Properties

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

Related objects

Downloads

Learn more

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: yes
Analytic conductor: \(3.70573159530\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = 2\sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{2} + 4 q^{4} + \beta q^{5} -\beta q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})\) \( q -2 q^{2} + 4 q^{4} + \beta q^{5} -\beta q^{7} -8 q^{8} + 9 q^{9} -2 \beta q^{10} + 2 \beta q^{14} + 16 q^{16} + 17 q^{17} -18 q^{18} + 30 q^{19} + 4 \beta q^{20} -\beta q^{23} + 43 q^{25} -4 \beta q^{28} -7 \beta q^{29} + 7 \beta q^{31} -32 q^{32} -34 q^{34} -68 q^{35} + 36 q^{36} + \beta q^{37} -60 q^{38} -8 \beta q^{40} -50 q^{43} + 9 \beta q^{45} + 2 \beta q^{46} + 19 q^{49} -86 q^{50} + 8 \beta q^{56} + 14 \beta q^{58} -18 q^{59} -7 \beta q^{61} -14 \beta q^{62} -9 \beta q^{63} + 64 q^{64} -66 q^{67} + 68 q^{68} + 136 q^{70} -17 \beta q^{71} -72 q^{72} -2 \beta q^{74} + 120 q^{76} + 7 \beta q^{79} + 16 \beta q^{80} + 81 q^{81} + 30 q^{83} + 17 \beta q^{85} + 100 q^{86} -110 q^{89} -18 \beta q^{90} -4 \beta q^{92} + 30 \beta q^{95} -38 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 18 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 18 q^{9} + 32 q^{16} + 34 q^{17} - 36 q^{18} + 60 q^{19} + 86 q^{25} - 64 q^{32} - 68 q^{34} - 136 q^{35} + 72 q^{36} - 120 q^{38} - 100 q^{43} + 38 q^{49} - 172 q^{50} - 36 q^{59} + 128 q^{64} - 132 q^{67} + 136 q^{68} + 272 q^{70} - 144 q^{72} + 240 q^{76} + 162 q^{81} + 60 q^{83} + 200 q^{86} - 220 q^{89} - 76 q^{98} + O(q^{100}) \)

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
−1.56155
2.56155
−2.00000 0 4.00000 −8.24621 0 8.24621 −8.00000 9.00000 16.4924
67.2 −2.00000 0 4.00000 8.24621 0 −8.24621 −8.00000 9.00000 −16.4924
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
136.e odd 2 1 CM by \(\Q(\sqrt{-34}) \)
8.d odd 2 1 inner
17.b even 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.a 2
4.b odd 2 1 544.3.e.c 2
8.b even 2 1 544.3.e.c 2
8.d odd 2 1 inner 136.3.e.a 2
17.b even 2 1 inner 136.3.e.a 2
68.d odd 2 1 544.3.e.c 2
136.e odd 2 1 CM 136.3.e.a 2
136.h even 2 1 544.3.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.e.a 2 1.a even 1 1 trivial
136.3.e.a 2 8.d odd 2 1 inner
136.3.e.a 2 17.b even 2 1 inner
136.3.e.a 2 136.e odd 2 1 CM
544.3.e.c 2 4.b odd 2 1
544.3.e.c 2 8.b even 2 1
544.3.e.c 2 68.d odd 2 1
544.3.e.c 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 \(S_{3}^{\mathrm{new}}(136, [\chi])\):

\( T_{3} \)
\( T_{5}^{2} - 68 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -68 + T^{2} \)
$7$ \( -68 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -17 + T )^{2} \)
$19$ \( ( -30 + T )^{2} \)
$23$ \( -68 + T^{2} \)
$29$ \( -3332 + T^{2} \)
$31$ \( -3332 + T^{2} \)
$37$ \( -68 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 50 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( ( 18 + T )^{2} \)
$61$ \( -3332 + T^{2} \)
$67$ \( ( 66 + T )^{2} \)
$71$ \( -19652 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( -3332 + T^{2} \)
$83$ \( ( -30 + T )^{2} \)
$89$ \( ( 110 + T )^{2} \)
$97$ \( T^{2} \)
show more
show less