LMFDB - Embedded newform 416.2.w.a.225.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(225,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("416.225");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.w (of order $6$, degree $2$, 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:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			225.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	416.225
Dual form			416.2.w.a.257.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 1.50000i) q^{3} -3.46410i q^{5} +(-2.59808 + 1.50000i) q^{7} +O(q^{10})$	\(q+(-0.866025 + 1.50000i) q^{3} -3.46410i q^{5} +(-2.59808 + 1.50000i) q^{7} +(4.33013 + 2.50000i) q^{11} +(-1.00000 + 3.46410i) q^{13} +(5.19615 + 3.00000i) q^{15} +(3.50000 + 6.06218i) q^{17} +(4.33013 - 2.50000i) q^{19} -5.19615i q^{21} +(-2.59808 + 4.50000i) q^{23} -7.00000 q^{25} -5.19615 q^{27} +(2.50000 - 4.33013i) q^{29} +2.00000i q^{31} +(-7.50000 + 4.33013i) q^{33} +(5.19615 + 9.00000i) q^{35} +(4.50000 + 2.59808i) q^{37} +(-4.33013 - 4.50000i) q^{39} +(-1.50000 - 0.866025i) q^{41} +(-2.59808 - 4.50000i) q^{43} +4.00000i q^{47} +(1.00000 - 1.73205i) q^{49} -12.1244 q^{51} +4.00000 q^{53} +(8.66025 - 15.0000i) q^{55} +8.66025i q^{57} +(-6.06218 + 3.50000i) q^{59} +(-1.50000 - 2.59808i) q^{61} +(12.0000 + 3.46410i) q^{65} +(2.59808 + 1.50000i) q^{67} +(-4.50000 - 7.79423i) q^{69} +(6.06218 - 3.50000i) q^{71} -3.46410i q^{73} +(6.06218 - 10.5000i) q^{75} -15.0000 q^{77} +3.46410 q^{79} +(4.50000 - 7.79423i) q^{81} -14.0000i q^{83} +(21.0000 - 12.1244i) q^{85} +(4.33013 + 7.50000i) q^{87} +(1.50000 + 0.866025i) q^{89} +(-2.59808 - 10.5000i) q^{91} +(-3.00000 - 1.73205i) q^{93} +(-8.66025 - 15.0000i) q^{95} +(7.50000 - 4.33013i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 4 q^{13} + 14 q^{17} - 28 q^{25} + 10 q^{29} - 30 q^{33} + 18 q^{37} - 6 q^{41} + 4 q^{49} + 16 q^{53} - 6 q^{61} + 48 q^{65} - 18 q^{69} - 60 q^{77} + 18 q^{81} + 84 q^{85} + 6 q^{89} - 12 q^{93} + 30 q^{97}+O(q^{100}) $ 4 * q - 4 * q^13 + 14 * q^17 - 28 * q^25 + 10 * q^29 - 30 * q^33 + 18 * q^37 - 6 * q^41 + 4 * q^49 + 16 * q^53 - 6 * q^61 + 48 * q^65 - 18 * q^69 - 60 * q^77 + 18 * q^81 + 84 * q^85 + 6 * q^89 - 12 * q^93 + 30 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.866025	+	1.50000i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$3$	−0.866025	+	1.50000i	−0.500000	+	0.866025i	−1.00000			$\pi$
$4$		0			0
$5$		−	3.46410i		−	1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$		−	3.46410i		−	1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$		0			0
$7$	−2.59808	+	1.50000i	−0.981981	+	0.566947i	−0.902867	−	0.429919i	$-0.858542\pi$
$7$	−2.59808	+	1.50000i	−0.981981	+	0.566947i	−0.0791130	+	0.996866i	$0.525209\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.33013	+	2.50000i	1.30558	+	0.753778i	0.981356	−	0.192201i	$-0.0615626\pi$
$11$	4.33013	+	2.50000i	1.30558	+	0.753778i	0.324227	+	0.945979i	$0.394896\pi$
$12$		0			0
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$14$		0			0
$15$	5.19615	+	3.00000i	1.34164	+	0.774597i
$16$		0			0
$17$	3.50000	+	6.06218i	0.848875	+	1.47029i	0.882213	+	0.470850i	$0.156053\pi$
$17$	3.50000	+	6.06218i	0.848875	+	1.47029i	−0.0333386	+	0.999444i	$0.510614\pi$
$18$		0			0
$19$	4.33013	−	2.50000i	0.993399	−	0.573539i	0.0871106	−	0.996199i	$-0.472237\pi$
$19$	4.33013	−	2.50000i	0.993399	−	0.573539i	0.906289	+	0.422659i	$0.138903\pi$
$20$		0			0
$21$		−	5.19615i		−	1.13389i
$22$		0			0
$23$	−2.59808	+	4.50000i	−0.541736	+	0.938315i	0.457068	+	0.889432i	$0.348900\pi$
$23$	−2.59808	+	4.50000i	−0.541736	+	0.938315i	−0.998805	+	0.0488832i	$0.984434\pi$
$24$		0			0
$25$	−7.00000			−1.40000
$26$		0			0
$27$	−5.19615			−1.00000
$28$		0			0
$29$	2.50000	−	4.33013i	0.464238	−	0.804084i	−0.534928	−	0.844897i	$-0.679661\pi$
$29$	2.50000	−	4.33013i	0.464238	−	0.804084i	0.999167	+	0.0408130i	$0.0129948\pi$
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$		0			0
$33$	−7.50000	+	4.33013i	−1.30558	+	0.753778i
$34$		0			0
$35$	5.19615	+	9.00000i	0.878310	+	1.52128i
$36$		0			0
$37$	4.50000	+	2.59808i	0.739795	+	0.427121i	0.821995	−	0.569495i	$-0.192861\pi$
$37$	4.50000	+	2.59808i	0.739795	+	0.427121i	−0.0821995	+	0.996616i	$0.526194\pi$
$38$		0			0
$39$	−4.33013	−	4.50000i	−0.693375	−	0.720577i
$40$		0			0
$41$	−1.50000	−	0.866025i	−0.234261	−	0.135250i	0.378275	−	0.925693i	$-0.376517\pi$
$41$	−1.50000	−	0.866025i	−0.234261	−	0.135250i	−0.612536	+	0.790443i	$0.709851\pi$
$42$		0			0
$43$	−2.59808	−	4.50000i	−0.396203	−	0.686244i	0.597051	−	0.802203i	$-0.296339\pi$
$43$	−2.59808	−	4.50000i	−0.396203	−	0.686244i	−0.993254	+	0.115960i	$0.963006\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			4.00000i			0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$			4.00000i			0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$		0			0
$49$	1.00000	−	1.73205i	0.142857	−	0.247436i
$50$		0			0
$51$	−12.1244			−1.69775
$52$		0			0
$53$	4.00000			0.549442			0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000			0.549442			0.274721	+	0.961524i	$0.411414\pi$
