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Properties

Label 18.26.a.e.1.1
Level $18$
Weight $26$
Character 18.1
Self dual yes
Analytic conductor $71.279$
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] = [18,26,Mod(1,18)] 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("18.1"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-8192,0,33554432,-741953100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(71.2794203914\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{5}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(163.829\) of defining polynomial
Character \(\chi\) \(=\) 18.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} +1.67772e7 q^{4} -8.78994e8 q^{5} +3.00388e10 q^{7} -6.87195e10 q^{8} +3.60036e12 q^{10} -5.73599e12 q^{11} -1.07343e13 q^{13} -1.23039e14 q^{14} +2.81475e14 q^{16} -2.97473e15 q^{17} -5.42738e15 q^{19} -1.47471e16 q^{20} +2.34946e16 q^{22} +1.04540e17 q^{23} +4.74607e17 q^{25} +4.39677e16 q^{26} +5.03967e17 q^{28} -3.09182e18 q^{29} -4.26809e18 q^{31} -1.15292e18 q^{32} +1.21845e19 q^{34} -2.64039e19 q^{35} -4.51028e19 q^{37} +2.22305e19 q^{38} +6.04040e19 q^{40} +7.56724e19 q^{41} -1.39036e20 q^{43} -9.62339e19 q^{44} -4.28196e20 q^{46} -4.67328e20 q^{47} -4.38741e20 q^{49} -1.94399e21 q^{50} -1.80092e20 q^{52} +1.24315e21 q^{53} +5.04190e21 q^{55} -2.06425e21 q^{56} +1.26641e22 q^{58} +1.32468e22 q^{59} -2.36984e20 q^{61} +1.74821e22 q^{62} +4.72237e21 q^{64} +9.43539e21 q^{65} -5.36922e22 q^{67} -4.99076e22 q^{68} +1.08150e23 q^{70} +2.27032e23 q^{71} +2.78528e23 q^{73} +1.84741e23 q^{74} -9.10563e22 q^{76} -1.72302e23 q^{77} +6.36958e23 q^{79} -2.47415e23 q^{80} -3.09954e23 q^{82} -1.19662e24 q^{83} +2.61477e24 q^{85} +5.69491e23 q^{86} +3.94174e23 q^{88} +3.22134e24 q^{89} -3.22445e23 q^{91} +1.75389e24 q^{92} +1.91418e24 q^{94} +4.77063e24 q^{95} +3.58465e24 q^{97} +1.79708e24 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8192 q^{2} + 33554432 q^{4} - 741953100 q^{5} - 376536944 q^{7} - 137438953472 q^{8} + 3039039897600 q^{10} - 8323034610264 q^{11} - 106467053152292 q^{13} + 1542295322624 q^{14} + 562949953421312 q^{16}+ \cdots + 35\!\cdots\!16 q^{98}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 1.67772e7 0.500000
\(5\) −8.78994e8 −1.61013 −0.805065 − 0.593187i \(-0.797870\pi\)
−0.805065 + 0.593187i \(0.797870\pi\)
\(6\) 0 0
\(7\) 3.00388e10 0.820270 0.410135 − 0.912025i \(-0.365482\pi\)
0.410135 + 0.912025i \(0.365482\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) 0 0
\(10\) 3.60036e12 1.13853
\(11\) −5.73599e12 −0.551061 −0.275531 − 0.961292i \(-0.588854\pi\)
−0.275531 + 0.961292i \(0.588854\pi\)
\(12\) 0 0
\(13\) −1.07343e13 −0.127786 −0.0638928 − 0.997957i \(-0.520352\pi\)
−0.0638928 + 0.997957i \(0.520352\pi\)
\(14\) −1.23039e14 −0.580018
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) −2.97473e15 −1.23833 −0.619164 − 0.785262i \(-0.712528\pi\)
−0.619164 + 0.785262i \(0.712528\pi\)
\(18\) 0 0
\(19\) −5.42738e15 −0.562561 −0.281281 − 0.959626i \(-0.590759\pi\)
−0.281281 + 0.959626i \(0.590759\pi\)
\(20\) −1.47471e16 −0.805065
\(21\) 0 0
\(22\) 2.34946e16 0.389659
\(23\) 1.04540e17 0.994683 0.497341 − 0.867555i \(-0.334310\pi\)
0.497341 + 0.867555i \(0.334310\pi\)
\(24\) 0 0
\(25\) 4.74607e17 1.59252
