Properties

Label 136.4.b.b
Level $136$
Weight $4$
Character orbit 136.b
Analytic conductor $8.024$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 95 x^{6} + 756 x^{4} + 1780 x^{2} + 1152\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} -\beta_{4} q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{7} + ( -16 + \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} -\beta_{4} q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{7} + ( -16 + \beta_{5} ) q^{9} + ( -2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{11} + ( 5 + \beta_{7} ) q^{13} + ( 2 - 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{15} + ( 4 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{17} + ( 4 - 2 \beta_{3} - \beta_{5} + \beta_{7} ) q^{19} + ( 37 - 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{7} ) q^{21} + ( -5 \beta_{1} + \beta_{6} ) q^{23} + ( -63 + 3 \beta_{5} - \beta_{7} ) q^{25} + ( -6 \beta_{1} - 19 \beta_{2} - 3 \beta_{4} + 3 \beta_{6} ) q^{27} + ( 18 \beta_{2} + 5 \beta_{4} - 2 \beta_{6} ) q^{29} + ( 5 \beta_{1} + 14 \beta_{2} - 4 \beta_{4} + 3 \beta_{6} ) q^{31} + ( 99 + 2 \beta_{3} - 4 \beta_{5} + 3 \beta_{7} ) q^{33} + ( -130 + 8 \beta_{5} + 2 \beta_{7} ) q^{35} + ( 4 \beta_{1} - 28 \beta_{2} + \beta_{4} ) q^{37} + ( -10 \beta_{1} + 3 \beta_{2} - 21 \beta_{4} - 3 \beta_{6} ) q^{39} + ( 4 \beta_{1} + 2 \beta_{2} - 16 \beta_{4} - 6 \beta_{6} ) q^{41} + ( 2 - 2 \beta_{3} + 3 \beta_{5} - \beta_{7} ) q^{43} + ( 20 \beta_{1} + 10 \beta_{2} + 43 \beta_{4} + 2 \beta_{6} ) q^{45} + ( 42 + 2 \beta_{3} + \beta_{5} - 3 \beta_{7} ) q^{47} + ( -132 + 8 \beta_{3} + 10 \beta_{5} + \beta_{7} ) q^{49} + ( 46 - 2 \beta_{1} + 16 \beta_{2} - 4 \beta_{3} - 23 \beta_{4} - 8 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{51} + ( 60 + 6 \beta_{3} - 7 \beta_{5} - 3 \beta_{7} ) q^{53} + ( 174 + 6 \beta_{3} - 5 \beta_{5} + 7 \beta_{7} ) q^{55} + ( 8 \beta_{2} + 28 \beta_{4} - 4 \beta_{6} ) q^{57} + ( -10 + 2 \beta_{3} - 3 \beta_{5} + \beta_{7} ) q^{59} + ( -16 \beta_{1} - 74 \beta_{2} + 5 \beta_{4} - 6 \beta_{6} ) q^{61} + ( 15 \beta_{1} + 80 \beta_{2} + 70 \beta_{4} - \beta_{6} ) q^{63} + ( 20 \beta_{1} + 78 \beta_{2} - 16 \beta_{4} - 10 \beta_{6} ) q^{65} + ( -76 - 6 \beta_{3} + \beta_{5} - 9 \beta_{7} ) q^{67} + ( -25 - 8 \beta_{5} - 3 \beta_{7} ) q^{69} + ( 29 \beta_{1} - 31 \beta_{2} + 27 \beta_{4} - 6 \beta_{6} ) q^{71} + ( 20 \beta_{1} - 6 \beta_{2} - 26 \beta_{4} + 6 \beta_{6} ) q^{73} + ( -8 \beta_{1} - 151 \beta_{2} + 12 \beta_{4} + 12 \beta_{6} ) q^{75} + ( 211 + 8 \beta_{3} + 2 \beta_{5} + 3 \beta_{7} ) q^{77} + ( -5 \beta_{1} - 64 \beta_{2} - 16 \beta_{4} - 11 \beta_{6} ) q^{79} + ( 388 - 6 \beta_{3} - 10 \beta_{5} - 3 \beta_{7} ) q^{81} + ( 298 + 2 \beta_{3} - 15 \beta_{5} + 9 \beta_{7} ) q^{83} + ( -268 - 16 \beta_{1} + 94 \beta_{2} - 19 \beta_{4} + 11 \beta_{5} - 2 \beta_{6} + 