Properties

Label 2240.4.a.bo
Level $2240$
Weight $4$
Character orbit 2240.a
Self dual yes
Analytic conductor $132.164$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.164278413\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

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

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 −4.65685 0 5.00000 0 7.00000 0 −5.31371 0
1.2 0 6.65685 0 5.00000 0 7.00000 0 17.3137 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 2240.4.a.bo 2
4.b odd 2 1 2240.4.a.bn 2
8.b even 2 1 560.4.a.r 2
8.d odd 2 1 35.4.a.b 2
24.f even 2 1 315.4.a.f 2
40.e odd 2 1 175.4.a.c 2
40.k even 4 2 175.4.b.c 4
56.e even 2 1 245.4.a.k 2
56.k odd 6 2 245.4.e.h 4
56.m even 6 2 245.4.e.i 4
120.m even 2 1 1575.4.a.z 2
168.e odd 2 1 2205.4.a.u 2
280.n 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 8.d odd 2 1
175.4.a.c 2 40.e odd 2 1
175.4.b.c 4 40.k even 4 2
245.4.a.k 2 56.e even 2 1
245.4.e.h 4 56.k odd 6 2
245.4.e.i 4 56.m even 6 2
315.4.a.f 2 24.f even 2 1
560.4.a.r 2 8.b even 2 1
1225.4.a.m 2 280.n even 2 1
1575.4.a.z 2 120.m even 2 1
2205.4.a.u 2 168.e odd 2 1
2240.4.a.bn 2 4.b odd 2 1
2240.4.a.bo 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2240))\):

\( T_{3}^{2} - 2 T_{3} - 31 \)
\( T_{11}^{2} + 14 T_{11} - 1999 \)

Hecke characteristic polynomials

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