$54$		0			0
$55$	8.66025	−	15.0000i	1.16775	−	2.02260i
$56$		0			0
$57$			8.66025i			1.14708i
$58$		0			0
$59$	−6.06218	+	3.50000i	−0.789228	+	0.455661i	−0.839691	−	0.543065i	$-0.817264\pi$
$59$	−6.06218	+	3.50000i	−0.789228	+	0.455661i	0.0504625	+	0.998726i	$0.483930\pi$
$60$		0			0
$61$	−1.50000	−	2.59808i	−0.192055	−	0.332650i	0.753876	−	0.657017i	$-0.228182\pi$
$61$	−1.50000	−	2.59808i	−0.192055	−	0.332650i	−0.945931	+	0.324367i	$0.894849\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	12.0000	+	3.46410i	1.48842	+	0.429669i
$66$		0			0
$67$	2.59808	+	1.50000i	0.317406	+	0.183254i	0.650236	−	0.759733i	$-0.274670\pi$
$67$	2.59808	+	1.50000i	0.317406	+	0.183254i	−0.332830	+	0.942987i	$0.608004\pi$
$68$		0			0
$69$	−4.50000	−	7.79423i	−0.541736	−	0.938315i
$70$		0			0
$71$	6.06218	−	3.50000i	0.719448	−	0.415374i	−0.0951014	−	0.995468i	$-0.530318\pi$
$71$	6.06218	−	3.50000i	0.719448	−	0.415374i	0.814550	+	0.580094i	$0.196984\pi$
$72$		0			0
$73$		−	3.46410i		−	0.405442i	−0.979236	−	0.202721i	$-0.935021\pi$
$73$		−	3.46410i		−	0.405442i	0.979236	−	0.202721i	$-0.0649785\pi$
$74$		0			0
$75$	6.06218	−	10.5000i	0.700000	−	1.21244i
$76$		0			0
$77$	−15.0000			−1.70941
$78$		0			0
$79$	3.46410			0.389742			0.194871	−	0.980829i	$-0.437571\pi$
$79$	3.46410			0.389742			0.194871	+	0.980829i	$0.437571\pi$
$80$		0			0
$81$	4.50000	−	7.79423i	0.500000	−	0.866025i
$82$		0			0
$83$		−	14.0000i		−	1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$		−	14.0000i		−	1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$		0			0
$85$	21.0000	−	12.1244i	2.27777	−	1.31507i
$86$		0			0
$87$	4.33013	+	7.50000i	0.464238	+	0.804084i
$88$		0			0
$89$	1.50000	+	0.866025i	0.159000	+	0.0917985i	0.577389	−	0.816469i	$-0.304072\pi$
$89$	1.50000	+	0.866025i	0.159000	+	0.0917985i	−0.418389	+	0.908268i	$0.637405\pi$
$90$		0			0
$91$	−2.59808	−	10.5000i	−0.272352	−	1.10070i
$92$		0			0
$93$	−3.00000	−	1.73205i	−0.311086	−	0.179605i
$94$		0			0
$95$	−8.66025	−	15.0000i	−0.888523	−	1.53897i
$96$		0			0
$97$	7.50000	−	4.33013i	0.761510	−	0.439658i	−0.0683279	−	0.997663i	$-0.521766\pi$
$97$	7.50000	−	4.33013i	0.761510	−	0.439658i	0.829837	+	0.558005i	$0.188433\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.50000	+	14.7224i	−0.845782	+	1.46494i	0.0391591	+	0.999233i	$0.487532\pi$
$101$	−8.50000	+	14.7224i	−0.845782	+	1.46494i	−0.884941	+	0.465704i	$0.845801\pi$
$102$		0			0
$103$	−6.92820			−0.682656			−0.341328	−	0.939944i	$-0.610877\pi$
$103$	−6.92820			−0.682656			−0.341328	+	0.939944i	$0.610877\pi$
$104$		0			0
$105$	−18.0000			−1.75662
$106$		0			0
$107$	−4.33013	+	7.50000i	−0.418609	+	0.725052i	−0.995800	−	0.0915571i	$-0.970816\pi$
$107$	−4.33013	+	7.50000i	−0.418609	+	0.725052i	0.577191	+	0.816609i	$0.304149\pi$
$108$		0			0
$109$			10.3923i			0.995402i	0.867349	+	0.497701i	$0.165822\pi$
$109$			10.3923i			0.995402i	−0.867349	+	0.497701i	$0.834178\pi$
$110$		0			0
$111$	−7.79423	+	4.50000i	−0.739795	+	0.427121i
$112$		0			0
$113$	−2.50000	−	4.33013i	−0.235180	−	0.407344i	0.724145	−	0.689648i	$-0.242235\pi$
$113$	−2.50000	−	4.33013i	−0.235180	−	0.407344i	−0.959325	+	0.282304i	$0.908901\pi$
$114$		0			0
$115$	15.5885	+	9.00000i	1.45363	+	0.839254i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−18.1865	−	10.5000i	−1.66716	−	0.962533i
$120$		0			0
$121$	7.00000	+	12.1244i	0.636364	+	1.10221i
$122$		0			0
$123$	2.59808	−	1.50000i	0.234261	−	0.135250i
$124$		0			0
$125$			6.92820i			0.619677i
$126$		0			0
$127$	9.52628	−	16.5000i	0.845321	−	1.46414i	−0.0400219	−	0.999199i	$-0.512743\pi$
$127$	9.52628	−	16.5000i	0.845321	−	1.46414i	0.885342	−	0.464939i	$-0.153924\pi$
$128$		0			0
$129$	9.00000			0.792406
$130$		0			0
$131$	−13.8564			−1.21064			−0.605320	−	0.795982i	$-0.706955\pi$
$131$	−13.8564			−1.21064			−0.605320	+	0.795982i	$0.706955\pi$
$132$		0			0
$133$	−7.50000	+	12.9904i	−0.650332	+	1.12641i
$134$		0			0
$135$			18.0000i			1.54919i
$136$		0			0
$137$	−4.50000	+	2.59808i	−0.384461	+	0.221969i	−0.679757	−	0.733437i	$-0.737915\pi$
$137$	−4.50000	+	2.59808i	−0.384461	+	0.221969i	0.295296	+	0.955406i	$0.404582\pi$
$138$		0			0
$139$	−4.33013	−	7.50000i	−0.367277	−	0.636142i	0.621862	−	0.783127i	$-0.286376\pi$
$139$	−4.33013	−	7.50000i	−0.367277	−	0.636142i	−0.989139	+	0.146985i	$0.953043\pi$
$140$		0			0
$141$	−6.00000	−	3.46410i	−0.505291	−	0.291730i
$142$		0			0
$143$	−12.9904	+	12.5000i	−1.08631	+	1.04530i
$144$		0			0
$145$	−15.0000	−	8.66025i	−1.24568	−	0.719195i
$146$		0			0
$147$	1.73205	+	3.00000i	0.142857	+	0.247436i
$148$		0			0
$149$	7.50000	−	4.33013i	0.614424	−	0.354738i	−0.160271	−	0.987073i	$-0.551237\pi$
$149$	7.50000	−	4.33013i	0.614424	−	0.354738i	0.774695	+	0.632335i	$0.217903\pi$
$150$		0			0
$151$			2.00000i			0.162758i	0.996683	+	0.0813788i	$0.0259324\pi$
$151$			2.00000i			0.162758i	−0.996683	+	0.0813788i	$0.974068\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.92820			0.556487
$156$		0			0
$157$	−8.00000			−0.638470			−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000			−0.638470			−0.319235	+	0.947676i	$0.603426\pi$
