LMFDB - Embedded newform 3600.2.w.a.1457.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,2,Mod(593,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))

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

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

chi := DirichletCharacter("3600.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3600.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:	$28.7461447277$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 450)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1457.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3600.1457
Dual form			3600.2.w.a.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.00000 - 3.00000i) q^{7} +O(q^{10})$	$q+(-3.00000 - 3.00000i) q^{7} +4.24264i q^{11} +(3.00000 - 3.00000i) q^{13} +(-4.24264 + 4.24264i) q^{17} -2.00000i q^{19} +(-4.24264 - 4.24264i) q^{23} +8.48528 q^{29} -4.00000 q^{31} +(3.00000 + 3.00000i) q^{37} +4.24264i q^{41} +11.0000i q^{49} +(-4.24264 - 4.24264i) q^{53} -4.24264 q^{59} -10.0000 q^{61} +(6.00000 + 6.00000i) q^{67} +8.48528i q^{71} +(-6.00000 + 6.00000i) q^{73} +(12.7279 - 12.7279i) q^{77} +8.00000i q^{79} +(8.48528 + 8.48528i) q^{83} +4.24264 q^{89} -18.0000 q^{91} +(-12.0000 - 12.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{7}+O(q^{10}) $ 4 * q - 12 * q^7	$ 4 q - 12 q^{7} + 12 q^{13} - 16 q^{31} + 12 q^{37} - 40 q^{61} + 24 q^{67} - 24 q^{73} - 72 q^{91} - 48 q^{97}+O(q^{100}) $ 4 * q - 12 * q^7 + 12 * q^13 - 16 * q^31 + 12 * q^37 - 40 * q^61 + 24 * q^67 - 24 * q^73 - 72 * q^91 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.00000	−	3.00000i	−1.13389	−	1.13389i	−0.989524	−	0.144370i	$-0.953885\pi$
$7$	−3.00000	−	3.00000i	−1.13389	−	1.13389i	−0.144370	−	0.989524i	$-0.546115\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.24264i			1.27920i	0.768706	+	0.639602i	$0.220901\pi$
$11$			4.24264i			1.27920i	−0.768706	+	0.639602i	$0.779099\pi$
$12$		0			0
$13$	3.00000	−	3.00000i	0.832050	−	0.832050i	−0.155747	−	0.987797i	$-0.549778\pi$
$13$	3.00000	−	3.00000i	0.832050	−	0.832050i	0.987797	+	0.155747i	$0.0497784\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.24264	+	4.24264i	−1.02899	+	1.02899i	−0.0294245	+	0.999567i	$0.509367\pi$
$17$	−4.24264	+	4.24264i	−1.02899	+	1.02899i	−0.999567	+	0.0294245i	$0.990633\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.24264	−	4.24264i	−0.884652	−	0.884652i	0.109351	−	0.994003i	$-0.465123\pi$
$23$	−4.24264	−	4.24264i	−0.884652	−	0.884652i	−0.994003	+	0.109351i	$0.965123\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	8.48528			1.57568			0.787839	−	0.615882i	$-0.211200\pi$
$29$	8.48528			1.57568			0.787839	+	0.615882i	$0.211200\pi$
$30$		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
$35$		0			0
$36$		0			0
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	0.909312	−	0.416115i	$-0.136609\pi$
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	−0.416115	+	0.909312i	$0.636609\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.24264i			0.662589i	0.943527	+	0.331295i	$0.107485\pi$
$41$			4.24264i			0.662589i	−0.943527	+	0.331295i	$0.892515\pi$
$42$		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
$49$			11.0000i			1.57143i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.24264	−	4.24264i	−0.582772	−	0.582772i	0.352892	−	0.935664i	$-0.385198\pi$
$53$	−4.24264	−	4.24264i	−0.582772	−	0.582772i	−0.935664	+	0.352892i	$0.885198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.24264			−0.552345			−0.276172	−	0.961108i	$-0.589066\pi$
$59$	−4.24264			−0.552345			−0.276172	+	0.961108i	$0.589066\pi$
$60$		0			0
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.00000	+	6.00000i	0.733017	+	0.733017i	0.971216	−	0.238200i	$-0.0765572\pi$
$67$	6.00000	+	6.00000i	0.733017	+	0.733017i	−0.238200	+	0.971216i	$0.576557\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			8.48528i			1.00702i	0.863990	+	0.503509i	$0.167958\pi$
$71$			8.48528i			1.00702i	−0.863990	+	0.503509i	$0.832042\pi$
$72$		0			0
$73$	−6.00000	+	6.00000i	−0.702247	+	0.702247i	−0.964892	−	0.262646i	$-0.915405\pi$
$73$	−6.00000	+	6.00000i	−0.702247	+	0.702247i	0.262646	+	0.964892i	$0.415405\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.7279	−	12.7279i	1.45048	−	1.45048i
$78$		0			0
$79$			8.00000i			0.900070i	0.893011	+	0.450035i	$0.148589\pi$
$79$			8.00000i			0.900070i	−0.893011	+	0.450035i	$0.851411\pi$
