LMFDB - Embedded newform 5520.2.be.c.1471.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1471,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5520.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.be (of order $2$, degree $1$, 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:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.23
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.c.1471.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} +1.70195 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{5} +1.70195 q^{7} -1.00000 q^{9} +3.79849 q^{11} +1.61626 q^{13} -1.00000 q^{15} -2.05437i q^{17} -0.518940 q^{19} -1.70195i q^{21} +(-4.69289 + 0.988343i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -2.88604 q^{29} -4.16850i q^{31} -3.79849i q^{33} -1.70195i q^{35} -7.56990i q^{37} -1.61626i q^{39} +11.8923 q^{41} -4.43156 q^{43} +1.00000i q^{45} -0.599371i q^{47} -4.10336 q^{49} -2.05437 q^{51} +2.97105i q^{53} -3.79849i q^{55} +0.518940i q^{57} -9.67061i q^{59} +2.26190i q^{61} -1.70195 q^{63} -1.61626i q^{65} +9.80785 q^{67} +(0.988343 + 4.69289i) q^{69} -9.51609i q^{71} -7.31068 q^{73} +1.00000i q^{75} +6.46485 q^{77} +7.33976 q^{79} +1.00000 q^{81} +4.63473 q^{83} -2.05437 q^{85} +2.88604i q^{87} -16.0785i q^{89} +2.75080 q^{91} -4.16850 q^{93} +0.518940i q^{95} +14.4819i q^{97} -3.79849 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) $ 32 * q - 8 * q^7 - 32 * q^9	$ 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) $ 32 * q - 8 * q^7 - 32 * q^9 + 8 * q^11 - 8 * q^13 - 32 * q^15 - 32 * q^25 + 4 * q^29 + 20 * q^41 + 52 * q^49 - 4 * q^51 + 8 * q^63 + 32 * q^67 - 40 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^81 - 4 * q^85 - 48 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1841$	$4417$	$4831$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	1.70195			0.643278			0.321639	−	0.946862i	$-0.395766\pi$
$7$	1.70195			0.643278			0.321639	+	0.946862i	$0.395766\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	3.79849			1.14529			0.572644	−	0.819804i	$-0.305918\pi$
$11$	3.79849			1.14529			0.572644	+	0.819804i	$0.305918\pi$
$12$		0			0
$13$	1.61626			0.448270			0.224135	−	0.974558i	$-0.428044\pi$
$13$	1.61626			0.448270			0.224135	+	0.974558i	$0.428044\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$		−	2.05437i		−	0.498259i	−0.968470	−	0.249129i	$-0.919856\pi$
$17$		−	2.05437i		−	0.498259i	0.968470	−	0.249129i	$-0.0801444\pi$
$18$		0			0
$19$	−0.518940			−0.119053			−0.0595265	−	0.998227i	$-0.518959\pi$
$19$	−0.518940			−0.119053			−0.0595265	+	0.998227i	$0.518959\pi$
$20$		0			0
$21$		−	1.70195i		−	0.371397i
$22$		0			0
$23$	−4.69289	+	0.988343i	−0.978534	+	0.206084i
$23$	−4.69289	+	0.988343i	−0.978534	+	0.206084i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−2.88604			−0.535924			−0.267962	−	0.963429i	$-0.586350\pi$
$29$	−2.88604			−0.535924			−0.267962	+	0.963429i	$0.586350\pi$
$30$		0			0
$31$		−	4.16850i		−	0.748685i	−0.927291	−	0.374343i	$-0.877868\pi$
$31$		−	4.16850i		−	0.748685i	0.927291	−	0.374343i	$-0.122132\pi$
$32$		0			0
$33$		−	3.79849i		−	0.661232i
$34$		0			0
$35$		−	1.70195i		−	0.287683i
$36$		0			0
$37$		−	7.56990i		−	1.24448i	−0.782825	−	0.622242i	$-0.786222\pi$
$37$		−	7.56990i		−	1.24448i	0.782825	−	0.622242i	$-0.213778\pi$
$38$		0			0
$39$		−	1.61626i		−	0.258809i
$40$		0			0
$41$	11.8923			1.85727			0.928635	−	0.370995i	$-0.120983\pi$
$41$	11.8923			1.85727			0.928635	+	0.370995i	$0.120983\pi$
$42$		0			0
$43$	−4.43156			−0.675807			−0.337903	−	0.941181i	$-0.609718\pi$
$43$	−4.43156			−0.675807			−0.337903	+	0.941181i	$0.609718\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$		−	0.599371i		−	0.0874273i	−0.999044	−	0.0437136i	$-0.986081\pi$
$47$		−	0.599371i		−	0.0874273i	0.999044	−	0.0437136i	$-0.0139189\pi$
$48$		0			0
$49$	−4.10336			−0.586194
$50$		0			0
$51$	−2.05437			−0.287670
$52$		0			0
$53$			2.97105i			0.408106i	0.978960	+	0.204053i	$0.0654114\pi$
$53$			2.97105i			0.408106i	−0.978960	+	0.204053i	$0.934589\pi$
$54$		0			0
$55$		−	3.79849i		−	0.512188i
$56$		0			0
$57$			0.518940i			0.0687353i
$58$		0			0
$59$		−	9.67061i		−	1.25901i	−0.776998	−	0.629503i	$-0.783259\pi$
$59$		−	9.67061i		−	1.25901i	0.776998	−	0.629503i	$-0.216741\pi$
$60$		0			0
$61$			2.26190i			0.289607i	0.989460	+	0.144804i	$0.0462550\pi$
$61$			2.26190i			0.289607i	−0.989460	+	0.144804i	$0.953745\pi$
$62$		0			0
$63$	−1.70195			−0.214426
$64$		0			0
$65$		−	1.61626i		−	0.200472i
$66$		0			0
$67$	9.80785			1.19822			0.599110	−	0.800667i	$-0.295521\pi$
$67$	9.80785			1.19822			0.599110	+	0.800667i	$0.295521\pi$
$68$		0			0
$69$	0.988343	+	4.69289i	0.118982	+	0.564957i
$70$		0			0
$71$		−	9.51609i		−	1.12935i	−0.825313	−	0.564676i	$-0.809001\pi$
$71$		−	9.51609i		−	1.12935i	0.825313	−	0.564676i	$-0.190999\pi$
$72$		0			0
$73$	−7.31068			−0.855651			−0.427825	−	0.903861i	$-0.640720\pi$
$73$	−7.31068			−0.855651			−0.427825	+	0.903861i	$0.640720\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	6.46485			0.736738
$78$		0			0
$79$	7.33976			0.825787			0.412894	−	0.910779i	$-0.364518\pi$
