LMFDB - Embedded newform 624.4.d.c.287.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(287,624)]

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, 0]))

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

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

chi := DirichletCharacter("624.287");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.d (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			287.18
Character	$\chi$	$=$	624.287
Dual form			624.4.d.c.287.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.35002 + 4.63437i) q^{3} -6.29951i q^{5} +30.3725i q^{7} +(-15.9549 + 21.7817i) q^{9} +O(q^{10})$	\(q+(2.35002 + 4.63437i) q^{3} -6.29951i q^{5} +30.3725i q^{7} +(-15.9549 + 21.7817i) q^{9} -49.7342 q^{11} -13.0000 q^{13} +(29.1943 - 14.8039i) q^{15} -131.636i q^{17} +115.753i q^{19} +(-140.757 + 71.3758i) q^{21} -74.5015 q^{23} +85.3162 q^{25} +(-138.439 - 22.7534i) q^{27} -229.499i q^{29} -67.7555i q^{31} +(-116.876 - 230.487i) q^{33} +191.332 q^{35} +67.2638 q^{37} +(-30.5502 - 60.2469i) q^{39} -88.1630i q^{41} -334.659i q^{43} +(137.214 + 100.508i) q^{45} -97.2469 q^{47} -579.487 q^{49} +(610.051 - 309.347i) q^{51} +684.071i q^{53} +313.301i q^{55} +(-536.444 + 272.022i) q^{57} -33.3463 q^{59} -305.585 q^{61} +(-661.564 - 484.588i) q^{63} +81.8936i q^{65} +659.134i q^{67} +(-175.080 - 345.268i) q^{69} -497.775 q^{71} +131.917 q^{73} +(200.494 + 395.387i) q^{75} -1510.55i q^{77} -924.642i q^{79} +(-219.885 - 695.048i) q^{81} +597.746 q^{83} -829.243 q^{85} +(1063.59 - 539.327i) q^{87} +166.492i q^{89} -394.842i q^{91} +(314.004 - 159.226i) q^{93} +729.188 q^{95} -1454.15 q^{97} +(793.502 - 1083.30i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 84 q^{9}+O(q^{10}) $ 24 * q - 84 * q^9	$ 24 q - 84 q^{9} - 312 q^{13} - 300 q^{21} - 240 q^{25} + 240 q^{33} - 456 q^{37} + 1836 q^{45} - 1632 q^{49} + 168 q^{57} - 960 q^{61} + 2760 q^{69} + 1248 q^{73} - 468 q^{81} - 7704 q^{85} + 3336 q^{93} - 2496 q^{97}+O(q^{100}) $ 24 * q - 84 * q^9 - 312 * q^13 - 300 * q^21 - 240 * q^25 + 240 * q^33 - 456 * q^37 + 1836 * q^45 - 1632 * q^49 + 168 * q^57 - 960 * q^61 + 2760 * q^69 + 1248 * q^73 - 468 * q^81 - 7704 * q^85 + 3336 * q^93 - 2496 * q^97

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
$3$	2.35002	+	4.63437i	0.452261	+	0.891886i
$3$	2.35002	+	4.63437i	0.452261	+	0.891886i
$4$		0			0
$5$		−	6.29951i		−	0.563445i	−0.959496	−	0.281722i	$-0.909094\pi$
$5$		−	6.29951i		−	0.563445i	0.959496	−	0.281722i	$-0.0909057\pi$
$6$		0			0
$7$			30.3725i			1.63996i	0.572393	+	0.819979i	$0.306015\pi$
$7$			30.3725i			1.63996i	−0.572393	+	0.819979i	$0.693985\pi$
$8$		0			0
$9$	−15.9549	+	21.7817i	−0.590920	+	0.806730i
$10$		0			0
$11$	−49.7342			−1.36322			−0.681611	−	0.731715i	$-0.738720\pi$
$11$	−49.7342			−1.36322			−0.681611	+	0.731715i	$0.738720\pi$
$12$		0			0
$13$	−13.0000			−0.277350
$13$	−13.0000			−0.277350
$14$		0			0
$15$	29.1943	−	14.8039i	0.502529	−	0.254824i
$16$		0			0
$17$		−	131.636i		−	1.87803i	−0.343881	−	0.939013i	$-0.611742\pi$
$17$		−	131.636i		−	1.87803i	0.343881	−	0.939013i	$-0.388258\pi$
$18$		0			0
$19$			115.753i			1.39766i	0.715286	+	0.698832i	$0.246296\pi$
$19$			115.753i			1.39766i	−0.715286	+	0.698832i	$0.753704\pi$
$20$		0			0
$21$	−140.757	+	71.3758i	−1.46266	+	0.741689i
$22$		0			0
$23$	−74.5015			−0.675419			−0.337709	−	0.941250i	$-0.609652\pi$
$23$	−74.5015			−0.675419			−0.337709	+	0.941250i	$0.609652\pi$
$24$		0			0
$25$	85.3162			0.682530
$26$		0			0
$27$	−138.439	−	22.7534i	−0.986761	−	0.162181i
$28$		0			0
$29$		−	229.499i		−	1.46955i	−0.678311	−	0.734775i	$-0.737288\pi$
$29$		−	229.499i		−	1.46955i	0.678311	−	0.734775i	$-0.262712\pi$
$30$		0			0
$31$		−	67.7555i		−	0.392556i	−0.980548	−	0.196278i	$-0.937114\pi$
$31$		−	67.7555i		−	0.392556i	0.980548	−	0.196278i	$-0.0628855\pi$
$32$		0			0
$33$	−116.876	−	230.487i	−0.616531	−	1.21584i
$34$		0			0
$35$	191.332			0.924027
$36$		0			0
$37$	67.2638			0.298868			0.149434	−	0.988772i	$-0.452255\pi$
$37$	67.2638			0.298868			0.149434	+	0.988772i	$0.452255\pi$
$38$		0			0
$39$	−30.5502	−	60.2469i	−0.125435	−	0.247365i
$40$		0			0
$41$		−	88.1630i		−	0.335823i	−0.985802	−	0.167911i	$-0.946298\pi$
$41$		−	88.1630i		−	0.335823i	0.985802	−	0.167911i	$-0.0537023\pi$
$42$		0			0
$43$		−	334.659i		−	1.18686i	−0.804885	−	0.593431i	$-0.797773\pi$
$43$		−	334.659i		−	1.18686i	0.804885	−	0.593431i	$-0.202227\pi$
$44$		0			0
$45$	137.214	+	100.508i	0.454548	+	0.332951i
$46$		0			0
$47$	−97.2469			−0.301807			−0.150903	−	0.988549i	$-0.548218\pi$
$47$	−97.2469			−0.301807			−0.150903	+	0.988549i	$0.548218\pi$
$48$		0			0
$49$	−579.487			−1.68947
$50$		0			0
$51$	610.051	−	309.347i	1.67499	−	0.849358i
$52$		0			0
$53$			684.071i			1.77291i	0.462813	+	0.886456i	$0.346840\pi$
$53$			684.071i			1.77291i	−0.462813	+	0.886456i	$0.653160\pi$
$54$		0			0
$55$			313.301i			0.768100i
$56$		0			0
$57$	−536.444	+	272.022i	−1.24656	+	0.632108i
$58$		0			0
$59$	−33.3463			−0.0735816			−0.0367908	−	0.999323i	$-0.511714\pi$
$59$	−33.3463			−0.0735816			−0.0367908	+	0.999323i	$0.511714\pi$
$60$		0			0
$61$	−305.585			−0.641411			−0.320706	−	0.947179i	$-0.603920\pi$
$61$	−305.585			−0.641411			−0.320706	+	0.947179i	$0.603920\pi$
$62$		0			0
$63$	−661.564	−	484.588i	−1.32300	−	0.969085i
$64$		0			0
$65$			81.8936i			0.156272i
$66$		0			0
$67$			659.134i			1.20188i	0.799294	+	0.600940i	$0.205207\pi$
$67$			659.134i			1.20188i	−0.799294	+	0.600940i	$0.794793\pi$
$68$		0			0
$69$	−175.080	−	345.268i	−0.305465	−	0.602396i
$70$		0			0
$71$	−497.775			−0.832043			−0.416022	−	0.909355i	$-0.636576\pi$
$71$	−497.775			−0.832043			−0.416022	+	0.909355i	$0.636576\pi$
