Properties

Label 8046.2.a.m
Level 8046
Weight 2
Character orbit 8046.a
Self dual Yes
Analytic conductor 64.248
Analytic rank 1
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8046.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2476334663\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} -\beta_{1} q^{5} -\beta_{2} q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} -\beta_{1} q^{5} -\beta_{2} q^{7} + q^{8} -\beta_{1} q^{10} + ( \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{11} + ( -1 - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{13} -\beta_{2} q^{14} + q^{16} + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{17} + ( -2 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{19} -\beta_{1} q^{20} + ( \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{22} + ( -2 + \beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{23} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{25} + ( -1 - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{26} -\beta_{2} q^{28} + ( -3 - 2 \beta_{1} - 2 \beta_{4} - \beta_{6} - \beta_{10} - \beta_{11} ) q^{29} + ( -1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{31} + q^{32} + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{34} + ( -3 + 2 \beta_{1} + \beta_{3} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{35} + ( -2 + \beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{37} + ( -2 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{38} -\beta_{1} q^{40} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{41} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{43} + ( \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{44} + ( -2 + \beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{46} + ( \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{47} + ( -\beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{49} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{50} + ( -1 - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{52} + ( -2 - \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{53} + ( -3 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{55} -\beta_{2} q^{56} + ( -3 - 2 \beta_{1} - 2 \beta_{4} - \beta_{6} - \beta_{10} - \beta_{11} ) q^{58} + ( \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{59} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{61} + ( -1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{62} + q^{64} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{65} + ( -3 + \beta_{3} - \beta_{5} - \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{67} + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{68} + ( -3 + 2 \beta_{1} + \beta_{3} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{70} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{71} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{73} + ( -2 + \beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{74} + ( -2 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{76} + ( -1 + \beta_{2} - \beta_{4} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{77} + ( -\beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{79} -\beta_{1} q^{80} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{82} + ( -\beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{83} + ( -2 - 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{85} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{86} + ( \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{88} + ( -2 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} - \beta_{8} - 