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: | \(12\) |
Coefficient field: | \(\Q(\zeta_{21})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: |
\( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \)
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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
The \(q\)-expansion and trace form are shown below.
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.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
135.1 |
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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 | |||||||||||||||||||||||||||||||||||||||||
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 \(S_{1}^{\mathrm{new}}(3332, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - T^{11} + T^{9} - T^{8} + T^{6} - T^{4} + \cdots + 1 \)
$3$
\( T^{12} - 13 T^{11} + 77 T^{10} - 274 T^{9} + \cdots + 1 \)
$5$
\( T^{12} \)
$7$
\( (T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$11$
\( T^{12} + 8 T^{11} + 35 T^{10} + 104 T^{9} + \cdots + 1 \)
$13$
\( T^{12} + 5 T^{11} + 17 T^{10} + 38 T^{9} + \cdots + 1 \)
$17$
\( T^{12} - T^{11} + T^{9} - T^{8} + T^{6} - T^{4} + \cdots + 1 \)
$19$
\( T^{12} \)
$23$
\( T^{12} - 2 T^{11} - 6 T^{9} + 12 T^{8} + \cdots + 1 \)
$29$
\( T^{12} \)
$31$
\( (T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2} \)
$37$
\( T^{12} \)
$41$
\( T^{12} \)
$43$
\( T^{12} \)
$47$
\( T^{12} \)
$53$
\( T^{12} + T^{11} - T^{9} + 6 T^{8} - 21 T^{7} + \cdots + 1 \)
$59$
\( T^{12} \)
$61$
\( T^{12} \)
$67$
\( T^{12} \)
$71$
\( T^{12} - 2 T^{11} + 3 T^{10} - 4 T^{9} + \cdots + 1 \)
$73$
\( T^{12} \)
$79$
\( T^{12} + T^{11} + 7 T^{10} + 6 T^{9} + 34 T^{8} + \cdots + 1 \)
$83$
\( T^{12} \)
$89$
\( T^{12} + T^{11} + 6 T^{9} + 6 T^{8} + 7 T^{7} + \cdots + 1 \)
$97$
\( T^{12} \)
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