LMFDB - Embedded newform 180.3.l.c.73.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [180,3,Mod(37,180)] 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("180.37"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	180.l (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$4.90464475849$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{10})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			73.2
Root			$1.58114 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	180.73
Dual form			180.3.l.c.37.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.74342 - 1.58114i) q^{5} +(-1.00000 + 1.00000i) q^{7} +15.8114 q^{11} +(-6.00000 - 6.00000i) q^{13} +(15.8114 - 15.8114i) q^{17} +14.0000i q^{19} +(15.8114 + 15.8114i) q^{23} +(20.0000 - 15.0000i) q^{25} -15.8114i q^{29} +16.0000 q^{31} +(-3.16228 + 6.32456i) q^{35} +(-30.0000 + 30.0000i) q^{37} -31.6228 q^{41} +(-54.0000 - 54.0000i) q^{43} +(-47.4342 + 47.4342i) q^{47} +47.0000i q^{49} +(-63.2456 - 63.2456i) q^{53} +(75.0000 - 25.0000i) q^{55} +79.0569i q^{59} +54.0000 q^{61} +(-37.9473 - 18.9737i) q^{65} +(-34.0000 + 34.0000i) q^{67} -63.2456 q^{71} +(-65.0000 - 65.0000i) q^{73} +(-15.8114 + 15.8114i) q^{77} +108.000i q^{79} +(47.4342 + 47.4342i) q^{83} +(50.0000 - 100.000i) q^{85} -126.491i q^{89} +12.0000 q^{91} +(22.1359 + 66.4078i) q^{95} +(-69.0000 + 69.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} - 24 q^{13} + 80 q^{25} + 64 q^{31} - 120 q^{37} - 216 q^{43} + 300 q^{55} + 216 q^{61} - 136 q^{67} - 260 q^{73} + 200 q^{85} + 48 q^{91} - 276 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 24 * q^13 + 80 * q^25 + 64 * q^31 - 120 * q^37 - 216 * q^43 + 300 * q^55 + 216 * q^61 - 136 * q^67 - 260 * q^73 + 200 * q^85 + 48 * q^91 - 276 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$.

$n$	$37$	$91$	$101$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	4.74342	−	1.58114i	0.948683	−	0.316228i
$5$	4.74342	−	1.58114i	0.948683	−	0.316228i
$6$		0			0
$7$	−1.00000	+	1.00000i	−0.142857	+	0.142857i	−0.774918	−	0.632061i	$-0.782209\pi$
$7$	−1.00000	+	1.00000i	−0.142857	+	0.142857i	0.632061	+	0.774918i	$0.282209\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	15.8114			1.43740			0.718699	−	0.695321i	$-0.244738\pi$
$11$	15.8114			1.43740			0.718699	+	0.695321i	$0.244738\pi$
$12$		0			0
$13$	−6.00000	−	6.00000i	−0.461538	−	0.461538i	0.437621	−	0.899160i	$-0.355821\pi$
$13$	−6.00000	−	6.00000i	−0.461538	−	0.461538i	−0.899160	+	0.437621i	$0.855821\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	15.8114	−	15.8114i	0.930082	−	0.930082i	−0.0676289	−	0.997711i	$-0.521543\pi$
$17$	15.8114	−	15.8114i	0.930082	−	0.930082i	0.997711	+	0.0676289i	$0.0215434\pi$
$18$		0			0
$19$			14.0000i			0.736842i	0.929659	+	0.368421i	$0.120102\pi$
$19$			14.0000i			0.736842i	−0.929659	+	0.368421i	$0.879898\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	15.8114	+	15.8114i	0.687452	+	0.687452i	0.961668	−	0.274216i	$-0.0884184\pi$
$23$	15.8114	+	15.8114i	0.687452	+	0.687452i	−0.274216	+	0.961668i	$0.588418\pi$
$24$		0			0
$25$	20.0000	−	15.0000i	0.800000	−	0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	15.8114i		−	0.545220i	−0.962125	−	0.272610i	$-0.912113\pi$
$29$		−	15.8114i		−	0.545220i	0.962125	−	0.272610i	$-0.0878869\pi$
$30$		0			0
$31$	16.0000			0.516129			0.258065	−	0.966128i	$-0.416915\pi$
$31$	16.0000			0.516129			0.258065	+	0.966128i	$0.416915\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.16228	+	6.32456i	−0.0903508	+	0.180702i
$36$		0			0
$37$	−30.0000	+	30.0000i	−0.810811	+	0.810811i	−0.984755	−	0.173945i	$-0.944349\pi$
$37$	−30.0000	+	30.0000i	−0.810811	+	0.810811i	0.173945	+	0.984755i	$0.444349\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−31.6228			−0.771287			−0.385644	−	0.922648i	$-0.626021\pi$
$41$	−31.6228			−0.771287			−0.385644	+	0.922648i	$0.626021\pi$
$42$		0			0
$43$	−54.0000	−	54.0000i	−1.25581	−	1.25581i	−0.953073	−	0.302741i	$-0.902098\pi$
$43$	−54.0000	−	54.0000i	−1.25581	−	1.25581i	−0.302741	−	0.953073i	$-0.597902\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−47.4342	+	47.4342i	−1.00924	+	1.00924i	−0.00928062	+	0.999957i	$0.502954\pi$
$47$	−47.4342	+	47.4342i	−1.00924	+	1.00924i	−0.999957	+	0.00928062i	$0.997046\pi$
$48$		0			0
$49$			47.0000i			0.959184i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−63.2456	−	63.2456i	−1.19331	−	1.19331i	−0.976131	−	0.217181i	$-0.930314\pi$
$53$	−63.2456	−	63.2456i	−1.19331	−	1.19331i	−0.217181	−	0.976131i	$-0.569686\pi$
$54$		0			0
$55$	75.0000	−	25.0000i	1.36364	−	0.454545i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			79.0569i			1.33995i	0.742384	+	0.669974i	$0.233695\pi$
$59$			79.0569i			1.33995i	−0.742384	+	0.669974i	$0.766305\pi$
$60$		0			0
$61$	54.0000			0.885246			0.442623	−	0.896708i	$-0.354048\pi$
$61$	54.0000			0.885246			0.442623	+	0.896708i	$0.354048\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−37.9473	−	18.9737i	−0.583805	−	0.291903i
$66$		0			0
$67$	−34.0000	+	34.0000i	−0.507463	+	0.507463i	−0.913747	−	0.406284i	$-0.866824\pi$
$67$	−34.0000	+	34.0000i	−0.507463	+	0.507463i	0.406284	+	0.913747i	$0.366824\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−63.2456			−0.890782			−0.445391	−	0.895336i	$-0.646935\pi$
$71$	−63.2456			−0.890782			−0.445391	+	0.895336i	$0.646935\pi$
$72$		0			0
$73$	−65.0000	−	65.0000i	−0.890411	−	0.890411i	0.104151	−	0.994562i	$-0.466788\pi$
$73$	−65.0000	−	65.0000i	−0.890411	−	0.890411i	−0.994562	+	0.104151i	$0.966788\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−15.8114	+	15.8114i	−0.205343	+	0.205343i
$78$		0			0
$79$			108.000i			1.36709i	0.729909	+	0.683544i	$0.239562\pi$
