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Properties

Label 2.34.a.b.1.1
Level $2$
Weight $34$
Character 2.1
Self dual yes
Analytic conductor $13.797$
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] = [2,34,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, 34); 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, 34, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 34 \)
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: \(13.7965657762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 19957422 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5\cdot 11 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4467.87\) 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)
 
\(f(q)\) \(=\) \(q+65536.0 q^{2} -9.01727e7 q^{3} +4.29497e9 q^{4} -3.79693e11 q^{5} -5.90956e12 q^{6} +1.50920e14 q^{7} +2.81475e14 q^{8} +2.57205e15 q^{9} -2.48835e16 q^{10} +1.29789e17 q^{11} -3.87289e17 q^{12} +1.20879e18 q^{13} +9.89072e18 q^{14} +3.42379e19 q^{15} +1.84467e19 q^{16} +7.25972e19 q^{17} +1.68562e20 q^{18} +2.41894e21 q^{19} -1.63077e21 q^{20} -1.36089e22 q^{21} +8.50583e21 q^{22} -2.97560e22 q^{23} -2.53814e22 q^{24} +2.77511e22 q^{25} +7.92194e22 q^{26} +2.69347e23 q^{27} +6.48198e23 q^{28} -1.10824e24 q^{29} +2.24381e24 q^{30} -1.95357e24 q^{31} +1.20893e24 q^{32} -1.17034e25 q^{33} +4.75773e24 q^{34} -5.73034e25 q^{35} +1.10469e25 q^{36} +1.24500e26 q^{37} +1.58528e26 q^{38} -1.09000e26 q^{39} -1.06874e26 q^{40} +3.94109e26 q^{41} -8.91873e26 q^{42} +5.67060e25 q^{43} +5.57438e26 q^{44} -9.76588e26 q^{45} -1.95009e27 q^{46} -2.69133e26 q^{47} -1.66339e27 q^{48} +1.50460e28 q^{49} +1.81869e27 q^{50} -6.54628e27 q^{51} +5.19172e27 q^{52} -9.81792e26 q^{53} +1.76519e28 q^{54} -4.92798e28 q^{55} +4.24803e28 q^{56} -2.18122e29 q^{57} -7.26299e28 q^{58} +2.39411e29 q^{59} +1.47051e29 q^{60} +1.92507e29 q^{61} -1.28029e29 q^{62} +3.88175e29 q^{63} +7.92282e28 q^{64} -4.58969e29 q^{65} -7.66993e29 q^{66} -4.67531e29 q^{67} +3.11802e29 q^{68} +2.68318e30 q^{69} -3.75543e30 q^{70} -1.71647e30 q^{71} +7.23968e29 q^{72} -3.26801e30 q^{73} +8.15925e30 q^{74} -2.50239e30 q^{75} +1.03893e31 q^{76} +1.95878e31 q^{77} -7.14342e30 q^{78} -1.85271e31 q^{79} -7.00409e30 q^{80} -3.85859e31 q^{81} +2.58283e31 q^{82} +1.46644e31 q^{83} -5.84498e31 q^{84} -2.75646e31 q^{85} +3.71629e30 q^{86} +9.99334e31 q^{87} +3.65322e31 q^{88} +1.04135e32 q^{89} -6.40017e31 q^{90} +1.82431e32 q^{91} -1.27801e32 q^{92} +1.76158e32 q^{93} -1.76379e31 q^{94} -9.18453e32 q^{95} -1.09012e32 q^{96} -1.93007e32 q^{97} +9.86053e32 q^{98} +3.33823e32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 131072 q^{2} + 8356488 q^{3} + 8589934592 q^{4} - 5332476660 q^{5} + 547650797568 q^{6} + 132719095875856 q^{7} + 562949953421312 q^{8} + 67\!\cdots\!26 q^{9} - 349469190389760 q^{10} - 15\!\cdots\!76 q^{11}+ \cdots - 86\!\cdots\!88 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 65536.0 0.707107
