi$
$614$		0			0
$615$	−9.12855e6			−0.973226
$616$		0			0
$617$		−	6.25819e6i		−	0.661814i	−0.943663	−	0.330907i	$-0.892645\pi$
$617$		−	6.25819e6i		−	0.661814i	0.943663	−	0.330907i	$-0.107355\pi$
$618$		0			0
$619$			1.49773e7i			1.57111i	0.618791	+	0.785556i	$0.287623\pi$
$619$			1.49773e7i			1.57111i	−0.618791	+	0.785556i	$0.712377\pi$
$620$		0			0
$621$	2.76341e6			0.287552
$622$		0			0
$623$	8.57776e6			0.885429
$624$		0			0
$625$	−6.97401e6			−0.714139
$626$		0			0
$627$	3.07769e6			0.312648
$628$		0			0
$629$			1.60219e6i			0.161468i
$630$		0			0
$631$		−	9.45858e6i		−	0.945698i	−0.881143	−	0.472849i	$-0.843226\pi$
$631$		−	9.45858e6i		−	0.945698i	0.881143	−	0.472849i	$-0.156774\pi$
$632$		0			0
$633$	−2.90735e6			−0.288396
$634$		0			0
$635$		−	2.36768e7i		−	2.33018i
$636$		0			0
$637$	−4.75617e6	−	1.00853e7i	−0.464417	−	0.984779i
$638$		0			0
$639$			6.50537e6i			0.630260i
$640$		0			0
$641$	−8.37531e6			−0.805111			−0.402556	−	0.915395i	$-0.631878\pi$
$641$	−8.37531e6			−0.805111			−0.402556	+	0.915395i	$0.631878\pi$
$642$		0			0
$643$		−	627456.i		−	0.0598489i	−0.999552	−	0.0299244i	$-0.990473\pi$
$643$		−	627456.i		−	0.0598489i	0.999552	−	0.0299244i	$-0.00952667\pi$
$644$		0			0
$645$			1.35918e6i			0.128640i
$646$		0			0
$647$	−9.06494e6			−0.851343			−0.425671	−	0.904878i	$-0.639962\pi$
$647$	−9.06494e6			−0.851343			−0.425671	+	0.904878i	$0.639962\pi$
$648$		0			0
$649$	−3.06033e7			−2.85205
$650$		0			0
$651$	−4.53688e6			−0.419570
$652$		0			0
$653$	4.92251e6			0.451756			0.225878	−	0.974156i	$-0.427475\pi$
$653$	4.92251e6			0.451756			0.225878	+	0.974156i	$0.427475\pi$
$654$		0			0
$655$		−	8.48734e6i		−	0.772980i
$656$		0			0
$657$			474997.i			0.0429317i
$658$		0			0
$659$	−1.00268e7			−0.899389			−0.449694	−	0.893182i	$-0.648467\pi$
$659$	−1.00268e7			−0.899389			−0.449694	+	0.893182i	$0.648467\pi$
$660$		0			0
$661$		−	97629.5i		−	0.00869116i	−0.999991	−	0.00434558i	$-0.998617\pi$
$661$		−	97629.5i		−	0.00869116i	0.999991	−	0.00434558i	$-0.00138324\pi$
$662$		0			0
$663$	−2.16090e6	−	4.58211e6i	−0.190920	−	0.404839i
$664$		0			0
$665$			6.80523e6i			0.596745i
$666$		0			0
$667$	1.41588e7			1.23228
$668$		0			0
$669$		−	1.08654e7i		−	0.938601i
$670$		0			0
$671$		−	7.08543e6i		−	0.607519i
$672$		0			0
$673$	7.82943e6			0.666334			0.333167	−	0.942868i	$-0.391883\pi$
$673$	7.82943e6			0.666334			0.333167	+	0.942868i	$0.391883\pi$
$674$		0			0
$675$	2.80671e6			0.237103
$676$		0			0
$677$	−6.24465e6			−0.523644			−0.261822	−	0.965116i	$-0.584323\pi$
$677$	−6.24465e6			−0.523644			−0.261822	+	0.965116i	$0.584323\pi$
$678$		0			0
$679$	1.45983e7			1.21514
$680$		0			0
$681$			7.96730e6i			0.658329i
$682$		0			0
$683$		−	1.22265e7i		−	1.00288i	−0.865192	−	0.501441i	$-0.832803\pi$
$683$		−	1.22265e7i		−	1.00288i	0.865192	−	0.501441i	$-0.167197\pi$
$684$		0			0
