LMFDB - Embedded newform 7942.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.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:	$63.4171892853$
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} - x^{7} - 19x^{6} + 14x^{5} + 116x^{4} - 65x^{3} - 235x^{2} + 120x + 80 $ x^8 - x^7 - 19x^6 + 14x^5 + 116x^4 - 65x^3 - 235x^2 + 120x + 80
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.24873$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.24873 q^{3} +1.00000 q^{4} +1.63070 q^{5} +3.24873 q^{6} -1.84467 q^{7} -1.00000 q^{8} +7.55426 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.24873 q^{3} +1.00000 q^{4} +1.63070 q^{5} +3.24873 q^{6} -1.84467 q^{7} -1.00000 q^{8} +7.55426 q^{9} -1.63070 q^{10} +1.00000 q^{11} -3.24873 q^{12} -7.17229 q^{13} +1.84467 q^{14} -5.29770 q^{15} +1.00000 q^{16} -0.522036 q^{17} -7.55426 q^{18} +1.63070 q^{20} +5.99285 q^{21} -1.00000 q^{22} +4.05075 q^{23} +3.24873 q^{24} -2.34083 q^{25} +7.17229 q^{26} -14.7956 q^{27} -1.84467 q^{28} -7.91752 q^{29} +5.29770 q^{30} +8.24130 q^{31} -1.00000 q^{32} -3.24873 q^{33} +0.522036 q^{34} -3.00810 q^{35} +7.55426 q^{36} +10.4305 q^{37} +23.3008 q^{39} -1.63070 q^{40} +9.06744 q^{41} -5.99285 q^{42} -6.11537 q^{43} +1.00000 q^{44} +12.3187 q^{45} -4.05075 q^{46} +2.80985 q^{47} -3.24873 q^{48} -3.59718 q^{49} +2.34083 q^{50} +1.69596 q^{51} -7.17229 q^{52} +0.238025 q^{53} +14.7956 q^{54} +1.63070 q^{55} +1.84467 q^{56} +7.91752 q^{58} +7.94508 q^{59} -5.29770 q^{60} -5.82986 q^{61} -8.24130 q^{62} -13.9351 q^{63} +1.00000 q^{64} -11.6958 q^{65} +3.24873 q^{66} -7.80410 q^{67} -0.522036 q^{68} -13.1598 q^{69} +3.00810 q^{70} +9.04332 q^{71} -7.55426 q^{72} -2.02182 q^{73} -10.4305 q^{74} +7.60471 q^{75} -1.84467 q^{77} -23.3008 q^{78} -9.86872 q^{79} +1.63070 q^{80} +25.4040 q^{81} -9.06744 q^{82} -0.302948 q^{83} +5.99285 q^{84} -0.851283 q^{85} +6.11537 q^{86} +25.7219 q^{87} -1.00000 q^{88} +2.73814 q^{89} -12.3187 q^{90} +13.2305 q^{91} +4.05075 q^{92} -26.7738 q^{93} -2.80985 q^{94} +3.24873 q^{96} -8.29725 q^{97} +3.59718 q^{98} +7.55426 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 3 q^{5} + q^{6} - 8 q^{8} + 15 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 3 * q^5 + q^6 - 8 * q^8 + 15 * q^9	$ 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 3 q^{5} + q^{6} - 8 q^{8} + 15 q^{9} + 3 q^{10} + 8 q^{11} - q^{12} - 3 q^{13} - 36 q^{15} + 8 q^{16} - 4 q^{17} - 15 q^{18} - 3 q^{20} - 3 q^{21} - 8 q^{22} - q^{23} + q^{24} + 5 q^{25} + 3 q^{26} - 10 q^{27} + 4 q^{29} + 36 q^{30} + 3 q^{31} - 8 q^{32} - q^{33} + 4 q^{34} + 3 q^{35} + 15 q^{36} + 26 q^{37} + 14 q^{39} + 3 q^{40} - 17 q^{41} + 3 q^{42} - 6 q^{43} + 8 q^{44} + 3 q^{45} + q^{46} - q^{48} + 36 q^{49} - 5 q^{50} + 12 q^{51} - 3 q^{52} - 49 q^{53} + 10 q^{54} - 3 q^{55} - 4 q^{58} - 15 q^{59} - 36 q^{60} - 14 q^{61} - 3 q^{62} + 17 q^{63} + 8 q^{64} + q^{66} - 5 q^{67} - 4 q^{68} - 4 q^{69} - 3 q^{70} + q^{71} - 15 q^{72} - 10 q^{73} - 26 q^{74} + 29 q^{75} - 14 q^{78} - 12 q^{79} - 3 q^{80} + 17 q^{82} - 7 q^{83} - 3 q^{84} - 20 q^{85} + 6 q^{86} + 45 q^{87} - 8 q^{88} - 8 q^{89} - 3 q^{90} - 17 q^{91} - q^{92} - 33 q^{93} + q^{96} - 54 q^{97} - 36 q^{98} + 15 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 3 * q^5 + q^6 - 8 * q^8 + 15 * q^9 + 3 * q^10 + 8 * q^11 - q^12 - 3 * q^13 - 36 * q^15 + 8 * q^16 - 4 * q^17 - 15 * q^18 - 3 * q^20 - 3 * q^21 - 8 * q^22 - q^23 + q^24 + 5 * q^25 + 3 * q^26 - 10 * q^27 + 4 * q^29 + 36 * q^30 + 3 * q^31 - 8 * q^32 - q^33 + 4 * q^34 + 3 * q^35 + 15 * q^36 + 26 * q^37 + 14 * q^39 + 3 * q^40 - 17 * q^41 + 3 * q^42 - 6 * q^43 + 8 * q^44 + 3 * q^45 + q^46 - q^48 + 36 * q^49 - 5 * q^50 + 12 * q^51 - 3 * q^52 - 49 * q^53 + 10 * q^54 - 3 * q^55 - 4 * q^58 - 15 * q^59 - 36 * q^60 - 14 * q^61 - 3 * q^62 + 17 * q^63 + 8 * q^64 + q^66 - 5 * q^67 - 4 * q^68 - 4 * q^69 - 3 * q^70 + q^71 - 15 * q^72 - 10 * q^73 - 26 * q^74 + 29 * q^75 - 14 * q^78 - 12 * q^79 - 3 * q^80 + 17 * q^82 - 7 * q^83 - 3 * q^84 - 20 * q^85 + 6 * q^86 + 45 * q^87 - 8 * q^88 - 8 * q^89 - 3 * q^90 - 17 * q^91 - q^92 - 33 * q^93 + q^96 - 54 * q^97 - 36 * q^98 + 15 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.24873	−1.87566	−0.937828	−	0.347100i	$-0.887166\pi$
$3$	−3.24873	−1.87566	−0.937828	+	0.347100i	$0.887166\pi$
$4$	1.00000	0.500000
$5$	1.63070	0.729270	0.364635	−	0.931151i	$-0.381194\pi$
$5$	1.63070	0.729270	0.364635	+	0.931151i	$0.381194\pi$
$6$	3.24873	1.32629
$7$	−1.84467	−0.697221	−0.348610	−	0.937268i	$-0.613346\pi$
$7$	−1.84467	−0.697221	−0.348610	+	0.937268i	$0.613346\pi$
$8$	−1.00000	−0.353553
$9$	7.55426	2.51809
$10$	−1.63070	−0.515672
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−3.24873	−0.937828
$13$	−7.17229	−1.98924	−0.994618	−	0.103612i	$-0.966960\pi$
$13$	−7.17229	−1.98924	−0.994618	+	0.103612i	$0.966960\pi$
$14$	1.84467	0.493009
$15$	−5.29770	−1.36786
$16$	1.00000	0.250000
$17$	−0.522036	−0.126612	−0.0633062	−	0.997994i	$-0.520164\pi$
$17$	−0.522036	−0.126612	−0.0633062	+	0.997994i	$0.520164\pi$
$18$	−7.55426	−1.78056
$19$	0	0
$19$	0	0
$20$	1.63070	0.364635
$21$	5.99285	1.30775
$22$	−1.00000	−0.213201
$23$	4.05075	0.844640	0.422320	−	0.906447i	$-0.361216\pi$
$23$	4.05075	0.844640	0.422320	+	0.906447i	$0.361216\pi$
$24$	3.24873	0.663145
$25$	−2.34083	−0.468165
$26$	7.17229	1.40660
$27$	−14.7956	−2.84741
$28$	−1.84467	−0.348610
$29$	−7.91752	−1.47025	−0.735123	−	0.677934i	$-0.762876\pi$
$29$	−7.91752	−1.47025	−0.735123	+	0.677934i	$0.762876\pi$
$30$	5.29770	0.967223
$31$	8.24130	1.48018	0.740091	−	0.672507i	$-0.234783\pi$
$31$	8.24130	1.48018	0.740091	+	0.672507i	$0.234783\pi$
$32$	−1.00000	−0.176777
$33$	−3.24873	−0.565532
$34$	0.522036	0.0895285
$35$	−3.00810	−0.508462
$36$	7.55426	1.25904
$37$	10.4305	1.71477	0.857383	−	0.514678i	$-0.172089\pi$
$37$	10.4305	1.71477	0.857383	+	0.514678i	$0.172089\pi$
$38$	0	0
$39$	23.3008	3.73112
$40$	−1.63070	−0.257836
