LMFDB - Embedded newform 624.4.n.c.623.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(623,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("624.623");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.n (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:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			623.15
Character	$\chi$	$=$	624.623
Dual form			624.4.n.c.623.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.74324 + 4.41301i) q^{3} -14.3044 q^{5} +18.8811 q^{7} +(-11.9493 + 24.2119i) q^{9} +O(q^{10})$	\(q+(2.74324 + 4.41301i) q^{3} -14.3044 q^{5} +18.8811 q^{7} +(-11.9493 + 24.2119i) q^{9} -26.9955i q^{11} +(45.5551 + 11.0331i) q^{13} +(-39.2404 - 63.1253i) q^{15} +26.8616i q^{17} +108.163 q^{19} +(51.7953 + 83.3223i) q^{21} +14.6582 q^{23} +79.6154 q^{25} +(-139.627 + 13.6869i) q^{27} +249.370i q^{29} +29.9344 q^{31} +(119.131 - 74.0552i) q^{33} -270.082 q^{35} -214.557i q^{37} +(76.2794 + 231.302i) q^{39} -177.585 q^{41} +319.109i q^{43} +(170.927 - 346.336i) q^{45} +394.021i q^{47} +13.4949 q^{49} +(-118.540 + 73.6878i) q^{51} +130.961i q^{53} +386.154i q^{55} +(296.717 + 477.323i) q^{57} -38.7221i q^{59} +25.9096 q^{61} +(-225.615 + 457.146i) q^{63} +(-651.638 - 157.822i) q^{65} +56.9292 q^{67} +(40.2109 + 64.6866i) q^{69} +382.294i q^{71} +1033.27i q^{73} +(218.404 + 351.343i) q^{75} -509.704i q^{77} +760.906i q^{79} +(-443.431 - 578.628i) q^{81} -1012.19i q^{83} -384.238i q^{85} +(-1100.47 + 684.083i) q^{87} -300.915 q^{89} +(860.130 + 208.317i) q^{91} +(82.1174 + 132.101i) q^{93} -1547.20 q^{95} -1292.09i q^{97} +(653.612 + 322.576i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 172 q^{9}+O(q^{10}) $ 24 * q + 172 * q^9	$ 24 q + 172 q^{9} + 96 q^{13} + 1392 q^{25} - 1152 q^{49} - 96 q^{61} + 2640 q^{69} - 5396 q^{81}+O(q^{100}) $ 24 * q + 172 * q^9 + 96 * q^13 + 1392 * q^25 - 1152 * q^49 - 96 * q^61 + 2640 * q^69 - 5396 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	2.74324	+	4.41301i	0.527937	+	0.849284i
$3$	2.74324	+	4.41301i	0.527937	+	0.849284i
$4$		0			0
$5$	−14.3044			−1.27942			−0.639711	−	0.768615i	$-0.720946\pi$
$5$	−14.3044			−1.27942			−0.639711	+	0.768615i	$0.720946\pi$
$6$		0			0
$7$	18.8811			1.01948			0.509741	−	0.860328i	$-0.329741\pi$
$7$	18.8811			1.01948			0.509741	+	0.860328i	$0.329741\pi$
$8$		0			0
$9$	−11.9493	+	24.2119i	−0.442565	+	0.896736i
$10$		0			0
$11$		−	26.9955i		−	0.739950i	−0.929042	−	0.369975i	$-0.879366\pi$
$11$		−	26.9955i		−	0.739950i	0.929042	−	0.369975i	$-0.120634\pi$
$12$		0			0
$13$	45.5551	+	11.0331i	0.971902	+	0.235388i
$13$	45.5551	+	11.0331i	0.971902	+	0.235388i
$14$		0			0
$15$	−39.2404	−	63.1253i	−0.675455	−	1.08659i
$16$		0			0
$17$			26.8616i			0.383229i	0.981470	+	0.191614i	$0.0613723\pi$
$17$			26.8616i			0.383229i	−0.981470	+	0.191614i	$0.938628\pi$
$18$		0			0
$19$	108.163			1.30601			0.653006	−	0.757352i	$-0.273507\pi$
$19$	108.163			1.30601			0.653006	+	0.757352i	$0.273507\pi$
$20$		0			0
$21$	51.7953	+	83.3223i	0.538222	+	0.865829i
$22$		0			0
$23$	14.6582			0.132889			0.0664443	−	0.997790i	$-0.478835\pi$
$23$	14.6582			0.132889			0.0664443	+	0.997790i	$0.478835\pi$
$24$		0			0
$25$	79.6154			0.636923
$26$		0			0
$27$	−139.627	+	13.6869i	−0.995230	+	0.0975573i
$28$		0			0
$29$			249.370i			1.59679i	0.602134	+	0.798395i	$0.294317\pi$
$29$			249.370i			1.59679i	−0.602134	+	0.798395i	$0.705683\pi$
$30$		0			0
$31$	29.9344			0.173432			0.0867159	−	0.996233i	$-0.472363\pi$
$31$	29.9344			0.173432			0.0867159	+	0.996233i	$0.472363\pi$
$32$		0			0
$33$	119.131	−	74.0552i	0.628428	−	0.390647i
$34$		0			0
$35$	−270.082			−1.30435
$36$		0			0
$37$		−	214.557i		−	0.953323i	−0.879087	−	0.476661i	$-0.841847\pi$
$37$		−	214.557i		−	0.953323i	0.879087	−	0.476661i	$-0.158153\pi$
$38$		0			0
$39$	76.2794	+	231.302i	0.313192	+	0.949690i
$40$		0			0
$41$	−177.585			−0.676443			−0.338221	−	0.941067i	$-0.609825\pi$
$41$	−177.585			−0.676443			−0.338221	+	0.941067i	$0.609825\pi$
$42$		0			0
$43$			319.109i			1.13171i	0.824504	+	0.565856i	$0.191454\pi$
$43$			319.109i			1.13171i	−0.824504	+	0.565856i	$0.808546\pi$
$44$		0			0
$45$	170.927	−	346.336i	0.566228	−	1.14731i
$46$		0			0
$47$			394.021i			1.22285i	0.791303	+	0.611424i	$0.209403\pi$
$47$			394.021i			1.22285i	−0.791303	+	0.611424i	$0.790597\pi$
$48$		0			0
$49$	13.4949			0.0393437
$50$		0			0
$51$	−118.540	+	73.6878i	−0.325470	+	0.202321i
$52$		0			0
$53$			130.961i			0.339412i	0.985495	+	0.169706i	$0.0542818\pi$
$53$			130.961i			0.339412i	−0.985495	+	0.169706i	$0.945718\pi$
$54$		0			0
$55$			386.154i			0.946709i
$56$		0			0
$57$	296.717	+	477.323i	0.689492	+	1.10918i
$58$		0			0
$59$		−	38.7221i		−	0.0854438i	−0.999087	−	0.0427219i	$-0.986397\pi$
$59$		−	38.7221i		−	0.0854438i	0.999087	−	0.0427219i	$-0.0136029\pi$
$60$		0			0
$61$	25.9096			0.0543834			0.0271917	−	0.999630i	$-0.491344\pi$
$61$	25.9096			0.0543834			0.0271917	+	0.999630i	$0.491344\pi$
$62$		0			0
$63$	−225.615	+	457.146i	−0.451187	+	0.914207i
$64$		0			0
$65$	−651.638	−	157.822i	−1.24347	−	0.301161i
$66$		0			0
$67$	56.9292			0.103806			0.0519030	−	0.998652i	$-0.483471\pi$
$67$	56.9292			0.103806			0.0519030	+	0.998652i	$0.483471\pi$
$68$		0			0
$69$	40.2109	+	64.6866i	0.0701569	+	0.112860i
$70$		0			0
$71$			382.294i			0.639014i	0.947584	+	0.319507i	$0.103517\pi$
$71$			382.294i			0.639014i	−0.947584	+	0.319507i	$0.896483\pi$
$72$		0			0
$73$			1033.27i			1.65665i	0.560248	+	0.828325i	$0.310706\pi$
$73$			1033.27i			1.65665i	−0.560248	+	0.828325i	$0.689294\pi$
$74$		0			0
$75$	218.404	+	351.343i	0.336255	+	0.540928i
$76$		0			0
$77$		−	509.704i		−	0.754366i
$78$		0			0
$79$			760.906i			1.08365i	0.840490	+	0.541827i	$0.182267\pi$
