Newspace parameters
Level: | \( N \) | \(=\) | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 390.t (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.11416567883\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(i)\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
Defining polynomial: |
\( x^{12} + 6x^{10} - 24x^{9} + 18x^{8} + 40x^{7} - 82x^{6} + 12x^{5} + 228x^{4} - 284x^{3} + 124x^{2} - 16x + 2 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 6x^{10} - 24x^{9} + 18x^{8} + 40x^{7} - 82x^{6} + 12x^{5} + 228x^{4} - 284x^{3} + 124x^{2} - 16x + 2 \)
:
\(\beta_{1}\) | \(=\) |
\( ( 47412266 \nu^{11} + 206226196 \nu^{10} + 498918647 \nu^{9} + 275321548 \nu^{8} - 2696897549 \nu^{7} + 1486501171 \nu^{6} + \cdots - 957747072 ) / 13913005925 \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 478873536 \nu^{11} - 47412266 \nu^{10} - 3079467412 \nu^{9} + 10994046217 \nu^{8} - 8895045196 \nu^{7} - 16458043891 \nu^{6} + \cdots + 16498970487 ) / 13913005925 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 562969191 \nu^{11} - 625242846 \nu^{10} - 3845754997 \nu^{9} + 9502475052 \nu^{8} + 1876460724 \nu^{7} - 24232626521 \nu^{6} + \cdots - 278134328 ) / 13913005925 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 65564 \nu^{11} + 54701 \nu^{10} - 348926 \nu^{9} + 1928638 \nu^{8} - 2217584 \nu^{7} - 2535511 \nu^{6} + 7789630 \nu^{5} - 3089126 \nu^{4} + \cdots + 1239927 ) / 1476181 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 642662513 \nu^{11} + 232175953 \nu^{10} + 3952388821 \nu^{9} - 13970332336 \nu^{8} + 6804053268 \nu^{7} + 28228005853 \nu^{6} + \cdots - 4015649046 ) / 13913005925 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 1158219438 \nu^{11} + 651192603 \nu^{10} + 7299225171 \nu^{9} - 23748128936 \nu^{8} + 7624490093 \nu^{7} + 50974131203 \nu^{6} + \cdots - 2779767646 ) / 13913005925 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 2007824523 \nu^{11} + 642662513 \nu^{10} + 12279123091 \nu^{9} - 44235399731 \nu^{8} + 22170509078 \nu^{7} + 87117034188 \nu^{6} + \cdots - 2642262891 ) / 13913005925 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 2024040388 \nu^{11} - 481621272 \nu^{10} + 11262741596 \nu^{9} - 52099333786 \nu^{8} + 42254553243 \nu^{7} + 89411069453 \nu^{6} + \cdots + 4694601129 ) / 13913005925 \)
|
\(\beta_{9}\) | \(=\) |
\( ( 2281630388 \nu^{11} + 1661658328 \nu^{10} + 14577758021 \nu^{9} - 44298602736 \nu^{8} + 6809317318 \nu^{7} + 102670257953 \nu^{6} + \cdots + 27875512029 ) / 13913005925 \)
