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Properties

Label 3332.1.cc.b
Level $3332$
Weight $1$
Character orbit 3332.cc
Analytic conductor $1.663$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -68
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,1,Mod(135,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 32, 21]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.135");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.cc (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: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q - \zeta_{42}^{17} q^{2} + ( - \zeta_{42}^{11} + 1) q^{3} - \zeta_{42}^{13} q^{4} + ( - \zeta_{42}^{17} - \zeta_{42}^{7}) q^{6} - \zeta_{42}^{9} q^{7} - \zeta_{42}^{9} q^{8} + ( - \zeta_{42}^{11} - \zeta_{42} + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{42}^{17} q^{2} + ( - \zeta_{42}^{11} + 1) q^{3} - \zeta_{42}^{13} q^{4} + ( - \zeta_{42}^{17} - \zeta_{42}^{7}) q^{6} - \zeta_{42}^{9} q^{7} - \zeta_{42}^{9} q^{8} + ( - \zeta_{42}^{11} - \zeta_{42} + 1) q^{9} + (\zeta_{42}^{14} + \zeta_{42}^{6}) q^{11} + ( - \zeta_{42}^{13} - \zeta_{42}^{3}) q^{12} + (\zeta_{42}^{20} - \zeta_{42}^{7}) q^{13} - \zeta_{42}^{5} q^{14} - \zeta_{42}^{5} q^{16} - \zeta_{42} q^{17} + (\zeta_{42}^{18} - \zeta_{42}^{17} - \zeta_{42}^{7}) q^{18} + (\zeta_{42}^{20} - \zeta_{42}^{9}) q^{21} + (\zeta_{42}^{10} + \zeta_{42}^{2}) q^{22} + ( - \zeta_{42}^{17} + \zeta_{42}^{2}) q^{23} + (\zeta_{42}^{20} - \zeta_{42}^{9}) q^{24} + \zeta_{42}^{4} q^{25} + (\zeta_{42}^{16} - \zeta_{42}^{3}) q^{26} + (\zeta_{42}^{12} - \zeta_{42}^{11} + \zeta_{42} + 1) q^{27} - \zeta_{42} q^{28} + ( - \zeta_{42}^{19} + \zeta_{42}^{16}) q^{31} - \zeta_{42} q^{32} + ( - \zeta_{42}^{17} + \zeta_{42}^{14} + \zeta_{42}^{6} + \zeta_{42}^{4}) q^{33} + \zeta_{42}^{18} q^{34} + (\zeta_{42}^{14} - \zeta_{42}^{13} - \zeta_{42}^{3}) q^{36} + (\zeta_{42}^{20} + \zeta_{42}^{18} + \zeta_{42}^{10} - \zeta_{42}^{7}) q^{39} + (\zeta_{42}^{16} - \zeta_{42}^{5}) q^{42} + ( - \zeta_{42}^{19} + \zeta_{42}^{6}) q^{44} + ( - \zeta_{42}^{19} - \zeta_{42}^{13}) q^{46} + (\zeta_{42}^{16} - \zeta_{42}^{5}) q^{48} + \zeta_{42}^{18} q^{49} + q^{50} + (\zeta_{42}^{12} - \zeta_{42}) q^{51} + (\zeta_{42}^{20} + \zeta_{42}^{12}) q^{52} + ( - \zeta_{42}^{15} - \zeta_{42}^{11}) q^{53} + (\zeta_{42}^{18} - \zeta_{42}^{17} + \zeta_{42}^{8} - \zeta_{42}^{7}) q^{54} + \zeta_{42}^{18} q^{56} + ( - \zeta_{42}^{15} + \zeta_{42}^{12}) q^{62} + (\zeta_{42}^{20} + \zeta_{42}^{10} - \zeta_{42}^{9}) q^{63} + \zeta_{42}^{18} q^{64} + ( - \zeta_{42}^{13} + \zeta_{42}^{10} + \zeta_{42}^{2} + 1) q^{66} + \zeta_{42}^{14} q^{68} + ( - \zeta_{42}^{17} - \zeta_{42}^{13} - \zeta_{42}^{7} + \zeta_{42}^{2}) q^{69} + (\zeta_{42}^{8} + \zeta_{42}^{4}) q^{71} + (\zeta_{42}^{20} + \zeta_{42}^{10} - \zeta_{42}^{9}) q^{72} + ( - \zeta_{42}^{15} + \zeta_{42}^{4}) q^{75} + ( - \zeta_{42}^{15} + \zeta_{42}^{2}) q^{77} + (\zeta_{42}^{16} + \zeta_{42}^{14} + \zeta_{42}^{6} - \zeta_{42}^{3}) q^{78} + (\zeta_{42}^{4} - \zeta_{42}^{3}) q^{79} + (\zeta_{42}^{12} - \zeta_{42}^{11} + \zeta_{42}^{2} + \zeta_{42} + 