LMFDB - Embedded newform 234.2.b.b.181.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,2,Mod(181,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, 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("234.181");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 234 = 2 \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	234.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:	$1.86849940730$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	234.181
Dual form			234.2.b.b.181.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +3.00000i q^{5} +3.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +3.00000i q^{5} +3.00000i q^{7} +1.00000i q^{8} +3.00000 q^{10} +(2.00000 + 3.00000i) q^{13} +3.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -6.00000i q^{19} -3.00000i q^{20} +6.00000 q^{23} -4.00000 q^{25} +(3.00000 - 2.00000i) q^{26} -3.00000i q^{28} -1.00000i q^{32} +3.00000i q^{34} -9.00000 q^{35} +3.00000i q^{37} -6.00000 q^{38} -3.00000 q^{40} -1.00000 q^{43} -6.00000i q^{46} -3.00000i q^{47} -2.00000 q^{49} +4.00000i q^{50} +(-2.00000 - 3.00000i) q^{52} +6.00000 q^{53} -3.00000 q^{56} +6.00000i q^{59} -8.00000 q^{61} -1.00000 q^{64} +(-9.00000 + 6.00000i) q^{65} -12.0000i q^{67} +3.00000 q^{68} +9.00000i q^{70} -15.0000i q^{71} +6.00000i q^{73} +3.00000 q^{74} +6.00000i q^{76} +10.0000 q^{79} +3.00000i q^{80} -6.00000i q^{83} -9.00000i q^{85} +1.00000i q^{86} +6.00000i q^{89} +(-9.00000 + 6.00000i) q^{91} -6.00000 q^{92} -3.00000 q^{94} +18.0000 q^{95} -12.0000i q^{97} +2.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 6 q^{10} + 4 q^{13} + 6 q^{14} + 2 q^{16} - 6 q^{17} + 12 q^{23} - 8 q^{25} + 6 q^{26} - 18 q^{35} - 12 q^{38} - 6 q^{40} - 2 q^{43} - 4 q^{49} - 4 q^{52} + 12 q^{53} - 6 q^{56} - 16 q^{61} - 2 q^{64} - 18 q^{65} + 6 q^{68} + 6 q^{74} + 20 q^{79} - 18 q^{91} - 12 q^{92} - 6 q^{94} + 36 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^10 + 4 * q^13 + 6 * q^14 + 2 * q^16 - 6 * q^17 + 12 * q^23 - 8 * q^25 + 6 * q^26 - 18 * q^35 - 12 * q^38 - 6 * q^40 - 2 * q^43 - 4 * q^49 - 4 * q^52 + 12 * q^53 - 6 * q^56 - 16 * q^61 - 2 * q^64 - 18 * q^65 + 6 * q^68 + 6 * q^74 + 20 * q^79 - 18 * q^91 - 12 * q^92 - 6 * q^94 + 36 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/234\mathbb{Z}\right)^\times$.

$n$	$145$	$209$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			3.00000i			1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$			3.00000i			1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$		0			0
$7$			3.00000i			1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$			3.00000i			1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	3.00000			0.948683
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$14$	3.00000			0.801784
$15$		0			0
$16$	1.00000			0.250000
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		−	3.00000i		−	0.670820i
$21$		0			0
$22$		0			0
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$	3.00000	−	2.00000i	0.588348	−	0.392232i
$27$		0			0
$28$		−	3.00000i		−	0.566947i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			3.00000i			0.514496i
$35$	−9.00000			−1.52128
$36$		0			0
$37$			3.00000i			0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$			3.00000i			0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	−6.00000			−0.973329
$39$		0			0
$40$	−3.00000			−0.474342
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$		0			0
$45$		0			0
$46$		−	6.00000i		−	0.884652i
$47$		−	3.00000i		−	0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$		−	3.00000i		−	0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$		0			0
$49$	−2.00000			−0.285714
$50$			4.00000i			0.565685i
$51$		0			0
$52$	−2.00000	−	3.00000i	−0.277350	−	0.416025i
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		0			0
$56$	−3.00000			−0.400892
$57$		0			0
$58$		0			0
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−9.00000	+	6.00000i	−1.11631	+	0.744208i
$66$		0			0
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	3.00000			0.363803
$69$		0			0
$70$			9.00000i			1.07571i
$71$		−	15.0000i		−	1.78017i	−0.455792	−	0.890086i	$-0.650644\pi$
$71$		−	15.0000i		−	1.78017i	0.455792	−	0.890086i	$-0.349356\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	3.00000			0.348743
$75$		0			0
$76$			6.00000i			0.688247i
$77$		0			0
$78$		0			0
$79$	10.0000			1.12509			0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000			1.12509			0.562544	+	0.826767i	$0.309823\pi$
$80$			3.00000i			0.335410i
$81$		0			0
$82$		0			0
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$		0			0
$85$		−	9.00000i		−	0.976187i
$86$			1.00000i			0.107833i
$87$		0			0
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$	−9.00000	+	6.00000i	−0.943456	+	0.628971i
$92$	−6.00000			−0.625543
$93$		0			0
$94$	−3.00000			−0.309426
$95$	18.0000			1.84676
$96$		0			0
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$			2.00000i			0.202031i
$99$		0			0
$100$	4.00000			0.400000
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$	−3.00000	+	2.00000i	−0.294174	+	0.196116i
$105$		0			0
$106$		−	6.00000i		−	0.582772i
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$			9.00000i			0.862044i	0.902342	+	0.431022i	$0.141847\pi$
