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Properties

Label 460.2.i.a
Level $460$
Weight $2$
Character orbit 460.i
Analytic conductor $3.673$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.11574317056.3
Defining polynomial: \( x^{8} + 45x^{4} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{3} + (\beta_{6} - \beta_1) q^{5} - \beta_{2} q^{7} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{3} + (\beta_{6} - \beta_1) q^{5} - \beta_{2} q^{7} - \beta_{3} q^{9} + (\beta_{2} + \beta_1) q^{11} + (\beta_{7} + \beta_{6} + 2 \beta_{5}) q^{13} + ( - \beta_{7} - \beta_{6} - \beta_{2} + \beta_1) q^{15} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{2}) q^{17} + ( - 2 \beta_{7} + \beta_{2} - \beta_1) q^{19} + (\beta_{2} + \beta_1) q^{21} + ( - \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{3} + 2 \beta_1 - 2) q^{23} + (\beta_{7} + \beta_{6} - 2 \beta_{4} - \beta_{3} - 2) q^{25} + (4 \beta_{3} - 4) q^{27} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 5 \beta_{3}) q^{29} + (3 \beta_{5} + 3 \beta_{4} + 1) q^{31} - 2 \beta_1 q^{33} + (\beta_{5} + \beta_{4} - \beta_{3} - 2) q^{35} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{2}) q^{37} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4}) q^{39} + (\beta_{5} + \beta_{4} + 5) q^{41} + ( - \beta_{7} - \beta_{6} - 4 \beta_1) q^{43} + ( - \beta_{7} - \beta_{2}) q^{45} + (2 \beta_{7} + 2 \beta_{6} - 4 \beta_{4} + \beta_{3} - 1) q^{47} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 4 \beta_{3}) q^{49} + ( - 4 \beta_{6} + \beta_{2} + \beta_1) q^{51} + (2 \beta_{7} + 2 \beta_{6} + \beta_1) q^{53} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} + 3 \beta_{3} + 1) q^{55} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{2}) q^{57} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3}) q^{59} + (2 \beta_{6} + 3 \beta_{2} + 3 \beta_1) q^{61} + \beta_1 q^{63} + (\beta_{7} + 3 \beta_{6} - 4 \beta_{2} + 2 \beta_1) q^{65} + (2 \beta_{7} - 2 \beta_{6} - 3 \beta_{2}) q^{67} + (2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{69} + ( - \beta_{5} - \beta_{4} + 5) q^{71} + ( - 2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + 3 \beta_{3} + 3) q^{73} + (2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 1) q^{75} + (\beta_{7} + \beta_{6} - 2 \beta_{4} - 3 \beta_{3} + 3) q^{77} + (6 \beta_{7} + 3 \beta_{2} - 3 \beta_1) q^{79} + 5 q^{81} - \beta_1 q^{83} + (2 \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + 7 \beta_{3} - 6) q^{85} + ( - \beta_{7} - \beta_{6} + 2 \beta_{4} + 5 \beta_{3} - 5) q^{87} + (8 \beta_{7} + 5 \beta_{2} - 5 \beta_1) q^{89} + ( - 2 \beta_{6} + 4 \beta_{2} + 4 \beta_1) q^{91} + ( - 3 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} - \beta_{3} - 1) q^{93} + (3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} + 5 \beta_{3} + 5) q^{95} + (5 \beta_{7} - 5 \beta_{6} + 6 \beta_{2}) q^{97} + (\beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} - 4 q^{13} - 14 q^{23} - 12 q^{25} - 32 q^{27} - 4 q^{31} - 20 q^{35} + 36 q^{41} + 12 q^{55} + 44 q^{71} + 32 q^{73} + 28 q^{77} + 40 q^{81} - 44 q^{85} - 44 q^{87} + 4 q^{93} + 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 45x^{4} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 47\nu^{3} ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 47\nu^{2} ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} + 2\nu^{5} + 2\nu^{4} + 127\nu^{2} + 94\nu + 38 ) / 28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} - 2\nu^{5} + 2\nu^{4} - 127\nu^{2} - 94\nu + 38 ) / 28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} + 2\nu^{5} + 315\nu^{3} + 94\nu ) / 28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} + 2\nu^{5} - 315\nu^{3} + 94\nu ) / 28 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 7\beta_{4} - 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{7} + 7\beta_{6} - 47\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -47\beta_{7} - 47\beta_{6} - 47\beta_{5} + 47\beta_{4} + 127\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -47\beta_{7} + 47\beta_{6} - 315\beta_{2} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(277\) \(281\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1
