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Properties

Label 49.6.a.a.1.1
Level $49$
Weight $6$
Character 49.1
Self dual yes
Analytic conductor $7.859$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [49,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-10,14] 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: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q-10.0000 q^{2} +14.0000 q^{3} +68.0000 q^{4} +56.0000 q^{5} -140.000 q^{6} -360.000 q^{8} -47.0000 q^{9} -560.000 q^{10} +232.000 q^{11} +952.000 q^{12} +140.000 q^{13} +784.000 q^{15} +1424.00 q^{16} +1722.00 q^{17} +470.000 q^{18} +98.0000 q^{19} +3808.00 q^{20} -2320.00 q^{22} +1824.00 q^{23} -5040.00 q^{24} +11.0000 q^{25} -1400.00 q^{26} -4060.00 q^{27} +3418.00 q^{29} -7840.00 q^{30} +7644.00 q^{31} -2720.00 q^{32} +3248.00 q^{33} -17220.0 q^{34} -3196.00 q^{36} -10398.0 q^{37} -980.000 q^{38} +1960.00 q^{39} -20160.0 q^{40} +17962.0 q^{41} +10880.0 q^{43} +15776.0 q^{44} -2632.00 q^{45} -18240.0 q^{46} -9324.00 q^{47} +19936.0 q^{48} -110.000 q^{50} +24108.0 q^{51} +9520.00 q^{52} +2262.00 q^{53} +40600.0 q^{54} +12992.0 q^{55} +1372.00 q^{57} -34180.0 q^{58} +2730.00 q^{59} +53312.0 q^{60} -25648.0 q^{61} -76440.0 q^{62} -18368.0 q^{64} +7840.00 q^{65} -32480.0 q^{66} -48404.0 q^{67} +117096. q^{68} +25536.0 q^{69} -58560.0 q^{71} +16920.0 q^{72} -68082.0 q^{73} +103980. q^{74} +154.000 q^{75} +6664.00 q^{76} -19600.0 q^{78} +31784.0 q^{79} +79744.0 q^{80} -45419.0 q^{81} -179620. q^{82} +20538.0 q^{83} +96432.0 q^{85} -108800. q^{86} +47852.0 q^{87} -83520.0 q^{88} +50582.0 q^{89} +26320.0 q^{90} +124032. q^{92} +107016. q^{93} +93240.0 q^{94} +5488.00 q^{95} -38080.0 q^{96} +58506.0 q^{97} -10904.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.0000 −1.76777 −0.883883 − 0.467707i \(-0.845080\pi\)
−0.883883 + 0.467707i \(0.845080\pi\)
\(3\) 14.0000 0.898100 0.449050 − 0.893507i \(-0.351762\pi\)
0.449050 + 0.893507i \(0.351762\pi\)
\(4\) 68.0000 2.12500
\(5\) 56.0000 1.00176 0.500879 − 0.865517i \(-0.333010\pi\)
0.500879 + 0.865517i \(0.333010\pi\)
\(6\) −140.000 −1.58763
\(7\) 0 0
\(8\) −360.000 −1.98874
\(9\) −47.0000 −0.193416
\(10\) −560.000 −1.77088
\(11\) 232.000 0.578104 0.289052 − 0.957313i \(-0.406660\pi\)
0.289052 + 0.957313i \(0.406660\pi\)
\(12\) 952.000 1.90846
\(13\) 140.000 0.229757 0.114879 − 0.993380i \(-0.463352\pi\)
0.114879 + 0.993380i \(0.463352\pi\)
\(14\) 0 0
\(15\) 784.000 0.899680
\(16\) 1424.00 1.39062
\(17\) 1722.00 1.44514 0.722572 − 0.691296i \(-0.242960\pi\)
0.722572 + 0.691296i \(0.242960\pi\)
\(18\) 470.000 0.341914
\(19\) 98.0000 0.0622791 0.0311395 − 0.999515i \(-0.490086\pi\)
0.0311395 + 0.999515i \(0.490086\pi\)
\(20\) 3808.00 2.12874
\(21\) 0 0
\(22\) −2320.00 −1.02195
\(23\) 1824.00 0.718961 0.359480 − 0.933153i \(-0.382954\pi\)
0.359480 + 0.933153i \(0.382954\pi\)
\(24\) −5040.00 −1.78609
\(25\) 11.0000 0.00352000
\(26\) −1400.00 −0.406158
\(27\) −4060.00 −1.07181
\(28\) 0 0
\(29\) 3418.00 0.754705 0.377352 − 0.926070i \(-0.376835\pi\)
0.377352 + 0.926070i \(0.376835\pi\)
\(30\) −7840.00 −1.59042
\(31\) 7644.00 1.42862 0.714310 − 0.699830i \(-0.246741\pi\)
0.714310 + 0.699830i \(0.246741\pi\)
\(32\) −2720.00 −0.469563
\(33\) 3248.00 0.519196
\(34\) −17220.0 −2.55468
\(35\) 0 0
\(36\) −3196.00 −0.411008
\(37\) −10398.0 −1.24866 −0.624332 − 0.781159i \(-0.714629\pi\)
−0.624332 + 0.781159i \(0.714629\pi\)
