Properties

Label 3040.1.cn.a.189.7
Level $3040$
Weight $1$
Character 3040.189
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 189.7
Root \(0.995185 - 0.0980171i\) of defining polynomial
Character \(\chi\) \(=\) 3040.189
Dual form 3040.1.cn.a.949.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.881921 + 0.471397i) q^{2} +(1.76820 + 0.732410i) q^{3} +(0.555570 + 0.831470i) q^{4} +(-0.382683 - 0.923880i) q^{5} +(1.21415 + 1.47945i) q^{6} +(0.0980171 + 0.995185i) q^{8} +(1.88298 + 1.88298i) q^{9} +(0.0980171 - 0.995185i) q^{10} +(-1.81225 + 0.750661i) q^{11} +(0.373380 + 1.87711i) q^{12} +(0.485544 - 1.17221i) q^{13} -1.91388i q^{15} +(-0.382683 + 0.923880i) q^{16} +(0.773010 + 2.54827i) q^{18} +(0.382683 - 0.923880i) q^{19} +(0.555570 - 0.831470i) q^{20} +(-1.95213 - 0.192268i) q^{22} +(-0.555570 + 1.83147i) q^{24} +(-0.707107 + 0.707107i) q^{25} +(0.980785 - 0.804910i) q^{26} +(1.21795 + 2.94040i) q^{27} +(0.902197 - 1.68789i) q^{30} +(-0.773010 + 0.634393i) q^{32} -3.75421 q^{33} +(-0.519514 + 2.61177i) q^{36} +(-0.360791 - 0.871028i) q^{37} +(0.773010 - 0.634393i) q^{38} +(1.71707 - 1.71707i) q^{39} +(0.881921 - 0.471397i) q^{40} +(-1.63099 - 1.08979i) q^{44} +(1.01906 - 2.46024i) q^{45} +(-1.35332 + 1.35332i) q^{48} -1.00000i q^{49} +(-0.956940 + 0.290285i) q^{50} +(1.24441 - 0.247528i) q^{52} +(-0.536376 + 0.222174i) q^{53} +(-0.311956 + 3.16734i) q^{54} +(1.38704 + 1.38704i) q^{55} +(1.35332 - 1.35332i) q^{57} +(1.59133 - 1.06330i) q^{60} +(0.360480 + 0.149316i) q^{61} +(-0.980785 + 0.195090i) q^{64} -1.26879 q^{65} +(-3.31092 - 1.76972i) q^{66} +(-1.83886 - 0.761681i) q^{67} +(-1.68935 + 2.05848i) q^{72} +(0.0924099 - 0.938254i) q^{74} +(-1.76820 + 0.732410i) q^{75} +(0.980785 - 0.195090i) q^{76} +(2.32374 - 0.704900i) q^{78} +1.00000 q^{80} +3.42831i q^{81} +(-0.924678 - 1.72995i) q^{88} +(2.05848 - 1.68935i) q^{90} -1.00000 q^{95} +(-1.83147 + 0.555570i) q^{96} +1.54602 q^{97} +(0.471397 - 0.881921i) q^{98} +(-4.82592 - 1.99896i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 32 q^{66} + 32 q^{80} - 32 q^{95} - 32 q^{96} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(1217\) \(1921\) \(2661\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{3}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.881921 + 0.471397i 0.881921 + 0.471397i
\(3\) 1.76820 + 0.732410i 1.76820 + 0.732410i 0.995185 + 0.0980171i \(0.0312500\pi\)
0.773010 + 0.634393i \(0.218750\pi\)
\(4\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(5\) −0.382683 − 0.923880i −0.382683 − 0.923880i
\(6\) 1.21415 + 1.47945i 1.21415 + 1.47945i
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0.0980171 + 0.995185i 0.0980171 + 0.995185i
\(9\) 1.88298 + 1.88298i 1.88298 + 1.88298i
\(10\) 0.0980171 − 0.995185i 0.0980171 − 0.995185i
\(11\) −1.81225 + 0.750661i −1.81225 + 0.750661i −0.831470 + 0.555570i \(0.812500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(12\) 0.373380 + 1.87711i 0.373380 + 1.87711i
\(13\) 0.485544 − 1.17221i 0.485544 − 1.17221i −0.471397 − 0.881921i \(-0.656250\pi\)
0.956940 − 0.290285i \(-0.0937500\pi\)
\(14\) 0 0
\(15\) − 1.91388i − 1.91388i
\(16\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0.773010 + 2.54827i 0.773010 + 2.54827i
\(19\) 0.382683 − 0.923880i 0.382683 − 0.923880i
\(20\) 0.555570 − 0.831470i 0.555570 − 0.831470i
\(21\) 0 0
\(22\) −1.95213 − 0.192268i −1.95213 − 0.192268i
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) −0.555570 + 1.83147i −0.555570 + 1.83147i
\(25\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(26\) 0.980785 − 0.804910i 0.980785 − 0.804910i