$41$	9.06744	1.41610	0.708048	−	0.706164i	$-0.249576\pi$
$41$	9.06744	1.41610	0.708048	+	0.706164i	$0.249576\pi$
$42$	−5.99285	−0.924716
$43$	−6.11537	−0.932586	−0.466293	−	0.884630i	$-0.654411\pi$
$43$	−6.11537	−0.932586	−0.466293	+	0.884630i	$0.654411\pi$
$44$	1.00000	0.150756
$45$	12.3187	1.83636
$46$	−4.05075	−0.597251
$47$	2.80985	0.409859	0.204929	−	0.978777i	$-0.434304\pi$
$47$	2.80985	0.409859	0.204929	+	0.978777i	$0.434304\pi$
$48$	−3.24873	−0.468914
$49$	−3.59718	−0.513883
$50$	2.34083	0.331043
$51$	1.69596	0.237481
$52$	−7.17229	−0.994618
$53$	0.238025	0.0326953	0.0163476	−	0.999866i	$-0.494796\pi$
$53$	0.238025	0.0326953	0.0163476	+	0.999866i	$0.494796\pi$
$54$	14.7956	2.01342
$55$	1.63070	0.219883
$56$	1.84467	0.246505
$57$	0	0
$58$	7.91752	1.03962
$59$	7.94508	1.03436	0.517181	−	0.855876i	$-0.326981\pi$
$59$	7.94508	1.03436	0.517181	+	0.855876i	$0.326981\pi$
$60$	−5.29770	−0.683930
$61$	−5.82986	−0.746437	−0.373219	−	0.927743i	$-0.621746\pi$
$61$	−5.82986	−0.746437	−0.373219	+	0.927743i	$0.621746\pi$
$62$	−8.24130	−1.04665
$63$	−13.9351	−1.75566
$64$	1.00000	0.125000
$65$	−11.6958	−1.45069
$66$	3.24873	0.399891
$67$	−7.80410	−0.953423	−0.476711	−	0.879060i	$-0.658171\pi$
$67$	−7.80410	−0.953423	−0.476711	+	0.879060i	$0.658171\pi$
$68$	−0.522036	−0.0633062
$69$	−13.1598	−1.58426
$70$	3.00810	0.359537
$71$	9.04332	1.07324	0.536622	−	0.843822i	$-0.319700\pi$
$71$	9.04332	1.07324	0.536622	+	0.843822i	$0.319700\pi$
$72$	−7.55426	−0.890278
$73$	−2.02182	−0.236636	−0.118318	−	0.992976i	$-0.537750\pi$
$73$	−2.02182	−0.236636	−0.118318	+	0.992976i	$0.537750\pi$
$74$	−10.4305	−1.21252
$75$	7.60471	0.878117
$76$	0	0
$77$	−1.84467	−0.210220
$78$	−23.3008	−2.63830
$79$	−9.86872	−1.11032	−0.555159	−	0.831744i	$-0.687343\pi$
$79$	−9.86872	−1.11032	−0.555159	+	0.831744i	$0.687343\pi$
$80$	1.63070	0.182318
$81$	25.4040	2.82267
$82$	−9.06744	−1.00133
$83$	−0.302948	−0.0332529	−0.0166264	−	0.999862i	$-0.505293\pi$
$83$	−0.302948	−0.0332529	−0.0166264	+	0.999862i	$0.505293\pi$
$84$	5.99285	0.653873
$85$	−0.851283	−0.0923346
$86$	6.11537	0.659438
$87$	25.7219	2.75768
$88$	−1.00000	−0.106600
$89$	2.73814	0.290242	0.145121	−	0.989414i	$-0.453643\pi$
$89$	2.73814	0.290242	0.145121	+	0.989414i	$0.453643\pi$
$90$	−12.3187	−1.29851
$91$	13.2305	1.38694
$92$	4.05075	0.422320
$93$	−26.7738	−2.77631
$94$	−2.80985	−0.289814
$95$	0	0
$96$	3.24873	0.331572
$97$	−8.29725	−0.842458	−0.421229	−	0.906954i	$-0.638401\pi$
$97$	−8.29725	−0.842458	−0.421229	+	0.906954i	$0.638401\pi$
$98$	3.59718	0.363370
$99$	7.55426	0.759231
$100$	−2.34083	−0.234083
$101$	−14.6174	−1.45448	−0.727242	−	0.686381i	$-0.759198\pi$
$101$	−14.6174	−1.45448	−0.727242	+	0.686381i	$0.759198\pi$
$102$	−1.69596	−0.167925
$103$	−3.44284	−0.339233	−0.169616	−	0.985510i	$-0.554253\pi$
$103$	−3.44284	−0.339233	−0.169616	+	0.985510i	$0.554253\pi$
$104$	7.17229	0.703301
$105$	9.77252	0.953700
$106$	−0.238025	−0.0231190
$107$	11.8145	1.14216	0.571078	−	0.820896i	$-0.306525\pi$
$107$	11.8145	1.14216	0.571078	+	0.820896i	$0.306525\pi$
$108$	−14.7956	−1.42370
$109$	−9.25431	−0.886402	−0.443201	−	0.896422i	$-0.646157\pi$
$109$	−9.25431	−0.886402	−0.443201	+	0.896422i	$0.646157\pi$
$110$	−1.63070	−0.155481
$111$	−33.8860	−3.21631
$112$	−1.84467	−0.174305
$113$	14.8386	1.39589	0.697947	−	0.716149i	$-0.254097\pi$
$113$	14.8386	1.39589	0.697947	+	0.716149i	$0.254097\pi$
$114$	0	0
$115$	6.60555	0.615971
$116$	−7.91752	−0.735123
$117$	−54.1813	−5.00907
$118$	−7.94508	−0.731404
$119$	0.962986	0.0882768
$120$	5.29770	0.483612
$121$	1.00000	0.0909091
$122$	5.82986	0.527811
$123$	−29.4577	−2.65611
$124$	8.24130	0.740091
$125$	−11.9707	−1.07069
$126$	13.9351	1.24144
$127$	3.11531	0.276439	0.138220	−	0.990402i	$-0.455862\pi$
$127$	3.11531	0.276439	0.138220	+	0.990402i	$0.455862\pi$
$128$	−1.00000	−0.0883883
$129$	19.8672	1.74921
$130$	11.6958	1.02579
$131$	8.47128	0.740139	0.370069	−	0.929004i	$-0.379334\pi$
$131$	8.47128	0.740139	0.370069	+	0.929004i	$0.379334\pi$
$132$	−3.24873	−0.282766
$133$	0	0
$134$	7.80410	0.674172
$135$	−24.1271	−2.07653
$136$	0.522036	0.0447642
$137$	9.47713	0.809686	0.404843	−	0.914386i	$-0.367326\pi$
$137$	9.47713	0.809686	0.404843	+	0.914386i	$0.367326\pi$
$138$	13.1598	1.12024
$139$	16.2926	1.38192	0.690959	−	0.722894i	$-0.257189\pi$
$139$	16.2926	1.38192	0.690959	+	0.722894i	$0.257189\pi$
$140$	−3.00810	−0.254231
$141$	−9.12844	−0.768754
$142$	−9.04332	−0.758899
$143$	−7.17229	−0.599777
$144$	7.55426	0.629521
$145$	−12.9111	−1.07221
$146$	2.02182	0.167327
$147$	11.6863	0.963868
$148$	10.4305	0.857383
$149$	16.2130	1.32822	0.664112	−	0.747633i	$-0.268810\pi$
$149$	16.2130	1.32822	0.664112	+	0.747633i	$0.268810\pi$
$150$	−7.60471	−0.620922
$151$	6.83442	0.556178	0.278089	−	0.960555i	$-0.410299\pi$
$151$	6.83442	0.556178	0.278089	+	0.960555i	$0.410299\pi$
$152$	0	0
$153$	−3.94360	−0.318821
$154$	1.84467	0.148648
$155$	13.4391	1.07945
$156$	23.3008	1.86556
$157$	−9.78113	−0.780619	−0.390310	−	0.920684i	$-0.627632\pi$
$157$	−9.78113	−0.780619	−0.390310	+	0.920684i	$0.627632\pi$
$158$	9.86872	0.785113
$159$	−0.773280	−0.0613251
$160$	−1.63070	−0.128918
$161$	−7.47231	−0.588901
$162$	−25.4040	−1.99593
$163$	11.5889	0.907713	0.453856	−	0.891075i	$-0.350048\pi$
$163$	11.5889	0.907713	0.453856	+	0.891075i	$0.350048\pi$
$164$	9.06744	0.708048
$165$	−5.29770	−0.412425
$166$	0.302948	0.0235133
$167$	−1.99497	−0.154376	−0.0771878	−	0.997017i	$-0.524594\pi$
$167$	−1.99497	−0.154376	−0.0771878	+	0.997017i	$0.524594\pi$
$168$	−5.99285	−0.462358
$169$	38.4418	2.95706
$170$	0.851283	0.0652904
$171$	0	0
$172$	−6.11537	−0.466293
$173$	−19.5671	−1.48766	−0.743829	−	0.668370i	$-0.766992\pi$
$173$	−19.5671	−1.48766	−0.743829	+	0.668370i	$0.766992\pi$
$174$	−25.7219	−1.94997
$175$	4.31806	0.326414
$176$	1.00000	0.0753778
$177$	−25.8114	−1.94011
$178$	−2.73814	−0.205232
$179$	−1.60082	−0.119651	−0.0598254	−	0.998209i	$-0.519054\pi$
