Newspace parameters
| Level: | \( N \) | \(=\) | \( 6845 = 5 \cdot 37^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6845.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(54.6576001836\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
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| Defining polynomial: |
\( x^{14} - 2 x^{13} - 22 x^{12} + 42 x^{11} + 187 x^{10} - 336 x^{9} - 771 x^{8} + 1264 x^{7} + 1599 x^{6} + \cdots - 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 185) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 2 x^{13} - 22 x^{12} + 42 x^{11} + 187 x^{10} - 336 x^{9} - 771 x^{8} + 1264 x^{7} + 1599 x^{6} + \cdots - 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 502 \nu^{13} - 203 \nu^{12} - 12898 \nu^{11} + 5196 \nu^{10} + 122158 \nu^{9} - 49647 \nu^{8} + \cdots - 97947 ) / 51207 \)
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| \(\beta_{4}\) | \(=\) |
\( ( - 2001 \nu^{13} + 14988 \nu^{12} + 31827 \nu^{11} - 301568 \nu^{10} - 135280 \nu^{9} + \cdots - 466530 ) / 187759 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 267 \nu^{13} - 618 \nu^{12} - 5296 \nu^{11} + 9428 \nu^{10} + 39675 \nu^{9} - 45617 \nu^{8} + \cdots + 4518 ) / 17069 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 4540 \nu^{13} + 3604 \nu^{12} + 95294 \nu^{11} - 68957 \nu^{10} - 720077 \nu^{9} + 504218 \nu^{8} + \cdots - 163894 ) / 187759 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 7269 \nu^{13} + 12222 \nu^{12} + 168539 \nu^{11} - 237496 \nu^{10} - 1485961 \nu^{9} + \cdots - 431394 ) / 187759 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 1996 \nu^{13} - 3022 \nu^{12} + 47533 \nu^{11} + 76905 \nu^{10} - 445600 \nu^{9} - 725439 \nu^{8} + \cdots - 213597 ) / 43329 \)
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| \(\beta_{9}\) | \(=\) |
\( ( - 9554 \nu^{13} - 4161 \nu^{12} + 226840 \nu^{11} + 116343 \nu^{10} - 2077701 \nu^{9} + \cdots - 420003 ) / 187759 \)
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| \(\beta_{10}\) | \(=\) |
\( ( 9554 \nu^{13} + 4161 \nu^{12} - 226840 \nu^{11} - 116343 \nu^{10} + 2077701 \nu^{9} + 1212211 \nu^{8} + \cdots + 232244 ) / 187759 \)
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| \(\beta_{11}\) | \(=\) |
\( ( - 1055 \nu^{13} + 1601 \nu^{12} + 22367 \nu^{11} - 31342 \nu^{10} - 180358 \nu^{9} + 228228 \nu^{8} + \cdots + 55794 ) / 14443 \)
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| \(\beta_{12}\) | \(=\) |
\( ( - 70039 \nu^{13} + 19040 \nu^{12} + 1629994 \nu^{11} - 187170 \nu^{10} - 14697745 \nu^{9} + \cdots - 2509632 ) / 563277 \)
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| \(\beta_{13}\) | \(=\) |
\( ( 27809 \nu^{13} - 20256 \nu^{12} - 612905 \nu^{11} + 360060 \nu^{10} + 5142432 \nu^{9} + \cdots + 797370 ) / 187759 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{10} + \beta_{9} + 5\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{13} + \beta_{11} + \beta_{9} + \beta_{6} + 7\beta_{2} + 16 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{13} + \beta_{11} + 9 \beta_{10} + 9 \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 10 \)
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| \(\nu^{6}\) | \(=\) |
\( 11 \beta_{13} + \beta_{12} + 10 \beta_{11} + \beta_{10} + 11 \beta_{9} + 12 \beta_{6} + \beta_{5} + \cdots + 97 \)
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| \(\nu^{7}\) | \(=\) |
\( 13 \beta_{13} + 13 \beta_{11} + 68 \beta_{10} + 67 \beta_{9} + 13 \beta_{8} + 2 \beta_{7} + 14 \beta_{6} + \cdots + 88 \)
