Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6958,2,Mod(1,6958)] 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("6958.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6958, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6958 = 2 \cdot 7^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6958.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,2,3,4,2,0,3,-1,4,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(55.5599097264\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 994)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 6958.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.48119 q^{3} +1.00000 q^{4} -0.481194 q^{5} +2.48119 q^{6} +1.00000 q^{8} +3.15633 q^{9} -0.481194 q^{10} +1.28726 q^{11} +2.48119 q^{12} +6.15633 q^{13} -1.19394 q^{15} +1.00000 q^{16} -2.15633 q^{17} +3.15633 q^{18} +4.48119 q^{19} -0.481194 q^{20} +1.28726 q^{22} +8.70052 q^{23} +2.48119 q^{24} -4.76845 q^{25} +6.15633 q^{26} +0.387873 q^{27} -8.70052 q^{29} -1.19394 q^{30} +4.80606 q^{31} +1.00000 q^{32} +3.19394 q^{33} -2.15633 q^{34} +3.15633 q^{36} -10.3127 q^{37} +4.48119 q^{38} +15.2750 q^{39} -0.481194 q^{40} +5.38058 q^{41} -2.57452 q^{43} +1.28726 q^{44} -1.51881 q^{45} +8.70052 q^{46} -0.962389 q^{47} +2.48119 q^{48} -4.76845 q^{50} -5.35026 q^{51} +6.15633 q^{52} +1.67513 q^{53} +0.387873 q^{54} -0.619421 q^{55} +11.1187 q^{57} -8.70052 q^{58} -1.11871 q^{59} -1.19394 q^{60} +7.76845 q^{61} +4.80606 q^{62} +1.00000 q^{64} -2.96239 q^{65} +3.19394 q^{66} +4.51151 q^{67} -2.15633 q^{68} +21.5877 q^{69} +1.00000 q^{71} +3.15633 q^{72} +14.0508 q^{73} -10.3127 q^{74} -11.8315 q^{75} +4.48119 q^{76} +15.2750 q^{78} -6.88717 q^{79} -0.481194 q^{80} -8.50659 q^{81} +5.38058 q^{82} +9.18172 q^{83} +1.03761 q^{85} -2.57452 q^{86} -21.5877 q^{87} +1.28726 q^{88} +9.73813 q^{89} -1.51881 q^{90} +8.70052 q^{92} +11.9248 q^{93} -0.962389 q^{94} -2.15633 q^{95} +2.48119 q^{96} -15.2447 q^{97} +4.06300 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 4 q^{5} + 2 q^{6} + 3 q^{8} - q^{9} + 4 q^{10} - 2 q^{11} + 2 q^{12} + 8 q^{13} - 4 q^{15} + 3 q^{16} + 4 q^{17} - q^{18} + 8 q^{19} + 4 q^{20} - 2 q^{22} + 6 q^{23}+ \cdots + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.48119 1.43252 0.716259 − 0.697834i \(-0.245853\pi\)
0.716259 + 0.697834i \(0.245853\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.481194 −0.215197 −0.107598 − 0.994194i \(-0.534316\pi\)
−0.107598 + 0.994194i \(0.534316\pi\)
\(6\) 2.48119 1.01294
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 3.15633 1.05211
\(10\) −0.481194 −0.152167
\(11\) 1.28726 0.388123 0.194061 − 0.980989i \(-0.437834\pi\)
0.194061 + 0.980989i \(0.437834\pi\)
\(12\) 2.48119 0.716259
\(13\) 6.15633 1.70746 0.853729 − 0.520718i \(-0.174336\pi\)
0.853729 + 0.520718i \(0.174336\pi\)
\(14\) 0 0
\(15\) −1.19394 −0.308273
\(16\) 1.00000 0.250000
\(17\) −2.15633 −0.522986 −0.261493 − 0.965205i \(-0.584215\pi\)
−0.261493 + 0.965205i \(0.584215\pi\)
\(18\) 3.15633 0.743953
\(19\) 4.48119 1.02806 0.514028 − 0.857773i \(-0.328153\pi\)
0.514028 + 0.857773i \(0.328153\pi\)
\(20\) −0.481194 −0.107598
\(21\) 0 0
\(22\) 1.28726 0.274444
\(23\) 8.70052 1.81418 0.907092 − 0.420932i \(-0.138297\pi\)
0.907092 + 0.420932i \(0.138297\pi\)
