LMFDB - Embedded newform 460.4.c.a.369.26

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [460,4,Mod(369,460)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("460.369"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(460, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 460 = 2^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	460.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$27.1408786026$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.26
Character	$\chi$	$=$	460.369
Dual form			460.4.c.a.369.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.05177i q^{3} +(11.1311 - 1.04775i) q^{5} +20.6013i q^{7} -9.62397 q^{9} +64.4013 q^{11} +18.5748i q^{13} +(6.34073 + 67.3631i) q^{15} -45.0640i q^{17} +62.2720 q^{19} -124.674 q^{21} -23.0000i q^{23} +(122.804 - 23.3252i) q^{25} +105.156i q^{27} +36.3576 q^{29} +157.921 q^{31} +389.742i q^{33} +(21.5849 + 229.316i) q^{35} -241.047i q^{37} -112.410 q^{39} -311.927 q^{41} -100.060i q^{43} +(-107.126 + 10.0835i) q^{45} -316.708i q^{47} -81.4132 q^{49} +272.717 q^{51} +693.712i q^{53} +(716.860 - 67.4763i) q^{55} +376.856i q^{57} -682.801 q^{59} -466.470 q^{61} -198.266i q^{63} +(19.4616 + 206.758i) q^{65} -34.0816i q^{67} +139.191 q^{69} -239.804 q^{71} +329.505i q^{73} +(141.159 + 743.185i) q^{75} +1326.75i q^{77} +38.1289 q^{79} -896.226 q^{81} +532.627i q^{83} +(-47.2156 - 501.613i) q^{85} +220.028i q^{87} -1465.95 q^{89} -382.664 q^{91} +955.704i q^{93} +(693.159 - 65.2453i) q^{95} +1063.54i q^{97} -619.796 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 12 q^{5} - 220 q^{9} - 8 q^{11} - 168 q^{15} + 144 q^{19} - 32 q^{21} + 100 q^{25} - 76 q^{29} + 652 q^{31} + 320 q^{35} - 1128 q^{39} - 560 q^{41} + 1208 q^{45} - 2224 q^{49} + 3192 q^{51} - 1304 q^{55}+ \cdots + 1952 q^{99}+O(q^{100}) $ 32 * q - 12 * q^5 - 220 * q^9 - 8 * q^11 - 168 * q^15 + 144 * q^19 - 32 * q^21 + 100 * q^25 - 76 * q^29 + 652 * q^31 + 320 * q^35 - 1128 * q^39 - 560 * q^41 + 1208 * q^45 - 2224 * q^49 + 3192 * q^51 - 1304 * q^55 - 376 * q^59 + 1000 * q^61 + 636 * q^65 + 552 * q^69 - 732 * q^71 - 2608 * q^75 - 1088 * q^79 + 2160 * q^81 + 1932 * q^85 + 3016 * q^89 - 96 * q^91 + 32 * q^95 + 1952 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/460\mathbb{Z}\right)^\times$.

$n$	$231$	$277$	$281$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			6.05177i			1.16466i	0.812951	+	0.582332i	$0.197860\pi$
$3$			6.05177i			1.16466i	−0.812951	+	0.582332i	$0.802140\pi$
$4$		0			0
$5$	11.1311	−	1.04775i	0.995599	−	0.0937133i
$5$	11.1311	−	1.04775i	0.995599	−	0.0937133i
$6$		0			0
$7$			20.6013i			1.11237i	0.831060	+	0.556183i	$0.187734\pi$
$7$			20.6013i			1.11237i	−0.831060	+	0.556183i	$0.812266\pi$
$8$		0			0
$9$	−9.62397			−0.356443
$10$		0			0
$11$	64.4013			1.76525			0.882624	−	0.470080i	$-0.155775\pi$
$11$	64.4013			1.76525			0.882624	+	0.470080i	$0.155775\pi$
$12$		0			0
$13$			18.5748i			0.396285i	0.980173	+	0.198143i	$0.0634909\pi$
$13$			18.5748i			0.396285i	−0.980173	+	0.198143i	$0.936509\pi$
$14$		0			0
$15$	6.34073	+	67.3631i	0.109145	+	1.15954i
$16$		0			0
$17$		−	45.0640i		−	0.642919i	−0.946923	−	0.321459i	$-0.895827\pi$
$17$		−	45.0640i		−	0.642919i	0.946923	−	0.321459i	$-0.104173\pi$
$18$		0			0
$19$	62.2720			0.751905			0.375952	−	0.926639i	$-0.377316\pi$
$19$	62.2720			0.751905			0.375952	+	0.926639i	$0.377316\pi$
$20$		0			0
$21$	−124.674			−1.29553
$22$		0			0
$23$		−	23.0000i		−	0.208514i
$23$		−	23.0000i		−	0.208514i
$24$		0			0
$25$	122.804	−	23.3252i	0.982436	−	0.186602i
$26$		0			0
$27$			105.156i			0.749528i
$28$		0			0
$29$	36.3576			0.232808			0.116404	−	0.993202i	$-0.462863\pi$
$29$	36.3576			0.232808			0.116404	+	0.993202i	$0.462863\pi$
$30$		0			0
$31$	157.921			0.914952			0.457476	−	0.889222i	$-0.348754\pi$
$31$	157.921			0.914952			0.457476	+	0.889222i	$0.348754\pi$
$32$		0			0
$33$			389.742i			2.05592i
$34$		0			0
$35$	21.5849	+	229.316i	0.104243	+	1.10747i
$36$		0			0
$37$		−	241.047i		−	1.07102i	−0.844528	−	0.535512i	$-0.820119\pi$
$37$		−	241.047i		−	1.07102i	0.844528	−	0.535512i	$-0.179881\pi$
$38$		0			0
$39$	−112.410			−0.461540
$40$		0			0
$41$	−311.927			−1.18816			−0.594082	−	0.804404i	$-0.702485\pi$
$41$	−311.927			−1.18816			−0.594082	+	0.804404i	$0.702485\pi$
$42$		0			0
$43$		−	100.060i		−	0.354860i	−0.984133	−	0.177430i	$-0.943222\pi$
$43$		−	100.060i		−	0.354860i	0.984133	−	0.177430i	$-0.0567784\pi$
$44$		0			0
$45$	−107.126	+	10.0835i	−0.354875	+	0.0334035i
$46$		0			0
$47$		−	316.708i		−	0.982907i	−0.870904	−	0.491454i	$-0.836466\pi$
$47$		−	316.708i		−	0.982907i	0.870904	−	0.491454i	$-0.163534\pi$
$48$		0			0
$49$	−81.4132			−0.237356
$50$		0			0
$51$	272.717			0.748785
$52$		0			0
$53$			693.712i			1.79790i	0.438051	+	0.898950i	$0.355669\pi$
$53$			693.712i			1.79790i	−0.438051	+	0.898950i	$0.644331\pi$
$54$		0			0
$55$	716.860	−	67.4763i	1.75748	−	0.165427i
$56$		0			0
$57$			376.856i			0.875717i
$58$		0			0
$59$	−682.801			−1.50666			−0.753331	−	0.657641i	$-0.771554\pi$
$59$	−682.801			−1.50666			−0.753331	+	0.657641i	$0.771554\pi$
$60$		0			0
$61$	−466.470			−0.979103			−0.489552	−	0.871974i	$-0.662840\pi$
$61$	−466.470			−0.979103			−0.489552	+	0.871974i	$0.662840\pi$
$62$		0			0
$63$		−	198.266i		−	0.396495i
$64$		0			0
$65$	19.4616	+	206.758i	0.0371372	+	0.394541i
$66$		0			0
$67$		−	34.0816i		−	0.0621452i	−0.999517	−	0.0310726i	$-0.990108\pi$
$67$		−	34.0816i		−	0.0621452i	0.999517	−	0.0310726i	$-0.00989231\pi$
$68$		0			0
