LMFDB - Embedded newform 189.2.c.c.188.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(188,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("189.188");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	189.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.50917259820$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			188.1
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	189.188
Dual form			189.2.c.c.188.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -1.73205 q^{5} +(1.00000 - 2.44949i) q^{7} -2.82843i q^{8} +O(q^{10})$	\(q-1.41421i q^{2} -1.73205 q^{5} +(1.00000 - 2.44949i) q^{7} -2.82843i q^{8} +2.44949i q^{10} -1.41421i q^{11} -2.44949i q^{13} +(-3.46410 - 1.41421i) q^{14} -4.00000 q^{16} +5.19615 q^{17} +7.34847i q^{19} -2.00000 q^{22} +2.82843i q^{23} -2.00000 q^{25} -3.46410 q^{26} +7.07107i q^{29} -2.44949i q^{31} -7.34847i q^{34} +(-1.73205 + 4.24264i) q^{35} +5.00000 q^{37} +10.3923 q^{38} +4.89898i q^{40} +8.66025 q^{41} +5.00000 q^{43} +4.00000 q^{46} -8.66025 q^{47} +(-5.00000 - 4.89898i) q^{49} +2.82843i q^{50} +11.3137i q^{53} +2.44949i q^{55} +(-6.92820 - 2.82843i) q^{56} +10.0000 q^{58} +8.66025 q^{59} +2.44949i q^{61} -3.46410 q^{62} -8.00000 q^{64} +4.24264i q^{65} +2.00000 q^{67} +(6.00000 + 2.44949i) q^{70} -14.1421i q^{71} -7.07107i q^{74} +(-3.46410 - 1.41421i) q^{77} -13.0000 q^{79} +6.92820 q^{80} -12.2474i q^{82} +1.73205 q^{83} -9.00000 q^{85} -7.07107i q^{86} -4.00000 q^{88} -10.3923 q^{89} +(-6.00000 - 2.44949i) q^{91} +12.2474i q^{94} -12.7279i q^{95} +17.1464i q^{97} +(-6.92820 + 7.07107i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} - 16 q^{16} - 8 q^{22} - 8 q^{25} + 20 q^{37} + 20 q^{43} + 16 q^{46} - 20 q^{49} + 40 q^{58} - 32 q^{64} + 8 q^{67} + 24 q^{70} - 52 q^{79} - 36 q^{85} - 16 q^{88} - 24 q^{91}+O(q^{100}) $ 4 * q + 4 * q^7 - 16 * q^16 - 8 * q^22 - 8 * q^25 + 20 * q^37 + 20 * q^43 + 16 * q^46 - 20 * q^49 + 40 * q^58 - 32 * q^64 + 8 * q^67 + 24 * q^70 - 52 * q^79 - 36 * q^85 - 16 * q^88 - 24 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	1.41421i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$		−	1.41421i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.73205			−0.774597			−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205			−0.774597			−0.387298	+	0.921954i	$0.626592\pi$
$6$		0			0
$7$	1.00000	−	2.44949i	0.377964	−	0.925820i
$7$	1.00000	−	2.44949i	0.377964	−	0.925820i
$8$		−	2.82843i		−	1.00000i
$9$		0			0
$10$			2.44949i			0.774597i
$11$		−	1.41421i		−	0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$		−	1.41421i		−	0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$		0			0
$13$		−	2.44949i		−	0.679366i	−0.940540	−	0.339683i	$-0.889680\pi$
$13$		−	2.44949i		−	0.679366i	0.940540	−	0.339683i	$-0.110320\pi$
$14$	−3.46410	−	1.41421i	−0.925820	−	0.377964i
$15$		0			0
$16$	−4.00000			−1.00000
$17$	5.19615			1.26025			0.630126	−	0.776493i	$-0.283003\pi$
$17$	5.19615			1.26025			0.630126	+	0.776493i	$0.283003\pi$
$18$		0			0
$19$			7.34847i			1.68585i	0.538028	+	0.842927i	$0.319170\pi$
$19$			7.34847i			1.68585i	−0.538028	+	0.842927i	$0.680830\pi$
$20$		0			0
$21$		0			0
$22$	−2.00000			−0.426401
$23$			2.82843i			0.589768i	0.955533	+	0.294884i	$0.0952810\pi$
$23$			2.82843i			0.589768i	−0.955533	+	0.294884i	$0.904719\pi$
$24$		0			0
$25$	−2.00000			−0.400000
$26$	−3.46410			−0.679366
$27$		0			0
$28$		0			0
$29$			7.07107i			1.31306i	0.754298	+	0.656532i	$0.227977\pi$
$29$			7.07107i			1.31306i	−0.754298	+	0.656532i	$0.772023\pi$
$30$		0			0
$31$		−	2.44949i		−	0.439941i	−0.975506	−	0.219971i	$-0.929404\pi$
$31$		−	2.44949i		−	0.439941i	0.975506	−	0.219971i	$-0.0705962\pi$
$32$		0			0
$33$		0			0
$34$		−	7.34847i		−	1.26025i
$35$	−1.73205	+	4.24264i	−0.292770	+	0.717137i
$36$		0			0
$37$	5.00000			0.821995			0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000			0.821995			0.410997	+	0.911636i	$0.365181\pi$
$38$	10.3923			1.68585
$39$		0			0
$40$			4.89898i			0.774597i
$41$	8.66025			1.35250			0.676252	−	0.736670i	$-0.263603\pi$
$41$	8.66025			1.35250			0.676252	+	0.736670i	$0.263603\pi$
$42$		0			0
$43$	5.00000			0.762493			0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000			0.762493			0.381246	+	0.924473i	$0.375495\pi$
$44$		0			0
$45$		0			0
$46$	4.00000			0.589768
$47$	−8.66025			−1.26323			−0.631614	−	0.775283i	$-0.717607\pi$
$47$	−8.66025			−1.26323			−0.631614	+	0.775283i	$0.717607\pi$
$48$		0			0
$49$	−5.00000	−	4.89898i	−0.714286	−	0.699854i
$50$			2.82843i			0.400000i
$51$		0			0
$52$		0			0
$53$			11.3137i			1.55406i	0.629465	+	0.777029i	$0.283274\pi$
$53$			11.3137i			1.55406i	−0.629465	+	0.777029i	$0.716726\pi$
$54$		0			0
$55$			2.44949i			0.330289i
$56$	−6.92820	−	2.82843i	−0.925820	−	0.377964i
$57$		0			0
$58$	10.0000			1.31306
$59$	8.66025			1.12747			0.563735	−	0.825956i	$-0.309364\pi$
$59$	8.66025			1.12747			0.563735	+	0.825956i	$0.309364\pi$
$60$		0			0
$61$			2.44949i			0.313625i	0.987628	+	0.156813i	$0.0501218\pi$
$61$			2.44949i			0.313625i	−0.987628	+	0.156813i	$0.949878\pi$
$62$	−3.46410			−0.439941
$63$		0			0
$64$	−8.00000			−1.00000
$65$			4.24264i			0.526235i
$66$		0			0
$67$	2.00000			0.244339			0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000			0.244339			0.122169	+	0.992509i	$0.461015\pi$
$68$		0			0
$69$		0			0
$70$	6.00000	+	2.44949i	0.717137	+	0.292770i
$71$		−	14.1421i		−	1.67836i	−0.543852	−	0.839181i	$-0.683035\pi$
$71$		−	14.1421i		−	1.67836i	0.543852	−	0.839181i	$-0.316965\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		−	7.07107i		−	0.821995i
