LMFDB - Embedded newform 6422.2.a.bc.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, 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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.316645497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 24x^{3} + 13x^{2} - 24x + 5 $ x^6 - 2x^5 - 12x^4 + 24x^3 + 13x^2 - 24*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.724358$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.724358 q^{3} +1.00000 q^{4} +1.67578 q^{5} -0.724358 q^{6} +2.46211 q^{7} -1.00000 q^{8} -2.47531 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.724358 q^{3} +1.00000 q^{4} +1.67578 q^{5} -0.724358 q^{6} +2.46211 q^{7} -1.00000 q^{8} -2.47531 q^{9} -1.67578 q^{10} +1.67578 q^{11} +0.724358 q^{12} -2.46211 q^{14} +1.21387 q^{15} +1.00000 q^{16} +6.14317 q^{17} +2.47531 q^{18} +1.00000 q^{19} +1.67578 q^{20} +1.78345 q^{21} -1.67578 q^{22} +4.29413 q^{23} -0.724358 q^{24} -2.19175 q^{25} -3.96608 q^{27} +2.46211 q^{28} +8.69427 q^{29} -1.21387 q^{30} +8.49888 q^{31} -1.00000 q^{32} +1.21387 q^{33} -6.14317 q^{34} +4.12596 q^{35} -2.47531 q^{36} -3.87978 q^{37} -1.00000 q^{38} -1.67578 q^{40} +4.26163 q^{41} -1.78345 q^{42} -4.20495 q^{43} +1.67578 q^{44} -4.14808 q^{45} -4.29413 q^{46} -1.32422 q^{47} +0.724358 q^{48} -0.938037 q^{49} +2.19175 q^{50} +4.44986 q^{51} +14.4960 q^{53} +3.96608 q^{54} +2.80825 q^{55} -2.46211 q^{56} +0.724358 q^{57} -8.69427 q^{58} -7.59208 q^{59} +1.21387 q^{60} -10.7850 q^{61} -8.49888 q^{62} -6.09446 q^{63} +1.00000 q^{64} -1.21387 q^{66} -8.53057 q^{67} +6.14317 q^{68} +3.11049 q^{69} -4.12596 q^{70} -9.85842 q^{71} +2.47531 q^{72} +6.30828 q^{73} +3.87978 q^{74} -1.58761 q^{75} +1.00000 q^{76} +4.12596 q^{77} +2.95411 q^{79} +1.67578 q^{80} +4.55305 q^{81} -4.26163 q^{82} -9.50430 q^{83} +1.78345 q^{84} +10.2946 q^{85} +4.20495 q^{86} +6.29776 q^{87} -1.67578 q^{88} +5.47753 q^{89} +4.14808 q^{90} +4.29413 q^{92} +6.15624 q^{93} +1.32422 q^{94} +1.67578 q^{95} -0.724358 q^{96} +8.82960 q^{97} +0.938037 q^{98} -4.14808 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 6 * q^8 + 10 * q^9	$ 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} + q^{14} + 9 q^{15} + 6 q^{16} - 2 q^{17} - 10 q^{18} + 6 q^{19} - 2 q^{20} - q^{21} + 2 q^{22} + 8 q^{23} - 2 q^{24} + 16 q^{25} - 4 q^{27} - q^{28} + 20 q^{29} - 9 q^{30} - 3 q^{31} - 6 q^{32} + 9 q^{33} + 2 q^{34} + 7 q^{35} + 10 q^{36} + 9 q^{37} - 6 q^{38} + 2 q^{40} - 3 q^{41} + q^{42} + 13 q^{43} - 2 q^{44} - 38 q^{45} - 8 q^{46} - 20 q^{47} + 2 q^{48} - 7 q^{49} - 16 q^{50} + 2 q^{51} + 25 q^{53} + 4 q^{54} + 46 q^{55} + q^{56} + 2 q^{57} - 20 q^{58} + 9 q^{60} + 6 q^{61} + 3 q^{62} - 46 q^{63} + 6 q^{64} - 9 q^{66} + 32 q^{67} - 2 q^{68} - 29 q^{69} - 7 q^{70} - 39 q^{71} - 10 q^{72} - 7 q^{73} - 9 q^{74} - 15 q^{75} + 6 q^{76} + 7 q^{77} + 18 q^{79} - 2 q^{80} + 54 q^{81} + 3 q^{82} + 7 q^{83} - q^{84} + 2 q^{85} - 13 q^{86} + 12 q^{87} + 2 q^{88} - 9 q^{89} + 38 q^{90} + 8 q^{92} + 47 q^{93} + 20 q^{94} - 2 q^{95} - 2 q^{96} + 4 q^{97} + 7 q^{98} - 38 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 6 * q^8 + 10 * q^9 + 2 * q^10 - 2 * q^11 + 2 * q^12 + q^14 + 9 * q^15 + 6 * q^16 - 2 * q^17 - 10 * q^18 + 6 * q^19 - 2 * q^20 - q^21 + 2 * q^22 + 8 * q^23 - 2 * q^24 + 16 * q^25 - 4 * q^27 - q^28 + 20 * q^29 - 9 * q^30 - 3 * q^31 - 6 * q^32 + 9 * q^33 + 2 * q^34 + 7 * q^35 + 10 * q^36 + 9 * q^37 - 6 * q^38 + 2 * q^40 - 3 * q^41 + q^42 + 13 * q^43 - 2 * q^44 - 38 * q^45 - 8 * q^46 - 20 * q^47 + 2 * q^48 - 7 * q^49 - 16 * q^50 + 2 * q^51 + 25 * q^53 + 4 * q^54 + 46 * q^55 + q^56 + 2 * q^57 - 20 * q^58 + 9 * q^60 + 6 * q^61 + 3 * q^62 - 46 * q^63 + 6 * q^64 - 9 * q^66 + 32 * q^67 - 2 * q^68 - 29 * q^69 - 7 * q^70 - 39 * q^71 - 10 * q^72 - 7 * q^73 - 9 * q^74 - 15 * q^75 + 6 * q^76 + 7 * q^77 + 18 * q^79 - 2 * q^80 + 54 * q^81 + 3 * q^82 + 7 * q^83 - q^84 + 2 * q^85 - 13 * q^86 + 12 * q^87 + 2 * q^88 - 9 * q^89 + 38 * q^90 + 8 * q^92 + 47 * q^93 + 20 * q^94 - 2 * q^95 - 2 * q^96 + 4 * q^97 + 7 * q^98 - 38 * 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
$3$	0.724358	0.418208	0.209104	−	0.977893i	$-0.432945\pi$
$3$	0.724358	0.418208	0.209104	+	0.977893i	$0.432945\pi$
$4$	1.00000	0.500000
$5$	1.67578	0.749433	0.374717	−	0.927139i	$-0.377740\pi$
$5$	1.67578	0.749433	0.374717	+	0.927139i	$0.377740\pi$
$6$	−0.724358	−0.295718
$7$	2.46211	0.930588	0.465294	−	0.885156i	$-0.345949\pi$
$7$	2.46211	0.930588	0.465294	+	0.885156i	$0.345949\pi$
$8$	−1.00000	−0.353553
$9$	−2.47531	−0.825102
$10$	−1.67578	−0.529929
$11$	1.67578	0.505268	0.252634	−	0.967562i	$-0.418703\pi$
$11$	1.67578	0.505268	0.252634	+	0.967562i	$0.418703\pi$
$12$	0.724358	0.209104
$13$	0	0
$13$	0	0
$14$	−2.46211	−0.658025
$15$	1.21387	0.313419
$16$	1.00000	0.250000
$17$	6.14317	1.48994	0.744969	−	0.667099i	$-0.232464\pi$
$17$	6.14317	1.48994	0.744969	+	0.667099i	$0.232464\pi$
$18$	2.47531	0.583435
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.67578	0.374717
$21$	1.78345	0.389180
$22$	−1.67578	−0.357278
$23$	4.29413	0.895387	0.447694	−	0.894187i	$-0.352246\pi$
$23$	4.29413	0.895387	0.447694	+	0.894187i	$0.352246\pi$
$24$	−0.724358	−0.147859
$25$	−2.19175	−0.438350
$26$	0	0
$27$	−3.96608	−0.763273
$28$	2.46211	0.465294
$29$	8.69427	1.61449	0.807243	−	0.590220i	$-0.200959\pi$
$29$	8.69427	1.61449	0.807243	+	0.590220i	$0.200959\pi$
$30$	−1.21387	−0.221621
$31$	8.49888	1.52644	0.763222	−	0.646136i	$-0.223616\pi$
$31$	8.49888	1.52644	0.763222	+	0.646136i	$0.223616\pi$
$32$	−1.00000	−0.176777
$33$	1.21387	0.211307
$34$	−6.14317	−1.05355
$35$	4.12596	0.697414
$36$	−2.47531	−0.412551
$37$	−3.87978	−0.637832	−0.318916	−	0.947783i	$-0.603319\pi$
$37$	−3.87978	−0.637832	−0.318916	+	0.947783i	$0.603319\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−1.67578	−0.264965
$41$	4.26163	0.665554	0.332777	−	0.943006i	$-0.392014\pi$
$41$	4.26163	0.665554	0.332777	+	0.943006i	$0.392014\pi$
