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 1.1
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.73205 q^{5} -1.00000 q^{8} +3.73205 q^{10} -4.19615 q^{11} -0.464102 q^{13} +1.00000 q^{16} -7.00000 q^{17} +2.73205 q^{19} -3.73205 q^{20} +4.19615 q^{22} +6.19615 q^{23} +8.92820 q^{25} +0.464102 q^{26} +8.46410 q^{29} +2.19615 q^{31} -1.00000 q^{32} +7.00000 q^{34} -6.66025 q^{37} -2.73205 q^{38} +3.73205 q^{40} -9.46410 q^{41} +5.46410 q^{43} -4.19615 q^{44} -6.19615 q^{46} -1.26795 q^{47} -8.92820 q^{50} -0.464102 q^{52} -2.53590 q^{53} +15.6603 q^{55} -8.46410 q^{58} +6.19615 q^{59} +9.92820 q^{61} -2.19615 q^{62} +1.00000 q^{64} +1.73205 q^{65} -3.26795 q^{67} -7.00000 q^{68} +13.4641 q^{71} -11.7321 q^{73} +6.66025 q^{74} +2.73205 q^{76} +15.1244 q^{79} -3.73205 q^{80} +9.46410 q^{82} +14.5885 q^{83} +26.1244 q^{85} -5.46410 q^{86} +4.19615 q^{88} +3.92820 q^{89} +6.19615 q^{92} +1.26795 q^{94} -10.1962 q^{95} +2.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} + 2 q^{11} + 6 q^{13} + 2 q^{16} - 14 q^{17} + 2 q^{19} - 4 q^{20} - 2 q^{22} + 2 q^{23} + 4 q^{25} - 6 q^{26} + 10 q^{29} - 6 q^{31} - 2 q^{32} + 14 q^{34}+ \cdots - 8 q^{97}+O(q^{100}) \) 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
\(5\) −3.73205 −1.66902 −0.834512 − 0.550990i \(-0.814250\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.73205 1.18018
\(11\) −4.19615 −1.26519 −0.632594 − 0.774484i \(-0.718010\pi\)
−0.632594 + 0.774484i \(0.718010\pi\)
\(12\) 0 0
\(13\) −0.464102 −0.128719 −0.0643593 − 0.997927i \(-0.520500\pi\)
−0.0643593 + 0.997927i \(0.520500\pi\)
\(14\) 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
\(19\) 2.73205 0.626775 0.313388 − 0.949625i \(-0.398536\pi\)
0.313388 + 0.949625i \(0.398536\pi\)
\(20\) −3.73205 −0.834512
\(21\) 0 0
\(22\) 4.19615 0.894623
\(23\) 6.19615 1.29199 0.645994 − 0.763343i \(-0.276443\pi\)
0.645994 + 0.763343i \(0.276443\pi\)
\(24\) 0 0
\(25\) 8.92820 1.78564
\(26\) 0.464102 0.0910178
\(27\) 0 0
\(28\) 0 0
\(29\) 8.46410 1.57174 0.785872 − 0.618389i \(-0.212214\pi\)
0.785872 + 0.618389i \(0.212214\pi\)
\(30\) 0 0
\(31\) 2.19615 0.394441 0.197220 − 0.980359i \(-0.436809\pi\)
0.197220 + 0.980359i \(0.436809\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.00000 1.20049
\(35\) 0 0
\(36\) 0 0
\(37\) −6.66025 −1.09494 −0.547470 − 0.836826i \(-0.684409\pi\)
−0.547470 + 0.836826i \(0.684409\pi\)
\(38\) −2.73205 −0.443197
\(39\) 0 0
\(40\) 3.73205 0.590089
\(41\) −9.46410 −1.47804 −0.739022 − 0.673681i \(-0.764712\pi\)
−0.739022 + 0.673681i \(0.764712\pi\)
\(42\) 0 0
\(43\) 5.46410 0.833268 0.416634 − 0.909074i \(-0.363210\pi\)