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| \(\nu^{8}\) | \(=\) |
\( 94 \beta_{13} + 15 \beta_{12} + 80 \beta_{11} + 17 \beta_{10} + 94 \beta_{9} + \beta_{8} - 2 \beta_{7} + \cdots + 623 \)
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| \(\nu^{9}\) | \(=\) |
\( 124 \beta_{13} - \beta_{12} + 123 \beta_{11} + 487 \beta_{10} + 475 \beta_{9} + 122 \beta_{8} + \cdots + 725 \)
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| \(\nu^{10}\) | \(=\) |
\( 734 \beta_{13} + 153 \beta_{12} + 595 \beta_{11} + 195 \beta_{10} + 741 \beta_{9} + 19 \beta_{8} + \cdots + 4130 \)
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| \(\nu^{11}\) | \(=\) |
\( 1053 \beta_{13} - 12 \beta_{12} + 1028 \beta_{11} + 3415 \beta_{10} + 3319 \beta_{9} + 1006 \beta_{8} + \cdots + 5723 \)
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| \(\nu^{12}\) | \(=\) |
\( 5498 \beta_{13} + 1333 \beta_{12} + 4296 \beta_{11} + 1888 \beta_{10} + 5633 \beta_{9} + 240 \beta_{8} + \cdots + 27911 \)
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| \(\nu^{13}\) | \(=\) |
\( 8456 \beta_{13} - 70 \beta_{12} + 8069 \beta_{11} + 23735 \beta_{10} + 23113 \beta_{9} + 7766 \beta_{8} + \cdots + 43925 \)
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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 |
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−2.54952 | −3.16727 | 4.50005 | −1.00000 | 8.07501 | −4.20071 | −6.37394 | 7.03157 | 2.54952 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.36864 | −2.14785 | 3.61047 | −1.00000 | 5.08750 | −1.64856 | −3.81462 | 1.61327 | 2.36864 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.21480 | 0.525972 | 2.90534 | −1.00000 | −1.16492 | 1.94620 | −2.00515 | −2.72335 | 2.21480 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.72834 | 1.66719 | 0.987175 | −1.00000 | −2.88148 | 0.0424322 | 1.75051 | −0.220468 | 1.72834 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.871378 | 1.99248 | −1.24070 | −1.00000 | −1.73620 | −3.34828 | 2.82387 | 0.969958 | 0.871378 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.688539 | −1.24487 | −1.52591 | −1.00000 | 0.857138 | 0.546116 | 2.42773 | −1.45031 | 0.688539 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.125387 | 0.313489 | −1.98428 | −1.00000 | −0.0393076 | 1.58176 | 0.499578 | −2.90172 | 0.125387 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.251643 | −3.23095 | −1.93668 | −1.00000 | −0.813045 | −0.255338 | −0.990637 | 7.43902 | −0.251643 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.41812 | −2.64977 | 0.0110645 | −1.00000 | −3.75769 | 2.94183 | −2.82055 | 4.02127 | −1.41812 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.52191 | 2.67106 | 0.316204 | −1.00000 | 4.06511 | 1.39372 | −2.56258 | 4.13458 | −1.52191 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.86451 | −0.0146673 | 1.47641 | −1.00000 | −0.0273473 | −4.57411 | −0.976246 | −2.99978 | −1.86451 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.15223 | −1.64645 | 2.63209 | −1.00000 | −3.54353 | 4.69073 | 1.36041 | −0.289214 | −2.15223 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.65216 | −0.127612 | 5.03395 | −1.00000 | −0.338448 | 3.45318 | 8.04652 | −2.98372 | −2.65216 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.68604 | 3.05923 | 5.21481 | −1.00000 | 8.21722 | −0.568970 | 8.63511 | 6.35890 | −2.68604 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(37\) | \( -1 \) |
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 | 6845.2.a.o | 14 | |
| 37.b | even | 2 | 1 | 6845.2.a.n | 14 | ||
| 37.g | odd | 12 | 2 | 185.2.m.a | ✓ | 28 | |
