LMFDB - Embedded newform 720.4.f.f.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,4,Mod(289,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("720.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	720.f (of order $2$, degree $1$, 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:	$42.4813752041$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	720.289
Dual form			720.4.f.f.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.00000 - 10.0000i) q^{5} +26.0000i q^{7} -28.0000 q^{11} +12.0000i q^{13} -64.0000i q^{17} -60.0000 q^{19} -58.0000i q^{23} +(-75.0000 - 100.000i) q^{25} +90.0000 q^{29} +128.000 q^{31} +(260.000 + 130.000i) q^{35} -236.000i q^{37} -242.000 q^{41} -362.000i q^{43} -226.000i q^{47} -333.000 q^{49} +108.000i q^{53} +(-140.000 + 280.000i) q^{55} +20.0000 q^{59} +542.000 q^{61} +(120.000 + 60.0000i) q^{65} -434.000i q^{67} -1128.00 q^{71} +632.000i q^{73} -728.000i q^{77} -720.000 q^{79} -478.000i q^{83} +(-640.000 - 320.000i) q^{85} -490.000 q^{89} -312.000 q^{91} +(-300.000 + 600.000i) q^{95} -1456.00i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{5} - 56 q^{11} - 120 q^{19} - 150 q^{25} + 180 q^{29} + 256 q^{31} + 520 q^{35} - 484 q^{41} - 666 q^{49} - 280 q^{55} + 40 q^{59} + 1084 q^{61} + 240 q^{65} - 2256 q^{71} - 1440 q^{79} - 1280 q^{85}+ \cdots - 600 q^{95}+O(q^{100}) $ 2 * q + 10 * q^5 - 56 * q^11 - 120 * q^19 - 150 * q^25 + 180 * q^29 + 256 * q^31 + 520 * q^35 - 484 * q^41 - 666 * q^49 - 280 * q^55 + 40 * q^59 + 1084 * q^61 + 240 * q^65 - 2256 * q^71 - 1440 * q^79 - 1280 * q^85 - 980 * q^89 - 624 * q^91 - 600 * q^95

Character values

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

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	5.00000	−	10.0000i	0.447214	−	0.894427i
$5$	5.00000	−	10.0000i	0.447214	−	0.894427i
$6$		0			0
$7$			26.0000i			1.40387i	0.712242	+	0.701934i	$0.247680\pi$
$7$			26.0000i			1.40387i	−0.712242	+	0.701934i	$0.752320\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−28.0000			−0.767483			−0.383742	−	0.923440i	$-0.625365\pi$
$11$	−28.0000			−0.767483			−0.383742	+	0.923440i	$0.625365\pi$
$12$		0			0
$13$			12.0000i			0.256015i	0.991773	+	0.128008i	$0.0408582\pi$
$13$			12.0000i			0.256015i	−0.991773	+	0.128008i	$0.959142\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	64.0000i		−	0.913075i	−0.889704	−	0.456538i	$-0.849089\pi$
$17$		−	64.0000i		−	0.913075i	0.889704	−	0.456538i	$-0.150911\pi$
$18$		0			0
$19$	−60.0000			−0.724471			−0.362235	−	0.932087i	$-0.617986\pi$
$19$	−60.0000			−0.724471			−0.362235	+	0.932087i	$0.617986\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	58.0000i		−	0.525819i	−0.964821	−	0.262909i	$-0.915318\pi$
$23$		−	58.0000i		−	0.525819i	0.964821	−	0.262909i	$-0.0846821\pi$
$24$		0			0
$25$	−75.0000	−	100.000i	−0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	90.0000			0.576296			0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000			0.576296			0.288148	+	0.957586i	$0.406961\pi$
$30$		0			0
$31$	128.000			0.741596			0.370798	−	0.928714i	$-0.379084\pi$
$31$	128.000			0.741596			0.370798	+	0.928714i	$0.379084\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	260.000	+	130.000i	1.25566	+	0.627829i
$36$		0			0
$37$		−	236.000i		−	1.04860i	−0.851534	−	0.524299i	$-0.824327\pi$
$37$		−	236.000i		−	1.04860i	0.851534	−	0.524299i	$-0.175673\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−242.000			−0.921806			−0.460903	−	0.887450i	$-0.652474\pi$
$41$	−242.000			−0.921806			−0.460903	+	0.887450i	$0.652474\pi$
$42$		0			0
$43$		−	362.000i		−	1.28383i	−0.766778	−	0.641913i	$-0.778141\pi$
$43$		−	362.000i		−	1.28383i	0.766778	−	0.641913i	$-0.221859\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	226.000i		−	0.701393i	−0.936489	−	0.350697i	$-0.885945\pi$
$47$		−	226.000i		−	0.701393i	0.936489	−	0.350697i	$-0.114055\pi$
$48$		0			0
$49$	−333.000			−0.970845
$50$		0			0
$51$		0			0
$52$		0			0
$53$			108.000i			0.279905i	0.990158	+	0.139952i	$0.0446949\pi$
$53$			108.000i			0.279905i	−0.990158	+	0.139952i	$0.955305\pi$
$54$		0			0
$55$	−140.000	+	280.000i	−0.343229	+	0.686458i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	20.0000			0.0441318			0.0220659	−	0.999757i	$-0.492976\pi$
$59$	20.0000			0.0441318			0.0220659	+	0.999757i	$0.492976\pi$
$60$		0			0
$61$	542.000			1.13764			0.568820	−	0.822462i	$-0.307400\pi$
$61$	542.000			1.13764			0.568820	+	0.822462i	$0.307400\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	120.000	+	60.0000i	0.228987	+	0.114494i
$66$		0			0
$67$		−	434.000i		−	0.791366i	−0.918387	−	0.395683i	$-0.870508\pi$
$67$		−	434.000i		−	0.791366i	0.918387	−	0.395683i	$-0.129492\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1128.00			−1.88548			−0.942739	−	0.333531i	$-0.891760\pi$
$71$	−1128.00			−1.88548			−0.942739	+	0.333531i	$0.891760\pi$
$72$		0			0
$73$			632.000i			1.01329i	0.862155	+	0.506644i	$0.169114\pi$
$73$			632.000i			1.01329i	−0.862155	+	0.506644i	$0.830886\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	728.000i		−	1.07745i
$78$		0			0
$79$	−720.000			−1.02540			−0.512698	−	0.858569i	$-0.671354\pi$
$79$	−720.000			−1.02540			−0.512698	+	0.858569i	$0.671354\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	478.000i		−	0.632136i	−0.948736	−	0.316068i	$-0.897637\pi$
