LMFDB - Embedded newform 580.2.d.c.521.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [580,2,Mod(521,580)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(580, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("580.521"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 580 = 2^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	580.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.63132331723$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 12x^{2} + 25 $ x^4 + 12*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			521.4
Root			$-1.63810i$ of defining polynomial
Character	$\chi$	$=$	580.521
Dual form			580.2.d.c.521.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{3} +1.00000 q^{5} +2.31662 q^{7} +1.00000 q^{9} +1.41421i q^{11} +1.41421i q^{15} -1.86199i q^{17} -1.86199i q^{19} +3.27620i q^{21} +0.316625 q^{23} +1.00000 q^{25} +5.65685i q^{27} +(3.31662 - 4.24264i) q^{29} +4.69042i q^{31} -2.00000 q^{33} +2.31662 q^{35} +7.51884i q^{37} -3.27620i q^{41} +7.96662i q^{43} +1.00000 q^{45} +4.24264i q^{47} -1.63325 q^{49} +2.63325 q^{51} -8.63325 q^{53} +1.41421i q^{55} +2.63325 q^{57} -2.63325 q^{59} -3.27620i q^{61} +2.31662 q^{63} +8.94987 q^{67} +0.447775i q^{69} -6.63325 q^{71} -13.6235i q^{73} +1.41421i q^{75} +3.27620i q^{77} -13.6235i q^{79} -5.00000 q^{81} -2.31662 q^{83} -1.86199i q^{85} +(6.00000 + 4.69042i) q^{87} -11.3137i q^{89} -6.63325 q^{93} -1.86199i q^{95} +4.24264i q^{97} +1.41421i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{7} + 4 q^{9} - 12 q^{23} + 4 q^{25} - 8 q^{33} - 4 q^{35} + 4 q^{45} + 20 q^{49} - 16 q^{51} - 8 q^{53} - 16 q^{57} + 16 q^{59} - 4 q^{63} - 4 q^{67} - 20 q^{81} + 4 q^{83} + 24 q^{87}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^7 + 4 * q^9 - 12 * q^23 + 4 * q^25 - 8 * q^33 - 4 * q^35 + 4 * q^45 + 20 * q^49 - 16 * q^51 - 8 * q^53 - 16 * q^57 + 16 * q^59 - 4 * q^63 - 4 * q^67 - 20 * q^81 + 4 * q^83 + 24 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/580\mathbb{Z}\right)^\times$.

$n$	$117$	$291$	$321$
$\chi(n)$	$1$	$1$	$-1$

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
$3$			1.41421i			0.816497i	0.912871	+	0.408248i	$0.133860\pi$
$3$			1.41421i			0.816497i	−0.912871	+	0.408248i	$0.866140\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	2.31662			0.875602			0.437801	−	0.899072i	$-0.355757\pi$
$7$	2.31662			0.875602			0.437801	+	0.899072i	$0.355757\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			1.41421i			0.426401i	0.977008	+	0.213201i	$0.0683888\pi$
$11$			1.41421i			0.426401i	−0.977008	+	0.213201i	$0.931611\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$			1.41421i			0.365148i
$16$		0			0
$17$		−	1.86199i		−	0.451599i	−0.974174	−	0.225799i	$-0.927501\pi$
$17$		−	1.86199i		−	0.451599i	0.974174	−	0.225799i	$-0.0724994\pi$
$18$		0			0
$19$		−	1.86199i		−	0.427169i	−0.976925	−	0.213585i	$-0.931486\pi$
$19$		−	1.86199i		−	0.427169i	0.976925	−	0.213585i	$-0.0685139\pi$
$20$		0			0
$21$			3.27620i			0.714926i
$22$		0			0
$23$	0.316625			0.0660208			0.0330104	−	0.999455i	$-0.489491\pi$
$23$	0.316625			0.0660208			0.0330104	+	0.999455i	$0.489491\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			5.65685i			1.08866i
$28$		0			0
$29$	3.31662	−	4.24264i	0.615882	−	0.787839i
$29$	3.31662	−	4.24264i	0.615882	−	0.787839i
$30$		0			0
$31$			4.69042i			0.842424i	0.906962	+	0.421212i	$0.138395\pi$
$31$			4.69042i			0.842424i	−0.906962	+	0.421212i	$0.861605\pi$
$32$		0			0
$33$	−2.00000			−0.348155
$34$		0			0
$35$	2.31662			0.391581
$36$		0			0
$37$			7.51884i			1.23609i	0.786143	+	0.618045i	$0.212075\pi$
$37$			7.51884i			1.23609i	−0.786143	+	0.618045i	$0.787925\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	3.27620i		−	0.511657i	−0.966722	−	0.255828i	$-0.917652\pi$
$41$		−	3.27620i		−	0.511657i	0.966722	−	0.255828i	$-0.0823482\pi$
$42$		0			0
$43$			7.96662i			1.21490i	0.794359	+	0.607449i	$0.207807\pi$
$43$			7.96662i			1.21490i	−0.794359	+	0.607449i	$0.792193\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$			4.24264i			0.618853i	0.950923	+	0.309426i	$0.100137\pi$
$47$			4.24264i			0.618853i	−0.950923	+	0.309426i	$0.899863\pi$
$48$		0			0
$49$	−1.63325			−0.233321
$50$		0			0
$51$	2.63325			0.368729
$52$		0			0
$53$	−8.63325			−1.18587			−0.592934	−	0.805251i	$-0.702031\pi$
$53$	−8.63325			−1.18587			−0.592934	+	0.805251i	$0.702031\pi$
$54$		0			0
$55$			1.41421i			0.190693i
$56$		0			0
$57$	2.63325			0.348782
$58$		0			0
$59$	−2.63325			−0.342820			−0.171410	−	0.985200i	$-0.554832\pi$
$59$	−2.63325			−0.342820			−0.171410	+	0.985200i	$0.554832\pi$
$60$		0			0
$61$		−	3.27620i		−	0.419475i	−0.977758	−	0.209737i	$-0.932739\pi$
$61$		−	3.27620i		−	0.419475i	0.977758	−	0.209737i	$-0.0672609\pi$
$62$		0			0
$63$	2.31662			0.291867
$64$		0			0
$65$		0			0
$66$		0			0
$67$	8.94987			1.09340			0.546701	−	0.837328i	$-0.315884\pi$
$67$	8.94987			1.09340			0.546701	+	0.837328i	$0.315884\pi$
$68$		0			0
$69$			0.447775i			0.0539058i
$70$		0			0
$71$	−6.63325			−0.787222			−0.393611	−	0.919277i	$-0.628774\pi$
$71$	−6.63325			−0.787222			−0.393611	+	0.919277i	$0.628774\pi$
$72$		0			0
$73$		−	13.6235i		−	1.59451i	−0.603645	−	0.797253i	$-0.706285\pi$
