LMFDB - Embedded newform 7942.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.4171892853$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.56155 q^{6} +3.56155 q^{7} +1.00000 q^{8} -0.561553 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.56155 q^{6} +3.56155 q^{7} +1.00000 q^{8} -0.561553 q^{9} +2.00000 q^{10} +1.00000 q^{11} +1.56155 q^{12} +3.56155 q^{13} +3.56155 q^{14} +3.12311 q^{15} +1.00000 q^{16} +3.56155 q^{17} -0.561553 q^{18} +2.00000 q^{20} +5.56155 q^{21} +1.00000 q^{22} +5.56155 q^{23} +1.56155 q^{24} -1.00000 q^{25} +3.56155 q^{26} -5.56155 q^{27} +3.56155 q^{28} -6.68466 q^{29} +3.12311 q^{30} -2.00000 q^{31} +1.00000 q^{32} +1.56155 q^{33} +3.56155 q^{34} +7.12311 q^{35} -0.561553 q^{36} -3.12311 q^{37} +5.56155 q^{39} +2.00000 q^{40} -2.00000 q^{41} +5.56155 q^{42} +1.00000 q^{44} -1.12311 q^{45} +5.56155 q^{46} +8.00000 q^{47} +1.56155 q^{48} +5.68466 q^{49} -1.00000 q^{50} +5.56155 q^{51} +3.56155 q^{52} +7.80776 q^{53} -5.56155 q^{54} +2.00000 q^{55} +3.56155 q^{56} -6.68466 q^{58} -4.68466 q^{59} +3.12311 q^{60} -10.2462 q^{61} -2.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +7.12311 q^{65} +1.56155 q^{66} -4.68466 q^{67} +3.56155 q^{68} +8.68466 q^{69} +7.12311 q^{70} -6.00000 q^{71} -0.561553 q^{72} +2.68466 q^{73} -3.12311 q^{74} -1.56155 q^{75} +3.56155 q^{77} +5.56155 q^{78} +4.00000 q^{79} +2.00000 q^{80} -7.00000 q^{81} -2.00000 q^{82} -2.24621 q^{83} +5.56155 q^{84} +7.12311 q^{85} -10.4384 q^{87} +1.00000 q^{88} +9.12311 q^{89} -1.12311 q^{90} +12.6847 q^{91} +5.56155 q^{92} -3.12311 q^{93} +8.00000 q^{94} +1.56155 q^{96} -1.12311 q^{97} +5.68466 q^{98} -0.561553 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 4 * q^5 - q^6 + 3 * q^7 + 2 * q^8 + 3 * q^9	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} + 2 q^{8} + 3 q^{9} + 4 q^{10} + 2 q^{11} - q^{12} + 3 q^{13} + 3 q^{14} - 2 q^{15} + 2 q^{16} + 3 q^{17} + 3 q^{18} + 4 q^{20} + 7 q^{21} + 2 q^{22} + 7 q^{23} - q^{24} - 2 q^{25} + 3 q^{26} - 7 q^{27} + 3 q^{28} - q^{29} - 2 q^{30} - 4 q^{31} + 2 q^{32} - q^{33} + 3 q^{34} + 6 q^{35} + 3 q^{36} + 2 q^{37} + 7 q^{39} + 4 q^{40} - 4 q^{41} + 7 q^{42} + 2 q^{44} + 6 q^{45} + 7 q^{46} + 16 q^{47} - q^{48} - q^{49} - 2 q^{50} + 7 q^{51} + 3 q^{52} - 5 q^{53} - 7 q^{54} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 2 q^{60} - 4 q^{61} - 4 q^{62} - 4 q^{63} + 2 q^{64} + 6 q^{65} - q^{66} + 3 q^{67} + 3 q^{68} + 5 q^{69} + 6 q^{70} - 12 q^{71} + 3 q^{72} - 7 q^{73} + 2 q^{74} + q^{75} + 3 q^{77} + 7 q^{78} + 8 q^{79} + 4 q^{80} - 14 q^{81} - 4 q^{82} + 12 q^{83} + 7 q^{84} + 6 q^{85} - 25 q^{87} + 2 q^{88} + 10 q^{89} + 6 q^{90} + 13 q^{91} + 7 q^{92} + 2 q^{93} + 16 q^{94} - q^{96} + 6 q^{97} - q^{98} + 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 4 * q^5 - q^6 + 3 * q^7 + 2 * q^8 + 3 * q^9 + 4 * q^10 + 2 * q^11 - q^12 + 3 * q^13 + 3 * q^14 - 2 * q^15 + 2 * q^16 + 3 * q^17 + 3 * q^18 + 4 * q^20 + 7 * q^21 + 2 * q^22 + 7 * q^23 - q^24 - 2 * q^25 + 3 * q^26 - 7 * q^27 + 3 * q^28 - q^29 - 2 * q^30 - 4 * q^31 + 2 * q^32 - q^33 + 3 * q^34 + 6 * q^35 + 3 * q^36 + 2 * q^37 + 7 * q^39 + 4 * q^40 - 4 * q^41 + 7 * q^42 + 2 * q^44 + 6 * q^45 + 7 * q^46 + 16 * q^47 - q^48 - q^49 - 2 * q^50 + 7 * q^51 + 3 * q^52 - 5 * q^53 - 7 * q^54 + 4 * q^55 + 3 * q^56 - q^58 + 3 * q^59 - 2 * q^60 - 4 * q^61 - 4 * q^62 - 4 * q^63 + 2 * q^64 + 6 * q^65 - q^66 + 3 * q^67 + 3 * q^68 + 5 * q^69 + 6 * q^70 - 12 * q^71 + 3 * q^72 - 7 * q^73 + 2 * q^74 + q^75 + 3 * q^77 + 7 * q^78 + 8 * q^79 + 4 * q^80 - 14 * q^81 - 4 * q^82 + 12 * q^83 + 7 * q^84 + 6 * q^85 - 25 * q^87 + 2 * q^88 + 10 * q^89 + 6 * q^90 + 13 * q^91 + 7 * q^92 + 2 * q^93 + 16 * q^94 - q^96 + 6 * q^97 - q^98 + 3 * 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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	1.56155	0.637501
$7$	3.56155	1.34614	0.673070	−	0.739579i	$-0.264975\pi$
$7$	3.56155	1.34614	0.673070	+	0.739579i	$0.264975\pi$
$8$	1.00000	0.353553
$9$	−0.561553	−0.187184
$10$	2.00000	0.632456
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.56155	0.450781
$13$	3.56155	0.987797	0.493899	−	0.869520i	$-0.335571\pi$
$13$	3.56155	0.987797	0.493899	+	0.869520i	$0.335571\pi$
$14$	3.56155	0.951865
$15$	3.12311	0.806382
$16$	1.00000	0.250000
$17$	3.56155	0.863803	0.431902	−	0.901921i	$-0.357843\pi$
$17$	3.56155	0.863803	0.431902	+	0.901921i	$0.357843\pi$
$18$	−0.561553	−0.132359
$19$	0	0
$19$	0	0
$20$	2.00000	0.447214
$21$	5.56155	1.21363
$22$	1.00000	0.213201
$23$	5.56155	1.15966	0.579832	−	0.814736i	$-0.303118\pi$
$23$	5.56155	1.15966	0.579832	+	0.814736i	$0.303118\pi$
$24$	1.56155	0.318751
$25$	−1.00000	−0.200000
$26$	3.56155	0.698478
$27$	−5.56155	−1.07032
$28$	3.56155	0.673070
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	3.12311	0.570198
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	1.56155	0.271831
$34$	3.56155	0.610801
$35$	7.12311	1.20402
$36$	−0.561553	−0.0935921
$37$	−3.12311	−0.513435	−0.256718	−	0.966486i	$-0.582641\pi$
$37$	−3.12311	−0.513435	−0.256718	+	0.966486i	$0.582641\pi$
$38$	0	0
$39$	5.56155	0.890561
$40$	2.00000	0.316228
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	5.56155	0.858166
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	1.00000	0.150756
$45$	−1.12311	−0.167423
$46$	5.56155	0.820006
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.56155	0.225391
$49$	5.68466	0.812094
$50$	−1.00000	−0.141421
$51$	5.56155	0.778773
$52$	3.56155	0.493899
$53$	7.80776	1.07248	0.536239	−	0.844066i	$-0.319844\pi$
$53$	7.80776	1.07248	0.536239	+	0.844066i	$0.319844\pi$
$54$	−5.56155	−0.756831
$55$	2.00000	0.269680
$56$	3.56155	0.475933
$57$	0	0
$58$	−6.68466	−0.877739
