Newspace parameters
Level: | \( N \) | \(=\) | \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1900.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(15.1715763840\) |
Analytic rank: | \(0\) |
Dimension: | \(20\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
Defining polynomial: |
\( x^{20} + 20 x^{18} + 261 x^{16} + 1994 x^{14} + 11074 x^{12} + 39211 x^{10} + 99376 x^{8} + 134299 x^{6} + 124617 x^{4} + 24768 x^{2} + 4096 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 380) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} + 20 x^{18} + 261 x^{16} + 1994 x^{14} + 11074 x^{12} + 39211 x^{10} + 99376 x^{8} + 134299 x^{6} + 124617 x^{4} + 24768 x^{2} + 4096 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( 257269301703 \nu^{18} + 4826223210105 \nu^{16} + 61286511972345 \nu^{14} + 438571056316700 \nu^{12} + \cdots - 63\!\cdots\!36 ) / 16\!\cdots\!37 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 71197437104086 \nu^{18} + \cdots - 55\!\cdots\!14 ) / 13\!\cdots\!71 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 58379368812683 \nu^{18} + \cdots - 23\!\cdots\!96 ) / 10\!\cdots\!68 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 46\!\cdots\!85 \nu^{19} + \cdots + 21\!\cdots\!40 \nu ) / 35\!\cdots\!76 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 63\!\cdots\!85 \nu^{18} + \cdots - 25\!\cdots\!84 ) / 88\!\cdots\!44 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 108802723986440 \nu^{18} + \cdots - 49\!\cdots\!91 ) / 13\!\cdots\!71 \)
|
\(\beta_{8}\) | \(=\) |
\( ( - 123113231093090 \nu^{18} + \cdots + 29\!\cdots\!75 ) / 13\!\cdots\!71 \)
|
\(\beta_{9}\) | \(=\) |
\( ( - 135352029273473 \nu^{19} + \cdots - 33\!\cdots\!89 \nu ) / 55\!\cdots\!84 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 149437719142871 \nu^{19} + \cdots - 22\!\cdots\!45 \nu ) / 55\!\cdots\!84 \)
|
\(\beta_{11}\) | \(=\) |
\( ( 54263059985435 \nu^{18} + \cdots + 12\!\cdots\!72 ) / 26\!\cdots\!92 \)
|
\(\beta_{12}\) | \(=\) |
\( ( 91\!\cdots\!17 \nu^{18} + \cdots + 22\!\cdots\!52 ) / 44\!\cdots\!72 \)
|
\(\beta_{13}\) | \(=\) |
\( ( 21\!\cdots\!13 \nu^{18} + \cdots + 91\!\cdots\!84 ) / 88\!\cdots\!44 \)
|
\(\beta_{14}\) | \(=\) |
\( ( 58379368812683 \nu^{19} + \cdots + 13\!\cdots\!64 \nu ) / 10\!\cdots\!68 \)
|
\(\beta_{15}\) | \(=\) |
\( ( - 352957477246353 \nu^{19} + \cdots + 15\!\cdots\!77 \nu ) / 27\!\cdots\!42 \)
|
\(\beta_{16}\) | \(=\) |
\( ( - 395664181329051 \nu^{19} + \cdots + 31\!\cdots\!37 \nu ) / 27\!\cdots\!42 \)
|
\(\beta_{17}\) | \(=\) |
\( ( - 86\!\cdots\!67 \nu^{19} + \cdots - 35\!\cdots\!92 \nu ) / 35\!\cdots\!76 \)
|
\(\beta_{18}\) | \(=\) |
