LMFDB - Embedded newform 5070.2.a.br.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.19806 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.19806 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -0.554958 q^{11} -1.00000 q^{12} -1.19806 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.45473 q^{17} +1.00000 q^{18} +5.40581 q^{19} -1.00000 q^{20} +1.19806 q^{21} -0.554958 q^{22} +7.96077 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.19806 q^{28} -6.89977 q^{29} +1.00000 q^{30} -3.04892 q^{31} +1.00000 q^{32} +0.554958 q^{33} -6.45473 q^{34} +1.19806 q^{35} +1.00000 q^{36} +7.96615 q^{37} +5.40581 q^{38} -1.00000 q^{40} -3.08815 q^{41} +1.19806 q^{42} +1.15883 q^{43} -0.554958 q^{44} -1.00000 q^{45} +7.96077 q^{46} -12.2620 q^{47} -1.00000 q^{48} -5.56465 q^{49} +1.00000 q^{50} +6.45473 q^{51} +8.47219 q^{53} -1.00000 q^{54} +0.554958 q^{55} -1.19806 q^{56} -5.40581 q^{57} -6.89977 q^{58} -1.45712 q^{59} +1.00000 q^{60} +2.76271 q^{61} -3.04892 q^{62} -1.19806 q^{63} +1.00000 q^{64} +0.554958 q^{66} +3.34481 q^{67} -6.45473 q^{68} -7.96077 q^{69} +1.19806 q^{70} -2.35690 q^{71} +1.00000 q^{72} -13.3448 q^{73} +7.96615 q^{74} -1.00000 q^{75} +5.40581 q^{76} +0.664874 q^{77} -0.990311 q^{79} -1.00000 q^{80} +1.00000 q^{81} -3.08815 q^{82} -5.88471 q^{83} +1.19806 q^{84} +6.45473 q^{85} +1.15883 q^{86} +6.89977 q^{87} -0.554958 q^{88} -13.7845 q^{89} -1.00000 q^{90} +7.96077 q^{92} +3.04892 q^{93} -12.2620 q^{94} -5.40581 q^{95} -1.00000 q^{96} -7.07606 q^{97} -5.56465 q^{98} -0.554958 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 8 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 8 * q^7 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 8 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 2 q^{11} - 3 q^{12} - 8 q^{14} + 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 3 q^{19} - 3 q^{20} + 8 q^{21} - 2 q^{22} + 11 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 8 q^{28} + 2 q^{29} + 3 q^{30} + 3 q^{32} + 2 q^{33} + 3 q^{34} + 8 q^{35} + 3 q^{36} + 8 q^{37} + 3 q^{38} - 3 q^{40} - 13 q^{41} + 8 q^{42} - 5 q^{43} - 2 q^{44} - 3 q^{45} + 11 q^{46} - 7 q^{47} - 3 q^{48} + 5 q^{49} + 3 q^{50} - 3 q^{51} + 19 q^{53} - 3 q^{54} + 2 q^{55} - 8 q^{56} - 3 q^{57} + 2 q^{58} - 23 q^{59} + 3 q^{60} - 9 q^{61} - 8 q^{63} + 3 q^{64} + 2 q^{66} - 13 q^{67} + 3 q^{68} - 11 q^{69} + 8 q^{70} - 3 q^{71} + 3 q^{72} - 17 q^{73} + 8 q^{74} - 3 q^{75} + 3 q^{76} + 3 q^{77} - 25 q^{79} - 3 q^{80} + 3 q^{81} - 13 q^{82} - 20 q^{83} + 8 q^{84} - 3 q^{85} - 5 q^{86} - 2 q^{87} - 2 q^{88} - 21 q^{89} - 3 q^{90} + 11 q^{92} - 7 q^{94} - 3 q^{95} - 3 q^{96} - 6 q^{97} + 5 q^{98} - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 8 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 - 2 * q^11 - 3 * q^12 - 8 * q^14 + 3 * q^15 + 3 * q^16 + 3 * q^17 + 3 * q^18 + 3 * q^19 - 3 * q^20 + 8 * q^21 - 2 * q^22 + 11 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 - 8 * q^28 + 2 * q^29 + 3 * q^30 + 3 * q^32 + 2 * q^33 + 3 * q^34 + 8 * q^35 + 3 * q^36 + 8 * q^37 + 3 * q^38 - 3 * q^40 - 13 * q^41 + 8 * q^42 - 5 * q^43 - 2 * q^44 - 3 * q^45 + 11 * q^46 - 7 * q^47 - 3 * q^48 + 5 * q^49 + 3 * q^50 - 3 * q^51 + 19 * q^53 - 3 * q^54 + 2 * q^55 - 8 * q^56 - 3 * q^57 + 2 * q^58 - 23 * q^59 + 3 * q^60 - 9 * q^61 - 8 * q^63 + 3 * q^64 + 2 * q^66 - 13 * q^67 + 3 * q^68 - 11 * q^69 + 8 * q^70 - 3 * q^71 + 3 * q^72 - 17 * q^73 + 8 * q^74 - 3 * q^75 + 3 * q^76 + 3 * q^77 - 25 * q^79 - 3 * q^80 + 3 * q^81 - 13 * q^82 - 20 * q^83 + 8 * q^84 - 3 * q^85 - 5 * q^86 - 2 * q^87 - 2 * q^88 - 21 * q^89 - 3 * q^90 + 11 * q^92 - 7 * q^94 - 3 * q^95 - 3 * q^96 - 6 * q^97 + 5 * q^98 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	−1.19806	−0.452825	−0.226412	−	0.974032i	$-0.572700\pi$
$7$	−1.19806	−0.452825	−0.226412	+	0.974032i	$0.572700\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−0.554958	−0.167326	−0.0836631	−	0.996494i	$-0.526662\pi$
$11$	−0.554958	−0.167326	−0.0836631	+	0.996494i	$0.526662\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−1.19806	−0.320196
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−6.45473	−1.56550	−0.782751	−	0.622335i	$-0.786184\pi$
$17$	−6.45473	−1.56550	−0.782751	+	0.622335i	$0.786184\pi$
$18$	1.00000	0.235702
$19$	5.40581	1.24018	0.620089	−	0.784531i	$-0.287096\pi$
$19$	5.40581	1.24018	0.620089	+	0.784531i	$0.287096\pi$
$20$	−1.00000	−0.223607
$21$	1.19806	0.261439
$22$	−0.554958	−0.118317
$23$	7.96077	1.65994	0.829968	−	0.557811i	$-0.188359\pi$
$23$	7.96077	1.65994	0.829968	+	0.557811i	$0.188359\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−1.19806	−0.226412
$29$	−6.89977	−1.28126	−0.640628	−	0.767852i	$-0.721326\pi$
$29$	−6.89977	−1.28126	−0.640628	+	0.767852i	$0.721326\pi$
$30$	1.00000	0.182574
$31$	−3.04892	−0.547602	−0.273801	−	0.961786i	$-0.588281\pi$
$31$	−3.04892	−0.547602	−0.273801	+	0.961786i	$0.588281\pi$
$32$	1.00000	0.176777
$33$	0.554958	0.0966058
$34$	−6.45473	−1.10698
$35$	1.19806	0.202509
$36$	1.00000	0.166667
$37$	7.96615	1.30963	0.654813	−	0.755791i	$-0.272747\pi$
$37$	7.96615	1.30963	0.654813	+	0.755791i	$0.272747\pi$
$38$	5.40581	0.876939
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−3.08815	−0.482287	−0.241144	−	0.970489i	$-0.577523\pi$
$41$	−3.08815	−0.482287	−0.241144	+	0.970489i	$0.577523\pi$
$42$	1.19806	0.184865
$43$	1.15883	0.176720	0.0883602	−	0.996089i	$-0.471837\pi$
$43$	1.15883	0.176720	0.0883602	+	0.996089i	$0.471837\pi$
$44$	−0.554958	−0.0836631
$45$	−1.00000	−0.149071
$46$	7.96077	1.17375
$47$	−12.2620	−1.78860	−0.894302	−	0.447465i	$-0.852327\pi$
$47$	−12.2620	−1.78860	−0.894302	+	0.447465i	$0.852327\pi$
$48$	−1.00000	−0.144338
$49$	−5.56465	−0.794950
$50$	1.00000	0.141421
$51$	6.45473	0.903843
$52$	0	0
$53$	8.47219	1.16374	0.581872	−	0.813280i	$-0.302320\pi$
