Newspace parameters
Level: | \( N \) | \(=\) | \( 370 = 2 \cdot 5 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 370.r (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.95446487479\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{12})\) |
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: | \( x^{12} - 2x^{11} - 4x^{10} + 6x^{8} + 44x^{7} + 56x^{6} + 32x^{5} + 92x^{4} - 16x^{3} + 36x^{2} - 24x + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \( x^{12} - 2x^{11} - 4x^{10} + 6x^{8} + 44x^{7} + 56x^{6} + 32x^{5} + 92x^{4} - 16x^{3} + 36x^{2} - 24x + 4 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( - 48023 \nu^{11} - 173427 \nu^{10} + 812758 \nu^{9} + 795518 \nu^{8} - 312576 \nu^{7} - 3181332 \nu^{6} - 14801278 \nu^{5} - 14727942 \nu^{4} + \cdots - 16072116 ) / 8800836 \) |
\(\beta_{3}\) | \(=\) | \( ( 2660983 \nu^{11} - 7621528 \nu^{10} - 6209960 \nu^{9} + 9898720 \nu^{8} + 18275158 \nu^{7} + 95878200 \nu^{6} + 40728518 \nu^{5} - 40308836 \nu^{4} + \cdots - 77578192 ) / 132012540 \) |
\(\beta_{4}\) | \(=\) | \( ( 1499 \nu^{11} - 1684 \nu^{10} - 8260 \nu^{9} - 6690 \nu^{8} + 8734 \nu^{7} + 77220 \nu^{6} + 146974 \nu^{5} + 127492 \nu^{4} + 163380 \nu^{3} + 66376 \nu^{2} + 66060 \nu - 22456 ) / 40260 \) |
\(\beta_{5}\) | \(=\) | \( ( - 5362957 \nu^{11} + 9940372 \nu^{10} + 23030390 \nu^{9} + 1380680 \nu^{8} - 29137702 \nu^{7} - 231395340 \nu^{6} - 335137622 \nu^{5} + \cdots + 32870848 ) / 132012540 \) |
\(\beta_{6}\) | \(=\) | \( ( - 8217712 \nu^{11} + 11072467 \nu^{10} + 42811220 \nu^{9} + 23030390 \nu^{8} - 47925592 \nu^{7} - 390717030 \nu^{6} - 691587212 \nu^{5} + \cdots + 35972008 ) / 132012540 \) |
\(\beta_{7}\) | \(=\) | \( ( - 5614 \nu^{11} + 9729 \nu^{10} + 24140 \nu^{9} + 8260 \nu^{8} - 26994 \nu^{7} - 255750 \nu^{6} - 391604 \nu^{5} - 326622 \nu^{4} - 643980 \nu^{3} - 73556 \nu^{2} + \cdots + 68676 ) / 40260 \) |
\(\beta_{8}\) | \(=\) | \( ( - 19394548 \nu^{11} + 36128113 \nu^{10} + 85199720 \nu^{9} + 6209960 \nu^{8} - 126266008 \nu^{7} - 871635270 \nu^{6} - 1181972888 \nu^{5} + \cdots + 277303252 ) / 132012540 \) |
\(\beta_{9}\) | \(=\) | \( ( 12537719 \nu^{11} - 22791104 \nu^{10} - 56749885 \nu^{9} - 5276635 \nu^{8} + 85217429 \nu^{7} + 561759990 \nu^{6} + 785090434 \nu^{5} + \cdots - 249596696 ) / 66006270 \) |
