Properties

Label 136.1.p.a
Level 136
Weight 1
Character orbit 136.p
Analytic conductor 0.068
Analytic rank 0
Dimension 4
Projective image \(D_{8}\)
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.p (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0678728417181\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{8}\)
Projective field Galois closure of 8.0.1680747204608.3

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{2} + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} -\zeta_{8}^{2} q^{4} + ( -\zeta_{8} + \zeta_{8}^{2} ) q^{6} + \zeta_{8} q^{8} + ( -1 + \zeta_{8} - \zeta_{8}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{8}^{3} q^{2} + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} -\zeta_{8}^{2} q^{4} + ( -\zeta_{8} + \zeta_{8}^{2} ) q^{6} + \zeta_{8} q^{8} + ( -1 + \zeta_{8} - \zeta_{8}^{2} ) q^{9} + ( -1 + \zeta_{8}^{3} ) q^{11} + ( 1 - \zeta_{8} ) q^{12} - q^{16} -\zeta_{8}^{3} q^{17} + ( -1 + \zeta_{8} - \zeta_{8}^{3} ) q^{18} + ( -\zeta_{8}^{2} - \zeta_{8}^{3} ) q^{22} + ( 1 + \zeta_{8}^{3} ) q^{24} -\zeta_{8} q^{25} + ( 1 - \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{27} -\zeta_{8}^{3} q^{32} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{33} + \zeta_{8}^{2} q^{34} + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{36} + ( \zeta_{8} + \zeta_{8}^{2} ) q^{41} + ( 1 + \zeta_{8}^{2} ) q^{43} + ( \zeta_{8} + \zeta_{8}^{2} ) q^{44} + ( -\zeta_{8}^{2} + \zeta_{8}^{3} ) q^{48} + \zeta_{8}^{3} q^{49} + q^{50} + ( \zeta_{8} - \zeta_{8}^{2} ) q^{51} + ( 1 + \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{54} + ( -1 - \zeta_{8}^{2} ) q^{59} + \zeta_{8}^{2} q^{64} + ( 1 - \zeta_{8}^{2} ) q^{66} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{67} -\zeta_{8} q^{68} + ( -\zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{72} + ( \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{73} + ( -1 - \zeta_{8}^{3} ) q^{75} + ( -\zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{81} + ( -1 - \zeta_{8} ) q^{82} + ( 1 - \zeta_{8}^{2} ) q^{83} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{86} + ( -1 - \zeta_{8} ) q^{88} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{89} + ( \zeta_{8} - \zeta_{8}^{2} ) q^{96} + ( -1 + \zeta_{8} ) q^{97} -\zeta_{8}^{2} q^{98} + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} - 4q^{11} + 4q^{12} - 4q^{16} - 4q^{18} + 4q^{24} + 4q^{27} - 4q^{36} + 4q^{43} + 4q^{50} + 4q^{54} - 4q^{59} + 4q^{66} - 4q^{75} - 4q^{82} + 4q^{83} - 4q^{88} - 4q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i 0.707107 0.292893i 1.00000i 0 −0.707107 0.292893i 0 0.707107 0.707107i −0.292893 + 0.292893i 0
43.1 −0.707107 + 0.707107i 0.707107 + 0.292893i 1.00000i 0 −0.707107 + 0.292893i 0 0.707107 + 0.707107i −0.292893 0.292893i 0
59.1 0.707107 0.707107i −0.707107 + 1.70711i 1.00000i 0 0.707107 + 1.70711i 0 −0.707107 0.707107i −1.70711 1.70711i 0
83.1 0.707107 + 0.707107i −0.707107 1.70711i 1.00000i 0 0.707107 1.70711i 0 −0.707107 + 0.707107i −1.70711 + 1.70711i 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.d even 8 1 inner
136.p odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.1.p.a 4
3.b odd 2 1 1224.1.bv.a 4
4.b odd 2 1 544.1.bl.a 4
