LMFDB - Embedded newform 507.4.b.i.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not 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:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 $ x^10 + 70x^8 + 1645x^6 + 14700x^4 + 44100x^2 + 27648
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$-5.04537i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.i.337.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.04537i q^{2} +3.00000 q^{3} -17.4557 q^{4} -20.1174i q^{5} -15.1361i q^{6} -15.4279i q^{7} +47.7076i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-5.04537i q^{2} +3.00000 q^{3} -17.4557 q^{4} -20.1174i q^{5} -15.1361i q^{6} -15.4279i q^{7} +47.7076i q^{8} +9.00000 q^{9} -101.500 q^{10} -26.9372i q^{11} -52.3672 q^{12} -77.8394 q^{14} -60.3522i q^{15} +101.057 q^{16} -23.2334 q^{17} -45.4083i q^{18} -45.0794i q^{19} +351.164i q^{20} -46.2837i q^{21} -135.908 q^{22} +142.010 q^{23} +143.123i q^{24} -279.710 q^{25} +27.0000 q^{27} +269.305i q^{28} +2.29068 q^{29} -304.499 q^{30} -37.7740i q^{31} -128.207i q^{32} -80.8116i q^{33} +117.221i q^{34} -310.369 q^{35} -157.102 q^{36} -313.840i q^{37} -227.442 q^{38} +959.753 q^{40} -5.86820i q^{41} -233.518 q^{42} +360.898 q^{43} +470.209i q^{44} -181.057i q^{45} -716.493i q^{46} +209.748i q^{47} +303.170 q^{48} +104.980 q^{49} +1411.24i q^{50} -69.7003 q^{51} +276.886 q^{53} -136.225i q^{54} -541.906 q^{55} +736.028 q^{56} -135.238i q^{57} -11.5573i q^{58} +543.189i q^{59} +1053.49i q^{60} +205.788 q^{61} -190.583 q^{62} -138.851i q^{63} +161.602 q^{64} -407.724 q^{66} +492.578i q^{67} +405.557 q^{68} +426.030 q^{69} +1565.93i q^{70} +826.859i q^{71} +429.369i q^{72} -66.1205i q^{73} -1583.44 q^{74} -839.129 q^{75} +786.894i q^{76} -415.584 q^{77} +317.642 q^{79} -2033.00i q^{80} +81.0000 q^{81} -29.6072 q^{82} +141.450i q^{83} +807.915i q^{84} +467.396i q^{85} -1820.86i q^{86} +6.87204 q^{87} +1285.11 q^{88} -641.320i q^{89} -913.497 q^{90} -2478.89 q^{92} -113.322i q^{93} +1058.26 q^{94} -906.880 q^{95} -384.621i q^{96} -1114.92i q^{97} -529.663i q^{98} -242.435i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9}+O(q^{10}) $ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9	$ 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36} - 1380 q^{38} + 2000 q^{40} - 180 q^{42} + 740 q^{43} + 1500 q^{48} - 1550 q^{49} - 630 q^{51} + 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} - 3140 q^{64} + 1740 q^{66} + 1200 q^{68} + 360 q^{69} - 4380 q^{74} - 2880 q^{75} - 4320 q^{77} + 1100 q^{79} + 810 q^{81} + 4780 q^{82} + 2970 q^{87} - 6340 q^{88} - 720 q^{90} - 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) $ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 - 80 * q^10 - 180 * q^12 - 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 - 960 * q^25 + 270 * q^27 + 990 * q^29 - 240 * q^30 - 120 * q^35 - 540 * q^36 - 1380 * q^38 + 2000 * q^40 - 180 * q^42 + 740 * q^43 + 1500 * q^48 - 1550 * q^49 - 630 * q^51 + 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 - 3140 * q^64 + 1740 * q^66 + 1200 * q^68 + 360 * q^69 - 4380 * q^74 - 2880 * q^75 - 4320 * q^77 + 1100 * q^79 + 810 * q^81 + 4780 * q^82 + 2970 * q^87 - 6340 * q^88 - 720 * q^90 - 1740 * q^92 + 6460 * q^94 + 2760 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 5.04537i	− 1.78381i	−0.452226	−	0.891903i	$-0.649370\pi$
$2$	− 5.04537i	− 1.78381i	0.452226	−	0.891903i	$-0.350630\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−17.4557	−2.18197
$5$	− 20.1174i	− 1.79935i	−0.436556	−	0.899677i	$-0.643802\pi$
$5$	− 20.1174i	− 1.79935i	0.436556	−	0.899677i	$-0.356198\pi$
$6$	− 15.1361i	− 1.02988i
$7$	− 15.4279i	− 0.833028i	−0.909129	−	0.416514i	$-0.863252\pi$
$7$	− 15.4279i	− 0.833028i	0.909129	−	0.416514i	$-0.136748\pi$
$8$	47.7076i	2.10840i
$9$	9.00000	0.333333
$10$	−101.500	−3.20970
$11$	− 26.9372i	− 0.738352i	−0.929359	−	0.369176i	$-0.879640\pi$
$11$	− 26.9372i	− 0.738352i	0.929359	−	0.369176i	$-0.120360\pi$
$12$	−52.3672	−1.25976
$13$	0	0
$13$	0	0
$14$	−77.8394	−1.48596
$15$	− 60.3522i	− 1.03886i
$16$	101.057	1.57901
$17$	−23.2334	−0.331467	−0.165733	−	0.986171i	$-0.552999\pi$
$17$	−23.2334	−0.331467	−0.165733	+	0.986171i	$0.552999\pi$
$18$	− 45.4083i	− 0.594602i
$19$	− 45.0794i	− 0.544312i	−0.962253	−	0.272156i	$-0.912263\pi$
$19$	− 45.0794i	− 0.544312i	0.962253	−	0.272156i	$-0.0877366\pi$
$20$	351.164i	3.92613i
$21$	− 46.2837i	− 0.480949i
$22$	−135.908	−1.31708
$23$	142.010	1.28744	0.643720	−	0.765261i	$-0.277390\pi$
$23$	142.010	1.28744	0.643720	+	0.765261i	$0.277390\pi$
$24$	143.123i	1.21729i
$25$	−279.710	−2.23768
$26$	0	0
$27$	27.0000	0.192450
$28$	269.305i	1.81764i
$29$	2.29068	0.0146679	0.00733394	−	0.999973i	$-0.497666\pi$
$29$	2.29068	0.0146679	0.00733394	+	0.999973i	$0.497666\pi$
$30$	−304.499	−1.85312
$31$	− 37.7740i	− 0.218852i	−0.993995	−	0.109426i	$-0.965099\pi$
$31$	− 37.7740i	− 0.218852i	0.993995	−	0.109426i	$-0.0349012\pi$
$32$	− 128.207i	− 0.708251i
$33$	− 80.8116i	− 0.426288i
$34$	117.221i	0.591273i
$35$	−310.369	−1.49891
$36$	−157.102	−0.727322
$37$	− 313.840i	− 1.39446i	−0.716849	−	0.697228i	$-0.754416\pi$
$37$	− 313.840i	− 1.39446i	0.716849	−	0.697228i	$-0.245584\pi$
$38$	−227.442	−0.970947
$39$	0	0
$40$	959.753	3.79376
$41$	− 5.86820i	− 0.0223527i	−0.999938	−	0.0111763i	$-0.996442\pi$
$41$	− 5.86820i	− 0.0223527i	0.999938	−	0.0111763i	$-0.00355761\pi$
$42$	−233.518	−0.857920
$43$	360.898	1.27992	0.639958	−	0.768410i	$-0.278952\pi$
$43$	360.898	1.27992	0.639958	+	0.768410i	$0.278952\pi$
$44$	470.209i	1.61106i
$45$	− 181.057i	− 0.599785i
$46$	− 716.493i	− 2.29655i
$47$	209.748i	0.650956i	0.945550	+	0.325478i	$0.105525\pi$
$47$	209.748i	0.650956i	−0.945550	+	0.325478i	$0.894475\pi$
$48$	303.170	0.911642
$49$	104.980	0.306064
$50$	1411.24i	3.99158i
$51$	−69.7003	−0.191372
$52$	0	0
$53$	276.886	0.717609	0.358804	−	0.933413i	$-0.383185\pi$
$53$	276.886	0.717609	0.358804	+	0.933413i	$0.383185\pi$
$54$	− 136.225i	− 0.343294i
