LMFDB - Embedded newform 507.4.b.f.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:	$4$
Coefficient field:	$\Q(i, \sqrt{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 49 $ x^4 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$-1.87083 - 1.87083i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.f.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.74166i q^{2} -3.00000 q^{3} +0.483315 q^{4} +19.4833i q^{5} +8.22497i q^{6} -7.48331i q^{7} -23.2583i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.74166i q^{2} -3.00000 q^{3} +0.483315 q^{4} +19.4833i q^{5} +8.22497i q^{6} -7.48331i q^{7} -23.2583i q^{8} +9.00000 q^{9} +53.4166 q^{10} -22.8999i q^{11} -1.44994 q^{12} -20.5167 q^{14} -58.4499i q^{15} -59.8999 q^{16} -67.0334 q^{17} -24.6749i q^{18} +16.5167i q^{19} +9.41657i q^{20} +22.4499i q^{21} -62.7836 q^{22} +175.600 q^{23} +69.7750i q^{24} -254.600 q^{25} -27.0000 q^{27} -3.61680i q^{28} +291.800 q^{29} -160.250 q^{30} +117.283i q^{31} -21.8418i q^{32} +68.6997i q^{33} +183.783i q^{34} +145.800 q^{35} +4.34983 q^{36} +154.766i q^{37} +45.2831 q^{38} +453.150 q^{40} -251.716i q^{41} +61.5501 q^{42} +502.566 q^{43} -11.0679i q^{44} +175.350i q^{45} -481.434i q^{46} +281.733i q^{47} +179.700 q^{48} +287.000 q^{49} +698.025i q^{50} +201.100 q^{51} +366.999 q^{53} +74.0247i q^{54} +446.166 q^{55} -174.049 q^{56} -49.5501i q^{57} -800.015i q^{58} +79.6663i q^{59} -28.2497i q^{60} -194.865 q^{61} +321.550 q^{62} -67.3498i q^{63} -539.082 q^{64} +188.351 q^{66} +400.082i q^{67} -32.3982 q^{68} -526.799 q^{69} -399.733i q^{70} +528.299i q^{71} -209.325i q^{72} +734.366i q^{73} +424.316 q^{74} +763.799 q^{75} +7.98276i q^{76} -171.367 q^{77} +113.266 q^{79} -1167.05i q^{80} +81.0000 q^{81} -690.118 q^{82} -933.466i q^{83} +10.8504i q^{84} -1306.03i q^{85} -1377.86i q^{86} -875.399 q^{87} -532.613 q^{88} -1190.91i q^{89} +480.749 q^{90} +84.8699 q^{92} -351.849i q^{93} +772.415 q^{94} -321.800 q^{95} +65.5253i q^{96} +557.165i q^{97} -786.856i q^{98} -206.099i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 28 q^{4} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 - 28 * q^4 + 36 * q^9	$ 4 q - 12 q^{3} - 28 q^{4} + 36 q^{9} + 64 q^{10} + 84 q^{12} - 112 q^{14} - 60 q^{16} - 328 q^{17} - 760 q^{22} - 16 q^{23} - 300 q^{25} - 108 q^{27} + 808 q^{29} - 192 q^{30} + 224 q^{35} - 252 q^{36} - 208 q^{38} + 1184 q^{40} + 336 q^{42} + 1232 q^{43} + 180 q^{48} + 1148 q^{49} + 984 q^{51} - 328 q^{53} + 288 q^{55} + 112 q^{56} + 1256 q^{61} + 1376 q^{62} + 388 q^{64} + 2280 q^{66} + 2744 q^{68} + 48 q^{69} + 1368 q^{74} + 900 q^{75} - 1344 q^{77} - 864 q^{79} + 324 q^{81} - 5664 q^{82} - 2424 q^{87} + 3048 q^{88} + 576 q^{90} + 5488 q^{92} + 1144 q^{94} - 928 q^{95}+O(q^{100}) $ 4 * q - 12 * q^3 - 28 * q^4 + 36 * q^9 + 64 * q^10 + 84 * q^12 - 112 * q^14 - 60 * q^16 - 328 * q^17 - 760 * q^22 - 16 * q^23 - 300 * q^25 - 108 * q^27 + 808 * q^29 - 192 * q^30 + 224 * q^35 - 252 * q^36 - 208 * q^38 + 1184 * q^40 + 336 * q^42 + 1232 * q^43 + 180 * q^48 + 1148 * q^49 + 984 * q^51 - 328 * q^53 + 288 * q^55 + 112 * q^56 + 1256 * q^61 + 1376 * q^62 + 388 * q^64 + 2280 * q^66 + 2744 * q^68 + 48 * q^69 + 1368 * q^74 + 900 * q^75 - 1344 * q^77 - 864 * q^79 + 324 * q^81 - 5664 * q^82 - 2424 * q^87 + 3048 * q^88 + 576 * q^90 + 5488 * q^92 + 1144 * q^94 - 928 * 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 $

$2$	− 2.74166i	− 0.969322i	−0.874702	−	0.484661i	$-0.838943\pi$
$2$	− 2.74166i	− 0.969322i	0.874702	−	0.484661i	$-0.161057\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0.483315	0.0604143
$5$	19.4833i	1.74264i	0.490715	+	0.871320i	$0.336736\pi$
$5$	19.4833i	1.74264i	−0.490715	+	0.871320i	$0.663264\pi$
$6$	8.22497i	0.559638i
$7$	− 7.48331i	− 0.404061i	−0.979379	−	0.202031i	$-0.935246\pi$
$7$	− 7.48331i	− 0.404061i	0.979379	−	0.202031i	$-0.0647540\pi$
$8$	− 23.2583i	− 1.02788i
$9$	9.00000	0.333333
$10$	53.4166	1.68918
$11$	− 22.8999i	− 0.627689i	−0.949474	−	0.313844i	$-0.898383\pi$
$11$	− 22.8999i	− 0.627689i	0.949474	−	0.313844i	$-0.101617\pi$
$12$	−1.44994	−0.0348802
$13$	0	0
$13$	0	0
$14$	−20.5167	−0.391665
$15$	− 58.4499i	− 1.00611i
$16$	−59.8999	−0.935936
$17$	−67.0334	−0.956352	−0.478176	−	0.878264i	$-0.658702\pi$
$17$	−67.0334	−0.956352	−0.478176	+	0.878264i	$0.658702\pi$
$18$	− 24.6749i	− 0.323107i
$19$	16.5167i	0.199431i	0.995016	+	0.0997155i	$0.0317933\pi$
$19$	16.5167i	0.199431i	−0.995016	+	0.0997155i	$0.968207\pi$
$20$	9.41657i	0.105280i
$21$	22.4499i	0.233285i
$22$	−62.7836	−0.608433
$23$	175.600	1.59196	0.795979	−	0.605324i	$-0.206956\pi$
$23$	175.600	1.59196	0.795979	+	0.605324i	$0.206956\pi$
$24$	69.7750i	0.593449i
$25$	−254.600	−2.03680
$26$	0	0
$27$	−27.0000	−0.192450
$28$	− 3.61680i	− 0.0244111i
$29$	291.800	1.86848	0.934239	−	0.356648i	$-0.116080\pi$
$29$	291.800	1.86848	0.934239	+	0.356648i	$0.116080\pi$
$30$	−160.250	−0.975249
$31$	117.283i	0.679505i	0.940515	+	0.339753i	$0.110343\pi$
$31$	117.283i	0.679505i	−0.940515	+	0.339753i	$0.889657\pi$
$32$	− 21.8418i	− 0.120660i
$33$	68.6997i	0.362396i
$34$	183.783i	0.927013i
$35$	145.800	0.704133
$36$	4.34983	0.0201381
$37$	154.766i	0.687661i	0.939032	+	0.343830i	$0.111724\pi$
$37$	154.766i	0.687661i	−0.939032	+	0.343830i	$0.888276\pi$
$38$	45.2831	0.193313
$39$	0	0
$40$	453.150	1.79123
$41$	− 251.716i	− 0.958815i	−0.877592	−	0.479407i	$-0.840852\pi$
$41$	− 251.716i	− 0.958815i	0.877592	−	0.479407i	$-0.159148\pi$
$42$	61.5501	0.226128
$43$	502.566	1.78234	0.891170	−	0.453669i	$-0.149885\pi$
$43$	502.566	1.78234	0.891170	+	0.453669i	$0.149885\pi$
$44$	− 11.0679i	− 0.0379214i
$45$	175.350i	0.580880i
$46$	− 481.434i	− 1.54312i
$47$	281.733i	0.874361i	0.899374	+	0.437181i	$0.144023\pi$
$47$	281.733i	0.874361i	−0.899374	+	0.437181i	$0.855977\pi$
$48$	179.700	0.540363
$49$	287.000	0.836735
$50$	698.025i	1.97431i
$51$	201.100	0.552150
$52$	0	0
$53$	366.999	0.951154	0.475577	−	0.879674i	$-0.342239\pi$
$53$	366.999	0.951154	0.475577	+	0.879674i	$0.342239\pi$
$54$	74.0247i	0.186546i
