Newspace parameters
| Level: | \( N \) | \(=\) | \( 2275 = 5^{2} \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2275.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(18.1659664598\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 3x^{6} - 6x^{5} + 22x^{4} - 31x^{2} + 12x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{7} - 3x^{6} - 6x^{5} + 22x^{4} - 31x^{2} + 12x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 8\nu^{4} + 5\nu^{3} + 12\nu^{2} - \nu \)
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| \(\beta_{3}\) | \(=\) |
\( -2\nu^{6} + 2\nu^{5} + 17\nu^{4} - 11\nu^{3} - 31\nu^{2} + 7\nu + 6 \)
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| \(\beta_{4}\) | \(=\) |
\( -2\nu^{6} + 3\nu^{5} + 16\nu^{4} - 19\nu^{3} - 25\nu^{2} + 19\nu + 1 \)
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| \(\beta_{5}\) | \(=\) |
\( 2\nu^{6} - 3\nu^{5} - 16\nu^{4} + 20\nu^{3} + 25\nu^{2} - 24\nu - 1 \)
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| \(\beta_{6}\) | \(=\) |
\( 4\nu^{6} - 5\nu^{5} - 33\nu^{4} + 31\nu^{3} + 55\nu^{2} - 31\nu - 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{3} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 5\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( -7\beta_{6} + 8\beta_{5} + \beta_{4} - 6\beta_{3} + 2\beta_{2} + 8 \)
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| \(\nu^{5}\) | \(=\) |
\( -\beta_{6} + 10\beta_{5} + 10\beta_{4} - \beta_{3} + 2\beta_{2} + 28\beta _1 + 1 \)
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| \(\nu^{6}\) | \(=\) |
\( -45\beta_{6} + 57\beta_{5} + 13\beta_{4} - 37\beta_{3} + 19\beta_{2} + 4\beta _1 + 41 \)
|
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 |
|
−2.35435 | 2.71231 | 3.54297 | 0 | −6.38573 | −1.00000 | −3.63269 | 4.35663 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.31857 | −1.63091 | −0.261378 | 0 | 2.15046 | −1.00000 | 2.98178 | −0.340140 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.126080 | 0.287532 | −1.98410 | 0 | −0.0362519 | −1.00000 | 0.502315 | −2.91733 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.628034 | 0.473449 | −1.60557 | 0 | 0.297342 | −1.00000 | −2.26442 | −2.77585 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.70895 | −2.66143 | 0.920509 | 0 | −4.54825 | −1.00000 | −1.84480 | 4.08322 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.76583 | 2.98211 | 1.11814 | 0 | 5.26589 | −1.00000 | −1.55721 | 5.89299 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.69619 | 0.836937 | 5.26943 | 0 | 2.25654 | −1.00000 | 8.81501 | −2.29954 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 2275.2.a.z | yes | 7 |
| 5.b | even | 2 | 1 | 2275.2.a.v | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2275.2.a.v | ✓ | 7 | 5.b | even | 2 | 1 | |
| 2275.2.a.z | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2275))\):
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\( T_{2}^{7} - 3T_{2}^{6} - 6T_{2}^{5} + 22T_{2}^{4} - 31T_{2}^{2} + 12T_{2} + 2 \)
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\( T_{3}^{7} - 3T_{3}^{6} - 9T_{3}^{5} + 28T_{3}^{4} + 10T_{3}^{3} - 47T_{3}^{2} + 26T_{3} - 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 3 T^{6} + \cdots + 2 \)
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| $3$ |
\( T^{7} - 3 T^{6} + \cdots - 4 \)
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| $5$ |
\( T^{7} \)
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| $7$ |
\( (T + 1)^{7} \)
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| $11$ |
\( T^{7} + 3 T^{6} + \cdots - 1346 \)
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| $13$ |
\( (T - 1)^{7} \)
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| $17$ |
\( T^{7} - 17 T^{6} + \cdots - 272 \)
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| $19$ |
\( T^{7} + 6 T^{6} + \cdots - 3768 \)
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| $23$ |
\( T^{7} - 17 T^{6} + \cdots - 21639 \)
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| $29$ |
\( T^{7} - 4 T^{6} + \cdots + 33487 \)
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| $31$ |
\( T^{7} - 3 T^{6} + \cdots + 38264 \)
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| $37$ |
\( T^{7} + 3 T^{6} + \cdots - 151132 \)
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| $41$ |
\( T^{7} + 16 T^{6} + \cdots - 66432 \)
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| $43$ |
\( T^{7} - 8 T^{6} + \cdots + 8 \)
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| $47$ |
\( T^{7} - T^{6} + \cdots - 25664 \)
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| $53$ |
\( T^{7} - 23 T^{6} + \cdots + 4 \)
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| $59$ |
\( T^{7} + 22 T^{6} + \cdots - 56976 \)
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| $61$ |
\( T^{7} - 13 T^{6} + \cdots - 52 \)
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| $67$ |
\( T^{7} - 12 T^{6} + \cdots - 77236 \)
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| $71$ |
\( T^{7} + 3 T^{6} + \cdots + 684 \)
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| $73$ |
\( T^{7} - 14 T^{6} + \cdots - 1536 \)
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| $79$ |
\( T^{7} - T^{6} + \cdots - 552723 \)
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| $83$ |
\( T^{7} - 33 T^{6} + \cdots - 238424 \)
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| $89$ |
\( T^{7} + 11 T^{6} + \cdots + 2970592 \)
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| $97$ |
\( T^{7} + 5 T^{6} + \cdots + 808496 \)
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