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 1.4
Root \(1.90903\) 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)
 
\(f(q)\) \(=\) \(q+0.264627 q^{2} +1.00000 q^{3} -1.92997 q^{4} +1.00000 q^{5} +0.264627 q^{6} +0.526121 q^{7} -1.03998 q^{8} +1.00000 q^{9} +0.264627 q^{10} +0.0609151 q^{11} -1.92997 q^{12} -1.26149 q^{13} +0.139226 q^{14} +1.00000 q^{15} +3.58474 q^{16} +1.96765 q^{17} +0.264627 q^{18} -6.64879 q^{19} -1.92997 q^{20} +0.526121 q^{21} +0.0161198 q^{22} -1.03998 q^{24} +1.00000 q^{25} -0.333825 q^{26} +1.00000 q^{27} -1.01540 q^{28} +3.75671 q^{29} +0.264627 q^{30} -8.07168 q^{31} +3.02857 q^{32} +0.0609151 q^{33} +0.520693 q^{34} +0.526121 q^{35} -1.92997 q^{36} +0.263789 q^{37} -1.75945 q^{38} -1.26149 q^{39} -1.03998 q^{40} +2.87130 q^{41} +0.139226 q^{42} +4.13011 q^{43} -0.117564 q^{44} +1.00000 q^{45} -9.43356 q^{47} +3.58474 q^{48} -6.72320 q^{49} +0.264627 q^{50} +1.96765 q^{51} +2.43465 q^{52} +1.86477 q^{53} +0.264627 q^{54} +0.0609151 q^{55} -0.547152 q^{56} -6.64879 q^{57} +0.994126 q^{58} -4.16246 q^{59} -1.92997 q^{60} -2.68460 q^{61} -2.13598 q^{62} +0.526121 q^{63} -6.36804 q^{64} -1.26149 q^{65} +0.0161198 q^{66} -10.5129 q^{67} -3.79752 q^{68} +0.139226 q^{70} +11.6882 q^{71} -1.03998 q^{72} -1.37060 q^{73} +0.0698057 q^{74} +1.00000 q^{75} +12.8320 q^{76} +0.0320487 q^{77} -0.333825 q^{78} +2.33560 q^{79} +3.58474 q^{80} +1.00000 q^{81} +0.759823 q^{82} +0.155784 q^{83} -1.01540 q^{84} +1.96765 q^{85} +1.09294 q^{86} +3.75671 q^{87} -0.0633502 q^{88} +9.39924 q^{89} +0.264627 q^{90} -0.663698 q^{91} -8.07168 q^{93} -2.49637 q^{94} -6.64879 q^{95} +3.02857 q^{96} -6.60460 q^{97} -1.77914 q^{98} +0.0609151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{7} + 6 q^{9} - 12 q^{11} + 4 q^{12} - 4 q^{13} - 8 q^{14} + 6 q^{15} - 14 q^{17} - 8 q^{19} + 4 q^{20} - 2 q^{21} - 12 q^{22} + 6 q^{25} + 24 q^{26} + 6 q^{27} - 4 q^{28}+ \cdots - 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.264627 0.187119 0.0935596 − 0.995614i \(-0.470175\pi\)
0.0935596 + 0.995614i \(0.470175\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.92997 −0.964986
\(5\) 1.00000 0.447214
\(6\) 0.264627 0.108033
\(7\) 0.526121 0.198855 0.0994275 − 0.995045i \(-0.468299\pi\)
0.0994275 + 0.995045i \(0.468299\pi\)
\(8\) −1.03998 −0.367687
\(9\) 1.00000 0.333333
\(10\) 0.264627 0.0836823
\(11\) 0.0609151 0.0183666 0.00918330 − 0.999958i \(-0.497077\pi\)
0.00918330 + 0.999958i \(0.497077\pi\)
\(12\) −1.92997 −0.557135
\(13\) −1.26149 −0.349876 −0.174938 − 0.984579i \(-0.555972\pi\)
−0.174938 + 0.984579i \(0.555972\pi\)
\(14\) 0.139226 0.0372096
\(15\) 1.00000 0.258199
\(16\) 3.58474 0.896185
\(17\) 1.96765 0.477226 0.238613 − 0.971115i \(-0.423307\pi\)
0.238613 + 0.971115i \(0.423307\pi\)
\(18\) 0.264627 0.0623731
\(19\) −6.64879 −1.52534 −0.762668 − 0.646790i \(-0.776111\pi\)
−0.762668 + 0.646790i \(0.776111\pi\)
\(20\) −1.92997 −0.431555
\(21\) 0.526121 0.114809
\(22\) 0.0161198 0.00343674
\(23\) 0 0
\(24\) −1.03998 −0.212284
\(25\) 1.00000 0.200000
\(26\) −0.333825 −0.0654684
\(27\) 1.00000 0.192450
\(28\) −1.01540 −0.191892
\(29\) 3.75671 0.697604 0.348802 − 0.937196i \(-0.386589\pi\)
