Newspace parameters
| Level: | \( N \) | \(=\) | \( 1445 = 5 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1445.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(85.2577599583\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{8} - 4x^{7} - 48x^{6} + 112x^{5} + 767x^{4} - 496x^{3} - 3404x^{2} + 576x + 4128 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 48x^{6} + 112x^{5} + 767x^{4} - 496x^{3} - 3404x^{2} + 576x + 4128 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 13 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{7} + 24\nu^{6} + 11\nu^{5} - 847\nu^{4} + 855\nu^{3} + 8597\nu^{2} - 4402\nu - 17296 ) / 428 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 19\nu^{7} - 121\nu^{6} - 693\nu^{5} + 3927\nu^{4} + 8944\nu^{3} - 34324\nu^{2} - 37040\nu + 81280 ) / 1712 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -31\nu^{7} + 265\nu^{6} + 973\nu^{5} - 10079\nu^{4} - 9164\nu^{3} + 106236\nu^{2} + 36736\nu - 236416 ) / 3424 \)
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| \(\beta_{6}\) | \(=\) |
\( ( -45\nu^{7} + 219\nu^{6} + 1799\nu^{5} - 5629\nu^{4} - 23188\nu^{3} + 17364\nu^{2} + 53216\nu + 320 ) / 3424 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 12\nu^{7} - 37\nu^{6} - 601\nu^{5} + 909\nu^{4} + 8887\nu^{3} - 650\nu^{2} - 21524\nu - 228 ) / 428 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 13 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 2\beta_{4} + 4\beta_{2} + 25\beta _1 + 26 \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} + 8\beta_{6} + 4\beta_{5} + 6\beta_{4} - 4\beta_{3} + 39\beta_{2} + 100\beta _1 + 337 \)
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| \(\nu^{5}\) | \(=\) |
\( 10\beta_{7} + 65\beta_{6} + 61\beta_{5} + 88\beta_{4} - 32\beta_{3} + 200\beta_{2} + 813\beta _1 + 1340 \)
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| \(\nu^{6}\) | \(=\) |
\( 132\beta_{7} + 506\beta_{6} + 330\beta_{5} + 412\beta_{4} - 292\beta_{3} + 1521\beta_{2} + 4186\beta _1 + 11221 \)
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| \(\nu^{7}\) | \(=\) |
\( 792 \beta_{7} + 3469 \beta_{6} + 3029 \beta_{5} + 3742 \beta_{4} - 2200 \beta_{3} + 8844 \beta_{2} + \cdots + 57650 \)
|
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 |
|
−5.49025 | −1.90849 | 22.1428 | −5.00000 | 10.4781 | 14.4921 | −77.6476 | −23.3576 | 27.4512 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.30190 | 9.17969 | 10.5063 | −5.00000 | −39.4901 | −13.6050 | −10.7820 | 57.2667 | 21.5095 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.944847 | 7.13877 | −7.10726 | −5.00000 | −6.74505 | 21.6633 | 14.2741 | 23.9620 | 4.72424 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.421239 | −9.69675 | −7.82256 | −5.00000 | 4.08465 | −27.4580 | 6.66508 | 67.0269 | 2.10620 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.42383 | 1.34922 | −2.12506 | −5.00000 | 3.27027 | −30.7032 | −24.5414 | −25.1796 | −12.1191 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.22172 | −7.57054 | 2.37948 | −5.00000 | −24.3902 | 26.0836 | −18.1077 | 30.3130 | −16.1086 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.13159 | 8.07129 | 9.07001 | −5.00000 | 33.3473 | 3.79835 | 4.42086 | 38.1458 | −20.6579 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.38110 | −4.56319 | 20.9562 | −5.00000 | −24.5550 | −26.2712 | 69.7188 | −6.17728 | −26.9055 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(17\) | \( +1 \) |
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 | 1445.4.a.r | yes | 8 |
| 17.b | even | 2 | 1 | 1445.4.a.q | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1445.4.a.q | ✓ | 8 | 17.b | even | 2 | 1 | |
| 1445.4.a.r | yes | 8 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1445))\):
|
\( T_{2}^{8} - 4T_{2}^{7} - 48T_{2}^{6} + 204T_{2}^{5} + 537T_{2}^{4} - 2648T_{2}^{3} + 54T_{2}^{2} + 4400T_{2} + 1632 \)
|
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\( T_{3}^{8} - 2T_{3}^{7} - 187T_{3}^{6} + 304T_{3}^{5} + 10999T_{3}^{4} - 9698T_{3}^{3} - 211785T_{3}^{2} - 61932T_{3} + 456236 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 4 T^{7} + \cdots + 1632 \)
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| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 456236 \)
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| $5$ |
\( (T + 5)^{8} \)
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| $7$ |
\( T^{8} + \cdots + 9372381748 \)
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| $11$ |
\( T^{8} + \cdots - 1550803898064 \)
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| $13$ |
\( T^{8} + \cdots + 11082701134528 \)
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| $17$ |
\( T^{8} \)
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| $19$ |
\( T^{8} + \cdots + 13\!\cdots\!96 \)
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| $23$ |
\( T^{8} + \cdots - 15\!\cdots\!84 \)
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| $29$ |
\( T^{8} + \cdots + 675341980498944 \)
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| $31$ |
\( T^{8} + \cdots - 54\!\cdots\!00 \)
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| $37$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
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| $41$ |
\( T^{8} + \cdots + 63\!\cdots\!49 \)
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| $43$ |
\( T^{8} + \cdots + 56\!\cdots\!96 \)
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| $47$ |
\( T^{8} + \cdots + 46\!\cdots\!56 \)
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| $53$ |
\( T^{8} + \cdots - 86\!\cdots\!72 \)
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| $59$ |
\( T^{8} + \cdots - 10\!\cdots\!56 \)
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| $61$ |
\( T^{8} + \cdots + 17\!\cdots\!56 \)
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| $67$ |
\( T^{8} + \cdots + 51\!\cdots\!96 \)
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| $71$ |
\( T^{8} + \cdots - 48\!\cdots\!32 \)
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| $73$ |
\( T^{8} + \cdots + 59\!\cdots\!04 \)
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| $79$ |
\( T^{8} + \cdots - 25\!\cdots\!04 \)
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| $83$ |
\( T^{8} + \cdots - 18\!\cdots\!64 \)
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| $89$ |
\( T^{8} + \cdots + 70\!\cdots\!29 \)
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| $97$ |
\( T^{8} + \cdots + 34\!\cdots\!44 \)
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