LMFDB - Embedded newform 294.3.c.a.97.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,3,Mod(97,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("294.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 294 = 2 \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	294.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.01091977219$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.3
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	294.97
Dual form			294.3.c.a.97.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +8.36308i q^{5} -2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +8.36308i q^{5} -2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +11.8272i q^{10} +6.00000 q^{11} -3.46410i q^{12} +17.8639i q^{13} +14.4853 q^{15} +4.00000 q^{16} +18.7554i q^{17} -4.24264 q^{18} -17.0233i q^{19} +16.7262i q^{20} +8.48528 q^{22} +13.4558 q^{23} -4.89898i q^{24} -44.9411 q^{25} +25.2633i q^{26} +5.19615i q^{27} +33.9411 q^{29} +20.4853 q^{30} -14.7479i q^{31} +5.65685 q^{32} -10.3923i q^{33} +26.5241i q^{34} -6.00000 q^{36} +5.97056 q^{37} -24.0746i q^{38} +30.9411 q^{39} +23.6544i q^{40} -35.2354i q^{41} +15.4853 q^{43} +12.0000 q^{44} -25.0892i q^{45} +19.0294 q^{46} -33.2061i q^{47} -6.92820i q^{48} -63.5563 q^{50} +32.4853 q^{51} +35.7277i q^{52} -34.5442 q^{53} +7.34847i q^{54} +50.1785i q^{55} -29.4853 q^{57} +48.0000 q^{58} -27.3647i q^{59} +28.9706 q^{60} -40.3805i q^{61} -20.8567i q^{62} +8.00000 q^{64} -149.397 q^{65} -14.6969i q^{66} -114.397 q^{67} +37.5108i q^{68} -23.3062i q^{69} +18.6030 q^{71} -8.48528 q^{72} +117.032i q^{73} +8.44365 q^{74} +77.8403i q^{75} -34.0467i q^{76} +43.7574 q^{78} -88.3381 q^{79} +33.4523i q^{80} +9.00000 q^{81} -49.8303i q^{82} -75.7601i q^{83} -156.853 q^{85} +21.8995 q^{86} -58.7878i q^{87} +16.9706 q^{88} -20.7846i q^{89} -35.4815i q^{90} +26.9117 q^{92} -25.5442 q^{93} -46.9606i q^{94} +142.368 q^{95} -9.79796i q^{96} -30.5826i q^{97} -18.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 12 q^{9}+O(q^{10}) $ 4 * q + 8 * q^4 - 12 * q^9	$ 4 q + 8 q^{4} - 12 q^{9} + 24 q^{11} + 24 q^{15} + 16 q^{16} - 48 q^{23} - 44 q^{25} + 48 q^{30} - 24 q^{36} - 44 q^{37} - 12 q^{39} + 28 q^{43} + 48 q^{44} + 144 q^{46} - 192 q^{50} + 96 q^{51} - 240 q^{53} - 84 q^{57} + 192 q^{58} + 48 q^{60} + 32 q^{64} - 360 q^{65} - 220 q^{67} + 312 q^{71} + 96 q^{74} + 192 q^{78} + 20 q^{79} + 36 q^{81} - 288 q^{85} + 48 q^{86} - 96 q^{92} - 204 q^{93} + 264 q^{95} - 72 q^{99}+O(q^{100}) $ 4 * q + 8 * q^4 - 12 * q^9 + 24 * q^11 + 24 * q^15 + 16 * q^16 - 48 * q^23 - 44 * q^25 + 48 * q^30 - 24 * q^36 - 44 * q^37 - 12 * q^39 + 28 * q^43 + 48 * q^44 + 144 * q^46 - 192 * q^50 + 96 * q^51 - 240 * q^53 - 84 * q^57 + 192 * q^58 + 48 * q^60 + 32 * q^64 - 360 * q^65 - 220 * q^67 + 312 * q^71 + 96 * q^74 + 192 * q^78 + 20 * q^79 + 36 * q^81 - 288 * q^85 + 48 * q^86 - 96 * q^92 - 204 * q^93 + 264 * q^95 - 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/294\mathbb{Z}\right)^\times$.

$n$	$197$	$199$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	2.00000	0.500000
$5$	8.36308i	1.67262i	0.548260	+	0.836308i	$0.315291\pi$
$5$	8.36308i	1.67262i	−0.548260	+	0.836308i	$0.684709\pi$
$6$	− 2.44949i	− 0.408248i
$7$	0	0
$7$	0	0
$8$	2.82843	0.353553
$9$	−3.00000	−0.333333
$10$	11.8272i	1.18272i
$11$	6.00000	0.545455	0.272727	−	0.962091i	$-0.412074\pi$
$11$	6.00000	0.545455	0.272727	+	0.962091i	$0.412074\pi$
$12$	− 3.46410i	− 0.288675i
$13$	17.8639i	1.37414i	0.726590	+	0.687072i	$0.241104\pi$
$13$	17.8639i	1.37414i	−0.726590	+	0.687072i	$0.758896\pi$
$14$	0	0
$15$	14.4853	0.965685
$16$	4.00000	0.250000
$17$	18.7554i	1.10326i	0.834090	+	0.551629i	$0.185994\pi$
$17$	18.7554i	1.10326i	−0.834090	+	0.551629i	$0.814006\pi$
$18$	−4.24264	−0.235702
$19$	− 17.0233i	− 0.895965i	−0.894042	−	0.447983i	$-0.852143\pi$
$19$	− 17.0233i	− 0.895965i	0.894042	−	0.447983i	$-0.147857\pi$
$20$	16.7262i	0.836308i
$21$	0	0
$22$	8.48528	0.385695
$23$	13.4558	0.585037	0.292518	−	0.956260i	$-0.405507\pi$
$23$	13.4558	0.585037	0.292518	+	0.956260i	$0.405507\pi$
$24$	− 4.89898i	− 0.204124i
$25$	−44.9411	−1.79765
$26$	25.2633i	0.971666i
$27$	5.19615i	0.192450i
$28$	0	0
$29$	33.9411	1.17038	0.585192	−	0.810895i	$-0.301019\pi$
$29$	33.9411	1.17038	0.585192	+	0.810895i	$0.301019\pi$
$30$	20.4853	0.682843
$31$	− 14.7479i	− 0.475740i	−0.971297	−	0.237870i	$-0.923551\pi$
$31$	− 14.7479i	− 0.475740i	0.971297	−	0.237870i	$-0.0764491\pi$
$32$	5.65685	0.176777
$33$	− 10.3923i	− 0.314918i
$34$	26.5241i	0.780121i
$35$	0	0
$36$	−6.00000	−0.166667
$37$	5.97056	0.161367	0.0806833	−	0.996740i	$-0.474290\pi$
$37$	5.97056	0.161367	0.0806833	+	0.996740i	$0.474290\pi$
$38$	− 24.0746i	− 0.633543i
$39$	30.9411	0.793362
$40$	23.6544i	0.591359i
$41$	− 35.2354i	− 0.859399i	−0.902972	−	0.429700i	$-0.858619\pi$
$41$	− 35.2354i	− 0.859399i	0.902972	−	0.429700i	$-0.141381\pi$
$42$	0	0
$43$	15.4853	0.360123	0.180061	−	0.983655i	$-0.442370\pi$
$43$	15.4853	0.360123	0.180061	+	0.983655i	$0.442370\pi$
$44$	12.0000	0.272727
$45$	− 25.0892i	− 0.557539i
$46$	19.0294	0.413683
$47$	− 33.2061i	− 0.706514i	−0.935526	−	0.353257i	$-0.885074\pi$
$47$	− 33.2061i	− 0.706514i	0.935526	−	0.353257i	$-0.114926\pi$
$48$	− 6.92820i	− 0.144338i
$49$	0	0
$50$	−63.5563	−1.27113
$51$	32.4853	0.636966
$52$	35.7277i	0.687072i
$53$	−34.5442	−0.651777	−0.325888	−	0.945408i	$-0.605663\pi$
$53$	−34.5442	−0.651777	−0.325888	+	0.945408i	$0.605663\pi$
$54$	7.34847i	0.136083i
$55$	50.1785i	0.912336i
$56$	0	0
$57$	−29.4853	−0.517286
$58$	48.0000	0.827586
$59$	− 27.3647i	− 0.463808i	−0.972739	−	0.231904i	$-0.925505\pi$
$59$	− 27.3647i	− 0.463808i	0.972739	−	0.231904i	$-0.0744955\pi$
$60$	28.9706	0.482843
$61$	− 40.3805i	− 0.661976i	−0.943635	−	0.330988i	$-0.892618\pi$
$61$	− 40.3805i	− 0.661976i	0.943635	−	0.330988i	$-0.107382\pi$
$62$	− 20.8567i	− 0.336399i
