LMFDB - Embedded newform 6422.2.a.k.1.2

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:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.2
Root			$0.445042$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.35690 q^{3} +1.00000 q^{4} -1.55496 q^{5} +2.35690 q^{6} -2.49396 q^{7} -1.00000 q^{8} +2.55496 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.35690 q^{3} +1.00000 q^{4} -1.55496 q^{5} +2.35690 q^{6} -2.49396 q^{7} -1.00000 q^{8} +2.55496 q^{9} +1.55496 q^{10} -2.29590 q^{11} -2.35690 q^{12} +2.49396 q^{14} +3.66487 q^{15} +1.00000 q^{16} -7.74094 q^{17} -2.55496 q^{18} +1.00000 q^{19} -1.55496 q^{20} +5.87800 q^{21} +2.29590 q^{22} +0.0217703 q^{23} +2.35690 q^{24} -2.58211 q^{25} +1.04892 q^{27} -2.49396 q^{28} +2.29590 q^{29} -3.66487 q^{30} +0.939001 q^{31} -1.00000 q^{32} +5.41119 q^{33} +7.74094 q^{34} +3.87800 q^{35} +2.55496 q^{36} +8.19567 q^{37} -1.00000 q^{38} +1.55496 q^{40} +11.7409 q^{41} -5.87800 q^{42} -11.4276 q^{43} -2.29590 q^{44} -3.97285 q^{45} -0.0217703 q^{46} +2.09783 q^{47} -2.35690 q^{48} -0.780167 q^{49} +2.58211 q^{50} +18.2446 q^{51} +0.664874 q^{53} -1.04892 q^{54} +3.57002 q^{55} +2.49396 q^{56} -2.35690 q^{57} -2.29590 q^{58} -1.78017 q^{59} +3.66487 q^{60} -7.16421 q^{61} -0.939001 q^{62} -6.37196 q^{63} +1.00000 q^{64} -5.41119 q^{66} +11.7017 q^{67} -7.74094 q^{68} -0.0513102 q^{69} -3.87800 q^{70} +8.14675 q^{71} -2.55496 q^{72} -11.0858 q^{73} -8.19567 q^{74} +6.08575 q^{75} +1.00000 q^{76} +5.72587 q^{77} +6.68963 q^{79} -1.55496 q^{80} -10.1371 q^{81} -11.7409 q^{82} +12.1371 q^{83} +5.87800 q^{84} +12.0368 q^{85} +11.4276 q^{86} -5.41119 q^{87} +2.29590 q^{88} +0.814019 q^{89} +3.97285 q^{90} +0.0217703 q^{92} -2.21313 q^{93} -2.09783 q^{94} -1.55496 q^{95} +2.35690 q^{96} -12.5700 q^{97} +0.780167 q^{98} -5.86592 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 5 q^{5} + 3 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 5 * q^5 + 3 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 5 q^{5} + 3 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} + 5 q^{10} + 7 q^{11} - 3 q^{12} - 2 q^{14} + 12 q^{15} + 3 q^{16} - 9 q^{17} - 8 q^{18} + 3 q^{19} - 5 q^{20} - 2 q^{21} - 7 q^{22} - 3 q^{23} + 3 q^{24} - 2 q^{25} - 6 q^{27} + 2 q^{28} - 7 q^{29} - 12 q^{30} - 7 q^{31} - 3 q^{32} + 9 q^{34} - 8 q^{35} + 8 q^{36} - 12 q^{37} - 3 q^{38} + 5 q^{40} + 21 q^{41} + 2 q^{42} - 18 q^{43} + 7 q^{44} - 18 q^{45} + 3 q^{46} - 12 q^{47} - 3 q^{48} - q^{49} + 2 q^{50} + 9 q^{51} + 3 q^{53} + 6 q^{54} - 14 q^{55} - 2 q^{56} - 3 q^{57} + 7 q^{58} - 4 q^{59} + 12 q^{60} - 10 q^{61} + 7 q^{62} + 10 q^{63} + 3 q^{64} + 8 q^{67} - 9 q^{68} - 32 q^{69} + 8 q^{70} - 3 q^{71} - 8 q^{72} + 4 q^{73} + 12 q^{74} - 19 q^{75} + 3 q^{76} + 28 q^{77} - 26 q^{79} - 5 q^{80} - 25 q^{81} - 21 q^{82} + 31 q^{83} - 2 q^{84} + 8 q^{85} + 18 q^{86} - 7 q^{88} + 17 q^{89} + 18 q^{90} - 3 q^{92} + 14 q^{93} + 12 q^{94} - 5 q^{95} + 3 q^{96} - 13 q^{97} + q^{98} + 21 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 5 * q^5 + 3 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9 + 5 * q^10 + 7 * q^11 - 3 * q^12 - 2 * q^14 + 12 * q^15 + 3 * q^16 - 9 * q^17 - 8 * q^18 + 3 * q^19 - 5 * q^20 - 2 * q^21 - 7 * q^22 - 3 * q^23 + 3 * q^24 - 2 * q^25 - 6 * q^27 + 2 * q^28 - 7 * q^29 - 12 * q^30 - 7 * q^31 - 3 * q^32 + 9 * q^34 - 8 * q^35 + 8 * q^36 - 12 * q^37 - 3 * q^38 + 5 * q^40 + 21 * q^41 + 2 * q^42 - 18 * q^43 + 7 * q^44 - 18 * q^45 + 3 * q^46 - 12 * q^47 - 3 * q^48 - q^49 + 2 * q^50 + 9 * q^51 + 3 * q^53 + 6 * q^54 - 14 * q^55 - 2 * q^56 - 3 * q^57 + 7 * q^58 - 4 * q^59 + 12 * q^60 - 10 * q^61 + 7 * q^62 + 10 * q^63 + 3 * q^64 + 8 * q^67 - 9 * q^68 - 32 * q^69 + 8 * q^70 - 3 * q^71 - 8 * q^72 + 4 * q^73 + 12 * q^74 - 19 * q^75 + 3 * q^76 + 28 * q^77 - 26 * q^79 - 5 * q^80 - 25 * q^81 - 21 * q^82 + 31 * q^83 - 2 * q^84 + 8 * q^85 + 18 * q^86 - 7 * q^88 + 17 * q^89 + 18 * q^90 - 3 * q^92 + 14 * q^93 + 12 * q^94 - 5 * q^95 + 3 * q^96 - 13 * q^97 + q^98 + 21 * 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
$3$	−2.35690	−1.36075	−0.680377	−	0.732862i	$-0.738184\pi$
$3$	−2.35690	−1.36075	−0.680377	+	0.732862i	$0.738184\pi$
$4$	1.00000	0.500000
$5$	−1.55496	−0.695398	−0.347699	−	0.937606i	$-0.613037\pi$
$5$	−1.55496	−0.695398	−0.347699	+	0.937606i	$0.613037\pi$
$6$	2.35690	0.962199
$7$	−2.49396	−0.942628	−0.471314	−	0.881965i	$-0.656220\pi$
$7$	−2.49396	−0.942628	−0.471314	+	0.881965i	$0.656220\pi$
$8$	−1.00000	−0.353553
$9$	2.55496	0.851653
$10$	1.55496	0.491721
$11$	−2.29590	−0.692239	−0.346119	−	0.938190i	$-0.612501\pi$
$11$	−2.29590	−0.692239	−0.346119	+	0.938190i	$0.612501\pi$
$12$	−2.35690	−0.680377
$13$	0	0
$13$	0	0
$14$	2.49396	0.666539
$15$	3.66487	0.946267
$16$	1.00000	0.250000
$17$	−7.74094	−1.87745	−0.938727	−	0.344662i	$-0.887993\pi$
$17$	−7.74094	−1.87745	−0.938727	+	0.344662i	$0.887993\pi$
$18$	−2.55496	−0.602209
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.55496	−0.347699
$21$	5.87800	1.28269
$22$	2.29590	0.489487
$23$	0.0217703	0.00453941	0.00226971	−	0.999997i	$-0.499278\pi$
$23$	0.0217703	0.00453941	0.00226971	+	0.999997i	$0.499278\pi$
$24$	2.35690	0.481099
$25$	−2.58211	−0.516421
$26$	0	0
$27$	1.04892	0.201864
$28$	−2.49396	−0.471314
$29$	2.29590	0.426337	0.213169	−	0.977015i	$-0.431622\pi$
$29$	2.29590	0.426337	0.213169	+	0.977015i	$0.431622\pi$
$30$	−3.66487	−0.669111
$31$	0.939001	0.168650	0.0843248	−	0.996438i	$-0.473127\pi$
$31$	0.939001	0.168650	0.0843248	+	0.996438i	$0.473127\pi$
$32$	−1.00000	−0.176777
$33$	5.41119	0.941967
$34$	7.74094	1.32756
$35$	3.87800	0.655502
$36$	2.55496	0.425826
$37$	8.19567	1.34736	0.673680	−	0.739023i	$-0.264713\pi$
$37$	8.19567	1.34736	0.673680	+	0.739023i	$0.264713\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	1.55496	0.245860
$41$	11.7409	1.83363	0.916813	−	0.399316i	$-0.130752\pi$
$41$	11.7409	1.83363	0.916813	+	0.399316i	$0.130752\pi$
