Properties

Label 1.26.a.a.1.1
Level $1$
Weight $26$
Character 1.1
Self dual yes
Analytic conductor $3.960$
Analytic rank $1$
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] = [1,26,Mod(1,1)] 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("1.1"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.95996779952\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1.1

$q$-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)
 
\(f(q)\) \(=\) \(q-48.0000 q^{2} -195804. q^{3} -3.35521e7 q^{4} -7.41990e8 q^{5} +9.39859e6 q^{6} +3.90806e10 q^{7} +3.22111e9 q^{8} -8.08949e11 q^{9} +3.56155e10 q^{10} +8.41952e12 q^{11} +6.56964e12 q^{12} -8.16510e13 q^{13} -1.87587e12 q^{14} +1.45285e14 q^{15} +1.12567e15 q^{16} -2.51990e15 q^{17} +3.88296e13 q^{18} -6.08206e15 q^{19} +2.48953e16 q^{20} -7.65214e15 q^{21} -4.04137e14 q^{22} -9.49953e16 q^{23} -6.30707e14 q^{24} +2.52526e17 q^{25} +3.91925e15 q^{26} +3.24298e17 q^{27} -1.31124e18 q^{28} -2.71247e17 q^{29} -6.97366e15 q^{30} +4.29167e18 q^{31} -1.62115e17 q^{32} -1.64857e18 q^{33} +1.20955e17 q^{34} -2.89974e19 q^{35} +2.71420e19 q^{36} +2.03015e19 q^{37} +2.91939e17 q^{38} +1.59876e19 q^{39} -2.39003e18 q^{40} -1.83744e20 q^{41} +3.67303e17 q^{42} +3.00902e20 q^{43} -2.82493e20 q^{44} +6.00232e20 q^{45} +4.55977e18 q^{46} -9.24361e20 q^{47} -2.20410e20 q^{48} +1.86224e20 q^{49} -1.21212e19 q^{50} +4.93407e20 q^{51} +2.73957e21 q^{52} -9.90292e20 q^{53} -1.55663e19 q^{54} -6.24719e21 q^{55} +1.25883e20 q^{56} +1.19089e21 q^{57} +1.30199e19 q^{58} +1.30526e22 q^{59} -4.87461e21 q^{60} +9.01545e21 q^{61} -2.06000e20 q^{62} -3.16142e22 q^{63} -3.77634e22 q^{64} +6.05842e22 q^{65} +7.91316e19 q^{66} -2.66891e22 q^{67} +8.45480e22 q^{68} +1.86005e22 q^{69} +1.39188e21 q^{70} -1.92391e23 q^{71} -2.60572e21 q^{72} +4.24046e22 q^{73} -9.74471e20 q^{74} -4.94455e22 q^{75} +2.04066e23 q^{76} +3.29040e23 q^{77} -7.67405e20 q^{78} -2.71681e23 q^{79} -8.35234e23 q^{80} +6.21915e23 q^{81} +8.81972e21 q^{82} -9.31454e23 q^{83} +2.56745e23 q^{84} +1.86974e24 q^{85} -1.44433e22 q^{86} +5.31112e22 q^{87} +2.71202e22 q^{88} -1.76364e24 q^{89} -2.88111e22 q^{90} -3.19097e24 q^{91} +3.18729e24 q^{92} -8.40325e23 q^{93} +4.43693e22 q^{94} +4.51282e24 q^{95} +3.17427e22 q^{96} +2.82924e24 q^{97} -8.93877e21 q^{98} -6.81096e24 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −48.0000 −0.00828641 −0.00414320 − 0.999991i \(-0.501319\pi\)
−0.00414320 + 0.999991i \(0.501319\pi\)
\(3\) −195804. −0.212719 −0.106359 − 0.994328i \(-0.533919\pi\)