0.348802 + 0.937196i \(0.386589\pi\)
\(30\) 0.264627 0.0483140
\(31\) −8.07168 −1.44972 −0.724858 − 0.688898i \(-0.758095\pi\)
−0.724858 + 0.688898i \(0.758095\pi\)
\(32\) 3.02857 0.535380
\(33\) 0.0609151 0.0106040
\(34\) 0.520693 0.0892981
\(35\) 0.526121 0.0889306
\(36\) −1.92997 −0.321662
\(37\) 0.263789 0.0433667 0.0216834 − 0.999765i \(-0.493097\pi\)
0.0216834 + 0.999765i \(0.493097\pi\)
\(38\) −1.75945 −0.285420
\(39\) −1.26149 −0.202001
\(40\) −1.03998 −0.164435
\(41\) 2.87130 0.448422 0.224211 − 0.974541i \(-0.428019\pi\)
0.224211 + 0.974541i \(0.428019\pi\)
\(42\) 0.139226 0.0214830
\(43\) 4.13011 0.629836 0.314918 − 0.949119i \(-0.398023\pi\)
0.314918 + 0.949119i \(0.398023\pi\)
\(44\) −0.117564 −0.0177235
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −9.43356 −1.37603 −0.688013 − 0.725699i \(-0.741517\pi\)
−0.688013 + 0.725699i \(0.741517\pi\)
\(48\) 3.58474 0.517413
\(49\) −6.72320 −0.960457
\(50\) 0.264627 0.0374238
\(51\) 1.96765 0.275526
\(52\) 2.43465 0.337625
\(53\) 1.86477 0.256145 0.128073 − 0.991765i \(-0.459121\pi\)
0.128073 + 0.991765i \(0.459121\pi\)
\(54\) 0.264627 0.0360111
\(55\) 0.0609151 0.00821379
\(56\) −0.547152 −0.0731163
\(57\) −6.64879 −0.880653
\(58\) 0.994126 0.130535
\(59\) −4.16246 −0.541906 −0.270953 − 0.962593i \(-0.587339\pi\)
−0.270953 + 0.962593i \(0.587339\pi\)
\(60\) −1.92997 −0.249158
\(61\) −2.68460 −0.343728 −0.171864 − 0.985121i \(-0.554979\pi\)
−0.171864 + 0.985121i \(0.554979\pi\)
\(62\) −2.13598 −0.271270
\(63\) 0.526121 0.0662850
\(64\) −6.36804 −0.796005
\(65\) −1.26149 −0.156469
\(66\) 0.0161198 0.00198420
\(67\) −10.5129 −1.28436 −0.642180 − 0.766554i \(-0.721970\pi\)
−0.642180 + 0.766554i \(0.721970\pi\)
\(68\) −3.79752 −0.460516
\(69\) 0 0
\(70\) 0.139226 0.0166406
\(71\) 11.6882 1.38713 0.693565 − 0.720394i \(-0.256039\pi\)
0.693565 + 0.720394i \(0.256039\pi\)
\(72\) −1.03998 −0.122562
\(73\) −1.37060 −0.160417 −0.0802084 − 0.996778i \(-0.525559\pi\)
−0.0802084 + 0.996778i \(0.525559\pi\)
\(74\) 0.0698057 0.00811475
\(75\) 1.00000 0.115470
\(76\) 12.8320 1.47193
\(77\) 0.0320487 0.00365229
\(78\) −0.333825 −0.0377982
\(79\) 2.33560 0.262776 0.131388 − 0.991331i \(-0.458057\pi\)
0.131388 + 0.991331i \(0.458057\pi\)
\(80\) 3.58474 0.400786
\(81\) 1.00000 0.111111
\(82\) 0.759823 0.0839085
\(83\) 0.155784 0.0170995 0.00854974 − 0.999963i \(-0.497278\pi\)
0.00854974 + 0.999963i \(0.497278\pi\)
\(84\) −1.01540 −0.110789
\(85\) 1.96765 0.213422
\(86\) 1.09294 0.117854
\(87\) 3.75671 0.402762
\(88\) −0.0633502 −0.00675315
\(89\) 9.39924 0.996317 0.498158 − 0.867086i \(-0.334010\pi\)
0.498158 + 0.867086i \(0.334010\pi\)
\(90\) 0.264627 0.0278941
\(91\) −0.663698 −0.0695745
\(92\) 0 0
\(93\) −8.07168 −0.836994
\(94\) −2.49637 −0.257481
\(95\) −6.64879 −0.682151
\(96\) 3.02857 0.309102
\(97\) −6.60460 −0.670596 −0.335298 − 0.942112i \(-0.608837\pi\)
−0.335298 + 0.942112i \(0.608837\pi\)
\(98\) −1.77914 −0.179720
\(99\) 0.0609151 0.00612220
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.1.4 yes 6
23.22 odd 2 7935.2.a.bf.1.4 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7935.2.a.bf.1.4 ✓ 6 23.22 odd 2
7935.2.a.bg.1.4 yes 6 1.1 even 1 trivial