Properties

Label 225.4.a.l
Level $225$
Weight $4$
Character orbit 225.a
Self dual yes
Analytic conductor $13.275$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
Defining polynomial: \(x^{2} - 10\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 2 q^{4} -15 q^{7} -6 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 2 q^{4} -15 q^{7} -6 \beta q^{8} -20 \beta q^{11} -35 q^{13} -15 \beta q^{14} -76 q^{16} + 28 \beta q^{17} + 91 q^{19} -200 q^{22} -36 \beta q^{23} -35 \beta q^{26} -30 q^{28} + 20 \beta q^{29} -147 q^{31} -28 \beta q^{32} + 280 q^{34} -370 q^{37} + 91 \beta q^{38} + 140 \beta q^{41} -335 q^{43} -40 \beta q^{44} -360 q^{46} -56 \beta q^{47} -118 q^{49} -70 q^{52} + 28 \beta q^{53} + 90 \beta q^{56} + 200 q^{58} + 280 \beta q^{59} + 427 q^{61} -147 \beta q^{62} + 328 q^{64} -15 q^{67} + 56 \beta q^{68} + 20 \beta q^{71} + 70 q^{73} -370 \beta q^{74} + 182 q^{76} + 300 \beta q^{77} -876 q^{79} + 1400 q^{82} -168 \beta q^{83} -335 \beta q^{86} + 1200 q^{88} + 525 q^{91} -72 \beta q^{92} -560 q^{94} + 1085 q^{97} -118 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - 30 q^{7} + O(q^{10}) \) \( 2 q + 4 q^{4} - 30 q^{7} - 70 q^{13} - 152 q^{16} + 182 q^{19} - 400 q^{22} - 60 q^{28} - 294 q^{31} + 560 q^{34} - 740 q^{37} - 670 q^{43} - 720 q^{46} - 236 q^{49} - 140 q^{52} + 400 q^{58} + 854 q^{61} + 656 q^{64} - 30 q^{67} + 140 q^{73} + 364 q^{76} - 1752 q^{79} + 2800 q^{82} + 2400 q^{88} + 1050 q^{91} - 1120 q^{94} + 2170 q^{97} + O(q^{100}) \)

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
−3.16228
3.16228
−3.16228 0 2.00000 0 0 −15.0000 18.9737 0 0
1.2 3.16228 0 2.00000 0 0 −15.0000 −18.9737 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.a.l 2
3.b odd 2 1 inner 225.4.a.l 2
5.b even 2 1 225.4.a.m yes 2
5.c odd 4 2 225.4.b.i 4
15.d odd 2 1 225.4.a.m yes 2
15.e even 4 2 225.4.b.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.a.l 2 1.a even 1 1 trivial
225.4.a.l 2 3.b odd 2 1 inner
225.4.a.m yes 2 5.b even 2 1
225.4.a.m yes 2 15.d odd 2 1
225.4.b.i 4 5.c odd 4 2
225.4.b.i 4 15.e even 4 2

Hecke kernels

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

\( T_{2}^{2} - 10 \)
\( T_{7} + 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -10 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 15 + T )^{2} \)
$11$ \( -4000 + T^{2} \)
$13$ \( ( 35 + T )^{2} \)
$17$ \( -7840 + T^{2} \)
$19$ \( ( -91 + T )^{2} \)
$23$ \( -12960 + T^{2} \)
$29$ \( -4000 + T^{2} \)
$31$ \( ( 147 + T )^{2} \)
$37$ \( ( 370 + T )^{2} \)
$41$ \( -196000 + T^{2} \)
$43$ \( ( 335 + T )^{2} \)
$47$ \( -31360 + T^{2} \)
$53$ \( -7840 + T^{2} \)
$59$ \( -784000 + T^{2} \)
$61$ \( ( -427 + T )^{2} \)
$67$ \( ( 15 + T )^{2} \)
$71$ \( -4000 + T^{2} \)
$73$ \( ( -70 + T )^{2} \)
$79$ \( ( 876 + T )^{2} \)
$83$ \( -282240 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -1085 + T )^{2} \)
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