LMFDB - Embedded newform 720.2.w.a.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,2,Mod(17,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))

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

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

chi := DirichletCharacter("720.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	720.w (of order $4$, degree $2$, not 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:	$5.74922894553$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	720.17
Dual form			720.2.w.a.593.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 2.12132i) q^{5} +(-2.00000 - 2.00000i) q^{7} +O(q^{10})$	$q+(0.707107 + 2.12132i) q^{5} +(-2.00000 - 2.00000i) q^{7} +2.82843i q^{11} +(-3.00000 + 3.00000i) q^{13} +(2.82843 - 2.82843i) q^{17} +8.00000i q^{19} +(2.82843 + 2.82843i) q^{23} +(-4.00000 + 3.00000i) q^{25} -1.41421 q^{29} -4.00000 q^{31} +(2.82843 - 5.65685i) q^{35} +(-3.00000 - 3.00000i) q^{37} +9.89949i q^{41} +1.00000i q^{49} +(2.82843 + 2.82843i) q^{53} +(-6.00000 + 2.00000i) q^{55} -2.82843 q^{59} +(-8.48528 - 4.24264i) q^{65} +(4.00000 + 4.00000i) q^{67} +5.65685i q^{71} +(1.00000 - 1.00000i) q^{73} +(5.65685 - 5.65685i) q^{77} -12.0000i q^{79} +(8.48528 + 8.48528i) q^{83} +(8.00000 + 4.00000i) q^{85} +9.89949 q^{89} +12.0000 q^{91} +(-16.9706 + 5.65685i) q^{95} +(-3.00000 - 3.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7}+O(q^{10}) $ 4 * q - 8 * q^7	$ 4 q - 8 q^{7} - 12 q^{13} - 16 q^{25} - 16 q^{31} - 12 q^{37} - 24 q^{55} + 16 q^{67} + 4 q^{73} + 32 q^{85} + 48 q^{91} - 12 q^{97}+O(q^{100}) $ 4 * q - 8 * q^7 - 12 * q^13 - 16 * q^25 - 16 * q^31 - 12 * q^37 - 24 * q^55 + 16 * q^67 + 4 * q^73 + 32 * q^85 + 48 * q^91 - 12 * q^97

Display 100 coefficients 10 coefficients

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
$5$	0.707107	+	2.12132i	0.316228	+	0.948683i
$5$	0.707107	+	2.12132i	0.316228	+	0.948683i
$6$		0			0
$7$	−2.00000	−	2.00000i	−0.755929	−	0.755929i	0.219650	−	0.975579i	$-0.429509\pi$
$7$	−2.00000	−	2.00000i	−0.755929	−	0.755929i	−0.975579	+	0.219650i	$0.929509\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.82843i			0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$			2.82843i			0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$	−3.00000	+	3.00000i	−0.832050	+	0.832050i	−0.987797	−	0.155747i	$-0.950222\pi$
$13$	−3.00000	+	3.00000i	−0.832050	+	0.832050i	0.155747	+	0.987797i	$0.450222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.82843	−	2.82843i	0.685994	−	0.685994i	−0.275350	−	0.961344i	$-0.588794\pi$
$17$	2.82843	−	2.82843i	0.685994	−	0.685994i	0.961344	+	0.275350i	$0.0887937\pi$
$18$		0			0
$19$			8.00000i			1.83533i	0.397360	+	0.917663i	$0.369927\pi$
$19$			8.00000i			1.83533i	−0.397360	+	0.917663i	$0.630073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.82843	+	2.82843i	0.589768	+	0.589768i	0.937568	−	0.347801i	$-0.113071\pi$
$23$	2.82843	+	2.82843i	0.589768	+	0.589768i	−0.347801	+	0.937568i	$0.613071\pi$
$24$		0			0
$25$	−4.00000	+	3.00000i	−0.800000	+	0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.41421			−0.262613			−0.131306	−	0.991342i	$-0.541917\pi$
$29$	−1.41421			−0.262613			−0.131306	+	0.991342i	$0.541917\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.82843	−	5.65685i	0.478091	−	0.956183i
$36$		0			0
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	0.416115	−	0.909312i	$-0.363391\pi$
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	−0.909312	+	0.416115i	$0.863391\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			9.89949i			1.54604i	0.634381	+	0.773021i	$0.281255\pi$
$41$			9.89949i			1.54604i	−0.634381	+	0.773021i	$0.718745\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.82843	+	2.82843i	0.388514	+	0.388514i	0.874157	−	0.485643i	$-0.161414\pi$
$53$	2.82843	+	2.82843i	0.388514	+	0.388514i	−0.485643	+	0.874157i	$0.661414\pi$
$54$		0			0
$55$	−6.00000	+	2.00000i	−0.809040	+	0.269680i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.82843			−0.368230			−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843			−0.368230			−0.184115	+	0.982905i	$0.558942\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.48528	−	4.24264i	−1.05247	−	0.526235i
$66$		0			0
$67$	4.00000	+	4.00000i	0.488678	+	0.488678i	0.907889	−	0.419211i	$-0.137693\pi$
$67$	4.00000	+	4.00000i	0.488678	+	0.488678i	−0.419211	+	0.907889i	$0.637693\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.65685i			0.671345i	0.941979	+	0.335673i	$0.108964\pi$
$71$			5.65685i			0.671345i	−0.941979	+	0.335673i	$0.891036\pi$
$72$		0			0
$73$	1.00000	−	1.00000i	0.117041	−	0.117041i	−0.646160	−	0.763202i	$-0.723626\pi$
$73$	1.00000	−	1.00000i	0.117041	−	0.117041i	0.763202	+	0.646160i	$0.223626\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.65685	−	5.65685i	0.644658	−	0.644658i
$78$		0			0
$79$		−	12.0000i		−	1.35011i	−0.737769	−	0.675053i	$-0.764121\pi$
$79$		−	12.0000i		−	1.35011i	0.737769	−	0.675053i	$-0.235879\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.48528	+	8.48528i	0.931381	+	0.931381i	0.997792	−	0.0664117i	$-0.0211551\pi$
$83$	8.48528	+	8.48528i	0.931381	+	0.931381i	−0.0664117	+	0.997792i	$0.521155\pi$
$84$		0			0
$85$	8.00000	+	4.00000i	0.867722	+	0.433861i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.89949			1.04934			0.524672	−	0.851304i	$-0.324188\pi$