$179$	−1.60082	−0.119651	−0.0598254	+	0.998209i	$0.519054\pi$
$180$	12.3187	0.918182
$181$	13.3150	0.989699	0.494850	−	0.868979i	$-0.335223\pi$
$181$	13.3150	0.989699	0.494850	+	0.868979i	$0.335223\pi$
$182$	−13.2305	−0.980712
$183$	18.9397	1.40006
$184$	−4.05075	−0.298626
$185$	17.0090	1.25053
$186$	26.7738	1.96315
$187$	−0.522036	−0.0381751
$188$	2.80985	0.204929
$189$	27.2930	1.98527
$190$	0	0
$191$	−21.7261	−1.57205	−0.786024	−	0.618196i	$-0.787864\pi$
$191$	−21.7261	−1.57205	−0.786024	+	0.618196i	$0.787864\pi$
$192$	−3.24873	−0.234457
$193$	−2.59084	−0.186493	−0.0932464	−	0.995643i	$-0.529724\pi$
$193$	−2.59084	−0.186493	−0.0932464	+	0.995643i	$0.529724\pi$
$194$	8.29725	0.595708
$195$	37.9966	2.72100
$196$	−3.59718	−0.256942
$197$	−6.18440	−0.440620	−0.220310	−	0.975430i	$-0.570707\pi$
$197$	−6.18440	−0.440620	−0.220310	+	0.975430i	$0.570707\pi$
$198$	−7.55426	−0.536858
$199$	12.8518	0.911039	0.455519	−	0.890226i	$-0.349454\pi$
$199$	12.8518	0.911039	0.455519	+	0.890226i	$0.349454\pi$
$200$	2.34083	0.165521
$201$	25.3534	1.78829
$202$	14.6174	1.02848
$203$	14.6052	1.02509
$204$	1.69596	0.118741
$205$	14.7863	1.03272
$206$	3.44284	0.239874
$207$	30.6004	2.12688
$208$	−7.17229	−0.497309
$209$	0	0
$210$	−9.77252	−0.674368
$211$	8.53605	0.587646	0.293823	−	0.955860i	$-0.405072\pi$
$211$	8.53605	0.587646	0.293823	+	0.955860i	$0.405072\pi$
$212$	0.238025	0.0163476
$213$	−29.3793	−2.01304
$214$	−11.8145	−0.807626
$215$	−9.97232	−0.680107
$216$	14.7956	1.00671
$217$	−15.2025	−1.03201
$218$	9.25431	0.626781
$219$	6.56834	0.443847
$220$	1.63070	0.109942
$221$	3.74420	0.251862
$222$	33.8860	2.27428
$223$	−2.00871	−0.134513	−0.0672565	−	0.997736i	$-0.521425\pi$
$223$	−2.00871	−0.134513	−0.0672565	+	0.997736i	$0.521425\pi$
$224$	1.84467	0.123252
$225$	−17.6832	−1.17888
$226$	−14.8386	−0.987047
$227$	7.39619	0.490902	0.245451	−	0.969409i	$-0.421064\pi$
$227$	7.39619	0.490902	0.245451	+	0.969409i	$0.421064\pi$
$228$	0	0
$229$	−23.8843	−1.57832	−0.789159	−	0.614189i	$-0.789483\pi$
$229$	−23.8843	−1.57832	−0.789159	+	0.614189i	$0.789483\pi$
$230$	−6.60555	−0.435557
$231$	5.99285	0.394300
$232$	7.91752	0.519811
$233$	−7.59013	−0.497246	−0.248623	−	0.968600i	$-0.579978\pi$
$233$	−7.59013	−0.497246	−0.248623	+	0.968600i	$0.579978\pi$
$234$	54.1813	3.54194
$235$	4.58201	0.298898
$236$	7.94508	0.517181
$237$	32.0608	2.08257
$238$	−0.962986	−0.0624211
$239$	25.8234	1.67038	0.835188	−	0.549964i	$-0.185359\pi$
$239$	25.8234	1.67038	0.835188	+	0.549964i	$0.185359\pi$
$240$	−5.29770	−0.341965
$241$	−24.7183	−1.59224	−0.796121	−	0.605137i	$-0.793118\pi$
$241$	−24.7183	−1.59224	−0.796121	+	0.605137i	$0.793118\pi$
$242$	−1.00000	−0.0642824
$243$	−38.1442	−2.44695
$244$	−5.82986	−0.373219
$245$	−5.86592	−0.374760
$246$	29.4577	1.87815
$247$	0	0
$248$	−8.24130	−0.523323
$249$	0.984197	0.0623709
$250$	11.9707	0.757091
$251$	−10.2543	−0.647244	−0.323622	−	0.946186i	$-0.604901\pi$
$251$	−10.2543	−0.647244	−0.323622	+	0.946186i	$0.604901\pi$
$252$	−13.9351	−0.877831
$253$	4.05075	0.254669
$254$	−3.11531	−0.195472
$255$	2.76559	0.173188
$256$	1.00000	0.0625000
$257$	−7.59712	−0.473895	−0.236948	−	0.971522i	$-0.576147\pi$
$257$	−7.59712	−0.473895	−0.236948	+	0.971522i	$0.576147\pi$
$258$	−19.8672	−1.23688
$259$	−19.2409	−1.19557
$260$	−11.6958	−0.725345
$261$	−59.8110	−3.70221
$262$	−8.47128	−0.523357
$263$	4.96245	0.305998	0.152999	−	0.988226i	$-0.451107\pi$
$263$	4.96245	0.305998	0.152999	+	0.988226i	$0.451107\pi$
$264$	3.24873	0.199946
$265$	0.388147	0.0238437
$266$	0	0
$267$	−8.89549	−0.544395
$268$	−7.80410	−0.476711
$269$	18.2396	1.11209	0.556043	−	0.831154i	$-0.312319\pi$
$269$	18.2396	1.11209	0.556043	+	0.831154i	$0.312319\pi$
$270$	24.1271	1.46833
$271$	9.67000	0.587410	0.293705	−	0.955896i	$-0.405112\pi$
$271$	9.67000	0.587410	0.293705	+	0.955896i	$0.405112\pi$
$272$	−0.522036	−0.0316531
$273$	−42.9824	−2.60142
$274$	−9.47713	−0.572534
$275$	−2.34083	−0.141157
$276$	−13.1598	−0.792128
$277$	4.22530	0.253874	0.126937	−	0.991911i	$-0.459485\pi$
$277$	4.22530	0.253874	0.126937	+	0.991911i	$0.459485\pi$
$278$	−16.2926	−0.977163
$279$	62.2569	3.72722
$280$	3.00810	0.179769
$281$	8.18877	0.488501	0.244250	−	0.969712i	$-0.421458\pi$
$281$	8.18877	0.488501	0.244250	+	0.969712i	$0.421458\pi$
$282$	9.12844	0.543591
$283$	−21.9782	−1.30647	−0.653233	−	0.757157i	$-0.726588\pi$
$283$	−21.9782	−1.30647	−0.653233	+	0.757157i	$0.726588\pi$
$284$	9.04332	0.536622
$285$	0	0
$286$	7.17229	0.424106
$287$	−16.7265	−0.987332
$288$	−7.55426	−0.445139
$289$	−16.7275	−0.983969
$290$	12.9111	0.758165
$291$	26.9555	1.58016
$292$	−2.02182	−0.118318
$293$	−0.848794	−0.0495871	−0.0247935	−	0.999693i	$-0.507893\pi$
$293$	−0.848794	−0.0495871	−0.0247935	+	0.999693i	$0.507893\pi$
$294$	−11.6863	−0.681558
$295$	12.9560	0.754329
$296$	−10.4305	−0.606262
$297$	−14.7956	−0.858525
$298$	−16.2130	−0.939196
$299$	−29.0532	−1.68019
$300$	7.60471	0.439058
$301$	11.2809	0.650218
$302$	−6.83442	−0.393277
$303$	47.4880	2.72811
$304$	0	0
$305$	−9.50674	−0.544354
$306$	3.94360	0.225440
$307$	−9.51189	−0.542872	−0.271436	−	0.962456i	$-0.587499\pi$
$307$	−9.51189	−0.542872	−0.271436	+	0.962456i	$0.587499\pi$
$308$	−1.84467	−0.105110
$309$	11.1849	0.636284
$310$	−13.4391	−0.763288
$311$	31.3898	1.77995	0.889975	−	0.456010i	$-0.150722\pi$
$311$	31.3898	1.77995	0.889975	+	0.456010i	$0.150722\pi$
$312$	−23.3008	−1.31915
$313$	8.96646	0.506814	0.253407	−	0.967360i	$-0.418449\pi$
$313$	8.96646	0.506814	0.253407	+	0.967360i	$0.418449\pi$
$314$	9.78113	0.551981
$315$	−22.7240	−1.28035
$316$	−9.86872	−0.555159
$317$	3.90005	0.219049	0.109524	−	0.993984i	$-0.465067\pi$
$317$	3.90005	0.219049	0.109524	+	0.993984i	$0.465067\pi$
$318$	0.773280	0.0433634
$319$	−7.91752	−0.443296
$320$	1.63070	0.0911588
$321$	−38.3823	−2.14229
$322$	7.47231	0.416416
$323$	0	0
$324$	25.4040	1.41133
$325$	16.7891	0.931291
$326$	−11.5889	−0.641850
$327$	30.0648	1.66259
$328$	−9.06744	−0.500666
