Properties

Label 912.6.a.y
Level $912$
Weight $6$
Character orbit 912.a
Self dual yes
Analytic conductor $146.270$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 912.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(146.270043669\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} - 4725 x^{4} + 92430 x^{3} + 1610577 x^{2} - 16081740 x - 24661341\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 456)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + ( -11 - \beta_{2} ) q^{5} + ( 25 + \beta_{4} ) q^{7} + 81 q^{9} +O(q^{10})\) \( q + 9 q^{3} + ( -11 - \beta_{2} ) q^{5} + ( 25 + \beta_{4} ) q^{7} + 81 q^{9} + ( 34 + \beta_{5} ) q^{11} + ( -49 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{13} + ( -99 - 9 \beta_{2} ) q^{15} + ( 223 - 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} ) q^{17} -361 q^{19} + ( 225 + 9 \beta_{4} ) q^{21} + ( -210 - 5 \beta_{1} - 31 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{23} + ( 771 - 2 \beta_{1} + 24 \beta_{2} - 4 \beta_{3} + 10 \beta_{4} - 3 \beta_{5} ) q^{25} + 729 q^{27} + ( 1221 + 11 \beta_{1} + 3 \beta_{2} + 27 \beta_{3} + 5 \beta_{5} ) q^{29} + ( -282 + 21 \beta_{1} - 31 \beta_{2} + 2 \beta_{3} - 9 \beta_{4} + 3 \beta_{5} ) q^{31} + ( 306 + 9 \beta_{5} ) q^{33} + ( -730 - 17 \beta_{1} - 76 \beta_{2} - 44 \beta_{3} + 6 \beta_{4} + 12 \beta_{5} ) q^{35} + ( 2362 + 12 \beta_{1} - 25 \beta_{2} - 30 \beta_{3} - 3 \beta_{4} - 22 \beta_{5} ) q^{37} + ( -441 + 9 \beta_{1} + 18 \beta_{2} + 9 \beta_{3} + 27 \beta_{4} + 9 \beta_{5} ) q^{39} + ( 2460 + 10 \beta_{1} + 20 \beta_{2} - 54 \beta_{3} - 42 \beta_{4} + 4 \beta_{5} ) q^{41} + ( 781 - 2 \beta_{1} + 19 \beta_{2} + 35 \beta_{3} + 18 \beta_{4} - 11 \beta_{5} ) q^{43} + ( -891 - 81 \beta_{2} ) q^{45} + ( 1790 + 21 \beta_{1} - 106 \beta_{2} + 83 \beta_{3} + 32 \beta_{4} - 37 \beta_{5} ) q^{47} + ( 6401 - 15 \beta_{1} - 52 \beta_{2} + 124 \beta_{3} + 28 \beta_{4} - 22 \beta_{5} ) q^{49} + ( 2007 - 36 \beta_{1} + 27 \beta_{2} - 27 \beta_{3} + 54 \beta_{4} + 27 \beta_{5} ) q^{51} + ( 7896 - 86 \beta_{1} - 108 \beta_{2} + 104 \beta_{3} + 2 \beta_{4} ) q^{53} + ( -119 + 17 \beta_{1} + 62 \beta_{2} - 46 \beta_{3} + 5 \beta_{4} - 41 \beta_{5} ) q^{55} -3249 q^{57} + ( 10251 - 4 \beta_{1} - 147 \beta_{2} + 11 \beta_{3} - 84 \beta_{5} ) q^{59} + ( -13608 + 40 \beta_{1} - 305 \beta_{2} - 189 \beta_{3} + 57 \beta_{4} + 32 \beta_{5} ) q^{61} + ( 2025 + 81 \beta_{4} ) q^{63} + ( -9915 - 4 \beta_{1} + 19 \beta_{2} - 269 \beta_{3} - 92 \beta_{4} - 4 \beta_{5} ) q^{65} + ( 7671 - 75 \beta_{1} + 