$42$	−5.87800	−0.906995
$43$	−11.4276	−1.74269	−0.871345	−	0.490671i	$-0.836752\pi$
$43$	−11.4276	−1.74269	−0.871345	+	0.490671i	$0.836752\pi$
$44$	−2.29590	−0.346119
$45$	−3.97285	−0.592238
$46$	−0.0217703	−0.00320985
$47$	2.09783	0.306001	0.153000	−	0.988226i	$-0.451106\pi$
$47$	2.09783	0.306001	0.153000	+	0.988226i	$0.451106\pi$
$48$	−2.35690	−0.340189
$49$	−0.780167	−0.111452
$50$	2.58211	0.365165
$51$	18.2446	2.55475
$52$	0	0
$53$	0.664874	0.0913275	0.0456638	−	0.998957i	$-0.485460\pi$
$53$	0.664874	0.0913275	0.0456638	+	0.998957i	$0.485460\pi$
$54$	−1.04892	−0.142740
$55$	3.57002	0.481382
$56$	2.49396	0.333269
$57$	−2.35690	−0.312178
$58$	−2.29590	−0.301466
$59$	−1.78017	−0.231758	−0.115879	−	0.993263i	$-0.536968\pi$
$59$	−1.78017	−0.231758	−0.115879	+	0.993263i	$0.536968\pi$
$60$	3.66487	0.473133
$61$	−7.16421	−0.917283	−0.458642	−	0.888621i	$-0.651664\pi$
$61$	−7.16421	−0.917283	−0.458642	+	0.888621i	$0.651664\pi$
$62$	−0.939001	−0.119253
$63$	−6.37196	−0.802792
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.41119	−0.666071
$67$	11.7017	1.42959	0.714796	−	0.699333i	$-0.246520\pi$
$67$	11.7017	1.42959	0.714796	+	0.699333i	$0.246520\pi$
$68$	−7.74094	−0.938727
$69$	−0.0513102	−0.00617703
$70$	−3.87800	−0.463510
$71$	8.14675	0.966842	0.483421	−	0.875388i	$-0.339394\pi$
$71$	8.14675	0.966842	0.483421	+	0.875388i	$0.339394\pi$
$72$	−2.55496	−0.301105
$73$	−11.0858	−1.29749	−0.648745	−	0.761006i	$-0.724706\pi$
$73$	−11.0858	−1.29749	−0.648745	+	0.761006i	$0.724706\pi$
$74$	−8.19567	−0.952727
$75$	6.08575	0.702722
$76$	1.00000	0.114708
$77$	5.72587	0.652524
$78$	0	0
$79$	6.68963	0.752642	0.376321	−	0.926489i	$-0.377189\pi$
$79$	6.68963	0.752642	0.376321	+	0.926489i	$0.377189\pi$
$80$	−1.55496	−0.173850
$81$	−10.1371	−1.12634
$82$	−11.7409	−1.29657
$83$	12.1371	1.33222	0.666108	−	0.745855i	$-0.267959\pi$
$83$	12.1371	1.33222	0.666108	+	0.745855i	$0.267959\pi$
$84$	5.87800	0.641343
$85$	12.0368	1.30558
$86$	11.4276	1.23227
$87$	−5.41119	−0.580140
$88$	2.29590	0.244743
$89$	0.814019	0.0862859	0.0431429	−	0.999069i	$-0.486263\pi$
$89$	0.814019	0.0862859	0.0431429	+	0.999069i	$0.486263\pi$
$90$	3.97285	0.418775
$91$	0	0
$92$	0.0217703	0.00226971
$93$	−2.21313	−0.229491
$94$	−2.09783	−0.216375
$95$	−1.55496	−0.159535
$96$	2.35690	0.240550
$97$	−12.5700	−1.27629	−0.638146	−	0.769915i	$-0.720298\pi$
$97$	−12.5700	−1.27629	−0.638146	+	0.769915i	$0.720298\pi$
$98$	0.780167	0.0788088
$99$	−5.86592	−0.589547
$100$	−2.58211	−0.258211
$101$	11.4819	1.14249	0.571245	−	0.820780i	$-0.306461\pi$
$101$	11.4819	1.14249	0.571245	+	0.820780i	$0.306461\pi$
$102$	−18.2446	−1.80648
$103$	9.48188	0.934277	0.467139	−	0.884184i	$-0.345285\pi$
$103$	9.48188	0.934277	0.467139	+	0.884184i	$0.345285\pi$
$104$	0	0
$105$	−9.14005	−0.891977
$106$	−0.664874	−0.0645783
$107$	−2.48858	−0.240580	−0.120290	−	0.992739i	$-0.538382\pi$
$107$	−2.48858	−0.240580	−0.120290	+	0.992739i	$0.538382\pi$
$108$	1.04892	0.100932
$109$	−5.92154	−0.567181	−0.283590	−	0.958945i	$-0.591526\pi$
$109$	−5.92154	−0.567181	−0.283590	+	0.958945i	$0.591526\pi$
$110$	−3.57002	−0.340388
$111$	−19.3163	−1.83343
$112$	−2.49396	−0.235657
$113$	12.0543	1.13397	0.566986	−	0.823727i	$-0.308109\pi$
$113$	12.0543	1.13397	0.566986	+	0.823727i	$0.308109\pi$
$114$	2.35690	0.220744
$115$	−0.0338518	−0.00315670
$116$	2.29590	0.213169
$117$	0	0
$118$	1.78017	0.163878
$119$	19.3056	1.76974
$120$	−3.66487	−0.334556
$121$	−5.72886	−0.520805
$122$	7.16421	0.648617
$123$	−27.6722	−2.49512
$124$	0.939001	0.0843248
$125$	11.7899	1.05452
$126$	6.37196	0.567659
$127$	−7.78017	−0.690378	−0.345189	−	0.938533i	$-0.612185\pi$
$127$	−7.78017	−0.690378	−0.345189	+	0.938533i	$0.612185\pi$
$128$	−1.00000	−0.0883883
$129$	26.9336	2.37137
$130$	0	0
$131$	2.79225	0.243960	0.121980	−	0.992533i	$-0.461076\pi$
$131$	2.79225	0.243960	0.121980	+	0.992533i	$0.461076\pi$
$132$	5.41119	0.470984
$133$	−2.49396	−0.216254
$134$	−11.7017	−1.01087
$135$	−1.63102	−0.140376
$136$	7.74094	0.663780
$137$	−3.06638	−0.261978	−0.130989	−	0.991384i	$-0.541815\pi$
$137$	−3.06638	−0.261978	−0.130989	+	0.991384i	$0.541815\pi$
$138$	0.0513102	0.00436782
$139$	3.56033	0.301984	0.150992	−	0.988535i	$-0.451753\pi$
$139$	3.56033	0.301984	0.150992	+	0.988535i	$0.451753\pi$
$140$	3.87800	0.327751
$141$	−4.94438	−0.416392
$142$	−8.14675	−0.683660
$143$	0	0
$144$	2.55496	0.212913
$145$	−3.57002	−0.296474
$146$	11.0858	0.917463
$147$	1.83877	0.151659
$148$	8.19567	0.673680
$149$	19.3207	1.58281	0.791405	−	0.611293i	$-0.209350\pi$
$149$	19.3207	1.58281	0.791405	+	0.611293i	$0.209350\pi$
$150$	−6.08575	−0.496900
$151$	−10.8334	−0.881609	−0.440805	−	0.897603i	$-0.645307\pi$
$151$	−10.8334	−0.881609	−0.440805	+	0.897603i	$0.645307\pi$
$152$	−1.00000	−0.0811107
$153$	−19.7778	−1.59894
$154$	−5.72587	−0.461404
$155$	−1.46011	−0.117279
$156$	0	0
$157$	15.1836	1.21178	0.605891	−	0.795548i	$-0.292817\pi$
$157$	15.1836	1.21178	0.605891	+	0.795548i	$0.292817\pi$
$158$	−6.68963	−0.532198
$159$	−1.56704	−0.124274
$160$	1.55496	0.122930
$161$	−0.0542942	−0.00427898
$162$	10.1371	0.796443
$163$	18.1129	1.41871	0.709356	−	0.704850i	$-0.248986\pi$
$163$	18.1129	1.41871	0.709356	+	0.704850i	$0.248986\pi$
$164$	11.7409	0.916813
$165$	−8.41417	−0.655043
$166$	−12.1371	−0.942019
$167$	10.8944	0.843034	0.421517	−	0.906821i	$-0.361498\pi$
$167$	10.8944	0.843034	0.421517	+	0.906821i	$0.361498\pi$
$168$	−5.87800	−0.453498
$169$	0	0
$170$	−12.0368	−0.923183
$171$	2.55496	0.195383
$172$	−11.4276	−0.871345
$173$	−12.0043	−0.912671	−0.456335	−	0.889808i	$-0.650838\pi$
$173$	−12.0043	−0.912671	−0.456335	+	0.889808i	$0.650838\pi$
$174$	5.41119	0.410221
$175$	6.43967	0.486793
$176$	−2.29590	−0.173060
$177$	4.19567	0.315366
$178$	−0.814019	−0.0610133
$179$	5.59850	0.418451	0.209226	−	0.977867i	$-0.432906\pi$
$179$	5.59850	0.418451	0.209226	+	0.977867i	$0.432906\pi$
$180$	−3.97285	−0.296119