$109$			9.00000i			0.862044i	−0.902342	+	0.431022i	$0.858153\pi$
$110$		0			0
$111$		0			0
$112$			3.00000i			0.283473i
$113$	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
$115$			18.0000i			1.67851i
$116$		0			0
$117$		0			0
$118$	6.00000			0.552345
$119$		−	9.00000i		−	0.825029i
$120$		0			0
$121$	11.0000			1.00000
$122$			8.00000i			0.724286i
$123$		0			0
$124$		0			0
$125$			3.00000i			0.268328i
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	6.00000	+	9.00000i	0.526235	+	0.789352i
$131$	3.00000			0.262111			0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000			0.262111			0.131056	+	0.991375i	$0.458163\pi$
$132$		0			0
$133$	18.0000			1.56080
$134$	−12.0000			−1.03664
$135$		0			0
$136$		−	3.00000i		−	0.257248i
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$		0			0
$139$	5.00000			0.424094			0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000			0.424094			0.212047	+	0.977259i	$0.431987\pi$
$140$	9.00000			0.760639
$141$		0			0
$142$	−15.0000			−1.25877
$143$		0			0
$144$		0			0
$145$		0			0
$146$	6.00000			0.496564
$147$		0			0
$148$		−	3.00000i		−	0.246598i
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$		−	15.0000i		−	1.22068i	−0.792139	−	0.610341i	$-0.791032\pi$
$151$		−	15.0000i		−	1.22068i	0.792139	−	0.610341i	$-0.208968\pi$
$152$	6.00000			0.486664
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−22.0000			−1.75579			−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000			−1.75579			−0.877896	+	0.478852i	$0.841053\pi$
$158$		−	10.0000i		−	0.795557i
$159$		0			0
$160$	3.00000			0.237171
$161$			18.0000i			1.41860i
$162$		0			0
$163$			6.00000i			0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$			6.00000i			0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$		0			0
$165$		0			0
$166$	−6.00000			−0.465690
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$	−9.00000			−0.690268
$171$		0			0
$172$	1.00000			0.0762493
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$		−	12.0000i		−	0.907115i
$176$		0			0
$177$		0			0
$178$	6.00000			0.449719
$179$	−15.0000			−1.12115			−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000			−1.12115			−0.560576	+	0.828103i	$0.689420\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$	6.00000	+	9.00000i	0.444750	+	0.667124i
$183$		0			0
$184$			6.00000i			0.442326i
$185$	−9.00000			−0.661693
$186$		0			0
$187$		0			0
$188$			3.00000i			0.218797i
$189$		0			0
$190$		−	18.0000i		−	1.30586i
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$			6.00000i			0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$			6.00000i			0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	−12.0000			−0.861550
$195$		0			0
$196$	2.00000			0.142857
$197$		−	3.00000i		−	0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$		−	3.00000i		−	0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		−	4.00000i		−	0.282843i
$201$		0			0
$202$			12.0000i			0.844317i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		−	14.0000i		−	0.975426i
$207$		0			0
$208$	2.00000	+	3.00000i	0.138675	+	0.208013i
$209$		0			0
$210$		0			0
$211$	−23.0000			−1.58339			−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000			−1.58339			−0.791693	+	0.610920i	$0.790800\pi$
$212$	−6.00000			−0.412082
$213$		0			0
$214$		−	12.0000i		−	0.820303i
$215$		−	3.00000i		−	0.204598i
$216$		0			0
$217$		0			0
$218$	9.00000			0.609557
$219$		0			0
$220$		0			0
$221$	−6.00000	−	9.00000i	−0.403604	−	0.605406i
$222$		0			0
$223$		−	9.00000i		−	0.602685i	−0.953516	−	0.301342i	$-0.902565\pi$
$223$		−	9.00000i		−	0.602685i	0.953516	−	0.301342i	$-0.0974347\pi$
$224$	3.00000			0.200446
$225$		0			0
$226$		−	6.00000i		−	0.399114i
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		0			0
$229$			9.00000i			0.594737i	0.954763	+	0.297368i	$0.0961089\pi$
$229$			9.00000i			0.594737i	−0.954763	+	0.297368i	$0.903891\pi$
$230$	18.0000			1.18688
$231$		0			0
$232$		0			0
$233$	21.0000			1.37576			0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000			1.37576			0.687878	+	0.725826i	$0.258542\pi$
$234$		0			0
$235$	9.00000			0.587095
$236$		−	6.00000i		−	0.390567i
$237$		0			0
$238$	−9.00000			−0.583383
$239$		−	9.00000i		−	0.582162i	−0.956698	−	0.291081i	$-0.905985\pi$
$239$		−	9.00000i		−	0.582162i	0.956698	−	0.291081i	$-0.0940149\pi$
$240$		0			0
$241$		−	30.0000i		−	1.93247i	−0.257663	−	0.966235i	$-0.582952\pi$
$241$		−	30.0000i		−	1.93247i	0.257663	−	0.966235i	$-0.417048\pi$
$242$		−	11.0000i		−	0.707107i
$243$		0			0
$244$	8.00000			0.512148
$245$		−	6.00000i		−	0.383326i
$246$		0			0
$247$	18.0000	−	12.0000i	1.14531	−	0.763542i
$248$		0			0
$249$		0			0
$250$	3.00000			0.189737
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$		0			0
$254$			2.00000i			0.125491i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−3.00000			−0.187135			−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000			−0.187135			−0.0935674	+	0.995613i	$0.529827\pi$
$258$		0			0
$259$	−9.00000			−0.559233
$260$	9.00000	−	6.00000i	0.558156	−	0.372104i
$261$		0			0
$262$		−	3.00000i		−	0.185341i
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$			18.0000i			1.10573i
