LMFDB - Embedded newform 720.2.x.e.127.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,2,Mod(127,720)] 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("720.127"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	720.x (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,8] 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:	$5.74922894553$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			127.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	720.127
Dual form			720.2.x.e.703.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 - 1.00000i) q^{5} +(2.82843 - 2.82843i) q^{7} +5.65685i q^{11} +(3.00000 - 3.00000i) q^{13} +(1.00000 + 1.00000i) q^{17} -5.65685 q^{19} +(-2.82843 - 2.82843i) q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000i q^{29} +(2.82843 - 8.48528i) q^{35} +(5.00000 + 5.00000i) q^{37} +(2.82843 + 2.82843i) q^{43} +(-2.82843 + 2.82843i) q^{47} -9.00000i q^{49} +(-1.00000 + 1.00000i) q^{53} +(5.65685 + 11.3137i) q^{55} +11.3137 q^{59} +4.00000 q^{61} +(3.00000 - 9.00000i) q^{65} +(-2.82843 + 2.82843i) q^{67} -5.65685i q^{71} +(-3.00000 + 3.00000i) q^{73} +(16.0000 + 16.0000i) q^{77} -5.65685 q^{79} +(-2.82843 - 2.82843i) q^{83} +(3.00000 + 1.00000i) q^{85} -8.00000i q^{89} -16.9706i q^{91} +(-11.3137 + 5.65685i) q^{95} +(-3.00000 - 3.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} + 12 q^{13} + 4 q^{17} + 12 q^{25} + 20 q^{37} - 4 q^{53} + 16 q^{61} + 12 q^{65} - 12 q^{73} + 64 q^{77} + 12 q^{85} - 12 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 12 * q^13 + 4 * q^17 + 12 * q^25 + 20 * q^37 - 4 * q^53 + 16 * q^61 + 12 * q^65 - 12 * q^73 + 64 * q^77 + 12 * q^85 - 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{4}\right)$	$1$

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
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$	2.82843	−	2.82843i	1.06904	−	1.06904i	0.0716124	−	0.997433i	$-0.477186\pi$
$7$	2.82843	−	2.82843i	1.06904	−	1.06904i	0.997433	−	0.0716124i	$-0.0228145\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.65685i			1.70561i	0.522233	+	0.852803i	$0.325099\pi$
$11$			5.65685i			1.70561i	−0.522233	+	0.852803i	$0.674901\pi$
$12$		0			0
$13$	3.00000	−	3.00000i	0.832050	−	0.832050i	−0.155747	−	0.987797i	$-0.549778\pi$
$13$	3.00000	−	3.00000i	0.832050	−	0.832050i	0.987797	+	0.155747i	$0.0497784\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	+	1.00000i	0.242536	+	0.242536i	0.817898	−	0.575363i	$-0.195139\pi$
$17$	1.00000	+	1.00000i	0.242536	+	0.242536i	−0.575363	+	0.817898i	$0.695139\pi$
$18$		0			0
$19$	−5.65685			−1.29777			−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685			−1.29777			−0.648886	+	0.760886i	$0.724765\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.82843	−	2.82843i	−0.589768	−	0.589768i	0.347801	−	0.937568i	$-0.386929\pi$
$23$	−2.82843	−	2.82843i	−0.589768	−	0.589768i	−0.937568	+	0.347801i	$0.886929\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.00000i		−	0.742781i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$		−	4.00000i		−	0.742781i	0.928477	−	0.371391i	$-0.121119\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.82843	−	8.48528i	0.478091	−	1.43427i
$36$		0			0
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	2.82843	+	2.82843i	0.431331	+	0.431331i	0.889081	−	0.457750i	$-0.151344\pi$
$43$	2.82843	+	2.82843i	0.431331	+	0.431331i	−0.457750	+	0.889081i	$0.651344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.82843	+	2.82843i	−0.412568	+	0.412568i	−0.882632	−	0.470064i	$-0.844231\pi$
$47$	−2.82843	+	2.82843i	−0.412568	+	0.412568i	0.470064	+	0.882632i	$0.344231\pi$
$48$		0			0
$49$		−	9.00000i		−	1.28571i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.00000	+	1.00000i	−0.137361	+	0.137361i	−0.772444	−	0.635083i	$-0.780966\pi$
$53$	−1.00000	+	1.00000i	−0.137361	+	0.137361i	0.635083	+	0.772444i	$0.280966\pi$
$54$		0			0
$55$	5.65685	+	11.3137i	0.762770	+	1.52554i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	11.3137			1.47292			0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137			1.47292			0.736460	+	0.676481i	$0.236496\pi$
$60$		0			0
$61$	4.00000			0.512148			0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000			0.512148			0.256074	+	0.966657i	$0.417571\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.00000	−	9.00000i	0.372104	−	1.11631i
$66$		0			0
$67$	−2.82843	+	2.82843i	−0.345547	+	0.345547i	−0.858448	−	0.512901i	$-0.828571\pi$
$67$	−2.82843	+	2.82843i	−0.345547	+	0.345547i	0.512901	+	0.858448i	$0.328571\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.65685i		−	0.671345i	−0.941979	−	0.335673i	$-0.891036\pi$
$71$		−	5.65685i		−	0.671345i	0.941979	−	0.335673i	$-0.108964\pi$
$72$		0			0
$73$	−3.00000	+	3.00000i	−0.351123	+	0.351123i	−0.860527	−	0.509404i	$-0.829866\pi$
$73$	−3.00000	+	3.00000i	−0.351123	+	0.351123i	0.509404	+	0.860527i	$0.329866\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	16.0000	+	16.0000i	1.82337	+	1.82337i
$78$		0			0
$79$	−5.65685			−0.636446			−0.318223	−	0.948016i	$-0.603086\pi$
$79$	−5.65685			−0.636446			−0.318223	+	0.948016i	$0.603086\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.82843	−	2.82843i	−0.310460	−	0.310460i	0.534628	−	0.845088i	$-0.320452\pi$
$83$	−2.82843	−	2.82843i	−0.310460	−	0.310460i	−0.845088	+	0.534628i	$0.820452\pi$
$84$		0			0
$85$	3.00000	+	1.00000i	0.325396	+	0.108465i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	8.00000i		−	0.847998i	−0.905663	−	0.423999i	$-0.860626\pi$
