LMFDB - Embedded newform 5520.2.be.c.1471.2

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			1471.2
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.c.1471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} +3.34851 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} +3.34851 q^{7} -1.00000 q^{9} -3.80844 q^{11} -4.01046 q^{13} -1.00000 q^{15} +3.43500i q^{17} -2.65201 q^{19} +3.34851i q^{21} +(-4.71918 + 0.853990i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +2.73385 q^{29} +5.29319i q^{31} -3.80844i q^{33} +3.34851i q^{35} -1.26784i q^{37} -4.01046i q^{39} -0.928703 q^{41} +6.49805 q^{43} -1.00000i q^{45} -1.22656i q^{47} +4.21251 q^{49} -3.43500 q^{51} -10.0810i q^{53} -3.80844i q^{55} -2.65201i q^{57} -15.1536i q^{59} -5.10695i q^{61} -3.34851 q^{63} -4.01046i q^{65} +3.01447 q^{67} +(-0.853990 - 4.71918i) q^{69} -4.10114i q^{71} -12.9930 q^{73} -1.00000i q^{75} -12.7526 q^{77} +11.9924 q^{79} +1.00000 q^{81} -12.4426 q^{83} -3.43500 q^{85} +2.73385i q^{87} +0.329105i q^{89} -13.4291 q^{91} -5.29319 q^{93} -2.65201i q^{95} -4.88722i q^{97} +3.80844 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	3.34851			1.26562			0.632809	−	0.774308i	$-0.281902\pi$
$7$	3.34851			1.26562			0.632809	+	0.774308i	$0.281902\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−3.80844			−1.14829			−0.574143	−	0.818755i	$-0.694665\pi$
$11$	−3.80844			−1.14829			−0.574143	+	0.818755i	$0.694665\pi$
$12$		0			0
$13$	−4.01046			−1.11230			−0.556151	−	0.831081i	$-0.687722\pi$
$13$	−4.01046			−1.11230			−0.556151	+	0.831081i	$0.687722\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$			3.43500i			0.833109i	0.909111	+	0.416554i	$0.136762\pi$
$17$			3.43500i			0.833109i	−0.909111	+	0.416554i	$0.863238\pi$
$18$		0			0
$19$	−2.65201			−0.608413			−0.304207	−	0.952606i	$-0.598391\pi$
$19$	−2.65201			−0.608413			−0.304207	+	0.952606i	$0.598391\pi$
$20$		0			0
$21$			3.34851i			0.730705i
$22$		0			0
$23$	−4.71918	+	0.853990i	−0.984018	+	0.178069i
$23$	−4.71918	+	0.853990i	−0.984018	+	0.178069i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	2.73385			0.507664			0.253832	−	0.967248i	$-0.418309\pi$
$29$	2.73385			0.507664			0.253832	+	0.967248i	$0.418309\pi$
$30$		0			0
$31$			5.29319i			0.950684i	0.879801	+	0.475342i	$0.157676\pi$
$31$			5.29319i			0.950684i	−0.879801	+	0.475342i	$0.842324\pi$
$32$		0			0
$33$		−	3.80844i		−	0.662964i
$34$		0			0
$35$			3.34851i			0.566001i
$36$		0			0
$37$		−	1.26784i		−	0.208431i	−0.994555	−	0.104216i	$-0.966767\pi$
$37$		−	1.26784i		−	0.208431i	0.994555	−	0.104216i	$-0.0332332\pi$
$38$		0			0
$39$		−	4.01046i		−	0.642188i
$40$		0			0
$41$	−0.928703			−0.145039			−0.0725195	−	0.997367i	$-0.523104\pi$
$41$	−0.928703			−0.145039			−0.0725195	+	0.997367i	$0.523104\pi$
$42$		0			0
$43$	6.49805			0.990944			0.495472	−	0.868624i	$-0.334995\pi$
$43$	6.49805			0.990944			0.495472	+	0.868624i	$0.334995\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$		−	1.22656i		−	0.178912i	−0.995991	−	0.0894562i	$-0.971487\pi$
$47$		−	1.22656i		−	0.178912i	0.995991	−	0.0894562i	$-0.0285129\pi$
$48$		0			0
$49$	4.21251			0.601788
$50$		0			0
$51$	−3.43500			−0.480996
$52$		0			0
$53$		−	10.0810i		−	1.38473i	−0.721548	−	0.692365i	$-0.756569\pi$
$53$		−	10.0810i		−	1.38473i	0.721548	−	0.692365i	$-0.243431\pi$
$54$		0			0
$55$		−	3.80844i		−	0.513530i
$56$		0			0
$57$		−	2.65201i		−	0.351267i
$58$		0			0
$59$		−	15.1536i		−	1.97283i	−0.164279	−	0.986414i	$-0.552530\pi$
$59$		−	15.1536i		−	1.97283i	0.164279	−	0.986414i	$-0.447470\pi$
$60$		0			0
$61$		−	5.10695i		−	0.653878i	−0.945045	−	0.326939i	$-0.893983\pi$
$61$		−	5.10695i		−	0.653878i	0.945045	−	0.326939i	$-0.106017\pi$
$62$		0			0
$63$	−3.34851			−0.421873
$64$		0			0
$65$		−	4.01046i		−	0.497437i
$66$		0			0
$67$	3.01447			0.368276			0.184138	−	0.982900i	$-0.441051\pi$
$67$	3.01447			0.368276			0.184138	+	0.982900i	$0.441051\pi$
$68$		0			0
$69$	−0.853990	−	4.71918i	−0.102808	−	0.568123i
$70$		0			0
$71$		−	4.10114i		−	0.486716i	−0.969937	−	0.243358i	$-0.921751\pi$
$71$		−	4.10114i		−	0.486716i	0.969937	−	0.243358i	$-0.0782490\pi$
$72$		0			0
$73$	−12.9930			−1.52072			−0.760360	−	0.649502i	$-0.774977\pi$
$73$	−12.9930			−1.52072			−0.760360	+	0.649502i	$0.774977\pi$
$74$		0			0
$75$		−	1.00000i		−	0.115470i
$76$		0			0
$77$	−12.7526			−1.45329
$78$		0			0
$79$	11.9924			1.34925			0.674624	−	0.738161i	$-0.264306\pi$
$79$	11.9924			1.34925			0.674624	+	0.738161i	$0.264306\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−12.4426			−1.36576			−0.682878	−	0.730532i	$-0.739272\pi$
$83$	−12.4426			−1.36576			−0.682878	+	0.730532i	$0.739272\pi$
$84$		0			0
$85$	−3.43500			−0.372578
$86$		0			0
$87$			2.73385i			0.293100i
$88$		0			0
$89$			0.329105i			0.0348850i	0.999848	+	0.0174425i	$0.00555241\pi$
$89$			0.329105i			0.0348850i	−0.999848	+	0.0174425i	$0.994448\pi$
$90$		0			0
$91$	−13.4291			−1.40775
$92$		0			0
$93$	−5.29319			−0.548878
$94$		0			0
$95$		−	2.65201i		−	0.272091i
$96$		0			0
