LMFDB - Embedded newform 76.9.c.b.37.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,9,Mod(37,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("76.37");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	76.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$30.9607743646$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 62471 x^{10} + 1417247091 x^{8} + 14731726590693 x^{6} + \cdots + 18\!\cdots\!00 $ x^12 + 62471x^10 + 1417247091x^8 + 14731726590693x^6 + 76361585248208952x^4 + 190911088284097639680*x^2 + 180929129574476434636800
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{4}\cdot 7 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.3
Root			$-76.7657i$ of defining polynomial
Character	$\chi$	$=$	76.37
Dual form			76.9.c.b.37.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-76.7657i q^{3} +510.215 q^{5} -1605.51 q^{7} +668.032 q^{9} +O(q^{10})$	$q-76.7657i q^{3} +510.215 q^{5} -1605.51 q^{7} +668.032 q^{9} +20628.0 q^{11} -8554.72i q^{13} -39167.0i q^{15} +78953.8 q^{17} +(30314.2 + 126746. i) q^{19} +123248. i q^{21} +48328.5 q^{23} -130306. q^{25} -554941. i q^{27} -1.17696e6i q^{29} +623808. i q^{31} -1.58352e6i q^{33} -819153. q^{35} -389526. i q^{37} -656709. q^{39} -4.04171e6i q^{41} +1.49327e6 q^{43} +340840. q^{45} +6.08681e6 q^{47} -3.18715e6 q^{49} -6.06094e6i q^{51} -4.25475e6i q^{53} +1.05247e7 q^{55} +(9.72976e6 - 2.32709e6i) q^{57} -2.34554e7i q^{59} -7.36127e6 q^{61} -1.07253e6 q^{63} -4.36475e6i q^{65} -4.78715e6i q^{67} -3.70997e6i q^{69} +2.18483e7i q^{71} -2.48363e7 q^{73} +1.00030e7i q^{75} -3.31184e7 q^{77} +2.31813e7i q^{79} -3.82175e7 q^{81} +7.59265e7 q^{83} +4.02834e7 q^{85} -9.03504e7 q^{87} -1.82086e7i q^{89} +1.37347e7i q^{91} +4.78871e7 q^{93} +(1.54667e7 + 6.46678e7i) q^{95} +1.38727e8i q^{97} +1.37802e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 568 q^{5} + 1734 q^{7} - 46210 q^{9}+O(q^{10}) $ 12 * q - 568 * q^5 + 1734 * q^7 - 46210 * q^9	$ 12 q - 568 q^{5} + 1734 q^{7} - 46210 q^{9} + 1976 q^{11} - 47446 q^{17} - 113178 q^{19} + 378950 q^{23} - 502656 q^{25} - 1876832 q^{35} - 1463394 q^{39} + 6234396 q^{43} - 507200 q^{45} + 4902224 q^{47} - 16736298 q^{49} - 38366664 q^{55} - 19396170 q^{57} + 16602912 q^{61} - 11274020 q^{63} + 111268614 q^{73} - 134425928 q^{77} + 193590584 q^{81} + 34030388 q^{83} + 387605748 q^{85} - 217889946 q^{87} - 474880524 q^{93} - 256537036 q^{95} + 276136420 q^{99}+O(q^{100}) $ 12 * q - 568 * q^5 + 1734 * q^7 - 46210 * q^9 + 1976 * q^11 - 47446 * q^17 - 113178 * q^19 + 378950 * q^23 - 502656 * q^25 - 1876832 * q^35 - 1463394 * q^39 + 6234396 * q^43 - 507200 * q^45 + 4902224 * q^47 - 16736298 * q^49 - 38366664 * q^55 - 19396170 * q^57 + 16602912 * q^61 - 11274020 * q^63 + 111268614 * q^73 - 134425928 * q^77 + 193590584 * q^81 + 34030388 * q^83 + 387605748 * q^85 - 217889946 * q^87 - 474880524 * q^93 - 256537036 * q^95 + 276136420 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	76.7657i		−	0.947724i	−0.880599	−	0.473862i	$-0.842859\pi$
$3$		−	76.7657i		−	0.947724i	0.880599	−	0.473862i	$-0.157141\pi$
$4$		0			0
$5$	510.215			0.816344			0.408172	−	0.912905i	$-0.366166\pi$
$5$	510.215			0.816344			0.408172	+	0.912905i	$0.366166\pi$
$6$		0			0
$7$	−1605.51			−0.668682			−0.334341	−	0.942452i	$-0.608514\pi$
$7$	−1605.51			−0.668682			−0.334341	+	0.942452i	$0.608514\pi$
$8$		0			0
$9$	668.032			0.101819
$10$		0			0
$11$	20628.0			1.40892			0.704460	−	0.709743i	$-0.251189\pi$
$11$	20628.0			1.40892			0.704460	+	0.709743i	$0.251189\pi$
$12$		0			0
$13$		−	8554.72i		−	0.299525i	−0.988722	−	0.149762i	$-0.952149\pi$
$13$		−	8554.72i		−	0.299525i	0.988722	−	0.149762i	$-0.0478508\pi$
$14$		0			0
$15$		−	39167.0i		−	0.773669i
$16$		0			0
$17$	78953.8			0.945317			0.472659	−	0.881246i	$-0.343294\pi$
$17$	78953.8			0.945317			0.472659	+	0.881246i	$0.343294\pi$
$18$		0			0
$19$	30314.2	+	126746.i	0.232611	+	0.972570i
$19$	30314.2	+	126746.i	0.232611	+	0.972570i
$20$		0			0
$21$			123248.i			0.633726i
$22$		0			0
$23$	48328.5			0.172700			0.0863500	−	0.996265i	$-0.472480\pi$
$23$	48328.5			0.172700			0.0863500	+	0.996265i	$0.472480\pi$
$24$		0			0
$25$	−130306.			−0.333582
$26$		0			0
$27$		−	554941.i		−	1.04422i
$28$		0			0
$29$		−	1.17696e6i		−	1.66407i	−0.554725	−	0.832034i	$-0.687176\pi$
$29$		−	1.17696e6i		−	1.66407i	0.554725	−	0.832034i	$-0.312824\pi$
$30$		0			0
$31$			623808.i			0.675467i	0.941242	+	0.337734i	$0.109660\pi$
$31$			623808.i			0.675467i	−0.941242	+	0.337734i	$0.890340\pi$
$32$		0			0
$33$		−	1.58352e6i		−	1.33527i
$34$		0			0
$35$	−819153.			−0.545875
$36$		0			0
$37$		−	389526.i		−	0.207840i	−0.994586	−	0.103920i	$-0.966861\pi$
$37$		−	389526.i		−	0.207840i	0.994586	−	0.103920i	$-0.0331387\pi$
$38$		0			0
$39$	−656709.			−0.283867
$40$		0			0
$41$		−	4.04171e6i		−	1.43031i	−0.698967	−	0.715154i	$-0.746357\pi$
$41$		−	4.04171e6i		−	1.43031i	0.698967	−	0.715154i	$-0.253643\pi$
$42$		0			0
$43$	1.49327e6			0.436782			0.218391	−	0.975861i	$-0.429919\pi$
$43$	1.49327e6			0.436782			0.218391	+	0.975861i	$0.429919\pi$
$44$		0			0
$45$	340840.			0.0831190
$46$		0			0
$47$	6.08681e6			1.24738			0.623689	−	0.781673i	$-0.285633\pi$
$47$	6.08681e6			1.24738			0.623689	+	0.781673i	$0.285633\pi$
$48$		0			0
$49$	−3.18715e6			−0.552864
$50$		0			0
$51$		−	6.06094e6i		−	0.895900i
$52$		0			0
$53$		−	4.25475e6i		−	0.539226i	−0.962969	−	0.269613i	$-0.913104\pi$
$53$		−	4.25475e6i		−	0.539226i	0.962969	−	0.269613i	$-0.0868958\pi$
$54$		0			0
$55$	1.05247e7			1.15016
$56$		0			0
$57$	9.72976e6	−	2.32709e6i	0.921728	−	0.220452i
$58$		0			0
$59$		−	2.34554e7i		−	1.93568i	−0.251561	−	0.967841i	$-0.580944\pi$
