LMFDB - Embedded newform 720.2.w.d.593.1

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 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	720.593
Dual form			720.2.w.d.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.12132 + 0.707107i) q^{5} +(2.00000 - 2.00000i) q^{7} +O(q^{10})$	$q+(-2.12132 + 0.707107i) q^{5} +(2.00000 - 2.00000i) q^{7} +2.82843i q^{11} +(1.00000 + 1.00000i) q^{13} +(2.82843 + 2.82843i) q^{17} +(2.82843 - 2.82843i) q^{23} +(4.00000 - 3.00000i) q^{25} +4.24264 q^{29} +4.00000 q^{31} +(-2.82843 + 5.65685i) q^{35} +(1.00000 - 1.00000i) q^{37} +1.41421i q^{41} +(8.00000 + 8.00000i) q^{43} +(5.65685 + 5.65685i) q^{47} -1.00000i q^{49} +(-2.82843 + 2.82843i) q^{53} +(-2.00000 - 6.00000i) q^{55} -8.48528 q^{59} +8.00000 q^{61} +(-2.82843 - 1.41421i) q^{65} +(-4.00000 + 4.00000i) q^{67} -5.65685i q^{71} +(1.00000 + 1.00000i) q^{73} +(5.65685 + 5.65685i) q^{77} -12.0000i q^{79} +(2.82843 - 2.82843i) q^{83} +(-8.00000 - 4.00000i) q^{85} -12.7279 q^{89} +4.00000 q^{91} +(-11.0000 + 11.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7}+O(q^{10}) $ 4 * q + 8 * q^7	$ 4 q + 8 q^{7} + 4 q^{13} + 16 q^{25} + 16 q^{31} + 4 q^{37} + 32 q^{43} - 8 q^{55} + 32 q^{61} - 16 q^{67} + 4 q^{73} - 32 q^{85} + 16 q^{91} - 44 q^{97}+O(q^{100}) $ 4 * q + 8 * q^7 + 4 * q^13 + 16 * q^25 + 16 * q^31 + 4 * q^37 + 32 * q^43 - 8 * q^55 + 32 * q^61 - 16 * q^67 + 4 * q^73 - 32 * q^85 + 16 * q^91 - 44 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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
$5$	−2.12132	+	0.707107i	−0.948683	+	0.316228i
$5$	−2.12132	+	0.707107i	−0.948683	+	0.316228i
$6$		0			0
$7$	2.00000	−	2.00000i	0.755929	−	0.755929i	−0.219650	−	0.975579i	$-0.570491\pi$
$7$	2.00000	−	2.00000i	0.755929	−	0.755929i	0.975579	+	0.219650i	$0.0704915\pi$
$8$		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
$13$	1.00000	+	1.00000i	0.277350	+	0.277350i	0.832050	−	0.554700i	$-0.187167\pi$
$13$	1.00000	+	1.00000i	0.277350	+	0.277350i	−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.82843	+	2.82843i	0.685994	+	0.685994i	0.961344	−	0.275350i	$-0.0887937\pi$
$17$	2.82843	+	2.82843i	0.685994	+	0.685994i	−0.275350	+	0.961344i	$0.588794\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.82843	−	2.82843i	0.589768	−	0.589768i	−0.347801	−	0.937568i	$-0.613071\pi$
$23$	2.82843	−	2.82843i	0.589768	−	0.589768i	0.937568	+	0.347801i	$0.113071\pi$
$24$		0			0
$25$	4.00000	−	3.00000i	0.800000	−	0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.24264			0.787839			0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264			0.787839			0.393919	+	0.919145i	$0.371119\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.82843	+	5.65685i	−0.478091	+	0.956183i
$36$		0			0
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	−0.620113	−	0.784512i	$-0.712913\pi$
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	0.784512	+	0.620113i	$0.212913\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.41421i			0.220863i	0.993884	+	0.110432i	$0.0352233\pi$
$41$			1.41421i			0.220863i	−0.993884	+	0.110432i	$0.964777\pi$
$42$		0			0
$43$	8.00000	+	8.00000i	1.21999	+	1.21999i	0.967635	+	0.252353i	$0.0812046\pi$
$43$	8.00000	+	8.00000i	1.21999	+	1.21999i	0.252353	+	0.967635i	$0.418795\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.65685	+	5.65685i	0.825137	+	0.825137i	0.986840	−	0.161703i	$-0.0516985\pi$
$47$	5.65685	+	5.65685i	0.825137	+	0.825137i	−0.161703	+	0.986840i	$0.551699\pi$
$48$		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	$-0.838586\pi$
$53$	−2.82843	+	2.82843i	−0.388514	+	0.388514i	0.485643	+	0.874157i	$0.338586\pi$
$54$		0			0
$55$	−2.00000	−	6.00000i	−0.269680	−	0.809040i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.48528			−1.10469			−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528			−1.10469			−0.552345	+	0.833616i	$0.686267\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.82843	−	1.41421i	−0.350823	−	0.175412i
$66$		0			0
$67$	−4.00000	+	4.00000i	−0.488678	+	0.488678i	−0.907889	−	0.419211i	$-0.862307\pi$
$67$	−4.00000	+	4.00000i	−0.488678	+	0.488678i	0.419211	+	0.907889i	$0.362307\pi$
$68$		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
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	0.763202	−	0.646160i	$-0.223626\pi$
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	−0.646160	+	0.763202i	$0.723626\pi$
$74$		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
$83$	2.82843	−	2.82843i	0.310460	−	0.310460i	−0.534628	−	0.845088i	$-0.679548\pi$
$83$	2.82843	−	2.82843i	0.310460	−	0.310460i	0.845088	+	0.534628i	$0.179548\pi$
$84$		0			0
$85$	−8.00000	−	4.00000i	−0.867722	−	0.433861i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.7279			−1.34916			−0.674579	−	0.738203i	$-0.735675\pi$
$89$	−12.7279			−1.34916			−0.674579	+	0.738203i	$0.735675\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−11.0000	+	11.0000i	−1.11688	+	1.11688i	−0.124684	+	0.992196i	$0.539792\pi$
$97$	−11.0000	+	11.0000i	−1.11688	+	1.11688i	−0.992196	+	0.124684i	$0.960208\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	15.5563i		−	1.54791i	−0.633238	−	0.773957i	$-0.718274\pi$