$97$		−	4.88722i		−	0.496222i	−0.968732	−	0.248111i	$-0.920190\pi$
$97$		−	4.88722i		−	0.496222i	0.968732	−	0.248111i	$-0.0798098\pi$
$98$		0			0
$99$	3.80844			0.382762
$100$		0			0
$101$	−10.6619			−1.06089			−0.530447	−	0.847718i	$-0.677976\pi$
$101$	−10.6619			−1.06089			−0.530447	+	0.847718i	$0.677976\pi$
$102$		0			0
$103$	17.5524			1.72949			0.864747	−	0.502208i	$-0.167479\pi$
$103$	17.5524			1.72949			0.864747	+	0.502208i	$0.167479\pi$
$104$		0			0
$105$	−3.34851			−0.326781
$106$		0			0
$107$	−9.06743			−0.876581			−0.438291	−	0.898833i	$-0.644416\pi$
$107$	−9.06743			−0.876581			−0.438291	+	0.898833i	$0.644416\pi$
$108$		0			0
$109$			1.91151i			0.183090i	0.995801	+	0.0915449i	$0.0291805\pi$
$109$			1.91151i			0.183090i	−0.995801	+	0.0915449i	$0.970819\pi$
$110$		0			0
$111$	1.26784			0.120338
$112$		0			0
$113$			9.47968i			0.891774i	0.895089	+	0.445887i	$0.147112\pi$
$113$			9.47968i			0.891774i	−0.895089	+	0.445887i	$0.852888\pi$
$114$		0			0
$115$	−0.853990	−	4.71918i	−0.0796350	−	0.440066i
$116$		0			0
$117$	4.01046			0.370767
$118$		0			0
$119$			11.5021i			1.05440i
$120$		0			0
$121$	3.50419			0.318563
$122$		0			0
$123$		−	0.928703i		−	0.0837384i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	14.5043i		−	1.28704i	−0.765427	−	0.643522i	$-0.777472\pi$
$127$		−	14.5043i		−	1.28704i	0.765427	−	0.643522i	$-0.222528\pi$
$128$		0			0
$129$			6.49805i			0.572122i
$130$		0			0
$131$		−	16.8636i		−	1.47338i	−0.676231	−	0.736690i	$-0.736388\pi$
$131$		−	16.8636i		−	1.47338i	0.676231	−	0.736690i	$-0.263612\pi$
$132$		0			0
$133$	−8.88028			−0.770018
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$			5.29294i			0.452206i	0.974103	+	0.226103i	$0.0725987\pi$
$137$			5.29294i			0.452206i	−0.974103	+	0.226103i	$0.927401\pi$
$138$		0			0
$139$		−	19.7415i		−	1.67445i	−0.546858	−	0.837225i	$-0.684176\pi$
$139$		−	19.7415i		−	1.67445i	0.546858	−	0.837225i	$-0.315824\pi$
$140$		0			0
$141$	1.22656			0.103295
$142$		0			0
$143$	15.2736			1.27724
$144$		0			0
$145$			2.73385i			0.227034i
$146$		0			0
$147$			4.21251i			0.347442i
$148$		0			0
$149$			11.2478i			0.921454i	0.887542	+	0.460727i	$0.152411\pi$
$149$			11.2478i			0.921454i	−0.887542	+	0.460727i	$0.847589\pi$
$150$		0			0
$151$		−	11.2361i		−	0.914382i	−0.889369	−	0.457191i	$-0.848856\pi$
$151$		−	11.2361i		−	0.914382i	0.889369	−	0.457191i	$-0.151144\pi$
$152$		0			0
$153$		−	3.43500i		−	0.277703i
$154$		0			0
$155$	−5.29319			−0.425159
$156$		0			0
$157$			2.61969i			0.209074i	0.994521	+	0.104537i	$0.0333361\pi$
$157$			2.61969i			0.209074i	−0.994521	+	0.104537i	$0.966664\pi$
$158$		0			0
$159$	10.0810			0.799474
$160$		0			0
$161$	−15.8022	+	2.85959i	−1.24539	+	0.225367i
$162$		0			0
$163$			16.4052i			1.28495i	0.766305	+	0.642477i	$0.222093\pi$
$163$			16.4052i			1.28495i	−0.766305	+	0.642477i	$0.777907\pi$
$164$		0			0
$165$	3.80844			0.296486
$166$		0			0
$167$			6.17814i			0.478079i	0.971010	+	0.239039i	$0.0768325\pi$
$167$			6.17814i			0.478079i	−0.971010	+	0.239039i	$0.923167\pi$
$168$		0			0
$169$	3.08380			0.237216
$170$		0			0
$171$	2.65201			0.202804
$172$		0			0
$173$	−24.5549			−1.86687			−0.933436	−	0.358744i	$-0.883205\pi$
$173$	−24.5549			−1.86687			−0.933436	+	0.358744i	$0.883205\pi$
$174$		0			0
$175$	−3.34851			−0.253124
$176$		0			0
$177$	15.1536			1.13901
$178$		0			0
$179$		−	11.9824i		−	0.895606i	−0.894132	−	0.447803i	$-0.852207\pi$
$179$		−	11.9824i		−	0.895606i	0.894132	−	0.447803i	$-0.147793\pi$
$180$		0			0
$181$			13.9533i			1.03714i	0.855036	+	0.518569i	$0.173535\pi$
$181$			13.9533i			1.03714i	−0.855036	+	0.518569i	$0.826465\pi$
$182$		0			0
$183$	5.10695			0.377517
$184$		0			0
$185$	1.26784			0.0932133
$186$		0			0
$187$		−	13.0820i		−	0.956648i
$188$		0			0
$189$		−	3.34851i		−	0.243568i
$190$		0			0
$191$	−17.2707			−1.24966			−0.624832	−	0.780759i	$-0.714833\pi$
$191$	−17.2707			−1.24966			−0.624832	+	0.780759i	$0.714833\pi$
$192$		0			0
$193$	3.94520			0.283982			0.141991	−	0.989868i	$-0.454650\pi$
$193$	3.94520			0.283982			0.141991	+	0.989868i	$0.454650\pi$
$194$		0			0
$195$	4.01046			0.287195
$196$		0			0
$197$	−5.70577			−0.406519			−0.203260	−	0.979125i	$-0.565154\pi$
$197$	−5.70577			−0.406519			−0.203260	+	0.979125i	$0.565154\pi$
$198$		0			0
$199$	−3.55709			−0.252156			−0.126078	−	0.992020i	$-0.540239\pi$
$199$	−3.55709			−0.252156			−0.126078	+	0.992020i	$0.540239\pi$
$200$		0			0
$201$			3.01447i			0.212624i
$202$		0			0
$203$	9.15433			0.642508
$204$		0			0
$205$		−	0.928703i		−	0.0648634i
$206$		0			0
$207$	4.71918	−	0.853990i	0.328006	−	0.0593564i
$208$		0			0
$209$	10.1000			0.698633
$210$		0			0
$211$			12.3538i			0.850468i	0.905083	+	0.425234i	$0.139808\pi$
$211$			12.3538i			0.850468i	−0.905083	+	0.425234i	$0.860192\pi$
$212$		0			0
$213$	4.10114			0.281006
$214$		0			0
$215$			6.49805i			0.443164i
$216$		0			0
$217$			17.7243i			1.20320i
$218$		0			0
$219$		−	12.9930i		−	0.877988i
$220$		0			0
$221$		−	13.7759i		−	0.926669i
$222$		0			0
$223$			3.60984i			0.241732i	0.992669	+	0.120866i	$0.0385672\pi$
