LMFDB - Embedded newform 5520.2.be.c.1471.5

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			1471.5
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.c.1471.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} +3.12483 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{5} +3.12483 q^{7} -1.00000 q^{9} -1.96967 q^{11} -2.52369 q^{13} -1.00000 q^{15} +3.66071i q^{17} -6.78607 q^{19} -3.12483i q^{21} +(2.20899 + 4.25680i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +2.23487 q^{29} +0.377839i q^{31} +1.96967i q^{33} -3.12483i q^{35} +5.26726i q^{37} +2.52369i q^{39} +5.40401 q^{41} +12.5558 q^{43} +1.00000i q^{45} +1.35089i q^{47} +2.76459 q^{49} +3.66071 q^{51} -3.17126i q^{53} +1.96967i q^{55} +6.78607i q^{57} -11.9962i q^{59} +4.62698i q^{61} -3.12483 q^{63} +2.52369i q^{65} +10.4936 q^{67} +(4.25680 - 2.20899i) q^{69} +15.1992i q^{71} +13.4156 q^{73} +1.00000i q^{75} -6.15489 q^{77} +5.38677 q^{79} +1.00000 q^{81} +14.7579 q^{83} +3.66071 q^{85} -2.23487i q^{87} -0.338033i q^{89} -7.88610 q^{91} +0.377839 q^{93} +6.78607i q^{95} -11.0882i q^{97} +1.96967 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	3.12483			1.18108			0.590538	−	0.807010i	$-0.298916\pi$
$7$	3.12483			1.18108			0.590538	+	0.807010i	$0.298916\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.96967			−0.593878			−0.296939	−	0.954896i	$-0.595966\pi$
$11$	−1.96967			−0.593878			−0.296939	+	0.954896i	$0.595966\pi$
$12$		0			0
$13$	−2.52369			−0.699945			−0.349972	−	0.936760i	$-0.613809\pi$
$13$	−2.52369			−0.699945			−0.349972	+	0.936760i	$0.613809\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$			3.66071i			0.887854i	0.896063	+	0.443927i	$0.146415\pi$
$17$			3.66071i			0.887854i	−0.896063	+	0.443927i	$0.853585\pi$
$18$		0			0
$19$	−6.78607			−1.55683			−0.778415	−	0.627750i	$-0.783976\pi$
$19$	−6.78607			−1.55683			−0.778415	+	0.627750i	$0.783976\pi$
$20$		0			0
$21$		−	3.12483i		−	0.681895i
$22$		0			0
$23$	2.20899	+	4.25680i	0.460606	+	0.887605i
$23$	2.20899	+	4.25680i	0.460606	+	0.887605i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	2.23487			0.415005			0.207503	−	0.978234i	$-0.433466\pi$
$29$	2.23487			0.415005			0.207503	+	0.978234i	$0.433466\pi$
$30$		0			0
$31$			0.377839i			0.0678619i	0.999424	+	0.0339309i	$0.0108026\pi$
$31$			0.377839i			0.0678619i	−0.999424	+	0.0339309i	$0.989197\pi$
$32$		0			0
$33$			1.96967i			0.342875i
$34$		0			0
$35$		−	3.12483i		−	0.528193i
$36$		0			0
$37$			5.26726i			0.865933i	0.901410	+	0.432966i	$0.142533\pi$
$37$			5.26726i			0.865933i	−0.901410	+	0.432966i	$0.857467\pi$
$38$		0			0
$39$			2.52369i			0.404113i
$40$		0			0
$41$	5.40401			0.843965			0.421982	−	0.906604i	$-0.361334\pi$
$41$	5.40401			0.843965			0.421982	+	0.906604i	$0.361334\pi$
$42$		0			0
$43$	12.5558			1.91475			0.957375	−	0.288849i	$-0.0932727\pi$
$43$	12.5558			1.91475			0.957375	+	0.288849i	$0.0932727\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$			1.35089i			0.197047i	0.995135	+	0.0985237i	$0.0314120\pi$
$47$			1.35089i			0.197047i	−0.995135	+	0.0985237i	$0.968588\pi$
$48$		0			0
$49$	2.76459			0.394941
$50$		0			0
$51$	3.66071			0.512603
$52$		0			0
$53$		−	3.17126i		−	0.435606i	−0.975993	−	0.217803i	$-0.930111\pi$
$53$		−	3.17126i		−	0.435606i	0.975993	−	0.217803i	$-0.0698891\pi$
$54$		0			0
$55$			1.96967i			0.265590i
$56$		0			0
$57$			6.78607i			0.898837i
$58$		0			0
$59$		−	11.9962i		−	1.56177i	−0.624675	−	0.780885i	$-0.714769\pi$
$59$		−	11.9962i		−	1.56177i	0.624675	−	0.780885i	$-0.285231\pi$
$60$		0			0
$61$			4.62698i			0.592424i	0.955122	+	0.296212i	$0.0957235\pi$
$61$			4.62698i			0.592424i	−0.955122	+	0.296212i	$0.904276\pi$
$62$		0			0
$63$	−3.12483			−0.393692
$64$		0			0
$65$			2.52369i			0.313025i
$66$		0			0
$67$	10.4936			1.28200			0.640998	−	0.767543i	$-0.278521\pi$
$67$	10.4936			1.28200			0.640998	+	0.767543i	$0.278521\pi$
$68$		0			0
$69$	4.25680	−	2.20899i	0.512459	−	0.265931i
$70$		0			0
$71$			15.1992i			1.80381i	0.431935	+	0.901905i	$0.357831\pi$
$71$			15.1992i			1.80381i	−0.431935	+	0.901905i	$0.642169\pi$
$72$		0			0
$73$	13.4156			1.57017			0.785087	−	0.619385i	$-0.212618\pi$
$73$	13.4156			1.57017			0.785087	+	0.619385i	$0.212618\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	−6.15489			−0.701415
$78$		0			0
$79$	5.38677			0.606059			0.303029	−	0.952981i	$-0.402002\pi$
$79$	5.38677			0.606059			0.303029	+	0.952981i	$0.402002\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	14.7579			1.61989			0.809943	−	0.586508i	$-0.199498\pi$
$83$	14.7579			1.61989			0.809943	+	0.586508i	$0.199498\pi$
$84$		0			0
$85$	3.66071			0.397060
$86$		0			0
$87$		−	2.23487i		−	0.239604i
$88$		0			0
$89$		−	0.338033i		−	0.0358315i	−0.999840	−	0.0179157i	$-0.994297\pi$
$89$		−	0.338033i		−	0.0358315i	0.999840	−	0.0179157i	$-0.00570306\pi$
$90$		0			0
$91$	−7.88610			−0.826688
$92$		0			0
$93$	0.377839			0.0391801
$94$		0			0
$95$			6.78607i			0.696236i
$96$		0			0
$97$		−	11.0882i		−	1.12584i	−0.826512	−	0.562919i	$-0.809678\pi$
