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: \( x^{6} - 2x^{5} - 4725x^{4} + 92430x^{3} + 1610577x^{2} - 16081740x - 24661341 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + 9 q^{3} + ( - \beta_{2} - 11) q^{5} + (\beta_{4} + 25) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + ( - \beta_{2} - 11) q^{5} + (\beta_{4} + 25) q^{7} + 81 q^{9} + (\beta_{5} + 34) q^{11} + (\beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 49) q^{13} + ( - 9 \beta_{2} - 99) q^{15} + (3 \beta_{5} + 6 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 223) q^{17} - 361 q^{19} + (9 \beta_{4} + 225) q^{21} + (\beta_{5} + 3 \beta_{4} + 4 \beta_{3} - 31 \beta_{2} - 5 \beta_1 - 210) q^{23} + ( - 3 \beta_{5} + 10 \beta_{4} - 4 \beta_{3} + 24 \beta_{2} - 2 \beta_1 + 771) q^{25} + 729 q^{27} + (5 \beta_{5} + 27 \beta_{3} + 3 \beta_{2} + 11 \beta_1 + 1221) q^{29} + (3 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} - 31 \beta_{2} + 21 \beta_1 - 282) q^{31} + (9 \beta_{5} + 306) q^{33} + (12 \beta_{5} + 6 \beta_{4} - 44 \beta_{3} - 76 \beta_{2} - 17 \beta_1 - 730) q^{35} + ( - 22 \beta_{5} - 3 \beta_{4} - 30 \beta_{3} - 25 \beta_{2} + 12 \beta_1 + 2362) q^{37} + (9 \beta_{5} + 27 \beta_{4} + 9 \beta_{3} + 18 \beta_{2} + 9 \beta_1 - 441) q^{39} + (4 \beta_{5} - 42 \beta_{4} - 54 \beta_{3} + 20 \beta_{2} + 10 \beta_1 + 2460) q^{41} + ( - 11 \beta_{5} + 18 \beta_{4} + 35 \beta_{3} + 19 \beta_{2} - 2 \beta_1 + 781) q^{43} + ( - 81 \beta_{2} - 891) q^{45} + ( - 37 \beta_{5} + 32 \beta_{4} + 83 \beta_{3} - 106 \beta_{2} + \cdots + 1790) q^{47}+ \cdots + (81 \beta_{5} + 2754) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 54 q^{3} - 65 q^{5} + 149 q^{7} + 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 54 q^{3} - 65 q^{5} + 149 q^{7} + 486 q^{9} + 203 q^{11} - 298 q^{13} - 585 q^{15} + 1319 q^{17} - 2166 q^{19} + 1341 q^{21} - 1234 q^{23} + 4589 q^{25} + 4374 q^{27} + 7356 q^{29} - 1632 q^{31} + 1827 q^{33} - 4383 q^{35} + 14204 q^{37} - 2682 q^{39} + 14734 q^{41} + 4693 q^{43} - 5265 q^{45} + 10955 q^{47} + 38561 q^{49} + 11871 q^{51} + 47500 q^{53} - 769 q^{55} - 19494 q^{57} + 61744 q^{59} - 81581 q^{61} + 12069 q^{63} - 59686 q^{65} + 45756 q^{67} - 11106 q^{69} + 10416 q^{71} - 54615 q^{73} + 41301 q^{75} - 29515 q^{77} + 145594 q^{79} + 39366 q^{81} + 160548 q^{83} - 53947 q^{85} + 66204 q^{87} - 97728 q^{89} + 418294 q^{91} - 14688 q^{93} + 23465 q^{95} - 760 q^{97} + 16443 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -4393\nu^{5} - 288577\nu^{4} + 17990538\nu^{3} + 820304928\nu^{2} - 23681399193\nu - 121791601863 ) / 828150480 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23423 \nu^{5} - 142247 \nu^{4} + 103907598 \nu^{3} - 1434178872 \nu^{2} - 30899430423 \nu + 62101268487 ) / 3312601920 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23423 \nu^{5} - 142247 \nu^{4} + 103907598 \nu^{3} - 1434178872 \nu^{2} - 17649022743 \nu + 58788666567 ) / 3312601920 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5543\nu^{5} - 64113\nu^{4} - 28829198\nu^{3} + 639322632\nu^{2} + 11916019503\nu - 53106861087 ) / 552100320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6041\nu^{5} + 3471\nu^{4} + 28357426\nu^{3} - 539262904\nu^{2} - 9893470161\nu + 97030518849 ) / 184033440 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -36\beta_{5} - 39\beta_{4} - 29\beta_{3} + 143\beta_{2} - 3\beta _1 + 12622 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1044\beta_{5} - 69\beta_{4} + 6971\beta_{3} - 12485\beta_{2} + 759\beta _1 - 335116 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -77850\beta_{5} - 80907\beta_{4} - 148531\beta_{3} + 410857\beta_{2} - 21057\beta _1 + 25100360 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3890574\beta_{5} + 1532274\beta_{4} + 15932809\beta_{3} - 33812113\beta_{2} + 1903236\beta _1 - 1272876977 ) / 4 \) Copy content Toggle raw display

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 \( T_{5}^{6} + 65T_{5}^{5} - 9557T_{5}^{4} - 445577T_{5}^{3} + 29529208T_{5}^{2} + 602356212T_{5} - 26557430352 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 65 T^{5} + \cdots - 26557430352 \) Copy content Toggle raw display
$7$ \( T^{6} - 149 T^{5} + \cdots - 143821013760 \) Copy content Toggle raw display
$11$ \( T^{6} - 203 T^{5} + \cdots + 1637954874112 \) Copy content Toggle raw display
$13$ \( T^{6} + 298 T^{5} + \cdots - 67\!\cdots\!72 \) Copy content Toggle raw display
$17$ \( T^{6} - 1319 T^{5} + \cdots + 11\!\cdots\!20 \) Copy content Toggle raw display
$19$ \( (T + 361)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 1234 T^{5} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{6} - 7356 T^{5} + \cdots + 86\!\cdots\!48 \) Copy content Toggle raw display
$31$ \( T^{6} + 1632 T^{5} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{6} - 14204 T^{5} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{6} - 14734 T^{5} + \cdots + 13\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( T^{6} - 4693 T^{5} + \cdots - 77\!\cdots\!60 \) Copy content Toggle raw display
$47$ \( T^{6} - 10955 T^{5} + \cdots + 60\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{6} - 47500 T^{5} + \cdots + 85\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{6} - 61744 T^{5} + \cdots - 49\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( T^{6} + 81581 T^{5} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} - 45756 T^{5} + \cdots + 85\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{6} - 10416 T^{5} + \cdots + 11\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{6} + 54615 T^{5} + \cdots + 21\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{6} - 145594 T^{5} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{6} - 160548 T^{5} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{6} + 97728 T^{5} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{6} + 760 T^{5} + \cdots - 53\!\cdots\!24 \) Copy content Toggle raw display
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