LMFDB - Embedded newform 77.2.b.b.76.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [77,2,Mod(76,77)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(77, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("77.76");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 77 = 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	77.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.614848095564$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			76.1
Root			$-1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	77.76
Dual form			77.2.b.b.76.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.23607i q^{3} +2.23607i q^{5} -3.16228 q^{6} +(-1.58114 + 2.12132i) q^{7} -2.82843i q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-1.41421i q^{2} -2.23607i q^{3} +2.23607i q^{5} -3.16228 q^{6} +(-1.58114 + 2.12132i) q^{7} -2.82843i q^{8} -2.00000 q^{9} +3.16228 q^{10} +(-3.00000 + 1.41421i) q^{11} +6.32456 q^{13} +(3.00000 + 2.23607i) q^{14} +5.00000 q^{15} -4.00000 q^{16} +2.82843i q^{18} -3.16228 q^{19} +(4.74342 + 3.53553i) q^{21} +(2.00000 + 4.24264i) q^{22} -3.00000 q^{23} -6.32456 q^{24} -8.94427i q^{26} -2.23607i q^{27} -1.41421i q^{29} -7.07107i q^{30} +6.70820i q^{31} +(3.16228 + 6.70820i) q^{33} +(-4.74342 - 3.53553i) q^{35} -1.00000 q^{37} +4.47214i q^{38} -14.1421i q^{39} +6.32456 q^{40} -9.48683 q^{41} +(5.00000 - 6.70820i) q^{42} +4.24264i q^{43} -4.47214i q^{45} +4.24264i q^{46} -4.47214i q^{47} +8.94427i q^{48} +(-2.00000 - 6.70820i) q^{49} -3.16228 q^{54} +(-3.16228 - 6.70820i) q^{55} +(6.00000 + 4.47214i) q^{56} +7.07107i q^{57} -2.00000 q^{58} +2.23607i q^{59} -3.16228 q^{61} +9.48683 q^{62} +(3.16228 - 4.24264i) q^{63} -8.00000 q^{64} +14.1421i q^{65} +(9.48683 - 4.47214i) q^{66} +11.0000 q^{67} +6.70820i q^{69} +(-5.00000 + 6.70820i) q^{70} +9.00000 q^{71} +5.65685i q^{72} -3.16228 q^{73} +1.41421i q^{74} +(1.74342 - 8.60003i) q^{77} -20.0000 q^{78} -8.48528i q^{79} -8.94427i q^{80} -11.0000 q^{81} +13.4164i q^{82} +9.48683 q^{83} +6.00000 q^{86} -3.16228 q^{87} +(4.00000 + 8.48528i) q^{88} +2.23607i q^{89} -6.32456 q^{90} +(-10.0000 + 13.4164i) q^{91} +15.0000 q^{93} -6.32456 q^{94} -7.07107i q^{95} -6.70820i q^{97} +(-9.48683 + 2.82843i) q^{98} +(6.00000 - 2.82843i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{9}+O(q^{10}) $ 4 * q - 8 * q^9	$ 4 q - 8 q^{9} - 12 q^{11} + 12 q^{14} + 20 q^{15} - 16 q^{16} + 8 q^{22} - 12 q^{23} - 4 q^{37} + 20 q^{42} - 8 q^{49} + 24 q^{56} - 8 q^{58} - 32 q^{64} + 44 q^{67} - 20 q^{70} + 36 q^{71} - 12 q^{77} - 80 q^{78} - 44 q^{81} + 24 q^{86} + 16 q^{88} - 40 q^{91} + 60 q^{93} + 24 q^{99}+O(q^{100}) $ 4 * q - 8 * q^9 - 12 * q^11 + 12 * q^14 + 20 * q^15 - 16 * q^16 + 8 * q^22 - 12 * q^23 - 4 * q^37 + 20 * q^42 - 8 * q^49 + 24 * q^56 - 8 * q^58 - 32 * q^64 + 44 * q^67 - 20 * q^70 + 36 * q^71 - 12 * q^77 - 80 * q^78 - 44 * q^81 + 24 * q^86 + 16 * q^88 - 40 * q^91 + 60 * q^93 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/77\mathbb{Z}\right)^\times$.

$n$	$45$	$57$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$		−	1.41421i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$		−	1.41421i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$3$		−	2.23607i		−	1.29099i	−0.763763	−	0.645497i	$-0.776650\pi$
$3$		−	2.23607i		−	1.29099i	0.763763	−	0.645497i	$-0.223350\pi$
$4$		0			0
$5$			2.23607i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$5$			2.23607i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$6$	−3.16228			−1.29099
$7$	−1.58114	+	2.12132i	−0.597614	+	0.801784i
$7$	−1.58114	+	2.12132i	−0.597614	+	0.801784i
$8$		−	2.82843i		−	1.00000i
$9$	−2.00000			−0.666667
$10$	3.16228			1.00000
$11$	−3.00000	+	1.41421i	−0.904534	+	0.426401i
$11$	−3.00000	+	1.41421i	−0.904534	+	0.426401i
$12$		0			0
$13$	6.32456			1.75412			0.877058	−	0.480384i	$-0.159503\pi$
$13$	6.32456			1.75412			0.877058	+	0.480384i	$0.159503\pi$
$14$	3.00000	+	2.23607i	0.801784	+	0.597614i
$15$	5.00000			1.29099
$16$	−4.00000			−1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$			2.82843i			0.666667i
$19$	−3.16228			−0.725476			−0.362738	−	0.931891i	$-0.618158\pi$
$19$	−3.16228			−0.725476			−0.362738	+	0.931891i	$0.618158\pi$
$20$		0			0
$21$	4.74342	+	3.53553i	1.03510	+	0.771517i
$22$	2.00000	+	4.24264i	0.426401	+	0.904534i
$23$	−3.00000			−0.625543			−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000			−0.625543			−0.312772	+	0.949828i	$0.601257\pi$
$24$	−6.32456			−1.29099
$25$		0			0
$26$		−	8.94427i		−	1.75412i
$27$		−	2.23607i		−	0.430331i
$28$		0			0
$29$		−	1.41421i		−	0.262613i	−0.991342	−	0.131306i	$-0.958083\pi$
$29$		−	1.41421i		−	0.262613i	0.991342	−	0.131306i	$-0.0419172\pi$
$30$		−	7.07107i		−	1.29099i
$31$			6.70820i			1.20483i	0.798183	+	0.602414i	$0.205795\pi$
$31$			6.70820i			1.20483i	−0.798183	+	0.602414i	$0.794205\pi$
$32$		0			0
$33$	3.16228	+	6.70820i	0.550482	+	1.16775i
$34$		0			0
$35$	−4.74342	−	3.53553i	−0.801784	−	0.597614i
$36$		0			0
$37$	−1.00000			−0.164399			−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000			−0.164399			−0.0821995	+	0.996616i	$0.526194\pi$
$38$			4.47214i			0.725476i
$39$		−	14.1421i		−	2.26455i
$40$	6.32456			1.00000
$41$	−9.48683			−1.48159			−0.740797	−	0.671729i	$-0.765552\pi$
$41$	−9.48683			−1.48159			−0.740797	+	0.671729i	$0.765552\pi$
$42$	5.00000	−	6.70820i	0.771517	−	1.03510i
$43$			4.24264i			0.646997i	0.946229	+	0.323498i	$0.104859\pi$
$43$			4.24264i			0.646997i	−0.946229	+	0.323498i	$0.895141\pi$
$44$		0			0
$45$		−	4.47214i		−	0.666667i
$46$			4.24264i			0.625543i
$47$		−	4.47214i		−	0.652328i	−0.945313	−	0.326164i	$-0.894244\pi$
$47$		−	4.47214i		−	0.652328i	0.945313	−	0.326164i	$-0.105756\pi$
$48$			8.94427i			1.29099i
$49$	−2.00000	−	6.70820i	−0.285714	−	0.958315i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−3.16228			−0.430331
$55$	−3.16228	−	6.70820i	−0.426401	−	0.904534i
$56$	6.00000	+	4.47214i	0.801784	+	0.597614i
$57$			7.07107i			0.936586i
$58$	−2.00000			−0.262613
$59$			2.23607i			0.291111i	0.989350	+	0.145556i	$0.0464970\pi$
$59$			2.23607i			0.291111i	−0.989350	+	0.145556i	$0.953503\pi$
$60$		0			0
$61$	−3.16228			−0.404888			−0.202444	−	0.979294i	$-0.564888\pi$
$61$	−3.16228			−0.404888			−0.202444	+	0.979294i	$0.564888\pi$
$62$	9.48683			1.20483
$63$	3.16228	−	4.24264i	0.398410	−	0.534522i
$64$	−8.00000			−1.00000
$65$			14.1421i			1.75412i
$66$	9.48683	−	4.47214i	1.16775	−	0.550482i
$67$	11.0000			1.34386			0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000			1.34386			0.671932	+	0.740613i	$0.265465\pi$
$68$		0			0