$72$		0			0
$73$	131.917			0.211503			0.105751	−	0.994393i	$-0.466275\pi$
$73$	131.917			0.211503			0.105751	+	0.994393i	$0.466275\pi$
$74$		0			0
$75$	200.494	+	395.387i	0.308681	+	0.608739i
$76$		0			0
$77$		−	1510.55i		−	2.23563i
$78$		0			0
$79$		−	924.642i		−	1.31684i	−0.752651	−	0.658420i	$-0.771225\pi$
$79$		−	924.642i		−	1.31684i	0.752651	−	0.658420i	$-0.228775\pi$
$80$		0			0
$81$	−219.885	−	695.048i	−0.301626	−	0.953426i
$82$		0			0
$83$	597.746			0.790495			0.395248	−	0.918575i	$-0.370659\pi$
$83$	597.746			0.790495			0.395248	+	0.918575i	$0.370659\pi$
$84$		0			0
$85$	−829.243			−1.05816
$86$		0			0
$87$	1063.59	−	539.327i	1.31067	−	0.664620i
$88$		0			0
$89$			166.492i			0.198294i	0.995073	+	0.0991468i	$0.0316113\pi$
$89$			166.492i			0.198294i	−0.995073	+	0.0991468i	$0.968389\pi$
$90$		0			0
$91$		−	394.842i		−	0.454843i
$92$		0			0
$93$	314.004	−	159.226i	0.350115	−	0.177538i
$94$		0			0
$95$	729.188			0.787507
$96$		0			0
$97$	−1454.15			−1.52213			−0.761065	−	0.648676i	$-0.775323\pi$
$97$	−1454.15			−1.52213			−0.761065	+	0.648676i	$0.775323\pi$
$98$		0			0
$99$	793.502	−	1083.30i	0.805555	−	1.09975i
$100$		0			0
$101$		−	654.193i		−	0.644501i	−0.946654	−	0.322251i	$-0.895561\pi$
$101$		−	654.193i		−	0.644501i	0.946654	−	0.322251i	$-0.104439\pi$
$102$		0			0
$103$		−	840.610i		−	0.804153i	−0.915606	−	0.402077i	$-0.868289\pi$
$103$		−	840.610i		−	0.804153i	0.915606	−	0.402077i	$-0.131711\pi$
$104$		0			0
$105$	449.632	+	886.702i	0.417901	+	0.824126i
$106$		0			0
$107$	−1821.69			−1.64589			−0.822943	−	0.568123i	$-0.807670\pi$
$107$	−1821.69			−1.64589			−0.822943	+	0.568123i	$0.807670\pi$
$108$		0			0
$109$	−735.825			−0.646599			−0.323299	−	0.946297i	$-0.604792\pi$
$109$	−735.825			−0.646599			−0.323299	+	0.946297i	$0.604792\pi$
$110$		0			0
$111$	158.071	+	311.726i	0.135166	+	0.266556i
$112$		0			0
$113$			1684.44i			1.40229i	0.713017	+	0.701146i	$0.247328\pi$
$113$			1684.44i			1.40229i	−0.713017	+	0.701146i	$0.752672\pi$
$114$		0			0
$115$			469.323i			0.380561i
$116$		0			0
$117$	207.413	−	283.162i	0.163892	−	0.223747i
$118$		0			0
$119$	3998.11			3.07989
$120$		0			0
$121$	1142.49			0.858372
$122$		0			0
$123$	408.580	−	207.184i	0.299516	−	0.151880i
$124$		0			0
$125$		−	1324.89i		−	0.948013i
$126$		0			0
$127$		−	1743.09i		−	1.21791i	−0.793206	−	0.608954i	$-0.791589\pi$
$127$		−	1743.09i		−	1.21791i	0.793206	−	0.608954i	$-0.208411\pi$
$128$		0			0
$129$	1550.94	−	786.454i	1.05854	−	0.536771i
$130$		0			0
$131$	−51.9538			−0.0346506			−0.0173253	−	0.999850i	$-0.505515\pi$
$131$	−51.9538			−0.0346506			−0.0173253	+	0.999850i	$0.505515\pi$
$132$		0			0
$133$	−3515.71			−2.29211
$134$		0			0
$135$	−143.335	+	872.096i	−0.0913803	+	0.555985i
$136$		0			0
$137$			1000.78i			0.624107i	0.950065	+	0.312053i	$0.101017\pi$
$137$			1000.78i			0.624107i	−0.950065	+	0.312053i	$0.898983\pi$
$138$		0			0
$139$			1128.95i			0.688893i	0.938806	+	0.344446i	$0.111933\pi$
$139$			1128.95i			0.688893i	−0.938806	+	0.344446i	$0.888067\pi$
$140$		0			0
$141$	−228.532	−	450.679i	−0.136495	−	0.269177i
$142$		0			0
$143$	646.545			0.378090
$144$		0			0
$145$	−1445.73			−0.828011
$146$		0			0
$147$	−1361.80	−	2685.56i	−0.764079	−	1.50681i
$148$		0			0
$149$		−	2737.68i		−	1.50523i	−0.658459	−	0.752617i	$-0.728791\pi$
$149$		−	2737.68i		−	1.50523i	0.658459	−	0.752617i	$-0.271209\pi$
$150$		0			0
$151$		−	2674.78i		−	1.44153i	−0.693181	−	0.720763i	$-0.743792\pi$
$151$		−	2674.78i		−	1.44153i	0.693181	−	0.720763i	$-0.256208\pi$
$152$		0			0
$153$	2867.26	+	2100.23i	1.51506	+	1.10976i
$154$		0			0
$155$	−426.826			−0.221184
$156$		0			0
$157$	−459.784			−0.233725			−0.116862	−	0.993148i	$-0.537284\pi$
$157$	−459.784			−0.233725			−0.116862	+	0.993148i	$0.537284\pi$
$158$		0			0
$159$	−3170.24	+	1607.58i	−1.58123	+	0.801818i
$160$		0			0
$161$		−	2262.79i		−	1.10766i
$162$		0			0
$163$			2262.52i			1.08720i	0.839343	+	0.543602i	$0.182940\pi$
$163$			2262.52i			1.08720i	−0.839343	+	0.543602i	$0.817060\pi$
$164$		0			0
$165$	−1451.95	+	736.262i	−0.685058	+	0.347382i
$166$		0			0
$167$	100.803			0.0467090			0.0233545	−	0.999727i	$-0.492565\pi$
$167$	100.803			0.0467090			0.0233545	+	0.999727i	$0.492565\pi$
$168$		0			0
$169$	169.000			0.0769231
$170$		0			0
$171$	−2521.30	−	1846.83i	−1.12754	−	0.825908i
$172$		0			0
$173$			4283.08i			1.88229i	0.338000	+	0.941146i	$0.390250\pi$
$173$			4283.08i			1.88229i	−0.338000	+	0.941146i	$0.609750\pi$
$174$		0			0
$175$			2591.26i			1.11932i
$176$		0			0
$177$	−78.3643	−	154.539i	−0.0332781	−	0.0656264i
$178$		0			0
$179$	−2891.55			−1.20740			−0.603699	−	0.797212i	$-0.706307\pi$
$179$	−2891.55			−1.20740			−0.603699	+	0.797212i	$0.706307\pi$
$180$		0			0
$181$	−1947.80			−0.799882			−0.399941	−	0.916541i	$-0.630969\pi$
$181$	−1947.80			−0.799882			−0.399941	+	0.916541i	$0.630969\pi$
$182$		0			0
$183$	−718.129	−	1416.19i	−0.290085	−	0.572066i
$184$		0			0
$185$		−	423.729i		−	0.168395i
$186$		0			0
$187$			6546.82i			2.56017i
$188$		0			0
$189$	691.077	−	4204.73i	0.265971	−	1.61825i
$190$		0			0
$191$	−4399.94			−1.66685			−0.833426	−	0.552631i	$-0.813624\pi$
$191$	−4399.94			−1.66685			−0.833426	+	0.552631i	$0.813624\pi$
$192$		0			0
$193$	1036.49			0.386571			0.193285	−	0.981143i	$-0.438086\pi$
$193$	1036.49			0.386571			0.193285	+	0.981143i	$0.438086\pi$
$194$		0			0
$195$	−379.526	+	192.451i	−0.139376	+	0.0706755i
$196$		0			0
$197$			3109.97i			1.12475i	0.826882	+	0.562375i	$0.190112\pi$
$197$			3109.97i			1.12475i	−0.826882	+	0.562375i	$0.809888\pi$
$198$		0			0
$199$			3204.26i			1.14143i	0.821149	+	0.570714i	$0.193333\pi$