$83$		−	478.000i		−	0.632136i	0.948736	−	0.316068i	$-0.102363\pi$
$84$		0			0
$85$	−640.000	−	320.000i	−0.816679	−	0.408340i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−490.000			−0.583594			−0.291797	−	0.956480i	$-0.594253\pi$
$89$	−490.000			−0.583594			−0.291797	+	0.956480i	$0.594253\pi$
$90$		0			0
$91$	−312.000			−0.359412
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−300.000	+	600.000i	−0.323993	+	0.647986i
$96$		0			0
$97$		−	1456.00i		−	1.52407i	−0.647538	−	0.762033i	$-0.724201\pi$
$97$		−	1456.00i		−	1.52407i	0.647538	−	0.762033i	$-0.275799\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	578.000			0.569437			0.284719	−	0.958611i	$-0.408100\pi$
$101$	578.000			0.569437			0.284719	+	0.958611i	$0.408100\pi$
$102$		0			0
$103$		−	1462.00i		−	1.39859i	−0.714831	−	0.699297i	$-0.753497\pi$
$103$		−	1462.00i		−	1.39859i	0.714831	−	0.699297i	$-0.246503\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	966.000i		−	0.872773i	−0.899759	−	0.436387i	$-0.856258\pi$
$107$		−	966.000i		−	0.872773i	0.899759	−	0.436387i	$-0.143742\pi$
$108$		0			0
$109$	−370.000			−0.325134			−0.162567	−	0.986698i	$-0.551977\pi$
$109$	−370.000			−0.325134			−0.162567	+	0.986698i	$0.551977\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			528.000i			0.439558i	0.975550	+	0.219779i	$0.0705336\pi$
$113$			528.000i			0.439558i	−0.975550	+	0.219779i	$0.929466\pi$
$114$		0			0
$115$	−580.000	−	290.000i	−0.470307	−	0.235153i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1664.00			1.28184
$120$		0			0
$121$	−547.000			−0.410969
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1375.00	+	250.000i	−0.983870	+	0.178885i
$126$		0			0
$127$		−	1534.00i		−	1.07181i	−0.844277	−	0.535907i	$-0.819970\pi$
$127$		−	1534.00i		−	1.07181i	0.844277	−	0.535907i	$-0.180030\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.0000			0.00800340			0.00400170	−	0.999992i	$-0.498726\pi$
$131$	12.0000			0.00800340			0.00400170	+	0.999992i	$0.498726\pi$
$132$		0			0
$133$		−	1560.00i		−	1.01706i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1224.00i		−	0.763309i	−0.924305	−	0.381655i	$-0.875354\pi$
$137$		−	1224.00i		−	0.763309i	0.924305	−	0.381655i	$-0.124646\pi$
$138$		0			0
$139$	3100.00			1.89164			0.945822	−	0.324685i	$-0.105258\pi$
$139$	3100.00			1.89164			0.945822	+	0.324685i	$0.105258\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	336.000i		−	0.196488i
$144$		0			0
$145$	450.000	−	900.000i	0.257727	−	0.515455i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	250.000			0.137455			0.0687275	−	0.997635i	$-0.478106\pi$
$149$	250.000			0.137455			0.0687275	+	0.997635i	$0.478106\pi$
$150$		0			0
$151$	−2152.00			−1.15978			−0.579892	−	0.814694i	$-0.696905\pi$
$151$	−2152.00			−1.15978			−0.579892	+	0.814694i	$0.696905\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	640.000	−	1280.00i	0.331652	−	0.663304i
$156$		0			0
$157$			524.000i			0.266368i	0.991091	+	0.133184i	$0.0425201\pi$
$157$			524.000i			0.266368i	−0.991091	+	0.133184i	$0.957480\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1508.00			0.738180
$162$		0			0
$163$			3518.00i			1.69050i	0.534373	+	0.845249i	$0.320548\pi$
$163$			3518.00i			1.69050i	−0.534373	+	0.845249i	$0.679452\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			534.000i			0.247438i	0.992317	+	0.123719i	$0.0394822\pi$
$167$			534.000i			0.247438i	−0.992317	+	0.123719i	$0.960518\pi$
$168$		0			0
$169$	2053.00			0.934456
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	4252.00i		−	1.86863i	−0.356444	−	0.934317i	$-0.616011\pi$
$173$		−	4252.00i		−	1.86863i	0.356444	−	0.934317i	$-0.383989\pi$
$174$		0			0
$175$	2600.00	−	1950.00i	1.12309	−	0.842321i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2500.00			−1.04390			−0.521952	−	0.852975i	$-0.674796\pi$
$179$	−2500.00			−1.04390			−0.521952	+	0.852975i	$0.674796\pi$
$180$		0			0
$181$	−2578.00			−1.05868			−0.529340	−	0.848410i	$-0.677561\pi$
$181$	−2578.00			−1.05868			−0.529340	+	0.848410i	$0.677561\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2360.00	−	1180.00i	−0.937895	−	0.468948i
$186$		0			0
$187$			1792.00i			0.700770i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−768.000			−0.290945			−0.145473	−	0.989362i	$-0.546470\pi$
$191$	−768.000			−0.290945			−0.145473	+	0.989362i	$0.546470\pi$
$192$		0			0
$193$		−	2608.00i		−	0.972684i	−0.873769	−	0.486342i	$-0.838331\pi$
$193$		−	2608.00i		−	0.972684i	0.873769	−	0.486342i	$-0.161669\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			5116.00i			1.85025i	0.379659	+	0.925127i	$0.376041\pi$
$197$			5116.00i			1.85025i	−0.379659	+	0.925127i	$0.623959\pi$
$198$		0			0
$199$	−3480.00			−1.23965			−0.619826	−	0.784739i	$-0.712797\pi$
$199$	−3480.00			−1.23965			−0.619826	+	0.784739i	$0.712797\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			2340.00i			0.809043i
$204$		0			0
$205$	−1210.00	+	2420.00i	−0.412244	+	0.824488i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1680.00			0.556019
$210$		0			0
$211$	−3132.00			−1.02188			−0.510938	−	0.859618i	$-0.670702\pi$
$211$	−3132.00			−1.02188			−0.510938	+	0.859618i	$0.670702\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3620.00	−	1810.00i	−1.14829	−	0.574144i
