Properties

Label 8046.2.a.p
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_{1} q^{5} + ( 1 - \beta_{5} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + \beta_{1} q^{5} + ( 1 - \beta_{5} ) q^{7} + q^{8} + \beta_{1} q^{10} + ( 1 - \beta_{2} - \beta_{3} - \beta_{8} ) q^{11} + ( 1 - \beta_{5} - \beta_{6} + \beta_{11} ) q^{13} + ( 1 - \beta_{5} ) q^{14} + q^{16} + ( \beta_{4} - \beta_{6} + \beta_{7} ) q^{17} + ( \beta_{4} - \beta_{10} ) q^{19} + \beta_{1} q^{20} + ( 1 - \beta_{2} - \beta_{3} - \beta_{8} ) q^{22} + ( 1 - \beta_{5} - \beta_{6} - \beta_{10} ) q^{23} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{25} + ( 1 - \beta_{5} - \beta_{6} + \beta_{11} ) q^{26} + ( 1 - \beta_{5} ) q^{28} + ( 3 + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{29} + ( \beta_{1} + \beta_{10} ) q^{31} + q^{32} + ( \beta_{4} - \beta_{6} + \beta_{7} ) q^{34} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{35} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{37} + ( \beta_{4} - \beta_{10} ) q^{38} + \beta_{1} q^{40} + ( 3 - \beta_{1} + \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{41} + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{43} + ( 1 - \beta_{2} - \beta_{3} - \beta_{8} ) q^{44} + ( 1 - \beta_{5} - \beta_{6} - \beta_{10} ) q^{46} + ( 2 + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{47} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - 2 \beta_{11} ) q^{49} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{50} + ( 1 - \beta_{5} - \beta_{6} + \beta_{11} ) q^{52} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{53} + ( \beta_{1} - 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{55} + ( 1 - \beta_{5} ) q^{56} + ( 3 + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{58} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{10} - \beta_{11} ) q^{59} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{61} + ( \beta_{1} + \beta_{10} ) q^{62} + q^{64} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{9} ) q^{65} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{67} + ( \beta_{4} - \beta_{6} + \beta_{7} ) q^{68} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{70} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{71} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{73} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{74} + ( \beta_{4} - \beta_{10} ) q^{76} + ( 6 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{10} ) q^{77} + ( 4 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{79} + \beta_{1} q^{80} + ( 3 - \beta_{1} + \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{82} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} ) q^{83} + ( -3 + 2 \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - \beta_{11} ) q^{85} + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{86} + ( 1 - \beta_{2} - \beta_{3} - \beta_{8} ) q^{88} + ( \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{10} ) q^{89} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{91} + ( 1 - \beta_{5} - \beta_{6} - \beta_{10} ) q^{92} + ( 2 + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{94} + ( 6 + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{95} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{97} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - 2 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 12q^{4} + 5q^{5} + 6q^{7} + 12q^{8} + O(q^{10}) \) \( 12q + 12q^{2} + 12q^{4} + 5q^{5} + 