LMFDB - Embedded newform 234.2.b.a.181.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [234,2,Mod(181,234)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,-2,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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 78)
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.a.181.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{5} -2.00000i q^{7} +1.00000i q^{8} -2.00000 q^{10} +(-3.00000 - 2.00000i) q^{13} -2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -6.00000i q^{19} +2.00000i q^{20} -4.00000 q^{23} +1.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} +2.00000i q^{28} +10.0000 q^{29} +10.0000i q^{31} -1.00000i q^{32} -2.00000i q^{34} -4.00000 q^{35} +8.00000i q^{37} -6.00000 q^{38} +2.00000 q^{40} -10.0000i q^{41} +4.00000 q^{43} +4.00000i q^{46} +12.0000i q^{47} +3.00000 q^{49} -1.00000i q^{50} +(3.00000 + 2.00000i) q^{52} +6.00000 q^{53} +2.00000 q^{56} -10.0000i q^{58} -4.00000i q^{59} +2.00000 q^{61} +10.0000 q^{62} -1.00000 q^{64} +(-4.00000 + 6.00000i) q^{65} -2.00000i q^{67} -2.00000 q^{68} +4.00000i q^{70} -4.00000i q^{73} +8.00000 q^{74} +6.00000i q^{76} -2.00000i q^{80} -10.0000 q^{82} +4.00000i q^{83} -4.00000i q^{85} -4.00000i q^{86} +6.00000i q^{89} +(-4.00000 + 6.00000i) q^{91} +4.00000 q^{92} +12.0000 q^{94} -12.0000 q^{95} -12.0000i q^{97} -3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{10} - 6 q^{13} - 4 q^{14} + 2 q^{16} + 4 q^{17} - 8 q^{23} + 2 q^{25} - 4 q^{26} + 20 q^{29} - 8 q^{35} - 12 q^{38} + 4 q^{40} + 8 q^{43} + 6 q^{49} + 6 q^{52} + 12 q^{53} + 4 q^{56}+ \cdots - 24 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^10 - 6 * q^13 - 4 * q^14 + 2 * q^16 + 4 * q^17 - 8 * q^23 + 2 * q^25 - 4 * q^26 + 20 * q^29 - 8 * q^35 - 12 * q^38 + 4 * q^40 + 8 * q^43 + 6 * q^49 + 6 * q^52 + 12 * q^53 + 4 * q^56 + 4 * q^61 + 20 * q^62 - 2 * q^64 - 8 * q^65 - 4 * q^68 + 16 * q^74 - 20 * q^82 - 8 * q^91 + 8 * q^92 + 24 * q^94 - 24 * q^95

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
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−2.00000			−0.632456
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$14$	−2.00000			−0.534522
$15$		0			0
$16$	1.00000			0.250000
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$			2.00000i			0.447214i
$21$		0			0
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$	−2.00000	+	3.00000i	−0.392232	+	0.588348i
$27$		0			0
$28$			2.00000i			0.377964i
$29$	10.0000			1.85695			0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000			1.85695			0.928477	+	0.371391i	$0.121119\pi$
$30$		0			0
$31$			10.0000i			1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$			10.0000i			1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	2.00000i		−	0.342997i
$35$	−4.00000			−0.676123
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	−6.00000			−0.973329
$39$		0			0
$40$	2.00000			0.316228
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$		0			0
$46$			4.00000i			0.589768i
$47$			12.0000i			1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$			12.0000i			1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		−	1.00000i		−	0.141421i
$51$		0			0
$52$	3.00000	+	2.00000i	0.416025	+	0.277350i
$53$	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
$56$	2.00000			0.267261
$57$		0			0
$58$		−	10.0000i		−	1.31306i
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$	10.0000			1.27000
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−4.00000	+	6.00000i	−0.496139	+	0.744208i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	−2.00000			−0.242536
$69$		0			0
$70$			4.00000i			0.478091i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	8.00000			0.929981
$75$		0			0
$76$			6.00000i			0.688247i
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		−	2.00000i		−	0.223607i
$81$		0			0
$82$	−10.0000			−1.10432
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$		−	4.00000i		−	0.433861i
$86$		−	4.00000i		−	0.431331i
$87$		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
$91$	−4.00000	+	6.00000i	−0.419314	+	0.628971i
$92$	4.00000			0.417029
$93$		0			0
$94$	12.0000			1.23771
$95$	−12.0000			−1.23117
$96$		0			0
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$		−	3.00000i		−	0.303046i
$99$		0			0
$100$	−1.00000			−0.100000
$101$	−2.00000			−0.199007			−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000			−0.199007			−0.0995037	+	0.995037i	$0.531726\pi$
$102$		0			0
$103$	−16.0000			−1.57653			−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000			−1.57653			−0.788263	+	0.615338i	$0.789020\pi$
$104$	2.00000	−	3.00000i	0.196116	−	0.294174i
$105$		0			0
$106$		−	6.00000i		−	0.582772i
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$		0			0
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$		0			0