$89$	9.89949			1.04934			0.524672	+	0.851304i	$0.324188\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−16.9706	+	5.65685i	−1.74114	+	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
$101$		−	1.41421i		−	0.140720i	−0.997522	−	0.0703598i	$-0.977585\pi$
$101$		−	1.41421i		−	0.140720i	0.997522	−	0.0703598i	$-0.0224147\pi$
$102$		0			0
$103$	2.00000	−	2.00000i	0.197066	−	0.197066i	−0.601675	−	0.798741i	$-0.705500\pi$
$103$	2.00000	−	2.00000i	0.197066	−	0.197066i	0.798741	+	0.601675i	$0.205500\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.1421	−	14.1421i	1.36717	−	1.36717i	0.502726	−	0.864446i	$-0.332330\pi$
$107$	14.1421	−	14.1421i	1.36717	−	1.36717i	0.864446	−	0.502726i	$-0.167670\pi$
$108$		0			0
$109$			8.00000i			0.766261i	0.923694	+	0.383131i	$0.125154\pi$
$109$			8.00000i			0.766261i	−0.923694	+	0.383131i	$0.874846\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.41421	−	1.41421i	−0.133038	−	0.133038i	0.637452	−	0.770490i	$-0.279988\pi$
$113$	−1.41421	−	1.41421i	−0.133038	−	0.133038i	−0.770490	+	0.637452i	$0.779988\pi$
$114$		0			0
$115$	−4.00000	+	8.00000i	−0.373002	+	0.746004i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−11.3137			−1.03713
$120$		0			0
$121$	3.00000			0.272727
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.19239	−	6.36396i	−0.822192	−	0.569210i
$126$		0			0
$127$	−14.0000	−	14.0000i	−1.24230	−	1.24230i	−0.959045	−	0.283254i	$-0.908586\pi$
$127$	−14.0000	−	14.0000i	−1.24230	−	1.24230i	−0.283254	−	0.959045i	$-0.591414\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	14.1421i		−	1.23560i	−0.786334	−	0.617802i	$-0.788023\pi$
$131$		−	14.1421i		−	1.23560i	0.786334	−	0.617802i	$-0.211977\pi$
$132$		0			0
$133$	16.0000	−	16.0000i	1.38738	−	1.38738i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.24264	+	4.24264i	−0.362473	+	0.362473i	−0.864723	−	0.502249i	$-0.832506\pi$
$137$	−4.24264	+	4.24264i	−0.362473	+	0.362473i	0.502249	+	0.864723i	$0.332506\pi$
$138$		0			0
$139$			4.00000i			0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$			4.00000i			0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.48528	−	8.48528i	−0.709575	−	0.709575i
$144$		0			0
$145$	−1.00000	−	3.00000i	−0.0830455	−	0.249136i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−21.2132			−1.73785			−0.868927	−	0.494941i	$-0.835190\pi$
$149$	−21.2132			−1.73785			−0.868927	+	0.494941i	$0.835190\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.82843	−	8.48528i	−0.227185	−	0.681554i
$156$		0			0
$157$	−9.00000	−	9.00000i	−0.718278	−	0.718278i	0.249974	−	0.968252i	$-0.419578\pi$
$157$	−9.00000	−	9.00000i	−0.718278	−	0.718278i	−0.968252	+	0.249974i	$0.919578\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	11.3137i		−	0.891645i
$162$		0			0
$163$	16.0000	−	16.0000i	1.25322	−	1.25322i	0.298947	−	0.954270i	$-0.403365\pi$
$163$	16.0000	−	16.0000i	1.25322	−	1.25322i	0.954270	−	0.298947i	$-0.0966354\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.82843	−	2.82843i	0.218870	−	0.218870i	−0.589152	−	0.808022i	$-0.700538\pi$
$167$	2.82843	−	2.82843i	0.218870	−	0.218870i	0.808022	+	0.589152i	$0.200538\pi$
$168$		0			0
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.89949	+	9.89949i	0.752645	+	0.752645i	0.974972	−	0.222327i	$-0.0713654\pi$
$173$	9.89949	+	9.89949i	0.752645	+	0.752645i	−0.222327	+	0.974972i	$0.571365\pi$
$174$		0			0
$175$	14.0000	+	2.00000i	1.05830	+	0.151186i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.82843			0.211407			0.105703	−	0.994398i	$-0.466291\pi$
$179$	2.82843			0.211407			0.105703	+	0.994398i	$0.466291\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.24264	−	8.48528i	0.311925	−	0.623850i
$186$		0			0
$187$	8.00000	+	8.00000i	0.585018	+	0.585018i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			22.6274i			1.63726i	0.574320	+	0.818631i	$0.305267\pi$
$191$			22.6274i			1.63726i	−0.574320	+	0.818631i	$0.694733\pi$
$192$		0			0
$193$	9.00000	−	9.00000i	0.647834	−	0.647834i	−0.304635	−	0.952469i	$-0.598534\pi$
$193$	9.00000	−	9.00000i	0.647834	−	0.647834i	0.952469	+	0.304635i	$0.0985345\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.7279	−	12.7279i	0.906827	−	0.906827i	−0.0891879	−	0.996015i	$-0.528427\pi$
$197$	12.7279	−	12.7279i	0.906827	−	0.906827i	0.996015	+	0.0891879i	$0.0284272\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.82843	+	2.82843i	0.198517	+	0.198517i
$204$		0			0
$205$	−21.0000	+	7.00000i	−1.46670	+	0.488901i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−22.6274			−1.56517
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	8.00000	+	8.00000i	0.543075	+	0.543075i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			16.9706i			1.14156i
$222$		0			0
$223$	−6.00000	+	6.00000i	−0.401790	+	0.401790i	−0.878863	−	0.477074i	$-0.841698\pi$
$223$	−6.00000	+	6.00000i	−0.401790	+	0.401790i	0.477074	+	0.878863i	$0.341698\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.65685	+	5.65685i	−0.375459	+	0.375459i	−0.869461	−	0.494002i	$-0.835534\pi$
$227$	−5.65685	+	5.65685i	−0.375459	+	0.375459i	0.494002	+	0.869461i	$0.335534\pi$
$228$		0			0
$229$		−	6.00000i		−	0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$		−	6.00000i		−	0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.48528	+	8.48528i	0.555889	+	0.555889i	0.928134	−	0.372245i	$-0.121412\pi$