\(38\) −980.000 −0.110095
\(39\) 1960.00 0.206345
\(40\) −20160.0 −1.99223
\(41\) 17962.0 1.66876 0.834382 − 0.551186i \(-0.185825\pi\)
0.834382 + 0.551186i \(0.185825\pi\)
\(42\) 0 0
\(43\) 10880.0 0.897342 0.448671 − 0.893697i \(-0.351898\pi\)
0.448671 + 0.893697i \(0.351898\pi\)
\(44\) 15776.0 1.22847
\(45\) −2632.00 −0.193756
\(46\) −18240.0 −1.27096
\(47\) −9324.00 −0.615684 −0.307842 − 0.951438i \(-0.599607\pi\)
−0.307842 + 0.951438i \(0.599607\pi\)
\(48\) 19936.0 1.24892
\(49\) 0 0
\(50\) −110.000 −0.00622254
\(51\) 24108.0 1.29788
\(52\) 9520.00 0.488235
\(53\) 2262.00 0.110612 0.0553061 − 0.998469i \(-0.482387\pi\)
0.0553061 + 0.998469i \(0.482387\pi\)
\(54\) 40600.0 1.89471
\(55\) 12992.0 0.579121
\(56\) 0 0
\(57\) 1372.00 0.0559329
\(58\) −34180.0 −1.33414
\(59\) 2730.00 0.102102 0.0510508 − 0.998696i \(-0.483743\pi\)
0.0510508 + 0.998696i \(0.483743\pi\)
\(60\) 53312.0 1.91182
\(61\) −25648.0 −0.882529 −0.441264 − 0.897377i \(-0.645470\pi\)
−0.441264 + 0.897377i \(0.645470\pi\)
\(62\) −76440.0 −2.52547
\(63\) 0 0
\(64\) −18368.0 −0.560547
\(65\) 7840.00 0.230161
\(66\) −32480.0 −0.917817
\(67\) −48404.0 −1.31733 −0.658664 − 0.752437i \(-0.728878\pi\)
−0.658664 + 0.752437i \(0.728878\pi\)
\(68\) 117096. 3.07093
\(69\) 25536.0 0.645699
\(70\) 0 0
\(71\) −58560.0 −1.37865 −0.689327 − 0.724450i \(-0.742094\pi\)
−0.689327 + 0.724450i \(0.742094\pi\)
\(72\) 16920.0 0.384653
\(73\) −68082.0 −1.49529 −0.747645 − 0.664099i \(-0.768815\pi\)
−0.747645 + 0.664099i \(0.768815\pi\)
\(74\) 103980. 2.20735
\(75\) 154.000 0.00316131
\(76\) 6664.00 0.132343
\(77\) 0 0
\(78\) −19600.0 −0.364770
\(79\) 31784.0 0.572982 0.286491 − 0.958083i \(-0.407511\pi\)
0.286491 + 0.958083i \(0.407511\pi\)
\(80\) 79744.0 1.39307
\(81\) −45419.0 −0.769175
\(82\) −179620. −2.94999
\(83\) 20538.0 0.327237 0.163619 − 0.986524i \(-0.447683\pi\)
0.163619 + 0.986524i \(0.447683\pi\)
\(84\) 0 0
\(85\) 96432.0 1.44768
\(86\) −108800. −1.58629
\(87\) 47852.0 0.677801
\(88\) −83520.0 −1.14970
\(89\) 50582.0 0.676894 0.338447 − 0.940985i \(-0.390098\pi\)
0.338447 + 0.940985i \(0.390098\pi\)
\(90\) 26320.0 0.342515
\(91\) 0 0
\(92\) 124032. 1.52779
\(93\) 107016. 1.28304
\(94\) 93240.0 1.08839
\(95\) 5488.00 0.0623886
\(96\) −38080.0 −0.421715
\(97\) 58506.0 0.631351 0.315676 − 0.948867i \(-0.397769\pi\)
0.315676 + 0.948867i \(0.397769\pi\)
\(98\) 0 0
\(99\) −10904.0 −0.111814
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 49.6.a.a.1.1 1
3.2 odd 2 441.6.a.k.1.1 1
4.3 odd 2 784.6.a.c.1.1 1
7.2 even 3 49.6.c.b.18.1 2
7.3 odd 6 49.6.c.c.30.1 2
7.4 even 3 49.6.c.b.30.1 2
7.5 odd 6 49.6.c.c.18.1 2
7.6 odd 2 7.6.a.a.1.1 ✓ 1
21.20 even 2 63.6.a.e.1.1 1
28.27 even 2 112.6.a.g.1.1 1
35.13 even 4 175.6.b.a.99.2 2
35.27 even 4 175.6.b.a.99.1 2
35.34 odd 2 175.6.a.b.1.1 1
56.13 odd 2 448.6.a.m.1.1 1
56.27 even 2 448.6.a.c.1.1 1
77.76 even 2 847.6.a.b.1.1 1
84.83 odd 2 1008.6.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 ✓ 1 7.6 odd 2
49.6.a.a.1.1 1 1.1 even 1 trivial
49.6.c.b.18.1 2 7.2 even 3
49.6.c.b.30.1 2 7.4 even 3
49.6.c.c.18.1 2 7.5 odd 6
49.6.c.c.30.1 2 7.3 odd 6
63.6.a.e.1.1 1 21.20 even 2
112.6.a.g.1.1 1 28.27 even 2
175.6.a.b.1.1 1 35.34 odd 2
175.6.b.a.99.1 2 35.27 even 4
175.6.b.a.99.2 2 35.13 even 4
441.6.a.k.1.1 1 3.2 odd 2
448.6.a.c.1.1 1 56.27 even 2
448.6.a.m.1.1 1 56.13 odd 2
784.6.a.c.1.1 1 4.3 odd 2
847.6.a.b.1.1 1 77.76 even 2
1008.6.a.y.1.1 1 84.83 odd 2