$80$		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
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.24264			0.449719			0.224860	−	0.974391i	$-0.427808\pi$
$89$	4.24264			0.449719			0.224860	+	0.974391i	$0.427808\pi$
$90$		0			0
$91$	−18.0000			−1.88691
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−12.0000	−	12.0000i	−1.21842	−	1.21842i	−0.968187	−	0.250229i	$-0.919494\pi$
$97$	−12.0000	−	12.0000i	−1.21842	−	1.21842i	−0.250229	−	0.968187i	$-0.580506\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			8.48528i			0.844317i	0.906522	+	0.422159i	$0.138727\pi$
$101$			8.48528i			0.844317i	−0.906522	+	0.422159i	$0.861273\pi$
$102$		0			0
$103$	3.00000	−	3.00000i	0.295599	−	0.295599i	−0.543688	−	0.839287i	$-0.682973\pi$
$103$	3.00000	−	3.00000i	0.295599	−	0.295599i	0.839287	+	0.543688i	$0.182973\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$		−	2.00000i		−	0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$		−	2.00000i		−	0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.7279	+	12.7279i	1.19734	+	1.19734i	0.974959	+	0.222383i	$0.0713835\pi$
$113$	12.7279	+	12.7279i	1.19734	+	1.19734i	0.222383	+	0.974959i	$0.428617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	25.4558			2.33353
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	9.00000	+	9.00000i	0.798621	+	0.798621i	0.982878	−	0.184257i	$-0.0589879\pi$
$127$	9.00000	+	9.00000i	0.798621	+	0.798621i	−0.184257	+	0.982878i	$0.558988\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			21.2132i			1.85341i	0.375794	+	0.926703i	$0.377370\pi$
$131$			21.2132i			1.85341i	−0.375794	+	0.926703i	$0.622630\pi$
$132$		0			0
$133$	−6.00000	+	6.00000i	−0.520266	+	0.520266i
$134$		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
$143$	12.7279	+	12.7279i	1.06436	+	1.06436i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		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
$155$		0			0
$156$		0			0
$157$	9.00000	+	9.00000i	0.718278	+	0.718278i	0.968252	−	0.249974i	$-0.0804222\pi$
$157$	9.00000	+	9.00000i	0.718278	+	0.718278i	−0.249974	+	0.968252i	$0.580422\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			25.4558i			2.00620i
$162$		0			0
$163$	−6.00000	+	6.00000i	−0.469956	+	0.469956i	−0.901900	−	0.431944i	$-0.857828\pi$
$163$	−6.00000	+	6.00000i	−0.469956	+	0.469956i	0.431944	+	0.901900i	$0.357828\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.24264	+	4.24264i	−0.328305	+	0.328305i	−0.851942	−	0.523636i	$-0.824575\pi$
$167$	−4.24264	+	4.24264i	−0.328305	+	0.328305i	0.523636	+	0.851942i	$0.324575\pi$
$168$		0			0
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.24264	−	4.24264i	−0.322562	−	0.322562i	0.527187	−	0.849749i	$-0.323247\pi$
$173$	−4.24264	−	4.24264i	−0.322562	−	0.322562i	−0.849749	+	0.527187i	$0.823247\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.24264			0.317110			0.158555	−	0.987350i	$-0.449317\pi$
$179$	4.24264			0.317110			0.158555	+	0.987350i	$0.449317\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−18.0000	−	18.0000i	−1.31629	−	1.31629i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	8.48528i		−	0.613973i	−0.951714	−	0.306987i	$-0.900679\pi$
$191$		−	8.48528i		−	0.613973i	0.951714	−	0.306987i	$-0.0993207\pi$
$192$		0			0
$193$	6.00000	−	6.00000i	0.431889	−	0.431889i	−0.457381	−	0.889271i	$-0.651213\pi$
$193$	6.00000	−	6.00000i	0.431889	−	0.431889i	0.889271	+	0.457381i	$0.151213\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.48528	+	8.48528i	−0.604551	+	0.604551i	−0.941517	−	0.336966i	$-0.890599\pi$
$197$	−8.48528	+	8.48528i	−0.604551	+	0.604551i	0.336966	+	0.941517i	$0.390599\pi$
$198$		0			0
$199$			20.0000i			1.41776i	0.705328	+	0.708881i	$0.250800\pi$
$199$			20.0000i			1.41776i	−0.705328	+	0.708881i	$0.749200\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−25.4558	−	25.4558i	−1.78665	−	1.78665i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.48528			0.586939
$210$		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
$217$	12.0000	+	12.0000i	0.814613	+	0.814613i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			25.4558i			1.71235i
$222$		0			0
$223$	−9.00000	+	9.00000i	−0.602685	+	0.602685i	−0.941024	−	0.338340i	$-0.890135\pi$
$223$	−9.00000	+	9.00000i	−0.602685	+	0.602685i	0.338340	+	0.941024i	$0.390135\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.48528	−	8.48528i	0.563188	−	0.563188i	−0.367024	−	0.930212i	$-0.619623\pi$