$158$		0			0
$159$	−3.46410	+	6.00000i	−0.274721	+	0.475831i
$160$		0			0
$161$		−	15.5885i		−	1.22854i
$162$		0			0
$163$	16.4545	−	9.50000i	1.28881	−	0.744097i	0.310372	−	0.950615i	$-0.399546\pi$
$163$	16.4545	−	9.50000i	1.28881	−	0.744097i	0.978443	+	0.206518i	$0.0662131\pi$
$164$		0			0
$165$	15.0000	+	25.9808i	1.16775	+	2.02260i
$166$		0			0
$167$	16.4545	+	9.50000i	1.27329	+	0.735132i	0.975605	−	0.219533i	$-0.0704535\pi$
$167$	16.4545	+	9.50000i	1.27329	+	0.735132i	0.297681	+	0.954665i	$0.403787\pi$
$168$		0			0
$169$	−11.0000	−	6.92820i	−0.846154	−	0.532939i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.50000	−	6.06218i	−0.266100	−	0.460899i	0.701751	−	0.712422i	$-0.252402\pi$
$173$	−3.50000	−	6.06218i	−0.266100	−	0.460899i	−0.967851	+	0.251523i	$0.919068\pi$
$174$		0			0
$175$	18.1865	−	10.5000i	1.37477	−	0.793725i
$176$		0			0
$177$		−	12.1244i		−	0.911322i
$178$		0			0
$179$	9.52628	−	16.5000i	0.712028	−	1.23327i	−0.252067	−	0.967710i	$-0.581110\pi$
$179$	9.52628	−	16.5000i	0.712028	−	1.23327i	0.964095	−	0.265558i	$-0.0855562\pi$
$180$		0			0
$181$	12.0000			0.891953			0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000			0.891953			0.445976	+	0.895045i	$0.352856\pi$
$182$		0			0
$183$	5.19615			0.384111
$184$		0			0
$185$	9.00000	−	15.5885i	0.661693	−	1.14609i
$186$		0			0
$187$			35.0000i			2.55945i
$188$		0			0
$189$	13.5000	−	7.79423i	0.981981	−	0.566947i
$190$		0			0
$191$	6.06218	+	10.5000i	0.438644	+	0.759753i	0.997585	−	0.0694538i	$-0.0221257\pi$
$191$	6.06218	+	10.5000i	0.438644	+	0.759753i	−0.558941	+	0.829207i	$0.688792\pi$
$192$		0			0
$193$	10.5000	+	6.06218i	0.755807	+	0.436365i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	10.5000	+	6.06218i	0.755807	+	0.436365i	−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$	−15.5885	+	15.0000i	−1.11631	+	1.07417i
$196$		0			0
$197$	1.50000	+	0.866025i	0.106871	+	0.0617018i	0.552483	−	0.833524i	$-0.313681\pi$
$197$	1.50000	+	0.866025i	0.106871	+	0.0617018i	−0.445612	+	0.895226i	$0.647014\pi$
$198$		0			0
$199$	0.866025	+	1.50000i	0.0613909	+	0.106332i	0.895087	−	0.445891i	$-0.147113\pi$
$199$	0.866025	+	1.50000i	0.0613909	+	0.106332i	−0.833696	+	0.552223i	$0.813780\pi$
$200$		0			0
$201$	−4.50000	+	2.59808i	−0.317406	+	0.183254i
$202$		0			0
$203$			15.0000i			1.05279i
$204$		0			0
$205$	−3.00000	+	5.19615i	−0.209529	+	0.362915i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	25.0000			1.72929
$210$		0			0
$211$	2.59808	−	4.50000i	0.178859	−	0.309793i	−0.762631	−	0.646834i	$-0.776093\pi$
$211$	2.59808	−	4.50000i	0.178859	−	0.309793i	0.941490	+	0.337041i	$0.109426\pi$
$212$		0			0
$213$			12.1244i			0.830747i
$214$		0			0
$215$	−15.5885	+	9.00000i	−1.06312	+	0.613795i
$216$		0			0
$217$	−3.00000	−	5.19615i	−0.203653	−	0.352738i
$218$		0			0
$219$	5.19615	+	3.00000i	0.351123	+	0.202721i
$220$		0			0
$221$	−24.5000	+	6.06218i	−1.64805	+	0.407786i
$222$		0			0
$223$	−11.2583	−	6.50000i	−0.753914	−	0.435272i	0.0731927	−	0.997318i	$-0.476681\pi$
$223$	−11.2583	−	6.50000i	−0.753914	−	0.435272i	−0.827106	+	0.562046i	$0.810015\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.9186	+	11.5000i	−1.32204	+	0.763282i	−0.984054	−	0.177868i	$-0.943080\pi$
$227$	−19.9186	+	11.5000i	−1.32204	+	0.763282i	−0.337989	+	0.941150i	$0.609747\pi$
$228$		0			0
$229$		−	24.2487i		−	1.60240i	−0.598397	−	0.801200i	$-0.704195\pi$
$229$		−	24.2487i		−	1.60240i	0.598397	−	0.801200i	$-0.295805\pi$
$230$		0			0
$231$	12.9904	−	22.5000i	0.854704	−	1.48039i
$232$		0			0
$233$	−8.00000			−0.524097			−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000			−0.524097			−0.262049	+	0.965055i	$0.584398\pi$
$234$		0			0
$235$	13.8564			0.903892
$236$		0			0
$237$	−3.00000	+	5.19615i	−0.194871	+	0.337526i
$238$		0			0
$239$		−	2.00000i		−	0.129369i	−0.997906	−	0.0646846i	$-0.979396\pi$
$239$		−	2.00000i		−	0.129369i	0.997906	−	0.0646846i	$-0.0206041\pi$
$240$		0			0
$241$	−7.50000	+	4.33013i	−0.483117	+	0.278928i	−0.721715	−	0.692191i	$-0.756646\pi$
$241$	−7.50000	+	4.33013i	−0.483117	+	0.278928i	0.238597	+	0.971119i	$0.423312\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.00000	−	3.46410i	−0.383326	−	0.221313i
$246$		0			0
$247$	4.33013	+	17.5000i	0.275519	+	1.11350i
$248$		0			0
$249$	21.0000	+	12.1244i	1.33082	+	0.768350i
$250$		0			0
$251$	7.79423	+	13.5000i	0.491967	+	0.852112i	0.999957	−	0.00925060i	$-0.00294460\pi$
$251$	7.79423	+	13.5000i	0.491967	+	0.852112i	−0.507990	+	0.861363i	$0.669611\pi$
$252$		0			0
$253$	−22.5000	+	12.9904i	−1.41456	+	0.816698i
$254$		0			0
$255$			42.0000i			2.63014i
$256$		0			0
$257$	11.5000	−	19.9186i	0.717350	−	1.24249i	−0.244696	−	0.969600i	$-0.578688\pi$
$257$	11.5000	−	19.9186i	0.717350	−	1.24249i	0.962046	−	0.272887i	$-0.0879786\pi$
$258$		0			0
$259$	−15.5885			−0.968620
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.866025	−	1.50000i	0.0534014	−	0.0924940i	−0.838089	−	0.545534i	$-0.816327\pi$
$263$	0.866025	−	1.50000i	0.0534014	−	0.0924940i	0.891490	+	0.453040i	$0.149660\pi$
$264$		0			0
