Properties

Label 3675.1.bx.a
Level $3675$
Weight $1$
Character orbit 3675.bx
Analytic conductor $1.834$
Analytic rank $0$
Dimension $24$
Projective image $D_{14}$
CM discriminant -3
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3675,1,Mod(482,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([14, 7, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3675.482");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3675.bx (of order \(28\), degree \(12\), 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 \(\Q(\zeta_{28})\)
Coefficient field: \(\Q(\zeta_{56})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{56}^{5} q^{3} - \zeta_{56}^{22} q^{4} - \zeta_{56}^{11} q^{7} + \zeta_{56}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{56}^{5} q^{3} - \zeta_{56}^{22} q^{4} - \zeta_{56}^{11} q^{7} + \zeta_{56}^{10} q^{9} + \zeta_{56}^{27} q^{12} + (\zeta_{56}^{13} - \zeta_{56}^{9}) q^{13} - \zeta_{56}^{16} q^{16} + (\zeta_{56}^{18} - \zeta_{56}^{10}) q^{19} + \zeta_{56}^{16} q^{21} - \zeta_{56}^{15} q^{27} - \zeta_{56}^{5} q^{28} + ( - \zeta_{56}^{20} - \zeta_{56}^{8}) q^{31} + \zeta_{56}^{4} q^{36} + ( - \zeta_{56}^{19} - \zeta_{56}^{7}) q^{37} + ( - \zeta_{56}^{18} + \zeta_{56}^{14}) q^{39} + (\zeta_{56}^{13} - \zeta_{56}^{5}) q^{43} + \zeta_{56}^{21} q^{48} + \zeta_{56}^{22} q^{49} + (\zeta_{56}^{7} - \zeta_{56}^{3}) q^{52} + ( - \zeta_{56}^{23} + \zeta_{56}^{15}) q^{57} + ( - \zeta_{56}^{12} - 1) q^{61} - \zeta_{56}^{21} q^{63} - \zeta_{56}^{10} q^{64} + ( - \zeta_{56}^{23} - \zeta_{56}^{19}) q^{67} + (\zeta_{56}^{25} + \zeta_{56}^{9}) q^{73} + (\zeta_{56}^{12} - \zeta_{56}^{4}) q^{76} + (\zeta_{56}^{22} + \zeta_{56}^{6}) q^{79} + \zeta_{56}^{20} q^{81} + \zeta_{56}^{10} q^{84} + ( - \zeta_{56}^{24} + \zeta_{56}^{20}) q^{91} + (\zeta_{56}^{25} + \zeta_{56}^{13}) q^{93} + (\zeta_{56}^{11} + \zeta_{56}^{3}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 4 q^{16} - 4 q^{21} + 4 q^{36} - 28 q^{61} + 4 q^{81} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1177\) \(1226\) \(2551\)
\(\chi(n)\) \(-\zeta_{56}^{14}\) \(-1\) \(-\zeta_{56}^{24}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
482.1
−0.846724 + 0.532032i
0.846724 0.532032i
−0.943883 + 0.330279i
0.943883 0.330279i
0.993712 0.111964i
−0.993712 + 0.111964i
0.993712 + 0.111964i
−0.993712 0.111964i
−0.943883 0.330279i
0.943883 + 0.330279i
−0.846724 0.532032i
0.846724 + 0.532032i
0.532032 + 0.846724i
−0.532032 0.846724i
0.330279 + 0.943883i
−0.330279 0.943883i
0.111964 + 0.993712i
−0.111964 0.993712i
0.111964 0.993712i
−0.111964 + 0.993712i
0 −0.943883 0.330279i −0.974928 0.222521i 0 0 0.993712 + 0.111964i 0 0.781831 + 0.623490i 0
482.2 0 0.943883 + 0.330279i −0.974928 0.222521i 0 0 −0.993712 0.111964i 0 0.781831 + 0.623490i 0
818.1 0 −0.111964 0.993712i −0.433884 + 0.900969i 0 0 −0.846724 + 0.532032i 0 −0.974928 + 0.222521i 0
818.2 0 0.111964 + 0.993712i −0.433884 + 0.900969i 0 0 0.846724 0.532032i 0 −0.974928 + 0.222521i 0
1007.1 0 −0.846724 + 0.532032i 0.781831 + 0.623490i 0 0 −0.330279 + 0.943883i 0 0.433884 0.900969i 0
1007.2 0 0.846724 0.532032i 0.781831 + 0.623490i 0 0 0.330279 0.943883i 0 0.433884 0.900969i 0
1343.1 0 −0.846724 0.532032i 0.781831 0.623490i 0 0 −0.330279 0.943883i 0 0.433884 + 0.900969i 0
1343.2 0 0.846724 + 0.532032i 0.781831 0.623490i 0 0 0.330279 + 0.943883i 0 0.433884 + 0.900969i 0
1532.1 0 −0.111964 + 0.993712i −0.433884 0.900969i 0 0 −0.846724 0.532032i 0 −0.974928 0.222521i 0
1532.2 0 0.111964 0.993712i −0.433884 0.900969i 0 0 0.846724 + 0.532032i 0 −0.974928 0.222521i 0
1868.1 0 −0.943883 + 0.330279i −0.974928 + 0.222521i 0 0 0.993712 0.111964i 0 0.781831 0.623490i 0
