Properties

Label 136.4.a.a
Level $136$
Weight $4$
Character orbit 136.a
Self dual yes
Analytic conductor $8.024$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + \beta ) q^{3} + ( -6 - 2 \beta ) q^{5} + ( -18 - 3 \beta ) q^{7} + ( -11 + 4 \beta ) q^{9} +O(q^{10})\) \( q + ( 2 + \beta ) q^{3} + ( -6 - 2 \beta ) q^{5} + ( -18 - 3 \beta ) q^{7} + ( -11 + 4 \beta ) q^{9} + ( -10 - 13 \beta ) q^{11} + ( -42 + 12 \beta ) q^{13} + ( -36 - 10 \beta ) q^{15} + 17 q^{17} + ( 16 + 38 \beta ) q^{19} + ( -72 - 24 \beta ) q^{21} + ( 22 + 25 \beta ) q^{23} + ( -41 + 24 \beta ) q^{25} + ( -28 - 30 \beta ) q^{27} + ( -198 + 26 \beta ) q^{29} + ( 58 - 37 \beta ) q^{31} + ( -176 - 36 \beta ) q^{33} + ( 180 + 54 \beta ) q^{35} + ( -70 - 90 \beta ) q^{37} + ( 60 - 18 \beta ) q^{39} + ( -30 + 4 \beta ) q^{41} + ( 320 + 18 \beta ) q^{43} + ( -30 - 2 \beta ) q^{45} + ( -248 - 60 \beta ) q^{47} + ( 89 + 108 \beta ) q^{49} + ( 34 + 17 \beta ) q^{51} + ( 118 - 28 \beta ) q^{53} + ( 372 + 98 \beta ) q^{55} + ( 488 + 92 \beta ) q^{57} + ( 288 - 86 \beta ) q^{59} + ( -174 - 146 \beta ) q^{61} + ( 54 - 39 \beta ) q^{63} + ( -36 + 12 \beta ) q^{65} + ( 764 - 16 \beta ) q^{67} + ( 344 + 72 \beta ) q^{69} + ( -438 + 151 \beta ) q^{71} + ( -190 + 44 \beta ) q^{73} + ( 206 + 7 \beta ) q^{75} + ( 648 + 264 \beta ) q^{77} + ( 86 + 173 \beta ) q^{79} + ( -119 - 196 \beta ) q^{81} + ( -512 - 226 \beta ) q^{83} + ( -102 - 34 \beta ) q^{85} + ( -84 - 146 \beta ) q^{87} + ( -422 - 332 \beta ) q^{89} + ( 324 - 90 \beta ) q^{91} + ( -328 - 16 \beta ) q^{93} + ( -1008 - 260 \beta ) q^{95} + ( 18 - 56 \beta ) q^{97} + ( -514 + 103 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} - 12 q^{5} - 36 q^{7} - 22 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{3} - 12 q^{5} - 36 q^{7} - 22 q^{9} - 20 q^{11} - 84 q^{13} - 72 q^{15} + 34 q^{17} + 32 q^{19} - 144 q^{21} + 44 q^{23} - 82 q^{25} - 56 q^{27} - 396 q^{29} + 116 q^{31} - 352 q^{33} + 360 q^{35} - 140 q^{37} + 120 q^{39} - 60 q^{41} + 640 q^{43} - 60 q^{45} - 496 q^{47} + 178 q^{49} + 68 q^{51} + 236 q^{53} + 744 q^{55} + 976 q^{57} + 576 q^{59} - 348 q^{61} + 108 q^{63} - 72 q^{65} + 1528 q^{67} + 688 q^{69} - 876 q^{71} - 380 q^{73} + 412 q^{75} + 1296 q^{77} + 172 q^{79} - 238 q^{81} - 1024 q^{83} - 204 q^{85} - 168 q^{87} - 844 q^{89} + 648 q^{91} - 656 q^{93} - 2016 q^{95} + 36 q^{97} - 1028 q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
0 −1.46410 0 0.928203 0 −7.60770 0 −24.8564 0
1.2 0 5.46410 0 −12.9282 0 −28.3923 0 2.85641 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.a 2
3.b odd 2 1 1224.4.a.d 2
4.b odd 2 1 272.4.a.f 2
8.b even 2 1 1088.4.a.n 2
8.d odd 2 1 1088.4.a.p 2
12.b even 2 1 2448.4.a.z 2
17.b even 2 1 2312.4.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.a.a 2 1.a even 1 1 trivial
272.4.a.f 2 4.b odd 2 1
1088.4.a.n 2 8.b even 2 1
1088.4.a.p 2 8.d odd 2 1
1224.4.a.d 2 3.b odd 2 1
2312.4.a.b 2 17.b even 2 1
2448.4.a.z 2 12.b even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -8 - 4 T + T^{2} \)
$5$ \( -12 + 12 T + T^{2} \)
$7$ \( 216 + 36 T + T^{2} \)
$11$ \( -1928 + 20 T + T^{2} \)
$13$ \( 36 + 84 T + T^{2} \)
$17$ \( ( -17 + T )^{2} \)
$19$ \( -17072 - 32 T + T^{2} \)
$23$ \( -7016 - 44 T + T^{2} \)
$29$ \( 31092 + 396 T + T^{2} \)
$31$ \( -13064 - 116 T + T^{2} \)
$37$ \( -92300 + 140 T + T^{2} \)
$41$ \( 708 + 60 T + T^{2} \)
$43$ \( 98512 - 640 T + T^{2} \)
$47$ \( 18304 + 496 T + T^{2} \)
$53$ \( 4516 - 236 T + T^{2} \)
$59$ \( -5808 - 576 T + T^{2} \)
$61$ \( -225516 + 348 T + T^{2} \)
$67$ \( 580624 - 1528 T + T^{2} \)
$71$ \( -81768 + 876 T + T^{2} \)
$73$ \( 12868 + 380 T + T^{2} \)
$79$ \( -351752 - 172 T + T^{2} \)
$83$ \( -350768 + 1024 T + T^{2} \)
$89$ \( -1144604 + 844 T + T^{2} \)
$97$ \( -37308 - 36 T + T^{2} \)
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