Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7935,2,Mod(1,7935)] 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("7935.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7935, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7935.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,6,4,6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.3612940039\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4507648.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.66745\) of defining polynomial
Character \(\chi\) \(=\) 7935.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.44785 q^{2} +1.00000 q^{3} +3.99195 q^{4} +1.00000 q^{5} -2.44785 q^{6} +1.75717 q^{7} -4.87599 q^{8} +1.00000 q^{9} -2.44785 q^{10} +1.55149 q^{11} +3.99195 q^{12} -5.20501 q^{13} -4.30127 q^{14} +1.00000 q^{15} +3.95177 q^{16} -2.47284 q^{17} -2.44785 q^{18} -1.85041 q^{19} +3.99195 q^{20} +1.75717 q^{21} -3.79781 q^{22} -4.87599 q^{24} +1.00000 q^{25} +12.7411 q^{26} +1.00000 q^{27} +7.01452 q^{28} -10.2897 q^{29} -2.44785 q^{30} -2.30178 q^{31} +0.0786478 q^{32} +1.55149 q^{33} +6.05314 q^{34} +1.75717 q^{35} +3.99195 q^{36} +6.56189 q^{37} +4.52951 q^{38} -5.20501 q^{39} -4.87599 q^{40} +3.29201 q^{41} -4.30127 q^{42} -4.74173 q^{43} +6.19348 q^{44} +1.00000 q^{45} +7.93359 q^{47} +3.95177 q^{48} -3.91237 q^{49} -2.44785 q^{50} -2.47284 q^{51} -20.7782 q^{52} -8.51059 q^{53} -2.44785 q^{54} +1.55149 q^{55} -8.56792 q^{56} -1.85041 q^{57} +25.1876 q^{58} +0.268887 q^{59} +3.99195 q^{60} +10.9948 q^{61} +5.63441 q^{62} +1.75717 q^{63} -8.09606 q^{64} -5.20501 q^{65} -3.79781 q^{66} +12.3016 q^{67} -9.87147 q^{68} -4.30127 q^{70} -8.79923 q^{71} -4.87599 q^{72} +16.6768 q^{73} -16.0625 q^{74} +1.00000 q^{75} -7.38673 q^{76} +2.72623 q^{77} +12.7411 q^{78} +6.52036 q^{79} +3.95177 q^{80} +1.00000 q^{81} -8.05834 q^{82} -2.69578 q^{83} +7.01452 q^{84} -2.47284 q^{85} +11.6070 q^{86} -10.2897 q^{87} -7.56506 q^{88} +3.86643 q^{89} -2.44785 q^{90} -9.14607 q^{91} -2.30178 q^{93} -19.4202 q^{94} -1.85041 q^{95} +0.0786478 q^{96} +9.26818 q^{97} +9.57688 q^{98} +1.55149 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{7} + 6 q^{9} - 12 q^{11} + 4 q^{12} - 4 q^{13} - 8 q^{14} + 6 q^{15} - 14 q^{17} - 8 q^{19} + 4 q^{20} - 2 q^{21} - 12 q^{22} + 6 q^{25} + 24 q^{26} + 6 q^{27} - 4 q^{28}+ \cdots - 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.44785 −1.73089 −0.865444 − 0.501005i \(-0.832964\pi\)
−0.865444 + 0.501005i \(0.832964\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.99195 1.99598
\(5\) 1.00000 0.447214
\(6\) −2.44785 −0.999329
\(7\) 1.75717 0.664146 0.332073 − 0.943254i \(-0.392252\pi\)
0.332073 + 0.943254i \(0.392252\pi\)
\(8\) −4.87599 −1.72392
\(9\) 1.00000 0.333333
\(10\) −2.44785 −0.774077
\(11\) 1.55149 0.467792 0.233896 − 0.972262i \(-0.424852\pi\)
0.233896 + 0.972262i \(0.424852\pi\)
\(12\) 3.99195 1.15238
\(13\) −5.20501 −1.44361 −0.721805 − 0.692096i \(-0.756687\pi\)
−0.721805 + 0.692096i \(0.756687\pi\)
\(14\) −4.30127 −1.14956
\(15\) 1.00000 0.258199
\(16\) 3.95177 0.987943
\(17\) −2.47284 −0.599753 −0.299876 − 0.953978i \(-0.596945\pi\)
−0.299876 + 0.953978i \(0.596945\pi\)
\(18\) −2.44785 −0.576963
\(19\) −1.85041 −0.424512 −0.212256 − 0.977214i \(-0.568081\pi\)
−0.212256 + 0.977214i \(0.568081\pi\)
\(20\) 3.99195 0.892627
\(21\) 1.75717 0.383445