$69$	139.191			0.242849
$70$		0			0
$71$	−239.804			−0.400838			−0.200419	−	0.979710i	$-0.564230\pi$
$71$	−239.804			−0.400838			−0.200419	+	0.979710i	$0.564230\pi$
$72$		0			0
$73$			329.505i			0.528297i	0.964482	+	0.264148i	$0.0850909\pi$
$73$			329.505i			0.528297i	−0.964482	+	0.264148i	$0.914909\pi$
$74$		0			0
$75$	141.159	+	743.185i	0.217329	+	1.14421i
$76$		0			0
$77$			1326.75i			1.96360i
$78$		0			0
$79$	38.1289			0.0543017			0.0271509	−	0.999631i	$-0.491357\pi$
$79$	38.1289			0.0543017			0.0271509	+	0.999631i	$0.491357\pi$
$80$		0			0
$81$	−896.226			−1.22939
$82$		0			0
$83$			532.627i			0.704378i	0.935929	+	0.352189i	$0.114563\pi$
$83$			532.627i			0.704378i	−0.935929	+	0.352189i	$0.885437\pi$
$84$		0			0
$85$	−47.2156	−	501.613i	−0.0602501	−	0.640089i
$86$		0			0
$87$			220.028i			0.271143i
$88$		0			0
$89$	−1465.95			−1.74596			−0.872979	−	0.487758i	$-0.837815\pi$
$89$	−1465.95			−1.74596			−0.872979	+	0.487758i	$0.837815\pi$
$90$		0			0
$91$	−382.664			−0.440814
$92$		0			0
$93$			955.704i			1.06561i
$94$		0			0
$95$	693.159	−	65.2453i	0.748596	−	0.0704635i
$96$		0			0
$97$			1063.54i			1.11326i	0.830761	+	0.556629i	$0.187906\pi$
$97$			1063.54i			1.11326i	−0.830761	+	0.556629i	$0.812094\pi$
$98$		0			0
$99$	−619.796			−0.629211
$100$		0			0
$101$	636.378			0.626950			0.313475	−	0.949596i	$-0.398507\pi$
$101$	636.378			0.626950			0.313475	+	0.949596i	$0.398507\pi$
$102$		0			0
$103$		−	1057.83i		−	1.01195i	−0.862549	−	0.505974i	$-0.831133\pi$
$103$		−	1057.83i		−	1.01195i	0.862549	−	0.505974i	$-0.168867\pi$
$104$		0			0
$105$	−1387.77	+	130.627i	−1.28983	+	0.121409i
$106$		0			0
$107$			1190.04i			1.07519i	0.843203	+	0.537595i	$0.180667\pi$
$107$			1190.04i			1.07519i	−0.843203	+	0.537595i	$0.819333\pi$
$108$		0			0
$109$	−357.479			−0.314131			−0.157065	−	0.987588i	$-0.550203\pi$
$109$	−357.479			−0.314131			−0.157065	+	0.987588i	$0.550203\pi$
$110$		0			0
$111$	1458.76			1.24738
$112$		0			0
$113$		−	1360.13i		−	1.13230i	−0.824302	−	0.566150i	$-0.808432\pi$
$113$		−	1360.13i		−	1.13230i	0.824302	−	0.566150i	$-0.191568\pi$
$114$		0			0
$115$	−24.0982	−	256.016i	−0.0195406	−	0.207597i
$116$		0			0
$117$		−	178.763i		−	0.141253i
$118$		0			0
$119$	928.376			0.715160
$120$		0			0
$121$	2816.53			2.11610
$122$		0			0
$123$		−	1887.71i		−	1.38381i
$124$		0			0
$125$	1342.51	−	388.304i	0.960625	−	0.277848i
$126$		0			0
$127$		−	1707.71i		−	1.19319i	−0.802544	−	0.596593i	$-0.796521\pi$
$127$		−	1707.71i		−	1.19319i	0.802544	−	0.596593i	$-0.203479\pi$
$128$		0			0
$129$	605.540			0.413293
$130$		0			0
$131$	723.428			0.482490			0.241245	−	0.970464i	$-0.422444\pi$
$131$	723.428			0.482490			0.241245	+	0.970464i	$0.422444\pi$
$132$		0			0
$133$			1282.88i			0.836392i
$134$		0			0
$135$	110.177	+	1170.50i	0.0702407	+	0.746229i
$136$		0			0
$137$		−	2938.38i		−	1.83243i	−0.400691	−	0.916213i	$-0.631230\pi$
$137$		−	2938.38i		−	1.83243i	0.400691	−	0.916213i	$-0.368770\pi$
$138$		0			0
$139$	−659.867			−0.402656			−0.201328	−	0.979524i	$-0.564526\pi$
$139$	−659.867			−0.402656			−0.201328	+	0.979524i	$0.564526\pi$
$140$		0			0
$141$	1916.65			1.14476
$142$		0			0
$143$			1196.24i			0.699542i
$144$		0			0
$145$	404.701	−	38.0936i	0.231784	−	0.0218172i
$146$		0			0
$147$		−	492.694i		−	0.276440i
$148$		0			0
$149$	−2339.30			−1.28620			−0.643098	−	0.765784i	$-0.722351\pi$
$149$	−2339.30			−1.28620			−0.643098	+	0.765784i	$0.722351\pi$
$150$		0			0
$151$	1691.70			0.911711			0.455855	−	0.890054i	$-0.349333\pi$
$151$	1691.70			0.911711			0.455855	+	0.890054i	$0.349333\pi$
$152$		0			0
$153$			433.694i			0.229164i
$154$		0			0
$155$	1757.84	−	165.462i	0.910925	−	0.0857432i
$156$		0			0
$157$		−	873.114i		−	0.443835i	−0.975065	−	0.221917i	$-0.928768\pi$
$157$		−	873.114i		−	0.443835i	0.975065	−	0.221917i	$-0.0712315\pi$
$158$		0			0
$159$	−4198.19			−2.09395
$160$		0			0
$161$	473.830			0.231944
$162$		0			0
$163$			2758.62i			1.32559i	0.748800	+	0.662796i	$0.230630\pi$
$163$			2758.62i			1.32559i	−0.748800	+	0.662796i	$0.769370\pi$
$164$		0			0
$165$	408.351	+	4338.27i	0.192667	+	2.04687i
$166$		0			0
$167$		−	3849.42i		−	1.78370i	−0.452336	−	0.891848i	$-0.649409\pi$
$167$		−	3849.42i		−	1.78370i	0.452336	−	0.891848i	$-0.350591\pi$
$168$		0			0
$169$	1851.98			0.842958
$170$		0			0
$171$	−599.304			−0.268011
$172$		0			0
$173$			1637.47i			0.719624i	0.933025	+	0.359812i	$0.117159\pi$
$173$			1637.47i			0.719624i	−0.933025	+	0.359812i	$0.882841\pi$
$174$		0			0
$175$	480.530	+	2529.93i	0.207569	+	1.09283i
$176$		0			0
$177$		−	4132.16i		−	1.75476i
$178$		0			0
$179$	899.051			0.375409			0.187705	−	0.982226i	$-0.439895\pi$
$179$	899.051			0.375409			0.187705	+	0.982226i	$0.439895\pi$
$180$		0			0
$181$	−3100.71			−1.27334			−0.636669	−	0.771137i	$-0.719688\pi$
$181$	−3100.71			−1.27334			−0.636669	+	0.771137i	$0.719688\pi$
$182$		0			0
$183$		−	2822.97i		−	1.14033i
$184$		0			0
$185$	−252.556	−	2683.13i	−0.100369	−	1.06631i
$186$		0			0
$187$		−	2902.18i		−	1.13491i
$188$		0			0
$189$	−2166.35			−0.833748
$190$		0			0
$191$	−3216.07			−1.21836			−0.609179	−	0.793033i	$-0.708501\pi$
$191$	−3216.07			−1.21836			−0.609179	+	0.793033i	$0.708501\pi$
$192$		0			0
$193$		−	2091.76i		−	0.780145i	−0.920784	−	0.390072i	$-0.872450\pi$
$193$		−	2091.76i		−	0.780145i	0.920784	−	0.390072i	$-0.127550\pi$
$194$		0			0
$195$	−1251.25	+	117.777i	−0.459508	+	0.0432524i
$196$		0			0
$197$		−	2987.50i		−	1.08046i	−0.841518	−	0.540229i	$-0.818337\pi$