$101$		−	15.5563i		−	1.54791i	0.633238	−	0.773957i	$-0.281726\pi$
$102$		0			0
$103$	−10.0000	−	10.0000i	−0.985329	−	0.985329i	0.0145647	−	0.999894i	$-0.495364\pi$
$103$	−10.0000	−	10.0000i	−0.985329	−	0.985329i	−0.999894	+	0.0145647i	$0.995364\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.82843	−	2.82843i	−0.273434	−	0.273434i	0.557047	−	0.830481i	$-0.311934\pi$
$107$	−2.82843	−	2.82843i	−0.273434	−	0.273434i	−0.830481	+	0.557047i	$0.811934\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.89949	−	9.89949i	0.931266	−	0.931266i	−0.0665190	−	0.997785i	$-0.521189\pi$
$113$	9.89949	−	9.89949i	0.931266	−	0.931266i	0.997785	+	0.0665190i	$0.0211893\pi$
$114$		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
$125$	−6.36396	+	9.19239i	−0.569210	+	0.822192i
$126$		0			0
$127$	−10.0000	+	10.0000i	−0.887357	+	0.887357i	−0.994268	−	0.106912i	$-0.965904\pi$
$127$	−10.0000	+	10.0000i	−0.887357	+	0.887357i	0.106912	+	0.994268i	$0.465904\pi$
$128$		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
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.07107	+	7.07107i	0.604122	+	0.604122i	0.941404	−	0.337282i	$-0.109507\pi$
$137$	7.07107	+	7.07107i	0.604122	+	0.604122i	−0.337282	+	0.941404i	$0.609507\pi$
$138$		0			0
$139$			12.0000i			1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$			12.0000i			1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.82843	+	2.82843i	−0.236525	+	0.236525i
$144$		0			0
$145$	−9.00000	+	3.00000i	−0.747409	+	0.249136i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.24264			−0.347571			−0.173785	−	0.984784i	$-0.555600\pi$
$149$	−4.24264			−0.347571			−0.173785	+	0.984784i	$0.555600\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.48528	+	2.82843i	−0.681554	+	0.227185i
$156$		0			0
$157$	−5.00000	+	5.00000i	−0.399043	+	0.399043i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−5.00000	+	5.00000i	−0.399043	+	0.399043i	0.478852	+	0.877896i	$0.341053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	11.3137i		−	0.891645i
$162$		0			0
$163$	8.00000	+	8.00000i	0.626608	+	0.626608i	0.947213	−	0.320605i	$-0.103886\pi$
$163$	8.00000	+	8.00000i	0.626608	+	0.626608i	−0.320605	+	0.947213i	$0.603886\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	14.1421	+	14.1421i	1.09435	+	1.09435i	0.995058	+	0.0992931i	$0.0316581\pi$
$167$	14.1421	+	14.1421i	1.09435	+	1.09435i	0.0992931	+	0.995058i	$0.468342\pi$
$168$		0			0
$169$		−	11.0000i		−	0.846154i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.89949	−	9.89949i	0.752645	−	0.752645i	−0.222327	−	0.974972i	$-0.571365\pi$
$173$	9.89949	−	9.89949i	0.752645	−	0.752645i	0.974972	+	0.222327i	$0.0713654\pi$
$174$		0			0
$175$	2.00000	−	14.0000i	0.151186	−	1.05830i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−25.4558			−1.90266			−0.951330	−	0.308175i	$-0.900282\pi$
$179$	−25.4558			−1.90266			−0.951330	+	0.308175i	$0.900282\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.41421	+	2.82843i	−0.103975	+	0.207950i
$186$		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	$-0.694733\pi$
$191$		−	22.6274i		−	1.63726i	0.574320	−	0.818631i	$-0.305267\pi$
$192$		0			0
$193$	1.00000	+	1.00000i	0.0719816	+	0.0719816i	0.742181	−	0.670199i	$-0.233791\pi$
$193$	1.00000	+	1.00000i	0.0719816	+	0.0719816i	−0.670199	+	0.742181i	$0.733791\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.89949	−	9.89949i	−0.705310	−	0.705310i	0.260235	−	0.965545i	$-0.416200\pi$
$197$	−9.89949	−	9.89949i	−0.705310	−	0.705310i	−0.965545	+	0.260235i	$0.916200\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.48528	−	8.48528i	0.595550	−	0.595550i
$204$		0			0
$205$	−1.00000	−	3.00000i	−0.0698430	−	0.209529i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		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
$215$	−22.6274	−	11.3137i	−1.54318	−	0.771589i
$216$		0			0
$217$	8.00000	−	8.00000i	0.543075	−	0.543075i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			5.65685i			0.380521i
$222$		0			0
$223$	−10.0000	−	10.0000i	−0.669650	−	0.669650i	0.287985	−	0.957635i	$-0.407015\pi$
$223$	−10.0000	−	10.0000i	−0.669650	−	0.669650i	−0.957635	+	0.287985i	$0.907015\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.65685	+	5.65685i	0.375459	+	0.375459i	0.869461	−	0.494002i	$-0.164466\pi$
$227$	5.65685	+	5.65685i	0.375459	+	0.375459i	−0.494002	+	0.869461i	$0.664466\pi$
$228$		0			0
$229$			6.00000i			0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$			6.00000i			0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.82843	+	2.82843i	−0.185296	+	0.185296i	−0.793659	−	0.608363i	$-0.791827\pi$
$233$	−2.82843	+	2.82843i	−0.185296	+	0.185296i	0.608363	+	0.793659i	$0.291827\pi$
$234$		0			0
$235$	−16.0000	−	8.00000i	−1.04372	−	0.521862i
$236$		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
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.707107	+	2.12132i	0.0451754	+	0.135526i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			19.7990i			1.24970i	0.780744	+	0.624851i	$0.214840\pi$
