LMFDB - Embedded newform 720.3.c.b.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,3,Mod(449,720)]

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, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("720.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	720.c (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:	$19.6185790339$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{23})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 24x^{2} + 121 $ x^4 + 24*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$-4.09827i$ of defining polynomial
Character	$\chi$	$=$	720.449
Dual form			720.3.c.b.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.79583 + 1.41421i) q^{5} -6.78233i q^{7} +O(q^{10})$	$q+(4.79583 + 1.41421i) q^{5} -6.78233i q^{7} +9.89949i q^{11} +20.3470i q^{13} +19.1833 q^{17} -12.0000 q^{19} -9.59166 q^{23} +(21.0000 + 13.5647i) q^{25} +8.48528i q^{29} +38.0000 q^{31} +(9.59166 - 32.5269i) q^{35} +6.78233i q^{37} -69.2965i q^{41} +67.8233i q^{43} +76.7333 q^{47} +3.00000 q^{49} +(-14.0000 + 47.4763i) q^{55} -83.4386i q^{59} -70.0000 q^{61} +(-28.7750 + 97.5807i) q^{65} +108.517i q^{67} +118.794i q^{71} +13.5647i q^{73} +67.1416 q^{77} -30.0000 q^{79} +134.283 q^{83} +(92.0000 + 27.1293i) q^{85} -32.5269i q^{89} +138.000 q^{91} +(-57.5500 - 16.9706i) q^{95} +94.9526i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 48 q^{19} + 84 q^{25} + 152 q^{31} + 12 q^{49} - 56 q^{55} - 280 q^{61} - 120 q^{79} + 368 q^{85} + 552 q^{91}+O(q^{100}) $ 4 * q - 48 * q^19 + 84 * q^25 + 152 * q^31 + 12 * q^49 - 56 * q^55 - 280 * q^61 - 120 * q^79 + 368 * q^85 + 552 * q^91

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	4.79583	+	1.41421i	0.959166	+	0.282843i
$5$	4.79583	+	1.41421i	0.959166	+	0.282843i
$6$		0			0
$7$		−	6.78233i		−	0.968904i	−0.874818	−	0.484452i	$-0.839019\pi$
$7$		−	6.78233i		−	0.968904i	0.874818	−	0.484452i	$-0.160981\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			9.89949i			0.899954i	0.893040	+	0.449977i	$0.148568\pi$
$11$			9.89949i			0.899954i	−0.893040	+	0.449977i	$0.851432\pi$
$12$		0			0
$13$			20.3470i			1.56515i	0.622554	+	0.782577i	$0.286095\pi$
$13$			20.3470i			1.56515i	−0.622554	+	0.782577i	$0.713905\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	19.1833			1.12843			0.564215	−	0.825628i	$-0.309179\pi$
$17$	19.1833			1.12843			0.564215	+	0.825628i	$0.309179\pi$
$18$		0			0
$19$	−12.0000			−0.631579			−0.315789	−	0.948829i	$-0.602269\pi$
$19$	−12.0000			−0.631579			−0.315789	+	0.948829i	$0.602269\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−9.59166			−0.417029			−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−9.59166			−0.417029			−0.208514	+	0.978019i	$0.566863\pi$
$24$		0			0
$25$	21.0000	+	13.5647i	0.840000	+	0.542586i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.48528i			0.292596i	0.989241	+	0.146298i	$0.0467358\pi$
$29$			8.48528i			0.292596i	−0.989241	+	0.146298i	$0.953264\pi$
$30$		0			0
$31$	38.0000			1.22581			0.612903	−	0.790158i	$-0.290002\pi$
$31$	38.0000			1.22581			0.612903	+	0.790158i	$0.290002\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9.59166	−	32.5269i	0.274048	−	0.929340i
$36$		0			0
$37$			6.78233i			0.183306i	0.995791	+	0.0916531i	$0.0292151\pi$
$37$			6.78233i			0.183306i	−0.995791	+	0.0916531i	$0.970785\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	69.2965i		−	1.69016i	−0.534642	−	0.845079i	$-0.679553\pi$
$41$		−	69.2965i		−	1.69016i	0.534642	−	0.845079i	$-0.320447\pi$
$42$		0			0
$43$			67.8233i			1.57729i	0.614851	+	0.788643i	$0.289216\pi$
$43$			67.8233i			1.57729i	−0.614851	+	0.788643i	$0.710784\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	76.7333			1.63262			0.816312	−	0.577612i	$-0.196015\pi$
$47$	76.7333			1.63262			0.816312	+	0.577612i	$0.196015\pi$
$48$		0			0
$49$	3.00000			0.0612245
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−14.0000	+	47.4763i	−0.254545	+	0.863206i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	83.4386i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$59$		−	83.4386i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$60$		0			0
$61$	−70.0000			−1.14754			−0.573770	−	0.819016i	$-0.694520\pi$
$61$	−70.0000			−1.14754			−0.573770	+	0.819016i	$0.694520\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−28.7750	+	97.5807i	−0.442692	+	1.50124i
$66$		0			0
$67$			108.517i			1.61966i	0.586664	+	0.809830i	$0.300441\pi$
$67$			108.517i			1.61966i	−0.586664	+	0.809830i	$0.699559\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			118.794i			1.67315i	0.547849	+	0.836577i	$0.315447\pi$
$71$			118.794i			1.67315i	−0.547849	+	0.836577i	$0.684553\pi$
$72$		0			0
$73$			13.5647i			0.185817i	0.995675	+	0.0929086i	$0.0296164\pi$
$73$			13.5647i			0.185817i	−0.995675	+	0.0929086i	$0.970384\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	67.1416			0.871969
$78$		0			0
$79$	−30.0000			−0.379747			−0.189873	−	0.981809i	$-0.560808\pi$
$79$	−30.0000			−0.379747			−0.189873	+	0.981809i	$0.560808\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	134.283			1.61787			0.808935	−	0.587897i	$-0.200044\pi$
$83$	134.283			1.61787			0.808935	+	0.587897i	$0.200044\pi$
