Properties

Label 1225.4.a.k
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.2773397570\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{2} + 2q^{3} + 8q^{4} + 8q^{6} - 23q^{9} + O(q^{10}) \) \( q + 4q^{2} + 2q^{3} + 8q^{4} + 8q^{6} - 23q^{9} + 32q^{11} + 16q^{12} - 38q^{13} - 64q^{16} + 26q^{17} - 92q^{18} - 100q^{19} + 128q^{22} + 78q^{23} - 152q^{26} - 100q^{27} - 50q^{29} + 108q^{31} - 256q^{32} + 64q^{33} + 104q^{34} - 184q^{36} - 266q^{37} - 400q^{38} - 76q^{39} - 22q^{41} - 442q^{43} + 256q^{44} + 312q^{46} - 514q^{47} - 128q^{48} + 52q^{51} - 304q^{52} - 2q^{53} - 400q^{54} - 200q^{57} - 200q^{58} - 500q^{59} + 518q^{61} + 432q^{62} - 512q^{64} + 256q^{66} - 126q^{67} + 208q^{68} + 156q^{69} + 412q^{71} - 878q^{73} - 1064q^{74} - 800q^{76} - 304q^{78} + 600q^{79} + 421q^{81} - 88q^{82} + 282q^{83} - 1768q^{86} - 100q^{87} + 150q^{89} + 624q^{92} + 216q^{93} - 2056q^{94} - 512q^{96} + 386q^{97} - 736q^{99} + O(q^{100}) \)

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
0
4.00000 2.00000 8.00000 0 8.00000 0 0 −23.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1225.4.a.k 1
5.b even 2 1 245.4.a.a 1
7.b odd 2 1 25.4.a.c 1
15.d odd 2 1 2205.4.a.q 1
21.c even 2 1 225.4.a.b 1
28.d even 2 1 400.4.a.m 1
35.c odd 2 1 5.4.a.a 1
35.f even 4 2 25.4.b.a 2
35.i odd 6 2 245.4.e.f 2
35.j even 6 2 245.4.e.g 2
56.e even 2 1 1600.4.a.s 1
56.h odd 2 1 1600.4.a.bi 1
105.g even 2 1 45.4.a.d 1
105.k odd 4 2 225.4.b.c 2
140.c even 2 1 80.4.a.d 1
140.j odd 4 2 400.4.c.k 2
280.c odd 2 1 320.4.a.g 1
280.n even 2 1 320.4.a.h 1
315.z even 6 2 405.4.e.c 2
315.bg odd 6 2 405.4.e.l 2
385.h even 2 1 605.4.a.d 1
420.o odd 2 1 720.4.a.u 1
455.h odd 2 1 845.4.a.b 1
560.be even 4 2 1280.4.d.l 2
560.bf odd 4 2 1280.4.d.e 2
595.b odd 2 1 1445.4.a.a 1
665.g even 2 1 1805.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 35.c odd 2 1
25.4.a.c 1 7.b odd 2 1
25.4.b.a 2 35.f even 4 2
45.4.a.d 1 105.g even 2 1
80.4.a.d 1 140.c even 2 1
225.4.a.b 1 21.c even 2 1
225.4.b.c 2 105.k odd 4 2
245.4.a.a 1 5.b even 2 1
245.4.e.f 2 35.i odd 6 2
245.4.e.g 2 35.j even 6 2
320.4.a.g 1 280.c odd 2 1
320.4.a.h 1 280.n even 2 1
400.4.a.m 1 28.d even 2 1
400.4.c.k 2 140.j odd 4 2
405.4.e.c 2 315.z even 6 2
405.4.e.l 2 315.bg odd 6 2
605.4.a.d 1 385.h even 2 1
720.4.a.u 1 420.o odd 2 1
845.4.a.b 1 455.h odd 2 1
1225.4.a.k 1 1.a even 1 1 trivial
1280.4.d.e 2 560.bf odd 4 2
1280.4.d.l 2 560.be even 4 2
1445.4.a.a 1 595.b odd 2 1
1600.4.a.s 1 56.e even 2 1
1600.4.a.bi 1 56.h odd 2 1
1805.4.a.h 1 665.g even 2 1
2205.4.a.q 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2} - 4 \)
\( T_{3} - 2 \)
\( T_{19} + 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( -2 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -32 + T \)
$13$ \( 38 + T \)
$17$ \( -26 + T \)
$19$ \( 100 + T \)
$23$ \( -78 + T \)
$29$ \( 50 + T \)
$31$ \( -108 + T \)
$37$ \( 266 + T \)
$41$ \( 22 + T \)
$43$ \( 442 + T \)
$47$ \( 514 + T \)
$53$ \( 2 + T \)
$59$ \( 500 + T \)
$61$ \( -518 + T \)
$67$ \( 126 + T \)
$71$ \( -412 + T \)
$73$ \( 878 + T \)
$79$ \( -600 + T \)
$83$ \( -282 + T \)
$89$ \( -150 + T \)
$97$ \( -386 + T \)
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