$42$	−1.78345	−0.275192
$43$	−4.20495	−0.641249	−0.320624	−	0.947206i	$-0.603893\pi$
$43$	−4.20495	−0.641249	−0.320624	+	0.947206i	$0.603893\pi$
$44$	1.67578	0.252634
$45$	−4.14808	−0.618359
$46$	−4.29413	−0.633134
$47$	−1.32422	−0.193157	−0.0965784	−	0.995325i	$-0.530790\pi$
$47$	−1.32422	−0.193157	−0.0965784	+	0.995325i	$0.530790\pi$
$48$	0.724358	0.104552
$49$	−0.938037	−0.134005
$50$	2.19175	0.309960
$51$	4.44986	0.623105
$52$	0	0
$53$	14.4960	1.99118	0.995590	−	0.0938114i	$-0.0299051\pi$
$53$	14.4960	1.99118	0.995590	+	0.0938114i	$0.0299051\pi$
$54$	3.96608	0.539715
$55$	2.80825	0.378665
$56$	−2.46211	−0.329013
$57$	0.724358	0.0959436
$58$	−8.69427	−1.14161
$59$	−7.59208	−0.988405	−0.494202	−	0.869347i	$-0.664540\pi$
$59$	−7.59208	−0.988405	−0.494202	+	0.869347i	$0.664540\pi$
$60$	1.21387	0.156710
$61$	−10.7850	−1.38088	−0.690441	−	0.723388i	$-0.742584\pi$
$61$	−10.7850	−1.38088	−0.690441	+	0.723388i	$0.742584\pi$
$62$	−8.49888	−1.07936
$63$	−6.09446	−0.767830
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.21387	−0.149417
$67$	−8.53057	−1.04217	−0.521087	−	0.853503i	$-0.674473\pi$
$67$	−8.53057	−1.04217	−0.521087	+	0.853503i	$0.674473\pi$
$68$	6.14317	0.744969
$69$	3.11049	0.374458
$70$	−4.12596	−0.493146
$71$	−9.85842	−1.16998	−0.584990	−	0.811041i	$-0.698901\pi$
$71$	−9.85842	−1.16998	−0.584990	+	0.811041i	$0.698901\pi$
$72$	2.47531	0.291718
$73$	6.30828	0.738328	0.369164	−	0.929364i	$-0.379644\pi$
$73$	6.30828	0.738328	0.369164	+	0.929364i	$0.379644\pi$
$74$	3.87978	0.451015
$75$	−1.58761	−0.183321
$76$	1.00000	0.114708
$77$	4.12596	0.470196
$78$	0	0
$79$	2.95411	0.332364	0.166182	−	0.986095i	$-0.446856\pi$
$79$	2.95411	0.332364	0.166182	+	0.986095i	$0.446856\pi$
$80$	1.67578	0.187358
$81$	4.55305	0.505895
$82$	−4.26163	−0.470618
$83$	−9.50430	−1.04323	−0.521616	−	0.853180i	$-0.674671\pi$
$83$	−9.50430	−1.04323	−0.521616	+	0.853180i	$0.674671\pi$
$84$	1.78345	0.194590
$85$	10.2946	1.11661
$86$	4.20495	0.453431
$87$	6.29776	0.675191
$88$	−1.67578	−0.178639
$89$	5.47753	0.580616	0.290308	−	0.956933i	$-0.406242\pi$
$89$	5.47753	0.580616	0.290308	+	0.956933i	$0.406242\pi$
$90$	4.14808	0.437246
$91$	0	0
$92$	4.29413	0.447694
$93$	6.15624	0.638372
$94$	1.32422	0.136582
$95$	1.67578	0.171932
$96$	−0.724358	−0.0739295
$97$	8.82960	0.896510	0.448255	−	0.893906i	$-0.352046\pi$
$97$	8.82960	0.896510	0.448255	+	0.893906i	$0.352046\pi$
$98$	0.938037	0.0947560
$99$	−4.14808	−0.416897
$100$	−2.19175	−0.219175
$101$	0.0708313	0.00704797	0.00352399	−	0.999994i	$-0.498878\pi$
$101$	0.0708313	0.00704797	0.00352399	+	0.999994i	$0.498878\pi$
$102$	−4.44986	−0.440602
$103$	−0.00240910	−0.000237375 0	−0.000118688	−	1.00000i	$-0.500038\pi$
$103$	−0.00240910	−0.000237375 0	−0.000118688		1.00000i	$0.500038\pi$
$104$	0	0
$105$	2.98867	0.291664
$106$	−14.4960	−1.40798
$107$	16.0583	1.55241	0.776206	−	0.630479i	$-0.217142\pi$
$107$	16.0583	1.55241	0.776206	+	0.630479i	$0.217142\pi$
$108$	−3.96608	−0.381636
$109$	−17.1897	−1.64647	−0.823235	−	0.567701i	$-0.807833\pi$
$109$	−17.1897	−1.64647	−0.823235	+	0.567701i	$0.807833\pi$
$110$	−2.80825	−0.267756
$111$	−2.81035	−0.266747
$112$	2.46211	0.232647
$113$	−10.4467	−0.982740	−0.491370	−	0.870951i	$-0.663504\pi$
$113$	−10.4467	−0.982740	−0.491370	+	0.870951i	$0.663504\pi$
$114$	−0.724358	−0.0678424
$115$	7.19603	0.671033
$116$	8.69427	0.807243
$117$	0	0
$118$	7.59208	0.698908
$119$	15.1251	1.38652
$120$	−1.21387	−0.110810
$121$	−8.19175	−0.744704
$122$	10.7850	0.976432
$123$	3.08694	0.278340
$124$	8.49888	0.763222
$125$	−12.0518	−1.07795
$126$	6.09446	0.542938
$127$	6.15236	0.545933	0.272967	−	0.962023i	$-0.411995\pi$
$127$	6.15236	0.545933	0.272967	+	0.962023i	$0.411995\pi$
$128$	−1.00000	−0.0883883
$129$	−3.04589	−0.268176
$130$	0	0
$131$	11.7568	1.02720	0.513598	−	0.858031i	$-0.328312\pi$
$131$	11.7568	1.02720	0.513598	+	0.858031i	$0.328312\pi$
$132$	1.21387	0.105654
$133$	2.46211	0.213492
$134$	8.53057	0.736929
$135$	−6.64630	−0.572022
$136$	−6.14317	−0.526773
$137$	−8.97801	−0.767044	−0.383522	−	0.923532i	$-0.625289\pi$
$137$	−8.97801	−0.767044	−0.383522	+	0.923532i	$0.625289\pi$
$138$	−3.11049	−0.264782
$139$	22.8014	1.93399	0.966997	−	0.254790i	$-0.0820062\pi$
$139$	22.8014	1.93399	0.966997	+	0.254790i	$0.0820062\pi$
$140$	4.12596	0.348707
$141$	−0.959207	−0.0807798
$142$	9.85842	0.827300
$143$	0	0
$144$	−2.47531	−0.206275
$145$	14.5697	1.20995
$146$	−6.30828	−0.522077
$147$	−0.679475	−0.0560421
$148$	−3.87978	−0.318916
$149$	−17.7029	−1.45028	−0.725138	−	0.688603i	$-0.758224\pi$
$149$	−17.7029	−1.45028	−0.725138	+	0.688603i	$0.758224\pi$
$150$	1.58761	0.129628
$151$	5.41882	0.440977	0.220488	−	0.975390i	$-0.429235\pi$
$151$	5.41882	0.440977	0.220488	+	0.975390i	$0.429235\pi$
$152$	−1.00000	−0.0811107
$153$	−15.2062	−1.22935
$154$	−4.12596	−0.332479
$155$	14.2423	1.14397
$156$	0	0
$157$	18.8681	1.50584	0.752920	−	0.658112i	$-0.228645\pi$
$157$	18.8681	1.50584	0.752920	+	0.658112i	$0.228645\pi$
$158$	−2.95411	−0.235017
$159$	10.5003	0.832728
$160$	−1.67578	−0.132482
$161$	10.5726	0.833237
$162$	−4.55305	−0.357722
$163$	8.21923	0.643779	0.321890	−	0.946777i	$-0.395682\pi$
$163$	8.21923	0.643779	0.321890	+	0.946777i	$0.395682\pi$
$164$	4.26163	0.332777
$165$	2.03418	0.158361
$166$	9.50430	0.737677
$167$	−21.2432	−1.64385	−0.821926	−	0.569594i	$-0.807100\pi$
$167$	−21.2432	−1.64385	−0.821926	+	0.569594i	$0.807100\pi$
$168$	−1.78345	−0.137596
$169$	0	0
$170$	−10.2946	−0.789562
$171$	−2.47531	−0.189291
$172$	−4.20495	−0.320624
$173$	−12.6653	−0.962926	−0.481463	−	0.876466i	$-0.659894\pi$
$173$	−12.6653	−0.962926	−0.481463	+	0.876466i	$0.659894\pi$
$174$	−6.29776	−0.477432
$175$	−5.39631	−0.407923
$176$	1.67578	0.126317
$177$	−5.49938	−0.413359
$178$	−5.47753	−0.410558
$179$	14.2262	1.06331	0.531657	−	0.846960i	$-0.321569\pi$
$179$	14.2262	1.06331	0.531657	+	0.846960i	$0.321569\pi$