$79$			108.000i			1.36709i	−0.729909	+	0.683544i	$0.760438\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	47.4342	+	47.4342i	0.571496	+	0.571496i	0.932546	−	0.361050i	$-0.117582\pi$
$83$	47.4342	+	47.4342i	0.571496	+	0.571496i	−0.361050	+	0.932546i	$0.617582\pi$
$84$		0			0
$85$	50.0000	−	100.000i	0.588235	−	1.17647i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	126.491i		−	1.42125i	−0.703572	−	0.710624i	$-0.748413\pi$
$89$		−	126.491i		−	1.42125i	0.703572	−	0.710624i	$-0.251587\pi$
$90$		0			0
$91$	12.0000			0.131868
$92$		0			0
$93$		0			0
$94$		0			0
$95$	22.1359	+	66.4078i	0.233010	+	0.699030i
$96$		0			0
$97$	−69.0000	+	69.0000i	−0.711340	+	0.711340i	−0.966816	−	0.255475i	$-0.917768\pi$
$97$	−69.0000	+	69.0000i	−0.711340	+	0.711340i	0.255475	+	0.966816i	$0.417768\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	79.0569			0.782742			0.391371	−	0.920233i	$-0.372001\pi$
$101$	79.0569			0.782742			0.391371	+	0.920233i	$0.372001\pi$
$102$		0			0
$103$	−95.0000	−	95.0000i	−0.922330	−	0.922330i	0.0748637	−	0.997194i	$-0.476148\pi$
$103$	−95.0000	−	95.0000i	−0.922330	−	0.922330i	−0.997194	+	0.0748637i	$0.976148\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	110.680	−	110.680i	1.03439	−	1.03439i	0.0350027	−	0.999387i	$-0.488856\pi$
$107$	110.680	−	110.680i	1.03439	−	1.03439i	0.999387	−	0.0350027i	$-0.0111440\pi$
$108$		0			0
$109$			46.0000i			0.422018i	0.977484	+	0.211009i	$0.0676750\pi$
$109$			46.0000i			0.422018i	−0.977484	+	0.211009i	$0.932325\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	79.0569	+	79.0569i	0.699619	+	0.699619i	0.964328	−	0.264709i	$-0.0852760\pi$
$113$	79.0569	+	79.0569i	0.699619	+	0.699619i	−0.264709	+	0.964328i	$0.585276\pi$
$114$		0			0
$115$	100.000	+	50.0000i	0.869565	+	0.434783i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			31.6228i			0.265738i
$120$		0			0
$121$	129.000			1.06612
$122$		0			0
$123$		0			0
$124$		0			0
$125$	71.1512	−	102.774i	0.569210	−	0.822192i
$126$		0			0
$127$	35.0000	−	35.0000i	0.275591	−	0.275591i	−0.555755	−	0.831346i	$-0.687571\pi$
$127$	35.0000	−	35.0000i	0.275591	−	0.275591i	0.831346	+	0.555755i	$0.187571\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	205.548			1.56907			0.784535	−	0.620085i	$-0.212902\pi$
$131$	205.548			1.56907			0.784535	+	0.620085i	$0.212902\pi$
$132$		0			0
$133$	−14.0000	−	14.0000i	−0.105263	−	0.105263i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	47.4342	−	47.4342i	0.346235	−	0.346235i	−0.512470	−	0.858705i	$-0.671270\pi$
$137$	47.4342	−	47.4342i	0.346235	−	0.346235i	0.858705	+	0.512470i	$0.171270\pi$
$138$		0			0
$139$			22.0000i			0.158273i	0.996864	+	0.0791367i	$0.0252164\pi$
$139$			22.0000i			0.158273i	−0.996864	+	0.0791367i	$0.974784\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−94.8683	−	94.8683i	−0.663415	−	0.663415i
$144$		0			0
$145$	−25.0000	−	75.0000i	−0.172414	−	0.517241i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			142.302i			0.955050i	0.878618	+	0.477525i	$0.158466\pi$
$149$			142.302i			0.955050i	−0.878618	+	0.477525i	$0.841534\pi$
$150$		0			0
$151$	−134.000			−0.887417			−0.443709	−	0.896171i	$-0.646337\pi$
$151$	−134.000			−0.887417			−0.443709	+	0.896171i	$0.646337\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	75.8947	−	25.2982i	0.489643	−	0.163214i
$156$		0			0
$157$	−144.000	+	144.000i	−0.917197	+	0.917197i	−0.996825	−	0.0796273i	$-0.974627\pi$
$157$	−144.000	+	144.000i	−0.917197	+	0.917197i	0.0796273	+	0.996825i	$0.474627\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−31.6228			−0.196415
$162$		0			0
$163$	−4.00000	−	4.00000i	−0.0245399	−	0.0245399i	0.694730	−	0.719270i	$-0.255524\pi$
$163$	−4.00000	−	4.00000i	−0.0245399	−	0.0245399i	−0.719270	+	0.694730i	$0.755524\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−110.680	+	110.680i	−0.662753	+	0.662753i	−0.956028	−	0.293275i	$-0.905255\pi$
$167$	−110.680	+	110.680i	−0.662753	+	0.662753i	0.293275	+	0.956028i	$0.405255\pi$
$168$		0			0
$169$		−	97.0000i		−	0.573964i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	15.8114	+	15.8114i	0.0913953	+	0.0913953i	0.751326	−	0.659931i	$-0.229414\pi$
$173$	15.8114	+	15.8114i	0.0913953	+	0.0913953i	−0.659931	+	0.751326i	$0.729414\pi$
$174$		0			0
$175$	−5.00000	+	35.0000i	−0.0285714	+	0.200000i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	268.794i		−	1.50164i	−0.660507	−	0.750820i	$-0.729659\pi$
$179$		−	268.794i		−	1.50164i	0.660507	−	0.750820i	$-0.270341\pi$
$180$		0			0
$181$	26.0000			0.143646			0.0718232	−	0.997417i	$-0.477118\pi$
$181$	26.0000			0.143646			0.0718232	+	0.997417i	$0.477118\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−94.8683	+	189.737i	−0.512802	+	1.02560i
$186$		0			0
$187$	250.000	−	250.000i	1.33690	−	1.33690i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−347.851			−1.82121			−0.910604	−	0.413281i	$-0.864383\pi$
$191$	−347.851			−1.82121			−0.910604	+	0.413281i	$0.864383\pi$
$192$		0			0
$193$	219.000	+	219.000i	1.13472	+	1.13472i	0.989383	+	0.145332i	$0.0464251\pi$
$193$	219.000	+	219.000i	1.13472	+	1.13472i	0.145332	+	0.989383i	$0.453575\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−94.8683	+	94.8683i	−0.481565	+	0.481565i	−0.905631	−	0.424066i	$-0.860602\pi$
$197$	−94.8683	+	94.8683i	−0.481565	+	0.481565i	0.424066	+	0.905631i	$0.360602\pi$
$198$		0			0
$199$			66.0000i			0.331658i	0.986154	+	0.165829i	$0.0530300\pi$
$199$			66.0000i			0.331658i	−0.986154	+	0.165829i	$0.946970\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.8114	+	15.8114i	0.0778886	+	0.0778886i
$204$		0			0