$84$		0			0
$85$	92.0000	+	27.1293i	1.08235	+	0.319168i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	32.5269i		−	0.365471i	−0.983162	−	0.182735i	$-0.941505\pi$
$89$		−	32.5269i		−	0.365471i	0.983162	−	0.182735i	$-0.0584952\pi$
$90$		0			0
$91$	138.000			1.51648
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−57.5500	−	16.9706i	−0.605789	−	0.178638i
$96$		0			0
$97$			94.9526i			0.978893i	0.872033	+	0.489446i	$0.162801\pi$
$97$			94.9526i			0.978893i	−0.872033	+	0.489446i	$0.837199\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	79.1960i		−	0.784118i	−0.919940	−	0.392059i	$-0.871763\pi$
$101$		−	79.1960i		−	0.784118i	0.919940	−	0.392059i	$-0.128237\pi$
$102$		0			0
$103$		−	88.1703i		−	0.856022i	−0.903773	−	0.428011i	$-0.859214\pi$
$103$		−	88.1703i		−	0.856022i	0.903773	−	0.428011i	$-0.140786\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−57.5500			−0.537850			−0.268925	−	0.963161i	$-0.586668\pi$
$107$	−57.5500			−0.537850			−0.268925	+	0.963161i	$0.586668\pi$
$108$		0			0
$109$	74.0000			0.678899			0.339450	−	0.940624i	$-0.389759\pi$
$109$	74.0000			0.678899			0.339450	+	0.940624i	$0.389759\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	172.650			1.52788			0.763938	−	0.645290i	$-0.223263\pi$
$113$	172.650			1.52788			0.763938	+	0.645290i	$0.223263\pi$
$114$		0			0
$115$	−46.0000	−	13.5647i	−0.400000	−	0.117954i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	130.108i		−	1.09334i
$120$		0			0
$121$	23.0000			0.190083
$122$		0			0
$123$		0			0
$124$		0			0
$125$	81.5291	+	94.7523i	0.652233	+	0.758018i
$126$		0			0
$127$		−	169.558i		−	1.33510i	−0.744563	−	0.667552i	$-0.767342\pi$
$127$		−	169.558i		−	1.33510i	0.744563	−	0.667552i	$-0.232658\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			165.463i			1.26308i	0.775345	+	0.631538i	$0.217576\pi$
$131$			165.463i			1.26308i	−0.775345	+	0.631538i	$0.782424\pi$
$132$		0			0
$133$			81.3880i			0.611940i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−115.100			−0.840146			−0.420073	−	0.907490i	$-0.637995\pi$
$137$	−115.100			−0.840146			−0.420073	+	0.907490i	$0.637995\pi$
$138$		0			0
$139$	−62.0000			−0.446043			−0.223022	−	0.974814i	$-0.571592\pi$
$139$	−62.0000			−0.446043			−0.223022	+	0.974814i	$0.571592\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−201.425			−1.40857
$144$		0			0
$145$	−12.0000	+	40.6940i	−0.0827586	+	0.280648i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	158.392i		−	1.06303i	−0.847048	−	0.531517i	$-0.821622\pi$
$149$		−	158.392i		−	1.06303i	0.847048	−	0.531517i	$-0.178378\pi$
$150$		0			0
$151$	−70.0000			−0.463576			−0.231788	−	0.972766i	$-0.574458\pi$
$151$	−70.0000			−0.463576			−0.231788	+	0.972766i	$0.574458\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	182.242	+	53.7401i	1.17575	+	0.346710i
$156$		0			0
$157$		−	47.4763i		−	0.302397i	−0.988503	−	0.151198i	$-0.951687\pi$
$157$		−	47.4763i		−	0.302397i	0.988503	−	0.151198i	$-0.0483132\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			65.0538i			0.404061i
$162$		0			0
$163$			94.9526i			0.582531i	0.956642	+	0.291266i	$0.0940764\pi$
$163$			94.9526i			0.582531i	−0.956642	+	0.291266i	$0.905924\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−201.425			−1.20614			−0.603069	−	0.797689i	$-0.706055\pi$
$167$	−201.425			−1.20614			−0.603069	+	0.797689i	$0.706055\pi$
$168$		0			0
$169$	−245.000			−1.44970
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−268.567			−1.55241			−0.776204	−	0.630482i	$-0.782857\pi$
$173$	−268.567			−1.55241			−0.776204	+	0.630482i	$0.782857\pi$
$174$		0			0
$175$	92.0000	−	142.429i	0.525714	−	0.813880i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	134.350i		−	0.750560i	−0.926911	−	0.375280i	$-0.877547\pi$
$179$		−	134.350i		−	0.750560i	0.926911	−	0.375280i	$-0.122453\pi$
$180$		0			0
$181$	−22.0000			−0.121547			−0.0607735	−	0.998152i	$-0.519357\pi$
$181$	−22.0000			−0.121547			−0.0607735	+	0.998152i	$0.519357\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9.59166	+	32.5269i	−0.0518468	+	0.175821i
$186$		0			0
$187$			189.905i			1.01554i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	195.161i		−	1.02179i	−0.859644	−	0.510894i	$-0.829314\pi$
$191$		−	195.161i		−	1.02179i	0.859644	−	0.510894i	$-0.170686\pi$
$192$		0			0
$193$		−	271.293i		−	1.40566i	−0.711356	−	0.702832i	$-0.751919\pi$
$193$		−	271.293i		−	1.40566i	0.711356	−	0.702832i	$-0.248081\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−163.058			−0.827707			−0.413853	−	0.910344i	$-0.635817\pi$
$197$	−163.058			−0.827707			−0.413853	+	0.910344i	$0.635817\pi$
$198$		0			0
$199$	−294.000			−1.47739			−0.738693	−	0.674042i	$-0.764557\pi$
$199$	−294.000			−1.47739			−0.738693	+	0.674042i	$0.764557\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	57.5500			0.283497
$204$		0			0
$205$	98.0000	−	332.334i	0.478049	−	1.62114i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	118.794i		−	0.568392i
$210$		0			0
$211$	42.0000			0.199052			0.0995261	−	0.995035i	$-0.468267\pi$
$211$	42.0000			0.199052			0.0995261	+	0.995035i	$0.468267\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−95.9166	+	325.269i	−0.446124	+	1.51288i
$216$		0			0
