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: \(1\)
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 1.5
Root \(-1.32131\) 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)
 
\(f(q)\) \(=\) \(q-1.32131 q^{2} -0.254137 q^{4} -3.90085 q^{7} +2.97842 q^{8} -2.89176 q^{11} +0.529101 q^{13} +5.15424 q^{14} -3.42714 q^{16} -3.91958 q^{17} +0.525175 q^{19} +3.82092 q^{22} -4.99607 q^{23} -0.699107 q^{26} +0.991353 q^{28} +7.35375 q^{29} +0.179843 q^{31} -1.42852 q^{32} +5.17899 q^{34} +1.00000 q^{37} -0.693920 q^{38} +0.810743 q^{41} +9.21734 q^{43} +0.734905 q^{44} +6.60137 q^{46} +10.9077 q^{47} +8.21667 q^{49} -0.134464 q^{52} +3.17670 q^{53} -11.6184 q^{56} -9.71660 q^{58} +3.78566 q^{59} +4.79033 q^{61} -0.237628 q^{62} +8.74179 q^{64} -13.1366 q^{67} +0.996113 q^{68} -8.58810 q^{71} +5.17413 q^{73} -1.32131 q^{74} -0.133467 q^{76} +11.2804 q^{77} +2.54676 q^{79} -1.07124 q^{82} +2.74363 q^{83} -12.1790 q^{86} -8.61288 q^{88} +8.46781 q^{89} -2.06395 q^{91} +1.26969 q^{92} -14.4124 q^{94} -17.6578 q^{97} -10.8568 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} - 20 q^{7} - 20 q^{13} + 20 q^{16} - 8 q^{19} - 12 q^{22} - 72 q^{28} + 8 q^{31} + 12 q^{34} + 16 q^{37} - 24 q^{43} - 4 q^{46} + 40 q^{49} - 52 q^{52} - 64 q^{58} - 8 q^{61} + 40 q^{64}+ \cdots - 28 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.32131 −0.934308 −0.467154 − 0.884176i \(-0.654721\pi\)
−0.467154 + 0.884176i \(0.654721\pi\)
\(3\) 0 0
\(4\) −0.254137 −0.127069
\(5\) 0 0
\(6\) 0 0
\(7\) −3.90085 −1.47438 −0.737192 − 0.675683i \(-0.763849\pi\)
−0.737192 + 0.675683i \(0.763849\pi\)
\(8\) 2.97842 1.05303
\(9\) 0 0
\(10\) 0 0
\(11\) −2.89176 −0.871900 −0.435950 − 0.899971i \(-0.643587\pi\)
−0.435950 + 0.899971i \(0.643587\pi\)
\(12\) 0 0
\(13\) 0.529101 0.146746 0.0733732 − 0.997305i \(-0.476624\pi\)
0.0733732 + 0.997305i \(0.476624\pi\)
\(14\) 5.15424 1.37753
\(15\) 0 0
\(16\) −3.42714 −0.856785
\(17\) −3.91958 −0.950639 −0.475319 − 0.879813i \(-0.657667\pi\)
−0.475319 + 0.879813i \(0.657667\pi\)
\(18\) 0 0
\(19\) 0.525175 0.120483 0.0602417 − 0.998184i \(-0.480813\pi\)
0.0602417 + 0.998184i \(0.480813\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.82092 0.814623
\(23\) −4.99607 −1.04175 −0.520877 − 0.853632i \(-0.674395\pi\)
−0.520877 + 0.853632i \(0.674395\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.699107 −0.137106
\(27\) 0 0
\(28\) 0.991353 0.187348
\(29\) 7.35375 1.36556 0.682779 − 0.730625i \(-0.260771\pi\)
0.682779 + 0.730625i \(0.260771\pi\)
\(30\) 0 0
\(31\) 0.179843 0.0323007 0.0161503 − 0.999870i \(-0.494859\pi\)
0.0161503 + 0.999870i \(0.494859\pi\)
\(32\) −1.42852 −0.252528
\(33\) 0 0
\(34\) 5.17899 0.888189
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −0.693920 −0.112569
\(39\) 0 0