$685$	1.14264e7			0.930432
$686$		0			0
$687$			4.26385e6i			0.344675i
$688$		0			0
$689$	−7.92935e6	−	1.68139e7i	−0.636341	−	1.34934i
$690$		0			0
$691$		−	1.33096e7i		−	1.06040i	−0.847874	−	0.530198i	$-0.822117\pi$
$691$		−	1.33096e7i		−	1.06040i	0.847874	−	0.530198i	$-0.177883\pi$
$692$		0			0
$693$	1.19340e7			0.943956
$694$		0			0
$695$		−	1.02273e7i		−	0.803150i
$696$		0			0
$697$		−	1.12191e7i		−	0.874731i
$698$		0			0
$699$	−1.52307e6			−0.117903
$700$		0			0
$701$	−8.08924e6			−0.621745			−0.310873	−	0.950452i	$-0.600621\pi$
$701$	−8.08924e6			−0.621745			−0.310873	+	0.950452i	$0.600621\pi$
$702$		0			0
$703$	754254.			0.0575611
$704$		0			0
$705$	−4.24492e6			−0.321660
$706$		0			0
$707$			3.21149e6i			0.241634i
$708$		0			0
$709$			1.44669e7i			1.08084i	0.841397	+	0.540418i	$0.181734\pi$
$709$			1.44669e7i			1.08084i	−0.841397	+	0.540418i	$0.818266\pi$
$710$		0			0
$711$	1.39731e6			0.103662
$712$		0			0
$713$			1.01986e7i			0.751304i
$714$		0			0
$715$	1.70688e7	+	3.61937e7i	1.24864	+	2.64769i
$716$		0			0
$717$			1.31882e7i			0.958048i
$718$		0			0
$719$	−1.10888e7			−0.799950			−0.399975	−	0.916526i	$-0.630981\pi$
$719$	−1.10888e7			−0.799950			−0.399975	+	0.916526i	$0.630981\pi$
$720$		0			0
$721$			1.92830e7i			1.38145i
$722$		0			0
$723$		−	314851.i		−	0.0224006i
$724$		0			0
$725$	1.43806e7			1.01609
$726$		0			0
$727$	1.58622e7			1.11308			0.556542	−	0.830819i	$-0.312128\pi$
$727$	1.58622e7			1.11308			0.556542	+	0.830819i	$0.312128\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−1.67044e6			−0.115621
$732$		0			0
$733$		−	1.25058e7i		−	0.859711i	−0.902898	−	0.429856i	$-0.858564\pi$
$733$		−	1.25058e7i		−	0.859711i	0.902898	−	0.429856i	$-0.141436\pi$
$734$		0			0
$735$			1.37548e7i			0.939150i
$736$		0			0
$737$	2.21526e7			1.50230
$738$		0			0
$739$		−	1.10338e7i		−	0.743217i	−0.928389	−	0.371609i	$-0.878806\pi$
$739$		−	1.10338e7i		−	0.743217i	0.928389	−	0.371609i	$-0.121194\pi$
$740$		0			0
$741$	−2.15710e6	+	1.01728e6i	−0.144319	+	0.0680602i
$742$		0			0
$743$		−	2.80392e7i		−	1.86335i	−0.363296	−	0.931674i	$-0.618349\pi$
$743$		−	2.80392e7i		−	1.86335i	0.363296	−	0.931674i	$-0.381651\pi$
$744$		0			0
$745$	2.07003e7			1.36643
$746$		0			0
$747$		−	6.05632e6i		−	0.397107i
$748$		0			0
$749$		−	2.50575e7i		−	1.63205i
$750$		0			0
$751$	−3.13111e6			−0.202581			−0.101291	−	0.994857i	$-0.532297\pi$
$751$	−3.13111e6			−0.202581			−0.101291	+	0.994857i	$0.532297\pi$
$752$		0			0
$753$	−5.66397e6			−0.364027
$754$		0			0
$755$	2.70837e7			1.72918
$756$		0			0
$757$	−4.09411e6			−0.259669			−0.129834	−	0.991536i	$-0.541445\pi$
$757$	−4.09411e6			−0.259669			−0.129834	+	0.991536i	$0.541445\pi$
$758$		0			0
$759$		−	2.68267e7i		−	1.69030i
$760$		0			0
$761$		−	7.05171e6i		−	0.441400i	−0.975342	−	0.220700i	$-0.929166\pi$
$761$		−	7.05171e6i		−	0.441400i	0.975342	−	0.220700i	$-0.0708342\pi$