$63$	0	0
$64$	8.00000	0.125000
$65$	−149.397	−2.29841
$66$	− 14.6969i	− 0.222681i
$67$	−114.397	−1.70742	−0.853709	−	0.520751i	$-0.825652\pi$
$67$	−114.397	−1.70742	−0.853709	+	0.520751i	$0.825652\pi$
$68$	37.5108i	0.551629i
$69$	− 23.3062i	− 0.337771i
$70$	0	0
$71$	18.6030	0.262015	0.131007	−	0.991381i	$-0.458179\pi$
$71$	18.6030	0.262015	0.131007	+	0.991381i	$0.458179\pi$
$72$	−8.48528	−0.117851
$73$	117.032i	1.60318i	0.597874	+	0.801590i	$0.296012\pi$
$73$	117.032i	1.60318i	−0.597874	+	0.801590i	$0.703988\pi$
$74$	8.44365	0.114103
$75$	77.8403i	1.03787i
$76$	− 34.0467i	− 0.447983i
$77$	0	0
$78$	43.7574	0.560992
$79$	−88.3381	−1.11820	−0.559102	−	0.829099i	$-0.688854\pi$
$79$	−88.3381	−1.11820	−0.559102	+	0.829099i	$0.688854\pi$
$80$	33.4523i	0.418154i
$81$	9.00000	0.111111
$82$	− 49.8303i	− 0.607687i
$83$	− 75.7601i	− 0.912772i	−0.889782	−	0.456386i	$-0.849144\pi$
$83$	− 75.7601i	− 0.912772i	0.889782	−	0.456386i	$-0.150856\pi$
$84$	0	0
$85$	−156.853	−1.84533
$86$	21.8995	0.254645
$87$	− 58.7878i	− 0.675721i
$88$	16.9706	0.192847
$89$	− 20.7846i	− 0.233535i	−0.993159	−	0.116767i	$-0.962747\pi$
$89$	− 20.7846i	− 0.233535i	0.993159	−	0.116767i	$-0.0372532\pi$
$90$	− 35.4815i	− 0.394239i
$91$	0	0
$92$	26.9117	0.292518
$93$	−25.5442	−0.274668
$94$	− 46.9606i	− 0.499581i
$95$	142.368	1.49861
$96$	− 9.79796i	− 0.102062i
$97$	− 30.5826i	− 0.315284i	−0.987496	−	0.157642i	$-0.949611\pi$
$97$	− 30.5826i	− 0.315284i	0.987496	−	0.157642i	$-0.0503892\pi$
$98$	0	0
$99$	−18.0000	−0.181818
$100$	−89.8823	−0.898823
$101$	− 127.968i	− 1.26701i	−0.773739	−	0.633504i	$-0.781616\pi$
$101$	− 127.968i	− 1.26701i	0.773739	−	0.633504i	$-0.218384\pi$
$102$	45.9411	0.450403
$103$	− 80.9563i	− 0.785983i	−0.919542	−	0.392992i	$-0.871440\pi$
$103$	− 80.9563i	− 0.785983i	0.919542	−	0.392992i	$-0.128560\pi$
$104$	50.5266i	0.485833i
$105$	0	0
$106$	−48.8528	−0.460876
$107$	169.456	1.58370	0.791850	−	0.610716i	$-0.209118\pi$
$107$	169.456	1.58370	0.791850	+	0.610716i	$0.209118\pi$
$108$	10.3923i	0.0962250i
$109$	178.941	1.64166	0.820831	−	0.571171i	$-0.193511\pi$
$109$	178.941	1.64166	0.820831	+	0.571171i	$0.193511\pi$
$110$	70.9631i	0.645119i
$111$	− 10.3413i	− 0.0931650i
$112$	0	0
$113$	−17.3970	−0.153955	−0.0769777	−	0.997033i	$-0.524527\pi$
$113$	−17.3970	−0.153955	−0.0769777	+	0.997033i	$0.524527\pi$
$114$	−41.6985	−0.365776
$115$	112.532i	0.978542i
$116$	67.8823	0.585192
$117$	− 53.5916i	− 0.458048i
$118$	− 38.6995i	− 0.327962i
$119$	0	0
$120$	40.9706	0.341421
$121$	−85.0000	−0.702479
$122$	− 57.1067i	− 0.468088i
$123$	−61.0294	−0.496174
$124$	− 29.4959i	− 0.237870i
$125$	− 166.769i	− 1.33415i
$126$	0	0
$127$	167.426	1.31832	0.659159	−	0.752004i	$-0.270912\pi$
$127$	167.426	1.31832	0.659159	+	0.752004i	$0.270912\pi$
$128$	11.3137	0.0883883
$129$	− 26.8213i	− 0.207917i
$130$	−211.279	−1.62522
$131$	− 1.78304i	− 0.0136110i	−0.999977	−	0.00680549i	$-0.997834\pi$
$131$	− 1.78304i	− 0.0136110i	0.999977	−	0.00680549i	$-0.00216627\pi$
$132$	− 20.7846i	− 0.157459i
$133$	0	0
$134$	−161.782	−1.20733
$135$	−43.4558	−0.321895
$136$	53.0482i	0.390061i
$137$	100.971	0.737011	0.368506	−	0.929625i	$-0.379870\pi$
$137$	100.971	0.737011	0.368506	+	0.929625i	$0.379870\pi$
$138$	− 32.9600i	− 0.238840i
$139$	140.542i	1.01110i	0.862799	+	0.505548i	$0.168710\pi$
$139$	140.542i	1.01110i	−0.862799	+	0.505548i	$0.831290\pi$
$140$	0	0
$141$	−57.5147	−0.407906
$142$	26.3087	0.185272
$143$	107.183i	0.749533i
$144$	−12.0000	−0.0833333
$145$	283.852i	1.95760i
$146$	165.508i	1.13362i
$147$	0	0
$148$	11.9411	0.0806833
$149$	182.912	1.22760	0.613798	−	0.789463i	$-0.289641\pi$
$149$	182.912	1.22760	0.613798	+	0.789463i	$0.289641\pi$
$150$	110.083i	0.733886i
$151$	−288.794	−1.91254	−0.956271	−	0.292481i	$-0.905519\pi$
$151$	−288.794	−1.91254	−0.956271	+	0.292481i	$0.905519\pi$
$152$	− 48.1493i	− 0.316771i
$153$	− 56.2662i	− 0.367753i
$154$	0	0
$155$	123.338	0.795730
$156$	61.8823	0.396681
$157$	− 187.061i	− 1.19147i	−0.803179	−	0.595737i	$-0.796860\pi$
$157$	− 187.061i	− 1.19147i	0.803179	−	0.595737i	$-0.203140\pi$
$158$	−124.929	−0.790689
$159$	59.8322i	0.376303i
$160$	47.3087i	0.295680i
$161$	0	0
$162$	12.7279	0.0785674
$163$	16.0589	0.0985207	0.0492604	−	0.998786i	$-0.484314\pi$
$163$	16.0589	0.0985207	0.0492604	+	0.998786i	$0.484314\pi$
$164$	− 70.4707i	− 0.429700i
$165$	86.9117	0.526738
$166$	− 107.141i	− 0.645427i
$167$	176.117i	1.05459i	0.849681	+	0.527297i	$0.176794\pi$
$167$	176.117i	1.05459i	−0.849681	+	0.527297i	$0.823206\pi$
$168$	0	0
$169$	−150.118	−0.888271
$170$	−221.823	−1.30484
$171$	51.0700i	0.298655i
$172$	30.9706	0.180061
$173$	− 231.152i	− 1.33614i	−0.744098	−	0.668070i	$-0.767121\pi$
$173$	− 231.152i	− 1.33614i	0.744098	−	0.668070i	$-0.232879\pi$
$174$	− 83.1384i	− 0.477807i
$175$	0	0
$176$	24.0000	0.136364
$177$	−47.3970	−0.267779
$178$	− 29.3939i	− 0.165134i
$179$	−85.2792	−0.476420	−0.238210	−	0.971214i	$-0.576561\pi$
$179$	−85.2792	−0.476420	−0.238210	+	0.971214i	$0.576561\pi$
$180$	− 50.1785i	− 0.278769i
$181$	5.58655i	0.0308649i	0.999881	+	0.0154325i	$0.00491250\pi$
$181$	5.58655i	0.0308649i	−0.999881	+	0.0154325i	$0.995087\pi$
$182$	0	0
$183$	−69.9411	−0.382192
$184$	38.0589	0.206842
$185$	49.9323i	0.269904i
$186$	−36.1249	−0.194220
$187$	112.532i	0.601777i
$188$	− 66.4123i	− 0.353257i
$189$	0	0
$190$	201.338	1.05967
$191$	185.397	0.970665	0.485332	−	0.874330i	$-0.338699\pi$
$191$	185.397	0.970665	0.485332	+	0.874330i	$0.338699\pi$
$192$	− 13.8564i	− 0.0721688i
$193$	227.794	1.18028	0.590140	−	0.807301i	$-0.299073\pi$
$193$	227.794	1.18028	0.590140	+	0.807301i	$0.299073\pi$
$194$	− 43.2503i	− 0.222940i
$195$	258.763i	1.32699i
$196$	0	0
$197$	123.161	0.625185	0.312593	−	0.949887i	$-0.398803\pi$
$197$	123.161	0.625185	0.312593	+	0.949887i	$0.398803\pi$
$198$	−25.4558	−0.128565
$199$	− 6.23188i	− 0.0313160i	−0.999877	−	0.0156580i	$-0.995016\pi$