$97$		−	11.0882i		−	1.12584i	0.826512	−	0.562919i	$-0.190322\pi$
$98$		0			0
$99$	1.96967			0.197959
$100$		0			0
$101$	6.64873			0.661573			0.330786	−	0.943706i	$-0.392686\pi$
$101$	6.64873			0.661573			0.330786	+	0.943706i	$0.392686\pi$
$102$		0			0
$103$	0.583656			0.0575093			0.0287547	−	0.999586i	$-0.490846\pi$
$103$	0.583656			0.0575093			0.0287547	+	0.999586i	$0.490846\pi$
$104$		0			0
$105$	−3.12483			−0.304953
$106$		0			0
$107$	2.04526			0.197723			0.0988615	−	0.995101i	$-0.468480\pi$
$107$	2.04526			0.197723			0.0988615	+	0.995101i	$0.468480\pi$
$108$		0			0
$109$		−	15.7438i		−	1.50798i	−0.656884	−	0.753992i	$-0.728126\pi$
$109$		−	15.7438i		−	1.50798i	0.656884	−	0.753992i	$-0.271874\pi$
$110$		0			0
$111$	5.26726			0.499946
$112$		0			0
$113$			12.6691i			1.19180i	0.803057	+	0.595902i	$0.203205\pi$
$113$			12.6691i			1.19180i	−0.803057	+	0.595902i	$0.796795\pi$
$114$		0			0
$115$	4.25680	−	2.20899i	0.396949	−	0.205989i
$116$		0			0
$117$	2.52369			0.233315
$118$		0			0
$119$			11.4391i			1.04862i
$120$		0			0
$121$	−7.12040			−0.647309
$122$		0			0
$123$		−	5.40401i		−	0.487263i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			21.2834i			1.88859i	0.329093	+	0.944297i	$0.393257\pi$
$127$			21.2834i			1.88859i	−0.329093	+	0.944297i	$0.606743\pi$
$128$		0			0
$129$		−	12.5558i		−	1.10548i
$130$		0			0
$131$			13.9512i			1.21892i	0.792816	+	0.609461i	$0.208614\pi$
$131$			13.9512i			1.21892i	−0.792816	+	0.609461i	$0.791386\pi$
$132$		0			0
$133$	−21.2053			−1.83874
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$		−	2.32226i		−	0.198404i	−0.995067	−	0.0992021i	$-0.968371\pi$
$137$		−	2.32226i		−	0.198404i	0.995067	−	0.0992021i	$-0.0316290\pi$
$138$		0			0
$139$		−	15.4464i		−	1.31014i	−0.755567	−	0.655071i	$-0.772639\pi$
$139$		−	15.4464i		−	1.31014i	0.755567	−	0.655071i	$-0.227361\pi$
$140$		0			0
$141$	1.35089			0.113765
$142$		0			0
$143$	4.97083			0.415682
$144$		0			0
$145$		−	2.23487i		−	0.185596i
$146$		0			0
$147$		−	2.76459i		−	0.228019i
$148$		0			0
$149$		−	4.84162i		−	0.396641i	−0.980137	−	0.198320i	$-0.936451\pi$
$149$		−	4.84162i		−	0.396641i	0.980137	−	0.198320i	$-0.0635487\pi$
$150$		0			0
$151$		−	19.8598i		−	1.61617i	−0.589069	−	0.808083i	$-0.700505\pi$
$151$		−	19.8598i		−	1.61617i	0.589069	−	0.808083i	$-0.299495\pi$
$152$		0			0
$153$		−	3.66071i		−	0.295951i
$154$		0			0
$155$	0.377839			0.0303488
$156$		0			0
$157$		−	16.2502i		−	1.29691i	−0.761255	−	0.648453i	$-0.775416\pi$
$157$		−	16.2502i		−	1.29691i	0.761255	−	0.648453i	$-0.224584\pi$
$158$		0			0
$159$	−3.17126			−0.251497
$160$		0			0
$161$	6.90273	+	13.3018i	0.544011	+	1.04833i
$162$		0			0
$163$			21.5017i			1.68415i	0.539363	+	0.842073i	$0.318665\pi$
$163$			21.5017i			1.68415i	−0.539363	+	0.842073i	$0.681335\pi$
$164$		0			0
$165$	1.96967			0.153339
$166$		0			0
$167$			13.7046i			1.06049i	0.847844	+	0.530245i	$0.177900\pi$
$167$			13.7046i			1.06049i	−0.847844	+	0.530245i	$0.822100\pi$
$168$		0			0
$169$	−6.63101			−0.510078
$170$		0			0
$171$	6.78607			0.518944
$172$		0			0
$173$	−12.6695			−0.963244			−0.481622	−	0.876379i	$-0.659952\pi$
$173$	−12.6695			−0.963244			−0.481622	+	0.876379i	$0.659952\pi$
$174$		0			0
$175$	−3.12483			−0.236215
$176$		0			0
$177$	−11.9962			−0.901688
$178$		0			0
$179$			19.8164i			1.48115i	0.671974	+	0.740575i	$0.265447\pi$
$179$			19.8164i			1.48115i	−0.671974	+	0.740575i	$0.734553\pi$
$180$		0			0
$181$		−	2.00943i		−	0.149360i	−0.997208	−	0.0746801i	$-0.976206\pi$
$181$		−	2.00943i		−	0.149360i	0.997208	−	0.0746801i	$-0.0237936\pi$
$182$		0			0
$183$	4.62698			0.342036
$184$		0			0
$185$	5.26726			0.387257
$186$		0			0
$187$		−	7.21040i		−	0.527277i
$188$		0			0
$189$			3.12483i			0.227298i
$190$		0			0
$191$	18.8616			1.36478			0.682388	−	0.730990i	$-0.260941\pi$
$191$	18.8616			1.36478			0.682388	+	0.730990i	$0.260941\pi$
$192$		0			0
$193$	−25.8785			−1.86277			−0.931387	−	0.364030i	$-0.881401\pi$
$193$	−25.8785			−1.86277			−0.931387	+	0.364030i	$0.881401\pi$
$194$		0			0
$195$	2.52369			0.180725
$196$		0			0
$197$	11.3975			0.812040			0.406020	−	0.913864i	$-0.366916\pi$
$197$	11.3975			0.812040			0.406020	+	0.913864i	$0.366916\pi$
$198$		0			0
$199$	10.1484			0.719398			0.359699	−	0.933068i	$-0.382879\pi$
$199$	10.1484			0.719398			0.359699	+	0.933068i	$0.382879\pi$
$200$		0			0
$201$		−	10.4936i		−	0.740160i
$202$		0			0
$203$	6.98361			0.490153
$204$		0			0
$205$		−	5.40401i		−	0.377432i
$206$		0			0
$207$	−2.20899	−	4.25680i	−0.153535	−	0.295868i
$208$		0			0
$209$	13.3663			0.924567
$210$		0			0
$211$			4.20971i			0.289808i	0.989446	+	0.144904i	$0.0462874\pi$
$211$			4.20971i			0.289808i	−0.989446	+	0.144904i	$0.953713\pi$
$212$		0			0
$213$	15.1992			1.04143
$214$		0			0
$215$		−	12.5558i		−	0.856302i
$216$		0			0
$217$			1.18068i			0.0801501i
$218$		0			0
$219$		−	13.4156i		−	0.906541i
$220$		0			0
$221$		−	9.23850i		−	0.621448i
$222$		0			0
$223$		−	20.2218i		−	1.35415i	−0.735912	−	0.677077i	$-0.763246\pi$
