LMFDB - Embedded newform 117.2.b.a.64.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,2,Mod(64,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("117.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	117.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.934249703649$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	117.64
Dual form			117.2.b.a.64.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{2} -1.00000 q^{4} -3.46410i q^{7} -1.73205i q^{8} +O(q^{10})$	$q-1.73205i q^{2} -1.00000 q^{4} -3.46410i q^{7} -1.73205i q^{8} +3.46410i q^{11} +(-1.00000 + 3.46410i) q^{13} -6.00000 q^{14} -5.00000 q^{16} +6.00000 q^{17} +3.46410i q^{19} +6.00000 q^{22} +5.00000 q^{25} +(6.00000 + 1.73205i) q^{26} +3.46410i q^{28} -6.00000 q^{29} -3.46410i q^{31} +5.19615i q^{32} -10.3923i q^{34} +6.92820i q^{37} +6.00000 q^{38} +6.92820i q^{41} -4.00000 q^{43} -3.46410i q^{44} -3.46410i q^{47} -5.00000 q^{49} -8.66025i q^{50} +(1.00000 - 3.46410i) q^{52} -6.00000 q^{53} -6.00000 q^{56} +10.3923i q^{58} -10.3923i q^{59} -2.00000 q^{61} -6.00000 q^{62} -1.00000 q^{64} -10.3923i q^{67} -6.00000 q^{68} -3.46410i q^{71} +12.0000 q^{74} -3.46410i q^{76} +12.0000 q^{77} -8.00000 q^{79} +12.0000 q^{82} +3.46410i q^{83} +6.92820i q^{86} +6.00000 q^{88} +6.92820i q^{89} +(12.0000 + 3.46410i) q^{91} -6.00000 q^{94} -13.8564i q^{97} +8.66025i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 2 q^{13} - 12 q^{14} - 10 q^{16} + 12 q^{17} + 12 q^{22} + 10 q^{25} + 12 q^{26} - 12 q^{29} + 12 q^{38} - 8 q^{43} - 10 q^{49} + 2 q^{52} - 12 q^{53} - 12 q^{56} - 4 q^{61} - 12 q^{62} - 2 q^{64} - 12 q^{68} + 24 q^{74} + 24 q^{77} - 16 q^{79} + 24 q^{82} + 12 q^{88} + 24 q^{91} - 12 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^13 - 12 * q^14 - 10 * q^16 + 12 * q^17 + 12 * q^22 + 10 * q^25 + 12 * q^26 - 12 * q^29 + 12 * q^38 - 8 * q^43 - 10 * q^49 + 2 * q^52 - 12 * q^53 - 12 * q^56 - 4 * q^61 - 12 * q^62 - 2 * q^64 - 12 * q^68 + 24 * q^74 + 24 * q^77 - 16 * q^79 + 24 * q^82 + 12 * q^88 + 24 * q^91 - 12 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$		−	1.73205i		−	1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$		−	1.73205i		−	1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		−	3.46410i		−	1.30931i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$		−	3.46410i		−	1.30931i	0.755929	−	0.654654i	$-0.227186\pi$
$8$		−	1.73205i		−	0.612372i
$9$		0			0
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$14$	−6.00000			−1.60357
$15$		0			0
$16$	−5.00000			−1.25000
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$		0			0
$22$	6.00000			1.27920
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$	6.00000	+	1.73205i	1.17670	+	0.339683i
$27$		0			0
$28$			3.46410i			0.654654i
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$		−	3.46410i		−	0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$		−	3.46410i		−	0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$			5.19615i			0.918559i
$33$		0			0
$34$		−	10.3923i		−	1.78227i
$35$		0			0
$36$		0			0
$37$			6.92820i			1.13899i	0.821995	+	0.569495i	$0.192861\pi$
$37$			6.92820i			1.13899i	−0.821995	+	0.569495i	$0.807139\pi$
$38$	6.00000			0.973329
$39$		0			0
$40$		0			0
$41$			6.92820i			1.08200i	0.841021	+	0.541002i	$0.181955\pi$
$41$			6.92820i			1.08200i	−0.841021	+	0.541002i	$0.818045\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		−	3.46410i		−	0.522233i
$45$		0			0
$46$		0			0
$47$		−	3.46410i		−	0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$		−	3.46410i		−	0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		−	8.66025i		−	1.22474i
$51$		0			0
$52$	1.00000	−	3.46410i	0.138675	−	0.480384i
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$		0			0
$56$	−6.00000			−0.801784
$57$		0			0
$58$			10.3923i			1.36458i
$59$		−	10.3923i		−	1.35296i	−0.736460	−	0.676481i	$-0.763504\pi$
$59$		−	10.3923i		−	1.35296i	0.736460	−	0.676481i	$-0.236496\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$	−6.00000			−0.762001
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$		−	10.3923i		−	1.26962i	−0.772667	−	0.634811i	$-0.781078\pi$
$67$		−	10.3923i		−	1.26962i	0.772667	−	0.634811i	$-0.218922\pi$
$68$	−6.00000			−0.727607
$69$		0			0
$70$		0			0
$71$		−	3.46410i		−	0.411113i	−0.978645	−	0.205557i	$-0.934100\pi$
$71$		−	3.46410i		−	0.411113i	0.978645	−	0.205557i	$-0.0659005\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	12.0000			1.39497
$75$		0			0
$76$		−	3.46410i		−	0.397360i
$77$	12.0000			1.36753
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$		0			0
$82$	12.0000			1.32518
$83$			3.46410i			0.380235i	0.981761	+	0.190117i	$0.0608868\pi$
$83$			3.46410i			0.380235i	−0.981761	+	0.190117i	$0.939113\pi$
$84$		0			0
$85$		0			0
$86$			6.92820i			0.747087i
$87$		0			0
$88$	6.00000			0.639602