$216$		0			0
$217$			3328.00i			1.04110i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	768.000			0.233761
$222$		0			0
$223$		−	62.0000i		−	0.0186181i	−0.999957	−	0.00930903i	$-0.997037\pi$
$223$		−	62.0000i		−	0.0186181i	0.999957	−	0.00930903i	$-0.00296320\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			5314.00i			1.55376i	0.629651	+	0.776878i	$0.283198\pi$
$227$			5314.00i			1.55376i	−0.629651	+	0.776878i	$0.716802\pi$
$228$		0			0
$229$	190.000			0.0548277			0.0274139	−	0.999624i	$-0.491273\pi$
$229$	190.000			0.0548277			0.0274139	+	0.999624i	$0.491273\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			2408.00i			0.677053i	0.940957	+	0.338526i	$0.109928\pi$
$233$			2408.00i			0.677053i	−0.940957	+	0.338526i	$0.890072\pi$
$234$		0			0
$235$	−2260.00	−	1130.00i	−0.627345	−	0.313673i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5680.00			1.53727			0.768637	−	0.639685i	$-0.220935\pi$
$239$	5680.00			1.53727			0.768637	+	0.639685i	$0.220935\pi$
$240$		0			0
$241$	−278.000			−0.0743052			−0.0371526	−	0.999310i	$-0.511829\pi$
$241$	−278.000			−0.0743052			−0.0371526	+	0.999310i	$0.511829\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1665.00	+	3330.00i	−0.434175	+	0.868351i
$246$		0			0
$247$		−	720.000i		−	0.185476i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3252.00			0.817787			0.408893	−	0.912582i	$-0.365915\pi$
$251$	3252.00			0.817787			0.408893	+	0.912582i	$0.365915\pi$
$252$		0			0
$253$			1624.00i			0.403557i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			1536.00i			0.372813i	0.982473	+	0.186407i	$0.0596842\pi$
$257$			1536.00i			0.372813i	−0.982473	+	0.186407i	$0.940316\pi$
$258$		0			0
$259$	6136.00			1.47209
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	4858.00i		−	1.13900i	−0.821991	−	0.569500i	$-0.807137\pi$
$263$		−	4858.00i		−	1.13900i	0.821991	−	0.569500i	$-0.192863\pi$
$264$		0			0
$265$	1080.00	+	540.000i	0.250354	+	0.125177i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2610.00			0.591578			0.295789	−	0.955253i	$-0.404417\pi$
$269$	2610.00			0.591578			0.295789	+	0.955253i	$0.404417\pi$
$270$		0			0
$271$	5168.00			1.15843			0.579213	−	0.815176i	$-0.303360\pi$
$271$	5168.00			1.15843			0.579213	+	0.815176i	$0.303360\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2100.00	+	2800.00i	0.460490	+	0.613987i
$276$		0			0
$277$			1924.00i			0.417336i	0.977987	+	0.208668i	$0.0669127\pi$
$277$			1924.00i			0.417336i	−0.977987	+	0.208668i	$0.933087\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3042.00			−0.645803			−0.322901	−	0.946433i	$-0.604658\pi$
$281$	−3042.00			−0.645803			−0.322901	+	0.946433i	$0.604658\pi$
$282$		0			0
$283$			1718.00i			0.360864i	0.983587	+	0.180432i	$0.0577496\pi$
$283$			1718.00i			0.360864i	−0.983587	+	0.180432i	$0.942250\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	6292.00i		−	1.29409i
$288$		0			0
$289$	817.000			0.166294
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	2292.00i		−	0.456997i	−0.973544	−	0.228498i	$-0.926618\pi$
$293$		−	2292.00i		−	0.456997i	0.973544	−	0.228498i	$-0.0733816\pi$
$294$		0			0
$295$	100.000	−	200.000i	0.0197364	−	0.0394727i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	696.000			0.134618
$300$		0			0
$301$	9412.00			1.80232
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2710.00	−	5420.00i	0.508768	−	1.01754i
$306$		0			0
$307$			5406.00i			1.00501i	0.864576	+	0.502503i	$0.167587\pi$
$307$			5406.00i			1.00501i	−0.864576	+	0.502503i	$0.832413\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5688.00			−1.03710			−0.518548	−	0.855048i	$-0.673527\pi$
$311$	−5688.00			−1.03710			−0.518548	+	0.855048i	$0.673527\pi$
$312$		0			0
$313$			7352.00i			1.32767i	0.747881	+	0.663833i	$0.231072\pi$
$313$			7352.00i			1.32767i	−0.747881	+	0.663833i	$0.768928\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	3484.00i		−	0.617290i	−0.951177	−	0.308645i	$-0.900124\pi$
$317$		−	3484.00i		−	0.617290i	0.951177	−	0.308645i	$-0.0998755\pi$
$318$		0			0
$319$	−2520.00			−0.442298
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3840.00i			0.661496i
$324$		0			0
$325$	1200.00	−	900.000i	0.204812	−	0.153609i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5876.00			0.984664
$330$		0			0
$331$	7868.00			1.30654			0.653269	−	0.757125i	$-0.273397\pi$
$331$	7868.00			1.30654			0.653269	+	0.757125i	$0.273397\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4340.00	−	2170.00i	−0.707819	−	0.353910i
$336$		0			0
$337$		−	656.000i		−	0.106037i	−0.998594	−	0.0530187i	$-0.983116\pi$
$337$		−	656.000i		−	0.106037i	0.998594	−	0.0530187i	$-0.0168843\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3584.00			−0.569163
$342$		0			0
$343$			260.000i			0.0409291i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			5754.00i			0.890176i	0.895487	+	0.445088i	$0.146828\pi$
$347$			5754.00i			0.890176i	−0.895487	+	0.445088i	$0.853172\pi$
$348$		0			0
$349$	3110.00			0.477004			0.238502	−	0.971142i	$-0.423344\pi$
$349$	3110.00			0.477004			0.238502	+	0.971142i	$0.423344\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			7808.00i			1.17727i	0.808397	+	0.588637i	$0.200335\pi$
