Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [13,0,0,16,0,0,-8,0,0,0,-10,0,-11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 21x^{11} + 165x^{9} - 2x^{8} - 600x^{7} + 8x^{6} + 1005x^{5} + 32x^{4} - 657x^{3} - 80x^{2} + 73x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 555)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.214651\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.214651 q^{2} -1.95393 q^{4} -4.41925 q^{7} +0.848712 q^{8} -5.57516 q^{11} -0.780716 q^{13} +0.948594 q^{14} +3.72567 q^{16} +4.10713 q^{17} -4.94347 q^{19} +1.19671 q^{22} +7.70902 q^{23} +0.167581 q^{26} +8.63488 q^{28} +1.53583 q^{29} -2.52604 q^{31} -2.49714 q^{32} -0.881597 q^{34} -1.00000 q^{37} +1.06112 q^{38} +11.2120 q^{41} +2.78517 q^{43} +10.8934 q^{44} -1.65475 q^{46} -2.53935 q^{47} +12.5297 q^{49} +1.52546 q^{52} +1.49186 q^{53} -3.75067 q^{56} -0.329667 q^{58} +3.84326 q^{59} +5.82961 q^{61} +0.542216 q^{62} -6.91533 q^{64} -1.94661 q^{67} -8.02502 q^{68} -9.36234 q^{71} +11.9612 q^{73} +0.214651 q^{74} +9.65917 q^{76} +24.6380 q^{77} +8.12132 q^{79} -2.40667 q^{82} -15.7827 q^{83} -0.597839 q^{86} -4.73170 q^{88} -0.343059 q^{89} +3.45018 q^{91} -15.0629 q^{92} +0.545073 q^{94} +12.1360 q^{97} -2.68952 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 16 q^{4} - 8 q^{7} - 10 q^{11} - 11 q^{13} - 8 q^{14} + 22 q^{16} + 2 q^{17} + 14 q^{19} - 14 q^{22} - 2 q^{23} - 16 q^{26} - 4 q^{28} - 23 q^{29} + 16 q^{31} - 10 q^{32} - 10 q^{34} - 13 q^{37}+ \cdots - 42 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.214651 −0.151781 −0.0758904 − 0.997116i \(-0.524180\pi\)
−0.0758904 + 0.997116i \(0.524180\pi\)
\(3\) 0 0
\(4\) −1.95393 −0.976963
\(5\) 0 0
\(6\) 0 0
\(7\) −4.41925 −1.67032 −0.835159 − 0.550008i \(-0.814625\pi\)
−0.835159 + 0.550008i \(0.814625\pi\)
\(8\) 0.848712 0.300065
\(9\) 0 0
\(10\) 0 0
\(11\) −5.57516 −1.68097 −0.840486 − 0.541833i \(-0.817731\pi\)
−0.840486 + 0.541833i \(0.817731\pi\)
\(12\) 0 0
\(13\) −0.780716 −0.216532 −0.108266 − 0.994122i \(-0.534530\pi\)
−0.108266 + 0.994122i \(0.534530\pi\)
\(14\) 0.948594 0.253522
\(15\) 0 0
\(16\) 3.72567 0.931418
\(17\) 4.10713 0.996125 0.498062 − 0.867141i \(-0.334045\pi\)
0.498062 + 0.867141i \(0.334045\pi\)
\(18\) 0 0
\(19\) −4.94347 −1.13411 −0.567055 − 0.823680i \(-0.691917\pi\)
−0.567055 + 0.823680i \(0.691917\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.19671 0.255140
\(23\) 7.70902 1.60744 0.803721 − 0.595006i \(-0.202850\pi\)
0.803721 + 0.595006i \(0.202850\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.167581 0.0328654
\(27\) 0 0
\(28\) 8.63488 1.63184
\(29\) 1.53583 0.285197 0.142598 − 0.989781i \(-0.454454\pi\)
0.142598 + 0.989781i \(0.454454\pi\)
\(30\) 0 0
\(31\) −2.52604 −0.453690 −0.226845 − 0.973931i \(-0.572841\pi\)
−0.226845 + 0.973931i \(0.572841\pi\)
\(32\) −2.49714 −0.441437
\(33\) 0 0