$199$			3204.26i			1.14143i	−0.821149	+	0.570714i	$0.806667\pi$
$200$		0			0
$201$	−3054.67	+	1548.97i	−1.07194	+	0.543564i
$202$		0			0
$203$	6970.46			2.41000
$204$		0			0
$205$	−555.383			−0.189218
$206$		0			0
$207$	1188.66	−	1622.77i	0.399119	−	0.544881i
$208$		0			0
$209$		−	5756.90i		−	1.90532i
$210$		0			0
$211$			6080.16i			1.98377i	0.127141	+	0.991885i	$0.459420\pi$
$211$			6080.16i			1.98377i	−0.127141	+	0.991885i	$0.540580\pi$
$212$		0			0
$213$	−1169.78	−	2306.88i	−0.376300	−	0.742088i
$214$		0			0
$215$	−2108.19			−0.668731
$216$		0			0
$217$	2057.90			0.643776
$218$		0			0
$219$	310.007	+	611.353i	0.0956545	+	0.188636i
$220$		0			0
$221$			1711.27i			0.520871i
$222$		0			0
$223$		−	1551.06i		−	0.465771i	−0.972504	−	0.232885i	$-0.925183\pi$
$223$		−	1551.06i		−	0.465771i	0.972504	−	0.232885i	$-0.0748167\pi$
$224$		0			0
$225$	−1361.21	+	1858.33i	−0.403321	+	0.550617i
$226$		0			0
$227$	−3599.16			−1.05236			−0.526178	−	0.850375i	$-0.676375\pi$
$227$	−3599.16			−1.05236			−0.526178	+	0.850375i	$0.676375\pi$
$228$		0			0
$229$	2939.97			0.848377			0.424189	−	0.905574i	$-0.360559\pi$
$229$	2939.97			0.848377			0.424189	+	0.905574i	$0.360559\pi$
$230$		0			0
$231$	7000.46	−	3549.82i	1.99392	−	1.01109i
$232$		0			0
$233$		−	93.6119i		−	0.0263207i	−0.999913	−	0.0131603i	$-0.995811\pi$
$233$		−	93.6119i		−	0.0263207i	0.999913	−	0.0131603i	$-0.00418919\pi$
$234$		0			0
$235$			612.608i			0.170052i
$236$		0			0
$237$	4285.14	−	2172.92i	1.17447	−	0.595555i
$238$		0			0
$239$	3276.54			0.886786			0.443393	−	0.896327i	$-0.353775\pi$
$239$	3276.54			0.886786			0.443393	+	0.896327i	$0.353775\pi$
$240$		0			0
$241$	6126.39			1.63749			0.818746	−	0.574156i	$-0.194670\pi$
$241$	6126.39			1.63749			0.818746	+	0.574156i	$0.194670\pi$
$242$		0			0
$243$	2704.38	−	2652.40i	0.713934	−	0.700213i
$244$		0			0
$245$			3650.48i			0.951921i
$246$		0			0
$247$		−	1504.79i		−	0.387642i
$248$		0			0
$249$	1404.71	+	2770.18i	0.357510	+	0.705032i
$250$		0			0
$251$	−4912.56			−1.23537			−0.617685	−	0.786425i	$-0.711929\pi$
$251$	−4912.56			−1.23537			−0.617685	+	0.786425i	$0.711929\pi$
$252$		0			0
$253$	3705.27			0.920745
$254$		0			0
$255$	−1948.73	−	3843.02i	−0.478566	−	0.943762i
$256$		0			0
$257$			4866.35i			1.18115i	0.806984	+	0.590573i	$0.201098\pi$
$257$			4866.35i			1.18115i	−0.806984	+	0.590573i	$0.798902\pi$
$258$		0			0
$259$			2042.97i			0.490130i
$260$		0			0
$261$	4998.89	+	3661.63i	1.18553	+	0.868387i
$262$		0			0
$263$	−1364.41			−0.319898			−0.159949	−	0.987125i	$-0.551133\pi$
$263$	−1364.41			−0.319898			−0.159949	+	0.987125i	$0.551133\pi$
$264$		0			0
$265$	4309.31			0.998938
$266$		0			0
$267$	−771.587	+	391.259i	−0.176855	+	0.0896804i
$268$		0			0
$269$			3268.01i			0.740720i	0.928888	+	0.370360i	$0.120766\pi$
$269$			3268.01i			0.740720i	−0.928888	+	0.370360i	$0.879234\pi$
$270$		0			0
$271$			1910.83i			0.428320i	0.976799	+	0.214160i	$0.0687014\pi$
$271$			1910.83i			0.428320i	−0.976799	+	0.214160i	$0.931299\pi$
$272$		0			0
$273$	1829.85	−	927.885i	0.405668	−	0.205708i
$274$		0			0
$275$	−4243.14			−0.930439
$276$		0			0
$277$	345.605			0.0749652			0.0374826	−	0.999297i	$-0.488066\pi$
$277$	345.605			0.0749652			0.0374826	+	0.999297i	$0.488066\pi$
$278$		0			0
$279$	1475.83	+	1081.03i	0.316687	+	0.231969i
$280$		0			0
$281$			3417.34i			0.725485i	0.931889	+	0.362742i	$0.118160\pi$
$281$			3417.34i			0.725485i	−0.931889	+	0.362742i	$0.881840\pi$
$282$		0			0
$283$			181.510i			0.0381260i	0.999818	+	0.0190630i	$0.00606831\pi$
$283$			181.510i			0.0381260i	−0.999818	+	0.0190630i	$0.993932\pi$
$284$		0			0
$285$	1713.60	+	3379.33i	0.356158	+	0.702366i
$286$		0			0
$287$	2677.73			0.550736
$288$		0			0
$289$	−12415.1			−2.52698
$290$		0			0
$291$	−3417.28	−	6739.08i	−0.688399	−	1.35757i
$292$		0			0
$293$		−	7900.66i		−	1.57529i	−0.616126	−	0.787647i	$-0.711299\pi$
$293$		−	7900.66i		−	1.57529i	0.616126	−	0.787647i	$-0.288701\pi$
$294$		0			0
$295$			210.065i			0.0414592i
$296$		0			0
$297$	6885.14	+	1131.62i	1.34517	+	0.221089i
$298$		0			0
$299$	968.519			0.187327
$300$		0			0
$301$	10164.4			1.94640
$302$		0			0
$303$	3031.77	−	1537.36i	0.574821	−	0.291483i
$304$		0			0
$305$			1925.03i			0.361400i
$306$		0			0
$307$			3793.89i			0.705305i	0.935754	+	0.352653i	$0.114720\pi$
$307$			3793.89i			0.705305i	−0.935754	+	0.352653i	$0.885280\pi$
$308$		0			0
$309$	3895.70	−	1975.45i	0.717213	−	0.363687i
$310$		0			0
$311$	2919.43			0.532301			0.266150	−	0.963931i	$-0.414248\pi$
$311$	2919.43			0.532301			0.266150	+	0.963931i	$0.414248\pi$
$312$		0			0
$313$	−169.191			−0.0305535			−0.0152768	−	0.999883i	$-0.504863\pi$
$313$	−169.191			−0.0305535			−0.0152768	+	0.999883i	$0.504863\pi$
$314$		0			0
$315$	−3052.67	+	4167.53i	−0.546026	+	0.745440i
$316$		0			0
$317$			5642.47i			0.999725i	0.866105	+	0.499863i	$0.166616\pi$
$317$			5642.47i			0.999725i	−0.866105	+	0.499863i	$0.833384\pi$
$318$		0			0
$319$			11414.0i			2.00332i
$320$		0			0
$321$	−4281.01	−	8442.42i	−0.744370	−	1.46794i
$322$		0			0
$323$	15237.3			2.62485
$324$		0			0
$325$	−1109.11			−0.189300
$326$		0			0
$327$	−1729.20	−	3410.09i	−0.292431	−	0.576692i
$328$		0			0
$329$		−	2953.63i		−	0.494951i
$330$		0			0
$331$			4763.10i			0.790948i	0.918477	+	0.395474i	$0.129420\pi$
$331$			4763.10i			0.790948i	−0.918477	+	0.395474i	$0.870580\pi$
$332$		0			0
$333$	−1073.18	+	1465.12i	−0.176607	+	0.241105i
$334$		0			0
$335$	4152.22			0.677194
$336$		0			0
$337$	3189.87			0.515618			0.257809	−	0.966196i	$-0.416999\pi$
$337$	3189.87			0.515618			0.257809	+	0.966196i	$0.416999\pi$
$338$		0			0
$339$	−7806.34	+	3958.47i	−1.25068	+	0.634202i
$340$		0			0
$341$			3369.77i			0.535141i