$353$			7808.00i			1.17727i	−0.808397	+	0.588637i	$0.799665\pi$
$354$		0			0
$355$	−5640.00	+	11280.0i	−0.843212	+	1.68642i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9240.00			1.35841			0.679204	−	0.733949i	$-0.262325\pi$
$359$	9240.00			1.35841			0.679204	+	0.733949i	$0.262325\pi$
$360$		0			0
$361$	−3259.00			−0.475142
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6320.00	+	3160.00i	0.906312	+	0.453156i
$366$		0			0
$367$		−	3214.00i		−	0.457137i	−0.973528	−	0.228569i	$-0.926595\pi$
$367$		−	3214.00i		−	0.457137i	0.973528	−	0.228569i	$-0.0734046\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2808.00			−0.392949
$372$		0			0
$373$		−	348.000i		−	0.0483077i	−0.999708	−	0.0241538i	$-0.992311\pi$
$373$		−	348.000i		−	0.0483077i	0.999708	−	0.0241538i	$-0.00768915\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			1080.00i			0.147541i
$378$		0			0
$379$	4940.00			0.669527			0.334764	−	0.942302i	$-0.391344\pi$
$379$	4940.00			0.669527			0.334764	+	0.942302i	$0.391344\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			6142.00i			0.819430i	0.912214	+	0.409715i	$0.134372\pi$
$383$			6142.00i			0.819430i	−0.912214	+	0.409715i	$0.865628\pi$
$384$		0			0
$385$	−7280.00	−	3640.00i	−0.963697	−	0.481848i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3050.00			0.397535			0.198768	−	0.980047i	$-0.436306\pi$
$389$	3050.00			0.397535			0.198768	+	0.980047i	$0.436306\pi$
$390$		0			0
$391$	−3712.00			−0.480112
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3600.00	+	7200.00i	−0.458571	+	0.917143i
$396$		0			0
$397$		−	5396.00i		−	0.682160i	−0.940034	−	0.341080i	$-0.889207\pi$
$397$		−	5396.00i		−	0.682160i	0.940034	−	0.341080i	$-0.110793\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−14482.0			−1.80348			−0.901741	−	0.432276i	$-0.857711\pi$
$401$	−14482.0			−1.80348			−0.901741	+	0.432276i	$0.857711\pi$
$402$		0			0
$403$			1536.00i			0.189860i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6608.00i			0.804782i
$408$		0			0
$409$	1090.00			0.131778			0.0658888	−	0.997827i	$-0.479012\pi$
$409$	1090.00			0.131778			0.0658888	+	0.997827i	$0.479012\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			520.000i			0.0619553i
$414$		0			0
$415$	−4780.00	−	2390.00i	−0.565400	−	0.282700i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7180.00			0.837150			0.418575	−	0.908182i	$-0.362530\pi$
$419$	7180.00			0.837150			0.418575	+	0.908182i	$0.362530\pi$
$420$		0			0
$421$	−8138.00			−0.942095			−0.471047	−	0.882108i	$-0.656124\pi$
$421$	−8138.00			−0.942095			−0.471047	+	0.882108i	$0.656124\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6400.00	+	4800.00i	−0.730460	+	0.547845i
$426$		0			0
$427$			14092.0i			1.59710i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−208.000			−0.0232460			−0.0116230	−	0.999932i	$-0.503700\pi$
$431$	−208.000			−0.0232460			−0.0116230	+	0.999932i	$0.503700\pi$
$432$		0			0
$433$			12992.0i			1.44193i	0.692971	+	0.720965i	$0.256301\pi$
$433$			12992.0i			1.44193i	−0.692971	+	0.720965i	$0.743699\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			3480.00i			0.380940i
$438$		0			0
$439$	1080.00			0.117416			0.0587080	−	0.998275i	$-0.481302\pi$
$439$	1080.00			0.117416			0.0587080	+	0.998275i	$0.481302\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	9078.00i		−	0.973609i	−0.873511	−	0.486805i	$-0.838162\pi$
$443$		−	9078.00i		−	0.973609i	0.873511	−	0.486805i	$-0.161838\pi$
$444$		0			0
$445$	−2450.00	+	4900.00i	−0.260991	+	0.521983i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	14310.0			1.50408			0.752039	−	0.659119i	$-0.229071\pi$
$449$	14310.0			1.50408			0.752039	+	0.659119i	$0.229071\pi$
$450$		0			0
$451$	6776.00			0.707471
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1560.00	+	3120.00i	−0.160734	+	0.321468i
$456$		0			0
$457$			2344.00i			0.239929i	0.992778	+	0.119965i	$0.0382781\pi$
$457$			2344.00i			0.239929i	−0.992778	+	0.119965i	$0.961722\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11382.0			−1.14992			−0.574959	−	0.818182i	$-0.694982\pi$
$461$	−11382.0			−1.14992			−0.574959	+	0.818182i	$0.694982\pi$
$462$		0			0
$463$		−	16062.0i		−	1.61223i	−0.591756	−	0.806117i	$-0.701565\pi$
$463$		−	16062.0i		−	1.61223i	0.591756	−	0.806117i	$-0.298435\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	17166.0i		−	1.70096i	−0.526008	−	0.850479i	$-0.676312\pi$
$467$		−	17166.0i		−	1.70096i	0.526008	−	0.850479i	$-0.323688\pi$
$468$		0			0
$469$	11284.0			1.11097
$470$		0			0
$471$		0			0
$472$		0			0
$473$			10136.0i			0.985315i
$474$		0			0
$475$	4500.00	+	6000.00i	0.434682	+	0.579577i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7520.00			−0.717323			−0.358661	−	0.933468i	$-0.616767\pi$
$479$	−7520.00			−0.717323			−0.358661	+	0.933468i	$0.616767\pi$
$480$		0			0
$481$	2832.00			0.268458
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−14560.0	−	7280.00i	−1.36317	−	0.681583i
$486$		0			0
$487$		−	11814.0i		−	1.09927i	−0.835406	−	0.549634i	$-0.814767\pi$
$487$		−	11814.0i		−	1.09927i	0.835406	−	0.549634i	$-0.185233\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	14052.0			1.29156			0.645782	−	0.763522i	$-0.276532\pi$
$491$	14052.0			1.29156			0.645782	+	0.763522i	$0.276532\pi$
