LMFDB - Embedded newform 1386.2.e.a.307.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(307,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1386.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.e (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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.6679465984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 61x^{4} + 88x^{2} + 36 $ x^8 + 14x^6 + 61x^4 + 88*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 462)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			307.6
Root			$2.62511i$ of defining polynomial
Character	$\chi$	$=$	1386.307
Dual form			1386.2.e.a.307.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -0.266088i q^{5} +(1.34926 - 2.27585i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} -0.266088i q^{5} +(1.34926 - 2.27585i) q^{7} -1.00000i q^{8} +0.266088 q^{10} +(-1.62511 + 2.89120i) q^{11} -2.55171 q^{13} +(2.27585 + 1.34926i) q^{14} +1.00000 q^{16} -4.96460 q^{17} +8.21482 q^{19} +0.266088i q^{20} +(-2.89120 - 1.62511i) q^{22} +5.23069 q^{23} +4.92920 q^{25} -2.55171i q^{26} +(-1.34926 + 2.27585i) q^{28} +5.25022i q^{29} -4.28562i q^{31} +1.00000i q^{32} -4.96460i q^{34} +(-0.605578 - 0.359021i) q^{35} +7.39702 q^{37} +8.21482i q^{38} -0.266088 q^{40} +6.21482 q^{41} -1.46782i q^{43} +(1.62511 - 2.89120i) q^{44} +5.23069i q^{46} -10.0485i q^{47} +(-3.35902 - 6.14142i) q^{49} +4.92920i q^{50} +2.55171 q^{52} +13.1989 q^{53} +(0.769313 + 0.432422i) q^{55} +(-2.27585 - 1.34926i) q^{56} -5.25022 q^{58} -9.10342i q^{59} -5.98047 q^{61} +4.28562 q^{62} -1.00000 q^{64} +0.678979i q^{65} +12.3536 q^{67} +4.96460 q^{68} +(0.359021 - 0.605578i) q^{70} -4.93708 q^{71} -2.96460 q^{73} +7.39702i q^{74} -8.21482 q^{76} +(4.38726 + 7.59947i) q^{77} +6.16633i q^{79} -0.266088i q^{80} +6.21482i q^{82} +10.6002 q^{83} +1.32102i q^{85} +1.46782 q^{86} +(2.89120 + 1.62511i) q^{88} -1.78239i q^{89} +(-3.44290 + 5.80731i) q^{91} -5.23069 q^{92} +10.0485 q^{94} -2.18587i q^{95} +0.146803i q^{97} +(6.14142 - 3.35902i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} - 4 q^{10} + 8 q^{11} + 8 q^{14} + 8 q^{16} - 12 q^{17} - 4 q^{19} + 4 q^{22} + 8 q^{23} - 16 q^{25} + 8 q^{35} + 16 q^{37} + 4 q^{40} - 20 q^{41} - 8 q^{44} - 12 q^{49} + 40 q^{55} - 8 q^{56} - 56 q^{61} + 20 q^{62} - 8 q^{64} + 16 q^{67} + 12 q^{68} - 12 q^{70} - 8 q^{71} + 4 q^{73} + 4 q^{76} - 4 q^{77} + 4 q^{83} + 24 q^{86} - 4 q^{88} + 20 q^{91} - 8 q^{92} + 20 q^{94} - 20 q^{98}+O(q^{100}) $ 8 * q - 8 * q^4 - 4 * q^10 + 8 * q^11 + 8 * q^14 + 8 * q^16 - 12 * q^17 - 4 * q^19 + 4 * q^22 + 8 * q^23 - 16 * q^25 + 8 * q^35 + 16 * q^37 + 4 * q^40 - 20 * q^41 - 8 * q^44 - 12 * q^49 + 40 * q^55 - 8 * q^56 - 56 * q^61 + 20 * q^62 - 8 * q^64 + 16 * q^67 + 12 * q^68 - 12 * q^70 - 8 * q^71 + 4 * q^73 + 4 * q^76 - 4 * q^77 + 4 * q^83 + 24 * q^86 - 4 * q^88 + 20 * q^91 - 8 * q^92 + 20 * q^94 - 20 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		−	0.266088i		−	0.118998i	−0.998228	−	0.0594991i	$-0.981050\pi$
$5$		−	0.266088i		−	0.118998i	0.998228	−	0.0594991i	$-0.0189504\pi$
$6$		0			0
$7$	1.34926	−	2.27585i	0.509971	−	0.860192i
$7$	1.34926	−	2.27585i	0.509971	−	0.860192i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	0.266088			0.0841445
$11$	−1.62511	+	2.89120i	−0.489989	+	0.871729i
$11$	−1.62511	+	2.89120i	−0.489989	+	0.871729i
$12$		0			0
$13$	−2.55171			−0.707716			−0.353858	−	0.935299i	$-0.615130\pi$
$13$	−2.55171			−0.707716			−0.353858	+	0.935299i	$0.615130\pi$
$14$	2.27585	+	1.34926i	0.608248	+	0.360604i
$15$		0			0
$16$	1.00000			0.250000
$17$	−4.96460			−1.20409			−0.602046	−	0.798461i	$-0.705648\pi$
$17$	−4.96460			−1.20409			−0.602046	+	0.798461i	$0.705648\pi$
$18$		0			0
$19$	8.21482			1.88461			0.942304	−	0.334758i	$-0.108655\pi$
$19$	8.21482			1.88461			0.942304	+	0.334758i	$0.108655\pi$
$20$			0.266088i			0.0594991i
$21$		0			0
$22$	−2.89120	−	1.62511i	−0.616405	−	0.346474i
$23$	5.23069			1.09067			0.545337	−	0.838217i	$-0.316402\pi$
$23$	5.23069			1.09067			0.545337	+	0.838217i	$0.316402\pi$
$24$		0			0
$25$	4.92920			0.985839
$26$		−	2.55171i		−	0.500431i
$27$		0			0
$28$	−1.34926	+	2.27585i	−0.254985	+	0.430096i
$29$			5.25022i			0.974941i	0.873139	+	0.487470i	$0.162080\pi$
$29$			5.25022i			0.974941i	−0.873139	+	0.487470i	$0.837920\pi$
$30$		0			0
$31$		−	4.28562i		−	0.769720i	−0.922975	−	0.384860i	$-0.874250\pi$
$31$		−	4.28562i		−	0.769720i	0.922975	−	0.384860i	$-0.125750\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	4.96460i		−	0.851422i
$35$	−0.605578	−	0.359021i	−0.102361	−	0.0606856i
$36$		0			0
$37$	7.39702			1.21606			0.608031	−	0.793913i	$-0.291959\pi$
$37$	7.39702			1.21606			0.608031	+	0.793913i	$0.291959\pi$
$38$			8.21482i			1.33262i
$39$		0			0
$40$	−0.266088			−0.0420722
$41$	6.21482			0.970591			0.485296	−	0.874350i	$-0.338712\pi$
$41$	6.21482			0.970591			0.485296	+	0.874350i	$0.338712\pi$
$42$		0			0
$43$		−	1.46782i		−	0.223841i	−0.993717	−	0.111921i	$-0.964300\pi$
$43$		−	1.46782i		−	0.223841i	0.993717	−	0.111921i	$-0.0357002\pi$
$44$	1.62511	−	2.89120i	0.244994	−	0.435864i
$45$		0			0
$46$			5.23069i			0.771223i
$47$		−	10.0485i		−	1.46572i	−0.680378	−	0.732861i	$-0.738185\pi$
$47$		−	10.0485i		−	1.46572i	0.680378	−	0.732861i	$-0.261815\pi$
$48$		0			0
$49$	−3.35902	−	6.14142i	−0.479860	−	0.877345i
$50$			4.92920i			0.697094i
$51$		0			0
$52$	2.55171			0.353858
$53$	13.1989			1.81301			0.906507	−	0.422190i	$-0.138739\pi$
$53$	13.1989			1.81301			0.906507	+	0.422190i	$0.138739\pi$
$54$		0			0
$55$	0.769313	+	0.432422i	0.103734	+	0.0583078i
$56$	−2.27585	−	1.34926i	−0.304124	−	0.180302i
$57$		0			0
$58$	−5.25022			−0.689387
$59$		−	9.10342i		−	1.18516i	−0.805510	−	0.592582i	$-0.798109\pi$
$59$		−	9.10342i		−	1.18516i	0.805510	−	0.592582i	$-0.201891\pi$
$60$		0			0
$61$	−5.98047			−0.765721			−0.382860	−	0.923806i	$-0.625061\pi$
$61$	−5.98047			−0.765721			−0.382860	+	0.923806i	$0.625061\pi$
$62$	4.28562			0.544274
$63$		0			0
$64$	−1.00000			−0.125000
$65$			0.678979i			0.0842170i
$66$		0			0
$67$	12.3536			1.50924			0.754618	−	0.656164i	$-0.227822\pi$
