Newspace parameters
Level: | \( N \) | \(=\) | \( 1875 = 3 \cdot 5^{4} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1875.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(14.9719503790\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
Defining polynomial: |
\( x^{16} + 25x^{14} + 246x^{12} + 1220x^{10} + 3281x^{8} + 4880x^{6} + 3936x^{4} + 1600x^{2} + 256 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 25x^{14} + 246x^{12} + 1220x^{10} + 3281x^{8} + 4880x^{6} + 3936x^{4} + 1600x^{2} + 256 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{15} + 25\nu^{13} + 246\nu^{11} + 1220\nu^{9} + 3281\nu^{7} + 4880\nu^{5} + 3936\nu^{3} + 1472\nu ) / 128 \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{14} + 25\nu^{12} + 246\nu^{10} + 1220\nu^{8} + 3281\nu^{6} + 4880\nu^{4} + 3872\nu^{2} + 1216 ) / 64 \)
|
\(\beta_{4}\) | \(=\) |
\( ( -3\nu^{15} - 75\nu^{13} - 738\nu^{11} - 3660\nu^{9} - 9843\nu^{7} - 14640\nu^{5} - 11680\nu^{3} - 3904\nu ) / 128 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 4\nu^{15} + 99\nu^{13} + 959\nu^{11} + 4634\nu^{9} + 11904\nu^{7} + 16239\nu^{5} + 10896\nu^{3} + 2752\nu ) / 32 \)
|
\(\beta_{6}\) | \(=\) |
\( ( -13\nu^{14} - 321\nu^{12} - 3098\nu^{10} - 14876\nu^{8} - 37773\nu^{6} - 50380\nu^{4} - 32416\nu^{2} - 7744 ) / 64 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 9\nu^{14} + 221\nu^{12} + 2114\nu^{10} + 9996\nu^{8} + 24681\nu^{6} + 31308\nu^{4} + 18592\nu^{2} + 4032 ) / 32 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 9\nu^{14} + 221\nu^{12} + 2114\nu^{10} + 9996\nu^{8} + 24681\nu^{6} + 31308\nu^{4} + 18624\nu^{2} + 4128 ) / 32 \)
|
\(\beta_{9}\) | \(=\) |
\( ( -5\nu^{14} - 123\nu^{12} - 1180\nu^{10} - 5608\nu^{8} - 13981\nu^{6} - 18078\nu^{4} - 11120\nu^{2} - 2528 ) / 16 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 21\nu^{15} + 519\nu^{13} + 5016\nu^{11} + 24144\nu^{9} + 61581\nu^{7} + 82858\nu^{5} + 54208\nu^{3} + 13248\nu ) / 64 \)
|
\(\beta_{11}\) | \(=\) |
\( ( 79 \nu^{15} + 1935 \nu^{13} + 18450 \nu^{11} + 86940 \nu^{9} + 214335 \nu^{7} + 273672 \nu^{5} + 166320 \nu^{3} + 37440 \nu ) / 128 \)
|
\(\beta_{12}\) | \(=\) |
\( ( 23\nu^{15} + 561\nu^{13} + 5316\nu^{11} + 24816\nu^{9} + 60319\nu^{7} + 75490\nu^{5} + 44712\nu^{3} + 9792\nu ) / 32 \)
|
\(\beta_{13}\) | \(=\) |
\( ( 23\nu^{14} + 561\nu^{12} + 5316\nu^{10} + 24816\nu^{8} + 60319\nu^{6} + 75490\nu^{4} + 44712\nu^{2} + 9824 ) / 32 \)
|
\(\beta_{14}\) | \(=\) |
\( ( 151 \nu^{15} + 3683 \nu^{13} + 34910 \nu^{11} + 163124 \nu^{9} + 397463 \nu^{7} + 500052 \nu^{5} + 298912 \nu^{3} + 66304 \nu ) / 128 \)
|
\(\beta_{15}\) | \(=\) |
\( ( -23\nu^{14} - 560\nu^{12} - 5295\nu^{10} - 24654\nu^{8} - 59763\nu^{6} - 74673\nu^{4} - 44268\nu^{2} - 9744 ) / 16 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{8} - \beta_{7} - 3 \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{4} + 3\beta_{2} - 4\beta_1 \)
|
\(\nu^{4}\) | \(=\) |
\( \beta_{9} - 7\beta_{8} + 8\beta_{7} + 2\beta_{3} + 15 \)
|
\(\nu^{5}\) | \(=\) |
\( \beta_{10} - 3\beta_{5} - 10\beta_{4} - 24\beta_{2} + 22\beta_1 \)