115 \beta_{2} - 135 \beta_{3} - 100 \beta_{4} + 45 \beta_{5} ) q^{67} + ( -1890 - 45 \beta_{1} - 279 \beta_{2} + 36 \beta_{3} + 27 \beta_{4} + 9 \beta_{5} ) q^{69} + ( 1683 - 9 \beta_{1} - 417 \beta_{2} - 81 \beta_{3} - 116 \beta_{4} + 125 \beta_{5} ) q^{71} + ( -9166 + 126 \beta_{1} - 162 \beta_{2} + 182 \beta_{3} + 92 \beta_{4} - 3 \beta_{5} ) q^{73} + ( 6939 - 18 \beta_{1} + 216 \beta_{2} - 36 \beta_{3} + 90 \beta_{4} - 27 \beta_{5} ) q^{75} + ( -4987 + 47 \beta_{1} - 154 \beta_{2} + 58 \beta_{3} - 55 \beta_{4} - 93 \beta_{5} ) q^{77} + ( 24253 - 20 \beta_{1} + 118 \beta_{2} + 121 \beta_{3} - 99 \beta_{4} + 6 \beta_{5} ) q^{79} + 6561 q^{81} + ( 26756 + 15 \beta_{1} - 40 \beta_{2} - 230 \beta_{3} - 266 \beta_{4} + 79 \beta_{5} ) q^{83} + ( -8978 - 162 \beta_{1} - 449 \beta_{2} - 39 \beta_{3} + 361 \beta_{4} - 34 \beta_{5} ) q^{85} + ( 10989 + 99 \beta_{1} + 27 \beta_{2} + 243 \beta_{3} + 45 \beta_{5} ) q^{87} + ( -16346 + 37 \beta_{1} + 46 \beta_{2} + 248 \beta_{3} - 288 \beta_{4} + 179 \beta_{5} ) q^{89} + ( 69567 + 151 \beta_{1} + 253 \beta_{2} + 653 \beta_{3} - 154 \beta_{4} - 187 \beta_{5} ) q^{91} + ( -2538 + 189 \beta_{1} - 279 \beta_{2} + 18 \beta_{3} - 81 \beta_{4} + 27 \beta_{5} ) q^{93} + ( 3971 + 361 \beta_{2} ) q^{95} + ( -109 - 83 \beta_{1} + 323 \beta_{2} + 325 \beta_{3} - 74 \beta_{4} + 99 \beta_{5} ) q^{97} + ( 2754 + 81 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 54q^{3} - 65q^{5} + 149q^{7} + 486q^{9} + O(q^{10}) \) \( 6q + 54q^{3} - 65q^{5} + 149q^{7} + 486q^{9} + 203q^{11} - 298q^{13} - 585q^{15} + 1319q^{17} - 2166q^{19} + 1341q^{21} - 1234q^{23} + 4589q^{25} + 4374q^{27} + 7356q^{29} - 1632q^{31} + 1827q^{33} - 4383q^{35} + 14204q^{37} - 2682q^{39} + 14734q^{41} + 4693q^{43} - 5265q^{45} + 10955q^{47} + 38561q^{49} + 11871q^{51} + 47500q^{53} - 769q^{55} - 19494q^{57} + 61744q^{59} - 81581q^{61} + 12069q^{63} - 59686q^{65} + 45756q^{67} - 11106q^{69} + 10416q^{71} - 54615q^{73} + 41301q^{75} - 29515q^{77} + 145594q^{79} + 39366q^{81} + 160548q^{83} - 53947q^{85} + 66204q^{87} - 97728q^{89} + 418294q^{91} - 14688q^{93} + 23465q^{95} - 760q^{97} + 16443q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 4725 x^{4} + 92430 x^{3} + 1610577 x^{2} - 16081740 x - 24661341\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -4393 \nu^{5} - 288577 \nu^{4} + 17990538 \nu^{3} + 820304928 \nu^{2} - 23681399193 \nu - 121791601863 \)\()/ 828150480 \)
\(\beta_{2}\)\(=\)\((\)\( -23423 \nu^{5} - 142247 \nu^{4} + 103907598 \nu^{3} - 1434178872 \nu^{2} - 30899430423 \nu + 62101268487 \)\()/ 3312601920 \)