$53$	8.47219	1.16374	0.581872	+	0.813280i	$0.302320\pi$
$54$	−1.00000	−0.136083
$55$	0.554958	0.0748305
$56$	−1.19806	−0.160098
$57$	−5.40581	−0.716017
$58$	−6.89977	−0.905985
$59$	−1.45712	−0.189701	−0.0948507	−	0.995492i	$-0.530237\pi$
$59$	−1.45712	−0.189701	−0.0948507	+	0.995492i	$0.530237\pi$
$60$	1.00000	0.129099
$61$	2.76271	0.353729	0.176864	−	0.984235i	$-0.443405\pi$
$61$	2.76271	0.353729	0.176864	+	0.984235i	$0.443405\pi$
$62$	−3.04892	−0.387213
$63$	−1.19806	−0.150942
$64$	1.00000	0.125000
$65$	0	0
$66$	0.554958	0.0683106
$67$	3.34481	0.408634	0.204317	−	0.978905i	$-0.434503\pi$
$67$	3.34481	0.408634	0.204317	+	0.978905i	$0.434503\pi$
$68$	−6.45473	−0.782751
$69$	−7.96077	−0.958364
$70$	1.19806	0.143196
$71$	−2.35690	−0.279712	−0.139856	−	0.990172i	$-0.544664\pi$
$71$	−2.35690	−0.279712	−0.139856	+	0.990172i	$0.544664\pi$
$72$	1.00000	0.117851
$73$	−13.3448	−1.56189	−0.780946	−	0.624598i	$-0.785263\pi$
$73$	−13.3448	−1.56189	−0.780946	+	0.624598i	$0.785263\pi$
$74$	7.96615	0.926046
$75$	−1.00000	−0.115470
$76$	5.40581	0.620089
$77$	0.664874	0.0757695
$78$	0	0
$79$	−0.990311	−0.111419	−0.0557094	−	0.998447i	$-0.517742\pi$
$79$	−0.990311	−0.111419	−0.0557094	+	0.998447i	$0.517742\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−3.08815	−0.341029
$83$	−5.88471	−0.645931	−0.322965	−	0.946411i	$-0.604680\pi$
$83$	−5.88471	−0.645931	−0.322965	+	0.946411i	$0.604680\pi$
$84$	1.19806	0.130719
$85$	6.45473	0.700114
$86$	1.15883	0.124960
$87$	6.89977	0.739733
$88$	−0.554958	−0.0591587
$89$	−13.7845	−1.46115	−0.730576	−	0.682831i	$-0.760748\pi$
$89$	−13.7845	−1.46115	−0.730576	+	0.682831i	$0.760748\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	7.96077	0.829968
$93$	3.04892	0.316158
$94$	−12.2620	−1.26473
$95$	−5.40581	−0.554625
$96$	−1.00000	−0.102062
$97$	−7.07606	−0.718465	−0.359233	−	0.933248i	$-0.616962\pi$
$97$	−7.07606	−0.718465	−0.359233	+	0.933248i	$0.616962\pi$
$98$	−5.56465	−0.562114
$99$	−0.554958	−0.0557754
$100$	1.00000	0.100000
$101$	4.09783	0.407750	0.203875	−	0.978997i	$-0.434646\pi$
$101$	4.09783	0.407750	0.203875	+	0.978997i	$0.434646\pi$
$102$	6.45473	0.639114
$103$	−4.16421	−0.410312	−0.205156	−	0.978729i	$-0.565770\pi$
$103$	−4.16421	−0.410312	−0.205156	+	0.978729i	$0.565770\pi$
$104$	0	0
$105$	−1.19806	−0.116919
$106$	8.47219	0.822892
$107$	−4.98254	−0.481680	−0.240840	−	0.970565i	$-0.577423\pi$
$107$	−4.98254	−0.481680	−0.240840	+	0.970565i	$0.577423\pi$
$108$	−1.00000	−0.0962250
$109$	−1.31336	−0.125797	−0.0628983	−	0.998020i	$-0.520034\pi$
$109$	−1.31336	−0.125797	−0.0628983	+	0.998020i	$0.520034\pi$
$110$	0.554958	0.0529132
$111$	−7.96615	−0.756113
$112$	−1.19806	−0.113206
$113$	3.55496	0.334422	0.167211	−	0.985921i	$-0.446524\pi$
$113$	3.55496	0.334422	0.167211	+	0.985921i	$0.446524\pi$
$114$	−5.40581	−0.506301
$115$	−7.96077	−0.742346
$116$	−6.89977	−0.640628
$117$	0	0
$118$	−1.45712	−0.134139
$119$	7.73317	0.708898
$120$	1.00000	0.0912871
$121$	−10.6920	−0.972002
$122$	2.76271	0.250124
$123$	3.08815	0.278449
$124$	−3.04892	−0.273801
$125$	−1.00000	−0.0894427
$126$	−1.19806	−0.106732
$127$	21.8756	1.94115	0.970573	−	0.240806i	$-0.0774118\pi$
$127$	21.8756	1.94115	0.970573	+	0.240806i	$0.0774118\pi$
$128$	1.00000	0.0883883
$129$	−1.15883	−0.102030
$130$	0	0
$131$	−21.6528	−1.89181	−0.945907	−	0.324439i	$-0.894825\pi$
$131$	−21.6528	−1.89181	−0.945907	+	0.324439i	$0.894825\pi$
$132$	0.554958	0.0483029
$133$	−6.47650	−0.561584
$134$	3.34481	0.288948
$135$	1.00000	0.0860663
$136$	−6.45473	−0.553489
$137$	−7.08815	−0.605581	−0.302791	−	0.953057i	$-0.597918\pi$
$137$	−7.08815	−0.605581	−0.302791	+	0.953057i	$0.597918\pi$
$138$	−7.96077	−0.677666
$139$	0.340502	0.0288810	0.0144405	−	0.999896i	$-0.495403\pi$
$139$	0.340502	0.0288810	0.0144405	+	0.999896i	$0.495403\pi$
$140$	1.19806	0.101255
$141$	12.2620	1.03265
$142$	−2.35690	−0.197786
$143$	0	0
$144$	1.00000	0.0833333
$145$	6.89977	0.572995
$146$	−13.3448	−1.10442
$147$	5.56465	0.458964
$148$	7.96615	0.654813
$149$	−7.40581	−0.606708	−0.303354	−	0.952878i	$-0.598106\pi$
$149$	−7.40581	−0.606708	−0.303354	+	0.952878i	$0.598106\pi$
$150$	−1.00000	−0.0816497
$151$	−10.4209	−0.848039	−0.424020	−	0.905653i	$-0.639381\pi$
$151$	−10.4209	−0.848039	−0.424020	+	0.905653i	$0.639381\pi$
$152$	5.40581	0.438469
$153$	−6.45473	−0.521834
$154$	0.664874	0.0535771
$155$	3.04892	0.244895
$156$	0	0
$157$	−19.0151	−1.51757	−0.758784	−	0.651343i	$-0.774206\pi$
$157$	−19.0151	−1.51757	−0.758784	+	0.651343i	$0.774206\pi$
$158$	−0.990311	−0.0787849
$159$	−8.47219	−0.671888
$160$	−1.00000	−0.0790569
$161$	−9.53750	−0.751660
$162$	1.00000	0.0785674
$163$	−5.65279	−0.442761	−0.221380	−	0.975188i	$-0.571056\pi$
$163$	−5.65279	−0.442761	−0.221380	+	0.975188i	$0.571056\pi$
$164$	−3.08815	−0.241144
$165$	−0.554958	−0.0432034
$166$	−5.88471	−0.456742
$167$	−2.15883	−0.167056	−0.0835278	−	0.996505i	$-0.526619\pi$
$167$	−2.15883	−0.167056	−0.0835278	+	0.996505i	$0.526619\pi$
$168$	1.19806	0.0924325
$169$	0	0
$170$	6.45473	0.495055
$171$	5.40581	0.413393
$172$	1.15883	0.0883602
$173$	21.6722	1.64770	0.823852	−	0.566805i	$-0.191821\pi$
$173$	21.6722	1.64770	0.823852	+	0.566805i	$0.191821\pi$
$174$	6.89977	0.523070
$175$	−1.19806	−0.0905650
$176$	−0.554958	−0.0418315
$177$	1.45712	0.109524
$178$	−13.7845	−1.03319
$179$	−0.956459	−0.0714891	−0.0357446	−	0.999361i	$-0.511380\pi$
$179$	−0.956459	−0.0714891	−0.0357446	+	0.999361i	$0.511380\pi$
$180$	−1.00000	−0.0745356
$181$	0.405813	0.0301639	0.0150819	−	0.999886i	$-0.495199\pi$
$181$	0.405813	0.0301639	0.0150819	+	0.999886i	$0.495199\pi$
$182$	0	0
$183$	−2.76271	−0.204225
$184$	7.96077	0.586876
$185$	−7.96615	−0.585683
$186$	3.04892	0.223557
$187$	3.58211	0.261949
$188$	−12.2620	−0.894302
$189$	1.19806	0.0871462
$190$	−5.40581	−0.392179
$191$	−19.0097	−1.37549	−0.687746	−	0.725951i	$-0.741400\pi$