$233$	8.48528	+	8.48528i	0.555889	+	0.555889i	−0.372245	+	0.928134i	$0.621412\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	16.9706			1.09773			0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706			1.09773			0.548867	+	0.835910i	$0.315059\pi$
$240$		0			0
$241$	22.0000			1.41714			0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000			1.41714			0.708572	+	0.705638i	$0.249340\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.12132	+	0.707107i	−0.135526	+	0.0451754i
$246$		0			0
$247$	−24.0000	−	24.0000i	−1.52708	−	1.52708i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			8.48528i			0.535586i	0.963476	+	0.267793i	$0.0862944\pi$
$251$			8.48528i			0.535586i	−0.963476	+	0.267793i	$0.913706\pi$
$252$		0			0
$253$	−8.00000	+	8.00000i	−0.502956	+	0.502956i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.89949	−	9.89949i	0.617514	−	0.617514i	−0.327379	−	0.944893i	$-0.606166\pi$
$257$	9.89949	−	9.89949i	0.617514	−	0.617514i	0.944893	+	0.327379i	$0.106166\pi$
$258$		0			0
$259$			12.0000i			0.745644i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−19.7990	−	19.7990i	−1.22086	−	1.22086i	−0.967327	−	0.253531i	$-0.918408\pi$
$263$	−19.7990	−	19.7990i	−1.22086	−	1.22086i	−0.253531	−	0.967327i	$-0.581592\pi$
$264$		0			0
$265$	−4.00000	+	8.00000i	−0.245718	+	0.491436i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.5563			0.948487			0.474244	−	0.880394i	$-0.342722\pi$
$269$	15.5563			0.948487			0.474244	+	0.880394i	$0.342722\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.48528	−	11.3137i	−0.511682	−	0.682242i
$276$		0			0
$277$	9.00000	+	9.00000i	0.540758	+	0.540758i	0.923751	−	0.382993i	$-0.125107\pi$
$277$	9.00000	+	9.00000i	0.540758	+	0.540758i	−0.382993	+	0.923751i	$0.625107\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			12.7279i			0.759284i	0.925133	+	0.379642i	$0.123953\pi$
$281$			12.7279i			0.759284i	−0.925133	+	0.379642i	$0.876047\pi$
$282$		0			0
$283$	−8.00000	+	8.00000i	−0.475551	+	0.475551i	−0.903705	−	0.428155i	$-0.859164\pi$
$283$	−8.00000	+	8.00000i	−0.475551	+	0.475551i	0.428155	+	0.903705i	$0.359164\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	19.7990	−	19.7990i	1.16870	−	1.16870i
$288$		0			0
$289$			1.00000i			0.0588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.7279	−	12.7279i	−0.743573	−	0.743573i	0.229691	−	0.973264i	$-0.426229\pi$
$293$	−12.7279	−	12.7279i	−0.743573	−	0.743573i	−0.973264	+	0.229691i	$0.926229\pi$
$294$		0			0
$295$	−2.00000	−	6.00000i	−0.116445	−	0.349334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.9706			−0.981433
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			22.6274i			1.28308i	0.767088	+	0.641542i	$0.221705\pi$
$311$			22.6274i			1.28308i	−0.767088	+	0.641542i	$0.778295\pi$
$312$		0			0
$313$	−21.0000	+	21.0000i	−1.18699	+	1.18699i	−0.209095	+	0.977895i	$0.567052\pi$
$313$	−21.0000	+	21.0000i	−1.18699	+	1.18699i	−0.977895	+	0.209095i	$0.932948\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.48528	+	8.48528i	−0.476581	+	0.476581i	−0.904036	−	0.427456i	$-0.859410\pi$
$317$	−8.48528	+	8.48528i	−0.476581	+	0.476581i	0.427456	+	0.904036i	$0.359410\pi$
$318$		0			0
$319$		−	4.00000i		−	0.223957i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	22.6274	+	22.6274i	1.25902	+	1.25902i
$324$		0			0
$325$	3.00000	−	21.0000i	0.166410	−	1.16487i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−16.0000			−0.879440			−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000			−0.879440			−0.439720	+	0.898135i	$0.644922\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.65685	+	11.3137i	−0.309067	+	0.618134i
$336$		0			0
$337$	−21.0000	−	21.0000i	−1.14394	−	1.14394i	−0.987722	−	0.156221i	$-0.950069\pi$
$337$	−21.0000	−	21.0000i	−1.14394	−	1.14394i	−0.156221	−	0.987722i	$-0.549931\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	11.3137i		−	0.612672i
$342$		0			0
$343$	−12.0000	+	12.0000i	−0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.3137	+	11.3137i	−0.607352	+	0.607352i	−0.942253	−	0.334901i	$-0.891297\pi$
$347$	−11.3137	+	11.3137i	−0.607352	+	0.607352i	0.334901	+	0.942253i	$0.391297\pi$
$348$		0			0
$349$			16.0000i			0.856460i	0.903670	+	0.428230i	$0.140863\pi$
$349$			16.0000i			0.856460i	−0.903670	+	0.428230i	$0.859137\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.48528	+	8.48528i	0.451626	+	0.451626i	0.895894	−	0.444268i	$-0.146536\pi$
$353$	8.48528	+	8.48528i	0.451626	+	0.451626i	−0.444268	+	0.895894i	$0.646536\pi$
$354$		0			0
$355$	−12.0000	+	4.00000i	−0.636894	+	0.212298i
$356$		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
$361$	−45.0000			−2.36842
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.82843	+	1.41421i	0.148047	+	0.0740233i
$366$		0			0
$367$	22.0000	+	22.0000i	1.14839	+	1.14839i	0.986869	+	0.161521i	$0.0516401\pi$
$367$	22.0000	+	22.0000i	1.14839	+	1.14839i	0.161521	+	0.986869i	$0.448360\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	11.3137i		−	0.587378i
$372$		0			0
$373$	−5.00000	+	5.00000i	−0.258890	+	0.258890i	−0.824603	−	0.565712i	$-0.808601\pi$
$373$	−5.00000	+	5.00000i	−0.258890	+	0.258890i	0.565712	+	0.824603i	$0.308601\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.24264	−	4.24264i	0.218507	−	0.218507i
$378$		0			0