7 \beta_{7} ) q^{85} + ( -814 + 10 \beta_{3} + 29 \beta_{5} + \beta_{7} ) q^{87} + ( 23 + 8 \beta_{3} + 22 \beta_{5} + \beta_{7} ) q^{89} + ( -20 \beta_{1} + 139 \beta_{2} + 21 \beta_{4} + \beta_{6} ) q^{91} + ( -509 - 8 \beta_{3} + 6 \beta_{5} + 7 \beta_{7} ) q^{93} + ( 32 \beta_{1} - 120 \beta_{2} - 12 \beta_{4} - 8 \beta_{6} ) q^{95} + ( -24 \beta_{1} + 140 \beta_{2} - 22 \beta_{4} + 16 \beta_{6} ) q^{97} + ( -10 \beta_{1} + 183 \beta_{2} - 70 \beta_{4} + 4 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 132 q^{9} + O(q^{10}) \) \( 8 q - 132 q^{9} + 44 q^{13} + 24 q^{15} + 28 q^{17} + 48 q^{19} + 308 q^{21} - 520 q^{25} + 812 q^{33} - 1064 q^{35} + 8 q^{43} + 312 q^{47} - 1124 q^{49} + 408 q^{51} + 472 q^{53} + 1416 q^{55} - 72 q^{59} - 624 q^{67} - 180 q^{69} + 1660 q^{77} + 3156 q^{81} + 2472 q^{83} - 2160 q^{85} - 6664 q^{87} + 68 q^{89} - 4036 q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 95 x^{6} + 756 x^{4} + 1780 x^{2} + 1152\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{7} - 284 \nu^{5} - 2159 \nu^{3} - 3402 \nu \)\()/172\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{6} + 327 \nu^{4} + 6158 \nu^{2} + 17420 \)\()/172\)
\(\beta_{4}\)\(=\)\((\)\( 59 \nu^{7} + 5485 \nu^{5} + 33588 \nu^{3} + 47900 \nu \)\()/1032\)
\(\beta_{5}\)\(=\)\((\)\( -41 \nu^{6} - 3781 \nu^{4} - 20634 \nu^{2} - 22844 \)\()/172\)
\(\beta_{6}\)\(=\)\((\)\( -109 \nu^{7} - 9989 \nu^{5} - 48702 \nu^{3} - 15160 \nu \)\()/1032\)
\(\beta_{7}\)\(=\)\((\)\( 45 \nu^{6} + 4217 \nu^{4} + 28386 \nu^{2} + 35292 \)\()/172\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/4\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{7} - 3 \beta_{5} + 4 \beta_{3} - 188\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{6} + 23 \beta_{4} + 33 \beta_{2} - 78 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(277 \beta_{7} + 279 \beta_{5} - 342 \beta_{3} + 14856\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(-619 \beta_{6} - 2075 \beta_{4} - 3053 \beta_{2} + 6708 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-24035 \beta_{7} - 24253 \beta_{5} + 29526 \beta_{3} - 1279856\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(53561 \beta_{6} + 179881 \beta_{4} + 265039 \beta_{2} - 580024 \beta_{1}\)\()/4\)

Character values

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

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
2.20783i
1.03229i
1.60125i
9.30031i
9.30031i
1.60125i
1.03229i
2.20783i
0 9.26065i 0 16.4090i 0 34.5010i 0 −58.7597 0
33.2 0 8.52350i 0 18.2701i 0 13.8757i 0 −45.6501 0
33.3 0 2.95309i 0 4.89575i 0 4.46235i 0 18.2793 0
33.4 0 2.62097i 0 11.5318i 0 23.0485i 0 20.1305 0
33.5 0 2.62097i 0 11.5318i 0 23.0485i 0 20.1305 0
33.6 0 2.95309i 0 4.89575i 0 4.46235i 0 18.2793 0
33.7 0 8.52350i 0 18.2701i 0 13.8757i 0 −45.6501 0
