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Properties

Label 3040.1.cn.a.2469.8
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 2469.8
Root \(0.290285 + 0.956940i\) of defining polynomial
Character \(\chi\) \(=\) 3040.2469
Dual form 3040.1.cn.a.1709.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.995185 - 0.0980171i) q^{2} +(-0.591637 - 1.42834i) q^{3} +(0.980785 - 0.195090i) q^{4} +(0.923880 + 0.382683i) q^{5} +(-0.728789 - 1.36347i) q^{6} +(0.956940 - 0.290285i) q^{8} +(-0.983006 + 0.983006i) q^{9} +(0.956940 + 0.290285i) q^{10} +(0.636379 - 1.53636i) q^{11} +(-0.858923 - 1.28547i) q^{12} +(-0.871028 + 0.360791i) q^{13} -1.54602i q^{15} +(0.923880 - 0.382683i) q^{16} +(-0.881921 + 1.07462i) q^{18} +(-0.923880 + 0.382683i) q^{19} +(0.980785 + 0.195090i) q^{20} +(0.482726 - 1.59133i) q^{22} +(-0.980785 - 1.19509i) q^{24} +(0.707107 + 0.707107i) q^{25} +(-0.831470 + 0.444430i) q^{26} +(0.557309 + 0.230845i) q^{27} +(-0.151537 - 1.53858i) q^{30} +(0.881921 - 0.471397i) q^{32} -2.57094 q^{33} +(-0.772343 + 1.15589i) q^{36} +(0.181112 + 0.0750191i) q^{37} +(-0.881921 + 0.471397i) q^{38} +(1.03066 + 1.03066i) q^{39} +(0.995185 + 0.0980171i) q^{40} +(0.324423 - 1.63099i) q^{44} +(-1.28436 + 0.531999i) q^{45} +(-1.09320 - 1.09320i) q^{48} +1.00000i q^{49} +(0.773010 + 0.634393i) q^{50} +(-0.783904 + 0.523788i) q^{52} +(0.485544 - 1.17221i) q^{53} +(0.577253 + 0.175108i) q^{54} +(1.17588 - 1.17588i) q^{55} +(1.09320 + 1.09320i) q^{57} +(-0.301614 - 1.51631i) q^{60} +(0.425215 + 1.02656i) q^{61} +(0.831470 - 0.555570i) q^{64} -0.942793 q^{65} +(-2.55856 + 0.251996i) q^{66} +(-0.222174 - 0.536376i) q^{67} +(-0.655327 + 1.22603i) q^{72} +(0.187593 + 0.0569057i) q^{74} +(0.591637 - 1.42834i) q^{75} +(-0.831470 + 0.555570i) q^{76} +(1.12672 + 0.924678i) q^{78} +1.00000 q^{80} +0.457578i q^{81} +(0.162997 - 1.65493i) q^{88} +(-1.22603 + 0.655327i) q^{90} -1.00000 q^{95} +(-1.19509 - 0.980785i) q^{96} -1.76384 q^{97} +(0.0980171 + 0.995185i) q^{98} +(0.884682 + 2.13581i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 32 q^{66} + 32 q^{80} - 32 q^{95} - 32 q^{96} - 32 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.995185 − 0.0980171i 0.995185 − 0.0980171i
\(3\) −0.591637 − 1.42834i −0.591637 − 1.42834i −0.881921 − 0.471397i \(-0.843750\pi\)
0.290285 − 0.956940i \(-0.406250\pi\)
\(4\) 0.980785 − 0.195090i 0.980785 − 0.195090i
\(5\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(6\) −0.728789 − 1.36347i −0.728789 − 1.36347i
\(7\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 0.956940 − 0.290285i 0.956940 − 0.290285i
\(9\) −0.983006 + 0.983006i −0.983006 + 0.983006i
\(10\) 0.956940 + 0.290285i 0.956940 + 0.290285i
\(11\) 0.636379 − 1.53636i 0.636379 − 1.53636i −0.195090 − 0.980785i \(-0.562500\pi\)
0.831470 − 0.555570i \(-0.187500\pi\)
\(12\) −0.858923 − 1.28547i −0.858923 − 1.28547i
\(13\) −0.871028 + 0.360791i −0.871028 + 0.360791i −0.773010 − 0.634393i \(-0.781250\pi\)
−0.0980171 + 0.995185i \(0.531250\pi\)
\(14\) 0 0
\(15\) − 1.54602i − 1.54602i
\(16\) 0.923880 − 0.382683i 0.923880 − 0.382683i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −0.881921 + 1.07462i −0.881921 + 1.07462i
\(19\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(20\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(21\) 0 0
\(22\) 0.482726 − 1.59133i 0.482726 − 1.59133i
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) −0.980785 − 1.19509i −0.980785 − 1.19509i
\(25\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(26\) −0.831470 + 0.444430i −0.831470 + 0.444430i