$266$		−	18.0000i		−	1.10365i
$267$		0			0
$268$			12.0000i			0.733017i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$			15.0000i			0.911185i	0.890188	+	0.455593i	$0.150573\pi$
$271$			15.0000i			0.911185i	−0.890188	+	0.455593i	$0.849427\pi$
$272$	−3.00000			−0.181902
$273$		0			0
$274$	−18.0000			−1.08742
$275$		0			0
$276$		0			0
$277$	8.00000			0.480673			0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000			0.480673			0.240337	+	0.970690i	$0.422742\pi$
$278$		−	5.00000i		−	0.299880i
$279$		0			0
$280$		−	9.00000i		−	0.537853i
$281$			30.0000i			1.78965i	0.446417	+	0.894825i	$0.352700\pi$
$281$			30.0000i			1.78965i	−0.446417	+	0.894825i	$0.647300\pi$
$282$		0			0
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$			15.0000i			0.890086i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		−	6.00000i		−	0.351123i
$293$			9.00000i			0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$			9.00000i			0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$		0			0
$295$	−18.0000			−1.04800
$296$	−3.00000			−0.174371
$297$		0			0
$298$	6.00000			0.347571
$299$	12.0000	+	18.0000i	0.693978	+	1.04097i
$300$		0			0
$301$		−	3.00000i		−	0.172917i
$302$	−15.0000			−0.863153
$303$		0			0
$304$		−	6.00000i		−	0.344124i
$305$		−	24.0000i		−	1.37424i
$306$		0			0
$307$			18.0000i			1.02731i	0.857996	+	0.513657i	$0.171710\pi$
$307$			18.0000i			1.02731i	−0.857996	+	0.513657i	$0.828290\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.0000			1.02069			0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000			1.02069			0.510343	+	0.859971i	$0.329518\pi$
$312$		0			0
$313$	19.0000			1.07394			0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000			1.07394			0.536972	+	0.843600i	$0.319568\pi$
$314$			22.0000i			1.24153i
$315$		0			0
$316$	−10.0000			−0.562544
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$		0			0
$320$		−	3.00000i		−	0.167705i
$321$		0			0
$322$	18.0000			1.00310
$323$			18.0000i			1.00155i
$324$		0			0
$325$	−8.00000	−	12.0000i	−0.443760	−	0.665640i
$326$	6.00000			0.332309
$327$		0			0
$328$		0			0
$329$	9.00000			0.496186
$330$		0			0
$331$			30.0000i			1.64895i	0.565899	+	0.824475i	$0.308529\pi$
$331$			30.0000i			1.64895i	−0.565899	+	0.824475i	$0.691471\pi$
$332$			6.00000i			0.329293i
$333$		0			0
$334$	12.0000			0.656611
$335$	36.0000			1.96689
$336$		0			0
$337$	13.0000			0.708155			0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000			0.708155			0.354078	+	0.935216i	$0.384795\pi$
$338$	12.0000	+	5.00000i	0.652714	+	0.271964i
$339$		0			0
$340$			9.00000i			0.488094i
$341$		0			0
$342$		0			0
$343$			15.0000i			0.809924i
$344$		−	1.00000i		−	0.0539164i
$345$		0			0
$346$		−	6.00000i		−	0.322562i
$347$	−33.0000			−1.77153			−0.885766	−	0.464131i	$-0.846367\pi$
$347$	−33.0000			−1.77153			−0.885766	+	0.464131i	$0.846367\pi$
$348$		0			0
$349$		−	21.0000i		−	1.12410i	−0.827102	−	0.562052i	$-0.810012\pi$
$349$		−	21.0000i		−	1.12410i	0.827102	−	0.562052i	$-0.189988\pi$
$350$	−12.0000			−0.641427
$351$		0			0
$352$		0			0
$353$		−	6.00000i		−	0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$		−	6.00000i		−	0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$		0			0
$355$	45.0000			2.38835
$356$		−	6.00000i		−	0.317999i
$357$		0			0
$358$			15.0000i			0.792775i
$359$		−	24.0000i		−	1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$		−	24.0000i		−	1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		−	2.00000i		−	0.105118i
$363$		0			0
$364$	9.00000	−	6.00000i	0.471728	−	0.314485i
$365$	−18.0000			−0.942163
$366$		0			0
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$	6.00000			0.312772
$369$		0			0
$370$			9.00000i			0.467888i
$371$			18.0000i			0.934513i
$372$		0			0
$373$	4.00000			0.207112			0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000			0.207112			0.103556	+	0.994624i	$0.466978\pi$
$374$		0			0
$375$		0			0
$376$	3.00000			0.154713
$377$		0			0
$378$		0			0
$379$		−	6.00000i		−	0.308199i	−0.988055	−	0.154100i	$-0.950752\pi$
$379$		−	6.00000i		−	0.308199i	0.988055	−	0.154100i	$-0.0492477\pi$
$380$	−18.0000			−0.923381
$381$		0			0
$382$			12.0000i			0.613973i
$383$			9.00000i			0.459879i	0.973205	+	0.229939i	$0.0738528\pi$
$383$			9.00000i			0.459879i	−0.973205	+	0.229939i	$0.926147\pi$
$384$		0			0
$385$		0			0
$386$	6.00000			0.305392
$387$		0			0
$388$			12.0000i			0.609208i
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	−18.0000			−0.910299
$392$		−	2.00000i		−	0.101015i
$393$		0			0
$394$	−3.00000			−0.151138
$395$			30.0000i			1.50946i
$396$		0			0
$397$			18.0000i			0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$			18.0000i			0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$			20.0000i			1.00251i
$399$		0			0
$400$	−4.00000			−0.200000
$401$		−	30.0000i		−	1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$		−	30.0000i		−	1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$		0			0
$403$		0			0
$404$	12.0000			0.597022
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	6.00000i		−	0.296681i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$		−	6.00000i		−	0.296681i	0.988936	−	0.148340i	$-0.0473931\pi$
