# 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 $2$

# Related objects

## 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: $$x^{4} - x^{3} + x^{2} - x + 1$$ 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{2}^{4} - 2 T_{2}^{3} + 4 T_{2}^{2} - 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 - 25 T + 15 T^{2} - 5 T^{3} + T^{4}$$
$7$ $$( -1 + T + T^{2} )^{2}$$
$11$ $$16 + 32 T + 24 T^{2} - 2 T^{3} + T^{4}$$
$13$ $$81 - 54 T + 36 T^{2} - 9 T^{3} + T^{4}$$
$17$ $$16 - 8 T + 24 T^{2} + 8 T^{3} + T^{4}$$
$19$ $$25 + 50 T + 40 T^{2} + 5 T^{3} + T^{4}$$
$23$ $$961 - 31 T + 51 T^{2} - 11 T^{3} + T^{4}$$
$29$ $$25 + 10 T^{2} + 5 T^{3} + T^{4}$$
$31$ $$81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4}$$
$37$ $$1 + 3 T + 19 T^{2} + 7 T^{3} + T^{4}$$
$41$ $$16 - 8 T + 24 T^{2} + 8 T^{3} + T^{4}$$
$43$ $$( -9 + 3 T + T^{2} )^{2}$$
$47$ $$1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$53$ $$361 + 209 T + 61 T^{2} + 9 T^{3} + T^{4}$$
$59$ $$2025 + 675 T + 90 T^{2} + T^{4}$$
$61$ $$1681 - 697 T + 139 T^{2} - 13 T^{3} + T^{4}$$
$67$ $$1936 + 528 T + 64 T^{2} + 2 T^{3} + T^{4}$$
$71$ $$841 + 87 T + 34 T^{2} + 8 T^{3} + T^{4}$$
$73$ $$6561 - 729 T + 81 T^{2} - 9 T^{3} + T^{4}$$
$79$ $$625 - 250 T + 100 T^{2} - 15 T^{3} + T^{4}$$
$83$ $$121 - 11 T + 31 T^{2} + 9 T^{3} + T^{4}$$
$89$ $$6400 - 1600 T + 240 T^{2} - 20 T^{3} + T^{4}$$
$97$ $$121 - 77 T + 34 T^{2} - 8 T^{3} + T^{4}$$