$342$		0			0
$343$		−	7182.68i		−	1.13069i
$344$		0			0
$345$	−2175.02	+	1102.92i	−0.339417	+	0.172113i
$346$		0			0
$347$	3093.34			0.478556			0.239278	−	0.970951i	$-0.423089\pi$
$347$	3093.34			0.478556			0.239278	+	0.970951i	$0.423089\pi$
$348$		0			0
$349$	−5847.03			−0.896803			−0.448402	−	0.893832i	$-0.648007\pi$
$349$	−5847.03			−0.896803			−0.448402	+	0.893832i	$0.648007\pi$
$350$		0			0
$351$	1799.70	+	295.794i	0.273678	+	0.0449810i
$352$		0			0
$353$		−	6848.04i		−	1.03253i	−0.856428	−	0.516267i	$-0.827321\pi$
$353$		−	6848.04i		−	1.03253i	0.856428	−	0.516267i	$-0.172679\pi$
$354$		0			0
$355$			3135.74i			0.468811i
$356$		0			0
$357$	9395.63	+	18528.8i	1.39291	+	2.74691i
$358$		0			0
$359$	−8193.94			−1.20462			−0.602311	−	0.798261i	$-0.705753\pi$
$359$	−8193.94			−1.20462			−0.602311	+	0.798261i	$0.705753\pi$
$360$		0			0
$361$	−6539.81			−0.953464
$362$		0			0
$363$	2684.88	+	5294.74i	0.388208	+	0.765570i
$364$		0			0
$365$		−	831.012i		−	0.119170i
$366$		0			0
$367$		−	8632.96i		−	1.22789i	−0.789348	−	0.613946i	$-0.789581\pi$
$367$		−	8632.96i		−	1.22789i	0.789348	−	0.613946i	$-0.210419\pi$
$368$		0			0
$369$	1920.34	+	1406.63i	0.270918	+	0.198445i
$370$		0			0
$371$	−20776.9			−2.90750
$372$		0			0
$373$	2725.01			0.378273			0.189136	−	0.981951i	$-0.439431\pi$
$373$	2725.01			0.378273			0.189136	+	0.981951i	$0.439431\pi$
$374$		0			0
$375$	6140.03	−	3113.51i	0.845519	−	0.428749i
$376$		0			0
$377$			2983.49i			0.407580i
$378$		0			0
$379$			7422.29i			1.00596i	0.864299	+	0.502978i	$0.167762\pi$
$379$			7422.29i			1.00596i	−0.864299	+	0.502978i	$0.832238\pi$
$380$		0			0
$381$	8078.14	−	4096.29i	1.08623	−	0.550812i
$382$		0			0
$383$	−4240.13			−0.565693			−0.282847	−	0.959165i	$-0.591279\pi$
$383$	−4240.13			−0.565693			−0.282847	+	0.959165i	$0.591279\pi$
$384$		0			0
$385$	−9515.73			−1.25965
$386$		0			0
$387$	7289.44	+	5339.44i	0.957476	+	0.701340i
$388$		0			0
$389$		−	495.116i		−	0.0645331i	−0.999479	−	0.0322665i	$-0.989727\pi$
$389$		−	495.116i		−	0.0645331i	0.999479	−	0.0322665i	$-0.0102725\pi$
$390$		0			0
$391$			9807.09i			1.26845i
$392$		0			0
$393$	−122.092	−	240.773i	−0.0156711	−	0.0309044i
$394$		0			0
$395$	−5824.79			−0.741967
$396$		0			0
$397$	−10292.3			−1.30114			−0.650572	−	0.759444i	$-0.725471\pi$
$397$	−10292.3			−1.30114			−0.650572	+	0.759444i	$0.725471\pi$
$398$		0			0
$399$	−8261.98	−	16293.1i	−1.03663	−	2.04430i
$400$		0			0
$401$		−	3009.15i		−	0.374738i	−0.982290	−	0.187369i	$-0.940004\pi$
$401$		−	3009.15i		−	0.374738i	0.982290	−	0.187369i	$-0.0599960\pi$
$402$		0			0
$403$			880.821i			0.108875i
$404$		0			0
$405$	−4378.46	+	1385.17i	−0.537203	+	0.169950i
$406$		0			0
$407$	−3345.31			−0.407423
$408$		0			0
$409$	−4456.30			−0.538753			−0.269376	−	0.963035i	$-0.586818\pi$
$409$	−4456.30			−0.538753			−0.269376	+	0.963035i	$0.586818\pi$
$410$		0			0
$411$	−4638.00	+	2351.85i	−0.556632	+	0.282259i
$412$		0			0
$413$		−	1012.81i		−	0.120671i
$414$		0			0
$415$		−	3765.50i		−	0.445401i
$416$		0			0
$417$	−5231.97	+	2653.05i	−0.614414	+	0.311559i
$418$		0			0
$419$	4854.63			0.566024			0.283012	−	0.959116i	$-0.408666\pi$
$419$	4854.63			0.566024			0.283012	+	0.959116i	$0.408666\pi$
$420$		0			0
$421$	8370.65			0.969028			0.484514	−	0.874784i	$-0.338996\pi$
$421$	8370.65			0.969028			0.484514	+	0.874784i	$0.338996\pi$
$422$		0			0
$423$	1551.56	−	2118.20i	0.178344	−	0.243477i
$424$		0			0
$425$		−	11230.7i		−	1.28181i
$426$		0			0
$427$		−	9281.36i		−	1.05189i
$428$		0			0
$429$	1519.39	+	2996.33i	0.170995	+	0.337213i
$430$		0			0
$431$	9939.53			1.11084			0.555418	−	0.831571i	$-0.312558\pi$
$431$	9939.53			1.11084			0.555418	+	0.831571i	$0.312558\pi$
$432$		0			0
$433$	−883.507			−0.0980569			−0.0490285	−	0.998797i	$-0.515613\pi$
$433$	−883.507			−0.0980569			−0.0490285	+	0.998797i	$0.515613\pi$
$434$		0			0
$435$	−3397.49	−	6700.07i	−0.374477	−	0.738491i
$436$		0			0
$437$		−	8623.79i		−	0.944008i
$438$		0			0
$439$		−	1780.58i		−	0.193582i	−0.995305	−	0.0967909i	$-0.969142\pi$
$439$		−	1780.58i		−	0.193582i	0.995305	−	0.0967909i	$-0.0308578\pi$
$440$		0			0
$441$	9245.62	−	12622.2i	0.998340	−	1.36294i
$442$		0			0
$443$	−8151.73			−0.874268			−0.437134	−	0.899396i	$-0.644006\pi$
$443$	−8151.73			−0.874268			−0.437134	+	0.899396i	$0.644006\pi$
$444$		0			0
$445$	1048.82			0.111728
$446$		0			0
$447$	12687.5	−	6433.60i	1.34250	−	0.680758i
$448$		0			0
$449$			11293.3i			1.18700i	0.804834	+	0.593501i	$0.202255\pi$
$449$			11293.3i			1.18700i	−0.804834	+	0.593501i	$0.797745\pi$
$450$		0			0
$451$			4384.72i			0.457801i
$452$		0			0
$453$	12395.9	−	6285.77i	1.28568	−	0.651946i
$454$		0			0
$455$	−2487.31			−0.256279
$456$		0			0
$457$	17878.3			1.83001			0.915004	−	0.403446i	$-0.132188\pi$
$457$	17878.3			1.83001			0.915004	+	0.403446i	$0.132188\pi$
$458$		0			0
$459$	−2995.17	+	18223.5i	−0.304581	+	1.85316i
$460$		0			0
$461$		−	4887.62i		−	0.493794i	−0.969042	−	0.246897i	$-0.920589\pi$
$461$		−	4887.62i		−	0.493794i	0.969042	−	0.246897i	$-0.0794109\pi$
$462$		0			0
$463$			2132.70i			0.214071i	0.994255	+	0.107036i	$0.0341359\pi$
$463$			2132.70i			0.214071i	−0.994255	+	0.107036i	$0.965864\pi$
$464$		0			0
$465$	−1003.05	−	1978.07i	−0.100033	−	0.197271i
$466$		0			0
$467$	3234.77			0.320530			0.160265	−	0.987074i	$-0.448765\pi$
$467$	3234.77			0.320530			0.160265	+	0.987074i	$0.448765\pi$
$468$		0			0
$469$	−20019.5			−1.97104
$470$		0			0
$471$	−1080.50	−	2130.81i	−0.105704	−	0.208456i
$472$		0			0
$473$			16644.0i			1.61795i
$474$		0			0
$475$			9875.63i			0.953947i
$476$		0			0
$477$	−14900.2	−	10914.2i	−1.43026	−	1.04765i
$478$		0			0
