Newspace parameters
Level: | \( N \) | \(=\) | \( 637 = 7^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 637.bc (of order \(12\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.08647060876\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 91) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | −2.60347 | − | 0.697597i | 0.657528 | + | 0.379624i | 4.55935 | + | 2.63234i | −0.654162 | + | 2.44137i | −1.44703 | − | 1.44703i | 0 | −6.22207 | − | 6.22207i | −1.21177 | − | 2.09885i | 3.40618 | − | 5.89968i | ||
31.2 | −1.84208 | − | 0.493585i | −2.29307 | − | 1.32391i | 1.41759 | + | 0.818448i | 0.885596 | − | 3.30509i | 3.57057 | + | 3.57057i | 0 | 0.489646 | + | 0.489646i | 2.00545 | + | 3.47355i | −3.26268 | + | 5.65113i | ||
31.3 | −1.58070 | − | 0.423548i | 1.71590 | + | 0.990678i | 0.587174 | + | 0.339005i | 0.742954 | − | 2.77274i | −2.29273 | − | 2.29273i | 0 | 1.52975 | + | 1.52975i | 0.462886 | + | 0.801742i | −2.34878 | + | 4.06820i | ||
31.4 | −0.746505 | − | 0.200025i | 0.421869 | + | 0.243566i | −1.21479 | − | 0.701360i | −0.472319 | + | 1.76272i | −0.266208 | − | 0.266208i | 0 | 1.85952 | + | 1.85952i | −1.38135 | − | 2.39257i | 0.705177 | − | 1.22140i | ||
31.5 | 0.694471 | + | 0.186083i | −1.44134 | − | 0.832160i | −1.28439 | − | 0.741542i | 0.501414 | − | 1.87130i | −0.846120 | − | 0.846120i | 0 | −1.77076 | − | 1.77076i | −0.115021 | − | 0.199221i | 0.696434 | − | 1.20626i | ||
31.6 | 0.788958 | + | 0.211401i | 2.60482 | + | 1.50389i | −1.15429 | − | 0.666428i | −0.814012 | + | 3.03793i | 1.73717 | + | 1.73717i | 0 | −1.92491 | − | 1.92491i | 3.02338 | + | 5.23665i | −1.28444 | + | 2.22472i | ||
31.7 | 1.53492 | + | 0.411280i | −0.436133 | − | 0.251802i | 0.454770 | + | 0.262561i | 0.0206096 | − | 0.0769162i | −0.565867 | − | 0.565867i | 0 | −1.65723 | − | 1.65723i | −1.37319 | − | 2.37844i | 0.0632682 | − | 0.109584i | ||
31.8 | 2.38839 | + | 0.639966i | 1.77043 | + | 1.02216i | 3.56278 | + | 2.05697i | 0.423894 | − | 1.58199i | 3.57432 | + | 3.57432i | 0 | 3.69606 | + | 3.69606i | 0.589612 | + | 1.02124i | 2.02484 | − | 3.50713i | ||
411.1 | −2.60347 | + | 0.697597i | 0.657528 | − | 0.379624i | 4.55935 | − | 2.63234i | −0.654162 | − | 2.44137i | −1.44703 | + | 1.44703i | 0 | −6.22207 | + | 6.22207i | −1.21177 | + | 2.09885i | 3.40618 | + | 5.89968i | ||
411.2 | −1.84208 | + | 0.493585i | −2.29307 | + | 1.32391i | 1.41759 | − | 0.818448i | 0.885596 | + | 3.30509i | 3.57057 | − | 3.57057i | 0 | 0.489646 | − | 0.489646i | 2.00545 | − | 3.47355i | −3.26268 | − | 5.65113i | ||
411.3 | −1.58070 | + | 0.423548i | 1.71590 | − | 0.990678i | 0.587174 | − | 0.339005i | 0.742954 | + | 2.77274i | −2.29273 | + | 2.29273i | 0 | 1.52975 | − | 1.52975i | 0.462886 | − | 0.801742i | −2.34878 | − | 4.06820i | ||
411.4 | −0.746505 | + | 0.200025i | 0.421869 | − | 0.243566i | −1.21479 | + | 0.701360i | −0.472319 | − | 1.76272i | −0.266208 | + | 0.266208i | 0 | 1.85952 | − | 1.85952i | −1.38135 | + | 2.39257i | 0.705177 | + | 1.22140i | ||
411.5 | 0.694471 | − | 0.186083i | −1.44134 | + | 0.832160i | −1.28439 | + | 0.741542i | 0.501414 | + | 1.87130i | −0.846120 | + | 0.846120i | 0 | −1.77076 | + | 1.77076i | −0.115021 | + | 0.199221i | 0.696434 | + | 1.20626i | ||
411.6 | 0.788958 | − | 0.211401i | 2.60482 | − | 1.50389i | −1.15429 | + | 0.666428i | −0.814012 | − | 3.03793i | 1.73717 | − | 1.73717i | 0 | −1.92491 | + | 1.92491i | 3.02338 | − | 5.23665i | −1.28444 | − | 2.22472i | ||