$227$	8.48528	−	8.48528i	0.563188	−	0.563188i	0.930212	+	0.367024i	$0.119623\pi$
$228$		0			0
$229$			14.0000i			0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$			14.0000i			0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.7279	−	12.7279i	−0.833834	−	0.833834i	0.154205	−	0.988039i	$-0.450718\pi$
$233$	−12.7279	−	12.7279i	−0.833834	−	0.833834i	−0.988039	+	0.154205i	$0.950718\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	25.4558			1.64660			0.823301	−	0.567605i	$-0.192130\pi$
$239$	25.4558			1.64660			0.823301	+	0.567605i	$0.192130\pi$
$240$		0			0
$241$	−28.0000			−1.80364			−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000			−1.80364			−0.901819	+	0.432113i	$0.857768\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.00000	−	6.00000i	−0.381771	−	0.381771i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			12.7279i			0.803379i	0.915776	+	0.401690i	$0.131577\pi$
$251$			12.7279i			0.803379i	−0.915776	+	0.401690i	$0.868423\pi$
$252$		0			0
$253$	18.0000	−	18.0000i	1.13165	−	1.13165i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.24264	+	4.24264i	−0.264649	+	0.264649i	−0.826940	−	0.562291i	$-0.809920\pi$
$257$	−4.24264	+	4.24264i	−0.264649	+	0.264649i	0.562291	+	0.826940i	$0.309920\pi$
$258$		0			0
$259$		−	18.0000i		−	1.11847i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.7279	−	12.7279i	−0.784837	−	0.784837i	0.195805	−	0.980643i	$-0.437268\pi$
$263$	−12.7279	−	12.7279i	−0.784837	−	0.784837i	−0.980643	+	0.195805i	$0.937268\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.48528			−0.517357			−0.258678	−	0.965964i	$-0.583287\pi$
$269$	−8.48528			−0.517357			−0.258678	+	0.965964i	$0.583287\pi$
$270$		0			0
$271$	−20.0000			−1.21491			−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000			−1.21491			−0.607457	+	0.794353i	$0.707810\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.00000	−	9.00000i	−0.540758	−	0.540758i	0.382993	−	0.923751i	$-0.374893\pi$
$277$	−9.00000	−	9.00000i	−0.540758	−	0.540758i	−0.923751	+	0.382993i	$0.874893\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	12.7279i		−	0.759284i	−0.925133	−	0.379642i	$-0.876047\pi$
$281$		−	12.7279i		−	0.759284i	0.925133	−	0.379642i	$-0.123953\pi$
$282$		0			0
$283$	18.0000	−	18.0000i	1.06999	−	1.06999i	0.0726300	−	0.997359i	$-0.476861\pi$
$283$	18.0000	−	18.0000i	1.06999	−	1.06999i	0.997359	−	0.0726300i	$-0.0231392\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.7279	−	12.7279i	0.751305	−	0.751305i
$288$		0			0
$289$		−	19.0000i		−	1.11765i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	8.48528	+	8.48528i	0.495715	+	0.495715i	0.910101	−	0.414386i	$-0.136004\pi$
$293$	8.48528	+	8.48528i	0.495715	+	0.495715i	−0.414386	+	0.910101i	$0.636004\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−25.4558			−1.47215
$300$		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
$311$		−	8.48528i		−	0.481156i	−0.970630	−	0.240578i	$-0.922663\pi$
$311$		−	8.48528i		−	0.481156i	0.970630	−	0.240578i	$-0.0773370\pi$
$312$		0			0
$313$	6.00000	−	6.00000i	0.339140	−	0.339140i	−0.516904	−	0.856044i	$-0.672915\pi$
$313$	6.00000	−	6.00000i	0.339140	−	0.339140i	0.856044	+	0.516904i	$0.172915\pi$
$314$		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
$319$			36.0000i			2.01561i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.48528	+	8.48528i	0.472134	+	0.472134i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−26.0000			−1.42909			−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000			−1.42909			−0.714545	+	0.699590i	$0.753366\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	6.00000	+	6.00000i	0.326841	+	0.326841i	0.851384	−	0.524543i	$-0.175764\pi$
$337$	6.00000	+	6.00000i	0.326841	+	0.326841i	−0.524543	+	0.851384i	$0.675764\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	16.9706i		−	0.919007i
$342$		0			0
$343$	12.0000	−	12.0000i	0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−25.4558	+	25.4558i	−1.36654	+	1.36654i	−0.501223	+	0.865318i	$0.667117\pi$
$347$	−25.4558	+	25.4558i	−1.36654	+	1.36654i	−0.865318	+	0.501223i	$0.832883\pi$
$348$		0			0
$349$			26.0000i			1.39175i	0.718164	+	0.695874i	$0.244983\pi$
$349$			26.0000i			1.39175i	−0.718164	+	0.695874i	$0.755017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.7279	−	12.7279i	−0.677439	−	0.677439i	0.281981	−	0.959420i	$-0.409008\pi$