$55$	−541.906	−1.32856
$56$	736.028	1.75636
$57$	− 135.238i	− 0.314259i
$58$	− 11.5573i	− 0.0261647i
$59$	543.189i	1.19860i	0.800526	+	0.599298i	$0.204553\pi$
$59$	543.189i	1.19860i	−0.800526	+	0.599298i	$0.795447\pi$
$60$	1053.49i	2.26675i
$61$	205.788	0.431942	0.215971	−	0.976400i	$-0.430708\pi$
$61$	205.788	0.431942	0.215971	+	0.976400i	$0.430708\pi$
$62$	−190.583	−0.390389
$63$	− 138.851i	− 0.277676i
$64$	161.602	0.315629
$65$	0	0
$66$	−407.724	−0.760415
$67$	492.578i	0.898179i	0.893487	+	0.449090i	$0.148252\pi$
$67$	492.578i	0.898179i	−0.893487	+	0.449090i	$0.851748\pi$
$68$	405.557	0.723249
$69$	426.030	0.743304
$70$	1565.93i	2.67377i
$71$	826.859i	1.38211i	0.722800	+	0.691057i	$0.242855\pi$
$71$	826.859i	1.38211i	−0.722800	+	0.691057i	$0.757145\pi$
$72$	429.369i	0.702800i
$73$	− 66.1205i	− 0.106011i	−0.998594	−	0.0530056i	$-0.983120\pi$
$73$	− 66.1205i	− 0.106011i	0.998594	−	0.0530056i	$-0.0168801\pi$
$74$	−1583.44	−2.48744
$75$	−839.129	−1.29192
$76$	786.894i	1.18767i
$77$	−415.584	−0.615068
$78$	0	0
$79$	317.642	0.452374	0.226187	−	0.974084i	$-0.427374\pi$
$79$	317.642	0.452374	0.226187	+	0.974084i	$0.427374\pi$
$80$	− 2033.00i	− 2.84120i
$81$	81.0000	0.111111
$82$	−29.6072	−0.0398728
$83$	141.450i	0.187063i	0.995616	+	0.0935313i	$0.0298155\pi$
$83$	141.450i	0.187063i	−0.995616	+	0.0935313i	$0.970184\pi$
$84$	807.915i	1.04941i
$85$	467.396i	0.596426i
$86$	− 1820.86i	− 2.28312i
$87$	6.87204	0.00846851
$88$	1285.11	1.55674
$89$	− 641.320i	− 0.763818i	−0.924200	−	0.381909i	$-0.875267\pi$
$89$	− 641.320i	− 0.763818i	0.924200	−	0.381909i	$-0.124733\pi$
$90$	−913.497	−1.06990
$91$	0	0
$92$	−2478.89	−2.80915
$93$	− 113.322i	− 0.126354i
$94$	1058.26	1.16118
$95$	−906.880	−0.979410
$96$	− 384.621i	− 0.408909i
$97$	− 1114.92i	− 1.16704i	−0.812097	−	0.583522i	$-0.801674\pi$
$97$	− 1114.92i	− 1.16704i	0.812097	−	0.583522i	$-0.198326\pi$
$98$	− 529.663i	− 0.545960i
$99$	− 242.435i	− 0.246117i
$100$	4882.53	4.88253
$101$	−1589.91	−1.56635	−0.783177	−	0.621799i	$-0.786402\pi$
$101$	−1589.91	−1.56635	−0.783177	+	0.621799i	$0.786402\pi$
$102$	351.664i	0.341371i
$103$	−527.502	−0.504625	−0.252312	−	0.967646i	$-0.581191\pi$
$103$	−527.502	−0.504625	−0.252312	+	0.967646i	$0.581191\pi$
$104$	0	0
$105$	−931.107	−0.865398
$106$	− 1396.99i	− 1.28008i
$107$	1751.96	1.58289	0.791443	−	0.611243i	$-0.209330\pi$
$107$	1751.96	1.58289	0.791443	+	0.611243i	$0.209330\pi$
$108$	−471.305	−0.419920
$109$	967.122i	0.849848i	0.905229	+	0.424924i	$0.139699\pi$
$109$	967.122i	0.849848i	−0.905229	+	0.424924i	$0.860301\pi$
$110$	2734.12i	2.36989i
$111$	− 941.519i	− 0.805090i
$112$	− 1559.09i	− 1.31536i
$113$	1910.97	1.59088	0.795439	−	0.606033i	$-0.207240\pi$
$113$	1910.97	1.59088	0.795439	+	0.606033i	$0.207240\pi$
$114$	−682.326	−0.560577
$115$	− 2856.87i	− 2.31656i
$116$	−39.9855	−0.0320048
$117$	0	0
$118$	2740.59	2.13806
$119$	358.443i	0.276121i
$120$	2879.26	2.19033
$121$	605.387	0.454836
$122$	− 1038.28i	− 0.770501i
$123$	− 17.6046i	− 0.0129053i
$124$	659.372i	0.477527i
$125$	3112.35i	2.22702i
$126$	−700.555	−0.495320
$127$	−1233.11	−0.861584	−0.430792	−	0.902451i	$-0.641766\pi$
$127$	−1233.11	−0.861584	−0.430792	+	0.902451i	$0.641766\pi$
$128$	− 1841.00i	− 1.27127i
$129$	1082.69	0.738960
$130$	0	0
$131$	−1274.90	−0.850292	−0.425146	−	0.905125i	$-0.639777\pi$
$131$	−1274.90	−0.850292	−0.425146	+	0.905125i	$0.639777\pi$
$132$	1410.63i	0.930145i
$133$	−695.480	−0.453427
$134$	2485.24	1.60218
$135$	− 543.170i	− 0.346286i
$136$	− 1108.41i	− 0.698864i
$137$	− 2031.32i	− 1.26677i	−0.773837	−	0.633385i	$-0.781665\pi$
$137$	− 2031.32i	− 1.26677i	0.773837	−	0.633385i	$-0.218335\pi$
$138$	− 2149.48i	− 1.32591i
$139$	−1445.66	−0.882156	−0.441078	−	0.897469i	$-0.645404\pi$
$139$	−1445.66	−0.882156	−0.441078	+	0.897469i	$0.645404\pi$
$140$	5417.72	3.27058
$141$	629.245i	0.375830i
$142$	4171.81	2.46542
$143$	0	0
$144$	909.510	0.526337
$145$	− 46.0825i	− 0.0263927i
$146$	−333.602	−0.189103
$147$	314.940	0.176706
$148$	5478.30i	3.04266i
$149$	966.318i	0.531301i	0.964069	+	0.265650i	$0.0855867\pi$
$149$	966.318i	0.531301i	−0.964069	+	0.265650i	$0.914413\pi$
$150$	4233.71i	2.30454i
$151$	− 1463.09i	− 0.788505i	−0.919002	−	0.394252i	$-0.871004\pi$
$151$	− 1463.09i	− 0.788505i	0.919002	−	0.394252i	$-0.128996\pi$
$152$	2150.63	1.14763
$153$	−209.101	−0.110489
$154$	2096.78i	1.09716i
$155$	−759.914	−0.393792
$156$	0	0
$157$	−66.0424	−0.0335717	−0.0167859	−	0.999859i	$-0.505343\pi$
$157$	−66.0424	−0.0335717	−0.0167859	+	0.999859i	$0.505343\pi$
$158$	− 1602.62i	− 0.806948i
$159$	830.659	0.414312
$160$	−2579.19	−1.27439
$161$	− 2190.92i	− 1.07247i
$162$	− 408.675i	− 0.198201i
$163$	− 3525.24i	− 1.69398i	−0.531612	−	0.846988i	$-0.678413\pi$
$163$	− 3525.24i	− 1.69398i	0.531612	−	0.846988i	$-0.321587\pi$
$164$	102.434i	0.0487728i
$165$	−1625.72	−0.767043
$166$	713.669	0.333683
$167$	− 260.652i	− 0.120777i	−0.998175	−	0.0603887i	$-0.980766\pi$
$167$	− 260.652i	− 0.120777i	0.998175	−	0.0603887i	$-0.0192340\pi$
$168$	2208.08	1.01403
$169$	0	0
$170$	2358.19	1.06391
$171$	− 405.715i	− 0.181437i
$172$	−6299.74	−2.79273
$173$	−911.753	−0.400689	−0.200345	−	0.979725i	$-0.564206\pi$
$173$	−911.753	−0.400689	−0.200345	+	0.979725i	$0.564206\pi$
$174$	− 34.6720i	− 0.0151062i
$175$	4315.33i	1.86405i
$176$	− 2722.18i	− 1.16587i
$177$	1629.57i	0.692009i
$178$	−3235.70	−1.36250
$179$	−2690.48	−1.12344	−0.561721	−	0.827327i	$-0.689861\pi$
$179$	−2690.48	−1.12344	−0.561721	+	0.827327i	$0.689861\pi$
$180$	3160.47i	1.30871i
$181$	−4773.85	−1.96043	−0.980213	−	0.197944i	$-0.936573\pi$
$181$	−4773.85	−1.96043	−0.980213	+	0.197944i	$0.936573\pi$
$182$	0	0
$183$	617.364	0.249382
$184$	6774.96i	2.71444i
$185$	−6313.63	−2.50912
$186$	−571.750	−0.225391
$187$	625.844i	0.244739i
$188$	− 3661.31i	− 1.42036i
$189$	− 416.553i	− 0.160316i
$190$	4575.54i	1.74708i
$191$	−2057.47	−0.779443	−0.389721	−	0.920933i	$-0.627429\pi$
$191$	−2057.47	−0.779443	−0.389721	+	0.920933i	$0.627429\pi$
