Properties

Label 3040.1.cn.a.949.1
Level $3040$
Weight $1$
Character 3040.949
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 949.1
Root \(0.773010 - 0.634393i\) of defining polynomial
Character \(\chi\) \(=\) 3040.949
Dual form 3040.1.cn.a.189.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.956940 - 0.290285i) q^{2} +(0.871028 - 0.360791i) q^{3} +(0.831470 + 0.555570i) q^{4} +(0.382683 - 0.923880i) q^{5} +(-0.938254 + 0.0924099i) q^{6} +(-0.634393 - 0.773010i) q^{8} +(-0.0785882 + 0.0785882i) q^{9} +(-0.634393 + 0.773010i) q^{10} +(0.360480 + 0.149316i) q^{11} +(0.924678 + 0.183930i) q^{12} +(-0.761681 - 1.83886i) q^{13} -0.942793i q^{15} +(0.382683 + 0.923880i) q^{16} +(0.0980171 - 0.0523913i) q^{18} +(-0.382683 - 0.923880i) q^{19} +(0.831470 - 0.555570i) q^{20} +(-0.301614 - 0.247528i) q^{22} +(-0.831470 - 0.444430i) q^{24} +(-0.707107 - 0.707107i) q^{25} +(0.195090 + 1.98079i) q^{26} +(-0.400890 + 0.967834i) q^{27} +(-0.273678 + 0.902197i) q^{30} +(-0.0980171 - 0.995185i) q^{32} +0.367860 q^{33} +(-0.109005 + 0.0216824i) q^{36} +(0.222174 - 0.536376i) q^{37} +(0.0980171 + 0.995185i) q^{38} +(-1.32689 - 1.32689i) q^{39} +(-0.956940 + 0.290285i) q^{40} +(0.216773 + 0.324423i) q^{44} +(0.0425316 + 0.102680i) q^{45} +(0.666656 + 0.666656i) q^{48} +1.00000i q^{49} +(0.471397 + 0.881921i) q^{50} +(0.388302 - 1.95213i) q^{52} +(-1.62958 - 0.674993i) q^{53} +(0.664575 - 0.809787i) q^{54} +(0.275899 - 0.275899i) q^{55} +(-0.666656 - 0.666656i) q^{57} +(0.523788 - 0.783904i) q^{60} +(1.81225 - 0.750661i) q^{61} +(-0.195090 + 0.980785i) q^{64} -1.99037 q^{65} +(-0.352020 - 0.106784i) q^{66} +(1.42834 - 0.591637i) q^{67} +(0.110605 + 0.0108937i) q^{72} +(-0.368309 + 0.448786i) q^{74} +(-0.871028 - 0.360791i) q^{75} +(0.195090 - 0.980785i) q^{76} +(0.884579 + 1.65493i) q^{78} +1.00000 q^{80} +0.876507i q^{81} +(-0.113263 - 0.373380i) q^{88} +(-0.0108937 - 0.110605i) q^{90} -1.00000 q^{95} +(-0.444430 - 0.831470i) q^{96} +0.196034 q^{97} +(0.290285 - 0.956940i) q^{98} +(-0.0400639 + 0.0165950i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 32 q^{66} + 32 q^{80} - 32 q^{95} - 32 q^{96} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(1217\) \(1921\) \(2661\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{5}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.956940 − 0.290285i −0.956940 − 0.290285i
\(3\) 0.871028 − 0.360791i 0.871028 − 0.360791i 0.0980171 − 0.995185i \(-0.468750\pi\)
0.773010 + 0.634393i \(0.218750\pi\)
\(4\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(5\) 0.382683 − 0.923880i 0.382683 − 0.923880i
\(6\) −0.938254 + 0.0924099i −0.938254 + 0.0924099i
\(7\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) −0.634393 − 0.773010i −0.634393 − 0.773010i
\(9\) −0.0785882 + 0.0785882i −0.0785882 + 0.0785882i
\(10\) −0.634393 + 0.773010i −0.634393 + 0.773010i
\(11\) 0.360480 + 0.149316i 0.360480 + 0.149316i 0.555570 − 0.831470i \(-0.312500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(12\) 0.924678 + 0.183930i 0.924678 + 0.183930i
\(13\) −0.761681 − 1.83886i −0.761681 − 1.83886i −0.471397 − 0.881921i \(-0.656250\pi\)
−0.290285 − 0.956940i \(-0.593750\pi\)
\(14\) 0 0
\(15\) − 0.942793i − 0.942793i
\(16\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0.0980171 − 0.0523913i 0.0980171 − 0.0523913i
\(19\) −0.382683 − 0.923880i −0.382683 − 0.923880i
\(20\) 0.831470 − 0.555570i 0.831470 − 0.555570i
\(21\) 0 0
\(22\) −0.301614 − 0.247528i −0.301614 − 0.247528i
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) −0.831470 − 0.444430i −0.831470 − 0.444430i
