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Properties

Label 2.26.a.b.1.1
Level $2$
Weight $26$
Character 2.1
Self dual yes
Analytic conductor $7.920$
Analytic rank $0$
Dimension $2$
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] = [2,26,Mod(1,2)] 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("2.1"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.91993559904\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{106705}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 26676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(163.829\) of defining polynomial
Character \(\chi\) \(=\) 2.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4096.00 q^{2} -1.37803e6 q^{3} +1.67772e7 q^{4} +8.78994e8 q^{5} -5.64442e9 q^{6} +3.00388e10 q^{7} +6.87195e10 q^{8} +1.05168e12 q^{9} +3.60036e12 q^{10} +5.73599e12 q^{11} -2.31195e13 q^{12} -1.07343e13 q^{13} +1.23039e14 q^{14} -1.21128e15 q^{15} +2.81475e14 q^{16} +2.97473e15 q^{17} +4.30769e15 q^{18} -5.42738e15 q^{19} +1.47471e16 q^{20} -4.13944e16 q^{21} +2.34946e16 q^{22} -1.04540e17 q^{23} -9.46976e16 q^{24} +4.74607e17 q^{25} -4.39677e16 q^{26} -2.81659e17 q^{27} +5.03967e17 q^{28} +3.09182e18 q^{29} -4.96141e18 q^{30} -4.26809e18 q^{31} +1.15292e18 q^{32} -7.90437e18 q^{33} +1.21845e19 q^{34} +2.64039e19 q^{35} +1.76443e19 q^{36} -4.51028e19 q^{37} -2.22305e19 q^{38} +1.47922e19 q^{39} +6.04040e19 q^{40} -7.56724e19 q^{41} -1.69551e20 q^{42} -1.39036e20 q^{43} +9.62339e19 q^{44} +9.24421e20 q^{45} -4.28196e20 q^{46} +4.67328e20 q^{47} -3.87881e20 q^{48} -4.38741e20 q^{49} +1.94399e21 q^{50} -4.09927e21 q^{51} -1.80092e20 q^{52} -1.24315e21 q^{53} -1.15368e21 q^{54} +5.04190e21 q^{55} +2.06425e21 q^{56} +7.47909e21 q^{57} +1.26641e22 q^{58} -1.32468e22 q^{59} -2.03219e22 q^{60} -2.36984e20 q^{61} -1.74821e22 q^{62} +3.15912e22 q^{63} +4.72237e21 q^{64} -9.43539e21 q^{65} -3.23763e22 q^{66} -5.36922e22 q^{67} +4.99076e22 q^{68} +1.44059e23 q^{69} +1.08150e23 q^{70} -2.27032e23 q^{71} +7.22710e22 q^{72} +2.78528e23 q^{73} -1.84741e23 q^{74} -6.54024e23 q^{75} -9.10563e22 q^{76} +1.72302e23 q^{77} +6.05889e22 q^{78} +6.36958e23 q^{79} +2.47415e23 q^{80} -5.02942e23 q^{81} -3.09954e23 q^{82} +1.19662e24 q^{83} -6.94482e23 q^{84} +2.61477e24 q^{85} -5.69491e23 q^{86} -4.26063e24 q^{87} +3.94174e23 q^{88} -3.22134e24 q^{89} +3.78643e24 q^{90} -3.22445e23 q^{91} -1.75389e24 q^{92} +5.88156e24 q^{93} +1.91418e24 q^{94} -4.77063e24 q^{95} -1.58876e24 q^{96} +3.58465e24 q^{97} -1.79708e24 q^{98} +6.03243e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8192 q^{2} + 379848 q^{3} + 33554432 q^{4} + 741953100 q^{5} + 1555857408 q^{6} - 376536944 q^{7} + 137438953472 q^{8} + 3294531432666 q^{9} + 3039039897600 q^{10} + 8323034610264 q^{11} + 6372791943168 q^{12}+ \cdots + 11\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4096.00 0.707107