$410$		0			0
$411$		0			0
$412$	−14.0000			−0.689730
$413$	−18.0000			−0.885722
$414$		0			0
$415$	18.0000			0.883585
$416$	3.00000	−	2.00000i	0.147087	−	0.0980581i
$417$		0			0
$418$		0			0
$419$	−15.0000			−0.732798			−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000			−0.732798			−0.366399	+	0.930458i	$0.619409\pi$
$420$		0			0
$421$		−	15.0000i		−	0.731055i	−0.930800	−	0.365528i	$-0.880889\pi$
$421$		−	15.0000i		−	0.731055i	0.930800	−	0.365528i	$-0.119111\pi$
$422$			23.0000i			1.11962i
$423$		0			0
$424$			6.00000i			0.291386i
$425$	12.0000			0.582086
$426$		0			0
$427$		−	24.0000i		−	1.16144i
$428$	−12.0000			−0.580042
$429$		0			0
$430$	−3.00000			−0.144673
$431$			15.0000i			0.722525i	0.932464	+	0.361262i	$0.117654\pi$
$431$			15.0000i			0.722525i	−0.932464	+	0.361262i	$0.882346\pi$
$432$		0			0
$433$	−11.0000			−0.528626			−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000			−0.528626			−0.264313	+	0.964437i	$0.585145\pi$
$434$		0			0
$435$		0			0
$436$		−	9.00000i		−	0.431022i
$437$		−	36.0000i		−	1.72211i
$438$		0			0
$439$	−10.0000			−0.477274			−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000			−0.477274			−0.238637	+	0.971109i	$0.576701\pi$
$440$		0			0
$441$		0			0
$442$	−9.00000	+	6.00000i	−0.428086	+	0.285391i
$443$	21.0000			0.997740			0.498870	−	0.866677i	$-0.333748\pi$
$443$	21.0000			0.997740			0.498870	+	0.866677i	$0.333748\pi$
$444$		0			0
$445$	−18.0000			−0.853282
$446$	−9.00000			−0.426162
$447$		0			0
$448$		−	3.00000i		−	0.141737i
$449$		−	24.0000i		−	1.13263i	−0.824189	−	0.566315i	$-0.808369\pi$
$449$		−	24.0000i		−	1.13263i	0.824189	−	0.566315i	$-0.191631\pi$
$450$		0			0
$451$		0			0
$452$	−6.00000			−0.282216
$453$		0			0
$454$	12.0000			0.563188
$455$	−18.0000	−	27.0000i	−0.843853	−	1.26578i
$456$		0			0
$457$			18.0000i			0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	9.00000			0.420542
$459$		0			0
$460$		−	18.0000i		−	0.839254i
$461$			15.0000i			0.698620i	0.937007	+	0.349310i	$0.113584\pi$
$461$			15.0000i			0.698620i	−0.937007	+	0.349310i	$0.886416\pi$
$462$		0			0
$463$		−	24.0000i		−	1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$		−	24.0000i		−	1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$		0			0
$465$		0			0
$466$		−	21.0000i		−	0.972806i
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$	36.0000			1.66233
$470$		−	9.00000i		−	0.415139i
$471$		0			0
$472$	−6.00000			−0.276172
$473$		0			0
$474$		0			0
$475$			24.0000i			1.10120i
$476$			9.00000i			0.412514i
$477$		0			0
$478$	−9.00000			−0.411650
$479$		−	39.0000i		−	1.78196i	−0.454047	−	0.890978i	$-0.650020\pi$
$479$		−	39.0000i		−	1.78196i	0.454047	−	0.890978i	$-0.349980\pi$
$480$		0			0
$481$	−9.00000	+	6.00000i	−0.410365	+	0.273576i
$482$	−30.0000			−1.36646
$483$		0			0
$484$	−11.0000			−0.500000
$485$	36.0000			1.63468
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		−	8.00000i		−	0.362143i
$489$		0			0
$490$	−6.00000			−0.271052
$491$	−27.0000			−1.21849			−0.609246	−	0.792981i	$-0.708528\pi$
$491$	−27.0000			−1.21849			−0.609246	+	0.792981i	$0.708528\pi$
$492$		0			0
$493$		0			0
$494$	−12.0000	−	18.0000i	−0.539906	−	0.809858i
$495$		0			0
$496$		0			0
$497$	45.0000			2.01853
$498$		0			0
$499$		−	36.0000i		−	1.61158i	−0.592200	−	0.805791i	$-0.701741\pi$
$499$		−	36.0000i		−	1.61158i	0.592200	−	0.805791i	$-0.298259\pi$
$500$		−	3.00000i		−	0.134164i
$501$		0			0
$502$			12.0000i			0.535586i
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$		−	36.0000i		−	1.60198i
$506$		0			0
$507$		0			0
$508$	2.00000			0.0887357
$509$			6.00000i			0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$			6.00000i			0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$		0			0
$511$	−18.0000			−0.796273
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			3.00000i			0.132324i
$515$			42.0000i			1.85074i
$516$		0			0
$517$		0			0
$518$			9.00000i			0.395437i
$519$		0			0
$520$	−6.00000	−	9.00000i	−0.263117	−	0.394676i
$521$	−27.0000			−1.18289			−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000			−1.18289			−0.591446	+	0.806345i	$0.701443\pi$
$522$		0			0
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$	−3.00000			−0.131056
$525$		0			0
$526$			24.0000i			1.04645i
$527$		0			0
$528$		0			0
$529$	13.0000			0.565217
$530$	18.0000			0.781870
$531$		0			0
$532$	−18.0000			−0.780399
$533$		0			0
$534$		0			0
$535$			36.0000i			1.55642i
$536$	12.0000			0.518321
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	15.0000i		−	0.644900i	−0.946586	−	0.322450i	$-0.895494\pi$
$541$		−	15.0000i		−	0.644900i	0.946586	−	0.322450i	$-0.104506\pi$
$542$	15.0000			0.644305
$543$		0			0
$544$			3.00000i			0.128624i
$545$	−27.0000			−1.15655
$546$		0			0
$547$	−37.0000			−1.58201			−0.791003	−	0.611812i	$-0.790441\pi$
$547$	−37.0000			−1.58201			−0.791003	+	0.611812i	$0.790441\pi$
$548$			18.0000i			0.768922i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			30.0000i			1.27573i
$554$		−	8.00000i		−	0.339887i
$555$		0			0
$556$	−5.00000			−0.212047
$557$			27.0000i			1.14403i	0.820244	+	0.572013i	$0.193837\pi$