$217$		−	257.729i		−	1.18769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			390.323i			1.76617i
$222$		0			0
$223$		−	196.688i		−	0.882007i	−0.897505	−	0.441004i	$-0.854623\pi$
$223$		−	196.688i		−	0.882007i	0.897505	−	0.441004i	$-0.145377\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	268.567			1.18311			0.591556	−	0.806264i	$-0.298514\pi$
$227$	268.567			1.18311			0.591556	+	0.806264i	$0.298514\pi$
$228$		0			0
$229$	−422.000			−1.84279			−0.921397	−	0.388622i	$-0.872951\pi$
$229$	−422.000			−1.84279			−0.921397	+	0.388622i	$0.872951\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−211.017			−0.905651			−0.452825	−	0.891599i	$-0.649584\pi$
$233$	−211.017			−0.905651			−0.452825	+	0.891599i	$0.649584\pi$
$234$		0			0
$235$	368.000	+	108.517i	1.56596	+	0.461776i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			70.7107i			0.295861i	0.988998	+	0.147930i	$0.0472611\pi$
$239$			70.7107i			0.295861i	−0.988998	+	0.147930i	$0.952739\pi$
$240$		0			0
$241$	280.000			1.16183			0.580913	−	0.813966i	$-0.302696\pi$
$241$	280.000			1.16183			0.580913	+	0.813966i	$0.302696\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	14.3875	+	4.24264i	0.0587245	+	0.0173169i
$246$		0			0
$247$		−	244.164i		−	0.988518i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	80.6102i		−	0.321156i	−0.987023	−	0.160578i	$-0.948664\pi$
$251$		−	80.6102i		−	0.321156i	0.987023	−	0.160578i	$-0.0513358\pi$
$252$		0			0
$253$		−	94.9526i		−	0.375307i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	46.0000			0.177606
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−67.1416			−0.255291			−0.127646	−	0.991820i	$-0.540742\pi$
$263$	−67.1416			−0.255291			−0.127646	+	0.991820i	$0.540742\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	263.044i		−	0.977858i	−0.872324	−	0.488929i	$-0.837388\pi$
$269$		−	263.044i		−	0.977858i	0.872324	−	0.488929i	$-0.162612\pi$
$270$		0			0
$271$	−322.000			−1.18819			−0.594096	−	0.804394i	$-0.702490\pi$
$271$	−322.000			−1.18819			−0.594096	+	0.804394i	$0.702490\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−134.283	+	207.889i	−0.488303	+	0.755961i
$276$		0			0
$277$		−	33.9116i		−	0.122425i	−0.998125	−	0.0612124i	$-0.980503\pi$
$277$		−	33.9116i		−	0.122425i	0.998125	−	0.0612124i	$-0.0194967\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			326.683i			1.16257i	0.813699	+	0.581287i	$0.197451\pi$
$281$			326.683i			1.16257i	−0.813699	+	0.581287i	$0.802549\pi$
$282$		0			0
$283$		−	108.517i		−	0.383453i	−0.981448	−	0.191727i	$-0.938591\pi$
$283$		−	108.517i		−	0.383453i	0.981448	−	0.191727i	$-0.0614087\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−469.991			−1.63760
$288$		0			0
$289$	79.0000			0.273356
$290$		0			0
$291$		0			0
$292$		0			0
$293$	201.425			0.687457			0.343729	−	0.939069i	$-0.388310\pi$
$293$	201.425			0.687457			0.343729	+	0.939069i	$0.388310\pi$
$294$		0			0
$295$	118.000	−	400.157i	0.400000	−	1.35647i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	195.161i		−	0.652714i
$300$		0			0
$301$	460.000			1.52824
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−335.708	−	98.9949i	−1.10068	−	0.324574i
$306$		0			0
$307$		−	162.776i		−	0.530215i	−0.964219	−	0.265107i	$-0.914593\pi$
$307$		−	162.776i		−	0.530215i	0.964219	−	0.265107i	$-0.0854074\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	178.191i		−	0.572961i	−0.958086	−	0.286481i	$-0.907515\pi$
$311$		−	178.191i		−	0.572961i	0.958086	−	0.286481i	$-0.0924854\pi$
$312$		0			0
$313$			54.2586i			0.173350i	0.996237	+	0.0866751i	$0.0276242\pi$
$313$			54.2586i			0.173350i	−0.996237	+	0.0866751i	$0.972376\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	278.158			0.877471			0.438735	−	0.898616i	$-0.355427\pi$
$317$	278.158			0.877471			0.438735	+	0.898616i	$0.355427\pi$
$318$		0			0
$319$	−84.0000			−0.263323
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−230.200			−0.712693
$324$		0			0
$325$	−276.000	+	427.287i	−0.849231	+	1.31473i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	520.431i		−	1.58186i
$330$		0			0
$331$	−40.0000			−0.120846			−0.0604230	−	0.998173i	$-0.519245\pi$
$331$	−40.0000			−0.120846			−0.0604230	+	0.998173i	$0.519245\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−153.467	+	520.431i	−0.458109	+	1.55352i
$336$		0			0
$337$			311.987i			0.925778i	0.886416	+	0.462889i	$0.153187\pi$
$337$			311.987i			0.925778i	−0.886416	+	0.462889i	$0.846813\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			376.181i			1.10317i
$342$		0			0
$343$		−	352.681i		−	1.02822i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	402.850			1.16095			0.580475	−	0.814278i	$-0.302867\pi$
$347$	402.850			1.16095			0.580475	+	0.814278i	$0.302867\pi$
$348$		0			0
$349$	−406.000			−1.16332			−0.581662	−	0.813431i	$-0.697597\pi$
$349$	−406.000			−1.16332			−0.581662	+	0.813431i	$0.697597\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	537.133			1.52162			0.760812	−	0.648973i	$-0.224801\pi$
$353$	537.133			1.52162			0.760812	+	0.648973i	$0.224801\pi$
$354$		0			0
$355$	−168.000	+	569.716i	−0.473239	+	1.60483i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	161.220i		−	0.449082i	−0.974465	−	0.224541i	$-0.927912\pi$
