Newspace parameters
| Level: | \( N \) | \(=\) | \( 2023 = 7 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2023.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(119.360863942\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 3x^{6} - 49x^{5} + 69x^{4} + 753x^{3} - 122x^{2} - 3621x - 2536 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 119) |
| 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
\( x^{7} - 3x^{6} - 49x^{5} + 69x^{4} + 753x^{3} - 122x^{2} - 3621x - 2536 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( -119\nu^{6} + 490\nu^{5} + 4678\nu^{4} - 12834\nu^{3} - 51049\nu^{2} + 78837\nu + 155733 ) / 10291 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 589\nu^{6} - 3636\nu^{5} - 17533\nu^{4} + 100190\nu^{3} + 137827\nu^{2} - 608483\nu - 430864 ) / 20582 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -827\nu^{6} + 4616\nu^{5} + 26889\nu^{4} - 125858\nu^{3} - 219343\nu^{2} + 724993\nu + 454182 ) / 20582 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -960\nu^{6} + 5769\nu^{5} + 31512\nu^{4} - 164416\nu^{3} - 304244\nu^{2} + 1031032\nu + 1054405 ) / 10291 \)
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| \(\beta_{6}\) | \(=\) |
\( ( -1028\nu^{6} + 6049\nu^{5} + 32715\nu^{4} - 164399\nu^{3} - 293721\nu^{2} + 965821\nu + 893471 ) / 10291 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - \beta_{2} + 2\beta _1 + 14 \)
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| \(\nu^{3}\) | \(=\) |
\( 3\beta_{6} - \beta_{5} + \beta_{4} + 7\beta_{3} - 4\beta_{2} + 21\beta _1 + 27 \)
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| \(\nu^{4}\) | \(=\) |
\( 8\beta_{6} + 2\beta_{5} + 32\beta_{4} + 62\beta_{3} - 43\beta_{2} + 84\beta _1 + 343 \)
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| \(\nu^{5}\) | \(=\) |
\( 128\beta_{6} - 21\beta_{5} + 84\beta_{4} + 388\beta_{3} - 268\beta_{2} + 656\beta _1 + 1363 \)
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| \(\nu^{6}\) | \(=\) |
\( 518\beta_{6} + 100\beta_{5} + 1067\beta_{4} + 2851\beta_{3} - 2020\beta_{2} + 3543\beta _1 + 11487 \)
|
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 |
|
−5.60698 | 5.36988 | 23.4382 | −12.6276 | −30.1088 | 7.00000 | −86.5615 | 1.83559 | 70.8026 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.18404 | 1.42027 | 2.13811 | 11.4559 | −4.52221 | 7.00000 | 18.6645 | −24.9828 | −36.4761 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.11723 | −8.38428 | −3.51733 | −10.3793 | 17.7515 | 7.00000 | 24.3849 | 43.2962 | 21.9754 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.82736 | 10.3277 | −4.66075 | −9.18809 | 18.8724 | 7.00000 | −23.1358 | 79.6609 | −16.7900 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 3.77278 | −4.49237 | 6.23387 | −19.9390 | −16.9487 | 7.00000 | −6.66322 | −6.81863 | −75.2256 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.93035 | −9.43251 | 7.44766 | 8.85857 | −37.0731 | 7.00000 | −2.17089 | 61.9722 | 34.8173 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.37776 | 0.191325 | 20.9203 | −3.18044 | 1.02890 | 7.00000 | 69.4820 | −26.9634 | −17.1036 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 2023.4.a.g | 7 | |
| 17.b | even | 2 | 1 | 119.4.a.d | ✓ | 7 | |
| 51.c | odd | 2 | 1 | 1071.4.a.o | 7 | ||
| 68.d | odd | 2 | 1 | 1904.4.a.p | 7 | ||
| 119.d | odd | 2 | 1 | 833.4.a.f | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 119.4.a.d | ✓ | 7 | 17.b | even | 2 | 1 | |
| 833.4.a.f | 7 | 119.d | odd | 2 | 1 | ||
| 1071.4.a.o | 7 | 51.c | odd | 2 | 1 | ||
| 1904.4.a.p | 7 | 68.d | odd | 2 | 1 | ||
| 2023.4.a.g | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2023))\):
|
\( T_{2}^{7} - 4T_{2}^{6} - 46T_{2}^{5} + 186T_{2}^{4} + 514T_{2}^{3} - 2037T_{2}^{2} - 1586T_{2} + 5508 \)
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\( T_{3}^{7} + 5T_{3}^{6} - 146T_{3}^{5} - 685T_{3}^{4} + 4670T_{3}^{3} + 14223T_{3}^{2} - 30871T_{3} + 5354 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 4 T^{6} + \cdots + 5508 \)
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| $3$ |
\( T^{7} + 5 T^{6} + \cdots + 5354 \)
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| $5$ |
\( T^{7} + 35 T^{6} + \cdots + 7749964 \)
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| $7$ |
\( (T - 7)^{7} \)
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| $11$ |
\( T^{7} + \cdots - 34663527936 \)
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| $13$ |
\( T^{7} + \cdots + 13756569984 \)
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| $17$ |
\( T^{7} \)
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| $19$ |
\( T^{7} + \cdots - 3100949967360 \)
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| $23$ |
\( T^{7} + \cdots + 426071334454272 \)
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| $29$ |
\( T^{7} + \cdots - 394911043338240 \)
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| $31$ |
\( T^{7} + \cdots - 229799266689312 \)
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| $37$ |
\( T^{7} + \cdots + 20\!\cdots\!28 \)
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| $41$ |
\( T^{7} + \cdots + 39\!\cdots\!78 \)
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| $43$ |
\( T^{7} + \cdots - 26\!\cdots\!12 \)
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| $47$ |
\( T^{7} + \cdots + 20\!\cdots\!36 \)
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| $53$ |
\( T^{7} + \cdots + 33\!\cdots\!18 \)
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| $59$ |
\( T^{7} + \cdots - 22\!\cdots\!20 \)
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| $61$ |
\( T^{7} + \cdots - 72\!\cdots\!28 \)
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| $67$ |
\( T^{7} + \cdots - 52\!\cdots\!68 \)
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| $71$ |
\( T^{7} + \cdots - 16\!\cdots\!96 \)
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| $73$ |
\( T^{7} + \cdots - 44\!\cdots\!34 \)
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| $79$ |
\( T^{7} + \cdots + 165111753621760 \)
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| $83$ |
\( T^{7} + \cdots - 31\!\cdots\!24 \)
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| $89$ |
\( T^{7} + \cdots - 11\!\cdots\!40 \)
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| $97$ |
\( T^{7} + \cdots - 36\!\cdots\!26 \)
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