$59$	−4.68466	−0.609891	−0.304945	−	0.952370i	$-0.598638\pi$
$59$	−4.68466	−0.609891	−0.304945	+	0.952370i	$0.598638\pi$
$60$	3.12311	0.403191
$61$	−10.2462	−1.31189	−0.655946	−	0.754807i	$-0.727730\pi$
$61$	−10.2462	−1.31189	−0.655946	+	0.754807i	$0.727730\pi$
$62$	−2.00000	−0.254000
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	7.12311	0.883513
$66$	1.56155	0.192214
$67$	−4.68466	−0.572322	−0.286161	−	0.958182i	$-0.592379\pi$
$67$	−4.68466	−0.572322	−0.286161	+	0.958182i	$0.592379\pi$
$68$	3.56155	0.431902
$69$	8.68466	1.04551
$70$	7.12311	0.851374
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−0.561553	−0.0661796
$73$	2.68466	0.314216	0.157108	−	0.987581i	$-0.449783\pi$
$73$	2.68466	0.314216	0.157108	+	0.987581i	$0.449783\pi$
$74$	−3.12311	−0.363054
$75$	−1.56155	−0.180313
$76$	0	0
$77$	3.56155	0.405877
$78$	5.56155	0.629722
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	2.00000	0.223607
$81$	−7.00000	−0.777778
$82$	−2.00000	−0.220863
$83$	−2.24621	−0.246554	−0.123277	−	0.992372i	$-0.539340\pi$
$83$	−2.24621	−0.246554	−0.123277	+	0.992372i	$0.539340\pi$
$84$	5.56155	0.606815
$85$	7.12311	0.772609
$86$	0	0
$87$	−10.4384	−1.11912
$88$	1.00000	0.106600
$89$	9.12311	0.967047	0.483524	−	0.875331i	$-0.339357\pi$
$89$	9.12311	0.967047	0.483524	+	0.875331i	$0.339357\pi$
$90$	−1.12311	−0.118386
$91$	12.6847	1.32971
$92$	5.56155	0.579832
$93$	−3.12311	−0.323851
$94$	8.00000	0.825137
$95$	0	0
$96$	1.56155	0.159375
$97$	−1.12311	−0.114034	−0.0570170	−	0.998373i	$-0.518159\pi$
$97$	−1.12311	−0.114034	−0.0570170	+	0.998373i	$0.518159\pi$
$98$	5.68466	0.574237
$99$	−0.561553	−0.0564382
$100$	−1.00000	−0.100000
$101$	−15.1231	−1.50481	−0.752403	−	0.658703i	$-0.771105\pi$
$101$	−15.1231	−1.50481	−0.752403	+	0.658703i	$0.771105\pi$
$102$	5.56155	0.550676
$103$	−11.3693	−1.12025	−0.560126	−	0.828407i	$-0.689247\pi$
$103$	−11.3693	−1.12025	−0.560126	+	0.828407i	$0.689247\pi$
$104$	3.56155	0.349239
$105$	11.1231	1.08550
$106$	7.80776	0.758357
$107$	−18.9309	−1.83012	−0.915058	−	0.403322i	$-0.867855\pi$
$107$	−18.9309	−1.83012	−0.915058	+	0.403322i	$0.867855\pi$
$108$	−5.56155	−0.535161
$109$	18.6847	1.78967	0.894833	−	0.446401i	$-0.147295\pi$
$109$	18.6847	1.78967	0.894833	+	0.446401i	$0.147295\pi$
$110$	2.00000	0.190693
$111$	−4.87689	−0.462894
$112$	3.56155	0.336535
$113$	−5.12311	−0.481941	−0.240971	−	0.970532i	$-0.577466\pi$
$113$	−5.12311	−0.481941	−0.240971	+	0.970532i	$0.577466\pi$
$114$	0	0
$115$	11.1231	1.03723
$116$	−6.68466	−0.620655
$117$	−2.00000	−0.184900
$118$	−4.68466	−0.431258
$119$	12.6847	1.16280
$120$	3.12311	0.285099
$121$	1.00000	0.0909091
$122$	−10.2462	−0.927648
$123$	−3.12311	−0.281601
$124$	−2.00000	−0.179605
$125$	−12.0000	−1.07331
$126$	−2.00000	−0.178174
$127$	14.2462	1.26415	0.632073	−	0.774909i	$-0.282204\pi$
$127$	14.2462	1.26415	0.632073	+	0.774909i	$0.282204\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	7.12311	0.624738
$131$	−7.12311	−0.622349	−0.311174	−	0.950353i	$-0.600722\pi$
$131$	−7.12311	−0.622349	−0.311174	+	0.950353i	$0.600722\pi$
$132$	1.56155	0.135916
$133$	0	0
$134$	−4.68466	−0.404693
$135$	−11.1231	−0.957325
$136$	3.56155	0.305401
$137$	8.43845	0.720945	0.360473	−	0.932770i	$-0.382615\pi$
$137$	8.43845	0.720945	0.360473	+	0.932770i	$0.382615\pi$
$138$	8.68466	0.739287
$139$	−17.3693	−1.47325	−0.736623	−	0.676303i	$-0.763581\pi$
$139$	−17.3693	−1.47325	−0.736623	+	0.676303i	$0.763581\pi$
$140$	7.12311	0.602012
$141$	12.4924	1.05205
$142$	−6.00000	−0.503509
$143$	3.56155	0.297832
$144$	−0.561553	−0.0467961
$145$	−13.3693	−1.11026
$146$	2.68466	0.222184
$147$	8.87689	0.732154
$148$	−3.12311	−0.256718
$149$	−15.1231	−1.23893	−0.619467	−	0.785023i	$-0.712651\pi$
$149$	−15.1231	−1.23893	−0.619467	+	0.785023i	$0.712651\pi$
$150$	−1.56155	−0.127500
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	3.56155	0.286998
$155$	−4.00000	−0.321288
$156$	5.56155	0.445281
$157$	18.4924	1.47586	0.737928	−	0.674879i	$-0.235804\pi$
$157$	18.4924	1.47586	0.737928	+	0.674879i	$0.235804\pi$
$158$	4.00000	0.318223
$159$	12.1922	0.966907
$160$	2.00000	0.158114
$161$	19.8078	1.56107
$162$	−7.00000	−0.549972
$163$	−13.3693	−1.04717	−0.523583	−	0.851975i	$-0.675405\pi$
$163$	−13.3693	−1.04717	−0.523583	+	0.851975i	$0.675405\pi$
$164$	−2.00000	−0.156174
$165$	3.12311	0.243133
$166$	−2.24621	−0.174340
$167$	25.3693	1.96314	0.981568	−	0.191111i	$-0.0612092\pi$
$167$	25.3693	1.96314	0.981568	+	0.191111i	$0.0612092\pi$
$168$	5.56155	0.429083
$169$	−0.315342	−0.0242570
$170$	7.12311	0.546317
$171$	0	0
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−10.4384	−0.791337
$175$	−3.56155	−0.269228
$176$	1.00000	0.0753778
$177$	−7.31534	−0.549855
$178$	9.12311	0.683806
$179$	−22.2462	−1.66276	−0.831380	−	0.555704i	$-0.812449\pi$
$179$	−22.2462	−1.66276	−0.831380	+	0.555704i	$0.812449\pi$
$180$	−1.12311	−0.0837114
$181$	19.1231	1.42141	0.710705	−	0.703491i	$-0.248376\pi$
$181$	19.1231	1.42141	0.710705	+	0.703491i	$0.248376\pi$
$182$	12.6847	0.940249
$183$	−16.0000	−1.18275
$184$	5.56155	0.410003
$185$	−6.24621	−0.459231
$186$	−3.12311	−0.228997
$187$	3.56155	0.260447
$188$	8.00000	0.583460
$189$	−19.8078	−1.44080
$190$	0	0
$191$	−20.6847	−1.49669	−0.748345	−	0.663310i	$-0.769151\pi$
$191$	−20.6847	−1.49669	−0.748345	+	0.663310i	$0.769151\pi$
$192$	1.56155	0.112695
$193$	1.12311	0.0808429	0.0404215	−	0.999183i	$-0.487130\pi$
$193$	1.12311	0.0808429	0.0404215	+	0.999183i	$0.487130\pi$
$194$	−1.12311	−0.0806343
$195$	11.1231	0.796542
$196$	5.68466	0.406047
$197$	1.75379	0.124952	0.0624761	−	0.998046i	$-0.480100\pi$
$197$	1.75379	0.124952	0.0624761	+	0.998046i	$0.480100\pi$
$198$	−0.561553	−0.0399078
