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Properties

Label 3420.1.bn.b
Level $3420$
Weight $1$
Character orbit 3420.bn
Analytic conductor $1.707$
Analytic rank $0$
Dimension $16$
Projective image $D_{24}$
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] = [3420,1,Mod(1139,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 3, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.1139");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3420.bn (of order \(6\), degree \(2\), 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: \(1.70680234320\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{24}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{48}^{9} q^{2} + \zeta_{48}^{19} q^{3} + \zeta_{48}^{18} q^{4} - \zeta_{48}^{20} q^{5} + \zeta_{48}^{4} q^{6} + \zeta_{48}^{3} q^{8} - \zeta_{48}^{14} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{48}^{9} q^{2} + \zeta_{48}^{19} q^{3} + \zeta_{48}^{18} q^{4} - \zeta_{48}^{20} q^{5} + \zeta_{48}^{4} q^{6} + \zeta_{48}^{3} q^{8} - \zeta_{48}^{14} q^{9} - \zeta_{48}^{5} q^{10} + ( - \zeta_{48}^{6} + \zeta_{48}^{2}) q^{11} - \zeta_{48}^{13} q^{12} + (\zeta_{48}^{9} + \zeta_{48}^{7}) q^{13} + \zeta_{48}^{15} q^{15} - \zeta_{48}^{12} q^{16} + \zeta_{48}^{23} q^{18} + \zeta_{48}^{12} q^{19} + \zeta_{48}^{14} q^{20} + (\zeta_{48}^{15} - \zeta_{48}^{11}) q^{22} + \zeta_{48}^{22} q^{24} - \zeta_{48}^{16} q^{25} + ( - \zeta_{48}^{18} - \zeta_{48}^{16}) q^{26} + \zeta_{48}^{9} q^{27} + q^{30} + \zeta_{48}^{21} q^{32} + (\zeta_{48}^{21} + \zeta_{48}) q^{33} + \zeta_{48}^{8} q^{36} + (\zeta_{48}^{13} - \zeta_{48}^{11}) q^{37} - \zeta_{48}^{21} q^{38} + ( - \zeta_{48}^{4} - \zeta_{48}^{2}) q^{39} - \zeta_{48}^{23} q^{40} + (\zeta_{48}^{20} + 1) q^{44} - \zeta_{48}^{10} q^{45} + \zeta_{48}^{7} q^{48} + \zeta_{48}^{8} q^{49} - \zeta_{48} q^{50} + ( - \zeta_{48}^{3} - \zeta_{48}) q^{52} + ( - \zeta_{48}^{19} - \zeta_{48}^{5}) q^{53} - \zeta_{48}^{18} q^{54} + ( - \zeta_{48}^{22} - \zeta_{48}^{2}) q^{55} - \zeta_{48}^{7} q^{57} - \zeta_{48}^{9} q^{60} + (\zeta_{48}^{22} + \zeta_{48}^{10}) q^{61} + \zeta_{48}^{6} q^{64} + (\zeta_{48}^{5} + \zeta_{48}^{3}) q^{65} + ( - \zeta_{48}^{10} + \zeta_{48}^{6}) q^{66} + (\zeta_{48}^{23} + \zeta_{48}^{17}) q^{67} - \zeta_{48}^{17} q^{72} + ( - \zeta_{48}^{22} + \zeta_{48}^{20}) q^{74} + \zeta_{48}^{11} q^{75} - \zeta_{48}^{6} q^{76} + (\zeta_{48}^{13} + \zeta_{48}^{11}) q^{78} - \zeta_{48}^{8} q^{80} - \zeta_{48}^{4} q^{81} + ( - \zeta_{48}^{9} + \zeta_{48}^{5}) q^{88} + \zeta_{48}^{19} q^{90} + \zeta_{48}^{8} q^{95} - \zeta_{48}^{16} q^{96} + ( - \zeta_{48}^{19} - \zeta_{48}^{13}) q^{97} - \zeta_{48}^{17} q^{98} + (\zeta_{48}^{20} - \zeta_{48}^{16}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{25} + 8 q^{26} + 16 q^{30} + 8 q^{36} + 16 q^{44} + 8 q^{49} - 8 q^{80} + 8 q^{95} + 8 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(-1\) \(\zeta_{48}^{8}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1139.1