$89$		−	8.00000i		−	0.847998i	0.905663	−	0.423999i	$-0.139374\pi$
$90$		0			0
$91$		−	16.9706i		−	1.77900i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.3137	+	5.65685i	−1.16076	+	0.580381i
$96$		0			0
$97$	−3.00000	−	3.00000i	−0.304604	−	0.304604i	0.538208	−	0.842812i	$-0.319101\pi$
$97$	−3.00000	−	3.00000i	−0.304604	−	0.304604i	−0.842812	+	0.538208i	$0.819101\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−18.0000			−1.79107			−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000			−1.79107			−0.895533	+	0.444994i	$0.853206\pi$
$102$		0			0
$103$	−2.82843	−	2.82843i	−0.278693	−	0.278693i	0.553894	−	0.832587i	$-0.313141\pi$
$103$	−2.82843	−	2.82843i	−0.278693	−	0.278693i	−0.832587	+	0.553894i	$0.813141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.48528	+	8.48528i	−0.820303	+	0.820303i	−0.986151	−	0.165848i	$-0.946964\pi$
$107$	−8.48528	+	8.48528i	−0.820303	+	0.820303i	0.165848	+	0.986151i	$0.446964\pi$
$108$		0			0
$109$			10.0000i			0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$			10.0000i			0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.0000	+	11.0000i	−1.03479	+	1.03479i	−0.0354205	+	0.999372i	$0.511277\pi$
$113$	−11.0000	+	11.0000i	−1.03479	+	1.03479i	−0.999372	+	0.0354205i	$0.988723\pi$
$114$		0			0
$115$	−8.48528	−	2.82843i	−0.791257	−	0.263752i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.65685			0.518563
$120$		0			0
$121$	−21.0000			−1.90909
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$	8.48528	−	8.48528i	0.752947	−	0.752947i	−0.222081	−	0.975028i	$-0.571285\pi$
$127$	8.48528	−	8.48528i	0.752947	−	0.752947i	0.975028	+	0.222081i	$0.0712850\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			5.65685i			0.494242i	0.968985	+	0.247121i	$0.0794845\pi$
$131$			5.65685i			0.494242i	−0.968985	+	0.247121i	$0.920516\pi$
$132$		0			0
$133$	−16.0000	+	16.0000i	−1.38738	+	1.38738i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.00000	−	5.00000i	−0.427179	−	0.427179i	0.460487	−	0.887666i	$-0.347675\pi$
$137$	−5.00000	−	5.00000i	−0.427179	−	0.427179i	−0.887666	+	0.460487i	$0.847675\pi$
$138$		0			0
$139$	16.9706			1.43942			0.719712	−	0.694273i	$-0.244274\pi$
$139$	16.9706			1.43942			0.719712	+	0.694273i	$0.244274\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	16.9706	+	16.9706i	1.41915	+	1.41915i
$144$		0			0
$145$	−4.00000	−	8.00000i	−0.332182	−	0.664364i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			18.0000i			1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$			18.0000i			1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$		0			0
$151$			11.3137i			0.920697i	0.887738	+	0.460348i	$0.152275\pi$
$151$			11.3137i			0.920697i	−0.887738	+	0.460348i	$0.847725\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	11.0000	+	11.0000i	0.877896	+	0.877896i	0.993317	−	0.115421i	$-0.0368217\pi$
$157$	11.0000	+	11.0000i	0.877896	+	0.877896i	−0.115421	+	0.993317i	$0.536822\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−16.0000			−1.26098
$162$		0			0
$163$	8.48528	+	8.48528i	0.664619	+	0.664619i	0.956465	−	0.291847i	$-0.0942697\pi$
$163$	8.48528	+	8.48528i	0.664619	+	0.664619i	−0.291847	+	0.956465i	$0.594270\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.48528	−	8.48528i	0.656611	−	0.656611i	−0.297966	−	0.954577i	$-0.596308\pi$
$167$	8.48528	−	8.48528i	0.656611	−	0.656611i	0.954577	+	0.297966i	$0.0963081\pi$
$168$		0			0
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.00000	+	5.00000i	−0.380143	+	0.380143i	−0.871154	−	0.491011i	$-0.836628\pi$
$173$	−5.00000	+	5.00000i	−0.380143	+	0.380143i	0.491011	+	0.871154i	$0.336628\pi$
$174$		0			0
$175$	−2.82843	−	19.7990i	−0.213809	−	1.49666i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−22.6274			−1.69125			−0.845626	−	0.533775i	$-0.820773\pi$
$179$	−22.6274			−1.69125			−0.845626	+	0.533775i	$0.820773\pi$
$180$		0			0
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.0000	+	5.00000i	1.10282	+	0.367607i
$186$		0			0
$187$	−5.65685	+	5.65685i	−0.413670	+	0.413670i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	16.9706i		−	1.22795i	−0.789327	−	0.613973i	$-0.789570\pi$
$191$		−	16.9706i		−	1.22795i	0.789327	−	0.613973i	$-0.210430\pi$
$192$		0			0
$193$	−5.00000	+	5.00000i	−0.359908	+	0.359908i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−5.00000	+	5.00000i	−0.359908	+	0.359908i	0.503871	+	0.863779i	$0.331909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	11.0000	+	11.0000i	0.783718	+	0.783718i	0.980456	−	0.196738i	$-0.0630350\pi$
$197$	11.0000	+	11.0000i	0.783718	+	0.783718i	−0.196738	+	0.980456i	$0.563035\pi$
$198$		0			0
$199$	−16.9706			−1.20301			−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706			−1.20301			−0.601506	+	0.798869i	$0.705432\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−11.3137	−	11.3137i	−0.794067	−	0.794067i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	32.0000i		−	2.21349i
$210$		0			0
$211$			22.6274i			1.55774i	0.627188	+	0.778868i	$0.284206\pi$
$211$			22.6274i			1.55774i	−0.627188	+	0.778868i	$0.715794\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.48528	+	2.82843i	0.578691	+	0.192897i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.00000			0.403604
$222$		0			0
$223$	−8.48528	−	8.48528i	−0.568216	−	0.568216i	0.363412	−	0.931629i	$-0.381612\pi$
