Newspace parameters
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 24 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(23.4642826142\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} - 3 x^{5} - 27997302 x^{4} - 8334207232 x^{3} + 155343730039680 x^{2} + \cdots - 52\!\cdots\!92 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{4}\cdot 5\cdot 7^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3 x^{5} - 27997302 x^{4} - 8334207232 x^{3} + 155343730039680 x^{2} + \cdots - 52\!\cdots\!92 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5384921 \nu^{5} + 47470977237 \nu^{4} - 306117999883302 \nu^{3} + \cdots + 37\!\cdots\!28 ) / 91\!\cdots\!76 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5384921 \nu^{5} + 47470977237 \nu^{4} - 306117999883302 \nu^{3} + \cdots - 82\!\cdots\!32 ) / 91\!\cdots\!76 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 895595161 \nu^{5} + 2308020934251 \nu^{4} + \cdots + 51\!\cdots\!80 ) / 91\!\cdots\!76 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1666574431 \nu^{5} + 1680075892605 \nu^{4} + \cdots - 36\!\cdots\!96 ) / 91\!\cdots\!76 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 451\beta _1 + 9332210 \)
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| \(\nu^{3}\) | \(=\) |
\( -1004\beta_{5} + 1611\beta_{4} - 18\beta_{3} - 42775\beta_{2} + 16907685\beta _1 + 4200652260 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2642380 \beta_{5} + 5140977 \beta_{4} + 22741304 \beta_{3} + 14494373 \beta_{2} + \cdots + 157731853492144 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 33780646188 \beta_{5} + 46260455013 \beta_{4} + 7208346460 \beta_{3} - 1060821568939 \beta_{2} + \cdots + 10\!\cdots\!24 \)
|
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 |
|
−4999.62 | −458525. | 1.66076e7 | 1.79255e8 | 2.29245e9 | −1.97733e9 | −4.10921e10 | 1.16102e11 | −8.96207e11 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3240.78 | −36314.3 | 2.11404e6 | −6.24332e7 | 1.17686e8 | −1.97733e9 | 2.03345e10 | −9.28245e10 | 2.02332e11 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −676.641 | 513256. | −7.93077e6 | −1.76618e8 | −3.47290e8 | −1.97733e9 | 1.10424e10 | 1.69289e11 | 1.19507e11 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 890.125 | −64762.8 | −7.59629e6 | 1.51631e8 | −5.76470e7 | −1.97733e9 | −1.42285e10 | −8.99490e10 | 1.34970e11 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1834.42 | −386098. | −5.02351e6 | −1.52140e8 | −7.08266e8 | −1.97733e9 | −2.46035e10 | 5.49283e10 | −2.79088e11 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4077.50 | 561795. | 8.23738e6 | 1.31743e8 | 2.29072e9 | −1.97733e9 | −6.16637e8 | 2.21471e11 | 5.37182e11 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( +1 \) |
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 | 7.24.a.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 63.24.a.e | 6 | ||
| 7.b | odd | 2 | 1 | 49.24.a.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.24.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 49.24.a.d | 6 | 7.b | odd | 2 | 1 | ||
| 63.24.a.e | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 2115 T_{2}^{5} - 26133462 T_{2}^{4} - 30209992704 T_{2}^{3} + 143562378079104 T_{2}^{2} + \cdots - 72\!\cdots\!80 \)
acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(7))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 72\!\cdots\!80 \)
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| $3$ |
\( T^{6} + \cdots + 12\!\cdots\!96 \)
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| $5$ |
\( T^{6} + \cdots - 60\!\cdots\!00 \)
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| $7$ |
\( (T + 1977326743)^{6} \)
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| $11$ |
\( T^{6} + \cdots + 81\!\cdots\!76 \)
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| $13$ |
\( T^{6} + \cdots + 49\!\cdots\!20 \)
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| $17$ |
\( T^{6} + \cdots + 43\!\cdots\!08 \)
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| $19$ |
\( T^{6} + \cdots - 20\!\cdots\!00 \)
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| $23$ |
\( T^{6} + \cdots + 55\!\cdots\!36 \)
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| $29$ |
\( T^{6} + \cdots - 11\!\cdots\!00 \)
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| $31$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 12\!\cdots\!56 \)
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| $41$ |
\( T^{6} + \cdots - 26\!\cdots\!24 \)
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| $43$ |
\( T^{6} + \cdots + 30\!\cdots\!40 \)
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| $47$ |
\( T^{6} + \cdots - 46\!\cdots\!04 \)
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| $53$ |
\( T^{6} + \cdots + 31\!\cdots\!68 \)
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| $59$ |
\( T^{6} + \cdots + 45\!\cdots\!00 \)
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| $61$ |
\( T^{6} + \cdots - 85\!\cdots\!56 \)
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| $67$ |
\( T^{6} + \cdots - 86\!\cdots\!92 \)
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| $71$ |
\( T^{6} + \cdots + 84\!\cdots\!80 \)
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| $73$ |
\( T^{6} + \cdots - 17\!\cdots\!44 \)
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| $79$ |
\( T^{6} + \cdots - 97\!\cdots\!00 \)
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| $83$ |
\( T^{6} + \cdots + 15\!\cdots\!32 \)
|
| $89$ |
\( T^{6} + \cdots - 19\!\cdots\!00 \)
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| $97$ |
\( T^{6} + \cdots - 23\!\cdots\!52 \)
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