$379$			20.0000i			1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$			20.0000i			1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	16.9706	+	16.9706i	0.867155	+	0.867155i	0.992157	−	0.125001i	$-0.0398935\pi$
$383$	16.9706	+	16.9706i	0.867155	+	0.867155i	−0.125001	+	0.992157i	$0.539894\pi$
$384$		0			0
$385$	16.0000	+	8.00000i	0.815436	+	0.407718i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	21.2132			1.07555			0.537776	−	0.843088i	$-0.319265\pi$
$389$	21.2132			1.07555			0.537776	+	0.843088i	$0.319265\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$	25.4558	−	8.48528i	1.28082	−	0.426941i
$396$		0			0
$397$	7.00000	+	7.00000i	0.351320	+	0.351320i	0.860601	−	0.509281i	$-0.170088\pi$
$397$	7.00000	+	7.00000i	0.351320	+	0.351320i	−0.509281	+	0.860601i	$0.670088\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			12.7279i			0.635602i	0.948157	+	0.317801i	$0.102944\pi$
$401$			12.7279i			0.635602i	−0.948157	+	0.317801i	$0.897056\pi$
$402$		0			0
$403$	12.0000	−	12.0000i	0.597763	−	0.597763i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.48528	−	8.48528i	0.420600	−	0.420600i
$408$		0			0
$409$			24.0000i			1.18672i	0.804936	+	0.593362i	$0.202200\pi$
$409$			24.0000i			1.18672i	−0.804936	+	0.593362i	$0.797800\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.65685	+	5.65685i	0.278356	+	0.278356i
$414$		0			0
$415$	−12.0000	+	24.0000i	−0.589057	+	1.17811i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−25.4558			−1.24360			−0.621800	−	0.783176i	$-0.713598\pi$
$419$	−25.4558			−1.24360			−0.621800	+	0.783176i	$0.713598\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.82843	+	19.7990i	−0.137199	+	0.960392i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	39.5980i		−	1.90737i	−0.300811	−	0.953684i	$-0.597257\pi$
$431$		−	39.5980i		−	1.90737i	0.300811	−	0.953684i	$-0.402743\pi$
$432$		0			0
$433$	23.0000	−	23.0000i	1.10531	−	1.10531i	0.111551	−	0.993759i	$-0.464418\pi$
$433$	23.0000	−	23.0000i	1.10531	−	1.10531i	0.993759	−	0.111551i	$-0.0355818\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−22.6274	+	22.6274i	−1.08242	+	1.08242i
$438$		0			0
$439$		−	8.00000i		−	0.381819i	−0.981608	−	0.190910i	$-0.938856\pi$
$439$		−	8.00000i		−	0.381819i	0.981608	−	0.190910i	$-0.0611437\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.9706	+	16.9706i	0.806296	+	0.806296i	0.984071	−	0.177775i	$-0.0568900\pi$
$443$	16.9706	+	16.9706i	0.806296	+	0.806296i	−0.177775	+	0.984071i	$0.556890\pi$
$444$		0			0
$445$	7.00000	+	21.0000i	0.331832	+	0.995495i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.41421			−0.0667409			−0.0333704	−	0.999443i	$-0.510624\pi$
$449$	−1.41421			−0.0667409			−0.0333704	+	0.999443i	$0.510624\pi$
$450$		0			0
$451$	−28.0000			−1.31847
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.48528	+	25.4558i	0.397796	+	1.19339i
$456$		0			0
$457$	−15.0000	−	15.0000i	−0.701670	−	0.701670i	0.263099	−	0.964769i	$-0.415256\pi$
$457$	−15.0000	−	15.0000i	−0.701670	−	0.701670i	−0.964769	+	0.263099i	$0.915256\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			18.3848i			0.856264i	0.903716	+	0.428132i	$0.140828\pi$
$461$			18.3848i			0.856264i	−0.903716	+	0.428132i	$0.859172\pi$
$462$		0			0
$463$	−14.0000	+	14.0000i	−0.650635	+	0.650635i	−0.953146	−	0.302511i	$-0.902175\pi$
$463$	−14.0000	+	14.0000i	−0.650635	+	0.650635i	0.302511	+	0.953146i	$0.402175\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19.7990	+	19.7990i	−0.916188	+	0.916188i	−0.996750	−	0.0805616i	$-0.974329\pi$
$467$	−19.7990	+	19.7990i	−0.916188	+	0.916188i	0.0805616	+	0.996750i	$0.474329\pi$
$468$		0			0
$469$		−	16.0000i		−	0.738811i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−24.0000	−	32.0000i	−1.10120	−	1.46826i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.65685			0.258468			0.129234	−	0.991614i	$-0.458748\pi$
$479$	5.65685			0.258468			0.129234	+	0.991614i	$0.458748\pi$
$480$		0			0
$481$	18.0000			0.820729
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.24264	−	8.48528i	0.192648	−	0.385297i
$486$		0			0
$487$	18.0000	+	18.0000i	0.815658	+	0.815658i	0.985476	−	0.169818i	$-0.0543179\pi$
$487$	18.0000	+	18.0000i	0.815658	+	0.815658i	−0.169818	+	0.985476i	$0.554318\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	2.82843i		−	0.127645i	−0.997961	−	0.0638226i	$-0.979671\pi$
$491$		−	2.82843i		−	0.127645i	0.997961	−	0.0638226i	$-0.0203292\pi$
$492$		0			0
$493$	−4.00000	+	4.00000i	−0.180151	+	0.180151i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	11.3137	−	11.3137i	0.507489	−	0.507489i
$498$		0			0
$499$		−	32.0000i		−	1.43252i	−0.697835	−	0.716258i	$-0.745853\pi$
$499$		−	32.0000i		−	1.43252i	0.697835	−	0.716258i	$-0.254147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16.9706	−	16.9706i	−0.756680	−	0.756680i	0.219037	−	0.975717i	$-0.429709\pi$
$503$	−16.9706	−	16.9706i	−0.756680	−	0.756680i	−0.975717	+	0.219037i	$0.929709\pi$
$504$		0			0
$505$	3.00000	−	1.00000i	0.133498	−	0.0444994i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	24.0416			1.06563			0.532813	−	0.846233i	$-0.321135\pi$
$509$	24.0416			1.06563			0.532813	+	0.846233i	$0.321135\pi$
$510$		0			0
$511$	−4.00000			−0.176950
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.65685	+	2.82843i	0.249271	+	0.124635i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	29.6985i		−	1.30111i	−0.759457	−	0.650557i	$-0.774535\pi$