3 \beta_{11} ) q^{89} + ( 1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} ) q^{91} + ( -2 + \beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{92} + ( \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{94} + ( 4 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{95} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{97} + ( -\beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 12q^{4} - 5q^{5} - 6q^{7} + 12q^{8} + O(q^{10}) \) \( 12q + 12q^{2} + 12q^{4} - 5q^{5} - 6q^{7} + 12q^{8} - 5q^{10} - 10q^{11} - q^{13} - 6q^{14} + 12q^{16} - 6q^{17} - 10q^{19} - 5q^{20} - 10q^{22} - 15q^{23} + 7q^{25} - q^{26} - 6q^{28} - 33q^{29} - 6q^{31} + 12q^{32} - 6q^{34} - 16q^{35} - 13q^{37} - 10q^{38} - 5q^{40} - 20q^{41} - 11q^{43} - 10q^{44} - 15q^{46} - 15q^{47} + 2q^{49} + 7q^{50} - q^{52} - 4q^{53} - 17q^{55} - 6q^{56} - 33q^{58} - 10q^{59} - 12q^{61} - 6q^{62} + 12q^{64} - 40q^{65} - 19q^{67} - 6q^{68} - 16q^{70} - 47q^{71} - 2q^{73} - 13q^{74} - 10q^{76} + 6q^{77} - 15q^{79} - 5q^{80} - 20q^{82} - 18q^{83} - 25q^{85} - 11q^{86} - 10q^{88} - 24q^{89} - 3q^{91} - 15q^{92} - 15q^{94} + 3q^{95} - 25q^{97} + 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{11} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} - 2249 x^{3} + 761 x^{2} + 910 x - 375\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-32197 \nu^{11} + 5673372 \nu^{10} - 27408721 \nu^{9} - 109537587 \nu^{8} + 609086107 \nu^{7} + 410288818 \nu^{6} - 3495027303 \nu^{5} + 447932378 \nu^{4} + 6463643167 \nu^{3} - 2256579470 \nu^{2} - 3729321842 \nu + 1592878652\)\()/5855999\)
\(\beta_{3}\)\(=\)\((\)\(-337779 \nu^{11} - 12051765 \nu^{10} + 77892894 \nu^{9} + 222792211 \nu^{8} - 1586491494 \nu^{7} - 664646379 \nu^{6} + 8970420017 \nu^{5} - 2097844062 \nu^{4} - 16685502287 \nu^{3} + 6698494966 \nu^{2} + 9660687071 \nu - 4301878920\)\()/29279995\)
\(\beta_{4}\)\(=\)\((\)\(-4626967 \nu^{11} + 27982820 \nu^{10} + 68317507 \nu^{9} - 610246892 \nu^{8} + 137663433 \nu^{7} + 3479105328 \nu^{6} - 3268300794 \nu^{5} - 6124911496 \nu^{4} + 8040696879 \nu^{3} + 2382541943 \nu^{2} - 5435421687 \nu + 1309434950\)\()/29279995\)
\(\beta_{5}\)\(=\)\((\)\(-5834388 \nu^{11} + 38266875 \nu^{10} + 69519048 \nu^{9} - 818338273 \nu^{8} + 526825367 \nu^{7} + 4419720832 \nu^{6} - 6005098721 \nu^{5} - 6551037399 \nu^{4} + 13075477756 \nu^{3} + 594551152 \nu^{2} - 8170221548 \nu + 2800717475\)\()/29279995\)
\(\beta_{6}\)\(=\)\((\)\(-11071809 \nu^{11} + 33647510 \nu^{10} + 329349654 \nu^{9} - 795142464 \nu^{8} - 3330885579 \nu^{7} + 5470397091 \nu^{6} + 13696849342 \nu^{5} - 15262968602 \nu^{4} - 22956311877 \nu^{3} + 17416329301 \nu^{2} + 12966000711 \nu - 6906022130\)\()/29279995\)
\(\beta_{7}\)\(=\)\((\)\(-15600107 \nu^{11} + 74241930 \nu^{10} + 327537312 \nu^{9} - 1640249962 \nu^{8} - 1697104197 \nu^{7} + 9679352383 \nu^{6} + 1916697621 \nu^{5} - 19399390581 \nu^{4} + 804575664 \nu^{3} + 13154737748 \nu^{2} - 1591143977 \nu - 1289750025\)\()/29279995\)
\(\beta_{8}\)\(=\)\((\)\(15915726 \nu^{11} - 100688050 \nu^{10} - 207068641 \nu^{9} + 2153825106 \nu^{8} - 1047579084 \nu^{7} - 11652307114 \nu^{6} + 14014610012 \nu^{5} + 17633361118 \nu^{4} - 30654436132 \nu^{3} - 3005054784 \nu^{2} + 18959758726 \nu - 6112777815\)\()/29279995\)
\(\beta_{9}\)\(=\)\((\)\(-3317302 \nu^{11} + 16732001 \nu^{10} + 64707528 \nu^{9} - 366093260 \nu^{8} - 254489192 \nu^{7} + 2107854700 \nu^{6} - 179845605 \nu^{5} - 3967835268 \nu^{4} + 1171309316 \nu^{3} + 2366997820 \nu^{2} - 842422122 \nu - 38973036\)\()/5855999\)