|
\(\beta_{10}\) | \(=\) |
\( ( - 511017007 \nu^{11} - 120344967 \nu^{10} - 3187847414 \nu^{9} + 11403406359 \nu^{8} - 7135610307 \nu^{7} - 20604017287 \nu^{6} + \cdots + 1016773599 ) / 2782601185 \)
|
\(\beta_{11}\) | \(=\) |
\( ( 771058311 \nu^{11} + 121340051 \nu^{10} + 4638436737 \nu^{9} - 17846443417 \nu^{8} + 10922555446 \nu^{7} + 32171693006 \nu^{6} + \cdots - 7445776057 ) / 2782601185 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{6} - \beta_{5} + \beta_{3} + \beta_1 ) / 2 \)
|
\(\nu^{2}\) | \(=\) |
\( ( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 3\beta_{5} - \beta_{4} - 3\beta_{3} - 4\beta_{2} - 4\beta _1 - 1 ) / 2 \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{10} - \beta_{9} - \beta_{8} + 4\beta_{7} - 5\beta_{6} + 5\beta_{5} - \beta_{4} - 2\beta_{3} + \beta_{2} + 3\beta _1 + 3 \)
|
\(\nu^{4}\) | \(=\) |
\( 5 \beta_{11} - 5 \beta_{10} - 3 \beta_{9} + 3 \beta_{8} - 13 \beta_{7} + 13 \beta_{6} + 5 \beta_{5} + 17 \beta_{4} + 20 \beta_{3} + 5 \beta_{2} + 15 \beta _1 + 3 \)
|
\(\nu^{5}\) | \(=\) |
\( - 13 \beta_{11} + 17 \beta_{9} + 13 \beta_{8} - 25 \beta_{7} + 27 \beta_{6} - 72 \beta_{5} - 25 \beta_{4} - 44 \beta_{3} - 28 \beta_{2} - 72 \beta _1 - 45 \)
|
\(\nu^{6}\) | \(=\) |
\( - 27 \beta_{11} + 59 \beta_{10} - 27 \beta_{9} - 59 \beta_{8} + 234 \beta_{7} - 218 \beta_{6} + 130 \beta_{5} - 100 \beta_{4} - 130 \beta_{3} + 100 \)
|
\(\nu^{7}\) | \(=\) |
\( 218 \beta_{11} - 157 \beta_{10} - 157 \beta_{9} - 301 \beta_{7} + 301 \beta_{6} + 519 \beta_{5} + 564 \beta_{4} + 867 \beta_{3} + 346 \beta_{2} + 867 \beta _1 + 301 \)
|
\(\nu^{8}\) | \(=\) |
\( - 301 \beta_{11} - 301 \beta_{10} + 710 \beta_{9} + 710 \beta_{8} - 1925 \beta_{7} + 1889 \beta_{6} - 2961 \beta_{5} - 301 \beta_{4} - 710 \beta_{3} - 914 \beta_{2} - 2298 \beta _1 - 2036 \)
|
\(\nu^{9}\) | \(=\) |
\( - 1889 \beta_{11} + 2660 \beta_{10} - 1889 \beta_{8} + 8736 \beta_{7} - 8557 \beta_{6} + 1674 \beta_{5} - 6076 \beta_{4} - 8557 \beta_{3} - 1735 \beta_{2} - 4334 \beta _1 + 1735 \)
|
\(\nu^{10}\) | \(=\) |
\( 8557 \beta_{11} - 3563 \beta_{10} - 8557 \beta_{9} - 3563 \beta_{8} + 31302 \beta_{5} + 19663 \beta_{4} + 31302 \beta_{3} + 15794 \beta_{2} + 39260 \beta _1 + 19663 \)
|
\(\nu^{11}\) | \(=\) |
\( - 22745 \beta_{10} + 22745 \beta_{9} + 32134 \beta_{8} - 105363 \beta_{7} + 103116 \beta_{6} - 103116 \beta_{5} + 20910 \beta_{4} + 20017 \beta_{3} - 20910 \beta_{2} - 52151 \beta _1 - 73229 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).
