Properties

Label 136.3.t.b
Level $136$
Weight $3$
Character orbit 136.t
Analytic conductor $3.706$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.70573159530\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{3} - 8 q^{7} - 16 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{3} - 8 q^{7} - 16 q^{9} + 24 q^{11} - 48 q^{13} - 96 q^{15} - 40 q^{19} + 80 q^{21} + 48 q^{23} + 48 q^{25} + 224 q^{27} + 24 q^{29} + 88 q^{31} + 32 q^{35} - 176 q^{37} - 120 q^{39} - 352 q^{43} + 264 q^{45} - 48 q^{47} - 208 q^{49} + 400 q^{51} - 472 q^{53} - 208 q^{55} + 24 q^{57} - 576 q^{59} - 632 q^{63} - 32 q^{65} + 160 q^{69} - 160 q^{71} + 256 q^{73} + 1128 q^{75} - 208 q^{77} + 1000 q^{79} + 24 q^{81} + 312 q^{83} + 1240 q^{85} - 664 q^{87} + 720 q^{89} + 664 q^{91} - 432 q^{93} + 736 q^{95} - 288 q^{97} + 16 q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 0 −3.07751 + 4.60583i 0 1.22413 + 6.15412i 0 0.478359 2.40487i 0 −8.29839 20.0341i 0
41.2 0 −1.26339 + 1.89080i 0 −0.944701 4.74933i 0 −2.41180 + 12.1249i 0 1.46519 + 3.53728i 0
41.3 0 −0.708293 + 1.06004i 0 −0.482121 2.42379i 0 2.22606 11.1911i 0 2.82215 + 6.81328i 0
41.4 0 1.26753 1.89699i 0 1.74040 + 8.74956i 0 −0.705386 + 3.54621i 0 1.45221 + 3.50595i 0
41.5 0 2.38532 3.56988i 0 −0.996509 5.00979i 0 −0.0460348 + 0.231433i 0 −3.61016 8.71570i 0
57.1 0 −3.88441 + 0.772657i 0 2.37425 3.55331i 0 2.98990 + 4.47471i 0 6.17671 2.55848i 0
57.2 0 −1.73192 + 0.344500i 0 −0.638729 + 0.955926i 0 −5.52260 8.26516i 0 −5.43405 + 2.25086i 0
57.3 0 1.63930 0.326077i 0 −4.61793 + 6.91122i 0 2.96901 + 4.44344i 0 −5.73394 + 2.37508i 0
57.4 0 1.67722 0.333620i 0 3.22552 4.82733i 0 5.51299 + 8.25077i 0 −5.61314 + 2.32504i 0
57.5 0 5.47199 1.08845i 0 0.963457 1.44192i 0 −5.64274 8.44495i 0 20.4430 8.46777i 0
65.1 0 −1.10801 + 5.57033i 0 −3.32830 + 2.22390i 0 −0.387063 0.258627i 0 −21.4860 8.89979i 0
65.2 0 −0.262425 + 1.31930i 0 6.57938 4.39620i 0 −6.01468 4.01888i 0 6.64323 + 2.75172i 0
65.3 0 −0.0253220 + 0.127302i 0 −2.11962 + 1.41628i 0 5.83745 + 3.90046i 0 8.29935 + 3.43770i 0
65.4 0 0.657237 3.30415i 0 −5.77013 + 3.85548i 0 −10.9767 7.33440i 0 −2.17054 0.899066i 0
65.5 0 0.980550 4.92956i 0 3.33211 2.22644i 0 9.23443 + 6.17025i 0 −15.0241 6.22320i 0
73.1 0 −3.07751 4.60583i 0 1.22413 6.15412i 0 0.478359 + 2.40487i 0 −8.29839 + 20.0341i 0
73.2 0 −1.26339 1.89080i 0 −0.944701 + 4.74933i 0 −2.41180 12.1249i 0 1.46519 3.53728i 0
73.3 0 −0.708293 1.06004i 0 −0.482121 + 2.42379i 0 2.22606 + 11.1911i 0 2.82215 6.81328i 0
73.4 0 1.26753 + 1.89699i 0 1.74040 8.74956i 0 −0.705386 3.54621i 0 1.45221 3.50595i 0
73.5 0 2.38532 + 3.56988i 0 −0.996509 + 5.00979i 0 −0.0460348 0.231433i 0 −3.61016 + 8.71570i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.3.t.b 40
4.b odd 2 1 272.3.bh.g 40
17.e odd 16 1 inner 136.3.t.b 40
68.i even 16 1 272.3.bh.g 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.t.b 40 1.a even 1 1 trivial
136.3.t.b 40 17.e odd 16 1 inner
272.3.bh.g 40 4.b odd 2 1
272.3.bh.g 40 68.i even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(37\!\cdots\!52\)\( T_{3}^{22} - \)\(55\!\cdots\!72\)\( T_{3}^{21} + \)\(87\!\cdots\!92\)\( T_{3}^{20} + \)\(26\!\cdots\!08\)\( T_{3}^{19} + \)\(86\!\cdots\!60\)\( T_{3}^{18} - \)\(12\!\cdots\!60\)\( T_{3}^{17} - \)\(57\!\cdots\!84\)\( T_{3}^{16} + \)\(91\!\cdots\!20\)\( T_{3}^{15} + \)\(46\!\cdots\!16\)\( T_{3}^{14} - \)\(50\!\cdots\!08\)\( T_{3}^{13} - \)\(51\!\cdots\!44\)\( T_{3}^{12} + \)\(10\!\cdots\!80\)\( T_{3}^{11} + \)\(39\!\cdots\!32\)\( T_{3}^{10} + \)\(24\!\cdots\!88\)\( T_{3}^{9} - \)\(48\!\cdots\!48\)\( T_{3}^{8} - \)\(12\!\cdots\!88\)\( T_{3}^{7} - \)\(84\!\cdots\!20\)\( T_{3}^{6} + \)\(70\!\cdots\!64\)\( T_{3}^{5} + \)\(18\!\cdots\!32\)\( T_{3}^{4} + \)\(13\!\cdots\!80\)\( T_{3}^{3} + \)\(42\!\cdots\!12\)\( T_{3}^{2} + \)\(37\!\cdots\!40\)\( T_{3} + \)\(55\!\cdots\!92\)\( \)">\(T_{3}^{40} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(136, [\chi])\).