$223$			3.60984i			0.241732i	−0.992669	+	0.120866i	$0.961433\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	10.4598			0.694241			0.347120	−	0.937821i	$-0.387160\pi$
$227$	10.4598			0.694241			0.347120	+	0.937821i	$0.387160\pi$
$228$		0			0
$229$		−	6.63659i		−	0.438558i	−0.975662	−	0.219279i	$-0.929629\pi$
$229$		−	6.63659i		−	0.438558i	0.975662	−	0.219279i	$-0.0703705\pi$
$230$		0			0
$231$		−	12.7526i		−	0.839059i
$232$		0			0
$233$	−28.4233			−1.86207			−0.931036	−	0.364927i	$-0.881094\pi$
$233$	−28.4233			−1.86207			−0.931036	+	0.364927i	$0.881094\pi$
$234$		0			0
$235$	1.22656			0.0800120
$236$		0			0
$237$			11.9924i			0.778989i
$238$		0			0
$239$			21.4035i			1.38448i	0.721667	+	0.692240i	$0.243376\pi$
$239$			21.4035i			1.38448i	−0.721667	+	0.692240i	$0.756624\pi$
$240$		0			0
$241$		−	17.7463i		−	1.14314i	−0.820554	−	0.571570i	$-0.806335\pi$
$241$		−	17.7463i		−	1.14314i	0.820554	−	0.571570i	$-0.193665\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$			4.21251i			0.269128i
$246$		0			0
$247$	10.6358			0.676739
$248$		0			0
$249$		−	12.4426i		−	0.788520i
$250$		0			0
$251$	−3.15845			−0.199360			−0.0996798	−	0.995020i	$-0.531782\pi$
$251$	−3.15845			−0.199360			−0.0996798	+	0.995020i	$0.531782\pi$
$252$		0			0
$253$	17.9727	−	3.25237i	1.12993	−	0.204475i
$254$		0			0
$255$		−	3.43500i		−	0.215108i
$256$		0			0
$257$	−5.33676			−0.332898			−0.166449	−	0.986050i	$-0.553230\pi$
$257$	−5.33676			−0.332898			−0.166449	+	0.986050i	$0.553230\pi$
$258$		0			0
$259$		−	4.24537i		−	0.263794i
$260$		0			0
$261$	−2.73385			−0.169221
$262$		0			0
$263$	12.1518			0.749312			0.374656	−	0.927164i	$-0.377761\pi$
$263$	12.1518			0.749312			0.374656	+	0.927164i	$0.377761\pi$
$264$		0			0
$265$	10.0810			0.619270
$266$		0			0
$267$	−0.329105			−0.0201409
$268$		0			0
$269$	−1.21138			−0.0738589			−0.0369295	−	0.999318i	$-0.511758\pi$
$269$	−1.21138			−0.0738589			−0.0369295	+	0.999318i	$0.511758\pi$
$270$		0			0
$271$		−	12.8041i		−	0.777794i	−0.921281	−	0.388897i	$-0.872856\pi$
$271$		−	12.8041i		−	0.777794i	0.921281	−	0.388897i	$-0.127144\pi$
$272$		0			0
$273$		−	13.4291i		−	0.812764i
$274$		0			0
$275$	3.80844			0.229657
$276$		0			0
$277$	−25.9384			−1.55849			−0.779243	−	0.626721i	$-0.784396\pi$
$277$	−25.9384			−1.55849			−0.779243	+	0.626721i	$0.784396\pi$
$278$		0			0
$279$		−	5.29319i		−	0.316895i
$280$		0			0
$281$			12.0016i			0.715954i	0.933730	+	0.357977i	$0.116533\pi$
$281$			12.0016i			0.715954i	−0.933730	+	0.357977i	$0.883467\pi$
$282$		0			0
$283$	17.9652			1.06792			0.533959	−	0.845510i	$-0.320704\pi$
$283$	17.9652			1.06792			0.533959	+	0.845510i	$0.320704\pi$
$284$		0			0
$285$	2.65201			0.157092
$286$		0			0
$287$	−3.10977			−0.183564
$288$		0			0
$289$	5.20080			0.305930
$290$		0			0
$291$	4.88722			0.286494
$292$		0			0
$293$		−	24.4657i		−	1.42930i	−0.699481	−	0.714651i	$-0.746585\pi$
$293$		−	24.4657i		−	1.42930i	0.699481	−	0.714651i	$-0.253415\pi$
$294$		0			0
$295$	15.1536			0.882275
$296$		0			0
$297$			3.80844i			0.220988i
$298$		0			0
$299$	18.9261	−	3.42489i	1.09453	−	0.198067i
$300$		0			0
$301$	21.7588			1.25416
$302$		0			0
$303$		−	10.6619i		−	0.612507i
$304$		0			0
$305$	5.10695			0.292423
$306$		0			0
$307$		−	33.9420i		−	1.93717i	−0.248677	−	0.968587i	$-0.579996\pi$
$307$		−	33.9420i		−	1.93717i	0.248677	−	0.968587i	$-0.420004\pi$
$308$		0			0
$309$			17.5524i			0.998523i
$310$		0			0
$311$		−	3.90724i		−	0.221559i	−0.993845	−	0.110780i	$-0.964665\pi$
$311$		−	3.90724i		−	0.221559i	0.993845	−	0.110780i	$-0.0353348\pi$
$312$		0			0
$313$			34.7059i			1.96169i	0.194782	+	0.980847i	$0.437600\pi$
$313$			34.7059i			1.96169i	−0.194782	+	0.980847i	$0.562400\pi$
$314$		0			0
$315$		−	3.34851i		−	0.188667i
$316$		0			0
$317$	−16.0862			−0.903492			−0.451746	−	0.892147i	$-0.649199\pi$
$317$	−16.0862			−0.903492			−0.451746	+	0.892147i	$0.649199\pi$
$318$		0			0
$319$	−10.4117			−0.582944
$320$		0			0
$321$		−	9.06743i		−	0.506094i
$322$		0			0
$323$		−	9.10965i		−	0.506874i
$324$		0			0
$325$	4.01046			0.222460
$326$		0			0
$327$	−1.91151			−0.105707
$328$		0			0
$329$		−	4.10715i		−	0.226435i
$330$		0			0
$331$			3.21976i			0.176974i	0.996077	+	0.0884871i	$0.0282032\pi$
$331$			3.21976i			0.176974i	−0.996077	+	0.0884871i	$0.971797\pi$
$332$		0			0
$333$			1.26784i			0.0694771i
$334$		0			0
$335$			3.01447i			0.164698i
$336$		0			0
$337$		−	3.42112i		−	0.186360i	−0.995649	−	0.0931801i	$-0.970297\pi$
$337$		−	3.42112i		−	0.186360i	0.995649	−	0.0931801i	$-0.0297032\pi$
$338$		0			0
$339$	−9.47968			−0.514866
$340$		0			0
$341$		−	20.1588i		−	1.09166i
$342$		0			0
$343$	−9.33392			−0.503984
$344$		0			0
$345$	4.71918	−	0.853990i	0.254072	−	0.0459773i
$346$		0			0
$347$			14.3395i			0.769784i	0.922962	+	0.384892i	$0.125761\pi$
$347$			14.3395i			0.769784i	−0.922962	+	0.384892i	$0.874239\pi$
$348$		0			0
$349$	−17.8040			−0.953025			−0.476512	−	0.879168i	$-0.658099\pi$
$349$	−17.8040			−0.953025			−0.476512	+	0.879168i	$0.658099\pi$
$350$		0			0
$351$			4.01046i			0.214063i