$75$		0			0
$76$		0			0
$77$	−3.46410	−	1.41421i	−0.394771	−	0.161165i
$78$		0			0
$79$	−13.0000			−1.46261			−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000			−1.46261			−0.731307	+	0.682048i	$0.761089\pi$
$80$	6.92820			0.774597
$81$		0			0
$82$		−	12.2474i		−	1.35250i
$83$	1.73205			0.190117			0.0950586	−	0.995472i	$-0.469696\pi$
$83$	1.73205			0.190117			0.0950586	+	0.995472i	$0.469696\pi$
$84$		0			0
$85$	−9.00000			−0.976187
$86$		−	7.07107i		−	0.762493i
$87$		0			0
$88$	−4.00000			−0.426401
$89$	−10.3923			−1.10158			−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923			−1.10158			−0.550791	+	0.834643i	$0.685674\pi$
$90$		0			0
$91$	−6.00000	−	2.44949i	−0.628971	−	0.256776i
$92$		0			0
$93$		0			0
$94$			12.2474i			1.26323i
$95$		−	12.7279i		−	1.30586i
$96$		0			0
$97$			17.1464i			1.74096i	0.492207	+	0.870478i	$0.336190\pi$
$97$			17.1464i			1.74096i	−0.492207	+	0.870478i	$0.663810\pi$
$98$	−6.92820	+	7.07107i	−0.699854	+	0.714286i
$99$		0			0
$100$		0			0
$101$	−3.46410			−0.344691			−0.172345	−	0.985037i	$-0.555135\pi$
$101$	−3.46410			−0.344691			−0.172345	+	0.985037i	$0.555135\pi$
$102$		0			0
$103$		−	9.79796i		−	0.965422i	−0.875780	−	0.482711i	$-0.839652\pi$
$103$		−	9.79796i		−	0.965422i	0.875780	−	0.482711i	$-0.160348\pi$
$104$	−6.92820			−0.679366
$105$		0			0
$106$	16.0000			1.55406
$107$		−	14.1421i		−	1.36717i	−0.729870	−	0.683586i	$-0.760419\pi$
$107$		−	14.1421i		−	1.36717i	0.729870	−	0.683586i	$-0.239581\pi$
$108$		0			0
$109$	−7.00000			−0.670478			−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000			−0.670478			−0.335239	+	0.942133i	$0.608817\pi$
$110$	3.46410			0.330289
$111$		0			0
$112$	−4.00000	+	9.79796i	−0.377964	+	0.925820i
$113$			7.07107i			0.665190i	0.943070	+	0.332595i	$0.107924\pi$
$113$			7.07107i			0.665190i	−0.943070	+	0.332595i	$0.892076\pi$
$114$		0			0
$115$		−	4.89898i		−	0.456832i
$116$		0			0
$117$		0			0
$118$		−	12.2474i		−	1.12747i
$119$	5.19615	−	12.7279i	0.476331	−	1.16677i
$120$		0			0
$121$	9.00000			0.818182
$122$	3.46410			0.313625
$123$		0			0
$124$		0			0
$125$	12.1244			1.08444
$126$		0			0
$127$	5.00000			0.443678			0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000			0.443678			0.221839	+	0.975083i	$0.428794\pi$
$128$			11.3137i			1.00000i
$129$		0			0
$130$	6.00000			0.526235
$131$	−17.3205			−1.51330			−0.756650	−	0.653820i	$-0.773165\pi$
$131$	−17.3205			−1.51330			−0.756650	+	0.653820i	$0.773165\pi$
$132$		0			0
$133$	18.0000	+	7.34847i	1.56080	+	0.637193i
$134$		−	2.82843i		−	0.244339i
$135$		0			0
$136$		−	14.6969i		−	1.26025i
$137$		−	1.41421i		−	0.120824i	−0.998174	−	0.0604122i	$-0.980758\pi$
$137$		−	1.41421i		−	0.120824i	0.998174	−	0.0604122i	$-0.0192415\pi$
$138$		0			0
$139$			12.2474i			1.03882i	0.854527	+	0.519408i	$0.173847\pi$
$139$			12.2474i			1.03882i	−0.854527	+	0.519408i	$0.826153\pi$
$140$		0			0
$141$		0			0
$142$	−20.0000			−1.67836
$143$	−3.46410			−0.289683
$144$		0			0
$145$		−	12.2474i		−	1.01710i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			7.07107i			0.579284i	0.957135	+	0.289642i	$0.0935363\pi$
$149$			7.07107i			0.579284i	−0.957135	+	0.289642i	$0.906464\pi$
$150$		0			0
$151$	5.00000			0.406894			0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000			0.406894			0.203447	+	0.979086i	$0.434786\pi$
$152$	20.7846			1.68585
$153$		0			0
$154$	−2.00000	+	4.89898i	−0.161165	+	0.394771i
$155$			4.24264i			0.340777i
$156$		0			0
$157$		−	17.1464i		−	1.36843i	−0.729279	−	0.684217i	$-0.760144\pi$
$157$		−	17.1464i		−	1.36843i	0.729279	−	0.684217i	$-0.239856\pi$
$158$			18.3848i			1.46261i
$159$		0			0
$160$		0			0
$161$	6.92820	+	2.82843i	0.546019	+	0.222911i
$162$		0			0
$163$	5.00000			0.391630			0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000			0.391630			0.195815	+	0.980641i	$0.437265\pi$
$164$		0			0
$165$		0			0
$166$		−	2.44949i		−	0.190117i
$167$	−12.1244			−0.938211			−0.469105	−	0.883142i	$-0.655424\pi$
$167$	−12.1244			−0.938211			−0.469105	+	0.883142i	$0.655424\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$			12.7279i			0.976187i
$171$		0			0
$172$		0			0
$173$	6.92820			0.526742			0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820			0.526742			0.263371	+	0.964695i	$0.415166\pi$
$174$		0			0
$175$	−2.00000	+	4.89898i	−0.151186	+	0.370328i
$176$			5.65685i			0.426401i
$177$		0			0
$178$			14.6969i			1.10158i
$179$			7.07107i			0.528516i	0.964452	+	0.264258i	$0.0851271\pi$
$179$			7.07107i			0.528516i	−0.964452	+	0.264258i	$0.914873\pi$
$180$		0			0
$181$			14.6969i			1.09241i	0.837650	+	0.546207i	$0.183929\pi$
$181$			14.6969i			1.09241i	−0.837650	+	0.546207i	$0.816071\pi$
$182$	−3.46410	+	8.48528i	−0.256776	+	0.628971i
$183$		0			0
$184$	8.00000			0.589768
$185$	−8.66025			−0.636715
$186$		0			0
$187$		−	7.34847i		−	0.537373i
$188$		0			0
$189$		0			0
$190$	−18.0000			−1.30586
$191$		−	14.1421i		−	1.02329i	−0.859197	−	0.511645i	$-0.829036\pi$
$191$		−	14.1421i		−	1.02329i	0.859197	−	0.511645i	$-0.170964\pi$
$192$		0			0
$193$	−1.00000			−0.0719816			−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000			−0.0719816			−0.0359908	+	0.999352i	$0.511459\pi$
$194$	24.2487			1.74096
$195$		0			0
$196$		0			0
$197$		−	1.41421i		−	0.100759i	−0.998730	−	0.0503793i	$-0.983957\pi$
$197$		−	1.41421i		−	0.100759i	0.998730	−	0.0503793i	$-0.0160430\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			5.65685i			0.400000i
$201$		0			0
$202$			4.89898i			0.344691i
$203$	17.3205	+	7.07107i	1.21566	+	0.496292i
$204$		0			0