$89$			6.92820i			0.734388i	0.930144	+	0.367194i	$0.119682\pi$
$89$			6.92820i			0.734388i	−0.930144	+	0.367194i	$0.880318\pi$
$90$		0			0
$91$	12.0000	+	3.46410i	1.25794	+	0.363137i
$92$		0			0
$93$		0			0
$94$	−6.00000			−0.618853
$95$		0			0
$96$		0			0
$97$		−	13.8564i		−	1.40690i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$		−	13.8564i		−	1.40690i	0.710742	−	0.703452i	$-0.248359\pi$
$98$			8.66025i			0.874818i
$99$		0			0
$100$	−5.00000			−0.500000
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$	6.00000	+	1.73205i	0.588348	+	0.169842i
$105$		0			0
$106$			10.3923i			1.00939i
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$			6.92820i			0.663602i	0.943349	+	0.331801i	$0.107656\pi$
$109$			6.92820i			0.663602i	−0.943349	+	0.331801i	$0.892344\pi$
$110$		0			0
$111$		0			0
$112$			17.3205i			1.63663i
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$		0			0
$116$	6.00000			0.557086
$117$		0			0
$118$	−18.0000			−1.65703
$119$		−	20.7846i		−	1.90532i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$			3.46410i			0.313625i
$123$		0			0
$124$			3.46410i			0.311086i
$125$		0			0
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$			12.1244i			1.07165i
$129$		0			0
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	12.0000			1.04053
$134$	−18.0000			−1.55496
$135$		0			0
$136$		−	10.3923i		−	0.891133i
$137$			20.7846i			1.77575i	0.460086	+	0.887875i	$0.347819\pi$
$137$			20.7846i			1.77575i	−0.460086	+	0.887875i	$0.652181\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		0			0
$142$	−6.00000			−0.503509
$143$	−12.0000	−	3.46410i	−1.00349	−	0.289683i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		−	6.92820i		−	0.569495i
$149$		−	13.8564i		−	1.13516i	−0.823318	−	0.567581i	$-0.807880\pi$
$149$		−	13.8564i		−	1.13516i	0.823318	−	0.567581i	$-0.192120\pi$
$150$		0			0
$151$			10.3923i			0.845714i	0.906196	+	0.422857i	$0.138973\pi$
$151$			10.3923i			0.845714i	−0.906196	+	0.422857i	$0.861027\pi$
$152$	6.00000			0.486664
$153$		0			0
$154$		−	20.7846i		−	1.67487i
$155$		0			0
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$			13.8564i			1.10236i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			3.46410i			0.271329i	0.990755	+	0.135665i	$0.0433170\pi$
$163$			3.46410i			0.271329i	−0.990755	+	0.135665i	$0.956683\pi$
$164$		−	6.92820i		−	0.541002i
$165$		0			0
$166$	6.00000			0.465690
$167$		−	17.3205i		−	1.34030i	−0.742225	−	0.670151i	$-0.766230\pi$
$167$		−	17.3205i		−	1.34030i	0.742225	−	0.670151i	$-0.233770\pi$
$168$		0			0
$169$	−11.0000	−	6.92820i	−0.846154	−	0.532939i
$170$		0			0
$171$		0			0
$172$	4.00000			0.304997
$173$	18.0000			1.36851			0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000			1.36851			0.684257	+	0.729241i	$0.260127\pi$
$174$		0			0
$175$		−	17.3205i		−	1.30931i
$176$		−	17.3205i		−	1.30558i
$177$		0			0
$178$	12.0000			0.899438
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	6.00000	−	20.7846i	0.444750	−	1.54066i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$			20.7846i			1.51992i
$188$			3.46410i			0.252646i
$189$		0			0
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	−24.0000			−1.72310
$195$		0			0
$196$	5.00000			0.357143
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	16.0000			1.13421			0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000			1.13421			0.567105	+	0.823646i	$0.308063\pi$
$200$		−	8.66025i		−	0.612372i
$201$		0			0
$202$			10.3923i			0.731200i
$203$			20.7846i			1.45879i
$204$		0			0
$205$		0			0
$206$		−	13.8564i		−	0.965422i
$207$		0			0
$208$	5.00000	−	17.3205i	0.346688	−	1.20096i
$209$	−12.0000			−0.830057
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$	6.00000			0.412082
$213$		0			0
$214$			20.7846i			1.42081i
$215$		0			0
$216$		0			0
$217$	−12.0000			−0.814613
$218$	12.0000			0.812743
$219$		0			0
$220$		0			0
$221$	−6.00000	+	20.7846i	−0.403604	+	1.39812i
$222$		0			0
$223$		−	3.46410i		−	0.231973i	−0.993251	−	0.115987i	$-0.962997\pi$
$223$		−	3.46410i		−	0.231973i	0.993251	−	0.115987i	$-0.0370030\pi$
$224$	18.0000			1.20268
$225$		0			0
$226$		−	10.3923i		−	0.691286i
$227$			17.3205i			1.14960i	0.818293	+	0.574801i	$0.194921\pi$
$227$			17.3205i			1.14960i	−0.818293	+	0.574801i	$0.805079\pi$
$228$		0			0
$229$		−	6.92820i		−	0.457829i	−0.973447	−	0.228914i	$-0.926482\pi$
$229$		−	6.92820i		−	0.457829i	0.973447	−	0.228914i	$-0.0735176\pi$
$230$		0			0
$231$		0			0
$232$			10.3923i			0.682288i
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$		0			0
$236$			10.3923i			0.676481i
$237$		0			0
$238$	−36.0000			−2.33353
$239$			10.3923i			0.672222i	0.941822	+	0.336111i	$0.109112\pi$
$239$			10.3923i			0.672222i	−0.941822	+	0.336111i	$0.890888\pi$
$240$		0			0
$241$			13.8564i			0.892570i	0.894891	+	0.446285i	$0.147253\pi$