$352$		0			0
$353$	7.05096			0.375285			0.187642	−	0.982237i	$-0.439915\pi$
$353$	7.05096			0.375285			0.187642	+	0.982237i	$0.439915\pi$
$354$		0			0
$355$	4.10114			0.217666
$356$		0			0
$357$	−11.5021			−0.608757
$358$		0			0
$359$	11.2467			0.593579			0.296790	−	0.954943i	$-0.404084\pi$
$359$	11.2467			0.593579			0.296790	+	0.954943i	$0.404084\pi$
$360$		0			0
$361$	−11.9668			−0.629834
$362$		0			0
$363$			3.50419i			0.183922i
$364$		0			0
$365$		−	12.9930i		−	0.680086i
$366$		0			0
$367$	−35.8800			−1.87292			−0.936461	−	0.350772i	$-0.885919\pi$
$367$	−35.8800			−1.87292			−0.936461	+	0.350772i	$0.885919\pi$
$368$		0			0
$369$	0.928703			0.0483464
$370$		0			0
$371$		−	33.7563i		−	1.75254i
$372$		0			0
$373$		−	9.99825i		−	0.517690i	−0.965919	−	0.258845i	$-0.916658\pi$
$373$		−	9.99825i		−	0.517690i	0.965919	−	0.258845i	$-0.0833418\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	−10.9640			−0.564675
$378$		0			0
$379$	−24.3987			−1.25328			−0.626638	−	0.779311i	$-0.715569\pi$
$379$	−24.3987			−1.25328			−0.626638	+	0.779311i	$0.715569\pi$
$380$		0			0
$381$	14.5043			0.743075
$382$		0			0
$383$	−8.66954			−0.442993			−0.221496	−	0.975161i	$-0.571094\pi$
$383$	−8.66954			−0.442993			−0.221496	+	0.975161i	$0.571094\pi$
$384$		0			0
$385$		−	12.7526i		−	0.649932i
$386$		0			0
$387$	−6.49805			−0.330315
$388$		0			0
$389$			6.29587i			0.319213i	0.987181	+	0.159607i	$0.0510226\pi$
$389$			6.29587i			0.319213i	−0.987181	+	0.159607i	$0.948977\pi$
$390$		0			0
$391$	−2.93345	−	16.2104i	−0.148351	−	0.819794i
$392$		0			0
$393$	16.8636			0.850656
$394$		0			0
$395$			11.9924i			0.603402i
$396$		0			0
$397$	6.23934			0.313143			0.156572	−	0.987667i	$-0.449956\pi$
$397$	6.23934			0.313143			0.156572	+	0.987667i	$0.449956\pi$
$398$		0			0
$399$		−	8.88028i		−	0.444570i
$400$		0			0
$401$			25.7877i			1.28777i	0.765120	+	0.643887i	$0.222679\pi$
$401$			25.7877i			1.28777i	−0.765120	+	0.643887i	$0.777321\pi$
$402$		0			0
$403$		−	21.2281i		−	1.05745i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$			4.82848i			0.239339i
$408$		0			0
$409$	19.7280			0.975485			0.487743	−	0.872987i	$-0.337820\pi$
$409$	19.7280			0.975485			0.487743	+	0.872987i	$0.337820\pi$
$410$		0			0
$411$	−5.29294			−0.261082
$412$		0			0
$413$		−	50.7419i		−	2.49685i
$414$		0			0
$415$		−	12.4426i		−	0.610785i
$416$		0			0
$417$	19.7415			0.966744
$418$		0			0
$419$	0.455871			0.0222707			0.0111354	−	0.999938i	$-0.496455\pi$
$419$	0.455871			0.0222707			0.0111354	+	0.999938i	$0.496455\pi$
$420$		0			0
$421$		−	10.9952i		−	0.535873i	−0.963437	−	0.267936i	$-0.913658\pi$
$421$		−	10.9952i		−	0.535873i	0.963437	−	0.267936i	$-0.0863417\pi$
$422$		0			0
$423$			1.22656i			0.0596375i
$424$		0			0
$425$		−	3.43500i		−	0.166622i
$426$		0			0
$427$		−	17.1007i		−	0.827559i
$428$		0			0
$429$			15.2736i			0.737416i
$430$		0			0
$431$	4.29748			0.207003			0.103501	−	0.994629i	$-0.466995\pi$
$431$	4.29748			0.207003			0.103501	+	0.994629i	$0.466995\pi$
$432$		0			0
$433$			0.952676i			0.0457827i	0.999738	+	0.0228914i	$0.00728718\pi$
$433$			0.952676i			0.0457827i	−0.999738	+	0.0228914i	$0.992713\pi$
$434$		0			0
$435$	−2.73385			−0.131078
$436$		0			0
$437$	12.5153	−	2.26479i	0.598689	−	0.108340i
$438$		0			0
$439$			25.4726i			1.21574i	0.794037	+	0.607870i	$0.207976\pi$
$439$			25.4726i			1.21574i	−0.794037	+	0.607870i	$0.792024\pi$
$440$		0			0
$441$	−4.21251			−0.200596
$442$		0			0
$443$		−	15.7371i		−	0.747694i	−0.927490	−	0.373847i	$-0.878039\pi$
$443$		−	15.7371i		−	0.747694i	0.927490	−	0.373847i	$-0.121961\pi$
$444$		0			0
$445$	−0.329105			−0.0156011
$446$		0			0
$447$	−11.2478			−0.532002
$448$		0			0
$449$	−22.3906			−1.05668			−0.528338	−	0.849034i	$-0.677185\pi$
$449$	−22.3906			−1.05668			−0.528338	+	0.849034i	$0.677185\pi$
$450$		0			0
$451$	3.53691			0.166546
$452$		0			0
$453$	11.2361			0.527918
$454$		0			0
$455$		−	13.4291i		−	0.629564i
$456$		0			0
$457$			9.16148i			0.428556i	0.976773	+	0.214278i	$0.0687398\pi$
$457$			9.16148i			0.428556i	−0.976773	+	0.214278i	$0.931260\pi$
$458$		0			0
$459$	3.43500			0.160332
$460$		0			0
$461$	−16.8603			−0.785264			−0.392632	−	0.919696i	$-0.628435\pi$
$461$	−16.8603			−0.785264			−0.392632	+	0.919696i	$0.628435\pi$
$462$		0			0
$463$		−	20.5902i		−	0.956906i	−0.878113	−	0.478453i	$-0.841198\pi$
$463$		−	20.5902i		−	0.956906i	0.878113	−	0.478453i	$-0.158802\pi$
$464$		0			0
$465$		−	5.29319i		−	0.245466i
$466$		0			0
$467$	22.7394			1.05225			0.526126	−	0.850406i	$-0.323644\pi$
$467$	22.7394			1.05225			0.526126	+	0.850406i	$0.323644\pi$
$468$		0			0
$469$	10.0940			0.466097
$470$		0			0
$471$	−2.61969			−0.120709
$472$		0			0
$473$	−24.7474			−1.13789
$474$		0			0
$475$	2.65201			0.121683
$476$		0			0
$477$			10.0810i			0.461576i
$478$		0			0
$479$	−17.6618			−0.806987			−0.403494	−	0.914982i	$-0.632204\pi$
$479$	−17.6618			−0.806987			−0.403494	+	0.914982i	$0.632204\pi$
$480$		0			0
$481$			5.08461i			0.231838i
$482$		0			0
$483$	−2.85959	−	15.8022i	−0.130116	−	0.719026i
$484$		0			0