$223$	−8.48528	−	8.48528i	−0.568216	−	0.568216i	−0.931629	+	0.363412i	$0.881612\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	19.7990	−	19.7990i	1.31411	−	1.31411i	0.395744	−	0.918361i	$-0.370487\pi$
$227$	19.7990	−	19.7990i	1.31411	−	1.31411i	0.918361	−	0.395744i	$-0.129513\pi$
$228$		0			0
$229$			12.0000i			0.792982i	0.918039	+	0.396491i	$0.129772\pi$
$229$			12.0000i			0.792982i	−0.918039	+	0.396491i	$0.870228\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−15.0000	+	15.0000i	−0.982683	+	0.982683i	−0.999853	−	0.0171699i	$-0.994534\pi$
$233$	−15.0000	+	15.0000i	−0.982683	+	0.982683i	0.0171699	+	0.999853i	$0.494534\pi$
$234$		0			0
$235$	−2.82843	+	8.48528i	−0.184506	+	0.553519i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.3137			−0.731823			−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137			−0.731823			−0.365911	+	0.930650i	$0.619243\pi$
$240$		0			0
$241$	8.00000			0.515325			0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000			0.515325			0.257663	+	0.966235i	$0.417048\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.00000	−	18.0000i	−0.574989	−	1.14998i
$246$		0			0
$247$	−16.9706	+	16.9706i	−1.07981	+	1.07981i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			5.65685i			0.357057i	0.983935	+	0.178529i	$0.0571337\pi$
$251$			5.65685i			0.357057i	−0.983935	+	0.178529i	$0.942866\pi$
$252$		0			0
$253$	16.0000	−	16.0000i	1.00591	−	1.00591i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	21.0000	+	21.0000i	1.30994	+	1.30994i	0.921452	+	0.388492i	$0.127004\pi$
$257$	21.0000	+	21.0000i	1.30994	+	1.30994i	0.388492	+	0.921452i	$0.372996\pi$
$258$		0			0
$259$	28.2843			1.75750
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.82843	−	2.82843i	−0.174408	−	0.174408i	0.614505	−	0.788913i	$-0.289356\pi$
$263$	−2.82843	−	2.82843i	−0.174408	−	0.174408i	−0.788913	+	0.614505i	$0.789356\pi$
$264$		0			0
$265$	−1.00000	+	3.00000i	−0.0614295	+	0.184289i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	6.00000i		−	0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$		−	6.00000i		−	0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$		0			0
$271$			11.3137i			0.687259i	0.939105	+	0.343629i	$0.111656\pi$
$271$			11.3137i			0.687259i	−0.939105	+	0.343629i	$0.888344\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	22.6274	+	16.9706i	1.36448	+	1.02336i
$276$		0			0
$277$	3.00000	+	3.00000i	0.180253	+	0.180253i	0.791466	−	0.611213i	$-0.209318\pi$
$277$	3.00000	+	3.00000i	0.180253	+	0.180253i	−0.611213	+	0.791466i	$0.709318\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.00000			−0.477240			−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000			−0.477240			−0.238620	+	0.971113i	$0.576695\pi$
$282$		0			0
$283$	−14.1421	−	14.1421i	−0.840663	−	0.840663i	0.148282	−	0.988945i	$-0.452626\pi$
$283$	−14.1421	−	14.1421i	−0.840663	−	0.840663i	−0.988945	+	0.148282i	$0.952626\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	15.0000i		−	0.882353i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.00000	−	5.00000i	0.292103	−	0.292103i	−0.545807	−	0.837911i	$-0.683777\pi$
$293$	5.00000	−	5.00000i	0.292103	−	0.292103i	0.837911	+	0.545807i	$0.183777\pi$
$294$		0			0
$295$	22.6274	−	11.3137i	1.31742	−	0.658710i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.9706			−0.981433
$300$		0			0
$301$	16.0000			0.922225
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.00000	−	4.00000i	0.458079	−	0.229039i
$306$		0			0
$307$	2.82843	−	2.82843i	0.161427	−	0.161427i	−0.621772	−	0.783199i	$-0.713587\pi$
$307$	2.82843	−	2.82843i	0.161427	−	0.161427i	0.783199	+	0.621772i	$0.213587\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			5.65685i			0.320771i	0.987054	+	0.160385i	$0.0512737\pi$
$311$			5.65685i			0.320771i	−0.987054	+	0.160385i	$0.948726\pi$
$312$		0			0
$313$	−9.00000	+	9.00000i	−0.508710	+	0.508710i	−0.914130	−	0.405420i	$-0.867125\pi$
$313$	−9.00000	+	9.00000i	−0.508710	+	0.508710i	0.405420	+	0.914130i	$0.367125\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.0000	−	11.0000i	−0.617822	−	0.617822i	0.327151	−	0.944972i	$-0.393912\pi$
$317$	−11.0000	−	11.0000i	−0.617822	−	0.617822i	−0.944972	+	0.327151i	$0.893912\pi$
$318$		0			0
$319$	22.6274			1.26689
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.65685	−	5.65685i	−0.314756	−	0.314756i
$324$		0			0
$325$	−3.00000	−	21.0000i	−0.166410	−	1.16487i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			16.0000i			0.882109i
$330$		0			0
$331$		−	22.6274i		−	1.24372i	−0.783130	−	0.621858i	$-0.786378\pi$
$331$		−	22.6274i		−	1.24372i	0.783130	−	0.621858i	$-0.213622\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.82843	+	8.48528i	−0.154533	+	0.463600i
$336$		0			0
$337$	1.00000	+	1.00000i	0.0544735	+	0.0544735i	0.733819	−	0.679345i	$-0.237736\pi$
$337$	1.00000	+	1.00000i	0.0544735	+	0.0544735i	−0.679345	+	0.733819i	$0.737736\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−5.65685	−	5.65685i	−0.305441	−	0.305441i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.82843	+	2.82843i	−0.151838	+	0.151838i	−0.778938	−	0.627100i	$-0.784242\pi$
$347$	−2.82843	+	2.82843i	−0.151838	+	0.151838i	0.627100	+	0.778938i	$0.284242\pi$
$348$		0			0
$349$		−	28.0000i		−	1.49881i	−0.662114	−	0.749403i	$-0.730341\pi$
$349$		−	28.0000i		−	1.49881i	0.662114	−	0.749403i	$-0.269659\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.0000	−	11.0000i	0.585471	−	0.585471i	−0.350931	−	0.936401i	$-0.614135\pi$
