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Properties

Label 3040.1.cn.a.2469.6
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 2469.6
Root \(0.471397 - 0.881921i\) of defining polynomial
Character \(\chi\) \(=\) 3040.2469
Dual form 3040.1.cn.a.1709.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.634393 - 0.773010i) q^{2} +(0.761681 + 1.83886i) q^{3} +(-0.195090 - 0.980785i) q^{4} +(-0.923880 - 0.382683i) q^{5} +(1.90466 + 0.577774i) q^{6} +(-0.881921 - 0.471397i) q^{8} +(-2.09415 + 2.09415i) q^{9} +(-0.881921 + 0.471397i) q^{10} +(-0.425215 + 1.02656i) q^{11} +(1.65493 - 1.10579i) q^{12} +(-1.76820 + 0.732410i) q^{13} -1.99037i q^{15} +(-0.923880 + 0.382683i) q^{16} +(0.290285 + 2.94731i) q^{18} +(0.923880 - 0.382683i) q^{19} +(-0.195090 + 0.980785i) q^{20} +(0.523788 + 0.979938i) q^{22} +(0.195090 - 1.98079i) q^{24} +(0.707107 + 0.707107i) q^{25} +(-0.555570 + 1.83147i) q^{26} +(-3.60706 - 1.49409i) q^{27} +(-1.53858 - 1.26268i) q^{30} +(-0.290285 + 0.956940i) q^{32} -2.21158 q^{33} +(2.46246 + 1.64536i) q^{36} +(-1.42834 - 0.591637i) q^{37} +(0.290285 - 0.956940i) q^{38} +(-2.69360 - 2.69360i) q^{39} +(0.634393 + 0.773010i) q^{40} +(1.08979 + 0.216773i) q^{44} +(2.73613 - 1.13334i) q^{45} +(-1.40740 - 1.40740i) q^{48} +1.00000i q^{49} +(0.995185 - 0.0980171i) q^{50} +(1.06330 + 1.59133i) q^{52} +(0.0750191 - 0.181112i) q^{53} +(-3.44324 + 1.84045i) q^{54} +(0.785695 - 0.785695i) q^{55} +(1.40740 + 1.40740i) q^{57} +(-1.95213 + 0.388302i) q^{60} +(0.636379 + 1.53636i) q^{61} +(0.555570 + 0.831470i) q^{64} +1.91388 q^{65} +(-1.40301 + 1.70957i) q^{66} +(0.360791 + 0.871028i) q^{67} +(2.83405 - 0.859699i) q^{72} +(-1.36347 + 0.728789i) q^{74} +(-0.761681 + 1.83886i) q^{75} +(-0.555570 - 0.831470i) q^{76} +(-3.79099 + 0.373380i) q^{78} +1.00000 q^{80} -4.80933i q^{81} +(0.858923 - 0.704900i) q^{88} +(0.859699 - 2.83405i) q^{90} -1.00000 q^{95} +(-1.98079 + 0.195090i) q^{96} +0.580569 q^{97} +(0.773010 + 0.634393i) q^{98} +(-1.25930 - 3.04023i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 32 q^{66} + 32 q^{80} - 32 q^{95} - 32 q^{96} - 32 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.634393 − 0.773010i 0.634393 − 0.773010i
\(3\) 0.761681 + 1.83886i 0.761681 + 1.83886i 0.471397 + 0.881921i \(0.343750\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(4\) −0.195090 − 0.980785i −0.195090 − 0.980785i
\(5\) −0.923880 − 0.382683i −0.923880 − 0.382683i
\(6\) 1.90466 + 0.577774i 1.90466 + 0.577774i
\(7\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) −0.881921 − 0.471397i −0.881921 − 0.471397i
\(9\) −2.09415 + 2.09415i −2.09415 + 2.09415i
\(10\) −0.881921 + 0.471397i −0.881921 + 0.471397i
\(11\) −0.425215 + 1.02656i −0.425215 + 1.02656i 0.555570 + 0.831470i \(0.312500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(12\) 1.65493 − 1.10579i 1.65493 − 1.10579i
\(13\) −1.76820 + 0.732410i −1.76820 + 0.732410i −0.773010 + 0.634393i \(0.781250\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(14\) 0 0
\(15\) − 1.99037i − 1.99037i
\(16\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0.290285 + 2.94731i 0.290285 + 2.94731i
\(19\) 0.923880 − 0.382683i 0.923880 − 0.382683i
\(20\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(21\) 0 0
\(22\) 0.523788 + 0.979938i 0.523788 + 0.979938i
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0.195090 − 1.98079i 0.195090 − 1.98079i
\(25\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(26\) −0.555570 + 1.83147i −0.555570 + 1.83147i