$55$	446.166	1.09384
$56$	−174.049	−0.415328
$57$	− 49.5501i	− 0.115141i
$58$	− 800.015i	− 1.81116i
$59$	79.6663i	0.175791i	0.996130	+	0.0878955i	$0.0280142\pi$
$59$	79.6663i	0.175791i	−0.996130	+	0.0878955i	$0.971986\pi$
$60$	− 28.2497i	− 0.0607837i
$61$	−194.865	−0.409016	−0.204508	−	0.978865i	$-0.565559\pi$
$61$	−194.865	−0.409016	−0.204508	+	0.978865i	$0.565559\pi$
$62$	321.550	0.658660
$63$	− 67.3498i	− 0.134687i
$64$	−539.082	−1.05289
$65$	0	0
$66$	188.351	0.351279
$67$	400.082i	0.729519i	0.931102	+	0.364759i	$0.118849\pi$
$67$	400.082i	0.729519i	−0.931102	+	0.364759i	$0.881151\pi$
$68$	−32.3982	−0.0577774
$69$	−526.799	−0.919117
$70$	− 399.733i	− 0.682532i
$71$	528.299i	0.883065i	0.897245	+	0.441532i	$0.145565\pi$
$71$	528.299i	0.883065i	−0.897245	+	0.441532i	$0.854435\pi$
$72$	− 209.325i	− 0.342628i
$73$	734.366i	1.17741i	0.808347	+	0.588706i	$0.200362\pi$
$73$	734.366i	1.17741i	−0.808347	+	0.588706i	$0.799638\pi$
$74$	424.316	0.666565
$75$	763.799	1.17594
$76$	7.98276i	0.0120485i
$77$	−171.367	−0.253625
$78$	0	0
$79$	113.266	0.161309	0.0806545	−	0.996742i	$-0.474299\pi$
$79$	113.266	0.161309	0.0806545	+	0.996742i	$0.474299\pi$
$80$	− 1167.05i	− 1.63100i
$81$	81.0000	0.111111
$82$	−690.118	−0.929400
$83$	− 933.466i	− 1.23447i	−0.786778	−	0.617236i	$-0.788252\pi$
$83$	− 933.466i	− 1.23447i	0.786778	−	0.617236i	$-0.211748\pi$
$84$	10.8504i	0.0140937i
$85$	− 1306.03i	− 1.66658i
$86$	− 1377.86i	− 1.72766i
$87$	−875.399	−1.07877
$88$	−532.613	−0.645191
$89$	− 1190.91i	− 1.41839i	−0.705012	−	0.709195i	$-0.749059\pi$
$89$	− 1190.91i	− 1.41839i	0.705012	−	0.709195i	$-0.250941\pi$
$90$	480.749	0.563060
$91$	0	0
$92$	84.8699	0.0961771
$93$	− 351.849i	− 0.392313i
$94$	772.415	0.847538
$95$	−321.800	−0.347536
$96$	65.5253i	0.0696630i
$97$	557.165i	0.583211i	0.956539	+	0.291606i	$0.0941895\pi$
$97$	557.165i	0.583211i	−0.956539	+	0.291606i	$0.905811\pi$
$98$	− 786.856i	− 0.811066i
$99$	− 206.099i	− 0.209230i
$100$	−123.052	−0.123052
$101$	286.766	0.282518	0.141259	−	0.989973i	$-0.454885\pi$
$101$	286.766	0.282518	0.141259	+	0.989973i	$0.454885\pi$
$102$	− 551.348i	− 0.535211i
$103$	1911.36	1.82847	0.914234	−	0.405187i	$-0.132794\pi$
$103$	1911.36	1.82847	0.914234	+	0.405187i	$0.132794\pi$
$104$	0	0
$105$	−437.399	−0.406531
$106$	− 1006.19i	− 0.921975i
$107$	834.334	0.753814	0.376907	−	0.926251i	$-0.376988\pi$
$107$	834.334	0.753814	0.376907	+	0.926251i	$0.376988\pi$
$108$	−13.0495	−0.0116267
$109$	− 1077.66i	− 0.946986i	−0.880798	−	0.473493i	$-0.842993\pi$
$109$	− 1077.66i	− 0.946986i	0.880798	−	0.473493i	$-0.157007\pi$
$110$	− 1223.23i	− 1.06028i
$111$	− 464.299i	− 0.397021i
$112$	448.250i	0.378175i
$113$	−166.065	−0.138248	−0.0691241	−	0.997608i	$-0.522020\pi$
$113$	−166.065	−0.138248	−0.0691241	+	0.997608i	$0.522020\pi$
$114$	−135.849	−0.111609
$115$	3421.26i	2.77421i
$116$	141.031	0.112883
$117$	0	0
$118$	218.418	0.170398
$119$	501.632i	0.386424i
$120$	−1359.45	−1.03417
$121$	806.595	0.606007
$122$	534.254i	0.396468i
$123$	755.147i	0.553572i
$124$	56.6847i	0.0410519i
$125$	− 2525.03i	− 1.80676i
$126$	−184.650	−0.130555
$127$	−1296.16	−0.905637	−0.452819	−	0.891603i	$-0.649581\pi$
$127$	−1296.16	−0.905637	−0.452819	+	0.891603i	$0.649581\pi$
$128$	1303.24i	0.899934i
$129$	−1507.70	−1.02903
$130$	0	0
$131$	−197.201	−0.131523	−0.0657617	−	0.997835i	$-0.520948\pi$
$131$	−197.201	−0.131523	−0.0657617	+	0.997835i	$0.520948\pi$
$132$	33.2036i	0.0218939i
$133$	123.600	0.0805823
$134$	1096.89	0.707139
$135$	− 526.049i	− 0.335371i
$136$	1559.09i	0.983018i
$137$	546.915i	0.341066i	0.985352	+	0.170533i	$0.0545490\pi$
$137$	546.915i	0.341066i	−0.985352	+	0.170533i	$0.945451\pi$
$138$	1444.30i	0.890921i
$139$	609.666	0.372023	0.186012	−	0.982548i	$-0.440444\pi$
$139$	609.666	0.372023	0.186012	+	0.982548i	$0.440444\pi$
$140$	70.4672	0.0425397
$141$	− 845.199i	− 0.504813i
$142$	1448.42	0.855974
$143$	0	0
$144$	−539.099	−0.311979
$145$	5685.23i	3.25609i
$146$	2013.38	1.14129
$147$	−861.000	−0.483089
$148$	74.8009i	0.0415446i
$149$	− 2165.08i	− 1.19040i	−0.803576	−	0.595202i	$-0.797072\pi$
$149$	− 2165.08i	− 1.19040i	0.803576	−	0.595202i	$-0.202928\pi$
$150$	− 2094.07i	− 1.13987i
$151$	846.549i	0.456233i	0.973634	+	0.228116i	$0.0732567\pi$
$151$	846.549i	0.456233i	−0.973634	+	0.228116i	$0.926743\pi$
$152$	384.151	0.204992
$153$	−603.300	−0.318784
$154$	469.830i	0.245844i
$155$	−2285.06	−1.18413
$156$	0	0
$157$	1653.60	0.840581	0.420291	−	0.907390i	$-0.361928\pi$
$157$	1653.60	0.840581	0.420291	+	0.907390i	$0.361928\pi$
$158$	− 310.536i	− 0.156360i
$159$	−1101.00	−0.549149
$160$	425.550	0.210267
$161$	− 1314.07i	− 0.643248i
$162$	− 222.074i	− 0.107702i
$163$	2866.51i	1.37744i	0.725027	+	0.688720i	$0.241827\pi$
$163$	2866.51i	1.37744i	−0.725027	+	0.688720i	$0.758173\pi$
$164$	− 121.658i	− 0.0579262i
$165$	−1338.50	−0.631526
$166$	−2559.24	−1.19660
$167$	− 729.066i	− 0.337825i	−0.985631	−	0.168913i	$-0.945974\pi$
$167$	− 729.066i	− 0.337825i	0.985631	−	0.168913i	$-0.0540255\pi$
$168$	522.148	0.239789
$169$	0	0
$170$	−3580.69	−1.61545
$171$	148.650i	0.0664770i
$172$	242.898	0.107679
$173$	3834.83	1.68530	0.842650	−	0.538462i	$-0.180995\pi$
$173$	3834.83	1.68530	0.842650	+	0.538462i	$0.180995\pi$
$174$	2400.05i	1.04567i
$175$	1905.25i	0.822990i
$176$	1371.70i	0.587476i
$177$	− 238.999i	− 0.101493i
$178$	−3265.08	−1.37488
$179$	283.862	0.118530	0.0592649	−	0.998242i	$-0.481124\pi$
$179$	283.862	0.118530	0.0592649	+	0.998242i	$0.481124\pi$
$180$	84.7492i	0.0350935i
$181$	−2363.60	−0.970634	−0.485317	−	0.874338i	$-0.661296\pi$
$181$	−2363.60	−0.970634	−0.485317	+	0.874338i	$0.661296\pi$
$182$	0	0
$183$	584.596	0.236145
$184$	− 4084.15i	− 1.63635i
$185$	−3015.36	−1.19835
$186$	−964.650	−0.380277
$187$	1535.06i	0.600291i
$188$	136.166i	0.0528240i
$189$	202.049i	0.0777616i
$190$	882.265i	0.336875i
$191$	2514.26	0.952491	0.476246	−	0.879312i	$-0.341997\pi$
$191$	2514.26	0.952491	0.476246	+	0.879312i	$0.341997\pi$
$192$	1617.25	0.607889
