Newspace parameters
Level: | \( N \) | \(=\) | \( 380 = 2^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 380.l (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.03431527681\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(i)\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 4 x^{11} + 28 x^{10} - 64 x^{9} + 236 x^{8} - 420 x^{7} + 946 x^{6} - 1216 x^{5} + 1896 x^{4} - 1564 x^{3} + 2284 x^{2} - 1088 x + 1370 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
Coefficient ring index: | \( 2^{6} \) |
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 \( x^{12} - 4 x^{11} + 28 x^{10} - 64 x^{9} + 236 x^{8} - 420 x^{7} + 946 x^{6} - 1216 x^{5} + 1896 x^{4} - 1564 x^{3} + 2284 x^{2} - 1088 x + 1370 \) :
\(\beta_{1}\) | \(=\) | \( ( 43066472536 \nu^{11} - 449979675808 \nu^{10} + 2364419690616 \nu^{9} - 10578246250319 \nu^{8} + 27902022591136 \nu^{7} + \cdots - 279575999635621 ) / 79197078563257 \) |
\(\beta_{2}\) | \(=\) | \( ( - 20904914650254 \nu^{11} + 151211473768106 \nu^{10} - 794968673642910 \nu^{9} + \cdots + 61\!\cdots\!02 ) / 22\!\cdots\!17 \) |
\(\beta_{3}\) | \(=\) | \( ( - 1434891834382 \nu^{11} + 4349786383365 \nu^{10} - 30559554581954 \nu^{9} + 42878862093020 \nu^{8} + \cdots - 614674855329394 ) / 11\!\cdots\!43 \) |
\(\beta_{4}\) | \(=\) | \( ( - 36029778947630 \nu^{11} + 211750342123647 \nu^{10} + \cdots + 82\!\cdots\!06 ) / 22\!\cdots\!17 \) |
\(\beta_{5}\) | \(=\) | \( ( 109895826 \nu^{11} - 371224964 \nu^{10} + 2654024378 \nu^{9} - 4749330090 \nu^{8} + 18447551030 \nu^{7} - 26443913423 \nu^{6} + \cdots + 32432735911 ) / 57697329013 \) |
\(\beta_{6}\) | \(=\) | \( ( 46589648822398 \nu^{11} - 179791490723729 \nu^{10} + \cdots + 68\!\cdots\!68 ) / 22\!\cdots\!17 \) |
\(\beta_{7}\) | \(=\) | \( ( 176200650116 \nu^{11} - 1329132544436 \nu^{10} + 6501003581929 \nu^{9} - 24026915922758 \nu^{8} + 54318387857268 \nu^{7} + \cdots - 625898181929096 ) / 79197078563257 \) |
\(\beta_{8}\) | \(=\) | \( ( 60614681294844 \nu^{11} - 176924960309206 \nu^{10} + \cdots - 21\!\cdots\!53 ) / 22\!\cdots\!17 \) |
\(\beta_{9}\) | \(=\) | \( ( 256791075958 \nu^{11} - 2213799473497 \nu^{10} + 10545830020631 \nu^{9} - 43417051770468 \nu^{8} + 98549420880028 \nu^{7} + \cdots - 965449441292380 ) / 79197078563257 \) |
\(\beta_{10}\) | \(=\) | \( ( - 115790335988434 \nu^{11} + 395417420255441 \nu^{10} + \cdots + 88\!\cdots\!30 ) / 22\!\cdots\!17 \) |
\(\beta_{11}\) | \(=\) | \( ( - 120425382922218 \nu^{11} + 468739736789324 \nu^{10} + \cdots - 68\!\cdots\!02 ) / 22\!\cdots\!17 \) |
\(\nu\) | \(=\) | \( ( -\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 2\beta_{6} - 3\beta_{5} + 2\beta_{2} - \beta _1 - 7 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( - \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 3 \beta_{6} + 12 \beta_{5} + 10 \beta_{4} + 15 \beta_{3} + 2 \beta_{2} - 4 \beta _1 - 18 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 8 \beta_{11} - 16 \beta_{10} - 4 \beta_{9} - 30 \beta_{8} + 4 \beta_{7} - 37 \beta_{6} + 84 \beta_{5} + 16 \beta_{4} + 29 \beta_{3} - 24 \beta_{2} + 4 \beta _1 + 20 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 2 \beta_{11} - 10 \beta_{10} - 55 \beta_{9} - 21 \beta_{8} + 5 \beta_{7} - 150 \beta_{6} + 37 \beta_{5} - 85 \beta_{4} - 116 \beta_{3} - 105 \beta_{2} + 89 \beta _1 + 273 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( - 32 \beta_{11} + 99 \beta_{10} - 51 \beta_{9} + 183 \beta_{8} + 8 \beta_{7} + 99 \beta_{6} - 498 \beta_{5} - 210 \beta_{4} - 325 \beta_{3} + 60 \beta_{2} + 93 \beta _1 + 238 \) |