$79$			760.906i			1.08365i	−0.840490	+	0.541827i	$0.817733\pi$
$80$		0			0
$81$	−443.431	−	578.628i	−0.608273	−	0.793728i
$82$		0			0
$83$		−	1012.19i		−	1.33858i	−0.743000	−	0.669291i	$-0.766598\pi$
$83$		−	1012.19i		−	1.33858i	0.743000	−	0.669291i	$-0.233402\pi$
$84$		0			0
$85$		−	384.238i		−	0.490312i
$86$		0			0
$87$	−1100.47	+	684.083i	−1.35613	+	0.843005i
$88$		0			0
$89$	−300.915			−0.358392			−0.179196	−	0.983813i	$-0.557350\pi$
$89$	−300.915			−0.358392			−0.179196	+	0.983813i	$0.557350\pi$
$90$		0			0
$91$	860.130	+	208.317i	0.990836	+	0.239974i
$92$		0			0
$93$	82.1174	+	132.101i	0.0915610	+	0.147293i
$94$		0			0
$95$	−1547.20			−1.67094
$96$		0			0
$97$		−	1292.09i		−	1.35250i	−0.736674	−	0.676248i	$-0.763605\pi$
$97$		−	1292.09i		−	1.35250i	0.736674	−	0.676248i	$-0.236395\pi$
$98$		0			0
$99$	653.612	+	322.576i	0.663540	+	0.327476i
$100$		0			0
$101$			260.334i			0.256477i	0.991743	+	0.128239i	$0.0409323\pi$
$101$			260.334i			0.256477i	−0.991743	+	0.128239i	$0.959068\pi$
$102$		0			0
$103$			1318.32i			1.26114i	0.776132	+	0.630571i	$0.217179\pi$
$103$			1318.32i			1.26114i	−0.776132	+	0.630571i	$0.782821\pi$
$104$		0			0
$105$	−740.900	−	1191.87i	−0.688614	−	1.10776i
$106$		0			0
$107$	−1830.39			−1.65374			−0.826871	−	0.562392i	$-0.809881\pi$
$107$	−1830.39			−1.65374			−0.826871	+	0.562392i	$0.809881\pi$
$108$		0			0
$109$			1053.11i			0.925411i	0.886512	+	0.462706i	$0.153121\pi$
$109$			1053.11i			0.925411i	−0.886512	+	0.462706i	$0.846879\pi$
$110$		0			0
$111$	946.841	−	588.581i	0.809641	−	0.503294i
$112$		0			0
$113$		−	1466.85i		−	1.22115i	−0.791959	−	0.610574i	$-0.790939\pi$
$113$		−	1466.85i		−	1.22115i	0.791959	−	0.610574i	$-0.209061\pi$
$114$		0			0
$115$	−209.676			−0.170021
$116$		0			0
$117$	−811.483	+	971.138i	−0.641210	+	0.767365i
$118$		0			0
$119$			507.175i			0.390695i
$120$		0			0
$121$	602.242			0.452474
$122$		0			0
$123$	−487.159	−	783.685i	−0.357119	−	0.574492i
$124$		0			0
$125$	649.199			0.464529
$126$		0			0
$127$			872.193i			0.609406i	0.952447	+	0.304703i	$0.0985573\pi$
$127$			872.193i			0.609406i	−0.952447	+	0.304703i	$0.901443\pi$
$128$		0			0
$129$	−1408.23	+	875.392i	−0.961144	+	0.597472i
$130$		0			0
$131$	2017.82			1.34578			0.672891	−	0.739742i	$-0.265052\pi$
$131$	2017.82			1.34578			0.672891	+	0.739742i	$0.265052\pi$
$132$		0			0
$133$	2042.23			1.33146
$134$		0			0
$135$	1997.28	−	195.783i	1.27332	−	0.124817i
$136$		0			0
$137$	272.737			0.170084			0.0850420	−	0.996377i	$-0.472898\pi$
$137$	272.737			0.170084			0.0850420	+	0.996377i	$0.472898\pi$
$138$		0			0
$139$			513.170i			0.313140i	0.987667	+	0.156570i	$0.0500437\pi$
$139$			513.170i			0.313140i	−0.987667	+	0.156570i	$0.949956\pi$
$140$		0			0
$141$	−1738.82	+	1080.89i	−1.03854	+	0.645587i
$142$		0			0
$143$	297.845	−	1229.78i	0.174175	−	0.719159i
$144$		0			0
$145$		−	3567.09i		−	2.04297i
$146$		0			0
$147$	37.0197	+	59.5530i	0.0207710	+	0.0334139i
$148$		0			0
$149$	2511.82			1.38105			0.690525	−	0.723308i	$-0.257379\pi$
$149$	2511.82			1.38105			0.690525	+	0.723308i	$0.257379\pi$
$150$		0			0
$151$	−3544.60			−1.91030			−0.955150	−	0.296124i	$-0.904306\pi$
$151$	−3544.60			−1.91030			−0.955150	+	0.296124i	$0.904306\pi$
$152$		0			0
$153$	−650.369	−	320.976i	−0.343655	−	0.169604i
$154$		0			0
$155$	−428.194			−0.221893
$156$		0			0
$157$	770.278			0.391559			0.195780	−	0.980648i	$-0.437276\pi$
$157$	770.278			0.391559			0.195780	+	0.980648i	$0.437276\pi$
$158$		0			0
$159$	−577.930	+	359.257i	−0.288257	+	0.179188i
$160$		0			0
$161$	276.762			0.135478
$162$		0			0
$163$	368.032			0.176850			0.0884249	−	0.996083i	$-0.471817\pi$
$163$	368.032			0.176850			0.0884249	+	0.996083i	$0.471817\pi$
$164$		0			0
$165$	−1704.10	+	1059.31i	−0.804025	+	0.499803i
$166$		0			0
$167$			4050.92i			1.87706i	0.345192	+	0.938532i	$0.387814\pi$
$167$			4050.92i			1.87706i	−0.345192	+	0.938532i	$0.612186\pi$
$168$		0			0
$169$	1953.54	+	1005.23i	0.889185	+	0.457547i
$170$		0			0
$171$	−1292.46	+	2618.82i	−0.577995	+	1.17115i
$172$		0			0
$173$			2913.71i			1.28049i	0.768170	+	0.640246i	$0.221168\pi$
$173$			2913.71i			1.28049i	−0.768170	+	0.640246i	$0.778832\pi$
$174$		0			0
$175$	1503.22			0.649332
$176$		0			0
$177$	170.881	−	106.224i	0.0725660	−	0.0451089i
$178$		0			0
$179$	2118.24			0.884494			0.442247	−	0.896893i	$-0.354182\pi$
$179$	2118.24			0.884494			0.442247	+	0.896893i	$0.354182\pi$
$180$		0			0
$181$	1909.33			0.784086			0.392043	−	0.919947i	$-0.371768\pi$
$181$	1909.33			0.784086			0.392043	+	0.919947i	$0.371768\pi$
$182$		0			0
$183$	71.0763	+	114.339i	0.0287110	+	0.0461869i
$184$		0			0
$185$			3069.10i			1.21970i
$186$		0			0
$187$	725.142			0.283570
$188$		0			0
$189$	−2636.31	+	258.424i	−1.01462	+	0.0994579i
$190$		0			0
$191$	−2588.85			−0.980747			−0.490373	−	0.871512i	$-0.663140\pi$
$191$	−2588.85			−0.980747			−0.490373	+	0.871512i	$0.663140\pi$
$192$		0			0
$193$			4932.27i			1.83955i	0.392451	+	0.919773i	$0.371627\pi$
$193$			4932.27i			1.83955i	−0.392451	+	0.919773i	$0.628373\pi$
$194$		0			0
$195$	−1091.13	−	3308.63i	−0.400705	−	1.21506i
$196$		0			0
$197$	1661.50			0.600899			0.300449	−	0.953798i	$-0.402863\pi$
$197$	1661.50			0.600899			0.300449	+	0.953798i	$0.402863\pi$
$198$		0			0
$199$		−	321.467i		−	0.114513i	−0.998359	−	0.0572567i	$-0.981765\pi$
$199$		−	321.467i		−	0.114513i	0.998359	−	0.0572567i	$-0.0182354\pi$
$200$		0			0
$201$	156.170	+	251.229i	0.0548031	+	0.0881608i
$202$		0			0
$203$			4708.38i			1.62790i
$204$		0			0
$205$	2540.25			0.865456
$206$		0			0
$207$	−175.154	+	354.902i	−0.0588119	+	0.119166i
$208$		0			0
$209$		−	2919.91i		−	0.966384i
$210$		0			0