$199$	−0.684658	−0.0485341	−0.0242671	−	0.999706i	$-0.507725\pi$
$199$	−0.684658	−0.0485341	−0.0242671	+	0.999706i	$0.507725\pi$
$200$	−1.00000	−0.0707107
$201$	−7.31534	−0.515984
$202$	−15.1231	−1.06406
$203$	−23.8078	−1.67098
$204$	5.56155	0.389387
$205$	−4.00000	−0.279372
$206$	−11.3693	−0.792138
$207$	−3.12311	−0.217071
$208$	3.56155	0.246949
$209$	0	0
$210$	11.1231	0.767567
$211$	12.6847	0.873248	0.436624	−	0.899644i	$-0.356174\pi$
$211$	12.6847	0.873248	0.436624	+	0.899644i	$0.356174\pi$
$212$	7.80776	0.536239
$213$	−9.36932	−0.641975
$214$	−18.9309	−1.29409
$215$	0	0
$216$	−5.56155	−0.378416
$217$	−7.12311	−0.483548
$218$	18.6847	1.26548
$219$	4.19224	0.283285
$220$	2.00000	0.134840
$221$	12.6847	0.853262
$222$	−4.87689	−0.327316
$223$	24.7386	1.65662	0.828311	−	0.560269i	$-0.189302\pi$
$223$	24.7386	1.65662	0.828311	+	0.560269i	$0.189302\pi$
$224$	3.56155	0.237966
$225$	0.561553	0.0374369
$226$	−5.12311	−0.340784
$227$	17.1771	1.14008	0.570041	−	0.821616i	$-0.306927\pi$
$227$	17.1771	1.14008	0.570041	+	0.821616i	$0.306927\pi$
$228$	0	0
$229$	21.1231	1.39585	0.697927	−	0.716169i	$-0.254106\pi$
$229$	21.1231	1.39585	0.697927	+	0.716169i	$0.254106\pi$
$230$	11.1231	0.733436
$231$	5.56155	0.365923
$232$	−6.68466	−0.438869
$233$	20.7386	1.35863	0.679317	−	0.733845i	$-0.262276\pi$
$233$	20.7386	1.35863	0.679317	+	0.733845i	$0.262276\pi$
$234$	−2.00000	−0.130744
$235$	16.0000	1.04372
$236$	−4.68466	−0.304945
$237$	6.24621	0.405735
$238$	12.6847	0.822224
$239$	−4.93087	−0.318951	−0.159476	−	0.987202i	$-0.550980\pi$
$239$	−4.93087	−0.318951	−0.159476	+	0.987202i	$0.550980\pi$
$240$	3.12311	0.201596
$241$	−21.6155	−1.39238	−0.696189	−	0.717858i	$-0.745123\pi$
$241$	−21.6155	−1.39238	−0.696189	+	0.717858i	$0.745123\pi$
$242$	1.00000	0.0642824
$243$	5.75379	0.369106
$244$	−10.2462	−0.655946
$245$	11.3693	0.726359
$246$	−3.12311	−0.199122
$247$	0	0
$248$	−2.00000	−0.127000
$249$	−3.50758	−0.222284
$250$	−12.0000	−0.758947
$251$	8.87689	0.560305	0.280152	−	0.959956i	$-0.409615\pi$
$251$	8.87689	0.560305	0.280152	+	0.959956i	$0.409615\pi$
$252$	−2.00000	−0.125988
$253$	5.56155	0.349652
$254$	14.2462	0.893887
$255$	11.1231	0.696556
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	−11.1231	−0.691156
$260$	7.12311	0.441756
$261$	3.75379	0.232354
$262$	−7.12311	−0.440067
$263$	17.1231	1.05586	0.527928	−	0.849289i	$-0.322969\pi$
$263$	17.1231	1.05586	0.527928	+	0.849289i	$0.322969\pi$
$264$	1.56155	0.0961069
$265$	15.6155	0.959254
$266$	0	0
$267$	14.2462	0.871854
$268$	−4.68466	−0.286161
$269$	−11.1231	−0.678188	−0.339094	−	0.940753i	$-0.610120\pi$
$269$	−11.1231	−0.678188	−0.339094	+	0.940753i	$0.610120\pi$
$270$	−11.1231	−0.676931
$271$	13.3153	0.808849	0.404425	−	0.914571i	$-0.367472\pi$
$271$	13.3153	0.808849	0.404425	+	0.914571i	$0.367472\pi$
$272$	3.56155	0.215951
$273$	19.8078	1.19882
$274$	8.43845	0.509785
$275$	−1.00000	−0.0603023
$276$	8.68466	0.522755
$277$	−24.4924	−1.47161	−0.735804	−	0.677195i	$-0.763195\pi$
$277$	−24.4924	−1.47161	−0.735804	+	0.677195i	$0.763195\pi$
$278$	−17.3693	−1.04174
$279$	1.12311	0.0672386
$280$	7.12311	0.425687
$281$	0.630683	0.0376234	0.0188117	−	0.999823i	$-0.494012\pi$
$281$	0.630683	0.0376234	0.0188117	+	0.999823i	$0.494012\pi$
$282$	12.4924	0.743913
$283$	−13.3693	−0.794723	−0.397362	−	0.917662i	$-0.630074\pi$
$283$	−13.3693	−0.794723	−0.397362	+	0.917662i	$0.630074\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	3.56155	0.210599
$287$	−7.12311	−0.420464
$288$	−0.561553	−0.0330898
$289$	−4.31534	−0.253844
$290$	−13.3693	−0.785073
$291$	−1.75379	−0.102809
$292$	2.68466	0.157108
$293$	24.9309	1.45648	0.728238	−	0.685324i	$-0.240339\pi$
$293$	24.9309	1.45648	0.728238	+	0.685324i	$0.240339\pi$
$294$	8.87689	0.517711
$295$	−9.36932	−0.545503
$296$	−3.12311	−0.181527
$297$	−5.56155	−0.322714
$298$	−15.1231	−0.876058
$299$	19.8078	1.14551
$300$	−1.56155	−0.0901563
$301$	0	0
$302$	−4.00000	−0.230174
$303$	−23.6155	−1.35668
$304$	0	0
$305$	−20.4924	−1.17339
$306$	−2.00000	−0.114332
$307$	5.75379	0.328386	0.164193	−	0.986428i	$-0.447498\pi$
$307$	5.75379	0.328386	0.164193	+	0.986428i	$0.447498\pi$
$308$	3.56155	0.202938
$309$	−17.7538	−1.00998
$310$	−4.00000	−0.227185
$311$	15.8078	0.896376	0.448188	−	0.893939i	$-0.352070\pi$
$311$	15.8078	0.896376	0.448188	+	0.893939i	$0.352070\pi$
$312$	5.56155	0.314861
$313$	3.56155	0.201311	0.100655	−	0.994921i	$-0.467906\pi$
$313$	3.56155	0.201311	0.100655	+	0.994921i	$0.467906\pi$
$314$	18.4924	1.04359
$315$	−4.00000	−0.225374
$316$	4.00000	0.225018
$317$	−30.0540	−1.68800	−0.844000	−	0.536344i	$-0.819805\pi$
$317$	−30.0540	−1.68800	−0.844000	+	0.536344i	$0.819805\pi$
$318$	12.1922	0.683707
$319$	−6.68466	−0.374269
$320$	2.00000	0.111803
$321$	−29.5616	−1.64996
$322$	19.8078	1.10384
$323$	0	0
$324$	−7.00000	−0.388889
$325$	−3.56155	−0.197559
$326$	−13.3693	−0.740458
$327$	29.1771	1.61350
$328$	−2.00000	−0.110432
$329$	28.4924	1.57084
$330$	3.12311	0.171921
$331$	10.4384	0.573749	0.286874	−	0.957968i	$-0.407384\pi$
$331$	10.4384	0.573749	0.286874	+	0.957968i	$0.407384\pi$
$332$	−2.24621	−0.123277
$333$	1.75379	0.0961070
$334$	25.3693	1.38815
$335$	−9.36932	−0.511900
$336$	5.56155	0.303408
$337$	−2.87689	−0.156714	−0.0783572	−	0.996925i	$-0.524967\pi$
$337$	−2.87689	−0.156714	−0.0783572	+	0.996925i	$0.524967\pi$
$338$	−0.315342	−0.0171523
$339$	−8.00000	−0.434500
$340$	7.12311	0.386305
$341$	−2.00000	−0.108306
$342$	0	0
$343$	−4.68466	−0.252948
$344$	0	0
$345$	17.3693	0.935133
$346$	−18.0000	−0.967686
$347$	−31.6155	−1.69721	−0.848605	−	0.529027i	$-0.822557\pi$
$347$	−31.6155	−1.69721	−0.848605	+	0.529027i	$0.822557\pi$
$348$	−10.4384	−0.559560
$349$	−19.6155	−1.05000	−0.524998	−	0.851104i	$-0.675934\pi$