| 185.p | even | 12 | 2 | 925.2.m.c | 28 | ||
| 185.q | odd | 12 | 2 | 925.2.n.d | 28 | ||
| 185.u | even | 12 | 2 | 925.2.m.d | 28 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 185.2.m.a | ✓ | 28 | 37.g | odd | 12 | 2 | |
| 925.2.m.c | 28 | 185.p | even | 12 | 2 | ||
| 925.2.m.d | 28 | 185.u | even | 12 | 2 | ||
| 925.2.n.d | 28 | 185.q | odd | 12 | 2 | ||
| 6845.2.a.n | 14 | 37.b | even | 2 | 1 | ||
| 6845.2.a.o | 14 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6845))\):
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\( T_{2}^{14} - 2 T_{2}^{13} - 22 T_{2}^{12} + 42 T_{2}^{11} + 187 T_{2}^{10} - 336 T_{2}^{9} - 771 T_{2}^{8} + \cdots - 27 \)
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\( T_{7}^{14} - 2 T_{7}^{13} - 50 T_{7}^{12} + 112 T_{7}^{11} + 828 T_{7}^{10} - 2154 T_{7}^{9} - 4810 T_{7}^{8} + \cdots + 73 \)
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\( T_{17}^{14} - 6 T_{17}^{13} - 119 T_{17}^{12} + 704 T_{17}^{11} + 5265 T_{17}^{10} - 30528 T_{17}^{9} + \cdots - 7050123 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} - 2 T^{13} + \cdots - 27 \)
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| $3$ |
\( T^{14} + 4 T^{13} + \cdots + 1 \)
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| $5$ |
\( (T + 1)^{14} \)
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| $7$ |
\( T^{14} - 2 T^{13} + \cdots + 73 \)
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| $11$ |
\( T^{14} + 2 T^{13} + \cdots + 283293 \)
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| $13$ |
\( T^{14} - 2 T^{13} + \cdots + 2432733 \)
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| $17$ |
\( T^{14} - 6 T^{13} + \cdots - 7050123 \)
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| $19$ |
\( T^{14} - 20 T^{13} + \cdots - 207567 \)
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| $23$ |
\( T^{14} - 14 T^{13} + \cdots - 90787923 \)
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| $29$ |
\( T^{14} + 8 T^{13} + \cdots + 33587577 \)
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| $31$ |
\( T^{14} - 20 T^{13} + \cdots - 8709663 \)
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| $37$ |
\( T^{14} \)
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| $41$ |
\( T^{14} + \cdots - 3313087371 \)
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| $43$ |
\( T^{14} + \cdots + 7664902197 \)
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| $47$ |
\( T^{14} + 22 T^{13} + \cdots + 17909541 \)
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| $53$ |
\( T^{14} + \cdots - 9606711699 \)
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| $59$ |
\( T^{14} - 44 T^{13} + \cdots + 2439081 \)
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| $61$ |
\( T^{14} + \cdots + 241817517 \)
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| $67$ |
\( T^{14} + \cdots + 241193754217 \)
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| $71$ |
\( T^{14} + \cdots + 240406215057 \)
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| $73$ |
\( T^{14} + \cdots + 13235662453 \)
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| $79$ |
\( T^{14} - 465 T^{12} + \cdots - 7438251 \)
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| $83$ |
\( T^{14} + \cdots + 923860953 \)
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| $89$ |
\( T^{14} + \cdots + 4888385073 \)
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| $97$ |
\( T^{14} + \cdots - 92493285159 \)
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