$73$		−	13.6235i		−	1.59451i	0.603645	−	0.797253i	$-0.293715\pi$
$74$		0			0
$75$			1.41421i			0.163299i
$76$		0			0
$77$			3.27620i			0.373358i
$78$		0			0
$79$		−	13.6235i		−	1.53276i	−0.642387	−	0.766380i	$-0.722056\pi$
$79$		−	13.6235i		−	1.53276i	0.642387	−	0.766380i	$-0.277944\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$	−2.31662			−0.254283			−0.127141	−	0.991885i	$-0.540580\pi$
$83$	−2.31662			−0.254283			−0.127141	+	0.991885i	$0.540580\pi$
$84$		0			0
$85$		−	1.86199i		−	0.201961i
$86$		0			0
$87$	6.00000	+	4.69042i	0.643268	+	0.502865i
$88$		0			0
$89$		−	11.3137i		−	1.19925i	−0.800281	−	0.599625i	$-0.795316\pi$
$89$		−	11.3137i		−	1.19925i	0.800281	−	0.599625i	$-0.204684\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−6.63325			−0.687836
$94$		0			0
$95$		−	1.86199i		−	0.191036i
$96$		0			0
$97$			4.24264i			0.430775i	0.976529	+	0.215387i	$0.0691014\pi$
$97$			4.24264i			0.430775i	−0.976529	+	0.215387i	$0.930899\pi$
$98$		0			0
$99$			1.41421i			0.142134i
$100$		0			0
$101$		−	12.2093i		−	1.21487i	−0.794371	−	0.607433i	$-0.792199\pi$
$101$		−	12.2093i		−	1.21487i	0.794371	−	0.607433i	$-0.207801\pi$
$102$		0			0
$103$	−0.949874			−0.0935939			−0.0467970	−	0.998904i	$-0.514901\pi$
$103$	−0.949874			−0.0935939			−0.0467970	+	0.998904i	$0.514901\pi$
$104$		0			0
$105$			3.27620i			0.319725i
$106$		0			0
$107$	5.68338			0.549433			0.274716	−	0.961525i	$-0.411416\pi$
$107$	5.68338			0.549433			0.274716	+	0.961525i	$0.411416\pi$
$108$		0			0
$109$	−4.63325			−0.443785			−0.221892	−	0.975071i	$-0.571223\pi$
$109$	−4.63325			−0.443785			−0.221892	+	0.975071i	$0.571223\pi$
$110$		0			0
$111$	−10.6332			−1.00926
$112$		0			0
$113$		−	12.7279i		−	1.19734i	−0.800995	−	0.598671i	$-0.795696\pi$
$113$		−	12.7279i		−	1.19734i	0.800995	−	0.598671i	$-0.204304\pi$
$114$		0			0
$115$	0.316625			0.0295254
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	4.31353i		−	0.395421i
$120$		0			0
$121$	9.00000			0.818182
$122$		0			0
$123$	4.63325			0.417766
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$			0.518663i			0.0460239i	0.999735	+	0.0230120i	$0.00732558\pi$
$127$			0.518663i			0.0460239i	−0.999735	+	0.0230120i	$0.992674\pi$
$128$		0			0
$129$	−11.2665			−0.991960
$130$		0			0
$131$		−	9.45172i		−	0.825801i	−0.910776	−	0.412900i	$-0.864516\pi$
$131$		−	9.45172i		−	0.825801i	0.910776	−	0.412900i	$-0.135484\pi$
$132$		0			0
$133$		−	4.31353i		−	0.374030i
$134$		0			0
$135$			5.65685i			0.486864i
$136$		0			0
$137$		−	0.966438i		−	0.0825684i	−0.999147	−	0.0412842i	$-0.986855\pi$
$137$		−	0.966438i		−	0.0825684i	0.999147	−	0.0412842i	$-0.0131449\pi$
$138$		0			0
$139$	−3.36675			−0.285564			−0.142782	−	0.989754i	$-0.545605\pi$
$139$	−3.36675			−0.285564			−0.142782	+	0.989754i	$0.545605\pi$
$140$		0			0
$141$	−6.00000			−0.505291
$142$		0			0
$143$		0			0
$144$		0			0
$145$	3.31662	−	4.24264i	0.275431	−	0.352332i
$146$		0			0
$147$		−	2.30976i		−	0.190506i
$148$		0			0
$149$	−15.2665			−1.25068			−0.625340	−	0.780352i	$-0.715040\pi$
$149$	−15.2665			−1.25068			−0.625340	+	0.780352i	$0.715040\pi$
$150$		0			0
$151$	2.00000			0.162758			0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000			0.162758			0.0813788	+	0.996683i	$0.474068\pi$
$152$		0			0
$153$		−	1.86199i		−	0.150533i
$154$		0			0
$155$			4.69042i			0.376743i
$156$		0			0
$157$			13.6235i			1.08727i	0.839321	+	0.543636i	$0.182953\pi$
$157$			13.6235i			1.08727i	−0.839321	+	0.543636i	$0.817047\pi$
$158$		0			0
$159$		−	12.2093i		−	0.968257i
$160$		0			0
$161$	0.733501			0.0578080
$162$		0			0
$163$			3.34709i			0.262164i	0.991372	+	0.131082i	$0.0418452\pi$
$163$			3.34709i			0.262164i	−0.991372	+	0.131082i	$0.958155\pi$
$164$		0			0
$165$	−2.00000			−0.155700
$166$		0			0
$167$	4.31662			0.334030			0.167015	−	0.985954i	$-0.446587\pi$
$167$	4.31662			0.334030			0.167015	+	0.985954i	$0.446587\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		−	1.86199i		−	0.142390i
$172$		0			0
$173$	−12.6332			−0.960488			−0.480244	−	0.877135i	$-0.659452\pi$
$173$	−12.6332			−0.960488			−0.480244	+	0.877135i	$0.659452\pi$
$174$		0			0
$175$	2.31662			0.175120
$176$		0			0
$177$		−	3.72398i		−	0.279911i
$178$		0			0
$179$	−4.63325			−0.346305			−0.173153	−	0.984895i	$-0.555395\pi$
$179$	−4.63325			−0.346305			−0.173153	+	0.984895i	$0.555395\pi$
$180$		0			0
$181$	5.26650			0.391456			0.195728	−	0.980658i	$-0.437293\pi$
$181$	5.26650			0.391456			0.195728	+	0.980658i	$0.437293\pi$
$182$		0			0
$183$	4.63325			0.342500
$184$		0			0
$185$			7.51884i			0.552796i
$186$		0			0
$187$	2.63325			0.192562
$188$		0			0
$189$			13.1048i			0.953235i
$190$		0			0
$191$		−	12.7279i		−	0.920960i	−0.887670	−	0.460480i	$-0.847677\pi$
$191$		−	12.7279i		−	0.920960i	0.887670	−	0.460480i	$-0.152323\pi$
$192$		0			0
$193$		−	0.966438i		−	0.0695658i	−0.999395	−	0.0347829i	$-0.988926\pi$
$193$		−	0.966438i		−	0.0695658i	0.999395	−	0.0347829i	$-0.0110740\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.36675			−0.667353			−0.333677	−	0.942688i	$-0.608289\pi$
$197$	−9.36675			−0.667353			−0.333677	+	0.942688i	$0.608289\pi$
$198$		0			0
$199$	−10.0000			−0.708881			−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000			−0.708881			−0.354441	+	0.935079i	$0.615329\pi$