$241$			13.8564i			0.892570i	−0.894891	+	0.446285i	$0.852747\pi$
$242$			1.73205i			0.111340i
$243$		0			0
$244$	2.00000			0.128037
$245$		0			0
$246$		0			0
$247$	−12.0000	−	3.46410i	−0.763542	−	0.220416i
$248$	−6.00000			−0.381000
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$		0			0
$254$			13.8564i			0.869428i
$255$		0			0
$256$	19.0000			1.18750
$257$	−18.0000			−1.12281			−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000			−1.12281			−0.561405	+	0.827541i	$0.689739\pi$
$258$		0			0
$259$	24.0000			1.49129
$260$		0			0
$261$		0			0
$262$		−	20.7846i		−	1.28408i
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$		0			0
$266$		−	20.7846i		−	1.27439i
$267$		0			0
$268$			10.3923i			0.634811i
$269$	−6.00000			−0.365826			−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000			−0.365826			−0.182913	+	0.983129i	$0.558553\pi$
$270$		0			0
$271$			10.3923i			0.631288i	0.948878	+	0.315644i	$0.102220\pi$
$271$			10.3923i			0.631288i	−0.948878	+	0.315644i	$0.897780\pi$
$272$	−30.0000			−1.81902
$273$		0			0
$274$	36.0000			2.17484
$275$			17.3205i			1.04447i
$276$		0			0
$277$	−10.0000			−0.600842			−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000			−0.600842			−0.300421	+	0.953807i	$0.597127\pi$
$278$			6.92820i			0.415526i
$279$		0			0
$280$		0			0
$281$		−	6.92820i		−	0.413302i	−0.978415	−	0.206651i	$-0.933744\pi$
$281$		−	6.92820i		−	0.413302i	0.978415	−	0.206651i	$-0.0662565\pi$
$282$		0			0
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$			3.46410i			0.205557i
$285$		0			0
$286$	−6.00000	+	20.7846i	−0.354787	+	1.22902i
$287$	24.0000			1.41668
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	27.7128i		−	1.61900i	−0.587120	−	0.809500i	$-0.699738\pi$
$293$		−	27.7128i		−	1.61900i	0.587120	−	0.809500i	$-0.300262\pi$
$294$		0			0
$295$		0			0
$296$	12.0000			0.697486
$297$		0			0
$298$	−24.0000			−1.39028
$299$		0			0
$300$		0			0
$301$			13.8564i			0.798670i
$302$	18.0000			1.03578
$303$		0			0
$304$		−	17.3205i		−	0.993399i
$305$		0			0
$306$		0			0
$307$		−	10.3923i		−	0.593120i	−0.955014	−	0.296560i	$-0.904160\pi$
$307$		−	10.3923i		−	0.593120i	0.955014	−	0.296560i	$-0.0958395\pi$
$308$	−12.0000			−0.683763
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$		−	24.2487i		−	1.36843i
$315$		0			0
$316$	8.00000			0.450035
$317$		−	13.8564i		−	0.778253i	−0.921184	−	0.389127i	$-0.872777\pi$
$317$		−	13.8564i		−	0.778253i	0.921184	−	0.389127i	$-0.127223\pi$
$318$		0			0
$319$		−	20.7846i		−	1.16371i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			20.7846i			1.15649i
$324$		0			0
$325$	−5.00000	+	17.3205i	−0.277350	+	0.960769i
$326$	6.00000			0.332309
$327$		0			0
$328$	12.0000			0.662589
$329$	−12.0000			−0.661581
$330$		0			0
$331$			3.46410i			0.190404i	0.995458	+	0.0952021i	$0.0303497\pi$
$331$			3.46410i			0.190404i	−0.995458	+	0.0952021i	$0.969650\pi$
$332$		−	3.46410i		−	0.190117i
$333$		0			0
$334$	−30.0000			−1.64153
$335$		0			0
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$	−12.0000	+	19.0526i	−0.652714	+	1.03632i
$339$		0			0
$340$		0			0
$341$	12.0000			0.649836
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$			6.92820i			0.373544i
$345$		0			0
$346$		−	31.1769i		−	1.67608i
$347$	−36.0000			−1.93258			−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000			−1.93258			−0.966291	+	0.257454i	$0.917117\pi$
$348$		0			0
$349$		−	6.92820i		−	0.370858i	−0.982658	−	0.185429i	$-0.940632\pi$
$349$		−	6.92820i		−	0.370858i	0.982658	−	0.185429i	$-0.0593675\pi$
$350$	−30.0000			−1.60357
$351$		0			0
$352$	−18.0000			−0.959403
$353$		−	34.6410i		−	1.84376i	−0.387481	−	0.921878i	$-0.626655\pi$
$353$		−	34.6410i		−	1.84376i	0.387481	−	0.921878i	$-0.373345\pi$
$354$		0			0
$355$		0			0
$356$		−	6.92820i		−	0.367194i
$357$		0			0
$358$		−	20.7846i		−	1.09850i
$359$		−	17.3205i		−	0.914141i	−0.889430	−	0.457071i	$-0.848899\pi$
$359$		−	17.3205i		−	0.914141i	0.889430	−	0.457071i	$-0.151101\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$			17.3205i			0.910346i
$363$		0			0
$364$	−12.0000	−	3.46410i	−0.628971	−	0.181568i
$365$		0			0
$366$		0			0
$367$	−16.0000			−0.835193			−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000			−0.835193			−0.417597	+	0.908633i	$0.637127\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			20.7846i			1.07908i
$372$		0			0
$373$	22.0000			1.13912			0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000			1.13912			0.569558	+	0.821951i	$0.307114\pi$
$374$	36.0000			1.86152
$375$		0			0
$376$	−6.00000			−0.309426
$377$	6.00000	−	20.7846i	0.309016	−	1.07046i
$378$		0			0
$379$			17.3205i			0.889695i	0.895606	+	0.444847i	$0.146742\pi$
$379$			17.3205i			0.889695i	−0.895606	+	0.444847i	$0.853258\pi$
$380$		0			0
$381$		0			0
$382$			41.5692i			2.12687i
$383$		−	3.46410i		−	0.177007i	−0.996076	−	0.0885037i	$-0.971792\pi$