$353$	−12.7279	−	12.7279i	−0.677439	−	0.677439i	−0.959420	+	0.281981i	$0.909008\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.9706			0.895672			0.447836	−	0.894116i	$-0.352195\pi$
$359$	16.9706			0.895672			0.447836	+	0.894116i	$0.352195\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	3.00000	+	3.00000i	0.156599	+	0.156599i	0.781058	−	0.624459i	$-0.214680\pi$
$367$	3.00000	+	3.00000i	0.156599	+	0.156599i	−0.624459	+	0.781058i	$0.714680\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			25.4558i			1.32160i
$372$		0			0
$373$	−15.0000	+	15.0000i	−0.776671	+	0.776671i	−0.979263	−	0.202593i	$-0.935063\pi$
$373$	−15.0000	+	15.0000i	−0.776671	+	0.776671i	0.202593	+	0.979263i	$0.435063\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	25.4558	−	25.4558i	1.31104	−	1.31104i
$378$		0			0
$379$		−	20.0000i		−	1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$		−	20.0000i		−	1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$		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
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$	36.0000			1.82060
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−27.0000	−	27.0000i	−1.35509	−	1.35509i	−0.879862	−	0.475229i	$-0.842365\pi$
$397$	−27.0000	−	27.0000i	−1.35509	−	1.35509i	−0.475229	−	0.879862i	$-0.657635\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	12.7279i		−	0.635602i	−0.948157	−	0.317801i	$-0.897056\pi$
$401$		−	12.7279i		−	0.635602i	0.948157	−	0.317801i	$-0.102944\pi$
$402$		0			0
$403$	−12.0000	+	12.0000i	−0.597763	+	0.597763i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.7279	+	12.7279i	−0.630900	+	0.630900i
$408$		0			0
$409$			14.0000i			0.692255i	0.938187	+	0.346128i	$0.112504\pi$
$409$			14.0000i			0.692255i	−0.938187	+	0.346128i	$0.887496\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	12.7279	+	12.7279i	0.626300	+	0.626300i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−38.1838			−1.86540			−0.932700	−	0.360654i	$-0.882553\pi$
$419$	−38.1838			−1.86540			−0.932700	+	0.360654i	$0.882553\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	30.0000	+	30.0000i	1.45180	+	1.45180i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	16.9706i		−	0.817443i	−0.912659	−	0.408722i	$-0.865975\pi$
$431$		−	16.9706i		−	0.817443i	0.912659	−	0.408722i	$-0.134025\pi$
$432$		0			0
$433$	−18.0000	+	18.0000i	−0.865025	+	0.865025i	−0.991917	−	0.126892i	$-0.959500\pi$
$433$	−18.0000	+	18.0000i	−0.865025	+	0.865025i	0.126892	+	0.991917i	$0.459500\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.48528	+	8.48528i	−0.405906	+	0.405906i
$438$		0			0
$439$		−	28.0000i		−	1.33637i	−0.743996	−	0.668184i	$-0.767072\pi$
$439$		−	28.0000i		−	1.33637i	0.743996	−	0.668184i	$-0.232928\pi$
$440$		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
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	29.6985			1.40156			0.700779	−	0.713378i	$-0.252836\pi$
$449$	29.6985			1.40156			0.700779	+	0.713378i	$0.252836\pi$
$450$		0			0
$451$	−18.0000			−0.847587
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			16.9706i			0.790398i	0.918596	+	0.395199i	$0.129324\pi$
$461$			16.9706i			0.790398i	−0.918596	+	0.395199i	$0.870676\pi$
$462$		0			0
$463$	−21.0000	+	21.0000i	−0.975953	+	0.975953i	−0.999718	−	0.0237648i	$-0.992435\pi$
$463$	−21.0000	+	21.0000i	−0.975953	+	0.975953i	0.0237648	+	0.999718i	$0.492435\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.48528	−	8.48528i	0.392652	−	0.392652i	−0.482980	−	0.875632i	$-0.660445\pi$
$467$	8.48528	−	8.48528i	0.392652	−	0.392652i	0.875632	+	0.482980i	$0.160445\pi$
$468$		0			0
$469$		−	36.0000i		−	1.66233i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8.48528			0.387702			0.193851	−	0.981031i	$-0.437902\pi$
$479$	8.48528			0.387702			0.193851	+	0.981031i	$0.437902\pi$
$480$		0			0
$481$	18.0000			0.820729
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−3.00000	−	3.00000i	−0.135943	−	0.135943i	0.635861	−	0.771804i	$-0.280645\pi$
$487$	−3.00000	−	3.00000i	−0.135943	−	0.135943i	−0.771804	+	0.635861i	$0.780645\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	4.24264i		−	0.191468i	−0.995407	−	0.0957338i	$-0.969480\pi$
$491$		−	4.24264i		−	0.191468i	0.995407	−	0.0957338i	$-0.0305198\pi$
$492$		0			0
$493$	−36.0000	+	36.0000i	−1.62136	+	1.62136i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	25.4558	−	25.4558i	1.14185	−	1.14185i