$205$	−150.000	+	50.0000i	−0.731707	+	0.243902i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			221.359i			1.05914i
$210$		0			0
$211$	−106.000			−0.502370			−0.251185	−	0.967939i	$-0.580820\pi$
$211$	−106.000			−0.502370			−0.251185	+	0.967939i	$0.580820\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−341.526	−	170.763i	−1.58849	−	0.794246i
$216$		0			0
$217$	−16.0000	+	16.0000i	−0.0737327	+	0.0737327i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−189.737			−0.858537
$222$		0			0
$223$	−51.0000	−	51.0000i	−0.228700	−	0.228700i	0.583450	−	0.812149i	$-0.301703\pi$
$223$	−51.0000	−	51.0000i	−0.228700	−	0.228700i	−0.812149	+	0.583450i	$0.801703\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	31.6228	−	31.6228i	0.139307	−	0.139307i	−0.634014	−	0.773321i	$-0.718594\pi$
$227$	31.6228	−	31.6228i	0.139307	−	0.139307i	0.773321	+	0.634014i	$0.218594\pi$
$228$		0			0
$229$		−	366.000i		−	1.59825i	−0.601163	−	0.799127i	$-0.705296\pi$
$229$		−	366.000i		−	1.59825i	0.601163	−	0.799127i	$-0.294704\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−47.4342	−	47.4342i	−0.203580	−	0.203580i	0.597952	−	0.801532i	$-0.295981\pi$
$233$	−47.4342	−	47.4342i	−0.203580	−	0.203580i	−0.801532	+	0.597952i	$0.795981\pi$
$234$		0			0
$235$	−150.000	+	300.000i	−0.638298	+	1.27660i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	4.00000			0.0165975			0.00829876	−	0.999966i	$-0.497358\pi$
$241$	4.00000			0.0165975			0.00829876	+	0.999966i	$0.497358\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	74.3135	+	222.941i	0.303321	+	0.909962i
$246$		0			0
$247$	84.0000	−	84.0000i	0.340081	−	0.340081i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	142.302			0.566942			0.283471	−	0.958981i	$-0.408514\pi$
$251$	142.302			0.566942			0.283471	+	0.958981i	$0.408514\pi$
$252$		0			0
$253$	250.000	+	250.000i	0.988142	+	0.988142i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−110.680	+	110.680i	−0.430660	+	0.430660i	−0.888853	−	0.458193i	$-0.848497\pi$
$257$	−110.680	+	110.680i	−0.430660	+	0.430660i	0.458193	+	0.888853i	$0.348497\pi$
$258$		0			0
$259$		−	60.0000i		−	0.231660i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−205.548	−	205.548i	−0.781552	−	0.781552i	0.198541	−	0.980093i	$-0.436380\pi$
$263$	−205.548	−	205.548i	−0.781552	−	0.781552i	−0.980093	+	0.198541i	$0.936380\pi$
$264$		0			0
$265$	−400.000	−	200.000i	−1.50943	−	0.754717i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			79.0569i			0.293892i	0.989145	+	0.146946i	$0.0469444\pi$
$269$			79.0569i			0.293892i	−0.989145	+	0.146946i	$0.953056\pi$
$270$		0			0
$271$	−258.000			−0.952030			−0.476015	−	0.879437i	$-0.657919\pi$
$271$	−258.000			−0.952030			−0.476015	+	0.879437i	$0.657919\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	316.228	−	237.171i	1.14992	−	0.862439i
$276$		0			0
$277$	114.000	−	114.000i	0.411552	−	0.411552i	−0.470727	−	0.882279i	$-0.656008\pi$
$277$	114.000	−	114.000i	0.411552	−	0.411552i	0.882279	+	0.470727i	$0.156008\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	284.605			1.01283			0.506415	−	0.862290i	$-0.330971\pi$
$281$	284.605			1.01283			0.506415	+	0.862290i	$0.330971\pi$
$282$		0			0
$283$	314.000	+	314.000i	1.10954	+	1.10954i	0.993211	+	0.116330i	$0.0371130\pi$
$283$	314.000	+	314.000i	1.10954	+	1.10954i	0.116330	+	0.993211i	$0.462887\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	31.6228	−	31.6228i	0.110184	−	0.110184i
$288$		0			0
$289$		−	211.000i		−	0.730104i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	189.737	+	189.737i	0.647565	+	0.647565i	0.952404	−	0.304839i	$-0.0986025\pi$
$293$	189.737	+	189.737i	0.647565	+	0.647565i	−0.304839	+	0.952404i	$0.598603\pi$
$294$		0			0
$295$	125.000	+	375.000i	0.423729	+	1.27119i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	189.737i		−	0.634571i
$300$		0			0
$301$	108.000			0.358804
$302$		0			0
$303$		0			0
$304$		0			0
$305$	256.144	−	85.3815i	0.839818	−	0.279939i
$306$		0			0
$307$	366.000	−	366.000i	1.19218	−	1.19218i	0.215729	−	0.976453i	$-0.430787\pi$
$307$	366.000	−	366.000i	1.19218	−	1.19218i	0.976453	−	0.215729i	$-0.0692128\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	126.491			0.406724			0.203362	−	0.979104i	$-0.434813\pi$
$311$	126.491			0.406724			0.203362	+	0.979104i	$0.434813\pi$
$312$		0			0
$313$	201.000	+	201.000i	0.642173	+	0.642173i	0.951089	−	0.308917i	$-0.0999664\pi$
$313$	201.000	+	201.000i	0.642173	+	0.642173i	−0.308917	+	0.951089i	$0.599966\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	237.171	−	237.171i	0.748173	−	0.748173i	−0.225963	−	0.974136i	$-0.572553\pi$
$317$	237.171	−	237.171i	0.748173	−	0.748173i	0.974136	+	0.225963i	$0.0725528\pi$
$318$		0			0
$319$		−	250.000i		−	0.783699i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	221.359	+	221.359i	0.685323	+	0.685323i
$324$		0			0
$325$	−210.000	−	30.0000i	−0.646154	−	0.0923077i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	94.8683i		−	0.288354i
$330$		0			0
$331$	−638.000			−1.92749			−0.963746	−	0.266821i	$-0.914027\pi$
$331$	−638.000			−1.92749			−0.963746	+	0.266821i	$0.914027\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−107.517	+	215.035i	−0.320948	+	0.641895i
$336$		0			0
$337$	−291.000	+	291.000i	−0.863501	+	0.863501i	−0.991743	−	0.128241i	$-0.959067\pi$
$337$	−291.000	+	291.000i	−0.863501	+	0.863501i	0.128241	+	0.991743i	$0.459067\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	252.982			0.741883
$342$		0			0
$343$	−96.0000	−	96.0000i	−0.279883	−	0.279883i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	442.719	−	442.719i	1.27585	−	1.27585i	0.332876	−	0.942970i	$-0.391981\pi$
$347$	442.719	−	442.719i	1.27585	−	1.27585i	0.942970	−	0.332876i	$-0.108019\pi$