$112$		−	2.00000i		−	0.188982i
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$			8.00000i			0.746004i
$116$	−10.0000			−0.928477
$117$		0			0
$118$	−4.00000			−0.368230
$119$		−	4.00000i		−	0.366679i
$120$		0			0
$121$	11.0000			1.00000
$122$		−	2.00000i		−	0.181071i
$123$		0			0
$124$		−	10.0000i		−	0.898027i
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	6.00000	+	4.00000i	0.526235	+	0.350823i
$131$	8.00000			0.698963			0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000			0.698963			0.349482	+	0.936943i	$0.386358\pi$
$132$		0			0
$133$	−12.0000			−1.04053
$134$	−2.00000			−0.172774
$135$		0			0
$136$			2.00000i			0.171499i
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$		0			0
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$	4.00000			0.338062
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		−	20.0000i		−	1.66091i
$146$	−4.00000			−0.331042
$147$		0			0
$148$		−	8.00000i		−	0.657596i
$149$		−	14.0000i		−	1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$		−	14.0000i		−	1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$		0			0
$151$			10.0000i			0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$	6.00000			0.486664
$153$		0			0
$154$		0			0
$155$	20.0000			1.60644
$156$		0			0
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		0			0
$159$		0			0
$160$	−2.00000			−0.158114
$161$			8.00000i			0.630488i
$162$		0			0
$163$		−	14.0000i		−	1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$		−	14.0000i		−	1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$			10.0000i			0.780869i
$165$		0			0
$166$	4.00000			0.310460
$167$			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
$170$	−4.00000			−0.306786
$171$		0			0
$172$	−4.00000			−0.304997
$173$	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
$175$		−	2.00000i		−	0.151186i
$176$		0			0
$177$		0			0
$178$	6.00000			0.449719
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$	6.00000	+	4.00000i	0.444750	+	0.296500i
$183$		0			0
$184$		−	4.00000i		−	0.294884i
$185$	16.0000			1.17634
$186$		0			0
$187$		0			0
$188$		−	12.0000i		−	0.875190i
$189$		0			0
$190$			12.0000i			0.870572i
$191$	−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
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	−12.0000			−0.861550
$195$		0			0
$196$	−3.00000			−0.214286
$197$			22.0000i			1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$			22.0000i			1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$			1.00000i			0.0707107i
$201$		0			0
$202$			2.00000i			0.140720i
$203$		−	20.0000i		−	1.40372i
$204$		0			0
$205$	−20.0000			−1.39686
$206$			16.0000i			1.11477i
$207$		0			0
$208$	−3.00000	−	2.00000i	−0.208013	−	0.138675i
$209$		0			0
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$	−6.00000			−0.412082
$213$		0			0
$214$			8.00000i			0.546869i
$215$		−	8.00000i		−	0.545595i
$216$		0			0
$217$	20.0000			1.35769
$218$	4.00000			0.270914
$219$		0			0
$220$		0			0
$221$	−6.00000	−	4.00000i	−0.403604	−	0.269069i
$222$		0			0
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	−2.00000			−0.133631
$225$		0			0
$226$			14.0000i			0.931266i
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$			4.00000i			0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$			4.00000i			0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	8.00000			0.527504
$231$		0			0
$232$			10.0000i			0.656532i
$233$	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
$235$	24.0000			1.56559
$236$			4.00000i			0.260378i
$237$		0			0
$238$	−4.00000			−0.259281
$239$			16.0000i			1.03495i	0.855697	+	0.517477i	$0.173129\pi$
$239$			16.0000i			1.03495i	−0.855697	+	0.517477i	$0.826871\pi$
$240$		0			0
$241$			20.0000i			1.28831i	0.764894	+	0.644157i	$0.222792\pi$
$241$			20.0000i			1.28831i	−0.764894	+	0.644157i	$0.777208\pi$
$242$		−	11.0000i		−	0.707107i
$243$		0			0
$244$	−2.00000			−0.128037
$245$		−	6.00000i		−	0.383326i
$246$		0			0
$247$	−12.0000	+	18.0000i	−0.763542	+	1.14531i
$248$	−10.0000			−0.635001
$249$		0			0
$250$	−12.0000			−0.758947
$251$	28.0000			1.76734			0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000			1.76734			0.883672	+	0.468106i	$0.155064\pi$
$252$		0			0
$253$		0			0
$254$		−	8.00000i		−	0.501965i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−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
$259$	16.0000			0.994192
$260$	4.00000	−	6.00000i	0.248069	−	0.372104i
$261$		0			0
$262$		−	8.00000i		−	0.494242i
$263$	−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
$265$		−	12.0000i		−	0.737154i
$266$			12.0000i			0.735767i
$267$		0			0
$268$			2.00000i			0.122169i
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$		−	10.0000i		−	0.607457i	−0.952759	−	0.303728i	$-0.901768\pi$