$181$	10.4112	0.773858	0.386929	−	0.922110i	$-0.373536\pi$
$181$	10.4112	0.773858	0.386929	+	0.922110i	$0.373536\pi$
$182$	0	0
$183$	16.8853	1.24820
$184$	−0.0217703	−0.00160493
$185$	−12.7439	−0.936952
$186$	2.21313	0.162274
$187$	17.7724	1.29965
$188$	2.09783	0.153000
$189$	−2.61596	−0.190283
$190$	1.55496	0.112809
$191$	17.8291	1.29007	0.645034	−	0.764154i	$-0.276843\pi$
$191$	17.8291	1.29007	0.645034	+	0.764154i	$0.276843\pi$
$192$	−2.35690	−0.170094
$193$	5.24698	0.377686	0.188843	−	0.982007i	$-0.439526\pi$
$193$	5.24698	0.377686	0.188843	+	0.982007i	$0.439526\pi$
$194$	12.5700	0.902475
$195$	0	0
$196$	−0.780167	−0.0557262
$197$	−15.4276	−1.09917	−0.549585	−	0.835438i	$-0.685214\pi$
$197$	−15.4276	−1.09917	−0.549585	+	0.835438i	$0.685214\pi$
$198$	5.86592	0.416873
$199$	12.2174	0.866071	0.433036	−	0.901377i	$-0.357442\pi$
$199$	12.2174	0.866071	0.433036	+	0.901377i	$0.357442\pi$
$200$	2.58211	0.182582
$201$	−27.5797	−1.94532
$202$	−11.4819	−0.807862
$203$	−5.72587	−0.401878
$204$	18.2446	1.27738
$205$	−18.2567	−1.27510
$206$	−9.48188	−0.660634
$207$	0.0556221	0.00386600
$208$	0	0
$209$	−2.29590	−0.158811
$210$	9.14005	0.630723
$211$	−25.0640	−1.72548	−0.862738	−	0.505651i	$-0.831252\pi$
$211$	−25.0640	−1.72548	−0.862738	+	0.505651i	$0.831252\pi$
$212$	0.664874	0.0456638
$213$	−19.2010	−1.31563
$214$	2.48858	0.170116
$215$	17.7694	1.21186
$216$	−1.04892	−0.0713698
$217$	−2.34183	−0.158974
$218$	5.92154	0.401057
$219$	26.1280	1.76556
$220$	3.57002	0.240691
$221$	0	0
$222$	19.3163	1.29643
$223$	−19.5211	−1.30723	−0.653615	−	0.756827i	$-0.726748\pi$
$223$	−19.5211	−1.30723	−0.653615	+	0.756827i	$0.726748\pi$
$224$	2.49396	0.166635
$225$	−6.59717	−0.439811
$226$	−12.0543	−0.801840
$227$	−14.1957	−0.942200	−0.471100	−	0.882080i	$-0.656143\pi$
$227$	−14.1957	−0.942200	−0.471100	+	0.882080i	$0.656143\pi$
$228$	−2.35690	−0.156089
$229$	−13.6993	−0.905276	−0.452638	−	0.891694i	$-0.649517\pi$
$229$	−13.6993	−0.905276	−0.452638	+	0.891694i	$0.649517\pi$
$230$	0.0338518	0.00223212
$231$	−13.4953	−0.887925
$232$	−2.29590	−0.150733
$233$	−13.9705	−0.915235	−0.457618	−	0.889149i	$-0.651297\pi$
$233$	−13.9705	−0.915235	−0.457618	+	0.889149i	$0.651297\pi$
$234$	0	0
$235$	−3.26205	−0.212792
$236$	−1.78017	−0.115879
$237$	−15.7668	−1.02416
$238$	−19.3056	−1.25140
$239$	23.8237	1.54103	0.770514	−	0.637423i	$-0.220000\pi$
$239$	23.8237	1.54103	0.770514	+	0.637423i	$0.220000\pi$
$240$	3.66487	0.236567
$241$	8.88471	0.572314	0.286157	−	0.958183i	$-0.407622\pi$
$241$	8.88471	0.572314	0.286157	+	0.958183i	$0.407622\pi$
$242$	5.72886	0.368265
$243$	20.7453	1.33081
$244$	−7.16421	−0.458642
$245$	1.21313	0.0775039
$246$	27.6722	1.76431
$247$	0	0
$248$	−0.939001	−0.0596266
$249$	−28.6058	−1.81282
$250$	−11.7899	−0.745656
$251$	8.63533	0.545057	0.272529	−	0.962148i	$-0.412140\pi$
$251$	8.63533	0.545057	0.272529	+	0.962148i	$0.412140\pi$
$252$	−6.37196	−0.401396
$253$	−0.0499823	−0.00314236
$254$	7.78017	0.488171
$255$	−28.3696	−1.77657
$256$	1.00000	0.0625000
$257$	29.3599	1.83142	0.915709	−	0.401841i	$-0.131630\pi$
$257$	29.3599	1.83142	0.915709	+	0.401841i	$0.131630\pi$
$258$	−26.9336	−1.67681
$259$	−20.4397	−1.27006
$260$	0	0
$261$	5.86592	0.363091
$262$	−2.79225	−0.172506
$263$	−19.9608	−1.23083	−0.615417	−	0.788202i	$-0.711012\pi$
$263$	−19.9608	−1.23083	−0.615417	+	0.788202i	$0.711012\pi$
$264$	−5.41119	−0.333036
$265$	−1.03385	−0.0635090
$266$	2.49396	0.152914
$267$	−1.91856	−0.117414
$268$	11.7017	0.714796
$269$	9.64742	0.588213	0.294107	−	0.955773i	$-0.404978\pi$
$269$	9.64742	0.588213	0.294107	+	0.955773i	$0.404978\pi$
$270$	1.63102	0.0992609
$271$	25.3840	1.54197	0.770985	−	0.636853i	$-0.219764\pi$
$271$	25.3840	1.54197	0.770985	+	0.636853i	$0.219764\pi$
$272$	−7.74094	−0.469363
$273$	0	0
$274$	3.06638	0.185247
$275$	5.92825	0.357487
$276$	−0.0513102	−0.00308851
$277$	−23.7453	−1.42671	−0.713357	−	0.700801i	$-0.752826\pi$
$277$	−23.7453	−1.42671	−0.713357	+	0.700801i	$0.752826\pi$
$278$	−3.56033	−0.213535
$279$	2.39911	0.143631
$280$	−3.87800	−0.231755
$281$	−15.8213	−0.943821	−0.471910	−	0.881647i	$-0.656435\pi$
$281$	−15.8213	−0.943821	−0.471910	+	0.881647i	$0.656435\pi$
$282$	4.94438	0.294433
$283$	−16.0301	−0.952892	−0.476446	−	0.879204i	$-0.658075\pi$
$283$	−16.0301	−0.952892	−0.476446	+	0.879204i	$0.658075\pi$
$284$	8.14675	0.483421
$285$	3.66487	0.217088
$286$	0	0
$287$	−29.2814	−1.72843
$288$	−2.55496	−0.150552
$289$	42.9221	2.52483
$290$	3.57002	0.209639
$291$	29.6262	1.73672
$292$	−11.0858	−0.648745
$293$	−14.4940	−0.846746	−0.423373	−	0.905955i	$-0.639154\pi$
$293$	−14.4940	−0.846746	−0.423373	+	0.905955i	$0.639154\pi$
$294$	−1.83877	−0.107239
$295$	2.76809	0.161164
$296$	−8.19567	−0.476364
$297$	−2.40821	−0.139738
$298$	−19.3207	−1.11922
$299$	0	0
$300$	6.08575	0.351361
$301$	28.4999	1.64271
$302$	10.8334	0.623392
$303$	−27.0616	−1.55465
$304$	1.00000	0.0573539
$305$	11.1400	0.637877
$306$	19.7778	1.13062
$307$	16.5676	0.945565	0.472782	−	0.881179i	$-0.343250\pi$
$307$	16.5676	0.945565	0.472782	+	0.881179i	$0.343250\pi$
$308$	5.72587	0.326262
$309$	−22.3478	−1.27132
$310$	1.46011	0.0829285
$311$	11.5211	0.653302	0.326651	−	0.945145i	$-0.394080\pi$
$311$	11.5211	0.653302	0.326651	+	0.945145i	$0.394080\pi$
$312$	0	0
$313$	−19.6528	−1.11084	−0.555421	−	0.831569i	$-0.687443\pi$
$313$	−19.6528	−1.11084	−0.555421	+	0.831569i	$0.687443\pi$
$314$	−15.1836	−0.856859
$315$	9.90813	0.558260
$316$	6.68963	0.376321
$317$	13.3840	0.751723	0.375861	−	0.926676i	$-0.377347\pi$
$317$	13.3840	0.751723	0.375861	+	0.926676i	$0.377347\pi$
$318$	1.56704	0.0878752
$319$	−5.27114	−0.295127
$320$	−1.55496	−0.0869248
$321$	5.86533	0.327371
$322$	0.0542942	0.00302569
$323$	−7.74094	−0.430717
$324$	−10.1371	−0.563170
$325$	0	0
$326$	−18.1129	−1.00318
$327$	13.9565	0.771794
$328$	−11.7409	−0.648285
$329$	−5.23191	−0.288445
$330$	8.41417	0.463185
$331$	−23.8431	−1.31053	−0.655267	−	0.755397i	$-0.727444\pi$
