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Properties

Label 135.4.a.g
Level $135$
Weight $4$
Character orbit 135.a
Self dual yes
Analytic conductor $7.965$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.96525785077\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5637.1
Defining polynomial: \(x^{3} - x^{2} - 23 x + 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} -5 q^{5} + ( 14 + 2 \beta_{1} ) q^{7} + ( 9 + 8 \beta_{1} + \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} -5 q^{5} + ( 14 + 2 \beta_{1} ) q^{7} + ( 9 + 8 \beta_{1} + \beta_{2} ) q^{8} -5 \beta_{1} q^{10} + ( -12 + 2 \beta_{1} - 4 \beta_{2} ) q^{11} + ( 14 - 10 \beta_{1} - 4 \beta_{2} ) q^{13} + ( 30 + 16 \beta_{1} + 2 \beta_{2} ) q^{14} + ( 58 + 17 \beta_{1} ) q^{16} + ( 3 + 2 \beta_{1} + 8 \beta_{2} ) q^{17} + ( 59 + 14 \beta_{1} - 4 \beta_{2} ) q^{19} + ( -35 - 5 \beta_{1} - 5 \beta_{2} ) q^{20} + ( 54 - 42 \beta_{1} + 2 \beta_{2} ) q^{22} + ( 39 - 32 \beta_{1} - 4 \beta_{2} ) q^{23} + 25 q^{25} + ( -126 - 28 \beta_{1} - 10 \beta_{2} ) q^{26} + ( 116 + 46 \beta_{1} + 16 \beta_{2} ) q^{28} + ( -42 - 42 \beta_{1} + 8 \beta_{2} ) q^{29} + ( 83 - 30 \beta_{1} + 8 \beta_{2} ) q^{31} + ( 183 + 11 \beta_{1} + 9 \beta_{2} ) q^{32} + ( -18 + 69 \beta_{1} + 2 \beta_{2} ) q^{34} + ( -70 - 10 \beta_{1} ) q^{35} + ( 50 - 64 \beta_{1} - 8 \beta_{2} ) q^{37} + ( 234 + 41 \beta_{1} + 14 \beta_{2} ) q^{38} + ( -45 - 40 \beta_{1} - 5 \beta_{2} ) q^{40} + ( 132 - 42 \beta_{1} - 16 \beta_{2} ) q^{41} + ( -16 + 78 \beta_{1} - 8 \beta_{2} ) q^{43} + ( -546 + 12 \beta_{1} - 10 \beta_{2} ) q^{44} + ( -456 - 25 \beta_{1} - 32 \beta_{2} ) q^{46} + ( 168 - 28 \beta_{1} - 4 \beta_{2} ) q^{47} + ( -87 + 60 \beta_{1} + 4 \beta_{2} ) q^{49} + 25 \beta_{1} q^{50} + ( -472 - 154 \beta_{1} + 4 \beta_{2} ) q^{52} + ( -165 - 14 \beta_{1} - 12 \beta_{2} ) q^{53} + ( 60 - 10 \beta_{1} + 20 \beta_{2} ) q^{55} + ( 354 + 162 \beta_{1} + 30 \beta_{2} ) q^{56} + ( -678 - 20 \beta_{1} - 42 \beta_{2} ) q^{58} + ( -78 + 82 \beta_{1} + 12 \beta_{2} ) q^{59} + ( 173 + 60 \beta_{1} + 16 \beta_{2} ) q^{61} + ( -498 + 117 \beta_{1} - 30 \beta_{2} ) q^{62} + ( -353 + 130 \beta_{1} + 11 \beta_{2} ) q^{64} + ( -70 + 50 \beta_{1} + 20 \beta_{2} ) q^{65} + ( 302 - 52 \beta_{1} + 24 \beta_{2} ) q^{67} + ( 999 + 51 \beta_{1} + 5 \beta_{2} ) q^{68} + ( -150 - 80 \beta_{1} - 10 \beta_{2} ) q^{70} + ( 192 - 34 \beta_{1} + 60 \beta_{2} ) q^{71} + ( 404 + 22 \beta_{1} + 60 \beta_{2} ) q^{73} + ( -912 - 78 \beta_{1} - 64 \beta_{2} ) q^{74} + ( 59 + 275 \beta_{1} + 73 \beta_{2} ) q^{76} + ( -60 - 56 \beta_{1} - 52 \beta_{2} ) q^{77} + ( 221 - 22 \beta_{1} - 12 \beta_{2} ) q^{79} + ( -290 - 85 \beta_{1} ) q^{80} + ( -534 - 38 \beta_{1} - 42 \beta_{2} ) q^{82} + ( 489 - 120 \beta_{1} - 60 \beta_{2} ) q^{83} + ( -15 - 10 \beta_{1} - 40 \beta_{2} ) q^{85} + ( 1218 - 2 \beta_{1} + 78 \beta_{2} ) q^{86} + ( -192 - 278 \beta_{1} - 4 \beta_{2} ) q^{88} + ( -756 + 66 \beta_{1} + 48 \beta_{2} ) q^{89} + ( -56 - 196 \beta_{1} - 76 \beta_{2} ) q^{91} + ( -495 - 481 \beta_{1} + 7 \beta_{2} ) q^{92} + ( -396 + 108 \beta_{1} - 28 \beta_{2} ) q^{94} + ( -295 - 70 \beta_{1} + 20 \beta_{2} ) q^{95} + ( 440 + 64 \beta_{1} + 8 \beta_{2} ) q^{97} + ( 876 + 5 \beta_{1} + 60 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 23 q^{4} - 15 q^{5} + 44 q^{7} + 36 q^{8} + O(q^{10}) \) \( 3 q + q^{2} + 23 q^{4} - 15 q^{5} + 44 q^{7} + 36 q^{8} - 5 q^{10} - 38 q^{11} + 28 q^{13} + 108 q^{14} + 191 q^{16} + 19 q^{17} + 187 q^{19} - 115 q^{20} + 122 q^{22} + 81 q^{23} + 75 q^{25} - 416 q^{26} + 410 q^{28} - 160 q^{29} + 227 q^{31} + 569 q^{32} + 17 q^{34} - 220 q^{35} + 78 q^{37} + 757 q^{38} - 180 q^{40} + 338 q^{41} + 22 q^{43} - 1636 q^{44} - 1425 q^{46} + 472 q^{47} - 197 q^{49} + 25 q^{50} - 1566 q^{52} - 521 q^{53} + 190 q^{55} + 1254 q^{56} - 2096 q^{58} - 140 q^{59} + 595 q^{61} - 1407 q^{62} - 918 q^{64} - 140 q^{65} + 878 q^{67} + 3053 q^{68} - 540 q^{70} + 602 q^{71} + 1294 q^{73} - 2878 q^{74} + 525 q^{76} - 288 q^{77} + 629 q^{79} - 955 q^{80} - 1682 q^{82} + 1287 q^{83} - 95 q^{85} + 3730 q^{86} - 858 q^{88} - 2154 q^{89} - 440 q^{91} - 1959 q^{92} - 1108 q^{94} - 935 q^{95} + 1392 q^{97} + 2693 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 23 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 15 \)