$205$	−15.0000			−1.04765
$206$	−13.8564			−0.965422
$207$		0			0
$208$			9.79796i			0.679366i
$209$	10.3923			0.718851
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$	−20.0000			−1.36717
$215$	−8.66025			−0.590624
$216$		0			0
$217$	−6.00000	−	2.44949i	−0.407307	−	0.166282i
$218$			9.89949i			0.670478i
$219$		0			0
$220$		0			0
$221$		−	12.7279i		−	0.856173i
$222$		0			0
$223$		−	12.2474i		−	0.820150i	−0.912052	−	0.410075i	$-0.865503\pi$
$223$		−	12.2474i		−	0.820150i	0.912052	−	0.410075i	$-0.134497\pi$
$224$		0			0
$225$		0			0
$226$	10.0000			0.665190
$227$	−13.8564			−0.919682			−0.459841	−	0.888001i	$-0.652094\pi$
$227$	−13.8564			−0.919682			−0.459841	+	0.888001i	$0.652094\pi$
$228$		0			0
$229$			4.89898i			0.323734i	0.986813	+	0.161867i	$0.0517515\pi$
$229$			4.89898i			0.323734i	−0.986813	+	0.161867i	$0.948248\pi$
$230$	−6.92820			−0.456832
$231$		0			0
$232$	20.0000			1.31306
$233$			7.07107i			0.463241i	0.972806	+	0.231621i	$0.0744028\pi$
$233$			7.07107i			0.463241i	−0.972806	+	0.231621i	$0.925597\pi$
$234$		0			0
$235$	15.0000			0.978492
$236$		0			0
$237$		0			0
$238$	−18.0000	−	7.34847i	−1.16677	−	0.476331i
$239$		−	18.3848i		−	1.18921i	−0.804017	−	0.594606i	$-0.797308\pi$
$239$		−	18.3848i		−	1.18921i	0.804017	−	0.594606i	$-0.202692\pi$
$240$		0			0
$241$			2.44949i			0.157786i	0.996883	+	0.0788928i	$0.0251385\pi$
$241$			2.44949i			0.157786i	−0.996883	+	0.0788928i	$0.974862\pi$
$242$		−	12.7279i		−	0.818182i
$243$		0			0
$244$		0			0
$245$	8.66025	+	8.48528i	0.553283	+	0.542105i
$246$		0			0
$247$	18.0000			1.14531
$248$	−6.92820			−0.439941
$249$		0			0
$250$		−	17.1464i		−	1.08444i
$251$	−5.19615			−0.327978			−0.163989	−	0.986462i	$-0.552436\pi$
$251$	−5.19615			−0.327978			−0.163989	+	0.986462i	$0.552436\pi$
$252$		0			0
$253$	4.00000			0.251478
$254$		−	7.07107i		−	0.443678i
$255$		0			0
$256$		0			0
$257$	3.46410			0.216085			0.108042	−	0.994146i	$-0.465542\pi$
$257$	3.46410			0.216085			0.108042	+	0.994146i	$0.465542\pi$
$258$		0			0
$259$	5.00000	−	12.2474i	0.310685	−	0.761019i
$260$		0			0
$261$		0			0
$262$			24.4949i			1.51330i
$263$		−	1.41421i		−	0.0872041i	−0.999049	−	0.0436021i	$-0.986117\pi$
$263$		−	1.41421i		−	0.0872041i	0.999049	−	0.0436021i	$-0.0138834\pi$
$264$		0			0
$265$		−	19.5959i		−	1.20377i
$266$	10.3923	−	25.4558i	0.637193	−	1.56080i
$267$		0			0
$268$		0			0
$269$	−25.9808			−1.58408			−0.792038	−	0.610472i	$-0.790980\pi$
$269$	−25.9808			−1.58408			−0.792038	+	0.610472i	$0.790980\pi$
$270$		0			0
$271$			14.6969i			0.892775i	0.894840	+	0.446388i	$0.147290\pi$
$271$			14.6969i			0.892775i	−0.894840	+	0.446388i	$0.852710\pi$
$272$	−20.7846			−1.26025
$273$		0			0
$274$	−2.00000			−0.120824
$275$			2.82843i			0.170561i
$276$		0			0
$277$	5.00000			0.300421			0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000			0.300421			0.150210	+	0.988654i	$0.452005\pi$
$278$	17.3205			1.03882
$279$		0			0
$280$	12.0000	+	4.89898i	0.717137	+	0.292770i
$281$		−	14.1421i		−	0.843649i	−0.906677	−	0.421825i	$-0.861390\pi$
$281$		−	14.1421i		−	0.843649i	0.906677	−	0.421825i	$-0.138610\pi$
$282$		0			0
$283$			12.2474i			0.728035i	0.931392	+	0.364018i	$0.118595\pi$
$283$			12.2474i			0.728035i	−0.931392	+	0.364018i	$0.881405\pi$
$284$		0			0
$285$		0			0
$286$			4.89898i			0.289683i
$287$	8.66025	−	21.2132i	0.511199	−	1.25218i
$288$		0			0
$289$	10.0000			0.588235
$290$	−17.3205			−1.01710
$291$		0			0
$292$		0			0
$293$	19.0526			1.11306			0.556531	−	0.830827i	$-0.312132\pi$
$293$	19.0526			1.11306			0.556531	+	0.830827i	$0.312132\pi$
$294$		0			0
$295$	−15.0000			−0.873334
$296$		−	14.1421i		−	0.821995i
$297$		0			0
$298$	10.0000			0.579284
$299$	6.92820			0.400668
$300$		0			0
$301$	5.00000	−	12.2474i	0.288195	−	0.705931i
$302$		−	7.07107i		−	0.406894i
$303$		0			0
$304$		−	29.3939i		−	1.68585i
$305$		−	4.24264i		−	0.242933i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	6.00000			0.340777
$311$	−12.1244			−0.687509			−0.343755	−	0.939060i	$-0.611699\pi$
$311$	−12.1244			−0.687509			−0.343755	+	0.939060i	$0.611699\pi$
$312$		0			0
$313$			24.4949i			1.38453i	0.721642	+	0.692267i	$0.243388\pi$
$313$			24.4949i			1.38453i	−0.721642	+	0.692267i	$0.756612\pi$
$314$	−24.2487			−1.36843
$315$		0			0
$316$		0			0
$317$		−	5.65685i		−	0.317721i	−0.987301	−	0.158860i	$-0.949218\pi$
$317$		−	5.65685i		−	0.317721i	0.987301	−	0.158860i	$-0.0507819\pi$
$318$		0			0
$319$	10.0000			0.559893
$320$	13.8564			0.774597
$321$		0			0
$322$	4.00000	−	9.79796i	0.222911	−	0.546019i
$323$			38.1838i			2.12460i
$324$		0			0
$325$			4.89898i			0.271746i
$326$		−	7.07107i		−	0.391630i
$327$		0			0
$328$		−	24.4949i		−	1.35250i
$329$	−8.66025	+	21.2132i	−0.477455	+	1.16952i
$330$		0			0
$331$	−25.0000			−1.37412			−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000			−1.37412			−0.687062	+	0.726599i	$0.741100\pi$
$332$		0			0
$333$		0			0
$334$			17.1464i			0.938211i
$335$	−3.46410			−0.189264
$336$		0			0
$337$	−7.00000			−0.381314			−0.190657	−	0.981657i	$-0.561062\pi$
$337$	−7.00000			−0.381314			−0.190657	+	0.981657i	$0.561062\pi$
$338$		−	9.89949i		−	0.538462i
$339$		0			0
$340$		0			0
$341$	−3.46410			−0.187592
$342$		0			0
$343$	−17.0000	+	7.34847i	−0.917914	+	0.396780i
$344$		−	14.1421i		−	0.762493i
$345$		0			0
$346$		−	9.79796i		−	0.526742i
$347$			7.07107i			0.379595i	0.981823	+	0.189797i	$0.0607831\pi$
$347$			7.07107i			0.379595i	−0.981823	+	0.189797i	$0.939217\pi$
$348$		0			0
$349$		−	19.5959i		−	1.04895i	−0.851427	−	0.524473i	$-0.824262\pi$
