Newspace parameters
Level: | \( N \) | \(=\) | \( 1875 = 3 \cdot 5^{4} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 1875.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(110.628581261\) |
Analytic rank: | \(0\) |
Dimension: | \(14\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
Defining polynomial: |
\( x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \cdots + 42064 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{2}\cdot 5^{3} \) |
Twist minimal: | no (minimal twist has level 75) |
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
\( x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \cdots + 42064 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 154777191 \nu^{13} + 214678293 \nu^{12} + 12768694032 \nu^{11} - 14915595999 \nu^{10} - 404069408515 \nu^{9} + \cdots - 29392108806288 ) / 9573422384000 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 39842017881 \nu^{13} + 18793858963 \nu^{12} + 3381917381312 \nu^{11} + 617153824991 \nu^{10} - 107039813741365 \nu^{9} + \cdots - 61\!\cdots\!08 ) / 20\!\cdots\!00 \)
|
\(\beta_{5}\) | \(=\) |
\( ( - 214678293 \nu^{13} - 231741561 \nu^{12} + 15999036336 \nu^{11} + 15269104723 \nu^{10} - 450292207345 \nu^{9} + \cdots - 6510547762224 ) / 9573422384000 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 464331573 \nu^{13} - 644034879 \nu^{12} - 38306082096 \nu^{11} + 44746787997 \nu^{10} + 1212208225545 \nu^{9} + \cdots + 59456059266864 ) / 9573422384000 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 52774069869 \nu^{13} - 146787187287 \nu^{12} - 4038749648088 \nu^{11} + 10590731904141 \nu^{10} + 114153083031385 \nu^{9} + \cdots - 24\!\cdots\!08 ) / 679712989264000 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 198268707541 \nu^{13} - 330612632143 \nu^{12} - 15446080755832 \nu^{11} + 24130842178949 \nu^{10} + 453486628821265 \nu^{9} + \cdots - 30\!\cdots\!12 ) / 20\!\cdots\!00 \)
|
\(\beta_{9}\) | \(=\) |
\( ( - 24723383429 \nu^{13} - 19698303673 \nu^{12} + 1994905817228 \nu^{11} + 1654834981079 \nu^{10} - 61648333032485 \nu^{9} + \cdots - 91\!\cdots\!52 ) / 203913896779200 \)
|
\(\beta_{10}\) | \(=\) |
\( ( - 8146109293 \nu^{13} + 6363639039 \nu^{12} + 663666222536 \nu^{11} - 440450615677 \nu^{10} - 20765197179345 \nu^{9} + \cdots - 212680826375024 ) / 53661551784000 \)
|
\(\beta_{11}\) | \(=\) |
\( ( - 2089067853 \nu^{13} + 1186234119 \nu^{12} + 166990685056 \nu^{11} - 90007487317 \nu^{10} - 5108559447745 \nu^{9} + \cdots - 228794009350704 ) / 9573422384000 \)
|
\(\beta_{12}\) | \(=\) |
\( ( 268898217 \nu^{13} - 60162011 \nu^{12} - 21662421664 \nu^{11} + 3229526593 \nu^{10} + 662126520605 \nu^{9} - 31650082226 \nu^{8} + \cdots + 16216865165616 ) / 957342238400 \)
|
\(\beta_{13}\) | \(=\) |
\( ( - 629711814493 \nu^{13} + 211929067839 \nu^{12} + 51141259110536 \nu^{11} - 12148429797277 \nu^{10} + \cdots - 10\!\cdots\!24 ) / 20\!\cdots\!00 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{6} + 3\beta_{3} + 19\beta _1 + 3 \)
|
\(\nu^{4}\) | \(=\) |
\( - \beta_{13} - \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{8} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + 23 \beta_{2} - 3 \beta _1 + 229 \)