$197$		−	2987.50i		−	1.08046i	0.841518	−	0.540229i	$-0.181663\pi$
$198$		0			0
$199$	1253.58			0.446554			0.223277	−	0.974755i	$-0.428325\pi$
$199$	1253.58			0.446554			0.223277	+	0.974755i	$0.428325\pi$
$200$		0			0
$201$	206.254			0.0723783
$202$		0			0
$203$			749.013i			0.258968i
$204$		0			0
$205$	−3472.10	+	326.820i	−1.18294	+	0.111347i
$206$		0			0
$207$			221.351i			0.0743236i
$208$		0			0
$209$	4010.40			1.32730
$210$		0			0
$211$	2399.55			0.782900			0.391450	−	0.920199i	$-0.371974\pi$
$211$	2399.55			0.782900			0.391450	+	0.920199i	$0.371974\pi$
$212$		0			0
$213$		−	1451.24i		−	0.466842i
$214$		0			0
$215$	−104.837	−	1113.78i	−0.0332551	−	0.353299i
$216$		0			0
$217$			3253.38i			1.01776i
$218$		0			0
$219$	−1994.09			−0.615289
$220$		0			0
$221$	837.052			0.254779
$222$		0			0
$223$		−	1649.43i		−	0.495310i	−0.968848	−	0.247655i	$-0.920340\pi$
$223$		−	1649.43i		−	0.495310i	0.968848	−	0.247655i	$-0.0796600\pi$
$224$		0			0
$225$	−1181.87	+	224.481i	−0.350183	+	0.0665130i
$226$		0			0
$227$		−	6300.47i		−	1.84219i	−0.389341	−	0.921094i	$-0.627297\pi$
$227$		−	6300.47i		−	1.84219i	0.389341	−	0.921094i	$-0.372703\pi$
$228$		0			0
$229$	467.631			0.134943			0.0674714	−	0.997721i	$-0.478507\pi$
$229$	467.631			0.134943			0.0674714	+	0.997721i	$0.478507\pi$
$230$		0			0
$231$	−8029.19			−2.28694
$232$		0			0
$233$			6394.67i			1.79798i	0.437973	+	0.898988i	$0.355697\pi$
$233$			6394.67i			1.79798i	−0.437973	+	0.898988i	$0.644303\pi$
$234$		0			0
$235$	−331.830	−	3525.32i	−0.0921115	−	0.978582i
$236$		0			0
$237$			230.747i			0.0632433i
$238$		0			0
$239$	−354.204			−0.0958642			−0.0479321	−	0.998851i	$-0.515263\pi$
$239$	−354.204			−0.0958642			−0.0479321	+	0.998851i	$0.515263\pi$
$240$		0			0
$241$	−2563.61			−0.685216			−0.342608	−	0.939479i	$-0.611310\pi$
$241$	−2563.61			−0.685216			−0.342608	+	0.939479i	$0.611310\pi$
$242$		0			0
$243$		−	2584.55i		−	0.682301i
$244$		0			0
$245$	−906.221	+	85.3004i	−0.236312	+	0.0222434i
$246$		0			0
$247$			1156.69i			0.297969i
$248$		0			0
$249$	−3223.34			−0.820364
$250$		0			0
$251$	5345.77			1.34431			0.672155	−	0.740410i	$-0.265369\pi$
$251$	5345.77			1.34431			0.672155	+	0.740410i	$0.265369\pi$
$252$		0			0
$253$		−	1481.23i		−	0.368080i
$254$		0			0
$255$	3035.65	−	285.738i	0.745489	−	0.0701711i
$256$		0			0
$257$			79.3498i			0.0192595i	0.999954	+	0.00962977i	$0.00306530\pi$
$257$			79.3498i			0.0192595i	−0.999954	+	0.00962977i	$0.996935\pi$
$258$		0			0
$259$	4965.88			1.19137
$260$		0			0
$261$	−349.904			−0.0829829
$262$		0			0
$263$		−	1926.93i		−	0.451786i	−0.974152	−	0.225893i	$-0.927470\pi$
$263$		−	1926.93i		−	0.451786i	0.974152	−	0.225893i	$-0.0725300\pi$
$264$		0			0
$265$	726.835	+	7721.81i	0.168487	+	1.78999i
$266$		0			0
$267$		−	8871.59i		−	2.03345i
$268$		0			0
$269$	2895.57			0.656305			0.328152	−	0.944625i	$-0.393574\pi$
$269$	2895.57			0.656305			0.328152	+	0.944625i	$0.393574\pi$
$270$		0			0
$271$	2595.37			0.581762			0.290881	−	0.956759i	$-0.406052\pi$
$271$	2595.37			0.581762			0.290881	+	0.956759i	$0.406052\pi$
$272$		0			0
$273$		−	2315.80i		−	0.513400i
$274$		0			0
$275$	7908.77	−	1502.18i	1.73424	−	0.329399i
$276$		0			0
$277$		−	1101.87i		−	0.239007i	−0.992834	−	0.119504i	$-0.961870\pi$
$277$		−	1101.87i		−	0.239007i	0.992834	−	0.119504i	$-0.0381303\pi$
$278$		0			0
$279$	−1519.83			−0.326128
$280$		0			0
$281$	4486.44			0.952451			0.476226	−	0.879323i	$-0.342005\pi$
$281$	4486.44			0.952451			0.476226	+	0.879323i	$0.342005\pi$
$282$		0			0
$283$		−	3569.01i		−	0.749666i	−0.927092	−	0.374833i	$-0.877700\pi$
$283$		−	3569.01i		−	0.749666i	0.927092	−	0.374833i	$-0.122300\pi$
$284$		0			0
$285$	394.850	+	4194.84i	0.0820663	+	0.871863i
$286$		0			0
$287$		−	6426.09i		−	1.32167i
$288$		0			0
$289$	2882.24			0.586655
$290$		0			0
$291$	−6436.30			−1.29657
$292$		0			0
$293$			6465.03i			1.28905i	0.764584	+	0.644524i	$0.222944\pi$
$293$			6465.03i			1.28905i	−0.764584	+	0.644524i	$0.777056\pi$
$294$		0			0
$295$	−7600.35	+	715.403i	−1.50003	+	0.141194i
$296$		0			0
$297$			6772.17i			1.32310i
$298$		0			0
$299$	427.219			0.0826312
$300$		0			0
$301$	2061.36			0.394734
$302$		0			0
$303$			3851.21i			0.730186i
$304$		0			0
$305$	−5192.34	+	488.742i	−0.974794	+	0.0917550i
$306$		0			0
$307$			539.666i			0.100327i	0.998741	+	0.0501634i	$0.0159742\pi$
$307$			539.666i			0.100327i	−0.998741	+	0.0501634i	$0.984026\pi$
$308$		0			0
$309$	6401.72			1.17858
$310$		0			0
$311$	−4416.91			−0.805337			−0.402668	−	0.915346i	$-0.631917\pi$
$311$	−4416.91			−0.805337			−0.402668	+	0.915346i	$0.631917\pi$
$312$		0			0
$313$		−	2463.01i		−	0.444785i	−0.974957	−	0.222393i	$-0.928613\pi$
$313$		−	2463.01i		−	0.444785i	0.974957	−	0.222393i	$-0.0713867\pi$
$314$		0			0
$315$	−207.733	−	2206.93i	−0.0371569	−	0.394750i
$316$		0			0
$317$		−	9157.93i		−	1.62259i	−0.584638	−	0.811295i	$-0.698763\pi$
$317$		−	9157.93i		−	1.62259i	0.584638	−	0.811295i	$-0.301237\pi$
$318$		0			0
$319$	2341.48			0.410964
$320$		0			0
$321$	−7201.85			−1.25224
$322$		0			0
$323$		−	2806.23i		−	0.483414i
$324$		0			0
$325$	433.260	+	2281.06i	0.0739476	+	0.389325i
$326$		0			0
$327$		−	2163.38i		−	0.365857i
$328$		0			0
$329$	6524.60			1.09335
$330$		0			0
$331$	−7722.52			−1.28238			−0.641190	−	0.767382i	$-0.721559\pi$
$331$	−7722.52			−1.28238			−0.641190	+	0.767382i	$0.721559\pi$
$332$		0			0
$333$			2319.83i			0.381759i
$334$		0			0
$335$	−35.7089	−	379.367i	−0.00582383	−	0.0618717i
$336$		0			0
$337$			3470.18i			0.560927i	0.959865	+	0.280464i	$0.0904882\pi$