$349$		−	19.5959i		−	1.04895i	0.851427	−	0.524473i	$-0.175738\pi$
$350$	6.92820	+	2.82843i	0.370328	+	0.151186i
$351$		0			0
$352$		0			0
$353$	−8.66025			−0.460939			−0.230469	−	0.973080i	$-0.574026\pi$
$353$	−8.66025			−0.460939			−0.230469	+	0.973080i	$0.574026\pi$
$354$		0			0
$355$			24.4949i			1.30005i
$356$		0			0
$357$		0			0
$358$	10.0000			0.528516
$359$			11.3137i			0.597115i	0.954392	+	0.298557i	$0.0965054\pi$
$359$			11.3137i			0.597115i	−0.954392	+	0.298557i	$0.903495\pi$
$360$		0			0
$361$	−35.0000			−1.84211
$362$	20.7846			1.09241
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	4.89898i		−	0.255725i	−0.991792	−	0.127862i	$-0.959188\pi$
$367$		−	4.89898i		−	0.255725i	0.991792	−	0.127862i	$-0.0408116\pi$
$368$		−	11.3137i		−	0.589768i
$369$		0			0
$370$			12.2474i			0.636715i
$371$	27.7128	+	11.3137i	1.43878	+	0.587378i
$372$		0			0
$373$	−25.0000			−1.29445			−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000			−1.29445			−0.647225	+	0.762299i	$0.724071\pi$
$374$	−10.3923			−0.537373
$375$		0			0
$376$			24.4949i			1.26323i
$377$	17.3205			0.892052
$378$		0			0
$379$	23.0000			1.18143			0.590715	−	0.806880i	$-0.298846\pi$
$379$	23.0000			1.18143			0.590715	+	0.806880i	$0.298846\pi$
$380$		0			0
$381$		0			0
$382$	−20.0000			−1.02329
$383$	−1.73205			−0.0885037			−0.0442518	−	0.999020i	$-0.514090\pi$
$383$	−1.73205			−0.0885037			−0.0442518	+	0.999020i	$0.514090\pi$
$384$		0			0
$385$	6.00000	+	2.44949i	0.305788	+	0.124838i
$386$			1.41421i			0.0719816i
$387$		0			0
$388$		0			0
$389$			32.5269i			1.64918i	0.565731	+	0.824590i	$0.308594\pi$
$389$			32.5269i			1.64918i	−0.565731	+	0.824590i	$0.691406\pi$
$390$		0			0
$391$			14.6969i			0.743256i
$392$	−13.8564	+	14.1421i	−0.699854	+	0.714286i
$393$		0			0
$394$	−2.00000			−0.100759
$395$	22.5167			1.13294
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	8.00000			0.400000
$401$			19.7990i			0.988714i	0.869259	+	0.494357i	$0.164597\pi$
$401$			19.7990i			0.988714i	−0.869259	+	0.494357i	$0.835403\pi$
$402$		0			0
$403$	−6.00000			−0.298881
$404$		0			0
$405$		0			0
$406$	10.0000	−	24.4949i	0.496292	−	1.21566i
$407$		−	7.07107i		−	0.350500i
$408$		0			0
$409$			19.5959i			0.968956i	0.874804	+	0.484478i	$0.160990\pi$
$409$			19.5959i			0.968956i	−0.874804	+	0.484478i	$0.839010\pi$
$410$			21.2132i			1.04765i
$411$		0			0
$412$		0			0
$413$	8.66025	−	21.2132i	0.426143	−	1.04383i
$414$		0			0
$415$	−3.00000			−0.147264
$416$		0			0
$417$		0			0
$418$		−	14.6969i		−	0.718851i
$419$	−1.73205			−0.0846162			−0.0423081	−	0.999105i	$-0.513471\pi$
$419$	−1.73205			−0.0846162			−0.0423081	+	0.999105i	$0.513471\pi$
$420$		0			0
$421$	−4.00000			−0.194948			−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000			−0.194948			−0.0974740	+	0.995238i	$0.531076\pi$
$422$			22.6274i			1.10149i
$423$		0			0
$424$	32.0000			1.55406
$425$	−10.3923			−0.504101
$426$		0			0
$427$	6.00000	+	2.44949i	0.290360	+	0.118539i
$428$		0			0
$429$		0			0
$430$			12.2474i			0.590624i
$431$		−	5.65685i		−	0.272481i	−0.990676	−	0.136241i	$-0.956498\pi$
$431$		−	5.65685i		−	0.272481i	0.990676	−	0.136241i	$-0.0435020\pi$
$432$		0			0
$433$		−	36.7423i		−	1.76572i	−0.469632	−	0.882862i	$-0.655613\pi$
$433$		−	36.7423i		−	1.76572i	0.469632	−	0.882862i	$-0.344387\pi$
$434$	−3.46410	+	8.48528i	−0.166282	+	0.407307i
$435$		0			0
$436$		0			0
$437$	−20.7846			−0.994263
$438$		0			0
$439$		−	19.5959i		−	0.935262i	−0.883924	−	0.467631i	$-0.845108\pi$
$439$		−	19.5959i		−	0.935262i	0.883924	−	0.467631i	$-0.154892\pi$
$440$	6.92820			0.330289
$441$		0			0
$442$	−18.0000			−0.856173
$443$			32.5269i			1.54540i	0.634771	+	0.772700i	$0.281094\pi$
$443$			32.5269i			1.54540i	−0.634771	+	0.772700i	$0.718906\pi$
$444$		0			0
$445$	18.0000			0.853282
$446$	−17.3205			−0.820150
$447$		0			0
$448$	−8.00000	+	19.5959i	−0.377964	+	0.925820i
$449$			24.0416i			1.13459i	0.823513	+	0.567297i	$0.192011\pi$
$449$			24.0416i			1.13459i	−0.823513	+	0.567297i	$0.807989\pi$
$450$		0			0
$451$		−	12.2474i		−	0.576710i
$452$		0			0
$453$		0			0
$454$			19.5959i			0.919682i
$455$	10.3923	+	4.24264i	0.487199	+	0.198898i
$456$		0			0
$457$	32.0000			1.49690			0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000			1.49690			0.748448	+	0.663193i	$0.230799\pi$
$458$	6.92820			0.323734
$459$		0			0
$460$		0			0
$461$	12.1244			0.564688			0.282344	−	0.959313i	$-0.408888\pi$
$461$	12.1244			0.564688			0.282344	+	0.959313i	$0.408888\pi$
$462$		0			0
$463$	11.0000			0.511213			0.255607	−	0.966781i	$-0.417725\pi$
$463$	11.0000			0.511213			0.255607	+	0.966781i	$0.417725\pi$
$464$		−	28.2843i		−	1.31306i
$465$		0			0
$466$	10.0000			0.463241
$467$	20.7846			0.961797			0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846			0.961797			0.480899	+	0.876776i	$0.340311\pi$
$468$		0			0
$469$	2.00000	−	4.89898i	0.0923514	−	0.226214i
$470$		−	21.2132i		−	0.978492i
$471$		0			0
$472$		−	24.4949i		−	1.12747i
$473$		−	7.07107i		−	0.325128i
$474$		0			0
$475$		−	14.6969i		−	0.674342i
$476$		0			0
$477$		0			0
$478$	−26.0000			−1.18921
$479$	32.9090			1.50365			0.751825	−	0.659363i	$-0.229174\pi$
$479$	32.9090			1.50365			0.751825	+	0.659363i	$0.229174\pi$
$480$		0			0
$481$		−	12.2474i		−	0.558436i
$482$	3.46410			0.157786
$483$		0			0
$484$		0			0
$485$		−	29.6985i		−	1.34854i
$486$		0			0
$487$	−22.0000			−0.996915			−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000			−0.996915			−0.498458	+	0.866914i	$0.666100\pi$
$488$	6.92820			0.313625
$489$		0			0
$490$	12.0000	−	12.2474i	0.542105	−	0.553283i
