Properties

 Label 3025.2.a.x Level $3025$ Weight $2$ Character orbit 3025.a Self dual yes Analytic conductor $24.155$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3025 = 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3025.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$24.1547466114$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.105840.1 Defining polynomial: $$x^{4} - 9 x^{2} + 15$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{2} ) q^{3} + ( 3 + \beta_{2} ) q^{4} + ( -2 \beta_{1} - \beta_{3} ) q^{6} -\beta_{1} q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( 3 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{2} ) q^{3} + ( 3 + \beta_{2} ) q^{4} + ( -2 \beta_{1} - \beta_{3} ) q^{6} -\beta_{1} q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( 3 + \beta_{2} ) q^{9} + ( -8 - 3 \beta_{2} ) q^{12} + \beta_{3} q^{13} + ( -5 - \beta_{2} ) q^{14} + ( 4 + 3 \beta_{2} ) q^{16} + ( -\beta_{1} - \beta_{3} ) q^{17} + ( 4 \beta_{1} + \beta_{3} ) q^{18} + ( -2 \beta_{1} - \beta_{3} ) q^{19} + ( 2 \beta_{1} + \beta_{3} ) q^{21} + ( -4 + \beta_{2} ) q^{23} + ( -7 \beta_{1} - \beta_{3} ) q^{24} + 3 \beta_{2} q^{26} -5 q^{27} + ( -4 \beta_{1} - \beta_{3} ) q^{28} + ( 3 \beta_{1} - \beta_{3} ) q^{29} -5 q^{31} + ( 3 \beta_{1} + \beta_{3} ) q^{32} + ( -5 - 4 \beta_{2} ) q^{34} + ( 14 + 5 \beta_{2} ) q^{36} + ( -3 + 2 \beta_{2} ) q^{37} + ( -10 - 5 \beta_{2} ) q^{38} + ( -3 \beta_{1} + \beta_{3} ) q^{39} + ( 2 \beta_{1} + \beta_{3} ) q^{41} + ( 10 + 5 \beta_{2} ) q^{42} -2 \beta_{1} q^{43} + ( -3 \beta_{1} + \beta_{3} ) q^{46} + ( -7 - 2 \beta_{2} ) q^{47} + ( -19 - 4 \beta_{2} ) q^{48} + ( -2 + \beta_{2} ) q^{49} + 5 \beta_{1} q^{51} + ( 3 \beta_{1} + \beta_{3} ) q^{52} + ( -4 + \beta_{2} ) q^{53} -5 \beta_{1} q^{54} + ( -10 - 5 \beta_{2} ) q^{56} + ( 7 \beta_{1} + \beta_{3} ) q^{57} + 15 q^{58} + ( -5 - 4 \beta_{2} ) q^{59} + ( -\beta_{1} + 2 \beta_{3} ) q^{61} -5 \beta_{1} q^{62} + ( -4 \beta_{1} - \beta_{3} ) q^{63} + 7 q^{64} + ( -2 - 2 \beta_{2} ) q^{67} + ( -7 \beta_{1} - 2 \beta_{3} ) q^{68} + ( -1 + 4 \beta_{2} ) q^{69} + ( 2 + 4 \beta_{2} ) q^{71} + ( 11 \beta_{1} + 3 \beta_{3} ) q^{72} + ( -\beta_{1} - 2 \beta_{3} ) q^{73} + ( -\beta_{1} + 2 \beta_{3} ) q^{74} + ( -11 \beta_{1} - 3 \beta_{3} ) q^{76} -15 q^{78} + ( \beta_{1} + 3 \beta_{3} ) q^{79} + ( -4 + 2 \beta_{2} ) q^{81} + ( 10 + 5 \beta_{2} ) q^{82} + ( -\beta_{1} + \beta_{3} ) q^{83} + ( 11 \beta_{1} + 3 \beta_{3} ) q^{84} + ( -10 - 2 \beta_{2} ) q^{86} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{87} + ( 11 + \beta_{2} ) q^{89} -3 \beta_{2} q^{91} + ( -7 - 2 \beta_{2} ) q^{92} + ( 5 + 5 \beta_{2} ) q^{93} + ( -9 \beta_{1} - 2 \beta_{3} ) q^{94} + ( -9 \beta_{1} - 2 \beta_{3} ) q^{96} + ( -8 - 3 \beta_{2} ) q^{97} + ( -\beta_{1} + \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 10 q^{4} + 10 q^{9} + O(q^{10})$$ $$4 q - 2 q^{3} + 10 q^{4} + 10 q^{9} - 26 q^{12} - 18 q^{14} + 10 q^{16} - 18 q^{23} - 6 q^{26} - 20 q^{27} - 20 q^{31} - 12 q^{34} + 46 q^{36} - 16 q^{37} - 30 q^{38} + 30 q^{42} - 24 q^{47} - 68 q^{48} - 10 q^{49} - 18 q^{53} - 30 q^{56} + 60 q^{58} - 12 q^{59} + 28 q^{64} - 4 q^{67} - 12 q^{69} - 60 q^{78} - 20 q^{81} + 30 q^{82} - 36 q^{86} + 42 q^{89} + 6 q^{91} - 24 q^{92} + 10 q^{93} - 26 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9 x^{2} + 15$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6 \beta_{1}$$

