LMFDB - Embedded newform 91.2.e.a.79.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,2,Mod(53,91)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("91.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 91 = 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	91.e (of order $3$, 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:	$0.726638658394$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			79.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	91.79
Dual form			91.2.e.a.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(0.500000 + 2.59808i) q^{7} -3.00000 q^{8} +(1.50000 - 2.59808i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(0.500000 + 2.59808i) q^{7} -3.00000 q^{8} +(1.50000 - 2.59808i) q^{9} +(1.50000 + 2.59808i) q^{11} -1.00000 q^{13} +(-2.50000 - 0.866025i) q^{14} +(0.500000 - 0.866025i) q^{16} +(-3.50000 - 6.06218i) q^{17} +(1.50000 + 2.59808i) q^{18} +(3.50000 - 6.06218i) q^{19} -3.00000 q^{22} +(3.00000 - 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +(0.500000 - 0.866025i) q^{26} +(-2.00000 + 1.73205i) q^{28} -5.00000 q^{29} +(-2.50000 - 4.33013i) q^{32} +7.00000 q^{34} +3.00000 q^{36} +(-4.00000 + 6.92820i) q^{37} +(3.50000 + 6.06218i) q^{38} +2.00000 q^{43} +(-1.50000 + 2.59808i) q^{44} +(3.00000 + 5.19615i) q^{46} +(-3.50000 + 6.06218i) q^{47} +(-6.50000 + 2.59808i) q^{49} -5.00000 q^{50} +(-0.500000 - 0.866025i) q^{52} +(1.50000 + 2.59808i) q^{53} +(-1.50000 - 7.79423i) q^{56} +(2.50000 - 4.33013i) q^{58} +(3.50000 + 6.06218i) q^{59} +(3.50000 - 6.06218i) q^{61} +(7.50000 + 2.59808i) q^{63} +7.00000 q^{64} +(1.50000 + 2.59808i) q^{67} +(3.50000 - 6.06218i) q^{68} -5.00000 q^{71} +(-4.50000 + 7.79423i) q^{72} +(-7.00000 - 12.1244i) q^{73} +(-4.00000 - 6.92820i) q^{74} +7.00000 q^{76} +(-6.00000 + 5.19615i) q^{77} +(3.00000 - 5.19615i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-1.00000 + 1.73205i) q^{86} +(-4.50000 - 7.79423i) q^{88} +(-0.500000 - 2.59808i) q^{91} +6.00000 q^{92} +(-3.50000 - 6.06218i) q^{94} -14.0000 q^{97} +(1.00000 - 6.92820i) q^{98} +9.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + q^{4} + q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) $ 2 * q - q^2 + q^4 + q^7 - 6 * q^8 + 3 * q^9	$ 2 q - q^{2} + q^{4} + q^{7} - 6 q^{8} + 3 q^{9} + 3 q^{11} - 2 q^{13} - 5 q^{14} + q^{16} - 7 q^{17} + 3 q^{18} + 7 q^{19} - 6 q^{22} + 6 q^{23} + 5 q^{25} + q^{26} - 4 q^{28} - 10 q^{29} - 5 q^{32} + 14 q^{34} + 6 q^{36} - 8 q^{37} + 7 q^{38} + 4 q^{43} - 3 q^{44} + 6 q^{46} - 7 q^{47} - 13 q^{49} - 10 q^{50} - q^{52} + 3 q^{53} - 3 q^{56} + 5 q^{58} + 7 q^{59} + 7 q^{61} + 15 q^{63} + 14 q^{64} + 3 q^{67} + 7 q^{68} - 10 q^{71} - 9 q^{72} - 14 q^{73} - 8 q^{74} + 14 q^{76} - 12 q^{77} + 6 q^{79} - 9 q^{81} - 2 q^{86} - 9 q^{88} - q^{91} + 12 q^{92} - 7 q^{94} - 28 q^{97} + 2 q^{98} + 18 q^{99}+O(q^{100}) $ 2 * q - q^2 + q^4 + q^7 - 6 * q^8 + 3 * q^9 + 3 * q^11 - 2 * q^13 - 5 * q^14 + q^16 - 7 * q^17 + 3 * q^18 + 7 * q^19 - 6 * q^22 + 6 * q^23 + 5 * q^25 + q^26 - 4 * q^28 - 10 * q^29 - 5 * q^32 + 14 * q^34 + 6 * q^36 - 8 * q^37 + 7 * q^38 + 4 * q^43 - 3 * q^44 + 6 * q^46 - 7 * q^47 - 13 * q^49 - 10 * q^50 - q^52 + 3 * q^53 - 3 * q^56 + 5 * q^58 + 7 * q^59 + 7 * q^61 + 15 * q^63 + 14 * q^64 + 3 * q^67 + 7 * q^68 - 10 * q^71 - 9 * q^72 - 14 * q^73 - 8 * q^74 + 14 * q^76 - 12 * q^77 + 6 * q^79 - 9 * q^81 - 2 * q^86 - 9 * q^88 - q^91 + 12 * q^92 - 7 * q^94 - 28 * q^97 + 2 * q^98 + 18 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$.

$n$	$15$	$66$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i	−0.986869	−	0.161521i	$-0.948360\pi$
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i	0.633316	+	0.773893i	$0.281693\pi$
$3$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$	−3.00000			−1.06066
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$		0			0
$11$	1.50000	+	2.59808i	0.452267	+	0.783349i	0.998526	−	0.0542666i	$-0.0172821\pi$
$11$	1.50000	+	2.59808i	0.452267	+	0.783349i	−0.546259	+	0.837616i	$0.683949\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$	−2.50000	−	0.866025i	−0.668153	−	0.231455i
$15$		0			0
$16$	0.500000	−	0.866025i	0.125000	−	0.216506i
$17$	−3.50000	−	6.06218i	−0.848875	−	1.47029i	−0.882213	−	0.470850i	$-0.843947\pi$
$17$	−3.50000	−	6.06218i	−0.848875	−	1.47029i	0.0333386	−	0.999444i	$-0.489386\pi$
$18$	1.50000	+	2.59808i	0.353553	+	0.612372i
$19$	3.50000	−	6.06218i	0.802955	−	1.39076i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	3.50000	−	6.06218i	0.802955	−	1.39076i	0.917663	−	0.397360i	$-0.130073\pi$
$20$		0			0
$21$		0			0
$22$	−3.00000			−0.639602
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	$-0.618211\pi$
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	0.988436	−	0.151642i	$-0.0484560\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$	0.500000	−	0.866025i	0.0980581	−	0.169842i
$27$		0			0
$28$	−2.00000	+	1.73205i	−0.377964	+	0.327327i
$29$	−5.00000			−0.928477			−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000			−0.928477			−0.464238	+	0.885710i	$0.653672\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	−2.50000	−	4.33013i	−0.441942	−	0.765466i
$33$		0			0
$34$	7.00000			1.20049
$35$		0			0
$36$	3.00000			0.500000
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	0.323640	+	0.946180i	$0.395093\pi$
