Properties

Label 49.22.a.a.1.1
Level $49$
Weight $22$
Character 49.1
Self dual yes
Analytic conductor $136.944$
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] = [49,22,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, 22); 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, 22, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-288,128844] 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: \(136.943898701\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
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)
 
\(f(q)\) \(=\) \(q-288.000 q^{2} +128844. q^{3} -2.01421e6 q^{4} -2.16410e7 q^{5} -3.71071e7 q^{6} +1.18407e9 q^{8} +6.14042e9 q^{9} +6.23259e9 q^{10} -9.47249e10 q^{11} -2.59519e11 q^{12} +8.06218e10 q^{13} -2.78831e12 q^{15} +3.88309e12 q^{16} -3.05228e12 q^{17} -1.76844e12 q^{18} +7.92079e12 q^{19} +4.35894e13 q^{20} +2.72808e13 q^{22} -7.38454e13 q^{23} +1.52561e14 q^{24} -8.50644e12 q^{25} -2.32191e13 q^{26} -5.56597e14 q^{27} -4.25303e15 q^{29} +8.03032e14 q^{30} -1.90054e15 q^{31} -3.60151e15 q^{32} -1.22047e16 q^{33} +8.79057e14 q^{34} -1.23681e16 q^{36} +2.21914e16 q^{37} -2.28119e15 q^{38} +1.03876e16 q^{39} -2.56244e16 q^{40} +2.06228e16 q^{41} -1.93606e17 q^{43} +1.90796e17 q^{44} -1.32885e17 q^{45} +2.12675e16 q^{46} -1.46961e17 q^{47} +5.00313e17 q^{48} +2.44986e15 q^{50} -3.93268e17 q^{51} -1.62389e17 q^{52} +2.03827e18 q^{53} +1.60300e17 q^{54} +2.04994e18 q^{55} +1.02055e18 q^{57} +1.22487e18 q^{58} +5.97588e18 q^{59} +5.61623e18 q^{60} -6.19062e18 q^{61} +5.47356e17 q^{62} -7.10619e18 q^{64} -1.74473e18 q^{65} +3.51496e18 q^{66} +1.69613e19 q^{67} +6.14793e18 q^{68} -9.51454e18 q^{69} -5.63276e18 q^{71} +7.27070e18 q^{72} +4.32848e19 q^{73} -6.39113e18 q^{74} -1.09600e18 q^{75} -1.59541e19 q^{76} -2.99164e18 q^{78} -5.12649e19 q^{79} -8.40337e19 q^{80} -1.35945e20 q^{81} -5.93937e18 q^{82} -4.89119e19 q^{83} +6.60543e19 q^{85} +5.57585e19 q^{86} -5.47978e20 q^{87} -1.12161e20 q^{88} +5.04303e20 q^{89} +3.82708e19 q^{90} +1.48740e20 q^{92} -2.44873e20 q^{93} +4.23246e19 q^{94} -1.71413e20 q^{95} -4.64033e20 q^{96} -8.08275e20 q^{97} -5.81651e20 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −288.000 −0.198874 −0.0994369 − 0.995044i \(-0.531704\pi\)
−0.0994369 + 0.995044i \(0.531704\pi\)
\(3\) 128844. 1.25977 0.629885 − 0.776689i \(-0.283102\pi\)
0.629885 + 0.776689i \(0.283102\pi\)
\(4\) −2.01421e6 −0.960449
\(5\) −2.16410e7 −0.991040 −0.495520 − 0.868596i \(-0.665023\pi\)
−0.495520 + 0.868596i \(0.665023\pi\)
\(6\) −3.71071e7 −0.250535
\(7\) 0 0
\(8\) 1.18407e9 0.389882
\(9\) 6.14042e9 0.587019
\(10\) 6.23259e9 0.197092
\(11\) −9.47249e10 −1.10114 −0.550568 − 0.834790i \(-0.685589\pi\)
−0.550568 + 0.834790i \(0.685589\pi\)
\(12\) −2.59519e11 −1.20994
\(13\) 8.06218e10 0.162199 0.0810993 − 0.996706i \(-0.474157\pi\)
0.0810993 + 0.996706i \(0.474157\pi\)
\(14\) 0 0
\(15\) −2.78831e12 −1.24848
\(16\) 3.88309e12 0.882912
\(17\) −3.05228e12 −0.367207 −0.183604 − 0.983000i \(-0.558776\pi\)
−0.183604 + 0.983000i \(0.558776\pi\)
\(18\) −1.76844e12 −0.116743
\(19\) 7.92079e12 0.296385 0.148192 − 0.988959i \(-0.452655\pi\)
0.148192 + 0.988959i \(0.452655\pi\)
\(20\) 4.35894e13 0.951844
\(21\) 0 0
\(22\) 2.72808e13 0.218987
\(23\) −7.38454e13 −0.371690 −0.185845 − 0.982579i \(-0.559502\pi\)
−0.185845 + 0.982579i \(0.559502\pi\)
\(24\) 1.52561e14 0.491161
\(25\) −8.50644e12 −0.0178393
\(26\) −2.32191e13 −0.0322571
\(27\) −5.56597e14 −0.520261
\(28\) 0 0
\(29\) −4.25303e15 −1.87724 −0.938620 − 0.344954i \(-0.887895\pi\)
−0.938620 + 0.344954i \(0.887895\pi\)