$265$		−	13.8564i		−	0.851192i
$266$		0			0
$267$	−2.59808	+	1.50000i	−0.159000	+	0.0917985i
$268$		0			0
$269$	−0.500000	−	0.866025i	−0.0304855	−	0.0528025i	0.850380	−	0.526169i	$-0.176372\pi$
$269$	−0.500000	−	0.866025i	−0.0304855	−	0.0528025i	−0.880866	+	0.473366i	$0.843039\pi$
$270$		0			0
$271$	−7.79423	−	4.50000i	−0.473466	−	0.273356i	0.244224	−	0.969719i	$-0.421467\pi$
$271$	−7.79423	−	4.50000i	−0.473466	−	0.273356i	−0.717689	+	0.696363i	$0.754800\pi$
$272$		0			0
$273$	18.0000	+	5.19615i	1.08941	+	0.314485i
$274$		0			0
$275$	−30.3109	−	17.5000i	−1.82782	−	1.05529i
$276$		0			0
$277$	−4.50000	−	7.79423i	−0.270379	−	0.468310i	0.698580	−	0.715532i	$-0.253816\pi$
$277$	−4.50000	−	7.79423i	−0.270379	−	0.468310i	−0.968959	+	0.247222i	$0.920482\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			24.2487i			1.44656i	0.690557	+	0.723278i	$0.257366\pi$
$281$			24.2487i			1.44656i	−0.690557	+	0.723278i	$0.742634\pi$
$282$		0			0
$283$	16.4545	−	28.5000i	0.978117	−	1.69415i	0.308879	−	0.951101i	$-0.400046\pi$
$283$	16.4545	−	28.5000i	0.978117	−	1.69415i	0.669238	−	0.743048i	$-0.266621\pi$
$284$		0			0
$285$	30.0000			1.77705
$286$		0			0
$287$	5.19615			0.306719
$288$		0			0
$289$	−16.0000	+	27.7128i	−0.941176	+	1.63017i
$290$		0			0
$291$			15.0000i			0.879316i
$292$		0			0
$293$	−22.5000	+	12.9904i	−1.31446	+	0.758906i	−0.982832	−	0.184503i	$-0.940933\pi$
$293$	−22.5000	+	12.9904i	−1.31446	+	0.758906i	−0.331632	+	0.943409i	$0.607599\pi$
$294$		0			0
$295$	12.1244	+	21.0000i	0.705907	+	1.22267i
$296$		0			0
$297$	−22.5000	−	12.9904i	−1.30558	−	0.753778i
$298$		0			0
$299$	−12.9904	−	13.5000i	−0.751253	−	0.780725i
$300$		0			0
$301$	13.5000	+	7.79423i	0.778127	+	0.449252i
$302$		0			0
$303$	−14.7224	−	25.5000i	−0.845782	−	1.46494i
$304$		0			0
$305$	−9.00000	+	5.19615i	−0.515339	+	0.297531i
$306$		0			0
$307$		−	6.00000i		−	0.342438i	−0.985233	−	0.171219i	$-0.945229\pi$
$307$		−	6.00000i		−	0.342438i	0.985233	−	0.171219i	$-0.0547706\pi$
$308$		0			0
$309$	6.00000	−	10.3923i	0.341328	−	0.591198i
$310$		0			0
$311$	20.7846			1.17859			0.589294	−	0.807919i	$-0.299406\pi$
$311$	20.7846			1.17859			0.589294	+	0.807919i	$0.299406\pi$
$312$		0			0
$313$	12.0000			0.678280			0.339140	−	0.940736i	$-0.389864\pi$
$313$	12.0000			0.678280			0.339140	+	0.940736i	$0.389864\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			6.92820i			0.389127i	0.980890	+	0.194563i	$0.0623290\pi$
$317$			6.92820i			0.389127i	−0.980890	+	0.194563i	$0.937671\pi$
$318$		0			0
$319$	21.6506	−	12.5000i	1.21220	−	0.699866i
$320$		0			0
$321$	−7.50000	−	12.9904i	−0.418609	−	0.725052i
$322$		0			0
$323$	30.3109	+	17.5000i	1.68654	+	0.973726i
$324$		0			0
$325$	7.00000	−	24.2487i	0.388290	−	1.34508i
$326$		0			0
$327$	−15.5885	−	9.00000i	−0.862044	−	0.497701i
$328$		0			0
$329$	−6.00000	−	10.3923i	−0.330791	−	0.572946i
$330$		0			0
$331$	12.9904	−	7.50000i	0.714016	−	0.412237i	−0.0985303	−	0.995134i	$-0.531414\pi$
$331$	12.9904	−	7.50000i	0.714016	−	0.412237i	0.812546	+	0.582897i	$0.198081\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.19615	−	9.00000i	0.283896	−	0.491723i
$336$		0			0
$337$	−28.0000			−1.52526			−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000			−1.52526			−0.762629	+	0.646837i	$0.776092\pi$
$338$		0			0
$339$	8.66025			0.470360
$340$		0			0
$341$	−5.00000	+	8.66025i	−0.270765	+	0.468979i
$342$		0			0
$343$		−	15.0000i		−	0.809924i
$344$		0			0
$345$	−27.0000	+	15.5885i	−1.45363	+	0.839254i
$346$		0			0
$347$	7.79423	+	13.5000i	0.418416	+	0.724718i	0.995780	−	0.0917687i	$-0.0292521\pi$
$347$	7.79423	+	13.5000i	0.418416	+	0.724718i	−0.577364	+	0.816487i	$0.695919\pi$
$348$		0			0
$349$	−13.5000	−	7.79423i	−0.722638	−	0.417215i	0.0930846	−	0.995658i	$-0.470327\pi$
$349$	−13.5000	−	7.79423i	−0.722638	−	0.417215i	−0.815723	+	0.578443i	$0.803661\pi$
$350$		0			0
$351$	5.19615	−	18.0000i	0.277350	−	0.960769i
$352$		0			0
$353$	13.5000	+	7.79423i	0.718532	+	0.414845i	0.814212	−	0.580567i	$-0.197169\pi$
$353$	13.5000	+	7.79423i	0.718532	+	0.414845i	−0.0956798	+	0.995412i	$0.530502\pi$
$354$		0			0
$355$	−12.1244	−	21.0000i	−0.643494	−	1.11456i
$356$		0			0
$357$	31.5000	−	18.1865i	1.66716	−	0.962533i
$358$		0			0
$359$		−	22.0000i		−	1.16112i	−0.814219	−	0.580558i	$-0.802835\pi$
$359$		−	22.0000i		−	1.16112i	0.814219	−	0.580558i	$-0.197165\pi$
$360$		0			0
$361$	3.00000	−	5.19615i	0.157895	−	0.273482i
$362$		0			0
$363$	−24.2487			−1.27273
$364$		0			0
$365$	−12.0000			−0.628109
$366$		0			0
$367$	−6.06218	+	10.5000i	−0.316443	+	0.548096i	−0.979743	−	0.200258i	$-0.935822\pi$
$367$	−6.06218	+	10.5000i	−0.316443	+	0.548096i	0.663300	+	0.748354i	$0.269155\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−10.3923	+	6.00000i	−0.539542	+	0.311504i
$372$		0			0
$373$	1.50000	+	2.59808i	0.0776671	+	0.134523i	0.902243	−	0.431228i	$-0.141920\pi$
$373$	1.50000	+	2.59808i	0.0776671	+	0.134523i	−0.824576	+	0.565751i	$0.808586\pi$
$374$		0			0
$375$	−10.3923	−	6.00000i	−0.536656	−	0.309839i
$376$		0			0
$377$	12.5000	+	12.9904i	0.643783	+	0.669039i