$485$	4.88722			0.221917
$486$		0			0
$487$		−	30.1379i		−	1.36568i	−0.730568	−	0.682840i	$-0.760745\pi$
$487$		−	30.1379i		−	1.36568i	0.730568	−	0.682840i	$-0.239255\pi$
$488$		0			0
$489$	−16.4052			−0.741868
$490$		0			0
$491$			35.7563i			1.61366i	0.590785	+	0.806829i	$0.298818\pi$
$491$			35.7563i			1.61366i	−0.590785	+	0.806829i	$0.701182\pi$
$492$		0			0
$493$			9.39077i			0.422939i
$494$		0			0
$495$			3.80844i			0.171177i
$496$		0			0
$497$		−	13.7327i		−	0.615997i
$498$		0			0
$499$		−	1.38923i		−	0.0621903i	−0.999516	−	0.0310951i	$-0.990101\pi$
$499$		−	1.38923i		−	0.0621903i	0.999516	−	0.0310951i	$-0.00989948\pi$
$500$		0			0
$501$	−6.17814			−0.276019
$502$		0			0
$503$	19.9672			0.890295			0.445147	−	0.895457i	$-0.353151\pi$
$503$	19.9672			0.890295			0.445147	+	0.895457i	$0.353151\pi$
$504$		0			0
$505$		−	10.6619i		−	0.474446i
$506$		0			0
$507$			3.08380i			0.136957i
$508$		0			0
$509$	−21.3968			−0.948395			−0.474198	−	0.880418i	$-0.657262\pi$
$509$	−21.3968			−0.948395			−0.474198	+	0.880418i	$0.657262\pi$
$510$		0			0
$511$	−43.5073			−1.92465
$512$		0			0
$513$			2.65201i			0.117089i
$514$		0			0
$515$			17.5524i			0.773453i
$516$		0			0
$517$			4.67128i			0.205443i
$518$		0			0
$519$		−	24.5549i		−	1.07784i
$520$		0			0
$521$		−	40.4491i		−	1.77211i	−0.463581	−	0.886054i	$-0.653436\pi$
$521$		−	40.4491i		−	1.77211i	0.463581	−	0.886054i	$-0.346564\pi$
$522$		0			0
$523$	33.8959			1.48216			0.741082	−	0.671414i	$-0.234313\pi$
$523$	33.8959			1.48216			0.741082	+	0.671414i	$0.234313\pi$
$524$		0			0
$525$		−	3.34851i		−	0.146141i
$526$		0			0
$527$	−18.1821			−0.792024
$528$		0			0
$529$	21.5414	−	8.06027i	0.936583	−	0.350447i
$530$		0			0
$531$			15.1536i			0.657609i
$532$		0			0
$533$	3.72453			0.161327
$534$		0			0
$535$		−	9.06743i		−	0.392019i
$536$		0			0
$537$	11.9824			0.517078
$538$		0			0
$539$	−16.0431			−0.691025
$540$		0			0
$541$	−13.7424			−0.590834			−0.295417	−	0.955368i	$-0.595459\pi$
$541$	−13.7424			−0.590834			−0.295417	+	0.955368i	$0.595459\pi$
$542$		0			0
$543$	−13.9533			−0.598792
$544$		0			0
$545$	−1.91151			−0.0818803
$546$		0			0
$547$			22.3514i			0.955676i	0.878448	+	0.477838i	$0.158579\pi$
$547$			22.3514i			0.955676i	−0.878448	+	0.477838i	$0.841421\pi$
$548$		0			0
$549$			5.10695i			0.217959i
$550$		0			0
$551$	−7.25021			−0.308869
$552$		0			0
$553$	40.1566			1.70763
$554$		0			0
$555$			1.26784i			0.0538167i
$556$		0			0
$557$			24.9665i			1.05787i	0.848664	+	0.528933i	$0.177408\pi$
$557$			24.9665i			1.05787i	−0.848664	+	0.528933i	$0.822592\pi$
$558$		0			0
$559$	−26.0602			−1.10223
$560$		0			0
$561$	13.0820			0.552321
$562$		0			0
$563$	33.3847			1.40700			0.703499	−	0.710697i	$-0.251620\pi$
$563$	33.3847			1.40700			0.703499	+	0.710697i	$0.251620\pi$
$564$		0			0
$565$	−9.47968			−0.398813
$566$		0			0
$567$	3.34851			0.140624
$568$		0			0
$569$		−	9.74274i		−	0.408437i	−0.978925	−	0.204219i	$-0.934535\pi$
$569$		−	9.74274i		−	0.408437i	0.978925	−	0.204219i	$-0.0654653\pi$
$570$		0			0
$571$	−16.9759			−0.710420			−0.355210	−	0.934786i	$-0.615591\pi$
$571$	−16.9759			−0.710420			−0.355210	+	0.934786i	$0.615591\pi$
$572$		0			0
$573$		−	17.2707i		−	0.721494i
$574$		0			0
$575$	4.71918	−	0.853990i	0.196804	−	0.0356138i
$576$		0			0
$577$	19.1108			0.795592			0.397796	−	0.917474i	$-0.369775\pi$
$577$	19.1108			0.795592			0.397796	+	0.917474i	$0.369775\pi$
$578$		0			0
$579$			3.94520i			0.163957i
$580$		0			0
$581$	−41.6643			−1.72853
$582$		0			0
$583$			38.3928i			1.59007i
$584$		0			0
$585$			4.01046i			0.165812i
$586$		0			0
$587$		−	20.4134i		−	0.842550i	−0.906933	−	0.421275i	$-0.861583\pi$
$587$		−	20.4134i		−	0.842550i	0.906933	−	0.421275i	$-0.138417\pi$
$588$		0			0
$589$		−	14.0376i		−	0.578409i
$590$		0			0
$591$		−	5.70577i		−	0.234704i
$592$		0			0
$593$	40.1285			1.64788			0.823940	−	0.566677i	$-0.191771\pi$
$593$	40.1285			1.64788			0.823940	+	0.566677i	$0.191771\pi$
$594$		0			0
$595$	−11.5021			−0.471541
$596$		0			0
$597$		−	3.55709i		−	0.145582i
$598$		0			0
$599$		−	4.23805i		−	0.173162i	−0.996245	−	0.0865809i	$-0.972406\pi$
$599$		−	4.23805i		−	0.173162i	0.996245	−	0.0865809i	$-0.0275941\pi$
$600$		0			0
$601$	−37.2617			−1.51994			−0.759968	−	0.649960i	$-0.774786\pi$
$601$	−37.2617			−1.51994			−0.759968	+	0.649960i	$0.774786\pi$
$602$		0			0
$603$	−3.01447			−0.122759
$604$		0			0
$605$			3.50419i			0.142466i
$606$		0			0
$607$			4.56947i			0.185469i	0.995691	+	0.0927346i	$0.0295608\pi$
$607$			4.56947i			0.185469i	−0.995691	+	0.0927346i	$0.970439\pi$
$608$		0			0
$609$			9.15433i			0.370952i
$610$		0			0
$611$			4.91908i			0.199005i
$612$		0			0
$613$			6.85443i			0.276848i	0.990373	+	0.138424i	$0.0442036\pi$
$613$			6.85443i			0.276848i	−0.990373	+	0.138424i	$0.955796\pi$
$614$		0			0
$615$	0.928703			0.0374489
$616$		0			0
$617$		−	1.07697i		−	0.0433574i	−0.999765	−	0.0216787i	$-0.993099\pi$
$617$		−	1.07697i		−	0.0433574i	0.999765	−	0.0216787i	$-0.00690108\pi$
$618$		0			0
$619$	39.6508			1.59370			0.796850	−	0.604177i	$-0.206498\pi$
