Newspace parameters
Level: | \( N \) | \(=\) | \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 360.k (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(21.2406876021\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
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: | \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{14}\cdot 5^{4} \) |
Twist minimal: | no (minimal twist has level 40) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) :
\(\beta_{1}\) | \(=\) | \( ( 2 \nu^{11} - 5 \nu^{10} + 10 \nu^{9} - 27 \nu^{8} + 30 \nu^{7} + 55 \nu^{6} - 64 \nu^{5} - 132 \nu^{4} + 384 \nu^{3} - 1168 \nu^{2} + 3008 \nu - 5888 ) / 960 \) |
\(\beta_{2}\) | \(=\) | \( ( 35 \nu^{11} + 72 \nu^{10} - 203 \nu^{9} + 168 \nu^{8} + 223 \nu^{7} + 484 \nu^{6} + 644 \nu^{5} - 6288 \nu^{4} + 944 \nu^{3} + 24128 \nu^{2} - 22528 \nu - 16384 ) / 7680 \) |
\(\beta_{3}\) | \(=\) | \( ( - 93 \nu^{11} + 204 \nu^{10} - 379 \nu^{9} + 388 \nu^{8} - 817 \nu^{7} - 1672 \nu^{6} + 4708 \nu^{5} + 4640 \nu^{4} - 20048 \nu^{3} + 34432 \nu^{2} - 96256 \nu + 191488 ) / 15360 \) |
\(\beta_{4}\) | \(=\) | \( ( - 3 \nu^{11} + 4 \nu^{10} + 3 \nu^{9} + 12 \nu^{8} - 55 \nu^{7} - 72 \nu^{6} + 132 \nu^{5} + 288 \nu^{4} - 368 \nu^{3} - 384 \nu^{2} - 2304 \nu + 7712 ) / 480 \) |
\(\beta_{5}\) | \(=\) | \( ( - 107 \nu^{11} + 204 \nu^{10} - 29 \nu^{9} + 292 \nu^{8} - 407 \nu^{7} - 1952 \nu^{6} + 5756 \nu^{5} + 15104 \nu^{4} - 14896 \nu^{3} - 1792 \nu^{2} - 33792 \nu + 134144 ) / 15360 \) |
\(\beta_{6}\) | \(=\) | \( ( 15 \nu^{11} - 4 \nu^{10} - 35 \nu^{9} - 28 \nu^{8} + 263 \nu^{7} - 248 \nu^{6} + 200 \nu^{5} - 1088 \nu^{4} - 128 \nu^{3} + 7232 \nu^{2} - 2240 \nu - 23424 ) / 1920 \) |
\(\beta_{7}\) | \(=\) | \( ( 23 \nu^{11} - 60 \nu^{10} + 81 \nu^{9} - 116 \nu^{8} + 51 \nu^{7} + 480 \nu^{6} - 684 \nu^{5} - 1024 \nu^{4} + 5616 \nu^{3} - 11520 \nu^{2} + 22528 \nu - 30720 ) / 3072 \) |
\(\beta_{8}\) | \(=\) | \( ( 171 \nu^{11} - 476 \nu^{10} + 925 \nu^{9} - 2068 \nu^{8} + 1047 \nu^{7} + 4048 \nu^{6} - 6652 \nu^{5} - 16448 \nu^{4} + 33840 \nu^{3} - 103936 \nu^{2} + 354304 \nu - 404480 ) / 15360 \) |
\(\beta_{9}\) | \(=\) | \( ( 103 \nu^{11} - 304 \nu^{10} + 49 \nu^{9} + 352 \nu^{8} + 467 \nu^{7} + 412 \nu^{6} - 4108 \nu^{5} - 13360 \nu^{4} + 49648 \nu^{3} - 12352 \nu^{2} - 13824 \nu - 96768 ) / 7680 \) |
