Properties

Label 136.1.j.a
Level 136
Weight 1
Character orbit 136.j
Analytic conductor 0.068
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM discriminant -8
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 136.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0678728417181\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.314432.1
Artin image $C_4\wr C_2$
Artin field Galois closure of 8.0.20123648.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -i q^{2} + ( -1 - i ) q^{3} - q^{4} + ( -1 + i ) q^{6} + i q^{8} + i q^{9} +O(q^{10})\) \( q -i q^{2} + ( -1 - i ) q^{3} - q^{4} + ( -1 + i ) q^{6} + i q^{8} + i q^{9} + ( 1 - i ) q^{11} + ( 1 + i ) q^{12} + q^{16} -i q^{17} + q^{18} + 2 i q^{19} + ( -1 - i ) q^{22} + ( 1 - i ) q^{24} + i q^{25} -i q^{32} -2 q^{33} - q^{34} -i q^{36} + 2 q^{38} + ( -1 + i ) q^{41} + ( -1 + i ) q^{44} + ( -1 - i ) q^{48} -i q^{49} + q^{50} + ( -1 + i ) q^{51} + ( 2 - 2 i ) q^{57} - q^{64} + 2 i q^{66} + i q^{68} - q^{72} + ( -1 - i ) q^{73} + ( 1 - i ) q^{75} -2 i q^{76} + q^{81} + ( 1 + i ) q^{82} + ( 1 + i ) q^{88} + ( -1 + i ) q^{96} + ( 1 + i ) q^{97} - q^{98} + ( 1 + i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{4} - 2q^{6} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} - 2q^{6} + 2q^{11} + 2q^{12} + 2q^{16} + 2q^{18} - 2q^{22} + 2q^{24} - 4q^{33} - 2q^{34} + 4q^{38} - 2q^{41} - 2q^{44} - 2q^{48} + 2q^{50} - 2q^{51} + 4q^{57} - 2q^{64} - 2q^{72} - 2q^{73} + 2q^{75} + 2q^{81} + 2q^{82} + 2q^{88} - 2q^{96} + 2q^{97} - 2q^{98} + 2q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
1.00000i
1.00000i
1.00000i −1.00000 1.00000i −1.00000 0 −1.00000 + 1.00000i 0 1.00000i 1.00000i 0
123.1 1.00000i −1.00000 + 1.00000i −1.00000 0 −1.00000 1.00000i 0 1.00000i 1.00000i 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.c even 4 1 inner
136.j odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.1.j.a 2
3.b odd 2 1 1224.1.s.a 2
4.b odd 2 1 544.1.n.a 2
5.b even 2 1 3400.1.y.a 2
5.c odd 4 1 3400.1.bc.a 2
5.c odd 4 1 3400.1.bc.b 2
8.b even 2 1 544.1.n.a 2
8.d odd 2 1 CM 136.1.j.a 2
17.b even 2 1 2312.1.j.b 2
17.c even 4 1 inner 136.1.j.a 2
17.c even 4 1 2312.1.j.b 2
17.d even 8 2 2312.1.e.a 2
17.d even 8 2 2312.1.f.b 2
17.e odd 16 8 2312.1.p.e 8
24.f even 2 1 1224.1.s.a 2
40.e odd 2 1 3400.1.y.a 2
40.k even 4 1 3400.1.bc.a 2
40.k even 4 1 3400.1.bc.b 2
51.f odd 4 1 1224.1.s.a 2
68.f odd 4 1 544.1.n.a 2
85.f odd 4 1 3400.1.bc.a 2
85.i odd 4 1 3400.1.bc.b 2
85.j even 4 1 3400.1.y.a 2
136.e odd 2 1 2312.1.j.b 2
136.i even 4 1 544.1.n.a 2
136.j odd 4 1 inner 136.1.j.a 2
136.j odd 4 1 2312.1.j.b 2
136.p odd 8 2 2312.1.e.a 2
136.p odd 8 2 2312.1.f.b 2
136.s even 16 8 2312.1.p.e 8
408.q even 4 1 1224.1.s.a 2
680.t even 4 1 3400.1.bc.a 2
680.bc odd 4 1 3400.1.y.a 2
680.bl even 4 1 3400.1.bc.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.j.a 2 1.a even 1 1 trivial
136.1.j.a 2 8.d odd 2 1 CM
136.1.j.a 2 17.c even 4 1 inner
136.1.j.a 2 136.j odd 4 1 inner
544.1.n.a 2 4.b odd 2 1
544.1.n.a 2 8.b even 2 1
544.1.n.a 2 68.f odd 4 1
544.1.n.a 2 136.i even 4 1
1224.1.s.a 2 3.b odd 2 1
1224.1.s.a 2 24.f even 2 1
1224.1.s.a 2 51.f odd 4 1
1224.1.s.a 2 408.q even 4 1
2312.1.e.a 2 17.d even 8 2
2312.1.e.a 2 136.p odd 8 2
2312.1.f.b 2 17.d even 8 2
2312.1.f.b 2 136.p odd 8 2
2312.1.j.b 2 17.b even 2 1
2312.1.j.b 2 17.c even 4 1
2312.1.j.b 2 136.e odd 2 1
2312.1.j.b 2 136.j odd 4 1
2312.1.p.e 8 17.e odd 16 8
2312.1.p.e 8 136.s even 16 8
3400.1.y.a 2 5.b even 2 1
3400.1.y.a 2 40.e odd 2 1
3400.1.y.a 2 85.j even 4 1
3400.1.y.a 2 680.bc odd 4 1
3400.1.bc.a 2 5.c odd 4 1
3400.1.bc.a 2 40.k even 4 1
3400.1.bc.a 2 85.f odd 4 1
3400.1.bc.a 2 680.t even 4 1
3400.1.bc.b 2 5.c odd 4 1
3400.1.bc.b 2 40.k even 4 1
3400.1.bc.b 2 85.i odd 4 1
3400.1.bc.b 2 680.bl even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(136, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
$5$ \( 1 + T^{4} \)
$7$ \( 1 + T^{4} \)
$11$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
$13$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$17$ \( 1 + T^{2} \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( 1 + T^{4} \)
$29$ \( 1 + T^{4} \)
$31$ \( 1 + T^{4} \)
$37$ \( 1 + T^{4} \)
$41$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
$43$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$47$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$53$ \( ( 1 + T^{2} )^{2} \)
$59$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$61$ \( 1 + T^{4} \)
$67$ \( ( 1 + T^{2} )^{2} \)
$71$ \( 1 + T^{4} \)
$73$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
$79$ \( 1 + T^{4} \)
$83$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$89$ \( ( 1 + T^{2} )^{2} \)
$97$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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