|
\(\nu^{5}\) | \(=\) |
\( 3 \beta_{13} + \beta_{12} + 3 \beta_{11} - 5 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 4 \beta_{7} + 34 \beta_{6} - 15 \beta_{5} - 4 \beta_{4} + 78 \beta_{3} - 2 \beta_{2} + 400 \beta _1 + 60 \)
|
\(\nu^{6}\) | \(=\) |
\( - 41 \beta_{13} - 38 \beta_{11} + 28 \beta_{10} + 76 \beta_{9} - 34 \beta_{8} + \beta_{7} - 74 \beta_{6} - 9 \beta_{5} - 72 \beta_{4} + 141 \beta_{3} + 515 \beta_{2} - 164 \beta _1 + 4850 \)
|
\(\nu^{7}\) | \(=\) |
\( 122 \beta_{13} + 49 \beta_{12} + 146 \beta_{11} - 202 \beta_{10} + 106 \beta_{9} + 128 \beta_{8} - 180 \beta_{7} + 931 \beta_{6} - 724 \beta_{5} - 174 \beta_{4} + 1565 \beta_{3} - 104 \beta_{2} + 8783 \beta _1 + 636 \)
|
\(\nu^{8}\) | \(=\) |
\( - 1257 \beta_{13} - 9 \beta_{12} - 1152 \beta_{11} + 672 \beta_{10} + 2204 \beta_{9} - 872 \beta_{8} + 23 \beta_{7} - 2125 \beta_{6} + 421 \beta_{5} - 2046 \beta_{4} + 6944 \beta_{3} + 11581 \beta_{2} + \cdots + 107016 \)
|
\(\nu^{9}\) | \(=\) |
\( 3623 \beta_{13} + 1534 \beta_{12} + 5048 \beta_{11} - 6120 \beta_{10} + 3642 \beta_{9} + 4046 \beta_{8} - 5813 \beta_{7} + 23756 \beta_{6} - 24641 \beta_{5} - 5368 \beta_{4} + 27815 \beta_{3} - 3748 \beta_{2} + \cdots - 5982 \)
|
\(\nu^{10}\) | \(=\) |
\( - 34186 \beta_{13} - 247 \beta_{12} - 32506 \beta_{11} + 15550 \beta_{10} + 58426 \beta_{9} - 20316 \beta_{8} - 212 \beta_{7} - 56078 \beta_{6} + 26260 \beta_{5} - 53402 \beta_{4} + 239954 \beta_{3} + \cdots + 2413393 \)
|
\(\nu^{11}\) | \(=\) |
\( 95312 \beta_{13} + 39355 \beta_{12} + 152581 \beta_{11} - 166261 \beta_{10} + 104498 \beta_{9} + 114853 \beta_{8} - 163757 \beta_{7} + 587178 \beta_{6} - 732762 \beta_{5} - 144636 \beta_{4} + \cdots - 620070 \)
|
\(\nu^{12}\) | \(=\) |
\( - 873549 \beta_{13} + 1297 \beta_{12} - 886792 \beta_{11} + 355354 \beta_{10} + 1487098 \beta_{9} - 455066 \beta_{8} - 37205 \beta_{7} - 1425812 \beta_{6} + 998693 \beta_{5} + \cdots + 55193599 \)
|
\(\nu^{13}\) | \(=\) |
\( 2359274 \beta_{13} + 891798 \beta_{12} + 4304259 \beta_{11} - 4273639 \beta_{10} + 2722752 \beta_{9} + 3092949 \beta_{8} - 4288261 \beta_{7} + 14271698 \beta_{6} - 20388722 \beta_{5} + \cdots - 25133913 \)
|
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 |
|
−4.81720 | −3.00000 | 15.2054 | 0 | 14.4516 | −1.13492 | −34.7100 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.2 | −4.65416 | −3.00000 | 13.6612 | 0 | 13.9625 | 26.0445 | −26.3483 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.3 | −4.24928 | −3.00000 | 10.0564 | 0 | 12.7478 | 28.2853 | −8.73812 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.4 | −3.90684 | −3.00000 | 7.26339 | 0 | 11.7205 | −22.0918 | 2.87782 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.5 | −1.73740 | −3.00000 | −4.98145 | 0 | 5.21219 | −7.66213 | 22.5539 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.6 | −1.33976 | −3.00000 | −6.20504 | 0 | 4.01928 | 12.2101 | 19.0313 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.7 | 0.134967 | −3.00000 | −7.98178 | 0 | −0.404902 | 17.3099 | −2.15702 