# Properties

 Label 4225.2.a.br Level $4225$ Weight $2$ Character orbit 4225.a Self dual yes Analytic conductor $33.737$ Analytic rank $0$ Dimension $6$ 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: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.199374400.1 Defining polynomial: $$x^{6} - 8 x^{4} + 10 x^{2} - 1$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{5} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 2 + \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{5} ) q^{7} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{8} + ( 1 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{5} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 2 + \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{5} ) q^{7} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{8} + ( 1 + \beta_{3} ) q^{9} -\beta_{3} q^{11} + ( 3 \beta_{1} + \beta_{4} + \beta_{5} ) q^{12} + ( 4 + \beta_{2} ) q^{14} + ( 3 + \beta_{2} + \beta_{3} ) q^{16} + ( \beta_{4} - \beta_{5} ) q^{17} + ( \beta_{1} + \beta_{4} ) q^{18} + ( 2 - \beta_{3} ) q^{19} + ( 2 \beta_{2} - \beta_{3} ) q^{21} -\beta_{4} q^{22} + ( -\beta_{1} + \beta_{5} ) q^{23} + ( 6 + 2 \beta_{2} + \beta_{3} ) q^{24} + ( -\beta_{1} - \beta_{5} ) q^{27} + ( 5 \beta_{1} + \beta_{4} - 3 \beta_{5} ) q^{28} + 3 q^{29} + ( -2 - 2 \beta_{2} ) q^{31} + ( 2 \beta_{1} + \beta_{5} ) q^{32} + ( -\beta_{1} + 3 \beta_{5} ) q^{33} + ( -1 + \beta_{2} + \beta_{3} ) q^{34} + ( 1 + 2 \beta_{2} - \beta_{3} ) q^{36} + ( -\beta_{4} + \beta_{5} ) q^{37} + ( 2 \beta_{1} - \beta_{4} ) q^{38} + ( -3 - 2 \beta_{2} ) q^{41} + ( 6 \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{42} + ( \beta_{1} - 3 \beta_{5} ) q^{43} + ( -\beta_{2} + \beta_{3} ) q^{44} + ( -2 - \beta_{2} ) q^{46} + ( 4 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( 6 \beta_{1} + \beta_{4} - 4 \beta_{5} ) q^{48} + ( 1 + \beta_{3} ) q^{49} + ( 2 + \beta_{3} ) q^{51} + ( \beta_{1} - \beta_{4} - 4 \beta_{5} ) q^{53} + ( -4 - \beta_{2} ) q^{54} + ( 4 + 4 \beta_{2} + \beta_{3} ) q^{56} + ( \beta_{1} + \beta_{5} ) q^{57} + 3 \beta_{1} q^{58} + ( 2 \beta_{2} + \beta_{3} ) q^{59} + ( -1 - 2 \beta_{3} ) q^{61} + ( -8 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{62} + ( 2 \beta_{4} + 4 \beta_{5} ) q^{63} + ( 1 - 2 \beta_{3} ) q^{64} -\beta_{2} q^{66} + ( -3 \beta_{1} - \beta_{5} ) q^{67} + ( 2 \beta_{1} + \beta_{5} ) q^{68} + ( -4 - \beta_{3} ) q^{69} + ( 2 + \beta_{3} ) q^{71} + ( 5 \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{72} + ( 5 \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{73} + ( 1 - \beta_{2} - \beta_{3} ) q^{74} + ( 2 + \beta_{2} + \beta_{3} ) q^{76} + ( \beta_{1} - 2 \beta_{4} - 3 \beta_{5} ) q^{77} + ( 8 - 2 \beta_{2} ) q^{79} + ( -3 - 2 \beta_{2} - 2 \beta_{3} ) q^{81} + ( -9 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{82} + ( -4 \beta_{1} - 2 \beta_{4} + 4 \beta_{5} ) q^{83} + ( 16 + 3 \beta_{2} + 3 \beta_{3} ) q^{84} + \beta_{2} q^{86} + ( 3 \beta_{1} - 3 \beta_{5} ) q^{87} + ( -3 \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{88} + ( 2 - 4 \beta_{2} - \beta_{3} ) q^{89} + ( -3 \beta_{1} - \beta_{4} - \beta_{5} ) q^{92} + ( -6 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{93} + ( 10 + 2 \beta_{2} - 2 \beta_{3} ) q^{94} + ( 2 + 3 \beta_{2} - \beta_{3} ) q^{96} + ( 5 \beta_{1} + 2 \beta_{4} - 3 \beta_{5} ) q^{97} + ( \beta_{1} + \beta_{4} ) q^{98} + ( -8 + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 4q^{4} + 10q^{6} + 6q^{9} + O(q^{10})$$ $$6q + 4q^{4} + 10q^{6} + 6q^{9} + 22q^{14} + 16q^{16} + 12q^{19} - 4q^{21} + 32q^{24} + 18q^{29} - 8q^{31} - 8q^{34} + 2q^{36} - 14q^{41} + 2q^{44} - 10q^{46} + 6q^{49} + 12q^{51} - 22q^{54} + 16q^{56} - 4q^{59} - 6q^{61} + 6q^{64} + 2q^{66} - 24q^{69} + 12q^{71} + 8q^{74} + 10q^{76} + 52q^{79} - 14q^{81} + 90q^{84} - 2q^{86} + 20q^{89} + 56q^{94} + 6q^{96} - 52q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 8 x^{4} + 10 x^{2} - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 7 \nu^{2} + 4$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - 7 \nu^{3} + 4 \nu$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 8 \nu^{3} + 10 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} + \beta_{4} + 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} + 7 \beta_{2} + 17$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{5} + 8 \beta_{4} + 38 \beta_{1}$$