$251$			19.7990i			1.24970i	−0.780744	+	0.624851i	$0.785160\pi$
$252$		0			0
$253$	8.00000	+	8.00000i	0.502956	+	0.502956i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.41421	−	1.41421i	−0.0882162	−	0.0882162i	0.661622	−	0.749838i	$-0.269869\pi$
$257$	−1.41421	−	1.41421i	−0.0882162	−	0.0882162i	−0.749838	+	0.661622i	$0.769869\pi$
$258$		0			0
$259$		−	4.00000i		−	0.248548i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.82843	−	2.82843i	0.174408	−	0.174408i	−0.614505	−	0.788913i	$-0.710644\pi$
$263$	2.82843	−	2.82843i	0.174408	−	0.174408i	0.788913	+	0.614505i	$0.210644\pi$
$264$		0			0
$265$	4.00000	−	8.00000i	0.245718	−	0.491436i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.7279			−0.776035			−0.388018	−	0.921652i	$-0.626840\pi$
$269$	−12.7279			−0.776035			−0.388018	+	0.921652i	$0.626840\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.48528	+	11.3137i	0.511682	+	0.682242i
$276$		0			0
$277$	−11.0000	+	11.0000i	−0.660926	+	0.660926i	−0.955598	−	0.294672i	$-0.904789\pi$
$277$	−11.0000	+	11.0000i	−0.660926	+	0.660926i	0.294672	+	0.955598i	$0.404789\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			9.89949i			0.590554i	0.955412	+	0.295277i	$0.0954120\pi$
$281$			9.89949i			0.590554i	−0.955412	+	0.295277i	$0.904588\pi$
$282$		0			0
$283$	8.00000	+	8.00000i	0.475551	+	0.475551i	0.903705	−	0.428155i	$-0.140836\pi$
$283$	8.00000	+	8.00000i	0.475551	+	0.475551i	−0.428155	+	0.903705i	$0.640836\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.82843	+	2.82843i	0.166957	+	0.166957i
$288$		0			0
$289$		−	1.00000i		−	0.0588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.89949	−	9.89949i	0.578335	−	0.578335i	−0.356110	−	0.934444i	$-0.615897\pi$
$293$	9.89949	−	9.89949i	0.578335	−	0.578335i	0.934444	+	0.356110i	$0.115897\pi$
$294$		0			0
$295$	18.0000	−	6.00000i	1.04800	−	0.349334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.65685			0.327144
$300$		0			0
$301$	32.0000			1.84445
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−16.9706	+	5.65685i	−0.971732	+	0.323911i
$306$		0			0
$307$	8.00000	−	8.00000i	0.456584	−	0.456584i	−0.440948	−	0.897532i	$-0.645358\pi$
$307$	8.00000	−	8.00000i	0.456584	−	0.456584i	0.897532	+	0.440948i	$0.145358\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			11.3137i			0.641542i	0.947157	+	0.320771i	$0.103942\pi$
$311$			11.3137i			0.641542i	−0.947157	+	0.320771i	$0.896058\pi$
$312$		0			0
$313$	19.0000	+	19.0000i	1.07394	+	1.07394i	0.997038	+	0.0769051i	$0.0245038\pi$
$313$	19.0000	+	19.0000i	1.07394	+	1.07394i	0.0769051	+	0.997038i	$0.475496\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	19.7990	+	19.7990i	1.11202	+	1.11202i	0.992877	+	0.119145i	$0.0380154\pi$
$317$	19.7990	+	19.7990i	1.11202	+	1.11202i	0.119145	+	0.992877i	$0.461985\pi$
$318$		0			0
$319$			12.0000i			0.671871i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	7.00000	+	1.00000i	0.388290	+	0.0554700i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	22.6274			1.24749
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.65685	−	11.3137i	0.309067	−	0.618134i
$336$		0			0
$337$	−5.00000	+	5.00000i	−0.272367	+	0.272367i	−0.830053	−	0.557685i	$-0.811690\pi$
$337$	−5.00000	+	5.00000i	−0.272367	+	0.272367i	0.557685	+	0.830053i	$0.311690\pi$
$338$		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	0.334901	−	0.942253i	$-0.391297\pi$
$347$	−11.3137	−	11.3137i	−0.607352	−	0.607352i	−0.942253	+	0.334901i	$0.891297\pi$
$348$		0			0
$349$		−	24.0000i		−	1.28469i	−0.766415	−	0.642345i	$-0.777962\pi$
$349$		−	24.0000i		−	1.28469i	0.766415	−	0.642345i	$-0.222038\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.82843	+	2.82843i	−0.150542	+	0.150542i	−0.778360	−	0.627818i	$-0.783948\pi$
$353$	−2.82843	+	2.82843i	−0.150542	+	0.150542i	0.627818	+	0.778360i	$0.283948\pi$
$354$		0			0
$355$	4.00000	+	12.0000i	0.212298	+	0.636894i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.82843	−	1.41421i	−0.148047	−	0.0740233i
$366$		0			0
$367$	2.00000	−	2.00000i	0.104399	−	0.104399i	−0.652978	−	0.757377i	$-0.726481\pi$
$367$	2.00000	−	2.00000i	0.104399	−	0.104399i	0.757377	+	0.652978i	$0.226481\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			11.3137i			0.587378i
$372$		0			0
$373$	−17.0000	−	17.0000i	−0.880227	−	0.880227i	0.113331	−	0.993557i	$-0.463848\pi$
$373$	−17.0000	−	17.0000i	−0.880227	−	0.880227i	−0.993557	+	0.113331i	$0.963848\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.24264	+	4.24264i	0.218507	+	0.218507i
$378$		0			0
$379$		−	36.0000i		−	1.84920i	−0.380945	−	0.924598i	$-0.624401\pi$
$379$		−	36.0000i		−	1.84920i	0.380945	−	0.924598i	$-0.375599\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−22.6274	+	22.6274i	−1.15621	+	1.15621i	−0.170923	+	0.985284i	$0.554675\pi$
$383$	−22.6274	+	22.6274i	−1.15621	+	1.15621i	−0.985284	+	0.170923i	$0.945325\pi$
$384$		0			0
$385$	−16.0000	−	8.00000i	−0.815436	−	0.407718i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4.24264			0.215110			0.107555	−	0.994199i	$-0.465698\pi$