$337$			3470.18i			0.560927i	−0.959865	+	0.280464i	$0.909512\pi$
$338$		0			0
$339$	8231.18			1.31875
$340$		0			0
$341$	10170.3			1.61512
$342$		0			0
$343$			5389.03i			0.848338i
$344$		0			0
$345$	1549.35	−	145.837i	0.241781	−	0.0227582i
$346$		0			0
$347$			11775.4i			1.82172i	0.412720	+	0.910858i	$0.364579\pi$
$347$			11775.4i			1.82172i	−0.412720	+	0.910858i	$0.635421\pi$
$348$		0			0
$349$	6964.25			1.06816			0.534080	−	0.845434i	$-0.320658\pi$
$349$	6964.25			1.06816			0.534080	+	0.845434i	$0.320658\pi$
$350$		0			0
$351$	−1953.24			−0.297027
$352$		0			0
$353$			2080.69i			0.313722i	0.987621	+	0.156861i	$0.0501375\pi$
$353$			2080.69i			0.313722i	−0.987621	+	0.156861i	$0.949862\pi$
$354$		0			0
$355$	−2669.29	+	251.254i	−0.399074	+	0.0375639i
$356$		0			0
$357$			5618.32i			0.832922i
$358$		0			0
$359$	−8440.17			−1.24082			−0.620411	−	0.784277i	$-0.713034\pi$
$359$	−8440.17			−1.24082			−0.620411	+	0.784277i	$0.713034\pi$
$360$		0			0
$361$	−2981.19			−0.434640
$362$		0			0
$363$			17045.0i			2.46455i
$364$		0			0
$365$	345.238	+	3667.77i	0.0495085	+	0.525972i
$366$		0			0
$367$		−	5135.52i		−	0.730441i	−0.930921	−	0.365220i	$-0.880994\pi$
$367$		−	5135.52i		−	0.730441i	0.930921	−	0.365220i	$-0.119006\pi$
$368$		0			0
$369$	3001.97			0.423513
$370$		0			0
$371$	−14291.4			−1.99992
$372$		0			0
$373$			6957.26i			0.965772i	0.875683	+	0.482886i	$0.160412\pi$
$373$			6957.26i			0.965772i	−0.875683	+	0.482886i	$0.839588\pi$
$374$		0			0
$375$	2349.93	+	8124.59i	0.323600	+	1.11881i
$376$		0			0
$377$			675.333i			0.0922584i
$378$		0			0
$379$	11669.8			1.58163			0.790814	−	0.612056i	$-0.209657\pi$
$379$	11669.8			1.58163			0.790814	+	0.612056i	$0.209657\pi$
$380$		0			0
$381$	10334.7			1.38966
$382$		0			0
$383$			2704.67i			0.360841i	0.983590	+	0.180420i	$0.0577458\pi$
$383$			2704.67i			0.360841i	−0.983590	+	0.180420i	$0.942254\pi$
$384$		0			0
$385$	1390.10	+	14768.2i	0.184016	+	1.95496i
$386$		0			0
$387$			962.973i			0.126488i
$388$		0			0
$389$	−9970.83			−1.29959			−0.649796	−	0.760109i	$-0.725146\pi$
$389$	−9970.83			−1.29959			−0.649796	+	0.760109i	$0.725146\pi$
$390$		0			0
$391$	−1036.47			−0.134058
$392$		0			0
$393$			4378.02i			0.561939i
$394$		0			0
$395$	424.418	−	39.9494i	0.0540627	−	0.00508879i
$396$		0			0
$397$			332.278i			0.0420064i	0.999779	+	0.0210032i	$0.00668602\pi$
$397$			332.278i			0.0420064i	−0.999779	+	0.0210032i	$0.993314\pi$
$398$		0			0
$399$	−7763.73			−0.974117
$400$		0			0
$401$	11227.7			1.39822			0.699111	−	0.715013i	$-0.253579\pi$
$401$	11227.7			1.39822			0.699111	+	0.715013i	$0.253579\pi$
$402$		0			0
$403$			2933.35i			0.362582i
$404$		0			0
$405$	−9976.02	+	939.018i	−1.22398	+	0.115210i
$406$		0			0
$407$		−	15523.7i		−	1.89062i
$408$		0			0
$409$	−12596.8			−1.52292			−0.761459	−	0.648213i	$-0.775517\pi$
$409$	−12596.8			−1.52292			−0.761459	+	0.648213i	$0.775517\pi$
$410$		0			0
$411$	17782.4			2.13416
$412$		0			0
$413$		−	14066.6i		−	1.67596i
$414$		0			0
$415$	558.058	+	5928.74i	0.0660096	+	0.701278i
$416$		0			0
$417$		−	3993.37i		−	0.468959i
$418$		0			0
$419$	9555.86			1.11416			0.557082	−	0.830458i	$-0.311921\pi$
$419$	9555.86			1.11416			0.557082	+	0.830458i	$0.311921\pi$
$420$		0			0
$421$	−7223.15			−0.836187			−0.418094	−	0.908404i	$-0.637302\pi$
$421$	−7223.15			−0.836187			−0.418094	+	0.908404i	$0.637302\pi$
$422$		0			0
$423$			3047.99i			0.350351i
$424$		0			0
$425$	−1051.13	−	5534.06i	−0.119970	−	0.631626i
$426$		0			0
$427$		−	9609.87i		−	1.08912i
$428$		0			0
$429$	−7239.37			−0.814732
$430$		0			0
$431$	−23.3962			−0.00261474			−0.00130737	−	0.999999i	$-0.500416\pi$
$431$	−23.3962			−0.00261474			−0.00130737	+	0.999999i	$0.500416\pi$
$432$		0			0
$433$			1834.11i			0.203560i	0.994807	+	0.101780i	$0.0324539\pi$
$433$			1834.11i			0.203560i	−0.994807	+	0.101780i	$0.967546\pi$
$434$		0			0
$435$	230.534	+	2449.16i	0.0254097	+	0.269950i
$436$		0			0
$437$		−	1432.26i		−	0.156783i
$438$		0			0
$439$	1275.02			0.138618			0.0693088	−	0.997595i	$-0.477921\pi$
$439$	1275.02			0.138618			0.0693088	+	0.997595i	$0.477921\pi$
$440$		0			0
$441$	783.518			0.0846040
$442$		0			0
$443$		−	11846.8i		−	1.27056i	−0.772283	−	0.635278i	$-0.780885\pi$
$443$		−	11846.8i		−	1.27056i	0.772283	−	0.635278i	$-0.219115\pi$
$444$		0			0
$445$	−16317.7	+	1535.94i	−1.73827	+	0.163620i
$446$		0			0
$447$		−	14156.9i		−	1.49799i
$448$		0			0
$449$	−558.915			−0.0587457			−0.0293729	−	0.999569i	$-0.509351\pi$
$449$	−558.915			−0.0587457			−0.0293729	+	0.999569i	$0.509351\pi$
$450$		0			0
$451$	−20088.5			−2.09740
$452$		0			0
$453$			10237.8i			1.06184i
$454$		0			0
$455$	−4259.48	+	400.935i	−0.438874	+	0.0413102i
$456$		0			0
$457$			12985.6i			1.32919i	0.747202	+	0.664597i	$0.231397\pi$
$457$			12985.6i			1.32919i	−0.747202	+	0.664597i	$0.768603\pi$
$458$		0			0
$459$	4738.74			0.481885
$460$		0			0
$461$	−12836.9			−1.29691			−0.648455	−	0.761253i	$-0.724585\pi$
$461$	−12836.9			−1.29691			−0.648455	+	0.761253i	$0.724585\pi$
$462$		0			0
$463$			6615.69i			0.664055i	0.943270	+	0.332027i	$0.107733\pi$
$463$			6615.69i			0.664055i	−0.943270	+	0.332027i	$0.892267\pi$
$464$		0			0
$465$	1001.34	+	10638.1i	0.0998620	+	1.06092i
$466$		0			0
$467$		−	9792.44i		−	0.970321i	−0.874425	−	0.485161i	$-0.838761\pi$
$467$		−	9792.44i		−	0.970321i	0.874425	−	0.485161i	$-0.161239\pi$
$468$		0			0
$469$	702.124			0.0691281
$470$		0			0
$471$	5283.89			0.516919
$472$		0			0
$473$		−	6443.99i		−	0.626416i