$359$		−	161.220i		−	0.449082i	0.974465	−	0.224541i	$-0.0720882\pi$
$360$		0			0
$361$	−217.000			−0.601108
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−19.1833	+	65.0538i	−0.0525571	+	0.178230i
$366$		0			0
$367$		−	250.946i		−	0.683777i	−0.939740	−	0.341889i	$-0.888933\pi$
$367$		−	250.946i		−	0.683777i	0.939740	−	0.341889i	$-0.111067\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$			101.735i			0.272748i	0.990657	+	0.136374i	$0.0435449\pi$
$373$			101.735i			0.272748i	−0.990657	+	0.136374i	$0.956455\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−172.650			−0.457957
$378$		0			0
$379$	−538.000			−1.41953			−0.709763	−	0.704441i	$-0.751198\pi$
$379$	−538.000			−1.41953			−0.709763	+	0.704441i	$0.751198\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	268.567			0.701218			0.350609	−	0.936522i	$-0.385975\pi$
$383$	268.567			0.701218			0.350609	+	0.936522i	$0.385975\pi$
$384$		0			0
$385$	322.000	+	94.9526i	0.836364	+	0.246630i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	494.975i		−	1.27243i	−0.771512	−	0.636214i	$-0.780499\pi$
$389$		−	494.975i		−	1.27243i	0.771512	−	0.636214i	$-0.219501\pi$
$390$		0			0
$391$	−184.000			−0.470588
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−143.875	−	42.4264i	−0.364240	−	0.107409i
$396$		0			0
$397$		−	278.076i		−	0.700442i	−0.936667	−	0.350221i	$-0.886106\pi$
$397$		−	278.076i		−	0.700442i	0.936667	−	0.350221i	$-0.113894\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			247.487i			0.617175i	0.951196	+	0.308588i	$0.0998563\pi$
$401$			247.487i			0.617175i	−0.951196	+	0.308588i	$0.900144\pi$
$402$		0			0
$403$			773.186i			1.91857i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−67.1416			−0.164967
$408$		0			0
$409$	−242.000			−0.591687			−0.295844	−	0.955236i	$-0.595601\pi$
$409$	−242.000			−0.591687			−0.295844	+	0.955236i	$0.595601\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−565.908			−1.37024
$414$		0			0
$415$	644.000	+	189.905i	1.55181	+	0.457603i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	790.545i		−	1.88674i	−0.331738	−	0.943372i	$-0.607635\pi$
$419$		−	790.545i		−	1.88674i	0.331738	−	0.943372i	$-0.392365\pi$
$420$		0			0
$421$	358.000			0.850356			0.425178	−	0.905110i	$-0.360211\pi$
$421$	358.000			0.850356			0.425178	+	0.905110i	$0.360211\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	402.850	+	260.215i	0.947882	+	0.612271i
$426$		0			0
$427$			474.763i			1.11186i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			14.1421i			0.0328124i	0.999865	+	0.0164062i	$0.00522249\pi$
$431$			14.1421i			0.0328124i	−0.999865	+	0.0164062i	$0.994778\pi$
$432$		0			0
$433$			257.729i			0.595216i	0.954688	+	0.297608i	$0.0961888\pi$
$433$			257.729i			0.595216i	−0.954688	+	0.297608i	$0.903811\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	115.100			0.263387
$438$		0			0
$439$	690.000			1.57175			0.785877	−	0.618383i	$-0.212212\pi$
$439$	690.000			1.57175			0.785877	+	0.618383i	$0.212212\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	402.850			0.909368			0.454684	−	0.890653i	$-0.349752\pi$
$443$	402.850			0.909368			0.454684	+	0.890653i	$0.349752\pi$
$444$		0			0
$445$	46.0000	−	155.994i	0.103371	−	0.350547i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	80.6102i		−	0.179533i	−0.995963	−	0.0897663i	$-0.971388\pi$
$449$		−	80.6102i		−	0.179533i	0.995963	−	0.0897663i	$-0.0286120\pi$
$450$		0			0
$451$	686.000			1.52106
$452$		0			0
$453$		0			0
$454$		0			0
$455$	661.825	+	195.161i	1.45456	+	0.428926i
$456$		0			0
$457$		−	339.116i		−	0.742049i	−0.928623	−	0.371025i	$-0.879007\pi$
$457$		−	339.116i		−	0.742049i	0.928623	−	0.371025i	$-0.120993\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	644.881i		−	1.39888i	−0.714694	−	0.699438i	$-0.753434\pi$
$461$		−	644.881i		−	1.39888i	0.714694	−	0.699438i	$-0.246566\pi$
$462$		0			0
$463$			264.511i			0.571298i	0.958334	+	0.285649i	$0.0922091\pi$
$463$			264.511i			0.571298i	−0.958334	+	0.285649i	$0.907791\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	460.400			0.985867			0.492933	−	0.870067i	$-0.335925\pi$
$467$	460.400			0.985867			0.492933	+	0.870067i	$0.335925\pi$
$468$		0			0
$469$	736.000			1.56930
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−671.416			−1.41949
$474$		0			0
$475$	−252.000	−	162.776i	−0.530526	−	0.342686i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			79.1960i			0.165336i	0.996577	+	0.0826680i	$0.0263441\pi$
$479$			79.1960i			0.165336i	−0.996577	+	0.0826680i	$0.973656\pi$
$480$		0			0
$481$	−138.000			−0.286902
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−134.283	+	455.377i	−0.276873	+	0.938921i
$486$		0			0
$487$			793.533i			1.62943i	0.579861	+	0.814715i	$0.303107\pi$
$487$			793.533i			1.62943i	−0.579861	+	0.814715i	$0.696893\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			564.271i			1.14923i	0.818424	+	0.574614i	$0.194848\pi$
$491$			564.271i			1.14923i	−0.818424	+	0.574614i	$0.805152\pi$
$492$		0			0
$493$			162.776i			0.330174i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	805.700			1.62113
$498$		0			0
$499$	72.0000			0.144289			0.0721443	−	0.997394i	$-0.477016\pi$