$223$		−	20.2218i		−	1.35415i	0.735912	−	0.677077i	$-0.236754\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−29.8101			−1.97857			−0.989284	−	0.146005i	$-0.953359\pi$
$227$	−29.8101			−1.97857			−0.989284	+	0.146005i	$0.953359\pi$
$228$		0			0
$229$		−	12.3772i		−	0.817909i	−0.912555	−	0.408955i	$-0.865893\pi$
$229$		−	12.3772i		−	0.817909i	0.912555	−	0.408955i	$-0.134107\pi$
$230$		0			0
$231$			6.15489i			0.404962i
$232$		0			0
$233$	−13.5820			−0.889789			−0.444895	−	0.895583i	$-0.646759\pi$
$233$	−13.5820			−0.889789			−0.444895	+	0.895583i	$0.646759\pi$
$234$		0			0
$235$	1.35089			0.0881222
$236$		0			0
$237$		−	5.38677i		−	0.349908i
$238$		0			0
$239$			8.77772i			0.567784i	0.958856	+	0.283892i	$0.0916257\pi$
$239$			8.77772i			0.567784i	−0.958856	+	0.283892i	$0.908374\pi$
$240$		0			0
$241$			13.8466i			0.891935i	0.895049	+	0.445968i	$0.147140\pi$
$241$			13.8466i			0.891935i	−0.895049	+	0.445968i	$0.852860\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$		−	2.76459i		−	0.176623i
$246$		0			0
$247$	17.1259			1.08970
$248$		0			0
$249$		−	14.7579i		−	0.935242i
$250$		0			0
$251$	26.2244			1.65527			0.827635	−	0.561267i	$-0.189686\pi$
$251$	26.2244			1.65527			0.827635	+	0.561267i	$0.189686\pi$
$252$		0			0
$253$	−4.35098	−	8.38449i	−0.273544	−	0.527129i
$254$		0			0
$255$		−	3.66071i		−	0.229243i
$256$		0			0
$257$	−3.73811			−0.233177			−0.116588	−	0.993180i	$-0.537196\pi$
$257$	−3.73811			−0.233177			−0.116588	+	0.993180i	$0.537196\pi$
$258$		0			0
$259$			16.4593i			1.02273i
$260$		0			0
$261$	−2.23487			−0.138335
$262$		0			0
$263$	19.3159			1.19107			0.595536	−	0.803328i	$-0.296940\pi$
$263$	19.3159			1.19107			0.595536	+	0.803328i	$0.296940\pi$
$264$		0			0
$265$	−3.17126			−0.194809
$266$		0			0
$267$	−0.338033			−0.0206873
$268$		0			0
$269$	−32.5176			−1.98264			−0.991318	−	0.131489i	$-0.958024\pi$
$269$	−32.5176			−1.98264			−0.991318	+	0.131489i	$0.958024\pi$
$270$		0			0
$271$			19.1381i			1.16255i	0.813706	+	0.581277i	$0.197447\pi$
$271$			19.1381i			1.16255i	−0.813706	+	0.581277i	$0.802553\pi$
$272$		0			0
$273$			7.88610i			0.477289i
$274$		0			0
$275$	1.96967			0.118776
$276$		0			0
$277$	9.32487			0.560277			0.280139	−	0.959960i	$-0.409620\pi$
$277$	9.32487			0.560277			0.280139	+	0.959960i	$0.409620\pi$
$278$		0			0
$279$		−	0.377839i		−	0.0226206i
$280$		0			0
$281$			12.4885i			0.745001i	0.928032	+	0.372500i	$0.121499\pi$
$281$			12.4885i			0.745001i	−0.928032	+	0.372500i	$0.878501\pi$
$282$		0			0
$283$	−10.1536			−0.603567			−0.301783	−	0.953377i	$-0.597582\pi$
$283$	−10.1536			−0.603567			−0.301783	+	0.953377i	$0.597582\pi$
$284$		0			0
$285$	6.78607			0.401972
$286$		0			0
$287$	16.8866			0.996787
$288$		0			0
$289$	3.59917			0.211716
$290$		0			0
$291$	−11.0882			−0.650003
$292$		0			0
$293$			24.8954i			1.45440i	0.686424	+	0.727201i	$0.259179\pi$
$293$			24.8954i			1.45440i	−0.686424	+	0.727201i	$0.740821\pi$
$294$		0			0
$295$	−11.9962			−0.698444
$296$		0			0
$297$		−	1.96967i		−	0.114292i
$298$		0			0
$299$	−5.57480	−	10.7428i	−0.322399	−	0.621274i
$300$		0			0
$301$	39.2349			2.26146
$302$		0			0
$303$		−	6.64873i		−	0.381959i
$304$		0			0
$305$	4.62698			0.264940
$306$		0			0
$307$			14.7305i			0.840712i	0.907359	+	0.420356i	$0.138095\pi$
$307$			14.7305i			0.840712i	−0.907359	+	0.420356i	$0.861905\pi$
$308$		0			0
$309$		−	0.583656i		−	0.0332030i
$310$		0			0
$311$		−	2.02347i		−	0.114740i	−0.998353	−	0.0573702i	$-0.981728\pi$
$311$		−	2.02347i		−	0.114740i	0.998353	−	0.0573702i	$-0.0182715\pi$
$312$		0			0
$313$		−	19.6539i		−	1.11091i	−0.831548	−	0.555453i	$-0.812545\pi$
$313$		−	19.6539i		−	1.11091i	0.831548	−	0.555453i	$-0.187455\pi$
$314$		0			0
$315$			3.12483i			0.176064i
$316$		0			0
$317$	0.0518500			0.00291219			0.00145609	−	0.999999i	$-0.499537\pi$
$317$	0.0518500			0.00291219			0.00145609	+	0.999999i	$0.499537\pi$
$318$		0			0
$319$	−4.40196			−0.246463
$320$		0			0
$321$		−	2.04526i		−	0.114155i
$322$		0			0
$323$		−	24.8419i		−	1.38224i
$324$		0			0
$325$	2.52369			0.139989
$326$		0			0
$327$	−15.7438			−0.870635
$328$		0			0
$329$			4.22130i			0.232728i
$330$		0			0
$331$			3.22601i			0.177317i	0.996062	+	0.0886586i	$0.0282580\pi$
$331$			3.22601i			0.177317i	−0.996062	+	0.0886586i	$0.971742\pi$
$332$		0			0
$333$		−	5.26726i		−	0.288644i
$334$		0			0
$335$		−	10.4936i		−	0.573326i
$336$		0			0
$337$		−	32.7479i		−	1.78389i	−0.452141	−	0.891946i	$-0.649340\pi$
$337$		−	32.7479i		−	1.78389i	0.452141	−	0.891946i	$-0.350660\pi$
$338$		0			0
$339$	12.6691			0.688089
$340$		0			0
$341$		−	0.744218i		−	0.0403017i
$342$		0			0
$343$	−13.2350			−0.714621
$344$		0			0
$345$	−2.20899	−	4.25680i	−0.118928	−	0.229179i
$346$		0			0
$347$			24.5999i			1.32059i	0.751006	+	0.660295i	$0.229569\pi$
$347$			24.5999i			1.32059i	−0.751006	+	0.660295i	$0.770431\pi$
$348$		0			0
$349$	29.6353			1.58634			0.793171	−	0.608999i	$-0.208429\pi$
$349$	29.6353			1.58634			0.793171	+	0.608999i	$0.208429\pi$
$350$		0			0
$351$		−	2.52369i		−	0.134704i
$352$		0			0
$353$	31.6740			1.68583			0.842917	−	0.538043i	$-0.180836\pi$
