LMFDB - Embedded newform 480.4.h.a.191.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,4,Mod(191,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("480.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.h (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:	$28.3209168028$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.8
Character	$\chi$	$=$	480.191
Dual form			480.4.h.a.191.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-3.26404 + 4.04302i) q^{3} -5.00000i q^{5} -14.4221i q^{7} +(-5.69203 - 26.3932i) q^{9} +44.8665 q^{11} -42.0732 q^{13} +(20.2151 + 16.3202i) q^{15} +133.779i q^{17} -140.796i q^{19} +(58.3090 + 47.0745i) q^{21} -98.4863 q^{23} -25.0000 q^{25} +(125.287 + 63.1355i) q^{27} +135.968i q^{29} +295.061i q^{31} +(-146.446 + 181.396i) q^{33} -72.1107 q^{35} -329.068 q^{37} +(137.329 - 170.103i) q^{39} -249.262i q^{41} -345.344i q^{43} +(-131.966 + 28.4602i) q^{45} -137.582 q^{47} +135.002 q^{49} +(-540.873 - 436.662i) q^{51} +64.8334i q^{53} -224.332i q^{55} +(569.241 + 459.564i) q^{57} -325.168 q^{59} -833.723 q^{61} +(-380.646 + 82.0913i) q^{63} +210.366i q^{65} -197.315i q^{67} +(321.464 - 398.182i) q^{69} -471.780 q^{71} -592.769 q^{73} +(81.6011 - 101.076i) q^{75} -647.070i q^{77} -728.542i q^{79} +(-664.202 + 300.462i) q^{81} +278.801 q^{83} +668.897 q^{85} +(-549.722 - 443.806i) q^{87} +1137.49i q^{89} +606.786i q^{91} +(-1192.94 - 963.093i) q^{93} -703.980 q^{95} -462.063 q^{97} +(-255.381 - 1184.17i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 20 q^{9} - 72 q^{11} - 72 q^{13} - 20 q^{15} - 68 q^{21} - 96 q^{23} - 600 q^{25} - 168 q^{27} - 80 q^{33} - 504 q^{37} + 456 q^{39} - 220 q^{45} - 432 q^{47} - 816 q^{49} - 1240 q^{51} + 40 q^{57}+ \cdots - 3160 q^{99}+O(q^{100}) $ 24 * q + 20 * q^9 - 72 * q^11 - 72 * q^13 - 20 * q^15 - 68 * q^21 - 96 * q^23 - 600 * q^25 - 168 * q^27 - 80 * q^33 - 504 * q^37 + 456 * q^39 - 220 * q^45 - 432 * q^47 - 816 * q^49 - 1240 * q^51 + 40 * q^57 + 2184 * q^59 - 1080 * q^61 + 1720 * q^63 - 1548 * q^69 - 816 * q^71 - 1440 * q^73 - 2304 * q^81 + 2016 * q^83 + 2144 * q^87 - 112 * q^93 - 1800 * q^95 + 720 * q^97 - 3160 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−3.26404	+	4.04302i	−0.628166	+	0.778080i
$3$	−3.26404	+	4.04302i	−0.628166	+	0.778080i
$4$		0			0
$5$		−	5.00000i		−	0.447214i
$5$		−	5.00000i		−	0.447214i
$6$		0			0
$7$		−	14.4221i		−	0.778723i	−0.921085	−	0.389361i	$-0.872696\pi$
$7$		−	14.4221i		−	0.778723i	0.921085	−	0.389361i	$-0.127304\pi$
$8$		0			0
$9$	−5.69203	−	26.3932i	−0.210816	−	0.977526i
$10$		0			0
$11$	44.8665			1.22979			0.614897	−	0.788607i	$-0.289197\pi$
$11$	44.8665			1.22979			0.614897	+	0.788607i	$0.289197\pi$
$12$		0			0
$13$	−42.0732			−0.897616			−0.448808	−	0.893628i	$-0.648151\pi$
$13$	−42.0732			−0.897616			−0.448808	+	0.893628i	$0.648151\pi$
$14$		0			0
$15$	20.2151	+	16.3202i	0.347968	+	0.280924i
$16$		0			0
$17$			133.779i			1.90860i	0.298845	+	0.954302i	$0.403399\pi$
$17$			133.779i			1.90860i	−0.298845	+	0.954302i	$0.596601\pi$
$18$		0			0
$19$		−	140.796i		−	1.70004i	−0.526748	−	0.850022i	$-0.676589\pi$
$19$		−	140.796i		−	1.70004i	0.526748	−	0.850022i	$-0.323411\pi$
$20$		0			0
$21$	58.3090	+	47.0745i	0.605908	+	0.489167i
$22$		0			0
$23$	−98.4863			−0.892862			−0.446431	−	0.894818i	$-0.647305\pi$
$23$	−98.4863			−0.892862			−0.446431	+	0.894818i	$0.647305\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		0			0
$27$	125.287	+	63.1355i	0.893020	+	0.450016i
$28$		0			0
$29$			135.968i			0.870644i	0.900275	+	0.435322i	$0.143365\pi$
$29$			135.968i			0.870644i	−0.900275	+	0.435322i	$0.856635\pi$
$30$		0			0
$31$			295.061i			1.70950i	0.519038	+	0.854751i	$0.326290\pi$
$31$			295.061i			1.70950i	−0.519038	+	0.854751i	$0.673710\pi$
$32$		0			0
$33$	−146.446	+	181.396i	−0.772515	+	0.956878i
$34$		0			0
$35$	−72.1107			−0.348255
$36$		0			0
$37$	−329.068			−1.46212			−0.731061	−	0.682312i	$-0.760975\pi$
$37$	−329.068			−1.46212			−0.731061	+	0.682312i	$0.760975\pi$
$38$		0			0
$39$	137.329	−	170.103i	0.563851	−	0.698417i
$40$		0			0
$41$		−	249.262i		−	0.949468i	−0.880129	−	0.474734i	$-0.842544\pi$
$41$		−	249.262i		−	0.949468i	0.880129	−	0.474734i	$-0.157456\pi$
$42$		0			0
$43$		−	345.344i		−	1.22475i	−0.790566	−	0.612377i	$-0.790214\pi$
$43$		−	345.344i		−	1.22475i	0.790566	−	0.612377i	$-0.209786\pi$
$44$		0			0
$45$	−131.966	+	28.4602i	−0.437163	+	0.0942798i
$46$		0			0
$47$	−137.582			−0.426988			−0.213494	−	0.976944i	$-0.568484\pi$
$47$	−137.582			−0.426988			−0.213494	+	0.976944i	$0.568484\pi$
$48$		0			0
$49$	135.002			0.393591
$50$		0			0
$51$	−540.873	−	436.662i	−1.48505	−	1.19892i
$52$		0			0
$53$			64.8334i			0.168029i	0.996465	+	0.0840147i	$0.0267743\pi$
$53$			64.8334i			0.168029i	−0.996465	+	0.0840147i	$0.973226\pi$
$54$		0			0
$55$		−	224.332i		−	0.549981i
$56$		0			0
$57$	569.241	+	459.564i	1.32277	+	1.06791i
$58$		0			0
$59$	−325.168			−0.717514			−0.358757	−	0.933431i	$-0.616799\pi$
$59$	−325.168			−0.717514			−0.358757	+	0.933431i	$0.616799\pi$
$60$		0			0
$61$	−833.723			−1.74996			−0.874978	−	0.484162i	$-0.839124\pi$
$61$	−833.723			−1.74996			−0.874978	+	0.484162i	$0.839124\pi$
$62$		0			0
$63$	−380.646	+	82.0913i	−0.761221	+	0.164167i
$64$		0			0
$65$			210.366i			0.401426i
$66$		0			0
$67$		−	197.315i		−	0.359789i	−0.983686	−	0.179895i	$-0.942424\pi$
$67$		−	197.315i		−	0.359789i	0.983686	−	0.179895i	$-0.0575757\pi$
$68$		0			0
$69$	321.464	−	398.182i	0.560865	−	0.694718i
$70$		0			0
$71$	−471.780			−0.788591			−0.394296	−	0.918984i	$-0.629011\pi$
$71$	−471.780			−0.788591			−0.394296	+	0.918984i	$0.629011\pi$
$72$		0			0
$73$	−592.769			−0.950388			−0.475194	−	0.879881i	$-0.657622\pi$