$389$	4.24264			0.215110			0.107555	+	0.994199i	$0.465698\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.48528	+	25.4558i	0.426941	+	1.28082i
$396$		0			0
$397$	19.0000	−	19.0000i	0.953583	−	0.953583i	−0.0453868	−	0.998969i	$-0.514452\pi$
$397$	19.0000	−	19.0000i	0.953583	−	0.953583i	0.998969	+	0.0453868i	$0.0144520\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	24.0416i		−	1.20058i	−0.799782	−	0.600291i	$-0.795051\pi$
$401$		−	24.0416i		−	1.20058i	0.799782	−	0.600291i	$-0.204949\pi$
$402$		0			0
$403$	4.00000	+	4.00000i	0.199254	+	0.199254i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.82843	+	2.82843i	0.140200	+	0.140200i
$408$		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
$413$	−16.9706	+	16.9706i	−0.835067	+	0.835067i
$414$		0			0
$415$	−4.00000	+	8.00000i	−0.196352	+	0.392705i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.48528			−0.414533			−0.207267	−	0.978285i	$-0.566457\pi$
$419$	−8.48528			−0.414533			−0.207267	+	0.978285i	$0.566457\pi$
$420$		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
$425$	19.7990	+	2.82843i	0.960392	+	0.137199i
$426$		0			0
$427$	16.0000	−	16.0000i	0.774294	−	0.774294i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	5.65685i		−	0.272481i	−0.990676	−	0.136241i	$-0.956498\pi$
$431$		−	5.65685i		−	0.272481i	0.990676	−	0.136241i	$-0.0435020\pi$
$432$		0			0
$433$	−17.0000	−	17.0000i	−0.816968	−	0.816968i	0.168700	−	0.985668i	$-0.446043\pi$
$433$	−17.0000	−	17.0000i	−0.816968	−	0.816968i	−0.985668	+	0.168700i	$0.946043\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			24.0000i			1.14546i	0.819745	+	0.572729i	$0.194115\pi$
$439$			24.0000i			1.14546i	−0.819745	+	0.572729i	$0.805885\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	28.2843	−	28.2843i	1.34383	−	1.34383i	0.451612	−	0.892215i	$-0.350849\pi$
$443$	28.2843	−	28.2843i	1.34383	−	1.34383i	0.892215	−	0.451612i	$-0.149151\pi$
$444$		0			0
$445$	27.0000	−	9.00000i	1.27992	−	0.426641i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.7279			−0.600668			−0.300334	−	0.953834i	$-0.597098\pi$
$449$	−12.7279			−0.600668			−0.300334	+	0.953834i	$0.597098\pi$
$450$		0			0
$451$	−4.00000			−0.188353
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.48528	+	2.82843i	−0.397796	+	0.132599i
$456$		0			0
$457$	25.0000	−	25.0000i	1.16945	−	1.16945i	0.187112	−	0.982339i	$-0.440087\pi$
$457$	25.0000	−	25.0000i	1.16945	−	1.16945i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			9.89949i			0.461065i	0.973065	+	0.230533i	$0.0740469\pi$
$461$			9.89949i			0.461065i	−0.973065	+	0.230533i	$0.925953\pi$
$462$		0			0
$463$	−10.0000	−	10.0000i	−0.464739	−	0.464739i	0.435466	−	0.900205i	$-0.356584\pi$
$463$	−10.0000	−	10.0000i	−0.464739	−	0.464739i	−0.900205	+	0.435466i	$0.856584\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.82843	−	2.82843i	−0.130884	−	0.130884i	0.638630	−	0.769514i	$-0.279501\pi$
$467$	−2.82843	−	2.82843i	−0.130884	−	0.130884i	−0.769514	+	0.638630i	$0.779501\pi$
$468$		0			0
$469$			16.0000i			0.738811i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−22.6274	+	22.6274i	−1.04041	+	1.04041i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.9706			−0.775405			−0.387702	−	0.921785i	$-0.626731\pi$
$479$	−16.9706			−0.775405			−0.387702	+	0.921785i	$0.626731\pi$
$480$		0			0
$481$	2.00000			0.0911922
$482$		0			0
$483$		0			0
$484$		0			0
$485$	15.5563	−	31.1127i	0.706377	−	1.41275i
$486$		0			0
$487$	−10.0000	+	10.0000i	−0.453143	+	0.453143i	−0.896396	−	0.443253i	$-0.853824\pi$
$487$	−10.0000	+	10.0000i	−0.453143	+	0.453143i	0.443253	+	0.896396i	$0.353824\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	14.1421i		−	0.638226i	−0.947717	−	0.319113i	$-0.896615\pi$
$491$		−	14.1421i		−	0.638226i	0.947717	−	0.319113i	$-0.103385\pi$
$492$		0			0
$493$	12.0000	+	12.0000i	0.540453	+	0.540453i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11.3137	−	11.3137i	−0.507489	−	0.507489i
$498$		0			0
$499$		−	24.0000i		−	1.07439i	−0.843459	−	0.537194i	$-0.819484\pi$
$499$		−	24.0000i		−	1.07439i	0.843459	−	0.537194i	$-0.180516\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−22.6274	+	22.6274i	−1.00891	+	1.00891i	−0.00894668	+	0.999960i	$0.502848\pi$
$503$	−22.6274	+	22.6274i	−1.00891	+	1.00891i	−0.999960	+	0.00894668i	$0.997152\pi$
$504$		0			0
$505$	11.0000	+	33.0000i	0.489494	+	1.46848i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.24264			−0.188052			−0.0940259	−	0.995570i	$-0.529974\pi$
$509$	−4.24264			−0.188052			−0.0940259	+	0.995570i	$0.529974\pi$
$510$		0			0
$511$	4.00000			0.176950
$512$		0			0
$513$		0			0
$514$		0			0
$515$	28.2843	+	14.1421i	1.24635	+	0.623177i
$516$		0			0
$517$	−16.0000	+	16.0000i	−0.703679	+	0.703679i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			18.3848i			0.805452i	0.915321	+	0.402726i	$0.131937\pi$
$521$			18.3848i			0.805452i	−0.915321	+	0.402726i	$0.868063\pi$
$522$		0			0
$523$	8.00000	+	8.00000i	0.349816	+	0.349816i	0.860041	−	0.510225i	$-0.170438\pi$
$523$	8.00000	+	8.00000i	0.349816	+	0.349816i	−0.510225	+	0.860041i	$0.670438\pi$