$69$			6.70820i			0.807573i
$70$	−5.00000	+	6.70820i	−0.597614	+	0.801784i
$71$	9.00000			1.06810			0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000			1.06810			0.534052	+	0.845452i	$0.320669\pi$
$72$			5.65685i			0.666667i
$73$	−3.16228			−0.370117			−0.185058	−	0.982728i	$-0.559247\pi$
$73$	−3.16228			−0.370117			−0.185058	+	0.982728i	$0.559247\pi$
$74$			1.41421i			0.164399i
$75$		0			0
$76$		0			0
$77$	1.74342	−	8.60003i	0.198681	−	0.980064i
$78$	−20.0000			−2.26455
$79$		−	8.48528i		−	0.954669i	−0.878722	−	0.477334i	$-0.841603\pi$
$79$		−	8.48528i		−	0.954669i	0.878722	−	0.477334i	$-0.158397\pi$
$80$		−	8.94427i		−	1.00000i
$81$	−11.0000			−1.22222
$82$			13.4164i			1.48159i
$83$	9.48683			1.04132			0.520658	−	0.853766i	$-0.325687\pi$
$83$	9.48683			1.04132			0.520658	+	0.853766i	$0.325687\pi$
$84$		0			0
$85$		0			0
$86$	6.00000			0.646997
$87$	−3.16228			−0.339032
$88$	4.00000	+	8.48528i	0.426401	+	0.904534i
$89$			2.23607i			0.237023i	0.992953	+	0.118511i	$0.0378122\pi$
$89$			2.23607i			0.237023i	−0.992953	+	0.118511i	$0.962188\pi$
$90$	−6.32456			−0.666667
$91$	−10.0000	+	13.4164i	−1.04828	+	1.40642i
$92$		0			0
$93$	15.0000			1.55543
$94$	−6.32456			−0.652328
$95$		−	7.07107i		−	0.725476i
$96$		0			0
$97$		−	6.70820i		−	0.681115i	−0.940224	−	0.340557i	$-0.889384\pi$
$97$		−	6.70820i		−	0.681115i	0.940224	−	0.340557i	$-0.110616\pi$
$98$	−9.48683	+	2.82843i	−0.958315	+	0.285714i
$99$	6.00000	−	2.82843i	0.603023	−	0.284268i
$100$		0			0
$101$	9.48683			0.943975			0.471988	−	0.881605i	$-0.343537\pi$
$101$	9.48683			0.943975			0.471988	+	0.881605i	$0.343537\pi$
$102$		0			0
$103$		−	13.4164i		−	1.32196i	−0.750404	−	0.660979i	$-0.770141\pi$
$103$		−	13.4164i		−	1.32196i	0.750404	−	0.660979i	$-0.229859\pi$
$104$		−	17.8885i		−	1.75412i
$105$	−7.90569	+	10.6066i	−0.771517	+	1.03510i
$106$		0			0
$107$			2.82843i			0.273434i	0.990610	+	0.136717i	$0.0436552\pi$
$107$			2.82843i			0.273434i	−0.990610	+	0.136717i	$0.956345\pi$
$108$		0			0
$109$		−	12.7279i		−	1.21911i	−0.792742	−	0.609557i	$-0.791347\pi$
$109$		−	12.7279i		−	1.21911i	0.792742	−	0.609557i	$-0.208653\pi$
$110$	−9.48683	+	4.47214i	−0.904534	+	0.426401i
$111$			2.23607i			0.212238i
$112$	6.32456	−	8.48528i	0.597614	−	0.801784i
$113$	−3.00000			−0.282216			−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000			−0.282216			−0.141108	+	0.989994i	$0.545067\pi$
$114$	10.0000			0.936586
$115$		−	6.70820i		−	0.625543i
$116$		0			0
$117$	−12.6491			−1.16941
$118$	3.16228			0.291111
$119$		0			0
$120$		−	14.1421i		−	1.29099i
$121$	7.00000	−	8.48528i	0.636364	−	0.771389i
$122$			4.47214i			0.404888i
$123$			21.2132i			1.91273i
$124$		0			0
$125$			11.1803i			1.00000i
$126$	−6.00000	−	4.47214i	−0.534522	−	0.398410i
$127$		−	12.7279i		−	1.12942i	−0.825289	−	0.564710i	$-0.808988\pi$
$127$		−	12.7279i		−	1.12942i	0.825289	−	0.564710i	$-0.191012\pi$
$128$			11.3137i			1.00000i
$129$	9.48683			0.835269
$130$	20.0000			1.75412
$131$	18.9737			1.65774			0.828868	−	0.559444i	$-0.188985\pi$
$131$	18.9737			1.65774			0.828868	+	0.559444i	$0.188985\pi$
$132$		0			0
$133$	5.00000	−	6.70820i	0.433555	−	0.581675i
$134$		−	15.5563i		−	1.34386i
$135$	5.00000			0.430331
$136$		0			0
$137$	−15.0000			−1.28154			−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000			−1.28154			−0.640768	+	0.767734i	$0.721384\pi$
$138$	9.48683			0.807573
$139$	−12.6491			−1.07288			−0.536442	−	0.843937i	$-0.680232\pi$
$139$	−12.6491			−1.07288			−0.536442	+	0.843937i	$0.680232\pi$
$140$		0			0
$141$	−10.0000			−0.842152
$142$		−	12.7279i		−	1.06810i
$143$	−18.9737	+	8.94427i	−1.58666	+	0.747958i
$144$	8.00000			0.666667
$145$	3.16228			0.262613
$146$			4.47214i			0.370117i
$147$	−15.0000	+	4.47214i	−1.23718	+	0.368856i
$148$		0			0
$149$			19.7990i			1.62200i	0.585049	+	0.810998i	$0.301075\pi$
$149$			19.7990i			1.62200i	−0.585049	+	0.810998i	$0.698925\pi$
$150$		0			0
$151$			16.9706i			1.38104i	0.723311	+	0.690522i	$0.242619\pi$
$151$			16.9706i			1.38104i	−0.723311	+	0.690522i	$0.757381\pi$
$152$			8.94427i			0.725476i
$153$		0			0
$154$	−12.1623	−	2.46556i	−0.980064	−	0.198681i
$155$	−15.0000			−1.20483
$156$		0			0
$157$			6.70820i			0.535373i	0.963506	+	0.267686i	$0.0862591\pi$
$157$			6.70820i			0.535373i	−0.963506	+	0.267686i	$0.913741\pi$
$158$	−12.0000			−0.954669
$159$		0			0
$160$		0			0
$161$	4.74342	−	6.36396i	0.373834	−	0.501550i
$162$			15.5563i			1.22222i
$163$	−10.0000			−0.783260			−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000			−0.783260			−0.391630	+	0.920123i	$0.628089\pi$
$164$		0			0
$165$	−15.0000	+	7.07107i	−1.16775	+	0.550482i
$166$		−	13.4164i		−	1.04132i
$167$	−9.48683			−0.734113			−0.367057	−	0.930199i	$-0.619634\pi$
$167$	−9.48683			−0.734113			−0.367057	+	0.930199i	$0.619634\pi$
$168$	10.0000	−	13.4164i	0.771517	−	1.03510i
$169$	27.0000			2.07692
$170$		0			0
$171$	6.32456			0.483651
$172$		0			0
$173$	9.48683			0.721271			0.360635	−	0.932707i	$-0.382560\pi$
$173$	9.48683			0.721271			0.360635	+	0.932707i	$0.382560\pi$
$174$			4.47214i			0.339032i
$175$		0			0
$176$	12.0000	−	5.65685i	0.904534	−	0.426401i
$177$	5.00000			0.375823
$178$	3.16228			0.237023
$179$	9.00000			0.672692			0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000			0.672692			0.336346	+	0.941739i	$0.390809\pi$
$180$		0			0
$181$			20.1246i			1.49585i	0.663783	+	0.747925i	$0.268950\pi$
$181$			20.1246i			1.49585i	−0.663783	+	0.747925i	$0.731050\pi$
$182$	18.9737	+	14.1421i	1.40642	+	1.04828i
$183$			7.07107i			0.522708i
$184$			8.48528i			0.625543i
$185$		−	2.23607i		−	0.164399i
$186$		−	21.2132i		−	1.55543i
$187$		0			0
$188$		0			0
$189$	4.74342	+	3.53553i	0.345033	+	0.257172i
$190$	−10.0000			−0.725476
$191$	3.00000			0.217072			0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000			0.217072			0.108536	+	0.994092i	$0.465384\pi$
$192$			17.8885i			1.29099i
$193$		−	4.24264i		−	0.305392i	−0.988273	−	0.152696i	$-0.951204\pi$
$193$		−	4.24264i		−	0.305392i	0.988273	−	0.152696i	$-0.0487955\pi$
$194$	−9.48683			−0.681115
$195$	31.6228			2.26455
$196$		0			0
$197$		−	5.65685i		−	0.403034i	−0.979485	−	0.201517i	$-0.935413\pi$
$197$		−	5.65685i		−	0.403034i	0.979485	−	0.201517i	$-0.0645872\pi$
$198$	−4.00000	−	8.48528i	−0.284268	−	0.603023i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		−	24.5967i		−	1.73492i
$202$		−	13.4164i		−	0.943975i
$203$	3.00000	+	2.23607i	0.210559	+	0.156941i
$204$		0			0
$205$		−	21.2132i		−	1.48159i
$206$	−18.9737			−1.32196
$207$	6.00000			0.417029
$208$	−25.2982			−1.75412
$209$	9.48683	−	4.47214i	0.656218	−	0.309344i