$199$	− 6.23188i	− 0.0313160i	0.999877	−	0.0156580i	$-0.00498430\pi$
$200$	−127.113	−0.635563
$201$	198.141i	0.985778i
$202$	− 180.974i	− 0.895910i
$203$	0	0
$204$	64.9706	0.318483
$205$	294.676	1.43744
$206$	− 114.489i	− 0.555774i
$207$	−40.3675	−0.195012
$208$	71.4555i	0.343536i
$209$	− 102.140i	− 0.488708i
$210$	0	0
$211$	−124.912	−0.591999	−0.295999	−	0.955188i	$-0.595653\pi$
$211$	−124.912	−0.591999	−0.295999	+	0.955188i	$0.595653\pi$
$212$	−69.0883	−0.325888
$213$	− 32.2214i	− 0.151274i
$214$	239.647	1.11984
$215$	129.505i	0.602347i
$216$	14.6969i	0.0680414i
$217$	0	0
$218$	253.061	1.16083
$219$	202.706	0.925596
$220$	100.357i	0.456168i
$221$	−335.044	−1.51603
$222$	− 14.6248i	− 0.0658776i
$223$	228.631i	1.02525i	0.858613	+	0.512625i	$0.171327\pi$
$223$	228.631i	1.02525i	−0.858613	+	0.512625i	$0.828673\pi$
$224$	0	0
$225$	134.823	0.599215
$226$	−24.6030	−0.108863
$227$	169.537i	0.746859i	0.927659	+	0.373430i	$0.121818\pi$
$227$	169.537i	0.746859i	−0.927659	+	0.373430i	$0.878182\pi$
$228$	−58.9706	−0.258643
$229$	34.6920i	0.151493i	0.997127	+	0.0757467i	$0.0241340\pi$
$229$	34.6920i	0.151493i	−0.997127	+	0.0757467i	$0.975866\pi$
$230$	159.145i	0.691934i
$231$	0	0
$232$	96.0000	0.413793
$233$	−254.485	−1.09221	−0.546106	−	0.837716i	$-0.683890\pi$
$233$	−254.485	−1.09221	−0.546106	+	0.837716i	$0.683890\pi$
$234$	− 75.7900i	− 0.323889i
$235$	277.706	1.18173
$236$	− 54.7293i	− 0.231904i
$237$	153.006i	0.645595i
$238$	0	0
$239$	197.147	0.824884	0.412442	−	0.910984i	$-0.364676\pi$
$239$	197.147	0.824884	0.412442	+	0.910984i	$0.364676\pi$
$240$	57.9411	0.241421
$241$	− 88.4701i	− 0.367096i	−0.983011	−	0.183548i	$-0.941242\pi$
$241$	− 88.4701i	− 0.367096i	0.983011	−	0.183548i	$-0.0587582\pi$
$242$	−120.208	−0.496728
$243$	− 15.5885i	− 0.0641500i
$244$	− 80.7611i	− 0.330988i
$245$	0	0
$246$	−86.3087	−0.350848
$247$	304.103	1.23118
$248$	− 41.7134i	− 0.168199i
$249$	−131.220	−0.526989
$250$	− 235.847i	− 0.943389i
$251$	215.903i	0.860172i	0.902788	+	0.430086i	$0.141517\pi$
$251$	215.903i	0.860172i	−0.902788	+	0.430086i	$0.858483\pi$
$252$	0	0
$253$	80.7351	0.319111
$254$	236.777	0.932192
$255$	271.677i	1.06540i
$256$	16.0000	0.0625000
$257$	− 4.30463i	− 0.0167495i	−0.999965	−	0.00837477i	$-0.997334\pi$
$257$	− 4.30463i	− 0.0167495i	0.999965	−	0.00837477i	$-0.00266580\pi$
$258$	− 37.9310i	− 0.147020i
$259$	0	0
$260$	−298.794	−1.14921
$261$	−101.823	−0.390128
$262$	− 2.52160i	− 0.00962441i
$263$	282.676	1.07481	0.537407	−	0.843323i	$-0.319404\pi$
$263$	282.676	1.07481	0.537407	+	0.843323i	$0.319404\pi$
$264$	− 29.3939i	− 0.111340i
$265$	− 288.896i	− 1.09017i
$266$	0	0
$267$	−36.0000	−0.134831
$268$	−228.794	−0.853709
$269$	381.934i	1.41983i	0.704288	+	0.709914i	$0.251266\pi$
$269$	381.934i	1.41983i	−0.704288	+	0.709914i	$0.748734\pi$
$270$	−61.4558	−0.227614
$271$	− 84.3271i	− 0.311170i	−0.987822	−	0.155585i	$-0.950274\pi$
$271$	− 84.3271i	− 0.311170i	0.987822	−	0.155585i	$-0.0497263\pi$
$272$	75.0215i	0.275814i
$273$	0	0
$274$	142.794	0.521146
$275$	−269.647	−0.980534
$276$	− 46.6124i	− 0.168886i
$277$	−137.118	−0.495010	−0.247505	−	0.968887i	$-0.579611\pi$
$277$	−137.118	−0.495010	−0.247505	+	0.968887i	$0.579611\pi$
$278$	198.757i	0.714953i
$279$	44.2438i	0.158580i
$280$	0	0
$281$	−325.103	−1.15695	−0.578474	−	0.815701i	$-0.696352\pi$
$281$	−325.103	−1.15695	−0.578474	+	0.815701i	$0.696352\pi$
$282$	−81.3381	−0.288433
$283$	− 194.575i	− 0.687545i	−0.939053	−	0.343773i	$-0.888295\pi$
$283$	− 194.575i	− 0.687545i	0.939053	−	0.343773i	$-0.111705\pi$
$284$	37.2061	0.131007
$285$	− 246.588i	− 0.865220i
$286$	151.580i	0.530000i
$287$	0	0
$288$	−16.9706	−0.0589256
$289$	−62.7645	−0.217178
$290$	401.428i	1.38423i
$291$	−52.9706	−0.182029
$292$	234.064i	0.801590i
$293$	239.702i	0.818095i	0.912513	+	0.409048i	$0.134139\pi$
$293$	239.702i	0.818095i	−0.912513	+	0.409048i	$0.865861\pi$
$294$	0	0
$295$	228.853	0.775772
$296$	16.8873	0.0570517
$297$	31.1769i	0.104973i
$298$	258.676	0.868041
$299$	240.373i	0.803924i
$300$	155.681i	0.518935i
$301$	0	0
$302$	−408.416	−1.35237
$303$	−221.647	−0.731507
$304$	− 68.0933i	− 0.223991i
$305$	337.706	1.10723
$306$	− 79.5724i	− 0.260040i
$307$	− 540.272i	− 1.75984i	−0.475120	−	0.879921i	$-0.657595\pi$
$307$	− 540.272i	− 1.75984i	0.475120	−	0.879921i	$-0.342405\pi$
$308$	0	0
$309$	−140.220	−0.453788
$310$	174.426	0.562666
$311$	− 404.196i	− 1.29966i	−0.760078	−	0.649832i	$-0.774839\pi$
$311$	− 404.196i	− 1.29966i	0.760078	−	0.649832i	$-0.225161\pi$
$312$	87.5147	0.280496
$313$	− 131.296i	− 0.419477i	−0.977757	−	0.209739i	$-0.932739\pi$
$313$	− 131.296i	− 0.419477i	0.977757	−	0.209739i	$-0.0672613\pi$
$314$	− 264.545i	− 0.842500i
$315$	0	0
$316$	−176.676	−0.559102
$317$	−93.9411	−0.296344	−0.148172	−	0.988962i	$-0.547339\pi$
$317$	−93.9411	−0.296344	−0.148172	+	0.988962i	$0.547339\pi$
$318$	84.6156i	0.266087i
$319$	203.647	0.638391
$320$	66.9046i	0.209077i
$321$	− 293.506i	− 0.914349i
$322$	0	0
$323$	319.279	0.988481
$324$	18.0000	0.0555556
$325$	− 802.822i	− 2.47022i
$326$	22.7107	0.0696647
$327$	− 309.935i	− 0.947814i
$328$	− 99.6607i	− 0.303843i
$329$	0	0
$330$	122.912	0.372460
$331$	−261.368	−0.789630	−0.394815	−	0.918761i	$-0.629191\pi$
$331$	−261.368	−0.789630	−0.394815	+	0.918761i	$0.629191\pi$
$332$	− 151.520i	− 0.456386i
$333$	−17.9117	−0.0537889
$334$	249.067i	0.745710i
$335$	− 956.711i	− 2.85585i
$336$	0	0
$337$	136.265	0.404347	0.202173	−	0.979350i	$-0.435200\pi$
$337$	136.265	0.404347	0.202173	+	0.979350i	$0.435200\pi$
$338$	−212.299	−0.628102
$339$	30.1324i	0.0888862i
$340$	−313.706	−0.922664
$341$	− 88.4876i	− 0.259494i
$342$	72.2239i	0.211181i
$343$	0	0
$344$	43.7990	0.127323
$345$	194.912	0.564961
$346$	− 326.899i	− 0.944794i
$347$	−322.191	−0.928504	−0.464252	−	0.885703i	$-0.653677\pi$
$347$	−322.191	−0.928504	−0.464252	+	0.885703i	$0.653677\pi$