$211$		−	3569.40i		−	1.16459i	−0.812979	−	0.582293i	$-0.802156\pi$
$211$		−	3569.40i		−	1.16459i	0.812979	−	0.582293i	$-0.197844\pi$
$212$		0			0
$213$	−1687.07	+	1048.73i	−0.542704	+	0.337359i
$214$		0			0
$215$		−	4564.65i		−	1.44794i
$216$		0			0
$217$	565.194			0.176811
$218$		0			0
$219$	−4559.84	+	2834.52i	−1.40697	+	0.874607i
$220$		0			0
$221$	−296.367	+	1223.68i	−0.0902073	+	0.372461i
$222$		0			0
$223$	1054.78			0.316742			0.158371	−	0.987380i	$-0.449376\pi$
$223$	1054.78			0.316742			0.158371	+	0.987380i	$0.449376\pi$
$224$		0			0
$225$	−951.344	+	1927.64i	−0.281880	+	0.571152i
$226$		0			0
$227$		−	2065.79i		−	0.604015i	−0.953305	−	0.302008i	$-0.902343\pi$
$227$		−	2065.79i		−	0.604015i	0.953305	−	0.302008i	$-0.0976568\pi$
$228$		0			0
$229$		−	5995.77i		−	1.73018i	−0.501614	−	0.865091i	$-0.667260\pi$
$229$		−	5995.77i		−	1.73018i	0.501614	−	0.865091i	$-0.332740\pi$
$230$		0			0
$231$	2249.33	−	1398.24i	0.640671	−	0.398258i
$232$		0			0
$233$		−	1628.45i		−	0.457868i	−0.973442	−	0.228934i	$-0.926476\pi$
$233$		−	1628.45i		−	0.457868i	0.973442	−	0.228934i	$-0.0735240\pi$
$234$		0			0
$235$		−	5636.23i		−	1.56454i
$236$		0			0
$237$	−3357.88	+	2087.35i	−0.920329	+	0.572101i
$238$		0			0
$239$		−	6373.35i		−	1.72493i	−0.506119	−	0.862464i	$-0.668920\pi$
$239$		−	6373.35i		−	1.72493i	0.506119	−	0.862464i	$-0.331080\pi$
$240$		0			0
$241$		−	2345.30i		−	0.626863i	−0.949611	−	0.313432i	$-0.898521\pi$
$241$		−	2345.30i		−	0.626863i	0.949611	−	0.313432i	$-0.101479\pi$
$242$		0			0
$243$	1337.05	−	3544.18i	0.352971	−	0.935634i
$244$		0			0
$245$	−193.036			−0.0503372
$246$		0			0
$247$	4927.37	+	1193.37i	1.26932	+	0.307419i
$248$		0			0
$249$	4466.80	−	2776.68i	1.13684	−	0.706687i
$250$		0			0
$251$	5814.56			1.46220			0.731100	−	0.682271i	$-0.239007\pi$
$251$	5814.56			1.46220			0.731100	+	0.682271i	$0.239007\pi$
$252$		0			0
$253$		−	395.705i		−	0.0983310i
$254$		0			0
$255$	1695.65	−	1054.06i	0.416413	−	0.258854i
$256$		0			0
$257$		−	0.665597i		−	0.000161552i	−1.00000		8.07758e-5i	$-0.999974\pi$
$257$		−	0.665597i		−	0.000161552i	1.00000		8.07758e-5i	$-2.57117e-5\pi$
$258$		0			0
$259$		−	4051.06i		−	0.971895i
$260$		0			0
$261$	−6037.73	−	2979.79i	−1.43190	−	0.706683i
$262$		0			0
$263$	5890.61			1.38111			0.690553	−	0.723282i	$-0.257367\pi$
$263$	5890.61			1.38111			0.690553	+	0.723282i	$0.257367\pi$
$264$		0			0
$265$		−	1873.31i		−	0.434251i
$266$		0			0
$267$	−825.482	−	1327.94i	−0.189209	−	0.304377i
$268$		0			0
$269$		−	6578.70i		−	1.49112i	−0.666440	−	0.745559i	$-0.732183\pi$
$269$		−	6578.70i		−	1.49112i	0.666440	−	0.745559i	$-0.267817\pi$
$270$		0			0
$271$	−3405.11			−0.763269			−0.381634	−	0.924313i	$-0.624639\pi$
$271$	−3405.11			−0.763269			−0.381634	+	0.924313i	$0.624639\pi$
$272$		0			0
$273$	1440.24	+	4367.22i	0.319294	+	0.968192i
$274$		0			0
$275$		−	2149.26i		−	0.471291i
$276$		0			0
$277$	7683.12			1.66655			0.833274	−	0.552860i	$-0.186464\pi$
$277$	7683.12			1.66655			0.833274	+	0.552860i	$0.186464\pi$
$278$		0			0
$279$	−357.694	+	724.769i	−0.0767548	+	0.155523i
$280$		0			0
$281$	−7793.40			−1.65450			−0.827251	−	0.561833i	$-0.810096\pi$
$281$	−7793.40			−1.65450			−0.827251	+	0.561833i	$0.810096\pi$
$282$		0			0
$283$		−	6095.14i		−	1.28028i	−0.768260	−	0.640138i	$-0.778877\pi$
$283$		−	6095.14i		−	1.28028i	0.768260	−	0.640138i	$-0.221123\pi$
$284$		0			0
$285$	−4244.35	−	6827.81i	−0.882152	−	1.41910i
$286$		0			0
$287$	−3353.00			−0.689621
$288$		0			0
$289$	4191.46			0.853136
$290$		0			0
$291$	5702.02	−	3544.52i	1.14865	−	0.714033i
$292$		0			0
$293$	5533.46			1.10330			0.551652	−	0.834074i	$-0.313998\pi$
$293$	5533.46			1.10330			0.551652	+	0.834074i	$0.313998\pi$
$294$		0			0
$295$			553.895i			0.109319i
$296$		0			0
$297$	369.485	+	3769.30i	0.0721875	+	0.736421i
$298$		0			0
$299$	667.755	+	161.726i	0.129155	+	0.0312804i
$300$		0			0
$301$			6025.11i			1.15376i
$302$		0			0
$303$	−1148.85	+	714.159i	−0.217822	+	0.135404i
$304$		0			0
$305$	−370.621			−0.0695793
$306$		0			0
$307$	−6872.70			−1.27767			−0.638836	−	0.769343i	$-0.720584\pi$
$307$	−6872.70			−1.27767			−0.638836	+	0.769343i	$0.720584\pi$
$308$		0			0
$309$	−5817.74	+	3616.46i	−1.07107	+	0.665804i
$310$		0			0
$311$	−559.112			−0.101943			−0.0509716	−	0.998700i	$-0.516232\pi$
$311$	−559.112			−0.101943			−0.0509716	+	0.998700i	$0.516232\pi$
$312$		0			0
$313$	−1659.94			−0.299761			−0.149881	−	0.988704i	$-0.547889\pi$
$313$	−1659.94			−0.299761			−0.149881	+	0.988704i	$0.547889\pi$
$314$		0			0
$315$	3227.28	−	6539.20i	0.577259	−	1.16966i
$316$		0			0
$317$	−6469.54			−1.14626			−0.573132	−	0.819463i	$-0.694272\pi$
$317$	−6469.54			−1.14626			−0.573132	+	0.819463i	$0.694272\pi$
$318$		0			0
$319$	6731.88			1.18155
$320$		0			0
$321$	−5021.20	−	8077.52i	−0.873071	−	1.40450i
$322$		0			0
$323$			2905.42i			0.500501i
$324$		0			0
$325$	3626.89	+	878.407i	0.619027	+	0.149924i
$326$		0			0
$327$	−4647.39	+	2888.94i	−0.785936	+	0.488559i
$328$		0			0
$329$			7439.54i			1.24667i
$330$		0			0
$331$	−4753.78			−0.789399			−0.394700	−	0.918810i	$-0.629151\pi$
$331$	−4753.78			−0.789399			−0.394700	+	0.918810i	$0.629151\pi$
$332$		0			0
$333$	5194.83	+	2563.80i	0.854879	+	0.421907i
$334$		0			0
$335$	−814.337			−0.132812
$336$		0			0
$337$	1531.82			0.247606			0.123803	−	0.992307i	$-0.460491\pi$
$337$	1531.82			0.247606			0.123803	+	0.992307i	$0.460491\pi$
$338$		0			0
$339$	6473.22	−	4023.93i	1.03710	−	0.644689i
$340$		0			0
$341$		−	808.095i		−	0.128331i
$342$		0			0
$343$	−6221.41			−0.979372
$344$		0			0
$345$	−575.192	−	925.302i	−0.0897603	−	0.144396i
$346$		0			0
$347$	7224.61			1.11769			0.558844	−	0.829273i	$-0.311245\pi$
$347$	7224.61			1.11769			0.558844	+	0.829273i	$0.311245\pi$
