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Properties

Label 3528.1.fb.a
Level $3528$
Weight $1$
Character orbit 3528.fb
Analytic conductor $1.761$
Analytic rank $0$
Dimension $24$
Projective image $D_{42}$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3528,1,Mod(397,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 21, 0, 29]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.397");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.fb (of order \(42\), 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: \(1.76070136457\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{42}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{84}^{31} q^{2} - \zeta_{84}^{20} q^{4} + (\zeta_{84}^{41} - \zeta_{84}^{33}) q^{5} - \zeta_{84}^{26} q^{7} - \zeta_{84}^{9} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{84}^{31} q^{2} - \zeta_{84}^{20} q^{4} + (\zeta_{84}^{41} - \zeta_{84}^{33}) q^{5} - \zeta_{84}^{26} q^{7} - \zeta_{84}^{9} q^{8} + (\zeta_{84}^{30} - \zeta_{84}^{22}) q^{10} + ( - \zeta_{84}^{39} + \zeta_{84}^{37}) q^{11} - \zeta_{84}^{15} q^{14} + \zeta_{84}^{40} q^{16} + (\zeta_{84}^{19} - \zeta_{84}^{11}) q^{20} + ( - \zeta_{84}^{28} + \zeta_{84}^{26}) q^{22} + ( - \zeta_{84}^{40} + \cdots - \zeta_{84}^{24}) q^{25} + \cdots + \zeta_{84}^{41} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 2 q^{4} + 2 q^{7} + 6 q^{10} + 2 q^{16} + 10 q^{22} + 4 q^{25} - 2 q^{28} - 6 q^{31} - 22 q^{40} + 2 q^{49} - 14 q^{55} - 16 q^{58} + 4 q^{64} + 8 q^{70} + 2 q^{79} - 2 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(-\zeta_{84}^{4}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
397.1
−0.294755 + 0.955573i
0.294755 − 0.955573i
0.930874 + 0.365341i
−0.930874 − 0.365341i
−0.294755 − 0.955573i
0.294755 + 0.955573i
0.680173 − 0.733052i
−0.680173 + 0.733052i
0.563320 − 0.826239i
−0.563320 + 0.826239i
0.563320 + 0.826239i
−0.563320 − 0.826239i
−0.997204 + 0.0747301i
0.997204 − 0.0747301i
0.930874 − 0.365341i
−0.930874 + 0.365341i
0.680173 + 0.733052i
−0.680173 − 0.733052i
0.149042 + 0.988831i
−0.149042 − 0.988831i
−0.149042 − 0.988831i 0 −0.955573 + 0.294755i −0.139129 + 1.85654i 0 0.0747301 + 0.997204i 0.433884 + 0.900969i 0 1.85654 − 0.139129i
397.2 0.149042 + 0.988831i 0 −0.955573 + 0.294755i 0.139129 − 1.85654i 0 0.0747301 + 0.997204i −0.433884 − 0.900969i 0 1.85654 − 0.139129i
829.1 −0.563320 + 0.826239i 0 −0.365341 − 0.930874i −1.90580 + 0.587862i 0 0.955573 + 0.294755i 0.974928 + 0.222521i 0 0.587862 − 1.90580i
829.2 0.563320 − 0.826239i 0 −0.365341 − 0.930874i 1.90580 − 0.587862i 0 0.955573 + 0.294755i −0.974928 − 0.222521i 0 0.587862 − 1.90580i
1333.1 −0.149042 + 0.988831i 0 −0.955573 − 0.294755i −0.139129 − 1.85654i 0 0.0747301 − 0.997204i 0.433884 − 0.900969i 0 1.85654 + 0.139129i
1333.2 0.149042 − 0.988831i 0 −0.955573 − 0.294755i 0.139129 + 1.85654i 0 0.0747301 − 0.997204i −0.433884 + 0.900969i 0 1.85654 + 0.139129i
1405.1 −0.930874 + 0.365341i 0 0.733052 − 0.680173i −0.246289 + 0.167917i 0 0.826239 + 0.563320i −0.433884 + 0.900969i 0 0.167917 − 0.246289i
1405.2 0.930874 − 0.365341i 0 0.733052 − 0.680173i 0.246289 − 0.167917i 0 0.826239 + 0.563320i 0.433884 − 0.900969i 0 0.167917 − 0.246289i
1837.1 −0.294755 − 0.955573i 0 −0.826239 + 0.563320i −1.34515 − 0.202749i 0 −0.988831 + 0.149042i 0.781831 + 0.623490i 0 0.202749 + 1.34515i
1837.2 0.294755 + 0.955573i 0 −0.826239 + 0.563320i 1.34515 + 0.202749i 0 −0.988831 + 0.149042i −0.781831 − 0.623490i 0 0.202749 + 1.34515i
1909.1 −0.294755 + 0.955573i 0 −0.826239 − 0.563320i −1.34515 + 0.202749i 0 −0.988831 − 0.149042i 0.781831 − 0.623490i 0 0.202749 − 1.34515i
1909.2 0.294755 − 0.955573i 0 −0.826239 − 0.563320i 1.34515 − 0.202749i 0 −0.988831 − 0.149042i −0.781831 + 0.623490i 0 0.202749 − 1.34515i