$79$	7.33976			0.825787			0.412894	+	0.910779i	$0.364518\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	4.63473			0.508728			0.254364	−	0.967109i	$-0.418134\pi$
$83$	4.63473			0.508728			0.254364	+	0.967109i	$0.418134\pi$
$84$		0			0
$85$	−2.05437			−0.222828
$86$		0			0
$87$			2.88604i			0.309416i
$88$		0			0
$89$		−	16.0785i		−	1.70432i	−0.523283	−	0.852159i	$-0.675293\pi$
$89$		−	16.0785i		−	1.70432i	0.523283	−	0.852159i	$-0.324707\pi$
$90$		0			0
$91$	2.75080			0.288362
$92$		0			0
$93$	−4.16850			−0.432253
$94$		0			0
$95$			0.518940i			0.0532421i
$96$		0			0
$97$			14.4819i			1.47041i	0.677842	+	0.735207i	$0.262915\pi$
$97$			14.4819i			1.47041i	−0.677842	+	0.735207i	$0.737085\pi$
$98$		0			0
$99$	−3.79849			−0.381762
$100$		0			0
$101$	0.619325			0.0616251			0.0308126	−	0.999525i	$-0.490191\pi$
$101$	0.619325			0.0616251			0.0308126	+	0.999525i	$0.490191\pi$
$102$		0			0
$103$	3.97808			0.391972			0.195986	−	0.980607i	$-0.437209\pi$
$103$	3.97808			0.391972			0.195986	+	0.980607i	$0.437209\pi$
$104$		0			0
$105$	−1.70195			−0.166094
$106$		0			0
$107$	9.38912			0.907681			0.453840	−	0.891083i	$-0.350054\pi$
$107$	9.38912			0.907681			0.453840	+	0.891083i	$0.350054\pi$
$108$		0			0
$109$			1.83752i			0.176003i	0.996120	+	0.0880014i	$0.0280480\pi$
$109$			1.83752i			0.176003i	−0.996120	+	0.0880014i	$0.971952\pi$
$110$		0			0
$111$	−7.56990			−0.718503
$112$		0			0
$113$			5.28710i			0.497368i	0.968585	+	0.248684i	$0.0799981\pi$
$113$			5.28710i			0.497368i	−0.968585	+	0.248684i	$0.920002\pi$
$114$		0			0
$115$	0.988343	+	4.69289i	0.0921634	+	0.437614i
$116$		0			0
$117$	−1.61626			−0.149423
$118$		0			0
$119$		−	3.49645i		−	0.320519i
$120$		0			0
$121$	3.42851			0.311683
$122$		0			0
$123$		−	11.8923i		−	1.07230i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	17.3539i		−	1.53991i	−0.638099	−	0.769954i	$-0.720279\pi$
$127$		−	17.3539i		−	1.53991i	0.638099	−	0.769954i	$-0.279721\pi$
$128$		0			0
$129$			4.43156i			0.390177i
$130$		0			0
$131$		−	7.24393i		−	0.632905i	−0.948608	−	0.316453i	$-0.897508\pi$
$131$		−	7.24393i		−	0.632905i	0.948608	−	0.316453i	$-0.102492\pi$
$132$		0			0
$133$	−0.883212			−0.0765842
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$			2.88747i			0.246693i	0.992364	+	0.123347i	$0.0393627\pi$
$137$			2.88747i			0.246693i	−0.992364	+	0.123347i	$0.960637\pi$
$138$		0			0
$139$			15.7354i			1.33466i	0.744764	+	0.667328i	$0.232562\pi$
$139$			15.7354i			1.33466i	−0.744764	+	0.667328i	$0.767438\pi$
$140$		0			0
$141$	−0.599371			−0.0504762
$142$		0			0
$143$	6.13935			0.513398
$144$		0			0
$145$			2.88604i			0.239673i
$146$		0			0
$147$			4.10336i			0.338439i
$148$		0			0
$149$			15.1946i			1.24479i	0.782704	+	0.622394i	$0.213840\pi$
$149$			15.1946i			1.24479i	−0.782704	+	0.622394i	$0.786160\pi$
$150$		0			0
$151$		−	0.629232i		−	0.0512062i	−0.999672	−	0.0256031i	$-0.991849\pi$
$151$		−	0.629232i		−	0.0512062i	0.999672	−	0.0256031i	$-0.00815061\pi$
$152$		0			0
$153$			2.05437i			0.166086i
$154$		0			0
$155$	−4.16850			−0.334822
$156$		0			0
$157$			5.67705i			0.453078i	0.974002	+	0.226539i	$0.0727411\pi$
$157$			5.67705i			0.453078i	−0.974002	+	0.226539i	$0.927259\pi$
$158$		0			0
$159$	2.97105			0.235620
$160$		0			0
$161$	−7.98707	+	1.68211i	−0.629469	+	0.132569i
$162$		0			0
$163$		−	4.41526i		−	0.345830i	−0.984937	−	0.172915i	$-0.944681\pi$
$163$		−	4.41526i		−	0.345830i	0.984937	−	0.172915i	$-0.0553185\pi$
$164$		0			0
$165$	−3.79849			−0.295712
$166$		0			0
$167$		−	19.0290i		−	1.47251i	−0.676704	−	0.736255i	$-0.736592\pi$
$167$		−	19.0290i		−	1.47251i	0.676704	−	0.736255i	$-0.263408\pi$
$168$		0			0
$169$	−10.3877			−0.799054
$170$		0			0
$171$	0.518940			0.0396843
$172$		0			0
$173$	−4.16476			−0.316640			−0.158320	−	0.987388i	$-0.550608\pi$
$173$	−4.16476			−0.316640			−0.158320	+	0.987388i	$0.550608\pi$
$174$		0			0
$175$	−1.70195			−0.128656
$176$		0			0
$177$	−9.67061			−0.726887
$178$		0			0
$179$		−	10.0952i		−	0.754554i	−0.926101	−	0.377277i	$-0.876861\pi$
$179$		−	10.0952i		−	0.754554i	0.926101	−	0.377277i	$-0.123139\pi$
$180$		0			0
$181$		−	9.38599i		−	0.697655i	−0.937187	−	0.348828i	$-0.886580\pi$
$181$		−	9.38599i		−	0.697655i	0.937187	−	0.348828i	$-0.113420\pi$
$182$		0			0
$183$	2.26190			0.167205
$184$		0			0
$185$	−7.56990			−0.556550
$186$		0			0
$187$		−	7.80351i		−	0.570649i
$188$		0			0
$189$			1.70195i			0.123799i
$190$		0			0
$191$	14.6961			1.06337			0.531687	−	0.846941i	$-0.321558\pi$
$191$	14.6961			1.06337			0.531687	+	0.846941i	$0.321558\pi$
$192$		0			0
$193$	−4.30526			−0.309899			−0.154950	−	0.987922i	$-0.549522\pi$
$193$	−4.30526			−0.309899			−0.154950	+	0.987922i	$0.549522\pi$
$194$		0			0
$195$	−1.61626			−0.115743
$196$		0			0
$197$	−3.85449			−0.274621			−0.137310	−	0.990528i	$-0.543846\pi$
$197$	−3.85449			−0.274621			−0.137310	+	0.990528i	$0.543846\pi$
$198$		0			0
$199$	2.53494			0.179697			0.0898486	−	0.995955i	$-0.471362\pi$
$199$	2.53494			0.179697			0.0898486	+	0.995955i	$0.471362\pi$
$200$		0			0
$201$		−	9.80785i		−	0.691793i
$202$		0			0
$203$	−4.91190			−0.344748
$204$		0			0
$205$		−	11.8923i		−	0.830596i
$206$		0			0
$207$	4.69289	−	0.988343i	0.326178	−	0.0686946i
$208$		0			0
$209$	−1.97119			−0.136350
$210$		0			0
