Newspace parameters
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(69.9198174990\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
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| Defining polynomial: |
\( x^{8} - 2 x^{7} + 7389 x^{6} + 349270 x^{5} + 54790705 x^{4} + 1232793330 x^{3} + 23637715200 x^{2} + \cdots + 344545520400 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 7) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2 x^{7} + 7389 x^{6} + 349270 x^{5} + 54790705 x^{4} + 1232793330 x^{3} + 23637715200 x^{2} + \cdots + 344545520400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 3056562322949 \nu^{7} + 17021850444868 \nu^{6} + \cdots - 47\!\cdots\!00 ) / 25\!\cdots\!00 \)
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| \(\beta_{2}\) | \(=\) |
\( ( - 223349910391 \nu^{7} - 13374773182279 \nu^{6} + \cdots + 29\!\cdots\!10 ) / 55\!\cdots\!90 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 113454788831 \nu^{7} + 2853863377241 \nu^{6} - 852474011356347 \nu^{5} + \cdots + 18\!\cdots\!70 ) / 61\!\cdots\!10 \)
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| \(\beta_{4}\) | \(=\) |
\( ( - 18\!\cdots\!29 \nu^{7} + \cdots - 18\!\cdots\!00 ) / 47\!\cdots\!00 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 879941176841 \nu^{7} - 21756297955093 \nu^{6} + \cdots - 32\!\cdots\!50 ) / 11\!\cdots\!98 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 10\!\cdots\!97 \nu^{7} + \cdots - 10\!\cdots\!00 ) / 28\!\cdots\!40 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 13\!\cdots\!27 \nu^{7} + \cdots - 20\!\cdots\!00 ) / 20\!\cdots\!00 \)
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| \(\nu\) | \(=\) |
\( ( 3\beta_{7} - 18\beta_{6} + 18\beta_{5} + 103\beta_{4} + 103\beta_{3} + 582\beta_1 ) / 1120 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -55\beta_{7} - 62\beta_{6} + 221\beta_{4} + 55\beta_{2} + 413698\beta _1 - 413698 ) / 112 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -3508\beta_{5} - 15039\beta_{3} - 349\beta_{2} - 3825446 ) / 28 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 44151\beta_{7} + 98373\beta_{6} - 98373\beta_{5} - 434384\beta_{4} - 434384\beta_{3} - 395154876\beta_1 ) / 14 \)
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| \(\nu^{5}\) | \(=\) |
\( ( - 145097 \beta_{7} + 29156588 \beta_{6} - 123552587 \beta_{4} + 145097 \beta_{2} - 47137826398 \beta _1 + 47137826398 ) / 28 \)
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| \(\nu^{6}\) | \(=\) |
\( ( 301013368\beta_{5} + 1315808607\beta_{3} - 85931543\beta_{2} + 947318161118 ) / 4 \)
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| \(\nu^{7}\) | \(=\) |
\( ( - 14695153737 \beta_{7} - 254500483872 \beta_{6} + 254500483872 \beta_{5} + \cdots + 495800068519962 \beta_1 ) / 28 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.1 |
|
−169.804 | − | 294.109i | 1695.62 | − | 2936.90i | −41282.6 | + | 71503.6i | −37789.9 | − | 65454.0i | −1.15169e6 | 0 | 1.69115e7 | 1.42421e6 | + | 2.46680e6i | −1.28337e7 | + | 2.22287e7i | ||||||||||||||||||||||||||||||
| 18.2 | −46.0231 | − | 79.7143i | 2665.52 | − | 4616.82i | 12147.8 | − | 21040.5i | 152560. | + | 264242.i | −490702. | 0 | −5.25248e6 | −7.03555e6 | − | 1.21859e7i | 1.40426e7 | − | 2.43225e7i | |||||||||||||||||||||||||||||||
| 18.3 | 32.4808 | + | 56.2585i | −2158.53 | + | 3738.69i | 14274.0 | − | 24723.3i | −39479.0 | − | 68379.6i | −280444. | 0 | 3.98319e6 | −2.14408e6 | − | 3.71365e6i | 2.56462e6 | − | 4.44206e6i | |||||||||||||||||||||||||||||||
| 18.4 | 136.846 | + | 237.024i | 2074.39 | − | 3592.95i | −21069.6 | + | 36493.7i | −71406.2 | − | 123679.i | 1.13549e6 | 0 | −2.56484e6 | −1.43176e6 | − | 2.47989e6i | 1.95433e7 | − | 3.38500e7i | |||||||||||||||||||||||||||||||
| 30.1 | −169.804 | + | 294.109i | 1695.62 | + | 2936.90i | −41282.6 | − | 71503.6i | −37789.9 | + | 65454.0i | −1.15169e6 | 0 | 1.69115e7 | 1.42421e6 | − | 2.46680e6i | −1.28337e7 | − | 2.22287e7i | |||||||||||||||||||||||||||||||
| 30.2 | −46.0231 | + | 79.7143i | 2665.52 | + | 4616.82i | 12147.8 | + | 21040.5i | 152560. | − | 264242.i | −490702. | 0 | −5.25248e6 | −7.03555e6 | + | 1.21859e7i | 1.40426e7 | + | 2.43225e7i | |||||||||||||||||||||||||||||||