$348$		0			0
$349$			2.00000i			0.00573066i	0.999996	+	0.00286533i	$0.000912064\pi$
$349$			2.00000i			0.00573066i	−0.999996	+	0.00286533i	$0.999088\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	142.302	+	142.302i	0.403123	+	0.403123i	0.879332	−	0.476209i	$-0.157989\pi$
$353$	142.302	+	142.302i	0.403123	+	0.403123i	−0.476209	+	0.879332i	$0.657989\pi$
$354$		0			0
$355$	−300.000	+	100.000i	−0.845070	+	0.281690i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			158.114i			0.440429i	0.975452	+	0.220214i	$0.0706757\pi$
$359$			158.114i			0.440429i	−0.975452	+	0.220214i	$0.929324\pi$
$360$		0			0
$361$	165.000			0.457064
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−411.096	−	205.548i	−1.12629	−	0.563145i
$366$		0			0
$367$	299.000	−	299.000i	0.814714	−	0.814714i	−0.170623	−	0.985336i	$-0.554578\pi$
$367$	299.000	−	299.000i	0.814714	−	0.814714i	0.985336	+	0.170623i	$0.0545779\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	126.491			0.340946
$372$		0			0
$373$	316.000	+	316.000i	0.847185	+	0.847185i	0.989781	−	0.142596i	$-0.0455450\pi$
$373$	316.000	+	316.000i	0.847185	+	0.847185i	−0.142596	+	0.989781i	$0.545545\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−94.8683	+	94.8683i	−0.251640	+	0.251640i
$378$		0			0
$379$		−	286.000i		−	0.754617i	−0.926088	−	0.377309i	$-0.876850\pi$
$379$		−	286.000i		−	0.754617i	0.926088	−	0.377309i	$-0.123150\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−142.302	−	142.302i	−0.371547	−	0.371547i	0.496493	−	0.868040i	$-0.334621\pi$
$383$	−142.302	−	142.302i	−0.371547	−	0.371547i	−0.868040	+	0.496493i	$0.834621\pi$
$384$		0			0
$385$	−50.0000	+	100.000i	−0.129870	+	0.259740i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	332.039i		−	0.853571i	−0.904353	−	0.426786i	$-0.859646\pi$
$389$		−	332.039i		−	0.853571i	0.904353	−	0.426786i	$-0.140354\pi$
$390$		0			0
$391$	500.000			1.27877
$392$		0			0
$393$		0			0
$394$		0			0
$395$	170.763	+	512.289i	0.432311	+	1.29693i
$396$		0			0
$397$	−116.000	+	116.000i	−0.292191	+	0.292191i	−0.837945	−	0.545754i	$-0.816243\pi$
$397$	−116.000	+	116.000i	−0.292191	+	0.292191i	0.545754	+	0.837945i	$0.316243\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−379.473			−0.946318			−0.473159	−	0.880977i	$-0.656886\pi$
$401$	−379.473			−0.946318			−0.473159	+	0.880977i	$0.656886\pi$
$402$		0			0
$403$	−96.0000	−	96.0000i	−0.238213	−	0.238213i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−474.342	+	474.342i	−1.16546	+	1.16546i
$408$		0			0
$409$		−	156.000i		−	0.381418i	−0.981647	−	0.190709i	$-0.938921\pi$
$409$		−	156.000i		−	0.381418i	0.981647	−	0.190709i	$-0.0610787\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−79.0569	−	79.0569i	−0.191421	−	0.191421i
$414$		0			0
$415$	300.000	+	150.000i	0.722892	+	0.361446i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			142.302i			0.339624i	0.985476	+	0.169812i	$0.0543161\pi$
$419$			142.302i			0.339624i	−0.985476	+	0.169812i	$0.945684\pi$
$420$		0			0
$421$	−142.000			−0.337292			−0.168646	−	0.985677i	$-0.553939\pi$
$421$	−142.000			−0.337292			−0.168646	+	0.985677i	$0.553939\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	79.0569	−	553.399i	0.186016	−	1.30211i
$426$		0			0
$427$	−54.0000	+	54.0000i	−0.126464	+	0.126464i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−600.833			−1.39404			−0.697022	−	0.717050i	$-0.745492\pi$
$431$	−600.833			−1.39404			−0.697022	+	0.717050i	$0.745492\pi$
$432$		0			0
$433$	−31.0000	−	31.0000i	−0.0715935	−	0.0715935i	0.670403	−	0.741997i	$-0.266121\pi$
$433$	−31.0000	−	31.0000i	−0.0715935	−	0.0715935i	−0.741997	+	0.670403i	$0.766121\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−221.359	+	221.359i	−0.506543	+	0.506543i
$438$		0			0
$439$			766.000i			1.74487i	0.488726	+	0.872437i	$0.337462\pi$
$439$			766.000i			1.74487i	−0.488726	+	0.872437i	$0.662538\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−94.8683	−	94.8683i	−0.214150	−	0.214150i	0.591878	−	0.806028i	$-0.298387\pi$
$443$	−94.8683	−	94.8683i	−0.214150	−	0.214150i	−0.806028	+	0.591878i	$0.798387\pi$
$444$		0			0
$445$	−200.000	−	600.000i	−0.449438	−	1.34831i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			411.096i			0.915582i	0.889060	+	0.457791i	$0.151359\pi$
$449$			411.096i			0.915582i	−0.889060	+	0.457791i	$0.848641\pi$
$450$		0			0
$451$	−500.000			−1.10865
$452$		0			0
$453$		0			0
$454$		0			0
$455$	56.9210	−	18.9737i	0.125101	−	0.0417004i
$456$		0			0
$457$	195.000	−	195.000i	0.426696	−	0.426696i	−0.460805	−	0.887501i	$-0.652439\pi$
$457$	195.000	−	195.000i	0.426696	−	0.426696i	0.887501	+	0.460805i	$0.152439\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−268.794			−0.583066			−0.291533	−	0.956561i	$-0.594165\pi$
$461$	−268.794			−0.583066			−0.291533	+	0.956561i	$0.594165\pi$
$462$		0			0
$463$	−295.000	−	295.000i	−0.637149	−	0.637149i	0.312702	−	0.949851i	$-0.398766\pi$
$463$	−295.000	−	295.000i	−0.637149	−	0.637149i	−0.949851	+	0.312702i	$0.898766\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−363.662	+	363.662i	−0.778719	+	0.778719i	−0.979613	−	0.200894i	$-0.935615\pi$
$467$	−363.662	+	363.662i	−0.778719	+	0.778719i	0.200894	+	0.979613i	$0.435615\pi$
$468$		0			0
$469$		−	68.0000i		−	0.144989i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−853.815	−	853.815i	−1.80511	−	1.80511i
$474$		0			0
$475$	210.000	+	280.000i	0.442105	+	0.589474i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			505.964i			1.05629i	0.849153	+	0.528147i	$0.177113\pi$
$479$			505.964i			1.05629i	−0.849153	+	0.528147i	$0.822887\pi$
$480$		0			0
$481$	360.000			0.748441