−0.106359 + 0.994328i \(0.533919\pi\)
\(4\) −3.35521e7 −0.999931
\(5\) −7.41990e8 −1.35917 −0.679584 − 0.733598i \(-0.737840\pi\)
−0.679584 + 0.733598i \(0.737840\pi\)
\(6\) 9.39859e6 0.00176267
\(7\) 3.90806e10 1.06718 0.533588 − 0.845745i \(-0.320843\pi\)
0.533588 + 0.845745i \(0.320843\pi\)
\(8\) 3.22111e9 0.0165722
\(9\) −8.08949e11 −0.954751
\(10\) 3.56155e10 0.0112626
\(11\) 8.41952e12 0.808870 0.404435 − 0.914567i \(-0.367468\pi\)
0.404435 + 0.914567i \(0.367468\pi\)
\(12\) 6.56964e12 0.212704
\(13\) −8.16510e13 −0.972008 −0.486004 − 0.873957i \(-0.661546\pi\)
−0.486004 + 0.873957i \(0.661546\pi\)
\(14\) −1.87587e12 −0.00884305
\(15\) 1.45285e14 0.289120
\(16\) 1.12567e15 0.999794
\(17\) −2.51990e15 −1.04899 −0.524496 − 0.851413i \(-0.675746\pi\)
−0.524496 + 0.851413i \(0.675746\pi\)
\(18\) 3.88296e13 0.00791145
\(19\) −6.08206e15 −0.630421 −0.315210 − 0.949022i \(-0.602075\pi\)
−0.315210 + 0.949022i \(0.602075\pi\)
\(20\) 2.48953e16 1.35907
\(21\) −7.65214e15 −0.227008
\(22\) −4.04137e14 −0.00670262
\(23\) −9.49953e16 −0.903866 −0.451933 − 0.892052i \(-0.649265\pi\)
−0.451933 + 0.892052i \(0.649265\pi\)
\(24\) −6.30707e14 −0.00352523
\(25\) 2.52526e17 0.847336
\(26\) 3.91925e15 0.00805445
\(27\) 3.24298e17 0.415812
\(28\) −1.31124e18 −1.06710
\(29\) −2.71247e17 −0.142361 −0.0711803 − 0.997463i \(-0.522677\pi\)
−0.0711803 + 0.997463i \(0.522677\pi\)
\(30\) −6.97366e15 −0.00239577
\(31\) 4.29167e18 0.978599 0.489299 − 0.872116i \(-0.337253\pi\)
0.489299 + 0.872116i \(0.337253\pi\)
\(32\) −1.62115e17 −0.0248569
\(33\) −1.64857e18 −0.172062
\(34\) 1.20955e17 0.00869237
\(35\) −2.89974e19 −1.45047
\(36\) 2.71420e19 0.954685
\(37\) 2.03015e19 0.506998 0.253499 − 0.967336i \(-0.418419\pi\)
0.253499 + 0.967336i \(0.418419\pi\)
\(38\) 2.91939e17 0.00522392
\(39\) 1.59876e19 0.206764
\(40\) −2.39003e18 −0.0225245
\(41\) −1.83744e20 −1.27179 −0.635895 − 0.771775i \(-0.719369\pi\)
−0.635895 + 0.771775i \(0.719369\pi\)
\(42\) 3.67303e17 0.00188108
\(43\) 3.00902e20 1.14834 0.574168 − 0.818737i \(-0.305326\pi\)
0.574168 + 0.818737i \(0.305326\pi\)
\(44\) −2.82493e20 −0.808814
\(45\) 6.00232e20 1.29767
\(46\) 4.55977e18 0.00748980
\(47\) −9.24361e20 −1.16043 −0.580214 − 0.814464i \(-0.697031\pi\)
−0.580214 + 0.814464i \(0.697031\pi\)
\(48\) −2.20410e20 −0.212675
\(49\) 1.86224e20 0.138863
\(50\) −1.21212e19 −0.00702137
\(51\) 4.93407e20 0.223140
\(52\) 2.73957e21 0.971941
\(53\) −9.90292e20 −0.276895 −0.138447 − 0.990370i \(-0.544211\pi\)
