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Properties

Label 3380.1.cg.a
Level $3380$
Weight $1$
Character orbit 3380.cg
Analytic conductor $1.687$
Analytic rank $0$
Dimension $24$
Projective image $D_{52}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3380,1,Mod(47,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(52))
 
chi = DirichletCharacter(H, H._module([26, 13, 21]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3380.47");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3380.cg (of order \(52\), degree \(24\), 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: \(1.68683974270\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{52})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{52}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{52} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{52}^{23} q^{2} - \zeta_{52}^{20} q^{4} - \zeta_{52}^{22} q^{5} + \zeta_{52}^{17} q^{8} + \zeta_{52}^{23} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{52}^{23} q^{2} - \zeta_{52}^{20} q^{4} - \zeta_{52}^{22} q^{5} + \zeta_{52}^{17} q^{8} + \zeta_{52}^{23} q^{9} + \zeta_{52}^{19} q^{10} + \zeta_{52}^{22} q^{13} - \zeta_{52}^{14} q^{16} + ( - \zeta_{52}^{11} + \zeta_{52}^{10}) q^{17} - \zeta_{52}^{20} q^{18} - \zeta_{52}^{16} q^{20} - \zeta_{52}^{18} q^{25} - \zeta_{52}^{19} q^{26} + ( - \zeta_{52}^{15} + \zeta_{52}^{5}) q^{29} + \zeta_{52}^{11} q^{32} + (\zeta_{52}^{8} - \zeta_{52}^{7}) q^{34} + \zeta_{52}^{17} q^{36} + (\zeta_{52}^{3} + \zeta_{52}) q^{37} + \zeta_{52}^{13} q^{40} + ( - \zeta_{52}^{21} + \zeta_{52}^{2}) q^{41} + \zeta_{52}^{19} q^{45} + \zeta_{52}^{8} q^{49} + \zeta_{52}^{15} q^{50} + \zeta_{52}^{16} q^{52} + (\zeta_{52}^{13} - \zeta_{52}^{8}) q^{53} + (\zeta_{52}^{12} - \zeta_{52}^{2}) q^{58} + ( - \zeta_{52}^{17} + \zeta_{52}) q^{61} - \zeta_{52}^{8} q^{64} + \zeta_{52}^{18} q^{65} + ( - \zeta_{52}^{5} + \zeta_{52}^{4}) q^{68} - \zeta_{52}^{14} q^{72} + \zeta_{52}^{3} q^{73} + (\zeta_{52}^{24} - 1) q^{74} - \zeta_{52}^{10} q^{80} - \zeta_{52}^{20} q^{81} + (\zeta_{52}^{25} + \zeta_{52}^{18}) q^{82} + ( - \zeta_{52}^{7} + \zeta_{52}^{6}) q^{85} + ( - \zeta_{52}^{12} - \zeta_{52}) q^{89} - \zeta_{52}^{16} q^{90} + (\zeta_{52}^{16} - \zeta_{52}^{12}) q^{97} - \zeta_{52}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 2 q^{4} - 2 q^{5} + 2 q^{13} - 2 q^{16} + 2 q^{17} + 2 q^{18} + 2 q^{20} - 2 q^{25} - 2 q^{34} + 2 q^{41} - 2 q^{49} - 2 q^{52} + 2 q^{53} - 4 q^{58} + 2 q^{64} + 2 q^{65} - 2 q^{68} - 2 q^{72} - 26 q^{74} - 2 q^{80} + 2 q^{81} + 2 q^{82} + 2 q^{85} + 2 q^{89} + 2 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(\zeta_{52}^{13}\) \(-1\) \(-\zeta_{52}^{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.935016 − 0.354605i
0.464723 − 0.885456i
0.992709 + 0.120537i
−0.663123 + 0.748511i
−0.992709 + 0.120537i
0.822984 − 0.568065i
0.935016 − 0.354605i
−0.935016 + 0.354605i
−0.822984 + 0.568065i
0.992709 − 0.120537i
0.663123 − 0.748511i
−0.992709 − 0.120537i
−0.464723 + 0.885456i
0.935016 + 0.354605i
0.239316 − 0.970942i
−0.822984 − 0.568065i
0.663123 + 0.748511i
−0.239316 − 0.970942i
−0.464723 − 0.885456i
0.464723 + 0.885456i
0.464723 − 0.885456i 0 −0.568065 − 0.822984i 0.120537 − 0.992709i 0 0 −0.992709 + 0.120537i 0.464723 − 0.885456i −0.822984 − 0.568065i
83.1 0.992709 + 0.120537i 0 0.970942 + 0.239316i −0.354605 − 0.935016i 0 0 0.935016 + 0.354605i 0.992709 + 0.120537i −0.239316 − 0.970942i
307.1 −0.935016 + 0.354605i 0 0.748511 − 0.663123i 0.885456 − 0.464723i 0 0 −0.464723 + 0.885456i −0.935016 + 0.354605i −0.663123 + 0.748511i
343.1 −0.822984 + 0.568065i 0 0.354605 − 0.935016i −0.970942 − 0.239316i 0 0 0.239316 + 0.970942i −0.822984 + 0.568065i 0.935016 − 0.354605i
567.1 0.935016 + 0.354605i 0 0.748511 + 0.663123i 0.885456 + 0.464723i 0 0 0.464723 + 0.885456i 0.935016 + 0.354605i 0.663123 + 0.748511i
603.1 0.239316 − 0.970942i 0 −0.885456 − 0.464723i −0.748511 + 0.663123i 0 0 −0.663123 + 0.748511i 0.239316 − 0.970942i 0.464723 + 0.885456i
827.1 −0.464723 − 0.885456i 0 −0.568065 + 0.822984i 0.120537 + 0.992709i 0 0 0.992709 + 0.120537i −0.464723 − 0.885456i 0.822984 − 0.568065i
