LMFDB - Embedded newform 6422.2.a.k.1.1

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			1.1
Root			$-1.24698$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.69202 q^{3} +1.00000 q^{4} -3.24698 q^{5} +2.69202 q^{6} +3.60388 q^{7} -1.00000 q^{8} +4.24698 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.69202 q^{3} +1.00000 q^{4} -3.24698 q^{5} +2.69202 q^{6} +3.60388 q^{7} -1.00000 q^{8} +4.24698 q^{9} +3.24698 q^{10} +5.15883 q^{11} -2.69202 q^{12} -3.60388 q^{14} +8.74094 q^{15} +1.00000 q^{16} +1.40581 q^{17} -4.24698 q^{18} +1.00000 q^{19} -3.24698 q^{20} -9.70171 q^{21} -5.15883 q^{22} +5.43296 q^{23} +2.69202 q^{24} +5.54288 q^{25} -3.35690 q^{27} +3.60388 q^{28} -5.15883 q^{29} -8.74094 q^{30} -6.85086 q^{31} -1.00000 q^{32} -13.8877 q^{33} -1.40581 q^{34} -11.7017 q^{35} +4.24698 q^{36} -9.42758 q^{37} -1.00000 q^{38} +3.24698 q^{40} +2.59419 q^{41} +9.70171 q^{42} -12.7681 q^{43} +5.15883 q^{44} -13.7899 q^{45} -5.43296 q^{46} -6.71379 q^{47} -2.69202 q^{48} +5.98792 q^{49} -5.54288 q^{50} -3.78448 q^{51} +5.74094 q^{53} +3.35690 q^{54} -16.7506 q^{55} -3.60388 q^{56} -2.69202 q^{57} +5.15883 q^{58} +4.98792 q^{59} +8.74094 q^{60} +9.08575 q^{61} +6.85086 q^{62} +15.3056 q^{63} +1.00000 q^{64} +13.8877 q^{66} +0.176292 q^{67} +1.40581 q^{68} -14.6256 q^{69} +11.7017 q^{70} -5.07069 q^{71} -4.24698 q^{72} +9.92154 q^{73} +9.42758 q^{74} -14.9215 q^{75} +1.00000 q^{76} +18.5918 q^{77} -17.0315 q^{79} -3.24698 q^{80} -3.70410 q^{81} -2.59419 q^{82} +5.70410 q^{83} -9.70171 q^{84} -4.56465 q^{85} +12.7681 q^{86} +13.8877 q^{87} -5.15883 q^{88} +11.6528 q^{89} +13.7899 q^{90} +5.43296 q^{92} +18.4426 q^{93} +6.71379 q^{94} -3.24698 q^{95} +2.69202 q^{96} +7.75063 q^{97} -5.98792 q^{98} +21.9095 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.69202	−1.55424	−0.777120	−	0.629353i	$-0.783320\pi$
$3$	−2.69202	−1.55424	−0.777120	+	0.629353i	$0.783320\pi$
$4$	1.00000	0.500000
$5$	−3.24698	−1.45209	−0.726047	−	0.687645i	$-0.758644\pi$
$5$	−3.24698	−1.45209	−0.726047	+	0.687645i	$0.758644\pi$
$6$	2.69202	1.09901
$7$	3.60388	1.36214	0.681068	−	0.732220i	$-0.261516\pi$
$7$	3.60388	1.36214	0.681068	+	0.732220i	$0.261516\pi$
$8$	−1.00000	−0.353553
$9$	4.24698	1.41566
$10$	3.24698	1.02679
$11$	5.15883	1.55545	0.777723	−	0.628607i	$-0.216374\pi$
$11$	5.15883	1.55545	0.777723	+	0.628607i	$0.216374\pi$
$12$	−2.69202	−0.777120
$13$	0	0
$13$	0	0
$14$	−3.60388	−0.963176
$15$	8.74094	2.25690
$16$	1.00000	0.250000
$17$	1.40581	0.340960	0.170480	−	0.985361i	$-0.445468\pi$
$17$	1.40581	0.340960	0.170480	+	0.985361i	$0.445468\pi$
$18$	−4.24698	−1.00102
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.24698	−0.726047
$21$	−9.70171	−2.11709
$22$	−5.15883	−1.09987
$23$	5.43296	1.13285	0.566425	−	0.824113i	$-0.308326\pi$
$23$	5.43296	1.13285	0.566425	+	0.824113i	$0.308326\pi$
$24$	2.69202	0.549507
$25$	5.54288	1.10858
$26$	0	0
$27$	−3.35690	−0.646035
$28$	3.60388	0.681068
$29$	−5.15883	−0.957971	−0.478986	−	0.877823i	$-0.658995\pi$
$29$	−5.15883	−0.957971	−0.478986	+	0.877823i	$0.658995\pi$
$30$	−8.74094	−1.59587
$31$	−6.85086	−1.23045	−0.615225	−	0.788352i	$-0.710935\pi$
$31$	−6.85086	−1.23045	−0.615225	+	0.788352i	$0.710935\pi$
$32$	−1.00000	−0.176777
$33$	−13.8877	−2.41754
$34$	−1.40581	−0.241095
$35$	−11.7017	−1.97795
$36$	4.24698	0.707830
$37$	−9.42758	−1.54989	−0.774943	−	0.632032i	$-0.782221\pi$
$37$	−9.42758	−1.54989	−0.774943	+	0.632032i	$0.782221\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	3.24698	0.513393
$41$	2.59419	0.405144	0.202572	−	0.979267i	$-0.435070\pi$
$41$	2.59419	0.405144	0.202572	+	0.979267i	$0.435070\pi$
$42$	9.70171	1.49701
$43$	−12.7681	−1.94711	−0.973557	−	0.228442i	$-0.926637\pi$
$43$	−12.7681	−1.94711	−0.973557	+	0.228442i	$0.926637\pi$
$44$	5.15883	0.777723
$45$	−13.7899	−2.05567
$46$	−5.43296	−0.801046
$47$	−6.71379	−0.979307	−0.489654	−	0.871917i	$-0.662877\pi$
$47$	−6.71379	−0.979307	−0.489654	+	0.871917i	$0.662877\pi$
$48$	−2.69202	−0.388560
$49$	5.98792	0.855417
$50$	−5.54288	−0.783881
$51$	−3.78448	−0.529933
$52$	0	0
$53$	5.74094	0.788579	0.394289	−	0.918986i	$-0.370991\pi$
$53$	5.74094	0.788579	0.394289	+	0.918986i	$0.370991\pi$
$54$	3.35690	0.456816
$55$	−16.7506	−2.25865
$56$	−3.60388	−0.481588
$57$	−2.69202	−0.356567
$58$	5.15883	0.677388
$59$	4.98792	0.649372	0.324686	−	0.945822i	$-0.394741\pi$
$59$	4.98792	0.649372	0.324686	+	0.945822i	$0.394741\pi$
$60$	8.74094	1.12845
$61$	9.08575	1.16331	0.581656	−	0.813435i	$-0.302405\pi$
$61$	9.08575	1.16331	0.581656	+	0.813435i	$0.302405\pi$
$62$	6.85086	0.870059
$63$	15.3056	1.92832
$64$	1.00000	0.125000
$65$	0	0
$66$	13.8877	1.70946
$67$	0.176292	0.0215375	0.0107687	−	0.999942i	$-0.496572\pi$
$67$	0.176292	0.0215375	0.0107687	+	0.999942i	$0.496572\pi$
$68$	1.40581	0.170480
$69$	−14.6256	−1.76072
$70$	11.7017	1.39862
$71$	−5.07069	−0.601780	−0.300890	−	0.953659i	$-0.597284\pi$
$71$	−5.07069	−0.601780	−0.300890	+	0.953659i	$0.597284\pi$
$72$	−4.24698	−0.500511
$73$	9.92154	1.16123	0.580614	−	0.814179i	$-0.302812\pi$
$73$	9.92154	1.16123	0.580614	+	0.814179i	$0.302812\pi$
$74$	9.42758	1.09593
$75$	−14.9215	−1.72299
$76$	1.00000	0.114708
$77$	18.5918	2.11873
$78$	0	0
$79$	−17.0315	−1.91619	−0.958094	−	0.286453i	$-0.907524\pi$
$79$	−17.0315	−1.91619	−0.958094	+	0.286453i	$0.907524\pi$
