# Properties

 Label 4225.2.a.bk Level $4225$ Weight $2$ Character orbit 4225.a Self dual yes Analytic conductor $33.737$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4225 = 5^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.7367948540$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{7})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + q^{3} -\beta_{2} q^{4} + \beta_{3} q^{6} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} -2 q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} + q^{3} -\beta_{2} q^{4} + \beta_{3} q^{6} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} -2 q^{9} + ( \beta_{1} - \beta_{3} ) q^{11} -\beta_{2} q^{12} + ( -1 + \beta_{2} ) q^{14} + ( 1 + \beta_{2} ) q^{16} + ( 1 + 2 \beta_{2} ) q^{17} -2 \beta_{3} q^{18} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( -\beta_{1} - \beta_{3} ) q^{21} + ( -3 + \beta_{2} ) q^{22} + ( -1 - 2 \beta_{2} ) q^{23} + ( \beta_{1} + \beta_{3} ) q^{24} -5 q^{27} + ( \beta_{1} - 2 \beta_{3} ) q^{28} + ( 1 + 2 \beta_{2} ) q^{29} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{31} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{32} + ( \beta_{1} - \beta_{3} ) q^{33} + ( -2 \beta_{1} - 5 \beta_{3} ) q^{34} + 2 \beta_{2} q^{36} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{37} + ( -1 + \beta_{2} ) q^{38} + ( \beta_{1} - \beta_{3} ) q^{41} + ( -1 + \beta_{2} ) q^{42} + ( -7 - 2 \beta_{2} ) q^{43} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{44} + ( 2 \beta_{1} + 5 \beta_{3} ) q^{46} + 4 \beta_{3} q^{47} + ( 1 + \beta_{2} ) q^{48} -4 q^{49} + ( 1 + 2 \beta_{2} ) q^{51} + ( 4 + 2 \beta_{2} ) q^{53} -5 \beta_{3} q^{54} -3 q^{56} + ( -\beta_{1} - \beta_{3} ) q^{57} + ( -2 \beta_{1} - 5 \beta_{3} ) q^{58} + ( 7 \beta_{1} + 3 \beta_{3} ) q^{59} + ( -5 + 2 \beta_{2} ) q^{61} + ( 10 - 4 \beta_{2} ) q^{62} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{63} + ( -7 + 2 \beta_{2} ) q^{64} + ( -3 + \beta_{2} ) q^{66} + ( -\beta_{1} - 7 \beta_{3} ) q^{67} + ( -10 + \beta_{2} ) q^{68} + ( -1 - 2 \beta_{2} ) q^{69} + ( -3 \beta_{1} + \beta_{3} ) q^{71} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{72} + ( -9 + 3 \beta_{2} ) q^{74} + ( \beta_{1} - 2 \beta_{3} ) q^{76} + ( -1 - 2 \beta_{2} ) q^{77} -6 q^{79} + q^{81} + ( -3 + \beta_{2} ) q^{82} + ( 4 \beta_{1} + 6 \beta_{3} ) q^{83} + ( \beta_{1} - 2 \beta_{3} ) q^{84} + ( 2 \beta_{1} - \beta_{3} ) q^{86} + ( 1 + 2 \beta_{2} ) q^{87} + ( 1 + 2 \beta_{2} ) q^{88} + ( 5 \beta_{1} + 3 \beta_{3} ) q^{89} + ( 10 - \beta_{2} ) q^{92} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{93} + ( 8 - 4 \beta_{2} ) q^{94} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{96} + ( 5 \beta_{1} - \beta_{3} ) q^{97} -4 \beta_{3} q^{98} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + 2q^{4} - 8q^{9} + O(q^{10})$$ $$4q + 4q^{3} + 2q^{4} - 8q^{9} + 2q^{12} - 6q^{14} + 2q^{16} - 14q^{22} - 20q^{27} - 4q^{36} - 6q^{38} - 6q^{42} - 24q^{43} + 2q^{48} - 16q^{49} + 12q^{53} - 12q^{56} - 24q^{61} + 48q^{62} - 32q^{64} - 14q^{66} - 42q^{68} - 42q^{74} - 24q^{79} + 4q^{81} - 14q^{82} + 42q^{92} + 40q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.456850 2.18890 −2.18890 −0.456850