$557$			27.0000i			1.14403i	−0.820244	+	0.572013i	$0.806163\pi$
$558$		0			0
$559$	−2.00000	−	3.00000i	−0.0845910	−	0.126886i
$560$	−9.00000			−0.380319
$561$		0			0
$562$	30.0000			1.26547
$563$	−39.0000			−1.64365			−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000			−1.64365			−0.821827	+	0.569737i	$0.807045\pi$
$564$		0			0
$565$			18.0000i			0.757266i
$566$		−	4.00000i		−	0.168133i
$567$		0			0
$568$	15.0000			0.629386
$569$	−45.0000			−1.88650			−0.943249	−	0.332086i	$-0.892248\pi$
$569$	−45.0000			−1.88650			−0.943249	+	0.332086i	$0.892248\pi$
$570$		0			0
$571$	−23.0000			−0.962520			−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000			−0.962520			−0.481260	+	0.876578i	$0.659821\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.0000			−1.00087
$576$		0			0
$577$		−	42.0000i		−	1.74848i	−0.485491	−	0.874241i	$-0.661359\pi$
$577$		−	42.0000i		−	1.74848i	0.485491	−	0.874241i	$-0.338641\pi$
$578$			8.00000i			0.332756i
$579$		0			0
$580$		0			0
$581$	18.0000			0.746766
$582$		0			0
$583$		0			0
$584$	−6.00000			−0.248282
$585$		0			0
$586$	9.00000			0.371787
$587$		−	18.0000i		−	0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$		−	18.0000i		−	0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$		0			0
$589$		0			0
$590$			18.0000i			0.741048i
$591$		0			0
$592$			3.00000i			0.123299i
$593$		−	36.0000i		−	1.47834i	−0.673517	−	0.739171i	$-0.735217\pi$
$593$		−	36.0000i		−	1.47834i	0.673517	−	0.739171i	$-0.264783\pi$
$594$		0			0
$595$	27.0000			1.10689
$596$		−	6.00000i		−	0.245770i
$597$		0			0
$598$	18.0000	−	12.0000i	0.736075	−	0.490716i
$599$	30.0000			1.22577			0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000			1.22577			0.612883	+	0.790173i	$0.290010\pi$
$600$		0			0
$601$	37.0000			1.50926			0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000			1.50926			0.754631	+	0.656150i	$0.227816\pi$
$602$	−3.00000			−0.122271
$603$		0			0
$604$			15.0000i			0.610341i
$605$			33.0000i			1.34164i
$606$		0			0
$607$	−22.0000			−0.892952			−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000			−0.892952			−0.446476	+	0.894795i	$0.647321\pi$
$608$	−6.00000			−0.243332
$609$		0			0
$610$	−24.0000			−0.971732
$611$	9.00000	−	6.00000i	0.364101	−	0.242734i
$612$		0			0
$613$			6.00000i			0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$			6.00000i			0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	18.0000			0.726421
$615$		0			0
$616$		0			0
$617$			12.0000i			0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$			12.0000i			0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$		0			0
$619$			24.0000i			0.964641i	0.875995	+	0.482321i	$0.160206\pi$
$619$			24.0000i			0.964641i	−0.875995	+	0.482321i	$0.839794\pi$
$620$		0			0
$621$		0			0
$622$		−	18.0000i		−	0.721734i
$623$	−18.0000			−0.721155
$624$		0			0
$625$	−29.0000			−1.16000
$626$		−	19.0000i		−	0.759393i
$627$		0			0
$628$	22.0000			0.877896
$629$		−	9.00000i		−	0.358854i
$630$		0			0
$631$		−	15.0000i		−	0.597141i	−0.954388	−	0.298570i	$-0.903490\pi$
$631$		−	15.0000i		−	0.597141i	0.954388	−	0.298570i	$-0.0965097\pi$
$632$			10.0000i			0.397779i
$633$		0			0
$634$	−18.0000			−0.714871
$635$		−	6.00000i		−	0.238103i
$636$		0			0
$637$	−4.00000	−	6.00000i	−0.158486	−	0.237729i
$638$		0			0
$639$		0			0
$640$	−3.00000			−0.118585
$641$	18.0000			0.710957			0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000			0.710957			0.355479	+	0.934684i	$0.384318\pi$
$642$		0			0
$643$			36.0000i			1.41970i	0.704352	+	0.709851i	$0.251238\pi$
$643$			36.0000i			1.41970i	−0.704352	+	0.709851i	$0.748762\pi$
$644$		−	18.0000i		−	0.709299i
$645$		0			0
$646$	18.0000			0.708201
$647$	42.0000			1.65119			0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000			1.65119			0.825595	+	0.564263i	$0.190840\pi$
$648$		0			0
$649$		0			0
$650$	−12.0000	+	8.00000i	−0.470679	+	0.313786i
$651$		0			0
$652$		−	6.00000i		−	0.234978i
$653$	36.0000			1.40879			0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000			1.40879			0.704394	+	0.709809i	$0.251219\pi$
$654$		0			0
$655$			9.00000i			0.351659i
$656$		0			0
$657$		0			0
$658$		−	9.00000i		−	0.350857i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$			30.0000i			1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$			30.0000i			1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$	30.0000			1.16598
$663$		0			0
$664$	6.00000			0.232845
$665$			54.0000i			2.09403i
$666$		0			0
$667$		0			0
$668$		−	12.0000i		−	0.464294i
$669$		0			0
$670$		−	36.0000i		−	1.39080i
$671$		0			0
$672$		0			0
$673$	−1.00000			−0.0385472			−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000			−0.0385472			−0.0192736	+	0.999814i	$0.506135\pi$
$674$		−	13.0000i		−	0.500741i
$675$		0			0
$676$	5.00000	−	12.0000i	0.192308	−	0.461538i
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	36.0000			1.38155
$680$	9.00000			0.345134
$681$		0			0
$682$		0			0
$683$		−	6.00000i		−	0.229584i	−0.993390	−	0.114792i	$-0.963380\pi$
$683$		−	6.00000i		−	0.229584i	0.993390	−	0.114792i	$-0.0366201\pi$
$684$		0			0
$685$	54.0000			2.06323
$686$	15.0000			0.572703
$687$		0			0
$688$	−1.00000			−0.0381246