$192$	484.806	0.182228
$193$	− 729.873i	− 0.272215i	−0.990694	−	0.136107i	$-0.956541\pi$
$193$	− 729.873i	− 0.272215i	0.990694	−	0.136107i	$-0.0434592\pi$
$194$	−5625.20	−2.08178
$195$	0	0
$196$	−1832.50	−0.667822
$197$	− 1701.11i	− 0.615225i	−0.951512	−	0.307613i	$-0.900470\pi$
$197$	− 1701.11i	− 0.615225i	0.951512	−	0.307613i	$-0.0995301\pi$
$198$	−1223.17	−0.439026
$199$	1840.88	0.655761	0.327881	−	0.944719i	$-0.393666\pi$
$199$	1840.88	0.655761	0.327881	+	0.944719i	$0.393666\pi$
$200$	− 13344.3i	− 4.71792i
$201$	1477.73i	0.518564i
$202$	8021.66i	2.79407i
$203$	− 35.3404i	− 0.0122188i
$204$	1216.67	0.417568
$205$	−118.053	−0.0402204
$206$	2661.44i	0.900153i
$207$	1278.09	0.429147
$208$	0	0
$209$	−1214.31	−0.401894
$210$	4697.78i	1.54370i
$211$	142.970	0.0466467	0.0233234	−	0.999728i	$-0.492575\pi$
$211$	142.970	0.0466467	0.0233234	+	0.999728i	$0.492575\pi$
$212$	−4833.25	−1.56580
$213$	2480.58i	0.797964i
$214$	− 8839.31i	− 2.82356i
$215$	− 7260.32i	− 2.30302i
$216$	1288.11i	0.405762i
$217$	−582.773	−0.182310
$218$	4879.48	1.51597
$219$	− 198.361i	− 0.0612056i
$220$	9459.37	2.89887
$221$	0	0
$222$	−4750.31	−1.43613
$223$	2338.71i	0.702295i	0.936320	+	0.351148i	$0.114208\pi$
$223$	2338.71i	0.702295i	−0.936320	+	0.351148i	$0.885792\pi$
$224$	−1977.96	−0.589993
$225$	−2517.39	−0.745892
$226$	− 9641.57i	− 2.83782i
$227$	− 3279.36i	− 0.958850i	−0.877583	−	0.479425i	$-0.840845\pi$
$227$	− 3279.36i	− 0.958850i	0.877583	−	0.479425i	$-0.159155\pi$
$228$	2360.68i	0.685702i
$229$	1143.72i	0.330041i	0.986290	+	0.165021i	$0.0527690\pi$
$229$	1143.72i	0.330041i	−0.986290	+	0.165021i	$0.947231\pi$
$230$	−14414.0	−4.13230
$231$	−1246.75	−0.355110
$232$	109.283i	0.0309258i
$233$	4238.17	1.19164	0.595819	−	0.803118i	$-0.296827\pi$
$233$	4238.17	1.19164	0.595819	+	0.803118i	$0.296827\pi$
$234$	0	0
$235$	4219.59	1.17130
$236$	− 9481.75i	− 2.61529i
$237$	952.927	0.261178
$238$	1808.48	0.492547
$239$	3310.03i	0.895849i	0.894072	+	0.447924i	$0.147837\pi$
$239$	3310.03i	0.895849i	−0.894072	+	0.447924i	$0.852163\pi$
$240$	− 6098.99i	− 1.64037i
$241$	5702.86i	1.52429i	0.647407	+	0.762145i	$0.275854\pi$
$241$	5702.86i	1.52429i	−0.647407	+	0.762145i	$0.724146\pi$
$242$	− 3054.40i	− 0.811340i
$243$	243.000	0.0641500
$244$	−3592.18	−0.942483
$245$	− 2111.93i	− 0.550718i
$246$	−88.8217	−0.0230206
$247$	0	0
$248$	1802.11	0.461427
$249$	424.351i	0.108001i
$250$	15703.0	3.97257
$251$	3910.23	0.983313	0.491657	−	0.870789i	$-0.336392\pi$
$251$	3910.23	0.983313	0.491657	+	0.870789i	$0.336392\pi$
$252$	2423.75i	0.605880i
$253$	− 3825.35i	− 0.950585i
$254$	6221.51i	1.53690i
$255$	1402.19i	0.344347i
$256$	−7995.69	−1.95207
$257$	−6972.80	−1.69242	−0.846209	−	0.532851i	$-0.821121\pi$
$257$	−6972.80	−1.69242	−0.846209	+	0.532851i	$0.821121\pi$
$258$	− 5462.59i	− 1.31816i
$259$	−4841.88	−1.16162
$260$	0	0
$261$	20.6161	0.00488930
$262$	6432.32i	1.51676i
$263$	281.691	0.0660449	0.0330224	−	0.999455i	$-0.489487\pi$
$263$	281.691	0.0660449	0.0330224	+	0.999455i	$0.489487\pi$
$264$	3855.33	0.898785
$265$	− 5570.23i	− 1.29123i
$266$	3508.95i	0.808826i
$267$	− 1923.96i	− 0.440990i
$268$	− 8598.31i	− 1.95980i
$269$	4333.13	0.982139	0.491070	−	0.871120i	$-0.336606\pi$
$269$	4333.13	0.982139	0.491070	+	0.871120i	$0.336606\pi$
$270$	−2740.49	−0.617707
$271$	− 428.596i	− 0.0960715i	−0.998846	−	0.0480357i	$-0.984704\pi$
$271$	− 428.596i	− 0.0960715i	0.998846	−	0.0480357i	$-0.0152961\pi$
$272$	−2347.89	−0.523390
$273$	0	0
$274$	−10248.8	−2.25967
$275$	7534.59i	1.65219i
$276$	−7436.67	−1.62187
$277$	−8938.75	−1.93891	−0.969454	−	0.245274i	$-0.921122\pi$
$277$	−8938.75	−1.93891	−0.969454	+	0.245274i	$0.921122\pi$
$278$	7293.91i	1.57360i
$279$	− 339.966i	− 0.0729506i
$280$	− 14807.0i	− 3.16031i
$281$	− 775.819i	− 0.164703i	−0.996603	−	0.0823514i	$-0.973757\pi$
$281$	− 775.819i	− 0.164703i	0.996603	−	0.0823514i	$-0.0262430\pi$
$282$	3174.77	0.670408
$283$	4014.53	0.843248	0.421624	−	0.906771i	$-0.361460\pi$
$283$	4014.53	0.843248	0.421624	+	0.906771i	$0.361460\pi$
$284$	− 14433.4i	− 3.01573i
$285$	−2720.64	−0.565463
$286$	0	0
$287$	−90.5340	−0.0186204
$288$	− 1153.86i	− 0.236084i
$289$	−4373.21	−0.890130
$290$	−232.503	−0.0470795
$291$	− 3344.77i	− 0.673793i
$292$	1154.18i	0.231313i
$293$	4956.21i	0.988208i	0.869403	+	0.494104i	$0.164504\pi$
$293$	4956.21i	0.988208i	−0.869403	+	0.494104i	$0.835496\pi$
$294$	− 1588.99i	− 0.315210i
$295$	10927.5	2.15670
$296$	14972.5	2.94007
$297$	− 727.304i	− 0.142096i
$298$	4875.43	0.947738
$299$	0	0
$300$	14647.6	2.81893
$301$	− 5567.89i	− 1.06621i
$302$	−7381.80	−1.40654
$303$	−4769.72	−0.904334
$304$	− 4555.58i	− 0.859474i
$305$	− 4139.92i	− 0.777217i
$306$	1054.99i	0.197091i
$307$	3894.90i	0.724084i	0.932162	+	0.362042i	$0.117920\pi$
$307$	3894.90i	0.724084i	−0.932162	+	0.362042i	$0.882080\pi$
$308$	7254.33	1.34206
$309$	−1582.51	−0.291345
$310$	3834.04i	0.702448i
$311$	−3097.44	−0.564758	−0.282379	−	0.959303i	$-0.591124\pi$
$311$	−3097.44	−0.564758	−0.282379	+	0.959303i	$0.591124\pi$
$312$	0	0
$313$	4487.36	0.810353	0.405177	−	0.914238i	$-0.367210\pi$
$313$	4487.36	0.810353	0.405177	+	0.914238i	$0.367210\pi$
$314$	333.208i	0.0598854i
$315$	−2793.32	−0.499638
$316$	−5544.68	−0.987065
$317$	6820.62i	1.20847i	0.796807	+	0.604233i	$0.206521\pi$
$317$	6820.62i	1.20847i	−0.796807	+	0.604233i	$0.793479\pi$
$318$	− 4190.98i	− 0.739052i
$319$	− 61.7045i	− 0.0108301i
$320$	− 3251.01i	− 0.567928i
$321$	5255.89	0.913880
$322$	−11054.0	−1.91309
$323$	1047.35i	0.180421i
$324$	−1413.91	−0.242441
$325$	0	0
$326$	−17786.1	−3.02173
$327$	2901.37i	0.490660i
$328$	279.958	0.0471283
$329$	3235.98	0.542265
$330$	8202.35i	1.36826i
$331$	5012.96i	0.832438i	0.909264	+	0.416219i	$0.136645\pi$
$331$	5012.96i	0.832438i	−0.909264	+	0.416219i	$0.863355\pi$
$332$	− 2469.12i	− 0.408164i
$333$	− 2824.56i	− 0.464819i
$334$	−1315.08	−0.215444
$335$	9909.39	1.61614
$336$	− 4677.28i	− 0.759424i
$337$	−3220.79	−0.520616	−0.260308	−	0.965526i	$-0.583824\pi$