$73$	−592.769			−0.950388			−0.475194	+	0.879881i	$0.657622\pi$
$74$		0			0
$75$	81.6011	−	101.076i	0.125633	−	0.155616i
$76$		0			0
$77$		−	647.070i		−	0.957669i
$78$		0			0
$79$		−	728.542i		−	1.03756i	−0.854907	−	0.518781i	$-0.826386\pi$
$79$		−	728.542i		−	1.03756i	0.854907	−	0.518781i	$-0.173614\pi$
$80$		0			0
$81$	−664.202	+	300.462i	−0.911113	+	0.412156i
$82$		0			0
$83$	278.801			0.368703			0.184352	−	0.982860i	$-0.440981\pi$
$83$	278.801			0.368703			0.184352	+	0.982860i	$0.440981\pi$
$84$		0			0
$85$	668.897			0.853553
$86$		0			0
$87$	−549.722	−	443.806i	−0.677430	−	0.546908i
$88$		0			0
$89$			1137.49i			1.35477i	0.735631	+	0.677383i	$0.236886\pi$
$89$			1137.49i			1.35477i	−0.735631	+	0.677383i	$0.763114\pi$
$90$		0			0
$91$			606.786i			0.698994i
$92$		0			0
$93$	−1192.94	−	963.093i	−1.33013	−	1.07385i
$94$		0			0
$95$	−703.980			−0.760282
$96$		0			0
$97$	−462.063			−0.483663			−0.241832	−	0.970318i	$-0.577748\pi$
$97$	−462.063			−0.483663			−0.241832	+	0.970318i	$0.577748\pi$
$98$		0			0
$99$	−255.381	−	1184.17i	−0.259261	−	1.20216i
$100$		0			0
$101$			318.476i			0.313758i	0.987618	+	0.156879i	$0.0501433\pi$
$101$			318.476i			0.313758i	−0.987618	+	0.156879i	$0.949857\pi$
$102$		0			0
$103$			1084.14i			1.03712i	0.855040	+	0.518562i	$0.173532\pi$
$103$			1084.14i			1.03712i	−0.855040	+	0.518562i	$0.826468\pi$
$104$		0			0
$105$	235.373	−	291.545i	0.218762	−	0.270970i
$106$		0			0
$107$	−1372.64			−1.24017			−0.620084	−	0.784535i	$-0.712902\pi$
$107$	−1372.64			−1.24017			−0.620084	+	0.784535i	$0.712902\pi$
$108$		0			0
$109$	−1117.24			−0.981766			−0.490883	−	0.871225i	$-0.663326\pi$
$109$	−1117.24			−0.981766			−0.490883	+	0.871225i	$0.663326\pi$
$110$		0			0
$111$	1074.09	−	1330.43i	0.918455	−	1.13765i
$112$		0			0
$113$			466.146i			0.388065i	0.980995	+	0.194032i	$0.0621567\pi$
$113$			466.146i			0.388065i	−0.980995	+	0.194032i	$0.937843\pi$
$114$		0			0
$115$			492.432i			0.399300i
$116$		0			0
$117$	239.482	+	1110.45i	0.189232	+	0.877443i
$118$		0			0
$119$	1929.38			1.48627
$120$		0			0
$121$	681.999			0.512396
$122$		0			0
$123$	1007.77	+	813.602i	0.738762	+	0.596423i
$124$		0			0
$125$			125.000i			0.0894427i
$126$		0			0
$127$			340.558i			0.237950i	0.992897	+	0.118975i	$0.0379608\pi$
$127$			340.558i			0.237950i	−0.992897	+	0.118975i	$0.962039\pi$
$128$		0			0
$129$	1396.23	+	1127.22i	0.952956	+	0.769348i
$130$		0			0
$131$	1226.37			0.817930			0.408965	−	0.912550i	$-0.365890\pi$
$131$	1226.37			0.817930			0.408965	+	0.912550i	$0.365890\pi$
$132$		0			0
$133$	−2030.58			−1.32386
$134$		0			0
$135$	315.678	−	626.436i	0.201253	−	0.399371i
$136$		0			0
$137$			2054.30i			1.28110i	0.767918	+	0.640549i	$0.221293\pi$
$137$			2054.30i			1.28110i	−0.767918	+	0.640549i	$0.778707\pi$
$138$		0			0
$139$		−	1331.26i		−	0.812345i	−0.913796	−	0.406172i	$-0.866863\pi$
$139$		−	1331.26i		−	0.812345i	0.913796	−	0.406172i	$-0.133137\pi$
$140$		0			0
$141$	449.075	−	556.248i	0.268219	−	0.332231i
$142$		0			0
$143$	−1887.68			−1.10388
$144$		0			0
$145$	679.841			0.389364
$146$		0			0
$147$	−440.652	+	545.815i	−0.247240	+	0.306245i
$148$		0			0
$149$		−	3011.30i		−	1.65567i	−0.560968	−	0.827837i	$-0.689571\pi$
$149$		−	3011.30i		−	1.65567i	0.560968	−	0.827837i	$-0.310429\pi$
$150$		0			0
$151$			1293.28i			0.696989i	0.937311	+	0.348494i	$0.113307\pi$
$151$			1293.28i			0.696989i	−0.937311	+	0.348494i	$0.886693\pi$
$152$		0			0
$153$	3530.86	−	761.476i	1.86571	−	0.402364i
$154$		0			0
$155$	1475.31			0.764513
$156$		0			0
$157$	−584.560			−0.297153			−0.148576	−	0.988901i	$-0.547469\pi$
$157$	−584.560			−0.297153			−0.148576	+	0.988901i	$0.547469\pi$
$158$		0			0
$159$	−262.123	−	211.619i	−0.130740	−	0.105550i
$160$		0			0
$161$			1420.38i			0.695292i
$162$		0			0
$163$			2997.44i			1.44035i	0.693790	+	0.720177i	$0.255940\pi$
$163$			2997.44i			1.44035i	−0.693790	+	0.720177i	$0.744060\pi$
$164$		0			0
$165$	906.980	+	732.230i	0.427929	+	0.345479i
$166$		0			0
$167$	−401.720			−0.186144			−0.0930720	−	0.995659i	$-0.529669\pi$
$167$	−401.720			−0.186144			−0.0930720	+	0.995659i	$0.529669\pi$
$168$		0			0
$169$	−426.845			−0.194285
$170$		0			0
$171$	−3716.06	+	801.416i	−1.66184	+	0.358396i
$172$		0			0
$173$			90.7400i			0.0398777i	0.999801	+	0.0199388i	$0.00634715\pi$
$173$			90.7400i			0.0398777i	−0.999801	+	0.0199388i	$0.993653\pi$
$174$		0			0
$175$			360.554i			0.155745i
$176$		0			0
$177$	1061.36	−	1314.66i	0.450717	−	0.558283i
$178$		0			0
$179$	3103.41			1.29587			0.647933	−	0.761697i	$-0.275634\pi$
$179$	3103.41			1.29587			0.647933	+	0.761697i	$0.275634\pi$
$180$		0			0
$181$	−2135.81			−0.877091			−0.438546	−	0.898709i	$-0.644506\pi$
$181$	−2135.81			−0.877091			−0.438546	+	0.898709i	$0.644506\pi$
$182$		0			0
$183$	2721.31	−	3370.76i	1.09926	−	1.36161i
$184$		0			0
$185$			1645.34i			0.653881i
$186$		0			0
$187$			6002.20i			2.34719i
$188$		0			0
$189$	910.550	−	1806.91i	0.350438	−	0.695415i
$190$		0			0
$191$	3520.17			1.33356			0.666782	−	0.745253i	$-0.267671\pi$
$191$	3520.17			1.33356			0.666782	+	0.745253i	$0.267671\pi$
$192$		0			0
$193$	−2066.06			−0.770561			−0.385281	−	0.922799i	$-0.625895\pi$
$193$	−2066.06			−0.770561			−0.385281	+	0.922799i	$0.625895\pi$
$194$		0			0
$195$	−850.514	−	686.644i	−0.312342	−	0.252162i
$196$		0			0
$197$		−	1341.14i		−	0.485037i	−0.970147	−	0.242518i	$-0.922027\pi$
$197$		−	1341.14i		−	0.485037i	0.970147	−	0.242518i	$-0.0779735\pi$
$198$		0			0
$199$		−	1149.61i		−	0.409515i	−0.978813	−	0.204757i	$-0.934359\pi$
$199$		−	1149.61i		−	0.409515i	0.978813	−	0.204757i	$-0.0656406\pi$
$200$		0			0
$201$	797.750	+	644.046i	0.279945	+	0.226007i
$202$		0			0