$353$	31.6740			1.68583			0.842917	+	0.538043i	$0.180836\pi$
$354$		0			0
$355$	15.1992			0.806688
$356$		0			0
$357$	11.4391			0.605423
$358$		0			0
$359$	4.92427			0.259893			0.129947	−	0.991521i	$-0.458519\pi$
$359$	4.92427			0.259893			0.129947	+	0.991521i	$0.458519\pi$
$360$		0			0
$361$	27.0507			1.42372
$362$		0			0
$363$			7.12040i			0.373724i
$364$		0			0
$365$		−	13.4156i		−	0.702203i
$366$		0			0
$367$	19.7214			1.02945			0.514725	−	0.857355i	$-0.327894\pi$
$367$	19.7214			1.02945			0.514725	+	0.857355i	$0.327894\pi$
$368$		0			0
$369$	−5.40401			−0.281322
$370$		0			0
$371$		−	9.90967i		−	0.514484i
$372$		0			0
$373$		−	6.38425i		−	0.330564i	−0.986246	−	0.165282i	$-0.947147\pi$
$373$		−	6.38425i		−	0.330564i	0.986246	−	0.165282i	$-0.0528534\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	−5.64012			−0.290481
$378$		0			0
$379$	2.70427			0.138909			0.0694546	−	0.997585i	$-0.477874\pi$
$379$	2.70427			0.138909			0.0694546	+	0.997585i	$0.477874\pi$
$380$		0			0
$381$	21.2834			1.09038
$382$		0			0
$383$	30.4327			1.55504			0.777519	−	0.628860i	$-0.216478\pi$
$383$	30.4327			1.55504			0.777519	+	0.628860i	$0.216478\pi$
$384$		0			0
$385$			6.15489i			0.313682i
$386$		0			0
$387$	−12.5558			−0.638250
$388$		0			0
$389$			24.3346i			1.23381i	0.787037	+	0.616906i	$0.211614\pi$
$389$			24.3346i			1.23381i	−0.787037	+	0.616906i	$0.788386\pi$
$390$		0			0
$391$	−15.5829	+	8.08648i	−0.788063	+	0.408951i
$392$		0			0
$393$	13.9512			0.703745
$394$		0			0
$395$		−	5.38677i		−	0.271038i
$396$		0			0
$397$	9.96983			0.500371			0.250186	−	0.968198i	$-0.419508\pi$
$397$	9.96983			0.500371			0.250186	+	0.968198i	$0.419508\pi$
$398$		0			0
$399$			21.2053i			1.06159i
$400$		0			0
$401$			14.6150i			0.729839i	0.931039	+	0.364920i	$0.118904\pi$
$401$			14.6150i			0.729839i	−0.931039	+	0.364920i	$0.881096\pi$
$402$		0			0
$403$		−	0.953547i		−	0.0474996i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$		−	10.3748i		−	0.514258i
$408$		0			0
$409$	5.77841			0.285724			0.142862	−	0.989743i	$-0.454369\pi$
$409$	5.77841			0.285724			0.142862	+	0.989743i	$0.454369\pi$
$410$		0			0
$411$	−2.32226			−0.114549
$412$		0			0
$413$		−	37.4861i		−	1.84457i
$414$		0			0
$415$		−	14.7579i		−	0.724435i
$416$		0			0
$417$	−15.4464			−0.756411
$418$		0			0
$419$	25.4826			1.24491			0.622453	−	0.782657i	$-0.286136\pi$
$419$	25.4826			1.24491			0.622453	+	0.782657i	$0.286136\pi$
$420$		0			0
$421$		−	8.35143i		−	0.407024i	−0.979072	−	0.203512i	$-0.934764\pi$
$421$		−	8.35143i		−	0.407024i	0.979072	−	0.203512i	$-0.0652356\pi$
$422$		0			0
$423$		−	1.35089i		−	0.0656824i
$424$		0			0
$425$		−	3.66071i		−	0.177571i
$426$		0			0
$427$			14.4585i			0.699698i
$428$		0			0
$429$		−	4.97083i		−	0.239994i
$430$		0			0
$431$	−2.81003			−0.135354			−0.0676771	−	0.997707i	$-0.521559\pi$
$431$	−2.81003			−0.135354			−0.0676771	+	0.997707i	$0.521559\pi$
$432$		0			0
$433$			21.5271i			1.03453i	0.855826	+	0.517263i	$0.173049\pi$
$433$			21.5271i			1.03453i	−0.855826	+	0.517263i	$0.826951\pi$
$434$		0			0
$435$	−2.23487			−0.107154
$436$		0			0
$437$	−14.9904	−	28.8870i	−0.717086	−	1.38185i
$438$		0			0
$439$			14.5927i			0.696471i	0.937407	+	0.348235i	$0.113219\pi$
$439$			14.5927i			0.696471i	−0.937407	+	0.348235i	$0.886781\pi$
$440$		0			0
$441$	−2.76459			−0.131647
$442$		0			0
$443$		−	19.8921i		−	0.945101i	−0.881304	−	0.472550i	$-0.843333\pi$
$443$		−	19.8921i		−	0.945101i	0.881304	−	0.472550i	$-0.156667\pi$
$444$		0			0
$445$	−0.338033			−0.0160243
$446$		0			0
$447$	−4.84162			−0.229001
$448$		0			0
$449$	−20.8817			−0.985470			−0.492735	−	0.870180i	$-0.664003\pi$
$449$	−20.8817			−0.985470			−0.492735	+	0.870180i	$0.664003\pi$
$450$		0			0
$451$	−10.6441			−0.501212
$452$		0			0
$453$	−19.8598			−0.933094
$454$		0			0
$455$			7.88610i			0.369706i
$456$		0			0
$457$			2.71008i			0.126772i	0.997989	+	0.0633861i	$0.0201900\pi$
$457$			2.71008i			0.126772i	−0.997989	+	0.0633861i	$0.979810\pi$
$458$		0			0
$459$	−3.66071			−0.170868
$460$		0			0
$461$	−23.9568			−1.11578			−0.557889	−	0.829915i	$-0.688389\pi$
$461$	−23.9568			−1.11578			−0.557889	+	0.829915i	$0.688389\pi$
$462$		0			0
$463$		−	26.9255i		−	1.25133i	−0.780090	−	0.625667i	$-0.784827\pi$
$463$		−	26.9255i		−	1.25133i	0.780090	−	0.625667i	$-0.215173\pi$
$464$		0			0
$465$		−	0.377839i		−	0.0175219i
$466$		0			0
$467$	−10.6351			−0.492134			−0.246067	−	0.969253i	$-0.579138\pi$
$467$	−10.6351			−0.492134			−0.246067	+	0.969253i	$0.579138\pi$
$468$		0			0
$469$	32.7907			1.51413
$470$		0			0
$471$	−16.2502			−0.748769
$472$		0			0
$473$	−24.7309			−1.13713
$474$		0			0
$475$	6.78607			0.311366
$476$		0			0
$477$			3.17126i			0.145202i
$478$		0			0
$479$	32.4647			1.48335			0.741676	−	0.670759i	$-0.234031\pi$
$479$	32.4647			1.48335			0.741676	+	0.670759i	$0.234031\pi$
$480$		0			0
$481$		−	13.2929i		−	0.606105i
$482$		0			0
$483$	13.3018	−	6.90273i	0.605253	−	0.314085i
$484$		0			0
$485$	−11.0882			−0.503490
$486$		0			0
$487$			24.0618i			1.09034i	0.838324	+	0.545172i	$0.183536\pi$