$349$	−19.6155	−1.05000	−0.524998	+	0.851104i	$0.675934\pi$
$350$	−3.56155	−0.190373
$351$	−19.8078	−1.05726
$352$	1.00000	0.0533002
$353$	−23.1771	−1.23359	−0.616796	−	0.787123i	$-0.711570\pi$
$353$	−23.1771	−1.23359	−0.616796	+	0.787123i	$0.711570\pi$
$354$	−7.31534	−0.388806
$355$	−12.0000	−0.636894
$356$	9.12311	0.483524
$357$	19.8078	1.04834
$358$	−22.2462	−1.17575
$359$	12.9309	0.682465	0.341233	−	0.939979i	$-0.389156\pi$
$359$	12.9309	0.682465	0.341233	+	0.939979i	$0.389156\pi$
$360$	−1.12311	−0.0591929
$361$	0	0
$362$	19.1231	1.00509
$363$	1.56155	0.0819603
$364$	12.6847	0.664857
$365$	5.36932	0.281043
$366$	−16.0000	−0.836333
$367$	13.7538	0.717942	0.358971	−	0.933349i	$-0.383128\pi$
$367$	13.7538	0.717942	0.358971	+	0.933349i	$0.383128\pi$
$368$	5.56155	0.289916
$369$	1.12311	0.0584665
$370$	−6.24621	−0.324725
$371$	27.8078	1.44371
$372$	−3.12311	−0.161925
$373$	−7.56155	−0.391522	−0.195761	−	0.980652i	$-0.562718\pi$
$373$	−7.56155	−0.391522	−0.195761	+	0.980652i	$0.562718\pi$
$374$	3.56155	0.184164
$375$	−18.7386	−0.967659
$376$	8.00000	0.412568
$377$	−23.8078	−1.22616
$378$	−19.8078	−1.01880
$379$	−2.43845	−0.125255	−0.0626273	−	0.998037i	$-0.519948\pi$
$379$	−2.43845	−0.125255	−0.0626273	+	0.998037i	$0.519948\pi$
$380$	0	0
$381$	22.2462	1.13971
$382$	−20.6847	−1.05832
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	1.56155	0.0796877
$385$	7.12311	0.363027
$386$	1.12311	0.0571646
$387$	0	0
$388$	−1.12311	−0.0570170
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	11.1231	0.563240
$391$	19.8078	1.00172
$392$	5.68466	0.287119
$393$	−11.1231	−0.561086
$394$	1.75379	0.0883546
$395$	8.00000	0.402524
$396$	−0.561553	−0.0282191
$397$	−26.9848	−1.35433	−0.677165	−	0.735831i	$-0.736792\pi$
$397$	−26.9848	−1.35433	−0.677165	+	0.735831i	$0.736792\pi$
$398$	−0.684658	−0.0343188
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	24.2462	1.21080	0.605399	−	0.795922i	$-0.293014\pi$
$401$	24.2462	1.21080	0.605399	+	0.795922i	$0.293014\pi$
$402$	−7.31534	−0.364856
$403$	−7.12311	−0.354827
$404$	−15.1231	−0.752403
$405$	−14.0000	−0.695666
$406$	−23.8078	−1.18156
$407$	−3.12311	−0.154807
$408$	5.56155	0.275338
$409$	0.246211	0.0121744	0.00608718	−	0.999981i	$-0.498062\pi$
$409$	0.246211	0.0121744	0.00608718	+	0.999981i	$0.498062\pi$
$410$	−4.00000	−0.197546
$411$	13.1771	0.649977
$412$	−11.3693	−0.560126
$413$	−16.6847	−0.820998
$414$	−3.12311	−0.153492
$415$	−4.49242	−0.220524
$416$	3.56155	0.174619
$417$	−27.1231	−1.32822
$418$	0	0
$419$	−23.1231	−1.12964	−0.564819	−	0.825215i	$-0.691054\pi$
$419$	−23.1231	−1.12964	−0.564819	+	0.825215i	$0.691054\pi$
$420$	11.1231	0.542752
$421$	1.06913	0.0521062	0.0260531	−	0.999661i	$-0.491706\pi$
$421$	1.06913	0.0521062	0.0260531	+	0.999661i	$0.491706\pi$
$422$	12.6847	0.617480
$423$	−4.49242	−0.218429
$424$	7.80776	0.379179
$425$	−3.56155	−0.172761
$426$	−9.36932	−0.453945
$427$	−36.4924	−1.76599
$428$	−18.9309	−0.915058
$429$	5.56155	0.268514
$430$	0	0
$431$	25.8617	1.24572	0.622858	−	0.782335i	$-0.285971\pi$
$431$	25.8617	1.24572	0.622858	+	0.782335i	$0.285971\pi$
$432$	−5.56155	−0.267580
$433$	−31.3693	−1.50751	−0.753757	−	0.657154i	$-0.771760\pi$
$433$	−31.3693	−1.50751	−0.753757	+	0.657154i	$0.771760\pi$
$434$	−7.12311	−0.341920
$435$	−20.8769	−1.00097
$436$	18.6847	0.894833
$437$	0	0
$438$	4.19224	0.200313
$439$	−31.6155	−1.50893	−0.754463	−	0.656342i	$-0.772103\pi$
$439$	−31.6155	−1.50893	−0.754463	+	0.656342i	$0.772103\pi$
$440$	2.00000	0.0953463
$441$	−3.19224	−0.152011
$442$	12.6847	0.603348
$443$	−17.8617	−0.848637	−0.424318	−	0.905513i	$-0.639486\pi$
$443$	−17.8617	−0.848637	−0.424318	+	0.905513i	$0.639486\pi$
$444$	−4.87689	−0.231447
$445$	18.2462	0.864953
$446$	24.7386	1.17141
$447$	−23.6155	−1.11698
$448$	3.56155	0.168268
$449$	17.6155	0.831328	0.415664	−	0.909518i	$-0.363549\pi$
$449$	17.6155	0.831328	0.415664	+	0.909518i	$0.363549\pi$
$450$	0.561553	0.0264719
$451$	−2.00000	−0.0941763
$452$	−5.12311	−0.240971
$453$	−6.24621	−0.293473
$454$	17.1771	0.806160
$455$	25.3693	1.18933
$456$	0	0
$457$	32.4384	1.51741	0.758703	−	0.651436i	$-0.225833\pi$
$457$	32.4384	1.51741	0.758703	+	0.651436i	$0.225833\pi$
$458$	21.1231	0.987018
$459$	−19.8078	−0.924547
$460$	11.1231	0.518617
$461$	−22.7386	−1.05904	−0.529522	−	0.848296i	$-0.677629\pi$
$461$	−22.7386	−1.05904	−0.529522	+	0.848296i	$0.677629\pi$
$462$	5.56155	0.258747
$463$	36.4924	1.69595	0.847973	−	0.530039i	$-0.177823\pi$
$463$	36.4924	1.69595	0.847973	+	0.530039i	$0.177823\pi$
$464$	−6.68466	−0.310327
$465$	−6.24621	−0.289661
$466$	20.7386	0.960699
$467$	26.2462	1.21453	0.607265	−	0.794499i	$-0.292267\pi$
$467$	26.2462	1.21453	0.607265	+	0.794499i	$0.292267\pi$
$468$	−2.00000	−0.0924500
$469$	−16.6847	−0.770426
$470$	16.0000	0.738025
$471$	28.8769	1.33058
$472$	−4.68466	−0.215629
$473$	0	0
$474$	6.24621	0.286898
$475$	0	0
$476$	12.6847	0.581400
$477$	−4.38447	−0.200751
$478$	−4.93087	−0.225533
$479$	11.3693	0.519477	0.259739	−	0.965679i	$-0.416364\pi$
$479$	11.3693	0.519477	0.259739	+	0.965679i	$0.416364\pi$
$480$	3.12311	0.142550
$481$	−11.1231	−0.507170
$482$	−21.6155	−0.984560
$483$	30.9309	1.40740
$484$	1.00000	0.0454545
$485$	−2.24621	−0.101995
$486$	5.75379	0.260997
$487$	−22.8769	−1.03665	−0.518326	−	0.855183i	$-0.673444\pi$
$487$	−22.8769	−1.03665	−0.518326	+	0.855183i	$0.673444\pi$
$488$	−10.2462	−0.463824
$489$	−20.8769	−0.944086
$490$	11.3693	0.513613
$491$	−14.6307	−0.660273	−0.330137	−	0.943933i	$-0.607095\pi$
$491$	−14.6307	−0.660273	−0.330137	+	0.943933i	$0.607095\pi$
$492$	−3.12311	−0.140800
$493$	−23.8078	−1.07225
$494$	0	0
$495$	−1.12311	−0.0504798
$496$	−2.00000	−0.0898027
