Newspace parameters
Level: | \( N \) | \(=\) | \( 76 = 2^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 76.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.07085000914\) |
Analytic rank: | \(0\) |
Dimension: | \(14\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{12} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) 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 \( x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{12} - 2 \nu^{11} + \nu^{10} + 14 \nu^{9} - 42 \nu^{8} + 28 \nu^{7} + 132 \nu^{6} - 440 \nu^{5} + 528 \nu^{4} + 448 \nu^{3} - 640 \nu^{2} + 3584 \nu + 1024 ) / 1024 \) |
\(\beta_{2}\) | \(=\) | \( ( - \nu^{13} + 14 \nu^{12} - 9 \nu^{11} + 94 \nu^{10} - 286 \nu^{9} + 844 \nu^{8} - 980 \nu^{7} - 1880 \nu^{6} + 8592 \nu^{5} - 8832 \nu^{4} - 3968 \nu^{3} + 59392 \nu^{2} - 119808 \nu + 114688 ) / 20480 \) |
\(\beta_{3}\) | \(=\) | \( ( - \nu^{13} + 2 \nu^{12} - \nu^{11} - 14 \nu^{10} + 42 \nu^{9} - 28 \nu^{8} - 132 \nu^{7} + 440 \nu^{6} - 528 \nu^{5} - 448 \nu^{4} + 2688 \nu^{3} - 3584 \nu^{2} - 1024 \nu + 8192 ) / 4096 \) |
\(\beta_{4}\) | \(=\) | \( ( - \nu^{13} + 3 \nu^{11} - 16 \nu^{10} + 14 \nu^{9} + 56 \nu^{8} - 188 \nu^{7} + 176 \nu^{6} + 352 \nu^{5} - 1504 \nu^{4} + 1792 \nu^{3} + 1792 \nu^{2} + 4096 ) / 2048 \) |
\(\beta_{5}\) | \(=\) | \( ( - \nu^{13} + 3 \nu^{11} - 16 \nu^{10} + 14 \nu^{9} + 56 \nu^{8} - 188 \nu^{7} + 176 \nu^{6} + 352 \nu^{5} - 1504 \nu^{4} + 1792 \nu^{3} + 1792 \nu^{2} - 8192 \nu + 6144 ) / 2048 \) |
\(\beta_{6}\) | \(=\) | \( ( - 9 \nu^{13} + 6 \nu^{12} - \nu^{11} + 86 \nu^{10} - 254 \nu^{9} + 476 \nu^{8} - 20 \nu^{7} - 3320 \nu^{6} + 4688 \nu^{5} - 2048 \nu^{4} - 16512 \nu^{3} + 48128 \nu^{2} - 64512 \nu - 22528 ) / 10240 \) |
\(\beta_{7}\) | \(=\) | \( ( 21 \nu^{13} + 6 \nu^{12} - 91 \nu^{11} + 246 \nu^{10} - 354 \nu^{9} - 244 \nu^{8} + 2100 \nu^{7} - 1560 \nu^{6} - 112 \nu^{5} + 3392 \nu^{4} - 10112 \nu^{3} + 4608 \nu^{2} - 13312 \nu + 172032 ) / 20480 \) |
\(\beta_{8}\) | \(=\) | \( ( 9 \nu^{13} - 166 \nu^{12} - 159 \nu^{11} + 1354 \nu^{10} - 2146 \nu^{9} - 1116 \nu^{8} + 14740 \nu^{7} - 31240 \nu^{6} + 2992 \nu^{5} + 98048 \nu^{4} - 217728 \nu^{3} + \cdots - 1052672 ) / 20480 \) |
\(\beta_{9}\) | \(=\) | \( ( 19 \nu^{13} - 126 \nu^{12} - 189 \nu^{11} + 1074 \nu^{10} - 1006 \nu^{9} - 2556 \nu^{8} + 15020 \nu^{7} - 22280 \nu^{6} - 11408 \nu^{5} + 107968 \nu^{4} - 167808 \nu^{3} + \cdots - 991232 ) / 20480 \) |
\(\beta_{10}\) | \(=\) | \( ( 39 \nu^{13} + 114 \nu^{12} - 569 \nu^{11} + 674 \nu^{10} + 954 \nu^{9} - 7036 \nu^{8} + 12380 \nu^{7} + 1400 \nu^{6} - 49168 \nu^{5} + 99648 \nu^{4} - 57728 \nu^{3} - 163328 \nu^{2} + \cdots - 69632 ) / 20480 \) |
\(\beta_{11}\) | \(=\) | \( ( 11 \nu^{13} - 38 \nu^{12} + 27 \nu^{11} + 106 \nu^{10} - 318 \nu^{9} + 180 \nu^{8} + 1036 \nu^{7} - 3176 \nu^{6} + 2288 \nu^{5} + 4288 \nu^{4} - 11904 \nu^{3} + 2560 \nu^{2} + 46080 \nu - 77824 ) / 4096 \) |
\(\beta_{12}\) | \(=\) | \( ( - 3 \nu^{13} + 9 \nu^{12} - \nu^{11} - 39 \nu^{10} + 112 \nu^{9} - 82 \nu^{8} - 328 \nu^{7} + 1012 \nu^{6} - 952 \nu^{5} - 1168 \nu^{4} + 4672 \nu^{3} - 4736 \nu^{2} - 7680 \nu + 12288 ) / 1024 \) |