$331$	−23.8431	−1.31053	−0.655267	+	0.755397i	$0.727444\pi$
$332$	12.1371	0.666108
$333$	20.9396	1.14748
$334$	−10.8944	−0.596115
$335$	−18.1957	−0.994136
$336$	5.87800	0.320671
$337$	−21.2814	−1.15927	−0.579636	−	0.814875i	$-0.696805\pi$
$337$	−21.2814	−1.15927	−0.579636	+	0.814875i	$0.696805\pi$
$338$	0	0
$339$	−28.4107	−1.54306
$340$	12.0368	0.652789
$341$	−2.15585	−0.116746
$342$	−2.55496	−0.138156
$343$	19.4034	1.04769
$344$	11.4276	0.616134
$345$	0.0797853	0.00429550
$346$	12.0043	0.645356
$347$	−32.4155	−1.74016	−0.870078	−	0.492915i	$-0.835931\pi$
$347$	−32.4155	−1.74016	−0.870078	+	0.492915i	$0.835931\pi$
$348$	−5.41119	−0.290070
$349$	3.83877	0.205485	0.102742	−	0.994708i	$-0.467238\pi$
$349$	3.83877	0.205485	0.102742	+	0.994708i	$0.467238\pi$
$350$	−6.43967	−0.344215
$351$	0	0
$352$	2.29590	0.122372
$353$	−13.7366	−0.731127	−0.365563	−	0.930786i	$-0.619124\pi$
$353$	−13.7366	−0.731127	−0.365563	+	0.930786i	$0.619124\pi$
$354$	−4.19567	−0.222997
$355$	−12.6679	−0.672340
$356$	0.814019	0.0431429
$357$	−45.5013	−2.40818
$358$	−5.59850	−0.295890
$359$	7.69096	0.405913	0.202957	−	0.979188i	$-0.434945\pi$
$359$	7.69096	0.405913	0.202957	+	0.979188i	$0.434945\pi$
$360$	3.97285	0.209388
$361$	1.00000	0.0526316
$362$	−10.4112	−0.547200
$363$	13.5023	0.708688
$364$	0	0
$365$	17.2379	0.902272
$366$	−16.8853	−0.882609
$367$	−25.2325	−1.31713	−0.658563	−	0.752526i	$-0.728835\pi$
$367$	−25.2325	−1.31713	−0.658563	+	0.752526i	$0.728835\pi$
$368$	0.0217703	0.00113485
$369$	29.9976	1.56161
$370$	12.7439	0.662525
$371$	−1.65817	−0.0860879
$372$	−2.21313	−0.114745
$373$	−8.87502	−0.459531	−0.229766	−	0.973246i	$-0.573796\pi$
$373$	−8.87502	−0.459531	−0.229766	+	0.973246i	$0.573796\pi$
$374$	−17.7724	−0.918989
$375$	−27.7875	−1.43494
$376$	−2.09783	−0.108188
$377$	0	0
$378$	2.61596	0.134550
$379$	6.67025	0.342628	0.171314	−	0.985216i	$-0.445199\pi$
$379$	6.67025	0.342628	0.171314	+	0.985216i	$0.445199\pi$
$380$	−1.55496	−0.0797677
$381$	18.3370	0.939435
$382$	−17.8291	−0.912215
$383$	6.02475	0.307851	0.153925	−	0.988082i	$-0.450808\pi$
$383$	6.02475	0.307851	0.153925	+	0.988082i	$0.450808\pi$
$384$	2.35690	0.120275
$385$	−8.90349	−0.453764
$386$	−5.24698	−0.267064
$387$	−29.1970	−1.48417
$388$	−12.5700	−0.638146
$389$	−15.7888	−0.800523	−0.400262	−	0.916401i	$-0.631081\pi$
$389$	−15.7888	−0.800523	−0.400262	+	0.916401i	$0.631081\pi$
$390$	0	0
$391$	−0.168522	−0.00852254
$392$	0.780167	0.0394044
$393$	−6.58104	−0.331970
$394$	15.4276	0.777230
$395$	−10.4021	−0.523386
$396$	−5.86592	−0.294774
$397$	1.55496	0.0780411	0.0390206	−	0.999238i	$-0.487576\pi$
$397$	1.55496	0.0780411	0.0390206	+	0.999238i	$0.487576\pi$
$398$	−12.2174	−0.612405
$399$	5.87800	0.294268
$400$	−2.58211	−0.129105
$401$	−7.21313	−0.360206	−0.180103	−	0.983648i	$-0.557643\pi$
$401$	−7.21313	−0.360206	−0.180103	+	0.983648i	$0.557643\pi$
$402$	27.5797	1.37555
$403$	0	0
$404$	11.4819	0.571245
$405$	15.7627	0.783255
$406$	5.72587	0.284170
$407$	−18.8164	−0.932695
$408$	−18.2446	−0.903242
$409$	15.1836	0.750780	0.375390	−	0.926867i	$-0.377509\pi$
$409$	15.1836	0.750780	0.375390	+	0.926867i	$0.377509\pi$
$410$	18.2567	0.901633
$411$	7.22713	0.356488
$412$	9.48188	0.467139
$413$	4.43967	0.218462
$414$	−0.0556221	−0.00273368
$415$	−18.8726	−0.926421
$416$	0	0
$417$	−8.39134	−0.410926
$418$	2.29590	0.112296
$419$	−5.16421	−0.252288	−0.126144	−	0.992012i	$-0.540260\pi$
$419$	−5.16421	−0.252288	−0.126144	+	0.992012i	$0.540260\pi$
$420$	−9.14005	−0.445989
$421$	−17.2271	−0.839599	−0.419799	−	0.907617i	$-0.637900\pi$
$421$	−17.2271	−0.839599	−0.419799	+	0.907617i	$0.637900\pi$
$422$	25.0640	1.22010
$423$	5.35988	0.260606
$424$	−0.664874	−0.0322892
$425$	19.9879	0.969556
$426$	19.2010	0.930294
$427$	17.8672	0.864657
$428$	−2.48858	−0.120290
$429$	0	0
$430$	−17.7694	−0.856917
$431$	29.3884	1.41559	0.707794	−	0.706419i	$-0.249691\pi$
$431$	29.3884	1.41559	0.707794	+	0.706419i	$0.249691\pi$
$432$	1.04892	0.0504661
$433$	0.987918	0.0474763	0.0237382	−	0.999718i	$-0.492443\pi$
$433$	0.987918	0.0474763	0.0237382	+	0.999718i	$0.492443\pi$
$434$	2.34183	0.112411
$435$	8.41417	0.403429
$436$	−5.92154	−0.283590
$437$	0.0217703	0.00104141
$438$	−26.1280	−1.24844
$439$	10.2064	0.487126	0.243563	−	0.969885i	$-0.421684\pi$
$439$	10.2064	0.487126	0.243563	+	0.969885i	$0.421684\pi$
$440$	−3.57002	−0.170194
$441$	−1.99330	−0.0949188
$442$	0	0
$443$	1.50604	0.0715542	0.0357771	−	0.999360i	$-0.488609\pi$
$443$	1.50604	0.0715542	0.0357771	+	0.999360i	$0.488609\pi$
$444$	−19.3163	−0.916713
$445$	−1.26577	−0.0600031
$446$	19.5211	0.924351
$447$	−45.5368	−2.15381
$448$	−2.49396	−0.117828
$449$	−7.46681	−0.352381	−0.176190	−	0.984356i	$-0.556377\pi$
$449$	−7.46681	−0.352381	−0.176190	+	0.984356i	$0.556377\pi$
$450$	6.59717	0.310994
$451$	−26.9560	−1.26931
$452$	12.0543	0.566986
$453$	25.5332	1.19965
$454$	14.1957	0.666236
$455$	0	0
$456$	2.35690	0.110372
$457$	12.9530	0.605916	0.302958	−	0.953004i	$-0.402026\pi$
$457$	12.9530	0.605916	0.302958	+	0.953004i	$0.402026\pi$
$458$	13.6993	0.640127
$459$	−8.11960	−0.378991
$460$	−0.0338518	−0.00157835
$461$	−0.559270	−0.0260478	−0.0130239	−	0.999915i	$-0.504146\pi$
$461$	−0.559270	−0.0260478	−0.0130239	+	0.999915i	$0.504146\pi$
$462$	13.4953	0.627858
$463$	29.5254	1.37216	0.686081	−	0.727525i	$-0.259329\pi$
$463$	29.5254	1.37216	0.686081	+	0.727525i	$0.259329\pi$
$464$	2.29590	0.106584
$465$	3.44132	0.159587
$466$	13.9705	0.647169
$467$	−9.23191	−0.427202	−0.213601	−	0.976921i	$-0.568519\pi$
$467$	−9.23191	−0.427202	−0.213601	+	0.976921i	$0.568519\pi$
$468$	0	0
$469$	−29.1836	−1.34757
$470$	3.26205	0.150467
$471$	−35.7861	−1.64894
$472$	1.78017	0.0819388
$473$	26.2366	1.20636
$474$	15.7668	0.724191
$475$	−2.58211	−0.118475
$476$	19.3056	0.884870
$477$	1.69873	0.0777793
$478$	−23.8237	−1.08967
$479$	−16.3370	−0.746459	−0.373229	−	0.927739i	$-0.621750\pi$
$479$	−16.3370	−0.746459	−0.373229	+	0.927739i	$0.621750\pi$