$203$	1960.95			0.677990
$204$		0			0
$205$	−1246.31			−0.424615
$206$		0			0
$207$	560.587	+	2599.37i	0.188230	+	0.872795i
$208$		0			0
$209$		−	6317.02i		−	2.09070i
$210$		0			0
$211$		−	1622.24i		−	0.529286i	−0.964346	−	0.264643i	$-0.914746\pi$
$211$		−	1622.24i		−	0.529286i	0.964346	−	0.264643i	$-0.0852541\pi$
$212$		0			0
$213$	1539.91	−	1907.42i	0.495366	−	0.613587i
$214$		0			0
$215$	−1726.72			−0.547726
$216$		0			0
$217$	4255.42			1.33123
$218$		0			0
$219$	1934.82	−	2396.58i	0.597001	−	0.739478i
$220$		0			0
$221$		−	5628.53i		−	1.71319i
$222$		0			0
$223$			985.075i			0.295810i	0.989002	+	0.147905i	$0.0472529\pi$
$223$			985.075i			0.295810i	−0.989002	+	0.147905i	$0.952747\pi$
$224$		0			0
$225$	142.301	+	659.830i	0.0421632	+	0.195505i
$226$		0			0
$227$	−896.149			−0.262024			−0.131012	−	0.991381i	$-0.541823\pi$
$227$	−896.149			−0.262024			−0.131012	+	0.991381i	$0.541823\pi$
$228$		0			0
$229$	3529.31			1.01844			0.509221	−	0.860636i	$-0.329934\pi$
$229$	3529.31			1.01844			0.509221	+	0.860636i	$0.329934\pi$
$230$		0			0
$231$	2616.12	+	2112.07i	0.745143	+	0.601575i
$232$		0			0
$233$		−	3045.50i		−	0.856299i	−0.903708	−	0.428149i	$-0.859166\pi$
$233$		−	3045.50i		−	0.856299i	0.903708	−	0.428149i	$-0.140834\pi$
$234$		0			0
$235$			687.911i			0.190955i
$236$		0			0
$237$	2945.51	+	2377.99i	0.807306	+	0.651761i
$238$		0			0
$239$	−3021.00			−0.817625			−0.408813	−	0.912618i	$-0.634057\pi$
$239$	−3021.00			−0.817625			−0.408813	+	0.912618i	$0.634057\pi$
$240$		0			0
$241$	4055.85			1.08407			0.542034	−	0.840356i	$-0.317654\pi$
$241$	4055.85			1.08407			0.542034	+	0.840356i	$0.317654\pi$
$242$		0			0
$243$	953.209	−	3666.10i	0.251640	−	0.967821i
$244$		0			0
$245$		−	675.009i		−	0.176019i
$246$		0			0
$247$			5923.74i			1.52599i
$248$		0			0
$249$	−910.019	+	1127.20i	−0.231607	+	0.286881i
$250$		0			0
$251$	−1851.94			−0.465711			−0.232855	−	0.972511i	$-0.574807\pi$
$251$	−1851.94			−0.465711			−0.232855	+	0.972511i	$0.574807\pi$
$252$		0			0
$253$	−4418.73			−1.09804
$254$		0			0
$255$	−2183.31	+	2704.36i	−0.536173	+	0.664132i
$256$		0			0
$257$			4407.29i			1.06972i	0.844939	+	0.534862i	$0.179636\pi$
$257$			4407.29i			1.06972i	−0.844939	+	0.534862i	$0.820364\pi$
$258$		0			0
$259$			4745.87i			1.13859i
$260$		0			0
$261$	3588.64	−	773.936i	0.851077	−	0.183546i
$262$		0			0
$263$	−5276.74			−1.23718			−0.618588	−	0.785715i	$-0.712295\pi$
$263$	−5276.74			−1.23718			−0.618588	+	0.785715i	$0.712295\pi$
$264$		0			0
$265$	324.167			0.0751450
$266$		0			0
$267$	−4598.91	−	3712.83i	−1.05412	−	0.851017i
$268$		0			0
$269$			341.373i			0.0773751i	0.999251	+	0.0386875i	$0.0123177\pi$
$269$			341.373i			0.0773751i	−0.999251	+	0.0386875i	$0.987682\pi$
$270$		0			0
$271$			2686.79i			0.602255i	0.953584	+	0.301127i	$0.0973629\pi$
$271$			2686.79i			0.602255i	−0.953584	+	0.301127i	$0.902637\pi$
$272$		0			0
$273$	−2453.25	−	1980.58i	−0.543873	−	0.439084i
$274$		0			0
$275$	−1121.66			−0.245959
$276$		0			0
$277$	5760.38			1.24949			0.624743	−	0.780830i	$-0.285204\pi$
$277$	5760.38			1.24949			0.624743	+	0.780830i	$0.285204\pi$
$278$		0			0
$279$	7787.61	−	1679.50i	1.67108	−	0.360391i
$280$		0			0
$281$		−	2432.21i		−	0.516347i	−0.966099	−	0.258174i	$-0.916879\pi$
$281$		−	2432.21i		−	0.516347i	0.966099	−	0.258174i	$-0.0831206\pi$
$282$		0			0
$283$		−	1769.84i		−	0.371754i	−0.982573	−	0.185877i	$-0.940487\pi$
$283$		−	1769.84i		−	0.371754i	0.982573	−	0.185877i	$-0.0595126\pi$
$284$		0			0
$285$	2297.82	−	2846.21i	0.477583	−	0.591560i
$286$		0			0
$287$	−3594.89			−0.739372
$288$		0			0
$289$	−12983.9			−2.64277
$290$		0			0
$291$	1508.19	−	1868.13i	0.303821	−	0.376329i
$292$		0			0
$293$		−	865.261i		−	0.172522i	−0.996273	−	0.0862612i	$-0.972508\pi$
$293$		−	865.261i		−	0.172522i	0.996273	−	0.0862612i	$-0.0274920\pi$
$294$		0			0
$295$			1625.84i			0.320882i
$296$		0			0
$297$	5621.20	+	2832.67i	1.09823	+	0.553428i
$298$		0			0
$299$	4143.64			0.801447
$300$		0			0
$301$	−4980.59			−0.953743
$302$		0			0
$303$	−1287.61	−	1039.52i	−0.244129	−	0.197092i
$304$		0			0
$305$			4168.62i			0.782604i
$306$		0			0
$307$		−	4820.25i		−	0.896112i	−0.894006	−	0.448056i	$-0.852117\pi$
$307$		−	4820.25i		−	0.896112i	0.894006	−	0.448056i	$-0.147883\pi$
$308$		0			0
$309$	−4383.21	−	3538.69i	−0.806965	−	0.651485i
$310$		0			0
$311$	3467.58			0.632245			0.316123	−	0.948718i	$-0.397619\pi$
$311$	3467.58			0.632245			0.316123	+	0.948718i	$0.397619\pi$
$312$		0			0
$313$	8018.37			1.44800			0.724002	−	0.689798i	$-0.242301\pi$
$313$	8018.37			1.44800			0.724002	+	0.689798i	$0.242301\pi$
$314$		0			0
$315$	410.457	+	1903.23i	0.0734178	+	0.340429i
$316$		0			0
$317$			7460.45i			1.32183i	0.750460	+	0.660916i	$0.229832\pi$
$317$			7460.45i			1.32183i	−0.750460	+	0.660916i	$0.770168\pi$
$318$		0			0
$319$			6100.41i			1.07071i
$320$		0			0
$321$	4480.36	−	5549.61i	0.779031	−	0.964950i
$322$		0			0
$323$	18835.6			3.24471
$324$		0			0
$325$	1051.83			0.179523
$326$		0			0
$327$	3646.73	−	4517.04i	0.616712	−	0.763892i
$328$		0			0
$329$			1984.23i			0.332505i
$330$		0			0
$331$			3446.80i			0.572367i	0.958175	+	0.286183i	$0.0923866\pi$
$331$			3446.80i			0.572367i	−0.958175	+	0.286183i	$0.907613\pi$
$332$		0			0
$333$	1873.07	+	8685.17i	0.308239	+	1.42926i
$334$		0			0
$335$	−986.576			−0.160903
$336$		0			0
$337$	−3122.33			−0.504701			−0.252350	−	0.967636i	$-0.581204\pi$
$337$	−3122.33			−0.504701			−0.252350	+	0.967636i	$0.581204\pi$
$338$		0			0
$339$	−1884.64	−	1521.52i	−0.301945	−	0.243769i
$340$		0			0
$341$			13238.4i			2.10234i
$342$		0			0
$343$		−	6893.81i		−	1.08522i
$344$		0			0
$345$	−1990.91	−	1607.32i	−0.310687	−	0.250826i