$80$	−3.24698	−0.363023
$81$	−3.70410	−0.411567
$82$	−2.59419	−0.286480
$83$	5.70410	0.626107	0.313053	−	0.949736i	$-0.398648\pi$
$83$	5.70410	0.626107	0.313053	+	0.949736i	$0.398648\pi$
$84$	−9.70171	−1.05854
$85$	−4.56465	−0.495105
$86$	12.7681	1.37682
$87$	13.8877	1.48892
$88$	−5.15883	−0.549934
$89$	11.6528	1.23519	0.617597	−	0.786495i	$-0.288106\pi$
$89$	11.6528	1.23519	0.617597	+	0.786495i	$0.288106\pi$
$90$	13.7899	1.45358
$91$	0	0
$92$	5.43296	0.566425
$93$	18.4426	1.91241
$94$	6.71379	0.692475
$95$	−3.24698	−0.333133
$96$	2.69202	0.274753
$97$	7.75063	0.786957	0.393478	−	0.919334i	$-0.371272\pi$
$97$	7.75063	0.786957	0.393478	+	0.919334i	$0.371272\pi$
$98$	−5.98792	−0.604871
$99$	21.9095	2.20198
$100$	5.54288	0.554288
$101$	−6.81163	−0.677782	−0.338891	−	0.940826i	$-0.610052\pi$
$101$	−6.81163	−0.677782	−0.338891	+	0.940826i	$0.610052\pi$
$102$	3.78448	0.374719
$103$	−8.81163	−0.868235	−0.434118	−	0.900856i	$-0.642940\pi$
$103$	−8.81163	−0.868235	−0.434118	+	0.900856i	$0.642940\pi$
$104$	0	0
$105$	31.5013	3.07421
$106$	−5.74094	−0.557609
$107$	−11.6189	−1.12325	−0.561623	−	0.827393i	$-0.689823\pi$
$107$	−11.6189	−1.12325	−0.561623	+	0.827393i	$0.689823\pi$
$108$	−3.35690	−0.323017
$109$	−1.16421	−0.111511	−0.0557556	−	0.998444i	$-0.517757\pi$
$109$	−1.16421	−0.111511	−0.0557556	+	0.998444i	$0.517757\pi$
$110$	16.7506	1.59711
$111$	25.3793	2.40889
$112$	3.60388	0.340534
$113$	−7.57971	−0.713039	−0.356520	−	0.934288i	$-0.616037\pi$
$113$	−7.57971	−0.713039	−0.356520	+	0.934288i	$0.616037\pi$
$114$	2.69202	0.252131
$115$	−17.6407	−1.64500
$116$	−5.15883	−0.478986
$117$	0	0
$118$	−4.98792	−0.459175
$119$	5.06638	0.464434
$120$	−8.74094	−0.797935
$121$	15.6136	1.41941
$122$	−9.08575	−0.822585
$123$	−6.98361	−0.629691
$124$	−6.85086	−0.615225
$125$	−1.76271	−0.157661
$126$	−15.3056	−1.36353
$127$	−1.01208	−0.0898077	−0.0449039	−	0.998991i	$-0.514298\pi$
$127$	−1.01208	−0.0898077	−0.0449039	+	0.998991i	$0.514298\pi$
$128$	−1.00000	−0.0883883
$129$	34.3720	3.02628
$130$	0	0
$131$	8.21983	0.718170	0.359085	−	0.933305i	$-0.383089\pi$
$131$	8.21983	0.718170	0.359085	+	0.933305i	$0.383089\pi$
$132$	−13.8877	−1.20877
$133$	3.60388	0.312496
$134$	−0.176292	−0.0152293
$135$	10.8998	0.938103
$136$	−1.40581	−0.120547
$137$	4.37196	0.373522	0.186761	−	0.982405i	$-0.440201\pi$
$137$	4.37196	0.373522	0.186761	+	0.982405i	$0.440201\pi$
$138$	14.6256	1.24502
$139$	−9.97584	−0.846139	−0.423070	−	0.906097i	$-0.639047\pi$
$139$	−9.97584	−0.846139	−0.423070	+	0.906097i	$0.639047\pi$
$140$	−11.7017	−0.988975
$141$	18.0737	1.52208
$142$	5.07069	0.425523
$143$	0	0
$144$	4.24698	0.353915
$145$	16.7506	1.39106
$146$	−9.92154	−0.821113
$147$	−16.1196	−1.32952
$148$	−9.42758	−0.774943
$149$	−16.9312	−1.38706	−0.693530	−	0.720427i	$-0.743946\pi$
$149$	−16.9312	−1.38706	−0.693530	+	0.720427i	$0.743946\pi$
$150$	14.9215	1.21834
$151$	−8.10321	−0.659430	−0.329715	−	0.944081i	$-0.606953\pi$
$151$	−8.10321	−0.659430	−0.329715	+	0.944081i	$0.606953\pi$
$152$	−1.00000	−0.0811107
$153$	5.97046	0.482683
$154$	−18.5918	−1.49817
$155$	22.2446	1.78673
$156$	0	0
$157$	−14.6353	−1.16803	−0.584013	−	0.811744i	$-0.698518\pi$
$157$	−14.6353	−1.16803	−0.584013	+	0.811744i	$0.698518\pi$
$158$	17.0315	1.35495
$159$	−15.4547	−1.22564
$160$	3.24698	0.256696
$161$	19.5797	1.54310
$162$	3.70410	0.291022
$163$	−12.7114	−0.995634	−0.497817	−	0.867282i	$-0.665865\pi$
$163$	−12.7114	−0.995634	−0.497817	+	0.867282i	$0.665865\pi$
$164$	2.59419	0.202572
$165$	45.0930	3.51049
$166$	−5.70410	−0.442724
$167$	15.9541	1.23456	0.617281	−	0.786742i	$-0.288234\pi$
$167$	15.9541	1.23456	0.617281	+	0.786742i	$0.288234\pi$
$168$	9.70171	0.748503
$169$	0	0
$170$	4.56465	0.350092
$171$	4.24698	0.324775
$172$	−12.7681	−0.973557
$173$	−20.4480	−1.55463	−0.777317	−	0.629109i	$-0.783420\pi$
$173$	−20.4480	−1.55463	−0.777317	+	0.629109i	$0.783420\pi$
$174$	−13.8877	−1.05282
$175$	19.9758	1.51003
$176$	5.15883	0.388862
$177$	−13.4276	−1.00928
$178$	−11.6528	−0.873414
$179$	18.1129	1.35382	0.676911	−	0.736065i	$-0.263318\pi$
$179$	18.1129	1.35382	0.676911	+	0.736065i	$0.263318\pi$
$180$	−13.7899	−1.02784
$181$	−8.88769	−0.660617	−0.330308	−	0.943873i	$-0.607153\pi$
$181$	−8.88769	−0.660617	−0.330308	+	0.943873i	$0.607153\pi$
$182$	0	0
$183$	−24.4590	−1.80806
$184$	−5.43296	−0.400523
$185$	30.6112	2.25058
$186$	−18.4426	−1.35228
$187$	7.25236	0.530345
$188$	−6.71379	−0.489654
$189$	−12.0978	−0.879988
$190$	3.24698	0.235561
$191$	6.65519	0.481552	0.240776	−	0.970581i	$-0.422598\pi$
$191$	6.65519	0.481552	0.240776	+	0.970581i	$0.422598\pi$
$192$	−2.69202	−0.194280
$193$	2.19806	0.158220	0.0791100	−	0.996866i	$-0.474792\pi$
$193$	2.19806	0.158220	0.0791100	+	0.996866i	$0.474792\pi$
$194$	−7.75063	−0.556463
$195$	0	0
$196$	5.98792	0.427708
$197$	−16.7681	−1.19468	−0.597338	−	0.801989i	$-0.703775\pi$
$197$	−16.7681	−1.19468	−0.597338	+	0.801989i	$0.703775\pi$
$198$	−21.9095	−1.55704
$199$	0.00537681	0.000381152 0	0.000190576	−	1.00000i	$-0.499939\pi$
$199$	0.00537681	0.000381152 0	0.000190576		1.00000i	$0.499939\pi$
$200$	−5.54288	−0.391941
$201$	−0.474582	−0.0334744
$202$	6.81163	0.479264
$203$	−18.5918	−1.30489
$204$	−3.78448	−0.264967
$205$	−8.42327	−0.588307
$206$	8.81163	0.613935
$207$	23.0737	1.60373
$208$	0	0
$209$	5.15883	0.356844
$210$	−31.5013	−2.17379
