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 = [5,1,0,1,0,0,0,0,0,0,16,0,-3] 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: \(5\)
Coefficient field: 5.5.65657.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 5x^{3} + 2x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 925)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.71737\) 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)
 
\(f(q)\) \(=\) \(q+0.531633 q^{2} -1.71737 q^{4} +2.93944 q^{7} -1.97628 q^{8} +5.92562 q^{11} -1.35031 q^{13} +1.56270 q^{14} +2.38408 q^{16} -6.16935 q^{17} +5.46850 q^{19} +3.15026 q^{22} -1.87203 q^{23} -0.717868 q^{26} -5.04809 q^{28} +7.96110 q^{29} +0.229916 q^{31} +5.22001 q^{32} -3.27983 q^{34} +1.00000 q^{37} +2.90724 q^{38} +3.07758 q^{41} -8.13251 q^{43} -10.1765 q^{44} -0.995233 q^{46} +9.06340 q^{47} +1.64029 q^{49} +2.31897 q^{52} -2.63638 q^{53} -5.80914 q^{56} +4.23238 q^{58} +11.7344 q^{59} +4.90190 q^{61} +0.122231 q^{62} -1.99303 q^{64} -4.03427 q^{67} +10.5950 q^{68} +9.03377 q^{71} -16.0541 q^{73} +0.531633 q^{74} -9.39142 q^{76} +17.4180 q^{77} -5.53201 q^{79} +1.63615 q^{82} -1.13714 q^{83} -4.32351 q^{86} -11.7107 q^{88} -8.14306 q^{89} -3.96914 q^{91} +3.21496 q^{92} +4.81841 q^{94} +17.9941 q^{97} +0.872030 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} + q^{4} + 16 q^{11} - 3 q^{13} + 5 q^{14} - 11 q^{16} - 2 q^{17} - 11 q^{19} + 19 q^{22} + 7 q^{23} + 4 q^{26} - 7 q^{28} + 15 q^{29} - 13 q^{31} + q^{32} + 13 q^{34} + 5 q^{37} + 2 q^{38}+ \cdots - 12 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.531633 0.375921 0.187961 − 0.982177i \(-0.439812\pi\)
0.187961 + 0.982177i \(0.439812\pi\)
\(3\) 0 0
\(4\) −1.71737 −0.858683
\(5\) 0 0
\(6\) 0 0
\(7\) 2.93944 1.11100 0.555501 − 0.831516i \(-0.312526\pi\)
0.555501 + 0.831516i \(0.312526\pi\)
\(8\) −1.97628 −0.698719
\(9\) 0 0
\(10\) 0 0
\(11\) 5.92562 1.78664 0.893321 − 0.449419i \(-0.148369\pi\)
0.893321 + 0.449419i \(0.148369\pi\)
\(12\) 0 0
\(13\) −1.35031 −0.374508 −0.187254 − 0.982312i \(-0.559959\pi\)
−0.187254 + 0.982312i \(0.559959\pi\)
\(14\) 1.56270 0.417650
\(15\) 0 0
\(16\) 2.38408 0.596020
\(17\) −6.16935 −1.49629 −0.748144 − 0.663537i \(-0.769055\pi\)
−0.748144 + 0.663537i \(0.769055\pi\)
\(18\) 0 0
\(19\) 5.46850 1.25456 0.627280 − 0.778793i \(-0.284168\pi\)
0.627280 + 0.778793i \(0.284168\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.15026 0.671637
\(23\) −1.87203 −0.390345 −0.195173 − 0.980769i \(-0.562527\pi\)
−0.195173 + 0.980769i \(0.562527\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.717868 −0.140786
\(27\) 0 0
\(28\) −5.04809 −0.953999
\(29\) 7.96110 1.47834 0.739169 − 0.673520i \(-0.235218\pi\)
0.739169 + 0.673520i \(0.235218\pi\)
\(30\) 0 0
\(31\) 0.229916 0.0412942 0.0206471 − 0.999787i \(-0.493427\pi\)