$383$		−	3.46410i		−	0.177007i	0.996076	−	0.0885037i	$-0.0282085\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$			13.8564i			0.703452i
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$		0			0
$392$			8.66025i			0.437409i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	34.6410i		−	1.73858i	−0.494300	−	0.869291i	$-0.664576\pi$
$397$		−	34.6410i		−	1.73858i	0.494300	−	0.869291i	$-0.335424\pi$
$398$		−	27.7128i		−	1.38912i
$399$		0			0
$400$	−25.0000			−1.25000
$401$		−	6.92820i		−	0.345978i	−0.984924	−	0.172989i	$-0.944657\pi$
$401$		−	6.92820i		−	0.345978i	0.984924	−	0.172989i	$-0.0553425\pi$
$402$		0			0
$403$	12.0000	+	3.46410i	0.597763	+	0.172559i
$404$	6.00000			0.298511
$405$		0			0
$406$	36.0000			1.78665
$407$	−24.0000			−1.18964
$408$		0			0
$409$			27.7128i			1.37031i	0.728397	+	0.685155i	$0.240266\pi$
$409$			27.7128i			1.37031i	−0.728397	+	0.685155i	$0.759734\pi$
$410$		0			0
$411$		0			0
$412$	−8.00000			−0.394132
$413$	−36.0000			−1.77144
$414$		0			0
$415$		0			0
$416$	−18.0000	−	5.19615i	−0.882523	−	0.254762i
$417$		0			0
$418$			20.7846i			1.01661i
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$		−	34.6410i		−	1.68830i	−0.536107	−	0.844150i	$-0.680106\pi$
$421$		−	34.6410i		−	1.68830i	0.536107	−	0.844150i	$-0.319894\pi$
$422$			34.6410i			1.68630i
$423$		0			0
$424$			10.3923i			0.504695i
$425$	30.0000			1.45521
$426$		0			0
$427$			6.92820i			0.335279i
$428$	12.0000			0.580042
$429$		0			0
$430$		0			0
$431$			24.2487i			1.16802i	0.811747	+	0.584010i	$0.198517\pi$
$431$			24.2487i			1.16802i	−0.811747	+	0.584010i	$0.801483\pi$
$432$		0			0
$433$	34.0000			1.63394			0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000			1.63394			0.816968	+	0.576683i	$0.195653\pi$
$434$			20.7846i			0.997693i
$435$		0			0
$436$		−	6.92820i		−	0.331801i
$437$		0			0
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$		0			0
$441$		0			0
$442$	36.0000	+	10.3923i	1.71235	+	0.494312i
$443$	36.0000			1.71041			0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000			1.71041			0.855206	+	0.518289i	$0.173431\pi$
$444$		0			0
$445$		0			0
$446$	−6.00000			−0.284108
$447$		0			0
$448$			3.46410i			0.163663i
$449$		−	6.92820i		−	0.326962i	−0.986546	−	0.163481i	$-0.947728\pi$
$449$		−	6.92820i		−	0.326962i	0.986546	−	0.163481i	$-0.0522723\pi$
$450$		0			0
$451$	−24.0000			−1.13012
$452$	−6.00000			−0.282216
$453$		0			0
$454$	30.0000			1.40797
$455$		0			0
$456$		0			0
$457$		−	27.7128i		−	1.29635i	−0.761491	−	0.648175i	$-0.775532\pi$
$457$		−	27.7128i		−	1.29635i	0.761491	−	0.648175i	$-0.224468\pi$
$458$	−12.0000			−0.560723
$459$		0			0
$460$		0			0
$461$			13.8564i			0.645357i	0.946509	+	0.322679i	$0.104583\pi$
$461$			13.8564i			0.645357i	−0.946509	+	0.322679i	$0.895417\pi$
$462$		0			0
$463$		−	17.3205i		−	0.804952i	−0.915430	−	0.402476i	$-0.868150\pi$
$463$		−	17.3205i		−	0.804952i	0.915430	−	0.402476i	$-0.131850\pi$
$464$	30.0000			1.39272
$465$		0			0
$466$		−	10.3923i		−	0.481414i
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	−36.0000			−1.66233
$470$		0			0
$471$		0			0
$472$	−18.0000			−0.828517
$473$		−	13.8564i		−	0.637118i
$474$		0			0
$475$			17.3205i			0.794719i
$476$			20.7846i			0.952661i
$477$		0			0
$478$	18.0000			0.823301
$479$			10.3923i			0.474837i	0.971408	+	0.237418i	$0.0763012\pi$
$479$			10.3923i			0.474837i	−0.971408	+	0.237418i	$0.923699\pi$
$480$		0			0
$481$	−24.0000	−	6.92820i	−1.09431	−	0.315899i
$482$	24.0000			1.09317
$483$		0			0
$484$	1.00000			0.0454545
$485$		0			0
$486$		0			0
$487$			38.1051i			1.72671i	0.504599	+	0.863354i	$0.331640\pi$
$487$			38.1051i			1.72671i	−0.504599	+	0.863354i	$0.668360\pi$
$488$			3.46410i			0.156813i
$489$		0			0
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	−36.0000			−1.62136
$494$	−6.00000	+	20.7846i	−0.269953	+	0.935144i
$495$		0			0
$496$			17.3205i			0.777714i
$497$	−12.0000			−0.538274
$498$		0			0
$499$		−	10.3923i		−	0.465223i	−0.972570	−	0.232612i	$-0.925273\pi$
$499$		−	10.3923i		−	0.465223i	0.972570	−	0.232612i	$-0.0747271\pi$
$500$		0			0
$501$		0			0
$502$		−	20.7846i		−	0.927663i
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	8.00000			0.354943
$509$			41.5692i			1.84252i	0.388943	+	0.921262i	$0.372840\pi$
$509$			41.5692i			1.84252i	−0.388943	+	0.921262i	$0.627160\pi$
$510$		0			0
$511$		0			0
$512$		−	8.66025i		−	0.382733i
$513$		0			0
$514$			31.1769i			1.37515i
$515$		0			0
$516$		0			0
$517$	12.0000			0.527759
$518$		−	41.5692i		−	1.82645i
$519$		0			0
$520$		0			0
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−12.0000			−0.524222
$525$		0			0
$526$		−	41.5692i		−	1.81250i
$527$		−	20.7846i		−	0.905392i
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$	−12.0000			−0.520266
$533$	−24.0000	−	6.92820i	−1.03956	−	0.300094i
