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: \(1\)
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 1.2
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 \(-0.519083\pi\)
−0.0599153 + 0.998203i \(0.519083\pi\)
\(6\) 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 \(-0.116200\pi\)
0.934105 + 0.356998i \(0.116200\pi\)
\(12\) 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 \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 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 \(-0.644129\pi\)
−0.437479 + 0.899229i \(0.644129\pi\)
\(24\) 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 \(-0.454452\pi\)
0.142605 + 0.989780i \(0.454452\pi\)
\(30\) 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 \(-0.563451\pi\)
−0.198020 + 0.980198i \(0.563451\pi\)
\(42\) 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 \(-0.612162\pi\)
−0.345120 + 0.938558i \(0.612162\pi\)
\(48\) 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 \(-0.725230\pi\)
−0.649997 + 0.759937i \(0.725230\pi\)
\(54\) 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 \(-0.588064\pi\)
−0.273146 + 0.961973i \(0.588064\pi\)
\(60\) 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 \(-0.373223\pi\)
0.387834 + 0.921729i \(0.373223\pi\)
\(72\) 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 \(-0.864234\pi\)
−0.910410 + 0.413707i \(0.864234\pi\)
\(84\) 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 \(-0.676381\pi\)
−0.526194 + 0.850365i \(0.676381\pi\)
\(90\) 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.bg.1.2 2
3.2 odd 2 7938.2.a.bt.1.1 2
7.6 odd 2 1134.2.a.l.1.1 ✓ 2
21.20 even 2 1134.2.a.m.1.2 yes 2
28.27 even 2 9072.2.a.bp.1.1 2
63.13 odd 6 1134.2.f.s.379.2 4
63.20 even 6 1134.2.f.r.757.1 4
63.34 odd 6 1134.2.f.s.757.2 4
63.41 even 6 1134.2.f.r.379.1 4
84.83 odd 2 9072.2.a.y.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.1 ✓ 2 7.6 odd 2
1134.2.a.m.1.2 yes 2 21.20 even 2
1134.2.f.r.379.1 4 63.41 even 6
1134.2.f.r.757.1 4 63.20 even 6
1134.2.f.s.379.2 4 63.13 odd 6
1134.2.f.s.757.2 4 63.34 odd 6
7938.2.a.bg.1.2 2 1.1 even 1 trivial
7938.2.a.bt.1.1 2 3.2 odd 2
9072.2.a.y.1.2 2 84.83 odd 2
9072.2.a.bp.1.1 2 28.27 even 2