Properties

Label 136.4.b.a
Level $136$
Weight $4$
Character orbit 136.b
Analytic conductor $8.024$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Defining polynomial: \( x^{6} + 81x^{4} + 222x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} - 7) q^{9} + (\beta_{2} - \beta_1) q^{11} + (\beta_{5} - 12) q^{13} + (\beta_{5} - \beta_{3} - 4) q^{15} + (\beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 21) q^{17} + ( - \beta_{5} + 3 \beta_{3} - 4) q^{19} + (5 \beta_{3} - 34) q^{21} + ( - 7 \beta_{4} - 2 \beta_{2} + 3 \beta_1) q^{23} + (3 \beta_{5} - \beta_{3} - 19) q^{25} + ( - 10 \beta_{4} - 7 \beta_{2} - 2 \beta_1) q^{27} + ( - 9 \beta_{4} + 6 \beta_{2} + 4 \beta_1) q^{29} + (15 \beta_{4} + 6 \beta_{2} + 3 \beta_1) q^{31} + (\beta_{5} + 4 \beta_{3} - 38) q^{33} + ( - 2 \beta_{5} - 4 \beta_{3} + 68) q^{35} + (9 \beta_{4} + 12 \beta_{2} - 28 \beta_1) q^{37} + (18 \beta_{4} - \beta_{2} + 2 \beta_1) q^{39} + ( - 2 \beta_{4} - 2 \beta_{2} - 24 \beta_1) q^{41} + (\beta_{5} - 13 \beta_{3} + 32) q^{43} + (\beta_{4} + 6 \beta_{2} - 24 \beta_1) q^{45} + ( - 5 \beta_{5} + \beta_{3} - 12) q^{47} + ( - 3 \beta_{5} + 6 \beta_{3} - 3) q^{49} + ( - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 15 \beta_{2} + 47 \beta_1 + 76) q^{51} + ( - \beta_{5} - 7 \beta_{3} + 154) q^{53} + (\beta_{5} - 5 \beta_{3} - 76) q^{55} + ( - 48 \beta_{4} - 20 \beta_{2} + 68 \beta_1) q^{57} + ( - 9 \beta_{5} - 11 \beta_{3} + 280) q^{59} + ( - 5 \beta_{4} - 14 \beta_{2} - 52 \beta_1) q^{61} + ( - 23 \beta_{4} - 8 \beta_{2} + 97 \beta_1) q^{63} + (66 \beta_{4} + 10 \beta_{2} + 72 \beta_1) q^{65} + ( - 7 \beta_{5} + 5 \beta_{3} - 52) q^{67} + (5 \beta_{5} - 16 \beta_{3} + 82) q^{69} + (7 \beta_{4} - 19 \beta_{2} - 117 \beta_1) q^{71} + ( - 8 \beta_{4} - 22 \beta_{2} + 44 \beta_1) q^{73} + (64 \beta_{4} + 4 \beta_{2} - 29 \beta_1) q^{75} + ( - 5 \beta_{5} + 2 \beta_{3} - 278) q^{77} + ( - 27 \beta_{4} + 34 \beta_{2} - 25 \beta_1) q^{79} + (3 \beta_{5} - 2 \beta_{3} - 269) q^{81} + ( - \beta_{5} - 23 \beta_{3} + 216) q^{83} + ( - 5 \beta_{5} + 63 \beta_{4} + 7 \beta_{3} + 22 \beta_{2} + 24 \beta_1 + 224) q^{85} + (15 \beta_{5} + 5 \beta_{3} + 76) q^{87} + ( - 3 \beta_{5} - 14 \beta_{3} + 4) q^{89} + ( - 20 \beta_{4} + 21 \beta_{2} - 22 \beta_1) q^{91} + ( - 9 \beta_{5} + 30 \beta_{3} + 138) q^{93} + ( - 68 \beta_{4} + 8 \beta_{2} - 144 \beta_1) q^{95} + ( - 70 \beta_{4} + 8 \beta_{2} + 116 \beta_1) q^{97} + ( - 22 \beta_{4} - 2 \beta_{2} + 73 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 42 q^{9} - 72 q^{13} - 24 q^{15} - 126 q^{17} - 24 q^{19} - 204 q^{21} - 114 q^{25} - 228 q^{33} + 408 q^{35} + 192 q^{43} - 72 q^{47} - 18 q^{49} + 456 q^{51} + 924 q^{53} - 456 q^{55} + 1680 q^{59} - 312 q^{67} + 492 q^{69} - 1668 q^{77} - 1614 q^{81} + 1296 q^{83} + 1344 q^{85} + 456 q^{87} + 24 q^{89} + 828 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 81x^{4} + 222x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 85\nu^{3} + 562\nu ) / 68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 85\nu^{3} - 426\nu ) / 34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{4} + 391\nu^{2} + 362 ) / 17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{5} + 1207\nu^{3} + 2650\nu ) / 68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{4} - 731\nu^{2} - 1386 ) / 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{5} - 9\beta_{3} - 216 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{4} - 85\beta_{2} - 110\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 391\beta_{5} + 731\beta_{3} + 16312 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 340\beta_{4} + 6663\beta_{2} + 8498\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