\(3\) −9.01727e7 −1.20941 −0.604706 − 0.796449i \(-0.706709\pi\)
−0.604706 + 0.796449i \(0.706709\pi\)
\(4\) 4.29497e9 0.500000
\(5\) −3.79693e11 −1.11283 −0.556413 − 0.830906i \(-0.687823\pi\)
−0.556413 + 0.830906i \(0.687823\pi\)
\(6\) −5.90956e12 −0.855183
\(7\) 1.50920e14 1.71645 0.858223 − 0.513276i \(-0.171568\pi\)
0.858223 + 0.513276i \(0.171568\pi\)
\(8\) 2.81475e14 0.353553
\(9\) 2.57205e15 0.462677
\(10\) −2.48835e16 −0.786886
\(11\) 1.29789e17 0.851642 0.425821 − 0.904807i \(-0.359985\pi\)
0.425821 + 0.904807i \(0.359985\pi\)
\(12\) −3.87289e17 −0.604706
\(13\) 1.20879e18 0.503832 0.251916 − 0.967749i \(-0.418939\pi\)
0.251916 + 0.967749i \(0.418939\pi\)
\(14\) 9.89072e18 1.21371
\(15\) 3.42379e19 1.34586
\(16\) 1.84467e19 0.250000
\(17\) 7.25972e19 0.361837 0.180918 − 0.983498i \(-0.442093\pi\)
0.180918 + 0.983498i \(0.442093\pi\)
\(18\) 1.68562e20 0.327162
\(19\) 2.41894e21 1.92394 0.961968 − 0.273162i \(-0.0880694\pi\)
0.961968 + 0.273162i \(0.0880694\pi\)
\(20\) −1.63077e21 −0.556413
\(21\) −1.36089e22 −2.07589
\(22\) 8.50583e21 0.602202
\(23\) −2.97560e22 −1.01173 −0.505866 − 0.862612i \(-0.668827\pi\)
−0.505866 + 0.862612i \(0.668827\pi\)
\(24\) −2.53814e22 −0.427592
\(25\) 2.77511e22 0.238380
\(26\) 7.92194e22 0.356263
\(27\) 2.69347e23 0.649845
\(28\) 6.48198e23 0.858223
\(29\) −1.10824e24 −0.822373 −0.411186 − 0.911551i \(-0.634885\pi\)
−0.411186 + 0.911551i \(0.634885\pi\)
\(30\) 2.24381e24 0.951670
\(31\) −1.95357e24 −0.482348 −0.241174 − 0.970482i \(-0.577532\pi\)
−0.241174 + 0.970482i \(0.577532\pi\)
\(32\) 1.20893e24 0.176777
\(33\) −1.17034e25 −1.02999
\(34\) 4.75773e24 0.255857
\(35\) −5.73034e25 −1.91011
\(36\) 1.10469e25 0.231339
\(37\) 1.24500e26 1.65898 0.829491 − 0.558520i \(-0.188630\pi\)
0.829491 + 0.558520i \(0.188630\pi\)
\(38\) 1.58528e26 1.36043
\(39\) −1.09000e26 −0.609341
\(40\) −1.06874e26 −0.393443
\(41\) 3.94109e26 0.965345 0.482672 − 0.875801i \(-0.339666\pi\)
0.482672 + 0.875801i \(0.339666\pi\)
\(42\) −8.91873e26 −1.46788
\(43\) 5.67060e25 0.0632994 0.0316497 − 0.999499i \(-0.489924\pi\)
0.0316497 + 0.999499i \(0.489924\pi\)
\(44\) 5.57438e26 0.425821
\(45\) −9.76588e26 −0.514879
\(46\) −1.95009e27 −0.715403