## 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
 −2.54574 −1.18733 −0.330837 0.330837 1.18733 2.54574
−2.54574 −2.15293 4.48079 0 5.48079 −2.93855 −6.31544 1.63509 0
1.2 −1.18733 −0.345110 −0.590239 0 0.409761 −2.02956 3.07548 −2.88090 0
1.3 −0.330837 2.69180 −1.89055 0 −0.890547 −3.35348 1.28714 4.24581 0
1.4 0.330837 −2.69180 −1.89055 0 −0.890547 3.35348 −1.28714 4.24581 0
1.5 1.18733 0.345110 −0.590239 0 0.409761 2.02956 −3.07548 −2.88090 0
1.6 2.54574 2.15293 4.48079 0 5.48079 2.93855 6.31544 1.63509 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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
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 4225.2.a.br 6
5.b even 2 1 inner 4225.2.a.br 6
5.c odd 4 2 845.2.b.d 6
13.b even 2 1 4225.2.a.bq 6
13.c even 3 2 325.2.e.e 12
65.d even 2 1 4225.2.a.bq 6
65.f even 4 2 845.2.d.d 12
65.h odd 4 2 845.2.b.e 6
65.k even 4 2 845.2.d.d 12
65.n even 6 2 325.2.e.e 12
65.o even 12 4 845.2.l.f 24
65.q odd 12 4 65.2.n.a 12
65.r odd 12 4 845.2.n.e 12
65.t even 12 4 845.2.l.f 24
195.bl even 12 4 585.2.bs.a 12
260.bj even 12 4 1040.2.dh.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 65.q odd 12 4
325.2.e.e 12 13.c even 3 2
325.2.e.e 12 65.n even 6 2
585.2.bs.a 12 195.bl even 12 4
845.2.b.d 6 5.c odd 4 2
845.2.b.e 6 65.h odd 4 2
845.2.d.d 12 65.f even 4 2
845.2.d.d 12 65.k even 4 2
845.2.l.f 24 65.o even 12 4
845.2.l.f 24 65.t even 12 4
845.2.n.e 12 65.r odd 12 4
1040.2.dh.a 12 260.bj even 12 4
4225.2.a.bq 6 13.b even 2 1
4225.2.a.bq 6 65.d even 2 1
4225.2.a.br 6 1.a even 1 1 trivial
4225.2.a.br 6 5.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4225))$$:

 $$T_{2}^{6} - 8 T_{2}^{4} + 10 T_{2}^{2} - 1$$ $$T_{3}^{6} - 12 T_{3}^{4} + 35 T_{3}^{2} - 4$$ $$T_{7}^{6} - 24 T_{7}^{4} + 179 T_{7}^{2} - 400$$ $$T_{11}^{3} - 13 T_{11} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 10 T^{2} - 8 T^{4} + T^{6}$$
$3$ $$-4 + 35 T^{2} - 12 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$-400 + 179 T^{2} - 24 T^{4} + T^{6}$$
$11$ $$( -8 - 13 T + T^{3} )^{2}$$
$13$ $$T^{6}$$
$17$ $$-169 + 163 T^{2} - 35 T^{4} + T^{6}$$
$19$ $$( 10 - T - 6 T^{2} + T^{3} )^{2}$$
$23$ $$-4 + 35 T^{2} - 12 T^{4} + T^{6}$$
$29$ $$( -3 + T )^{6}$$
$31$ $$( 40 - 40 T + 4 T^{2} + T^{3} )^{2}$$
$37$ $$-169 + 163 T^{2} - 35 T^{4} + T^{6}$$
$41$ $$( 5 - 29 T + 7 T^{2} + T^{3} )^{2}$$
$43$ $$-256 + 283 T^{2} - 80 T^{4} + T^{6}$$
$47$ $$-270400 + 14640 T^{2} - 236 T^{4} + T^{6}$$
$53$ $$-400 + 1040 T^{2} - 171 T^{4} + T^{6}$$
$59$ $$( -136 - 55 T + 2 T^{2} + T^{3} )^{2}$$
$61$ $$( -115 - 49 T + 3 T^{2} + T^{3} )^{2}$$
$67$ $$-20164 + 2603 T^{2} - 100 T^{4} + T^{6}$$
$71$ $$( 26 - T - 6 T^{2} + T^{3} )^{2}$$
$73$ $$-250000 + 13900 T^{2} - 215 T^{4} + T^{6}$$
$79$ $$( -160 + 180 T - 26 T^{2} + T^{3} )^{2}$$
$83$ $$-640000 + 23600 T^{2} - 276 T^{4} + T^{6}$$
$89$ $$( 1586 - 157 T - 10 T^{2} + T^{3} )^{2}$$
$97$ $$-204304 + 14363 T^{2} - 280 T^{4} + T^{6}$$