$200$		0			0
$201$			12.6570i			0.892758i
$202$		0			0
$203$	7.68338	−	9.82861i	0.539267	−	0.689833i
$204$		0			0
$205$		−	3.27620i		−	0.228820i
$206$		0			0
$207$	0.316625			0.0220069
$208$		0			0
$209$	2.63325			0.182146
$210$		0			0
$211$			19.2803i			1.32731i	0.748038	+	0.663656i	$0.230996\pi$
$211$			19.2803i			1.32731i	−0.748038	+	0.663656i	$0.769004\pi$
$212$		0			0
$213$		−	9.38083i		−	0.642764i
$214$		0			0
$215$			7.96662i			0.543319i
$216$		0			0
$217$			10.8659i			0.737628i
$218$		0			0
$219$	19.2665			1.30191
$220$		0			0
$221$		0			0
$222$		0			0
$223$	3.68338			0.246657			0.123329	−	0.992366i	$-0.460643\pi$
$223$	3.68338			0.246657			0.123329	+	0.992366i	$0.460643\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−8.31662			−0.551994			−0.275997	−	0.961159i	$-0.589008\pi$
$227$	−8.31662			−0.551994			−0.275997	+	0.961159i	$0.589008\pi$
$228$		0			0
$229$		−	8.48528i		−	0.560723i	−0.959894	−	0.280362i	$-0.909546\pi$
$229$		−	8.48528i		−	0.560723i	0.959894	−	0.280362i	$-0.0904544\pi$
$230$		0			0
$231$	−4.63325			−0.304845
$232$		0			0
$233$	1.36675			0.0895388			0.0447694	−	0.998997i	$-0.485745\pi$
$233$	1.36675			0.0895388			0.0447694	+	0.998997i	$0.485745\pi$
$234$		0			0
$235$			4.24264i			0.276759i
$236$		0			0
$237$	19.2665			1.25149
$238$		0			0
$239$	−19.8997			−1.28721			−0.643604	−	0.765359i	$-0.722562\pi$
$239$	−19.8997			−1.28721			−0.643604	+	0.765359i	$0.722562\pi$
$240$		0			0
$241$	18.0000			1.15948			0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000			1.15948			0.579741	+	0.814801i	$0.303154\pi$
$242$		0			0
$243$			9.89949i			0.635053i
$244$		0			0
$245$	−1.63325			−0.104344
$246$		0			0
$247$		0			0
$248$		0			0
$249$		−	3.27620i		−	0.207621i
$250$		0			0
$251$			10.7950i			0.681377i	0.940176	+	0.340689i	$0.110660\pi$
$251$			10.7950i			0.681377i	−0.940176	+	0.340689i	$0.889340\pi$
$252$		0			0
$253$			0.447775i			0.0281514i
$254$		0			0
$255$	2.63325			0.164900
$256$		0			0
$257$	0.633250			0.0395010			0.0197505	−	0.999805i	$-0.493713\pi$
$257$	0.633250			0.0395010			0.0197505	+	0.999805i	$0.493713\pi$
$258$		0			0
$259$			17.4183i			1.08232i
$260$		0			0
$261$	3.31662	−	4.24264i	0.205294	−	0.262613i
$262$		0			0
$263$		−	1.41421i		−	0.0872041i	−0.999049	−	0.0436021i	$-0.986117\pi$
$263$		−	1.41421i		−	0.0872041i	0.999049	−	0.0436021i	$-0.0138834\pi$
$264$		0			0
$265$	−8.63325			−0.530336
$266$		0			0
$267$	16.0000			0.979184
$268$		0			0
$269$		−	7.00018i		−	0.426808i	−0.976964	−	0.213404i	$-0.931545\pi$
$269$		−	7.00018i		−	0.426808i	0.976964	−	0.213404i	$-0.0684551\pi$
$270$		0			0
$271$		−	5.13819i		−	0.312123i	−0.987747	−	0.156061i	$-0.950120\pi$
$271$		−	5.13819i		−	0.312123i	0.987747	−	0.156061i	$-0.0498798\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			1.41421i			0.0852803i
$276$		0			0
$277$	8.63325			0.518722			0.259361	−	0.965780i	$-0.416488\pi$
$277$	8.63325			0.518722			0.259361	+	0.965780i	$0.416488\pi$
$278$		0			0
$279$			4.69042i			0.280808i
$280$		0			0
$281$	20.6332			1.23088			0.615438	−	0.788185i	$-0.288979\pi$
$281$	20.6332			1.23088			0.615438	+	0.788185i	$0.288979\pi$
$282$		0			0
$283$	29.5831			1.75853			0.879267	−	0.476329i	$-0.158033\pi$
$283$	29.5831			1.75853			0.879267	+	0.476329i	$0.158033\pi$
$284$		0			0
$285$	2.63325			0.155980
$286$		0			0
$287$		−	7.58973i		−	0.448008i
$288$		0			0
$289$	13.5330			0.796059
$290$		0			0
$291$	−6.00000			−0.351726
$292$		0			0
$293$			1.41421i			0.0826192i	0.999146	+	0.0413096i	$0.0131530\pi$
$293$			1.41421i			0.0826192i	−0.999146	+	0.0413096i	$0.986847\pi$
$294$		0			0
$295$	−2.63325			−0.153314
$296$		0			0
$297$	−8.00000			−0.464207
$298$		0			0
$299$		0			0
$300$		0			0
$301$			18.4557i			1.06377i
$302$		0			0
$303$	17.2665			0.991934
$304$		0			0
$305$		−	3.27620i		−	0.187595i
$306$		0			0
$307$			12.7279i			0.726421i	0.931707	+	0.363210i	$0.118319\pi$
$307$			12.7279i			0.726421i	−0.931707	+	0.363210i	$0.881681\pi$
$308$		0			0
$309$		−	1.34333i		−	0.0764191i
$310$		0			0
$311$		−	31.0418i		−	1.76022i	−0.474770	−	0.880110i	$-0.657469\pi$
$311$		−	31.0418i		−	1.76022i	0.474770	−	0.880110i	$-0.342531\pi$
$312$		0			0
$313$	−22.5330			−1.27364			−0.636820	−	0.771012i	$-0.719751\pi$
$313$	−22.5330			−1.27364			−0.636820	+	0.771012i	$0.719751\pi$
$314$		0			0
$315$	2.31662			0.130527
$316$		0			0
$317$			5.72774i			0.321702i	0.986979	+	0.160851i	$0.0514239\pi$
$317$			5.72774i			0.321702i	−0.986979	+	0.160851i	$0.948576\pi$
$318$		0			0
$319$	6.00000	+	4.69042i	0.335936	+	0.262613i
$320$		0			0
$321$			8.03751i			0.448610i
$322$		0			0
$323$	−3.46700			−0.192909
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	6.55240i		−	0.362349i
$328$		0			0
$329$			9.82861i			0.541869i
$330$		0			0
$331$			9.45172i			0.519514i	0.965674	+	0.259757i	$0.0836424\pi$
$331$			9.45172i			0.519514i	−0.965674	+	0.259757i	$0.916358\pi$
$332$		0			0
$333$			7.51884i			0.412030i
$334$		0			0
$335$	8.94987			0.488984
$336$		0			0
$337$			22.1088i			1.20434i	0.798368	+	0.602170i	$0.205697\pi$
$337$			22.1088i			1.20434i	−0.798368	+	0.602170i	$0.794303\pi$
$338$		0			0
$339$	18.0000			0.977626
$340$		0			0
$341$	−6.63325			−0.359211
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$			0.447775i			0.0241074i