$329$	−5.18325	−0.285762
$330$	5.29770	0.291629
$331$	−27.8616	−1.53141	−0.765705	−	0.643192i	$-0.777610\pi$
$331$	−27.8616	−1.53141	−0.765705	+	0.643192i	$0.777610\pi$
$332$	−0.302948	−0.0166264
$333$	78.7948	4.31793
$334$	1.99497	0.109160
$335$	−12.7261	−0.695303
$336$	5.99285	0.326937
$337$	25.9158	1.41173	0.705863	−	0.708348i	$-0.250559\pi$
$337$	25.9158	1.41173	0.705863	+	0.708348i	$0.250559\pi$
$338$	−38.4418	−2.09096
$339$	−48.2065	−2.61822
$340$	−0.851283	−0.0461673
$341$	8.24130	0.446291
$342$	0	0
$343$	19.5483	1.05551
$344$	6.11537	0.329719
$345$	−21.4597	−1.15535
$346$	19.5671	1.05193
$347$	20.9691	1.12568	0.562841	−	0.826566i	$-0.309709\pi$
$347$	20.9691	1.12568	0.562841	+	0.826566i	$0.309709\pi$
$348$	25.7219	1.37884
$349$	12.9618	0.693827	0.346913	−	0.937897i	$-0.387230\pi$
$349$	12.9618	0.693827	0.346913	+	0.937897i	$0.387230\pi$
$350$	−4.31806	−0.230810
$351$	106.118	5.66416
$352$	−1.00000	−0.0533002
$353$	17.8129	0.948084	0.474042	−	0.880502i	$-0.342795\pi$
$353$	17.8129	0.948084	0.474042	+	0.880502i	$0.342795\pi$
$354$	25.8114	1.37186
$355$	14.7469	0.782685
$356$	2.73814	0.145121
$357$	−3.12848	−0.165577
$358$	1.60082	0.0846058
$359$	−22.7506	−1.20073	−0.600366	−	0.799725i	$-0.704978\pi$
$359$	−22.7506	−1.20073	−0.600366	+	0.799725i	$0.704978\pi$
$360$	−12.3187	−0.649253
$361$	0	0
$362$	−13.3150	−0.699823
$363$	−3.24873	−0.170514
$364$	13.2305	0.693468
$365$	−3.29697	−0.172571
$366$	−18.9397	−0.989992
$367$	14.6824	0.766413	0.383207	−	0.923663i	$-0.374820\pi$
$367$	14.6824	0.766413	0.383207	+	0.923663i	$0.374820\pi$
$368$	4.05075	0.211160
$369$	68.4978	3.56585
$370$	−17.0090	−0.884257
$371$	−0.439078	−0.0227958
$372$	−26.7738	−1.38816
$373$	−15.7441	−0.815200	−0.407600	−	0.913161i	$-0.633634\pi$
$373$	−15.7441	−0.815200	−0.407600	+	0.913161i	$0.633634\pi$
$374$	0.522036	0.0269939
$375$	38.8895	2.00824
$376$	−2.80985	−0.144907
$377$	56.7867	2.92467
$378$	−27.2930	−1.40380
$379$	−5.78266	−0.297035	−0.148518	−	0.988910i	$-0.547450\pi$
$379$	−5.78266	−0.297035	−0.148518	+	0.988910i	$0.547450\pi$
$380$	0	0
$381$	−10.1208	−0.518505
$382$	21.7261	1.11161
$383$	−4.81045	−0.245802	−0.122901	−	0.992419i	$-0.539220\pi$
$383$	−4.81045	−0.245802	−0.122901	+	0.992419i	$0.539220\pi$
$384$	3.24873	0.165786
$385$	−3.00810	−0.153307
$386$	2.59084	0.131870
$387$	−46.1971	−2.34833
$388$	−8.29725	−0.421229
$389$	−31.4155	−1.59283	−0.796415	−	0.604750i	$-0.793273\pi$
$389$	−31.4155	−1.59283	−0.796415	+	0.604750i	$0.793273\pi$
$390$	−37.9966	−1.92403
$391$	−2.11464	−0.106942
$392$	3.59718	0.181685
$393$	−27.5209	−1.38825
$394$	6.18440	0.311565
$395$	−16.0929	−0.809722
$396$	7.55426	0.379616
$397$	−21.1914	−1.06357	−0.531783	−	0.846880i	$-0.678478\pi$
$397$	−21.1914	−1.06357	−0.531783	+	0.846880i	$0.678478\pi$
$398$	−12.8518	−0.644202
$399$	0	0
$400$	−2.34083	−0.117041
$401$	19.1096	0.954288	0.477144	−	0.878825i	$-0.341672\pi$
$401$	19.1096	0.954288	0.477144	+	0.878825i	$0.341672\pi$
$402$	−25.3534	−1.26451
$403$	−59.1090	−2.94443
$404$	−14.6174	−0.727242
$405$	41.4263	2.05849
$406$	−14.6052	−0.724845
$407$	10.4305	0.517022
$408$	−1.69596	−0.0839623
$409$	−3.72787	−0.184331	−0.0921657	−	0.995744i	$-0.529379\pi$
$409$	−3.72787	−0.184331	−0.0921657	+	0.995744i	$0.529379\pi$
$410$	−14.7863	−0.730241
$411$	−30.7886	−1.51869
$412$	−3.44284	−0.169616
$413$	−14.6561	−0.721178
$414$	−30.6004	−1.50393
$415$	−0.494017	−0.0242503
$416$	7.17229	0.351650
$417$	−52.9302	−2.59200
$418$	0	0
$419$	33.1189	1.61797	0.808983	−	0.587832i	$-0.200018\pi$
$419$	33.1189	1.61797	0.808983	+	0.587832i	$0.200018\pi$
$420$	9.77252	0.476850
$421$	−11.4895	−0.559965	−0.279983	−	0.960005i	$-0.590329\pi$
$421$	−11.4895	−0.559965	−0.279983	+	0.960005i	$0.590329\pi$
$422$	−8.53605	−0.415529
$423$	21.2263	1.03206
$424$	−0.238025	−0.0115595
$425$	1.22200	0.0592755
$426$	29.3793	1.42343
$427$	10.7542	0.520432
$428$	11.8145	0.571078
$429$	23.3008	1.12498
$430$	9.97232	0.480908
$431$	−19.1013	−0.920077	−0.460039	−	0.887899i	$-0.652164\pi$
$431$	−19.1013	−0.920077	−0.460039	+	0.887899i	$0.652164\pi$
$432$	−14.7956	−0.711852
$433$	−12.6714	−0.608947	−0.304473	−	0.952521i	$-0.598480\pi$
$433$	−12.6714	−0.608947	−0.304473	+	0.952521i	$0.598480\pi$
$434$	15.2025	0.729743
$435$	41.9446	2.01109
$436$	−9.25431	−0.443201
$437$	0	0
$438$	−6.56834	−0.313847
$439$	−34.4415	−1.64380	−0.821902	−	0.569628i	$-0.807087\pi$
$439$	−34.4415	−1.64380	−0.821902	+	0.569628i	$0.807087\pi$
$440$	−1.63070	−0.0777405
$441$	−27.1740	−1.29400
$442$	−3.74420	−0.178093
$443$	21.1955	1.00703	0.503515	−	0.863987i	$-0.332040\pi$
$443$	21.1955	1.00703	0.503515	+	0.863987i	$0.332040\pi$
$444$	−33.8860	−1.60816
$445$	4.46508	0.211665
$446$	2.00871	0.0951150
$447$	−52.6718	−2.49129
$448$	−1.84467	−0.0871526
$449$	15.0352	0.709553	0.354777	−	0.934951i	$-0.384557\pi$
$449$	15.0352	0.709553	0.354777	+	0.934951i	$0.384557\pi$
$450$	17.6832	0.833594
$451$	9.06744	0.426969
$452$	14.8386	0.697947
$453$	−22.2032	−1.04320
$454$	−7.39619	−0.347120
$455$	21.5750	1.01145
$456$	0	0
$457$	10.6213	0.496842	0.248421	−	0.968652i	$-0.420088\pi$
$457$	10.6213	0.496842	0.248421	+	0.968652i	$0.420088\pi$
$458$	23.8843	1.11604
$459$	7.72382	0.360517
$460$	6.60555	0.307986
$461$	2.98979	0.139249	0.0696243	−	0.997573i	$-0.477820\pi$
$461$	2.98979	0.139249	0.0696243	+	0.997573i	$0.477820\pi$
$462$	−5.99285	−0.278812
$463$	−10.2765	−0.477591	−0.238795	−	0.971070i	$-0.576753\pi$
$463$	−10.2765	−0.477591	−0.238795	+	0.971070i	$0.576753\pi$
$464$	−7.91752	−0.367562
$465$	−43.6599	−2.02468
$466$	7.59013	0.351606
$467$	−31.0911	−1.43872	−0.719361	−	0.694636i	$-0.755565\pi$
$467$	−31.0911	−1.43872	−0.719361	+	0.694636i	$0.755565\pi$
$468$	−54.1813	−2.50453
$469$	14.3960	0.664746
$470$	−4.58201	−0.211352
$471$	31.7763	1.46417
$472$	−7.94508	−0.365702
$473$	−6.11537	−0.281185
$474$	−32.0608	−1.47260
$475$	0	0
$476$	0.962986	0.0441384
$477$	1.79810	0.0823295
$478$	−25.8234	−1.18113