$271$		−	10.0000i		−	0.607457i	0.952759	−	0.303728i	$-0.0982315\pi$
$272$	2.00000			0.121268
$273$		0			0
$274$	2.00000			0.120824
$275$		0			0
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$			20.0000i			1.19952i
$279$		0			0
$280$		−	4.00000i		−	0.239046i
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$		0			0
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−20.0000			−1.18056
$288$		0			0
$289$	−13.0000			−0.764706
$290$	−20.0000			−1.17444
$291$		0			0
$292$			4.00000i			0.234082i
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$		0			0
$295$	−8.00000			−0.465778
$296$	−8.00000			−0.464991
$297$		0			0
$298$	−14.0000			−0.810998
$299$	12.0000	+	8.00000i	0.693978	+	0.462652i
$300$		0			0
$301$		−	8.00000i		−	0.461112i
$302$	10.0000			0.575435
$303$		0			0
$304$		−	6.00000i		−	0.344124i
$305$		−	4.00000i		−	0.229039i
$306$		0			0
$307$		−	2.00000i		−	0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$		−	2.00000i		−	0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$		0			0
$309$		0			0
$310$		−	20.0000i		−	1.13592i
$311$	28.0000			1.58773			0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000			1.58773			0.793867	+	0.608091i	$0.208065\pi$
$312$		0			0
$313$	−26.0000			−1.46961			−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000			−1.46961			−0.734803	+	0.678280i	$0.762726\pi$
$314$			2.00000i			0.112867i
$315$		0			0
$316$		0			0
$317$		−	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
$320$			2.00000i			0.111803i
$321$		0			0
$322$	8.00000			0.445823
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$	−3.00000	−	2.00000i	−0.166410	−	0.110940i
$326$	−14.0000			−0.775388
$327$		0			0
$328$	10.0000			0.552158
$329$	24.0000			1.32316
$330$		0			0
$331$		−	10.0000i		−	0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$		−	10.0000i		−	0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$		−	4.00000i		−	0.219529i
$333$		0			0
$334$	12.0000			0.656611
$335$	−4.00000			−0.218543
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$	12.0000	−	5.00000i	0.652714	−	0.271964i
$339$		0			0
$340$			4.00000i			0.216930i
$341$		0			0
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$			4.00000i			0.215666i
$345$		0			0
$346$		−	6.00000i		−	0.322562i
$347$	12.0000			0.644194			0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000			0.644194			0.322097	+	0.946707i	$0.395612\pi$
$348$		0			0
$349$		−	16.0000i		−	0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$		−	16.0000i		−	0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$	−2.00000			−0.106904
$351$		0			0
$352$		0			0
$353$		−	26.0000i		−	1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$		−	26.0000i		−	1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$		0			0
$355$		0			0
$356$		−	6.00000i		−	0.317999i
$357$		0			0
$358$		0			0
$359$		−	4.00000i		−	0.211112i	−0.994413	−	0.105556i	$-0.966338\pi$
$359$		−	4.00000i		−	0.211112i	0.994413	−	0.105556i	$-0.0336622\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		−	22.0000i		−	1.15629i
$363$		0			0
$364$	4.00000	−	6.00000i	0.209657	−	0.314485i
$365$	−8.00000			−0.418739
$366$		0			0
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$	−4.00000			−0.208514
$369$		0			0
$370$		−	16.0000i		−	0.831800i
$371$		−	12.0000i		−	0.623009i
$372$		0			0
$373$	−6.00000			−0.310668			−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000			−0.310668			−0.155334	+	0.987862i	$0.549645\pi$
$374$		0			0
$375$		0			0
$376$	−12.0000			−0.618853
$377$	−30.0000	−	20.0000i	−1.54508	−	1.03005i
$378$		0			0
$379$			34.0000i			1.74646i	0.487306	+	0.873231i	$0.337980\pi$
$379$			34.0000i			1.74646i	−0.487306	+	0.873231i	$0.662020\pi$
$380$	12.0000			0.615587
$381$		0			0
$382$			12.0000i			0.613973i
$383$			4.00000i			0.204390i	0.994764	+	0.102195i	$0.0325866\pi$
$383$			4.00000i			0.204390i	−0.994764	+	0.102195i	$0.967413\pi$
$384$		0			0
$385$		0			0
$386$	16.0000			0.814379
$387$		0			0
$388$			12.0000i			0.609208i
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$	−8.00000			−0.404577
$392$			3.00000i			0.151523i
$393$		0			0
$394$	22.0000			1.10834
$395$		0			0
$396$		0			0
$397$			8.00000i			0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$			8.00000i			0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			0.0500000
$401$			30.0000i			1.49813i	0.662497	+	0.749064i	$0.269497\pi$
$401$			30.0000i			1.49813i	−0.662497	+	0.749064i	$0.730503\pi$
$402$		0			0
$403$	20.0000	−	30.0000i	0.996271	−	1.49441i
$404$	2.00000			0.0995037
$405$		0			0
$406$	−20.0000			−0.992583
$407$		0			0
$408$		0			0
$409$			4.00000i			0.197787i	0.995098	+	0.0988936i	$0.0315304\pi$
$409$			4.00000i			0.197787i	−0.995098	+	0.0988936i	$0.968470\pi$
$410$			20.0000i			0.987730i
$411$		0			0
$412$	16.0000			0.788263
