Newspace parameters
| Level: | \( N \) | \(=\) | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1911.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(112.752650021\) |
| Analytic rank: | \(0\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{13} - x^{12} - 79 x^{11} + 56 x^{10} + 2342 x^{9} - 1043 x^{8} - 32595 x^{7} + 7442 x^{6} + \cdots - 201600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 273) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - x^{12} - 79 x^{11} + 56 x^{10} + 2342 x^{9} - 1043 x^{8} - 32595 x^{7} + 7442 x^{6} + \cdots - 201600 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 20\nu + 8 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 16069 \nu^{12} - 2098 \nu^{11} - 1287949 \nu^{10} - 160683 \nu^{9} + 38626109 \nu^{8} + \cdots + 5593144800 ) / 1684320 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 1701 \nu^{12} - 276 \nu^{11} - 135765 \nu^{10} - 14665 \nu^{9} + 4053495 \nu^{8} + 1373538 \nu^{7} + \cdots + 586130400 ) / 153120 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 50245 \nu^{12} + 19999 \nu^{11} + 3979579 \nu^{10} - 403482 \nu^{9} - 117774824 \nu^{8} + \cdots - 15904511520 ) / 3368640 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 27702 \nu^{12} - 11633 \nu^{11} - 2190556 \nu^{10} + 263363 \nu^{9} + 64717401 \nu^{8} + \cdots + 8814012480 ) / 1684320 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 73705 \nu^{12} + 31209 \nu^{11} + 5831219 \nu^{10} - 728512 \nu^{9} - 172359554 \nu^{8} + \cdots - 23497178400 ) / 3368640 \)
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| \(\beta_{9}\) | \(=\) |
\( ( 83351 \nu^{12} - 32951 \nu^{11} - 6600885 \nu^{10} + 662080 \nu^{9} + 195330070 \nu^{8} + \cdots + 26783610720 ) / 3368640 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 202511 \nu^{12} + 70957 \nu^{11} + 16058761 \nu^{10} - 959038 \nu^{9} - 475889296 \nu^{8} + \cdots - 65769858720 ) / 3368640 \)
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| \(\beta_{11}\) | \(=\) |
\( ( - 124941 \nu^{12} + 46749 \nu^{11} + 9900023 \nu^{10} - 797344 \nu^{9} - 293150658 \nu^{8} + \cdots - 40219934880 ) / 1684320 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 66899 \nu^{12} - 24366 \nu^{11} - 5301537 \nu^{10} + 385431 \nu^{9} + 156984577 \nu^{8} + \cdots + 21466022640 ) / 842160 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 20\beta _1 + 4 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + 29\beta_{2} + 5\beta _1 + 240 \)
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| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{11} + 5 \beta_{9} + 3 \beta_{8} + 3 \beta_{7} - 2 \beta_{6} + \beta_{5} + 37 \beta_{3} + \cdots + 180 \)
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| \(\nu^{6}\) | \(=\) |
\( 2 \beta_{12} + 52 \beta_{11} + 2 \beta_{10} + 44 \beta_{9} - 38 \beta_{8} + 70 \beta_{7} - 12 \beta_{6} + \cdots + 5700 \)
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| \(\nu^{7}\) | \(=\) |
\( - 4 \beta_{12} + 133 \beta_{11} - 16 \beta_{10} + 257 \beta_{9} + 163 \beta_{8} + 241 \beta_{7} + \cdots + 6532 \)
|
| \(\nu^{8}\) | \(=\) |
\( 108 \beta_{12} + 1983 \beta_{11} + 152 \beta_{10} + 1562 \beta_{9} - 1134 \beta_{8} + 3196 \beta_{7} + \cdots + 147372 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 162 \beta_{12} + 5933 \beta_{11} - 858 \beta_{10} + 9884 \beta_{9} + 6282 \beta_{8} + 11664 \beta_{7} + \cdots + 226360 \)
|
| \(\nu^{10}\) | \(=\) |
\( 4322 \beta_{12} + 68111 \beta_{11} + 6842 \beta_{10} + 52116 \beta_{9} - 30870 \beta_{8} + \cdots + 4000344 \)
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| \(\nu^{11}\) | \(=\) |
\( - 3526 \beta_{12} + 227556 \beta_{11} - 32014 \beta_{10} + 342523 \beta_{9} + 213103 \beta_{8} + \cdots + 7697692 \)
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| \(\nu^{12}\) | \(=\) |
\( 155162 \beta_{12} + 2232727 \beta_{11} + 250510 \beta_{10} + 1697749 \beta_{9} - 796499 \beta_{8} + \cdots + 112068948 \)
|
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 |
|