6q^{7} + 12q^{8} + 5q^{10} + 6q^{11} + 3q^{13} + 6q^{14} + 12q^{16} + 6q^{17} + 8q^{19} + 5q^{20} + 6q^{22} + 11q^{23} + 11q^{25} + 3q^{26} + 6q^{28} + 29q^{29} + 2q^{31} + 12q^{32} + 6q^{34} + 4q^{35} + 5q^{37} + 8q^{38} + 5q^{40} + 22q^{41} + 9q^{43} + 6q^{44} + 11q^{46} + 15q^{47} + 14q^{49} + 11q^{50} + 3q^{52} + 12q^{53} + 13q^{55} + 6q^{56} + 29q^{58} + 34q^{59} - 4q^{61} + 2q^{62} + 12q^{64} + 12q^{65} + q^{67} + 6q^{68} + 4q^{70} + 21q^{71} - 2q^{73} + 5q^{74} + 8q^{76} + 34q^{77} + 9q^{79} + 5q^{80} + 22q^{82} + 10q^{83} + 5q^{85} + 9q^{86} + 6q^{88} - 2q^{89} + 17q^{91} + 11q^{92} + 15q^{94} + 69q^{95} - 13q^{97} + 14q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} - 3333 x^{3} + 4805 x^{2} - 224 x - 553\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(39753790711 \nu^{11} - 198200510956 \nu^{10} - 967064971419 \nu^{9} + 5714838266656 \nu^{8} + 5759628424968 \nu^{7} - 53850580778217 \nu^{6} + 7072339723081 \nu^{5} + 189124537678876 \nu^{4} - 106414471211396 \nu^{3} - 177474875785820 \nu^{2} + 99233618007315 \nu + 17827377683725\)\()/ 7247830497003 \)
\(\beta_{3}\)\(=\)\((\)\(80534503868 \nu^{11} - 351696237482 \nu^{10} - 2108871784872 \nu^{9} + 10131881046614 \nu^{8} + 15896674895505 \nu^{7} - 95422133728626 \nu^{6} - 23898794925802 \nu^{5} + 335975682458783 \nu^{4} - 90437371366366 \nu^{3} - 327059747358370 \nu^{2} + 92295543950502 \nu + 42821455278854\)\()/ 7247830497003 \)
\(\beta_{4}\)\(=\)\((\)\(-82493402629 \nu^{11} + 263830494832 \nu^{10} + 2411694496521 \nu^{9} - 7581331666612 \nu^{8} - 23166629157747 \nu^{7} + 71744854896003 \nu^{6} + 84556908978098 \nu^{5} - 260250839987092 \nu^{4} - 94448906762320 \nu^{3} + 286297569815276 \nu^{2} + 36333927662739 \nu - 37905558611308\)\()/ 7247830497003 \)
\(\beta_{5}\)\(=\)\((\)\(111684601157 \nu^{11} - 439380667640 \nu^{10} - 3018114108333 \nu^{9} + 12565700557217 \nu^{8} + 24513574501275 \nu^{7} - 117331699982658 \nu^{6} - 53450947539877 \nu^{5} + 411530092181318 \nu^{4} - 68568106482415 \nu^{3} - 416256290933614 \nu^{2} + 100786856841096 \nu + 78764454042977\)\()/ 7247830497003 \)
\(\beta_{6}\)\(=\)\((\)\(126477824479 \nu^{11} - 465007218553 \nu^{10} - 3547935397485 \nu^{9} + 13375681875472 \nu^{8} + 31285095157641 \nu^{7} - 125748469351326 \nu^{6} - 90255138053357 \nu^{5} + 441397343073052 \nu^{4} + 6081982929931 \nu^{3} - 427114040209208 \nu^{2} + 79483670481720 \nu + 52979135676436\)\()/ 7247830497003 \)
\(\beta_{7}\)\(=\)\((\)\(-233966606153 \nu^{11} + 906289859315 \nu^{10} + 6340426420320 \nu^{9} - 25919528717069 \nu^{8} - 51693664769592 \nu^{7} + 241755270906525 \nu^{6} + 112822364381107 \nu^{5} - 841128794307659 \nu^{4} + 152460063612613 \nu^{3} + 807815616826948 \nu^{2} - 241244419540353 \nu - 116176930526963\)\()/ 7247830497003 \)
\(\beta_{8}\)\(=\)\((\)\(-264188216893 \nu^{11} + 1052591859745 \nu^{10} + 7122140689275 \nu^{9} - 30110515868080 \nu^{8} - 57550285913988 \nu^{7} + 280571600276976 \nu^{6} + 122706115800737 \nu^{5} - 970382672201788 \nu^{4} + 176585646871547 \nu^{3} + 895224641034665 \nu^{2} - 259184076615561 \nu - 75969812453317\)\()/ 7247830497003 \)
\(\beta_{9}\)\(=\)\((\)\(265780092172 \nu^{11} - 1045158248329 \nu^{10} - 7226529730506 \nu^{9} + 29933540825302 \nu^{8} + 59704300931763 \nu^{7} - 279842271576060 \nu^{6} - 140534451251591 \nu^{5} + 978788449854322 \nu^{4} - 122917256587556 \nu^{3} - 959282014883780 \nu^{2} + 236241029700939 \nu + 135370428068599\)\()/ 7247830497003 \)