$497$	−21.3693	−0.958545
$498$	−3.50758	−0.157178
$499$	26.2462	1.17494	0.587471	−	0.809245i	$-0.300124\pi$
$499$	26.2462	1.17494	0.587471	+	0.809245i	$0.300124\pi$
$500$	−12.0000	−0.536656
$501$	39.6155	1.76989
$502$	8.87689	0.396195
$503$	9.31534	0.415351	0.207675	−	0.978198i	$-0.433410\pi$
$503$	9.31534	0.415351	0.207675	+	0.978198i	$0.433410\pi$
$504$	−2.00000	−0.0890871
$505$	−30.2462	−1.34594
$506$	5.56155	0.247241
$507$	−0.492423	−0.0218693
$508$	14.2462	0.632073
$509$	6.63068	0.293900	0.146950	−	0.989144i	$-0.453054\pi$
$509$	6.63068	0.293900	0.146950	+	0.989144i	$0.453054\pi$
$510$	11.1231	0.492539
$511$	9.56155	0.422978
$512$	1.00000	0.0441942
$513$	0	0
$514$	22.0000	0.970378
$515$	−22.7386	−1.00198
$516$	0	0
$517$	8.00000	0.351840
$518$	−11.1231	−0.488721
$519$	−28.1080	−1.23380
$520$	7.12311	0.312369
$521$	−1.12311	−0.0492042	−0.0246021	−	0.999697i	$-0.507832\pi$
$521$	−1.12311	−0.0492042	−0.0246021	+	0.999697i	$0.507832\pi$
$522$	3.75379	0.164299
$523$	6.05398	0.264722	0.132361	−	0.991202i	$-0.457744\pi$
$523$	6.05398	0.264722	0.132361	+	0.991202i	$0.457744\pi$
$524$	−7.12311	−0.311174
$525$	−5.56155	−0.242726
$526$	17.1231	0.746603
$527$	−7.12311	−0.310287
$528$	1.56155	0.0679579
$529$	7.93087	0.344820
$530$	15.6155	0.678295
$531$	2.63068	0.114162
$532$	0	0
$533$	−7.12311	−0.308536
$534$	14.2462	0.616494
$535$	−37.8617	−1.63691
$536$	−4.68466	−0.202346
$537$	−34.7386	−1.49908
$538$	−11.1231	−0.479551
$539$	5.68466	0.244856
$540$	−11.1231	−0.478662
$541$	43.2311	1.85865	0.929324	−	0.369265i	$-0.120391\pi$
$541$	43.2311	1.85865	0.929324	+	0.369265i	$0.120391\pi$
$542$	13.3153	0.571943
$543$	29.8617	1.28149
$544$	3.56155	0.152700
$545$	37.3693	1.60073
$546$	19.8078	0.847694
$547$	−6.73863	−0.288123	−0.144062	−	0.989569i	$-0.546016\pi$
$547$	−6.73863	−0.288123	−0.144062	+	0.989569i	$0.546016\pi$
$548$	8.43845	0.360473
$549$	5.75379	0.245566
$550$	−1.00000	−0.0426401
$551$	0	0
$552$	8.68466	0.369644
$553$	14.2462	0.605811
$554$	−24.4924	−1.04058
$555$	−9.75379	−0.414025
$556$	−17.3693	−0.736623
$557$	8.87689	0.376126	0.188063	−	0.982157i	$-0.439779\pi$
$557$	8.87689	0.376126	0.188063	+	0.982157i	$0.439779\pi$
$558$	1.12311	0.0475449
$559$	0	0
$560$	7.12311	0.301006
$561$	5.56155	0.234809
$562$	0.630683	0.0266038
$563$	−6.73863	−0.284000	−0.142000	−	0.989867i	$-0.545353\pi$
$563$	−6.73863	−0.284000	−0.142000	+	0.989867i	$0.545353\pi$
$564$	12.4924	0.526026
$565$	−10.2462	−0.431061
$566$	−13.3693	−0.561954
$567$	−24.9309	−1.04700
$568$	−6.00000	−0.251754
$569$	−3.36932	−0.141249	−0.0706246	−	0.997503i	$-0.522499\pi$
$569$	−3.36932	−0.141249	−0.0706246	+	0.997503i	$0.522499\pi$
$570$	0	0
$571$	35.6155	1.49046	0.745232	−	0.666806i	$-0.232339\pi$
$571$	35.6155	1.49046	0.745232	+	0.666806i	$0.232339\pi$
$572$	3.56155	0.148916
$573$	−32.3002	−1.34936
$574$	−7.12311	−0.297313
$575$	−5.56155	−0.231933
$576$	−0.561553	−0.0233980
$577$	3.94602	0.164275	0.0821376	−	0.996621i	$-0.473825\pi$
$577$	3.94602	0.164275	0.0821376	+	0.996621i	$0.473825\pi$
$578$	−4.31534	−0.179495
$579$	1.75379	0.0728850
$580$	−13.3693	−0.555131
$581$	−8.00000	−0.331896
$582$	−1.75379	−0.0726969
$583$	7.80776	0.323365
$584$	2.68466	0.111092
$585$	−4.00000	−0.165380
$586$	24.9309	1.02988
$587$	−7.50758	−0.309871	−0.154935	−	0.987925i	$-0.549517\pi$
$587$	−7.50758	−0.309871	−0.154935	+	0.987925i	$0.549517\pi$
$588$	8.87689	0.366077
$589$	0	0
$590$	−9.36932	−0.385729
$591$	2.73863	0.112652
$592$	−3.12311	−0.128359
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	−5.56155	−0.228193
$595$	25.3693	1.04004
$596$	−15.1231	−0.619467
$597$	−1.06913	−0.0437566
$598$	19.8078	0.810000
$599$	−39.8617	−1.62871	−0.814353	−	0.580370i	$-0.802908\pi$
$599$	−39.8617	−1.62871	−0.814353	+	0.580370i	$0.802908\pi$
$600$	−1.56155	−0.0637501
$601$	3.36932	0.137437	0.0687187	−	0.997636i	$-0.478109\pi$
$601$	3.36932	0.137437	0.0687187	+	0.997636i	$0.478109\pi$
$602$	0	0
$603$	2.63068	0.107130
$604$	−4.00000	−0.162758
$605$	2.00000	0.0813116
$606$	−23.6155	−0.959315
$607$	−27.6155	−1.12088	−0.560440	−	0.828195i	$-0.689368\pi$
$607$	−27.6155	−1.12088	−0.560440	+	0.828195i	$0.689368\pi$
$608$	0	0
$609$	−37.1771	−1.50649
$610$	−20.4924	−0.829714
$611$	28.4924	1.15268
$612$	−2.00000	−0.0808452
$613$	1.75379	0.0708349	0.0354174	−	0.999373i	$-0.488724\pi$
$613$	1.75379	0.0708349	0.0354174	+	0.999373i	$0.488724\pi$
$614$	5.75379	0.232204
$615$	−6.24621	−0.251872
$616$	3.56155	0.143499
$617$	−4.73863	−0.190770	−0.0953851	−	0.995440i	$-0.530408\pi$
$617$	−4.73863	−0.190770	−0.0953851	+	0.995440i	$0.530408\pi$
$618$	−17.7538	−0.714162
$619$	−26.2462	−1.05492	−0.527462	−	0.849579i	$-0.676856\pi$
$619$	−26.2462	−1.05492	−0.527462	+	0.849579i	$0.676856\pi$
$620$	−4.00000	−0.160644
$621$	−30.9309	−1.24121
$622$	15.8078	0.633834
$623$	32.4924	1.30178
$624$	5.56155	0.222640
$625$	−19.0000	−0.760000
$626$	3.56155	0.142348
$627$	0	0
$628$	18.4924	0.737928
$629$	−11.1231	−0.443507
$630$	−4.00000	−0.159364
$631$	36.4924	1.45274	0.726370	−	0.687304i	$-0.241206\pi$
$631$	36.4924	1.45274	0.726370	+	0.687304i	$0.241206\pi$
$632$	4.00000	0.159111
$633$	19.8078	0.787288
$634$	−30.0540	−1.19360
$635$	28.4924	1.13069
$636$	12.1922	0.483454
$637$	20.2462	0.802184
$638$	−6.68466	−0.264648
$639$	3.36932	0.133288
$640$	2.00000	0.0790569
$641$	9.61553	0.379791	0.189895	−	0.981804i	$-0.439185\pi$
$641$	9.61553	0.379791	0.189895	+	0.981804i	$0.439185\pi$
$642$	−29.5616	−1.16670
$643$	−37.3693	−1.47370	−0.736851	−	0.676055i	$-0.763688\pi$
$643$	−37.3693	−1.47370	−0.736851	+	0.676055i	$0.763688\pi$
$644$	19.8078	0.780535
$645$	0	0
$646$	0	0
$647$	22.5464	0.886390	0.443195	−	0.896425i	$-0.353845\pi$