$762$		0			0
$763$	4.44185e7			2.76219
$764$		0			0
$765$			6.24930e6i			0.386081i
$766$		0			0
$767$	2.14493e7	−	1.01154e7i	1.31651	−	0.620861i
$768$		0			0
$769$			1.49719e7i			0.912977i	0.889729	+	0.456488i	$0.150893\pi$
$769$			1.49719e7i			0.912977i	−0.889729	+	0.456488i	$0.849107\pi$
$770$		0			0
$771$	7.46773e6			0.452431
$772$		0			0
$773$		−	2.67527e7i		−	1.61035i	−0.593040	−	0.805173i	$-0.702072\pi$
$773$		−	2.67527e7i		−	1.61035i	0.593040	−	0.805173i	$-0.297928\pi$
$774$		0			0
$775$			1.03584e7i			0.619494i
$776$		0			0
$777$	2.92467e6			0.173790
$778$		0			0
$779$	−5.28153e6			−0.311829
$780$		0			0
$781$	6.31530e7			3.70481
$782$		0			0
$783$	2.72292e6			0.158719
$784$		0			0
$785$			3.39976e7i			1.96913i
$786$		0			0
$787$		−	1.16234e7i		−	0.668956i	−0.942404	−	0.334478i	$-0.891440\pi$
$787$		−	1.16234e7i		−	0.668956i	0.942404	−	0.334478i	$-0.108560\pi$
$788$		0			0
$789$	−1.88645e6			−0.107883
$790$		0			0
$791$			2.59077e7i			1.47227i
$792$		0			0
$793$	2.34196e6	+	4.96604e6i	0.132250	+	0.280432i
$794$		0			0
$795$			2.29316e7i			1.28682i
$796$		0			0
$797$	−2.45692e7			−1.37008			−0.685040	−	0.728505i	$-0.740215\pi$
$797$	−2.45692e7			−1.37008			−0.685040	+	0.728505i	$0.740215\pi$
$798$		0			0
$799$		−	5.21705e6i		−	0.289106i
$800$		0			0
$801$		−	3.70822e6i		−	0.204214i
$802$		0			0
$803$	4.61119e6			0.252362
$804$		0			0
$805$	5.93179e7			3.22623
$806$		0			0
$807$	−1.51902e7			−0.821068
$808$		0			0
$809$	2.45251e7			1.31747			0.658734	−	0.752376i	$-0.271092\pi$
$809$	2.45251e7			1.31747			0.658734	+	0.752376i	$0.271092\pi$
$810$		0			0
$811$			6.07437e6i			0.324302i	0.986766	+	0.162151i	$0.0518431\pi$
$811$			6.07437e6i			0.324302i	−0.986766	+	0.162151i	$0.948157\pi$
$812$		0			0
$813$		−	1.27910e7i		−	0.678703i
$814$		0			0
$815$	−1.61140e7			−0.849783
$816$		0			0
$817$			786385.i			0.0412174i
$818$		0			0
$819$	−8.36429e6	+	3.94456e6i	−0.435732	+	0.205489i
$820$		0			0
$821$		−	7.25192e6i		−	0.375487i	−0.982218	−	0.187744i	$-0.939883\pi$
$821$		−	7.25192e6i		−	0.375487i	0.982218	−	0.187744i	$-0.0601174\pi$
$822$		0			0
$823$	−3.62080e7			−1.86340			−0.931699	−	0.363233i	$-0.881673\pi$
$823$	−3.62080e7			−1.86340			−0.931699	+	0.363233i	$0.881673\pi$
$824$		0			0
$825$		−	2.72470e7i		−	1.39375i
$826$		0			0
$827$		−	1.58515e7i		−	0.805946i	−0.915212	−	0.402973i	$-0.867977\pi$
$827$		−	1.58515e7i		−	0.805946i	0.915212	−	0.402973i	$-0.132023\pi$
$828$		0			0
$829$	−8.06150e6			−0.407408			−0.203704	−	0.979033i	$-0.565298\pi$
$829$	−8.06150e6			−0.407408			−0.203704	+	0.979033i	$0.565298\pi$
$830$		0			0
$831$	3.85765e6			0.193785
$832$		0			0
$833$	−1.69047e7			−0.844104
$834$		0			0
$835$	−2.99698e7			−1.48754
$836$		0			0
$837$			1.96132e6i			0.0967688i
$838$		0			0
$839$		−	1.60489e7i		−	0.787118i	−0.919299	−	0.393559i	$-0.871244\pi$