$59$		−	2.34554e7i		−	1.93568i	0.251561	−	0.967841i	$-0.419056\pi$
$60$		0			0
$61$	−7.36127e6			−0.531659			−0.265830	−	0.964020i	$-0.585646\pi$
$61$	−7.36127e6			−0.531659			−0.265830	+	0.964020i	$0.585646\pi$
$62$		0			0
$63$	−1.07253e6			−0.0680843
$64$		0			0
$65$		−	4.36475e6i		−	0.244515i
$66$		0			0
$67$		−	4.78715e6i		−	0.237563i	−0.992920	−	0.118781i	$-0.962101\pi$
$67$		−	4.78715e6i		−	0.237563i	0.992920	−	0.118781i	$-0.0378987\pi$
$68$		0			0
$69$		−	3.70997e6i		−	0.163672i
$70$		0			0
$71$			2.18483e7i			0.859773i	0.902883	+	0.429886i	$0.141446\pi$
$71$			2.18483e7i			0.859773i	−0.902883	+	0.429886i	$0.858554\pi$
$72$		0			0
$73$	−2.48363e7			−0.874573			−0.437286	−	0.899322i	$-0.644060\pi$
$73$	−2.48363e7			−0.874573			−0.437286	+	0.899322i	$0.644060\pi$
$74$		0			0
$75$			1.00030e7i			0.316144i
$76$		0			0
$77$	−3.31184e7			−0.942120
$78$		0			0
$79$			2.31813e7i			0.595154i	0.954698	+	0.297577i	$0.0961785\pi$
$79$			2.31813e7i			0.595154i	−0.954698	+	0.297577i	$0.903821\pi$
$80$		0			0
$81$	−3.82175e7			−0.887814
$82$		0			0
$83$	7.59265e7			1.59986			0.799928	−	0.600096i	$-0.204871\pi$
$83$	7.59265e7			1.59986			0.799928	+	0.600096i	$0.204871\pi$
$84$		0			0
$85$	4.02834e7			0.771704
$86$		0			0
$87$	−9.03504e7			−1.57708
$88$		0			0
$89$		−	1.82086e7i		−	0.290213i	−0.989416	−	0.145107i	$-0.953648\pi$
$89$		−	1.82086e7i		−	0.290213i	0.989416	−	0.145107i	$-0.0463525\pi$
$90$		0			0
$91$			1.37347e7i			0.200287i
$92$		0			0
$93$	4.78871e7			0.640157
$94$		0			0
$95$	1.54667e7	+	6.46678e7i	0.189891	+	0.793952i
$96$		0			0
$97$			1.38727e8i			1.56701i	0.621383	+	0.783507i	$0.286571\pi$
$97$			1.38727e8i			1.56701i	−0.621383	+	0.783507i	$0.713429\pi$
$98$		0			0
$99$	1.37802e7			0.143454
$100$		0			0
$101$	1.66194e7			0.159709			0.0798546	−	0.996807i	$-0.474554\pi$
$101$	1.66194e7			0.159709			0.0798546	+	0.996807i	$0.474554\pi$
$102$		0			0
$103$			7.55699e7i			0.671429i	0.941964	+	0.335714i	$0.108978\pi$
$103$			7.55699e7i			0.671429i	−0.941964	+	0.335714i	$0.891022\pi$
$104$		0			0
$105$			6.28828e7i			0.517339i
$106$		0			0
$107$			9.05527e7i			0.690822i	0.938452	+	0.345411i	$0.112260\pi$
$107$			9.05527e7i			0.690822i	−0.938452	+	0.345411i	$0.887740\pi$
$108$		0			0
$109$		−	6.12370e7i		−	0.433819i	−0.976192	−	0.216909i	$-0.930402\pi$
$109$		−	6.12370e7i		−	0.433819i	0.976192	−	0.216909i	$-0.0695976\pi$
$110$		0			0
$111$	−2.99023e7			−0.196975
$112$		0			0
$113$			2.53259e7i			0.155328i	0.996980	+	0.0776642i	$0.0247462\pi$
$113$			2.53259e7i			0.155328i	−0.996980	+	0.0776642i	$0.975254\pi$
$114$		0			0
$115$	2.46579e7			0.140983
$116$		0			0
$117$		−	5.71483e6i		−	0.0304972i
$118$		0			0
$119$	−1.26761e8			−0.632117
$120$		0			0
$121$	2.11156e8			0.985058
$122$		0			0
$123$	−3.10264e8			−1.35554
$124$		0			0
$125$	−2.65787e8			−1.08866
$126$		0			0
$127$			4.33998e8i			1.66830i	0.551541	+	0.834148i	$0.314040\pi$
$127$			4.33998e8i			1.66830i	−0.551541	+	0.834148i	$0.685960\pi$
$128$		0			0
$129$		−	1.14632e8i		−	0.413949i
$130$		0			0
$131$	2.34373e8			0.795832			0.397916	−	0.917422i	$-0.369734\pi$
$131$	2.34373e8			0.795832			0.397916	+	0.917422i	$0.369734\pi$
$132$		0			0
$133$	−4.86695e7	−	2.03492e8i	−0.155543	−	0.650340i
$134$		0			0
$135$		−	2.83139e8i		−	0.852443i
$136$		0			0
$137$	2.00110e8			0.568050			0.284025	−	0.958817i	$-0.408330\pi$
$137$	2.00110e8			0.568050			0.284025	+	0.958817i	$0.408330\pi$
$138$		0			0
$139$	−3.88159e8			−1.03980			−0.519900	−	0.854227i	$-0.674031\pi$
$139$	−3.88159e8			−1.03980			−0.519900	+	0.854227i	$0.674031\pi$
$140$		0			0
$141$		−	4.67258e8i		−	1.18217i
$142$		0			0
$143$		−	1.76467e8i		−	0.422006i
$144$		0			0
$145$		−	6.00505e8i		−	1.35845i
$146$		0			0
$147$			2.44664e8i			0.523963i
$148$		0			0
$149$	8.60288e8			1.74541			0.872707	−	0.488244i	$-0.162362\pi$
$149$	8.60288e8			1.74541			0.872707	+	0.488244i	$0.162362\pi$
$150$		0			0
$151$		−	3.30372e8i		−	0.635471i	−0.948179	−	0.317736i	$-0.897078\pi$
$151$		−	3.30372e8i		−	0.635471i	0.948179	−	0.317736i	$-0.102922\pi$
$152$		0			0
$153$	5.27437e7			0.0962509
$154$		0			0
$155$			3.18276e8i			0.551414i
$156$		0			0
$157$	4.15204e8			0.683380			0.341690	−	0.939813i	$-0.389001\pi$
$157$	4.15204e8			0.683380			0.341690	+	0.939813i	$0.389001\pi$
$158$		0			0
$159$	−3.26619e8			−0.511038
$160$		0			0
$161$	−7.75917e7			−0.115481
$162$		0			0
$163$	4.65528e8			0.659471			0.329735	−	0.944073i	$-0.393040\pi$
$163$	4.65528e8			0.659471			0.329735	+	0.944073i	$0.393040\pi$
$164$		0			0
$165$		−	8.07937e8i		−	1.09004i
$166$		0			0
$167$		−	3.24103e7i		−	0.0416693i	−0.999783	−	0.0208347i	$-0.993368\pi$
$167$		−	3.24103e7i		−	0.0416693i	0.999783	−	0.0208347i	$-0.00663236\pi$
$168$		0			0
$169$	7.42547e8			0.910285
$170$		0			0
$171$	2.02508e7	+	8.46705e7i	0.0236842	+	0.0990257i
$172$		0			0
$173$		−	7.58338e7i		−	0.0846600i	−0.999104	−	0.0423300i	$-0.986522\pi$
$173$		−	7.58338e7i		−	0.0846600i	0.999104	−	0.0423300i	$-0.0134781\pi$
$174$		0			0
$175$	2.09206e8			0.223061
$176$		0			0
$177$	−1.80057e9			−1.83449
$178$		0			0
$179$		−	2.69607e8i		−	0.262615i	−0.991342	−	0.131307i	$-0.958083\pi$
$179$		−	2.69607e8i		−	0.262615i	0.991342	−	0.131307i	$-0.0419175\pi$
$180$		0			0
$181$		−	1.29574e9i		−	1.20727i	−0.797261	−	0.603635i	$-0.793718\pi$
$181$		−	1.29574e9i		−	1.20727i	0.797261	−	0.603635i	$-0.206282\pi$
$182$		0			0
$183$			5.65093e8i			0.503866i
$184$		0			0
$185$		−	1.98742e8i		−	0.169669i
$186$		0			0
$187$	1.62866e9			1.33188
$188$		0			0
$189$			8.90962e8i			0.698251i
$190$		0			0
$191$	−2.69213e8			−0.202284			−0.101142	−	0.994872i	$-0.532250\pi$
$191$	−2.69213e8			−0.202284			−0.101142	+	0.994872i	$0.532250\pi$
$192$		0			0
