Newspace parameters
| Level: | \( N \) | \(=\) | \( 123 = 3 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 123.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(7.25723493071\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 34x^{4} - 26x^{3} + 269x^{2} + 258x - 272 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 34x^{4} - 26x^{3} + 269x^{2} + 258x - 272 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 6\nu^{4} + 12\nu^{3} - 88\nu^{2} + 7\nu + 134 ) / 14 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} + 45\nu^{3} - 57\nu^{2} - 175\nu + 72 ) / 14 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} + 59\nu^{3} - 85\nu^{2} - 399\nu + 212 ) / 14 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 2\beta_{2} + 18\beta _1 + 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{5} - 5\beta_{4} + 4\beta_{3} + 23\beta_{2} + 44\beta _1 + 195 \)
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| \(\nu^{5}\) | \(=\) |
\( 30\beta_{5} - 42\beta_{4} + 10\beta_{3} + 74\beta_{2} + 399\beta _1 + 480 \)
|
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 |
|
−4.27916 | 3.00000 | 10.3112 | 15.7767 | −12.8375 | 5.96642 | −9.89000 | 9.00000 | −67.5112 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.17178 | 3.00000 | −3.28338 | −11.3872 | −6.51533 | −6.57196 | 24.5050 | 9.00000 | 24.7304 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.342985 | 3.00000 | −7.88236 | 9.86667 | 1.02895 | 16.4245 | −5.44740 | 9.00000 | 3.38412 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.93430 | 3.00000 | 0.610133 | 6.30631 | 8.80291 | 18.1286 | −21.6841 | 9.00000 | 18.5046 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.29831 | 3.00000 | 10.4755 | 21.2228 | 12.8949 | −27.1762 | 10.6404 | 9.00000 | 91.2221 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.87534 | 3.00000 | 15.7689 | −4.78533 | 14.6260 | 7.22865 | 37.8761 | 9.00000 | −23.3301 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(41\) | \( -1 \) |
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 | 123.4.a.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | 369.4.a.f | 6 | ||
| 4.b | odd | 2 | 1 | 1968.4.a.s | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 123.4.a.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 369.4.a.f | 6 | 3.b | odd | 2 | 1 | ||
| 1968.4.a.s | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 6T_{2}^{5} - 19T_{2}^{4} + 142T_{2}^{3} + 2T_{2}^{2} - 588T_{2} + 196 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(123))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 6 T^{5} + \cdots + 196 \)
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| $3$ |
\( (T - 3)^{6} \)
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| $5$ |
\( T^{6} - 37 T^{5} + \cdots + 1135256 \)
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| $7$ |
\( T^{6} - 14 T^{5} + \cdots + 2293568 \)
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| $11$ |
\( T^{6} - 50 T^{5} + \cdots - 81911864 \)
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| $13$ |
\( T^{6} - 27 T^{5} + \cdots - 978161912 \)
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| $17$ |
\( T^{6} + \cdots + 19797717802 \)
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| $19$ |
\( T^{6} + \cdots + 1289742832 \)
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| $23$ |
\( T^{6} + \cdots + 50890186048 \)
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| $29$ |
\( T^{6} + \cdots + 2531498501216 \)
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| $31$ |
\( T^{6} + \cdots - 2218959458272 \)
|
| $37$ |
\( T^{6} + \cdots + 884867511276 \)
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| $41$ |
\( (T - 41)^{6} \)
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| $43$ |
\( T^{6} + \cdots - 17814848169328 \)
|
| $47$ |
\( T^{6} + \cdots + 96987904976768 \)
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| $53$ |
\( T^{6} + \cdots - 270637221082112 \)
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| $59$ |
\( T^{6} + \cdots - 586668983577344 \)
|
| $61$ |
\( T^{6} + \cdots - 499608027642676 \)
|
| $67$ |
\( T^{6} + \cdots - 869522980253504 \)
|
| $71$ |
\( T^{6} + \cdots + 12\!\cdots\!48 \)
|
| $73$ |
\( T^{6} + \cdots - 420232761125654 \)
|
| $79$ |
\( T^{6} + \cdots - 822986137536512 \)
|
| $83$ |
\( T^{6} + \cdots + 41\!\cdots\!88 \)
|
| $89$ |
\( T^{6} + \cdots - 65\!\cdots\!48 \)
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| $97$ |
\( T^{6} + \cdots - 13\!\cdots\!16 \)
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