$619$	39.6508			1.59370			0.796850	+	0.604177i	$0.206498\pi$
$620$		0			0
$621$	0.853990	+	4.71918i	0.0342694	+	0.189374i
$622$		0			0
$623$			1.10201i			0.0441511i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			10.1000i			0.403356i
$628$		0			0
$629$	4.35502			0.173646
$630$		0			0
$631$	−12.7755			−0.508583			−0.254291	−	0.967128i	$-0.581842\pi$
$631$	−12.7755			−0.508583			−0.254291	+	0.967128i	$0.581842\pi$
$632$		0			0
$633$	−12.3538			−0.491018
$634$		0			0
$635$	14.5043			0.575584
$636$		0			0
$637$	−16.8941			−0.669370
$638$		0			0
$639$			4.10114i			0.162239i
$640$		0			0
$641$			6.23062i			0.246095i	0.992401	+	0.123047i	$0.0392667\pi$
$641$			6.23062i			0.246095i	−0.992401	+	0.123047i	$0.960733\pi$
$642$		0			0
$643$	23.6102			0.931095			0.465547	−	0.885023i	$-0.345857\pi$
$643$	23.6102			0.931095			0.465547	+	0.885023i	$0.345857\pi$
$644$		0			0
$645$	−6.49805			−0.255861
$646$		0			0
$647$			17.3269i			0.681189i	0.940210	+	0.340595i	$0.110628\pi$
$647$			17.3269i			0.681189i	−0.940210	+	0.340595i	$0.889372\pi$
$648$		0			0
$649$			57.7114i			2.26537i
$650$		0			0
$651$	−17.7243			−0.694669
$652$		0			0
$653$	7.82842			0.306350			0.153175	−	0.988199i	$-0.451050\pi$
$653$	7.82842			0.306350			0.153175	+	0.988199i	$0.451050\pi$
$654$		0			0
$655$	16.8636			0.658915
$656$		0			0
$657$	12.9930			0.506906
$658$		0			0
$659$	−11.1695			−0.435103			−0.217551	−	0.976049i	$-0.569807\pi$
$659$	−11.1695			−0.435103			−0.217551	+	0.976049i	$0.569807\pi$
$660$		0			0
$661$			15.4691i			0.601680i	0.953675	+	0.300840i	$0.0972670\pi$
$661$			15.4691i			0.601680i	−0.953675	+	0.300840i	$0.902733\pi$
$662$		0			0
$663$	13.7759			0.535012
$664$		0			0
$665$		−	8.88028i		−	0.344363i
$666$		0			0
$667$	−12.9016	+	2.33468i	−0.499550	+	0.0903993i
$668$		0			0
$669$	−3.60984			−0.139564
$670$		0			0
$671$			19.4495i			0.750839i
$672$		0			0
$673$	47.5831			1.83420			0.917098	−	0.398663i	$-0.130526\pi$
$673$	47.5831			1.83420			0.917098	+	0.398663i	$0.130526\pi$
$674$		0			0
$675$			1.00000i			0.0384900i
$676$		0			0
$677$			49.1013i			1.88712i	0.331206	+	0.943558i	$0.392544\pi$
$677$			49.1013i			1.88712i	−0.331206	+	0.943558i	$0.607456\pi$
$678$		0			0
$679$		−	16.3649i		−	0.628028i
$680$		0			0
$681$			10.4598i			0.400820i
$682$		0			0
$683$		−	36.1616i		−	1.38368i	−0.722049	−	0.691842i	$-0.756800\pi$
$683$		−	36.1616i		−	1.38368i	0.722049	−	0.691842i	$-0.243200\pi$
$684$		0			0
$685$	−5.29294			−0.202233
$686$		0			0
$687$	6.63659			0.253202
$688$		0			0
$689$			40.4294i			1.54024i
$690$		0			0
$691$			0.284481i			0.0108222i	0.999985	+	0.00541108i	$0.00172241\pi$
$691$			0.284481i			0.0108222i	−0.999985	+	0.00541108i	$0.998278\pi$
$692$		0			0
$693$	12.7526			0.484431
$694$		0			0
$695$	19.7415			0.748837
$696$		0			0
$697$		−	3.19009i		−	0.120833i
$698$		0			0
$699$		−	28.4233i		−	1.07507i
$700$		0			0
$701$			8.87200i			0.335091i	0.985864	+	0.167545i	$0.0535841\pi$
$701$			8.87200i			0.335091i	−0.985864	+	0.167545i	$0.946416\pi$
$702$		0			0
$703$			3.36232i			0.126812i
$704$		0			0
$705$			1.22656i			0.0461950i
$706$		0			0
$707$	−35.7013			−1.34269
$708$		0			0
$709$		−	40.1502i		−	1.50787i	−0.656948	−	0.753936i	$-0.728153\pi$
$709$		−	40.1502i		−	1.50787i	0.656948	−	0.753936i	$-0.271847\pi$
$710$		0			0
$711$	−11.9924			−0.449750
$712$		0			0
$713$	−4.52033	−	24.9795i	−0.169288	−	0.935490i
$714$		0			0
$715$			15.2736i			0.571200i
$716$		0			0
$717$	−21.4035			−0.799330
$718$		0			0
$719$			41.0118i			1.52948i	0.644337	+	0.764741i	$0.277133\pi$
$719$			41.0118i			1.52948i	−0.644337	+	0.764741i	$0.722867\pi$
$720$		0			0
$721$	58.7745			2.18888
$722$		0			0
$723$	17.7463			0.659992
$724$		0			0
$725$	−2.73385			−0.101533
$726$		0			0
$727$	32.7737			1.21551			0.607755	−	0.794124i	$-0.292070\pi$
$727$	32.7737			1.21551			0.607755	+	0.794124i	$0.292070\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			22.3208i			0.825564i
$732$		0			0
$733$			22.7817i			0.841462i	0.907186	+	0.420731i	$0.138226\pi$
$733$			22.7817i			0.841462i	−0.907186	+	0.420731i	$0.861774\pi$
$734$		0			0
$735$	−4.21251			−0.155381
$736$		0			0
$737$	−11.4804			−0.422887
$738$		0			0
$739$			42.0012i			1.54504i	0.634990	+	0.772520i	$0.281004\pi$
$739$			42.0012i			1.54504i	−0.634990	+	0.772520i	$0.718996\pi$
$740$		0			0
$741$			10.6358i			0.390715i
$742$		0			0
$743$	−7.19669			−0.264021			−0.132010	−	0.991248i	$-0.542143\pi$
$743$	−7.19669			−0.264021			−0.132010	+	0.991248i	$0.542143\pi$
$744$		0			0
$745$	−11.2478			−0.412087
$746$		0			0
$747$	12.4426			0.455252
$748$		0			0
$749$	−30.3624			−1.10942
$750$		0			0
$751$	3.66405			0.133703			0.0668515	−	0.997763i	$-0.478705\pi$
$751$	3.66405			0.133703			0.0668515	+	0.997763i	$0.478705\pi$
$752$		0			0
$753$		−	3.15845i		−	0.115100i
$754$		0			0
$755$	11.2361			0.408924
$756$		0			0
$757$			15.3288i			0.557134i	0.960417	+	0.278567i	$0.0898594\pi$
$757$			15.3288i			0.557134i	−0.960417	+	0.278567i	$0.910141\pi$
$758$		0			0
$759$	3.25237	+	17.9727i	0.118053	+	0.652368i