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−218.197	+	436.394i	−0.449891	+	0.899782i
$486$		0			0
$487$	−129.000	+	129.000i	−0.264887	+	0.264887i	−0.827036	−	0.562149i	$-0.809975\pi$
$487$	−129.000	+	129.000i	−0.264887	+	0.264887i	0.562149	+	0.827036i	$0.309975\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	173.925			0.354227			0.177113	−	0.984190i	$-0.443324\pi$
$491$	173.925			0.354227			0.177113	+	0.984190i	$0.443324\pi$
$492$		0			0
$493$	−250.000	−	250.000i	−0.507099	−	0.507099i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	63.2456	−	63.2456i	0.127255	−	0.127255i
$498$		0			0
$499$			226.000i			0.452906i	0.974022	+	0.226453i	$0.0727129\pi$
$499$			226.000i			0.452906i	−0.974022	+	0.226453i	$0.927287\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	521.776	+	521.776i	1.03733	+	1.03733i	0.999276	+	0.0380519i	$0.0121152\pi$
$503$	521.776	+	521.776i	1.03733	+	1.03733i	0.0380519	+	0.999276i	$0.487885\pi$
$504$		0			0
$505$	375.000	−	125.000i	0.742574	−	0.247525i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	964.495i		−	1.89488i	−0.319931	−	0.947441i	$-0.603660\pi$
$509$		−	964.495i		−	1.89488i	0.319931	−	0.947441i	$-0.396340\pi$
$510$		0			0
$511$	130.000			0.254403
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−600.833	−	300.416i	−1.16667	−	0.583333i
$516$		0			0
$517$	−750.000	+	750.000i	−1.45068	+	1.45068i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	94.8683			0.182089			0.0910445	−	0.995847i	$-0.470979\pi$
$521$	94.8683			0.182089			0.0910445	+	0.995847i	$0.470979\pi$
$522$		0			0
$523$	−54.0000	−	54.0000i	−0.103250	−	0.103250i	0.653594	−	0.756845i	$-0.273260\pi$
$523$	−54.0000	−	54.0000i	−0.103250	−	0.103250i	−0.756845	+	0.653594i	$0.773260\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	252.982	−	252.982i	0.480042	−	0.480042i
$528$		0			0
$529$		−	29.0000i		−	0.0548204i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	189.737	+	189.737i	0.355979	+	0.355979i
$534$		0			0
$535$	350.000	−	700.000i	0.654206	−	1.30841i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			743.135i			1.37873i
$540$		0			0
$541$	634.000			1.17190			0.585952	−	0.810346i	$-0.300721\pi$
$541$	634.000			1.17190			0.585952	+	0.810346i	$0.300721\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	72.7324	+	218.197i	0.133454	+	0.400362i
$546$		0			0
$547$	330.000	−	330.000i	0.603291	−	0.603291i	−0.337894	−	0.941184i	$-0.609714\pi$
$547$	330.000	−	330.000i	0.603291	−	0.603291i	0.941184	+	0.337894i	$0.109714\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	221.359			0.401741
$552$		0			0
$553$	−108.000	−	108.000i	−0.195298	−	0.195298i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	569.210	−	569.210i	1.02192	−	1.02192i	0.0221667	−	0.999754i	$-0.492944\pi$
$557$	569.210	−	569.210i	1.02192	−	1.02192i	0.999754	−	0.0221667i	$-0.00705645\pi$
$558$		0			0
$559$			648.000i			1.15921i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	379.473	+	379.473i	0.674020	+	0.674020i	0.958640	−	0.284620i	$-0.0918675\pi$
$563$	379.473	+	379.473i	0.674020	+	0.674020i	−0.284620	+	0.958640i	$0.591868\pi$
$564$		0			0
$565$	500.000	+	250.000i	0.884956	+	0.442478i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	695.701i		−	1.22267i	−0.791371	−	0.611337i	$-0.790632\pi$
$569$		−	695.701i		−	1.22267i	0.791371	−	0.611337i	$-0.209368\pi$
$570$		0			0
$571$	494.000			0.865149			0.432574	−	0.901598i	$-0.357605\pi$
$571$	494.000			0.865149			0.432574	+	0.901598i	$0.357605\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	553.399	+	79.0569i	0.962432	+	0.137490i
$576$		0			0
$577$	−299.000	+	299.000i	−0.518198	+	0.518198i	−0.917026	−	0.398828i	$-0.869417\pi$
$577$	−299.000	+	299.000i	−0.518198	+	0.518198i	0.398828	+	0.917026i	$0.369417\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−94.8683			−0.163285
$582$		0			0
$583$	−1000.00	−	1000.00i	−1.71527	−	1.71527i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−47.4342	+	47.4342i	−0.0808078	+	0.0808078i	−0.746355	−	0.665548i	$-0.768198\pi$
$587$	−47.4342	+	47.4342i	−0.0808078	+	0.0808078i	0.665548	+	0.746355i	$0.268198\pi$
$588$		0			0
$589$			224.000i			0.380306i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	300.416	+	300.416i	0.506604	+	0.506604i	0.913482	−	0.406878i	$-0.133383\pi$
$593$	300.416	+	300.416i	0.506604	+	0.506604i	−0.406878	+	0.913482i	$0.633383\pi$
$594$		0			0
$595$	50.0000	+	150.000i	0.0840336	+	0.252101i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	917.061i		−	1.53099i	−0.643444	−	0.765493i	$-0.722495\pi$
$599$		−	917.061i		−	1.53099i	0.643444	−	0.765493i	$-0.277505\pi$
$600$		0			0
$601$	−574.000			−0.955075			−0.477537	−	0.878611i	$-0.658471\pi$
$601$	−574.000			−0.955075			−0.477537	+	0.878611i	$0.658471\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	611.901	−	203.967i	1.01141	−	0.337135i
$606$		0			0
$607$	−765.000	+	765.000i	−1.26030	+	1.26030i	−0.309347	+	0.950949i	$0.600111\pi$
$607$	−765.000	+	765.000i	−1.26030	+	1.26030i	−0.950949	+	0.309347i	$0.899889\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	569.210			0.931604
$612$		0			0
$613$	−540.000	−	540.000i	−0.880914	−	0.880914i	0.112714	−	0.993627i	$-0.464046\pi$
$613$	−540.000	−	540.000i	−0.880914	−	0.880914i	−0.993627	+	0.112714i	$0.964046\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	332.039	−	332.039i	0.538151	−	0.538151i	−0.384835	−	0.922986i	$-0.625742\pi$
$617$	332.039	−	332.039i	0.538151	−	0.538151i	0.922986	+	0.384835i	$0.125742\pi$
$618$		0			0
$619$		−	174.000i		−	0.281099i	−0.990074	−	0.140549i	$-0.955113\pi$
$619$		−	174.000i		−	0.281099i	0.990074	−	0.140549i	$-0.0448868\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	126.491	+	126.491i	0.203035	+	0.203035i
