Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7938,2,Mod(1,7938)] 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("7938.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7938, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7938.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,0,2,4,0,0,2,0,4,-2,0,6,0,0,2,14,0,2,4,0,-2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.3852491245\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1134)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7938.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +0.267949 q^{5} +1.00000 q^{8} +0.267949 q^{10} -6.19615 q^{11} +6.46410 q^{13} +1.00000 q^{16} +7.00000 q^{17} -0.732051 q^{19} +0.267949 q^{20} -6.19615 q^{22} +4.19615 q^{23} -4.92820 q^{25} +6.46410 q^{26} -1.53590 q^{29} -8.19615 q^{31} +1.00000 q^{32} +7.00000 q^{34} +10.6603 q^{37} -0.732051 q^{38} +0.267949 q^{40} +2.53590 q^{41} -1.46410 q^{43} -6.19615 q^{44} +4.19615 q^{46} +4.73205 q^{47} -4.92820 q^{50} +6.46410 q^{52} +9.46410 q^{53} -1.66025 q^{55} -1.53590 q^{58} +4.19615 q^{59} -3.92820 q^{61} -8.19615 q^{62} +1.00000 q^{64} +1.73205 q^{65} -6.73205 q^{67} +7.00000 q^{68} -6.53590 q^{71} -8.26795 q^{73} +10.6603 q^{74} -0.732051 q^{76} -9.12436 q^{79} +0.267949 q^{80} +2.53590 q^{82} +16.5885 q^{83} +1.87564 q^{85} -1.46410 q^{86} -6.19615 q^{88} +9.92820 q^{89} +4.19615 q^{92} +4.73205 q^{94} -0.196152 q^{95} -10.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{8} + 4 q^{10} - 2 q^{11} + 6 q^{13} + 2 q^{16} + 14 q^{17} + 2 q^{19} + 4 q^{20} - 2 q^{22} - 2 q^{23} + 4 q^{25} + 6 q^{26} - 10 q^{29} - 6 q^{31} + 2 q^{32} + 14 q^{34}+ \cdots - 8 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0.267949 0.119831 0.0599153 − 0.998203i \(-0.480917\pi\)
0.0599153 + 0.998203i \(0.480917\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0.267949 0.0847330
\(11\) −6.19615 −1.86821 −0.934105 − 0.356998i \(-0.883800\pi\)
−0.934105 + 0.356998i \(0.883800\pi\)
\(12\) 0 0
\(13\) 6.46410 1.79282 0.896410 − 0.443227i \(-0.146166\pi\)
0.896410 + 0.443227i \(0.146166\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 − 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 0 0
\(19\) −0.732051 −0.167944 −0.0839720 − 0.996468i \(-0.526761\pi\)
−0.0839720 + 0.996468i \(0.526761\pi\)
\(20\) 0.267949 0.0599153
\(21\) 0 0
\(22\) −6.19615 −1.32102
\(23\) 4.19615 0.874958 0.437479 − 0.899229i \(-0.355871\pi\)
0.437479 + 0.899229i \(0.355871\pi\)
\(24\) 0 0
\(25\) −4.92820 −0.985641
\(26\) 6.46410 1.26771
\(27\) 0 0
\(28\) 0 0
\(29\) −1.53590 −0.285209 −0.142605 − 0.989780i \(-0.545548\pi\)
−0.142605 + 0.989780i \(0.545548\pi\)
\(30\) 0 0
\(31\) −8.19615 −1.47207 −0.736036 − 0.676942i \(-0.763305\pi\)
−0.736036 + 0.676942i \(0.763305\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 7.00000 1.20049
\(35\) 0 0
\(36\) 0 0
\(37\) 10.6603 1.75253 0.876267 − 0.481825i \(-0.160026\pi\)