$521$		−	29.6985i		−	1.30111i	0.759457	−	0.650557i	$-0.225465\pi$
$522$		0			0
$523$	24.0000	−	24.0000i	1.04945	−	1.04945i	0.0507346	−	0.998712i	$-0.483844\pi$
$523$	24.0000	−	24.0000i	1.04945	−	1.04945i	0.998712	−	0.0507346i	$-0.0161562\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11.3137	+	11.3137i	−0.492833	+	0.492833i
$528$		0			0
$529$		−	7.00000i		−	0.304348i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−29.6985	−	29.6985i	−1.28638	−	1.28638i
$534$		0			0
$535$	40.0000	+	20.0000i	1.72935	+	0.864675i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.82843			−0.121829
$540$		0			0
$541$	−8.00000			−0.343947			−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000			−0.343947			−0.171973	+	0.985102i	$0.555014\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−16.9706	+	5.65685i	−0.726939	+	0.242313i
$546$		0			0
$547$	−12.0000	−	12.0000i	−0.513083	−	0.513083i	0.402387	−	0.915470i	$-0.368181\pi$
$547$	−12.0000	−	12.0000i	−0.513083	−	0.513083i	−0.915470	+	0.402387i	$0.868181\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	11.3137i		−	0.481980i
$552$		0			0
$553$	−24.0000	+	24.0000i	−1.02058	+	1.02058i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8.48528	−	8.48528i	0.359533	−	0.359533i	−0.504108	−	0.863641i	$-0.668179\pi$
$557$	8.48528	−	8.48528i	0.359533	−	0.359533i	0.863641	+	0.504108i	$0.168179\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16.9706	−	16.9706i	−0.715224	−	0.715224i	0.252399	−	0.967623i	$-0.418780\pi$
$563$	−16.9706	−	16.9706i	−0.715224	−	0.715224i	−0.967623	+	0.252399i	$0.918780\pi$
$564$		0			0
$565$	2.00000	−	4.00000i	0.0841406	−	0.168281i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.3848			0.770730			0.385365	−	0.922764i	$-0.374076\pi$
$569$	18.3848			0.770730			0.385365	+	0.922764i	$0.374076\pi$
$570$		0			0
$571$	−8.00000			−0.334790			−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000			−0.334790			−0.167395	+	0.985890i	$0.553535\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−19.7990	−	2.82843i	−0.825675	−	0.117954i
$576$		0			0
$577$	31.0000	+	31.0000i	1.29055	+	1.29055i	0.934448	+	0.356098i	$0.115893\pi$
$577$	31.0000	+	31.0000i	1.29055	+	1.29055i	0.356098	+	0.934448i	$0.384107\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	33.9411i		−	1.40812i
$582$		0			0
$583$	−8.00000	+	8.00000i	−0.331326	+	0.331326i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−25.4558	+	25.4558i	−1.05068	+	1.05068i	−0.0520296	+	0.998646i	$0.516569\pi$
$587$	−25.4558	+	25.4558i	−1.05068	+	1.05068i	−0.998646	+	0.0520296i	$0.983431\pi$
$588$		0			0
$589$		−	32.0000i		−	1.31854i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24.0416	−	24.0416i	−0.987271	−	0.987271i	0.0126486	−	0.999920i	$-0.495974\pi$
$593$	−24.0416	−	24.0416i	−0.987271	−	0.987271i	−0.999920	+	0.0126486i	$0.995974\pi$
$594$		0			0
$595$	−8.00000	−	24.0000i	−0.327968	−	0.983904i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	39.5980			1.61793			0.808965	−	0.587857i	$-0.200028\pi$
$599$	39.5980			1.61793			0.808965	+	0.587857i	$0.200028\pi$
$600$		0			0
$601$	8.00000			0.326327			0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000			0.326327			0.163163	+	0.986599i	$0.447830\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.12132	+	6.36396i	0.0862439	+	0.258732i
$606$		0			0
$607$	6.00000	+	6.00000i	0.243532	+	0.243532i	0.818310	−	0.574777i	$-0.194911\pi$
$607$	6.00000	+	6.00000i	0.243532	+	0.243532i	−0.574777	+	0.818310i	$0.694911\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−3.00000	+	3.00000i	−0.121169	+	0.121169i	−0.765091	−	0.643922i	$-0.777306\pi$
$613$	−3.00000	+	3.00000i	−0.121169	+	0.121169i	0.643922	+	0.765091i	$0.277306\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−25.4558	+	25.4558i	−1.02481	+	1.02481i	−0.0251295	+	0.999684i	$0.508000\pi$
$617$	−25.4558	+	25.4558i	−1.02481	+	1.02481i	−0.999684	+	0.0251295i	$0.992000\pi$
$618$		0			0
$619$			12.0000i			0.482321i	0.970485	+	0.241160i	$0.0775280\pi$
$619$			12.0000i			0.482321i	−0.970485	+	0.241160i	$0.922472\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−19.7990	−	19.7990i	−0.793230	−	0.793230i
$624$		0			0
$625$	7.00000	−	24.0000i	0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−16.9706			−0.676661
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	19.7990	−	39.5980i	0.785699	−	1.57140i
$636$		0			0
$637$	−3.00000	−	3.00000i	−0.118864	−	0.118864i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	18.3848i		−	0.726155i	−0.931759	−	0.363078i	$-0.881726\pi$
$641$		−	18.3848i		−	0.726155i	0.931759	−	0.363078i	$-0.118274\pi$
$642$		0			0
$643$	−12.0000	+	12.0000i	−0.473234	+	0.473234i	−0.902959	−	0.429726i	$-0.858610\pi$
$643$	−12.0000	+	12.0000i	−0.473234	+	0.473234i	0.429726	+	0.902959i	$0.358610\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11.3137	−	11.3137i	0.444788	−	0.444788i	−0.448830	−	0.893617i	$-0.648159\pi$
$647$	11.3137	−	11.3137i	0.444788	−	0.444788i	0.893617	+	0.448830i	$0.148159\pi$
$648$		0			0
$649$		−	8.00000i		−	0.314027i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.4558	+	25.4558i	0.996164	+	0.996164i	0.999993	−	0.00382851i	$-0.00121866\pi$
$653$	25.4558	+	25.4558i	0.996164	+	0.996164i	−0.00382851	+	0.999993i	$0.501219\pi$