\(27\) 0.557309 + 0.230845i 0.557309 + 0.230845i
\(28\) 0 0
\(29\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(30\) −0.151537 − 1.53858i −0.151537 − 1.53858i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.881921 − 0.471397i 0.881921 − 0.471397i
\(33\) −2.57094 −2.57094
\(34\) 0 0
\(35\) 0 0
\(36\) −0.772343 + 1.15589i −0.772343 + 1.15589i
\(37\) 0.181112 + 0.0750191i 0.181112 + 0.0750191i 0.471397 − 0.881921i \(-0.343750\pi\)
−0.290285 + 0.956940i \(0.593750\pi\)
\(38\) −0.881921 + 0.471397i −0.881921 + 0.471397i
\(39\) 1.03066 + 1.03066i 1.03066 + 1.03066i
\(40\) 0.995185 + 0.0980171i 0.995185 + 0.0980171i
\(41\) 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
\(45\) −1.28436 + 0.531999i −1.28436 + 0.531999i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.09320 − 1.09320i −1.09320 − 1.09320i
\(49\) 1.00000i 1.00000i
\(50\) 0.773010 + 0.634393i 0.773010 + 0.634393i
\(51\) 0 0
\(52\) −0.783904 + 0.523788i −0.783904 + 0.523788i
\(53\) 0.485544 − 1.17221i 0.485544 − 1.17221i −0.471397 − 0.881921i \(-0.656250\pi\)
0.956940 − 0.290285i \(-0.0937500\pi\)
\(54\) 0.577253 + 0.175108i 0.577253 + 0.175108i
\(55\) 1.17588 − 1.17588i 1.17588 − 1.17588i
\(56\) 0 0
\(57\) 1.09320 + 1.09320i 1.09320 + 1.09320i
\(58\) 0 0
\(59\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(60\) −0.301614 − 1.51631i −0.301614 − 1.51631i
\(61\) 0.425215 + 1.02656i 0.425215 + 1.02656i 0.980785 + 0.195090i \(0.0625000\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.831470 − 0.555570i 0.831470 − 0.555570i
\(65\) −0.942793 −0.942793
\(66\) −2.55856 + 0.251996i −2.55856 + 0.251996i
\(67\) −0.222174 − 0.536376i −0.222174 − 0.536376i 0.773010 − 0.634393i \(-0.218750\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(72\) −0.655327 + 1.22603i −0.655327 + 1.22603i
\(73\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0.187593 + 0.0569057i 0.187593 + 0.0569057i
\(75\) 0.591637 − 1.42834i 0.591637 − 1.42834i
\(76\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(77\) 0 0
\(78\) 1.12672 + 0.924678i 1.12672 + 0.924678i
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 1.00000
\(81\) 0.457578i 0.457578i
\(82\) 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
\(88\) 0.162997 − 1.65493i 0.162997 − 1.65493i
\(89\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) −1.22603 + 0.655327i −1.22603 + 0.655327i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −1.00000
\(96\) −1.19509 − 0.980785i −1.19509 − 0.980785i
\(97\) −1.76384 −1.76384 −0.881921 − 0.471397i \(-0.843750\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(98\) 0.0980171 + 0.995185i 0.0980171 + 0.995185i
\(99\) 0.884682 + 2.13581i 0.884682 + 2.13581i
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.2469.8 yes 32
5.4 even 2 inner 3040.1.cn.a.2469.1 yes 32
19.18 odd 2 inner 3040.1.cn.a.2469.1 yes 32
32.13 even 8 inner 3040.1.cn.a.1709.8 yes 32
95.94 odd 2 CM 3040.1.cn.a.2469.8 yes 32
160.109 even 8 inner 3040.1.cn.a.1709.1 ✓ 32
608.493 odd 8 inner 3040.1.cn.a.1709.1 ✓ 32
3040.1709 odd 8 inner 3040.1.cn.a.1709.8 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.1.cn.a.1709.1 ✓ 32 160.109 even 8 inner
3040.1.cn.a.1709.1 ✓ 32 608.493 odd 8 inner
3040.1.cn.a.1709.8 yes 32 32.13 even 8 inner
3040.1.cn.a.1709.8 yes 32 3040.1709 odd 8 inner
3040.1.cn.a.2469.1 yes 32 5.4 even 2 inner
3040.1.cn.a.2469.1 yes 32 19.18 odd 2 inner
3040.1.cn.a.2469.8 yes 32 1.1 even 1 trivial
3040.1.cn.a.2469.8 yes 32 95.94 odd 2 CM