\(\beta_{10}\) | \(=\) | \( ( - 59 \nu^{11} + 134 \nu^{10} - 189 \nu^{9} + 314 \nu^{8} - 407 \nu^{7} - 1182 \nu^{6} + 2580 \nu^{5} + 4216 \nu^{4} - 11920 \nu^{3} + 39456 \nu^{2} - 73600 \nu + 109312 ) / 3840 \) |
\(\beta_{11}\) | \(=\) | \( ( 173 \nu^{11} - 352 \nu^{10} + 459 \nu^{9} - 1424 \nu^{8} + 1633 \nu^{7} + 1956 \nu^{6} - 9108 \nu^{5} - 9616 \nu^{4} + 42512 \nu^{3} - 55488 \nu^{2} + 207616 \nu - 421376 ) / 7680 \) |
\(\nu\) | \(=\) | \( ( \beta_{8} + \beta_{5} - \beta_{4} - 5\beta _1 + 3 ) / 10 \) |
\(\nu^{2}\) | \(=\) | \( ( 2\beta_{10} + \beta_{8} + \beta_{7} - \beta_{4} - 2\beta_{3} - 3\beta _1 + 2 ) / 10 \) |
\(\nu^{3}\) | \(=\) | \( ( 3\beta_{11} + 3\beta_{10} + 3\beta_{9} - 4\beta_{6} + 8\beta_{5} + 2\beta_{3} + 5\beta_{2} + 7\beta _1 + 27 ) / 20 \) |
\(\nu^{4}\) | \(=\) | \( ( \beta_{11} + \beta_{10} - \beta_{9} - 2\beta_{8} + 6\beta_{7} + 2\beta_{5} - 2\beta_{4} - \beta_{2} - 5\beta _1 + 3 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( - 7 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 8 \beta_{8} + 18 \beta_{7} + 6 \beta_{6} + 16 \beta_{5} - 3 \beta_{4} + 12 \beta_{3} + 5 \beta_{2} + 17 \beta _1 - 94 ) / 10 \) |
\(\nu^{6}\) | \(=\) | \( ( - 17 \beta_{11} + 23 \beta_{10} - 7 \beta_{9} - 16 \beta_{8} + 72 \beta_{7} - 44 \beta_{6} + 2 \beta_{5} - 64 \beta_{4} - 8 \beta_{3} + 25 \beta_{2} + 127 \beta _1 + 29 ) / 20 \) |
\(\nu^{7}\) | \(=\) | \( ( - 33 \beta_{11} - 37 \beta_{10} + 27 \beta_{9} - 24 \beta_{8} - 164 \beta_{7} + 24 \beta_{6} + 80 \beta_{5} - 86 \beta_{4} - 98 \beta_{3} - 35 \beta_{2} + 339 \beta _1 + 1513 ) / 20 \) |
\(\nu^{8}\) | \(=\) | \( ( - 11 \beta_{11} - 7 \beta_{10} + 3 \beta_{9} + 9 \beta_{8} - 7 \beta_{7} + 2 \beta_{6} + 16 \beta_{5} - 15 \beta_{4} - 40 \beta_{3} - \beta_{2} - 88 \beta _1 - 117 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( - 139 \beta_{11} + 237 \beta_{10} + 21 \beta_{9} + 96 \beta_{8} - 432 \beta_{7} + 12 \beta_{6} - 56 \beta_{5} + 544 \beta_{4} - 1442 \beta_{3} + 155 \beta_{2} + 1025 \beta _1 + 589 ) / 20 \) |
\(\nu^{10}\) | \(=\) | \( ( 387 \beta_{11} - 109 \beta_{10} - 123 \beta_{9} - 270 \beta_{8} - 1046 \beta_{7} - 416 \beta_{6} + 510 \beta_{5} + 970 \beta_{4} - 1096 \beta_{3} + 1325 \beta_{2} + 1097 \beta _1 + 3321 ) / 20 \) |
\(\nu^{11}\) | \(=\) | \( ( 223 \beta_{11} - 99 \beta_{10} - 467 \beta_{9} - 564 \beta_{8} + 1690 \beta_{7} + 586 \beta_{6} - 200 \beta_{5} + 859 \beta_{4} - 596 \beta_{3} + 635 \beta_{2} - 1805 \beta _1 + 4026 ) / 10 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).