$378$		0			0
$379$	−6.06218	−	3.50000i	−0.311393	−	0.179783i	0.336157	−	0.941806i	$-0.390873\pi$
$379$	−6.06218	−	3.50000i	−0.311393	−	0.179783i	−0.647550	+	0.762023i	$0.724206\pi$
$380$		0			0
$381$	16.5000	+	28.5788i	0.845321	+	1.46414i
$382$		0			0
$383$	−6.06218	+	3.50000i	−0.309763	+	0.178842i	−0.646820	−	0.762642i	$-0.723902\pi$
$383$	−6.06218	+	3.50000i	−0.309763	+	0.178842i	0.337058	+	0.941484i	$0.390568\pi$
$384$		0			0
$385$			51.9615i			2.64820i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	14.0000			0.709828			0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000			0.709828			0.354914	+	0.934899i	$0.384510\pi$
$390$		0			0
$391$	−36.3731			−1.83947
$392$		0			0
$393$	12.0000	−	20.7846i	0.605320	−	1.04844i
$394$		0			0
$395$		−	12.0000i		−	0.603786i
$396$		0			0
$397$	28.5000	−	16.4545i	1.43037	−	0.825827i	0.433225	−	0.901286i	$-0.357376\pi$
$397$	28.5000	−	16.4545i	1.43037	−	0.825827i	0.997149	+	0.0754589i	$0.0240422\pi$
$398$		0			0
$399$	−12.9904	−	22.5000i	−0.650332	−	1.12641i
$400$		0			0
$401$	28.5000	+	16.4545i	1.42322	+	0.821698i	0.996573	−	0.0827195i	$-0.0263606\pi$
$401$	28.5000	+	16.4545i	1.42322	+	0.821698i	0.426649	+	0.904417i	$0.359694\pi$
$402$		0			0
$403$	−6.92820	−	2.00000i	−0.345118	−	0.0996271i
$404$		0			0
$405$	−27.0000	−	15.5885i	−1.34164	−	0.774597i
$406$		0			0
$407$	12.9904	+	22.5000i	0.643909	+	1.11528i
$408$		0			0
$409$	−13.5000	+	7.79423i	−0.667532	+	0.385400i	−0.795141	−	0.606425i	$-0.792603\pi$
$409$	−13.5000	+	7.79423i	−0.667532	+	0.385400i	0.127609	+	0.991825i	$0.459270\pi$
$410$		0			0
$411$		−	9.00000i		−	0.443937i
$412$		0			0
$413$	10.5000	−	18.1865i	0.516671	−	0.894901i
$414$		0			0
$415$	−48.4974			−2.38064
$416$		0			0
$417$	15.0000			0.734553
$418$		0			0
$419$	14.7224	−	25.5000i	0.719238	−	1.24576i	−0.242064	−	0.970260i	$-0.577824\pi$
$419$	14.7224	−	25.5000i	0.719238	−	1.24576i	0.961302	−	0.275496i	$-0.0888422\pi$
$420$		0			0
$421$		−	31.1769i		−	1.51947i	−0.650233	−	0.759735i	$-0.725329\pi$
$421$		−	31.1769i		−	1.51947i	0.650233	−	0.759735i	$-0.274671\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−24.5000	−	42.4352i	−1.18842	−	2.05841i
$426$		0			0
$427$	7.79423	+	4.50000i	0.377189	+	0.217770i
$428$		0			0
$429$	−7.50000	−	30.3109i	−0.362103	−	1.46342i
$430$		0			0
$431$	32.0429	+	18.5000i	1.54345	+	0.891114i	0.998617	+	0.0525716i	$0.0167418\pi$
$431$	32.0429	+	18.5000i	1.54345	+	0.891114i	0.544837	+	0.838542i	$0.316592\pi$
$432$		0			0
$433$	−16.5000	−	28.5788i	−0.792939	−	1.37341i	−0.924139	−	0.382055i	$-0.875216\pi$
$433$	−16.5000	−	28.5788i	−0.792939	−	1.37341i	0.131200	−	0.991356i	$-0.458117\pi$
$434$		0			0
$435$	25.9808	−	15.0000i	1.24568	−	0.719195i
$436$		0			0
$437$			25.9808i			1.24283i
$438$		0			0
$439$	2.59808	−	4.50000i	0.123999	−	0.214773i	−0.797342	−	0.603528i	$-0.793761\pi$
$439$	2.59808	−	4.50000i	0.123999	−	0.214773i	0.921341	+	0.388755i	$0.127095\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.8564			−0.658338			−0.329169	−	0.944271i	$-0.606769\pi$
$443$	−13.8564			−0.658338			−0.329169	+	0.944271i	$0.606769\pi$
$444$		0			0
$445$	3.00000	−	5.19615i	0.142214	−	0.246321i
$446$		0			0
$447$			15.0000i			0.709476i
$448$		0			0
$449$	−7.50000	+	4.33013i	−0.353947	+	0.204351i	−0.666422	−	0.745575i	$-0.732175\pi$
$449$	−7.50000	+	4.33013i	−0.353947	+	0.204351i	0.312475	+	0.949926i	$0.398842\pi$
$450$		0			0
$451$	−4.33013	−	7.50000i	−0.203898	−	0.353161i
$452$		0			0
$453$	−3.00000	−	1.73205i	−0.140952	−	0.0813788i
$454$		0			0
$455$	−36.3731	+	9.00000i	−1.70520	+	0.421927i
$456$		0			0
$457$	31.5000	+	18.1865i	1.47351	+	0.850730i	0.999555	−	0.0298202i	$-0.00949348\pi$
$457$	31.5000	+	18.1865i	1.47351	+	0.850730i	0.473953	+	0.880550i	$0.342827\pi$
$458$		0			0
$459$	−18.1865	−	31.5000i	−0.848875	−	1.47029i
$460$		0			0
$461$	4.50000	−	2.59808i	0.209586	−	0.121004i	−0.391533	−	0.920164i	$-0.628055\pi$
$461$	4.50000	−	2.59808i	0.209586	−	0.121004i	0.601119	+	0.799160i	$0.294722\pi$
$462$		0			0
$463$		−	18.0000i		−	0.836531i	−0.908325	−	0.418265i	$-0.862638\pi$
$463$		−	18.0000i		−	0.836531i	0.908325	−	0.418265i	$-0.137362\pi$
$464$		0			0
$465$	−6.00000	+	10.3923i	−0.278243	+	0.481932i
$466$		0			0
$467$	38.1051			1.76329			0.881647	−	0.471909i	$-0.156435\pi$
$467$	38.1051			1.76329			0.881647	+	0.471909i	$0.156435\pi$
$468$		0			0
$469$	−9.00000			−0.415581
$470$		0			0
$471$	6.92820	−	12.0000i	0.319235	−	0.552931i
$472$		0			0
$473$		−	25.9808i		−	1.19460i
$474$		0			0
$475$	−30.3109	+	17.5000i	−1.39076	+	0.802955i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	14.7224	+	8.50000i	0.672685	+	0.388375i	0.797093	−	0.603856i	$-0.206370\pi$
$479$	14.7224	+	8.50000i	0.672685	+	0.388375i	−0.124408	+	0.992231i	$0.539703\pi$
$480$		0			0
$481$	−13.5000	+	12.9904i	−0.615547	+	0.592310i
$482$		0			0
$483$	23.3827	+	13.5000i	1.06395	+	0.614271i
$484$		0			0
$485$	−15.0000	−	25.9808i	−0.681115	−	1.17973i
$486$		0			0
$487$	19.9186	−	11.5000i	0.902597	−	0.521115i	0.0245553	−	0.999698i	$-0.492183\pi$
