Properties

Label 136.2.n.b
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: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8} + 1) q^{3} + (\zeta_{8}^{3} + \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{5} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} + 1) q^{7} + ( - 2 \zeta_{8}^{2} + \zeta_{8} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8} + 1) q^{3} + (\zeta_{8}^{3} + \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{5} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} + 1) q^{7} + ( - 2 \zeta_{8}^{2} + \zeta_{8} - 2) q^{9} + ( - 3 \zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8} + 3) q^{11} + ( - \zeta_{8}^{3} - 4 \zeta_{8}^{2} - \zeta_{8}) q^{13} + ( - 2 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 4) q^{15} + (2 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 2 \zeta_{8}) q^{17} + (2 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4) q^{19} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{21} + ( - \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 3 \zeta_{8} + 1) q^{23} + ( - \zeta_{8}^{2} - 5 \zeta_{8} - 1) q^{25} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{27} + (3 \zeta_{8} - 3) q^{29} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} + 1) q^{31} + (2 \zeta_{8}^{3} - 2 \zeta_{8} + 8) q^{33} + (2 \zeta_{8}^{3} - 2 \zeta_{8} - 6) q^{35} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{37} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{39} + ( - 5 \zeta_{8}^{3} - 5) q^{41} + ( - 4 \zeta_{8}^{2} - 6 \zeta_{8} - 4) q^{43} + ( - 5 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 4 \zeta_{8} + 5) q^{45} + (2 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 2 \zeta_{8}) q^{47} + ( - 3 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{49} + (7 \zeta_{8}^{3} - \zeta_{8}^{2} - 3 \zeta_{8} + 3) q^{51} + (4 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3) q^{53} + (10 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 10 \zeta_{8}) q^{55} + (2 \zeta_{8}^{3} - 10 \zeta_{8}^{2} + 10 \zeta_{8} - 2) q^{57} + (2 \zeta_{8}^{2} + 10 \zeta_{8} + 2) q^{59} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8} + 2) q^{61} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + \zeta_{8} - 1) q^{63} + ( - 7 \zeta_{8}^{3} + 7 \zeta_{8}^{2} + 7 \zeta_{8} + 7) q^{65} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{67} + (6 \zeta_{8}^{3} - 6 \zeta_{8} + 8) q^{69} + (7 \zeta_{8}^{3} - 7 \zeta_{8}^{2} - 5 \zeta_{8} - 5) q^{71} + (8 \zeta_{8}^{3} + 8 \zeta_{8}^{2} - \zeta_{8} + 1) q^{73} + (3 \zeta_{8}^{3} - 5 \zeta_{8}^{2} - 5 \zeta_{8} + 3) q^{75} + (6 \zeta_{8}^{2} + 4 \zeta_{8} + 6) q^{77} + ( - 3 \zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8} + 3) q^{79} + (2 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 2 \zeta_{8}) q^{81} + ( - 10 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{83} + ( - 12 \zeta_{8}^{2} - \zeta_{8} - 5) q^{85} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}^{2} - 6) q^{87} + (3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{89} + ( - 6 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4 \zeta_{8} + 6) q^{91} + (2 \zeta_{8}^{2} + 2) q^{93} + ( - 8 \zeta_{8}^{3} + 10 \zeta_{8}^{2} + 10 \zeta_{8} - 8) q^{95} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{97} + (9 \zeta_{8}^{3} - 9 \zeta_{8}^{2} - \zeta_{8} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 8 q^{5} + 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 8 q^{5} + 4 q^{7} - 8 q^{9} + 12 q^{11} - 16 q^{15} + 16 q^{19} + 4 q^{23} - 4 q^{25} - 8 q^{27} - 12 q^{29} + 4 q^{31} + 32 q^{33} - 24 q^{35} - 8 q^{37} - 8 q^{39} - 20 q^{41} - 16 q^{43} + 20 q^{45} - 8 q^{49} + 12 q^{51} + 12 q^{53} - 8 q^{57} + 8 q^{59} + 8 q^{61} - 4 q^{63} + 28 q^{65} + 32 q^{69} - 20 q^{71} + 4 q^{73} + 12 q^{75} + 24 q^{77} + 12 q^{79} - 8 q^{83} - 20 q^{85} - 24 q^{87} + 24 q^{91} + 8 q^{93} - 32 q^{95} - 8 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 1.00000 + 0.414214i 0 −1.29289 + 3.12132i 0 1.00000 + 2.41421i 0 −1.29289 1.29289i 0
25.1 0 1.00000 2.41421i 0 −2.70711 1.12132i 0 1.00000 0.414214i 0 −2.70711 2.70711i 0
49.1 0 1.00000 + 2.41421i 0 −2.70711 + 1.12132i 0 1.00000 + 0.414214i 0 −2.70711 + 2.70711i 0
121.1 0 1.00000 0.414214i 0 −1.29289 3.12132i 0 1.00000 2.41421i 0 −1.29289 + 1.29289i 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.b 4
3.b odd 2 1 1224.2.bq.b 4
4.b odd 2 1 272.2.v.a 4
17.d even 8 1 inner 136.2.n.b 4
17.e odd 16 2 2312.2.a.t 4
17.e odd 16 2 2312.2.b.i 4
51.g odd 8 1 1224.2.bq.b 4
68.g odd 8 1 272.2.v.a 4
68.i even 16 2 4624.2.a.bo 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.n.b 4 1.a even 1 1 trivial
136.2.n.b 4 17.d even 8 1 inner
272.2.v.a 4 4.b odd 2 1
272.2.v.a 4 68.g odd 8 1
1224.2.bq.b 4 3.b odd 2 1
1224.2.bq.b 4 51.g odd 8 1
2312.2.a.t 4 17.e odd 16 2
2312.2.b.i 4 17.e odd 16 2
4624.2.a.bo 4 68.i even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} + 12T_{3}^{2} - 16T_{3} + 8 \) acting on \(S_{2}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + 12 T^{2} - 16 T + 8 \) Copy content Toggle raw display
$5$ \( T^{4} + 8 T^{3} + 34 T^{2} + 84 T + 98 \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + 12 T^{2} - 16 T + 8 \) Copy content Toggle raw display
$11$ \( T^{4} - 12 T^{3} + 68 T^{2} + \cdots + 392 \) Copy content Toggle raw display
$13$ \( T^{4} + 36T^{2} + 196 \) Copy content Toggle raw display
$17$ \( T^{4} + 2T^{2} + 289 \) Copy content Toggle raw display
$19$ \( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} + 36 T^{2} + 32 T + 8 \) Copy content Toggle raw display
$29$ \( T^{4} + 12 T^{3} + 54 T^{2} + \cdots + 162 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + 4 T^{2} + 8 \) Copy content Toggle raw display
$37$ \( T^{4} + 8 T^{3} + 114 T^{2} - 28 T + 2 \) Copy content Toggle raw display
$41$ \( T^{4} + 20 T^{3} + 150 T^{2} + \cdots + 1250 \) Copy content Toggle raw display
$43$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + 72 T^{2} - 24 T + 4 \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 8464 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + 18 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 20 T^{3} + 108 T^{2} + \cdots + 17672 \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + 102 T^{2} + \cdots + 12482 \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} + 68 T^{2} + \cdots + 392 \) Copy content Toggle raw display
$83$ \( T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 8464 \) Copy content Toggle raw display
$89$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 8 T^{3} + 18 T^{2} - 4 T + 2 \) Copy content Toggle raw display
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