\(n\) | \(131\) | \(157\) | \(301\) |
\(\chi(n)\) | \(1\) | \(\beta_{4}\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
307.1 |
|
− | 1.00000i | −0.707107 | + | 0.707107i | −1.00000 | −2.20289 | + | 0.383740i | 0.707107 | + | 0.707107i | 1.52873 | 1.00000i | − | 1.00000i | 0.383740 | + | 2.20289i | ||||||||||||||||||||||||||||||||||||||||||||
307.2 | − | 1.00000i | −0.707107 | + | 0.707107i | −1.00000 | 1.04698 | − | 1.97581i | 0.707107 | + | 0.707107i | 2.54214 | 1.00000i | − | 1.00000i | −1.97581 | − | 1.04698i | |||||||||||||||||||||||||||||||||||||||||||||
307.3 | − | 1.00000i | −0.707107 | + | 0.707107i | −1.00000 | 1.57013 | + | 1.59207i | 0.707107 | + | 0.707107i | −1.24244 | 1.00000i | − | 1.00000i | 1.59207 | − | 1.57013i | |||||||||||||||||||||||||||||||||||||||||||||
307.4 | − | 1.00000i | 0.707107 | − | 0.707107i | −1.00000 | −2.23563 | − | 0.0440169i | −0.707107 | − | 0.707107i | −4.35328 | 1.00000i | − | 1.00000i | −0.0440169 | + | 2.23563i | |||||||||||||||||||||||||||||||||||||||||||||
307.5 | − | 1.00000i | 0.707107 | − | 0.707107i | −1.00000 | −0.623976 | − | 2.14724i | −0.707107 | − | 0.707107i | 1.64083 | 1.00000i | − | 1.00000i | −2.14724 | + | 0.623976i | |||||||||||||||||||||||||||||||||||||||||||||
307.6 | − | 1.00000i | 0.707107 | − | 0.707107i | −1.00000 | 0.445397 | + | 2.19126i | −0.707107 | − | 0.707107i | −0.115977 | 1.00000i | − | 1.00000i | 2.19126 | − | 0.445397i | |||||||||||||||||||||||||||||||||||||||||||||
343.1 | 1.00000i | −0.707107 | − | 0.707107i | −1.00000 | −2.20289 | − | 0.383740i | 0.707107 | − | 0.707107i | 1.52873 | − | 1.00000i | 1.00000i | 0.383740 | − | 2.20289i | ||||||||||||||||||||||||||||||||||||||||||||||
343.2 | 1.00000i | −0.707107 | − | 0.707107i | −1.00000 | 1.04698 | + | 1.97581i | 0.707107 | − | 0.707107i | 2.54214 | − | 1.00000i | 1.00000i | −1.97581 | + | 1.04698i | ||||||||||||||||||||||||||||||||||||||||||||||
343.3 | 1.00000i | −0.707107 | − | 0.707107i | −1.00000 | 1.57013 | − | 1.59207i | 0.707107 | − | 0.707107i | −1.24244 | − | 1.00000i | 1.00000i | 1.59207 | + | 1.57013i | ||||||||||||||||||||||||||||||||||||||||||||||
343.4 | 1.00000i | 0.707107 | + | 0.707107i | −1.00000 | −2.23563 | + | 0.0440169i | −0.707107 | + | 0.707107i | −4.35328 | − | 1.00000i | 1.00000i | −0.0440169 | − | 2.23563i | ||||||||||||||||||||||||||||||||||||||||||||||
343.5 | 1.00000i | 0.707107 | + | 0.707107i | −1.00000 | −0.623976 | + | 2.14724i | −0.707107 | + | 0.707107i | 1.64083 | − | 1.00000i | 1.00000i | −2.14724 | − | 0.623976i | ||||||||||||||||||||||||||||||||||||||||||||||
343.6 | 1.00000i | 0.707107 | + | 0.707107i | −1.00000 | 0.445397 | − | 2.19126i | −0.707107 | + | 0.707107i | −0.115977 | − | 1.00000i | 1.00000i | 2.19126 | + | 0.445397i | ||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 390.2.t.a | yes | 12 |