$487$	19.9186	−	11.5000i	0.902597	−	0.521115i	0.878042	+	0.478584i	$0.158850\pi$
$488$		0			0
$489$			32.9090i			1.48819i
$490$		0			0
$491$	−6.06218	+	10.5000i	−0.273582	+	0.473858i	−0.969776	−	0.243995i	$-0.921542\pi$
$491$	−6.06218	+	10.5000i	−0.273582	+	0.473858i	0.696194	+	0.717853i	$0.254875\pi$
$492$		0			0
$493$	35.0000			1.57632
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−10.5000	+	18.1865i	−0.470989	+	0.815778i
$498$		0			0
$499$		−	6.00000i		−	0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$		−	6.00000i		−	0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$		0			0
$501$	−28.5000	+	16.4545i	−1.27329	+	0.735132i
$502$		0			0
$503$	−4.33013	−	7.50000i	−0.193071	−	0.334408i	0.753196	−	0.657797i	$-0.228511\pi$
$503$	−4.33013	−	7.50000i	−0.193071	−	0.334408i	−0.946266	+	0.323388i	$0.895178\pi$
$504$		0			0
$505$	51.0000	+	29.4449i	2.26947	+	1.31028i
$506$		0			0
$507$	19.9186	−	10.5000i	0.884615	−	0.466321i
$508$		0			0
$509$	−19.5000	−	11.2583i	−0.864322	−	0.499017i	0.00113503	−	0.999999i	$-0.499639\pi$
$509$	−19.5000	−	11.2583i	−0.864322	−	0.499017i	−0.865457	+	0.500983i	$0.832972\pi$
$510$		0			0
$511$	5.19615	+	9.00000i	0.229864	+	0.398137i
$512$		0			0
$513$	−22.5000	+	12.9904i	−0.993399	+	0.573539i
$514$		0			0
$515$			24.0000i			1.05757i
$516$		0			0
$517$	−10.0000	+	17.3205i	−0.439799	+	0.761755i
$518$		0			0
$519$	12.1244			0.532200
$520$		0			0
$521$	4.00000			0.175243			0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000			0.175243			0.0876216	+	0.996154i	$0.472073\pi$
$522$		0			0
$523$	−6.06218	+	10.5000i	−0.265081	+	0.459133i	−0.967585	−	0.252547i	$-0.918732\pi$
$523$	−6.06218	+	10.5000i	−0.265081	+	0.459133i	0.702504	+	0.711680i	$0.252065\pi$
$524$		0			0
$525$			36.3731i			1.58745i
$526$		0			0
$527$	−12.1244	+	7.00000i	−0.528145	+	0.304925i
$528$		0			0
$529$	−2.00000	−	3.46410i	−0.0869565	−	0.150613i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.50000	−	4.33013i	0.194917	−	0.187559i
$534$		0			0
$535$	25.9808	+	15.0000i	1.12325	+	0.648507i
$536$		0			0
$537$	16.5000	+	28.5788i	0.712028	+	1.23327i
$538$		0			0
$539$	8.66025	−	5.00000i	0.373024	−	0.215365i
$540$		0			0
$541$		−	20.7846i		−	0.893600i	−0.894634	−	0.446800i	$-0.852564\pi$
$541$		−	20.7846i		−	0.893600i	0.894634	−	0.446800i	$-0.147436\pi$
$542$		0			0
$543$	−10.3923	+	18.0000i	−0.445976	+	0.772454i
$544$		0			0
$545$	36.0000			1.54207
$546$		0			0
$547$	6.92820			0.296229			0.148114	−	0.988970i	$-0.452680\pi$
$547$	6.92820			0.296229			0.148114	+	0.988970i	$0.452680\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	25.0000i		−	1.06504i
$552$		0			0
$553$	−9.00000	+	5.19615i	−0.382719	+	0.220963i
$554$		0			0
$555$	15.5885	+	27.0000i	0.661693	+	1.14609i
$556$		0			0
$557$	−34.5000	−	19.9186i	−1.46181	−	0.843978i	−0.462717	−	0.886506i	$-0.653125\pi$
$557$	−34.5000	−	19.9186i	−1.46181	−	0.843978i	−0.999095	+	0.0425287i	$0.986459\pi$
$558$		0			0
$559$	18.1865	−	4.50000i	0.769208	−	0.190330i
$560$		0			0
$561$	−52.5000	−	30.3109i	−2.21655	−	1.27973i
$562$		0			0
$563$	−4.33013	−	7.50000i	−0.182493	−	0.316087i	0.760236	−	0.649647i	$-0.225083\pi$
$563$	−4.33013	−	7.50000i	−0.182493	−	0.316087i	−0.942729	+	0.333560i	$0.891750\pi$
$564$		0			0
$565$	−15.0000	+	8.66025i	−0.631055	+	0.364340i
$566$		0			0
$567$			27.0000i			1.13389i
$568$		0			0
$569$	−15.5000	+	26.8468i	−0.649794	+	1.12548i	0.333378	+	0.942793i	$0.391811\pi$
$569$	−15.5000	+	26.8468i	−0.649794	+	1.12548i	−0.983172	+	0.182683i	$0.941522\pi$
$570$		0			0
$571$	−27.7128			−1.15975			−0.579873	−	0.814707i	$-0.696898\pi$
$571$	−27.7128			−1.15975			−0.579873	+	0.814707i	$0.696898\pi$
$572$		0			0
$573$	−21.0000			−0.877288
$574$		0			0
$575$	18.1865	−	31.5000i	0.758431	−	1.31364i
$576$		0			0
$577$		−	24.2487i		−	1.00949i	−0.863269	−	0.504744i	$-0.831587\pi$
$577$		−	24.2487i		−	1.00949i	0.863269	−	0.504744i	$-0.168413\pi$
$578$		0			0
$579$	−18.1865	+	10.5000i	−0.755807	+	0.436365i
$580$		0			0
$581$	21.0000	+	36.3731i	0.871227	+	1.50901i
$582$		0			0
$583$	17.3205	+	10.0000i	0.717342	+	0.414158i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.4545	+	9.50000i	0.679149	+	0.392107i	0.799534	−	0.600620i	$-0.205080\pi$
$587$	16.4545	+	9.50000i	0.679149	+	0.392107i	−0.120385	+	0.992727i	$0.538413\pi$
$588$		0			0
$589$	5.00000	+	8.66025i	0.206021	+	0.356840i
$590$		0			0
$591$	−2.59808	+	1.50000i	−0.106871	+	0.0617018i
$592$		0			0
$593$		−	38.1051i		−	1.56479i	−0.622783	−	0.782395i	$-0.713998\pi$
$593$		−	38.1051i		−	1.56479i	0.622783	−	0.782395i	$-0.286002\pi$
$594$		0			0
$595$	−36.3731	+	63.0000i	−1.49115	+	2.58275i
$596$		0			0
$597$	−3.00000			−0.122782
$598$		0			0
$599$	−17.3205			−0.707697			−0.353848	−	0.935303i	$-0.615127\pi$
$599$	−17.3205			−0.707697			−0.353848	+	0.935303i	$0.615127\pi$
$600$		0			0
$601$	19.5000	−	33.7750i	0.795422	−	1.37771i	−0.127150	−	0.991884i	$-0.540583\pi$
$601$	19.5000	−	33.7750i	0.795422	−	1.37771i	0.922571	−	0.385827i	$-0.126084\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	42.0000	−	24.2487i	1.70754	−	0.985850i
$606$		0			0
