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Properties

Label 3040.1.cn.a
Level $3040$
Weight $1$
Character orbit 3040.cn
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 / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3040,1,Mod(189,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 3, 4, 4]))
 
N = Newforms(chi, 1, names="a")
 
//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);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
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: \(\Q(\zeta_{64})\)
comment: defining polynomial
 
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 \(\mathbb{Q}[x]/(x^{32} + \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{64}^{27} q^{2} + ( - \zeta_{64}^{31} - \zeta_{64}^{25}) q^{3} - \zeta_{64}^{22} q^{4} + \zeta_{64}^{20} q^{5} + ( - \zeta_{64}^{26} - \zeta_{64}^{20}) q^{6} - \zeta_{64}^{17} q^{8} + ( - \zeta_{64}^{30} + \cdots - \zeta_{64}^{18}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{64}^{27} q^{2} + ( - \zeta_{64}^{31} - \zeta_{64}^{25}) q^{3} - \zeta_{64}^{22} q^{4} + \zeta_{64}^{20} q^{5} + ( - \zeta_{64}^{26} - \zeta_{64}^{20}) q^{6} - \zeta_{64}^{17} q^{8} + ( - \zeta_{64}^{30} + \cdots - \zeta_{64}^{18}) q^{9} + \cdots + (\zeta_{64}^{30} + \zeta_{64}^{26} + \cdots - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 32 q^{66} + 32 q^{80} - 32 q^{95} - 32 q^{96} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(191\) \(1217\) \(1921\) \(2661\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\zeta_{64}^{8}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
189.1
0.773010 + 0.634393i
−0.995185 + 0.0980171i
−0.0980171 − 0.995185i
0.634393 − 0.773010i
−0.634393 + 0.773010i
0.0980171 + 0.995185i
0.995185 − 0.0980171i
−0.773010 − 0.634393i
0.773010 − 0.634393i
−0.995185 − 0.0980171i
−0.0980171 + 0.995185i
0.634393 + 0.773010i
−0.634393 − 0.773010i
0.0980171 − 0.995185i
0.995185 + 0.0980171i
−0.773010 + 0.634393i
−0.290285 + 0.956940i
0.881921 − 0.471397i
−0.471397 − 0.881921i
−0.956940 − 0.290285i
−0.956940 + 0.290285i 0.871028 + 0.360791i 0.831470 − 0.555570i 0.382683 + 0.923880i −0.938254 − 0.0924099i 0 −0.634393 + 0.773010i −0.0785882 − 0.0785882i −0.634393 − 0.773010i
189.2 −0.881921 − 0.471397i −1.76820 − 0.732410i 0.555570 + 0.831470i −0.382683 − 0.923880i 1.21415 + 1.47945i 0 −0.0980171 − 0.995185i 1.88298 + 1.88298i −0.0980171 + 0.995185i
189.3 −0.471397 + 0.881921i 0.536376 + 0.222174i −0.555570 − 0.831470i −0.382683 − 0.923880i −0.448786 + 0.368309i 0 0.995185 − 0.0980171i −0.468769 − 0.468769i 0.995185 + 0.0980171i
189.4 −0.290285 − 0.956940i 1.62958 + 0.674993i −0.831470 + 0.555570i 0.382683 + 0.923880i 0.172887 − 1.75535i 0 0.773010 + 0.634393i 1.49280 + 1.49280i 0.773010 − 0.634393i
189.5 0.290285 + 0.956940i −1.62958 − 0.674993i −0.831470 + 0.555570i 0.382683 + 0.923880i 0.172887 − 1.75535i 0 −0.773010 − 0.634393i 1.49280 + 1.49280i −0.773010 + 0.634393i
189.6 0.471397 − 0.881921i −0.536376 − 0.222174i −0.555570 − 0.831470i −0.382683 − 0.923880i −0.448786 + 0.368309i 0 −0.995185 + 0.0980171i −0.468769 − 0.468769i −0.995185 − 0.0980171i
189.7 0.881921 + 0.471397i 1.76820 + 0.732410i 0.555570 + 0.831470i −0.382683 − 0.923880i 1.21415 + 1.47945i 0 0.0980171 + 0.995185i 1.88298 + 1.88298i 0.0980171 − 0.995185i
189.8 0.956940 − 0.290285i −0.871028 − 0.360791i 0.831470 − 0.555570i 0.382683 + 0.923880i −0.938254 − 0.0924099i 0 0.634393 − 0.773010i −0.0785882 − 0.0785882i 0.634393 + 0.773010i
949.1 −0.956940 − 0.290285i 0.871028 − 0.360791i 0.831470 + 0.555570i 0.382683 − 0.923880i −0.938254 + 0.0924099i 0 −0.634393 − 0.773010i −0.0785882 + 0.0785882i −0.634393 + 0.773010i
949.2 −0.881921 + 0.471397i −1.76820 + 0.732410i 0.555570 − 0.831470i −0.382683 + 0.923880i 1.21415 − 1.47945i 0 −0.0980171 + 0.995185i 1.88298 − 1.88298i −0.0980171 − 0.995185i
949.3 −0.471397 − 0.881921i 0.536376 − 0.222174i −0.555570 + 0.831470i −0.382683 + 0.923880i −0.448786 − 0.368309i 0 0.995185 + 0.0980171i −0.468769 + 0.468769i 0.995185 − 0.0980171i