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

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

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
1.58186i
0.572006i
8.84141i
8.84141i
0.572006i
1.58186i
0 8.27144i 0 6.42821i 0 24.9151i 0 −41.4167 0
33.2 0 4.49442i 0 18.9829i 0 7.78768i 0 6.80022 0
33.3 0 3.65834i 0 5.50702i 0 18.8836i 0 13.6165 0
33.4 0 3.65834i 0 5.50702i 0 18.8836i 0 13.6165 0
33.5 0 4.49442i 0 18.9829i 0 7.78768i 0 6.80022 0
33.6 0 8.27144i 0 6.42821i 0 24.9151i 0 −41.4167 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.6
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.a 6
3.b odd 2 1 1224.4.c.c 6
4.b odd 2 1 272.4.b.e 6
17.b even 2 1 inner 136.4.b.a 6
17.c even 4 2 2312.4.a.h 6
51.c odd 2 1 1224.4.c.c 6
68.d odd 2 1 272.4.b.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.b.a 6 1.a even 1 1 trivial
136.4.b.a 6 17.b even 2 1 inner
272.4.b.e 6 4.b odd 2 1
272.4.b.e 6 68.d odd 2 1
1224.4.c.c 6 3.b odd 2 1
1224.4.c.c 6 51.c odd 2 1
2312.4.a.h 6 17.c even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 102T_{3}^{4} + 2568T_{3}^{2} + 18496 \) acting on \(S_{4}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 102 T^{4} + 2568 T^{2} + \cdots + 18496 \) Copy content Toggle raw display
$5$ \( T^{6} + 432 T^{4} + 27072 T^{2} + \cdots + 451584 \) Copy content Toggle raw display
$7$ \( T^{6} + 1038 T^{4} + \cdots + 13424896 \) Copy content Toggle raw display
$11$ \( T^{6} + 1062 T^{4} + \cdots + 25482304 \) Copy content Toggle raw display
$13$ \( (T^{3} + 36 T^{2} - 3108 T + 28016)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 126 T^{5} + \cdots + 118587876497 \) Copy content Toggle raw display
$19$ \( (T^{3} + 12 T^{2} - 13968 T + 178496)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 19422 T^{4} + \cdots + 3953894400 \) Copy content Toggle raw display
$29$ \( T^{6} + 92208 T^{4} + \cdots + 1770986054656 \) Copy content Toggle raw display
$31$ \( T^{6} + 93582 T^{4} + \cdots + 10987474749696 \) Copy content Toggle raw display
$37$ \( T^{6} + 205104 T^{4} + \cdots + 7252378264576 \) Copy content Toggle raw display
$41$ \( T^{6} + 62496 T^{4} + \cdots + 8454416891904 \) Copy content Toggle raw display
$43$ \( (T^{3} - 96 T^{2} - 162864 T - 20641152)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + 36 T^{2} - 92928 T - 11699200)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 462 T^{2} + 29052 T + 1630776)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 840 T^{2} - 82032 T + 116758016)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + 428592 T^{4} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{3} + 156 T^{2} - 215568 T - 49226176)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 1739790 T^{4} + \cdots + 47\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{6} + 616896 T^{4} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{6} + 1893870 T^{4} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{3} - 648 T^{2} - 321456 T - 10110464)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 12 T^{2} - 174948 T + 2334896)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 3573984 T^{4} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
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