Properties

Label 225.10.a.f
Level $225$
Weight $10$
Character orbit 225.a
Self dual yes
Analytic conductor $115.883$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(115.883063137\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 22q^{2} - 28q^{4} + 5988q^{7} - 11880q^{8} + O(q^{10}) \) \( q + 22q^{2} - 28q^{4} + 5988q^{7} - 11880q^{8} + 14648q^{11} - 37906q^{13} + 131736q^{14} - 247024q^{16} - 441098q^{17} + 441820q^{19} + 322256q^{22} + 2264136q^{23} - 833932q^{26} - 167664q^{28} + 1049350q^{29} - 7910568q^{31} + 648032q^{32} - 9704156q^{34} + 20992558q^{37} + 9720040q^{38} - 13285562q^{41} + 23130764q^{43} - 410144q^{44} + 49810992q^{46} - 13873688q^{47} - 4497463q^{49} + 1061368q^{52} - 57635174q^{53} - 71137440q^{56} + 23085700q^{58} + 32042120q^{59} + 110664022q^{61} - 174032496q^{62} + 140732992q^{64} + 118568268q^{67} + 12350744q^{68} - 276679712q^{71} + 264023294q^{73} + 461836276q^{74} - 12370960q^{76} + 87712224q^{77} + 448202760q^{79} - 292282364q^{82} + 851015796q^{83} + 508876808q^{86} - 174018240q^{88} - 189894930q^{89} - 226981128q^{91} - 63395808q^{92} - 305221136q^{94} + 1014149278q^{97} - 98944186q^{98} + O(q^{100}) \)

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
22.0000 0 −28.0000 0 0 5988.00 −11880.0 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 225.10.a.f 1
3.b odd 2 1 75.10.a.a 1
5.b even 2 1 45.10.a.a 1
5.c odd 4 2 225.10.b.b 2
15.d odd 2 1 15.10.a.b 1
15.e even 4 2 75.10.b.b 2
60.h even 2 1 240.10.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.b 1 15.d odd 2 1
45.10.a.a 1 5.b even 2 1
75.10.a.a 1 3.b odd 2 1
75.10.b.b 2 15.e even 4 2
225.10.a.f 1 1.a even 1 1 trivial
225.10.b.b 2 5.c odd 4 2
240.10.a.g 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} - 22 \)
\( T_{7} - 5988 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 22 T + 512 T^{2} \)
$3$ 1
$5$ 1
$7$ \( 1 - 5988 T + 40353607 T^{2} \)
$11$ \( 1 - 14648 T + 2357947691 T^{2} \)
$13$ \( 1 + 37906 T + 10604499373 T^{2} \)
$17$ \( 1 + 441098 T + 118587876497 T^{2} \)
$19$ \( 1 - 441820 T + 322687697779 T^{2} \)
$23$ \( 1 - 2264136 T + 1801152661463 T^{2} \)
$29$ \( 1 - 1049350 T + 14507145975869 T^{2} \)
$31$ \( 1 + 7910568 T + 26439622160671 T^{2} \)
$37$ \( 1 - 20992558 T + 129961739795077 T^{2} \)
$41$ \( 1 + 13285562 T + 327381934393961 T^{2} \)
$43$ \( 1 - 23130764 T + 502592611936843 T^{2} \)
$47$ \( 1 + 13873688 T + 1119130473102767 T^{2} \)
$53$ \( 1 + 57635174 T + 3299763591802133 T^{2} \)
$59$ \( 1 - 32042120 T + 8662995818654939 T^{2} \)
$61$ \( 1 - 110664022 T + 11694146092834141 T^{2} \)
$67$ \( 1 - 118568268 T + 27206534396294947 T^{2} \)
$71$ \( 1 + 276679712 T + 45848500718449031 T^{2} \)
$73$ \( 1 - 264023294 T + 58871586708267913 T^{2} \)
$79$ \( 1 - 448202760 T + 119851595982618319 T^{2} \)
$83$ \( 1 - 851015796 T + 186940255267540403 T^{2} \)
$89$ \( 1 + 189894930 T + 350356403707485209 T^{2} \)
$97$ \( 1 - 1014149278 T + 760231058654565217 T^{2} \)
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