$647$	22.5464	0.886390	0.443195	+	0.896425i	$0.353845\pi$
$648$	−7.00000	−0.274986
$649$	−4.68466	−0.183889
$650$	−3.56155	−0.139696
$651$	−11.1231	−0.435949
$652$	−13.3693	−0.523583
$653$	25.6155	1.00241	0.501207	−	0.865328i	$-0.332890\pi$
$653$	25.6155	1.00241	0.501207	+	0.865328i	$0.332890\pi$
$654$	29.1771	1.14091
$655$	−14.2462	−0.556646
$656$	−2.00000	−0.0780869
$657$	−1.50758	−0.0588162
$658$	28.4924	1.11075
$659$	−47.4233	−1.84735	−0.923675	−	0.383178i	$-0.874830\pi$
$659$	−47.4233	−1.84735	−0.923675	+	0.383178i	$0.874830\pi$
$660$	3.12311	0.121567
$661$	22.4384	0.872754	0.436377	−	0.899764i	$-0.356261\pi$
$661$	22.4384	0.872754	0.436377	+	0.899764i	$0.356261\pi$
$662$	10.4384	0.405702
$663$	19.8078	0.769270
$664$	−2.24621	−0.0871699
$665$	0	0
$666$	1.75379	0.0679579
$667$	−37.1771	−1.43950
$668$	25.3693	0.981568
$669$	38.6307	1.49355
$670$	−9.36932	−0.361968
$671$	−10.2462	−0.395551
$672$	5.56155	0.214542
$673$	18.8769	0.727651	0.363825	−	0.931467i	$-0.381470\pi$
$673$	18.8769	0.727651	0.363825	+	0.931467i	$0.381470\pi$
$674$	−2.87689	−0.110814
$675$	5.56155	0.214064
$676$	−0.315342	−0.0121285
$677$	39.1771	1.50570	0.752849	−	0.658194i	$-0.228679\pi$
$677$	39.1771	1.50570	0.752849	+	0.658194i	$0.228679\pi$
$678$	−8.00000	−0.307238
$679$	−4.00000	−0.153506
$680$	7.12311	0.273159
$681$	26.8229	1.02786
$682$	−2.00000	−0.0765840
$683$	4.49242	0.171898	0.0859489	−	0.996300i	$-0.472608\pi$
$683$	4.49242	0.171898	0.0859489	+	0.996300i	$0.472608\pi$
$684$	0	0
$685$	16.8769	0.644833
$686$	−4.68466	−0.178861
$687$	32.9848	1.25845
$688$	0	0
$689$	27.8078	1.05939
$690$	17.3693	0.661239
$691$	−43.6155	−1.65921	−0.829606	−	0.558349i	$-0.811435\pi$
$691$	−43.6155	−1.65921	−0.829606	+	0.558349i	$0.811435\pi$
$692$	−18.0000	−0.684257
$693$	−2.00000	−0.0759737
$694$	−31.6155	−1.20011
$695$	−34.7386	−1.31771
$696$	−10.4384	−0.395668
$697$	−7.12311	−0.269807
$698$	−19.6155	−0.742459
$699$	32.3845	1.22489
$700$	−3.56155	−0.134614
$701$	24.9848	0.943665	0.471832	−	0.881688i	$-0.343593\pi$
$701$	24.9848	0.943665	0.471832	+	0.881688i	$0.343593\pi$
$702$	−19.8078	−0.747596
$703$	0	0
$704$	1.00000	0.0376889
$705$	24.9848	0.940984
$706$	−23.1771	−0.872281
$707$	−53.8617	−2.02568
$708$	−7.31534	−0.274927
$709$	−9.50758	−0.357065	−0.178532	−	0.983934i	$-0.557135\pi$
$709$	−9.50758	−0.357065	−0.178532	+	0.983934i	$0.557135\pi$
$710$	−12.0000	−0.450352
$711$	−2.24621	−0.0842395
$712$	9.12311	0.341903
$713$	−11.1231	−0.416564
$714$	19.8078	0.741287
$715$	7.12311	0.266389
$716$	−22.2462	−0.831380
$717$	−7.69981	−0.287555
$718$	12.9309	0.482576
$719$	−6.43845	−0.240114	−0.120057	−	0.992767i	$-0.538308\pi$
$719$	−6.43845	−0.240114	−0.120057	+	0.992767i	$0.538308\pi$
$720$	−1.12311	−0.0418557
$721$	−40.4924	−1.50802
$722$	0	0
$723$	−33.7538	−1.25532
$724$	19.1231	0.710705
$725$	6.68466	0.248262
$726$	1.56155	0.0579547
$727$	−39.4233	−1.46213	−0.731064	−	0.682308i	$-0.760976\pi$
$727$	−39.4233	−1.46213	−0.731064	+	0.682308i	$0.760976\pi$
$728$	12.6847	0.470125
$729$	29.9848	1.11055
$730$	5.36932	0.198727
$731$	0	0
$732$	−16.0000	−0.591377
$733$	−48.1080	−1.77691	−0.888454	−	0.458966i	$-0.848220\pi$
$733$	−48.1080	−1.77691	−0.888454	+	0.458966i	$0.848220\pi$
$734$	13.7538	0.507662
$735$	17.7538	0.654858
$736$	5.56155	0.205002
$737$	−4.68466	−0.172562
$738$	1.12311	0.0413421
$739$	20.9848	0.771940	0.385970	−	0.922511i	$-0.373867\pi$
$739$	20.9848	0.771940	0.385970	+	0.922511i	$0.373867\pi$
$740$	−6.24621	−0.229615
$741$	0	0
$742$	27.8078	1.02086
$743$	25.7538	0.944815	0.472407	−	0.881380i	$-0.343385\pi$
$743$	25.7538	0.944815	0.472407	+	0.881380i	$0.343385\pi$
$744$	−3.12311	−0.114499
$745$	−30.2462	−1.10814
$746$	−7.56155	−0.276848
$747$	1.26137	0.0461510
$748$	3.56155	0.130223
$749$	−67.4233	−2.46359
$750$	−18.7386	−0.684238
$751$	41.2311	1.50454	0.752271	−	0.658853i	$-0.228958\pi$
$751$	41.2311	1.50454	0.752271	+	0.658853i	$0.228958\pi$
$752$	8.00000	0.291730
$753$	13.8617	0.505150
$754$	−23.8078	−0.867028
$755$	−8.00000	−0.291150
$756$	−19.8078	−0.720401
$757$	−9.50758	−0.345559	−0.172779	−	0.984961i	$-0.555275\pi$
$757$	−9.50758	−0.345559	−0.172779	+	0.984961i	$0.555275\pi$
$758$	−2.43845	−0.0885684
$759$	8.68466	0.315233
$760$	0	0
$761$	−22.1922	−0.804468	−0.402234	−	0.915537i	$-0.631766\pi$
$761$	−22.1922	−0.804468	−0.402234	+	0.915537i	$0.631766\pi$
$762$	22.2462	0.805895
$763$	66.5464	2.40914
$764$	−20.6847	−0.748345
$765$	−4.00000	−0.144620
$766$	2.00000	0.0722629
$767$	−16.6847	−0.602448
$768$	1.56155	0.0563477
$769$	−1.31534	−0.0474324	−0.0237162	−	0.999719i	$-0.507550\pi$
$769$	−1.31534	−0.0474324	−0.0237162	+	0.999719i	$0.507550\pi$
$770$	7.12311	0.256699
$771$	34.3542	1.23723
$772$	1.12311	0.0404215
$773$	−9.06913	−0.326194	−0.163097	−	0.986610i	$-0.552148\pi$
$773$	−9.06913	−0.326194	−0.163097	+	0.986610i	$0.552148\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	−1.12311	−0.0403171
$777$	−17.3693	−0.623121
$778$	18.0000	0.645331
$779$	0	0
$780$	11.1231	0.398271
$781$	−6.00000	−0.214697
$782$	19.8078	0.708324
$783$	37.1771	1.32860
$784$	5.68466	0.203024
$785$	36.9848	1.32005
$786$	−11.1231	−0.396748
$787$	−3.31534	−0.118179	−0.0590896	−	0.998253i	$-0.518820\pi$
$787$	−3.31534	−0.118179	−0.0590896	+	0.998253i	$0.518820\pi$
$788$	1.75379	0.0624761
$789$	26.7386	0.951921
$790$	8.00000	0.284627
$791$	−18.2462	−0.648761
$792$	−0.561553	−0.0199539
$793$	−36.4924	−1.29588
$794$	−26.9848	−0.957656
$795$	24.3845	0.864828
$796$	−0.684658	−0.0242671
$797$	35.8078	1.26838	0.634188	−	0.773179i	$-0.281335\pi$
$797$	35.8078	1.26838	0.634188	+	0.773179i	$0.281335\pi$
$798$	0	0
$799$	28.4924	1.00799
$800$	−1.00000	−0.0353553
$801$	−5.12311	−0.181016
$802$	24.2462	0.856163