$524$		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
$533$	−1.41421	+	1.41421i	−0.0612564	+	0.0612564i
$534$		0			0
$535$	8.00000	+	4.00000i	0.345870	+	0.172935i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.82843			0.121829
$540$		0			0
$541$	−16.0000			−0.687894			−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000			−0.687894			−0.343947	+	0.938989i	$0.611764\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	20.0000	−	20.0000i	0.855138	−	0.855138i	−0.135622	−	0.990761i	$-0.543303\pi$
$547$	20.0000	−	20.0000i	0.855138	−	0.855138i	0.990761	+	0.135622i	$0.0433034\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−24.0000	−	24.0000i	−1.02058	−	1.02058i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.82843	+	2.82843i	0.119844	+	0.119844i	0.764485	−	0.644641i	$-0.222993\pi$
$557$	2.82843	+	2.82843i	0.119844	+	0.119844i	−0.644641	+	0.764485i	$0.722993\pi$
$558$		0			0
$559$			16.0000i			0.676728i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	28.2843	−	28.2843i	1.19204	−	1.19204i	0.215546	−	0.976494i	$-0.430847\pi$
$563$	28.2843	−	28.2843i	1.19204	−	1.19204i	0.976494	−	0.215546i	$-0.0691532\pi$
$564$		0			0
$565$	−14.0000	+	28.0000i	−0.588984	+	1.17797i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	29.6985			1.24503			0.622513	−	0.782610i	$-0.286112\pi$
$569$	29.6985			1.24503			0.622513	+	0.782610i	$0.286112\pi$
$570$		0			0
$571$	−32.0000			−1.33916			−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000			−1.33916			−0.669579	+	0.742741i	$0.733526\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.82843	−	19.7990i	0.117954	−	0.825675i
$576$		0			0
$577$	−17.0000	+	17.0000i	−0.707719	+	0.707719i	−0.966055	−	0.258336i	$-0.916826\pi$
$577$	−17.0000	+	17.0000i	−0.707719	+	0.707719i	0.258336	+	0.966055i	$0.416826\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	11.3137i		−	0.469372i
$582$		0			0
$583$	−8.00000	−	8.00000i	−0.331326	−	0.331326i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−19.7990	−	19.7990i	−0.817192	−	0.817192i	0.168508	−	0.985700i	$-0.446105\pi$
$587$	−19.7990	−	19.7990i	−0.817192	−	0.817192i	−0.985700	+	0.168508i	$0.946105\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.89949	−	9.89949i	0.406524	−	0.406524i	−0.474001	−	0.880524i	$-0.657191\pi$
$593$	9.89949	−	9.89949i	0.406524	−	0.406524i	0.880524	+	0.474001i	$0.157191\pi$
$594$		0			0
$595$	−24.0000	+	8.00000i	−0.983904	+	0.327968i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16.9706			0.693398			0.346699	−	0.937976i	$-0.387302\pi$
$599$	16.9706			0.693398			0.346699	+	0.937976i	$0.387302\pi$
$600$		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
$605$	−6.36396	+	2.12132i	−0.258732	+	0.0862439i
$606$		0			0
$607$	−22.0000	+	22.0000i	−0.892952	+	0.892952i	−0.994800	−	0.101848i	$-0.967525\pi$
$607$	−22.0000	+	22.0000i	−0.892952	+	0.892952i	0.101848	+	0.994800i	$0.467525\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			11.3137i			0.457704i
$612$		0			0
$613$	1.00000	+	1.00000i	0.0403896	+	0.0403896i	0.727013	−	0.686624i	$-0.240908\pi$
$613$	1.00000	+	1.00000i	0.0403896	+	0.0403896i	−0.686624	+	0.727013i	$0.740908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14.1421	−	14.1421i	−0.569341	−	0.569341i	0.362603	−	0.931944i	$-0.381888\pi$
$617$	−14.1421	−	14.1421i	−0.569341	−	0.569341i	−0.931944	+	0.362603i	$0.881888\pi$
$618$		0			0
$619$		−	12.0000i		−	0.482321i	−0.970485	−	0.241160i	$-0.922472\pi$
$619$		−	12.0000i		−	0.482321i	0.970485	−	0.241160i	$-0.0775280\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−25.4558	+	25.4558i	−1.01987	+	1.01987i
$624$		0			0
$625$	7.00000	−	24.0000i	0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.65685			0.225554
$630$		0			0
$631$	−20.0000			−0.796187			−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000			−0.796187			−0.398094	+	0.917345i	$0.630328\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.1421	−	28.2843i	0.561214	−	1.12243i
$636$		0			0
$637$	1.00000	−	1.00000i	0.0396214	−	0.0396214i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	15.5563i		−	0.614439i	−0.951639	−	0.307219i	$-0.900601\pi$
$641$		−	15.5563i		−	0.614439i	0.951639	−	0.307219i	$-0.0993986\pi$
$642$		0			0
$643$	−28.0000	−	28.0000i	−1.10421	−	1.10421i	−0.993897	−	0.110316i	$-0.964814\pi$
$643$	−28.0000	−	28.0000i	−1.10421	−	1.10421i	−0.110316	−	0.993897i	$-0.535186\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−28.2843	−	28.2843i	−1.11197	−	1.11197i	−0.992884	−	0.119085i	$-0.962004\pi$
$647$	−28.2843	−	28.2843i	−1.11197	−	1.11197i	−0.119085	−	0.992884i	$-0.537996\pi$
$648$		0			0
$649$		−	24.0000i		−	0.942082i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.82843	+	2.82843i	−0.110685	+	0.110685i	−0.760280	−	0.649595i	$-0.774938\pi$
$653$	−2.82843	+	2.82843i	−0.110685	+	0.110685i	0.649595	+	0.760280i	$0.274938\pi$
$654$		0			0
$655$	10.0000	+	30.0000i	0.390732	+	1.17220i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	8.48528			0.330540			0.165270	−	0.986248i	$-0.447151\pi$
$659$	8.48528			0.330540			0.165270	+	0.986248i	$0.447151\pi$
$660$		0			0