$534$		0			0
$535$		0			0
$536$	−18.0000			−0.777482
$537$		0			0
$538$			10.3923i			0.448044i
$539$		−	17.3205i		−	0.746047i
$540$		0			0
$541$			6.92820i			0.297867i	0.988847	+	0.148933i	$0.0475840\pi$
$541$			6.92820i			0.297867i	−0.988847	+	0.148933i	$0.952416\pi$
$542$	18.0000			0.773166
$543$		0			0
$544$			31.1769i			1.33670i
$545$		0			0
$546$		0			0
$547$	20.0000			0.855138			0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000			0.855138			0.427569	+	0.903983i	$0.359370\pi$
$548$		−	20.7846i		−	0.887875i
$549$		0			0
$550$	30.0000			1.27920
$551$		−	20.7846i		−	0.885454i
$552$		0			0
$553$			27.7128i			1.17847i
$554$			17.3205i			0.735878i
$555$		0			0
$556$	4.00000			0.169638
$557$		−	13.8564i		−	0.587115i	−0.955941	−	0.293557i	$-0.905161\pi$
$557$		−	13.8564i		−	0.587115i	0.955941	−	0.293557i	$-0.0948392\pi$
$558$		0			0
$559$	4.00000	−	13.8564i	0.169182	−	0.586064i
$560$		0			0
$561$		0			0
$562$	−12.0000			−0.506189
$563$	12.0000			0.505740			0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000			0.505740			0.252870	+	0.967500i	$0.418626\pi$
$564$		0			0
$565$		0			0
$566$		−	6.92820i		−	0.291214i
$567$		0			0
$568$	−6.00000			−0.251754
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$	12.0000	+	3.46410i	0.501745	+	0.144841i
$573$		0			0
$574$		−	41.5692i		−	1.73507i
$575$		0			0
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		−	32.9090i		−	1.36883i
$579$		0			0
$580$		0			0
$581$	12.0000			0.497844
$582$		0			0
$583$		−	20.7846i		−	0.860811i
$584$		0			0
$585$		0			0
$586$	−48.0000			−1.98286
$587$		−	10.3923i		−	0.428936i	−0.976731	−	0.214468i	$-0.931198\pi$
$587$		−	10.3923i		−	0.428936i	0.976731	−	0.214468i	$-0.0688018\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$		0			0
$591$		0			0
$592$		−	34.6410i		−	1.42374i
$593$		−	6.92820i		−	0.284507i	−0.989830	−	0.142254i	$-0.954565\pi$
$593$		−	6.92820i		−	0.284507i	0.989830	−	0.142254i	$-0.0454349\pi$
$594$		0			0
$595$		0			0
$596$			13.8564i			0.567581i
$597$		0			0
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$	24.0000			0.978167
$603$		0			0
$604$		−	10.3923i		−	0.422857i
$605$		0			0
$606$		0			0
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$	−18.0000			−0.729996
$609$		0			0
$610$		0			0
$611$	12.0000	+	3.46410i	0.485468	+	0.140143i
$612$		0			0
$613$		−	20.7846i		−	0.839482i	−0.907644	−	0.419741i	$-0.862121\pi$
$613$		−	20.7846i		−	0.839482i	0.907644	−	0.419741i	$-0.137879\pi$
$614$	−18.0000			−0.726421
$615$		0			0
$616$		−	20.7846i		−	0.837436i
$617$			6.92820i			0.278919i	0.990228	+	0.139459i	$0.0445365\pi$
$617$			6.92820i			0.278919i	−0.990228	+	0.139459i	$0.955464\pi$
$618$		0			0
$619$			31.1769i			1.25311i	0.779379	+	0.626553i	$0.215535\pi$
$619$			31.1769i			1.25311i	−0.779379	+	0.626553i	$0.784465\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	24.0000			0.961540
$624$		0			0
$625$	25.0000			1.00000
$626$		−	17.3205i		−	0.692267i
$627$		0			0
$628$	−14.0000			−0.558661
$629$			41.5692i			1.65747i
$630$		0			0
$631$			38.1051i			1.51694i	0.651707	+	0.758470i	$0.274053\pi$
$631$			38.1051i			1.51694i	−0.651707	+	0.758470i	$0.725947\pi$
$632$			13.8564i			0.551178i
$633$		0			0
$634$	−24.0000			−0.953162
$635$		0			0
$636$		0			0
$637$	5.00000	−	17.3205i	0.198107	−	0.686264i
$638$	−36.0000			−1.42525
$639$		0			0
$640$		0			0
$641$	6.00000			0.236986			0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000			0.236986			0.118493	+	0.992955i	$0.462194\pi$
$642$		0			0
$643$		−	10.3923i		−	0.409832i	−0.978780	−	0.204916i	$-0.934308\pi$
$643$		−	10.3923i		−	0.409832i	0.978780	−	0.204916i	$-0.0656922\pi$
$644$		0			0
$645$		0			0
$646$	36.0000			1.41640
$647$	24.0000			0.943537			0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		0			0
$649$	36.0000			1.41312
$650$	30.0000	+	8.66025i	1.17670	+	0.339683i
$651$		0			0
$652$		−	3.46410i		−	0.135665i
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$		0			0
$655$		0			0
$656$		−	34.6410i		−	1.35250i
$657$		0			0
$658$			20.7846i			0.810268i
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$		−	20.7846i		−	0.808428i	−0.914665	−	0.404214i	$-0.867545\pi$
$661$		−	20.7846i		−	0.808428i	0.914665	−	0.404214i	$-0.132455\pi$
$662$	6.00000			0.233197
$663$		0			0
$664$	6.00000			0.232845
$665$		0			0
$666$		0			0
$667$		0			0
$668$			17.3205i			0.670151i
$669$		0			0
$670$		0			0
$671$		−	6.92820i		−	0.267460i
$672$		0			0
$673$	−46.0000			−1.77317			−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000			−1.77317			−0.886585	+	0.462566i	$0.846929\pi$
$674$			24.2487i			0.934025i
$675$		0			0
$676$	11.0000	+	6.92820i	0.423077	+	0.266469i
$677$	−6.00000			−0.230599			−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000			−0.230599			−0.115299	+	0.993331i	$0.536783\pi$
$678$		0			0
$679$	−48.0000			−1.84207