$492$		0			0
$493$		−	5760.00i		−	0.526202i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	29328.0i		−	2.64696i
$498$		0			0
$499$	7620.00			0.683603			0.341802	−	0.939772i	$-0.388963\pi$
$499$	7620.00			0.683603			0.341802	+	0.939772i	$0.388963\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	1818.00i		−	0.161154i	−0.996748	−	0.0805772i	$-0.974324\pi$
$503$		−	1818.00i		−	0.161154i	0.996748	−	0.0805772i	$-0.0256763\pi$
$504$		0			0
$505$	2890.00	−	5780.00i	0.254660	−	0.509320i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17850.0			1.55440			0.777198	−	0.629256i	$-0.216640\pi$
$509$	17850.0			1.55440			0.777198	+	0.629256i	$0.216640\pi$
$510$		0			0
$511$	−16432.0			−1.42252
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−14620.0	−	7310.00i	−1.25094	−	0.625470i
$516$		0			0
$517$			6328.00i			0.538308i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	19238.0			1.61772			0.808860	−	0.588001i	$-0.200085\pi$
$521$	19238.0			1.61772			0.808860	+	0.588001i	$0.200085\pi$
$522$		0			0
$523$			6278.00i			0.524891i	0.964947	+	0.262445i	$0.0845289\pi$
$523$			6278.00i			0.524891i	−0.964947	+	0.262445i	$0.915471\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	8192.00i		−	0.677133i
$528$		0			0
$529$	8803.00			0.723514
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	2904.00i		−	0.235997i
$534$		0			0
$535$	−9660.00	−	4830.00i	−0.780632	−	0.390316i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9324.00			0.745108
$540$		0			0
$541$	−9818.00			−0.780238			−0.390119	−	0.920764i	$-0.627566\pi$
$541$	−9818.00			−0.780238			−0.390119	+	0.920764i	$0.627566\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1850.00	+	3700.00i	−0.145404	+	0.290808i
$546$		0			0
$547$		−	12514.0i		−	0.978172i	−0.872236	−	0.489086i	$-0.837330\pi$
$547$		−	12514.0i		−	0.978172i	0.872236	−	0.489086i	$-0.162670\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5400.00			−0.417509
$552$		0			0
$553$		−	18720.0i		−	1.43952i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			10596.0i			0.806045i	0.915190	+	0.403022i	$0.132040\pi$
$557$			10596.0i			0.806045i	−0.915190	+	0.403022i	$0.867960\pi$
$558$		0			0
$559$	4344.00			0.328679
$560$		0			0
$561$		0			0
$562$		0			0
$563$			14002.0i			1.04816i	0.851669	+	0.524080i	$0.175591\pi$
$563$			14002.0i			1.04816i	−0.851669	+	0.524080i	$0.824409\pi$
$564$		0			0
$565$	5280.00	+	2640.00i	0.393153	+	0.196576i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7330.00			−0.540052			−0.270026	−	0.962853i	$-0.587032\pi$
$569$	−7330.00			−0.540052			−0.270026	+	0.962853i	$0.587032\pi$
$570$		0			0
$571$	−5812.00			−0.425963			−0.212981	−	0.977056i	$-0.568317\pi$
$571$	−5812.00			−0.425963			−0.212981	+	0.977056i	$0.568317\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5800.00	+	4350.00i	−0.420655	+	0.315491i
$576$		0			0
$577$		−	16736.0i		−	1.20750i	−0.797173	−	0.603751i	$-0.793672\pi$
$577$		−	16736.0i		−	1.20750i	0.797173	−	0.603751i	$-0.206328\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12428.0			0.887436
$582$		0			0
$583$		−	3024.00i		−	0.214822i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			7434.00i			0.522716i	0.965242	+	0.261358i	$0.0841702\pi$
$587$			7434.00i			0.522716i	−0.965242	+	0.261358i	$0.915830\pi$
$588$		0			0
$589$	−7680.00			−0.537265
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	25872.0i		−	1.79163i	−0.444429	−	0.895814i	$-0.646593\pi$
$593$		−	25872.0i		−	1.79163i	0.444429	−	0.895814i	$-0.353407\pi$
$594$		0			0
$595$	8320.00	−	16640.0i	0.573255	−	1.14651i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3720.00			0.253748			0.126874	−	0.991919i	$-0.459506\pi$
$599$	3720.00			0.253748			0.126874	+	0.991919i	$0.459506\pi$
$600$		0			0
$601$	−12958.0			−0.879481			−0.439740	−	0.898125i	$-0.644930\pi$
$601$	−12958.0			−0.879481			−0.439740	+	0.898125i	$0.644930\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2735.00	+	5470.00i	−0.183791	+	0.367582i
$606$		0			0
$607$		−	7214.00i		−	0.482384i	−0.970477	−	0.241192i	$-0.922462\pi$
$607$		−	7214.00i		−	0.482384i	0.970477	−	0.241192i	$-0.0775384\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2712.00			0.179568
$612$		0			0
$613$		−	4828.00i		−	0.318109i	−0.987270	−	0.159055i	$-0.949155\pi$
$613$		−	4828.00i		−	0.318109i	0.987270	−	0.159055i	$-0.0508446\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			27656.0i			1.80452i	0.431193	+	0.902260i	$0.358093\pi$
$617$			27656.0i			1.80452i	−0.431193	+	0.902260i	$0.641907\pi$
$618$		0			0
$619$	−21220.0			−1.37787			−0.688937	−	0.724821i	$-0.741922\pi$
$619$	−21220.0			−1.37787			−0.688937	+	0.724821i	$0.741922\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	12740.0i		−	0.819289i
$624$		0			0
$625$	−4375.00	+	15000.0i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15104.0			−0.957450
$630$		0			0
$631$	−17672.0			−1.11491			−0.557457	−	0.830206i	$-0.688223\pi$
$631$	−17672.0			−1.11491			−0.557457	+	0.830206i	$0.688223\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−15340.0	−	7670.00i	−0.958660	−	0.479330i
$636$		0			0
$637$		−	3996.00i		−	0.248551i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7322.00			−0.451173			−0.225586	−	0.974223i	$-0.572430\pi$