949.4 −0.290285 + 0.956940i 1.62958 − 0.674993i −0.831470 − 0.555570i 0.382683 − 0.923880i 0.172887 + 1.75535i 0 0.773010 − 0.634393i 1.49280 − 1.49280i 0.773010 + 0.634393i
949.5 0.290285 − 0.956940i −1.62958 + 0.674993i −0.831470 − 0.555570i 0.382683 − 0.923880i 0.172887 + 1.75535i 0 −0.773010 + 0.634393i 1.49280 − 1.49280i −0.773010 − 0.634393i
949.6 0.471397 + 0.881921i −0.536376 + 0.222174i −0.555570 + 0.831470i −0.382683 + 0.923880i −0.448786 − 0.368309i 0 −0.995185 − 0.0980171i −0.468769 + 0.468769i −0.995185 + 0.0980171i
949.7 0.881921 − 0.471397i 1.76820 − 0.732410i 0.555570 − 0.831470i −0.382683 + 0.923880i 1.21415 − 1.47945i 0 0.0980171 − 0.995185i 1.88298 − 1.88298i 0.0980171 + 0.995185i
949.8 0.956940 + 0.290285i −0.871028 + 0.360791i 0.831470 + 0.555570i 0.382683 − 0.923880i −0.938254 + 0.0924099i 0 0.634393 + 0.773010i −0.0785882 + 0.0785882i 0.634393 − 0.773010i
1709.1 −0.995185 − 0.0980171i 0.591637 − 1.42834i 0.980785 + 0.195090i 0.923880 − 0.382683i −0.728789 + 1.36347i 0 −0.956940 − 0.290285i −0.983006 − 0.983006i −0.956940 + 0.290285i
1709.2 −0.773010 + 0.634393i −0.0750191 + 0.181112i 0.195090 − 0.980785i −0.923880 + 0.382683i −0.0569057 − 0.187593i 0 0.471397 + 0.881921i 0.679933 + 0.679933i 0.471397 − 0.881921i
1709.3 −0.634393 − 0.773010i −0.761681 + 1.83886i −0.195090 + 0.980785i −0.923880 + 0.382683i 1.90466 − 0.577774i 0 0.881921 − 0.471397i −2.09415 − 2.09415i 0.881921 + 0.471397i
1709.4 −0.0980171 + 0.995185i −0.485544 + 1.17221i −0.980785 − 0.195090i 0.923880 − 0.382683i −1.11897 − 0.598102i 0 0.290285 − 0.956940i −0.431207 − 0.431207i 0.290285 + 0.956940i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 189.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
19.b odd 2 1 inner
32.g even 8 1 inner
160.z even 8 1 inner
608.w odd 8 1 inner
3040.cn odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.1.cn.a ✓ 32
5.b even 2 1 inner 3040.1.cn.a ✓ 32
19.b odd 2 1 inner 3040.1.cn.a ✓ 32
32.g even 8 1 inner 3040.1.cn.a ✓ 32
95.d odd 2 1 CM 3040.1.cn.a ✓ 32
160.z even 8 1 inner 3040.1.cn.a ✓ 32
608.w odd 8 1 inner 3040.1.cn.a ✓ 32
3040.cn odd 8 1 inner 3040.1.cn.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.1.cn.a ✓ 32 1.a even 1 1 trivial
3040.1.cn.a ✓ 32 5.b even 2 1 inner
3040.1.cn.a ✓ 32 19.b odd 2 1 inner
3040.1.cn.a ✓ 32 32.g even 8 1 inner
3040.1.cn.a ✓ 32 95.d odd 2 1 CM
3040.1.cn.a ✓ 32 160.z even 8 1 inner
3040.1.cn.a ✓ 32 608.w odd 8 1 inner
3040.1.cn.a ✓ 32 3040.cn odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3040, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{32} + 1 \) Copy content Toggle raw display
$3$ \( T^{32} + 280 T^{24} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{8} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{32} \) Copy content Toggle raw display
$11$ \( (T^{16} + 16 T^{10} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{32} + 280 T^{24} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{32} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{32} \) Copy content Toggle raw display
$29$ \( T^{32} \) Copy content Toggle raw display
$31$ \( T^{32} \) Copy content Toggle raw display
$37$ \( T^{32} + 280 T^{24} + \cdots + 4 \) Copy content Toggle raw display
$41$ \( T^{32} \) Copy content Toggle raw display
$43$ \( T^{32} \) Copy content Toggle raw display
$47$ \( T^{32} \) Copy content Toggle raw display
$53$ \( T^{32} + 280 T^{24} + \cdots + 4 \) Copy content Toggle raw display
$59$ \( T^{32} \) Copy content Toggle raw display
$61$ \( (T^{16} + 16 T^{10} + \cdots + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{32} + 280 T^{24} + \cdots + 4 \) Copy content Toggle raw display
$71$ \( T^{32} \) Copy content Toggle raw display
$73$ \( T^{32} \) Copy content Toggle raw display
$79$ \( T^{32} \) Copy content Toggle raw display
$83$ \( T^{32} \) Copy content Toggle raw display
$89$ \( T^{32} \) Copy content Toggle raw display
$97$ \( (T^{16} - 16 T^{14} + \cdots + 2)^{2} \) Copy content Toggle raw display
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