\(27\) 1.21795 + 2.94040i 1.21795 + 2.94040i
\(28\) 0 0
\(29\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(30\) 0.902197 − 1.68789i 0.902197 − 1.68789i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −0.773010 + 0.634393i −0.773010 + 0.634393i
\(33\) −3.75421 −3.75421
\(34\) 0 0
\(35\) 0 0
\(36\) −0.519514 + 2.61177i −0.519514 + 2.61177i
\(37\) −0.360791 − 0.871028i −0.360791 − 0.871028i −0.995185 − 0.0980171i \(-0.968750\pi\)
0.634393 − 0.773010i \(-0.281250\pi\)
\(38\) 0.773010 − 0.634393i 0.773010 − 0.634393i
\(39\) 1.71707 − 1.71707i 1.71707 − 1.71707i
\(40\) 0.881921 − 0.471397i 0.881921 − 0.471397i
\(41\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(42\) 0 0
\(43\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(44\) −1.63099 − 1.08979i −1.63099 − 1.08979i
\(45\) 1.01906 − 2.46024i 1.01906 − 2.46024i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.35332 + 1.35332i −1.35332 + 1.35332i
\(49\) − 1.00000i − 1.00000i
\(50\) −0.956940 + 0.290285i −0.956940 + 0.290285i
\(51\) 0 0
\(52\) 1.24441 − 0.247528i 1.24441 − 0.247528i
\(53\) −0.536376 + 0.222174i −0.536376 + 0.222174i −0.634393 − 0.773010i \(-0.718750\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(54\) −0.311956 + 3.16734i −0.311956 + 3.16734i
\(55\) 1.38704 + 1.38704i 1.38704 + 1.38704i
\(56\) 0 0
\(57\) 1.35332 − 1.35332i 1.35332 − 1.35332i
\(58\) 0 0
\(59\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(60\) 1.59133 − 1.06330i 1.59133 − 1.06330i
\(61\) 0.360480 + 0.149316i 0.360480 + 0.149316i 0.555570 − 0.831470i \(-0.312500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(65\) −1.26879 −1.26879
\(66\) −3.31092 − 1.76972i −3.31092 − 1.76972i
\(67\) −1.83886 − 0.761681i −1.83886 − 0.761681i −0.956940 − 0.290285i \(-0.906250\pi\)
−0.881921 − 0.471397i \(-0.843750\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(72\) −1.68935 + 2.05848i −1.68935 + 2.05848i
\(73\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0.0924099 − 0.938254i 0.0924099 − 0.938254i
\(75\) −1.76820 + 0.732410i −1.76820 + 0.732410i
\(76\) 0.980785 − 0.195090i 0.980785 − 0.195090i
\(77\) 0 0
\(78\) 2.32374 − 0.704900i 2.32374 − 0.704900i
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 1.00000
\(81\) 3.42831i 3.42831i
\(82\) 0 0
\(83\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −0.924678 − 1.72995i −0.924678 − 1.72995i
\(89\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) 2.05848 − 1.68935i 2.05848 − 1.68935i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −1.00000
\(96\) −1.83147 + 0.555570i −1.83147 + 0.555570i
\(97\) 1.54602 1.54602 0.773010 − 0.634393i \(-0.218750\pi\)
0.773010 + 0.634393i \(0.218750\pi\)
\(98\) 0.471397 − 0.881921i 0.471397 − 0.881921i
\(99\) −4.82592 − 1.99896i −4.82592 − 1.99896i
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.189.7 yes 32
5.4 even 2 inner 3040.1.cn.a.189.2 ✓ 32
19.18 odd 2 inner 3040.1.cn.a.189.2 ✓ 32
32.21 even 8 inner 3040.1.cn.a.949.7 yes 32
95.94 odd 2 CM 3040.1.cn.a.189.7 yes 32
160.149 even 8 inner 3040.1.cn.a.949.2 yes 32
608.341 odd 8 inner 3040.1.cn.a.949.2 yes 32
3040.949 odd 8 inner 3040.1.cn.a.949.7 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.1.cn.a.189.2 ✓ 32 5.4 even 2 inner
3040.1.cn.a.189.2 ✓ 32 19.18 odd 2 inner
3040.1.cn.a.189.7 yes 32 1.1 even 1 trivial
3040.1.cn.a.189.7 yes 32 95.94 odd 2 CM
3040.1.cn.a.949.2 yes 32 160.149 even 8 inner
3040.1.cn.a.949.2 yes 32 608.341 odd 8 inner
3040.1.cn.a.949.7 yes 32 32.21 even 8 inner
3040.1.cn.a.949.7 yes 32 3040.949 odd 8 inner