$348$		0			0
$349$		−	9618.02i		−	1.47519i	−0.675244	−	0.737594i	$-0.735962\pi$
$349$		−	9618.02i		−	1.47519i	0.675244	−	0.737594i	$-0.264038\pi$
$350$		0			0
$351$	−6511.73	−	917.013i	−0.990229	−	0.139449i
$352$		0			0
$353$	−11955.4			−1.80262			−0.901309	−	0.433178i	$-0.857392\pi$
$353$	−11955.4			−1.80262			−0.901309	+	0.433178i	$0.857392\pi$
$354$		0			0
$355$		−	5468.49i		−	0.817569i
$356$		0			0
$357$	−2238.17	+	1391.30i	−0.331811	+	0.206262i
$358$		0			0
$359$			10908.0i			1.60363i	0.597571	+	0.801816i	$0.296132\pi$
$359$			10908.0i			1.60363i	−0.597571	+	0.801816i	$0.703868\pi$
$360$		0			0
$361$	4840.18			0.705669
$362$		0			0
$363$	1652.10	+	2657.70i	0.238878	+	0.384278i
$364$		0			0
$365$		−	14780.3i		−	2.11956i
$366$		0			0
$367$		−	4279.23i		−	0.608648i	−0.952569	−	0.304324i	$-0.901569\pi$
$367$		−	4279.23i		−	0.608648i	0.952569	−	0.304324i	$-0.0984305\pi$
$368$		0			0
$369$	2122.01	−	4299.67i	0.299370	−	0.606591i
$370$		0			0
$371$			2472.68i			0.346024i
$372$		0			0
$373$	−27.3835			−0.00380124			−0.00190062	−	0.999998i	$-0.500605\pi$
$373$	−27.3835			−0.00380124			−0.00190062	+	0.999998i	$0.500605\pi$
$374$		0			0
$375$	1780.91	+	2864.92i	0.245242	+	0.394517i
$376$		0			0
$377$	−2751.34	+	11360.1i	−0.375865	+	1.55192i
$378$		0			0
$379$	−4145.18			−0.561803			−0.280902	−	0.959737i	$-0.590633\pi$
$379$	−4145.18			−0.561803			−0.280902	+	0.959737i	$0.590633\pi$
$380$		0			0
$381$	−3848.99	+	2392.64i	−0.517559	+	0.321728i
$382$		0			0
$383$			7597.39i			1.01360i	0.862064	+	0.506800i	$0.169172\pi$
$383$			7597.39i			1.01360i	−0.862064	+	0.506800i	$0.830828\pi$
$384$		0			0
$385$			7291.00i			0.965153i
$386$		0			0
$387$	−7726.22	−	3813.11i	−1.01485	−	0.500856i
$388$		0			0
$389$			5073.24i			0.661243i	0.943763	+	0.330622i	$0.107258\pi$
$389$			5073.24i			0.661243i	−0.943763	+	0.330622i	$0.892742\pi$
$390$		0			0
$391$			393.741i			0.0509268i
$392$		0			0
$393$	5535.36	+	8904.64i	0.710488	+	1.14295i
$394$		0			0
$395$		−	10884.3i		−	1.38645i
$396$		0			0
$397$			1370.84i			0.173301i	0.996239	+	0.0866504i	$0.0276163\pi$
$397$			1370.84i			0.173301i	−0.996239	+	0.0866504i	$0.972384\pi$
$398$		0			0
$399$	5602.33	+	9012.37i	0.702925	+	1.13078i
$400$		0			0
$401$	11246.2			1.40051			0.700257	−	0.713890i	$-0.253069\pi$
$401$	11246.2			1.40051			0.700257	+	0.713890i	$0.253069\pi$
$402$		0			0
$403$	1363.67	+	330.271i	0.168559	+	0.0408237i
$404$		0			0
$405$	6343.00	+	8276.92i	0.778238	+	1.01551i
$406$		0			0
$407$	−5792.07			−0.705411
$408$		0			0
$409$		−	5689.43i		−	0.687834i	−0.939000	−	0.343917i	$-0.888246\pi$
$409$		−	5689.43i		−	0.687834i	0.939000	−	0.343917i	$-0.111754\pi$
$410$		0			0
$411$	748.184	+	1203.59i	0.0897937	+	0.144450i
$412$		0			0
$413$		−	731.114i		−	0.0871084i
$414$		0			0
$415$			14478.8i			1.71261i
$416$		0			0
$417$	−2264.62	+	1407.75i	−0.265945	+	0.165318i
$418$		0			0
$419$	10844.9			1.26446			0.632230	−	0.774780i	$-0.282140\pi$
$419$	10844.9			1.26446			0.632230	+	0.774780i	$0.282140\pi$
$420$		0			0
$421$		−	820.199i		−	0.0949503i	−0.998872	−	0.0474751i	$-0.984883\pi$
$421$		−	820.199i		−	0.0949503i	0.998872	−	0.0474751i	$-0.0151175\pi$
$422$		0			0
$423$	−9539.99	−	4708.26i	−1.09657	−	0.541190i
$424$		0			0
$425$			2138.59i			0.244087i
$426$		0			0
$427$	489.201			0.0554429
$428$		0			0
$429$	6244.11	−	2059.20i	0.702723	−	0.231746i
$430$		0			0
$431$		−	4171.52i		−	0.466206i	−0.972452	−	0.233103i	$-0.925112\pi$
$431$		−	4171.52i		−	0.466206i	0.972452	−	0.233103i	$-0.0748880\pi$
$432$		0			0
$433$	7230.83			0.802521			0.401260	−	0.915964i	$-0.368572\pi$
$433$	7230.83			0.802521			0.401260	+	0.915964i	$0.368572\pi$
$434$		0			0
$435$	15741.6	−	9785.39i	1.73506	−	1.07856i
$436$		0			0
$437$	1585.47			0.173554
$438$		0			0
$439$			4267.12i			0.463915i	0.972726	+	0.231957i	$0.0745130\pi$
$439$			4267.12i			0.463915i	−0.972726	+	0.231957i	$0.925487\pi$
$440$		0			0
$441$	−161.254	+	326.736i	−0.0174121	+	0.0352809i
$442$		0			0
$443$	5687.17			0.609945			0.304973	−	0.952361i	$-0.401353\pi$
$443$	5687.17			0.609945			0.304973	+	0.952361i	$0.401353\pi$
$444$		0			0
$445$	4304.40			0.458536
$446$		0			0
$447$	6890.54	+	11084.7i	0.729108	+	1.17290i
$448$		0			0
$449$	16477.7			1.73192			0.865960	−	0.500113i	$-0.166708\pi$
$449$	16477.7			1.73192			0.865960	+	0.500113i	$0.166708\pi$
$450$		0			0
$451$			4794.00i			0.500534i
$452$		0			0
$453$	−9723.68	−	15642.3i	−1.00852	−	1.62239i
$454$		0			0
$455$	−12303.6	−	2979.85i	−1.26770	−	0.307028i
$456$		0			0
$457$			1845.18i			0.188871i	0.995531	+	0.0944353i	$0.0301045\pi$
$457$			1845.18i			0.188871i	−0.995531	+	0.0944353i	$0.969895\pi$
$458$		0			0
$459$	−367.652	−	3750.60i	−0.0373867	−	0.381401i
$460$		0			0
$461$	−6294.13			−0.635894			−0.317947	−	0.948109i	$-0.602993\pi$
$461$	−6294.13			−0.635894			−0.317947	+	0.948109i	$0.602993\pi$
$462$		0			0
$463$	7489.19			0.751732			0.375866	−	0.926674i	$-0.377345\pi$
$463$	7489.19			0.751732			0.375866	+	0.926674i	$0.377345\pi$
$464$		0			0
$465$	−1174.64	−	1889.62i	−0.117145	−	0.188450i
$466$		0			0
$467$	−11587.2			−1.14816			−0.574080	−	0.818800i	$-0.694640\pi$
$467$	−11587.2			−1.14816			−0.574080	+	0.818800i	$0.694640\pi$
$468$		0			0
$469$	1074.88			0.105828
$470$		0			0
$471$	2113.06	+	3399.24i	0.206719	+	0.332545i
$472$		0			0
$473$	8614.50			0.837410
$474$		0			0
$475$	8611.42			0.831830
$476$		0			0
$477$	−3170.80	−	1564.88i	−0.304363	−	0.150212i
$478$		0			0
$479$		−	7022.38i		−	0.669855i	−0.942244	−	0.334928i	$-0.891288\pi$
$479$		−	7022.38i		−	0.669855i	0.942244	−	0.334928i	$-0.108712\pi$
$480$		0			0
$481$	2367.24	−	9774.17i	0.224401	−	0.926536i
$482$		0			0