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	−0.981236	+	0.192809i	$0.938240\pi$
$38$	3.50000	+	6.06218i	0.567775	+	0.983415i
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	2.00000			0.304997			0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000			0.304997			0.152499	+	0.988304i	$0.451268\pi$
$44$	−1.50000	+	2.59808i	−0.226134	+	0.391675i
$45$		0			0
$46$	3.00000	+	5.19615i	0.442326	+	0.766131i
$47$	−3.50000	+	6.06218i	−0.510527	+	0.884260i	0.489398	+	0.872060i	$0.337217\pi$
$47$	−3.50000	+	6.06218i	−0.510527	+	0.884260i	−0.999926	+	0.0121990i	$0.996117\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$	−5.00000			−0.707107
$51$		0			0
$52$	−0.500000	−	0.866025i	−0.0693375	−	0.120096i
$53$	1.50000	+	2.59808i	0.206041	+	0.356873i	0.950464	−	0.310835i	$-0.100609\pi$
$53$	1.50000	+	2.59808i	0.206041	+	0.356873i	−0.744423	+	0.667708i	$0.767275\pi$
$54$		0			0
$55$		0			0
$56$	−1.50000	−	7.79423i	−0.200446	−	1.04155i
$57$		0			0
$58$	2.50000	−	4.33013i	0.328266	−	0.568574i
$59$	3.50000	+	6.06218i	0.455661	+	0.789228i	0.998726	−	0.0504625i	$-0.0160695\pi$
$59$	3.50000	+	6.06218i	0.455661	+	0.789228i	−0.543065	+	0.839691i	$0.682736\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$		0			0
$63$	7.50000	+	2.59808i	0.944911	+	0.327327i
$64$	7.00000			0.875000
$65$		0			0
$66$		0			0
$67$	1.50000	+	2.59808i	0.183254	+	0.317406i	0.942987	−	0.332830i	$-0.108004\pi$
$67$	1.50000	+	2.59808i	0.183254	+	0.317406i	−0.759733	+	0.650236i	$0.774670\pi$
$68$	3.50000	−	6.06218i	0.424437	−	0.735147i
$69$		0			0
$70$		0			0
$71$	−5.00000			−0.593391			−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000			−0.593391			−0.296695	+	0.954972i	$0.595885\pi$
$72$	−4.50000	+	7.79423i	−0.530330	+	0.918559i
$73$	−7.00000	−	12.1244i	−0.819288	−	1.41905i	−0.906208	−	0.422833i	$-0.861036\pi$
$73$	−7.00000	−	12.1244i	−0.819288	−	1.41905i	0.0869195	−	0.996215i	$-0.472298\pi$
$74$	−4.00000	−	6.92820i	−0.464991	−	0.805387i
$75$		0			0
$76$	7.00000			0.802955
$77$	−6.00000	+	5.19615i	−0.683763	+	0.592157i
$78$		0			0
$79$	3.00000	−	5.19615i	0.337526	−	0.584613i	−0.646440	−	0.762964i	$-0.723743\pi$
$79$	3.00000	−	5.19615i	0.337526	−	0.584613i	0.983967	+	0.178352i	$0.0570765\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	−1.00000	+	1.73205i	−0.107833	+	0.186772i
$87$		0			0
$88$	−4.50000	−	7.79423i	−0.479702	−	0.830868i
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	−0.500000	−	2.59808i	−0.0524142	−	0.272352i
$92$	6.00000			0.625543
$93$		0			0
$94$	−3.50000	−	6.06218i	−0.360997	−	0.625266i
$95$		0			0
$96$		0			0
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$	1.00000	−	6.92820i	0.101015	−	0.699854i
$99$	9.00000			0.904534
$100$	−2.50000	+	4.33013i	−0.250000	+	0.433013i
$101$	7.00000	+	12.1244i	0.696526	+	1.20642i	0.969664	+	0.244443i	$0.0786053\pi$
$101$	7.00000	+	12.1244i	0.696526	+	1.20642i	−0.273138	+	0.961975i	$0.588061\pi$
$102$		0			0
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	0.282194	+	0.959357i	$0.408938\pi$
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	−0.971925	+	0.235291i	$0.924396\pi$
$104$	3.00000			0.294174
$105$		0			0
$106$	−3.00000			−0.291386
$107$	−4.00000	+	6.92820i	−0.386695	+	0.669775i	−0.992003	−	0.126217i	$-0.959717\pi$
$107$	−4.00000	+	6.92820i	−0.386695	+	0.669775i	0.605308	+	0.795991i	$0.293050\pi$
$108$		0			0
$109$	−2.00000	−	3.46410i	−0.191565	−	0.331801i	0.754204	−	0.656640i	$-0.228023\pi$
$109$	−2.00000	−	3.46410i	−0.191565	−	0.331801i	−0.945769	+	0.324840i	$0.894690\pi$
$110$		0			0
$111$		0			0
$112$	2.50000	+	0.866025i	0.236228	+	0.0818317i
$113$	9.00000			0.846649			0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000			0.846649			0.423324	+	0.905978i	$0.360863\pi$
$114$		0			0
$115$		0			0
$116$	−2.50000	−	4.33013i	−0.232119	−	0.402042i
$117$	−1.50000	+	2.59808i	−0.138675	+	0.240192i
$118$	−7.00000			−0.644402
$119$	14.0000	−	12.1244i	1.28338	−	1.11144i
$120$		0			0
$121$	1.00000	−	1.73205i	0.0909091	−	0.157459i
$122$	3.50000	+	6.06218i	0.316875	+	0.548844i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−6.00000	+	5.19615i	−0.534522	+	0.462910i
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.50000	−	2.59808i	0.132583	−	0.229640i
$129$		0			0
$130$		0			0
$131$	−7.00000	+	12.1244i	−0.611593	+	1.05931i	0.379379	+	0.925241i	$0.376138\pi$
$131$	−7.00000	+	12.1244i	−0.611593	+	1.05931i	−0.990972	+	0.134069i	$0.957196\pi$
$132$		0			0
$133$	17.5000	+	6.06218i	1.51744	+	0.525657i
$134$	−3.00000			−0.259161
$135$		0			0
$136$	10.5000	+	18.1865i	0.900368	+	1.55948i
$137$	−2.00000	−	3.46410i	−0.170872	−	0.295958i	0.767853	−	0.640626i	$-0.221325\pi$
$137$	−2.00000	−	3.46410i	−0.170872	−	0.295958i	−0.938725	+	0.344668i	$0.887992\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	2.50000	−	4.33013i	0.209795	−	0.363376i
$143$	−1.50000	−	2.59808i	−0.125436	−	0.217262i
$144$	−1.50000	−	2.59808i	−0.125000	−	0.216506i
$145$		0			0
$146$	14.0000			1.15865
$147$		0			0
$148$	−8.00000			−0.657596
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	−0.716578	−	0.697507i	$-0.754293\pi$
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	0.962348	+	0.271821i	$0.0876260\pi$
$150$		0			0
$151$	1.50000	+	2.59808i	0.122068	+	0.211428i	0.920583	−	0.390547i	$-0.127714\pi$
$151$	1.50000	+	2.59808i	0.122068	+	0.211428i	−0.798515	+	0.601975i	$0.794381\pi$