$413$	−8.00000			−0.393654
$414$		0			0
$415$	8.00000			0.392705
$416$	−2.00000	+	3.00000i	−0.0980581	+	0.147087i
$417$		0			0
$418$		0			0
$419$	−40.0000			−1.95413			−0.977064	−	0.212946i	$-0.931694\pi$
$419$	−40.0000			−1.95413			−0.977064	+	0.212946i	$0.931694\pi$
$420$		0			0
$421$			20.0000i			0.974740i	0.873195	+	0.487370i	$0.162044\pi$
$421$			20.0000i			0.974740i	−0.873195	+	0.487370i	$0.837956\pi$
$422$		−	12.0000i		−	0.584151i
$423$		0			0
$424$			6.00000i			0.291386i
$425$	2.00000			0.0970143
$426$		0			0
$427$		−	4.00000i		−	0.193574i
$428$	8.00000			0.386695
$429$		0			0
$430$	−8.00000			−0.385794
$431$			20.0000i			0.963366i	0.876346	+	0.481683i	$0.159974\pi$
$431$			20.0000i			0.963366i	−0.876346	+	0.481683i	$0.840026\pi$
$432$		0			0
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$		−	20.0000i		−	0.960031i
$435$		0			0
$436$		−	4.00000i		−	0.191565i
$437$			24.0000i			1.14808i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$	−4.00000	+	6.00000i	−0.190261	+	0.285391i
$443$	16.0000			0.760183			0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000			0.760183			0.380091	+	0.924949i	$0.375893\pi$
$444$		0			0
$445$	12.0000			0.568855
$446$	−14.0000			−0.662919
$447$		0			0
$448$			2.00000i			0.0944911i
$449$			6.00000i			0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$			6.00000i			0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$		0			0
$451$		0			0
$452$	14.0000			0.658505
$453$		0			0
$454$	−8.00000			−0.375459
$455$	12.0000	+	8.00000i	0.562569	+	0.375046i
$456$		0			0
$457$			28.0000i			1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$			28.0000i			1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$	4.00000			0.186908
$459$		0			0
$460$		−	8.00000i		−	0.373002i
$461$			30.0000i			1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$			30.0000i			1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$		0			0
$463$			6.00000i			0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$			6.00000i			0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$	10.0000			0.464238
$465$		0			0
$466$		−	6.00000i		−	0.277945i
$467$	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
$469$	−4.00000			−0.184703
$470$		−	24.0000i		−	1.10704i
$471$		0			0
$472$	4.00000			0.184115
$473$		0			0
$474$		0			0
$475$		−	6.00000i		−	0.275299i
$476$			4.00000i			0.183340i
$477$		0			0
$478$	16.0000			0.731823
$479$		−	24.0000i		−	1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$		−	24.0000i		−	1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$		0			0
$481$	16.0000	−	24.0000i	0.729537	−	1.09431i
$482$	20.0000			0.910975
$483$		0			0
$484$	−11.0000			−0.500000
$485$	−24.0000			−1.08978
$486$		0			0
$487$			18.0000i			0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$			18.0000i			0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$			2.00000i			0.0905357i
$489$		0			0
$490$	−6.00000			−0.271052
$491$	28.0000			1.26362			0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000			1.26362			0.631811	+	0.775122i	$0.282312\pi$
$492$		0			0
$493$	20.0000			0.900755
$494$	18.0000	+	12.0000i	0.809858	+	0.539906i
$495$		0			0
$496$			10.0000i			0.449013i
$497$		0			0
$498$		0			0
$499$			14.0000i			0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$			12.0000i			0.536656i
$501$		0			0
$502$		−	28.0000i		−	1.24970i
$503$	−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
$505$			4.00000i			0.177998i
$506$		0			0
$507$		0			0
$508$	−8.00000			−0.354943
$509$			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
$511$	−8.00000			−0.353899
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			18.0000i			0.793946i
$515$			32.0000i			1.41009i
$516$		0			0
$517$		0			0
$518$		−	16.0000i		−	0.703000i
$519$		0			0
$520$	−6.00000	−	4.00000i	−0.263117	−	0.175412i
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$	−8.00000			−0.349482
$525$		0			0
$526$			24.0000i			1.04645i
$527$			20.0000i			0.871214i
$528$		0			0
$529$	−7.00000			−0.304348
$530$	−12.0000			−0.521247
$531$		0			0
$532$	12.0000			0.520266
$533$	−20.0000	+	30.0000i	−0.866296	+	1.29944i
$534$		0			0
$535$			16.0000i			0.691740i
$536$	2.00000			0.0863868
$537$		0			0
$538$		−	10.0000i		−	0.431131i
$539$		0			0
$540$		0			0
$541$		−	20.0000i		−	0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$	−10.0000			−0.429537
$543$		0			0
$544$		−	2.00000i		−	0.0857493i
$545$	8.00000			0.342682
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$		−	2.00000i		−	0.0854358i
$549$		0			0
$550$		0			0
$551$		−	60.0000i		−	2.55609i
$552$		0			0
$553$		0			0
$554$			2.00000i			0.0849719i
$555$		0			0
$556$	20.0000			0.848189
$557$		−	18.0000i		−	0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$		−	18.0000i		−	0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$		0			0