$191$	−19.0097	−1.37549	−0.687746	+	0.725951i	$0.741400\pi$
$192$	−1.00000	−0.0721688
$193$	−0.599564	−0.0431575	−0.0215788	−	0.999767i	$-0.506869\pi$
$193$	−0.599564	−0.0431575	−0.0215788	+	0.999767i	$0.506869\pi$
$194$	−7.07606	−0.508032
$195$	0	0
$196$	−5.56465	−0.397475
$197$	−3.78448	−0.269633	−0.134816	−	0.990871i	$-0.543044\pi$
$197$	−3.78448	−0.269633	−0.134816	+	0.990871i	$0.543044\pi$
$198$	−0.554958	−0.0394392
$199$	−10.2349	−0.725533	−0.362766	−	0.931880i	$-0.618168\pi$
$199$	−10.2349	−0.725533	−0.362766	+	0.931880i	$0.618168\pi$
$200$	1.00000	0.0707107
$201$	−3.34481	−0.235925
$202$	4.09783	0.288323
$203$	8.26636	0.580185
$204$	6.45473	0.451922
$205$	3.08815	0.215685
$206$	−4.16421	−0.290134
$207$	7.96077	0.553312
$208$	0	0
$209$	−3.00000	−0.207514
$210$	−1.19806	−0.0826742
$211$	−8.71379	−0.599882	−0.299941	−	0.953958i	$-0.596967\pi$
$211$	−8.71379	−0.599882	−0.299941	+	0.953958i	$0.596967\pi$
$212$	8.47219	0.581872
$213$	2.35690	0.161492
$214$	−4.98254	−0.340600
$215$	−1.15883	−0.0790318
$216$	−1.00000	−0.0680414
$217$	3.65279	0.247968
$218$	−1.31336	−0.0889516
$219$	13.3448	0.901759
$220$	0.554958	0.0374153
$221$	0	0
$222$	−7.96615	−0.534653
$223$	15.0694	1.00912	0.504559	−	0.863377i	$-0.331655\pi$
$223$	15.0694	1.00912	0.504559	+	0.863377i	$0.331655\pi$
$224$	−1.19806	−0.0800489
$225$	1.00000	0.0666667
$226$	3.55496	0.236472
$227$	20.4577	1.35783	0.678913	−	0.734219i	$-0.262451\pi$
$227$	20.4577	1.35783	0.678913	+	0.734219i	$0.262451\pi$
$228$	−5.40581	−0.358009
$229$	−18.7657	−1.24007	−0.620036	−	0.784573i	$-0.712882\pi$
$229$	−18.7657	−1.24007	−0.620036	+	0.784573i	$0.712882\pi$
$230$	−7.96077	−0.524918
$231$	−0.664874	−0.0437455
$232$	−6.89977	−0.452992
$233$	23.6353	1.54840	0.774201	−	0.632940i	$-0.218152\pi$
$233$	23.6353	1.54840	0.774201	+	0.632940i	$0.218152\pi$
$234$	0	0
$235$	12.2620	0.799888
$236$	−1.45712	−0.0948507
$237$	0.990311	0.0643276
$238$	7.73317	0.501267
$239$	−28.7506	−1.85972	−0.929862	−	0.367909i	$-0.880074\pi$
$239$	−28.7506	−1.85972	−0.929862	+	0.367909i	$0.880074\pi$
$240$	1.00000	0.0645497
$241$	3.89440	0.250860	0.125430	−	0.992102i	$-0.459969\pi$
$241$	3.89440	0.250860	0.125430	+	0.992102i	$0.459969\pi$
$242$	−10.6920	−0.687309
$243$	−1.00000	−0.0641500
$244$	2.76271	0.176864
$245$	5.56465	0.355512
$246$	3.08815	0.196893
$247$	0	0
$248$	−3.04892	−0.193606
$249$	5.88471	0.372928
$250$	−1.00000	−0.0632456
$251$	−24.3327	−1.53587	−0.767934	−	0.640529i	$-0.778715\pi$
$251$	−24.3327	−1.53587	−0.767934	+	0.640529i	$0.778715\pi$
$252$	−1.19806	−0.0754708
$253$	−4.41789	−0.277751
$254$	21.8756	1.37260
$255$	−6.45473	−0.404211
$256$	1.00000	0.0625000
$257$	30.4426	1.89896	0.949480	−	0.313827i	$-0.101611\pi$
$257$	30.4426	1.89896	0.949480	+	0.313827i	$0.101611\pi$
$258$	−1.15883	−0.0721458
$259$	−9.54394	−0.593032
$260$	0	0
$261$	−6.89977	−0.427085
$262$	−21.6528	−1.33771
$263$	−9.33944	−0.575894	−0.287947	−	0.957646i	$-0.592973\pi$
$263$	−9.33944	−0.575894	−0.287947	+	0.957646i	$0.592973\pi$
$264$	0.554958	0.0341553
$265$	−8.47219	−0.520442
$266$	−6.47650	−0.397100
$267$	13.7845	0.843596
$268$	3.34481	0.204317
$269$	26.7821	1.63293	0.816466	−	0.577393i	$-0.195930\pi$
$269$	26.7821	1.63293	0.816466	+	0.577393i	$0.195930\pi$
$270$	1.00000	0.0608581
$271$	2.58748	0.157178	0.0785892	−	0.996907i	$-0.474958\pi$
$271$	2.58748	0.157178	0.0785892	+	0.996907i	$0.474958\pi$
$272$	−6.45473	−0.391376
$273$	0	0
$274$	−7.08815	−0.428211
$275$	−0.554958	−0.0334652
$276$	−7.96077	−0.479182
$277$	11.0532	0.664124	0.332062	−	0.943258i	$-0.392256\pi$
$277$	11.0532	0.664124	0.332062	+	0.943258i	$0.392256\pi$
$278$	0.340502	0.0204220
$279$	−3.04892	−0.182534
$280$	1.19806	0.0715979
$281$	−10.5090	−0.626916	−0.313458	−	0.949602i	$-0.601487\pi$
$281$	−10.5090	−0.626916	−0.313458	+	0.949602i	$0.601487\pi$
$282$	12.2620	0.730194
$283$	−1.07069	−0.0636458	−0.0318229	−	0.999494i	$-0.510131\pi$
$283$	−1.07069	−0.0636458	−0.0318229	+	0.999494i	$0.510131\pi$
$284$	−2.35690	−0.139856
$285$	5.40581	0.320213
$286$	0	0
$287$	3.69979	0.218392
$288$	1.00000	0.0589256
$289$	24.6635	1.45080
$290$	6.89977	0.405169
$291$	7.07606	0.414806
$292$	−13.3448	−0.780946
$293$	−27.4142	−1.60155	−0.800777	−	0.598963i	$-0.795580\pi$
$293$	−27.4142	−1.60155	−0.800777	+	0.598963i	$0.795580\pi$
$294$	5.56465	0.324537
$295$	1.45712	0.0848370
$296$	7.96615	0.463023
$297$	0.554958	0.0322019
$298$	−7.40581	−0.429007
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−1.38835	−0.0800234
$302$	−10.4209	−0.599654
$303$	−4.09783	−0.235414
$304$	5.40581	0.310045
$305$	−2.76271	−0.158192
$306$	−6.45473	−0.368992
$307$	−24.4969	−1.39811	−0.699057	−	0.715066i	$-0.746397\pi$
$307$	−24.4969	−1.39811	−0.699057	+	0.715066i	$0.746397\pi$
$308$	0.664874	0.0378847
$309$	4.16421	0.236894
$310$	3.04892	0.173167
$311$	16.4403	0.932241	0.466121	−	0.884721i	$-0.345651\pi$
$311$	16.4403	0.932241	0.466121	+	0.884721i	$0.345651\pi$
$312$	0	0
$313$	−26.0320	−1.47142	−0.735709	−	0.677298i	$-0.763151\pi$
$313$	−26.0320	−1.47142	−0.735709	+	0.677298i	$0.763151\pi$
$314$	−19.0151	−1.07308
$315$	1.19806	0.0675032
$316$	−0.990311	−0.0557094
$317$	−19.3545	−1.08706	−0.543529	−	0.839391i	$-0.682912\pi$
$317$	−19.3545	−1.08706	−0.543529	+	0.839391i	$0.682912\pi$
$318$	−8.47219	−0.475097
$319$	3.82908	0.214388
$320$	−1.00000	−0.0559017
$321$	4.98254	0.278098
$322$	−9.53750	−0.531504
$323$	−34.8931	−1.94150
$324$	1.00000	0.0555556
$325$	0	0
$326$	−5.65279	−0.313079
$327$	1.31336	0.0726287
$328$	−3.08815	−0.170514
$329$	14.6907	0.809924
$330$	−0.554958	−0.0305494
$331$	0.173899	0.00955836	0.00477918	−	0.999989i	$-0.498479\pi$
$331$	0.173899	0.00955836	0.00477918	+	0.999989i	$0.498479\pi$
$332$	−5.88471	−0.322965
$333$	7.96615	0.436542
$334$	−2.15883	−0.118126
$335$	−3.34481	−0.182747
$336$	1.19806	0.0653597
$337$	−14.3793	−0.783288	−0.391644	−	0.920117i	$-0.628094\pi$