0.0206471 + 0.999787i \(0.493427\pi\)
\(32\) 5.22001 0.922775
\(33\) 0 0
\(34\) −3.27983 −0.562487
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 2.90724 0.471616
\(39\) 0 0
\(40\) 0 0
\(41\) 3.07758 0.480638 0.240319 − 0.970694i \(-0.422748\pi\)
0.240319 + 0.970694i \(0.422748\pi\)
\(42\) 0 0
\(43\) −8.13251 −1.24020 −0.620098 − 0.784524i \(-0.712907\pi\)
−0.620098 + 0.784524i \(0.712907\pi\)
\(44\) −10.1765 −1.53416
\(45\) 0 0
\(46\) −0.995233 −0.146739
\(47\) 9.06340 1.32203 0.661017 − 0.750371i \(-0.270125\pi\)
0.661017 + 0.750371i \(0.270125\pi\)
\(48\) 0 0
\(49\) 1.64029 0.234326
\(50\) 0 0
\(51\) 0 0
\(52\) 2.31897 0.321583
\(53\) −2.63638 −0.362135 −0.181067 − 0.983471i \(-0.557955\pi\)
−0.181067 + 0.983471i \(0.557955\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −5.80914 −0.776278
\(57\) 0 0
\(58\) 4.23238 0.555739
\(59\) 11.7344 1.52769 0.763843 − 0.645402i \(-0.223310\pi\)
0.763843 + 0.645402i \(0.223310\pi\)
\(60\) 0 0
\(61\) 4.90190 0.627624 0.313812 − 0.949485i \(-0.398394\pi\)
0.313812 + 0.949485i \(0.398394\pi\)
\(62\) 0.122231 0.0155234
\(63\) 0 0
\(64\) −1.99303 −0.249128
\(65\) 0 0
\(66\) 0 0
\(67\) −4.03427 −0.492865 −0.246432 − 0.969160i \(-0.579258\pi\)
−0.246432 + 0.969160i \(0.579258\pi\)
\(68\) 10.5950 1.28484
\(69\) 0 0
\(70\) 0 0
\(71\) 9.03377 1.07211 0.536056 − 0.844183i \(-0.319914\pi\)
0.536056 + 0.844183i \(0.319914\pi\)
\(72\) 0 0
\(73\) −16.0541 −1.87899 −0.939497 − 0.342557i \(-0.888707\pi\)
−0.939497 + 0.342557i \(0.888707\pi\)
\(74\) 0.531633 0.0618011
\(75\) 0 0
\(76\) −9.39142 −1.07727
\(77\) 17.4180 1.98496
\(78\) 0 0
\(79\) −5.53201 −0.622399 −0.311200 − 0.950345i \(-0.600731\pi\)
−0.311200 + 0.950345i \(0.600731\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.63615 0.180682
\(83\) −1.13714 −0.124818 −0.0624088 − 0.998051i \(-0.519878\pi\)
−0.0624088 + 0.998051i \(0.519878\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.32351 −0.466217
\(87\) 0 0
\(88\) −11.7107 −1.24836
\(89\) −8.14306 −0.863163 −0.431581 − 0.902074i \(-0.642044\pi\)
−0.431581 + 0.902074i \(0.642044\pi\)
\(90\) 0 0
\(91\) −3.96914 −0.416079
\(92\) 3.21496 0.335183
\(93\) 0 0
\(94\) 4.81841 0.496981
\(95\) 0 0
\(96\) 0 0
\(97\) 17.9941 1.82702 0.913511 − 0.406815i \(-0.133361\pi\)
0.913511 + 0.406815i \(0.133361\pi\)
\(98\) 0.872030 0.0880883
\(99\) 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.cd.1.3 5
3.2 odd 2 925.2.a.g.1.3 ✓ 5
5.4 even 2 8325.2.a.cb.1.3 5
15.2 even 4 925.2.b.h.149.5 10
15.8 even 4 925.2.b.h.149.6 10
15.14 odd 2 925.2.a.i.1.3 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
925.2.a.g.1.3 ✓ 5 3.2 odd 2
925.2.a.i.1.3 yes 5 15.14 odd 2
925.2.b.h.149.5 10 15.2 even 4
925.2.b.h.149.6 10 15.8 even 4
8325.2.a.cb.1.3 5 5.4 even 2
8325.2.a.cd.1.3 5 1.1 even 1 trivial