Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1656,2,Mod(1,1656)] 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("1656.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1656, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1656.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,0,0,2,0,0,0,-4,0,6] 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: \(13.2232265747\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 552)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 1656.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.42864 q^{5} +0.622216 q^{7} -5.80642 q^{11} +2.00000 q^{13} -1.37778 q^{17} +6.42864 q^{19} +1.00000 q^{23} +14.6128 q^{25} +0.755569 q^{29} -1.24443 q^{31} +2.75557 q^{35} +9.05086 q^{37} +9.61285 q^{41} -5.18421 q^{43} +5.24443 q^{47} -6.61285 q^{49} -12.0415 q^{53} -25.7146 q^{55} +12.8573 q^{59} +2.94914 q^{61} +8.85728 q^{65} +5.18421 q^{67} -6.10171 q^{71} +2.00000 q^{73} -3.61285 q^{77} -8.62222 q^{79} -11.9081 q^{83} -6.10171 q^{85} -6.23506 q^{89} +1.24443 q^{91} +28.4701 q^{95} -2.85728 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{7} - 4 q^{11} + 6 q^{13} - 4 q^{17} + 6 q^{19} + 3 q^{23} + 17 q^{25} + 2 q^{29} - 4 q^{31} + 8 q^{35} + 14 q^{37} + 2 q^{41} - 2 q^{43} + 16 q^{47} + 7 q^{49} + 4 q^{53} - 24 q^{55} + 12 q^{59}+ \cdots + 18 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 4.42864 1.98055 0.990274 − 0.139132i \(-0.0444311\pi\)
0.990274 + 0.139132i \(0.0444311\pi\)
\(6\) 0 0
\(7\) 0.622216 0.235175 0.117588 − 0.993063i \(-0.462484\pi\)
0.117588 + 0.993063i \(0.462484\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.80642 −1.75070 −0.875351 − 0.483487i \(-0.839370\pi\)
−0.875351 + 0.483487i \(0.839370\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.37778 −0.334162 −0.167081 − 0.985943i \(-0.553434\pi\)
−0.167081 + 0.985943i \(0.553434\pi\)
\(18\) 0 0
\(19\) 6.42864 1.47483 0.737416 − 0.675439i \(-0.236046\pi\)
0.737416 + 0.675439i \(0.236046\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 14.6128 2.92257
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.755569 0.140306 0.0701528 − 0.997536i \(-0.477651\pi\)
0.0701528 + 0.997536i \(0.477651\pi\)
\(30\) 0 0
\(31\) −1.24443 −0.223506 −0.111753 − 0.993736i \(-0.535647\pi\)
−0.111753 + 0.993736i \(0.535647\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.75557 0.465776
\(36\) 0 0
\(37\) 9.05086 1.48795 0.743976 − 0.668207i \(-0.232938\pi\)
0.743976 + 0.668207i \(0.232938\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.61285 1.50127 0.750637 − 0.660715i \(-0.229747\pi\)
0.750637 + 0.660715i \(0.229747\pi\)
\(42\) 0 0
\(43\) −5.18421 −0.790584 −0.395292 − 0.918555i \(-0.629357\pi\)
−0.395292 + 0.918555i \(0.629357\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.24443 0.764979 0.382489 − 0.923960i \(-0.375067\pi\)
0.382489 + 0.923960i \(0.375067\pi\)
\(48\) 0 0
\(49\) −6.61285 −0.944693
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0415 −1.65403 −0.827013 − 0.562183i \(-0.809962\pi\)
−0.827013 + 0.562183i \(0.809962\pi\)
\(54\) 0 0
\(55\) −25.7146 −3.46735
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.8573 1.67388 0.836938 − 0.547298i \(-0.184344\pi\)
0.836938 + 0.547298i \(0.184344\pi\)
\(60\) 0 0
\(61\) 2.94914 0.377599 0.188800 − 0.982016i \(-0.439540\pi\)
0.188800 + 0.982016i \(0.439540\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.85728 1.09861
\(66\) 0 0
\(67\) 5.18421 0.633352 0.316676 − 0.948534i \(-0.397433\pi\)
0.316676 + 0.948534i \(0.397433\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.10171 −0.724140 −0.362070 − 0.932151i \(-0.617930\pi\)
−0.362070 + 0.932151i \(0.617930\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.61285 −0.411722
\(78\) 0 0
\(79\) −8.62222 −0.970075 −0.485038 − 0.874493i \(-0.661194\pi\)
−0.485038 + 0.874493i \(0.661194\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.9081 −1.30709 −0.653544 − 0.756889i \(-0.726718\pi\)
−0.653544 + 0.756889i \(0.726718\pi\)
\(84\) 0 0
\(85\) −6.10171 −0.661823
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.23506 −0.660915 −0.330458 − 0.943821i \(-0.607203\pi\)
−0.330458 + 0.943821i \(0.607203\pi\)
\(90\) 0 0
\(91\) 1.24443 0.130452
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 28.4701 2.92097
\(96\) 0 0
\(97\) −2.85728 −0.290113 −0.145056 − 0.989423i \(-0.546336\pi\)
−0.145056 + 0.989423i \(0.546336\pi\)
\(98\) 0 0
\(99\) 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 1656.2.a.n.1.3 3
3.2 odd 2 552.2.a.g.1.1 ✓ 3
4.3 odd 2 3312.2.a.bf.1.3 3
12.11 even 2 1104.2.a.o.1.1 3
24.5 odd 2 4416.2.a.bp.1.3 3
24.11 even 2 4416.2.a.bs.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
552.2.a.g.1.1 ✓ 3 3.2 odd 2
1104.2.a.o.1.1 3 12.11 even 2
1656.2.a.n.1.3 3 1.1 even 1 trivial
3312.2.a.bf.1.3 3 4.3 odd 2
4416.2.a.bp.1.3 3 24.5 odd 2
4416.2.a.bs.1.3 3 24.11 even 2