$607$	21.6506	+	37.5000i	0.878772	+	1.52208i	0.852689	+	0.522418i	$0.174970\pi$
$607$	21.6506	+	37.5000i	0.878772	+	1.52208i	0.0260828	+	0.999660i	$0.491697\pi$
$608$		0			0
$609$	−22.5000	−	12.9904i	−0.911746	−	0.526397i
$610$		0			0
$611$	−13.8564	−	4.00000i	−0.560570	−	0.161823i
$612$		0			0
$613$	34.5000	+	19.9186i	1.39344	+	0.804504i	0.993695	−	0.112121i	$-0.0357645\pi$
$613$	34.5000	+	19.9186i	1.39344	+	0.804504i	0.399747	+	0.916625i	$0.369098\pi$
$614$		0			0
$615$	−5.19615	−	9.00000i	−0.209529	−	0.362915i
$616$		0			0
$617$	−4.50000	+	2.59808i	−0.181163	+	0.104595i	−0.587839	−	0.808978i	$-0.700021\pi$
$617$	−4.50000	+	2.59808i	−0.181163	+	0.104595i	0.406676	+	0.913573i	$0.366688\pi$
$618$		0			0
$619$		−	6.00000i		−	0.241160i	−0.992704	−	0.120580i	$-0.961525\pi$
$619$		−	6.00000i		−	0.241160i	0.992704	−	0.120580i	$-0.0384755\pi$
$620$		0			0
$621$	13.5000	−	23.3827i	0.541736	−	0.938315i
$622$		0			0
$623$	−5.19615			−0.208179
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$	−21.6506	+	37.5000i	−0.864643	+	1.49761i
$628$		0			0
$629$			36.3731i			1.45029i
$630$		0			0
$631$	−4.33013	+	2.50000i	−0.172380	+	0.0995234i	−0.583707	−	0.811964i	$-0.698398\pi$
$631$	−4.33013	+	2.50000i	−0.172380	+	0.0995234i	0.411328	+	0.911487i	$0.365065\pi$
$632$		0			0
$633$	4.50000	+	7.79423i	0.178859	+	0.309793i
$634$		0			0
$635$	−57.1577	−	33.0000i	−2.26823	−	1.30957i
$636$		0			0
$637$	5.00000	+	5.19615i	0.198107	+	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.500000	−	0.866025i	−0.0197488	−	0.0342059i	0.855982	−	0.517005i	$-0.172953\pi$
$641$	−0.500000	−	0.866025i	−0.0197488	−	0.0342059i	−0.875731	+	0.482800i	$0.839620\pi$
$642$		0			0
$643$	−7.79423	+	4.50000i	−0.307374	+	0.177463i	−0.645751	−	0.763548i	$-0.723456\pi$
$643$	−7.79423	+	4.50000i	−0.307374	+	0.177463i	0.338377	+	0.941011i	$0.390122\pi$
$644$		0			0
$645$		−	31.1769i		−	1.22759i
$646$		0			0
$647$	−7.79423	+	13.5000i	−0.306423	+	0.530740i	−0.977577	−	0.210578i	$-0.932465\pi$
$647$	−7.79423	+	13.5000i	−0.306423	+	0.530740i	0.671154	+	0.741318i	$0.265799\pi$
$648$		0			0
$649$	−35.0000			−1.37387
$650$		0			0
$651$	10.3923			0.407307
$652$		0			0
$653$	−2.50000	+	4.33013i	−0.0978326	+	0.169451i	−0.910787	−	0.412876i	$-0.864524\pi$
$653$	−2.50000	+	4.33013i	−0.0978326	+	0.169451i	0.812955	+	0.582327i	$0.197858\pi$
$654$		0			0
$655$			48.0000i			1.87552i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.1865	+	31.5000i	0.708447	+	1.22707i	0.965433	+	0.260651i	$0.0839371\pi$
$659$	18.1865	+	31.5000i	0.708447	+	1.22707i	−0.256986	+	0.966415i	$0.582730\pi$
$660$		0			0
$661$	−1.50000	−	0.866025i	−0.0583432	−	0.0336845i	0.470545	−	0.882376i	$-0.344057\pi$
$661$	−1.50000	−	0.866025i	−0.0583432	−	0.0336845i	−0.528888	+	0.848692i	$0.677391\pi$
$662$		0			0
$663$	12.1244	−	42.0000i	0.470871	−	1.63114i
$664$		0			0
$665$	45.0000	+	25.9808i	1.74503	+	1.00749i
$666$		0			0
$667$	12.9904	+	22.5000i	0.502990	+	0.871203i
$668$		0			0
$669$	19.5000	−	11.2583i	0.753914	−	0.435272i
$670$		0			0
$671$		−	15.0000i		−	0.579069i
$672$		0			0
$673$	20.5000	−	35.5070i	0.790217	−	1.36870i	−0.135615	−	0.990762i	$-0.543301\pi$
$673$	20.5000	−	35.5070i	0.790217	−	1.36870i	0.925832	−	0.377934i	$-0.123365\pi$
$674$		0			0
$675$	36.3731			1.40000
$676$		0			0
$677$	−8.00000			−0.307465			−0.153732	−	0.988113i	$-0.549129\pi$
$677$	−8.00000			−0.307465			−0.153732	+	0.988113i	$0.549129\pi$
$678$		0			0
$679$	−12.9904	+	22.5000i	−0.498525	+	0.863471i
$680$		0			0
$681$		−	39.8372i		−	1.52656i
$682$		0			0
$683$	35.5070	−	20.5000i	1.35864	−	0.784411i	0.369199	−	0.929350i	$-0.379632\pi$
$683$	35.5070	−	20.5000i	1.35864	−	0.784411i	0.989440	+	0.144940i	$0.0462988\pi$
$684$		0			0
$685$	9.00000	+	15.5885i	0.343872	+	0.595604i
$686$		0			0
$687$	36.3731	+	21.0000i	1.38772	+	0.801200i
$688$		0			0
$689$	−4.00000	+	13.8564i	−0.152388	+	0.527887i
$690$		0			0
$691$	−12.9904	−	7.50000i	−0.494177	−	0.285313i	0.232128	−	0.972685i	$-0.425431\pi$
$691$	−12.9904	−	7.50000i	−0.494177	−	0.285313i	−0.726306	+	0.687372i	$0.758764\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−25.9808	+	15.0000i	−0.985506	+	0.568982i
$696$		0			0
$697$		−	12.1244i		−	0.459243i
$698$		0			0
$699$	6.92820	−	12.0000i	0.262049	−	0.453882i
$700$		0			0
$701$	8.00000			0.302156			0.151078	−	0.988522i	$-0.451726\pi$
$701$	8.00000			0.302156			0.151078	+	0.988522i	$0.451726\pi$
$702$		0			0
$703$	25.9808			0.979883
$704$		0			0
$705$	−12.0000	+	20.7846i	−0.451946	+	0.782794i
$706$		0			0
$707$		−	51.0000i		−	1.91805i
$708$		0			0
$709$	−1.50000	+	0.866025i	−0.0563337	+	0.0325243i	−0.527902	−	0.849305i	$-0.677021\pi$
$709$	−1.50000	+	0.866025i	−0.0563337	+	0.0325243i	0.471569	+	0.881829i	$0.343688\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.00000	−	5.19615i	−0.337053	−	0.194597i
$714$		0			0
$715$	43.3013	+	45.0000i	1.61938	+	1.68290i
$716$		0			0
$717$	3.00000	+	1.73205i	0.112037	+	0.0646846i
$718$		0			0
$719$	6.06218	+	10.5000i	0.226081	+	0.391584i	0.956643	−	0.291262i	$-0.0940752\pi$
$719$	6.06218	+	10.5000i	0.226081	+	0.391584i	−0.730562	+	0.682846i	$0.760742\pi$