$483$	759.225	+	1221.35i	0.0715237	+	0.115059i
$484$		0			0
$485$			18482.6i			1.73042i
$486$		0			0
$487$	−10281.2			−0.956642			−0.478321	−	0.878185i	$-0.658754\pi$
$487$	−10281.2			−0.956642			−0.478321	+	0.878185i	$0.658754\pi$
$488$		0			0
$489$	1009.60	+	1624.13i	0.0933656	+	0.150196i
$490$		0			0
$491$	−14168.2			−1.30224			−0.651120	−	0.758975i	$-0.725701\pi$
$491$	−14168.2			−1.30224			−0.651120	+	0.758975i	$0.725701\pi$
$492$		0			0
$493$	−6698.48			−0.611936
$494$		0			0
$495$	−9349.52	−	4614.25i	−0.848949	−	0.418980i
$496$		0			0
$497$			7218.13i			0.651463i
$498$		0			0
$499$	−482.676			−0.0433017			−0.0216508	−	0.999766i	$-0.506892\pi$
$499$	−482.676			−0.0433017			−0.0216508	+	0.999766i	$0.506892\pi$
$500$		0			0
$501$	−17876.7	+	11112.7i	−1.59416	+	0.990972i
$502$		0			0
$503$	−7557.18			−0.669897			−0.334948	−	0.942236i	$-0.608719\pi$
$503$	−7557.18			−0.669897			−0.334948	+	0.942236i	$0.608719\pi$
$504$		0			0
$505$		−	3723.91i		−	0.328143i
$506$		0			0
$507$	922.937	+	11378.6i	0.0808463	+	0.996727i
$508$		0			0
$509$	−5505.83			−0.479453			−0.239727	−	0.970840i	$-0.577058\pi$
$509$	−5505.83			−0.479453			−0.239727	+	0.970840i	$0.577058\pi$
$510$		0			0
$511$			19509.3i			1.68893i
$512$		0			0
$513$	−15102.4	+	1480.41i	−1.29978	+	0.127411i
$514$		0			0
$515$		−	18857.7i		−	1.61353i
$516$		0			0
$517$	10636.8			0.904847
$518$		0			0
$519$	−12858.2	+	7993.00i	−1.08750	+	0.676019i
$520$		0			0
$521$		−	16485.7i		−	1.38628i	−0.720805	−	0.693138i	$-0.756228\pi$
$521$		−	16485.7i		−	1.38628i	0.720805	−	0.693138i	$-0.243772\pi$
$522$		0			0
$523$		−	7179.63i		−	0.600274i	−0.953896	−	0.300137i	$-0.902968\pi$
$523$		−	7179.63i		−	0.600274i	0.953896	−	0.300137i	$-0.0970323\pi$
$524$		0			0
$525$	4123.71	+	6633.74i	0.342806	+	0.551467i
$526$		0			0
$527$			804.086i			0.0664640i
$528$		0			0
$529$	−11952.1			−0.982341
$530$		0			0
$531$	937.534	+	462.700i	0.0766206	+	0.0378144i
$532$		0			0
$533$	−8089.92	−	1959.32i	−0.657436	−	0.159226i
$534$		0			0
$535$	26182.6			2.11583
$536$		0			0
$537$	5810.83	+	9347.79i	0.466957	+	0.751186i
$538$		0			0
$539$		−	364.301i		−	0.0291123i
$540$		0			0
$541$			1388.69i			0.110359i	0.998476	+	0.0551797i	$0.0175732\pi$
$541$			1388.69i			0.110359i	−0.998476	+	0.0551797i	$0.982427\pi$
$542$		0			0
$543$	5237.76	+	8425.90i	0.413948	+	0.665911i
$544$		0			0
$545$		−	15064.1i		−	1.18399i
$546$		0			0
$547$		−	9480.60i		−	0.741062i	−0.928820	−	0.370531i	$-0.879176\pi$
$547$		−	9480.60i		−	0.741062i	0.928820	−	0.370531i	$-0.120824\pi$
$548$		0			0
$549$	−309.601	+	627.321i	−0.0240682	+	0.0487676i
$550$		0			0
$551$			26972.6i			2.08543i
$552$		0			0
$553$			14366.7i			1.10477i
$554$		0			0
$555$	−13544.0	+	8419.30i	−1.03587	+	0.643926i
$556$		0			0
$557$	19947.1			1.51739			0.758695	−	0.651446i	$-0.225837\pi$
$557$	19947.1			1.51739			0.758695	+	0.651446i	$0.225837\pi$
$558$		0			0
$559$	−3520.77	+	14537.0i	−0.266391	+	1.09991i
$560$		0			0
$561$	1989.24	+	3200.06i	0.149707	+	0.240831i
$562$		0			0
$563$	1860.65			0.139284			0.0696420	−	0.997572i	$-0.477814\pi$
$563$	1860.65			0.139284			0.0696420	+	0.997572i	$0.477814\pi$
$564$		0			0
$565$			20982.4i			1.56237i
$566$		0			0
$567$	−8372.45	−	10925.1i	−0.620123	−	0.809192i
$568$		0			0
$569$			1503.41i			0.110766i	0.998465	+	0.0553831i	$0.0176380\pi$
$569$			1503.41i			0.110766i	−0.998465	+	0.0553831i	$0.982362\pi$
$570$		0			0
$571$			25481.9i			1.86757i	0.357834	+	0.933785i	$0.383515\pi$
$571$			25481.9i			1.86757i	−0.357834	+	0.933785i	$0.616485\pi$
$572$		0			0
$573$	−7101.84	−	11424.6i	−0.517773	−	0.832932i
$574$		0			0
$575$	1167.02			0.0846399
$576$		0			0
$577$		−	7838.15i		−	0.565523i	−0.959190	−	0.282761i	$-0.908750\pi$
$577$		−	7838.15i		−	0.565523i	0.959190	−	0.282761i	$-0.0912504\pi$
$578$		0			0
$579$	−21766.1	+	13530.4i	−1.56230	+	0.971164i
$580$		0			0
$581$		−	19111.2i		−	1.36466i
$582$		0			0
$583$	3535.35			0.251148
$584$		0			0
$585$	11607.8	−	13891.5i	0.820379	−	0.981785i
$586$		0			0
$587$			13096.5i			0.920870i	0.887693	+	0.460435i	$0.152307\pi$
$587$			13096.5i			0.920870i	−0.887693	+	0.460435i	$0.847693\pi$
$588$		0			0
$589$	3237.79			0.226504
$590$		0			0
$591$	4557.90	+	7332.21i	0.317237	+	0.510333i
$592$		0			0
$593$	3283.82			0.227403			0.113702	−	0.993515i	$-0.463729\pi$
$593$	3283.82			0.227403			0.113702	+	0.993515i	$0.463729\pi$
$594$		0			0
$595$		−	7254.83i		−	0.499864i
$596$		0			0
$597$	1418.64	−	881.861i	0.0972544	−	0.0604559i
$598$		0			0
$599$	−2609.41			−0.177993			−0.0889963	−	0.996032i	$-0.528366\pi$
$599$	−2609.41			−0.177993			−0.0889963	+	0.996032i	$0.528366\pi$
$600$		0			0
$601$	−16006.3			−1.08637			−0.543187	−	0.839612i	$-0.682783\pi$
$601$	−16006.3			−1.08637			−0.543187	+	0.839612i	$0.682783\pi$
$602$		0			0
$603$	−680.261	+	1378.36i	−0.0459409	+	0.0930867i
$604$		0			0
$605$	−8614.71			−0.578905
$606$		0			0
$607$			20497.6i			1.37063i	0.728248	+	0.685314i	$0.240335\pi$
$607$			20497.6i			1.37063i	−0.728248	+	0.685314i	$0.759665\pi$
$608$		0			0
$609$	−20778.1	+	12916.2i	−1.38255	+	0.859428i
$610$		0			0
$611$	−4347.29	+	17949.7i	−0.287844	+	1.18849i
$612$		0			0
$613$		−	11568.5i		−	0.762230i	−0.924528	−	0.381115i	$-0.875540\pi$
$613$		−	11568.5i		−	0.762230i	0.924528	−	0.381115i	$-0.124460\pi$
$614$		0			0
$615$	6968.51	+	11210.1i	0.456906	+	0.735018i
$616$		0			0
$617$	15376.4			1.00329			0.501646	−	0.865073i	$-0.332728\pi$
$617$	15376.4			1.00329			0.501646	+	0.865073i	$0.332728\pi$
$618$		0			0
$619$	15839.0			1.02847			0.514237	−	0.857648i	$-0.328075\pi$
$619$	15839.0			1.02847			0.514237	+	0.857648i	$0.328075\pi$
$620$		0			0
$621$	−2046.67	+	200.625i	−0.132255	+	0.0129643i
$622$		0			0
$623$	−5681.60			−0.365375