$210$	15.0000	+	11.1803i	1.03510	+	0.771517i
$211$			21.2132i			1.46038i	0.683246	+	0.730189i	$0.260568\pi$
$211$			21.2132i			1.46038i	−0.683246	+	0.730189i	$0.739432\pi$
$212$		0			0
$213$		−	20.1246i		−	1.37892i
$214$	4.00000			0.273434
$215$	−9.48683			−0.646997
$216$	−6.32456			−0.430331
$217$	−14.2302	−	10.6066i	−0.966012	−	0.720023i
$218$	−18.0000			−1.21911
$219$			7.07107i			0.477818i
$220$		0			0
$221$		0			0
$222$	3.16228			0.212238
$223$		−	6.70820i		−	0.449215i	−0.974449	−	0.224607i	$-0.927890\pi$
$223$		−	6.70820i		−	0.449215i	0.974449	−	0.224607i	$-0.0721099\pi$
$224$		0			0
$225$		0			0
$226$			4.24264i			0.282216i
$227$	9.48683			0.629663			0.314832	−	0.949148i	$-0.398052\pi$
$227$	9.48683			0.629663			0.314832	+	0.949148i	$0.398052\pi$
$228$		0			0
$229$			6.70820i			0.443291i	0.975127	+	0.221645i	$0.0711427\pi$
$229$			6.70820i			0.443291i	−0.975127	+	0.221645i	$0.928857\pi$
$230$	−9.48683			−0.625543
$231$	−19.2302	−	3.89840i	−1.26526	−	0.256496i
$232$	−4.00000			−0.262613
$233$		−	18.3848i		−	1.20443i	−0.798335	−	0.602213i	$-0.794286\pi$
$233$		−	18.3848i		−	1.20443i	0.798335	−	0.602213i	$-0.205714\pi$
$234$			17.8885i			1.16941i
$235$	10.0000			0.652328
$236$		0			0
$237$	−18.9737			−1.23247
$238$		0			0
$239$			7.07107i			0.457389i	0.973498	+	0.228695i	$0.0734457\pi$
$239$			7.07107i			0.457389i	−0.973498	+	0.228695i	$0.926554\pi$
$240$	−20.0000			−1.29099
$241$	−12.6491			−0.814801			−0.407400	−	0.913250i	$-0.633565\pi$
$241$	−12.6491			−0.814801			−0.407400	+	0.913250i	$0.633565\pi$
$242$	−12.0000	−	9.89949i	−0.771389	−	0.636364i
$243$			17.8885i			1.14755i
$244$		0			0
$245$	15.0000	−	4.47214i	0.958315	−	0.285714i
$246$	30.0000			1.91273
$247$	−20.0000			−1.27257
$248$	18.9737			1.20483
$249$		−	21.2132i		−	1.34433i
$250$	15.8114			1.00000
$251$		−	11.1803i		−	0.705697i	−0.935681	−	0.352848i	$-0.885213\pi$
$251$		−	11.1803i		−	0.705697i	0.935681	−	0.352848i	$-0.114787\pi$
$252$		0			0
$253$	9.00000	−	4.24264i	0.565825	−	0.266733i
$254$	−18.0000			−1.12942
$255$		0			0
$256$		0			0
$257$			22.3607i			1.39482i	0.716672	+	0.697410i	$0.245665\pi$
$257$			22.3607i			1.39482i	−0.716672	+	0.697410i	$0.754335\pi$
$258$		−	13.4164i		−	0.835269i
$259$	1.58114	−	2.12132i	0.0982472	−	0.131812i
$260$		0			0
$261$			2.82843i			0.175075i
$262$		−	26.8328i		−	1.65774i
$263$			11.3137i			0.697633i	0.937191	+	0.348817i	$0.113416\pi$
$263$			11.3137i			0.697633i	−0.937191	+	0.348817i	$0.886584\pi$
$264$	18.9737	−	8.94427i	1.16775	−	0.550482i
$265$		0			0
$266$	−9.48683	−	7.07107i	−0.581675	−	0.433555i
$267$	5.00000			0.305995
$268$		0			0
$269$		−	17.8885i		−	1.09068i	−0.838214	−	0.545342i	$-0.816400\pi$
$269$		−	17.8885i		−	1.09068i	0.838214	−	0.545342i	$-0.183600\pi$
$270$		−	7.07107i		−	0.430331i
$271$	25.2982			1.53676			0.768379	−	0.639995i	$-0.221064\pi$
$271$	25.2982			1.53676			0.768379	+	0.639995i	$0.221064\pi$
$272$		0			0
$273$	30.0000	+	22.3607i	1.81568	+	1.35333i
$274$			21.2132i			1.28154i
$275$		0			0
$276$		0			0
$277$			4.24264i			0.254916i	0.991844	+	0.127458i	$0.0406817\pi$
$277$			4.24264i			0.254916i	−0.991844	+	0.127458i	$0.959318\pi$
$278$			17.8885i			1.07288i
$279$		−	13.4164i		−	0.803219i
$280$	−10.0000	+	13.4164i	−0.597614	+	0.801784i
$281$			24.0416i			1.43420i	0.696969	+	0.717102i	$0.254532\pi$
$281$			24.0416i			1.43420i	−0.696969	+	0.717102i	$0.745468\pi$
$282$			14.1421i			0.842152i
$283$	6.32456			0.375956			0.187978	−	0.982173i	$-0.439807\pi$
$283$	6.32456			0.375956			0.187978	+	0.982173i	$0.439807\pi$
$284$		0			0
$285$	−15.8114			−0.936586
$286$	12.6491	+	26.8328i	0.747958	+	1.58666i
$287$	15.0000	−	20.1246i	0.885422	−	1.18792i
$288$		0			0
$289$	−17.0000			−1.00000
$290$		−	4.47214i		−	0.262613i
$291$	−15.0000			−0.879316
$292$		0			0
$293$	−9.48683			−0.554227			−0.277113	−	0.960837i	$-0.589378\pi$
$293$	−9.48683			−0.554227			−0.277113	+	0.960837i	$0.589378\pi$
$294$	6.32456	+	21.2132i	0.368856	+	1.23718i
$295$	−5.00000			−0.291111
$296$			2.82843i			0.164399i
$297$	3.16228	+	6.70820i	0.183494	+	0.389249i
$298$	28.0000			1.62200
$299$	−18.9737			−1.09728
$300$		0			0
$301$	−9.00000	−	6.70820i	−0.518751	−	0.386654i
$302$	24.0000			1.38104
$303$		−	21.2132i		−	1.21867i
$304$	12.6491			0.725476
$305$		−	7.07107i		−	0.404888i
$306$		0			0
$307$	6.32456			0.360961			0.180481	−	0.983579i	$-0.442235\pi$
$307$	6.32456			0.360961			0.180481	+	0.983579i	$0.442235\pi$
$308$		0			0
$309$	−30.0000			−1.70664
$310$			21.2132i			1.20483i
$311$		−	4.47214i		−	0.253592i	−0.991929	−	0.126796i	$-0.959531\pi$
$311$		−	4.47214i		−	0.253592i	0.991929	−	0.126796i	$-0.0404693\pi$
$312$	−40.0000			−2.26455
$313$		−	6.70820i		−	0.379170i	−0.981864	−	0.189585i	$-0.939286\pi$
$313$		−	6.70820i		−	0.379170i	0.981864	−	0.189585i	$-0.0607143\pi$
$314$	9.48683			0.535373
$315$	9.48683	+	7.07107i	0.534522	+	0.398410i
$316$		0			0
$317$	21.0000			1.17948			0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000			1.17948			0.589739	+	0.807594i	$0.299231\pi$
$318$		0			0
$319$	2.00000	+	4.24264i	0.111979	+	0.237542i
$320$		−	17.8885i		−	1.00000i
$321$	6.32456			0.353002
$322$	−9.00000	−	6.70820i	−0.501550	−	0.373834i
$323$		0			0
$324$		0			0
$325$		0			0
$326$			14.1421i			0.783260i
$327$	−28.4605			−1.57387
$328$			26.8328i			1.48159i
$329$	9.48683	+	7.07107i	0.523026	+	0.389841i
$330$	10.0000	+	21.2132i	0.550482	+	1.16775i
$331$	−19.0000			−1.04433			−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000			−1.04433			−0.522167	+	0.852843i	$0.674876\pi$
$332$		0			0
$333$	2.00000			0.109599
$334$			13.4164i			0.734113i
$335$			24.5967i			1.34386i
$336$	−18.9737	−	14.1421i	−1.03510	−	0.771517i
$337$		−	16.9706i		−	0.924445i	−0.886764	−	0.462223i	$-0.847052\pi$
$337$		−	16.9706i		−	0.924445i	0.886764	−	0.462223i	$-0.152948\pi$
$338$		−	38.1838i		−	2.07692i
$339$			6.70820i			0.364340i
$340$		0			0
$341$	−9.48683	−	20.1246i	−0.513741	−	1.08981i
$342$		−	8.94427i		−	0.483651i
$343$	17.3925	+	6.36396i	0.939108	+	0.343622i
$344$	12.0000			0.646997
$345$	−15.0000			−0.807573
$346$		−	13.4164i		−	0.721271i
$347$		−	1.41421i		−	0.0759190i	−0.999279	−	0.0379595i	$-0.987914\pi$
$347$		−	1.41421i		−	0.0759190i	0.999279	−	0.0379595i	$-0.0120858\pi$
$348$		0			0
$349$	−12.6491			−0.677091			−0.338546	−	0.940950i	$-0.609935\pi$
$349$	−12.6491			−0.677091			−0.338546	+	0.940950i	$0.609935\pi$
$350$		0			0
$351$		−	14.1421i		−	0.754851i
$352$		0			0
$353$			15.6525i			0.833097i	0.909113	+	0.416549i	$0.136760\pi$
$353$			15.6525i			0.833097i	−0.909113	+	0.416549i	$0.863240\pi$
$354$		−	7.07107i		−	0.375823i
