Properties

Label 414.3.b.c
Level $414$
Weight $3$
Character orbit 414.b
Analytic conductor $11.281$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 414.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.2806829445\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1358954496.3
Defining polynomial: \( x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 138)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + 2 q^{4} + (\beta_{6} - 2 \beta_{3} + \beta_{2}) q^{5} + (2 \beta_{7} - \beta_{6} + 3 \beta_{3} + 2 \beta_{2}) q^{7} + 2 \beta_{5} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + 2 q^{4} + (\beta_{6} - 2 \beta_{3} + \beta_{2}) q^{5} + (2 \beta_{7} - \beta_{6} + 3 \beta_{3} + 2 \beta_{2}) q^{7} + 2 \beta_{5} q^{8} + ( - \beta_{7} - 2 \beta_{6} + 3 \beta_{3} - \beta_{2}) q^{10} + ( - 2 \beta_{7} + 2 \beta_{3} + 4 \beta_{2}) q^{11} + ( - 2 \beta_{5} + 3 \beta_{4} + 2 \beta_1 + 2) q^{13} + (4 \beta_{7} + \beta_{6} - \beta_{3} + 5 \beta_{2}) q^{14} + 4 q^{16} + ( - 6 \beta_{7} + 4 \beta_{6} + 3 \beta_{3}) q^{17} + (11 \beta_{7} + 7 \beta_{6} - 4 \beta_{3} + 3 \beta_{2}) q^{19} + (2 \beta_{6} - 4 \beta_{3} + 2 \beta_{2}) q^{20} + ( - 6 \beta_{6} + 2 \beta_{3} + 6 \beta_{2}) q^{22} + (5 \beta_{7} - 8 \beta_{6} - 5 \beta_{5} - 4 \beta_{4} + 5 \beta_{3} + 7 \beta_{2} + \cdots - 2) q^{23}+ \cdots + (5 \beta_{5} - 14 \beta_{4} - 18 \beta_1 - 28) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} + 16 q^{13} + 32 q^{16} - 16 q^{23} + 72 q^{25} - 32 q^{26} + 144 q^{29} - 128 q^{31} + 112 q^{35} + 16 q^{41} - 80 q^{46} + 112 q^{47} + 40 q^{49} + 160 q^{50} + 32 q^{52} - 64 q^{55} + 128 q^{58} - 80 q^{59} + 96 q^{62} + 64 q^{64} - 144 q^{70} - 32 q^{71} + 64 q^{73} - 224 q^{77} + 48 q^{85} - 32 q^{92} - 16 q^{94} - 112 q^{95} - 224 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} + 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{5} + 5\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} + 5\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{6} - 6\nu^{4} - 7\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} + 6\nu^{4} + 9\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} + 7\nu^{5} + 13\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} + 7\nu^{5} + 15\nu^{3} + 11\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} - 3\beta_{3} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{5} - 2\beta_{4} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} + 5\beta_{6} + 11\beta_{3} - 9\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 17\beta_{5} + 15\beta_{4} - 12\beta _1 - 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 22\beta_{7} - 20\beta_{6} - 43\beta_{3} + 29\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(235\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
1.21752i
1.98289i
1.98289i
1.21752i
1.58671i
0.261052i
0.261052i
1.58671i
−1.41421 0 2.00000 6.00464i 0 4.69017i −2.82843 0 8.49185i
91.2 −1.41421 0 2.00000 4.92225i 0 5.13851i −2.82843 0 6.96111i
91.3 −1.41421 0 2.00000 4.92225i 0 5.13851i −2.82843 0 6.96111i
91.4 −1.41421 0 2.00000 6.00464i 0 4.69017i −2.82843 0 8.49185i
91.5 1.41421 0 2.00000 1.69484i 0 4.18388i 2.82843 0 2.39686i
91.6 1.41421 0 2.00000 0.918288i 0 10.4925i 2.82843 0 1.29866i
91.7 1.41421 0 2.00000 0.918288i 0 10.4925i 2.82843 0 1.29866i
91.8 1.41421 0 2.00000 1.69484i 0 4.18388i 2.82843 0 2.39686i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.3.b.c 8
3.b odd 2 1 138.3.b.a 8
12.b even 2 1 1104.3.c.c 8
23.b odd 2 1 inner 414.3.b.c 8
69.c even 2 1 138.3.b.a 8
276.h odd 2 1 1104.3.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.3.b.a 8 3.b odd 2 1
138.3.b.a 8 69.c even 2 1
414.3.b.c 8 1.a even 1 1 trivial
414.3.b.c 8 23.b odd 2 1 inner
1104.3.c.c 8 12.b even 2 1
1104.3.c.c 8 276.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 64T_{5}^{6} + 1100T_{5}^{4} + 3392T_{5}^{2} + 2116 \) acting on \(S_{3}^{\mathrm{new}}(414, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 64 T^{6} + 1100 T^{4} + \cdots + 2116 \) Copy content Toggle raw display
$7$ \( T^{8} + 176 T^{6} + 8684 T^{4} + \cdots + 1119364 \) Copy content Toggle raw display
$11$ \( T^{8} + 512 T^{6} + 43712 T^{4} + \cdots + 2262016 \) Copy content Toggle raw display
$13$ \( (T^{4} - 8 T^{3} - 124 T^{2} + 1136 T - 956)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 1168 T^{6} + \cdots + 1138792516 \) Copy content Toggle raw display
$19$ \( T^{8} + 2144 T^{6} + \cdots + 14491825924 \) Copy content Toggle raw display
$23$ \( T^{8} + 16 T^{7} + \cdots + 78310985281 \) Copy content Toggle raw display
$29$ \( (T^{4} - 72 T^{3} + 1160 T^{2} + \cdots - 6128)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 64 T^{3} + 816 T^{2} + 2560 T + 1600)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 7168 T^{6} + \cdots + 2955978735616 \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} - 2568 T^{2} + \cdots + 1208464)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 6704 T^{6} + \cdots + 26854687876 \) Copy content Toggle raw display
$47$ \( (T^{4} - 56 T^{3} - 412 T^{2} + \cdots + 256036)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 842259883497796 \) Copy content Toggle raw display
$59$ \( (T^{4} + 40 T^{3} - 5788 T^{2} + \cdots + 6720292)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 24704 T^{6} + \cdots + 138674176 \) Copy content Toggle raw display
$67$ \( T^{8} + 12192 T^{6} + \cdots + 15376496004 \) Copy content Toggle raw display
$71$ \( (T^{4} + 16 T^{3} - 5704 T^{2} + \cdots - 812912)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 32 T^{3} - 7872 T^{2} + \cdots + 590848)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 8800 T^{6} + \cdots + 7772598291844 \) Copy content Toggle raw display
$83$ \( T^{8} + 14912 T^{6} + \cdots + 5440313672704 \) Copy content Toggle raw display
$89$ \( T^{8} + 40448 T^{6} + \cdots + 69\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 380951699424256 \) Copy content Toggle raw display
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