$803$	2.68466	0.0947395
$804$	−7.31534	−0.257992
$805$	39.6155	1.39626
$806$	−7.12311	−0.250901
$807$	−17.3693	−0.611429
$808$	−15.1231	−0.532029
$809$	36.9309	1.29842	0.649210	−	0.760609i	$-0.275100\pi$
$809$	36.9309	1.29842	0.649210	+	0.760609i	$0.275100\pi$
$810$	−14.0000	−0.491910
$811$	−20.6847	−0.726337	−0.363168	−	0.931724i	$-0.618305\pi$
$811$	−20.6847	−0.726337	−0.363168	+	0.931724i	$0.618305\pi$
$812$	−23.8078	−0.835489
$813$	20.7926	0.729229
$814$	−3.12311	−0.109465
$815$	−26.7386	−0.936613
$816$	5.56155	0.194693
$817$	0	0
$818$	0.246211	0.00860857
$819$	−7.12311	−0.248901
$820$	−4.00000	−0.139686
$821$	44.9848	1.56998	0.784991	−	0.619507i	$-0.212668\pi$
$821$	44.9848	1.56998	0.784991	+	0.619507i	$0.212668\pi$
$822$	13.1771	0.459603
$823$	14.9309	0.520457	0.260229	−	0.965547i	$-0.416202\pi$
$823$	14.9309	0.520457	0.260229	+	0.965547i	$0.416202\pi$
$824$	−11.3693	−0.396069
$825$	−1.56155	−0.0543663
$826$	−16.6847	−0.580534
$827$	−21.0691	−0.732645	−0.366323	−	0.930488i	$-0.619383\pi$
$827$	−21.0691	−0.732645	−0.366323	+	0.930488i	$0.619383\pi$
$828$	−3.12311	−0.108535
$829$	33.1771	1.15229	0.576144	−	0.817348i	$-0.304557\pi$
$829$	33.1771	1.15229	0.576144	+	0.817348i	$0.304557\pi$
$830$	−4.49242	−0.155934
$831$	−38.2462	−1.32675
$832$	3.56155	0.123475
$833$	20.2462	0.701490
$834$	−27.1231	−0.939196
$835$	50.7386	1.75588
$836$	0	0
$837$	11.1231	0.384471
$838$	−23.1231	−0.798774
$839$	−1.12311	−0.0387739	−0.0193870	−	0.999812i	$-0.506171\pi$
$839$	−1.12311	−0.0387739	−0.0193870	+	0.999812i	$0.506171\pi$
$840$	11.1231	0.383784
$841$	15.6847	0.540850
$842$	1.06913	0.0368447
$843$	0.984845	0.0339199
$844$	12.6847	0.436624
$845$	−0.630683	−0.0216962
$846$	−4.49242	−0.154453
$847$	3.56155	0.122376
$848$	7.80776	0.268120
$849$	−20.8769	−0.716493
$850$	−3.56155	−0.122160
$851$	−17.3693	−0.595413
$852$	−9.36932	−0.320988
$853$	38.3542	1.31322	0.656611	−	0.754230i	$-0.271989\pi$
$853$	38.3542	1.31322	0.656611	+	0.754230i	$0.271989\pi$
$854$	−36.4924	−1.24874
$855$	0	0
$856$	−18.9309	−0.647044
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	5.56155	0.189868
$859$	−41.8617	−1.42830	−0.714152	−	0.699991i	$-0.753188\pi$
$859$	−41.8617	−1.42830	−0.714152	+	0.699991i	$0.753188\pi$
$860$	0	0
$861$	−11.1231	−0.379074
$862$	25.8617	0.880854
$863$	16.2462	0.553027	0.276514	−	0.961010i	$-0.410821\pi$
$863$	16.2462	0.553027	0.276514	+	0.961010i	$0.410821\pi$
$864$	−5.56155	−0.189208
$865$	−36.0000	−1.22404
$866$	−31.3693	−1.06597
$867$	−6.73863	−0.228856
$868$	−7.12311	−0.241774
$869$	4.00000	0.135691
$870$	−20.8769	−0.707793
$871$	−16.6847	−0.565338
$872$	18.6847	0.632742
$873$	0.630683	0.0213454
$874$	0	0
$875$	−42.7386	−1.44483
$876$	4.19224	0.141643
$877$	40.4384	1.36551	0.682755	−	0.730648i	$-0.260782\pi$
$877$	40.4384	1.36551	0.682755	+	0.730648i	$0.260782\pi$
$878$	−31.6155	−1.06697
$879$	38.9309	1.31311
$880$	2.00000	0.0674200
$881$	−28.7386	−0.968229	−0.484115	−	0.875005i	$-0.660858\pi$
$881$	−28.7386	−0.968229	−0.484115	+	0.875005i	$0.660858\pi$
$882$	−3.19224	−0.107488
$883$	−54.7386	−1.84210	−0.921051	−	0.389442i	$-0.872668\pi$
$883$	−54.7386	−1.84210	−0.921051	+	0.389442i	$0.872668\pi$
$884$	12.6847	0.426631
$885$	−14.6307	−0.491805
$886$	−17.8617	−0.600077
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	−4.87689	−0.163658
$889$	50.7386	1.70172
$890$	18.2462	0.611614
$891$	−7.00000	−0.234509
$892$	24.7386	0.828311
$893$	0	0
$894$	−23.6155	−0.789821
$895$	−44.4924	−1.48722
$896$	3.56155	0.118983
$897$	30.9309	1.03275
$898$	17.6155	0.587838
$899$	13.3693	0.445892
$900$	0.561553	0.0187184
$901$	27.8078	0.926411
$902$	−2.00000	−0.0665927
$903$	0	0
$904$	−5.12311	−0.170392
$905$	38.2462	1.27135
$906$	−6.24621	−0.207516
$907$	−56.7926	−1.88577	−0.942884	−	0.333122i	$-0.891898\pi$
$907$	−56.7926	−1.88577	−0.942884	+	0.333122i	$0.891898\pi$
$908$	17.1771	0.570041
$909$	8.49242	0.281676
$910$	25.3693	0.840985
$911$	46.9848	1.55668	0.778339	−	0.627845i	$-0.216063\pi$
$911$	46.9848	1.55668	0.778339	+	0.627845i	$0.216063\pi$
$912$	0	0
$913$	−2.24621	−0.0743387
$914$	32.4384	1.07297
$915$	−32.0000	−1.05789
$916$	21.1231	0.697927
$917$	−25.3693	−0.837769
$918$	−19.8078	−0.653754
$919$	28.4384	0.938098	0.469049	−	0.883172i	$-0.344597\pi$
$919$	28.4384	0.938098	0.469049	+	0.883172i	$0.344597\pi$
$920$	11.1231	0.366718
$921$	8.98485	0.296061
$922$	−22.7386	−0.748857
$923$	−21.3693	−0.703380
$924$	5.56155	0.182962
$925$	3.12311	0.102687
$926$	36.4924	1.19922
$927$	6.38447	0.209694
$928$	−6.68466	−0.219435
$929$	−16.9309	−0.555484	−0.277742	−	0.960656i	$-0.589586\pi$
$929$	−16.9309	−0.555484	−0.277742	+	0.960656i	$0.589586\pi$
$930$	−6.24621	−0.204821
$931$	0	0
$932$	20.7386	0.679317
$933$	24.6847	0.808139
$934$	26.2462	0.858802
$935$	7.12311	0.232950
$936$	−2.00000	−0.0653720
$937$	−54.3002	−1.77391	−0.886955	−	0.461856i	$-0.847184\pi$
$937$	−54.3002	−1.77391	−0.886955	+	0.461856i	$0.847184\pi$
$938$	−16.6847	−0.544773
$939$	5.56155	0.181494
$940$	16.0000	0.521862
$941$	−20.4384	−0.666274	−0.333137	−	0.942878i	$-0.608107\pi$
$941$	−20.4384	−0.666274	−0.333137	+	0.942878i	$0.608107\pi$
$942$	28.8769	0.940860
$943$	−11.1231	−0.362218
$944$	−4.68466	−0.152473
$945$	−39.6155	−1.28869
$946$	0	0
$947$	28.9848	0.941881	0.470940	−	0.882165i	$-0.343915\pi$
$947$	28.9848	0.941881	0.470940	+	0.882165i	$0.343915\pi$
$948$	6.24621	0.202868
$949$	9.56155	0.310381
$950$	0	0
$951$	−46.9309	−1.52184
$952$	12.6847	0.411112
$953$	9.12311	0.295526	0.147763	−	0.989023i	$-0.452793\pi$
$953$	9.12311	0.295526	0.147763	+	0.989023i	$0.452793\pi$
$954$	−4.38447	−0.141953
$955$	−41.3693	−1.33868
$956$	−4.93087	−0.159476
$957$	−10.4384	−0.337427
$958$	11.3693	0.367326