$654$		0			0
$655$	30.0000	−	10.0000i	1.17220	−	0.390732i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.82843			0.110180			0.0550899	−	0.998481i	$-0.482455\pi$
$659$	2.82843			0.110180			0.0550899	+	0.998481i	$0.482455\pi$
$660$		0			0
$661$	32.0000			1.24466			0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000			1.24466			0.622328	+	0.782757i	$0.286187\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	45.2548	+	22.6274i	1.75491	+	0.877454i
$666$		0			0
$667$	−4.00000	−	4.00000i	−0.154881	−	0.154881i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	25.0000	−	25.0000i	0.963679	−	0.963679i	−0.0356839	−	0.999363i	$-0.511361\pi$
$673$	25.0000	−	25.0000i	0.963679	−	0.963679i	0.999363	+	0.0356839i	$0.0113610\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14.1421	+	14.1421i	−0.543526	+	0.543526i	−0.924561	−	0.381034i	$-0.875568\pi$
$677$	−14.1421	+	14.1421i	−0.543526	+	0.543526i	0.381034	+	0.924561i	$0.375568\pi$
$678$		0			0
$679$			12.0000i			0.460518i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8.48528	+	8.48528i	0.324680	+	0.324680i	0.850559	−	0.525879i	$-0.176264\pi$
$683$	8.48528	+	8.48528i	0.324680	+	0.324680i	−0.525879	+	0.850559i	$0.676264\pi$
$684$		0			0
$685$	−12.0000	−	6.00000i	−0.458496	−	0.229248i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−16.9706			−0.646527
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.48528	+	2.82843i	−0.321865	+	0.107288i
$696$		0			0
$697$	28.0000	+	28.0000i	1.06058	+	1.06058i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			35.3553i			1.33535i	0.744452	+	0.667676i	$0.232711\pi$
$701$			35.3553i			1.33535i	−0.744452	+	0.667676i	$0.767289\pi$
$702$		0			0
$703$	24.0000	−	24.0000i	0.905177	−	0.905177i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.82843	+	2.82843i	−0.106374	+	0.106374i
$708$		0			0
$709$		−	26.0000i		−	0.976450i	−0.872718	−	0.488225i	$-0.837644\pi$
$709$		−	26.0000i		−	0.976450i	0.872718	−	0.488225i	$-0.162356\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−11.3137	−	11.3137i	−0.423702	−	0.423702i
$714$		0			0
$715$	12.0000	−	24.0000i	0.448775	−	0.897549i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22.6274			0.843860			0.421930	−	0.906628i	$-0.361353\pi$
$719$	22.6274			0.843860			0.421930	+	0.906628i	$0.361353\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.65685	−	4.24264i	0.210090	−	0.157568i
$726$		0			0
$727$	6.00000	+	6.00000i	0.222528	+	0.222528i	0.809562	−	0.587034i	$-0.199705\pi$
$727$	6.00000	+	6.00000i	0.222528	+	0.222528i	−0.587034	+	0.809562i	$0.699705\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−11.0000	+	11.0000i	−0.406294	+	0.406294i	−0.880444	−	0.474150i	$-0.842755\pi$
$733$	−11.0000	+	11.0000i	−0.406294	+	0.406294i	0.474150	+	0.880444i	$0.342755\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.3137	+	11.3137i	−0.416746	+	0.416746i
$738$		0			0
$739$		−	8.00000i		−	0.294285i	−0.989115	−	0.147142i	$-0.952992\pi$
$739$		−	8.00000i		−	0.294285i	0.989115	−	0.147142i	$-0.0470076\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−28.2843	−	28.2843i	−1.03765	−	1.03765i	−0.999263	−	0.0383863i	$-0.987778\pi$
$743$	−28.2843	−	28.2843i	−1.03765	−	1.03765i	−0.0383863	−	0.999263i	$-0.512222\pi$
$744$		0			0
$745$	−15.0000	−	45.0000i	−0.549557	−	1.64867i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−56.5685			−2.06697
$750$		0			0
$751$	12.0000			0.437886			0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000			0.437886			0.218943	+	0.975738i	$0.429739\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.65685	+	16.9706i	0.205874	+	0.617622i
$756$		0			0
$757$	15.0000	+	15.0000i	0.545184	+	0.545184i	0.925044	−	0.379860i	$-0.124028\pi$
$757$	15.0000	+	15.0000i	0.545184	+	0.545184i	−0.379860	+	0.925044i	$0.624028\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	41.0122i		−	1.48669i	−0.668908	−	0.743345i	$-0.733238\pi$
$761$		−	41.0122i		−	1.48669i	0.668908	−	0.743345i	$-0.266762\pi$
$762$		0			0
$763$	16.0000	−	16.0000i	0.579239	−	0.579239i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.48528	−	8.48528i	0.306386	−	0.306386i
$768$		0			0
$769$		−	40.0000i		−	1.44244i	−0.692708	−	0.721218i	$-0.743582\pi$
$769$		−	40.0000i		−	1.44244i	0.692708	−	0.721218i	$-0.256418\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.48528	−	8.48528i	−0.305194	−	0.305194i	0.537848	−	0.843042i	$-0.319238\pi$
$773$	−8.48528	−	8.48528i	−0.305194	−	0.305194i	−0.843042	+	0.537848i	$0.819238\pi$
$774$		0			0
$775$	16.0000	−	12.0000i	0.574737	−	0.431053i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−79.1960			−2.83749
$780$		0			0
$781$	−16.0000			−0.572525
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.7279	−	25.4558i	0.454279	−	0.908558i
$786$		0			0
$787$	20.0000	+	20.0000i	0.712923	+	0.712923i	0.967146	−	0.254223i	$-0.0818196\pi$
$787$	20.0000	+	20.0000i	0.712923	+	0.712923i	−0.254223	+	0.967146i	$0.581820\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			5.65685i			0.201135i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21.2132	−	21.2132i	0.751410	−	0.751410i	−0.223332	−	0.974742i	$-0.571693\pi$
$797$	21.2132	−	21.2132i	0.751410	−	0.751410i	0.974742	+	0.223332i	$0.0716935\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.82843	+	2.82843i	0.0998130	+	0.0998130i