$479$	3.27108	0.149460	0.0747298	−	0.997204i	$-0.476191\pi$
$479$	3.27108	0.149460	0.0747298	+	0.997204i	$0.476191\pi$
$480$	5.29770	0.241806
$481$	−74.8107	−3.41108
$482$	24.7183	1.12589
$483$	24.2755	1.10458
$484$	1.00000	0.0454545
$485$	−13.5303	−0.614380
$486$	38.1442	1.73026
$487$	13.8579	0.627963	0.313981	−	0.949429i	$-0.398337\pi$
$487$	13.8579	0.627963	0.313981	+	0.949429i	$0.398337\pi$
$488$	5.82986	0.263905
$489$	−37.6492	−1.70256
$490$	5.86592	0.264995
$491$	17.3244	0.781841	0.390920	−	0.920425i	$-0.372157\pi$
$491$	17.3244	0.781841	0.390920	+	0.920425i	$0.372157\pi$
$492$	−29.4577	−1.32805
$493$	4.13323	0.186151
$494$	0	0
$495$	12.3187	0.553685
$496$	8.24130	0.370045
$497$	−16.6820	−0.748289
$498$	−0.984197	−0.0441029
$499$	−26.0723	−1.16716	−0.583579	−	0.812057i	$-0.698348\pi$
$499$	−26.0723	−1.16716	−0.583579	+	0.812057i	$0.698348\pi$
$500$	−11.9707	−0.535344
$501$	6.48113	0.289556
$502$	10.2543	0.457671
$503$	−10.7577	−0.479662	−0.239831	−	0.970815i	$-0.577092\pi$
$503$	−10.7577	−0.479662	−0.239831	+	0.970815i	$0.577092\pi$
$504$	13.9351	0.620720
$505$	−23.8365	−1.06071
$506$	−4.05075	−0.180078
$507$	−124.887	−5.54642
$508$	3.11531	0.138220
$509$	−29.9394	−1.32704	−0.663519	−	0.748159i	$-0.730938\pi$
$509$	−29.9394	−1.32704	−0.663519	+	0.748159i	$0.730938\pi$
$510$	−2.76559	−0.122462
$511$	3.72959	0.164987
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	7.59712	0.335095
$515$	−5.61423	−0.247392
$516$	19.8672	0.874605
$517$	2.80985	0.123577
$518$	19.2409	0.845396
$519$	63.5682	2.79033
$520$	11.6958	0.512896
$521$	9.50825	0.416564	0.208282	−	0.978069i	$-0.433213\pi$
$521$	9.50825	0.416564	0.208282	+	0.978069i	$0.433213\pi$
$522$	59.8110	2.61786
$523$	0.758833	0.0331815	0.0165907	−	0.999862i	$-0.494719\pi$
$523$	0.758833	0.0331815	0.0165907	+	0.999862i	$0.494719\pi$
$524$	8.47128	0.370069
$525$	−14.0282	−0.612241
$526$	−4.96245	−0.216373
$527$	−4.30226	−0.187409
$528$	−3.24873	−0.141383
$529$	−6.59140	−0.286582
$530$	−0.388147	−0.0168600
$531$	60.0192	2.60461
$532$	0	0
$533$	−65.0343	−2.81695
$534$	8.89549	0.384945
$535$	19.2660	0.832940
$536$	7.80410	0.337086
$537$	5.20063	0.224424
$538$	−18.2396	−0.786363
$539$	−3.59718	−0.154942
$540$	−24.1271	−1.03826
$541$	−24.5075	−1.05366	−0.526830	−	0.849970i	$-0.676620\pi$
$541$	−24.5075	−1.05366	−0.526830	+	0.849970i	$0.676620\pi$
$542$	−9.67000	−0.415362
$543$	−43.2570	−1.85633
$544$	0.522036	0.0223821
$545$	−15.0910	−0.646426
$546$	42.9824	1.83948
$547$	−16.7322	−0.715418	−0.357709	−	0.933833i	$-0.616442\pi$
$547$	−16.7322	−0.715418	−0.357709	+	0.933833i	$0.616442\pi$
$548$	9.47713	0.404843
$549$	−44.0403	−1.87959
$550$	2.34083	0.0998131
$551$	0	0
$552$	13.1598	0.560119
$553$	18.2046	0.774137
$554$	−4.22530	−0.179516
$555$	−55.2577	−2.34556
$556$	16.2926	0.690959
$557$	11.6106	0.491958	0.245979	−	0.969275i	$-0.420891\pi$
$557$	11.6106	0.491958	0.245979	+	0.969275i	$0.420891\pi$
$558$	−62.2569	−2.63554
$559$	43.8612	1.85513
$560$	−3.00810	−0.127116
$561$	1.69596	0.0716033
$562$	−8.18877	−0.345422
$563$	−25.8653	−1.09009	−0.545046	−	0.838406i	$-0.683488\pi$
$563$	−25.8653	−1.09009	−0.545046	+	0.838406i	$0.683488\pi$
$564$	−9.12844	−0.384377
$565$	24.1972	1.01798
$566$	21.9782	0.923811
$567$	−46.8621	−1.96802
$568$	−9.04332	−0.379449
$569$	−14.9984	−0.628764	−0.314382	−	0.949297i	$-0.601797\pi$
$569$	−14.9984	−0.628764	−0.314382	+	0.949297i	$0.601797\pi$
$570$	0	0
$571$	11.7631	0.492271	0.246136	−	0.969235i	$-0.420839\pi$
$571$	11.7631	0.492271	0.246136	+	0.969235i	$0.420839\pi$
$572$	−7.17229	−0.299889
$573$	70.5824	2.94862
$574$	16.7265	0.698149
$575$	−9.48211	−0.395431
$576$	7.55426	0.314761
$577$	−21.2216	−0.883465	−0.441733	−	0.897147i	$-0.645636\pi$
$577$	−21.2216	−0.883465	−0.441733	+	0.897147i	$0.645636\pi$
$578$	16.7275	0.695771
$579$	8.41695	0.349796
$580$	−12.9111	−0.536103
$581$	0.558840	0.0231846
$582$	−26.9555	−1.11734
$583$	0.238025	0.00985799
$584$	2.02182	0.0836634
$585$	−88.3534	−3.65296
$586$	0.848794	0.0350634
$587$	−36.5850	−1.51003	−0.755013	−	0.655710i	$-0.772370\pi$
$587$	−36.5850	−1.51003	−0.755013	+	0.655710i	$0.772370\pi$
$588$	11.6863	0.481934
$589$	0	0
$590$	−12.9560	−0.533391
$591$	20.0914	0.826452
$592$	10.4305	0.428692
$593$	−38.6516	−1.58723	−0.793615	−	0.608420i	$-0.791804\pi$
$593$	−38.6516	−1.58723	−0.793615	+	0.608420i	$0.791804\pi$
$594$	14.7956	0.607069
$595$	1.57034	0.0643776
$596$	16.2130	0.664112
$597$	−41.7520	−1.70879
$598$	29.0532	1.18807
$599$	−13.6153	−0.556305	−0.278152	−	0.960537i	$-0.589722\pi$
$599$	−13.6153	−0.556305	−0.278152	+	0.960537i	$0.589722\pi$
$600$	−7.60471	−0.310461
$601$	−32.3026	−1.31765	−0.658826	−	0.752296i	$-0.728946\pi$
$601$	−32.3026	−1.31765	−0.658826	+	0.752296i	$0.728946\pi$
$602$	−11.2809	−0.459774
$603$	−58.9542	−2.40080
$604$	6.83442	0.278089
$605$	1.63070	0.0662973
$606$	−47.4880	−1.92907
$607$	38.3367	1.55604	0.778020	−	0.628240i	$-0.216224\pi$
$607$	38.3367	1.55604	0.778020	+	0.628240i	$0.216224\pi$
$608$	0	0
$609$	−47.4485	−1.92271
$610$	9.50674	0.384917
$611$	−20.1530	−0.815305
$612$	−3.94360	−0.159410
$613$	−36.2927	−1.46585	−0.732923	−	0.680311i	$-0.761845\pi$
$613$	−36.2927	−1.46585	−0.732923	+	0.680311i	$0.761845\pi$
$614$	9.51189	0.383869
$615$	−48.0366	−1.93702
$616$	1.84467	0.0743240
$617$	−47.6911	−1.91997	−0.959986	−	0.280048i	$-0.909650\pi$
$617$	−47.6911	−1.91997	−0.959986	+	0.280048i	$0.909650\pi$
$618$	−11.1849	−0.449921
$619$	32.2248	1.29522	0.647612	−	0.761970i	$-0.275768\pi$
$619$	32.2248	1.29522	0.647612	+	0.761970i	$0.275768\pi$
$620$	13.4391	0.539726
$621$	−59.9332	−2.40503
$622$	−31.3898	−1.25861
$623$	−5.05097	−0.202363
$624$	23.3008	0.932780
$625$	−7.81641	−0.312656
$626$	−8.96646	−0.358372
$627$	0	0
$628$	−9.78113	−0.390310
$629$	−5.44511	−0.217111
$630$	22.7240	0.905345
$631$	−50.0515	−1.99252	−0.996260	−	0.0864115i	$-0.972460\pi$
$631$	−50.0515	−1.99252	−0.996260	+	0.0864115i	$0.972460\pi$
$632$	9.86872	0.392557
