Properties

Label 7220.2.a.p
Level $7220$
Weight $2$
Character orbit 7220.a
Self dual yes
Analytic conductor $57.652$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7220 = 2^{2} \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7220.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.6519902594\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.133593.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + 3 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 380)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + q^{5} -\beta_{3} q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + q^{5} -\beta_{3} q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{11} + ( 2 + \beta_{1} + \beta_{3} ) q^{13} -\beta_{1} q^{15} + ( -\beta_{1} - \beta_{2} ) q^{17} + ( 2 + \beta_{2} ) q^{21} + ( \beta_{2} + \beta_{3} ) q^{23} + q^{25} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{27} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{29} + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{31} + ( 6 + \beta_{1} + \beta_{2} ) q^{33} -\beta_{3} q^{35} + ( 6 + 2 \beta_{1} - \beta_{3} ) q^{37} + ( -6 - 3 \beta_{1} - 2 \beta_{2} ) q^{39} + ( -2 - \beta_{3} ) q^{41} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{43} + ( 1 + \beta_{1} + \beta_{2} ) q^{45} + ( -4 - 2 \beta_{2} - \beta_{3} ) q^{47} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{49} + ( 2 + 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{51} + ( 2 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{53} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{55} + ( 3 - \beta_{1} - 2 \beta_{3} ) q^{59} + ( -3 + 2 \beta_{3} ) q^{61} + ( 2 - 5 \beta_{1} + 2 \beta_{3} ) q^{63} + ( 2 + \beta_{1} + \beta_{3} ) q^{65} + 2 \beta_{2} q^{67} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{69} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{71} + ( 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{73} -\beta_{1} q^{75} + ( -6 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{77} + ( 3 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{79} + ( 5 + 4 \beta_{1} + \beta_{3} ) q^{81} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{83} + ( -\beta_{1} - \beta_{2} ) q^{85} + ( -2 \beta_{1} + \beta_{2} ) q^{87} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{89} + ( -12 + 2 \beta_{1} - 3 \beta_{3} ) q^{91} + ( 8 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( -\beta_{1} - 2 \beta_{2} ) q^{97} + ( -5 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{3} + 4q^{5} + 5q^{9} + O(q^{10}) \) \( 4q - q^{3} + 4q^{5} + 5q^{9} + 2q^{11} + 9q^{13} - q^{15} - q^{17} + 8q^{21} + 4q^{25} - 10q^{27} + 5q^{29} + 10q^{31} + 25q^{33} + 26q^{37} - 27q^{39} - 8q^{41} - 7q^{43} + 5q^{45} - 16q^{47} + 10q^{49} + 12q^{51} + 5q^{53} + 2q^{55} + 11q^{59} - 12q^{61} + 3q^{63} + 9q^{65} - 3q^{69} + 14q^{71} + 4q^{73} - q^{75} - 22q^{77} + 13q^{79} + 24q^{81} + 5q^{83} - q^{85} - 2q^{87} + 5q^{89} - 46q^{91} + 28q^{93} - q^{97} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 8 x^{2} + 3 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 7 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 8 \beta_{1} + 2\)

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
3.16505
0.709218
−0.353450
−2.52082
0 −3.16505 0 1.00000 0 −1.53315 0 7.01755 0
1.2 0 −0.709218 0 1.00000 0 3.11079 0 −2.49701 0
1.3 0 0.353450 0 1.00000 0 −4.30507 0 −2.87507 0
1.4 0 2.52082 0 1.00000 0 2.72743 0 3.35453 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(19\) \(-1\)

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 7220.2.a.p 4
19.b odd 2 1 7220.2.a.r 4
19.d odd 6 2 380.2.i.c 8
57.f even 6 2 3420.2.t.w 8
76.f even 6 2 1520.2.q.m 8
95.h odd 6 2 1900.2.i.d 8
95.l even 12 4 1900.2.s.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.c 8 19.d odd 6 2
1520.2.q.m 8 76.f even 6 2
1900.2.i.d 8 95.h odd 6 2
1900.2.s.d 16 95.l even 12 4
3420.2.t.w 8 57.f even 6 2
7220.2.a.p 4 1.a even 1 1 trivial
7220.2.a.r 4 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7220))\):

\( T_{3}^{4} + T_{3}^{3} - 8 T_{3}^{2} - 3 T_{3} + 2 \)
\( T_{7}^{4} - 19 T_{7}^{2} + 11 T_{7} + 56 \)
\( T_{13}^{4} - 9 T_{13}^{3} - 5 T_{13}^{2} + 129 T_{13} + 52 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 2 - 3 T - 8 T^{2} + T^{3} + T^{4} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( 56 + 11 T - 19 T^{2} + T^{4} \)
$11$ \( 267 + 21 T - 35 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( 52 + 129 T - 5 T^{2} - 9 T^{3} + T^{4} \)
$17$ \( 192 + 9 T - 34 T^{2} + T^{3} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( -54 + 99 T - 39 T^{2} + T^{4} \)
$29$ \( -273 + 318 T - 59 T^{2} - 5 T^{3} + T^{4} \)
$31$ \( -277 + 303 T - 29 T^{2} - 10 T^{3} + T^{4} \)
$37$ \( 414 - 609 T + 217 T^{2} - 26 T^{3} + T^{4} \)
$41$ \( 18 - 33 T + 5 T^{2} + 8 T^{3} + T^{4} \)
$43$ \( -178 - 177 T - 32 T^{2} + 7 T^{3} + T^{4} \)
$47$ \( -1176 - 987 T - 25 T^{2} + 16 T^{3} + T^{4} \)
$53$ \( 1218 + 603 T - 118 T^{2} - 5 T^{3} + T^{4} \)
$59$ \( 1587 + 462 T - 55 T^{2} - 11 T^{3} + T^{4} \)
$61$ \( 29 - 436 T - 22 T^{2} + 12 T^{3} + T^{4} \)
$67$ \( 3296 - 56 T - 124 T^{2} + T^{4} \)
$71$ \( -2247 + 669 T - T^{2} - 14 T^{3} + T^{4} \)
$73$ \( 558 - 75 T - 137 T^{2} - 4 T^{3} + T^{4} \)
$79$ \( -3016 + 1812 T - 128 T^{2} - 13 T^{3} + T^{4} \)
$83$ \( -1302 + 669 T - 82 T^{2} - 5 T^{3} + T^{4} \)
$89$ \( 1176 + 84 T - 122 T^{2} - 5 T^{3} + T^{4} \)
$97$ \( 3654 - 51 T - 122 T^{2} + T^{3} + T^{4} \)
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