\(\beta_{3}\)\(=\)\((\)\( -23423 \nu^{5} - 142247 \nu^{4} + 103907598 \nu^{3} - 1434178872 \nu^{2} - 17649022743 \nu + 58788666567 \)\()/ 3312601920 \)
\(\beta_{4}\)\(=\)\((\)\( 5543 \nu^{5} - 64113 \nu^{4} - 28829198 \nu^{3} + 639322632 \nu^{2} + 11916019503 \nu - 53106861087 \)\()/ 552100320 \)
\(\beta_{5}\)\(=\)\((\)\( -6041 \nu^{5} + 3471 \nu^{4} + 28357426 \nu^{3} - 539262904 \nu^{2} - 9893470161 \nu + 97030518849 \)\()/ 184033440 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-36 \beta_{5} - 39 \beta_{4} - 29 \beta_{3} + 143 \beta_{2} - 3 \beta_{1} + 12622\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(1044 \beta_{5} - 69 \beta_{4} + 6971 \beta_{3} - 12485 \beta_{2} + 759 \beta_{1} - 335116\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-77850 \beta_{5} - 80907 \beta_{4} - 148531 \beta_{3} + 410857 \beta_{2} - 21057 \beta_{1} + 25100360\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(3890574 \beta_{5} + 1532274 \beta_{4} + 15932809 \beta_{3} - 33812113 \beta_{2} + 1903236 \beta_{1} - 1272876977\)\()/4\)

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
−74.1084
34.8797
−1.36301
49.1493
−15.6548
9.09727
0 9.00000 0 −86.8767 0 −2.94760 0 81.0000 0
1.2 0 9.00000 0 −80.1366 0 221.037 0 81.0000 0
1.3 0 9.00000 0 −41.5770 0 −98.3255 0 81.0000 0
1.4 0 9.00000 0 28.9245 0 −210.915 0 81.0000 0
1.5 0 9.00000 0 46.6057 0 58.6437 0 81.0000 0
1.6 0 9.00000 0 68.0602 0 181.508 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(19\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.6.a.y 6
4.b odd 2 1 456.6.a.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.6.a.f 6 4.b odd 2 1
912.6.a.y 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 65 T_{5}^{5} - 9557 T_{5}^{4} - 445577 T_{5}^{3} + 29529208 T_{5}^{2} + 602356212 T_{5} - 26557430352 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( -9 + T )^{6} \)
$5$ \( -26557430352 + 602356212 T + 29529208 T^{2} - 445577 T^{3} - 9557 T^{4} + 65 T^{5} + T^{6} \)
$7$ \( -143821013760 - 47041668096 T + 616968276 T^{2} + 7618369 T^{3} - 58601 T^{4} - 149 T^{5} + T^{6} \)
$11$ \( 1637954874112 + 267853720384 T + 4031719924 T^{2} - 17133601 T^{3} - 251657 T^{4} - 203 T^{5} + T^{6} \)
$13$ \( -6708606782009472 + 15213557591328 T + 264801371792 T^{2} - 343769584 T^{3} - 1307040 T^{4} + 298 T^{5} + T^{6} \)
$17$ \( 112483698699536520 - 1696792136443948 T + 4545676438234 T^{2} + 4944257071 T^{3} - 4786951 T^{4} - 1319 T^{5} + T^{6} \)
$19$ \( ( 361 + T )^{6} \)