$353$	11.0000	−	11.0000i	0.585471	−	0.585471i	0.936401	+	0.350931i	$0.114135\pi$
$354$		0			0
$355$	−5.65685	−	11.3137i	−0.300235	−	0.600469i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	11.3137			0.597115			0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137			0.597115			0.298557	+	0.954392i	$0.403495\pi$
$360$		0			0
$361$	13.0000			0.684211
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.00000	+	9.00000i	−0.157027	+	0.471082i
$366$		0			0
$367$	−8.48528	+	8.48528i	−0.442928	+	0.442928i	−0.892995	−	0.450067i	$-0.851400\pi$
$367$	−8.48528	+	8.48528i	−0.442928	+	0.442928i	0.450067	+	0.892995i	$0.351400\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			5.65685i			0.293689i
$372$		0			0
$373$	9.00000	−	9.00000i	0.466002	−	0.466002i	−0.434614	−	0.900617i	$-0.643115\pi$
$373$	9.00000	−	9.00000i	0.466002	−	0.466002i	0.900617	+	0.434614i	$0.143115\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.0000	−	12.0000i	−0.618031	−	0.618031i
$378$		0			0
$379$	−16.9706			−0.871719			−0.435860	−	0.900015i	$-0.643556\pi$
$379$	−16.9706			−0.871719			−0.435860	+	0.900015i	$0.643556\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.1421	+	14.1421i	0.722629	+	0.722629i	0.969140	−	0.246511i	$-0.0792840\pi$
$383$	14.1421	+	14.1421i	0.722629	+	0.722629i	−0.246511	+	0.969140i	$0.579284\pi$
$384$		0			0
$385$	48.0000	+	16.0000i	2.44631	+	0.815436i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	2.00000i		−	0.101404i	−0.998714	−	0.0507020i	$-0.983854\pi$
$389$		−	2.00000i		−	0.101404i	0.998714	−	0.0507020i	$-0.0161459\pi$
$390$		0			0
$391$		−	5.65685i		−	0.286079i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.3137	+	5.65685i	−0.569254	+	0.284627i
$396$		0			0
$397$	9.00000	+	9.00000i	0.451697	+	0.451697i	0.895918	−	0.444220i	$-0.146519\pi$
$397$	9.00000	+	9.00000i	0.451697	+	0.451697i	−0.444220	+	0.895918i	$0.646519\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−28.2843	+	28.2843i	−1.40200	+	1.40200i
$408$		0			0
$409$			10.0000i			0.494468i	0.968956	+	0.247234i	$0.0795217\pi$
$409$			10.0000i			0.494468i	−0.968956	+	0.247234i	$0.920478\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	32.0000	−	32.0000i	1.57462	−	1.57462i
$414$		0			0
$415$	−8.48528	−	2.82843i	−0.416526	−	0.138842i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.3137			−0.552711			−0.276355	−	0.961056i	$-0.589127\pi$
$419$	−11.3137			−0.552711			−0.276355	+	0.961056i	$0.589127\pi$
$420$		0			0
$421$	−4.00000			−0.194948			−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000			−0.194948			−0.0974740	+	0.995238i	$0.531076\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.00000	−	1.00000i	0.339550	−	0.0485071i
$426$		0			0
$427$	11.3137	−	11.3137i	0.547509	−	0.547509i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	16.9706i		−	0.817443i	−0.912659	−	0.408722i	$-0.865975\pi$
$431$		−	16.9706i		−	0.817443i	0.912659	−	0.408722i	$-0.134025\pi$
$432$		0			0
$433$	21.0000	−	21.0000i	1.00920	−	1.00920i	0.00923827	−	0.999957i	$-0.497059\pi$
$433$	21.0000	−	21.0000i	1.00920	−	1.00920i	0.999957	−	0.00923827i	$-0.00294067\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	16.0000	+	16.0000i	0.765384	+	0.765384i
$438$		0			0
$439$	−5.65685			−0.269987			−0.134993	−	0.990846i	$-0.543101\pi$
$439$	−5.65685			−0.269987			−0.134993	+	0.990846i	$0.543101\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	25.4558	+	25.4558i	1.20944	+	1.20944i	0.971207	+	0.238236i	$0.0765693\pi$
$443$	25.4558	+	25.4558i	1.20944	+	1.20944i	0.238236	+	0.971207i	$0.423431\pi$
$444$		0			0
$445$	−8.00000	−	16.0000i	−0.379236	−	0.758473i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			2.00000i			0.0943858i	0.998886	+	0.0471929i	$0.0150276\pi$
$449$			2.00000i			0.0943858i	−0.998886	+	0.0471929i	$0.984972\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−16.9706	−	33.9411i	−0.795592	−	1.59118i
$456$		0			0
$457$	−17.0000	−	17.0000i	−0.795226	−	0.795226i	0.187112	−	0.982339i	$-0.440087\pi$
$457$	−17.0000	−	17.0000i	−0.795226	−	0.795226i	−0.982339	+	0.187112i	$0.940087\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.0000			−0.465746			−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000			−0.465746			−0.232873	+	0.972507i	$0.574813\pi$
$462$		0			0
$463$	25.4558	+	25.4558i	1.18303	+	1.18303i	0.978954	+	0.204079i	$0.0654199\pi$
$463$	25.4558	+	25.4558i	1.18303	+	1.18303i	0.204079	+	0.978954i	$0.434580\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	25.4558	−	25.4558i	1.17796	−	1.17796i	0.197692	−	0.980264i	$-0.436655\pi$
$467$	25.4558	−	25.4558i	1.17796	−	1.17796i	0.980264	−	0.197692i	$-0.0633445\pi$
$468$		0			0
$469$			16.0000i			0.738811i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−16.0000	+	16.0000i	−0.735681	+	0.735681i
$474$		0			0
$475$	−16.9706	+	22.6274i	−0.778663	+	1.03822i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	22.6274			1.03387			0.516937	−	0.856024i	$-0.327072\pi$
$479$	22.6274			1.03387			0.516937	+	0.856024i	$0.327072\pi$
$480$		0			0
$481$	30.0000			1.36788
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.00000	−	3.00000i	−0.408669	−	0.136223i
$486$		0			0
$487$	19.7990	−	19.7990i	0.897178	−	0.897178i	−0.0980078	−	0.995186i	$-0.531247\pi$
$487$	19.7990	−	19.7990i	0.897178	−	0.897178i	0.995186	+	0.0980078i	$0.0312470\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	28.2843i		−	1.27645i	−0.769849	−	0.638226i	$-0.779669\pi$
$491$		−	28.2843i		−	1.27645i	0.769849	−	0.638226i	$-0.220331\pi$