$487$			24.0618i			1.09034i	−0.838324	+	0.545172i	$0.816464\pi$
$488$		0			0
$489$	21.5017			0.972343
$490$		0			0
$491$		−	0.570628i		−	0.0257521i	−0.999917	−	0.0128760i	$-0.995901\pi$
$491$		−	0.570628i		−	0.0257521i	0.999917	−	0.0128760i	$-0.00409868\pi$
$492$		0			0
$493$			8.18123i			0.368464i
$494$		0			0
$495$		−	1.96967i		−	0.0885301i
$496$		0			0
$497$			47.4949i			2.13044i
$498$		0			0
$499$		−	22.2429i		−	0.995727i	−0.867255	−	0.497864i	$-0.834118\pi$
$499$		−	22.2429i		−	0.995727i	0.867255	−	0.497864i	$-0.165882\pi$
$500$		0			0
$501$	13.7046			0.612275
$502$		0			0
$503$	32.1596			1.43392			0.716962	−	0.697112i	$-0.245532\pi$
$503$	32.1596			1.43392			0.716962	+	0.697112i	$0.245532\pi$
$504$		0			0
$505$		−	6.64873i		−	0.295864i
$506$		0			0
$507$			6.63101i			0.294493i
$508$		0			0
$509$	5.93120			0.262896			0.131448	−	0.991323i	$-0.458037\pi$
$509$	5.93120			0.262896			0.131448	+	0.991323i	$0.458037\pi$
$510$		0			0
$511$	41.9215			1.85450
$512$		0			0
$513$		−	6.78607i		−	0.299612i
$514$		0			0
$515$		−	0.583656i		−	0.0257189i
$516$		0			0
$517$		−	2.66080i		−	0.117022i
$518$		0			0
$519$			12.6695i			0.556129i
$520$		0			0
$521$		−	0.545006i		−	0.0238771i	−0.999929	−	0.0119386i	$-0.996200\pi$
$521$		−	0.545006i		−	0.0238771i	0.999929	−	0.0119386i	$-0.00380025\pi$
$522$		0			0
$523$	−32.0697			−1.40231			−0.701155	−	0.713009i	$-0.747332\pi$
$523$	−32.0697			−1.40231			−0.701155	+	0.713009i	$0.747332\pi$
$524$		0			0
$525$			3.12483i			0.136379i
$526$		0			0
$527$	−1.38316			−0.0602514
$528$		0			0
$529$	−13.2407	+	18.8065i	−0.575684	+	0.817672i
$530$		0			0
$531$			11.9962i			0.520590i
$532$		0			0
$533$	−13.6380			−0.590728
$534$		0			0
$535$		−	2.04526i		−	0.0884244i
$536$		0			0
$537$	19.8164			0.855142
$538$		0			0
$539$	−5.44533			−0.234547
$540$		0			0
$541$	−36.4749			−1.56818			−0.784089	−	0.620648i	$-0.786870\pi$
$541$	−36.4749			−1.56818			−0.784089	+	0.620648i	$0.786870\pi$
$542$		0			0
$543$	−2.00943			−0.0862331
$544$		0			0
$545$	−15.7438			−0.674391
$546$		0			0
$547$		−	0.286412i		−	0.0122461i	−0.999981	−	0.00612304i	$-0.998051\pi$
$547$		−	0.286412i		−	0.0122461i	0.999981	−	0.00612304i	$-0.00194904\pi$
$548$		0			0
$549$		−	4.62698i		−	0.197475i
$550$		0			0
$551$	−15.1660			−0.646093
$552$		0			0
$553$	16.8328			0.715802
$554$		0			0
$555$		−	5.26726i		−	0.223583i
$556$		0			0
$557$			7.77446i			0.329414i	0.986343	+	0.164707i	$0.0526679\pi$
$557$			7.77446i			0.329414i	−0.986343	+	0.164707i	$0.947332\pi$
$558$		0			0
$559$	−31.6870			−1.34022
$560$		0			0
$561$	−7.21040			−0.304423
$562$		0			0
$563$	−13.1915			−0.555955			−0.277977	−	0.960588i	$-0.589664\pi$
$563$	−13.1915			−0.555955			−0.277977	+	0.960588i	$0.589664\pi$
$564$		0			0
$565$	12.6691			0.532991
$566$		0			0
$567$	3.12483			0.131231
$568$		0			0
$569$		−	18.9198i		−	0.793159i	−0.918000	−	0.396580i	$-0.870197\pi$
$569$		−	18.9198i		−	0.793159i	0.918000	−	0.396580i	$-0.129803\pi$
$570$		0			0
$571$	2.35834			0.0986933			0.0493466	−	0.998782i	$-0.484286\pi$
$571$	2.35834			0.0986933			0.0493466	+	0.998782i	$0.484286\pi$
$572$		0			0
$573$		−	18.8616i		−	0.787954i
$574$		0			0
$575$	−2.20899	−	4.25680i	−0.0921212	−	0.177521i
$576$		0			0
$577$	39.1515			1.62990			0.814949	−	0.579533i	$-0.196765\pi$
$577$	39.1515			1.62990			0.814949	+	0.579533i	$0.196765\pi$
$578$		0			0
$579$			25.8785i			1.07547i
$580$		0			0
$581$	46.1159			1.91321
$582$		0			0
$583$			6.24634i			0.258697i
$584$		0			0
$585$		−	2.52369i		−	0.104342i
$586$		0			0
$587$		−	28.8938i		−	1.19258i	−0.802771	−	0.596288i	$-0.796642\pi$
$587$		−	28.8938i		−	1.19258i	0.802771	−	0.596288i	$-0.203358\pi$
$588$		0			0
$589$		−	2.56404i		−	0.105649i
$590$		0			0
$591$		−	11.3975i		−	0.468832i
$592$		0			0
$593$	6.95439			0.285583			0.142791	−	0.989753i	$-0.454392\pi$
$593$	6.95439			0.285583			0.142791	+	0.989753i	$0.454392\pi$
$594$		0			0
$595$	11.4391			0.468958
$596$		0			0
$597$		−	10.1484i		−	0.415344i
$598$		0			0
$599$			39.5319i			1.61523i	0.589711	+	0.807614i	$0.299242\pi$
$599$			39.5319i			1.61523i	−0.589711	+	0.807614i	$0.700758\pi$
$600$		0			0
$601$	23.3570			0.952751			0.476376	−	0.879242i	$-0.341950\pi$
$601$	23.3570			0.952751			0.476376	+	0.879242i	$0.341950\pi$
$602$		0			0
$603$	−10.4936			−0.427332
$604$		0			0
$605$			7.12040i			0.289485i
$606$		0			0
$607$			6.94207i			0.281770i	0.990026	+	0.140885i	$0.0449947\pi$
$607$			6.94207i			0.281770i	−0.990026	+	0.140885i	$0.955005\pi$
$608$		0			0
$609$		−	6.98361i		−	0.282990i
$610$		0			0
$611$		−	3.40922i		−	0.137922i
$612$		0			0
$613$		−	48.1058i		−	1.94298i	−0.237089	−	0.971488i	$-0.576193\pi$
$613$		−	48.1058i		−	1.94298i	0.237089	−	0.971488i	$-0.423807\pi$
$614$		0			0
$615$	−5.40401			−0.217911
$616$		0			0
$617$		−	4.62309i		−	0.186119i	−0.995661	−	0.0930593i	$-0.970335\pi$
$617$		−	4.62309i		−	0.186119i	0.995661	−	0.0930593i	$-0.0296646\pi$
$618$		0			0
$619$	−34.8837			−1.40210			−0.701048	−	0.713114i	$-0.747284\pi$
$619$	−34.8837			−1.40210			−0.701048	+	0.713114i	$0.747284\pi$