$337$	−3220.79	−0.520616	−0.260308	+	0.965526i	$0.583824\pi$
$338$	0	0
$339$	5732.92	0.918494
$340$	− 8158.74i	− 1.30138i
$341$	−1017.52	−0.161590
$342$	−2046.98	−0.323649
$343$	− 6911.39i	− 1.08799i
$344$	17217.6i	2.69858i
$345$	− 8570.62i	− 1.33747i
$346$	4600.13i	0.714753i
$347$	−3360.71	−0.519921	−0.259960	−	0.965619i	$-0.583710\pi$
$347$	−3360.71	−0.519921	−0.259960	+	0.965619i	$0.583710\pi$
$348$	−119.957	−0.0184780
$349$	− 4591.32i	− 0.704205i	−0.935961	−	0.352102i	$-0.885467\pi$
$349$	− 4591.32i	− 0.704205i	0.935961	−	0.352102i	$-0.114533\pi$
$350$	21772.4	3.32510
$351$	0	0
$352$	−3453.54	−0.522938
$353$	− 1740.09i	− 0.262367i	−0.991358	−	0.131183i	$-0.958122\pi$
$353$	− 1740.09i	− 0.262367i	0.991358	−	0.131183i	$-0.0418777\pi$
$354$	8221.76	1.23441
$355$	16634.2	2.48691
$356$	11194.7i	1.66662i
$357$	1075.33i	0.159419i
$358$	13574.5i	2.00400i
$359$	1425.49i	0.209567i	0.994495	+	0.104784i	$0.0334150\pi$
$359$	1425.49i	0.209567i	−0.994495	+	0.104784i	$0.966585\pi$
$360$	8637.78	1.26459
$361$	4826.85	0.703725
$362$	24085.8i	3.49702i
$363$	1816.16	0.262600
$364$	0	0
$365$	−1330.17	−0.190752
$366$	− 3114.83i	− 0.444849i
$367$	10849.2	1.54312	0.771559	−	0.636158i	$-0.219477\pi$
$367$	10849.2	1.54312	0.771559	+	0.636158i	$0.219477\pi$
$368$	14351.1	2.03288
$369$	− 52.8138i	− 0.00745089i
$370$	31854.6i	4.47579i
$371$	− 4271.77i	− 0.597788i
$372$	1978.12i	0.275700i
$373$	−494.749	−0.0686786	−0.0343393	−	0.999410i	$-0.510933\pi$
$373$	−494.749	−0.0686786	−0.0343393	+	0.999410i	$0.510933\pi$
$374$	3157.61	0.436567
$375$	9337.06i	1.28577i
$376$	−10006.6	−1.37248
$377$	0	0
$378$	−2101.66	−0.285973
$379$	− 12643.3i	− 1.71357i	−0.515671	−	0.856786i	$-0.672457\pi$
$379$	− 12643.3i	− 1.71357i	0.515671	−	0.856786i	$-0.327543\pi$
$380$	15830.3	2.13704
$381$	−3699.34	−0.497436
$382$	10380.7i	1.39037i
$383$	2321.51i	0.309722i	0.987936	+	0.154861i	$0.0494930\pi$
$383$	2321.51i	0.309722i	−0.987936	+	0.154861i	$0.950507\pi$
$384$	− 5522.99i	− 0.733969i
$385$	8360.47i	1.10673i
$386$	−3682.48	−0.485578
$387$	3248.08	0.426639
$388$	19461.8i	2.54645i
$389$	10477.6	1.36564	0.682821	−	0.730586i	$-0.260753\pi$
$389$	10477.6	1.36564	0.682821	+	0.730586i	$0.260753\pi$
$390$	0	0
$391$	−3299.38	−0.426744
$392$	5008.35i	0.645306i
$393$	−3824.69	−0.490916
$394$	−8582.75	−1.09744
$395$	− 6390.14i	− 0.813982i
$396$	4231.88i	0.537020i
$397$	1766.85i	0.223364i	0.993744	+	0.111682i	$0.0356238\pi$
$397$	1766.85i	0.223364i	−0.993744	+	0.111682i	$0.964376\pi$
$398$	− 9287.91i	− 1.16975i
$399$	−2086.44	−0.261786
$400$	−28266.5	−3.53332
$401$	5001.65i	0.622869i	0.950268	+	0.311434i	$0.100809\pi$
$401$	5001.65i	0.622869i	−0.950268	+	0.311434i	$0.899191\pi$
$402$	7455.71	0.925018
$403$	0	0
$404$	27753.0	3.41773
$405$	− 1629.51i	− 0.199928i
$406$	−178.305	−0.0217959
$407$	−8453.96	−1.02960
$408$	− 3325.24i	− 0.403490i
$409$	− 11208.6i	− 1.35508i	−0.735485	−	0.677541i	$-0.763046\pi$
$409$	− 11208.6i	− 1.35508i	0.735485	−	0.677541i	$-0.236954\pi$
$410$	595.620i	0.0717453i
$411$	− 6093.96i	− 0.731370i
$412$	9207.94	1.10107
$413$	8380.26	0.998464
$414$	− 6448.44i	− 0.765515i
$415$	2845.61	0.336592
$416$	0	0
$417$	−4336.99	−0.509313
$418$	6126.66i	0.716901i
$419$	3285.19	0.383036	0.191518	−	0.981489i	$-0.438659\pi$
$419$	3285.19	0.383036	0.191518	+	0.981489i	$0.438659\pi$
$420$	16253.2	1.88827
$421$	− 13289.9i	− 1.53850i	−0.638948	−	0.769250i	$-0.720630\pi$
$421$	− 13289.9i	− 1.53850i	0.638948	−	0.769250i	$-0.279370\pi$
$422$	− 721.336i	− 0.0832087i
$423$	1887.74i	0.216985i
$424$	13209.6i	1.51301i
$425$	6498.61	0.741715
$426$	12515.4	1.42341
$427$	− 3174.88i	− 0.359820i
$428$	−30581.8	−3.45380
$429$	0	0
$430$	−36631.0	−4.10815
$431$	− 13692.3i	− 1.53024i	−0.643889	−	0.765119i	$-0.722680\pi$
$431$	− 13692.3i	− 1.53024i	0.643889	−	0.765119i	$-0.277320\pi$
$432$	2728.53	0.303881
$433$	10002.5	1.11014	0.555070	−	0.831804i	$-0.312692\pi$
$433$	10002.5	1.11014	0.555070	+	0.831804i	$0.312692\pi$
$434$	2940.30i	0.325205i
$435$	− 138.248i	− 0.0152378i
$436$	− 16881.8i	− 1.85434i
$437$	− 6401.73i	− 0.700769i
$438$	−1000.81	−0.109179
$439$	−4487.21	−0.487842	−0.243921	−	0.969795i	$-0.578434\pi$
$439$	−4487.21	−0.487842	−0.243921	+	0.969795i	$0.578434\pi$
$440$	− 25853.1i	− 2.80113i
$441$	944.821	0.102021
$442$	0	0
$443$	2035.35	0.218290	0.109145	−	0.994026i	$-0.465189\pi$
$443$	2035.35	0.218290	0.109145	+	0.994026i	$0.465189\pi$
$444$	16434.9i	1.75668i
$445$	−12901.7	−1.37438
$446$	11799.7	1.25276
$447$	2898.95i	0.306747i
$448$	− 2493.18i	− 0.262928i
$449$	− 3575.58i	− 0.375818i	−0.982186	−	0.187909i	$-0.939829\pi$
$449$	− 3575.58i	− 0.375818i	0.982186	−	0.187909i	$-0.0601710\pi$
$450$	12701.1i	1.33053i
$451$	−158.073	−0.0165041
$452$	−33357.4	−3.47124
$453$	− 4389.26i	− 0.455243i
$454$	−16545.6	−1.71040
$455$	0	0
$456$	6451.90	0.662583
$457$	8058.60i	0.824869i	0.910987	+	0.412435i	$0.135321\pi$
$457$	8058.60i	0.824869i	−0.910987	+	0.412435i	$0.864679\pi$
$458$	5770.51	0.588730
$459$	−627.303	−0.0637908
$460$	49868.8i	5.05466i
$461$	11692.1i	1.18125i	0.806945	+	0.590626i	$0.201119\pi$
$461$	11692.1i	1.18125i	−0.806945	+	0.590626i	$0.798881\pi$
$462$	6290.33i	0.633447i
$463$	− 1732.40i	− 0.173891i	−0.996213	−	0.0869455i	$-0.972289\pi$
$463$	− 1732.40i	− 0.173891i	0.996213	−	0.0869455i	$-0.0277106\pi$
$464$	231.489	0.0231607
$465$	−2279.74	−0.227356
$466$	− 21383.1i	− 2.12565i
$467$	−10769.9	−1.06718	−0.533588	−	0.845745i	$-0.679157\pi$
$467$	−10769.9	−1.06718	−0.533588	+	0.845745i	$0.679157\pi$
$468$	0	0
$469$	7599.44	0.748208
$470$	− 21289.4i	− 2.08937i
$471$	−198.127	−0.0193826
$472$	−25914.2	−2.52712
$473$	− 9721.58i	− 0.945029i
$474$	− 4807.87i	− 0.465892i
$475$	12609.1i	1.21799i
$476$	− 6256.88i	− 0.602487i
$477$	2491.98	0.239203
$478$	16700.3	1.59802
$479$	− 7911.64i	− 0.754680i	−0.926075	−	0.377340i	$-0.876839\pi$
$479$	− 7911.64i	− 0.754680i	0.926075	−	0.377340i	$-0.123161\pi$
$480$	−7737.57	−0.735772
$481$	0	0