$337$	−14.3793	−0.783288	−0.391644	+	0.920117i	$0.628094\pi$
$338$	0	0
$339$	−3.55496	−0.193079
$340$	6.45473	0.350057
$341$	1.69202	0.0916281
$342$	5.40581	0.292313
$343$	15.0532	0.812798
$344$	1.15883	0.0624801
$345$	7.96077	0.428594
$346$	21.6722	1.16510
$347$	−26.3400	−1.41401	−0.707003	−	0.707210i	$-0.749953\pi$
$347$	−26.3400	−1.41401	−0.707003	+	0.707210i	$0.749953\pi$
$348$	6.89977	0.369867
$349$	8.44371	0.451982	0.225991	−	0.974129i	$-0.427438\pi$
$349$	8.44371	0.451982	0.225991	+	0.974129i	$0.427438\pi$
$350$	−1.19806	−0.0640391
$351$	0	0
$352$	−0.554958	−0.0295794
$353$	10.9420	0.582383	0.291192	−	0.956665i	$-0.405948\pi$
$353$	10.9420	0.582383	0.291192	+	0.956665i	$0.405948\pi$
$354$	1.45712	0.0774452
$355$	2.35690	0.125091
$356$	−13.7845	−0.730576
$357$	−7.73317	−0.409283
$358$	−0.956459	−0.0505505
$359$	−8.07606	−0.426238	−0.213119	−	0.977026i	$-0.568362\pi$
$359$	−8.07606	−0.426238	−0.213119	+	0.977026i	$0.568362\pi$
$360$	−1.00000	−0.0527046
$361$	10.2228	0.538043
$362$	0.405813	0.0213291
$363$	10.6920	0.561186
$364$	0	0
$365$	13.3448	0.698500
$366$	−2.76271	−0.144409
$367$	17.1575	0.895615	0.447807	−	0.894130i	$-0.352205\pi$
$367$	17.1575	0.895615	0.447807	+	0.894130i	$0.352205\pi$
$368$	7.96077	0.414984
$369$	−3.08815	−0.160762
$370$	−7.96615	−0.414140
$371$	−10.1502	−0.526973
$372$	3.04892	0.158079
$373$	2.72455	0.141072	0.0705358	−	0.997509i	$-0.477529\pi$
$373$	2.72455	0.141072	0.0705358	+	0.997509i	$0.477529\pi$
$374$	3.58211	0.185226
$375$	1.00000	0.0516398
$376$	−12.2620	−0.632367
$377$	0	0
$378$	1.19806	0.0616217
$379$	26.8998	1.38175	0.690874	−	0.722975i	$-0.257226\pi$
$379$	26.8998	1.38175	0.690874	+	0.722975i	$0.257226\pi$
$380$	−5.40581	−0.277312
$381$	−21.8756	−1.12072
$382$	−19.0097	−0.972620
$383$	−29.9511	−1.53043	−0.765214	−	0.643776i	$-0.777367\pi$
$383$	−29.9511	−1.53043	−0.765214	+	0.643776i	$0.777367\pi$
$384$	−1.00000	−0.0510310
$385$	−0.664874	−0.0338851
$386$	−0.599564	−0.0305170
$387$	1.15883	0.0589068
$388$	−7.07606	−0.359233
$389$	10.0140	0.507730	0.253865	−	0.967240i	$-0.418298\pi$
$389$	10.0140	0.507730	0.253865	+	0.967240i	$0.418298\pi$
$390$	0	0
$391$	−51.3846	−2.59863
$392$	−5.56465	−0.281057
$393$	21.6528	1.09224
$394$	−3.78448	−0.190659
$395$	0.990311	0.0498280
$396$	−0.554958	−0.0278877
$397$	−17.0489	−0.855661	−0.427830	−	0.903859i	$-0.640722\pi$
$397$	−17.0489	−0.855661	−0.427830	+	0.903859i	$0.640722\pi$
$398$	−10.2349	−0.513029
$399$	6.47650	0.324231
$400$	1.00000	0.0500000
$401$	−0.975246	−0.0487015	−0.0243507	−	0.999703i	$-0.507752\pi$
$401$	−0.975246	−0.0487015	−0.0243507	+	0.999703i	$0.507752\pi$
$402$	−3.34481	−0.166824
$403$	0	0
$404$	4.09783	0.203875
$405$	−1.00000	−0.0496904
$406$	8.26636	0.410252
$407$	−4.42088	−0.219135
$408$	6.45473	0.319557
$409$	37.7439	1.86632	0.933158	−	0.359465i	$-0.117041\pi$
$409$	37.7439	1.86632	0.933158	+	0.359465i	$0.117041\pi$
$410$	3.08815	0.152513
$411$	7.08815	0.349632
$412$	−4.16421	−0.205156
$413$	1.74572	0.0859015
$414$	7.96077	0.391251
$415$	5.88471	0.288869
$416$	0	0
$417$	−0.340502	−0.0166745
$418$	−3.00000	−0.146735
$419$	−21.5700	−1.05376	−0.526882	−	0.849938i	$-0.676639\pi$
$419$	−21.5700	−1.05376	−0.526882	+	0.849938i	$0.676639\pi$
$420$	−1.19806	−0.0584595
$421$	−33.2403	−1.62003	−0.810016	−	0.586408i	$-0.800542\pi$
$421$	−33.2403	−1.62003	−0.810016	+	0.586408i	$0.800542\pi$
$422$	−8.71379	−0.424181
$423$	−12.2620	−0.596201
$424$	8.47219	0.411446
$425$	−6.45473	−0.313100
$426$	2.35690	0.114192
$427$	−3.30990	−0.160177
$428$	−4.98254	−0.240840
$429$	0	0
$430$	−1.15883	−0.0558839
$431$	−20.9041	−1.00691	−0.503457	−	0.864020i	$-0.667939\pi$
$431$	−20.9041	−1.00691	−0.503457	+	0.864020i	$0.667939\pi$
$432$	−1.00000	−0.0481125
$433$	36.5864	1.75823	0.879115	−	0.476609i	$-0.158134\pi$
$433$	36.5864	1.75823	0.879115	+	0.476609i	$0.158134\pi$
$434$	3.65279	0.175340
$435$	−6.89977	−0.330819
$436$	−1.31336	−0.0628983
$437$	43.0344	2.05862
$438$	13.3448	0.637640
$439$	27.5036	1.31268	0.656339	−	0.754466i	$-0.272104\pi$
$439$	27.5036	1.31268	0.656339	+	0.754466i	$0.272104\pi$
$440$	0.554958	0.0264566
$441$	−5.56465	−0.264983
$442$	0	0
$443$	−4.34614	−0.206491	−0.103246	−	0.994656i	$-0.532923\pi$
$443$	−4.34614	−0.206491	−0.103246	+	0.994656i	$0.532923\pi$
$444$	−7.96615	−0.378057
$445$	13.7845	0.653447
$446$	15.0694	0.713555
$447$	7.40581	0.350283
$448$	−1.19806	−0.0566031
$449$	−0.423272	−0.0199754	−0.00998771	−	0.999950i	$-0.503179\pi$
$449$	−0.423272	−0.0199754	−0.00998771	+	0.999950i	$0.503179\pi$
$450$	1.00000	0.0471405
$451$	1.71379	0.0806993
$452$	3.55496	0.167211
$453$	10.4209	0.489616
$454$	20.4577	0.960128
$455$	0	0
$456$	−5.40581	−0.253150
$457$	18.6396	0.871926	0.435963	−	0.899965i	$-0.356408\pi$
$457$	18.6396	0.871926	0.435963	+	0.899965i	$0.356408\pi$
$458$	−18.7657	−0.876863
$459$	6.45473	0.301281
$460$	−7.96077	−0.371173
$461$	−8.22223	−0.382947	−0.191474	−	0.981498i	$-0.561327\pi$
$461$	−8.22223	−0.382947	−0.191474	+	0.981498i	$0.561327\pi$
$462$	−0.664874	−0.0309328
$463$	−34.1118	−1.58531	−0.792656	−	0.609670i	$-0.791302\pi$
$463$	−34.1118	−1.58531	−0.792656	+	0.609670i	$0.791302\pi$
$464$	−6.89977	−0.320314
$465$	−3.04892	−0.141390
$466$	23.6353	1.09489
$467$	13.9245	0.644350	0.322175	−	0.946680i	$-0.395586\pi$
$467$	13.9245	0.644350	0.322175	+	0.946680i	$0.395586\pi$
$468$	0	0
$469$	−4.00730	−0.185040
$470$	12.2620	0.565606
$471$	19.0151	0.876168
$472$	−1.45712	−0.0670695
$473$	−0.643104	−0.0295700
$474$	0.990311	0.0454865
$475$	5.40581	0.248036
$476$	7.73317	0.354449
$477$	8.47219	0.387915
$478$	−28.7506	−1.31502
$479$	−1.79225	−0.0818899	−0.0409450	−	0.999161i	$-0.513037\pi$
$479$	−1.79225	−0.0818899	−0.0409450	+	0.999161i	$0.513037\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	3.89440	0.177385
$483$	9.53750	0.433971
$484$	−10.6920	−0.486001
$485$	7.07606	0.321308