$641$	−7322.00			−0.451173			−0.225586	+	0.974223i	$0.572430\pi$
$642$		0			0
$643$			8238.00i			0.505249i	0.967564	+	0.252624i	$0.0812937\pi$
$643$			8238.00i			0.505249i	−0.967564	+	0.252624i	$0.918706\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	6426.00i		−	0.390467i	−0.980757	−	0.195233i	$-0.937454\pi$
$647$		−	6426.00i		−	0.390467i	0.980757	−	0.195233i	$-0.0625465\pi$
$648$		0			0
$649$	−560.000			−0.0338705
$650$		0			0
$651$		0			0
$652$		0			0
$653$			5908.00i			0.354055i	0.984206	+	0.177027i	$0.0566482\pi$
$653$			5908.00i			0.354055i	−0.984206	+	0.177027i	$0.943352\pi$
$654$		0			0
$655$	60.0000	−	120.000i	0.00357923	−	0.00715845i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	26780.0			1.58301			0.791503	−	0.611166i	$-0.209299\pi$
$659$	26780.0			1.58301			0.791503	+	0.611166i	$0.209299\pi$
$660$		0			0
$661$	−24538.0			−1.44390			−0.721950	−	0.691945i	$-0.756754\pi$
$661$	−24538.0			−1.44390			−0.721950	+	0.691945i	$0.756754\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−15600.0	−	7800.00i	−0.909687	−	0.454844i
$666$		0			0
$667$		−	5220.00i		−	0.303027i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−15176.0			−0.873119
$672$		0			0
$673$		−	28848.0i		−	1.65232i	−0.563439	−	0.826158i	$-0.690522\pi$
$673$		−	28848.0i		−	1.65232i	0.563439	−	0.826158i	$-0.309478\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	26884.0i		−	1.52620i	−0.646282	−	0.763099i	$-0.723677\pi$
$677$		−	26884.0i		−	1.52620i	0.646282	−	0.763099i	$-0.276323\pi$
$678$		0			0
$679$	37856.0			2.13959
$680$		0			0
$681$		0			0
$682$		0			0
$683$			14282.0i			0.800125i	0.916488	+	0.400063i	$0.131012\pi$
$683$			14282.0i			0.800125i	−0.916488	+	0.400063i	$0.868988\pi$
$684$		0			0
$685$	−12240.0	−	6120.00i	−0.682725	−	0.341362i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1296.00			−0.0716599
$690$		0			0
$691$	3428.00			0.188723			0.0943613	−	0.995538i	$-0.469919\pi$
$691$	3428.00			0.188723			0.0943613	+	0.995538i	$0.469919\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	15500.0	−	31000.0i	0.845969	−	1.69194i
$696$		0			0
$697$			15488.0i			0.841678i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−26942.0			−1.45162			−0.725810	−	0.687895i	$-0.758535\pi$
$701$	−26942.0			−1.45162			−0.725810	+	0.687895i	$0.758535\pi$
$702$		0			0
$703$			14160.0i			0.759679i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			15028.0i			0.799415i
$708$		0			0
$709$	1950.00			0.103292			0.0516458	−	0.998665i	$-0.483553\pi$
$709$	1950.00			0.103292			0.0516458	+	0.998665i	$0.483553\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	7424.00i		−	0.389945i
$714$		0			0
$715$	−3360.00	−	1680.00i	−0.175744	−	0.0878719i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12080.0			−0.626576			−0.313288	−	0.949658i	$-0.601430\pi$
$719$	−12080.0			−0.626576			−0.313288	+	0.949658i	$0.601430\pi$
$720$		0			0
$721$	38012.0			1.96344
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6750.00	−	9000.00i	−0.345778	−	0.461037i
$726$		0			0
$727$			17226.0i			0.878785i	0.898295	+	0.439393i	$0.144806\pi$
$727$			17226.0i			0.878785i	−0.898295	+	0.439393i	$0.855194\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−23168.0			−1.17223
$732$		0			0
$733$		−	788.000i		−	0.0397073i	−0.999803	−	0.0198536i	$-0.993680\pi$
$733$		−	788.000i		−	0.0397073i	0.999803	−	0.0198536i	$-0.00632003\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			12152.0i			0.607360i
$738$		0			0
$739$	−2060.00			−0.102542			−0.0512709	−	0.998685i	$-0.516327\pi$
$739$	−2060.00			−0.102542			−0.0512709	+	0.998685i	$0.516327\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	3258.00i		−	0.160867i	−0.996760	−	0.0804337i	$-0.974369\pi$
$743$		−	3258.00i		−	0.160867i	0.996760	−	0.0804337i	$-0.0256305\pi$
$744$		0			0
$745$	1250.00	−	2500.00i	0.0614718	−	0.122944i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	25116.0			1.22526
$750$		0			0
$751$	4528.00			0.220012			0.110006	−	0.993931i	$-0.464913\pi$
$751$	4528.00			0.220012			0.110006	+	0.993931i	$0.464913\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10760.0	+	21520.0i	−0.518671	+	1.03734i
$756$		0			0
$757$		−	18236.0i		−	0.875560i	−0.899082	−	0.437780i	$-0.855765\pi$
$757$		−	18236.0i		−	0.875560i	0.899082	−	0.437780i	$-0.144235\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18678.0			0.889720			0.444860	−	0.895600i	$-0.353253\pi$
$761$	18678.0			0.889720			0.444860	+	0.895600i	$0.353253\pi$
$762$		0			0
$763$		−	9620.00i		−	0.456445i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			240.000i			0.0112984i
$768$		0			0
$769$	−27390.0			−1.28441			−0.642203	−	0.766534i	$-0.721980\pi$
$769$	−27390.0			−1.28441			−0.642203	+	0.766534i	$0.721980\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	9252.00i		−	0.430493i	−0.976560	−	0.215247i	$-0.930944\pi$
$773$		−	9252.00i		−	0.430493i	0.976560	−	0.215247i	$-0.0690555\pi$
$774$		0			0
$775$	−9600.00	−	12800.0i	−0.444958	−	0.593277i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14520.0			0.667822
$780$		0			0
$781$	31584.0			1.44707
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5240.00	+	2620.00i	0.238247	+	0.119123i
$786$		0			0
$787$			5726.00i			0.259352i	0.991556	+	0.129676i	$0.0413937\pi$
