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Properties

Label 3675.1.cl.a
Level $3675$
Weight $1$
Character orbit 3675.cl
Analytic conductor $1.834$
Analytic rank $0$
Dimension $24$
Projective image $D_{21}$
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] = [3675,1,Mod(74,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 21, 16]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3675.74");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3675.cl (of order \(42\), degree \(12\), 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: \(1.83406392143\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{42})\)
Coefficient field: \(\Q(\zeta_{84})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{84}^{25} q^{3} - \zeta_{84}^{40} q^{4} - \zeta_{84}^{13} q^{7} - \zeta_{84}^{8} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{84}^{25} q^{3} - \zeta_{84}^{40} q^{4} - \zeta_{84}^{13} q^{7} - \zeta_{84}^{8} q^{9} + \zeta_{84}^{23} q^{12} + (\zeta_{84}^{37} + \zeta_{84}^{17}) q^{13} - \zeta_{84}^{38} q^{16} + (\zeta_{84}^{22} + \zeta_{84}^{6}) q^{19} - \zeta_{84}^{38} q^{21} - \zeta_{84}^{33} q^{27} - \zeta_{84}^{11} q^{28} + (\zeta_{84}^{40} + \zeta_{84}^{16}) q^{31} - \zeta_{84}^{6} q^{36} + (\zeta_{84}^{39} + \zeta_{84}^{7}) q^{37} + ( - \zeta_{84}^{20} - 1) q^{39} + ( - \zeta_{84}^{39} + \zeta_{84}^{9}) q^{43} + \zeta_{84}^{21} q^{48} + \zeta_{84}^{26} q^{49} + (\zeta_{84}^{35} + \zeta_{84}^{15}) q^{52} + (\zeta_{84}^{31} - \zeta_{84}^{5}) q^{57} + (\zeta_{84}^{4} + 1) q^{61} + \zeta_{84}^{21} q^{63} - \zeta_{84}^{36} q^{64} + (\zeta_{84}^{11} + \zeta_{84}^{3}) q^{67} + ( - \zeta_{84}^{27} + \zeta_{84}^{17}) q^{73} + (\zeta_{84}^{20} + \zeta_{84}^{4}) q^{76} + ( - \zeta_{84}^{16} - \zeta_{84}^{12}) q^{79} + \zeta_{84}^{16} q^{81} - \zeta_{84}^{36} q^{84} + ( - \zeta_{84}^{30} + \zeta_{84}^{8}) q^{91} + (\zeta_{84}^{41} - \zeta_{84}^{23}) q^{93} + ( - \zeta_{84}^{41} - \zeta_{84}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 2 q^{4} - 2 q^{9} + 2 q^{16} + 2 q^{19} + 2 q^{21} + 4 q^{31} - 4 q^{36} - 26 q^{39} - 2 q^{49} + 26 q^{61} + 4 q^{64} + 4 q^{76} + 2 q^{79} + 2 q^{81} + 4 q^{84} - 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3675\mathbb{Z}\right)^\times\).

\(n\) \(1177\) \(1226\) \(2551\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{84}^{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
−0.294755 + 0.955573i
0.294755 − 0.955573i
−0.294755 − 0.955573i
0.294755 + 0.955573i
0.680173 − 0.733052i
−0.680173 + 0.733052i
−0.563320 + 0.826239i
0.563320 − 0.826239i
0.930874 − 0.365341i
−0.930874 + 0.365341i
0.997204 − 0.0747301i
−0.997204 + 0.0747301i
0.997204 + 0.0747301i
−0.997204 − 0.0747301i
0.680173 + 0.733052i
−0.680173 − 0.733052i
0.149042 − 0.988831i
−0.149042 + 0.988831i
−0.563320 − 0.826239i
0.563320 + 0.826239i
0 −0.930874 + 0.365341i −0.826239 + 0.563320i 0 0 −0.680173 + 0.733052i 0 0.733052 − 0.680173i 0
74.2 0 0.930874 − 0.365341i −0.826239 + 0.563320i 0 0 0.680173 − 0.733052i 0 0.733052 − 0.680173i 0
149.1 0 −0.930874 − 0.365341i −0.826239 − 0.563320i 0 0 −0.680173 − 0.733052i 0 0.733052 + 0.680173i 0
149.2 0 0.930874 + 0.365341i −0.826239 − 0.563320i 0 0 0.680173 + 0.733052i 0 0.733052 + 0.680173i 0
599.1 0 −0.149042 − 0.988831i −0.0747301 + 0.997204i 0 0 0.294755 − 0.955573i 0 −0.955573 + 0.294755i 0
599.2 0 0.149042 + 0.988831i −0.0747301 + 0.997204i 0 0 −0.294755 + 0.955573i 0 −0.955573 + 0.294755i 0
674.1 0 −0.680173 − 0.733052i −0.365341 + 0.930874i 0 0 0.997204 − 0.0747301i 0 −0.0747301 + 0.997204i 0
674.2 0 0.680173 + 0.733052i −0.365341 + 0.930874i 0 0 −0.997204 + 0.0747301i 0 −0.0747301 + 0.997204i 0
1124.1 0 −0.997204 − 0.0747301i 0.733052 + 0.680173i 0 0 −0.149042 − 0.988831i 0 0.988831 + 0.149042i 0
1124.2 0 0.997204 + 0.0747301i 0.733052 + 0.680173i 0 0 0.149042 + 0.988831i 0 0.988831 + 0.149042i 0
1199.1 0 −0.294755 − 0.955573i 0.988831 + 0.149042i 0 0 −0.563320 + 0.826239i 0 −0.826239 + 0.563320i 0
