Properties

Label 156.3.j.a
Level $156$
Weight $3$
Character orbit 156.j
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.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.25069212402\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1579585536.10
Defining polynomial: \( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{5} + \beta_{4} - \beta_1 + 1) q^{5} + (2 \beta_{6} + 2 \beta_{5} + \beta_{2} - \beta_1 - 2) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{5} + \beta_{4} - \beta_1 + 1) q^{5} + (2 \beta_{6} + 2 \beta_{5} + \beta_{2} - \beta_1 - 2) q^{7} + 3 q^{9} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{3} - 3 \beta_1) q^{11} + (3 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{13} + ( - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{15} + (6 \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} - 1) q^{17} + ( - 3 \beta_{7} + 10 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 10) q^{19} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{3} - 2 \beta_1 - 1) q^{21} + (6 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} - 2 \beta_{4} + 10 \beta_1 + 2) q^{23} + ( - 2 \beta_{7} + 4 \beta_{6} - 7 \beta_{5} + 4 \beta_{4} - 12 \beta_1 - 4) q^{25} + 3 \beta_{2} q^{27} + ( - 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 8 \beta_{2} + 3) q^{29} + ( - 2 \beta_{7} - 4 \beta_{4} - 2 \beta_{3} + 7 \beta_{2} - 3 \beta_1) q^{31} + ( - 5 \beta_{7} - \beta_{6} + 4 \beta_{5} + 5 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{33} + (5 \beta_{6} + 5 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} - 27) q^{35} + ( - 7 \beta_{7} - 10 \beta_{6} - 9 \beta_{5} + 7 \beta_{3} + 3 \beta_{2} + 13 \beta_1 + 9) q^{37} + ( - 3 \beta_{7} - \beta_{6} - 4 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} + 5 \beta_{2} + \cdots + 12) q^{39}+ \cdots + (6 \beta_{7} + 9 \beta_{6} - 6 \beta_{3} - 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{5} - 8 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{5} - 8 q^{7} + 24 q^{9} + 12 q^{11} + 24 q^{13} + 12 q^{15} + 88 q^{19} + 24 q^{29} - 16 q^{31} - 36 q^{33} - 216 q^{35} + 32 q^{37} + 72 q^{39} - 180 q^{41} + 36 q^{45} + 36 q^{47} - 72 q^{53} - 240 q^{55} + 24 q^{57} - 228 q^{59} - 192 q^{61} - 24 q^{63} + 132 q^{65} + 16 q^{67} + 36 q^{71} + 160 q^{73} + 48 q^{79} + 72 q^{81} + 12 q^{83} + 24 q^{85} - 120 q^{87} + 60 q^{89} + 112 q^{91} + 120 q^{93} + 416 q^{97} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 46\nu^{6} + 150\nu^{5} - 27\nu^{4} - 720\nu^{3} + 1768\nu^{2} - 2792\nu + 1881 ) / 935 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 118\nu^{7} + 607\nu^{6} - 490\nu^{5} + 384\nu^{4} + 8710\nu^{3} - 19856\nu^{2} + 35024\nu + 91398 ) / 68510 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3448 \nu^{7} - 21163 \nu^{6} - 45670 \nu^{5} + 200494 \nu^{4} - 126360 \nu^{3} - 635936 \nu^{2} + 1327644 \nu - 2421122 ) / 753610 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3448 \nu^{7} + 21163 \nu^{6} + 45670 \nu^{5} - 200494 \nu^{4} + 126360 \nu^{3} + 635936 \nu^{2} + 179576 \nu + 2421122 ) / 753610 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 86\nu^{7} - 227\nu^{6} + 162\nu^{5} + 758\nu^{4} - 1794\nu^{3} + 8124\nu^{2} - 8408\nu + 13662 ) / 8866 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3777 \nu^{7} - 11052 \nu^{6} - 45875 \nu^{5} + 222466 \nu^{4} - 199615 \nu^{3} - 521764 \nu^{2} + 1944346 \nu - 3038343 ) / 376805 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -1123\nu^{7} + 3803\nu^{6} - 3465\nu^{5} - 12944\nu^{4} + 51805\nu^{3} - 107134\nu^{2} + 75416\nu - 40743 ) / 34255 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 4\beta_{5} + \beta_{4} - \beta_{3} - 4\beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 