\(\beta_{10}\)\(=\)\((\)\(-116959146335 \nu^{11} + 438173460485 \nu^{10} + 3253152757098 \nu^{9} - 12571233750104 \nu^{8} - 28273973320353 \nu^{7} + 117880451200146 \nu^{6} + 79382248663654 \nu^{5} - 414144051045197 \nu^{4} - 2612399983634 \nu^{3} + 408108251887606 \nu^{2} - 56374695744096 \nu - 57643004413031\)\()/ 2415943499001 \)
\(\beta_{11}\)\(=\)\((\)\(376299976007 \nu^{11} - 1405118384054 \nu^{10} - 10441796596398 \nu^{9} + 40199942268095 \nu^{8} + 90410848803024 \nu^{7} - 375246438055287 \nu^{6} - 252528829047475 \nu^{5} + 1308736876472261 \nu^{4} + 10639560659018 \nu^{3} - 1271104499897389 \nu^{2} + 158243198361369 \nu + 171474493496885\)\()/ 7247830497003 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(-\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{3} - 3 \beta_{2} + 8 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{11} + 17 \beta_{10} - 12 \beta_{8} + 15 \beta_{7} + 18 \beta_{6} + 19 \beta_{5} - 13 \beta_{4} + \beta_{3} + 14 \beta_{1} + 61\)
\(\nu^{5}\)\(=\)\(-22 \beta_{11} - 15 \beta_{10} - 6 \beta_{9} - 18 \beta_{8} - \beta_{7} - 16 \beta_{6} + 9 \beta_{5} - 2 \beta_{4} + 36 \beta_{3} - 61 \beta_{2} + 86 \beta_{1} + 16\)
\(\nu^{6}\)\(=\)\(55 \beta_{11} + 270 \beta_{10} - 4 \beta_{9} - 149 \beta_{8} + 211 \beta_{7} + 292 \beta_{6} + 286 \beta_{5} - 168 \beta_{4} + 28 \beta_{3} + 3 \beta_{2} + 190 \beta_{1} + 742\)
\(\nu^{7}\)\(=\)\(-334 \beta_{11} - 165 \beta_{10} + 18 \beta_{9} - 258 \beta_{8} - 31 \beta_{7} - 224 \beta_{6} + 23 \beta_{5} - 63 \beta_{4} + 588 \beta_{3} - 986 \beta_{2} + 1062 \beta_{1} + 437\)
\(\nu^{8}\)\(=\)\(1032 \beta_{11} + 4150 \beta_{10} - 61 \beta_{9} - 1902 \beta_{8} + 2902 \beta_{7} + 4487 \beta_{6} + 4003 \beta_{5} - 2315 \beta_{4} + 664 \beta_{3} + 37 \beta_{2} + 2643 \beta_{1} + 9899\)
\(\nu^{9}\)\(=\)\(-4527 \beta_{11} - 1300 \beta_{10} + 1303 \beta_{9} - 3491 \beta_{8} - 706 \beta_{7} - 2872 \beta_{6} - 945 \beta_{5} - 1461 \beta_{4} + 9351 \beta_{3} - 14872 \beta_{2} + 14198 \beta_{1} + 8796\)
\(\nu^{10}\)\(=\)\(17114 \beta_{11} + 62852 \beta_{10} - 375 \beta_{9} - 24898 \beta_{8} + 39584 \beta_{7} + 67054 \beta_{6} + 54592 \beta_{5} - 33297 \beta_{4} + 13472 \beta_{3} - 329 \beta_{2} + 37757 \beta_{1} + 138910\)
\(\nu^{11}\)\(=\)\(-58540 \beta_{11} - 1621 \beta_{10} + 28961 \beta_{9} - 46940 \beta_{8} - 13406 \beta_{7} - 33588 \beta_{6} - 27030 \beta_{5} - 29463 \beta_{4} + 146119 \beta_{3} - 218819 \beta_{2} + 198673 \beta_{1} + 158591\)

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
−3.72799
−2.78478
−2.38357
−1.09224
−0.302744
0.521020
1.20508
1.33540
2.27289
2.74383
3.31930
3.89381
1.00000 0 1.00000 −3.72799 0 −2.68331 1.00000 0 −3.72799
1.2 1.00000 0 1.00000 −2.78478 0 2.18432 1.00000 0 −2.78478
1.3 1.00000 0 1.00000 −2.38357 0 4.24377 1.00000 0 −2.38357
1.4 1.00000 0 1.00000 −1.09224 0 −1.25253 1.00000 0 −1.09224
1.5 1.00000 0 1.00000 −0.302744 0 −1.13869 1.00000 0 −0.302744
1.6 1.00000 0 1.00000 0.521020 0 −3.80450 1.00000 0 0.521020
1.7 1.00000 0 1.00000 1.20508 0 4.22185 1.00000 0 1.20508
1.8 1.00000 0 1.00000 1.33540 0 2.26685 1.00000 0 1.33540
1.9 1.00000 0 1.00000 2.27289 0 3.85060 1.00000 0 2.27289
1.10 1.00000 0 1.00000 2.74383 0 0.881706 1.00000 0 2.74383
1.11 1.00000 0 1.00000 3.31930 0 −3.41918 1.00000 0 3.31930
1.12 1.00000 0 1.00000 3.89381 0 0.649105 1.00000 0 3.89381
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

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