$67$	12.3536			1.50924			0.754618	+	0.656164i	$0.227822\pi$
$68$	4.96460			0.602046
$69$		0			0
$70$	0.359021	−	0.605578i	0.0429112	−	0.0723804i
$71$	−4.93708			−0.585924			−0.292962	−	0.956124i	$-0.594641\pi$
$71$	−4.93708			−0.585924			−0.292962	+	0.956124i	$0.594641\pi$
$72$		0			0
$73$	−2.96460			−0.346980			−0.173490	−	0.984836i	$-0.555504\pi$
$73$	−2.96460			−0.346980			−0.173490	+	0.984836i	$0.555504\pi$
$74$			7.39702i			0.859886i
$75$		0			0
$76$	−8.21482			−0.942304
$77$	4.38726	+	7.59947i	0.499974	+	0.866040i
$78$		0			0
$79$			6.16633i			0.693767i	0.937908	+	0.346883i	$0.112760\pi$
$79$			6.16633i			0.693767i	−0.937908	+	0.346883i	$0.887240\pi$
$80$		−	0.266088i		−	0.0297496i
$81$		0			0
$82$			6.21482i			0.686312i
$83$	10.6002			1.16352			0.581761	−	0.813360i	$-0.302364\pi$
$83$	10.6002			1.16352			0.581761	+	0.813360i	$0.302364\pi$
$84$		0			0
$85$			1.32102i			0.143285i
$86$	1.46782			0.158279
$87$		0			0
$88$	2.89120	+	1.62511i	0.308203	+	0.173237i
$89$		−	1.78239i		−	0.188933i	−0.995528	−	0.0944667i	$-0.969885\pi$
$89$		−	1.78239i		−	0.188933i	0.995528	−	0.0944667i	$-0.0301146\pi$
$90$		0			0
$91$	−3.44290	+	5.80731i	−0.360914	+	0.608772i
$92$	−5.23069			−0.545337
$93$		0			0
$94$	10.0485			1.03642
$95$		−	2.18587i		−	0.224265i
$96$		0			0
$97$			0.146803i			0.0149056i	0.999972	+	0.00745279i	$0.00237232\pi$
$97$			0.146803i			0.0149056i	−0.999972	+	0.00745279i	$0.997628\pi$
$98$	6.14142	−	3.35902i	0.620377	−	0.339312i
$99$		0			0
$100$	−4.92920			−0.492920
$101$	13.7824			1.37140			0.685700	−	0.727885i	$-0.259496\pi$
$101$	13.7824			1.37140			0.685700	+	0.727885i	$0.259496\pi$
$102$		0			0
$103$			19.8504i			1.95592i	0.208795	+	0.977959i	$0.433046\pi$
$103$			19.8504i			1.95592i	−0.208795	+	0.977959i	$0.566954\pi$
$104$			2.55171i			0.250216i
$105$		0			0
$106$			13.1989i			1.28199i
$107$			13.7824i			1.33239i	0.745776	+	0.666197i	$0.232079\pi$
$107$			13.7824i			1.33239i	−0.745776	+	0.666197i	$0.767921\pi$
$108$		0			0
$109$			0.365842i			0.0350413i	0.999847	+	0.0175207i	$0.00557729\pi$
$109$			0.365842i			0.0350413i	−0.999847	+	0.0175207i	$0.994423\pi$
$110$	−0.432422	+	0.769313i	−0.0412298	+	0.0733511i
$111$		0			0
$112$	1.34926	−	2.27585i	0.127493	−	0.215048i
$113$	−12.2828			−1.15547			−0.577736	−	0.816224i	$-0.696064\pi$
$113$	−12.2828			−1.15547			−0.577736	+	0.816224i	$0.696064\pi$
$114$		0			0
$115$		−	1.39182i		−	0.129788i
$116$		−	5.25022i		−	0.487470i
$117$		0			0
$118$	9.10342			0.838037
$119$	−6.69851	+	11.2987i	−0.614051	+	1.03575i
$120$		0			0
$121$	−5.71804	−	9.39702i	−0.519822	−	0.854275i
$122$		−	5.98047i		−	0.541446i
$123$		0			0
$124$			4.28562i			0.384860i
$125$		−	2.64204i		−	0.236311i
$126$		0			0
$127$			10.0247i			0.889551i	0.895642	+	0.444775i	$0.146716\pi$
$127$			10.0247i			0.889551i	−0.895642	+	0.444775i	$0.853284\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−0.678979			−0.0595504
$131$	−20.0363			−1.75058			−0.875289	−	0.483601i	$-0.839329\pi$
$131$	−20.0363			−1.75058			−0.875289	+	0.483601i	$0.839329\pi$
$132$		0			0
$133$	11.0839	−	18.6957i	0.961095	−	1.62112i
$134$			12.3536i			1.06719i
$135$		0			0
$136$			4.96460i			0.425711i
$137$	7.17942			0.613379			0.306689	−	0.951810i	$-0.400779\pi$
$137$	7.17942			0.613379			0.306689	+	0.951810i	$0.400779\pi$
$138$		0			0
$139$	−16.7861			−1.42377			−0.711887	−	0.702294i	$-0.752159\pi$
$139$	−16.7861			−1.42377			−0.711887	+	0.702294i	$0.752159\pi$
$140$	0.605578	+	0.359021i	0.0511807	+	0.0303428i
$141$		0			0
$142$		−	4.93708i		−	0.414311i
$143$	4.14680	−	7.37749i	0.346773	−	0.616937i
$144$		0			0
$145$	1.39702			0.116016
$146$		−	2.96460i		−	0.245352i
$147$		0			0
$148$	−7.39702			−0.608031
$149$		−	17.5648i		−	1.43896i	−0.694511	−	0.719482i	$-0.744379\pi$
$149$		−	17.5648i		−	1.43896i	0.694511	−	0.719482i	$-0.255621\pi$
$150$		0			0
$151$			15.0131i			1.22175i	0.791728	+	0.610874i	$0.209182\pi$
$151$			15.0131i			1.22175i	−0.791728	+	0.610874i	$0.790818\pi$
$152$		−	8.21482i		−	0.666310i
$153$		0			0
$154$	−7.59947	+	4.38726i	−0.612383	+	0.353535i
$155$	−1.14035			−0.0915953
$156$		0			0
$157$			10.6393i			0.849105i	0.905403	+	0.424552i	$0.139569\pi$
$157$			10.6393i			0.849105i	−0.905403	+	0.424552i	$0.860431\pi$
$158$	−6.16633			−0.490567
$159$		0			0
$160$	0.266088			0.0210361
$161$	7.05753	−	11.9043i	0.556211	−	0.938189i
$162$		0			0
$163$	0.385373			0.0301848			0.0150924	−	0.999886i	$-0.495196\pi$
$163$	0.385373			0.0301848			0.0150924	+	0.999886i	$0.495196\pi$
$164$	−6.21482			−0.485296
$165$		0			0
$166$			10.6002i			0.822734i
$167$	−0.967388			−0.0748587			−0.0374294	−	0.999299i	$-0.511917\pi$
$167$	−0.967388			−0.0748587			−0.0374294	+	0.999299i	$0.511917\pi$
$168$		0			0
$169$	−6.48879			−0.499138
$170$	−1.32102			−0.101318
$171$		0			0
$172$			1.46782i			0.111921i
$173$	−0.385373			−0.0292994			−0.0146497	−	0.999893i	$-0.504663\pi$
$173$	−0.385373			−0.0292994			−0.0146497	+	0.999893i	$0.504663\pi$
$174$		0			0
$175$	6.65074	−	11.2181i	0.502749	−	0.848011i
$176$	−1.62511	+	2.89120i	−0.122497	+	0.217932i
$177$		0			0
$178$	1.78239			0.133596
$179$	6.69994			0.500777			0.250389	−	0.968145i	$-0.419442\pi$
$179$	6.69994			0.500777			0.250389	+	0.968145i	$0.419442\pi$
$180$		0			0
$181$		−	26.1650i		−	1.94483i	−0.233263	−	0.972414i	$-0.574940\pi$
$181$		−	26.1650i		−	1.94483i	0.233263	−	0.972414i	$-0.425060\pi$
$182$	−5.80731	−	3.44290i	−0.430467	−	0.255205i
$183$		0			0
$184$		−	5.23069i		−	0.385611i
$185$		−	1.96826i		−	0.144709i
$186$		0			0
$187$	8.06801	−	14.3536i	0.589992	−	1.04964i
$188$			10.0485i			0.732861i
$189$		0			0
$190$	2.18587			0.158579
$191$	−10.6277			−0.768994			−0.384497	−	0.923126i	$-0.625625\pi$
$191$	−10.6277			−0.768994			−0.384497	+	0.923126i	$0.625625\pi$
$192$		0			0
$193$		−	14.4614i		−	1.04095i	−0.853876	−	0.520476i	$-0.825754\pi$
$193$		−	14.4614i		−	1.04095i	0.853876	−	0.520476i	$-0.174246\pi$
$194$	−0.146803			−0.0105398
$195$		0			0
$196$	3.35902	+	6.14142i	0.239930	+	0.438673i
$197$		−	1.35276i		−	0.0963802i	−0.998838	−	0.0481901i	$-0.984655\pi$
$197$		−	1.35276i		−	0.0963802i	0.998838	−	0.0481901i	$-0.0153453\pi$
$198$		0			0
$199$			8.52419i			0.604264i	0.953266	+	0.302132i	$0.0976983\pi$