$680$		0			0
$681$		0			0
$682$		−	20.7846i		−	0.795884i
$683$			31.1769i			1.19295i	0.802631	+	0.596476i	$0.203433\pi$
$683$			31.1769i			1.19295i	−0.802631	+	0.596476i	$0.796567\pi$
$684$		0			0
$685$		0			0
$686$	−12.0000			−0.458162
$687$		0			0
$688$	20.0000			0.762493
$689$	6.00000	−	20.7846i	0.228582	−	0.791831i
$690$		0			0
$691$			45.0333i			1.71315i	0.516024	+	0.856574i	$0.327412\pi$
$691$			45.0333i			1.71315i	−0.516024	+	0.856574i	$0.672588\pi$
$692$	−18.0000			−0.684257
$693$		0			0
$694$			62.3538i			2.36692i
$695$		0			0
$696$		0			0
$697$			41.5692i			1.57455i
$698$	−12.0000			−0.454207
$699$		0			0
$700$			17.3205i			0.654654i
$701$	42.0000			1.58632			0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000			1.58632			0.793159	+	0.609015i	$0.208435\pi$
$702$		0			0
$703$	−24.0000			−0.905177
$704$		−	3.46410i		−	0.130558i
$705$		0			0
$706$	−60.0000			−2.25813
$707$			20.7846i			0.781686i
$708$		0			0
$709$		−	6.92820i		−	0.260194i	−0.991501	−	0.130097i	$-0.958471\pi$
$709$		−	6.92820i		−	0.260194i	0.991501	−	0.130097i	$-0.0415289\pi$
$710$		0			0
$711$		0			0
$712$	12.0000			0.449719
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−12.0000			−0.448461
$717$		0			0
$718$	−30.0000			−1.11959
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$		−	27.7128i		−	1.03208i
$722$		−	12.1244i		−	0.451222i
$723$		0			0
$724$	10.0000			0.371647
$725$	−30.0000			−1.11417
$726$		0			0
$727$	16.0000			0.593407			0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000			0.593407			0.296704	+	0.954970i	$0.404113\pi$
$728$	6.00000	−	20.7846i	0.222375	−	0.770329i
$729$		0			0
$730$		0			0
$731$	−24.0000			−0.887672
$732$		0			0
$733$			34.6410i			1.27950i	0.768585	+	0.639748i	$0.220961\pi$
$733$			34.6410i			1.27950i	−0.768585	+	0.639748i	$0.779039\pi$
$734$			27.7128i			1.02290i
$735$		0			0
$736$		0			0
$737$	36.0000			1.32608
$738$		0			0
$739$		−	38.1051i		−	1.40172i	−0.713299	−	0.700860i	$-0.752800\pi$
$739$		−	38.1051i		−	1.40172i	0.713299	−	0.700860i	$-0.247200\pi$
$740$		0			0
$741$		0			0
$742$	36.0000			1.32160
$743$		−	3.46410i		−	0.127086i	−0.997979	−	0.0635428i	$-0.979760\pi$
$743$		−	3.46410i		−	0.127086i	0.997979	−	0.0635428i	$-0.0202399\pi$
$744$		0			0
$745$		0			0
$746$		−	38.1051i		−	1.39513i
$747$		0			0
$748$		−	20.7846i		−	0.759961i
$749$			41.5692i			1.51891i
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$			17.3205i			0.631614i
$753$		0			0
$754$	−36.0000	−	10.3923i	−1.31104	−	0.378465i
$755$		0			0
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$	30.0000			1.08965
$759$		0			0
$760$		0			0
$761$			48.4974i			1.75803i	0.476794	+	0.879015i	$0.341799\pi$
$761$			48.4974i			1.75803i	−0.476794	+	0.879015i	$0.658201\pi$
$762$		0			0
$763$	24.0000			0.868858
$764$	24.0000			0.868290
$765$		0			0
$766$	−6.00000			−0.216789
$767$	36.0000	+	10.3923i	1.29988	+	0.375244i
$768$		0			0
$769$			27.7128i			0.999350i	0.866213	+	0.499675i	$0.166547\pi$
$769$			27.7128i			0.999350i	−0.866213	+	0.499675i	$0.833453\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			13.8564i			0.498380i	0.968455	+	0.249190i	$0.0801644\pi$
$773$			13.8564i			0.498380i	−0.968455	+	0.249190i	$0.919836\pi$
$774$		0			0
$775$		−	17.3205i		−	0.622171i
$776$	−24.0000			−0.861550
$777$		0			0
$778$		−	31.1769i		−	1.11775i
$779$	−24.0000			−0.859889
$780$		0			0
$781$	12.0000			0.429394
$782$		0			0
$783$		0			0
$784$	25.0000			0.892857
$785$		0			0
$786$		0			0
$787$		−	10.3923i		−	0.370446i	−0.982697	−	0.185223i	$-0.940699\pi$
$787$		−	10.3923i		−	0.370446i	0.982697	−	0.185223i	$-0.0593007\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	20.7846i		−	0.739016i
$792$		0			0
$793$	2.00000	−	6.92820i	0.0710221	−	0.246028i
$794$	−60.0000			−2.12932
$795$		0			0
$796$	−16.0000			−0.567105
$797$	42.0000			1.48772			0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000			1.48772			0.743858	+	0.668338i	$0.232994\pi$
$798$		0			0
$799$		−	20.7846i		−	0.735307i
$800$			25.9808i			0.918559i
$801$		0			0
$802$	−12.0000			−0.423735
$803$		0			0
$804$		0			0
$805$		0			0
$806$	6.00000	−	20.7846i	0.211341	−	0.732107i
$807$		0			0
$808$			10.3923i			0.365600i
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$		0			0
$811$		−	38.1051i		−	1.33805i	−0.743239	−	0.669026i	$-0.766712\pi$
$811$		−	38.1051i		−	1.33805i	0.743239	−	0.669026i	$-0.233288\pi$
$812$		−	20.7846i		−	0.729397i
$813$		0			0
$814$			41.5692i			1.45700i
$815$		0			0
$816$		0			0
$817$		−	13.8564i		−	0.484774i
$818$	48.0000			1.67828
$819$		0			0
$820$		0			0
$821$		−	13.8564i		−	0.483592i	−0.970327	−	0.241796i	$-0.922264\pi$
$821$		−	13.8564i		−	0.483592i	0.970327	−	0.241796i	$-0.0777365\pi$
$822$		0			0
$823$	−40.0000			−1.39431			−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000			−1.39431			−0.697156	+	0.716919i	$0.745552\pi$