$559$	−12.0000	−	8.00000i	−0.507546	−	0.338364i
$560$	−4.00000			−0.169031
$561$		0			0
$562$	10.0000			0.421825
$563$	16.0000			0.674320			0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000			0.674320			0.337160	+	0.941447i	$0.390534\pi$
$564$		0			0
$565$			28.0000i			1.17797i
$566$		−	4.00000i		−	0.168133i
$567$		0			0
$568$		0			0
$569$	−10.0000			−0.419222			−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000			−0.419222			−0.209611	+	0.977785i	$0.567220\pi$
$570$		0			0
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$		0			0
$574$			20.0000i			0.834784i
$575$	−4.00000			−0.166812
$576$		0			0
$577$			8.00000i			0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$			8.00000i			0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$			13.0000i			0.540729i
$579$		0			0
$580$			20.0000i			0.830455i
$581$	8.00000			0.331896
$582$		0			0
$583$		0			0
$584$	4.00000			0.165521
$585$		0			0
$586$	14.0000			0.578335
$587$		−	28.0000i		−	1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$		−	28.0000i		−	1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$		0			0
$589$	60.0000			2.47226
$590$			8.00000i			0.329355i
$591$		0			0
$592$			8.00000i			0.328798i
$593$		−	26.0000i		−	1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$		−	26.0000i		−	1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$		0			0
$595$	−8.00000			−0.327968
$596$			14.0000i			0.573462i
$597$		0			0
$598$	8.00000	−	12.0000i	0.327144	−	0.490716i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$	−8.00000			−0.326056
$603$		0			0
$604$		−	10.0000i		−	0.406894i
$605$		−	22.0000i		−	0.894427i
$606$		0			0
$607$	−32.0000			−1.29884			−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000			−1.29884			−0.649420	+	0.760430i	$0.724988\pi$
$608$	−6.00000			−0.243332
$609$		0			0
$610$	−4.00000			−0.161955
$611$	24.0000	−	36.0000i	0.970936	−	1.45640i
$612$		0			0
$613$			16.0000i			0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$			16.0000i			0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	−2.00000			−0.0807134
$615$		0			0
$616$		0			0
$617$			22.0000i			0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$			22.0000i			0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$		0			0
$619$		−	26.0000i		−	1.04503i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$		−	26.0000i		−	1.04503i	0.852631	−	0.522514i	$-0.175006\pi$
$620$	−20.0000			−0.803219
$621$		0			0
$622$		−	28.0000i		−	1.12270i
$623$	12.0000			0.480770
$624$		0			0
$625$	−19.0000			−0.760000
$626$			26.0000i			1.03917i
$627$		0			0
$628$	2.00000			0.0798087
$629$			16.0000i			0.637962i
$630$		0			0
$631$			10.0000i			0.398094i	0.979990	+	0.199047i	$0.0637846\pi$
$631$			10.0000i			0.398094i	−0.979990	+	0.199047i	$0.936215\pi$
$632$		0			0
$633$		0			0
$634$	−18.0000			−0.714871
$635$		−	16.0000i		−	0.634941i
$636$		0			0
$637$	−9.00000	−	6.00000i	−0.356593	−	0.237729i
$638$		0			0
$639$		0			0
$640$	2.00000			0.0790569
$641$	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
$643$			6.00000i			0.236617i	0.992977	+	0.118308i	$0.0377472\pi$
$643$			6.00000i			0.236617i	−0.992977	+	0.118308i	$0.962253\pi$
$644$		−	8.00000i		−	0.315244i
$645$		0			0
$646$	−12.0000			−0.472134
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		0			0
$649$		0			0
$650$	−2.00000	+	3.00000i	−0.0784465	+	0.117670i
$651$		0			0
$652$			14.0000i			0.548282i
$653$	26.0000			1.01746			0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000			1.01746			0.508729	+	0.860927i	$0.330115\pi$
$654$		0			0
$655$		−	16.0000i		−	0.625172i
$656$		−	10.0000i		−	0.390434i
$657$		0			0
$658$		−	24.0000i		−	0.935617i
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$		0			0
$661$		−	40.0000i		−	1.55582i	−0.628376	−	0.777910i	$-0.716280\pi$
$661$		−	40.0000i		−	1.55582i	0.628376	−	0.777910i	$-0.283720\pi$
$662$	−10.0000			−0.388661
$663$		0			0
$664$	−4.00000			−0.155230
$665$			24.0000i			0.930680i
$666$		0			0
$667$	−40.0000			−1.54881
$668$		−	12.0000i		−	0.464294i
$669$		0			0
$670$			4.00000i			0.154533i
$671$		0			0
$672$		0			0
$673$	−6.00000			−0.231283			−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000			−0.231283			−0.115642	+	0.993291i	$0.536892\pi$
$674$			2.00000i			0.0770371i
$675$		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
$679$	−24.0000			−0.921035
$680$	4.00000			0.153393
$681$		0			0
$682$		0			0
$683$			24.0000i			0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$			24.0000i			0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$		0			0
$685$	4.00000			0.152832
$686$	−20.0000			−0.763604
$687$		0			0
$688$	4.00000			0.152499
$689$	−18.0000	−	12.0000i	−0.685745	−	0.457164i
$690$		0			0
$691$		−	10.0000i		−	0.380418i	−0.981744	−	0.190209i	$-0.939083\pi$