33.8 0 9.26065i 0 16.4090i 0 34.5010i 0 −58.7597 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.b.b 8
3.b odd 2 1 1224.4.c.e 8
4.b odd 2 1 272.4.b.f 8
17.b even 2 1 inner 136.4.b.b 8
17.c even 4 2 2312.4.a.k 8
51.c odd 2 1 1224.4.c.e 8
68.d odd 2 1 272.4.b.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.b.b 8 1.a even 1 1 trivial
136.4.b.b 8 17.b even 2 1 inner
272.4.b.f 8 4.b odd 2 1
272.4.b.f 8 68.d odd 2 1
1224.4.c.e 8 3.b odd 2 1
1224.4.c.e 8 51.c odd 2 1
2312.4.a.k 8 17.c even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 174 T_{3}^{6} + 8760 T_{3}^{4} + 106624 T_{3}^{2} + 373248 \) acting on \(S_{4}^{\mathrm{new}}(136, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 373248 + 106624 T^{2} + 8760 T^{4} + 174 T^{6} + T^{8} \)
$5$ \( 286466048 + 16028160 T^{2} + 187712 T^{4} + 760 T^{6} + T^{8} \)
$7$ \( 2424307712 + 140939264 T^{2} + 1001912 T^{4} + 1934 T^{6} + T^{8} \)
$11$ \( 1063158272 + 764940416 T^{2} + 13918712 T^{4} + 7406 T^{6} + T^{8} \)
$13$ \( ( 8525216 + 70680 T - 5836 T^{2} - 22 T^{3} + T^{4} )^{2} \)
$17$ \( 582622237229761 - 3320460541916 T + 42192470612 T^{2} - 1708171492 T^{3} + 9642774 T^{4} - 347684 T^{5} + 1748 T^{6} - 28 T^{7} + T^{8} \)
$19$ \( ( 44946176 + 382336 T - 21056 T^{2} - 24 T^{3} + T^{4} )^{2} \)
$23$ \( 2935871578112 + 126066546688 T^{2} + 134935032 T^{4} + 36734 T^{6} + T^{8} \)
$29$ \( 147297439325233152 + 36222945874432 T^{2} + 3126815040 T^{4} + 106232 T^{6} + T^{8} \)
$31$ \( 321310420423933952 + 83917512114176 T^{2} + 6678611960 T^{4} + 155918 T^{6} + T^{8} \)
$37$ \( 548861080203763712 + 175406389281280 T^{2} + 9074395456 T^{4} + 166072 T^{6} + T^{8} \)
$41$ \( 55409237072210296832 + 3227605845606400 T^{2} + 58679507968 T^{4} + 419392 T^{6} + T^{8} \)
$43$ \( ( 112195072 + 4732224 T - 60592 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$47$ \( ( 134217728 + 3407872 T - 57344 T^{2} - 156 T^{3} + T^{4} )^{2} \)
$53$ \( ( -14761769616 + 165833136 T - 400992 T^{2} - 236 T^{3} + T^{4} )^{2} \)
$59$ \( ( 70465536 - 5698880 T - 60112 T^{2} + 36 T^{3} + T^{4} )^{2} \)
$61$ \( \)\(18\!\cdots\!28\)\( + 216217434642342400 T^{2} + 907789046080 T^{4} + 1597240 T^{6} + T^{8} \)
$67$ \( ( 30967766784 + 70501504 T - 663552 T^{2} + 312 T^{3} + T^{4} )^{2} \)
$71$ \( \)\(14\!\cdots\!28\)\( + 195987755820328960 T^{2} + 887074814648 T^{4} + 1607086 T^{6} + T^{8} \)
$73$ \( \)\(21\!\cdots\!68\)\( + 84967502970355712 T^{2} + 772622417920 T^{4} + 1779648 T^{6} + T^{8} \)
$79$ \( 17195215357908099072 + 8819538685493248 T^{2} + 628706103544 T^{4} + 1578638 T^{6} + T^{8} \)
$83$ \( ( -54225864704 + 540066368 T - 567888 T^{2} - 1236 T^{3} + T^{4} )^{2} \)
$89$ \( ( 21605388512 + 368883272 T - 1202908 T^{2} - 34 T^{3} + T^{4} )^{2} \)
$97$ \( \)\(13\!\cdots\!92\)\( + 6103257558949298176 T^{2} + 9052262153216 T^{4} + 5185024 T^{6} + T^{8} \)
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