$193$		−	8.49964e8i		−	0.612592i	−0.951936	−	0.306296i	$-0.900910\pi$
$193$		−	8.49964e8i		−	0.612592i	0.951936	−	0.306296i	$-0.0990897\pi$
$194$		0			0
$195$	−3.35063e8			−0.231733
$196$		0			0
$197$	−1.33835e9			−0.888599			−0.444300	−	0.895878i	$-0.646547\pi$
$197$	−1.33835e9			−0.888599			−0.444300	+	0.895878i	$0.646547\pi$
$198$		0			0
$199$	−2.49225e9			−1.58920			−0.794601	−	0.607133i	$-0.792320\pi$
$199$	−2.49225e9			−1.58920			−0.794601	+	0.607133i	$0.792320\pi$
$200$		0			0
$201$	−3.67489e8			−0.225144
$202$		0			0
$203$			1.88962e9i			1.11273i
$204$		0			0
$205$		−	2.06214e9i		−	1.16762i
$206$		0			0
$207$	3.22850e7			0.0175841
$208$		0			0
$209$	6.25321e8	+	2.61452e9i	0.327731	+	1.37027i
$210$		0			0
$211$			1.64455e9i			0.829695i	0.909891	+	0.414847i	$0.136165\pi$
$211$			1.64455e9i			0.829695i	−0.909891	+	0.414847i	$0.863835\pi$
$212$		0			0
$213$	1.67720e9			0.814828
$214$		0			0
$215$	7.61889e8			0.356564
$216$		0			0
$217$		−	1.00153e9i		−	0.451673i
$218$		0			0
$219$			1.90658e9i			0.828854i
$220$		0			0
$221$		−	6.75428e8i		−	0.283146i
$222$		0			0
$223$			4.88374e9i			1.97485i	0.158103	+	0.987423i	$0.449462\pi$
$223$			4.88374e9i			1.97485i	−0.158103	+	0.987423i	$0.550538\pi$
$224$		0			0
$225$	−8.70483e7			−0.0339649
$226$		0			0
$227$			1.70581e9i			0.642430i	0.947006	+	0.321215i	$0.104091\pi$
$227$			1.70581e9i			0.642430i	−0.947006	+	0.321215i	$0.895909\pi$
$228$		0			0
$229$	−3.49102e9			−1.26944			−0.634718	−	0.772744i	$-0.718884\pi$
$229$	−3.49102e9			−1.26944			−0.634718	+	0.772744i	$0.718884\pi$
$230$		0			0
$231$			2.54236e9i			0.892870i
$232$		0			0
$233$	1.04122e9			0.353279			0.176639	−	0.984276i	$-0.443477\pi$
$233$	1.04122e9			0.353279			0.176639	+	0.984276i	$0.443477\pi$
$234$		0			0
$235$	3.10558e9			1.01829
$236$		0			0
$237$	1.77953e9			0.564042
$238$		0			0
$239$	−2.03040e9			−0.622287			−0.311143	−	0.950363i	$-0.600712\pi$
$239$	−2.03040e9			−0.622287			−0.311143	+	0.950363i	$0.600712\pi$
$240$		0			0
$241$			4.07452e9i			1.20784i	0.797046	+	0.603919i	$0.206395\pi$
$241$			4.07452e9i			1.20784i	−0.797046	+	0.603919i	$0.793605\pi$
$242$		0			0
$243$		−	7.07179e8i		−	0.202817i
$244$		0			0
$245$	−1.62613e9			−0.451328
$246$		0			0
$247$	1.08428e9	−	2.59329e8i	0.291309	−	0.0696728i
$248$		0			0
$249$		−	5.82855e9i		−	1.51622i
$250$		0			0
$251$	3.46879e9			0.873944			0.436972	−	0.899475i	$-0.356051\pi$
$251$	3.46879e9			0.873944			0.436972	+	0.899475i	$0.356051\pi$
$252$		0			0
$253$	9.96921e8			0.243321
$254$		0			0
$255$		−	3.09238e9i		−	0.731363i
$256$		0			0
$257$		−	3.77200e9i		−	0.864647i	−0.901719	−	0.432323i	$-0.857694\pi$
$257$		−	3.77200e9i		−	0.864647i	0.901719	−	0.432323i	$-0.142306\pi$
$258$		0			0
$259$			6.25387e8i			0.138979i
$260$		0			0
$261$		−	7.86249e8i		−	0.169433i
$262$		0			0
$263$	−5.06037e9			−1.05769			−0.528846	−	0.848718i	$-0.677375\pi$
$263$	−5.06037e9			−1.05769			−0.528846	+	0.848718i	$0.677375\pi$
$264$		0			0
$265$		−	2.17084e9i		−	0.440194i
$266$		0			0
$267$	−1.39780e9			−0.275042
$268$		0			0
$269$			8.20471e9i			1.56695i	0.621425	+	0.783473i	$0.286554\pi$
$269$			8.20471e9i			1.56695i	−0.621425	+	0.783473i	$0.713446\pi$
$270$		0			0
$271$	−6.84652e9			−1.26938			−0.634692	−	0.772765i	$-0.718873\pi$
$271$	−6.84652e9			−1.26938			−0.634692	+	0.772765i	$0.718873\pi$
$272$		0			0
$273$	1.05435e9			0.189817
$274$		0			0
$275$	−2.68795e9			−0.469991
$276$		0			0
$277$	8.12858e8			0.138069			0.0690344	−	0.997614i	$-0.478008\pi$
$277$	8.12858e8			0.138069			0.0690344	+	0.997614i	$0.478008\pi$
$278$		0			0
$279$			4.16724e8i			0.0687751i
$280$		0			0
$281$		−	2.68412e9i		−	0.430503i	−0.976559	−	0.215251i	$-0.930943\pi$
$281$		−	2.68412e9i		−	0.430503i	0.976559	−	0.215251i	$-0.0690571\pi$
$282$		0			0
$283$	−3.69558e9			−0.576152			−0.288076	−	0.957607i	$-0.593016\pi$
$283$	−3.69558e9			−0.576152			−0.288076	+	0.957607i	$0.593016\pi$
$284$		0			0
$285$	4.96427e9	−	1.18731e9i	0.752447	−	0.179964i
$286$		0			0
$287$			6.48899e9i			0.956421i
$288$		0			0
$289$	−7.42049e8			−0.106375
$290$		0			0
$291$	1.06494e10			1.48510
$292$		0			0
$293$		−	9.52736e9i		−	1.29271i	−0.763035	−	0.646357i	$-0.776292\pi$
$293$		−	9.52736e9i		−	1.29271i	0.763035	−	0.646357i	$-0.223708\pi$
$294$		0			0
$295$		−	1.19673e10i		−	1.58018i
$296$		0			0
$297$		−	1.14473e10i		−	1.47122i
$298$		0			0
$299$		−	4.13437e8i		−	0.0517279i
$300$		0			0
$301$	−2.39745e9			−0.292068
$302$		0			0
$303$		−	1.27580e9i		−	0.151360i
$304$		0			0
$305$	−3.75583e9			−0.434017
$306$		0			0
$307$		−	6.34754e9i		−	0.714582i	−0.933993	−	0.357291i	$-0.883700\pi$
$307$		−	6.34754e9i		−	0.714582i	0.933993	−	0.357291i	$-0.116300\pi$
$308$		0			0
$309$	5.80117e9			0.636329
$310$		0			0
$311$	1.06236e9			0.113562			0.0567808	−	0.998387i	$-0.481916\pi$
$311$	1.06236e9			0.113562			0.0567808	+	0.998387i	$0.481916\pi$
$312$		0			0
$313$	1.74238e10			1.81537			0.907684	−	0.419654i	$-0.137849\pi$
$313$	1.74238e10			1.81537			0.907684	+	0.419654i	$0.137849\pi$
$314$		0			0
$315$	−5.47220e8			−0.0555802
$316$		0			0
$317$			1.55485e10i			1.53975i	0.638192	+	0.769877i	$0.279682\pi$
$317$			1.55485e10i			1.53975i	−0.638192	+	0.769877i	$0.720318\pi$
$318$		0			0
$319$		−	2.42784e10i		−	2.34454i
$320$		0			0
$321$	6.95134e9			0.654709
$322$		0			0
$323$	2.39342e9	+	1.00071e10i	0.219892	+	0.919387i
$324$		0			0
$325$			1.11473e9i			0.0999161i
$326$		0			0
$327$	−4.70090e9			−0.411140
$328$		0			0
$329$	−9.77240e9			−0.834099
$330$		0			0
$331$		−	1.13997e10i		−	0.949686i	−0.880070	−	0.474843i	$-0.842505\pi$
$331$		−	1.13997e10i		−	0.949686i	0.880070	−	0.474843i	$-0.157495\pi$
$332$		0			0
$333$		−	2.60216e8i		−	0.0211620i
$334$		0			0
