Newspace parameters
| Level: | \( N \) | \(=\) | \( 145 = 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 145.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(23.2556538729\) |
| Analytic rank: | \(0\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{13} - 4 x^{12} - 342 x^{11} + 1270 x^{10} + 44810 x^{9} - 155844 x^{8} - 2796324 x^{7} + \cdots - 11370826136 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - 4 x^{12} - 342 x^{11} + 1270 x^{10} + 44810 x^{9} - 155844 x^{8} - 2796324 x^{7} + \cdots - 11370826136 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 54 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 11038736241983 \nu^{12} - 144062210847433 \nu^{11} + \cdots + 13\!\cdots\!68 ) / 38\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1581601242739 \nu^{12} - 71525802097605 \nu^{11} + 603782913300555 \nu^{10} + \cdots + 78\!\cdots\!08 ) / 38\!\cdots\!16 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 17289597636481 \nu^{12} + 154722207699319 \nu^{11} + \cdots + 16\!\cdots\!48 ) / 38\!\cdots\!60 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 8674735415599 \nu^{12} + 140464938810599 \nu^{11} + \cdots + 73\!\cdots\!88 ) / 19\!\cdots\!80 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 17289597636481 \nu^{12} + 154722207699319 \nu^{11} + \cdots + 18\!\cdots\!48 ) / 19\!\cdots\!80 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 49004056169457 \nu^{12} + 704345711676673 \nu^{11} + \cdots + 44\!\cdots\!32 ) / 95\!\cdots\!40 \)
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| \(\beta_{9}\) | \(=\) |
\( ( - 103295500304381 \nu^{12} + 670203609061349 \nu^{11} + \cdots - 11\!\cdots\!84 ) / 19\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 292545803137877 \nu^{12} - 321037265561581 \nu^{11} + \cdots + 13\!\cdots\!44 ) / 38\!\cdots\!60 \)
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| \(\beta_{11}\) | \(=\) |
\( ( 183796681175259 \nu^{12} - 504878992363139 \nu^{11} + \cdots + 67\!\cdots\!92 ) / 12\!\cdots\!20 \)
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| \(\beta_{12}\) | \(=\) |
\( ( 653081008433457 \nu^{12} + 799775182906071 \nu^{11} + \cdots + 37\!\cdots\!60 ) / 38\!\cdots\!60 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 54 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 2\beta_{5} + 2\beta_{2} + 81\beta _1 + 3 \)
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| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + \beta_{9} + 3 \beta_{8} + \beta_{7} + 4 \beta_{6} + \cdots + 4352 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4 \beta_{12} + 6 \beta_{11} + 6 \beta_{10} + 32 \beta_{9} - 6 \beta_{8} + 129 \beta_{7} - 24 \beta_{6} + \cdots + 2385 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 318 \beta_{12} + 513 \beta_{11} + 537 \beta_{10} + 229 \beta_{9} + 503 \beta_{8} + 259 \beta_{7} + \cdots + 394844 \)
|
| \(\nu^{7}\) | \(=\) |
\( 712 \beta_{12} + 1424 \beta_{11} + 1200 \beta_{10} + 6608 \beta_{9} - 1230 \beta_{8} + 14981 \beta_{7} + \cdots + 485443 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 38318 \beta_{12} + 66837 \beta_{11} + 74805 \beta_{10} + 39403 \beta_{9} + 64121 \beta_{8} + \cdots + 37947622 \)
|
| \(\nu^{9}\) | \(=\) |
\( 87344 \beta_{12} + 248564 \beta_{11} + 189796 \beta_{10} + 997802 \beta_{9} - 168164 \beta_{8} + \cdots + 75207585 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 4228966 \beta_{12} + 7989901 \beta_{11} + 9561749 \beta_{10} + 5911549 \beta_{9} + 7472243 \beta_{8} + \cdots + 3793860772 \)
|
| \(\nu^{11}\) | \(=\) |
\( 8966840 \beta_{12} + 37717304 \beta_{11} + 27874584 \beta_{10} + 133737492 \beta_{9} + \cdots + 10449371899 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 451825414 \beta_{12} + 925461857 \beta_{11} + 1172808049 \beta_{10} + 820265987 \beta_{9} + \cdots + 391044983570 \)
|
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 |
|
