Newspace parameters
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.39795672093\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
| Defining polynomial: |
\( x^{6} - x^{5} + 238188x^{4} - 14589496x^{3} + 11212054600x^{2} - 101757597480x + 81251686776288 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3\cdot 7^{3} \) |
| 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
\( x^{6} - x^{5} + 238188x^{4} - 14589496x^{3} + 11212054600x^{2} - 101757597480x + 81251686776288 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 316322973 \nu^{5} - 1880954567990 \nu^{4} - 229888629934646 \nu^{3} + \cdots + 67\!\cdots\!71 ) / 78\!\cdots\!45 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 829712739 \nu^{5} - 436403341435 \nu^{4} - 197574368898172 \nu^{3} + \cdots - 23\!\cdots\!88 ) / 15\!\cdots\!90 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 214467963392 \nu^{5} + 2208641196555 \nu^{4} + \cdots + 34\!\cdots\!96 ) / 10\!\cdots\!30 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 29099137708946 \nu^{5} + \cdots + 57\!\cdots\!28 ) / 98\!\cdots\!70 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 36542487068476 \nu^{5} + \cdots - 31\!\cdots\!08 ) / 98\!\cdots\!70 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 8\beta_{4} - 105\beta_{3} + 132\beta_{2} + 12\beta _1 + 108 ) / 672 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 43\beta_{5} - 160\beta_{4} + 5285\beta_{3} - 70276\beta_{2} + 7540\beta _1 - 17787180 ) / 224 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4319\beta_{5} - 435680\beta_{4} + 6648369\beta_{3} - 1152820\beta_{2} - 1586684\beta _1 + 1607875428 ) / 224 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 6442253 \beta_{5} + 79955072 \beta_{4} - 1576024835 \beta_{3} + 8005050140 \beta_{2} - 1545103820 \beta _1 + 2564427752724 ) / 224 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2655070297 \beta_{5} + 71240832448 \beta_{4} - 1193406700087 \beta_{3} - 1339548073876 \beta_{2} + 438107594404 \beta _1 - 621325934265084 ) / 224 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 6.1 |
|
−108.657 | − | 1048.40i | 7710.33 | − | 23251.4i | 113916.i | −98544.2 | − | 64267.7i | −392722. | −567695. | 2.52642e6i | ||||||||||||||||||||||||||||||||
| 6.2 | −108.657 | 1048.40i | 7710.33 | 23251.4i | − | 113916.i | −98544.2 | + | 64267.7i | −392722. | −567695. | − | 2.52642e6i | |||||||||||||||||||||||||||||||||
| 6.3 | 17.4333 | − | 949.924i | −3792.08 | 18388.3i | − | 16560.3i | −116360. | − | 17366.5i | −137515. | −370914. | 320568.i | |||||||||||||||||||||||||||||||||
| 6.4 | 17.4333 | 949.924i | −3792.08 | − | 18388.3i | 16560.3i | −116360. | + | 17366.5i | −137515. | −370914. | − | 320568.i | |||||||||||||||||||||||||||||||||
| 6.5 | 91.2237 | − | 658.189i | 4225.75 | − | 20289.4i | − | 60042.4i | 57579.4 | + | 102596.i | 11836.7 | 98227.8 | − | 1.85087e6i | |||||||||||||||||||||||||||||||
| 6.6 | 91.2237 | 658.189i | 4225.75 | 20289.4i | 60042.4i | 57579.4 | − | 102596.i | 11836.7 | 98227.8 | 1.85087e6i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7.13.b.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 63.13.d.d | 6 | ||
| 4.b | odd | 2 | 1 | 112.13.c.b | 6 | ||
| 7.b | odd | 2 | 1 | inner | 7.13.b.b | ✓ | 6 |
| 7.c | even | 3 | 2 | 49.13.d.b | 12 | ||
| 7.d | odd | 6 | 2 | 49.13.d.b | 12 | ||
| 21.c | even | 2 | 1 | 63.13.d.d | 6 | ||
| 28.d | even | 2 | 1 | 112.13.c.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.13.b.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 7.13.b.b | ✓ | 6 | 7.b | odd | 2 | 1 | inner |
| 49.13.d.b | 12 | 7.c | even | 3 | 2 | ||
| 49.13.d.b | 12 | 7.d | odd | 6 | 2 | ||
| 63.13.d.d | 6 | 3.b | odd | 2 | 1 | ||
| 63.13.d.d | 6 | 21.c | even | 2 | 1 | ||
| 112.13.c.b | 6 | 4.b | odd | 2 | 1 | ||
| 112.13.c.b | 6 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 10216T_{2} + 172800 \)
acting on \(S_{13}^{\mathrm{new}}(7, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{3} - 10216 T + 172800)^{2} \)
$3$
\( T^{6} + 2434704 T^{4} + \cdots + 42\!\cdots\!00 \)
$5$
\( T^{6} + 1290415440 T^{4} + \cdots + 75\!\cdots\!00 \)
$7$
\( T^{6} + 314650 T^{5} + \cdots + 26\!\cdots\!01 \)
$11$
\( (T^{3} - 852390 T^{2} + \cdots + 22\!\cdots\!48)^{2} \)
$13$
\( T^{6} + 65493616180944 T^{4} + \cdots + 33\!\cdots\!00 \)
$17$
\( T^{6} + \cdots + 24\!\cdots\!00 \)
$19$
\( T^{6} + \cdots + 62\!\cdots\!00 \)
$23$
\( (T^{3} + 133367850 T^{2} + \cdots - 23\!\cdots\!00)^{2} \)
$29$
\( (T^{3} - 1045092726 T^{2} + \cdots + 41\!\cdots\!92)^{2} \)
$31$
\( T^{6} + \cdots + 37\!\cdots\!00 \)
$37$
\( (T^{3} + 2412433450 T^{2} + \cdots - 10\!\cdots\!00)^{2} \)
$41$
\( T^{6} + \cdots + 49\!\cdots\!00 \)
$43$
\( (T^{3} + 14362094650 T^{2} + \cdots + 36\!\cdots\!00)^{2} \)
$47$
\( T^{6} + \cdots + 20\!\cdots\!00 \)
$53$
\( (T^{3} + 52260928650 T^{2} + \cdots + 49\!\cdots\!00)^{2} \)
$59$
\( T^{6} + \cdots + 27\!\cdots\!00 \)
$61$
\( T^{6} + \cdots + 30\!\cdots\!00 \)
$67$
\( (T^{3} + 64873301850 T^{2} + \cdots - 76\!\cdots\!00)^{2} \)
$71$
\( (T^{3} + 7508802330 T^{2} + \cdots - 18\!\cdots\!12)^{2} \)
$73$
\( T^{6} + \cdots + 76\!\cdots\!00 \)
$79$
\( (T^{3} + 86263815834 T^{2} + \cdots - 66\!\cdots\!68)^{2} \)
$83$
\( T^{6} + \cdots + 13\!\cdots\!00 \)
$89$
\( T^{6} + \cdots + 34\!\cdots\!00 \)
$97$
\( T^{6} + \cdots + 30\!\cdots\!00 \)
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