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Properties

Label 2541.1.bh.c
Level $2541$
Weight $1$
Character orbit 2541.bh
Analytic conductor $1.268$
Analytic rank $0$
Dimension $16$
Projective image $D_{6}$
CM discriminant -3
Inner twists $32$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2541,1,Mod(632,2541)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2541, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 10, 12]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2541.632");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2541.bh (of order \(30\), degree \(8\), not 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: \(1.26812419710\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{30})\)
Coefficient field: \(\Q(\zeta_{60})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.3480151059.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{60}^{28} q^{3} - \zeta_{60}^{22} q^{4} - \zeta_{60}^{7} q^{7} - \zeta_{60}^{26} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{60}^{28} q^{3} - \zeta_{60}^{22} q^{4} - \zeta_{60}^{7} q^{7} - \zeta_{60}^{26} q^{9} - \zeta_{60}^{20} q^{12} + (\zeta_{60}^{11} + \zeta_{60}) q^{13} - \zeta_{60}^{14} q^{16} - \zeta_{60}^{5} q^{21} + \zeta_{60}^{28} q^{25} - \zeta_{60}^{24} q^{27} + \zeta_{60}^{29} q^{28} + \zeta_{60}^{16} q^{31} - \zeta_{60}^{18} q^{36} - \zeta_{60}^{2} q^{37} + ( - \zeta_{60}^{29} + \zeta_{60}^{9}) q^{39} + (\zeta_{60}^{25} - \zeta_{60}^{5}) q^{43} - \zeta_{60}^{12} q^{48} + \zeta_{60}^{14} q^{49} + ( - \zeta_{60}^{23} + \zeta_{60}^{3}) q^{52} + ( - \zeta_{60}^{19} - \zeta_{60}^{9}) q^{61} - \zeta_{60}^{3} q^{63} - \zeta_{60}^{6} q^{64} + \zeta_{60}^{10} q^{67} + ( - \zeta_{60}^{27} - \zeta_{60}^{17}) q^{73} + \zeta_{60}^{26} q^{75} - \zeta_{60}^{22} q^{81} + \zeta_{60}^{27} q^{84} + ( - \zeta_{60}^{18} - \zeta_{60}^{8}) q^{91} + \zeta_{60}^{14} q^{93} + \zeta_{60}^{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{3} + 2 q^{4} + 2 q^{9} + 8 q^{12} + 2 q^{16} + 2 q^{25} + 4 q^{27} + 2 q^{31} - 4 q^{36} + 2 q^{37} + 4 q^{48} - 2 q^{49} - 4 q^{64} + 16 q^{67} - 2 q^{75} + 2 q^{81} - 6 q^{91} - 2 q^{93} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2541\mathbb{Z}\right)^\times\).

\(n\) \(848\) \(1816\) \(2059\)
\(\chi(n)\) \(-1\) \(\zeta_{60}^{20}\) \(\zeta_{60}^{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
632.1
0.743145 + 0.669131i
−0.743145 − 0.669131i
0.743145 − 0.669131i
−0.743145 + 0.669131i
0.994522 − 0.104528i
−0.994522 + 0.104528i
−0.207912 − 0.978148i
0.207912 + 0.978148i
−0.406737 − 0.913545i
0.406737 + 0.913545i
−0.406737 + 0.913545i
0.406737 − 0.913545i
−0.207912 + 0.978148i
0.207912 − 0.978148i
0.994522 + 0.104528i
−0.994522 − 0.104528i
0 0.104528 − 0.994522i 0.913545 + 0.406737i 0 0 −0.406737 + 0.913545i 0 −0.978148 − 0.207912i 0
632.2 0 0.104528 − 0.994522i 0.913545 + 0.406737i 0 0 0.406737 − 0.913545i 0 −0.978148 − 0.207912i 0
977.1 0 0.104528 + 0.994522i 0.913545 − 0.406737i 0 0 −0.406737 − 0.913545i 0 −0.978148 + 0.207912i 0
977.2 0 0.104528 + 0.994522i 0.913545 − 0.406737i 0 0 0.406737 + 0.913545i 0 −0.978148 + 0.207912i 0
1334.1 0 0.978148 + 0.207912i 0.669131 + 0.743145i 0 0 −0.743145 + 0.669131i 0 0.913545 + 0.406737i 0
1334.2 0 0.978148 + 0.207912i 0.669131 + 0.743145i 0 0 0.743145 − 0.669131i 0 0.913545 + 0.406737i 0
1703.1 0 −0.913545 − 0.406737i −0.104528 + 0.994522i 0 0 −0.994522 − 0.104528i 0 0.669131 + 0.743145i 0
1703.2 0 −0.913545 − 0.406737i −0.104528 + 0.994522i 0 0 0.994522 + 0.104528i 0 0.669131 + 0.743145i 0