$824$		−	13.8564i		−	0.482711i
$825$		0			0
$826$			62.3538i			2.16957i
$827$		−	24.2487i		−	0.843210i	−0.906780	−	0.421605i	$-0.861467\pi$
$827$		−	24.2487i		−	0.843210i	0.906780	−	0.421605i	$-0.138533\pi$
$828$		0			0
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$		0			0
$832$	1.00000	−	3.46410i	0.0346688	−	0.120096i
$833$	−30.0000			−1.03944
$834$		0			0
$835$		0			0
$836$	12.0000			0.415029
$837$		0			0
$838$		−	20.7846i		−	0.717992i
$839$		−	3.46410i		−	0.119594i	−0.998211	−	0.0597970i	$-0.980955\pi$
$839$		−	3.46410i		−	0.119594i	0.998211	−	0.0597970i	$-0.0190453\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$	−60.0000			−2.06774
$843$		0			0
$844$	20.0000			0.688428
$845$		0			0
$846$		0			0
$847$			3.46410i			0.119028i
$848$	30.0000			1.03020
$849$		0			0
$850$		−	51.9615i		−	1.78227i
$851$		0			0
$852$		0			0
$853$		−	20.7846i		−	0.711651i	−0.934552	−	0.355826i	$-0.884200\pi$
$853$		−	20.7846i		−	0.711651i	0.934552	−	0.355826i	$-0.115800\pi$
$854$	12.0000			0.410632
$855$		0			0
$856$			20.7846i			0.710403i
$857$	−42.0000			−1.43469			−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000			−1.43469			−0.717346	+	0.696717i	$0.754643\pi$
$858$		0			0
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$		0			0
$861$		0			0
$862$	42.0000			1.43053
$863$		−	31.1769i		−	1.06127i	−0.847599	−	0.530637i	$-0.821953\pi$
$863$		−	31.1769i		−	1.06127i	0.847599	−	0.530637i	$-0.178047\pi$
$864$		0			0
$865$		0			0
$866$		−	58.8897i		−	2.00115i
$867$		0			0
$868$	12.0000			0.407307
$869$		−	27.7128i		−	0.940093i
$870$		0			0
$871$	36.0000	+	10.3923i	1.21981	+	0.352130i
$872$	12.0000			0.406371
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			48.4974i			1.63764i	0.574049	+	0.818821i	$0.305372\pi$
$877$			48.4974i			1.63764i	−0.574049	+	0.818821i	$0.694628\pi$
$878$		−	13.8564i		−	0.467631i
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$	6.00000	−	20.7846i	0.201802	−	0.699062i
$885$		0			0
$886$		−	62.3538i		−	2.09482i
$887$	−48.0000			−1.61168			−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000			−1.61168			−0.805841	+	0.592132i	$0.798286\pi$
$888$		0			0
$889$			27.7128i			0.929458i
$890$		0			0
$891$		0			0
$892$			3.46410i			0.115987i
$893$	12.0000			0.401565
$894$		0			0
$895$		0			0
$896$	42.0000			1.40312
$897$		0			0
$898$	−12.0000			−0.400445
$899$			20.7846i			0.693206i
$900$		0			0
$901$	−36.0000			−1.19933
$902$			41.5692i			1.38410i
$903$		0			0
$904$		−	10.3923i		−	0.345643i
$905$		0			0
$906$		0			0
$907$	−44.0000			−1.46100			−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000			−1.46100			−0.730498	+	0.682915i	$0.760712\pi$
$908$		−	17.3205i		−	0.574801i
$909$		0			0
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$	−12.0000			−0.397142
$914$	−48.0000			−1.58770
$915$		0			0
$916$			6.92820i			0.228914i
$917$		−	41.5692i		−	1.37274i
$918$		0			0
$919$	−32.0000			−1.05558			−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000			−1.05558			−0.527791	+	0.849374i	$0.676980\pi$
$920$		0			0
$921$		0			0
$922$	24.0000			0.790398
$923$	12.0000	+	3.46410i	0.394985	+	0.114022i
$924$		0			0
$925$			34.6410i			1.13899i
$926$	−30.0000			−0.985861
$927$		0			0
$928$		−	31.1769i		−	1.02343i
$929$			20.7846i			0.681921i	0.940078	+	0.340960i	$0.110752\pi$
$929$			20.7846i			0.681921i	−0.940078	+	0.340960i	$0.889248\pi$
$930$		0			0
$931$		−	17.3205i		−	0.567657i
$932$	−6.00000			−0.196537
$933$		0			0
$934$			20.7846i			0.680093i
$935$		0			0
$936$		0			0
$937$	−38.0000			−1.24141			−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000			−1.24141			−0.620703	+	0.784046i	$0.713153\pi$
$938$			62.3538i			2.03592i
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$			51.9615i			1.69120i
$945$		0			0
$946$	−24.0000			−0.780307
$947$		−	51.9615i		−	1.68852i	−0.535932	−	0.844261i	$-0.680040\pi$
$947$		−	51.9615i		−	1.68852i	0.535932	−	0.844261i	$-0.319960\pi$
$948$		0			0
$949$		0			0
$950$	30.0000			0.973329
$951$		0			0
$952$	−36.0000			−1.16677
$953$	−42.0000			−1.36051			−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000			−1.36051			−0.680257	+	0.732974i	$0.738132\pi$
$954$		0			0
$955$		0			0
$956$		−	10.3923i		−	0.336111i
$957$		0			0
$958$	18.0000			0.581554
$959$	72.0000			2.32500
$960$		0			0
$961$	19.0000			0.612903
$962$	−12.0000	+	41.5692i	−0.386896	+	1.34025i
$963$		0			0
$964$		−	13.8564i		−	0.446285i
$965$		0			0
$966$		0			0
$967$			10.3923i			0.334194i	0.985940	+	0.167097i	$0.0534393\pi$
$967$			10.3923i			0.334194i	−0.985940	+	0.167097i	$0.946561\pi$
$968$			1.73205i			0.0556702i
$969$		0			0
$970$		0			0
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$		0			0
$973$			13.8564i			0.444216i
$974$	66.0000			2.11478
$975$		0			0
$976$	10.0000			0.320092