$689$	12.0000	+	18.0000i	0.457164	+	0.685745i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	−6.00000			−0.228086
$693$		0			0
$694$			33.0000i			1.25266i
$695$			15.0000i			0.568982i
$696$		0			0
$697$		0			0
$698$	−21.0000			−0.794862
$699$		0			0
$700$			12.0000i			0.453557i
$701$	−42.0000			−1.58632			−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000			−1.58632			−0.793159	+	0.609015i	$0.791565\pi$
$702$		0			0
$703$	18.0000			0.678883
$704$		0			0
$705$		0			0
$706$	−6.00000			−0.225813
$707$		−	36.0000i		−	1.35392i
$708$		0			0
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$		−	45.0000i		−	1.68882i
$711$		0			0
$712$	−6.00000			−0.224860
$713$		0			0
$714$		0			0
$715$		0			0
$716$	15.0000			0.560576
$717$		0			0
$718$	−24.0000			−0.895672
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$			42.0000i			1.56416i
$722$			17.0000i			0.632674i
$723$		0			0
$724$	−2.00000			−0.0743294
$725$		0			0
$726$		0			0
$727$	28.0000			1.03846			0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000			1.03846			0.519231	+	0.854634i	$0.326218\pi$
$728$	−6.00000	−	9.00000i	−0.222375	−	0.333562i
$729$		0			0
$730$			18.0000i			0.666210i
$731$	3.00000			0.110959
$732$		0			0
$733$		−	9.00000i		−	0.332423i	−0.986090	−	0.166211i	$-0.946847\pi$
$733$		−	9.00000i		−	0.332423i	0.986090	−	0.166211i	$-0.0531534\pi$
$734$		−	8.00000i		−	0.295285i
$735$		0			0
$736$		−	6.00000i		−	0.221163i
$737$		0			0
$738$		0			0
$739$		−	36.0000i		−	1.32428i	−0.749380	−	0.662141i	$-0.769648\pi$
$739$		−	36.0000i		−	1.32428i	0.749380	−	0.662141i	$-0.230352\pi$
$740$	9.00000			0.330847
$741$		0			0
$742$	18.0000			0.660801
$743$			39.0000i			1.43077i	0.698730	+	0.715386i	$0.253749\pi$
$743$			39.0000i			1.43077i	−0.698730	+	0.715386i	$0.746251\pi$
$744$		0			0
$745$	−18.0000			−0.659469
$746$		−	4.00000i		−	0.146450i
$747$		0			0
$748$		0			0
$749$			36.0000i			1.31541i
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		−	3.00000i		−	0.109399i
$753$		0			0
$754$		0			0
$755$	45.0000			1.63772
$756$		0			0
$757$	−2.00000			−0.0726912			−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000			−0.0726912			−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−6.00000			−0.217930
$759$		0			0
$760$			18.0000i			0.652929i
$761$			30.0000i			1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		0			0
$763$	−27.0000			−0.977466
$764$	12.0000			0.434145
$765$		0			0
$766$	9.00000			0.325183
$767$	−18.0000	+	12.0000i	−0.649942	+	0.433295i
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$		−	6.00000i		−	0.215945i
$773$		−	21.0000i		−	0.755318i	−0.925945	−	0.377659i	$-0.876729\pi$
$773$		−	21.0000i		−	0.755318i	0.925945	−	0.377659i	$-0.123271\pi$
$774$		0			0
$775$		0			0
$776$	12.0000			0.430775
$777$		0			0
$778$		−	30.0000i		−	1.07555i
$779$		0			0
$780$		0			0
$781$		0			0
$782$			18.0000i			0.643679i
$783$		0			0
$784$	−2.00000			−0.0714286
$785$		−	66.0000i		−	2.35564i
$786$		0			0
$787$		−	12.0000i		−	0.427754i	−0.976861	−	0.213877i	$-0.931391\pi$
$787$		−	12.0000i		−	0.427754i	0.976861	−	0.213877i	$-0.0686091\pi$
$788$			3.00000i			0.106871i
$789$		0			0
$790$	30.0000			1.06735
$791$			18.0000i			0.640006i
$792$		0			0
$793$	−16.0000	−	24.0000i	−0.568177	−	0.852265i
$794$	18.0000			0.638796
$795$		0			0
$796$	20.0000			0.708881
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$			9.00000i			0.318397i
$800$			4.00000i			0.141421i
$801$		0			0
$802$	−30.0000			−1.05934
$803$		0			0
$804$		0			0
$805$	−54.0000			−1.90325
$806$		0			0
$807$		0			0
$808$		−	12.0000i		−	0.422159i
$809$	−15.0000			−0.527372			−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000			−0.527372			−0.263686	+	0.964609i	$0.584938\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−18.0000			−0.630512
$816$		0			0
$817$			6.00000i			0.209913i
$818$	−6.00000			−0.209785
$819$		0			0
$820$		0			0
$821$		−	45.0000i		−	1.57051i	−0.619172	−	0.785255i	$-0.712532\pi$
$821$		−	45.0000i		−	1.57051i	0.619172	−	0.785255i	$-0.287468\pi$
$822$		0			0
$823$	14.0000			0.488009			0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000			0.488009			0.244005	+	0.969774i	$0.421539\pi$
$824$			14.0000i			0.487713i
$825$		0			0
$826$			18.0000i			0.626300i
$827$		−	18.0000i		−	0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$		−	18.0000i		−	0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$		0			0
$829$	−20.0000			−0.694629			−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000			−0.694629			−0.347314	+	0.937749i	$0.612906\pi$
$830$		−	18.0000i		−	0.624789i
$831$		0			0
$832$	−2.00000	−	3.00000i	−0.0693375	−	0.104006i
$833$	6.00000			0.207888
$834$		0			0
$835$	−36.0000			−1.24583
$836$		0			0
$837$		0			0
$838$			15.0000i			0.518166i
$839$		−	24.0000i		−	0.828572i	−0.910147	−	0.414286i	$-0.864031\pi$
$839$		−	24.0000i		−	0.828572i	0.910147	−	0.414286i	$-0.135969\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	−15.0000			−0.516934
$843$		0			0
$844$	23.0000			0.791693
$845$	−36.0000	−	15.0000i	−1.23844	−	0.516016i
$846$		0			0
$847$			33.0000i			1.13389i
$848$	6.00000			0.206041
$849$		0			0
$850$		−	12.0000i		−	0.411597i
$851$			18.0000i			0.617032i
$852$		0			0