3.b | odd | 2 | 1 | 1170.2.w.f | 12 | ||
5.b | even | 2 | 1 | 1950.2.t.b | 12 | ||
5.c | odd | 4 | 1 | 390.2.j.a | ✓ | 12 | |
5.c | odd | 4 | 1 | 1950.2.j.c | 12 | ||
13.d | odd | 4 | 1 | 390.2.j.a | ✓ | 12 | |
15.e | even | 4 | 1 | 1170.2.m.f | 12 | ||
39.f | even | 4 | 1 | 1170.2.m.f | 12 | ||
65.f | even | 4 | 1 | inner | 390.2.t.a | yes | 12 |
65.g | odd | 4 | 1 | 1950.2.j.c | 12 | ||
65.k | even | 4 | 1 | 1950.2.t.b | 12 | ||
195.u | odd | 4 | 1 | 1170.2.w.f | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
390.2.j.a | ✓ | 12 | 5.c | odd | 4 | 1 | |
390.2.j.a | ✓ | 12 | 13.d | odd | 4 | 1 | |
390.2.t.a | yes | 12 | 1.a | even | 1 | 1 | trivial |
390.2.t.a | yes | 12 | 65.f | even | 4 | 1 | inner |
1170.2.m.f | 12 | 15.e | even | 4 | 1 | ||
1170.2.m.f | 12 | 39.f | even | 4 | 1 | ||
1170.2.w.f | 12 | 3.b | odd | 2 | 1 | ||
1170.2.w.f | 12 | 195.u | odd | 4 | 1 | ||
1950.2.j.c | 12 | 5.c | odd | 4 | 1 | ||
1950.2.j.c | 12 | 65.g | odd | 4 | 1 | ||
1950.2.t.b | 12 | 5.b | even | 2 | 1 | ||
1950.2.t.b | 12 | 65.k | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} - 16T_{7}^{4} + 20T_{7}^{3} + 24T_{7}^{2} - 32T_{7} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{6} \)
$3$
\( (T^{4} + 1)^{3} \)
$5$
\( T^{12} + 4 T^{11} + 10 T^{10} + \cdots + 15625 \)
$7$
\( (T^{6} - 16 T^{4} + 20 T^{3} + 24 T^{2} + \cdots - 4)^{2} \)
$11$
\( T^{12} - 4 T^{11} + 8 T^{10} + \cdots + 141376 \)
$13$
\( T^{12} - 20 T^{11} + 188 T^{10} + \cdots + 4826809 \)
$17$
\( T^{12} - 8 T^{11} + 32 T^{10} + \cdots + 446224 \)
$19$
\( T^{12} + 4 T^{11} + 8 T^{10} + \cdots + 23309584 \)
$23$
\( T^{12} + 104 T^{9} + 5768 T^{8} + \cdots + 38416 \)
$29$
\( T^{12} + 332 T^{10} + \cdots + 932203024 \)
$31$
\( (T^{2} - 2 T + 2)^{6} \)
$37$
\( (T^{6} + 8 T^{5} - 118 T^{4} - 840 T^{3} + \cdots + 25144)^{2} \)
$41$
\( T^{12} - 8 T^{11} + \cdots + 3237154816 \)
$43$
\( T^{12} - 16 T^{11} + \cdots + 289272064 \)
$47$
\( (T^{6} - 16 T^{5} - 24 T^{4} + 800 T^{3} + \cdots - 9248)^{2} \)
$53$
\( T^{12} + 16 T^{11} + 128 T^{10} + \cdots + 1024 \)
$59$
\( T^{12} - 20 T^{11} + 200 T^{10} + \cdots + 94789696 \)
$61$
\( (T^{6} - 8 T^{5} - 160 T^{4} + 480 T^{3} + \cdots - 33056)^{2} \)
$67$
\( T^{12} + 428 T^{10} + \cdots + 15038607424 \)
$71$
\( T^{12} + 32 T^{11} + \cdots + 209295270144 \)
$73$
\( T^{12} + 536 T^{10} + \cdots + 43930384 \)
$79$
\( T^{12} + 368 T^{10} + \cdots + 485409024 \)
$83$
\( (T^{6} - 16 T^{5} - 32 T^{4} + 520 T^{3} + \cdots - 1424)^{2} \)
$89$
\( T^{12} - 16 T^{11} + \cdots + 24551129344 \)
$97$
\( T^{12} + 840 T^{10} + \cdots + 2393122368784 \)
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