$498$		0			0
$499$		−	22.0000i		−	0.984855i	−0.870353	−	0.492428i	$-0.836110\pi$
$499$		−	22.0000i		−	0.984855i	0.870353	−	0.492428i	$-0.163890\pi$
$500$		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
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	25.4558			1.12831			0.564155	−	0.825669i	$-0.309202\pi$
$509$	25.4558			1.12831			0.564155	+	0.825669i	$0.309202\pi$
$510$		0			0
$511$	36.0000			1.59255
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	12.7279i		−	0.557620i	−0.960346	−	0.278810i	$-0.910060\pi$
$521$		−	12.7279i		−	0.557620i	0.960346	−	0.278810i	$-0.0899400\pi$
$522$		0			0
$523$	6.00000	−	6.00000i	0.262362	−	0.262362i	−0.563651	−	0.826013i	$-0.690604\pi$
$523$	6.00000	−	6.00000i	0.262362	−	0.262362i	0.826013	+	0.563651i	$0.190604\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	16.9706	−	16.9706i	0.739249	−	0.739249i
$528$		0			0
$529$			13.0000i			0.565217i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12.7279	+	12.7279i	0.551308	+	0.551308i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−46.6690			−2.01018
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	12.0000	+	12.0000i	0.513083	+	0.513083i	0.915470	−	0.402387i	$-0.131819\pi$
$547$	12.0000	+	12.0000i	0.513083	+	0.513083i	−0.402387	+	0.915470i	$0.631819\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	16.9706i		−	0.722970i
$552$		0			0
$553$	24.0000	−	24.0000i	1.02058	−	1.02058i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.7279	+	12.7279i	−0.539299	+	0.539299i	−0.923323	−	0.384024i	$-0.874538\pi$
$557$	−12.7279	+	12.7279i	−0.539299	+	0.539299i	0.384024	+	0.923323i	$0.374538\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	25.4558	+	25.4558i	1.07284	+	1.07284i	0.997130	+	0.0757057i	$0.0241210\pi$
$563$	25.4558	+	25.4558i	1.07284	+	1.07284i	0.0757057	+	0.997130i	$0.475879\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−46.6690			−1.95647			−0.978234	−	0.207504i	$-0.933466\pi$
$569$	−46.6690			−1.95647			−0.978234	+	0.207504i	$0.933466\pi$
$570$		0			0
$571$	22.0000			0.920671			0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000			0.920671			0.460336	+	0.887745i	$0.347729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	24.0000	+	24.0000i	0.999133	+	0.999133i	1.00000		0.000866551i	$-0.000275832\pi$
$577$	24.0000	+	24.0000i	0.999133	+	0.999133i	−0.000866551		1.00000i	$0.500276\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	50.9117i		−	2.11217i
$582$		0			0
$583$	18.0000	−	18.0000i	0.745484	−	0.745484i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.9706	−	16.9706i	0.700450	−	0.700450i	−0.264057	−	0.964507i	$-0.585061\pi$
$587$	16.9706	−	16.9706i	0.700450	−	0.700450i	0.964507	+	0.264057i	$0.0850607\pi$
$588$		0			0
$589$			8.00000i			0.329634i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4.24264	+	4.24264i	0.174224	+	0.174224i	0.788833	−	0.614608i	$-0.210686\pi$
$593$	4.24264	+	4.24264i	0.174224	+	0.174224i	−0.614608	+	0.788833i	$0.710686\pi$
$594$		0			0
$595$		0			0
$596$		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
$601$	28.0000			1.14214			0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000			1.14214			0.571072	+	0.820900i	$0.306528\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−21.0000	−	21.0000i	−0.852364	−	0.852364i	0.138060	−	0.990424i	$-0.455913\pi$
$607$	−21.0000	−	21.0000i	−0.852364	−	0.852364i	−0.990424	+	0.138060i	$0.955913\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	33.0000	−	33.0000i	1.33286	−	1.33286i	0.430055	−	0.902803i	$-0.358494\pi$
$613$	33.0000	−	33.0000i	1.33286	−	1.33286i	0.902803	−	0.430055i	$-0.141506\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.24264	+	4.24264i	−0.170802	+	0.170802i	−0.787332	−	0.616530i	$-0.788538\pi$
$617$	−4.24264	+	4.24264i	−0.170802	+	0.170802i	0.616530	+	0.787332i	$0.288538\pi$
$618$		0			0
$619$		−	28.0000i		−	1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$		−	28.0000i		−	1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.7279	−	12.7279i	−0.509933	−	0.509933i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−25.4558			−1.01499
$630$		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
$635$		0			0
$636$		0			0
$637$	33.0000	+	33.0000i	1.30751	+	1.30751i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			4.24264i			0.167574i	0.996484	+	0.0837871i	$0.0267016\pi$