$720$		0			0
$721$	18.0000	−	10.3923i	0.670355	−	0.387030i
$722$		0			0
$723$		−	15.0000i		−	0.557856i
$724$		0			0
$725$	−17.5000	+	30.3109i	−0.649934	+	1.12572i
$726$		0			0
$727$	−41.5692			−1.54172			−0.770859	−	0.637006i	$-0.780172\pi$
$727$	−41.5692			−1.54172			−0.770859	+	0.637006i	$0.780172\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	18.1865	−	31.5000i	0.672653	−	1.16507i
$732$		0			0
$733$			48.4974i			1.79129i	0.444766	+	0.895647i	$0.353287\pi$
$733$			48.4974i			1.79129i	−0.444766	+	0.895647i	$0.646713\pi$
$734$		0			0
$735$	10.3923	−	6.00000i	0.383326	−	0.221313i
$736$		0			0
$737$	7.50000	+	12.9904i	0.276266	+	0.478507i
$738$		0			0
$739$	−18.1865	−	10.5000i	−0.669002	−	0.386249i	0.126696	−	0.991942i	$-0.459563\pi$
$739$	−18.1865	−	10.5000i	−0.669002	−	0.386249i	−0.795699	+	0.605693i	$0.792896\pi$
$740$		0			0
$741$	−30.0000	−	8.66025i	−1.10208	−	0.318142i
$742$		0			0
$743$	11.2583	+	6.50000i	0.413028	+	0.238462i	0.692090	−	0.721811i	$-0.256690\pi$
$743$	11.2583	+	6.50000i	0.413028	+	0.238462i	−0.279062	+	0.960273i	$0.590023\pi$
$744$		0			0
$745$	−15.0000	−	25.9808i	−0.549557	−	0.951861i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	25.9808i		−	0.949316i
$750$		0			0
$751$	−6.06218	+	10.5000i	−0.221212	+	0.383150i	−0.955176	−	0.296038i	$-0.904335\pi$
$751$	−6.06218	+	10.5000i	−0.221212	+	0.383150i	0.733964	+	0.679188i	$0.237668\pi$
$752$		0			0
$753$	−27.0000			−0.983935
$754$		0			0
$755$	6.92820			0.252143
$756$		0			0
$757$	−26.5000	+	45.8993i	−0.963159	+	1.66824i	−0.248677	+	0.968587i	$0.579996\pi$
$757$	−26.5000	+	45.8993i	−0.963159	+	1.66824i	−0.714482	+	0.699654i	$0.753338\pi$
$758$		0			0
$759$		−	45.0000i		−	1.63340i
$760$		0			0
$761$	40.5000	−	23.3827i	1.46812	−	0.847622i	0.468761	−	0.883325i	$-0.344700\pi$
$761$	40.5000	−	23.3827i	1.46812	−	0.847622i	0.999362	+	0.0357031i	$0.0113671\pi$
$762$		0			0
$763$	−15.5885	−	27.0000i	−0.564340	−	0.977466i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.06218	−	24.5000i	−0.218893	−	0.884644i
$768$		0			0
$769$	−25.5000	−	14.7224i	−0.919554	−	0.530904i	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−25.5000	−	14.7224i	−0.919554	−	0.530904i	−0.883493	+	0.468445i	$0.844814\pi$
$770$		0			0
$771$	19.9186	+	34.5000i	0.717350	+	1.24249i
$772$		0			0
$773$	−1.50000	+	0.866025i	−0.0539513	+	0.0311488i	−0.526733	−	0.850031i	$-0.676583\pi$
$773$	−1.50000	+	0.866025i	−0.0539513	+	0.0311488i	0.472782	+	0.881180i	$0.343250\pi$
$774$		0			0
$775$		−	14.0000i		−	0.502895i
$776$		0			0
$777$	13.5000	−	23.3827i	0.484310	−	0.838849i
$778$		0			0
$779$	−8.66025			−0.310286
$780$		0			0
$781$	35.0000			1.25240
$782$		0			0
$783$	−12.9904	+	22.5000i	−0.464238	+	0.804084i
$784$		0			0
$785$			27.7128i			0.989113i
$786$		0			0
$787$	−7.79423	+	4.50000i	−0.277834	+	0.160408i	−0.632443	−	0.774607i	$-0.717948\pi$
$787$	−7.79423	+	4.50000i	−0.277834	+	0.160408i	0.354608	+	0.935015i	$0.384614\pi$
$788$		0			0
$789$	1.50000	+	2.59808i	0.0534014	+	0.0924940i
$790$		0			0
$791$	12.9904	+	7.50000i	0.461885	+	0.266669i

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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.w (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 1.50000i) q^{3} -3.46410i q^{5} +(-2.59808 + 1.50000i) q^{7} +O(q^{10})\)	\(q+(-0.866025 + 1.50000i) q^{3} -3.46410i q^{5} +(-2.59808 + 1.50000i) q^{7} +(4.33013 + 2.50000i) q^{11} +(-1.00000 + 3.46410i) q^{13} +(5.19615 + 3.00000i) q^{15} +(3.50000 + 6.06218i) q^{17} +(4.33013 - 2.50000i) q^{19} -5.19615i q^{21} +(-2.59808 + 4.50000i) q^{23} -7.00000 q^{25} -5.19615 q^{27} +(2.50000 - 4.33013i) q^{29} +2.00000i q^{31} +(-7.50000 + 4.33013i) q^{33} +(5.19615 + 9.00000i) q^{35} +(4.50000 + 2.59808i) q^{37} +(-4.33013 - 4.50000i) q^{39} +(-1.50000 - 0.866025i) q^{41} +(-2.59808 - 4.50000i) q^{43} +4.00000i q^{47} +(1.00000 - 1.73205i) q^{49} -12.1244 q^{51} +4.00000 q^{53} +(8.66025 - 15.0000i) q^{55} +8.66025i q^{57} +(-6.06218 + 3.50000i) q^{59} +(-1.50000 - 2.59808i) q^{61} +(12.0000 + 3.46410i) q^{65} +(2.59808 + 1.50000i) q^{67} +(-4.50000 - 7.79423i) q^{69} +(6.06218 - 3.50000i) q^{71} -3.46410i q^{73} +(6.06218 - 10.5000i) q^{75} -15.0000 q^{77} +3.46410 q^{79} +(4.50000 - 7.79423i) q^{81} -14.0000i q^{83} +(21.0000 - 12.1244i) q^{85} +(4.33013 + 7.50000i) q^{87} +(1.50000 + 0.866025i) q^{89} +(-2.59808 - 10.5000i) q^{91} +(-3.00000 - 1.73205i) q^{93} +(-8.66025 - 15.0000i) q^{95} +(7.50000 - 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 4 q^{13} + 14 q^{17} - 28 q^{25} + 10 q^{29} - 30 q^{33} + 18 q^{37} - 6 q^{41} + 4 q^{49} + 16 q^{53} - 6 q^{61} + 48 q^{65} - 18 q^{69} - 60 q^{77} + 18 q^{81} + 84 q^{85} + 6 q^{89} - 12 q^{93} + 30 q^{97}+O(q^{100}) \) 4 * q - 4 * q^13 + 14 * q^17 - 28 * q^25 + 10 * q^29 - 30 * q^33 + 18 * q^37 - 6 * q^41 + 4 * q^49 + 16 * q^53 - 6 * q^61 + 48 * q^65 - 18 * q^69 - 60 * q^77 + 18 * q^81 + 84 * q^85 + 6 * q^89 - 12 * q^93 + 30 * q^97

Introduction

L-functions

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