$348$	− 117.576i	− 0.337861i
$349$	− 346.495i	− 0.992821i	−0.868088	−	0.496411i	$-0.834651\pi$
$349$	− 346.495i	− 0.992821i	0.868088	−	0.496411i	$-0.165349\pi$
$350$	0	0
$351$	−92.8234	−0.264454
$352$	33.9411	0.0964237
$353$	620.591i	1.75805i	0.476777	+	0.879025i	$0.341805\pi$
$353$	620.591i	1.75805i	−0.476777	+	0.879025i	$0.658195\pi$
$354$	−67.0294	−0.189349
$355$	155.579i	0.438250i
$356$	− 41.5692i	− 0.116767i
$357$	0	0
$358$	−120.603	−0.336880
$359$	20.2355	0.0563663	0.0281831	−	0.999603i	$-0.491028\pi$
$359$	20.2355	0.0563663	0.0281831	+	0.999603i	$0.491028\pi$
$360$	− 70.9631i	− 0.197120i
$361$	71.2061	0.197247
$362$	7.90058i	0.0218248i
$363$	147.224i	0.405577i
$364$	0	0
$365$	−978.749	−2.68151
$366$	−98.9117	−0.270251
$367$	311.574i	0.848975i	0.905434	+	0.424488i	$0.139546\pi$
$367$	311.574i	0.848975i	−0.905434	+	0.424488i	$0.860454\pi$
$368$	53.8234	0.146259
$369$	105.706i	0.286466i
$370$	70.6149i	0.190851i
$371$	0	0
$372$	−51.0883	−0.137334
$373$	−681.382	−1.82676	−0.913380	−	0.407107i	$-0.866538\pi$
$373$	−681.382	−1.82676	−0.913380	+	0.407107i	$0.866538\pi$
$374$	159.145i	0.425521i
$375$	−288.853	−0.770274
$376$	− 93.9211i	− 0.249790i
$377$	606.320i	1.60828i
$378$	0	0
$379$	−624.779	−1.64849	−0.824246	−	0.566231i	$-0.808401\pi$
$379$	−624.779	−1.64849	−0.824246	+	0.566231i	$0.808401\pi$
$380$	284.735	0.749303
$381$	− 289.991i	− 0.761131i
$382$	262.191	0.686364
$383$	138.300i	0.361098i	0.983566	+	0.180549i	$0.0577874\pi$
$383$	138.300i	0.361098i	−0.983566	+	0.180549i	$0.942213\pi$
$384$	− 19.5959i	− 0.0510310i
$385$	0	0
$386$	322.149	0.834584
$387$	−46.4558	−0.120041
$388$	− 61.1651i	− 0.157642i
$389$	−563.574	−1.44878	−0.724388	−	0.689393i	$-0.757877\pi$
$389$	−563.574	−1.44878	−0.724388	+	0.689393i	$0.757877\pi$
$390$	365.946i	0.938324i
$391$	252.370i	0.645446i
$392$	0	0
$393$	−3.08831	−0.00785830
$394$	174.177	0.442073
$395$	− 738.779i	− 1.87033i
$396$	−36.0000	−0.0909091
$397$	− 453.338i	− 1.14191i	−0.820981	−	0.570955i	$-0.806573\pi$
$397$	− 453.338i	− 1.14191i	0.820981	−	0.570955i	$-0.193427\pi$
$398$	− 8.81321i	− 0.0221438i
$399$	0	0
$400$	−179.765	−0.449411
$401$	−275.750	−0.687656	−0.343828	−	0.939033i	$-0.611724\pi$
$401$	−275.750	−0.687656	−0.343828	+	0.939033i	$0.611724\pi$
$402$	280.214i	0.697050i
$403$	263.455	0.653734
$404$	− 255.936i	− 0.633504i
$405$	75.2677i	0.185846i
$406$	0	0
$407$	35.8234	0.0880181
$408$	91.8823	0.225202
$409$	435.831i	1.06560i	0.846240	+	0.532801i	$0.178861\pi$
$409$	435.831i	1.06560i	−0.846240	+	0.532801i	$0.821139\pi$
$410$	416.735	1.01643
$411$	− 174.886i	− 0.425514i
$412$	− 161.913i	− 0.392992i
$413$	0	0
$414$	−57.0883	−0.137894
$415$	633.588	1.52672
$416$	101.053i	0.242917i
$417$	243.426	0.583756
$418$	− 144.448i	− 0.345569i
$419$	301.257i	0.718991i	0.933147	+	0.359496i	$0.117051\pi$
$419$	301.257i	0.718991i	−0.933147	+	0.359496i	$0.882949\pi$
$420$	0	0
$421$	−203.794	−0.484071	−0.242036	−	0.970267i	$-0.577815\pi$
$421$	−203.794	−0.484071	−0.242036	+	0.970267i	$0.577815\pi$
$422$	−176.652	−0.418606
$423$	99.6184i	0.235505i
$424$	−97.7056	−0.230438
$425$	− 842.888i	− 1.98327i
$426$	− 45.5679i	− 0.106967i
$427$	0	0
$428$	338.912	0.791850
$429$	185.647	0.432743
$430$	183.147i	0.425924i
$431$	−395.720	−0.918144	−0.459072	−	0.888399i	$-0.651818\pi$
$431$	−395.720	−0.918144	−0.459072	+	0.888399i	$0.651818\pi$
$432$	20.7846i	0.0481125i
$433$	44.2685i	0.102237i	0.998693	+	0.0511184i	$0.0162786\pi$
$433$	44.2685i	0.102237i	−0.998693	+	0.0511184i	$0.983721\pi$
$434$	0	0
$435$	491.647	1.13022
$436$	357.882	0.820831
$437$	− 229.063i	− 0.524172i
$438$	286.669	0.654496
$439$	397.862i	0.906291i	0.891437	+	0.453146i	$0.149698\pi$
$439$	397.862i	0.906291i	−0.891437	+	0.453146i	$0.850302\pi$
$440$	141.926i	0.322560i
$441$	0	0
$442$	−473.823	−1.07200
$443$	−118.544	−0.267594	−0.133797	−	0.991009i	$-0.542717\pi$
$443$	−118.544	−0.267594	−0.133797	+	0.991009i	$0.542717\pi$
$444$	− 20.6826i	− 0.0465825i
$445$	173.823	0.390614
$446$	323.333i	0.724961i
$447$	− 316.812i	− 0.708752i
$448$	0	0
$449$	713.897	1.58997	0.794985	−	0.606629i	$-0.207479\pi$
$449$	713.897	1.58997	0.794985	+	0.606629i	$0.207479\pi$
$450$	190.669	0.423709
$451$	− 211.412i	− 0.468763i
$452$	−34.7939	−0.0769777
$453$	500.206i	1.10421i
$454$	239.762i	0.528109i
$455$	0	0
$456$	−83.3970	−0.182888
$457$	−125.177	−0.273909	−0.136955	−	0.990577i	$-0.543731\pi$
$457$	−125.177	−0.273909	−0.136955	+	0.990577i	$0.543731\pi$
$458$	49.0619i	0.107122i
$459$	−97.4558	−0.212322
$460$	225.065i	0.489271i
$461$	− 655.767i	− 1.42249i	−0.702945	−	0.711244i	$-0.748132\pi$
$461$	− 655.767i	− 1.42249i	0.702945	−	0.711244i	$-0.251868\pi$
$462$	0	0
$463$	869.396	1.87775	0.938873	−	0.344265i	$-0.111872\pi$
$463$	869.396	1.87775	0.938873	+	0.344265i	$0.111872\pi$
$464$	135.765	0.292596
$465$	− 213.628i	− 0.459415i
$466$	−359.897	−0.772310
$467$	− 267.372i	− 0.572532i	−0.958150	−	0.286266i	$-0.907586\pi$
$467$	− 267.372i	− 0.572532i	0.958150	−	0.286266i	$-0.0924141\pi$
$468$	− 107.183i	− 0.229024i
$469$	0	0
$470$	392.735	0.835607
$471$	−324.000	−0.687898
$472$	− 77.3989i	− 0.163981i
$473$	92.9117	0.196431
$474$	216.383i	0.456505i
$475$	765.048i	1.61063i
$476$	0	0
$477$	103.632	0.217259
$478$	278.808	0.583281
$479$	− 271.737i	− 0.567300i	−0.958928	−	0.283650i	$-0.908455\pi$
$479$	− 271.737i	− 0.567300i	0.958928	−	0.283650i	$-0.0915454\pi$
$480$	81.9411	0.170711
$481$	106.657i	0.221741i
$482$	− 125.116i	− 0.259576i
$483$	0	0
$484$	−170.000	−0.351240
$485$	255.765	0.527349
$486$	− 22.0454i	− 0.0453609i
$487$	−561.514	−1.15301	−0.576503	−	0.817095i	$-0.695583\pi$
$487$	−561.514	−1.15301	−0.576503	+	0.817095i	$0.695583\pi$
$488$	− 114.213i	− 0.234044i
$489$	− 27.8148i	− 0.0568810i
$490$	0	0
$491$	−406.441	−0.827781	−0.413891	−	0.910327i	$-0.635830\pi$
$491$	−406.441	−0.827781	−0.413891	+	0.910327i	$0.635830\pi$