$180$	−4.14808	−0.309179
$181$	23.1779	1.72280	0.861399	−	0.507929i	$-0.169589\pi$
$181$	23.1779	1.72280	0.861399	+	0.507929i	$0.169589\pi$
$182$	0	0
$183$	−7.81223	−0.577497
$184$	−4.29413	−0.316567
$185$	−6.50167	−0.478012
$186$	−6.15624	−0.451397
$187$	10.2946	0.752818
$188$	−1.32422	−0.0965784
$189$	−9.76491	−0.710293
$190$	−1.67578	−0.121574
$191$	15.6675	1.13366	0.566830	−	0.823834i	$-0.308169\pi$
$191$	15.6675	1.13366	0.566830	+	0.823834i	$0.308169\pi$
$192$	0.724358	0.0522760
$193$	−3.17581	−0.228600	−0.114300	−	0.993446i	$-0.536462\pi$
$193$	−3.17581	−0.228600	−0.114300	+	0.993446i	$0.536462\pi$
$194$	−8.82960	−0.633928
$195$	0	0
$196$	−0.938037	−0.0670026
$197$	14.5756	1.03847	0.519235	−	0.854631i	$-0.326217\pi$
$197$	14.5756	1.03847	0.519235	+	0.854631i	$0.326217\pi$
$198$	4.14808	0.294791
$199$	−12.7146	−0.901312	−0.450656	−	0.892698i	$-0.648810\pi$
$199$	−12.7146	−0.901312	−0.450656	+	0.892698i	$0.648810\pi$
$200$	2.19175	0.154980
$201$	−6.17919	−0.435846
$202$	−0.0708313	−0.00498367
$203$	21.4062	1.50242
$204$	4.44986	0.311552
$205$	7.14157	0.498789
$206$	0.00240910	0.000167850 0
$207$	−10.6293	−0.738785
$208$	0	0
$209$	1.67578	0.115916
$210$	−2.98867	−0.206238
$211$	22.0652	1.51903	0.759514	−	0.650491i	$-0.225437\pi$
$211$	22.0652	1.51903	0.759514	+	0.650491i	$0.225437\pi$
$212$	14.4960	0.995590
$213$	−7.14103	−0.489295
$214$	−16.0583	−1.09772
$215$	−7.04658	−0.480573
$216$	3.96608	0.269858
$217$	20.9251	1.42049
$218$	17.1897	1.16423
$219$	4.56945	0.308775
$220$	2.80825	0.189332
$221$	0	0
$222$	2.81035	0.188618
$223$	13.0403	0.873243	0.436621	−	0.899645i	$-0.356175\pi$
$223$	13.0403	0.873243	0.436621	+	0.899645i	$0.356175\pi$
$224$	−2.46211	−0.164506
$225$	5.42525	0.361683
$226$	10.4467	0.694902
$227$	8.19169	0.543702	0.271851	−	0.962339i	$-0.412364\pi$
$227$	8.19169	0.543702	0.271851	+	0.962339i	$0.412364\pi$
$228$	0.724358	0.0479718
$229$	6.66258	0.440276	0.220138	−	0.975469i	$-0.429349\pi$
$229$	6.66258	0.440276	0.220138	+	0.975469i	$0.429349\pi$
$230$	−7.19603	−0.474492
$231$	2.98867	0.196640
$232$	−8.69427	−0.570807
$233$	−11.4360	−0.749196	−0.374598	−	0.927187i	$-0.622219\pi$
$233$	−11.4360	−0.749196	−0.374598	+	0.927187i	$0.622219\pi$
$234$	0	0
$235$	−2.21910	−0.144758
$236$	−7.59208	−0.494202
$237$	2.13984	0.138997
$238$	−15.1251	−0.980417
$239$	−14.1793	−0.917184	−0.458592	−	0.888647i	$-0.651646\pi$
$239$	−14.1793	−0.917184	−0.458592	+	0.888647i	$0.651646\pi$
$240$	1.21387	0.0783548
$241$	−7.14344	−0.460149	−0.230075	−	0.973173i	$-0.573897\pi$
$241$	−7.14344	−0.460149	−0.230075	+	0.973173i	$0.573897\pi$
$242$	8.19175	0.526586
$243$	15.1963	0.974842
$244$	−10.7850	−0.690441
$245$	−1.57195	−0.100428
$246$	−3.08694	−0.196816
$247$	0	0
$248$	−8.49888	−0.539680
$249$	−6.88452	−0.436289
$250$	12.0518	0.762224
$251$	12.4610	0.786532	0.393266	−	0.919425i	$-0.371345\pi$
$251$	12.4610	0.786532	0.393266	+	0.919425i	$0.371345\pi$
$252$	−6.09446	−0.383915
$253$	7.19603	0.452410
$254$	−6.15236	−0.386033
$255$	7.45700	0.466976
$256$	1.00000	0.0625000
$257$	−0.165949	−0.0103516	−0.00517581	−	0.999987i	$-0.501648\pi$
$257$	−0.165949	−0.0103516	−0.00517581	+	0.999987i	$0.501648\pi$
$258$	3.04589	0.189629
$259$	−9.55243	−0.593559
$260$	0	0
$261$	−21.5210	−1.33211
$262$	−11.7568	−0.726338
$263$	−11.0654	−0.682322	−0.341161	−	0.940005i	$-0.610820\pi$
$263$	−11.0654	−0.682322	−0.341161	+	0.940005i	$0.610820\pi$
$264$	−1.21387	−0.0747084
$265$	24.2922	1.49226
$266$	−2.46211	−0.150961
$267$	3.96769	0.242819
$268$	−8.53057	−0.521087
$269$	−19.4742	−1.18736	−0.593682	−	0.804699i	$-0.702326\pi$
$269$	−19.4742	−1.18736	−0.593682	+	0.804699i	$0.702326\pi$
$270$	6.64630	0.404481
$271$	9.05564	0.550091	0.275046	−	0.961431i	$-0.411307\pi$
$271$	9.05564	0.550091	0.275046	+	0.961431i	$0.411307\pi$
$272$	6.14317	0.372485
$273$	0	0
$274$	8.97801	0.542382
$275$	−3.67290	−0.221484
$276$	3.11049	0.187229
$277$	−23.0387	−1.38426	−0.692131	−	0.721772i	$-0.743328\pi$
$277$	−23.0387	−1.38426	−0.692131	+	0.721772i	$0.743328\pi$
$278$	−22.8014	−1.36754
$279$	−21.0373	−1.25947
$280$	−4.12596	−0.246573
$281$	−21.0203	−1.25397	−0.626984	−	0.779032i	$-0.715711\pi$
$281$	−21.0203	−1.25397	−0.626984	+	0.779032i	$0.715711\pi$
$282$	0.959207	0.0571199
$283$	−23.8066	−1.41516	−0.707579	−	0.706634i	$-0.750213\pi$
$283$	−23.8066	−1.41516	−0.707579	+	0.706634i	$0.750213\pi$
$284$	−9.85842	−0.584990
$285$	1.21387	0.0719033
$286$	0	0
$287$	10.4926	0.619357
$288$	2.47531	0.145859
$289$	20.7386	1.21992
$290$	−14.5697	−0.855563
$291$	6.39579	0.374928
$292$	6.30828	0.369164
$293$	−3.54605	−0.207163	−0.103581	−	0.994621i	$-0.533030\pi$
$293$	−3.54605	−0.207163	−0.103581	+	0.994621i	$0.533030\pi$
$294$	0.679475	0.0396278
$295$	−12.7227	−0.740743
$296$	3.87978	0.225508
$297$	−6.64630	−0.385657
$298$	17.7029	1.02550
$299$	0	0
$300$	−1.58761	−0.0916607
$301$	−10.3530	−0.596738
$302$	−5.41882	−0.311818
$303$	0.0513072	0.00294752
$304$	1.00000	0.0573539
$305$	−18.0734	−1.03488
$306$	15.2062	0.869282
$307$	2.25251	0.128558	0.0642789	−	0.997932i	$-0.479525\pi$
$307$	2.25251	0.128558	0.0642789	+	0.997932i	$0.479525\pi$
$308$	4.12596	0.235098
$309$	−0.00174505	−9.92723e−5 0
$310$	−14.2423	−0.808908
$311$	−31.6962	−1.79732	−0.898662	−	0.438641i	$-0.855460\pi$
$311$	−31.6962	−1.79732	−0.898662	+	0.438641i	$0.855460\pi$
$312$	0	0
$313$	26.9827	1.52515	0.762575	−	0.646899i	$-0.223935\pi$
$313$	26.9827	1.52515	0.762575	+	0.646899i	$0.223935\pi$
$314$	−18.8681	−1.06479
$315$	−10.2130	−0.575438
$316$	2.95411	0.166182
$317$	27.3422	1.53569	0.767845	−	0.640635i	$-0.221329\pi$
$317$	27.3422	1.53569	0.767845	+	0.640635i	$0.221329\pi$
$318$	−10.5003	−0.588828
$319$	14.5697	0.815747
$320$	1.67578	0.0936792
$321$	11.6319	0.649232
$322$	−10.5726	−0.589187
$323$	6.14317	0.341815
$324$	4.55305	0.252947
$325$	0	0
$326$	−8.21923	−0.455221
$327$	−12.4515	−0.688568
$328$	−4.26163	−0.235309
$329$	−3.26036	−0.179749
