Properties

Label 8046.2.a.k
Level 8046
Weight 2
Character orbit 8046.a
Self dual Yes
Analytic conductor 64.248
Analytic rank 0
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: \(0\)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{1} q^{5} + ( -1 - \beta_{9} ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta_{1} q^{5} + ( -1 - \beta_{9} ) q^{7} - q^{8} -\beta_{1} q^{10} + ( 1 + \beta_{6} - \beta_{7} - \beta_{9} ) q^{11} + ( 1 + \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{13} + ( 1 + \beta_{9} ) q^{14} + q^{16} + ( 1 + \beta_{2} - \beta_{5} - \beta_{6} ) q^{17} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{11} ) q^{19} + \beta_{1} q^{20} + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{22} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{23} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{25} + ( -1 - \beta_{1} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{26} + ( -1 - \beta_{9} ) q^{28} + ( 2 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{10} ) q^{29} + ( -2 + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{31} - q^{32} + ( -1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{34} + ( 4 - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{35} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{8} - \beta_{11} ) q^{37} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{11} ) q^{38} -\beta_{1} q^{40} + ( 3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{10} + \beta_{11} ) q^{41} + ( -4 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 3 \beta_{11} ) q^{43} + ( 1 + \beta_{6} - \beta_{7} - \beta_{9} ) q^{44} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{46} + ( -\beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} ) q^{47} + ( 3 - \beta_{2} - 2 \beta_{4} - \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{49} + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} ) q^{50} + ( 1 + \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{52} + ( 1 - \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{11} ) q^{53} + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{55} + ( 1 + \beta_{9} ) q^{56} + ( -2 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} ) q^{58} + ( 2 - \beta_{4} - 3 \beta_{7} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{59} + ( -\beta_{2} - \beta_{4} + \beta_{6} - 4 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{61} + ( 2 - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{11} ) q^{62} + q^{64} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{65} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{67} + ( 1 + \beta_{2} - \beta_{5} - \beta_{6} ) q^{68} + ( -4 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{70} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{71} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{73} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} + \beta_{11} ) q^{74} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{11} ) q^{76} + ( 3 - 2 \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} ) q^{77} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{79} + \beta_{1} q^{80} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{10} - \beta_{11} ) q^{82} + ( 2 - 2 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 3 \beta_{11} ) q^{83} + ( -2 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{85} + ( 4 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - 3 \beta_{11} ) q^{86} + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{88} + ( 1 - 2 \beta_{2} - \beta_{5} + \beta_{6} - 3 \beta_{7} - 3 \beta_{9} - \beta_{10} ) q^{89} + ( -5 - \beta_{1} - 3 \beta_{2} - 3 \beta_{7} + \beta_{8} - 