LMFDB - Embedded newform 6027.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.a (trivial)

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:	yes
Analytic conductor:	$48.1258372982$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 8x^{6} + 14x^{5} + 18x^{4} - 24x^{3} - 10x^{2} + 10x - 1 $ x^8 - 2x^7 - 8x^6 + 14x^5 + 18x^4 - 24x^3 - 10x^2 + 10*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.28873$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.28873 q^{2} -1.00000 q^{3} +3.23826 q^{4} +2.92705 q^{5} +2.28873 q^{6} -2.83405 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.28873 q^{2} -1.00000 q^{3} +3.23826 q^{4} +2.92705 q^{5} +2.28873 q^{6} -2.83405 q^{8} +1.00000 q^{9} -6.69922 q^{10} -0.608757 q^{11} -3.23826 q^{12} -3.41992 q^{13} -2.92705 q^{15} +0.00982553 q^{16} -2.88681 q^{17} -2.28873 q^{18} +2.02704 q^{19} +9.47856 q^{20} +1.39328 q^{22} -3.71591 q^{23} +2.83405 q^{24} +3.56763 q^{25} +7.82726 q^{26} -1.00000 q^{27} -2.68366 q^{29} +6.69922 q^{30} -0.845922 q^{31} +5.64560 q^{32} +0.608757 q^{33} +6.60711 q^{34} +3.23826 q^{36} +4.12496 q^{37} -4.63933 q^{38} +3.41992 q^{39} -8.29540 q^{40} +1.00000 q^{41} +0.717546 q^{43} -1.97132 q^{44} +2.92705 q^{45} +8.50470 q^{46} +5.20024 q^{47} -0.00982553 q^{48} -8.16531 q^{50} +2.88681 q^{51} -11.0746 q^{52} +4.40902 q^{53} +2.28873 q^{54} -1.78186 q^{55} -2.02704 q^{57} +6.14216 q^{58} +8.32320 q^{59} -9.47856 q^{60} +5.34616 q^{61} +1.93608 q^{62} -12.9409 q^{64} -10.0103 q^{65} -1.39328 q^{66} +7.48135 q^{67} -9.34825 q^{68} +3.71591 q^{69} -10.5145 q^{71} -2.83405 q^{72} -4.75212 q^{73} -9.44089 q^{74} -3.56763 q^{75} +6.56408 q^{76} -7.82726 q^{78} +0.682550 q^{79} +0.0287598 q^{80} +1.00000 q^{81} -2.28873 q^{82} +1.24939 q^{83} -8.44983 q^{85} -1.64227 q^{86} +2.68366 q^{87} +1.72525 q^{88} +3.56714 q^{89} -6.69922 q^{90} -12.0331 q^{92} +0.845922 q^{93} -11.9019 q^{94} +5.93324 q^{95} -5.64560 q^{96} -13.5915 q^{97} -0.608757 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 2 * q^2 - 8 * q^3 + 4 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^8 + 8 * q^9	$ 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 4 q^{13} - 2 q^{15} + 8 q^{17} - 2 q^{18} + 6 q^{19} + 4 q^{20} - 14 q^{22} - 12 q^{23} + 6 q^{24} - 4 q^{25} + 4 q^{26} - 8 q^{27} - 4 q^{29} + 2 q^{30} - 10 q^{31} - 4 q^{32} + 2 q^{33} + 4 q^{34} + 4 q^{36} - 20 q^{37} - 18 q^{38} - 4 q^{39} + 12 q^{40} + 8 q^{41} - 8 q^{43} + 20 q^{44} + 2 q^{45} - 12 q^{46} + 24 q^{47} - 22 q^{50} - 8 q^{51} - 30 q^{52} - 36 q^{53} + 2 q^{54} + 4 q^{55} - 6 q^{57} + 14 q^{58} + 10 q^{59} - 4 q^{60} - 22 q^{61} + 30 q^{62} - 24 q^{64} + 8 q^{65} + 14 q^{66} - 14 q^{67} + 38 q^{68} + 12 q^{69} - 10 q^{71} - 6 q^{72} - 12 q^{73} - 2 q^{74} + 4 q^{75} + 32 q^{76} - 4 q^{78} + 16 q^{79} - 14 q^{80} + 8 q^{81} - 2 q^{82} + 24 q^{83} - 44 q^{85} + 36 q^{86} + 4 q^{87} - 34 q^{88} + 2 q^{89} - 2 q^{90} - 48 q^{92} + 10 q^{93} - 34 q^{94} - 24 q^{95} + 4 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) $ 8 * q - 2 * q^2 - 8 * q^3 + 4 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^8 + 8 * q^9 - 2 * q^10 - 2 * q^11 - 4 * q^12 + 4 * q^13 - 2 * q^15 + 8 * q^17 - 2 * q^18 + 6 * q^19 + 4 * q^20 - 14 * q^22 - 12 * q^23 + 6 * q^24 - 4 * q^25 + 4 * q^26 - 8 * q^27 - 4 * q^29 + 2 * q^30 - 10 * q^31 - 4 * q^32 + 2 * q^33 + 4 * q^34 + 4 * q^36 - 20 * q^37 - 18 * q^38 - 4 * q^39 + 12 * q^40 + 8 * q^41 - 8 * q^43 + 20 * q^44 + 2 * q^45 - 12 * q^46 + 24 * q^47 - 22 * q^50 - 8 * q^51 - 30 * q^52 - 36 * q^53 + 2 * q^54 + 4 * q^55 - 6 * q^57 + 14 * q^58 + 10 * q^59 - 4 * q^60 - 22 * q^61 + 30 * q^62 - 24 * q^64 + 8 * q^65 + 14 * q^66 - 14 * q^67 + 38 * q^68 + 12 * q^69 - 10 * q^71 - 6 * q^72 - 12 * q^73 - 2 * q^74 + 4 * q^75 + 32 * q^76 - 4 * q^78 + 16 * q^79 - 14 * q^80 + 8 * q^81 - 2 * q^82 + 24 * q^83 - 44 * q^85 + 36 * q^86 + 4 * q^87 - 34 * q^88 + 2 * q^89 - 2 * q^90 - 48 * q^92 + 10 * q^93 - 34 * q^94 - 24 * q^95 + 4 * q^96 + 16 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−2.28873	−1.61837	−0.809187	−	0.587552i	$-0.800092\pi$
$2$	−2.28873	−1.61837	−0.809187	+	0.587552i	$0.800092\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.23826	1.61913
$5$	2.92705	1.30902	0.654508	−	0.756055i	$-0.272876\pi$
$5$	2.92705	1.30902	0.654508	+	0.756055i	$0.272876\pi$
$6$	2.28873	0.934368
$7$	0	0
$7$	0	0
$8$	−2.83405	−1.00199
$9$	1.00000	0.333333
$10$	−6.69922	−2.11848
$11$	−0.608757	−0.183547	−0.0917736	−	0.995780i	$-0.529254\pi$
$11$	−0.608757	−0.183547	−0.0917736	+	0.995780i	$0.529254\pi$
$12$	−3.23826	−0.934806
$13$	−3.41992	−0.948516	−0.474258	−	0.880386i	$-0.657284\pi$
$13$	−3.41992	−0.948516	−0.474258	+	0.880386i	$0.657284\pi$
$14$	0	0
$15$	−2.92705	−0.755761
$16$	0.00982553	0.00245638
$17$	−2.88681	−0.700154	−0.350077	−	0.936721i	$-0.613845\pi$
$17$	−2.88681	−0.700154	−0.350077	+	0.936721i	$0.613845\pi$
$18$	−2.28873	−0.539458
$19$	2.02704	0.465034	0.232517	−	0.972592i	$-0.425304\pi$
$19$	2.02704	0.465034	0.232517	+	0.972592i	$0.425304\pi$
$20$	9.47856	2.11947
$21$	0	0
$22$	1.39328	0.297048
$23$	−3.71591	−0.774821	−0.387411	−	0.921907i	$-0.626630\pi$
$23$	−3.71591	−0.774821	−0.387411	+	0.921907i	$0.626630\pi$
$24$	2.83405	0.578497
$25$	3.56763	0.713525
$26$	7.82726	1.53505
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.68366	−0.498343	−0.249172	−	0.968459i	$-0.580158\pi$
$29$	−2.68366	−0.498343	−0.249172	+	0.968459i	$0.580158\pi$
$30$	6.69922	1.22310
$31$	−0.845922	−0.151932	−0.0759661	−	0.997110i	$-0.524204\pi$
$31$	−0.845922	−0.151932	−0.0759661	+	0.997110i	$0.524204\pi$
$32$	5.64560	0.998011
$33$	0.608757	0.105971
$34$	6.60711	1.13311
$35$	0	0
$36$	3.23826	0.539711
$37$	4.12496	0.678139	0.339069	−	0.940761i	$-0.389888\pi$
$37$	4.12496	0.678139	0.339069	+	0.940761i	$0.389888\pi$
