Newspace parameters
| Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 722.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(42.5993790241\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{8} - 4x^{7} - 70x^{6} + 224x^{5} + 1679x^{4} - 3736x^{3} - 15650x^{2} + 17556x + 46141 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 19 \) |
| 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
\( x^{8} - 4x^{7} - 70x^{6} + 224x^{5} + 1679x^{4} - 3736x^{3} - 15650x^{2} + 17556x + 46141 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} + 54\nu^{4} - 113\nu^{3} - 744\nu^{2} + 1637\nu + 1501 ) / 836 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{7} - 7\nu^{6} - 190\nu^{5} + 275\nu^{4} + 3690\nu^{3} - 2169\nu^{2} - 19965\nu - 3002 ) / 1672 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{6} - 13\nu^{5} + 228\nu^{4} + 431\nu^{3} - 4300\nu^{2} - 589\nu + 16691 ) / 1672 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} + 98\nu^{4} - 201\nu^{3} - 2372\nu^{2} + 2473\nu + 11665 ) / 836 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 4\nu^{6} + 51\nu^{5} - 167\nu^{4} - 631\nu^{3} + 1545\nu^{2} - 136\nu - 1501 ) / 836 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 7\nu^{6} - 21\nu^{5} - 356\nu^{4} + 747\nu^{3} + 4812\nu^{2} - 5189\nu - 14203 ) / 1672 \)
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| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{6} + 9\nu^{5} + 173\nu^{4} - 361\nu^{3} - 3057\nu^{2} + 3239\nu + 15404 ) / 836 \)
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| \(\nu\) | \(=\) |
\( ( 4\beta_{7} + 8\beta_{6} - 3\beta_{4} + 19\beta _1 + 2 ) / 19 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -56\beta_{7} - 36\beta_{6} + 23\beta_{4} + 19\beta _1 + 371 ) / 19 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 60\beta_{7} + 120\beta_{6} + 114\beta_{5} - 26\beta_{4} - 76\beta_{3} + 76\beta_{2} + 570\beta _1 + 353 ) / 19 \)
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| \(\nu^{4}\) | \(=\) |
\( ( - 2028 \beta_{7} - 1244 \beta_{6} + 228 \beta_{5} + 1217 \beta_{4} - 152 \beta_{3} + 152 \beta_{2} + \cdots + 10006 ) / 19 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 280 \beta_{7} + 1852 \beta_{6} + 4674 \beta_{5} + 1329 \beta_{4} - 4560 \beta_{3} + 3116 \beta_{2} + \cdots + 17525 ) / 19 \)
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| \(\nu^{6}\) | \(=\) |
\( ( - 67240 \beta_{7} - 35300 \beta_{6} + 13452 \beta_{5} + 50620 \beta_{4} - 13300 \beta_{3} + \cdots + 308779 ) / 19 \)
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| \(\nu^{7}\) | \(=\) |
\( ( - 40928 \beta_{7} + 28572 \beta_{6} + 166288 \beta_{5} + 119369 \beta_{4} - 212420 \beta_{3} + \cdots + 779550 ) / 19 \)
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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 |
|
2.00000 | −8.87755 | 4.00000 | −12.9854 | −17.7551 | 22.1836 | 8.00000 | 51.8108 | −25.9709 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −8.71426 | 4.00000 | 9.53711 | −17.4285 | −27.1225 | 8.00000 | 48.9383 | 19.0742 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −6.19215 | 4.00000 | −0.275591 | −12.3843 | −21.0524 | 8.00000 | 11.3427 | −0.551183 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | −0.112693 | 4.00000 | −12.6298 | −0.225386 | 14.7720 | 8.00000 | −26.9873 | −25.2597 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 1.00596 | 4.00000 | −5.62903 | 2.01191 | 33.8378 | 8.00000 | −25.9881 | −11.2581 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 2.53495 | 4.00000 | 11.7705 | 5.06991 | −27.8492 | 8.00000 | −20.5740 | 23.5410 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.00000 | 2.74508 | 4.00000 | 1.68636 | 5.49016 | −0.654888 | 8.00000 | −19.4645 | 3.37272 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.00000 | 7.61065 | 4.00000 | −9.47410 | 15.2213 | −8.11445 | 8.00000 | 30.9221 | −18.9482 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 722.4.a.s | yes | 8 |
| 19.b | odd | 2 | 1 | 722.4.a.r | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 722.4.a.r | ✓ | 8 | 19.b | odd | 2 | 1 | |
| 722.4.a.s | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(722))\):
|
\( T_{3}^{8} + 10T_{3}^{7} - 83T_{3}^{6} - 790T_{3}^{5} + 2119T_{3}^{4} + 11610T_{3}^{3} - 36882T_{3}^{2} + 21220T_{3} + 2876 \)
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\( T_{5}^{8} + 18T_{5}^{7} - 179T_{5}^{6} - 4256T_{5}^{5} + 1049T_{5}^{4} + 254612T_{5}^{3} + 638604T_{5}^{2} - 1499024T_{5} - 456304 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{8} \)
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| $3$ |
\( T^{8} + 10 T^{7} + \cdots + 2876 \)
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| $5$ |
\( T^{8} + 18 T^{7} + \cdots - 456304 \)
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| $7$ |
\( T^{8} + 14 T^{7} + \cdots - 937005364 \)
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| $11$ |
\( T^{8} + \cdots - 7865689912324 \)
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| $13$ |
\( T^{8} + \cdots + 13602132727441 \)
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| $17$ |
\( T^{8} + \cdots + 19832275922176 \)
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| $19$ |
\( T^{8} \)
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| $23$ |
\( T^{8} + \cdots - 13\!\cdots\!44 \)
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| $29$ |
\( T^{8} + \cdots - 12\!\cdots\!99 \)
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| $31$ |
\( T^{8} + \cdots + 87\!\cdots\!56 \)
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| $37$ |
\( T^{8} + \cdots + 87\!\cdots\!61 \)
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| $41$ |
\( T^{8} + \cdots + 42\!\cdots\!01 \)
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| $43$ |
\( T^{8} + \cdots - 20\!\cdots\!84 \)
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| $47$ |
\( T^{8} + \cdots - 22\!\cdots\!84 \)
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| $53$ |
\( T^{8} + \cdots + 30\!\cdots\!01 \)
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| $59$ |
\( T^{8} + \cdots - 38\!\cdots\!84 \)
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| $61$ |
\( T^{8} + \cdots + 16\!\cdots\!21 \)
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| $67$ |
\( T^{8} + \cdots - 46\!\cdots\!44 \)
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| $71$ |
\( T^{8} + \cdots - 70\!\cdots\!04 \)
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| $73$ |
\( T^{8} + \cdots - 11\!\cdots\!79 \)
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| $79$ |
\( T^{8} + \cdots - 10\!\cdots\!00 \)
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| $83$ |
\( T^{8} + \cdots - 65\!\cdots\!24 \)
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| $89$ |
\( T^{8} + \cdots + 20\!\cdots\!81 \)
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| $97$ |
\( T^{8} + \cdots + 96\!\cdots\!01 \)
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