$661$	−40.0000			−1.55582			−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000			−1.55582			−0.777910	+	0.628376i	$0.783720\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	12.0000	−	12.0000i	0.464642	−	0.464642i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			22.6274i			0.873522i
$672$		0			0
$673$	1.00000	+	1.00000i	0.0385472	+	0.0385472i	0.726118	−	0.687570i	$-0.241323\pi$
$673$	1.00000	+	1.00000i	0.0385472	+	0.0385472i	−0.687570	+	0.726118i	$0.741323\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−31.1127	−	31.1127i	−1.19576	−	1.19576i	−0.975425	−	0.220334i	$-0.929285\pi$
$677$	−31.1127	−	31.1127i	−1.19576	−	1.19576i	−0.220334	−	0.975425i	$-0.570715\pi$
$678$		0			0
$679$			44.0000i			1.68857i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	2.82843	−	2.82843i	0.108227	−	0.108227i	−0.650920	−	0.759147i	$-0.725617\pi$
$683$	2.82843	−	2.82843i	0.108227	−	0.108227i	0.759147	+	0.650920i	$0.225617\pi$
$684$		0			0
$685$	−20.0000	−	10.0000i	−0.764161	−	0.382080i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.65685			−0.215509
$690$		0			0
$691$	−8.00000			−0.304334			−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000			−0.304334			−0.152167	+	0.988355i	$0.548625\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.48528	−	25.4558i	−0.321865	−	0.965595i
$696$		0			0
$697$	−4.00000	+	4.00000i	−0.151511	+	0.151511i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	7.07107i		−	0.267071i	−0.991044	−	0.133535i	$-0.957367\pi$
$701$		−	7.07107i		−	0.267071i	0.991044	−	0.133535i	$-0.0426329\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−31.1127	−	31.1127i	−1.17011	−	1.17011i
$708$		0			0
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	11.3137	−	11.3137i	0.423702	−	0.423702i
$714$		0			0
$715$	4.00000	−	8.00000i	0.149592	−	0.299183i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−40.0000			−1.48968
$722$		0			0
$723$		0			0
$724$		0			0
$725$	16.9706	−	12.7279i	0.630271	−	0.472703i
$726$		0			0
$727$	2.00000	−	2.00000i	0.0741759	−	0.0741759i	−0.669046	−	0.743221i	$-0.733297\pi$
$727$	2.00000	−	2.00000i	0.0741759	−	0.0741759i	0.743221	+	0.669046i	$0.233297\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			45.2548i			1.67381i
$732$		0			0
$733$	1.00000	+	1.00000i	0.0369358	+	0.0369358i	0.725333	−	0.688398i	$-0.241686\pi$
$733$	1.00000	+	1.00000i	0.0369358	+	0.0369358i	−0.688398	+	0.725333i	$0.741686\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.3137	−	11.3137i	−0.416746	−	0.416746i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−22.6274	+	22.6274i	−0.830119	+	0.830119i	−0.987533	−	0.157413i	$-0.949684\pi$
$743$	−22.6274	+	22.6274i	−0.830119	+	0.830119i	0.157413	+	0.987533i	$0.449684\pi$
$744$		0			0
$745$	9.00000	−	3.00000i	0.329734	−	0.109911i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−11.3137			−0.413394
$750$		0			0
$751$	4.00000			0.145962			0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000			0.145962			0.0729810	+	0.997333i	$0.476749\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.9706	−	5.65685i	0.617622	−	0.205874i
$756$		0			0
$757$	19.0000	−	19.0000i	0.690567	−	0.690567i	−0.271790	−	0.962357i	$-0.587616\pi$
$757$	19.0000	−	19.0000i	0.690567	−	0.690567i	0.962357	+	0.271790i	$0.0876156\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			52.3259i			1.89681i	0.317058	+	0.948406i	$0.397305\pi$
$761$			52.3259i			1.89681i	−0.317058	+	0.948406i	$0.602695\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.48528	−	8.48528i	−0.306386	−	0.306386i
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2.82843	+	2.82843i	−0.101731	+	0.101731i	−0.756141	−	0.654409i	$-0.772917\pi$
$773$	−2.82843	+	2.82843i	−0.101731	+	0.101731i	0.654409	+	0.756141i	$0.272917\pi$
$774$		0			0
$775$	16.0000	−	12.0000i	0.574737	−	0.431053i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	16.0000			0.572525
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.07107	−	14.1421i	0.252377	−	0.504754i
$786$		0			0
$787$	−4.00000	+	4.00000i	−0.142585	+	0.142585i	−0.774796	−	0.632211i	$-0.782147\pi$
$787$	−4.00000	+	4.00000i	−0.142585	+	0.142585i	0.632211	+	0.774796i	$0.282147\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	39.5980i		−	1.40794i
$792$		0			0
$793$	8.00000	+	8.00000i	0.284088	+	0.284088i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.41421	−	1.41421i	−0.0500940	−	0.0500940i	0.681616	−	0.731710i	$-0.261277\pi$
$797$	−1.41421	−	1.41421i	−0.0500940	−	0.0500940i	−0.731710	+	0.681616i	$0.761277\pi$
$798$		0			0
$799$			32.0000i			1.13208i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.82843	+	2.82843i	−0.0998130	+	0.0998130i
$804$		0			0
$805$	8.00000	+	24.0000i	0.281963	+	0.845889i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	12.7279			0.447490			0.223745	−	0.974648i	$-0.428172\pi$
$809$	12.7279			0.447490			0.223745	+	0.974648i	$0.428172\pi$
$810$		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
$815$	−22.6274	−	11.3137i	−0.792604	−	0.396302i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			35.3553i			1.23391i	0.786998	+	0.616955i	$0.211634\pi$