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
 −2.60601 −1.48617 1.48617 2.60601
−2.60601 −2.79129 4.79129 0 7.27412 2.60601 −7.27412 4.79129 0
1.2 −1.48617 1.79129 0.208712 0 −2.66216 1.48617 2.66216 0.208712 0
1.3 1.48617 1.79129 0.208712 0 2.66216 −1.48617 −2.66216 0.208712 0
1.4 2.60601 −2.79129 4.79129 0 −7.27412 −2.60601 7.27412 4.79129 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
$$5$$ $$1$$
$$11$$ $$1$$

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3025.2.a.x 4
5.b even 2 1 3025.2.a.bc yes 4
11.b odd 2 1 inner 3025.2.a.x 4
55.d odd 2 1 3025.2.a.bc yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3025.2.a.x 4 1.a even 1 1 trivial
3025.2.a.x 4 11.b odd 2 1 inner
3025.2.a.bc yes 4 5.b even 2 1
3025.2.a.bc yes 4 55.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3025))$$:

 $$T_{2}^{4} - 9 T_{2}^{2} + 15$$ $$T_{3}^{2} + T_{3} - 5$$ $$T_{19}^{4} - 60 T_{19}^{2} + 375$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$15 - 9 T^{2} + T^{4}$$
$3$ $$( -5 + T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$15 - 9 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$135 - 36 T^{2} + T^{4}$$
$17$ $$375 - 39 T^{2} + T^{4}$$
$19$ $$375 - 60 T^{2} + T^{4}$$
$23$ $$( 15 + 9 T + T^{2} )^{2}$$
$29$ $$3375 - 135 T^{2} + T^{4}$$
$31$ $$( 5 + T )^{4}$$
$37$ $$( -5 + 8 T + T^{2} )^{2}$$
$41$ $$375 - 60 T^{2} + T^{4}$$
$43$ $$240 - 36 T^{2} + T^{4}$$
$47$ $$( 15 + 12 T + T^{2} )^{2}$$
$53$ $$( 15 + 9 T + T^{2} )^{2}$$
$59$ $$( -75 + 6 T + T^{2} )^{2}$$
$61$ $$375 - 165 T^{2} + T^{4}$$
$67$ $$( -20 + 2 T + T^{2} )^{2}$$
$71$ $$( -84 + T^{2} )^{2}$$
$73$ $$4335 - 141 T^{2} + T^{4}$$
$79$ $$18375 - 315 T^{2} + T^{4}$$
$83$ $$15 - 51 T^{2} + T^{4}$$
$89$ $$( 105 - 21 T + T^{2} )^{2}$$
$97$ $$( -5 + 13 T + T^{2} )^{2}$$