Newspace parameters
Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 825.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(132.316651346\) |
Analytic rank: | \(0\) |
Dimension: | \(13\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
Defining polynomial: |
\( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{9}\cdot 3^{2}\cdot 5^{7} \) |
Twist minimal: | no (minimal twist has level 165) |
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
\( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 47 \)
|
\(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 78\nu + 47 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 700402408295 \nu^{12} + 4192326267667 \nu^{11} + 204218759138159 \nu^{10} + \cdots - 10\!\cdots\!00 ) / 89\!\cdots\!40 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 766585621025 \nu^{12} - 1865244494613 \nu^{11} - 224871623640281 \nu^{10} + 405732383014815 \nu^{9} + \cdots + 42\!\cdots\!40 ) / 89\!\cdots\!40 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 891142389387 \nu^{12} - 3560118843527 \nu^{11} - 258925322555763 \nu^{10} + 840913100582581 \nu^{9} + \cdots + 10\!\cdots\!40 ) / 89\!\cdots\!40 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 960809696913 \nu^{12} - 3695209656645 \nu^{11} - 277342519722953 \nu^{10} + 870845035605551 \nu^{9} + \cdots + 10\!\cdots\!40 ) / 89\!\cdots\!40 \)
|
\(\beta_{8}\) | \(=\) |
\( ( - 38192377867 \nu^{12} + 223973388087 \nu^{11} + 11044992841811 \nu^{10} - 56730273953229 \nu^{9} + \cdots - 58\!\cdots\!00 ) / 27\!\cdots\!20 \)
|
\(\beta_{9}\) | \(=\) |
\( ( 741184491385 \nu^{12} - 2928178310573 \nu^{11} - 213498624550193 \nu^{10} + 689953296380007 \nu^{9} + \cdots + 63\!\cdots\!20 ) / 44\!\cdots\!20 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 3500985580757 \nu^{12} - 14835282370393 \nu^{11} + \cdots + 44\!\cdots\!20 ) / 89\!\cdots\!40 \)
|
\(\beta_{11}\) | \(=\) |
\( ( - 438710012089 \nu^{12} + 1585086479789 \nu^{11} + 127508098559345 \nu^{10} - 369800452473191 \nu^{9} + \cdots - 62\!\cdots\!00 ) / 11\!\cdots\!80 \)
|
\(\beta_{12}\) | \(=\) |
\( ( - 6007694208307 \nu^{12} + 26162256334671 \nu^{11} + \cdots - 79\!\cdots\!00 ) / 89\!\cdots\!40 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 47 \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 80\beta _1 + 47 \)
|
\(\nu^{4}\) | \(=\) |
\( -\beta_{11} - \beta_{7} - 2\beta_{6} - \beta_{5} + 4\beta_{3} + 116\beta_{2} + 188\beta _1 + 3758 \)
|
\(\nu^{5}\) | \(=\) |
\( 3 \beta_{12} + 11 \beta_{10} - 4 \beta_{9} - \beta_{8} + 3 \beta_{7} - 12 \beta_{6} - 4 \beta_{5} + 7 \beta_{4} + 144 \beta_{3} + 401 \beta_{2} + 7639 \beta _1 + 8798 \)
|
\(\nu^{6}\) | \(=\) |
\( 2 \beta_{12} - 135 \beta_{11} + 64 \beta_{10} + 24 \beta_{9} + 46 \beta_{8} - 213 \beta_{7} - 368 \beta_{6} - 173 \beta_{5} + 878 \beta_{3} + 12992 \beta_{2} + 28982 \beta _1 + 359206 \)
|
\(\nu^{7}\) | \(=\) |
\( 521 \beta_{12} - 86 \beta_{11} + 2307 \beta_{10} - 684 \beta_{9} - 79 \beta_{8} - 49 \beta_{7} - 2774 \beta_{6} - 976 \beta_{5} + 1519 \beta_{4} + 18556 \beta_{3} + 64781 \beta_{2} + 799607 \beta _1 + 1355512 \)
|
\(\nu^{8}\) | \(=\) |
\( 720 \beta_{12} - 15391 \beta_{11} + 15724 \beta_{10} + 3328 \beta_{9} + 9976 \beta_{8} - 30583 \beta_{7} - 53826 \beta_{6} - 24803 \beta_{5} + 1596 \beta_{4} + 143254 \beta_{3} + 1498742 \beta_{2} + \cdots + 37618998 \)
|
\(\nu^{9}\) | \(=\) |
\( 68397 \beta_{12} - 25726 \beta_{11} + 354129 \beta_{10} - 88412 \beta_{9} + 4617 \beta_{8} - 77729 \beta_{7} - 467588 \beta_{6} - 167014 \beta_{5} + 232837 \beta_{4} + 2368506 \beta_{3} + \cdots + 196361842 \)
|
\(\nu^{10}\) | \(=\) |
\( 158440 \beta_{12} - 1705535 \beta_{11} + 2743638 \beta_{10} + 284656 \beta_{9} + 1549900 \beta_{8} - 3866735 \beta_{7} - 7335528 \beta_{6} - 3344661 \beta_{5} + 501198 \beta_{4} + \cdots + 4192972336 \)
|
\(\nu^{11}\) | \(=\) |
\( 8245237 \beta_{12} - 5090222 \beta_{11} + 48984123 \beta_{10} - 10562556 \beta_{9} + 2811285 \beta_{8} - 17696197 \beta_{7} - 69879050 \beta_{6} - 25223916 \beta_{5} + \cdots + 27556846220 \)