\( ( - 44\!\cdots\!75 \nu^{19} + \cdots - 12\!\cdots\!08 \nu ) / 17\!\cdots\!88 \)
|
\(\beta_{19}\) | \(=\) |
\( ( - 99\!\cdots\!71 \nu^{19} + \cdots - 25\!\cdots\!64 \nu ) / 17\!\cdots\!88 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( -\beta_{11} - 4\beta_{4} - \beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{19} - \beta_{18} + \beta_{16} - \beta_{15} + 6\beta_{14} - 6\beta_1 \)
|
\(\nu^{4}\) | \(=\) |
\( -2\beta_{13} - \beta_{12} + 9\beta_{11} + 2\beta_{7} - \beta_{6} + 25\beta_{4} - 25 \)
|
\(\nu^{5}\) | \(=\) |
\( -10\beta_{19} + 11\beta_{18} + 3\beta_{17} - 40\beta_{14} + 3\beta_{10} + 3\beta_{9} + 3\beta_{5} \)
|
\(\nu^{6}\) | \(=\) |
\( -16\beta_{8} - 27\beta_{7} - 8\beta_{3} + 72\beta_{2} + 177 \)
|
\(\nu^{7}\) | \(=\) |
\( -88\beta_{16} + 99\beta_{15} - 40\beta_{10} - 46\beta_{9} + 283\beta_1 \)
|
\(\nu^{8}\) | \(=\) |
\( 279 \beta_{13} + 47 \beta_{12} - 569 \beta_{11} + 174 \beta_{8} + 174 \beta_{6} - 1329 \beta_{4} + 47 \beta_{3} - 569 \beta_{2} \)
|
\(\nu^{9}\) | \(=\) |
\( 743 \beta_{19} - 848 \beta_{18} - 395 \beta_{17} + 743 \beta_{16} - 848 \beta_{15} + 2083 \beta_{14} - 511 \beta_{5} - 2083 \beta_1 \)
|
\(\nu^{10}\) | \(=\) |
\( -2613\beta_{13} - 221\beta_{12} + 4522\beta_{11} + 2613\beta_{7} - 1649\beta_{6} + 10318\beta_{4} - 10318 \)
|
\(\nu^{11}\) | \(=\) |
\( - 6171 \beta_{19} + 7135 \beta_{18} + 3519 \beta_{17} - 15785 \beta_{14} + 3519 \beta_{10} + 5005 \beta_{9} + 5005 \beta_{5} \)
|
\(\nu^{12}\) | \(=\) |
\( -14695\beta_{8} - 23316\beta_{7} - 644\beta_{3} + 36226\beta_{2} + 81771 \)
|
\(\nu^{13}\) | \(=\) |
\( -50921\beta_{16} + 59542\beta_{15} - 30034\beta_{10} - 45988\beta_{9} + 122286\beta_1 \)
|
\(\nu^{14}\) | \(=\) |
\( 202439 \beta_{13} - 2400 \beta_{12} - 292291 \beta_{11} + 126943 \beta_{8} + 126943 \beta_{6} - 656616 \beta_{4} - 2400 \beta_{3} - 292291 \beta_{2} \)
|
\(\nu^{15}\) | \(=\) |
\( 419234 \beta_{19} - 494730 \beta_{18} - 251486 \beta_{17} + 419234 \beta_{16} - 494730 \beta_{15} + 963263 \beta_{14} - 407278 \beta_{5} - 963263 \beta_1 \)
|
\(\nu^{16}\) | \(=\) |
\( - 1728520 \beta_{13} + 68340 \beta_{12} + 2371957 \beta_{11} + 1728520 \beta_{7} - 1077998 \beta_{6} + 5318130 \beta_{4} - 5318130 \)
|
\(\nu^{17}\) | \(=\) |
\( - 3449955 \beta_{19} + 4100477 \beta_{18} + 2087656 \beta_{17} - 7683002 \beta_{14} + 2087656 \beta_{10} + 3525380 \beta_{9} + 3525380 \beta_{5} \)
|
\(\nu^{18}\) | \(=\) |
\( -9062991\beta_{8} - 14601192\beta_{7} + 862627\beta_{3} + 19333911\beta_{2} + 43320969 \)
|
\(\nu^{19}\) | \(=\) |
\( -28396902\beta_{16} + 33935103\beta_{15} - 17263355\beta_{10} - 30065011\beta_{9} + 61849398\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).