$152$	−10.5000	+	18.1865i	−0.851662	+	1.47512i
$153$	−21.0000			−1.69775
$154$	−1.50000	−	7.79423i	−0.120873	−	0.628077i
$155$		0			0
$156$		0			0
$157$	−3.50000	−	6.06218i	−0.279330	−	0.483814i	0.691888	−	0.722005i	$-0.256779\pi$
$157$	−3.50000	−	6.06218i	−0.279330	−	0.483814i	−0.971219	+	0.238190i	$0.923446\pi$
$158$	3.00000	+	5.19615i	0.238667	+	0.413384i
$159$		0			0
$160$		0			0
$161$	15.0000	+	5.19615i	1.18217	+	0.409514i
$162$	9.00000			0.707107
$163$	6.50000	−	11.2583i	0.509119	−	0.881820i	−0.490825	−	0.871258i	$-0.663305\pi$
$163$	6.50000	−	11.2583i	0.509119	−	0.881820i	0.999944	−	0.0105623i	$-0.00336213\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.00000			−0.541676			−0.270838	−	0.962625i	$-0.587301\pi$
$167$	−7.00000			−0.541676			−0.270838	+	0.962625i	$0.587301\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−10.5000	−	18.1865i	−0.802955	−	1.39076i
$172$	1.00000	+	1.73205i	0.0762493	+	0.132068i
$173$	−3.50000	+	6.06218i	−0.266100	+	0.460899i	−0.967851	−	0.251523i	$-0.919068\pi$
$173$	−3.50000	+	6.06218i	−0.266100	+	0.460899i	0.701751	+	0.712422i	$0.252402\pi$
$174$		0			0
$175$	−10.0000	+	8.66025i	−0.755929	+	0.654654i
$176$	3.00000			0.226134
$177$		0			0
$178$		0			0
$179$	5.00000	+	8.66025i	0.373718	+	0.647298i	0.990134	−	0.140122i	$-0.0447496\pi$
$179$	5.00000	+	8.66025i	0.373718	+	0.647298i	−0.616417	+	0.787420i	$0.711416\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$	2.50000	+	0.866025i	0.185312	+	0.0641941i
$183$		0			0
$184$	−9.00000	+	15.5885i	−0.663489	+	1.14920i
$185$		0			0
$186$		0			0
$187$	10.5000	−	18.1865i	0.767836	−	1.32993i
$188$	−7.00000			−0.510527
$189$		0			0
$190$		0			0
$191$	10.0000	−	17.3205i	0.723575	−	1.25327i	−0.235983	−	0.971757i	$-0.575831\pi$
$191$	10.0000	−	17.3205i	0.723575	−	1.25327i	0.959558	−	0.281511i	$-0.0908356\pi$
$192$		0			0
$193$	−2.00000	−	3.46410i	−0.143963	−	0.249351i	0.785022	−	0.619467i	$-0.212651\pi$
$193$	−2.00000	−	3.46410i	−0.143963	−	0.249351i	−0.928986	+	0.370116i	$0.879318\pi$
$194$	7.00000	−	12.1244i	0.502571	−	0.870478i
$195$		0			0
$196$	−5.50000	−	4.33013i	−0.392857	−	0.309295i
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$	−4.50000	+	7.79423i	−0.319801	+	0.553912i
$199$	7.00000	+	12.1244i	0.496217	+	0.859473i	0.999990	−	0.00436292i	$-0.00138876\pi$
$199$	7.00000	+	12.1244i	0.496217	+	0.859473i	−0.503774	+	0.863836i	$0.668055\pi$
$200$	−7.50000	−	12.9904i	−0.530330	−	0.918559i
$201$		0			0
$202$	−14.0000			−0.985037
$203$	−2.50000	−	12.9904i	−0.175466	−	0.911746i
$204$		0			0
$205$		0			0
$206$	−7.00000	−	12.1244i	−0.487713	−	0.844744i
$207$	−9.00000	−	15.5885i	−0.625543	−	1.08347i
$208$	−0.500000	+	0.866025i	−0.0346688	+	0.0600481i
$209$	21.0000			1.45260
$210$		0			0
$211$	−26.0000			−1.78991			−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000			−1.78991			−0.894957	+	0.446153i	$0.852794\pi$
$212$	−1.50000	+	2.59808i	−0.103020	+	0.178437i
$213$		0			0
$214$	−4.00000	−	6.92820i	−0.273434	−	0.473602i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	4.00000			0.270914
$219$		0			0
$220$		0			0
$221$	3.50000	+	6.06218i	0.235435	+	0.407786i
$222$		0			0
$223$	21.0000			1.40626			0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000			1.40626			0.703132	+	0.711059i	$0.251784\pi$
$224$	10.0000	−	8.66025i	0.668153	−	0.578638i
$225$	15.0000			1.00000
$226$	−4.50000	+	7.79423i	−0.299336	+	0.518464i
$227$	14.0000	+	24.2487i	0.929213	+	1.60944i	0.784642	+	0.619949i	$0.212847\pi$
$227$	14.0000	+	24.2487i	0.929213	+	1.60944i	0.144571	+	0.989494i	$0.453820\pi$
$228$		0			0
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	−0.999088	−	0.0426906i	$-0.986407\pi$
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	0.536515	+	0.843891i	$0.319740\pi$
$230$		0			0
$231$		0			0
$232$	15.0000			0.984798
$233$	13.5000	−	23.3827i	0.884414	−	1.53185i	0.0380310	−	0.999277i	$-0.487891\pi$
$233$	13.5000	−	23.3827i	0.884414	−	1.53185i	0.846383	−	0.532574i	$-0.178775\pi$
$234$	−1.50000	−	2.59808i	−0.0980581	−	0.169842i
$235$		0			0
$236$	−3.50000	+	6.06218i	−0.227831	+	0.394614i
$237$		0			0
$238$	3.50000	+	18.1865i	0.226871	+	1.17886i
$239$	−19.0000			−1.22901			−0.614504	−	0.788914i	$-0.710644\pi$
$239$	−19.0000			−1.22901			−0.614504	+	0.788914i	$0.710644\pi$
$240$		0			0
$241$	−14.0000	−	24.2487i	−0.901819	−	1.56200i	−0.825131	−	0.564942i	$-0.808899\pi$
$241$	−14.0000	−	24.2487i	−0.901819	−	1.56200i	−0.0766885	−	0.997055i	$-0.524435\pi$
$242$	1.00000	+	1.73205i	0.0642824	+	0.111340i
$243$		0			0
$244$	7.00000			0.448129
$245$		0			0
$246$		0			0
$247$	−3.50000	+	6.06218i	−0.222700	+	0.385727i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14.0000			−0.883672			−0.441836	−	0.897096i	$-0.645673\pi$
$251$	−14.0000			−0.883672			−0.441836	+	0.897096i	$0.645673\pi$
$252$	1.50000	+	7.79423i	0.0944911	+	0.490990i
$253$	18.0000			1.13165
$254$	−1.00000	+	1.73205i	−0.0627456	+	0.108679i
$255$		0			0
$256$	8.50000	+	14.7224i	0.531250	+	0.920152i
$257$	7.00000	−	12.1244i	0.436648	−	0.756297i	−0.560781	−	0.827964i	$-0.689499\pi$
$257$	7.00000	−	12.1244i	0.436648	−	0.756297i	0.997429	+	0.0716680i	$0.0228322\pi$
$258$		0			0
$259$	−20.0000	−	6.92820i	−1.24274	−	0.430498i
$260$		0			0
$261$	−7.50000	+	12.9904i	−0.464238	+	0.804084i
$262$	−7.00000	−	12.1244i	−0.432461	−	0.749045i
