Newspace parameters
| Level: | \( N \) | \(=\) | \( 845 = 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 845.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(49.8566139549\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 40x^{5} + 64x^{4} + 409x^{3} - 568x^{2} - 480x + 96 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 65) |
| 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
\( x^{7} - 2x^{6} - 40x^{5} + 64x^{4} + 409x^{3} - 568x^{2} - 480x + 96 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 20\nu + 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 6\nu^{5} - 16\nu^{4} + 120\nu^{3} - 47\nu^{2} - 204\nu - 32 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 10\nu^{5} - 8\nu^{4} - 176\nu^{3} + 495\nu^{2} - 144\nu - 256 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 6\nu^{5} - 14\nu^{4} + 114\nu^{3} - 91\nu^{2} - 110\nu + 30 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 20\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - 4\beta_{4} + 3\beta_{3} + 25\beta_{2} + 13\beta _1 + 239 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{6} + 2\beta_{5} - 22\beta_{4} + 32\beta_{3} + 52\beta_{2} + 445\beta _1 + 190 \)
|
| \(\nu^{6}\) | \(=\) |
\( 52\beta_{6} + 12\beta_{5} - 188\beta_{4} + 120\beta_{3} + 639\beta_{2} + 682\beta _1 + 5320 \)
|
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 |
|
−5.37688 | −0.936724 | 20.9108 | −5.00000 | 5.03665 | −0.201771 | −69.4197 | −26.1225 | 26.8844 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.65500 | 8.59394 | 5.35901 | −5.00000 | −31.4108 | −19.4590 | 9.65282 | 46.8559 | 18.2750 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.10835 | −3.08338 | −3.55484 | −5.00000 | 6.50086 | 17.4384 | 24.3617 | −17.4928 | 10.5418 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.170066 | −8.43460 | −7.97108 | −5.00000 | 1.43444 | −32.7758 | 2.71614 | 44.1424 | 0.850329 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.742335 | 5.13862 | −7.44894 | −5.00000 | 3.81458 | 9.97127 | −11.4683 | −0.594575 | −3.71168 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.27643 | −7.17327 | 10.2878 | −5.00000 | −30.6760 | 5.20875 | 9.78383 | 24.4558 | −21.3821 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.29153 | 5.89541 | 10.4172 | −5.00000 | 25.3003 | −34.1818 | 10.3735 | 7.75582 | −21.4576 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 845.4.a.i | 7 | |
| 13.b | even | 2 | 1 | 845.4.a.l | 7 | ||
| 13.d | odd | 4 | 2 | 65.4.c.a | ✓ | 14 | |
| 39.f | even | 4 | 2 | 585.4.b.e | 14 | ||
| 52.f | even | 4 | 2 | 1040.4.k.d | 14 | ||
| 65.f | even | 4 | 2 | 325.4.d.d | 14 | ||
| 65.g | odd | 4 | 2 | 325.4.c.e | 14 | ||
| 65.k | even | 4 | 2 | 325.4.d.c | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.4.c.a | ✓ | 14 | 13.d | odd | 4 | 2 | |
| 325.4.c.e | 14 | 65.g | odd | 4 | 2 | ||
| 325.4.d.c | 14 | 65.k | even | 4 | 2 | ||
| 325.4.d.d | 14 | 65.f | even | 4 | 2 | ||
| 585.4.b.e | 14 | 39.f | even | 4 | 2 | ||
| 845.4.a.i | 7 | 1.a | even | 1 | 1 | trivial | |
| 845.4.a.l | 7 | 13.b | even | 2 | 1 | ||
| 1040.4.k.d | 14 | 52.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 2T_{2}^{6} - 40T_{2}^{5} - 64T_{2}^{4} + 409T_{2}^{3} + 568T_{2}^{2} - 480T_{2} - 96 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(845))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 2 T^{6} + \cdots - 96 \)
|
| $3$ |
\( T^{7} - 134 T^{5} + \cdots - 45496 \)
|
| $5$ |
\( (T + 5)^{7} \)
|
| $7$ |
\( T^{7} + 54 T^{6} + \cdots - 3984000 \)
|
| $11$ |
\( T^{7} - 4 T^{6} + \cdots + 250531200 \)
|
| $13$ |
\( T^{7} \)
|
| $17$ |
\( T^{7} + \cdots - 44413747200 \)
|
| $19$ |
\( T^{7} + \cdots + 153425796096 \)
|
| $23$ |
\( T^{7} + \cdots + 5699551522752 \)
|
| $29$ |
\( T^{7} + \cdots + 440579141474400 \)
|
| $31$ |
\( T^{7} + \cdots + 195148355290200 \)
|
| $37$ |
\( T^{7} + \cdots + 877152531948576 \)
|
| $41$ |
\( T^{7} + \cdots - 50\!\cdots\!00 \)
|
| $43$ |
\( T^{7} + \cdots + 97\!\cdots\!24 \)
|
| $47$ |
\( T^{7} + \cdots + 21\!\cdots\!16 \)
|
| $53$ |
\( T^{7} + \cdots - 405893642720256 \)
|
| $59$ |
\( T^{7} + \cdots + 84\!\cdots\!72 \)
|
| $61$ |
\( T^{7} + \cdots - 35\!\cdots\!44 \)
|
| $67$ |
\( T^{7} + \cdots + 14\!\cdots\!84 \)
|
| $71$ |
\( T^{7} + \cdots - 11\!\cdots\!00 \)
|
| $73$ |
\( T^{7} + \cdots - 65\!\cdots\!12 \)
|
| $79$ |
\( T^{7} + \cdots - 47\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots + 34\!\cdots\!12 \)
|
| $89$ |
\( T^{7} + \cdots + 15\!\cdots\!72 \)
|
| $97$ |
\( T^{7} + \cdots - 13\!\cdots\!72 \)
|
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