$346$		0			0
$347$	−19.5831			−1.05128			−0.525639	−	0.850708i	$-0.676174\pi$
$347$	−19.5831			−1.05128			−0.525639	+	0.850708i	$0.676174\pi$
$348$		0			0
$349$	11.8997			0.636979			0.318489	−	0.947926i	$-0.396824\pi$
$349$	11.8997			0.636979			0.318489	+	0.947926i	$0.396824\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	20.6332			1.09820			0.549099	−	0.835758i	$-0.314971\pi$
$353$	20.6332			1.09820			0.549099	+	0.835758i	$0.314971\pi$
$354$		0			0
$355$	−6.63325			−0.352056
$356$		0			0
$357$	6.10025			0.322860
$358$		0			0
$359$			11.2428i			0.593373i	0.954975	+	0.296687i	$0.0958817\pi$
$359$			11.2428i			0.593373i	−0.954975	+	0.296687i	$0.904118\pi$
$360$		0			0
$361$	15.5330			0.817526
$362$		0			0
$363$			12.7279i			0.668043i
$364$		0			0
$365$		−	13.6235i		−	0.713085i
$366$		0			0
$367$		−	11.8324i		−	0.617645i	−0.951120	−	0.308822i	$-0.900065\pi$
$367$		−	11.8324i		−	0.617645i	0.951120	−	0.308822i	$-0.0999349\pi$
$368$		0			0
$369$		−	3.27620i		−	0.170552i
$370$		0			0
$371$	−20.0000			−1.03835
$372$		0			0
$373$	−2.00000			−0.103556			−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000			−0.103556			−0.0517780	+	0.998659i	$0.516489\pi$
$374$		0			0
$375$			1.41421i			0.0730297i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			16.4519i			0.845077i	0.906345	+	0.422539i	$0.138861\pi$
$379$			16.4519i			0.845077i	−0.906345	+	0.422539i	$0.861139\pi$
$380$		0			0
$381$	−0.733501			−0.0375784
$382$		0			0
$383$	−6.94987			−0.355122			−0.177561	−	0.984110i	$-0.556821\pi$
$383$	−6.94987			−0.355122			−0.177561	+	0.984110i	$0.556821\pi$
$384$		0			0
$385$			3.27620i			0.166971i
$386$		0			0
$387$			7.96662i			0.404966i
$388$		0			0
$389$		−	1.34333i		−	0.0681093i	−0.999420	−	0.0340546i	$-0.989158\pi$
$389$		−	1.34333i		−	0.0681093i	0.999420	−	0.0340546i	$-0.0108420\pi$
$390$		0			0
$391$		−	0.589552i		−	0.0298149i
$392$		0			0
$393$	13.3668			0.674263
$394$		0			0
$395$		−	13.6235i		−	0.685471i
$396$		0			0
$397$	14.6332			0.734422			0.367211	−	0.930138i	$-0.380313\pi$
$397$	14.6332			0.734422			0.367211	+	0.930138i	$0.380313\pi$
$398$		0			0
$399$	6.10025			0.305395
$400$		0			0
$401$	5.26650			0.262996			0.131498	−	0.991316i	$-0.458021\pi$
$401$	5.26650			0.262996			0.131498	+	0.991316i	$0.458021\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−5.00000			−0.248452
$406$		0			0
$407$	−10.6332			−0.527071
$408$		0			0
$409$		−	0.447775i		−	0.0221411i	−0.999939	−	0.0110705i	$-0.996476\pi$
$409$		−	0.447775i		−	0.0221411i	0.999939	−	0.0110705i	$-0.00352393\pi$
$410$		0			0
$411$	1.36675			0.0674168
$412$		0			0
$413$	−6.10025			−0.300174
$414$		0			0
$415$	−2.31662			−0.113719
$416$		0			0
$417$		−	4.76130i		−	0.233162i
$418$		0			0
$419$	−10.6332			−0.519468			−0.259734	−	0.965680i	$-0.583635\pi$
$419$	−10.6332			−0.519468			−0.259734	+	0.965680i	$0.583635\pi$
$420$		0			0
$421$		−	6.10463i		−	0.297521i	−0.988873	−	0.148761i	$-0.952472\pi$
$421$		−	6.10463i		−	0.297521i	0.988873	−	0.148761i	$-0.0475284\pi$
$422$		0			0
$423$			4.24264i			0.206284i
$424$		0			0
$425$		−	1.86199i		−	0.0903197i
$426$		0			0
$427$		−	7.58973i		−	0.367293i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−37.8997			−1.82557			−0.912783	−	0.408444i	$-0.866071\pi$
$431$	−37.8997			−1.82557			−0.912783	+	0.408444i	$0.866071\pi$
$432$		0			0
$433$			4.69042i			0.225407i	0.993629	+	0.112703i	$0.0359510\pi$
$433$			4.69042i			0.225407i	−0.993629	+	0.112703i	$0.964049\pi$
$434$		0			0
$435$	6.00000	+	4.69042i	0.287678	+	0.224888i
$436$		0			0
$437$		−	0.589552i		−	0.0282021i
$438$		0			0
$439$	−19.8997			−0.949763			−0.474882	−	0.880050i	$-0.657509\pi$
$439$	−19.8997			−0.949763			−0.474882	+	0.880050i	$0.657509\pi$
$440$		0			0
$441$	−1.63325			−0.0777738
$442$		0			0
$443$			29.5567i			1.40428i	0.712038	+	0.702141i	$0.247772\pi$
$443$			29.5567i			1.40428i	−0.712038	+	0.702141i	$0.752228\pi$
$444$		0			0
$445$		−	11.3137i		−	0.536321i
$446$		0			0
$447$		−	21.5901i		−	1.02118i
$448$		0			0
$449$			29.6276i			1.39821i	0.715018	+	0.699106i	$0.246419\pi$
$449$			29.6276i			1.39821i	−0.715018	+	0.699106i	$0.753581\pi$
$450$		0			0
$451$	4.63325			0.218171
$452$		0			0
$453$			2.82843i			0.132891i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.36675			0.157490			0.0787450	−	0.996895i	$-0.474909\pi$
$457$	3.36675			0.157490			0.0787450	+	0.996895i	$0.474909\pi$
$458$		0			0
$459$	10.5330			0.491638
$460$		0			0
$461$			23.5230i			1.09557i	0.836618	+	0.547787i	$0.184530\pi$
$461$			23.5230i			1.09557i	−0.836618	+	0.547787i	$0.815470\pi$
$462$		0			0
$463$	−33.5831			−1.56074			−0.780370	−	0.625318i	$-0.784969\pi$
$463$	−33.5831			−1.56074			−0.780370	+	0.625318i	$0.784969\pi$
$464$		0			0
$465$	−6.63325			−0.307610
$466$		0			0
$467$		−	41.9077i		−	1.93926i	−0.244578	−	0.969630i	$-0.578650\pi$
$467$		−	41.9077i		−	1.93926i	0.244578	−	0.969630i	$-0.421350\pi$
$468$		0			0
$469$	20.7335			0.957384
$470$		0			0
$471$	−19.2665			−0.887753
$472$		0			0
$473$	−11.2665			−0.518034
$474$		0			0
$475$		−	1.86199i		−	0.0854339i
$476$		0			0
$477$	−8.63325			−0.395289
$478$		0			0
$479$		−	40.8704i		−	1.86742i	−0.358034	−	0.933709i	$-0.616553\pi$