$335$		−	2.44248e9i		−	0.193933i
$336$		0			0
$337$			1.42106e10i			1.10177i	0.834581	+	0.550886i	$0.185710\pi$
$337$			1.42106e10i			1.10177i	−0.834581	+	0.550886i	$0.814290\pi$
$338$		0			0
$339$	1.94416e9			0.147209
$340$		0			0
$341$			1.28679e10i			0.951680i
$342$		0			0
$343$	1.43724e10			1.03837
$344$		0			0
$345$		−	1.89288e9i		−	0.133613i
$346$		0			0
$347$	3.20839e9			0.221294			0.110647	−	0.993860i	$-0.464708\pi$
$347$	3.20839e9			0.221294			0.110647	+	0.993860i	$0.464708\pi$
$348$		0			0
$349$	−2.86679e10			−1.93239			−0.966193	−	0.257822i	$-0.916995\pi$
$349$	−2.86679e10			−1.93239			−0.966193	+	0.257822i	$0.916995\pi$
$350$		0			0
$351$	−4.74737e9			−0.312770
$352$		0			0
$353$	1.14990e10			0.740564			0.370282	−	0.928919i	$-0.379261\pi$
$353$	1.14990e10			0.740564			0.370282	+	0.928919i	$0.379261\pi$
$354$		0			0
$355$			1.11473e10i			0.701870i
$356$		0			0
$357$			9.73088e9i			0.599072i
$358$		0			0
$359$	−1.88189e10			−1.13297			−0.566483	−	0.824073i	$-0.691696\pi$
$359$	−1.88189e10			−1.13297			−0.566483	+	0.824073i	$0.691696\pi$
$360$		0			0
$361$	−1.51457e10	+	7.68441e9i	−0.891784	+	0.452462i
$362$		0			0
$363$		−	1.62095e10i		−	0.933564i
$364$		0			0
$365$	−1.26719e10			−0.713952
$366$		0			0
$367$	5.48640e9			0.302429			0.151214	−	0.988501i	$-0.451682\pi$
$367$	5.48640e9			0.302429			0.151214	+	0.988501i	$0.451682\pi$
$368$		0			0
$369$		−	2.69999e9i		−	0.145632i
$370$		0			0
$371$			6.83103e9i			0.360571i
$372$		0			0
$373$			1.07376e10i			0.554717i	0.960766	+	0.277359i	$0.0894590\pi$
$373$			1.07376e10i			0.554717i	−0.960766	+	0.277359i	$0.910541\pi$
$374$		0			0
$375$			2.04033e10i			1.03175i
$376$		0			0
$377$	−1.00686e10			−0.498429
$378$		0			0
$379$		−	1.00275e10i		−	0.485999i	−0.970026	−	0.242999i	$-0.921869\pi$
$379$		−	1.00275e10i		−	0.485999i	0.970026	−	0.242999i	$-0.0781313\pi$
$380$		0			0
$381$	3.33162e10			1.58108
$382$		0			0
$383$			2.60628e10i			1.21123i	0.795759	+	0.605614i	$0.207072\pi$
$383$			2.60628e10i			1.21123i	−0.795759	+	0.605614i	$0.792928\pi$
$384$		0			0
$385$	−1.68975e10			−0.769094
$386$		0			0
$387$	9.97552e8			0.0444725
$388$		0			0
$389$	−4.62877e9			−0.202147			−0.101074	−	0.994879i	$-0.532228\pi$
$389$	−4.62877e9			−0.202147			−0.101074	+	0.994879i	$0.532228\pi$
$390$		0			0
$391$	3.81572e9			0.163256
$392$		0			0
$393$		−	1.79918e10i		−	0.754229i
$394$		0			0
$395$			1.18275e10i			0.485851i
$396$		0			0
$397$	1.50261e10			0.604900			0.302450	−	0.953165i	$-0.402195\pi$
$397$	1.50261e10			0.604900			0.302450	+	0.953165i	$0.402195\pi$
$398$		0			0
$399$	−1.56212e10	+	3.73615e9i	−0.616343	+	0.147412i
$400$		0			0
$401$			2.39010e10i			0.924354i	0.886788	+	0.462177i	$0.152932\pi$
$401$			2.39010e10i			0.924354i	−0.886788	+	0.462177i	$0.847068\pi$
$402$		0			0
$403$	5.33651e9			0.202319
$404$		0			0
$405$	−1.94991e10			−0.724762
$406$		0			0
$407$		−	8.03516e9i		−	0.292831i
$408$		0			0
$409$		−	1.13350e10i		−	0.405070i	−0.979275	−	0.202535i	$-0.935082\pi$
$409$		−	1.13350e10i		−	0.405070i	0.979275	−	0.202535i	$-0.0649180\pi$
$410$		0			0
$411$		−	1.53616e10i		−	0.538355i
$412$		0			0
$413$			3.76577e10i			1.29436i
$414$		0			0
$415$	3.87388e10			1.30603
$416$		0			0
$417$			2.97972e10i			0.985444i
$418$		0			0
$419$	4.27255e10			1.38622			0.693108	−	0.720833i	$-0.256241\pi$
$419$	4.27255e10			1.38622			0.693108	+	0.720833i	$0.256241\pi$
$420$		0			0
$421$			2.30587e10i			0.734018i	0.930217	+	0.367009i	$0.119618\pi$
$421$			2.30587e10i			0.734018i	−0.930217	+	0.367009i	$0.880382\pi$
$422$		0			0
$423$	4.06618e9			0.127006
$424$		0			0
$425$	−1.02881e10			−0.315341
$426$		0			0
$427$	1.18186e10			0.355511
$428$		0			0
$429$	−1.35466e10			−0.399946
$430$		0			0
$431$			3.89542e10i			1.12887i	0.825476	+	0.564437i	$0.190907\pi$
$431$			3.89542e10i			1.12887i	−0.825476	+	0.564437i	$0.809093\pi$
$432$		0			0
$433$		−	3.87503e10i		−	1.10236i	−0.834386	−	0.551180i	$-0.814178\pi$
$433$		−	3.87503e10i		−	1.10236i	0.834386	−	0.551180i	$-0.185822\pi$
$434$		0			0
$435$	−4.60981e10			−1.28744
$436$		0			0
$437$	1.46504e9	+	6.12546e9i	0.0401720	+	0.167963i
$438$		0			0
$439$		−	2.73862e10i		−	0.737350i	−0.929558	−	0.368675i	$-0.879812\pi$
$439$		−	2.73862e10i		−	0.737350i	0.929558	−	0.368675i	$-0.120188\pi$
$440$		0			0
$441$	−2.12912e9			−0.0562919
$442$		0			0
$443$	2.70145e10			0.701426			0.350713	−	0.936483i	$-0.385939\pi$
$443$	2.70145e10			0.701426			0.350713	+	0.936483i	$0.385939\pi$
$444$		0			0
$445$		−	9.29031e9i		−	0.236914i
$446$		0			0
$447$		−	6.60406e10i		−	1.65417i
$448$		0			0
$449$			3.01192e10i			0.741067i	0.928819	+	0.370534i	$0.120825\pi$
$449$			3.01192e10i			0.741067i	−0.928819	+	0.370534i	$0.879175\pi$
$450$		0			0
$451$		−	8.33724e10i		−	2.01519i
$452$		0			0
$453$	−2.53613e10			−0.602252
$454$		0			0
$455$			7.00763e9i			0.163503i
$456$		0			0
$457$	1.10301e10			0.252881			0.126440	−	0.991974i	$-0.459645\pi$
$457$	1.10301e10			0.252881			0.126440	+	0.991974i	$0.459645\pi$
$458$		0			0
$459$		−	4.38148e10i		−	0.987119i
$460$		0			0
$461$	−2.55069e10			−0.564746			−0.282373	−	0.959305i	$-0.591122\pi$
$461$	−2.55069e10			−0.564746			−0.282373	+	0.959305i	$0.591122\pi$
$462$		0			0
$463$	2.66774e10			0.580523			0.290262	−	0.956947i	$-0.406258\pi$
$463$	2.66774e10			0.580523			0.290262	+	0.956947i	$0.406258\pi$
$464$		0			0
$465$	2.44327e10			0.522588
$466$		0			0
$467$	−6.46264e10			−1.35876			−0.679379	−	0.733788i	$-0.737751\pi$
$467$	−6.46264e10			−1.35876			−0.679379	+	0.733788i	$0.737751\pi$
$468$		0			0
$469$			7.68580e9i			0.158854i
$470$		0			0
$471$		−	3.18734e10i		−	0.647656i
$472$		0			0
$473$	3.08032e10			0.615391
$474$		0			0
$475$	−3.95010e9	−	1.65158e10i	−0.0775951	−	0.324432i
