Properties

Label 560.4.a.r
Level $560$
Weight $4$
Character orbit 560.a
Self dual yes
Analytic conductor $33.041$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.0410696032\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{3} - 5 q^{5} + 7 q^{7} + ( - 2 \beta + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{3} - 5 q^{5} + 7 q^{7} + ( - 2 \beta + 6) q^{9} + (8 \beta + 7) q^{11} + (\beta + 25) q^{13} + ( - 5 \beta + 5) q^{15} + (11 \beta - 25) q^{17} + ( - 11 \beta - 18) q^{19} + (7 \beta - 7) q^{21} + (17 \beta - 122) q^{23} + 25 q^{25} + ( - 19 \beta - 43) q^{27} + (6 \beta - 13) q^{29} + (45 \beta + 60) q^{31} + ( - \beta + 249) q^{33} - 35 q^{35} + ( - 15 \beta + 282) q^{37} + (24 \beta + 7) q^{39} + (31 \beta - 164) q^{41} + ( - 17 \beta + 130) q^{43} + (10 \beta - 30) q^{45} + (33 \beta + 175) q^{47} + 49 q^{49} + ( - 36 \beta + 377) q^{51} + (32 \beta - 28) q^{53} + ( - 40 \beta - 35) q^{55} + ( - 7 \beta - 334) q^{57} + 616 q^{59} + ( - 27 \beta + 168) q^{61} + ( - 14 \beta + 42) q^{63} + ( - 5 \beta - 125) q^{65} + (16 \beta + 76) q^{67} + ( - 139 \beta + 666) q^{69} + 952 q^{71} + ( - 86 \beta + 338) q^{73} + (25 \beta - 25) q^{75} + (56 \beta + 49) q^{77} + ( - 62 \beta - 507) q^{79} + (30 \beta - 727) q^{81} + ( - 150 \beta + 188) q^{83} + ( - 55 \beta + 125) q^{85} + ( - 19 \beta + 205) q^{87} + (11 \beta - 108) q^{89} + (7 \beta + 175) q^{91} + (15 \beta + 1380) q^{93} + (55 \beta + 90) q^{95} + (55 \beta + 1371) q^{97} + (34 \beta - 470) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 10 q^{5} + 14 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 10 q^{5} + 14 q^{7} + 12 q^{9} + 14 q^{11} + 50 q^{13} + 10 q^{15} - 50 q^{17} - 36 q^{19} - 14 q^{21} - 244 q^{23} + 50 q^{25} - 86 q^{27} - 26 q^{29} + 120 q^{31} + 498 q^{33} - 70 q^{35} + 564 q^{37} + 14 q^{39} - 328 q^{41} + 260 q^{43} - 60 q^{45} + 350 q^{47} + 98 q^{49} + 754 q^{51} - 56 q^{53} - 70 q^{55} - 668 q^{57} + 1232 q^{59} + 336 q^{61} + 84 q^{63} - 250 q^{65} + 152 q^{67} + 1332 q^{69} + 1904 q^{71} + 676 q^{73} - 50 q^{75} + 98 q^{77} - 1014 q^{79} - 1454 q^{81} + 376 q^{83} + 250 q^{85} + 410 q^{87} - 216 q^{89} + 350 q^{91} + 2760 q^{93} + 180 q^{95} + 2742 q^{97} - 940 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −6.65685 0 −5.00000 0 7.00000 0 17.3137 0
1.2 0 4.65685 0 −5.00000 0 7.00000 0 −5.31371 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

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 560.4.a.r 2
4.b odd 2 1 35.4.a.b 2
8.b even 2 1 2240.4.a.bo 2
8.d odd 2 1 2240.4.a.bn 2
12.b even 2 1 315.4.a.f 2
20.d odd 2 1 175.4.a.c 2
20.e even 4 2 175.4.b.c 4
28.d even 2 1 245.4.a.k 2
28.f even 6 2 245.4.e.i 4
28.g odd 6 2 245.4.e.h 4
60.h even 2 1 1575.4.a.z 2
84.h odd 2 1 2205.4.a.u 2
140.c even 2 1 1225.4.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 4.b odd 2 1
175.4.a.c 2 20.d odd 2 1
175.4.b.c 4 20.e even 4 2
245.4.a.k 2 28.d even 2 1
245.4.e.h 4 28.g odd 6 2
245.4.e.i 4 28.f even 6 2
315.4.a.f 2 12.b even 2 1
560.4.a.r 2 1.a even 1 1 trivial
1225.4.a.m 2 140.c even 2 1
1575.4.a.z 2 60.h even 2 1
2205.4.a.u 2 84.h odd 2 1
2240.4.a.bn 2 8.d odd 2 1
2240.4.a.bo 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(560))\):

\( T_{3}^{2} + 2T_{3} - 31 \) Copy content Toggle raw display
\( T_{11}^{2} - 14T_{11} - 1999 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 31 \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 14T - 1999 \) Copy content Toggle raw display
$13$ \( T^{2} - 50T + 593 \) Copy content Toggle raw display
$17$ \( T^{2} + 50T - 3247 \) Copy content Toggle raw display
$19$ \( T^{2} + 36T - 3548 \) Copy content Toggle raw display
$23$ \( T^{2} + 244T + 5636 \) Copy content Toggle raw display
$29$ \( T^{2} + 26T - 983 \) Copy content Toggle raw display
$31$ \( T^{2} - 120T - 61200 \) Copy content Toggle raw display
$37$ \( T^{2} - 564T + 72324 \) Copy content Toggle raw display
$41$ \( T^{2} + 328T - 3856 \) Copy content Toggle raw display
$43$ \( T^{2} - 260T + 7652 \) Copy content Toggle raw display
$47$ \( T^{2} - 350T - 4223 \) Copy content Toggle raw display
$53$ \( T^{2} + 56T - 31984 \) Copy content Toggle raw display
$59$ \( (T - 616)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 336T + 4896 \) Copy content Toggle raw display
$67$ \( T^{2} - 152T - 2416 \) Copy content Toggle raw display
$71$ \( (T - 952)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 676T - 122428 \) Copy content Toggle raw display
$79$ \( T^{2} + 1014 T + 134041 \) Copy content Toggle raw display
$83$ \( T^{2} - 376T - 684656 \) Copy content Toggle raw display
$89$ \( T^{2} + 216T + 7792 \) Copy content Toggle raw display
$97$ \( T^{2} - 2742 T + 1782841 \) Copy content Toggle raw display
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