\(\beta_{10}\) | \(=\) | \( ( - 15731201 \nu^{11} + 24513101 \nu^{10} + 73188085 \nu^{9} + 35555635 \nu^{8} - 80700356 \nu^{7} - 738220065 \nu^{6} - 1202248126 \nu^{5} + \cdots + 164311814 ) / 66006270 \) |
\(\beta_{11}\) | \(=\) | \( ( - 16622963 \nu^{11} + 31633798 \nu^{10} + 68509300 \nu^{9} + 8136630 \nu^{8} - 92738998 \nu^{7} - 739730970 \nu^{6} - 1013258878 \nu^{5} + \cdots + 249278272 ) / 44004180 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{10} - \beta_{7} - 2\beta_{6} - \beta_{3} + \beta_{2} + \beta _1 + 1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{11} + \beta_{10} - \beta_{8} - 2\beta_{7} - \beta_{6} - 3\beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} + 2\beta _1 + 2 \) |
\(\nu^{4}\) | \(=\) | \( 6 \beta_{11} + 7 \beta_{10} - \beta_{9} - 8 \beta_{8} - 16 \beta_{7} - 9 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + 7 \beta_{2} + 7 \beta _1 + 6 \) |
\(\nu^{5}\) | \(=\) | \( 13 \beta_{11} + 18 \beta_{10} - 9 \beta_{9} - 24 \beta_{8} - 34 \beta_{7} - 21 \beta_{6} - 16 \beta_{5} + 16 \beta_{4} + 8 \beta_{3} + 13 \beta_{2} + 9 \beta _1 + 3 \) |
\(\nu^{6}\) | \(=\) | \( 54 \beta_{11} + 54 \beta_{10} - 42 \beta_{9} - 112 \beta_{8} - 124 \beta_{7} - 70 \beta_{6} - 34 \beta_{5} + 34 \beta_{4} + 36 \beta_{3} + 42 \beta_{2} + 2 \beta _1 - 8 \) |
\(\nu^{7}\) | \(=\) | \( 152 \beta_{11} + 124 \beta_{10} - 124 \beta_{9} - 310 \beta_{8} - 310 \beta_{7} - 148 \beta_{6} - 112 \beta_{5} + 124 \beta_{4} + 188 \beta_{3} + 76 \beta_{2} - 62 \beta _1 - 72 \) |
\(\nu^{8}\) | \(=\) | \( 436 \beta_{11} + 326 \beta_{10} - 436 \beta_{9} - 996 \beta_{8} - 874 \beta_{7} - 388 \beta_{6} - 224 \beta_{5} + 310 \beta_{4} + 636 \beta_{3} + 110 \beta_{2} - 334 \beta _1 - 438 \) |
\(\nu^{9}\) | \(=\) | \( 1080 \beta_{11} + 636 \beta_{10} - 1272 \beta_{9} - 2662 \beta_{8} - 1940 \beta_{7} - 722 \beta_{6} - 498 \beta_{5} + 874 \beta_{4} + 2076 \beta_{3} - 1510 \beta _1 - 1582 \) |
\(\nu^{10}\) | \(=\) | \( 2644 \beta_{11} + 942 \beta_{10} - 3586 \beta_{9} - 7104 \beta_{8} - 4084 \beta_{7} - 942 \beta_{6} - 722 \beta_{5} + 1940 \beta_{4} + 6248 \beta_{3} - 942 \beta_{2} - 5306 \beta _1 - 5524 \) |
\(\nu^{11}\) | \(=\) | \( 5306 \beta_{11} - 9130 \beta_{9} - 16376 \beta_{8} - 6028 \beta_{7} + 722 \beta_{6} + 4084 \beta_{4} + 17716 \beta_{3} - 5306 \beta_{2} - 17298 \beta _1 - 17098 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).