$193$	− 2420.73i	− 0.902839i	−0.892312	−	0.451420i	$-0.850918\pi$
$193$	− 2420.73i	− 0.902839i	0.892312	−	0.451420i	$-0.149082\pi$
$194$	1527.55	0.565320
$195$	0	0
$196$	138.711	0.0505508
$197$	− 4633.65i	− 1.67581i	−0.545819	−	0.837903i	$-0.683781\pi$
$197$	− 4633.65i	− 1.67581i	0.545819	−	0.837903i	$-0.316219\pi$
$198$	−565.053	−0.202811
$199$	−3054.17	−1.08796	−0.543980	−	0.839098i	$-0.683083\pi$
$199$	−3054.17	−1.08796	−0.543980	+	0.839098i	$0.683083\pi$
$200$	5921.56i	2.09359i
$201$	− 1200.25i	− 0.421188i
$202$	− 786.215i	− 0.273851i
$203$	− 2183.63i	− 0.754979i
$204$	97.1947	0.0333578
$205$	4904.26	1.67087
$206$	− 5240.30i	− 1.77237i
$207$	1580.40	0.530653
$208$	0	0
$209$	378.230	0.125181
$210$	1199.20i	0.394060i
$211$	−4031.60	−1.31539	−0.657694	−	0.753285i	$-0.728468\pi$
$211$	−4031.60	−1.31539	−0.657694	+	0.753285i	$0.728468\pi$
$212$	177.376	0.0574634
$213$	− 1584.90i	− 0.509838i
$214$	− 2287.46i	− 0.730689i
$215$	9791.66i	3.10598i
$216$	627.975i	0.197816i
$217$	877.666	0.274562
$218$	−2954.59	−0.917935
$219$	− 2203.10i	− 0.679779i
$220$	215.638	0.0660834
$221$	0	0
$222$	−1272.95	−0.384841
$223$	3784.95i	1.13659i	0.822826	+	0.568294i	$0.192396\pi$
$223$	3784.95i	1.13659i	−0.822826	+	0.568294i	$0.807604\pi$
$224$	−163.449	−0.0487539
$225$	−2291.40	−0.678932
$226$	455.292i	0.134007i
$227$	2013.83i	0.588821i	0.955679	+	0.294411i	$0.0951233\pi$
$227$	2013.83i	0.588821i	−0.955679	+	0.294411i	$0.904877\pi$
$228$	− 23.9483i	− 0.00695620i
$229$	3050.73i	0.880340i	0.897915	+	0.440170i	$0.145082\pi$
$229$	3050.73i	0.880340i	−0.897915	+	0.440170i	$0.854918\pi$
$230$	9379.93	2.68910
$231$	514.101	0.146430
$232$	− 6786.78i	− 1.92058i
$233$	−5587.49	−1.57103	−0.785513	−	0.618846i	$-0.787601\pi$
$233$	−5587.49	−1.57103	−0.785513	+	0.618846i	$0.787601\pi$
$234$	0	0
$235$	−5489.09	−1.52370
$236$	38.5039i	0.0106203i
$237$	−339.798	−0.0931317
$238$	1375.30	0.374570
$239$	− 1335.69i	− 0.361501i	−0.983529	−	0.180750i	$-0.942147\pi$
$239$	− 1335.69i	− 0.361501i	0.983529	−	0.180750i	$-0.0578526\pi$
$240$	3501.15i	0.941658i
$241$	571.558i	0.152769i	0.997078	+	0.0763845i	$0.0243376\pi$
$241$	571.558i	0.152769i	−0.997078	+	0.0763845i	$0.975662\pi$
$242$	− 2211.41i	− 0.587416i
$243$	−243.000	−0.0641500
$244$	−94.1813	−0.0247104
$245$	5591.71i	1.45813i
$246$	2070.36	0.536590
$247$	0	0
$248$	2727.81	0.698452
$249$	2800.40i	0.712723i
$250$	−6922.76	−1.75134
$251$	−4088.60	−1.02817	−0.514084	−	0.857740i	$-0.671868\pi$
$251$	−4088.60	−1.02817	−0.514084	+	0.857740i	$0.671868\pi$
$252$	− 32.5512i	− 0.00813703i
$253$	− 4021.21i	− 0.999254i
$254$	3553.64i	0.877854i
$255$	3918.10i	0.962199i
$256$	−739.607	−0.180568
$257$	−3050.23	−0.740342	−0.370171	−	0.928964i	$-0.620701\pi$
$257$	−3050.23	−0.740342	−0.370171	+	0.928964i	$0.620701\pi$
$258$	4133.59i	0.997466i
$259$	1158.17	0.277857
$260$	0	0
$261$	2626.20	0.622826
$262$	540.659i	0.127489i
$263$	5770.99	1.35306	0.676530	−	0.736415i	$-0.263483\pi$
$263$	5770.99	1.35306	0.676530	+	0.736415i	$0.263483\pi$
$264$	1597.84	0.372501
$265$	7150.35i	1.65752i
$266$	− 338.868i	− 0.0781102i
$267$	3572.74i	0.818908i
$268$	193.365i	0.0440734i
$269$	−2079.40	−0.471314	−0.235657	−	0.971836i	$-0.575724\pi$
$269$	−2079.40	−0.471314	−0.235657	+	0.971836i	$0.575724\pi$
$270$	−1442.25	−0.325083
$271$	− 6012.00i	− 1.34761i	−0.738908	−	0.673807i	$-0.764658\pi$
$271$	− 6012.00i	− 1.34761i	0.738908	−	0.673807i	$-0.235342\pi$
$272$	4015.29	0.895084
$273$	0	0
$274$	1499.45	0.330603
$275$	5830.30i	1.27847i
$276$	−254.610	−0.0555279
$277$	735.201	0.159473	0.0797364	−	0.996816i	$-0.474592\pi$
$277$	735.201	0.159473	0.0797364	+	0.996816i	$0.474592\pi$
$278$	− 1671.50i	− 0.360610i
$279$	1055.55i	0.226502i
$280$	− 3391.06i	− 0.723767i
$281$	1902.92i	0.403981i	0.979387	+	0.201990i	$0.0647410\pi$
$281$	1902.92i	0.403981i	−0.979387	+	0.201990i	$0.935259\pi$
$282$	−2317.25	−0.489326
$283$	−2125.71	−0.446502	−0.223251	−	0.974761i	$-0.571667\pi$
$283$	−2125.71	−0.446502	−0.223251	+	0.974761i	$0.571667\pi$
$284$	255.335i	0.0533498i
$285$	965.399	0.200650
$286$	0	0
$287$	−1883.67	−0.387420
$288$	− 196.576i	− 0.0402200i
$289$	−419.527	−0.0853913
$290$	15586.9	3.15620
$291$	− 1671.49i	− 0.336717i
$292$	354.930i	0.0711325i
$293$	1641.03i	0.327200i	0.986527	+	0.163600i	$0.0523107\pi$
$293$	1641.03i	0.327200i	−0.986527	+	0.163600i	$0.947689\pi$
$294$	2360.57i	0.468269i
$295$	−1552.16	−0.306341
$296$	3599.61	0.706835
$297$	618.297i	0.120799i
$298$	−5935.91	−1.15389
$299$	0	0
$300$	369.155	0.0710439
$301$	− 3760.86i	− 0.720174i
$302$	2320.95	0.442237
$303$	−860.299	−0.163112
$304$	− 989.348i	− 0.186655i
$305$	− 3796.62i	− 0.712767i
$306$	1654.04i	0.309004i
$307$	3373.27i	0.627111i	0.949570	+	0.313555i	$0.101520\pi$
$307$	3373.27i	0.627111i	−0.949570	+	0.313555i	$0.898480\pi$
$308$	−82.8242	−0.0153226
$309$	−5734.09	−1.05567
$310$	6264.86i	1.14781i
$311$	868.525	0.158359	0.0791793	−	0.996860i	$-0.474770\pi$
$311$	868.525	0.158359	0.0791793	+	0.996860i	$0.474770\pi$
$312$	0	0
$313$	−4343.19	−0.784319	−0.392159	−	0.919897i	$-0.628272\pi$
$313$	−4343.19	−0.784319	−0.392159	+	0.919897i	$0.628272\pi$
$314$	− 4533.59i	− 0.814794i
$315$	1312.20	0.234711
$316$	54.7431	0.00974537
$317$	− 3277.65i	− 0.580730i	−0.956916	−	0.290365i	$-0.906223\pi$
$317$	− 3277.65i	− 0.580730i	0.956916	−	0.290365i	$-0.0937767\pi$
$318$	3018.56i	0.532302i
$319$	− 6682.18i	− 1.17282i
$320$	− 10503.1i	− 1.83482i
$321$	−2503.00	−0.435215
$322$	−3602.72	−0.623515
$323$	− 1107.17i	− 0.190726i
$324$	39.1485	0.00671271
$325$	0	0
$326$	7859.00	1.33518
$327$	3232.99i	0.546743i
$328$	−5854.49	−0.985550
$329$	2108.30	0.353295
$330$	3669.70i	0.612153i
$331$	5589.62i	0.928197i	0.885784	+	0.464099i	$0.153622\pi$
$331$	5589.62i	0.928197i	−0.885784	+	0.464099i	$0.846378\pi$
$332$	− 451.158i	− 0.0745798i
$333$	1392.90i	0.229220i
$334$	−1998.85	−0.327461
$335$	−7794.92	−1.27129
$336$	− 1344.75i	− 0.218340i
$337$	−901.544	−0.145728	−0.0728638	−	0.997342i	$-0.523214\pi$
$337$	−901.544	−0.145728	−0.0728638	+	0.997342i	$0.523214\pi$
$338$	0	0