$355$			20.1246i			1.06810i
$356$		0			0
$357$		0			0
$358$		−	12.7279i		−	0.672692i
$359$		−	5.65685i		−	0.298557i	−0.988795	−	0.149279i	$-0.952305\pi$
$359$		−	5.65685i		−	0.298557i	0.988795	−	0.149279i	$-0.0476951\pi$
$360$	−12.6491			−0.666667
$361$	−9.00000			−0.473684
$362$	28.4605			1.49585
$363$	−18.9737	−	15.6525i	−0.995859	−	0.821542i
$364$		0			0
$365$		−	7.07107i		−	0.370117i
$366$	10.0000			0.522708
$367$			33.5410i			1.75083i	0.483375	+	0.875413i	$0.339411\pi$
$367$			33.5410i			1.75083i	−0.483375	+	0.875413i	$0.660589\pi$
$368$	12.0000			0.625543
$369$	18.9737			0.987730
$370$	−3.16228			−0.164399
$371$		0			0
$372$		0			0
$373$		−	4.24264i		−	0.219676i	−0.993950	−	0.109838i	$-0.964967\pi$
$373$		−	4.24264i		−	0.219676i	0.993950	−	0.109838i	$-0.0350331\pi$
$374$		0			0
$375$	25.0000			1.29099
$376$	−12.6491			−0.652328
$377$		−	8.94427i		−	0.460653i
$378$	5.00000	−	6.70820i	0.257172	−	0.345033i
$379$	17.0000			0.873231			0.436616	−	0.899648i	$-0.356177\pi$
$379$	17.0000			0.873231			0.436616	+	0.899648i	$0.356177\pi$
$380$		0			0
$381$	−28.4605			−1.45808
$382$		−	4.24264i		−	0.217072i
$383$		−	24.5967i		−	1.25684i	−0.777876	−	0.628418i	$-0.783703\pi$
$383$		−	24.5967i		−	1.25684i	0.777876	−	0.628418i	$-0.216297\pi$
$384$	25.2982			1.29099
$385$	19.2302	+	3.89840i	0.980064	+	0.198681i
$386$	−6.00000			−0.305392
$387$		−	8.48528i		−	0.431331i
$388$		0			0
$389$	3.00000			0.152106			0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000			0.152106			0.0760530	+	0.997104i	$0.475768\pi$
$390$		−	44.7214i		−	2.26455i
$391$		0			0
$392$	−18.9737	+	5.65685i	−0.958315	+	0.285714i
$393$		−	42.4264i		−	2.14013i
$394$	−8.00000			−0.403034
$395$	18.9737			0.954669
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$	−15.0000	−	11.1803i	−0.750939	−	0.559717i
$400$		0			0
$401$	24.0000			1.19850			0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000			1.19850			0.599251	+	0.800561i	$0.295465\pi$
$402$	−34.7851			−1.73492
$403$			42.4264i			2.11341i
$404$		0			0
$405$		−	24.5967i		−	1.22222i
$406$	3.16228	−	4.24264i	0.156941	−	0.210559i
$407$	3.00000	−	1.41421i	0.148704	−	0.0701000i
$408$		0			0
$409$	−12.6491			−0.625458			−0.312729	−	0.949842i	$-0.601243\pi$
$409$	−12.6491			−0.625458			−0.312729	+	0.949842i	$0.601243\pi$
$410$	−30.0000			−1.48159
$411$			33.5410i			1.65446i
$412$		0			0
$413$	−4.74342	−	3.53553i	−0.233408	−	0.173972i
$414$		−	8.48528i		−	0.417029i
$415$			21.2132i			1.04132i
$416$		0			0
$417$			28.2843i			1.38509i
$418$	−6.32456	−	13.4164i	−0.309344	−	0.656218i
$419$			22.3607i			1.09239i	0.837658	+	0.546195i	$0.183924\pi$
$419$			22.3607i			1.09239i	−0.837658	+	0.546195i	$0.816076\pi$
$420$		0			0
$421$	−4.00000			−0.194948			−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000			−0.194948			−0.0974740	+	0.995238i	$0.531076\pi$
$422$	30.0000			1.46038
$423$			8.94427i			0.434885i
$424$		0			0
$425$		0			0
$426$	−28.4605			−1.37892
$427$	5.00000	−	6.70820i	0.241967	−	0.324633i
$428$		0			0
$429$	20.0000	+	42.4264i	0.965609	+	2.04837i
$430$			13.4164i			0.646997i
$431$		−	35.3553i		−	1.70301i	−0.524349	−	0.851503i	$-0.675691\pi$
$431$		−	35.3553i		−	1.70301i	0.524349	−	0.851503i	$-0.324309\pi$
$432$			8.94427i			0.430331i
$433$		−	20.1246i		−	0.967127i	−0.875309	−	0.483564i	$-0.839342\pi$
$433$		−	20.1246i		−	0.967127i	0.875309	−	0.483564i	$-0.160658\pi$
$434$	−15.0000	+	20.1246i	−0.720023	+	0.966012i
$435$		−	7.07107i		−	0.339032i
$436$		0			0
$437$	9.48683			0.453817
$438$	10.0000			0.477818
$439$	15.8114			0.754636			0.377318	−	0.926084i	$-0.376846\pi$
$439$	15.8114			0.754636			0.377318	+	0.926084i	$0.376846\pi$
$440$	−18.9737	+	8.94427i	−0.904534	+	0.426401i
$441$	4.00000	+	13.4164i	0.190476	+	0.638877i
$442$		0			0
$443$	−15.0000			−0.712672			−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000			−0.712672			−0.356336	+	0.934358i	$0.615974\pi$
$444$		0			0
$445$	−5.00000			−0.237023
$446$	−9.48683			−0.449215
$447$	44.2719			2.09399
$448$	12.6491	−	16.9706i	0.597614	−	0.801784i
$449$	−27.0000			−1.27421			−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000			−1.27421			−0.637104	+	0.770778i	$0.719868\pi$
$450$		0			0
$451$	28.4605	−	13.4164i	1.34015	−	0.631754i
$452$		0			0
$453$	37.9473			1.78292
$454$		−	13.4164i		−	0.629663i
$455$	−30.0000	−	22.3607i	−1.40642	−	1.04828i
$456$	20.0000			0.936586
$457$		−	8.48528i		−	0.396925i	−0.980109	−	0.198462i	$-0.936405\pi$
$457$		−	8.48528i		−	0.396925i	0.980109	−	0.198462i	$-0.0635948\pi$
$458$	9.48683			0.443291
$459$		0			0
$460$		0			0
$461$	9.48683			0.441846			0.220923	−	0.975291i	$-0.429093\pi$
$461$	9.48683			0.441846			0.220923	+	0.975291i	$0.429093\pi$
$462$	−5.51317	+	27.1957i	−0.256496	+	1.26526i
$463$	−25.0000			−1.16185			−0.580924	−	0.813958i	$-0.697309\pi$
$463$	−25.0000			−1.16185			−0.580924	+	0.813958i	$0.697309\pi$
$464$			5.65685i			0.262613i
$465$			33.5410i			1.55543i
$466$	−26.0000			−1.20443
$467$		−	11.1803i		−	0.517364i	−0.965962	−	0.258682i	$-0.916712\pi$
$467$		−	11.1803i		−	0.517364i	0.965962	−	0.258682i	$-0.0832882\pi$
$468$		0			0
$469$	−17.3925	+	23.3345i	−0.803112	+	1.07749i
$470$		−	14.1421i		−	0.652328i
$471$	15.0000			0.691164
$472$	6.32456			0.291111
$473$	−6.00000	−	12.7279i	−0.275880	−	0.585230i
$474$			26.8328i			1.23247i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	10.0000			0.457389
$479$	−18.9737			−0.866929			−0.433464	−	0.901171i	$-0.642709\pi$
$479$	−18.9737			−0.866929			−0.433464	+	0.901171i	$0.642709\pi$
$480$		0			0
$481$	−6.32456			−0.288375
$482$			17.8885i			0.814801i
$483$	−14.2302	−	10.6066i	−0.647499	−	0.482617i
$484$		0			0
$485$	15.0000			0.681115
$486$	25.2982			1.14755
$487$	29.0000			1.31412			0.657058	−	0.753840i	$-0.271801\pi$
$487$	29.0000			1.31412			0.657058	+	0.753840i	$0.271801\pi$
$488$			8.94427i			0.404888i
$489$			22.3607i			1.01118i
$490$	−6.32456	−	21.2132i	−0.285714	−	0.958315i
$491$			32.5269i			1.46792i	0.679193	+	0.733959i	$0.262330\pi$
$491$			32.5269i			1.46792i	−0.679193	+	0.733959i	$0.737670\pi$
$492$		0			0
$493$		0			0
$494$			28.2843i			1.27257i
$495$	6.32456	+	13.4164i	0.284268	+	0.603023i
$496$		−	26.8328i		−	1.20483i
$497$	−14.2302	+	19.0919i	−0.638314	+	0.856388i
$498$	−30.0000			−1.34433
$499$	−16.0000			−0.716258			−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000			−0.716258			−0.358129	+	0.933672i	$0.616585\pi$
$500$		0			0
$501$			21.2132i			0.947736i
$502$	−15.8114			−0.705697
$503$	−28.4605			−1.26899			−0.634495	−	0.772927i	$-0.718792\pi$
$503$	−28.4605			−1.26899			−0.634495	+	0.772927i	$0.718792\pi$