$633$	−27.7313	−1.10222
$634$	−3.90005	−0.154891
$635$	5.08013	0.201599
$636$	−0.773280	−0.0306625
$637$	25.8000	1.02224
$638$	7.91752	0.313458
$639$	68.3156	2.70252
$640$	−1.63070	−0.0644590
$641$	10.1398	0.400498	0.200249	−	0.979745i	$-0.435825\pi$
$641$	10.1398	0.400498	0.200249	+	0.979745i	$0.435825\pi$
$642$	38.3823	1.51483
$643$	17.3882	0.685723	0.342861	−	0.939386i	$-0.388604\pi$
$643$	17.3882	0.685723	0.342861	+	0.939386i	$0.388604\pi$
$644$	−7.47231	−0.294450
$645$	32.3974	1.27565
$646$	0	0
$647$	−41.3464	−1.62549	−0.812747	−	0.582617i	$-0.802029\pi$
$647$	−41.3464	−1.62549	−0.812747	+	0.582617i	$0.802029\pi$
$648$	−25.4040	−0.997964
$649$	7.94508	0.311872
$650$	−16.7891	−0.658522
$651$	49.3888	1.93570
$652$	11.5889	0.453856
$653$	−25.7317	−1.00696	−0.503480	−	0.864007i	$-0.667947\pi$
$653$	−25.7317	−1.00696	−0.503480	+	0.864007i	$0.667947\pi$
$654$	−30.0648	−1.17563
$655$	13.8141	0.539761
$656$	9.06744	0.354024
$657$	−15.2733	−0.595869
$658$	5.18325	0.202064
$659$	−12.5253	−0.487917	−0.243959	−	0.969786i	$-0.578446\pi$
$659$	−12.5253	−0.487917	−0.243959	+	0.969786i	$0.578446\pi$
$660$	−5.29770	−0.206213
$661$	41.3814	1.60955	0.804776	−	0.593579i	$-0.202285\pi$
$661$	41.3814	1.60955	0.804776	+	0.593579i	$0.202285\pi$
$662$	27.8616	1.08287
$663$	−12.1639	−0.472406
$664$	0.302948	0.0117567
$665$	0	0
$666$	−78.7948	−3.05324
$667$	−32.0719	−1.24183
$668$	−1.99497	−0.0771878
$669$	6.52575	0.252300
$670$	12.7261	0.491653
$671$	−5.82986	−0.225059
$672$	−5.99285	−0.231179
$673$	5.88277	0.226764	0.113382	−	0.993551i	$-0.463832\pi$
$673$	5.88277	0.226764	0.113382	+	0.993551i	$0.463832\pi$
$674$	−25.9158	−0.998241
$675$	34.6338	1.33306
$676$	38.4418	1.47853
$677$	39.9480	1.53533	0.767664	−	0.640853i	$-0.221419\pi$
$677$	39.9480	1.53533	0.767664	+	0.640853i	$0.221419\pi$
$678$	48.2065	1.85136
$679$	15.3057	0.587379
$680$	0.851283	0.0326452
$681$	−24.0282	−0.920764
$682$	−8.24130	−0.315576
$683$	−46.2326	−1.76904	−0.884520	−	0.466502i	$-0.845514\pi$
$683$	−46.2326	−1.76904	−0.884520	+	0.466502i	$0.845514\pi$
$684$	0	0
$685$	15.4543	0.590480
$686$	−19.5483	−0.746359
$687$	77.5936	2.96038
$688$	−6.11537	−0.233146
$689$	−1.70719	−0.0650386
$690$	21.4597	0.816956
$691$	−6.34804	−0.241491	−0.120745	−	0.992684i	$-0.538528\pi$
$691$	−6.34804	−0.241491	−0.120745	+	0.992684i	$0.538528\pi$
$692$	−19.5671	−0.743829
$693$	−13.9351	−0.529352
$694$	−20.9691	−0.795977
$695$	26.5683	1.00779
$696$	−25.7219	−0.974986
$697$	−4.73353	−0.179295
$698$	−12.9618	−0.490610
$699$	24.6583	0.932663
$700$	4.31806	0.163207
$701$	0.255306	0.00964278	0.00482139	−	0.999988i	$-0.498465\pi$
$701$	0.255306	0.00964278	0.00482139	+	0.999988i	$0.498465\pi$
$702$	−106.118	−4.00517
$703$	0	0
$704$	1.00000	0.0376889
$705$	−14.8857	−0.560629
$706$	−17.8129	−0.670396
$707$	26.9643	1.01410
$708$	−25.8114	−0.970053
$709$	−14.9069	−0.559841	−0.279920	−	0.960023i	$-0.590308\pi$
$709$	−14.9069	−0.559841	−0.279920	+	0.960023i	$0.590308\pi$
$710$	−14.7469	−0.553442
$711$	−74.5509	−2.79588
$712$	−2.73814	−0.102616
$713$	33.3835	1.25022
$714$	3.12848	0.117081
$715$	−11.6958	−0.437400
$716$	−1.60082	−0.0598254
$717$	−83.8933	−3.13305
$718$	22.7506	0.849046
$719$	−17.4874	−0.652169	−0.326085	−	0.945341i	$-0.605729\pi$
$719$	−17.4874	−0.652169	−0.326085	+	0.945341i	$0.605729\pi$
$720$	12.3187	0.459091
$721$	6.35091	0.236520
$722$	0	0
$723$	80.3030	2.98650
$724$	13.3150	0.494850
$725$	18.5335	0.688318
$726$	3.24873	0.120572
$727$	−34.7702	−1.28955	−0.644777	−	0.764371i	$-0.723050\pi$
$727$	−34.7702	−1.28955	−0.644777	+	0.764371i	$0.723050\pi$
$728$	−13.2305	−0.490356
$729$	47.7081	1.76697
$730$	3.29697	0.122026
$731$	3.19245	0.118077
$732$	18.9397	0.700030
$733$	−39.7863	−1.46954	−0.734771	−	0.678316i	$-0.762710\pi$
$733$	−39.7863	−1.46954	−0.734771	+	0.678316i	$0.762710\pi$
$734$	−14.6824	−0.541936
$735$	19.0568	0.702920
$736$	−4.05075	−0.149313
$737$	−7.80410	−0.287468
$738$	−68.4978	−2.52144
$739$	10.4235	0.383435	0.191717	−	0.981450i	$-0.438594\pi$
$739$	10.4235	0.383435	0.191717	+	0.981450i	$0.438594\pi$
$740$	17.0090	0.625264
$741$	0	0
$742$	0.439078	0.0161191
$743$	2.44533	0.0897106	0.0448553	−	0.998993i	$-0.485717\pi$
$743$	2.44533	0.0897106	0.0448553	+	0.998993i	$0.485717\pi$
$744$	26.7738	0.981574
$745$	26.4386	0.968634
$746$	15.7441	0.576433
$747$	−2.28855	−0.0837336
$748$	−0.522036	−0.0190875
$749$	−21.7940	−0.796334
$750$	−38.8895	−1.42004
$751$	−26.8688	−0.980455	−0.490228	−	0.871594i	$-0.663086\pi$
$751$	−26.8688	−0.980455	−0.490228	+	0.871594i	$0.663086\pi$
$752$	2.80985	0.102465
$753$	33.3134	1.21401
$754$	−56.7867	−2.06805
$755$	11.1449	0.405604
$756$	27.2930	0.992635
$757$	20.1868	0.733700	0.366850	−	0.930280i	$-0.380436\pi$
$757$	20.1868	0.733700	0.366850	+	0.930280i	$0.380436\pi$
$758$	5.78266	0.210036
$759$	−13.1598	−0.477671
$760$	0	0
$761$	17.3147	0.627659	0.313829	−	0.949479i	$-0.398388\pi$
$761$	17.3147	0.627659	0.313829	+	0.949479i	$0.398388\pi$
$762$	10.1208	0.366639
$763$	17.0712	0.618018
$764$	−21.7261	−0.786024
$765$	−6.43081	−0.232507
$766$	4.81045	0.173808
$767$	−56.9844	−2.05759
$768$	−3.24873	−0.117229
$769$	−35.1287	−1.26677	−0.633387	−	0.773836i	$-0.718336\pi$
$769$	−35.1287	−1.26677	−0.633387	+	0.773836i	$0.718336\pi$
$770$	3.00810	0.108405
$771$	24.6810	0.888865
$772$	−2.59084	−0.0932464
$773$	−2.46147	−0.0885331	−0.0442665	−	0.999020i	$-0.514095\pi$
$773$	−2.46147	−0.0885331	−0.0442665	+	0.999020i	$0.514095\pi$
$774$	46.1971	1.66052
$775$	−19.2914	−0.692969
$776$	8.29725	0.297854
$777$	62.5085	2.24248
$778$	31.4155	1.12630
$779$	0	0
$780$	37.9966	1.36050
$781$	9.04332	0.323596
$782$	2.11464	0.0756194
$783$	117.144	4.18639
$784$	−3.59718	−0.128471
$785$	−15.9501	−0.569282
$786$	27.5209	0.981638
$787$	3.72732	0.132865	0.0664323	−	0.997791i	$-0.478838\pi$
$787$	3.72732	0.132865	0.0664323	+	0.997791i	$0.478838\pi$
$788$	−6.18440	−0.220310
$789$	−16.1217	−0.573947
$790$	16.0929	0.572560
$791$	−27.3723	−0.973247