$346$		0			0
$347$	−7557.98			−1.16926			−0.584631	−	0.811300i	$-0.698761\pi$
$347$	−7557.98			−1.16926			−0.584631	+	0.811300i	$0.698761\pi$
$348$		0			0
$349$	−5253.77			−0.805811			−0.402905	−	0.915242i	$-0.632000\pi$
$349$	−5253.77			−0.805811			−0.402905	+	0.915242i	$0.632000\pi$
$350$		0			0
$351$	−5271.24	−	2656.31i	−0.801589	−	0.403942i
$352$		0			0
$353$		−	5382.98i		−	0.811635i	−0.913954	−	0.405817i	$-0.866987\pi$
$353$		−	5382.98i		−	0.811635i	0.913954	−	0.405817i	$-0.133013\pi$
$354$		0			0
$355$			2358.90i			0.352669i
$356$		0			0
$357$	−6297.60	+	7800.54i	−0.933625	+	1.15644i
$358$		0			0
$359$	2471.10			0.363285			0.181643	−	0.983365i	$-0.441859\pi$
$359$	2471.10			0.363285			0.181643	+	0.983365i	$0.441859\pi$
$360$		0			0
$361$	−12964.5			−1.89015
$362$		0			0
$363$	−2226.07	+	2757.33i	−0.321869	+	0.398685i
$364$		0			0
$365$			2963.84i			0.425027i
$366$		0			0
$367$			8326.08i			1.18424i	0.805848	+	0.592122i	$0.201710\pi$
$367$			8326.08i			1.18424i	−0.805848	+	0.592122i	$0.798290\pi$
$368$		0			0
$369$	−6578.82	+	1418.81i	−0.928130	+	0.200163i
$370$		0			0
$371$	935.037			0.130848
$372$		0			0
$373$	4237.46			0.588224			0.294112	−	0.955771i	$-0.404976\pi$
$373$	4237.46			0.588224			0.294112	+	0.955771i	$0.404976\pi$
$374$		0			0
$375$	−505.378	−	408.006i	−0.0695936	−	0.0561848i
$376$		0			0
$377$		−	5720.62i		−	0.781504i
$378$		0			0
$379$		−	4157.49i		−	0.563472i	−0.959492	−	0.281736i	$-0.909090\pi$
$379$		−	4157.49i		−	0.563472i	0.959492	−	0.281736i	$-0.0909103\pi$
$380$		0			0
$381$	−1376.88	−	1111.60i	−0.185144	−	0.149472i
$382$		0			0
$383$	961.981			0.128342			0.0641709	−	0.997939i	$-0.479560\pi$
$383$	961.981			0.128342			0.0641709	+	0.997939i	$0.479560\pi$
$384$		0			0
$385$	−3235.35			−0.428283
$386$		0			0
$387$	−9114.72	+	1965.71i	−1.19723	+	0.258198i
$388$		0			0
$389$		−	11662.8i		−	1.52012i	−0.649851	−	0.760061i	$-0.725169\pi$
$389$		−	11662.8i		−	1.52012i	0.649851	−	0.760061i	$-0.274831\pi$
$390$		0			0
$391$		−	13175.4i		−	1.70412i
$392$		0			0
$393$	−4002.94	+	4958.26i	−0.513795	+	0.636415i
$394$		0			0
$395$	−3642.71			−0.464012
$396$		0			0
$397$	12670.6			1.60181			0.800907	−	0.598789i	$-0.204351\pi$
$397$	12670.6			1.60181			0.800907	+	0.598789i	$0.204351\pi$
$398$		0			0
$399$	6627.90	−	8209.68i	0.831604	−	1.03007i
$400$		0			0
$401$			404.911i			0.0504247i	0.999682	+	0.0252123i	$0.00802619\pi$
$401$			404.911i			0.0504247i	−0.999682	+	0.0252123i	$0.991974\pi$
$402$		0			0
$403$		−	12414.2i		−	1.53448i
$404$		0			0
$405$	1502.31	+	3321.01i	0.184322	+	0.407462i
$406$		0			0
$407$	−14764.1			−1.79811
$408$		0			0
$409$	−13515.2			−1.63395			−0.816974	−	0.576675i	$-0.804350\pi$
$409$	−13515.2			−1.63395			−0.816974	+	0.576675i	$0.804350\pi$
$410$		0			0
$411$	−8305.56	−	6705.31i	−0.996796	−	0.804741i
$412$		0			0
$413$			4689.62i			0.558744i
$414$		0			0
$415$		−	1394.00i		−	0.164889i
$416$		0			0
$417$	5382.31	+	4345.29i	0.632069	+	0.510287i
$418$		0			0
$419$	2392.20			0.278918			0.139459	−	0.990228i	$-0.455464\pi$
$419$	2392.20			0.278918			0.139459	+	0.990228i	$0.455464\pi$
$420$		0			0
$421$	−3223.75			−0.373197			−0.186598	−	0.982436i	$-0.559746\pi$
$421$	−3223.75			−0.373197			−0.186598	+	0.982436i	$0.559746\pi$
$422$		0			0
$423$	783.123	+	3631.24i	0.0900159	+	0.417392i
$424$		0			0
$425$		−	3344.48i		−	0.381721i
$426$		0			0
$427$			12024.1i			1.36273i
$428$		0			0
$429$	6161.46	−	7631.91i	0.693422	−	0.858909i
$430$		0			0
$431$	−7467.21			−0.834531			−0.417266	−	0.908785i	$-0.637011\pi$
$431$	−7467.21			−0.834531			−0.417266	+	0.908785i	$0.637011\pi$
$432$		0			0
$433$	7058.81			0.783429			0.391714	−	0.920087i	$-0.371882\pi$
$433$	7058.81			0.783429			0.391714	+	0.920087i	$0.371882\pi$
$434$		0			0
$435$	−2219.03	+	2748.61i	−0.244585	+	0.302956i
$436$		0			0
$437$			13866.5i			1.51790i
$438$		0			0
$439$		−	14738.2i		−	1.60231i	−0.598455	−	0.801157i	$-0.704218\pi$
$439$		−	14738.2i		−	1.60231i	0.598455	−	0.801157i	$-0.295782\pi$
$440$		0			0
$441$	−768.435	−	3563.13i	−0.0829754	−	0.384746i
$442$		0			0
$443$	−12124.9			−1.30039			−0.650196	−	0.759767i	$-0.725313\pi$
$443$	−12124.9			−1.30039			−0.650196	+	0.759767i	$0.725313\pi$
$444$		0			0
$445$	5687.47			0.605870
$446$		0			0
$447$	12174.8	+	9829.03i	1.28825	+	1.04004i
$448$		0			0
$449$		−	2844.47i		−	0.298973i	−0.988764	−	0.149487i	$-0.952238\pi$
$449$		−	2844.47i		−	0.298973i	0.988764	−	0.149487i	$-0.0477621\pi$
$450$		0			0
$451$		−	11183.5i		−	1.16765i
$452$		0			0
$453$	−5228.74	−	4221.31i	−0.542313	−	0.437824i
$454$		0			0
$455$	3033.93			0.312600
$456$		0			0
$457$	−9582.44			−0.980848			−0.490424	−	0.871484i	$-0.663158\pi$
$457$	−9582.44			−0.980848			−0.490424	+	0.871484i	$0.663158\pi$
$458$		0			0
$459$	−8446.23	+	16760.8i	−0.858903	+	1.70442i
$460$		0			0
$461$			16696.5i			1.68684i	0.537257	+	0.843419i	$0.319461\pi$
$461$			16696.5i			1.68684i	−0.537257	+	0.843419i	$0.680539\pi$
$462$		0			0
$463$		−	9088.87i		−	0.912302i	−0.889902	−	0.456151i	$-0.849228\pi$
$463$		−	9088.87i		−	0.912302i	0.889902	−	0.456151i	$-0.150772\pi$
$464$		0			0
$465$	−4815.47	+	5964.70i	−0.480241	+	0.594852i
$466$		0			0
$467$	2840.80			0.281492			0.140746	−	0.990046i	$-0.455050\pi$
$467$	2840.80			0.281492			0.140746	+	0.990046i	$0.455050\pi$
$468$		0			0
$469$	−2845.71			−0.280176
$470$		0			0
$471$	1908.03	−	2363.39i	0.186661	−	0.231209i
$472$		0			0
$473$		−	15494.3i		−	1.50620i
$474$		0			0
$475$			3519.90i			0.340009i
$476$		0			0
$477$	1711.16	−	369.034i	0.164253	−	0.0354233i
$478$		0			0
$479$	−10410.2			−0.993017			−0.496509	−	0.868032i	$-0.665385\pi$
$479$	−10410.2			−0.993017			−0.496509	+	0.868032i	$0.665385\pi$
$480$		0			0
