Properties

Label 156.3.d.a
Level $156$
Weight $3$
Character orbit 156.d
Analytic conductor $4.251$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 156.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.25069212402\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{7} + 15x^{6} + 10x^{5} + 97x^{4} + 252x^{3} + 700x^{2} + 1696x + 3792 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + \beta_{3} q^{5} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + \beta_{3} q^{5} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{9} + (\beta_{6} - \beta_{4} - \beta_{2} + 1) q^{11} - \beta_{5} q^{13} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{15} + ( - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 1) q^{17} + (2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_1 - 2) q^{19} + ( - 2 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_1 + 2) q^{21} + ( - 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2) q^{23} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{25} + ( - \beta_{7} - 5 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{27} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 2) q^{29} + (10 \beta_{5} + 12) q^{31} + (4 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + \cdots + 3) q^{33}+ \cdots + (16 \beta_{7} - 9 \beta_{6} + 4 \beta_{5} + 7 \beta_{4} + 5 \beta_{2} - 2 \beta_1 + 37) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} - 8 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{3} - 8 q^{7} - 22 q^{9} + 4 q^{15} - 24 q^{19} + 16 q^{21} - 28 q^{25} + 36 q^{27} + 96 q^{31} + 4 q^{33} - 96 q^{37} + 76 q^{43} - 136 q^{45} + 204 q^{49} - 62 q^{51} - 80 q^{55} + 184 q^{57} - 104 q^{61} + 36 q^{63} - 384 q^{67} - 8 q^{73} - 100 q^{75} + 328 q^{79} + 50 q^{81} - 64 q^{85} - 204 q^{87} - 52 q^{91} + 72 q^{93} + 416 q^{97} + 284 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 15x^{6} + 10x^{5} + 97x^{4} + 252x^{3} + 700x^{2} + 1696x + 3792 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 28\nu^{6} - 95\nu^{5} + 516\nu^{4} - 553\nu^{3} + 3758\nu^{2} + 736\nu + 13836 ) / 2916 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} - 4\nu^{6} + 11\nu^{5} - 408\nu^{4} + 511\nu^{3} - 1676\nu^{2} - 1972\nu - 1560 ) / 2916 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 14\nu^{6} + 11\nu^{5} - 156\nu^{4} - 263\nu^{3} - 272\nu^{2} - 1144\nu - 6204 ) / 2916 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 14\nu^{5} + 18\nu^{4} - 7\nu^{3} - 139\nu^{2} + 280\nu + 102 ) / 486 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} - 52\nu^{6} + 179\nu^{5} - 138\nu^{4} + 109\nu^{3} - 494\nu^{2} + 2444\nu + 132 ) / 2916 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\nu^{7} - 40\nu^{6} + 263\nu^{5} - 228\nu^{4} + 2005\nu^{3} + 718\nu^{2} + 9224\nu + 13560 ) / 2916 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{3} + \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{5} - 2\beta_{4} + 2\beta_{3} + \beta_{2} - 4\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 6\beta_{6} + 13\beta_{5} - 2\beta_{4} - 3\beta_{3} + 2\beta_{2} - 19\beta _1 - 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{7} + 12\beta_{6} + 19\beta_{5} + 32\beta_{4} - 17\beta_{3} - \beta_{2} - 45\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 29\beta_{7} - 84\beta_{6} - 18\beta_{5} + 104\beta_{4} - 112\beta_{3} - 63\beta_{2} - 30\beta _1 + 333 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15\beta_{7} - 396\beta_{6} - 465\beta_{5} - 54\beta_{4} - 417\beta_{3} - 348\beta_{2} + 337\beta _1 + 1359 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
2.33932 + 2.68444i
2.33932 2.68444i
1.12162 + 2.99753i
1.12162 2.99753i
−0.621622 + 2.52395i
−0.621622 2.52395i
−1.83932 + 0.968627i
−1.83932 0.968627i
0 −1.33932 2.68444i 0 5.74349i 0 −9.51181 0 −5.41242 + 7.19067i 0
53.2 0 −1.33932 + 2.68444i 0 5.74349i 0 −9.51181 0 −5.41242 7.19067i 0
53.3 0 −0.121622 2.99753i 0 0.347906i 0 10.5746 0 −8.97042 + 0.729130i 0
53.4 0 −0.121622 + 2.99753i 0 0.347906i 0 10.5746 0 −8.97042 0.729130i 0
53.5 0 1.62162 2.52395i 0 7.04754i 0 −8.96906 0 −3.74069 8.18580i 0
53.6 0 1.62162 + 2.52395i 0 7.04754i 0 −8.96906 0 −3.74069 + 8.18580i 0
53.7 0 2.83932 0.968627i 0 5.58779i 0 3.90626 0 7.12352 5.50049i 0
53.8 0 2.83932 + 0.968627i 0 5.58779i 0 3.90626 0 7.12352 + 5.50049i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.d.a 8
3.b odd 2 1 inner 156.3.d.a 8
4.b odd 2 1 624.3.f.a 8
12.b even 2 1 624.3.f.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.d.a 8 1.a even 1 1 trivial
156.3.d.a 8 3.b odd 2 1 inner
624.3.f.a 8 4.b odd 2 1
624.3.f.a 8 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(156, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + 29 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} + 114 T^{6} + 4233 T^{4} + \cdots + 6192 \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} - 141 T^{2} - 472 T + 3524)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 612 T^{6} + \cdots + 128397312 \) Copy content Toggle raw display
$13$ \( (T^{2} - 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + 2342 T^{6} + \cdots + 72223488 \) Copy content Toggle raw display
$19$ \( (T^{4} + 12 T^{3} - 1216 T^{2} + \cdots - 46224)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 2584 T^{6} + \cdots + 2054356992 \) Copy content Toggle raw display
$29$ \( T^{8} + 4288 T^{6} + \cdots + 120750538752 \) Copy content Toggle raw display
$31$ \( (T^{2} - 24 T - 1156)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 48 T^{3} - 673 T^{2} + 2784 T - 3708)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 6380 T^{6} + \cdots + 406257120000 \) Copy content Toggle raw display
$43$ \( (T^{4} - 38 T^{3} - 5943 T^{2} + \cdots - 4083628)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 7098 T^{6} + \cdots + 181027138608 \) Copy content Toggle raw display
$53$ \( T^{8} + 8840 T^{6} + \cdots + 722234880000 \) Copy content Toggle raw display
$59$ \( T^{8} + 8244 T^{6} + \cdots + 717961310208 \) Copy content Toggle raw display
$61$ \( (T^{4} + 52 T^{3} - 1296 T^{2} + \cdots + 7856)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 192 T^{3} + 10456 T^{2} + \cdots + 953488)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 16218 T^{6} + \cdots + 12770012803248 \) Copy content Toggle raw display
$73$ \( (T^{4} + 4 T^{3} - 732 T^{2} - 1472 T + 68032)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 164 T^{3} + 8412 T^{2} + \cdots - 966848)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 36036 T^{6} + \cdots + 28\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{8} + 29420 T^{6} + \cdots + 10668283884288 \) Copy content Toggle raw display
$97$ \( (T^{4} - 208 T^{3} - 6204 T^{2} + \cdots + 57925568)^{2} \) Copy content Toggle raw display
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