$482$	28773.0	2.71904
$483$	− 6572.75i	− 0.619193i
$484$	−10567.5	−0.992438
$485$	−22429.3	−2.09993
$486$	− 1226.02i	− 0.114431i
$487$	− 10964.8i	− 1.02025i	−0.860100	−	0.510125i	$-0.829599\pi$
$487$	− 10964.8i	− 1.02025i	0.860100	−	0.510125i	$-0.170401\pi$
$488$	9817.66i	0.910707i
$489$	− 10575.7i	− 0.978018i
$490$	−10655.4	−0.982375
$491$	9139.37	0.840029	0.420014	−	0.907518i	$-0.362025\pi$
$491$	9139.37	0.840029	0.420014	+	0.907518i	$0.362025\pi$
$492$	307.301i	0.0281590i
$493$	−53.2204	−0.00486192
$494$	0	0
$495$	−4877.16	−0.442852
$496$	− 3817.31i	− 0.345569i
$497$	12756.7	1.15134
$498$	2141.01	0.192652
$499$	− 12577.5i	− 1.12835i	−0.825655	−	0.564175i	$-0.809194\pi$
$499$	− 12577.5i	− 1.12835i	0.825655	−	0.564175i	$-0.190806\pi$
$500$	− 54328.4i	− 4.85928i
$501$	− 781.955i	− 0.0697309i
$502$	− 19728.6i	− 1.75404i
$503$	13214.6	1.17139	0.585696	−	0.810531i	$-0.300821\pi$
$503$	13214.6	1.17139	0.585696	+	0.810531i	$0.300821\pi$
$504$	6624.25	0.585452
$505$	31984.8i	2.81842i
$506$	−19300.3	−1.69566
$507$	0	0
$508$	21524.9	1.87995
$509$	21719.8i	1.89138i	0.325073	+	0.945689i	$0.394611\pi$
$509$	21719.8i	1.89138i	−0.325073	+	0.945689i	$0.605389\pi$
$510$	7074.56	0.614248
$511$	−1020.10	−0.0883103
$512$	25613.2i	2.21085i
$513$	− 1217.14i	− 0.104753i
$514$	35180.4i	3.01895i
$515$	10612.0i	0.907999i
$516$	−18899.2	−1.61239
$517$	5650.03	0.480635
$518$	24429.1i	2.07211i
$519$	−2735.26	−0.231338
$520$	0	0
$521$	−4627.05	−0.389088	−0.194544	−	0.980894i	$-0.562323\pi$
$521$	−4627.05	−0.389088	−0.194544	+	0.980894i	$0.562323\pi$
$522$	− 104.016i	− 0.00872156i
$523$	13784.0	1.15245	0.576224	−	0.817292i	$-0.304526\pi$
$523$	13784.0	1.15245	0.576224	+	0.817292i	$0.304526\pi$
$524$	22254.3	1.85531
$525$	12946.0i	1.07621i
$526$	− 1421.23i	− 0.117811i
$527$	877.619i	0.0725421i
$528$	− 8166.55i	− 0.673113i
$529$	7999.85	0.657504
$530$	−28103.9	−2.30331
$531$	4888.70i	0.399532i
$532$	12140.1	0.989362
$533$	0	0
$534$	−9707.09	−0.786642
$535$	− 35245.0i	− 2.84817i
$536$	−23499.7	−1.89372
$537$	−8071.45	−0.648620
$538$	− 21862.2i	− 1.75195i
$539$	− 2827.87i	− 0.225983i
$540$	9481.42i	0.755584i
$541$	− 454.638i	− 0.0361302i	−0.999837	−	0.0180651i	$-0.994249\pi$
$541$	− 454.638i	− 0.0361302i	0.999837	−	0.0180651i	$-0.00575061\pi$
$542$	−2162.43	−0.171373
$543$	−14321.5	−1.13185
$544$	2978.69i	0.234762i
$545$	19456.0	1.52918
$546$	0	0
$547$	11611.4	0.907621	0.453810	−	0.891098i	$-0.350064\pi$
$547$	11611.4	0.907621	0.453810	+	0.891098i	$0.350064\pi$
$548$	35458.2i	2.76405i
$549$	1852.09	0.143981
$550$	38014.8	2.94719
$551$	− 103.263i	− 0.00798390i
$552$	20324.9i	1.56718i
$553$	− 4900.55i	− 0.376840i
$554$	45099.3i	3.45864i
$555$	−18940.9	−1.44864
$556$	25235.1	1.92483
$557$	− 4883.93i	− 0.371524i	−0.982595	−	0.185762i	$-0.940525\pi$
$557$	− 4883.93i	− 0.371524i	0.982595	−	0.185762i	$-0.0594754\pi$
$558$	−1715.25	−0.130130
$559$	0	0
$560$	−31364.9	−2.36680
$561$	1877.53i	0.141300i
$562$	−3914.29	−0.293798
$563$	−19617.3	−1.46851	−0.734256	−	0.678873i	$-0.762469\pi$
$563$	−19617.3	−1.46851	−0.734256	+	0.678873i	$0.762469\pi$
$564$	− 10983.9i	− 0.820048i
$565$	− 38443.8i	− 2.86256i
$566$	− 20254.8i	− 1.50419i
$567$	− 1249.66i	− 0.0925587i
$568$	−39447.5	−2.91405
$569$	−7475.19	−0.550749	−0.275374	−	0.961337i	$-0.588802\pi$
$569$	−7475.19	−0.550749	−0.275374	+	0.961337i	$0.588802\pi$
$570$	13726.6i	1.00868i
$571$	7799.56	0.571631	0.285816	−	0.958285i	$-0.407735\pi$
$571$	7799.56	0.571631	0.285816	+	0.958285i	$0.407735\pi$
$572$	0	0
$573$	−6172.42	−0.450011
$574$	456.777i	0.0332152i
$575$	−39721.6	−2.88088
$576$	1454.42	0.105210
$577$	13136.1i	0.947771i	0.880587	+	0.473885i	$0.157149\pi$
$577$	13136.1i	0.947771i	−0.880587	+	0.473885i	$0.842851\pi$
$578$	22064.4i	1.58782i
$579$	− 2189.62i	− 0.157163i
$580$	804.404i	0.0575880i
$581$	2182.28	0.155828
$582$	−16875.6	−1.20192
$583$	− 7458.54i	− 0.529848i
$584$	3154.45	0.223514
$585$	0	0
$586$	25005.9	1.76277
$587$	22150.7i	1.55751i	0.627330	+	0.778754i	$0.284148\pi$
$587$	22150.7i	1.55751i	−0.627330	+	0.778754i	$0.715852\pi$
$588$	−5497.51	−0.385567
$589$	−1702.83	−0.119124
$590$	− 55133.4i	− 3.84713i
$591$	− 5103.34i	− 0.355201i
$592$	− 31715.6i	− 2.20186i
$593$	22770.3i	1.57684i	0.615138	+	0.788419i	$0.289100\pi$
$593$	22770.3i	1.57684i	−0.615138	+	0.788419i	$0.710900\pi$
$594$	−3669.52	−0.253472
$595$	7210.94	0.496840
$596$	− 16867.8i	− 1.15928i
$597$	5522.64	0.378604
$598$	0	0
$599$	−7214.11	−0.492088	−0.246044	−	0.969259i	$-0.579131\pi$
$599$	−7214.11	−0.492088	−0.246044	+	0.969259i	$0.579131\pi$
$600$	− 40032.8i	− 2.72389i
$601$	27276.7	1.85132	0.925658	−	0.378360i	$-0.123512\pi$
$601$	27276.7	1.85132	0.925658	+	0.378360i	$0.123512\pi$
$602$	−28092.1	−1.90191
$603$	4433.20i	0.299393i
$604$	25539.2i	1.72049i
$605$	− 12178.8i	− 0.818412i
$606$	24065.0i	1.61316i
$607$	11566.2	0.773403	0.386701	−	0.922205i	$-0.373614\pi$
$607$	11566.2	0.773403	0.386701	+	0.922205i	$0.373614\pi$
$608$	−5779.50	−0.385509
$609$	− 106.021i	− 0.00705450i
$610$	−20887.4	−1.38640
$611$	0	0
$612$	3650.01	0.241083
$613$	24476.6i	1.61272i	0.591422	+	0.806362i	$0.298567\pi$
$613$	24476.6i	1.61272i	−0.591422	+	0.806362i	$0.701433\pi$
$614$	19651.2	1.29163
$615$	−354.159	−0.0232212
$616$	− 19826.5i	− 1.29681i
$617$	2423.28i	0.158116i	0.996870	+	0.0790581i	$0.0251913\pi$
$617$	2423.28i	0.158116i	−0.996870	+	0.0790581i	$0.974809\pi$
$618$	7984.33i	0.519704i
$619$	− 17223.4i	− 1.11836i	−0.829045	−	0.559182i	$-0.811115\pi$
$619$	− 17223.4i	− 1.11836i	0.829045	−	0.559182i	$-0.188885\pi$
$620$	13264.8	0.859240
$621$	3834.27	0.247768
$622$	15627.7i	1.00742i
$623$	−9894.22	−0.636282
$624$	0	0
$625$	27648.7	1.76952
$626$	− 22640.4i	− 1.44551i
$627$	−3642.94	−0.232033
$628$	1152.82	0.0732523
$629$	7291.57i	0.462216i
$630$	14093.3i	0.891257i
$631$	− 9242.58i	− 0.583108i	−0.956554	−	0.291554i	$-0.905828\pi$
$631$	− 9242.58i	− 0.583108i	0.956554	−	0.291554i	$-0.0941724\pi$
$632$	15154.0i	0.953786i
$633$	428.910	0.0269315