$499$	72.0000			0.144289			0.0721443	+	0.997394i	$0.477016\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−230.200			−0.457654			−0.228827	−	0.973467i	$-0.573489\pi$
$503$	−230.200			−0.457654			−0.228827	+	0.973467i	$0.573489\pi$
$504$		0			0
$505$	112.000	−	379.810i	0.221782	−	0.752100i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			492.146i			0.966889i	0.875375	+	0.483444i	$0.160614\pi$
$509$			492.146i			0.966889i	−0.875375	+	0.483444i	$0.839386\pi$
$510$		0			0
$511$	92.0000			0.180039
$512$		0			0
$513$		0			0
$514$		0			0
$515$	124.692	−	422.850i	0.242120	−	0.821068i
$516$		0			0
$517$			759.621i			1.46929i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			69.2965i			0.133007i	0.997786	+	0.0665033i	$0.0211843\pi$
$521$			69.2965i			0.133007i	−0.997786	+	0.0665033i	$0.978816\pi$
$522$		0			0
$523$			339.116i			0.648406i	0.945987	+	0.324203i	$0.105096\pi$
$523$			339.116i			0.648406i	−0.945987	+	0.324203i	$0.894904\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	728.966			1.38324
$528$		0			0
$529$	−437.000			−0.826087
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1409.97			2.64536
$534$		0			0
$535$	−276.000	−	81.3880i	−0.515888	−	0.152127i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			29.6985i			0.0550992i
$540$		0			0
$541$	−590.000			−1.09057			−0.545287	−	0.838250i	$-0.683579\pi$
$541$	−590.000			−1.09057			−0.545287	+	0.838250i	$0.683579\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	354.892	+	104.652i	0.651177	+	0.192022i
$546$		0			0
$547$		−	230.599i		−	0.421571i	−0.977532	−	0.210785i	$-0.932398\pi$
$547$		−	230.599i		−	0.421571i	0.977532	−	0.210785i	$-0.0676021\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	101.823i		−	0.184797i
$552$		0			0
$553$			203.470i			0.367938i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−690.600			−1.23986			−0.619928	−	0.784659i	$-0.712838\pi$
$557$	−690.600			−1.23986			−0.619928	+	0.784659i	$0.712838\pi$
$558$		0			0
$559$	−1380.00			−2.46869
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.1833			−0.0340734			−0.0170367	−	0.999855i	$-0.505423\pi$
$563$	−19.1833			−0.0340734			−0.0170367	+	0.999855i	$0.505423\pi$
$564$		0			0
$565$	828.000	+	244.164i	1.46549	+	0.432148i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	108.894i		−	0.191379i	−0.995411	−	0.0956893i	$-0.969494\pi$
$569$		−	108.894i		−	0.191379i	0.995411	−	0.0956893i	$-0.0305055\pi$
$570$		0			0
$571$	704.000			1.23292			0.616462	−	0.787384i	$-0.288565\pi$
$571$	704.000			1.23292			0.616462	+	0.787384i	$0.288565\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−201.425	−	130.108i	−0.350304	−	0.226274i
$576$		0			0
$577$			1125.87i			1.95124i	0.219462	+	0.975621i	$0.429570\pi$
$577$			1125.87i			1.95124i	−0.219462	+	0.975621i	$0.570430\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	910.754i		−	1.56756i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−978.350			−1.66669			−0.833347	−	0.552750i	$-0.813578\pi$
$587$	−978.350			−1.66669			−0.833347	+	0.552750i	$0.813578\pi$
$588$		0			0
$589$	−456.000			−0.774194
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$	184.000	−	623.974i	0.309244	−	1.04870i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	296.985i		−	0.495801i	−0.968785	−	0.247901i	$-0.920259\pi$
$599$		−	296.985i		−	0.495801i	0.968785	−	0.247901i	$-0.0797406\pi$
$600$		0			0
$601$	20.0000			0.0332779			0.0166389	−	0.999862i	$-0.494703\pi$
$601$	20.0000			0.0332779			0.0166389	+	0.999862i	$0.494703\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	110.304	+	32.5269i	0.182321	+	0.0537635i
$606$		0			0
$607$			6.78233i			0.0111735i	0.999984	+	0.00558676i	$0.00177833\pi$
$607$			6.78233i			0.0111735i	−0.999984	+	0.00558676i	$0.998222\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1561.29i			2.55531i
$612$		0			0
$613$		−	47.4763i		−	0.0774491i	−0.999250	−	0.0387246i	$-0.987671\pi$
$613$		−	47.4763i		−	0.0774491i	0.999250	−	0.0387246i	$-0.0123295\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−38.3667			−0.0621826			−0.0310913	−	0.999517i	$-0.509898\pi$
$617$	−38.3667			−0.0621826			−0.0310913	+	0.999517i	$0.509898\pi$
$618$		0			0
$619$	−178.000			−0.287561			−0.143780	−	0.989610i	$-0.545926\pi$
$619$	−178.000			−0.287561			−0.143780	+	0.989610i	$0.545926\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−220.608			−0.354106
$624$		0			0
$625$	257.000	+	569.716i	0.411200	+	0.911545i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			130.108i			0.206848i
$630$		0			0
$631$	154.000			0.244057			0.122029	−	0.992527i	$-0.461060\pi$
$631$	154.000			0.244057			0.122029	+	0.992527i	$0.461060\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	239.792	−	813.173i	0.377625	−	1.28059i
$636$		0			0
$637$			61.0410i			0.0958257i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	807.516i		−	1.25978i	−0.776686	−	0.629888i	$-0.783101\pi$
$641$		−	807.516i		−	1.25978i	0.776686	−	0.629888i	$-0.216899\pi$
$642$		0			0
$643$		−	1017.35i		−	1.58219i	−0.611692	−	0.791096i	$-0.709511\pi$
$643$		−	1017.35i		−	1.58219i	0.611692	−	0.791096i	$-0.290489\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−805.700			−1.24529			−0.622643	−	0.782506i	$-0.713941\pi$
