Properties

Label 225.4.b.b
Level $225$
Weight $4$
Character orbit 225.b
Analytic conductor $13.275$
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: \( k \) \(=\) \( 4 \)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 45)
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.

\(f(q)\) \(=\) \( q + 5 i q^{2} -17 q^{4} -30 i q^{7} -45 i q^{8} +O(q^{10})\) \( q + 5 i q^{2} -17 q^{4} -30 i q^{7} -45 i q^{8} + 50 q^{11} + 20 i q^{13} + 150 q^{14} + 89 q^{16} -10 i q^{17} + 44 q^{19} + 250 i q^{22} -120 i q^{23} -100 q^{26} + 510 i q^{28} + 50 q^{29} + 108 q^{31} + 85 i q^{32} + 50 q^{34} -40 i q^{37} + 220 i q^{38} + 400 q^{41} -280 i q^{43} -850 q^{44} + 600 q^{46} -280 i q^{47} -557 q^{49} -340 i q^{52} + 610 i q^{53} -1350 q^{56} + 250 i q^{58} -50 q^{59} -518 q^{61} + 540 i q^{62} + 287 q^{64} -180 i q^{67} + 170 i q^{68} + 700 q^{71} + 410 i q^{73} + 200 q^{74} -748 q^{76} -1500 i q^{77} + 516 q^{79} + 2000 i q^{82} -660 i q^{83} + 1400 q^{86} -2250 i q^{88} + 1500 q^{89} + 600 q^{91} + 2040 i q^{92} + 1400 q^{94} -1630 i q^{97} -2785 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 34 q^{4} + O(q^{10}) \) \( 2 q - 34 q^{4} + 100 q^{11} + 300 q^{14} + 178 q^{16} + 88 q^{19} - 200 q^{26} + 100 q^{29} + 216 q^{31} + 100 q^{34} + 800 q^{41} - 1700 q^{44} + 1200 q^{46} - 1114 q^{49} - 2700 q^{56} - 100 q^{59} - 1036 q^{61} + 574 q^{64} + 1400 q^{71} + 400 q^{74} - 1496 q^{76} + 1032 q^{79} + 2800 q^{86} + 3000 q^{89} + 1200 q^{91} + 2800 q^{94} + O(q^{100}) \)

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
5.00000i 0 −17.0000 0 0 30.0000i 45.0000i 0 0
199.2 5.00000i 0 −17.0000 0 0 30.0000i 45.0000i 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 225.4.b.b 2
3.b odd 2 1 225.4.b.a 2
5.b even 2 1 inner 225.4.b.b 2
5.c odd 4 1 45.4.a.e yes 1
5.c odd 4 1 225.4.a.a 1
15.d odd 2 1 225.4.b.a 2
15.e even 4 1 45.4.a.a 1
15.e even 4 1 225.4.a.h 1
20.e even 4 1 720.4.a.o 1
35.f even 4 1 2205.4.a.t 1
45.k odd 12 2 405.4.e.b 2
45.l even 12 2 405.4.e.n 2
60.l odd 4 1 720.4.a.bc 1
105.k odd 4 1 2205.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 15.e even 4 1
45.4.a.e yes 1 5.c odd 4 1
225.4.a.a 1 5.c odd 4 1
225.4.a.h 1 15.e even 4 1
225.4.b.a 2 3.b odd 2 1
225.4.b.a 2 15.d odd 2 1
225.4.b.b 2 1.a even 1 1 trivial
225.4.b.b 2 5.b even 2 1 inner
405.4.e.b 2 45.k odd 12 2
405.4.e.n 2 45.l even 12 2
720.4.a.o 1 20.e even 4 1
720.4.a.bc 1 60.l odd 4 1
2205.4.a.a 1 105.k odd 4 1
2205.4.a.t 1 35.f even 4 1

Hecke kernels

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

\( T_{2}^{2} + 25 \)
\( T_{7}^{2} + 900 \)
\( T_{11} - 50 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 900 + T^{2} \)
$11$ \( ( -50 + T )^{2} \)
$13$ \( 400 + T^{2} \)
$17$ \( 100 + T^{2} \)
$19$ \( ( -44 + T )^{2} \)
$23$ \( 14400 + T^{2} \)
$29$ \( ( -50 + T )^{2} \)
$31$ \( ( -108 + T )^{2} \)
$37$ \( 1600 + T^{2} \)
$41$ \( ( -400 + T )^{2} \)
$43$ \( 78400 + T^{2} \)
$47$ \( 78400 + T^{2} \)
$53$ \( 372100 + T^{2} \)
$59$ \( ( 50 + T )^{2} \)
$61$ \( ( 518 + T )^{2} \)
$67$ \( 32400 + T^{2} \)
$71$ \( ( -700 + T )^{2} \)
$73$ \( 168100 + T^{2} \)
$79$ \( ( -516 + T )^{2} \)
$83$ \( 435600 + T^{2} \)
$89$ \( ( -1500 + T )^{2} \)
$97$ \( 2656900 + T^{2} \)
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