|
\(\nu^{12}\) | \(=\) |
\( 28040720 \beta_{12} - 190804435 \beta_{11} + 419463104 \beta_{10} + 13305632 \beta_{9} + 213106496 \beta_{8} - 471239355 \beta_{7} - 972828342 \beta_{6} + \cdots + 489536497526 \)
|
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 |
|
−10.4306 | 9.00000 | 76.7973 | 0 | −93.8753 | 21.9028 | −467.262 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.2 | −7.81495 | 9.00000 | 29.0734 | 0 | −70.3345 | 11.0326 | 22.8710 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.3 | −7.39855 | 9.00000 | 22.7385 | 0 | −66.5869 | 150.852 | 68.5217 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.4 | −3.62206 | 9.00000 | −18.8807 | 0 | −32.5985 | −168.040 | 184.293 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.5 | −3.33747 | 9.00000 | −20.8613 | 0 | −30.0373 | 103.473 | 176.423 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.6 | −0.886559 | 9.00000 | −31.2140 | 0 | −7.97903 | 224.774 | 56.0430 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.7 | 1.31741 | 9.00000 | −30.2644 | 0 | 11.8567 | −87.4786 | −82.0279 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.8 | 4.97394 | 9.00000 | −7.25990 | 0 | 44.7655 | −155.412 | −195.276 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.9 | 5.09558 | 9.00000 | −6.03505 | 0 | 45.8602 | 170.277 | −193.811 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.10 | 5.72893 | 9.00000 | 0.820586 | 0 | 51.5603 | −158.171 | −178.625 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.11 | 8.57523 | 9.00000 | 41.5346 | 0 | 77.1771 | 178.462 | 81.7612 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.12 | 10.0460 | 9.00000 | 68.9227 | 0 | 90.4143 | −29.0524 | 370.927 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.13 | 10.7531 | 9.00000 | 83.6283 | 0 | 96.7775 | 41.3813 | 555.162 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(5\) | \(-1\) |
\(11\) | \(-1\) |
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 | 825.6.a.y | 13 | |
5.b | even | 2 | 1 | 825.6.a.v | 13 | ||
5.c | odd | 4 | 2 | 165.6.c.b | ✓ | 26 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
165.6.c.b | ✓ | 26 | 5.c | odd | 4 | 2 | |
825.6.a.v | 13 | 5.b | even | 2 | 1 | ||
825.6.a.y | 13 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{13} - 13 T_{2}^{12} - 228 T_{2}^{11} + 3286 T_{2}^{10} + 16399 T_{2}^{9} - 292621 T_{2}^{8} - 348208 T_{2}^{7} + 11270528 T_{2}^{6} - 3851488 T_{2}^{5} - 183538464 T_{2}^{4} + \cdots - 1145312512 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{13} - 13 T^{12} + \cdots - 1145312512 \)
$3$
\( (T - 9)^{13} \)
$5$
\( T^{13} \)
$7$
\( T^{13} - 304 T^{12} + \cdots + 11\!\cdots\!48 \)
$11$
\( (T - 121)^{13} \)
$13$
\( T^{13} - 986 T^{12} + \cdots - 51\!\cdots\!12 \)
$17$
\( T^{13} - 1476 T^{12} + \cdots - 48\!\cdots\!96 \)
$19$
\( T^{13} - 270 T^{12} + \cdots + 58\!\cdots\!00 \)
$23$
\( T^{13} - 9084 T^{12} + \cdots + 44\!\cdots\!72 \)
$29$
\( T^{13} - 11952 T^{12} + \cdots + 36\!\cdots\!00 \)
$31$
\( T^{13} - 19096 T^{12} + \cdots + 22\!\cdots\!56 \)
$37$
\( T^{13} - 39964 T^{12} + \cdots - 29\!\cdots\!12 \)
$41$
\( T^{13} - 35184 T^{12} + \cdots + 32\!\cdots\!28 \)
$43$
\( T^{13} + 96 T^{12} + \cdots - 16\!\cdots\!00 \)
$47$
\( T^{13} - 34984 T^{12} + \cdots + 12\!\cdots\!12 \)
$53$
\( T^{13} - 22984 T^{12} + \cdots + 45\!\cdots\!48 \)
$59$
\( T^{13} + 9192 T^{12} + \cdots - 13\!\cdots\!00 \)
$61$
\( T^{13} - 5438 T^{12} + \cdots + 21\!\cdots\!08 \)
$67$
\( T^{13} - 71508 T^{12} + \cdots + 11\!\cdots\!48 \)
$71$
\( T^{13} - 101700 T^{12} + \cdots + 27\!\cdots\!00 \)
$73$
\( T^{13} - 77390 T^{12} + \cdots + 28\!\cdots\!92 \)
$79$
\( T^{13} - 93954 T^{12} + \cdots - 24\!\cdots\!00 \)
$83$
\( T^{13} - 185918 T^{12} + \cdots + 40\!\cdots\!96 \)
$89$
\( T^{13} + 18418 T^{12} + \cdots - 97\!\cdots\!00 \)
$97$
\( T^{13} - 94312 T^{12} + \cdots - 87\!\cdots\!52 \)
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