\(3\) −1.37803e6 −1.49707 −0.748537 − 0.663093i \(-0.769243\pi\)
−0.748537 + 0.663093i \(0.769243\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 8.78994e8 1.61013 0.805065 − 0.593187i \(-0.202130\pi\)
0.805065 + 0.593187i \(0.202130\pi\)
\(6\) −5.64442e9 −1.05859
\(7\) 3.00388e10 0.820270 0.410135 − 0.912025i \(-0.365482\pi\)
0.410135 + 0.912025i \(0.365482\pi\)
\(8\) 6.87195e10 0.353553
\(9\) 1.05168e12 1.24123
\(10\) 3.60036e12 1.13853
\(11\) 5.73599e12 0.551061 0.275531 − 0.961292i \(-0.411146\pi\)
0.275531 + 0.961292i \(0.411146\pi\)
\(12\) −2.31195e13 −0.748537
\(13\) −1.07343e13 −0.127786 −0.0638928 − 0.997957i \(-0.520352\pi\)
−0.0638928 + 0.997957i \(0.520352\pi\)
\(14\) 1.23039e14 0.580018
\(15\) −1.21128e15 −2.41048
\(16\) 2.81475e14 0.250000
\(17\) 2.97473e15 1.23833 0.619164 − 0.785262i \(-0.287472\pi\)
0.619164 + 0.785262i \(0.287472\pi\)
\(18\) 4.30769e15 0.877683
\(19\) −5.42738e15 −0.562561 −0.281281 − 0.959626i \(-0.590759\pi\)
−0.281281 + 0.959626i \(0.590759\pi\)
\(20\) 1.47471e16 0.805065
\(21\) −4.13944e16 −1.22800
\(22\) 2.34946e16 0.389659
\(23\) −1.04540e17 −0.994683 −0.497341 − 0.867555i \(-0.665690\pi\)
−0.497341 + 0.867555i \(0.665690\pi\)
\(24\) −9.46976e16 −0.529296
\(25\) 4.74607e17 1.59252
\(26\) −4.39677e16 −0.0903581
\(27\) −2.81659e17 −0.361141
\(28\) 5.03967e17 0.410135
\(29\) 3.09182e18 1.62270 0.811352 − 0.584558i \(-0.198732\pi\)
0.811352 + 0.584558i \(0.198732\pi\)
\(30\) −4.96141e18 −1.70447
\(31\) −4.26809e18 −0.973222 −0.486611 − 0.873619i \(-0.661767\pi\)
−0.486611 + 0.873619i \(0.661767\pi\)
\(32\) 1.15292e18 0.176777
\(33\) −7.90437e18 −0.824980
\(34\) 1.21845e19 0.875630
\(35\) 2.64039e19 1.32074
\(36\) 1.76443e19 0.620616
\(37\) −4.51028e19 −1.12637 −0.563186 − 0.826330i \(-0.690424\pi\)
−0.563186 + 0.826330i \(0.690424\pi\)
\(38\) −2.22305e19 −0.397791
\(39\) 1.47922e19 0.191305
\(40\) 6.04040e19 0.569267
\(41\) −7.56724e19 −0.523769 −0.261884 − 0.965099i \(-0.584344\pi\)
−0.261884 + 0.965099i \(0.584344\pi\)
\(42\) −1.69551e20 −0.868330
\(43\) −1.39036e20 −0.530605 −0.265302 − 0.964165i \(-0.585472\pi\)
−0.265302 + 0.964165i \(0.585472\pi\)
\(44\) 9.62339e19 0.275531
\(45\) 9.24421e20 1.99854
\(46\) −4.28196e20 −0.703347
\(47\) 4.67328e20 0.586676 0.293338 − 0.956009i \(-0.405234\pi\)
0.293338 + 0.956009i \(0.405234\pi\)
\(48\) −3.87881e20 −0.374269