1) q^{81} + (\zeta_{42}^{12} - \zeta_{42}) q^{84} + ( - \zeta_{42}^{15} + \zeta_{42}^{2}) q^{88} + (\zeta_{42}^{16} + \zeta_{42}^{6}) q^{89} + (\zeta_{42}^{16} + \zeta_{42}^{8}) q^{91} + ( - \zeta_{42}^{15} - \zeta_{42}^{9}) q^{92} + ( - \zeta_{42}^{19} + \zeta_{42}^{16} - \zeta_{42}^{9} + \zeta_{42}^{6}) q^{93} + (\zeta_{42}^{12} - \zeta_{42}) q^{96} + \zeta_{42}^{14} q^{98} + ( - \zeta_{42}^{17} - \zeta_{42}^{15} + \zeta_{42}^{14} - \zeta_{42}^{7} + \zeta_{42}^{6} + \zeta_{42}^{4}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} + 13 q^{3} + q^{4} - 5 q^{6} - 2 q^{7} - 2 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{2} + 13 q^{3} + q^{4} - 5 q^{6} - 2 q^{7} - 2 q^{8} + 14 q^{9} - 8 q^{11} - q^{12} - 5 q^{13} + q^{14} + q^{16} + q^{17} - 7 q^{18} - q^{21} + 2 q^{22} + 2 q^{23} - q^{24} + q^{25} - q^{26} + 12 q^{27} + q^{28} + 2 q^{31} + q^{32} - 6 q^{33} - 2 q^{34} - 7 q^{36} - 6 q^{39} + 2 q^{42} - q^{44} + 2 q^{46} + 2 q^{48} - 2 q^{49} + 12 q^{50} - q^{51} - q^{52} - q^{53} - 6 q^{54} - 2 q^{56} - 4 q^{62} - 2 q^{64} + 15 q^{66} - 6 q^{68} - 3 q^{69} + 2 q^{71} - q^{75} - q^{77} - 9 q^{78} - q^{79} + 13 q^{81} - q^{84} - q^{88} - q^{89} + 2 q^{91} - 4 q^{92} - 2 q^{93} - q^{96} - 6 q^{98} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(-\zeta_{42}^{11}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
135.1
−0.988831 − 0.149042i
−0.988831 + 0.149042i
0.365341 − 0.930874i
−0.733052 − 0.680173i
0.826239 + 0.563320i
0.0747301 − 0.997204i
−0.733052 + 0.680173i
0.826239 − 0.563320i
0.955573 + 0.294755i
0.955573 − 0.294755i
0.365341 + 0.930874i
0.0747301 + 0.997204i
0.826239 − 0.563320i 1.07473 − 0.997204i 0.365341 − 0.930874i 0 0.326239 − 1.42935i −0.222521 − 0.974928i −0.222521 − 0.974928i 0.0858993 − 1.14625i 0
543.1 0.826239 + 0.563320i 1.07473 + 0.997204i 0.365341 + 0.930874i 0 0.326239 + 1.42935i −0.222521 + 0.974928i −0.222521 + 0.974928i 0.0858993 + 1.14625i 0
611.1 0.0747301 − 0.997204i 1.82624 − 0.563320i −0.988831 − 0.149042i 0 −0.425270 − 1.86323i −0.222521 + 0.974928i −0.222521 + 0.974928i 2.19158 − 1.49419i 0
1019.1 −0.988831 − 0.149042i 1.36534 − 0.930874i 0.955573 + 0.294755i 0 −1.48883 + 0.716983i −0.900969 − 0.433884i −0.900969 − 0.433884i 0.632289 − 1.61105i 0
1087.1 −0.733052 − 0.680173i 1.95557 + 0.294755i 0.0747301 + 0.997204i 0 −1.23305 − 1.54620i 0.623490 − 0.781831i 0.623490 − 0.781831i 2.78181 + 0.858075i 0
1495.1 0.955573 − 0.294755i 0.266948 + 0.680173i 0.826239 − 0.563320i 0 0.455573 + 0.571270i 0.623490 − 0.781831i 0.623490 − 0.781831i 0.341678 − 0.317031i 0
1563.1 −0.988831 + 0.149042i 1.36534 + 0.930874i 0.955573 − 0.294755i 0 −1.48883 − 0.716983i −0.900969 + 0.433884i −0.900969 + 0.433884i 0.632289 + 1.61105i 0