$339$	498.194	0.0798176
$340$	− 631.225i	− 0.100685i
$341$	2685.77	0.426518
$342$	407.548	0.0644376
$343$	− 4714.49i	− 0.742153i
$344$	− 11688.9i	− 1.83204i
$345$	− 10263.8i	− 1.60169i
$346$	− 10513.8i	− 1.63360i
$347$	−812.318	−0.125670	−0.0628350	−	0.998024i	$-0.520014\pi$
$347$	−812.318	−0.125670	−0.0628350	+	0.998024i	$0.520014\pi$
$348$	−423.093	−0.0651730
$349$	− 4437.96i	− 0.680683i	−0.940302	−	0.340342i	$-0.889457\pi$
$349$	− 4437.96i	− 0.680683i	0.940302	−	0.340342i	$-0.110543\pi$
$350$	5223.54	0.797743
$351$	0	0
$352$	−500.174	−0.0757368
$353$	7115.35i	1.07284i	0.843952	+	0.536419i	$0.180223\pi$
$353$	7115.35i	1.07284i	−0.843952	+	0.536419i	$0.819777\pi$
$354$	−655.253	−0.0983794
$355$	−10293.0	−1.53886
$356$	− 575.587i	− 0.0856911i
$357$	− 1504.90i	− 0.223102i
$358$	− 778.253i	− 0.114894i
$359$	− 4693.98i	− 0.690081i	−0.938588	−	0.345040i	$-0.887865\pi$
$359$	− 4693.98i	− 0.690081i	0.938588	−	0.345040i	$-0.112135\pi$
$360$	4078.35	0.597077
$361$	6586.20	0.960227
$362$	6480.17i	0.940857i
$363$	−2419.79	−0.349878
$364$	0	0
$365$	−14307.9	−2.05181
$366$	− 1602.76i	− 0.228901i
$367$	9243.98	1.31480	0.657400	−	0.753542i	$-0.271656\pi$
$367$	9243.98	1.31480	0.657400	+	0.753542i	$0.271656\pi$
$368$	−10518.4	−1.48997
$369$	− 2265.44i	− 0.319605i
$370$	8267.09i	1.16158i
$371$	− 2746.37i	− 0.384324i
$372$	− 170.054i	− 0.0237013i
$373$	−4311.99	−0.598569	−0.299285	−	0.954164i	$-0.596748\pi$
$373$	−4311.99	−0.598569	−0.299285	+	0.954164i	$0.596748\pi$
$374$	4208.60	0.581876
$375$	7575.09i	1.04314i
$376$	6552.64	0.898741
$377$	0	0
$378$	553.951	0.0753760
$379$	− 2382.73i	− 0.322936i	−0.986878	−	0.161468i	$-0.948377\pi$
$379$	− 2382.73i	− 0.322936i	0.986878	−	0.161468i	$-0.0516229\pi$
$380$	−155.531	−0.0209962
$381$	3888.49	0.522870
$382$	− 6893.25i	− 0.923271i
$383$	4845.81i	0.646499i	0.946314	+	0.323250i	$0.104775\pi$
$383$	4845.81i	0.646499i	−0.946314	+	0.323250i	$0.895225\pi$
$384$	− 3909.73i	− 0.519577i
$385$	− 3338.80i	− 0.441976i
$386$	−6636.81	−0.875142
$387$	4523.10	0.594113
$388$	269.286i	0.0352343i
$389$	−9561.50	−1.24624	−0.623120	−	0.782127i	$-0.714135\pi$
$389$	−9561.50	−1.24624	−0.623120	+	0.782127i	$0.714135\pi$
$390$	0	0
$391$	−11771.0	−1.52247
$392$	− 6675.14i	− 0.860066i
$393$	591.604	0.0759350
$394$	−12703.9	−1.62440
$395$	2206.79i	0.281103i
$396$	− 99.6107i	− 0.0126405i
$397$	7440.11i	0.940575i	0.882513	+	0.470287i	$0.155850\pi$
$397$	7440.11i	0.940575i	−0.882513	+	0.470287i	$0.844150\pi$
$398$	8373.48i	1.05458i
$399$	−370.799	−0.0465242
$400$	15250.5	1.90631
$401$	8687.80i	1.08192i	0.841050	+	0.540958i	$0.181938\pi$
$401$	8687.80i	1.08192i	−0.841050	+	0.540958i	$0.818062\pi$
$402$	−3290.66	−0.408267
$403$	0	0
$404$	138.598	0.0170681
$405$	1578.15i	0.193627i
$406$	−5986.76	−0.731818
$407$	3544.13	0.431637
$408$	− 4677.26i	− 0.567546i
$409$	2556.10i	0.309024i	0.987991	+	0.154512i	$0.0493805\pi$
$409$	2556.10i	0.309024i	−0.987991	+	0.154512i	$0.950619\pi$
$410$	− 13445.8i	− 1.61961i
$411$	− 1640.74i	− 0.196915i
$412$	923.790	0.110466
$413$	596.168	0.0710303
$414$	− 4332.90i	− 0.514374i
$415$	18187.0	2.15124
$416$	0	0
$417$	−1829.00	−0.214788
$418$	− 1036.98i	− 0.121340i
$419$	−3347.46	−0.390296	−0.195148	−	0.980774i	$-0.562519\pi$
$419$	−3347.46	−0.390296	−0.195148	+	0.980774i	$0.562519\pi$
$420$	−211.402	−0.0245603
$421$	− 1854.48i	− 0.214684i	−0.994222	−	0.107342i	$-0.965766\pi$
$421$	− 1854.48i	− 0.214684i	0.994222	−	0.107342i	$-0.0342340\pi$
$422$	11053.3i	1.27503i
$423$	2535.60i	0.291454i
$424$	− 8535.79i	− 0.977675i
$425$	17066.7	1.94789
$426$	−4345.25	−0.494197
$427$	1458.24i	0.165267i
$428$	403.246	0.0455412
$429$	0	0
$430$	26845.4	3.01069
$431$	− 14043.1i	− 1.56945i	−0.619844	−	0.784725i	$-0.712804\pi$
$431$	− 14043.1i	− 1.56945i	0.619844	−	0.784725i	$-0.287196\pi$
$432$	1617.30	0.180121
$433$	−3086.47	−0.342555	−0.171278	−	0.985223i	$-0.554790\pi$
$433$	−3086.47	−0.342555	−0.171278	+	0.985223i	$0.554790\pi$
$434$	− 2406.26i	− 0.266139i
$435$	− 17055.7i	− 1.87990i
$436$	− 520.851i	− 0.0572116i
$437$	2900.32i	0.317486i
$438$	−6040.14	−0.658925
$439$	−2837.68	−0.308508	−0.154254	−	0.988031i	$-0.549297\pi$
$439$	−2837.68	−0.308508	−0.154254	+	0.988031i	$0.549297\pi$
$440$	− 10377.1i	− 1.12434i
$441$	2583.00	0.278912
$442$	0	0
$443$	18309.4	1.96367	0.981834	−	0.189744i	$-0.0607658\pi$
$443$	18309.4	1.96367	0.981834	+	0.189744i	$0.0607658\pi$
$444$	− 224.403i	− 0.0239858i
$445$	23203.0	2.47174
$446$	10377.0	1.10172
$447$	6495.24i	0.687281i
$448$	4034.12i	0.425433i
$449$	− 13861.2i	− 1.45690i	−0.685098	−	0.728451i	$-0.740241\pi$
$449$	− 13861.2i	− 1.45690i	0.685098	−	0.728451i	$-0.259759\pi$
$450$	6282.22i	0.658104i
$451$	−5764.26	−0.601837
$452$	−80.2614	−0.00835217
$453$	− 2539.65i	− 0.263406i
$454$	5521.23	0.570758
$455$	0	0
$456$	−1152.45	−0.118352
$457$	− 8990.36i	− 0.920243i	−0.887856	−	0.460122i	$-0.847806\pi$
$457$	− 8990.36i	− 0.920243i	0.887856	−	0.460122i	$-0.152194\pi$
$458$	8364.05	0.853333
$459$	1809.90	0.184050
$460$	1653.55i	0.167602i
$461$	− 3406.90i	− 0.344198i	−0.985080	−	0.172099i	$-0.944945\pi$
$461$	− 3406.90i	− 0.344198i	0.985080	−	0.172099i	$-0.0550549\pi$
$462$	− 1409.49i	− 0.141938i
$463$	7498.45i	0.752662i	0.926485	+	0.376331i	$0.122814\pi$
$463$	7498.45i	0.752662i	−0.926485	+	0.376331i	$0.877186\pi$
$464$	−17478.8	−1.74878
$465$	6855.19	0.683660
$466$	15319.0i	1.52283i
$467$	−7711.38	−0.764112	−0.382056	−	0.924139i	$-0.624784\pi$
$467$	−7711.38	−0.764112	−0.382056	+	0.924139i	$0.624784\pi$
$468$	0	0
$469$	2993.94	0.294770
$470$	15049.2i	1.47695i
$471$	−4960.79	−0.485310
$472$	1852.91	0.180693
$473$	− 11508.7i	− 1.11875i
$474$	931.608i	0.0902747i
$475$	− 4205.14i	− 0.406200i
$476$	242.446i	0.0233456i
$477$	3302.99	0.317051
$478$	−3662.01	−0.350411
$479$	9439.82i	0.900451i	0.892915	+	0.450226i	$0.148656\pi$
$479$	9439.82i	0.900451i	−0.892915	+	0.450226i	$0.851344\pi$
$480$	−1276.65	−0.121398
$481$	0	0
$482$	1567.02	0.148082
$483$	3942.20i	0.371380i
$484$	389.839	0.0366115
$485$	−10855.4	−1.01633