$486$	−1.00000	−0.0453609
$487$	0.126310	0.00572364	0.00286182	−	0.999996i	$-0.499089\pi$
$487$	0.126310	0.00572364	0.00286182	+	0.999996i	$0.499089\pi$
$488$	2.76271	0.125062
$489$	5.65279	0.255628
$490$	5.56465	0.251385
$491$	−17.6823	−0.797993	−0.398996	−	0.916953i	$-0.630641\pi$
$491$	−17.6823	−0.797993	−0.398996	+	0.916953i	$0.630641\pi$
$492$	3.08815	0.139224
$493$	44.5362	2.00581
$494$	0	0
$495$	0.554958	0.0249435
$496$	−3.04892	−0.136900
$497$	2.82371	0.126661
$498$	5.88471	0.263700
$499$	21.1535	0.946959	0.473479	−	0.880805i	$-0.342998\pi$
$499$	21.1535	0.946959	0.473479	+	0.880805i	$0.342998\pi$
$500$	−1.00000	−0.0447214
$501$	2.15883	0.0964496
$502$	−24.3327	−1.08602
$503$	−29.7187	−1.32509	−0.662546	−	0.749022i	$-0.730524\pi$
$503$	−29.7187	−1.32509	−0.662546	+	0.749022i	$0.730524\pi$
$504$	−1.19806	−0.0533659
$505$	−4.09783	−0.182351
$506$	−4.41789	−0.196399
$507$	0	0
$508$	21.8756	0.970573
$509$	−0.786872	−0.0348775	−0.0174387	−	0.999848i	$-0.505551\pi$
$509$	−0.786872	−0.0348775	−0.0174387	+	0.999848i	$0.505551\pi$
$510$	−6.45473	−0.285820
$511$	15.9879	0.707264
$512$	1.00000	0.0441942
$513$	−5.40581	−0.238672
$514$	30.4426	1.34277
$515$	4.16421	0.183497
$516$	−1.15883	−0.0510148
$517$	6.80492	0.299280
$518$	−9.54394	−0.419337
$519$	−21.6722	−0.951303
$520$	0	0
$521$	24.1782	1.05927	0.529633	−	0.848227i	$-0.322330\pi$
$521$	24.1782	1.05927	0.529633	+	0.848227i	$0.322330\pi$
$522$	−6.89977	−0.301995
$523$	23.0750	1.00900	0.504500	−	0.863412i	$-0.331677\pi$
$523$	23.0750	1.00900	0.504500	+	0.863412i	$0.331677\pi$
$524$	−21.6528	−0.945907
$525$	1.19806	0.0522877
$526$	−9.33944	−0.407219
$527$	19.6799	0.857272
$528$	0.554958	0.0241515
$529$	40.3739	1.75539
$530$	−8.47219	−0.368008
$531$	−1.45712	−0.0632338
$532$	−6.47650	−0.280792
$533$	0	0
$534$	13.7845	0.596513
$535$	4.98254	0.215414
$536$	3.34481	0.144474
$537$	0.956459	0.0412743
$538$	26.7821	1.15466
$539$	3.08815	0.133016
$540$	1.00000	0.0430331
$541$	24.6122	1.05816	0.529081	−	0.848571i	$-0.322537\pi$
$541$	24.6122	1.05816	0.529081	+	0.848571i	$0.322537\pi$
$542$	2.58748	0.111142
$543$	−0.405813	−0.0174151
$544$	−6.45473	−0.276744
$545$	1.31336	0.0562580
$546$	0	0
$547$	−0.492894	−0.0210746	−0.0105373	−	0.999944i	$-0.503354\pi$
$547$	−0.492894	−0.0210746	−0.0105373	+	0.999944i	$0.503354\pi$
$548$	−7.08815	−0.302791
$549$	2.76271	0.117910
$550$	−0.554958	−0.0236635
$551$	−37.2989	−1.58899
$552$	−7.96077	−0.338833
$553$	1.18645	0.0504532
$554$	11.0532	0.469607
$555$	7.96615	0.338144
$556$	0.340502	0.0144405
$557$	12.6028	0.533998	0.266999	−	0.963697i	$-0.413968\pi$
$557$	12.6028	0.533998	0.266999	+	0.963697i	$0.413968\pi$
$558$	−3.04892	−0.129071
$559$	0	0
$560$	1.19806	0.0506274
$561$	−3.58211	−0.151237
$562$	−10.5090	−0.443296
$563$	24.5743	1.03568	0.517842	−	0.855476i	$-0.326735\pi$
$563$	24.5743	1.03568	0.517842	+	0.855476i	$0.326735\pi$
$564$	12.2620	0.516325
$565$	−3.55496	−0.149558
$566$	−1.07069	−0.0450044
$567$	−1.19806	−0.0503139
$568$	−2.35690	−0.0988932
$569$	18.9414	0.794065	0.397032	−	0.917805i	$-0.370040\pi$
$569$	18.9414	0.794065	0.397032	+	0.917805i	$0.370040\pi$
$570$	5.40581	0.226425
$571$	−17.2978	−0.723891	−0.361946	−	0.932199i	$-0.617887\pi$
$571$	−17.2978	−0.723891	−0.361946	+	0.932199i	$0.617887\pi$
$572$	0	0
$573$	19.0097	0.794141
$574$	3.69979	0.154426
$575$	7.96077	0.331987
$576$	1.00000	0.0416667
$577$	42.5628	1.77191	0.885957	−	0.463767i	$-0.153503\pi$
$577$	42.5628	1.77191	0.885957	+	0.463767i	$0.153503\pi$
$578$	24.6635	1.02587
$579$	0.599564	0.0249170
$580$	6.89977	0.286497
$581$	7.05025	0.292493
$582$	7.07606	0.293312
$583$	−4.70171	−0.194725
$584$	−13.3448	−0.552212
$585$	0	0
$586$	−27.4142	−1.13247
$587$	−43.2737	−1.78609	−0.893047	−	0.449963i	$-0.851437\pi$
$587$	−43.2737	−1.78609	−0.893047	+	0.449963i	$0.851437\pi$
$588$	5.56465	0.229482
$589$	−16.4819	−0.679124
$590$	1.45712	0.0599888
$591$	3.78448	0.155673
$592$	7.96615	0.327407
$593$	−19.8672	−0.815850	−0.407925	−	0.913015i	$-0.633748\pi$
$593$	−19.8672	−0.815850	−0.407925	+	0.913015i	$0.633748\pi$
$594$	0.554958	0.0227702
$595$	−7.73317	−0.317029
$596$	−7.40581	−0.303354
$597$	10.2349	0.418886
$598$	0	0
$599$	22.3448	0.912984	0.456492	−	0.889727i	$-0.349106\pi$
$599$	22.3448	0.912984	0.456492	+	0.889727i	$0.349106\pi$
$600$	−1.00000	−0.0408248
$601$	38.3696	1.56513	0.782564	−	0.622571i	$-0.213912\pi$
$601$	38.3696	1.56513	0.782564	+	0.622571i	$0.213912\pi$
$602$	−1.38835	−0.0565851
$603$	3.34481	0.136211
$604$	−10.4209	−0.424020
$605$	10.6920	0.434692
$606$	−4.09783	−0.166463
$607$	34.9778	1.41970	0.709852	−	0.704351i	$-0.248762\pi$
$607$	34.9778	1.41970	0.709852	+	0.704351i	$0.248762\pi$
$608$	5.40581	0.219235
$609$	−8.26636	−0.334970
$610$	−2.76271	−0.111859
$611$	0	0
$612$	−6.45473	−0.260917
$613$	−28.1648	−1.13757	−0.568783	−	0.822488i	$-0.692586\pi$
$613$	−28.1648	−1.13757	−0.568783	+	0.822488i	$0.692586\pi$
$614$	−24.4969	−0.988616
$615$	−3.08815	−0.124526
$616$	0.664874	0.0267886
$617$	24.0640	0.968779	0.484390	−	0.874852i	$-0.339042\pi$
$617$	24.0640	0.968779	0.484390	+	0.874852i	$0.339042\pi$
$618$	4.16421	0.167509
$619$	14.7651	0.593460	0.296730	−	0.954961i	$-0.404104\pi$
$619$	14.7651	0.593460	0.296730	+	0.954961i	$0.404104\pi$
$620$	3.04892	0.122447
$621$	−7.96077	−0.319455
$622$	16.4403	0.659194
$623$	16.5147	0.661646
$624$	0	0
$625$	1.00000	0.0400000
$626$	−26.0320	−1.04045
$627$	3.00000	0.119808
$628$	−19.0151	−0.758784
$629$	−51.4193	−2.05022
$630$	1.19806	0.0477319
$631$	22.9788	0.914772	0.457386	−	0.889268i	$-0.348786\pi$
$631$	22.9788	0.914772	0.457386	+	0.889268i	$0.348786\pi$
$632$	−0.990311	−0.0393925
$633$	8.71379	0.346342
$634$	−19.3545	−0.768666
$635$	−21.8756	−0.868107
$636$	−8.47219	−0.335944
$637$	0	0
$638$	3.82908	0.151595
$639$	−2.35690	−0.0932374
$640$	−1.00000	−0.0395285
$641$	−5.34913	−0.211278	−0.105639	−	0.994405i	$-0.533689\pi$