$624$		0			0
$625$	175.000	−	600.000i	0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			948.683i			1.50824i
$630$		0			0
$631$	−236.000			−0.374010			−0.187005	−	0.982359i	$-0.559878\pi$
$631$	−236.000			−0.374010			−0.187005	+	0.982359i	$0.559878\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	110.680	−	221.359i	0.174299	−	0.348598i
$636$		0			0
$637$	282.000	−	282.000i	0.442700	−	0.442700i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	695.701			1.08534			0.542669	−	0.839947i	$-0.317414\pi$
$641$	695.701			1.08534			0.542669	+	0.839947i	$0.317414\pi$
$642$		0			0
$643$	540.000	+	540.000i	0.839813	+	0.839813i	0.988834	−	0.149021i	$-0.0476121\pi$
$643$	540.000	+	540.000i	0.839813	+	0.839813i	−0.149021	+	0.988834i	$0.547612\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−110.680	+	110.680i	−0.171066	+	0.171066i	−0.787448	−	0.616382i	$-0.788598\pi$
$647$	−110.680	+	110.680i	−0.171066	+	0.171066i	0.616382	+	0.787448i	$0.288598\pi$
$648$		0			0
$649$			1250.00i			1.92604i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−521.776	−	521.776i	−0.799044	−	0.799044i	0.183901	−	0.982945i	$-0.441128\pi$
$653$	−521.776	−	521.776i	−0.799044	−	0.799044i	−0.982945	+	0.183901i	$0.941128\pi$
$654$		0			0
$655$	975.000	−	325.000i	1.48855	−	0.496183i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			15.8114i			0.0239930i	0.999928	+	0.0119965i	$0.00381870\pi$
$659$			15.8114i			0.0239930i	−0.999928	+	0.0119965i	$0.996181\pi$
$660$		0			0
$661$	878.000			1.32829			0.664145	−	0.747604i	$-0.268796\pi$
$661$	878.000			1.32829			0.664145	+	0.747604i	$0.268796\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−88.5438	−	44.2719i	−0.133149	−	0.0665743i
$666$		0			0
$667$	250.000	−	250.000i	0.374813	−	0.374813i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	853.815			1.27245
$672$		0			0
$673$	−425.000	−	425.000i	−0.631501	−	0.631501i	0.316944	−	0.948444i	$-0.397343\pi$
$673$	−425.000	−	425.000i	−0.631501	−	0.631501i	−0.948444	+	0.316944i	$0.897343\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.8114	−	15.8114i	0.0233551	−	0.0233551i	−0.695333	−	0.718688i	$-0.744743\pi$
$677$	15.8114	−	15.8114i	0.0233551	−	0.0233551i	0.718688	+	0.695333i	$0.244743\pi$
$678$		0			0
$679$		−	138.000i		−	0.203240i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−521.776	−	521.776i	−0.763947	−	0.763947i	0.213086	−	0.977033i	$-0.431648\pi$
$683$	−521.776	−	521.776i	−0.763947	−	0.763947i	−0.977033	+	0.213086i	$0.931648\pi$
$684$		0			0
$685$	150.000	−	300.000i	0.218978	−	0.437956i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			758.947i			1.10152i
$690$		0			0
$691$	318.000			0.460203			0.230101	−	0.973167i	$-0.426094\pi$
$691$	318.000			0.460203			0.230101	+	0.973167i	$0.426094\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	34.7851	+	104.355i	0.0500504	+	0.150151i
$696$		0			0
$697$	−500.000	+	500.000i	−0.717360	+	0.717360i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−173.925			−0.248110			−0.124055	−	0.992275i	$-0.539590\pi$
$701$	−173.925			−0.248110			−0.124055	+	0.992275i	$0.539590\pi$
$702$		0			0
$703$	−420.000	−	420.000i	−0.597440	−	0.597440i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−79.0569	+	79.0569i	−0.111820	+	0.111820i
$708$		0			0
$709$			734.000i			1.03526i	0.855604	+	0.517630i	$0.173186\pi$
$709$			734.000i			1.03526i	−0.855604	+	0.517630i	$0.826814\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	252.982	+	252.982i	0.354814	+	0.354814i
$714$		0			0
$715$	−600.000	−	300.000i	−0.839161	−	0.419580i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	442.719i		−	0.615743i	−0.951428	−	0.307871i	$-0.900383\pi$
$719$		−	442.719i		−	0.615743i	0.951428	−	0.307871i	$-0.0996166\pi$
$720$		0			0
$721$	190.000			0.263523
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−237.171	−	316.228i	−0.327132	−	0.436176i
$726$		0			0
$727$	−429.000	+	429.000i	−0.590096	+	0.590096i	−0.937657	−	0.347561i	$-0.887010\pi$
$727$	−429.000	+	429.000i	−0.590096	+	0.590096i	0.347561	+	0.937657i	$0.387010\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1707.63			−2.33602
$732$		0			0
$733$	−50.0000	−	50.0000i	−0.0682128	−	0.0682128i	0.672177	−	0.740390i	$-0.265359\pi$
$733$	−50.0000	−	50.0000i	−0.0682128	−	0.0682128i	−0.740390	+	0.672177i	$0.765359\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−537.587	+	537.587i	−0.729426	+	0.729426i
$738$		0			0
$739$		−	278.000i		−	0.376184i	−0.982151	−	0.188092i	$-0.939770\pi$
$739$		−	278.000i		−	0.376184i	0.982151	−	0.188092i	$-0.0602303\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.8114	−	15.8114i	−0.0212805	−	0.0212805i	0.696386	−	0.717667i	$-0.254790\pi$
$743$	−15.8114	−	15.8114i	−0.0212805	−	0.0212805i	−0.717667	+	0.696386i	$0.754790\pi$
$744$		0			0
$745$	225.000	+	675.000i	0.302013	+	0.906040i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			221.359i			0.295540i
$750$		0			0
$751$	48.0000			0.0639148			0.0319574	−	0.999489i	$-0.489826\pi$
$751$	48.0000			0.0639148			0.0319574	+	0.999489i	$0.489826\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−635.618	+	211.873i	−0.841878	+	0.280626i
$756$		0			0
$757$	414.000	−	414.000i	0.546896	−	0.546896i	−0.378646	−	0.925542i	$-0.623610\pi$
$757$	414.000	−	414.000i	0.546896	−	0.546896i	0.925542	+	0.378646i	$0.123610\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1170.04			−1.53751			−0.768753	−	0.639545i	$-0.779123\pi$
$761$	−1170.04			−1.53751			−0.768753	+	0.639545i	$0.779123\pi$
$762$		0			0
$763$	−46.0000	−	46.0000i	−0.0602883	−	0.0602883i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	474.342	−	474.342i	0.618438	−	0.618438i