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.8 | 0.574607 | −3.00000 | −7.66983 | 0 | −1.72382 | 2.67744 | −9.00399 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.9 | 0.955230 | −3.00000 | −7.08754 | 0 | −2.86569 | −12.4836 | −14.4121 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.10 | 2.33957 | −3.00000 | −2.52642 | 0 | −7.01870 | 32.9322 | −24.6273 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.11 | 2.93781 | −3.00000 | 0.630743 | 0 | −8.81344 | −18.9115 | −21.6495 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.12 | 4.01755 | −3.00000 | 8.14069 | 0 | −12.0526 | −1.75849 | 0.565245 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.13 | 4.79301 | −3.00000 | 14.9730 | 0 | −14.3790 | −0.140520 | 33.4216 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1.14 | 4.95189 | −3.00000 | 16.5212 | 0 | −14.8557 | −28.2766 | 42.1962 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(5\) | \(1\) |
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 | 1875.4.a.f | 14 | |
5.b | even | 2 | 1 | 1875.4.a.g | 14 | ||
25.e | even | 10 | 2 | 75.4.g.b | ✓ | 28 | |
75.h | odd | 10 | 2 | 225.4.h.a | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.4.g.b | ✓ | 28 | 25.e | even | 10 | 2 | |
225.4.h.a | 28 | 75.h | odd | 10 | 2 | ||
1875.4.a.f | 14 | 1.a | even | 1 | 1 | trivial | |
1875.4.a.g | 14 | 5.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} - 81 T_{2}^{12} + 7 T_{2}^{11} + 2512 T_{2}^{10} - 517 T_{2}^{9} - 36970 T_{2}^{8} + 12987 T_{2}^{7} + 257291 T_{2}^{6} - 125779 T_{2}^{5} - 718713 T_{2}^{4} + 371750 T_{2}^{3} + 579848 T_{2}^{2} + \cdots + 42064 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1875))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{14} - 81 T^{12} + 7 T^{11} + \cdots + 42064 \)
$3$
\( (T + 3)^{14} \)
$5$
\( T^{14} \)
$7$
\( T^{14} - 27 T^{13} + \cdots + 4350562367600 \)
$11$
\( T^{14} + 33 T^{13} + \cdots + 57\!\cdots\!96 \)
$13$
\( T^{14} - 188 T^{13} + \cdots - 21\!\cdots\!44 \)
$17$
\( T^{14} - 146 T^{13} + \cdots + 29\!\cdots\!76 \)
$19$
\( T^{14} + 184 T^{13} + \cdots + 37\!\cdots\!00 \)
$23$
\( T^{14} - 164 T^{13} + \cdots + 33\!\cdots\!00 \)
$29$
\( T^{14} - 252 T^{13} + \cdots + 69\!\cdots\!00 \)
$31$
\( T^{14} + 889 T^{13} + \cdots - 98\!\cdots\!00 \)
$37$
\( T^{14} - 642 T^{13} + \cdots - 50\!\cdots\!00 \)
$41$
\( T^{14} + 164 T^{13} + \cdots + 73\!\cdots\!00 \)
$43$
\( T^{14} - 696 T^{13} + \cdots - 55\!\cdots\!16 \)
$47$
\( T^{14} + 92 T^{13} + \cdots - 35\!\cdots\!84 \)
$53$
\( T^{14} - 949 T^{13} + \cdots - 14\!\cdots\!75 \)
$59$
\( T^{14} + 81 T^{13} + \cdots - 44\!\cdots\!00 \)
$61$
\( T^{14} + 496 T^{13} + \cdots - 10\!\cdots\!96 \)
$67$
\( T^{14} - 1926 T^{13} + \cdots - 22\!\cdots\!44 \)
$71$
\( T^{14} - 2498 T^{13} + \cdots + 16\!\cdots\!04 \)
$73$
\( T^{14} + 1026 T^{13} + \cdots + 33\!\cdots\!00 \)
$79$
\( T^{14} + 695 T^{13} + \cdots + 54\!\cdots\!00 \)
$83$
\( T^{14} - 5315 T^{13} + \cdots + 23\!\cdots\!96 \)
$89$
\( T^{14} + 1424 T^{13} + \cdots - 11\!\cdots\!00 \)
$97$
\( T^{14} - 291 T^{13} + \cdots + 17\!\cdots\!71 \)
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