$504$	−12.0000	−	8.94427i	−0.534522	−	0.398410i
$505$			21.2132i			0.943975i
$506$	−6.00000	−	12.7279i	−0.266733	−	0.565825i
$507$		−	60.3738i		−	2.68130i
$508$		0			0
$509$			15.6525i			0.693784i	0.937905	+	0.346892i	$0.112763\pi$
$509$			15.6525i			0.693784i	−0.937905	+	0.346892i	$0.887237\pi$
$510$		0			0
$511$	5.00000	−	6.70820i	0.221187	−	0.296753i
$512$			22.6274i			1.00000i
$513$			7.07107i			0.312195i
$514$	31.6228			1.39482
$515$	30.0000			1.32196
$516$		0			0
$517$	6.32456	+	13.4164i	0.278154	+	0.590053i
$518$	−3.00000	−	2.23607i	−0.131812	−	0.0982472i
$519$		−	21.2132i		−	0.931156i
$520$	40.0000			1.75412
$521$		−	11.1803i		−	0.489820i	−0.969546	−	0.244910i	$-0.921242\pi$
$521$		−	11.1803i		−	0.489820i	0.969546	−	0.244910i	$-0.0787583\pi$
$522$	4.00000			0.175075
$523$	−31.6228			−1.38277			−0.691384	−	0.722488i	$-0.742999\pi$
$523$	−31.6228			−1.38277			−0.691384	+	0.722488i	$0.742999\pi$
$524$		0			0
$525$		0			0
$526$	16.0000			0.697633
$527$		0			0
$528$	−12.6491	−	26.8328i	−0.550482	−	1.16775i
$529$	−14.0000			−0.608696
$530$		0			0
$531$		−	4.47214i		−	0.194074i
$532$		0			0
$533$	−60.0000			−2.59889
$534$		−	7.07107i		−	0.305995i
$535$	−6.32456			−0.273434
$536$		−	31.1127i		−	1.34386i
$537$		−	20.1246i		−	0.868441i
$538$	−25.2982			−1.09068
$539$	15.4868	+	17.2962i	0.667065	+	0.744999i
$540$		0			0
$541$			25.4558i			1.09443i	0.836991	+	0.547216i	$0.184312\pi$
$541$			25.4558i			1.09443i	−0.836991	+	0.547216i	$0.815688\pi$
$542$		−	35.7771i		−	1.53676i
$543$	45.0000			1.93113
$544$		0			0
$545$	28.4605			1.21911
$546$	31.6228	−	42.4264i	1.35333	−	1.81568i
$547$		−	8.48528i		−	0.362804i	−0.983409	−	0.181402i	$-0.941936\pi$
$547$		−	8.48528i		−	0.362804i	0.983409	−	0.181402i	$-0.0580636\pi$
$548$		0			0
$549$	6.32456			0.269925
$550$		0			0
$551$			4.47214i			0.190519i
$552$	18.9737			0.807573
$553$	18.0000	+	13.4164i	0.765438	+	0.570524i
$554$	6.00000			0.254916
$555$	−5.00000			−0.212238
$556$		0			0
$557$		−	22.6274i		−	0.958754i	−0.877609	−	0.479377i	$-0.840863\pi$
$557$		−	22.6274i		−	0.958754i	0.877609	−	0.479377i	$-0.159137\pi$
$558$	−18.9737			−0.803219
$559$			26.8328i			1.13491i
$560$	18.9737	+	14.1421i	0.801784	+	0.597614i
$561$		0			0
$562$	34.0000			1.43420
$563$	−37.9473			−1.59929			−0.799645	−	0.600473i	$-0.794979\pi$
$563$	−37.9473			−1.59929			−0.799645	+	0.600473i	$0.794979\pi$
$564$		0			0
$565$		−	6.70820i		−	0.282216i
$566$		−	8.94427i		−	0.375956i
$567$	17.3925	−	23.3345i	0.730417	−	0.979958i
$568$		−	25.4558i		−	1.06810i
$569$			15.5563i			0.652156i	0.945343	+	0.326078i	$0.105727\pi$
$569$			15.5563i			0.652156i	−0.945343	+	0.326078i	$0.894273\pi$
$570$			22.3607i			0.936586i
$571$		−	16.9706i		−	0.710196i	−0.934829	−	0.355098i	$-0.884448\pi$
$571$		−	16.9706i		−	0.710196i	0.934829	−	0.355098i	$-0.115552\pi$
$572$		0			0
$573$		−	6.70820i		−	0.280239i
$574$	−28.4605	−	21.2132i	−1.18792	−	0.885422i
$575$		0			0
$576$	16.0000			0.666667
$577$		−	20.1246i		−	0.837799i	−0.908033	−	0.418899i	$-0.862416\pi$
$577$		−	20.1246i		−	0.837799i	0.908033	−	0.418899i	$-0.137584\pi$
$578$			24.0416i			1.00000i
$579$	−9.48683			−0.394259
$580$		0			0
$581$	−15.0000	+	20.1246i	−0.622305	+	0.834910i
$582$			21.2132i			0.879316i
$583$		0			0
$584$			8.94427i			0.370117i
$585$		−	28.2843i		−	1.16941i
$586$			13.4164i			0.554227i
$587$		−	4.47214i		−	0.184585i	−0.995732	−	0.0922924i	$-0.970581\pi$
$587$		−	4.47214i		−	0.184585i	0.995732	−	0.0922924i	$-0.0294195\pi$
$588$		0			0
$589$		−	21.2132i		−	0.874075i
$590$			7.07107i			0.291111i
$591$	−12.6491			−0.520315
$592$	4.00000			0.164399
$593$	−28.4605			−1.16873			−0.584366	−	0.811490i	$-0.698657\pi$
$593$	−28.4605			−1.16873			−0.584366	+	0.811490i	$0.698657\pi$
$594$	9.48683	−	4.47214i	0.389249	−	0.183494i
$595$		0			0
$596$		0			0
$597$		0			0
$598$			26.8328i			1.09728i
$599$	−12.0000			−0.490307			−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000			−0.490307			−0.245153	+	0.969484i	$0.578838\pi$
$600$		0			0
$601$	15.8114			0.644960			0.322480	−	0.946576i	$-0.395483\pi$
$601$	15.8114			0.644960			0.322480	+	0.946576i	$0.395483\pi$
$602$	−9.48683	+	12.7279i	−0.386654	+	0.518751i
$603$	−22.0000			−0.895909
$604$		0			0
$605$	18.9737	+	15.6525i	0.771389	+	0.636364i
$606$	−30.0000			−1.21867
$607$	6.32456			0.256706			0.128353	−	0.991729i	$-0.459031\pi$
$607$	6.32456			0.256706			0.128353	+	0.991729i	$0.459031\pi$
$608$		0			0
$609$	5.00000	−	6.70820i	0.202610	−	0.271830i
$610$	−10.0000			−0.404888
$611$		−	28.2843i		−	1.14426i
$612$		0			0
$613$		−	25.4558i		−	1.02815i	−0.857745	−	0.514076i	$-0.828135\pi$
$613$		−	25.4558i		−	1.02815i	0.857745	−	0.514076i	$-0.171865\pi$
$614$		−	8.94427i		−	0.360961i
$615$	−47.4342			−1.91273
$616$	−24.3246	−	4.93113i	−0.980064	−	0.198681i
$617$	24.0000			0.966204			0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000			0.966204			0.483102	+	0.875564i	$0.339510\pi$
$618$			42.4264i			1.70664i
$619$		−	46.9574i		−	1.88738i	−0.330833	−	0.943689i	$-0.607330\pi$
$619$		−	46.9574i		−	1.88738i	0.330833	−	0.943689i	$-0.392670\pi$
$620$		0			0
$621$			6.70820i			0.269191i
$622$	−6.32456			−0.253592
$623$	−4.74342	−	3.53553i	−0.190041	−	0.141648i
$624$			56.5685i			2.26455i
$625$	−25.0000			−1.00000
$626$	−9.48683			−0.379170
$627$	−10.0000	−	21.2132i	−0.399362	−	0.847174i
$628$		0			0
$629$		0			0
$630$	10.0000	−	13.4164i	0.398410	−	0.534522i
$631$	11.0000			0.437903			0.218952	−	0.975736i	$-0.429736\pi$
$631$	11.0000			0.437903			0.218952	+	0.975736i	$0.429736\pi$
$632$	−24.0000			−0.954669
$633$	47.4342			1.88534
$634$		−	29.6985i		−	1.17948i
$635$	28.4605			1.12942
$636$		0			0
$637$	−12.6491	−	42.4264i	−0.501176	−	1.68100i
$638$	6.00000	−	2.82843i	0.237542	−	0.111979i
$639$	−18.0000			−0.712069
$640$	−25.2982			−1.00000
$641$	21.0000			0.829450			0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000			0.829450			0.414725	+	0.909947i	$0.363878\pi$
$642$		−	8.94427i		−	0.353002i
$643$			6.70820i			0.264546i	0.991213	+	0.132273i	$0.0422275\pi$
$643$			6.70820i			0.264546i	−0.991213	+	0.132273i	$0.957772\pi$
$644$		0			0
$645$			21.2132i			0.835269i
$646$		0			0
$647$			29.0689i			1.14282i	0.820666	+	0.571408i	$0.193603\pi$
$647$			29.0689i			1.14282i	−0.820666	+	0.571408i	$0.806397\pi$
$648$			31.1127i			1.22222i
$649$	−3.16228	−	6.70820i	−0.124130	−	0.263320i
$650$		0			0
$651$	−23.7171	+	31.8198i	−0.929546	+	1.24712i
$652$		0			0
$653$	15.0000			0.586995			0.293498	−	0.955960i	$-0.405181\pi$
$653$	15.0000			0.586995			0.293498	+	0.955960i	$0.405181\pi$
$654$			40.2492i			1.57387i