5 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{91} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{92} + ( \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{94} + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 4 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{95} + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{97} + ( -3 + \beta_{2} + 2 \beta_{4} + \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{4} + 3q^{5} - 6q^{7} - 12q^{8} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{4} + 3q^{5} - 6q^{7} - 12q^{8} - 3q^{10} + 14q^{11} - 3q^{13} + 6q^{14} + 12q^{16} + 8q^{17} - 4q^{19} + 3q^{20} - 14q^{22} + 13q^{23} + 11q^{25} + 3q^{26} - 6q^{28} + 23q^{29} - 14q^{31} - 12q^{32} - 8q^{34} + 32q^{35} - 19q^{37} + 4q^{38} - 3q^{40} + 30q^{41} - 15q^{43} + 14q^{44} - 13q^{46} - q^{47} + 14q^{49} - 11q^{50} - 3q^{52} + 16q^{53} - 7q^{55} + 6q^{56} - 23q^{58} + 26q^{59} - 16q^{61} + 14q^{62} + 12q^{64} + 8q^{65} - 39q^{67} + 8q^{68} - 32q^{70} + 15q^{71} - 2q^{73} + 19q^{74} - 4q^{76} + 34q^{77} - 13q^{79} + 3q^{80} - 30q^{82} + 6q^{83} - 11q^{85} + 15q^{86} - 14q^{88} + 18q^{89} - 35q^{91} + 13q^{92} + q^{94} + 51q^{95} + 19q^{97} - 14q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 31 x^{10} + 82 x^{9} + 334 x^{8} - 684 x^{7} - 1561 x^{6} + 1551 x^{5} + 3573 x^{4} + 345 x^{3} - 1607 x^{2} - 594 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(24748733 \nu^{11} + 373555153 \nu^{10} - 2862765831 \nu^{9} - 8579109025 \nu^{8} + 62879646690 \nu^{7} + 50310595815 \nu^{6} - 465585432656 \nu^{5} - 38007995419 \nu^{4} + 994832840887 \nu^{3} + 294511464920 \nu^{2} - 403274368788 \nu - 135751602546\)\()/ 24466366011 \)
\(\beta_{3}\)\(=\)\((\)\(-142183724 \nu^{11} + 838314338 \nu^{10} + 2550850410 \nu^{9} - 22317146501 \nu^{8} + 3959704842 \nu^{7} + 175851726345 \nu^{6} - 208981623823 \nu^{5} - 377751239522 \nu^{4} + 495900208592 \nu^{3} + 403759068040 \nu^{2} - 184654740606 \nu - 100388062134\)\()/ 24466366011 \)
\(\beta_{4}\)\(=\)\((\)\(-743676686 \nu^{11} + 3071881331 \nu^{10} + 19932549972 \nu^{9} - 83789564729 \nu^{8} - 164381452443 \nu^{7} + 697617356013 \nu^{6} + 479743706735 \nu^{5} - 1672929464267 \nu^{4} - 1150914398347 \nu^{3} + 745303063093 \nu^{2} + 651960038514 \nu + 65302848768\)\()/ 24466366011 \)
\(\beta_{5}\)\(=\)\((\)\(260555422 \nu^{11} - 1008357826 \nu^{10} - 7145245871 \nu^{9} + 27363214940 \nu^{8} + 61692870961 \nu^{7} - 226011640501 \nu^{6} - 196533632656 \nu^{5} + 528630702028 \nu^{4} + 427946138875 \nu^{3} - 193999453699 \nu^{2} - 177941248499 \nu + 915421136\)\()/ 8155455337 \)
\(\beta_{6}\)\(=\)\((\)\(791457988 \nu^{11} - 2873219941 \nu^{10} - 22855402146 \nu^{9} + 79587640096 \nu^{8} + 218268434445 \nu^{7} - 686060598954 \nu^{6} - 843237849712 \nu^{5} + 1806674571358 \nu^{4} + 1819129337198 \nu^{3} - 899515205951 \nu^{2} - 830466305577 \nu - 29055232476\)\()/ 24466366011 \)
\(\beta_{7}\)\(=\)\((\)\(905897063 \nu^{11} - 3376570442 \nu^{10} - 25430053989 \nu^{9} + 92407418951 \nu^{8} + 230188534851 \nu^{7} - 779842057185 \nu^{6} - 806208285203 \nu^{5} + 1958545237151 \nu^{4} + 1735102707136 \nu^{3} - 954029651272 \nu^{2} - 857897373063 \nu - 7334114235\)\()/ 24466366011 \)
\(\beta_{8}\)\(=\)\((\)\(-303077175 \nu^{11} + 1063333243 \nu^{10} + 8694135210 \nu^{9} - 28833755646 \nu^{8} - 82150526984 \nu^{7} + 238485257763 \nu^{6} + 313671881091 \nu^{5} - 556795852562 \nu^{4} - 689122894818 \nu^{3} + 140774506663 \nu^{2} + 293198015341 \nu + 44724727668\)\()/ 8155455337 \)
