Properties

Label 136.4.s.a
Level $136$
Weight $4$
Character orbit 136.s
Analytic conductor $8.024$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.s (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{16}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{16}^{5} + 2 \zeta_{16}) q^{2} + ( - 5 \zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{4} + 5 \zeta_{16}^{2} - \zeta_{16} - 1) q^{3} - 8 \zeta_{16}^{6} q^{4} + ( - 10 \zeta_{16}^{7} + 4 \zeta_{16}^{5} - 10 \zeta_{16}^{4} + 10 \zeta_{16}^{3} - 4 \zeta_{16}^{2} + \cdots + 10) q^{6} + \cdots + ( - 13 \zeta_{16}^{6} + 13 \zeta_{16}^{4} - 10 \zeta_{16}^{2} + 27 \zeta_{16} - 10) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{16}^{5} + 2 \zeta_{16}) q^{2} + ( - 5 \zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{4} + 5 \zeta_{16}^{2} - \zeta_{16} - 1) q^{3} - 8 \zeta_{16}^{6} q^{4} + ( - 10 \zeta_{16}^{7} + 4 \zeta_{16}^{5} - 10 \zeta_{16}^{4} + 10 \zeta_{16}^{3} - 4 \zeta_{16}^{2} + \cdots + 10) q^{6} + \cdots + (717 \zeta_{16}^{7} - 1340 \zeta_{16}^{6} + 728 \zeta_{16}^{5} + 728 \zeta_{16}^{4} + \cdots - 192) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9} + 320 q^{12} + 144 q^{22} - 256 q^{24} + 496 q^{27} - 720 q^{34} - 640 q^{36} - 1440 q^{38} + 160 q^{41} + 208 q^{43} - 1600 q^{44} + 512 q^{48} + 1880 q^{51} - 176 q^{54} + 2840 q^{57} - 920 q^{59} + 5232 q^{66} + 3456 q^{72} - 3312 q^{73} - 2632 q^{81} - 3472 q^{83} - 5120 q^{96} - 5488 q^{98} - 1536 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{16}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.923880 + 0.382683i
−0.923880 + 0.382683i
0.382683 + 0.923880i
−0.382683 0.923880i
0.923880 0.382683i
−0.923880 0.382683i
−0.382683 + 0.923880i
0.382683 0.923880i
2.61313 1.08239i 6.61374 + 1.31555i 5.65685 5.65685i 0 18.7065 3.72095i 0 8.65914 20.9050i 17.0661 + 7.06900i 0
11.1 −2.61313 1.08239i −1.54267 7.75551i 5.65685 + 5.65685i 0 −4.36332 + 21.9359i 0 −8.65914 20.9050i −32.8234 + 13.5959i 0
27.1 −1.08239 + 2.61313i −3.92868 2.62506i −5.65685 5.65685i 0 11.1120 7.42479i 0 20.9050 8.65914i −1.78887 4.31871i 0
75.1 1.08239 2.61313i −5.14239 + 7.69613i −5.65685 5.65685i 0 14.5449 + 21.7679i 0 −20.9050 + 8.65914i −22.4538 54.2082i 0
91.1 2.61313 + 1.08239i 6.61374 1.31555i 5.65685 + 5.65685i 0 18.7065 + 3.72095i 0 8.65914 + 20.9050i 17.0661 7.06900i 0
99.1 −2.61313 + 1.08239i −1.54267 + 7.75551i 5.65685 5.65685i 0 −4.36332 21.9359i 0 −8.65914 + 20.9050i −32.8234 13.5959i 0
107.1 1.08239 + 2.61313i −5.14239 7.69613i −5.65685 + 5.65685i 0 14.5449 21.7679i 0 −20.9050 8.65914i −22.4538 + 54.2082i 0
131.1 −1.08239 2.61313i −3.92868 + 2.62506i −5.65685 + 5.65685i 0 11.1120 + 7.42479i 0 20.9050 + 8.65914i −1.78887 + 4.31871i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.1
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.e odd 16 1 inner
136.s even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.s.a 8
8.d odd 2 1 CM 136.4.s.a 8
17.e odd 16 1 inner 136.4.s.a 8
136.s even 16 1 inner 136.4.s.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.s.a 8 1.a even 1 1 trivial
136.4.s.a 8 8.d odd 2 1 CM
136.4.s.a 8 17.e odd 16 1 inner
136.4.s.a 8 136.s even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 8T_{3}^{7} + 72T_{3}^{6} - 480T_{3}^{5} - 4174T_{3}^{4} - 36832T_{3}^{3} + 45332T_{3}^{2} + 1253240T_{3} + 5438402 \) acting on \(S_{4}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4096 \) Copy content Toggle raw display
$3$ \( T^{8} + 8 T^{7} + 72 T^{6} + \cdots + 5438402 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 6476 T^{6} + \cdots + 221297408642 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 582622237229761 \) Copy content Toggle raw display
$19$ \( T^{8} + 38160 T^{6} + \cdots + 37949591784976 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} - 160 T^{7} + \cdots + 11\!\cdots\!02 \) Copy content Toggle raw display
$43$ \( (T^{4} - 104 T^{3} + 80322 T^{2} + \cdots + 774053858)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 460 T^{3} + 410758 T^{2} + \cdots + 233236802)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 1094692 T^{2} + \cdots + 121606351778)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 3312 T^{7} + \cdots + 71\!\cdots\!82 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1736 T^{3} + \cdots + 12454523138)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 11402977910412 T^{4} + \cdots + 31\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{8} + 3507988 T^{6} + \cdots + 18\!\cdots\!22 \) Copy content Toggle raw display
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