Display \(\nu^j\) in terms of \(\beta_i\)

\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 15\)

Display \(\beta_i\) in terms of \(\nu^j\)

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
−4.45938
0.258712
5.20067
−4.45938 0 11.8861 −5.00000 0 5.08123 −17.3296 0 22.2969
1.2 0.258712 0 −7.93307 −5.00000 0 14.5174 −4.12208 0 −1.29356
1.3 5.20067 0 19.0470 −5.00000 0 24.4013 57.4517 0 −26.0034
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)

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 135.4.a.g yes 3
3.b odd 2 1 135.4.a.f ✓ 3
4.b odd 2 1 2160.4.a.be 3
5.b even 2 1 675.4.a.q 3
5.c odd 4 2 675.4.b.k 6
9.c even 3 2 405.4.e.r 6
9.d odd 6 2 405.4.e.t 6
12.b even 2 1 2160.4.a.bm 3
15.d odd 2 1 675.4.a.r 3
15.e even 4 2 675.4.b.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.f ✓ 3 3.b odd 2 1
135.4.a.g yes 3 1.a even 1 1 trivial
405.4.e.r 6 9.c even 3 2
405.4.e.t 6 9.d odd 6 2
675.4.a.q 3 5.b even 2 1
675.4.a.r 3 15.d odd 2 1
675.4.b.k 6 5.c odd 4 2
675.4.b.l 6 15.e even 4 2
2160.4.a.be 3 4.b odd 2 1
2160.4.a.bm 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 23 T_{2} + 6 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(135))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 6 - 23 T - T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( 5 + T )^{3} \)
$7$ \( -1800 + 552 T - 44 T^{2} + T^{3} \)
$11$ \( -83280 - 2612 T + 38 T^{2} + T^{3} \)
$13$ \( 100120 - 4576 T - 28 T^{2} + T^{3} \)
$17$ \( 553887 - 11477 T - 19 T^{2} + T^{3} \)
$19$ \( 525871 + 3587 T - 187 T^{2} + T^{3} \)
$23$ \( 2043981 - 23301 T - 81 T^{2} + T^{3} \)
$29$ \( -7892760 - 47768 T + 160 T^{2} + T^{3} \)
$31$ \( -246321 - 17973 T - 227 T^{2} + T^{3} \)
$37$ \( 13637080 - 99924 T - 78 T^{2} + T^{3} \)
$41$ \( 12116640 - 42812 T - 338 T^{2} + T^{3} \)
$43$ \( 18464560 - 159916 T - 22 T^{2} + T^{3} \)
$47$ \( 283200 + 54208 T - 472 T^{2} + T^{3} \)
$53$ \( -939789 + 61387 T + 521 T^{2} + T^{3} \)
$59$ \( -34131480 - 166448 T + 140 T^{2} + T^{3} \)
$61$ \( 1782607 - 2749 T - 595 T^{2} + T^{3} \)
$67$ \( 11295000 + 75948 T - 878 T^{2} + T^{3} \)
$71$ \( 280550880 - 583652 T - 602 T^{2} + T^{3} \)
$73$ \( 404091280 - 95908 T - 1294 T^{2} + T^{3} \)
$79$ \( -2010303 + 97059 T - 629 T^{2} + T^{3} \)
$83$ \( 346404411 - 365877 T - 1287 T^{2} + T^{3} \)
$89$ \( 74325600 + 1057572 T + 2154 T^{2} + T^{3} \)
$97$ \( -63595520 + 543936 T - 1392 T^{2} + T^{3} \)
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