Properties

Label 3332.1.cc.c
Level $3332$
Weight $1$
Character orbit 3332.cc
Analytic conductor $1.663$
Analytic rank $0$
Dimension $24$
Projective image $D_{42}$
CM discriminant -68
Inner twists $8$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{42})\)
Coefficient field: \(\Q(\zeta_{84})\)
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: \(D_{42}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{84}^{40} q^{2} + ( - \zeta_{84}^{37} + \zeta_{84}^{21}) q^{3} - \zeta_{84}^{38} q^{4} + ( - \zeta_{84}^{35} + \zeta_{84}^{19}) q^{6} - \zeta_{84}^{15} q^{7} - \zeta_{84}^{36} q^{8} + ( - \zeta_{84}^{32} + \zeta_{84}^{16} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{84}^{40} q^{2} + ( - \zeta_{84}^{37} + \zeta_{84}^{21}) q^{3} - \zeta_{84}^{38} q^{4} + ( - \zeta_{84}^{35} + \zeta_{84}^{19}) q^{6} - \zeta_{84}^{15} q^{7} - \zeta_{84}^{36} q^{8} + ( - \zeta_{84}^{32} + \zeta_{84}^{16} - 1) q^{9} + ( - \zeta_{84}^{7} + \zeta_{84}^{3}) q^{11} + ( - \zeta_{84}^{33} + \zeta_{84}^{17}) q^{12} + (\zeta_{84}^{14} + \zeta_{84}^{10}) q^{13} - \zeta_{84}^{13} q^{14} - \zeta_{84}^{34} q^{16} + \zeta_{84}^{32} q^{17} + (\zeta_{84}^{40} - \zeta_{84}^{30} + \zeta_{84}^{14}) q^{18} + ( - \zeta_{84}^{36} - \zeta_{84}^{10}) q^{21} + ( - \zeta_{84}^{5} + \zeta_{84}) q^{22} + ( - \zeta_{84}^{19} - \zeta_{84}) q^{23} + ( - \zeta_{84}^{31} + \zeta_{84}^{15}) q^{24} - \zeta_{84}^{2} q^{25} + (\zeta_{84}^{12} + \zeta_{84}^{8}) q^{26} + (\zeta_{84}^{37} - \zeta_{84}^{27} - \zeta_{84}^{21} - \zeta_{84}^{11}) q^{27} - \zeta_{84}^{11} q^{28} + ( - \zeta_{84}^{41} + \zeta_{84}^{29}) q^{31} - \zeta_{84}^{32} q^{32} + ( - \zeta_{84}^{40} - \zeta_{84}^{28} + \zeta_{84}^{24} - \zeta_{84}^{2}) q^{33} + \zeta_{84}^{30} q^{34} + (\zeta_{84}^{38} - \zeta_{84}^{28} + \zeta_{84}^{12}) q^{36} + (\zeta_{84}^{35} + \zeta_{84}^{31} + \zeta_{84}^{9} + \zeta_{84}^{5}) q^{39} + ( - \zeta_{84}^{34} - \zeta_{84}^{8}) q^{42} + ( - \zeta_{84}^{41} - \zeta_{84}^{3}) q^{44} + (\zeta_{84}^{41} - \zeta_{84}^{17}) q^{46} + ( - \zeta_{84}^{29} + \zeta_{84}^{13}) q^{48} + \zeta_{84}^{30} q^{49} - q^{50} + (\zeta_{84}^{27} - \zeta_{84}^{11}) q^{51} + (\zeta_{84}^{10} + \zeta_{84}^{6}) q^{52} + ( - \zeta_{84}^{18} + \zeta_{84}^{16}) q^{53} + (\zeta_{84}^{35} - \zeta_{84}^{25} - \zeta_{84}^{19} + \zeta_{84}^{9}) q^{54} - \zeta_{84}^{9} q^{56} + ( - \zeta_{84}^{39} + \zeta_{84}^{27}) q^{62} + ( - \zeta_{84}^{31} + \zeta_{84}^{15} - \zeta_{84}^{5}) q^{63} - \zeta_{84}^{30} q^{64} + ( - \zeta_{84}^{38} - \zeta_{84}^{26} + \zeta_{84}^{22} - 1) q^{66} + \zeta_{84}^{28} q^{68} + ( - \zeta_{84}^{40} + \zeta_{84}^{38} - \zeta_{84}^{22} - \zeta_{84}^{14}) q^{69} + (\zeta_{84}^{25} + \zeta_{84}^{23}) q^{71} + (\zeta_{84}^{36} - \zeta_{84}^{26} + \zeta_{84}^{10}) q^{72} + (\zeta_{84}^{39} - \zeta_{84}^{23}) q^{75} + (\zeta_{84}^{22} - \zeta_{84}^{18}) q^{77} + (\zeta_{84}^{33} + \zeta_{84}^{29} + \zeta_{84}^{7} + \zeta_{84}^{3}) q^{78} + (\zeta_{84}^{33} + \zeta_{84}^{23}) q^{79} + (\zeta_{84}^{32} - \zeta_{84}^{22} - \zeta_{84}^{16} - \zeta_{84}^{6} + 1) q^{81} + ( - \zeta_{84}^{32} - \zeta_{84}^{6}) q^{84} + ( - \zeta_{84}^{39} - \zeta_{84}) q^{88} + ( - \zeta_{84}^{24} - \zeta_{84}^{8}) q^{89} + ( - \zeta_{84}^{29} - \zeta_{84}^{25}) q^{91} + (\zeta_{84}^{39} - \zeta_{84}^{15}) q^{92} + ( - \zeta_{84}^{36} + \zeta_{84}^{24} + \zeta_{84}^{20} - \zeta_{84}^{8}) q^{93} + ( - \zeta_{84}^{27} + \zeta_{84}^{11}) q^{96} + \zeta_{84}^{28} q^{98} + (\zeta_{84}^{39} - \zeta_{84}^{35} - \zeta_{84}^{23} + \zeta_{84}^{19} + \zeta_{84}^{7} - \zeta_{84}^{3}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9} + 