\(47\) −2.69133e26 −0.0692392 −0.0346196 − 0.999401i \(-0.511022\pi\)
−0.0346196 + 0.999401i \(0.511022\pi\)
\(48\) −1.66339e27 −0.302353
\(49\) 1.50460e28 1.94619
\(50\) 1.81869e27 0.168560
\(51\) −6.54628e27 −0.437609
\(52\) 5.19172e27 0.251916
\(53\) −9.81792e26 −0.0347911 −0.0173955 − 0.999849i \(-0.505537\pi\)
−0.0173955 + 0.999849i \(0.505537\pi\)
\(54\) 1.76519e28 0.459509
\(55\) −4.92798e28 −0.947729
\(56\) 4.24803e28 0.606856
\(57\) −2.18122e29 −2.32683
\(58\) −7.26299e28 −0.581505
\(59\) 2.39411e29 1.44573 0.722864 − 0.690991i \(-0.242825\pi\)
0.722864 + 0.690991i \(0.242825\pi\)
\(60\) 1.47051e29 0.672932
\(61\) 1.92507e29 0.670661 0.335331 − 0.942101i \(-0.391152\pi\)
0.335331 + 0.942101i \(0.391152\pi\)
\(62\) −1.28029e29 −0.341071
\(63\) 3.88175e29 0.794161
\(64\) 7.92282e28 0.125000
\(65\) −4.58969e29 −0.560677
\(66\) −7.66993e29 −0.728310
\(67\) −4.67531e29 −0.346399 −0.173199 − 0.984887i \(-0.555411\pi\)
−0.173199 + 0.984887i \(0.555411\pi\)
\(68\) 3.11802e29 0.180918
\(69\) 2.68318e30 1.22360
\(70\) −3.75543e30 −1.35065
\(71\) −1.71647e30 −0.488511 −0.244256 − 0.969711i \(-0.578544\pi\)
−0.244256 + 0.969711i \(0.578544\pi\)
\(72\) 7.23968e29 0.163581
\(73\) −3.26801e30 −0.588107 −0.294053 − 0.955789i \(-0.595004\pi\)
−0.294053 + 0.955789i \(0.595004\pi\)
\(74\) 8.15925e30 1.17308
\(75\) −2.50239e30 −0.288300
\(76\) 1.03893e31 0.961968
\(77\) 1.95878e31 1.46180
\(78\) −7.14342e30 −0.430869
\(79\) −1.85271e31 −0.905652 −0.452826 − 0.891599i \(-0.649584\pi\)
−0.452826 + 0.891599i \(0.649584\pi\)
\(80\) −7.00409e30 −0.278206
\(81\) −3.85859e31 −1.24861
\(82\) 2.58283e31 0.682602
\(83\) 1.46644e31 0.317304 0.158652 − 0.987335i \(-0.449285\pi\)
0.158652 + 0.987335i \(0.449285\pi\)
\(84\) −5.84498e31 −1.03795
\(85\) −2.75646e31 −0.402661
\(86\) 3.71629e30 0.0447594
\(87\) 9.99334e31 0.994587
\(88\) 3.65322e31 0.301101
\(89\) 1.04135e32 0.712298 0.356149 − 0.934429i \(-0.384089\pi\)
0.356149 + 0.934429i \(0.384089\pi\)
\(90\) −6.40017e31 −0.364074
\(91\) 1.82431e32 0.864801
\(92\) −1.27801e32 −0.505866
\(93\) 1.76158e32 0.583357
\(94\) −1.76379e31 −0.0489595
\(95\) −9.18453e32 −2.14100
\(96\) −1.09012e32 −0.213796
\(97\) −1.93007e32 −0.319036 −0.159518 − 0.987195i \(-0.550994\pi\)
−0.159518 + 0.987195i \(0.550994\pi\)
\(98\) 9.86053e32 1.37616
\(99\) 3.33823e32 0.394036
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 2.34.a.b.1.1 ✓ 2
3.2 odd 2 18.34.a.e.1.2 2
4.3 odd 2 16.34.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.34.a.b.1.1 ✓ 2 1.1 even 1 trivial
16.34.a.c.1.2 2 4.3 odd 2
18.34.a.e.1.2 2 3.2 odd 2