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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: \( x^{4} + 1 \) Copy content Toggle raw display
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.

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

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 \( T_{3}^{4} + 4T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 2 \) 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
$3$ \( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + 22 T^{2} + 12 T + 2 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 17)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} + 22 T^{2} + \cdots + 1058 \) Copy content Toggle raw display
$29$ \( T^{4} + 12 T^{3} + 54 T^{2} + \cdots + 162 \) Copy content Toggle raw display
$31$ \( T^{4} - 12 T^{3} + 54 T^{2} + \cdots + 162 \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + 134 T^{2} + \cdots + 578 \) Copy content Toggle raw display
$43$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$47$ \( T^{4} + 216T^{2} + 8464 \) Copy content Toggle raw display
$53$ \( T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 1156 \) Copy content Toggle raw display
$59$ \( T^{4} - 28 T^{3} + 392 T^{2} + \cdots + 8836 \) Copy content Toggle raw display
$61$ \( T^{4} + 28 T^{3} + 438 T^{2} + \cdots + 15842 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T - 16)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} + 134 T^{2} + \cdots + 578 \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + 102 T^{2} + \cdots + 12482 \) Copy content Toggle raw display
$79$ \( T^{4} - 36 T^{3} + 662 T^{2} + \cdots + 37538 \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$89$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$97$ \( T^{4} + 28 T^{3} + 214 T^{2} + \cdots + 1058 \) Copy content Toggle raw display
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