$211$	1.35450	0.0932478	0.0466239	−	0.998913i	$-0.485154\pi$
$211$	1.35450	0.0932478	0.0466239	+	0.998913i	$0.485154\pi$
$212$	5.74094	0.394289
$213$	13.6504	0.935310
$214$	11.6189	0.794254
$215$	41.4577	2.82739
$216$	3.35690	0.228408
$217$	−24.6896	−1.67604
$218$	1.16421	0.0788503
$219$	−26.7090	−1.80483
$220$	−16.7506	−1.12933
$221$	0	0
$222$	−25.3793	−1.70334
$223$	−3.60627	−0.241494	−0.120747	−	0.992683i	$-0.538529\pi$
$223$	−3.60627	−0.241494	−0.120747	+	0.992683i	$0.538529\pi$
$224$	−3.60388	−0.240794
$225$	23.5405	1.56937
$226$	7.57971	0.504195
$227$	3.42758	0.227497	0.113748	−	0.993510i	$-0.463714\pi$
$227$	3.42758	0.227497	0.113748	+	0.993510i	$0.463714\pi$
$228$	−2.69202	−0.178283
$229$	16.8062	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$229$	16.8062	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$230$	17.6407	1.16319
$231$	−50.0495	−3.29302
$232$	5.15883	0.338694
$233$	−4.80731	−0.314938	−0.157469	−	0.987524i	$-0.550333\pi$
$233$	−4.80731	−0.314938	−0.157469	+	0.987524i	$0.550333\pi$
$234$	0	0
$235$	21.7995	1.42205
$236$	4.98792	0.324686
$237$	45.8491	2.97822
$238$	−5.06638	−0.328404
$239$	27.8780	1.80328	0.901639	−	0.432489i	$-0.142365\pi$
$239$	27.8780	1.80328	0.901639	+	0.432489i	$0.142365\pi$
$240$	8.74094	0.564225
$241$	20.7289	1.33526	0.667632	−	0.744492i	$-0.267308\pi$
$241$	20.7289	1.33526	0.667632	+	0.744492i	$0.267308\pi$
$242$	−15.6136	−1.00368
$243$	20.0422	1.28571
$244$	9.08575	0.581656
$245$	−19.4426	−1.24215
$246$	6.98361	0.445258
$247$	0	0
$248$	6.85086	0.435030
$249$	−15.3556	−0.973120
$250$	1.76271	0.111484
$251$	4.54825	0.287083	0.143542	−	0.989644i	$-0.454151\pi$
$251$	4.54825	0.287083	0.143542	+	0.989644i	$0.454151\pi$
$252$	15.3056	0.964161
$253$	28.0277	1.76209
$254$	1.01208	0.0635036
$255$	12.2881	0.769512
$256$	1.00000	0.0625000
$257$	−4.51334	−0.281534	−0.140767	−	0.990043i	$-0.544957\pi$
$257$	−4.51334	−0.281534	−0.140767	+	0.990043i	$0.544957\pi$
$258$	−34.3720	−2.13990
$259$	−33.9758	−2.11116
$260$	0	0
$261$	−21.9095	−1.35616
$262$	−8.21983	−0.507823
$263$	−17.5821	−1.08416	−0.542080	−	0.840327i	$-0.682363\pi$
$263$	−17.5821	−1.08416	−0.542080	+	0.840327i	$0.682363\pi$
$264$	13.8877	0.854728
$265$	−18.6407	−1.14509
$266$	−3.60388	−0.220968
$267$	−31.3696	−1.91979
$268$	0.176292	0.0107687
$269$	17.7560	1.08260	0.541301	−	0.840829i	$-0.317932\pi$
$269$	17.7560	1.08260	0.541301	+	0.840829i	$0.317932\pi$
$270$	−10.8998	−0.663339
$271$	15.9022	0.965988	0.482994	−	0.875624i	$-0.339549\pi$
$271$	15.9022	0.965988	0.482994	+	0.875624i	$0.339549\pi$
$272$	1.40581	0.0852399
$273$	0	0
$274$	−4.37196	−0.264120
$275$	28.5948	1.72433
$276$	−14.6256	−0.880360
$277$	−23.0422	−1.38447	−0.692236	−	0.721671i	$-0.743374\pi$
$277$	−23.0422	−1.38447	−0.692236	+	0.721671i	$0.743374\pi$
$278$	9.97584	0.598311
$279$	−29.0954	−1.74190
$280$	11.7017	0.699311
$281$	−0.895461	−0.0534187	−0.0267093	−	0.999643i	$-0.508503\pi$
$281$	−0.895461	−0.0534187	−0.0267093	+	0.999643i	$0.508503\pi$
$282$	−18.0737	−1.07627
$283$	27.9952	1.66414	0.832071	−	0.554669i	$-0.187155\pi$
$283$	27.9952	1.66414	0.832071	+	0.554669i	$0.187155\pi$
$284$	−5.07069	−0.300890
$285$	8.74094	0.517769
$286$	0	0
$287$	9.34913	0.551861
$288$	−4.24698	−0.250256
$289$	−15.0237	−0.883746
$290$	−16.7506	−0.983631
$291$	−20.8649	−1.22312
$292$	9.92154	0.580614
$293$	−8.39612	−0.490507	−0.245253	−	0.969459i	$-0.578871\pi$
$293$	−8.39612	−0.490507	−0.245253	+	0.969459i	$0.578871\pi$
$294$	16.1196	0.940114
$295$	−16.1957	−0.942948
$296$	9.42758	0.547967
$297$	−17.3177	−1.00487
$298$	16.9312	0.980800
$299$	0	0
$300$	−14.9215	−0.861496
$301$	−46.0146	−2.65224
$302$	8.10321	0.466287
$303$	18.3370	1.05344
$304$	1.00000	0.0573539
$305$	−29.5013	−1.68924
$306$	−5.97046	−0.341308
$307$	−22.7332	−1.29745	−0.648725	−	0.761023i	$-0.724698\pi$
$307$	−22.7332	−1.29745	−0.648725	+	0.761023i	$0.724698\pi$
$308$	18.5918	1.05937
$309$	23.7211	1.34945
$310$	−22.2446	−1.26341
$311$	−4.39373	−0.249146	−0.124573	−	0.992210i	$-0.539756\pi$
$311$	−4.39373	−0.249146	−0.124573	+	0.992210i	$0.539756\pi$
$312$	0	0
$313$	−12.5332	−0.708418	−0.354209	−	0.935166i	$-0.615250\pi$
$313$	−12.5332	−0.708418	−0.354209	+	0.935166i	$0.615250\pi$
$314$	14.6353	0.825920
$315$	−49.6969	−2.80010
$316$	−17.0315	−0.958094
$317$	3.90217	0.219167	0.109584	−	0.993978i	$-0.465048\pi$
$317$	3.90217	0.219167	0.109584	+	0.993978i	$0.465048\pi$
$318$	15.4547	0.866658
$319$	−26.6136	−1.49007
$320$	−3.24698	−0.181512
$321$	31.2784	1.74579
$322$	−19.5797	−1.09113
$323$	1.40581	0.0782215
$324$	−3.70410	−0.205784
$325$	0	0
$326$	12.7114	0.704019
$327$	3.13408	0.173315
$328$	−2.59419	−0.143240
$329$	−24.1957	−1.33395
$330$	−45.0930	−2.48229
$331$	−14.3284	−0.787561	−0.393781	−	0.919204i	$-0.628833\pi$
$331$	−14.3284	−0.787561	−0.393781	+	0.919204i	$0.628833\pi$
$332$	5.70410	0.313053
$333$	−40.0388	−2.19411
$334$	−15.9541	−0.872968
$335$	−0.572417	−0.0312745
$336$	−9.70171	−0.529272
$337$	17.3491	0.945067	0.472534	−	0.881313i	$-0.343340\pi$
$337$	17.3491	0.945067	0.472534	+	0.881313i	$0.343340\pi$
$338$	0	0
$339$	20.4047	1.10823
$340$	−4.56465	−0.247553
$341$	−35.3424	−1.91390
$342$	−4.24698	−0.229650
$343$	−3.64742	−0.196942
$344$	12.7681	0.688409
$345$	47.4892	2.55673
$346$	20.4480	1.09929
$347$	−21.5603	−1.15742	−0.578710	−	0.815534i	$-0.696444\pi$