$634$	34412.5	2.15567
$635$	24807.0i	1.55029i
$636$	−14499.8	−0.904014
$637$	0	0
$638$	−311.322	−0.0193187
$639$	7441.73i	0.460705i
$640$	−37036.1	−2.28747
$641$	11598.8	0.714706	0.357353	−	0.933969i	$-0.383679\pi$
$641$	11598.8	0.714706	0.357353	+	0.933969i	$0.383679\pi$
$642$	− 26517.9i	− 1.63018i
$643$	25363.9i	1.55561i	0.628507	+	0.777804i	$0.283666\pi$
$643$	25363.9i	1.55561i	−0.628507	+	0.777804i	$0.716334\pi$
$644$	38244.0i	2.34010i
$645$	− 21781.0i	− 1.32965i
$646$	5284.26	0.321837
$647$	6590.09	0.400438	0.200219	−	0.979751i	$-0.435835\pi$
$647$	6590.09	0.400438	0.200219	+	0.979751i	$0.435835\pi$
$648$	3864.32i	0.234267i
$649$	14632.0	0.884985
$650$	0	0
$651$	−1748.32	−0.105257
$652$	61535.6i	3.69620i
$653$	15698.2	0.940765	0.470382	−	0.882463i	$-0.344116\pi$
$653$	15698.2	0.940765	0.470382	+	0.882463i	$0.344116\pi$
$654$	14638.5	0.875243
$655$	25647.6i	1.52998i
$656$	− 593.021i	− 0.0352951i
$657$	− 595.084i	− 0.0353371i
$658$	− 16326.7i	− 0.967295i
$659$	2840.81	0.167925	0.0839624	−	0.996469i	$-0.473242\pi$
$659$	2840.81	0.167925	0.0839624	+	0.996469i	$0.473242\pi$
$660$	28378.1	1.67366
$661$	− 20819.1i	− 1.22507i	−0.790445	−	0.612533i	$-0.790151\pi$
$661$	− 20819.1i	− 1.22507i	0.790445	−	0.612533i	$-0.209849\pi$
$662$	25292.2	1.48491
$663$	0	0
$664$	−6748.26	−0.394403
$665$	13991.3i	0.815876i
$666$	−14250.9	−0.829147
$667$	325.300	0.0188840
$668$	4549.87i	0.263532i
$669$	7016.14i	0.405470i
$670$	− 49996.5i	− 2.88289i
$671$	− 5543.36i	− 0.318925i
$672$	−5933.89	−0.340632
$673$	−31264.5	−1.79073	−0.895364	−	0.445335i	$-0.853084\pi$
$673$	−31264.5	−1.79073	−0.895364	+	0.445335i	$0.853084\pi$
$674$	16250.1i	0.928679i
$675$	−7552.16	−0.430641
$676$	0	0
$677$	−27953.2	−1.58689	−0.793447	−	0.608639i	$-0.791716\pi$
$677$	−27953.2	−1.58689	−0.793447	+	0.608639i	$0.791716\pi$
$678$	− 28924.7i	− 1.63842i
$679$	−17200.9	−0.972180
$680$	−22298.4	−1.25750
$681$	− 9838.09i	− 0.553592i
$682$	5133.79i	0.288245i
$683$	34709.9i	1.94456i	0.233811	+	0.972282i	$0.424880\pi$
$683$	34709.9i	1.94456i	−0.233811	+	0.972282i	$0.575120\pi$
$684$	7082.05i	0.395890i
$685$	−40864.9	−2.27937
$686$	−34870.5	−1.94076
$687$	3431.17i	0.190549i
$688$	36471.1	2.02100
$689$	0	0
$690$	−43241.9	−2.38578
$691$	− 11860.8i	− 0.652974i	−0.945202	−	0.326487i	$-0.894135\pi$
$691$	− 11860.8i	− 0.652974i	0.945202	−	0.326487i	$-0.105865\pi$
$692$	15915.3	0.874291
$693$	−3740.26	−0.205023
$694$	16956.0i	0.927438i
$695$	29083.0i	1.58731i
$696$	327.849i	0.0178550i
$697$	136.338i	0.00740916i
$698$	−23164.9	−1.25617
$699$	12714.5	0.687993
$700$	− 75327.2i	− 4.06729i
$701$	−16100.5	−0.867486	−0.433743	−	0.901037i	$-0.642807\pi$
$701$	−16100.5	−0.867486	−0.433743	+	0.901037i	$0.642807\pi$
$702$	0	0
$703$	−14147.7	−0.759019
$704$	− 4353.10i	− 0.233045i
$705$	12658.8	0.676251
$706$	−8779.38	−0.468012
$707$	24528.9i	1.30482i
$708$	− 28445.3i	− 1.50994i
$709$	28465.5i	1.50782i	0.656978	+	0.753910i	$0.271834\pi$
$709$	28465.5i	1.50782i	−0.656978	+	0.753910i	$0.728166\pi$
$710$	− 83925.9i	− 4.43617i
$711$	2858.78	0.150791
$712$	30595.9	1.61043
$713$	− 5364.28i	− 0.281759i
$714$	5425.43	0.284372
$715$	0	0
$716$	46964.4	2.45131
$717$	9930.08i	0.517218i
$718$	7192.14	0.373828
$719$	27822.1	1.44310	0.721550	−	0.692362i	$-0.243430\pi$
$719$	27822.1	1.44310	0.721550	+	0.692362i	$0.243430\pi$
$720$	− 18297.0i	− 0.947067i
$721$	8138.25i	0.420367i
$722$	− 24353.2i	− 1.25531i
$723$	17108.6i	0.880049i
$724$	83331.0	4.27758
$725$	−640.725	−0.0328220
$726$	− 9163.20i	− 0.468427i
$727$	−866.153	−0.0441869	−0.0220934	−	0.999756i	$-0.507033\pi$
$727$	−866.153	−0.0441869	−0.0220934	+	0.999756i	$0.507033\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	6711.20i	0.340264i
$731$	−8384.90	−0.424250
$732$	−10776.5	−0.544143
$733$	− 23120.1i	− 1.16502i	−0.812823	−	0.582511i	$-0.802070\pi$
$733$	− 23120.1i	− 1.16502i	0.812823	−	0.582511i	$-0.197930\pi$
$734$	− 54738.3i	− 2.75262i
$735$	− 6335.78i	− 0.317957i
$736$	− 18206.7i	− 0.911831i
$737$	13268.7	0.663172
$738$	−266.465	−0.0132909
$739$	− 20765.1i	− 1.03364i	−0.856095	−	0.516819i	$-0.827116\pi$
$739$	− 20765.1i	− 1.03364i	0.856095	−	0.516819i	$-0.172884\pi$
$740$	110209.	5.47482
$741$	0	0
$742$	−21552.7	−1.06634
$743$	− 36995.8i	− 1.82671i	−0.407166	−	0.913354i	$-0.633483\pi$
$743$	− 36995.8i	− 1.82671i	0.407166	−	0.913354i	$-0.366517\pi$
$744$	5406.32	0.266405
$745$	19439.8	0.955999
$746$	2496.19i	0.122509i
$747$	1273.05i	0.0623542i
$748$	− 10924.6i	− 0.534013i
$749$	− 27029.1i	− 1.31859i
$750$	47108.9	2.29356
$751$	34429.1	1.67289	0.836443	−	0.548055i	$-0.184631\pi$
$751$	34429.1	1.67289	0.836443	+	0.548055i	$0.184631\pi$
$752$	21196.5i	1.02787i
$753$	11730.7	0.567716
$754$	0	0
$755$	−29433.5	−1.41880
$756$	7271.24i	0.349805i
$757$	4268.88	0.204960	0.102480	−	0.994735i	$-0.467322\pi$
$757$	4268.88	0.204960	0.102480	+	0.994735i	$0.467322\pi$
$758$	−63790.2	−3.05668
$759$	− 11476.1i	− 0.548820i
$760$	− 43265.1i	− 2.06499i
$761$	7342.11i	0.349739i	0.984592	+	0.174869i	$0.0559503\pi$
$761$	7342.11i	0.349739i	−0.984592	+	0.174869i	$0.944050\pi$
$762$	18664.5i	0.887329i
$763$	14920.7	0.707947
$764$	35914.7	1.70072
$765$	4206.57i	0.198809i
$766$	11712.9	0.552484
$767$	0	0
$768$	−23987.1	−1.12703
$769$	− 14728.5i	− 0.690666i	−0.938480	−	0.345333i	$-0.887766\pi$
$769$	− 14728.5i	− 0.690666i	0.938480	−	0.345333i	$-0.112234\pi$
$770$	42181.7	1.97418
$771$	−20918.4	−0.977118
$772$	12740.5i	0.593963i
$773$	9937.41i	0.462385i	0.972908	+	0.231193i	$0.0742628\pi$
$773$	9937.41i	0.462385i	−0.972908	+	0.231193i	$0.925737\pi$
$774$	− 16387.8i	− 0.761041i
$775$	10565.7i	0.489719i
$776$	53190.3	2.46059
$777$	−14525.7	−0.670663
$778$	− 52863.3i	− 2.43604i
$779$	−264.535	−0.0121668
$780$	0	0
$781$	22273.3	1.02049
$782$	16646.6i	0.761229i
$783$	61.8484	0.00282284
$784$	10608.9	0.483279
$785$	1328.60i	0.0604074i
$786$	19297.0i	0.875700i
$787$	− 36305.0i	− 1.64439i	−0.569207	−	0.822194i	$-0.692750\pi$