$479$	3368.70			0.321335			0.160668	−	0.987009i	$-0.448635\pi$
$479$	3368.70			0.321335			0.160668	+	0.987009i	$0.448635\pi$
$480$		0			0
$481$	−874.429			−0.0828909
$482$		0			0
$483$	10486.6	−	5317.60i	0.987905	−	0.500951i
$484$		0			0
$485$			9160.43i			0.857636i
$486$		0			0
$487$		−	17106.6i		−	1.59173i	−0.605474	−	0.795865i	$-0.707016\pi$
$487$		−	17106.6i		−	1.59173i	0.605474	−	0.795865i	$-0.292984\pi$
$488$		0			0
$489$	−10485.4	+	5316.96i	−0.969662	+	0.491700i
$490$		0			0
$491$	3678.12			0.338068			0.169034	−	0.985610i	$-0.445935\pi$
$491$	3678.12			0.338068			0.169034	+	0.985610i	$0.445935\pi$
$492$		0			0
$493$	−30210.4			−2.75985
$494$		0			0
$495$	−6824.23	−	4998.67i	−0.619649	−	0.453886i
$496$		0			0
$497$		−	15118.7i		−	1.36452i
$498$		0			0
$499$		−	9157.62i		−	0.821546i	−0.911738	−	0.410773i	$-0.865259\pi$
$499$		−	9157.62i		−	0.821546i	0.911738	−	0.410773i	$-0.134741\pi$
$500$		0			0
$501$	236.890	+	467.161i	0.0211247	+	0.0416591i
$502$		0			0
$503$	12489.8			1.10714			0.553571	−	0.832802i	$-0.313265\pi$
$503$	12489.8			1.10714			0.553571	+	0.832802i	$0.313265\pi$
$504$		0			0
$505$	−4121.09			−0.363141
$506$		0			0
$507$	397.153	+	783.209i	0.0347893	+	0.0686066i
$508$		0			0
$509$		−	5135.54i		−	0.447208i	−0.974680	−	0.223604i	$-0.928218\pi$
$509$		−	5135.54i		−	0.447208i	0.974680	−	0.223604i	$-0.0717823\pi$
$510$		0			0
$511$			4006.64i			0.346856i
$512$		0			0
$513$	2633.78	−	16024.7i	0.226675	−	1.37916i
$514$		0			0
$515$	−5295.43			−0.453096
$516$		0			0
$517$	4836.50			0.411430
$518$		0			0
$519$	−19849.4	+	10065.3i	−1.67879	+	0.851287i
$520$		0			0
$521$		−	18718.9i		−	1.57407i	−0.616909	−	0.787034i	$-0.711615\pi$
$521$		−	18718.9i		−	1.57407i	0.616909	−	0.787034i	$-0.288385\pi$
$522$		0			0
$523$		−	4366.03i		−	0.365035i	−0.983203	−	0.182518i	$-0.941575\pi$
$523$		−	4366.03i		−	0.365035i	0.983203	−	0.182518i	$-0.0584246\pi$
$524$		0			0
$525$	−12008.9	+	6089.51i	−0.998306	+	0.506225i
$526$		0			0
$527$	−8919.07			−0.737231
$528$		0			0
$529$	−6616.53			−0.543809
$530$		0			0
$531$	532.035	−	726.339i	0.0434809	−	0.0593605i
$532$		0			0
$533$			1146.12i			0.0931405i
$534$		0			0
$535$			11475.8i			0.927367i
$536$		0			0
$537$	−6795.18	−	13400.5i	−0.546059	−	1.07686i
$538$		0			0
$539$	28820.3			2.30311
$540$		0			0
$541$	−24364.1			−1.93622			−0.968110	−	0.250527i	$-0.919396\pi$
$541$	−24364.1			−1.93622			−0.968110	+	0.250527i	$0.919396\pi$
$542$		0			0
$543$	−4577.36	−	9026.83i	−0.361755	−	0.713404i
$544$		0			0
$545$			4635.33i			0.364323i
$546$		0			0
$547$			18260.7i			1.42737i	0.700468	+	0.713684i	$0.252975\pi$
$547$			18260.7i			1.42737i	−0.700468	+	0.713684i	$0.747025\pi$
$548$		0			0
$549$	4875.56	−	6656.15i	0.379023	−	0.517446i
$550$		0			0
$551$	26565.3			2.05394
$552$		0			0
$553$	28083.7			2.15956
$554$		0			0
$555$	1963.72	−	995.769i	0.150189	−	0.0761586i
$556$		0			0
$557$		−	2978.65i		−	0.226588i	−0.993562	−	0.113294i	$-0.963860\pi$
$557$		−	2978.65i		−	0.226588i	0.993562	−	0.113294i	$-0.0361402\pi$
$558$		0			0
$559$			4350.57i			0.329176i
$560$		0			0
$561$	−30340.4	+	15385.1i	−2.28338	+	1.15786i
$562$		0			0
$563$	−19911.1			−1.49050			−0.745251	−	0.666784i	$-0.767670\pi$
$563$	−19911.1			−1.49050			−0.745251	+	0.666784i	$0.767670\pi$
$564$		0			0
$565$	10611.2			0.790115
$566$		0			0
$567$	21110.3	−	6678.46i	1.56358	−	0.494654i
$568$		0			0
$569$			13916.9i			1.02536i	0.858581	+	0.512679i	$0.171347\pi$
$569$			13916.9i			1.02536i	−0.858581	+	0.512679i	$0.828653\pi$
$570$		0			0
$571$		−	3978.95i		−	0.291618i	−0.989313	−	0.145809i	$-0.953421\pi$
$571$		−	3978.95i		−	0.291618i	0.989313	−	0.145809i	$-0.0465785\pi$
$572$		0			0
$573$	−10339.9	−	20391.0i	−0.753852	−	1.48664i
$574$		0			0
$575$	−6356.19			−0.460993
$576$		0			0
$577$	22699.9			1.63780			0.818899	−	0.573937i	$-0.194585\pi$
$577$	22699.9			1.63780			0.818899	+	0.573937i	$0.194585\pi$
$578$		0			0
$579$	2435.77	+	4803.48i	0.174831	+	0.344777i
$580$		0			0
$581$			18155.0i			1.29638i
$582$		0			0
$583$		−	34021.7i		−	2.41687i
$584$		0			0
$585$	−1783.78	−	1306.60i	−0.126069	−	0.0923440i
$586$		0			0
$587$	−1027.30			−0.0722339			−0.0361170	−	0.999348i	$-0.511499\pi$
$587$	−1027.30			−0.0722339			−0.0361170	+	0.999348i	$0.511499\pi$
$588$		0			0
$589$	7842.91			0.548662
$590$		0			0
$591$	−14412.7	+	7308.47i	−1.00315	+	0.508680i
$592$		0			0
$593$		−	9999.15i		−	0.692438i	−0.938154	−	0.346219i	$-0.887465\pi$
$593$		−	9999.15i		−	0.692438i	0.938154	−	0.346219i	$-0.112535\pi$
$594$		0			0
$595$		−	25186.1i		−	1.73535i
$596$		0			0
$597$	−14849.8	+	7530.07i	−1.01802	+	0.516223i
$598$		0			0
$599$	4386.77			0.299229			0.149615	−	0.988744i	$-0.452197\pi$
$599$	4386.77			0.299229			0.149615	+	0.988744i	$0.452197\pi$
$600$		0			0
$601$	−16661.6			−1.13085			−0.565425	−	0.824800i	$-0.691288\pi$
$601$	−16661.6			−1.13085			−0.565425	+	0.824800i	$0.691288\pi$
$602$		0			0
$603$	−14357.1	−	10516.4i	−0.969593	−	0.710216i
$604$		0			0
$605$		−	7197.14i		−	0.483645i
$606$		0			0
$607$			10951.9i			0.732327i	0.930550	+	0.366164i	$0.119329\pi$
$607$			10951.9i			0.732327i	−0.930550	+	0.366164i	$0.880671\pi$
$608$		0			0
$609$	16380.7	+	32303.7i	1.08995	+	2.14945i
$610$		0			0
$611$	1264.21			0.0837062
$612$		0			0
$613$	12356.0			0.814115			0.407057	−	0.913403i	$-0.366555\pi$
$613$	12356.0			0.814115			0.407057	+	0.913403i	$0.366555\pi$
$614$		0			0
$615$	−1305.16	−	2573.85i	−0.0855758	−	0.168761i
$616$		0			0
$617$			10779.1i			0.703322i	0.936127	+	0.351661i	$0.114383\pi$
$617$			10779.1i			0.703322i	−0.936127	+	0.351661i	$0.885617\pi$
$618$		0			0
$619$		−	26350.2i		−	1.71099i	−0.517811	−	0.855495i	$-0.673253\pi$