$211$		−	21.3867i		−	1.47232i	−0.676808	−	0.736159i	$-0.736637\pi$
$211$		−	21.3867i		−	1.47232i	0.676808	−	0.736159i	$-0.263363\pi$
$212$		0			0
$213$	−9.51609			−0.652032
$214$		0			0
$215$			4.43156i			0.302230i
$216$		0			0
$217$		−	7.09459i		−	0.481612i
$218$		0			0
$219$			7.31068i			0.494010i
$220$		0			0
$221$		−	3.32040i		−	0.223354i
$222$		0			0
$223$		−	22.4725i		−	1.50487i	−0.658667	−	0.752435i	$-0.728879\pi$
$223$		−	22.4725i		−	1.50487i	0.658667	−	0.752435i	$-0.271121\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	0.856466			0.0568456			0.0284228	−	0.999596i	$-0.490952\pi$
$227$	0.856466			0.0568456			0.0284228	+	0.999596i	$0.490952\pi$
$228$		0			0
$229$		−	6.28764i		−	0.415499i	−0.978182	−	0.207749i	$-0.933386\pi$
$229$		−	6.28764i		−	0.415499i	0.978182	−	0.207749i	$-0.0666138\pi$
$230$		0			0
$231$		−	6.46485i		−	0.425356i
$232$		0			0
$233$	25.5189			1.67180			0.835900	−	0.548882i	$-0.184947\pi$
$233$	25.5189			1.67180			0.835900	+	0.548882i	$0.184947\pi$
$234$		0			0
$235$	−0.599371			−0.0390987
$236$		0			0
$237$		−	7.33976i		−	0.476769i
$238$		0			0
$239$		−	17.9194i		−	1.15911i	−0.814934	−	0.579553i	$-0.803227\pi$
$239$		−	17.9194i		−	1.15911i	0.814934	−	0.579553i	$-0.196773\pi$
$240$		0			0
$241$			7.79751i			0.502282i	0.967951	+	0.251141i	$0.0808058\pi$
$241$			7.79751i			0.502282i	−0.967951	+	0.251141i	$0.919194\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$			4.10336i			0.262154i
$246$		0			0
$247$	−0.838743			−0.0533679
$248$		0			0
$249$		−	4.63473i		−	0.293714i
$250$		0			0
$251$	−13.1223			−0.828273			−0.414136	−	0.910215i	$-0.635916\pi$
$251$	−13.1223			−0.828273			−0.414136	+	0.910215i	$0.635916\pi$
$252$		0			0
$253$	−17.8259	+	3.75421i	−1.12070	+	0.236025i
$254$		0			0
$255$			2.05437i			0.128650i
$256$		0			0
$257$	23.3047			1.45370			0.726852	−	0.686794i	$-0.240982\pi$
$257$	23.3047			1.45370			0.726852	+	0.686794i	$0.240982\pi$
$258$		0			0
$259$		−	12.8836i		−	0.800549i
$260$		0			0
$261$	2.88604			0.178641
$262$		0			0
$263$	−10.9363			−0.674361			−0.337181	−	0.941440i	$-0.609473\pi$
$263$	−10.9363			−0.674361			−0.337181	+	0.941440i	$0.609473\pi$
$264$		0			0
$265$	2.97105			0.182510
$266$		0			0
$267$	−16.0785			−0.983988
$268$		0			0
$269$	−24.0819			−1.46830			−0.734150	−	0.678987i	$-0.762419\pi$
$269$	−24.0819			−1.46830			−0.734150	+	0.678987i	$0.762419\pi$
$270$		0			0
$271$			25.0959i			1.52447i	0.647301	+	0.762234i	$0.275898\pi$
$271$			25.0959i			1.52447i	−0.647301	+	0.762234i	$0.724102\pi$
$272$		0			0
$273$		−	2.75080i		−	0.166486i
$274$		0			0
$275$	−3.79849			−0.229057
$276$		0			0
$277$	−19.8548			−1.19296			−0.596480	−	0.802628i	$-0.703435\pi$
$277$	−19.8548			−1.19296			−0.596480	+	0.802628i	$0.703435\pi$
$278$		0			0
$279$			4.16850i			0.249562i
$280$		0			0
$281$		−	31.3138i		−	1.86803i	−0.357240	−	0.934013i	$-0.616282\pi$
$281$		−	31.3138i		−	1.86803i	0.357240	−	0.934013i	$-0.383718\pi$
$282$		0			0
$283$	−20.8530			−1.23958			−0.619790	−	0.784768i	$-0.712782\pi$
$283$	−20.8530			−1.23958			−0.619790	+	0.784768i	$0.712782\pi$
$284$		0			0
$285$	0.518940			0.0307394
$286$		0			0
$287$	20.2402			1.19474
$288$		0			0
$289$	12.7796			0.751738
$290$		0			0
$291$	14.4819			0.848944
$292$		0			0
$293$			16.4218i			0.959373i	0.877440	+	0.479686i	$0.159250\pi$
$293$			16.4218i			0.959373i	−0.877440	+	0.479686i	$0.840750\pi$
$294$		0			0
$295$	−9.67061			−0.563044
$296$		0			0
$297$			3.79849i			0.220411i
$298$		0			0
$299$	−7.58493	+	1.59742i	−0.438648	+	0.0923812i
$300$		0			0
$301$	−7.54231			−0.434732
$302$		0			0
$303$		−	0.619325i		−	0.0355793i
$304$		0			0
$305$	2.26190			0.129516
$306$		0			0
$307$			9.40135i			0.536563i	0.963341	+	0.268282i	$0.0864558\pi$
$307$			9.40135i			0.536563i	−0.963341	+	0.268282i	$0.913544\pi$
$308$		0			0
$309$		−	3.97808i		−	0.226305i
$310$		0			0
$311$		−	10.4152i		−	0.590593i	−0.955406	−	0.295297i	$-0.904582\pi$
$311$		−	10.4152i		−	0.590593i	0.955406	−	0.295297i	$-0.0954184\pi$
$312$		0			0
$313$			17.4733i			0.987649i	0.869562	+	0.493824i	$0.164401\pi$
$313$			17.4733i			0.987649i	−0.869562	+	0.493824i	$0.835599\pi$
$314$		0			0
$315$			1.70195i			0.0958942i
$316$		0			0
$317$	−0.464007			−0.0260612			−0.0130306	−	0.999915i	$-0.504148\pi$
$317$	−0.464007			−0.0260612			−0.0130306	+	0.999915i	$0.504148\pi$
$318$		0			0
$319$	−10.9626			−0.613787
$320$		0			0
$321$		−	9.38912i		−	0.524050i
$322$		0			0
$323$			1.06610i			0.0593192i
$324$		0			0
$325$	−1.61626			−0.0896540
$326$		0			0
$327$	1.83752			0.101615
$328$		0			0
$329$		−	1.02010i		−	0.0562400i
$330$		0			0
$331$		−	6.64361i		−	0.365166i	−0.983190	−	0.182583i	$-0.941554\pi$
$331$		−	6.64361i		−	0.365166i	0.983190	−	0.182583i	$-0.0584458\pi$
$332$		0			0
$333$			7.56990i			0.414828i
$334$		0			0
$335$		−	9.80785i		−	0.535860i
$336$		0			0
$337$			0.953089i			0.0519181i	0.999663	+	0.0259590i	$0.00826395\pi$
$337$			0.953089i			0.0519181i	−0.999663	+	0.0259590i	$0.991736\pi$
$338$		0			0
$339$	5.28710			0.287156
$340$		0			0
$341$		−	15.8340i		−	0.857459i
$342$		0			0
$343$	−18.8974			−1.02036
$344$		0			0
$345$	4.69289	−	0.988343i	0.252656	−	0.0532106i
$346$		0			0