$38$	−4.63933	−0.752599
$39$	3.41992	0.547626
$40$	−8.29540	−1.31162
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	0.717546	0.109425	0.0547124	−	0.998502i	$-0.482576\pi$
$43$	0.717546	0.109425	0.0547124	+	0.998502i	$0.482576\pi$
$44$	−1.97132	−0.297187
$45$	2.92705	0.436339
$46$	8.50470	1.25395
$47$	5.20024	0.758532	0.379266	−	0.925288i	$-0.376176\pi$
$47$	5.20024	0.758532	0.379266	+	0.925288i	$0.376176\pi$
$48$	−0.00982553	−0.00141819
$49$	0	0
$50$	−8.16531	−1.15475
$51$	2.88681	0.404234
$52$	−11.0746	−1.53577
$53$	4.40902	0.605625	0.302813	−	0.953050i	$-0.402074\pi$
$53$	4.40902	0.605625	0.302813	+	0.953050i	$0.402074\pi$
$54$	2.28873	0.311456
$55$	−1.78186	−0.240266
$56$	0	0
$57$	−2.02704	−0.268488
$58$	6.14216	0.806505
$59$	8.32320	1.08359	0.541794	−	0.840511i	$-0.317745\pi$
$59$	8.32320	1.08359	0.541794	+	0.840511i	$0.317745\pi$
$60$	−9.47856	−1.22368
$61$	5.34616	0.684506	0.342253	−	0.939608i	$-0.388810\pi$
$61$	5.34616	0.684506	0.342253	+	0.939608i	$0.388810\pi$
$62$	1.93608	0.245883
$63$	0	0
$64$	−12.9409	−1.61761
$65$	−10.0103	−1.24162
$66$	−1.39328	−0.171501
$67$	7.48135	0.913993	0.456996	−	0.889469i	$-0.348925\pi$
$67$	7.48135	0.913993	0.456996	+	0.889469i	$0.348925\pi$
$68$	−9.34825	−1.13364
$69$	3.71591	0.447343
$70$	0	0
$71$	−10.5145	−1.24785	−0.623923	−	0.781486i	$-0.714462\pi$
$71$	−10.5145	−1.24785	−0.623923	+	0.781486i	$0.714462\pi$
$72$	−2.83405	−0.333996
$73$	−4.75212	−0.556194	−0.278097	−	0.960553i	$-0.589704\pi$
$73$	−4.75212	−0.556194	−0.278097	+	0.960553i	$0.589704\pi$
$74$	−9.44089	−1.09748
$75$	−3.56763	−0.411954
$76$	6.56408	0.752952
$77$	0	0
$78$	−7.82726	−0.886263
$79$	0.682550	0.0767929	0.0383964	−	0.999263i	$-0.487775\pi$
$79$	0.682550	0.0767929	0.0383964	+	0.999263i	$0.487775\pi$
$80$	0.0287598	0.00321545
$81$	1.00000	0.111111
$82$	−2.28873	−0.252747
$83$	1.24939	0.137139	0.0685694	−	0.997646i	$-0.478157\pi$
$83$	1.24939	0.137139	0.0685694	+	0.997646i	$0.478157\pi$
$84$	0	0
$85$	−8.44983	−0.916513
$86$	−1.64227	−0.177090
$87$	2.68366	0.287719
$88$	1.72525	0.183912
$89$	3.56714	0.378116	0.189058	−	0.981966i	$-0.439457\pi$
$89$	3.56714	0.378116	0.189058	+	0.981966i	$0.439457\pi$
$90$	−6.69922	−0.706159
$91$	0	0
$92$	−12.0331	−1.25454
$93$	0.845922	0.0877181
$94$	−11.9019	−1.22759
$95$	5.93324	0.608738
$96$	−5.64560	−0.576202
$97$	−13.5915	−1.38001	−0.690006	−	0.723804i	$-0.742392\pi$
$97$	−13.5915	−1.38001	−0.690006	+	0.723804i	$0.742392\pi$
$98$	0	0
$99$	−0.608757	−0.0611824
$100$	11.5529	1.15529
$101$	7.00621	0.697144	0.348572	−	0.937282i	$-0.386667\pi$
$101$	7.00621	0.697144	0.348572	+	0.937282i	$0.386667\pi$
$102$	−6.60711	−0.654201
$103$	−12.9840	−1.27935	−0.639677	−	0.768644i	$-0.720932\pi$
$103$	−12.9840	−1.27935	−0.639677	+	0.768644i	$0.720932\pi$
$104$	9.69222	0.950400
$105$	0	0
$106$	−10.0910	−0.980127
$107$	10.9971	1.06313	0.531566	−	0.847017i	$-0.321604\pi$
$107$	10.9971	1.06313	0.531566	+	0.847017i	$0.321604\pi$
$108$	−3.23826	−0.311602
$109$	3.47454	0.332801	0.166400	−	0.986058i	$-0.446786\pi$
$109$	3.47454	0.332801	0.166400	+	0.986058i	$0.446786\pi$
$110$	4.07819	0.388841
$111$	−4.12496	−0.391523
$112$	0	0
$113$	−15.1226	−1.42261	−0.711307	−	0.702882i	$-0.751896\pi$
$113$	−15.1226	−1.42261	−0.711307	+	0.702882i	$0.751896\pi$
$114$	4.63933	0.434513
$115$	−10.8767	−1.01425
$116$	−8.69040	−0.806883
$117$	−3.41992	−0.316172
$118$	−19.0495	−1.75365
$119$	0	0
$120$	8.29540	0.757263
$121$	−10.6294	−0.966310
$122$	−12.2359	−1.10779
$123$	−1.00000	−0.0901670
$124$	−2.73932	−0.245998
$125$	−4.19263	−0.375001
$126$	0	0
$127$	−5.54115	−0.491698	−0.245849	−	0.969308i	$-0.579067\pi$
$127$	−5.54115	−0.491698	−0.245849	+	0.969308i	$0.579067\pi$
$128$	18.3269	1.61989
$129$	−0.717546	−0.0631764
$130$	22.9108	2.00941
$131$	0.469897	0.0410551	0.0205276	−	0.999789i	$-0.493465\pi$
$131$	0.469897	0.0410551	0.0205276	+	0.999789i	$0.493465\pi$
$132$	1.97132	0.171581
$133$	0	0
$134$	−17.1228	−1.47918
$135$	−2.92705	−0.251920
$136$	8.18135	0.701545
$137$	−4.21932	−0.360481	−0.180241	−	0.983623i	$-0.557688\pi$
$137$	−4.21932	−0.360481	−0.180241	+	0.983623i	$0.557688\pi$
$138$	−8.50470	−0.723968
$139$	−13.0636	−1.10804	−0.554021	−	0.832503i	$-0.686907\pi$
$139$	−13.0636	−1.10804	−0.554021	+	0.832503i	$0.686907\pi$
$140$	0	0
$141$	−5.20024	−0.437939
$142$	24.0649	2.01948
$143$	2.08190	0.174097
$144$	0.00982553	0.000818794 0
$145$	−7.85521	−0.652340
$146$	10.8763	0.900129
$147$	0	0
$148$	13.3577	1.09800
$149$	18.7649	1.53728	0.768639	−	0.639683i	$-0.220934\pi$
$149$	18.7649	1.53728	0.768639	+	0.639683i	$0.220934\pi$
$150$	8.16531	0.666695
$151$	−12.1409	−0.988010	−0.494005	−	0.869459i	$-0.664468\pi$
$151$	−12.1409	−0.988010	−0.494005	+	0.869459i	$0.664468\pi$
$152$	−5.74472	−0.465958
$153$	−2.88681	−0.233385
$154$	0	0
$155$	−2.47606	−0.198882
$156$	11.0746	0.886678
$157$	−9.83418	−0.784853	−0.392427	−	0.919783i	$-0.628364\pi$
$157$	−9.83418	−0.784853	−0.392427	+	0.919783i	$0.628364\pi$
$158$	−1.56217	−0.124280
$159$	−4.40902	−0.349658
$160$	16.5250	1.30641
$161$	0	0
$162$	−2.28873	−0.179819
$163$	−9.56943	−0.749536	−0.374768	−	0.927119i	$-0.622278\pi$
$163$	−9.56943	−0.749536	−0.374768	+	0.927119i	$0.622278\pi$
$164$	3.23826	0.252866
$165$	1.78186	0.138718
$166$	−2.85952	−0.221942
$167$	−0.912899	−0.0706422	−0.0353211	−	0.999376i	$-0.511245\pi$
$167$	−0.912899	−0.0706422	−0.0353211	+	0.999376i	$0.511245\pi$
$168$	0	0
$169$	−1.30413	−0.100318
$170$	19.3393	1.48326
$171$	2.02704	0.155011
$172$	2.32360	0.177173
$173$	3.91474	0.297632	0.148816	−	0.988865i	$-0.452454\pi$
$173$	3.91474	0.297632	0.148816	+	0.988865i	$0.452454\pi$
$174$	−6.14216	−0.465636
$175$	0	0
$176$	−0.00598136	−0.000450862 0
$177$	−8.32320	−0.625610
$178$	−8.16421	−0.611933