$839$		−	1.60489e7i		−	0.787118i	0.919299	−	0.393559i	$-0.128756\pi$
$840$		0			0
$841$	−6.55987e6			−0.319820
$842$		0			0
$843$		−	2.19004e6i		−	0.106141i
$844$		0			0
$845$	−2.39264e7	−	1.97257e7i	−1.15275	−	0.950365i
$846$		0			0
$847$		−	8.56772e7i		−	4.10352i
$848$		0			0
$849$	−1.38461e7			−0.659262
$850$		0			0
$851$		−	6.57446e6i		−	0.311197i
$852$		0			0
$853$		−	1.48875e7i		−	0.700566i	−0.936644	−	0.350283i	$-0.886085\pi$
$853$		−	1.48875e7i		−	0.700566i	0.936644	−	0.350283i	$-0.113915\pi$
$854$		0			0
$855$	2.94195e6			0.137632
$856$		0			0
$857$	−1.30374e7			−0.606369			−0.303185	−	0.952932i	$-0.598050\pi$
$857$	−1.30374e7			−0.606369			−0.303185	+	0.952932i	$0.598050\pi$
$858$		0			0
$859$	1.33964e7			0.619448			0.309724	−	0.950826i	$-0.399763\pi$
$859$	1.33964e7			0.619448			0.309724	+	0.950826i	$0.399763\pi$
$860$		0			0
$861$	−2.04795e7			−0.941482
$862$		0			0
$863$		−	2.38224e7i		−	1.08882i	−0.838818	−	0.544412i	$-0.816753\pi$
$863$		−	2.38224e7i		−	1.08882i	0.838818	−	0.544412i	$-0.183247\pi$
$864$		0			0
$865$			4.80433e7i			2.18319i
$866$		0			0
$867$	5.09826e6			0.230343
$868$		0			0
$869$		−	1.35648e7i		−	0.609347i
$870$		0			0
$871$	−1.55263e7	+	7.32215e6i	−0.693464	+	0.327034i
$872$		0			0
$873$		−	6.31093e6i		−	0.280258i
$874$		0			0
$875$	1.13463e7			0.500995
$876$		0			0
$877$		−	2.72239e7i		−	1.19523i	−0.801783	−	0.597616i	$-0.796115\pi$
$877$		−	2.72239e7i		−	1.19523i	0.801783	−	0.597616i	$-0.203885\pi$
$878$		0			0
$879$		−	9.19221e6i		−	0.401280i
$880$		0			0
$881$	1.21240e7			0.526268			0.263134	−	0.964759i	$-0.415244\pi$
$881$	1.21240e7			0.526268			0.263134	+	0.964759i	$0.415244\pi$
$882$		0			0
$883$	5.51350e6			0.237972			0.118986	−	0.992896i	$-0.462036\pi$
$883$	5.51350e6			0.237972			0.118986	+	0.992896i	$0.462036\pi$
$884$		0			0
$885$	−2.92536e7			−1.25551
$886$		0			0
$887$	−2.79700e7			−1.19367			−0.596835	−	0.802364i	$-0.703575\pi$
$887$	−2.79700e7			−1.19367			−0.596835	+	0.802364i	$0.703575\pi$
$888$		0			0
$889$		−	5.31180e7i		−	2.25417i
$890$		0			0
$891$		−	5.15914e6i		−	0.217712i
$892$		0			0
$893$	−2.45600e6			−0.103062
$894$		0			0
$895$			6.09799e7i			2.54466i
$896$		0			0
$897$	8.86710e6	+	1.88023e7i	0.367960	+	0.780244i
$898$		0			0
$899$			1.00491e7i			0.414696i
$900$		0			0
$901$	−2.81831e7			−1.15658
$902$		0			0
$903$			3.04927e6i			0.124445i
$904$		0			0
$905$			1.50367e6i			0.0610282i
$906$		0			0
$907$	−5.29278e6			−0.213632			−0.106816	−	0.994279i	$-0.534066\pi$
$907$	−5.29278e6			−0.213632			−0.106816	+	0.994279i	$0.534066\pi$
$908$		0			0
$909$	1.38835e6			0.0557299
$910$		0			0
$911$	2.21671e7			0.884939			0.442469	−	0.896784i	$-0.354103\pi$
$911$	2.21671e7			0.884939			0.442469	+	0.896784i	$0.354103\pi$
$912$		0			0
$913$	−5.87937e7			−2.33429
$914$		0			0
$915$		−	6.77292e6i		−	0.267438i
$916$		0			0
