Properties

Label 225.2.h.b
Level $225$
Weight $2$
Character orbit 225.h
Analytic conductor $1.797$
Analytic rank $0$
Dimension $4$
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.h (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{2} + \zeta_{10}) q^{2} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{4} + (2 \zeta_{10}^{2} - \zeta_{10} + 2) q^{5} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{7} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{2} + \zeta_{10}) q^{2} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{4} + (2 \zeta_{10}^{2} - \zeta_{10} + 2) q^{5} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{7} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 2) q^{10} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 2) q^{11} + (3 \zeta_{10}^{2} + 3) q^{13} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{14} + ( - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{16} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2 \zeta_{10}) q^{17} + ( - 3 \zeta_{10}^{3} - \zeta_{10}^{2} - 3 \zeta_{10}) q^{19} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 3) q^{20} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{22} + ( - 7 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 7) q^{23} + 5 \zeta_{10}^{2} q^{25} + 3 q^{26} + ( - 2 \zeta_{10}^{3} - \zeta_{10} + 1) q^{28} + ( - 2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{29} - 3 \zeta_{10}^{2} q^{31} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 5) q^{32} + (4 \zeta_{10}^{2} - 6 \zeta_{10} + 4) q^{34} + ( - \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{35} + ( - 2 \zeta_{10}^{2} - \zeta_{10} - 2) q^{37} + ( - \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{38} + 5 \zeta_{10}^{3} q^{40} + ( - 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{41} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2}) q^{43} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{44} + ( - 2 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{46} + ( - \zeta_{10}^{3} - \zeta_{10} + 1) q^{47} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 5) q^{49} + (5 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{50} + (6 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 6) q^{52} + (3 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{53} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 2) q^{55} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} - \zeta_{10}) q^{56} + ( - 3 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 3 \zeta_{10}) q^{58} + (3 \zeta_{10}^{2} - 9 \zeta_{10} + 3) q^{59} + ( - \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} + 1) q^{61} + ( - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{62} + (\zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 1) q^{64} + (3 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 3 \zeta_{10}) q^{65} + (2 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 2 \zeta_{10}) q^{67} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2}) q^{68} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 1) q^{70} + ( - 5 \zeta_{10}^{3} + \zeta_{10} - 1) q^{71} + ( - 9 \zeta_{10}^{3} + 9 \zeta_{10}^{2} - 9 \zeta_{10} + 9) q^{73} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 2) q^{74} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 7) q^{76} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{77} + ( - 5 \zeta_{10} + 5) q^{79} + (3 \zeta_{10}^{3} - 9 \zeta_{10}^{2} + 3 \zeta_{10}) q^{80} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{82} + ( - 2 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 2 \zeta_{10}) q^{83} + ( - 2 \zeta_{10}^{3} + 6 \zeta_{10} - 6) q^{85} + ( - 3 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 3) q^{86} + ( - 6 \zeta_{10}^{2} + 8 \zeta_{10} - 6) q^{88} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} + 4) q^{89} + ( - 3 \zeta_{10}^{2} - 3 \zeta_{10} - 3) q^{91} + (7 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 7 \zeta_{10}) q^{92} - \zeta_{10}^{2} q^{94} + ( - 10 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{95} + ( - \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{97} + ( - \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + 5 q^{5} - 2 q^{7} + 5 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} + 5 q^{5} - 2 q^{7} + 5 q^{8} + 10 q^{10} + 2 q^{11} + 9 q^{13} - q^{14} - 6 q^{16} - 8 q^{17} - 5 q^{19} - 5 q^{20} - 14 q^{22} + 11 q^{23} - 5 q^{25} + 12 q^{26} + q^{28} - 5 q^{29} + 3 q^{31} - 18 q^{32} + 6 q^{34} - 5 q^{35} - 7 q^{37} - 5 q^{38} + 5 q^{40} - 8 q^{41} - 6 q^{43} - 6 q^{44} + 13 q^{46} + 2 q^{47} - 22 q^{49} + 10 q^{50} - 12 q^{52} - 9 q^{53} + 20 q^{55} - 10 q^{58} + 13 q^{61} - 6 q^{62} + 3 q^{64} - 2 q^{67} + 4 q^{68} - 8 q^{71} + 9 q^{73} - 6 q^{74} + 20 q^{76} + 4 q^{77} + 15 q^{79} + 15 q^{80} - 4 q^{82} - 9 q^{83} - 20 q^{85} - 3 q^{86} - 10 q^{88} + 20 q^{89} - 12 q^{91} + 12 q^{92} + q^{94} + 5 q^{95} + 8 q^{97} - 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0.500000 1.53884i 0 −0.500000 0.363271i 0.690983 + 2.12663i 0 0.618034 1.80902 1.31433i 0 3.61803
91.1 0.500000 + 0.363271i 0 −0.500000 1.53884i 1.80902 1.31433i 0 −1.61803 0.690983 2.12663i 0 1.38197
136.1 0.500000 0.363271i 0 −0.500000 + 1.53884i 1.80902 + 1.31433i 0 −1.61803 0.690983 + 2.12663i 0 1.38197
181.1 0.500000 + 1.53884i 0 −0.500000 + 0.363271i 0.690983 2.12663i 0 0.618034 1.80902 + 1.31433i 0 3.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.h.b 4
3.b odd 2 1 25.2.d.a 4
12.b even 2 1 400.2.u.b 4
15.d odd 2 1 125.2.d.a 4
15.e even 4 2 125.2.e.a 8
25.d even 5 1 inner 225.2.h.b 4
25.d even 5 1 5625.2.a.f 2
25.e even 10 1 5625.2.a.d 2
75.h odd 10 1 125.2.d.a 4
75.h odd 10 1 625.2.a.c 2
75.h odd 10 2 625.2.d.b 4
75.j odd 10 1 25.2.d.a 4
75.j odd 10 1 625.2.a.b 2
75.j odd 10 2 625.2.d.h 4
75.l even 20 2 125.2.e.a 8
75.l even 20 2 625.2.b.a 4
75.l even 20 4 625.2.e.c 8
300.n even 10 1 400.2.u.b 4
300.n even 10 1 10000.2.a.c 2
300.r even 10 1 10000.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 3.b odd 2 1
25.2.d.a 4 75.j odd 10 1
125.2.d.a 4 15.d odd 2 1
125.2.d.a 4 75.h odd 10 1
125.2.e.a 8 15.e even 4 2
125.2.e.a 8 75.l even 20 2
225.2.h.b 4 1.a even 1 1 trivial
225.2.h.b 4 25.d even 5 1 inner
400.2.u.b 4 12.b even 2 1
400.2.u.b 4 300.n even 10 1
625.2.a.b 2 75.j odd 10 1
625.2.a.c 2 75.h odd 10 1
625.2.b.a 4 75.l even 20 2
625.2.d.b 4 75.h odd 10 2
625.2.d.h 4 75.j odd 10 2
625.2.e.c 8 75.l even 20 4
5625.2.a.d 2 25.e even 10 1
5625.2.a.f 2 25.d even 5 1
10000.2.a.c 2 300.n even 10 1
10000.2.a.l 2 300.r even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 5 T^{3} + 15 T^{2} - 25 T + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + 24 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$13$ \( T^{4} - 9 T^{3} + 36 T^{2} - 54 T + 81 \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + 24 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} + 40 T^{2} + 50 T + 25 \) Copy content Toggle raw display
$23$ \( T^{4} - 11 T^{3} + 51 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$29$ \( T^{4} + 5 T^{3} + 10 T^{2} + 25 \) Copy content Toggle raw display
$31$ \( T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$37$ \( T^{4} + 7 T^{3} + 19 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$41$ \( T^{4} + 8 T^{3} + 24 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$43$ \( (T^{2} + 3 T - 9)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$53$ \( T^{4} + 9 T^{3} + 61 T^{2} + 209 T + 361 \) Copy content Toggle raw display
$59$ \( T^{4} + 90 T^{2} + 675 T + 2025 \) Copy content Toggle raw display
$61$ \( T^{4} - 13 T^{3} + 139 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} + 64 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} + 34 T^{2} + 87 T + 841 \) Copy content Toggle raw display
$73$ \( T^{4} - 9 T^{3} + 81 T^{2} + \cdots + 6561 \) Copy content Toggle raw display
$79$ \( T^{4} - 15 T^{3} + 100 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$83$ \( T^{4} + 9 T^{3} + 31 T^{2} - 11 T + 121 \) Copy content Toggle raw display
$89$ \( T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400 \) Copy content Toggle raw display
$97$ \( T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121 \) Copy content Toggle raw display
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