$199$			8.52419i			0.604264i	−0.953266	+	0.302132i	$0.902302\pi$
$200$		−	4.92920i		−	0.348547i
$201$		0			0
$202$			13.7824i			0.969726i
$203$	11.9487	+	7.08388i	0.838636	+	0.497191i
$204$		0			0
$205$		−	1.65369i		−	0.115499i
$206$	−19.8504			−1.38304
$207$		0			0
$208$	−2.55171			−0.176929
$209$	−13.3500	+	23.7507i	−0.923437	+	1.64287i
$210$		0			0
$211$		−	10.8258i		−	0.745278i	−0.927976	−	0.372639i	$-0.878453\pi$
$211$		−	10.8258i		−	0.745278i	0.927976	−	0.372639i	$-0.121547\pi$
$212$	−13.1989			−0.906507
$213$		0			0
$214$	−13.7824			−0.942145
$215$	−0.390570			−0.0266367
$216$		0			0
$217$	−9.75344	−	5.78239i	−0.662107	−	0.392534i
$218$	−0.365842			−0.0247780
$219$		0			0
$220$	−0.769313	−	0.432422i	−0.0518671	−	0.0291539i
$221$	12.6682			0.852156
$222$		0			0
$223$			1.38903i			0.0930166i	0.998918	+	0.0465083i	$0.0148094\pi$
$223$			1.38903i			0.0930166i	−0.998918	+	0.0465083i	$0.985191\pi$
$224$	2.27585	+	1.34926i	0.152062	+	0.0901509i
$225$		0			0
$226$		−	12.2828i		−	0.817042i
$227$	−4.04705			−0.268612			−0.134306	−	0.990940i	$-0.542881\pi$
$227$	−4.04705			−0.268612			−0.134306	+	0.990940i	$0.542881\pi$
$228$		0			0
$229$		−	15.2265i		−	1.00619i	−0.864230	−	0.503097i	$-0.832194\pi$
$229$		−	15.2265i		−	1.00619i	0.864230	−	0.503097i	$-0.167806\pi$
$230$	1.39182			0.0917741
$231$		0			0
$232$	5.25022			0.344694
$233$		−	10.2936i		−	0.674357i	−0.941441	−	0.337178i	$-0.890527\pi$
$233$		−	10.2936i		−	0.674357i	0.941441	−	0.337178i	$-0.109473\pi$
$234$		0			0
$235$	−2.67378			−0.174418
$236$			9.10342i			0.592582i
$237$		0			0
$238$	−11.2987	−	6.69851i	−0.732386	−	0.434200i
$239$			1.14248i			0.0739007i	0.999317	+	0.0369504i	$0.0117643\pi$
$239$			1.14248i			0.0739007i	−0.999317	+	0.0369504i	$0.988236\pi$
$240$		0			0
$241$	0.503225			0.0324156			0.0162078	−	0.999869i	$-0.494841\pi$
$241$	0.503225			0.0324156			0.0162078	+	0.999869i	$0.494841\pi$
$242$	9.39702	−	5.71804i	0.604063	−	0.367570i
$243$		0			0
$244$	5.98047			0.382860
$245$	−1.63416	+	0.893796i	−0.104403	+	0.0571025i
$246$		0			0
$247$	−20.9618			−1.33377
$248$	−4.28562			−0.272137
$249$		0			0
$250$	2.64204			0.167097
$251$			7.03261i			0.443895i	0.975059	+	0.221947i	$0.0712413\pi$
$251$			7.03261i			0.443895i	−0.975059	+	0.221947i	$0.928759\pi$
$252$		0			0
$253$	−8.50044	+	15.1229i	−0.534418	+	0.950771i
$254$	−10.0247			−0.629007
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	3.85320i		−	0.240356i	−0.992752	−	0.120178i	$-0.961653\pi$
$257$		−	3.85320i		−	0.240356i	0.992752	−	0.120178i	$-0.0383465\pi$
$258$		0			0
$259$	9.98047	−	16.8345i	0.620156	−	1.04605i
$260$		−	0.678979i		−	0.0421085i
$261$		0			0
$262$		−	20.0363i		−	1.23785i
$263$		−	6.48234i		−	0.399718i	−0.979825	−	0.199859i	$-0.935952\pi$
$263$		−	6.48234i		−	0.399718i	0.979825	−	0.199859i	$-0.0640484\pi$
$264$		0			0
$265$		−	3.51208i		−	0.215746i
$266$	18.6957	+	11.0839i	1.14631	+	0.679597i
$267$		0			0
$268$	−12.3536			−0.754618
$269$			20.7983i			1.26809i	0.773295	+	0.634046i	$0.218607\pi$
$269$			20.7983i			1.26809i	−0.773295	+	0.634046i	$0.781393\pi$
$270$		0			0
$271$	22.7955			1.38473			0.692363	−	0.721549i	$-0.256570\pi$
$271$	22.7955			1.38473			0.692363	+	0.721549i	$0.256570\pi$
$272$	−4.96460			−0.301023
$273$		0			0
$274$			7.17942i			0.433724i
$275$	−8.01048	+	14.2513i	−0.483050	+	0.859385i
$276$		0			0
$277$			17.3277i			1.04112i	0.853826	+	0.520559i	$0.174276\pi$
$277$			17.3277i			1.04112i	−0.853826	+	0.520559i	$0.825724\pi$
$278$		−	16.7861i		−	1.00676i
$279$		0			0
$280$	−0.359021	+	0.605578i	−0.0214556	+	0.0361902i
$281$			5.17942i			0.308978i	0.987995	+	0.154489i	$0.0493731\pi$
$281$			5.17942i			0.308978i	−0.987995	+	0.154489i	$0.950627\pi$
$282$		0			0
$283$	−12.4534			−0.740277			−0.370139	−	0.928977i	$-0.620690\pi$
$283$	−12.4534			−0.740277			−0.370139	+	0.928977i	$0.620690\pi$
$284$	4.93708			0.292962
$285$		0			0
$286$	7.37749	+	4.14680i	0.436240	+	0.245206i
$287$	8.38537	−	14.1440i	0.494973	−	0.834895i
$288$		0			0
$289$	7.64724			0.449838
$290$			1.39702i			0.0820359i
$291$		0			0
$292$	2.96460			0.173490
$293$	−11.4570			−0.669328			−0.334664	−	0.942338i	$-0.608623\pi$
$293$	−11.4570			−0.669328			−0.334664	+	0.942338i	$0.608623\pi$
$294$		0			0
$295$	−2.42231			−0.141032
$296$		−	7.39702i		−	0.429943i
$297$		0			0
$298$	17.5648			1.01750
$299$	−13.3472			−0.771888
$300$		0			0
$301$	−3.34055	−	1.98047i	−0.192546	−	0.114152i
$302$	−15.0131			−0.863906
$303$		0			0
$304$	8.21482			0.471152
$305$			1.59133i			0.0911194i
$306$		0			0
$307$	4.78606			0.273155			0.136577	−	0.990629i	$-0.456390\pi$
$307$	4.78606			0.273155			0.136577	+	0.990629i	$0.456390\pi$
$308$	−4.38726	−	7.59947i	−0.249987	−	0.433020i
$309$		0			0
$310$		−	1.14035i		−	0.0647677i
$311$			15.8807i			0.900513i	0.892899	+	0.450256i	$0.148667\pi$
$311$			15.8807i			0.900513i	−0.892899	+	0.450256i	$0.851333\pi$
$312$		0			0
$313$			30.6524i			1.73258i	0.499543	+	0.866289i	$0.333501\pi$
$313$			30.6524i			1.73258i	−0.499543	+	0.866289i	$0.666499\pi$
$314$	−10.6393			−0.600408
$315$		0			0
$316$		−	6.16633i		−	0.346883i
$317$	6.69851			0.376226			0.188113	−	0.982147i	$-0.439763\pi$
$317$	6.69851			0.376226			0.188113	+	0.982147i	$0.439763\pi$
$318$		0			0
$319$	−15.1794	−	8.53218i	−0.849884	−	0.477710i
$320$			0.266088i			0.0148748i
$321$		0			0
$322$	11.9043	+	7.05753i	0.663399	+	0.393301i
$323$	−40.7833			−2.26924
$324$		0			0
$325$	−12.5779			−0.697695
$326$			0.385373i			0.0213439i
$327$		0			0
$328$		−	6.21482i		−	0.343156i
$329$	−22.8689	−	13.5580i	−1.26080	−	0.747475i
$330$		0			0
$331$	31.5330			1.73321			0.866607	−	0.498992i	$-0.166296\pi$
$331$	31.5330			1.73321			0.866607	+	0.498992i	$0.166296\pi$
$332$	−10.6002			−0.581761
$333$		0			0
$334$		−	0.967388i		−	0.0529331i
$335$		−	3.28716i		−	0.179596i
$336$		0			0
$337$		−	22.0652i		−	1.20197i	−0.799261	−	0.600985i	$-0.794775\pi$
$337$		−	22.0652i		−	1.20197i	0.799261	−	0.600985i	$-0.205225\pi$
$338$		−	6.48879i		−	0.352944i
$339$		0			0
$340$		−	1.32102i		−	0.0716424i
$341$	12.3906	+	6.96460i	0.670987	+	0.377154i
$342$		0			0
$343$	−18.5091	−	0.641696i	−0.999400	−	0.0346483i