$486$	666.223i	0.0621821i
$487$	− 6156.20i	− 0.572821i	−0.958107	−	0.286411i	$-0.907538\pi$
$487$	− 6156.20i	− 0.572821i	0.958107	−	0.286411i	$-0.0924622\pi$
$488$	4532.25i	0.420420i
$489$	− 8599.54i	− 0.795265i
$490$	15330.6	1.41340
$491$	−3842.74	−0.353198	−0.176599	−	0.984283i	$-0.556510\pi$
$491$	−3842.74	−0.353198	−0.176599	+	0.984283i	$0.556510\pi$
$492$	364.974i	0.0334437i
$493$	−19560.3	−1.78692
$494$	0	0
$495$	4015.49	0.364612
$496$	− 7025.24i	− 0.635973i
$497$	3953.43	0.356812
$498$	7677.73	0.690858
$499$	− 12842.4i	− 1.15211i	−0.817410	−	0.576056i	$-0.804591\pi$
$499$	− 12842.4i	− 1.15211i	0.817410	−	0.576056i	$-0.195409\pi$
$500$	− 1220.38i	− 0.109154i
$501$	2187.20i	0.195043i
$502$	11209.5i	0.996627i
$503$	8580.11	0.760573	0.380287	−	0.924869i	$-0.375825\pi$
$503$	8580.11	0.760573	0.380287	+	0.924869i	$0.375825\pi$
$504$	−1566.45	−0.138443
$505$	5587.16i	0.492327i
$506$	−11024.8	−0.968599
$507$	0	0
$508$	−626.455	−0.0547135
$509$	− 43.5957i	− 0.00379635i	−0.999998	−	0.00189818i	$-0.999396\pi$
$509$	− 43.5957i	− 0.00379635i	0.999998	−	0.00189818i	$-0.000604209\pi$
$510$	10742.1	0.932681
$511$	5495.49	0.475746
$512$	12453.7i	1.07496i
$513$	− 445.951i	− 0.0383805i
$514$	8362.68i	0.717630i
$515$	37239.7i	3.18636i
$516$	−728.693	−0.0621684
$517$	6451.66	0.548827
$518$	− 3175.29i	− 0.269333i
$519$	−11504.5	−0.973008
$520$	0	0
$521$	11368.1	0.955939	0.477969	−	0.878377i	$-0.341373\pi$
$521$	11368.1	0.955939	0.477969	+	0.878377i	$0.341373\pi$
$522$	− 7200.14i	− 0.603719i
$523$	−5229.53	−0.437230	−0.218615	−	0.975811i	$-0.570154\pi$
$523$	−5229.53	−0.437230	−0.218615	+	0.975811i	$0.570154\pi$
$524$	−95.3103	−0.00794590
$525$	− 5715.75i	− 0.475154i
$526$	− 15822.1i	− 1.31155i
$527$	− 7861.88i	− 0.649846i
$528$	− 4115.10i	− 0.339180i
$529$	18668.2	1.53433
$530$	19603.8	1.60667
$531$	716.997i	0.0585970i
$532$	59.7375	0.00486832
$533$	0	0
$534$	9795.24	0.793786
$535$	16255.6i	1.31363i
$536$	9305.24	0.749860
$537$	−851.586	−0.0684333
$538$	5701.01i	0.456855i
$539$	− 6572.27i	− 0.525209i
$540$	− 254.247i	− 0.0202612i
$541$	6567.99i	0.521959i	0.965344	+	0.260980i	$0.0840455\pi$
$541$	6567.99i	0.521959i	−0.965344	+	0.260980i	$0.915954\pi$
$542$	−16482.9	−1.30627
$543$	7090.79	0.560396
$544$	1464.13i	0.115393i
$545$	20996.5	1.65026
$546$	0	0
$547$	−13675.7	−1.06897	−0.534487	−	0.845177i	$-0.679495\pi$
$547$	−13675.7	−1.06897	−0.534487	+	0.845177i	$0.679495\pi$
$548$	264.332i	0.0206053i
$549$	−1753.79	−0.136339
$550$	15984.7	1.23925
$551$	4819.57i	0.372632i
$552$	12252.5i	0.944745i
$553$	− 847.604i	− 0.0651786i
$554$	− 2015.67i	− 0.154581i
$555$	9046.09	0.691865
$556$	294.661	0.0224755
$557$	− 4527.96i	− 0.344445i	−0.985058	−	0.172222i	$-0.944905\pi$
$557$	− 4527.96i	− 0.344445i	0.985058	−	0.172222i	$-0.0550947\pi$
$558$	2893.95	0.219553
$559$	0	0
$560$	−8733.39	−0.659023
$561$	− 4605.17i	− 0.346578i
$562$	5217.15	0.391588
$563$	−18441.8	−1.38051	−0.690256	−	0.723566i	$-0.742502\pi$
$563$	−18441.8	−1.38051	−0.690256	+	0.723566i	$0.742502\pi$
$564$	− 408.497i	− 0.0304979i
$565$	− 3235.49i	− 0.240917i
$566$	5827.96i	0.432804i
$567$	− 606.148i	− 0.0448957i
$568$	12287.4	0.907687
$569$	13553.5	0.998578	0.499289	−	0.866436i	$-0.333595\pi$
$569$	13553.5	0.998578	0.499289	+	0.866436i	$0.333595\pi$
$570$	− 2646.79i	− 0.194495i
$571$	−14815.5	−1.08583	−0.542915	−	0.839788i	$-0.682679\pi$
$571$	−14815.5	−1.08583	−0.542915	+	0.839788i	$0.682679\pi$
$572$	0	0
$573$	−7542.79	−0.549921
$574$	5164.37i	0.375534i
$575$	−44707.6	−3.24249
$576$	−4851.74	−0.350965
$577$	21596.2i	1.55816i	0.626923	+	0.779081i	$0.284314\pi$
$577$	21596.2i	1.55816i	−0.626923	+	0.779081i	$0.715686\pi$
$578$	1150.20i	0.0827716i
$579$	7262.19i	0.521254i
$580$	2747.75i	0.196714i
$581$	−6985.42	−0.498802
$582$	−4582.66	−0.326387
$583$	− 8404.23i	− 0.597029i
$584$	17080.1	1.21024
$585$	0	0
$586$	4499.13	0.317163
$587$	− 918.801i	− 0.0646047i	−0.999478	−	0.0323024i	$-0.989716\pi$
$587$	− 918.801i	− 0.0646047i	0.999478	−	0.0323024i	$-0.0102839\pi$
$588$	−416.134	−0.0291855
$589$	−1937.13	−0.135514
$590$	4255.50i	0.296943i
$591$	13900.9i	0.967527i
$592$	− 9270.49i	− 0.643606i
$593$	− 19816.0i	− 1.37226i	−0.727481	−	0.686128i	$-0.759309\pi$
$593$	− 19816.0i	− 1.37226i	0.727481	−	0.686128i	$-0.240691\pi$
$594$	1695.16	0.117093
$595$	−9773.45	−0.673399
$596$	− 1046.42i	− 0.0719175i
$597$	9162.50	0.628134
$598$	0	0
$599$	−5141.86	−0.350736	−0.175368	−	0.984503i	$-0.556111\pi$
$599$	−5141.86	−0.350736	−0.175368	+	0.984503i	$0.556111\pi$
$600$	− 17764.7i	− 1.20873i
$601$	12380.9	0.840312	0.420156	−	0.907452i	$-0.361975\pi$
$601$	12380.9	0.840312	0.420156	+	0.907452i	$0.361975\pi$
$602$	−10311.0	−0.698081
$603$	3600.74i	0.243173i
$604$	409.150i	0.0275630i
$605$	15715.1i	1.05605i
$606$	2358.65i	0.158108i
$607$	−23717.0	−1.58590	−0.792951	−	0.609286i	$-0.791456\pi$
$607$	−23717.0	−1.58590	−0.792951	+	0.609286i	$0.791456\pi$
$608$	360.754	0.0240633
$609$	6550.89i	0.435887i
$610$	−10409.0	−0.690901
$611$	0	0
$612$	−291.584	−0.0192591
$613$	− 26157.1i	− 1.72345i	−0.507373	−	0.861726i	$-0.669383\pi$
$613$	− 26157.1i	− 1.72345i	0.507373	−	0.861726i	$-0.330617\pi$
$614$	9248.36	0.607872
$615$	−14712.8	−0.964677
$616$	3985.71i	0.260696i
$617$	23613.9i	1.54077i	0.637576	+	0.770387i	$0.279937\pi$
$617$	23613.9i	1.54077i	−0.637576	+	0.770387i	$0.720063\pi$
$618$	15720.9i	1.02328i
$619$	− 23345.4i	− 1.51588i	−0.652324	−	0.757940i	$-0.726206\pi$
$619$	− 23345.4i	− 1.51588i	0.652324	−	0.757940i	$-0.273794\pi$
$620$	−1104.40	−0.0715387
$621$	−4741.19	−0.306372
$622$	− 2381.20i	− 0.153501i
$623$	−8911.99	−0.573116
$624$	0	0
$625$	17371.0	1.11174
$626$	11907.5i	0.760258i
$627$	−1134.69	−0.0722730
$628$	799.207	0.0507832
$629$	− 10374.5i	− 0.657645i
$630$	− 3597.60i	− 0.227511i
$631$	− 15245.7i	− 0.961841i	−0.876764	−	0.480921i	$-0.840302\pi$
$631$	− 15245.7i	− 0.961841i	0.876764	−	0.480921i	$-0.159698\pi$
$632$	− 2634.38i	− 0.165807i
$633$	12094.8	0.759439
$634$	−8986.20	−0.562914
$635$	− 25253.6i	− 1.57820i
$636$	−532.128	−0.0331765