$330$	−2.03418	−0.111978
$331$	−3.88358	−0.213461	−0.106730	−	0.994288i	$-0.534038\pi$
$331$	−3.88358	−0.213461	−0.106730	+	0.994288i	$0.534038\pi$
$332$	−9.50430	−0.521616
$333$	9.60364	0.526276
$334$	21.2432	1.16238
$335$	−14.2954	−0.781041
$336$	1.78345	0.0972950
$337$	10.4961	0.571761	0.285880	−	0.958265i	$-0.407714\pi$
$337$	10.4961	0.571761	0.285880	+	0.958265i	$0.407714\pi$
$338$	0	0
$339$	−7.56713	−0.410990
$340$	10.2946	0.558305
$341$	14.2423	0.771263
$342$	2.47531	0.133849
$343$	−19.5443	−1.05529
$344$	4.20495	0.226716
$345$	5.21250	0.280632
$346$	12.6653	0.680891
$347$	−21.2209	−1.13920	−0.569598	−	0.821923i	$-0.692901\pi$
$347$	−21.2209	−1.13920	−0.569598	+	0.821923i	$0.692901\pi$
$348$	6.29776	0.337596
$349$	5.19372	0.278014	0.139007	−	0.990291i	$-0.455609\pi$
$349$	5.19372	0.278014	0.139007	+	0.990291i	$0.455609\pi$
$350$	5.39631	0.288445
$351$	0	0
$352$	−1.67578	−0.0893196
$353$	10.4972	0.558708	0.279354	−	0.960188i	$-0.409880\pi$
$353$	10.4972	0.558708	0.279354	+	0.960188i	$0.409880\pi$
$354$	5.49938	0.292289
$355$	−16.5206	−0.876821
$356$	5.47753	0.290308
$357$	10.9560	0.579854
$358$	−14.2262	−0.751877
$359$	18.4451	0.973494	0.486747	−	0.873543i	$-0.338183\pi$
$359$	18.4451	0.973494	0.486747	+	0.873543i	$0.338183\pi$
$360$	4.14808	0.218623
$361$	1.00000	0.0526316
$362$	−23.1779	−1.21820
$363$	−5.93376	−0.311442
$364$	0	0
$365$	10.5713	0.553328
$366$	7.81223	0.408352
$367$	−10.8674	−0.567271	−0.283636	−	0.958932i	$-0.591541\pi$
$367$	−10.8674	−0.567271	−0.283636	+	0.958932i	$0.591541\pi$
$368$	4.29413	0.223847
$369$	−10.5488	−0.549150
$370$	6.50167	0.338006
$371$	35.6907	1.85297
$372$	6.15624	0.319186
$373$	−14.9684	−0.775034	−0.387517	−	0.921863i	$-0.626667\pi$
$373$	−14.9684	−0.775034	−0.387517	+	0.921863i	$0.626667\pi$
$374$	−10.2946	−0.532323
$375$	−8.72983	−0.450807
$376$	1.32422	0.0682912
$377$	0	0
$378$	9.76491	0.502253
$379$	9.04086	0.464398	0.232199	−	0.972668i	$-0.425408\pi$
$379$	9.04086	0.464398	0.232199	+	0.972668i	$0.425408\pi$
$380$	1.67578	0.0859659
$381$	4.45651	0.228314
$382$	−15.6675	−0.801619
$383$	16.9009	0.863598	0.431799	−	0.901970i	$-0.357879\pi$
$383$	16.9009	0.863598	0.431799	+	0.901970i	$0.357879\pi$
$384$	−0.724358	−0.0369647
$385$	6.91421	0.352381
$386$	3.17581	0.161644
$387$	10.4085	0.529095
$388$	8.82960	0.448255
$389$	0.559064	0.0283457	0.0141728	−	0.999900i	$-0.495488\pi$
$389$	0.559064	0.0283457	0.0141728	+	0.999900i	$0.495488\pi$
$390$	0	0
$391$	26.3796	1.33407
$392$	0.938037	0.0473780
$393$	8.51614	0.429582
$394$	−14.5756	−0.734309
$395$	4.95045	0.249084
$396$	−4.14808	−0.208449
$397$	8.43814	0.423498	0.211749	−	0.977324i	$-0.432084\pi$
$397$	8.43814	0.423498	0.211749	+	0.977324i	$0.432084\pi$
$398$	12.7146	0.637324
$399$	1.78345	0.0892840
$400$	−2.19175	−0.109587
$401$	−26.3948	−1.31809	−0.659046	−	0.752103i	$-0.729040\pi$
$401$	−26.3948	−1.31809	−0.659046	+	0.752103i	$0.729040\pi$
$402$	6.17919	0.308190
$403$	0	0
$404$	0.0708313	0.00352399
$405$	7.62993	0.379134
$406$	−21.4062	−1.06237
$407$	−6.50167	−0.322276
$408$	−4.44986	−0.220301
$409$	22.1518	1.09534	0.547669	−	0.836695i	$-0.315515\pi$
$409$	22.1518	1.09534	0.547669	+	0.836695i	$0.315515\pi$
$410$	−7.14157	−0.352697
$411$	−6.50330	−0.320784
$412$	−0.00240910	−0.000118688 0
$413$	−18.6925	−0.919798
$414$	10.6293	0.522400
$415$	−15.9272	−0.781834
$416$	0	0
$417$	16.5164	0.808812
$418$	−1.67578	−0.0819653
$419$	−5.68094	−0.277532	−0.138766	−	0.990325i	$-0.544314\pi$
$419$	−5.68094	−0.277532	−0.138766	+	0.990325i	$0.544314\pi$
$420$	2.98867	0.145832
$421$	37.4889	1.82710	0.913549	−	0.406728i	$-0.133330\pi$
$421$	37.4889	1.82710	0.913549	+	0.406728i	$0.133330\pi$
$422$	−22.0652	−1.07411
$423$	3.27784	0.159374
$424$	−14.4960	−0.703988
$425$	−13.4643	−0.653114
$426$	7.14103	0.345984
$427$	−26.5539	−1.28503
$428$	16.0583	0.776206
$429$	0	0
$430$	7.04658	0.339816
$431$	29.3102	1.41182	0.705911	−	0.708300i	$-0.250538\pi$
$431$	29.3102	1.41182	0.705911	+	0.708300i	$0.250538\pi$
$432$	−3.96608	−0.190818
$433$	4.54709	0.218519	0.109260	−	0.994013i	$-0.465152\pi$
$433$	4.54709	0.218519	0.109260	+	0.994013i	$0.465152\pi$
$434$	−20.9251	−1.00444
$435$	10.5537	0.506011
$436$	−17.1897	−0.823235
$437$	4.29413	0.205416
$438$	−4.56945	−0.218337
$439$	0.449965	0.0214757	0.0107378	−	0.999942i	$-0.496582\pi$
$439$	0.449965	0.0214757	0.0107378	+	0.999942i	$0.496582\pi$
$440$	−2.80825	−0.133878
$441$	2.32193	0.110568
$442$	0	0
$443$	7.83836	0.372412	0.186206	−	0.982511i	$-0.440381\pi$
$443$	7.83836	0.372412	0.186206	+	0.982511i	$0.440381\pi$
$444$	−2.81035	−0.133373
$445$	9.17915	0.435133
$446$	−13.0403	−0.617476
$447$	−12.8232	−0.606518
$448$	2.46211	0.116324
$449$	2.51676	0.118773	0.0593865	−	0.998235i	$-0.481086\pi$
$449$	2.51676	0.118773	0.0593865	+	0.998235i	$0.481086\pi$
$450$	−5.42525	−0.255749
$451$	7.14157	0.336283
$452$	−10.4467	−0.491370
$453$	3.92516	0.184420
$454$	−8.19169	−0.384455
$455$	0	0
$456$	−0.724358	−0.0339212
$457$	20.7252	0.969483	0.484741	−	0.874658i	$-0.338914\pi$
$457$	20.7252	0.969483	0.484741	+	0.874658i	$0.338914\pi$
$458$	−6.66258	−0.311322
$459$	−24.3643	−1.13723
$460$	7.19603	0.335517
$461$	−11.9187	−0.555108	−0.277554	−	0.960710i	$-0.589524\pi$
$461$	−11.9187	−0.555108	−0.277554	+	0.960710i	$0.589524\pi$
$462$	−2.98867	−0.139046
$463$	−26.3989	−1.22686	−0.613431	−	0.789748i	$-0.710211\pi$
$463$	−26.3989	−1.22686	−0.613431	+	0.789748i	$0.710211\pi$
$464$	8.69427	0.403621
$465$	10.3165	0.478417
$466$	11.4360	0.529761
$467$	−38.1584	−1.76576	−0.882880	−	0.469598i	$-0.844399\pi$
$467$	−38.1584	−1.76576	−0.882880	+	0.469598i	$0.844399\pi$
$468$	0	0
$469$	−21.0032	−0.969836
$470$	2.21910	0.102359
$471$	13.6673	0.629755
$472$	7.59208	0.349454
$473$	−7.04658	−0.324002
$474$	−2.13984	−0.0982859
$475$	−2.19175	−0.100564
$476$	15.1251	0.693260
$477$	−35.8820	−1.64293
$478$	14.1793	0.648547
$479$	−7.16427	−0.327344	−0.163672	−	0.986515i	$-0.552334\pi$