|
\(\nu^{6}\) | \(=\) |
\( -14\beta_{9} + 46\beta_{8} - 59\beta_{7} + 2\beta_{6} - 20\beta_{3} - 90 \)
|
\(\nu^{7}\) | \(=\) |
\( -\beta_{12} + 2\beta_{11} - 16\beta_{10} + 42\beta_{5} + 79\beta_{4} + 171\beta_{2} - 136\beta_1 \)
|
\(\nu^{8}\) | \(=\) |
\( \beta_{15} + 3\beta_{13} + 139\beta_{9} - 307\beta_{8} + 427\beta_{7} - 32\beta_{6} + 158\beta_{3} + 577 \)
|
\(\nu^{9}\) | \(=\) |
\( - 2 \beta_{14} + 23 \beta_{12} - 38 \beta_{11} + 172 \beta_{10} - 412 \beta_{5} - 585 \beta_{4} - 1199 \beta_{2} + 887 \beta_1 \)
|
\(\nu^{10}\) | \(=\) |
\( -25\beta_{15} - 71\beta_{13} - 1207\beta_{9} + 2086\beta_{8} - 3060\beta_{7} + 344\beta_{6} - 1170\beta_{3} - 3814 \)
|
\(\nu^{11}\) | \(=\) |
\( 50 \beta_{14} - 329 \beta_{12} + 466 \beta_{11} - 1576 \beta_{10} + 3524 \beta_{5} + 4230 \beta_{4} + 8402 \beta_{2} - 5971 \beta_1 \)
|
\(\nu^{12}\) | \(=\) |
\( 379 \beta_{15} + 1037 \beta_{13} + 9796 \beta_{9} - 14373 \beta_{8} + 21798 \beta_{7} - 3152 \beta_{6} + 8460 \beta_{3} + 25657 \)
|
\(\nu^{13}\) | \(=\) |
\( - 758 \beta_{14} + 3787 \beta_{12} - 4742 \beta_{11} + 13327 \beta_{10} - 28177 \beta_{5} - 30258 \beta_{4} - 59004 \beta_{2} + 41067 \beta_1 \)
|
\(\nu^{14}\) | \(=\) |
\( - 4545 \beta_{15} - 12119 \beta_{13} - 76504 \beta_{9} + 100071 \beta_{8} - 154719 \beta_{7} + 26654 \beta_{6} - 60516 \beta_{3} - 174631 \)
|
\(\nu^{15}\) | \(=\) |
\( 9090 \beta_{14} - 38520 \beta_{12} + 43712 \beta_{11} - 107703 \beta_{10} + 216999 \beta_{5} + 215235 \beta_{4} + 415377 \beta_{2} - 286821 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).
\(n\) | \(626\) | \(1252\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1249.1 |
|
− | 2.69767i | − | 1.00000i | −5.27745 | 0 | −2.69767 | − | 3.56649i | 8.84149i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.2 | − | 2.59716i | 1.00000i | −4.74525 | 0 | 2.59716 | − | 3.28414i | 7.12986i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.3 | − | 2.23365i | 1.00000i | −2.98921 | 0 | 2.23365 | 1.03143i | 2.20956i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.4 | − | 1.52260i | − | 1.00000i | −0.318310 | 0 | −1.52260 | − | 0.990985i | − | 2.56054i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.5 | − | 1.31354i | 1.00000i | 0.274605 | 0 | 1.31354 | − | 4.19091i | − | 2.98779i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.6 | − | 0.895394i | − | 1.00000i | 1.19827 | 0 | −0.895394 | 5.08992i | − | 2.86371i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.7 | − | 0.770071i | − | 1.00000i | 1.40699 | 0 | −0.770071 | 3.98808i | − | 2.62363i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.8 | − | 0.741379i | 1.00000i | 1.45036 | 0 | 0.741379 | − | 1.03586i | − | 2.55802i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.9 | 0.741379i | − | 1.00000i | 1.45036 | 0 | 0.741379 | 1.03586i | 2.55802i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.10 | 0.770071i | 1.00000i | 1.40699 | 0 | −0.770071 | − | 3.98808i | 2.62363i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.11 | 0.895394i | 