$655$			42.4264i			1.65774i
$656$	37.9473			1.48159
$657$	6.32456			0.246744
$658$	10.0000	−	13.4164i	0.389841	−	0.523026i
$659$		−	14.1421i		−	0.550899i	−0.961315	−	0.275450i	$-0.911173\pi$
$659$		−	14.1421i		−	0.550899i	0.961315	−	0.275450i	$-0.0888267\pi$
$660$		0			0
$661$			46.9574i			1.82643i	0.407476	+	0.913216i	$0.366409\pi$
$661$			46.9574i			1.82643i	−0.407476	+	0.913216i	$0.633591\pi$
$662$			26.8701i			1.04433i
$663$		0			0
$664$		−	26.8328i		−	1.04132i
$665$	15.0000	+	11.1803i	0.581675	+	0.433555i
$666$		−	2.82843i		−	0.109599i
$667$			4.24264i			0.164276i
$668$		0			0
$669$	−15.0000			−0.579934
$670$	34.7851			1.34386
$671$	9.48683	−	4.47214i	0.366235	−	0.172645i
$672$		0			0
$673$			29.6985i			1.14479i	0.819977	+	0.572396i	$0.193986\pi$
$673$			29.6985i			1.14479i	−0.819977	+	0.572396i	$0.806014\pi$
$674$	−24.0000			−0.924445
$675$		0			0
$676$		0			0
$677$	37.9473			1.45843			0.729217	−	0.684282i	$-0.239884\pi$
$677$	37.9473			1.45843			0.729217	+	0.684282i	$0.239884\pi$
$678$	9.48683			0.364340
$679$	14.2302	+	10.6066i	0.546107	+	0.407044i
$680$		0			0
$681$		−	21.2132i		−	0.812892i
$682$	−28.4605	+	13.4164i	−1.08981	+	0.513741i
$683$	18.0000			0.688751			0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000			0.688751			0.344375	+	0.938832i	$0.388091\pi$
$684$		0			0
$685$		−	33.5410i		−	1.28154i
$686$	9.00000	−	24.5967i	0.343622	−	0.939108i
$687$	15.0000			0.572286
$688$		−	16.9706i		−	0.646997i
$689$		0			0
$690$			21.2132i			0.807573i
$691$		−	6.70820i		−	0.255192i	−0.991826	−	0.127596i	$-0.959274\pi$
$691$		−	6.70820i		−	0.255192i	0.991826	−	0.127596i	$-0.0407261\pi$
$692$		0			0
$693$	−3.48683	+	17.2001i	−0.132454	+	0.653376i
$694$	−2.00000			−0.0759190
$695$		−	28.2843i		−	1.07288i
$696$			8.94427i			0.339032i
$697$		0			0
$698$			17.8885i			0.677091i
$699$	−41.1096			−1.55491
$700$		0			0
$701$			7.07107i			0.267071i	0.991044	+	0.133535i	$0.0426329\pi$
$701$			7.07107i			0.267071i	−0.991044	+	0.133535i	$0.957367\pi$
$702$	−20.0000			−0.754851
$703$	3.16228			0.119268
$704$	24.0000	−	11.3137i	0.904534	−	0.426401i
$705$		−	22.3607i		−	0.842152i
$706$	22.1359			0.833097
$707$	−15.0000	+	20.1246i	−0.564133	+	0.756864i
$708$		0			0
$709$	41.0000			1.53979			0.769894	−	0.638172i	$-0.220309\pi$
$709$	41.0000			1.53979			0.769894	+	0.638172i	$0.220309\pi$
$710$	28.4605			1.06810
$711$			16.9706i			0.636446i
$712$	6.32456			0.237023
$713$		−	20.1246i		−	0.753673i
$714$		0			0
$715$	−20.0000	−	42.4264i	−0.747958	−	1.58666i
$716$		0			0
$717$	15.8114			0.590487
$718$	−8.00000			−0.298557
$719$		−	51.4296i		−	1.91800i	−0.283408	−	0.959000i	$-0.591465\pi$
$719$		−	51.4296i		−	1.91800i	0.283408	−	0.959000i	$-0.408535\pi$
$720$			17.8885i			0.666667i
$721$	28.4605	+	21.2132i	1.05992	+	0.790021i
$722$			12.7279i			0.473684i
$723$			28.2843i			1.05190i
$724$		0			0
$725$		0			0
$726$	−22.1359	+	26.8328i	−0.821542	+	0.995859i
$727$			33.5410i			1.24397i	0.783030	+	0.621984i	$0.213673\pi$
$727$			33.5410i			1.24397i	−0.783030	+	0.621984i	$0.786327\pi$
$728$	37.9473	+	28.2843i	1.40642	+	1.04828i
$729$	7.00000			0.259259
$730$	−10.0000			−0.370117
$731$		0			0
$732$		0			0
$733$	−12.6491			−0.467206			−0.233603	−	0.972332i	$-0.575052\pi$
$733$	−12.6491			−0.467206			−0.233603	+	0.972332i	$0.575052\pi$
$734$	47.4342			1.75083
$735$	−10.0000	−	33.5410i	−0.368856	−	1.23718i
$736$		0			0
$737$	−33.0000	+	15.5563i	−1.21557	+	0.573025i
$738$		−	26.8328i		−	0.987730i
$739$		−	50.9117i		−	1.87282i	−0.350912	−	0.936408i	$-0.614128\pi$
$739$		−	50.9117i		−	1.87282i	0.350912	−	0.936408i	$-0.385872\pi$
$740$		0			0
$741$			44.7214i			1.64288i
$742$		0			0
$743$			19.7990i			0.726354i	0.931720	+	0.363177i	$0.118308\pi$
$743$			19.7990i			0.726354i	−0.931720	+	0.363177i	$0.881692\pi$
$744$		−	42.4264i		−	1.55543i
$745$	−44.2719			−1.62200
$746$	−6.00000			−0.219676
$747$	−18.9737			−0.694210
$748$		0			0
$749$	−6.00000	−	4.47214i	−0.219235	−	0.163408i
$750$		−	35.3553i		−	1.29099i
$751$	−7.00000			−0.255434			−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000			−0.255434			−0.127717	+	0.991811i	$0.540765\pi$
$752$			17.8885i			0.652328i
$753$	−25.0000			−0.911051
$754$	−12.6491			−0.460653
$755$	−37.9473			−1.38104
$756$		0			0
$757$	44.0000			1.59921			0.799604	−	0.600528i	$-0.205043\pi$
$757$	44.0000			1.59921			0.799604	+	0.600528i	$0.205043\pi$
$758$		−	24.0416i		−	0.873231i
$759$	−9.48683	−	20.1246i	−0.344350	−	0.730477i
$760$	−20.0000			−0.725476
$761$	−37.9473			−1.37559			−0.687795	−	0.725905i	$-0.741421\pi$
$761$	−37.9473			−1.37559			−0.687795	+	0.725905i	$0.741421\pi$
$762$			40.2492i			1.45808i
$763$	27.0000	+	20.1246i	0.977466	+	0.728560i
$764$		0			0
$765$		0			0
$766$	−34.7851			−1.25684
$767$			14.1421i			0.510643i
$768$		0			0
$769$	6.32456			0.228069			0.114035	−	0.993477i	$-0.463623\pi$
$769$	6.32456			0.228069			0.114035	+	0.993477i	$0.463623\pi$
$770$	5.51317	−	27.1957i	0.198681	−	0.980064i
$771$	50.0000			1.80071
$772$		0			0
$773$			8.94427i			0.321703i	0.986979	+	0.160852i	$0.0514240\pi$
$773$			8.94427i			0.321703i	−0.986979	+	0.160852i	$0.948576\pi$
$774$	−12.0000			−0.431331
$775$		0			0
$776$	−18.9737			−0.681115
$777$	−4.74342	−	3.53553i	−0.170169	−	0.126837i
$778$		−	4.24264i		−	0.152106i
$779$	30.0000			1.07486
$780$		0			0
$781$	−27.0000	+	12.7279i	−0.966136	+	0.455441i
$782$		0			0
$783$	−3.16228			−0.113011
$784$	8.00000	+	26.8328i	0.285714	+	0.958315i
$785$	−15.0000			−0.535373
$786$	−60.0000			−2.14013
$787$	−22.1359			−0.789061			−0.394531	−	0.918883i	$-0.629093\pi$
$787$	−22.1359			−0.789061			−0.394531	+	0.918883i	$0.629093\pi$
$788$		0			0
$789$	25.2982			0.900641
$790$		−	26.8328i		−	0.954669i
$791$	4.74342	−	6.36396i	0.168656	−	0.226276i
$792$	−8.00000	−	16.9706i	−0.284268	−	0.603023i
$793$	−20.0000			−0.710221
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	38.0132i		−	1.34650i	−0.739417	−	0.673248i	$-0.764899\pi$
$797$		−	38.0132i		−	1.34650i	0.739417	−	0.673248i	$-0.235101\pi$
$798$	−15.8114	+	21.2132i	−0.559717	+	0.750939i
$799$		0			0
$800$		0			0
$801$		−	4.47214i		−	0.158015i
$802$		−	33.9411i		−	1.19850i
$803$	9.48683	−	4.47214i	0.334783	−	0.157818i
$804$		0			0
$805$	14.2302	+	10.6066i	0.501550	+	0.373834i
$806$	60.0000			2.11341
$807$	−40.0000			−1.40807
$808$		−	26.8328i		−	0.943975i
$809$			41.0122i			1.44191i	0.692981	+	0.720956i	$0.256297\pi$
$809$			41.0122i			1.44191i	−0.692981	+	0.720956i	$0.743703\pi$
$810$	−34.7851			−1.22222
$811$	25.2982			0.888341			0.444170	−	0.895942i	$-0.353498\pi$
$811$	25.2982			0.888341			0.444170	+	0.895942i	$0.353498\pi$