$347$			12.9528i			0.695343i	0.937616	+	0.347672i	$0.113028\pi$
$347$			12.9528i			0.695343i	−0.937616	+	0.347672i	$0.886972\pi$
$348$		0			0
$349$	−15.4203			−0.825427			−0.412714	−	0.910861i	$-0.635419\pi$
$349$	−15.4203			−0.825427			−0.412714	+	0.910861i	$0.635419\pi$
$350$		0			0
$351$			1.61626i			0.0862696i
$352$		0			0
$353$	8.77844			0.467229			0.233615	−	0.972329i	$-0.424945\pi$
$353$	8.77844			0.467229			0.233615	+	0.972329i	$0.424945\pi$
$354$		0			0
$355$	−9.51609			−0.505061
$356$		0			0
$357$	−3.49645			−0.185052
$358$		0			0
$359$	32.4004			1.71003			0.855015	−	0.518604i	$-0.173548\pi$
$359$	32.4004			1.71003			0.855015	+	0.518604i	$0.173548\pi$
$360$		0			0
$361$	−18.7307			−0.985826
$362$		0			0
$363$		−	3.42851i		−	0.179950i
$364$		0			0
$365$			7.31068i			0.382659i
$366$		0			0
$367$	20.0531			1.04676			0.523381	−	0.852099i	$-0.324670\pi$
$367$	20.0531			1.04676			0.523381	+	0.852099i	$0.324670\pi$
$368$		0			0
$369$	−11.8923			−0.619090
$370$		0			0
$371$			5.05660i			0.262525i
$372$		0			0
$373$			6.56772i			0.340063i	0.985439	+	0.170032i	$0.0543870\pi$
$373$			6.56772i			0.340063i	−0.985439	+	0.170032i	$0.945613\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	−4.66459			−0.240239
$378$		0			0
$379$	−23.3165			−1.19769			−0.598843	−	0.800866i	$-0.704373\pi$
$379$	−23.3165			−1.19769			−0.598843	+	0.800866i	$0.704373\pi$
$380$		0			0
$381$	−17.3539			−0.889067
$382$		0			0
$383$	−28.5675			−1.45973			−0.729864	−	0.683592i	$-0.760417\pi$
$383$	−28.5675			−1.45973			−0.729864	+	0.683592i	$0.760417\pi$
$384$		0			0
$385$		−	6.46485i		−	0.329479i
$386$		0			0
$387$	4.43156			0.225269
$388$		0			0
$389$			11.2986i			0.572864i	0.958101	+	0.286432i	$0.0924692\pi$
$389$			11.2986i			0.572864i	−0.958101	+	0.286432i	$0.907531\pi$
$390$		0			0
$391$	2.03042	+	9.64094i	0.102683	+	0.487563i
$392$		0			0
$393$	−7.24393			−0.365408
$394$		0			0
$395$		−	7.33976i		−	0.369303i
$396$		0			0
$397$	−37.0073			−1.85734			−0.928672	−	0.370903i	$-0.879048\pi$
$397$	−37.0073			−1.85734			−0.928672	+	0.370903i	$0.879048\pi$
$398$		0			0
$399$			0.883212i			0.0442159i
$400$		0			0
$401$			8.19438i			0.409208i	0.978845	+	0.204604i	$0.0655906\pi$
$401$			8.19438i			0.409208i	−0.978845	+	0.204604i	$0.934409\pi$
$402$		0			0
$403$		−	6.73739i		−	0.335613i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$		−	28.7542i		−	1.42529i
$408$		0			0
$409$	−0.924307			−0.0457040			−0.0228520	−	0.999739i	$-0.507275\pi$
$409$	−0.924307			−0.0457040			−0.0228520	+	0.999739i	$0.507275\pi$
$410$		0			0
$411$	2.88747			0.142428
$412$		0			0
$413$		−	16.4589i		−	0.809890i
$414$		0			0
$415$		−	4.63473i		−	0.227510i
$416$		0			0
$417$	15.7354			0.770564
$418$		0			0
$419$	15.8431			0.773985			0.386993	−	0.922083i	$-0.373514\pi$
$419$	15.8431			0.773985			0.386993	+	0.922083i	$0.373514\pi$
$420$		0			0
$421$		−	32.5195i		−	1.58490i	−0.609936	−	0.792451i	$-0.708805\pi$
$421$		−	32.5195i		−	1.58490i	0.609936	−	0.792451i	$-0.291195\pi$
$422$		0			0
$423$			0.599371i			0.0291424i
$424$		0			0
$425$			2.05437i			0.0996517i
$426$		0			0
$427$			3.84966i			0.186298i
$428$		0			0
$429$		−	6.13935i		−	0.296410i
$430$		0			0
$431$	16.9821			0.817997			0.408998	−	0.912535i	$-0.365878\pi$
$431$	16.9821			0.817997			0.408998	+	0.912535i	$0.365878\pi$
$432$		0			0
$433$			7.08151i			0.340315i	0.985417	+	0.170158i	$0.0544277\pi$
$433$			7.08151i			0.340315i	−0.985417	+	0.170158i	$0.945572\pi$
$434$		0			0
$435$	2.88604			0.138375
$436$		0			0
$437$	2.43533	−	0.512891i	0.116497	−	0.0245349i
$438$		0			0
$439$			21.1603i			1.00993i	0.863141	+	0.504964i	$0.168494\pi$
$439$			21.1603i			1.00993i	−0.863141	+	0.504964i	$0.831506\pi$
$440$		0			0
$441$	4.10336			0.195398
$442$		0			0
$443$		−	18.1298i		−	0.861372i	−0.902502	−	0.430686i	$-0.858272\pi$
$443$		−	18.1298i		−	0.861372i	0.902502	−	0.430686i	$-0.141728\pi$
$444$		0			0
$445$	−16.0785			−0.762194
$446$		0			0
$447$	15.1946			0.718678
$448$		0			0
$449$	24.4026			1.15163			0.575815	−	0.817580i	$-0.304685\pi$
$449$	24.4026			1.15163			0.575815	+	0.817580i	$0.304685\pi$
$450$		0			0
$451$	45.1729			2.12711
$452$		0			0
$453$	−0.629232			−0.0295639
$454$		0			0
$455$		−	2.75080i		−	0.128960i
$456$		0			0
$457$			17.7769i			0.831570i	0.909463	+	0.415785i	$0.136493\pi$
$457$			17.7769i			0.831570i	−0.909463	+	0.415785i	$0.863507\pi$
$458$		0			0
$459$	2.05437			0.0958899
$460$		0			0
$461$	25.6259			1.19352			0.596759	−	0.802420i	$-0.296455\pi$
$461$	25.6259			1.19352			0.596759	+	0.802420i	$0.296455\pi$
$462$		0			0
$463$		−	6.31020i		−	0.293260i	−0.989191	−	0.146630i	$-0.953157\pi$
$463$		−	6.31020i		−	0.293260i	0.989191	−	0.146630i	$-0.0468426\pi$
$464$		0			0
$465$			4.16850i			0.193310i
$466$		0			0
$467$	−4.70115			−0.217543			−0.108772	−	0.994067i	$-0.534692\pi$
$467$	−4.70115			−0.217543			−0.108772	+	0.994067i	$0.534692\pi$
$468$		0			0
$469$	16.6925			0.770788
$470$		0			0
$471$	5.67705			0.261585
$472$		0			0
$473$	−16.8332			−0.773993
$474$		0			0
$475$	0.518940			0.0238106
$476$		0			0
$477$		−	2.97105i		−	0.136035i
$478$		0			0
$479$	7.56647			0.345721			0.172860	−	0.984946i	$-0.444699\pi$
$479$	7.56647			0.345721			0.172860	+	0.984946i	$0.444699\pi$