5.b even 2 1 3400.1.ce.a 4
5.c odd 4 1 3400.1.br.a 4
5.c odd 4 1 3400.1.br.b 4
8.b even 2 1 544.1.bl.a 4
8.d odd 2 1 CM 136.1.p.a 4
17.b even 2 1 2312.1.p.b 4
17.c even 4 1 2312.1.p.a 4
17.c even 4 1 2312.1.p.d 4
17.d even 8 1 inner 136.1.p.a 4
17.d even 8 1 2312.1.p.a 4
17.d even 8 1 2312.1.p.b 4
17.d even 8 1 2312.1.p.d 4
17.e odd 16 2 2312.1.e.b 4
17.e odd 16 2 2312.1.f.c 4
17.e odd 16 4 2312.1.j.c 8
24.f even 2 1 1224.1.bv.a 4
40.e odd 2 1 3400.1.ce.a 4
40.k even 4 1 3400.1.br.a 4
40.k even 4 1 3400.1.br.b 4
51.g odd 8 1 1224.1.bv.a 4
68.g odd 8 1 544.1.bl.a 4
85.k odd 8 1 3400.1.br.a 4
85.m even 8 1 3400.1.ce.a 4
85.n odd 8 1 3400.1.br.b 4
136.e odd 2 1 2312.1.p.b 4
136.j odd 4 1 2312.1.p.a 4
136.j odd 4 1 2312.1.p.d 4
136.o even 8 1 544.1.bl.a 4
136.p odd 8 1 inner 136.1.p.a 4
136.p odd 8 1 2312.1.p.a 4
136.p odd 8 1 2312.1.p.b 4
136.p odd 8 1 2312.1.p.d 4
136.s even 16 2 2312.1.e.b 4
136.s even 16 2 2312.1.f.c 4
136.s even 16 4 2312.1.j.c 8
408.bd even 8 1 1224.1.bv.a 4
680.bq odd 8 1 3400.1.ce.a 4
680.bw even 8 1 3400.1.br.a 4
680.bz even 8 1 3400.1.br.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.p.a 4 1.a even 1 1 trivial
136.1.p.a 4 8.d odd 2 1 CM
136.1.p.a 4 17.d even 8 1 inner
136.1.p.a 4 136.p odd 8 1 inner
544.1.bl.a 4 4.b odd 2 1
544.1.bl.a 4 8.b even 2 1
544.1.bl.a 4 68.g odd 8 1
544.1.bl.a 4 136.o even 8 1
1224.1.bv.a 4 3.b odd 2 1
1224.1.bv.a 4 24.f even 2 1
1224.1.bv.a 4 51.g odd 8 1
1224.1.bv.a 4 408.bd even 8 1
2312.1.e.b 4 17.e odd 16 2
2312.1.e.b 4 136.s even 16 2
2312.1.f.c 4 17.e odd 16 2
2312.1.f.c 4 136.s even 16 2
2312.1.j.c 8 17.e odd 16 4
2312.1.j.c 8 136.s even 16 4
2312.1.p.a 4 17.c even 4 1
2312.1.p.a 4 17.d even 8 1
2312.1.p.a 4 136.j odd 4 1
2312.1.p.a 4 136.p odd 8 1
2312.1.p.b 4 17.b even 2 1
2312.1.p.b 4 17.d even 8 1
2312.1.p.b 4 136.e odd 2 1
2312.1.p.b 4 136.p odd 8 1
2312.1.p.d 4 17.c even 4 1
2312.1.p.d 4 17.d even 8 1
2312.1.p.d 4 136.j odd 4 1
2312.1.p.d 4 136.p odd 8 1
3400.1.br.a 4 5.c odd 4 1
3400.1.br.a 4 40.k even 4 1
3400.1.br.a 4 85.k odd 8 1
3400.1.br.a 4 680.bw even 8 1
3400.1.br.b 4 5.c odd 4 1
3400.1.br.b 4 40.k even 4 1
3400.1.br.b 4 85.n odd 8 1
3400.1.br.b 4 680.bz even 8 1
3400.1.ce.a 4 5.b even 2 1
3400.1.ce.a 4 40.e odd 2 1
3400.1.ce.a 4 85.m even 8 1
3400.1.ce.a 4 680.bq odd 8 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
$5$ \( 1 + T^{8} \)
$7$ \( 1 + T^{8} \)
$11$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
$13$ \( ( 1 + T^{2} )^{4} \)
$17$ \( 1 + T^{4} \)
$19$ \( ( 1 + T^{4} )^{2} \)
$23$ \( 1 + T^{8} \)
$29$ \( 1 + T^{8} \)
$31$ \( 1 + T^{8} \)
$37$ \( 1 + T^{8} \)
$41$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
$43$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
$47$ \( ( 1 + T^{2} )^{4} \)
$53$ \( ( 1 + T^{4} )^{2} \)
$59$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
$61$ \( 1 + T^{8} \)
$67$ \( ( 1 + T^{4} )^{2} \)
$71$ \( 1 + T^{8} \)
$73$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
$79$ \( 1 + T^{8} \)
$83$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
$89$ \( ( 1 + T^{4} )^{2} \)
$97$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
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