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 = [7,-1,0,7,0,0,0,0,0,0,-16,0,-1] 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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 9x^{4} + 26x^{3} - 23x^{2} - 9x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 925)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.74907\) 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)
 
\(f(q)\) \(=\) \(q-1.74907 q^{2} +1.05925 q^{4} +0.963428 q^{7} +1.64544 q^{8} -2.88433 q^{11} -1.65840 q^{13} -1.68510 q^{14} -4.99649 q^{16} -6.95992 q^{17} -2.39544 q^{19} +5.04490 q^{22} +0.0129671 q^{23} +2.90067 q^{26} +1.02051 q^{28} +7.63082 q^{29} +2.66242 q^{31} +5.44835 q^{32} +12.1734 q^{34} +1.00000 q^{37} +4.18979 q^{38} -1.52856 q^{41} +4.22095 q^{43} -3.05523 q^{44} -0.0226804 q^{46} +11.0761 q^{47} -6.07181 q^{49} -1.75667 q^{52} +9.51515 q^{53} +1.58526 q^{56} -13.3469 q^{58} +5.18662 q^{59} -0.854073 q^{61} -4.65677 q^{62} +0.463427 q^{64} +9.33337 q^{67} -7.37231 q^{68} -2.93088 q^{71} -3.04319 q^{73} -1.74907 q^{74} -2.53737 q^{76} -2.77884 q^{77} +8.24087 q^{79} +2.67356 q^{82} +11.2635 q^{83} -7.38275 q^{86} -4.74597 q^{88} -8.75646 q^{89} -1.59775 q^{91} +0.0137354 q^{92} -19.3729 q^{94} -2.99392 q^{97} +10.6200 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} + 7 q^{4} - 16 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 4 q^{17} + 9 q^{19} + q^{22} + q^{23} - 24 q^{26} + 7 q^{28} - 3 q^{29} + 11 q^{31} - 21 q^{32} - 23 q^{34} + 7 q^{37} + 32 q^{38}+ \cdots - 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.74907 −1.23678 −0.618390 − 0.785871i \(-0.712215\pi\)
−0.618390 + 0.785871i \(0.712215\pi\)
\(3\) 0 0
\(4\) 1.05925 0.529626
\(5\) 0 0
\(6\) 0 0
\(7\) 0.963428 0.364142 0.182071 − 0.983285i \(-0.441720\pi\)
0.182071 + 0.983285i \(0.441720\pi\)
\(8\) 1.64544 0.581749
\(9\) 0 0
\(10\) 0 0
\(11\) −2.88433 −0.869657 −0.434829 − 0.900513i \(-0.643191\pi\)
−0.434829 + 0.900513i \(0.643191\pi\)
\(12\) 0 0
\(13\) −1.65840 −0.459958 −0.229979 − 0.973196i \(-0.573866\pi\)
−0.229979 + 0.973196i \(0.573866\pi\)
\(14\) −1.68510 −0.450363
\(15\) 0 0
\(16\) −4.99649 −1.24912
\(17\) −6.95992 −1.68803 −0.844014 − 0.536321i \(-0.819814\pi\)
−0.844014 + 0.536321i \(0.819814\pi\)
\(18\) 0 0
\(19\) −2.39544 −0.549551 −0.274776 − 0.961508i \(-0.588604\pi\)
−0.274776 + 0.961508i \(0.588604\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.04490 1.07558
\(23\) 0.0129671 0.00270383 0.00135191 − 0.999999i \(-0.499570\pi\)
0.00135191 + 0.999999i \(0.499570\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.90067 0.568867
\(27\) 0 0
\(28\) 1.02051 0.192859
\(29\) 7.63082 1.41701 0.708504 − 0.705707i \(-0.249371\pi\)
0.708504 + 0.705707i \(0.249371\pi\)
\(30\) 0 0
\(31\) 2.66242 0.478186 0.239093 − 0.970997i \(-0.423150\pi\)