$480$		0			0
$481$		−	12.2349i		−	0.557865i
$482$		0			0
$483$	1.68211	+	7.98707i	0.0765388	+	0.363424i
$484$		0			0
$485$	14.4819			0.657589
$486$		0			0
$487$			7.14323i			0.323691i	0.986816	+	0.161845i	$0.0517446\pi$
$487$			7.14323i			0.323691i	−0.986816	+	0.161845i	$0.948255\pi$
$488$		0			0
$489$	−4.41526			−0.199665
$490$		0			0
$491$		−	7.13437i		−	0.321970i	−0.986957	−	0.160985i	$-0.948533\pi$
$491$		−	7.13437i		−	0.321970i	0.986957	−	0.160985i	$-0.0514670\pi$
$492$		0			0
$493$			5.92900i			0.267029i
$494$		0			0
$495$			3.79849i			0.170729i
$496$		0			0
$497$		−	16.1959i		−	0.726487i
$498$		0			0
$499$		−	9.56615i		−	0.428239i	−0.976807	−	0.214120i	$-0.931312\pi$
$499$		−	9.56615i		−	0.428239i	0.976807	−	0.214120i	$-0.0686883\pi$
$500$		0			0
$501$	−19.0290			−0.850154
$502$		0			0
$503$	−19.2171			−0.856849			−0.428424	−	0.903578i	$-0.640931\pi$
$503$	−19.2171			−0.856849			−0.428424	+	0.903578i	$0.640931\pi$
$504$		0			0
$505$		−	0.619325i		−	0.0275596i
$506$		0			0
$507$			10.3877i			0.461334i
$508$		0			0
$509$	−26.7890			−1.18740			−0.593701	−	0.804686i	$-0.702334\pi$
$509$	−26.7890			−1.18740			−0.593701	+	0.804686i	$0.702334\pi$
$510$		0			0
$511$	−12.4424			−0.550421
$512$		0			0
$513$		−	0.518940i		−	0.0229118i
$514$		0			0
$515$		−	3.97808i		−	0.175295i
$516$		0			0
$517$		−	2.27670i		−	0.100129i
$518$		0			0
$519$			4.16476i			0.182812i
$520$		0			0
$521$		−	6.43941i		−	0.282116i	−0.990001	−	0.141058i	$-0.954950\pi$
$521$		−	6.43941i		−	0.282116i	0.990001	−	0.141058i	$-0.0450503\pi$
$522$		0			0
$523$	33.4197			1.46134			0.730672	−	0.682729i	$-0.239207\pi$
$523$	33.4197			1.46134			0.730672	+	0.682729i	$0.239207\pi$
$524$		0			0
$525$			1.70195i			0.0742793i
$526$		0			0
$527$	−8.56366			−0.373039
$528$		0			0
$529$	21.0464	−	9.27636i	0.915059	−	0.403320i
$530$		0			0
$531$			9.67061i			0.419669i
$532$		0			0
$533$	19.2211			0.832559
$534$		0			0
$535$		−	9.38912i		−	0.405927i
$536$		0			0
$537$	−10.0952			−0.435642
$538$		0			0
$539$	−15.5865			−0.671360
$540$		0			0
$541$	−27.0647			−1.16360			−0.581800	−	0.813332i	$-0.697651\pi$
$541$	−27.0647			−1.16360			−0.581800	+	0.813332i	$0.697651\pi$
$542$		0			0
$543$	−9.38599			−0.402791
$544$		0			0
$545$	1.83752			0.0787108
$546$		0			0
$547$			18.1055i			0.774133i	0.922052	+	0.387067i	$0.126512\pi$
$547$			18.1055i			0.774133i	−0.922052	+	0.387067i	$0.873488\pi$
$548$		0			0
$549$		−	2.26190i		−	0.0965357i
$550$		0			0
$551$	1.49768			0.0638034
$552$		0			0
$553$	12.4919			0.531211
$554$		0			0
$555$			7.56990i			0.321325i
$556$		0			0
$557$			17.8186i			0.754999i	0.926010	+	0.377500i	$0.123216\pi$
$557$			17.8186i			0.754999i	−0.926010	+	0.377500i	$0.876784\pi$
$558$		0			0
$559$	−7.16256			−0.302944
$560$		0			0
$561$	−7.80351			−0.329464
$562$		0			0
$563$	−13.6780			−0.576459			−0.288229	−	0.957561i	$-0.593067\pi$
$563$	−13.6780			−0.576459			−0.288229	+	0.957561i	$0.593067\pi$
$564$		0			0
$565$	5.28710			0.222430
$566$		0			0
$567$	1.70195			0.0714753
$568$		0			0
$569$			40.3074i			1.68977i	0.534945	+	0.844887i	$0.320332\pi$
$569$			40.3074i			1.68977i	−0.534945	+	0.844887i	$0.679668\pi$
$570$		0			0
$571$	0.681491			0.0285195			0.0142598	−	0.999898i	$-0.495461\pi$
$571$	0.681491			0.0285195			0.0142598	+	0.999898i	$0.495461\pi$
$572$		0			0
$573$		−	14.6961i		−	0.613939i
$574$		0			0
$575$	4.69289	−	0.988343i	0.195707	−	0.0412167i
$576$		0			0
$577$	−17.5651			−0.731243			−0.365621	−	0.930764i	$-0.619144\pi$
$577$	−17.5651			−0.731243			−0.365621	+	0.930764i	$0.619144\pi$
$578$		0			0
$579$			4.30526i			0.178920i
$580$		0			0
$581$	7.88810			0.327253
$582$		0			0
$583$			11.2855i			0.467398i
$584$		0			0
$585$			1.61626i			0.0668242i
$586$		0			0
$587$			33.6825i			1.39022i	0.718901	+	0.695112i	$0.244645\pi$
$587$			33.6825i			1.39022i	−0.718901	+	0.695112i	$0.755355\pi$
$588$		0			0
$589$			2.16320i			0.0891332i
$590$		0			0
$591$			3.85449i			0.158552i
$592$		0			0
$593$	32.0500			1.31614			0.658068	−	0.752958i	$-0.271374\pi$
$593$	32.0500			1.31614			0.658068	+	0.752958i	$0.271374\pi$
$594$		0			0
$595$	−3.49645			−0.143340
$596$		0			0
$597$		−	2.53494i		−	0.103748i
$598$		0			0
$599$		−	35.9423i		−	1.46856i	−0.678846	−	0.734281i	$-0.737520\pi$
$599$		−	35.9423i		−	1.46856i	0.678846	−	0.734281i	$-0.262480\pi$
$600$		0			0
$601$	13.9387			0.568572			0.284286	−	0.958740i	$-0.408244\pi$
$601$	13.9387			0.568572			0.284286	+	0.958740i	$0.408244\pi$
$602$		0			0
$603$	−9.80785			−0.399407
$604$		0			0
$605$		−	3.42851i		−	0.139389i
$606$		0			0
$607$		−	30.7906i		−	1.24975i	−0.780723	−	0.624877i	$-0.785149\pi$
$607$		−	30.7906i		−	1.24975i	0.780723	−	0.624877i	$-0.214851\pi$
$608$		0			0
$609$			4.91190i			0.199040i
$610$		0			0
$611$		−	0.968740i		−	0.0391910i
$612$		0			0
$613$		−	8.74935i		−	0.353383i	−0.984266	−	0.176691i	$-0.943461\pi$
$613$		−	8.74935i		−	0.353383i	0.984266	−	0.176691i	$-0.0565395\pi$
$614$		0			0
$615$	−11.8923			−0.479545
$616$		0			0
$617$			36.0508i			1.45135i	0.688036	+	0.725676i	$0.258473\pi$
$617$			36.0508i			1.45135i	−0.688036	+	0.725676i	$0.741527\pi$
$618$		0			0
$619$	14.5946			0.586606			0.293303	−	0.956020i	$-0.405246\pi$
