Properties

Label 1089.6.a.j
Level $1089$
Weight $6$
Character orbit 1089.a
Self dual yes
Analytic conductor $174.658$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(174.657979776\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
Defining polynomial: \(x^{2} - x - 44\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 - \beta ) q^{2} + ( 16 + 5 \beta ) q^{4} + ( -24 - 10 \beta ) q^{5} + ( 138 + 10 \beta ) q^{7} + ( -188 + \beta ) q^{8} +O(q^{10})\) \( q + ( -2 - \beta ) q^{2} + ( 16 + 5 \beta ) q^{4} + ( -24 - 10 \beta ) q^{5} + ( 138 + 10 \beta ) q^{7} + ( -188 + \beta ) q^{8} + ( 488 + 54 \beta ) q^{10} + ( 64 + 38 \beta ) q^{13} + ( -716 - 168 \beta ) q^{14} + ( -180 + 25 \beta ) q^{16} + ( -370 - 60 \beta ) q^{17} + ( 744 - 12 \beta ) q^{19} + ( -2584 - 330 \beta ) q^{20} + ( 1502 + 366 \beta ) q^{23} + ( 1851 + 580 \beta ) q^{25} + ( -1800 - 178 \beta ) q^{26} + ( 4408 + 900 \beta ) q^{28} + ( 3506 - 412 \beta ) q^{29} + ( -3592 - 344 \beta ) q^{31} + ( 5276 + 73 \beta ) q^{32} + ( 3380 + 550 \beta ) q^{34} + ( -7712 - 1720 \beta ) q^{35} + ( -15026 + 136 \beta ) q^{37} + ( -960 - 708 \beta ) q^{38} + ( 4072 + 1846 \beta ) q^{40} + ( -3246 + 712 \beta ) q^{41} + ( 9076 - 1496 \beta ) q^{43} + ( -19108 - 2600 \beta ) q^{46} + ( -2662 - 2526 \beta ) q^{47} + ( 6637 + 2860 \beta ) q^{49} + ( -29222 - 3591 \beta ) q^{50} + ( 9384 + 1118 \beta ) q^{52} + ( -8192 + 2206 \beta ) q^{53} + ( -25504 - 1732 \beta ) q^{56} + ( 11116 - 2270 \beta ) q^{58} + ( -7912 - 1476 \beta ) q^{59} + ( 308 + 2330 \beta ) q^{61} + ( 22320 + 4624 \beta ) q^{62} + ( -8004 - 6295 \beta ) q^{64} + ( -18256 - 1932 \beta ) q^{65} + ( 17268 - 3200 \beta ) q^{67} + ( -19120 - 3110 \beta ) q^{68} + ( 91104 + 12872 \beta ) q^{70} + ( 18454 - 3098 \beta ) q^{71} + ( -34090 + 7536 \beta ) q^{73} + ( 24068 + 14618 \beta ) q^{74} + ( 9264 + 3468 \beta ) q^{76} + ( 3806 - 9482 \beta ) q^{79} + ( -6680 + 950 \beta ) q^{80} + ( -24836 + 1110 \beta ) q^{82} + ( -27852 - 2592 \beta ) q^{83} + ( 35280 + 5740 \beta ) q^{85} + ( 47672 - 4588 \beta ) q^{86} + ( -53722 + 15056 \beta ) q^{89} + ( 25552 + 6264 \beta ) q^{91} + ( 104552 + 15196 \beta ) q^{92} + ( 116468 + 10240 \beta ) q^{94} + ( -12576 - 7032 \beta ) q^{95} + ( -7090 + 21300 \beta ) q^{97} + ( -139114 - 15217 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5q^{2} + 37q^{4} - 58q^{5} + 286q^{7} - 375q^{8} + O(q^{10}) \) \( 2q - 5q^{2} + 37q^{4} - 58q^{5} + 286q^{7} - 375q^{8} + 1030q^{10} + 166q^{13} - 1600q^{14} - 335q^{16} - 800q^{17} + 1476q^{19} - 5498q^{20} + 3370q^{23} + 4282q^{25} - 3778q^{26} + 9716q^{28} + 6600q^{29} - 7528q^{31} + 10625q^{32} + 7310q^{34} - 17144q^{35} - 29916q^{37} - 2628q^{38} + 9990q^{40} - 5780q^{41} + 16656q^{43} - 40816q^{46} - 7850q^{47} + 16134q^{49} - 62035q^{50} + 19886q^{52} - 14178q^{53} - 52740q^{56} + 19962q^{58} - 17300q^{59} + 2946q^{61} + 49264q^{62} - 22303q^{64} - 38444q^{65} + 31336q^{67} - 41350q^{68} + 195080q^{70} + 33810q^{71} - 60644q^{73} + 62754q^{74} + 21996q^{76} - 1870q^{79} - 12410q^{80} - 48562q^{82} - 58296q^{83} + 76300q^{85} + 90756q^{86} - 92388q^{89} + 57368q^{91} + 224300q^{92} + 243176q^{94} - 32184q^{95} + 7120q^{97} - 293445q^{98} + O(q^{100}) \)

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
7.15207
−6.15207
−9.15207 0 51.7603 −95.5207 0 209.521 −180.848 0 874.212
1.2 4.15207 0 −14.7603 37.5207 0 76.4793 −194.152 0 155.788
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)

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 1089.6.a.j 2
3.b odd 2 1 363.6.a.j 2
11.b odd 2 1 99.6.a.f 2
33.d even 2 1 33.6.a.c 2
132.d odd 2 1 528.6.a.s 2
165.d even 2 1 825.6.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.c 2 33.d even 2 1
99.6.a.f 2 11.b odd 2 1
363.6.a.j 2 3.b odd 2 1
528.6.a.s 2 132.d odd 2 1
825.6.a.e 2 165.d even 2 1
1089.6.a.j 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2}^{2} + 5 T_{2} - 38 \)
\( T_{5}^{2} + 58 T_{5} - 3584 \)
\( T_{7}^{2} - 286 T_{7} + 16024 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -38 + 5 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -3584 + 58 T + T^{2} \)
$7$ \( 16024 - 286 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -57008 - 166 T + T^{2} \)
$17$ \( 700 + 800 T + T^{2} \)
$19$ \( 538272 - 1476 T + T^{2} \)
$23$ \( -3088328 - 3370 T + T^{2} \)
$29$ \( 3378828 - 6600 T + T^{2} \)
$31$ \( 8931328 + 7528 T + T^{2} \)
$37$ \( 222923316 + 29916 T + T^{2} \)
$41$ \( -14080172 + 5780 T + T^{2} \)
$43$ \( -29676624 - 16656 T + T^{2} \)
$47$ \( -266939288 + 7850 T + T^{2} \)
$53$ \( -165085872 + 14178 T + T^{2} \)
$59$ \( -21579488 + 17300 T + T^{2} \)
$61$ \( -238059096 - 2946 T + T^{2} \)
$67$ \( -207633776 - 31336 T + T^{2} \)
$71$ \( -138914952 - 33810 T + T^{2} \)
$73$ \( -1593591164 + 60644 T + T^{2} \)
$79$ \( -3977569112 + 1870 T + T^{2} \)
$83$ \( 552313872 + 58296 T + T^{2} \)
$89$ \( -7896843132 + 92388 T + T^{2} \)
$97$ \( -20063108900 - 7120 T + T^{2} \)
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