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 = [4,0,0,4,0,0,-8,0,0,0,0,0,-8] 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: \(4\)
Coefficient field: \(\Q(\sqrt{3 + \sqrt{6}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 333)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.33441\) 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)
 
\(f(q)\) \(=\) \(q-2.33441 q^{2} +3.44949 q^{4} -2.00000 q^{7} -3.38371 q^{8} -2.00000 q^{13} +4.66883 q^{14} +1.00000 q^{16} -1.04930 q^{17} -2.89898 q^{19} +5.71812 q^{23} +4.66883 q^{26} -6.89898 q^{28} -8.28836 q^{29} +6.89898 q^{31} +4.43300 q^{32} +2.44949 q^{34} -1.00000 q^{37} +6.76742 q^{38} +2.09859 q^{41} -6.89898 q^{43} -13.3485 q^{46} +11.4362 q^{47} -3.00000 q^{49} -6.89898 q^{52} -4.19718 q^{53} +6.76742 q^{56} +19.3485 q^{58} +9.91530 q^{59} +11.7980 q^{61} -16.1051 q^{62} -12.3485 q^{64} -2.00000 q^{67} -3.61953 q^{68} +7.23907 q^{71} -0.898979 q^{73} +2.33441 q^{74} -10.0000 q^{76} -12.6969 q^{79} -4.89898 q^{82} +7.23907 q^{83} +16.1051 q^{86} +6.18977 q^{89} +4.00000 q^{91} +19.7246 q^{92} -26.6969 q^{94} +7.79796 q^{97} +7.00324 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 8 q^{7} - 8 q^{13} + 4 q^{16} + 8 q^{19} - 8 q^{28} + 8 q^{31} - 4 q^{37} - 8 q^{43} - 24 q^{46} - 12 q^{49} - 8 q^{52} + 48 q^{58} + 8 q^{61} - 20 q^{64} - 8 q^{67} + 16 q^{73} - 40 q^{76}+ \cdots - 8 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.33441 −1.65068 −0.825340 − 0.564636i \(-0.809017\pi\)
−0.825340 + 0.564636i \(0.809017\pi\)
\(3\) 0 0
\(4\) 3.44949 1.72474
\(5\) 0 0
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −3.38371 −1.19632
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.66883 1.24780
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.04930 −0.254491 −0.127246 − 0.991871i \(-0.540614\pi\)
−0.127246 + 0.991871i \(0.540614\pi\)
\(18\) 0 0
\(19\) −2.89898 −0.665072 −0.332536 − 0.943091i \(-0.607904\pi\)
−0.332536 + 0.943091i \(0.607904\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.71812 1.19231 0.596156 − 0.802869i \(-0.296694\pi\)
0.596156 + 0.802869i \(0.296694\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.66883 0.915633
\(27\) 0 0
\(28\) −6.89898 −1.30378
\(29\) −8.28836 −1.53911 −0.769555 − 0.638580i \(-0.779522\pi\)
−0.769555 + 0.638580i \(0.779522\pi\)
\(30\) 0 0
\(31\) 6.89898 1.23909 0.619547 − 0.784960i \(-0.287316\pi\)
0.619547 + 0.784960i \(0.287316\pi\)
\(32\) 4.43300 0.783652
\(33\) 0 0
\(34\) 2.44949 0.420084
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 6.76742 1.09782
\(39\) 0 0
\(40\) 0 0
\(41\) 2.09859 0.327745 0.163872 − 0.986482i \(-0.447601\pi\)
0.163872 + 0.986482i \(0.447601\pi\)
\(42\) 0 0
\(43\) −6.89898 −1.05208 −0.526042 − 0.850458i \(-0.676325\pi\)
−0.526042 + 0.850458i \(0.676325\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −13.3485 −1.96812
\(47\) 11.4362 1.66815 0.834074 − 0.551652i \(-0.186003\pi\)
0.834074 + 0.551652i \(0.186003\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −6.89898 −0.956716
\(53\) −4.19718 −0.576527 −0.288264 − 0.957551i \(-0.593078\pi\)
−0.288264 + 0.957551i \(0.593078\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 6.76742 0.904334
\(57\) 0 0
\(58\) 19.3485 2.54058
\(59\) 9.91530 1.29086 0.645431 − 0.763818i \(-0.276678\pi\)
0.645431 + 0.763818i \(0.276678\pi\)
\(60\) 0 0
\(61\) 11.7980 1.51057 0.755287 − 0.655394i \(-0.227498\pi\)
0.755287 + 0.655394i \(0.227498\pi\)
\(62\) −16.1051 −2.04535
\(63\) 0 0
\(64\) −12.3485 −1.54356
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −3.61953 −0.438933
\(69\) 0 0
\(70\) 0 0
\(71\) 7.23907 0.859119 0.429560 − 0.903039i \(-0.358669\pi\)
0.429560 + 0.903039i \(0.358669\pi\)
\(72\) 0 0
\(73\) −0.898979 −0.105218 −0.0526088 − 0.998615i \(-0.516754\pi\)
−0.0526088 + 0.998615i \(0.516754\pi\)
\(74\) 2.33441 0.271370
\(75\) 0 0
\(76\) −10.0000 −1.14708
\(77\) 0 0
\(78\) 0 0
\(79\) −12.6969 −1.42852 −0.714259 − 0.699882i \(-0.753236\pi\)
−0.714259 + 0.699882i \(0.753236\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −4.89898 −0.541002
\(83\) 7.23907 0.794591 0.397295 − 0.917691i \(-0.369949\pi\)
0.397295 + 0.917691i \(0.369949\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 16.1051 1.73665
\(87\) 0 0
\(88\) 0 0
\(89\) 6.18977 0.656114 0.328057 − 0.944658i \(-0.393606\pi\)
0.328057 + 0.944658i \(0.393606\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 19.7246 2.05643
\(93\) 0 0
\(94\) −26.6969 −2.75358
\(95\) 0 0
\(96\) 0 0
\(97\) 7.79796 0.791763 0.395881 − 0.918302i \(-0.370439\pi\)
0.395881 + 0.918302i \(0.370439\pi\)
\(98\) 7.00324 0.707434
\(99\) 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.bu.1.1 4
3.2 odd 2 inner 8325.2.a.bu.1.4 4
5.4 even 2 333.2.a.f.1.4 yes 4
15.14 odd 2 333.2.a.f.1.1 ✓ 4
20.19 odd 2 5328.2.a.bq.1.2 4
60.59 even 2 5328.2.a.bq.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
333.2.a.f.1.1 ✓ 4 15.14 odd 2
333.2.a.f.1.4 yes 4 5.4 even 2
5328.2.a.bq.1.2 4 20.19 odd 2
5328.2.a.bq.1.3 4 60.59 even 2
8325.2.a.bu.1.1 4 1.1 even 1 trivial
8325.2.a.bu.1.4 4 3.2 odd 2 inner