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	0.952517	+	0.304487i	$0.0984850\pi$
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	−0.212565	+	0.977147i	$0.568182\pi$
$264$		0			0
$265$		0			0
$266$	−14.0000	+	12.1244i	−0.858395	+	0.743392i
$267$		0			0
$268$	−1.50000	+	2.59808i	−0.0916271	+	0.158703i
$269$	10.5000	+	18.1865i	0.640196	+	1.10885i	0.985389	+	0.170321i	$0.0544803\pi$
$269$	10.5000	+	18.1865i	0.640196	+	1.10885i	−0.345192	+	0.938532i	$0.612186\pi$
$270$		0			0
$271$	3.50000	−	6.06218i	0.212610	−	0.368251i	−0.739921	−	0.672694i	$-0.765137\pi$
$271$	3.50000	−	6.06218i	0.212610	−	0.368251i	0.952531	+	0.304443i	$0.0984703\pi$
$272$	−7.00000			−0.424437
$273$		0			0
$274$	4.00000			0.241649
$275$	−7.50000	+	12.9904i	−0.452267	+	0.783349i
$276$		0			0
$277$	8.50000	+	14.7224i	0.510716	+	0.884585i	0.999923	+	0.0124177i	$0.00395278\pi$
$277$	8.50000	+	14.7224i	0.510716	+	0.884585i	−0.489207	+	0.872167i	$0.662714\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$		0			0
$283$	7.00000	+	12.1244i	0.416107	+	0.720718i	0.995544	−	0.0942988i	$-0.0300609\pi$
$283$	7.00000	+	12.1244i	0.416107	+	0.720718i	−0.579437	+	0.815017i	$0.696728\pi$
$284$	−2.50000	−	4.33013i	−0.148348	−	0.256946i
$285$		0			0
$286$	3.00000			0.177394
$287$		0			0
$288$	−15.0000			−0.883883
$289$	−16.0000	+	27.7128i	−0.941176	+	1.63017i
$290$		0			0
$291$		0			0
$292$	7.00000	−	12.1244i	0.409644	−	0.709524i
$293$	14.0000			0.817889			0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000			0.817889			0.408944	+	0.912559i	$0.365897\pi$
$294$		0			0
$295$		0			0
$296$	12.0000	−	20.7846i	0.697486	−	1.20808i
$297$		0			0
$298$	3.00000	+	5.19615i	0.173785	+	0.301005i
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$	1.00000	+	5.19615i	0.0576390	+	0.299501i
$302$	−3.00000			−0.172631
$303$		0			0
$304$	−3.50000	−	6.06218i	−0.200739	−	0.347690i
$305$		0			0
$306$	10.5000	−	18.1865i	0.600245	−	1.03965i
$307$	21.0000			1.19853			0.599267	−	0.800549i	$-0.295459\pi$
$307$	21.0000			1.19853			0.599267	+	0.800549i	$0.295459\pi$
$308$	−7.50000	−	2.59808i	−0.427352	−	0.148039i
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	7.00000	−	12.1244i	0.395663	−	0.685309i	−0.597522	−	0.801852i	$-0.703848\pi$
$313$	7.00000	−	12.1244i	0.395663	−	0.685309i	0.993186	+	0.116543i	$0.0371814\pi$
$314$	7.00000			0.395033
$315$		0			0
$316$	6.00000			0.337526
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	−0.769395	−	0.638774i	$-0.779442\pi$
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	0.937892	+	0.346929i	$0.112775\pi$
$318$		0			0
$319$	−7.50000	−	12.9904i	−0.419919	−	0.727322i
$320$		0			0
$321$		0			0
$322$	−12.0000	+	10.3923i	−0.668734	+	0.579141i
$323$	−49.0000			−2.72643
$324$	4.50000	−	7.79423i	0.250000	−	0.433013i
$325$	−2.50000	−	4.33013i	−0.138675	−	0.240192i
$326$	6.50000	+	11.2583i	0.360002	+	0.623541i
$327$		0			0
$328$		0			0
$329$	−17.5000	−	6.06218i	−0.964806	−	0.334219i
$330$		0			0
$331$	10.0000	−	17.3205i	0.549650	−	0.952021i	−0.448649	−	0.893708i	$-0.648095\pi$
$331$	10.0000	−	17.3205i	0.549650	−	0.952021i	0.998298	−	0.0583130i	$-0.0185721\pi$
$332$		0			0
$333$	12.0000	+	20.7846i	0.657596	+	1.13899i
$334$	3.50000	−	6.06218i	0.191511	−	0.331708i
$335$		0			0
$336$		0			0
$337$	23.0000			1.25289			0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000			1.25289			0.626445	+	0.779466i	$0.284509\pi$
$338$	−0.500000	+	0.866025i	−0.0271964	+	0.0471056i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	21.0000			1.13555
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$	−6.00000			−0.323498
$345$		0			0
$346$	−3.50000	−	6.06218i	−0.188161	−	0.325905i
$347$	−2.00000	−	3.46410i	−0.107366	−	0.185963i	0.807337	−	0.590091i	$-0.200908\pi$
$347$	−2.00000	−	3.46410i	−0.107366	−	0.185963i	−0.914702	+	0.404128i	$0.867575\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$	−2.50000	−	12.9904i	−0.133631	−	0.694365i
$351$		0			0
$352$	7.50000	−	12.9904i	0.399751	−	0.692390i
$353$	−7.00000	−	12.1244i	−0.372572	−	0.645314i	0.617388	−	0.786659i	$-0.288191\pi$
$353$	−7.00000	−	12.1244i	−0.372572	−	0.645314i	−0.989960	+	0.141344i	$0.954858\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−10.0000			−0.528516
$359$	−4.00000	+	6.92820i	−0.211112	+	0.365657i	−0.952063	−	0.305903i	$-0.901042\pi$
$359$	−4.00000	+	6.92820i	−0.211112	+	0.365657i	0.740951	+	0.671559i	$0.234375\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$	3.50000	−	6.06218i	0.183956	−	0.318621i
$363$		0			0
$364$	2.00000	−	1.73205i	0.104828	−	0.0907841i
$365$		0			0
$366$		0			0
$367$	−7.00000	−	12.1244i	−0.365397	−	0.632886i	0.623443	−	0.781869i	$-0.285733\pi$
$367$	−7.00000	−	12.1244i	−0.365397	−	0.632886i	−0.988840	+	0.148983i	$0.952400\pi$
$368$	−3.00000	−	5.19615i	−0.156386	−	0.270868i
$369$		0			0
$370$		0			0
$371$	−6.00000	+	5.19615i	−0.311504	+	0.269771i
$372$		0			0
$373$	−7.50000	+	12.9904i	−0.388335	+	0.672616i	−0.992226	−	0.124451i	$-0.960283\pi$
$373$	−7.50000	+	12.9904i	−0.388335	+	0.672616i	0.603890	+	0.797067i	$0.293616\pi$
$374$	10.5000	+	18.1865i	0.542942	+	0.940403i
$375$		0			0
$376$	10.5000	−	18.1865i	0.541496	−	0.937899i
$377$	5.00000			0.257513
$378$		0			0
$379$	−12.0000			−0.616399			−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000			−0.616399			−0.308199	+	0.951322i	$0.599726\pi$
$380$		0			0
$381$		0			0
