Properties

Label 136.4.a.c
Level $136$
Weight $4$
Character orbit 136.a
Self dual yes
Analytic conductor $8.024$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1556.1
Defining polynomial: \(x^{3} - x^{2} - 9 x + 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + ( 3 - \beta_{2} ) q^{3} + ( 1 - \beta_{1} ) q^{5} + ( 4 - \beta_{1} + \beta_{2} ) q^{7} + ( 22 + 3 \beta_{1} - 6 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 3 - \beta_{2} ) q^{3} + ( 1 - \beta_{1} ) q^{5} + ( 4 - \beta_{1} + \beta_{2} ) q^{7} + ( 22 + 3 \beta_{1} - 6 \beta_{2} ) q^{9} + ( 23 + 2 \beta_{1} + \beta_{2} ) q^{11} + ( 17 + \beta_{1} - 2 \beta_{2} ) q^{13} + ( 20 - 2 \beta_{1} + 6 \beta_{2} ) q^{15} + 17 q^{17} + ( 38 + 10 \beta_{2} ) q^{19} + ( -11 - 5 \beta_{1} + 6 \beta_{2} ) q^{21} + ( 18 - 11 \beta_{1} + 17 \beta_{2} ) q^{23} + ( -25 - 8 \beta_{1} + 8 \beta_{2} ) q^{25} + ( 174 + 24 \beta_{1} - 34 \beta_{2} ) q^{27} + ( -25 + 5 \beta_{1} + 16 \beta_{2} ) q^{29} + ( 104 + 5 \beta_{1} - 17 \beta_{2} ) q^{31} + ( -5 + \beta_{1} - 34 \beta_{2} ) q^{33} + ( 86 - 12 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -185 - 15 \beta_{1} + 28 \beta_{2} ) q^{37} + ( 114 + 8 \beta_{1} - 30 \beta_{2} ) q^{39} + ( 8 + 30 \beta_{1} - 24 \beta_{2} ) q^{41} + ( 14 - 12 \beta_{1} + 22 \beta_{2} ) q^{43} + ( -173 + 5 \beta_{1} + 12 \beta_{2} ) q^{45} + ( 16 - 20 \beta_{1} - 12 \beta_{2} ) q^{47} + ( -222 - 13 \beta_{1} + 2 \beta_{2} ) q^{49} + ( 51 - 17 \beta_{2} ) q^{51} + ( -196 + 30 \beta_{1} + 40 \beta_{2} ) q^{53} + ( -192 - 10 \beta_{1} - 22 \beta_{2} ) q^{55} + ( -286 - 30 \beta_{1} - 8 \beta_{2} ) q^{57} + ( 26 + 54 \beta_{2} ) q^{59} + ( -197 + 45 \beta_{1} - 68 \beta_{2} ) q^{61} + ( -296 - \beta_{1} + 37 \beta_{2} ) q^{63} + ( -48 - 8 \beta_{1} + 4 \beta_{2} ) q^{65} + ( -112 + 12 \beta_{1} - 8 \beta_{2} ) q^{67} + ( -439 - 73 \beta_{1} + 110 \beta_{2} ) q^{69} + ( -102 + 63 \beta_{1} - 25 \beta_{2} ) q^{71} + ( -206 - 52 \beta_{1} + 92 \beta_{2} ) q^{73} + ( -259 - 40 \beta_{1} + 105 \beta_{2} ) q^{75} + ( -49 + \beta_{1} + 18 \beta_{2} ) q^{77} + ( -278 + 81 \beta_{1} + 85 \beta_{2} ) q^{79} + ( 880 + 69 \beta_{1} - 282 \beta_{2} ) q^{81} + ( 322 - 20 \beta_{1} - 70 \beta_{2} ) q^{83} + ( 17 - 17 \beta_{1} ) q^{85} + ( -800 - 38 \beta_{1} + 38 \beta_{2} ) q^{87} + ( 601 + 37 \beta_{1} - 50 \beta_{2} ) q^{89} + ( -60 - 10 \beta_{1} + 22 \beta_{2} ) q^{91} + ( 907 + 61 \beta_{1} - 190 \beta_{2} ) q^{93} + ( -132 - 48 \beta_{1} - 60 \beta_{2} ) q^{95} + ( 258 + 40 \beta_{1} + 24 \beta_{2} ) q^{97} + ( 707 + 50 \beta_{1} - 131 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 8 q^{3} + 2 q^{5} + 12 q^{7} + 63 q^{9} + O(q^{10}) \) \( 3 q + 8 q^{3} + 2 q^{5} + 12 q^{7} + 63 q^{9} + 72 q^{11} + 50 q^{13} + 64 q^{15} + 51 