$347$	−21.5603	−1.15742	−0.578710	+	0.815534i	$0.696444\pi$
$348$	13.8877	0.744458
$349$	−14.1196	−0.755805	−0.377903	−	0.925845i	$-0.623355\pi$
$349$	−14.1196	−0.755805	−0.377903	+	0.925845i	$0.623355\pi$
$350$	−19.9758	−1.06775
$351$	0	0
$352$	−5.15883	−0.274967
$353$	3.85384	0.205119	0.102560	−	0.994727i	$-0.467297\pi$
$353$	3.85384	0.205119	0.102560	+	0.994727i	$0.467297\pi$
$354$	13.4276	0.713668
$355$	16.4644	0.873841
$356$	11.6528	0.617597
$357$	−13.6388	−0.721841
$358$	−18.1129	−0.957297
$359$	26.6219	1.40505	0.702526	−	0.711658i	$-0.252056\pi$
$359$	26.6219	1.40505	0.702526	+	0.711658i	$0.252056\pi$
$360$	13.7899	0.726789
$361$	1.00000	0.0526316
$362$	8.88769	0.467127
$363$	−42.0320	−2.20611
$364$	0	0
$365$	−32.2150	−1.68621
$366$	24.4590	1.27849
$367$	8.99223	0.469391	0.234695	−	0.972069i	$-0.424591\pi$
$367$	8.99223	0.469391	0.234695	+	0.972069i	$0.424591\pi$
$368$	5.43296	0.283213
$369$	11.0175	0.573546
$370$	−30.6112	−1.59140
$371$	20.6896	1.07415
$372$	18.4426	0.956207
$373$	−27.5036	−1.42408	−0.712042	−	0.702136i	$-0.752230\pi$
$373$	−27.5036	−1.42408	−0.712042	+	0.702136i	$0.752230\pi$
$374$	−7.25236	−0.375010
$375$	4.74525	0.245044
$376$	6.71379	0.346237
$377$	0	0
$378$	12.0978	0.622245
$379$	−3.48188	−0.178852	−0.0894260	−	0.995993i	$-0.528503\pi$
$379$	−3.48188	−0.178852	−0.0894260	+	0.995993i	$0.528503\pi$
$380$	−3.24698	−0.166567
$381$	2.72455	0.139583
$382$	−6.65519	−0.340509
$383$	−22.7724	−1.16362	−0.581808	−	0.813326i	$-0.697654\pi$
$383$	−22.7724	−1.16362	−0.581808	+	0.813326i	$0.697654\pi$
$384$	2.69202	0.137377
$385$	−60.3672	−3.07660
$386$	−2.19806	−0.111878
$387$	−54.2258	−2.75645
$388$	7.75063	0.393478
$389$	−25.9081	−1.31359	−0.656797	−	0.754067i	$-0.728089\pi$
$389$	−25.9081	−1.31359	−0.656797	+	0.754067i	$0.728089\pi$
$390$	0	0
$391$	7.63773	0.386256
$392$	−5.98792	−0.302436
$393$	−22.1280	−1.11621
$394$	16.7681	0.844764
$395$	55.3008	2.78249
$396$	21.9095	1.10099
$397$	3.24698	0.162961	0.0814806	−	0.996675i	$-0.474035\pi$
$397$	3.24698	0.162961	0.0814806	+	0.996675i	$0.474035\pi$
$398$	−0.00537681	−0.000269515 0
$399$	−9.70171	−0.485693
$400$	5.54288	0.277144
$401$	13.4426	0.671294	0.335647	−	0.941988i	$-0.391045\pi$
$401$	13.4426	0.671294	0.335647	+	0.941988i	$0.391045\pi$
$402$	0.474582	0.0236700
$403$	0	0
$404$	−6.81163	−0.338891
$405$	12.0271	0.597634
$406$	18.5918	0.922695
$407$	−48.6353	−2.41076
$408$	3.78448	0.187360
$409$	−14.6353	−0.723671	−0.361835	−	0.932242i	$-0.617850\pi$
$409$	−14.6353	−0.723671	−0.361835	+	0.932242i	$0.617850\pi$
$410$	8.42327	0.415996
$411$	−11.7694	−0.580542
$412$	−8.81163	−0.434118
$413$	17.9758	0.884533
$414$	−23.0737	−1.13401
$415$	−18.5211	−0.909165
$416$	0	0
$417$	26.8552	1.31510
$418$	−5.15883	−0.252327
$419$	11.0858	0.541574	0.270787	−	0.962639i	$-0.412716\pi$
$419$	11.0858	0.541574	0.270787	+	0.962639i	$0.412716\pi$
$420$	31.5013	1.53710
$421$	1.76941	0.0862360	0.0431180	−	0.999070i	$-0.486271\pi$
$421$	1.76941	0.0862360	0.0431180	+	0.999070i	$0.486271\pi$
$422$	−1.35450	−0.0659362
$423$	−28.5133	−1.38637
$424$	−5.74094	−0.278805
$425$	7.79225	0.377980
$426$	−13.6504	−0.661364
$427$	32.7439	1.58459
$428$	−11.6189	−0.561623
$429$	0	0
$430$	−41.4577	−1.99927
$431$	28.3502	1.36558	0.682790	−	0.730614i	$-0.260766\pi$
$431$	28.3502	1.36558	0.682790	+	0.730614i	$0.260766\pi$
$432$	−3.35690	−0.161509
$433$	−11.2078	−0.538610	−0.269305	−	0.963055i	$-0.586794\pi$
$433$	−11.2078	−0.538610	−0.269305	+	0.963055i	$0.586794\pi$
$434$	24.6896	1.18514
$435$	−45.0930	−2.16205
$436$	−1.16421	−0.0557556
$437$	5.43296	0.259894
$438$	26.7090	1.27621
$439$	−37.8732	−1.80759	−0.903795	−	0.427966i	$-0.859230\pi$
$439$	−37.8732	−1.80759	−0.903795	+	0.427966i	$0.859230\pi$
$440$	16.7506	0.798555
$441$	25.4306	1.21098
$442$	0	0
$443$	7.60388	0.361271	0.180636	−	0.983550i	$-0.442185\pi$
$443$	7.60388	0.361271	0.180636	+	0.983550i	$0.442185\pi$
$444$	25.3793	1.20445
$445$	−37.8364	−1.79362
$446$	3.60627	0.170762
$447$	45.5792	2.15582
$448$	3.60388	0.170267
$449$	−11.1860	−0.527899	−0.263950	−	0.964536i	$-0.585025\pi$
$449$	−11.1860	−0.527899	−0.263950	+	0.964536i	$0.585025\pi$
$450$	−23.5405	−1.10971
$451$	13.3830	0.630180
$452$	−7.57971	−0.356520
$453$	21.8140	1.02491
$454$	−3.42758	−0.160864
$455$	0	0
$456$	2.69202	0.126065
$457$	6.82238	0.319137	0.159569	−	0.987187i	$-0.448990\pi$
$457$	6.82238	0.319137	0.159569	+	0.987187i	$0.448990\pi$
$458$	−16.8062	−0.785304
$459$	−4.71917	−0.220272
$460$	−17.6407	−0.822502
$461$	−10.6950	−0.498116	−0.249058	−	0.968489i	$-0.580121\pi$
$461$	−10.6950	−0.498116	−0.249058	+	0.968489i	$0.580121\pi$
$462$	50.0495	2.32851
$463$	22.0543	1.02495	0.512475	−	0.858702i	$-0.328729\pi$
$463$	22.0543	1.02495	0.512475	+	0.858702i	$0.328729\pi$
$464$	−5.15883	−0.239493
$465$	−59.8829	−2.77700
$466$	4.80731	0.222695
$467$	−28.1957	−1.30474	−0.652370	−	0.757901i	$-0.726225\pi$
$467$	−28.1957	−1.30474	−0.652370	+	0.757901i	$0.726225\pi$
$468$	0	0
$469$	0.635334	0.0293370
$470$	−21.7995	−1.00554
$471$	39.3986	1.81539
$472$	−4.98792	−0.229588
$473$	−65.8684	−3.02863
$474$	−45.8491	−2.10592
$475$	5.54288	0.254325
$476$	5.06638	0.232217
$477$	24.3817	1.11636
$478$	−27.8780	−1.27511
$479$	−0.724545	−0.0331053	−0.0165527	−	0.999863i	$-0.505269\pi$
$479$	−0.724545	−0.0331053	−0.0165527	+	0.999863i	$0.505269\pi$
$480$	−8.74094	−0.398967
$481$	0	0
