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Properties

Label 49.8.a.g.1.7
Level $49$
Weight $8$
Character 49.1
Self dual yes
Analytic conductor $15.307$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,8,Mod(1,49)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("49.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,30,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.3068662487\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 574x^{6} - 2240x^{5} + 95697x^{4} + 624960x^{3} - 2980216x^{2} - 17427200x + 41900096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{6} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(2.08214\) of defining polynomial
Character \(\chi\) \(=\) 49.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+20.4398 q^{2} -60.9835 q^{3} +289.787 q^{4} +267.917 q^{5} -1246.49 q^{6} +3306.90 q^{8} +1531.99 q^{9} +5476.19 q^{10} +6858.69 q^{11} -17672.2 q^{12} +553.232 q^{13} -16338.6 q^{15} +30499.9 q^{16} +18782.4 q^{17} +31313.7 q^{18} +26526.1 q^{19} +77639.0 q^{20} +140190. q^{22} -14206.0 q^{23} -201667. q^{24} -6345.24 q^{25} +11308.0 q^{26} +39944.7 q^{27} -218240. q^{29} -333958. q^{30} -177622. q^{31} +200128. q^{32} -418267. q^{33} +383909. q^{34} +443952. q^{36} +166649. q^{37} +542189. q^{38} -33738.0 q^{39} +885977. q^{40} -392711. q^{41} -349297. q^{43} +1.98756e6 q^{44} +410447. q^{45} -290369. q^{46} -998513. q^{47} -1.85999e6 q^{48} -129696. q^{50} -1.14542e6 q^{51} +160319. q^{52} +558894. q^{53} +816463. q^{54} +1.83756e6 q^{55} -1.61766e6 q^{57} -4.46080e6 q^{58} +2.86220e6 q^{59} -4.73470e6 q^{60} -804192. q^{61} -3.63057e6 q^{62} +186612. q^{64} +148220. q^{65} -8.54931e6 q^{66} +1.01152e6 q^{67} +5.44289e6 q^{68} +866335. q^{69} +1.38526e6 q^{71} +5.06615e6 q^{72} -584471. q^{73} +3.40629e6 q^{74} +386955. q^{75} +7.68693e6 q^{76} -689600. q^{78} +3.78559e6 q^{79} +8.17144e6 q^{80} -5.78644e6 q^{81} -8.02694e6 q^{82} -3.66559e6 q^{83} +5.03213e6 q^{85} -7.13958e6 q^{86} +1.33091e7 q^{87} +2.26810e7 q^{88} -9.21773e6 q^{89} +8.38948e6 q^{90} -4.11673e6 q^{92} +1.08320e7 q^{93} -2.04094e7 q^{94} +7.10681e6 q^{95} -1.22045e7 q^{96} -8.21720e6 q^{97} +1.05075e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 30 q^{2} + 458 q^{4} + 4290 q^{8} + 7600 q^{9} + 17760 q^{11} + 29216 q^{15} + 33762 q^{16} + 244850 q^{18} + 430660 q^{22} + 96000 q^{23} + 468992 q^{25} + 236280 q^{29} - 193752 q^{30} + 1064370 q^{32}+ \cdots + 36092800 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 20.4398 1.80664 0.903322 − 0.428963i \(-0.141121\pi\)
0.903322 + 0.428963i \(0.141121\pi\)
\(3\) −60.9835 −1.30403 −0.652016 − 0.758205i \(-0.726077\pi\)
−0.652016 + 0.758205i \(0.726077\pi\)
\(4\) 289.787 2.26396
\(5\) 267.917 0.958531 0.479265 − 0.877670i \(-0.340903\pi\)
0.479265 + 0.877670i \(0.340903\pi\)
\(6\) −1246.49 −2.35592
\(7\) 0 0
\(8\) 3306.90 2.28353
\(9\) 1531.99 0.700499
\(10\) 5476.19 1.73172
\(11\) 6858.69 1.55370 0.776849 − 0.629687i \(-0.216817\pi\)
0.776849 + 0.629687i \(0.216817\pi\)
\(12\) −17672.2 −2.95228
\(13\) 553.232 0.0698402 0.0349201 − 0.999390i \(-0.488882\pi\)
0.0349201 + 0.999390i \(0.488882\pi\)
\(14\) 0 0
\(15\) −16338.6 −1.24995
\(16\) 30499.9 1.86156
\(17\) 18782.4 0.927213 0.463606 − 0.886041i \(-0.346555\pi\)
0.463606 + 0.886041i \(0.346555\pi\)
\(18\) 31313.7 1.26555
\(19\) 26526.1 0.887229 0.443615 − 0.896218i \(-0.353696\pi\)
0.443615 + 0.896218i \(0.353696\pi\)
\(20\) 77639.0 2.17008
\(21\) 0 0
\(22\) 140190. 2.80698
\(23\) −14206.0 −0.243459 −0.121730 − 0.992563i \(-0.538844\pi\)
−0.121730 + 0.992563i \(0.538844\pi\)