$492$		0			0
$493$	4.00000	−	4.00000i	0.180151	−	0.180151i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.0000	−	16.0000i	−0.717698	−	0.717698i
$498$		0			0
$499$	−28.2843			−1.26618			−0.633089	−	0.774079i	$-0.718213\pi$
$499$	−28.2843			−1.26618			−0.633089	+	0.774079i	$0.718213\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−31.1127	−	31.1127i	−1.38725	−	1.38725i	−0.831052	−	0.556195i	$-0.812261\pi$
$503$	−31.1127	−	31.1127i	−1.38725	−	1.38725i	−0.556195	−	0.831052i	$-0.687739\pi$
$504$		0			0
$505$	−36.0000	+	18.0000i	−1.60198	+	0.800989i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			20.0000i			0.886484i	0.896402	+	0.443242i	$0.146172\pi$
$509$			20.0000i			0.886484i	−0.896402	+	0.443242i	$0.853828\pi$
$510$		0			0
$511$			16.9706i			0.750733i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.48528	−	2.82843i	−0.373906	−	0.124635i
$516$		0			0
$517$	−16.0000	−	16.0000i	−0.703679	−	0.703679i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10.0000			0.438108			0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000			0.438108			0.219054	+	0.975713i	$0.429703\pi$
$522$		0			0
$523$	25.4558	+	25.4558i	1.11311	+	1.11311i	0.992728	+	0.120378i	$0.0384107\pi$
$523$	25.4558	+	25.4558i	1.11311	+	1.11311i	0.120378	+	0.992728i	$0.461589\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	7.00000i		−	0.304348i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−8.48528	+	25.4558i	−0.366851	+	1.10055i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	50.9117			2.19292
$540$		0			0
$541$	−22.0000			−0.945854			−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000			−0.945854			−0.472927	+	0.881102i	$0.656803\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.0000	+	20.0000i	0.428353	+	0.856706i
$546$		0			0
$547$	−8.48528	+	8.48528i	−0.362804	+	0.362804i	−0.864844	−	0.502040i	$-0.832583\pi$
$547$	−8.48528	+	8.48528i	−0.362804	+	0.362804i	0.502040	+	0.864844i	$0.332583\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			22.6274i			0.963960i
$552$		0			0
$553$	−16.0000	+	16.0000i	−0.680389	+	0.680389i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17.0000	−	17.0000i	−0.720313	−	0.720313i	0.248356	−	0.968669i	$-0.420110\pi$
$557$	−17.0000	−	17.0000i	−0.720313	−	0.720313i	−0.968669	+	0.248356i	$0.920110\pi$
$558$		0			0
$559$	16.9706			0.717778
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.48528	+	8.48528i	0.357612	+	0.357612i	0.862932	−	0.505320i	$-0.168626\pi$
$563$	8.48528	+	8.48528i	0.357612	+	0.357612i	−0.505320	+	0.862932i	$0.668626\pi$
$564$		0			0
$565$	−11.0000	+	33.0000i	−0.462773	+	1.38832i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	22.0000i		−	0.922288i	−0.887325	−	0.461144i	$-0.847439\pi$
$569$		−	22.0000i		−	0.922288i	0.887325	−	0.461144i	$-0.152561\pi$
$570$		0			0
$571$		−	11.3137i		−	0.473464i	−0.971575	−	0.236732i	$-0.923924\pi$
$571$		−	11.3137i		−	0.473464i	0.971575	−	0.236732i	$-0.0760763\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−19.7990	+	2.82843i	−0.825675	+	0.117954i
$576$		0			0
$577$	23.0000	+	23.0000i	0.957503	+	0.957503i	0.999133	−	0.0416305i	$-0.0132552\pi$
$577$	23.0000	+	23.0000i	0.957503	+	0.957503i	−0.0416305	+	0.999133i	$0.513255\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−16.0000			−0.663792
$582$		0			0
$583$	−5.65685	−	5.65685i	−0.234283	−	0.234283i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.1421	+	14.1421i	−0.583708	+	0.583708i	−0.935920	−	0.352212i	$-0.885430\pi$
$587$	−14.1421	+	14.1421i	−0.583708	+	0.583708i	0.352212	+	0.935920i	$0.385430\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−21.0000	+	21.0000i	−0.862367	+	0.862367i	−0.991613	−	0.129246i	$-0.958744\pi$
$593$	−21.0000	+	21.0000i	−0.862367	+	0.862367i	0.129246	+	0.991613i	$0.458744\pi$
$594$		0			0
$595$	11.3137	−	5.65685i	0.463817	−	0.231908i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	11.3137			0.462266			0.231133	−	0.972922i	$-0.425757\pi$
$599$	11.3137			0.462266			0.231133	+	0.972922i	$0.425757\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−42.0000	+	21.0000i	−1.70754	+	0.853771i
$606$		0			0
$607$	8.48528	−	8.48528i	0.344407	−	0.344407i	−0.513614	−	0.858021i	$-0.671694\pi$
$607$	8.48528	−	8.48528i	0.344407	−	0.344407i	0.858021	+	0.513614i	$0.171694\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			16.9706i			0.686555i
$612$		0			0
$613$	3.00000	−	3.00000i	0.121169	−	0.121169i	−0.643922	−	0.765091i	$-0.722694\pi$
$613$	3.00000	−	3.00000i	0.121169	−	0.121169i	0.765091	+	0.643922i	$0.222694\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−23.0000	−	23.0000i	−0.925945	−	0.925945i	0.0714958	−	0.997441i	$-0.477223\pi$
$617$	−23.0000	−	23.0000i	−0.925945	−	0.925945i	−0.997441	+	0.0714958i	$0.977223\pi$
$618$		0			0
$619$	−39.5980			−1.59158			−0.795789	−	0.605575i	$-0.792943\pi$
$619$	−39.5980			−1.59158			−0.795789	+	0.605575i	$0.792943\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−22.6274	−	22.6274i	−0.906548	−	0.906548i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			10.0000i			0.398726i
$630$		0			0
$631$		−	22.6274i		−	0.900783i	−0.892831	−	0.450392i	$-0.851284\pi$
$631$		−	22.6274i		−	0.900783i	0.892831	−	0.450392i	$-0.148716\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.48528	−	25.4558i	0.336728	−	1.01018i
$636$		0			0
$637$	−27.0000	−	27.0000i	−1.06978	−	1.06978i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	48.0000			1.89589			0.947943	−	0.318440i	$-0.103159\pi$