$344$	−1.46782			−0.0791397
$345$		0			0
$346$		−	0.385373i		−	0.0207178i
$347$		−	27.9365i		−	1.49971i	−0.661602	−	0.749855i	$-0.730123\pi$
$347$		−	27.9365i		−	1.49971i	0.661602	−	0.749855i	$-0.269877\pi$
$348$		0			0
$349$	−1.19375			−0.0638999			−0.0319500	−	0.999489i	$-0.510172\pi$
$349$	−1.19375			−0.0638999			−0.0319500	+	0.999489i	$0.510172\pi$
$350$	11.2181	+	6.65074i	0.599634	+	0.355497i
$351$		0			0
$352$	−2.89120	−	1.62511i	−0.154101	−	0.0866186i
$353$		−	33.4413i		−	1.77990i	−0.456058	−	0.889950i	$-0.650739\pi$
$353$		−	33.4413i		−	1.77990i	0.456058	−	0.889950i	$-0.349261\pi$
$354$		0			0
$355$			1.31370i			0.0697239i
$356$			1.78239i			0.0944667i
$357$		0			0
$358$			6.69994i			0.354103i
$359$		−	8.45618i		−	0.446300i	−0.974784	−	0.223150i	$-0.928366\pi$
$359$		−	8.45618i		−	0.446300i	0.974784	−	0.223150i	$-0.0716340\pi$
$360$		0			0
$361$	48.4832			2.55175
$362$	26.1650			1.37520
$363$		0			0
$364$	3.44290	−	5.80731i	0.180457	−	0.304386i
$365$			0.788845i			0.0412900i
$366$		0			0
$367$			21.4831i			1.12141i	0.828016	+	0.560705i	$0.189470\pi$
$367$			21.4831i			1.12141i	−0.828016	+	0.560705i	$0.810530\pi$
$368$	5.23069			0.272668
$369$		0			0
$370$	1.96826			0.102325
$371$	17.8087	−	30.0389i	0.924584	−	1.55954i
$372$		0			0
$373$		−	9.94141i		−	0.514747i	−0.966312	−	0.257373i	$-0.917143\pi$
$373$		−	9.94141i		−	0.514747i	0.966312	−	0.257373i	$-0.0828570\pi$
$374$	14.3536	+	8.06801i	0.742209	+	0.417187i
$375$		0			0
$376$	−10.0485			−0.518211
$377$		−	13.3970i		−	0.689982i
$378$		0			0
$379$	−21.3862			−1.09854			−0.549269	−	0.835646i	$-0.685094\pi$
$379$	−21.3862			−1.09854			−0.549269	+	0.835646i	$0.685094\pi$
$380$			2.18587i			0.112133i
$381$		0			0
$382$		−	10.6277i		−	0.543761i
$383$		−	0.706497i		−	0.0361003i	−0.999837	−	0.0180501i	$-0.994254\pi$
$383$		−	0.706497i		−	0.0361003i	0.999837	−	0.0180501i	$-0.00574585\pi$
$384$		0			0
$385$	2.02213	−	1.16740i	0.103057	−	0.0594960i
$386$	14.4614			0.736065
$387$		0			0
$388$		−	0.146803i		−	0.00745279i
$389$	5.30149			0.268796			0.134398	−	0.990927i	$-0.457090\pi$
$389$	5.30149			0.268796			0.134398	+	0.990927i	$0.457090\pi$
$390$		0			0
$391$	−25.9683			−1.31327
$392$	−6.14142	+	3.35902i	−0.310188	+	0.169656i
$393$		0			0
$394$	1.35276			0.0681511
$395$	1.64079			0.0825570
$396$		0			0
$397$		−	24.9646i		−	1.25294i	−0.779447	−	0.626469i	$-0.784500\pi$
$397$		−	24.9646i		−	1.25294i	0.779447	−	0.626469i	$-0.215500\pi$
$398$	−8.52419			−0.427279
$399$		0			0
$400$	4.92920			0.246460
$401$	11.6146			0.580007			0.290003	−	0.957026i	$-0.406344\pi$
$401$	11.6146			0.580007			0.290003	+	0.957026i	$0.406344\pi$
$402$		0			0
$403$			10.9356i			0.544743i
$404$	−13.7824			−0.685700
$405$		0			0
$406$	−7.08388	+	11.9487i	−0.351567	+	0.593005i
$407$	−12.0210	+	21.3862i	−0.595857	+	1.06008i
$408$		0			0
$409$	−15.2032			−0.751748			−0.375874	−	0.926671i	$-0.622658\pi$
$409$	−15.2032			−0.751748			−0.375874	+	0.926671i	$0.622658\pi$
$410$	1.65369			0.0816699
$411$		0			0
$412$		−	19.8504i		−	0.977959i
$413$	−20.7180	−	12.2828i	−1.01947	−	0.604399i
$414$		0			0
$415$		−	2.82058i		−	0.138457i
$416$		−	2.55171i		−	0.125108i
$417$		0			0
$418$	−23.7507	−	13.3500i	−1.16168	−	0.652968i
$419$		−	13.9683i		−	0.682394i	−0.939992	−	0.341197i	$-0.889168\pi$
$419$		−	13.9683i		−	0.682394i	0.939992	−	0.341197i	$-0.110832\pi$
$420$		0			0
$421$	−4.53950			−0.221242			−0.110621	−	0.993863i	$-0.535284\pi$
$421$	−4.53950			−0.221242			−0.110621	+	0.993863i	$0.535284\pi$
$422$	10.8258			0.526991
$423$		0			0
$424$		−	13.1989i		−	0.640997i
$425$	−24.4715			−1.18704
$426$		0			0
$427$	−8.06918	+	13.6107i	−0.390495	+	0.658667i
$428$		−	13.7824i		−	0.666197i
$429$		0			0
$430$		−	0.390570i		−	0.0188350i
$431$			11.5829i			0.557928i	0.960302	+	0.278964i	$0.0899910\pi$
$431$			11.5829i			0.557928i	−0.960302	+	0.278964i	$0.910009\pi$
$432$		0			0
$433$			20.9227i			1.00548i	0.864437	+	0.502742i	$0.167675\pi$
$433$			20.9227i			1.00548i	−0.864437	+	0.502742i	$0.832325\pi$
$434$	5.78239	−	9.75344i	0.277564	−	0.468180i
$435$		0			0
$436$		−	0.365842i		−	0.0175207i
$437$	42.9691			2.05549
$438$		0			0
$439$	9.63703			0.459950			0.229975	−	0.973197i	$-0.426136\pi$
$439$	9.63703			0.459950			0.229975	+	0.973197i	$0.426136\pi$
$440$	0.432422	−	0.769313i	0.0206149	−	0.0366756i
$441$		0			0
$442$			12.6682i			0.602565i
$443$	−26.9935			−1.28250			−0.641251	−	0.767331i	$-0.721584\pi$
$443$	−26.9935			−1.28250			−0.641251	+	0.767331i	$0.721584\pi$
$444$		0			0
$445$	−0.474274			−0.0224827
$446$	−1.38903			−0.0657727
$447$		0			0
$448$	−1.34926	+	2.27585i	−0.0637463	+	0.107524i
$449$	−17.6799			−0.834364			−0.417182	−	0.908823i	$-0.636982\pi$
$449$	−17.6799			−0.834364			−0.417182	+	0.908823i	$0.636982\pi$
$450$		0			0
$451$	−10.0998	+	17.9683i	−0.475579	+	0.846092i
$452$	12.2828			0.577736
$453$		0			0
$454$		−	4.04705i		−	0.189937i
$455$	1.54526	+	0.916116i	0.0724428	+	0.0429482i
$456$		0			0
$457$			35.4622i			1.65885i	0.558615	+	0.829427i	$0.311333\pi$
$457$			35.4622i			1.65885i	−0.558615	+	0.829427i	$0.688667\pi$
$458$	15.2265			0.711486
$459$		0			0
$460$			1.39182i			0.0648941i
$461$	25.0507			1.16673			0.583364	−	0.812211i	$-0.301736\pi$
$461$	25.0507			1.16673			0.583364	+	0.812211i	$0.301736\pi$
$462$		0			0
$463$	−7.04858			−0.327576			−0.163788	−	0.986496i	$-0.552371\pi$
$463$	−7.04858			−0.327576			−0.163788	+	0.986496i	$0.552371\pi$
$464$			5.25022i			0.243735i
$465$		0			0
$466$	10.2936			0.476842
$467$			6.93007i			0.320685i	0.987061	+	0.160343i	$0.0512599\pi$
$467$			6.93007i			0.320685i	−0.987061	+	0.160343i	$0.948740\pi$
$468$		0			0
$469$	16.6682	−	28.1151i	0.769666	−	1.29823i
$470$		−	2.67378i		−	0.123332i
$471$		0			0
$472$	−9.10342			−0.419019
$473$	4.24377	+	2.38537i	0.195129	+	0.109680i
$474$		0			0
$475$	40.4924			1.85792
$476$	6.69851	−	11.2987i	0.307026	−	0.517875i
$477$		0			0
$478$	−1.14248			−0.0522557
$479$	−11.7614			−0.537393			−0.268697	−	0.963225i	$-0.586593\pi$
$479$	−11.7614			−0.537393			−0.268697	+	0.963225i	$0.586593\pi$
$480$		0			0
$481$	−18.8750			−0.860627
$482$			0.503225i			0.0229213i
$483$		0			0