$474$		0			0
$475$	7647.28	−	1452.51i	0.738698	−	0.140307i
$476$		0			0
$477$		−	6676.27i		−	0.640850i
$478$		0			0
$479$	2085.71			0.198953			0.0994764	−	0.995040i	$-0.468283\pi$
$479$	2085.71			0.198953			0.0994764	+	0.995040i	$0.468283\pi$
$480$		0			0
$481$	4477.39			0.424431
$482$		0			0
$483$			2867.51i			0.270137i
$484$		0			0
$485$	1114.32	+	11838.4i	0.104327	+	1.10836i
$486$		0			0
$487$			14137.0i			1.31542i	0.753272	+	0.657709i	$0.228474\pi$
$487$			14137.0i			1.31542i	−0.753272	+	0.657709i	$0.771526\pi$
$488$		0			0
$489$	−16694.5			−1.54387
$490$		0			0
$491$	−2927.63			−0.269087			−0.134544	−	0.990908i	$-0.542957\pi$
$491$	−2927.63			−0.269087			−0.134544	+	0.990908i	$0.542957\pi$
$492$		0			0
$493$		−	1638.42i		−	0.149677i
$494$		0			0
$495$	−6899.04	+	649.390i	−0.626442	+	0.0589655i
$496$		0			0
$497$		−	4940.28i		−	0.445879i
$498$		0			0
$499$	10660.6			0.956384			0.478192	−	0.878255i	$-0.341292\pi$
$499$	10660.6			0.956384			0.478192	+	0.878255i	$0.341292\pi$
$500$		0			0
$501$	23295.8			2.07741
$502$		0			0
$503$			14813.7i			1.31314i	0.754264	+	0.656571i	$0.227994\pi$
$503$			14813.7i			1.31314i	−0.754264	+	0.656571i	$0.772006\pi$
$504$		0			0
$505$	7083.61	−	666.763i	0.624191	−	0.0587536i
$506$		0			0
$507$			11207.8i			0.981763i
$508$		0			0
$509$	−699.201			−0.0608872			−0.0304436	−	0.999536i	$-0.509692\pi$
$509$	−699.201			−0.0608872			−0.0304436	+	0.999536i	$0.509692\pi$
$510$		0			0
$511$	−6788.23			−0.587659
$512$		0			0
$513$			6548.27i			0.563573i
$514$		0			0
$515$	−1108.33	−	11774.8i	−0.0948331	−	1.00750i
$516$		0			0
$517$		−	20396.4i		−	1.73508i
$518$		0			0
$519$	−9909.63			−0.838120
$520$		0			0
$521$	−17670.4			−1.48590			−0.742952	−	0.669345i	$-0.766575\pi$
$521$	−17670.4			−1.48590			−0.742952	+	0.669345i	$0.766575\pi$
$522$		0			0
$523$			20531.1i			1.71656i	0.513181	+	0.858280i	$0.328467\pi$
$523$			20531.1i			1.71656i	−0.513181	+	0.858280i	$0.671533\pi$
$524$		0			0
$525$	−15310.6	+	2908.06i	−1.27278	+	0.241749i
$526$		0			0
$527$		−	7116.56i		−	0.588240i
$528$		0			0
$529$	−529.000			−0.0434783
$530$		0			0
$531$	6571.26			0.537040
$532$		0			0
$533$		−	5793.96i		−	0.470852i
$534$		0			0
$535$	1246.86	+	13246.5i	0.100760	+	1.07046i
$536$		0			0
$537$			5440.86i			0.437226i
$538$		0			0
$539$	−5243.11			−0.418992
$540$		0			0
$541$	14922.9			1.18593			0.592963	−	0.805230i	$-0.297958\pi$
$541$	14922.9			1.18593			0.592963	+	0.805230i	$0.297958\pi$
$542$		0			0
$543$		−	18764.8i		−	1.48301i
$544$		0			0
$545$	−3979.14	+	374.547i	−0.312748	+	0.0294382i
$546$		0			0
$547$			18004.4i			1.40733i	0.710530	+	0.703667i	$0.248455\pi$
$547$			18004.4i			1.40733i	−0.710530	+	0.703667i	$0.751545\pi$
$548$		0			0
$549$	4489.29			0.348995
$550$		0			0
$551$	2264.06			0.175049
$552$		0			0
$553$			785.504i			0.0604033i
$554$		0			0
$555$	16237.7	−	1528.41i	1.24189	−	0.116896i
$556$		0			0
$557$		−	13928.8i		−	1.05958i	−0.848130	−	0.529788i	$-0.822272\pi$
$557$		−	13928.8i		−	1.05958i	0.848130	−	0.529788i	$-0.177728\pi$
$558$		0			0
$559$	1858.59			0.140626
$560$		0			0
$561$	17563.3			1.32179
$562$		0			0
$563$		−	13981.6i		−	1.04663i	−0.852138	−	0.523317i	$-0.824694\pi$
$563$		−	13981.6i		−	1.04663i	0.852138	−	0.523317i	$-0.175306\pi$
$564$		0			0
$565$	−1425.07	−	15139.8i	−0.106112	−	1.12732i
$566$		0			0
$567$		−	18463.4i		−	1.36753i
$568$		0			0
$569$	−1959.28			−0.144354			−0.0721768	−	0.997392i	$-0.522995\pi$
$569$	−1959.28			−0.144354			−0.0721768	+	0.997392i	$0.522995\pi$
$570$		0			0
$571$	25217.7			1.84821			0.924105	−	0.382138i	$-0.124812\pi$
$571$	25217.7			1.84821			0.924105	+	0.382138i	$0.124812\pi$
$572$		0			0
$573$		−	19462.9i		−	1.41898i
$574$		0			0
$575$	−536.480	−	2824.50i	−0.0389092	−	0.204852i
$576$		0			0
$577$		−	3014.07i		−	0.217465i	−0.994071	−	0.108732i	$-0.965321\pi$
$577$		−	3014.07i		−	0.217465i	0.994071	−	0.108732i	$-0.0346792\pi$
$578$		0			0
$579$	12658.8			0.908607
$580$		0			0
$581$	−10972.8			−0.783526
$582$		0			0
$583$			44676.0i			3.17374i
$584$		0			0
$585$	−187.298	−	1989.83i	−0.0132373	−	0.140632i
$586$		0			0
$587$			22159.8i			1.55815i	0.626933	+	0.779073i	$0.284310\pi$
$587$			22159.8i			1.55815i	−0.626933	+	0.779073i	$0.715690\pi$
$588$		0			0
$589$	9834.08			0.687956
$590$		0			0
$591$	18079.7			1.25837
$592$		0			0
$593$		−	7046.06i		−	0.487937i	−0.969783	−	0.243969i	$-0.921551\pi$
$593$		−	7046.06i		−	0.487937i	0.969783	−	0.243969i	$-0.0784494\pi$
$594$		0			0
$595$	10333.9	−	972.703i	0.712013	−	0.0670201i
$596$		0			0
$597$			7586.40i			0.520085i
$598$		0			0
$599$	−1288.33			−0.0878791			−0.0439395	−	0.999034i	$-0.513991\pi$
$599$	−1288.33			−0.0878791			−0.0439395	+	0.999034i	$0.513991\pi$
$600$		0			0
$601$	−24289.7			−1.64858			−0.824292	−	0.566165i	$-0.808427\pi$
$601$	−24289.7			−1.64858			−0.824292	+	0.566165i	$0.808427\pi$
$602$		0			0
$603$			328.000i			0.0221512i
$604$		0			0
$605$	31351.2	−	2951.01i	2.10679	−	0.198307i
$606$		0			0
$607$		−	24513.6i		−	1.63917i	−0.572959	−	0.819584i	$-0.694205\pi$
$607$		−	24513.6i		−	1.63917i	0.572959	−	0.819584i	$-0.305795\pi$
$608$		0			0
$609$	−4532.86			−0.301610
$610$		0			0
$611$	5882.78			0.389512
$612$		0			0
$613$		−	11399.8i		−	0.751112i	−0.926800	−	0.375556i	$-0.877452\pi$
$613$		−	11399.8i		−	0.751112i	0.926800	−	0.375556i	$-0.122548\pi$
$614$		0			0
$615$	−1977.84	−	21012.3i	−0.129682	−	1.37772i
$616$		0			0
$617$			2015.09i			0.131482i	0.997837	+	0.0657412i	$0.0209412\pi$
$617$			2015.09i			0.131482i	−0.997837	+	0.0657412i	$0.979059\pi$