$619$		−	26350.2i		−	1.71099i	0.517811	−	0.855495i	$-0.326747\pi$
$620$		0			0
$621$	10313.9	+	1695.16i	0.666477	+	0.109540i
$622$		0			0
$623$	−5056.78			−0.325193
$624$		0			0
$625$	2318.39			0.148377
$626$		0			0
$627$	26679.6	−	13528.8i	1.69933	−	0.861704i
$628$		0			0
$629$		−	8854.34i		−	0.561281i
$630$		0			0
$631$			3980.40i			0.251121i	0.992086	+	0.125560i	$0.0400729\pi$
$631$			3980.40i			0.251121i	−0.992086	+	0.125560i	$0.959927\pi$
$632$		0			0
$633$	−28177.7	+	14288.5i	−1.76930	+	0.897181i
$634$		0			0
$635$	−10980.6			−0.686224
$636$		0			0
$637$	7533.33			0.468573
$638$		0			0
$639$	7941.93	−	10842.4i	0.491671	−	0.671234i
$640$		0			0
$641$		−	6966.24i		−	0.429251i	−0.976696	−	0.214625i	$-0.931147\pi$
$641$		−	6966.24i		−	0.429251i	0.976696	−	0.214625i	$-0.0688531\pi$
$642$		0			0
$643$		−	2808.80i		−	0.172268i	−0.996284	−	0.0861338i	$-0.972549\pi$
$643$		−	2808.80i		−	0.172268i	0.996284	−	0.0861338i	$-0.0274513\pi$
$644$		0			0
$645$	−4954.27	−	9770.13i	−0.302441	−	0.596432i
$646$		0			0
$647$	7435.63			0.451815			0.225908	−	0.974149i	$-0.427465\pi$
$647$	7435.63			0.451815			0.225908	+	0.974149i	$0.427465\pi$
$648$		0			0
$649$	1658.45			0.100308
$650$		0			0
$651$	4836.10	+	9537.08i	0.291155	+	0.574175i
$652$		0			0
$653$		−	6604.10i		−	0.395771i	−0.980225	−	0.197886i	$-0.936593\pi$
$653$		−	6604.10i		−	0.395771i	0.980225	−	0.197886i	$-0.0634074\pi$
$654$		0			0
$655$			327.283i			0.0195237i
$656$		0			0
$657$	−2104.72	+	2873.38i	−0.124981	+	0.170626i
$658$		0			0
$659$	12394.4			0.732651			0.366326	−	0.930487i	$-0.380616\pi$
$659$	12394.4			0.732651			0.366326	+	0.930487i	$0.380616\pi$
$660$		0			0
$661$	−28308.8			−1.66578			−0.832892	−	0.553435i	$-0.813317\pi$
$661$	−28308.8			−1.66578			−0.832892	+	0.553435i	$0.813317\pi$
$662$		0			0
$663$	−7930.66	+	4021.51i	−0.464557	+	0.235569i
$664$		0			0
$665$			22147.2i			1.29148i
$666$		0			0
$667$			17098.0i			0.992562i
$668$		0			0
$669$	7188.21	−	3645.02i	0.415414	−	0.210650i
$670$		0			0
$671$	15198.0			0.874386
$672$		0			0
$673$	−4483.16			−0.256780			−0.128390	−	0.991724i	$-0.540981\pi$
$673$	−4483.16			−0.256780			−0.128390	+	0.991724i	$0.540981\pi$
$674$		0			0
$675$	−11811.1	−	1941.24i	−0.673494	−	0.110694i
$676$		0			0
$677$		−	380.067i		−	0.0215763i	−0.999942	−	0.0107882i	$-0.996566\pi$
$677$		−	380.067i		−	0.0215763i	0.999942	−	0.0107882i	$-0.00343405\pi$
$678$		0			0
$679$		−	44166.1i		−	2.49623i
$680$		0			0
$681$	−8458.09	−	16679.9i	−0.475939	−	0.938581i
$682$		0			0
$683$	9027.22			0.505735			0.252868	−	0.967501i	$-0.418626\pi$
$683$	9027.22			0.505735			0.252868	+	0.967501i	$0.418626\pi$
$684$		0			0
$685$	6304.44			0.351650
$686$		0			0
$687$	6908.97	+	13624.9i	0.383688	+	0.756656i
$688$		0			0
$689$		−	8892.92i		−	0.491717i
$690$		0			0
$691$		−	7227.76i		−	0.397912i	−0.980008	−	0.198956i	$-0.936245\pi$
$691$		−	7227.76i		−	0.397912i	0.980008	−	0.198956i	$-0.0637550\pi$
$692$		0			0
$693$	32902.4	+	24100.6i	1.80355	+	1.32108i
$694$		0			0
$695$	7111.82			0.388153
$696$		0			0
$697$	−11605.4			−0.630684
$698$		0			0
$699$	433.833	−	219.989i	0.0234750	−	0.0119038i
$700$		0			0
$701$			67.4040i			0.00363169i	0.999998	+	0.00181584i	$0.000578001\pi$
$701$			67.4040i			0.00363169i	−0.999998	+	0.00181584i	$0.999422\pi$
$702$		0			0
$703$			7786.00i			0.417716i
$704$		0			0
$705$	−2839.05	+	1439.64i	−0.151667	+	0.0769077i
$706$		0			0
$707$	19869.4			1.05696
$708$		0			0
$709$	12746.5			0.675184			0.337592	−	0.941293i	$-0.390388\pi$
$709$	12746.5			0.675184			0.337592	+	0.941293i	$0.390388\pi$
$710$		0			0
$711$	20140.3	+	14752.5i	1.06233	+	0.778148i
$712$		0			0
$713$			5047.88i			0.265140i
$714$		0			0
$715$		−	4072.91i		−	0.213033i
$716$		0			0
$717$	7699.93	+	15184.7i	0.401059	+	0.790912i
$718$		0			0
$719$	14932.2			0.774516			0.387258	−	0.921971i	$-0.373422\pi$
$719$	14932.2			0.774516			0.387258	+	0.921971i	$0.373422\pi$
$720$		0			0
$721$	25531.4			1.31878
$722$		0			0
$723$	14397.1	+	28392.0i	0.740573	+	1.46046i
$724$		0			0
$725$		−	19580.0i		−	1.00301i
$726$		0			0
$727$		−	17681.4i		−	0.902019i	−0.892519	−	0.451010i	$-0.851064\pi$
$727$		−	17681.4i		−	0.902019i	0.892519	−	0.451010i	$-0.148936\pi$
$728$		0			0
$729$	18647.6	+	6299.91i	0.947394	+	0.320068i
$730$		0			0
$731$	−44053.2			−2.22896
$732$		0			0
$733$	−24510.3			−1.23507			−0.617536	−	0.786543i	$-0.711869\pi$
$733$	−24510.3			−1.23507			−0.617536	+	0.786543i	$0.711869\pi$
$734$		0			0
$735$	−16917.7	+	8578.68i	−0.849005	+	0.430516i
$736$		0			0
$737$		−	32781.5i		−	1.63843i
$738$		0			0
$739$			19488.0i			0.970066i	0.874496	+	0.485033i	$0.161192\pi$
$739$			19488.0i			0.970066i	−0.874496	+	0.485033i	$0.838808\pi$
$740$		0			0
$741$	6973.77	−	3536.28i	0.345733	−	0.175315i
$742$		0			0
$743$	−35876.6			−1.77145			−0.885724	−	0.464212i	$-0.846338\pi$
$743$	−35876.6			−1.77145			−0.885724	+	0.464212i	$0.846338\pi$
$744$		0			0
$745$	−17246.1			−0.848116
$746$		0			0
$747$	−9536.95	+	13019.9i	−0.467120	+	0.637716i
$748$		0			0
$749$		−	55329.4i		−	2.69919i
$750$		0			0
$751$			11923.4i			0.579351i	0.957125	+	0.289676i	$0.0935474\pi$
$751$			11923.4i			0.579351i	−0.957125	+	0.289676i	$0.906453\pi$
$752$		0			0
$753$	−11544.6	−	22766.6i	−0.558709	−	1.10181i
$754$		0			0
$755$	−16849.8			−0.812221
$756$		0			0
$757$	−18215.6			−0.874580			−0.437290	−	0.899321i	$-0.644062\pi$
$757$	−18215.6			−0.874580			−0.437290	+	0.899321i	$0.644062\pi$
$758$		0			0
$759$	8707.45	+	17171.6i	0.416417	+	0.821200i
$760$		0			0
$761$			28834.9i			1.37354i	0.726875	+	0.686770i	$0.240972\pi$
$761$			28834.9i			1.37354i	−0.726875	+	0.686770i	$0.759028\pi$
$762$		0			0
$763$		−	22348.8i		−	1.06040i