$812$		0			0
$813$		−	56.5685i		−	1.98395i
$814$	−2.00000	−	4.24264i	−0.0701000	−	0.148704i
$815$		−	22.3607i		−	0.783260i
$816$		0			0
$817$		−	13.4164i		−	0.469381i
$818$			17.8885i			0.625458i
$819$	20.0000	−	26.8328i	0.698857	−	0.937614i
$820$		0			0
$821$		−	14.1421i		−	0.493564i	−0.969071	−	0.246782i	$-0.920627\pi$
$821$		−	14.1421i		−	0.493564i	0.969071	−	0.246782i	$-0.0793731\pi$
$822$	47.4342			1.65446
$823$	−43.0000			−1.49889			−0.749443	−	0.662069i	$-0.769679\pi$
$823$	−43.0000			−1.49889			−0.749443	+	0.662069i	$0.769679\pi$
$824$	−37.9473			−1.32196
$825$		0			0
$826$	−5.00000	+	6.70820i	−0.173972	+	0.233408i
$827$		−	22.6274i		−	0.786832i	−0.919360	−	0.393416i	$-0.871293\pi$
$827$		−	22.6274i		−	0.786832i	0.919360	−	0.393416i	$-0.128707\pi$
$828$		0			0
$829$		−	20.1246i		−	0.698957i	−0.936944	−	0.349478i	$-0.886359\pi$
$829$		−	20.1246i		−	0.698957i	0.936944	−	0.349478i	$-0.113641\pi$
$830$	30.0000			1.04132
$831$	9.48683			0.329095
$832$	−50.5964			−1.75412
$833$		0			0
$834$	40.0000			1.38509
$835$		−	21.2132i		−	0.734113i
$836$		0			0
$837$	15.0000			0.518476
$838$	31.6228			1.09239
$839$			15.6525i			0.540383i	0.962807	+	0.270192i	$0.0870871\pi$
$839$			15.6525i			0.540383i	−0.962807	+	0.270192i	$0.912913\pi$
$840$	30.0000	+	22.3607i	1.03510	+	0.771517i
$841$	27.0000			0.931034
$842$			5.65685i			0.194948i
$843$	53.7587			1.85155
$844$		0			0
$845$			60.3738i			2.07692i
$846$	12.6491			0.434885
$847$	6.93203	+	28.2657i	0.238187	+	0.971219i
$848$		0			0
$849$		−	14.1421i		−	0.485357i
$850$		0			0
$851$	3.00000			0.102839
$852$		0			0
$853$	25.2982			0.866195			0.433097	−	0.901347i	$-0.357421\pi$
$853$	25.2982			0.866195			0.433097	+	0.901347i	$0.357421\pi$
$854$	−9.48683	−	7.07107i	−0.324633	−	0.241967i
$855$			14.1421i			0.483651i
$856$	8.00000			0.273434
$857$	−18.9737			−0.648128			−0.324064	−	0.946035i	$-0.605049\pi$
$857$	−18.9737			−0.648128			−0.324064	+	0.946035i	$0.605049\pi$
$858$	60.0000	−	28.2843i	2.04837	−	0.965609i
$859$			6.70820i			0.228881i	0.993430	+	0.114440i	$0.0365075\pi$
$859$			6.70820i			0.228881i	−0.993430	+	0.114440i	$0.963492\pi$
$860$		0			0
$861$	−45.0000	−	33.5410i	−1.53360	−	1.14307i
$862$	−50.0000			−1.70301
$863$	18.0000			0.612727			0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000			0.612727			0.306364	+	0.951915i	$0.400888\pi$
$864$		0			0
$865$			21.2132i			0.721271i
$866$	−28.4605			−0.967127
$867$			38.0132i			1.29099i
$868$		0			0
$869$	12.0000	+	25.4558i	0.407072	+	0.863530i
$870$	−10.0000			−0.339032
$871$	69.5701			2.35729
$872$	−36.0000			−1.21911
$873$			13.4164i			0.454077i
$874$		−	13.4164i		−	0.453817i
$875$	−23.7171	−	17.6777i	−0.801784	−	0.597614i
$876$		0			0
$877$		−	4.24264i		−	0.143264i	−0.997431	−	0.0716319i	$-0.977179\pi$
$877$		−	4.24264i		−	0.143264i	0.997431	−	0.0716319i	$-0.0228207\pi$
$878$		−	22.3607i		−	0.754636i
$879$			21.2132i			0.715504i
$880$	12.6491	+	26.8328i	0.426401	+	0.904534i
$881$		−	51.4296i		−	1.73271i	−0.499432	−	0.866353i	$-0.666458\pi$
$881$		−	51.4296i		−	1.73271i	0.499432	−	0.866353i	$-0.333542\pi$
$882$	18.9737	−	5.65685i	0.638877	−	0.190476i
$883$	8.00000			0.269221			0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000			0.269221			0.134611	+	0.990899i	$0.457022\pi$
$884$		0			0
$885$			11.1803i			0.375823i
$886$			21.2132i			0.712672i
$887$	18.9737			0.637073			0.318537	−	0.947911i	$-0.396809\pi$
$887$	18.9737			0.637073			0.318537	+	0.947911i	$0.396809\pi$
$888$	6.32456			0.212238
$889$	27.0000	+	20.1246i	0.905551	+	0.674958i
$890$			7.07107i			0.237023i
$891$	33.0000	−	15.5563i	1.10554	−	0.521157i
$892$		0			0
$893$			14.1421i			0.473249i
$894$		−	62.6099i		−	2.09399i
$895$			20.1246i			0.672692i
$896$	−24.0000	−	17.8885i	−0.801784	−	0.597614i
$897$			42.4264i			1.41658i
$898$			38.1838i			1.27421i
$899$	9.48683			0.316404
$900$		0			0
$901$		0			0
$902$	−18.9737	−	40.2492i	−0.631754	−	1.34015i
$903$	−15.0000	+	20.1246i	−0.499169	+	0.669705i
$904$			8.48528i			0.282216i
$905$	−45.0000			−1.49585
$906$		−	53.6656i		−	1.78292i
$907$	−10.0000			−0.332045			−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000			−0.332045			−0.166022	+	0.986122i	$0.553092\pi$
$908$		0			0
$909$	−18.9737			−0.629317
$910$	−31.6228	+	42.4264i	−1.04828	+	1.40642i
$911$	−6.00000			−0.198789			−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000			−0.198789			−0.0993944	+	0.995048i	$0.531691\pi$
$912$		−	28.2843i		−	0.936586i
$913$	−28.4605	+	13.4164i	−0.941905	+	0.444018i
$914$	−12.0000			−0.396925
$915$	−15.8114			−0.522708
$916$		0			0
$917$	−30.0000	+	40.2492i	−0.990687	+	1.32915i
$918$		0			0
$919$			12.7279i			0.419855i	0.977717	+	0.209928i	$0.0673229\pi$
$919$			12.7279i			0.419855i	−0.977717	+	0.209928i	$0.932677\pi$
$920$	−18.9737			−0.625543
$921$		−	14.1421i		−	0.465999i
$922$		−	13.4164i		−	0.441846i
$923$	56.9210			1.87358
$924$		0			0
$925$		0			0
$926$			35.3553i			1.16185i
$927$			26.8328i			0.881305i
$928$		0			0
$929$			35.7771i			1.17381i	0.809656	+	0.586904i	$0.199653\pi$
$929$			35.7771i			1.17381i	−0.809656	+	0.586904i	$0.800347\pi$
$930$	47.4342			1.55543
$931$	6.32456	+	21.2132i	0.207279	+	0.695235i
$932$		0			0
$933$	−10.0000			−0.327385
$934$	−15.8114			−0.517364
$935$		0			0
$936$			35.7771i			1.16941i
$937$	−50.5964			−1.65291			−0.826457	−	0.563000i	$-0.809647\pi$
$937$	−50.5964			−1.65291			−0.826457	+	0.563000i	$0.809647\pi$
$938$	33.0000	+	24.5967i	1.07749	+	0.803112i
$939$	−15.0000			−0.489506
$940$		0			0
$941$	−37.9473			−1.23705			−0.618524	−	0.785766i	$-0.712269\pi$
$941$	−37.9473			−1.23705			−0.618524	+	0.785766i	$0.712269\pi$
$942$		−	21.2132i		−	0.691164i
$943$	28.4605			0.926801
$944$		−	8.94427i		−	0.291111i
$945$	−7.90569	+	10.6066i	−0.257172	+	0.345033i
$946$	−18.0000	+	8.48528i	−0.585230	+	0.275880i
$947$	−15.0000			−0.487435			−0.243717	−	0.969846i	$-0.578367\pi$
$947$	−15.0000			−0.487435			−0.243717	+	0.969846i	$0.578367\pi$
$948$		0			0
$949$	−20.0000			−0.649227
$950$		0			0
$951$		−	46.9574i		−	1.52270i
$952$		0			0
$953$		−	9.89949i		−	0.320676i	−0.987062	−	0.160338i	$-0.948742\pi$
$953$		−	9.89949i		−	0.320676i	0.987062	−	0.160338i	$-0.0512584\pi$
$954$		0			0
$955$			6.70820i			0.217072i
$956$		0			0
$957$	9.48683	−	4.47214i	0.306666	−	0.144564i
$958$			26.8328i			0.866929i
$959$	23.7171	−	31.8198i	0.765865	−	1.02752i
$960$	−40.0000			−1.29099
$961$	−14.0000			−0.451613
$962$			8.94427i			0.288375i
$963$		−	5.65685i		−	0.182290i
$964$		0			0
$965$	9.48683			0.305392
$966$	−15.0000	+	20.1246i	−0.482617	+	0.647499i
