Newspace parameters
Level: | \( N \) | \(=\) | \( 784 = 2^{4} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 784.bg (of order \(21\), degree \(12\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.26027151847\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{21})\) |
Twist minimal: | no (minimal twist has level 98) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{21}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
65.1 | 0 | −0.0519131 | − | 0.692733i | 0 | 1.52046 | − | 1.03663i | 0 | −2.03934 | + | 1.68555i | 0 | 2.48931 | − | 0.375203i | 0 | ||||||||||
65.2 | 0 | 0.250133 | + | 3.33779i | 0 | −1.09780 | + | 0.748464i | 0 | −2.60370 | − | 0.469838i | 0 | −8.11181 | + | 1.22266i | 0 | ||||||||||
81.1 | 0 | −1.52870 | − | 0.471543i | 0 | −0.144243 | + | 0.133838i | 0 | −2.44775 | + | 1.00425i | 0 | −0.364133 | − | 0.248262i | 0 | ||||||||||
81.2 | 0 | 1.08331 | + | 0.334156i | 0 | 2.80532 | − | 2.60296i | 0 | 2.61464 | + | 0.404557i | 0 | −1.41682 | − | 0.965972i | 0 | ||||||||||
193.1 | 0 | −0.0519131 | + | 0.692733i | 0 | 1.52046 | + | 1.03663i | 0 | −2.03934 | − | 1.68555i | 0 | 2.48931 | + | 0.375203i | 0 | ||||||||||
193.2 | 0 | 0.250133 | − | 3.33779i | 0 | −1.09780 | − | 0.748464i | 0 | −2.60370 | + | 0.469838i | 0 | −8.11181 | − | 1.22266i | 0 | ||||||||||
289.1 | 0 | −0.520118 | − | 0.354611i | 0 | −0.0807922 | + | 1.07810i | 0 | 2.64324 | − | 0.115218i | 0 | −0.951249 | − | 2.42374i | 0 | ||||||||||
289.2 | 0 | 1.46985 | + | 1.00212i | 0 | 0.169008 | − | 2.25526i | 0 | −2.40601 | − | 1.10050i | 0 | 0.0601727 | + | 0.153317i | 0 | ||||||||||
305.1 | 0 | −1.17911 | + | 0.177723i | 0 | −1.10349 | + | 2.81164i | 0 | −1.42353 | + | 2.23015i | 0 | −1.50800 | + | 0.465156i | 0 | ||||||||||
305.2 | 0 | −0.532240 | + | 0.0802223i | 0 | 0.459870 | − | 1.17173i | 0 | 1.68832 | − | 2.03705i | 0 | −2.58987 | + | 0.798871i | 0 | ||||||||||
401.1 | 0 | −1.17911 | − | 0.177723i | 0 | −1.10349 | − | 2.81164i | 0 | −1.42353 | − | 2.23015i | 0 | −1.50800 | − | 0.465156i | 0 | ||||||||||
401.2 | 0 | −0.532240 | − | 0.0802223i | 0 | 0.459870 | + | 1.17173i | 0 | 1.68832 | + | 2.03705i | 0 | −2.58987 | − | 0.798871i | 0 | ||||||||||
417.1 | 0 | −1.04951 | − | 2.67410i | 0 | 0.926902 | + | 0.139708i | 0 | 2.29875 | + | 1.30987i | 0 | −3.85019 | + | 3.57245i | 0 | ||||||||||
417.2 | 0 | 0.692327 | + | 1.76402i | 0 | −4.01065 | − | 0.604508i | 0 | 2.17740 | − | 1.50297i | 0 | −0.433294 | + | 0.402038i | 0 | ||||||||||
513.1 | 0 | −1.52870 | + | 0.471543i | 0 | −0.144243 | − | 0.133838i | 0 | −2.44775 | − | 1.00425i | 0 | −0.364133 | + | 0.248262i | 0 | ||||||||||
513.2 | 0 | 1.08331 | − | 0.334156i | 0 | 2.80532 | + | 2.60296i | 0 | 2.61464 | − | 0.404557i | 0 | −1.41682 | + | 0.965972i | 0 | ||||||||||