$760$		0			0
$761$	52.0017			1.88506			0.942531	−	0.334120i	$-0.108439\pi$
$761$	52.0017			1.88506			0.942531	+	0.334120i	$0.108439\pi$
$762$		0			0
$763$			6.40072i			0.231722i
$764$		0			0
$765$	3.43500			0.124193
$766$		0			0
$767$			60.7728i			2.19438i
$768$		0			0
$769$		−	13.3406i		−	0.481074i	−0.970640	−	0.240537i	$-0.922676\pi$
$769$		−	13.3406i		−	0.481074i	0.970640	−	0.240537i	$-0.0773236\pi$
$770$		0			0
$771$		−	5.33676i		−	0.192199i
$772$		0			0
$773$			23.5703i			0.847766i	0.905717	+	0.423883i	$0.139333\pi$
$773$			23.5703i			0.847766i	−0.905717	+	0.423883i	$0.860667\pi$
$774$		0			0
$775$		−	5.29319i		−	0.190137i
$776$		0			0
$777$	4.24537			0.152302
$778$		0			0
$779$	2.46293			0.0882437
$780$		0			0
$781$			15.6189i			0.558890i
$782$		0			0
$783$		−	2.73385i		−	0.0976999i
$784$		0			0
$785$	−2.61969			−0.0935009
$786$		0			0
$787$	2.86992			0.102301			0.0511507	−	0.998691i	$-0.483711\pi$
$787$	2.86992			0.102301			0.0511507	+	0.998691i	$0.483711\pi$
$788$		0			0
$789$			12.1518i			0.432616i
$790$		0			0
$791$			31.7428i			1.12864i
$792$		0			0
$793$			20.4812i			0.727310i
$794$		0			0
$795$			10.0810i			0.357536i
$796$		0			0
$797$			7.88351i			0.279248i	0.990205	+	0.139624i	$0.0445894\pi$
$797$			7.88351i			0.279248i	−0.990205	+	0.139624i	$0.955411\pi$
$798$		0			0
$799$	4.21323			0.149053
$800$		0			0
$801$		−	0.329105i		−	0.0116283i
$802$		0			0
$803$	49.4831			1.74622
$804$		0			0
$805$	−2.85959	−	15.8022i	−0.100787	−	0.556956i
$806$		0			0
$807$		−	1.21138i		−	0.0426425i
$808$		0			0
$809$	9.74636			0.342664			0.171332	−	0.985213i	$-0.445193\pi$
$809$	9.74636			0.342664			0.171332	+	0.985213i	$0.445193\pi$
$810$		0			0
$811$		−	23.9370i		−	0.840542i	−0.907399	−	0.420271i	$-0.861935\pi$
$811$		−	23.9370i		−	0.840542i	0.907399	−	0.420271i	$-0.138065\pi$
$812$		0			0
$813$	12.8041			0.449060
$814$		0			0
$815$	−16.4052			−0.574649
$816$		0			0
$817$	−17.2329			−0.602903
$818$		0			0
$819$	13.4291			0.469250
$820$		0			0
$821$	−45.6615			−1.59360			−0.796799	−	0.604244i	$-0.793475\pi$
$821$	−45.6615			−1.59360			−0.796799	+	0.604244i	$0.793475\pi$
$822$		0			0
$823$			3.14057i			0.109473i	0.998501	+	0.0547367i	$0.0174319\pi$
$823$			3.14057i			0.109473i	−0.998501	+	0.0547367i	$0.982568\pi$
$824$		0			0
$825$			3.80844i			0.132593i
$826$		0			0
$827$	−18.4428			−0.641320			−0.320660	−	0.947194i	$-0.603905\pi$
$827$	−18.4428			−0.641320			−0.320660	+	0.947194i	$0.603905\pi$
$828$		0			0
$829$	−35.7912			−1.24308			−0.621541	−	0.783382i	$-0.713493\pi$
$829$	−35.7912			−1.24308			−0.621541	+	0.783382i	$0.713493\pi$
$830$		0			0
$831$		−	25.9384i		−	0.899793i
$832$		0			0
$833$			14.4700i			0.501355i
$834$		0			0
$835$	−6.17814			−0.213803
$836$		0			0
$837$	5.29319			0.182959
$838$		0			0
$839$	10.7645			0.371633			0.185817	−	0.982584i	$-0.440507\pi$
$839$	10.7645			0.371633			0.185817	+	0.982584i	$0.440507\pi$
$840$		0			0
$841$	−21.5260			−0.742278
$842$		0			0
$843$	−12.0016			−0.413356
$844$		0			0
$845$			3.08380i			0.106086i
$846$		0			0
$847$	11.7338			0.403179
$848$		0			0
$849$			17.9652i			0.616563i
$850$		0			0
$851$	1.08272	+	5.98316i	0.0371152	+	0.205100i
$852$		0			0
$853$	−52.8988			−1.81122			−0.905611	−	0.424110i	$-0.860587\pi$
$853$	−52.8988			−1.81122			−0.905611	+	0.424110i	$0.860587\pi$
$854$		0			0
$855$			2.65201i			0.0906969i
$856$		0			0
$857$	1.96495			0.0671215			0.0335608	−	0.999437i	$-0.489315\pi$
$857$	1.96495			0.0671215			0.0335608	+	0.999437i	$0.489315\pi$
$858$		0			0
$859$		−	11.1376i		−	0.380009i	−0.981783	−	0.190005i	$-0.939150\pi$
$859$		−	11.1376i		−	0.380009i	0.981783	−	0.190005i	$-0.0608503\pi$
$860$		0			0
$861$		−	3.10977i		−	0.105981i
$862$		0			0
$863$		−	3.27846i		−	0.111600i	−0.998442	−	0.0558000i	$-0.982229\pi$
$863$		−	3.27846i		−	0.111600i	0.998442	−	0.0558000i	$-0.0177709\pi$
$864$		0			0
$865$		−	24.5549i		−	0.834891i
$866$		0			0
$867$			5.20080i			0.176629i
$868$		0			0
$869$	−45.6722			−1.54932
$870$		0			0
$871$	−12.0894			−0.409634
$872$		0			0
$873$			4.88722i			0.165407i
$874$		0			0
$875$		−	3.34851i		−	0.113200i
$876$		0			0
$877$	−36.0512			−1.21736			−0.608680	−	0.793415i	$-0.708301\pi$
$877$	−36.0512			−1.21736			−0.608680	+	0.793415i	$0.708301\pi$
$878$		0			0
$879$	24.4657			0.825208
$880$		0			0
$881$			31.0847i			1.04727i	0.851943	+	0.523634i	$0.175424\pi$
$881$			31.0847i			1.04727i	−0.851943	+	0.523634i	$0.824576\pi$
$882$		0			0
$883$			45.7713i			1.54033i	0.637846	+	0.770164i	$0.279826\pi$
$883$			45.7713i			1.54033i	−0.637846	+	0.770164i	$0.720174\pi$
$884$		0			0
$885$			15.1536i			0.509382i
$886$		0			0
$887$		−	15.6422i		−	0.525213i	−0.964903	−	0.262606i	$-0.915418\pi$
$887$		−	15.6422i		−	0.525213i	0.964903	−	0.262606i	$-0.0845821\pi$
$888$		0			0
$889$		−	48.5676i		−	1.62891i
$890$		0			0
$891$	−3.80844			−0.127587
$892$		0			0
$893$			3.25285i			0.108853i
$894$		0			0
$895$	11.9824			0.400527
$896$		0			0
$897$	3.42489	+	18.9261i	0.114354	+	0.631924i
$898$		0			0
$899$			14.4708i			0.482628i