q^{17} + 124 q^{19} - 32 q^{21} + 60 q^{23} - 75 q^{25} + 512 q^{27} - 54 q^{29} + 300 q^{31} - 48 q^{33} + 248 q^{35} - 542 q^{37} + 320 q^{39} + 30 q^{41} + 52 q^{43} - 502 q^{45} + 16 q^{47} - 677 q^{49} + 136 q^{51} - 518 q^{53} - 608 q^{55} - 896 q^{57} + 132 q^{59} - 614 q^{61} - 852 q^{63} - 148 q^{65} - 332 q^{67} - 1280 q^{69} - 268 q^{71} - 578 q^{73} - 712 q^{75} - 128 q^{77} - 668 q^{79} + 2427 q^{81} + 876 q^{83} + 34 q^{85} - 2400 q^{87} + 1790 q^{89} - 168 q^{91} + 2592 q^{93} - 504 q^{95} + 838 q^{97} + 2040 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 9 x + 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} + 2 \nu - 13 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{2} - \beta_{1} + 25\)\()/4\)

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
2.80979
−3.08060
1.27082
0 −5.40937 0 −9.23914 0 2.17022 0 2.26124 0
1.2 0 3.18096 0 14.3224 0 17.1415 0 −16.8815 0
1.3 0 10.2284 0 −3.08327 0 −7.31168 0 77.6202 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.c 3
3.b odd 2 1 1224.4.a.f 3
4.b odd 2 1 272.4.a.g 3
8.b even 2 1 1088.4.a.s 3
8.d odd 2 1 1088.4.a.z 3
12.b even 2 1 2448.4.a.bf 3
17.b even 2 1 2312.4.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.a.c 3 1.a even 1 1 trivial
272.4.a.g 3 4.b odd 2 1
1088.4.a.s 3 8.b even 2 1
1088.4.a.z 3 8.d odd 2 1
1224.4.a.f 3 3.b odd 2 1
2312.4.a.c 3 17.b even 2 1
2448.4.a.bf 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 8 T_{3}^{2} - 40 T_{3} + 176 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(136))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 176 - 40 T - 8 T^{2} + T^{3} \)
$5$ \( -408 - 148 T - 2 T^{2} + T^{3} \)
$7$ \( 272 - 104 T - 12 T^{2} + T^{3} \)
$11$ \( 4752 + 952 T - 72 T^{2} + T^{3} \)
$13$ \( -1496 + 556 T - 50 T^{2} + T^{3} \)
$17$ \( ( -17 + T )^{3} \)
$19$ \( 151488 - 1008 T - 124 T^{2} + T^{3} \)
$23$ \( 1168976 - 23624 T - 60 T^{2} + T^{3} \)
$29$ \( -1826616 - 23156 T + 54 T^{2} + T^{3} \)
$31$ \( -122448 + 13528 T - 300 T^{2} + T^{3} \)
$37$ \( -451992 + 40876 T + 542 T^{2} + T^{3} \)
$41$ \( 13345816 - 127188 T - 30 T^{2} + T^{3} \)
$43$ \( 2509632 - 34800 T - 52 T^{2} + T^{3} \)
$47$ \( 1737728 - 82560 T - 16 T^{2} + T^{3} \)
$53$ \( -97851384 - 213492 T + 518 T^{2} + T^{3} \)
$59$ \( 2838592 - 173040 T - 132 T^{2} + T^{3} \)
$61$ \( -115549176 - 280820 T + 614 T^{2} + T^{3} \)
$67$ \( 78912 + 16944 T + 332 T^{2} + T^{3} \)
$71$ \( 104748848 - 514696 T + 268 T^{2} + T^{3} \)
$73$ \( 17940632 - 530900 T + 578 T^{2} + T^{3} \)
$79$ \( -976074672 - 1678088 T + 668 T^{2} + T^{3} \)
$83$ \( 211499712 - 186608 T - 876 T^{2} + T^{3} \)
$89$ \( -72949736 + 818796 T - 1790 T^{2} + T^{3} \)
$97$ \( 60169976 - 96500 T - 838 T^{2} + T^{3} \)
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