$382$	10.0000	+	17.3205i	0.511645	+	0.886194i
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$	4.00000			0.203595
$387$	3.00000	−	5.19615i	0.152499	−	0.264135i
$388$	−7.00000	−	12.1244i	−0.355371	−	0.615521i
$389$	1.50000	+	2.59808i	0.0760530	+	0.131728i	0.901544	−	0.432688i	$-0.142435\pi$
$389$	1.50000	+	2.59808i	0.0760530	+	0.131728i	−0.825491	+	0.564416i	$0.809102\pi$
$390$		0			0
$391$	−42.0000			−2.12403
$392$	19.5000	−	7.79423i	0.984899	−	0.393668i
$393$		0			0
$394$	−1.00000	+	1.73205i	−0.0503793	+	0.0872595i
$395$		0			0
$396$	4.50000	+	7.79423i	0.226134	+	0.391675i
$397$	−7.00000	+	12.1244i	−0.351320	+	0.608504i	−0.986481	−	0.163876i	$-0.947600\pi$
$397$	−7.00000	+	12.1244i	−0.351320	+	0.608504i	0.635161	+	0.772380i	$0.280934\pi$
$398$	−14.0000			−0.701757
$399$		0			0
$400$	5.00000			0.250000
$401$	−11.0000	+	19.0526i	−0.549314	+	0.951439i	0.449008	+	0.893528i	$0.351777\pi$
$401$	−11.0000	+	19.0526i	−0.549314	+	0.951439i	−0.998322	+	0.0579116i	$0.981556\pi$
$402$		0			0
$403$		0			0
$404$	−7.00000	+	12.1244i	−0.348263	+	0.603209i
$405$		0			0
$406$	12.5000	+	4.33013i	0.620365	+	0.214901i
$407$	−24.0000			−1.18964
$408$		0			0
$409$	14.0000	+	24.2487i	0.692255	+	1.19902i	0.971097	+	0.238685i	$0.0767162\pi$
$409$	14.0000	+	24.2487i	0.692255	+	1.19902i	−0.278842	+	0.960337i	$0.589950\pi$
$410$		0			0
$411$		0			0
$412$	−14.0000			−0.689730
$413$	−14.0000	+	12.1244i	−0.688895	+	0.596601i
$414$	18.0000			0.884652
$415$		0			0
$416$	2.50000	+	4.33013i	0.122573	+	0.212302i
$417$		0			0
$418$	−10.5000	+	18.1865i	−0.513572	+	0.889532i
$419$	14.0000			0.683945			0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000			0.683945			0.341972	+	0.939710i	$0.388905\pi$
$420$		0			0
$421$	30.0000			1.46211			0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000			1.46211			0.731055	+	0.682318i	$0.239028\pi$
$422$	13.0000	−	22.5167i	0.632830	−	1.09609i
$423$	10.5000	+	18.1865i	0.510527	+	0.884260i
$424$	−4.50000	−	7.79423i	−0.218539	−	0.378521i
$425$	17.5000	−	30.3109i	0.848875	−	1.47029i
$426$		0			0
$427$	17.5000	+	6.06218i	0.846884	+	0.293369i
$428$	−8.00000			−0.386695
$429$		0			0
$430$		0			0
$431$	12.0000	+	20.7846i	0.578020	+	1.00116i	0.995706	+	0.0925683i	$0.0295076\pi$
$431$	12.0000	+	20.7846i	0.578020	+	1.00116i	−0.417687	+	0.908591i	$0.637159\pi$
$432$		0			0
$433$	21.0000			1.00920			0.504598	−	0.863355i	$-0.331641\pi$
$433$	21.0000			1.00920			0.504598	+	0.863355i	$0.331641\pi$
$434$		0			0
$435$		0			0
$436$	2.00000	−	3.46410i	0.0957826	−	0.165900i
$437$	−21.0000	−	36.3731i	−1.00457	−	1.73996i
$438$		0			0
$439$	−7.00000	+	12.1244i	−0.334092	+	0.578664i	−0.983310	−	0.181938i	$-0.941763\pi$
$439$	−7.00000	+	12.1244i	−0.334092	+	0.578664i	0.649218	+	0.760602i	$0.275096\pi$
$440$		0			0
$441$	−3.00000	+	20.7846i	−0.142857	+	0.989743i
$442$	−7.00000			−0.332956
$443$	10.0000	−	17.3205i	0.475114	−	0.822922i	−0.524479	−	0.851423i	$-0.675740\pi$
$443$	10.0000	−	17.3205i	0.475114	−	0.822922i	0.999594	+	0.0285009i	$0.00907336\pi$
$444$		0			0
$445$		0			0
$446$	−10.5000	+	18.1865i	−0.497189	+	0.861157i
$447$		0			0
$448$	3.50000	+	18.1865i	0.165359	+	0.859233i
$449$	−12.0000			−0.566315			−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000			−0.566315			−0.283158	+	0.959073i	$0.591382\pi$
$450$	−7.50000	+	12.9904i	−0.353553	+	0.612372i
$451$		0			0
$452$	4.50000	+	7.79423i	0.211662	+	0.366610i
$453$		0			0
$454$	−28.0000			−1.31411
$455$		0			0
$456$		0			0
$457$	3.00000	−	5.19615i	0.140334	−	0.243066i	−0.787288	−	0.616585i	$-0.788516\pi$
$457$	3.00000	−	5.19615i	0.140334	−	0.243066i	0.927622	+	0.373519i	$0.121849\pi$
$458$	−7.00000	−	12.1244i	−0.327089	−	0.566534i
$459$		0			0
$460$		0			0
$461$	−28.0000			−1.30409			−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000			−1.30409			−0.652045	+	0.758180i	$0.726089\pi$
$462$		0			0
$463$	16.0000			0.743583			0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000			0.743583			0.371792	+	0.928316i	$0.378744\pi$
$464$	−2.50000	+	4.33013i	−0.116060	+	0.201021i
$465$		0			0
$466$	13.5000	+	23.3827i	0.625375	+	1.08318i
$467$	−7.00000	+	12.1244i	−0.323921	+	0.561048i	−0.981293	−	0.192518i	$-0.938335\pi$
$467$	−7.00000	+	12.1244i	−0.323921	+	0.561048i	0.657372	+	0.753566i	$0.271668\pi$
$468$	−3.00000			−0.138675
$469$	−6.00000	+	5.19615i	−0.277054	+	0.239936i
$470$		0			0
$471$		0			0
$472$	−10.5000	−	18.1865i	−0.483302	−	0.837103i
$473$	3.00000	+	5.19615i	0.137940	+	0.238919i
$474$		0			0
$475$	35.0000			1.60591
$476$	17.5000	+	6.06218i	0.802111	+	0.277859i
$477$	9.00000			0.412082
$478$	9.50000	−	16.4545i	0.434520	−	0.752611i
$479$	−3.50000	−	6.06218i	−0.159919	−	0.276988i	0.774920	−	0.632059i	$-0.217790\pi$
$479$	−3.50000	−	6.06218i	−0.159919	−	0.276988i	−0.934839	+	0.355071i	$0.884457\pi$
$480$		0			0
$481$	4.00000	−	6.92820i	0.182384	−	0.315899i
$482$	28.0000			1.27537
$483$		0			0
$484$	2.00000			0.0909091
$485$		0			0
$486$		0			0
$487$	−12.5000	−	21.6506i	−0.566429	−	0.981084i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−12.5000	−	21.6506i	−0.566429	−	0.981084i	0.430486	−	0.902597i	$-0.358342\pi$
$488$	−10.5000	+	18.1865i	−0.475313	+	0.823266i
$489$		0			0
$490$		0			0
$491$	30.0000			1.35388			0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000			1.35388			0.676941	+	0.736038i	$0.263305\pi$