2341.1 −0.680173 − 0.733052i 0 −0.0747301 + 0.997204i 0.215372 − 0.548760i 0 0.365341 + 0.930874i 0.781831 − 0.623490i 0 −0.548760 + 0.215372i
2341.2 0.680173 + 0.733052i 0 −0.0747301 + 0.997204i −0.215372 + 0.548760i 0 0.365341 + 0.930874i −0.781831 + 0.623490i 0 −0.548760 + 0.215372i
2413.1 −0.563320 − 0.826239i 0 −0.365341 + 0.930874i −1.90580 − 0.587862i 0 0.955573 − 0.294755i 0.974928 − 0.222521i 0 0.587862 + 1.90580i
2413.2 0.563320 + 0.826239i 0 −0.365341 + 0.930874i 1.90580 + 0.587862i 0 0.955573 − 0.294755i −0.974928 + 0.222521i 0 0.587862 + 1.90580i
2845.1 −0.930874 − 0.365341i 0 0.733052 + 0.680173i −0.246289 − 0.167917i 0 0.826239 − 0.563320i −0.433884 − 0.900969i 0 0.167917 + 0.246289i
2845.2 0.930874 + 0.365341i 0 0.733052 + 0.680173i 0.246289 + 0.167917i 0 0.826239 − 0.563320i 0.433884 + 0.900969i 0 0.167917 + 0.246289i
2917.1 −0.997204 − 0.0747301i 0 0.988831 + 0.149042i 0.825886 + 0.766310i 0 −0.733052 + 0.680173i −0.974928 − 0.222521i 0 −0.766310 − 0.825886i
2917.2 0.997204 + 0.0747301i 0 0.988831 + 0.149042i −0.825886 − 0.766310i 0 −0.733052 + 0.680173i 0.974928 + 0.222521i 0 −0.766310 − 0.825886i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 397.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
49.h odd 42 1 inner
147.o even 42 1 inner
392.bf odd 42 1 inner
1176.cc even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.fb.a ✓ 24
3.b odd 2 1 inner 3528.1.fb.a ✓ 24
8.b even 2 1 inner 3528.1.fb.a ✓ 24
24.h odd 2 1 CM 3528.1.fb.a ✓ 24
49.h odd 42 1 inner 3528.1.fb.a ✓ 24
147.o even 42 1 inner 3528.1.fb.a ✓ 24
392.bf odd 42 1 inner 3528.1.fb.a ✓ 24
1176.cc even 42 1 inner 3528.1.fb.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.fb.a ✓ 24 1.a even 1 1 trivial
3528.1.fb.a ✓ 24 3.b odd 2 1 inner
3528.1.fb.a ✓ 24 8.b even 2 1 inner
3528.1.fb.a ✓ 24 24.h odd 2 1 CM
3528.1.fb.a ✓ 24 49.h odd 42 1 inner
3528.1.fb.a ✓ 24 147.o even 42 1 inner
3528.1.fb.a ✓ 24 392.bf odd 42 1 inner
3528.1.fb.a ✓ 24 1176.cc even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3528, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} + T^{22} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{24} \) Copy content Toggle raw display
$5$ \( T^{24} - 3 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{12} - T^{11} + T^{9} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{24} + T^{22} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{24} \) Copy content Toggle raw display
$17$ \( T^{24} \) Copy content Toggle raw display
$19$ \( T^{24} \) Copy content Toggle raw display
$23$ \( T^{24} \) Copy content Toggle raw display
$29$ \( T^{24} - 2 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{12} + 3 T^{11} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{24} \) Copy content Toggle raw display
$41$ \( T^{24} \) Copy content Toggle raw display
$43$ \( T^{24} \) Copy content Toggle raw display
$47$ \( T^{24} \) Copy content Toggle raw display
$53$ \( T^{24} + T^{22} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{24} - 3 T^{22} + \cdots + 531441 \) Copy content Toggle raw display
$61$ \( T^{24} \) Copy content Toggle raw display
$67$ \( T^{24} \) Copy content Toggle raw display
$71$ \( T^{24} \) Copy content Toggle raw display
$73$ \( (T^{12} + 14 T^{9} + \cdots + 49)^{2} \) Copy content Toggle raw display
$79$ \( (T^{12} - T^{11} + 7 T^{10} + \cdots + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{24} - T^{22} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{24} \) Copy content Toggle raw display
$97$ \( (T^{12} + 11 T^{10} + \cdots + 1)^{2} \) Copy content Toggle raw display
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