# Properties

 Label 4225.2.a.bl Level $4225$ Weight $2$ Character orbit 4225.a Self dual yes Analytic conductor $33.737$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

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## 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: $$4$$ Coefficient field: 4.4.4752.1 Defining polynomial: $$x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{3} q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{6} + ( 3 + \beta_{2} - \beta_{3} ) q^{7} + ( 1 + \beta_{2} + \beta_{3} ) q^{8} + 2 \beta_{1} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{3} q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{6} + ( 3 + \beta_{2} - \beta_{3} ) q^{7} + ( 1 + \beta_{2} + \beta_{3} ) q^{8} + 2 \beta_{1} q^{9} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{11} + ( 1 + 3 \beta_{1} + 2 \beta_{2} ) q^{12} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{14} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{16} + ( -2 \beta_{2} + \beta_{3} ) q^{17} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{18} + ( -4 - \beta_{2} ) q^{19} + ( -2 + \beta_{2} + 2 \beta_{3} ) q^{21} + ( -3 - \beta_{1} + \beta_{3} ) q^{22} + ( 4 - 2 \beta_{1} - \beta_{3} ) q^{23} + ( 4 + 4 \beta_{1} + \beta_{2} ) q^{24} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{27} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{28} + ( 1 + 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -2 + 2 \beta_{2} ) q^{31} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{32} + ( 5 - 3 \beta_{2} - \beta_{3} ) q^{33} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{34} + ( 2 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{36} + ( -1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{37} + ( 1 - 5 \beta_{1} - \beta_{3} ) q^{38} + ( -2 + \beta_{2} ) q^{41} + ( -1 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{42} + ( -2 \beta_{2} + \beta_{3} ) q^{43} + ( -2 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{44} + ( -4 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{46} + ( 4 - 4 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 5 + 3 \beta_{1} + \beta_{3} ) q^{48} + ( 6 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{49} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( 2 + 2 \beta_{1} ) q^{53} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{54} + ( 3 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{56} + ( -1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{57} + ( 2 + 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{58} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{59} + ( 7 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -2 + 2 \beta_{3} ) q^{62} + ( -2 + 6 \beta_{1} - 2 \beta_{2} ) q^{63} + ( 3 + 2 \beta_{2} - 4 \beta_{3} ) q^{64} + ( 3 + \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{66} + ( 7 - \beta_{2} + \beta_{3} ) q^{67} + ( -3 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{68} + ( -3 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{69} + ( 2 + 3 \beta_{2} - 6 \beta_{3} ) q^{71} + ( -2 + 6 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{72} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -3 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{74} + ( -2 - 5 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{76} + ( -4 - 4 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{77} + ( -6 \beta_{1} + 2 \beta_{3} ) q^{79} + ( -1 - 2 \beta_{1} + 4 \beta_{2} ) q^{81} + ( -1 - \beta_{1} + \beta_{3} ) q^{82} + ( -4 + 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( 4 + 5 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{84} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{86} + ( 2 + 6 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{87} + ( 6 - 2 \beta_{2} - 3 \beta_{3} ) q^{88} + ( 8 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{89} + ( -3 - 3 \beta_{1} - 2 \beta_{3} ) q^{92} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{93} + ( -6 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{94} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{96} + ( 1 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{97} + ( -8 + 4 \beta_{1} - 6 \beta_{2} ) q^{98} + ( -6 - 2 \beta_{1} + 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{6} + 10q^{7} + 6q^{8} + 4q^{9} + O(q^{10})$$ $$4q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{6} + 10q^{7} + 6q^{8} + 4q^{9} + 10q^{12} + 2q^{14} + 2q^{16} + 2q^{17} + 20q^{18} - 16q^{19} - 4q^{21} - 12q^{22} + 10q^{23} + 24q^{24} + 2q^{27} + 8q^{28} + 8q^{29} - 8q^{31} + 4q^{32} + 18q^{33} + 4q^{34} + 20q^{36} - 2q^{37} - 8q^{38} - 8q^{41} + 4q^{42} + 2q^{43} - 12q^{44} - 16q^{46} + 8q^{47} + 28q^{48} + 12q^{49} + 4q^{51} + 12q^{53} + 16q^{54} + 12q^{56} - 14q^{57} + 22q^{58} - 12q^{59} + 28q^{61} - 4q^{62} + 4q^{63} + 4q^{64} + 6q^{66} + 30q^{67} - 14q^{68} - 16q^{69} - 4q^{71} + 12q^{72} - 8q^{73} - 10q^{74} - 20q^{76} - 18q^{77} - 8q^{79} - 8q^{81} - 4q^{82} - 12q^{83} + 28q^{84} + 4q^{86} + 22q^{87} + 18q^{88} + 12q^{89} - 22q^{92} + 8q^{93} - 32q^{94} - 4q^{96} + 2q^{97} - 24q^{98} - 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$