$479$		−	40.8704i		−	1.86742i	0.358034	−	0.933709i	$-0.383447\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$			1.03733i			0.0472000i
$484$		0			0
$485$			4.24264i			0.192648i
$486$		0			0
$487$	−35.5831			−1.61243			−0.806213	−	0.591626i	$-0.798486\pi$
$487$	−35.5831			−1.61243			−0.806213	+	0.591626i	$0.798486\pi$
$488$		0			0
$489$	−4.73350			−0.214056
$490$		0			0
$491$		−	34.7658i		−	1.56896i	−0.620155	−	0.784479i	$-0.712930\pi$
$491$		−	34.7658i		−	1.56896i	0.620155	−	0.784479i	$-0.287070\pi$
$492$		0			0
$493$	−7.89975	−	6.17552i	−0.355787	−	0.278131i
$494$		0			0
$495$			1.41421i			0.0635642i
$496$		0			0
$497$	−15.3668			−0.689293
$498$		0			0
$499$	35.7995			1.60261			0.801303	−	0.598259i	$-0.204141\pi$
$499$	35.7995			1.60261			0.801303	+	0.598259i	$0.204141\pi$
$500$		0			0
$501$			6.10463i			0.272735i
$502$		0			0
$503$			4.24264i			0.189170i	0.995517	+	0.0945850i	$0.0301524\pi$
$503$			4.24264i			0.189170i	−0.995517	+	0.0945850i	$0.969848\pi$
$504$		0			0
$505$		−	12.2093i		−	0.543305i
$506$		0			0
$507$		−	18.3848i		−	0.816497i
$508$		0			0
$509$	42.5330			1.88524			0.942621	−	0.333865i	$-0.108353\pi$
$509$	42.5330			1.88524			0.942621	+	0.333865i	$0.108353\pi$
$510$		0			0
$511$		−	31.5605i		−	1.39615i
$512$		0			0
$513$	10.5330			0.465043
$514$		0			0
$515$	−0.949874			−0.0418565
$516$		0			0
$517$	−6.00000			−0.263880
$518$		0			0
$519$		−	17.8661i		−	0.784235i
$520$		0			0
$521$	2.73350			0.119757			0.0598784	−	0.998206i	$-0.480929\pi$
$521$	2.73350			0.119757			0.0598784	+	0.998206i	$0.480929\pi$
$522$		0			0
$523$	0.949874			0.0415351			0.0207676	−	0.999784i	$-0.493389\pi$
$523$	0.949874			0.0415351			0.0207676	+	0.999784i	$0.493389\pi$
$524$		0			0
$525$			3.27620i			0.142985i
$526$		0			0
$527$	8.73350			0.380437
$528$		0			0
$529$	−22.8997			−0.995641
$530$		0			0
$531$	−2.63325			−0.114273
$532$		0			0
$533$		0			0
$534$		0			0
$535$	5.68338			0.245714
$536$		0			0
$537$		−	6.55240i		−	0.282757i
$538$		0			0
$539$		−	2.30976i		−	0.0994886i
$540$		0			0
$541$			24.5603i			1.05593i	0.849266	+	0.527965i	$0.177045\pi$
$541$			24.5603i			1.05593i	−0.849266	+	0.527965i	$0.822955\pi$
$542$		0			0
$543$			7.44795i			0.319622i
$544$		0			0
$545$	−4.63325			−0.198467
$546$		0			0
$547$	−34.9499			−1.49435			−0.747174	−	0.664628i	$-0.768590\pi$
$547$	−34.9499			−1.49435			−0.747174	+	0.664628i	$0.768590\pi$
$548$		0			0
$549$		−	3.27620i		−	0.139825i
$550$		0			0
$551$	−7.89975	−	6.17552i	−0.336541	−	0.263086i
$552$		0			0
$553$		−	31.5605i		−	1.34209i
$554$		0			0
$555$	−10.6332			−0.451356
$556$		0			0
$557$	−7.89975			−0.334723			−0.167362	−	0.985896i	$-0.553525\pi$
$557$	−7.89975			−0.334723			−0.167362	+	0.985896i	$0.553525\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$			3.72398i			0.157226i
$562$		0			0
$563$			19.2803i			0.812569i	0.913747	+	0.406284i	$0.133176\pi$
$563$			19.2803i			0.812569i	−0.913747	+	0.406284i	$0.866824\pi$
$564$		0			0
$565$		−	12.7279i		−	0.535468i
$566$		0			0
$567$	−11.5831			−0.486445
$568$		0			0
$569$			12.7988i			0.536554i	0.963342	+	0.268277i	$0.0864543\pi$
$569$			12.7988i			0.536554i	−0.963342	+	0.268277i	$0.913546\pi$
$570$		0			0
$571$	44.6332			1.86784			0.933922	−	0.357478i	$-0.116363\pi$
$571$	44.6332			1.86784			0.933922	+	0.357478i	$0.116363\pi$
$572$		0			0
$573$	18.0000			0.751961
$574$		0			0
$575$	0.316625			0.0132042
$576$		0			0
$577$		−	41.3182i		−	1.72010i	−0.510211	−	0.860049i	$-0.670433\pi$
$577$		−	41.3182i		−	1.72010i	0.510211	−	0.860049i	$-0.329567\pi$
$578$		0			0
$579$	1.36675			0.0568002
$580$		0			0
$581$	−5.36675			−0.222650
$582$		0			0
$583$		−	12.2093i		−	0.505656i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−30.2164			−1.24716			−0.623582	−	0.781758i	$-0.714323\pi$
$587$	−30.2164			−1.24716			−0.623582	+	0.781758i	$0.714323\pi$
$588$		0			0
$589$	8.73350			0.359858
$590$		0			0
$591$		−	13.2466i		−	0.544892i
$592$		0			0
$593$	38.5330			1.58236			0.791180	−	0.611583i	$-0.209467\pi$
$593$	38.5330			1.58236			0.791180	+	0.611583i	$0.209467\pi$
$594$		0			0
$595$		−	4.31353i		−	0.176837i
$596$		0			0
$597$		−	14.1421i		−	0.578799i
$598$		0			0
$599$		−	14.6608i		−	0.599024i	−0.954093	−	0.299512i	$-0.903176\pi$
$599$		−	14.6608i		−	0.599024i	0.954093	−	0.299512i	$-0.0968239\pi$
$600$		0			0
$601$			25.9036i			1.05663i	0.849048	+	0.528315i	$0.177176\pi$
$601$			25.9036i			1.05663i	−0.849048	+	0.528315i	$0.822824\pi$
$602$		0			0
$603$	8.94987			0.364467
$604$		0			0
$605$	9.00000			0.365902
$606$		0			0
$607$			6.03374i			0.244902i	0.992475	+	0.122451i	$0.0390754\pi$
$607$			6.03374i			0.244902i	−0.992475	+	0.122451i	$0.960925\pi$
$608$		0			0
$609$	13.8997	+	10.8659i	0.563246	+	0.440310i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−30.0000			−1.21169			−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000			−1.21169			−0.605844	+	0.795583i	$0.707165\pi$
$614$		0			0
$615$	4.63325			0.186831
$616$		0			0
$617$		−	6.17552i		−	0.248617i	−0.992244	−	0.124308i	$-0.960329\pi$
$617$		−	6.17552i		−	0.248617i	0.992244	−	0.124308i	$-0.0396712\pi$
$618$		0			0
$619$			3.34709i			0.134531i	0.997735	+	0.0672655i	$0.0214274\pi$
$619$			3.34709i			0.134531i	−0.997735	+	0.0672655i	$0.978573\pi$
$620$		0			0
$621$			1.79110i			0.0718744i