$792$	−7.55426	−0.268429
$793$	41.8135	1.48484
$794$	21.1914	0.752055
$795$	−1.26099	−0.0447225
$796$	12.8518	0.455519
$797$	−27.9329	−0.989433	−0.494716	−	0.869054i	$-0.664728\pi$
$797$	−27.9329	−0.989433	−0.494716	+	0.869054i	$0.664728\pi$
$798$	0	0
$799$	−1.46684	−0.0518932
$800$	2.34083	0.0827607
$801$	20.6846	0.730855
$802$	−19.1096	−0.674783
$803$	−2.02182	−0.0713484
$804$	25.3534	0.894147
$805$	−12.1851	−0.429468
$806$	59.1090	2.08203
$807$	−59.2554	−2.08589
$808$	14.6174	0.514238
$809$	−3.20397	−0.112646	−0.0563228	−	0.998413i	$-0.517938\pi$
$809$	−3.20397	−0.112646	−0.0563228	+	0.998413i	$0.517938\pi$
$810$	−41.4263	−1.45557
$811$	33.5543	1.17825	0.589126	−	0.808041i	$-0.299472\pi$
$811$	33.5543	1.17825	0.589126	+	0.808041i	$0.299472\pi$
$812$	14.6052	0.512543
$813$	−31.4152	−1.10178
$814$	−10.4305	−0.365590
$815$	18.8980	0.661968
$816$	1.69596	0.0593703
$817$	0	0
$818$	3.72787	0.130342
$819$	99.9468	3.49242
$820$	14.7863	0.516358
$821$	35.8207	1.25015	0.625075	−	0.780565i	$-0.285068\pi$
$821$	35.8207	1.25015	0.625075	+	0.780565i	$0.285068\pi$
$822$	30.7886	1.07388
$823$	19.6877	0.686269	0.343134	−	0.939286i	$-0.388511\pi$
$823$	19.6877	0.686269	0.343134	+	0.939286i	$0.388511\pi$
$824$	3.44284	0.119937
$825$	7.60471	0.264762
$826$	14.6561	0.509950
$827$	23.7618	0.826279	0.413139	−	0.910668i	$-0.364432\pi$
$827$	23.7618	0.826279	0.413139	+	0.910668i	$0.364432\pi$
$828$	30.6004	1.06344
$829$	−33.5519	−1.16530	−0.582652	−	0.812722i	$-0.697985\pi$
$829$	−33.5519	−1.16530	−0.582652	+	0.812722i	$0.697985\pi$
$830$	0.494017	0.0171476
$831$	−13.7269	−0.476179
$832$	−7.17229	−0.248654
$833$	1.87786	0.0650640
$834$	52.9302	1.83282
$835$	−3.25320	−0.112582
$836$	0	0
$837$	−121.935	−4.21468
$838$	−33.1189	−1.14407
$839$	−18.9122	−0.652923	−0.326461	−	0.945211i	$-0.605856\pi$
$839$	−18.9122	−0.652923	−0.326461	+	0.945211i	$0.605856\pi$
$840$	−9.77252	−0.337184
$841$	33.6871	1.16162
$842$	11.4895	0.395955
$843$	−26.6031	−0.916259
$844$	8.53605	0.293823
$845$	62.6869	2.15649
$846$	−21.2263	−0.729776
$847$	−1.84467	−0.0633837
$848$	0.238025	0.00817382
$849$	71.4011	2.45048
$850$	−1.22200	−0.0419141
$851$	42.2515	1.44836
$852$	−29.3793	−1.00652
$853$	1.95220	0.0668420	0.0334210	−	0.999441i	$-0.489360\pi$
$853$	1.95220	0.0668420	0.0334210	+	0.999441i	$0.489360\pi$
$854$	−10.7542	−0.368001
$855$	0	0
$856$	−11.8145	−0.403813
$857$	−35.3340	−1.20699	−0.603493	−	0.797368i	$-0.706225\pi$
$857$	−35.3340	−1.20699	−0.603493	+	0.797368i	$0.706225\pi$
$858$	−23.3008	−0.795478
$859$	36.4982	1.24530	0.622652	−	0.782499i	$-0.286055\pi$
$859$	36.4982	1.24530	0.622652	+	0.782499i	$0.286055\pi$
$860$	−9.97232	−0.340053
$861$	54.3398	1.85189
$862$	19.1013	0.650593
$863$	38.3550	1.30562	0.652810	−	0.757521i	$-0.273590\pi$
$863$	38.3550	1.30562	0.652810	+	0.757521i	$0.273590\pi$
$864$	14.7956	0.503355
$865$	−31.9080	−1.08490
$866$	12.6714	0.430590
$867$	54.3431	1.84559
$868$	−15.2025	−0.516006
$869$	−9.86872	−0.334773
$870$	−41.9446	−1.42206
$871$	55.9733	1.89658
$872$	9.25431	0.313390
$873$	−62.6796	−2.12138
$874$	0	0
$875$	22.0820	0.746506
$876$	6.56834	0.221924
$877$	34.0428	1.14954	0.574772	−	0.818314i	$-0.305091\pi$
$877$	34.0428	1.14954	0.574772	+	0.818314i	$0.305091\pi$
$878$	34.4415	1.16235
$879$	2.75751	0.0930083
$880$	1.63070	0.0549708
$881$	−25.0525	−0.844041	−0.422021	−	0.906586i	$-0.638679\pi$
$881$	−25.0525	−0.844041	−0.422021	+	0.906586i	$0.638679\pi$
$882$	27.1740	0.914998
$883$	3.68732	0.124088	0.0620442	−	0.998073i	$-0.480238\pi$
$883$	3.68732	0.124088	0.0620442	+	0.998073i	$0.480238\pi$
$884$	3.74420	0.125931
$885$	−42.0907	−1.41486
$886$	−21.1955	−0.712077
$887$	−37.2761	−1.25161	−0.625804	−	0.779981i	$-0.715229\pi$
$887$	−37.2761	−1.25161	−0.625804	+	0.779981i	$0.715229\pi$
$888$	33.8860	1.13714
$889$	−5.74673	−0.192739
$890$	−4.46508	−0.149670
$891$	25.4040	0.851067
$892$	−2.00871	−0.0672565
$893$	0	0
$894$	52.6718	1.76161
$895$	−2.61045	−0.0872577
$896$	1.84467	0.0616262
$897$	94.3860	3.15146
$898$	−15.0352	−0.501730
$899$	−65.2507	−2.17623
$900$	−17.6832	−0.589440
$901$	−0.124258	−0.00413963
$902$	−9.06744	−0.301913
$903$	−36.6485	−1.21959
$904$	−14.8386	−0.493523
$905$	21.7128	0.721758
$906$	22.2032	0.737652
$907$	34.5118	1.14595	0.572973	−	0.819574i	$-0.305790\pi$
$907$	34.5118	1.14595	0.572973	+	0.819574i	$0.305790\pi$
$908$	7.39619	0.245451
$909$	−110.423	−3.66252
$910$	−21.5750	−0.715204
$911$	−3.71073	−0.122942	−0.0614710	−	0.998109i	$-0.519579\pi$
$911$	−3.71073	−0.122942	−0.0614710	+	0.998109i	$0.519579\pi$
$912$	0	0
$913$	−0.302948	−0.0100261
$914$	−10.6213	−0.351320
$915$	30.8849	1.02102
$916$	−23.8843	−0.789159
$917$	−15.6267	−0.516040
$918$	−7.72382	−0.254924
$919$	−29.2213	−0.963921	−0.481961	−	0.876193i	$-0.660075\pi$
$919$	−29.2213	−0.963921	−0.481961	+	0.876193i	$0.660075\pi$
$920$	−6.60555	−0.217779
$921$	30.9016	1.01824
$922$	−2.98979	−0.0984636
$923$	−64.8613	−2.13494
$924$	5.99285	0.197150
$925$	−24.4160	−0.802794
$926$	10.2765	0.337708
$927$	−26.0081	−0.854217
$928$	7.91752	0.259905
$929$	−24.6116	−0.807480	−0.403740	−	0.914874i	$-0.632290\pi$
$929$	−24.6116	−0.807480	−0.403740	+	0.914874i	$0.632290\pi$
$930$	43.6599	1.43167
$931$	0	0
$932$	−7.59013	−0.248623
$933$	−101.977	−3.33857
$934$	31.0911	1.01733
$935$	−0.851283	−0.0278399
$936$	54.1813	1.77097
$937$	55.2201	1.80396	0.901980	−	0.431777i	$-0.142113\pi$
$937$	55.2201	1.80396	0.901980	+	0.431777i	$0.142113\pi$
$938$	−14.3960	−0.470046
$939$	−29.1296	−0.950609
$940$	4.58201	0.149449
$941$	−19.4053	−0.632595	−0.316298	−	0.948660i	$-0.602440\pi$
$941$	−19.4053	−0.632595	−0.316298	+	0.948660i	$0.602440\pi$
$942$	−31.7763	−1.03533
$943$	36.7300	1.19609
$944$	7.94508	0.258590
$945$	44.5066	1.44780
$946$	6.11537	0.198828
$947$	6.04609	0.196471	0.0982357	−	0.995163i	$-0.468680\pi$
$947$	6.04609	0.196471	0.0982357	+	0.995163i	$0.468680\pi$
$948$	32.0608	1.04129
$949$	14.5011	0.470724
$950$	0	0