\(25\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(26\) 0.195090 + 1.98079i 0.195090 + 1.98079i
\(27\) −0.400890 + 0.967834i −0.400890 + 0.967834i
\(28\) 0 0
\(29\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(30\) −0.273678 + 0.902197i −0.273678 + 0.902197i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −0.0980171 − 0.995185i −0.0980171 − 0.995185i
\(33\) 0.367860 0.367860
\(34\) 0 0
\(35\) 0 0
\(36\) −0.109005 + 0.0216824i −0.109005 + 0.0216824i
\(37\) 0.222174 − 0.536376i 0.222174 − 0.536376i −0.773010 − 0.634393i \(-0.781250\pi\)
0.995185 + 0.0980171i \(0.0312500\pi\)
\(38\) 0.0980171 + 0.995185i 0.0980171 + 0.995185i
\(39\) −1.32689 − 1.32689i −1.32689 − 1.32689i
\(40\) −0.956940 + 0.290285i −0.956940 + 0.290285i
\(41\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(44\) 0.216773 + 0.324423i 0.216773 + 0.324423i
\(45\) 0.0425316 + 0.102680i 0.0425316 + 0.102680i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0.666656 + 0.666656i 0.666656 + 0.666656i
\(49\) 1.00000i 1.00000i
\(50\) 0.471397 + 0.881921i 0.471397 + 0.881921i
\(51\) 0 0
\(52\) 0.388302 − 1.95213i 0.388302 − 1.95213i
\(53\) −1.62958 − 0.674993i −1.62958 − 0.674993i −0.634393 − 0.773010i \(-0.718750\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(54\) 0.664575 − 0.809787i 0.664575 − 0.809787i
\(55\) 0.275899 − 0.275899i 0.275899 − 0.275899i
\(56\) 0 0
\(57\) −0.666656 − 0.666656i −0.666656 − 0.666656i
\(58\) 0 0
\(59\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(60\) 0.523788 − 0.783904i 0.523788 − 0.783904i
\(61\) 1.81225 − 0.750661i 1.81225 − 0.750661i 0.831470 − 0.555570i \(-0.187500\pi\)
0.980785 − 0.195090i \(-0.0625000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(65\) −1.99037 −1.99037
\(66\) −0.352020 − 0.106784i −0.352020 − 0.106784i
\(67\) 1.42834 − 0.591637i 1.42834 − 0.591637i 0.471397 − 0.881921i \(-0.343750\pi\)
0.956940 + 0.290285i \(0.0937500\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(72\) 0.110605 + 0.0108937i 0.110605 + 0.0108937i
\(73\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) −0.368309 + 0.448786i −0.368309 + 0.448786i
\(75\) −0.871028 − 0.360791i −0.871028 − 0.360791i
\(76\) 0.195090 − 0.980785i 0.195090 − 0.980785i
\(77\) 0 0
\(78\) 0.884579 + 1.65493i 0.884579 + 1.65493i
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 1.00000
\(81\) 0.876507i 0.876507i
\(82\) 0 0
\(83\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −0.113263 − 0.373380i −0.113263 − 0.373380i
\(89\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) −0.0108937 − 0.110605i −0.0108937 − 0.110605i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −1.00000
\(96\) −0.444430 − 0.831470i −0.444430 − 0.831470i
\(97\) 0.196034 0.196034 0.0980171 − 0.995185i \(-0.468750\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(98\) 0.290285 − 0.956940i 0.290285 − 0.956940i
\(99\) −0.0400639 + 0.0165950i −0.0400639 + 0.0165950i
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.949.1 yes 32
5.4 even 2 inner 3040.1.cn.a.949.8 yes 32
19.18 odd 2 inner 3040.1.cn.a.949.8 yes 32
32.29 even 8 inner 3040.1.cn.a.189.1 ✓ 32
95.94 odd 2 CM 3040.1.cn.a.949.1 yes 32
160.29 even 8 inner 3040.1.cn.a.189.8 yes 32
608.189 odd 8 inner 3040.1.cn.a.189.8 yes 32
3040.189 odd 8 inner 3040.1.cn.a.189.1 ✓ 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.1.cn.a.189.1 ✓ 32 32.29 even 8 inner
3040.1.cn.a.189.1 ✓ 32 3040.189 odd 8 inner
3040.1.cn.a.189.8 yes 32 160.29 even 8 inner
3040.1.cn.a.189.8 yes 32 608.189 odd 8 inner
3040.1.cn.a.949.1 yes 32 1.1 even 1 trivial
3040.1.cn.a.949.1 yes 32 95.94 odd 2 CM
3040.1.cn.a.949.8 yes 32 5.4 even 2 inner
3040.1.cn.a.949.8 yes 32 19.18 odd 2 inner