0.416634 + 0.909074i \(0.363210\pi\)
\(44\) −4.19615 −0.632594
\(45\) 0 0
\(46\) −6.19615 −0.913573
\(47\) −1.26795 −0.184949 −0.0924747 − 0.995715i \(-0.529478\pi\)
−0.0924747 + 0.995715i \(0.529478\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −8.92820 −1.26264
\(51\) 0 0
\(52\) −0.464102 −0.0643593
\(53\) −2.53590 −0.348332 −0.174166 − 0.984716i \(-0.555723\pi\)
−0.174166 + 0.984716i \(0.555723\pi\)
\(54\) 0 0
\(55\) 15.6603 2.11163
\(56\) 0 0
\(57\) 0 0
\(58\) −8.46410 −1.11139
\(59\) 6.19615 0.806670 0.403335 − 0.915052i \(-0.367851\pi\)
0.403335 + 0.915052i \(0.367851\pi\)
\(60\) 0 0
\(61\) 9.92820 1.27118 0.635588 − 0.772028i \(-0.280758\pi\)
0.635588 + 0.772028i \(0.280758\pi\)
\(62\) −2.19615 −0.278912
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.73205 0.214834
\(66\) 0 0
\(67\) −3.26795 −0.399244 −0.199622 − 0.979873i \(-0.563971\pi\)
−0.199622 + 0.979873i \(0.563971\pi\)
\(68\) −7.00000 −0.848875
\(69\) 0 0
\(70\) 0 0
\(71\) 13.4641 1.59789 0.798947 − 0.601401i \(-0.205391\pi\)
0.798947 + 0.601401i \(0.205391\pi\)
\(72\) 0 0
\(73\) −11.7321 −1.37313 −0.686566 − 0.727067i \(-0.740883\pi\)
−0.686566 + 0.727067i \(0.740883\pi\)
\(74\) 6.66025 0.774239
\(75\) 0 0
\(76\) 2.73205 0.313388
\(77\) 0 0
\(78\) 0 0
\(79\) 15.1244 1.70162 0.850811 − 0.525471i \(-0.176111\pi\)
0.850811 + 0.525471i \(0.176111\pi\)
\(80\) −3.73205 −0.417256
\(81\) 0 0
\(82\) 9.46410 1.04514
\(83\) 14.5885 1.60129 0.800646 − 0.599138i \(-0.204490\pi\)
0.800646 + 0.599138i \(0.204490\pi\)
\(84\) 0 0
\(85\) 26.1244 2.83358
\(86\) −5.46410 −0.589209
\(87\) 0 0
\(88\) 4.19615 0.447311
\(89\) 3.92820 0.416389 0.208194 − 0.978087i \(-0.433241\pi\)
0.208194 + 0.978087i \(0.433241\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.19615 0.645994
\(93\) 0 0
\(94\) 1.26795 0.130779
\(95\) −10.1962 −1.04610
\(96\) 0 0
\(97\) 2.92820 0.297314 0.148657 − 0.988889i \(-0.452505\pi\)
0.148657 + 0.988889i \(0.452505\pi\)
\(98\) 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.1.1 2
3.2 odd 2 7938.2.a.bt.1.2 2
7.6 odd 2 1134.2.a.l.1.2 ✓ 2
21.20 even 2 1134.2.a.m.1.1 yes 2
28.27 even 2 9072.2.a.bp.1.2 2
63.13 odd 6 1134.2.f.s.379.1 4
63.20 even 6 1134.2.f.r.757.2 4
63.34 odd 6 1134.2.f.s.757.1 4
63.41 even 6 1134.2.f.r.379.2 4
84.83 odd 2 9072.2.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.2 ✓ 2 7.6 odd 2
1134.2.a.m.1.1 yes 2 21.20 even 2
1134.2.f.r.379.2 4 63.41 even 6
1134.2.f.r.757.2 4 63.20 even 6
1134.2.f.s.379.1 4 63.13 odd 6
1134.2.f.s.757.1 4 63.34 odd 6
7938.2.a.bg.1.1 2 1.1 even 1 trivial
7938.2.a.bt.1.2 2 3.2 odd 2
9072.2.a.y.1.1 2 84.83 odd 2
9072.2.a.bp.1.2 2 28.27 even 2