$768$		0			0
$769$		−	1154.00i		−	1.50065i	−0.661069	−	0.750325i	$-0.729897\pi$
$769$		−	1154.00i		−	1.50065i	0.661069	−	0.750325i	$-0.270103\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	711.512	+	711.512i	0.920456	+	0.920456i	0.997061	−	0.0766055i	$-0.0244082\pi$
$773$	711.512	+	711.512i	0.920456	+	0.920456i	−0.0766055	+	0.997061i	$0.524408\pi$
$774$		0			0
$775$	320.000	−	240.000i	0.412903	−	0.309677i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	442.719i		−	0.568317i
$780$		0			0
$781$	−1000.00			−1.28041
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−455.368	+	910.736i	−0.580087	+	1.16017i
$786$		0			0
$787$	−446.000	+	446.000i	−0.566709	+	0.566709i	−0.931205	−	0.364496i	$-0.881241\pi$
$787$	−446.000	+	446.000i	−0.566709	+	0.566709i	0.364496	+	0.931205i	$0.381241\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−158.114			−0.199891
$792$		0			0
$793$	−324.000	−	324.000i	−0.408575	−	0.408575i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1090.99	+	1090.99i	−1.36887	+	1.36887i	−0.506804	+	0.862061i	$0.669173\pi$
$797$	−1090.99	+	1090.99i	−1.36887	+	1.36887i	−0.862061	+	0.506804i	$0.830827\pi$
$798$		0			0
$799$			1500.00i			1.87735i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1027.74	−	1027.74i	−1.27988	−	1.27988i
$804$		0			0
$805$	−150.000	+	50.0000i	−0.186335	+	0.0621118i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1075.17i		−	1.32902i	−0.747281	−	0.664508i	$-0.768641\pi$
$809$		−	1075.17i		−	1.32902i	0.747281	−	0.664508i	$-0.231359\pi$
$810$		0			0
$811$	386.000			0.475956			0.237978	−	0.971271i	$-0.423515\pi$
$811$	386.000			0.475956			0.237978	+	0.971271i	$0.423515\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−25.2982	−	12.6491i	−0.0310408	−	0.0155204i
$816$		0			0
$817$	756.000	−	756.000i	0.925337	−	0.925337i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	300.416			0.365915			0.182958	−	0.983121i	$-0.441433\pi$
$821$	300.416			0.365915			0.182958	+	0.983121i	$0.441433\pi$
$822$		0			0
$823$	281.000	+	281.000i	0.341434	+	0.341434i	0.856906	−	0.515472i	$-0.172384\pi$
$823$	281.000	+	281.000i	0.341434	+	0.341434i	−0.515472	+	0.856906i	$0.672384\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−379.473	+	379.473i	−0.458855	+	0.458855i	−0.898280	−	0.439424i	$-0.855182\pi$
$827$	−379.473	+	379.473i	−0.458855	+	0.458855i	0.439424	+	0.898280i	$0.355182\pi$
$828$		0			0
$829$		−	458.000i		−	0.552473i	−0.961090	−	0.276236i	$-0.910913\pi$
$829$		−	458.000i		−	0.552473i	0.961090	−	0.276236i	$-0.0890873\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	743.135	+	743.135i	0.892119	+	0.892119i
$834$		0			0
$835$	−350.000	+	700.000i	−0.419162	+	0.838323i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			347.851i			0.414601i	0.978277	+	0.207301i	$0.0664678\pi$
$839$			347.851i			0.414601i	−0.978277	+	0.207301i	$0.933532\pi$
$840$		0			0
$841$	591.000			0.702735
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−153.370	−	460.111i	−0.181504	−	0.544511i
$846$		0			0
$847$	−129.000	+	129.000i	−0.152302	+	0.152302i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−948.683			−1.11479
$852$		0			0
$853$	304.000	+	304.000i	0.356389	+	0.356389i	0.862480	−	0.506091i	$-0.168910\pi$
$853$	304.000	+	304.000i	0.356389	+	0.356389i	−0.506091	+	0.862480i	$0.668910\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	47.4342	−	47.4342i	0.0553491	−	0.0553491i	−0.678890	−	0.734240i	$-0.737539\pi$
$857$	47.4342	−	47.4342i	0.0553491	−	0.0553491i	0.734240	+	0.678890i	$0.237539\pi$
$858$		0			0
$859$		−	1082.00i		−	1.25960i	−0.776756	−	0.629802i	$-0.783136\pi$
$859$		−	1082.00i		−	1.25960i	0.776756	−	0.629802i	$-0.216864\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−648.267	−	648.267i	−0.751178	−	0.751178i	0.223521	−	0.974699i	$-0.428245\pi$
$863$	−648.267	−	648.267i	−0.751178	−	0.751178i	−0.974699	+	0.223521i	$0.928245\pi$
$864$		0			0
$865$	100.000	+	50.0000i	0.115607	+	0.0578035i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1707.63i			1.96505i
$870$		0			0
$871$	408.000			0.468427
$872$		0			0
$873$		0			0
$874$		0			0
$875$	31.6228	+	173.925i	0.0361403	+	0.198772i
$876$		0			0
$877$	84.0000	−	84.0000i	0.0957811	−	0.0957811i	−0.657593	−	0.753374i	$-0.728425\pi$
$877$	84.0000	−	84.0000i	0.0957811	−	0.0957811i	0.753374	+	0.657593i	$0.228425\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	822.192			0.933249			0.466624	−	0.884456i	$-0.345470\pi$
$881$	822.192			0.933249			0.466624	+	0.884456i	$0.345470\pi$
$882$		0			0
$883$	1096.00	+	1096.00i	1.24122	+	1.24122i	0.959494	+	0.281729i	$0.0909080\pi$
$883$	1096.00	+	1096.00i	1.24122	+	1.24122i	0.281729	+	0.959494i	$0.409092\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1185.85	−	1185.85i	1.33693	−	1.33693i	0.437906	−	0.899021i	$-0.355720\pi$
$887$	1185.85	−	1185.85i	1.33693	−	1.33693i	0.899021	−	0.437906i	$-0.144280\pi$
$888$		0			0
$889$			70.0000i			0.0787402i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−664.078	−	664.078i	−0.743649	−	0.743649i
$894$		0			0
$895$	−425.000	−	1275.00i	−0.474860	−	1.42458i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	252.982i		−	0.281404i
$900$		0			0
$901$	−2000.00			−2.21976
$902$		0			0
$903$		0			0
$904$		0			0
$905$	123.329	−	41.1096i	0.136275	−	0.0454250i
$906$		0			0
$907$	280.000	−	280.000i	0.308710	−	0.308710i	−0.535699	−	0.844409i	$-0.679952\pi$
$907$	280.000	−	280.000i	0.308710	−	0.308710i	0.844409	+	0.535699i	$0.179952\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	980.306			1.07608			0.538038	−	0.842920i	$-0.319166\pi$