$479$	−7.16427	−0.327344	−0.163672	+	0.986515i	$0.552334\pi$
$480$	−1.21387	−0.0554052
$481$	0	0
$482$	7.14344	0.325375
$483$	7.65834	0.348467
$484$	−8.19175	−0.372352
$485$	14.7965	0.671874
$486$	−15.1963	−0.689318
$487$	−10.5383	−0.477538	−0.238769	−	0.971076i	$-0.576744\pi$
$487$	−10.5383	−0.477538	−0.238769	+	0.971076i	$0.576744\pi$
$488$	10.7850	0.488216
$489$	5.95366	0.269234
$490$	1.57195	0.0710133
$491$	16.8818	0.761867	0.380933	−	0.924603i	$-0.375603\pi$
$491$	16.8818	0.761867	0.380933	+	0.924603i	$0.375603\pi$
$492$	3.08694	0.139170
$493$	53.4104	2.40548
$494$	0	0
$495$	−6.95128	−0.312437
$496$	8.49888	0.381611
$497$	−24.2725	−1.08877
$498$	6.88452	0.308503
$499$	−29.9948	−1.34275	−0.671376	−	0.741117i	$-0.734297\pi$
$499$	−29.9948	−1.34275	−0.671376	+	0.741117i	$0.734297\pi$
$500$	−12.0518	−0.538974
$501$	−15.3877	−0.687473
$502$	−12.4610	−0.556162
$503$	−36.2493	−1.61628	−0.808138	−	0.588993i	$-0.799525\pi$
$503$	−36.2493	−1.61628	−0.808138	+	0.588993i	$0.799525\pi$
$504$	6.09446	0.271469
$505$	0.118698	0.00528199
$506$	−7.19603	−0.319902
$507$	0	0
$508$	6.15236	0.272967
$509$	−12.5906	−0.558067	−0.279033	−	0.960281i	$-0.590014\pi$
$509$	−12.5906	−0.558067	−0.279033	+	0.960281i	$0.590014\pi$
$510$	−7.45700	−0.330202
$511$	15.5316	0.687080
$512$	−1.00000	−0.0441942
$513$	−3.96608	−0.175107
$514$	0.165949	0.00731970
$515$	−0.00403712	−0.000177897 0
$516$	−3.04589	−0.134088
$517$	−2.21910	−0.0975959
$518$	9.55243	0.419709
$519$	−9.17422	−0.402704
$520$	0	0
$521$	13.0417	0.571367	0.285683	−	0.958324i	$-0.407779\pi$
$521$	13.0417	0.571367	0.285683	+	0.958324i	$0.407779\pi$
$522$	21.5210	0.941947
$523$	25.2319	1.10331	0.551657	−	0.834071i	$-0.313996\pi$
$523$	25.2319	1.10331	0.551657	+	0.834071i	$0.313996\pi$
$524$	11.7568	0.513598
$525$	−3.90886	−0.170597
$526$	11.0654	0.482474
$527$	52.2101	2.27431
$528$	1.21387	0.0528268
$529$	−4.56048	−0.198282
$530$	−24.2922	−1.05518
$531$	18.7927	0.815534
$532$	2.46211	0.106746
$533$	0	0
$534$	−3.96769	−0.171699
$535$	26.9102	1.16343
$536$	8.53057	0.368464
$537$	10.3049	0.444687
$538$	19.4742	0.839594
$539$	−1.57195	−0.0677086
$540$	−6.64630	−0.286011
$541$	23.9375	1.02915	0.514576	−	0.857445i	$-0.327949\pi$
$541$	23.9375	1.02915	0.514576	+	0.857445i	$0.327949\pi$
$542$	−9.05564	−0.388973
$543$	16.7891	0.720488
$544$	−6.14317	−0.263386
$545$	−28.8061	−1.23392
$546$	0	0
$547$	5.83345	0.249420	0.124710	−	0.992193i	$-0.460200\pi$
$547$	5.83345	0.249420	0.124710	+	0.992193i	$0.460200\pi$
$548$	−8.97801	−0.383522
$549$	26.6963	1.13937
$550$	3.67290	0.156613
$551$	8.69427	0.370388
$552$	−3.11049	−0.132391
$553$	7.27333	0.309294
$554$	23.0387	0.978821
$555$	−4.70954	−0.199909
$556$	22.8014	0.966997
$557$	8.83375	0.374298	0.187149	−	0.982332i	$-0.440075\pi$
$557$	8.83375	0.374298	0.187149	+	0.982332i	$0.440075\pi$
$558$	21.0373	0.890581
$559$	0	0
$560$	4.12596	0.174354
$561$	7.45700	0.314835
$562$	21.0203	0.886690
$563$	16.4317	0.692514	0.346257	−	0.938140i	$-0.387452\pi$
$563$	16.4317	0.692514	0.346257	+	0.938140i	$0.387452\pi$
$564$	−0.959207	−0.0403899
$565$	−17.5064	−0.736498
$566$	23.8066	1.00067
$567$	11.2101	0.470780
$568$	9.85842	0.413650
$569$	−9.84114	−0.412562	−0.206281	−	0.978493i	$-0.566136\pi$
$569$	−9.84114	−0.412562	−0.206281	+	0.978493i	$0.566136\pi$
$570$	−1.21387	−0.0508433
$571$	−16.0578	−0.672000	−0.336000	−	0.941862i	$-0.609074\pi$
$571$	−16.0578	−0.672000	−0.336000	+	0.941862i	$0.609074\pi$
$572$	0	0
$573$	11.3489	0.474106
$574$	−10.4926	−0.437952
$575$	−9.41164	−0.392493
$576$	−2.47531	−0.103138
$577$	22.0567	0.918234	0.459117	−	0.888376i	$-0.348166\pi$
$577$	22.0567	0.918234	0.459117	+	0.888376i	$0.348166\pi$
$578$	−20.7386	−0.862611
$579$	−2.30042	−0.0956023
$580$	14.5697	0.604975
$581$	−23.4006	−0.970820
$582$	−6.39579	−0.265114
$583$	24.2922	1.00608
$584$	−6.30828	−0.261038
$585$	0	0
$586$	3.54605	0.146486
$587$	−2.13366	−0.0880657	−0.0440328	−	0.999030i	$-0.514021\pi$
$587$	−2.13366	−0.0880657	−0.0440328	+	0.999030i	$0.514021\pi$
$588$	−0.679475	−0.0280211
$589$	8.49888	0.350190
$590$	12.7227	0.523785
$591$	10.5580	0.434297
$592$	−3.87978	−0.159458
$593$	−8.36233	−0.343400	−0.171700	−	0.985149i	$-0.554926\pi$
$593$	−8.36233	−0.343400	−0.171700	+	0.985149i	$0.554926\pi$
$594$	6.64630	0.272701
$595$	25.3465	1.03910
$596$	−17.7029	−0.725138
$597$	−9.20990	−0.376936
$598$	0	0
$599$	35.5061	1.45074	0.725371	−	0.688358i	$-0.241668\pi$
$599$	35.5061	1.45074	0.725371	+	0.688358i	$0.241668\pi$
$600$	1.58761	0.0648139
$601$	−10.0760	−0.411007	−0.205504	−	0.978656i	$-0.565883\pi$
$601$	−10.0760	−0.411007	−0.205504	+	0.978656i	$0.565883\pi$
$602$	10.3530	0.421958
$603$	21.1158	0.859900
$604$	5.41882	0.220488
$605$	−13.7276	−0.558106
$606$	−0.0513072	−0.00208421
$607$	41.9312	1.70194	0.850968	−	0.525217i	$-0.176016\pi$
$607$	41.9312	1.70194	0.850968	+	0.525217i	$0.176016\pi$
$608$	−1.00000	−0.0405554
$609$	15.5058	0.628325
$610$	18.0734	0.731770
$611$	0	0
$612$	−15.2062	−0.614675
$613$	−28.5706	−1.15396	−0.576978	−	0.816760i	$-0.695768\pi$
$613$	−28.5706	−1.15396	−0.576978	+	0.816760i	$0.695768\pi$
$614$	−2.25251	−0.0909041
$615$	5.17305	0.208598
$616$	−4.12596	−0.166240
$617$	−1.29364	−0.0520798	−0.0260399	−	0.999661i	$-0.508290\pi$
$617$	−1.29364	−0.0520798	−0.0260399	+	0.999661i	$0.508290\pi$
$618$	0.00174505	7.01961e−5 0
$619$	17.9280	0.720586	0.360293	−	0.932839i	$-0.382677\pi$
$619$	17.9280	0.720586	0.360293	+	0.932839i	$0.382677\pi$
$620$	14.2423	0.571984
$621$	−17.0309	−0.683425
$622$	31.6962	1.27090
$623$	13.4862	0.540315
$624$	0	0
$625$	−9.23750	−0.369500
$626$	−26.9827	−1.07844
$627$	1.21387	0.0484772
$628$	18.8681	0.752920
$629$	−23.8342	−0.950330
$630$	10.2130	0.406896
$631$	−28.0599	−1.11705	−0.558523	−	0.829489i	$-0.688632\pi$
$631$	−28.0599	−1.11705	−0.558523	+	0.829489i	$0.688632\pi$
$632$	−2.95411	−0.117508
$633$	15.9831	0.635270
$634$	−27.3422	−1.08590