$622$		0			0
$623$		−	26.2096i		−	1.05007i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			3.72398i			0.148721i
$628$		0			0
$629$	14.0000			0.558217
$630$		0			0
$631$	33.2665			1.32432			0.662159	−	0.749363i	$-0.269640\pi$
$631$	33.2665			1.32432			0.662159	+	0.749363i	$0.269640\pi$
$632$		0			0
$633$	−27.2665			−1.08375
$634$		0			0
$635$			0.518663i			0.0205825i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−6.63325			−0.262407
$640$		0			0
$641$			39.9040i			1.57611i	0.615603	+	0.788056i	$0.288912\pi$
$641$			39.9040i			1.57611i	−0.615603	+	0.788056i	$0.711088\pi$
$642$		0			0
$643$	8.31662			0.327976			0.163988	−	0.986462i	$-0.447564\pi$
$643$	8.31662			0.327976			0.163988	+	0.986462i	$0.447564\pi$
$644$		0			0
$645$	−11.2665			−0.443618
$646$		0			0
$647$	−6.31662			−0.248332			−0.124166	−	0.992261i	$-0.539626\pi$
$647$	−6.31662			−0.248332			−0.124166	+	0.992261i	$0.539626\pi$
$648$		0			0
$649$		−	3.72398i		−	0.146179i
$650$		0			0
$651$	−15.3668			−0.602270
$652$		0			0
$653$			34.3180i			1.34297i	0.741019	+	0.671484i	$0.234343\pi$
$653$			34.3180i			1.34297i	−0.741019	+	0.671484i	$0.765657\pi$
$654$		0			0
$655$		−	9.45172i		−	0.369309i
$656$		0			0
$657$		−	13.6235i		−	0.531502i
$658$		0			0
$659$			34.7658i			1.35428i	0.735853	+	0.677141i	$0.236781\pi$
$659$			34.7658i			1.35428i	−0.735853	+	0.677141i	$0.763219\pi$
$660$		0			0
$661$	−42.5330			−1.65434			−0.827171	−	0.561950i	$-0.810051\pi$
$661$	−42.5330			−1.65434			−0.827171	+	0.561950i	$0.810051\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	4.31353i		−	0.167271i
$666$		0			0
$667$	1.05013	−	1.34333i	0.0406610	−	0.0520138i
$668$		0			0
$669$			5.20908i			0.201395i
$670$		0			0
$671$	4.63325			0.178865
$672$		0			0
$673$	−31.8997			−1.22964			−0.614822	−	0.788666i	$-0.710772\pi$
$673$	−31.8997			−1.22964			−0.614822	+	0.788666i	$0.710772\pi$
$674$		0			0
$675$			5.65685i			0.217732i
$676$		0			0
$677$		−	20.1759i		−	0.775422i	−0.921781	−	0.387711i	$-0.873266\pi$
$677$		−	20.1759i		−	0.775422i	0.921781	−	0.387711i	$-0.126734\pi$
$678$		0			0
$679$			9.82861i			0.377187i
$680$		0			0
$681$		−	11.7615i		−	0.450701i
$682$		0			0
$683$	23.6834			0.906219			0.453110	−	0.891455i	$-0.350315\pi$
$683$	23.6834			0.906219			0.453110	+	0.891455i	$0.350315\pi$
$684$		0			0
$685$		−	0.966438i		−	0.0369257i
$686$		0			0
$687$	12.0000			0.457829
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−27.8997			−1.06136			−0.530678	−	0.847573i	$-0.678063\pi$
$691$	−27.8997			−1.06136			−0.530678	+	0.847573i	$0.678063\pi$
$692$		0			0
$693$			3.27620i			0.124453i
$694$		0			0
$695$	−3.36675			−0.127708
$696$		0			0
$697$	−6.10025			−0.231063
$698$		0			0
$699$			1.93288i			0.0731081i
$700$		0			0
$701$	51.7995			1.95644			0.978220	−	0.207571i	$-0.0665557\pi$
$701$	51.7995			1.95644			0.978220	+	0.207571i	$0.0665557\pi$
$702$		0			0
$703$	14.0000			0.528020
$704$		0			0
$705$	−6.00000			−0.225973
$706$		0			0
$707$		−	28.2843i		−	1.06374i
$708$		0			0
$709$	20.0000			0.751116			0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000			0.751116			0.375558	+	0.926799i	$0.377451\pi$
$710$		0			0
$711$		−	13.6235i		−	0.510920i
$712$		0			0
$713$			1.48510i			0.0556175i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	28.1425i		−	1.05100i
$718$		0			0
$719$	−20.7335			−0.773229			−0.386615	−	0.922241i	$-0.626356\pi$
$719$	−20.7335			−0.773229			−0.386615	+	0.922241i	$0.626356\pi$
$720$		0			0
$721$	−2.20050			−0.0819510
$722$		0			0
$723$			25.4558i			0.946713i
$724$		0			0
$725$	3.31662	−	4.24264i	0.123176	−	0.157568i
$726$		0			0
$727$		−	23.0043i		−	0.853182i	−0.904445	−	0.426591i	$-0.859714\pi$
$727$		−	23.0043i		−	0.853182i	0.904445	−	0.426591i	$-0.140286\pi$
$728$		0			0
$729$	−29.0000			−1.07407
$730$		0			0
$731$	14.8338			0.548646
$732$		0			0
$733$		−	40.4226i		−	1.49304i	−0.665361	−	0.746522i	$-0.731722\pi$
$733$		−	40.4226i		−	1.49304i	0.665361	−	0.746522i	$-0.268278\pi$
$734$		0			0
$735$		−	2.30976i		−	0.0851969i
$736$		0			0
$737$			12.6570i			0.466228i
$738$		0			0
$739$			20.7654i			0.763869i	0.924189	+	0.381934i	$0.124742\pi$
$739$			20.7654i			0.763869i	−0.924189	+	0.381934i	$0.875258\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			26.7283i			0.980566i	0.871563	+	0.490283i	$0.163106\pi$
$743$			26.7283i			0.980566i	−0.871563	+	0.490283i	$0.836894\pi$
$744$		0			0
$745$	−15.2665			−0.559321
$746$		0			0
$747$	−2.31662			−0.0847609
$748$		0			0
$749$	13.1662			0.481084
$750$		0			0
$751$			47.1168i			1.71932i	0.510869	+	0.859659i	$0.329324\pi$
$751$			47.1168i			1.71932i	−0.510869	+	0.859659i	$0.670676\pi$
$752$		0			0
$753$	−15.2665			−0.556342
$754$		0			0
$755$	2.00000			0.0727875
$756$		0			0
$757$			24.7954i			0.901204i	0.892725	+	0.450602i	$0.148791\pi$
$757$			24.7954i			0.901204i	−0.892725	+	0.450602i	$0.851209\pi$
$758$		0			0
$759$	−0.633250			−0.0229855
$760$		0			0
$761$	−17.3668			−0.629544			−0.314772	−	0.949167i	$-0.601928\pi$
$761$	−17.3668			−0.629544			−0.314772	+	0.949167i	$0.601928\pi$
$762$		0			0
$763$	−10.7335			−0.388579
$764$		0			0
$765$		−	1.86199i		−	0.0673203i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			25.4558i			0.917961i	0.888446	+	0.458981i	$0.151785\pi$