$900$		0			0
$901$	34.6281			1.15363
$902$		0			0
$903$			21.7588i			0.724087i
$904$		0			0
$905$	−13.9533			−0.463822
$906$		0			0
$907$	16.3806			0.543909			0.271954	−	0.962310i	$-0.412330\pi$
$907$	16.3806			0.543909			0.271954	+	0.962310i	$0.412330\pi$
$908$		0			0
$909$	10.6619			0.353631
$910$		0			0
$911$	−31.7937			−1.05337			−0.526686	−	0.850060i	$-0.676566\pi$
$911$	−31.7937			−1.05337			−0.526686	+	0.850060i	$0.676566\pi$
$912$		0			0
$913$	47.3870			1.56828
$914$		0			0
$915$			5.10695i			0.168831i
$916$		0			0
$917$		−	56.4679i		−	1.86473i
$918$		0			0
$919$	−46.6308			−1.53821			−0.769104	−	0.639123i	$-0.779297\pi$
$919$	−46.6308			−1.53821			−0.769104	+	0.639123i	$0.779297\pi$
$920$		0			0
$921$	33.9420			1.11843
$922$		0			0
$923$			16.4475i			0.541375i
$924$		0			0
$925$			1.26784i			0.0416862i
$926$		0			0
$927$	−17.5524			−0.576498
$928$		0			0
$929$	−37.5701			−1.23264			−0.616318	−	0.787498i	$-0.711376\pi$
$929$	−37.5701			−1.23264			−0.616318	+	0.787498i	$0.711376\pi$
$930$		0			0
$931$	−11.1716			−0.366136
$932$		0			0
$933$	3.90724			0.127917
$934$		0			0
$935$	13.0820			0.427826
$936$		0			0
$937$			20.4844i			0.669196i	0.942361	+	0.334598i	$0.108601\pi$
$937$			20.4844i			0.669196i	−0.942361	+	0.334598i	$0.891399\pi$
$938$		0			0
$939$	−34.7059			−1.13258
$940$		0			0
$941$			13.2639i			0.432389i	0.976350	+	0.216195i	$0.0693646\pi$
$941$			13.2639i			0.432389i	−0.976350	+	0.216195i	$0.930635\pi$
$942$		0			0
$943$	4.38272	−	0.793103i	0.142721	−	0.0258270i
$944$		0			0
$945$	3.34851			0.108927
$946$		0			0
$947$		−	6.18691i		−	0.201048i	−0.994935	−	0.100524i	$-0.967948\pi$
$947$		−	6.18691i		−	0.201048i	0.994935	−	0.100524i	$-0.0320519\pi$
$948$		0			0
$949$	52.1081			1.69150
$950$		0			0
$951$		−	16.0862i		−	0.521631i
$952$		0			0
$953$		−	24.1764i		−	0.783149i	−0.920146	−	0.391575i	$-0.871930\pi$
$953$		−	24.1764i		−	0.783149i	0.920146	−	0.391575i	$-0.128070\pi$
$954$		0			0
$955$		−	17.2707i		−	0.558867i
$956$		0			0
$957$		−	10.4117i		−	0.336563i
$958$		0			0
$959$			17.7235i			0.572320i
$960$		0			0
$961$	2.98218			0.0961994
$962$		0			0
$963$	9.06743			0.292194
$964$		0			0
$965$			3.94520i			0.127000i
$966$		0			0
$967$		−	15.4189i		−	0.495837i	−0.968781	−	0.247919i	$-0.920253\pi$
$967$		−	15.4189i		−	0.495837i	0.968781	−	0.247919i	$-0.0797466\pi$
$968$		0			0
$969$	9.10965			0.292644
$970$		0			0
$971$	−41.7152			−1.33870			−0.669352	−	0.742945i	$-0.733428\pi$
$971$	−41.7152			−1.33870			−0.669352	+	0.742945i	$0.733428\pi$
$972$		0			0
$973$		−	66.1045i		−	2.11921i
$974$		0			0
$975$			4.01046i			0.128438i
$976$		0			0
$977$		−	9.68123i		−	0.309730i	−0.987936	−	0.154865i	$-0.950506\pi$
$977$		−	9.68123i		−	0.309730i	0.987936	−	0.154865i	$-0.0494943\pi$
$978$		0			0
$979$		−	1.25338i		−	0.0400580i
$980$		0			0
$981$		−	1.91151i		−	0.0610299i
$982$		0			0
$983$	18.4414			0.588190			0.294095	−	0.955776i	$-0.404982\pi$
$983$	18.4414			0.588190			0.294095	+	0.955776i	$0.404982\pi$
$984$		0			0
$985$		−	5.70577i		−	0.181801i
$986$		0			0
$987$	4.10715			0.130732
$988$		0			0
$989$	−30.6655	+	5.54927i	−0.975107	+	0.176457i
$990$		0			0
$991$		−	4.52059i		−	0.143601i	−0.997419	−	0.0718007i	$-0.977125\pi$
$991$		−	4.52059i		−	0.143601i	0.997419	−	0.0718007i	$-0.0228746\pi$
$992$		0			0
$993$	−3.21976			−0.102176
$994$		0			0
$995$		−	3.55709i		−	0.112767i
$996$		0			0
$997$	56.8827			1.80149			0.900747	−	0.434344i	$-0.143020\pi$
$997$	56.8827			1.80149			0.900747	+	0.434344i	$0.143020\pi$
$998$		0			0
$999$	−1.26784			−0.0401126

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.c.1471.2	yes	32
4.3	odd	2		5520.2.be.d.1471.1	yes	32
23.22	odd	2		5520.2.be.d.1471.2	yes	32
92.91	even	2	inner	5520.2.be.c.1471.1	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.c.1471.1	✓	32	92.91	even	2	inner
5520.2.be.c.1471.2	yes	32	1.1	even	1	trivial
5520.2.be.d.1471.1	yes	32	4.3	odd	2
5520.2.be.d.1471.2	yes	32	23.22	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} +3.34851 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} +3.34851 q^{7} -1.00000 q^{9} -3.80844 q^{11} -4.01046 q^{13} -1.00000 q^{15} +3.43500i q^{17} -2.65201 q^{19} +3.34851i q^{21} +(-4.71918 + 0.853990i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +2.73385 q^{29} +5.29319i q^{31} -3.80844i q^{33} +3.34851i q^{35} -1.26784i q^{37} -4.01046i q^{39} -0.928703 q^{41} +6.49805 q^{43} -1.00000i q^{45} -1.22656i q^{47} +4.21251 q^{49} -3.43500 q^{51} -10.0810i q^{53} -3.80844i q^{55} -2.65201i q^{57} -15.1536i q^{59} -5.10695i q^{61} -3.34851 q^{63} -4.01046i q^{65} +3.01447 q^{67} +(-0.853990 - 4.71918i) q^{69} -4.10114i q^{71} -12.9930 q^{73} -1.00000i q^{75} -12.7526 q^{77} +11.9924 q^{79} +1.00000 q^{81} -12.4426 q^{83} -3.43500 q^{85} +2.73385i q^{87} +0.329105i q^{89} -13.4291 q^{91} -5.29319 q^{93} -2.65201i q^{95} -4.88722i q^{97} +3.80844 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) \) 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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