−0.138447 + 0.990370i \(0.544211\pi\)
\(54\) −1.55663e19 −0.00344559
\(55\) −6.24719e21 −1.09939
\(56\) 1.25883e20 0.0176855
\(57\) 1.19089e21 0.134102
\(58\) 1.30199e19 0.00117966
\(59\) 1.30526e22 0.955093 0.477547 − 0.878606i \(-0.341526\pi\)
0.477547 + 0.878606i \(0.341526\pi\)
\(60\) −4.87461e21 −0.289101
\(61\) 9.01545e21 0.434875 0.217438 − 0.976074i \(-0.430230\pi\)
0.217438 + 0.976074i \(0.430230\pi\)
\(62\) −2.06000e20 −0.00810907
\(63\) −3.16142e22 −1.01889
\(64\) −3.77634e22 −0.999588
\(65\) 6.05842e22 1.32112
\(66\) 7.91316e19 0.00142577
\(67\) −2.66891e22 −0.398473 −0.199236 − 0.979951i \(-0.563846\pi\)
−0.199236 + 0.979951i \(0.563846\pi\)
\(68\) 8.45480e22 1.04892
\(69\) 1.86005e22 0.192269
\(70\) 1.39188e21 0.0120192
\(71\) −1.92391e23 −1.39141 −0.695704 − 0.718329i \(-0.744907\pi\)
−0.695704 + 0.718329i \(0.744907\pi\)
\(72\) −2.60572e21 −0.0158224
\(73\) 4.24046e22 0.216709 0.108355 − 0.994112i \(-0.465442\pi\)
0.108355 + 0.994112i \(0.465442\pi\)
\(74\) −9.74471e20 −0.00420119
\(75\) −4.94455e22 −0.180244
\(76\) 2.04066e23 0.630377
\(77\) 3.29040e23 0.863206
\(78\) −7.67405e20 −0.00171333
\(79\) −2.71681e23 −0.517274 −0.258637 − 0.965975i \(-0.583273\pi\)
−0.258637 + 0.965975i \(0.583273\pi\)
\(80\) −8.35234e23 −1.35889
\(81\) 6.21915e23 0.866300
\(82\) 8.81972e21 0.0105386
\(83\) −9.31454e23 −0.956501 −0.478251 − 0.878223i \(-0.658729\pi\)
−0.478251 + 0.878223i \(0.658729\pi\)
\(84\) 2.56745e23 0.226993
\(85\) 1.86974e24 1.42575
\(86\) −1.44433e22 −0.00951558
\(87\) 5.31112e22 0.0302828
\(88\) 2.71202e22 0.0134048
\(89\) −1.76364e24 −0.756892 −0.378446 − 0.925623i \(-0.623541\pi\)
−0.378446 + 0.925623i \(0.623541\pi\)
\(90\) −2.88111e22 −0.0107530
\(91\) −3.19097e24 −1.03730
\(92\) 3.18729e24 0.903804
\(93\) −8.40325e23 −0.208166
\(94\) 4.43693e22 0.00961578
\(95\) 4.51282e24 0.856847
\(96\) 3.17427e22 0.00528754
\(97\) 2.82924e24 0.414022 0.207011 − 0.978339i \(-0.433626\pi\)
0.207011 + 0.978339i \(0.433626\pi\)
\(98\) −8.93877e21 −0.00115067
\(99\) −6.81096e24 −0.772269
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 1.26.a.a.1.1 ✓ 1
3.2 odd 2 9.26.a.a.1.1 1
4.3 odd 2 16.26.a.b.1.1 1
5.2 odd 4 25.26.b.a.24.1 2
5.3 odd 4 25.26.b.a.24.2 2
5.4 even 2 25.26.a.a.1.1 1
7.6 odd 2 49.26.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.26.a.a.1.1 ✓ 1 1.1 even 1 trivial
9.26.a.a.1.1 1 3.2 odd 2
16.26.a.b.1.1 1 4.3 odd 2
25.26.a.a.1.1 1 5.4 even 2
25.26.b.a.24.1 2 5.2 odd 4
25.26.b.a.24.2 2 5.3 odd 4
49.26.a.a.1.1 1 7.6 odd 2