Properties

Label 225.10.a.c
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

\(f(q)\) \(=\) \( q - 4q^{2} - 496q^{4} + 7680q^{7} + 4032q^{8} + O(q^{10}) \) \( q - 4q^{2} - 496q^{4} + 7680q^{7} + 4032q^{8} + 86404q^{11} + 149978q^{13} - 30720q^{14} + 237824q^{16} - 207622q^{17} + 716284q^{19} - 345616q^{22} + 1369920q^{23} - 599912q^{26} - 3809280q^{28} + 3194402q^{29} - 2349000q^{31} - 3015680q^{32} + 830488q^{34} - 18735710q^{37} - 2865136q^{38} + 29282630q^{41} + 1516724q^{43} - 42856384q^{44} - 5479680q^{46} + 615752q^{47} + 18628793q^{49} - 74389088q^{52} + 4747430q^{53} + 30965760q^{56} - 12777608q^{58} - 60616076q^{59} - 126745682q^{61} + 9396000q^{62} - 109703168q^{64} + 111182652q^{67} + 102980512q^{68} + 175551608q^{71} + 61233350q^{73} + 74942840q^{74} - 355276864q^{76} + 663582720q^{77} + 234431160q^{79} - 117130520q^{82} + 118910388q^{83} - 6066896q^{86} + 348380928q^{88} + 316534326q^{89} + 1151831040q^{91} - 679480320q^{92} - 2463008q^{94} - 242912258q^{97} - 74515172q^{98} + O(q^{100}) \)

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 −496.000 0 0 7680.00 4032.00 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.c 1
3.b odd 2 1 75.10.a.c 1
5.b even 2 1 45.10.a.b 1
5.c odd 4 2 225.10.b.e 2
15.d odd 2 1 15.10.a.a 1
15.e even 4 2 75.10.b.d 2
60.h even 2 1 240.10.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.a 1 15.d odd 2 1
45.10.a.b 1 5.b even 2 1
75.10.a.c 1 3.b odd 2 1
75.10.b.d 2 15.e even 4 2
225.10.a.c 1 1.a even 1 1 trivial
225.10.b.e 2 5.c odd 4 2
240.10.a.c 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 \(S_{10}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} + 4 \)
\( T_{7} - 7680 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 512 T^{2} \)
$3$ 1
$5$ 1
$7$ \( 1 - 7680 T + 40353607 T^{2} \)
$11$ \( 1 - 86404 T + 2357947691 T^{2} \)
$13$ \( 1 - 149978 T + 10604499373 T^{2} \)
$17$ \( 1 + 207622 T + 118587876497 T^{2} \)
$19$ \( 1 - 716284 T + 322687697779 T^{2} \)
$23$ \( 1 - 1369920 T + 1801152661463 T^{2} \)
$29$ \( 1 - 3194402 T + 14507145975869 T^{2} \)
$31$ \( 1 + 2349000 T + 26439622160671 T^{2} \)
$37$ \( 1 + 18735710 T + 129961739795077 T^{2} \)
$41$ \( 1 - 29282630 T + 327381934393961 T^{2} \)
$43$ \( 1 - 1516724 T + 502592611936843 T^{2} \)
$47$ \( 1 - 615752 T + 1119130473102767 T^{2} \)
$53$ \( 1 - 4747430 T + 3299763591802133 T^{2} \)
$59$ \( 1 + 60616076 T + 8662995818654939 T^{2} \)
$61$ \( 1 + 126745682 T + 11694146092834141 T^{2} \)
$67$ \( 1 - 111182652 T + 27206534396294947 T^{2} \)
$71$ \( 1 - 175551608 T + 45848500718449031 T^{2} \)
$73$ \( 1 - 61233350 T + 58871586708267913 T^{2} \)
$79$ \( 1 - 234431160 T + 119851595982618319 T^{2} \)
$83$ \( 1 - 118910388 T + 186940255267540403 T^{2} \)
$89$ \( 1 - 316534326 T + 350356403707485209 T^{2} \)
$97$ \( 1 + 242912258 T + 760231058654565217 T^{2} \)
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