\(\beta_{13}\) | \(=\) | \( ( - 43 \nu^{13} - 18 \nu^{12} + 293 \nu^{11} - 578 \nu^{10} - 98 \nu^{9} + 3292 \nu^{8} - 8140 \nu^{7} + 3400 \nu^{6} + 22416 \nu^{5} - 56256 \nu^{4} + 50816 \nu^{3} + 78336 \nu^{2} + \cdots + 57344 ) / 10240 \) |
\(\nu\) | \(=\) | \( ( -\beta_{5} + \beta_{4} + 1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{5} - 2\beta_{3} + \beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{13} + \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta _1 - 5 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( \beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 3 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \beta _1 + 8 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 3 \beta_{13} + 4 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + \beta_{9} + \beta_{8} + 3 \beta_{7} - 7 \beta_{6} - 5 \beta_{5} + 6 \beta_{4} - 4 \beta_{3} + 11 \beta_{2} - 7 \beta _1 + 18 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( \beta_{13} + 2 \beta_{12} - 5 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + 7 \beta_{8} + 9 \beta_{7} - 17 \beta_{6} - 5 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} + \beta _1 - 20 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( 7 \beta_{13} + 4 \beta_{12} - \beta_{11} - \beta_{10} + 17 \beta_{9} - 11 \beta_{8} + 39 \beta_{7} + 9 \beta_{6} + 15 \beta_{5} + 2 \beta_{4} - 40 \beta_{3} - 25 \beta_{2} + 9 \beta _1 + 2 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( - 23 \beta_{13} - 54 \beta_{12} - 53 \beta_{11} - 35 \beta_{10} + 29 \beta_{9} - 33 \beta_{8} - 23 \beta_{7} + 39 \beta_{6} - \beta_{5} - 34 \beta_{4} + 82 \beta_{3} - 53 \beta_{2} + 29 \beta _1 - 188 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( - 69 \beta_{13} + 52 \beta_{12} + 19 \beta_{11} - 61 \beta_{10} + 29 \beta_{9} - 15 \beta_{8} - 37 \beta_{7} + 5 \beta_{6} + 159 \beta_{5} - 34 \beta_{4} - 56 \beta_{3} + 59 \beta_{2} - 11 \beta _1 + 254 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( - 75 \beta_{13} + 210 \beta_{12} + 127 \beta_{11} - 103 \beta_{10} - 55 \beta_{9} + 115 \beta_{8} + 149 \beta_{7} - 101 \beta_{6} - 133 \beta_{5} + 38 \beta_{4} + 490 \beta_{3} + 255 \beta_{2} - 31 \beta _1 - 316 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( 23 \beta_{13} + 340 \beta_{12} + 223 \beta_{11} - 97 \beta_{10} + 33 \beta_{9} + 5 \beta_{8} + 311 \beta_{7} + 57 \beta_{6} - 189 \beta_{5} - 130 \beta_{4} - 888 \beta_{3} - 89 \beta_{2} + 281 \beta _1 + 1886 ) / 2 \) |
\(\nu^{12}\) | \(=\) | \( ( 137 \beta_{13} - 86 \beta_{12} - 85 \beta_{11} + 61 \beta_{10} + 317 \beta_{9} - 481 \beta_{8} + 201 \beta_{7} + 775 \beta_{6} - 1513 \beta_{5} + 414 \beta_{4} - 1118 \beta_{3} - 373 \beta_{2} + 213 \beta _1 - 2684 ) / 2 \) |
\(\nu^{13}\) | \(=\) | \( ( - 781 \beta_{13} - 620 \beta_{12} - 405 \beta_{11} - 453 \beta_{10} - 75 \beta_{9} - 519 \beta_{8} - 1645 \beta_{7} + 189 \beta_{6} + 1567 \beta_{5} - 1290 \beta_{4} - 4648 \beta_{3} - 109 \beta_{2} + 909 \beta _1 - 6618 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).