\(n\) | \(77\) | \(401\) | \(951\) |
\(\chi(n)\) | \(1\) | \(-1 + \beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
201.1 |
|
0 | −1.43632 | + | 2.48777i | 0 | 0 | 0 | 3.54568 | 0 | −2.62601 | − | 4.54838i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
201.2 | 0 | −1.21562 | + | 2.10552i | 0 | 0 | 0 | 0.663818 | 0 | −1.45548 | − | 2.52097i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
201.3 | 0 | −1.00667 | + | 1.74361i | 0 | 0 | 0 | −1.34403 | 0 | −0.526784 | − | 0.912416i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
201.4 | 0 | −0.628167 | + | 1.08802i | 0 | 0 | 0 | −4.97100 | 0 | 0.710812 | + | 1.23116i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
201.5 | 0 | −0.226426 | + | 0.392182i | 0 | 0 | 0 | 2.54366 | 0 | 1.39746 | + | 2.42048i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
201.6 | 0 | 0.226426 | − | 0.392182i | 0 | 0 | 0 | −2.54366 | 0 | 1.39746 | + | 2.42048i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
201.7 | 0 | 0.628167 | − | 1.08802i | 0 | 0 | 0 | 4.97100 | 0 | 0.710812 | + | 1.23116i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
201.8 | 0 | 1.00667 | − | 1.74361i | 0 | 0 | 0 | 1.34403 | 0 | −0.526784 | − | 0.912416i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
201.9 | 0 | 1.21562 | − | 2.10552i | 0 | 0 | 0 | −0.663818 | 0 | −1.45548 | − | 2.52097i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
201.10 | 0 | 1.43632 | − | 2.48777i | 0 | 0 | 0 | −3.54568 | 0 | −2.62601 | − | 4.54838i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
501.1 | 0 | −1.43632 | − | 2.48777i | 0 | 0 | 0 | 3.54568 | 0 | −2.62601 | + | 4.54838i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
501.2 | 0 | −1.21562 | − | 2.10552i | 0 | 0 | 0 | 0.663818 | 0 | −1.45548 | + | 2.52097i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
501.3 | 0 | −1.00667 | − | 1.74361i | 0 | 0 | 0 | −1.34403 | 0 | −0.526784 | + | 0.912416i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
501.4 | 0 | −0.628167 | − | 1.08802i | 0 | 0 | 0 | −4.97100 | 0 | 0.710812 | − | 1.23116i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
501.5 | 0 | −0.226426 | − | 0.392182i | 0 | 0 | 0 | 2.54366 | 0 | 1.39746 | − | 2.42048i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
501.6 | 0 | 0.226426 | + | 0.392182i | 0 | 0 | 0 | −2.54366 | 0 | 1.39746 | − | 2.42048i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
501.7 | 0 | 0.628167 | + | 1.08802i | 0 | 0 | 0 | 4.97100 | 0 | 0.710812 | − | 1.23116i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
501.8 | 0 | 1.00667 | + | 1.74361i | 0 | 0 | 0 | 1.34403 | 0 | −0.526784 | + | 0.912416i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
501.9 | 0 | 1.21562 | + | 2.10552i | 0 | 0 | 0 | −0.663818 | 0 | −1.45548 | + | 2.52097i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