$641$	48.0000			1.89589			0.947943	+	0.318440i	$0.103159\pi$
$642$		0			0
$643$	−19.7990	−	19.7990i	−0.780796	−	0.780796i	0.199169	−	0.979965i	$-0.436176\pi$
$643$	−19.7990	−	19.7990i	−0.780796	−	0.780796i	−0.979965	+	0.199169i	$0.936176\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.48528	+	8.48528i	−0.333591	+	0.333591i	−0.853948	−	0.520358i	$-0.825799\pi$
$647$	−8.48528	+	8.48528i	−0.333591	+	0.333591i	0.520358	+	0.853948i	$0.325799\pi$
$648$		0			0
$649$			64.0000i			2.51222i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	23.0000	−	23.0000i	0.900060	−	0.900060i	−0.0953813	−	0.995441i	$-0.530407\pi$
$653$	23.0000	−	23.0000i	0.900060	−	0.900060i	0.995441	+	0.0953813i	$0.0304070\pi$
$654$		0			0
$655$	5.65685	+	11.3137i	0.221032	+	0.442063i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−11.3137			−0.440720			−0.220360	−	0.975419i	$-0.570723\pi$
$659$	−11.3137			−0.440720			−0.220360	+	0.975419i	$0.570723\pi$
$660$		0			0
$661$	−28.0000			−1.08907			−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000			−1.08907			−0.544537	+	0.838737i	$0.683295\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−16.0000	+	48.0000i	−0.620453	+	1.86136i
$666$		0			0
$667$	−11.3137	+	11.3137i	−0.438069	+	0.438069i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			22.6274i			0.873522i
$672$		0			0
$673$	−35.0000	+	35.0000i	−1.34915	+	1.34915i	−0.462566	+	0.886585i	$0.653071\pi$
$673$	−35.0000	+	35.0000i	−1.34915	+	1.34915i	−0.886585	+	0.462566i	$0.846929\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.00000	−	5.00000i	−0.192166	−	0.192166i	0.604466	−	0.796631i	$-0.293387\pi$
$677$	−5.00000	−	5.00000i	−0.192166	−	0.192166i	−0.796631	+	0.604466i	$0.793387\pi$
$678$		0			0
$679$	−16.9706			−0.651270
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.48528	−	8.48528i	−0.324680	−	0.324680i	0.525879	−	0.850559i	$-0.323736\pi$
$683$	−8.48528	−	8.48528i	−0.324680	−	0.324680i	−0.850559	+	0.525879i	$0.823736\pi$
$684$		0			0
$685$	−15.0000	−	5.00000i	−0.573121	−	0.191040i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			6.00000i			0.228582i
$690$		0			0
$691$		−	22.6274i		−	0.860788i	−0.902641	−	0.430394i	$-0.858375\pi$
$691$		−	22.6274i		−	0.860788i	0.902641	−	0.430394i	$-0.141625\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	33.9411	−	16.9706i	1.28746	−	0.643730i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	28.0000			1.05755			0.528773	−	0.848763i	$-0.322652\pi$
$701$	28.0000			1.05755			0.528773	+	0.848763i	$0.322652\pi$
$702$		0			0
$703$	−28.2843	−	28.2843i	−1.06676	−	1.06676i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−50.9117	+	50.9117i	−1.91473	+	1.91473i
$708$		0			0
$709$		−	28.0000i		−	1.05156i	−0.850620	−	0.525781i	$-0.823773\pi$
$709$		−	28.0000i		−	1.05156i	0.850620	−	0.525781i	$-0.176227\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	50.9117	+	16.9706i	1.90399	+	0.634663i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−11.3137			−0.421930			−0.210965	−	0.977494i	$-0.567661\pi$
$719$	−11.3137			−0.421930			−0.210965	+	0.977494i	$0.567661\pi$
$720$		0			0
$721$	−16.0000			−0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−16.0000	−	12.0000i	−0.594225	−	0.445669i
$726$		0			0
$727$	31.1127	−	31.1127i	1.15391	−	1.15391i	0.168144	−	0.985763i	$-0.446223\pi$
$727$	31.1127	−	31.1127i	1.15391	−	1.15391i	0.985763	−	0.168144i	$-0.0537772\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			5.65685i			0.209226i
$732$		0			0
$733$	17.0000	−	17.0000i	0.627909	−	0.627909i	−0.319632	−	0.947542i	$-0.603559\pi$
$733$	17.0000	−	17.0000i	0.627909	−	0.627909i	0.947542	+	0.319632i	$0.103559\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−16.0000	−	16.0000i	−0.589368	−	0.589368i
$738$		0			0
$739$	28.2843			1.04045			0.520227	−	0.854028i	$-0.325847\pi$
$739$	28.2843			1.04045			0.520227	+	0.854028i	$0.325847\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	25.4558	+	25.4558i	0.933884	+	0.933884i	0.997946	−	0.0640616i	$-0.0204054\pi$
$743$	25.4558	+	25.4558i	0.933884	+	0.933884i	−0.0640616	+	0.997946i	$0.520405\pi$
$744$		0			0
$745$	18.0000	+	36.0000i	0.659469	+	1.31894i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			48.0000i			1.75388i
$750$		0			0
$751$		−	33.9411i		−	1.23853i	−0.785182	−	0.619265i	$-0.787431\pi$
$751$		−	33.9411i		−	1.23853i	0.785182	−	0.619265i	$-0.212569\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.3137	+	22.6274i	0.411748	+	0.823496i
$756$		0			0
$757$	−5.00000	−	5.00000i	−0.181728	−	0.181728i	0.610380	−	0.792108i	$-0.291017\pi$
$757$	−5.00000	−	5.00000i	−0.181728	−	0.181728i	−0.792108	+	0.610380i	$0.791017\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$	28.2843	+	28.2843i	1.02396	+	1.02396i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	33.9411	−	33.9411i	1.22554	−	1.22554i
$768$		0			0
$769$			8.00000i			0.288487i	0.989542	+	0.144244i	$0.0460749\pi$
$769$			8.00000i			0.288487i	−0.989542	+	0.144244i	$0.953925\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	17.0000	−	17.0000i	0.611448	−	0.611448i	−0.331876	−	0.943323i	$-0.607681\pi$
$773$	17.0000	−	17.0000i	0.611448	−	0.611448i	0.943323	+	0.331876i	$0.107681\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	32.0000			1.14505
$782$		0			0
$783$		0			0
$784$		0			0
$785$	33.0000	+	11.0000i	1.17782	+	0.392607i
$786$		0			0