| 30.3 | 32.4808 | − | 56.2585i | −2158.53 | − | 3738.69i | 14274.0 | + | 24723.3i | −39479.0 | + | 68379.6i | −280444. | 0 | 3.98319e6 | −2.14408e6 | + | 3.71365e6i | 2.56462e6 | + | 4.44206e6i | |||||||||||||||||||||||||||||||
| 30.4 | 136.846 | − | 237.024i | 2074.39 | + | 3592.95i | −21069.6 | − | 36493.7i | −71406.2 | + | 123679.i | 1.13549e6 | 0 | −2.56484e6 | −1.43176e6 | + | 2.47989e6i | 1.95433e7 | + | 3.38500e7i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 49.16.c.g | 8 | |
| 7.b | odd | 2 | 1 | 49.16.c.f | 8 | ||
| 7.c | even | 3 | 1 | 49.16.a.d | 4 | ||
| 7.c | even | 3 | 1 | inner | 49.16.c.g | 8 | |
| 7.d | odd | 6 | 1 | 7.16.a.b | ✓ | 4 | |
| 7.d | odd | 6 | 1 | 49.16.c.f | 8 | ||
| 21.g | even | 6 | 1 | 63.16.a.e | 4 | ||
| 28.f | even | 6 | 1 | 112.16.a.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.16.a.b | ✓ | 4 | 7.d | odd | 6 | 1 | |
| 49.16.a.d | 4 | 7.c | even | 3 | 1 | ||
| 49.16.c.f | 8 | 7.b | odd | 2 | 1 | ||
| 49.16.c.f | 8 | 7.d | odd | 6 | 1 | ||
| 49.16.c.g | 8 | 1.a | even | 1 | 1 | trivial | |
| 49.16.c.g | 8 | 7.c | even | 3 | 1 | inner | |
| 63.16.a.e | 4 | 21.g | even | 6 | 1 | ||
| 112.16.a.f | 4 | 28.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{16}^{\mathrm{new}}(49, [\chi])\):
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\( T_{2}^{8} + 93 T_{2}^{7} + 105791 T_{2}^{6} - 3211038 T_{2}^{5} + 9151566276 T_{2}^{4} + \cdots + 30\!\cdots\!00 \)
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\( T_{3}^{8} - 8554 T_{3}^{7} + 74470456 T_{3}^{6} - 307410645192 T_{3}^{5} + \cdots + 10\!\cdots\!76 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 30\!\cdots\!00 \)
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| $3$ |
\( T^{8} + \cdots + 10\!\cdots\!76 \)
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| $5$ |
\( T^{8} + \cdots + 67\!\cdots\!00 \)
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| $7$ |
\( T^{8} \)
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| $11$ |
\( T^{8} + \cdots + 24\!\cdots\!36 \)
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| $13$ |
\( (T^{4} + \cdots + 30\!\cdots\!20)^{2} \)
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| $17$ |
\( T^{8} + \cdots + 17\!\cdots\!36 \)
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| $19$ |
\( T^{8} + \cdots + 35\!\cdots\!00 \)
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| $23$ |
\( T^{8} + \cdots + 22\!\cdots\!36 \)
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| $29$ |
\( (T^{4} + \cdots + 39\!\cdots\!00)^{2} \)
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| $31$ |
\( T^{8} + \cdots + 16\!\cdots\!64 \)
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| $37$ |
\( T^{8} + \cdots + 75\!\cdots\!76 \)
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| $41$ |
\( (T^{4} + \cdots - 75\!\cdots\!36)^{2} \)
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| $43$ |
\( (T^{4} + \cdots + 59\!\cdots\!60)^{2} \)
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| $47$ |
\( T^{8} + \cdots + 10\!\cdots\!64 \)
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| $53$ |
\( T^{8} + \cdots + 86\!\cdots\!24 \)
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| $59$ |
\( T^{8} + \cdots + 46\!\cdots\!00 \)
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| $61$ |
\( T^{8} + \cdots + 29\!\cdots\!96 \)
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| $67$ |
\( T^{8} + \cdots + 47\!\cdots\!24 \)
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| $71$ |
\( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \)
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| $73$ |
\( T^{8} + \cdots + 27\!\cdots\!04 \)
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| $79$ |
\( T^{8} + \cdots + 85\!\cdots\!00 \)
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| $83$ |
\( (T^{4} + \cdots + 15\!\cdots\!88)^{2} \)
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| $89$ |
\( T^{8} + \cdots + 59\!\cdots\!00 \)
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| $97$ |
\( (T^{4} + \cdots + 10\!\cdots\!68)^{2} \)
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