863.1 0.464723 + 0.885456i 0 −0.568065 + 0.822984i 0.120537 + 0.992709i 0 0 −0.992709 − 0.120537i 0.464723 + 0.885456i −0.822984 + 0.568065i
1087.1 −0.239316 + 0.970942i 0 −0.885456 − 0.464723i −0.748511 + 0.663123i 0 0 0.663123 − 0.748511i −0.239316 + 0.970942i −0.464723 − 0.885456i
1123.1 −0.935016 − 0.354605i 0 0.748511 + 0.663123i 0.885456 + 0.464723i 0 0 −0.464723 − 0.885456i −0.935016 − 0.354605i −0.663123 − 0.748511i
1347.1 0.822984 − 0.568065i 0 0.354605 − 0.935016i −0.970942 − 0.239316i 0 0 −0.239316 − 0.970942i 0.822984 − 0.568065i −0.935016 + 0.354605i
1383.1 0.935016 − 0.354605i 0 0.748511 − 0.663123i 0.885456 − 0.464723i 0 0 0.464723 − 0.885456i 0.935016 − 0.354605i 0.663123 − 0.748511i
1607.1 −0.992709 − 0.120537i 0 0.970942 + 0.239316i −0.354605 − 0.935016i 0 0 −0.935016 − 0.354605i −0.992709 − 0.120537i 0.239316 + 0.970942i
1643.1 −0.464723 + 0.885456i 0 −0.568065 − 0.822984i 0.120537 − 0.992709i 0 0 0.992709 − 0.120537i −0.464723 + 0.885456i 0.822984 + 0.568065i
1867.1 0.663123 + 0.748511i 0 −0.120537 + 0.992709i 0.568065 − 0.822984i 0 0 −0.822984 + 0.568065i 0.663123 + 0.748511i 0.992709 − 0.120537i
1903.1 −0.239316 − 0.970942i 0 −0.885456 + 0.464723i −0.748511 − 0.663123i 0 0 0.663123 + 0.748511i −0.239316 − 0.970942i −0.464723 + 0.885456i
2163.1 0.822984 + 0.568065i 0 0.354605 + 0.935016i −0.970942 + 0.239316i 0 0 −0.239316 + 0.970942i 0.822984 + 0.568065i −0.935016 − 0.354605i
2387.1 −0.663123 + 0.748511i 0 −0.120537 − 0.992709i 0.568065 + 0.822984i 0 0 0.822984 + 0.568065i −0.663123 + 0.748511i −0.992709 − 0.120537i
2423.1 −0.992709 + 0.120537i 0 0.970942 − 0.239316i −0.354605 + 0.935016i 0 0 −0.935016 + 0.354605i −0.992709 + 0.120537i 0.239316 − 0.970942i
2647.1 0.992709 − 0.120537i 0 0.970942 − 0.239316i −0.354605 + 0.935016i 0 0 0.935016 − 0.354605i 0.992709 − 0.120537i −0.239316 + 0.970942i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
845.be even 52 1 inner
3380.cg odd 52 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.1.cg.a yes 24
4.b odd 2 1 CM 3380.1.cg.a yes 24
5.c odd 4 1 3380.1.bz.a ✓ 24
20.e even 4 1 3380.1.bz.a ✓ 24
169.j odd 52 1 3380.1.bz.a ✓ 24
676.s even 52 1 3380.1.bz.a ✓ 24
845.be even 52 1 inner 3380.1.cg.a yes 24
3380.cg odd 52 1 inner 3380.1.cg.a yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3380.1.bz.a ✓ 24 5.c odd 4 1
3380.1.bz.a ✓ 24 20.e even 4 1
3380.1.bz.a ✓ 24 169.j odd 52 1
3380.1.bz.a ✓ 24 676.s even 52 1
3380.1.cg.a yes 24 1.a even 1 1 trivial
3380.1.cg.a yes 24 4.b odd 2 1 CM
3380.1.cg.a yes 24 845.be even 52 1 inner
3380.1.cg.a yes 24 3380.cg odd 52 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} - T^{22} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{24} \) Copy content Toggle raw display
$5$ \( (T^{12} + T^{11} + T^{10} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{24} \) Copy content Toggle raw display
$11$ \( T^{24} \) Copy content Toggle raw display
$13$ \( (T^{12} - T^{11} + T^{10} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{24} - 2 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{24} \) Copy content Toggle raw display
$23$ \( T^{24} \) Copy content Toggle raw display
$29$ \( T^{24} - 4 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{24} \) Copy content Toggle raw display
$37$ \( T^{24} + 117 T^{14} + \cdots + 169 \) Copy content Toggle raw display
$41$ \( T^{24} - 2 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{24} \) Copy content Toggle raw display
$47$ \( T^{24} \) Copy content Toggle raw display
$53$ \( T^{24} - 2 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{24} \) Copy content Toggle raw display
$61$ \( T^{24} + 26 T^{20} + \cdots + 169 \) Copy content Toggle raw display
$67$ \( T^{24} \) Copy content Toggle raw display
$71$ \( T^{24} \) Copy content Toggle raw display
$73$ \( T^{24} - 4 T^{22} + \cdots + 16777216 \) Copy content Toggle raw display
$79$ \( T^{24} \) Copy content Toggle raw display
$83$ \( T^{24} \) Copy content Toggle raw display
$89$ \( T^{24} - 2 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( (T^{12} + 13 T^{8} + \cdots + 13)^{2} \) Copy content Toggle raw display
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