6\beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} + \beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{7} + 5\beta_{6} + 13\beta_{5} - 3\beta_{4} - 7\beta_{3} + \beta_{2} + 4\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{7} + 2\beta_{6} + 7\beta_{5} + \beta_{4} + 2\beta_{3} + 34\beta_{2} + 12\beta _1 - 55 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{6} - 4\beta_{5} - 22\beta_{4} - 70\beta_{3} + 22\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -59\beta_{7} - 37\beta_{6} - 99\beta_{5} - 11\beta_{4} + 11\beta_{3} + 195\beta_{2} - 37\beta _1 - 331 \) Copy content Toggle raw display

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
−2.59436 + 0.0368949i
2.22833 + 1.32913i
0.252411 1.79004i
1.11361 + 1.42401i
−2.59436 0.0368949i
2.22833 1.32913i
0.252411 + 1.79004i
1.11361 1.42401i
0 −1.73205 0 −0.658261 0.658261i 0 1.58447 1.58447i 0 3.00000 0
73.2 0 −1.73205 0 1.92621 + 1.92621i 0 −3.58447 + 3.58447i 0 3.00000 0
73.3 0 1.73205 0 −0.848026 0.848026i 0 5.42810 5.42810i 0 3.00000 0
73.4 0 1.73205 0 5.58008 + 5.58008i 0 −7.42810 + 7.42810i 0 3.00000 0
109.1 0 −1.73205 0 −0.658261 + 0.658261i 0 1.58447 + 1.58447i 0 3.00000 0
109.2 0 −1.73205 0 1.92621 1.92621i 0 −3.58447 3.58447i 0 3.00000 0
109.3 0 1.73205 0 −0.848026 + 0.848026i 0 5.42810 + 5.42810i 0 3.00000 0
109.4 0 1.73205 0 5.58008 5.58008i 0 −7.42810 7.42810i 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.j.a 8
3.b odd 2 1 468.3.m.d 8
4.b odd 2 1 624.3.ba.d 8
13.d odd 4 1 inner 156.3.j.a 8
39.f even 4 1 468.3.m.d 8
52.f even 4 1 624.3.ba.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.j.a 8 1.a even 1 1 trivial
156.3.j.a 8 13.d odd 4 1 inner
468.3.m.d 8 3.b odd 2 1
468.3.m.d 8 39.f even 4 1
624.3.ba.d 8 4.b odd 2 1
624.3.ba.d 8 52.f even 4 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
$3$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 576 \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 839056 \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 90782784 \) Copy content Toggle raw display
$13$ \( T^{8} - 24 T^{7} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{8} + 1296 T^{6} + \cdots + 427993344 \) Copy content Toggle raw display
$19$ \( T^{8} - 88 T^{7} + 3872 T^{6} + \cdots + 2972176 \) Copy content Toggle raw display
$23$ \( T^{8} + 2880 T^{6} + \cdots + 8422834176 \) Copy content Toggle raw display
$29$ \( (T^{4} - 12 T^{3} - 960 T^{2} + \cdots - 75504)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 10061584 \) Copy content Toggle raw display
$37$ \( T^{8} - 32 T^{7} + \cdots + 149561639824 \) Copy content Toggle raw display
$41$ \( T^{8} + 180 T^{7} + \cdots + 28199813184 \) Copy content Toggle raw display
$43$ \( T^{8} + 5088 T^{6} + \cdots + 619986161664 \) Copy content Toggle raw display
$47$ \( T^{8} - 36 T^{7} + \cdots + 4080820170816 \) Copy content Toggle raw display
$53$ \( (T^{4} + 36 T^{3} - 5184 T^{2} + \cdots - 976944)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 228 T^{7} + \cdots + 4746925417536 \) Copy content Toggle raw display
$61$ \( (T^{4} + 96 T^{3} + 1152 T^{2} + \cdots - 249024)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 16 T^{7} + \cdots + 929274256 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 448998131059776 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 601874565729424 \) Copy content Toggle raw display
$79$ \( (T^{4} - 24 T^{3} - 7296 T^{2} + \cdots + 613248)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 12 T^{7} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{8} - 60 T^{7} + \cdots + 1057162337856 \) Copy content Toggle raw display
$97$ \( T^{8} - 416 T^{7} + \cdots + 63914204772496 \) Copy content Toggle raw display
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