$620$		0			0
$621$	−4.25680	+	2.20899i	−0.170820	+	0.0886437i
$622$		0			0
$623$		−	1.05630i		−	0.0423197i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	13.3663i		−	0.533799i
$628$		0			0
$629$	−19.2819			−0.768821
$630$		0			0
$631$	−1.52455			−0.0606915			−0.0303458	−	0.999539i	$-0.509661\pi$
$631$	−1.52455			−0.0606915			−0.0303458	+	0.999539i	$0.509661\pi$
$632$		0			0
$633$	4.20971			0.167321
$634$		0			0
$635$	21.2834			0.844605
$636$		0			0
$637$	−6.97695			−0.276437
$638$		0			0
$639$		−	15.1992i		−	0.601270i
$640$		0			0
$641$			33.0371i			1.30489i	0.757837	+	0.652443i	$0.226256\pi$
$641$			33.0371i			1.30489i	−0.757837	+	0.652443i	$0.773744\pi$
$642$		0			0
$643$	10.4205			0.410943			0.205472	−	0.978663i	$-0.434127\pi$
$643$	10.4205			0.410943			0.205472	+	0.978663i	$0.434127\pi$
$644$		0			0
$645$	−12.5558			−0.494386
$646$		0			0
$647$		−	6.28523i		−	0.247098i	−0.992338	−	0.123549i	$-0.960572\pi$
$647$		−	6.28523i		−	0.247098i	0.992338	−	0.123549i	$-0.0394276\pi$
$648$		0			0
$649$			23.6285i			0.927500i
$650$		0			0
$651$	1.18068			0.0462747
$652$		0			0
$653$	21.7826			0.852419			0.426209	−	0.904625i	$-0.359849\pi$
$653$	21.7826			0.852419			0.426209	+	0.904625i	$0.359849\pi$
$654$		0			0
$655$	13.9512			0.545119
$656$		0			0
$657$	−13.4156			−0.523391
$658$		0			0
$659$	−18.2120			−0.709438			−0.354719	−	0.934973i	$-0.615424\pi$
$659$	−18.2120			−0.709438			−0.354719	+	0.934973i	$0.615424\pi$
$660$		0			0
$661$			17.9091i			0.696583i	0.937386	+	0.348291i	$0.113238\pi$
$661$			17.9091i			0.696583i	−0.937386	+	0.348291i	$0.886762\pi$
$662$		0			0
$663$	−9.23850			−0.358793
$664$		0			0
$665$			21.2053i			0.822308i
$666$		0			0
$667$	4.93681	+	9.51341i	0.191154	+	0.368361i
$668$		0			0
$669$	−20.2218			−0.781821
$670$		0			0
$671$		−	9.11362i		−	0.351828i
$672$		0			0
$673$	−40.4411			−1.55889			−0.779444	−	0.626471i	$-0.784499\pi$
$673$	−40.4411			−1.55889			−0.779444	+	0.626471i	$0.784499\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$			12.0827i			0.464375i	0.972671	+	0.232187i	$0.0745882\pi$
$677$			12.0827i			0.464375i	−0.972671	+	0.232187i	$0.925412\pi$
$678$		0			0
$679$		−	34.6489i		−	1.32970i
$680$		0			0
$681$			29.8101i			1.14233i
$682$		0			0
$683$			44.2598i			1.69356i	0.531947	+	0.846778i	$0.321461\pi$
$683$			44.2598i			1.69356i	−0.531947	+	0.846778i	$0.678539\pi$
$684$		0			0
$685$	−2.32226			−0.0887290
$686$		0			0
$687$	−12.3772			−0.472220
$688$		0			0
$689$			8.00327i			0.304900i
$690$		0			0
$691$			17.0262i			0.647705i	0.946108	+	0.323853i	$0.104978\pi$
$691$			17.0262i			0.647705i	−0.946108	+	0.323853i	$0.895022\pi$
$692$		0			0
$693$	6.15489			0.233805
$694$		0			0
$695$	−15.4464			−0.585914
$696$		0			0
$697$			19.7825i			0.749317i
$698$		0			0
$699$			13.5820i			0.513720i
$700$		0			0
$701$			15.2474i			0.575887i	0.957647	+	0.287943i	$0.0929715\pi$
$701$			15.2474i			0.575887i	−0.957647	+	0.287943i	$0.907029\pi$
$702$		0			0
$703$		−	35.7440i		−	1.34811i
$704$		0			0
$705$		−	1.35089i		−	0.0508774i
$706$		0			0
$707$	20.7762			0.781368
$708$		0			0
$709$		−	17.8601i		−	0.670751i	−0.942085	−	0.335375i	$-0.891137\pi$
$709$		−	17.8601i		−	0.670751i	0.942085	−	0.335375i	$-0.108863\pi$
$710$		0			0
$711$	−5.38677			−0.202020
$712$		0			0
$713$	−1.60839	+	0.834642i	−0.0602345	+	0.0312576i
$714$		0			0
$715$		−	4.97083i		−	0.185898i
$716$		0			0
$717$	8.77772			0.327810
$718$		0			0
$719$		−	11.5363i		−	0.430231i	−0.976589	−	0.215116i	$-0.930987\pi$
$719$		−	11.5363i		−	0.430231i	0.976589	−	0.215116i	$-0.0690129\pi$
$720$		0			0
$721$	1.82383			0.0679229
$722$		0			0
$723$	13.8466			0.514959
$724$		0			0
$725$	−2.23487			−0.0830011
$726$		0			0
$727$	0.409166			0.0151751			0.00758756	−	0.999971i	$-0.497585\pi$
$727$	0.409166			0.0151751			0.00758756	+	0.999971i	$0.497585\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			45.9634i			1.70002i
$732$		0			0
$733$		−	28.6370i		−	1.05773i	−0.848705	−	0.528866i	$-0.822617\pi$
$733$		−	28.6370i		−	1.05773i	0.848705	−	0.528866i	$-0.177383\pi$
$734$		0			0
$735$	−2.76459			−0.101973
$736$		0			0
$737$	−20.6689			−0.761348
$738$		0			0
$739$		−	6.08323i		−	0.223775i	−0.993721	−	0.111888i	$-0.964310\pi$
$739$		−	6.08323i		−	0.223775i	0.993721	−	0.111888i	$-0.0356897\pi$
$740$		0			0
$741$		−	17.1259i		−	0.629136i
$742$		0			0
$743$	−12.0715			−0.442860			−0.221430	−	0.975176i	$-0.571072\pi$
$743$	−12.0715			−0.442860			−0.221430	+	0.975176i	$0.571072\pi$
$744$		0			0
$745$	−4.84162			−0.177383
$746$		0			0
$747$	−14.7579			−0.539962
$748$		0			0
$749$	6.39111			0.233526
$750$		0			0
$751$	−49.1905			−1.79499			−0.897493	−	0.441029i	$-0.854614\pi$
$751$	−49.1905			−1.79499			−0.897493	+	0.441029i	$0.854614\pi$
$752$		0			0
$753$		−	26.2244i		−	0.955670i
$754$		0			0
$755$	−19.8598			−0.722771
$756$		0			0
$757$			21.0572i			0.765335i	0.923886	+	0.382668i	$0.124995\pi$
$757$			21.0572i			0.765335i	−0.923886	+	0.382668i	$0.875005\pi$
$758$		0			0
$759$	−8.38449	+	4.35098i	−0.304338	+	0.157931i
$760$		0			0