\(n\) | \(261\) | \(297\) |
\(\chi(n)\) | \(\beta_{6}\) | \(-\beta_{6} - \beta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 |
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0.866025 | + | 0.500000i | −0.658815 | − | 0.176529i | 0.500000 | + | 0.866025i | −1.32150 | + | 1.80379i | −0.482286 | − | 0.482286i | −4.10264 | − | 1.09930i | 1.00000i | −2.19520 | − | 1.26740i | −2.04634 | + | 0.901374i | ||||||||||||||||||||||||||||||||||||||
23.2 | 0.866025 | + | 0.500000i | 1.62090 | + | 0.434319i | 0.500000 | + | 0.866025i | −2.05890 | + | 0.872317i | 1.18658 | + | 1.18658i | 2.36181 | + | 0.632846i | 1.00000i | −0.159392 | − | 0.0920251i | −2.21922 | − | 0.274000i | |||||||||||||||||||||||||||||||||||||||
23.3 | 0.866025 | + | 0.500000i | 2.26997 | + | 0.608236i | 0.500000 | + | 0.866025i | 0.514372 | − | 2.17610i | 1.66173 | + | 1.66173i | −0.991227 | − | 0.265598i | 1.00000i | 2.18472 | + | 1.26135i | 1.53351 | − | 1.62737i | |||||||||||||||||||||||||||||||||||||||
177.1 | 0.866025 | − | 0.500000i | −0.658815 | + | 0.176529i | 0.500000 | − | 0.866025i | −1.32150 | − | 1.80379i | −0.482286 | + | 0.482286i | −4.10264 | + | 1.09930i | − | 1.00000i | −2.19520 | + | 1.26740i | −2.04634 | − | 0.901374i | ||||||||||||||||||||||||||||||||||||||
177.2 | 0.866025 | − | 0.500000i | 1.62090 | − | 0.434319i | 0.500000 | − | 0.866025i | −2.05890 | − | 0.872317i | 1.18658 | − | 1.18658i | 2.36181 | − | 0.632846i | − | 1.00000i | −0.159392 | + | 0.0920251i | −2.21922 | + | 0.274000i | ||||||||||||||||||||||||||||||||||||||
177.3 | 0.866025 | − | 0.500000i | 2.26997 | − | 0.608236i | 0.500000 | − | 0.866025i | 0.514372 | + | 2.17610i | 1.66173 | − | 1.66173i | −0.991227 | + | 0.265598i | − | 1.00000i | 2.18472 | − | 1.26135i | 1.53351 | + | 1.62737i | ||||||||||||||||||||||||||||||||||||||
193.1 | −0.866025 | + | 0.500000i | −0.745286 | − | 2.78144i | 0.500000 | − | 0.866025i | −2.22784 | + | 0.191679i | 2.03616 | + | 2.03616i | 1.31478 | + | 4.90683i | 1.00000i | −4.58291 | + | 2.64594i | 1.83352 | − | 1.27992i | |||||||||||||||||||||||||||||||||||||||
193.2 | −0.866025 | + | 0.500000i | −0.290531 | − | 1.08428i | 0.500000 | − | 0.866025i | 1.13365 | − | 1.92739i | 0.793745 | + | 0.793745i | 0.304145 | + | 1.13508i | 1.00000i | 1.50683 | − | 0.869970i | −0.0180717 | + | 2.23599i | |||||||||||||||||||||||||||||||||||||||
193.3 | −0.866025 | + | 0.500000i | 0.803766 | + | 2.99969i | 0.500000 | − | 0.866025i | −0.0397852 | + | 2.23571i | −2.19593 | − | 2.19593i | −0.886874 | − | 3.30986i | 1.00000i | −5.75405 | + | 3.32210i | −1.08340 | − | 1.95608i | |||||||||||||||||||||||||||||||||||||||
347.1 | −0.866025 | − | 0.500000i | −0.745286 | + | 2.78144i | 0.500000 | + | 0.866025i | −2.22784 | − | 0.191679i | 2.03616 | − | 2.03616i | 1.31478 | − | 4.90683i | − | 1.00000i | −4.58291 | − | 2.64594i | 1.83352 | + | 1.27992i | ||||||||||||||||||||||||||||||||||||||