$635$	10.3100	0.409141
$636$	10.5003	0.416364
$637$	0	0
$638$	−14.5697	−0.576821
$639$	24.4026	0.965352
$640$	−1.67578	−0.0662412
$641$	29.1376	1.15086	0.575432	−	0.817849i	$-0.304834\pi$
$641$	29.1376	1.15086	0.575432	+	0.817849i	$0.304834\pi$
$642$	−11.6319	−0.459076
$643$	−12.0546	−0.475385	−0.237693	−	0.971340i	$-0.576391\pi$
$643$	−12.0546	−0.475385	−0.237693	+	0.971340i	$0.576391\pi$
$644$	10.5726	0.416618
$645$	−5.10425	−0.200980
$646$	−6.14317	−0.241700
$647$	8.53069	0.335376	0.167688	−	0.985840i	$-0.446370\pi$
$647$	8.53069	0.335376	0.167688	+	0.985840i	$0.446370\pi$
$648$	−4.55305	−0.178861
$649$	−12.7227	−0.499409
$650$	0	0
$651$	15.1573	0.594062
$652$	8.21923	0.321890
$653$	17.8290	0.697704	0.348852	−	0.937178i	$-0.386572\pi$
$653$	17.8290	0.697704	0.348852	+	0.937178i	$0.386572\pi$
$654$	12.4515	0.486891
$655$	19.7019	0.769816
$656$	4.26163	0.166389
$657$	−15.6149	−0.609196
$658$	3.26036	0.127102
$659$	14.5386	0.566344	0.283172	−	0.959069i	$-0.408613\pi$
$659$	14.5386	0.566344	0.283172	+	0.959069i	$0.408613\pi$
$660$	2.03418	0.0791804
$661$	−15.3014	−0.595154	−0.297577	−	0.954698i	$-0.596178\pi$
$661$	−15.3014	−0.595154	−0.297577	+	0.954698i	$0.596178\pi$
$662$	3.88358	0.150940
$663$	0	0
$664$	9.50430	0.368838
$665$	4.12596	0.159998
$666$	−9.60364	−0.372133
$667$	37.3343	1.44559
$668$	−21.2432	−0.821926
$669$	9.44584	0.365197
$670$	14.2954	0.552279
$671$	−18.0734	−0.697716
$672$	−1.78345	−0.0687979
$673$	31.5995	1.21807	0.609036	−	0.793143i	$-0.291557\pi$
$673$	31.5995	1.21807	0.609036	+	0.793143i	$0.291557\pi$
$674$	−10.4961	−0.404296
$675$	8.69265	0.334580
$676$	0	0
$677$	−37.9340	−1.45792	−0.728962	−	0.684555i	$-0.759997\pi$
$677$	−37.9340	−1.45792	−0.728962	+	0.684555i	$0.759997\pi$
$678$	7.56713	0.290614
$679$	21.7394	0.834282
$680$	−10.2946	−0.394781
$681$	5.93372	0.227381
$682$	−14.2423	−0.545366
$683$	12.0669	0.461728	0.230864	−	0.972986i	$-0.425845\pi$
$683$	12.0669	0.461728	0.230864	+	0.972986i	$0.425845\pi$
$684$	−2.47531	−0.0946457
$685$	−15.0452	−0.574848
$686$	19.5443	0.746204
$687$	4.82610	0.184127
$688$	−4.20495	−0.160312
$689$	0	0
$690$	−5.21250	−0.198437
$691$	−34.1048	−1.29741	−0.648704	−	0.761041i	$-0.724689\pi$
$691$	−34.1048	−1.29741	−0.648704	+	0.761041i	$0.724689\pi$
$692$	−12.6653	−0.481463
$693$	−10.2130	−0.387960
$694$	21.2209	0.805533
$695$	38.2103	1.44940
$696$	−6.29776	−0.238716
$697$	26.1799	0.991635
$698$	−5.19372	−0.196585
$699$	−8.28374	−0.313320
$700$	−5.39631	−0.203962
$701$	2.27337	0.0858640	0.0429320	−	0.999078i	$-0.486330\pi$
$701$	2.27337	0.0858640	0.0429320	+	0.999078i	$0.486330\pi$
$702$	0	0
$703$	−3.87978	−0.146329
$704$	1.67578	0.0631585
$705$	−1.60742	−0.0605391
$706$	−10.4972	−0.395066
$707$	0.174394	0.00655876
$708$	−5.49938	−0.206680
$709$	37.4352	1.40591	0.702953	−	0.711236i	$-0.251864\pi$
$709$	37.4352	1.40591	0.702953	+	0.711236i	$0.251864\pi$
$710$	16.5206	0.620006
$711$	−7.31233	−0.274234
$712$	−5.47753	−0.205279
$713$	36.4953	1.36676
$714$	−10.9560	−0.410019
$715$	0	0
$716$	14.2262	0.531657
$717$	−10.2709	−0.383574
$718$	−18.4451	−0.688364
$719$	7.04960	0.262906	0.131453	−	0.991322i	$-0.458036\pi$
$719$	7.04960	0.262906	0.131453	+	0.991322i	$0.458036\pi$
$720$	−4.14808	−0.154590
$721$	−0.00593145	−0.000220899 0
$722$	−1.00000	−0.0372161
$723$	−5.17441	−0.192438
$724$	23.1779	0.861399
$725$	−19.0556	−0.707709
$726$	5.93376	0.220222
$727$	−28.0658	−1.04090	−0.520452	−	0.853891i	$-0.674236\pi$
$727$	−28.0658	−1.04090	−0.520452	+	0.853891i	$0.674236\pi$
$728$	0	0
$729$	−2.65160	−0.0982074
$730$	−10.5713	−0.391262
$731$	−25.8317	−0.955421
$732$	−7.81223	−0.288748
$733$	3.58379	0.132370	0.0661851	−	0.997807i	$-0.478917\pi$
$733$	3.58379	0.132370	0.0661851	+	0.997807i	$0.478917\pi$
$734$	10.8674	0.401121
$735$	−1.13865	−0.0419998
$736$	−4.29413	−0.158284
$737$	−14.2954	−0.526577
$738$	10.5488	0.388308
$739$	−5.21815	−0.191953	−0.0959763	−	0.995384i	$-0.530597\pi$
$739$	−5.21815	−0.191953	−0.0959763	+	0.995384i	$0.530597\pi$
$740$	−6.50167	−0.239006
$741$	0	0
$742$	−35.6907	−1.31025
$743$	−41.8910	−1.53683	−0.768416	−	0.639951i	$-0.778955\pi$
$743$	−41.8910	−1.53683	−0.768416	+	0.639951i	$0.778955\pi$
$744$	−6.15624	−0.225699
$745$	−29.6662	−1.08689
$746$	14.9684	0.548032
$747$	23.5261	0.860773
$748$	10.2946	0.376409
$749$	39.5372	1.44466
$750$	8.72983	0.318768
$751$	7.80567	0.284833	0.142416	−	0.989807i	$-0.454513\pi$
$751$	7.80567	0.284833	0.142416	+	0.989807i	$0.454513\pi$
$752$	−1.32422	−0.0482892
$753$	9.02624	0.328934
$754$	0	0
$755$	9.08076	0.330483
$756$	−9.76491	−0.355146
$757$	−9.16453	−0.333090	−0.166545	−	0.986034i	$-0.553261\pi$
$757$	−9.16453	−0.333090	−0.166545	+	0.986034i	$0.553261\pi$
$758$	−9.04086	−0.328379
$759$	5.21250	0.189202
$760$	−1.67578	−0.0607871
$761$	4.85450	0.175975	0.0879877	−	0.996122i	$-0.471956\pi$
$761$	4.85450	0.175975	0.0879877	+	0.996122i	$0.471956\pi$
$762$	−4.45651	−0.161442
$763$	−42.3227	−1.53219
$764$	15.6675	0.566830
$765$	−25.4824	−0.921317
$766$	−16.9009	−0.610656
$767$	0	0
$768$	0.724358	0.0261380
$769$	13.2382	0.477381	0.238690	−	0.971096i	$-0.423282\pi$
$769$	13.2382	0.477381	0.238690	+	0.971096i	$0.423282\pi$
$770$	−6.91421	−0.249171
$771$	−0.120207	−0.00432913
$772$	−3.17581	−0.114300
$773$	36.4623	1.31146	0.655729	−	0.754997i	$-0.272362\pi$
$773$	36.4623	1.31146	0.655729	+	0.754997i	$0.272362\pi$
$774$	−10.4085	−0.374127
$775$	−18.6274	−0.669116
$776$	−8.82960	−0.316964
$777$	−6.91938	−0.248231
$778$	−0.559064	−0.0200434
$779$	4.26163	0.152689
$780$	0	0
$781$	−16.5206	−0.591153
$782$	−26.3796	−0.943331
$783$	−34.4822	−1.23229
$784$	−0.938037	−0.0335013
$785$	31.6189	1.12853
$786$	−8.51614	−0.303761
$787$	−52.6861	−1.87806	−0.939028	−	0.343841i	$-0.888272\pi$
$787$	−52.6861	−1.87806	−0.939028	+	0.343841i	$0.888272\pi$
$788$	14.5756	0.519235
$789$	−8.01531	−0.285353
$790$	−4.95045	−0.176129
$791$	−25.7208	−0.914527
$792$	4.14808	0.147395
$793$	0	0