$481$	13845.0			1.31242
$482$		0			0
$483$	−5742.64	−	4636.20i	−0.540992	−	0.436758i
$484$		0			0
$485$			2310.31i			0.216301i
$486$		0			0
$487$		−	15786.9i		−	1.46893i	−0.678644	−	0.734467i	$-0.737432\pi$
$487$		−	15786.9i		−	1.46893i	0.678644	−	0.734467i	$-0.262568\pi$
$488$		0			0
$489$	−12118.7	−	9783.78i	−1.12071	−	0.904781i
$490$		0			0
$491$	12936.6			1.18905			0.594523	−	0.804079i	$-0.297341\pi$
$491$	12936.6			1.18905			0.594523	+	0.804079i	$0.297341\pi$
$492$		0			0
$493$	−18189.7			−1.66171
$494$		0			0
$495$	−5920.85	+	1276.91i	−0.537621	+	0.115945i
$496$		0			0
$497$			6804.08i			0.614094i
$498$		0			0
$499$		−	12533.0i		−	1.12435i	−0.827017	−	0.562177i	$-0.809964\pi$
$499$		−	12533.0i		−	1.12435i	0.827017	−	0.562177i	$-0.190036\pi$
$500$		0			0
$501$	1311.23	−	1624.16i	0.116929	−	0.144835i
$502$		0			0
$503$	−640.555			−0.0567812			−0.0283906	−	0.999597i	$-0.509038\pi$
$503$	−640.555			−0.0567812			−0.0283906	+	0.999597i	$0.509038\pi$
$504$		0			0
$505$	1592.38			0.140317
$506$		0			0
$507$	1393.24	−	1725.74i	0.122043	−	0.151170i
$508$		0			0
$509$		−	6917.78i		−	0.602407i	−0.953560	−	0.301204i	$-0.902612\pi$
$509$		−	6917.78i		−	0.602407i	0.953560	−	0.301204i	$-0.0973884\pi$
$510$		0			0
$511$			8549.00i			0.740089i
$512$		0			0
$513$	8889.23	−	17639.9i	0.765047	−	1.51817i
$514$		0			0
$515$	5420.71			0.463816
$516$		0			0
$517$	−6172.83			−0.525108
$518$		0			0
$519$	−366.864	−	296.179i	−0.0310280	−	0.0250498i
$520$		0			0
$521$			253.276i			0.0212980i	0.999943	+	0.0106490i	$0.00338974\pi$
$521$			253.276i			0.0212980i	−0.999943	+	0.0106490i	$0.996610\pi$
$522$		0			0
$523$			8978.68i			0.750689i	0.926885	+	0.375344i	$0.122476\pi$
$523$			8978.68i			0.750689i	−0.926885	+	0.375344i	$0.877524\pi$
$524$		0			0
$525$	−1457.73	−	1176.86i	−0.121182	−	0.0978333i
$526$		0			0
$527$	−39473.1			−3.26276
$528$		0			0
$529$	−2467.44			−0.202798
$530$		0			0
$531$	1850.87	+	8582.23i	0.151263	+	0.701388i
$532$		0			0
$533$			10487.3i			0.852258i
$534$		0			0
$535$			6863.20i			0.554620i
$536$		0			0
$537$	−10129.7	+	12547.2i	−0.814018	+	1.00829i
$538$		0			0
$539$	6057.05			0.484037
$540$		0			0
$541$	23716.2			1.88473			0.942364	−	0.334590i	$-0.108598\pi$
$541$	23716.2			1.88473			0.942364	+	0.334590i	$0.108598\pi$
$542$		0			0
$543$	6971.38	−	8635.13i	0.550959	−	0.682447i
$544$		0			0
$545$			5586.22i			0.439059i
$546$		0			0
$547$			9271.56i			0.724722i	0.932038	+	0.362361i	$0.118029\pi$
$547$			9271.56i			0.724722i	−0.932038	+	0.362361i	$0.881971\pi$
$548$		0			0
$549$	4745.58	+	22004.6i	0.368919	+	1.71063i
$550$		0			0
$551$	19143.8			1.48013
$552$		0			0
$553$	−10507.1			−0.807973
$554$		0			0
$555$	−6652.15	−	5370.47i	−0.508771	−	0.410745i
$556$		0			0
$557$		−	19268.7i		−	1.46578i	−0.680345	−	0.732892i	$-0.738170\pi$
$557$		−	19268.7i		−	1.46578i	0.680345	−	0.732892i	$-0.261830\pi$
$558$		0			0
$559$			14529.7i			1.09936i
$560$		0			0
$561$	−24267.0	−	19591.5i	−1.82630	−	1.47442i
$562$		0			0
$563$	9228.75			0.690844			0.345422	−	0.938447i	$-0.387736\pi$
$563$	9228.75			0.690844			0.345422	+	0.938447i	$0.387736\pi$
$564$		0			0
$565$	2330.73			0.173548
$566$		0			0
$567$	4333.30	+	9579.21i	0.320955	+	0.709504i
$568$		0			0
$569$			20222.1i			1.48991i	0.667118	+	0.744953i	$0.267528\pi$
$569$			20222.1i			1.48991i	−0.667118	+	0.744953i	$0.732472\pi$
$570$		0			0
$571$			5905.92i			0.432846i	0.976300	+	0.216423i	$0.0694391\pi$
$571$			5905.92i			0.432846i	−0.976300	+	0.216423i	$0.930561\pi$
$572$		0			0
$573$	−11490.0	+	14232.1i	−0.837699	+	1.03762i
$574$		0			0
$575$	2462.16			0.178572
$576$		0			0
$577$	25069.6			1.80877			0.904384	−	0.426720i	$-0.140331\pi$
$577$	25069.6			1.80877			0.904384	+	0.426720i	$0.140331\pi$
$578$		0			0
$579$	6743.71	−	8353.13i	0.484040	−	0.599558i
$580$		0			0
$581$		−	4020.91i		−	0.287118i
$582$		0			0
$583$			2908.84i			0.206642i
$584$		0			0
$585$	5552.23	−	1197.41i	0.392404	−	0.0846271i
$586$		0			0
$587$	−7484.02			−0.526232			−0.263116	−	0.964764i	$-0.584750\pi$
$587$	−7484.02			−0.526232			−0.263116	+	0.964764i	$0.584750\pi$
$588$		0			0
$589$	41543.5			2.90623
$590$		0			0
$591$	5422.26	+	4377.54i	0.377397	+	0.304683i
$592$		0			0
$593$			4020.39i			0.278411i	0.990264	+	0.139206i	$0.0444549\pi$
$593$			4020.39i			0.278411i	−0.990264	+	0.139206i	$0.955545\pi$
$594$		0			0
$595$		−	9646.92i		−	0.664681i
$596$		0			0
$597$	4647.88	+	3752.37i	0.318635	+	0.257243i
$598$		0			0
$599$	15236.3			1.03930			0.519649	−	0.854380i	$-0.326063\pi$
$599$	15236.3			1.03930			0.519649	+	0.854380i	$0.326063\pi$
$600$		0			0
$601$	17173.1			1.16557			0.582784	−	0.812627i	$-0.301963\pi$
$601$	17173.1			1.16557			0.582784	+	0.812627i	$0.301963\pi$
$602$		0			0
$603$	−5207.78	+	1123.12i	−0.351703	+	0.0758494i
$604$		0			0
$605$		−	3409.99i		−	0.229150i
$606$		0			0
$607$		−	14868.7i		−	0.994237i	−0.867683	−	0.497118i	$-0.834391\pi$
$607$		−	14868.7i		−	0.994237i	0.867683	−	0.497118i	$-0.165609\pi$
$608$		0			0
$609$	−6400.64	+	7928.18i	−0.425890	+	0.527530i
$610$		0			0
$611$	5788.53			0.383271
$612$		0			0
$613$	−2566.79			−0.169122			−0.0845608	−	0.996418i	$-0.526949\pi$
$613$	−2566.79			−0.169122			−0.0845608	+	0.996418i	$0.526949\pi$
$614$		0			0
$615$	4068.01	−	5038.86i	0.266729	−	0.330384i
$616$		0			0
$617$		−	25317.1i		−	1.65191i	−0.563734	−	0.825956i	$-0.690636\pi$
$617$		−	25317.1i		−	1.65191i	0.563734	−	0.825956i	$-0.309364\pi$
$618$		0			0
$619$		−	24229.3i		−	1.57327i	−0.617416	−	0.786637i	$-0.711820\pi$
$619$		−	24229.3i		−	1.57327i	0.617416	−	0.786637i	$-0.288180\pi$
$620$		0			0
$621$	−12339.1	−	6217.99i	−0.797344	−	0.401802i
$622$		0			0
$623$	16405.1			1.05499