$179$	16.6937	1.24775	0.623874	−	0.781525i	$-0.285558\pi$
$179$	16.6937	1.24775	0.623874	+	0.781525i	$0.285558\pi$
$180$	9.47856	0.706490
$181$	−7.61276	−0.565852	−0.282926	−	0.959142i	$-0.591305\pi$
$181$	−7.61276	−0.565852	−0.282926	+	0.959142i	$0.591305\pi$
$182$	0	0
$183$	−5.34616	−0.395200
$184$	10.5311	0.776360
$185$	12.0740	0.887695
$186$	−1.93608	−0.141961
$187$	1.75736	0.128511
$188$	16.8397	1.22816
$189$	0	0
$190$	−13.5796	−0.985165
$191$	3.70107	0.267800	0.133900	−	0.990995i	$-0.457250\pi$
$191$	3.70107	0.267800	0.133900	+	0.990995i	$0.457250\pi$
$192$	12.9409	0.933928
$193$	−25.6969	−1.84971	−0.924853	−	0.380326i	$-0.875812\pi$
$193$	−25.6969	−1.84971	−0.924853	+	0.380326i	$0.875812\pi$
$194$	31.1073	2.23337
$195$	10.0103	0.716851
$196$	0	0
$197$	−22.0996	−1.57453	−0.787266	−	0.616614i	$-0.788504\pi$
$197$	−22.0996	−1.57453	−0.787266	+	0.616614i	$0.788504\pi$
$198$	1.39328	0.0990159
$199$	1.01277	0.0717935	0.0358968	−	0.999356i	$-0.488571\pi$
$199$	1.01277	0.0717935	0.0358968	+	0.999356i	$0.488571\pi$
$200$	−10.1108	−0.714943
$201$	−7.48135	−0.527694
$202$	−16.0353	−1.12824
$203$	0	0
$204$	9.34825	0.654508
$205$	2.92705	0.204434
$206$	29.7169	2.07047
$207$	−3.71591	−0.258274
$208$	−0.0336026	−0.00232992
$209$	−1.23397	−0.0853557
$210$	0	0
$211$	13.5270	0.931236	0.465618	−	0.884986i	$-0.345832\pi$
$211$	13.5270	0.931236	0.465618	+	0.884986i	$0.345832\pi$
$212$	14.2776	0.980587
$213$	10.5145	0.720444
$214$	−25.1694	−1.72055
$215$	2.10029	0.143239
$216$	2.83405	0.192832
$217$	0	0
$218$	−7.95227	−0.538596
$219$	4.75212	0.321119
$220$	−5.77014	−0.389023
$221$	9.87266	0.664107
$222$	9.44089	0.633631
$223$	−14.5693	−0.975634	−0.487817	−	0.872946i	$-0.662207\pi$
$223$	−14.5693	−0.975634	−0.487817	+	0.872946i	$0.662207\pi$
$224$	0	0
$225$	3.56763	0.237842
$226$	34.6115	2.30232
$227$	6.91389	0.458891	0.229445	−	0.973322i	$-0.426309\pi$
$227$	6.91389	0.458891	0.229445	+	0.973322i	$0.426309\pi$
$228$	−6.56408	−0.434717
$229$	10.7145	0.708037	0.354019	−	0.935238i	$-0.384815\pi$
$229$	10.7145	0.708037	0.354019	+	0.935238i	$0.384815\pi$
$230$	24.8937	1.64144
$231$	0	0
$232$	7.60562	0.499333
$233$	−18.4752	−1.21035	−0.605176	−	0.796092i	$-0.706897\pi$
$233$	−18.4752	−1.21035	−0.605176	+	0.796092i	$0.706897\pi$
$234$	7.82726	0.511684
$235$	15.2214	0.992932
$236$	26.9527	1.75447
$237$	−0.682550	−0.0443364
$238$	0	0
$239$	12.2275	0.790930	0.395465	−	0.918481i	$-0.370583\pi$
$239$	12.2275	0.790930	0.395465	+	0.918481i	$0.370583\pi$
$240$	−0.0287598	−0.00185644
$241$	15.5057	0.998812	0.499406	−	0.866368i	$-0.333552\pi$
$241$	15.5057	0.998812	0.499406	+	0.866368i	$0.333552\pi$
$242$	24.3278	1.56385
$243$	−1.00000	−0.0641500
$244$	17.3123	1.10831
$245$	0	0
$246$	2.28873	0.145924
$247$	−6.93231	−0.441092
$248$	2.39738	0.152234
$249$	−1.24939	−0.0791771
$250$	9.59579	0.606891
$251$	−8.42100	−0.531529	−0.265765	−	0.964038i	$-0.585624\pi$
$251$	−8.42100	−0.531529	−0.265765	+	0.964038i	$0.585624\pi$
$252$	0	0
$253$	2.26209	0.142216
$254$	12.6822	0.795751
$255$	8.44983	0.529149
$256$	−16.0635	−1.00397
$257$	11.3723	0.709382	0.354691	−	0.934984i	$-0.384586\pi$
$257$	11.3723	0.709382	0.354691	+	0.934984i	$0.384586\pi$
$258$	1.64227	0.102243
$259$	0	0
$260$	−32.4159	−2.01035
$261$	−2.68366	−0.166114
$262$	−1.07547	−0.0664425
$263$	−15.9552	−0.983839	−0.491919	−	0.870641i	$-0.663704\pi$
$263$	−15.9552	−0.983839	−0.491919	+	0.870641i	$0.663704\pi$
$264$	−1.72525	−0.106182
$265$	12.9054	0.792773
$266$	0	0
$267$	−3.56714	−0.218305
$268$	24.2266	1.47988
$269$	−10.9890	−0.670010	−0.335005	−	0.942216i	$-0.608738\pi$
$269$	−10.9890	−0.670010	−0.335005	+	0.942216i	$0.608738\pi$
$270$	6.69922	0.407701
$271$	3.75539	0.228123	0.114062	−	0.993474i	$-0.463614\pi$
$271$	3.75539	0.228123	0.114062	+	0.993474i	$0.463614\pi$
$272$	−0.0283644	−0.00171985
$273$	0	0
$274$	9.65687	0.583393
$275$	−2.17182	−0.130965
$276$	12.0331	0.724308
$277$	−19.9041	−1.19592	−0.597961	−	0.801525i	$-0.704022\pi$
$277$	−19.9041	−1.19592	−0.597961	+	0.801525i	$0.704022\pi$
$278$	29.8990	1.79322
$279$	−0.845922	−0.0506440
$280$	0	0
$281$	5.65900	0.337588	0.168794	−	0.985651i	$-0.446013\pi$
$281$	5.65900	0.337588	0.168794	+	0.985651i	$0.446013\pi$
$282$	11.9019	0.708748
$283$	23.3837	1.39001	0.695007	−	0.719003i	$-0.255401\pi$
$283$	23.3837	1.39001	0.695007	+	0.719003i	$0.255401\pi$
$284$	−34.0488	−2.02043
$285$	−5.93324	−0.351455
$286$	−4.76490	−0.281754
$287$	0	0
$288$	5.64560	0.332670
$289$	−8.66634	−0.509785
$290$	17.9784	1.05573
$291$	13.5915	0.796750
$292$	−15.3886	−0.900551
$293$	14.4856	0.846256	0.423128	−	0.906070i	$-0.360932\pi$
$293$	14.4856	0.846256	0.423128	+	0.906070i	$0.360932\pi$
$294$	0	0
$295$	24.3624	1.41843
$296$	−11.6903	−0.679486
$297$	0.608757	0.0353237
$298$	−42.9476	−2.48789
$299$	12.7081	0.734930
$300$	−11.5529	−0.667008
$301$	0	0
$302$	27.7871	1.59897
$303$	−7.00621	−0.402496
$304$	0.0199167	0.00114230
$305$	15.6485	0.896030
$306$	6.60711	0.377703
$307$	−6.70376	−0.382604	−0.191302	−	0.981531i	$-0.561271\pi$
$307$	−6.70376	−0.382604	−0.191302	+	0.981531i	$0.561271\pi$
$308$	0	0
$309$	12.9840	0.738635
$310$	5.66702	0.321865
$311$	−7.41192	−0.420291	−0.210146	−	0.977670i	$-0.567394\pi$
$311$	−7.41192	−0.420291	−0.210146	+	0.977670i	$0.567394\pi$
$312$	−9.69222	−0.548714
$313$	−16.7688	−0.947829	−0.473914	−	0.880571i	$-0.657159\pi$
$313$	−16.7688	−0.947829	−0.473914	+	0.880571i	$0.657159\pi$
$314$	22.5077	1.27019
$315$	0	0
$316$	2.21028	0.124338
$317$	−6.67784	−0.375065	−0.187532	−	0.982258i	$-0.560049\pi$
$317$	−6.67784	−0.375065	−0.187532	+	0.982258i	$0.560049\pi$
$318$	10.0910	0.565877
$319$	1.63370	0.0914695
$320$	−37.8786	−2.11748
$321$	−10.9971	−0.613800
$322$	0	0
$323$	−5.85167	−0.325596
$324$	3.23826	0.179904
$325$	−12.2010	−0.676790
$326$	21.9018	1.21303