\(\nu^{7}\) | \(=\) | \( ( - 100 \beta_{11} + 586 \beta_{10} + 532 \beta_{9} + 1074 \beta_{8} - 70 \beta_{7} + 2615 \beta_{6} - 2858 \beta_{5} - 20 \beta_{4} - 77 \beta_{3} + 1742 \beta_{2} - 934 \beta _1 - 2648 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( 176 \beta_{11} - 616 \beta_{10} + 1388 \beta_{9} - 1120 \beta_{8} - 292 \beta_{7} + 1768 \beta_{6} + 3024 \beta_{5} + 2876 \beta_{4} + 4400 \beta_{3} + 1164 \beta_{2} - 2527 \beta _1 - 6723 \) |
\(\nu^{9}\) | \(=\) | \( 993 \beta_{11} - 5524 \beta_{10} - 348 \beta_{9} - 10000 \beta_{8} - 13512 \beta_{6} + 26866 \beta_{5} + 7641 \beta_{4} + 11633 \beta_{3} - 8961 \beta_{2} + 584 \beta _1 + 1838 \) |
\(\nu^{10}\) | \(=\) | \( 877 \beta_{11} - 5469 \beta_{10} - 19989 \beta_{9} - 9929 \beta_{8} + 4117 \beta_{7} - 57259 \beta_{6} + 26452 \beta_{5} - 21000 \beta_{4} - 32160 \beta_{3} - 37564 \beta_{2} + 36399 \beta _1 + 97170 \) |
\(\nu^{11}\) | \(=\) | \( ( - 23101 \beta_{11} + 124939 \beta_{10} - 88121 \beta_{9} + 226350 \beta_{8} + 17347 \beta_{7} + 85400 \beta_{6} - 607538 \beta_{5} - 316704 \beta_{4} - 481316 \beta_{3} + 56994 \beta_{2} + \cdots + 427540 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).
\(n\) | \(21\) | \(77\) | \(191\) |
\(\chi(n)\) | \(-1\) | \(\beta_{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
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37.1 |
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0 | −2.25348 | − | 2.25348i | 0 | −1.48119 | + | 1.67513i | 0 | −1.48119 | − | 1.48119i | 0 | 7.15633i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
37.2 | 0 | −1.63846 | − | 1.63846i | 0 | 2.17009 | + | 0.539189i | 0 | 2.17009 | + | 2.17009i | 0 | 2.36910i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
37.3 | 0 | −1.49576 | − | 1.49576i | 0 | 0.311108 | − | 2.21432i | 0 | 0.311108 | + | 0.311108i | 0 | 1.47457i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
37.4 | 0 | 1.49576 | + | 1.49576i | 0 | 0.311108 | − | 2.21432i | 0 | 0.311108 | + | 0.311108i | 0 | 1.47457i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
37.5 | 0 | 1.63846 | + | 1.63846i | 0 | 2.17009 | + | 0.539189i | 0 | 2.17009 | + | 2.17009i | 0 | 2.36910i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
37.6 | 0 | 2.25348 | + | 2.25348i | 0 | −1.48119 | + | 1.67513i | 0 | −1.48119 | − | 1.48119i | 0 | 7.15633i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
113.1 | 0 | −2.25348 | + | 2.25348i | 0 | −1.48119 | − | 1.67513i | 0 | −1.48119 | + | 1.48119i | 0 | − | 7.15633i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
113.2 | 0 | −1.63846 | + | 1.63846i | 0 | 2.17009 | − | 0.539189i | 0 | 2.17009 | − | 2.17009i | 0 | − | 2.36910i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
113.3 | 0 | −1.49576 | + | 1.49576i | 0 | 0.311108 | + | 2.21432i | 0 | 0.311108 | − | 0.311108i | 0 | − | 1.47457i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
113.4 | 0 | 1.49576 | − | 1.49576i | 0 | 0.311108 | + | 2.21432i | 0 | 0.311108 | − | 0.311108i | 0 | − | 1.47457i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