$821$			35.3553i			1.23391i	−0.786998	+	0.616955i	$0.788366\pi$
$822$		0			0
$823$	−10.0000	−	10.0000i	−0.348578	−	0.348578i	0.511002	−	0.859580i	$-0.329275\pi$
$823$	−10.0000	−	10.0000i	−0.348578	−	0.348578i	−0.859580	+	0.511002i	$0.829275\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22.6274	+	22.6274i	0.786832	+	0.786832i	0.980974	−	0.194141i	$-0.0621920\pi$
$827$	22.6274	+	22.6274i	0.786832	+	0.786832i	−0.194141	+	0.980974i	$0.562192\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.82843	−	2.82843i	0.0979992	−	0.0979992i
$834$		0			0
$835$	−40.0000	−	20.0000i	−1.38426	−	0.692129i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	33.9411			1.17178			0.585889	−	0.810391i	$-0.300745\pi$
$839$	33.9411			1.17178			0.585889	+	0.810391i	$0.300745\pi$
$840$		0			0
$841$	−11.0000			−0.379310
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.77817	+	23.3345i	0.267577	+	0.802732i
$846$		0			0
$847$	6.00000	−	6.00000i	0.206162	−	0.206162i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	5.65685i		−	0.193914i
$852$		0			0
$853$	−17.0000	−	17.0000i	−0.582069	−	0.582069i	0.353402	−	0.935471i	$-0.385025\pi$
$853$	−17.0000	−	17.0000i	−0.582069	−	0.582069i	−0.935471	+	0.353402i	$0.885025\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−31.1127	−	31.1127i	−1.06279	−	1.06279i	−0.997892	−	0.0648976i	$-0.979328\pi$
$857$	−31.1127	−	31.1127i	−1.06279	−	1.06279i	−0.0648976	−	0.997892i	$-0.520672\pi$
$858$		0			0
$859$			12.0000i			0.409435i	0.978821	+	0.204717i	$0.0656275\pi$
$859$			12.0000i			0.409435i	−0.978821	+	0.204717i	$0.934372\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.82843	−	2.82843i	0.0962808	−	0.0962808i	−0.657326	−	0.753607i	$-0.728312\pi$
$863$	2.82843	−	2.82843i	0.0962808	−	0.0962808i	0.753607	+	0.657326i	$0.228312\pi$
$864$		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
$871$	−8.00000			−0.271070
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5.65685	+	31.1127i	0.191237	+	1.05180i
$876$		0			0
$877$	13.0000	−	13.0000i	0.438979	−	0.438979i	−0.452689	−	0.891668i	$-0.649535\pi$
$877$	13.0000	−	13.0000i	0.438979	−	0.438979i	0.891668	+	0.452689i	$0.149535\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	7.07107i		−	0.238230i	−0.992880	−	0.119115i	$-0.961994\pi$
$881$		−	7.07107i		−	0.238230i	0.992880	−	0.119115i	$-0.0380058\pi$
$882$		0			0
$883$	8.00000	+	8.00000i	0.269221	+	0.269221i	0.828786	−	0.559565i	$-0.189032\pi$
$883$	8.00000	+	8.00000i	0.269221	+	0.269221i	−0.559565	+	0.828786i	$0.689032\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	14.1421	+	14.1421i	0.474846	+	0.474846i	0.903479	−	0.428632i	$-0.141004\pi$
$887$	14.1421	+	14.1421i	0.474846	+	0.474846i	−0.428632	+	0.903479i	$0.641004\pi$
$888$		0			0
$889$			40.0000i			1.34156i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	54.0000	−	18.0000i	1.80502	−	0.601674i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	16.9706			0.566000
$900$		0			0
$901$	−16.0000			−0.533037
$902$		0			0
$903$		0			0
$904$		0			0
$905$	33.9411	−	11.3137i	1.12824	−	0.376080i
$906$		0			0
$907$	20.0000	−	20.0000i	0.664089	−	0.664089i	−0.292252	−	0.956341i	$-0.594405\pi$
$907$	20.0000	−	20.0000i	0.664089	−	0.664089i	0.956341	+	0.292252i	$0.0944047\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	22.6274i		−	0.749680i	−0.927090	−	0.374840i	$-0.877698\pi$
$911$		−	22.6274i		−	0.749680i	0.927090	−	0.374840i	$-0.122302\pi$
$912$		0			0
$913$	8.00000	+	8.00000i	0.264761	+	0.264761i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−28.2843	−	28.2843i	−0.934029	−	0.934029i
$918$		0			0
$919$		−	36.0000i		−	1.18753i	−0.804638	−	0.593765i	$-0.797641\pi$
$919$		−	36.0000i		−	1.18753i	0.804638	−	0.593765i	$-0.202359\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.65685	−	5.65685i	0.186198	−	0.186198i
$924$		0			0
$925$	1.00000	−	7.00000i	0.0328798	−	0.230159i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−46.6690			−1.53116			−0.765581	−	0.643340i	$-0.777548\pi$
$929$	−46.6690			−1.53116			−0.765581	+	0.643340i	$0.777548\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11.3137	−	22.6274i	0.369998	−	0.739996i
$936$		0			0
$937$	37.0000	−	37.0000i	1.20874	−	1.20874i	0.237301	−	0.971436i	$-0.423737\pi$
$937$	37.0000	−	37.0000i	1.20874	−	1.20874i	0.971436	−	0.237301i	$-0.0762628\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1.41421i			0.0461020i	0.999734	+	0.0230510i	$0.00733802\pi$
$941$			1.41421i			0.0461020i	−0.999734	+	0.0230510i	$0.992662\pi$
$942$		0			0
$943$	4.00000	+	4.00000i	0.130258	+	0.130258i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.65685	+	5.65685i	0.183823	+	0.183823i	0.793019	−	0.609196i	$-0.208508\pi$
$947$	5.65685	+	5.65685i	0.183823	+	0.183823i	−0.609196	+	0.793019i	$0.708508\pi$
$948$		0			0
$949$			2.00000i			0.0649227i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15.5563	+	15.5563i	−0.503920	+	0.503920i	−0.912654	−	0.408734i	$-0.865970\pi$
$953$	−15.5563	+	15.5563i	−0.503920	+	0.503920i	0.408734	+	0.912654i	$0.365970\pi$
$954$		0			0