$641$	−5.34913	−0.211278	−0.105639	+	0.994405i	$0.533689\pi$
$642$	4.98254	0.196645
$643$	−39.2984	−1.54978	−0.774889	−	0.632097i	$-0.782194\pi$
$643$	−39.2984	−1.54978	−0.774889	+	0.632097i	$0.782194\pi$
$644$	−9.53750	−0.375830
$645$	1.15883	0.0456290
$646$	−34.8931	−1.37285
$647$	0.328684	0.0129219	0.00646095	−	0.999979i	$-0.497943\pi$
$647$	0.328684	0.0129219	0.00646095	+	0.999979i	$0.497943\pi$
$648$	1.00000	0.0392837
$649$	0.808643	0.0317420
$650$	0	0
$651$	−3.65279	−0.143164
$652$	−5.65279	−0.221380
$653$	41.6426	1.62960	0.814801	−	0.579741i	$-0.196846\pi$
$653$	41.6426	1.62960	0.814801	+	0.579741i	$0.196846\pi$
$654$	1.31336	0.0513563
$655$	21.6528	0.846045
$656$	−3.08815	−0.120572
$657$	−13.3448	−0.520631
$658$	14.6907	0.572703
$659$	−4.24698	−0.165439	−0.0827194	−	0.996573i	$-0.526361\pi$
$659$	−4.24698	−0.165439	−0.0827194	+	0.996573i	$0.526361\pi$
$660$	−0.554958	−0.0216017
$661$	−29.7071	−1.15547	−0.577736	−	0.816224i	$-0.696064\pi$
$661$	−29.7071	−1.15547	−0.577736	+	0.816224i	$0.696064\pi$
$662$	0.173899	0.00675878
$663$	0	0
$664$	−5.88471	−0.228371
$665$	6.47650	0.251148
$666$	7.96615	0.308682
$667$	−54.9275	−2.12680
$668$	−2.15883	−0.0835278
$669$	−15.0694	−0.582615
$670$	−3.34481	−0.129221
$671$	−1.53319	−0.0591881
$672$	1.19806	0.0462163
$673$	−21.7004	−0.836488	−0.418244	−	0.908335i	$-0.637354\pi$
$673$	−21.7004	−0.836488	−0.418244	+	0.908335i	$0.637354\pi$
$674$	−14.3793	−0.553868
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−12.5168	−0.481059	−0.240530	−	0.970642i	$-0.577321\pi$
$677$	−12.5168	−0.481059	−0.240530	+	0.970642i	$0.577321\pi$
$678$	−3.55496	−0.136527
$679$	8.47757	0.325339
$680$	6.45473	0.247528
$681$	−20.4577	−0.783941
$682$	1.69202	0.0647909
$683$	3.82908	0.146516	0.0732579	−	0.997313i	$-0.476660\pi$
$683$	3.82908	0.146516	0.0732579	+	0.997313i	$0.476660\pi$
$684$	5.40581	0.206696
$685$	7.08815	0.270824
$686$	15.0532	0.574735
$687$	18.7657	0.715956
$688$	1.15883	0.0441801
$689$	0	0
$690$	7.96077	0.303061
$691$	−27.4239	−1.04325	−0.521626	−	0.853174i	$-0.674675\pi$
$691$	−27.4239	−1.04325	−0.521626	+	0.853174i	$0.674675\pi$
$692$	21.6722	0.823852
$693$	0.664874	0.0252565
$694$	−26.3400	−0.999854
$695$	−0.340502	−0.0129160
$696$	6.89977	0.261535
$697$	19.9332	0.755022
$698$	8.44371	0.319599
$699$	−23.6353	−0.893970
$700$	−1.19806	−0.0452825
$701$	31.5803	1.19277	0.596386	−	0.802698i	$-0.296603\pi$
$701$	31.5803	1.19277	0.596386	+	0.802698i	$0.296603\pi$
$702$	0	0
$703$	43.0635	1.62417
$704$	−0.554958	−0.0209158
$705$	−12.2620	−0.461815
$706$	10.9420	0.411807
$707$	−4.90946	−0.184639
$708$	1.45712	0.0547621
$709$	−28.9788	−1.08832	−0.544161	−	0.838981i	$-0.683152\pi$
$709$	−28.9788	−1.08832	−0.544161	+	0.838981i	$0.683152\pi$
$710$	2.35690	0.0884527
$711$	−0.990311	−0.0371396
$712$	−13.7845	−0.516595
$713$	−24.2717	−0.908984
$714$	−7.73317	−0.289407
$715$	0	0
$716$	−0.956459	−0.0357446
$717$	28.7506	1.07371
$718$	−8.07606	−0.301396
$719$	27.0670	1.00943	0.504714	−	0.863287i	$-0.331598\pi$
$719$	27.0670	1.00943	0.504714	+	0.863287i	$0.331598\pi$
$720$	−1.00000	−0.0372678
$721$	4.98898	0.185799
$722$	10.2228	0.380454
$723$	−3.89440	−0.144834
$724$	0.405813	0.0150819
$725$	−6.89977	−0.256251
$726$	10.6920	0.396818
$727$	−26.6612	−0.988807	−0.494404	−	0.869232i	$-0.664614\pi$
$727$	−26.6612	−0.988807	−0.494404	+	0.869232i	$0.664614\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	13.3448	0.493914
$731$	−7.47996	−0.276656
$732$	−2.76271	−0.102113
$733$	51.2984	1.89475	0.947375	−	0.320126i	$-0.103725\pi$
$733$	51.2984	1.89475	0.947375	+	0.320126i	$0.103725\pi$
$734$	17.1575	0.633295
$735$	−5.56465	−0.205255
$736$	7.96077	0.293438
$737$	−1.85623	−0.0683752
$738$	−3.08815	−0.113676
$739$	−28.6456	−1.05375	−0.526873	−	0.849944i	$-0.676636\pi$
$739$	−28.6456	−1.05375	−0.526873	+	0.849944i	$0.676636\pi$
$740$	−7.96615	−0.292841
$741$	0	0
$742$	−10.1502	−0.372626
$743$	43.6674	1.60200	0.801000	−	0.598664i	$-0.204301\pi$
$743$	43.6674	1.60200	0.801000	+	0.598664i	$0.204301\pi$
$744$	3.04892	0.111779
$745$	7.40581	0.271328
$746$	2.72455	0.0997527
$747$	−5.88471	−0.215310
$748$	3.58211	0.130975
$749$	5.96940	0.218117
$750$	1.00000	0.0365148
$751$	−20.1535	−0.735410	−0.367705	−	0.929942i	$-0.619856\pi$
$751$	−20.1535	−0.735410	−0.367705	+	0.929942i	$0.619856\pi$
$752$	−12.2620	−0.447151
$753$	24.3327	0.886734
$754$	0	0
$755$	10.4209	0.379255
$756$	1.19806	0.0435731
$757$	−41.8562	−1.52129	−0.760645	−	0.649168i	$-0.775117\pi$
$757$	−41.8562	−1.52129	−0.760645	+	0.649168i	$0.775117\pi$
$758$	26.8998	0.977044
$759$	4.41789	0.160359
$760$	−5.40581	−0.196089
$761$	32.8665	1.19141	0.595705	−	0.803203i	$-0.296873\pi$
$761$	32.8665	1.19141	0.595705	+	0.803203i	$0.296873\pi$
$762$	−21.8756	−0.792470
$763$	1.57348	0.0569639
$764$	−19.0097	−0.687746
$765$	6.45473	0.233371
$766$	−29.9511	−1.08218
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	37.8877	1.36627	0.683133	−	0.730294i	$-0.260617\pi$
$769$	37.8877	1.36627	0.683133	+	0.730294i	$0.260617\pi$
$770$	−0.664874	−0.0239604
$771$	−30.4426	−1.09637
$772$	−0.599564	−0.0215788
$773$	12.4523	0.447879	0.223940	−	0.974603i	$-0.428108\pi$
$773$	12.4523	0.447879	0.223940	+	0.974603i	$0.428108\pi$
$774$	1.15883	0.0416534
$775$	−3.04892	−0.109520
$776$	−7.07606	−0.254016
$777$	9.54394	0.342387
$778$	10.0140	0.359019
$779$	−16.6939	−0.598122
$780$	0	0
$781$	1.30798	0.0468032
$782$	−51.3846	−1.83751
$783$	6.89977	0.246578
$784$	−5.56465	−0.198737
$785$	19.0151	0.678677
$786$	21.6528	0.772330
$787$	6.20536	0.221197	0.110599	−	0.993865i	$-0.464723\pi$
$787$	6.20536	0.221197	0.110599	+	0.993865i	$0.464723\pi$
$788$	−3.78448	−0.134816
$789$	9.33944	0.332493
$790$	0.990311	0.0352337
$791$	−4.25906	−0.151435
$792$	−0.554958	−0.0197196
$793$	0	0
$794$	−17.0489	−0.605043
$795$	8.47219	0.300478
$796$	−10.2349	−0.362766
