Newspace parameters
| Level: | \( N \) | \(=\) | \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 700.e (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(112.268673869\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
| Defining polynomial: |
\( x^{6} + 998x^{4} + 249001x^{2} + 44100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 140) |
| 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
\( x^{6} + 998x^{4} + 249001x^{2} + 44100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 499\nu ) / 210 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 499\nu^{2} ) / 210 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 709\nu^{2} + 69930 ) / 210 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 832\nu^{3} + 165957\nu ) / 210 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - 333 \)
|
| \(\nu^{3}\) | \(=\) |
\( 210\beta_{2} - 499\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -499\beta_{4} + 709\beta_{3} + 166167 \)
|
| \(\nu^{5}\) | \(=\) |
\( 210\beta_{5} - 174720\beta_{2} + 249211\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(351\) | \(477\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 449.1 |
|
0 | − | 24.5458i | 0 | 0 | 0 | 49.0000i | 0 | −359.498 | 0 | |||||||||||||||||||||||||||||||||||
| 449.2 | 0 | − | 20.1248i | 0 | 0 | 0 | − | 49.0000i | 0 | −162.009 | 0 | |||||||||||||||||||||||||||||||||||
| 449.3 | 0 | − | 1.57901i | 0 | 0 | 0 | 49.0000i | 0 | 240.507 | 0 | ||||||||||||||||||||||||||||||||||||
| 449.4 | 0 | 1.57901i | 0 | 0 | 0 | − | 49.0000i | 0 | 240.507 | 0 | ||||||||||||||||||||||||||||||||||||
| 449.5 | 0 | 20.1248i | 0 | 0 | 0 | 49.0000i | 0 | −162.009 | 0 | |||||||||||||||||||||||||||||||||||||
| 449.6 | 0 | 24.5458i | 0 | 0 | 0 | − | 49.0000i | 0 | −359.498 | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 700.6.e.g | 6 | |
| 5.b | even | 2 | 1 | inner | 700.6.e.g | 6 | |
| 5.c | odd | 4 | 1 | 140.6.a.d | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 700.6.a.i | 3 | ||
| 20.e | even | 4 | 1 | 560.6.a.t | 3 | ||
| 35.f | even | 4 | 1 | 980.6.a.h | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.6.a.d | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 560.6.a.t | 3 | 20.e | even | 4 | 1 | ||
| 700.6.a.i | 3 | 5.c | odd | 4 | 1 | ||
| 700.6.e.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 700.6.e.g | 6 | 5.b | even | 2 | 1 | inner | |
| 980.6.a.h | 3 | 35.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 1010T_{3}^{4} + 246529T_{3}^{2} + 608400 \)
acting on \(S_{6}^{\mathrm{new}}(700, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( T^{6} + 1010 T^{4} + 246529 T^{2} + \cdots + 608400 \)
$5$
\( T^{6} \)
$7$
\( (T^{2} + 2401)^{3} \)
$11$
\( (T^{3} + 14 T^{2} - 147231 T + 12436740)^{2} \)
$13$
\( T^{6} + 850758 T^{4} + \cdots + 37708614058756 \)
$17$
\( T^{6} + 1668310 T^{4} + \cdots + 77\!\cdots\!00 \)
$19$
\( (T^{3} + 2328 T^{2} + \cdots - 11429521264)^{2} \)
$23$
\( T^{6} + 9508744 T^{4} + \cdots + 43\!\cdots\!16 \)
$29$
\( (T^{3} + 4092 T^{2} + \cdots - 17580868722)^{2} \)
$31$
\( (T^{3} - 5888 T^{2} + \cdots + 211802104832)^{2} \)
$37$
\( T^{6} + 47359404 T^{4} + \cdots + 22\!\cdots\!00 \)
$41$
\( (T^{3} - 11450 T^{2} + \cdots + 478579953600)^{2} \)
$43$
\( T^{6} + 370031384 T^{4} + \cdots + 56\!\cdots\!76 \)
$47$
\( T^{6} + 214248754 T^{4} + \cdots + 30\!\cdots\!36 \)
$53$
\( T^{6} + 379440036 T^{4} + \cdots + 55\!\cdots\!00 \)
$59$
\( (T^{3} + 12388 T^{2} + \cdots - 41176040028480)^{2} \)
$61$
\( (T^{3} - 27182 T^{2} + \cdots - 169955356480)^{2} \)
$67$
\( T^{6} + 4971185168 T^{4} + \cdots + 23\!\cdots\!00 \)
$71$
\( (T^{3} - 81992 T^{2} + \cdots + 113425504819200)^{2} \)
$73$
\( T^{6} + 13818436332 T^{4} + \cdots + 12\!\cdots\!00 \)
$79$
\( (T^{3} - 15926 T^{2} + \cdots + 274092525845520)^{2} \)
$83$
\( T^{6} + 6449648688 T^{4} + \cdots + 16\!\cdots\!00 \)
$89$
\( (T^{3} - 95710 T^{2} + \cdots + 305457269205600)^{2} \)
$97$
\( T^{6} + 28176239622 T^{4} + \cdots + 17\!\cdots\!24 \)
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