$492$	−122.059	−0.248087
$493$	636.579i	1.29124i
$494$	430.066	0.870579
$495$	− 150.535i	− 0.304112i
$496$	− 58.9917i	− 0.118935i
$497$	0	0
$498$	−185.574	−0.372638
$499$	−371.426	−0.744341	−0.372171	−	0.928164i	$-0.621386\pi$
$499$	−371.426	−0.744341	−0.372171	+	0.928164i	$0.621386\pi$
$500$	− 333.538i	− 0.667077i
$501$	305.044	0.608870
$502$	305.333i	0.608234i
$503$	− 64.6292i	− 0.128488i	−0.997934	−	0.0642438i	$-0.979536\pi$
$503$	− 64.6292i	− 0.128488i	0.997934	−	0.0642438i	$-0.0204635\pi$
$504$	0	0
$505$	1070.21	2.11922
$506$	114.177	0.225646
$507$	260.012i	0.512843i
$508$	334.853	0.659159
$509$	1006.77i	1.97794i	0.148125	+	0.988969i	$0.452676\pi$
$509$	1006.77i	1.97794i	−0.148125	+	0.988969i	$0.547324\pi$
$510$	384.209i	0.753352i
$511$	0	0
$512$	22.6274	0.0441942
$513$	88.4558	0.172429
$514$	− 6.08767i	− 0.0118437i
$515$	677.044	1.31465
$516$	− 53.6426i	− 0.103959i
$517$	− 199.237i	− 0.385371i
$518$	0	0
$519$	−400.368	−0.771421
$520$	−422.558	−0.812612
$521$	372.153i	0.714306i	0.934046	+	0.357153i	$0.116253\pi$
$521$	372.153i	0.714306i	−0.934046	+	0.357153i	$0.883747\pi$
$522$	−144.000	−0.275862
$523$	− 637.284i	− 1.21852i	−0.792972	−	0.609258i	$-0.791467\pi$
$523$	− 637.284i	− 1.21852i	0.792972	−	0.609258i	$-0.208533\pi$
$524$	− 3.56608i	− 0.00680549i
$525$	0	0
$526$	399.765	0.760009
$527$	276.603	0.524863
$528$	− 41.5692i	− 0.0787296i
$529$	−347.940	−0.657732
$530$	− 408.560i	− 0.770868i
$531$	82.0940i	0.154603i
$532$	0	0
$533$	629.440	1.18094
$534$	−50.9117	−0.0953402
$535$	1417.17i	2.64892i
$536$	−323.563	−0.603663
$537$	147.708i	0.275061i
$538$	540.136i	1.00397i
$539$	0	0
$540$	−86.9117	−0.160948
$541$	220.823	0.408176	0.204088	−	0.978953i	$-0.434577\pi$
$541$	220.823	0.408176	0.204088	+	0.978953i	$0.434577\pi$
$542$	− 119.257i	− 0.220031i
$543$	9.67619	0.0178199
$544$	106.096i	0.195030i
$545$	1496.50i	2.74587i
$546$	0	0
$547$	−160.676	−0.293741	−0.146870	−	0.989156i	$-0.546920\pi$
$547$	−160.676	−0.293741	−0.146870	+	0.989156i	$0.546920\pi$
$548$	201.941	0.368506
$549$	121.142i	0.220659i
$550$	−381.338	−0.693342
$551$	− 577.791i	− 1.04862i
$552$	− 65.9199i	− 0.119420i
$553$	0	0
$554$	−193.914	−0.350025
$555$	86.4853	0.155829
$556$	281.085i	0.505548i
$557$	−474.353	−0.851622	−0.425811	−	0.904812i	$-0.640011\pi$
$557$	−474.353	−0.851622	−0.425811	+	0.904812i	$0.640011\pi$
$558$	62.5701i	0.112133i
$559$	276.627i	0.494860i
$560$	0	0
$561$	194.912	0.347436
$562$	−459.765	−0.818086
$563$	496.868i	0.882537i	0.897375	+	0.441269i	$0.145471\pi$
$563$	496.868i	0.882537i	−0.897375	+	0.441269i	$0.854529\pi$
$564$	−115.029	−0.203953
$565$	− 145.492i	− 0.257508i
$566$	− 275.171i	− 0.486168i
$567$	0	0
$568$	52.6173	0.0926361
$569$	785.294	1.38013	0.690065	−	0.723748i	$-0.257582\pi$
$569$	785.294	1.38013	0.690065	+	0.723748i	$0.257582\pi$
$570$	− 348.728i	− 0.611803i
$571$	−715.043	−1.25226	−0.626132	−	0.779717i	$-0.715363\pi$
$571$	−715.043	−1.25226	−0.626132	+	0.779717i	$0.715363\pi$
$572$	214.366i	0.374766i
$573$	− 321.117i	− 0.560414i
$574$	0	0
$575$	−604.721	−1.05169
$576$	−24.0000	−0.0416667
$577$	− 772.630i	− 1.33905i	−0.742791	−	0.669524i	$-0.766498\pi$
$577$	− 772.630i	− 1.33905i	0.742791	−	0.669524i	$-0.233502\pi$
$578$	−88.7624	−0.153568
$579$	− 394.551i	− 0.681435i
$580$	567.705i	0.978801i
$581$	0	0
$582$	−74.9117	−0.128714
$583$	−207.265	−0.355514
$584$	331.017i	0.566810i
$585$	448.191	0.766138
$586$	338.990i	0.578481i
$587$	436.477i	0.743572i	0.928318	+	0.371786i	$0.121254\pi$
$587$	436.477i	0.743572i	−0.928318	+	0.371786i	$0.878746\pi$
$588$	0	0
$589$	−251.059	−0.426246
$590$	323.647	0.548554
$591$	− 213.322i	− 0.360951i
$592$	23.8823	0.0403416
$593$	834.152i	1.40666i	0.710861	+	0.703332i	$0.248305\pi$
$593$	834.152i	1.40666i	−0.710861	+	0.703332i	$0.751695\pi$
$594$	44.0908i	0.0742270i
$595$	0	0
$596$	365.823	0.613798
$597$	−10.7939	−0.0180803
$598$	339.939i	0.568460i
$599$	873.588	1.45841	0.729205	−	0.684295i	$-0.239890\pi$
$599$	873.588	1.45841	0.729205	+	0.684295i	$0.239890\pi$
$600$	220.166i	0.366943i
$601$	− 198.982i	− 0.331085i	−0.986203	−	0.165542i	$-0.947063\pi$
$601$	− 198.982i	− 0.331085i	0.986203	−	0.165542i	$-0.0529375\pi$
$602$	0	0
$603$	343.191	0.569139
$604$	−577.588	−0.956271
$605$	− 710.862i	− 1.17498i
$606$	−313.456	−0.517254
$607$	158.950i	0.261861i	0.991392	+	0.130930i	$0.0417964\pi$
$607$	158.950i	0.261861i	−0.991392	+	0.130930i	$0.958204\pi$
$608$	− 96.2985i	− 0.158386i
$609$	0	0
$610$	477.588	0.782931
$611$	593.190	0.970851
$612$	− 112.532i	− 0.183876i
$613$	−714.735	−1.16596	−0.582981	−	0.812486i	$-0.698114\pi$
$613$	−714.735	−1.16596	−0.582981	+	0.812486i	$0.698114\pi$
$614$	− 764.059i	− 1.24440i
$615$	− 510.394i	− 0.829909i
$616$	0	0
$617$	−639.381	−1.03627	−0.518137	−	0.855298i	$-0.673374\pi$
$617$	−639.381	−1.03627	−0.518137	+	0.855298i	$0.673374\pi$
$618$	−198.302	−0.320876
$619$	− 172.025i	− 0.277908i	−0.990299	−	0.138954i	$-0.955626\pi$
$619$	− 172.025i	− 0.277908i	0.990299	−	0.138954i	$-0.0443740\pi$
$620$	246.676	0.397865
$621$	69.9186i	0.112590i
$622$	− 571.619i	− 0.919002i
$623$	0	0
$624$	123.765	0.198341
$625$	271.177	0.433883
$626$	− 185.681i	− 0.296615i
$627$	−176.912	−0.282156
$628$	− 374.123i	− 0.595737i
$629$	111.980i	0.178029i
$630$	0	0
$631$	−1141.06	−1.80833	−0.904166	−	0.427180i	$-0.859507\pi$
$631$	−1141.06	−1.80833	−0.904166	+	0.427180i	$0.859507\pi$
$632$	−249.858	−0.395345
$633$	216.353i	0.341791i
$634$	−132.853	−0.209547
$635$	1400.20i	2.20504i
$636$	119.664i	0.188152i
$637$	0	0
$638$	288.000	0.451411
$639$	−55.8091	−0.0873382
$640$	94.6175i	0.147840i
$641$	229.103	0.357414	0.178707	−	0.983902i	$-0.442809\pi$
$641$	229.103	0.357414	0.178707	+	0.983902i	$0.442809\pi$
$642$	− 415.080i	− 0.646543i
$643$	− 707.670i	− 1.10058i	−0.834975	−	0.550288i	$-0.814518\pi$
$643$	− 707.670i	− 1.10058i	0.834975	−	0.550288i	$-0.185482\pi$
$644$	0	0