$484$	5.71804	+	9.39702i	0.259911	+	0.427137i
$485$	0.0390625			0.00177374
$486$		0			0
$487$	5.14248			0.233028			0.116514	−	0.993189i	$-0.462828\pi$
$487$	5.14248			0.233028			0.116514	+	0.993189i	$0.462828\pi$
$488$			5.98047i			0.270723i
$489$		0			0
$490$	−0.893796	−	1.63416i	−0.0403776	−	0.0738237i
$491$		−	0.424436i		−	0.0191545i	−0.999954	−	0.00957726i	$-0.996951\pi$
$491$		−	0.424436i		−	0.0191545i	0.999954	−	0.00957726i	$-0.00304858\pi$
$492$		0			0
$493$		−	26.0652i		−	1.17392i
$494$		−	20.9618i		−	0.943116i
$495$		0			0
$496$		−	4.28562i		−	0.192430i
$497$	−6.66138	+	11.2361i	−0.298804	+	0.504007i
$498$		0			0
$499$	−28.5974			−1.28020			−0.640098	−	0.768293i	$-0.721106\pi$
$499$	−28.5974			−1.28020			−0.640098	+	0.768293i	$0.721106\pi$
$500$			2.64204i			0.118156i
$501$		0			0
$502$	−7.03261			−0.313881
$503$	−3.83955			−0.171197			−0.0855986	−	0.996330i	$-0.527280\pi$
$503$	−3.83955			−0.171197			−0.0855986	+	0.996330i	$0.527280\pi$
$504$		0			0
$505$		−	3.66733i		−	0.163194i
$506$	−15.1229	−	8.50044i	−0.672297	−	0.377890i
$507$		0			0
$508$		−	10.0247i		−	0.444775i
$509$		−	5.27254i		−	0.233701i	−0.993150	−	0.116851i	$-0.962720\pi$
$509$		−	5.27254i		−	0.233701i	0.993150	−	0.116851i	$-0.0372799\pi$
$510$		0			0
$511$	−4.00000	+	6.74699i	−0.176950	+	0.298469i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	3.85320			0.169957
$515$	5.28196			0.232751
$516$		0			0
$517$	29.0521	+	16.3299i	1.27771	+	0.718187i
$518$	16.8345	+	9.98047i	0.739667	+	0.438517i
$519$		0			0
$520$	0.678979			0.0297752
$521$		−	21.8794i		−	0.958552i	−0.877664	−	0.479276i	$-0.840899\pi$
$521$		−	21.8794i		−	0.958552i	0.877664	−	0.479276i	$-0.159101\pi$
$522$		0			0
$523$	−36.1123			−1.57908			−0.789540	−	0.613699i	$-0.789681\pi$
$523$	−36.1123			−1.57908			−0.789540	+	0.613699i	$0.789681\pi$
$524$	20.0363			0.875289
$525$		0			0
$526$	6.48234			0.282643
$527$			21.2764i			0.926814i
$528$		0			0
$529$	4.36008			0.189569
$530$	3.51208			0.152555
$531$		0			0
$532$	−11.0839	+	18.6957i	−0.480547	+	0.810562i
$533$	−15.8584			−0.686903
$534$		0			0
$535$	3.66733			0.158553
$536$		−	12.3536i		−	0.533596i
$537$		0			0
$538$	−20.7983			−0.896677
$539$	23.2148	+	0.268877i	0.999933	+	0.0115814i
$540$		0			0
$541$			8.58345i			0.369031i	0.982830	+	0.184516i	$0.0590716\pi$
$541$			8.58345i			0.369031i	−0.982830	+	0.184516i	$0.940928\pi$
$542$			22.7955i			0.979150i
$543$		0			0
$544$		−	4.96460i		−	0.212855i
$545$	0.0973463			0.00416986
$546$		0			0
$547$			37.2237i			1.59157i	0.605580	+	0.795785i	$0.292941\pi$
$547$			37.2237i			1.59157i	−0.605580	+	0.795785i	$0.707059\pi$
$548$	−7.17942			−0.306689
$549$		0			0
$550$	−14.2513	−	8.01048i	−0.607677	−	0.341568i
$551$			43.1296i			1.83738i
$552$		0			0
$553$	14.0337	+	8.31996i	0.596773	+	0.353801i
$554$	−17.3277			−0.736181
$555$		0			0
$556$	16.7861			0.711887
$557$			10.9018i			0.461923i	0.972963	+	0.230962i	$0.0741872\pi$
$557$			10.9018i			0.461923i	−0.972963	+	0.230962i	$0.925813\pi$
$558$		0			0
$559$			3.74546i			0.158416i
$560$	−0.605578	−	0.359021i	−0.0255903	−	0.0151714i
$561$		0			0
$562$	−5.17942			−0.218480
$563$	15.0406			0.633886			0.316943	−	0.948445i	$-0.397344\pi$
$563$	15.0406			0.633886			0.316943	+	0.948445i	$0.397344\pi$
$564$		0			0
$565$			3.26832i			0.137499i
$566$		−	12.4534i		−	0.523455i
$567$		0			0
$568$			4.93708i			0.207155i
$569$			20.2986i			0.850961i	0.904967	+	0.425481i	$0.139895\pi$
$569$			20.2986i			0.850961i	−0.904967	+	0.425481i	$0.860105\pi$
$570$		0			0
$571$			14.1995i			0.594231i	0.954842	+	0.297115i	$0.0960246\pi$
$571$			14.1995i			0.594231i	−0.954842	+	0.297115i	$0.903975\pi$
$572$	−4.14680	+	7.37749i	−0.173387	+	0.308468i
$573$		0			0
$574$	14.1440	+	8.38537i	0.590360	+	0.349999i
$575$	25.7831			1.07523
$576$		0			0
$577$		−	7.08245i		−	0.294846i	−0.989074	−	0.147423i	$-0.952902\pi$
$577$		−	7.08245i		−	0.294846i	0.989074	−	0.147423i	$-0.0470979\pi$
$578$			7.64724i			0.318083i
$579$		0			0
$580$	−1.39702			−0.0580081
$581$	14.3024	−	24.1245i	0.593362	−	1.00085i
$582$		0			0
$583$	−21.4497	+	38.1608i	−0.888357	+	1.58046i
$584$			2.96460i			0.122676i
$585$		0			0
$586$		−	11.4570i		−	0.473286i
$587$			5.83223i			0.240722i	0.992730	+	0.120361i	$0.0384052\pi$
$587$			5.83223i			0.240722i	−0.992730	+	0.120361i	$0.961595\pi$
$588$		0			0
$589$		−	35.2056i		−	1.45062i
$590$		−	2.42231i		−	0.0997250i
$591$		0			0
$592$	7.39702			0.304016
$593$	12.4353			0.510656			0.255328	−	0.966854i	$-0.417817\pi$
$593$	12.4353			0.510656			0.255328	+	0.966854i	$0.417817\pi$
$594$		0			0
$595$	3.00645	+	1.78239i	0.123252	+	0.0730710i
$596$			17.5648i			0.719482i
$597$		0			0
$598$		−	13.3472i		−	0.545807i
$599$	−20.1663			−0.823974			−0.411987	−	0.911190i	$-0.635165\pi$
$599$	−20.1663			−0.823974			−0.411987	+	0.911190i	$0.635165\pi$
$600$		0			0
$601$	18.6075			0.759016			0.379508	−	0.925188i	$-0.376093\pi$
$601$	18.6075			0.759016			0.379508	+	0.925188i	$0.376093\pi$
$602$	1.98047	−	3.34055i	0.0807179	−	0.136151i
$603$		0			0
$604$		−	15.0131i		−	0.610874i
$605$	−2.50044	+	1.52150i	−0.101657	+	0.0618579i
$606$		0			0
$607$	21.5401			0.874284			0.437142	−	0.899392i	$-0.355991\pi$
$607$	21.5401			0.874284			0.437142	+	0.899392i	$0.355991\pi$
$608$			8.21482i			0.333155i
$609$		0			0
$610$	−1.59133			−0.0644311
$611$			25.6408i			1.03732i
$612$		0			0
$613$			8.87705i			0.358541i	0.983800	+	0.179270i	$0.0573737\pi$
$613$			8.87705i			0.358541i	−0.983800	+	0.179270i	$0.942626\pi$
$614$			4.78606i			0.193149i
$615$		0			0
$616$	7.59947	−	4.38726i	0.306191	−	0.176768i
$617$	−15.7716			−0.634941			−0.317471	−	0.948268i	$-0.602834\pi$
$617$	−15.7716			−0.634941			−0.317471	+	0.948268i	$0.602834\pi$
$618$		0			0
$619$		−	25.7433i		−	1.03471i	−0.855770	−	0.517356i	$-0.826916\pi$
$619$		−	25.7433i		−	1.03471i	0.855770	−	0.517356i	$-0.173084\pi$
$620$	1.14035			0.0457977
$621$		0			0
$622$	−15.8807			−0.636759
$623$	−4.05647	−	2.40490i	−0.162519	−	0.0963505i
$624$		0			0
$625$	23.9430			0.957719
$626$	−30.6524			−1.22512
$627$		0			0
$628$		−	10.6393i		−	0.424552i
$629$	−36.7232			−1.46425
$630$		0			0