0.130526 + 0.991445i
0.793353 + 0.608761i
−0.608761 + 0.793353i
0.991445 − 0.130526i
−0.991445 + 0.130526i
0.608761 − 0.793353i
−0.793353 − 0.608761i
−0.130526 − 0.991445i
0.793353 − 0.608761i
0.130526 − 0.991445i
0.991445 + 0.130526i
−0.608761 − 0.793353i
0.608761 + 0.793353i
−0.991445 − 0.130526i
−0.130526 + 0.991445i
−0.793353 + 0.608761i
−0.923880 − 0.382683i −0.608761 + 0.793353i 0.707107 + 0.707107i 0.866025 + 0.500000i 0.866025 − 0.500000i 0 −0.382683 − 0.923880i −0.258819 − 0.965926i −0.608761 − 0.793353i
1139.2 −0.923880 + 0.382683i 0.991445 − 0.130526i 0.707107 − 0.707107i −0.866025 − 0.500000i −0.866025 + 0.500000i 0 −0.382683 + 0.923880i 0.965926 − 0.258819i 0.991445 + 0.130526i
1139.3 −0.382683 − 0.923880i −0.130526 − 0.991445i −0.707107 + 0.707107i −0.866025 − 0.500000i −0.866025 + 0.500000i 0 0.923880 + 0.382683i −0.965926 + 0.258819i −0.130526 + 0.991445i
1139.4 −0.382683 + 0.923880i −0.793353 − 0.608761i −0.707107 − 0.707107i 0.866025 + 0.500000i 0.866025 − 0.500000i 0 0.923880 − 0.382683i 0.258819 + 0.965926i −0.793353 + 0.608761i
1139.5 0.382683 − 0.923880i 0.793353 + 0.608761i −0.707107 − 0.707107i 0.866025 + 0.500000i 0.866025 − 0.500000i 0 −0.923880 + 0.382683i 0.258819 + 0.965926i 0.793353 − 0.608761i
1139.6 0.382683 + 0.923880i 0.130526 + 0.991445i −0.707107 + 0.707107i −0.866025 − 0.500000i −0.866025 + 0.500000i 0 −0.923880 − 0.382683i −0.965926 + 0.258819i 0.130526 − 0.991445i
1139.7 0.923880 − 0.382683i −0.991445 + 0.130526i 0.707107 − 0.707107i −0.866025 − 0.500000i −0.866025 + 0.500000i 0 0.382683 − 0.923880i 0.965926 − 0.258819i −0.991445 − 0.130526i
1139.8 0.923880 + 0.382683i 0.608761 − 0.793353i 0.707107 + 0.707107i 0.866025 + 0.500000i 0.866025 − 0.500000i 0 0.382683 + 0.923880i −0.258819 − 0.965926i 0.608761 + 0.793353i
2279.1 −0.923880 − 0.382683i 0.991445 + 0.130526i 0.707107 + 0.707107i −0.866025 + 0.500000i −0.866025 − 0.500000i 0 −0.382683 − 0.923880i 0.965926 + 0.258819i 0.991445 − 0.130526i
2279.2 −0.923880 + 0.382683i −0.608761 − 0.793353i 0.707107 − 0.707107i 0.866025 − 0.500000i 0.866025 + 0.500000i 0 −0.382683 + 0.923880i −0.258819 + 0.965926i −0.608761 + 0.793353i
2279.3 −0.382683 − 0.923880i −0.793353 + 0.608761i −0.707107 + 0.707107i 0.866025 − 0.500000i 0.866025 + 0.500000i 0 0.923880 + 0.382683i 0.258819 − 0.965926i −0.793353 − 0.608761i