1199.2 0 0.294755 + 0.955573i 0.988831 + 0.149042i 0 0 0.563320 − 0.826239i 0 −0.826239 + 0.563320i 0
1649.1 0 −0.294755 + 0.955573i 0.988831 − 0.149042i 0 0 −0.563320 − 0.826239i 0 −0.826239 − 0.563320i 0
1649.2 0 0.294755 − 0.955573i 0.988831 − 0.149042i 0 0 0.563320 + 0.826239i 0 −0.826239 − 0.563320i 0
1724.1 0 −0.149042 + 0.988831i −0.0747301 − 0.997204i 0 0 0.294755 + 0.955573i 0 −0.955573 − 0.294755i 0
1724.2 0 0.149042 − 0.988831i −0.0747301 − 0.997204i 0 0 −0.294755 − 0.955573i 0 −0.955573 − 0.294755i 0
2249.1 0 −0.563320 + 0.826239i −0.955573 + 0.294755i 0 0 −0.930874 − 0.365341i 0 −0.365341 − 0.930874i 0
2249.2 0 0.563320 − 0.826239i −0.955573 + 0.294755i 0 0 0.930874 + 0.365341i 0 −0.365341 − 0.930874i 0
2699.1 0 −0.680173 + 0.733052i −0.365341 − 0.930874i 0 0 0.997204 + 0.0747301i 0 −0.0747301 − 0.997204i 0
2699.2 0 0.680173 − 0.733052i −0.365341 − 0.930874i 0 0 −0.997204 − 0.0747301i 0 −0.0747301 − 0.997204i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.2
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
49.g even 21 1 inner
147.n odd 42 1 inner
245.t even 42 1 inner
735.bq odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3675.1.cl.a 24
3.b odd 2 1 CM 3675.1.cl.a 24
5.b even 2 1 inner 3675.1.cl.a 24
5.c odd 4 1 3675.1.cg.a ✓ 12
5.c odd 4 1 3675.1.cg.b yes 12
15.d odd 2 1 inner 3675.1.cl.a 24
15.e even 4 1 3675.1.cg.a ✓ 12
15.e even 4 1 3675.1.cg.b yes 12
49.g even 21 1 inner 3675.1.cl.a 24
147.n odd 42 1 inner 3675.1.cl.a 24
245.t even 42 1 inner 3675.1.cl.a 24
245.w odd 84 1 3675.1.cg.a ✓ 12
245.w odd 84 1 3675.1.cg.b yes 12
735.bq odd 42 1 inner 3675.1.cl.a 24
735.bt even 84 1 3675.1.cg.a ✓ 12
735.bt even 84 1 3675.1.cg.b yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3675.1.cg.a ✓ 12 5.c odd 4 1
3675.1.cg.a ✓ 12 15.e even 4 1
3675.1.cg.a ✓ 12 245.w odd 84 1
3675.1.cg.a ✓ 12 735.bt even 84 1
3675.1.cg.b yes 12 5.c odd 4 1
3675.1.cg.b yes 12 15.e even 4 1
3675.1.cg.b yes 12 245.w odd 84 1
3675.1.cg.b yes 12 735.bt even 84 1
3675.1.cl.a 24 1.a even 1 1 trivial
3675.1.cl.a 24 3.b odd 2 1 CM
3675.1.cl.a 24 5.b even 2 1 inner
3675.1.cl.a 24 15.d odd 2 1 inner
3675.1.cl.a 24 49.g even 21 1 inner
3675.1.cl.a 24 147.n odd 42 1 inner
3675.1.cl.a 24 245.t even 42 1 inner
3675.1.cl.a 24 735.bq odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3675, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} \) Copy content Toggle raw display
$3$ \( T^{24} + T^{22} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{24} \) Copy content Toggle raw display
$7$ \( T^{24} + T^{22} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{24} \) Copy content Toggle raw display
$13$ \( T^{24} - 2 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{24} \) Copy content Toggle raw display
$19$ \( (T^{12} - T^{11} + 7 T^{10} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{24} \) Copy content Toggle raw display
$29$ \( T^{24} \) Copy content Toggle raw display
$31$ \( (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{4} \) Copy content Toggle raw display
$37$ \( T^{24} - 6 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{24} \) Copy content Toggle raw display
$43$ \( (T^{12} - 4 T^{10} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{24} \) Copy content Toggle raw display
$53$ \( T^{24} \) Copy content Toggle raw display
$59$ \( T^{24} \) Copy content Toggle raw display
$61$ \( (T^{12} - 13 T^{11} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{24} - 13 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{24} \) Copy content Toggle raw display
$73$ \( T^{24} + T^{22} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( (T^{12} - T^{11} + 7 T^{10} + \cdots + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{24} \) Copy content Toggle raw display
$89$ \( T^{24} \) Copy content Toggle raw display
$97$ \( (T^{12} + 13 T^{10} + \cdots + 1)^{2} \) Copy content Toggle raw display
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