113.5 | 0 | 1.63846 | − | 1.63846i | 0 | 2.17009 | − | 0.539189i | 0 | 2.17009 | − | 2.17009i | 0 | − | 2.36910i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
113.6 | 0 | 2.25348 | − | 2.25348i | 0 | −1.48119 | − | 1.67513i | 0 | −1.48119 | + | 1.48119i | 0 | − | 7.15633i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
19.b | odd | 2 | 1 | inner |
95.g | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 380.2.l.b | ✓ | 12 |
3.b | odd | 2 | 1 | 3420.2.bb.d | 12 | ||
5.b | even | 2 | 1 | 1900.2.l.b | 12 | ||
5.c | odd | 4 | 1 | inner | 380.2.l.b | ✓ | 12 |
5.c | odd | 4 | 1 | 1900.2.l.b | 12 | ||
15.e | even | 4 | 1 | 3420.2.bb.d | 12 | ||
19.b | odd | 2 | 1 | inner | 380.2.l.b | ✓ | 12 |
57.d | even | 2 | 1 | 3420.2.bb.d | 12 | ||
95.d | odd | 2 | 1 | 1900.2.l.b | 12 | ||
95.g | even | 4 | 1 | inner | 380.2.l.b | ✓ | 12 |
95.g | even | 4 | 1 | 1900.2.l.b | 12 | ||
285.j | odd | 4 | 1 | 3420.2.bb.d | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
380.2.l.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
380.2.l.b | ✓ | 12 | 5.c | odd | 4 | 1 | inner |
380.2.l.b | ✓ | 12 | 19.b | odd | 2 | 1 | inner |
380.2.l.b | ✓ | 12 | 95.g | even | 4 | 1 | inner |
1900.2.l.b | 12 | 5.b | even | 2 | 1 | ||
1900.2.l.b | 12 | 5.c | odd | 4 | 1 | ||
1900.2.l.b | 12 | 95.d | odd | 2 | 1 | ||
1900.2.l.b | 12 | 95.g | even | 4 | 1 | ||
3420.2.bb.d | 12 | 3.b | odd | 2 | 1 | ||
3420.2.bb.d | 12 | 15.e | even | 4 | 1 | ||
3420.2.bb.d | 12 | 57.d | even | 2 | 1 | ||
3420.2.bb.d | 12 | 285.j | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 152T_{3}^{8} + 5616T_{3}^{4} + 59536 \)
acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( T^{12} + 152 T^{8} + 5616 T^{4} + \cdots + 59536 \)
$5$
\( (T^{6} - 2 T^{5} + 3 T^{4} - 12 T^{3} + \cdots + 125)^{2} \)
$7$
\( (T^{6} - 2 T^{5} + 2 T^{4} + 8 T^{3} + 36 T^{2} + \cdots + 8)^{2} \)
$11$
\( (T^{3} - 2 T^{2} - 4 T + 4)^{4} \)
$13$
\( T^{12} + 1816 T^{8} + 408272 T^{4} + \cdots + 59536 \)
$17$
\( (T^{6} + 2 T^{5} + 2 T^{4} + 16 T^{3} + \cdots + 1352)^{2} \)
$19$
\( T^{12} - 6 T^{10} + 71 T^{8} + \cdots + 47045881 \)
$23$
\( (T^{6} - 2 T^{5} + 2 T^{4} - 40 T^{3} + \cdots + 5000)^{2} \)
$29$
\( (T^{6} - 40 T^{4} + 496 T^{2} - 1952)^{2} \)
$31$
\( (T^{6} + 96 T^{4} + 2336 T^{2} + \cdots + 1952)^{2} \)
$37$
\( T^{12} + 20536 T^{8} + 54727472 T^{4} + \cdots + 59536 \)
$41$
\( (T^{6} + 144 T^{4} + 2432 T^{2} + \cdots + 7808)^{2} \)
$43$
\( (T^{6} + 14 T^{5} + 98 T^{4} + 200 T^{3} + \cdots + 14792)^{2} \)
$47$
\( (T^{6} - 10 T^{5} + 50 T^{4} + 232 T^{3} + \cdots + 2312)^{2} \)
$53$
\( T^{12} + 29016 T^{8} + \cdots + 37210000 \)
$59$
\( (T^{6} - 448 T^{4} + 63808 T^{2} + \cdots - 2818688)^{2} \)
$61$
\( (T^{3} - 4 T^{2} - 4 T + 20)^{4} \)
$67$
\( T^{12} + 16568 T^{8} + 33883600 T^{4} + \cdots + 59536 \)
$71$
\( (T^{6} + 104 T^{4} + 1088 T^{2} + \cdots + 1952)^{2} \)
$73$
\( (T^{6} + 38 T^{5} + 722 T^{4} + \cdots + 390728)^{2} \)
$79$
\( (T^{6} - 184 T^{4} + 4032 T^{2} + \cdots - 7808)^{2} \)
$83$
\( (T^{6} - 42 T^{5} + 882 T^{4} + \cdots + 803912)^{2} \)
$89$
\( (T^{6} - 288 T^{4} + 18960 T^{2} + \cdots - 329888)^{2} \)
$97$
\( T^{12} + 140856 T^{8} + \cdots + 46306881767056 \)
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