\(30\) 8.03032e14 0.248290
\(31\) −1.90054e15 −0.416466 −0.208233 − 0.978079i \(-0.566771\pi\)
−0.208233 + 0.978079i \(0.566771\pi\)
\(32\) −3.60151e15 −0.565470
\(33\) −1.22047e16 −1.38718
\(34\) 8.79057e14 0.0730279
\(35\) 0 0
\(36\) −1.23681e16 −0.563802
\(37\) 2.21914e16 0.758695 0.379347 − 0.925254i \(-0.376149\pi\)
0.379347 + 0.925254i \(0.376149\pi\)
\(38\) −2.28119e15 −0.0589431
\(39\) 1.03876e16 0.204333
\(40\) −2.56244e16 −0.386389
\(41\) 2.06228e16 0.239948 0.119974 − 0.992777i \(-0.461719\pi\)
0.119974 + 0.992777i \(0.461719\pi\)
\(42\) 0 0
\(43\) −1.93606e17 −1.36615 −0.683077 − 0.730346i \(-0.739359\pi\)
−0.683077 + 0.730346i \(0.739359\pi\)
\(44\) 1.90796e17 1.05759
\(45\) −1.32885e17 −0.581759
\(46\) 2.12675e16 0.0739195
\(47\) −1.46961e17 −0.407543 −0.203771 − 0.979019i \(-0.565320\pi\)
−0.203771 + 0.979019i \(0.565320\pi\)
\(48\) 5.00313e17 1.11227
\(49\) 0 0
\(50\) 2.44986e15 0.00354777
\(51\) −3.93268e17 −0.462596
\(52\) −1.62389e17 −0.155784
\(53\) 2.03827e18 1.60090 0.800450 − 0.599399i \(-0.204594\pi\)
0.800450 + 0.599399i \(0.204594\pi\)
\(54\) 1.60300e17 0.103466
\(55\) 2.04994e18 1.09127
\(56\) 0 0
\(57\) 1.02055e18 0.373376
\(58\) 1.22487e18 0.373334
\(59\) 5.97588e18 1.52214 0.761072 − 0.648667i \(-0.224673\pi\)
0.761072 + 0.648667i \(0.224673\pi\)
\(60\) 5.61623e18 1.19910
\(61\) −6.19062e18 −1.11114 −0.555572 − 0.831468i \(-0.687501\pi\)
−0.555572 + 0.831468i \(0.687501\pi\)
\(62\) 5.47356e17 0.0828242
\(63\) 0 0
\(64\) −7.10619e18 −0.770455
\(65\) −1.74473e18 −0.160745
\(66\) 3.51496e18 0.275873
\(67\) 1.69613e19 1.13677 0.568387 − 0.822761i \(-0.307568\pi\)
0.568387 + 0.822761i \(0.307568\pi\)
\(68\) 6.14793e18 0.352684
\(69\) −9.51454e18 −0.468244
\(70\) 0 0
\(71\) −5.63276e18 −0.205357 −0.102678 − 0.994715i \(-0.532741\pi\)
−0.102678 + 0.994715i \(0.532741\pi\)
\(72\) 7.27070e18 0.228868
\(73\) 4.32848e19 1.17881 0.589407 − 0.807837i \(-0.299362\pi\)
0.589407 + 0.807837i \(0.299362\pi\)
\(74\) −6.39113e18 −0.150885
\(75\) −1.09600e18 −0.0224734
\(76\) −1.59541e19 −0.284662
\(77\) 0 0
\(78\) −2.99164e18 −0.0406365
\(79\) −5.12649e19 −0.609166 −0.304583 − 0.952486i \(-0.598517\pi\)
−0.304583 + 0.952486i \(0.598517\pi\)
\(80\) −8.40337e19 −0.875001
\(81\) −1.35945e20 −1.24243
\(82\) −5.93937e18 −0.0477194
\(83\) −4.89119e19 −0.346014 −0.173007 − 0.984921i \(-0.555348\pi\)
−0.173007 + 0.984921i \(0.555348\pi\)
\(84\) 0 0
\(85\) 6.60543e19 0.363917
\(86\) 5.57585e19 0.271692
\(87\) −5.47978e20 −2.36489
\(88\) −1.12161e20 −0.429313
\(89\) 5.04303e20 1.71434 0.857170 − 0.515034i \(-0.172221\pi\)
0.857170 + 0.515034i \(0.172221\pi\)
\(90\) 3.82708e19 0.115697
\(91\) 0 0
\(92\) 1.48740e20 0.356990
\(93\) −2.44873e20 −0.524651
\(94\) 4.23246e19 0.0810496
\(95\) −1.71413e20 −0.293729
\(96\) −4.64033e20 −0.712362
\(97\) −8.08275e20 −1.11290 −0.556450 − 0.830881i \(-0.687837\pi\)
−0.556450 + 0.830881i \(0.687837\pi\)
\(98\) 0 0
\(99\) −5.81651e20 −0.646388
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 49.22.a.a.1.1 1
7.6 odd 2 1.22.a.a.1.1 ✓ 1
21.20 even 2 9.22.a.c.1.1 1
28.27 even 2 16.22.a.c.1.1 1
35.13 even 4 25.22.b.a.24.2 2
35.27 even 4 25.22.b.a.24.1 2
35.34 odd 2 25.22.a.a.1.1 1
56.13 odd 2 64.22.a.g.1.1 1
56.27 even 2 64.22.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.22.a.a.1.1 ✓ 1 7.6 odd 2
9.22.a.c.1.1 1 21.20 even 2
16.22.a.c.1.1 1 28.27 even 2
25.22.a.a.1.1 1 35.34 odd 2
25.22.b.a.24.1 2 35.27 even 4
25.22.b.a.24.2 2 35.13 even 4
49.22.a.a.1.1 1 1.1 even 1 trivial
64.22.a.a.1.1 1 56.27 even 2
64.22.a.g.1.1 1 56.13 odd 2