$764$		0			0
$765$	13230.4	−	18062.3i	0.625291	−	0.853653i
$766$		0			0
$767$	433.502			0.0204079
$768$		0			0
$769$	30545.3			1.43237			0.716184	−	0.697912i	$-0.245887\pi$
$769$	30545.3			1.43237			0.716184	+	0.697912i	$0.245887\pi$
$770$		0			0
$771$	−22552.5	+	11436.0i	−1.05345	+	0.534186i
$772$		0			0
$773$			11668.5i			0.542933i	0.962448	+	0.271466i	$0.0875085\pi$
$773$			11668.5i			0.542933i	−0.962448	+	0.271466i	$0.912491\pi$
$774$		0			0
$775$		−	5780.64i		−	0.267931i
$776$		0			0
$777$	−9467.87	+	4801.00i	−0.437140	+	0.221667i
$778$		0			0
$779$	10205.1			0.469368
$780$		0			0
$781$	24756.5			1.13426
$782$		0			0
$783$	−5221.89	+	31771.6i	−0.238334	+	1.45009i
$784$		0			0
$785$			2896.41i			0.131691i
$786$		0			0
$787$		−	2836.81i		−	0.128489i	−0.997934	−	0.0642447i	$-0.979536\pi$
$787$		−	2836.81i		−	0.128489i	0.997934	−	0.0642447i	$-0.0204638\pi$
$788$		0			0
$789$	−3206.39	−	6323.19i	−0.144677	−	0.285312i
$790$		0			0
$791$	−51160.7			−2.29970
$792$		0			0
$793$	3972.60			0.177896
$794$		0			0
$795$	10126.9	+	19970.9i	0.451781	+	0.890939i
$796$		0			0
$797$			15560.7i			0.691578i	0.938312	+	0.345789i	$0.112389\pi$
$797$			15560.7i			0.691578i	−0.938312	+	0.345789i	$0.887611\pi$
$798$		0			0
$799$			12801.2i			0.566801i
$800$		0			0
$801$	−3626.48	−	2656.36i	−0.159969	−	0.117176i
$802$		0			0
$803$	−6560.79			−0.288325
$804$		0			0
$805$	−14254.5			−0.624105
$806$		0			0
$807$	−15145.2	+	7679.86i	−0.660638	+	0.334999i
$808$		0			0
$809$		−	29813.0i		−	1.29563i	−0.761796	−	0.647817i	$-0.775682\pi$
$809$		−	29813.0i		−	1.29563i	0.761796	−	0.647817i	$-0.224318\pi$
$810$		0			0
$811$			28060.8i			1.21498i	0.794329	+	0.607488i	$0.207823\pi$
$811$			28060.8i			1.21498i	−0.794329	+	0.607488i	$0.792177\pi$
$812$		0			0
$813$	−8855.51	+	4490.48i	−0.382013	+	0.193712i
$814$		0			0
$815$	14252.8			0.612580
$816$		0			0
$817$	38737.9			1.65883
$818$		0			0
$819$	8600.33	+	6299.65i	0.366935	+	0.268776i
$820$		0			0
$821$			29065.7i			1.23557i	0.786348	+	0.617784i	$0.211969\pi$
$821$			29065.7i			1.23557i	−0.786348	+	0.617784i	$0.788031\pi$
$822$		0			0
$823$			770.979i			0.0326545i	0.999867	+	0.0163272i	$0.00519735\pi$
$823$			770.979i			0.0326545i	−0.999867	+	0.0163272i	$0.994803\pi$
$824$		0			0
$825$	−9971.44	−	19664.3i	−0.420801	−	0.829845i
$826$		0			0
$827$	−33508.6			−1.40896			−0.704479	−	0.709724i	$-0.748819\pi$
$827$	−33508.6			−1.40896			−0.704479	+	0.709724i	$0.748819\pi$
$828$		0			0
$829$	−7765.41			−0.325336			−0.162668	−	0.986681i	$-0.552010\pi$
$829$	−7765.41			−0.325336			−0.162668	+	0.986681i	$0.552010\pi$
$830$		0			0
$831$	812.176	+	1601.66i	0.0339038	+	0.0668604i
$832$		0			0
$833$			76281.4i			3.17286i
$834$		0			0
$835$		−	635.012i		−	0.0263180i
$836$		0			0
$837$	−1541.67	+	9379.98i	−0.0636653	+	0.387359i
$838$		0			0
$839$	−14793.9			−0.608749			−0.304375	−	0.952552i	$-0.598447\pi$
$839$	−14793.9			−0.608749			−0.304375	+	0.952552i	$0.598447\pi$
$840$		0			0
$841$	−28280.9			−1.15958
$842$		0			0
$843$	−15837.2	+	8030.80i	−0.647050	+	0.328108i
$844$		0			0
$845$		−	1064.62i		−	0.0433419i
$846$		0			0
$847$			34700.3i			1.40770i
$848$		0			0
$849$	−841.185	+	426.551i	−0.0340040	+	0.0172429i
$850$		0			0
$851$	−5011.25			−0.201861
$852$		0			0
$853$	−17424.1			−0.699403			−0.349701	−	0.936861i	$-0.613717\pi$
$853$	−17424.1			−0.699403			−0.349701	+	0.936861i	$0.613717\pi$
$854$		0			0
$855$	−11634.1	+	15883.0i	−0.465354	+	0.635305i
$856$		0			0
$857$			26938.0i			1.07373i	0.843669	+	0.536864i	$0.180391\pi$
$857$			26938.0i			1.07373i	−0.843669	+	0.536864i	$0.819609\pi$
$858$		0			0
$859$		−	38964.0i		−	1.54766i	−0.633396	−	0.773828i	$-0.718340\pi$
$859$		−	38964.0i		−	1.54766i	0.633396	−	0.773828i	$-0.281660\pi$
$860$		0			0
$861$	6292.70	+	12409.6i	0.249076	+	0.491193i
$862$		0			0
$863$	−27534.9			−1.08609			−0.543047	−	0.839702i	$-0.682730\pi$
$863$	−27534.9			−1.08609			−0.543047	+	0.839702i	$0.682730\pi$
$864$		0			0
$865$	26981.3			1.06057
$866$		0			0
$867$	−29175.6	−	57536.1i	−1.14286	−	2.25378i
$868$		0			0
$869$			45986.4i			1.79514i
$870$		0			0
$871$		−	8568.74i		−	0.333342i
$872$		0			0
$873$	23200.8	−	31673.9i	0.899458	−	1.22795i
$874$		0			0
$875$	40240.1			1.55470
$876$		0			0
$877$	19279.0			0.742311			0.371155	−	0.928571i	$-0.378962\pi$
$877$	19279.0			0.742311			0.371155	+	0.928571i	$0.378962\pi$
$878$		0			0
$879$	36614.6	−	18566.7i	1.40498	−	0.712444i
$880$		0			0
$881$		−	36794.3i		−	1.40707i	−0.710660	−	0.703536i	$-0.751603\pi$
$881$		−	36794.3i		−	1.40707i	0.710660	−	0.703536i	$-0.248397\pi$
$882$		0			0
$883$		−	30271.2i		−	1.15369i	−0.816854	−	0.576844i	$-0.804284\pi$
$883$		−	30271.2i		−	1.15369i	0.816854	−	0.576844i	$-0.195716\pi$
$884$		0			0
$885$	−973.521	+	493.656i	−0.0369769	+	0.0187504i
$886$		0			0
$887$	40089.3			1.51755			0.758775	−	0.651352i	$-0.225798\pi$
$887$	40089.3			1.51755			0.758775	+	0.651352i	$0.225798\pi$
$888$		0			0
$889$	52942.0			1.99732
$890$		0			0
$891$	10935.8	+	34567.7i	0.411183	+	1.29973i
$892$		0			0
$893$		−	11256.6i		−	0.421825i
$894$		0			0
$895$			18215.3i			0.680303i
$896$		0			0
$897$	2276.04	+	4488.48i	0.0847209	+	0.167075i
$898$		0			0
$899$	−15549.8			−0.576881
$900$		0			0
$901$	90048.4			3.32958
$902$		0			0
$903$	23886.5	+	47105.7i	0.880282	+	1.73597i
$904$		0			0
$905$			12270.2i			0.450690i
$906$		0			0
$907$			13815.3i			0.505765i	0.967497	+	0.252883i	$0.0813786\pi$
$907$			13815.3i			0.505765i	−0.967497	+	0.252883i	$0.918621\pi$
$908$		0			0
$909$	14249.4	+	10437.5i	0.519938	+	0.380849i
$910$		0			0
$911$	−4192.43			−0.152471			−0.0762356	−	0.997090i	$-0.524290\pi$
$911$	−4192.43			−0.152471			−0.0762356	+	0.997090i	$0.524290\pi$