10 q^{13} + 2 q^{16} + 2 q^{17} + 10 q^{18} + 6 q^{21} + 2 q^{25} - 2 q^{26} - 2 q^{32} + 8 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{49} - 24 q^{50} + 2 q^{52} - 2 q^{53} - 4 q^{64} - 22 q^{66} - 12 q^{68} - 14 q^{69} - 4 q^{72} - 6 q^{77} + 30 q^{81} - 6 q^{84} + 2 q^{89} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(\zeta_{84}^{16}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
135.1
−0.294755 + 0.955573i
0.294755 0.955573i
−0.294755 0.955573i
0.294755 + 0.955573i
0.680173 0.733052i
−0.680173 + 0.733052i
−0.997204 + 0.0747301i
0.997204 0.0747301i
0.930874 0.365341i
−0.930874 + 0.365341i
−0.149042 + 0.988831i
0.149042 0.988831i
−0.997204 0.0747301i
0.997204 + 0.0747301i
0.930874 + 0.365341i
−0.930874 0.365341i
−0.563320 + 0.826239i
0.563320 0.826239i
−0.563320 0.826239i
0.563320 + 0.826239i
−0.826239 + 0.563320i −0.997204 + 0.925270i 0.365341 0.930874i 0 0.302705 1.32624i 0.974928 0.222521i 0.222521 + 0.974928i 0.0635609 0.848162i 0
135.2 −0.826239 + 0.563320i 0.997204 0.925270i 0.365341 0.930874i 0 −0.302705 + 1.32624i −0.974928 + 0.222521i 0.222521 + 0.974928i 0.0635609 0.848162i 0
543.1 −0.826239 0.563320i −0.997204 0.925270i 0.365341 + 0.930874i 0 0.302705 + 1.32624i 0.974928 + 0.222521i 0.222521 0.974928i 0.0635609 + 0.848162i 0
543.2 −0.826239 0.563320i 0.997204 + 0.925270i 0.365341 + 0.930874i 0 −0.302705 1.32624i −0.974928 0.222521i 0.222521 0.974928i 0.0635609 + 0.848162i 0
611.1 −0.0747301 + 0.997204i −0.563320 + 0.173761i −0.988831 0.149042i 0 −0.131178 0.574730i −0.974928 0.222521i 0.222521 0.974928i −0.539102 + 0.367554i 0
611.2 −0.0747301 + 0.997204i 0.563320 0.173761i −0.988831 0.149042i 0 0.131178 + 0.574730i 0.974928 + 0.222521i 0.222521 0.974928i −0.539102 + 0.367554i 0
1019.1 0.988831 + 0.149042i −0.930874 + 0.634659i 0.955573 + 0.294755i 0 −1.01507 + 0.488831i 0.433884 0.900969i 0.900969 + 0.433884i 0.0983929 0.250701i 0
1019.2 0.988831 + 0.149042i 0.930874 0.634659i 0.955573 + 0.294755i 0 1.01507 0.488831i −0.433884 + 0.900969i 0.900969 + 0.433884i 0.0983929 0.250701i 0
1087.1 0.733052 + 0.680173i −0.294755 0.0444272i 0.0747301 + 0.997204i 0 −0.185853 0.233052i −0.781831 0.623490i −0.623490 + 0.781831i −0.870666 0.268565i 0
1087.2 0.733052 + 0.680173i 0.294755 + 0.0444272i 0.0747301 + 0.997204i 0 0.185853 + 0.233052i 0.781831 + 0.623490i −0.623490 + 0.781831i −0.870666 0.268565i 0
1495.1 −0.955573 + 0.294755i −0.680173 1.73305i 0.826239 0.563320i 0 1.16078 + 1.45557i −0.781831 0.623490i −0.623490 + 0.781831i −1.80778 + 1.67738i 0
1495.2 −0.955573 + 0.294755i 0.680173 + 1.73305i 0.826239 0.563320i 0 −1.16078 1.45557i 0.781831 + 0.623490i −0.623490 + 0.781831i −1.80778 + 1.67738i 0
1563.1 0.988831 0.149042i −0.930874 0.634659i 0.955573 0.294755i 0 −1.01507 0.488831i 0.433884 + 0.900969i 0.900969 0.433884i 0.0983929 + 0.250701i 0
1563.2 0.988831 0.149042i 0.930874 + 0.634659i 0.955573 0.294755i 0 1.01507 + 0.488831i −0.433884 0.900969i 0.900969 0.433884i 0.0983929 + 0.250701i 0
1971.1 0.733052 0.680173i −0.294755 + 0.0444272i 0.0747301 0.997204i 0 −0.185853 + 0.233052i −0.781831 + 0.623490i −0.623490 0.781831i −0.870666 + 0.268565i 0
1971.2 0.733052 0.680173i 0.294755 0.0444272i 0.0747301 0.997204i 0 0.185853 0.233052i 0.781831 0.623490i −0.623490 0.781831i −0.870666 + 0.268565i 0