## 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
 −1.49551 −0.219687 1.21969 2.49551
−1.49551 −0.0947876 0.236543 0 0.141756 4.82684 2.63726 −2.99102 0
1.2 −0.219687 1.60020 −1.95174 0 −0.351542 −0.332247 0.868145 −0.439374 0
1.3 1.21969 −2.33225 −0.512364 0 −2.84461 3.60020 −3.06430 2.43937 0
1.4 2.49551 2.82684 4.22756 0 7.05440 1.90521 5.55889 4.99102 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4225.2.a.bl 4
5.b even 2 1 845.2.a.l 4
13.b even 2 1 4225.2.a.bi 4
13.f odd 12 2 325.2.n.d 8
15.d odd 2 1 7605.2.a.cj 4
65.d even 2 1 845.2.a.m 4
65.g odd 4 2 845.2.c.g 8
65.l even 6 2 845.2.e.m 8
65.n even 6 2 845.2.e.n 8
65.o even 12 2 325.2.m.b 8
65.s odd 12 2 65.2.m.a 8
65.s odd 12 2 845.2.m.g 8
65.t even 12 2 325.2.m.c 8
195.e odd 2 1 7605.2.a.cf 4
195.bh even 12 2 585.2.bu.c 8
260.bc even 12 2 1040.2.da.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 65.s odd 12 2
325.2.m.b 8 65.o even 12 2
325.2.m.c 8 65.t even 12 2
325.2.n.d 8 13.f odd 12 2
585.2.bu.c 8 195.bh even 12 2
845.2.a.l 4 5.b even 2 1
845.2.a.m 4 65.d even 2 1
845.2.c.g 8 65.g odd 4 2
845.2.e.m 8 65.l even 6 2
845.2.e.n 8 65.n even 6 2
845.2.m.g 8 65.s odd 12 2
1040.2.da.b 8 260.bc even 12 2
4225.2.a.bi 4 13.b even 2 1
4225.2.a.bl 4 1.a even 1 1 trivial
7605.2.a.cf 4 195.e odd 2 1
7605.2.a.cj 4 15.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{4} - 2 T_{2}^{3} - 3 T_{2}^{2} + 4 T_{2} + 1$$ $$T_{3}^{4} - 2 T_{3}^{3} - 6 T_{3}^{2} + 10 T_{3} + 1$$ $$T_{7}^{4} - 10 T_{7}^{3} + 30 T_{7}^{2} - 22 T_{7} - 11$$ $$T_{11}^{4} - 30 T_{11}^{2} + 33$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T - 3 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$1 + 10 T - 6 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$-11 - 22 T + 30 T^{2} - 10 T^{3} + T^{4}$$
$11$ $$33 - 30 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$13 + 10 T - 18 T^{2} - 2 T^{3} + T^{4}$$
$19$ $$( 13 + 8 T + T^{2} )^{2}$$
$23$ $$-299 + 146 T + 6 T^{2} - 10 T^{3} + T^{4}$$
$29$ $$1 + 40 T - 18 T^{2} - 8 T^{3} + T^{4}$$
$31$ $$( -8 + 4 T + T^{2} )^{2}$$
$37$ $$1 + 38 T - 54 T^{2} + 2 T^{3} + T^{4}$$
$41$ $$( 1 + 4 T + T^{2} )^{2}$$
$43$ $$13 + 10 T - 18 T^{2} - 2 T^{3} + T^{4}$$
$47$ $$-1328 + 736 T - 72 T^{2} - 8 T^{3} + T^{4}$$
$53$ $$-48 + 36 T^{2} - 12 T^{3} + T^{4}$$
$59$ $$-3 + 12 T + 30 T^{2} + 12 T^{3} + T^{4}$$
$61$ $$1261 - 964 T + 258 T^{2} - 28 T^{3} + T^{4}$$
$67$ $$2769 - 1578 T + 330 T^{2} - 30 T^{3} + T^{4}$$
$71$ $$10477 - 428 T - 210 T^{2} + 4 T^{3} + T^{4}$$
$73$ $$-1712 - 832 T - 84 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$4432 - 640 T - 132 T^{2} + 8 T^{3} + T^{4}$$
$83$ $$-192 - 288 T - 24 T^{2} + 12 T^{3} + T^{4}$$
$89$ $$8853 + 2148 T - 234 T^{2} - 12 T^{3} + T^{4}$$
$97$ $$-443 - 374 T - 90 T^{2} - 2 T^{3} + T^{4}$$
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