Newspace parameters
| Level: | \( N \) | \(=\) | \( 7105 = 5 \cdot 7^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7105.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(56.7337106361\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{7} - x^{6} - 11x^{5} + 8x^{4} + 30x^{3} - 17x^{2} - 20x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1015) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 11x^{5} + 8x^{4} + 30x^{3} - 17x^{2} - 20x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 13\nu^{4} - 10\nu^{3} + 42\nu^{2} + 15\nu - 30 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{6} + \nu^{5} + 35\nu^{4} - 2\nu^{3} - 102\nu^{2} - 9\nu + 62 ) / 4 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{6} + 3\nu^{5} + 53\nu^{4} - 14\nu^{3} - 126\nu^{2} + \nu + 54 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{6} + \nu^{5} + 79\nu^{4} + 10\nu^{3} - 210\nu^{2} - 49\nu + 110 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7\nu^{6} - \nu^{5} - 79\nu^{4} - 10\nu^{3} + 214\nu^{2} + 49\nu - 122 ) / 4 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 2\beta_{2} + 5\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( 9\beta_{6} + 9\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + \beta _1 + 17 \)
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| \(\nu^{5}\) | \(=\) |
\( 3\beta_{6} + 11\beta_{5} - 9\beta_{4} + 3\beta_{3} + 20\beta_{2} + 32\beta _1 + 14 \)
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| \(\nu^{6}\) | \(=\) |
\( 72\beta_{6} + 74\beta_{5} - 14\beta_{4} + 23\beta_{3} + 17\beta_{2} + 16\beta _1 + 121 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.77653 | −0.768424 | 5.70911 | 1.00000 | 2.13355 | 0 | −10.2985 | −2.40952 | −2.77653 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.25469 | 2.55866 | 3.08364 | 1.00000 | −5.76899 | 0 | −2.44327 | 3.54673 | −2.25469 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.61034 | −2.93471 | 0.593195 | 1.00000 | 4.72588 | 0 | 2.26543 | 5.61251 | −1.61034 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.670735 | 1.55877 | −1.55011 | 1.00000 | −1.04552 | 0 | 2.38119 | −0.570221 | −0.670735 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.867803 | −1.78557 | −1.24692 | 1.00000 | −1.54952 | 0 | −2.81769 | 0.188244 | 0.867803 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.23295 | −0.698490 | −0.479834 | 1.00000 | −0.861204 | 0 | −3.05751 | −2.51211 | 1.23295 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.21154 | 1.06975 | 2.89092 | 1.00000 | 2.36581 | 0 | 1.97031 | −1.85563 | 2.21154 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(29\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7105.2.a.r | 7 | |
| 7.b | odd | 2 | 1 | 1015.2.a.j | ✓ | 7 | |
| 21.c | even | 2 | 1 | 9135.2.a.bg | 7 | ||
| 35.c | odd | 2 | 1 | 5075.2.a.z | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1015.2.a.j | ✓ | 7 | 7.b | odd | 2 | 1 | |
| 5075.2.a.z | 7 | 35.c | odd | 2 | 1 | ||
| 7105.2.a.r | 7 | 1.a | even | 1 | 1 | trivial | |
| 9135.2.a.bg | 7 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7105))\):
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\( T_{2}^{7} + 3T_{2}^{6} - 7T_{2}^{5} - 22T_{2}^{4} + 12T_{2}^{3} + 39T_{2}^{2} - 8T_{2} - 16 \)
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\( T_{3}^{7} + T_{3}^{6} - 11T_{3}^{5} - 8T_{3}^{4} + 30T_{3}^{3} + 17T_{3}^{2} - 20T_{3} - 12 \)
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\( T_{17}^{7} - 16T_{17}^{6} + 81T_{17}^{5} - 89T_{17}^{4} - 420T_{17}^{3} + 1200T_{17}^{2} - 800T_{17} - 64 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 3 T^{6} + \cdots - 16 \)
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| $3$ |
\( T^{7} + T^{6} + \cdots - 12 \)
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| $5$ |
\( (T - 1)^{7} \)
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| $7$ |
\( T^{7} \)
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| $11$ |
\( T^{7} + 5 T^{6} + \cdots - 96 \)
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| $13$ |
\( T^{7} - 3 T^{6} + \cdots + 736 \)
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| $17$ |
\( T^{7} - 16 T^{6} + \cdots - 64 \)
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| $19$ |
\( T^{7} + 7 T^{6} + \cdots - 10016 \)
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| $23$ |
\( T^{7} + 22 T^{6} + \cdots + 16 \)
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| $29$ |
\( (T + 1)^{7} \)
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| $31$ |
\( T^{7} + 7 T^{6} + \cdots + 7168 \)
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| $37$ |
\( T^{7} + 5 T^{6} + \cdots + 15616 \)
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| $41$ |
\( T^{7} - 17 T^{6} + \cdots + 252052 \)
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| $43$ |
\( T^{7} + 3 T^{6} + \cdots - 277616 \)
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| $47$ |
\( T^{7} + 8 T^{6} + \cdots + 117228 \)
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| $53$ |
\( T^{7} + 23 T^{6} + \cdots - 11124 \)
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| $59$ |
\( T^{7} + 4 T^{6} + \cdots - 274048 \)
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| $61$ |
\( T^{7} + 14 T^{6} + \cdots - 135936 \)
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| $67$ |
\( T^{7} + 29 T^{6} + \cdots - 126992 \)
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| $71$ |
\( T^{7} + 13 T^{6} + \cdots + 2488128 \)
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| $73$ |
\( T^{7} - 16 T^{6} + \cdots + 1656 \)
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| $79$ |
\( T^{7} - 14 T^{6} + \cdots + 23328 \)
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| $83$ |
\( T^{7} - 2 T^{6} + \cdots - 250816 \)
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| $89$ |
\( T^{7} - 27 T^{6} + \cdots + 306036 \)
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| $97$ |
\( T^{7} + 4 T^{6} + \cdots + 282184 \)
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