$327$	−3.47454	−0.192143
$328$	−2.83405	−0.156484
$329$	0	0
$330$	−4.07819	−0.224497
$331$	−22.2105	−1.22080	−0.610399	−	0.792094i	$-0.708991\pi$
$331$	−22.2105	−1.22080	−0.610399	+	0.792094i	$0.708991\pi$
$332$	4.04587	0.222046
$333$	4.12496	0.226046
$334$	2.08937	0.114325
$335$	21.8983	1.19643
$336$	0	0
$337$	−8.06165	−0.439146	−0.219573	−	0.975596i	$-0.570466\pi$
$337$	−8.06165	−0.439146	−0.219573	+	0.975596i	$0.570466\pi$
$338$	2.98480	0.162352
$339$	15.1226	0.821346
$340$	−27.3628	−1.48396
$341$	0.514961	0.0278867
$342$	−4.63933	−0.250866
$343$	0	0
$344$	−2.03356	−0.109642
$345$	10.8767	0.585580
$346$	−8.95976	−0.481680
$347$	7.12163	0.382309	0.191154	−	0.981560i	$-0.438777\pi$
$347$	7.12163	0.382309	0.191154	+	0.981560i	$0.438777\pi$
$348$	8.69040	0.465854
$349$	13.7173	0.734272	0.367136	−	0.930167i	$-0.380338\pi$
$349$	13.7173	0.734272	0.367136	+	0.930167i	$0.380338\pi$
$350$	0	0
$351$	3.41992	0.182542
$352$	−3.43680	−0.183182
$353$	−22.1845	−1.18076	−0.590380	−	0.807126i	$-0.701022\pi$
$353$	−22.1845	−1.18076	−0.590380	+	0.807126i	$0.701022\pi$
$354$	19.0495	1.01247
$355$	−30.7766	−1.63345
$356$	11.5513	0.612220
$357$	0	0
$358$	−38.2074	−2.01932
$359$	6.04012	0.318785	0.159393	−	0.987215i	$-0.449046\pi$
$359$	6.04012	0.318785	0.159393	+	0.987215i	$0.449046\pi$
$360$	−8.29540	−0.437206
$361$	−14.8911	−0.783743
$362$	17.4235	0.915760
$363$	10.6294	0.557900
$364$	0	0
$365$	−13.9097	−0.728067
$366$	12.2359	0.639581
$367$	1.35316	0.0706345	0.0353172	−	0.999376i	$-0.488756\pi$
$367$	1.35316	0.0706345	0.0353172	+	0.999376i	$0.488756\pi$
$368$	−0.0365108	−0.00190326
$369$	1.00000	0.0520579
$370$	−27.6340	−1.43662
$371$	0	0
$372$	2.73932	0.142027
$373$	−29.5094	−1.52794	−0.763970	−	0.645251i	$-0.776753\pi$
$373$	−29.5094	−1.52794	−0.763970	+	0.645251i	$0.776753\pi$
$374$	−4.02212	−0.207979
$375$	4.19263	0.216507
$376$	−14.7377	−0.760039
$377$	9.17791	0.472686
$378$	0	0
$379$	−5.78403	−0.297106	−0.148553	−	0.988904i	$-0.547462\pi$
$379$	−5.78403	−0.297106	−0.148553	+	0.988904i	$0.547462\pi$
$380$	19.2134	0.985627
$381$	5.54115	0.283882
$382$	−8.47073	−0.433400
$383$	8.68937	0.444006	0.222003	−	0.975046i	$-0.428740\pi$
$383$	8.68937	0.444006	0.222003	+	0.975046i	$0.428740\pi$
$384$	−18.3269	−0.935242
$385$	0	0
$386$	58.8132	2.99351
$387$	0.717546	0.0364749
$388$	−44.0130	−2.23442
$389$	−29.0515	−1.47297	−0.736484	−	0.676455i	$-0.763516\pi$
$389$	−29.0515	−1.47297	−0.736484	+	0.676455i	$0.763516\pi$
$390$	−22.9108	−1.16013
$391$	10.7271	0.542494
$392$	0	0
$393$	−0.469897	−0.0237032
$394$	50.5799	2.54818
$395$	1.99786	0.100523
$396$	−1.97132	−0.0990623
$397$	23.2522	1.16699	0.583497	−	0.812115i	$-0.301684\pi$
$397$	23.2522	1.16699	0.583497	+	0.812115i	$0.301684\pi$
$398$	−2.31796	−0.116189
$399$	0	0
$400$	0.0350538	0.00175269
$401$	3.15628	0.157617	0.0788086	−	0.996890i	$-0.474888\pi$
$401$	3.15628	0.157617	0.0788086	+	0.996890i	$0.474888\pi$
$402$	17.1228	0.854006
$403$	2.89299	0.144110
$404$	22.6880	1.12877
$405$	2.92705	0.145446
$406$	0	0
$407$	−2.51110	−0.124470
$408$	−8.18135	−0.405037
$409$	−5.39404	−0.266718	−0.133359	−	0.991068i	$-0.542576\pi$
$409$	−5.39404	−0.266718	−0.133359	+	0.991068i	$0.542576\pi$
$410$	−6.69922	−0.330851
$411$	4.21932	0.208124
$412$	−42.0457	−2.07144
$413$	0	0
$414$	8.50470	0.417983
$415$	3.65704	0.179517
$416$	−19.3075	−0.946629
$417$	13.0636	0.639728
$418$	2.82423	0.138137
$419$	−18.6107	−0.909193	−0.454596	−	0.890698i	$-0.650216\pi$
$419$	−18.6107	−0.909193	−0.454596	+	0.890698i	$0.650216\pi$
$420$	0	0
$421$	−13.8411	−0.674575	−0.337287	−	0.941402i	$-0.609509\pi$
$421$	−13.8411	−0.674575	−0.337287	+	0.941402i	$0.609509\pi$
$422$	−30.9596	−1.50709
$423$	5.20024	0.252844
$424$	−12.4954	−0.606828
$425$	−10.2990	−0.499577
$426$	−24.0649	−1.16595
$427$	0	0
$428$	35.6116	1.72135
$429$	−2.08190	−0.100515
$430$	−4.80700	−0.231814
$431$	−6.67726	−0.321632	−0.160816	−	0.986984i	$-0.551413\pi$
$431$	−6.67726	−0.321632	−0.160816	+	0.986984i	$0.551413\pi$
$432$	−0.00982553	−0.000472731 0
$433$	−33.1341	−1.59232	−0.796161	−	0.605084i	$-0.793139\pi$
$433$	−33.1341	−1.59232	−0.796161	+	0.605084i	$0.793139\pi$
$434$	0	0
$435$	7.85521	0.376628
$436$	11.2515	0.538848
$437$	−7.53229	−0.360318
$438$	−10.8763	−0.519690
$439$	−4.72295	−0.225414	−0.112707	−	0.993628i	$-0.535952\pi$
$439$	−4.72295	−0.225414	−0.112707	+	0.993628i	$0.535952\pi$
$440$	5.04988	0.240744
$441$	0	0
$442$	−22.5958	−1.07477
$443$	6.79271	0.322731	0.161366	−	0.986895i	$-0.448410\pi$
$443$	6.79271	0.322731	0.161366	+	0.986895i	$0.448410\pi$
$444$	−13.3577	−0.633928
$445$	10.4412	0.494960
$446$	33.3452	1.57894
$447$	−18.7649	−0.887548
$448$	0	0
$449$	−4.12828	−0.194826	−0.0974128	−	0.995244i	$-0.531057\pi$
$449$	−4.12828	−0.194826	−0.0974128	+	0.995244i	$0.531057\pi$
$450$	−8.16531	−0.384917
$451$	−0.608757	−0.0286652
$452$	−48.9709	−2.30340
$453$	12.1409	0.570428
$454$	−15.8240	−0.742657
$455$	0	0
$456$	5.74472	0.269021
$457$	15.7297	0.735806	0.367903	−	0.929864i	$-0.380076\pi$
$457$	15.7297	0.735806	0.367903	+	0.929864i	$0.380076\pi$
$458$	−24.5227	−1.14587
$459$	2.88681	0.134745
$460$	−35.2215	−1.64221
$461$	19.0027	0.885043	0.442522	−	0.896758i	$-0.354084\pi$
$461$	19.0027	0.885043	0.442522	+	0.896758i	$0.354084\pi$
$462$	0	0
$463$	9.20764	0.427915	0.213958	−	0.976843i	$-0.431365\pi$
$463$	9.20764	0.427915	0.213958	+	0.976843i	$0.431365\pi$
$464$	−0.0263684	−0.00122412
$465$	2.47606	0.114824
$466$	42.2847	1.95880
$467$	−28.8693	−1.33591	−0.667955	−	0.744202i	$-0.732830\pi$
$467$	−28.8693	−1.33591	−0.667955	+	0.744202i	$0.732830\pi$
$468$	−11.0746	−0.511924
$469$	0	0
$470$	−34.8375	−1.60693
$471$	9.83418	0.453135
$472$	−23.5883	−1.08574
$473$	−0.436811	−0.0200846
$474$	1.56217	0.0717528
$475$	7.23171	0.331814
$476$	0	0
$477$	4.40902	0.201875
