Properties

Label 225.2.e.a
Level $225$
Weight $2$
Character orbit 225.e
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + ( - \zeta_{6} + 2) q^{3} + \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 1) q^{6} + (3 \zeta_{6} - 3) q^{7} + 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + ( - \zeta_{6} + 2) q^{3} + \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 1) q^{6} + (3 \zeta_{6} - 3) q^{7} + 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + ( - 2 \zeta_{6} + 2) q^{11} + (\zeta_{6} + 1) q^{12} - 2 \zeta_{6} q^{13} + 3 \zeta_{6} q^{14} + ( - \zeta_{6} + 1) q^{16} - 4 q^{17} - 3 \zeta_{6} q^{18} - 8 q^{19} + (6 \zeta_{6} - 3) q^{21} - 2 \zeta_{6} q^{22} + 3 \zeta_{6} q^{23} + ( - 3 \zeta_{6} + 6) q^{24} - 2 q^{26} + ( - 6 \zeta_{6} + 3) q^{27} - 3 q^{28} + ( - \zeta_{6} + 1) q^{29} + 5 \zeta_{6} q^{32} + ( - 4 \zeta_{6} + 2) q^{33} + (4 \zeta_{6} - 4) q^{34} + 3 q^{36} + 4 q^{37} + (8 \zeta_{6} - 8) q^{38} + ( - 2 \zeta_{6} - 2) q^{39} - 5 \zeta_{6} q^{41} + (3 \zeta_{6} + 3) q^{42} + (8 \zeta_{6} - 8) q^{43} + 2 q^{44} + 3 q^{46} + ( - 7 \zeta_{6} + 7) q^{47} + ( - 2 \zeta_{6} + 1) q^{48} - 2 \zeta_{6} q^{49} + (4 \zeta_{6} - 8) q^{51} + ( - 2 \zeta_{6} + 2) q^{52} + 2 q^{53} + ( - 3 \zeta_{6} - 3) q^{54} + (9 \zeta_{6} - 9) q^{56} + (8 \zeta_{6} - 16) q^{57} - \zeta_{6} q^{58} + 14 \zeta_{6} q^{59} + (7 \zeta_{6} - 7) q^{61} + 9 \zeta_{6} q^{63} + 7 q^{64} + ( - 2 \zeta_{6} - 2) q^{66} - 3 \zeta_{6} q^{67} - 4 \zeta_{6} q^{68} + (3 \zeta_{6} + 3) q^{69} + 2 q^{71} + ( - 9 \zeta_{6} + 9) q^{72} - 4 q^{73} + ( - 4 \zeta_{6} + 4) q^{74} - 8 \zeta_{6} q^{76} + 6 \zeta_{6} q^{77} + (2 \zeta_{6} - 4) q^{78} + ( - 6 \zeta_{6} + 6) q^{79} - 9 \zeta_{6} q^{81} - 5 q^{82} + ( - 9 \zeta_{6} + 9) q^{83} + (3 \zeta_{6} - 6) q^{84} + 8 \zeta_{6} q^{86} + ( - 2 \zeta_{6} + 1) q^{87} + ( - 6 \zeta_{6} + 6) q^{88} - 15 q^{89} + 6 q^{91} + (3 \zeta_{6} - 3) q^{92} - 7 \zeta_{6} q^{94} + (5 \zeta_{6} + 5) q^{96} + ( - 2 \zeta_{6} + 2) q^{97} - 2 q^{98} - 6 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} + q^{4} - 3 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 3 q^{3} + q^{4} - 3 q^{7} + 6 q^{8} + 3 q^{9} + 2 q^{11} + 3 q^{12} - 2 q^{13} + 3 q^{14} + q^{16} - 8 q^{17} - 3 q^{18} - 16 q^{19} - 2 q^{22} + 3 q^{23} + 9 q^{24} - 4 q^{26} - 6 q^{28} + q^{29} + 5 q^{32} - 4 q^{34} + 6 