Properties

Label 4056.2.c.p
Level $4056$
Weight $2$
Character orbit 4056.c
Analytic conductor $32.387$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.649638144.4
Defining polynomial: \( x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 312)
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.

\(f(q)\) \(=\) \( q + q^{3} - \beta_1 q^{5} + ( - \beta_{4} - \beta_1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_1 q^{5} + ( - \beta_{4} - \beta_1) q^{7} + q^{9} + ( - \beta_{7} + \beta_{4} + \beta_1) q^{11} - \beta_1 q^{15} + ( - \beta_{2} - 3) q^{17} + (\beta_{7} - \beta_{4} + \beta_1) q^{19} + ( - \beta_{4} - \beta_1) q^{21} + ( - \beta_{3} - \beta_{2} + 1) q^{23} + (2 \beta_{6} - \beta_{3} - 1) q^{25} + q^{27} + (\beta_{6} - 2) q^{29} + (\beta_{7} - 2 \beta_{5} - 2 \beta_1) q^{31} + ( - \beta_{7} + \beta_{4} + \beta_1) q^{33} + (2 \beta_{6} - \beta_{3} - \beta_{2} - 3) q^{35} + (\beta_{5} - \beta_{4} + \beta_1) q^{37} + ( - 3 \beta_{5} - \beta_{4} - 3 \beta_1) q^{41} + (\beta_{6} - \beta_{3} + \beta_{2} + 2) q^{43} - \beta_1 q^{45} + ( - \beta_{7} + 2 \beta_{5} - \beta_{4} - \beta_1) q^{47} + ( - \beta_{3} - 2 \beta_{2} - 4) q^{49} + ( - \beta_{2} - 3) q^{51} + (\beta_{3} - 2 \beta_{2}) q^{53} + ( - 3 \beta_{6} + 2 \beta_{3} + \beta_{2} + 3) q^{55} + (\beta_{7} - \beta_{4} + \beta_1) q^{57} + ( - \beta_{7} + 3 \beta_{5} - \beta_1) q^{59} + (\beta_{6} - 2 \beta_{3} - 3 \beta_{2}) q^{61} + ( - \beta_{4} - \beta_1) q^{63} + (4 \beta_{5} - \beta_{4} + \beta_1) q^{67} + ( - \beta_{3} - \beta_{2} + 1) q^{69} + ( - 2 \beta_{7} - \beta_{5} + 3 \beta_{4}) q^{71} + (3 \beta_{5} + 2 \beta_1) q^{73} + (2 \beta_{6} - \beta_{3} - 1) q^{75} + ( - 2 \beta_{6} - 2 \beta_{2} + 8) q^{77} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{79} + q^{81} + (2 \beta_{7} - \beta_{5} + \beta_{4}) q^{83} + ( - \beta_{5} + \beta_{4} + 3 \beta_1) q^{85} + (\beta_{6} - 2) q^{87} + ( - 3 \beta_{7} + \beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{89} + (\beta_{7} - 2 \beta_{5} - 2 \beta_1) q^{93} + ( - \beta_{6} - \beta_{2} + 9) q^{95} + ( - \beta_{7} + 6 \beta_{5}) q^{97} + ( - \beta_{7} + \beta_{4} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 8 q^{9} - 24 q^{17} + 4 q^{23} - 4 q^{25} + 8 q^{27} - 12 q^{29} - 20 q^{35} + 16 q^{43} - 36 q^{49} - 24 q^{51} + 4 q^{53} + 20 q^{55} - 4 q^{61} + 4 q^{69} - 4 q^{75} + 56 q^{77} - 12 q^{79} + 8 q^{81} - 12 q^{87} + 68 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{7} + 26\nu^{6} + 16\nu^{5} - 260\nu^{4} - 56\nu^{3} + 1092\nu^{2} + 55\nu - 1508 ) / 364 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} - 39\nu^{6} + 57\nu^{5} + 390\nu^{4} - 245\nu^{3} - 1092\nu^{2} + 668\nu + 351 ) / 364 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{7} + 48\nu^{5} + 196\nu^{3} - 1655\nu + 182 ) / 364 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{7} - 39\nu^{6} - 73\nu^{5} + 572\nu^{4} + 301\nu^{3} - 2730\nu^{2} - 359\nu + 3991 ) / 364 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 62\nu^{3} + 79\nu ) / 26 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -19\nu^{7} + 26\nu^{6} + 162\nu^{5} - 260\nu^{4} - 294\nu^{3} + 728\nu^{2} - 319\nu - 52 ) / 364 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23\nu^{7} + 26\nu^{6} - 244\nu^{5} - 260\nu^{4} + 1036\nu^{3} + 1092\nu^{2} - 1271\nu - 1508 ) / 364 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{6} - \beta_{5} - 3\beta_{3} + 2\beta_{2} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2} + 3\beta _1 + 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{7} + 15\beta_{6} + \beta_{5} - 7\beta_{3} + 10\beta_{2} - 11\beta _1 - 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} + 4\beta_{5} + 4\beta_{4} + 3\beta_{3} + 6\beta_{2} + 15\beta _1 + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 38\beta_{7} + 27\beta_{6} + 32\beta_{5} - 7\beta_{3} + 18\beta_{2} - 38\beta _1 - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 16\beta_{7} - 9\beta_{6} + 26\beta_{5} + 40\beta_{4} + 9\beta_{3} + 18\beta_{2} + 108\beta _1 + 52 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 461\beta_{7} + 63\beta_{6} + 547\beta_{5} + \beta_{3} + 42\beta_{2} - 461\beta _1 - 32 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times\).