\(26\) 4.39677e16 0.0903581
\(27\) 0 0
\(28\) 5.03967e17 0.410135
\(29\) −3.09182e18 −1.62270 −0.811352 − 0.584558i \(-0.801268\pi\)
−0.811352 + 0.584558i \(0.801268\pi\)
\(30\) 0 0
\(31\) −4.26809e18 −0.973222 −0.486611 − 0.873619i \(-0.661767\pi\)
−0.486611 + 0.873619i \(0.661767\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) 0 0
\(34\) 1.21845e19 0.875630
\(35\) −2.64039e19 −1.32074
\(36\) 0 0
\(37\) −4.51028e19 −1.12637 −0.563186 − 0.826330i \(-0.690424\pi\)
−0.563186 + 0.826330i \(0.690424\pi\)
\(38\) 2.22305e19 0.397791
\(39\) 0 0
\(40\) 6.04040e19 0.569267
\(41\) 7.56724e19 0.523769 0.261884 − 0.965099i \(-0.415656\pi\)
0.261884 + 0.965099i \(0.415656\pi\)
\(42\) 0 0
\(43\) −1.39036e20 −0.530605 −0.265302 − 0.964165i \(-0.585472\pi\)
−0.265302 + 0.964165i \(0.585472\pi\)
\(44\) −9.62339e19 −0.275531
\(45\) 0 0
\(46\) −4.28196e20 −0.703347
\(47\) −4.67328e20 −0.586676 −0.293338 − 0.956009i \(-0.594766\pi\)
−0.293338 + 0.956009i \(0.594766\pi\)
\(48\) 0 0
\(49\) −4.38741e20 −0.327158
\(50\) −1.94399e21 −1.12608
\(51\) 0 0
\(52\) −1.80092e20 −0.0638928
\(53\) 1.24315e21 0.347595 0.173797 − 0.984781i \(-0.444396\pi\)
0.173797 + 0.984781i \(0.444396\pi\)
\(54\) 0 0
\(55\) 5.04190e21 0.887280
\(56\) −2.06425e21 −0.290009
\(57\) 0 0
\(58\) 1.26641e22 1.14743
\(59\) 1.32468e22 0.969303 0.484652 − 0.874707i \(-0.338946\pi\)
0.484652 + 0.874707i \(0.338946\pi\)
\(60\) 0 0
\(61\) −2.36984e20 −0.0114313 −0.00571565 − 0.999984i \(-0.501819\pi\)
−0.00571565 + 0.999984i \(0.501819\pi\)
\(62\) 1.74821e22 0.688172
\(63\) 0 0
\(64\) 4.72237e21 0.125000
\(65\) 9.43539e21 0.205752
\(66\) 0 0
\(67\) −5.36922e22 −0.801634 −0.400817 − 0.916158i \(-0.631274\pi\)
−0.400817 + 0.916158i \(0.631274\pi\)
\(68\) −4.99076e22 −0.619164
\(69\) 0 0
\(70\) 1.08150e23 0.933905
\(71\) 2.27032e23 1.64194 0.820971 − 0.570970i \(-0.193433\pi\)
0.820971 + 0.570970i \(0.193433\pi\)
\(72\) 0 0
\(73\) 2.78528e23 1.42342 0.711710 − 0.702473i \(-0.247921\pi\)
0.711710 + 0.702473i \(0.247921\pi\)
\(74\) 1.84741e23 0.796465
\(75\) 0 0
\(76\) −9.10563e22 −0.281281
\(77\) −1.72302e23 −0.452019
\(78\) 0 0
\(79\) 6.36958e23 1.21275 0.606376 − 0.795178i \(-0.292622\pi\)
0.606376 + 0.795178i \(0.292622\pi\)
\(80\) −2.47415e23 −0.402532
\(81\) 0 0
\(82\) −3.09954e23 −0.370360
\(83\) −1.19662e24 −1.22880 −0.614398 − 0.788996i \(-0.710601\pi\)
−0.614398 + 0.788996i \(0.710601\pi\)
\(84\) 0 0
\(85\) 2.61477e24 1.99387
\(86\) 5.69491e23 0.375194
\(87\) 0 0
\(88\) 3.94174e23 0.194830
\(89\) 3.22134e24 1.38249 0.691244 − 0.722621i \(-0.257063\pi\)
0.691244 + 0.722621i \(0.257063\pi\)
\(90\) 0 0
\(91\) −3.22445e23 −0.104819
\(92\) 1.75389e24 0.497341
\(93\) 0 0
\(94\) 1.91418e24 0.414843
\(95\) 4.77063e24 0.905797
\(96\) 0 0
\(97\) 3.58465e24 0.524566 0.262283 − 0.964991i \(-0.415525\pi\)
0.262283 + 0.964991i \(0.415525\pi\)
\(98\) 1.79708e24 0.231335
\(99\) 0 0
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 18.26.a.e.1.1 2
3.2 odd 2 2.26.a.b.1.1 ✓ 2
12.11 even 2 16.26.a.c.1.2 2
15.2 even 4 50.26.b.e.49.4 4
15.8 even 4 50.26.b.e.49.1 4
15.14 odd 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 3.2 odd 2
16.26.a.c.1.2 2 12.11 even 2
18.26.a.e.1.1 2 1.1 even 1 trivial
50.26.a.c.1.2 2 15.14 odd 2
50.26.b.e.49.1 4 15.8 even 4
50.26.b.e.49.4 4 15.2 even 4