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	25539.8	+	20619.0i	1.62673	+	1.31331i
$628$		0			0
$629$		−	44022.6i		−	2.79061i
$630$		0			0
$631$			10560.6i			0.666261i	0.942881	+	0.333131i	$0.108105\pi$
$631$			10560.6i			0.666261i	−0.942881	+	0.333131i	$0.891895\pi$
$632$		0			0
$633$	6558.73	+	5295.05i	0.411827	+	0.332479i
$634$		0			0
$635$	1702.79			0.106414
$636$		0			0
$637$	−5679.96			−0.353294
$638$		0			0
$639$	2685.39	+	12451.8i	0.166248	+	0.770868i
$640$		0			0
$641$			5599.55i			0.345037i	0.985006	+	0.172519i	$0.0551905\pi$
$641$			5599.55i			0.345037i	−0.985006	+	0.172519i	$0.944810\pi$
$642$		0			0
$643$		−	5597.59i		−	0.343308i	−0.985157	−	0.171654i	$-0.945089\pi$
$643$		−	5597.59i		−	0.343308i	0.985157	−	0.171654i	$-0.0549112\pi$
$644$		0			0
$645$	5636.08	−	6981.16i	0.344063	−	0.426175i
$646$		0			0
$647$	27855.2			1.69258			0.846290	−	0.532722i	$-0.178831\pi$
$647$	27855.2			1.69258			0.846290	+	0.532722i	$0.178831\pi$
$648$		0			0
$649$	−14589.1			−0.882395
$650$		0			0
$651$	−13889.9	+	17204.7i	−0.836232	+	1.03580i
$652$		0			0
$653$			4511.63i			0.270373i	0.990820	+	0.135187i	$0.0431634\pi$
$653$			4511.63i			0.270373i	−0.990820	+	0.135187i	$0.956837\pi$
$654$		0			0
$655$		−	6131.87i		−	0.365789i
$656$		0			0
$657$	3374.06	+	15645.1i	0.200357	+	0.929029i
$658$		0			0
$659$	17264.4			1.02053			0.510263	−	0.860018i	$-0.329548\pi$
$659$	17264.4			1.02053			0.510263	+	0.860018i	$0.329548\pi$
$660$		0			0
$661$	−22656.6			−1.33319			−0.666595	−	0.745420i	$-0.732249\pi$
$661$	−22656.6			−1.33319			−0.666595	+	0.745420i	$0.732249\pi$
$662$		0			0
$663$	22756.2	+	18371.8i	1.33300	+	1.07617i
$664$		0			0
$665$			10152.9i			0.592049i
$666$		0			0
$667$		−	13391.0i		−	0.777365i
$668$		0			0
$669$	−3982.68	−	3215.33i	−0.230163	−	0.185817i
$670$		0			0
$671$	−37406.2			−2.15209
$672$		0			0
$673$	10868.2			0.622495			0.311248	−	0.950329i	$-0.399253\pi$
$673$	10868.2			0.622495			0.311248	+	0.950329i	$0.399253\pi$
$674$		0			0
$675$	−3132.18	−	1578.39i	−0.178604	−	0.0900033i
$676$		0			0
$677$			8676.98i			0.492590i	0.969195	+	0.246295i	$0.0792132\pi$
$677$			8676.98i			0.492590i	−0.969195	+	0.246295i	$0.920787\pi$
$678$		0			0
$679$			6663.93i			0.376640i
$680$		0			0
$681$	2925.07	−	3623.15i	0.164595	−	0.203876i
$682$		0			0
$683$	−22090.2			−1.23756			−0.618782	−	0.785563i	$-0.712374\pi$
$683$	−22090.2			−1.23756			−0.618782	+	0.785563i	$0.712374\pi$
$684$		0			0
$685$	10271.5			0.572924
$686$		0			0
$687$	−11519.8	+	14269.1i	−0.639750	+	0.792428i
$688$		0			0
$689$		−	2727.75i		−	0.150826i
$690$		0			0
$691$			3408.19i			0.187632i	0.995590	+	0.0938160i	$0.0299065\pi$
$691$			3408.19i			0.187632i	−0.995590	+	0.0938160i	$0.970093\pi$
$692$		0			0
$693$	−17078.3	+	3683.15i	−0.936146	+	0.201892i
$694$		0			0
$695$	−6656.30			−0.363292
$696$		0			0
$697$	33346.1			1.81216
$698$		0			0
$699$	12313.0	+	9940.66i	0.666269	+	0.537897i
$700$		0			0
$701$			15940.1i			0.858843i	0.903104	+	0.429421i	$0.141282\pi$
$701$			15940.1i			0.858843i	−0.903104	+	0.429421i	$0.858718\pi$
$702$		0			0
$703$			46331.5i			2.48567i
$704$		0			0
$705$	−2781.24	−	2245.37i	−0.148578	−	0.119951i
$706$		0			0
$707$	4593.11			0.244330
$708$		0			0
$709$	29390.4			1.55681			0.778407	−	0.627760i	$-0.216028\pi$
$709$	29390.4			1.55681			0.778407	+	0.627760i	$0.216028\pi$
$710$		0			0
$711$	−19228.6	+	4146.89i	−1.01424	+	0.218735i
$712$		0			0
$713$		−	29059.5i		−	1.52635i
$714$		0			0
$715$			9438.38i			0.493672i
$716$		0			0
$717$	9860.69	−	12214.0i	0.513604	−	0.636178i
$718$		0			0
$719$	−8719.04			−0.452247			−0.226123	−	0.974099i	$-0.572605\pi$
$719$	−8719.04			−0.452247			−0.226123	+	0.974099i	$0.572605\pi$
$720$		0			0
$721$	15635.7			0.807631
$722$		0			0
$723$	−13238.5	+	16397.9i	−0.680974	+	0.843492i
$724$		0			0
$725$		−	3399.21i		−	0.174129i
$726$		0			0
$727$			1864.65i			0.0951254i	0.998868	+	0.0475627i	$0.0151454\pi$
$727$			1864.65i			0.0951254i	−0.998868	+	0.0475627i	$0.984855\pi$
$728$		0			0
$729$	11710.8	+	15820.2i	0.594971	+	0.803747i
$730$		0			0
$731$	46199.8			2.33757
$732$		0			0
$733$	8671.73			0.436968			0.218484	−	0.975841i	$-0.429889\pi$
$733$	8671.73			0.436968			0.218484	+	0.975841i	$0.429889\pi$
$734$		0			0
$735$	2729.08	+	2203.26i	0.136957	+	0.110569i
$736$		0			0
$737$		−	8852.83i		−	0.442467i
$738$		0			0
$739$		−	12190.9i		−	0.606832i	−0.952858	−	0.303416i	$-0.901873\pi$
$739$		−	12190.9i		−	0.606832i	0.952858	−	0.303416i	$-0.0981271\pi$
$740$		0			0
$741$	−23949.8	−	19335.3i	−1.18734	−	0.958572i
$742$		0			0
$743$	7417.63			0.366254			0.183127	−	0.983089i	$-0.441378\pi$
$743$	7417.63			0.366254			0.183127	+	0.983089i	$0.441378\pi$
$744$		0			0
$745$	−15056.5			−0.740440
$746$		0			0
$747$	−1586.94	−	7358.45i	−0.0777286	−	0.360417i
$748$		0			0
$749$			19796.4i			0.965747i
$750$		0			0
$751$		−	26901.6i		−	1.30713i	−0.756872	−	0.653563i	$-0.773273\pi$
$751$		−	26901.6i		−	1.30713i	0.756872	−	0.653563i	$-0.226727\pi$
$752$		0			0
$753$	6044.81	−	7487.43i	0.292543	−	0.362360i
$754$		0			0
$755$	6466.38			0.311703
$756$		0			0
$757$	2029.01			0.0974184			0.0487092	−	0.998813i	$-0.484489\pi$
$757$	2029.01			0.0974184			0.0487092	+	0.998813i	$0.484489\pi$
$758$		0			0
$759$	14422.9	−	17865.0i	0.689749	−	0.854360i
$760$		0			0
$761$			16579.6i			0.789765i	0.918732	+	0.394883i	$0.129215\pi$
$761$			16579.6i			0.789765i	−0.918732	+	0.394883i	$0.870785\pi$
$762$		0			0
$763$			16113.0i			0.764523i
$764$		0			0
$765$	−3807.38	−	17654.3i	−0.179943	−	0.834370i
$766$		0			0
$767$	13680.9			0.644052
$768$		0			0
$769$	28853.0			1.35301			0.676506	−	0.736437i	$-0.263493\pi$
$769$	28853.0			1.35301			0.676506	+	0.736437i	$0.263493\pi$