347.2 | −0.866025 | − | 0.500000i | −0.290531 | + | 1.08428i | 0.500000 | + | 0.866025i | 1.13365 | + | 1.92739i | 0.793745 | − | 0.793745i | 0.304145 | − | 1.13508i | − | 1.00000i | 1.50683 | + | 0.869970i | −0.0180717 | − | 2.23599i | ||||||||||||||||||||||||||||||||||||||
347.3 | −0.866025 | − | 0.500000i | 0.803766 | − | 2.99969i | 0.500000 | + | 0.866025i | −0.0397852 | − | 2.23571i | −2.19593 | + | 2.19593i | −0.886874 | + | 3.30986i | − | 1.00000i | −5.75405 | − | 3.32210i | −1.08340 | + | 1.95608i | ||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
185.u | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 370.2.r.d | yes | 12 |
5.c | odd | 4 | 1 | 370.2.q.d | ✓ | 12 | |
37.g | odd | 12 | 1 | 370.2.q.d | ✓ | 12 | |
185.u | even | 12 | 1 | inner | 370.2.r.d | yes | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
370.2.q.d | ✓ | 12 | 5.c | odd | 4 | 1 | |
370.2.q.d | ✓ | 12 | 37.g | odd | 12 | 1 | |
370.2.r.d | yes | 12 | 1.a | even | 1 | 1 | trivial |
370.2.r.d | yes | 12 | 185.u | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 6 T_{3}^{11} + 27 T_{3}^{10} - 96 T_{3}^{9} + 242 T_{3}^{8} - 534 T_{3}^{7} + 855 T_{3}^{6} - 444 T_{3}^{5} - 86 T_{3}^{4} - 378 T_{3}^{3} - 405 T_{3}^{2} + 972 T_{3} + 729 \)
acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{3} \)
$3$
\( T^{12} - 6 T^{11} + 27 T^{10} - 96 T^{9} + \cdots + 729 \)
$5$
\( T^{12} + 8 T^{11} + 37 T^{10} + \cdots + 15625 \)
$7$
\( T^{12} + 4 T^{11} + 20 T^{10} + \cdots + 47524 \)
$11$
\( T^{12} + 36 T^{10} + 428 T^{8} + \cdots + 324 \)
$13$
\( T^{12} + 6 T^{11} - 21 T^{10} + \cdots + 97969 \)
$17$
\( T^{12} - 12 T^{11} + 106 T^{10} + \cdots + 47524 \)
$19$
\( T^{12} - 8 T^{11} + 56 T^{10} + \cdots + 30976 \)
$23$
\( T^{12} + 56 T^{10} + 972 T^{8} + \cdots + 5476 \)
$29$
\( T^{12} + 4 T^{11} + 8 T^{10} + \cdots + 238144 \)
$31$
\( T^{12} + 2 T^{11} + 2 T^{10} - 170 T^{9} + \cdots + 529 \)
$37$
\( T^{12} - 12 T^{11} + \cdots + 2565726409 \)
$41$
\( T^{12} - 12 T^{11} - 23 T^{10} + \cdots + 1390041 \)
$43$
\( T^{12} + 486 T^{10} + \cdots + 134361101809 \)
$47$
\( T^{12} + 28 T^{11} + \cdots + 16333351204 \)
$53$
\( T^{12} - 14 T^{11} + \cdots + 24894212841 \)
$59$
\( T^{12} + 18 T^{10} + 140 T^{9} + \cdots + 1747684 \)
$61$
\( T^{12} - 24 T^{11} + \cdots + 898081024 \)
$67$
\( T^{12} - 18 T^{11} + \cdots + 47924215056 \)
$71$
\( T^{12} + 304 T^{10} + \cdots + 165408143616 \)
$73$
\( T^{12} - 28 T^{11} + \cdots + 573985764 \)
$79$
\( T^{12} - 56 T^{11} + \cdots + 4859205264 \)
$83$
\( T^{12} - 14 T^{11} + \cdots + 6754209856 \)
$89$
\( T^{12} - 2 T^{11} + \cdots + 2304384016 \)
$97$
\( (T^{6} + 22 T^{5} - 78 T^{4} + \cdots + 484152)^{2} \)
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