Properties

Label 225.4.a.b
Level $225$
Weight $4$
Character orbit 225.a
Self dual yes
Analytic conductor $13.275$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.2754297513\)
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

\(f(q)\) \(=\) \( q - 4q^{2} + 8q^{4} - 6q^{7} + O(q^{10}) \) \( q - 4q^{2} + 8q^{4} - 6q^{7} - 32q^{11} + 38q^{13} + 24q^{14} - 64q^{16} + 26q^{17} + 100q^{19} + 128q^{22} - 78q^{23} - 152q^{26} - 48q^{28} + 50q^{29} - 108q^{31} + 256q^{32} - 104q^{34} - 266q^{37} - 400q^{38} - 22q^{41} - 442q^{43} - 256q^{44} + 312q^{46} - 514q^{47} - 307q^{49} + 304q^{52} + 2q^{53} - 200q^{58} - 500q^{59} - 518q^{61} + 432q^{62} - 512q^{64} - 126q^{67} + 208q^{68} - 412q^{71} + 878q^{73} + 1064q^{74} + 800q^{76} + 192q^{77} + 600q^{79} + 88q^{82} + 282q^{83} + 1768q^{86} + 150q^{89} - 228q^{91} - 624q^{92} + 2056q^{94} - 386q^{97} + 1228q^{98} + O(q^{100}) \)

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 0 8.00000 0 0 −6.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

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 225.4.a.b 1
3.b odd 2 1 25.4.a.c 1
5.b even 2 1 45.4.a.d 1
5.c odd 4 2 225.4.b.c 2
12.b even 2 1 400.4.a.m 1
15.d odd 2 1 5.4.a.a 1
15.e even 4 2 25.4.b.a 2
20.d odd 2 1 720.4.a.u 1
21.c even 2 1 1225.4.a.k 1
24.f even 2 1 1600.4.a.s 1
24.h odd 2 1 1600.4.a.bi 1
35.c odd 2 1 2205.4.a.q 1
45.h odd 6 2 405.4.e.l 2
45.j even 6 2 405.4.e.c 2
60.h even 2 1 80.4.a.d 1
60.l odd 4 2 400.4.c.k 2
105.g even 2 1 245.4.a.a 1
105.o odd 6 2 245.4.e.f 2
105.p even 6 2 245.4.e.g 2
120.i odd 2 1 320.4.a.g 1
120.m even 2 1 320.4.a.h 1
165.d even 2 1 605.4.a.d 1
195.e odd 2 1 845.4.a.b 1
240.t even 4 2 1280.4.d.l 2
240.bm odd 4 2 1280.4.d.e 2
255.h odd 2 1 1445.4.a.a 1
285.b 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 15.d odd 2 1
25.4.a.c 1 3.b odd 2 1
25.4.b.a 2 15.e even 4 2
45.4.a.d 1 5.b even 2 1
80.4.a.d 1 60.h even 2 1
225.4.a.b 1 1.a even 1 1 trivial
225.4.b.c 2 5.c odd 4 2
245.4.a.a 1 105.g even 2 1
245.4.e.f 2 105.o odd 6 2
245.4.e.g 2 105.p even 6 2
320.4.a.g 1 120.i odd 2 1
320.4.a.h 1 120.m even 2 1
400.4.a.m 1 12.b even 2 1
400.4.c.k 2 60.l odd 4 2
405.4.e.c 2 45.j even 6 2
405.4.e.l 2 45.h odd 6 2
605.4.a.d 1 165.d even 2 1
720.4.a.u 1 20.d odd 2 1
845.4.a.b 1 195.e odd 2 1
1225.4.a.k 1 21.c even 2 1
1280.4.d.e 2 240.bm odd 4 2
1280.4.d.l 2 240.t even 4 2
1445.4.a.a 1 255.h odd 2 1
1600.4.a.s 1 24.f even 2 1
1600.4.a.bi 1 24.h odd 2 1
1805.4.a.h 1 285.b even 2 1
2205.4.a.q 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} + 4 \)
\( T_{7} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 8 T^{2} \)
$3$ 1
$5$ 1
$7$ \( 1 + 6 T + 343 T^{2} \)
$11$ \( 1 + 32 T + 1331 T^{2} \)
$13$ \( 1 - 38 T + 2197 T^{2} \)
$17$ \( 1 - 26 T + 4913 T^{2} \)
$19$ \( 1 - 100 T + 6859 T^{2} \)
$23$ \( 1 + 78 T + 12167 T^{2} \)
$29$ \( 1 - 50 T + 24389 T^{2} \)
$31$ \( 1 + 108 T + 29791 T^{2} \)
$37$ \( 1 + 266 T + 50653 T^{2} \)
$41$ \( 1 + 22 T + 68921 T^{2} \)
$43$ \( 1 + 442 T + 79507 T^{2} \)
$47$ \( 1 + 514 T + 103823 T^{2} \)
$53$ \( 1 - 2 T + 148877 T^{2} \)
$59$ \( 1 + 500 T + 205379 T^{2} \)
$61$ \( 1 + 518 T + 226981 T^{2} \)
$67$ \( 1 + 126 T + 300763 T^{2} \)
$71$ \( 1 + 412 T + 357911 T^{2} \)
$73$ \( 1 - 878 T + 389017 T^{2} \)
$79$ \( 1 - 600 T + 493039 T^{2} \)
$83$ \( 1 - 282 T + 571787 T^{2} \)
$89$ \( 1 - 150 T + 704969 T^{2} \)
$97$ \( 1 + 386 T + 912673 T^{2} \)
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