$480$	−3.66487	−0.167278
$481$	0	0
$482$	−8.88471	−0.404687
$483$	0.127966	0.00582264
$484$	−5.72886	−0.260403
$485$	19.5459	0.887532
$486$	−20.7453	−0.941024
$487$	14.8334	0.672165	0.336083	−	0.941832i	$-0.390898\pi$
$487$	14.8334	0.672165	0.336083	+	0.941832i	$0.390898\pi$
$488$	7.16421	0.324309
$489$	−42.6902	−1.93052
$490$	−1.21313	−0.0548035
$491$	15.3793	0.694056	0.347028	−	0.937855i	$-0.387191\pi$
$491$	15.3793	0.694056	0.347028	+	0.937855i	$0.387191\pi$
$492$	−27.6722	−1.24756
$493$	−17.7724	−0.800429
$494$	0	0
$495$	9.12126	0.409970
$496$	0.939001	0.0421624
$497$	−20.3177	−0.911372
$498$	28.6058	1.28186
$499$	−40.2543	−1.80203	−0.901014	−	0.433789i	$-0.857176\pi$
$499$	−40.2543	−1.80203	−0.901014	+	0.433789i	$0.857176\pi$
$500$	11.7899	0.527258
$501$	−25.6770	−1.14716
$502$	−8.63533	−0.385414
$503$	8.74392	0.389872	0.194936	−	0.980816i	$-0.437550\pi$
$503$	8.74392	0.389872	0.194936	+	0.980816i	$0.437550\pi$
$504$	6.37196	0.283830
$505$	−17.8538	−0.794485
$506$	0.0499823	0.00222198
$507$	0	0
$508$	−7.78017	−0.345189
$509$	−36.3129	−1.60954	−0.804770	−	0.593587i	$-0.797711\pi$
$509$	−36.3129	−1.60954	−0.804770	+	0.593587i	$0.797711\pi$
$510$	28.3696	1.25623
$511$	27.6474	1.22305
$512$	−1.00000	−0.0441942
$513$	1.04892	0.0463108
$514$	−29.3599	−1.29501
$515$	−14.7439	−0.649695
$516$	26.9336	1.18569
$517$	−4.81641	−0.211826
$518$	20.4397	0.898067
$519$	28.2929	1.24192
$520$	0	0
$521$	9.92154	0.434671	0.217335	−	0.976097i	$-0.430263\pi$
$521$	9.92154	0.434671	0.217335	+	0.976097i	$0.430263\pi$
$522$	−5.86592	−0.256744
$523$	4.90408	0.214441	0.107220	−	0.994235i	$-0.465805\pi$
$523$	4.90408	0.214441	0.107220	+	0.994235i	$0.465805\pi$
$524$	2.79225	0.121980
$525$	−15.1776	−0.662406
$526$	19.9608	0.870331
$527$	−7.26875	−0.316632
$528$	5.41119	0.235492
$529$	−22.9995	−0.999979
$530$	1.03385	0.0449077
$531$	−4.54825	−0.197377
$532$	−2.49396	−0.108127
$533$	0	0
$534$	1.91856	0.0830242
$535$	3.86964	0.167299
$536$	−11.7017	−0.505437
$537$	−13.1951	−0.569410
$538$	−9.64742	−0.415930
$539$	1.79118	0.0771518
$540$	−1.63102	−0.0701880
$541$	−40.8605	−1.75673	−0.878366	−	0.477989i	$-0.841366\pi$
$541$	−40.8605	−1.75673	−0.878366	+	0.477989i	$0.841366\pi$
$542$	−25.3840	−1.09034
$543$	−24.5381	−1.05303
$544$	7.74094	0.331890
$545$	9.20775	0.394417
$546$	0	0
$547$	2.48858	0.106404	0.0532020	−	0.998584i	$-0.483057\pi$
$547$	2.48858	0.106404	0.0532020	+	0.998584i	$0.483057\pi$
$548$	−3.06638	−0.130989
$549$	−18.3043	−0.781207
$550$	−5.92825	−0.252781
$551$	2.29590	0.0978085
$552$	0.0513102	0.00218391
$553$	−16.6837	−0.709461
$554$	23.7453	1.00884
$555$	30.0361	1.27496
$556$	3.56033	0.150992
$557$	−15.6752	−0.664178	−0.332089	−	0.943248i	$-0.607753\pi$
$557$	−15.6752	−0.664178	−0.332089	+	0.943248i	$0.607753\pi$
$558$	−2.39911	−0.101562
$559$	0	0
$560$	3.87800	0.163876
$561$	−41.8877	−1.76850
$562$	15.8213	0.667382
$563$	8.91185	0.375590	0.187795	−	0.982208i	$-0.439866\pi$
$563$	8.91185	0.375590	0.187795	+	0.982208i	$0.439866\pi$
$564$	−4.94438	−0.208196
$565$	−18.7439	−0.788563
$566$	16.0301	0.673797
$567$	25.2814	1.06172
$568$	−8.14675	−0.341830
$569$	−31.7512	−1.33108	−0.665540	−	0.746362i	$-0.731799\pi$
$569$	−31.7512	−1.33108	−0.665540	+	0.746362i	$0.731799\pi$
$570$	−3.66487	−0.153505
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	−42.0213	−1.75546
$574$	29.2814	1.22218
$575$	−0.0562131	−0.00234425
$576$	2.55496	0.106457
$577$	−28.3129	−1.17868	−0.589340	−	0.807885i	$-0.700612\pi$
$577$	−28.3129	−1.17868	−0.589340	+	0.807885i	$0.700612\pi$
$578$	−42.9221	−1.78533
$579$	−12.3666	−0.513938
$580$	−3.57002	−0.148237
$581$	−30.2693	−1.25578
$582$	−29.6262	−1.22805
$583$	−1.52648	−0.0632205
$584$	11.0858	0.458732
$585$	0	0
$586$	14.4940	0.598740
$587$	19.6474	0.810936	0.405468	−	0.914109i	$-0.367109\pi$
$587$	19.6474	0.810936	0.405468	+	0.914109i	$0.367109\pi$
$588$	1.83877	0.0758297
$589$	0.939001	0.0386909
$590$	−2.76809	−0.113960
$591$	36.3612	1.49570
$592$	8.19567	0.336840
$593$	2.27413	0.0933872	0.0466936	−	0.998909i	$-0.485132\pi$
$593$	2.27413	0.0933872	0.0466936	+	0.998909i	$0.485132\pi$
$594$	2.40821	0.0988099
$595$	−30.0194	−1.23067
$596$	19.3207	0.791405
$597$	−28.7952	−1.17851
$598$	0	0
$599$	32.3370	1.32126	0.660628	−	0.750714i	$-0.270290\pi$
$599$	32.3370	1.32126	0.660628	+	0.750714i	$0.270290\pi$
$600$	−6.08575	−0.248450
$601$	8.13062	0.331655	0.165827	−	0.986155i	$-0.446971\pi$
$601$	8.13062	0.331655	0.165827	+	0.986155i	$0.446971\pi$
$602$	−28.4999	−1.16157
$603$	29.8974	1.21752
$604$	−10.8334	−0.440805
$605$	8.90813	0.362167
$606$	27.0616	1.09930
$607$	−1.19913	−0.0486711	−0.0243355	−	0.999704i	$-0.507747\pi$
$607$	−1.19913	−0.0486711	−0.0243355	+	0.999704i	$0.507747\pi$
$608$	−1.00000	−0.0405554
$609$	13.4953	0.546857
$610$	−11.1400	−0.451047
$611$	0	0
$612$	−19.7778	−0.799469
$613$	−16.2145	−0.654896	−0.327448	−	0.944869i	$-0.606189\pi$
$613$	−16.2145	−0.654896	−0.327448	+	0.944869i	$0.606189\pi$
$614$	−16.5676	−0.668615
$615$	43.0291	1.73510
$616$	−5.72587	−0.230702
$617$	−36.8418	−1.48319	−0.741597	−	0.670846i	$-0.765931\pi$
$617$	−36.8418	−1.48319	−0.741597	+	0.670846i	$0.765931\pi$
$618$	22.3478	0.898960
$619$	5.24591	0.210851	0.105426	−	0.994427i	$-0.466380\pi$
$619$	5.24591	0.210851	0.105426	+	0.994427i	$0.466380\pi$
$620$	−1.46011	−0.0586393
$621$	0.0228352	0.000916345 0
$622$	−11.5211	−0.461954
$623$	−2.03013	−0.0813355
$624$	0	0
$625$	−5.42221	−0.216888
$626$	19.6528	0.785484
$627$	5.41119	0.216102
$628$	15.1836	0.605891
$629$	−63.4422	−2.52961
$630$	−9.90813	−0.394749
$631$	5.53617	0.220392	0.110196	−	0.993910i	$-0.464852\pi$
$631$	5.53617	0.220392	0.110196	+	0.993910i	$0.464852\pi$
$632$	−6.68963	−0.266099
$633$	59.0732	2.34795
$634$	−13.3840	−0.531548
$635$	12.0978	0.480088
$636$	−1.56704	−0.0621372
$637$	0	0
$638$	5.27114	0.208687
$639$	20.8146	0.823413
$640$	1.55496	0.0614651
