Newspace parameters
Level: | \( N \) | \(=\) | \( 370 = 2 \cdot 5 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 370.g (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.95446487479\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).
\(n\) | \(261\) | \(297\) |
\(\chi(n)\) | \(i\) | \(i\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 |
|
− | 1.00000i | −1.00000 | + | 1.00000i | −1.00000 | 2.00000 | − | 1.00000i | 1.00000 | + | 1.00000i | 1.00000 | − | 1.00000i | 1.00000i | 1.00000i | −1.00000 | − | 2.00000i | |||||||||||||
327.1 | 1.00000i | −1.00000 | − | 1.00000i | −1.00000 | 2.00000 | + | 1.00000i | 1.00000 | − | 1.00000i | 1.00000 | + | 1.00000i | − | 1.00000i | − | 1.00000i | −1.00000 | + | 2.00000i | |||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
185.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 370.2.g.a | ✓ | 2 |
5.c | odd | 4 | 1 | 370.2.h.b | yes | 2 | |
37.d | odd | 4 | 1 | 370.2.h.b | yes | 2 | |
185.f | even | 4 | 1 | inner | 370.2.g.a | ✓ | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
370.2.g.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
370.2.g.a | ✓ | 2 | 185.f | even | 4 | 1 | inner |
370.2.h.b | yes | 2 | 5.c | odd | 4 | 1 | |
370.2.h.b | yes | 2 | 37.d | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 2T_{3} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 1 \)
$3$
\( T^{2} + 2T + 2 \)
$5$
\( T^{2} - 4T + 5 \)
$7$
\( T^{2} - 2T + 2 \)
$11$
\( T^{2} + 4 \)
$13$
\( T^{2} + 4 \)
$17$
\( (T - 4)^{2} \)
$19$
\( T^{2} - 10T + 50 \)
$23$
\( T^{2} \)
$29$
\( T^{2} - 6T + 18 \)
$31$
\( T^{2} - 14T + 98 \)
$37$
\( T^{2} + 12T + 37 \)
$41$
\( T^{2} \)
$43$
\( T^{2} + 16 \)
$47$
\( T^{2} + 14T + 98 \)
$53$
\( T^{2} - 2T + 2 \)
$59$
\( T^{2} - 2T + 2 \)
$61$
\( T^{2} + 6T + 18 \)
$67$
\( T^{2} + 6T + 18 \)
$71$
\( (T + 8)^{2} \)
$73$
\( T^{2} + 18T + 162 \)
$79$
\( T^{2} - 2T + 2 \)
$83$
\( T^{2} + 10T + 50 \)
$89$
\( T^{2} + 10T + 50 \)
$97$
\( (T - 8)^{2} \)
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