\(n\) \(1015\) \(2029\) \(2705\) \(3889\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
−2.34138 0.500000i
−1.42055 0.500000i
1.42055 + 0.500000i
2.34138 + 0.500000i
2.34138 0.500000i
1.42055 0.500000i
−1.42055 + 0.500000i
−2.34138 + 0.500000i
0 1.00000 0 4.20740i 0 3.55539i 0 1.00000 0
337.2 0 1.00000 0 1.55452i 0 2.96046i 0 1.00000 0
337.3 0 1.00000 0 1.28657i 0 1.96046i 0 1.00000 0
337.4 0 1.00000 0 0.475353i 0 4.55539i 0 1.00000 0
337.5 0 1.00000 0 0.475353i 0 4.55539i 0 1.00000 0
337.6 0 1.00000 0 1.28657i 0 1.96046i 0 1.00000 0
337.7 0 1.00000 0 1.55452i 0 2.96046i 0 1.00000 0
337.8 0 1.00000 0 4.20740i 0 3.55539i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4056.2.c.p 8
13.b even 2 1 inner 4056.2.c.p 8
13.c even 3 1 312.2.bf.b 8
13.d odd 4 1 4056.2.a.bd 4
13.d odd 4 1 4056.2.a.be 4
13.e even 6 1 312.2.bf.b 8
39.h odd 6 1 936.2.bi.c 8
39.i odd 6 1 936.2.bi.c 8
52.f even 4 1 8112.2.a.cq 4
52.f even 4 1 8112.2.a.cs 4
52.i odd 6 1 624.2.bv.g 8
52.j odd 6 1 624.2.bv.g 8
156.p even 6 1 1872.2.by.m 8
156.r even 6 1 1872.2.by.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bf.b 8 13.c even 3 1
312.2.bf.b 8 13.e even 6 1
624.2.bv.g 8 52.i odd 6 1
624.2.bv.g 8 52.j odd 6 1
936.2.bi.c 8 39.h odd 6 1
936.2.bi.c 8 39.i odd 6 1
1872.2.by.m 8 156.p even 6 1
1872.2.by.m 8 156.r even 6 1
4056.2.a.bd 4 13.d odd 4 1
4056.2.a.be 4 13.d odd 4 1
4056.2.c.p 8 1.a even 1 1 trivial
4056.2.c.p 8 13.b even 2 1 inner
8112.2.a.cq 4 52.f even 4 1
8112.2.a.cs 4 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4056, [\chi])\):

\( T_{5}^{8} + 22T_{5}^{6} + 81T_{5}^{4} + 88T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} + 46T_{7}^{6} + 717T_{7}^{4} + 4432T_{7}^{2} + 8836 \) Copy content Toggle raw display
\( T_{11}^{8} + 64T_{11}^{6} + 1236T_{11}^{4} + 6880T_{11}^{2} + 10816 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 22 T^{6} + 81 T^{4} + 88 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 46 T^{6} + 717 T^{4} + \cdots + 8836 \) Copy content Toggle raw display
$11$ \( T^{8} + 64 T^{6} + 1236 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 12 T^{3} + 33 T^{2} - 48)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 112 T^{6} + 3636 T^{4} + \cdots + 141376 \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{3} - 30 T^{2} - 32 T - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} - 15 T^{2} - 36 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 118 T^{6} + 4473 T^{4} + \cdots + 141376 \) Copy content Toggle raw display
$37$ \( T^{8} + 82 T^{6} + 1425 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$41$ \( T^{8} + 210 T^{6} + 13809 T^{4} + \cdots + 3154176 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} - 45 T^{2} + 190 T + 694)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 160 T^{6} + 9300 T^{4} + \cdots + 2096704 \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} - 147 T^{2} - 248 T + 208)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 184 T^{6} + 9744 T^{4} + \cdots + 262144 \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{3} - 198 T^{2} - 430 T + 6481)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 190 T^{6} + 3693 T^{4} + \cdots + 45796 \) Copy content Toggle raw display
$71$ \( T^{8} + 432 T^{6} + 48756 T^{4} + \cdots + 1272384 \) Copy content Toggle raw display
$73$ \( T^{8} + 124 T^{6} + 4518 T^{4} + \cdots + 185761 \) Copy content Toggle raw display
$79$ \( (T^{4} + 6 T^{3} - 63 T^{2} - 108 T + 216)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 312 T^{6} + 22692 T^{4} + \cdots + 1557504 \) Copy content Toggle raw display
$89$ \( T^{8} + 480 T^{6} + 68160 T^{4} + \cdots + 589824 \) Copy content Toggle raw display
$97$ \( T^{8} + 414 T^{6} + \cdots + 33039504 \) Copy content Toggle raw display
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