$631$	−44.2330			−1.76089			−0.880444	−	0.474151i	$-0.842755\pi$
$631$	−44.2330			−1.76089			−0.880444	+	0.474151i	$0.842755\pi$
$632$	6.16633			0.245284
$633$		0			0
$634$			6.69851i			0.266032i
$635$	2.66746			0.105855
$636$		0			0
$637$	8.57124	+	15.6711i	0.339605	+	0.620911i
$638$	8.53218	−	15.1794i	0.337792	−	0.600959i
$639$		0			0
$640$	−0.266088			−0.0105181
$641$	7.39702			0.292165			0.146082	−	0.989272i	$-0.453334\pi$
$641$	7.39702			0.292165			0.146082	+	0.989272i	$0.453334\pi$
$642$		0			0
$643$			41.8159i			1.64906i	0.565820	+	0.824529i	$0.308560\pi$
$643$			41.8159i			1.64906i	−0.565820	+	0.824529i	$0.691440\pi$
$644$	−7.05753	+	11.9043i	−0.278106	+	0.469094i
$645$		0			0
$646$		−	40.7833i		−	1.60460i
$647$			5.88804i			0.231483i	0.993279	+	0.115741i	$0.0369244\pi$
$647$			5.88804i			0.231483i	−0.993279	+	0.115741i	$0.963076\pi$
$648$		0			0
$649$	26.3198	+	14.7940i	1.03314	+	0.580717i
$650$		−	12.5779i		−	0.493345i
$651$		0			0
$652$	−0.385373			−0.0150924
$653$	−4.90534			−0.191961			−0.0959804	−	0.995383i	$-0.530599\pi$
$653$	−4.90534			−0.191961			−0.0959804	+	0.995383i	$0.530599\pi$
$654$		0			0
$655$			5.33141i			0.208316i
$656$	6.21482			0.242648
$657$		0			0
$658$	13.5580	−	22.8689i	0.528545	−	0.891522i
$659$			25.7189i			1.00187i	0.865486	+	0.500933i	$0.167010\pi$
$659$			25.7189i			1.00187i	−0.865486	+	0.500933i	$0.832990\pi$
$660$		0			0
$661$			2.70273i			0.105124i	0.998618	+	0.0525621i	$0.0167387\pi$
$661$			2.70273i			0.105124i	−0.998618	+	0.0525621i	$0.983261\pi$
$662$			31.5330i			1.22557i
$663$		0			0
$664$		−	10.6002i		−	0.411367i
$665$	−4.97471	−	2.94929i	−0.192911	−	0.114369i
$666$		0			0
$667$			27.4622i			1.06334i
$668$	0.967388			0.0374294
$669$		0			0
$670$	3.28716			0.126994
$671$	9.71891	−	17.2907i	0.375194	−	0.667501i
$672$		0			0
$673$		−	39.0166i		−	1.50398i	−0.659174	−	0.751990i	$-0.729094\pi$
$673$		−	39.0166i		−	1.50398i	0.659174	−	0.751990i	$-0.270906\pi$
$674$	22.0652			0.849921
$675$		0			0
$676$	6.48879			0.249569
$677$	44.7906			1.72144			0.860721	−	0.509077i	$-0.170013\pi$
$677$	44.7906			1.72144			0.860721	+	0.509077i	$0.170013\pi$
$678$		0			0
$679$	0.334102	+	0.198074i	0.0128217	+	0.00760140i
$680$	1.32102			0.0506588
$681$		0			0
$682$	−6.96460	+	12.3906i	−0.266688	+	0.474459i
$683$	4.57124			0.174914			0.0874568	−	0.996168i	$-0.472126\pi$
$683$	4.57124			0.174914			0.0874568	+	0.996168i	$0.472126\pi$
$684$		0			0
$685$		−	1.91036i		−	0.0729910i
$686$	0.641696	−	18.5091i	0.0245001	−	0.706682i
$687$		0			0
$688$		−	1.46782i		−	0.0559603i
$689$	−33.6799			−1.28310
$690$		0			0
$691$		−	29.1086i		−	1.10734i	−0.832735	−	0.553672i	$-0.813226\pi$
$691$		−	29.1086i		−	1.10734i	0.832735	−	0.553672i	$-0.186774\pi$
$692$	0.385373			0.0146497
$693$		0			0
$694$	27.9365			1.06046
$695$			4.46657i			0.169427i
$696$		0			0
$697$	−30.8541			−1.16868
$698$		−	1.19375i		−	0.0451841i
$699$		0			0
$700$	−6.65074	+	11.2181i	−0.251375	+	0.424006i
$701$			12.1678i			0.459570i	0.973241	+	0.229785i	$0.0738023\pi$
$701$			12.1678i			0.459570i	−0.973241	+	0.229785i	$0.926198\pi$
$702$		0			0
$703$	60.7652			2.29180
$704$	1.62511	−	2.89120i	0.0612486	−	0.108966i
$705$		0			0
$706$	33.4413			1.25858
$707$	18.5960	−	31.3667i	0.699373	−	1.17967i
$708$		0			0
$709$	11.2945			0.424173			0.212087	−	0.977251i	$-0.431974\pi$
$709$	11.2945			0.424173			0.212087	+	0.977251i	$0.431974\pi$
$710$	−1.31370			−0.0493022
$711$		0			0
$712$	−1.78239			−0.0667980
$713$		−	22.4167i		−	0.839513i
$714$		0			0
$715$	−1.96306	−	1.10342i	−0.0734144	−	0.0412654i
$716$	−6.69994			−0.250389
$717$		0			0
$718$	8.45618			0.315582
$719$			47.5180i			1.77212i	0.463567	+	0.886062i	$0.346569\pi$
$719$			47.5180i			1.77212i	−0.463567	+	0.886062i	$0.653431\pi$
$720$		0			0
$721$	45.1766	+	26.7833i	1.68247	+	0.997461i
$722$			48.4832i			1.80436i
$723$		0			0
$724$			26.1650i			0.972414i
$725$			25.8794i			0.961135i
$726$		0			0
$727$			9.38903i			0.348220i	0.984726	+	0.174110i	$0.0557048\pi$
$727$			9.38903i			0.348220i	−0.984726	+	0.174110i	$0.944295\pi$
$728$	5.80731	+	3.44290i	0.215233	+	0.127603i
$729$		0			0
$730$	−0.788845			−0.0291964
$731$			7.28716i			0.269525i
$732$		0			0
$733$	−23.1333			−0.854449			−0.427225	−	0.904145i	$-0.640509\pi$
$733$	−23.1333			−0.854449			−0.427225	+	0.904145i	$0.640509\pi$
$734$	−21.4831			−0.792957
$735$		0			0
$736$			5.23069i			0.192806i
$737$	−20.0760	+	35.7168i	−0.739509	+	1.31564i
$738$		0			0
$739$		−	35.0978i		−	1.29109i	−0.763720	−	0.645547i	$-0.776629\pi$
$739$		−	35.0978i		−	1.29109i	0.763720	−	0.645547i	$-0.223371\pi$
$740$			1.96826i			0.0723547i
$741$		0			0
$742$	30.0389	+	17.8087i	1.10276	+	0.653780i
$743$		−	24.9618i		−	0.915760i	−0.889014	−	0.457880i	$-0.848609\pi$
$743$		−	24.9618i		−	0.915760i	0.889014	−	0.457880i	$-0.151391\pi$
$744$		0			0
$745$	−4.67378			−0.171234
$746$	9.94141			0.363981
$747$		0			0
$748$	−8.06801	+	14.3536i	−0.294996	+	0.524821i
$749$	31.3667	+	18.5960i	1.14611	+	0.679482i
$750$		0			0
$751$	−22.7550			−0.830341			−0.415170	−	0.909744i	$-0.636278\pi$
$751$	−22.7550			−0.830341			−0.415170	+	0.909744i	$0.636278\pi$
$752$		−	10.0485i		−	0.366430i
$753$		0			0
$754$	13.3970			0.487891
$755$	3.99480			0.145386
$756$		0			0
$757$	−27.9627			−1.01632			−0.508160	−	0.861262i	$-0.669674\pi$
$757$	−27.9627			−1.01632			−0.508160	+	0.861262i	$0.669674\pi$
$758$		−	21.3862i		−	0.776783i
$759$		0			0
$760$	−2.18587			−0.0792897
$761$	16.6654			0.604121			0.302060	−	0.953289i	$-0.402326\pi$
$761$	16.6654			0.604121			0.302060	+	0.953289i	$0.402326\pi$
$762$		0			0
$763$	0.832603	+	0.493615i	0.0301423	+	0.0178700i
$764$	10.6277			0.384497
$765$		0			0
$766$	0.706497			0.0255268
$767$			23.2293i			0.838760i
$768$		0			0
$769$	−21.2740			−0.767159			−0.383580	−	0.923508i	$-0.625309\pi$
$769$	−21.2740			−0.767159			−0.383580	+	0.923508i	$0.625309\pi$
$770$	1.16740	+	2.02213i	0.0420701	+	0.0728725i
$771$		0			0
$772$			14.4614i			0.520476i
$773$			25.0759i			0.901917i	0.892545	+	0.450959i	$0.148918\pi$
$773$			25.0759i			0.901917i	−0.892545	+	0.450959i	$0.851082\pi$
$774$		0			0
$775$		−	21.1247i		−	0.758820i
$776$	0.146803			0.00526991
$777$		0			0