$787$	14.1421	−	14.1421i	0.504113	−	0.504113i	−0.408601	−	0.912713i	$-0.633983\pi$
$787$	14.1421	−	14.1421i	0.504113	−	0.504113i	0.912713	+	0.408601i	$0.133983\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			62.2254i			2.21248i
$792$		0			0
$793$	12.0000	−	12.0000i	0.426132	−	0.426132i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−27.0000	−	27.0000i	−0.956389	−	0.956389i	0.0426989	−	0.999088i	$-0.486404\pi$
$797$	−27.0000	−	27.0000i	−0.956389	−	0.956389i	−0.999088	+	0.0426989i	$0.986404\pi$
$798$		0			0
$799$	−5.65685			−0.200125
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−16.9706	−	16.9706i	−0.598878	−	0.598878i
$804$		0			0
$805$	−32.0000	+	16.0000i	−1.12785	+	0.563926i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			32.0000i			1.12506i	0.826777	+	0.562530i	$0.190172\pi$
$809$			32.0000i			1.12506i	−0.826777	+	0.562530i	$0.809828\pi$
$810$		0			0
$811$			33.9411i			1.19183i	0.803046	+	0.595917i	$0.203211\pi$
$811$			33.9411i			1.19183i	−0.803046	+	0.595917i	$0.796789\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	25.4558	+	8.48528i	0.891679	+	0.297226i
$816$		0			0
$817$	−16.0000	−	16.0000i	−0.559769	−	0.559769i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−52.0000			−1.81481			−0.907406	−	0.420255i	$-0.861941\pi$
$821$	−52.0000			−1.81481			−0.907406	+	0.420255i	$0.861941\pi$
$822$		0			0
$823$	19.7990	+	19.7990i	0.690149	+	0.690149i	0.962265	−	0.272115i	$-0.0877232\pi$
$823$	19.7990	+	19.7990i	0.690149	+	0.690149i	−0.272115	+	0.962265i	$0.587723\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.82843	−	2.82843i	0.0983540	−	0.0983540i	−0.656218	−	0.754572i	$-0.727845\pi$
$827$	2.82843	−	2.82843i	0.0983540	−	0.0983540i	0.754572	+	0.656218i	$0.227845\pi$
$828$		0			0
$829$			38.0000i			1.31979i	0.751356	+	0.659897i	$0.229400\pi$
$829$			38.0000i			1.31979i	−0.751356	+	0.659897i	$0.770600\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.00000	−	9.00000i	0.311832	−	0.311832i
$834$		0			0
$835$	8.48528	−	25.4558i	0.293645	−	0.880936i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−11.3137			−0.390593			−0.195296	−	0.980744i	$-0.562567\pi$
$839$	−11.3137			−0.390593			−0.195296	+	0.980744i	$0.562567\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.00000	−	10.0000i	−0.172005	−	0.344010i
$846$		0			0
$847$	−59.3970	+	59.3970i	−2.04090	+	2.04090i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	28.2843i		−	0.969572i
$852$		0			0
$853$	−21.0000	+	21.0000i	−0.719026	+	0.719026i	−0.968406	−	0.249380i	$-0.919773\pi$
$853$	−21.0000	+	21.0000i	−0.719026	+	0.719026i	0.249380	+	0.968406i	$0.419773\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5.00000	−	5.00000i	−0.170797	−	0.170797i	0.616533	−	0.787329i	$-0.288537\pi$
$857$	−5.00000	−	5.00000i	−0.170797	−	0.170797i	−0.787329	+	0.616533i	$0.788537\pi$
$858$		0			0
$859$	−39.5980			−1.35107			−0.675533	−	0.737330i	$-0.736086\pi$
$859$	−39.5980			−1.35107			−0.675533	+	0.737330i	$0.736086\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−19.7990	−	19.7990i	−0.673965	−	0.673965i	0.284662	−	0.958628i	$-0.408119\pi$
$863$	−19.7990	−	19.7990i	−0.673965	−	0.673965i	−0.958628	+	0.284662i	$0.908119\pi$
$864$		0			0
$865$	−5.00000	+	15.0000i	−0.170005	+	0.510015i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	32.0000i		−	1.08553i
$870$		0			0
$871$			16.9706i			0.575026i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−25.4558	−	36.7696i	−0.860565	−	1.24304i
$876$		0			0
$877$	−29.0000	−	29.0000i	−0.979260	−	0.979260i	0.0205288	−	0.999789i	$-0.493465\pi$
$877$	−29.0000	−	29.0000i	−0.979260	−	0.979260i	−0.999789	+	0.0205288i	$0.993465\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−24.0000			−0.808581			−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000			−0.808581			−0.404290	+	0.914631i	$0.632481\pi$
$882$		0			0
$883$	−36.7696	−	36.7696i	−1.23739	−	1.23739i	−0.961062	−	0.276332i	$-0.910881\pi$
$883$	−36.7696	−	36.7696i	−1.23739	−	1.23739i	−0.276332	−	0.961062i	$-0.589119\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	25.4558	−	25.4558i	0.854724	−	0.854724i	−0.135987	−	0.990711i	$-0.543421\pi$
$887$	25.4558	−	25.4558i	0.854724	−	0.854724i	0.990711	+	0.135987i	$0.0434205\pi$
$888$		0			0
$889$		−	48.0000i		−	1.60987i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	16.0000	−	16.0000i	0.535420	−	0.535420i
$894$		0			0
$895$	−45.2548	+	22.6274i	−1.51270	+	0.756351i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−2.00000			−0.0666297
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−36.0000	+	18.0000i	−1.19668	+	0.598340i
$906$		0			0
$907$	31.1127	−	31.1127i	1.03308	−	1.03308i	0.0336464	−	0.999434i	$-0.489288\pi$
$907$	31.1127	−	31.1127i	1.03308	−	1.03308i	0.999434	−	0.0336464i	$-0.0107120\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	39.5980i		−	1.31194i	−0.754787	−	0.655970i	$-0.772260\pi$
$911$		−	39.5980i		−	1.31194i	0.754787	−	0.655970i	$-0.227740\pi$
$912$		0			0
$913$	16.0000	−	16.0000i	0.529523	−	0.529523i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	16.0000	+	16.0000i	0.528367	+	0.528367i
$918$		0			0
$919$	16.9706			0.559807			0.279904	−	0.960028i	$-0.409697\pi$
$919$	16.9706			0.559807			0.279904	+	0.960028i	$0.409697\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−16.9706	−	16.9706i	−0.558593	−	0.558593i
$924$		0			0
$925$	35.0000	−	5.00000i	1.15079	−	0.164399i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			30.0000i			0.984268i	0.870519	+	0.492134i	$0.163783\pi$