$647$	−805.700			−1.24529			−0.622643	+	0.782506i	$0.713941\pi$
$648$		0			0
$649$	826.000			1.27273
$650$		0			0
$651$		0			0
$652$		0			0
$653$	67.1416			0.102820			0.0514101	−	0.998678i	$-0.483628\pi$
$653$	67.1416			0.102820			0.0514101	+	0.998678i	$0.483628\pi$
$654$		0			0
$655$	−234.000	+	793.533i	−0.357252	+	1.21150i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	524.673i		−	0.796166i	−0.917349	−	0.398083i	$-0.869676\pi$
$659$		−	524.673i		−	0.796166i	0.917349	−	0.398083i	$-0.130324\pi$
$660$		0			0
$661$	74.0000			0.111952			0.0559758	−	0.998432i	$-0.482173\pi$
$661$	74.0000			0.111952			0.0559758	+	0.998432i	$0.482173\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−115.100	+	390.323i	−0.173083	+	0.586952i
$666$		0			0
$667$		−	81.3880i		−	0.122021i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	692.965i		−	1.03273i
$672$		0			0
$673$			691.798i			1.02793i	0.857811	+	0.513966i	$0.171824\pi$
$673$			691.798i			1.02793i	−0.857811	+	0.513966i	$0.828176\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	278.158			0.410869			0.205434	−	0.978671i	$-0.434139\pi$
$677$	278.158			0.410869			0.205434	+	0.978671i	$0.434139\pi$
$678$		0			0
$679$	644.000			0.948454
$680$		0			0
$681$		0			0
$682$		0			0
$683$	422.033			0.617911			0.308955	−	0.951077i	$-0.400021\pi$
$683$	422.033			0.617911			0.308955	+	0.951077i	$0.400021\pi$
$684$		0			0
$685$	−552.000	−	162.776i	−0.805839	−	0.237629i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	68.0000			0.0984081			0.0492041	−	0.998789i	$-0.484332\pi$
$691$	68.0000			0.0984081			0.0492041	+	0.998789i	$0.484332\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−297.342	−	87.6812i	−0.427830	−	0.126160i
$696$		0			0
$697$		−	1329.34i		−	1.90723i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			130.108i			0.185603i	0.995685	+	0.0928015i	$0.0295822\pi$
$701$			130.108i			0.185603i	−0.995685	+	0.0928015i	$0.970418\pi$
$702$		0			0
$703$		−	81.3880i		−	0.115772i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−537.133			−0.759736
$708$		0			0
$709$	−210.000			−0.296192			−0.148096	−	0.988973i	$-0.547314\pi$
$709$	−210.000			−0.296192			−0.148096	+	0.988973i	$0.547314\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−364.483			−0.511197
$714$		0			0
$715$	−966.000	−	284.858i	−1.35105	−	0.398403i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			121.622i			0.169155i	0.996417	+	0.0845774i	$0.0269540\pi$
$719$			121.622i			0.169155i	−0.996417	+	0.0845774i	$0.973046\pi$
$720$		0			0
$721$	−598.000			−0.829404
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−115.100	+	178.191i	−0.158759	+	0.245781i
$726$		0			0
$727$			20.3470i			0.0279876i	0.999902	+	0.0139938i	$0.00445451\pi$
$727$			20.3470i			0.0279876i	−0.999902	+	0.0139938i	$0.995545\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			1301.08i			1.77986i
$732$		0			0
$733$		−	1417.51i		−	1.93384i	−0.255074	−	0.966922i	$-0.582100\pi$
$733$		−	1417.51i		−	1.93384i	0.255074	−	0.966922i	$-0.417900\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1074.27			−1.45762
$738$		0			0
$739$	−416.000			−0.562923			−0.281461	−	0.959573i	$-0.590819\pi$
$739$	−416.000			−0.562923			−0.281461	+	0.959573i	$0.590819\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	805.700			1.08439			0.542194	−	0.840254i	$-0.317594\pi$
$743$	805.700			1.08439			0.542194	+	0.840254i	$0.317594\pi$
$744$		0			0
$745$	224.000	−	759.621i	0.300671	−	1.01963i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			390.323i			0.521125i
$750$		0			0
$751$	−86.0000			−0.114514			−0.0572570	−	0.998359i	$-0.518235\pi$
$751$	−86.0000			−0.114514			−0.0572570	+	0.998359i	$0.518235\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−335.708	−	98.9949i	−0.444647	−	0.131119i
$756$		0			0
$757$			47.4763i			0.0627164i	0.999508	+	0.0313582i	$0.00998326\pi$
$757$			47.4763i			0.0627164i	−0.999508	+	0.0313582i	$0.990017\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	369.110i		−	0.485033i	−0.970147	−	0.242516i	$-0.922027\pi$
$761$		−	369.110i		−	0.485033i	0.970147	−	0.242516i	$-0.0779728\pi$
$762$		0			0
$763$		−	501.892i		−	0.657788i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1697.72			2.21346
$768$		0			0
$769$	384.000			0.499350			0.249675	−	0.968330i	$-0.419676\pi$
$769$	384.000			0.499350			0.249675	+	0.968330i	$0.419676\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−201.425			−0.260576			−0.130288	−	0.991476i	$-0.541590\pi$
$773$	−201.425			−0.260576			−0.130288	+	0.991476i	$0.541590\pi$
$774$		0			0
$775$	798.000	+	515.457i	1.02968	+	0.665106i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			831.558i			1.06747i
$780$		0			0
$781$	−1176.00			−1.50576
$782$		0			0
$783$		0			0
$784$		0			0
$785$	67.1416	−	227.688i	0.0855308	−	0.290049i
$786$		0			0
$787$		−	474.763i		−	0.603257i	−0.953426	−	0.301628i	$-0.902470\pi$
$787$		−	474.763i		−	0.603257i	0.953426	−	0.301628i	$-0.0975302\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1170.97i		−	1.48037i
$792$		0			0
$793$		−	1424.29i		−	1.79608i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−498.766			−0.625805			−0.312902	−	0.949785i	$-0.601301\pi$
$797$	−498.766			−0.625805			−0.312902	+	0.949785i	$0.601301\pi$