$476$		0			0
$477$		−	2.84231e9i		−	0.0549032i
$478$		0			0
$479$	−1.72804e10			−0.328255			−0.164127	−	0.986439i	$-0.552481\pi$
$479$	−1.72804e10			−0.328255			−0.164127	+	0.986439i	$0.552481\pi$
$480$		0			0
$481$	−3.33229e9			−0.0622533
$482$		0			0
$483$			5.95638e9i			0.109445i
$484$		0			0
$485$			7.07804e10i			1.27922i
$486$		0			0
$487$		−	5.68154e10i		−	1.01007i	−0.863100	−	0.505034i	$-0.831480\pi$
$487$		−	5.68154e10i		−	1.01007i	0.863100	−	0.505034i	$-0.168520\pi$
$488$		0			0
$489$		−	3.57366e10i		−	0.624996i
$490$		0			0
$491$	1.29539e10			0.222883			0.111441	−	0.993771i	$-0.464453\pi$
$491$	1.29539e10			0.222883			0.111441	+	0.993771i	$0.464453\pi$
$492$		0			0
$493$		−	9.29258e10i		−	1.57307i
$494$		0			0
$495$	7.03085e9			0.117108
$496$		0			0
$497$		−	3.50775e10i		−	0.574915i
$498$		0			0
$499$	−6.54388e10			−1.05544			−0.527719	−	0.849419i	$-0.676953\pi$
$499$	−6.54388e10			−1.05544			−0.527719	+	0.849419i	$0.676953\pi$
$500$		0			0
$501$	−2.48800e9			−0.0394910
$502$		0			0
$503$	3.72073e10			0.581241			0.290620	−	0.956838i	$-0.406138\pi$
$503$	3.72073e10			0.581241			0.290620	+	0.956838i	$0.406138\pi$
$504$		0			0
$505$	8.47947e9			0.130378
$506$		0			0
$507$		−	5.70022e10i		−	0.862699i
$508$		0			0
$509$			2.19572e10i			0.327119i	0.986533	+	0.163559i	$0.0522976\pi$
$509$			2.19572e10i			0.327119i	−0.986533	+	0.163559i	$0.947702\pi$
$510$		0			0
$511$	3.98749e10			0.584811
$512$		0			0
$513$	7.03368e10	−	1.68226e10i	1.01558	−	0.242898i
$514$		0			0
$515$			3.85569e10i			0.548117i
$516$		0			0
$517$	1.25559e11			1.75746
$518$		0			0
$519$	−5.82143e9			−0.0802343
$520$		0			0
$521$			3.60269e10i			0.488962i	0.969654	+	0.244481i	$0.0786176\pi$
$521$			3.60269e10i			0.488962i	−0.969654	+	0.244481i	$0.921382\pi$
$522$		0			0
$523$			1.05558e11i			1.41086i	0.708780	+	0.705430i	$0.249246\pi$
$523$			1.05558e11i			1.41086i	−0.708780	+	0.705430i	$0.750754\pi$
$524$		0			0
$525$		−	1.60599e10i		−	0.211400i
$526$		0			0
$527$			4.92521e10i			0.638531i
$528$		0			0
$529$	−7.59753e10			−0.970175
$530$		0			0
$531$		−	1.56689e10i		−	0.197088i
$532$		0			0
$533$	−3.45757e10			−0.428412
$534$		0			0
$535$			4.62013e10i			0.563948i
$536$		0			0
$537$	−2.06966e10			−0.248886
$538$		0			0
$539$	−6.57446e10			−0.778942
$540$		0			0
$541$	1.13182e11			1.32126			0.660630	−	0.750712i	$-0.270289\pi$
$541$	1.13182e11			1.32126			0.660630	+	0.750712i	$0.270289\pi$
$542$		0			0
$543$	−9.94685e10			−1.14416
$544$		0			0
$545$		−	3.12441e10i		−	0.354145i
$546$		0			0
$547$			1.70701e11i			1.90672i	0.301829	+	0.953362i	$0.402403\pi$
$547$			1.70701e11i			1.90672i	−0.301829	+	0.953362i	$0.597597\pi$
$548$		0			0
$549$	−4.91756e9			−0.0541328
$550$		0			0
$551$	1.49176e11	−	3.56787e10i	1.61842	−	0.387081i
$552$		0			0
$553$		−	3.72177e10i		−	0.397969i
$554$		0			0
$555$	−1.52566e10			−0.160800
$556$		0			0
$557$	−3.21341e10			−0.333845			−0.166923	−	0.985970i	$-0.553383\pi$
$557$	−3.21341e10			−0.333845			−0.166923	+	0.985970i	$0.553383\pi$
$558$		0			0
$559$		−	1.27745e10i		−	0.130827i
$560$		0			0
$561$		−	1.25025e11i		−	1.26225i
$562$		0			0
$563$			1.21729e11i			1.21160i	0.795616	+	0.605801i	$0.207147\pi$
$563$			1.21729e11i			1.21160i	−0.795616	+	0.605801i	$0.792853\pi$
$564$		0			0
$565$			1.29216e10i			0.126801i
$566$		0			0
$567$	6.13584e10			0.593666
$568$		0			0
$569$			4.87665e10i			0.465235i	0.972568	+	0.232617i	$0.0747290\pi$
$569$			4.87665e10i			0.465235i	−0.972568	+	0.232617i	$0.925271\pi$
$570$		0			0
$571$	−1.10443e11			−1.03895			−0.519475	−	0.854486i	$-0.673872\pi$
$571$	−1.10443e11			−1.03895			−0.519475	+	0.854486i	$0.673872\pi$
$572$		0			0
$573$			2.06663e10i			0.191710i
$574$		0			0
$575$	−6.29748e9			−0.0576097
$576$		0			0
$577$	−5.34282e10			−0.482022			−0.241011	−	0.970522i	$-0.577479\pi$
$577$	−5.34282e10			−0.482022			−0.241011	+	0.970522i	$0.577479\pi$
$578$		0			0
$579$	−6.52481e10			−0.580568
$580$		0			0
$581$	−1.21900e11			−1.06980
$582$		0			0
$583$		−	8.77671e10i		−	0.759727i
$584$		0			0
$585$		−	2.91579e9i		−	0.0248962i
$586$		0			0
$587$	−1.12376e11			−0.946498			−0.473249	−	0.880929i	$-0.656919\pi$
$587$	−1.12376e11			−0.946498			−0.473249	+	0.880929i	$0.656919\pi$
$588$		0			0
$589$	−7.90654e10	+	1.89102e10i	−0.656939	+	0.157121i
$590$		0			0
$591$			1.02740e11i			0.842147i
$592$		0			0
$593$	1.98113e11			1.60212			0.801060	−	0.598584i	$-0.204270\pi$
$593$	1.98113e11			1.60212			0.801060	+	0.598584i	$0.204270\pi$
$594$		0			0
$595$	−6.46753e10			−0.516025
$596$		0			0
$597$			1.91319e11i			1.50612i
$598$		0			0
$599$			1.46071e11i			1.13463i	0.823500	+	0.567317i	$0.192018\pi$
$599$			1.46071e11i			1.13463i	−0.823500	+	0.567317i	$0.807982\pi$
$600$		0			0
$601$			2.83242e10i			0.217100i	0.994091	+	0.108550i	$0.0346207\pi$
$601$			2.83242e10i			0.217100i	−0.994091	+	0.108550i	$0.965379\pi$
$602$		0			0
$603$		−	3.19797e9i		−	0.0241883i
$604$		0			0
$605$	1.07735e11			0.804146
$606$		0			0
$607$			1.81302e11i			1.33552i	0.744379	+	0.667758i	$0.232746\pi$
$607$			1.81302e11i			1.33552i	−0.744379	+	0.667758i	$0.767254\pi$
$608$		0			0
$609$	1.45058e11			1.05456
$610$		0			0
$611$		−	5.20709e10i		−	0.373620i
$612$		0			0
$613$	−2.21255e11			−1.56693			−0.783466	−	0.621434i	$-0.786550\pi$
$613$	−2.21255e11			−1.56693			−0.783466	+	0.621434i	$0.786550\pi$
$614$		0			0
$615$	−1.58302e11			−1.10659
$616$		0			0
$617$	−2.38147e11			−1.64326			−0.821628	−	0.570024i	$-0.806934\pi$
$617$	−2.38147e11			−1.64326			−0.821628	+	0.570024i	$0.806934\pi$
$618$		0			0
$619$	1.87142e11			1.27470			0.637352	−	0.770572i	$-0.280030\pi$
$619$	1.87142e11			1.27470			0.637352	+	0.770572i	$0.280030\pi$
$620$		0			0