$787$			5726.00i			0.259352i	−0.991556	+	0.129676i	$0.958606\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−13728.0			−0.617082
$792$		0			0
$793$			6504.00i			0.291253i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			27236.0i			1.21048i	0.796045	+	0.605238i	$0.206922\pi$
$797$			27236.0i			1.21048i	−0.796045	+	0.605238i	$0.793078\pi$
$798$		0			0
$799$	−14464.0			−0.640425
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	17696.0i		−	0.777682i
$804$		0			0
$805$	7540.00	−	15080.0i	0.330124	−	0.660249i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	10950.0			0.475873			0.237937	−	0.971281i	$-0.423529\pi$
$809$	10950.0			0.475873			0.237937	+	0.971281i	$0.423529\pi$
$810$		0			0
$811$	8828.00			0.382236			0.191118	−	0.981567i	$-0.438789\pi$
$811$	8828.00			0.382236			0.191118	+	0.981567i	$0.438789\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	35180.0	+	17590.0i	1.51203	+	0.756013i
$816$		0			0
$817$			21720.0i			0.930094i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	16058.0			0.682616			0.341308	−	0.939951i	$-0.389130\pi$
$821$	16058.0			0.682616			0.341308	+	0.939951i	$0.389130\pi$
$822$		0			0
$823$		−	41862.0i		−	1.77305i	−0.462684	−	0.886523i	$-0.653113\pi$
$823$		−	41862.0i		−	1.77305i	0.462684	−	0.886523i	$-0.346887\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			12154.0i			0.511047i	0.966803	+	0.255524i	$0.0822478\pi$
$827$			12154.0i			0.511047i	−0.966803	+	0.255524i	$0.917752\pi$
$828$		0			0
$829$	15390.0			0.644773			0.322386	−	0.946608i	$-0.395515\pi$
$829$	15390.0			0.644773			0.322386	+	0.946608i	$0.395515\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			21312.0i			0.886455i
$834$		0			0
$835$	5340.00	+	2670.00i	0.221315	+	0.110658i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4280.00			0.176117			0.0880584	−	0.996115i	$-0.471934\pi$
$839$	4280.00			0.176117			0.0880584	+	0.996115i	$0.471934\pi$
$840$		0			0
$841$	−16289.0			−0.667883
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10265.0	−	20530.0i	0.417901	−	0.835803i
$846$		0			0
$847$		−	14222.0i		−	0.576947i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13688.0			−0.551373
$852$		0			0
$853$			14452.0i			0.580102i	0.957011	+	0.290051i	$0.0936723\pi$
$853$			14452.0i			0.580102i	−0.957011	+	0.290051i	$0.906328\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	22584.0i		−	0.900181i	−0.892983	−	0.450090i	$-0.851392\pi$
$857$		−	22584.0i		−	0.900181i	0.892983	−	0.450090i	$-0.148608\pi$
$858$		0			0
$859$	−26740.0			−1.06212			−0.531058	−	0.847336i	$-0.678205\pi$
$859$	−26740.0			−1.06212			−0.531058	+	0.847336i	$0.678205\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	498.000i		−	0.0196432i	−0.999952	−	0.00982162i	$-0.996874\pi$
$863$		−	498.000i		−	0.0196432i	0.999952	−	0.00982162i	$-0.00312637\pi$
$864$		0			0
$865$	−42520.0	−	21260.0i	−1.67136	−	0.835678i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	20160.0			0.786975
$870$		0			0
$871$	5208.00			0.202602
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6500.00	−	35750.0i	−0.251132	−	1.38122i
$876$		0			0
$877$			13244.0i			0.509941i	0.966949	+	0.254970i	$0.0820657\pi$
$877$			13244.0i			0.509941i	−0.966949	+	0.254970i	$0.917934\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−40842.0			−1.56186			−0.780932	−	0.624616i	$-0.785255\pi$
$881$	−40842.0			−1.56186			−0.780932	+	0.624616i	$0.785255\pi$
$882$		0			0
$883$			12078.0i			0.460314i	0.973154	+	0.230157i	$0.0739239\pi$
$883$			12078.0i			0.460314i	−0.973154	+	0.230157i	$0.926076\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			18294.0i			0.692506i	0.938141	+	0.346253i	$0.112546\pi$
$887$			18294.0i			0.692506i	−0.938141	+	0.346253i	$0.887454\pi$
$888$		0			0
$889$	39884.0			1.50469
$890$		0			0
$891$		0			0
$892$		0			0
$893$			13560.0i			0.508139i
$894$		0			0
$895$	−12500.0	+	25000.0i	−0.466848	+	0.933696i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11520.0			0.427379
$900$		0			0
$901$	6912.00			0.255574
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12890.0	+	25780.0i	−0.473456	+	0.946913i
$906$		0			0
$907$			22566.0i			0.826121i	0.910704	+	0.413060i	$0.135540\pi$
$907$			22566.0i			0.826121i	−0.910704	+	0.413060i	$0.864460\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6768.00			−0.246140			−0.123070	−	0.992398i	$-0.539274\pi$
$911$	−6768.00			−0.246140			−0.123070	+	0.992398i	$0.539274\pi$
$912$		0			0
$913$			13384.0i			0.485154i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			312.000i			0.0112357i
$918$		0			0
$919$	22200.0			0.796856			0.398428	−	0.917200i	$-0.369556\pi$
$919$	22200.0			0.796856			0.398428	+	0.917200i	$0.369556\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	13536.0i		−	0.482712i
$924$		0			0
$925$	−23600.0	+	17700.0i	−0.838879	+	0.629159i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6330.00			−0.223553			−0.111776	−	0.993733i	$-0.535654\pi$
$929$	−6330.00			−0.223553			−0.111776	+	0.993733i	$0.535654\pi$
$930$		0			0
$931$	19980.0			0.703349
$932$		0			0
$933$		0			0
$934$		0			0
$935$	17920.0	+	8960.00i	0.626788	+	0.313394i
$936$		0			0
$937$			19544.0i			0.681403i	0.940172	+	0.340702i	$0.110665\pi$