$619$	14.5946			0.586606			0.293303	+	0.956020i	$0.405246\pi$
$620$		0			0
$621$	−0.988343	−	4.69289i	−0.0396608	−	0.188319i
$622$		0			0
$623$		−	27.3649i		−	1.09635i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			1.97119i			0.0787217i
$628$		0			0
$629$	−15.5514			−0.620075
$630$		0			0
$631$	33.5062			1.33386			0.666931	−	0.745119i	$-0.267607\pi$
$631$	33.5062			1.33386			0.666931	+	0.745119i	$0.267607\pi$
$632$		0			0
$633$	−21.3867			−0.850044
$634$		0			0
$635$	−17.3539			−0.688668
$636$		0			0
$637$	−6.63209			−0.262773
$638$		0			0
$639$			9.51609i			0.376451i
$640$		0			0
$641$			1.14299i			0.0451454i	0.999745	+	0.0225727i	$0.00718572\pi$
$641$			1.14299i			0.0451454i	−0.999745	+	0.0225727i	$0.992814\pi$
$642$		0			0
$643$	−9.64698			−0.380440			−0.190220	−	0.981741i	$-0.560920\pi$
$643$	−9.64698			−0.380440			−0.190220	+	0.981741i	$0.560920\pi$
$644$		0			0
$645$	4.43156			0.174493
$646$		0			0
$647$			11.2536i			0.442424i	0.975226	+	0.221212i	$0.0710013\pi$
$647$			11.2536i			0.442424i	−0.975226	+	0.221212i	$0.928999\pi$
$648$		0			0
$649$		−	36.7337i		−	1.44192i
$650$		0			0
$651$	−7.09459			−0.278059
$652$		0			0
$653$	−18.3899			−0.719653			−0.359826	−	0.933019i	$-0.617164\pi$
$653$	−18.3899			−0.719653			−0.359826	+	0.933019i	$0.617164\pi$
$654$		0			0
$655$	−7.24393			−0.283044
$656$		0			0
$657$	7.31068			0.285217
$658$		0			0
$659$	32.3893			1.26171			0.630853	−	0.775902i	$-0.282705\pi$
$659$	32.3893			1.26171			0.630853	+	0.775902i	$0.282705\pi$
$660$		0			0
$661$			13.3014i			0.517363i	0.965963	+	0.258682i	$0.0832881\pi$
$661$			13.3014i			0.517363i	−0.965963	+	0.258682i	$0.916712\pi$
$662$		0			0
$663$	−3.32040			−0.128954
$664$		0			0
$665$			0.883212i			0.0342495i
$666$		0			0
$667$	13.5439	−	2.85240i	0.524420	−	0.110445i
$668$		0			0
$669$	−22.4725			−0.868837
$670$		0			0
$671$			8.59182i			0.331683i
$672$		0			0
$673$	−6.92259			−0.266846			−0.133423	−	0.991059i	$-0.542597\pi$
$673$	−6.92259			−0.266846			−0.133423	+	0.991059i	$0.542597\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$			14.2078i			0.546049i	0.962007	+	0.273025i	$0.0880241\pi$
$677$			14.2078i			0.546049i	−0.962007	+	0.273025i	$0.911976\pi$
$678$		0			0
$679$			24.6475i			0.945885i
$680$		0			0
$681$		−	0.856466i		−	0.0328198i
$682$		0			0
$683$		−	27.7980i		−	1.06366i	−0.846851	−	0.531830i	$-0.821505\pi$
$683$		−	27.7980i		−	1.06366i	0.846851	−	0.531830i	$-0.178495\pi$
$684$		0			0
$685$	2.88747			0.110325
$686$		0			0
$687$	−6.28764			−0.239888
$688$		0			0
$689$			4.80200i			0.182942i
$690$		0			0
$691$			16.3975i			0.623791i	0.950116	+	0.311895i	$0.100964\pi$
$691$			16.3975i			0.623791i	−0.950116	+	0.311895i	$0.899036\pi$
$692$		0			0
$693$	−6.46485			−0.245579
$694$		0			0
$695$	15.7354			0.596876
$696$		0			0
$697$		−	24.4313i		−	0.925401i
$698$		0			0
$699$		−	25.5189i		−	0.965214i
$700$		0			0
$701$		−	21.5725i		−	0.814780i	−0.913254	−	0.407390i	$-0.866439\pi$
$701$		−	21.5725i		−	0.814780i	0.913254	−	0.407390i	$-0.133561\pi$
$702$		0			0
$703$			3.92833i			0.148160i
$704$		0			0
$705$			0.599371i			0.0225736i
$706$		0			0
$707$	1.05406			0.0396421
$708$		0			0
$709$		−	20.0927i		−	0.754597i	−0.926092	−	0.377299i	$-0.876853\pi$
$709$		−	20.0927i		−	0.754597i	0.926092	−	0.377299i	$-0.123147\pi$
$710$		0			0
$711$	−7.33976			−0.275262
$712$		0			0
$713$	4.11991	+	19.5623i	0.154292	+	0.732614i
$714$		0			0
$715$		−	6.13935i		−	0.229599i
$716$		0			0
$717$	−17.9194			−0.669210
$718$		0			0
$719$		−	14.2012i		−	0.529616i	−0.964301	−	0.264808i	$-0.914691\pi$
$719$		−	14.2012i		−	0.529616i	0.964301	−	0.264808i	$-0.0853085\pi$
$720$		0			0
$721$	6.77051			0.252147
$722$		0			0
$723$	7.79751			0.289993
$724$		0			0
$725$	2.88604			0.107185
$726$		0			0
$727$	−17.1754			−0.637002			−0.318501	−	0.947923i	$-0.603179\pi$
$727$	−17.1754			−0.637002			−0.318501	+	0.947923i	$0.603179\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			9.10408i			0.336726i
$732$		0			0
$733$			31.9438i			1.17987i	0.807450	+	0.589936i	$0.200847\pi$
$733$			31.9438i			1.17987i	−0.807450	+	0.589936i	$0.799153\pi$
$734$		0			0
$735$	4.10336			0.151355
$736$		0			0
$737$	37.2550			1.37231
$738$		0			0
$739$			29.5581i			1.08731i	0.839308	+	0.543656i	$0.182960\pi$
$739$			29.5581i			1.08731i	−0.839308	+	0.543656i	$0.817040\pi$
$740$		0			0
$741$			0.838743i			0.0308120i
$742$		0			0
$743$	9.15177			0.335746			0.167873	−	0.985809i	$-0.446310\pi$
$743$	9.15177			0.335746			0.167873	+	0.985809i	$0.446310\pi$
$744$		0			0
$745$	15.1946			0.556686
$746$		0			0
$747$	−4.63473			−0.169576
$748$		0			0
$749$	15.9798			0.583891
$750$		0			0
$751$	−0.853249			−0.0311355			−0.0155678	−	0.999879i	$-0.504956\pi$
$751$	−0.853249			−0.0311355			−0.0155678	+	0.999879i	$0.504956\pi$
$752$		0			0
$753$			13.1223i			0.478204i
$754$		0			0
$755$	−0.629232			−0.0229001
$756$		0			0
$757$		−	44.7472i		−	1.62636i	−0.582010	−	0.813182i	$-0.697733\pi$
$757$		−	44.7472i		−	1.62636i	0.582010	−	0.813182i	$-0.302267\pi$
$758$		0			0
$759$	3.75421	+	17.8259i	0.136269	+	0.647038i
$760$		0			0
$761$	40.5354			1.46941			0.734704	−	0.678388i	$-0.237321\pi$