$967$			46.6690i			1.50078i	0.660998	+	0.750388i	$0.270133\pi$
$967$			46.6690i			1.50078i	−0.660998	+	0.750388i	$0.729867\pi$
$968$	−24.0000	−	19.7990i	−0.771389	−	0.636364i
$969$		0			0
$970$		−	21.2132i		−	0.681115i
$971$			29.0689i			0.932865i	0.884557	+	0.466432i	$0.154461\pi$
$971$			29.0689i			0.932865i	−0.884557	+	0.466432i	$0.845539\pi$
$972$		0			0
$973$	20.0000	−	26.8328i	0.641171	−	0.860221i
$974$		−	41.0122i		−	1.31412i
$975$		0			0
$976$	12.6491			0.404888
$977$	−21.0000			−0.671850			−0.335925	−	0.941889i	$-0.609049\pi$
$977$	−21.0000			−0.671850			−0.335925	+	0.941889i	$0.609049\pi$
$978$	31.6228			1.01118
$979$	−3.16228	−	6.70820i	−0.101067	−	0.214395i
$980$		0			0
$981$			25.4558i			0.812743i
$982$	46.0000			1.46792
$983$			29.0689i			0.927153i	0.886057	+	0.463577i	$0.153434\pi$
$983$			29.0689i			0.927153i	−0.886057	+	0.463577i	$0.846566\pi$
$984$	60.0000			1.91273
$985$	12.6491			0.403034
$986$		0			0
$987$	15.8114	−	21.2132i	0.503282	−	0.675224i
$988$		0			0
$989$		−	12.7279i		−	0.404724i
$990$	18.9737	−	8.94427i	0.603023	−	0.284268i
$991$	−46.0000			−1.46124			−0.730619	−	0.682785i	$-0.760768\pi$
$991$	−46.0000			−1.46124			−0.730619	+	0.682785i	$0.760768\pi$
$992$		0			0
$993$			42.4853i			1.34823i
$994$	27.0000	+	20.1246i	0.856388	+	0.638314i
$995$		0			0
$996$		0			0
$997$	53.7587			1.70256			0.851278	−	0.524715i	$-0.175828\pi$
$997$	53.7587			1.70256			0.851278	+	0.524715i	$0.175828\pi$
$998$			22.6274i			0.716258i
$999$			2.23607i			0.0707461i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.2.b.b.76.1	✓	4
3.2	odd	2		693.2.c.b.307.3		4
4.3	odd	2		1232.2.e.c.769.4		4
7.2	even	3		539.2.i.b.472.2		8
7.3	odd	6		539.2.i.b.362.4		8
7.4	even	3		539.2.i.b.362.3		8
7.5	odd	6		539.2.i.b.472.1		8
7.6	odd	2	inner	77.2.b.b.76.2	yes	4
11.2	odd	10		847.2.l.g.524.3		16
11.3	even	5		847.2.l.g.475.2		16
11.4	even	5		847.2.l.g.699.3		16
11.5	even	5		847.2.l.g.118.4		16
11.6	odd	10		847.2.l.g.118.2		16
11.7	odd	10		847.2.l.g.699.1		16
11.8	odd	10		847.2.l.g.475.4		16
11.9	even	5		847.2.l.g.524.1		16
11.10	odd	2	inner	77.2.b.b.76.3	yes	4
21.20	even	2		693.2.c.b.307.4		4
28.27	even	2		1232.2.e.c.769.1		4
33.32	even	2		693.2.c.b.307.1		4
44.43	even	2		1232.2.e.c.769.3		4
77.6	even	10		847.2.l.g.118.1		16
77.10	even	6		539.2.i.b.362.2		8
77.13	even	10		847.2.l.g.524.4		16
77.20	odd	10		847.2.l.g.524.2		16
77.27	odd	10		847.2.l.g.118.3		16
77.32	odd	6		539.2.i.b.362.1		8
77.41	even	10		847.2.l.g.475.3		16
77.48	odd	10		847.2.l.g.699.4		16
77.54	even	6		539.2.i.b.472.3		8
77.62	even	10		847.2.l.g.699.2		16
77.65	odd	6		539.2.i.b.472.4		8
77.69	odd	10		847.2.l.g.475.1		16
77.76	even	2	inner	77.2.b.b.76.4	yes	4
231.230	odd	2		693.2.c.b.307.2		4
308.307	odd	2		1232.2.e.c.769.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.b.b.76.1	✓	4	1.1	even	1	trivial
77.2.b.b.76.2	yes	4	7.6	odd	2	inner
77.2.b.b.76.3	yes	4	11.10	odd	2	inner
77.2.b.b.76.4	yes	4	77.76	even	2	inner
539.2.i.b.362.1		8	77.32	odd	6
539.2.i.b.362.2		8	77.10	even	6
539.2.i.b.362.3		8	7.4	even	3
539.2.i.b.362.4		8	7.3	odd	6
539.2.i.b.472.1		8	7.5	odd	6
539.2.i.b.472.2		8	7.2	even	3
539.2.i.b.472.3		8	77.54	even	6
539.2.i.b.472.4		8	77.65	odd	6
693.2.c.b.307.1		4	33.32	even	2
693.2.c.b.307.2		4	231.230	odd	2
693.2.c.b.307.3		4	3.2	odd	2
693.2.c.b.307.4		4	21.20	even	2
847.2.l.g.118.1		16	77.6	even	10
847.2.l.g.118.2		16	11.6	odd	10
847.2.l.g.118.3		16	77.27	odd	10
847.2.l.g.118.4		16	11.5	even	5
847.2.l.g.475.1		16	77.69	odd	10
847.2.l.g.475.2		16	11.3	even	5
847.2.l.g.475.3		16	77.41	even	10
847.2.l.g.475.4		16	11.8	odd	10
847.2.l.g.524.1		16	11.9	even	5
847.2.l.g.524.2		16	77.20	odd	10
847.2.l.g.524.3		16	11.2	odd	10
847.2.l.g.524.4		16	77.13	even	10
847.2.l.g.699.1		16	11.7	odd	10
847.2.l.g.699.2		16	77.62	even	10
847.2.l.g.699.3		16	11.4	even	5
847.2.l.g.699.4		16	77.48	odd	10
1232.2.e.c.769.1		4	28.27	even	2
1232.2.e.c.769.2		4	308.307	odd	2
1232.2.e.c.769.3		4	44.43	even	2
1232.2.e.c.769.4		4	4.3	odd	2

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:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	77.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.23607i q^{3} +2.23607i q^{5} -3.16228 q^{6} +(-1.58114 + 2.12132i) q^{7} -2.82843i q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} -2.23607i q^{3} +2.23607i q^{5} -3.16228 q^{6} +(-1.58114 + 2.12132i) q^{7} -2.82843i q^{8} -2.00000 q^{9} +3.16228 q^{10} +(-3.00000 + 1.41421i) q^{11} +6.32456 q^{13} +(3.00000 + 2.23607i) q^{14} +5.00000 q^{15} -4.00000 q^{16} +2.82843i q^{18} -3.16228 q^{19} +(4.74342 + 3.53553i) q^{21} +(2.00000 + 4.24264i) q^{22} -3.00000 q^{23} -6.32456 q^{24} -8.94427i q^{26} -2.23607i q^{27} -1.41421i q^{29} -7.07107i q^{30} +6.70820i q^{31} +(3.16228 + 6.70820i) q^{33} +(-4.74342 - 3.53553i) q^{35} -1.00000 q^{37} +4.47214i q^{38} -14.1421i q^{39} +6.32456 q^{40} -9.48683 q^{41} +(5.00000 - 6.70820i) q^{42} +4.24264i q^{43} -4.47214i q^{45} +4.24264i q^{46} -4.47214i q^{47} +8.94427i q^{48} +(-2.00000 - 6.70820i) q^{49} -3.16228 q^{54} +(-3.16228 - 6.70820i) q^{55} +(6.00000 + 4.47214i) q^{56} +7.07107i q^{57} -2.00000 q^{58} +2.23607i q^{59} -3.16228 q^{61} +9.48683 q^{62} +(3.16228 - 4.24264i) q^{63} -8.00000 q^{64} +14.1421i q^{65} +(9.48683 - 4.47214i) q^{66} +11.0000 q^{67} +6.70820i q^{69} +(-5.00000 + 6.70820i) q^{70} +9.00000 q^{71} +5.65685i q^{72} -3.16228 q^{73} +1.41421i q^{74} +(1.74342 - 8.60003i) q^{77} -20.0000 q^{78} -8.48528i q^{79} -8.94427i q^{80} -11.0000 q^{81} +13.4164i q^{82} +9.48683 q^{83} +6.00000 q^{86} -3.16228 q^{87} +(4.00000 + 8.48528i) q^{88} +2.23607i q^{89} -6.32456 q^{90} +(-10.0000 + 13.4164i) q^{91} +15.0000 q^{93} -6.32456 q^{94} -7.07107i q^{95} -6.70820i q^{97} +(-9.48683 + 2.82843i) q^{98} +(6.00000 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{9}+O(q^{10}) \) 4 * q - 8 * q^9	\( 4 q - 8 q^{9} - 12 q^{11} + 12 q^{14} + 20 q^{15} - 16 q^{16} + 8 q^{22} - 12 q^{23} - 4 q^{37} + 20 q^{42} - 8 q^{49} + 24 q^{56} - 8 q^{58} - 32 q^{64} + 44 q^{67} - 20 q^{70} + 36 q^{71} - 12 q^{77} - 80 q^{78} - 44 q^{81} + 24 q^{86} + 16 q^{88} - 40 q^{91} + 60 q^{93} + 24 q^{99}+O(q^{100}) \) 4 * q - 8 * q^9 - 12 * q^11 + 12 * q^14 + 20 * q^15 - 16 * q^16 + 8 * q^22 - 12 * q^23 - 4 * q^37 + 20 * q^42 - 8 * q^49 + 24 * q^56 - 8 * q^58 - 32 * q^64 + 44 * q^67 - 20 * q^70 + 36 * q^71 - 12 * q^77 - 80 * q^78 - 44 * q^81 + 24 * q^86 + 16 * q^88 - 40 * q^91 + 60 * q^93 + 24 * q^99

Introduction

L-functions

Modular forms

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Database

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