Newspace parameters
Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 945.be (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.54586299101\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 315) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).
\(n\) | \(136\) | \(596\) | \(757\) |
\(\chi(n)\) | \(\zeta_{6}\) | \(1 - \zeta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
206.1 |
|
1.50000 | + | 0.866025i | 0 | 0.500000 | + | 0.866025i | −1.00000 | 0 | −2.50000 | + | 0.866025i | − | 1.73205i | 0 | −1.50000 | − | 0.866025i | |||||||||||||||
656.1 | 1.50000 | − | 0.866025i | 0 | 0.500000 | − | 0.866025i | −1.00000 | 0 | −2.50000 | − | 0.866025i | 1.73205i | 0 | −1.50000 | + | 0.866025i | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 945.2.be.a | 2 | |
3.b | odd | 2 | 1 | 315.2.be.a | yes | 2 | |
7.d | odd | 6 | 1 | 945.2.t.a | 2 | ||
9.c | even | 3 | 1 | 315.2.t.a | ✓ | 2 | |
9.d | odd | 6 | 1 | 945.2.t.a | 2 | ||
21.g | even | 6 | 1 | 315.2.t.a | ✓ | 2 | |
63.k | odd | 6 | 1 | 315.2.be.a | yes | 2 | |
63.s | even | 6 | 1 | inner | 945.2.be.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.t.a | ✓ | 2 | 9.c | even | 3 | 1 | |
315.2.t.a | ✓ | 2 | 21.g | even | 6 | 1 | |
315.2.be.a | yes | 2 | 3.b | odd | 2 | 1 | |
315.2.be.a | yes | 2 | 63.k | odd | 6 | 1 | |
945.2.t.a | 2 | 7.d | odd | 6 | 1 | ||
945.2.t.a | 2 | 9.d | odd | 6 | 1 | ||
945.2.be.a | 2 | 1.a | even | 1 | 1 | trivial | |
945.2.be.a | 2 | 63.s | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3T_{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 3T + 3 \)
$3$
\( T^{2} \)
$5$
\( (T + 1)^{2} \)
$7$
\( T^{2} + 5T + 7 \)
$11$
\( T^{2} + 12 \)
$13$
\( T^{2} + 12T + 48 \)
$17$
\( T^{2} - 6T + 36 \)
$19$
\( T^{2} \)
$23$
\( T^{2} + 27 \)
$29$
\( T^{2} \)
$31$
\( T^{2} + 6T + 12 \)
$37$
\( T^{2} - 2T + 4 \)
$41$
\( T^{2} - 6T + 36 \)
$43$
\( T^{2} + T + 1 \)
$47$
\( T^{2} + 9T + 81 \)
$53$
\( T^{2} - 6T + 12 \)
$59$
\( T^{2} \)
$61$
\( T^{2} + 21T + 147 \)
$67$
\( T^{2} + 5T + 25 \)
$71$
\( T^{2} + 48 \)
$73$
\( T^{2} + 18T + 108 \)
$79$
\( T^{2} - 10T + 100 \)
$83$
\( T^{2} + 12T + 144 \)
$89$
\( T^{2} - 15T + 225 \)
$97$
\( T^{2} + 12T + 48 \)
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