$478$	−27.9853	−1.28002
$479$	17.4001	0.795033	0.397517	−	0.917595i	$-0.369872\pi$
$479$	17.4001	0.795033	0.397517	+	0.917595i	$0.369872\pi$
$480$	−16.5250	−0.754258
$481$	−14.1070	−0.643225
$482$	−35.4884	−1.61645
$483$	0	0
$484$	−34.4209	−1.56458
$485$	−39.7831	−1.80646
$486$	2.28873	0.103819
$487$	−29.6791	−1.34489	−0.672443	−	0.740149i	$-0.734755\pi$
$487$	−29.6791	−1.34489	−0.672443	+	0.740149i	$0.734755\pi$
$488$	−15.1513	−0.685866
$489$	9.56943	0.432745
$490$	0	0
$491$	38.7906	1.75059	0.875297	−	0.483585i	$-0.160666\pi$
$491$	38.7906	1.75059	0.875297	+	0.483585i	$0.160666\pi$
$492$	−3.23826	−0.145992
$493$	7.74721	0.348917
$494$	15.8662	0.713852
$495$	−1.78186	−0.0800888
$496$	−0.00831164	−0.000373204 0
$497$	0	0
$498$	2.85952	0.128138
$499$	−6.03265	−0.270058	−0.135029	−	0.990842i	$-0.543113\pi$
$499$	−6.03265	−0.270058	−0.135029	+	0.990842i	$0.543113\pi$
$500$	−13.5769	−0.607175
$501$	0.912899	0.0407853
$502$	19.2734	0.860212
$503$	11.5833	0.516473	0.258237	−	0.966082i	$-0.416859\pi$
$503$	11.5833	0.516473	0.258237	+	0.966082i	$0.416859\pi$
$504$	0	0
$505$	20.5075	0.912573
$506$	−5.17730	−0.230159
$507$	1.30413	0.0579186
$508$	−17.9437	−0.796124
$509$	39.4142	1.74700	0.873501	−	0.486822i	$-0.161844\pi$
$509$	39.4142	1.74700	0.873501	+	0.486822i	$0.161844\pi$
$510$	−19.3393	−0.856361
$511$	0	0
$512$	0.111163	0.00491276
$513$	−2.02704	−0.0894959
$514$	−26.0280	−1.14804
$515$	−38.0049	−1.67470
$516$	−2.32360	−0.102291
$517$	−3.16568	−0.139226
$518$	0	0
$519$	−3.91474	−0.171838
$520$	28.3696	1.24409
$521$	−25.2288	−1.10530	−0.552648	−	0.833415i	$-0.686383\pi$
$521$	−25.2288	−1.10530	−0.552648	+	0.833415i	$0.686383\pi$
$522$	6.14216	0.268835
$523$	−10.4796	−0.458242	−0.229121	−	0.973398i	$-0.573585\pi$
$523$	−10.4796	−0.458242	−0.229121	+	0.973398i	$0.573585\pi$
$524$	1.52165	0.0664737
$525$	0	0
$526$	36.5170	1.59222
$527$	2.44201	0.106376
$528$	0.00598136	0.000260305 0
$529$	−9.19200	−0.399652
$530$	−29.5370	−1.28300
$531$	8.32320	0.361196
$532$	0	0
$533$	−3.41992	−0.148133
$534$	8.16421	0.353300
$535$	32.1892	1.39166
$536$	−21.2025	−0.915809
$537$	−16.6937	−0.720388
$538$	25.1508	1.08433
$539$	0	0
$540$	−9.47856	−0.407892
$541$	−31.8941	−1.37123	−0.685617	−	0.727962i	$-0.740468\pi$
$541$	−31.8941	−1.37123	−0.685617	+	0.727962i	$0.740468\pi$
$542$	−8.59505	−0.369189
$543$	7.61276	0.326695
$544$	−16.2978	−0.698761
$545$	10.1702	0.435642
$546$	0	0
$547$	−29.5699	−1.26432	−0.632158	−	0.774840i	$-0.717831\pi$
$547$	−29.5699	−1.26432	−0.632158	+	0.774840i	$0.717831\pi$
$548$	−13.6633	−0.583666
$549$	5.34616	0.228169
$550$	4.97069	0.211951
$551$	−5.43988	−0.231747
$552$	−10.5311	−0.448232
$553$	0	0
$554$	45.5551	1.93545
$555$	−12.0740	−0.512511
$556$	−42.3034	−1.79406
$557$	−16.1913	−0.686049	−0.343025	−	0.939326i	$-0.611451\pi$
$557$	−16.1913	−0.686049	−0.343025	+	0.939326i	$0.611451\pi$
$558$	1.93608	0.0819610
$559$	−2.45395	−0.103791
$560$	0	0
$561$	−1.75736	−0.0741960
$562$	−12.9519	−0.546343
$563$	−44.0238	−1.85538	−0.927692	−	0.373347i	$-0.878210\pi$
$563$	−44.0238	−1.85538	−0.927692	+	0.373347i	$0.878210\pi$
$564$	−16.8397	−0.709081
$565$	−44.2646	−1.86222
$566$	−53.5188	−2.24956
$567$	0	0
$568$	29.7987	1.25032
$569$	−1.55627	−0.0652423	−0.0326211	−	0.999468i	$-0.510385\pi$
$569$	−1.55627	−0.0652423	−0.0326211	+	0.999468i	$0.510385\pi$
$570$	13.5796	0.568785
$571$	2.77742	0.116231	0.0581156	−	0.998310i	$-0.481491\pi$
$571$	2.77742	0.116231	0.0581156	+	0.998310i	$0.481491\pi$
$572$	6.74175	0.281887
$573$	−3.70107	−0.154614
$574$	0	0
$575$	−13.2570	−0.552854
$576$	−12.9409	−0.539204
$577$	13.0836	0.544677	0.272339	−	0.962201i	$-0.412203\pi$
$577$	13.0836	0.544677	0.272339	+	0.962201i	$0.412203\pi$
$578$	19.8349	0.825022
$579$	25.6969	1.06793
$580$	−25.4372	−1.05622
$581$	0	0
$582$	−31.1073	−1.28944
$583$	−2.68402	−0.111161
$584$	13.4677	0.557299
$585$	−10.0103	−0.413874
$586$	−33.1535	−1.36956
$587$	43.8023	1.80792	0.903958	−	0.427622i	$-0.140648\pi$
$587$	43.8023	1.80792	0.903958	+	0.427622i	$0.140648\pi$
$588$	0	0
$589$	−1.71472	−0.0706537
$590$	−55.7589	−2.29556
$591$	22.0996	0.909056
$592$	0.0405299	0.00166577
$593$	−8.07068	−0.331423	−0.165712	−	0.986174i	$-0.552992\pi$
$593$	−8.07068	−0.331423	−0.165712	+	0.986174i	$0.552992\pi$
$594$	−1.39328	−0.0571669
$595$	0	0
$596$	60.7656	2.48906
$597$	−1.01277	−0.0414500
$598$	−29.0854	−1.18939
$599$	−41.8204	−1.70874	−0.854368	−	0.519669i	$-0.826055\pi$
$599$	−41.8204	−1.70874	−0.854368	+	0.519669i	$0.826055\pi$
$600$	10.1108	0.412772
$601$	4.99552	0.203772	0.101886	−	0.994796i	$-0.467512\pi$
$601$	4.99552	0.203772	0.101886	+	0.994796i	$0.467512\pi$
$602$	0	0
$603$	7.48135	0.304664
$604$	−39.3153	−1.59972
$605$	−31.1128	−1.26492
$606$	16.0353	0.651389
$607$	−29.7455	−1.20733	−0.603667	−	0.797237i	$-0.706294\pi$
$607$	−29.7455	−1.20733	−0.603667	+	0.797237i	$0.706294\pi$
$608$	11.4439	0.464110
$609$	0	0
$610$	−35.8151	−1.45011
$611$	−17.7844	−0.719480
$612$	−9.34825	−0.377880
$613$	1.91945	0.0775259	0.0387630	−	0.999248i	$-0.487658\pi$
$613$	1.91945	0.0775259	0.0387630	+	0.999248i	$0.487658\pi$
$614$	15.3431	0.619196
$615$	−2.92705	−0.118030
$616$	0	0
$617$	−12.6186	−0.508004	−0.254002	−	0.967204i	$-0.581747\pi$
$617$	−12.6186	−0.508004	−0.254002	+	0.967204i	$0.581747\pi$
$618$	−29.7169	−1.19539
$619$	33.3246	1.33943	0.669714	−	0.742619i	$-0.266417\pi$
$619$	33.3246	1.33943	0.669714	+	0.742619i	$0.266417\pi$
$620$	−8.01813	−0.322016
$621$	3.71591	0.149114
$622$	16.9638	0.680188
$623$	0	0
$624$	0.0336026	0.00134518
$625$	−30.1102	−1.20441
$626$	38.3792	1.53394
$627$	1.23397	0.0492801
$628$	−31.8457	−1.27078
$629$	−11.9080	−0.474801
$630$	0	0
$631$	0.318323	0.0126722	0.00633612	−	0.999980i	$-0.497983\pi$
$631$	0.318323	0.0126722	0.00633612	+	0.999980i	$0.497983\pi$