1.00000i | 1.19827 | 0 | −0.895394 | − | 5.08992i | 2.86371i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.12 | 1.31354i | − | 1.00000i | 0.274605 | 0 | 1.31354 | 4.19091i | 2.98779i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.13 | 1.52260i | 1.00000i | −0.318310 | 0 | −1.52260 | 0.990985i | 2.56054i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.14 | 2.23365i | − | 1.00000i | −2.98921 | 0 | 2.23365 | − | 1.03143i | − | 2.20956i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.15 | 2.59716i | − | 1.00000i | −4.74525 | 0 | 2.59716 | 3.28414i | − | 7.12986i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1249.16 | 2.69767i | 1.00000i | −5.27745 | 0 | −2.69767 | 3.56649i | − | 8.84149i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1875.2.b.g | 16 | |
5.b | even | 2 | 1 | inner | 1875.2.b.g | 16 | |
5.c | odd | 4 | 1 | 1875.2.a.n | ✓ | 8 | |
5.c | odd | 4 | 1 | 1875.2.a.o | yes | 8 | |
15.e | even | 4 | 1 | 5625.2.a.u | 8 | ||
15.e | even | 4 | 1 | 5625.2.a.bc | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1875.2.a.n | ✓ | 8 | 5.c | odd | 4 | 1 | |
1875.2.a.o | yes | 8 | 5.c | odd | 4 | 1 | |
1875.2.b.g | 16 | 1.a | even | 1 | 1 | trivial | |
1875.2.b.g | 16 | 5.b | even | 2 | 1 | inner | |
5625.2.a.u | 8 | 15.e | even | 4 | 1 | ||
5625.2.a.bc | 8 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 25T_{2}^{14} + 246T_{2}^{12} + 1220T_{2}^{10} + 3281T_{2}^{8} + 4880T_{2}^{6} + 3936T_{2}^{4} + 1600T_{2}^{2} + 256 \)
acting on \(S_{2}^{\mathrm{new}}(1875, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} + 25 T^{14} + 246 T^{12} + \cdots + 256 \)
$3$
\( (T^{2} + 1)^{8} \)
$5$
\( T^{16} \)
$7$
\( T^{16} + 86 T^{14} + 2941 T^{12} + \cdots + 1113025 \)
$11$
\( (T^{8} - 12 T^{7} + 2 T^{6} + 455 T^{5} + \cdots + 16)^{2} \)
$13$
\( T^{16} + 110 T^{14} + \cdots + 76230361 \)
$17$
\( T^{16} + 177 T^{14} + \cdots + 74926336 \)
$19$
\( (T^{8} + 16 T^{7} + 51 T^{6} - 554 T^{5} + \cdots - 14975)^{2} \)
$23$
\( T^{16} + 224 T^{14} + \cdots + 824838400 \)
$29$
\( (T^{8} + 2 T^{7} - 126 T^{6} - 257 T^{5} + \cdots + 57520)^{2} \)
$31$
\( (T^{8} - 13 T^{7} - 96 T^{6} + \cdots + 801025)^{2} \)
$37$
\( T^{16} + 126 T^{14} + 5021 T^{12} + \cdots + 625 \)
$41$
\( (T^{8} + 12 T^{7} - 6 T^{6} - 487 T^{5} + \cdots - 48080)^{2} \)
$43$
\( T^{16} + 388 T^{14} + \cdots + 5530748161 \)
$47$
\( T^{16} + 367 T^{14} + \cdots + 65445918976 \)
$53$
\( T^{16} + 334 T^{14} + \cdots + 824838400 \)
$59$
\( (T^{8} + 14 T^{7} - 164 T^{6} + \cdots + 11920)^{2} \)
$61$
\( (T^{8} - 10 T^{7} - 186 T^{6} + \cdots - 1093919)^{2} \)
$67$
\( T^{16} + \cdots + 408210546216841 \)
$71$
\( (T^{8} - 21 T^{7} - 18 T^{6} + 2102 T^{5} + \cdots + 67696)^{2} \)
$73$
\( T^{16} + 319 T^{14} + \cdots + 144137919025 \)
$79$
\( (T^{8} + 10 T^{7} - 320 T^{6} + \cdots + 6951025)^{2} \)
$83$
\( T^{16} + 465 T^{14} + \cdots + 12937697536 \)
$89$
\( (T^{8} - 9 T^{7} - 294 T^{6} + \cdots - 12105680)^{2} \)
$97$
\( T^{16} + 402 T^{14} + \cdots + 161554959721 \)
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