$853$		−	39.0000i		−	1.33533i	−0.744460	−	0.667667i	$-0.767293\pi$
$853$		−	39.0000i		−	1.33533i	0.744460	−	0.667667i	$-0.232707\pi$
$854$	−24.0000			−0.821263
$855$		0			0
$856$			12.0000i			0.410152i
$857$	−18.0000			−0.614868			−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000			−0.614868			−0.307434	+	0.951569i	$0.599470\pi$
$858$		0			0
$859$	40.0000			1.36478			0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000			1.36478			0.682391	+	0.730987i	$0.260940\pi$
$860$			3.00000i			0.102299i
$861$		0			0
$862$	15.0000			0.510902
$863$			39.0000i			1.32758i	0.747921	+	0.663788i	$0.231052\pi$
$863$			39.0000i			1.32758i	−0.747921	+	0.663788i	$0.768948\pi$
$864$		0			0
$865$			18.0000i			0.612018i
$866$			11.0000i			0.373795i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	36.0000	−	24.0000i	1.21981	−	0.813209i
$872$	−9.00000			−0.304778
$873$		0			0
$874$	−36.0000			−1.21772
$875$	−9.00000			−0.304256
$876$		0			0
$877$			3.00000i			0.101303i	0.998716	+	0.0506514i	$0.0161297\pi$
$877$			3.00000i			0.101303i	−0.998716	+	0.0506514i	$0.983870\pi$
$878$			10.0000i			0.337484i
$879$		0			0
$880$		0			0
$881$	33.0000			1.11180			0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000			1.11180			0.555899	+	0.831250i	$0.312374\pi$
$882$		0			0
$883$	−11.0000			−0.370179			−0.185090	−	0.982722i	$-0.559258\pi$
$883$	−11.0000			−0.370179			−0.185090	+	0.982722i	$0.559258\pi$
$884$	6.00000	+	9.00000i	0.201802	+	0.302703i
$885$		0			0
$886$		−	21.0000i		−	0.705509i
$887$	12.0000			0.402921			0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000			0.402921			0.201460	+	0.979497i	$0.435431\pi$
$888$		0			0
$889$		−	6.00000i		−	0.201234i
$890$			18.0000i			0.603361i
$891$		0			0
$892$			9.00000i			0.301342i
$893$	−18.0000			−0.602347
$894$		0			0
$895$		−	45.0000i		−	1.50418i
$896$	−3.00000			−0.100223
$897$		0			0
$898$	−24.0000			−0.800890
$899$		0			0
$900$		0			0
$901$	−18.0000			−0.599667
$902$		0			0
$903$		0			0
$904$			6.00000i			0.199557i
$905$			6.00000i			0.199447i
$906$		0			0
$907$	−17.0000			−0.564476			−0.282238	−	0.959344i	$-0.591077\pi$
$907$	−17.0000			−0.564476			−0.282238	+	0.959344i	$0.591077\pi$
$908$		−	12.0000i		−	0.398234i
$909$		0			0
$910$	−27.0000	+	18.0000i	−0.895041	+	0.596694i
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$		0			0
$914$	18.0000			0.595387
$915$		0			0
$916$		−	9.00000i		−	0.297368i
$917$			9.00000i			0.297206i
$918$		0			0
$919$	−20.0000			−0.659739			−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000			−0.659739			−0.329870	+	0.944027i	$0.607005\pi$
$920$	−18.0000			−0.593442
$921$		0			0
$922$	15.0000			0.493999
$923$	45.0000	−	30.0000i	1.48119	−	0.987462i
$924$		0			0
$925$		−	12.0000i		−	0.394558i
$926$	−24.0000			−0.788689
$927$		0			0
$928$		0			0
$929$			36.0000i			1.18112i	0.806993	+	0.590561i	$0.201093\pi$
$929$			36.0000i			1.18112i	−0.806993	+	0.590561i	$0.798907\pi$
$930$		0			0
$931$			12.0000i			0.393284i
$932$	−21.0000			−0.687878
$933$		0			0
$934$		−	12.0000i		−	0.392652i
$935$		0			0
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		−	36.0000i		−	1.17544i
$939$		0			0
$940$	−9.00000			−0.293548
$941$			45.0000i			1.46696i	0.679712	+	0.733479i	$0.262105\pi$
$941$			45.0000i			1.46696i	−0.679712	+	0.733479i	$0.737895\pi$
$942$		0			0
$943$		0			0
$944$			6.00000i			0.195283i
$945$		0			0
$946$		0			0
$947$		−	48.0000i		−	1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$		−	48.0000i		−	1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$		0			0
$949$	−18.0000	+	12.0000i	−0.584305	+	0.389536i
$950$	24.0000			0.778663
$951$		0			0
$952$	9.00000			0.291692
$953$	−9.00000			−0.291539			−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000			−0.291539			−0.145769	+	0.989319i	$0.546566\pi$
$954$		0			0
$955$		−	36.0000i		−	1.16493i
$956$			9.00000i			0.291081i
$957$		0			0
$958$	−39.0000			−1.26003
$959$	54.0000			1.74375
$960$		0			0
$961$	31.0000			1.00000
$962$	6.00000	+	9.00000i	0.193448	+	0.290172i
$963$		0			0
$964$			30.0000i			0.966235i
$965$	−18.0000			−0.579441
$966$		0			0
$967$			3.00000i			0.0964735i	0.998836	+	0.0482367i	$0.0153602\pi$
$967$			3.00000i			0.0964735i	−0.998836	+	0.0482367i	$0.984640\pi$
$968$			11.0000i			0.353553i
$969$		0			0
$970$		−	36.0000i		−	1.15589i
$971$	−27.0000			−0.866471			−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000			−0.866471			−0.433236	+	0.901281i	$0.642628\pi$
$972$		0			0
$973$			15.0000i			0.480878i
$974$	−12.0000			−0.384505
$975$		0			0
$976$	−8.00000			−0.256074
$977$			12.0000i			0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$			12.0000i			0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$		0			0
$979$		0			0
$980$			6.00000i			0.191663i
$981$		0			0
$982$			27.0000i			0.861605i
$983$			9.00000i			0.287055i	0.989646	+	0.143528i	$0.0458446\pi$
$983$			9.00000i			0.287055i	−0.989646	+	0.143528i	$0.954155\pi$
$984$		0			0
$985$	9.00000			0.286764
$986$		0			0
$987$		0			0
$988$	−18.0000	+	12.0000i	−0.572656	+	0.381771i
$989$	−6.00000			−0.190789
$990$		0			0
$991$	2.00000			0.0635321			0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000			0.0635321			0.0317660	+	0.999495i	$0.489887\pi$