$798$		0			0
$799$	1472.00			1.84230
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−134.283			−0.167227
$804$		0			0
$805$	−92.0000	+	311.987i	−0.114286	+	0.387562i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1480.68i			1.83026i	0.403157	+	0.915131i	$0.367913\pi$
$809$			1480.68i			1.83026i	−0.403157	+	0.915131i	$0.632087\pi$
$810$		0			0
$811$	602.000			0.742293			0.371147	−	0.928574i	$-0.378965\pi$
$811$	602.000			0.742293			0.371147	+	0.928574i	$0.378965\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−134.283	+	455.377i	−0.164765	+	0.558744i
$816$		0			0
$817$		−	813.880i		−	0.996181i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			1405.73i			1.71221i	0.516798	+	0.856107i	$0.327124\pi$
$821$			1405.73i			1.71221i	−0.516798	+	0.856107i	$0.672876\pi$
$822$		0			0
$823$			1281.86i			1.55755i	0.627306	+	0.778773i	$0.284158\pi$
$823$			1281.86i			1.55755i	−0.627306	+	0.778773i	$0.715842\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1170.18			−1.41497			−0.707487	−	0.706727i	$-0.750171\pi$
$827$	−1170.18			−1.41497			−0.707487	+	0.706727i	$0.750171\pi$
$828$		0			0
$829$	1074.00			1.29554			0.647768	−	0.761837i	$-0.275702\pi$
$829$	1074.00			1.29554			0.647768	+	0.761837i	$0.275702\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	57.5500			0.0690876
$834$		0			0
$835$	−966.000	−	284.858i	−1.15689	−	0.341147i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			743.876i			0.886623i	0.896368	+	0.443311i	$0.146196\pi$
$839$			743.876i			0.886623i	−0.896368	+	0.443311i	$0.853804\pi$
$840$		0			0
$841$	769.000			0.914388
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1174.98	−	346.482i	−1.39051	−	0.410038i
$846$		0			0
$847$		−	155.994i		−	0.184172i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	65.0538i		−	0.0764440i
$852$		0			0
$853$			169.558i			0.198779i	0.995049	+	0.0993894i	$0.0316889\pi$
$853$			169.558i			0.198779i	−0.995049	+	0.0993894i	$0.968311\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	115.100			0.134306			0.0671528	−	0.997743i	$-0.478608\pi$
$857$	115.100			0.134306			0.0671528	+	0.997743i	$0.478608\pi$
$858$		0			0
$859$	−154.000			−0.179278			−0.0896391	−	0.995974i	$-0.528571\pi$
$859$	−154.000			−0.179278			−0.0896391	+	0.995974i	$0.528571\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1064.67			−1.23369			−0.616845	−	0.787085i	$-0.711589\pi$
$863$	−1064.67			−1.23369			−0.616845	+	0.787085i	$0.711589\pi$
$864$		0			0
$865$	−1288.00	−	379.810i	−1.48902	−	0.439087i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	296.985i		−	0.341755i
$870$		0			0
$871$	−2208.00			−2.53502
$872$		0			0
$873$		0			0
$874$		0			0
$875$	642.641	−	552.958i	0.734447	−	0.631951i
$876$		0			0
$877$		−	1390.38i		−	1.58538i	−0.609625	−	0.792690i	$-0.708680\pi$
$877$		−	1390.38i		−	1.58538i	0.609625	−	0.792690i	$-0.291320\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	861.256i		−	0.977589i	−0.872399	−	0.488795i	$-0.837437\pi$
$881$		−	861.256i		−	0.977589i	0.872399	−	0.488795i	$-0.162563\pi$
$882$		0			0
$883$		−	1017.35i		−	1.15215i	−0.817396	−	0.576076i	$-0.804583\pi$
$883$		−	1017.35i		−	1.15215i	0.817396	−	0.576076i	$-0.195417\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	604.275			0.681257			0.340628	−	0.940198i	$-0.389360\pi$
$887$	604.275			0.681257			0.340628	+	0.940198i	$0.389360\pi$
$888$		0			0
$889$	−1150.00			−1.29359
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−920.800			−1.03113
$894$		0			0
$895$	190.000	−	644.321i	0.212291	−	0.719912i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			322.441i			0.358666i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−105.508	−	31.1127i	−0.116584	−	0.0343787i
$906$		0			0
$907$		−	935.962i		−	1.03193i	−0.856609	−	0.515966i	$-0.827433\pi$
$907$		−	935.962i		−	1.03193i	0.856609	−	0.515966i	$-0.172567\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	763.675i		−	0.838282i	−0.907921	−	0.419141i	$-0.862331\pi$
$911$		−	763.675i		−	0.838282i	0.907921	−	0.419141i	$-0.137669\pi$
$912$		0			0
$913$			1329.34i			1.45601i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1122.22			1.22380
$918$		0			0
$919$	54.0000			0.0587595			0.0293798	−	0.999568i	$-0.490647\pi$
$919$	54.0000			0.0587595			0.0293798	+	0.999568i	$0.490647\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2417.10			−2.61874
$924$		0			0
$925$	−92.0000	+	142.429i	−0.0994595	+	0.153977i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	148.492i		−	0.159841i	−0.996801	−	0.0799206i	$-0.974533\pi$
$929$		−	148.492i		−	0.159841i	0.996801	−	0.0799206i	$-0.0254667\pi$
$930$		0			0
$931$	−36.0000			−0.0386681
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−268.567	+	910.754i	−0.287237	+	0.974068i
$936$		0			0
$937$			786.750i			0.839648i	0.907606	+	0.419824i	$0.137908\pi$
$937$			786.750i			0.839648i	−0.907606	+	0.419824i	$0.862092\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			342.240i			0.363698i	0.983326	+	0.181849i	$0.0582082\pi$
$941$			342.240i			0.363698i	−0.983326	+	0.181849i	$0.941792\pi$
$942$		0			0
$943$			664.668i			0.704844i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	402.850			0.425396			0.212698	−	0.977118i	$-0.431775\pi$