$23$ \( 10912197866748678144 + 45742188138549760 T + 26100108896512 T^{2} - 26840454464 T^{3} - 16647152 T^{4} + 1234 T^{5} + T^{6} \)
$29$ \( \)\(86\!\cdots\!48\)\( - 5798286151586715072 T + 247015746026736 T^{2} + 470208142688 T^{3} - 63158820 T^{4} - 7356 T^{5} + T^{6} \)
$31$ \( 1775258940705673216 + 21145538634270528 T - 151177840715920 T^{2} - 402954089040 T^{3} - 89208160 T^{4} + 1632 T^{5} + T^{6} \)
$37$ \( \)\(44\!\cdots\!00\)\( - 21855154949371187200 T - 3937790841698560 T^{2} + 2549548661904 T^{3} - 152187540 T^{4} - 14204 T^{5} + T^{6} \)
$41$ \( \)\(13\!\cdots\!32\)\( - \)\(13\!\cdots\!52\)\( T + 4295040987543424 T^{2} + 3521404106744 T^{3} - 218995972 T^{4} - 14734 T^{5} + T^{6} \)
$43$ \( -\)\(77\!\cdots\!60\)\( - 220901908593157840 T + 2310347035035788 T^{2} + 13046528417 T^{3} - 121424401 T^{4} - 4693 T^{5} + T^{6} \)
$47$ \( \)\(60\!\cdots\!28\)\( - \)\(19\!\cdots\!76\)\( T + 109163360832510938 T^{2} + 12059614729919 T^{3} - 875485039 T^{4} - 10955 T^{5} + T^{6} \)
$53$ \( \)\(85\!\cdots\!12\)\( - \)\(16\!\cdots\!40\)\( T - 27461508012574992 T^{2} + 76411039032672 T^{3} - 1362911364 T^{4} - 47500 T^{5} + T^{6} \)
$59$ \( -\)\(49\!\cdots\!12\)\( + \)\(19\!\cdots\!56\)\( T - 2508318902055701248 T^{2} + 107702950243712 T^{3} - 523832560 T^{4} - 61744 T^{5} + T^{6} \)
$61$ \( \)\(42\!\cdots\!00\)\( + \)\(31\!\cdots\!64\)\( T - 2680188236475503230 T^{2} - 214064110340925 T^{3} - 1397314919 T^{4} + 81581 T^{5} + T^{6} \)
$67$ \( \)\(85\!\cdots\!76\)\( - \)\(19\!\cdots\!56\)\( T + 405388370893277440 T^{2} + 112565725121280 T^{3} - 2579706560 T^{4} - 45756 T^{5} + T^{6} \)
$71$ \( \)\(11\!\cdots\!80\)\( - \)\(24\!\cdots\!80\)\( T + 12054624853039017984 T^{2} + 91249375845248 T^{3} - 7810260288 T^{4} - 10416 T^{5} + T^{6} \)
$73$ \( \)\(21\!\cdots\!08\)\( + \)\(23\!\cdots\!72\)\( T + 3166952202416264678 T^{2} - 249447316672487 T^{3} - 4851012187 T^{4} + 54615 T^{5} + T^{6} \)
$79$ \( \)\(13\!\cdots\!88\)\( + \)\(79\!\cdots\!52\)\( T - 565134851002588160 T^{2} - 91150843755968 T^{3} + 6759148176 T^{4} - 145594 T^{5} + T^{6} \)
$83$ \( \)\(12\!\cdots\!00\)\( + \)\(18\!\cdots\!40\)\( T - 42414970839760802304 T^{2} + 896709308291392 T^{3} + 1059380944 T^{4} - 160548 T^{5} + T^{6} \)
$89$ \( \)\(37\!\cdots\!84\)\( + \)\(30\!\cdots\!08\)\( T + 28929446906634701616 T^{2} - 1821987516761856 T^{3} - 18458090156 T^{4} + 97728 T^{5} + T^{6} \)
$97$ \( -\)\(53\!\cdots\!24\)\( - \)\(18\!\cdots\!28\)\( T + 55323449017729409200 T^{2} - 3353906992256 T^{3} - 14254958316 T^{4} + 760 T^{5} + T^{6} \)
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