Newspace parameters
| Level: | \( N \) | \(=\) | \( 2175 = 3 \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2175.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(128.329154262\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{7} - x^{6} - 35x^{5} + 18x^{4} + 329x^{3} - 167x^{2} - 767x + 638 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 435) |
| 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
\( x^{7} - x^{6} - 35x^{5} + 18x^{4} + 329x^{3} - 167x^{2} - 767x + 638 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 10 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 4\nu^{5} - 23\nu^{4} + 67\nu^{3} + 128\nu^{2} - 131\nu - 94 ) / 20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{6} - 2\nu^{5} - 99\nu^{4} + \nu^{3} + 814\nu^{2} + 157\nu - 1262 ) / 20 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} - \nu^{5} + 38\nu^{4} + 43\nu^{3} - 363\nu^{2} - 304\nu + 744 ) / 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 17\nu^{6} + 2\nu^{5} - 581\nu^{4} - 381\nu^{3} + 4946\nu^{2} + 3183\nu - 8898 ) / 40 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 10 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} + 18\beta _1 + 4 \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} + 6\beta_{5} - \beta_{4} - 2\beta_{3} + 24\beta_{2} + 42\beta _1 + 166 \)
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| \(\nu^{5}\) | \(=\) |
\( 6\beta_{6} + 38\beta_{5} + 19\beta_{4} - 32\beta_{3} + 69\beta_{2} + 388\beta _1 + 246 \)
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| \(\nu^{6}\) | \(=\) |
\( 70\beta_{6} + 223\beta_{5} - 14\beta_{4} - 87\beta_{3} + 566\beta_{2} + 1315\beta _1 + 3348 \)
|
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 |
|
−3.96612 | −3.00000 | 7.73008 | 0 | 11.8983 | 32.8717 | 1.07055 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.52131 | −3.00000 | 4.39964 | 0 | 10.5639 | 8.86571 | 12.6780 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.08033 | −3.00000 | −3.67221 | 0 | 6.24100 | −15.0555 | 24.2821 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.10304 | −3.00000 | −6.78329 | 0 | −3.30913 | −1.72550 | −16.3066 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.26184 | −3.00000 | −6.40776 | 0 | −3.78552 | 13.4311 | −18.1803 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.07921 | −3.00000 | 1.48151 | 0 | −9.23762 | −23.1532 | −20.0718 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.12367 | −3.00000 | 18.2520 | 0 | −15.3710 | 21.7657 | 52.5281 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 2175.4.a.m | 7 | |
| 5.b | even | 2 | 1 | 435.4.a.j | ✓ | 7 | |
| 15.d | odd | 2 | 1 | 1305.4.a.m | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 435.4.a.j | ✓ | 7 | 5.b | even | 2 | 1 | |
| 1305.4.a.m | 7 | 15.d | odd | 2 | 1 | ||
| 2175.4.a.m | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2175))\):
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\( T_{2}^{7} - T_{2}^{6} - 35T_{2}^{5} + 18T_{2}^{4} + 329T_{2}^{3} - 167T_{2}^{2} - 767T_{2} + 638 \)
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\( T_{7}^{7} - 37T_{7}^{6} - 605T_{7}^{5} + 28229T_{7}^{4} - 7048T_{7}^{3} - 4672484T_{7}^{2} + 21806208T_{7} + 51243488 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - T^{6} + \cdots + 638 \)
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| $3$ |
\( (T + 3)^{7} \)
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| $5$ |
\( T^{7} \)
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| $7$ |
\( T^{7} - 37 T^{6} + \cdots + 51243488 \)
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| $11$ |
\( T^{7} + 11 T^{6} + \cdots - 344198304 \)
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| $13$ |
\( T^{7} + \cdots - 724528410064 \)
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| $17$ |
\( T^{7} + \cdots + 1088280942608 \)
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| $19$ |
\( T^{7} + \cdots + 206799749120 \)
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| $23$ |
\( T^{7} + \cdots + 199055640559616 \)
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| $29$ |
\( (T + 29)^{7} \)
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| $31$ |
\( T^{7} + \cdots - 2046710061056 \)
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| $37$ |
\( T^{7} + \cdots - 48\!\cdots\!52 \)
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| $41$ |
\( T^{7} + \cdots + 24\!\cdots\!48 \)
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| $43$ |
\( T^{7} + \cdots + 24\!\cdots\!48 \)
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| $47$ |
\( T^{7} + \cdots + 41\!\cdots\!36 \)
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| $53$ |
\( T^{7} + \cdots + 40\!\cdots\!68 \)
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| $59$ |
\( T^{7} + \cdots - 19\!\cdots\!80 \)
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| $61$ |
\( T^{7} + \cdots + 36\!\cdots\!16 \)
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| $67$ |
\( T^{7} + \cdots - 43\!\cdots\!52 \)
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| $71$ |
\( T^{7} + \cdots + 57\!\cdots\!52 \)
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| $73$ |
\( T^{7} + \cdots + 24\!\cdots\!56 \)
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| $79$ |
\( T^{7} + \cdots + 16\!\cdots\!60 \)
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| $83$ |
\( T^{7} + \cdots - 20\!\cdots\!24 \)
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| $89$ |
\( T^{7} + \cdots - 53\!\cdots\!00 \)
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| $97$ |
\( T^{7} + \cdots - 27\!\cdots\!76 \)
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