$618$		0			0
$619$	−5263.62			−0.341782			−0.170891	−	0.985290i	$-0.554665\pi$
$619$	−5263.62			−0.341782			−0.170891	+	0.985290i	$0.554665\pi$
$620$		0			0
$621$	2418.58			0.156287
$622$		0			0
$623$		−	30200.4i		−	1.94214i
$624$		0			0
$625$	14536.9	−	5728.88i	0.930359	−	0.366649i
$626$		0			0
$627$			24270.0i			1.54586i
$628$		0			0
$629$	−10862.5			−0.688581
$630$		0			0
$631$	11571.7			0.730050			0.365025	−	0.930998i	$-0.381060\pi$
$631$	11571.7			0.730050			0.365025	+	0.930998i	$0.381060\pi$
$632$		0			0
$633$			14521.5i			0.911816i
$634$		0			0
$635$	−1789.25	−	19008.7i	−0.111817	−	1.18793i
$636$		0			0
$637$		−	1512.23i		−	0.0940608i
$638$		0			0
$639$	2307.87			0.142876
$640$		0			0
$641$	4848.02			0.298729			0.149364	−	0.988782i	$-0.452277\pi$
$641$	4848.02			0.298729			0.149364	+	0.988782i	$0.452277\pi$
$642$		0			0
$643$			776.608i			0.0476305i	0.999716	+	0.0238153i	$0.00758135\pi$
$643$			776.608i			0.0476305i	−0.999716	+	0.0238153i	$0.992419\pi$
$644$		0			0
$645$	6740.35	−	634.452i	0.411474	−	0.0387311i
$646$		0			0
$647$			912.712i			0.0554597i	0.999615	+	0.0277298i	$0.00882781\pi$
$647$			912.712i			0.0554597i	−0.999615	+	0.0277298i	$0.991172\pi$
$648$		0			0
$649$	−43973.3			−2.65963
$650$		0			0
$651$	−19688.7			−1.18535
$652$		0			0
$653$		−	24378.6i		−	1.46096i	−0.682934	−	0.730480i	$-0.739297\pi$
$653$		−	24378.6i		−	1.46096i	0.682934	−	0.730480i	$-0.260703\pi$
$654$		0			0
$655$	8052.58	−	757.970i	0.480367	−	0.0452158i
$656$		0			0
$657$		−	3171.15i		−	0.188308i
$658$		0			0
$659$	28185.8			1.66610			0.833052	−	0.553194i	$-0.186591\pi$
$659$	28185.8			1.66610			0.833052	+	0.553194i	$0.186591\pi$
$660$		0			0
$661$	−19575.0			−1.15186			−0.575929	−	0.817499i	$-0.695360\pi$
$661$	−19575.0			−1.15186			−0.575929	+	0.817499i	$0.695360\pi$
$662$		0			0
$663$			5065.65i			0.296732i
$664$		0			0
$665$	1344.14	+	14280.0i	0.0783811	+	0.832712i
$666$		0			0
$667$		−	836.225i		−	0.0485438i
$668$		0			0
$669$	9982.00			0.576871
$670$		0			0
$671$	−30041.3			−1.72836
$672$		0			0
$673$		−	16634.6i		−	0.952773i	−0.879236	−	0.476387i	$-0.841946\pi$
$673$		−	16634.6i		−	0.952773i	0.879236	−	0.476387i	$-0.158054\pi$
$674$		0			0
$675$	2452.78	+	12913.6i	0.139863	+	0.736363i
$676$		0			0
$677$		−	22180.9i		−	1.25920i	−0.776918	−	0.629602i	$-0.783218\pi$
$677$		−	22180.9i		−	1.25920i	0.776918	−	0.629602i	$-0.216782\pi$
$678$		0			0
$679$	−21910.3			−1.23835
$680$		0			0
$681$	38129.0			2.14553
$682$		0			0
$683$		−	12085.9i		−	0.677095i	−0.940949	−	0.338547i	$-0.890064\pi$
$683$		−	12085.9i		−	0.677095i	0.940949	−	0.338547i	$-0.109936\pi$
$684$		0			0
$685$	−3078.67	−	32707.5i	−0.171723	−	1.82436i
$686$		0			0
$687$			2829.99i			0.157163i
$688$		0			0
$689$	−12885.5			−0.712482
$690$		0			0
$691$	−33047.9			−1.81940			−0.909698	−	0.415271i	$-0.863687\pi$
$691$	−33047.9			−1.81940			−0.909698	+	0.415271i	$0.863687\pi$
$692$		0			0
$693$		−	12768.6i		−	0.699912i
$694$		0			0
$695$	−7345.07	+	691.374i	−0.400884	+	0.0377343i
$696$		0			0
$697$			14056.6i			0.763893i
$698$		0			0
$699$	−38699.1			−2.09404
$700$		0			0
$701$	−16463.3			−0.887032			−0.443516	−	0.896266i	$-0.646269\pi$
$701$	−16463.3			−0.887032			−0.443516	+	0.896266i	$0.646269\pi$
$702$		0			0
$703$		−	15010.5i		−	0.805307i
$704$		0			0
$705$	21334.5	−	2008.16i	1.13972	−	0.107279i
$706$		0			0
$707$			13110.2i			0.697397i
$708$		0			0
$709$	10234.4			0.542119			0.271059	−	0.962563i	$-0.412626\pi$
$709$	10234.4			0.542119			0.271059	+	0.962563i	$0.412626\pi$
$710$		0			0
$711$	−366.951			−0.0193555
$712$		0			0
$713$		−	3632.19i		−	0.190781i
$714$		0			0
$715$	1253.36	+	13315.5i	0.0655564	+	0.696463i
$716$		0			0
$717$		−	2143.56i		−	0.111650i
$718$		0			0
$719$	17791.3			0.922813			0.461406	−	0.887189i	$-0.347345\pi$
$719$	17791.3			0.922813			0.461406	+	0.887189i	$0.347345\pi$
$720$		0			0
$721$	21792.6			1.12566
$722$		0			0
$723$		−	15514.4i		−	0.798046i
$724$		0			0
$725$	4464.87	−	848.049i	0.228719	−	0.0434424i
$726$		0			0
$727$		−	7191.32i		−	0.366866i	−0.983032	−	0.183433i	$-0.941279\pi$
$727$		−	7191.32i		−	0.366866i	0.983032	−	0.183433i	$-0.0587210\pi$
$728$		0			0
$729$	−8556.98			−0.434740
$730$		0			0
$731$	−4509.10			−0.228146
$732$		0			0
$733$		−	21919.3i		−	1.10451i	−0.833674	−	0.552257i	$-0.813767\pi$
$733$		−	21919.3i		−	1.10451i	0.833674	−	0.552257i	$-0.186233\pi$
$734$		0			0
$735$	−516.219	−	5484.25i	−0.0259061	−	0.275224i
$736$		0			0
$737$		−	2194.90i		−	0.109702i
$738$		0			0
$739$	20508.9			1.02088			0.510440	−	0.859913i	$-0.329482\pi$
$739$	20508.9			1.02088			0.510440	+	0.859913i	$0.329482\pi$
$740$		0			0
$741$	−7000.01			−0.347034
$742$		0			0
$743$			23705.1i			1.17047i	0.810865	+	0.585234i	$0.198997\pi$
$743$			23705.1i			1.17047i	−0.810865	+	0.585234i	$0.801003\pi$
$744$		0			0
$745$	−26039.1	+	2451.00i	−1.28054	+	0.120534i
$746$		0			0
$747$		−	5125.98i		−	0.251071i
$748$		0			0
$749$	−24516.3			−1.19600
$750$		0			0
$751$	−13277.9			−0.645163			−0.322582	−	0.946542i	$-0.604551\pi$
$751$	−13277.9			−0.645163			−0.322582	+	0.946542i	$0.604551\pi$
$752$		0			0
$753$			32351.4i			1.56567i
$754$		0			0
$755$	18830.5	−	1772.47i	0.907698	−	0.0854395i
$756$		0			0
$757$		−	23877.9i		−	1.14644i	−0.819400	−	0.573222i	$-0.805693\pi$
$757$		−	23877.9i		−	1.14644i	0.819400	−	0.573222i	$-0.194307\pi$
$758$		0			0
$759$	8964.07			0.428689
$760$		0			0
$761$	10405.3			0.495651			0.247825	−	0.968805i	$-0.420284\pi$