411.7 | 1.53492 | − | 0.411280i | −0.436133 | + | 0.251802i | 0.454770 | − | 0.262561i | 0.0206096 | + | 0.0769162i | −0.565867 | + | 0.565867i | 0 | −1.65723 | + | 1.65723i | −1.37319 | + | 2.37844i | 0.0632682 | + | 0.109584i | ||
411.8 | 2.38839 | − | 0.639966i | 1.77043 | − | 1.02216i | 3.56278 | − | 2.05697i | 0.423894 | + | 1.58199i | 3.57432 | − | 3.57432i | 0 | 3.69606 | − | 3.69606i | 0.589612 | − | 1.02124i | 2.02484 | + | 3.50713i | ||
460.1 | −0.639966 | − | 2.38839i | 1.77043 | − | 1.02216i | −3.56278 | + | 2.05697i | 1.58199 | − | 0.423894i | −3.57432 | − | 3.57432i | 0 | 3.69606 | + | 3.69606i | 0.589612 | − | 1.02124i | −2.02484 | − | 3.50713i | ||
460.2 | −0.411280 | − | 1.53492i | −0.436133 | + | 0.251802i | −0.454770 | + | 0.262561i | 0.0769162 | − | 0.0206096i | 0.565867 | + | 0.565867i | 0 | −1.65723 | − | 1.65723i | −1.37319 | + | 2.37844i | −0.0632682 | − | 0.109584i | ||
460.3 | −0.211401 | − | 0.788958i | 2.60482 | − | 1.50389i | 1.15429 | − | 0.666428i | −3.03793 | + | 0.814012i | −1.73717 | − | 1.73717i | 0 | −1.92491 | − | 1.92491i | 3.02338 | − | 5.23665i | 1.28444 | + | 2.22472i | ||
460.4 | −0.186083 | − | 0.694471i | −1.44134 | + | 0.832160i | 1.28439 | − | 0.741542i | 1.87130 | − | 0.501414i | 0.846120 | + | 0.846120i | 0 | −1.77076 | − | 1.77076i | −0.115021 | + | 0.199221i | −0.696434 | − | 1.20626i | ||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
13.d | odd | 4 | 1 | inner |
91.bb | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 637.2.bc.b | 32 | |
7.b | odd | 2 | 1 | 91.2.bb.a | ✓ | 32 | |
7.c | even | 3 | 1 | 91.2.bb.a | ✓ | 32 | |
7.c | even | 3 | 1 | 637.2.i.a | 32 | ||
7.d | odd | 6 | 1 | 637.2.i.a | 32 | ||
7.d | odd | 6 | 1 | inner | 637.2.bc.b | 32 | |
13.d | odd | 4 | 1 | inner | 637.2.bc.b | 32 | |
21.c | even | 2 | 1 | 819.2.fn.e | 32 | ||
21.h | odd | 6 | 1 | 819.2.fn.e | 32 | ||
91.i | even | 4 | 1 | 91.2.bb.a | ✓ | 32 | |
91.z | odd | 12 | 1 | 91.2.bb.a | ✓ | 32 | |
91.z | odd | 12 | 1 | 637.2.i.a | 32 | ||
91.bb | even | 12 | 1 | 637.2.i.a | 32 | ||
91.bb | even | 12 | 1 | inner | 637.2.bc.b | 32 | |
273.o | odd | 4 | 1 | 819.2.fn.e | 32 | ||
273.cd | even | 12 | 1 | 819.2.fn.e | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.2.bb.a | ✓ | 32 | 7.b | odd | 2 | 1 | |
91.2.bb.a | ✓ | 32 | 7.c | even | 3 | 1 | |
91.2.bb.a | ✓ | 32 | 91.i | even | 4 | 1 | |
91.2.bb.a | ✓ | 32 | 91.z | odd | 12 | 1 | |
637.2.i.a | 32 | 7.c | even | 3 | 1 | ||
637.2.i.a | 32 | 7.d | odd | 6 | 1 | ||
637.2.i.a | 32 | 91.z | odd | 12 | 1 | ||
637.2.i.a | 32 | 91.bb | even | 12 | 1 | ||
637.2.bc.b | 32 | 1.a | even | 1 | 1 | trivial | |
637.2.bc.b | 32 | 7.d | odd | 6 | 1 | inner | |
637.2.bc.b | 32 | 13.d | odd | 4 | 1 | inner | |
637.2.bc.b | 32 | 91.bb | even | 12 | 1 | inner | |
819.2.fn.e | 32 | 21.c | even | 2 | 1 | ||
819.2.fn.e | 32 | 21.h | odd | 6 | 1 | ||
819.2.fn.e | 32 | 273.o | odd | 4 | 1 | ||
819.2.fn.e | 32 | 273.cd | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{32} + 2 T_{2}^{31} + 2 T_{2}^{30} + 12 T_{2}^{29} - 37 T_{2}^{28} - 118 T_{2}^{27} - 90 T_{2}^{26} - 516 T_{2}^{25} + 1641 T_{2}^{24} + 4354 T_{2}^{23} + 2866 T_{2}^{22} + 17542 T_{2}^{21} - 1650 T_{2}^{20} - 49422 T_{2}^{19} + \cdots + 50625 \)
acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\).