$641$			4.24264i			0.167574i	−0.996484	+	0.0837871i	$0.973298\pi$
$642$		0			0
$643$	−18.0000	+	18.0000i	−0.709851	+	0.709851i	−0.966504	−	0.256653i	$-0.917380\pi$
$643$	−18.0000	+	18.0000i	−0.709851	+	0.709851i	0.256653	+	0.966504i	$0.417380\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−16.9706	+	16.9706i	−0.667182	+	0.667182i	−0.957063	−	0.289881i	$-0.906384\pi$
$647$	−16.9706	+	16.9706i	−0.667182	+	0.667182i	0.289881	+	0.957063i	$0.406384\pi$
$648$		0			0
$649$		−	18.0000i		−	0.706562i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−16.9706	−	16.9706i	−0.664109	−	0.664109i	0.292237	−	0.956346i	$-0.405601\pi$
$653$	−16.9706	−	16.9706i	−0.664109	−	0.664109i	−0.956346	+	0.292237i	$0.905601\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−38.1838			−1.48743			−0.743714	−	0.668498i	$-0.766938\pi$
$659$	−38.1838			−1.48743			−0.743714	+	0.668498i	$0.766938\pi$
$660$		0			0
$661$	22.0000			0.855701			0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000			0.855701			0.427850	+	0.903850i	$0.359271\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−36.0000	−	36.0000i	−1.39393	−	1.39393i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	42.4264i		−	1.63785i
$672$		0			0
$673$	30.0000	−	30.0000i	1.15642	−	1.15642i	0.171174	−	0.985241i	$-0.445244\pi$
$673$	30.0000	−	30.0000i	1.15642	−	1.15642i	0.985241	−	0.171174i	$-0.0547561\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$			72.0000i			2.76311i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−33.9411	−	33.9411i	−1.29872	−	1.29872i	−0.929237	−	0.369484i	$-0.879534\pi$
$683$	−33.9411	−	33.9411i	−1.29872	−	1.29872i	−0.369484	−	0.929237i	$-0.620466\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−25.4558			−0.969790
$690$		0			0
$691$	−10.0000			−0.380418			−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000			−0.380418			−0.190209	+	0.981744i	$0.560917\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−18.0000	−	18.0000i	−0.681799	−	0.681799i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	42.4264i		−	1.60242i	−0.598381	−	0.801212i	$-0.704189\pi$
$701$		−	42.4264i		−	1.60242i	0.598381	−	0.801212i	$-0.295811\pi$
$702$		0			0
$703$	6.00000	−	6.00000i	0.226294	−	0.226294i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	25.4558	−	25.4558i	0.957366	−	0.957366i
$708$		0			0
$709$		−	46.0000i		−	1.72757i	−0.503864	−	0.863783i	$-0.668089\pi$
$709$		−	46.0000i		−	1.72757i	0.503864	−	0.863783i	$-0.331911\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	16.9706	+	16.9706i	0.635553	+	0.635553i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	33.9411			1.26579			0.632895	−	0.774237i	$-0.281866\pi$
$719$	33.9411			1.26579			0.632895	+	0.774237i	$0.281866\pi$
$720$		0			0
$721$	−18.0000			−0.670355
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	9.00000	+	9.00000i	0.333792	+	0.333792i	0.854024	−	0.520233i	$-0.174155\pi$
$727$	9.00000	+	9.00000i	0.333792	+	0.333792i	−0.520233	+	0.854024i	$0.674155\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−9.00000	+	9.00000i	−0.332423	+	0.332423i	−0.853506	−	0.521083i	$-0.825528\pi$
$733$	−9.00000	+	9.00000i	−0.332423	+	0.332423i	0.521083	+	0.853506i	$0.325528\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−25.4558	+	25.4558i	−0.937678	+	0.937678i
$738$		0			0
$739$		−	38.0000i		−	1.39785i	−0.715194	−	0.698926i	$-0.753662\pi$
$739$		−	38.0000i		−	1.39785i	0.715194	−	0.698926i	$-0.246338\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−15.0000	−	15.0000i	−0.545184	−	0.545184i	0.379860	−	0.925044i	$-0.375972\pi$
$757$	−15.0000	−	15.0000i	−0.545184	−	0.545184i	−0.925044	+	0.379860i	$0.875972\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	29.6985i		−	1.07657i	−0.842763	−	0.538285i	$-0.819073\pi$
$761$		−	29.6985i		−	1.07657i	0.842763	−	0.538285i	$-0.180927\pi$
$762$		0			0
$763$	−6.00000	+	6.00000i	−0.217215	+	0.217215i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.7279	+	12.7279i	−0.459579	+	0.459579i
$768$		0			0
$769$			40.0000i			1.44244i	0.692708	+	0.721218i	$0.256418\pi$
$769$			40.0000i			1.44244i	−0.692708	+	0.721218i	$0.743582\pi$
$770$		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
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.48528			0.304017
$780$		0			0
$781$	−36.0000			−1.28818
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−30.0000	−	30.0000i	−1.06938	−	1.06938i	−0.997406	−	0.0719783i	$-0.977069\pi$