$761$	−44.4485			−1.61126			−0.805628	−	0.592422i	$-0.798172\pi$
$761$	−44.4485			−1.61126			−0.805628	+	0.592422i	$0.798172\pi$
$762$		0			0
$763$		−	49.1968i		−	1.78104i
$764$		0			0
$765$	−3.66071			−0.132353
$766$		0			0
$767$			30.2746i			1.09315i
$768$		0			0
$769$			27.8869i			1.00563i	0.864395	+	0.502814i	$0.167702\pi$
$769$			27.8869i			1.00563i	−0.864395	+	0.502814i	$0.832298\pi$
$770$		0			0
$771$			3.73811i			0.134625i
$772$		0			0
$773$		−	4.96861i		−	0.178709i	−0.996000	−	0.0893543i	$-0.971520\pi$
$773$		−	4.96861i		−	0.178709i	0.996000	−	0.0893543i	$-0.0284803\pi$
$774$		0			0
$775$		−	0.377839i		−	0.0135724i
$776$		0			0
$777$	16.4593			0.590475
$778$		0			0
$779$	−36.6720			−1.31391
$780$		0			0
$781$		−	29.9373i		−	1.07124i
$782$		0			0
$783$			2.23487i			0.0798678i
$784$		0			0
$785$	−16.2502			−0.579994
$786$		0			0
$787$	−21.1512			−0.753957			−0.376979	−	0.926222i	$-0.623037\pi$
$787$	−21.1512			−0.753957			−0.376979	+	0.926222i	$0.623037\pi$
$788$		0			0
$789$		−	19.3159i		−	0.687666i
$790$		0			0
$791$			39.5887i			1.40761i
$792$		0			0
$793$		−	11.6770i		−	0.414664i
$794$		0			0
$795$			3.17126i			0.112473i
$796$		0			0
$797$			4.80963i			0.170366i	0.996365	+	0.0851830i	$0.0271475\pi$
$797$			4.80963i			0.170366i	−0.996365	+	0.0851830i	$0.972853\pi$
$798$		0			0
$799$	−4.94522			−0.174949
$800$		0			0
$801$			0.338033i			0.0119438i
$802$		0			0
$803$	−26.4243			−0.932492
$804$		0			0
$805$	13.3018	−	6.90273i	0.468827	−	0.243289i
$806$		0			0
$807$			32.5176i			1.14468i
$808$		0			0
$809$	−29.7916			−1.04742			−0.523709	−	0.851897i	$-0.675452\pi$
$809$	−29.7916			−1.04742			−0.523709	+	0.851897i	$0.675452\pi$
$810$		0			0
$811$			36.0674i			1.26650i	0.773949	+	0.633249i	$0.218279\pi$
$811$			36.0674i			1.26650i	−0.773949	+	0.633249i	$0.781721\pi$
$812$		0			0
$813$	19.1381			0.671201
$814$		0			0
$815$	21.5017			0.753173
$816$		0			0
$817$	−85.2049			−2.98094
$818$		0			0
$819$	7.88610			0.275563
$820$		0			0
$821$	28.3189			0.988336			0.494168	−	0.869366i	$-0.335473\pi$
$821$	28.3189			0.988336			0.494168	+	0.869366i	$0.335473\pi$
$822$		0			0
$823$		−	11.8822i		−	0.414189i	−0.978321	−	0.207094i	$-0.933599\pi$
$823$		−	11.8822i		−	0.414189i	0.978321	−	0.207094i	$-0.0664007\pi$
$824$		0			0
$825$		−	1.96967i		−	0.0685751i
$826$		0			0
$827$	24.8754			0.865003			0.432501	−	0.901633i	$-0.357631\pi$
$827$	24.8754			0.865003			0.432501	+	0.901633i	$0.357631\pi$
$828$		0			0
$829$	−0.572815			−0.0198947			−0.00994734	−	0.999951i	$-0.503166\pi$
$829$	−0.572815			−0.0198947			−0.00994734	+	0.999951i	$0.503166\pi$
$830$		0			0
$831$		−	9.32487i		−	0.323476i
$832$		0			0
$833$			10.1204i			0.350650i
$834$		0			0
$835$	13.7046			0.474266
$836$		0			0
$837$	−0.377839			−0.0130600
$838$		0			0
$839$	32.7025			1.12902			0.564509	−	0.825427i	$-0.309066\pi$
$839$	32.7025			1.12902			0.564509	+	0.825427i	$0.309066\pi$
$840$		0			0
$841$	−24.0053			−0.827770
$842$		0			0
$843$	12.4885			0.430126
$844$		0			0
$845$			6.63101i			0.228114i
$846$		0			0
$847$	−22.2501			−0.764522
$848$		0			0
$849$			10.1536i			0.348469i
$850$		0			0
$851$	−22.4217	+	11.6353i	−0.768606	+	0.398854i
$852$		0			0
$853$	−27.5299			−0.942606			−0.471303	−	0.881971i	$-0.656216\pi$
$853$	−27.5299			−0.942606			−0.471303	+	0.881971i	$0.656216\pi$
$854$		0			0
$855$		−	6.78607i		−	0.232079i
$856$		0			0
$857$	−15.5228			−0.530249			−0.265125	−	0.964214i	$-0.585413\pi$
$857$	−15.5228			−0.530249			−0.265125	+	0.964214i	$0.585413\pi$
$858$		0			0
$859$			24.2757i			0.828277i	0.910214	+	0.414138i	$0.135917\pi$
$859$			24.2757i			0.828277i	−0.910214	+	0.414138i	$0.864083\pi$
$860$		0			0
$861$		−	16.8866i		−	0.575495i
$862$		0			0
$863$		−	47.7816i		−	1.62651i	−0.581910	−	0.813253i	$-0.697695\pi$
$863$		−	47.7816i		−	1.62651i	0.581910	−	0.813253i	$-0.302305\pi$
$864$		0			0
$865$			12.6695i			0.430776i
$866$		0			0
$867$		−	3.59917i		−	0.122234i
$868$		0			0
$869$	−10.6102			−0.359925
$870$		0			0
$871$	−26.4825			−0.897326
$872$		0			0
$873$			11.0882i			0.375279i
$874$		0			0
$875$			3.12483i			0.105639i
$876$		0			0
$877$	25.7834			0.870644			0.435322	−	0.900275i	$-0.356635\pi$
$877$	25.7834			0.870644			0.435322	+	0.900275i	$0.356635\pi$
$878$		0			0
$879$	24.8954			0.839700
$880$		0			0
$881$		−	9.85252i		−	0.331940i	−0.986131	−	0.165970i	$-0.946925\pi$
$881$		−	9.85252i		−	0.331940i	0.986131	−	0.165970i	$-0.0530754\pi$
$882$		0			0
$883$			2.69581i			0.0907212i	0.998971	+	0.0453606i	$0.0144437\pi$
$883$			2.69581i			0.0907212i	−0.998971	+	0.0453606i	$0.985556\pi$
$884$		0			0
$885$			11.9962i			0.403247i
$886$		0			0
$887$		−	14.9520i		−	0.502038i	−0.967982	−	0.251019i	$-0.919234\pi$
$887$		−	14.9520i		−	0.502038i	0.967982	−	0.251019i	$-0.0807656\pi$
$888$		0			0
$889$			66.5070i			2.23057i
$890$		0			0
$891$	−1.96967			−0.0659864
$892$		0			0
$893$		−	9.16722i		−	0.306769i
$894$		0			0
$895$	19.8164			0.662390
$896$		0			0
$897$	−10.7428	+	5.57480i	−0.358693	+	0.186137i
$898$		0			0
$899$			0.844422i			0.0281631i
$900$		0			0