$778$			5.30149i			0.190068i
$779$	51.0536			1.82918
$780$		0			0
$781$	8.02329	−	14.2741i	0.287096	−	0.510767i
$782$		−	25.9683i		−	0.928623i
$783$		0			0
$784$	−3.35902	−	6.14142i	−0.119965	−	0.219336i
$785$	2.83098			0.101042
$786$		0			0
$787$	16.3753			0.583715			0.291858	−	0.956462i	$-0.405727\pi$
$787$	16.3753			0.583715			0.291858	+	0.956462i	$0.405727\pi$
$788$			1.35276i			0.0481901i
$789$		0			0
$790$			1.64079i			0.0583766i
$791$	−16.5727	+	27.9539i	−0.589256	+	0.993927i
$792$		0			0
$793$	15.2604			0.541913
$794$	24.9646			0.885960
$795$		0			0
$796$		−	8.52419i		−	0.302132i
$797$		−	0.250116i		−	0.00885955i	−0.999990	−	0.00442977i	$-0.998590\pi$
$797$		−	0.250116i		−	0.00885955i	0.999990	−	0.00442977i	$-0.00141005\pi$
$798$		0			0
$799$			49.8867i			1.76486i
$800$			4.92920i			0.174273i
$801$		0			0
$802$			11.6146i			0.410127i
$803$	4.81780	−	8.57124i	0.170016	−	0.302472i
$804$		0			0
$805$	−3.16759	−	1.87793i	−0.111643	−	0.0661882i
$806$	−10.9356			−0.385192
$807$		0			0
$808$		−	13.7824i		−	0.484863i
$809$		−	7.20596i		−	0.253348i	−0.991944	−	0.126674i	$-0.959570\pi$
$809$		−	7.20596i		−	0.253348i	0.991944	−	0.126674i	$-0.0404302\pi$
$810$		0			0
$811$	−25.7479			−0.904130			−0.452065	−	0.891985i	$-0.649312\pi$
$811$	−25.7479			−0.904130			−0.452065	+	0.891985i	$0.649312\pi$
$812$	−11.9487	−	7.08388i	−0.419318	−	0.248596i
$813$		0			0
$814$	−21.3862	−	12.0210i	−0.749588	−	0.421335i
$815$		−	0.102543i		−	0.00359194i
$816$		0			0
$817$		−	12.0579i		−	0.421853i
$818$		−	15.2032i		−	0.531566i
$819$		0			0
$820$			1.65369i			0.0577493i
$821$		−	11.2060i		−	0.391091i	−0.980695	−	0.195545i	$-0.937352\pi$
$821$		−	11.2060i		−	0.391091i	0.980695	−	0.195545i	$-0.0626477\pi$
$822$		0			0
$823$	−19.7949			−0.690007			−0.345004	−	0.938601i	$-0.612122\pi$
$823$	−19.7949			−0.690007			−0.345004	+	0.938601i	$0.612122\pi$
$824$	19.8504			0.691522
$825$		0			0
$826$	12.2828	−	20.7180i	0.427374	−	0.720873i
$827$			10.9460i			0.380631i	0.981723	+	0.190316i	$0.0609511\pi$
$827$			10.9460i			0.380631i	−0.981723	+	0.190316i	$0.939049\pi$
$828$		0			0
$829$			1.51275i			0.0525399i	0.999655	+	0.0262699i	$0.00836295\pi$
$829$			1.51275i			0.0525399i	−0.999655	+	0.0262699i	$0.991637\pi$
$830$	2.82058			0.0979039
$831$		0			0
$832$	2.55171			0.0884645
$833$	16.6762	+	30.4897i	0.577796	+	1.05640i
$834$		0			0
$835$			0.257410i			0.00890806i
$836$	13.3500	−	23.7507i	0.461718	−	0.821434i
$837$		0			0
$838$	13.9683			0.482526
$839$			3.94420i			0.136169i	0.997680	+	0.0680844i	$0.0216887\pi$
$839$			3.94420i			0.136169i	−0.997680	+	0.0680844i	$0.978311\pi$
$840$		0			0
$841$	1.43521			0.0494901
$842$		−	4.53950i		−	0.156441i
$843$		0			0
$844$			10.8258i			0.372639i
$845$			1.72659i			0.0593965i
$846$		0			0
$847$	−29.1013	+	0.334448i	−0.999934	+	0.0114918i
$848$	13.1989			0.453254
$849$		0			0
$850$		−	24.4715i		−	0.839365i
$851$	38.6915			1.32633
$852$		0			0
$853$	−13.8444			−0.474025			−0.237012	−	0.971507i	$-0.576168\pi$
$853$	−13.8444			−0.474025			−0.237012	+	0.971507i	$0.576168\pi$
$854$	−13.6107	−	8.06918i	−0.465748	−	0.276122i
$855$		0			0
$856$	13.7824			0.471073
$857$	−6.07879			−0.207647			−0.103824	−	0.994596i	$-0.533108\pi$
$857$	−6.07879			−0.207647			−0.103824	+	0.994596i	$0.533108\pi$
$858$		0			0
$859$		−	55.6533i		−	1.89887i	−0.313969	−	0.949433i	$-0.601659\pi$
$859$		−	55.6533i		−	1.89887i	0.313969	−	0.949433i	$-0.398341\pi$
$860$	0.390570			0.0133183
$861$		0			0
$862$	−11.5829			−0.394515
$863$	−32.6539			−1.11155			−0.555775	−	0.831333i	$-0.687579\pi$
$863$	−32.6539			−1.11155			−0.555775	+	0.831333i	$0.687579\pi$
$864$		0			0
$865$			0.102543i			0.00348658i
$866$	−20.9227			−0.710984
$867$		0			0
$868$	9.75344	+	5.78239i	0.331053	+	0.196267i
$869$	−17.8281	−	10.0210i	−0.604776	−	0.339938i
$870$		0			0
$871$	−31.5229			−1.06811
$872$	0.365842			0.0123890
$873$		0			0
$874$			42.9691i			1.45345i
$875$	−6.01290	−	3.56479i	−0.203273	−	0.120512i
$876$		0			0
$877$			37.9324i			1.28089i	0.768006	+	0.640443i	$0.221249\pi$
$877$			37.9324i			1.28089i	−0.768006	+	0.640443i	$0.778751\pi$
$878$			9.63703i			0.325234i
$879$		0			0
$880$	0.769313	+	0.432422i	0.0259335	+	0.0145770i
$881$		−	16.5820i		−	0.558662i	−0.960195	−	0.279331i	$-0.909887\pi$
$881$		−	16.5820i		−	0.558662i	0.960195	−	0.279331i	$-0.0901127\pi$
$882$		0			0
$883$	21.1425			0.711501			0.355751	−	0.934581i	$-0.384225\pi$
$883$	21.1425			0.711501			0.355751	+	0.934581i	$0.384225\pi$
$884$	−12.6682			−0.426078
$885$		0			0
$886$		−	26.9935i		−	0.906866i
$887$	39.9944			1.34288			0.671441	−	0.741058i	$-0.265676\pi$
$887$	39.9944			1.34288			0.671441	+	0.741058i	$0.265676\pi$
$888$		0			0
$889$	22.8148	+	13.5259i	0.765184	+	0.453645i
$890$		−	0.474274i		−	0.0158977i
$891$		0			0
$892$		−	1.38903i		−	0.0465083i
$893$		−	82.5464i		−	2.76231i
$894$		0			0
$895$		−	1.78278i		−	0.0595916i
$896$	−2.27585	−	1.34926i	−0.0760309	−	0.0450755i
$897$		0			0
$898$		−	17.6799i		−	0.589984i
$899$	22.5004			0.750432
$900$		0			0
$901$	−65.5275			−2.18304
$902$	−17.9683	−	10.0998i	−0.598278	−	0.336285i
$903$		0			0
$904$			12.2828i			0.408521i
$905$	−6.96219			−0.231431
$906$		0			0
$907$	8.01810			0.266237			0.133118	−	0.991100i	$-0.457501\pi$
$907$	8.01810			0.266237			0.133118	+	0.991100i	$0.457501\pi$
$908$	4.04705			0.134306
$909$		0			0
$910$	−0.916116	+	1.54526i	−0.0303690	+	0.0512248i
$911$	−19.0629			−0.631583			−0.315791	−	0.948829i	$-0.602270\pi$
$911$	−19.0629			−0.631583			−0.315791	+	0.948829i	$0.602270\pi$
$912$		0			0
$913$	−17.2265	+	30.6472i	−0.570113	+	1.01428i
$914$	−35.4622			−1.17299
$915$		0			0
$916$			15.2265i			0.503097i
$917$	−27.0340	+	45.5996i	−0.892743	+	1.50583i
$918$		0			0
$919$			3.51177i			0.115843i	0.998321	+	0.0579214i	$0.0184473\pi$
$919$			3.51177i			0.115843i	−0.998321	+	0.0579214i	$0.981553\pi$
$920$	−1.39182			−0.0458871
$921$		0			0
$922$			25.0507i			0.825001i
$923$	12.5980			0.414668
$924$		0			0
$925$	36.4614			1.19884
$926$		−	7.04858i		−	0.231631i
$927$		0			0
$928$	−5.25022			−0.172347
$929$		−	45.2446i		−	1.48443i	−0.670163	−	0.742214i	$-0.733776\pi$
$929$		−	45.2446i		−	1.48443i	0.670163	−	0.742214i	$-0.266224\pi$
