Newspace parameters
| Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 605.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(97.0322109869\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{7} - x^{6} - 165x^{5} + 145x^{4} + 7100x^{3} - 4720x^{2} - 71936x + 96176 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| 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
\( x^{7} - x^{6} - 165x^{5} + 145x^{4} + 7100x^{3} - 4720x^{2} - 71936x + 96176 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 47 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 173\nu^{6} - 295\nu^{5} - 26315\nu^{4} + 31915\nu^{3} + 918270\nu^{2} - 165620\nu - 4422248 ) / 139920 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -19\nu^{6} - 35\nu^{5} + 3281\nu^{4} + 1415\nu^{3} - 148798\nu^{2} + 113788\nu + 1284168 ) / 9328 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 397\nu^{6} + 1345\nu^{5} - 72115\nu^{4} - 214285\nu^{3} + 3545670\nu^{2} + 7805660\nu - 34382632 ) / 139920 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 6\nu^{6} + 25\nu^{5} - 905\nu^{4} - 3585\nu^{3} + 32955\nu^{2} + 111340\nu - 232836 ) / 1060 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 47 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} - 2\beta_{4} - \beta_{3} + 79\beta _1 - 2 \)
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| \(\nu^{4}\) | \(=\) |
\( 5\beta_{6} - 10\beta_{5} + 3\beta_{4} + 5\beta_{3} + 113\beta_{2} + 2\beta _1 + 3697 \)
|
| \(\nu^{5}\) | \(=\) |
\( 29\beta_{6} - 131\beta_{5} - 267\beta_{4} - 272\beta_{3} - 56\beta_{2} + 7197\beta _1 - 292 \)
|
| \(\nu^{6}\) | \(=\) |
\( 810\beta_{6} - 1560\beta_{5} + 370\beta_{4} + 1290\beta_{3} + 11785\beta_{2} - 1040\beta _1 + 338311 \)
|
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 |
|
−10.1821 | −11.1325 | 71.6760 | 25.0000 | 113.352 | 3.03241 | −403.986 | −119.068 | −254.553 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −6.33233 | 16.0826 | 8.09835 | 25.0000 | −101.840 | 39.3177 | 151.353 | 15.6497 | −158.308 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.44550 | −27.0662 | −12.2375 | 25.0000 | 120.323 | 254.325 | 196.658 | 489.582 | −111.138 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.52774 | 14.3093 | −29.6660 | 25.0000 | 21.8609 | 3.10893 | −94.2098 | −38.2440 | 38.1936 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.93615 | −17.6251 | −23.3790 | 25.0000 | −51.7500 | −136.783 | −162.601 | 67.6438 | 73.4039 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 7.43427 | −6.40574 | 23.2684 | 25.0000 | −47.6220 | 194.792 | −64.9133 | −201.966 | 185.857 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 10.0618 | 4.83767 | 69.2398 | 25.0000 | 48.6757 | −224.792 | 374.700 | −219.597 | 251.545 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 605.6.a.g | yes | 7 |
| 11.b | odd | 2 | 1 | 605.6.a.f | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 605.6.a.f | ✓ | 7 | 11.b | odd | 2 | 1 | |
| 605.6.a.g | yes | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - T_{2}^{6} - 165T_{2}^{5} + 145T_{2}^{4} + 7100T_{2}^{3} - 4720T_{2}^{2} - 71936T_{2} + 96176 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(605))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - T^{6} + \cdots + 96176 \)
|
| $3$ |
\( T^{7} + 27 T^{6} + \cdots - 37873143 \)
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| $5$ |
\( (T - 25)^{7} \)
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| $7$ |
\( T^{7} + \cdots - 564625832067 \)
|
| $11$ |
\( T^{7} \)
|
| $13$ |
\( T^{7} + \cdots + 10\!\cdots\!24 \)
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| $17$ |
\( T^{7} + \cdots - 18\!\cdots\!00 \)
|
| $19$ |
\( T^{7} + \cdots - 28\!\cdots\!44 \)
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| $23$ |
\( T^{7} + \cdots - 12\!\cdots\!52 \)
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| $29$ |
\( T^{7} + \cdots - 14\!\cdots\!64 \)
|
| $31$ |
\( T^{7} + \cdots + 21\!\cdots\!68 \)
|
| $37$ |
\( T^{7} + \cdots - 17\!\cdots\!72 \)
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| $41$ |
\( T^{7} + \cdots + 13\!\cdots\!25 \)
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| $43$ |
\( T^{7} + \cdots - 33\!\cdots\!75 \)
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| $47$ |
\( T^{7} + \cdots + 10\!\cdots\!73 \)
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| $53$ |
\( T^{7} + \cdots - 14\!\cdots\!44 \)
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| $59$ |
\( T^{7} + \cdots - 58\!\cdots\!12 \)
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| $61$ |
\( T^{7} + \cdots + 83\!\cdots\!79 \)
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| $67$ |
\( T^{7} + \cdots - 44\!\cdots\!97 \)
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| $71$ |
\( T^{7} + \cdots + 42\!\cdots\!88 \)
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| $73$ |
\( T^{7} + \cdots + 12\!\cdots\!72 \)
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| $79$ |
\( T^{7} + \cdots - 77\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots + 32\!\cdots\!44 \)
|
| $89$ |
\( T^{7} + \cdots - 18\!\cdots\!69 \)
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| $97$ |
\( T^{7} + \cdots - 63\!\cdots\!88 \)
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