$992$		0			0
$993$		0			0
$994$		−	45.0000i		−	1.42731i
$995$		−	60.0000i		−	1.90213i
$996$		0			0
$997$	8.00000			0.253363			0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000			0.253363			0.126681	+	0.991943i	$0.459567\pi$
$998$	−36.0000			−1.13956
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	234.2.b.b.181.1		2
3.2	odd	2		26.2.b.a.25.2	yes	2
4.3	odd	2		1872.2.c.f.1585.2		2
12.11	even	2		208.2.f.a.129.1		2
13.5	odd	4		3042.2.a.g.1.1		1
13.8	odd	4		3042.2.a.j.1.1		1
13.12	even	2	inner	234.2.b.b.181.2		2
15.2	even	4		650.2.c.a.649.2		2
15.8	even	4		650.2.c.d.649.1		2
15.14	odd	2		650.2.d.b.51.1		2
21.2	odd	6		1274.2.n.d.753.2		4
21.5	even	6		1274.2.n.c.753.2		4
21.11	odd	6		1274.2.n.d.961.1		4
21.17	even	6		1274.2.n.c.961.1		4
21.20	even	2		1274.2.d.c.883.2		2
24.5	odd	2		832.2.f.d.129.2		2
24.11	even	2		832.2.f.b.129.2		2
39.2	even	12		338.2.c.b.191.1		2
39.5	even	4		338.2.a.d.1.1		1
39.8	even	4		338.2.a.b.1.1		1
39.11	even	12		338.2.c.f.191.1		2
39.17	odd	6		338.2.e.c.23.1		4
39.20	even	12		338.2.c.f.315.1		2
39.23	odd	6		338.2.e.c.147.2		4
39.29	odd	6		338.2.e.c.147.1		4
39.32	even	12		338.2.c.b.315.1		2
39.35	odd	6		338.2.e.c.23.2		4
39.38	odd	2		26.2.b.a.25.1	✓	2
52.51	odd	2		1872.2.c.f.1585.1		2
156.47	odd	4		2704.2.a.k.1.1		1
156.83	odd	4		2704.2.a.j.1.1		1
156.155	even	2		208.2.f.a.129.2		2
195.38	even	4		650.2.c.a.649.1		2
195.44	even	4		8450.2.a.h.1.1		1
195.77	even	4		650.2.c.d.649.2		2
195.164	even	4		8450.2.a.u.1.1		1
195.194	odd	2		650.2.d.b.51.2		2
273.38	even	6		1274.2.n.c.961.2		4
273.116	odd	6		1274.2.n.d.961.2		4
273.194	even	6		1274.2.n.c.753.1		4
273.233	odd	6		1274.2.n.d.753.1		4
273.272	even	2		1274.2.d.c.883.1		2
312.77	odd	2		832.2.f.d.129.1		2
312.155	even	2		832.2.f.b.129.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.b.a.25.1	✓	2	39.38	odd	2
26.2.b.a.25.2	yes	2	3.2	odd	2
208.2.f.a.129.1		2	12.11	even	2
208.2.f.a.129.2		2	156.155	even	2
234.2.b.b.181.1		2	1.1	even	1	trivial
234.2.b.b.181.2		2	13.12	even	2	inner
338.2.a.b.1.1		1	39.8	even	4
338.2.a.d.1.1		1	39.5	even	4
338.2.c.b.191.1		2	39.2	even	12
338.2.c.b.315.1		2	39.32	even	12
338.2.c.f.191.1		2	39.11	even	12
338.2.c.f.315.1		2	39.20	even	12
338.2.e.c.23.1		4	39.17	odd	6
338.2.e.c.23.2		4	39.35	odd	6
338.2.e.c.147.1		4	39.29	odd	6
338.2.e.c.147.2		4	39.23	odd	6
650.2.c.a.649.1		2	195.38	even	4
650.2.c.a.649.2		2	15.2	even	4
650.2.c.d.649.1		2	15.8	even	4
650.2.c.d.649.2		2	195.77	even	4
650.2.d.b.51.1		2	15.14	odd	2
650.2.d.b.51.2		2	195.194	odd	2
832.2.f.b.129.1		2	312.155	even	2
832.2.f.b.129.2		2	24.11	even	2
832.2.f.d.129.1		2	312.77	odd	2
832.2.f.d.129.2		2	24.5	odd	2
1274.2.d.c.883.1		2	273.272	even	2
1274.2.d.c.883.2		2	21.20	even	2
1274.2.n.c.753.1		4	273.194	even	6
1274.2.n.c.753.2		4	21.5	even	6
1274.2.n.c.961.1		4	21.17	even	6
1274.2.n.c.961.2		4	273.38	even	6
1274.2.n.d.753.1		4	273.233	odd	6
1274.2.n.d.753.2		4	21.2	odd	6
1274.2.n.d.961.1		4	21.11	odd	6
1274.2.n.d.961.2		4	273.116	odd	6
1872.2.c.f.1585.1		2	52.51	odd	2
1872.2.c.f.1585.2		2	4.3	odd	2
2704.2.a.j.1.1		1	156.83	odd	4
2704.2.a.k.1.1		1	156.47	odd	4
3042.2.a.g.1.1		1	13.5	odd	4
3042.2.a.j.1.1		1	13.8	odd	4
8450.2.a.h.1.1		1	195.44	even	4
8450.2.a.u.1.1		1	195.164	even	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:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	234.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +3.00000i q^{5} +3.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +3.00000i q^{5} +3.00000i q^{7} +1.00000i q^{8} +3.00000 q^{10} +(2.00000 + 3.00000i) q^{13} +3.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -6.00000i q^{19} -3.00000i q^{20} +6.00000 q^{23} -4.00000 q^{25} +(3.00000 - 2.00000i) q^{26} -3.00000i q^{28} -1.00000i q^{32} +3.00000i q^{34} -9.00000 q^{35} +3.00000i q^{37} -6.00000 q^{38} -3.00000 q^{40} -1.00000 q^{43} -6.00000i q^{46} -3.00000i q^{47} -2.00000 q^{49} +4.00000i q^{50} +(-2.00000 - 3.00000i) q^{52} +6.00000 q^{53} -3.00000 q^{56} +6.00000i q^{59} -8.00000 q^{61} -1.00000 q^{64} +(-9.00000 + 6.00000i) q^{65} -12.0000i q^{67} +3.00000 q^{68} +9.00000i q^{70} -15.0000i q^{71} +6.00000i q^{73} +3.00000 q^{74} +6.00000i q^{76} +10.0000 q^{79} +3.00000i q^{80} -6.00000i q^{83} -9.00000i q^{85} +1.00000i q^{86} +6.00000i q^{89} +(-9.00000 + 6.00000i) q^{91} -6.00000 q^{92} -3.00000 q^{94} +18.0000 q^{95} -12.0000i q^{97} +2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 6 q^{10} + 4 q^{13} + 6 q^{14} + 2 q^{16} - 6 q^{17} + 12 q^{23} - 8 q^{25} + 6 q^{26} - 18 q^{35} - 12 q^{38} - 6 q^{40} - 2 q^{43} - 4 q^{49} - 4 q^{52} + 12 q^{53} - 6 q^{56} - 16 q^{61} - 2 q^{64} - 18 q^{65} + 6 q^{68} + 6 q^{74} + 20 q^{79} - 18 q^{91} - 12 q^{92} - 6 q^{94} + 36 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^10 + 4 * q^13 + 6 * q^14 + 2 * q^16 - 6 * q^17 + 12 * q^23 - 8 * q^25 + 6 * q^26 - 18 * q^35 - 12 * q^38 - 6 * q^40 - 2 * q^43 - 4 * q^49 - 4 * q^52 + 12 * q^53 - 6 * q^56 - 16 * q^61 - 2 * q^64 - 18 * q^65 + 6 * q^68 + 6 * q^74 + 20 * q^79 - 18 * q^91 - 12 * q^92 - 6 * q^94 + 36 * q^95

Introduction

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