$691$		−	10.0000i		−	0.380418i	0.981744	−	0.190209i	$-0.0609166\pi$
$692$	−6.00000			−0.228086
$693$		0			0
$694$		−	12.0000i		−	0.455514i
$695$			40.0000i			1.51729i
$696$		0			0
$697$		−	20.0000i		−	0.757554i
$698$	−16.0000			−0.605609
$699$		0			0
$700$			2.00000i			0.0755929i
$701$	−22.0000			−0.830929			−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000			−0.830929			−0.415464	+	0.909610i	$0.636381\pi$
$702$		0			0
$703$	48.0000			1.81035
$704$		0			0
$705$		0			0
$706$	−26.0000			−0.978523
$707$			4.00000i			0.150435i
$708$		0			0
$709$		−	36.0000i		−	1.35201i	−0.736898	−	0.676004i	$-0.763710\pi$
$709$		−	36.0000i		−	1.35201i	0.736898	−	0.676004i	$-0.236290\pi$
$710$		0			0
$711$		0			0
$712$	−6.00000			−0.224860
$713$		−	40.0000i		−	1.49801i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−4.00000			−0.149279
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$			32.0000i			1.19174i
$722$			17.0000i			0.632674i
$723$		0			0
$724$	−22.0000			−0.817624
$725$	10.0000			0.371391
$726$		0			0
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$	−6.00000	−	4.00000i	−0.222375	−	0.148250i
$729$		0			0
$730$			8.00000i			0.296093i
$731$	8.00000			0.295891
$732$		0			0
$733$		−	44.0000i		−	1.62518i	−0.582838	−	0.812589i	$-0.698058\pi$
$733$		−	44.0000i		−	1.62518i	0.582838	−	0.812589i	$-0.301942\pi$
$734$		−	8.00000i		−	0.295285i
$735$		0			0
$736$			4.00000i			0.147442i
$737$		0			0
$738$		0			0
$739$		−	26.0000i		−	0.956425i	−0.878244	−	0.478213i	$-0.841285\pi$
$739$		−	26.0000i		−	0.956425i	0.878244	−	0.478213i	$-0.158715\pi$
$740$	−16.0000			−0.588172
$741$		0			0
$742$	−12.0000			−0.440534
$743$		−	16.0000i		−	0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$		−	16.0000i		−	0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$		0			0
$745$	−28.0000			−1.02584
$746$			6.00000i			0.219676i
$747$		0			0
$748$		0			0
$749$			16.0000i			0.584627i
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$			12.0000i			0.437595i
$753$		0			0
$754$	−20.0000	+	30.0000i	−0.728357	+	1.09254i
$755$	20.0000			0.727875
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$	34.0000			1.23494
$759$		0			0
$760$		−	12.0000i		−	0.435286i
$761$		−	30.0000i		−	1.08750i	−0.839248	−	0.543750i	$-0.817004\pi$
$761$		−	30.0000i		−	1.08750i	0.839248	−	0.543750i	$-0.182996\pi$
$762$		0			0
$763$	8.00000			0.289619
$764$	12.0000			0.434145
$765$		0			0
$766$	4.00000			0.144526
$767$	−8.00000	+	12.0000i	−0.288863	+	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
$772$		−	16.0000i		−	0.575853i
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$		0			0
$775$			10.0000i			0.359211i
$776$	12.0000			0.430775
$777$		0			0
$778$			30.0000i			1.07555i
$779$	−60.0000			−2.14972
$780$		0			0
$781$		0			0
$782$			8.00000i			0.286079i
$783$		0			0
$784$	3.00000			0.107143
$785$			4.00000i			0.142766i
$786$		0			0
$787$			38.0000i			1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$			38.0000i			1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$		−	22.0000i		−	0.783718i
$789$		0			0
$790$		0			0
$791$			28.0000i			0.995565i
$792$		0			0
$793$	−6.00000	−	4.00000i	−0.213066	−	0.142044i
$794$	8.00000			0.283909
$795$		0			0
$796$		0			0
$797$	2.00000			0.0708436			0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000			0.0708436			0.0354218	+	0.999372i	$0.488723\pi$
$798$		0			0
$799$			24.0000i			0.849059i
$800$		−	1.00000i		−	0.0353553i
$801$		0			0
$802$	30.0000			1.05934
$803$		0			0
$804$		0			0
$805$	16.0000			0.563926
$806$	−30.0000	−	20.0000i	−1.05670	−	0.704470i
$807$		0			0
$808$		−	2.00000i		−	0.0703598i
$809$	50.0000			1.75791			0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000			1.75791			0.878953	+	0.476908i	$0.158243\pi$
$810$		0			0
$811$			10.0000i			0.351147i	0.984466	+	0.175574i	$0.0561780\pi$
$811$			10.0000i			0.351147i	−0.984466	+	0.175574i	$0.943822\pi$
$812$			20.0000i			0.701862i
$813$		0			0
$814$		0			0
$815$	−28.0000			−0.980797
$816$		0			0
$817$		−	24.0000i		−	0.839654i
$818$	4.00000			0.139857
$819$		0			0
$820$	20.0000			0.698430
$821$		−	30.0000i		−	1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$		−	30.0000i		−	1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$		0			0
$823$	24.0000			0.836587			0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000			0.836587			0.418294	+	0.908312i	$0.362628\pi$
$824$		−	16.0000i		−	0.557386i
$825$		0			0
$826$			8.00000i			0.278356i
$827$		−	48.0000i		−	1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$		−	48.0000i		−	1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$		0			0
$829$	−30.0000			−1.04194			−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000			−1.04194			−0.520972	+	0.853574i	$0.674430\pi$
$830$		−	8.00000i		−	0.277684i