0.876267 + 0.481825i \(0.160026\pi\)
\(38\) −0.732051 −0.118754
\(39\) 0 0
\(40\) 0.267949 0.0423665
\(41\) 2.53590 0.396041 0.198020 − 0.980198i \(-0.436549\pi\)
0.198020 + 0.980198i \(0.436549\pi\)
\(42\) 0 0
\(43\) −1.46410 −0.223273 −0.111637 − 0.993749i \(-0.535609\pi\)
−0.111637 + 0.993749i \(0.535609\pi\)
\(44\) −6.19615 −0.934105
\(45\) 0 0
\(46\) 4.19615 0.618689
\(47\) 4.73205 0.690241 0.345120 − 0.938558i \(-0.387838\pi\)
0.345120 + 0.938558i \(0.387838\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −4.92820 −0.696953
\(51\) 0 0
\(52\) 6.46410 0.896410
\(53\) 9.46410 1.29999 0.649997 − 0.759937i \(-0.274770\pi\)
0.649997 + 0.759937i \(0.274770\pi\)
\(54\) 0 0
\(55\) −1.66025 −0.223869
\(56\) 0 0
\(57\) 0 0
\(58\) −1.53590 −0.201673
\(59\) 4.19615 0.546293 0.273146 − 0.961973i \(-0.411936\pi\)
0.273146 + 0.961973i \(0.411936\pi\)
\(60\) 0 0
\(61\) −3.92820 −0.502955 −0.251477 − 0.967863i \(-0.580916\pi\)
−0.251477 + 0.967863i \(0.580916\pi\)
\(62\) −8.19615 −1.04091
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.73205 0.214834
\(66\) 0 0
\(67\) −6.73205 −0.822451 −0.411225 − 0.911534i \(-0.634899\pi\)
−0.411225 + 0.911534i \(0.634899\pi\)
\(68\) 7.00000 0.848875
\(69\) 0 0
\(70\) 0 0
\(71\) −6.53590 −0.775668 −0.387834 − 0.921729i \(-0.626777\pi\)
−0.387834 + 0.921729i \(0.626777\pi\)
\(72\) 0 0
\(73\) −8.26795 −0.967690 −0.483845 − 0.875154i \(-0.660760\pi\)
−0.483845 + 0.875154i \(0.660760\pi\)
\(74\) 10.6603 1.23923
\(75\) 0 0
\(76\) −0.732051 −0.0839720
\(77\) 0 0
\(78\) 0 0
\(79\) −9.12436 −1.02657 −0.513285 − 0.858218i \(-0.671572\pi\)
−0.513285 + 0.858218i \(0.671572\pi\)
\(80\) 0.267949 0.0299576
\(81\) 0 0
\(82\) 2.53590 0.280043
\(83\) 16.5885 1.82082 0.910410 − 0.413707i \(-0.135766\pi\)
0.910410 + 0.413707i \(0.135766\pi\)
\(84\) 0 0
\(85\) 1.87564 0.203442
\(86\) −1.46410 −0.157878
\(87\) 0 0
\(88\) −6.19615 −0.660512
\(89\) 9.92820 1.05239 0.526194 − 0.850365i \(-0.323619\pi\)
0.526194 + 0.850365i \(0.323619\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.19615 0.437479
\(93\) 0 0
\(94\) 4.73205 0.488074
\(95\) −0.196152 −0.0201248
\(96\) 0 0
\(97\) −10.9282 −1.10959 −0.554795 − 0.831987i \(-0.687203\pi\)
−0.554795 + 0.831987i \(0.687203\pi\)
\(98\) 0 0
\(99\) 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 7938.2.a.bt.1.1 2
3.2 odd 2 7938.2.a.bg.1.2 2
7.6 odd 2 1134.2.a.m.1.2 yes 2
21.20 even 2 1134.2.a.l.1.1 ✓ 2
28.27 even 2 9072.2.a.y.1.2 2
63.13 odd 6 1134.2.f.r.379.1 4
63.20 even 6 1134.2.f.s.757.2 4
63.34 odd 6 1134.2.f.r.757.1 4
63.41 even 6 1134.2.f.s.379.2 4
84.83 odd 2 9072.2.a.bp.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.1 ✓ 2 21.20 even 2
1134.2.a.m.1.2 yes 2 7.6 odd 2
1134.2.f.r.379.1 4 63.13 odd 6
1134.2.f.r.757.1 4 63.34 odd 6
1134.2.f.s.379.2 4 63.41 even 6
1134.2.f.s.757.2 4 63.20 even 6
7938.2.a.bg.1.2 2 3.2 odd 2
7938.2.a.bt.1.1 2 1.1 even 1 trivial
9072.2.a.y.1.2 2 28.27 even 2
9072.2.a.bp.1.1 2 84.83 odd 2