\(\beta_{10}\)\(=\)\((\)\(17078923 \nu^{11} - 69687500 \nu^{10} - 418005468 \nu^{9} + 1573561298 \nu^{8} + 3159684708 \nu^{7} - 9796137762 \nu^{6} - 9615020874 \nu^{5} + 22410652829 \nu^{4} + 13297311074 \nu^{3} - 19695978447 \nu^{2} - 6697290927 \nu + 5205009495\)\()/29279995\)
\(\beta_{11}\)\(=\)\((\)\(5445786 \nu^{11} - 29080114 \nu^{10} - 98087514 \nu^{9} + 632612308 \nu^{8} + 238661630 \nu^{7} - 3586733161 \nu^{6} + 1341737333 \nu^{5} + 6428622786 \nu^{4} - 3951467976 \nu^{3} - 3278532055 \nu^{2} + 2627858133 \nu - 395658693\)\()/5855999\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{3} + \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + 3 \beta_{10} - \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 14 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{11} - 13 \beta_{10} - 11 \beta_{9} + 19 \beta_{8} + 21 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} - 16 \beta_{3} + 17 \beta_{2} + 25 \beta_{1} + 60\)
\(\nu^{5}\)\(=\)\(-15 \beta_{11} + 51 \beta_{10} - 23 \beta_{9} + 44 \beta_{8} + 79 \beta_{7} + 27 \beta_{6} - 16 \beta_{5} + 23 \beta_{4} - 13 \beta_{3} + 27 \beta_{2} + 222 \beta_{1} + 49\)
\(\nu^{6}\)\(=\)\(34 \beta_{11} - 162 \beta_{10} - 138 \beta_{9} + 341 \beta_{8} + 378 \beta_{7} - 11 \beta_{6} - 24 \beta_{5} + 61 \beta_{4} - 258 \beta_{3} + 280 \beta_{2} + 506 \beta_{1} + 923\)
\(\nu^{7}\)\(=\)\(-223 \beta_{11} + 787 \beta_{10} - 406 \beta_{9} + 914 \beta_{8} + 1435 \beta_{7} + 498 \beta_{6} - 222 \beta_{5} + 450 \beta_{4} - 384 \beta_{3} + 578 \beta_{2} + 3665 \beta_{1} + 1285\)
\(\nu^{8}\)\(=\)\(451 \beta_{11} - 1964 \beta_{10} - 1962 \beta_{9} + 6113 \beta_{8} + 6668 \beta_{7} + 77 \beta_{6} - 348 \beta_{5} + 1390 \beta_{4} - 4316 \beta_{3} + 4693 \beta_{2} + 9664 \beta_{1} + 15112\)
\(\nu^{9}\)\(=\)\(-3384 \beta_{11} + 12202 \beta_{10} - 6570 \beta_{9} + 18455 \beta_{8} + 25796 \beta_{7} + 8546 \beta_{6} - 2832 \beta_{5} + 8544 \beta_{4} - 8748 \beta_{3} + 11538 \beta_{2} + 61610 \beta_{1} + 28417\)
\(\nu^{10}\)\(=\)\(5285 \beta_{11} - 22019 \beta_{10} - 29192 \beta_{9} + 110188 \beta_{8} + 117443 \beta_{7} + 6505 \beta_{6} - 3650 \beta_{5} + 29179 \beta_{4} - 73887 \beta_{3} + 79863 \beta_{2} + 180002 \beta_{1} + 253681\)
\(\nu^{11}\)\(=\)\(-52344 \beta_{11} + 193515 \beta_{10} - 102076 \beta_{9} + 364000 \beta_{8} + 462496 \beta_{7} + 145903 \beta_{6} - 31755 \beta_{5} + 161388 \beta_{4} - 182355 \beta_{3} + 222808 \beta_{2} + 1046987 \beta_{1} + 583486\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.25305
4.11015
1.99250
1.53008
1.09665
1.06735
0.448758
−1.12518
−1.17050
−1.20126
−2.26764
−3.73396
1.00000 0 1.00000 −4.25305 0 2.10378 1.00000 0 −4.25305
1.2 1.00000 0 1.00000 −4.11015 0 −3.78475 1.00000 0 −4.11015
1.3 1.00000 0 1.00000 −1.99250 0 −0.0206178 1.00000 0 −1.99250
1.4 1.00000 0 1.00000 −1.53008 0 5.04291 1.00000 0 −1.53008
1.5 1.00000 0 1.00000 −1.09665 0 −2.72925 1.00000 0 −1.09665
1.6 1.00000 0 1.00000 −1.06735 0 2.41300 1.00000 0 −1.06735
1.7 1.00000 0 1.00000 −0.448758 0 −1.52942 1.00000 0 −0.448758
1.8 1.00000 0 1.00000 1.12518 0 −0.686663 1.00000 0 1.12518
1.9 1.00000 0 1.00000 1.17050 0 0.506529 1.00000 0 1.17050
1.10 1.00000 0 1.00000 1.20126 0 −4.29383 1.00000 0 1.20126
1.11 1.00000 0 1.00000 2.26764 0 −0.397853 1.00000 0 2.26764
1.12 1.00000 0 1.00000 3.73396 0 −2.62384 1.00000 0 3.73396
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(149\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\(T_{5}^{12} + \cdots\)
\(T_{11}^{12} + \cdots\)