$769$			25.4558i			0.917961i	−0.888446	+	0.458981i	$0.848215\pi$
$770$		0			0
$771$			0.895550i			0.0322525i
$772$		0			0
$773$			20.6237i			0.741781i	0.928677	+	0.370891i	$0.120948\pi$
$773$			20.6237i			0.741781i	−0.928677	+	0.370891i	$0.879052\pi$
$774$		0			0
$775$			4.69042i			0.168485i
$776$		0			0
$777$	−24.6332			−0.883713
$778$		0			0
$779$	−6.10025			−0.218564
$780$		0			0
$781$		−	9.38083i		−	0.335673i
$782$		0			0
$783$	24.0000	+	18.7617i	0.857690	+	0.670487i
$784$		0			0
$785$			13.6235i			0.486243i
$786$		0			0
$787$	−14.9499			−0.532905			−0.266453	−	0.963848i	$-0.585852\pi$
$787$	−14.9499			−0.532905			−0.266453	+	0.963848i	$0.585852\pi$
$788$		0			0
$789$	2.00000			0.0712019
$790$		0			0
$791$		−	29.4858i		−	1.04840i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		−	12.2093i		−	0.433018i
$796$		0			0
$797$			45.6317i			1.61636i	0.588937	+	0.808179i	$0.299547\pi$
$797$			45.6317i			1.61636i	−0.588937	+	0.808179i	$0.700453\pi$
$798$		0			0
$799$	7.89975			0.279473
$800$		0			0
$801$		−	11.3137i		−	0.399750i
$802$		0			0
$803$	19.2665			0.679900
$804$		0			0
$805$	0.733501			0.0258525
$806$		0			0
$807$	9.89975			0.348488
$808$		0			0
$809$			13.1048i			0.460741i	0.973103	+	0.230370i	$0.0739937\pi$
$809$			13.1048i			0.460741i	−0.973103	+	0.230370i	$0.926006\pi$
$810$		0			0
$811$	25.8997			0.909463			0.454732	−	0.890629i	$-0.349735\pi$
$811$	25.8997			0.909463			0.454732	+	0.890629i	$0.349735\pi$
$812$		0			0
$813$	7.26650			0.254847
$814$		0			0
$815$			3.34709i			0.117243i
$816$		0			0
$817$	14.8338			0.518967
$818$		0			0
$819$		0			0
$820$		0			0
$821$	27.2665			0.951607			0.475804	−	0.879552i	$-0.342157\pi$
$821$	27.2665			0.951607			0.475804	+	0.879552i	$0.342157\pi$
$822$		0			0
$823$		−	16.4519i		−	0.573477i	−0.958009	−	0.286739i	$-0.907429\pi$
$823$		−	16.4519i		−	0.573477i	0.958009	−	0.286739i	$-0.0925711\pi$
$824$		0			0
$825$	−2.00000			−0.0696311
$826$		0			0
$827$			8.86217i			0.308168i	0.988058	+	0.154084i	$0.0492426\pi$
$827$			8.86217i			0.308168i	−0.988058	+	0.154084i	$0.950757\pi$
$828$		0			0
$829$			30.5231i			1.06011i	0.847962	+	0.530056i	$0.177829\pi$
$829$			30.5231i			1.06011i	−0.847962	+	0.530056i	$0.822171\pi$
$830$		0			0
$831$			12.2093i			0.423534i
$832$		0			0
$833$			3.04109i			0.105368i
$834$		0			0
$835$	4.31662			0.149383
$836$		0			0
$837$	−26.5330			−0.917115
$838$		0			0
$839$			4.24264i			0.146472i	0.997315	+	0.0732361i	$0.0233327\pi$
$839$			4.24264i			0.146472i	−0.997315	+	0.0732361i	$0.976667\pi$
$840$		0			0
$841$	−7.00000	−	28.1425i	−0.241379	−	0.970431i
$842$		0			0
$843$			29.1798i			1.00501i
$844$		0			0
$845$	−13.0000			−0.447214
$846$		0			0
$847$	20.8496			0.716402
$848$		0			0
$849$			41.8369i			1.43584i
$850$		0			0
$851$			2.38065i			0.0816077i
$852$		0			0
$853$		−	40.4226i		−	1.38404i	−0.721876	−	0.692022i	$-0.756720\pi$
$853$		−	40.4226i		−	1.38404i	0.721876	−	0.692022i	$-0.243280\pi$
$854$		0			0
$855$		−	1.86199i		−	0.0636787i
$856$		0			0
$857$	23.3668			0.798193			0.399096	−	0.916909i	$-0.369324\pi$
$857$	23.3668			0.798193			0.399096	+	0.916909i	$0.369324\pi$
$858$		0			0
$859$			23.0043i			0.784897i	0.919774	+	0.392448i	$0.128372\pi$
$859$			23.0043i			0.784897i	−0.919774	+	0.392448i	$0.871628\pi$
$860$		0			0
$861$	10.7335			0.365797
$862$		0			0
$863$	55.5831			1.89207			0.946036	−	0.324062i	$-0.105049\pi$
$863$	55.5831			1.89207			0.946036	+	0.324062i	$0.105049\pi$
$864$		0			0
$865$	−12.6332			−0.429543
$866$		0			0
$867$			19.1385i			0.649979i
$868$		0			0
$869$	19.2665			0.653571
$870$		0			0
$871$		0			0
$872$		0			0
$873$			4.24264i			0.143592i
$874$		0			0
$875$	2.31662			0.0783162
$876$		0			0
$877$	19.1662			0.647198			0.323599	−	0.946194i	$-0.395107\pi$
$877$	19.1662			0.647198			0.323599	+	0.946194i	$0.395107\pi$
$878$		0			0
$879$	−2.00000			−0.0674583
$880$		0			0
$881$			40.0458i			1.34918i	0.738195	+	0.674588i	$0.235679\pi$
$881$			40.0458i			1.34918i	−0.738195	+	0.674588i	$0.764321\pi$
$882$		0			0
$883$	30.9499			1.04155			0.520773	−	0.853695i	$-0.325644\pi$
$883$	30.9499			1.04155			0.520773	+	0.853695i	$0.325644\pi$
$884$		0			0
$885$		−	3.72398i		−	0.125180i
$886$		0			0
$887$			57.8410i			1.94211i	0.238856	+	0.971055i	$0.423228\pi$
$887$			57.8410i			1.94211i	−0.238856	+	0.971055i	$0.576772\pi$
$888$		0			0
$889$			1.20155i			0.0402986i
$890$		0			0
$891$		−	7.07107i		−	0.236890i
$892$		0			0
$893$	7.89975			0.264355
$894$		0			0
$895$	−4.63325			−0.154872
$896$		0			0
$897$		0			0
$898$		0			0
$899$	19.8997	+	15.5563i	0.663694	+	0.518833i
$900$		0			0
$901$			16.0750i			0.535536i
$902$		0			0
$903$	−26.1003			−0.868562
$904$		0			0
$905$	5.26650			0.175064
$906$		0			0
$907$		−	34.3180i		−	1.13951i	−0.821814	−	0.569755i	$-0.807038\pi$
$907$		−	34.3180i		−	1.13951i	0.821814	−	0.569755i	$-0.192962\pi$
$908$		0			0
$909$		−	12.2093i		−	0.404956i
$910$		0			0
$911$			8.41439i			0.278781i	0.990237	+	0.139391i	$0.0445144\pi$
$911$			8.41439i			0.278781i	−0.990237	+	0.139391i	$0.955486\pi$
$912$		0			0
$913$		−	3.27620i		−	0.108426i
$914$		0			0
$915$	4.63325			0.153171
$916$		0			0
$917$		−	21.8961i		−	0.723073i
$918$		0			0
$919$	13.3668			0.440928			0.220464	−	0.975395i	$-0.429243\pi$