$794$	−8.43814	−0.299459
$795$	17.5962	0.624074
$796$	−12.7146	−0.450656
$797$	2.04192	0.0723284	0.0361642	−	0.999346i	$-0.488486\pi$
$797$	2.04192	0.0723284	0.0361642	+	0.999346i	$0.488486\pi$
$798$	−1.78345	−0.0631333
$799$	−8.13489	−0.287792
$800$	2.19175	0.0774900
$801$	−13.5585	−0.479068
$802$	26.3948	0.932032
$803$	10.5713	0.373053
$804$	−6.17919	−0.217923
$805$	17.7174	0.624456
$806$	0	0
$807$	−14.1063	−0.496566
$808$	−0.0708313	−0.00249183
$809$	−28.0271	−0.985380	−0.492690	−	0.870205i	$-0.663986\pi$
$809$	−28.0271	−0.985380	−0.492690	+	0.870205i	$0.663986\pi$
$810$	−7.62993	−0.268088
$811$	−25.8716	−0.908475	−0.454237	−	0.890881i	$-0.650088\pi$
$811$	−25.8716	−0.908475	−0.454237	+	0.890881i	$0.650088\pi$
$812$	21.4062	0.751211
$813$	6.55953	0.230053
$814$	6.50167	0.227883
$815$	13.7736	0.482470
$816$	4.44986	0.155776
$817$	−4.20495	−0.147112
$818$	−22.1518	−0.774521
$819$	0	0
$820$	7.14157	0.249394
$821$	27.5492	0.961474	0.480737	−	0.876865i	$-0.340369\pi$
$821$	27.5492	0.961474	0.480737	+	0.876865i	$0.340369\pi$
$822$	6.50330	0.226829
$823$	14.6411	0.510356	0.255178	−	0.966894i	$-0.417866\pi$
$823$	14.6411	0.510356	0.255178	+	0.966894i	$0.417866\pi$
$824$	0.00240910	8.39248e−5 0
$825$	−2.66049	−0.0926265
$826$	18.6925	0.650395
$827$	−20.6600	−0.718417	−0.359209	−	0.933257i	$-0.616953\pi$
$827$	−20.6600	−0.718417	−0.359209	+	0.933257i	$0.616953\pi$
$828$	−10.6293	−0.369393
$829$	30.1214	1.04616	0.523079	−	0.852284i	$-0.324783\pi$
$829$	30.1214	1.04616	0.523079	+	0.852284i	$0.324783\pi$
$830$	15.9272	0.552840
$831$	−16.6883	−0.578910
$832$	0	0
$833$	−5.76252	−0.199660
$834$	−16.5164	−0.571917
$835$	−35.5991	−1.23196
$836$	1.67578	0.0579582
$837$	−33.7073	−1.16509
$838$	5.68094	0.196245
$839$	−8.15391	−0.281504	−0.140752	−	0.990045i	$-0.544952\pi$
$839$	−8.15391	−0.281504	−0.140752	+	0.990045i	$0.544952\pi$
$840$	−2.98867	−0.103119
$841$	46.5903	1.60656
$842$	−37.4889	−1.29195
$843$	−15.2263	−0.524420
$844$	22.0652	0.759514
$845$	0	0
$846$	−3.27784	−0.112694
$847$	−20.1689	−0.693013
$848$	14.4960	0.497795
$849$	−17.2445	−0.591831
$850$	13.4643	0.461821
$851$	−16.6603	−0.571106
$852$	−7.14103	−0.244648
$853$	−5.05129	−0.172953	−0.0864765	−	0.996254i	$-0.527561\pi$
$853$	−5.05129	−0.172953	−0.0864765	+	0.996254i	$0.527561\pi$
$854$	26.5539	0.908656
$855$	−4.14808	−0.141861
$856$	−16.0583	−0.548860
$857$	−17.4149	−0.594883	−0.297441	−	0.954740i	$-0.596133\pi$
$857$	−17.4149	−0.594883	−0.297441	+	0.954740i	$0.596133\pi$
$858$	0	0
$859$	15.9600	0.544549	0.272274	−	0.962220i	$-0.412224\pi$
$859$	15.9600	0.544549	0.272274	+	0.962220i	$0.412224\pi$
$860$	−7.04658	−0.240287
$861$	7.60038	0.259020
$862$	−29.3102	−0.998309
$863$	−10.4340	−0.355177	−0.177588	−	0.984105i	$-0.556830\pi$
$863$	−10.4340	−0.355177	−0.177588	+	0.984105i	$0.556830\pi$
$864$	3.96608	0.134929
$865$	−21.2243	−0.721649
$866$	−4.54709	−0.154516
$867$	15.0222	0.510179
$868$	20.9251	0.710246
$869$	4.95045	0.167933
$870$	−10.5537	−0.357804
$871$	0	0
$872$	17.1897	0.582115
$873$	−21.8559	−0.739712
$874$	−4.29413	−0.145251
$875$	−29.6728	−1.00313
$876$	4.56945	0.154388
$877$	−32.0189	−1.08120	−0.540601	−	0.841279i	$-0.681803\pi$
$877$	−32.0189	−1.08120	−0.540601	+	0.841279i	$0.681803\pi$
$878$	−0.449965	−0.0151856
$879$	−2.56861	−0.0866372
$880$	2.80825	0.0946662
$881$	34.5306	1.16337	0.581683	−	0.813416i	$-0.302394\pi$
$881$	34.5306	1.16337	0.581683	+	0.813416i	$0.302394\pi$
$882$	−2.32193	−0.0781834
$883$	15.5981	0.524917	0.262459	−	0.964943i	$-0.415467\pi$
$883$	15.5981	0.524917	0.262459	+	0.964943i	$0.415467\pi$
$884$	0	0
$885$	−9.21578	−0.309785
$886$	−7.83836	−0.263335
$887$	37.9416	1.27395	0.636977	−	0.770883i	$-0.280185\pi$
$887$	37.9416	1.27395	0.636977	+	0.770883i	$0.280185\pi$
$888$	2.81035	0.0943092
$889$	15.1478	0.508039
$890$	−9.17915	−0.307686
$891$	7.62993	0.255612
$892$	13.0403	0.436621
$893$	−1.32422	−0.0443132
$894$	12.8232	0.428873
$895$	23.8400	0.796884
$896$	−2.46211	−0.0822532
$897$	0	0
$898$	−2.51676	−0.0839852
$899$	73.8916	2.46442
$900$	5.42525	0.180842
$901$	89.0515	2.96674
$902$	−7.14157	−0.237788
$903$	−7.49930	−0.249561
$904$	10.4467	0.347451
$905$	38.8411	1.29112
$906$	−3.92516	−0.130405
$907$	−53.8294	−1.78738	−0.893689	−	0.448687i	$-0.851892\pi$
$907$	−53.8294	−1.78738	−0.893689	+	0.448687i	$0.851892\pi$
$908$	8.19169	0.271851
$909$	−0.175329	−0.00581529
$910$	0	0
$911$	−33.8377	−1.12109	−0.560547	−	0.828123i	$-0.689409\pi$
$911$	−33.8377	−1.12109	−0.560547	+	0.828123i	$0.689409\pi$
$912$	0.724358	0.0239859
$913$	−15.9272	−0.527112
$914$	−20.7252	−0.685528
$915$	−13.0916	−0.432795
$916$	6.66258	0.220138
$917$	28.9465	0.955897
$918$	24.3643	0.804143
$919$	−52.3462	−1.72674	−0.863370	−	0.504571i	$-0.831651\pi$
$919$	−52.3462	−1.72674	−0.863370	+	0.504571i	$0.831651\pi$
$920$	−7.19603	−0.237246
$921$	1.63163	0.0537640
$922$	11.9187	0.392520
$923$	0	0
$924$	2.98867	0.0983200
$925$	8.50350	0.279593
$926$	26.3989	0.867523
$927$	0.00596325	0.000195859 0
$928$	−8.69427	−0.285403
$929$	−42.2265	−1.38541	−0.692703	−	0.721223i	$-0.743580\pi$
$929$	−42.2265	−1.38541	−0.692703	+	0.721223i	$0.743580\pi$
$930$	−10.3165	−0.338292
$931$	−0.938037	−0.0307429
$932$	−11.4360	−0.374598
$933$	−22.9594	−0.751656
$934$	38.1584	1.24858
$935$	17.2516	0.564187
$936$	0	0
$937$	23.0654	0.753515	0.376758	−	0.926312i	$-0.377039\pi$
$937$	23.0654	0.753515	0.376758	+	0.926312i	$0.377039\pi$
$938$	21.0032	0.685777
$939$	19.5451	0.637831
$940$	−2.21910	−0.0723791
$941$	0.345669	0.0112685	0.00563425	−	0.999984i	$-0.498207\pi$
$941$	0.345669	0.0112685	0.00563425	+	0.999984i	$0.498207\pi$
$942$	−13.6673	−0.445304
$943$	18.3000	0.595929
$944$	−7.59208	−0.247101
$945$	−16.3639	−0.532317
$946$	7.04658	0.229104
$947$	−5.86618	−0.190625	−0.0953126	−	0.995447i	$-0.530385\pi$
$947$	−5.86618	−0.190625	−0.0953126	+	0.995447i	$0.530385\pi$
$948$	2.13984	0.0694986
$949$	0	0
$950$	2.19175	0.0711097