\(n\) | \(181\) | \(217\) | \(271\) | \(281\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
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181.1 |
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−2.74276 | − | 0.690860i | 0 | 7.04543 | + | 3.78972i | 5.00000i | 0 | −14.6308 | −16.7057 | − | 15.2617i | 0 | 3.45430 | − | 13.7138i | ||||||||||||||||||||||||||||||||||||||||||||||
181.2 | −2.74276 | + | 0.690860i | 0 | 7.04543 | − | 3.78972i | − | 5.00000i | 0 | −14.6308 | −16.7057 | + | 15.2617i | 0 | 3.45430 | + | 13.7138i | ||||||||||||||||||||||||||||||||||||||||||||||
181.3 | −2.20360 | − | 1.77318i | 0 | 1.71169 | + | 7.81474i | 5.00000i | 0 | 21.5703 | 10.0850 | − | 20.2557i | 0 | 8.86588 | − | 11.0180i | |||||||||||||||||||||||||||||||||||||||||||||||
181.4 | −2.20360 | + | 1.77318i | 0 | 1.71169 | − | 7.81474i | − | 5.00000i | 0 | 21.5703 | 10.0850 | + | 20.2557i | 0 | 8.86588 | + | 11.0180i | ||||||||||||||||||||||||||||||||||||||||||||||
181.5 | −1.52528 | − | 2.38191i | 0 | −3.34703 | + | 7.26618i | − | 5.00000i | 0 | 5.13620 | 22.4126 | − | 3.11063i | 0 | −11.9096 | + | 7.62641i | ||||||||||||||||||||||||||||||||||||||||||||||
181.6 | −1.52528 | + | 2.38191i | 0 | −3.34703 | − | 7.26618i | 5.00000i | 0 | 5.13620 | 22.4126 | + | 3.11063i | 0 | −11.9096 | − | 7.62641i | |||||||||||||||||||||||||||||||||||||||||||||||
181.7 | 0.337480 | − | 2.80822i | 0 | −7.77221 | − | 1.89544i | 5.00000i | 0 | 9.93501 | −7.94578 | + | 21.1864i | 0 | 14.0411 | + | 1.68740i | |||||||||||||||||||||||||||||||||||||||||||||||
181.8 | 0.337480 | + | 2.80822i | 0 | −7.77221 | + | 1.89544i | − | 5.00000i | 0 | 9.93501 | −7.94578 | − | 21.1864i | 0 | 14.0411 | − | 1.68740i | ||||||||||||||||||||||||||||||||||||||||||||||
181.9 | 2.54175 | − | 1.24077i | 0 | 4.92097 | − | 6.30746i | 5.00000i | 0 | 26.6173 | 4.68175 | − | 22.1378i | 0 | 6.20386 | + | 12.7087i | |||||||||||||||||||||||||||||||||||||||||||||||
181.10 | 2.54175 | + | 1.24077i | 0 | 4.92097 | + | 6.30746i | − | 5.00000i | 0 | 26.6173 | 4.68175 | + | 22.1378i | 0 | 6.20386 | − | 12.7087i | ||||||||||||||||||||||||||||||||||||||||||||||
181.11 | 2.59241 | − | 1.13111i | 0 | 5.44116 | − | 5.86462i | − | 5.00000i | 0 | −34.6280 | 7.47214 | − | 21.3581i | 0 | −5.65557 | − | 12.9620i | ||||||||||||||||||||||||||||||||||||||||||||||
181.12 | 2.59241 | + | 1.13111i | 0 | 5.44116 | + | 5.86462i | 5.00000i | 0 | −34.6280 | 7.47214 | + | 21.3581i | 0 | −5.65557 | + | 12.9620i | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 360.4.k.c | 12 | |
3.b | odd | 2 | 1 | 40.4.d.a | ✓ | 12 | |
4.b | odd | 2 | 1 | 1440.4.k.c | 12 | ||
8.b | even | 2 | 1 | inner | 360.4.k.c | 12 | |
8.d | odd | 2 | 1 | 1440.4.k.c | 12 | ||
12.b | even | 2 | 1 | 160.4.d.a | 12 | ||
15.d | odd | 2 | 1 | 200.4.d.b | 12 | ||
15.e | even | 4 | 1 | 200.4.f.b | 12 | ||
15.e | even | 4 | 1 | 200.4.f.c | 12 | ||
24.f | even | 2 | 1 | 160.4.d.a | 12 | ||
24.h | odd | 2 | 1 | 40.4.d.a | ✓ | 12 | |
48.i | odd | 4 | 1 | 1280.4.a.bb | 6 | ||
48.i | odd | 4 | 1 | 1280.4.a.bc | 6 | ||
48.k | even | 4 | 1 | 1280.4.a.ba | 6 | ||
48.k | even | 4 | 1 | 1280.4.a.bd | 6 | ||