2279.4 −0.382683 + 0.923880i −0.130526 + 0.991445i −0.707107 − 0.707107i −0.866025 + 0.500000i −0.866025 − 0.500000i 0 0.923880 − 0.382683i −0.965926 − 0.258819i −0.130526 − 0.991445i
2279.5 0.382683 − 0.923880i 0.130526 − 0.991445i −0.707107 − 0.707107i −0.866025 + 0.500000i −0.866025 − 0.500000i 0 −0.923880 + 0.382683i −0.965926 − 0.258819i 0.130526 + 0.991445i
2279.6 0.382683 + 0.923880i 0.793353 − 0.608761i −0.707107 + 0.707107i 0.866025 − 0.500000i 0.866025 + 0.500000i 0 −0.923880 − 0.382683i 0.258819 − 0.965926i 0.793353 + 0.608761i
2279.7 0.923880 − 0.382683i 0.608761 + 0.793353i 0.707107 − 0.707107i 0.866025 − 0.500000i 0.866025 + 0.500000i 0 0.382683 − 0.923880i −0.258819 + 0.965926i 0.608761 − 0.793353i
2279.8 0.923880 + 0.382683i −0.991445 − 0.130526i 0.707107 + 0.707107i −0.866025 + 0.500000i −0.866025 − 0.500000i 0 0.382683 + 0.923880i 0.965926 + 0.258819i −0.991445 + 0.130526i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1139.8
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
36.h even 6 1 inner
180.n even 6 1 inner
684.bh odd 6 1 inner
3420.bn odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3420.1.bn.b yes 16
4.b odd 2 1 3420.1.bn.a ✓ 16
5.b even 2 1 inner 3420.1.bn.b yes 16
9.d odd 6 1 3420.1.bn.a ✓ 16
19.b odd 2 1 inner 3420.1.bn.b yes 16
20.d odd 2 1 3420.1.bn.a ✓ 16
36.h even 6 1 inner 3420.1.bn.b yes 16
45.h odd 6 1 3420.1.bn.a ✓ 16
76.d even 2 1 3420.1.bn.a ✓ 16
95.d odd 2 1 CM 3420.1.bn.b yes 16
171.l even 6 1 3420.1.bn.a ✓ 16
180.n even 6 1 inner 3420.1.bn.b yes 16
380.d even 2 1 3420.1.bn.a ✓ 16
684.bh odd 6 1 inner 3420.1.bn.b yes 16
855.bl even 6 1 3420.1.bn.a ✓ 16
3420.bn odd 6 1 inner 3420.1.bn.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3420.1.bn.a ✓ 16 4.b odd 2 1
3420.1.bn.a ✓ 16 9.d odd 6 1
3420.1.bn.a ✓ 16 20.d odd 2 1
3420.1.bn.a ✓ 16 45.h odd 6 1
3420.1.bn.a ✓ 16 76.d even 2 1
3420.1.bn.a ✓ 16 171.l even 6 1
3420.1.bn.a ✓ 16 380.d even 2 1
3420.1.bn.a ✓ 16 855.bl even 6 1
3420.1.bn.b yes 16 1.a even 1 1 trivial
3420.1.bn.b yes 16 5.b even 2 1 inner
3420.1.bn.b yes 16 19.b odd 2 1 inner
3420.1.bn.b yes 16 36.h even 6 1 inner
3420.1.bn.b yes 16 95.d odd 2 1 CM
3420.1.bn.b yes 16 180.n even 6 1 inner
3420.1.bn.b yes 16 684.bh odd 6 1 inner
3420.1.bn.b yes 16 3420.bn odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{131}^{2} - T_{131} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3420, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{8} + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 4 T^{6} + 15 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( (T^{8} - 8 T^{6} + 20 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{8} + 8 T^{6} + 20 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
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