$770$		0			0
$771$	−17818.8	−	14385.6i	−0.832330	−	0.671964i
$772$		0			0
$773$		−	29347.9i		−	1.36555i	−0.730627	−	0.682776i	$-0.760772\pi$
$773$		−	29347.9i		−	1.36555i	0.730627	−	0.682776i	$-0.239228\pi$
$774$		0			0
$775$		−	7376.53i		−	0.341901i
$776$		0			0
$777$	−19187.7	−	15490.7i	−0.885912	−	0.715221i
$778$		0			0
$779$	−35095.1			−1.61414
$780$		0			0
$781$	−21167.1			−0.969806
$782$		0			0
$783$	−8584.43	+	17035.1i	−0.391804	+	0.777503i
$784$		0			0
$785$			2922.80i			0.132891i
$786$		0			0
$787$		−	5150.59i		−	0.233289i	−0.993174	−	0.116645i	$-0.962786\pi$
$787$		−	5150.59i		−	0.233289i	0.993174	−	0.116645i	$-0.0372139\pi$
$788$		0			0
$789$	17223.5	−	21334.0i	0.777152	−	0.962622i
$790$		0			0
$791$	6722.82			0.302195
$792$		0			0
$793$	35077.4			1.57079
$794$		0			0
$795$	−1058.10	+	1310.61i	−0.0472035	+	0.0584688i
$796$		0			0
$797$		−	8587.65i		−	0.381669i	−0.981622	−	0.190835i	$-0.938881\pi$
$797$		−	8587.65i		−	0.381669i	0.981622	−	0.190835i	$-0.0611194\pi$
$798$		0			0
$799$		−	18405.7i		−	0.814951i
$800$		0			0
$801$	30022.1	−	6474.65i	1.32432	−	0.285606i
$802$		0			0
$803$	−26595.4			−1.16878
$804$		0			0
$805$	7101.92			0.310944
$806$		0			0
$807$	−1380.18	−	1114.26i	−0.0602040	−	0.0486044i
$808$		0			0
$809$			33494.4i			1.45562i	0.685777	+	0.727812i	$0.259463\pi$
$809$			33494.4i			1.45562i	−0.685777	+	0.727812i	$0.740537\pi$
$810$		0			0
$811$			14657.7i			0.634649i	0.948317	+	0.317324i	$0.102784\pi$
$811$			14657.7i			0.634649i	−0.948317	+	0.317324i	$0.897216\pi$
$812$		0			0
$813$	−10862.8	−	8769.81i	−0.468602	−	0.378316i
$814$		0			0
$815$	14987.2			0.644146
$816$		0			0
$817$	−48623.0			−2.08213
$818$		0			0
$819$	16015.0	−	3453.85i	0.683284	−	0.147359i
$820$		0			0
$821$		−	21764.5i		−	0.925197i	−0.886568	−	0.462598i	$-0.846917\pi$
$821$		−	21764.5i		−	0.925197i	0.886568	−	0.462598i	$-0.153083\pi$
$822$		0			0
$823$		−	40675.3i		−	1.72279i	−0.507939	−	0.861393i	$-0.669592\pi$
$823$		−	40675.3i		−	1.72279i	0.507939	−	0.861393i	$-0.330408\pi$
$824$		0			0
$825$	3661.15	−	4534.90i	0.154503	−	0.191376i
$826$		0			0
$827$	−2336.34			−0.0982378			−0.0491189	−	0.998793i	$-0.515641\pi$
$827$	−2336.34			−0.0982378			−0.0491189	+	0.998793i	$0.515641\pi$
$828$		0			0
$829$	−7687.68			−0.322080			−0.161040	−	0.986948i	$-0.551485\pi$
$829$	−7687.68			−0.322080			−0.161040	+	0.986948i	$0.551485\pi$
$830$		0			0
$831$	−18802.1	+	23289.3i	−0.784884	+	0.972200i
$832$		0			0
$833$			18060.5i			0.751210i
$834$		0			0
$835$			2008.60i			0.0832461i
$836$		0			0
$837$	−18628.9	+	36967.4i	−0.769304	+	1.52662i
$838$		0			0
$839$	−48190.7			−1.98299			−0.991495	−	0.130147i	$-0.958455\pi$
$839$	−48190.7			−1.98299			−0.991495	+	0.130147i	$0.958455\pi$
$840$		0			0
$841$	5901.63			0.241979
$842$		0			0
$843$	9833.48	+	7938.84i	0.401759	+	0.324351i
$844$		0			0
$845$			2134.23i			0.0868871i
$846$		0			0
$847$		−	9835.88i		−	0.399014i
$848$		0			0
$849$	7155.52	+	5776.85i	0.289254	+	0.233523i
$850$		0			0
$851$	32408.7			1.30547
$852$		0			0
$853$	−23473.3			−0.942216			−0.471108	−	0.882076i	$-0.656146\pi$
$853$	−23473.3			−0.942216			−0.471108	+	0.882076i	$0.656146\pi$
$854$		0			0
$855$	4007.08	+	18580.3i	0.160280	+	0.743196i
$856$		0			0
$857$		−	37662.1i		−	1.50118i	−0.660767	−	0.750591i	$-0.729769\pi$
$857$		−	37662.1i		−	1.50118i	0.660767	−	0.750591i	$-0.270231\pi$
$858$		0			0
$859$			18961.8i			0.753166i	0.926383	+	0.376583i	$0.122901\pi$
$859$			18961.8i			0.753166i	−0.926383	+	0.376583i	$0.877099\pi$
$860$		0			0
$861$	11733.9	−	14534.2i	0.464448	−	0.575291i
$862$		0			0
$863$	−18661.2			−0.736077			−0.368039	−	0.929811i	$-0.619971\pi$
$863$	−18661.2			−0.736077			−0.368039	+	0.929811i	$0.619971\pi$
$864$		0			0
$865$	453.700			0.0178338
$866$		0			0
$867$	42380.1	−	52494.2i	1.66009	−	2.05628i
$868$		0			0
$869$		−	32687.1i		−	1.27599i
$870$		0			0
$871$			8301.69i			0.322953i
$872$		0			0
$873$	2630.08	+	12195.3i	0.101964	+	0.472793i
$874$		0			0
$875$	1802.77			0.0696511
$876$		0			0
$877$	−36118.3			−1.39068			−0.695341	−	0.718680i	$-0.744746\pi$
$877$	−36118.3			−1.39068			−0.695341	+	0.718680i	$0.744746\pi$
$878$		0			0
$879$	3498.27	+	2824.25i	0.134236	+	0.108373i
$880$		0			0
$881$			1414.90i			0.0541082i	0.999634	+	0.0270541i	$0.00861263\pi$
$881$			1414.90i			0.0541082i	−0.999634	+	0.0270541i	$0.991387\pi$
$882$		0			0
$883$		−	30126.3i		−	1.14817i	−0.818797	−	0.574083i	$-0.805359\pi$
$883$		−	30126.3i		−	1.14817i	0.818797	−	0.574083i	$-0.194641\pi$
$884$		0			0
$885$	−6573.31	−	5306.82i	−0.249672	−	0.201567i
$886$		0			0
$887$	−16362.4			−0.619386			−0.309693	−	0.950837i	$-0.600226\pi$
$887$	−16362.4			−0.619386			−0.309693	+	0.950837i	$0.600226\pi$
$888$		0			0
$889$	4911.57			0.185297
$890$		0			0
$891$	−29800.4	+	13480.7i	−1.12048	+	0.506868i
$892$		0			0
$893$			19371.0i			0.725898i
$894$		0			0
$895$		−	15517.1i		−	0.579529i
$896$		0			0
$897$	−13525.0	+	16752.8i	−0.503441	+	0.623590i
$898$		0			0
$899$	−40119.0			−1.48837
$900$		0			0
$901$	−8673.37			−0.320701
$902$		0			0
$903$	16256.9	−	20136.6i	0.599109	−	0.742088i
$904$		0			0
$905$			10679.1i			0.392247i
$906$		0			0
$907$			29712.8i			1.08776i	0.839163	+	0.543880i	$0.183045\pi$
$907$			29712.8i			1.08776i	−0.839163	+	0.543880i	$0.816955\pi$
$908$		0			0
$909$	8405.60	−	1812.78i	0.306706	−	0.0661452i
$910$		0			0
$911$	25605.9			0.931243			0.465622	−	0.884984i	$-0.345831\pi$
$911$	25605.9			0.931243			0.465622	+	0.884984i	$0.345831\pi$
$912$		0			0
$913$	12508.8			0.453430
$914$		0			0
$915$	−16853.8	−	13606.5i	−0.608928	−	0.491605i
$916$		0			0
$917$		−	17686.9i		−	0.636940i
$918$		0			0
$919$			382.697i			0.0137367i	0.999976	+	0.00686835i	$0.00218628\pi$