$797$	33.0627	1.17114	0.585570	−	0.810622i	$-0.300871\pi$
$797$	33.0627	1.17114	0.585570	+	0.810622i	$0.300871\pi$
$798$	6.47650	0.229266
$799$	79.1482	2.80006
$800$	1.00000	0.0353553
$801$	−13.7845	−0.487051
$802$	−0.975246	−0.0344371
$803$	7.40581	0.261345
$804$	−3.34481	−0.117963
$805$	9.53750	0.336153
$806$	0	0
$807$	−26.7821	−0.942774
$808$	4.09783	0.144161
$809$	−52.5454	−1.84740	−0.923699	−	0.383120i	$-0.874850\pi$
$809$	−52.5454	−1.84740	−0.923699	+	0.383120i	$0.874850\pi$
$810$	−1.00000	−0.0351364
$811$	9.96508	0.349921	0.174961	−	0.984575i	$-0.444020\pi$
$811$	9.96508	0.349921	0.174961	+	0.984575i	$0.444020\pi$
$812$	8.26636	0.290092
$813$	−2.58748	−0.0907470
$814$	−4.42088	−0.154952
$815$	5.65279	0.198009
$816$	6.45473	0.225961
$817$	6.26444	0.219165
$818$	37.7439	1.31969
$819$	0	0
$820$	3.08815	0.107843
$821$	−28.6950	−1.00146	−0.500731	−	0.865603i	$-0.666936\pi$
$821$	−28.6950	−1.00146	−0.500731	+	0.865603i	$0.666936\pi$
$822$	7.08815	0.247227
$823$	37.4053	1.30387	0.651934	−	0.758276i	$-0.273958\pi$
$823$	37.4053	1.30387	0.651934	+	0.758276i	$0.273958\pi$
$824$	−4.16421	−0.145067
$825$	0.554958	0.0193212
$826$	1.74572	0.0607415
$827$	36.3666	1.26459	0.632295	−	0.774728i	$-0.282113\pi$
$827$	36.3666	1.26459	0.632295	+	0.774728i	$0.282113\pi$
$828$	7.96077	0.276656
$829$	−32.6370	−1.13353	−0.566765	−	0.823880i	$-0.691805\pi$
$829$	−32.6370	−1.13353	−0.566765	+	0.823880i	$0.691805\pi$
$830$	5.88471	0.204261
$831$	−11.0532	−0.383432
$832$	0	0
$833$	35.9183	1.24450
$834$	−0.340502	−0.0117906
$835$	2.15883	0.0747095
$836$	−3.00000	−0.103757
$837$	3.04892	0.105386
$838$	−21.5700	−0.745124
$839$	11.6810	0.403273	0.201637	−	0.979460i	$-0.435374\pi$
$839$	11.6810	0.403273	0.201637	+	0.979460i	$0.435374\pi$
$840$	−1.19806	−0.0413371
$841$	18.6069	0.641616
$842$	−33.2403	−1.14554
$843$	10.5090	0.361950
$844$	−8.71379	−0.299941
$845$	0	0
$846$	−12.2620	−0.421578
$847$	12.8097	0.440147
$848$	8.47219	0.290936
$849$	1.07069	0.0367459
$850$	−6.45473	−0.221395
$851$	63.4167	2.17390
$852$	2.35690	0.0807459
$853$	45.1756	1.54678	0.773391	−	0.633930i	$-0.218559\pi$
$853$	45.1756	1.54678	0.773391	+	0.633930i	$0.218559\pi$
$854$	−3.30990	−0.113262
$855$	−5.40581	−0.184875
$856$	−4.98254	−0.170300
$857$	−16.1602	−0.552021	−0.276010	−	0.961155i	$-0.589012\pi$
$857$	−16.1602	−0.552021	−0.276010	+	0.961155i	$0.589012\pi$
$858$	0	0
$859$	24.9071	0.849818	0.424909	−	0.905236i	$-0.360306\pi$
$859$	24.9071	0.849818	0.424909	+	0.905236i	$0.360306\pi$
$860$	−1.15883	−0.0395159
$861$	−3.69979	−0.126089
$862$	−20.9041	−0.711996
$863$	25.9030	0.881749	0.440875	−	0.897569i	$-0.354668\pi$
$863$	25.9030	0.881749	0.440875	+	0.897569i	$0.354668\pi$
$864$	−1.00000	−0.0340207
$865$	−21.6722	−0.736876
$866$	36.5864	1.24326
$867$	−24.6635	−0.837618
$868$	3.65279	0.123984
$869$	0.549581	0.0186433
$870$	−6.89977	−0.233924
$871$	0	0
$872$	−1.31336	−0.0444758
$873$	−7.07606	−0.239488
$874$	43.0344	1.45566
$875$	1.19806	0.0405019
$876$	13.3448	0.450879
$877$	21.5875	0.728957	0.364479	−	0.931212i	$-0.381247\pi$
$877$	21.5875	0.728957	0.364479	+	0.931212i	$0.381247\pi$
$878$	27.5036	0.928203
$879$	27.4142	0.924657
$880$	0.554958	0.0187076
$881$	0.468140	0.0157720	0.00788602	−	0.999969i	$-0.497490\pi$
$881$	0.468140	0.0157720	0.00788602	+	0.999969i	$0.497490\pi$
$882$	−5.56465	−0.187371
$883$	−43.6034	−1.46737	−0.733686	−	0.679489i	$-0.762202\pi$
$883$	−43.6034	−1.46737	−0.733686	+	0.679489i	$0.762202\pi$
$884$	0	0
$885$	−1.45712	−0.0489807
$886$	−4.34614	−0.146012
$887$	24.1038	0.809326	0.404663	−	0.914466i	$-0.367389\pi$
$887$	24.1038	0.809326	0.404663	+	0.914466i	$0.367389\pi$
$888$	−7.96615	−0.267326
$889$	−26.2083	−0.879000
$890$	13.7845	0.462057
$891$	−0.554958	−0.0185918
$892$	15.0694	0.504559
$893$	−66.2863	−2.21819
$894$	7.40581	0.247687
$895$	0.956459	0.0319709
$896$	−1.19806	−0.0400245
$897$	0	0
$898$	−0.423272	−0.0141248
$899$	21.0368	0.701618
$900$	1.00000	0.0333333
$901$	−54.6857	−1.82184
$902$	1.71379	0.0570630
$903$	1.38835	0.0462016
$904$	3.55496	0.118236
$905$	−0.405813	−0.0134897
$906$	10.4209	0.346211
$907$	55.0635	1.82835	0.914177	−	0.405315i	$-0.132838\pi$
$907$	55.0635	1.82835	0.914177	+	0.405315i	$0.132838\pi$
$908$	20.4577	0.678913
$909$	4.09783	0.135917
$910$	0	0
$911$	7.17151	0.237603	0.118801	−	0.992918i	$-0.462095\pi$
$911$	7.17151	0.237603	0.118801	+	0.992918i	$0.462095\pi$
$912$	−5.40581	−0.179004
$913$	3.26577	0.108081
$914$	18.6396	0.616545
$915$	2.76271	0.0913323
$916$	−18.7657	−0.620036
$917$	25.9414	0.856660
$918$	6.45473	0.213038
$919$	−32.1081	−1.05915	−0.529574	−	0.848263i	$-0.677648\pi$
$919$	−32.1081	−1.05915	−0.529574	+	0.848263i	$0.677648\pi$
$920$	−7.96077	−0.262459
$921$	24.4969	0.807202
$922$	−8.22223	−0.270785
$923$	0	0
$924$	−0.664874	−0.0218728
$925$	7.96615	0.261925
$926$	−34.1118	−1.12098
$927$	−4.16421	−0.136771
$928$	−6.89977	−0.226496
$929$	−3.46011	−0.113522	−0.0567612	−	0.998388i	$-0.518077\pi$
$929$	−3.46011	−0.113522	−0.0567612	+	0.998388i	$0.518077\pi$
$930$	−3.04892	−0.0999779
$931$	−30.0814	−0.985879
$932$	23.6353	0.774201
$933$	−16.4403	−0.538230
$934$	13.9245	0.455624
$935$	−3.58211	−0.117147
$936$	0	0
$937$	13.3918	0.437491	0.218746	−	0.975782i	$-0.429803\pi$
$937$	13.3918	0.437491	0.218746	+	0.975782i	$0.429803\pi$
$938$	−4.00730	−0.130843
$939$	26.0320	0.849524
$940$	12.2620	0.399944
$941$	−47.0060	−1.53235	−0.766175	−	0.642632i	$-0.777843\pi$
$941$	−47.0060	−1.53235	−0.766175	+	0.642632i	$0.777843\pi$
$942$	19.0151	0.619544
$943$	−24.5840	−0.800566
$944$	−1.45712	−0.0474253
$945$	−1.19806	−0.0389730
$946$	−0.643104	−0.0209091
$947$	−44.9687	−1.46129	−0.730643	−	0.682760i	$-0.760779\pi$
$947$	−44.9687	−1.46129	−0.730643	+	0.682760i	$0.760779\pi$
$948$	0.990311	0.0321638
$949$	0	0
$950$	5.40581	0.175388
$951$	19.3545	0.627613
$952$	7.73317	0.250633