$491$			15.5563i			0.702048i	0.936366	+	0.351024i	$0.114166\pi$
$491$			15.5563i			0.702048i	−0.936366	+	0.351024i	$0.885834\pi$
$492$		0			0
$493$			36.7423i			1.65479i
$494$		−	25.4558i		−	1.14531i
$495$		0			0
$496$			9.79796i			0.439941i
$497$	−34.6410	−	14.1421i	−1.55386	−	0.634361i
$498$		0			0
$499$	17.0000			0.761025			0.380512	−	0.924776i	$-0.375748\pi$
$499$	17.0000			0.761025			0.380512	+	0.924776i	$0.375748\pi$
$500$		0			0
$501$		0			0
$502$			7.34847i			0.327978i
$503$	−36.3731			−1.62179			−0.810897	−	0.585188i	$-0.801021\pi$
$503$	−36.3731			−1.62179			−0.810897	+	0.585188i	$0.801021\pi$
$504$		0			0
$505$	6.00000			0.266996
$506$		−	5.65685i		−	0.251478i
$507$		0			0
$508$		0			0
$509$	8.66025			0.383859			0.191930	−	0.981409i	$-0.438526\pi$
$509$	8.66025			0.383859			0.191930	+	0.981409i	$0.438526\pi$
$510$		0			0
$511$		0			0
$512$			22.6274i			1.00000i
$513$		0			0
$514$		−	4.89898i		−	0.216085i
$515$			16.9706i			0.747812i
$516$		0			0
$517$			12.2474i			0.538642i
$518$	−17.3205	−	7.07107i	−0.761019	−	0.310685i
$519$		0			0
$520$	12.0000			0.526235
$521$	−5.19615			−0.227648			−0.113824	−	0.993501i	$-0.536310\pi$
$521$	−5.19615			−0.227648			−0.113824	+	0.993501i	$0.536310\pi$
$522$		0			0
$523$		−	36.7423i		−	1.60663i	−0.595554	−	0.803315i	$-0.703067\pi$
$523$		−	36.7423i		−	1.60663i	0.595554	−	0.803315i	$-0.296933\pi$
$524$		0			0
$525$		0			0
$526$	−2.00000			−0.0872041
$527$		−	12.7279i		−	0.554437i
$528$		0			0
$529$	15.0000			0.652174
$530$	−27.7128			−1.20377
$531$		0			0
$532$		0			0
$533$		−	21.2132i		−	0.918846i
$534$		0			0
$535$			24.4949i			1.05901i
$536$		−	5.65685i		−	0.244339i
$537$		0			0
$538$			36.7423i			1.58408i
$539$	−6.92820	+	7.07107i	−0.298419	+	0.304572i
$540$		0			0
$541$	5.00000			0.214967			0.107483	−	0.994207i	$-0.465721\pi$
$541$	5.00000			0.214967			0.107483	+	0.994207i	$0.465721\pi$
$542$	20.7846			0.892775
$543$		0			0
$544$		0			0
$545$	12.1244			0.519350
$546$		0			0
$547$	−13.0000			−0.555840			−0.277920	−	0.960604i	$-0.589645\pi$
$547$	−13.0000			−0.555840			−0.277920	+	0.960604i	$0.589645\pi$
$548$		0			0
$549$		0			0
$550$	4.00000			0.170561
$551$	−51.9615			−2.21364
$552$		0			0
$553$	−13.0000	+	31.8434i	−0.552816	+	1.35412i
$554$		−	7.07107i		−	0.300421i
$555$		0			0
$556$		0			0
$557$		−	14.1421i		−	0.599222i	−0.954062	−	0.299611i	$-0.903143\pi$
$557$		−	14.1421i		−	0.599222i	0.954062	−	0.299611i	$-0.0968568\pi$
$558$		0			0
$559$		−	12.2474i		−	0.518012i
$560$	6.92820	−	16.9706i	0.292770	−	0.717137i
$561$		0			0
$562$	−20.0000			−0.843649
$563$	34.6410			1.45994			0.729972	−	0.683477i	$-0.239533\pi$
$563$	34.6410			1.45994			0.729972	+	0.683477i	$0.239533\pi$
$564$		0			0
$565$		−	12.2474i		−	0.515254i
$566$	17.3205			0.728035
$567$		0			0
$568$	−40.0000			−1.67836
$569$		−	39.5980i		−	1.66003i	−0.557738	−	0.830017i	$-0.688331\pi$
$569$		−	39.5980i		−	1.66003i	0.557738	−	0.830017i	$-0.311669\pi$
$570$		0			0
$571$	−19.0000			−0.795125			−0.397563	−	0.917575i	$-0.630144\pi$
$571$	−19.0000			−0.795125			−0.397563	+	0.917575i	$0.630144\pi$
$572$		0			0
$573$		0			0
$574$	−30.0000	−	12.2474i	−1.25218	−	0.511199i
$575$		−	5.65685i		−	0.235907i
$576$		0			0
$577$		−	29.3939i		−	1.22368i	−0.790980	−	0.611842i	$-0.790429\pi$
$577$		−	29.3939i		−	1.22368i	0.790980	−	0.611842i	$-0.209571\pi$
$578$		−	14.1421i		−	0.588235i
$579$		0			0
$580$		0			0
$581$	1.73205	−	4.24264i	0.0718576	−	0.176014i
$582$		0			0
$583$	16.0000			0.662652
$584$		0			0
$585$		0			0
$586$		−	26.9444i		−	1.11306i
$587$	−13.8564			−0.571915			−0.285958	−	0.958242i	$-0.592312\pi$
$587$	−13.8564			−0.571915			−0.285958	+	0.958242i	$0.592312\pi$
$588$		0			0
$589$	18.0000			0.741677
$590$			21.2132i			0.873334i
$591$		0			0
$592$	−20.0000			−0.821995
$593$	−15.5885			−0.640141			−0.320071	−	0.947394i	$-0.603707\pi$
$593$	−15.5885			−0.640141			−0.320071	+	0.947394i	$0.603707\pi$
$594$		0			0
$595$	−9.00000	+	22.0454i	−0.368964	+	0.903774i
$596$		0			0
$597$		0			0
$598$		−	9.79796i		−	0.400668i
$599$		−	18.3848i		−	0.751182i	−0.926786	−	0.375591i	$-0.877440\pi$
$599$		−	18.3848i		−	0.751182i	0.926786	−	0.375591i	$-0.122560\pi$
$600$		0			0
$601$			46.5403i			1.89842i	0.314645	+	0.949209i	$0.398114\pi$
$601$			46.5403i			1.89842i	−0.314645	+	0.949209i	$0.601886\pi$
$602$	−17.3205	−	7.07107i	−0.705931	−	0.288195i
$603$		0			0
$604$		0			0
$605$	−15.5885			−0.633761
$606$		0			0
$607$			12.2474i			0.497109i	0.968618	+	0.248554i	$0.0799554\pi$
$607$			12.2474i			0.497109i	−0.968618	+	0.248554i	$0.920045\pi$
$608$		0			0
$609$		0			0
$610$	−6.00000			−0.242933
$611$			21.2132i			0.858194i
$612$		0			0
$613$	−4.00000			−0.161558			−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000			−0.161558			−0.0807792	+	0.996732i	$0.525741\pi$
$614$		0			0
$615$		0			0
$616$	−4.00000	+	9.79796i	−0.161165	+	0.394771i
$617$		−	48.0833i		−	1.93576i	−0.251414	−	0.967880i	$-0.580896\pi$
$617$		−	48.0833i		−	1.93576i	0.251414	−	0.967880i	$-0.419104\pi$
$618$		0			0
$619$		−	26.9444i		−	1.08299i	−0.840705	−	0.541493i	$-0.817859\pi$
$619$		−	26.9444i		−	1.08299i	0.840705	−	0.541493i	$-0.182141\pi$
$620$		0			0
$621$		0			0
$622$			17.1464i			0.687509i
$623$	−10.3923	+	25.4558i	−0.416359	+	1.01987i
$624$		0			0
$625$	−11.0000			−0.440000
$626$	34.6410			1.38453
$627$		0			0
$628$		0			0
$629$	25.9808			1.03592
$630$		0			0
$631$	29.0000			1.15447			0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000			1.15447			0.577236	+	0.816577i	$0.304131\pi$
$632$			36.7696i			1.46261i
$633$		0			0
$634$	−8.00000			−0.317721