$492$		0			0
$493$	17.5000	+	30.3109i	0.788160	+	1.36513i
$494$	−3.50000	−	6.06218i	−0.157472	−	0.272750i
$495$		0			0
$496$		0			0
$497$	−2.50000	−	12.9904i	−0.112140	−	0.582698i
$498$		0			0
$499$	−4.00000	+	6.92820i	−0.179065	+	0.310149i	−0.941560	−	0.336844i	$-0.890640\pi$
$499$	−4.00000	+	6.92820i	−0.179065	+	0.310149i	0.762496	+	0.646993i	$0.223974\pi$
$500$		0			0
$501$		0			0
$502$	7.00000	−	12.1244i	0.312425	−	0.541136i
$503$	−28.0000			−1.24846			−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000			−1.24846			−0.624229	+	0.781241i	$0.714587\pi$
$504$	−22.5000	−	7.79423i	−1.00223	−	0.347183i
$505$		0			0
$506$	−9.00000	+	15.5885i	−0.400099	+	0.692991i
$507$		0			0
$508$	1.00000	+	1.73205i	0.0443678	+	0.0768473i
$509$	14.0000	−	24.2487i	0.620539	−	1.07481i	−0.368846	−	0.929490i	$-0.620247\pi$
$509$	14.0000	−	24.2487i	0.620539	−	1.07481i	0.989385	−	0.145315i	$-0.0464195\pi$
$510$		0			0
$511$	28.0000	−	24.2487i	1.23865	−	1.07270i
$512$	−11.0000			−0.486136
$513$		0			0
$514$	7.00000	+	12.1244i	0.308757	+	0.534782i
$515$		0			0
$516$		0			0
$517$	−21.0000			−0.923579
$518$	16.0000	−	13.8564i	0.703000	−	0.608816i
$519$		0			0
$520$		0			0
$521$	7.00000	+	12.1244i	0.306676	+	0.531178i	0.977633	−	0.210318i	$-0.0674500\pi$
$521$	7.00000	+	12.1244i	0.306676	+	0.531178i	−0.670957	+	0.741496i	$0.734117\pi$
$522$	−7.50000	−	12.9904i	−0.328266	−	0.568574i
$523$	7.00000	−	12.1244i	0.306089	−	0.530161i	−0.671414	−	0.741082i	$-0.734313\pi$
$523$	7.00000	−	12.1244i	0.306089	−	0.530161i	0.977503	+	0.210921i	$0.0676463\pi$
$524$	−14.0000			−0.611593
$525$		0			0
$526$	−24.0000			−1.04645
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$	21.0000			0.911322
$532$	3.50000	+	18.1865i	0.151744	+	0.788486i
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−4.50000	−	7.79423i	−0.194370	−	0.336659i
$537$		0			0
$538$	−21.0000			−0.905374
$539$	−16.5000	−	12.9904i	−0.710705	−	0.559535i
$540$		0			0
$541$	−4.00000	+	6.92820i	−0.171973	+	0.297867i	−0.939110	−	0.343617i	$-0.888348\pi$
$541$	−4.00000	+	6.92820i	−0.171973	+	0.297867i	0.767136	+	0.641484i	$0.221681\pi$
$542$	3.50000	+	6.06218i	0.150338	+	0.260393i
$543$		0			0
$544$	−17.5000	+	30.3109i	−0.750306	+	1.29957i
$545$		0			0
$546$		0			0
$547$	2.00000			0.0855138			0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000			0.0855138			0.0427569	+	0.999086i	$0.486386\pi$
$548$	2.00000	−	3.46410i	0.0854358	−	0.147979i
$549$	−10.5000	−	18.1865i	−0.448129	−	0.776182i
$550$	−7.50000	−	12.9904i	−0.319801	−	0.553912i
$551$	−17.5000	+	30.3109i	−0.745525	+	1.29129i
$552$		0			0
$553$	15.0000	+	5.19615i	0.637865	+	0.220963i
$554$	−17.0000			−0.722261
$555$		0			0
$556$		0			0
$557$	−9.00000	−	15.5885i	−0.381342	−	0.660504i	0.609912	−	0.792469i	$-0.291205\pi$
$557$	−9.00000	−	15.5885i	−0.381342	−	0.660504i	−0.991254	+	0.131965i	$0.957871\pi$
$558$		0			0
$559$	−2.00000			−0.0845910
$560$		0			0
$561$		0			0
$562$	6.00000	−	10.3923i	0.253095	−	0.438373i
$563$	−14.0000	−	24.2487i	−0.590030	−	1.02196i	−0.994228	−	0.107290i	$-0.965783\pi$
$563$	−14.0000	−	24.2487i	−0.590030	−	1.02196i	0.404198	−	0.914671i	$-0.367551\pi$
$564$		0			0
$565$		0			0
$566$	−14.0000			−0.588464
$567$	18.0000	−	15.5885i	0.755929	−	0.654654i
$568$	15.0000			0.629386
$569$	−0.500000	+	0.866025i	−0.0209611	+	0.0363057i	−0.876316	−	0.481737i	$-0.840006\pi$
$569$	−0.500000	+	0.866025i	−0.0209611	+	0.0363057i	0.855355	+	0.518043i	$0.173339\pi$
$570$		0			0
$571$	−2.00000	−	3.46410i	−0.0836974	−	0.144968i	0.821138	−	0.570730i	$-0.193340\pi$
$571$	−2.00000	−	3.46410i	−0.0836974	−	0.144968i	−0.904835	+	0.425762i	$0.860006\pi$
$572$	1.50000	−	2.59808i	0.0627182	−	0.108631i
$573$		0			0
$574$		0			0
$575$	30.0000			1.25109
$576$	10.5000	−	18.1865i	0.437500	−	0.757772i
$577$	−7.00000	−	12.1244i	−0.291414	−	0.504744i	0.682730	−	0.730670i	$-0.260792\pi$
$577$	−7.00000	−	12.1244i	−0.291414	−	0.504744i	−0.974144	+	0.225927i	$0.927459\pi$
$578$	−16.0000	−	27.7128i	−0.665512	−	1.15270i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−4.50000	+	7.79423i	−0.186371	+	0.322804i
$584$	21.0000	+	36.3731i	0.868986	+	1.50513i
$585$		0			0
$586$	−7.00000	+	12.1244i	−0.289167	+	0.500853i
$587$	21.0000			0.866763			0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000			0.866763			0.433381	+	0.901211i	$0.357320\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	4.00000	+	6.92820i	0.164399	+	0.284747i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$	6.00000			0.245770
$597$		0			0
$598$	−3.00000	−	5.19615i	−0.122679	−	0.212486i
$599$	−16.0000	−	27.7128i	−0.653742	−	1.13231i	−0.982208	−	0.187799i	$-0.939865\pi$
$599$	−16.0000	−	27.7128i	−0.653742	−	1.13231i	0.328465	−	0.944516i	$-0.393469\pi$
$600$		0			0
$601$	−7.00000			−0.285536			−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000			−0.285536			−0.142768	+	0.989756i	$0.545600\pi$
$602$	−5.00000	−	1.73205i	−0.203785	−	0.0705931i
$603$	9.00000			0.366508
$604$	−1.50000	+	2.59808i	−0.0610341	+	0.105714i
$605$		0			0
$606$		0			0
$607$	−7.00000	+	12.1244i	−0.284121	+	0.492112i	−0.972396	−	0.233338i	$-0.925035\pi$
$607$	−7.00000	+	12.1244i	−0.284121	+	0.492112i	0.688274	+	0.725450i	$0.258368\pi$
$608$	−35.0000			−1.41944
$609$		0			0
$610$		0			0
$611$	3.50000	−	6.06218i	0.141595	−	0.245249i
$612$	−10.5000	−	18.1865i	−0.424437	−	0.735147i