$632$	−1.93438	−0.0769454
$633$	−13.5270	−0.537649
$634$	15.2837	0.606995
$635$	−16.2192	−0.643641
$636$	−14.2776	−0.566142
$637$	0	0
$638$	−3.73908	−0.148032
$639$	−10.5145	−0.415948
$640$	53.6439	2.12046
$641$	38.6465	1.52645	0.763223	−	0.646135i	$-0.223616\pi$
$641$	38.6465	1.52645	0.763223	+	0.646135i	$0.223616\pi$
$642$	25.1694	0.993358
$643$	42.9801	1.69497	0.847485	−	0.530819i	$-0.178116\pi$
$643$	42.9801	1.69497	0.847485	+	0.530819i	$0.178116\pi$
$644$	0	0
$645$	−2.10029	−0.0826990
$646$	13.3929	0.526935
$647$	46.8808	1.84307	0.921536	−	0.388292i	$-0.126935\pi$
$647$	46.8808	1.84307	0.921536	+	0.388292i	$0.126935\pi$
$648$	−2.83405	−0.111332
$649$	−5.06680	−0.198889
$650$	27.9247	1.09530
$651$	0	0
$652$	−30.9883	−1.21360
$653$	4.47205	0.175005	0.0875024	−	0.996164i	$-0.472111\pi$
$653$	4.47205	0.175005	0.0875024	+	0.996164i	$0.472111\pi$
$654$	7.95227	0.310958
$655$	1.37541	0.0537418
$656$	0.00982553	0.000383623 0
$657$	−4.75212	−0.185398
$658$	0	0
$659$	−50.5709	−1.96996	−0.984982	−	0.172657i	$-0.944765\pi$
$659$	−50.5709	−1.96996	−0.984982	+	0.172657i	$0.944765\pi$
$660$	5.77014	0.224602
$661$	−19.5720	−0.761263	−0.380632	−	0.924727i	$-0.624293\pi$
$661$	−19.5720	−0.761263	−0.380632	+	0.924727i	$0.624293\pi$
$662$	50.8337	1.97571
$663$	−9.87266	−0.383422
$664$	−3.54084	−0.137411
$665$	0	0
$666$	−9.44089	−0.365827
$667$	9.97224	0.386127
$668$	−2.95621	−0.114379
$669$	14.5693	0.563283
$670$	−50.1192	−1.93627
$671$	−3.25451	−0.125639
$672$	0	0
$673$	19.5808	0.754784	0.377392	−	0.926054i	$-0.376821\pi$
$673$	19.5808	0.754784	0.377392	+	0.926054i	$0.376821\pi$
$674$	18.4509	0.710702
$675$	−3.56763	−0.137318
$676$	−4.22313	−0.162428
$677$	−2.92171	−0.112291	−0.0561453	−	0.998423i	$-0.517881\pi$
$677$	−2.92171	−0.112291	−0.0561453	+	0.998423i	$0.517881\pi$
$678$	−34.6115	−1.32924
$679$	0	0
$680$	23.9472	0.918334
$681$	−6.91389	−0.264941
$682$	−1.17860	−0.0451311
$683$	−26.8964	−1.02916	−0.514581	−	0.857442i	$-0.672052\pi$
$683$	−26.8964	−1.02916	−0.514581	+	0.857442i	$0.672052\pi$
$684$	6.56408	0.250984
$685$	−12.3502	−0.471876
$686$	0	0
$687$	−10.7145	−0.408785
$688$	0.00705028	0.000268789 0
$689$	−15.0785	−0.574445
$690$	−24.8937	−0.947687
$691$	20.1551	0.766738	0.383369	−	0.923595i	$-0.374764\pi$
$691$	20.1551	0.766738	0.383369	+	0.923595i	$0.374764\pi$
$692$	12.6770	0.481906
$693$	0	0
$694$	−16.2994	−0.618718
$695$	−38.2379	−1.45044
$696$	−7.60562	−0.288290
$697$	−2.88681	−0.109346
$698$	−31.3952	−1.18833
$699$	18.4752	0.698797
$700$	0	0
$701$	−31.3666	−1.18470	−0.592351	−	0.805680i	$-0.701800\pi$
$701$	−31.3666	−1.18470	−0.592351	+	0.805680i	$0.701800\pi$
$702$	−7.82726	−0.295421
$703$	8.36144	0.315358
$704$	7.87786	0.296908
$705$	−15.2214	−0.573269
$706$	50.7741	1.91091
$707$	0	0
$708$	−26.9527	−1.01294
$709$	−39.8398	−1.49621	−0.748107	−	0.663578i	$-0.769037\pi$
$709$	−39.8398	−1.49621	−0.748107	+	0.663578i	$0.769037\pi$
$710$	70.4391	2.64353
$711$	0.682550	0.0255976
$712$	−10.1094	−0.378867
$713$	3.14337	0.117720
$714$	0	0
$715$	6.09383	0.227896
$716$	54.0587	2.02027
$717$	−12.2275	−0.456644
$718$	−13.8242	−0.515913
$719$	−36.5431	−1.36283	−0.681414	−	0.731899i	$-0.738634\pi$
$719$	−36.5431	−1.36283	−0.681414	+	0.731899i	$0.738634\pi$
$720$	0.0287598	0.00107182
$721$	0	0
$722$	34.0817	1.26839
$723$	−15.5057	−0.576665
$724$	−24.6521	−0.916189
$725$	−9.57429	−0.355580
$726$	−24.3278	−0.902890
$727$	−9.59428	−0.355832	−0.177916	−	0.984046i	$-0.556936\pi$
$727$	−9.59428	−0.355832	−0.177916	+	0.984046i	$0.556936\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	31.8355	1.17828
$731$	−2.07142	−0.0766142
$732$	−17.3123	−0.639881
$733$	24.6790	0.911539	0.455769	−	0.890098i	$-0.349364\pi$
$733$	24.6790	0.911539	0.455769	+	0.890098i	$0.349364\pi$
$734$	−3.09702	−0.114313
$735$	0	0
$736$	−20.9786	−0.773280
$737$	−4.55433	−0.167761
$738$	−2.28873	−0.0842491
$739$	−13.7444	−0.505597	−0.252799	−	0.967519i	$-0.581351\pi$
$739$	−13.7444	−0.505597	−0.252799	+	0.967519i	$0.581351\pi$
$740$	39.0987	1.43729
$741$	6.93231	0.254665
$742$	0	0
$743$	35.8201	1.31411	0.657055	−	0.753842i	$-0.271802\pi$
$743$	35.8201	1.31411	0.657055	+	0.753842i	$0.271802\pi$
$744$	−2.39738	−0.0878923
$745$	54.9257	2.01232
$746$	67.5390	2.47278
$747$	1.24939	0.0457129
$748$	5.69081	0.208077
$749$	0	0
$750$	−9.59579	−0.350389
$751$	−27.6537	−1.00910	−0.504549	−	0.863383i	$-0.668341\pi$
$751$	−27.6537	−1.00910	−0.504549	+	0.863383i	$0.668341\pi$
$752$	0.0510951	0.00186325
$753$	8.42100	0.306878
$754$	−21.0057	−0.764983
$755$	−35.5369	−1.29332
$756$	0	0
$757$	−53.2987	−1.93718	−0.968588	−	0.248673i	$-0.920006\pi$
$757$	−53.2987	−1.93718	−0.968588	+	0.248673i	$0.920006\pi$
$758$	13.2381	0.480828
$759$	−2.26209	−0.0821086
$760$	−16.8151	−0.609947
$761$	45.7807	1.65955	0.829775	−	0.558098i	$-0.188469\pi$
$761$	45.7807	1.65955	0.829775	+	0.558098i	$0.188469\pi$
$762$	−12.6822	−0.459427
$763$	0	0
$764$	11.9850	0.433603
$765$	−8.44983	−0.305504
$766$	−19.8876	−0.718568
$767$	−28.4647	−1.02780
$768$	16.0635	0.579643
$769$	38.4188	1.38542	0.692708	−	0.721218i	$-0.256417\pi$
$769$	38.4188	1.38542	0.692708	+	0.721218i	$0.256417\pi$
$770$	0	0
$771$	−11.3723	−0.409562
$772$	−83.2134	−2.99492
$773$	−3.82589	−0.137608	−0.0688039	−	0.997630i	$-0.521918\pi$
$773$	−3.82589	−0.137608	−0.0688039	+	0.997630i	$0.521918\pi$
$774$	−1.64227	−0.0590301
$775$	−3.01793	−0.108407
$776$	38.5191	1.38275
$777$	0	0
$778$	66.4909	2.38381
$779$	2.02704	0.0726262
$780$	32.4159	1.16068
$781$	6.40079	0.229038
$782$	−24.5514	−0.877958
$783$	2.68366	0.0959062
$784$	0	0
$785$	−28.7852	−1.02739
$786$	1.07547	0.0383606
$787$	37.4452	1.33478	0.667389	−	0.744710i	$-0.267412\pi$
$787$	37.4452	1.33478	0.667389	+	0.744710i	$0.267412\pi$
$788$	−71.5644	−2.54937