$635$	−8.66025			−0.343672
$636$		0			0
$637$	−12.0000	+	12.2474i	−0.475457	+	0.485262i
$638$		−	14.1421i		−	0.559893i
$639$		0			0
$640$		−	19.5959i		−	0.774597i
$641$		−	1.41421i		−	0.0558581i	−0.999610	−	0.0279290i	$-0.991109\pi$
$641$		−	1.41421i		−	0.0558581i	0.999610	−	0.0279290i	$-0.00889125\pi$
$642$		0			0
$643$		−	24.4949i		−	0.965984i	−0.875625	−	0.482992i	$-0.839550\pi$
$643$		−	24.4949i		−	0.965984i	0.875625	−	0.482992i	$-0.160450\pi$
$644$		0			0
$645$		0			0
$646$	54.0000			2.12460
$647$	20.7846			0.817127			0.408564	−	0.912730i	$-0.366030\pi$
$647$	20.7846			0.817127			0.408564	+	0.912730i	$0.366030\pi$
$648$		0			0
$649$		−	12.2474i		−	0.480754i
$650$	6.92820			0.271746
$651$		0			0
$652$		0			0
$653$		−	18.3848i		−	0.719452i	−0.933058	−	0.359726i	$-0.882870\pi$
$653$		−	18.3848i		−	0.719452i	0.933058	−	0.359726i	$-0.117130\pi$
$654$		0			0
$655$	30.0000			1.17220
$656$	−34.6410			−1.35250
$657$		0			0
$658$	30.0000	+	12.2474i	1.16952	+	0.477455i
$659$			11.3137i			0.440720i	0.975419	+	0.220360i	$0.0707231\pi$
$659$			11.3137i			0.440720i	−0.975419	+	0.220360i	$0.929277\pi$
$660$		0			0
$661$		−	17.1464i		−	0.666919i	−0.942764	−	0.333459i	$-0.891784\pi$
$661$		−	17.1464i		−	0.666919i	0.942764	−	0.333459i	$-0.108216\pi$
$662$			35.3553i			1.37412i
$663$		0			0
$664$		−	4.89898i		−	0.190117i
$665$	−31.1769	−	12.7279i	−1.20899	−	0.493568i
$666$		0			0
$667$	−20.0000			−0.774403
$668$		0			0
$669$		0			0
$670$			4.89898i			0.189264i
$671$	3.46410			0.133730
$672$		0			0
$673$	−40.0000			−1.54189			−0.770943	−	0.636904i	$-0.780215\pi$
$673$	−40.0000			−1.54189			−0.770943	+	0.636904i	$0.780215\pi$
$674$			9.89949i			0.381314i
$675$		0			0
$676$		0			0
$677$	17.3205			0.665681			0.332841	−	0.942983i	$-0.391993\pi$
$677$	17.3205			0.665681			0.332841	+	0.942983i	$0.391993\pi$
$678$		0			0
$679$	42.0000	+	17.1464i	1.61181	+	0.658020i
$680$			25.4558i			0.976187i
$681$		0			0
$682$			4.89898i			0.187592i
$683$			24.0416i			0.919927i	0.887938	+	0.459964i	$0.152138\pi$
$683$			24.0416i			0.919927i	−0.887938	+	0.459964i	$0.847862\pi$
$684$		0			0
$685$			2.44949i			0.0935902i
$686$	10.3923	+	24.0416i	0.396780	+	0.917914i
$687$		0			0
$688$	−20.0000			−0.762493
$689$	27.7128			1.05577
$690$		0			0
$691$		−	26.9444i		−	1.02501i	−0.858683	−	0.512506i	$-0.828717\pi$
$691$		−	26.9444i		−	1.02501i	0.858683	−	0.512506i	$-0.171283\pi$
$692$		0			0
$693$		0			0
$694$	10.0000			0.379595
$695$		−	21.2132i		−	0.804663i
$696$		0			0
$697$	45.0000			1.70450
$698$	−27.7128			−1.04895
$699$		0			0
$700$		0			0
$701$		−	1.41421i		−	0.0534141i	−0.999643	−	0.0267071i	$-0.991498\pi$
$701$		−	1.41421i		−	0.0534141i	0.999643	−	0.0267071i	$-0.00850213\pi$
$702$		0			0
$703$			36.7423i			1.38576i
$704$			11.3137i			0.426401i
$705$		0			0
$706$			12.2474i			0.460939i
$707$	−3.46410	+	8.48528i	−0.130281	+	0.319122i
$708$		0			0
$709$	5.00000			0.187779			0.0938895	−	0.995583i	$-0.470070\pi$
$709$	5.00000			0.187779			0.0938895	+	0.995583i	$0.470070\pi$
$710$	34.6410			1.30005
$711$		0			0
$712$			29.3939i			1.10158i
$713$	6.92820			0.259463
$714$		0			0
$715$	6.00000			0.224387
$716$		0			0
$717$		0			0
$718$	16.0000			0.597115
$719$	15.5885			0.581351			0.290676	−	0.956822i	$-0.406120\pi$
$719$	15.5885			0.581351			0.290676	+	0.956822i	$0.406120\pi$
$720$		0			0
$721$	−24.0000	−	9.79796i	−0.893807	−	0.364895i
$722$			49.4975i			1.84211i
$723$		0			0
$724$		0			0
$725$		−	14.1421i		−	0.525226i
$726$		0			0
$727$		−	12.2474i		−	0.454233i	−0.973868	−	0.227116i	$-0.927070\pi$
$727$		−	12.2474i		−	0.454233i	0.973868	−	0.227116i	$-0.0729298\pi$
$728$	−6.92820	+	16.9706i	−0.256776	+	0.628971i
$729$		0			0
$730$		0			0
$731$	25.9808			0.960933
$732$		0			0
$733$		−	39.1918i		−	1.44758i	−0.690018	−	0.723792i	$-0.742398\pi$
$733$		−	39.1918i		−	1.44758i	0.690018	−	0.723792i	$-0.257602\pi$
$734$	−6.92820			−0.255725
$735$		0			0
$736$		0			0
$737$		−	2.82843i		−	0.104186i
$738$		0			0
$739$	−22.0000			−0.809283			−0.404642	−	0.914475i	$-0.632604\pi$
$739$	−22.0000			−0.809283			−0.404642	+	0.914475i	$0.632604\pi$
$740$		0			0
$741$		0			0
$742$	16.0000	−	39.1918i	0.587378	−	1.43878i
$743$			7.07107i			0.259412i	0.991552	+	0.129706i	$0.0414034\pi$
$743$			7.07107i			0.259412i	−0.991552	+	0.129706i	$0.958597\pi$
$744$		0			0
$745$		−	12.2474i		−	0.448712i
$746$			35.3553i			1.29445i
$747$		0			0
$748$		0			0
$749$	−34.6410	−	14.1421i	−1.26576	−	0.516742i
$750$		0			0
$751$	−34.0000			−1.24068			−0.620339	−	0.784334i	$-0.713005\pi$
$751$	−34.0000			−1.24068			−0.620339	+	0.784334i	$0.713005\pi$
$752$	34.6410			1.26323
$753$		0			0
$754$		−	24.4949i		−	0.892052i
$755$	−8.66025			−0.315179
$756$		0			0
$757$	47.0000			1.70824			0.854122	−	0.520073i	$-0.174095\pi$
$757$	47.0000			1.70824			0.854122	+	0.520073i	$0.174095\pi$
$758$		−	32.5269i		−	1.18143i
$759$		0			0
$760$	−36.0000			−1.30586
$761$	−12.1244			−0.439508			−0.219754	−	0.975555i	$-0.570525\pi$
$761$	−12.1244			−0.439508			−0.219754	+	0.975555i	$0.570525\pi$
$762$		0			0
$763$	−7.00000	+	17.1464i	−0.253417	+	0.620742i
$764$		0			0
$765$		0			0
$766$			2.44949i			0.0885037i
$767$		−	21.2132i		−	0.765964i
$768$		0			0
$769$			4.89898i			0.176662i	0.996091	+	0.0883309i	$0.0281533\pi$
$769$			4.89898i			0.176662i	−0.996091	+	0.0883309i	$0.971847\pi$
$770$	3.46410	−	8.48528i	0.124838	−	0.305788i
$771$		0			0
$772$		0			0
$773$	−15.5885			−0.560678			−0.280339	−	0.959901i	$-0.590447\pi$
$773$	−15.5885			−0.560678			−0.280339	+	0.959901i	$0.590447\pi$
$774$		0			0
$775$			4.89898i			0.175977i