$919$	13.3668			0.440928			0.220464	+	0.975395i	$0.429243\pi$
$920$		0			0
$921$	−18.0000			−0.593120
$922$		0			0
$923$		0			0
$924$		0			0
$925$			7.51884i			0.247218i
$926$		0			0
$927$	−0.949874			−0.0311980
$928$		0			0
$929$	−11.2665			−0.369642			−0.184821	−	0.982772i	$-0.559171\pi$
$929$	−11.2665			−0.369642			−0.184821	+	0.982772i	$0.559171\pi$
$930$		0			0
$931$			3.04109i			0.0996678i
$932$		0			0
$933$	43.8997			1.43721
$934$		0			0
$935$	2.63325			0.0861165
$936$		0			0
$937$	41.2665			1.34812			0.674059	−	0.738678i	$-0.264549\pi$
$937$	41.2665			1.34812			0.674059	+	0.738678i	$0.264549\pi$
$938$		0			0
$939$		−	31.8665i		−	1.03992i
$940$		0			0
$941$	−28.5330			−0.930149			−0.465075	−	0.885272i	$-0.653972\pi$
$941$	−28.5330			−0.930149			−0.465075	+	0.885272i	$0.653972\pi$
$942$		0			0
$943$		−	1.03733i		−	0.0337800i
$944$		0			0
$945$			13.1048i			0.426299i
$946$		0			0
$947$		−	43.6988i		−	1.42002i	−0.704191	−	0.710011i	$-0.748690\pi$
$947$		−	43.6988i		−	1.42002i	0.704191	−	0.710011i	$-0.251310\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−8.10025			−0.262669
$952$		0			0
$953$	16.7335			0.542051			0.271026	−	0.962572i	$-0.412637\pi$
$953$	16.7335			0.542051			0.271026	+	0.962572i	$0.412637\pi$
$954$		0			0
$955$		−	12.7279i		−	0.411866i
$956$		0			0
$957$	−6.63325	+	8.48528i	−0.214423	+	0.274290i
$958$		0			0
$959$		−	2.23888i		−	0.0722971i
$960$		0			0
$961$	9.00000			0.290323
$962$		0			0
$963$	5.68338			0.183144
$964$		0			0
$965$		−	0.966438i		−	0.0311108i
$966$		0			0
$967$			30.5940i			0.983838i	0.870641	+	0.491919i	$0.163704\pi$
$967$			30.5940i			0.983838i	−0.870641	+	0.491919i	$0.836296\pi$
$968$		0			0
$969$		−	4.90308i		−	0.157510i
$970$		0			0
$971$		−	42.3555i		−	1.35925i	−0.733558	−	0.679627i	$-0.762142\pi$
$971$		−	42.3555i		−	1.35925i	0.733558	−	0.679627i	$-0.237858\pi$
$972$		0			0
$973$	−7.79950			−0.250040
$974$		0			0
$975$		0			0
$976$		0			0
$977$	37.8997			1.21252			0.606260	−	0.795266i	$-0.292669\pi$
$977$	37.8997			1.21252			0.606260	+	0.795266i	$0.292669\pi$
$978$		0			0
$979$	16.0000			0.511362
$980$		0			0
$981$	−4.63325			−0.147928
$982$		0			0
$983$			22.2505i			0.709682i	0.934927	+	0.354841i	$0.115465\pi$
$983$			22.2505i			0.709682i	−0.934927	+	0.354841i	$0.884535\pi$
$984$		0			0
$985$	−9.36675			−0.298449
$986$		0			0
$987$	−13.8997			−0.442434
$988$		0			0
$989$			2.52243i			0.0802086i
$990$		0			0
$991$	8.73350			0.277429			0.138714	−	0.990332i	$-0.455703\pi$
$991$	8.73350			0.277429			0.138714	+	0.990332i	$0.455703\pi$
$992$		0			0
$993$	−13.3668			−0.424181
$994$		0			0
$995$	−10.0000			−0.317021
$996$		0			0
$997$		−	25.0790i		−	0.794259i	−0.917763	−	0.397129i	$-0.870006\pi$
$997$		−	25.0790i		−	0.794259i	0.917763	−	0.397129i	$-0.129994\pi$
$998$		0			0
$999$	−42.5330			−1.34568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	580.2.d.c.521.4	yes	4
3.2	odd	2		5220.2.l.d.3421.3		4
4.3	odd	2		2320.2.g.h.1681.1		4
5.2	odd	4		2900.2.f.c.1449.7		8
5.3	odd	4		2900.2.f.c.1449.2		8
5.4	even	2		2900.2.d.d.1101.1		4
29.28	even	2	inner	580.2.d.c.521.2	✓	4
87.86	odd	2		5220.2.l.d.3421.4		4
116.115	odd	2		2320.2.g.h.1681.3		4
145.28	odd	4		2900.2.f.c.1449.6		8
145.57	odd	4		2900.2.f.c.1449.3		8
145.144	even	2		2900.2.d.d.1101.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.d.c.521.2	✓	4	29.28	even	2	inner
580.2.d.c.521.4	yes	4	1.1	even	1	trivial
2320.2.g.h.1681.1		4	4.3	odd	2
2320.2.g.h.1681.3		4	116.115	odd	2
2900.2.d.d.1101.1		4	5.4	even	2
2900.2.d.d.1101.3		4	145.144	even	2
2900.2.f.c.1449.2		8	5.3	odd	4
2900.2.f.c.1449.3		8	145.57	odd	4
2900.2.f.c.1449.6		8	145.28	odd	4
2900.2.f.c.1449.7		8	5.2	odd	4
5220.2.l.d.3421.3		4	3.2	odd	2
5220.2.l.d.3421.4		4	87.86	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 580 = 2^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	580.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{3} +1.00000 q^{5} +2.31662 q^{7} +1.00000 q^{9} +1.41421i q^{11} +1.41421i q^{15} -1.86199i q^{17} -1.86199i q^{19} +3.27620i q^{21} +0.316625 q^{23} +1.00000 q^{25} +5.65685i q^{27} +(3.31662 - 4.24264i) q^{29} +4.69042i q^{31} -2.00000 q^{33} +2.31662 q^{35} +7.51884i q^{37} -3.27620i q^{41} +7.96662i q^{43} +1.00000 q^{45} +4.24264i q^{47} -1.63325 q^{49} +2.63325 q^{51} -8.63325 q^{53} +1.41421i q^{55} +2.63325 q^{57} -2.63325 q^{59} -3.27620i q^{61} +2.31662 q^{63} +8.94987 q^{67} +0.447775i q^{69} -6.63325 q^{71} -13.6235i q^{73} +1.41421i q^{75} +3.27620i q^{77} -13.6235i q^{79} -5.00000 q^{81} -2.31662 q^{83} -1.86199i q^{85} +(6.00000 + 4.69042i) q^{87} -11.3137i q^{89} -6.63325 q^{93} -1.86199i q^{95} +4.24264i q^{97} +1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{7} + 4 q^{9} - 12 q^{23} + 4 q^{25} - 8 q^{33} - 4 q^{35} + 4 q^{45} + 20 q^{49} - 16 q^{51} - 8 q^{53} - 16 q^{57} + 16 q^{59} - 4 q^{63} - 4 q^{67} - 20 q^{81} + 4 q^{83} + 24 q^{87}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^7 + 4 * q^9 - 12 * q^23 + 4 * q^25 - 8 * q^33 - 4 * q^35 + 4 * q^45 + 20 * q^49 - 16 * q^51 - 8 * q^53 - 16 * q^57 + 16 * q^59 - 4 * q^63 - 4 * q^67 - 20 * q^81 + 4 * q^83 + 24 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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