\(\beta_{9}\)\(=\)\((\)\(-329344029 \nu^{11} + 1140734420 \nu^{10} + 9678326681 \nu^{9} - 31578991963 \nu^{8} - 94978496783 \nu^{7} + 271887620614 \nu^{6} + 378855759913 \nu^{5} - 709008838650 \nu^{4} - 783228359697 \nu^{3} + 297129010822 \nu^{2} + 318441901306 \nu + 17837198367\)\()/ 8155455337 \)
\(\beta_{10}\)\(=\)\((\)\(-353735657 \nu^{11} + 1292153510 \nu^{10} + 10017640191 \nu^{9} - 35438802905 \nu^{8} - 91893899224 \nu^{7} + 299777725695 \nu^{6} + 328556990398 \nu^{5} - 756307087494 \nu^{4} - 702253132145 \nu^{3} + 389355266298 \nu^{2} + 322371833162 \nu - 27070193104\)\()/ 8155455337 \)
\(\beta_{11}\)\(=\)\((\)\(701406468 \nu^{11} - 2905422747 \nu^{10} - 18475689520 \nu^{9} + 78645513257 \nu^{8} + 146244224743 \nu^{7} - 647726374616 \nu^{6} - 374724551769 \nu^{5} + 1522325564529 \nu^{4} + 836181281764 \nu^{3} - 700613773249 \nu^{2} - 384310270718 \nu - 1717939244\)\()/ 8155455337 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + 2 \beta_{10} + \beta_{9} + 3 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{3} + 10 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-3 \beta_{11} + 16 \beta_{10} + 3 \beta_{9} - 14 \beta_{8} + \beta_{7} + 20 \beta_{6} - 6 \beta_{5} + \beta_{4} - 16 \beta_{3} - 2 \beta_{2} + 15 \beta_{1} + 68\)
\(\nu^{5}\)\(=\)\(-19 \beta_{11} + 43 \beta_{10} + 23 \beta_{9} - 3 \beta_{8} + 50 \beta_{7} + 35 \beta_{6} + 45 \beta_{5} + 8 \beta_{4} - 24 \beta_{3} - \beta_{2} + 127 \beta_{1} + 31\)
\(\nu^{6}\)\(=\)\(-62 \beta_{11} + 258 \beta_{10} + 85 \beta_{9} - 184 \beta_{8} + 55 \beta_{7} + 335 \beta_{6} + 4 \beta_{5} + 41 \beta_{4} - 239 \beta_{3} - 26 \beta_{2} + 238 \beta_{1} + 846\)
\(\nu^{7}\)\(=\)\(-313 \beta_{11} + 786 \beta_{10} + 465 \beta_{9} - 93 \beta_{8} + 795 \beta_{7} + 780 \beta_{6} + 809 \beta_{5} + 225 \beta_{4} - 459 \beta_{3} - 9 \beta_{2} + 1763 \beta_{1} + 576\)
\(\nu^{8}\)\(=\)\(-1051 \beta_{11} + 4141 \beta_{10} + 1808 \beta_{9} - 2429 \beta_{8} + 1441 \beta_{7} + 5432 \beta_{6} + 1061 \beta_{5} + 1011 \beta_{4} - 3568 \beta_{3} - 225 \beta_{2} + 3995 \beta_{1} + 11067\)
\(\nu^{9}\)\(=\)\(-4900 \beta_{11} + 13689 \beta_{10} + 8697 \beta_{9} - 2068 \beta_{8} + 12789 \beta_{7} + 14973 \beta_{6} + 13495 \beta_{5} + 4850 \beta_{4} - 8212 \beta_{3} + 193 \beta_{2} + 25829 \beta_{1} + 11256\)
\(\nu^{10}\)\(=\)\(-16949 \beta_{11} + 66601 \beta_{10} + 34473 \beta_{9} - 32858 \beta_{8} + 30539 \beta_{7} + 87444 \beta_{6} + 27494 \beta_{5} + 20849 \beta_{4} - 54040 \beta_{3} - 667 \beta_{2} + 68339 \beta_{1} + 151126\)
\(\nu^{11}\)\(=\)\(-75864 \beta_{11} + 233383 \beta_{10} + 155287 \beta_{9} - 40971 \beta_{8} + 207800 \beta_{7} + 269383 \beta_{6} + 219090 \beta_{5} + 93810 \beta_{4} - 143006 \beta_{3} + 9222 \beta_{2} + 392614 \beta_{1} + 215351\)

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
−3.46379
−3.43402
−1.28984
−1.11280
−0.815716
−0.539392
−0.00512154
0.761360
2.60206
2.95427
3.27568
4.06732
−1.00000 0 1.00000 −3.46379 0 −3.72396 −1.00000 0 3.46379
1.2 −1.00000 0 1.00000 −3.43402 0 −1.90397 −1.00000 0 3.43402
1.3 −1.00000 0 1.00000 −1.28984 0 3.20787 −1.00000 0 1.28984
1.4 −1.00000 0 1.00000 −1.11280 0 −0.851837 −1.00000 0 1.11280
1.5 −1.00000 0 1.00000 −0.815716 0 −4.13057 −1.00000 0 0.815716
1.6 −1.00000 0 1.00000 −0.539392 0 0.739124 −1.00000 0 0.539392
1.7 −1.00000 0 1.00000 −0.00512154 0 −2.98814 −1.00000 0 0.00512154
1.8 −1.00000 0 1.00000 0.761360 0 1.23946 −1.00000 0 −0.761360
1.9 −1.00000 0 1.00000 2.60206 0 −4.35416 −1.00000 0 −2.60206
1.10 −1.00000 0 1.00000 2.95427 0 0.794893 −1.00000 0 −2.95427
1.11 −1.00000 0 1.00000 3.27568 0 4.39969 −1.00000 0 −3.27568
1.12 −1.00000 0 1.00000 4.06732 0 1.57160 −1.00000 0 −4.06732
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

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