$804$		0			0
$805$	24.0000	−	8.00000i	0.845889	−	0.281963i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.41421			0.0497211			0.0248606	−	0.999691i	$-0.492086\pi$
$809$	1.41421			0.0497211			0.0248606	+	0.999691i	$0.492086\pi$
$810$		0			0
$811$	−20.0000			−0.702295			−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000			−0.702295			−0.351147	+	0.936320i	$0.614208\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	45.2548	+	22.6274i	1.58521	+	0.792604i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	41.0122i		−	1.43134i	−0.698441	−	0.715668i	$-0.746123\pi$
$821$		−	41.0122i		−	1.43134i	0.698441	−	0.715668i	$-0.253877\pi$
$822$		0			0
$823$	−38.0000	+	38.0000i	−1.32460	+	1.32460i	−0.414587	+	0.910010i	$0.636074\pi$
$823$	−38.0000	+	38.0000i	−1.32460	+	1.32460i	−0.910010	+	0.414587i	$0.863926\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−33.9411	+	33.9411i	−1.18025	+	1.18025i	−0.200569	+	0.979680i	$0.564279\pi$
$827$	−33.9411	+	33.9411i	−1.18025	+	1.18025i	−0.979680	+	0.200569i	$0.935721\pi$
$828$		0			0
$829$			56.0000i			1.94496i	0.232986	+	0.972480i	$0.425151\pi$
$829$			56.0000i			1.94496i	−0.232986	+	0.972480i	$0.574849\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.82843	+	2.82843i	0.0979992	+	0.0979992i
$834$		0			0
$835$	8.00000	+	4.00000i	0.276851	+	0.138426i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	45.2548			1.56237			0.781185	−	0.624299i	$-0.214615\pi$
$839$	45.2548			1.56237			0.781185	+	0.624299i	$0.214615\pi$
$840$		0			0
$841$	−27.0000			−0.931034
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.6066	−	3.53553i	0.364878	−	0.121626i
$846$		0			0
$847$	−6.00000	−	6.00000i	−0.206162	−	0.206162i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	16.9706i		−	0.581743i
$852$		0			0
$853$	19.0000	−	19.0000i	0.650548	−	0.650548i	−0.302577	−	0.953125i	$-0.597847\pi$
$853$	19.0000	−	19.0000i	0.650548	−	0.650548i	0.953125	+	0.302577i	$0.0978470\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.48528	+	8.48528i	−0.289852	+	0.289852i	−0.837022	−	0.547170i	$-0.815705\pi$
$857$	−8.48528	+	8.48528i	−0.289852	+	0.289852i	0.547170	+	0.837022i	$0.315705\pi$
$858$		0			0
$859$		−	44.0000i		−	1.50126i	−0.660722	−	0.750630i	$-0.729750\pi$
$859$		−	44.0000i		−	1.50126i	0.660722	−	0.750630i	$-0.270250\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−31.1127	−	31.1127i	−1.05909	−	1.05909i	−0.998141	−	0.0609476i	$-0.980588\pi$
$863$	−31.1127	−	31.1127i	−1.05909	−	1.05909i	−0.0609476	−	0.998141i	$-0.519412\pi$
$864$		0			0
$865$	−14.0000	+	28.0000i	−0.476014	+	0.952029i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	33.9411			1.15137
$870$		0			0
$871$	−24.0000			−0.813209
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5.65685	+	31.1127i	0.191237	+	1.05180i
$876$		0			0
$877$	−15.0000	−	15.0000i	−0.506514	−	0.506514i	0.406941	−	0.913455i	$-0.366596\pi$
$877$	−15.0000	−	15.0000i	−0.506514	−	0.506514i	−0.913455	+	0.406941i	$0.866596\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			52.3259i			1.76290i	0.472273	+	0.881452i	$0.343434\pi$
$881$			52.3259i			1.76290i	−0.472273	+	0.881452i	$0.656566\pi$
$882$		0			0
$883$	−16.0000	+	16.0000i	−0.538443	+	0.538443i	−0.923071	−	0.384629i	$-0.874330\pi$
$883$	−16.0000	+	16.0000i	−0.538443	+	0.538443i	0.384629	+	0.923071i	$0.374330\pi$
$884$		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
$889$			56.0000i			1.87818i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	2.00000	+	6.00000i	0.0668526	+	0.200558i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.65685			0.188667
$900$		0			0
$901$	16.0000			0.533037
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.65685	+	16.9706i	0.188040	+	0.564121i
$906$		0			0
$907$	−4.00000	−	4.00000i	−0.132818	−	0.132818i	0.637573	−	0.770390i	$-0.279939\pi$
$907$	−4.00000	−	4.00000i	−0.132818	−	0.132818i	−0.770390	+	0.637573i	$0.779939\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	11.3137i		−	0.374840i	−0.982280	−	0.187420i	$-0.939987\pi$
$911$		−	11.3137i		−	0.374840i	0.982280	−	0.187420i	$-0.0600125\pi$
$912$		0			0
$913$	−24.0000	+	24.0000i	−0.794284	+	0.794284i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−28.2843	+	28.2843i	−0.934029	+	0.934029i
$918$		0			0
$919$		−	20.0000i		−	0.659739i	−0.944027	−	0.329870i	$-0.892995\pi$
$919$		−	20.0000i		−	0.659739i	0.944027	−	0.329870i	$-0.107005\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−16.9706	−	16.9706i	−0.558593	−	0.558593i
$924$		0			0
$925$	21.0000	+	3.00000i	0.690476	+	0.0986394i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	32.5269			1.06717			0.533587	−	0.845745i	$-0.320844\pi$
$929$	32.5269			1.06717			0.533587	+	0.845745i	$0.320844\pi$
$930$		0			0
$931$	−8.00000			−0.262189
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−11.3137	+	22.6274i	−0.369998	+	0.739996i
$936$		0			0
$937$	−11.0000	−	11.0000i	−0.359354	−	0.359354i	0.504221	−	0.863575i	$-0.331780\pi$
$937$	−11.0000	−	11.0000i	−0.359354	−	0.359354i	−0.863575	+	0.504221i	$0.831780\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	29.6985i		−	0.968143i	−0.875028	−	0.484071i	$-0.839157\pi$
$941$		−	29.6985i		−	0.968143i	0.875028	−	0.484071i	$-0.160843\pi$
$942$		0			0
