Properties

Label 136.3.e.b
Level $136$
Weight $3$
Character orbit 136.e
Analytic conductor $3.706$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 136.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.70573159530\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + \beta q^{3} + 4 q^{4} + 2 \beta q^{6} + 8 q^{8} -23 q^{9} +O(q^{10})\) \( q + 2 q^{2} + \beta q^{3} + 4 q^{4} + 2 \beta q^{6} + 8 q^{8} -23 q^{9} -3 \beta q^{11} + 4 \beta q^{12} + 16 q^{16} + ( -1 + 3 \beta ) q^{17} -46 q^{18} + 34 q^{19} -6 \beta q^{22} + 8 \beta q^{24} -25 q^{25} -14 \beta q^{27} + 32 q^{32} + 96 q^{33} + ( -2 + 6 \beta ) q^{34} -92 q^{36} + 68 q^{38} -12 \beta q^{41} + 14 q^{43} -12 \beta q^{44} + 16 \beta q^{48} -49 q^{49} -50 q^{50} + ( -96 - \beta ) q^{51} -28 \beta q^{54} + 34 \beta q^{57} -82 q^{59} + 64 q^{64} + 192 q^{66} -62 q^{67} + ( -4 + 12 \beta ) q^{68} -184 q^{72} + 6 \beta q^{73} -25 \beta q^{75} + 136 q^{76} + 241 q^{81} -24 \beta q^{82} + 158 q^{83} + 28 q^{86} -24 \beta q^{88} -146 q^{89} + 32 \beta q^{96} + 30 \beta q^{97} -98 q^{98} + 69 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} - 46 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} - 46 q^{9} + 32 q^{16} - 2 q^{17} - 92 q^{18} + 68 q^{19} - 50 q^{25} + 64 q^{32} + 192 q^{33} - 4 q^{34} - 184 q^{36} + 136 q^{38} + 28 q^{43} - 98 q^{49} - 100 q^{50} - 192 q^{51} - 164 q^{59} + 128 q^{64} + 384 q^{66} - 124 q^{67} - 8 q^{68} - 368 q^{72} + 272 q^{76} + 482 q^{81} + 316 q^{83} + 56 q^{86} - 292 q^{89} - 196 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
1.41421i
1.41421i
2.00000 5.65685i 4.00000 0 11.3137i 0 8.00000 −23.0000 0
67.2 2.00000 5.65685i 4.00000 0 11.3137i 0 8.00000 −23.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
17.b even 2 1 inner
136.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.3.e.b 2
4.b odd 2 1 544.3.e.a 2
8.b even 2 1 544.3.e.a 2
8.d odd 2 1 CM 136.3.e.b 2
17.b even 2 1 inner 136.3.e.b 2
68.d odd 2 1 544.3.e.a 2
136.e odd 2 1 inner 136.3.e.b 2
136.h even 2 1 544.3.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.e.b 2 1.a even 1 1 trivial
136.3.e.b 2 8.d odd 2 1 CM
136.3.e.b 2 17.b even 2 1 inner
136.3.e.b 2 136.e odd 2 1 inner
544.3.e.a 2 4.b odd 2 1
544.3.e.a 2 8.b even 2 1
544.3.e.a 2 68.d odd 2 1
544.3.e.a 2 136.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(136, [\chi])\):

\( T_{3}^{2} + 32 \)
\( T_{5} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{2} \)
$3$ \( 32 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 288 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 289 + 2 T + T^{2} \)
$19$ \( ( -34 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( 4608 + T^{2} \)
$43$ \( ( -14 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( ( 82 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( ( 62 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 1152 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( -158 + T )^{2} \)
$89$ \( ( 146 + T )^{2} \)
$97$ \( 28800 + T^{2} \)
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