2447.1 −0.365341 + 0.930874i −0.149042 + 1.98883i −0.733052 0.680173i 0 −1.79690 0.865341i −0.433884 0.900969i 0.900969 0.433884i −2.94440 0.443797i 0
2447.2 −0.365341 + 0.930874i 0.149042 1.98883i −0.733052 0.680173i 0 1.79690 + 0.865341i 0.433884 + 0.900969i 0.900969 0.433884i −2.94440 0.443797i 0
2515.1 −0.365341 0.930874i −0.149042 1.98883i −0.733052 + 0.680173i 0 −1.79690 + 0.865341i −0.433884 + 0.900969i 0.900969 + 0.433884i −2.94440 + 0.443797i 0
2515.2 −0.365341 0.930874i 0.149042 + 1.98883i −0.733052 + 0.680173i 0 1.79690 0.865341i 0.433884 0.900969i 0.900969 + 0.433884i −2.94440 + 0.443797i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2991.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
4.b odd 2 1 inner
17.b even 2 1 inner
49.g even 21 1 inner
196.o odd 42 1 inner
833.z even 42 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.c 24
4.b odd 2 1 inner 3332.1.cc.c 24
17.b even 2 1 inner 3332.1.cc.c 24
49.g even 21 1 inner 3332.1.cc.c 24
68.d odd 2 1 CM 3332.1.cc.c 24
196.o odd 42 1 inner 3332.1.cc.c 24
833.z even 42 1 inner 3332.1.cc.c 24
3332.cc odd 42 1 inner 3332.1.cc.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.cc.c 24 1.a even 1 1 trivial
3332.1.cc.c 24 4.b odd 2 1 inner
3332.1.cc.c 24 17.b even 2 1 inner
3332.1.cc.c 24 49.g even 21 1 inner
3332.1.cc.c 24 68.d odd 2 1 CM
3332.1.cc.c 24 196.o odd 42 1 inner
3332.1.cc.c 24 833.z even 42 1 inner
3332.1.cc.c 24 3332.cc odd 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 11 T_{3}^{22} + 49 T_{3}^{20} + 118 T_{3}^{18} + 192 T_{3}^{16} + 336 T_{3}^{14} + 421 T_{3}^{12} - 168 T_{3}^{10} + 999 T_{3}^{8} - 736 T_{3}^{6} + 231 T_{3}^{4} - 26 T_{3}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{12} + T^{11} - T^{9} - T^{8} + T^{6} - T^{4} - T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{24} + 11 T^{22} + 49 T^{20} + 118 T^{18} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{24} \) Copy content Toggle raw display
$7$ \( (T^{12} - T^{10} + T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{24} - 10 T^{22} + 49 T^{20} - 134 T^{18} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{12} - 5 T^{11} + 17 T^{10} - 38 T^{9} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{12} - T^{11} + T^{9} - T^{8} + T^{6} - T^{4} + T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{24} \) Copy content Toggle raw display
$23$ \( T^{24} + 70 T^{18} - 98 T^{16} + \cdots + 2401 \) Copy content Toggle raw display
$29$ \( T^{24} \) Copy content Toggle raw display
$31$ \( (T^{12} + 7 T^{10} + 35 T^{8} + 84 T^{6} + \cdots + 49)^{2} \) 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
$53$ \( (T^{12} + T^{11} - T^{9} + 6 T^{8} - 21 T^{7} + \cdots + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{24} \) Copy content Toggle raw display
$61$ \( T^{24} \) Copy content Toggle raw display
$67$ \( T^{24} \) Copy content Toggle raw display
$71$ \( T^{24} + 6 T^{22} + 13 T^{20} + 38 T^{18} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{24} \) Copy content Toggle raw display
$79$ \( T^{24} + 11 T^{22} + 77 T^{20} + 328 T^{18} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{24} \) Copy content Toggle raw display
$89$ \( (T^{12} - T^{11} - 6 T^{9} + 6 T^{8} - 7 T^{7} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{24} \) Copy content Toggle raw display
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