Properties

Label 225.2.b.b
Level $225$
Weight $2$
Character orbit 225.b
Analytic conductor $1.797$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} + 3 i q^{8} +O(q^{10})\) \( q + i q^{2} + q^{4} + 3 i q^{8} + 4 q^{11} + 2 i q^{13} - q^{16} -2 i q^{17} -4 q^{19} + 4 i q^{22} -2 q^{26} -2 q^{29} + 5 i q^{32} + 2 q^{34} -10 i q^{37} -4 i q^{38} -10 q^{41} -4 i q^{43} + 4 q^{44} -8 i q^{47} + 7 q^{49} + 2 i q^{52} -10 i q^{53} -2 i q^{58} -4 q^{59} -2 q^{61} -7 q^{64} + 12 i q^{67} -2 i q^{68} + 8 q^{71} -10 i q^{73} + 10 q^{74} -4 q^{76} -10 i q^{82} + 12 i q^{83} + 4 q^{86} + 12 i q^{88} -6 q^{89} + 8 q^{94} + 2 i q^{97} + 7 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + O(q^{10}) \) \( 2q + 2q^{4} + 8q^{11} - 2q^{16} - 8q^{19} - 4q^{26} - 4q^{29} + 4q^{34} - 20q^{41} + 8q^{44} + 14q^{49} - 8q^{59} - 4q^{61} - 14q^{64} + 16q^{71} + 20q^{74} - 8q^{76} + 8q^{86} - 12q^{89} + 16q^{94} + O(q^{100}) \)

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
1.00000i 0 1.00000 0 0 0 3.00000i 0 0
199.2 1.00000i 0 1.00000 0 0 0 3.00000i 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.b.b 2
3.b odd 2 1 75.2.b.b 2
4.b odd 2 1 3600.2.f.e 2
5.b even 2 1 inner 225.2.b.b 2
5.c odd 4 1 45.2.a.a 1
5.c odd 4 1 225.2.a.b 1
12.b even 2 1 1200.2.f.h 2
15.d odd 2 1 75.2.b.b 2
15.e even 4 1 15.2.a.a 1
15.e even 4 1 75.2.a.b 1
20.d odd 2 1 3600.2.f.e 2
20.e even 4 1 720.2.a.c 1
20.e even 4 1 3600.2.a.u 1
24.f even 2 1 4800.2.f.c 2
24.h odd 2 1 4800.2.f.bf 2
35.f even 4 1 2205.2.a.i 1
40.i odd 4 1 2880.2.a.y 1
40.k even 4 1 2880.2.a.bc 1
45.k odd 12 2 405.2.e.c 2
45.l even 12 2 405.2.e.f 2
55.e even 4 1 5445.2.a.c 1
60.h even 2 1 1200.2.f.h 2
60.l odd 4 1 240.2.a.d 1
60.l odd 4 1 1200.2.a.e 1
65.h odd 4 1 7605.2.a.g 1
105.k odd 4 1 735.2.a.c 1
105.k odd 4 1 3675.2.a.j 1
105.w odd 12 2 735.2.i.d 2
105.x even 12 2 735.2.i.e 2
120.i odd 2 1 4800.2.f.bf 2
120.m even 2 1 4800.2.f.c 2
120.q odd 4 1 960.2.a.a 1
120.q odd 4 1 4800.2.a.bz 1
120.w even 4 1 960.2.a.l 1
120.w even 4 1 4800.2.a.t 1
165.l odd 4 1 1815.2.a.d 1
165.l odd 4 1 9075.2.a.g 1
195.s even 4 1 2535.2.a.j 1
240.z odd 4 1 3840.2.k.r 2
240.bb even 4 1 3840.2.k.m 2
240.bd odd 4 1 3840.2.k.r 2
240.bf even 4 1 3840.2.k.m 2
255.o even 4 1 4335.2.a.c 1
285.j odd 4 1 5415.2.a.j 1
345.l odd 4 1 7935.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 15.e even 4 1
45.2.a.a 1 5.c odd 4 1
75.2.a.b 1 15.e even 4 1
75.2.b.b 2 3.b odd 2 1
75.2.b.b 2 15.d odd 2 1
225.2.a.b 1 5.c odd 4 1
225.2.b.b 2 1.a even 1 1 trivial
225.2.b.b 2 5.b even 2 1 inner
240.2.a.d 1 60.l odd 4 1
405.2.e.c 2 45.k odd 12 2
405.2.e.f 2 45.l even 12 2
720.2.a.c 1 20.e even 4 1
735.2.a.c 1 105.k odd 4 1
735.2.i.d 2 105.w odd 12 2
735.2.i.e 2 105.x even 12 2
960.2.a.a 1 120.q odd 4 1
960.2.a.l 1 120.w even 4 1
1200.2.a.e 1 60.l odd 4 1
1200.2.f.h 2 12.b even 2 1
1200.2.f.h 2 60.h even 2 1
1815.2.a.d 1 165.l odd 4 1
2205.2.a.i 1 35.f even 4 1
2535.2.a.j 1 195.s even 4 1
2880.2.a.y 1 40.i odd 4 1
2880.2.a.bc 1 40.k even 4 1
3600.2.a.u 1 20.e even 4 1
3600.2.f.e 2 4.b odd 2 1
3600.2.f.e 2 20.d odd 2 1
3675.2.a.j 1 105.k odd 4 1
3840.2.k.m 2 240.bb even 4 1
3840.2.k.m 2 240.bf even 4 1
3840.2.k.r 2 240.z odd 4 1
3840.2.k.r 2 240.bd odd 4 1
4335.2.a.c 1 255.o even 4 1
4800.2.a.t 1 120.w even 4 1
4800.2.a.bz 1 120.q odd 4 1
4800.2.f.c 2 24.f even 2 1
4800.2.f.c 2 120.m even 2 1
4800.2.f.bf 2 24.h odd 2 1
4800.2.f.bf 2 120.i odd 2 1
5415.2.a.j 1 285.j odd 4 1
5445.2.a.c 1 55.e even 4 1
7605.2.a.g 1 65.h odd 4 1
7935.2.a.d 1 345.l odd 4 1
9075.2.a.g 1 165.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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