1868.2 0 0.943883 0.330279i −0.974928 + 0.222521i 0 0 −0.993712 + 0.111964i 0 0.781831 0.623490i 0
2393.1 0 −0.330279 + 0.943883i 0.974928 + 0.222521i 0 0 −0.111964 + 0.993712i 0 −0.781831 0.623490i 0
2393.2 0 0.330279 0.943883i 0.974928 + 0.222521i 0 0 0.111964 0.993712i 0 −0.781831 0.623490i 0
2582.1 0 −0.993712 + 0.111964i 0.433884 0.900969i 0 0 −0.532032 0.846724i 0 0.974928 0.222521i 0
2582.2 0 0.993712 0.111964i 0.433884 0.900969i 0 0 0.532032 + 0.846724i 0 0.974928 0.222521i 0
2918.1 0 −0.532032 0.846724i −0.781831 0.623490i 0 0 0.943883 + 0.330279i 0 −0.433884 + 0.900969i 0
2918.2 0 0.532032 + 0.846724i −0.781831 0.623490i 0 0 −0.943883 0.330279i 0 −0.433884 + 0.900969i 0
3107.1 0 −0.532032 + 0.846724i −0.781831 + 0.623490i 0 0 0.943883 0.330279i 0 −0.433884 0.900969i 0
3107.2 0 0.532032 0.846724i −0.781831 + 0.623490i 0 0 −0.943883 + 0.330279i 0 −0.433884 0.900969i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 482.2
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
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner
49.f odd 14 1 inner
147.k even 14 1 inner
245.o odd 14 1 inner
245.s even 28 2 inner
735.ba even 14 1 inner
735.bk odd 28 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3675.1.bx.a 24
3.b odd 2 1 CM 3675.1.bx.a 24
5.b even 2 1 inner 3675.1.bx.a 24
5.c odd 4 2 inner 3675.1.bx.a 24
15.d odd 2 1 inner 3675.1.bx.a 24
15.e even 4 2 inner 3675.1.bx.a 24
49.f odd 14 1 inner 3675.1.bx.a 24
147.k even 14 1 inner 3675.1.bx.a 24
245.o odd 14 1 inner 3675.1.bx.a 24
245.s even 28 2 inner 3675.1.bx.a 24
735.ba even 14 1 inner 3675.1.bx.a 24
735.bk odd 28 2 inner 3675.1.bx.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3675.1.bx.a 24 1.a even 1 1 trivial
3675.1.bx.a 24 3.b odd 2 1 CM
3675.1.bx.a 24 5.b even 2 1 inner
3675.1.bx.a 24 5.c odd 4 2 inner
3675.1.bx.a 24 15.d odd 2 1 inner
3675.1.bx.a 24 15.e even 4 2 inner
3675.1.bx.a 24 49.f odd 14 1 inner
3675.1.bx.a 24 147.k even 14 1 inner
3675.1.bx.a 24 245.o odd 14 1 inner
3675.1.bx.a 24 245.s even 28 2 inner
3675.1.bx.a 24 735.ba even 14 1 inner
3675.1.bx.a 24 735.bk odd 28 2 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
$3$ \( T^{24} - T^{20} + T^{16} - T^{12} + T^{8} - T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{24} \) Copy content Toggle raw display
$7$ \( T^{24} - T^{20} + T^{16} - T^{12} + T^{8} - T^{4} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{24} \) Copy content Toggle raw display
$13$ \( T^{24} - 16 T^{20} + 60 T^{16} + 223 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{24} \) Copy content Toggle raw display
$19$ \( (T^{6} - 7 T^{4} + 14 T^{2} - 7)^{4} \) Copy content Toggle raw display
$23$ \( T^{24} \) Copy content Toggle raw display
$29$ \( T^{24} \) Copy content Toggle raw display
$31$ \( (T^{6} + 7 T^{4} + 14 T^{2} + 7)^{4} \) Copy content Toggle raw display
$37$ \( T^{24} + 7 T^{20} + 49 T^{16} + \cdots + 2401 \) Copy content Toggle raw display
$41$ \( T^{24} \) Copy content Toggle raw display
$43$ \( T^{24} - 28 T^{20} + 294 T^{16} + \cdots + 2401 \) 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
$61$ \( (T^{6} + 7 T^{5} + 21 T^{4} + 35 T^{3} + \cdots + 7)^{4} \) Copy content Toggle raw display
$67$ \( (T^{12} + 21 T^{8} + 98 T^{4} + 49)^{2} \) Copy content Toggle raw display
$71$ \( T^{24} \) Copy content Toggle raw display
$73$ \( T^{24} + 12 T^{20} + 102 T^{16} - 50 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{4} \) Copy content Toggle raw display
$83$ \( T^{24} \) Copy content Toggle raw display
$89$ \( T^{24} \) Copy content Toggle raw display
$97$ \( (T^{12} + 13 T^{8} + 26 T^{4} + 1)^{2} \) Copy content Toggle raw display
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