$776$	48.4974			1.74096
$777$		0			0
$778$	46.0000			1.64918
$779$			63.6396i			2.28013i
$780$		0			0
$781$	−20.0000			−0.715656
$782$	20.7846			0.743256
$783$		0			0
$784$	20.0000	+	19.5959i	0.714286	+	0.699854i
$785$			29.6985i			1.05998i
$786$		0			0
$787$			4.89898i			0.174630i	0.996181	+	0.0873149i	$0.0278286\pi$
$787$			4.89898i			0.174630i	−0.996181	+	0.0873149i	$0.972171\pi$
$788$		0			0
$789$		0			0
$790$		−	31.8434i		−	1.13294i
$791$	17.3205	+	7.07107i	0.615846	+	0.251418i
$792$		0			0
$793$	6.00000			0.213066
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−38.1051			−1.34975			−0.674876	−	0.737931i	$-0.735803\pi$
$797$	−38.1051			−1.34975			−0.674876	+	0.737931i	$0.735803\pi$
$798$		0			0
$799$	−45.0000			−1.59199
$800$		0			0
$801$		0			0
$802$	28.0000			0.988714
$803$		0			0
$804$		0			0
$805$	−12.0000	−	4.89898i	−0.422944	−	0.172666i
$806$			8.48528i			0.298881i
$807$		0			0
$808$			9.79796i			0.344691i
$809$		−	14.1421i		−	0.497211i	−0.968605	−	0.248606i	$-0.920028\pi$
$809$		−	14.1421i		−	0.497211i	0.968605	−	0.248606i	$-0.0799723\pi$
$810$		0			0
$811$			36.7423i			1.29020i	0.764099	+	0.645099i	$0.223184\pi$
$811$			36.7423i			1.29020i	−0.764099	+	0.645099i	$0.776816\pi$
$812$		0			0
$813$		0			0
$814$	−10.0000			−0.350500
$815$	−8.66025			−0.303355
$816$		0			0
$817$			36.7423i			1.28545i
$818$	27.7128			0.968956
$819$		0			0
$820$		0			0
$821$			36.7696i			1.28327i	0.767012	+	0.641633i	$0.221743\pi$
$821$			36.7696i			1.28327i	−0.767012	+	0.641633i	$0.778257\pi$
$822$		0			0
$823$	−55.0000			−1.91718			−0.958590	−	0.284791i	$-0.908076\pi$
$823$	−55.0000			−1.91718			−0.958590	+	0.284791i	$0.908076\pi$
$824$	−27.7128			−0.965422
$825$		0			0
$826$	−30.0000	−	12.2474i	−1.04383	−	0.426143i
$827$		−	26.8701i		−	0.934363i	−0.884161	−	0.467182i	$-0.845269\pi$
$827$		−	26.8701i		−	0.934363i	0.884161	−	0.467182i	$-0.154731\pi$
$828$		0			0
$829$			36.7423i			1.27611i	0.769989	+	0.638057i	$0.220262\pi$
$829$			36.7423i			1.27611i	−0.769989	+	0.638057i	$0.779738\pi$
$830$			4.24264i			0.147264i
$831$		0			0
$832$			19.5959i			0.679366i
$833$	−25.9808	−	25.4558i	−0.900180	−	0.881993i
$834$		0			0
$835$	21.0000			0.726735
$836$		0			0
$837$		0			0
$838$			2.44949i			0.0846162i
$839$	−19.0526			−0.657767			−0.328884	−	0.944370i	$-0.606672\pi$
$839$	−19.0526			−0.657767			−0.328884	+	0.944370i	$0.606672\pi$
$840$		0			0
$841$	−21.0000			−0.724138
$842$			5.65685i			0.194948i
$843$		0			0
$844$		0			0
$845$	−12.1244			−0.417091
$846$		0			0
$847$	9.00000	−	22.0454i	0.309244	−	0.757489i
$848$		−	45.2548i		−	1.55406i
$849$		0			0
$850$			14.6969i			0.504101i
$851$			14.1421i			0.484786i
$852$		0			0
$853$		−	48.9898i		−	1.67738i	−0.544610	−	0.838689i	$-0.683322\pi$
$853$		−	48.9898i		−	1.67738i	0.544610	−	0.838689i	$-0.316678\pi$
$854$	3.46410	−	8.48528i	0.118539	−	0.290360i
$855$		0			0
$856$	−40.0000			−1.36717
$857$	22.5167			0.769154			0.384577	−	0.923093i	$-0.374347\pi$
$857$	22.5167			0.769154			0.384577	+	0.923093i	$0.374347\pi$
$858$		0			0
$859$			19.5959i			0.668604i	0.942466	+	0.334302i	$0.108501\pi$
$859$			19.5959i			0.668604i	−0.942466	+	0.334302i	$0.891499\pi$
$860$		0			0
$861$		0			0
$862$	−8.00000			−0.272481
$863$		−	39.5980i		−	1.34793i	−0.738763	−	0.673965i	$-0.764590\pi$
$863$		−	39.5980i		−	1.34793i	0.738763	−	0.673965i	$-0.235410\pi$
$864$		0			0
$865$	−12.0000			−0.408012
$866$	−51.9615			−1.76572
$867$		0			0
$868$		0			0
$869$			18.3848i			0.623661i
$870$		0			0
$871$		−	4.89898i		−	0.165996i
$872$			19.7990i			0.670478i
$873$		0			0
$874$			29.3939i			0.994263i
$875$	12.1244	−	29.6985i	0.409878	−	1.00399i
$876$		0			0
$877$	−43.0000			−1.45201			−0.726003	−	0.687691i	$-0.758624\pi$
$877$	−43.0000			−1.45201			−0.726003	+	0.687691i	$0.758624\pi$
$878$	−27.7128			−0.935262
$879$		0			0
$880$		−	9.79796i		−	0.330289i
$881$	51.9615			1.75063			0.875314	−	0.483555i	$-0.160655\pi$
$881$	51.9615			1.75063			0.875314	+	0.483555i	$0.160655\pi$
$882$		0			0
$883$	5.00000			0.168263			0.0841317	−	0.996455i	$-0.473188\pi$
$883$	5.00000			0.168263			0.0841317	+	0.996455i	$0.473188\pi$
$884$		0			0
$885$		0			0
$886$	46.0000			1.54540
$887$	29.4449			0.988662			0.494331	−	0.869274i	$-0.335413\pi$
$887$	29.4449			0.988662			0.494331	+	0.869274i	$0.335413\pi$
$888$		0			0
$889$	5.00000	−	12.2474i	0.167695	−	0.410766i
$890$		−	25.4558i		−	0.853282i
$891$		0			0
$892$		0			0
$893$		−	63.6396i		−	2.12962i
$894$		0			0
$895$		−	12.2474i		−	0.409387i
$896$	27.7128	+	11.3137i	0.925820	+	0.377964i
$897$		0			0
$898$	34.0000			1.13459
$899$	17.3205			0.577671
$900$		0			0
$901$			58.7878i			1.95850i
$902$	−17.3205			−0.576710
$903$		0			0
$904$	20.0000			0.665190
$905$		−	25.4558i		−	0.846181i
$906$		0			0
$907$	23.0000			0.763702			0.381851	−	0.924224i	$-0.375287\pi$
$907$	23.0000			0.763702			0.381851	+	0.924224i	$0.375287\pi$
$908$		0			0
$909$		0			0
$910$	6.00000	−	14.6969i	0.198898	−	0.487199i
$911$			7.07107i			0.234275i	0.993116	+	0.117137i	$0.0373718\pi$
$911$			7.07107i			0.234275i	−0.993116	+	0.117137i	$0.962628\pi$
$912$		0			0
$913$		−	2.44949i		−	0.0810663i
$914$		−	45.2548i		−	1.49690i
$915$		0			0
$916$		0			0
$917$	−17.3205	+	42.4264i	−0.571974	+	1.40104i
$918$		0			0
$919$	5.00000			0.164935			0.0824674	−	0.996594i	$-0.473720\pi$
$919$	5.00000			0.164935			0.0824674	+	0.996594i	$0.473720\pi$
$920$	−13.8564			−0.456832
$921$		0			0
$922$		−	17.1464i		−	0.564688i
$923$	−34.6410			−1.14022
$924$		0			0
$925$	−10.0000			−0.328798
$926$		−	15.5563i		−	0.511213i