$951$	19.8055	0.642239
$952$	−15.1251	−0.490209
$953$	39.0267	1.26420	0.632100	−	0.774887i	$-0.282193\pi$
$953$	39.0267	1.26420	0.632100	+	0.774887i	$0.282193\pi$
$954$	35.8820	1.16172
$955$	26.2554	0.849603
$956$	−14.1793	−0.458592
$957$	10.5537	0.341152
$958$	7.16427	0.231467
$959$	−22.1048	−0.713802
$960$	1.21387	0.0391774
$961$	41.2310	1.33003
$962$	0	0
$963$	−39.7491	−1.28090
$964$	−7.14344	−0.230075
$965$	−5.32197	−0.171320
$966$	−7.65834	−0.246403
$967$	36.7380	1.18141	0.590707	−	0.806886i	$-0.298849\pi$
$967$	36.7380	1.18141	0.590707	+	0.806886i	$0.298849\pi$
$968$	8.19175	0.263293
$969$	4.44986	0.142950
$970$	−14.7965	−0.475087
$971$	28.7183	0.921613	0.460807	−	0.887501i	$-0.347560\pi$
$971$	28.7183	0.921613	0.460807	+	0.887501i	$0.347560\pi$
$972$	15.1963	0.487421
$973$	56.1396	1.79975
$974$	10.5383	0.337670
$975$	0	0
$976$	−10.7850	−0.345221
$977$	−37.3335	−1.19440	−0.597202	−	0.802091i	$-0.703721\pi$
$977$	−37.3335	−1.19440	−0.597202	+	0.802091i	$0.703721\pi$
$978$	−5.95366	−0.190377
$979$	9.17915	0.293367
$980$	−1.57195	−0.0502140
$981$	42.5496	1.35851
$982$	−16.8818	−0.538721
$983$	−57.0943	−1.82103	−0.910513	−	0.413481i	$-0.864313\pi$
$983$	−57.0943	−1.82103	−0.910513	+	0.413481i	$0.864313\pi$
$984$	−3.08694	−0.0984082
$985$	24.4256	0.778264
$986$	−53.4104	−1.70093
$987$	−2.36167	−0.0751727
$988$	0	0
$989$	−18.0566	−0.574166
$990$	6.95128	0.220926
$991$	−19.7638	−0.627816	−0.313908	−	0.949453i	$-0.601638\pi$
$991$	−19.7638	−0.627816	−0.313908	+	0.949453i	$0.601638\pi$
$992$	−8.49888	−0.269840
$993$	−2.81310	−0.0892711
$994$	24.2725	0.769876
$995$	−21.3069	−0.675473
$996$	−6.88452	−0.218144
$997$	11.6721	0.369660	0.184830	−	0.982770i	$-0.440826\pi$
$997$	11.6721	0.369660	0.184830	+	0.982770i	$0.440826\pi$
$998$	29.9948	0.949469
$999$	15.3875	0.486840

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bc.1.4		6
13.4	even	6		494.2.g.f.419.3	yes	12
13.10	even	6		494.2.g.f.191.3	✓	12
13.12	even	2		6422.2.a.bd.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.f.191.3	✓	12	13.10	even	6
494.2.g.f.419.3	yes	12	13.4	even	6
6422.2.a.bc.1.4		6	1.1	even	1	trivial
6422.2.a.bd.1.4		6	13.12	even	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.724358 q^{3} +1.00000 q^{4} +1.67578 q^{5} -0.724358 q^{6} +2.46211 q^{7} -1.00000 q^{8} -2.47531 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.724358 q^{3} +1.00000 q^{4} +1.67578 q^{5} -0.724358 q^{6} +2.46211 q^{7} -1.00000 q^{8} -2.47531 q^{9} -1.67578 q^{10} +1.67578 q^{11} +0.724358 q^{12} -2.46211 q^{14} +1.21387 q^{15} +1.00000 q^{16} +6.14317 q^{17} +2.47531 q^{18} +1.00000 q^{19} +1.67578 q^{20} +1.78345 q^{21} -1.67578 q^{22} +4.29413 q^{23} -0.724358 q^{24} -2.19175 q^{25} -3.96608 q^{27} +2.46211 q^{28} +8.69427 q^{29} -1.21387 q^{30} +8.49888 q^{31} -1.00000 q^{32} +1.21387 q^{33} -6.14317 q^{34} +4.12596 q^{35} -2.47531 q^{36} -3.87978 q^{37} -1.00000 q^{38} -1.67578 q^{40} +4.26163 q^{41} -1.78345 q^{42} -4.20495 q^{43} +1.67578 q^{44} -4.14808 q^{45} -4.29413 q^{46} -1.32422 q^{47} +0.724358 q^{48} -0.938037 q^{49} +2.19175 q^{50} +4.44986 q^{51} +14.4960 q^{53} +3.96608 q^{54} +2.80825 q^{55} -2.46211 q^{56} +0.724358 q^{57} -8.69427 q^{58} -7.59208 q^{59} +1.21387 q^{60} -10.7850 q^{61} -8.49888 q^{62} -6.09446 q^{63} +1.00000 q^{64} -1.21387 q^{66} -8.53057 q^{67} +6.14317 q^{68} +3.11049 q^{69} -4.12596 q^{70} -9.85842 q^{71} +2.47531 q^{72} +6.30828 q^{73} +3.87978 q^{74} -1.58761 q^{75} +1.00000 q^{76} +4.12596 q^{77} +2.95411 q^{79} +1.67578 q^{80} +4.55305 q^{81} -4.26163 q^{82} -9.50430 q^{83} +1.78345 q^{84} +10.2946 q^{85} +4.20495 q^{86} +6.29776 q^{87} -1.67578 q^{88} +5.47753 q^{89} +4.14808 q^{90} +4.29413 q^{92} +6.15624 q^{93} +1.32422 q^{94} +1.67578 q^{95} -0.724358 q^{96} +8.82960 q^{97} +0.938037 q^{98} -4.14808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 6 * q^8 + 10 * q^9	\( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} + q^{14} + 9 q^{15} + 6 q^{16} - 2 q^{17} - 10 q^{18} + 6 q^{19} - 2 q^{20} - q^{21} + 2 q^{22} + 8 q^{23} - 2 q^{24} + 16 q^{25} - 4 q^{27} - q^{28} + 20 q^{29} - 9 q^{30} - 3 q^{31} - 6 q^{32} + 9 q^{33} + 2 q^{34} + 7 q^{35} + 10 q^{36} + 9 q^{37} - 6 q^{38} + 2 q^{40} - 3 q^{41} + q^{42} + 13 q^{43} - 2 q^{44} - 38 q^{45} - 8 q^{46} - 20 q^{47} + 2 q^{48} - 7 q^{49} - 16 q^{50} + 2 q^{51} + 25 q^{53} + 4 q^{54} + 46 q^{55} + q^{56} + 2 q^{57} - 20 q^{58} + 9 q^{60} + 6 q^{61} + 3 q^{62} - 46 q^{63} + 6 q^{64} - 9 q^{66} + 32 q^{67} - 2 q^{68} - 29 q^{69} - 7 q^{70} - 39 q^{71} - 10 q^{72} - 7 q^{73} - 9 q^{74} - 15 q^{75} + 6 q^{76} + 7 q^{77} + 18 q^{79} - 2 q^{80} + 54 q^{81} + 3 q^{82} + 7 q^{83} - q^{84} + 2 q^{85} - 13 q^{86} + 12 q^{87} + 2 q^{88} - 9 q^{89} + 38 q^{90} + 8 q^{92} + 47 q^{93} + 20 q^{94} - 2 q^{95} - 2 q^{96} + 4 q^{97} + 7 q^{98} - 38 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 6 * q^8 + 10 * q^9 + 2 * q^10 - 2 * q^11 + 2 * q^12 + q^14 + 9 * q^15 + 6 * q^16 - 2 * q^17 - 10 * q^18 + 6 * q^19 - 2 * q^20 - q^21 + 2 * q^22 + 8 * q^23 - 2 * q^24 + 16 * q^25 - 4 * q^27 - q^28 + 20 * q^29 - 9 * q^30 - 3 * q^31 - 6 * q^32 + 9 * q^33 + 2 * q^34 + 7 * q^35 + 10 * q^36 + 9 * q^37 - 6 * q^38 + 2 * q^40 - 3 * q^41 + q^42 + 13 * q^43 - 2 * q^44 - 38 * q^45 - 8 * q^46 - 20 * q^47 + 2 * q^48 - 7 * q^49 - 16 * q^50 + 2 * q^51 + 25 * q^53 + 4 * q^54 + 46 * q^55 + q^56 + 2 * q^57 - 20 * q^58 + 9 * q^60 + 6 * q^61 + 3 * q^62 - 46 * q^63 + 6 * q^64 - 9 * q^66 + 32 * q^67 - 2 * q^68 - 29 * q^69 - 7 * q^70 - 39 * q^71 - 10 * q^72 - 7 * q^73 - 9 * q^74 - 15 * q^75 + 6 * q^76 + 7 * q^77 + 18 * q^79 - 2 * q^80 + 54 * q^81 + 3 * q^82 + 7 * q^83 - q^84 + 2 * q^85 - 13 * q^86 + 12 * q^87 + 2 * q^88 - 9 * q^89 + 38 * q^90 + 8 * q^92 + 47 * q^93 + 20 * q^94 - 2 * q^95 - 2 * q^96 + 4 * q^97 + 7 * q^98 - 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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