−2.18890 1.00000 2.79129 0 −2.18890 1.73205 −1.73205 −2.00000 0
1.2 −0.456850 1.00000 −1.79129 0 −0.456850 −1.73205 1.73205 −2.00000 0
1.3 0.456850 1.00000 −1.79129 0 0.456850 1.73205 −1.73205 −2.00000 0
1.4 2.18890 1.00000 2.79129 0 2.18890 −1.73205 1.73205 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4225.2.a.bk 4
5.b even 2 1 4225.2.a.bj 4
5.c odd 4 2 845.2.b.f 8
13.b even 2 1 inner 4225.2.a.bk 4
13.f odd 12 2 325.2.n.c 4
65.d even 2 1 4225.2.a.bj 4
65.f even 4 2 845.2.d.c 8
65.h odd 4 2 845.2.b.f 8
65.k even 4 2 845.2.d.c 8
65.o even 12 2 65.2.l.a 8
65.o even 12 2 845.2.l.c 8
65.q odd 12 2 845.2.n.c 8
65.q odd 12 2 845.2.n.d 8
65.r odd 12 2 845.2.n.c 8
65.r odd 12 2 845.2.n.d 8
65.s odd 12 2 325.2.n.b 4
65.t even 12 2 65.2.l.a 8
65.t even 12 2 845.2.l.c 8
195.bc odd 12 2 585.2.bf.a 8
195.bn odd 12 2 585.2.bf.a 8
260.be odd 12 2 1040.2.df.b 8
260.bl odd 12 2 1040.2.df.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.l.a 8 65.o even 12 2
65.2.l.a 8 65.t even 12 2
325.2.n.b 4 65.s odd 12 2
325.2.n.c 4 13.f odd 12 2
585.2.bf.a 8 195.bc odd 12 2
585.2.bf.a 8 195.bn odd 12 2
845.2.b.f 8 5.c odd 4 2
845.2.b.f 8 65.h odd 4 2
845.2.d.c 8 65.f even 4 2
845.2.d.c 8 65.k even 4 2
845.2.l.c 8 65.o even 12 2
845.2.l.c 8 65.t even 12 2
845.2.n.c 8 65.q odd 12 2
845.2.n.c 8 65.r odd 12 2
845.2.n.d 8 65.q odd 12 2
845.2.n.d 8 65.r odd 12 2
1040.2.df.b 8 260.be odd 12 2
1040.2.df.b 8 260.bl odd 12 2
4225.2.a.bj 4 5.b even 2 1
4225.2.a.bj 4 65.d even 2 1
4225.2.a.bk 4 1.a even 1 1 trivial
4225.2.a.bk 4 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4225))$$:

 $$T_{2}^{4} - 5 T_{2}^{2} + 1$$ $$T_{3} - 1$$ $$T_{7}^{2} - 3$$ $$T_{11}^{2} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T^{2} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$( -3 + T^{2} )^{2}$$
$11$ $$( -7 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -21 + T^{2} )^{2}$$
$19$ $$( -3 + T^{2} )^{2}$$
$23$ $$( -21 + T^{2} )^{2}$$
$29$ $$( -21 + T^{2} )^{2}$$
$31$ $$3600 - 132 T^{2} + T^{4}$$
$37$ $$( -63 + T^{2} )^{2}$$
$41$ $$( -7 + T^{2} )^{2}$$
$43$ $$( 15 + 12 T + T^{2} )^{2}$$
$47$ $$256 - 80 T^{2} + T^{4}$$
$53$ $$( -12 - 6 T + T^{2} )^{2}$$
$59$ $$2209 - 206 T^{2} + T^{4}$$
$61$ $$( 15 + 12 T + T^{2} )^{2}$$
$67$ $$225 - 222 T^{2} + T^{4}$$
$71$ $$625 - 62 T^{2} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 6 + T )^{4}$$
$83$ $$4624 - 164 T^{2} + T^{4}$$
$89$ $$1681 - 110 T^{2} + T^{4}$$
$97$ $$2601 - 150 T^{2} + T^{4}$$