\(22\) −3.79781 −0.809696
\(23\) 0 0
\(24\) −4.87599 −0.995307
\(25\) 1.00000 0.200000
\(26\) 12.7411 2.49873
\(27\) 1.00000 0.192450
\(28\) 7.01452 1.32562
\(29\) −10.2897 −1.91075 −0.955374 − 0.295399i \(-0.904548\pi\)
−0.955374 + 0.295399i \(0.904548\pi\)
\(30\) −2.44785 −0.446914
\(31\) −2.30178 −0.413412 −0.206706 − 0.978403i \(-0.566274\pi\)
−0.206706 + 0.978403i \(0.566274\pi\)
\(32\) 0.0786478 0.0139031
\(33\) 1.55149 0.270080
\(34\) 6.05314 1.03810
\(35\) 1.75717 0.297015
\(36\) 3.99195 0.665325
\(37\) 6.56189 1.07877 0.539384 − 0.842060i \(-0.318657\pi\)
0.539384 + 0.842060i \(0.318657\pi\)
\(38\) 4.52951 0.734784
\(39\) −5.20501 −0.833469
\(40\) −4.87599 −0.770962
\(41\) 3.29201 0.514126 0.257063 − 0.966395i \(-0.417245\pi\)
0.257063 + 0.966395i \(0.417245\pi\)
\(42\) −4.30127 −0.663700
\(43\) −4.74173 −0.723107 −0.361554 − 0.932351i \(-0.617754\pi\)
−0.361554 + 0.932351i \(0.617754\pi\)
\(44\) 6.19348 0.933702
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 7.93359 1.15723 0.578617 − 0.815600i \(-0.303593\pi\)
0.578617 + 0.815600i \(0.303593\pi\)
\(48\) 3.95177 0.570389
\(49\) −3.91237 −0.558910
\(50\) −2.44785 −0.346178
\(51\) −2.47284 −0.346267
\(52\) −20.7782 −2.88141
\(53\) −8.51059 −1.16902 −0.584510 − 0.811387i \(-0.698713\pi\)
−0.584510 + 0.811387i \(0.698713\pi\)
\(54\) −2.44785 −0.333110
\(55\) 1.55149 0.209203
\(56\) −8.56792 −1.14494
\(57\) −1.85041 −0.245092
\(58\) 25.1876 3.30729
\(59\) 0.268887 0.0350062 0.0175031 − 0.999847i \(-0.494428\pi\)
0.0175031 + 0.999847i \(0.494428\pi\)
\(60\) 3.99195 0.515359
\(61\) 10.9948 1.40774 0.703868 − 0.710330i \(-0.251454\pi\)
0.703868 + 0.710330i \(0.251454\pi\)
\(62\) 5.63441 0.715570
\(63\) 1.75717 0.221382
\(64\) −8.09606 −1.01201
\(65\) −5.20501 −0.645602
\(66\) −3.79781 −0.467478
\(67\) 12.3016 1.50288 0.751440 − 0.659801i \(-0.229360\pi\)
0.751440 + 0.659801i \(0.229360\pi\)
\(68\) −9.87147 −1.19709
\(69\) 0 0
\(70\) −4.30127 −0.514100
\(71\) −8.79923 −1.04428 −0.522138 − 0.852861i \(-0.674865\pi\)
−0.522138 + 0.852861i \(0.674865\pi\)
\(72\) −4.87599 −0.574641
\(73\) 16.6768 1.95187 0.975935 − 0.218061i \(-0.0699732\pi\)
0.975935 + 0.218061i \(0.0699732\pi\)
\(74\) −16.0625 −1.86723
\(75\) 1.00000 0.115470
\(76\) −7.38673 −0.847317
\(77\) 2.72623 0.310682
\(78\) 12.7411 1.44264
\(79\) 6.52036 0.733598 0.366799 − 0.930300i \(-0.380454\pi\)
0.366799 + 0.930300i \(0.380454\pi\)
\(80\) 3.95177 0.441822
\(81\) 1.00000 0.111111
\(82\) −8.05834 −0.889895
\(83\) −2.69578 −0.295901 −0.147950 − 0.988995i \(-0.547268\pi\)
−0.147950 + 0.988995i \(0.547268\pi\)
\(84\) 7.01452 0.765347
\(85\) −2.47284 −0.268218
\(86\) 11.6070 1.25162
\(87\) −10.2897 −1.10317
\(88\) −7.56506 −0.806438
\(89\) 3.86643 0.409841 0.204920 − 0.978779i \(-0.434306\pi\)
0.204920 + 0.978779i \(0.434306\pi\)
\(90\) −2.44785 −0.258026
\(91\) −9.14607 −0.958768
\(92\) 0 0
\(93\) −2.30178 −0.238684
\(94\) −19.4202 −2.00304
\(95\) −1.85041 −0.189848
\(96\) 0.0786478 0.00802696
\(97\) 9.26818 0.941041 0.470521 − 0.882389i \(-0.344066\pi\)
0.470521 + 0.882389i \(0.344066\pi\)
\(98\) 9.57688 0.967411
\(99\) 1.55149 0.155931
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 7935.2.a.bg.1.1 yes 6
23.22 odd 2 7935.2.a.bf.1.1 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7935.2.a.bf.1.1 ✓ 6 23.22 odd 2
7935.2.a.bg.1.1 yes 6 1.1 even 1 trivial