1.83051 + 1.83051i
0.386289 + 0.386289i
−0.386289 − 0.386289i
−1.83051 − 1.83051i
1.83051 − 1.83051i
0.386289 − 0.386289i
−0.386289 + 0.386289i
−1.83051 + 1.83051i
0 −1.00000 + 1.00000i 0 −1.83051 − 1.28422i 0 1.83051 − 1.83051i 0 1.00000i 0
137.2 0 −1.00000 + 1.00000i 0 −0.386289 + 2.20245i 0 0.386289 − 0.386289i 0 1.00000i 0
137.3 0 −1.00000 + 1.00000i 0 0.386289 − 2.20245i 0 −0.386289 + 0.386289i 0 1.00000i 0
137.4 0 −1.00000 + 1.00000i 0 1.83051 + 1.28422i 0 −1.83051 + 1.83051i 0 1.00000i 0
413.1 0 −1.00000 − 1.00000i 0 −1.83051 + 1.28422i 0 1.83051 + 1.83051i 0 − 1.00000i 0
413.2 0 −1.00000 − 1.00000i 0 −0.386289 − 2.20245i 0 0.386289 + 0.386289i 0 − 1.00000i 0
413.3 0 −1.00000 − 1.00000i 0 0.386289 + 2.20245i 0 −0.386289 − 0.386289i 0 − 1.00000i 0
413.4 0 −1.00000 − 1.00000i 0 1.83051 − 1.28422i 0 −1.83051 − 1.83051i 0 − 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 413.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.i.a ✓ 8
5.b even 2 1 2300.2.i.c 8
5.c odd 4 1 inner 460.2.i.a ✓ 8
5.c odd 4 1 2300.2.i.c 8
23.b odd 2 1 inner 460.2.i.a ✓ 8
115.c odd 2 1 2300.2.i.c 8
115.e even 4 1 inner 460.2.i.a ✓ 8
115.e even 4 1 2300.2.i.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.i.a ✓ 8 1.a even 1 1 trivial
460.2.i.a ✓ 8 5.c odd 4 1 inner
460.2.i.a ✓ 8 23.b odd 2 1 inner
460.2.i.a ✓ 8 115.e even 4 1 inner
2300.2.i.c 8 5.b even 2 1
2300.2.i.c 8 5.c odd 4 1
2300.2.i.c 8 115.c odd 2 1
2300.2.i.c 8 115.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(460, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{6} + 18 T^{4} + 150 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 45T^{4} + 4 \) Copy content Toggle raw display
$11$ \( (T^{4} + 14 T^{2} + 8)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} + 2 T^{2} - 40 T + 400)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 2109 T^{4} + 2500 \) Copy content Toggle raw display
$19$ \( (T^{4} - 58 T^{2} + 800)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 14 T^{7} + 98 T^{6} + \cdots + 279841 \) Copy content Toggle raw display
$29$ \( (T^{4} + 81 T^{2} + 400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + T - 92)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 2109 T^{4} + 2500 \) Copy content Toggle raw display
$41$ \( (T^{2} - 9 T + 10)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + 16500 T^{4} + \cdots + 17909824 \) Copy content Toggle raw display
$47$ \( (T^{4} + 6724)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 4125 T^{4} + \cdots + 1119364 \) Copy content Toggle raw display
$59$ \( (T^{4} + 21 T^{2} + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 202 T^{2} + 8192)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 13965 T^{4} + \cdots + 48469444 \) Copy content Toggle raw display
$71$ \( (T^{2} - 11 T + 20)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 16 T^{3} + 128 T^{2} + 800 T + 2500)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 234 T^{2} + 10368)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 45T^{4} + 4 \) Copy content Toggle raw display
$89$ \( (T^{4} - 478 T^{2} + 55112)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 69540 T^{4} + \cdots + 945685504 \) Copy content Toggle raw display
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