$624$		0			0
$625$	−19238.3			−1.23125
$626$		0			0
$627$	12885.6	−	8010.02i	0.820734	−	0.510190i
$628$		0			0
$629$	5763.34			0.365341
$630$		0			0
$631$	21666.1			1.36690			0.683449	−	0.729999i	$-0.260479\pi$
$631$	21666.1			1.36690			0.683449	+	0.729999i	$0.260479\pi$
$632$		0			0
$633$	15751.8	−	9791.72i	0.989063	−	0.614828i
$634$		0			0
$635$		−	12476.2i		−	0.779688i
$636$		0			0
$637$	614.761	+	148.891i	0.0382382	+	0.00926101i
$638$		0			0
$639$	−9256.07	−	4568.13i	−0.573027	−	0.282805i
$640$		0			0
$641$		−	25100.8i		−	1.54668i	−0.633991	−	0.773340i	$-0.718584\pi$
$641$		−	25100.8i		−	1.54668i	0.633991	−	0.773340i	$-0.281416\pi$
$642$		0			0
$643$	−18829.8			−1.15486			−0.577428	−	0.816441i	$-0.695944\pi$
$643$	−18829.8			−1.15486			−0.577428	+	0.816441i	$0.695944\pi$
$644$		0			0
$645$	20143.8	−	12521.9i	1.22971	−	0.764420i
$646$		0			0
$647$	26411.1			1.60484			0.802418	−	0.596763i	$-0.203547\pi$
$647$	26411.1			1.60484			0.802418	+	0.596763i	$0.203547\pi$
$648$		0			0
$649$	−1045.32			−0.0632241
$650$		0			0
$651$	1550.46	+	2494.21i	0.0933448	+	0.150162i
$652$		0			0
$653$			8363.95i			0.501235i	0.968086	+	0.250618i	$0.0806337\pi$
$653$			8363.95i			0.501235i	−0.968086	+	0.250618i	$0.919366\pi$
$654$		0			0
$655$	−28863.6			−1.72182
$656$		0			0
$657$	−25017.5	−	12346.8i	−1.48558	−	0.733175i
$658$		0			0
$659$	−16292.0			−0.963042			−0.481521	−	0.876435i	$-0.659916\pi$
$659$	−16292.0			−0.963042			−0.481521	+	0.876435i	$0.659916\pi$
$660$		0			0
$661$		−	9820.12i		−	0.577849i	−0.957352	−	0.288925i	$-0.906702\pi$
$661$		−	9820.12i		−	0.577849i	0.957352	−	0.288925i	$-0.0932977\pi$
$662$		0			0
$663$	−6213.12	+	2048.98i	−0.363948	+	0.120024i
$664$		0			0
$665$	−29212.8			−1.70350
$666$		0			0
$667$			3655.31i			0.212195i
$668$		0			0
$669$	2893.53	+	4654.77i	0.167220	+	0.269004i
$670$		0			0
$671$		−	699.443i		−	0.0402410i
$672$		0			0
$673$	−27402.8			−1.56954			−0.784769	−	0.619788i	$-0.787219\pi$
$673$	−27402.8			−1.56954			−0.784769	+	0.619788i	$0.787219\pi$
$674$		0			0
$675$	−11116.4	+	1089.69i	−0.633885	+	0.0621365i
$676$		0			0
$677$			25695.1i			1.45870i	0.684139	+	0.729352i	$0.260178\pi$
$677$			25695.1i			1.45870i	−0.684139	+	0.729352i	$0.739822\pi$
$678$		0			0
$679$		−	24396.1i		−	1.37885i
$680$		0			0
$681$	9116.36	−	5666.97i	0.512980	−	0.318882i
$682$		0			0
$683$			17767.8i			0.995409i	0.867347	+	0.497705i	$0.165824\pi$
$683$			17767.8i			0.995409i	−0.867347	+	0.497705i	$0.834176\pi$
$684$		0			0
$685$	−3901.34			−0.217609
$686$		0			0
$687$	26459.4	−	16447.9i	1.46942	−	0.913428i
$688$		0			0
$689$	−1444.91	+	5965.93i	−0.0798934	+	0.329875i
$690$		0			0
$691$	−12612.8			−0.694377			−0.347189	−	0.937795i	$-0.612864\pi$
$691$	−12612.8			−0.694377			−0.347189	+	0.937795i	$0.612864\pi$
$692$		0			0
$693$	12340.9	+	6090.58i	0.676468	+	0.333856i
$694$		0			0
$695$		−	7340.58i		−	0.400639i
$696$		0			0
$697$		−	4770.22i		−	0.259232i
$698$		0			0
$699$	7186.36	−	4467.23i	0.388860	−	0.241726i
$700$		0			0
$701$		−	15387.0i		−	0.829044i	−0.910039	−	0.414522i	$-0.863949\pi$
$701$		−	15387.0i		−	0.829044i	0.910039	−	0.414522i	$-0.136051\pi$
$702$		0			0
$703$		−	23207.1i		−	1.24505i
$704$		0			0
$705$	24872.7	−	15461.5i	1.32874	−	0.825979i
$706$		0			0
$707$			4915.38i			0.261474i
$708$		0			0
$709$		−	5700.35i		−	0.301948i	−0.988538	−	0.150974i	$-0.951759\pi$
$709$		−	5700.35i		−	0.301948i	0.988538	−	0.150974i	$-0.0482410\pi$
$710$		0			0
$711$	−18423.0	−	9092.26i	−0.971752	−	0.479587i
$712$		0			0
$713$	438.784			0.0230471
$714$		0			0
$715$	−4260.49	+	17591.3i	−0.222844	+	0.920108i
$716$		0			0
$717$	28125.6	−	17483.6i	1.46495	−	0.910653i
$718$		0			0
$719$	−24696.2			−1.28096			−0.640482	−	0.767973i	$-0.721265\pi$
$719$	−24696.2			−1.28096			−0.640482	+	0.767973i	$0.721265\pi$
$720$		0			0
$721$			24891.2i			1.28571i
$722$		0			0
$723$	10349.8	−	6433.73i	0.532385	−	0.330944i
$724$		0			0
$725$			19853.7i			1.01703i
$726$		0			0
$727$			11839.1i			0.603970i	0.953313	+	0.301985i	$0.0976493\pi$
$727$			11839.1i			0.603970i	−0.953313	+	0.301985i	$0.902351\pi$
$728$		0			0
$729$	19308.3	−	3822.12i	0.980965	−	0.194184i
$730$		0			0
$731$	−8571.75			−0.433704
$732$		0			0
$733$			2720.64i			0.137093i	0.997648	+	0.0685464i	$0.0218361\pi$
$733$			2720.64i			0.137093i	−0.997648	+	0.0685464i	$0.978164\pi$
$734$		0			0
$735$	−529.544	−	851.868i	−0.0265749	−	0.0427505i
$736$		0			0
$737$		−	1536.83i		−	0.0768113i
$738$		0			0
$739$	25290.0			1.25887			0.629437	−	0.777052i	$-0.283286\pi$
$739$	25290.0			1.25887			0.629437	+	0.777052i	$0.283286\pi$
$740$		0			0
$741$	8250.59	+	25018.2i	0.409033	+	1.24031i
$742$		0			0
$743$		−	36091.8i		−	1.78207i	−0.453935	−	0.891035i	$-0.649980\pi$
$743$		−	36091.8i		−	1.78207i	0.453935	−	0.891035i	$-0.350020\pi$
$744$		0			0
$745$	−35930.1			−1.76695
$746$		0			0
$747$	24507.0	+	12094.9i	1.20036	+	0.592409i
$748$		0			0
$749$	−34559.7			−1.68596
$750$		0			0
$751$		−	18107.9i		−	0.879850i	−0.898035	−	0.439925i	$-0.855005\pi$
$751$		−	18107.9i		−	0.879850i	0.898035	−	0.439925i	$-0.144995\pi$
$752$		0			0
$753$	15950.8	+	25659.7i	0.771949	+	1.24182i
$754$		0			0
$755$	50703.3			2.44408
$756$		0			0
$757$	1442.14			0.0692410			0.0346205	−	0.999401i	$-0.488978\pi$
$757$	1442.14			0.0692410			0.0346205	+	0.999401i	$0.488978\pi$
$758$		0			0
$759$	1746.25	−	1085.51i	0.0835109	−	0.0519126i
$760$		0			0
$761$	22361.2			1.06517			0.532585	−	0.846377i	$-0.321221\pi$
$761$	22361.2			1.06517			0.532585	+	0.846377i	$0.321221\pi$
$762$		0			0
$763$			19883.9i			0.943440i
$764$		0			0
$765$	9303.13	+	4591.36i	0.439680	+	0.216995i
$766$		0			0
$767$	427.226	−	1763.99i	0.0201124	−	0.0830429i
$768$		0			0