$917$		−	1.90410e7i		−	0.747767i
$918$		0			0
$919$	−2.94503e7			−1.15027			−0.575136	−	0.818058i	$-0.695051\pi$
$919$	−2.94503e7			−1.15027			−0.575136	+	0.818058i	$0.695051\pi$
$920$		0			0
$921$			4.82068e6i			0.187266i
$922$		0			0
$923$	−4.42627e7	+	2.08741e7i	−1.71015	+	0.806499i
$924$		0			0
$925$		−	6.67747e6i		−	0.256600i
$926$		0			0
$927$	8.33617e6			0.318616
$928$		0			0
$929$		−	3.97704e7i		−	1.51189i	−0.654635	−	0.755945i	$-0.727178\pi$
$929$		−	3.97704e7i		−	1.51189i	0.654635	−	0.755945i	$-0.272822\pi$
$930$		0			0
$931$			7.95814e6i			0.300911i
$932$		0			0
$933$	2.67825e7			1.00727
$934$		0			0
$935$	6.06671e7			2.26947
$936$		0			0
$937$	−4.09277e7			−1.52289			−0.761445	−	0.648229i	$-0.775510\pi$
$937$	−4.09277e7			−1.52289			−0.761445	+	0.648229i	$0.775510\pi$
$938$		0			0
$939$	−147708.			−0.00546689
$940$		0			0
$941$			3.05137e7i			1.12337i	0.827353	+	0.561683i	$0.189846\pi$
$941$			3.05137e7i			1.12337i	−0.827353	+	0.561683i	$0.810154\pi$
$942$		0			0
$943$			4.60365e7i			1.68587i
$944$		0			0
$945$	1.14076e7			0.415543
$946$		0			0
$947$		−	3.50278e7i		−	1.26922i	−0.772831	−	0.634611i	$-0.781160\pi$
$947$		−	3.50278e7i		−	1.26922i	0.772831	−	0.634611i	$-0.218840\pi$
$948$		0			0
$949$	−3.23190e6	+	1.52415e6i	−0.116491	+	0.0549366i
$950$		0			0
$951$			2.68548e7i			0.962878i
$952$		0			0
$953$	3.97551e7			1.41795			0.708974	−	0.705234i	$-0.249158\pi$
$953$	3.97551e7			1.41795			0.708974	+	0.705234i	$0.249158\pi$
$954$		0			0
$955$		−	6.00392e7i		−	2.13023i
$956$		0			0
$957$		−	2.64336e7i		−	0.932989i
$958$		0			0
$959$	2.56348e7			0.900084
$960$		0			0
$961$	2.13907e7			0.747166
$962$		0			0
$963$	−1.08325e7			−0.376413
$964$		0			0
$965$	−1.36570e7			−0.472104
$966$		0			0
$967$			1.45550e7i			0.500547i	0.968175	+	0.250274i	$0.0805206\pi$
$967$			1.45550e7i			0.500547i	−0.968175	+	0.250274i	$0.919479\pi$
$968$		0			0
$969$			3.61568e6i			0.123703i
$970$		0			0
$971$	1.10708e7			0.376818			0.188409	−	0.982091i	$-0.439667\pi$
$971$	1.10708e7			0.376818			0.188409	+	0.982091i	$0.439667\pi$
$972$		0			0
$973$		−	2.29444e7i		−	0.776954i
$974$		0			0
$975$	9.00602e6	+	1.90969e7i	0.303404	+	0.643357i
$976$		0			0
$977$			1.10774e7i			0.371280i	0.982618	+	0.185640i	$0.0594358\pi$
$977$			1.10774e7i			0.371280i	−0.982618	+	0.185640i	$0.940564\pi$
$978$		0			0
$979$	−3.59988e7			−1.20041
$980$		0			0
$981$		−	1.92024e7i		−	0.637065i
$982$		0			0
$983$			3.48354e7i			1.14984i	0.818210	+	0.574919i	$0.194967\pi$
$983$			3.48354e7i			1.14984i	−0.818210	+	0.574919i	$0.805033\pi$
$984$		0			0
$985$	−3.48660e6			−0.114502
$986$		0			0
$987$	−9.52331e6			−0.311168
$988$		0			0
$989$	6.85453e6			0.222837
$990$		0			0
$991$	−1.08055e7			−0.349512			−0.174756	−	0.984612i	$-0.555914\pi$
$991$	−1.08055e7			−0.349512			−0.174756	+	0.984612i	$0.555914\pi$
$992$		0			0
