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Properties

Label 3040.1.cn.a.2469.5
Level $3040$
Weight $1$
Character 3040.2469
Analytic conductor $1.517$
Analytic rank $0$
Dimension $32$
Projective image $D_{32}$
CM discriminant -95
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3040,1,Mod(189,3040)] 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("3040.189"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3040, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 3, 4, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3040.cn (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.51715763840\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{64})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{32} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{32}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{32} + \cdots)\)

Embedding invariants

Embedding label 2469.5
Root \(0.956940 - 0.290285i\) of defining polynomial
Character \(\chi\) \(=\) 3040.2469
Dual form 3040.1.cn.a.1709.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.0980171 + 0.995185i) q^{2} +(0.485544 + 1.17221i) q^{3} +(-0.980785 + 0.195090i) q^{4} +(0.923880 + 0.382683i) q^{5} +(-1.11897 + 0.598102i) q^{6} +(-0.290285 - 0.956940i) q^{8} +(-0.431207 + 0.431207i) q^{9} +(-0.290285 + 0.956940i) q^{10} +(-0.636379 + 1.53636i) q^{11} +(-0.704900 - 1.05496i) q^{12} +(1.62958 - 0.674993i) q^{13} +1.26879i q^{15} +(0.923880 - 0.382683i) q^{16} +(-0.471397 - 0.386865i) q^{18} +(-0.923880 + 0.382683i) q^{19} +(-0.980785 - 0.195090i) q^{20} +(-1.59133 - 0.482726i) q^{22} +(0.980785 - 0.804910i) q^{24} +(0.707107 + 0.707107i) q^{25} +(0.831470 + 1.55557i) q^{26} +(0.457372 + 0.189450i) q^{27} +(-1.26268 + 0.124363i) q^{30} +(0.471397 + 0.881921i) q^{32} -2.10991 q^{33} +(0.338797 - 0.507046i) q^{36} +(-1.83886 - 0.761681i) q^{37} +(-0.471397 - 0.881921i) q^{38} +(1.58246 + 1.58246i) q^{39} +(0.0980171 - 0.995185i) q^{40} +(0.324423 - 1.63099i) q^{44} +(-0.563400 + 0.233368i) q^{45} +(0.897168 + 0.897168i) q^{48} +1.00000i q^{49} +(-0.634393 + 0.773010i) q^{50} +(-1.46658 + 0.979938i) q^{52} +(0.591637 - 1.42834i) q^{53} +(-0.143707 + 0.473739i) q^{54} +(-1.17588 + 1.17588i) q^{55} +(-0.897168 - 0.897168i) q^{57} +(-0.247528 - 1.24441i) q^{60} +(-0.425215 - 1.02656i) q^{61} +(-0.831470 + 0.555570i) q^{64} +1.76384 q^{65} +(-0.206808 - 2.09976i) q^{66} +(-0.732410 - 1.76820i) q^{67} +(0.537813 + 0.287467i) q^{72} +(0.577774 - 1.90466i) q^{74} +(-0.485544 + 1.17221i) q^{75} +(0.831470 - 0.555570i) q^{76} +(-1.41973 + 1.72995i) q^{78} +1.00000 q^{80} +1.23794i q^{81} +(1.65493 + 0.162997i) q^{88} +(-0.287467 - 0.537813i) q^{90} -1.00000 q^{95} +(-0.804910 + 0.980785i) q^{96} -0.942793 q^{97} +(-0.995185 + 0.0980171i) q^{98} +(-0.388076 - 0.936899i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 32 q^{66} + 32 q^{80} - 32 q^{95} - 32 q^{96} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1217\) \(1921\) \(2661\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{1}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0980171 + 0.995185i 0.0980171 + 0.995185i
\(3\) 0.485544 + 1.17221i 0.485544 + 1.17221i 0.956940 + 0.290285i \(0.0937500\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(4\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(5\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(6\) −1.11897 + 0.598102i −1.11897 + 0.598102i
\(7\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) −0.290285 − 0.956940i −0.290285 − 0.956940i
\(9\) −0.431207 + 0.431207i −0.431207 + 0.431207i
\(10\) −0.290285 + 0.956940i −0.290285 + 0.956940i
\(11\) −0.636379 + 1.53636i −0.636379 + 1.53636i 0.195090 + 0.980785i \(0.437500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(12\) −0.704900 − 1.05496i −0.704900 − 1.05496i
\(13\) 1.62958 − 0.674993i 1.62958 − 0.674993i 0.634393 − 0.773010i \(-0.281250\pi\)
0.995185 + 0.0980171i \(0.0312500\pi\)
\(14\) 0 0
\(15\) 1.26879i 1.26879i
\(16\) 0.923880 − 0.382683i 0.923880 − 0.382683i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −0.471397 − 0.386865i −0.471397 − 0.386865i
