Properties

Label 225.4.b.g
Level $225$
Weight $4$
Character orbit 225.b
Analytic conductor $13.275$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{4} + 20 i q^{7} +O(q^{10})\) \( q + 8 q^{4} + 20 i q^{7} + 70 i q^{13} + 64 q^{16} -56 q^{19} + 160 i q^{28} + 308 q^{31} + 110 i q^{37} + 520 i q^{43} -57 q^{49} + 560 i q^{52} + 182 q^{61} + 512 q^{64} -880 i q^{67} -1190 i q^{73} -448 q^{76} -884 q^{79} -1400 q^{91} -1330 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 16q^{4} + O(q^{10}) \) \( 2q + 16q^{4} + 128q^{16} - 112q^{19} + 616q^{31} - 114q^{49} + 364q^{61} + 1024q^{64} - 896q^{76} - 1768q^{79} - 2800q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
0 0 8.00000 0 0 20.0000i 0 0 0
199.2 0 0 8.00000 0 0 20.0000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
15.d 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.b.g 2
3.b odd 2 1 CM 225.4.b.g 2
5.b even 2 1 inner 225.4.b.g 2
5.c odd 4 1 9.4.a.a 1
5.c odd 4 1 225.4.a.d 1
15.d odd 2 1 inner 225.4.b.g 2
15.e even 4 1 9.4.a.a 1
15.e even 4 1 225.4.a.d 1
20.e even 4 1 144.4.a.d 1
35.f even 4 1 441.4.a.f 1
35.k even 12 2 441.4.e.j 2
35.l odd 12 2 441.4.e.i 2
40.i odd 4 1 576.4.a.m 1
40.k even 4 1 576.4.a.l 1
45.k odd 12 2 81.4.c.b 2
45.l even 12 2 81.4.c.b 2
55.e even 4 1 1089.4.a.g 1
60.l odd 4 1 144.4.a.d 1
65.h odd 4 1 1521.4.a.g 1
105.k odd 4 1 441.4.a.f 1
105.w odd 12 2 441.4.e.j 2
105.x even 12 2 441.4.e.i 2
120.q odd 4 1 576.4.a.l 1
120.w even 4 1 576.4.a.m 1
165.l odd 4 1 1089.4.a.g 1
195.s even 4 1 1521.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 5.c odd 4 1
9.4.a.a 1 15.e even 4 1
81.4.c.b 2 45.k odd 12 2
81.4.c.b 2 45.l even 12 2
144.4.a.d 1 20.e even 4 1
144.4.a.d 1 60.l odd 4 1
225.4.a.d 1 5.c odd 4 1
225.4.a.d 1 15.e even 4 1
225.4.b.g 2 1.a even 1 1 trivial
225.4.b.g 2 3.b odd 2 1 CM
225.4.b.g 2 5.b even 2 1 inner
225.4.b.g 2 15.d odd 2 1 inner
441.4.a.f 1 35.f even 4 1
441.4.a.f 1 105.k odd 4 1
441.4.e.i 2 35.l odd 12 2
441.4.e.i 2 105.x even 12 2
441.4.e.j 2 35.k even 12 2
441.4.e.j 2 105.w odd 12 2
576.4.a.l 1 40.k even 4 1
576.4.a.l 1 120.q odd 4 1
576.4.a.m 1 40.i odd 4 1
576.4.a.m 1 120.w even 4 1
1089.4.a.g 1 55.e even 4 1
1089.4.a.g 1 165.l odd 4 1
1521.4.a.g 1 65.h odd 4 1
1521.4.a.g 1 195.s even 4 1

Hecke kernels

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

\( T_{2} \)
\( T_{7}^{2} + 400 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 400 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4900 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 56 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -308 + T )^{2} \)
$37$ \( 12100 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 270400 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -182 + T )^{2} \)
$67$ \( 774400 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 1416100 + T^{2} \)
$79$ \( ( 884 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 1768900 + T^{2} \)
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