$929$			30.0000i			0.984268i	−0.870519	+	0.492134i	$0.836217\pi$
$930$		0			0
$931$			50.9117i			1.66856i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.65685	+	16.9706i	−0.184999	+	0.554997i
$936$		0			0
$937$	−5.00000	−	5.00000i	−0.163343	−	0.163343i	0.620703	−	0.784046i	$-0.286847\pi$
$937$	−5.00000	−	5.00000i	−0.163343	−	0.163343i	−0.784046	+	0.620703i	$0.786847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	42.0000			1.36916			0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000			1.36916			0.684580	+	0.728937i	$0.259985\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19.7990	−	19.7990i	0.643381	−	0.643381i	−0.308004	−	0.951385i	$-0.599661\pi$
$947$	19.7990	−	19.7990i	0.643381	−	0.643381i	0.951385	+	0.308004i	$0.0996611\pi$
$948$		0			0
$949$			18.0000i			0.584305i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	5.00000	−	5.00000i	0.161966	−	0.161966i	−0.621471	−	0.783437i	$-0.713465\pi$
$953$	5.00000	−	5.00000i	0.161966	−	0.161966i	0.783437	+	0.621471i	$0.213465\pi$
$954$		0			0
$955$	−16.9706	−	33.9411i	−0.549155	−	1.09831i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−28.2843			−0.913347
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.00000	+	15.0000i	−0.160956	+	0.482867i
$966$		0			0
$967$	2.82843	−	2.82843i	0.0909561	−	0.0909561i	−0.660165	−	0.751121i	$-0.729514\pi$
$967$	2.82843	−	2.82843i	0.0909561	−	0.0909561i	0.751121	+	0.660165i	$0.229514\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	16.9706i		−	0.544611i	−0.962211	−	0.272306i	$-0.912214\pi$
$971$		−	16.9706i		−	0.544611i	0.962211	−	0.272306i	$-0.0877862\pi$
$972$		0			0
$973$	48.0000	−	48.0000i	1.53881	−	1.53881i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25.0000	−	25.0000i	−0.799821	−	0.799821i	0.183246	−	0.983067i	$-0.441339\pi$
$977$	−25.0000	−	25.0000i	−0.799821	−	0.799821i	−0.983067	+	0.183246i	$0.941339\pi$
$978$		0			0
$979$	45.2548			1.44635
$980$		0			0
$981$		0			0
$982$		0			0
$983$	25.4558	+	25.4558i	0.811915	+	0.811915i	0.984921	−	0.173006i	$-0.0553478\pi$
$983$	25.4558	+	25.4558i	0.811915	+	0.811915i	−0.173006	+	0.984921i	$0.555348\pi$
$984$		0			0
$985$	33.0000	+	11.0000i	1.05147	+	0.350489i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	16.0000i		−	0.508770i
$990$		0			0
$991$			33.9411i			1.07818i	0.842250	+	0.539088i	$0.181231\pi$
$991$			33.9411i			1.07818i	−0.842250	+	0.539088i	$0.818769\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−33.9411	+	16.9706i	−1.07601	+	0.538003i
$996$		0			0
$997$	7.00000	+	7.00000i	0.221692	+	0.221692i	0.809211	−	0.587519i	$-0.199895\pi$
$997$	7.00000	+	7.00000i	0.221692	+	0.221692i	−0.587519	+	0.809211i	$0.699895\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.2.x.e.127.2		4
3.2	odd	2		240.2.w.a.127.1	✓	4
4.3	odd	2	inner	720.2.x.e.127.1		4
5.2	odd	4		3600.2.x.d.2143.2		4
5.3	odd	4	inner	720.2.x.e.703.1		4
5.4	even	2		3600.2.x.d.3007.1		4
12.11	even	2		240.2.w.a.127.2	yes	4
15.2	even	4		1200.2.w.a.943.1		4
15.8	even	4		240.2.w.a.223.2	yes	4
15.14	odd	2		1200.2.w.a.607.2		4
20.3	even	4	inner	720.2.x.e.703.2		4
20.7	even	4		3600.2.x.d.2143.1		4
20.19	odd	2		3600.2.x.d.3007.2		4
24.5	odd	2		960.2.w.c.127.2		4
24.11	even	2		960.2.w.c.127.1		4
60.23	odd	4		240.2.w.a.223.1	yes	4
60.47	odd	4		1200.2.w.a.943.2		4
60.59	even	2		1200.2.w.a.607.1		4
120.53	even	4		960.2.w.c.703.1		4
120.83	odd	4		960.2.w.c.703.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.w.a.127.1	✓	4	3.2	odd	2
240.2.w.a.127.2	yes	4	12.11	even	2
240.2.w.a.223.1	yes	4	60.23	odd	4
240.2.w.a.223.2	yes	4	15.8	even	4
720.2.x.e.127.1		4	4.3	odd	2	inner
720.2.x.e.127.2		4	1.1	even	1	trivial
720.2.x.e.703.1		4	5.3	odd	4	inner
720.2.x.e.703.2		4	20.3	even	4	inner
960.2.w.c.127.1		4	24.11	even	2
960.2.w.c.127.2		4	24.5	odd	2
960.2.w.c.703.1		4	120.53	even	4
960.2.w.c.703.2		4	120.83	odd	4
1200.2.w.a.607.1		4	60.59	even	2
1200.2.w.a.607.2		4	15.14	odd	2
1200.2.w.a.943.1		4	15.2	even	4
1200.2.w.a.943.2		4	60.47	odd	4
3600.2.x.d.2143.1		4	20.7	even	4
3600.2.x.d.2143.2		4	5.2	odd	4
3600.2.x.d.3007.1		4	5.4	even	2
3600.2.x.d.3007.2		4	20.19	odd	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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 1.00000i) q^{5} +(2.82843 - 2.82843i) q^{7} +5.65685i q^{11} +(3.00000 - 3.00000i) q^{13} +(1.00000 + 1.00000i) q^{17} -5.65685 q^{19} +(-2.82843 - 2.82843i) q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000i q^{29} +(2.82843 - 8.48528i) q^{35} +(5.00000 + 5.00000i) q^{37} +(2.82843 + 2.82843i) q^{43} +(-2.82843 + 2.82843i) q^{47} -9.00000i q^{49} +(-1.00000 + 1.00000i) q^{53} +(5.65685 + 11.3137i) q^{55} +11.3137 q^{59} +4.00000 q^{61} +(3.00000 - 9.00000i) q^{65} +(-2.82843 + 2.82843i) q^{67} -5.65685i q^{71} +(-3.00000 + 3.00000i) q^{73} +(16.0000 + 16.0000i) q^{77} -5.65685 q^{79} +(-2.82843 - 2.82843i) q^{83} +(3.00000 + 1.00000i) q^{85} -8.00000i q^{89} -16.9706i q^{91} +(-11.3137 + 5.65685i) q^{95} +(-3.00000 - 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} + 12 q^{13} + 4 q^{17} + 12 q^{25} + 20 q^{37} - 4 q^{53} + 16 q^{61} + 12 q^{65} - 12 q^{73} + 64 q^{77} + 12 q^{85} - 12 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 12 * q^13 + 4 * q^17 + 12 * q^25 + 20 * q^37 - 4 * q^53 + 16 * q^61 + 12 * q^65 - 12 * q^73 + 64 * q^77 + 12 * q^85 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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