$947$	402.850			0.425396			0.212698	+	0.977118i	$0.431775\pi$
$948$		0			0
$949$	−276.000			−0.290832
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1074.27			−1.12725			−0.563623	−	0.826032i	$-0.690593\pi$
$953$	−1074.27			−1.12725			−0.563623	+	0.826032i	$0.690593\pi$
$954$		0			0
$955$	276.000	−	935.962i	0.289005	−	0.980064i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			780.646i			0.814021i
$960$		0			0
$961$	483.000			0.502601
$962$		0			0
$963$		0			0
$964$		0			0
$965$	383.667	−	1301.08i	0.397582	−	1.34827i
$966$		0			0
$967$		−	400.157i		−	0.413813i	−0.978361	−	0.206907i	$-0.933660\pi$
$967$		−	400.157i		−	0.413813i	0.978361	−	0.206907i	$-0.0663396\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	683.065i		−	0.703466i	−0.936100	−	0.351733i	$-0.885593\pi$
$971$		−	683.065i		−	0.703466i	0.936100	−	0.351733i	$-0.114407\pi$
$972$		0			0
$973$			420.504i			0.432173i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1611.40			1.64933			0.824667	−	0.565618i	$-0.191363\pi$
$977$	1611.40			1.64933			0.824667	+	0.565618i	$0.191363\pi$
$978$		0			0
$979$	322.000			0.328907
$980$		0			0
$981$		0			0
$982$		0			0
$983$	959.166			0.975754			0.487877	−	0.872912i	$-0.337771\pi$
$983$	959.166			0.975754			0.487877	+	0.872912i	$0.337771\pi$
$984$		0			0
$985$	−782.000	−	230.599i	−0.793909	−	0.234111i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	650.538i		−	0.657774i
$990$		0			0
$991$	−1054.00			−1.06357			−0.531786	−	0.846879i	$-0.678479\pi$
$991$	−1054.00			−1.06357			−0.531786	+	0.846879i	$0.678479\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1409.97	−	415.779i	−1.41706	−	0.417868i
$996$		0			0
$997$			1200.47i			1.20408i	0.798464	+	0.602042i	$0.205646\pi$
$997$			1200.47i			1.20408i	−0.798464	+	0.602042i	$0.794354\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.3.c.b.449.4		4
3.2	odd	2	inner	720.3.c.b.449.1		4
4.3	odd	2		180.3.b.a.89.4	yes	4
5.2	odd	4		3600.3.l.r.1601.4		4
5.3	odd	4		3600.3.l.r.1601.2		4
5.4	even	2	inner	720.3.c.b.449.2		4
8.3	odd	2		2880.3.c.c.449.1		4
8.5	even	2		2880.3.c.f.449.1		4
12.11	even	2		180.3.b.a.89.1	✓	4
15.2	even	4		3600.3.l.r.1601.3		4
15.8	even	4		3600.3.l.r.1601.1		4
15.14	odd	2	inner	720.3.c.b.449.3		4
20.3	even	4		900.3.g.c.701.3		4
20.7	even	4		900.3.g.c.701.1		4
20.19	odd	2		180.3.b.a.89.2	yes	4
24.5	odd	2		2880.3.c.f.449.4		4
24.11	even	2		2880.3.c.c.449.4		4
36.7	odd	6		1620.3.t.c.1349.1		8
36.11	even	6		1620.3.t.c.1349.4		8
36.23	even	6		1620.3.t.c.269.3		8
36.31	odd	6		1620.3.t.c.269.2		8
40.19	odd	2		2880.3.c.c.449.3		4
40.29	even	2		2880.3.c.f.449.3		4
60.23	odd	4		900.3.g.c.701.4		4
60.47	odd	4		900.3.g.c.701.2		4
60.59	even	2		180.3.b.a.89.3	yes	4
120.29	odd	2		2880.3.c.f.449.2		4
120.59	even	2		2880.3.c.c.449.2		4
180.59	even	6		1620.3.t.c.269.1		8
180.79	odd	6		1620.3.t.c.1349.3		8
180.119	even	6		1620.3.t.c.1349.2		8
180.139	odd	6		1620.3.t.c.269.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.b.a.89.1	✓	4	12.11	even	2
180.3.b.a.89.2	yes	4	20.19	odd	2
180.3.b.a.89.3	yes	4	60.59	even	2
180.3.b.a.89.4	yes	4	4.3	odd	2
720.3.c.b.449.1		4	3.2	odd	2	inner
720.3.c.b.449.2		4	5.4	even	2	inner
720.3.c.b.449.3		4	15.14	odd	2	inner
720.3.c.b.449.4		4	1.1	even	1	trivial
900.3.g.c.701.1		4	20.7	even	4
900.3.g.c.701.2		4	60.47	odd	4
900.3.g.c.701.3		4	20.3	even	4
900.3.g.c.701.4		4	60.23	odd	4
1620.3.t.c.269.1		8	180.59	even	6
1620.3.t.c.269.2		8	36.31	odd	6
1620.3.t.c.269.3		8	36.23	even	6
1620.3.t.c.269.4		8	180.139	odd	6
1620.3.t.c.1349.1		8	36.7	odd	6
1620.3.t.c.1349.2		8	180.119	even	6
1620.3.t.c.1349.3		8	180.79	odd	6
1620.3.t.c.1349.4		8	36.11	even	6
2880.3.c.c.449.1		4	8.3	odd	2
2880.3.c.c.449.2		4	120.59	even	2
2880.3.c.c.449.3		4	40.19	odd	2
2880.3.c.c.449.4		4	24.11	even	2
2880.3.c.f.449.1		4	8.5	even	2
2880.3.c.f.449.2		4	120.29	odd	2
2880.3.c.f.449.3		4	40.29	even	2
2880.3.c.f.449.4		4	24.5	odd	2
3600.3.l.r.1601.1		4	15.8	even	4
3600.3.l.r.1601.2		4	5.3	odd	4
3600.3.l.r.1601.3		4	15.2	even	4
3600.3.l.r.1601.4		4	5.2	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	720.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.79583 + 1.41421i) q^{5} -6.78233i q^{7} +O(q^{10})\)	\(q+(4.79583 + 1.41421i) q^{5} -6.78233i q^{7} +9.89949i q^{11} +20.3470i q^{13} +19.1833 q^{17} -12.0000 q^{19} -9.59166 q^{23} +(21.0000 + 13.5647i) q^{25} +8.48528i q^{29} +38.0000 q^{31} +(9.59166 - 32.5269i) q^{35} +6.78233i q^{37} -69.2965i q^{41} +67.8233i q^{43} +76.7333 q^{47} +3.00000 q^{49} +(-14.0000 + 47.4763i) q^{55} -83.4386i q^{59} -70.0000 q^{61} +(-28.7750 + 97.5807i) q^{65} +108.517i q^{67} +118.794i q^{71} +13.5647i q^{73} +67.1416 q^{77} -30.0000 q^{79} +134.283 q^{83} +(92.0000 + 27.1293i) q^{85} -32.5269i q^{89} +138.000 q^{91} +(-57.5500 - 16.9706i) q^{95} +94.9526i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 48 q^{19} + 84 q^{25} + 152 q^{31} + 12 q^{49} - 56 q^{55} - 280 q^{61} - 120 q^{79} + 368 q^{85} + 552 q^{91}+O(q^{100}) \) 4 * q - 48 * q^19 + 84 * q^25 + 152 * q^31 + 12 * q^49 - 56 * q^55 - 280 * q^61 - 120 * q^79 + 368 * q^85 + 552 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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