$482$	−20.7289	−0.944174
$483$	−52.7090	−2.39834
$484$	15.6136	0.709707
$485$	−25.1661	−1.14274
$486$	−20.0422	−0.909133
$487$	12.1032	0.548449	0.274224	−	0.961666i	$-0.411579\pi$
$487$	12.1032	0.548449	0.274224	+	0.961666i	$0.411579\pi$
$488$	−9.08575	−0.411293
$489$	34.2194	1.54745
$490$	19.4426	0.878329
$491$	−32.0629	−1.44698	−0.723490	−	0.690335i	$-0.757463\pi$
$491$	−32.0629	−1.44698	−0.723490	+	0.690335i	$0.757463\pi$
$492$	−6.98361	−0.314845
$493$	−7.25236	−0.326630
$494$	0	0
$495$	−71.1396	−3.19749
$496$	−6.85086	−0.307612
$497$	−18.2741	−0.819707
$498$	15.3556	0.688099
$499$	−11.4407	−0.512157	−0.256079	−	0.966656i	$-0.582431\pi$
$499$	−11.4407	−0.512157	−0.256079	+	0.966656i	$0.582431\pi$
$500$	−1.76271	−0.0788307
$501$	−42.9487	−1.91881
$502$	−4.54825	−0.202998
$503$	−34.6112	−1.54324	−0.771618	−	0.636086i	$-0.780552\pi$
$503$	−34.6112	−1.54324	−0.771618	+	0.636086i	$0.780552\pi$
$504$	−15.3056	−0.681765
$505$	22.1172	0.984203
$506$	−28.0277	−1.24598
$507$	0	0
$508$	−1.01208	−0.0449039
$509$	3.69096	0.163599	0.0817994	−	0.996649i	$-0.473933\pi$
$509$	3.69096	0.163599	0.0817994	+	0.996649i	$0.473933\pi$
$510$	−12.2881	−0.544127
$511$	35.7560	1.58175
$512$	−1.00000	−0.0441942
$513$	−3.35690	−0.148211
$514$	4.51334	0.199075
$515$	28.6112	1.26076
$516$	34.3720	1.51314
$517$	−34.6353	−1.52326
$518$	33.9758	1.49281
$519$	55.0465	2.41627
$520$	0	0
$521$	5.16421	0.226248	0.113124	−	0.993581i	$-0.463914\pi$
$521$	5.16421	0.226248	0.113124	+	0.993581i	$0.463914\pi$
$522$	21.9095	0.958951
$523$	3.17928	0.139020	0.0695100	−	0.997581i	$-0.477856\pi$
$523$	3.17928	0.139020	0.0695100	+	0.997581i	$0.477856\pi$
$524$	8.21983	0.359085
$525$	−53.7754	−2.34695
$526$	17.5821	0.766616
$527$	−9.63102	−0.419534
$528$	−13.8877	−0.604384
$529$	6.51706	0.283350
$530$	18.6407	0.809701
$531$	21.1836	0.919289
$532$	3.60388	0.156248
$533$	0	0
$534$	31.3696	1.35749
$535$	37.7265	1.63106
$536$	−0.176292	−0.00761465
$537$	−48.7603	−2.10416
$538$	−17.7560	−0.765516
$539$	30.8907	1.33056
$540$	10.8998	0.469052
$541$	−28.3134	−1.21729	−0.608643	−	0.793444i	$-0.708286\pi$
$541$	−28.3134	−1.21729	−0.608643	+	0.793444i	$0.708286\pi$
$542$	−15.9022	−0.683056
$543$	23.9259	1.02676
$544$	−1.40581	−0.0602737
$545$	3.78017	0.161925
$546$	0	0
$547$	11.6189	0.496790	0.248395	−	0.968659i	$-0.420097\pi$
$547$	11.6189	0.496790	0.248395	+	0.968659i	$0.420097\pi$
$548$	4.37196	0.186761
$549$	38.5870	1.64685
$550$	−28.5948	−1.21929
$551$	−5.15883	−0.219774
$552$	14.6256	0.622509
$553$	−61.3793	−2.61011
$554$	23.0422	0.978970
$555$	−82.4059	−3.49794
$556$	−9.97584	−0.423070
$557$	39.2218	1.66188	0.830939	−	0.556363i	$-0.187804\pi$
$557$	39.2218	1.66188	0.830939	+	0.556363i	$0.187804\pi$
$558$	29.0954	1.23171
$559$	0	0
$560$	−11.7017	−0.494488
$561$	−19.5235	−0.824283
$562$	0.895461	0.0377727
$563$	10.9390	0.461024	0.230512	−	0.973069i	$-0.425960\pi$
$563$	10.9390	0.461024	0.230512	+	0.973069i	$0.425960\pi$
$564$	18.0737	0.761039
$565$	24.6112	1.03540
$566$	−27.9952	−1.17673
$567$	−13.3491	−0.560611
$568$	5.07069	0.212761
$569$	37.3685	1.56657	0.783285	−	0.621663i	$-0.213543\pi$
$569$	37.3685	1.56657	0.783285	+	0.621663i	$0.213543\pi$
$570$	−8.74094	−0.366118
$571$	−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
$573$	−17.9159	−0.748448
$574$	−9.34913	−0.390225
$575$	30.1142	1.25585
$576$	4.24698	0.176957
$577$	11.6910	0.486701	0.243350	−	0.969938i	$-0.421753\pi$
$577$	11.6910	0.486701	0.243350	+	0.969938i	$0.421753\pi$
$578$	15.0237	0.624903
$579$	−5.91723	−0.245912
$580$	16.7506	0.695532
$581$	20.5569	0.852843
$582$	20.8649	0.864876
$583$	29.6165	1.22659
$584$	−9.92154	−0.410556
$585$	0	0
$586$	8.39612	0.346841
$587$	27.7560	1.14561	0.572806	−	0.819691i	$-0.305855\pi$
$587$	27.7560	1.14561	0.572806	+	0.819691i	$0.305855\pi$
$588$	−16.1196	−0.664761
$589$	−6.85086	−0.282285
$590$	16.1957	0.666765
$591$	45.1400	1.85681
$592$	−9.42758	−0.387471
$593$	−10.5918	−0.434953	−0.217476	−	0.976066i	$-0.569783\pi$
$593$	−10.5918	−0.434953	−0.217476	+	0.976066i	$0.569783\pi$
$594$	17.3177	0.710552
$595$	−16.4504	−0.674401
$596$	−16.9312	−0.693530
$597$	−0.0144745	−0.000592401 0
$598$	0	0
$599$	16.7245	0.683346	0.341673	−	0.939819i	$-0.389006\pi$
$599$	16.7245	0.683346	0.341673	+	0.939819i	$0.389006\pi$
$600$	14.9215	0.609169
$601$	40.5978	1.65602	0.828009	−	0.560715i	$-0.189474\pi$
$601$	40.5978	1.65602	0.828009	+	0.560715i	$0.189474\pi$
$602$	46.0146	1.87541
$603$	0.748709	0.0304898
$604$	−8.10321	−0.329715
$605$	−50.6969	−2.06112
$606$	−18.3370	−0.744892
$607$	21.1159	0.857067	0.428534	−	0.903526i	$-0.359030\pi$
$607$	21.1159	0.857067	0.428534	+	0.903526i	$0.359030\pi$
$608$	−1.00000	−0.0405554
$609$	50.0495	2.02811
$610$	29.5013	1.19447
$611$	0	0
$612$	5.97046	0.241342
$613$	−38.2107	−1.54332	−0.771659	−	0.636037i	$-0.780573\pi$
$613$	−38.2107	−1.54332	−0.771659	+	0.636037i	$0.780573\pi$
$614$	22.7332	0.917436
$615$	22.6756	0.914370
$616$	−18.5918	−0.749085
$617$	15.3250	0.616960	0.308480	−	0.951231i	$-0.400180\pi$
$617$	15.3250	0.616960	0.308480	+	0.951231i	$0.400180\pi$
$618$	−23.7211	−0.954202
$619$	25.8689	1.03976	0.519880	−	0.854240i	$-0.325977\pi$
$619$	25.8689	1.03976	0.519880	+	0.854240i	$0.325977\pi$
$620$	22.2446	0.893364
$621$	−18.2379	−0.731861
$622$	4.39373	0.176173
$623$	41.9952	1.68250
$624$	0	0
$625$	−21.9909	−0.879636
