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Properties

Label 3225.1.bi.a
Level $3225$
Weight $1$
Character orbit 3225.bi
Analytic conductor $1.609$
Analytic rank $0$
Dimension $12$
Projective image $D_{7}$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3225,1,Mod(299,3225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3225, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 7, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3225.299");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3225 = 3 \cdot 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3225.bi (of order \(14\), degree \(6\), 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: \(1.60948466574\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 129)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.170676802323.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{28}^{9} q^{3} + \zeta_{28}^{10} q^{4} + ( - \zeta_{28}^{13} - \zeta_{28}) q^{7} - \zeta_{28}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{28}^{9} q^{3} + \zeta_{28}^{10} q^{4} + ( - \zeta_{28}^{13} - \zeta_{28}) q^{7} - \zeta_{28}^{4} q^{9} - \zeta_{28}^{5} q^{12} + ( - \zeta_{28}^{11} + \zeta_{28}^{5}) q^{13} - \zeta_{28}^{6} q^{16} + ( - \zeta_{28}^{12} - \zeta_{28}^{8}) q^{19} + ( - \zeta_{28}^{10} + \zeta_{28}^{8}) q^{21} - \zeta_{28}^{13} q^{27} + ( - \zeta_{28}^{11} + \zeta_{28}^{9}) q^{28} + ( - \zeta_{28}^{10} + 1) q^{31} + q^{36} + (\zeta_{28}^{11} + \zeta_{28}^{3}) q^{37} + (\zeta_{28}^{6} - 1) q^{39} + \zeta_{28}^{9} q^{43} + \zeta_{28} q^{48} + ( - \zeta_{28}^{12} + \zeta_{28}^{2} - 1) q^{49} + (\zeta_{28}^{7} - \zeta_{28}) q^{52} + (\zeta_{28}^{7} + \zeta_{28}^{3}) q^{57} + (\zeta_{28}^{12} - \zeta_{28}^{6}) q^{61} + (\zeta_{28}^{5} - \zeta_{28}^{3}) q^{63} + \zeta_{28}^{2} q^{64} + (\zeta_{28}^{11} - \zeta_{28}^{9}) q^{67} + (\zeta_{28}^{13} - \zeta_{28}^{3}) q^{73} + (\zeta_{28}^{8} + \zeta_{28}^{4}) q^{76} + ( - \zeta_{28}^{8} + \zeta_{28}^{6}) q^{79} + \zeta_{28}^{8} q^{81} + (\zeta_{28}^{6} - \zeta_{28}^{4}) q^{84} + (\zeta_{28}^{12} + \cdots + \zeta_{28}^{4}) q^{91} + \cdots + ( - \zeta_{28}^{5} + \zeta_{28}^{3}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{4} + 2 q^{9} - 2 q^{16} + 4 q^{19} - 4 q^{21} + 10 q^{31} + 12 q^{36} - 10 q^{39} - 8 q^{49} - 4 q^{61} + 2 q^{64} - 4 q^{76} + 4 q^{79} - 2 q^{81} + 4 q^{84} - 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3225\mathbb{Z}\right)^\times\).