−5.58417 | 3.00000 | 23.1829 | 18.9283 | −16.7525 | 0 | −84.7842 | 9.00000 | −105.699 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.70294 | 3.00000 | 14.1177 | −5.85882 | −14.1088 | 0 | −28.7709 | 9.00000 | 27.5537 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.99039 | 3.00000 | 7.92322 | 5.64940 | −11.9712 | 0 | 0.306392 | 9.00000 | −22.5433 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.09669 | 3.00000 | 1.58951 | 13.6249 | −9.29008 | 0 | 19.8513 | 9.00000 | −42.1921 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.12882 | 3.00000 | −3.46812 | −9.36199 | −6.38646 | 0 | 24.4136 | 9.00000 | 19.9300 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.627484 | 3.00000 | −7.60626 | −18.3276 | −1.88245 | 0 | 9.79268 | 9.00000 | 11.5003 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.625075 | 3.00000 | −7.60928 | 6.73628 | −1.87523 | 0 | 9.75697 | 9.00000 | −4.21068 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.31861 | 3.00000 | −6.26127 | 5.43650 | 3.95583 | 0 | −18.8050 | 9.00000 | 7.16862 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.43661 | 3.00000 | −2.06295 | −5.79278 | 7.30982 | 0 | −24.5194 | 9.00000 | −14.1147 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 3.04612 | 3.00000 | 1.27888 | −11.6156 | 9.13837 | 0 | −20.4734 | 9.00000 | −35.3827 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 3.22563 | 3.00000 | 2.40468 | 21.4709 | 9.67689 | 0 | −18.0484 | 9.00000 | 69.2570 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 4.55760 | 3.00000 | 12.7717 | −0.133912 | 13.6728 | 0 | 21.7476 | 9.00000 | −0.610318 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 5.17100 | 3.00000 | 18.7393 | 18.2446 | 15.5130 | 0 | 55.5328 | 9.00000 | 94.3429 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 1911.4.a.ba | 13 | |
| 7.b | odd | 2 | 1 | 1911.4.a.z | 13 | ||
| 7.d | odd | 6 | 2 | 273.4.i.f | ✓ | 26 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.4.i.f | ✓ | 26 | 7.d | odd | 6 | 2 | |
| 1911.4.a.z | 13 | 7.b | odd | 2 | 1 | ||
| 1911.4.a.ba | 13 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1911))\):
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\( T_{2}^{13} + T_{2}^{12} - 79 T_{2}^{11} - 56 T_{2}^{10} + 2342 T_{2}^{9} + 1043 T_{2}^{8} + \cdots + 201600 \)
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\( T_{5}^{13} - 39 T_{5}^{12} - 275 T_{5}^{11} + 24835 T_{5}^{10} - 91896 T_{5}^{9} - 5141446 T_{5}^{8} + \cdots - 189323197920 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} + T^{12} + \cdots + 201600 \)
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| $3$ |
\( (T - 3)^{13} \)
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| $5$ |
\( T^{13} + \cdots - 189323197920 \)
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| $7$ |
\( T^{13} \)
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| $11$ |
\( T^{13} + \cdots + 21\!\cdots\!85 \)
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| $13$ |
\( (T + 13)^{13} \)
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| $17$ |
\( T^{13} + \cdots + 73\!\cdots\!98 \)
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| $19$ |
\( T^{13} + \cdots - 96\!\cdots\!16 \)
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| $23$ |
\( T^{13} + \cdots - 26\!\cdots\!92 \)
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| $29$ |
\( T^{13} + \cdots + 59\!\cdots\!31 \)
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| $31$ |
\( T^{13} + \cdots + 87\!\cdots\!04 \)
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| $37$ |
\( T^{13} + \cdots + 40\!\cdots\!60 \)
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| $41$ |
\( T^{13} + \cdots + 12\!\cdots\!84 \)
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| $43$ |
\( T^{13} + \cdots - 11\!\cdots\!48 \)
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| $47$ |
\( T^{13} + \cdots + 77\!\cdots\!20 \)
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| $53$ |
\( T^{13} + \cdots - 26\!\cdots\!65 \)
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| $59$ |
\( T^{13} + \cdots + 35\!\cdots\!75 \)
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| $61$ |
\( T^{13} + \cdots + 30\!\cdots\!20 \)
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| $67$ |
\( T^{13} + \cdots - 18\!\cdots\!52 \)
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| $71$ |
\( T^{13} + \cdots - 12\!\cdots\!38 \)
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| $73$ |
\( T^{13} + \cdots - 19\!\cdots\!96 \)
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| $79$ |
\( T^{13} + \cdots - 66\!\cdots\!92 \)
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| $83$ |
\( T^{13} + \cdots + 15\!\cdots\!28 \)
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| $89$ |
\( T^{13} + \cdots + 39\!\cdots\!00 \)
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| $97$ |
\( T^{13} + \cdots + 13\!\cdots\!00 \)
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