$919$			382.697i			0.0137367i	−0.999976	+	0.00686835i	$0.997814\pi$
$920$		0			0
$921$	19488.4	+	15733.5i	0.697246	+	0.562906i
$922$		0			0
$923$	19849.3			0.707852
$924$		0			0
$925$	8226.71			0.292424
$926$		0			0
$927$	28614.0	−	6170.97i	1.01381	−	0.218642i
$928$		0			0
$929$			22876.2i			0.807906i	0.914780	+	0.403953i	$0.132364\pi$
$929$			22876.2i			0.807906i	−0.914780	+	0.403953i	$0.867636\pi$
$930$		0			0
$931$		−	19007.7i		−	0.669122i
$932$		0			0
$933$	−11318.3	+	14019.5i	−0.397155	+	0.491937i
$934$		0			0
$935$	30011.0			1.04970
$936$		0			0
$937$	12304.9			0.429010			0.214505	−	0.976723i	$-0.431186\pi$
$937$	12304.9			0.429010			0.214505	+	0.976723i	$0.431186\pi$
$938$		0			0
$939$	−26172.3	+	32418.4i	−0.909586	+	1.12666i
$940$		0			0
$941$			18897.2i			0.654656i	0.944911	+	0.327328i	$0.106148\pi$
$941$			18897.2i			0.654656i	−0.944911	+	0.327328i	$0.893852\pi$
$942$		0			0
$943$			24548.9i			0.847744i
$944$		0			0
$945$	−9034.56	−	4552.75i	−0.310999	−	0.156721i
$946$		0			0
$947$	822.925			0.0282381			0.0141191	−	0.999900i	$-0.495506\pi$
$947$	822.925			0.0282381			0.0141191	+	0.999900i	$0.495506\pi$
$948$		0			0
$949$	24939.7			0.853084
$950$		0			0
$951$	−30162.8	−	24351.3i	−1.02849	−	0.830330i
$952$		0			0
$953$			42672.2i			1.45046i	0.688506	+	0.725230i	$0.258267\pi$
$953$			42672.2i			1.45046i	−0.688506	+	0.725230i	$0.741733\pi$
$954$		0			0
$955$		−	17600.9i		−	0.596388i
$956$		0			0
$957$	−24664.1	−	19912.0i	−0.833100	−	0.672585i
$958$		0			0
$959$	29627.3			0.997619
$960$		0			0
$961$	−57270.2			−1.92240
$962$		0			0
$963$	7813.11	+	36228.3i	0.261448	+	1.21230i
$964$		0			0
$965$			10330.3i			0.344605i
$966$		0			0
$967$		−	29990.1i		−	0.997327i	−0.866796	−	0.498663i	$-0.833824\pi$
$967$		−	29990.1i		−	0.997327i	0.866796	−	0.498663i	$-0.166176\pi$
$968$		0			0
$969$	−61480.2	+	76152.7i	−2.03821	+	2.52464i
$970$		0			0
$971$	−3401.57			−0.112422			−0.0562109	−	0.998419i	$-0.517902\pi$
$971$	−3401.57			−0.112422			−0.0562109	+	0.998419i	$0.517902\pi$
$972$		0			0
$973$	−19199.6			−0.632591
$974$		0			0
$975$	−3433.22	+	4252.57i	−0.112770	+	0.139683i
$976$		0			0
$977$			36808.1i			1.20532i	0.797999	+	0.602659i	$0.205892\pi$
$977$			36808.1i			1.20532i	−0.797999	+	0.602659i	$0.794108\pi$
$978$		0			0
$979$			51035.3i			1.66608i
$980$		0			0
$981$	6359.39	+	29487.6i	0.206972	+	0.959702i
$982$		0			0
$983$	7068.84			0.229360			0.114680	−	0.993402i	$-0.463416\pi$
$983$	7068.84			0.229360			0.114680	+	0.993402i	$0.463416\pi$
$984$		0			0
$985$	−6705.70			−0.216915
$986$		0			0
$987$	−8022.29	−	6476.62i	−0.258715	−	0.208868i
$988$		0			0
$989$			34011.6i			1.09354i
$990$		0			0
$991$		−	636.665i		−	0.0204080i	−0.999948	−	0.0102040i	$-0.996752\pi$
$991$		−	636.665i		−	0.0204080i	0.999948	−	0.0102040i	$-0.00324809\pi$
$992$		0			0
$993$	−13935.5	−	11250.5i	−0.445347	−	0.359541i
$994$		0			0
$995$	−5748.03			−0.183141
$996$		0			0
$997$	1275.41			0.0405143			0.0202571	−	0.999795i	$-0.493552\pi$
$997$	1275.41			0.0405143			0.0202571	+	0.999795i	$0.493552\pi$
$998$		0			0
$999$	−41228.1	−	20775.9i	−1.30570	−	0.657979i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.h.a.191.8	yes	24
3.2	odd	2		480.4.h.b.191.18	yes	24
4.3	odd	2		480.4.h.b.191.17	yes	24
8.3	odd	2		960.4.h.c.191.8		24
8.5	even	2		960.4.h.e.191.17		24
12.11	even	2	inner	480.4.h.a.191.7	✓	24
24.5	odd	2		960.4.h.c.191.7		24
24.11	even	2		960.4.h.e.191.18		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.h.a.191.7	✓	24	12.11	even	2	inner
480.4.h.a.191.8	yes	24	1.1	even	1	trivial
480.4.h.b.191.17	yes	24	4.3	odd	2
480.4.h.b.191.18	yes	24	3.2	odd	2
960.4.h.c.191.7		24	24.5	odd	2
960.4.h.c.191.8		24	8.3	odd	2
960.4.h.e.191.17		24	8.5	even	2
960.4.h.e.191.18		24	24.11	even	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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.26404 + 4.04302i) q^{3} -5.00000i q^{5} -14.4221i q^{7} +(-5.69203 - 26.3932i) q^{9} +44.8665 q^{11} -42.0732 q^{13} +(20.2151 + 16.3202i) q^{15} +133.779i q^{17} -140.796i q^{19} +(58.3090 + 47.0745i) q^{21} -98.4863 q^{23} -25.0000 q^{25} +(125.287 + 63.1355i) q^{27} +135.968i q^{29} +295.061i q^{31} +(-146.446 + 181.396i) q^{33} -72.1107 q^{35} -329.068 q^{37} +(137.329 - 170.103i) q^{39} -249.262i q^{41} -345.344i q^{43} +(-131.966 + 28.4602i) q^{45} -137.582 q^{47} +135.002 q^{49} +(-540.873 - 436.662i) q^{51} +64.8334i q^{53} -224.332i q^{55} +(569.241 + 459.564i) q^{57} -325.168 q^{59} -833.723 q^{61} +(-380.646 + 82.0913i) q^{63} +210.366i q^{65} -197.315i q^{67} +(321.464 - 398.182i) q^{69} -471.780 q^{71} -592.769 q^{73} +(81.6011 - 101.076i) q^{75} -647.070i q^{77} -728.542i q^{79} +(-664.202 + 300.462i) q^{81} +278.801 q^{83} +668.897 q^{85} +(-549.722 - 443.806i) q^{87} +1137.49i q^{89} +606.786i q^{91} +(-1192.94 - 963.093i) q^{93} -703.980 q^{95} -462.063 q^{97} +(-255.381 - 1184.17i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 20 q^{9} - 72 q^{11} - 72 q^{13} - 20 q^{15} - 68 q^{21} - 96 q^{23} - 600 q^{25} - 168 q^{27} - 80 q^{33} - 504 q^{37} + 456 q^{39} - 220 q^{45} - 432 q^{47} - 816 q^{49} - 1240 q^{51} + 40 q^{57}+ \cdots - 3160 q^{99}+O(q^{100}) \) 24 * q + 20 * q^9 - 72 * q^11 - 72 * q^13 - 20 * q^15 - 68 * q^21 - 96 * q^23 - 600 * q^25 - 168 * q^27 - 80 * q^33 - 504 * q^37 + 456 * q^39 - 220 * q^45 - 432 * q^47 - 816 * q^49 - 1240 * q^51 + 40 * q^57 + 2184 * q^59 - 1080 * q^61 + 1720 * q^63 - 1548 * q^69 - 816 * q^71 - 1440 * q^73 - 2304 * q^81 + 2016 * q^83 + 2144 * q^87 - 112 * q^93 - 1800 * q^95 + 720 * q^97 - 3160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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