$613$	−16.0000	−	27.7128i	−0.646234	−	1.11931i	−0.984015	−	0.178085i	$-0.943010\pi$
$613$	−16.0000	−	27.7128i	−0.646234	−	1.11931i	0.337781	−	0.941225i	$-0.390324\pi$
$614$	−10.5000	+	18.1865i	−0.423746	+	0.733949i
$615$		0			0
$616$	18.0000	−	15.5885i	0.725241	−	0.628077i
$617$	30.0000			1.20775			0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000			1.20775			0.603877	+	0.797077i	$0.293622\pi$
$618$		0			0
$619$	14.0000	+	24.2487i	0.562708	+	0.974638i	0.997259	+	0.0739910i	$0.0235736\pi$
$619$	14.0000	+	24.2487i	0.562708	+	0.974638i	−0.434551	+	0.900647i	$0.643093\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$	7.00000	+	12.1244i	0.279776	+	0.484587i
$627$		0			0
$628$	3.50000	−	6.06218i	0.139665	−	0.241907i
$629$	56.0000			2.23287
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$	−9.00000	+	15.5885i	−0.358001	+	0.620076i
$633$		0			0
$634$	3.00000	+	5.19615i	0.119145	+	0.206366i
$635$		0			0
$636$		0			0
$637$	6.50000	−	2.59808i	0.257539	−	0.102940i
$638$	15.0000			0.593856
$639$	−7.50000	+	12.9904i	−0.296695	+	0.513892i
$640$		0			0
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	0.631721	−	0.775196i	$-0.282349\pi$
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	−0.987200	+	0.159489i	$0.949015\pi$
$642$		0			0
$643$	−7.00000			−0.276053			−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000			−0.276053			−0.138027	+	0.990429i	$0.544076\pi$
$644$	3.00000	+	15.5885i	0.118217	+	0.614271i
$645$		0			0
$646$	24.5000	−	42.4352i	0.963940	−	1.66959i
$647$	21.0000	+	36.3731i	0.825595	+	1.42997i	0.901464	+	0.432855i	$0.142494\pi$
$647$	21.0000	+	36.3731i	0.825595	+	1.42997i	−0.0758684	+	0.997118i	$0.524173\pi$
$648$	13.5000	+	23.3827i	0.530330	+	0.918559i
$649$	−10.5000	+	18.1865i	−0.412161	+	0.713884i
$650$	5.00000			0.196116
$651$		0			0
$652$	13.0000			0.509119
$653$	3.00000	−	5.19615i	0.117399	−	0.203341i	−0.801337	−	0.598213i	$-0.795878\pi$
$653$	3.00000	−	5.19615i	0.117399	−	0.203341i	0.918736	+	0.394872i	$0.129211\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−42.0000			−1.63858
$658$	14.0000	−	12.1244i	0.545777	−	0.472657i
$659$	−40.0000			−1.55818			−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000			−1.55818			−0.779089	+	0.626913i	$0.784318\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$	10.0000	+	17.3205i	0.388661	+	0.673181i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−24.0000			−0.929981
$667$	−15.0000	+	25.9808i	−0.580802	+	1.00598i
$668$	−3.50000	−	6.06218i	−0.135419	−	0.234553i
$669$		0			0
$670$		0			0
$671$	21.0000			0.810696
$672$		0			0
$673$	−26.0000			−1.00223			−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000			−1.00223			−0.501113	+	0.865382i	$0.667076\pi$
$674$	−11.5000	+	19.9186i	−0.442963	+	0.767235i
$675$		0			0
$676$	0.500000	+	0.866025i	0.0192308	+	0.0333087i
$677$	17.5000	−	30.3109i	0.672580	−	1.16494i	−0.304590	−	0.952483i	$-0.598520\pi$
$677$	17.5000	−	30.3109i	0.672580	−	1.16494i	0.977170	−	0.212459i	$-0.0681471\pi$
$678$		0			0
$679$	−7.00000	−	36.3731i	−0.268635	−	1.39587i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.0000	+	20.7846i	0.459167	+	0.795301i	0.998917	−	0.0465244i	$-0.0148145\pi$
$683$	12.0000	+	20.7846i	0.459167	+	0.795301i	−0.539750	+	0.841825i	$0.681481\pi$
$684$	10.5000	−	18.1865i	0.401478	−	0.695379i
$685$		0			0
$686$	18.5000	−	0.866025i	0.706333	−	0.0330650i
$687$		0			0
$688$	1.00000	−	1.73205i	0.0381246	−	0.0660338i
$689$	−1.50000	−	2.59808i	−0.0571454	−	0.0989788i
$690$		0			0
$691$	−17.5000	+	30.3109i	−0.665731	+	1.15308i	0.313355	+	0.949636i	$0.398547\pi$
$691$	−17.5000	+	30.3109i	−0.665731	+	1.15308i	−0.979086	+	0.203445i	$0.934786\pi$
$692$	−7.00000			−0.266100
$693$	4.50000	+	23.3827i	0.170941	+	0.888235i
$694$	4.00000			0.151838
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−7.00000	+	12.1244i	−0.264954	+	0.458914i
$699$		0			0
$700$	−12.5000	−	4.33013i	−0.472456	−	0.163663i
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$	28.0000	+	48.4974i	1.05604	+	1.82911i
$704$	10.5000	+	18.1865i	0.395734	+	0.685431i
$705$		0			0
$706$	14.0000			0.526897
$707$	−28.0000	+	24.2487i	−1.05305	+	0.911967i
$708$		0			0
$709$	−25.0000	+	43.3013i	−0.938895	+	1.62621i	−0.171358	+	0.985209i	$0.554815\pi$
$709$	−25.0000	+	43.3013i	−0.938895	+	1.62621i	−0.767537	+	0.641004i	$0.778518\pi$
$710$		0			0
$711$	−9.00000	−	15.5885i	−0.337526	−	0.584613i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−5.00000	+	8.66025i	−0.186859	+	0.323649i
$717$		0			0
$718$	−4.00000	−	6.92820i	−0.149279	−	0.258558i
$719$	21.0000	−	36.3731i	0.783168	−	1.35649i	−0.146920	−	0.989148i	$-0.546936\pi$
$719$	21.0000	−	36.3731i	0.783168	−	1.35649i	0.930087	−	0.367338i	$-0.119731\pi$
$720$		0			0
$721$	−35.0000	−	12.1244i	−1.30347	−	0.451535i
$722$	30.0000			1.11648
$723$		0			0
$724$	−3.50000	−	6.06218i	−0.130076	−	0.225299i
$725$	−12.5000	−	21.6506i	−0.464238	−	0.804084i
$726$		0			0
$727$	28.0000			1.03846			0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000			1.03846			0.519231	+	0.854634i	$0.326218\pi$
$728$	1.50000	+	7.79423i	0.0555937	+	0.288873i
$729$	−27.0000			−1.00000
$730$		0			0
$731$	−7.00000	−	12.1244i	−0.258904	−	0.448435i
$732$		0			0
$733$	−21.0000	+	36.3731i	−0.775653	+	1.34347i	0.158774	+	0.987315i	$0.449246\pi$
$733$	−21.0000	+	36.3731i	−0.775653	+	1.34347i	−0.934427	+	0.356155i	$0.884088\pi$