$787$	−30.0000	−	30.0000i	−1.06938	−	1.06938i	−0.0719783	−	0.997406i	$-0.522931\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	76.3675i		−	2.71532i
$792$		0			0
$793$	−30.0000	+	30.0000i	−1.06533	+	1.06533i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−21.2132	+	21.2132i	−0.751410	+	0.751410i	−0.974742	−	0.223332i	$-0.928307\pi$
$797$	−21.2132	+	21.2132i	−0.751410	+	0.751410i	0.223332	+	0.974742i	$0.428307\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−25.4558	−	25.4558i	−0.898317	−	0.898317i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−29.6985			−1.04414			−0.522072	−	0.852902i	$-0.674841\pi$
$809$	−29.6985			−1.04414			−0.522072	+	0.852902i	$0.674841\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	8.48528i		−	0.296138i	−0.988977	−	0.148069i	$-0.952694\pi$
$821$		−	8.48528i		−	0.296138i	0.988977	−	0.148069i	$-0.0473058\pi$
$822$		0			0
$823$	3.00000	−	3.00000i	0.104573	−	0.104573i	−0.652884	−	0.757458i	$-0.726441\pi$
$823$	3.00000	−	3.00000i	0.104573	−	0.104573i	0.757458	+	0.652884i	$0.226441\pi$
$824$		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
$829$		−	34.0000i		−	1.18087i	−0.807086	−	0.590434i	$-0.798956\pi$
$829$		−	34.0000i		−	1.18087i	0.807086	−	0.590434i	$-0.201044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−46.6690	−	46.6690i	−1.61699	−	1.61699i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−16.9706			−0.585889			−0.292944	−	0.956129i	$-0.594635\pi$
$839$	−16.9706			−0.585889			−0.292944	+	0.956129i	$0.594635\pi$
$840$		0			0
$841$	43.0000			1.48276
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	21.0000	+	21.0000i	0.721569	+	0.721569i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	25.4558i		−	0.872615i
$852$		0			0
$853$	21.0000	−	21.0000i	0.719026	−	0.719026i	−0.249380	−	0.968406i	$-0.580227\pi$
$853$	21.0000	−	21.0000i	0.719026	−	0.719026i	0.968406	+	0.249380i	$0.0802267\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−29.6985	+	29.6985i	−1.01448	+	1.01448i	−0.0145873	+	0.999894i	$0.504643\pi$
$857$	−29.6985	+	29.6985i	−1.01448	+	1.01448i	−0.999894	+	0.0145873i	$0.995357\pi$
$858$		0			0
$859$		−	4.00000i		−	0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$		−	4.00000i		−	0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.24264	+	4.24264i	0.144421	+	0.144421i	0.775621	−	0.631199i	$-0.217437\pi$
$863$	4.24264	+	4.24264i	0.144421	+	0.144421i	−0.631199	+	0.775621i	$0.717437\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−33.9411			−1.15137
$870$		0			0
$871$	36.0000			1.21981
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		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
$881$			46.6690i			1.57232i	0.618023	+	0.786160i	$0.287934\pi$
$881$			46.6690i			1.57232i	−0.618023	+	0.786160i	$0.712066\pi$
$882$		0			0
$883$	6.00000	−	6.00000i	0.201916	−	0.201916i	−0.598904	−	0.800821i	$-0.704397\pi$
$883$	6.00000	−	6.00000i	0.201916	−	0.201916i	0.800821	+	0.598904i	$0.204397\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.24264	−	4.24264i	0.142454	−	0.142454i	−0.632283	−	0.774737i	$-0.717882\pi$
$887$	4.24264	−	4.24264i	0.142454	−	0.142454i	0.774737	+	0.632283i	$0.217882\pi$
$888$		0			0
$889$		−	54.0000i		−	1.81110i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−33.9411			−1.13200
$900$		0			0
$901$	36.0000			1.19933
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	24.0000	+	24.0000i	0.796907	+	0.796907i	0.982607	−	0.185700i	$-0.0594551\pi$
$907$	24.0000	+	24.0000i	0.796907	+	0.796907i	−0.185700	+	0.982607i	$0.559455\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	16.9706i		−	0.562260i	−0.959670	−	0.281130i	$-0.909291\pi$
$911$		−	16.9706i		−	0.562260i	0.959670	−	0.281130i	$-0.0907092\pi$
$912$		0			0
$913$	−36.0000	+	36.0000i	−1.19143	+	1.19143i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	63.6396	−	63.6396i	2.10157	−	2.10157i
$918$		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
$923$	25.4558	+	25.4558i	0.837889	+	0.837889i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.24264			−0.139197			−0.0695983	−	0.997575i	$-0.522172\pi$
$929$	−4.24264			−0.139197			−0.0695983	+	0.997575i	$0.522172\pi$
$930$		0			0
$931$	22.0000			0.721021
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−24.0000	−	24.0000i	−0.784046	−	0.784046i	0.196465	−	0.980511i	$-0.437054\pi$