$789$	15.9552	0.568019
$790$	−4.57255	−0.162684
$791$	0	0
$792$	1.72525	0.0613039
$793$	−18.2835	−0.649265
$794$	−53.2179	−1.88863
$795$	−12.9054	−0.457708
$796$	3.27962	0.116243
$797$	−47.4050	−1.67917	−0.839586	−	0.543227i	$-0.817202\pi$
$797$	−47.4050	−1.67917	−0.839586	+	0.543227i	$0.817202\pi$
$798$	0	0
$799$	−15.0121	−0.531089
$800$	20.1414	0.712106
$801$	3.56714	0.126039
$802$	−7.22386	−0.255083
$803$	2.89289	0.102088
$804$	−24.2266	−0.854406
$805$	0	0
$806$	−6.62126	−0.233224
$807$	10.9890	0.386831
$808$	−19.8559	−0.698529
$809$	−4.56961	−0.160659	−0.0803294	−	0.996768i	$-0.525597\pi$
$809$	−4.56961	−0.160659	−0.0803294	+	0.996768i	$0.525597\pi$
$810$	−6.69922	−0.235386
$811$	−39.9591	−1.40315	−0.701577	−	0.712594i	$-0.747520\pi$
$811$	−39.9591	−1.40315	−0.701577	+	0.712594i	$0.747520\pi$
$812$	0	0
$813$	−3.75539	−0.131707
$814$	5.74721	0.201440
$815$	−28.0102	−0.981155
$816$	0.0283644	0.000992953 0
$817$	1.45449	0.0508863
$818$	12.3455	0.431650
$819$	0	0
$820$	9.47856	0.331006
$821$	15.6877	0.547504	0.273752	−	0.961800i	$-0.411735\pi$
$821$	15.6877	0.547504	0.273752	+	0.961800i	$0.411735\pi$
$822$	−9.65687	−0.336822
$823$	29.8856	1.04175	0.520873	−	0.853634i	$-0.325606\pi$
$823$	29.8856	1.04175	0.520873	+	0.853634i	$0.325606\pi$
$824$	36.7973	1.28189
$825$	2.17182	0.0756130
$826$	0	0
$827$	−8.79265	−0.305750	−0.152875	−	0.988246i	$-0.548853\pi$
$827$	−8.79265	−0.305750	−0.152875	+	0.988246i	$0.548853\pi$
$828$	−12.0331	−0.418179
$829$	18.1178	0.629258	0.314629	−	0.949215i	$-0.398120\pi$
$829$	18.1178	0.629258	0.314629	+	0.949215i	$0.398120\pi$
$830$	−8.36996	−0.290525
$831$	19.9041	0.690466
$832$	44.2568	1.53433
$833$	0	0
$834$	−29.8990	−1.03532
$835$	−2.67210	−0.0924719
$836$	−3.99593	−0.138202
$837$	0.845922	0.0292394
$838$	42.5948	1.47141
$839$	21.3469	0.736977	0.368489	−	0.929632i	$-0.379875\pi$
$839$	21.3469	0.736977	0.368489	+	0.929632i	$0.379875\pi$
$840$	0	0
$841$	−21.7980	−0.751654
$842$	31.6785	1.09171
$843$	−5.65900	−0.194906
$844$	43.8039	1.50779
$845$	−3.81726	−0.131318
$846$	−11.9019	−0.409196
$847$	0	0
$848$	0.0433209	0.00148765
$849$	−23.3837	−0.802525
$850$	23.5717	0.808502
$851$	−15.3280	−0.525436
$852$	34.0488	1.16649
$853$	18.5830	0.636270	0.318135	−	0.948045i	$-0.396943\pi$
$853$	18.5830	0.636270	0.318135	+	0.948045i	$0.396943\pi$
$854$	0	0
$855$	5.93324	0.202913
$856$	−31.1664	−1.06524
$857$	−38.8453	−1.32693	−0.663465	−	0.748207i	$-0.730915\pi$
$857$	−38.8453	−1.32693	−0.663465	+	0.748207i	$0.730915\pi$
$858$	4.76490	0.162671
$859$	−10.3316	−0.352511	−0.176255	−	0.984344i	$-0.556398\pi$
$859$	−10.3316	−0.352511	−0.176255	+	0.984344i	$0.556398\pi$
$860$	6.80131	0.231923
$861$	0	0
$862$	15.2824	0.520521
$863$	−49.2323	−1.67589	−0.837944	−	0.545757i	$-0.816242\pi$
$863$	−49.2323	−1.67589	−0.837944	+	0.545757i	$0.816242\pi$
$864$	−5.64560	−0.192067
$865$	11.4586	0.389605
$866$	75.8348	2.57697
$867$	8.66634	0.294324
$868$	0	0
$869$	−0.415507	−0.0140951
$870$	−17.9784	−0.609525
$871$	−25.5856	−0.866937
$872$	−9.84701	−0.333462
$873$	−13.5915	−0.460004
$874$	17.2393	0.583130
$875$	0	0
$876$	15.3886	0.519933
$877$	−27.6814	−0.934734	−0.467367	−	0.884063i	$-0.654797\pi$
$877$	−27.6814	−0.934734	−0.467367	+	0.884063i	$0.654797\pi$
$878$	10.8095	0.364804
$879$	−14.4856	−0.488586
$880$	−0.0175077	−0.000590186 0
$881$	−14.1997	−0.478400	−0.239200	−	0.970970i	$-0.576885\pi$
$881$	−14.1997	−0.478400	−0.239200	+	0.970970i	$0.576885\pi$
$882$	0	0
$883$	24.5409	0.825869	0.412934	−	0.910761i	$-0.364504\pi$
$883$	24.5409	0.825869	0.412934	+	0.910761i	$0.364504\pi$
$884$	31.9703	1.07528
$885$	−24.3624	−0.818934
$886$	−15.5466	−0.522300
$887$	45.1211	1.51502	0.757510	−	0.652824i	$-0.226416\pi$
$887$	45.1211	1.51502	0.757510	+	0.652824i	$0.226416\pi$
$888$	11.6903	0.392301
$889$	0	0
$890$	−23.8970	−0.801031
$891$	−0.608757	−0.0203941
$892$	−47.1793	−1.57968
$893$	10.5411	0.352744
$894$	42.9476	1.43638
$895$	48.8634	1.63332
$896$	0	0
$897$	−12.7081	−0.424312
$898$	9.44850	0.315300
$899$	2.27017	0.0757143
$900$	11.5529	0.385097
$901$	−12.7280	−0.424031
$902$	1.39328	0.0463911
$903$	0	0
$904$	42.8581	1.42544
$905$	−22.2829	−0.740710
$906$	−27.7871	−0.923165
$907$	43.3718	1.44014	0.720068	−	0.693903i	$-0.244111\pi$
$907$	43.3718	1.44014	0.720068	+	0.693903i	$0.244111\pi$
$908$	22.3890	0.743005
$909$	7.00621	0.232381
$910$	0	0
$911$	6.89404	0.228410	0.114205	−	0.993457i	$-0.463568\pi$
$911$	6.89404	0.228410	0.114205	+	0.993457i	$0.463568\pi$
$912$	−0.0199167	−0.000659509 0
$913$	−0.760577	−0.0251714
$914$	−36.0011	−1.19081
$915$	−15.6485	−0.517323
$916$	34.6965	1.14641
$917$	0	0
$918$	−6.60711	−0.218067
$919$	−21.2218	−0.700043	−0.350022	−	0.936742i	$-0.613826\pi$
$919$	−21.2218	−0.700043	−0.350022	+	0.936742i	$0.613826\pi$
$920$	30.8250	1.01627
$921$	6.70376	0.220896
$922$	−43.4919	−1.43233
$923$	35.9589	1.18360
$924$	0	0
$925$	14.7163	0.483869
$926$	−21.0738	−0.692527
$927$	−12.9840	−0.426451
$928$	−15.1509	−0.497352
$929$	33.2194	1.08989	0.544947	−	0.838471i	$-0.316550\pi$
$929$	33.2194	1.08989	0.544947	+	0.838471i	$0.316550\pi$
$930$	−5.66702	−0.185829
$931$	0	0
$932$	−59.8276	−1.95972
$933$	7.41192	0.242655
$934$	66.0738	2.16200
$935$	5.14389	0.168223
$936$	9.69222	0.316800
$937$	48.1152	1.57186	0.785928	−	0.618318i	$-0.212186\pi$
$937$	48.1152	1.57186	0.785928	+	0.618318i	$0.212186\pi$
$938$	0	0
$939$	16.7688	0.547229
$940$	49.2908	1.60769
$941$	−39.1978	−1.27781	−0.638906	−	0.769285i	$-0.720613\pi$
$941$	−39.1978	−1.27781	−0.638906	+	0.769285i	$0.720613\pi$
$942$	−22.5077	−0.733342
$943$	−3.71591	−0.121007
$944$	0.0817798	0.00266171
$945$	0	0
$946$	0.999741	0.0325044
$947$	7.83468	0.254593	0.127296	−	0.991865i	$-0.459370\pi$
$947$	7.83468	0.254593	0.127296	+	0.991865i	$0.459370\pi$
$948$	−2.21028	−0.0717865