\(27\) −3.60706 − 1.49409i −3.60706 − 1.49409i
\(28\) 0 0
\(29\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(30\) −1.53858 − 1.26268i −1.53858 − 1.26268i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −0.290285 + 0.956940i −0.290285 + 0.956940i
\(33\) −2.21158 −2.21158
\(34\) 0 0
\(35\) 0 0
\(36\) 2.46246 + 1.64536i 2.46246 + 1.64536i
\(37\) −1.42834 − 0.591637i −1.42834 − 0.591637i −0.471397 − 0.881921i \(-0.656250\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(38\) 0.290285 − 0.956940i 0.290285 − 0.956940i
\(39\) −2.69360 − 2.69360i −2.69360 − 2.69360i
\(40\) 0.634393 + 0.773010i 0.634393 + 0.773010i
\(41\) 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 \(0.625000\pi\)
\(44\) 1.08979 + 0.216773i 1.08979 + 0.216773i
\(45\) 2.73613 − 1.13334i 2.73613 − 1.13334i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.40740 − 1.40740i −1.40740 − 1.40740i
\(49\) 1.00000i 1.00000i
\(50\) 0.995185 − 0.0980171i 0.995185 − 0.0980171i
\(51\) 0 0
\(52\) 1.06330 + 1.59133i 1.06330 + 1.59133i
\(53\) 0.0750191 − 0.181112i 0.0750191 − 0.181112i −0.881921 − 0.471397i \(-0.843750\pi\)
0.956940 + 0.290285i \(0.0937500\pi\)
\(54\) −3.44324 + 1.84045i −3.44324 + 1.84045i
\(55\) 0.785695 − 0.785695i 0.785695 − 0.785695i
\(56\) 0 0
\(57\) 1.40740 + 1.40740i 1.40740 + 1.40740i
\(58\) 0 0
\(59\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(60\) −1.95213 + 0.388302i −1.95213 + 0.388302i
\(61\) 0.636379 + 1.53636i 0.636379 + 1.53636i 0.831470 + 0.555570i \(0.187500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(65\) 1.91388 1.91388
\(66\) −1.40301 + 1.70957i −1.40301 + 1.70957i
\(67\) 0.360791 + 0.871028i 0.360791 + 0.871028i 0.995185 + 0.0980171i \(0.0312500\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(72\) 2.83405 − 0.859699i 2.83405 − 0.859699i
\(73\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) −1.36347 + 0.728789i −1.36347 + 0.728789i
\(75\) −0.761681 + 1.83886i −0.761681 + 1.83886i
\(76\) −0.555570 − 0.831470i −0.555570 − 0.831470i
\(77\) 0 0
\(78\) −3.79099 + 0.373380i −3.79099 + 0.373380i
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 1.00000
\(81\) − 4.80933i − 4.80933i
\(82\) 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
\(88\) 0.858923 − 0.704900i 0.858923 − 0.704900i
\(89\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) 0.859699 − 2.83405i 0.859699 − 2.83405i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −1.00000
\(96\) −1.98079 + 0.195090i −1.98079 + 0.195090i
\(97\) 0.580569 0.580569 0.290285 − 0.956940i \(-0.406250\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(98\) 0.773010 + 0.634393i 0.773010 + 0.634393i
\(99\) −1.25930 − 3.04023i −1.25930 − 3.04023i
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.2469.6 yes 32
5.4 even 2 inner 3040.1.cn.a.2469.3 yes 32
19.18 odd 2 inner 3040.1.cn.a.2469.3 yes 32
32.13 even 8 inner 3040.1.cn.a.1709.6 yes 32
95.94 odd 2 CM 3040.1.cn.a.2469.6 yes 32
160.109 even 8 inner 3040.1.cn.a.1709.3 ✓ 32
608.493 odd 8 inner 3040.1.cn.a.1709.3 ✓ 32
3040.1709 odd 8 inner 3040.1.cn.a.1709.6 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.1.cn.a.1709.3 ✓ 32 160.109 even 8 inner
3040.1.cn.a.1709.3 ✓ 32 608.493 odd 8 inner
3040.1.cn.a.1709.6 yes 32 32.13 even 8 inner
3040.1.cn.a.1709.6 yes 32 3040.1709 odd 8 inner
3040.1.cn.a.2469.3 yes 32 5.4 even 2 inner
3040.1.cn.a.2469.3 yes 32 19.18 odd 2 inner
3040.1.cn.a.2469.6 yes 32 1.1 even 1 trivial
3040.1.cn.a.2469.6 yes 32 95.94 odd 2 CM