\(n\) | \(21\) | \(39\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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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39.1 |
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−1.92254 | − | 0.551226i | 0.644704i | 3.39230 | + | 2.11950i | −2.32715 | 0.355377 | − | 1.23947i | 8.62924i | −5.35350 | − | 5.94475i | 8.58436 | 4.47404 | + | 1.28279i | ||||||||||||||||||||||||||||||||||||||||||||||||||
39.2 | −1.92254 | + | 0.551226i | − | 0.644704i | 3.39230 | − | 2.11950i | −2.32715 | 0.355377 | + | 1.23947i | − | 8.62924i | −5.35350 | + | 5.94475i | 8.58436 | 4.47404 | − | 1.28279i | |||||||||||||||||||||||||||||||||||||||||||||||||
39.3 | −1.89728 | − | 0.632718i | − | 5.34370i | 3.19934 | + | 2.40089i | 5.79268 | −3.38106 | + | 10.1385i | − | 5.87536i | −4.55095 | − | 6.57943i | −19.5551 | −10.9903 | − | 3.66514i | |||||||||||||||||||||||||||||||||||||||||||||||||
39.4 | −1.89728 | + | 0.632718i | 5.34370i | 3.19934 | − | 2.40089i | 5.79268 | −3.38106 | − | 10.1385i | 5.87536i | −4.55095 | + | 6.57943i | −19.5551 | −10.9903 | + | 3.66514i | |||||||||||||||||||||||||||||||||||||||||||||||||||
39.5 | −0.0607713 | − | 1.99908i | 5.37609i | −3.99261 | + | 0.242973i | −5.82257 | 10.7472 | − | 0.326712i | 5.45132i | 0.728358 | + | 7.96677i | −19.9023 | 0.353845 | + | 11.6398i | |||||||||||||||||||||||||||||||||||||||||||||||||||
39.6 | −0.0607713 | + | 1.99908i | − | 5.37609i | −3.99261 | − | 0.242973i | −5.82257 | 10.7472 | + | 0.326712i | − | 5.45132i | 0.728358 | − | 7.96677i | −19.9023 | 0.353845 | − | 11.6398i | |||||||||||||||||||||||||||||||||||||||||||||||||
39.7 | 0.645572 | − | 1.89294i | − | 0.820457i | −3.16647 | − | 2.44406i | −2.38184 | −1.55308 | − | 0.529664i | − | 12.3764i | −6.67066 | + | 4.41614i | 8.32685 | −1.53765 | + | 4.50868i | |||||||||||||||||||||||||||||||||||||||||||||||||
39.8 | 0.645572 | + | 1.89294i | 0.820457i | −3.16647 | + | 2.44406i | −2.38184 | −1.55308 | + | 0.529664i | 12.3764i | −6.67066 | − | 4.41614i | 8.32685 | −1.53765 | − | 4.50868i | |||||||||||||||||||||||||||||||||||||||||||||||||||
39.9 | 0.711746 | − | 1.86907i | − | 4.44946i | −2.98683 | − | 2.66060i | 4.97973 | −8.31634 | − | 3.16688i | 12.2628i | −7.09872 | + | 3.68892i | −10.7977 | 3.54430 | − | 9.30745i | ||||||||||||||||||||||||||||||||||||||||||||||||||
39.10 | 0.711746 | + | 1.86907i | 4.44946i | −2.98683 | + | 2.66060i | 4.97973 | −8.31634 | + | 3.16688i | − | 12.2628i | −7.09872 | − | 3.68892i | −10.7977 | 3.54430 | + | 9.30745i | ||||||||||||||||||||||||||||||||||||||||||||||||||
39.11 | 1.57398 | − | 1.23393i | 2.90118i | 0.954817 | − | 3.88437i | 3.66290 | 3.57986 | + | 4.56639i | 1.93414i | −3.29019 | − | 7.29209i | 0.583162 | 5.76533 | − | 4.51977i | |||||||||||||||||||||||||||||||||||||||||||||||||||