$769$			25805.9i			1.21012i	0.796179	+	0.605061i	$0.206851\pi$
$769$			25805.9i			1.21012i	−0.796179	+	0.605061i	$0.793149\pi$
$770$		0			0
$771$	2.93728	−	1.82589i	0.000137203	−	8.52891e-5i
$772$		0			0
$773$	4701.85			0.218776			0.109388	−	0.993999i	$-0.465111\pi$
$773$	4701.85			0.218776			0.109388	+	0.993999i	$0.465111\pi$
$774$		0			0
$775$	2383.24			0.110463
$776$		0			0
$777$	17877.4	−	11113.0i	0.825415	−	0.513100i
$778$		0			0
$779$	−19208.1			−0.883443
$780$		0			0
$781$	10320.2			0.472839
$782$		0			0
$783$	−3413.11	−	34818.8i	−0.155779	−	1.58917i
$784$		0			0
$785$	−11018.3			−0.500970
$786$		0			0
$787$	−32019.3			−1.45027			−0.725135	−	0.688606i	$-0.758223\pi$
$787$	−32019.3			−1.45027			−0.725135	+	0.688606i	$0.758223\pi$
$788$		0			0
$789$	16159.4	+	25995.3i	0.729137	+	1.17295i
$790$		0			0
$791$		−	27695.7i		−	1.24494i
$792$		0			0
$793$	1180.32	+	285.864i	0.0528553	+	0.0128012i
$794$		0			0
$795$	8266.93	−	5138.94i	0.368802	−	0.229257i
$796$		0			0
$797$			23495.2i			1.04422i	0.852878	+	0.522110i	$0.174855\pi$
$797$			23495.2i			1.04422i	−0.852878	+	0.522110i	$0.825145\pi$
$798$		0			0
$799$	−10584.0			−0.468630
$800$		0			0
$801$	3595.71	−	7285.72i	0.158612	−	0.321384i
$802$		0			0
$803$	27893.7			1.22584
$804$		0			0
$805$	−3958.91			−0.173333
$806$		0			0
$807$	29031.9	−	18047.0i	1.26638	−	0.787216i
$808$		0			0
$809$			26231.2i			1.13997i	0.821654	+	0.569987i	$0.193052\pi$
$809$			26231.2i			1.13997i	−0.821654	+	0.569987i	$0.806948\pi$
$810$		0			0
$811$	−4917.91			−0.212936			−0.106468	−	0.994316i	$-0.533954\pi$
$811$	−4917.91			−0.212936			−0.106468	+	0.994316i	$0.533954\pi$
$812$		0			0
$813$	−9341.04	−	15026.8i	−0.402958	−	0.648231i
$814$		0			0
$815$	−5264.48			−0.226266
$816$		0			0
$817$			34515.7i			1.47803i
$818$		0			0
$819$	−15321.7	+	18336.1i	−0.653702	+	0.782315i
$820$		0			0
$821$	−18401.9			−0.782253			−0.391127	−	0.920337i	$-0.627915\pi$
$821$	−18401.9			−0.782253			−0.391127	+	0.920337i	$0.627915\pi$
$822$		0			0
$823$		−	33459.7i		−	1.41717i	−0.705625	−	0.708586i	$-0.749334\pi$
$823$		−	33459.7i		−	1.41717i	0.705625	−	0.708586i	$-0.250666\pi$
$824$		0			0
$825$	9484.69	−	5895.93i	0.400260	−	0.248812i
$826$		0			0
$827$		−	29537.9i		−	1.24200i	−0.783811	−	0.620999i	$-0.786727\pi$
$827$		−	29537.9i		−	1.24200i	0.783811	−	0.620999i	$-0.213273\pi$
$828$		0			0
$829$	21269.6			0.891101			0.445551	−	0.895257i	$-0.353008\pi$
$829$	21269.6			0.891101			0.445551	+	0.895257i	$0.353008\pi$
$830$		0			0
$831$	21076.7	+	33905.7i	0.879833	+	1.41537i
$832$		0			0
$833$			362.493i			0.0150776i
$834$		0			0
$835$		−	57945.9i		−	2.40156i
$836$		0			0
$837$	−4179.65	+	409.710i	−0.172604	+	0.0169195i
$838$		0			0
$839$			3088.83i			0.127102i	0.997979	+	0.0635509i	$0.0202425\pi$
$839$			3088.83i			0.127102i	−0.997979	+	0.0635509i	$0.979757\pi$
$840$		0			0
$841$	−37796.6			−1.54974
$842$		0			0
$843$	−21379.2	−	34392.3i	−0.873473	−	1.40514i
$844$		0			0
$845$	−27944.2	−	14379.2i	−1.13764	−	0.585397i
$846$		0			0
$847$	11371.0			0.461289
$848$		0			0
$849$	26897.9	−	16720.4i	1.08732	−	0.675905i
$850$		0			0
$851$		−	3145.01i		−	0.126686i
$852$		0			0
$853$		−	10875.3i		−	0.436532i	−0.975889	−	0.218266i	$-0.929960\pi$
$853$		−	10875.3i		−	0.436532i	0.975889	−	0.218266i	$-0.0700401\pi$
$854$		0			0
$855$	18487.9	−	37460.7i	0.739501	−	1.49840i
$856$		0			0
$857$			19135.9i			0.762744i	0.924422	+	0.381372i	$0.124548\pi$
$857$			19135.9i			0.762744i	−0.924422	+	0.381372i	$0.875452\pi$
$858$		0			0
$859$		−	27566.6i		−	1.09495i	−0.836822	−	0.547474i	$-0.815589\pi$
$859$		−	27566.6i		−	1.09495i	0.836822	−	0.547474i	$-0.184411\pi$
$860$		0			0
$861$	−9198.09	−	14796.8i	−0.364077	−	0.585684i
$862$		0			0
$863$			46060.7i			1.81683i	0.418070	+	0.908415i	$0.362707\pi$
$863$			46060.7i			1.81683i	−0.418070	+	0.908415i	$0.637293\pi$
$864$		0			0
$865$		−	41678.8i		−	1.63829i
$866$		0			0
$867$	11498.2	+	18496.9i	0.450402	+	0.724554i
$868$		0			0
$869$	20541.0			0.801850
$870$		0			0
$871$	2593.42	+	628.107i	0.100889	+	0.0244347i
$872$		0			0
$873$	31284.0	+	15439.6i	1.21283	+	0.598568i
$874$		0			0
$875$	12257.6			0.473579
$876$		0			0
$877$		−	41554.7i		−	1.60000i	−0.599998	−	0.800002i	$-0.704832\pi$
$877$		−	41554.7i		−	1.60000i	0.599998	−	0.800002i	$-0.295168\pi$
$878$		0			0
$879$	15179.6	+	24419.2i	0.582475	+	0.937018i
$880$		0			0
$881$		−	22071.0i		−	0.844029i	−0.906589	−	0.422015i	$-0.861323\pi$
$881$		−	22071.0i		−	0.844029i	0.906589	−	0.422015i	$-0.138677\pi$
$882$		0			0
$883$			7238.80i			0.275883i	0.990440	+	0.137942i	$0.0440486\pi$
$883$			7238.80i			0.275883i	−0.990440	+	0.137942i	$0.955951\pi$
$884$		0			0
$885$	−2444.34	+	1519.47i	−0.0928426	+	0.0577134i
$886$		0			0
$887$	49447.0			1.87178			0.935890	−	0.352293i	$-0.114598\pi$
$887$	49447.0			1.87178			0.935890	+	0.352293i	$0.114598\pi$
$888$		0			0
$889$			16467.9i			0.621279i
$890$		0			0
$891$	−15620.4	+	11970.6i	−0.587319	+	0.450091i
$892$		0			0
$893$			42618.4i			1.59706i
$894$		0			0
$895$	−30300.1			−1.13164
$896$		0			0
$897$	1118.12	+	3390.46i	0.0416197	+	0.126203i
$898$		0			0
$899$			7464.76i			0.276934i
$900$		0			0
$901$	−3517.81			−0.130072
$902$		0			0
$903$	−26588.9	+	16528.3i	−0.979869	+	0.609112i
$904$		0			0
$905$	−27311.8			−1.00318
$906$		0			0
$907$			5231.82i			0.191532i	0.995404	+	0.0957661i	$0.0305301\pi$
$907$			5231.82i			0.191532i	−0.995404	+	0.0957661i	$0.969470\pi$
$908$		0			0
$909$	−6303.17	−	3110.79i	−0.229992	−	0.113508i
$910$		0			0
$911$	28413.3			1.03334			0.516671	−	0.856184i	$-0.327171\pi$
$911$	28413.3			1.03334			0.516671	+	0.856184i	$0.327171\pi$
$912$		0			0
$913$	−27324.6			−0.990484
$914$		0			0