$943$	−28.0000	+	28.0000i	−0.911805	+	0.911805i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5.65685	+	5.65685i	−0.183823	+	0.183823i	−0.793019	−	0.609196i	$-0.791492\pi$
$947$	−5.65685	+	5.65685i	−0.183823	+	0.183823i	0.609196	+	0.793019i	$0.291492\pi$
$948$		0			0
$949$			6.00000i			0.194768i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−4.24264	−	4.24264i	−0.137433	−	0.137433i	0.635044	−	0.772476i	$-0.280982\pi$
$953$	−4.24264	−	4.24264i	−0.137433	−	0.137433i	−0.772476	+	0.635044i	$0.780982\pi$
$954$		0			0
$955$	−48.0000	+	16.0000i	−1.55324	+	0.517748i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	16.9706			0.548008
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	25.4558	+	12.7279i	0.819453	+	0.409726i
$966$		0			0
$967$	−10.0000	−	10.0000i	−0.321578	−	0.321578i	0.527794	−	0.849372i	$-0.323019\pi$
$967$	−10.0000	−	10.0000i	−0.321578	−	0.321578i	−0.849372	+	0.527794i	$0.823019\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	36.7696i		−	1.17999i	−0.807406	−	0.589996i	$-0.799129\pi$
$971$		−	36.7696i		−	1.17999i	0.807406	−	0.589996i	$-0.200871\pi$
$972$		0			0
$973$	8.00000	−	8.00000i	0.256468	−	0.256468i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8.48528	+	8.48528i	−0.271468	+	0.271468i	−0.829691	−	0.558223i	$-0.811483\pi$
$977$	−8.48528	+	8.48528i	−0.271468	+	0.271468i	0.558223	+	0.829691i	$0.311483\pi$
$978$		0			0
$979$			28.0000i			0.894884i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	5.65685	+	5.65685i	0.180426	+	0.180426i	0.791541	−	0.611116i	$-0.209279\pi$
$983$	5.65685	+	5.65685i	0.180426	+	0.180426i	−0.611116	+	0.791541i	$0.709279\pi$
$984$		0			0
$985$	36.0000	+	18.0000i	1.14706	+	0.573528i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	13.0000	+	13.0000i	0.411714	+	0.411714i	0.882335	−	0.470621i	$-0.155970\pi$
$997$	13.0000	+	13.0000i	0.411714	+	0.411714i	−0.470621	+	0.882335i	$0.655970\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.2.w.a.17.2		4
3.2	odd	2	inner	720.2.w.a.17.1		4
4.3	odd	2		90.2.f.a.17.1	✓	4
5.2	odd	4		3600.2.w.g.593.2		4
5.3	odd	4	inner	720.2.w.a.593.1		4
5.4	even	2		3600.2.w.g.1457.2		4
8.3	odd	2		2880.2.w.l.2177.1		4
8.5	even	2		2880.2.w.c.2177.1		4
12.11	even	2		90.2.f.a.17.2	yes	4
15.2	even	4		3600.2.w.g.593.1		4
15.8	even	4	inner	720.2.w.a.593.2		4
15.14	odd	2		3600.2.w.g.1457.1		4
20.3	even	4		90.2.f.a.53.2	yes	4
20.7	even	4		450.2.f.b.143.1		4
20.19	odd	2		450.2.f.b.107.2		4
24.5	odd	2		2880.2.w.c.2177.2		4
24.11	even	2		2880.2.w.l.2177.2		4
36.7	odd	6		810.2.m.c.377.2		8
36.11	even	6		810.2.m.c.377.1		8
36.23	even	6		810.2.m.c.107.2		8
36.31	odd	6		810.2.m.c.107.1		8
40.3	even	4		2880.2.w.l.2753.2		4
40.13	odd	4		2880.2.w.c.2753.2		4
60.23	odd	4		90.2.f.a.53.1	yes	4
60.47	odd	4		450.2.f.b.143.2		4
60.59	even	2		450.2.f.b.107.1		4
120.53	even	4		2880.2.w.c.2753.1		4
120.83	odd	4		2880.2.w.l.2753.1		4
180.23	odd	12		810.2.m.c.593.2		8
180.43	even	12		810.2.m.c.53.2		8
180.83	odd	12		810.2.m.c.53.1		8
180.103	even	12		810.2.m.c.593.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.f.a.17.1	✓	4	4.3	odd	2
90.2.f.a.17.2	yes	4	12.11	even	2
90.2.f.a.53.1	yes	4	60.23	odd	4
90.2.f.a.53.2	yes	4	20.3	even	4
450.2.f.b.107.1		4	60.59	even	2
450.2.f.b.107.2		4	20.19	odd	2
450.2.f.b.143.1		4	20.7	even	4
450.2.f.b.143.2		4	60.47	odd	4
720.2.w.a.17.1		4	3.2	odd	2	inner
720.2.w.a.17.2		4	1.1	even	1	trivial
720.2.w.a.593.1		4	5.3	odd	4	inner
720.2.w.a.593.2		4	15.8	even	4	inner
810.2.m.c.53.1		8	180.83	odd	12
810.2.m.c.53.2		8	180.43	even	12
810.2.m.c.107.1		8	36.31	odd	6
810.2.m.c.107.2		8	36.23	even	6
810.2.m.c.377.1		8	36.11	even	6
810.2.m.c.377.2		8	36.7	odd	6
810.2.m.c.593.1		8	180.103	even	12
810.2.m.c.593.2		8	180.23	odd	12
2880.2.w.c.2177.1		4	8.5	even	2
2880.2.w.c.2177.2		4	24.5	odd	2
2880.2.w.c.2753.1		4	120.53	even	4
2880.2.w.c.2753.2		4	40.13	odd	4
2880.2.w.l.2177.1		4	8.3	odd	2
2880.2.w.l.2177.2		4	24.11	even	2
2880.2.w.l.2753.1		4	120.83	odd	4
2880.2.w.l.2753.2		4	40.3	even	4
3600.2.w.g.593.1		4	15.2	even	4
3600.2.w.g.593.2		4	5.2	odd	4
3600.2.w.g.1457.1		4	15.14	odd	2
3600.2.w.g.1457.2		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 2.12132i) q^{5} +(-2.00000 - 2.00000i) q^{7} +O(q^{10})\)	\(q+(0.707107 + 2.12132i) q^{5} +(-2.00000 - 2.00000i) q^{7} +2.82843i q^{11} +(-3.00000 + 3.00000i) q^{13} +(2.82843 - 2.82843i) q^{17} +8.00000i q^{19} +(2.82843 + 2.82843i) q^{23} +(-4.00000 + 3.00000i) q^{25} -1.41421 q^{29} -4.00000 q^{31} +(2.82843 - 5.65685i) q^{35} +(-3.00000 - 3.00000i) q^{37} +9.89949i q^{41} +1.00000i q^{49} +(2.82843 + 2.82843i) q^{53} +(-6.00000 + 2.00000i) q^{55} -2.82843 q^{59} +(-8.48528 - 4.24264i) q^{65} +(4.00000 + 4.00000i) q^{67} +5.65685i q^{71} +(1.00000 - 1.00000i) q^{73} +(5.65685 - 5.65685i) q^{77} -12.0000i q^{79} +(8.48528 + 8.48528i) q^{83} +(8.00000 + 4.00000i) q^{85} +9.89949 q^{89} +12.0000 q^{91} +(-16.9706 + 5.65685i) q^{95} +(-3.00000 - 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7}+O(q^{10}) \) 4 * q - 8 * q^7	\( 4 q - 8 q^{7} - 12 q^{13} - 16 q^{25} - 16 q^{31} - 12 q^{37} - 24 q^{55} + 16 q^{67} + 4 q^{73} + 32 q^{85} + 48 q^{91} - 12 q^{97}+O(q^{100}) \) 4 * q - 8 * q^7 - 12 * q^13 - 16 * q^25 - 16 * q^31 - 12 * q^37 - 24 * q^55 + 16 * q^67 + 4 * q^73 + 32 * q^85 + 48 * q^91 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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