39.12 | 1.57398 | + | 1.23393i | − | 2.90118i | 0.954817 | + | 3.88437i | 3.66290 | 3.57986 | − | 4.56639i | − | 1.93414i | −3.29019 | + | 7.29209i | 0.583162 | 5.76533 | + | 4.51977i | |||||||||||||||||||||||||||||||||||||||||||||||||
39.13 | 1.94929 | − | 0.447510i | − | 3.19988i | 3.59947 | − | 1.74465i | −3.90374 | −1.43198 | − | 6.23749i | 2.64664i | 6.23566 | − | 5.01164i | −1.23921 | −7.60953 | + | 1.74696i | ||||||||||||||||||||||||||||||||||||||||||||||||||
39.14 | 1.94929 | + | 0.447510i | 3.19988i | 3.59947 | + | 1.74465i | −3.90374 | −1.43198 | + | 6.23749i | − | 2.64664i | 6.23566 | + | 5.01164i | −1.23921 | −7.60953 | − | 1.74696i | ||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 76.3.b.b | ✓ | 14 |
3.b | odd | 2 | 1 | 684.3.g.b | 14 | ||
4.b | odd | 2 | 1 | inner | 76.3.b.b | ✓ | 14 |
8.b | even | 2 | 1 | 1216.3.d.d | 14 | ||
8.d | odd | 2 | 1 | 1216.3.d.d | 14 | ||
12.b | even | 2 | 1 | 684.3.g.b | 14 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
76.3.b.b | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
76.3.b.b | ✓ | 14 | 4.b | odd | 2 | 1 | inner |
684.3.g.b | 14 | 3.b | odd | 2 | 1 | ||
684.3.g.b | 14 | 12.b | even | 2 | 1 | ||
1216.3.d.d | 14 | 8.b | even | 2 | 1 | ||
1216.3.d.d | 14 | 8.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{14} + 97T_{3}^{12} + 3595T_{3}^{10} + 63443T_{3}^{8} + 539872T_{3}^{6} + 1940896T_{3}^{4} + 1665792T_{3}^{2} + 393984 \)
acting on \(S_{3}^{\mathrm{new}}(76, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{14} - 2 T^{13} + T^{12} + 14 T^{11} + \cdots + 16384 \)
$3$
\( T^{14} + 97 T^{12} + 3595 T^{10} + \cdots + 393984 \)
$5$
\( (T^{7} - 66 T^{5} - 28 T^{4} + 1337 T^{3} + \cdots - 13312)^{2} \)
$7$
\( T^{14} + 453 T^{12} + \cdots + 46106276299 \)
$11$
\( T^{14} + 880 T^{12} + \cdots + 94488337600 \)
$13$
\( (T^{7} - 27 T^{6} - 293 T^{5} + \cdots - 1265920)^{2} \)
$17$
\( (T^{7} - 17 T^{6} - 1107 T^{5} + \cdots - 69101055)^{2} \)
$19$
\( (T^{2} + 19)^{7} \)
$23$
\( T^{14} + \cdots + 683970816311296 \)
$29$
\( (T^{7} - 27 T^{6} - 2901 T^{5} + \cdots - 112127472)^{2} \)
$31$
\( T^{14} + 4048 T^{12} + \cdots + 11\!\cdots\!00 \)
$37$
\( (T^{7} - 50 T^{6} - 1508 T^{5} + \cdots + 914894720)^{2} \)
$41$
\( (T^{7} - 112 T^{6} + \cdots + 18882293760)^{2} \)
$43$
\( T^{14} + 13244 T^{12} + \cdots + 58\!\cdots\!24 \)
$47$
\( T^{14} + 8204 T^{12} + \cdots + 64\!\cdots\!04 \)
$53$
\( (T^{7} - 7 T^{6} - 6645 T^{5} + \cdots + 18228603200)^{2} \)
$59$
\( T^{14} + 30513 T^{12} + \cdots + 50\!\cdots\!00 \)
$61$
\( (T^{7} - 14 T^{6} + \cdots + 522558358600)^{2} \)
$67$
\( T^{14} + 27889 T^{12} + \cdots + 29\!\cdots\!76 \)
$71$
\( T^{14} + 37812 T^{12} + \cdots + 49\!\cdots\!00 \)
$73$
\( (T^{7} - 35 T^{6} - 5907 T^{5} + \cdots - 77971926925)^{2} \)
$79$
\( T^{14} + 38740 T^{12} + \cdots + 18\!\cdots\!00 \)
$83$
\( T^{14} + 62788 T^{12} + \cdots + 34\!\cdots\!76 \)
$89$
\( (T^{7} - 23640 T^{5} + \cdots - 88310345728)^{2} \)
$97$
\( (T^{7} - 154 T^{6} + \cdots + 4410892984320)^{2} \)
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