$645$	224.309	0.347765
$646$	451.529	0.698961
$647$	− 1179.37i	− 1.82283i	−0.411484	−	0.911417i	$-0.634989\pi$
$647$	− 1179.37i	− 1.82283i	0.411484	−	0.911417i	$-0.365011\pi$
$648$	25.4558	0.0392837
$649$	− 164.188i	− 0.252986i
$650$	− 1135.36i	− 1.74671i
$651$	0	0
$652$	32.1177	0.0492604
$653$	−154.764	−0.237004	−0.118502	−	0.992954i	$-0.537809\pi$
$653$	−154.764	−0.237004	−0.118502	+	0.992954i	$0.537809\pi$
$654$	− 438.314i	− 0.670206i
$655$	14.9117	0.0227659
$656$	− 140.941i	− 0.214850i
$657$	− 351.096i	− 0.534393i
$658$	0	0
$659$	591.308	0.897280	0.448640	−	0.893712i	$-0.351908\pi$
$659$	591.308	0.897280	0.448640	+	0.893712i	$0.351908\pi$
$660$	173.823	0.263369
$661$	− 162.167i	− 0.245337i	−0.992448	−	0.122668i	$-0.960855\pi$
$661$	− 162.167i	− 0.245337i	0.992448	−	0.122668i	$-0.0391451\pi$
$662$	−369.630	−0.558353
$663$	580.313i	0.875283i
$664$	− 214.282i	− 0.322714i
$665$	0	0
$666$	−25.3310	−0.0380345
$667$	456.706	0.684717
$668$	352.234i	0.527297i
$669$	396.000	0.591928
$670$	− 1352.99i	− 2.01939i
$671$	− 242.283i	− 0.361078i
$672$	0	0
$673$	42.3238	0.0628883	0.0314441	−	0.999506i	$-0.489989\pi$
$673$	42.3238	0.0628883	0.0314441	+	0.999506i	$0.489989\pi$
$674$	192.708	0.285916
$675$	− 233.521i	− 0.345957i
$676$	−300.235	−0.444135
$677$	497.354i	0.734643i	0.930094	+	0.367322i	$0.119725\pi$
$677$	497.354i	0.734643i	−0.930094	+	0.367322i	$0.880275\pi$
$678$	42.6137i	0.0628521i
$679$	0	0
$680$	−443.647	−0.652422
$681$	293.647	0.431199
$682$	− 125.140i	− 0.183490i
$683$	1216.16	1.78062	0.890308	−	0.455359i	$-0.150489\pi$
$683$	1216.16	1.78062	0.890308	+	0.455359i	$0.150489\pi$
$684$	102.140i	0.149328i
$685$	844.425i	1.23274i
$686$	0	0
$687$	60.0883	0.0874648
$688$	61.9411	0.0900307
$689$	− 617.092i	− 0.895635i
$690$	275.647	0.399488
$691$	− 1076.39i	− 1.55773i	−0.627191	−	0.778865i	$-0.715796\pi$
$691$	− 1076.39i	− 1.55773i	0.627191	−	0.778865i	$-0.284204\pi$
$692$	− 462.305i	− 0.668070i
$693$	0	0
$694$	−455.647	−0.656552
$695$	−1175.37	−1.69118
$696$	− 166.277i	− 0.238904i
$697$	660.853	0.948139
$698$	− 490.017i	− 0.702031i
$699$	440.781i	0.630589i
$700$	0	0
$701$	−695.897	−0.992720	−0.496360	−	0.868117i	$-0.665330\pi$
$701$	−695.897	−0.992720	−0.496360	+	0.868117i	$0.665330\pi$
$702$	−131.272	−0.186997
$703$	− 101.639i	− 0.144579i
$704$	48.0000	0.0681818
$705$	− 481.000i	− 0.682270i
$706$	877.649i	1.24313i
$707$	0	0
$708$	−94.7939	−0.133890
$709$	254.824	0.359414	0.179707	−	0.983720i	$-0.442485\pi$
$709$	254.824	0.359414	0.179707	+	0.983720i	$0.442485\pi$
$710$	220.021i	0.309889i
$711$	265.014	0.372735
$712$	− 58.7878i	− 0.0825671i
$713$	− 198.446i	− 0.278325i
$714$	0	0
$715$	−896.382	−1.25368
$716$	−170.558	−0.238210
$717$	− 341.469i	− 0.476247i
$718$	28.6173	0.0398570
$719$	− 1114.20i	− 1.54965i	−0.632175	−	0.774826i	$-0.717838\pi$
$719$	− 1114.20i	− 1.54965i	0.632175	−	0.774826i	$-0.282162\pi$
$720$	− 100.357i	− 0.139385i
$721$	0	0
$722$	100.701	0.139474
$723$	−153.235	−0.211943
$724$	11.1731i	0.0154325i
$725$	−1525.35	−2.10393
$726$	208.207i	0.286786i
$727$	398.345i	0.547930i	0.961740	+	0.273965i	$0.0883353\pi$
$727$	398.345i	0.547930i	−0.961740	+	0.273965i	$0.911665\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	−1384.16	−1.89611
$731$	290.432i	0.397308i
$732$	−139.882	−0.191096
$733$	945.139i	1.28941i	0.764431	+	0.644706i	$0.223020\pi$
$733$	945.139i	1.28941i	−0.764431	+	0.644706i	$0.776980\pi$
$734$	440.632i	0.600316i
$735$	0	0
$736$	76.1177	0.103421
$737$	−686.382	−0.931319
$738$	149.491i	0.202562i
$739$	192.632	0.260666	0.130333	−	0.991470i	$-0.458395\pi$
$739$	192.632	0.260666	0.130333	+	0.991470i	$0.458395\pi$
$740$	99.8646i	0.134952i
$741$	− 526.721i	− 0.710825i
$742$	0	0
$743$	−911.616	−1.22694	−0.613470	−	0.789718i	$-0.710227\pi$
$743$	−911.616	−1.22694	−0.613470	+	0.789718i	$0.710227\pi$
$744$	−72.2498	−0.0971099
$745$	1529.71i	2.05330i
$746$	−963.619	−1.29172
$747$	227.280i	0.304257i
$748$	225.065i	0.300889i
$749$	0	0
$750$	−408.500	−0.544666
$751$	−391.662	−0.521521	−0.260760	−	0.965404i	$-0.583973\pi$
$751$	−391.662	−0.521521	−0.260760	+	0.965404i	$0.583973\pi$
$752$	− 132.825i	− 0.176628i
$753$	373.955	0.496621
$754$	857.466i	1.13722i
$755$	− 2415.21i	− 3.19895i
$756$	0	0
$757$	−152.823	−0.201879	−0.100940	−	0.994893i	$-0.532185\pi$
$757$	−152.823	−0.201879	−0.100940	+	0.994893i	$0.532185\pi$
$758$	−883.571	−1.16566
$759$	− 139.837i	− 0.184239i
$760$	402.676	0.529837
$761$	− 126.245i	− 0.165893i	−0.996554	−	0.0829465i	$-0.973567\pi$
$761$	− 126.245i	− 0.165893i	0.996554	−	0.0829465i	$-0.0264330\pi$
$762$	− 410.109i	− 0.538201i
$763$	0	0
$764$	370.794	0.485332
$765$	470.558	0.615109
$766$	195.586i	0.255335i
$767$	488.839	0.637338
$768$	− 27.7128i	− 0.0360844i
$769$	369.148i	0.480037i	0.970768	+	0.240018i	$0.0771535\pi$
$769$	369.148i	0.480037i	−0.970768	+	0.240018i	$0.922847\pi$
$770$	0	0
$771$	−7.45584	−0.00967036
$772$	455.588	0.590140
$773$	1403.71i	1.81592i	0.419056	+	0.907961i	$0.362361\pi$
$773$	1403.71i	1.81592i	−0.419056	+	0.907961i	$0.637639\pi$
$774$	−65.6985	−0.0848818
$775$	662.788i	0.855211i
$776$	− 86.5006i	− 0.111470i
$777$	0	0
$778$	−797.013	−1.02444
$779$	−599.823	−0.769991
$780$	517.526i	0.663495i
$781$	111.618	0.142917
$782$	356.904i	0.456400i
$783$	176.363i	0.225240i
$784$	0	0
$785$	1564.41	1.99288
$786$	−4.36753	−0.00555666
$787$	− 226.507i	− 0.287810i	−0.989591	−	0.143905i	$-0.954034\pi$
$787$	− 226.507i	− 0.287810i	0.989591	−	0.143905i	$-0.0459660\pi$
$788$	246.323	0.312593
$789$	− 489.610i	− 0.620544i
$790$	− 1044.79i	− 1.32252i
$791$	0	0
$792$	−50.9117	−0.0642824
$793$	721.352	0.909650
$794$	− 641.117i	− 0.807453i
$795$	−500.382	−0.629411
$796$	− 12.4638i	− 0.0156580i
$797$	688.414i	0.863756i	0.901932	+	0.431878i	$0.142149\pi$
$797$	688.414i	0.863756i	−0.901932	+	0.431878i	$0.857851\pi$
$798$	0	0
$799$	622.794	0.779467
$800$	−254.225	−0.317782