$761$	10405.3			0.495651			0.247825	+	0.968805i	$0.420284\pi$
$762$		0			0
$763$		−	7364.52i		−	0.349428i
$764$		0			0
$765$	454.402	+	4827.51i	0.0214757	+	0.228156i
$766$		0			0
$767$		−	12682.9i		−	0.597068i
$768$		0			0
$769$	20163.4			0.945530			0.472765	−	0.881189i	$-0.343256\pi$
$769$	20163.4			0.945530			0.472765	+	0.881189i	$0.343256\pi$
$770$		0			0
$771$	−480.207			−0.0224309
$772$		0			0
$773$		−	17732.9i		−	0.825109i	−0.910933	−	0.412554i	$-0.864637\pi$
$773$		−	17732.9i		−	0.825109i	0.910933	−	0.412554i	$-0.135363\pi$
$774$		0			0
$775$	19393.4	−	3683.55i	0.898881	−	0.170732i
$776$		0			0
$777$			30052.4i			1.38755i
$778$		0			0
$779$	−19424.3			−0.893386
$780$		0			0
$781$	−15443.7			−0.707579
$782$		0			0
$783$			3823.21i			0.174496i
$784$		0			0
$785$	−914.802	−	9718.75i	−0.0415932	−	0.441882i
$786$		0			0
$787$		−	28474.5i		−	1.28972i	−0.764302	−	0.644858i	$-0.776917\pi$
$787$		−	28474.5i		−	1.28972i	0.764302	−	0.644858i	$-0.223083\pi$
$788$		0			0
$789$	11661.4			0.526180
$790$		0			0
$791$	28020.4			1.25953
$792$		0			0
$793$		−	8664.56i		−	0.388004i
$794$		0			0
$795$	−46730.6	+	4398.64i	−2.08474	+	0.196231i
$796$		0			0
$797$		−	180.323i		−	0.00801428i	−0.999992	−	0.00400714i	$-0.998724\pi$
$797$		−	180.323i		−	0.00801428i	0.999992	−	0.00400714i	$-0.00127552\pi$
$798$		0			0
$799$	−14272.1			−0.631930
$800$		0			0
$801$	14108.2			0.622335
$802$		0			0
$803$			21220.6i			0.932575i
$804$		0			0
$805$	5274.26	−	496.454i	0.230923	−	0.0217363i
$806$		0			0
$807$			17523.3i			0.764375i
$808$		0			0
$809$	−26491.8			−1.15130			−0.575651	−	0.817695i	$-0.695251\pi$
$809$	−26491.8			−1.15130			−0.575651	+	0.817695i	$0.695251\pi$
$810$		0			0
$811$	−23465.1			−1.01599			−0.507996	−	0.861359i	$-0.669614\pi$
$811$	−23465.1			−1.01599			−0.507996	+	0.861359i	$0.669614\pi$
$812$		0			0
$813$			15706.6i			0.677557i
$814$		0			0
$815$	2890.33	+	30706.5i	0.124226	+	1.31976i
$816$		0			0
$817$		−	6230.93i		−	0.266821i
$818$		0			0
$819$	3682.75			0.157125
$820$		0			0
$821$	23056.7			0.980127			0.490063	−	0.871687i	$-0.336974\pi$
$821$	23056.7			0.980127			0.490063	+	0.871687i	$0.336974\pi$
$822$		0			0
$823$			8271.27i			0.350326i	0.984539	+	0.175163i	$0.0560453\pi$
$823$			8271.27i			0.350326i	−0.984539	+	0.175163i	$0.943955\pi$
$824$		0			0
$825$	9090.83	+	47862.1i	0.383639	+	2.01981i
$826$		0			0
$827$		−	1740.84i		−	0.0731980i	−0.999330	−	0.0365990i	$-0.988348\pi$
$827$		−	1740.84i		−	0.0731980i	0.999330	−	0.0365990i	$-0.0116524\pi$
$828$		0			0
$829$	−8857.96			−0.371109			−0.185555	−	0.982634i	$-0.559408\pi$
$829$	−8857.96			−0.371109			−0.185555	+	0.982634i	$0.559408\pi$
$830$		0			0
$831$	6668.27			0.278363
$832$		0			0
$833$			3668.80i			0.152601i
$834$		0			0
$835$	−4033.22	−	42848.4i	−0.167156	−	1.77585i
$836$		0			0
$837$			16606.3i			0.685782i
$838$		0			0
$839$	−40285.0			−1.65768			−0.828838	−	0.559488i	$-0.810998\pi$
$839$	−40285.0			−1.65768			−0.828838	+	0.559488i	$0.810998\pi$
$840$		0			0
$841$	−23067.1			−0.945800
$842$		0			0
$843$			27150.9i			1.10929i
$844$		0			0
$845$	20614.6	−	1940.40i	0.839248	−	0.0789964i
$846$		0			0
$847$			58024.1i			2.35388i
$848$		0			0
$849$	21598.8			0.873109
$850$		0			0
$851$	−5544.08			−0.223324
$852$		0			0
$853$			18329.9i			0.735763i	0.929873	+	0.367881i	$0.119917\pi$
$853$			18329.9i			0.735763i	−0.929873	+	0.367881i	$0.880083\pi$
$854$		0			0
$855$	−6670.94	+	627.919i	−0.266832	+	0.0251162i
$856$		0			0
$857$		−	17355.1i		−	0.691760i	−0.938279	−	0.345880i	$-0.887580\pi$
$857$		−	17355.1i		−	0.691760i	0.938279	−	0.345880i	$-0.112420\pi$
$858$		0			0
$859$	23150.2			0.919530			0.459765	−	0.888041i	$-0.347934\pi$
$859$	23150.2			0.919530			0.459765	+	0.888041i	$0.347934\pi$
$860$		0			0
$861$	38889.2			1.53930
$862$		0			0
$863$		−	35000.5i		−	1.38057i	−0.723539	−	0.690283i	$-0.757486\pi$
$863$		−	35000.5i		−	1.38057i	0.723539	−	0.690283i	$-0.242514\pi$
$864$		0			0
$865$	1715.66	+	18227.0i	0.0674383	+	0.716457i
$866$		0			0
$867$			17442.7i			0.683257i
$868$		0			0
$869$	2455.55			0.0958560
$870$		0			0
$871$	633.057			0.0246272
$872$		0			0
$873$		−	10235.5i		−	0.396813i
$874$		0			0
$875$	7999.57	+	27657.5i	0.309068	+	1.06857i
$876$		0			0
$877$			42115.6i			1.62160i	0.585322	+	0.810801i	$0.300968\pi$
$877$			42115.6i			1.62160i	−0.585322	+	0.810801i	$0.699032\pi$
$878$		0			0
$879$	−39124.9			−1.50131
$880$		0			0
$881$	18338.4			0.701290			0.350645	−	0.936508i	$-0.385962\pi$
$881$	18338.4			0.701290			0.350645	+	0.936508i	$0.385962\pi$
$882$		0			0
$883$			29365.3i			1.11916i	0.828776	+	0.559581i	$0.189038\pi$
$883$			29365.3i			1.11916i	−0.828776	+	0.559581i	$0.810962\pi$
$884$		0			0
$885$	−4329.45	−	45995.6i	−0.164444	−	1.74703i
$886$		0			0
$887$			2104.59i			0.0796676i	0.999206	+	0.0398338i	$0.0126828\pi$
$887$			2104.59i			0.0796676i	−0.999206	+	0.0398338i	$0.987317\pi$
$888$		0			0
$889$	35181.0			1.32726
$890$		0			0
$891$	−57718.2			−2.17018
$892$		0			0
$893$		−	19722.1i		−	0.739053i
$894$		0			0
$895$	10007.5	−	941.978i	0.373757	−	0.0351809i
$896$		0			0
$897$			2585.44i			0.0962376i
$898$		0			0
$899$	5741.64			0.213008
$900$		0			0
$901$	31261.4			1.15590
$902$		0			0
$903$			12474.9i			0.459733i
$904$		0			0
$905$	−34514.4	+	3248.76i	−1.26773	+	0.119329i
$906$		0			0
$907$		−	26839.3i		−	0.982562i	−0.871001	−	0.491281i	$-0.836529\pi$
$907$		−	26839.3i		−	0.982562i	0.871001	−	0.491281i	$-0.163471\pi$
$908$		0			0
$909$	−6124.48			−0.223472
$910$		0			0
$911$	37094.3			1.34905			0.674527	−	0.738250i	$-0.264348\pi$