$993$			7.63439e6i			0.245698i
$994$		0			0
$995$		−	1.32811e7i		−	0.425283i
$996$		0			0
$997$	−2.76495e6			−0.0880945			−0.0440473	−	0.999029i	$-0.514025\pi$
$997$	−2.76495e6			−0.0880945			−0.0440473	+	0.999029i	$0.514025\pi$
$998$		0			0
$999$		−	1.26436e6i		−	0.0400826i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.6.c.f.337.7		8
4.3	odd	2		78.6.b.b.25.4	✓	8
12.11	even	2		234.6.b.d.181.5		8
13.12	even	2	inner	624.6.c.f.337.2		8
52.31	even	4		1014.6.a.t.1.4		4
52.47	even	4		1014.6.a.u.1.1		4
52.51	odd	2		78.6.b.b.25.5	yes	8
156.155	even	2		234.6.b.d.181.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.6.b.b.25.4	✓	8	4.3	odd	2
78.6.b.b.25.5	yes	8	52.51	odd	2
234.6.b.d.181.4		8	156.155	even	2
234.6.b.d.181.5		8	12.11	even	2
624.6.c.f.337.2		8	13.12	even	2	inner
624.6.c.f.337.7		8	1.1	even	1	trivial
1014.6.a.t.1.4		4	52.31	even	4
1014.6.a.u.1.1		4	52.47	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	624.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +83.5169i q^{5} +187.367i q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +83.5169i q^{5} +187.367i q^{7} +81.0000 q^{9} -786.334i q^{11} +(259.909 + 551.127i) q^{13} -751.652i q^{15} +923.787 q^{17} -434.886i q^{19} -1686.30i q^{21} -3790.69 q^{23} -3850.08 q^{25} -729.000 q^{27} -3735.14 q^{29} -2690.43i q^{31} +7077.00i q^{33} -15648.3 q^{35} +1734.37i q^{37} +(-2339.18 - 4960.14i) q^{39} -12144.6i q^{41} -1808.26 q^{43} +6764.87i q^{45} -5647.45i q^{47} -18299.4 q^{49} -8314.09 q^{51} -30508.2 q^{53} +65672.2 q^{55} +3913.97i q^{57} -38919.0i q^{59} +9010.71 q^{61} +15176.7i q^{63} +(-46028.4 + 21706.8i) q^{65} +28172.0i q^{67} +34116.2 q^{69} +80313.2i q^{71} +5864.17i q^{73} +34650.7 q^{75} +147333. q^{77} +17250.7 q^{79} +6561.00 q^{81} -74769.4i q^{83} +77151.9i q^{85} +33616.3 q^{87} -45780.6i q^{89} +(-103263. + 48698.3i) q^{91} +24213.9i q^{93} +36320.4 q^{95} -77912.8i q^{97} -63693.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{3} + 648 q^{9}+O(q^{10}) \) 8 * q - 72 * q^3 + 648 * q^9	\( 8 q - 72 q^{3} + 648 q^{9} - 52 q^{13} + 600 q^{17} + 1656 q^{23} - 10128 q^{25} - 5832 q^{27} - 3360 q^{29} - 27144 q^{35} + 468 q^{39} - 16944 q^{43} - 59480 q^{49} - 5400 q^{51} + 9576 q^{53} - 5424 q^{55} + 58344 q^{61} - 192888 q^{65} - 14904 q^{69} + 91152 q^{75} + 243960 q^{77} - 155968 q^{79} + 52488 q^{81} + 30240 q^{87} + 159280 q^{91} + 368232 q^{95}+O(q^{100}) \) 8 * q - 72 * q^3 + 648 * q^9 - 52 * q^13 + 600 * q^17 + 1656 * q^23 - 10128 * q^25 - 5832 * q^27 - 3360 * q^29 - 27144 * q^35 + 468 * q^39 - 16944 * q^43 - 59480 * q^49 - 5400 * q^51 + 9576 * q^53 - 5424 * q^55 + 58344 * q^61 - 192888 * q^65 - 14904 * q^69 + 91152 * q^75 + 243960 * q^77 - 155968 * q^79 + 52488 * q^81 + 30240 * q^87 + 159280 * q^91 + 368232 * q^95

Introduction

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