$761$	40.5354			1.46941			0.734704	+	0.678388i	$0.237321\pi$
$762$		0			0
$763$			3.12738i			0.113219i
$764$		0			0
$765$	2.05437			0.0742760
$766$		0			0
$767$		−	15.6302i		−	0.564375i
$768$		0			0
$769$			15.2957i			0.551579i	0.961218	+	0.275789i	$0.0889392\pi$
$769$			15.2957i			0.551579i	−0.961218	+	0.275789i	$0.911061\pi$
$770$		0			0
$771$		−	23.3047i		−	0.839297i
$772$		0			0
$773$			41.3703i			1.48799i	0.668187	+	0.743994i	$0.267071\pi$
$773$			41.3703i			1.48799i	−0.668187	+	0.743994i	$0.732929\pi$
$774$		0			0
$775$			4.16850i			0.149737i
$776$		0			0
$777$	−12.8836			−0.462197
$778$		0			0
$779$	−6.17141			−0.221114
$780$		0			0
$781$		−	36.1467i		−	1.29343i
$782$		0			0
$783$		−	2.88604i		−	0.103139i
$784$		0			0
$785$	5.67705			0.202623
$786$		0			0
$787$	11.4285			0.407383			0.203692	−	0.979035i	$-0.434706\pi$
$787$	11.4285			0.407383			0.203692	+	0.979035i	$0.434706\pi$
$788$		0			0
$789$			10.9363i			0.389343i
$790$		0			0
$791$			8.99839i			0.319946i
$792$		0			0
$793$			3.65583i			0.129822i
$794$		0			0
$795$		−	2.97105i		−	0.105372i
$796$		0			0
$797$		−	33.2097i		−	1.17635i	−0.808734	−	0.588175i	$-0.799847\pi$
$797$		−	33.2097i		−	1.17635i	0.808734	−	0.588175i	$-0.200153\pi$
$798$		0			0
$799$	−1.23133			−0.0435614
$800$		0			0
$801$			16.0785i			0.568106i
$802$		0			0
$803$	−27.7695			−0.979966
$804$		0			0
$805$	1.68211	+	7.98707i	0.0592867	+	0.281507i
$806$		0			0
$807$			24.0819i			0.847724i
$808$		0			0
$809$	55.0270			1.93465			0.967323	−	0.253546i	$-0.0815970\pi$
$809$	55.0270			1.93465			0.967323	+	0.253546i	$0.0815970\pi$
$810$		0			0
$811$			15.0471i			0.528377i	0.964471	+	0.264188i	$0.0851041\pi$
$811$			15.0471i			0.528377i	−0.964471	+	0.264188i	$0.914896\pi$
$812$		0			0
$813$	25.0959			0.880153
$814$		0			0
$815$	−4.41526			−0.154660
$816$		0			0
$817$	2.29972			0.0804568
$818$		0			0
$819$	−2.75080			−0.0961207
$820$		0			0
$821$	12.2209			0.426511			0.213255	−	0.976996i	$-0.431593\pi$
$821$	12.2209			0.426511			0.213255	+	0.976996i	$0.431593\pi$
$822$		0			0
$823$			22.4440i			0.782349i	0.920317	+	0.391174i	$0.127931\pi$
$823$			22.4440i			0.782349i	−0.920317	+	0.391174i	$0.872069\pi$
$824$		0			0
$825$			3.79849i			0.132246i
$826$		0			0
$827$	7.74646			0.269371			0.134685	−	0.990888i	$-0.456998\pi$
$827$	7.74646			0.269371			0.134685	+	0.990888i	$0.456998\pi$
$828$		0			0
$829$	0.186882			0.00649067			0.00324534	−	0.999995i	$-0.498967\pi$
$829$	0.186882			0.00649067			0.00324534	+	0.999995i	$0.498967\pi$
$830$		0			0
$831$			19.8548i			0.688756i
$832$		0			0
$833$			8.42982i			0.292076i
$834$		0			0
$835$	−19.0290			−0.658527
$836$		0			0
$837$	4.16850			0.144084
$838$		0			0
$839$	6.72736			0.232254			0.116127	−	0.993234i	$-0.462952\pi$
$839$	6.72736			0.232254			0.116127	+	0.993234i	$0.462952\pi$
$840$		0			0
$841$	−20.6708			−0.712785
$842$		0			0
$843$	−31.3138			−1.07850
$844$		0			0
$845$			10.3877i			0.357348i
$846$		0			0
$847$	5.83517			0.200499
$848$		0			0
$849$			20.8530i			0.715672i
$850$		0			0
$851$	7.48166	+	35.5247i	0.256468	+	1.21777i
$852$		0			0
$853$	−28.2109			−0.965924			−0.482962	−	0.875641i	$-0.660439\pi$
$853$	−28.2109			−0.965924			−0.482962	+	0.875641i	$0.660439\pi$
$854$		0			0
$855$		−	0.518940i		−	0.0177474i
$856$		0			0
$857$	31.9591			1.09170			0.545851	−	0.837882i	$-0.316206\pi$
$857$	31.9591			1.09170			0.545851	+	0.837882i	$0.316206\pi$
$858$		0			0
$859$			24.3487i			0.830767i	0.909646	+	0.415384i	$0.136353\pi$
$859$			24.3487i			0.830767i	−0.909646	+	0.415384i	$0.863647\pi$
$860$		0			0
$861$		−	20.2402i		−	0.689784i
$862$		0			0
$863$			2.27641i			0.0774899i	0.999249	+	0.0387449i	$0.0123360\pi$
$863$			2.27641i			0.0774899i	−0.999249	+	0.0387449i	$0.987664\pi$
$864$		0			0
$865$			4.16476i			0.141606i
$866$		0			0
$867$		−	12.7796i		−	0.434016i
$868$		0			0
$869$	27.8800			0.945764
$870$		0			0
$871$	15.8521			0.537126
$872$		0			0
$873$		−	14.4819i		−	0.490138i
$874$		0			0
$875$			1.70195i			0.0575365i
$876$		0			0
$877$	−3.39485			−0.114636			−0.0573179	−	0.998356i	$-0.518255\pi$
$877$	−3.39485			−0.114636			−0.0573179	+	0.998356i	$0.518255\pi$
$878$		0			0
$879$	16.4218			0.553894
$880$		0			0
$881$		−	32.6829i		−	1.10112i	−0.834797	−	0.550558i	$-0.814415\pi$
$881$		−	32.6829i		−	1.10112i	0.834797	−	0.550558i	$-0.185585\pi$
$882$		0			0
$883$			44.7629i			1.50639i	0.657797	+	0.753196i	$0.271489\pi$
$883$			44.7629i			1.50639i	−0.657797	+	0.753196i	$0.728511\pi$
$884$		0			0
$885$			9.67061i			0.325074i
$886$		0			0
$887$			5.00975i			0.168211i	0.996457	+	0.0841054i	$0.0268032\pi$
$887$			5.00975i			0.168211i	−0.996457	+	0.0841054i	$0.973197\pi$
$888$		0			0
$889$		−	29.5355i		−	0.990589i
$890$		0			0
$891$	3.79849			0.127254
$892$		0			0
$893$			0.311038i			0.0104085i
$894$		0			0
$895$	−10.0952			−0.337447
$896$		0			0
$897$	1.59742	+	7.58493i	0.0533363	+	0.253253i
$898$		0			0
$899$			12.0305i			0.401238i
$900$		0			0
$901$	6.10365			0.203342
$902$		0			0
$903$			7.54231i			0.250992i
$904$		0			0
$905$	−9.38599			−0.312001
$906$		0			0
$907$	37.4914			1.24488			0.622440	−	0.782668i	$-0.286141\pi$
$907$	37.4914			1.24488			0.622440	+	0.782668i	$0.286141\pi$
$908$		0			0