$641$	−4.10859	−0.162279	−0.0811397	−	0.996703i	$-0.525856\pi$
$641$	−4.10859	−0.162279	−0.0811397	+	0.996703i	$0.525856\pi$
$642$	−5.86533	−0.231486
$643$	18.5439	0.731302	0.365651	−	0.930752i	$-0.380846\pi$
$643$	18.5439	0.731302	0.365651	+	0.930752i	$0.380846\pi$
$644$	−0.0542942	−0.00213949
$645$	−41.8807	−1.64905
$646$	7.74094	0.304563
$647$	−45.4282	−1.78597	−0.892983	−	0.450091i	$-0.851392\pi$
$647$	−45.4282	−1.78597	−0.892983	+	0.450091i	$0.851392\pi$
$648$	10.1371	0.398221
$649$	4.08708	0.160432
$650$	0	0
$651$	5.51945	0.216324
$652$	18.1129	0.709356
$653$	6.67025	0.261027	0.130514	−	0.991447i	$-0.458337\pi$
$653$	6.67025	0.261027	0.130514	+	0.991447i	$0.458337\pi$
$654$	−13.9565	−0.545741
$655$	−4.34183	−0.169649
$656$	11.7409	0.458407
$657$	−28.3236	−1.10501
$658$	5.23191	0.203961
$659$	2.46548	0.0960416	0.0480208	−	0.998846i	$-0.484709\pi$
$659$	2.46548	0.0960416	0.0480208	+	0.998846i	$0.484709\pi$
$660$	−8.41417	−0.327521
$661$	32.3236	1.25724	0.628622	−	0.777711i	$-0.283619\pi$
$661$	32.3236	1.25724	0.628622	+	0.777711i	$0.283619\pi$
$662$	23.8431	0.926688
$663$	0	0
$664$	−12.1371	−0.471009
$665$	3.87800	0.150382
$666$	−20.9396	−0.811393
$667$	0.0499823	0.00193532
$668$	10.8944	0.421517
$669$	46.0092	1.77882
$670$	18.1957	0.702960
$671$	16.4483	0.634979
$672$	−5.87800	−0.226749
$673$	37.8926	1.46065	0.730326	−	0.683099i	$-0.239368\pi$
$673$	37.8926	1.46065	0.730326	+	0.683099i	$0.239368\pi$
$674$	21.2814	0.819730
$675$	−2.70841	−0.104247
$676$	0	0
$677$	25.2597	0.970807	0.485404	−	0.874290i	$-0.338673\pi$
$677$	25.2597	0.970807	0.485404	+	0.874290i	$0.338673\pi$
$678$	28.4107	1.09111
$679$	31.3491	1.20307
$680$	−12.0368	−0.461592
$681$	33.4577	1.28210
$682$	2.15585	0.0825518
$683$	−51.3879	−1.96630	−0.983151	−	0.182794i	$-0.941486\pi$
$683$	−51.3879	−1.96630	−0.983151	+	0.182794i	$0.941486\pi$
$684$	2.55496	0.0976913
$685$	4.76809	0.182179
$686$	−19.4034	−0.740826
$687$	32.2879	1.23186
$688$	−11.4276	−0.435673
$689$	0	0
$690$	−0.0797853	−0.00303737
$691$	−31.0398	−1.18081	−0.590405	−	0.807107i	$-0.701032\pi$
$691$	−31.0398	−1.18081	−0.590405	+	0.807107i	$0.701032\pi$
$692$	−12.0043	−0.456335
$693$	14.6294	0.555724
$694$	32.4155	1.23048
$695$	−5.53617	−0.209999
$696$	5.41119	0.205111
$697$	−90.8859	−3.44255
$698$	−3.83877	−0.145300
$699$	32.9269	1.24541
$700$	6.43967	0.243396
$701$	−10.1849	−0.384679	−0.192339	−	0.981328i	$-0.561607\pi$
$701$	−10.1849	−0.384679	−0.192339	+	0.981328i	$0.561607\pi$
$702$	0	0
$703$	8.19567	0.309106
$704$	−2.29590	−0.0865299
$705$	7.68830	0.289558
$706$	13.7366	0.516985
$707$	−28.6353	−1.07694
$708$	4.19567	0.157683
$709$	29.8431	1.12078	0.560390	−	0.828229i	$-0.310651\pi$
$709$	29.8431	1.12078	0.560390	+	0.828229i	$0.310651\pi$
$710$	12.6679	0.475416
$711$	17.0917	0.640990
$712$	−0.814019	−0.0305067
$713$	0.0204423	0.000765570 0
$714$	45.5013	1.70284
$715$	0	0
$716$	5.59850	0.209226
$717$	−56.1500	−2.09696
$718$	−7.69096	−0.287024
$719$	−21.9758	−0.819560	−0.409780	−	0.912184i	$-0.634395\pi$
$719$	−21.9758	−0.819560	−0.409780	+	0.912184i	$0.634395\pi$
$720$	−3.97285	−0.148059
$721$	−23.6474	−0.880676
$722$	−1.00000	−0.0372161
$723$	−20.9403	−0.778779
$724$	10.4112	0.386929
$725$	−5.92825	−0.220170
$726$	−13.5023	−0.501118
$727$	34.7590	1.28914	0.644570	−	0.764546i	$-0.277037\pi$
$727$	34.7590	1.28914	0.644570	+	0.764546i	$0.277037\pi$
$728$	0	0
$729$	−18.4832	−0.684563
$730$	−17.2379	−0.638003
$731$	88.4602	3.27182
$732$	16.8853	0.624099
$733$	−1.86102	−0.0687383	−0.0343691	−	0.999409i	$-0.510942\pi$
$733$	−1.86102	−0.0687383	−0.0343691	+	0.999409i	$0.510942\pi$
$734$	25.2325	0.931349
$735$	−2.85922	−0.105464
$736$	−0.0217703	−0.000802463 0
$737$	−26.8659	−0.989619
$738$	−29.9976	−1.10423
$739$	−34.6872	−1.27599	−0.637995	−	0.770040i	$-0.720236\pi$
$739$	−34.6872	−1.27599	−0.637995	+	0.770040i	$0.720236\pi$
$740$	−12.7439	−0.468476
$741$	0	0
$742$	1.65817	0.0608733
$743$	−13.8774	−0.509113	−0.254556	−	0.967058i	$-0.581929\pi$
$743$	−13.8774	−0.509113	−0.254556	+	0.967058i	$0.581929\pi$
$744$	2.21313	0.0811372
$745$	−30.0428	−1.10068
$746$	8.87502	0.324938
$747$	31.0097	1.13459
$748$	17.7724	0.649823
$749$	6.20642	0.226778
$750$	27.7875	1.01465
$751$	−5.84522	−0.213295	−0.106647	−	0.994297i	$-0.534012\pi$
$751$	−5.84522	−0.213295	−0.106647	+	0.994297i	$0.534012\pi$
$752$	2.09783	0.0765002
$753$	−20.3526	−0.741689
$754$	0	0
$755$	16.8455	0.613070
$756$	−2.61596	−0.0951414
$757$	37.3599	1.35787	0.678934	−	0.734199i	$-0.262442\pi$
$757$	37.3599	1.35787	0.678934	+	0.734199i	$0.262442\pi$
$758$	−6.67025	−0.242274
$759$	0.117803	0.00427598
$760$	1.55496	0.0564043
$761$	−39.4228	−1.42908	−0.714538	−	0.699597i	$-0.753363\pi$
$761$	−39.4228	−1.42908	−0.714538	+	0.699597i	$0.753363\pi$
$762$	−18.3370	−0.664281
$763$	14.7681	0.534641
$764$	17.8291	0.645034
$765$	30.7536	1.11190
$766$	−6.02475	−0.217683
$767$	0	0
$768$	−2.35690	−0.0850472
$769$	17.5362	0.632371	0.316185	−	0.948697i	$-0.397598\pi$
$769$	17.5362	0.632371	0.316185	+	0.948697i	$0.397598\pi$
$770$	8.90349	0.320860
$771$	−69.1982	−2.49211
$772$	5.24698	0.188843
$773$	−5.26205	−0.189263	−0.0946313	−	0.995512i	$-0.530167\pi$
$773$	−5.26205	−0.189263	−0.0946313	+	0.995512i	$0.530167\pi$
$774$	29.1970	1.04946
$775$	−2.42460	−0.0870942
$776$	12.5700	0.451238
$777$	48.1742	1.72824
$778$	15.7888	0.566056
$779$	11.7409	0.420663
$780$	0	0
$781$	−18.7041	−0.669285
$782$	0.168522	0.00602634
$783$	2.40821	0.0860623
$784$	−0.780167	−0.0278631
$785$	−23.6098	−0.842671
$786$	6.58104	0.234738
$787$	−24.2285	−0.863651	−0.431826	−	0.901957i	$-0.642130\pi$
$787$	−24.2285	−0.863651	−0.431826	+	0.901957i	$0.642130\pi$
$788$	−15.4276	−0.549585
$789$	47.0455	1.67486
$790$	10.4021	0.370090
$791$	−30.0629	−1.06891
$792$	5.86592	0.208436
$793$	0	0
$794$	−1.55496	−0.0551834
$795$	2.43668	0.0864202
$796$	12.2174	0.433036
$797$	−42.8829	−1.51899	−0.759495	−	0.650513i	$-0.774554\pi$