−9.88290 | 24.5061 | 65.6717 | 25.0000 | −242.192 | −94.3098 | −332.774 | 357.551 | −247.072 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.35764 | −12.3911 | 37.8501 | 25.0000 | 103.560 | 58.2801 | −48.8928 | −89.4601 | −208.941 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −8.00561 | −27.3159 | 32.0897 | 25.0000 | 218.680 | 244.685 | −0.718278 | 503.158 | −200.140 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −5.18100 | 13.1866 | −5.15721 | 25.0000 | −68.3200 | 137.284 | 192.512 | −69.1125 | −129.525 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −4.24767 | −17.1327 | −13.9573 | 25.0000 | 72.7739 | −82.2461 | 195.211 | 50.5283 | −106.192 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.59512 | 29.1351 | −29.4556 | 25.0000 | −46.4738 | 218.047 | 98.0288 | 605.852 | −39.8779 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.41189 | 4.67916 | −30.0066 | 25.0000 | −6.60648 | −217.629 | 87.5467 | −221.105 | −35.2974 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.98954 | −18.0770 | −16.0836 | 25.0000 | −72.1190 | −2.55988 | −191.831 | 83.7784 | 99.7386 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 4.48131 | 22.1935 | −11.9179 | 25.0000 | 99.4559 | −21.2578 | −196.810 | 249.552 | 112.033 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 8.83213 | 2.24444 | 46.0065 | 25.0000 | 19.8232 | 200.981 | 123.707 | −237.962 | 220.803 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 8.94882 | 25.1408 | 48.0815 | 25.0000 | 224.981 | 35.2643 | 143.910 | 389.060 | 223.721 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 10.3575 | −22.2201 | 75.2773 | 25.0000 | −230.144 | −103.079 | 448.244 | 250.731 | 258.937 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 11.0726 | 14.0510 | 90.6014 | 25.0000 | 155.580 | −57.4584 | 648.867 | −45.5696 | 276.814 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(29\) | \( +1 \) |
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 | 145.6.a.d | ✓ | 13 |
| 5.b | even | 2 | 1 | 725.6.a.e | 13 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 145.6.a.d | ✓ | 13 | 1.a | even | 1 | 1 | trivial |
| 725.6.a.e | 13 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{13} - 9 T_{2}^{12} - 312 T_{2}^{11} + 2470 T_{2}^{10} + 38535 T_{2}^{9} - 247473 T_{2}^{8} + \cdots + 5311067904 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(145))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} + \cdots + 5311067904 \)
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| $3$ |
\( T^{13} + \cdots + 18\!\cdots\!64 \)
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| $5$ |
\( (T - 25)^{13} \)
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| $7$ |
\( T^{13} + \cdots + 16\!\cdots\!84 \)
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| $11$ |
\( T^{13} + \cdots - 38\!\cdots\!76 \)
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| $13$ |
\( T^{13} + \cdots + 21\!\cdots\!52 \)
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| $17$ |
\( T^{13} + \cdots - 83\!\cdots\!88 \)
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| $19$ |
\( T^{13} + \cdots - 28\!\cdots\!80 \)
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| $23$ |
\( T^{13} + \cdots + 16\!\cdots\!72 \)
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| $29$ |
\( (T + 841)^{13} \)
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| $31$ |
\( T^{13} + \cdots - 88\!\cdots\!24 \)
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| $37$ |
\( T^{13} + \cdots - 73\!\cdots\!52 \)
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| $41$ |
\( T^{13} + \cdots - 15\!\cdots\!00 \)
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| $43$ |
\( T^{13} + \cdots - 30\!\cdots\!76 \)
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| $47$ |
\( T^{13} + \cdots - 32\!\cdots\!24 \)
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| $53$ |
\( T^{13} + \cdots - 15\!\cdots\!12 \)
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| $59$ |
\( T^{13} + \cdots - 11\!\cdots\!00 \)
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| $61$ |
\( T^{13} + \cdots - 12\!\cdots\!76 \)
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| $67$ |
\( T^{13} + \cdots + 22\!\cdots\!64 \)
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| $71$ |
\( T^{13} + \cdots - 27\!\cdots\!28 \)
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| $73$ |
\( T^{13} + \cdots + 25\!\cdots\!00 \)
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| $79$ |
\( T^{13} + \cdots + 81\!\cdots\!80 \)
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| $83$ |
\( T^{13} + \cdots + 20\!\cdots\!72 \)
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| $89$ |
\( T^{13} + \cdots - 46\!\cdots\!40 \)
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| $97$ |
\( T^{13} + \cdots + 59\!\cdots\!44 \)
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