$901$	11.6091			0.386755
$902$		0			0
$903$		−	39.2349i		−	1.30566i
$904$		0			0
$905$	−2.00943			−0.0667959
$906$		0			0
$907$	−13.2046			−0.438452			−0.219226	−	0.975674i	$-0.570353\pi$
$907$	−13.2046			−0.438452			−0.219226	+	0.975674i	$0.570353\pi$
$908$		0			0
$909$	−6.64873			−0.220524
$910$		0			0
$911$	1.55943			0.0516662			0.0258331	−	0.999666i	$-0.491776\pi$
$911$	1.55943			0.0516662			0.0258331	+	0.999666i	$0.491776\pi$
$912$		0			0
$913$	−29.0681			−0.962015
$914$		0			0
$915$		−	4.62698i		−	0.152963i
$916$		0			0
$917$			43.5952i			1.43964i
$918$		0			0
$919$	5.46921			0.180413			0.0902063	−	0.995923i	$-0.471247\pi$
$919$	5.46921			0.180413			0.0902063	+	0.995923i	$0.471247\pi$
$920$		0			0
$921$	14.7305			0.485386
$922$		0			0
$923$		−	38.3579i		−	1.26257i
$924$		0			0
$925$		−	5.26726i		−	0.173187i
$926$		0			0
$927$	−0.583656			−0.0191698
$928$		0			0
$929$	−10.6885			−0.350678			−0.175339	−	0.984508i	$-0.556102\pi$
$929$	−10.6885			−0.350678			−0.175339	+	0.984508i	$0.556102\pi$
$930$		0			0
$931$	−18.7607			−0.614857
$932$		0			0
$933$	−2.02347			−0.0662454
$934$		0			0
$935$	−7.21040			−0.235805
$936$		0			0
$937$		−	4.41143i		−	0.144115i	−0.997400	−	0.0720576i	$-0.977043\pi$
$937$		−	4.41143i		−	0.144115i	0.997400	−	0.0720576i	$-0.0229565\pi$
$938$		0			0
$939$	−19.6539			−0.641381
$940$		0			0
$941$		−	38.3983i		−	1.25175i	−0.779924	−	0.625875i	$-0.784742\pi$
$941$		−	38.3983i		−	1.25175i	0.779924	−	0.625875i	$-0.215258\pi$
$942$		0			0
$943$	11.9374	+	23.0038i	0.388735	+	0.749107i
$944$		0			0
$945$	3.12483			0.101651
$946$		0			0
$947$			21.6901i			0.704832i	0.935843	+	0.352416i	$0.114640\pi$
$947$			21.6901i			0.704832i	−0.935843	+	0.352416i	$0.885360\pi$
$948$		0			0
$949$	−33.8567			−1.09904
$950$		0			0
$951$		−	0.0518500i		−	0.00168135i
$952$		0			0
$953$			21.7352i			0.704072i	0.935986	+	0.352036i	$0.114511\pi$
$953$			21.7352i			0.704072i	−0.935986	+	0.352036i	$0.885489\pi$
$954$		0			0
$955$		−	18.8616i		−	0.610346i
$956$		0			0
$957$			4.40196i			0.142295i
$958$		0			0
$959$		−	7.25668i		−	0.234330i
$960$		0			0
$961$	30.8572			0.995395
$962$		0			0
$963$	−2.04526			−0.0659077
$964$		0			0
$965$			25.8785i			0.833058i
$966$		0			0
$967$		−	47.1981i		−	1.51779i	−0.651215	−	0.758894i	$-0.725740\pi$
$967$		−	47.1981i		−	1.51779i	0.651215	−	0.758894i	$-0.274260\pi$
$968$		0			0
$969$	−24.8419			−0.798036
$970$		0			0
$971$	11.1263			0.357059			0.178529	−	0.983935i	$-0.442866\pi$
$971$	11.1263			0.357059			0.178529	+	0.983935i	$0.442866\pi$
$972$		0			0
$973$		−	48.2673i		−	1.54738i
$974$		0			0
$975$		−	2.52369i		−	0.0808226i
$976$		0			0
$977$			59.5577i			1.90542i	0.303883	+	0.952709i	$0.401717\pi$
$977$			59.5577i			1.90542i	−0.303883	+	0.952709i	$0.598283\pi$
$978$		0			0
$979$			0.665814i			0.0212795i
$980$		0			0
$981$			15.7438i			0.502661i
$982$		0			0
$983$	37.3662			1.19180			0.595898	−	0.803060i	$-0.296796\pi$
$983$	37.3662			1.19180			0.595898	+	0.803060i	$0.296796\pi$
$984$		0			0
$985$		−	11.3975i		−	0.363155i
$986$		0			0
$987$	4.22130			0.134366
$988$		0			0
$989$	27.7357	+	53.4478i	0.881945	+	1.69954i
$990$		0			0
$991$		−	0.0206000i		−	0.000654379i	−1.00000		0.000327190i	$-0.999896\pi$
$991$		−	0.0206000i		−	0.000654379i	1.00000		0.000327190i	$-0.000104148\pi$
$992$		0			0
$993$	3.22601			0.102374
$994$		0			0
$995$		−	10.1484i		−	0.321724i
$996$		0			0
$997$	−13.0778			−0.414179			−0.207089	−	0.978322i	$-0.566399\pi$
$997$	−13.0778			−0.414179			−0.207089	+	0.978322i	$0.566399\pi$
$998$		0			0
$999$	−5.26726			−0.166649

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.c.1471.5	✓	32
4.3	odd	2		5520.2.be.d.1471.6	yes	32
23.22	odd	2		5520.2.be.d.1471.5	yes	32
92.91	even	2	inner	5520.2.be.c.1471.6	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.c.1471.5	✓	32	1.1	even	1	trivial
5520.2.be.c.1471.6	yes	32	92.91	even	2	inner
5520.2.be.d.1471.5	yes	32	23.22	odd	2
5520.2.be.d.1471.6	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} +3.12483 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{5} +3.12483 q^{7} -1.00000 q^{9} -1.96967 q^{11} -2.52369 q^{13} -1.00000 q^{15} +3.66071i q^{17} -6.78607 q^{19} -3.12483i q^{21} +(2.20899 + 4.25680i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +2.23487 q^{29} +0.377839i q^{31} +1.96967i q^{33} -3.12483i q^{35} +5.26726i q^{37} +2.52369i q^{39} +5.40401 q^{41} +12.5558 q^{43} +1.00000i q^{45} +1.35089i q^{47} +2.76459 q^{49} +3.66071 q^{51} -3.17126i q^{53} +1.96967i q^{55} +6.78607i q^{57} -11.9962i q^{59} +4.62698i q^{61} -3.12483 q^{63} +2.52369i q^{65} +10.4936 q^{67} +(4.25680 - 2.20899i) q^{69} +15.1992i q^{71} +13.4156 q^{73} +1.00000i q^{75} -6.15489 q^{77} +5.38677 q^{79} +1.00000 q^{81} +14.7579 q^{83} +3.66071 q^{85} -2.23487i q^{87} -0.338033i q^{89} -7.88610 q^{91} +0.377839 q^{93} +6.78607i q^{95} -11.0882i q^{97} +1.96967 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) \) 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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