$787$	− 36305.0i	− 1.64439i	0.569207	−	0.822194i	$-0.307250\pi$
$788$	29694.2i	1.34240i
$789$	845.072	0.0381310
$790$	−32240.6	−1.45199
$791$	− 29482.3i	− 1.32525i
$792$	11566.0	0.518914
$793$	0	0
$794$	8914.40	0.398439
$795$	− 16710.7i	− 0.745493i
$796$	−32133.9	−1.43085
$797$	9150.73	0.406694	0.203347	−	0.979107i	$-0.434818\pi$
$797$	9150.73	0.406694	0.203347	+	0.979107i	$0.434818\pi$
$798$	10526.9i	0.466976i
$799$	− 4873.17i	− 0.215770i
$800$	35860.7i	1.58484i
$801$	− 5771.88i	− 0.254606i
$802$	25235.1	1.11108
$803$	−1781.10	−0.0782736
$804$	− 25794.9i	− 1.13149i
$805$	−44075.5	−1.92976
$806$	0	0
$807$	12999.4	0.567038
$808$	− 75850.7i	− 3.30250i
$809$	30349.9	1.31897	0.659484	−	0.751719i	$-0.270775\pi$
$809$	30349.9	1.31897	0.659484	+	0.751719i	$0.270775\pi$
$810$	−8221.47	−0.356633
$811$	− 4238.72i	− 0.183529i	−0.995781	−	0.0917643i	$-0.970749\pi$
$811$	− 4238.72i	− 0.183529i	0.995781	−	0.0917643i	$-0.0292506\pi$
$812$	616.892i	0.0266609i
$813$	− 1285.79i	− 0.0554669i
$814$	42653.3i	1.83661i
$815$	−70918.6	−3.04806
$816$	−7043.68	−0.302179
$817$	− 16269.1i	− 0.696674i
$818$	−56551.4	−2.41721
$819$	0	0
$820$	2060.70	0.0877595
$821$	5934.48i	0.252271i	0.992013	+	0.126136i	$0.0402574\pi$
$821$	5934.48i	0.252271i	−0.992013	+	0.126136i	$0.959743\pi$
$822$	−30746.3	−1.30462
$823$	22459.6	0.951265	0.475633	−	0.879644i	$-0.342219\pi$
$823$	22459.6	0.951265	0.475633	+	0.879644i	$0.342219\pi$
$824$	− 25165.9i	− 1.06395i
$825$	22603.8i	0.953894i
$826$	− 42281.5i	− 1.78107i
$827$	− 20138.4i	− 0.846772i	−0.905949	−	0.423386i	$-0.860841\pi$
$827$	− 20138.4i	− 0.846772i	0.905949	−	0.423386i	$-0.139159\pi$
$828$	−22310.0	−0.936384
$829$	−5845.70	−0.244909	−0.122455	−	0.992474i	$-0.539077\pi$
$829$	−5845.70	−0.244909	−0.122455	+	0.992474i	$0.539077\pi$
$830$	− 14357.2i	− 0.600415i
$831$	−26816.2	−1.11943
$832$	0	0
$833$	−2439.05	−0.101450
$834$	21881.7i	0.908516i
$835$	−5243.63	−0.217322
$836$	21196.7	0.876919
$837$	− 1019.90i	− 0.0421180i
$838$	− 16575.0i	− 0.683261i
$839$	27779.9i	1.14311i	0.820564	+	0.571554i	$0.193659\pi$
$839$	27779.9i	1.14311i	−0.820564	+	0.571554i	$0.806341\pi$
$840$	− 44420.9i	− 1.82460i
$841$	−24383.8	−0.999785
$842$	−67052.3	−2.74439
$843$	− 2327.46i	− 0.0950911i
$844$	−2495.65	−0.101782
$845$	0	0
$846$	9524.32	0.387060
$847$	− 9339.85i	− 0.378891i
$848$	27981.2	1.13311
$849$	12043.6	0.486849
$850$	− 32787.9i	− 1.32308i
$851$	− 44568.4i	− 1.79528i
$852$	− 43300.3i	− 1.74113i
$853$	− 16480.5i	− 0.661526i	−0.943714	−	0.330763i	$-0.892694\pi$
$853$	− 16480.5i	− 0.661526i	0.943714	−	0.330763i	$-0.107306\pi$
$854$	−16018.4	−0.641849
$855$	−8161.92	−0.326470
$856$	83582.1i	3.33736i
$857$	45445.2	1.81141	0.905704	−	0.423910i	$-0.139343\pi$
$857$	45445.2	1.81141	0.905704	+	0.423910i	$0.139343\pi$
$858$	0	0
$859$	−28243.1	−1.12182	−0.560909	−	0.827877i	$-0.689548\pi$
$859$	−28243.1	−1.12182	−0.560909	+	0.827877i	$0.689548\pi$
$860$	126734.i	5.02512i
$861$	−271.602	−0.0107505
$862$	−69082.5	−2.72965
$863$	− 328.319i	− 0.0129503i	−0.999979	−	0.00647514i	$-0.997939\pi$
$863$	− 328.319i	− 0.0129503i	0.999979	−	0.00647514i	$-0.00206112\pi$
$864$	− 3461.59i	− 0.136303i
$865$	18342.1i	0.720982i
$866$	− 50466.4i	− 1.98027i
$867$	−13119.6	−0.513917
$868$	10172.7	0.397793
$869$	− 8556.40i	− 0.334011i
$870$	−697.510	−0.0271814
$871$	0	0
$872$	−46139.1	−1.79182
$873$	− 10034.3i	− 0.389015i
$874$	−32299.1	−1.25004
$875$	48017.0	1.85517
$876$	3462.54i	0.133549i
$877$	42480.0i	1.63563i	0.575482	+	0.817815i	$0.304815\pi$
$877$	42480.0i	1.63563i	−0.575482	+	0.817815i	$0.695185\pi$
$878$	22639.6i	0.870216i
$879$	14868.6i	0.570542i
$880$	−54763.3	−2.09781
$881$	5220.95	0.199657	0.0998287	−	0.995005i	$-0.468171\pi$
$881$	5220.95	0.199657	0.0998287	+	0.995005i	$0.468171\pi$
$882$	− 4766.97i	− 0.181987i
$883$	−11790.3	−0.449349	−0.224674	−	0.974434i	$-0.572132\pi$
$883$	−11790.3	−0.449349	−0.224674	+	0.974434i	$0.572132\pi$
$884$	0	0
$885$	32782.6	1.24517
$886$	− 10269.1i	− 0.389387i
$887$	−49352.3	−1.86819	−0.934097	−	0.357019i	$-0.883793\pi$
$887$	−49352.3	−1.86819	−0.934097	+	0.357019i	$0.883793\pi$
$888$	44917.6	1.69745
$889$	19024.4i	0.717724i
$890$	65093.8i	2.45163i
$891$	− 2181.91i	− 0.0820391i
$892$	− 40823.9i	− 1.53238i
$893$	9455.33	0.354323
$894$	14626.3	0.547177
$895$	54125.5i	2.02147i
$896$	−28402.7	−1.05900
$897$	0	0
$898$	−18040.1	−0.670386
$899$	− 86.5281i	− 0.00321009i
$900$	43942.8	1.62751
$901$	−6433.02	−0.237863
$902$	797.536i	0.0294402i
$903$	− 16703.7i	− 0.615574i
$904$	91168.1i	3.35421i
$905$	96037.3i	3.52750i
$906$	−22145.4	−0.812066
$907$	−6088.94	−0.222910	−0.111455	−	0.993769i	$-0.535551\pi$
$907$	−6088.94	−0.222910	−0.111455	+	0.993769i	$0.535551\pi$
$908$	57243.7i	2.09218i
$909$	−14309.2	−0.522118
$910$	0	0
$911$	−30301.7	−1.10202	−0.551010	−	0.834499i	$-0.685757\pi$
$911$	−30301.7	−1.10202	−0.551010	+	0.834499i	$0.685757\pi$
$912$	− 13666.7i	− 0.496218i
$913$	3810.28	0.138118
$914$	40658.6	1.47141
$915$	− 12419.8i	− 0.448726i
$916$	− 19964.5i	− 0.720139i
$917$	19669.0i	0.708317i
$918$	3164.97i	0.113790i
$919$	34695.8	1.24538	0.622692	−	0.782467i	$-0.286039\pi$
$919$	34695.8	1.24538	0.622692	+	0.782467i	$0.286039\pi$
$920$	136295.	4.88424
$921$	11684.7i	0.418050i
$922$	58991.2	2.10713
$923$	0	0
$924$	21763.0	0.774837
$925$	87783.9i	3.12034i
$926$	−8740.61	−0.310188
$927$	−4747.52	−0.168208
$928$	− 293.681i	− 0.0103885i
$929$	− 16990.8i	− 0.600054i	−0.953931	−	0.300027i	$-0.903004\pi$
$929$	− 16990.8i	− 0.600054i	0.953931	−	0.300027i	$-0.0969957\pi$
$930$	11502.1i	0.405559i
$931$	− 4732.44i	− 0.166594i
$932$	−73980.4	−2.60012
$933$	−9292.32	−0.326063
$934$	54338.0i	1.90363i
$935$	12590.3	0.440372
$936$	0	0
$937$	9307.86	0.324519	0.162260	−	0.986748i	$-0.448122\pi$
$937$	9307.86	0.324519	0.162260	+	0.986748i	$0.448122\pi$
$938$	− 38342.0i	− 1.33466i
$939$	13462.1	0.467858
$940$	−73656.0	−2.55574
$941$	52285.3i	1.81132i	0.424006	+	0.905659i	$0.360623\pi$
$941$	52285.3i	1.81132i	−0.424006	+	0.905659i	$0.639377\pi$