$977$		−	48.4974i		−	1.55157i	−0.630997	−	0.775785i	$-0.717354\pi$
$977$		−	48.4974i		−	1.55157i	0.630997	−	0.775785i	$-0.282646\pi$
$978$		0			0
$979$	−24.0000			−0.767043
$980$		0			0
$981$		0			0
$982$		−	20.7846i		−	0.663264i
$983$			51.9615i			1.65732i	0.559756	+	0.828658i	$0.310895\pi$
$983$			51.9615i			1.65732i	−0.559756	+	0.828658i	$0.689105\pi$
$984$		0			0
$985$		0			0
$986$			62.3538i			1.98575i
$987$		0			0
$988$	12.0000	+	3.46410i	0.381771	+	0.110208i
$989$		0			0
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$	18.0000			0.571501
$993$		0			0
$994$			20.7846i			0.659248i
$995$		0			0
$996$		0			0
$997$	38.0000			1.20347			0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000			1.20347			0.601736	+	0.798695i	$0.294476\pi$
$998$	−18.0000			−0.569780
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.b.a.64.1		2
3.2	odd	2		39.2.b.a.25.2	yes	2
4.3	odd	2		1872.2.c.e.1585.2		2
12.11	even	2		624.2.c.e.337.2		2
13.5	odd	4		1521.2.a.l.1.1		2
13.8	odd	4		1521.2.a.l.1.2		2
13.12	even	2	inner	117.2.b.a.64.2		2
15.2	even	4		975.2.h.f.649.2		4
15.8	even	4		975.2.h.f.649.3		4
15.14	odd	2		975.2.b.d.376.1		2
21.20	even	2		1911.2.c.d.883.2		2
24.5	odd	2		2496.2.c.k.961.1		2
24.11	even	2		2496.2.c.d.961.2		2
39.2	even	12		507.2.e.e.22.1		4
39.5	even	4		507.2.a.f.1.2		2
39.8	even	4		507.2.a.f.1.1		2
39.11	even	12		507.2.e.e.22.2		4
39.17	odd	6		507.2.j.a.361.1		2
39.20	even	12		507.2.e.e.484.2		4
39.23	odd	6		507.2.j.c.316.1		2
39.29	odd	6		507.2.j.a.316.1		2
39.32	even	12		507.2.e.e.484.1		4
39.35	odd	6		507.2.j.c.361.1		2
39.38	odd	2		39.2.b.a.25.1	✓	2
52.51	odd	2		1872.2.c.e.1585.1		2
156.47	odd	4		8112.2.a.bv.1.2		2
156.83	odd	4		8112.2.a.bv.1.1		2
156.155	even	2		624.2.c.e.337.1		2
195.38	even	4		975.2.h.f.649.1		4
195.77	even	4		975.2.h.f.649.4		4
195.194	odd	2		975.2.b.d.376.2		2
273.272	even	2		1911.2.c.d.883.1		2
312.77	odd	2		2496.2.c.k.961.2		2
312.155	even	2		2496.2.c.d.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.b.a.25.1	✓	2	39.38	odd	2
39.2.b.a.25.2	yes	2	3.2	odd	2
117.2.b.a.64.1		2	1.1	even	1	trivial
117.2.b.a.64.2		2	13.12	even	2	inner
507.2.a.f.1.1		2	39.8	even	4
507.2.a.f.1.2		2	39.5	even	4
507.2.e.e.22.1		4	39.2	even	12
507.2.e.e.22.2		4	39.11	even	12
507.2.e.e.484.1		4	39.32	even	12
507.2.e.e.484.2		4	39.20	even	12
507.2.j.a.316.1		2	39.29	odd	6
507.2.j.a.361.1		2	39.17	odd	6
507.2.j.c.316.1		2	39.23	odd	6
507.2.j.c.361.1		2	39.35	odd	6
624.2.c.e.337.1		2	156.155	even	2
624.2.c.e.337.2		2	12.11	even	2
975.2.b.d.376.1		2	15.14	odd	2
975.2.b.d.376.2		2	195.194	odd	2
975.2.h.f.649.1		4	195.38	even	4
975.2.h.f.649.2		4	15.2	even	4
975.2.h.f.649.3		4	15.8	even	4
975.2.h.f.649.4		4	195.77	even	4
1521.2.a.l.1.1		2	13.5	odd	4
1521.2.a.l.1.2		2	13.8	odd	4
1872.2.c.e.1585.1		2	52.51	odd	2
1872.2.c.e.1585.2		2	4.3	odd	2
1911.2.c.d.883.1		2	273.272	even	2
1911.2.c.d.883.2		2	21.20	even	2
2496.2.c.d.961.1		2	312.155	even	2
2496.2.c.d.961.2		2	24.11	even	2
2496.2.c.k.961.1		2	24.5	odd	2
2496.2.c.k.961.2		2	312.77	odd	2
8112.2.a.bv.1.1		2	156.83	odd	4
8112.2.a.bv.1.2		2	156.47	odd	4

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{2} -1.00000 q^{4} -3.46410i q^{7} -1.73205i q^{8} +O(q^{10})\)	\(q-1.73205i q^{2} -1.00000 q^{4} -3.46410i q^{7} -1.73205i q^{8} +3.46410i q^{11} +(-1.00000 + 3.46410i) q^{13} -6.00000 q^{14} -5.00000 q^{16} +6.00000 q^{17} +3.46410i q^{19} +6.00000 q^{22} +5.00000 q^{25} +(6.00000 + 1.73205i) q^{26} +3.46410i q^{28} -6.00000 q^{29} -3.46410i q^{31} +5.19615i q^{32} -10.3923i q^{34} +6.92820i q^{37} +6.00000 q^{38} +6.92820i q^{41} -4.00000 q^{43} -3.46410i q^{44} -3.46410i q^{47} -5.00000 q^{49} -8.66025i q^{50} +(1.00000 - 3.46410i) q^{52} -6.00000 q^{53} -6.00000 q^{56} +10.3923i q^{58} -10.3923i q^{59} -2.00000 q^{61} -6.00000 q^{62} -1.00000 q^{64} -10.3923i q^{67} -6.00000 q^{68} -3.46410i q^{71} +12.0000 q^{74} -3.46410i q^{76} +12.0000 q^{77} -8.00000 q^{79} +12.0000 q^{82} +3.46410i q^{83} +6.92820i q^{86} +6.00000 q^{88} +6.92820i q^{89} +(12.0000 + 3.46410i) q^{91} -6.00000 q^{94} -13.8564i q^{97} +8.66025i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 2 q^{13} - 12 q^{14} - 10 q^{16} + 12 q^{17} + 12 q^{22} + 10 q^{25} + 12 q^{26} - 12 q^{29} + 12 q^{38} - 8 q^{43} - 10 q^{49} + 2 q^{52} - 12 q^{53} - 12 q^{56} - 4 q^{61} - 12 q^{62} - 2 q^{64} - 12 q^{68} + 24 q^{74} + 24 q^{77} - 16 q^{79} + 24 q^{82} + 12 q^{88} + 24 q^{91} - 12 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^13 - 12 * q^14 - 10 * q^16 + 12 * q^17 + 12 * q^22 + 10 * q^25 + 12 * q^26 - 12 * q^29 + 12 * q^38 - 8 * q^43 - 10 * q^49 + 2 * q^52 - 12 * q^53 - 12 * q^56 - 4 * q^61 - 12 * q^62 - 2 * q^64 - 12 * q^68 + 24 * q^74 + 24 * q^77 - 16 * q^79 + 24 * q^82 + 12 * q^88 + 24 * q^91 - 12 * q^94

Introduction

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