\(n\) \(1076\) \(2452\) \(2626\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{28}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1
0.974928 + 0.222521i
−0.974928 − 0.222521i
−0.781831 + 0.623490i
0.781831 − 0.623490i
0.433884 − 0.900969i
−0.433884 + 0.900969i
−0.781831 − 0.623490i
0.781831 + 0.623490i
0.433884 + 0.900969i
−0.433884 − 0.900969i
0.974928 − 0.222521i
−0.974928 + 0.222521i
0 −0.433884 + 0.900969i −0.623490 + 0.781831i 0 0 − 0.445042i 0 −0.623490 − 0.781831i 0
299.2 0 0.433884 − 0.900969i −0.623490 + 0.781831i 0 0 0.445042i 0 −0.623490 − 0.781831i 0
1349.1 0 −0.974928 − 0.222521i 0.900969 − 0.433884i 0 0 − 1.24698i 0 0.900969 + 0.433884i 0
1349.2 0 0.974928 + 0.222521i 0.900969 − 0.433884i 0 0 1.24698i 0 0.900969 + 0.433884i 0
1724.1 0 −0.781831 + 0.623490i 0.222521 + 0.974928i 0 0 1.80194i 0 0.222521 − 0.974928i 0
1724.2 0 0.781831 − 0.623490i 0.222521 + 0.974928i 0 0 − 1.80194i 0 0.222521 − 0.974928i 0
2099.1 0 −0.974928 + 0.222521i 0.900969 + 0.433884i 0 0 1.24698i 0 0.900969 − 0.433884i 0
2099.2 0 0.974928 − 0.222521i 0.900969 + 0.433884i 0 0 − 1.24698i 0 0.900969 − 0.433884i 0
2849.1 0 −0.781831 − 0.623490i 0.222521 − 0.974928i 0 0 − 1.80194i 0 0.222521 + 0.974928i 0
2849.2 0 0.781831 + 0.623490i 0.222521 − 0.974928i 0 0 1.80194i 0 0.222521 + 0.974928i 0
3074.1 0 −0.433884 − 0.900969i −0.623490 − 0.781831i 0 0 0.445042i 0 −0.623490 + 0.781831i 0
3074.2 0 0.433884 + 0.900969i −0.623490 − 0.781831i 0 0 − 0.445042i 0 −0.623490 + 0.781831i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
15.d odd 2 1 inner
43.e even 7 1 inner
129.l odd 14 1 inner
215.p even 14 1 inner
645.ba odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3225.1.bi.a 12
3.b odd 2 1 CM 3225.1.bi.a 12
5.b even 2 1 inner 3225.1.bi.a 12
5.c odd 4 1 129.1.l.a ✓ 6
5.c odd 4 1 3225.1.bm.a 6
15.d odd 2 1 inner 3225.1.bi.a 12
15.e even 4 1 129.1.l.a ✓ 6
15.e even 4 1 3225.1.bm.a 6
20.e even 4 1 2064.1.cj.a 6
43.e even 7 1 inner 3225.1.bi.a 12
45.k odd 12 2 3483.1.bs.a 12
45.l even 12 2 3483.1.bs.a 12
60.l odd 4 1 2064.1.cj.a 6
129.l odd 14 1 inner 3225.1.bi.a 12
215.p even 14 1 inner 3225.1.bi.a 12
215.s odd 28 1 129.1.l.a ✓ 6
215.s odd 28 1 3225.1.bm.a 6
645.ba odd 14 1 inner 3225.1.bi.a 12
645.bk even 28 1 129.1.l.a ✓ 6
645.bk even 28 1 3225.1.bm.a 6
860.bj even 28 1 2064.1.cj.a 6
1935.ef even 84 2 3483.1.bs.a 12
1935.ej odd 84 2 3483.1.bs.a 12
2580.co odd 28 1 2064.1.cj.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.1.l.a ✓ 6 5.c odd 4 1
129.1.l.a ✓ 6 15.e even 4 1
129.1.l.a ✓ 6 215.s odd 28 1
129.1.l.a ✓ 6 645.bk even 28 1
2064.1.cj.a 6 20.e even 4 1
2064.1.cj.a 6 60.l odd 4 1
2064.1.cj.a 6 860.bj even 28 1
2064.1.cj.a 6 2580.co odd 28 1
3225.1.bi.a 12 1.a even 1 1 trivial
3225.1.bi.a 12 3.b odd 2 1 CM
3225.1.bi.a 12 5.b even 2 1 inner
3225.1.bi.a 12 15.d odd 2 1 inner
3225.1.bi.a 12 43.e even 7 1 inner
3225.1.bi.a 12 129.l odd 14 1 inner
3225.1.bi.a 12 215.p even 14 1 inner
3225.1.bi.a 12 645.ba odd 14 1 inner
3225.1.bm.a 6 5.c odd 4 1
3225.1.bm.a 6 15.e even 4 1
3225.1.bm.a 6 215.s odd 28 1
3225.1.bm.a 6 645.bk even 28 1
3483.1.bs.a 12 45.k odd 12 2
3483.1.bs.a 12 45.l even 12 2
3483.1.bs.a 12 1935.ef even 84 2
3483.1.bs.a 12 1935.ej odd 84 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3225, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 4 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} - 2 T^{5} + 4 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( (T^{6} - 5 T^{5} + 11 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( (T^{6} + 2 T^{5} + 4 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 4 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} - 4 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( (T^{3} - T^{2} - 2 T + 1)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} - 4 T^{10} + \cdots + 1 \) Copy content Toggle raw display
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