$937$	−24.0000	−	24.0000i	−0.784046	−	0.784046i	−0.980511	+	0.196465i	$0.937054\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			50.9117i			1.65967i	0.558006	+	0.829837i	$0.311567\pi$
$941$			50.9117i			1.65967i	−0.558006	+	0.829837i	$0.688433\pi$
$942$		0			0
$943$	18.0000	−	18.0000i	0.586161	−	0.586161i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−33.9411	+	33.9411i	−1.10294	+	1.10294i	−0.108884	+	0.994054i	$0.534728\pi$
$947$	−33.9411	+	33.9411i	−1.10294	+	1.10294i	−0.994054	+	0.108884i	$0.965272\pi$
$948$		0			0
$949$			36.0000i			1.16861i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	38.1838	+	38.1838i	1.23689	+	1.23689i	0.961264	+	0.275630i	$0.0888863\pi$
$953$	38.1838	+	38.1838i	1.23689	+	1.23689i	0.275630	+	0.961264i	$0.411114\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	25.4558			0.822012
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	15.0000	+	15.0000i	0.482367	+	0.482367i	0.905887	−	0.423520i	$-0.139205\pi$
$967$	15.0000	+	15.0000i	0.482367	+	0.482367i	−0.423520	+	0.905887i	$0.639205\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			29.6985i			0.953070i	0.879156	+	0.476535i	$0.158107\pi$
$971$			29.6985i			0.953070i	−0.879156	+	0.476535i	$0.841893\pi$
$972$		0			0
$973$	12.0000	−	12.0000i	0.384702	−	0.384702i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	12.7279	−	12.7279i	0.407202	−	0.407202i	−0.473560	−	0.880762i	$-0.657031\pi$
$977$	12.7279	−	12.7279i	0.407202	−	0.407202i	0.880762	+	0.473560i	$0.157031\pi$
$978$		0			0
$979$			18.0000i			0.575282i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	33.9411	+	33.9411i	1.08255	+	1.08255i	0.996271	+	0.0862831i	$0.0274990\pi$
$983$	33.9411	+	33.9411i	1.08255	+	1.08255i	0.0862831	+	0.996271i	$0.472501\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	52.0000			1.65183			0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000			1.65183			0.825917	+	0.563791i	$0.190658\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	27.0000	+	27.0000i	0.855099	+	0.855099i	0.990756	−	0.135657i	$-0.0433146\pi$
$997$	27.0000	+	27.0000i	0.855099	+	0.855099i	−0.135657	+	0.990756i	$0.543315\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.2.w.a.1457.2		4
3.2	odd	2	inner	3600.2.w.a.1457.1		4
4.3	odd	2		450.2.f.c.107.1	yes	4
5.2	odd	4		3600.2.w.h.593.2		4
5.3	odd	4	inner	3600.2.w.a.593.2		4
5.4	even	2		3600.2.w.h.1457.2		4
12.11	even	2		450.2.f.c.107.2	yes	4
15.2	even	4		3600.2.w.h.593.1		4
15.8	even	4	inner	3600.2.w.a.593.1		4
15.14	odd	2		3600.2.w.h.1457.1		4
20.3	even	4		450.2.f.c.143.2	yes	4
20.7	even	4		450.2.f.a.143.1	yes	4
20.19	odd	2		450.2.f.a.107.2	yes	4
60.23	odd	4		450.2.f.c.143.1	yes	4
60.47	odd	4		450.2.f.a.143.2	yes	4
60.59	even	2		450.2.f.a.107.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.f.a.107.1	✓	4	60.59	even	2
450.2.f.a.107.2	yes	4	20.19	odd	2
450.2.f.a.143.1	yes	4	20.7	even	4
450.2.f.a.143.2	yes	4	60.47	odd	4
450.2.f.c.107.1	yes	4	4.3	odd	2
450.2.f.c.107.2	yes	4	12.11	even	2
450.2.f.c.143.1	yes	4	60.23	odd	4
450.2.f.c.143.2	yes	4	20.3	even	4
3600.2.w.a.593.1		4	15.8	even	4	inner
3600.2.w.a.593.2		4	5.3	odd	4	inner
3600.2.w.a.1457.1		4	3.2	odd	2	inner
3600.2.w.a.1457.2		4	1.1	even	1	trivial
3600.2.w.h.593.1		4	15.2	even	4
3600.2.w.h.593.2		4	5.2	odd	4
3600.2.w.h.1457.1		4	15.14	odd	2
3600.2.w.h.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:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3600.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.00000 - 3.00000i) q^{7} +O(q^{10})\)	\(q+(-3.00000 - 3.00000i) q^{7} +4.24264i q^{11} +(3.00000 - 3.00000i) q^{13} +(-4.24264 + 4.24264i) q^{17} -2.00000i q^{19} +(-4.24264 - 4.24264i) q^{23} +8.48528 q^{29} -4.00000 q^{31} +(3.00000 + 3.00000i) q^{37} +4.24264i q^{41} +11.0000i q^{49} +(-4.24264 - 4.24264i) q^{53} -4.24264 q^{59} -10.0000 q^{61} +(6.00000 + 6.00000i) q^{67} +8.48528i q^{71} +(-6.00000 + 6.00000i) q^{73} +(12.7279 - 12.7279i) q^{77} +8.00000i q^{79} +(8.48528 + 8.48528i) q^{83} +4.24264 q^{89} -18.0000 q^{91} +(-12.0000 - 12.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{7}+O(q^{10}) \) 4 * q - 12 * q^7	\( 4 q - 12 q^{7} + 12 q^{13} - 16 q^{31} + 12 q^{37} - 40 q^{61} + 24 q^{67} - 24 q^{73} - 72 q^{91} - 48 q^{97}+O(q^{100}) \) 4 * q - 12 * q^7 + 12 * q^13 - 16 * q^31 + 12 * q^37 - 40 * q^61 + 24 * q^67 - 24 * q^73 - 72 * q^91 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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