Properties

Label 1480.2.a.g
Level $1480$
Weight $2$
Character orbit 1480.a
Self dual yes
Analytic conductor $11.818$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1480 = 2^{3} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1480.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.8178594991\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.583504.1
Defining polynomial: \(x^{5} - 8 x^{3} - 2 x^{2} + 15 x + 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} + q^{5} + ( -1 - \beta_{2} + \beta_{4} ) q^{7} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} + q^{5} + ( -1 - \beta_{2} + \beta_{4} ) q^{7} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{9} + ( -1 - \beta_{3} - \beta_{4} ) q^{11} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( -1 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{17} + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{19} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{21} + ( -2 + 2 \beta_{2} + 2 \beta_{3} ) q^{23} + q^{25} + ( -3 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{27} + ( 2 - 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{33} + ( -1 - \beta_{2} + \beta_{4} ) q^{35} - q^{37} + ( -3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{39} + ( 1 + \beta_{3} + \beta_{4} ) q^{41} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{45} + ( -3 - 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{47} + ( 3 - 3 \beta_{2} + 2 \beta_{3} ) q^{49} + ( -2 + 3 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - \beta_{4} ) q^{51} + ( -1 + 2 \beta_{1} - 5 \beta_{3} - \beta_{4} ) q^{53} + ( -1 - \beta_{3} - \beta_{4} ) q^{55} + ( -1 - 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{57} + ( -2 + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{59} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -2 + 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{63} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{65} + ( -6 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{67} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{69} + ( -3 + 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{71} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{73} + ( -1 + \beta_{1} ) q^{75} + ( -5 + 6 \beta_{2} + \beta_{3} - \beta_{4} ) q^{77} + ( -\beta_{2} + 3 \beta_{3} ) q^{79} + ( -2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{81} + ( -3 + 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{83} + ( -1 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{87} + ( 2 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} ) q^{89} + ( -2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{91} + ( -3 + 2 \beta_{1} - 5 \beta_{2} + \beta_{3} - \beta_{4} ) q^{93} + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{95} + ( -2 + 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} ) q^{97} + 2 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{3} + 5q^{5} - 5q^{7} + 6q^{9} + O(q^{10}) \) \( 5q - 5q^{3} + 5q^{5} - 5q^{7} + 6q^{9} - 7q^{11} - 6q^{13} - 5q^{15} - 12q^{19} - q^{21} - 6q^{23} + 5q^{25} - 17q^{27} + 6q^{29} - 10q^{31} + 3q^{33} - 5q^{35} - 5q^{37} + 4q^{39} + 7q^{41} - 22q^{43} + 6q^{45} - 13q^{47} + 14q^{49} - 10q^{51} - 11q^{53} - 7q^{55} - 8q^{57} - 10q^{59} - 10q^{63} - 6q^{65} - 24q^{67} + 26q^{69} - 13q^{71} - 13q^{73} - 5q^{75} - 19q^{77} + 2q^{79} + q^{81} - 13q^{83} - 26q^{87} + 8q^{89} - 8q^{91} - 20q^{93} - 12q^{95} - 20q^{97} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 8 x^{3} - 2 x^{2} + 15 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu - 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 6 \nu^{2} + 4 \nu + 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 6 \beta_{2} + 12\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.15365
−1.44679
−0.592644
1.84514
2.34794
0 −3.15365 0 1.00000 0 −0.579974 0 6.94552 0
1.2 0 −2.44679 0 1.00000 0 −4.02963 0 2.98676 0
1.3 0 −1.59264 0 1.00000 0 4.50235 0 −0.463486 0
1.4 0 0.845141 0 1.00000 0 −2.14219 0 −2.28574 0
1.5 0 1.34794 0 1.00000 0 −2.75056 0 −1.18306 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(37\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1480.2.a.g 5
4.b odd 2 1 2960.2.a.bb 5
5.b even 2 1 7400.2.a.r 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.a.g 5 1.a even 1 1 trivial
2960.2.a.bb 5 4.b odd 2 1
7400.2.a.r 5 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} + 5 T_{3}^{4} + 2 T_{3}^{3} - 16 T_{3}^{2} - 8 T_{3} + 14 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1480))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( 14 - 8 T - 16 T^{2} + 2 T^{3} + 5 T^{4} + T^{5} \)
$5$ \( ( -1 + T )^{5} \)
$7$ \( -62 - 160 T - 100 T^{2} - 12 T^{3} + 5 T^{4} + T^{5} \)
$11$ \( 32 + 16 T - 48 T^{2} + 7 T^{4} + T^{5} \)
$13$ \( -8 - 196 T - 172 T^{2} - 28 T^{3} + 6 T^{4} + T^{5} \)
$17$ \( 2896 + 1268 T - 100 T^{2} - 76 T^{3} + T^{5} \)
$19$ \( 32 - 46 T - 62 T^{2} + 26 T^{3} + 12 T^{4} + T^{5} \)
$23$ \( -224 - 880 T - 432 T^{2} - 48 T^{3} + 6 T^{4} + T^{5} \)
$29$ \( 224 - 880 T + 432 T^{2} - 48 T^{3} - 6 T^{4} + T^{5} \)
$31$ \( -1112 - 1206 T - 370 T^{2} - 10 T^{3} + 10 T^{4} + T^{5} \)
$37$ \( ( 1 + T )^{5} \)
$41$ \( -32 + 16 T + 48 T^{2} - 7 T^{4} + T^{5} \)
$43$ \( -32 + 240 T + 384 T^{2} + 152 T^{3} + 22 T^{4} + T^{5} \)
$47$ \( -1238 - 1228 T - 276 T^{2} + 24 T^{3} + 13 T^{4} + T^{5} \)
$53$ \( 58544 + 848 T - 2520 T^{2} - 192 T^{3} + 11 T^{4} + T^{5} \)
$59$ \( 1448 - 134 T - 486 T^{2} - 42 T^{3} + 10 T^{4} + T^{5} \)
$61$ \( -10816 + 3952 T + 144 T^{2} - 128 T^{3} + T^{5} \)
$67$ \( 992 - 4554 T - 1582 T^{2} + 50 T^{3} + 24 T^{4} + T^{5} \)
$71$ \( 99136 + 9344 T - 2624 T^{2} - 212 T^{3} + 13 T^{4} + T^{5} \)
$73$ \( 11632 + 2768 T - 912 T^{2} - 108 T^{3} + 13 T^{4} + T^{5} \)
$79$ \( -11456 + 2930 T + 346 T^{2} - 110 T^{3} - 2 T^{4} + T^{5} \)
$83$ \( -35434 + 22984 T - 1612 T^{2} - 254 T^{3} + 13 T^{4} + T^{5} \)
$89$ \( 1984 + 1008 T - 272 T^{2} - 144 T^{3} - 8 T^{4} + T^{5} \)
$97$ \( -31744 - 19808 T - 3536 T^{2} - 80 T^{3} + 20 T^{4} + T^{5} \)
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