$927$		0			0
$928$		0			0
$929$	32.9090			1.07971			0.539854	−	0.841759i	$-0.318479\pi$
$929$	32.9090			1.07971			0.539854	+	0.841759i	$0.318479\pi$
$930$		0			0
$931$	36.0000	−	36.7423i	1.17985	−	1.20418i
$932$		0			0
$933$		0			0
$934$		−	29.3939i		−	0.961797i
$935$			12.7279i			0.416248i
$936$		0			0
$937$		−	36.7423i		−	1.20032i	−0.799880	−	0.600160i	$-0.795104\pi$
$937$		−	36.7423i		−	1.20032i	0.799880	−	0.600160i	$-0.204896\pi$
$938$	−6.92820	−	2.82843i	−0.226214	−	0.0923514i
$939$		0			0
$940$		0			0
$941$	−43.3013			−1.41158			−0.705791	−	0.708421i	$-0.749408\pi$
$941$	−43.3013			−1.41158			−0.705791	+	0.708421i	$0.749408\pi$
$942$		0			0
$943$			24.4949i			0.797664i
$944$	−34.6410			−1.12747
$945$		0			0
$946$	−10.0000			−0.325128
$947$		−	5.65685i		−	0.183823i	−0.995767	−	0.0919115i	$-0.970702\pi$
$947$		−	5.65685i		−	0.183823i	0.995767	−	0.0919115i	$-0.0292977\pi$
$948$		0			0
$949$		0			0
$950$	−20.7846			−0.674342
$951$		0			0
$952$	−36.0000	−	14.6969i	−1.16677	−	0.476331i
$953$		−	56.5685i		−	1.83243i	−0.400681	−	0.916217i	$-0.631227\pi$
$953$		−	56.5685i		−	1.83243i	0.400681	−	0.916217i	$-0.368773\pi$
$954$		0			0
$955$			24.4949i			0.792636i
$956$		0			0
$957$		0			0
$958$		−	46.5403i		−	1.50365i
$959$	−3.46410	−	1.41421i	−0.111862	−	0.0456673i
$960$		0			0
$961$	25.0000			0.806452
$962$	−17.3205			−0.558436
$963$		0			0
$964$		0			0
$965$	1.73205			0.0557567
$966$		0			0
$967$	−10.0000			−0.321578			−0.160789	−	0.986989i	$-0.551404\pi$
$967$	−10.0000			−0.321578			−0.160789	+	0.986989i	$0.551404\pi$
$968$		−	25.4558i		−	0.818182i
$969$		0			0
$970$	−42.0000			−1.34854
$971$	−5.19615			−0.166752			−0.0833762	−	0.996518i	$-0.526570\pi$
$971$	−5.19615			−0.166752			−0.0833762	+	0.996518i	$0.526570\pi$
$972$		0			0
$973$	30.0000	+	12.2474i	0.961756	+	0.392635i
$974$			31.1127i			0.996915i
$975$		0			0
$976$		−	9.79796i		−	0.313625i
$977$			2.82843i			0.0904894i	0.998976	+	0.0452447i	$0.0144068\pi$
$977$			2.82843i			0.0904894i	−0.998976	+	0.0452447i	$0.985593\pi$
$978$		0			0
$979$			14.6969i			0.469716i
$980$		0			0
$981$		0			0
$982$	22.0000			0.702048
$983$	32.9090			1.04963			0.524816	−	0.851215i	$-0.324134\pi$
$983$	32.9090			1.04963			0.524816	+	0.851215i	$0.324134\pi$
$984$		0			0
$985$			2.44949i			0.0780472i
$986$	51.9615			1.65479
$987$		0			0
$988$		0			0
$989$			14.1421i			0.449694i
$990$		0			0
$991$	41.0000			1.30241			0.651204	−	0.758903i	$-0.274264\pi$
$991$	41.0000			1.30241			0.651204	+	0.758903i	$0.274264\pi$
$992$		0			0
$993$		0			0
$994$	−20.0000	+	48.9898i	−0.634361	+	1.55386i
$995$		0			0
$996$		0			0
$997$		−	12.2474i		−	0.387881i	−0.981013	−	0.193940i	$-0.937873\pi$
$997$		−	12.2474i		−	0.387881i	0.981013	−	0.193940i	$-0.0621268\pi$
$998$		−	24.0416i		−	0.761025i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.c.c.188.1	✓	4
3.2	odd	2	inner	189.2.c.c.188.4	yes	4
4.3	odd	2		3024.2.k.g.1889.2		4
7.6	odd	2	inner	189.2.c.c.188.2	yes	4
9.2	odd	6		567.2.o.e.377.3		8
9.4	even	3		567.2.o.e.188.4		8
9.5	odd	6		567.2.o.e.188.1		8
9.7	even	3		567.2.o.e.377.2		8
12.11	even	2		3024.2.k.g.1889.4		4
21.20	even	2	inner	189.2.c.c.188.3	yes	4
28.27	even	2		3024.2.k.g.1889.3		4
63.13	odd	6		567.2.o.e.188.3		8
63.20	even	6		567.2.o.e.377.4		8
63.34	odd	6		567.2.o.e.377.1		8
63.41	even	6		567.2.o.e.188.2		8
84.83	odd	2		3024.2.k.g.1889.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.c.c.188.1	✓	4	1.1	even	1	trivial
189.2.c.c.188.2	yes	4	7.6	odd	2	inner
189.2.c.c.188.3	yes	4	21.20	even	2	inner
189.2.c.c.188.4	yes	4	3.2	odd	2	inner
567.2.o.e.188.1		8	9.5	odd	6
567.2.o.e.188.2		8	63.41	even	6
567.2.o.e.188.3		8	63.13	odd	6
567.2.o.e.188.4		8	9.4	even	3
567.2.o.e.377.1		8	63.34	odd	6
567.2.o.e.377.2		8	9.7	even	3
567.2.o.e.377.3		8	9.2	odd	6
567.2.o.e.377.4		8	63.20	even	6
3024.2.k.g.1889.1		4	84.83	odd	2
3024.2.k.g.1889.2		4	4.3	odd	2
3024.2.k.g.1889.3		4	28.27	even	2
3024.2.k.g.1889.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -1.73205 q^{5} +(1.00000 - 2.44949i) q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -1.73205 q^{5} +(1.00000 - 2.44949i) q^{7} -2.82843i q^{8} +2.44949i q^{10} -1.41421i q^{11} -2.44949i q^{13} +(-3.46410 - 1.41421i) q^{14} -4.00000 q^{16} +5.19615 q^{17} +7.34847i q^{19} -2.00000 q^{22} +2.82843i q^{23} -2.00000 q^{25} -3.46410 q^{26} +7.07107i q^{29} -2.44949i q^{31} -7.34847i q^{34} +(-1.73205 + 4.24264i) q^{35} +5.00000 q^{37} +10.3923 q^{38} +4.89898i q^{40} +8.66025 q^{41} +5.00000 q^{43} +4.00000 q^{46} -8.66025 q^{47} +(-5.00000 - 4.89898i) q^{49} +2.82843i q^{50} +11.3137i q^{53} +2.44949i q^{55} +(-6.92820 - 2.82843i) q^{56} +10.0000 q^{58} +8.66025 q^{59} +2.44949i q^{61} -3.46410 q^{62} -8.00000 q^{64} +4.24264i q^{65} +2.00000 q^{67} +(6.00000 + 2.44949i) q^{70} -14.1421i q^{71} -7.07107i q^{74} +(-3.46410 - 1.41421i) q^{77} -13.0000 q^{79} +6.92820 q^{80} -12.2474i q^{82} +1.73205 q^{83} -9.00000 q^{85} -7.07107i q^{86} -4.00000 q^{88} -10.3923 q^{89} +(-6.00000 - 2.44949i) q^{91} +12.2474i q^{94} -12.7279i q^{95} +17.1464i q^{97} +(-6.92820 + 7.07107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} - 16 q^{16} - 8 q^{22} - 8 q^{25} + 20 q^{37} + 20 q^{43} + 16 q^{46} - 20 q^{49} + 40 q^{58} - 32 q^{64} + 8 q^{67} + 24 q^{70} - 52 q^{79} - 36 q^{85} - 16 q^{88} - 24 q^{91}+O(q^{100}) \) 4 * q + 4 * q^7 - 16 * q^16 - 8 * q^22 - 8 * q^25 + 20 * q^37 + 20 * q^43 + 16 * q^46 - 20 * q^49 + 40 * q^58 - 32 * q^64 + 8 * q^67 + 24 * q^70 - 52 * q^79 - 36 * q^85 - 16 * q^88 - 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more