$637$	0	0
$638$	−18320.3	−1.13684
$639$	4754.69i	0.294355i
$640$	−25391.5	−1.56826
$641$	−10192.7	−0.628063	−0.314032	−	0.949413i	$-0.601680\pi$
$641$	−10192.7	−0.628063	−0.314032	+	0.949413i	$0.601680\pi$
$642$	6862.37i	0.421863i
$643$	− 5506.31i	− 0.337710i	−0.985641	−	0.168855i	$-0.945993\pi$
$643$	− 5506.31i	− 0.337710i	0.985641	−	0.168855i	$-0.0540070\pi$
$644$	− 635.108i	− 0.0388614i
$645$	− 29375.0i	− 1.79324i
$646$	−3035.48	−0.184875
$647$	13297.5	0.808005	0.404003	−	0.914758i	$-0.367619\pi$
$647$	13297.5	0.808005	0.404003	+	0.914758i	$0.367619\pi$
$648$	− 1883.93i	− 0.114209i
$649$	1824.35	0.110342
$650$	0	0
$651$	−2633.00	−0.158518
$652$	1385.43i	0.0832171i
$653$	−12440.2	−0.745519	−0.372760	−	0.927928i	$-0.621588\pi$
$653$	−12440.2	−0.745519	−0.372760	+	0.927928i	$0.621588\pi$
$654$	8863.76	0.529970
$655$	− 3842.14i	− 0.229198i
$656$	15077.7i	0.897389i
$657$	6609.29i	0.392470i
$658$	− 5780.23i	− 0.342457i
$659$	−9562.87	−0.565276	−0.282638	−	0.959227i	$-0.591209\pi$
$659$	−9562.87	−0.565276	−0.282638	+	0.959227i	$0.591209\pi$
$660$	−646.915	−0.0381533
$661$	− 2409.69i	− 0.141795i	−0.997484	−	0.0708973i	$-0.977414\pi$
$661$	− 2409.69i	− 0.141795i	0.997484	−	0.0708973i	$-0.0225863\pi$
$662$	15324.8	0.899722
$663$	0	0
$664$	−21710.9	−1.26889
$665$	2408.13i	0.140426i
$666$	3818.85	0.222188
$667$	51239.9	2.97454
$668$	− 352.368i	− 0.0204095i
$669$	− 11354.9i	− 0.656209i
$670$	21371.0i	1.23229i
$671$	4462.40i	0.256735i
$672$	490.346	0.0281481
$673$	−7929.02	−0.454147	−0.227074	−	0.973878i	$-0.572916\pi$
$673$	−7929.02	−0.454147	−0.227074	+	0.973878i	$0.572916\pi$
$674$	2471.72i	0.141257i
$675$	6874.19	0.391982
$676$	0	0
$677$	−2628.26	−0.149206	−0.0746030	−	0.997213i	$-0.523769\pi$
$677$	−2628.26	−0.149206	−0.0746030	+	0.997213i	$0.523769\pi$
$678$	− 1365.88i	− 0.0773690i
$679$	4169.44	0.235653
$680$	−30376.1	−1.71305
$681$	− 6041.48i	− 0.339956i
$682$	− 7363.46i	− 0.413433i
$683$	− 10021.5i	− 0.561437i	−0.959790	−	0.280719i	$-0.909427\pi$
$683$	− 10021.5i	− 0.561437i	0.959790	−	0.280719i	$-0.0905728\pi$
$684$	71.8448i	0.00401616i
$685$	−10655.7	−0.594356
$686$	−12925.5	−0.719385
$687$	− 9152.18i	− 0.508264i
$688$	−30103.7	−1.66816
$689$	0	0
$690$	−28139.8	−1.55256
$691$	23987.2i	1.32057i	0.751014	+	0.660286i	$0.229565\pi$
$691$	23987.2i	1.32057i	−0.751014	+	0.660286i	$0.770435\pi$
$692$	1853.43	0.101816
$693$	−1542.30	−0.0845415
$694$	2227.10i	0.121815i
$695$	11878.3i	0.648303i
$696$	20360.3i	1.10885i
$697$	16873.4i	0.916964i
$698$	−12167.4	−0.659801
$699$	16762.5	0.907032
$700$	920.835i	0.0497204i
$701$	3763.71	0.202787	0.101393	−	0.994846i	$-0.467670\pi$
$701$	3763.71	0.202787	0.101393	+	0.994846i	$0.467670\pi$
$702$	0	0
$703$	−2556.23	−0.137141
$704$	12344.9i	0.660890i
$705$	16467.3	0.879707
$706$	19507.8	1.03993
$707$	− 2145.96i	− 0.114155i
$708$	− 115.512i	− 0.00613163i
$709$	36047.8i	1.90946i	0.297479	+	0.954728i	$0.403854\pi$
$709$	36047.8i	1.90946i	−0.297479	+	0.954728i	$0.596146\pi$
$710$	28219.9i	1.49166i
$711$	1019.39	0.0537696
$712$	−27698.7	−1.45794
$713$	20594.9i	1.08174i
$714$	−4125.91	−0.216258
$715$	0	0
$716$	137.195	0.00716090
$717$	4007.08i	0.208713i
$718$	−12869.3	−0.668910
$719$	3944.18	0.204580	0.102290	−	0.994755i	$-0.467383\pi$
$719$	3944.18	0.204580	0.102290	+	0.994755i	$0.467383\pi$
$720$	− 10503.4i	− 0.543667i
$721$	− 14303.3i	− 0.738812i
$722$	− 18057.1i	− 0.930770i
$723$	− 1714.68i	− 0.0882012i
$724$	−1142.36	−0.0586402
$725$	−74292.1	−3.80571
$726$	6634.22i	0.339145i
$727$	20447.8	1.04315	0.521573	−	0.853206i	$-0.325345\pi$
$727$	20447.8	1.04315	0.521573	+	0.853206i	$0.325345\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	39227.3i	1.98886i
$731$	−33688.7	−1.70454
$732$	282.544	0.0142666
$733$	− 13536.2i	− 0.682089i	−0.940047	−	0.341045i	$-0.889219\pi$
$733$	− 13536.2i	− 0.682089i	0.940047	−	0.341045i	$-0.110781\pi$
$734$	− 25343.8i	− 1.27447i
$735$	− 16775.1i	− 0.841851i
$736$	− 3835.40i	− 0.192085i
$737$	9161.83	0.457911
$738$	−6211.07	−0.309800
$739$	− 15839.1i	− 0.788433i	−0.919018	−	0.394217i	$-0.871016\pi$
$739$	− 15839.1i	− 0.788433i	0.919018	−	0.394217i	$-0.128984\pi$
$740$	−1457.37	−0.0723972
$741$	0	0
$742$	−7529.60	−0.372534
$743$	− 1664.92i	− 0.0822075i	−0.999155	−	0.0411037i	$-0.986913\pi$
$743$	− 1664.92i	− 0.0822075i	0.999155	−	0.0411037i	$-0.0130874\pi$
$744$	−8183.43	−0.403252
$745$	42182.9	2.07445
$746$	11822.0i	0.580206i
$747$	− 8401.19i	− 0.411491i
$748$	741.916i	0.0362662i
$749$	− 6243.58i	− 0.304587i
$750$	20768.3	1.01113
$751$	−22399.1	−1.08835	−0.544177	−	0.838970i	$-0.683158\pi$
$751$	−22399.1	−1.08835	−0.544177	+	0.838970i	$0.683158\pi$
$752$	− 16875.8i	− 0.818346i
$753$	12265.8	0.593613
$754$	0	0
$755$	−16493.6	−0.795050
$756$	97.6535i	0.00469792i
$757$	23798.9	1.14265	0.571326	−	0.820723i	$-0.306429\pi$
$757$	23798.9	1.14265	0.571326	+	0.820723i	$0.306429\pi$
$758$	−6532.64	−0.313029
$759$	12063.6i	0.576920i
$760$	7484.53i	0.357227i
$761$	− 13693.5i	− 0.652285i	−0.945321	−	0.326142i	$-0.894251\pi$
$761$	− 13693.5i	− 0.652285i	0.945321	−	0.326142i	$-0.105749\pi$
$762$	− 10660.9i	− 0.506829i
$763$	−8064.50	−0.382640
$764$	1215.18	0.0575441
$765$	− 11754.3i	− 0.555526i
$766$	13285.5	0.626666
$767$	0	0
$768$	2218.82	0.104251
$769$	16299.9i	0.764358i	0.924088	+	0.382179i	$0.124826\pi$
$769$	16299.9i	0.764358i	−0.924088	+	0.382179i	$0.875174\pi$
$770$	−9153.84	−0.428418
$771$	9150.68	0.427437
$772$	− 1169.97i	− 0.0545444i
$773$	33532.2i	1.56024i	0.625628	+	0.780122i	$0.284843\pi$
$773$	33532.2i	1.56024i	−0.625628	+	0.780122i	$0.715157\pi$
$774$	− 12400.8i	− 0.575887i
$775$	− 29860.2i	− 1.38401i
$776$	12958.7	0.599473
$777$	−3474.50	−0.160421
$778$	26214.3i	1.20801i
$779$	4157.51	0.191217
$780$	0	0
$781$	12098.0	0.554290
$782$	32272.1i	1.47577i
$783$	−7878.59	−0.359589
$784$	−17191.3	−0.783130
$785$	32217.5i	1.46483i
$786$	− 1621.98i	− 0.0736055i
$787$	− 16163.3i	− 0.732097i	−0.930596	−	0.366049i	$-0.880710\pi$
$787$	− 16163.3i	− 0.732097i	0.930596	−	0.366049i	$-0.119290\pi$