$801$	62.3538i	0.0778450i
$802$	−389.970	−0.486247
$803$	702.193i	0.874462i
$804$	396.283i	0.492889i
$805$	0	0
$806$	372.582	0.462260
$807$	661.529	0.819739
$808$	− 361.948i	− 0.447955i
$809$	25.2792	0.0312475	0.0156237	−	0.999878i	$-0.495027\pi$
$809$	25.2792	0.0312475	0.0156237	+	0.999878i	$0.495027\pi$
$810$	106.445i	0.131413i
$811$	− 1527.62i	− 1.88362i	−0.336145	−	0.941810i	$-0.609123\pi$
$811$	− 1527.62i	− 1.88362i	0.336145	−	0.941810i	$-0.390877\pi$
$812$	0	0
$813$	−146.059	−0.179654
$814$	50.6619	0.0622382
$815$	134.302i	0.164787i
$816$	129.941	0.159242
$817$	− 263.611i	− 0.322657i
$818$	616.359i	0.753495i
$819$	0	0
$820$	589.352	0.718722
$821$	−116.662	−0.142097	−0.0710487	−	0.997473i	$-0.522635\pi$
$821$	−116.662	−0.142097	−0.0710487	+	0.997473i	$0.522635\pi$
$822$	− 247.326i	− 0.300884i
$823$	125.911	0.152990	0.0764950	−	0.997070i	$-0.475627\pi$
$823$	125.911	0.152990	0.0764950	+	0.997070i	$0.475627\pi$
$824$	− 228.979i	− 0.277887i
$825$	467.042i	0.566111i
$826$	0	0
$827$	−1434.40	−1.73446	−0.867229	−	0.497910i	$-0.834101\pi$
$827$	−1434.40	−1.73446	−0.867229	+	0.497910i	$0.834101\pi$
$828$	−80.7351	−0.0975061
$829$	37.3228i	0.0450215i	0.999747	+	0.0225107i	$0.00716600\pi$
$829$	37.3228i	0.0450215i	−0.999747	+	0.0225107i	$0.992834\pi$
$830$	896.029	1.07955
$831$	237.495i	0.285794i
$832$	142.911i	0.171768i
$833$	0	0
$834$	344.257	0.412778
$835$	−1472.88	−1.76393
$836$	− 204.280i	− 0.244354i
$837$	76.6325	0.0915561
$838$	426.042i	0.508404i
$839$	− 3.07370i	− 0.00366353i	−0.999998	−	0.00183177i	$-0.999417\pi$
$839$	− 3.07370i	− 0.00366353i	0.999998	−	0.00183177i	$-0.000583069\pi$
$840$	0	0
$841$	311.000	0.369798
$842$	−288.208	−0.342290
$843$	563.094i	0.667965i
$844$	−249.823	−0.295999
$845$	− 1255.45i	− 1.48574i
$846$	140.882i	0.166527i
$847$	0	0
$848$	−138.177	−0.162944
$849$	−337.014	−0.396954
$850$	− 1192.02i	− 1.40238i
$851$	80.3390	0.0944054
$852$	− 64.4428i	− 0.0756371i
$853$	155.257i	0.182013i	0.995850	+	0.0910063i	$0.0290083\pi$
$853$	155.257i	0.182013i	−0.995850	+	0.0910063i	$0.970992\pi$
$854$	0	0
$855$	−427.103	−0.499535
$856$	479.294	0.559922
$857$	1603.86i	1.87148i	0.352686	+	0.935742i	$0.385269\pi$
$857$	1603.86i	1.87148i	−0.352686	+	0.935742i	$0.614731\pi$
$858$	262.544	0.305996
$859$	629.735i	0.733103i	0.930398	+	0.366551i	$0.119462\pi$
$859$	629.735i	0.733103i	−0.930398	+	0.366551i	$0.880538\pi$
$860$	259.009i	0.301174i
$861$	0	0
$862$	−559.632	−0.649226
$863$	1029.41	1.19283	0.596414	−	0.802677i	$-0.296592\pi$
$863$	1029.41	1.19283	0.596414	+	0.802677i	$0.296592\pi$
$864$	29.3939i	0.0340207i
$865$	1933.15	2.23485
$866$	62.6051i	0.0722923i
$867$	108.711i	0.125388i
$868$	0	0
$869$	−530.029	−0.609929
$870$	695.294	0.799188
$871$	− 2043.57i	− 2.34624i
$872$	506.122	0.580415
$873$	91.7477i	0.105095i
$874$	− 323.944i	− 0.370646i
$875$	0	0
$876$	405.411	0.462798
$877$	648.441	0.739385	0.369693	−	0.929154i	$-0.379463\pi$
$877$	648.441	0.739385	0.369693	+	0.929154i	$0.379463\pi$
$878$	562.662i	0.640845i
$879$	415.176	0.472327
$880$	200.714i	0.228084i
$881$	363.857i	0.413005i	0.978446	+	0.206502i	$0.0662082\pi$
$881$	363.857i	0.413005i	−0.978446	+	0.206502i	$0.933792\pi$
$882$	0	0
$883$	1536.16	1.73971	0.869853	−	0.493312i	$-0.164214\pi$
$883$	1536.16	1.73971	0.869853	+	0.493312i	$0.164214\pi$
$884$	−670.087	−0.758017
$885$	− 396.385i	− 0.447892i
$886$	−167.647	−0.189218
$887$	1125.51i	1.26889i	0.772966	+	0.634447i	$0.218772\pi$
$887$	1125.51i	1.26889i	−0.772966	+	0.634447i	$0.781228\pi$
$888$	− 29.2497i	− 0.0329388i
$889$	0	0
$890$	245.823	0.276206
$891$	54.0000	0.0606061
$892$	457.261i	0.512625i
$893$	−565.279	−0.633011
$894$	− 448.040i	− 0.501164i
$895$	− 713.197i	− 0.796868i
$896$	0	0
$897$	416.339	0.464146
$898$	1009.60	1.12428
$899$	− 500.561i	− 0.556798i
$900$	269.647	0.299608
$901$	− 647.889i	− 0.719078i
$902$	− 298.982i	− 0.331466i
$903$	0	0
$904$	−49.2061	−0.0544315
$905$	−46.7208	−0.0516252
$906$	707.398i	0.780792i
$907$	234.897	0.258982	0.129491	−	0.991581i	$-0.458666\pi$
$907$	234.897	0.258982	0.129491	+	0.991581i	$0.458666\pi$
$908$	339.074i	0.373430i
$909$	383.903i	0.422336i
$910$	0	0
$911$	224.278	0.246189	0.123095	−	0.992395i	$-0.460718\pi$
$911$	224.278	0.246189	0.123095	+	0.992395i	$0.460718\pi$
$912$	−117.941	−0.129321
$913$	− 454.561i	− 0.497876i
$914$	−177.026	−0.193683
$915$	− 584.923i	− 0.639260i
$916$	69.3840i	0.0757467i
$917$	0	0
$918$	−137.823	−0.150134
$919$	−932.161	−1.01432	−0.507161	−	0.861851i	$-0.669305\pi$
$919$	−932.161	−1.01432	−0.507161	+	0.861851i	$0.669305\pi$
$920$	318.289i	0.345967i
$921$	−935.778	−1.01605
$922$	− 927.394i	− 1.00585i
$923$	332.322i	0.360046i
$924$	0	0
$925$	−268.324	−0.290080
$926$	1229.51	1.32777
$927$	242.869i	0.261994i
$928$	192.000	0.206897
$929$	− 714.055i	− 0.768628i	−0.923203	−	0.384314i	$-0.874438\pi$
$929$	− 714.055i	− 0.768628i	0.923203	−	0.384314i	$-0.125562\pi$
$930$	− 302.115i	− 0.324855i
$931$	0	0
$932$	−508.971	−0.546106
$933$	−700.087	−0.750362
$934$	− 378.122i	− 0.404841i
$935$	−941.117	−1.00654
$936$	− 151.580i	− 0.161944i
$937$	1723.25i	1.83912i	0.392952	+	0.919559i	$0.371454\pi$
$937$	1723.25i	1.83912i	−0.392952	+	0.919559i	$0.628546\pi$
$938$	0	0
$939$	−227.412	−0.242185
$940$	555.411	0.590863
$941$	− 964.761i	− 1.02525i	−0.858612	−	0.512625i	$-0.828673\pi$
$941$	− 964.761i	− 1.02525i	0.858612	−	0.512625i	$-0.171327\pi$
$942$	−458.205	−0.486417
$943$	− 474.122i	− 0.502780i
$944$	− 109.459i	− 0.115952i
$945$	0	0
$946$	131.397	0.138897
$947$	1451.76	1.53301	0.766506	−	0.642237i	$-0.221993\pi$
$947$	1451.76	1.53301	0.766506	+	0.642237i	$0.221993\pi$
$948$	306.012i	0.322798i
$949$	−2090.65	−2.20300
$950$	1081.94i	1.13889i
$951$	162.711i	0.171094i
$952$	0	0
$953$	1147.43	1.20401	0.602007	−	0.798491i	$-0.294368\pi$
$953$	1147.43	1.20401	0.602007	+	0.798491i	$0.294368\pi$
$954$	146.558	0.153625