$912$		0			0
$913$	−29728.4			−1.07762
$914$		0			0
$915$	−8921.32	+	4523.85i	−0.322328	+	0.163447i
$916$		0			0
$917$		−	1577.96i		−	0.0568255i
$918$		0			0
$919$		−	6750.72i		−	0.242313i	−0.992633	−	0.121156i	$-0.961340\pi$
$919$		−	6750.72i		−	0.242313i	0.992633	−	0.121156i	$-0.0386603\pi$
$920$		0			0
$921$	−17582.3	+	8915.70i	−0.629052	+	0.318982i
$922$		0			0
$923$	6471.08			0.230767
$924$		0			0
$925$	5738.69			0.203986
$926$		0			0
$927$	18309.9	+	13411.8i	0.648734	+	0.475191i
$928$		0			0
$929$			30331.0i			1.07118i	0.844477	+	0.535591i	$0.179911\pi$
$929$			30331.0i			1.07118i	−0.844477	+	0.535591i	$0.820089\pi$
$930$		0			0
$931$		−	67077.4i		−	2.36130i
$932$		0			0
$933$	6860.70	+	13529.7i	0.240739	+	0.474752i
$934$		0			0
$935$	41241.7			1.44251
$936$		0			0
$937$	42703.4			1.48886			0.744429	−	0.667701i	$-0.232722\pi$
$937$	42703.4			1.48886			0.744429	+	0.667701i	$0.232722\pi$
$938$		0			0
$939$	−397.602	−	784.095i	−0.0138182	−	0.0272502i
$940$		0			0
$941$		−	30488.9i		−	1.05623i	−0.849174	−	0.528113i	$-0.822900\pi$
$941$		−	30488.9i		−	1.05623i	0.849174	−	0.528113i	$-0.177100\pi$
$942$		0			0
$943$			6568.27i			0.226821i
$944$		0			0
$945$	−26487.7	−	4353.45i	−0.911793	−	0.149860i
$946$		0			0
$947$	47690.3			1.63646			0.818229	−	0.574892i	$-0.194956\pi$
$947$	47690.3			1.63646			0.818229	+	0.574892i	$0.194956\pi$
$948$		0			0
$949$	−1714.92			−0.0586604
$950$		0			0
$951$	−26149.3	+	13259.9i	−0.891641	+	0.452137i
$952$		0			0
$953$		−	4062.11i		−	0.138074i	−0.997614	−	0.0690370i	$-0.978007\pi$
$953$		−	4062.11i		−	0.138074i	0.997614	−	0.0690370i	$-0.0219927\pi$
$954$		0			0
$955$			27717.5i			0.939179i
$956$		0			0
$957$	−52896.6	+	26823.0i	−1.78673	+	0.906024i
$958$		0			0
$959$	−30396.2			−1.02351
$960$		0			0
$961$	25200.2			0.845900
$962$		0			0
$963$	29064.9	−	39679.6i	0.972588	−	1.32779i
$964$		0			0
$965$		−	6529.37i		−	0.217811i
$966$		0			0
$967$		−	43725.5i		−	1.45410i	−0.686583	−	0.727052i	$-0.740890\pi$
$967$		−	43725.5i		−	1.45410i	0.686583	−	0.727052i	$-0.259110\pi$
$968$		0			0
$969$	35807.9	+	70615.4i	1.18712	+	2.34107i
$970$		0			0
$971$	−16739.6			−0.553243			−0.276621	−	0.960979i	$-0.589215\pi$
$971$	−16739.6			−0.553243			−0.276621	+	0.960979i	$0.589215\pi$
$972$		0			0
$973$	−34288.9			−1.12976
$974$		0			0
$975$	−2606.43	−	5140.04i	−0.0856128	−	0.168834i
$976$		0			0
$977$		−	40560.0i		−	1.32818i	−0.747654	−	0.664089i	$-0.768820\pi$
$977$		−	40560.0i		−	1.32818i	0.747654	−	0.664089i	$-0.231180\pi$
$978$		0			0
$979$		−	8280.36i		−	0.270318i
$980$		0			0
$981$	11740.0	−	16027.5i	0.382088	−	0.521630i
$982$		0			0
$983$	−237.351			−0.00770125			−0.00385063	−	0.999993i	$-0.501226\pi$
$983$	−237.351			−0.00770125			−0.00385063	+	0.999993i	$0.501226\pi$
$984$		0			0
$985$	19591.2			0.633735
$986$		0			0
$987$	13688.2	−	6941.08i	0.441440	−	0.223847i
$988$		0			0
$989$			24932.6i			0.801628i
$990$		0			0
$991$			42723.4i			1.36948i	0.728788	+	0.684740i	$0.240084\pi$
$991$			42723.4i			1.36948i	−0.728788	+	0.684740i	$0.759916\pi$
$992$		0			0
$993$	−22074.0	+	11193.4i	−0.705435	+	0.357715i
$994$		0			0
$995$	20185.3			0.643132
$996$		0			0
$997$	1517.98			0.0482195			0.0241097	−	0.999709i	$-0.492325\pi$
$997$	1517.98			0.0482195			0.0241097	+	0.999709i	$0.492325\pi$
$998$		0			0
$999$	−9311.91	−	1530.48i	−0.294911	−	0.0484707i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.d.c.287.18	yes	24
3.2	odd	2	inner	624.4.d.c.287.8	yes	24
4.3	odd	2	inner	624.4.d.c.287.7	✓	24
12.11	even	2	inner	624.4.d.c.287.17	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.4.d.c.287.7	✓	24	4.3	odd	2	inner
624.4.d.c.287.8	yes	24	3.2	odd	2	inner
624.4.d.c.287.17	yes	24	12.11	even	2	inner
624.4.d.c.287.18	yes	24	1.1	even	1	trivial

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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.35002 + 4.63437i) q^{3} -6.29951i q^{5} +30.3725i q^{7} +(-15.9549 + 21.7817i) q^{9} +O(q^{10})\)	\(q+(2.35002 + 4.63437i) q^{3} -6.29951i q^{5} +30.3725i q^{7} +(-15.9549 + 21.7817i) q^{9} -49.7342 q^{11} -13.0000 q^{13} +(29.1943 - 14.8039i) q^{15} -131.636i q^{17} +115.753i q^{19} +(-140.757 + 71.3758i) q^{21} -74.5015 q^{23} +85.3162 q^{25} +(-138.439 - 22.7534i) q^{27} -229.499i q^{29} -67.7555i q^{31} +(-116.876 - 230.487i) q^{33} +191.332 q^{35} +67.2638 q^{37} +(-30.5502 - 60.2469i) q^{39} -88.1630i q^{41} -334.659i q^{43} +(137.214 + 100.508i) q^{45} -97.2469 q^{47} -579.487 q^{49} +(610.051 - 309.347i) q^{51} +684.071i q^{53} +313.301i q^{55} +(-536.444 + 272.022i) q^{57} -33.3463 q^{59} -305.585 q^{61} +(-661.564 - 484.588i) q^{63} +81.8936i q^{65} +659.134i q^{67} +(-175.080 - 345.268i) q^{69} -497.775 q^{71} +131.917 q^{73} +(200.494 + 395.387i) q^{75} -1510.55i q^{77} -924.642i q^{79} +(-219.885 - 695.048i) q^{81} +597.746 q^{83} -829.243 q^{85} +(1063.59 - 539.327i) q^{87} +166.492i q^{89} -394.842i q^{91} +(314.004 - 159.226i) q^{93} +729.188 q^{95} -1454.15 q^{97} +(793.502 - 1083.30i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 84 q^{9}+O(q^{10}) \) 24 * q - 84 * q^9	\( 24 q - 84 q^{9} - 312 q^{13} - 300 q^{21} - 240 q^{25} + 240 q^{33} - 456 q^{37} + 1836 q^{45} - 1632 q^{49} + 168 q^{57} - 960 q^{61} + 2760 q^{69} + 1248 q^{73} - 468 q^{81} - 7704 q^{85} + 3336 q^{93} - 2496 q^{97}+O(q^{100}) \) 24 * q - 84 * q^9 - 312 * q^13 - 300 * q^21 - 240 * q^25 + 240 * q^33 - 456 * q^37 + 1836 * q^45 - 1632 * q^49 + 168 * q^57 - 960 * q^61 + 2760 * q^69 + 1248 * q^73 - 468 * q^81 - 7704 * q^85 + 3336 * q^93 - 2496 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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