$788$	− 2239.51i	− 0.101243i
$789$	−17313.0	−0.781189
$790$	6050.27	0.272480
$791$	1242.71i	0.0558607i
$792$	−4793.52	−0.215064
$793$	0	0
$794$	20398.2	0.911720
$795$	− 21451.1i	− 0.956970i
$796$	−1476.12	−0.0657284
$797$	39636.4	1.76160	0.880798	−	0.473492i	$-0.157007\pi$
$797$	39636.4	1.76160	0.880798	+	0.473492i	$0.157007\pi$
$798$	1016.60i	0.0450969i
$799$	− 18885.5i	− 0.836197i
$800$	5560.90i	0.245760i
$801$	− 10718.2i	− 0.472797i
$802$	23819.0	1.04872
$803$	16816.9	0.739048
$804$	− 580.096i	− 0.0254458i
$805$	25602.4	1.12095
$806$	0	0
$807$	6238.21	0.272113
$808$	− 6669.71i	− 0.290396i
$809$	−23811.2	−1.03481	−0.517403	−	0.855742i	$-0.673101\pi$
$809$	−23811.2	−1.03481	−0.517403	+	0.855742i	$0.673101\pi$
$810$	4326.74	0.187687
$811$	27218.6i	1.17851i	0.807946	+	0.589256i	$0.200579\pi$
$811$	27218.6i	1.17851i	−0.807946	+	0.589256i	$0.799421\pi$
$812$	− 1055.38i	− 0.0456116i
$813$	18036.0i	0.778045i
$814$	− 9716.80i	− 0.418395i
$815$	−55849.2	−2.40038
$816$	−12045.9	−0.516777
$817$	8300.73i	0.355454i
$818$	7007.94	0.299544
$819$	0	0
$820$	2370.30	0.100944
$821$	− 43094.8i	− 1.83193i	−0.401253	−	0.915967i	$-0.631425\pi$
$821$	− 43094.8i	− 1.83193i	0.401253	−	0.915967i	$-0.368575\pi$
$822$	−4498.36	−0.190874
$823$	−26541.1	−1.12414	−0.562068	−	0.827091i	$-0.689994\pi$
$823$	−26541.1	−1.12414	−0.562068	+	0.827091i	$0.689994\pi$
$824$	− 44455.1i	− 1.87945i
$825$	− 17490.9i	− 0.738127i
$826$	− 1634.49i	− 0.0688512i
$827$	44898.7i	1.88788i	0.330112	+	0.943942i	$0.392913\pi$
$827$	44898.7i	1.88788i	−0.330112	+	0.943942i	$0.607087\pi$
$828$	763.829	0.0320590
$829$	7137.48	0.299029	0.149514	−	0.988760i	$-0.452229\pi$
$829$	7137.48	0.299029	0.149514	+	0.988760i	$0.452229\pi$
$830$	− 49862.6i	− 2.08525i
$831$	−2205.60	−0.0920717
$832$	0	0
$833$	−19238.6	−0.800213
$834$	5014.49i	0.208198i
$835$	14204.6	0.588708
$836$	182.804	0.00756270
$837$	− 3166.64i	− 0.130771i
$838$	9177.59i	0.378323i
$839$	4387.17i	0.180527i	0.995918	+	0.0902634i	$0.0287709\pi$
$839$	4387.17i	0.180527i	−0.995918	+	0.0902634i	$0.971229\pi$
$840$	10173.2i	0.417867i
$841$	60758.1	2.49121
$842$	−5084.36	−0.208098
$843$	− 5708.76i	− 0.233239i
$844$	−1948.53	−0.0794683
$845$	0	0
$846$	6951.74	0.282513
$847$	− 6036.01i	− 0.244864i
$848$	−21983.2	−0.890219
$849$	6377.12	0.257788
$850$	− 46791.0i	− 1.88814i
$851$	27176.9i	1.09473i
$852$	− 766.004i	− 0.0308015i
$853$	9328.85i	0.374459i	0.982316	+	0.187230i	$0.0599508\pi$
$853$	9328.85i	0.374459i	−0.982316	+	0.187230i	$0.940049\pi$
$854$	3997.99	0.160197
$855$	−2896.20	−0.115845
$856$	− 19405.2i	− 0.774833i
$857$	5010.39	0.199710	0.0998552	−	0.995002i	$-0.468162\pi$
$857$	5010.39	0.199710	0.0998552	+	0.995002i	$0.468162\pi$
$858$	0	0
$859$	30233.4	1.20088	0.600438	−	0.799672i	$-0.294993\pi$
$859$	30233.4	1.20088	0.600438	+	0.799672i	$0.294993\pi$
$860$	4732.45i	0.187646i
$861$	5651.01	0.223677
$862$	−38501.4	−1.52130
$863$	4334.93i	0.170988i	0.996339	+	0.0854940i	$0.0272468\pi$
$863$	4334.93i	0.170988i	−0.996339	+	0.0854940i	$0.972753\pi$
$864$	589.728i	0.0232210i
$865$	74715.2i	2.93687i
$866$	8462.05i	0.332047i
$867$	1258.58	0.0493007
$868$	424.189	0.0165875
$869$	− 2593.78i	− 0.101252i
$870$	−46760.8	−1.82223
$871$	0	0
$872$	−25064.7	−0.973391
$873$	5014.48i	0.194404i
$874$	7951.69	0.307746
$875$	−18895.6	−0.730043
$876$	− 1064.79i	− 0.0410684i
$877$	34683.3i	1.33543i	0.744416	+	0.667716i	$0.232728\pi$
$877$	34683.3i	1.33543i	−0.744416	+	0.667716i	$0.767272\pi$
$878$	7779.95i	0.299044i
$879$	− 4923.08i	− 0.188909i
$880$	−26725.3	−1.02376
$881$	18269.2	0.698642	0.349321	−	0.937003i	$-0.386412\pi$
$881$	18269.2	0.698642	0.349321	+	0.937003i	$0.386412\pi$
$882$	− 7081.70i	− 0.270355i
$883$	14592.0	0.556128	0.278064	−	0.960563i	$-0.410307\pi$
$883$	14592.0	0.556128	0.278064	+	0.960563i	$0.410307\pi$
$884$	0	0
$885$	4656.49	0.176866
$886$	− 50198.0i	− 1.90343i
$887$	30459.3	1.15301	0.576507	−	0.817092i	$-0.304415\pi$
$887$	30459.3	1.15301	0.576507	+	0.817092i	$0.304415\pi$
$888$	−10798.8	−0.408091
$889$	9699.60i	0.365933i
$890$	− 63614.6i	− 2.39592i
$891$	− 1854.89i	− 0.0697432i
$892$	1829.32i	0.0686662i
$893$	−4653.30	−0.174375
$894$	17807.7	0.666196
$895$	5530.57i	0.206555i
$896$	9752.58	0.363628
$897$	0	0
$898$	−38002.6	−1.41221
$899$	34223.2i	1.26964i
$900$	−1107.47	−0.0410172
$901$	−24601.2	−0.909638
$902$	15803.6i	0.583374i
$903$	11282.6i	0.415793i
$904$	3862.39i	0.142103i
$905$	− 46050.7i	− 1.69147i
$906$	−6962.84	−0.255326
$907$	9364.89	0.342840	0.171420	−	0.985198i	$-0.445164\pi$
$907$	9364.89	0.342840	0.171420	+	0.985198i	$0.445164\pi$
$908$	973.313i	0.0355733i
$909$	2580.90	0.0941727
$910$	0	0
$911$	32479.8	1.18123	0.590616	−	0.806952i	$-0.298885\pi$
$911$	32479.8	1.18123	0.590616	+	0.806952i	$0.298885\pi$
$912$	2968.04i	0.107765i
$913$	−21376.3	−0.774864
$914$	−24648.5	−0.892012
$915$	11389.9i	0.411516i
$916$	1474.46i	0.0531851i
$917$	1475.72i	0.0531435i
$918$	− 4962.13i	− 0.178404i
$919$	295.958	0.0106232	0.00531161	−	0.999986i	$-0.498309\pi$
$919$	295.958	0.0106232	0.00531161	+	0.999986i	$0.498309\pi$
$920$	79572.9	2.85157
$921$	− 10119.8i	− 0.362062i
$922$	−9340.56	−0.333639
$923$	0	0
$924$	248.473	0.00884649
$925$	− 39403.5i	− 1.40062i
$926$	20558.2	0.729572
$927$	17202.3	0.609489
$928$	− 6373.42i	− 0.225450i
$929$	− 5620.38i	− 0.198492i	−0.995063	−	0.0992458i	$-0.968357\pi$
$929$	− 5620.38i	− 0.198492i	0.995063	−	0.0992458i	$-0.0316430\pi$
$930$	− 18794.6i	− 0.662687i
$931$	4740.29i	0.166871i
$932$	−2700.52	−0.0949125
$933$	−2605.58	−0.0914284
$934$	21142.0i	0.740670i
$935$	−29908.0	−1.04609
$936$	0	0
$937$	−32583.1	−1.13601	−0.568006	−	0.823024i	$-0.692285\pi$
$937$	−32583.1	−1.13601	−0.568006	+	0.823024i	$0.692285\pi$
$938$	− 8208.35i	− 0.285727i
$939$	13029.6	0.452827
$940$	−2652.96	−0.0920532
$941$	8812.99i	0.305308i	0.988280	+	0.152654i	$0.0487821\pi$
$941$	8812.99i	0.305308i	−0.988280	+	0.152654i	$0.951218\pi$
$942$	13600.8i	0.470422i
$943$	− 44201.2i	− 1.52639i
$944$	− 4772.00i	− 0.164529i
$945$	−3936.59	−0.135510