\(34\) −0.881597 −0.151193
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 1.06112 0.172136
\(39\) 0 0
\(40\) 0 0
\(41\) 11.2120 1.75102 0.875512 − 0.483197i \(-0.160524\pi\)
0.875512 + 0.483197i \(0.160524\pi\)
\(42\) 0 0
\(43\) 2.78517 0.424735 0.212367 − 0.977190i \(-0.431883\pi\)
0.212367 + 0.977190i \(0.431883\pi\)
\(44\) 10.8934 1.64225
\(45\) 0 0
\(46\) −1.65475 −0.243979
\(47\) −2.53935 −0.370402 −0.185201 − 0.982701i \(-0.559294\pi\)
−0.185201 + 0.982701i \(0.559294\pi\)
\(48\) 0 0
\(49\) 12.5297 1.78996
\(50\) 0 0
\(51\) 0 0
\(52\) 1.52546 0.211543
\(53\) 1.49186 0.204922 0.102461 − 0.994737i \(-0.467328\pi\)
0.102461 + 0.994737i \(0.467328\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.75067 −0.501204
\(57\) 0 0
\(58\) −0.329667 −0.0432874
\(59\) 3.84326 0.500349 0.250175 − 0.968201i \(-0.419512\pi\)
0.250175 + 0.968201i \(0.419512\pi\)
\(60\) 0 0
\(61\) 5.82961 0.746405 0.373202 − 0.927750i \(-0.378260\pi\)
0.373202 + 0.927750i \(0.378260\pi\)
\(62\) 0.542216 0.0688615
\(63\) 0 0
\(64\) −6.91533 −0.864417
\(65\) 0 0
\(66\) 0 0
\(67\) −1.94661 −0.237816 −0.118908 − 0.992905i \(-0.537939\pi\)
−0.118908 + 0.992905i \(0.537939\pi\)
\(68\) −8.02502 −0.973176
\(69\) 0 0
\(70\) 0 0
\(71\) −9.36234 −1.11110 −0.555552 − 0.831481i \(-0.687493\pi\)
−0.555552 + 0.831481i \(0.687493\pi\)
\(72\) 0 0
\(73\) 11.9612 1.39995 0.699975 − 0.714168i \(-0.253195\pi\)
0.699975 + 0.714168i \(0.253195\pi\)
\(74\) 0.214651 0.0249526
\(75\) 0 0
\(76\) 9.65917 1.10798
\(77\) 24.6380 2.80776
\(78\) 0 0
\(79\) 8.12132 0.913720 0.456860 − 0.889539i \(-0.348974\pi\)
0.456860 + 0.889539i \(0.348974\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.40667 −0.265772
\(83\) −15.7827 −1.73238 −0.866188 − 0.499718i \(-0.833437\pi\)
−0.866188 + 0.499718i \(0.833437\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.597839 −0.0644666
\(87\) 0 0
\(88\) −4.73170 −0.504401
\(89\) −0.343059 −0.0363642 −0.0181821 − 0.999835i \(-0.505788\pi\)
−0.0181821 + 0.999835i \(0.505788\pi\)
\(90\) 0 0
\(91\) 3.45018 0.361677
\(92\) −15.0629 −1.57041
\(93\) 0 0
\(94\) 0.545073 0.0562199
\(95\) 0 0
\(96\) 0 0
\(97\) 12.1360 1.23222 0.616111 − 0.787659i \(-0.288707\pi\)
0.616111 + 0.787659i \(0.288707\pi\)
\(98\) −2.68952 −0.271682
\(99\) 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 8325.2.a.cu.1.6 13
3.2 odd 2 2775.2.a.bg.1.8 13
5.2 odd 4 1665.2.c.f.334.12 26
5.3 odd 4 1665.2.c.f.334.15 26
5.4 even 2 8325.2.a.cv.1.8 13
15.2 even 4 555.2.c.c.334.15 yes 26
15.8 even 4 555.2.c.c.334.12 ✓ 26
15.14 odd 2 2775.2.a.bh.1.6 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.c.c.334.12 ✓ 26 15.8 even 4
555.2.c.c.334.15 yes 26 15.2 even 4
1665.2.c.f.334.12 26 5.2 odd 4
1665.2.c.f.334.15 26 5.3 odd 4
2775.2.a.bg.1.8 13 3.2 odd 2
2775.2.a.bh.1.6 13 15.14 odd 2
8325.2.a.cu.1.6 13 1.1 even 1 trivial
8325.2.a.cv.1.8 13 5.4 even 2