Newspace parameters
| Level: | \( N \) | \(=\) | \( 152 = 2^{3} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 152.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(216.894127752\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{18} - 8 x^{17} - 183998036 x^{16} - 16776585512 x^{15} + \cdots + 19\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{104}\cdot 3^{5}\cdot 5^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 8 x^{17} - 183998036 x^{16} - 16776585512 x^{15} + \cdots + 19\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 39\!\cdots\!21 \nu^{17} + \cdots - 60\!\cdots\!28 ) / 32\!\cdots\!72 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 39\!\cdots\!21 \nu^{17} + \cdots + 72\!\cdots\!08 ) / 32\!\cdots\!72 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 56\!\cdots\!77 \nu^{17} + \cdots - 56\!\cdots\!16 ) / 13\!\cdots\!88 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 18\!\cdots\!93 \nu^{17} + \cdots + 11\!\cdots\!24 ) / 69\!\cdots\!52 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 16\!\cdots\!09 \nu^{17} + \cdots - 11\!\cdots\!44 ) / 13\!\cdots\!88 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 38\!\cdots\!47 \nu^{17} + \cdots + 35\!\cdots\!44 ) / 16\!\cdots\!36 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 15\!\cdots\!65 \nu^{17} + \cdots + 75\!\cdots\!56 ) / 43\!\cdots\!96 \)
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| \(\beta_{9}\) | \(=\) |
\( ( 74\!\cdots\!29 \nu^{17} + \cdots - 39\!\cdots\!08 ) / 13\!\cdots\!88 \)
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| \(\beta_{10}\) | \(=\) |
\( ( - 16\!\cdots\!77 \nu^{17} + \cdots + 15\!\cdots\!28 ) / 13\!\cdots\!88 \)
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| \(\beta_{11}\) | \(=\) |
\( ( - 24\!\cdots\!69 \nu^{17} + \cdots + 74\!\cdots\!08 ) / 18\!\cdots\!84 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 17\!\cdots\!91 \nu^{17} + \cdots - 24\!\cdots\!24 ) / 13\!\cdots\!88 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 48\!\cdots\!17 \nu^{17} + \cdots - 63\!\cdots\!68 ) / 23\!\cdots\!84 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 32\!\cdots\!49 \nu^{17} + \cdots + 60\!\cdots\!32 ) / 13\!\cdots\!88 \)
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| \(\beta_{15}\) | \(=\) |
\( ( 28\!\cdots\!55 \nu^{17} + \cdots - 40\!\cdots\!64 ) / 82\!\cdots\!68 \)
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| \(\beta_{16}\) | \(=\) |
\( ( - 62\!\cdots\!99 \nu^{17} + \cdots + 64\!\cdots\!68 ) / 13\!\cdots\!88 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 64\!\cdots\!39 \nu^{17} + \cdots - 30\!\cdots\!64 ) / 13\!\cdots\!88 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 149\beta _1 + 20444163 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 3 \beta_{17} + 6 \beta_{16} + 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + \beta_{12} + \cdots + 3025691450 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4154 \beta_{17} + 19354 \beta_{16} + 16962 \beta_{15} - 11317 \beta_{14} - 1448 \beta_{13} + \cdots + 722068133653044 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 210888976 \beta_{17} + 376877524 \beta_{16} + 159568139 \beta_{15} - 40934502 \beta_{14} + \cdots + 51\!\cdots\!41 \)
|
| \(\nu^{6}\) | \(=\) |
\( 293149691590 \beta_{17} + 1673827196858 \beta_{16} + 1363106900796 \beta_{15} - 702402017771 \beta_{14} + \cdots + 30\!\cdots\!98 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 11\!\cdots\!48 \beta_{17} + \cdots + 40\!\cdots\!95 \)
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| \(\nu^{8}\) | \(=\) |
\( 18\!\cdots\!74 \beta_{17} + \cdots + 13\!\cdots\!36 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 58\!\cdots\!12 \beta_{17} + \cdots + 26\!\cdots\!33 \)
|
| \(\nu^{10}\) | \(=\) |
\( 11\!\cdots\!54 \beta_{17} + \cdots + 62\!\cdots\!86 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 28\!\cdots\!44 \beta_{17} + \cdots + 15\!\cdots\!75 \)
|
| \(\nu^{12}\) | \(=\) |
\( 61\!\cdots\!82 \beta_{17} + \cdots + 29\!\cdots\!92 \)
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| \(\nu^{13}\) | \(=\) |
\( - 14\!\cdots\!16 \beta_{17} + \cdots + 88\!\cdots\!53 \)
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| \(\nu^{14}\) | \(=\) |
\( 32\!\cdots\!58 \beta_{17} + \cdots + 14\!\cdots\!62 \)
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| \(\nu^{15}\) | \(=\) |
\( - 68\!\cdots\!84 \beta_{17} + \cdots + 49\!\cdots\!03 \)
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| \(\nu^{16}\) | \(=\) |