$946$	−31552.9	−1.08443
$947$	− 13426.8i	− 0.460732i	−0.973104	−	0.230366i	$-0.926008\pi$
$947$	− 13426.8i	− 0.460732i	0.973104	−	0.230366i	$-0.0739923\pi$
$948$	−164.229	−0.00562649
$949$	0	0
$950$	−11529.1	−0.393739
$951$	9832.96i	0.335285i
$952$	11667.1	0.397199
$953$	13394.6	0.455293	0.227647	−	0.973744i	$-0.426897\pi$
$953$	13394.6	0.455293	0.227647	+	0.973744i	$0.426897\pi$
$954$	− 9055.67i	− 0.307325i
$955$	48986.2i	1.65985i
$956$	− 645.560i	− 0.0218398i
$957$	20046.5i	0.677129i
$958$	25880.7	0.872828
$959$	4092.74	0.137812
$960$	31509.3i	1.05933i
$961$	16035.7	0.538273
$962$	0	0
$963$	7509.00	0.251271
$964$	276.243i	0.00922944i
$965$	47163.8	1.57332
$966$	10808.2	0.359986
$967$	45590.8i	1.51613i	0.652178	+	0.758066i	$0.273856\pi$
$967$	45590.8i	1.51613i	−0.652178	+	0.758066i	$0.726144\pi$
$968$	− 18760.1i	− 0.622904i
$969$	3321.51i	0.110116i
$970$	29761.8i	0.985149i
$971$	264.763	0.00875041	0.00437521	−	0.999990i	$-0.498607\pi$
$971$	264.763	0.00875041	0.00437521	+	0.999990i	$0.498607\pi$
$972$	−117.445	−0.00387558
$973$	− 4562.32i	− 0.150320i
$974$	−16878.2	−0.555248
$975$	0	0
$976$	11672.4	0.382812
$977$	610.521i	0.0199921i	0.999950	+	0.00999606i	$0.00318190\pi$
$977$	610.521i	0.0199921i	−0.999950	+	0.00999606i	$0.996818\pi$
$978$	−23577.0	−0.770868
$979$	−27271.8	−0.890308
$980$	2702.56i	0.0880918i
$981$	− 9698.98i	− 0.315662i
$982$	10535.5i	0.342363i
$983$	57829.7i	1.87638i	0.346121	+	0.938190i	$0.387499\pi$
$983$	57829.7i	1.87638i	−0.346121	+	0.938190i	$0.612501\pi$
$984$	17563.5	0.569007
$985$	90278.8	2.92033
$986$	53627.7i	1.73210i
$987$	−6324.89	−0.203975
$988$	0	0
$989$	88250.4	2.83741
$990$	− 11009.1i	− 0.353427i
$991$	−56780.7	−1.82008	−0.910039	−	0.414522i	$-0.863949\pi$
$991$	−56780.7	−1.82008	−0.910039	+	0.414522i	$0.863949\pi$
$992$	2561.67	0.0819890
$993$	− 16768.9i	− 0.535895i
$994$	− 10838.9i	− 0.345866i
$995$	− 59505.3i	− 1.89592i
$996$	1353.47i	0.0430587i
$997$	18616.6	0.591369	0.295684	−	0.955286i	$-0.404452\pi$
$997$	18616.6	0.591369	0.295684	+	0.955286i	$0.404452\pi$
$998$	−35209.4	−1.11677
$999$	− 4178.69i	− 0.132340i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.f.337.2		4
13.5	odd	4		39.4.a.b.1.1	✓	2
13.8	odd	4		507.4.a.f.1.2		2
13.12	even	2	inner	507.4.b.f.337.3		4
39.5	even	4		117.4.a.c.1.2		2
39.8	even	4		1521.4.a.s.1.1		2
52.31	even	4		624.4.a.r.1.2		2
65.44	odd	4		975.4.a.j.1.2		2
91.83	even	4		1911.4.a.h.1.1		2
104.5	odd	4		2496.4.a.bc.1.1		2
104.83	even	4		2496.4.a.s.1.1		2
156.83	odd	4		1872.4.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.b.1.1	✓	2	13.5	odd	4
117.4.a.c.1.2		2	39.5	even	4
507.4.a.f.1.2		2	13.8	odd	4
507.4.b.f.337.2		4	1.1	even	1	trivial
507.4.b.f.337.3		4	13.12	even	2	inner
624.4.a.r.1.2		2	52.31	even	4
975.4.a.j.1.2		2	65.44	odd	4
1521.4.a.s.1.1		2	39.8	even	4
1872.4.a.t.1.1		2	156.83	odd	4
1911.4.a.h.1.1		2	91.83	even	4
2496.4.a.s.1.1		2	104.83	even	4
2496.4.a.bc.1.1		2	104.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.74166i q^{2} -3.00000 q^{3} +0.483315 q^{4} +19.4833i q^{5} +8.22497i q^{6} -7.48331i q^{7} -23.2583i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.74166i q^{2} -3.00000 q^{3} +0.483315 q^{4} +19.4833i q^{5} +8.22497i q^{6} -7.48331i q^{7} -23.2583i q^{8} +9.00000 q^{9} +53.4166 q^{10} -22.8999i q^{11} -1.44994 q^{12} -20.5167 q^{14} -58.4499i q^{15} -59.8999 q^{16} -67.0334 q^{17} -24.6749i q^{18} +16.5167i q^{19} +9.41657i q^{20} +22.4499i q^{21} -62.7836 q^{22} +175.600 q^{23} +69.7750i q^{24} -254.600 q^{25} -27.0000 q^{27} -3.61680i q^{28} +291.800 q^{29} -160.250 q^{30} +117.283i q^{31} -21.8418i q^{32} +68.6997i q^{33} +183.783i q^{34} +145.800 q^{35} +4.34983 q^{36} +154.766i q^{37} +45.2831 q^{38} +453.150 q^{40} -251.716i q^{41} +61.5501 q^{42} +502.566 q^{43} -11.0679i q^{44} +175.350i q^{45} -481.434i q^{46} +281.733i q^{47} +179.700 q^{48} +287.000 q^{49} +698.025i q^{50} +201.100 q^{51} +366.999 q^{53} +74.0247i q^{54} +446.166 q^{55} -174.049 q^{56} -49.5501i q^{57} -800.015i q^{58} +79.6663i q^{59} -28.2497i q^{60} -194.865 q^{61} +321.550 q^{62} -67.3498i q^{63} -539.082 q^{64} +188.351 q^{66} +400.082i q^{67} -32.3982 q^{68} -526.799 q^{69} -399.733i q^{70} +528.299i q^{71} -209.325i q^{72} +734.366i q^{73} +424.316 q^{74} +763.799 q^{75} +7.98276i q^{76} -171.367 q^{77} +113.266 q^{79} -1167.05i q^{80} +81.0000 q^{81} -690.118 q^{82} -933.466i q^{83} +10.8504i q^{84} -1306.03i q^{85} -1377.86i q^{86} -875.399 q^{87} -532.613 q^{88} -1190.91i q^{89} +480.749 q^{90} +84.8699 q^{92} -351.849i q^{93} +772.415 q^{94} -321.800 q^{95} +65.5253i q^{96} +557.165i q^{97} -786.856i q^{98} -206.099i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 28 q^{4} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 - 28 * q^4 + 36 * q^9	\( 4 q - 12 q^{3} - 28 q^{4} + 36 q^{9} + 64 q^{10} + 84 q^{12} - 112 q^{14} - 60 q^{16} - 328 q^{17} - 760 q^{22} - 16 q^{23} - 300 q^{25} - 108 q^{27} + 808 q^{29} - 192 q^{30} + 224 q^{35} - 252 q^{36} - 208 q^{38} + 1184 q^{40} + 336 q^{42} + 1232 q^{43} + 180 q^{48} + 1148 q^{49} + 984 q^{51} - 328 q^{53} + 288 q^{55} + 112 q^{56} + 1256 q^{61} + 1376 q^{62} + 388 q^{64} + 2280 q^{66} + 2744 q^{68} + 48 q^{69} + 1368 q^{74} + 900 q^{75} - 1344 q^{77} - 864 q^{79} + 324 q^{81} - 5664 q^{82} - 2424 q^{87} + 3048 q^{88} + 576 q^{90} + 5488 q^{92} + 1144 q^{94} - 928 q^{95}+O(q^{100}) \) 4 * q - 12 * q^3 - 28 * q^4 + 36 * q^9 + 64 * q^10 + 84 * q^12 - 112 * q^14 - 60 * q^16 - 328 * q^17 - 760 * q^22 - 16 * q^23 - 300 * q^25 - 108 * q^27 + 808 * q^29 - 192 * q^30 + 224 * q^35 - 252 * q^36 - 208 * q^38 + 1184 * q^40 + 336 * q^42 + 1232 * q^43 + 180 * q^48 + 1148 * q^49 + 984 * q^51 - 328 * q^53 + 288 * q^55 + 112 * q^56 + 1256 * q^61 + 1376 * q^62 + 388 * q^64 + 2280 * q^66 + 2744 * q^68 + 48 * q^69 + 1368 * q^74 + 900 * q^75 - 1344 * q^77 - 864 * q^79 + 324 * q^81 - 5664 * q^82 - 2424 * q^87 + 3048 * q^88 + 576 * q^90 + 5488 * q^92 + 1144 * q^94 - 928 * q^95

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