$951$	−12.6702	−0.410860
$952$	−0.962986	−0.0312106
$953$	−50.9405	−1.65012	−0.825062	−	0.565042i	$-0.808860\pi$
$953$	−50.9405	−1.65012	−0.825062	+	0.565042i	$0.808860\pi$
$954$	−1.79810	−0.0582157
$955$	−35.4288	−1.14645
$956$	25.8234	0.835188
$957$	25.7219	0.831471
$958$	−3.27108	−0.105684
$959$	−17.4822	−0.564530
$960$	−5.29770	−0.170982
$961$	36.9190	1.19094
$962$	74.8107	2.41199
$963$	89.2501	2.87605
$964$	−24.7183	−0.796121
$965$	−4.22488	−0.136004
$966$	−24.2755	−0.781053
$967$	8.16016	0.262413	0.131206	−	0.991355i	$-0.458115\pi$
$967$	8.16016	0.262413	0.131206	+	0.991355i	$0.458115\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	13.5303	0.434432
$971$	−9.03916	−0.290080	−0.145040	−	0.989426i	$-0.546331\pi$
$971$	−9.03916	−0.290080	−0.145040	+	0.989426i	$0.546331\pi$
$972$	−38.1442	−1.22348
$973$	−30.0545	−0.963502
$974$	−13.8579	−0.444037
$975$	−54.5432	−1.74678
$976$	−5.82986	−0.186609
$977$	−23.3321	−0.746459	−0.373230	−	0.927739i	$-0.621750\pi$
$977$	−23.3321	−0.746459	−0.373230	+	0.927739i	$0.621750\pi$
$978$	37.6492	1.20389
$979$	2.73814	0.0875114
$980$	−5.86592	−0.187380
$981$	−69.9094	−2.23204
$982$	−17.3244	−0.552845
$983$	−21.4815	−0.685152	−0.342576	−	0.939490i	$-0.611299\pi$
$983$	−21.4815	−0.685152	−0.342576	+	0.939490i	$0.611299\pi$
$984$	29.4577	0.939077
$985$	−10.0849	−0.321331
$986$	−4.13323	−0.131629
$987$	16.8390	0.535991
$988$	0	0
$989$	−24.7719	−0.787700
$990$	−12.3187	−0.391514
$991$	−17.9178	−0.569177	−0.284588	−	0.958650i	$-0.591857\pi$
$991$	−17.9178	−0.569177	−0.284588	+	0.958650i	$0.591857\pi$
$992$	−8.24130	−0.261662
$993$	90.5148	2.87240
$994$	16.6820	0.529120
$995$	20.9574	0.664393
$996$	0.984197	0.0311855
$997$	32.9297	1.04289	0.521447	−	0.853284i	$-0.325392\pi$
$997$	32.9297	1.04289	0.521447	+	0.853284i	$0.325392\pi$
$998$	26.0723	0.825305
$999$	−154.325	−4.88264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.bn.1.1	✓	8
19.18	odd	2		7942.2.a.bq.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7942.2.a.bn.1.1	✓	8	1.1	even	1	trivial
7942.2.a.bq.1.8	yes	8	19.18	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:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.24873 q^{3} +1.00000 q^{4} +1.63070 q^{5} +3.24873 q^{6} -1.84467 q^{7} -1.00000 q^{8} +7.55426 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.24873 q^{3} +1.00000 q^{4} +1.63070 q^{5} +3.24873 q^{6} -1.84467 q^{7} -1.00000 q^{8} +7.55426 q^{9} -1.63070 q^{10} +1.00000 q^{11} -3.24873 q^{12} -7.17229 q^{13} +1.84467 q^{14} -5.29770 q^{15} +1.00000 q^{16} -0.522036 q^{17} -7.55426 q^{18} +1.63070 q^{20} +5.99285 q^{21} -1.00000 q^{22} +4.05075 q^{23} +3.24873 q^{24} -2.34083 q^{25} +7.17229 q^{26} -14.7956 q^{27} -1.84467 q^{28} -7.91752 q^{29} +5.29770 q^{30} +8.24130 q^{31} -1.00000 q^{32} -3.24873 q^{33} +0.522036 q^{34} -3.00810 q^{35} +7.55426 q^{36} +10.4305 q^{37} +23.3008 q^{39} -1.63070 q^{40} +9.06744 q^{41} -5.99285 q^{42} -6.11537 q^{43} +1.00000 q^{44} +12.3187 q^{45} -4.05075 q^{46} +2.80985 q^{47} -3.24873 q^{48} -3.59718 q^{49} +2.34083 q^{50} +1.69596 q^{51} -7.17229 q^{52} +0.238025 q^{53} +14.7956 q^{54} +1.63070 q^{55} +1.84467 q^{56} +7.91752 q^{58} +7.94508 q^{59} -5.29770 q^{60} -5.82986 q^{61} -8.24130 q^{62} -13.9351 q^{63} +1.00000 q^{64} -11.6958 q^{65} +3.24873 q^{66} -7.80410 q^{67} -0.522036 q^{68} -13.1598 q^{69} +3.00810 q^{70} +9.04332 q^{71} -7.55426 q^{72} -2.02182 q^{73} -10.4305 q^{74} +7.60471 q^{75} -1.84467 q^{77} -23.3008 q^{78} -9.86872 q^{79} +1.63070 q^{80} +25.4040 q^{81} -9.06744 q^{82} -0.302948 q^{83} +5.99285 q^{84} -0.851283 q^{85} +6.11537 q^{86} +25.7219 q^{87} -1.00000 q^{88} +2.73814 q^{89} -12.3187 q^{90} +13.2305 q^{91} +4.05075 q^{92} -26.7738 q^{93} -2.80985 q^{94} +3.24873 q^{96} -8.29725 q^{97} +3.59718 q^{98} +7.55426 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 3 q^{5} + q^{6} - 8 q^{8} + 15 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 3 * q^5 + q^6 - 8 * q^8 + 15 * q^9	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 3 q^{5} + q^{6} - 8 q^{8} + 15 q^{9} + 3 q^{10} + 8 q^{11} - q^{12} - 3 q^{13} - 36 q^{15} + 8 q^{16} - 4 q^{17} - 15 q^{18} - 3 q^{20} - 3 q^{21} - 8 q^{22} - q^{23} + q^{24} + 5 q^{25} + 3 q^{26} - 10 q^{27} + 4 q^{29} + 36 q^{30} + 3 q^{31} - 8 q^{32} - q^{33} + 4 q^{34} + 3 q^{35} + 15 q^{36} + 26 q^{37} + 14 q^{39} + 3 q^{40} - 17 q^{41} + 3 q^{42} - 6 q^{43} + 8 q^{44} + 3 q^{45} + q^{46} - q^{48} + 36 q^{49} - 5 q^{50} + 12 q^{51} - 3 q^{52} - 49 q^{53} + 10 q^{54} - 3 q^{55} - 4 q^{58} - 15 q^{59} - 36 q^{60} - 14 q^{61} - 3 q^{62} + 17 q^{63} + 8 q^{64} + q^{66} - 5 q^{67} - 4 q^{68} - 4 q^{69} - 3 q^{70} + q^{71} - 15 q^{72} - 10 q^{73} - 26 q^{74} + 29 q^{75} - 14 q^{78} - 12 q^{79} - 3 q^{80} + 17 q^{82} - 7 q^{83} - 3 q^{84} - 20 q^{85} + 6 q^{86} + 45 q^{87} - 8 q^{88} - 8 q^{89} - 3 q^{90} - 17 q^{91} - q^{92} - 33 q^{93} + q^{96} - 54 q^{97} - 36 q^{98} + 15 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 3 * q^5 + q^6 - 8 * q^8 + 15 * q^9 + 3 * q^10 + 8 * q^11 - q^12 - 3 * q^13 - 36 * q^15 + 8 * q^16 - 4 * q^17 - 15 * q^18 - 3 * q^20 - 3 * q^21 - 8 * q^22 - q^23 + q^24 + 5 * q^25 + 3 * q^26 - 10 * q^27 + 4 * q^29 + 36 * q^30 + 3 * q^31 - 8 * q^32 - q^33 + 4 * q^34 + 3 * q^35 + 15 * q^36 + 26 * q^37 + 14 * q^39 + 3 * q^40 - 17 * q^41 + 3 * q^42 - 6 * q^43 + 8 * q^44 + 3 * q^45 + q^46 - q^48 + 36 * q^49 - 5 * q^50 + 12 * q^51 - 3 * q^52 - 49 * q^53 + 10 * q^54 - 3 * q^55 - 4 * q^58 - 15 * q^59 - 36 * q^60 - 14 * q^61 - 3 * q^62 + 17 * q^63 + 8 * q^64 + q^66 - 5 * q^67 - 4 * q^68 - 4 * q^69 - 3 * q^70 + q^71 - 15 * q^72 - 10 * q^73 - 26 * q^74 + 29 * q^75 - 14 * q^78 - 12 * q^79 - 3 * q^80 + 17 * q^82 - 7 * q^83 - 3 * q^84 - 20 * q^85 + 6 * q^86 + 45 * q^87 - 8 * q^88 - 8 * q^89 - 3 * q^90 - 17 * q^91 - q^92 - 33 * q^93 + q^96 - 54 * q^97 - 36 * q^98 + 15 * q^99

Introduction

L-functions

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