Properties

Label 3040.1.cn.a.189.4
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 189.4
Root \(0.634393 - 0.773010i\) of defining polynomial
Character \(\chi\) \(=\) 3040.189
Dual form 3040.1.cn.a.949.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.290285 - 0.956940i) q^{2} +(1.62958 + 0.674993i) q^{3} +(-0.831470 + 0.555570i) q^{4} +(0.382683 + 0.923880i) q^{5} +(0.172887 - 1.75535i) q^{6} +(0.773010 + 0.634393i) q^{8} +(1.49280 + 1.49280i) q^{9} +(0.773010 - 0.634393i) q^{10} +(-0.360480 + 0.149316i) q^{11} +(-1.72995 + 0.344109i) q^{12} +(0.0750191 - 0.181112i) q^{13} +1.76384i q^{15} +(0.382683 - 0.923880i) q^{16} +(0.995185 - 1.86186i) q^{18} +(-0.382683 + 0.923880i) q^{19} +(-0.831470 - 0.555570i) q^{20} +(0.247528 + 0.301614i) q^{22} +(0.831470 + 1.55557i) q^{24} +(-0.707107 + 0.707107i) q^{25} +(-0.195090 - 0.0192147i) q^{26} +(0.750012 + 1.81069i) q^{27} +(1.68789 - 0.512016i) q^{30} +(-0.995185 - 0.0980171i) q^{32} -0.688217 q^{33} +(-2.07058 - 0.411863i) q^{36} +(-0.732410 - 1.76820i) q^{37} +(0.995185 + 0.0980171i) q^{38} +(0.244499 - 0.244499i) q^{39} +(-0.290285 + 0.956940i) q^{40} +(0.216773 - 0.324423i) q^{44} +(-0.807898 + 1.95044i) q^{45} +(1.24723 - 1.24723i) q^{48} -1.00000i q^{49} +(0.881921 + 0.471397i) q^{50} +(0.0382444 + 0.192268i) q^{52} +(0.871028 - 0.360791i) q^{53} +(1.51501 - 1.24333i) q^{54} +(-0.275899 - 0.275899i) q^{55} +(-1.24723 + 1.24723i) q^{57} +(-0.979938 - 1.46658i) q^{60} +(-1.81225 - 0.750661i) q^{61} +(0.195090 + 0.980785i) q^{64} +0.196034 q^{65} +(0.199779 + 0.658583i) q^{66} +(1.17221 + 0.485544i) q^{67} +(0.206928 + 2.10097i) q^{72} +(-1.47945 + 1.21415i) q^{74} +(-1.62958 + 0.674993i) q^{75} +(-0.195090 - 0.980785i) q^{76} +(-0.304945 - 0.162997i) q^{78} +1.00000 q^{80} +1.34577i q^{81} +(-0.373380 - 0.113263i) q^{88} +(2.10097 + 0.206928i) q^{90} -1.00000 q^{95} +(-1.55557 - 0.831470i) q^{96} +1.99037 q^{97} +(-0.956940 + 0.290285i) q^{98} +(-0.761024 - 0.315226i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 32 q^{66} + 32 q^{80} - 32 q^{95} - 32 q^{96} - 32 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.290285 − 0.956940i −0.290285 − 0.956940i
\(3\) 1.62958 + 0.674993i 1.62958 + 0.674993i 0.995185 − 0.0980171i \(-0.0312500\pi\)
0.634393 + 0.773010i \(0.281250\pi\)
\(4\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(5\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(6\) 0.172887 − 1.75535i 0.172887 − 1.75535i
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0.773010 + 0.634393i 0.773010 + 0.634393i
\(9\) 1.49280 + 1.49280i 1.49280 + 1.49280i
\(10\) 0.773010 − 0.634393i 0.773010 − 0.634393i
\(11\) −0.360480 + 0.149316i −0.360480 + 0.149316i −0.555570 − 0.831470i \(-0.687500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(12\) −1.72995 + 0.344109i −1.72995 + 0.344109i
\(13\) 0.0750191 − 0.181112i 0.0750191 − 0.181112i −0.881921 − 0.471397i \(-0.843750\pi\)
0.956940 + 0.290285i \(0.0937500\pi\)
\(14\) 0 0
\(15\) 1.76384i 1.76384i
\(16\) 0.382683 − 0.923880i 0.382683 − 0.923880i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0.995185 − 1.86186i 0.995185 − 1.86186i
\(19\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(20\) −0.831470 − 0.555570i −0.831470 − 0.555570i
\(21\) 0 0
\(22\) 0.247528 + 0.301614i 0.247528 + 0.301614i
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0.831470 + 1.55557i 0.831470 + 1.55557i
\(25\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(26\) −0.195090 − 0.0192147i −0.195090 − 0.0192147i