$621$		−	2.68195e10i		−	0.180337i
$622$		0			0
$623$			2.92340e10i			0.194060i
$624$		0			0
$625$	−8.47077e10			−0.555140
$626$		0			0
$627$	2.00706e11	−	4.80032e10i	1.29864	−	0.310599i
$628$		0			0
$629$		−	3.07546e10i		−	0.196475i
$630$		0			0
$631$	1.16636e11			0.735726			0.367863	−	0.929880i	$-0.380090\pi$
$631$	1.16636e11			0.735726			0.367863	+	0.929880i	$0.380090\pi$
$632$		0			0
$633$	1.26245e11			0.786322
$634$		0			0
$635$			2.21432e11i			1.36190i
$636$		0			0
$637$			2.72652e10i			0.165596i
$638$		0			0
$639$			1.45953e10i			0.0875408i
$640$		0			0
$641$		−	2.45497e11i		−	1.45416i	−0.686551	−	0.727082i	$-0.740876\pi$
$641$		−	2.45497e11i		−	1.45416i	0.686551	−	0.727082i	$-0.259124\pi$
$642$		0			0
$643$	1.48849e11			0.870769			0.435384	−	0.900245i	$-0.356612\pi$
$643$	1.48849e11			0.870769			0.435384	+	0.900245i	$0.356612\pi$
$644$		0			0
$645$		−	5.84869e10i		−	0.337925i
$646$		0			0
$647$	1.74880e11			0.997982			0.498991	−	0.866607i	$-0.333704\pi$
$647$	1.74880e11			0.997982			0.498991	+	0.866607i	$0.333704\pi$
$648$		0			0
$649$		−	4.83838e11i		−	2.72722i
$650$		0			0
$651$	−7.68830e10			−0.428061
$652$		0			0
$653$	−1.32391e11			−0.728123			−0.364061	−	0.931375i	$-0.618610\pi$
$653$	−1.32391e11			−0.728123			−0.364061	+	0.931375i	$0.618610\pi$
$654$		0			0
$655$	1.19580e11			0.649673
$656$		0			0
$657$	−1.65915e10			−0.0890478
$658$		0			0
$659$		−	7.31207e10i		−	0.387702i	−0.981031	−	0.193851i	$-0.937902\pi$
$659$		−	7.31207e10i		−	0.387702i	0.981031	−	0.193851i	$-0.0620979\pi$
$660$		0			0
$661$			2.49067e11i			1.30470i	0.757918	+	0.652350i	$0.226217\pi$
$661$			2.49067e11i			1.30470i	−0.757918	+	0.652350i	$0.773783\pi$
$662$		0			0
$663$	−5.18497e10			−0.268344
$664$		0			0
$665$	−2.48319e10	−	1.03825e11i	−0.126977	−	0.530901i
$666$		0			0
$667$		−	5.68809e10i		−	0.287385i
$668$		0			0
$669$	3.74904e11			1.87161
$670$		0			0
$671$	−1.51848e11			−0.749066
$672$		0			0
$673$		−	1.38164e11i		−	0.673493i	−0.941595	−	0.336747i	$-0.890673\pi$
$673$		−	1.38164e11i		−	0.673493i	0.941595	−	0.336747i	$-0.109327\pi$
$674$		0			0
$675$			7.23120e10i			0.348334i
$676$		0			0
$677$			1.61898e11i			0.770703i	0.922770	+	0.385352i	$0.125920\pi$
$677$			1.61898e11i			0.770703i	−0.922770	+	0.385352i	$0.874080\pi$
$678$		0			0
$679$		−	2.22726e11i		−	1.04783i
$680$		0			0
$681$	1.30947e11			0.608847
$682$		0			0
$683$		−	3.68514e11i		−	1.69344i	−0.532035	−	0.846722i	$-0.678573\pi$
$683$		−	3.68514e11i		−	1.69344i	0.532035	−	0.846722i	$-0.321427\pi$
$684$		0			0
$685$	1.02099e11			0.463724
$686$		0			0
$687$			2.67991e11i			1.20307i
$688$		0			0
$689$	−3.63982e10			−0.161511
$690$		0			0
$691$	−2.28782e11			−1.00348			−0.501742	−	0.865018i	$-0.667307\pi$
$691$	−2.28782e11			−1.00348			−0.501742	+	0.865018i	$0.667307\pi$
$692$		0			0
$693$	−2.21241e10			−0.0959253
$694$		0			0
$695$	−1.98044e11			−0.848835
$696$		0			0
$697$		−	3.19108e11i		−	1.35209i
$698$		0			0
$699$		−	7.99297e10i		−	0.334811i
$700$		0			0
$701$	−1.07947e11			−0.447031			−0.223516	−	0.974700i	$-0.571753\pi$
$701$	−1.07947e11			−0.447031			−0.223516	+	0.974700i	$0.571753\pi$
$702$		0			0
$703$	4.93710e10	−	1.18082e10i	0.202139	−	0.0483461i
$704$		0			0
$705$		−	2.38402e11i		−	0.965058i
$706$		0			0
$707$	−2.66825e10			−0.106795
$708$		0			0
$709$	−1.17450e11			−0.464801			−0.232401	−	0.972620i	$-0.574658\pi$
$709$	−1.17450e11			−0.464801			−0.232401	+	0.972620i	$0.574658\pi$
$710$		0			0
$711$			1.54858e10i			0.0605978i
$712$		0			0
$713$			3.01477e10i			0.116653i
$714$		0			0
$715$		−	9.00361e10i		−	0.344502i
$716$		0			0
$717$			1.55865e11i			0.589756i
$718$		0			0
$719$	2.32336e11			0.869363			0.434681	−	0.900584i	$-0.356861\pi$
$719$	2.32336e11			0.869363			0.434681	+	0.900584i	$0.356861\pi$
$720$		0			0
$721$		−	1.21328e11i		−	0.448972i
$722$		0			0
$723$	3.12784e11			1.14470
$724$		0			0
$725$			1.53365e11i			0.555104i
$726$		0			0
$727$	−4.58251e11			−1.64046			−0.820230	−	0.572034i	$-0.806155\pi$
$727$	−4.58251e11			−1.64046			−0.820230	+	0.572034i	$0.806155\pi$
$728$		0			0
$729$	−3.05032e11			−1.08003
$730$		0			0
$731$	1.17899e11			0.412897
$732$		0			0
$733$	−1.58938e11			−0.550571			−0.275285	−	0.961363i	$-0.588772\pi$
$733$	−1.58938e11			−0.550571			−0.275285	+	0.961363i	$0.588772\pi$
$734$		0			0
$735$			1.24831e11i			0.427734i
$736$		0			0
$737$		−	9.87494e10i		−	0.334707i
$738$		0			0
$739$	−6.27278e10			−0.210321			−0.105160	−	0.994455i	$-0.533536\pi$
$739$	−6.27278e10			−0.210321			−0.105160	+	0.994455i	$0.533536\pi$
$740$		0			0
$741$	−1.99076e10	−	8.32354e10i	−0.0660306	−	0.276080i
$742$		0			0
$743$			1.79464e11i			0.588873i	0.955671	+	0.294436i	$0.0951319\pi$
$743$			1.79464e11i			0.588873i	−0.955671	+	0.294436i	$0.904868\pi$
$744$		0			0
$745$	4.38932e11			1.42486
$746$		0			0
$747$	5.07213e10			0.162895
$748$		0			0
$749$		−	1.45383e11i		−	0.461940i
$750$		0			0
$751$			6.97388e10i			0.219237i	0.993974	+	0.109619i	$0.0349630\pi$
$751$			6.97388e10i			0.219237i	−0.993974	+	0.109619i	$0.965037\pi$
$752$		0			0
$753$		−	2.66284e11i		−	0.828258i
$754$		0			0
$755$		−	1.68561e11i		−	0.518763i
$756$		0			0
$757$	−2.79526e11			−0.851214			−0.425607	−	0.904908i	$-0.639939\pi$
$757$	−2.79526e11			−0.851214			−0.425607	+	0.904908i	$0.639939\pi$
$758$		0			0
$759$		−	7.65293e10i		−	0.230601i
$760$		0			0
$761$	4.48433e11			1.33709			0.668543	−	0.743674i	$-0.266918\pi$
$761$	4.48433e11			1.33709			0.668543	+	0.743674i	$0.266918\pi$
$762$		0			0
$763$			9.83164e10i			0.290087i
$764$		0			0
$765$	2.69106e10			0.0785738
$766$		0			0
$767$	−2.00654e11			−0.579785
$768$		0			0
$769$	6.46055e11			1.84741			0.923707	−	0.383101i	$-0.125144\pi$