$911$	37094.3			1.34905			0.674527	+	0.738250i	$0.264348\pi$
$912$		0			0
$913$			34301.9i			1.24340i
$914$		0			0
$915$	−2957.76	−	31422.8i	−0.106864	−	1.13531i
$916$		0			0
$917$			14903.6i			0.536705i
$918$		0			0
$919$	17240.0			0.618820			0.309410	−	0.950929i	$-0.399868\pi$
$919$	17240.0			0.618820			0.309410	+	0.950929i	$0.399868\pi$
$920$		0			0
$921$	−3265.93			−0.116847
$922$		0			0
$923$		−	4454.30i		−	0.158846i
$924$		0			0
$925$	−5622.48	−	29601.6i	−0.199855	−	1.05221i
$926$		0			0
$927$			10180.5i			0.360702i
$928$		0			0
$929$	55819.4			1.97134			0.985670	−	0.168685i	$-0.0539520\pi$
$929$	55819.4			1.97134			0.985670	+	0.168685i	$0.0539520\pi$
$930$		0			0
$931$	−5069.76			−0.178469
$932$		0			0
$933$		−	26730.1i		−	0.937947i
$934$		0			0
$935$	−3040.75	−	32304.6i	−0.106356	−	1.12992i
$936$		0			0
$937$		−	2884.16i		−	0.100556i	−0.998735	−	0.0502782i	$-0.983989\pi$
$937$		−	2884.16i		−	0.100556i	0.998735	−	0.0502782i	$-0.0160108\pi$
$938$		0			0
$939$	14905.6			0.518026
$940$		0			0
$941$	−11743.4			−0.406826			−0.203413	−	0.979093i	$-0.565203\pi$
$941$	−11743.4			−0.406826			−0.203413	+	0.979093i	$0.565203\pi$
$942$		0			0
$943$			7174.31i			0.247749i
$944$		0			0
$945$	−24113.9	+	2269.78i	−0.830079	+	0.0781333i
$946$		0			0
$947$			4799.90i			0.164705i	0.996603	+	0.0823527i	$0.0262434\pi$
$947$			4799.90i			0.164705i	−0.996603	+	0.0823527i	$0.973757\pi$
$948$		0			0
$949$	−6120.48			−0.209356
$950$		0			0
$951$	55421.7			1.88977
$952$		0			0
$953$		−	9114.43i		−	0.309806i	−0.987930	−	0.154903i	$-0.950493\pi$
$953$		−	9114.43i		−	0.309806i	0.987930	−	0.154903i	$-0.0495065\pi$
$954$		0			0
$955$	−35798.5	+	3369.62i	−1.21300	+	0.114176i
$956$		0			0
$957$			14170.1i			0.478635i
$958$		0			0
$959$	60534.3			2.03833
$960$		0			0
$961$	−4851.86			−0.162863
$962$		0			0
$963$		−	11452.9i		−	0.383245i
$964$		0			0
$965$	−2191.63	−	23283.6i	−0.0731100	−	0.776711i
$966$		0			0
$967$			44321.2i			1.47391i	0.675939	+	0.736957i	$0.263738\pi$
$967$			44321.2i			1.47391i	−0.675939	+	0.736957i	$0.736262\pi$
$968$		0			0
$969$	16982.6			0.563015
$970$		0			0
$971$	−5027.49			−0.166159			−0.0830793	−	0.996543i	$-0.526475\pi$
$971$	−5027.49			−0.166159			−0.0830793	+	0.996543i	$0.526475\pi$
$972$		0			0
$973$		−	13594.1i		−	0.447901i
$974$		0			0
$975$	−13804.5	+	2621.99i	−0.453433	+	0.0861241i
$976$		0			0
$977$			48350.6i			1.58329i	0.610982	+	0.791645i	$0.290775\pi$
$977$			48350.6i			1.58329i	−0.610982	+	0.791645i	$0.709225\pi$
$978$		0			0
$979$	−94409.0			−3.08205
$980$		0			0
$981$	3440.36			0.111970
$982$		0			0
$983$			35027.6i			1.13653i	0.822846	+	0.568264i	$0.192385\pi$
$983$			35027.6i			1.13653i	−0.822846	+	0.568264i	$0.807615\pi$
$984$		0			0
$985$	−3130.14	−	33254.2i	−0.101253	−	1.07570i
$986$		0			0
$987$			39485.4i			1.27339i
$988$		0			0
$989$	−2301.38			−0.0739935
$990$		0			0
$991$	−1395.17			−0.0447216			−0.0223608	−	0.999750i	$-0.507118\pi$
$991$	−1395.17			−0.0447216			−0.0223608	+	0.999750i	$0.507118\pi$
$992$		0			0
$993$		−	46734.9i		−	1.49354i
$994$		0			0
$995$	13953.8	−	1313.44i	0.444588	−	0.0418480i
$996$		0			0
$997$			37604.2i			1.19452i	0.802048	+	0.597260i	$0.203744\pi$
$997$			37604.2i			1.19452i	−0.802048	+	0.597260i	$0.796256\pi$
$998$		0			0
$999$	25347.5			0.802762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.4.c.a.369.26	yes	32
5.2	odd	4		2300.4.a.l.1.13		16
5.3	odd	4		2300.4.a.k.1.4		16
5.4	even	2	inner	460.4.c.a.369.7	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.c.a.369.7	✓	32	5.4	even	2	inner
460.4.c.a.369.26	yes	32	1.1	even	1	trivial
2300.4.a.k.1.4		16	5.3	odd	4
2300.4.a.l.1.13		16	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	460.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.05177i q^{3} +(11.1311 - 1.04775i) q^{5} +20.6013i q^{7} -9.62397 q^{9} +64.4013 q^{11} +18.5748i q^{13} +(6.34073 + 67.3631i) q^{15} -45.0640i q^{17} +62.2720 q^{19} -124.674 q^{21} -23.0000i q^{23} +(122.804 - 23.3252i) q^{25} +105.156i q^{27} +36.3576 q^{29} +157.921 q^{31} +389.742i q^{33} +(21.5849 + 229.316i) q^{35} -241.047i q^{37} -112.410 q^{39} -311.927 q^{41} -100.060i q^{43} +(-107.126 + 10.0835i) q^{45} -316.708i q^{47} -81.4132 q^{49} +272.717 q^{51} +693.712i q^{53} +(716.860 - 67.4763i) q^{55} +376.856i q^{57} -682.801 q^{59} -466.470 q^{61} -198.266i q^{63} +(19.4616 + 206.758i) q^{65} -34.0816i q^{67} +139.191 q^{69} -239.804 q^{71} +329.505i q^{73} +(141.159 + 743.185i) q^{75} +1326.75i q^{77} +38.1289 q^{79} -896.226 q^{81} +532.627i q^{83} +(-47.2156 - 501.613i) q^{85} +220.028i q^{87} -1465.95 q^{89} -382.664 q^{91} +955.704i q^{93} +(693.159 - 65.2453i) q^{95} +1063.54i q^{97} -619.796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 12 q^{5} - 220 q^{9} - 8 q^{11} - 168 q^{15} + 144 q^{19} - 32 q^{21} + 100 q^{25} - 76 q^{29} + 652 q^{31} + 320 q^{35} - 1128 q^{39} - 560 q^{41} + 1208 q^{45} - 2224 q^{49} + 3192 q^{51} - 1304 q^{55}+ \cdots + 1952 q^{99}+O(q^{100}) \) 32 * q - 12 * q^5 - 220 * q^9 - 8 * q^11 - 168 * q^15 + 144 * q^19 - 32 * q^21 + 100 * q^25 - 76 * q^29 + 652 * q^31 + 320 * q^35 - 1128 * q^39 - 560 * q^41 + 1208 * q^45 - 2224 * q^49 + 3192 * q^51 - 1304 * q^55 - 376 * q^59 + 1000 * q^61 + 636 * q^65 + 552 * q^69 - 732 * q^71 - 2608 * q^75 - 1088 * q^79 + 2160 * q^81 + 1932 * q^85 + 3016 * q^89 - 96 * q^91 + 32 * q^95 + 1952 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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