Properties

Label 3520.2.a.bn
Level $3520$
Weight $2$
Character orbit 3520.a
Self dual yes
Analytic conductor $28.107$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1073415115\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} -2 q^{7} + 5 q^{9} +O(q^{10})\) \( q + \beta q^{3} + q^{5} -2 q^{7} + 5 q^{9} - q^{11} + ( 4 - \beta ) q^{13} + \beta q^{15} + ( 4 + \beta ) q^{17} -2 \beta q^{21} -\beta q^{23} + q^{25} + 2 \beta q^{27} + ( -2 + 2 \beta ) q^{29} -\beta q^{33} -2 q^{35} + ( 2 + 2 \beta ) q^{37} + ( -8 + 4 \beta ) q^{39} + 6 q^{41} + 6 q^{43} + 5 q^{45} + \beta q^{47} -3 q^{49} + ( 8 + 4 \beta ) q^{51} + ( -6 - 2 \beta ) q^{53} - q^{55} + ( 4 - 2 \beta ) q^{59} + ( -2 + 4 \beta ) q^{61} -10 q^{63} + ( 4 - \beta ) q^{65} + ( -4 - 3 \beta ) q^{67} -8 q^{69} + 4 \beta q^{71} + ( -4 + \beta ) q^{73} + \beta q^{75} + 2 q^{77} + 4 q^{79} + q^{81} + 6 q^{83} + ( 4 + \beta ) q^{85} + ( 16 - 2 \beta ) q^{87} + ( -2 - 4 \beta ) q^{89} + ( -8 + 2 \beta ) q^{91} + ( -2 + 2 \beta ) q^{97} -5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 4 q^{7} + 10 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{5} - 4 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} + 8 q^{17} + 2 q^{25} - 4 q^{29} - 4 q^{35} + 4 q^{37} - 16 q^{39} + 12 q^{41} + 12 q^{43} + 10 q^{45} - 6 q^{49} + 16 q^{51} - 12 q^{53} - 2 q^{55} + 8 q^{59} - 4 q^{61} - 20 q^{63} + 8 q^{65} - 8 q^{67} - 16 q^{69} - 8 q^{73} + 4 q^{77} + 8 q^{79} + 2 q^{81} + 12 q^{83} + 8 q^{85} + 32 q^{87} - 4 q^{89} - 16 q^{91} - 4 q^{97} - 10 q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −2.82843 0 1.00000 0 −2.00000 0 5.00000 0
1.2 0 2.82843 0 1.00000 0 −2.00000 0 5.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(1\)

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 3520.2.a.bn 2
4.b odd 2 1 3520.2.a.bo 2
8.b even 2 1 55.2.a.b 2
8.d odd 2 1 880.2.a.m 2
24.f even 2 1 7920.2.a.ch 2
24.h odd 2 1 495.2.a.b 2
40.e odd 2 1 4400.2.a.bn 2
40.f even 2 1 275.2.a.c 2
40.i odd 4 2 275.2.b.d 4
40.k even 4 2 4400.2.b.q 4
56.h odd 2 1 2695.2.a.f 2
88.b odd 2 1 605.2.a.d 2
88.g even 2 1 9680.2.a.bn 2
88.o even 10 4 605.2.g.f 8
88.p odd 10 4 605.2.g.l 8
104.e even 2 1 9295.2.a.g 2
120.i odd 2 1 2475.2.a.x 2
120.w even 4 2 2475.2.c.l 4
264.m even 2 1 5445.2.a.y 2
440.o odd 2 1 3025.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 8.b even 2 1
275.2.a.c 2 40.f even 2 1
275.2.b.d 4 40.i odd 4 2
495.2.a.b 2 24.h odd 2 1
605.2.a.d 2 88.b odd 2 1
605.2.g.f 8 88.o even 10 4
605.2.g.l 8 88.p odd 10 4
880.2.a.m 2 8.d odd 2 1
2475.2.a.x 2 120.i odd 2 1
2475.2.c.l 4 120.w even 4 2
2695.2.a.f 2 56.h odd 2 1
3025.2.a.o 2 440.o odd 2 1
3520.2.a.bn 2 1.a even 1 1 trivial
3520.2.a.bo 2 4.b odd 2 1
4400.2.a.bn 2 40.e odd 2 1
4400.2.b.q 4 40.k even 4 2
5445.2.a.y 2 264.m even 2 1
7920.2.a.ch 2 24.f even 2 1
9295.2.a.g 2 104.e even 2 1
9680.2.a.bn 2 88.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3520))\):

\( T_{3}^{2} - 8 \)
\( T_{7} + 2 \)
\( T_{13}^{2} - 8 T_{13} + 8 \)
\( T_{17}^{2} - 8 T_{17} + 8 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -8 + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 8 - 8 T + T^{2} \)
$17$ \( 8 - 8 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( -8 + T^{2} \)
$29$ \( -28 + 4 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( -28 - 4 T + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( ( -6 + T )^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( 4 + 12 T + T^{2} \)
$59$ \( -16 - 8 T + T^{2} \)
$61$ \( -124 + 4 T + T^{2} \)
$67$ \( -56 + 8 T + T^{2} \)
$71$ \( -128 + T^{2} \)
$73$ \( 8 + 8 T + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( -124 + 4 T + T^{2} \)
$97$ \( -28 + 4 T + T^{2} \)
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