0.239093 + 0.970997i \(0.423150\pi\)
\(32\) 5.44835 0.963141
\(33\) 0 0
\(34\) 12.1734 2.08772
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 4.18979 0.679674
\(39\) 0 0
\(40\) 0 0
\(41\) −1.52856 −0.238721 −0.119360 − 0.992851i \(-0.538084\pi\)
−0.119360 + 0.992851i \(0.538084\pi\)
\(42\) 0 0
\(43\) 4.22095 0.643689 0.321845 − 0.946792i \(-0.395697\pi\)
0.321845 + 0.946792i \(0.395697\pi\)
\(44\) −3.05523 −0.460593
\(45\) 0 0
\(46\) −0.0226804 −0.00334404
\(47\) 11.0761 1.61562 0.807808 − 0.589446i \(-0.200654\pi\)
0.807808 + 0.589446i \(0.200654\pi\)
\(48\) 0 0
\(49\) −6.07181 −0.867401
\(50\) 0 0
\(51\) 0 0
\(52\) −1.75667 −0.243606
\(53\) 9.51515 1.30701 0.653503 − 0.756924i \(-0.273299\pi\)
0.653503 + 0.756924i \(0.273299\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.58526 0.211839
\(57\) 0 0
\(58\) −13.3469 −1.75253
\(59\) 5.18662 0.675240 0.337620 − 0.941282i \(-0.390378\pi\)
0.337620 + 0.941282i \(0.390378\pi\)
\(60\) 0 0
\(61\) −0.854073 −0.109353 −0.0546764 − 0.998504i \(-0.517413\pi\)
−0.0546764 + 0.998504i \(0.517413\pi\)
\(62\) −4.65677 −0.591411
\(63\) 0 0
\(64\) 0.463427 0.0579284
\(65\) 0 0
\(66\) 0 0
\(67\) 9.33337 1.14025 0.570126 − 0.821557i \(-0.306894\pi\)
0.570126 + 0.821557i \(0.306894\pi\)
\(68\) −7.37231 −0.894024
\(69\) 0 0
\(70\) 0 0
\(71\) −2.93088 −0.347831 −0.173916 − 0.984761i \(-0.555642\pi\)
−0.173916 + 0.984761i \(0.555642\pi\)
\(72\) 0 0
\(73\) −3.04319 −0.356179 −0.178089 − 0.984014i \(-0.556992\pi\)
−0.178089 + 0.984014i \(0.556992\pi\)
\(74\) −1.74907 −0.203325
\(75\) 0 0
\(76\) −2.53737 −0.291057
\(77\) −2.77884 −0.316678
\(78\) 0 0
\(79\) 8.24087 0.927171 0.463585 − 0.886052i \(-0.346563\pi\)
0.463585 + 0.886052i \(0.346563\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.67356 0.295245
\(83\) 11.2635 1.23633 0.618166 − 0.786047i \(-0.287876\pi\)
0.618166 + 0.786047i \(0.287876\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.38275 −0.796102
\(87\) 0 0
\(88\) −4.74597 −0.505923
\(89\) −8.75646 −0.928183 −0.464092 − 0.885787i \(-0.653619\pi\)
−0.464092 + 0.885787i \(0.653619\pi\)
\(90\) 0 0
\(91\) −1.59775 −0.167490
\(92\) 0.0137354 0.00143202
\(93\) 0 0
\(94\) −19.3729 −1.99816
\(95\) 0 0
\(96\) 0 0
\(97\) −2.99392 −0.303987 −0.151993 − 0.988382i \(-0.548569\pi\)
−0.151993 + 0.988382i \(0.548569\pi\)
\(98\) 10.6200 1.07278
\(99\) 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.cm.1.2 7
3.2 odd 2 925.2.a.k.1.6 yes 7
5.4 even 2 8325.2.a.cn.1.6 7
15.2 even 4 925.2.b.i.149.11 14
15.8 even 4 925.2.b.i.149.4 14
15.14 odd 2 925.2.a.j.1.2 ✓ 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
925.2.a.j.1.2 ✓ 7 15.14 odd 2
925.2.a.k.1.6 yes 7 3.2 odd 2
925.2.b.i.149.4 14 15.8 even 4
925.2.b.i.149.11 14 15.2 even 4
8325.2.a.cm.1.2 7 1.1 even 1 trivial
8325.2.a.cn.1.6 7 5.4 even 2