$955$	1550.49i	1.62355i
$956$	394.294	0.412442
$957$	− 352.727i	− 0.368575i
$958$	− 384.294i	− 0.401142i
$959$	0	0
$960$	115.882	0.120711
$961$	743.499	0.773672
$962$	150.836i	0.156794i
$963$	−508.368	−0.527900
$964$	− 176.940i	− 0.183548i
$965$	1905.06i	1.97415i
$966$	0	0
$967$	412.190	0.426257	0.213128	−	0.977024i	$-0.431635\pi$
$967$	412.190	0.426257	0.213128	+	0.977024i	$0.431635\pi$
$968$	−240.416	−0.248364
$969$	− 553.008i	− 0.570700i
$970$	361.706	0.372892
$971$	1004.12i	1.03411i	0.855952	+	0.517056i	$0.172972\pi$
$971$	1004.12i	1.03411i	−0.855952	+	0.517056i	$0.827028\pi$
$972$	− 31.1769i	− 0.0320750i
$973$	0	0
$974$	−794.101	−0.815298
$975$	−1390.53	−1.42618
$976$	− 161.522i	− 0.165494i
$977$	−1588.23	−1.62562	−0.812812	−	0.582527i	$-0.802064\pi$
$977$	−1588.23	−1.62562	−0.812812	+	0.582527i	$0.802064\pi$
$978$	− 39.3360i	− 0.0402209i
$979$	− 124.708i	− 0.127383i
$980$	0	0
$981$	−536.823	−0.547221
$982$	−574.794	−0.585330
$983$	− 833.533i	− 0.847948i	−0.905674	−	0.423974i	$-0.860635\pi$
$983$	− 833.533i	− 0.847948i	0.905674	−	0.423974i	$-0.139365\pi$
$984$	−172.617	−0.175424
$985$	1030.01i	1.04569i
$986$	900.259i	0.913041i
$987$	0	0
$988$	608.205	0.615592
$989$	208.368	0.210685
$990$	− 212.889i	− 0.215040i
$991$	66.8965	0.0675041	0.0337520	−	0.999430i	$-0.489254\pi$
$991$	66.8965	0.0675041	0.0337520	+	0.999430i	$0.489254\pi$
$992$	− 83.4269i	− 0.0840997i
$993$	452.702i	0.455893i
$994$	0	0
$995$	52.1177	0.0523796
$996$	−262.441	−0.263495
$997$	− 1464.91i	− 1.46931i	−0.678439	−	0.734657i	$-0.737343\pi$
$997$	− 1464.91i	− 1.46931i	0.678439	−	0.734657i	$-0.262657\pi$
$998$	−525.276	−0.526329
$999$	31.0240i	0.0310550i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.3.c.a.97.3		4
3.2	odd	2		882.3.c.b.685.1		4
4.3	odd	2		2352.3.f.e.97.4		4
7.2	even	3		42.3.g.a.31.1	yes	4
7.3	odd	6		42.3.g.a.19.1	✓	4
7.4	even	3		294.3.g.a.19.1		4
7.5	odd	6		294.3.g.a.31.1		4
7.6	odd	2	inner	294.3.c.a.97.4		4
21.2	odd	6		126.3.n.a.73.2		4
21.5	even	6		882.3.n.e.325.2		4
21.11	odd	6		882.3.n.e.19.2		4
21.17	even	6		126.3.n.a.19.2		4
21.20	even	2		882.3.c.b.685.2		4
28.3	even	6		336.3.bh.e.145.2		4
28.23	odd	6		336.3.bh.e.241.2		4
28.27	even	2		2352.3.f.e.97.1		4
35.2	odd	12		1050.3.q.a.199.4		8
35.3	even	12		1050.3.q.a.649.4		8
35.9	even	6		1050.3.p.a.451.2		4
35.17	even	12		1050.3.q.a.649.1		8
35.23	odd	12		1050.3.q.a.199.1		8
35.24	odd	6		1050.3.p.a.901.2		4
84.23	even	6		1008.3.cg.h.577.1		4
84.59	odd	6		1008.3.cg.h.145.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.3.g.a.19.1	✓	4	7.3	odd	6
42.3.g.a.31.1	yes	4	7.2	even	3
126.3.n.a.19.2		4	21.17	even	6
126.3.n.a.73.2		4	21.2	odd	6
294.3.c.a.97.3		4	1.1	even	1	trivial
294.3.c.a.97.4		4	7.6	odd	2	inner
294.3.g.a.19.1		4	7.4	even	3
294.3.g.a.31.1		4	7.5	odd	6
336.3.bh.e.145.2		4	28.3	even	6
336.3.bh.e.241.2		4	28.23	odd	6
882.3.c.b.685.1		4	3.2	odd	2
882.3.c.b.685.2		4	21.20	even	2
882.3.n.e.19.2		4	21.11	odd	6
882.3.n.e.325.2		4	21.5	even	6
1008.3.cg.h.145.1		4	84.59	odd	6
1008.3.cg.h.577.1		4	84.23	even	6
1050.3.p.a.451.2		4	35.9	even	6
1050.3.p.a.901.2		4	35.24	odd	6
1050.3.q.a.199.1		8	35.23	odd	12
1050.3.q.a.199.4		8	35.2	odd	12
1050.3.q.a.649.1		8	35.17	even	12
1050.3.q.a.649.4		8	35.3	even	12
2352.3.f.e.97.1		4	28.27	even	2
2352.3.f.e.97.4		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	294.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +8.36308i q^{5} -2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +8.36308i q^{5} -2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +11.8272i q^{10} +6.00000 q^{11} -3.46410i q^{12} +17.8639i q^{13} +14.4853 q^{15} +4.00000 q^{16} +18.7554i q^{17} -4.24264 q^{18} -17.0233i q^{19} +16.7262i q^{20} +8.48528 q^{22} +13.4558 q^{23} -4.89898i q^{24} -44.9411 q^{25} +25.2633i q^{26} +5.19615i q^{27} +33.9411 q^{29} +20.4853 q^{30} -14.7479i q^{31} +5.65685 q^{32} -10.3923i q^{33} +26.5241i q^{34} -6.00000 q^{36} +5.97056 q^{37} -24.0746i q^{38} +30.9411 q^{39} +23.6544i q^{40} -35.2354i q^{41} +15.4853 q^{43} +12.0000 q^{44} -25.0892i q^{45} +19.0294 q^{46} -33.2061i q^{47} -6.92820i q^{48} -63.5563 q^{50} +32.4853 q^{51} +35.7277i q^{52} -34.5442 q^{53} +7.34847i q^{54} +50.1785i q^{55} -29.4853 q^{57} +48.0000 q^{58} -27.3647i q^{59} +28.9706 q^{60} -40.3805i q^{61} -20.8567i q^{62} +8.00000 q^{64} -149.397 q^{65} -14.6969i q^{66} -114.397 q^{67} +37.5108i q^{68} -23.3062i q^{69} +18.6030 q^{71} -8.48528 q^{72} +117.032i q^{73} +8.44365 q^{74} +77.8403i q^{75} -34.0467i q^{76} +43.7574 q^{78} -88.3381 q^{79} +33.4523i q^{80} +9.00000 q^{81} -49.8303i q^{82} -75.7601i q^{83} -156.853 q^{85} +21.8995 q^{86} -58.7878i q^{87} +16.9706 q^{88} -20.7846i q^{89} -35.4815i q^{90} +26.9117 q^{92} -25.5442 q^{93} -46.9606i q^{94} +142.368 q^{95} -9.79796i q^{96} -30.5826i q^{97} -18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) 4 * q + 8 * q^4 - 12 * q^9	\( 4 q + 8 q^{4} - 12 q^{9} + 24 q^{11} + 24 q^{15} + 16 q^{16} - 48 q^{23} - 44 q^{25} + 48 q^{30} - 24 q^{36} - 44 q^{37} - 12 q^{39} + 28 q^{43} + 48 q^{44} + 144 q^{46} - 192 q^{50} + 96 q^{51} - 240 q^{53} - 84 q^{57} + 192 q^{58} + 48 q^{60} + 32 q^{64} - 360 q^{65} - 220 q^{67} + 312 q^{71} + 96 q^{74} + 192 q^{78} + 20 q^{79} + 36 q^{81} - 288 q^{85} + 48 q^{86} - 96 q^{92} - 204 q^{93} + 264 q^{95} - 72 q^{99}+O(q^{100}) \) 4 * q + 8 * q^4 - 12 * q^9 + 24 * q^11 + 24 * q^15 + 16 * q^16 - 48 * q^23 - 44 * q^25 + 48 * q^30 - 24 * q^36 - 44 * q^37 - 12 * q^39 + 28 * q^43 + 48 * q^44 + 144 * q^46 - 192 * q^50 + 96 * q^51 - 240 * q^53 - 84 * q^57 + 192 * q^58 + 48 * q^60 + 32 * q^64 - 360 * q^65 - 220 * q^67 + 312 * q^71 + 96 * q^74 + 192 * q^78 + 20 * q^79 + 36 * q^81 - 288 * q^85 + 48 * q^86 - 96 * q^92 - 204 * q^93 + 264 * q^95 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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