$959$	30.0540	0.970493
$960$	3.12311	0.100798
$961$	−27.0000	−0.870968
$962$	−11.1231	−0.358623
$963$	10.6307	0.342569
$964$	−21.6155	−0.696189
$965$	2.24621	0.0723081
$966$	30.9309	0.995184
$967$	−3.86174	−0.124185	−0.0620926	−	0.998070i	$-0.519777\pi$
$967$	−3.86174	−0.124185	−0.0620926	+	0.998070i	$0.519777\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−2.24621	−0.0721215
$971$	14.2462	0.457183	0.228591	−	0.973522i	$-0.426588\pi$
$971$	14.2462	0.457183	0.228591	+	0.973522i	$0.426588\pi$
$972$	5.75379	0.184553
$973$	−61.8617	−1.98320
$974$	−22.8769	−0.733023
$975$	−5.56155	−0.178112
$976$	−10.2462	−0.327973
$977$	−55.3693	−1.77142	−0.885711	−	0.464238i	$-0.846328\pi$
$977$	−55.3693	−1.77142	−0.885711	+	0.464238i	$0.846328\pi$
$978$	−20.8769	−0.667569
$979$	9.12311	0.291576
$980$	11.3693	0.363180
$981$	−10.4924	−0.334997
$982$	−14.6307	−0.466884
$983$	48.2462	1.53882	0.769408	−	0.638758i	$-0.220552\pi$
$983$	48.2462	1.53882	0.769408	+	0.638758i	$0.220552\pi$
$984$	−3.12311	−0.0995610
$985$	3.50758	0.111761
$986$	−23.8078	−0.758194
$987$	44.4924	1.41621
$988$	0	0
$989$	0	0
$990$	−1.12311	−0.0356946
$991$	−21.1231	−0.670998	−0.335499	−	0.942041i	$-0.608905\pi$
$991$	−21.1231	−0.670998	−0.335499	+	0.942041i	$0.608905\pi$
$992$	−2.00000	−0.0635001
$993$	16.3002	0.517271
$994$	−21.3693	−0.677794
$995$	−1.36932	−0.0434103
$996$	−3.50758	−0.111142
$997$	−53.3693	−1.69022	−0.845112	−	0.534590i	$-0.820466\pi$
$997$	−53.3693	−1.69022	−0.845112	+	0.534590i	$0.820466\pi$
$998$	26.2462	0.830809
$999$	17.3693	0.549541

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.x.1.2		2
19.18	odd	2		418.2.a.e.1.1	✓	2
57.56	even	2		3762.2.a.y.1.2		2
76.75	even	2		3344.2.a.k.1.2		2
209.208	even	2		4598.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.e.1.1	✓	2	19.18	odd	2
3344.2.a.k.1.2		2	76.75	even	2
3762.2.a.y.1.2		2	57.56	even	2
4598.2.a.bj.1.1		2	209.208	even	2
7942.2.a.x.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.56155 q^{6} +3.56155 q^{7} +1.00000 q^{8} -0.561553 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.56155 q^{6} +3.56155 q^{7} +1.00000 q^{8} -0.561553 q^{9} +2.00000 q^{10} +1.00000 q^{11} +1.56155 q^{12} +3.56155 q^{13} +3.56155 q^{14} +3.12311 q^{15} +1.00000 q^{16} +3.56155 q^{17} -0.561553 q^{18} +2.00000 q^{20} +5.56155 q^{21} +1.00000 q^{22} +5.56155 q^{23} +1.56155 q^{24} -1.00000 q^{25} +3.56155 q^{26} -5.56155 q^{27} +3.56155 q^{28} -6.68466 q^{29} +3.12311 q^{30} -2.00000 q^{31} +1.00000 q^{32} +1.56155 q^{33} +3.56155 q^{34} +7.12311 q^{35} -0.561553 q^{36} -3.12311 q^{37} +5.56155 q^{39} +2.00000 q^{40} -2.00000 q^{41} +5.56155 q^{42} +1.00000 q^{44} -1.12311 q^{45} +5.56155 q^{46} +8.00000 q^{47} +1.56155 q^{48} +5.68466 q^{49} -1.00000 q^{50} +5.56155 q^{51} +3.56155 q^{52} +7.80776 q^{53} -5.56155 q^{54} +2.00000 q^{55} +3.56155 q^{56} -6.68466 q^{58} -4.68466 q^{59} +3.12311 q^{60} -10.2462 q^{61} -2.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +7.12311 q^{65} +1.56155 q^{66} -4.68466 q^{67} +3.56155 q^{68} +8.68466 q^{69} +7.12311 q^{70} -6.00000 q^{71} -0.561553 q^{72} +2.68466 q^{73} -3.12311 q^{74} -1.56155 q^{75} +3.56155 q^{77} +5.56155 q^{78} +4.00000 q^{79} +2.00000 q^{80} -7.00000 q^{81} -2.00000 q^{82} -2.24621 q^{83} +5.56155 q^{84} +7.12311 q^{85} -10.4384 q^{87} +1.00000 q^{88} +9.12311 q^{89} -1.12311 q^{90} +12.6847 q^{91} +5.56155 q^{92} -3.12311 q^{93} +8.00000 q^{94} +1.56155 q^{96} -1.12311 q^{97} +5.68466 q^{98} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 4 * q^5 - q^6 + 3 * q^7 + 2 * q^8 + 3 * q^9	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} + 2 q^{8} + 3 q^{9} + 4 q^{10} + 2 q^{11} - q^{12} + 3 q^{13} + 3 q^{14} - 2 q^{15} + 2 q^{16} + 3 q^{17} + 3 q^{18} + 4 q^{20} + 7 q^{21} + 2 q^{22} + 7 q^{23} - q^{24} - 2 q^{25} + 3 q^{26} - 7 q^{27} + 3 q^{28} - q^{29} - 2 q^{30} - 4 q^{31} + 2 q^{32} - q^{33} + 3 q^{34} + 6 q^{35} + 3 q^{36} + 2 q^{37} + 7 q^{39} + 4 q^{40} - 4 q^{41} + 7 q^{42} + 2 q^{44} + 6 q^{45} + 7 q^{46} + 16 q^{47} - q^{48} - q^{49} - 2 q^{50} + 7 q^{51} + 3 q^{52} - 5 q^{53} - 7 q^{54} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 2 q^{60} - 4 q^{61} - 4 q^{62} - 4 q^{63} + 2 q^{64} + 6 q^{65} - q^{66} + 3 q^{67} + 3 q^{68} + 5 q^{69} + 6 q^{70} - 12 q^{71} + 3 q^{72} - 7 q^{73} + 2 q^{74} + q^{75} + 3 q^{77} + 7 q^{78} + 8 q^{79} + 4 q^{80} - 14 q^{81} - 4 q^{82} + 12 q^{83} + 7 q^{84} + 6 q^{85} - 25 q^{87} + 2 q^{88} + 10 q^{89} + 6 q^{90} + 13 q^{91} + 7 q^{92} + 2 q^{93} + 16 q^{94} - q^{96} + 6 q^{97} - q^{98} + 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 4 * q^5 - q^6 + 3 * q^7 + 2 * q^8 + 3 * q^9 + 4 * q^10 + 2 * q^11 - q^12 + 3 * q^13 + 3 * q^14 - 2 * q^15 + 2 * q^16 + 3 * q^17 + 3 * q^18 + 4 * q^20 + 7 * q^21 + 2 * q^22 + 7 * q^23 - q^24 - 2 * q^25 + 3 * q^26 - 7 * q^27 + 3 * q^28 - q^29 - 2 * q^30 - 4 * q^31 + 2 * q^32 - q^33 + 3 * q^34 + 6 * q^35 + 3 * q^36 + 2 * q^37 + 7 * q^39 + 4 * q^40 - 4 * q^41 + 7 * q^42 + 2 * q^44 + 6 * q^45 + 7 * q^46 + 16 * q^47 - q^48 - q^49 - 2 * q^50 + 7 * q^51 + 3 * q^52 - 5 * q^53 - 7 * q^54 + 4 * q^55 + 3 * q^56 - q^58 + 3 * q^59 - 2 * q^60 - 4 * q^61 - 4 * q^62 - 4 * q^63 + 2 * q^64 + 6 * q^65 - q^66 + 3 * q^67 + 3 * q^68 + 5 * q^69 + 6 * q^70 - 12 * q^71 + 3 * q^72 - 7 * q^73 + 2 * q^74 + q^75 + 3 * q^77 + 7 * q^78 + 8 * q^79 + 4 * q^80 - 14 * q^81 - 4 * q^82 + 12 * q^83 + 7 * q^84 + 6 * q^85 - 25 * q^87 + 2 * q^88 + 10 * q^89 + 6 * q^90 + 13 * q^91 + 7 * q^92 + 2 * q^93 + 16 * q^94 - q^96 + 6 * q^97 - q^98 + 3 * q^99

Introduction

L-functions

Modular forms

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Database

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