60.h | even | 2 | 1 | 800.4.d.d | 12 | ||
60.l | odd | 4 | 1 | 800.4.f.b | 12 | ||
60.l | odd | 4 | 1 | 800.4.f.c | 12 | ||
120.i | odd | 2 | 1 | 200.4.d.b | 12 | ||
120.m | even | 2 | 1 | 800.4.d.d | 12 | ||
120.q | odd | 4 | 1 | 800.4.f.b | 12 | ||
120.q | odd | 4 | 1 | 800.4.f.c | 12 | ||
120.w | even | 4 | 1 | 200.4.f.b | 12 | ||
120.w | even | 4 | 1 | 200.4.f.c | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
40.4.d.a | ✓ | 12 | 3.b | odd | 2 | 1 | |
40.4.d.a | ✓ | 12 | 24.h | odd | 2 | 1 | |
160.4.d.a | 12 | 12.b | even | 2 | 1 | ||
160.4.d.a | 12 | 24.f | even | 2 | 1 | ||
200.4.d.b | 12 | 15.d | odd | 2 | 1 | ||
200.4.d.b | 12 | 120.i | odd | 2 | 1 | ||
200.4.f.b | 12 | 15.e | even | 4 | 1 | ||
200.4.f.b | 12 | 120.w | even | 4 | 1 | ||
200.4.f.c | 12 | 15.e | even | 4 | 1 | ||
200.4.f.c | 12 | 120.w | even | 4 | 1 | ||
360.4.k.c | 12 | 1.a | even | 1 | 1 | trivial | |
360.4.k.c | 12 | 8.b | even | 2 | 1 | inner | |
800.4.d.d | 12 | 60.h | even | 2 | 1 | ||
800.4.d.d | 12 | 120.m | even | 2 | 1 | ||
800.4.f.b | 12 | 60.l | odd | 4 | 1 | ||
800.4.f.b | 12 | 120.q | odd | 4 | 1 | ||
800.4.f.c | 12 | 60.l | odd | 4 | 1 | ||
800.4.f.c | 12 | 120.q | odd | 4 | 1 | ||
1280.4.a.ba | 6 | 48.k | even | 4 | 1 | ||
1280.4.a.bb | 6 | 48.i | odd | 4 | 1 | ||
1280.4.a.bc | 6 | 48.i | odd | 4 | 1 | ||
1280.4.a.bd | 6 | 48.k | even | 4 | 1 | ||
1440.4.k.c | 12 | 4.b | odd | 2 | 1 | ||
1440.4.k.c | 12 | 8.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} - 14T_{7}^{5} - 1258T_{7}^{4} + 23408T_{7}^{3} + 166612T_{7}^{2} - 4186552T_{7} + 14843128 \)
acting on \(S_{4}^{\mathrm{new}}(360, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} + 2 T^{11} - 6 T^{10} + \cdots + 262144 \)
$3$
\( T^{12} \)
$5$
\( (T^{2} + 25)^{6} \)
$7$
\( (T^{6} - 14 T^{5} - 1258 T^{4} + \cdots + 14843128)^{2} \)
$11$
\( T^{12} + 10072 T^{10} + \cdots + 26\!\cdots\!00 \)
$13$
\( T^{12} + 11144 T^{10} + \cdots + 24\!\cdots\!00 \)
$17$
\( (T^{6} - 14500 T^{4} + \cdots - 7473839808)^{2} \)
$19$
\( T^{12} + 46312 T^{10} + \cdots + 17\!\cdots\!24 \)
$23$
\( (T^{6} + 302 T^{5} + 12942 T^{4} + \cdots + 881168216)^{2} \)
$29$
\( T^{12} + 92448 T^{10} + \cdots + 11\!\cdots\!00 \)
$31$
\( (T^{6} + 132 T^{5} + \cdots - 1437816300032)^{2} \)
$37$
\( T^{12} + 310360 T^{10} + \cdots + 21\!\cdots\!04 \)
$41$
\( (T^{6} + 20 T^{5} + \cdots - 71667547865600)^{2} \)
$43$
\( T^{12} + 338808 T^{10} + \cdots + 25\!\cdots\!04 \)
$47$
\( (T^{6} - 470 T^{5} + \cdots - 72048375466472)^{2} \)
$53$
\( T^{12} + 939912 T^{10} + \cdots + 86\!\cdots\!76 \)
$59$
\( T^{12} + 1889384 T^{10} + \cdots + 10\!\cdots\!24 \)
$61$
\( T^{12} + 627448 T^{10} + \cdots + 61\!\cdots\!00 \)
$67$
\( T^{12} + 1320888 T^{10} + \cdots + 87\!\cdots\!36 \)
$71$
\( (T^{6} - 796 T^{5} + \cdots - 36\!\cdots\!48)^{2} \)
$73$
\( (T^{6} - 216 T^{5} + \cdots - 378730163491776)^{2} \)
$79$
\( (T^{6} - 1008 T^{5} + \cdots - 44\!\cdots\!00)^{2} \)
$83$
\( T^{12} + 4256648 T^{10} + \cdots + 78\!\cdots\!24 \)
$89$
\( (T^{6} - 212 T^{5} + \cdots - 62\!\cdots\!00)^{2} \)
$97$
\( (T^{6} + 792 T^{5} + \cdots - 33\!\cdots\!28)^{2} \)
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