Newspace parameters
Level: | \( N \) | \(=\) | \( 3 \) |
Weight: | \( k \) | \(=\) | \( 25 \) |
Character orbit: | \([\chi]\) | \(=\) | 3.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.9490145677\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
Defining polynomial: |
\( x^{6} + 253102x^{4} + 17425276096x^{2} + 250659115499520 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2^{13}\cdot 3^{24}\cdot 5^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} + 253102x^{4} + 17425276096x^{2} + 250659115499520 \)
:
\(\beta_{1}\) | \(=\) |
\( 18\nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( 927 \nu^{5} - 22464 \nu^{4} + 158192658 \nu^{3} - 8746814592 \nu^{2} + 4583765471040 \nu - 519220416540672 ) / 1041302528 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 927 \nu^{5} + 162432 \nu^{4} - 158192658 \nu^{3} + 11341271808 \nu^{2} - 4574393748288 \nu - 624717182509056 ) / 520651264 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 129501 \nu^{5} + 252288 \nu^{4} + 36839351862 \nu^{3} + 46328530176 \nu^{2} + \cdots + 14\!\cdots\!32 ) / 520651264 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 927 \nu^{5} + 2217024 \nu^{4} + 158192658 \nu^{3} + 370146519936 \nu^{2} + 4583765471040 \nu + 96\!\cdots\!28 ) / 1041302528 \)
|
\(\nu\) | \(=\) |
\( ( \beta_1 ) / 18 \)
|
\(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 8\beta_{3} - 17\beta_{2} + 8\beta _1 - 27335016 ) / 324 \)
|
\(\nu^{3}\) | \(=\) |
\( ( -736\beta_{5} + 103\beta_{4} + 3393\beta_{3} - 21256\beta_{2} - 18690797\beta_1 ) / 2916 \)
|
\(\nu^{4}\) | \(=\) |
\( ( -250237\beta_{5} + 18272248\beta_{3} + 36794733\beta_{2} - 18272248\beta _1 + 42588282369096 ) / 4374 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 103238176\beta_{5} - 8788481\beta_{4} - 481592855\beta_{3} + 3026837432\beta_{2} + 1194462231243\beta_1 ) / 1458 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 |
|
− | 6843.54i | 162008. | − | 506145.i | −3.00569e7 | 3.79015e8i | −3.46383e9 | − | 1.10871e9i | −1.14171e10 | 9.08798e10i | −2.29937e11 | − | 1.63999e11i | 2.59380e12 | |||||||||||||||||||||||||||||
2.2 | − | 5372.56i | −24323.9 | + | 530884.i | −1.20872e7 | − | 5.28531e7i | 2.85221e9 | + | 1.30681e8i | 2.07202e10 | − | 2.51975e10i | −2.81246e11 | − | 2.58263e10i | −2.83957e11 | ||||||||||||||||||||||||||||
2.3 | − | 2511.29i | −446105. | − | 288825.i | 1.04706e7 | − | 3.68111e8i | −7.25325e8 | + | 1.12030e9i | −8.30906e9 | − | 6.84273e10i | 1.15589e11 | + | 2.57693e11i | −9.24434e11 | ||||||||||||||||||||||||||||
2.4 | 2511.29i | −446105. | + | 288825.i | 1.04706e7 | 3.68111e8i | −7.25325e8 | − | 1.12030e9i | −8.30906e9 | 6.84273e10i | 1.15589e11 | − | 2.57693e11i | −9.24434e11 | |||||||||||||||||||||||||||||||
2.5 | 5372.56i | −24323.9 | − | 530884.i | −1.20872e7 | 5.28531e7i | 2.85221e9 | − | 1.30681e8i | 2.07202e10 | 2.51975e10i | −2.81246e11 | + | 2.58263e10i | −2.83957e11 | |||||||||||||||||||||||||||||||
2.6 | 6843.54i | 162008. | + | 506145.i | −3.00569e7 | − | 3.79015e8i | −3.46383e9 | + | 1.10871e9i | −1.14171e10 | − | 9.08798e10i | −2.29937e11 | + | 1.63999e11i | 2.59380e12 | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3.25.b.b | ✓ | 6 |
3.b | odd | 2 | 1 | inner | 3.25.b.b | ✓ | 6 |
4.b | odd | 2 | 1 | 48.25.e.b | 6 | ||
12.b | even | 2 | 1 | 48.25.e.b | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3.25.b.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
3.25.b.b | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
48.25.e.b | 6 | 4.b | odd | 2 | 1 | ||
48.25.e.b | 6 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 82005048T_{2}^{4} + 1829235783453696T_{2}^{2} + 8525473984011546132480 \)
acting on \(S_{25}^{\mathrm{new}}(3, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 82005048 T^{4} + \cdots + 85\!\cdots\!80 \)
$3$
\( T^{6} + 616842 T^{5} + \cdots + 22\!\cdots\!41 \)
$5$
\( T^{6} + \cdots + 54\!\cdots\!00 \)
$7$
\( (T^{3} - 994032438 T^{2} + \cdots - 19\!\cdots\!00)^{2} \)
$11$
\( T^{6} + \cdots + 21\!\cdots\!00 \)
$13$
\( (T^{3} + 36771219031962 T^{2} + \cdots - 73\!\cdots\!00)^{2} \)
$17$
\( T^{6} + \cdots + 54\!\cdots\!20 \)
$19$
\( (T^{3} + \cdots + 24\!\cdots\!72)^{2} \)
$23$
\( T^{6} + \cdots + 10\!\cdots\!80 \)
$29$
\( T^{6} + \cdots + 72\!\cdots\!00 \)
$31$
\( (T^{3} + \cdots + 26\!\cdots\!72)^{2} \)
$37$
\( (T^{3} + \cdots - 85\!\cdots\!00)^{2} \)
$41$
\( T^{6} + \cdots + 12\!\cdots\!00 \)
$43$
\( (T^{3} + \cdots - 31\!\cdots\!00)^{2} \)
$47$
\( T^{6} + \cdots + 15\!\cdots\!20 \)
$53$
\( T^{6} + \cdots + 53\!\cdots\!20 \)
$59$
\( T^{6} + \cdots + 77\!\cdots\!00 \)
$61$
\( (T^{3} + \cdots - 18\!\cdots\!88)^{2} \)
$67$
\( (T^{3} + \cdots - 20\!\cdots\!00)^{2} \)
$71$
\( T^{6} + \cdots + 12\!\cdots\!00 \)
$73$
\( (T^{3} + \cdots + 14\!\cdots\!00)^{2} \)
$79$
\( (T^{3} + \cdots + 25\!\cdots\!72)^{2} \)
$83$
\( T^{6} + \cdots + 29\!\cdots\!20 \)
$89$
\( T^{6} + \cdots + 54\!\cdots\!00 \)
$97$
\( (T^{3} + \cdots - 61\!\cdots\!00)^{2} \)
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