$915$	−1016.70	−	1635.55i	−0.0367335	−	0.0590926i
$916$		0			0
$917$	38098.5			1.37200
$918$		0			0
$919$		−	8994.82i		−	0.322864i	−0.986884	−	0.161432i	$-0.948389\pi$
$919$		−	8994.82i		−	0.322864i	0.986884	−	0.161432i	$-0.0516112\pi$
$920$		0			0
$921$	−18853.5	−	30329.3i	−0.674531	−	1.08511i
$922$		0			0
$923$	−4217.91	+	17415.5i	−0.150416	+	0.621059i
$924$		0			0
$925$		−	17082.0i		−	0.607193i
$926$		0			0
$927$	−31918.9	−	15752.9i	−1.13091	−	0.558137i
$928$		0			0
$929$	31711.7			1.11994			0.559971	−	0.828512i	$-0.310812\pi$
$929$	31711.7			1.11994			0.559971	+	0.828512i	$0.310812\pi$
$930$		0			0
$931$	1459.64			0.0513833
$932$		0			0
$933$	−1533.78	−	2467.36i	−0.0538196	−	0.0865786i
$934$		0			0
$935$	−10372.7			−0.362806
$936$		0			0
$937$	8115.74			0.282956			0.141478	−	0.989941i	$-0.454815\pi$
$937$	8115.74			0.282956			0.141478	+	0.989941i	$0.454815\pi$
$938$		0			0
$939$	−4553.61	−	7325.32i	−0.158255	−	0.254582i
$940$		0			0
$941$	−11412.8			−0.395375			−0.197687	−	0.980265i	$-0.563343\pi$
$941$	−11412.8			−0.395375			−0.197687	+	0.980265i	$0.563343\pi$
$942$		0			0
$943$	−2603.07			−0.0898916
$944$		0			0
$945$	37710.7	−	3696.59i	1.29813	−	0.127249i
$946$		0			0
$947$			20158.5i			0.691724i	0.938285	+	0.345862i	$0.112414\pi$
$947$			20158.5i			0.691724i	−0.938285	+	0.345862i	$0.887586\pi$
$948$		0			0
$949$	−11400.2	+	47070.9i	−0.389955	+	1.61010i
$950$		0			0
$951$	−17747.5	−	28550.1i	−0.605155	−	0.973503i
$952$		0			0
$953$		−	34313.7i		−	1.16635i	−0.812348	−	0.583174i	$-0.801811\pi$
$953$		−	34313.7i		−	1.16635i	0.812348	−	0.583174i	$-0.198189\pi$
$954$		0			0
$955$	37031.9			1.25479
$956$		0			0
$957$	18467.2	+	29707.8i	0.623782	+	1.00347i
$958$		0			0
$959$	5149.57			0.173398
$960$		0			0
$961$	−28894.9			−0.969921
$962$		0			0
$963$	21871.8	−	44317.2i	0.731888	−	1.48297i
$964$		0			0
$965$		−	70553.0i		−	2.35356i
$966$		0			0
$967$	−33343.8			−1.10886			−0.554428	−	0.832232i	$-0.687063\pi$
$967$	−33343.8			−1.10886			−0.554428	+	0.832232i	$0.687063\pi$
$968$		0			0
$969$	−12821.6	+	7970.27i	−0.425068	+	0.264233i
$970$		0			0
$971$	−55349.0			−1.82928			−0.914641	−	0.404266i	$-0.867527\pi$
$971$	−55349.0			−1.82928			−0.914641	+	0.404266i	$0.867527\pi$
$972$		0			0
$973$			9689.19i			0.319241i
$974$		0			0
$975$	6073.02	+	18415.2i	0.199479	+	0.604879i
$976$		0			0
$977$	−1406.80			−0.0460671			−0.0230336	−	0.999735i	$-0.507332\pi$
$977$	−1406.80			−0.0460671			−0.0230336	+	0.999735i	$0.507332\pi$
$978$		0			0
$979$			8123.35i			0.265193i
$980$		0			0
$981$	−25497.8	−	12583.9i	−0.829850	−	0.409554i
$982$		0			0
$983$		−	20317.2i		−	0.659226i	−0.944116	−	0.329613i	$-0.893082\pi$
$983$		−	20317.2i		−	0.659226i	0.944116	−	0.329613i	$-0.106918\pi$
$984$		0			0
$985$	−23766.7			−0.768803
$986$		0			0
$987$	−32830.7	+	20408.5i	−1.05878	+	0.658164i
$988$		0			0
$989$			4677.55i			0.150392i
$990$		0			0
$991$		−	41950.7i		−	1.34471i	−0.740228	−	0.672356i	$-0.765283\pi$
$991$		−	41950.7i		−	1.34471i	0.740228	−	0.672356i	$-0.234717\pi$
$992$		0			0
$993$	−13040.8	−	20978.4i	−0.416753	−	0.670424i
$994$		0			0
$995$			4598.38i			0.146511i
$996$		0			0
$997$	−5739.30			−0.182313			−0.0911563	−	0.995837i	$-0.529056\pi$
$997$	−5739.30			−0.182313			−0.0911563	+	0.995837i	$0.529056\pi$
$998$		0			0
$999$	2936.62	+	29957.9i	0.0930036	+	0.948775i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.n.c.623.15	yes	24
3.2	odd	2	inner	624.4.n.c.623.12	yes	24
4.3	odd	2	inner	624.4.n.c.623.9	✓	24
12.11	even	2	inner	624.4.n.c.623.14	yes	24
13.12	even	2	inner	624.4.n.c.623.16	yes	24
39.38	odd	2	inner	624.4.n.c.623.11	yes	24
52.51	odd	2	inner	624.4.n.c.623.10	yes	24
156.155	even	2	inner	624.4.n.c.623.13	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.4.n.c.623.9	✓	24	4.3	odd	2	inner
624.4.n.c.623.10	yes	24	52.51	odd	2	inner
624.4.n.c.623.11	yes	24	39.38	odd	2	inner
624.4.n.c.623.12	yes	24	3.2	odd	2	inner
624.4.n.c.623.13	yes	24	156.155	even	2	inner
624.4.n.c.623.14	yes	24	12.11	even	2	inner
624.4.n.c.623.15	yes	24	1.1	even	1	trivial
624.4.n.c.623.16	yes	24	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.74324 + 4.41301i) q^{3} -14.3044 q^{5} +18.8811 q^{7} +(-11.9493 + 24.2119i) q^{9} +O(q^{10})\)	\(q+(2.74324 + 4.41301i) q^{3} -14.3044 q^{5} +18.8811 q^{7} +(-11.9493 + 24.2119i) q^{9} -26.9955i q^{11} +(45.5551 + 11.0331i) q^{13} +(-39.2404 - 63.1253i) q^{15} +26.8616i q^{17} +108.163 q^{19} +(51.7953 + 83.3223i) q^{21} +14.6582 q^{23} +79.6154 q^{25} +(-139.627 + 13.6869i) q^{27} +249.370i q^{29} +29.9344 q^{31} +(119.131 - 74.0552i) q^{33} -270.082 q^{35} -214.557i q^{37} +(76.2794 + 231.302i) q^{39} -177.585 q^{41} +319.109i q^{43} +(170.927 - 346.336i) q^{45} +394.021i q^{47} +13.4949 q^{49} +(-118.540 + 73.6878i) q^{51} +130.961i q^{53} +386.154i q^{55} +(296.717 + 477.323i) q^{57} -38.7221i q^{59} +25.9096 q^{61} +(-225.615 + 457.146i) q^{63} +(-651.638 - 157.822i) q^{65} +56.9292 q^{67} +(40.2109 + 64.6866i) q^{69} +382.294i q^{71} +1033.27i q^{73} +(218.404 + 351.343i) q^{75} -509.704i q^{77} +760.906i q^{79} +(-443.431 - 578.628i) q^{81} -1012.19i q^{83} -384.238i q^{85} +(-1100.47 + 684.083i) q^{87} -300.915 q^{89} +(860.130 + 208.317i) q^{91} +(82.1174 + 132.101i) q^{93} -1547.20 q^{95} -1292.09i q^{97} +(653.612 + 322.576i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 172 q^{9}+O(q^{10}) \) 24 * q + 172 * q^9	\( 24 q + 172 q^{9} + 96 q^{13} + 1392 q^{25} - 1152 q^{49} - 96 q^{61} + 2640 q^{69} - 5396 q^{81}+O(q^{100}) \) 24 * q + 172 * q^9 + 96 * q^13 + 1392 * q^25 - 1152 * q^49 - 96 * q^61 + 2640 * q^69 - 5396 * q^81

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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