$734$	14.0000			0.516749
$735$		0			0
$736$	−30.0000			−1.10581
$737$	−4.50000	+	7.79423i	−0.165760	+	0.287104i
$738$		0			0
$739$	−2.00000	−	3.46410i	−0.0735712	−	0.127429i	0.826893	−	0.562360i	$-0.190106\pi$
$739$	−2.00000	−	3.46410i	−0.0735712	−	0.127429i	−0.900464	+	0.434930i	$0.856773\pi$
$740$		0			0
$741$		0			0
$742$	−1.50000	−	7.79423i	−0.0550667	−	0.286135i
$743$	9.00000			0.330178			0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000			0.330178			0.165089	+	0.986279i	$0.447209\pi$
$744$		0			0
$745$		0			0
$746$	−7.50000	−	12.9904i	−0.274595	−	0.475612i
$747$		0			0
$748$	21.0000			0.767836
$749$	−20.0000	−	6.92820i	−0.730784	−	0.253151i
$750$		0			0
$751$	10.0000	−	17.3205i	0.364905	−	0.632034i	−0.623856	−	0.781540i	$-0.714435\pi$
$751$	10.0000	−	17.3205i	0.364905	−	0.632034i	0.988761	+	0.149505i	$0.0477681\pi$
$752$	3.50000	+	6.06218i	0.127632	+	0.221065i
$753$		0			0
$754$	−2.50000	+	4.33013i	−0.0910446	+	0.157694i
$755$		0			0
$756$		0			0
$757$	9.00000			0.327111			0.163555	−	0.986534i	$-0.447704\pi$
$757$	9.00000			0.327111			0.163555	+	0.986534i	$0.447704\pi$
$758$	6.00000	−	10.3923i	0.217930	−	0.377466i
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	8.00000	−	6.92820i	0.289619	−	0.250818i
$764$	20.0000			0.723575
$765$		0			0
$766$		0			0
$767$	−3.50000	−	6.06218i	−0.126378	−	0.218893i
$768$		0			0
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$		0			0
$772$	2.00000	−	3.46410i	0.0719816	−	0.124676i
$773$	−21.0000	−	36.3731i	−0.755318	−	1.30825i	−0.945216	−	0.326445i	$-0.894149\pi$
$773$	−21.0000	−	36.3731i	−0.755318	−	1.30825i	0.189899	−	0.981804i	$-0.439184\pi$
$774$	3.00000	+	5.19615i	0.107833	+	0.186772i
$775$		0			0
$776$	42.0000			1.50771
$777$		0			0
$778$	−3.00000			−0.107555
$779$		0			0
$780$		0			0
$781$	−7.50000	−	12.9904i	−0.268371	−	0.464832i
$782$	21.0000	−	36.3731i	0.750958	−	1.30070i
$783$		0			0
$784$	−1.00000	+	6.92820i	−0.0357143	+	0.247436i
$785$		0			0
$786$		0			0
$787$	−3.50000	−	6.06218i	−0.124762	−	0.216093i	0.796878	−	0.604140i	$-0.206483\pi$
$787$	−3.50000	−	6.06218i	−0.124762	−	0.216093i	−0.921640	+	0.388047i	$0.873150\pi$
$788$	1.00000	+	1.73205i

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:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	91.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(0.500000 + 2.59808i) q^{7} -3.00000 q^{8} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(0.500000 + 2.59808i) q^{7} -3.00000 q^{8} +(1.50000 - 2.59808i) q^{9} +(1.50000 + 2.59808i) q^{11} -1.00000 q^{13} +(-2.50000 - 0.866025i) q^{14} +(0.500000 - 0.866025i) q^{16} +(-3.50000 - 6.06218i) q^{17} +(1.50000 + 2.59808i) q^{18} +(3.50000 - 6.06218i) q^{19} -3.00000 q^{22} +(3.00000 - 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +(0.500000 - 0.866025i) q^{26} +(-2.00000 + 1.73205i) q^{28} -5.00000 q^{29} +(-2.50000 - 4.33013i) q^{32} +7.00000 q^{34} +3.00000 q^{36} +(-4.00000 + 6.92820i) q^{37} +(3.50000 + 6.06218i) q^{38} +2.00000 q^{43} +(-1.50000 + 2.59808i) q^{44} +(3.00000 + 5.19615i) q^{46} +(-3.50000 + 6.06218i) q^{47} +(-6.50000 + 2.59808i) q^{49} -5.00000 q^{50} +(-0.500000 - 0.866025i) q^{52} +(1.50000 + 2.59808i) q^{53} +(-1.50000 - 7.79423i) q^{56} +(2.50000 - 4.33013i) q^{58} +(3.50000 + 6.06218i) q^{59} +(3.50000 - 6.06218i) q^{61} +(7.50000 + 2.59808i) q^{63} +7.00000 q^{64} +(1.50000 + 2.59808i) q^{67} +(3.50000 - 6.06218i) q^{68} -5.00000 q^{71} +(-4.50000 + 7.79423i) q^{72} +(-7.00000 - 12.1244i) q^{73} +(-4.00000 - 6.92820i) q^{74} +7.00000 q^{76} +(-6.00000 + 5.19615i) q^{77} +(3.00000 - 5.19615i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-1.00000 + 1.73205i) q^{86} +(-4.50000 - 7.79423i) q^{88} +(-0.500000 - 2.59808i) q^{91} +6.00000 q^{92} +(-3.50000 - 6.06218i) q^{94} -14.0000 q^{97} +(1.00000 - 6.92820i) q^{98} +9.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + q^{4} + q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) 2 * q - q^2 + q^4 + q^7 - 6 * q^8 + 3 * q^9	\( 2 q - q^{2} + q^{4} + q^{7} - 6 q^{8} + 3 q^{9} + 3 q^{11} - 2 q^{13} - 5 q^{14} + q^{16} - 7 q^{17} + 3 q^{18} + 7 q^{19} - 6 q^{22} + 6 q^{23} + 5 q^{25} + q^{26} - 4 q^{28} - 10 q^{29} - 5 q^{32} + 14 q^{34} + 6 q^{36} - 8 q^{37} + 7 q^{38} + 4 q^{43} - 3 q^{44} + 6 q^{46} - 7 q^{47} - 13 q^{49} - 10 q^{50} - q^{52} + 3 q^{53} - 3 q^{56} + 5 q^{58} + 7 q^{59} + 7 q^{61} + 15 q^{63} + 14 q^{64} + 3 q^{67} + 7 q^{68} - 10 q^{71} - 9 q^{72} - 14 q^{73} - 8 q^{74} + 14 q^{76} - 12 q^{77} + 6 q^{79} - 9 q^{81} - 2 q^{86} - 9 q^{88} - q^{91} + 12 q^{92} - 7 q^{94} - 28 q^{97} + 2 q^{98} + 18 q^{99}+O(q^{100}) \) 2 * q - q^2 + q^4 + q^7 - 6 * q^8 + 3 * q^9 + 3 * q^11 - 2 * q^13 - 5 * q^14 + q^16 - 7 * q^17 + 3 * q^18 + 7 * q^19 - 6 * q^22 + 6 * q^23 + 5 * q^25 + q^26 - 4 * q^28 - 10 * q^29 - 5 * q^32 + 14 * q^34 + 6 * q^36 - 8 * q^37 + 7 * q^38 + 4 * q^43 - 3 * q^44 + 6 * q^46 - 7 * q^47 - 13 * q^49 - 10 * q^50 - q^52 + 3 * q^53 - 3 * q^56 + 5 * q^58 + 7 * q^59 + 7 * q^61 + 15 * q^63 + 14 * q^64 + 3 * q^67 + 7 * q^68 - 10 * q^71 - 9 * q^72 - 14 * q^73 - 8 * q^74 + 14 * q^76 - 12 * q^77 + 6 * q^79 - 9 * q^81 - 2 * q^86 - 9 * q^88 - q^91 + 12 * q^92 - 7 * q^94 - 28 * q^97 + 2 * q^98 + 18 * q^99

Introduction

L-functions

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