$930$		0			0
$931$	−27.5937	−	50.4506i	−0.904348	−	1.65345i
$932$			10.2936i			0.337178i
$933$		0			0
$934$	−6.93007			−0.226759
$935$	−3.81933	−	2.14680i	−0.124906	−	0.0702080i
$936$		0			0
$937$	−0.259076			−0.00846364			−0.00423182	−	0.999991i	$-0.501347\pi$
$937$	−0.259076			−0.00846364			−0.00423182	+	0.999991i	$0.501347\pi$
$938$	28.1151	+	16.6682i	0.917989	+	0.544236i
$939$		0			0
$940$	2.67378			0.0872092
$941$	−53.1896			−1.73393			−0.866966	−	0.498368i	$-0.833933\pi$
$941$	−53.1896			−1.73393			−0.866966	+	0.498368i	$0.833933\pi$
$942$		0			0
$943$	32.5078			1.05860
$944$		−	9.10342i		−	0.296291i
$945$		0			0
$946$	−2.38537	+	4.24377i	−0.0775552	+	0.137977i
$947$	38.6099			1.25465			0.627327	−	0.778756i	$-0.284149\pi$
$947$	38.6099			1.25465			0.627327	+	0.778756i	$0.284149\pi$
$948$		0			0
$949$	7.56479			0.245563
$950$			40.4924i			1.31375i
$951$		0			0
$952$	11.2987	+	6.69851i	0.366193	+	0.217100i
$953$			7.87129i			0.254976i	0.991840	+	0.127488i	$0.0406915\pi$
$953$			7.87129i			0.254976i	−0.991840	+	0.127488i	$0.959309\pi$
$954$		0			0
$955$			2.82791i			0.0915089i
$956$		−	1.14248i		−	0.0369504i
$957$		0			0
$958$		−	11.7614i		−	0.379995i
$959$	9.68686	−	16.3393i	0.312805	−	0.527623i
$960$		0			0
$961$	12.6335			0.407531
$962$		−	18.8750i		−	0.608556i
$963$		0			0
$964$	−0.503225			−0.0162078
$965$	−3.84800			−0.123872
$966$		0			0
$967$		−	27.4977i		−	0.884268i	−0.896949	−	0.442134i	$-0.854222\pi$
$967$		−	27.4977i		−	0.884268i	0.896949	−	0.442134i	$-0.145778\pi$
$968$	−9.39702	+	5.71804i	−0.302032	+	0.183785i
$969$		0			0
$970$			0.0390625i			0.00125422i
$971$			4.00287i			0.128458i	0.997935	+	0.0642291i	$0.0204588\pi$
$971$			4.00287i			0.128458i	−0.997935	+	0.0642291i	$0.979541\pi$
$972$		0			0
$973$	−22.6487	+	38.2026i	−0.726083	+	1.22472i
$974$			5.14248i			0.164776i
$975$		0			0
$976$	−5.98047			−0.191430
$977$	43.8107			1.40163			0.700814	−	0.713344i	$-0.252820\pi$
$977$	43.8107			1.40163			0.700814	+	0.713344i	$0.252820\pi$
$978$		0			0
$979$	5.15325	+	2.89658i	0.164699	+	0.0925753i
$980$	1.63416	−	0.893796i	0.0522013	−	0.0285513i
$981$		0			0
$982$	0.424436			0.0135443
$983$			56.5768i			1.80452i	0.431192	+	0.902260i	$0.358093\pi$
$983$			56.5768i			1.80452i	−0.431192	+	0.902260i	$0.641907\pi$
$984$		0			0
$985$	−0.359954			−0.0114691
$986$	26.0652			0.830086
$987$		0			0
$988$	20.9618			0.666884
$989$		−	7.67773i		−	0.244137i
$990$		0			0
$991$	0.731684			0.0232427			0.0116214	−	0.999932i	$-0.496301\pi$
$991$	0.731684			0.0232427			0.0116214	+	0.999932i	$0.496301\pi$
$992$	4.28562			0.136069
$993$		0			0
$994$	−11.2361	−	6.66138i	−0.356387	−	0.211286i
$995$	2.26819			0.0719063
$996$		0			0
$997$	23.1229			0.732311			0.366156	−	0.930554i	$-0.380674\pi$
$997$	23.1229			0.732311			0.366156	+	0.930554i	$0.380674\pi$
$998$		−	28.5974i		−	0.905235i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.e.a.307.6		8
3.2	odd	2		462.2.e.b.307.3	yes	8
7.6	odd	2		1386.2.e.e.307.7		8
11.10	odd	2		1386.2.e.e.307.2		8
12.11	even	2		3696.2.q.c.769.3		8
21.20	even	2		462.2.e.a.307.2	✓	8
33.32	even	2		462.2.e.a.307.7	yes	8
77.76	even	2	inner	1386.2.e.a.307.3		8
84.83	odd	2		3696.2.q.b.769.6		8
132.131	odd	2		3696.2.q.b.769.3		8
231.230	odd	2		462.2.e.b.307.6	yes	8
924.923	even	2		3696.2.q.c.769.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.e.a.307.2	✓	8	21.20	even	2
462.2.e.a.307.7	yes	8	33.32	even	2
462.2.e.b.307.3	yes	8	3.2	odd	2
462.2.e.b.307.6	yes	8	231.230	odd	2
1386.2.e.a.307.3		8	77.76	even	2	inner
1386.2.e.a.307.6		8	1.1	even	1	trivial
1386.2.e.e.307.2		8	11.10	odd	2
1386.2.e.e.307.7		8	7.6	odd	2
3696.2.q.b.769.3		8	132.131	odd	2
3696.2.q.b.769.6		8	84.83	odd	2
3696.2.q.c.769.3		8	12.11	even	2
3696.2.q.c.769.6		8	924.923	even	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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -0.266088i q^{5} +(1.34926 - 2.27585i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -0.266088i q^{5} +(1.34926 - 2.27585i) q^{7} -1.00000i q^{8} +0.266088 q^{10} +(-1.62511 + 2.89120i) q^{11} -2.55171 q^{13} +(2.27585 + 1.34926i) q^{14} +1.00000 q^{16} -4.96460 q^{17} +8.21482 q^{19} +0.266088i q^{20} +(-2.89120 - 1.62511i) q^{22} +5.23069 q^{23} +4.92920 q^{25} -2.55171i q^{26} +(-1.34926 + 2.27585i) q^{28} +5.25022i q^{29} -4.28562i q^{31} +1.00000i q^{32} -4.96460i q^{34} +(-0.605578 - 0.359021i) q^{35} +7.39702 q^{37} +8.21482i q^{38} -0.266088 q^{40} +6.21482 q^{41} -1.46782i q^{43} +(1.62511 - 2.89120i) q^{44} +5.23069i q^{46} -10.0485i q^{47} +(-3.35902 - 6.14142i) q^{49} +4.92920i q^{50} +2.55171 q^{52} +13.1989 q^{53} +(0.769313 + 0.432422i) q^{55} +(-2.27585 - 1.34926i) q^{56} -5.25022 q^{58} -9.10342i q^{59} -5.98047 q^{61} +4.28562 q^{62} -1.00000 q^{64} +0.678979i q^{65} +12.3536 q^{67} +4.96460 q^{68} +(0.359021 - 0.605578i) q^{70} -4.93708 q^{71} -2.96460 q^{73} +7.39702i q^{74} -8.21482 q^{76} +(4.38726 + 7.59947i) q^{77} +6.16633i q^{79} -0.266088i q^{80} +6.21482i q^{82} +10.6002 q^{83} +1.32102i q^{85} +1.46782 q^{86} +(2.89120 + 1.62511i) q^{88} -1.78239i q^{89} +(-3.44290 + 5.80731i) q^{91} -5.23069 q^{92} +10.0485 q^{94} -2.18587i q^{95} +0.146803i q^{97} +(6.14142 - 3.35902i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} - 4 q^{10} + 8 q^{11} + 8 q^{14} + 8 q^{16} - 12 q^{17} - 4 q^{19} + 4 q^{22} + 8 q^{23} - 16 q^{25} + 8 q^{35} + 16 q^{37} + 4 q^{40} - 20 q^{41} - 8 q^{44} - 12 q^{49} + 40 q^{55} - 8 q^{56} - 56 q^{61} + 20 q^{62} - 8 q^{64} + 16 q^{67} + 12 q^{68} - 12 q^{70} - 8 q^{71} + 4 q^{73} + 4 q^{76} - 4 q^{77} + 4 q^{83} + 24 q^{86} - 4 q^{88} + 20 q^{91} - 8 q^{92} + 20 q^{94} - 20 q^{98}+O(q^{100}) \) 8 * q - 8 * q^4 - 4 * q^10 + 8 * q^11 + 8 * q^14 + 8 * q^16 - 12 * q^17 - 4 * q^19 + 4 * q^22 + 8 * q^23 - 16 * q^25 + 8 * q^35 + 16 * q^37 + 4 * q^40 - 20 * q^41 - 8 * q^44 - 12 * q^49 + 40 * q^55 - 8 * q^56 - 56 * q^61 + 20 * q^62 - 8 * q^64 + 16 * q^67 + 12 * q^68 - 12 * q^70 - 8 * q^71 + 4 * q^73 + 4 * q^76 - 4 * q^77 + 4 * q^83 + 24 * q^86 - 4 * q^88 + 20 * q^91 - 8 * q^92 + 20 * q^94 - 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more