$769$	6.46055e11			1.84741			0.923707	+	0.383101i	$0.125144\pi$
$770$		0			0
$771$	−2.89560e11			−0.819447
$772$		0			0
$773$		−	3.80149e11i		−	1.06472i	−0.846517	−	0.532361i	$-0.821305\pi$
$773$		−	3.80149e11i		−	1.06472i	0.846517	−	0.532361i	$-0.178695\pi$
$774$		0			0
$775$		−	8.12857e10i		−	0.225324i
$776$		0			0
$777$	4.80082e10			0.131714
$778$		0			0
$779$	5.12271e11	−	1.22521e11i	1.39107	−	0.332706i
$780$		0			0
$781$			4.50686e11i			1.21135i
$782$		0			0
$783$	−6.53146e11			−1.73765
$784$		0			0
$785$	2.11843e11			0.557874
$786$		0			0
$787$			6.89832e11i			1.79823i	0.437717	+	0.899113i	$0.355787\pi$
$787$			6.89832e11i			1.79823i	−0.437717	+	0.899113i	$0.644213\pi$
$788$		0			0
$789$			3.88462e11i			1.00240i
$790$		0			0
$791$		−	4.06608e10i		−	0.103865i
$792$		0			0
$793$			6.29736e10i			0.159245i
$794$		0			0
$795$	−1.66646e11			−0.417183
$796$		0			0
$797$			7.68049e11i			1.90351i	0.306858	+	0.951755i	$0.400722\pi$
$797$			7.68049e11i			1.90351i	−0.306858	+	0.951755i	$0.599278\pi$
$798$		0			0
$799$	4.80577e11			1.17917
$800$		0			0
$801$		−	1.21639e10i		−	0.0295491i
$802$		0			0
$803$	−5.12324e11			−1.23220
$804$		0			0
$805$	−3.95885e10			−0.0942725
$806$		0			0
$807$	6.29840e11			1.48503
$808$		0			0
$809$	−7.20647e11			−1.68240			−0.841199	−	0.540726i	$-0.818150\pi$
$809$	−7.20647e11			−1.68240			−0.841199	+	0.540726i	$0.818150\pi$
$810$		0			0
$811$			7.37264e11i			1.70427i	0.523318	+	0.852137i	$0.324694\pi$
$811$			7.37264e11i			1.70427i	−0.523318	+	0.852137i	$0.675306\pi$
$812$		0			0
$813$			5.25578e11i			1.20303i
$814$		0			0
$815$	2.37519e11			0.538355
$816$		0			0
$817$	4.52672e10	+	1.89266e11i	0.101600	+	0.424801i
$818$		0			0
$819$			9.17518e9i			0.0203929i
$820$		0			0
$821$	1.82223e11			0.401080			0.200540	−	0.979685i	$-0.435730\pi$
$821$	1.82223e11			0.401080			0.200540	+	0.979685i	$0.435730\pi$
$822$		0			0
$823$	−6.16005e11			−1.34272			−0.671359	−	0.741133i	$-0.734289\pi$
$823$	−6.16005e11			−1.34272			−0.671359	+	0.741133i	$0.734289\pi$
$824$		0			0
$825$			2.06342e11i			0.445422i
$826$		0			0
$827$		−	6.39024e11i		−	1.36614i	−0.730353	−	0.683070i	$-0.760645\pi$
$827$		−	6.39024e11i		−	1.36614i	0.730353	−	0.683070i	$-0.239355\pi$
$828$		0			0
$829$			6.47140e11i			1.37019i	0.728454	+	0.685094i	$0.240239\pi$
$829$			6.47140e11i			1.37019i	−0.728454	+	0.685094i	$0.759761\pi$
$830$		0			0
$831$		−	6.23996e10i		−	0.130851i
$832$		0			0
$833$	−2.51638e11			−0.522632
$834$		0			0
$835$		−	1.65362e10i		−	0.0340165i
$836$		0			0
$837$	3.46177e11			0.705337
$838$		0			0
$839$		−	7.65171e11i		−	1.54422i	−0.635486	−	0.772112i	$-0.719200\pi$
$839$		−	7.65171e11i		−	1.54422i	0.635486	−	0.772112i	$-0.280800\pi$
$840$		0			0
$841$	−8.84997e11			−1.76912
$842$		0			0
$843$	−2.06048e11			−0.407998
$844$		0			0
$845$	3.78859e11			0.743106
$846$		0			0
$847$	−3.39012e11			−0.658691
$848$		0			0
$849$			2.83694e11i			0.546034i
$850$		0			0
$851$		−	1.88252e10i		−	0.0358940i
$852$		0			0
$853$	−7.93043e11			−1.49796			−0.748980	−	0.662592i	$-0.769456\pi$
$853$	−7.93043e11			−1.49796			−0.748980	+	0.662592i	$0.769456\pi$
$854$		0			0
$855$	1.03323e10	+	4.32002e10i	0.0193344	+	0.0808390i
$856$		0			0
$857$		−	4.44308e11i		−	0.823684i	−0.911255	−	0.411842i	$-0.864886\pi$
$857$		−	4.44308e11i		−	0.823684i	0.911255	−	0.411842i	$-0.135114\pi$
$858$		0			0
$859$	7.37212e11			1.35400			0.677002	−	0.735982i	$-0.263279\pi$
$859$	7.37212e11			1.35400			0.677002	+	0.735982i	$0.263279\pi$
$860$		0			0
$861$	4.98131e11			0.906424
$862$		0			0
$863$		−	9.27633e11i		−	1.67237i	−0.548446	−	0.836186i	$-0.684780\pi$
$863$		−	9.27633e11i		−	1.67237i	0.548446	−	0.836186i	$-0.315220\pi$
$864$		0			0
$865$		−	3.86915e10i		−	0.0691117i
$866$		0			0
$867$			5.69639e10i			0.100815i
$868$		0			0
$869$			4.78184e11i			0.838525i
$870$		0			0
$871$	−4.09528e10			−0.0711558
$872$		0			0
$873$			9.26738e10i			0.159551i
$874$		0			0
$875$	4.26722e11			0.727969
$876$		0			0
$877$			6.62111e11i			1.11926i	0.828741	+	0.559632i	$0.189058\pi$
$877$			6.62111e11i			1.11926i	−0.828741	+	0.559632i	$0.810942\pi$
$878$		0			0
$879$	−7.31374e11			−1.22514
$880$		0			0
$881$	−4.40753e11			−0.731631			−0.365815	−	0.930687i	$-0.619210\pi$
$881$	−4.40753e11			−0.731631			−0.365815	+	0.930687i	$0.619210\pi$
$882$		0			0
$883$	1.01575e12			1.67087			0.835437	−	0.549586i	$-0.185214\pi$
$883$	1.01575e12			1.67087			0.835437	+	0.549586i	$0.185214\pi$
$884$		0			0
$885$	−9.18676e11			−1.49758
$886$		0			0
$887$		−	2.06328e11i		−	0.333322i	−0.986014	−	0.166661i	$-0.946701\pi$
$887$		−	2.06328e11i		−	0.333322i	0.986014	−	0.166661i	$-0.0532986\pi$
$888$		0			0
$889$		−	6.96786e11i		−	1.11556i
$890$		0			0
$891$	−7.88351e11			−1.25086
$892$		0			0
$893$	1.84516e11	+	7.71480e11i	0.290154	+	1.21316i
$894$		0			0
$895$		−	1.37558e11i		−	0.214384i
$896$		0			0
$897$	−3.17378e10			−0.0490238
$898$		0			0
$899$	7.34200e11			1.12402
$900$		0			0
$901$		−	3.35929e11i		−	0.509740i
$902$		0			0
$903$			1.84042e11i			0.276800i
$904$		0			0
$905$		−	6.61107e11i		−	0.985548i
$906$		0			0
$907$			7.36817e11i			1.08876i	0.838840	+	0.544378i	$0.183234\pi$
$907$			7.36817e11i			1.08876i	−0.838840	+	0.544378i	$0.816766\pi$
$908$		0			0
$909$	1.11023e10			0.0162614
$910$		0			0
$911$		−	9.04637e11i		−	1.31341i	−0.754147	−	0.656706i	$-0.771949\pi$
$911$		−	9.04637e11i		−	1.31341i	0.754147	−	0.656706i	$-0.228051\pi$
$912$		0			0
$913$	1.56621e12			2.25407
$914$		0			0
$915$			2.88319e11i			0.411328i
$916$		0			0
$917$	−3.76286e11			−0.532159
$918$		0			0
$919$	7.52255e10			0.105464			0.0527319	−	0.998609i	$-0.483207\pi$
$919$	7.52255e10			0.105464			0.0527319	+	0.998609i	$0.483207\pi$
$920$		0			0