1775.1 0 −0.669131 − 0.743145i −0.978148 − 0.207912i 0 0 −0.207912 + 0.978148i 0 −0.104528 + 0.994522i 0
1775.2 0 −0.669131 − 0.743145i −0.978148 − 0.207912i 0 0 0.207912 − 0.978148i 0 −0.104528 + 0.994522i 0
2060.1 0 −0.669131 + 0.743145i −0.978148 + 0.207912i 0 0 −0.207912 − 0.978148i 0 −0.104528 − 0.994522i 0
2060.2 0 −0.669131 + 0.743145i −0.978148 + 0.207912i 0 0 0.207912 + 0.978148i 0 −0.104528 − 0.994522i 0
2447.1 0 −0.913545 + 0.406737i −0.104528 − 0.994522i 0 0 −0.994522 + 0.104528i 0 0.669131 − 0.743145i 0
2447.2 0 −0.913545 + 0.406737i −0.104528 − 0.994522i 0 0 0.994522 − 0.104528i 0 0.669131 − 0.743145i 0
2501.1 0 0.978148 − 0.207912i 0.669131 − 0.743145i 0 0 −0.743145 − 0.669131i 0 0.913545 − 0.406737i 0
2501.2 0 0.978148 − 0.207912i 0.669131 − 0.743145i 0 0 0.743145 + 0.669131i 0 0.913545 − 0.406737i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 632.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
21.h odd 6 1 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner
77.h odd 6 1 inner
77.m even 15 3 inner
77.o odd 30 3 inner
231.l even 6 1 inner
231.z odd 30 3 inner
231.be even 30 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2541.1.bh.c 16
3.b odd 2 1 CM 2541.1.bh.c 16
7.c even 3 1 inner 2541.1.bh.c 16
11.b odd 2 1 inner 2541.1.bh.c 16
11.c even 5 1 2541.1.q.c ✓ 4
11.c even 5 3 inner 2541.1.bh.c 16
11.d odd 10 1 2541.1.q.c ✓ 4
11.d odd 10 3 inner 2541.1.bh.c 16
21.h odd 6 1 inner 2541.1.bh.c 16
33.d even 2 1 inner 2541.1.bh.c 16
33.f even 10 1 2541.1.q.c ✓ 4
33.f even 10 3 inner 2541.1.bh.c 16
33.h odd 10 1 2541.1.q.c ✓ 4
33.h odd 10 3 inner 2541.1.bh.c 16
77.h odd 6 1 inner 2541.1.bh.c 16
77.m even 15 1 2541.1.q.c ✓ 4
77.m even 15 3 inner 2541.1.bh.c 16
77.o odd 30 1 2541.1.q.c ✓ 4
77.o odd 30 3 inner 2541.1.bh.c 16
231.l even 6 1 inner 2541.1.bh.c 16
231.z odd 30 1 2541.1.q.c ✓ 4
231.z odd 30 3 inner 2541.1.bh.c 16
231.be even 30 1 2541.1.q.c ✓ 4
231.be even 30 3 inner 2541.1.bh.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.1.q.c ✓ 4 11.c even 5 1
2541.1.q.c ✓ 4 11.d odd 10 1
2541.1.q.c ✓ 4 33.f even 10 1
2541.1.q.c ✓ 4 33.h odd 10 1
2541.1.q.c ✓ 4 77.m even 15 1
2541.1.q.c ✓ 4 77.o odd 30 1
2541.1.q.c ✓ 4 231.z odd 30 1
2541.1.q.c ✓ 4 231.be even 30 1
2541.1.bh.c 16 1.a even 1 1 trivial
2541.1.bh.c 16 3.b odd 2 1 CM
2541.1.bh.c 16 7.c even 3 1 inner
2541.1.bh.c 16 11.b odd 2 1 inner
2541.1.bh.c 16 11.c even 5 3 inner
2541.1.bh.c 16 11.d odd 10 3 inner
2541.1.bh.c 16 21.h odd 6 1 inner
2541.1.bh.c 16 33.d even 2 1 inner
2541.1.bh.c 16 33.f even 10 3 inner
2541.1.bh.c 16 33.h odd 10 3 inner
2541.1.bh.c 16 77.h odd 6 1 inner
2541.1.bh.c 16 77.m even 15 3 inner
2541.1.bh.c 16 77.o odd 30 3 inner
2541.1.bh.c 16 231.l even 6 1 inner
2541.1.bh.c 16 231.z odd 30 3 inner
2541.1.bh.c 16 231.be even 30 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{8} + 3T_{13}^{6} + 9T_{13}^{4} + 27T_{13}^{2} + 81 \) acting on \(S_{1}^{\mathrm{new}}(2541, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + T^{7} - T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + T^{14} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} + 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{2} - 3)^{8} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} - 3 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T + 4)^{8} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 3 T^{14} + \cdots + 6561 \) 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
$97$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
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