$626$	12.5332	0.500927
$627$	−13.8877	−0.554621
$628$	−14.6353	−0.584013
$629$	−13.2534	−0.528449
$630$	49.6969	1.97997
$631$	−32.3913	−1.28948	−0.644739	−	0.764402i	$-0.723034\pi$
$631$	−32.3913	−1.28948	−0.644739	+	0.764402i	$0.723034\pi$
$632$	17.0315	0.677475
$633$	−3.64635	−0.144929
$634$	−3.90217	−0.154975
$635$	3.28621	0.130409
$636$	−15.4547	−0.612820
$637$	0	0
$638$	26.6136	1.05364
$639$	−21.5351	−0.851916
$640$	3.24698	0.128348
$641$	35.1594	1.38871	0.694357	−	0.719631i	$-0.255689\pi$
$641$	35.1594	1.38871	0.694357	+	0.719631i	$0.255689\pi$
$642$	−31.2784	−1.23446
$643$	−15.6316	−0.616451	−0.308225	−	0.951313i	$-0.599735\pi$
$643$	−15.6316	−0.616451	−0.308225	+	0.951313i	$0.599735\pi$
$644$	19.5797	0.771549
$645$	−111.605	−4.39444
$646$	−1.40581	−0.0553110
$647$	6.41981	0.252389	0.126194	−	0.992006i	$-0.459724\pi$
$647$	6.41981	0.252389	0.126194	+	0.992006i	$0.459724\pi$
$648$	3.70410	0.145511
$649$	25.7318	1.01006
$650$	0	0
$651$	66.4650	2.60497
$652$	−12.7114	−0.497817
$653$	−3.48188	−0.136256	−0.0681282	−	0.997677i	$-0.521703\pi$
$653$	−3.48188	−0.136256	−0.0681282	+	0.997677i	$0.521703\pi$
$654$	−3.13408	−0.122552
$655$	−26.6896	−1.04285
$656$	2.59419	0.101286
$657$	42.1366	1.64390
$658$	24.1957	0.943245
$659$	−36.4674	−1.42057	−0.710284	−	0.703915i	$-0.751434\pi$
$659$	−36.4674	−1.42057	−0.710284	+	0.703915i	$0.751434\pi$
$660$	45.0930	1.75524
$661$	−38.1366	−1.48334	−0.741671	−	0.670764i	$-0.765966\pi$
$661$	−38.1366	−1.48334	−0.741671	+	0.670764i	$0.765966\pi$
$662$	14.3284	0.556890
$663$	0	0
$664$	−5.70410	−0.221362
$665$	−11.7017	−0.453773
$666$	40.0388	1.55147
$667$	−28.0277	−1.08524
$668$	15.9541	0.617281
$669$	9.70815	0.375339
$670$	0.572417	0.0221144
$671$	46.8719	1.80947
$672$	9.70171	0.374252
$673$	−29.2164	−1.12621	−0.563104	−	0.826386i	$-0.690393\pi$
$673$	−29.2164	−1.12621	−0.563104	+	0.826386i	$0.690393\pi$
$674$	−17.3491	−0.668263
$675$	−18.6069	−0.716178
$676$	0	0
$677$	−18.7821	−0.721854	−0.360927	−	0.932594i	$-0.617540\pi$
$677$	−18.7821	−0.721854	−0.360927	+	0.932594i	$0.617540\pi$
$678$	−20.4047	−0.783640
$679$	27.9323	1.07194
$680$	4.56465	0.175046
$681$	−9.22713	−0.353584
$682$	35.3424	1.35333
$683$	−20.8331	−0.797158	−0.398579	−	0.917134i	$-0.630496\pi$
$683$	−20.8331	−0.797158	−0.398579	+	0.917134i	$0.630496\pi$
$684$	4.24698	0.162387
$685$	−14.1957	−0.542389
$686$	3.64742	0.139259
$687$	−45.2428	−1.72612
$688$	−12.7681	−0.486779
$689$	0	0
$690$	−47.4892	−1.80788
$691$	19.7700	0.752086	0.376043	−	0.926602i	$-0.377284\pi$
$691$	19.7700	0.752086	0.376043	+	0.926602i	$0.377284\pi$
$692$	−20.4480	−0.777317
$693$	78.9590	2.99940
$694$	21.5603	0.818419
$695$	32.3913	1.22867
$696$	−13.8877	−0.526412
$697$	3.64694	0.138138
$698$	14.1196	0.534435
$699$	12.9414	0.489488
$700$	19.9758	0.755016
$701$	−23.0180	−0.869380	−0.434690	−	0.900580i	$-0.643142\pi$
$701$	−23.0180	−0.869380	−0.434690	+	0.900580i	$0.643142\pi$
$702$	0	0
$703$	−9.42758	−0.355568
$704$	5.15883	0.194431
$705$	−58.6848	−2.21020
$706$	−3.85384	−0.145041
$707$	−24.5483	−0.923232
$708$	−13.4276	−0.504639
$709$	20.3284	0.763450	0.381725	−	0.924276i	$-0.375330\pi$
$709$	20.3284	0.763450	0.381725	+	0.924276i	$0.375330\pi$
$710$	−16.4644	−0.617899
$711$	−72.3323	−2.71267
$712$	−11.6528	−0.436707
$713$	−37.2204	−1.39392
$714$	13.6388	0.510419
$715$	0	0
$716$	18.1129	0.676911
$717$	−75.0482	−2.80273
$718$	−26.6219	−0.993521
$719$	2.41550	0.0900830	0.0450415	−	0.998985i	$-0.485658\pi$
$719$	2.41550	0.0900830	0.0450415	+	0.998985i	$0.485658\pi$
$720$	−13.7899	−0.513918
$721$	−31.7560	−1.18266
$722$	−1.00000	−0.0372161
$723$	−55.8025	−2.07532
$724$	−8.88769	−0.330308
$725$	−28.5948	−1.06198
$726$	42.0320	1.55996
$727$	−30.6088	−1.13522	−0.567608	−	0.823299i	$-0.692131\pi$
$727$	−30.6088	−1.13522	−0.567608	+	0.823299i	$0.692131\pi$
$728$	0	0
$729$	−42.8418	−1.58673
$730$	32.2150	1.19233
$731$	−17.9495	−0.663888
$732$	−24.4590	−0.904032
$733$	−18.8304	−0.695517	−0.347759	−	0.937584i	$-0.613057\pi$
$733$	−18.8304	−0.695517	−0.347759	+	0.937584i	$0.613057\pi$
$734$	−8.99223	−0.331909
$735$	52.3400	1.93059
$736$	−5.43296	−0.200262
$737$	0.909461	0.0335004
$738$	−11.0175	−0.405558
$739$	8.01400	0.294800	0.147400	−	0.989077i	$-0.452910\pi$
$739$	8.01400	0.294800	0.147400	+	0.989077i	$0.452910\pi$
$740$	30.6112	1.12529
$741$	0	0
$742$	−20.6896	−0.759540
$743$	−51.4862	−1.88885	−0.944423	−	0.328734i	$-0.893378\pi$
$743$	−51.4862	−1.88885	−0.944423	+	0.328734i	$0.893378\pi$
$744$	−18.4426	−0.676140
$745$	54.9754	2.01414
$746$	27.5036	1.00698
$747$	24.2252	0.886354
$748$	7.25236	0.265172
$749$	−41.8732	−1.53001
$750$	−4.74525	−0.173272
$751$	51.0133	1.86150	0.930750	−	0.365656i	$-0.119155\pi$
$751$	51.0133	1.86150	0.930750	+	0.365656i	$0.119155\pi$
$752$	−6.71379	−0.244827
$753$	−12.2440	−0.446196
$754$	0	0
$755$	26.3110	0.957554
$756$	−12.0978	−0.439994
$757$	3.48666	0.126725	0.0633625	−	0.997991i	$-0.479818\pi$
$757$	3.48666	0.126725	0.0633625	+	0.997991i	$0.479818\pi$
$758$	3.48188	0.126467
$759$	−75.4513	−2.73871
$760$	3.24698	0.117780
$761$	−2.80300	−0.101609	−0.0508044	−	0.998709i	$-0.516179\pi$
$761$	−2.80300	−0.101609	−0.0508044	+	0.998709i	$0.516179\pi$
$762$	−2.72455	−0.0986999
$763$	−4.19567	−0.151893
$764$	6.65519	0.240776
$765$	−19.3860	−0.700901
$766$	22.7724	0.822800
$767$	0	0
$768$	−2.69202	−0.0971400