$921$	−4.87273e11			−0.677227
$922$		0			0
$923$	1.86906e11			0.257523
$924$		0			0
$925$			5.07575e10i			0.0693319i
$926$		0			0
$927$			5.04831e10i			0.0683639i
$928$		0			0
$929$	7.97856e11			1.07118			0.535589	−	0.844479i	$-0.320090\pi$
$929$	7.97856e11			1.07118			0.535589	+	0.844479i	$0.320090\pi$
$930$		0			0
$931$	−9.66158e10	−	4.03960e11i	−0.128603	−	0.537699i
$932$		0			0
$933$		−	8.15530e10i		−	0.107625i
$934$		0			0
$935$	8.30967e11			1.08727
$936$		0			0
$937$	−3.13213e11			−0.406332			−0.203166	−	0.979144i	$-0.565123\pi$
$937$	−3.13213e11			−0.406332			−0.203166	+	0.979144i	$0.565123\pi$
$938$		0			0
$939$		−	1.33755e12i		−	1.72047i
$940$		0			0
$941$		−	9.88446e11i		−	1.26065i	−0.776332	−	0.630325i	$-0.782922\pi$
$941$		−	9.88446e11i		−	1.26065i	0.776332	−	0.630325i	$-0.217078\pi$
$942$		0			0
$943$		−	1.95330e11i		−	0.247014i
$944$		0			0
$945$			4.54582e11i			0.570013i
$946$		0			0
$947$	1.16017e12			1.44252			0.721260	−	0.692665i	$-0.243564\pi$
$947$	1.16017e12			1.44252			0.721260	+	0.692665i	$0.243564\pi$
$948$		0			0
$949$			2.12468e11i			0.261956i
$950$		0			0
$951$	1.19359e12			1.45926
$952$		0			0
$953$		−	1.42909e11i		−	0.173256i	−0.996241	−	0.0866281i	$-0.972391\pi$
$953$		−	1.42909e11i		−	0.173256i	0.996241	−	0.0866281i	$-0.0276092\pi$
$954$		0			0
$955$	−1.37356e11			−0.165134
$956$		0			0
$957$	−1.86375e12			−2.22198
$958$		0			0
$959$	−3.21278e11			−0.379845
$960$		0			0
$961$	4.63754e11			0.543744
$962$		0			0
$963$			6.04921e10i			0.0703385i
$964$		0			0
$965$		−	4.33664e11i		−	0.500086i
$966$		0			0
$967$	−1.32338e12			−1.51349			−0.756743	−	0.653712i	$-0.773211\pi$
$967$	−1.32338e12			−1.51349			−0.756743	+	0.653712i	$0.773211\pi$
$968$		0			0
$969$	7.68202e11	−	1.83732e11i	0.871325	−	0.208397i
$970$		0			0
$971$		−	1.17801e11i		−	0.132518i	−0.997802	−	0.0662588i	$-0.978894\pi$
$971$		−	1.17801e11i		−	0.132518i	0.997802	−	0.0662588i	$-0.0211063\pi$
$972$		0			0
$973$	6.23191e11			0.695296
$974$		0			0
$975$	8.55729e10			0.0946929
$976$		0			0
$977$		−	1.06154e11i		−	0.116509i	−0.998302	−	0.0582543i	$-0.981447\pi$
$977$		−	1.06154e11i		−	0.116509i	0.998302	−	0.0582543i	$-0.0185534\pi$
$978$		0			0
$979$		−	3.75608e11i		−	0.408887i
$980$		0			0
$981$		−	4.09083e10i		−	0.0441708i
$982$		0			0
$983$		−	1.31899e12i		−	1.41263i	−0.707898	−	0.706315i	$-0.750356\pi$
$983$		−	1.31899e12i		−	1.41263i	0.707898	−	0.706315i	$-0.249644\pi$
$984$		0			0
$985$	−6.82848e11			−0.725403
$986$		0			0
$987$			7.50185e11i			0.790496i
$988$		0			0
$989$	7.21675e10			0.0754322
$990$		0			0
$991$			8.51804e11i			0.883171i	0.897219	+	0.441586i	$0.145584\pi$
$991$			8.51804e11i			0.883171i	−0.897219	+	0.441586i	$0.854416\pi$
$992$		0			0
$993$	−8.75103e11			−0.900041
$994$		0			0
$995$	−1.27158e12			−1.29733
$996$		0			0
$997$	9.51130e11			0.962630			0.481315	−	0.876548i	$-0.340159\pi$
$997$	9.51130e11			0.962630			0.481315	+	0.876548i	$0.340159\pi$
$998$		0			0
$999$	−2.16164e11			−0.217031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.9.c.b.37.3	✓	12
4.3	odd	2		304.9.e.c.113.10		12
19.18	odd	2	inner	76.9.c.b.37.10	yes	12
76.75	even	2		304.9.e.c.113.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.9.c.b.37.3	✓	12	1.1	even	1	trivial
76.9.c.b.37.10	yes	12	19.18	odd	2	inner
304.9.e.c.113.3		12	76.75	even	2
304.9.e.c.113.10		12	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:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	76.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-76.7657i q^{3} +510.215 q^{5} -1605.51 q^{7} +668.032 q^{9} +O(q^{10})\)	\(q-76.7657i q^{3} +510.215 q^{5} -1605.51 q^{7} +668.032 q^{9} +20628.0 q^{11} -8554.72i q^{13} -39167.0i q^{15} +78953.8 q^{17} +(30314.2 + 126746. i) q^{19} +123248. i q^{21} +48328.5 q^{23} -130306. q^{25} -554941. i q^{27} -1.17696e6i q^{29} +623808. i q^{31} -1.58352e6i q^{33} -819153. q^{35} -389526. i q^{37} -656709. q^{39} -4.04171e6i q^{41} +1.49327e6 q^{43} +340840. q^{45} +6.08681e6 q^{47} -3.18715e6 q^{49} -6.06094e6i q^{51} -4.25475e6i q^{53} +1.05247e7 q^{55} +(9.72976e6 - 2.32709e6i) q^{57} -2.34554e7i q^{59} -7.36127e6 q^{61} -1.07253e6 q^{63} -4.36475e6i q^{65} -4.78715e6i q^{67} -3.70997e6i q^{69} +2.18483e7i q^{71} -2.48363e7 q^{73} +1.00030e7i q^{75} -3.31184e7 q^{77} +2.31813e7i q^{79} -3.82175e7 q^{81} +7.59265e7 q^{83} +4.02834e7 q^{85} -9.03504e7 q^{87} -1.82086e7i q^{89} +1.37347e7i q^{91} +4.78871e7 q^{93} +(1.54667e7 + 6.46678e7i) q^{95} +1.38727e8i q^{97} +1.37802e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 568 q^{5} + 1734 q^{7} - 46210 q^{9}+O(q^{10}) \) 12 * q - 568 * q^5 + 1734 * q^7 - 46210 * q^9	\( 12 q - 568 q^{5} + 1734 q^{7} - 46210 q^{9} + 1976 q^{11} - 47446 q^{17} - 113178 q^{19} + 378950 q^{23} - 502656 q^{25} - 1876832 q^{35} - 1463394 q^{39} + 6234396 q^{43} - 507200 q^{45} + 4902224 q^{47} - 16736298 q^{49} - 38366664 q^{55} - 19396170 q^{57} + 16602912 q^{61} - 11274020 q^{63} + 111268614 q^{73} - 134425928 q^{77} + 193590584 q^{81} + 34030388 q^{83} + 387605748 q^{85} - 217889946 q^{87} - 474880524 q^{93} - 256537036 q^{95} + 276136420 q^{99}+O(q^{100}) \) 12 * q - 568 * q^5 + 1734 * q^7 - 46210 * q^9 + 1976 * q^11 - 47446 * q^17 - 113178 * q^19 + 378950 * q^23 - 502656 * q^25 - 1876832 * q^35 - 1463394 * q^39 + 6234396 * q^43 - 507200 * q^45 + 4902224 * q^47 - 16736298 * q^49 - 38366664 * q^55 - 19396170 * q^57 + 16602912 * q^61 - 11274020 * q^63 + 111268614 * q^73 - 134425928 * q^77 + 193590584 * q^81 + 34030388 * q^83 + 387605748 * q^85 - 217889946 * q^87 - 474880524 * q^93 - 256537036 * q^95 + 276136420 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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