\(24\) 2.48119 0.506472
\(25\) −4.76845 −0.953690
\(26\) 6.15633 1.20735
\(27\) 0.387873 0.0746462
\(28\) 0 0
\(29\) −8.70052 −1.61565 −0.807823 − 0.589425i \(-0.799354\pi\)
−0.807823 + 0.589425i \(0.799354\pi\)
\(30\) −1.19394 −0.217982
\(31\) 4.80606 0.863194 0.431597 − 0.902066i \(-0.357950\pi\)
0.431597 + 0.902066i \(0.357950\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.19394 0.555993
\(34\) −2.15633 −0.369807
\(35\) 0 0
\(36\) 3.15633 0.526054
\(37\) −10.3127 −1.69539 −0.847695 − 0.530484i \(-0.822010\pi\)
−0.847695 + 0.530484i \(0.822010\pi\)
\(38\) 4.48119 0.726946
\(39\) 15.2750 2.44596
\(40\) −0.481194 −0.0760835
\(41\) 5.38058 0.840305 0.420153 − 0.907453i \(-0.361977\pi\)
0.420153 + 0.907453i \(0.361977\pi\)
\(42\) 0 0
\(43\) −2.57452 −0.392610 −0.196305 − 0.980543i \(-0.562894\pi\)
−0.196305 + 0.980543i \(0.562894\pi\)
\(44\) 1.28726 0.194061
\(45\) −1.51881 −0.226410
\(46\) 8.70052 1.28282
\(47\) −0.962389 −0.140379 −0.0701894 − 0.997534i \(-0.522360\pi\)
−0.0701894 + 0.997534i \(0.522360\pi\)
\(48\) 2.48119 0.358130
\(49\) 0 0
\(50\) −4.76845 −0.674361
\(51\) −5.35026 −0.749186
\(52\) 6.15633 0.853729
\(53\) 1.67513 0.230097 0.115048 − 0.993360i \(-0.463298\pi\)
0.115048 + 0.993360i \(0.463298\pi\)
\(54\) 0.387873 0.0527828
\(55\) −0.619421 −0.0835227
\(56\) 0 0
\(57\) 11.1187 1.47271
\(58\) −8.70052 −1.14243
\(59\) −1.11871 −0.145644 −0.0728221 − 0.997345i \(-0.523201\pi\)
−0.0728221 + 0.997345i \(0.523201\pi\)
\(60\) −1.19394 −0.154137
\(61\) 7.76845 0.994648 0.497324 − 0.867565i \(-0.334316\pi\)
0.497324 + 0.867565i \(0.334316\pi\)
\(62\) 4.80606 0.610371
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.96239 −0.367439
\(66\) 3.19394 0.393146
\(67\) 4.51151 0.551169 0.275584 − 0.961277i \(-0.411129\pi\)
0.275584 + 0.961277i \(0.411129\pi\)
\(68\) −2.15633 −0.261493
\(69\) 21.5877 2.59885
\(70\) 0 0
\(71\) 1.00000 0.118678
\(72\) 3.15633 0.371976
\(73\) 14.0508 1.64452 0.822260 − 0.569112i \(-0.192713\pi\)
0.822260 + 0.569112i \(0.192713\pi\)
\(74\) −10.3127 −1.19882
\(75\) −11.8315 −1.36618
\(76\) 4.48119 0.514028
\(77\) 0 0
\(78\) 15.2750 1.72956
\(79\) −6.88717 −0.774867 −0.387433 − 0.921898i \(-0.626638\pi\)
−0.387433 + 0.921898i \(0.626638\pi\)
\(80\) −0.481194 −0.0537992
\(81\) −8.50659 −0.945176
\(82\) 5.38058 0.594186
\(83\) 9.18172 1.00782 0.503912 − 0.863755i \(-0.331893\pi\)
0.503912 + 0.863755i \(0.331893\pi\)
\(84\) 0 0
\(85\) 1.03761 0.112545
\(86\) −2.57452 −0.277617
\(87\) −21.5877 −2.31444
\(88\) 1.28726 0.137222
\(89\) 9.73813 1.03224 0.516120 − 0.856516i \(-0.327376\pi\)
0.516120 + 0.856516i \(0.327376\pi\)
\(90\) −1.51881 −0.160096
\(91\) 0 0
\(92\) 8.70052 0.907092
\(93\) 11.9248 1.23654
\(94\) −0.962389 −0.0992628
\(95\) −2.15633 −0.221234
\(96\) 2.48119 0.253236
\(97\) −15.2447 −1.54787 −0.773933 − 0.633267i \(-0.781714\pi\)
−0.773933 + 0.633267i \(0.781714\pi\)
\(98\) 0 0
\(99\) 4.06300 0.408347
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 6958.2.a.bd.1.3 3
7.6 odd 2 994.2.a.k.1.1 ✓ 3
21.20 even 2 8946.2.a.bj.1.1 3
28.27 even 2 7952.2.a.q.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
994.2.a.k.1.1 ✓ 3 7.6 odd 2
6958.2.a.bd.1.3 3 1.1 even 1 trivial
7952.2.a.q.1.3 3 28.27 even 2
8946.2.a.bj.1.1 3 21.20 even 2