\(27\) 0.750012 + 1.81069i 0.750012 + 1.81069i
\(28\) 0 0
\(29\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(30\) 1.68789 − 0.512016i 1.68789 − 0.512016i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −0.995185 − 0.0980171i −0.995185 − 0.0980171i
\(33\) −0.688217 −0.688217
\(34\) 0 0
\(35\) 0 0
\(36\) −2.07058 − 0.411863i −2.07058 − 0.411863i
\(37\) −0.732410 − 1.76820i −0.732410 − 1.76820i −0.634393 − 0.773010i \(-0.718750\pi\)
−0.0980171 − 0.995185i \(-0.531250\pi\)
\(38\) 0.995185 + 0.0980171i 0.995185 + 0.0980171i
\(39\) 0.244499 − 0.244499i 0.244499 − 0.244499i
\(40\) −0.290285 + 0.956940i −0.290285 + 0.956940i
\(41\) 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 \(0.875000\pi\)
\(44\) 0.216773 − 0.324423i 0.216773 − 0.324423i
\(45\) −0.807898 + 1.95044i −0.807898 + 1.95044i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.24723 − 1.24723i 1.24723 − 1.24723i
\(49\) − 1.00000i − 1.00000i
\(50\) 0.881921 + 0.471397i 0.881921 + 0.471397i
\(51\) 0 0
\(52\) 0.0382444 + 0.192268i 0.0382444 + 0.192268i
\(53\) 0.871028 − 0.360791i 0.871028 − 0.360791i 0.0980171 − 0.995185i \(-0.468750\pi\)
0.773010 + 0.634393i \(0.218750\pi\)
\(54\) 1.51501 − 1.24333i 1.51501 − 1.24333i
\(55\) −0.275899 − 0.275899i −0.275899 − 0.275899i
\(56\) 0 0
\(57\) −1.24723 + 1.24723i −1.24723 + 1.24723i
\(58\) 0 0
\(59\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(60\) −0.979938 − 1.46658i −0.979938 − 1.46658i
\(61\) −1.81225 − 0.750661i −1.81225 − 0.750661i −0.980785 − 0.195090i \(-0.937500\pi\)
−0.831470 − 0.555570i \(-0.812500\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(65\) 0.196034 0.196034
\(66\) 0.199779 + 0.658583i 0.199779 + 0.658583i
\(67\) 1.17221 + 0.485544i 1.17221 + 0.485544i 0.881921 − 0.471397i \(-0.156250\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(72\) 0.206928 + 2.10097i 0.206928 + 2.10097i
\(73\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) −1.47945 + 1.21415i −1.47945 + 1.21415i
\(75\) −1.62958 + 0.674993i −1.62958 + 0.674993i
\(76\) −0.195090 − 0.980785i −0.195090 − 0.980785i
\(77\) 0 0
\(78\) −0.304945 − 0.162997i −0.304945 − 0.162997i
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 1.00000
\(81\) 1.34577i 1.34577i
\(82\) 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
\(88\) −0.373380 − 0.113263i −0.373380 − 0.113263i
\(89\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) 2.10097 + 0.206928i 2.10097 + 0.206928i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −1.00000
\(96\) −1.55557 − 0.831470i −1.55557 − 0.831470i
\(97\) 1.99037 1.99037 0.995185 − 0.0980171i \(-0.0312500\pi\)
0.995185 + 0.0980171i \(0.0312500\pi\)
\(98\) −0.956940 + 0.290285i −0.956940 + 0.290285i
\(99\) −0.761024 − 0.315226i −0.761024 − 0.315226i
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.189.4 ✓ 32
5.4 even 2 inner 3040.1.cn.a.189.5 yes 32
19.18 odd 2 inner 3040.1.cn.a.189.5 yes 32
32.21 even 8 inner 3040.1.cn.a.949.4 yes 32
95.94 odd 2 CM 3040.1.cn.a.189.4 ✓ 32
160.149 even 8 inner 3040.1.cn.a.949.5 yes 32
608.341 odd 8 inner 3040.1.cn.a.949.5 yes 32
3040.949 odd 8 inner 3040.1.cn.a.949.4 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.1.cn.a.189.4 ✓ 32 1.1 even 1 trivial
3040.1.cn.a.189.4 ✓ 32 95.94 odd 2 CM
3040.1.cn.a.189.5 yes 32 5.4 even 2 inner
3040.1.cn.a.189.5 yes 32 19.18 odd 2 inner
3040.1.cn.a.949.4 yes 32 32.21 even 8 inner
3040.1.cn.a.949.4 yes 32 3040.949 odd 8 inner
3040.1.cn.a.949.5 yes 32 160.149 even 8 inner
3040.1.cn.a.949.5 yes 32 608.341 odd 8 inner