\( 16\!\cdots\!02 \beta_{17} + \cdots + 72\!\cdots\!28 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 33\!\cdots\!24 \beta_{17} + \cdots + 27\!\cdots\!85 \)
|
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 | −6923.44 | 0 | 293533. | 0 | 2.48398e6 | 0 | 3.35851e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −5951.97 | 0 | −116965. | 0 | −735505. | 0 | 2.10771e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −5936.18 | 0 | 3744.79 | 0 | −3.58063e6 | 0 | 2.08893e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −5466.66 | 0 | −346820. | 0 | −459410. | 0 | 1.55354e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −4058.32 | 0 | 210207. | 0 | 1.55839e6 | 0 | 2.12103e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −3025.69 | 0 | −33445.6 | 0 | 3.54779e6 | 0 | −5.19413e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −2693.75 | 0 | −160375. | 0 | 2.45639e6 | 0 | −7.09262e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | −2061.05 | 0 | 8451.44 | 0 | −581535. | 0 | −1.01010e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 483.460 | 0 | 288329. | 0 | −3.41308e6 | 0 | −1.41152e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 991.271 | 0 | −246518. | 0 | −1.67649e6 | 0 | −1.33663e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 1683.36 | 0 | 89102.1 | 0 | 102360. | 0 | −1.15152e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 1982.68 | 0 | 20830.7 | 0 | −1.35216e6 | 0 | −1.04179e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0 | 3034.66 | 0 | 229199. | 0 | 3.97139e6 | 0 | −5.13977e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 0 | 3347.73 | 0 | −269245. | 0 | 2.57310e6 | 0 | −3.14163e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 0 | 4329.84 | 0 | −407.705 | 0 | −4.07362e6 | 0 | 4.39861e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 0 | 5841.12 | 0 | −35947.8 | 0 | 1.30298e6 | 0 | 1.97698e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 0 | 6893.88 | 0 | 291059. | 0 | −123643. | 0 | 3.31767e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 0 | 7321.05 | 0 | −286200. | 0 | −1.39745e6 | 0 | 3.92488e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 152.16.a.d | ✓ | 18 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.16.a.d | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{18} + 208 T_{3}^{17} - 183977636 T_{3}^{16} - 52102955048 T_{3}^{15} + \cdots + 18\!\cdots\!24 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(152))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} \)
|
| $3$ |
\( T^{18} + \cdots + 18\!\cdots\!24 \)
|
| $5$ |
\( T^{18} + \cdots - 42\!\cdots\!00 \)
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| $7$ |
\( T^{18} + \cdots + 17\!\cdots\!00 \)
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| $11$ |
\( T^{18} + \cdots - 48\!\cdots\!00 \)
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| $13$ |
\( T^{18} + \cdots - 38\!\cdots\!84 \)
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| $17$ |
\( T^{18} + \cdots + 27\!\cdots\!00 \)
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| $19$ |
\( (T - 893871739)^{18} \)
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| $23$ |
\( T^{18} + \cdots + 33\!\cdots\!28 \)
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| $29$ |
\( T^{18} + \cdots + 76\!\cdots\!52 \)
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| $31$ |
\( T^{18} + \cdots - 78\!\cdots\!08 \)
|
| $37$ |
\( T^{18} + \cdots - 15\!\cdots\!32 \)
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| $41$ |
\( T^{18} + \cdots - 13\!\cdots\!32 \)
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| $43$ |
\( T^{18} + \cdots - 29\!\cdots\!08 \)
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| $47$ |
\( T^{18} + \cdots + 13\!\cdots\!48 \)
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| $53$ |
\( T^{18} + \cdots - 20\!\cdots\!00 \)
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| $59$ |
\( T^{18} + \cdots - 18\!\cdots\!24 \)
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| $61$ |
\( T^{18} + \cdots - 11\!\cdots\!00 \)
|
| $67$ |
\( T^{18} + \cdots + 77\!\cdots\!72 \)
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| $71$ |
\( T^{18} + \cdots - 47\!\cdots\!00 \)
|
| $73$ |
\( T^{18} + \cdots - 51\!\cdots\!88 \)
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| $79$ |
\( T^{18} + \cdots + 23\!\cdots\!00 \)
|
| $83$ |
\( T^{18} + \cdots - 64\!\cdots\!00 \)
|
| $89$ |
\( T^{18} + \cdots - 25\!\cdots\!84 \)
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| $97$ |
\( T^{18} + \cdots - 24\!\cdots\!68 \)
|
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