$955$	16.0000	+	48.0000i	0.517748	+	1.55324i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	28.2843			0.913347
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.82843	−	1.41421i	−0.0910503	−	0.0455251i
$966$		0			0
$967$	−22.0000	+	22.0000i	−0.707472	+	0.707472i	−0.966003	−	0.258531i	$-0.916762\pi$
$967$	−22.0000	+	22.0000i	−0.707472	+	0.707472i	0.258531	+	0.966003i	$0.416762\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			19.7990i			0.635380i	0.948195	+	0.317690i	$0.102907\pi$
$971$			19.7990i			0.635380i	−0.948195	+	0.317690i	$0.897093\pi$
$972$		0			0
$973$	24.0000	+	24.0000i	0.769405	+	0.769405i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	36.7696	+	36.7696i	1.17636	+	1.17636i	0.980664	+	0.195698i	$0.0626972\pi$
$977$	36.7696	+	36.7696i	1.17636	+	1.17636i	0.195698	+	0.980664i	$0.437303\pi$
$978$		0			0
$979$		−	36.0000i		−	1.15056i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−22.6274	+	22.6274i	−0.721703	+	0.721703i	−0.968952	−	0.247249i	$-0.920473\pi$
$983$	−22.6274	+	22.6274i	−0.721703	+	0.721703i	0.247249	+	0.968952i	$0.420473\pi$
$984$		0			0
$985$	28.0000	+	14.0000i	0.892154	+	0.446077i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	45.2548			1.43902
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	25.0000	−	25.0000i	0.791758	−	0.791758i	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	25.0000	−	25.0000i	0.791758	−	0.791758i	0.981780	+	0.190022i	$0.0608559\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.2.w.d.593.1		4
3.2	odd	2	inner	720.2.w.d.593.2		4
4.3	odd	2		45.2.f.a.8.2	yes	4
5.2	odd	4	inner	720.2.w.d.17.2		4
5.3	odd	4		3600.2.w.b.1457.2		4
5.4	even	2		3600.2.w.b.593.2		4
8.3	odd	2		2880.2.w.b.2753.2		4
8.5	even	2		2880.2.w.k.2753.2		4
12.11	even	2		45.2.f.a.8.1	✓	4
15.2	even	4	inner	720.2.w.d.17.1		4
15.8	even	4		3600.2.w.b.1457.1		4
15.14	odd	2		3600.2.w.b.593.1		4
20.3	even	4		225.2.f.a.107.2		4
20.7	even	4		45.2.f.a.17.1	yes	4
20.19	odd	2		225.2.f.a.143.1		4
24.5	odd	2		2880.2.w.k.2753.1		4
24.11	even	2		2880.2.w.b.2753.1		4
36.7	odd	6		405.2.m.a.53.2		8
36.11	even	6		405.2.m.a.53.1		8
36.23	even	6		405.2.m.a.188.2		8
36.31	odd	6		405.2.m.a.188.1		8
40.27	even	4		2880.2.w.b.2177.1		4
40.37	odd	4		2880.2.w.k.2177.1		4
60.23	odd	4		225.2.f.a.107.1		4
60.47	odd	4		45.2.f.a.17.2	yes	4
60.59	even	2		225.2.f.a.143.2		4
120.77	even	4		2880.2.w.k.2177.2		4
120.107	odd	4		2880.2.w.b.2177.2		4
180.7	even	12		405.2.m.a.377.2		8
180.47	odd	12		405.2.m.a.377.1		8
180.67	even	12		405.2.m.a.107.1		8
180.167	odd	12		405.2.m.a.107.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.f.a.8.1	✓	4	12.11	even	2
45.2.f.a.8.2	yes	4	4.3	odd	2
45.2.f.a.17.1	yes	4	20.7	even	4
45.2.f.a.17.2	yes	4	60.47	odd	4
225.2.f.a.107.1		4	60.23	odd	4
225.2.f.a.107.2		4	20.3	even	4
225.2.f.a.143.1		4	20.19	odd	2
225.2.f.a.143.2		4	60.59	even	2
405.2.m.a.53.1		8	36.11	even	6
405.2.m.a.53.2		8	36.7	odd	6
405.2.m.a.107.1		8	180.67	even	12
405.2.m.a.107.2		8	180.167	odd	12
405.2.m.a.188.1		8	36.31	odd	6
405.2.m.a.188.2		8	36.23	even	6
405.2.m.a.377.1		8	180.47	odd	12
405.2.m.a.377.2		8	180.7	even	12
720.2.w.d.17.1		4	15.2	even	4	inner
720.2.w.d.17.2		4	5.2	odd	4	inner
720.2.w.d.593.1		4	1.1	even	1	trivial
720.2.w.d.593.2		4	3.2	odd	2	inner
2880.2.w.b.2177.1		4	40.27	even	4
2880.2.w.b.2177.2		4	120.107	odd	4
2880.2.w.b.2753.1		4	24.11	even	2
2880.2.w.b.2753.2		4	8.3	odd	2
2880.2.w.k.2177.1		4	40.37	odd	4
2880.2.w.k.2177.2		4	120.77	even	4
2880.2.w.k.2753.1		4	24.5	odd	2
2880.2.w.k.2753.2		4	8.5	even	2
3600.2.w.b.593.1		4	15.14	odd	2
3600.2.w.b.593.2		4	5.4	even	2
3600.2.w.b.1457.1		4	15.8	even	4
3600.2.w.b.1457.2		4	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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)

\(f(q)\)	\(=\)	\(q+(-2.12132 + 0.707107i) q^{5} +(2.00000 - 2.00000i) q^{7} +O(q^{10})\)	\(q+(-2.12132 + 0.707107i) q^{5} +(2.00000 - 2.00000i) q^{7} +2.82843i q^{11} +(1.00000 + 1.00000i) q^{13} +(2.82843 + 2.82843i) q^{17} +(2.82843 - 2.82843i) q^{23} +(4.00000 - 3.00000i) q^{25} +4.24264 q^{29} +4.00000 q^{31} +(-2.82843 + 5.65685i) q^{35} +(1.00000 - 1.00000i) q^{37} +1.41421i q^{41} +(8.00000 + 8.00000i) q^{43} +(5.65685 + 5.65685i) q^{47} -1.00000i q^{49} +(-2.82843 + 2.82843i) q^{53} +(-2.00000 - 6.00000i) q^{55} -8.48528 q^{59} +8.00000 q^{61} +(-2.82843 - 1.41421i) q^{65} +(-4.00000 + 4.00000i) q^{67} -5.65685i q^{71} +(1.00000 + 1.00000i) q^{73} +(5.65685 + 5.65685i) q^{77} -12.0000i q^{79} +(2.82843 - 2.82843i) q^{83} +(-8.00000 - 4.00000i) q^{85} -12.7279 q^{89} +4.00000 q^{91} +(-11.0000 + 11.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7}+O(q^{10}) \) 4 * q + 8 * q^7	\( 4 q + 8 q^{7} + 4 q^{13} + 16 q^{25} + 16 q^{31} + 4 q^{37} + 32 q^{43} - 8 q^{55} + 32 q^{61} - 16 q^{67} + 4 q^{73} - 32 q^{85} + 16 q^{91} - 44 q^{97}+O(q^{100}) \) 4 * q + 8 * q^7 + 4 * q^13 + 16 * q^25 + 16 * q^31 + 4 * q^37 + 32 * q^43 - 8 * q^55 + 32 * q^61 - 16 * q^67 + 4 * q^73 - 32 * q^85 + 16 * q^91 - 44 * q^97

Introduction

L-functions

Modular forms

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