$769$	−20.3913	−0.735330	−0.367665	−	0.929958i	$-0.619843\pi$
$769$	−20.3913	−0.735330	−0.367665	+	0.929958i	$0.619843\pi$
$770$	60.3672	2.17548
$771$	12.1500	0.437572
$772$	2.19806	0.0791100
$773$	19.7995	0.712140	0.356070	−	0.934459i	$-0.384116\pi$
$773$	19.7995	0.712140	0.356070	+	0.934459i	$0.384116\pi$
$774$	54.2258	1.94911
$775$	−37.9734	−1.36405
$776$	−7.75063	−0.278231
$777$	91.4637	3.28124
$778$	25.9081	0.928852
$779$	2.59419	0.0929464
$780$	0	0
$781$	−26.1588	−0.936037
$782$	−7.63773	−0.273125
$783$	17.3177	0.618883
$784$	5.98792	0.213854
$785$	47.5206	1.69608
$786$	22.1280	0.789278
$787$	−47.8840	−1.70688	−0.853440	−	0.521192i	$-0.825488\pi$
$787$	−47.8840	−1.70688	−0.853440	+	0.521192i	$0.825488\pi$
$788$	−16.7681	−0.597338
$789$	47.3314	1.68504
$790$	−55.3008	−1.96751
$791$	−27.3163	−0.971257
$792$	−21.9095	−0.778519
$793$	0	0
$794$	−3.24698	−0.115231
$795$	50.1812	1.77974
$796$	0.00537681	0.000190576 0
$797$	17.4416	0.617813	0.308906	−	0.951092i	$-0.400037\pi$
$797$	17.4416	0.617813	0.308906	+	0.951092i	$0.400037\pi$
$798$	9.70171	0.343437
$799$	−9.43834	−0.333904
$800$	−5.54288	−0.195970
$801$	49.4892	1.74861
$802$	−13.4426	−0.474676
$803$	51.1836	1.80623
$804$	−0.474582	−0.0167372
$805$	−63.5749	−2.24072
$806$	0	0
$807$	−47.7995	−1.68262
$808$	6.81163	0.239632
$809$	16.2774	0.572282	0.286141	−	0.958188i	$-0.407627\pi$
$809$	16.2774	0.572282	0.286141	+	0.958188i	$0.407627\pi$
$810$	−12.0271	−0.422591
$811$	−53.5991	−1.88212	−0.941059	−	0.338242i	$-0.890168\pi$
$811$	−53.5991	−1.88212	−0.941059	+	0.338242i	$0.890168\pi$
$812$	−18.5918	−0.652444
$813$	−42.8090	−1.50138
$814$	48.6353	1.70467
$815$	41.2737	1.44575
$816$	−3.78448	−0.132483
$817$	−12.7681	−0.446699
$818$	14.6353	0.511712
$819$	0	0
$820$	−8.42327	−0.294153
$821$	23.5749	0.822771	0.411385	−	0.911462i	$-0.365045\pi$
$821$	23.5749	0.822771	0.411385	+	0.911462i	$0.365045\pi$
$822$	11.7694	0.410505
$823$	−34.0092	−1.18549	−0.592743	−	0.805391i	$-0.701955\pi$
$823$	−34.0092	−1.18549	−0.592743	+	0.805391i	$0.701955\pi$
$824$	8.81163	0.306968
$825$	−76.9778	−2.68002
$826$	−17.9758	−0.625459
$827$	−2.46980	−0.0858832	−0.0429416	−	0.999078i	$-0.513673\pi$
$827$	−2.46980	−0.0858832	−0.0429416	+	0.999078i	$0.513673\pi$
$828$	23.0737	0.801866
$829$	3.87369	0.134539	0.0672694	−	0.997735i	$-0.478571\pi$
$829$	3.87369	0.134539	0.0672694	+	0.997735i	$0.478571\pi$
$830$	18.5211	0.642877
$831$	62.0301	2.15180
$832$	0	0
$833$	8.41789	0.291663
$834$	−26.8552	−0.929918
$835$	−51.8025	−1.79270
$836$	5.15883	0.178422
$837$	22.9976	0.794914
$838$	−11.0858	−0.382951
$839$	−15.2895	−0.527851	−0.263925	−	0.964543i	$-0.585017\pi$
$839$	−15.2895	−0.527851	−0.263925	+	0.964543i	$0.585017\pi$
$840$	−31.5013	−1.08690
$841$	−2.38644	−0.0822909
$842$	−1.76941	−0.0609780
$843$	2.41060	0.0830254
$844$	1.35450	0.0466239
$845$	0	0
$846$	28.5133	0.980309
$847$	56.2693	1.93344
$848$	5.74094	0.197145
$849$	−75.3637	−2.58648
$850$	−7.79225	−0.267272
$851$	−51.2197	−1.75579
$852$	13.6504	0.467655
$853$	26.9071	0.921280	0.460640	−	0.887587i	$-0.347620\pi$
$853$	26.9071	0.921280	0.460640	+	0.887587i	$0.347620\pi$
$854$	−32.7439	−1.12047
$855$	−13.7899	−0.471603
$856$	11.6189	0.397127
$857$	16.4698	0.562598	0.281299	−	0.959620i	$-0.409235\pi$
$857$	16.4698	0.562598	0.281299	+	0.959620i	$0.409235\pi$
$858$	0	0
$859$	49.1982	1.67862	0.839310	−	0.543653i	$-0.182959\pi$
$859$	49.1982	1.67862	0.839310	+	0.543653i	$0.182959\pi$
$860$	41.4577	1.41370
$861$	−25.1680	−0.857725
$862$	−28.3502	−0.965611
$863$	42.6273	1.45105	0.725525	−	0.688196i	$-0.241597\pi$
$863$	42.6273	1.45105	0.725525	+	0.688196i	$0.241597\pi$
$864$	3.35690	0.114204
$865$	66.3943	2.25747
$866$	11.2078	0.380855
$867$	40.4441	1.37355
$868$	−24.6896	−0.838021
$869$	−87.8625	−2.98053
$870$	45.0930	1.52880
$871$	0	0
$872$	1.16421	0.0394251
$873$	32.9168	1.11406
$874$	−5.43296	−0.183773
$875$	−6.35258	−0.214757
$876$	−26.7090	−0.902414
$877$	18.8767	0.637420	0.318710	−	0.947852i	$-0.396750\pi$
$877$	18.8767	0.637420	0.318710	+	0.947852i	$0.396750\pi$
$878$	37.8732	1.27816
$879$	22.6025	0.762365
$880$	−16.7506	−0.564664
$881$	−4.42460	−0.149069	−0.0745343	−	0.997218i	$-0.523747\pi$
$881$	−4.42460	−0.149069	−0.0745343	+	0.997218i	$0.523747\pi$
$882$	−25.4306	−0.856292
$883$	−40.3236	−1.35700	−0.678499	−	0.734601i	$-0.737369\pi$
$883$	−40.3236	−1.35700	−0.678499	+	0.734601i	$0.737369\pi$
$884$	0	0
$885$	43.5991	1.46557
$886$	−7.60388	−0.255457
$887$	43.8491	1.47231	0.736154	−	0.676815i	$-0.236640\pi$
$887$	43.8491	1.47231	0.736154	+	0.676815i	$0.236640\pi$
$888$	−25.3793	−0.851672
$889$	−3.64742	−0.122330
$890$	37.8364	1.26828
$891$	−19.1089	−0.640171
$892$	−3.60627	−0.120747
$893$	−6.71379	−0.224668
$894$	−45.5792	−1.52440
$895$	−58.8122	−1.96588
$896$	−3.60388	−0.120397
$897$	0	0
$898$	11.1860	0.373281
$899$	35.3424	1.17874
$900$	23.5405	0.784683
$901$	8.07069	0.268874
$902$	−13.3830	−0.445604
$903$	123.872	4.12221
$904$	7.57971	0.252097
$905$	28.8582	0.959277
$906$	−21.8140	−0.724722
$907$	38.9922	1.29472	0.647358	−	0.762186i	$-0.275874\pi$
$907$	38.9922	1.29472	0.647358	+	0.762186i	$0.275874\pi$
$908$	3.42758	0.113748
$909$	−28.9288	−0.959509
$910$	0	0
$911$	12.2983	0.407461	0.203730	−	0.979027i	$-0.434693\pi$
$911$	12.2983	0.407461	0.203730	+	0.979027i	$0.434693\pi$
$912$	−2.69202	−0.0891417
$913$	29.4265	0.973876