501.10 | 0 | 1.43632 | + | 2.48777i | 0 | 0 | 0 | −3.54568 | 0 | −2.62601 | + | 4.54838i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
19.c | even | 3 | 1 | inner |
95.i | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1900.2.i.g | 20 | |
5.b | even | 2 | 1 | inner | 1900.2.i.g | 20 | |
5.c | odd | 4 | 2 | 380.2.r.a | ✓ | 20 | |
15.e | even | 4 | 2 | 3420.2.bj.c | 20 | ||
19.c | even | 3 | 1 | inner | 1900.2.i.g | 20 | |
95.i | even | 6 | 1 | inner | 1900.2.i.g | 20 | |
95.m | odd | 12 | 2 | 380.2.r.a | ✓ | 20 | |
285.v | even | 12 | 2 | 3420.2.bj.c | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
380.2.r.a | ✓ | 20 | 5.c | odd | 4 | 2 | |
380.2.r.a | ✓ | 20 | 95.m | odd | 12 | 2 | |
1900.2.i.g | 20 | 1.a | even | 1 | 1 | trivial | |
1900.2.i.g | 20 | 5.b | even | 2 | 1 | inner | |
1900.2.i.g | 20 | 19.c | even | 3 | 1 | inner | |
1900.2.i.g | 20 | 95.i | even | 6 | 1 | inner | |
3420.2.bj.c | 20 | 15.e | even | 4 | 2 | ||
3420.2.bj.c | 20 | 285.v | even | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} + 20 T_{3}^{18} + 261 T_{3}^{16} + 1994 T_{3}^{14} + 11074 T_{3}^{12} + 39211 T_{3}^{10} + 99376 T_{3}^{8} + 134299 T_{3}^{6} + 124617 T_{3}^{4} + 24768 T_{3}^{2} + 4096 \)
acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{20} \)
$3$
\( T^{20} + 20 T^{18} + 261 T^{16} + \cdots + 4096 \)
$5$
\( T^{20} \)
$7$
\( (T^{10} - 46 T^{8} + 651 T^{6} - 3285 T^{4} + \cdots - 1600)^{2} \)
$11$
\( (T^{5} - 27 T^{3} - 29 T^{2} + 107 T + 148)^{4} \)
$13$
\( T^{20} + 78 T^{18} + \cdots + 13051691536 \)
$17$
\( T^{20} + 98 T^{18} + \cdots + 9721171216 \)
$19$
\( (T^{10} + 7 T^{9} + 48 T^{8} + \cdots + 2476099)^{2} \)
$23$
\( T^{20} + 211 T^{18} + \cdots + 39691260010000 \)
$29$
\( (T^{10} - 8 T^{9} + 78 T^{8} - 38 T^{7} + \cdots + 28561)^{2} \)
$31$
\( (T^{5} - 2 T^{4} - 93 T^{3} + 251 T^{2} + \cdots + 388)^{4} \)
$37$
\( (T^{10} - 86 T^{8} + 2655 T^{6} + \cdots - 518400)^{2} \)
$41$
\( (T^{10} - 13 T^{9} + 186 T^{8} + \cdots + 1368900)^{2} \)
$43$
\( T^{20} + \cdots + 122963703210000 \)
$47$
\( T^{20} + 81 T^{18} + \cdots + 2998219536 \)
$53$
\( T^{20} + \cdots + 591687332741376 \)
$59$
\( (T^{10} + 2 T^{9} + 62 T^{8} + 82 T^{7} + \cdots + 1)^{2} \)
$61$
\( (T^{10} - T^{9} + 115 T^{8} - 450 T^{7} + \cdots + 3073009)^{2} \)
$67$
\( T^{20} + \cdots + 805854925357056 \)
$71$
\( (T^{10} + T^{9} + 164 T^{8} + \cdots + 640140601)^{2} \)
$73$
\( T^{20} + 227 T^{18} + \cdots + 25628906250000 \)
$79$
\( (T^{10} - 8 T^{9} + 213 T^{8} + \cdots + 233967616)^{2} \)
$83$
\( (T^{10} - 323 T^{8} + 26698 T^{6} + \cdots - 54464400)^{2} \)
$89$
\( (T^{10} - 20 T^{9} + 479 T^{8} + \cdots + 5314993216)^{2} \)
$97$
\( T^{20} + \cdots + 385136700010000 \)
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