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Properties

Label 136.2.n.a
Level $136$
Weight $2$
Character orbit 136.n
Analytic conductor $1.086$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 136.n (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.08596546749\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{8} ) q^{3} + ( 1 - \zeta_{8} ) q^{5} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} + ( 1 - \zeta_{8} + \zeta_{8}^{2} ) q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{8} ) q^{3} + ( 1 - \zeta_{8} ) q^{5} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} + ( 1 - \zeta_{8} + \zeta_{8}^{2} ) q^{9} + ( 1 - \zeta_{8}^{3} ) q^{11} + ( -1 + \zeta_{8}^{2} ) q^{15} + ( 1 - 4 \zeta_{8}^{2} ) q^{17} + ( -3 + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{19} + ( 3 \zeta_{8} + 4 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{21} + ( 1 + 4 \zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{23} + ( 1 + 3 \zeta_{8} + \zeta_{8}^{2} ) q^{25} + ( -1 + 3 \zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( -3 + 3 \zeta_{8} ) q^{29} + ( 3 + 3 \zeta_{8} ) q^{31} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{33} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{35} + ( 1 + \zeta_{8} ) q^{37} + ( -3 - 4 \zeta_{8} - 4 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{41} + ( 3 - 6 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{43} + ( 1 - 2 \zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{45} + ( 2 \zeta_{8} - 10 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{47} + ( -7 + 7 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{49} + ( -1 - \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{51} + ( 5 - 5 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{53} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{55} + ( 5 + 3 \zeta_{8} - 3 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{57} + ( 7 + 2 \zeta_{8} + 7 \zeta_{8}^{2} ) q^{59} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{61} + ( -\zeta_{8}^{2} - \zeta_{8}^{3} ) q^{63} + ( 4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{67} + ( -2 - 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{69} + ( 3 + 3 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{71} + ( 1 - \zeta_{8} + 8 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{73} + ( -1 - 4 \zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{75} + ( -3 - 4 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{77} + ( 9 - 4 \zeta_{8} + 4 \zeta_{8}^{2} - 9 \zeta_{8}^{3} ) q^{79} + ( -5 \zeta_{8} - 3 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{81} + ( -3 + 3 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{83} + ( 1 - \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{85} + ( 3 - 3 \zeta_{8}^{2} ) q^{87} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{89} + ( -3 - 6 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{93} + ( -1 + 3 \zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{95} + ( -7 + 7 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{97} + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{15} + 4 q^{17} - 12 q^{19} + 4 q^{23} + 4 q^{25} - 4 q^{27} - 12 q^{29} + 12 q^{31} - 8 q^{33} - 8 q^{35} + 4 q^{37} - 12 q^{41} + 12 q^{43} + 4 q^{45} - 28 q^{49} - 4 q^{51} + 20 q^{53} + 20 q^{57} + 28 q^{59} - 28 q^{61} + 16 q^{67} - 8 q^{69} + 12 q^{71} + 4 q^{73} - 4 q^{75} - 12 q^{77} + 36 q^{79} - 12 q^{83} + 4 q^{85} + 12 q^{87} - 12 q^{93} - 4 q^{95} - 28 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
0.707107 + 0.707107i
−0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 − 0.707107i
0 −1.70711 − 0.707107i 0 0.292893 − 0.707107i 0 −1.70711 − 4.12132i 0 0.292893 + 0.292893i 0
25.1 0 −0.292893 + 0.707107i 0 1.70711 + 0.707107i 0 −0.292893 + 0.121320i 0 1.70711 + 1.70711i 0
49.1 0 −0.292893 − 0.707107i 0 1.70711 − 0.707107i 0 −0.292893 − 0.121320i 0 1.70711 − 1.70711i 0
121.1 0 −1.70711 + 0.707107i 0 0.292893 + 0.707107i 0 −1.70711 + 4.12132i 0 0.292893 − 0.292893i 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
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.2.n.a ✓ 4
3.b odd 2 1 1224.2.bq.a 4
4.b odd 2 1 272.2.v.e 4
17.d even 8 1 inner 136.2.n.a ✓ 4
17.e odd 16 2 2312.2.a.s 4
17.e odd 16 2 2312.2.b.j 4
51.g odd 8 1 1224.2.bq.a 4
68.g odd 8 1 272.2.v.e 4
68.i even 16 2 4624.2.a.bm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.n.a ✓ 4 1.a even 1 1 trivial
136.2.n.a ✓ 4 17.d even 8 1 inner
272.2.v.e 4 4.b odd 2 1
272.2.v.e 4 68.g odd 8 1
1224.2.bq.a 4 3.b odd 2 1
1224.2.bq.a 4 51.g odd 8 1
2312.2.a.s 4 17.e odd 16 2
2312.2.b.j 4 17.e odd 16 2
4624.2.a.bm 4 68.i even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4 T_{3}^{3} + 6 T_{3}^{2} + 4 T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(136, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4} \)
$5$ \( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 2 + 12 T + 22 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 17 - 2 T + T^{2} )^{2} \)
$19$ \( 196 + 168 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$23$ \( 1058 - 276 T + 22 T^{2} - 4 T^{3} + T^{4} \)
$29$ \( 162 + 108 T + 54 T^{2} + 12 T^{3} + T^{4} \)
$31$ \( 162 - 108 T + 54 T^{2} - 12 T^{3} + T^{4} \)
$37$ \( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( 578 + 476 T + 134 T^{2} + 12 T^{3} + T^{4} \)
$43$ \( 324 + 216 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$47$ \( 8464 + 216 T^{2} + T^{4} \)
$53$ \( 1156 - 680 T + 200 T^{2} - 20 T^{3} + T^{4} \)
$59$ \( 8836 - 2632 T + 392 T^{2} - 28 T^{3} + T^{4} \)
$61$ \( 15842 + 3916 T + 438 T^{2} + 28 T^{3} + T^{4} \)
$67$ \( ( -16 - 8 T + T^{2} )^{2} \)
$71$ \( 578 - 476 T + 134 T^{2} - 12 T^{3} + T^{4} \)
$73$ \( 12482 - 2212 T + 102 T^{2} - 4 T^{3} + T^{4} \)
$79$ \( 37538 - 7124 T + 662 T^{2} - 36 T^{3} + T^{4} \)
$83$ \( 324 - 216 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$89$ \( 256 + 96 T^{2} + T^{4} \)
$97$ \( 1058 - 276 T + 214 T^{2} + 28 T^{3} + T^{4} \)
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