$914$	−6.82238	−0.225664
$915$	79.4180	2.62548
$916$	16.8062	0.555294
$917$	29.6233	0.978246
$918$	4.71917	0.155756
$919$	43.2006	1.42506	0.712528	−	0.701644i	$-0.247550\pi$
$919$	43.2006	1.42506	0.712528	+	0.701644i	$0.247550\pi$
$920$	17.6407	0.581597
$921$	61.1982	2.01655
$922$	10.6950	0.352221
$923$	0	0
$924$	−50.0495	−1.64651
$925$	−52.2559	−1.71816
$926$	−22.0543	−0.724749
$927$	−37.4228	−1.22913
$928$	5.15883	0.169347
$929$	49.1460	1.61243	0.806214	−	0.591624i	$-0.201513\pi$
$929$	49.1460	1.61243	0.806214	+	0.591624i	$0.201513\pi$
$930$	59.8829	1.96364
$931$	5.98792	0.196246
$932$	−4.80731	−0.157469
$933$	11.8280	0.387232
$934$	28.1957	0.922590
$935$	−23.5483	−0.770110
$936$	0	0
$937$	−13.7737	−0.449968	−0.224984	−	0.974362i	$-0.572233\pi$
$937$	−13.7737	−0.449968	−0.224984	+	0.974362i	$0.572233\pi$
$938$	−0.635334	−0.0207444
$939$	33.7396	1.10105
$940$	21.7995	0.711023
$941$	−25.9168	−0.844862	−0.422431	−	0.906395i	$-0.638823\pi$
$941$	−25.9168	−0.844862	−0.422431	+	0.906395i	$0.638823\pi$
$942$	−39.3986	−1.28368
$943$	14.0941	0.458967
$944$	4.98792	0.162343
$945$	39.2814	1.27782
$946$	65.8684	2.14157
$947$	20.0500	0.651537	0.325768	−	0.945450i	$-0.394377\pi$
$947$	20.0500	0.651537	0.325768	+	0.945450i	$0.394377\pi$
$948$	45.8491	1.48911
$949$	0	0
$950$	−5.54288	−0.179835
$951$	−10.5047	−0.340639
$952$	−5.06638	−0.164202
$953$	−29.6689	−0.961071	−0.480535	−	0.876975i	$-0.659558\pi$
$953$	−29.6689	−0.961071	−0.480535	+	0.876975i	$0.659558\pi$
$954$	−24.3817	−0.789385
$955$	−21.6093	−0.699259
$956$	27.8780	0.901639
$957$	71.6443	2.31593
$958$	0.724545	0.0234090
$959$	15.7560	0.508788
$960$	8.74094	0.282113
$961$	15.9342	0.514007
$962$	0	0
$963$	−49.3454	−1.59013
$964$	20.7289	0.667632
$965$	−7.13706	−0.229750
$966$	52.7090	1.69588
$967$	27.9603	0.899143	0.449571	−	0.893244i	$-0.351577\pi$
$967$	27.9603	0.899143	0.449571	+	0.893244i	$0.351577\pi$
$968$	−15.6136	−0.501839
$969$	−3.78448	−0.121575
$970$	25.1661	0.808036
$971$	−7.78448	−0.249816	−0.124908	−	0.992168i	$-0.539864\pi$
$971$	−7.78448	−0.249816	−0.124908	+	0.992168i	$0.539864\pi$
$972$	20.0422	0.642854
$973$	−35.9517	−1.15256
$974$	−12.1032	−0.387812
$975$	0	0
$976$	9.08575	0.290828
$977$	−24.7493	−0.791800	−0.395900	−	0.918294i	$-0.629567\pi$
$977$	−24.7493	−0.791800	−0.395900	+	0.918294i	$0.629567\pi$
$978$	−34.2194	−1.09421
$979$	60.1148	1.92128
$980$	−19.4426	−0.621073
$981$	−4.94438	−0.157862
$982$	32.0629	1.02317
$983$	−17.8649	−0.569800	−0.284900	−	0.958557i	$-0.591960\pi$
$983$	−17.8649	−0.569800	−0.284900	+	0.958557i	$0.591960\pi$
$984$	6.98361	0.222629
$985$	54.4456	1.73478
$986$	7.25236	0.230962
$987$	65.1353	2.07328
$988$	0	0
$989$	−69.3685	−2.20579
$990$	71.1396	2.26096
$991$	−45.0374	−1.43066	−0.715331	−	0.698786i	$-0.753724\pi$
$991$	−45.0374	−1.43066	−0.715331	+	0.698786i	$0.753724\pi$
$992$	6.85086	0.217515
$993$	38.5724	1.22406
$994$	18.2741	0.579620
$995$	−0.0174584	−0.000553468 0
$996$	−15.3556	−0.486560
$997$	−8.51547	−0.269688	−0.134844	−	0.990867i	$-0.543053\pi$
$997$	−8.51547	−0.269688	−0.134844	+	0.990867i	$0.543053\pi$
$998$	11.4407	0.362150
$999$	31.6474	1.00128

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.k.1.1		3
13.5	odd	4		494.2.d.b.77.4	yes	6
13.8	odd	4		494.2.d.b.77.1	✓	6
13.12	even	2		6422.2.a.t.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.d.b.77.1	✓	6	13.8	odd	4
494.2.d.b.77.4	yes	6	13.5	odd	4
6422.2.a.k.1.1		3	1.1	even	1	trivial
6422.2.a.t.1.1		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.69202 q^{3} +1.00000 q^{4} -3.24698 q^{5} +2.69202 q^{6} +3.60388 q^{7} -1.00000 q^{8} +4.24698 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.69202 q^{3} +1.00000 q^{4} -3.24698 q^{5} +2.69202 q^{6} +3.60388 q^{7} -1.00000 q^{8} +4.24698 q^{9} +3.24698 q^{10} +5.15883 q^{11} -2.69202 q^{12} -3.60388 q^{14} +8.74094 q^{15} +1.00000 q^{16} +1.40581 q^{17} -4.24698 q^{18} +1.00000 q^{19} -3.24698 q^{20} -9.70171 q^{21} -5.15883 q^{22} +5.43296 q^{23} +2.69202 q^{24} +5.54288 q^{25} -3.35690 q^{27} +3.60388 q^{28} -5.15883 q^{29} -8.74094 q^{30} -6.85086 q^{31} -1.00000 q^{32} -13.8877 q^{33} -1.40581 q^{34} -11.7017 q^{35} +4.24698 q^{36} -9.42758 q^{37} -1.00000 q^{38} +3.24698 q^{40} +2.59419 q^{41} +9.70171 q^{42} -12.7681 q^{43} +5.15883 q^{44} -13.7899 q^{45} -5.43296 q^{46} -6.71379 q^{47} -2.69202 q^{48} +5.98792 q^{49} -5.54288 q^{50} -3.78448 q^{51} +5.74094 q^{53} +3.35690 q^{54} -16.7506 q^{55} -3.60388 q^{56} -2.69202 q^{57} +5.15883 q^{58} +4.98792 q^{59} +8.74094 q^{60} +9.08575 q^{61} +6.85086 q^{62} +15.3056 q^{63} +1.00000 q^{64} +13.8877 q^{66} +0.176292 q^{67} +1.40581 q^{68} -14.6256 q^{69} +11.7017 q^{70} -5.07069 q^{71} -4.24698 q^{72} +9.92154 q^{73} +9.42758 q^{74} -14.9215 q^{75} +1.00000 q^{76} +18.5918 q^{77} -17.0315 q^{79} -3.24698 q^{80} -3.70410 q^{81} -2.59419 q^{82} +5.70410 q^{83} -9.70171 q^{84} -4.56465 q^{85} +12.7681 q^{86} +13.8877 q^{87} -5.15883 q^{88} +11.6528 q^{89} +13.7899 q^{90} +5.43296 q^{92} +18.4426 q^{93} +6.71379 q^{94} -3.24698 q^{95} +2.69202 q^{96} +7.75063 q^{97} -5.98792 q^{98} +21.9095 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 5 q^{5} + 3 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) 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

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