$831$		0			0
$832$	3.00000	+	2.00000i	0.104006	+	0.0693375i
$833$	6.00000			0.207888
$834$		0			0
$835$	24.0000			0.830554
$836$		0			0
$837$		0			0
$838$			40.0000i			1.38178i
$839$			16.0000i			0.552381i	0.961103	+	0.276191i	$0.0890721\pi$
$839$			16.0000i			0.552381i	−0.961103	+	0.276191i	$0.910928\pi$
$840$		0			0
$841$	71.0000			2.44828
$842$	20.0000			0.689246
$843$		0			0
$844$	−12.0000			−0.413057
$845$	24.0000	−	10.0000i	0.825625	−	0.344010i
$846$		0			0
$847$		−	22.0000i		−	0.755929i
$848$	6.00000			0.206041
$849$		0			0
$850$		−	2.00000i		−	0.0685994i
$851$		−	32.0000i		−	1.09695i
$852$		0			0
$853$			56.0000i			1.91740i	0.284413	+	0.958702i	$0.408201\pi$
$853$			56.0000i			1.91740i	−0.284413	+	0.958702i	$0.591799\pi$
$854$	−4.00000			−0.136877
$855$		0			0
$856$		−	8.00000i		−	0.273434i
$857$	22.0000			0.751506			0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000			0.751506			0.375753	+	0.926720i	$0.377384\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$			8.00000i			0.272798i
$861$		0			0
$862$	20.0000			0.681203
$863$			44.0000i			1.49778i	0.662696	+	0.748889i	$0.269412\pi$
$863$			44.0000i			1.49778i	−0.662696	+	0.748889i	$0.730588\pi$
$864$		0			0
$865$		−	12.0000i		−	0.408012i
$866$			26.0000i			0.883516i
$867$		0			0
$868$	−20.0000			−0.678844
$869$		0			0
$870$		0			0
$871$	−4.00000	+	6.00000i	−0.135535	+	0.203302i
$872$	−4.00000			−0.135457
$873$		0			0
$874$	24.0000			0.811812
$875$	−24.0000			−0.811348
$876$		0			0
$877$			8.00000i			0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$			8.00000i			0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−42.0000			−1.41502			−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000			−1.41502			−0.707508	+	0.706705i	$0.750181\pi$
$882$		0			0
$883$	−36.0000			−1.21150			−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000			−1.21150			−0.605748	+	0.795656i	$0.707126\pi$
$884$	6.00000	+	4.00000i	0.201802	+	0.134535i
$885$		0			0
$886$		−	16.0000i		−	0.537531i
$887$	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
$889$		−	16.0000i		−	0.536623i
$890$		−	12.0000i		−	0.402241i
$891$		0			0
$892$			14.0000i			0.468755i
$893$	72.0000			2.40939
$894$		0			0
$895$		0			0
$896$	2.00000			0.0668153
$897$		0			0
$898$	6.00000			0.200223
$899$			100.000i			3.33519i
$900$		0			0
$901$	12.0000			0.399778
$902$		0			0
$903$		0			0
$904$		−	14.0000i		−	0.465633i
$905$		−	44.0000i		−	1.46261i
$906$		0			0
$907$	28.0000			0.929725			0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000			0.929725			0.464862	+	0.885383i	$0.346104\pi$
$908$			8.00000i			0.265489i
$909$		0			0
$910$	8.00000	−	12.0000i	0.265197	−	0.397796i
$911$	−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
$914$	28.0000			0.926158
$915$		0			0
$916$		−	4.00000i		−	0.132164i
$917$		−	16.0000i		−	0.528367i
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	−8.00000			−0.263752
$921$		0			0
$922$	30.0000			0.987997
$923$		0			0
$924$		0			0
$925$			8.00000i			0.263038i
$926$	6.00000			0.197172
$927$		0			0
$928$		−	10.0000i		−	0.328266i
$929$			6.00000i			0.196854i	0.995144	+	0.0984268i	$0.0313810\pi$
$929$			6.00000i			0.196854i	−0.995144	+	0.0984268i	$0.968619\pi$
$930$		0			0
$931$		−	18.0000i		−	0.589926i
$932$	−6.00000			−0.196537
$933$		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	$0.510401\pi$
$938$			4.00000i			0.130605i
$939$		0			0
$940$	−24.0000			−0.782794
$941$			10.0000i			0.325991i	0.986627	+	0.162995i	$0.0521156\pi$
$941$			10.0000i			0.325991i	−0.986627	+	0.162995i	$0.947884\pi$
$942$		0			0
$943$			40.0000i			1.30258i
$944$		−	4.00000i		−	0.130189i
$945$		0			0
$946$		0			0
$947$			52.0000i			1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$			52.0000i			1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$		0			0
$949$	−8.00000	+	12.0000i	−0.259691	+	0.389536i
$950$	−6.00000			−0.194666
$951$		0			0
$952$	4.00000			0.129641
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$			24.0000i			0.776622i
$956$		−	16.0000i		−	0.517477i
$957$		0			0
$958$	−24.0000			−0.775405
$959$	4.00000			0.129167
$960$		0			0
$961$	−69.0000			−2.22581
$962$	−24.0000	−	16.0000i	−0.773791	−	0.515861i
$963$		0			0
$964$		−	20.0000i		−	0.644157i
$965$	32.0000			1.03012
$966$		0			0
$967$		−	22.0000i		−	0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$		−	22.0000i		−	0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$			11.0000i			0.353553i
$969$		0			0
$970$			24.0000i			0.770594i
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$			40.0000i			1.28234i
$974$	18.0000			0.576757
$975$		0			0
$976$	2.00000			0.0640184
$977$			42.0000i			1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$			42.0000i			1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$		0			0
$979$		0			0
