Properties

Label 136.4.a.b
Level $136$
Weight $4$
Character orbit 136.a
Self dual yes
Analytic conductor $8.024$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.02425976078\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.8396.1
Defining polynomial: \(x^{3} - x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} ) q^{3} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{5} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{7} + ( -4 - \beta_{1} - 6 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} ) q^{3} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{5} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{7} + ( -4 - \beta_{1} - 6 \beta_{2} ) q^{9} + ( -27 + 2 \beta_{1} + \beta_{2} ) q^{11} + ( -15 + 3 \beta_{1} + 2 \beta_{2} ) q^{13} + ( -46 + 4 \beta_{1} + 6 \beta_{2} ) q^{15} -17 q^{17} + ( -72 + 2 \beta_{1} + 6 \beta_{2} ) q^{19} + ( -5 - 5 \beta_{1} + 6 \beta_{2} ) q^{21} + ( -78 - \beta_{1} + \beta_{2} ) q^{23} + ( 49 - 22 \beta_{1} + 8 \beta_{2} ) q^{25} + ( -96 + 4 \beta_{1} ) q^{27} + ( -13 + 15 \beta_{1} + 18 \beta_{2} ) q^{29} + ( -8 - 25 \beta_{1} + 3 \beta_{2} ) q^{31} + ( 39 + 3 \beta_{1} - 34 \beta_{2} ) q^{33} + ( -146 + 20 \beta_{1} - 30 \beta_{2} ) q^{35} + ( 55 + 27 \beta_{1} + 38 \beta_{2} ) q^{37} + ( 44 + 4 \beta_{1} - 28 \beta_{2} ) q^{39} + ( 64 + 14 \beta_{1} - 40 \beta_{2} ) q^{41} + ( -20 - 26 \beta_{1} - 78 \beta_{2} ) q^{43} + ( 239 - 25 \beta_{1} - 26 \beta_{2} ) q^{45} + ( 22 + 30 \beta_{1} + 52 \beta_{2} ) q^{47} + ( 166 + 29 \beta_{1} + 66 \beta_{2} ) q^{49} + ( 17 - 17 \beta_{2} ) q^{51} + ( 142 - 48 \beta_{1} + 12 \beta_{2} ) q^{53} + ( 166 - 40 \beta_{1} + 78 \beta_{2} ) q^{55} + ( 194 - 2 \beta_{1} - 104 \beta_{2} ) q^{57} + ( -8 - 42 \beta_{1} + 86 \beta_{2} ) q^{59} + ( 183 - 25 \beta_{1} + 114 \beta_{2} ) q^{61} + ( 216 + 65 \beta_{1} - 3 \beta_{2} ) q^{63} + ( 148 - 32 \beta_{1} + 68 \beta_{2} ) q^{65} + ( -282 + 102 \beta_{1} + 60 \beta_{2} ) q^{67} + ( 105 - 3 \beta_{1} - 82 \beta_{2} ) q^{69} + ( 136 - 51 \beta_{1} - 59 \beta_{2} ) q^{71} + ( 266 + 56 \beta_{1} - 96 \beta_{2} ) q^{73} + ( 237 - 52 \beta_{1} + 31 \beta_{2} ) q^{75} + ( -285 + 63 \beta_{1} - 14 \beta_{2} ) q^{77} + ( 406 + 23 \beta_{1} - 43 \beta_{2} ) q^{79} + ( 184 + 35 \beta_{1} + 62 \beta_{2} ) q^{81} + ( -424 - 22 \beta_{1} - 102 \beta_{2} ) q^{83} + ( 51 - 17 \beta_{1} + 34 \beta_{2} ) q^{85} + ( 334 + 12 \beta_{1} - 118 \beta_{2} ) q^{87} + ( -827 + 43 \beta_{1} - 18 \beta_{2} ) q^{89} + ( -482 + 14 \beta_{1} - 44 \beta_{2} ) q^{91} + ( 199 - 53 \beta_{1} + 2 \beta_{2} ) q^{93} + ( 56 - 60 \beta_{1} + 188 \beta_{2} ) q^{95} + ( 618 - 152 \beta_{1} - 92 \beta_{2} ) q^{97} + ( -73 - 14 \beta_{1} + 179 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{3} - 8 q^{5} - 2 q^{7} - 5 q^{9} + O(q^{10}) \) \( 3 q - 4 q^{3} - 8 q^{5} - 2 q^{7} - 5 q^{9} - 84 q^{11} - 50 q^{13} - 148 q^{15} - 51 q^{17} - 224 q^{19} - 16 q^{21} - 234 q^{23} + 161 q^{25} - 292 q^{27} - 72 q^{29} - 2 q^{31} + 148 q^{33} - 428 q^{35} + 100 q^{37} + 156 q^{39} + 218 q^{41} + 44 q^{43} + 768 q^{45} - 16 q^{47} + 403 q^{49} + 68 q^{51} + 462 q^{53} + 460 q^{55} + 688 q^{57} - 68 q^{59} + 460 q^{61} + 586 q^{63} + 408 q^{65} - 1008 q^{67} + 400 q^{69} + 518 q^{71} + 838 q^{73} + 732 q^{75} - 904 q^{77} + 1238 q^{79} + 455 q^{81} - 1148 q^{83} + 136 q^{85} + 1108 q^{87} - 2506 q^{89} - 1416 q^{91} + 648 q^{93} + 40 q^{95} + 2098 q^{97} - 384 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 20 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 14 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(4 \beta_{2} + \beta_{1} + 29\)\()/2\)

