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 = [16,0,0,20,0,0,20,0,0,0,0,0,20] 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: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 26x^{14} + 271x^{12} - 1454x^{10} + 4306x^{8} - 7062x^{6} + 6123x^{4} - 2486x^{2} + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1665)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.70149\) 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)
 
\(f(q)\) \(=\) \(q-2.70149 q^{2} +5.29803 q^{4} +4.46991 q^{7} -8.90958 q^{8} +4.48103 q^{11} +6.42766 q^{13} -12.0754 q^{14} +13.4730 q^{16} +4.01670 q^{17} -3.60893 q^{19} -12.1054 q^{22} +4.64914 q^{23} -17.3642 q^{26} +23.6817 q^{28} -8.05215 q^{29} +0.0715520 q^{31} -18.5781 q^{32} -10.8511 q^{34} -1.00000 q^{37} +9.74946 q^{38} -2.11865 q^{41} +4.68948 q^{43} +23.7406 q^{44} -12.5596 q^{46} -2.61144 q^{47} +12.9801 q^{49} +34.0539 q^{52} -2.81729 q^{53} -39.8250 q^{56} +21.7528 q^{58} -4.66034 q^{59} +1.78497 q^{61} -0.193297 q^{62} +23.2424 q^{64} -1.23093 q^{67} +21.2806 q^{68} +0.214659 q^{71} +4.83217 q^{73} +2.70149 q^{74} -19.1202 q^{76} +20.0298 q^{77} +10.7365 q^{79} +5.72351 q^{82} +12.4806 q^{83} -12.6686 q^{86} -39.9241 q^{88} -6.23598 q^{89} +28.7311 q^{91} +24.6313 q^{92} +7.05478 q^{94} +3.68041 q^{97} -35.0655 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} + 20 q^{7} + 20 q^{13} + 20 q^{16} - 8 q^{19} + 12 q^{22} + 72 q^{28} + 8 q^{31} + 12 q^{34} - 16 q^{37} + 24 q^{43} - 4 q^{46} + 40 q^{49} + 52 q^{52} + 64 q^{58} - 8 q^{61} + 40 q^{64}+ \cdots + 28 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.70149 −1.91024 −0.955120 − 0.296220i \(-0.904274\pi\)
−0.955120 + 0.296220i \(0.904274\pi\)
\(3\) 0 0
\(4\) 5.29803 2.64901
\(5\) 0 0
\(6\) 0 0
\(7\) 4.46991 1.68947 0.844733 − 0.535188i \(-0.179759\pi\)
0.844733 + 0.535188i \(0.179759\pi\)
\(8\) −8.90958 −3.15001
\(9\) 0 0
\(10\) 0 0
\(11\) 4.48103 1.35108 0.675541 − 0.737323i \(-0.263910\pi\)
0.675541 + 0.737323i \(0.263910\pi\)
\(12\) 0 0
\(13\) 6.42766 1.78271 0.891356 − 0.453303i \(-0.149755\pi\)
0.891356 + 0.453303i \(0.149755\pi\)
\(14\) −12.0754 −3.22728
\(15\) 0 0
\(16\) 13.4730 3.36826
\(17\) 4.01670 0.974192 0.487096 − 0.873348i \(-0.338056\pi\)
0.487096 + 0.873348i \(0.338056\pi\)
\(18\) 0 0
\(19\) −3.60893 −0.827944 −0.413972 − 0.910290i \(-0.635859\pi\)
−0.413972 + 0.910290i \(0.635859\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −12.1054 −2.58089
\(23\) 4.64914 0.969413 0.484707 − 0.874677i \(-0.338926\pi\)
0.484707 + 0.874677i \(0.338926\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −17.3642 −3.40541
\(27\) 0 0
\(28\) 23.6817 4.47542
\(29\) −8.05215 −1.49525 −0.747623 − 0.664123i \(-0.768805\pi\)
−0.747623 + 0.664123i \(0.768805\pi\)
\(30\) 0 0
\(31\) 0.0715520 0.0128511 0.00642556 − 0.999979i \(-0.497955\pi\)
0.00642556 + 0.999979i \(0.497955\pi\)
\(32\) −18.5781 −3.28417