\(40\) 0 0
\(41\) 0.810743 0.126617 0.0633084 − 0.997994i \(-0.479835\pi\)
0.0633084 + 0.997994i \(0.479835\pi\)
\(42\) 0 0
\(43\) 9.21734 1.40563 0.702816 − 0.711372i \(-0.251926\pi\)
0.702816 + 0.711372i \(0.251926\pi\)
\(44\) 0.734905 0.110791
\(45\) 0 0
\(46\) 6.60137 0.973318
\(47\) 10.9077 1.59105 0.795523 − 0.605923i \(-0.207196\pi\)
0.795523 + 0.605923i \(0.207196\pi\)
\(48\) 0 0
\(49\) 8.21667 1.17381
\(50\) 0 0
\(51\) 0 0
\(52\) −0.134464 −0.0186469
\(53\) 3.17670 0.436354 0.218177 − 0.975909i \(-0.429989\pi\)
0.218177 + 0.975909i \(0.429989\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −11.6184 −1.55257
\(57\) 0 0
\(58\) −9.71660 −1.27585
\(59\) 3.78566 0.492851 0.246425 − 0.969162i \(-0.420744\pi\)
0.246425 + 0.969162i \(0.420744\pi\)
\(60\) 0 0
\(61\) 4.79033 0.613339 0.306670 − 0.951816i \(-0.400785\pi\)
0.306670 + 0.951816i \(0.400785\pi\)
\(62\) −0.237628 −0.0301788
\(63\) 0 0
\(64\) 8.74179 1.09272
\(65\) 0 0
\(66\) 0 0
\(67\) −13.1366 −1.60490 −0.802448 − 0.596722i \(-0.796470\pi\)
−0.802448 + 0.596722i \(0.796470\pi\)
\(68\) 0.996113 0.120796
\(69\) 0 0
\(70\) 0 0
\(71\) −8.58810 −1.01922 −0.509610 − 0.860405i \(-0.670210\pi\)
−0.509610 + 0.860405i \(0.670210\pi\)
\(72\) 0 0
\(73\) 5.17413 0.605587 0.302793 − 0.953056i \(-0.402081\pi\)
0.302793 + 0.953056i \(0.402081\pi\)
\(74\) −1.32131 −0.153599
\(75\) 0 0
\(76\) −0.133467 −0.0153097
\(77\) 11.2804 1.28552
\(78\) 0 0
\(79\) 2.54676 0.286533 0.143267 − 0.989684i \(-0.454239\pi\)
0.143267 + 0.989684i \(0.454239\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.07124 −0.118299
\(83\) 2.74363 0.301153 0.150576 − 0.988598i \(-0.451887\pi\)
0.150576 + 0.988598i \(0.451887\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.1790 −1.31329
\(87\) 0 0
\(88\) −8.61288 −0.918136
\(89\) 8.46781 0.897586 0.448793 − 0.893636i \(-0.351854\pi\)
0.448793 + 0.893636i \(0.351854\pi\)
\(90\) 0 0
\(91\) −2.06395 −0.216361
\(92\) 1.26969 0.132374
\(93\) 0 0
\(94\) −14.4124 −1.48653
\(95\) 0 0
\(96\) 0 0
\(97\) −17.6578 −1.79288 −0.896438 − 0.443170i \(-0.853854\pi\)
−0.896438 + 0.443170i \(0.853854\pi\)
\(98\) −10.8568 −1.09670
\(99\) 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.cw.1.5 16
3.2 odd 2 inner 8325.2.a.cw.1.12 16
5.2 odd 4 1665.2.c.g.334.9 ✓ 32
5.3 odd 4 1665.2.c.g.334.23 yes 32
5.4 even 2 8325.2.a.cx.1.12 16
15.2 even 4 1665.2.c.g.334.24 yes 32
15.8 even 4 1665.2.c.g.334.10 yes 32
15.14 odd 2 8325.2.a.cx.1.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1665.2.c.g.334.9 ✓ 32 5.2 odd 4
1665.2.c.g.334.10 yes 32 15.8 even 4
1665.2.c.g.334.23 yes 32 5.3 odd 4
1665.2.c.g.334.24 yes 32 15.2 even 4
8325.2.a.cw.1.5 16 1.1 even 1 trivial
8325.2.a.cw.1.12 16 3.2 odd 2 inner
8325.2.a.cx.1.5 16 15.14 odd 2
8325.2.a.cx.1.12 16 5.4 even 2