Newspace parameters
| Level: | \( N \) | \(=\) | \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 828.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(48.8535814848\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 194x^{4} + 484x^{3} + 7122x^{2} - 18036x + 6804 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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
\( x^{6} - 2x^{5} - 194x^{4} + 484x^{3} + 7122x^{2} - 18036x + 6804 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -305\nu^{5} + 4462\nu^{4} + 34564\nu^{3} - 939134\nu^{2} + 1746102\nu + 24289470 ) / 880254 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 373\nu^{5} - 326\nu^{4} - 64076\nu^{3} - 1748\nu^{2} + 1833282\nu + 3643812 ) / 293418 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4819\nu^{5} + 11816\nu^{4} + 1015580\nu^{3} - 2397394\nu^{2} - 43770846\nu + 67762440 ) / 2640762 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -1933\nu^{5} - 2506\nu^{4} + 349892\nu^{3} + 69368\nu^{2} - 9224778\nu + 9303660 ) / 880254 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1933\nu^{5} + 2506\nu^{4} - 349892\nu^{3} - 69368\nu^{2} + 12745794\nu - 11064168 ) / 880254 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 2 ) / 4 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 2\beta_{2} - \beta _1 + 65 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 131\beta_{5} + 118\beta_{4} + 48\beta_{3} + 35\beta_{2} - 42\beta _1 - 108 ) / 4 \)
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| \(\nu^{4}\) | \(=\) |
\( 170\beta_{5} - 68\beta_{4} + 51\beta_{3} - 354\beta_{2} - 59\beta _1 + 7571 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 9101\beta_{5} + 7559\beta_{4} + 4212\beta_{3} + 3942\beta_{2} - 3720\beta _1 - 19886 ) / 2 \)
|
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 |
|
0 | 0 | 0 | −20.6668 | 0 | 5.93238 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −10.0501 | 0 | 21.0723 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −8.04942 | 0 | −11.2672 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 4.31779 | 0 | −17.5188 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 8.64828 | 0 | 21.4788 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0 | 0 | 15.8003 | 0 | −23.6974 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(23\) | \( +1 \) |
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 | 828.4.a.h | ✓ | 6 |
| 3.b | odd | 2 | 1 | 828.4.a.i | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 828.4.a.h | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 828.4.a.i | yes | 6 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 10T_{5}^{5} - 418T_{5}^{4} - 2616T_{5}^{3} + 39228T_{5}^{2} + 136520T_{5} - 986424 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(828))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} + 10 T^{5} + \cdots - 986424 \)
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| $7$ |
\( T^{6} + 4 T^{5} + \cdots - 12559536 \)
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| $11$ |
\( T^{6} - 2 T^{5} + \cdots + 495766656 \)
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| $13$ |
\( T^{6} + \cdots - 10357236736 \)
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| $17$ |
\( T^{6} + \cdots - 40565727072 \)
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| $19$ |
\( T^{6} + \cdots + 39934428648 \)
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| $23$ |
\( (T + 23)^{6} \)
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| $29$ |
\( T^{6} + \cdots + 20060936140032 \)
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| $31$ |
\( T^{6} + \cdots - 7169385167616 \)
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| $37$ |
\( T^{6} + \cdots - 32370358064736 \)
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| $41$ |
\( T^{6} + \cdots - 54845857078272 \)
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| $43$ |
\( T^{6} + \cdots + 1398782399976 \)
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| $47$ |
\( T^{6} + \cdots + 573130723392 \)
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| $53$ |
\( T^{6} + \cdots + 10\!\cdots\!64 \)
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| $59$ |
\( T^{6} + \cdots + 15\!\cdots\!32 \)
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| $61$ |
\( T^{6} + \cdots + 391138335158176 \)
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| $67$ |
\( T^{6} + \cdots - 56\!\cdots\!72 \)
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| $71$ |
\( T^{6} + \cdots + 27\!\cdots\!12 \)
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| $73$ |
\( T^{6} + \cdots + 97\!\cdots\!00 \)
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| $79$ |
\( T^{6} + \cdots + 25\!\cdots\!04 \)
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| $83$ |
\( T^{6} + \cdots - 25\!\cdots\!72 \)
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| $89$ |
\( T^{6} + \cdots - 38\!\cdots\!72 \)
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| $97$ |
\( T^{6} + \cdots - 187093131265728 \)
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