q^{36} + 8 q^{37} - 8 q^{38} - 6 q^{39} - 5 q^{41} + 9 q^{42} - 8 q^{43} + 4 q^{44} + 6 q^{46} + 7 q^{47} - 2 q^{49} - 12 q^{51} + 2 q^{52} + 4 q^{53} - 9 q^{54} - 9 q^{56} - 24 q^{57} - q^{58} + 14 q^{59} - 7 q^{61} + 9 q^{63} + 14 q^{64} - 6 q^{66} - 3 q^{67} - 4 q^{68} + 9 q^{69} + 4 q^{71} + 9 q^{72} - 8 q^{73} + 4 q^{74} - 8 q^{76} + 6 q^{77} - 6 q^{78} + 6 q^{79} - 9 q^{81} - 10 q^{82} + 9 q^{83} - 9 q^{84} + 8 q^{86} + 6 q^{88} - 30 q^{89} + 12 q^{91} - 3 q^{92} - 7 q^{94} + 15 q^{96} + 2 q^{97} - 4 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 1.50000 + 0.866025i 0.500000 0.866025i 0 1.73205i −1.50000 2.59808i 3.00000 1.50000 + 2.59808i 0
151.1 0.500000 0.866025i 1.50000 0.866025i 0.500000 + 0.866025i 0 1.73205i −1.50000 + 2.59808i 3.00000 1.50000 2.59808i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.e.a 2
3.b odd 2 1 675.2.e.a 2
5.b even 2 1 45.2.e.a 2
5.c odd 4 2 225.2.k.a 4
9.c even 3 1 inner 225.2.e.a 2
9.c even 3 1 2025.2.a.b 1
9.d odd 6 1 675.2.e.a 2
9.d odd 6 1 2025.2.a.e 1
15.d odd 2 1 135.2.e.a 2
15.e even 4 2 675.2.k.a 4
20.d odd 2 1 720.2.q.d 2
45.h odd 6 1 135.2.e.a 2
45.h odd 6 1 405.2.a.b 1
45.j even 6 1 45.2.e.a 2
45.j even 6 1 405.2.a.e 1
45.k odd 12 2 225.2.k.a 4
45.k odd 12 2 2025.2.b.c 2
45.l even 12 2 675.2.k.a 4
45.l even 12 2 2025.2.b.d 2
60.h even 2 1 2160.2.q.a 2
180.n even 6 1 2160.2.q.a 2
180.n even 6 1 6480.2.a.x 1
180.p odd 6 1 720.2.q.d 2
180.p odd 6 1 6480.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.a 2 5.b even 2 1
45.2.e.a 2 45.j even 6 1
135.2.e.a 2 15.d odd 2 1
135.2.e.a 2 45.h odd 6 1
225.2.e.a 2 1.a even 1 1 trivial
225.2.e.a 2 9.c even 3 1 inner
225.2.k.a 4 5.c odd 4 2
225.2.k.a 4 45.k odd 12 2
405.2.a.b 1 45.h odd 6 1
405.2.a.e 1 45.j even 6 1
675.2.e.a 2 3.b odd 2 1
675.2.e.a 2 9.d odd 6 1
675.2.k.a 4 15.e even 4 2
675.2.k.a 4 45.l even 12 2
720.2.q.d 2 20.d odd 2 1
720.2.q.d 2 180.p odd 6 1
2025.2.a.b 1 9.c even 3 1
2025.2.a.e 1 9.d odd 6 1
2025.2.b.c 2 45.k odd 12 2
2025.2.b.d 2 45.l even 12 2
2160.2.q.a 2 60.h even 2 1
2160.2.q.a 2 180.n even 6 1
6480.2.a.k 1 180.p odd 6 1
6480.2.a.x 1 180.n even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$71$ \( (T - 2)^{2} \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( (T + 15)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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