$937$			19544.0i			0.681403i	−0.940172	+	0.340702i	$0.889335\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9898.00			0.342896			0.171448	−	0.985193i	$-0.445155\pi$
$941$	9898.00			0.342896			0.171448	+	0.985193i	$0.445155\pi$
$942$		0			0
$943$			14036.0i			0.484703i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	41406.0i		−	1.42082i	−0.703789	−	0.710409i	$-0.748510\pi$
$947$		−	41406.0i		−	1.42082i	0.703789	−	0.710409i	$-0.251490\pi$
$948$		0			0
$949$	−7584.00			−0.259417
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	25432.0i		−	0.864453i	−0.901765	−	0.432226i	$-0.857728\pi$
$953$		−	25432.0i		−	0.864453i	0.901765	−	0.432226i	$-0.142272\pi$
$954$		0			0
$955$	−3840.00	+	7680.00i	−0.130115	+	0.260229i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	31824.0			1.07159
$960$		0			0
$961$	−13407.0			−0.450035
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−26080.0	−	13040.0i	−0.869995	−	0.434997i
$966$		0			0
$967$			12106.0i			0.402588i	0.979531	+	0.201294i	$0.0645147\pi$
$967$			12106.0i			0.402588i	−0.979531	+	0.201294i	$0.935485\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	7812.00			0.258186			0.129093	−	0.991632i	$-0.458793\pi$
$971$	7812.00			0.258186			0.129093	+	0.991632i	$0.458793\pi$
$972$		0			0
$973$			80600.0i			2.65562i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			12576.0i			0.411814i	0.978572	+	0.205907i	$0.0660144\pi$
$977$			12576.0i			0.411814i	−0.978572	+	0.205907i	$0.933986\pi$
$978$		0			0
$979$	13720.0			0.447899
$980$		0			0
$981$		0			0
$982$		0			0
$983$			4342.00i			0.140883i	0.997516	+	0.0704417i	$0.0224409\pi$
$983$			4342.00i			0.140883i	−0.997516	+	0.0704417i	$0.977559\pi$
$984$		0			0
$985$	51160.0	+	25580.0i	1.65492	+	0.827458i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−20996.0			−0.675060
$990$		0			0
$991$	−26272.0			−0.842137			−0.421068	−	0.907029i	$-0.638345\pi$
$991$	−26272.0			−0.842137			−0.421068	+	0.907029i	$0.638345\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−17400.0	+	34800.0i	−0.554389	+	1.10878i
$996$		0			0
$997$		−	44796.0i		−	1.42297i	−0.702700	−	0.711486i	$-0.748022\pi$
$997$		−	44796.0i		−	1.42297i	0.702700	−	0.711486i	$-0.251978\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.4.f.f.289.1		2
3.2	odd	2		80.4.c.a.49.1		2
4.3	odd	2		90.4.c.b.19.2		2
5.4	even	2	inner	720.4.f.f.289.2		2
12.11	even	2		10.4.b.a.9.1	✓	2
15.2	even	4		400.4.a.h.1.1		1
15.8	even	4		400.4.a.n.1.1		1
15.14	odd	2		80.4.c.a.49.2		2
20.3	even	4		450.4.a.k.1.1		1
20.7	even	4		450.4.a.j.1.1		1
20.19	odd	2		90.4.c.b.19.1		2
24.5	odd	2		320.4.c.c.129.2		2
24.11	even	2		320.4.c.d.129.1		2
60.23	odd	4		50.4.a.b.1.1		1
60.47	odd	4		50.4.a.d.1.1		1
60.59	even	2		10.4.b.a.9.2	yes	2
84.83	odd	2		490.4.c.b.99.1		2
120.29	odd	2		320.4.c.c.129.1		2
120.53	even	4		1600.4.a.t.1.1		1
120.59	even	2		320.4.c.d.129.2		2
120.77	even	4		1600.4.a.bg.1.1		1
120.83	odd	4		1600.4.a.bh.1.1		1
120.107	odd	4		1600.4.a.u.1.1		1
420.83	even	4		2450.4.a.o.1.1		1
420.167	even	4		2450.4.a.bb.1.1		1
420.419	odd	2		490.4.c.b.99.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.4.b.a.9.1	✓	2	12.11	even	2
10.4.b.a.9.2	yes	2	60.59	even	2
50.4.a.b.1.1		1	60.23	odd	4
50.4.a.d.1.1		1	60.47	odd	4
80.4.c.a.49.1		2	3.2	odd	2
80.4.c.a.49.2		2	15.14	odd	2
90.4.c.b.19.1		2	20.19	odd	2
90.4.c.b.19.2		2	4.3	odd	2
320.4.c.c.129.1		2	120.29	odd	2
320.4.c.c.129.2		2	24.5	odd	2
320.4.c.d.129.1		2	24.11	even	2
320.4.c.d.129.2		2	120.59	even	2
400.4.a.h.1.1		1	15.2	even	4
400.4.a.n.1.1		1	15.8	even	4
450.4.a.j.1.1		1	20.7	even	4
450.4.a.k.1.1		1	20.3	even	4
490.4.c.b.99.1		2	84.83	odd	2
490.4.c.b.99.2		2	420.419	odd	2
720.4.f.f.289.1		2	1.1	even	1	trivial
720.4.f.f.289.2		2	5.4	even	2	inner
1600.4.a.t.1.1		1	120.53	even	4
1600.4.a.u.1.1		1	120.107	odd	4
1600.4.a.bg.1.1		1	120.77	even	4
1600.4.a.bh.1.1		1	120.83	odd	4
2450.4.a.o.1.1		1	420.83	even	4
2450.4.a.bb.1.1		1	420.167	even	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	720.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(5.00000 - 10.0000i) q^{5} +26.0000i q^{7} -28.0000 q^{11} +12.0000i q^{13} -64.0000i q^{17} -60.0000 q^{19} -58.0000i q^{23} +(-75.0000 - 100.000i) q^{25} +90.0000 q^{29} +128.000 q^{31} +(260.000 + 130.000i) q^{35} -236.000i q^{37} -242.000 q^{41} -362.000i q^{43} -226.000i q^{47} -333.000 q^{49} +108.000i q^{53} +(-140.000 + 280.000i) q^{55} +20.0000 q^{59} +542.000 q^{61} +(120.000 + 60.0000i) q^{65} -434.000i q^{67} -1128.00 q^{71} +632.000i q^{73} -728.000i q^{77} -720.000 q^{79} -478.000i q^{83} +(-640.000 - 320.000i) q^{85} -490.000 q^{89} -312.000 q^{91} +(-300.000 + 600.000i) q^{95} -1456.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{5} - 56 q^{11} - 120 q^{19} - 150 q^{25} + 180 q^{29} + 256 q^{31} + 520 q^{35} - 484 q^{41} - 666 q^{49} - 280 q^{55} + 40 q^{59} + 1084 q^{61} + 240 q^{65} - 2256 q^{71} - 1440 q^{79} - 1280 q^{85}+ \cdots - 600 q^{95}+O(q^{100}) \) 2 * q + 10 * q^5 - 56 * q^11 - 120 * q^19 - 150 * q^25 + 180 * q^29 + 256 * q^31 + 520 * q^35 - 484 * q^41 - 666 * q^49 - 280 * q^55 + 40 * q^59 + 1084 * q^61 + 240 * q^65 - 2256 * q^71 - 1440 * q^79 - 1280 * q^85 - 980 * q^89 - 624 * q^91 - 600 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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