$797$	−42.8829	−1.51899	−0.759495	+	0.650513i	$0.774554\pi$
$798$	−5.87800	−0.208079
$799$	−16.2392	−0.574502
$800$	2.58211	0.0912912
$801$	2.07979	0.0734856
$802$	7.21313	0.254704
$803$	25.4517	0.898173
$804$	−27.5797	−0.972661
$805$	0.0844251	0.00297559
$806$	0	0
$807$	−22.7380	−0.800414
$808$	−11.4819	−0.403931
$809$	−48.8044	−1.71587	−0.857937	−	0.513756i	$-0.828254\pi$
$809$	−48.8044	−1.71587	−0.857937	+	0.513756i	$0.828254\pi$
$810$	−15.7627	−0.553845
$811$	−3.47591	−0.122056	−0.0610279	−	0.998136i	$-0.519438\pi$
$811$	−3.47591	−0.122056	−0.0610279	+	0.998136i	$0.519438\pi$
$812$	−5.72587	−0.200939
$813$	−59.8275	−2.09824
$814$	18.8164	0.659515
$815$	−28.1648	−0.986570
$816$	18.2446	0.638688
$817$	−11.4276	−0.399801
$818$	−15.1836	−0.530882
$819$	0	0
$820$	−18.2567	−0.637551
$821$	−40.0844	−1.39896	−0.699478	−	0.714654i	$-0.746584\pi$
$821$	−40.0844	−1.39896	−0.699478	+	0.714654i	$0.746584\pi$
$822$	−7.22713	−0.252075
$823$	52.7174	1.83761	0.918806	−	0.394710i	$-0.129155\pi$
$823$	52.7174	1.83761	0.918806	+	0.394710i	$0.129155\pi$
$824$	−9.48188	−0.330317
$825$	−13.9723	−0.486452
$826$	−4.43967	−0.154476
$827$	14.4504	0.502490	0.251245	−	0.967923i	$-0.419160\pi$
$827$	14.4504	0.502490	0.251245	+	0.967923i	$0.419160\pi$
$828$	0.0556221	0.00193300
$829$	27.2760	0.947336	0.473668	−	0.880703i	$-0.342930\pi$
$829$	27.2760	0.947336	0.473668	+	0.880703i	$0.342930\pi$
$830$	18.8726	0.655078
$831$	55.9651	1.94141
$832$	0	0
$833$	6.03923	0.209247
$834$	8.39134	0.290568
$835$	−16.9403	−0.586244
$836$	−2.29590	−0.0794053
$837$	0.984935	0.0340443
$838$	5.16421	0.178395
$839$	43.0243	1.48536	0.742682	−	0.669645i	$-0.233554\pi$
$839$	43.0243	1.48536	0.742682	+	0.669645i	$0.233554\pi$
$840$	9.14005	0.315362
$841$	−23.7289	−0.818236
$842$	17.2271	0.593686
$843$	37.2892	1.28431
$844$	−25.0640	−0.862738
$845$	0	0
$846$	−5.35988	−0.184276
$847$	14.2875	0.490926
$848$	0.664874	0.0228319
$849$	37.7813	1.29665
$850$	−19.9879	−0.685580
$851$	0.178422	0.00611622
$852$	−19.2010	−0.657817
$853$	−22.8810	−0.783430	−0.391715	−	0.920087i	$-0.628118\pi$
$853$	−22.8810	−0.783430	−0.391715	+	0.920087i	$0.628118\pi$
$854$	−17.8672	−0.611405
$855$	−3.97285	−0.135869
$856$	2.48858	0.0850580
$857$	−0.450419	−0.0153860	−0.00769300	−	0.999970i	$-0.502449\pi$
$857$	−0.450419	−0.0153860	−0.00769300	+	0.999970i	$0.502449\pi$
$858$	0	0
$859$	−51.0482	−1.74174	−0.870871	−	0.491512i	$-0.836444\pi$
$859$	−51.0482	−1.74174	−0.870871	+	0.491512i	$0.836444\pi$
$860$	17.7694	0.605932
$861$	69.0133	2.35197
$862$	−29.3884	−1.00097
$863$	35.9084	1.22234	0.611168	−	0.791501i	$-0.290700\pi$
$863$	35.9084	1.22234	0.611168	+	0.791501i	$0.290700\pi$
$864$	−1.04892	−0.0356849
$865$	18.6662	0.634670
$866$	−0.987918	−0.0335708
$867$	−101.163	−3.43568
$868$	−2.34183	−0.0794869
$869$	−15.3587	−0.521008
$870$	−8.41417	−0.285267
$871$	0	0
$872$	5.92154	0.200529
$873$	−32.1159	−1.08696
$874$	−0.0217703	−0.000736390 0
$875$	−29.4034	−0.994017
$876$	26.1280	0.882782
$877$	32.4784	1.09672	0.548359	−	0.836243i	$-0.315253\pi$
$877$	32.4784	1.09672	0.548359	+	0.836243i	$0.315253\pi$
$878$	−10.2064	−0.344450
$879$	34.1608	1.15221
$880$	3.57002	0.120345
$881$	3.39804	0.114483	0.0572415	−	0.998360i	$-0.481770\pi$
$881$	3.39804	0.114483	0.0572415	+	0.998360i	$0.481770\pi$
$882$	1.99330	0.0671177
$883$	−5.81295	−0.195621	−0.0978107	−	0.995205i	$-0.531184\pi$
$883$	−5.81295	−0.195621	−0.0978107	+	0.995205i	$0.531184\pi$
$884$	0	0
$885$	−6.52409	−0.219305
$886$	−1.50604	−0.0505964
$887$	−17.7668	−0.596549	−0.298275	−	0.954480i	$-0.596411\pi$
$887$	−17.7668	−0.596549	−0.298275	+	0.954480i	$0.596411\pi$
$888$	19.3163	0.648214
$889$	19.4034	0.650770
$890$	1.26577	0.0424286
$891$	23.2737	0.779697
$892$	−19.5211	−0.653615
$893$	2.09783	0.0702014
$894$	45.5368	1.52298
$895$	−8.70543	−0.290990
$896$	2.49396	0.0833173
$897$	0	0
$898$	7.46681	0.249171
$899$	2.15585	0.0719016
$900$	−6.59717	−0.219906
$901$	−5.14675	−0.171463
$902$	26.9560	0.897536
$903$	−67.1714	−2.23532
$904$	−12.0543	−0.400920
$905$	−16.1890	−0.538139
$906$	−25.5332	−0.848283
$907$	4.76749	0.158302	0.0791510	−	0.996863i	$-0.474779\pi$
$907$	4.76749	0.158302	0.0791510	+	0.996863i	$0.474779\pi$
$908$	−14.1957	−0.471100
$909$	29.3357	0.973004
$910$	0	0
$911$	27.8780	0.923639	0.461820	−	0.886974i	$-0.347197\pi$
$911$	27.8780	0.923639	0.461820	+	0.886974i	$0.347197\pi$
$912$	−2.35690	−0.0780446
$913$	−27.8654	−0.922212
$914$	−12.9530	−0.428447
$915$	−26.2559	−0.867994
$916$	−13.6993	−0.452638
$917$	−6.96376	−0.229963
$918$	8.11960	0.267987
$919$	−35.0331	−1.15564	−0.577818	−	0.816166i	$-0.696096\pi$
$919$	−35.0331	−1.15564	−0.577818	+	0.816166i	$0.696096\pi$
$920$	0.0338518	0.00111606
$921$	−39.0482	−1.28668
$922$	0.559270	0.0184186
$923$	0	0
$924$	−13.4953	−0.443962
$925$	−21.1621	−0.695805
$926$	−29.5254	−0.970265
$927$	24.2258	0.795680
$928$	−2.29590	−0.0753665
$929$	20.7660	0.681309	0.340654	−	0.940189i	$-0.389351\pi$
$929$	20.7660	0.681309	0.340654	+	0.940189i	$0.389351\pi$
$930$	−3.44132	−0.112845
$931$	−0.780167	−0.0255690
$932$	−13.9705	−0.457618
$933$	−27.1540	−0.888984
$934$	9.23191	0.302077
$935$	−27.6353	−0.903772
$936$	0	0
$937$	32.6795	1.06759	0.533796	−	0.845613i	$-0.320765\pi$
$937$	32.6795	1.06759	0.533796	+	0.845613i	$0.320765\pi$
$938$	29.1836	0.952878
$939$	46.3196	1.51158
$940$	−3.26205	−0.106396
$941$	39.1159	1.27514	0.637571	−	0.770392i	$-0.279939\pi$
$941$	39.1159	1.27514	0.637571	+	0.770392i	$0.279939\pi$
$942$	35.7861	1.16598
$943$	0.255603	0.00832359
$944$	−1.78017	−0.0579395
$945$	4.06770	0.132322
$946$	−26.2366	−0.853024
$947$	54.9778	1.78654	0.893269	−	0.449523i	$-0.148406\pi$
$947$	54.9778	1.78654	0.893269	+	0.449523i	$0.148406\pi$
$948$	−15.7668	−0.512080
$949$	0	0
$950$	2.58211	0.0837746
$951$	−31.5448	−1.02291
$952$	−19.3056	−0.625698
$953$	−55.4663	−1.79673	−0.898365	−	0.439249i	$-0.855245\pi$
$953$	−55.4663	−1.79673	−0.898365	+	0.439249i	$0.855245\pi$