\(49\) −4.38741e20 −0.327158
\(50\) 1.94399e21 1.12608
\(51\) −4.09927e21 −1.85387
\(52\) −1.80092e20 −0.0638928
\(53\) −1.24315e21 −0.347595 −0.173797 − 0.984781i \(-0.555604\pi\)
−0.173797 + 0.984781i \(0.555604\pi\)
\(54\) −1.15368e21 −0.255365
\(55\) 5.04190e21 0.887280
\(56\) 2.06425e21 0.290009
\(57\) 7.47909e21 0.842196
\(58\) 1.26641e22 1.14743
\(59\) −1.32468e22 −0.969303 −0.484652 − 0.874707i \(-0.661054\pi\)
−0.484652 + 0.874707i \(0.661054\pi\)
\(60\) −2.03219e22 −1.20524
\(61\) −2.36984e20 −0.0114313 −0.00571565 − 0.999984i \(-0.501819\pi\)
−0.00571565 + 0.999984i \(0.501819\pi\)
\(62\) −1.74821e22 −0.688172
\(63\) 3.15912e22 1.01814
\(64\) 4.72237e21 0.125000
\(65\) −9.43539e21 −0.205752
\(66\) −3.23763e22 −0.583349
\(67\) −5.36922e22 −0.801634 −0.400817 − 0.916158i \(-0.631274\pi\)
−0.400817 + 0.916158i \(0.631274\pi\)
\(68\) 4.99076e22 0.619164
\(69\) 1.44059e23 1.48911
\(70\) 1.08150e23 0.933905
\(71\) −2.27032e23 −1.64194 −0.820971 − 0.570970i \(-0.806567\pi\)
−0.820971 + 0.570970i \(0.806567\pi\)
\(72\) 7.22710e22 0.438842
\(73\) 2.78528e23 1.42342 0.711710 − 0.702473i \(-0.247921\pi\)
0.711710 + 0.702473i \(0.247921\pi\)
\(74\) −1.84741e23 −0.796465
\(75\) −6.54024e23 −2.38412
\(76\) −9.10563e22 −0.281281
\(77\) 1.72302e23 0.452019
\(78\) 6.05889e22 0.135273
\(79\) 6.36958e23 1.21275 0.606376 − 0.795178i \(-0.292622\pi\)
0.606376 + 0.795178i \(0.292622\pi\)
\(80\) 2.47415e23 0.402532
\(81\) −5.02942e23 −0.700576
\(82\) −3.09954e23 −0.370360
\(83\) 1.19662e24 1.22880 0.614398 − 0.788996i \(-0.289399\pi\)
0.614398 + 0.788996i \(0.289399\pi\)
\(84\) −6.94482e23 −0.614002
\(85\) 2.61477e24 1.99387
\(86\) −5.69491e23 −0.375194
\(87\) −4.26063e24 −2.42931
\(88\) 3.94174e23 0.194830
\(89\) −3.22134e24 −1.38249 −0.691244 − 0.722621i \(-0.742937\pi\)
−0.691244 + 0.722621i \(0.742937\pi\)
\(90\) 3.78643e24 1.41318
\(91\) −3.22445e23 −0.104819
\(92\) −1.75389e24 −0.497341
\(93\) 5.88156e24 1.45699
\(94\) 1.91418e24 0.414843
\(95\) −4.77063e24 −0.905797
\(96\) −1.58876e24 −0.264648
\(97\) 3.58465e24 0.524566 0.262283 − 0.964991i \(-0.415525\pi\)
0.262283 + 0.964991i \(0.415525\pi\)
\(98\) −1.79708e24 −0.231335
\(99\) 6.03243e24 0.683994
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 2.26.a.b.1.1 ✓ 2
3.2 odd 2 18.26.a.e.1.1 2
4.3 odd 2 16.26.a.c.1.2 2
5.2 odd 4 50.26.b.e.49.4 4
5.3 odd 4 50.26.b.e.49.1 4
5.4 even 2 50.26.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.1 ✓ 2 1.1 even 1 trivial
16.26.a.c.1.2 2 4.3 odd 2
18.26.a.e.1.1 2 3.2 odd 2
50.26.a.c.1.2 2 5.4 even 2
50.26.b.e.49.1 4 5.3 odd 4
50.26.b.e.49.4 4 5.2 odd 4