$953$	−12.1691	−0.394196	−0.197098	−	0.980384i	$-0.563152\pi$
$953$	−12.1691	−0.394196	−0.197098	+	0.980384i	$0.563152\pi$
$954$	8.47219	0.274297
$955$	19.0097	0.615139
$956$	−28.7506	−0.929862
$957$	−3.82908	−0.123777
$958$	−1.79225	−0.0579049
$959$	8.49204	0.274222
$960$	1.00000	0.0322749
$961$	−21.7041	−0.700132
$962$	0	0
$963$	−4.98254	−0.160560
$964$	3.89440	0.125430
$965$	0.599564	0.0193006
$966$	9.53750	0.306864
$967$	−21.8799	−0.703611	−0.351805	−	0.936073i	$-0.614432\pi$
$967$	−21.8799	−0.703611	−0.351805	+	0.936073i	$0.614432\pi$
$968$	−10.6920	−0.343655
$969$	34.8931	1.12093
$970$	7.07606	0.227199
$971$	−36.6450	−1.17599	−0.587997	−	0.808863i	$-0.700083\pi$
$971$	−36.6450	−1.17599	−0.587997	+	0.808863i	$0.700083\pi$
$972$	−1.00000	−0.0320750
$973$	−0.407943	−0.0130781
$974$	0.126310	0.00404722
$975$	0	0
$976$	2.76271	0.0884322
$977$	−7.36957	−0.235773	−0.117887	−	0.993027i	$-0.537612\pi$
$977$	−7.36957	−0.235773	−0.117887	+	0.993027i	$0.537612\pi$
$978$	5.65279	0.180756
$979$	7.64981	0.244489
$980$	5.56465	0.177756
$981$	−1.31336	−0.0419322
$982$	−17.6823	−0.564266
$983$	−45.5612	−1.45318	−0.726588	−	0.687073i	$-0.758895\pi$
$983$	−45.5612	−1.45318	−0.726588	+	0.687073i	$0.758895\pi$
$984$	3.08815	0.0984465
$985$	3.78448	0.120584
$986$	44.5362	1.41832
$987$	−14.6907	−0.467610
$988$	0	0
$989$	9.22521	0.293345
$990$	0.554958	0.0176377
$991$	35.4693	1.12672	0.563360	−	0.826211i	$-0.309508\pi$
$991$	35.4693	1.12672	0.563360	+	0.826211i	$0.309508\pi$
$992$	−3.04892	−0.0968032
$993$	−0.173899	−0.00551852
$994$	2.82371	0.0895626
$995$	10.2349	0.324468
$996$	5.88471	0.186464
$997$	−3.74632	−0.118647	−0.0593235	−	0.998239i	$-0.518894\pi$
$997$	−3.74632	−0.118647	−0.0593235	+	0.998239i	$0.518894\pi$
$998$	21.1535	0.669601
$999$	−7.96615	−0.252038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.br.1.3	yes	3
13.5	odd	4		5070.2.b.u.1351.3		6
13.8	odd	4		5070.2.b.u.1351.4		6
13.12	even	2		5070.2.a.bm.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bm.1.1	✓	3	13.12	even	2
5070.2.a.br.1.3	yes	3	1.1	even	1	trivial
5070.2.b.u.1351.3		6	13.5	odd	4
5070.2.b.u.1351.4		6	13.8	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.19806 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.19806 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -0.554958 q^{11} -1.00000 q^{12} -1.19806 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.45473 q^{17} +1.00000 q^{18} +5.40581 q^{19} -1.00000 q^{20} +1.19806 q^{21} -0.554958 q^{22} +7.96077 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.19806 q^{28} -6.89977 q^{29} +1.00000 q^{30} -3.04892 q^{31} +1.00000 q^{32} +0.554958 q^{33} -6.45473 q^{34} +1.19806 q^{35} +1.00000 q^{36} +7.96615 q^{37} +5.40581 q^{38} -1.00000 q^{40} -3.08815 q^{41} +1.19806 q^{42} +1.15883 q^{43} -0.554958 q^{44} -1.00000 q^{45} +7.96077 q^{46} -12.2620 q^{47} -1.00000 q^{48} -5.56465 q^{49} +1.00000 q^{50} +6.45473 q^{51} +8.47219 q^{53} -1.00000 q^{54} +0.554958 q^{55} -1.19806 q^{56} -5.40581 q^{57} -6.89977 q^{58} -1.45712 q^{59} +1.00000 q^{60} +2.76271 q^{61} -3.04892 q^{62} -1.19806 q^{63} +1.00000 q^{64} +0.554958 q^{66} +3.34481 q^{67} -6.45473 q^{68} -7.96077 q^{69} +1.19806 q^{70} -2.35690 q^{71} +1.00000 q^{72} -13.3448 q^{73} +7.96615 q^{74} -1.00000 q^{75} +5.40581 q^{76} +0.664874 q^{77} -0.990311 q^{79} -1.00000 q^{80} +1.00000 q^{81} -3.08815 q^{82} -5.88471 q^{83} +1.19806 q^{84} +6.45473 q^{85} +1.15883 q^{86} +6.89977 q^{87} -0.554958 q^{88} -13.7845 q^{89} -1.00000 q^{90} +7.96077 q^{92} +3.04892 q^{93} -12.2620 q^{94} -5.40581 q^{95} -1.00000 q^{96} -7.07606 q^{97} -5.56465 q^{98} -0.554958 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 8 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 8 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 8 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 2 q^{11} - 3 q^{12} - 8 q^{14} + 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 3 q^{19} - 3 q^{20} + 8 q^{21} - 2 q^{22} + 11 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 8 q^{28} + 2 q^{29} + 3 q^{30} + 3 q^{32} + 2 q^{33} + 3 q^{34} + 8 q^{35} + 3 q^{36} + 8 q^{37} + 3 q^{38} - 3 q^{40} - 13 q^{41} + 8 q^{42} - 5 q^{43} - 2 q^{44} - 3 q^{45} + 11 q^{46} - 7 q^{47} - 3 q^{48} + 5 q^{49} + 3 q^{50} - 3 q^{51} + 19 q^{53} - 3 q^{54} + 2 q^{55} - 8 q^{56} - 3 q^{57} + 2 q^{58} - 23 q^{59} + 3 q^{60} - 9 q^{61} - 8 q^{63} + 3 q^{64} + 2 q^{66} - 13 q^{67} + 3 q^{68} - 11 q^{69} + 8 q^{70} - 3 q^{71} + 3 q^{72} - 17 q^{73} + 8 q^{74} - 3 q^{75} + 3 q^{76} + 3 q^{77} - 25 q^{79} - 3 q^{80} + 3 q^{81} - 13 q^{82} - 20 q^{83} + 8 q^{84} - 3 q^{85} - 5 q^{86} - 2 q^{87} - 2 q^{88} - 21 q^{89} - 3 q^{90} + 11 q^{92} - 7 q^{94} - 3 q^{95} - 3 q^{96} - 6 q^{97} + 5 q^{98} - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 8 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 - 2 * q^11 - 3 * q^12 - 8 * q^14 + 3 * q^15 + 3 * q^16 + 3 * q^17 + 3 * q^18 + 3 * q^19 - 3 * q^20 + 8 * q^21 - 2 * q^22 + 11 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 - 8 * q^28 + 2 * q^29 + 3 * q^30 + 3 * q^32 + 2 * q^33 + 3 * q^34 + 8 * q^35 + 3 * q^36 + 8 * q^37 + 3 * q^38 - 3 * q^40 - 13 * q^41 + 8 * q^42 - 5 * q^43 - 2 * q^44 - 3 * q^45 + 11 * q^46 - 7 * q^47 - 3 * q^48 + 5 * q^49 + 3 * q^50 - 3 * q^51 + 19 * q^53 - 3 * q^54 + 2 * q^55 - 8 * q^56 - 3 * q^57 + 2 * q^58 - 23 * q^59 + 3 * q^60 - 9 * q^61 - 8 * q^63 + 3 * q^64 + 2 * q^66 - 13 * q^67 + 3 * q^68 - 11 * q^69 + 8 * q^70 - 3 * q^71 + 3 * q^72 - 17 * q^73 + 8 * q^74 - 3 * q^75 + 3 * q^76 + 3 * q^77 - 25 * q^79 - 3 * q^80 + 3 * q^81 - 13 * q^82 - 20 * q^83 + 8 * q^84 - 3 * q^85 - 5 * q^86 - 2 * q^87 - 2 * q^88 - 21 * q^89 - 3 * q^90 + 11 * q^92 - 7 * q^94 - 3 * q^95 - 3 * q^96 - 6 * q^97 + 5 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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