\(24\) −201667. −2.97780
\(25\) −6345.24 −0.0812191
\(26\) 11308.0 0.126176
\(27\) 39944.7 0.390558
\(28\) 0 0
\(29\) −218240. −1.66166 −0.830830 − 0.556527i \(-0.812134\pi\)
−0.830830 + 0.556527i \(0.812134\pi\)
\(30\) −333958. −2.25822
\(31\) −177622. −1.07086 −0.535429 − 0.844580i \(-0.679850\pi\)
−0.535429 + 0.844580i \(0.679850\pi\)
\(32\) 200128. 1.07965
\(33\) −418267. −2.02607
\(34\) 383909. 1.67514
\(35\) 0 0
\(36\) 443952. 1.58590
\(37\) 166649. 0.540876 0.270438 − 0.962737i \(-0.412831\pi\)
0.270438 + 0.962737i \(0.412831\pi\)
\(38\) 542189. 1.60291
\(39\) −33738.0 −0.0910738
\(40\) 885977. 2.18883
\(41\) −392711. −0.889875 −0.444938 − 0.895562i \(-0.646774\pi\)
−0.444938 + 0.895562i \(0.646774\pi\)
\(42\) 0 0
\(43\) −349297. −0.669971 −0.334985 − 0.942223i \(-0.608731\pi\)
−0.334985 + 0.942223i \(0.608731\pi\)
\(44\) 1.98756e6 3.51751
\(45\) 410447. 0.671450
\(46\) −290369. −0.439844
\(47\) −998513. −1.40285 −0.701425 − 0.712743i \(-0.747452\pi\)
−0.701425 + 0.712743i \(0.747452\pi\)
\(48\) −1.85999e6 −2.42754
\(49\) 0 0
\(50\) −129696. −0.146734
\(51\) −1.14542e6 −1.20912
\(52\) 160319. 0.158116
\(53\) 558894. 0.515660 0.257830 − 0.966190i \(-0.416992\pi\)
0.257830 + 0.966190i \(0.416992\pi\)
\(54\) 816463. 0.705600
\(55\) 1.83756e6 1.48927
\(56\) 0 0
\(57\) −1.61766e6 −1.15698
\(58\) −4.46080e6 −3.00203
\(59\) 2.86220e6 1.81434 0.907169 − 0.420767i \(-0.138239\pi\)
0.907169 + 0.420767i \(0.138239\pi\)
\(60\) −4.73470e6 −2.82985
\(61\) −804192. −0.453634 −0.226817 − 0.973937i \(-0.572832\pi\)
−0.226817 + 0.973937i \(0.572832\pi\)
\(62\) −3.63057e6 −1.93466
\(63\) 0 0
\(64\) 186612. 0.0889836
\(65\) 148220. 0.0669439
\(66\) −8.54931e6 −3.66039
\(67\) 1.01152e6 0.410876 0.205438 − 0.978670i \(-0.434138\pi\)
0.205438 + 0.978670i \(0.434138\pi\)
\(68\) 5.44289e6 2.09917
\(69\) 866335. 0.317478
\(70\) 0 0
\(71\) 1.38526e6 0.459333 0.229666 − 0.973269i \(-0.426236\pi\)
0.229666 + 0.973269i \(0.426236\pi\)
\(72\) 5.06615e6 1.59961
\(73\) −584471. −0.175846 −0.0879231 − 0.996127i \(-0.528023\pi\)
−0.0879231 + 0.996127i \(0.528023\pi\)
\(74\) 3.40629e6 0.977171
\(75\) 386955. 0.105912
\(76\) 7.68693e6 2.00865
\(77\) 0 0
\(78\) −689600. −0.164538
\(79\) 3.78559e6 0.863850 0.431925 − 0.901910i \(-0.357835\pi\)
0.431925 + 0.901910i \(0.357835\pi\)
\(80\) 8.17144e6 1.78437
\(81\) −5.78644e6 −1.20980
\(82\) −8.02694e6 −1.60769
\(83\) −3.66559e6 −0.703673 −0.351837 − 0.936061i \(-0.614443\pi\)
−0.351837 + 0.936061i \(0.614443\pi\)
\(84\) 0 0
\(85\) 5.03213e6 0.888762
\(86\) −7.13958e6 −1.21040
\(87\) 1.33091e7 2.16686
\(88\) 2.26810e7 3.54792
\(89\) −9.21773e6 −1.38599 −0.692993 − 0.720944i \(-0.743709\pi\)
−0.692993 + 0.720944i \(0.743709\pi\)
\(90\) 8.38948e6 1.21307
\(91\) 0 0
\(92\) −4.11673e6 −0.551182
\(93\) 1.08320e7 1.39643
\(94\) −2.04094e7 −2.53445
\(95\) 7.10681e6 0.850437
\(96\) −1.22045e7 −1.40790
\(97\) −8.21720e6 −0.914161 −0.457081 − 0.889425i \(-0.651105\pi\)
−0.457081 + 0.889425i \(0.651105\pi\)
\(98\) 0 0
\(99\) 1.05075e7 1.08836
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 49.8.a.g.1.7 ✓ 8
3.2 odd 2 441.8.a.ba.1.1 8
7.2 even 3 49.8.c.h.18.2 16
7.3 odd 6 49.8.c.h.30.1 16
7.4 even 3 49.8.c.h.30.2 16
7.5 odd 6 49.8.c.h.18.1 16
7.6 odd 2 inner 49.8.a.g.1.8 yes 8
21.20 even 2 441.8.a.ba.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.8.a.g.1.7 ✓ 8 1.1 even 1 trivial
49.8.a.g.1.8 yes 8 7.6 odd 2 inner
49.8.c.h.18.1 16 7.5 odd 6
49.8.c.h.18.2 16 7.2 even 3
49.8.c.h.30.1 16 7.3 odd 6
49.8.c.h.30.2 16 7.4 even 3
441.8.a.ba.1.1 8 3.2 odd 2
441.8.a.ba.1.2 8 21.20 even 2