$980$			6.00000i			0.191663i
$981$		0			0
$982$		−	28.0000i		−	0.893516i
$983$			24.0000i			0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$			24.0000i			0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$		0			0
$985$	44.0000			1.40196
$986$		−	20.0000i		−	0.636930i
$987$		0			0
$988$	12.0000	−	18.0000i	0.381771	−	0.572656i
$989$	−16.0000			−0.508770
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$	10.0000			0.317500
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−42.0000			−1.33015			−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000			−1.33015			−0.665077	+	0.746775i	$0.731601\pi$
$998$	14.0000			0.443162
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	234.2.b.a.181.1		2
3.2	odd	2		78.2.b.a.25.2	yes	2
4.3	odd	2		1872.2.c.b.1585.1		2
12.11	even	2		624.2.c.a.337.2		2
13.5	odd	4		3042.2.a.c.1.1		1
13.8	odd	4		3042.2.a.n.1.1		1
13.12	even	2	inner	234.2.b.a.181.2		2
15.2	even	4		1950.2.f.d.649.1		2
15.8	even	4		1950.2.f.g.649.2		2
15.14	odd	2		1950.2.b.c.1351.1		2
21.20	even	2		3822.2.c.d.883.2		2
24.5	odd	2		2496.2.c.f.961.1		2
24.11	even	2		2496.2.c.m.961.1		2
39.2	even	12		1014.2.e.b.529.1		2
39.5	even	4		1014.2.a.g.1.1		1
39.8	even	4		1014.2.a.b.1.1		1
39.11	even	12		1014.2.e.e.529.1		2
39.17	odd	6		1014.2.i.c.361.1		4
39.20	even	12		1014.2.e.e.991.1		2
39.23	odd	6		1014.2.i.c.823.2		4
39.29	odd	6		1014.2.i.c.823.1		4
39.32	even	12		1014.2.e.b.991.1		2
39.35	odd	6		1014.2.i.c.361.2		4
39.38	odd	2		78.2.b.a.25.1	✓	2
52.51	odd	2		1872.2.c.b.1585.2		2
156.47	odd	4		8112.2.a.g.1.1		1
156.83	odd	4		8112.2.a.j.1.1		1
156.155	even	2		624.2.c.a.337.1		2
195.38	even	4		1950.2.f.d.649.2		2
195.77	even	4		1950.2.f.g.649.1		2
195.194	odd	2		1950.2.b.c.1351.2		2
273.272	even	2		3822.2.c.d.883.1		2
312.77	odd	2		2496.2.c.f.961.2		2
312.155	even	2		2496.2.c.m.961.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.b.a.25.1	✓	2	39.38	odd	2
78.2.b.a.25.2	yes	2	3.2	odd	2
234.2.b.a.181.1		2	1.1	even	1	trivial
234.2.b.a.181.2		2	13.12	even	2	inner
624.2.c.a.337.1		2	156.155	even	2
624.2.c.a.337.2		2	12.11	even	2
1014.2.a.b.1.1		1	39.8	even	4
1014.2.a.g.1.1		1	39.5	even	4
1014.2.e.b.529.1		2	39.2	even	12
1014.2.e.b.991.1		2	39.32	even	12
1014.2.e.e.529.1		2	39.11	even	12
1014.2.e.e.991.1		2	39.20	even	12
1014.2.i.c.361.1		4	39.17	odd	6
1014.2.i.c.361.2		4	39.35	odd	6
1014.2.i.c.823.1		4	39.29	odd	6
1014.2.i.c.823.2		4	39.23	odd	6
1872.2.c.b.1585.1		2	4.3	odd	2
1872.2.c.b.1585.2		2	52.51	odd	2
1950.2.b.c.1351.1		2	15.14	odd	2
1950.2.b.c.1351.2		2	195.194	odd	2
1950.2.f.d.649.1		2	15.2	even	4
1950.2.f.d.649.2		2	195.38	even	4
1950.2.f.g.649.1		2	195.77	even	4
1950.2.f.g.649.2		2	15.8	even	4
2496.2.c.f.961.1		2	24.5	odd	2
2496.2.c.f.961.2		2	312.77	odd	2
2496.2.c.m.961.1		2	24.11	even	2
2496.2.c.m.961.2		2	312.155	even	2
3042.2.a.c.1.1		1	13.5	odd	4
3042.2.a.n.1.1		1	13.8	odd	4
3822.2.c.d.883.1		2	273.272	even	2
3822.2.c.d.883.2		2	21.20	even	2
8112.2.a.g.1.1		1	156.47	odd	4
8112.2.a.j.1.1		1	156.83	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	234.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{5} -2.00000i q^{7} +1.00000i q^{8} -2.00000 q^{10} +(-3.00000 - 2.00000i) q^{13} -2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -6.00000i q^{19} +2.00000i q^{20} -4.00000 q^{23} +1.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} +2.00000i q^{28} +10.0000 q^{29} +10.0000i q^{31} -1.00000i q^{32} -2.00000i q^{34} -4.00000 q^{35} +8.00000i q^{37} -6.00000 q^{38} +2.00000 q^{40} -10.0000i q^{41} +4.00000 q^{43} +4.00000i q^{46} +12.0000i q^{47} +3.00000 q^{49} -1.00000i q^{50} +(3.00000 + 2.00000i) q^{52} +6.00000 q^{53} +2.00000 q^{56} -10.0000i q^{58} -4.00000i q^{59} +2.00000 q^{61} +10.0000 q^{62} -1.00000 q^{64} +(-4.00000 + 6.00000i) q^{65} -2.00000i q^{67} -2.00000 q^{68} +4.00000i q^{70} -4.00000i q^{73} +8.00000 q^{74} +6.00000i q^{76} -2.00000i q^{80} -10.0000 q^{82} +4.00000i q^{83} -4.00000i q^{85} -4.00000i q^{86} +6.00000i q^{89} +(-4.00000 + 6.00000i) q^{91} +4.00000 q^{92} +12.0000 q^{94} -12.0000 q^{95} -12.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{10} - 6 q^{13} - 4 q^{14} + 2 q^{16} + 4 q^{17} - 8 q^{23} + 2 q^{25} - 4 q^{26} + 20 q^{29} - 8 q^{35} - 12 q^{38} + 4 q^{40} + 8 q^{43} + 6 q^{49} + 6 q^{52} + 12 q^{53} + 4 q^{56}+ \cdots - 24 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^10 - 6 * q^13 - 4 * q^14 + 2 * q^16 + 4 * q^17 - 8 * q^23 + 2 * q^25 - 4 * q^26 + 20 * q^29 - 8 * q^35 - 12 * q^38 + 4 * q^40 + 8 * q^43 + 6 * q^49 + 6 * q^52 + 12 * q^53 + 4 * q^56 + 4 * q^61 + 20 * q^62 - 2 * q^64 - 8 * q^65 - 4 * q^68 + 16 * q^74 - 20 * q^82 - 8 * q^91 + 8 * q^92 + 24 * q^94 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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