1971.1 −0.733052 + 0.680173i 1.95557 − 0.294755i 0.0747301 − 0.997204i 0 −1.23305 + 1.54620i 0.623490 + 0.781831i 0.623490 + 0.781831i 2.78181 − 0.858075i 0
2447.1 0.365341 − 0.930874i 0.0111692 − 0.149042i −0.733052 − 0.680173i 0 −0.134659 − 0.0648483i −0.900969 + 0.433884i −0.900969 + 0.433884i 0.966742 + 0.145713i 0
2515.1 0.365341 + 0.930874i 0.0111692 + 0.149042i −0.733052 + 0.680173i 0 −0.134659 + 0.0648483i −0.900969 − 0.433884i −0.900969 − 0.433884i 0.966742 − 0.145713i 0
2923.1 0.0747301 + 0.997204i 1.82624 + 0.563320i −0.988831 + 0.149042i 0 −0.425270 + 1.86323i −0.222521 − 0.974928i −0.222521 − 0.974928i 2.19158 + 1.49419i 0
2991.1 0.955573 + 0.294755i 0.266948 − 0.680173i 0.826239 + 0.563320i 0 0.455573 − 0.571270i 0.623490 + 0.781831i 0.623490 + 0.781831i 0.341678 + 0.317031i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2991.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
49.g even 21 1 inner
3332.cc odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.cc.b yes 12
4.b odd 2 1 3332.1.cc.a ✓ 12
17.b even 2 1 3332.1.cc.a ✓ 12
49.g even 21 1 inner 3332.1.cc.b yes 12
68.d odd 2 1 CM 3332.1.cc.b yes 12
196.o odd 42 1 3332.1.cc.a ✓ 12
833.z even 42 1 3332.1.cc.a ✓ 12
3332.cc odd 42 1 inner 3332.1.cc.b yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.cc.a ✓ 12 4.b odd 2 1
3332.1.cc.a ✓ 12 17.b even 2 1
3332.1.cc.a ✓ 12 196.o odd 42 1
3332.1.cc.a ✓ 12 833.z even 42 1
3332.1.cc.b yes 12 1.a even 1 1 trivial
3332.1.cc.b yes 12 49.g even 21 1 inner
3332.1.cc.b yes 12 68.d odd 2 1 CM
3332.1.cc.b yes 12 3332.cc odd 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 13 T_{3}^{11} + 77 T_{3}^{10} - 274 T_{3}^{9} + 650 T_{3}^{8} - 1078 T_{3}^{7} + 1275 T_{3}^{6} - 1078 T_{3}^{5} + 643 T_{3}^{4} - 260 T_{3}^{3} + 63 T_{3}^{2} - 6 T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{11} + T^{9} - T^{8} + T^{6} - T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - 13 T^{11} + 77 T^{10} - 274 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 8 T^{11} + 35 T^{10} + 104 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} + 5 T^{11} + 17 T^{10} + 38 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} - T^{11} + T^{9} - T^{8} + T^{6} - T^{4} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} - 2 T^{11} - 6 T^{9} + 12 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( (T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} + T^{11} - T^{9} + 6 T^{8} - 21 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} - 2 T^{11} + 3 T^{10} - 4 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} + T^{11} + 7 T^{10} + 6 T^{9} + 34 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} + T^{11} + 6 T^{9} + 6 T^{8} + 7 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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