$942$	999.624i	0.0345749i
$943$	− 833.344i	− 0.0287777i
$944$	54892.8i	1.89260i
$945$	−8379.96	−0.288466
$946$	−49048.9	−1.68575
$947$	− 12348.5i	− 0.423731i	−0.977299	−	0.211865i	$-0.932046\pi$
$947$	− 12348.5i	− 0.423731i	0.977299	−	0.211865i	$-0.0679539\pi$
$948$	−16634.0	−0.569882
$949$	0	0
$950$	63617.7	2.17267
$951$	20461.8i	0.697709i
$952$	−17100.5	−0.582174
$953$	25631.0	0.871218	0.435609	−	0.900136i	$-0.356533\pi$
$953$	25631.0	0.871218	0.435609	+	0.900136i	$0.356533\pi$
$954$	− 12572.9i	− 0.426692i
$955$	41391.0i	1.40249i
$956$	− 57778.9i	− 1.95471i
$957$	− 185.114i	− 0.00625274i
$958$	−39917.1	−1.34620
$959$	−31339.0	−1.05525
$960$	− 9753.03i	− 0.327893i
$961$	28364.1	0.952104
$962$	0	0
$963$	15767.7	0.527629
$964$	− 99547.7i	− 3.32595i
$965$	−14683.1	−0.489811
$966$	−33161.9	−1.10452
$967$	− 11185.6i	− 0.371981i	−0.982552	−	0.185991i	$-0.940451\pi$
$967$	− 11185.6i	− 0.371981i	0.982552	−	0.185991i	$-0.0595494\pi$
$968$	28881.6i	0.958977i
$969$	3142.05i	0.104166i
$970$	113164.i	3.74586i
$971$	−23541.3	−0.778039	−0.389019	−	0.921230i	$-0.627186\pi$
$971$	−23541.3	−0.778039	−0.389019	+	0.921230i	$0.627186\pi$
$972$	−4241.74	−0.139973
$973$	22303.6i	0.734860i
$974$	−55321.3	−1.81993
$975$	0	0
$976$	20796.3	0.682041
$977$	24198.8i	0.792415i	0.918161	+	0.396207i	$0.129674\pi$
$977$	24198.8i	0.792415i	−0.918161	+	0.396207i	$0.870326\pi$
$978$	−53358.4	−1.74459
$979$	−17275.4	−0.563966
$980$	36865.2i	1.20165i
$981$	8704.10i	0.283283i
$982$	− 46111.5i	− 1.49845i
$983$	33757.4i	1.09532i	0.836702	+	0.547658i	$0.184480\pi$
$983$	33757.4i	1.09532i	−0.836702	+	0.547658i	$0.815520\pi$
$984$	839.874	0.0272096
$985$	−34222.0	−1.10701
$986$	268.516i	0.00867272i
$987$	9707.93	0.313077
$988$	0	0
$989$	51251.1	1.64782
$990$	24607.0i	0.789963i
$991$	25298.5	0.810933	0.405466	−	0.914110i	$-0.367109\pi$
$991$	25298.5	0.810933	0.405466	+	0.914110i	$0.367109\pi$
$992$	−4842.89	−0.155002
$993$	15038.9i	0.480608i
$994$	− 64362.2i	− 2.05377i
$995$	− 37033.7i	− 1.17995i
$996$	− 7407.36i	− 0.235654i
$997$	−28766.1	−0.913773	−0.456886	−	0.889525i	$-0.651035\pi$
$997$	−28766.1	−0.913773	−0.456886	+	0.889525i	$0.651035\pi$
$998$	−63458.2	−2.01276
$999$	− 8473.67i	− 0.268363i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.i.337.2		10
13.3	even	3		39.4.j.c.4.5	✓	10
13.4	even	6		39.4.j.c.10.5	yes	10
13.5	odd	4		507.4.a.r.1.2		10
13.8	odd	4		507.4.a.r.1.9		10
13.12	even	2	inner	507.4.b.i.337.9		10
39.5	even	4		1521.4.a.bk.1.9		10
39.8	even	4		1521.4.a.bk.1.2		10
39.17	odd	6		117.4.q.e.10.1		10
39.29	odd	6		117.4.q.e.82.1		10
52.3	odd	6		624.4.bv.h.433.1		10
52.43	odd	6		624.4.bv.h.49.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.c.4.5	✓	10	13.3	even	3
39.4.j.c.10.5	yes	10	13.4	even	6
117.4.q.e.10.1		10	39.17	odd	6
117.4.q.e.82.1		10	39.29	odd	6
507.4.a.r.1.2		10	13.5	odd	4
507.4.a.r.1.9		10	13.8	odd	4
507.4.b.i.337.2		10	1.1	even	1	trivial
507.4.b.i.337.9		10	13.12	even	2	inner
624.4.bv.h.49.5		10	52.43	odd	6
624.4.bv.h.433.1		10	52.3	odd	6
1521.4.a.bk.1.2		10	39.8	even	4
1521.4.a.bk.1.9		10	39.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.04537i q^{2} +3.00000 q^{3} -17.4557 q^{4} -20.1174i q^{5} -15.1361i q^{6} -15.4279i q^{7} +47.7076i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-5.04537i q^{2} +3.00000 q^{3} -17.4557 q^{4} -20.1174i q^{5} -15.1361i q^{6} -15.4279i q^{7} +47.7076i q^{8} +9.00000 q^{9} -101.500 q^{10} -26.9372i q^{11} -52.3672 q^{12} -77.8394 q^{14} -60.3522i q^{15} +101.057 q^{16} -23.2334 q^{17} -45.4083i q^{18} -45.0794i q^{19} +351.164i q^{20} -46.2837i q^{21} -135.908 q^{22} +142.010 q^{23} +143.123i q^{24} -279.710 q^{25} +27.0000 q^{27} +269.305i q^{28} +2.29068 q^{29} -304.499 q^{30} -37.7740i q^{31} -128.207i q^{32} -80.8116i q^{33} +117.221i q^{34} -310.369 q^{35} -157.102 q^{36} -313.840i q^{37} -227.442 q^{38} +959.753 q^{40} -5.86820i q^{41} -233.518 q^{42} +360.898 q^{43} +470.209i q^{44} -181.057i q^{45} -716.493i q^{46} +209.748i q^{47} +303.170 q^{48} +104.980 q^{49} +1411.24i q^{50} -69.7003 q^{51} +276.886 q^{53} -136.225i q^{54} -541.906 q^{55} +736.028 q^{56} -135.238i q^{57} -11.5573i q^{58} +543.189i q^{59} +1053.49i q^{60} +205.788 q^{61} -190.583 q^{62} -138.851i q^{63} +161.602 q^{64} -407.724 q^{66} +492.578i q^{67} +405.557 q^{68} +426.030 q^{69} +1565.93i q^{70} +826.859i q^{71} +429.369i q^{72} -66.1205i q^{73} -1583.44 q^{74} -839.129 q^{75} +786.894i q^{76} -415.584 q^{77} +317.642 q^{79} -2033.00i q^{80} +81.0000 q^{81} -29.6072 q^{82} +141.450i q^{83} +807.915i q^{84} +467.396i q^{85} -1820.86i q^{86} +6.87204 q^{87} +1285.11 q^{88} -641.320i q^{89} -913.497 q^{90} -2478.89 q^{92} -113.322i q^{93} +1058.26 q^{94} -906.880 q^{95} -384.621i q^{96} -1114.92i q^{97} -529.663i q^{98} -242.435i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9}+O(q^{10}) \) 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9	\( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36} - 1380 q^{38} + 2000 q^{40} - 180 q^{42} + 740 q^{43} + 1500 q^{48} - 1550 q^{49} - 630 q^{51} + 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} - 3140 q^{64} + 1740 q^{66} + 1200 q^{68} + 360 q^{69} - 4380 q^{74} - 2880 q^{75} - 4320 q^{77} + 1100 q^{79} + 810 q^{81} + 4780 q^{82} + 2970 q^{87} - 6340 q^{88} - 720 q^{90} - 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 - 80 * q^10 - 180 * q^12 - 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 - 960 * q^25 + 270 * q^27 + 990 * q^29 - 240 * q^30 - 120 * q^35 - 540 * q^36 - 1380 * q^38 + 2000 * q^40 - 180 * q^42 + 740 * q^43 + 1500 * q^48 - 1550 * q^49 - 630 * q^51 + 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 - 3140 * q^64 + 1740 * q^66 + 1200 * q^68 + 360 * q^69 - 4380 * q^74 - 2880 * q^75 - 4320 * q^77 + 1100 * q^79 + 810 * q^81 + 4780 * q^82 + 2970 * q^87 - 6340 * q^88 - 720 * q^90 - 1740 * q^92 + 6460 * q^94 + 2760 * q^95

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