$949$	16.2519	0.527558
$950$	−16.5514	−0.536998
$951$	6.67784	0.216544
$952$	0	0
$953$	−26.0595	−0.844150	−0.422075	−	0.906561i	$-0.638698\pi$
$953$	−26.0595	−0.844150	−0.422075	+	0.906561i	$0.638698\pi$
$954$	−10.0910	−0.326709
$955$	10.8332	0.350555
$956$	39.5958	1.28062
$957$	−1.63370	−0.0528099
$958$	−39.8242	−1.28666
$959$	0	0
$960$	37.8786	1.22253
$961$	−30.2844	−0.976917
$962$	32.2871	1.04098
$963$	10.9971	0.354378
$964$	50.2117	1.61721
$965$	−75.2162	−2.42130
$966$	0	0
$967$	−28.2924	−0.909821	−0.454911	−	0.890537i	$-0.650329\pi$
$967$	−28.2924	−0.909821	−0.454911	+	0.890537i	$0.650329\pi$
$968$	30.1243	0.968230
$969$	5.85167	0.187983
$970$	91.0527	2.92353
$971$	40.1883	1.28970	0.644851	−	0.764308i	$-0.276919\pi$
$971$	40.1883	1.28970	0.644851	+	0.764308i	$0.276919\pi$
$972$	−3.23826	−0.103867
$973$	0	0
$974$	67.9272	2.17653
$975$	12.2010	0.390745
$976$	0.0525289	0.00168141
$977$	15.5653	0.497979	0.248989	−	0.968506i	$-0.419902\pi$
$977$	15.5653	0.497979	0.248989	+	0.968506i	$0.419902\pi$
$978$	−21.9018	−0.700342
$979$	−2.17152	−0.0694021
$980$	0	0
$981$	3.47454	0.110934
$982$	−88.7810	−2.83312
$983$	−33.7929	−1.07783	−0.538913	−	0.842361i	$-0.681165\pi$
$983$	−33.7929	−1.07783	−0.538913	+	0.842361i	$0.681165\pi$
$984$	2.83405	0.0903461
$985$	−64.6867	−2.06109
$986$	−17.7312	−0.564678
$987$	0	0
$988$	−22.4487	−0.714187
$989$	−2.66634	−0.0847846
$990$	4.07819	0.129614
$991$	46.8354	1.48777	0.743887	−	0.668305i	$-0.232980\pi$
$991$	46.8354	1.48777	0.743887	+	0.668305i	$0.232980\pi$
$992$	−4.77574	−0.151630
$993$	22.2105	0.704829
$994$	0	0
$995$	2.96443	0.0939789
$996$	−4.04587	−0.128198
$997$	8.38234	0.265472	0.132736	−	0.991151i	$-0.457624\pi$
$997$	8.38234	0.265472	0.132736	+	0.991151i	$0.457624\pi$
$998$	13.8071	0.437055
$999$	−4.12496	−0.130508

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.z.1.2	✓	8
7.6	odd	2		6027.2.a.ba.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.z.1.2	✓	8	1.1	even	1	trivial
6027.2.a.ba.1.2	yes	8	7.6	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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.28873 q^{2} -1.00000 q^{3} +3.23826 q^{4} +2.92705 q^{5} +2.28873 q^{6} -2.83405 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.28873 q^{2} -1.00000 q^{3} +3.23826 q^{4} +2.92705 q^{5} +2.28873 q^{6} -2.83405 q^{8} +1.00000 q^{9} -6.69922 q^{10} -0.608757 q^{11} -3.23826 q^{12} -3.41992 q^{13} -2.92705 q^{15} +0.00982553 q^{16} -2.88681 q^{17} -2.28873 q^{18} +2.02704 q^{19} +9.47856 q^{20} +1.39328 q^{22} -3.71591 q^{23} +2.83405 q^{24} +3.56763 q^{25} +7.82726 q^{26} -1.00000 q^{27} -2.68366 q^{29} +6.69922 q^{30} -0.845922 q^{31} +5.64560 q^{32} +0.608757 q^{33} +6.60711 q^{34} +3.23826 q^{36} +4.12496 q^{37} -4.63933 q^{38} +3.41992 q^{39} -8.29540 q^{40} +1.00000 q^{41} +0.717546 q^{43} -1.97132 q^{44} +2.92705 q^{45} +8.50470 q^{46} +5.20024 q^{47} -0.00982553 q^{48} -8.16531 q^{50} +2.88681 q^{51} -11.0746 q^{52} +4.40902 q^{53} +2.28873 q^{54} -1.78186 q^{55} -2.02704 q^{57} +6.14216 q^{58} +8.32320 q^{59} -9.47856 q^{60} +5.34616 q^{61} +1.93608 q^{62} -12.9409 q^{64} -10.0103 q^{65} -1.39328 q^{66} +7.48135 q^{67} -9.34825 q^{68} +3.71591 q^{69} -10.5145 q^{71} -2.83405 q^{72} -4.75212 q^{73} -9.44089 q^{74} -3.56763 q^{75} +6.56408 q^{76} -7.82726 q^{78} +0.682550 q^{79} +0.0287598 q^{80} +1.00000 q^{81} -2.28873 q^{82} +1.24939 q^{83} -8.44983 q^{85} -1.64227 q^{86} +2.68366 q^{87} +1.72525 q^{88} +3.56714 q^{89} -6.69922 q^{90} -12.0331 q^{92} +0.845922 q^{93} -11.9019 q^{94} +5.93324 q^{95} -5.64560 q^{96} -13.5915 q^{97} -0.608757 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 2 * q^2 - 8 * q^3 + 4 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^8 + 8 * q^9	\( 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 4 q^{13} - 2 q^{15} + 8 q^{17} - 2 q^{18} + 6 q^{19} + 4 q^{20} - 14 q^{22} - 12 q^{23} + 6 q^{24} - 4 q^{25} + 4 q^{26} - 8 q^{27} - 4 q^{29} + 2 q^{30} - 10 q^{31} - 4 q^{32} + 2 q^{33} + 4 q^{34} + 4 q^{36} - 20 q^{37} - 18 q^{38} - 4 q^{39} + 12 q^{40} + 8 q^{41} - 8 q^{43} + 20 q^{44} + 2 q^{45} - 12 q^{46} + 24 q^{47} - 22 q^{50} - 8 q^{51} - 30 q^{52} - 36 q^{53} + 2 q^{54} + 4 q^{55} - 6 q^{57} + 14 q^{58} + 10 q^{59} - 4 q^{60} - 22 q^{61} + 30 q^{62} - 24 q^{64} + 8 q^{65} + 14 q^{66} - 14 q^{67} + 38 q^{68} + 12 q^{69} - 10 q^{71} - 6 q^{72} - 12 q^{73} - 2 q^{74} + 4 q^{75} + 32 q^{76} - 4 q^{78} + 16 q^{79} - 14 q^{80} + 8 q^{81} - 2 q^{82} + 24 q^{83} - 44 q^{85} + 36 q^{86} + 4 q^{87} - 34 q^{88} + 2 q^{89} - 2 q^{90} - 48 q^{92} + 10 q^{93} - 34 q^{94} - 24 q^{95} + 4 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 - 8 * q^3 + 4 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^8 + 8 * q^9 - 2 * q^10 - 2 * q^11 - 4 * q^12 + 4 * q^13 - 2 * q^15 + 8 * q^17 - 2 * q^18 + 6 * q^19 + 4 * q^20 - 14 * q^22 - 12 * q^23 + 6 * q^24 - 4 * q^25 + 4 * q^26 - 8 * q^27 - 4 * q^29 + 2 * q^30 - 10 * q^31 - 4 * q^32 + 2 * q^33 + 4 * q^34 + 4 * q^36 - 20 * q^37 - 18 * q^38 - 4 * q^39 + 12 * q^40 + 8 * q^41 - 8 * q^43 + 20 * q^44 + 2 * q^45 - 12 * q^46 + 24 * q^47 - 22 * q^50 - 8 * q^51 - 30 * q^52 - 36 * q^53 + 2 * q^54 + 4 * q^55 - 6 * q^57 + 14 * q^58 + 10 * q^59 - 4 * q^60 - 22 * q^61 + 30 * q^62 - 24 * q^64 + 8 * q^65 + 14 * q^66 - 14 * q^67 + 38 * q^68 + 12 * q^69 - 10 * q^71 - 6 * q^72 - 12 * q^73 - 2 * q^74 + 4 * q^75 + 32 * q^76 - 4 * q^78 + 16 * q^79 - 14 * q^80 + 8 * q^81 - 2 * q^82 + 24 * q^83 - 44 * q^85 + 36 * q^86 + 4 * q^87 - 34 * q^88 + 2 * q^89 - 2 * q^90 - 48 * q^92 + 10 * q^93 - 34 * q^94 - 24 * q^95 + 4 * q^96 + 16 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

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Database

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