529.1 | 0 | −0.520118 | + | 0.354611i | 0 | −0.0807922 | − | 1.07810i | 0 | 2.64324 | + | 0.115218i | 0 | −0.951249 | + | 2.42374i | 0 | ||||||||||
529.2 | 0 | 1.46985 | − | 1.00212i | 0 | 0.169008 | + | 2.25526i | 0 | −2.40601 | + | 1.10050i | 0 | 0.0601727 | − | 0.153317i | 0 | ||||||||||
625.1 | 0 | −2.11865 | + | 1.96582i | 0 | −0.596780 | − | 0.184082i | 0 | 2.13962 | + | 1.55628i | 0 | 0.400039 | − | 5.33815i | 0 | ||||||||||
625.2 | 0 | −0.0153718 | + | 0.0142629i | 0 | 1.15218 | + | 0.355400i | 0 | −2.64164 | − | 0.147471i | 0 | −0.224157 | + | 2.99117i | 0 | ||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
49.g | even | 21 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 784.2.bg.a | 24 | |
4.b | odd | 2 | 1 | 98.2.g.a | ✓ | 24 | |
12.b | even | 2 | 1 | 882.2.z.d | 24 | ||
28.d | even | 2 | 1 | 686.2.g.b | 24 | ||
28.f | even | 6 | 1 | 686.2.e.e | 24 | ||
28.f | even | 6 | 1 | 686.2.g.c | 24 | ||
28.g | odd | 6 | 1 | 686.2.e.f | 24 | ||
28.g | odd | 6 | 1 | 686.2.g.a | 24 | ||
49.g | even | 21 | 1 | inner | 784.2.bg.a | 24 | |
196.j | even | 14 | 1 | 686.2.g.c | 24 | ||
196.k | odd | 14 | 1 | 686.2.g.a | 24 | ||
196.o | odd | 42 | 1 | 98.2.g.a | ✓ | 24 | |
196.o | odd | 42 | 1 | 686.2.e.f | 24 | ||
196.o | odd | 42 | 1 | 4802.2.a.i | 12 | ||
196.p | even | 42 | 1 | 686.2.e.e | 24 | ||
196.p | even | 42 | 1 | 686.2.g.b | 24 | ||
196.p | even | 42 | 1 | 4802.2.a.k | 12 | ||
588.bb | even | 42 | 1 | 882.2.z.d | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
98.2.g.a | ✓ | 24 | 4.b | odd | 2 | 1 | |
98.2.g.a | ✓ | 24 | 196.o | odd | 42 | 1 | |
686.2.e.e | 24 | 28.f | even | 6 | 1 | ||
686.2.e.e | 24 | 196.p | even | 42 | 1 | ||
686.2.e.f | 24 | 28.g | odd | 6 | 1 | ||
686.2.e.f | 24 | 196.o | odd | 42 | 1 | ||
686.2.g.a | 24 | 28.g | odd | 6 | 1 | ||
686.2.g.a | 24 | 196.k | odd | 14 | 1 | ||
686.2.g.b | 24 | 28.d | even | 2 | 1 | ||
686.2.g.b | 24 | 196.p | even | 42 | 1 | ||
686.2.g.c | 24 | 28.f | even | 6 | 1 | ||
686.2.g.c | 24 | 196.j | even | 14 | 1 | ||
784.2.bg.a | 24 | 1.a | even | 1 | 1 | trivial | |
784.2.bg.a | 24 | 49.g | even | 21 | 1 | inner | |
882.2.z.d | 24 | 12.b | even | 2 | 1 | ||
882.2.z.d | 24 | 588.bb | even | 42 | 1 | ||
4802.2.a.i | 12 | 196.o | odd | 42 | 1 | ||
4802.2.a.k | 12 | 196.p | even | 42 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{24} + 7 T_{3}^{23} + 38 T_{3}^{22} + 140 T_{3}^{21} + 413 T_{3}^{20} + 861 T_{3}^{19} + 1387 T_{3}^{18} + 1778 T_{3}^{17} + 3090 T_{3}^{16} + 7469 T_{3}^{15} + 9807 T_{3}^{14} + 13650 T_{3}^{13} + 40489 T_{3}^{12} + 47299 T_{3}^{11} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\).