$911$	980.306			1.07608			0.538038	+	0.842920i	$0.319166\pi$
$912$		0			0
$913$	750.000	+	750.000i	0.821468	+	0.821468i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−205.548	+	205.548i	−0.224153	+	0.224153i
$918$		0			0
$919$		−	896.000i		−	0.974973i	−0.873131	−	0.487486i	$-0.837914\pi$
$919$		−	896.000i		−	0.974973i	0.873131	−	0.487486i	$-0.162086\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	379.473	+	379.473i	0.411130	+	0.411130i
$924$		0			0
$925$	−150.000	+	1050.00i	−0.162162	+	1.13514i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			411.096i			0.442515i	0.975215	+	0.221257i	$0.0710161\pi$
$929$			411.096i			0.442515i	−0.975215	+	0.221257i	$0.928984\pi$
$930$		0			0
$931$	−658.000			−0.706767
$932$		0			0
$933$		0			0
$934$		0			0
$935$	790.569	−	1581.14i	0.845529	−	1.69106i
$936$		0			0
$937$	289.000	−	289.000i	0.308431	−	0.308431i	−0.535870	−	0.844301i	$-0.680016\pi$
$937$	289.000	−	289.000i	0.308431	−	0.308431i	0.844301	+	0.535870i	$0.180016\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1280.72			1.36102			0.680511	−	0.732737i	$-0.261758\pi$
$941$	1280.72			1.36102			0.680511	+	0.732737i	$0.261758\pi$
$942$		0			0
$943$	−500.000	−	500.000i	−0.530223	−	0.530223i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−158.114	+	158.114i	−0.166963	+	0.166963i	−0.785643	−	0.618680i	$-0.787668\pi$
$947$	−158.114	+	158.114i	−0.166963	+	0.166963i	0.618680	+	0.785643i	$0.287668\pi$
$948$		0			0
$949$			780.000i			0.821918i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	142.302	+	142.302i	0.149321	+	0.149321i	0.777814	−	0.628494i	$-0.216328\pi$
$953$	142.302	+	142.302i	0.149321	+	0.149321i	−0.628494	+	0.777814i	$0.716328\pi$
$954$		0			0
$955$	−1650.00	+	550.000i	−1.72775	+	0.575916i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			94.8683i			0.0989242i
$960$		0			0
$961$	−705.000			−0.733611
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1385.08	+	692.539i	1.43531	+	0.717657i
$966$		0			0
$967$	−365.000	+	365.000i	−0.377456	+	0.377456i	−0.870184	−	0.492728i	$-0.836000\pi$
$967$	−365.000	+	365.000i	−0.377456	+	0.377456i	0.492728	+	0.870184i	$0.336000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−395.285			−0.407090			−0.203545	−	0.979066i	$-0.565246\pi$
$971$	−395.285			−0.407090			−0.203545	+	0.979066i	$0.565246\pi$
$972$		0			0
$973$	−22.0000	−	22.0000i	−0.0226105	−	0.0226105i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	142.302	−	142.302i	0.145653	−	0.145653i	−0.630520	−	0.776173i	$-0.717158\pi$
$977$	142.302	−	142.302i	0.145653	−	0.145653i	0.776173	+	0.630520i	$0.217158\pi$
$978$		0			0
$979$		−	2000.00i		−	2.04290i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	932.872	+	932.872i	0.949005	+	0.949005i	0.998761	−	0.0497564i	$-0.0158445\pi$
$983$	932.872	+	932.872i	0.949005	+	0.949005i	−0.0497564	+	0.998761i	$0.515844\pi$
$984$		0			0
$985$	−300.000	+	600.000i	−0.304569	+	0.609137i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1707.63i		−	1.72662i
$990$		0			0
$991$	1714.00			1.72957			0.864783	−	0.502146i	$-0.167456\pi$
$991$	1714.00			1.72957			0.864783	+	0.502146i	$0.167456\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	104.355	+	313.065i	0.104880	+	0.314639i
$996$		0			0
$997$	1270.00	−	1270.00i	1.27382	−	1.27382i	0.329755	−	0.944067i	$-0.393034\pi$
$997$	1270.00	−	1270.00i	1.27382	−	1.27382i	0.944067	−	0.329755i	$-0.106966\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.3.l.c.73.2	yes	4
3.2	odd	2	inner	180.3.l.c.73.1	yes	4
4.3	odd	2		720.3.bh.h.433.2		4
5.2	odd	4	inner	180.3.l.c.37.2	yes	4
5.3	odd	4		900.3.l.h.757.2		4
5.4	even	2		900.3.l.h.793.2		4
12.11	even	2		720.3.bh.h.433.1		4
15.2	even	4	inner	180.3.l.c.37.1	✓	4
15.8	even	4		900.3.l.h.757.1		4
15.14	odd	2		900.3.l.h.793.1		4
20.7	even	4		720.3.bh.h.577.2		4
60.47	odd	4		720.3.bh.h.577.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.l.c.37.1	✓	4	15.2	even	4	inner
180.3.l.c.37.2	yes	4	5.2	odd	4	inner
180.3.l.c.73.1	yes	4	3.2	odd	2	inner
180.3.l.c.73.2	yes	4	1.1	even	1	trivial
720.3.bh.h.433.1		4	12.11	even	2
720.3.bh.h.433.2		4	4.3	odd	2
720.3.bh.h.577.1		4	60.47	odd	4
720.3.bh.h.577.2		4	20.7	even	4
900.3.l.h.757.1		4	15.8	even	4
900.3.l.h.757.2		4	5.3	odd	4
900.3.l.h.793.1		4	15.14	odd	2
900.3.l.h.793.2		4	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	180.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.74342 - 1.58114i) q^{5} +(-1.00000 + 1.00000i) q^{7} +15.8114 q^{11} +(-6.00000 - 6.00000i) q^{13} +(15.8114 - 15.8114i) q^{17} +14.0000i q^{19} +(15.8114 + 15.8114i) q^{23} +(20.0000 - 15.0000i) q^{25} -15.8114i q^{29} +16.0000 q^{31} +(-3.16228 + 6.32456i) q^{35} +(-30.0000 + 30.0000i) q^{37} -31.6228 q^{41} +(-54.0000 - 54.0000i) q^{43} +(-47.4342 + 47.4342i) q^{47} +47.0000i q^{49} +(-63.2456 - 63.2456i) q^{53} +(75.0000 - 25.0000i) q^{55} +79.0569i q^{59} +54.0000 q^{61} +(-37.9473 - 18.9737i) q^{65} +(-34.0000 + 34.0000i) q^{67} -63.2456 q^{71} +(-65.0000 - 65.0000i) q^{73} +(-15.8114 + 15.8114i) q^{77} +108.000i q^{79} +(47.4342 + 47.4342i) q^{83} +(50.0000 - 100.000i) q^{85} -126.491i q^{89} +12.0000 q^{91} +(22.1359 + 66.4078i) q^{95} +(-69.0000 + 69.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 24 q^{13} + 80 q^{25} + 64 q^{31} - 120 q^{37} - 216 q^{43} + 300 q^{55} + 216 q^{61} - 136 q^{67} - 260 q^{73} + 200 q^{85} + 48 q^{91} - 276 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 24 * q^13 + 80 * q^25 + 64 * q^31 - 120 * q^37 - 216 * q^43 + 300 * q^55 + 216 * q^61 - 136 * q^67 - 260 * q^73 + 200 * q^85 + 48 * q^91 - 276 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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