$909$	−0.619325			−0.0205417
$910$		0			0
$911$	−38.3505			−1.27061			−0.635305	−	0.772261i	$-0.719126\pi$
$911$	−38.3505			−1.27061			−0.635305	+	0.772261i	$0.719126\pi$
$912$		0			0
$913$	17.6050			0.582640
$914$		0			0
$915$		−	2.26190i		−	0.0747763i
$916$		0			0
$917$		−	12.3288i		−	0.407134i
$918$		0			0
$919$	−31.0288			−1.02354			−0.511772	−	0.859121i	$-0.671011\pi$
$919$	−31.0288			−1.02354			−0.511772	+	0.859121i	$0.671011\pi$
$920$		0			0
$921$	9.40135			0.309785
$922$		0			0
$923$		−	15.3805i		−	0.506255i
$924$		0			0
$925$			7.56990i			0.248897i
$926$		0			0
$927$	−3.97808			−0.130657
$928$		0			0
$929$	45.2173			1.48353			0.741766	−	0.670658i	$-0.233988\pi$
$929$	45.2173			1.48353			0.741766	+	0.670658i	$0.233988\pi$
$930$		0			0
$931$	2.12940			0.0697881
$932$		0			0
$933$	−10.4152			−0.340979
$934$		0			0
$935$	−7.80351			−0.255202
$936$		0			0
$937$			24.8260i			0.811031i	0.914088	+	0.405515i	$0.132908\pi$
$937$			24.8260i			0.811031i	−0.914088	+	0.405515i	$0.867092\pi$
$938$		0			0
$939$	17.4733			0.570219
$940$		0			0
$941$			17.5941i			0.573553i	0.957998	+	0.286776i	$0.0925836\pi$
$941$			17.5941i			0.573553i	−0.957998	+	0.286776i	$0.907416\pi$
$942$		0			0
$943$	−55.8093	+	11.7537i	−1.81740	+	0.382753i
$944$		0			0
$945$	1.70195			0.0553645
$946$		0			0
$947$		−	11.6563i		−	0.378779i	−0.981902	−	0.189389i	$-0.939349\pi$
$947$		−	11.6563i		−	0.378779i	0.981902	−	0.189389i	$-0.0606508\pi$
$948$		0			0
$949$	−11.8160			−0.383563
$950$		0			0
$951$			0.464007i			0.0150465i
$952$		0			0
$953$		−	13.9507i		−	0.451907i	−0.974138	−	0.225953i	$-0.927450\pi$
$953$		−	13.9507i		−	0.451907i	0.974138	−	0.225953i	$-0.0725497\pi$
$954$		0			0
$955$		−	14.6961i		−	0.475555i
$956$		0			0
$957$			10.9626i			0.354370i
$958$		0			0
$959$			4.91434i			0.158692i
$960$		0			0
$961$	13.6236			0.439471
$962$		0			0
$963$	−9.38912			−0.302560
$964$		0			0
$965$			4.30526i			0.138591i
$966$		0			0
$967$			6.45709i			0.207646i	0.994596	+	0.103823i	$0.0331076\pi$
$967$			6.45709i			0.207646i	−0.994596	+	0.103823i	$0.966892\pi$
$968$		0			0
$969$	1.06610			0.0342480
$970$		0			0
$971$	53.5450			1.71834			0.859171	−	0.511689i	$-0.170980\pi$
$971$	53.5450			1.71834			0.859171	+	0.511689i	$0.170980\pi$
$972$		0			0
$973$			26.7808i			0.858554i
$974$		0			0
$975$			1.61626i			0.0517618i
$976$		0			0
$977$			51.3610i			1.64318i	0.570076	+	0.821592i	$0.306914\pi$
$977$			51.3610i			1.64318i	−0.570076	+	0.821592i	$0.693086\pi$
$978$		0			0
$979$		−	61.0740i		−	1.95193i
$980$		0			0
$981$		−	1.83752i		−	0.0586676i
$982$		0			0
$983$	−40.0176			−1.27636			−0.638181	−	0.769886i	$-0.720313\pi$
$983$	−40.0176			−1.27636			−0.638181	+	0.769886i	$0.720313\pi$
$984$		0			0
$985$			3.85449i			0.122814i
$986$		0			0
$987$	−1.02010			−0.0324702
$988$		0			0
$989$	20.7968	−	4.37990i	0.661300	−	0.139273i
$990$		0			0
$991$			15.3110i			0.486371i	0.969980	+	0.243186i	$0.0781924\pi$
$991$			15.3110i			0.486371i	−0.969980	+	0.243186i	$0.921808\pi$
$992$		0			0
$993$	−6.64361			−0.210829
$994$		0			0
$995$		−	2.53494i		−	0.0803631i
$996$		0			0
$997$	28.2781			0.895576			0.447788	−	0.894140i	$-0.352212\pi$
$997$	28.2781			0.895576			0.447788	+	0.894140i	$0.352212\pi$
$998$		0			0
$999$	7.56990			0.239501

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.c.1471.23	✓	32
4.3	odd	2		5520.2.be.d.1471.24	yes	32
23.22	odd	2		5520.2.be.d.1471.23	yes	32
92.91	even	2	inner	5520.2.be.c.1471.24	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.c.1471.23	✓	32	1.1	even	1	trivial
5520.2.be.c.1471.24	yes	32	92.91	even	2	inner
5520.2.be.d.1471.23	yes	32	23.22	odd	2
5520.2.be.d.1471.24	yes	32	4.3	odd	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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.be (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} +1.70195 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{5} +1.70195 q^{7} -1.00000 q^{9} +3.79849 q^{11} +1.61626 q^{13} -1.00000 q^{15} -2.05437i q^{17} -0.518940 q^{19} -1.70195i q^{21} +(-4.69289 + 0.988343i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -2.88604 q^{29} -4.16850i q^{31} -3.79849i q^{33} -1.70195i q^{35} -7.56990i q^{37} -1.61626i q^{39} +11.8923 q^{41} -4.43156 q^{43} +1.00000i q^{45} -0.599371i q^{47} -4.10336 q^{49} -2.05437 q^{51} +2.97105i q^{53} -3.79849i q^{55} +0.518940i q^{57} -9.67061i q^{59} +2.26190i q^{61} -1.70195 q^{63} -1.61626i q^{65} +9.80785 q^{67} +(0.988343 + 4.69289i) q^{69} -9.51609i q^{71} -7.31068 q^{73} +1.00000i q^{75} +6.46485 q^{77} +7.33976 q^{79} +1.00000 q^{81} +4.63473 q^{83} -2.05437 q^{85} +2.88604i q^{87} -16.0785i q^{89} +2.75080 q^{91} -4.16850 q^{93} +0.518940i q^{95} +14.4819i q^{97} -3.79849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) \) 32 * q - 8 * q^7 - 32 * q^9	\( 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) \) 32 * q - 8 * q^7 - 32 * q^9 + 8 * q^11 - 8 * q^13 - 32 * q^15 - 32 * q^25 + 4 * q^29 + 20 * q^41 + 52 * q^49 - 4 * q^51 + 8 * q^63 + 32 * q^67 - 40 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^81 - 4 * q^85 - 48 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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