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
0.395276
4.81129
−4.20657
0 −8.11952 0 11.0296 0 5.74786 0 38.9265 0
1.2 0 1.16862 0 1.28534 0 −30.0364 0 −25.6343 0
1.3 0 2.95089 0 −20.3149 0 22.2885 0 −18.2922 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)

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 136.4.a.b 3
3.b odd 2 1 1224.4.a.i 3
4.b odd 2 1 272.4.a.j 3
8.b even 2 1 1088.4.a.y 3
8.d odd 2 1 1088.4.a.u 3
12.b even 2 1 2448.4.a.bj 3
17.b even 2 1 2312.4.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.a.b 3 1.a even 1 1 trivial
272.4.a.j 3 4.b odd 2 1
1088.4.a.u 3 8.d odd 2 1
1088.4.a.y 3 8.b even 2 1
1224.4.a.i 3 3.b odd 2 1
2312.4.a.d 3 17.b even 2 1
2448.4.a.bj 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 4 T_{3}^{2} - 30 T_{3} + 28 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(136))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 28 - 30 T + 4 T^{2} + T^{3} \)
$5$ \( 288 - 236 T + 8 T^{2} + T^{3} \)
$7$ \( 3848 - 714 T + 2 T^{2} + T^{3} \)
$11$ \( 10972 + 2026 T + 84 T^{2} + T^{3} \)
$13$ \( -16048 + 64 T + 50 T^{2} + T^{3} \)
$17$ \( ( 17 + T )^{3} \)
$19$ \( 322592 + 15336 T + 224 T^{2} + T^{3} \)
$23$ \( 463504 + 18118 T + 234 T^{2} + T^{3} \)
$29$ \( -1862624 - 23340 T + 72 T^{2} + T^{3} \)
$31$ \( -1252344 - 52450 T + 2 T^{2} + T^{3} \)
$37$ \( -4014352 - 89196 T - 100 T^{2} + T^{3} \)
$41$ \( 7651496 - 66340 T - 218 T^{2} + T^{3} \)
$43$ \( -18647936 - 234152 T - 44 T^{2} + T^{3} \)
$47$ \( -7663872 - 141616 T + 16 T^{2} + T^{3} \)
$53$ \( 10502936 - 131316 T - 462 T^{2} + T^{3} \)
$59$ \( -81654208 - 465864 T + 68 T^{2} + T^{3} \)
$61$ \( 116249648 - 488892 T - 460 T^{2} + T^{3} \)
$67$ \( -534256128 - 528624 T + 1008 T^{2} + T^{3} \)
$71$ \( 93696432 - 192946 T - 518 T^{2} + T^{3} \)
$73$ \( 324704504 - 439796 T - 838 T^{2} + T^{3} \)
$79$ \( -7088608 + 385382 T - 1238 T^{2} + T^{3} \)
$83$ \( -158784832 + 71224 T + 1148 T^{2} + T^{3} \)
$89$ \( 456757648 + 1918096 T + 2506 T^{2} + T^{3} \)
$97$ \( 1961875944 - 468596 T - 2098 T^{2} + T^{3} \)
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