\(19\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(20\) −0.980785 − 0.195090i −0.980785 − 0.195090i
\(21\) 0 0
\(22\) −1.59133 − 0.482726i −1.59133 − 0.482726i
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0.980785 − 0.804910i 0.980785 − 0.804910i
\(25\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(26\) 0.831470 + 1.55557i 0.831470 + 1.55557i
\(27\) 0.457372 + 0.189450i 0.457372 + 0.189450i
\(28\) 0 0
\(29\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(30\) −1.26268 + 0.124363i −1.26268 + 0.124363i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.471397 + 0.881921i 0.471397 + 0.881921i
\(33\) −2.10991 −2.10991
\(34\) 0 0
\(35\) 0 0
\(36\) 0.338797 − 0.507046i 0.338797 − 0.507046i
\(37\) −1.83886 − 0.761681i −1.83886 − 0.761681i −0.956940 − 0.290285i \(-0.906250\pi\)
−0.881921 − 0.471397i \(-0.843750\pi\)
\(38\) −0.471397 − 0.881921i −0.471397 − 0.881921i
\(39\) 1.58246 + 1.58246i 1.58246 + 1.58246i
\(40\) 0.0980171 − 0.995185i 0.0980171 − 0.995185i
\(41\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(42\) 0 0
\(43\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(44\) 0.324423 − 1.63099i 0.324423 − 1.63099i
\(45\) −0.563400 + 0.233368i −0.563400 + 0.233368i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0.897168 + 0.897168i 0.897168 + 0.897168i
\(49\) 1.00000i 1.00000i
\(50\) −0.634393 + 0.773010i −0.634393 + 0.773010i
\(51\) 0 0
\(52\) −1.46658 + 0.979938i −1.46658 + 0.979938i
\(53\) 0.591637 − 1.42834i 0.591637 − 1.42834i −0.290285 − 0.956940i \(-0.593750\pi\)
0.881921 − 0.471397i \(-0.156250\pi\)
\(54\) −0.143707 + 0.473739i −0.143707 + 0.473739i
\(55\) −1.17588 + 1.17588i −1.17588 + 1.17588i
\(56\) 0 0
\(57\) −0.897168 − 0.897168i −0.897168 − 0.897168i
\(58\) 0 0
\(59\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(60\) −0.247528 − 1.24441i −0.247528 − 1.24441i
\(61\) −0.425215 − 1.02656i −0.425215 − 1.02656i −0.980785 − 0.195090i \(-0.937500\pi\)
0.555570 − 0.831470i \(-0.312500\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(65\) 1.76384 1.76384
\(66\) −0.206808 − 2.09976i −0.206808 − 2.09976i
\(67\) −0.732410 − 1.76820i −0.732410 − 1.76820i −0.634393 − 0.773010i \(-0.718750\pi\)
−0.0980171 − 0.995185i \(-0.531250\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(72\) 0.537813 + 0.287467i 0.537813 + 0.287467i
\(73\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0.577774 − 1.90466i 0.577774 − 1.90466i
\(75\) −0.485544 + 1.17221i −0.485544 + 1.17221i
\(76\) 0.831470 − 0.555570i 0.831470 − 0.555570i
\(77\) 0 0
\(78\) −1.41973 + 1.72995i −1.41973 + 1.72995i
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 1.00000
\(81\) 1.23794i 1.23794i
\(82\) 0 0
\(83\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 1.65493 + 0.162997i 1.65493 + 0.162997i
\(89\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) −0.287467 − 0.537813i −0.287467 − 0.537813i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −1.00000
\(96\) −0.804910 + 0.980785i −0.804910 + 0.980785i
\(97\) −0.942793 −0.942793 −0.471397 − 0.881921i \(-0.656250\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(98\) −0.995185 + 0.0980171i −0.995185 + 0.0980171i
\(99\) −0.388076 − 0.936899i −0.388076 − 0.936899i
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 3040.1.cn.a.2469.5 yes 32
5.4 even 2 inner 3040.1.cn.a.2469.4 yes 32
19.18 odd 2 inner 3040.1.cn.a.2469.4 yes 32
32.13 even 8 inner 3040.1.cn.a.1709.5 yes 32
95.94 odd 2 CM 3040.1.cn.a.2469.5 yes 32
160.109 even 8 inner 3040.1.cn.a.1709.4 ✓ 32
608.493 odd 8 inner 3040.1.cn.a.1709.4 ✓ 32
3040.1709 odd 8 inner 3040.1.cn.a.1709.5 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.1.cn.a.1709.4 ✓ 32 160.109 even 8 inner
3040.1.cn.a.1709.4 ✓ 32 608.493 odd 8 inner
3040.1.cn.a.1709.5 yes 32 32.13 even 8 inner
3040.1.cn.a.1709.5 yes 32 3040.1709 odd 8 inner
3040.1.cn.a.2469.4 yes 32 5.4 even 2 inner
3040.1.cn.a.2469.4 yes 32 19.18 odd 2 inner
3040.1.cn.a.2469.5 yes 32 1.1 even 1 trivial
3040.1.cn.a.2469.5 yes 32 95.94 odd 2 CM