$954$	−1.69873	−0.0549983
$955$	−27.7235	−0.897111
$956$	23.8237	0.770514
$957$	12.4235	0.401596
$958$	16.3370	0.527826
$959$	7.64742	0.246948
$960$	3.66487	0.118283
$961$	−30.1183	−0.971557
$962$	0	0
$963$	−6.35822	−0.204891
$964$	8.88471	0.286157
$965$	−8.15883	−0.262642
$966$	−0.127966	−0.00411723
$967$	−54.0253	−1.73734	−0.868669	−	0.495393i	$-0.835024\pi$
$967$	−54.0253	−1.73734	−0.868669	+	0.495393i	$0.835024\pi$
$968$	5.72886	0.184132
$969$	18.2446	0.586101
$970$	−19.5459	−0.627580
$971$	14.2446	0.457131	0.228565	−	0.973529i	$-0.426597\pi$
$971$	14.2446	0.457131	0.228565	+	0.973529i	$0.426597\pi$
$972$	20.7453	0.665404
$973$	−8.87933	−0.284658
$974$	−14.8334	−0.475293
$975$	0	0
$976$	−7.16421	−0.229321
$977$	−22.0847	−0.706552	−0.353276	−	0.935519i	$-0.614932\pi$
$977$	−22.0847	−0.706552	−0.353276	+	0.935519i	$0.614932\pi$
$978$	42.6902	1.36508
$979$	−1.86890	−0.0597304
$980$	1.21313	0.0387519
$981$	−15.1293	−0.483041
$982$	−15.3793	−0.490772
$983$	32.6262	1.04062	0.520308	−	0.853979i	$-0.325817\pi$
$983$	32.6262	1.04062	0.520308	+	0.853979i	$0.325817\pi$
$984$	27.6722	0.882157
$985$	23.9892	0.764361
$986$	17.7724	0.565988
$987$	12.3311	0.392502
$988$	0	0
$989$	−0.248782	−0.00791079
$990$	−9.12126	−0.289893
$991$	−1.71512	−0.0544826	−0.0272413	−	0.999629i	$-0.508672\pi$
$991$	−1.71512	−0.0544826	−0.0272413	+	0.999629i	$0.508672\pi$
$992$	−0.939001	−0.0298133
$993$	56.1957	1.78332
$994$	20.3177	0.644437
$995$	−18.9976	−0.602265
$996$	−28.6058	−0.906409
$997$	−53.9797	−1.70955	−0.854777	−	0.518996i	$-0.826306\pi$
$997$	−53.9797	−1.70955	−0.854777	+	0.518996i	$0.826306\pi$
$998$	40.2543	1.27423
$999$	8.59658	0.271984

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.k.1.2		3
13.5	odd	4		494.2.d.b.77.5	yes	6
13.8	odd	4		494.2.d.b.77.2	✓	6
13.12	even	2		6422.2.a.t.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.d.b.77.2	✓	6	13.8	odd	4
494.2.d.b.77.5	yes	6	13.5	odd	4
6422.2.a.k.1.2		3	1.1	even	1	trivial
6422.2.a.t.1.2		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.35690 q^{3} +1.00000 q^{4} -1.55496 q^{5} +2.35690 q^{6} -2.49396 q^{7} -1.00000 q^{8} +2.55496 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.35690 q^{3} +1.00000 q^{4} -1.55496 q^{5} +2.35690 q^{6} -2.49396 q^{7} -1.00000 q^{8} +2.55496 q^{9} +1.55496 q^{10} -2.29590 q^{11} -2.35690 q^{12} +2.49396 q^{14} +3.66487 q^{15} +1.00000 q^{16} -7.74094 q^{17} -2.55496 q^{18} +1.00000 q^{19} -1.55496 q^{20} +5.87800 q^{21} +2.29590 q^{22} +0.0217703 q^{23} +2.35690 q^{24} -2.58211 q^{25} +1.04892 q^{27} -2.49396 q^{28} +2.29590 q^{29} -3.66487 q^{30} +0.939001 q^{31} -1.00000 q^{32} +5.41119 q^{33} +7.74094 q^{34} +3.87800 q^{35} +2.55496 q^{36} +8.19567 q^{37} -1.00000 q^{38} +1.55496 q^{40} +11.7409 q^{41} -5.87800 q^{42} -11.4276 q^{43} -2.29590 q^{44} -3.97285 q^{45} -0.0217703 q^{46} +2.09783 q^{47} -2.35690 q^{48} -0.780167 q^{49} +2.58211 q^{50} +18.2446 q^{51} +0.664874 q^{53} -1.04892 q^{54} +3.57002 q^{55} +2.49396 q^{56} -2.35690 q^{57} -2.29590 q^{58} -1.78017 q^{59} +3.66487 q^{60} -7.16421 q^{61} -0.939001 q^{62} -6.37196 q^{63} +1.00000 q^{64} -5.41119 q^{66} +11.7017 q^{67} -7.74094 q^{68} -0.0513102 q^{69} -3.87800 q^{70} +8.14675 q^{71} -2.55496 q^{72} -11.0858 q^{73} -8.19567 q^{74} +6.08575 q^{75} +1.00000 q^{76} +5.72587 q^{77} +6.68963 q^{79} -1.55496 q^{80} -10.1371 q^{81} -11.7409 q^{82} +12.1371 q^{83} +5.87800 q^{84} +12.0368 q^{85} +11.4276 q^{86} -5.41119 q^{87} +2.29590 q^{88} +0.814019 q^{89} +3.97285 q^{90} +0.0217703 q^{92} -2.21313 q^{93} -2.09783 q^{94} -1.55496 q^{95} +2.35690 q^{96} -12.5700 q^{97} +0.780167 q^{98} -5.86592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 5 q^{5} + 3 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 5 * q^5 + 3 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 5 q^{5} + 3 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} + 5 q^{10} + 7 q^{11} - 3 q^{12} - 2 q^{14} + 12 q^{15} + 3 q^{16} - 9 q^{17} - 8 q^{18} + 3 q^{19} - 5 q^{20} - 2 q^{21} - 7 q^{22} - 3 q^{23} + 3 q^{24} - 2 q^{25} - 6 q^{27} + 2 q^{28} - 7 q^{29} - 12 q^{30} - 7 q^{31} - 3 q^{32} + 9 q^{34} - 8 q^{35} + 8 q^{36} - 12 q^{37} - 3 q^{38} + 5 q^{40} + 21 q^{41} + 2 q^{42} - 18 q^{43} + 7 q^{44} - 18 q^{45} + 3 q^{46} - 12 q^{47} - 3 q^{48} - q^{49} + 2 q^{50} + 9 q^{51} + 3 q^{53} + 6 q^{54} - 14 q^{55} - 2 q^{56} - 3 q^{57} + 7 q^{58} - 4 q^{59} + 12 q^{60} - 10 q^{61} + 7 q^{62} + 10 q^{63} + 3 q^{64} + 8 q^{67} - 9 q^{68} - 32 q^{69} + 8 q^{70} - 3 q^{71} - 8 q^{72} + 4 q^{73} + 12 q^{74} - 19 q^{75} + 3 q^{76} + 28 q^{77} - 26 q^{79} - 5 q^{80} - 25 q^{81} - 21 q^{82} + 31 q^{83} - 2 q^{84} + 8 q^{85} + 18 q^{86} - 7 q^{88} + 17 q^{89} + 18 q^{90} - 3 q^{92} + 14 q^{93} + 12 q^{94} - 5 q^{95} + 3 q^{96} - 13 q^{97} + q^{98} + 21 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 5 * q^5 + 3 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9 + 5 * q^10 + 7 * q^11 - 3 * q^12 - 2 * q^14 + 12 * q^15 + 3 * q^16 - 9 * q^17 - 8 * q^18 + 3 * q^19 - 5 * q^20 - 2 * q^21 - 7 * q^22 - 3 * q^23 + 3 * q^24 - 2 * q^25 - 6 * q^27 + 2 * q^28 - 7 * q^29 - 12 * q^30 - 7 * q^31 - 3 * q^32 + 9 * q^34 - 8 * q^35 + 8 * q^36 - 12 * q^37 - 3 * q^38 + 5 * q^40 + 21 * q^41 + 2 * q^42 - 18 * q^43 + 7 * q^44 - 18 * q^45 + 3 * q^46 - 12 * q^47 - 3 * q^48 - q^49 + 2 * q^50 + 9 * q^51 + 3 * q^53 + 6 * q^54 - 14 * q^55 - 2 * q^56 - 3 * q^57 + 7 * q^58 - 4 * q^59 + 12 * q^60 - 10 * q^61 + 7 * q^62 + 10 * q^63 + 3 * q^64 + 8 * q^67 - 9 * q^68 - 32 * q^69 + 8 * q^70 - 3 * q^71 - 8 * q^72 + 4 * q^73 + 12 * q^74 - 19 * q^75 + 3 * q^76 + 28 * q^77 - 26 * q^79 - 5 * q^80 - 25 * q^81 - 21 * q^82 + 31 * q^83 - 2 * q^84 + 8 * q^85 + 18 * q^86 - 7 * q^88 + 17 * q^89 + 18 * q^90 - 3 * q^92 + 14 * q^93 + 12 * q^94 - 5 * q^95 + 3 * q^96 - 13 * q^97 + q^98 + 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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