\(33\) 0 0
\(34\) −10.8511 −1.86094
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 9.74946 1.58157
\(39\) 0 0
\(40\) 0 0
\(41\) −2.11865 −0.330878 −0.165439 − 0.986220i \(-0.552904\pi\)
−0.165439 + 0.986220i \(0.552904\pi\)
\(42\) 0 0
\(43\) 4.68948 0.715138 0.357569 − 0.933887i \(-0.383606\pi\)
0.357569 + 0.933887i \(0.383606\pi\)
\(44\) 23.7406 3.57903
\(45\) 0 0
\(46\) −12.5596 −1.85181
\(47\) −2.61144 −0.380918 −0.190459 − 0.981695i \(-0.560998\pi\)
−0.190459 + 0.981695i \(0.560998\pi\)
\(48\) 0 0
\(49\) 12.9801 1.85430
\(50\) 0 0
\(51\) 0 0
\(52\) 34.0539 4.72243
\(53\) −2.81729 −0.386984 −0.193492 − 0.981102i \(-0.561981\pi\)
−0.193492 + 0.981102i \(0.561981\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −39.8250 −5.32184
\(57\) 0 0
\(58\) 21.7528 2.85628
\(59\) −4.66034 −0.606725 −0.303362 − 0.952875i \(-0.598109\pi\)
−0.303362 + 0.952875i \(0.598109\pi\)
\(60\) 0 0
\(61\) 1.78497 0.228542 0.114271 − 0.993450i \(-0.463547\pi\)
0.114271 + 0.993450i \(0.463547\pi\)
\(62\) −0.193297 −0.0245487
\(63\) 0 0
\(64\) 23.2424 2.90529
\(65\) 0 0
\(66\) 0 0
\(67\) −1.23093 −0.150383 −0.0751913 − 0.997169i \(-0.523957\pi\)
−0.0751913 + 0.997169i \(0.523957\pi\)
\(68\) 21.2806 2.58065
\(69\) 0 0
\(70\) 0 0
\(71\) 0.214659 0.0254753 0.0127377 − 0.999919i \(-0.495945\pi\)
0.0127377 + 0.999919i \(0.495945\pi\)
\(72\) 0 0
\(73\) 4.83217 0.565562 0.282781 − 0.959184i \(-0.408743\pi\)
0.282781 + 0.959184i \(0.408743\pi\)
\(74\) 2.70149 0.314041
\(75\) 0 0
\(76\) −19.1202 −2.19324
\(77\) 20.0298 2.28261
\(78\) 0 0
\(79\) 10.7365 1.20795 0.603973 − 0.797005i \(-0.293584\pi\)
0.603973 + 0.797005i \(0.293584\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 5.72351 0.632056
\(83\) 12.4806 1.36993 0.684963 − 0.728577i \(-0.259818\pi\)
0.684963 + 0.728577i \(0.259818\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.6686 −1.36609
\(87\) 0 0
\(88\) −39.9241 −4.25592
\(89\) −6.23598 −0.661012 −0.330506 − 0.943804i \(-0.607219\pi\)
−0.330506 + 0.943804i \(0.607219\pi\)
\(90\) 0 0
\(91\) 28.7311 3.01183
\(92\) 24.6313 2.56799
\(93\) 0 0
\(94\) 7.05478 0.727645
\(95\) 0 0
\(96\) 0 0
\(97\) 3.68041 0.373689 0.186845 − 0.982389i \(-0.440174\pi\)
0.186845 + 0.982389i \(0.440174\pi\)
\(98\) −35.0655 −3.54215
\(99\) 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.cx.1.1 16
3.2 odd 2 inner 8325.2.a.cx.1.16 16
5.2 odd 4 1665.2.c.g.334.1 ✓ 32
5.3 odd 4 1665.2.c.g.334.31 yes 32
5.4 even 2 8325.2.a.cw.1.16 16
15.2 even 4 1665.2.c.g.334.32 yes 32
15.8 even 4 1665.2.c.g.334.2 yes 32
15.14 odd 2 8325.2.a.cw.1.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1665.2.c.g.334.1 ✓ 32 5.2 odd 4
1665.2.c.g.334.2 yes 32 15.8 even 4
1665.2.c.g.334.31 yes 32 5.3 odd 4
1665.2.c.g.334.32 yes 32 15.2 even 4
8325.2.a.cw.1.1 16 15.14 odd 2
8325.2.a.cw.1.16 16 5.4 even 2
8325.2.a.cx.1.1 16 1.1 even 1 trivial
8325.2.a.cx.1.16 16 3.2 odd 2 inner