# Properties

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

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## 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: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 605) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{3} q^{3} + \beta_{2} q^{4} + q^{6} + ( \beta_{1} + 2 \beta_{3} ) q^{7} + \beta_{3} q^{8} + ( -1 - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{3} q^{3} + \beta_{2} q^{4} + q^{6} + ( \beta_{1} + 2 \beta_{3} ) q^{7} + \beta_{3} q^{8} + ( -1 - \beta_{2} ) q^{9} + ( \beta_{1} + 2 \beta_{3} ) q^{12} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{13} + \beta_{2} q^{14} + ( -1 - 2 \beta_{2} ) q^{16} -2 \beta_{3} q^{17} + ( -3 \beta_{1} - \beta_{3} ) q^{18} + ( -1 - 3 \beta_{2} ) q^{19} + ( -3 + 2 \beta_{2} ) q^{21} + ( -3 \beta_{1} + \beta_{3} ) q^{23} + ( -2 + \beta_{2} ) q^{24} + ( -3 - 3 \beta_{2} ) q^{26} + ( -\beta_{1} + 2 \beta_{3} ) q^{27} -3 \beta_{3} q^{28} + 4 \beta_{2} q^{29} + ( -7 + \beta_{2} ) q^{31} + ( -5 \beta_{1} - 4 \beta_{3} ) q^{32} + 2 q^{34} + ( -3 - \beta_{2} ) q^{36} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{37} + ( -7 \beta_{1} - 3 \beta_{3} ) q^{38} + ( 3 - 3 \beta_{2} ) q^{39} + \beta_{2} q^{41} + ( \beta_{1} + 2 \beta_{3} ) q^{42} + ( -5 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -7 - 3 \beta_{2} ) q^{46} + ( 4 \beta_{1} + 7 \beta_{3} ) q^{47} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{48} + ( -1 - 3 \beta_{2} ) q^{49} + ( 4 - 2 \beta_{2} ) q^{51} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{52} + ( 7 \beta_{1} + 3 \beta_{3} ) q^{53} + ( -4 - \beta_{2} ) q^{54} + ( 3 - 2 \beta_{2} ) q^{56} + ( -3 \beta_{1} - 5 \beta_{3} ) q^{57} + ( 8 \beta_{1} + 4 \beta_{3} ) q^{58} + ( -3 - \beta_{2} ) q^{59} + ( 10 - \beta_{2} ) q^{61} + ( -5 \beta_{1} + \beta_{3} ) q^{62} + ( -\beta_{1} + \beta_{3} ) q^{63} + ( -4 - \beta_{2} ) q^{64} + ( -8 \beta_{1} - \beta_{3} ) q^{67} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{68} + ( -5 + \beta_{2} ) q^{69} + ( -3 + 3 \beta_{2} ) q^{71} + ( \beta_{1} + \beta_{3} ) q^{72} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{73} + ( 6 + 4 \beta_{2} ) q^{74} + ( -9 - \beta_{2} ) q^{76} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{78} + ( 2 + 2 \beta_{2} ) q^{79} + ( -2 + 5 \beta_{2} ) q^{81} + ( 2 \beta_{1} + \beta_{3} ) q^{82} + ( 7 \beta_{1} + 7 \beta_{3} ) q^{83} + ( 6 - 3 \beta_{2} ) q^{84} + ( -12 - 5 \beta_{2} ) q^{86} + ( 4 \beta_{1} + 8 \beta_{3} ) q^{87} + ( -3 - 2 \beta_{2} ) q^{89} + ( -9 + 3 \beta_{2} ) q^{91} + ( -7 \beta_{1} - 5 \beta_{3} ) q^{92} + ( \beta_{1} + 9 \beta_{3} ) q^{93} + ( 1 + 4 \beta_{2} ) q^{94} + ( 3 - 4 \beta_{2} ) q^{96} + ( -\beta_{1} - 5 \beta_{3} ) q^{97} + ( -7 \beta_{1} - 3 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{6} - 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{6} - 4 q^{9} - 4 q^{16} - 4 q^{19} - 12 q^{21} - 8 q^{24} - 12 q^{26} - 28 q^{31} + 8 q^{34} - 12 q^{36} + 12 q^{39} - 28 q^{46} - 4 q^{49} + 16 q^{51} - 16 q^{54} + 12 q^{56} - 12 q^{59} + 40 q^{61} - 16 q^{64} - 20 q^{69} - 12 q^{71} + 24 q^{74} - 36 q^{76} + 8 q^{79} - 8 q^{81} + 24 q^{84} - 48 q^{86} - 12 q^{89} - 36 q^{91} + 4 q^{94} + 12 q^{96} + O(q^{100})$$

## 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
 −1.93185 −0.517638 0.517638 1.93185
−1.93185 −0.517638 1.73205 0 1.00000 −0.896575 0.517638 −2.73205 0
1.2 −0.517638 −1.93185 −1.73205 0 1.00000 3.34607 1.93185 0.732051 0
1.3 0.517638 1.93185 −1.73205 0 1.00000 −3.34607 −1.93185 0.732051 0
1.4 1.93185 0.517638 1.73205 0 1.00000 0.896575 −0.517638 −2.73205 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
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 3025.2.a.z 4
5.b even 2 1 inner 3025.2.a.z 4
5.c odd 4 2 605.2.b.e yes 4
11.b odd 2 1 3025.2.a.y 4
55.d odd 2 1 3025.2.a.y 4
55.e even 4 2 605.2.b.d 4
55.k odd 20 8 605.2.j.e 16
55.l even 20 8 605.2.j.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.d 4 55.e even 4 2
605.2.b.e yes 4 5.c odd 4 2
605.2.j.e 16 55.k odd 20 8
605.2.j.f 16 55.l even 20 8
3025.2.a.y 4 11.b odd 2 1
3025.2.a.y 4 55.d odd 2 1
3025.2.a.z 4 1.a even 1 1 trivial
3025.2.a.z 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3025))$$:

 $$T_{2}^{4} - 4 T_{2}^{2} + 1$$ $$T_{3}^{4} - 4 T_{3}^{2} + 1$$ $$T_{19}^{2} + 2 T_{19} - 26$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T^{2} + T^{4}$$
$3$ $$1 - 4 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$9 - 12 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$( -18 + T^{2} )^{2}$$
$17$ $$16 - 16 T^{2} + T^{4}$$
$19$ $$( -26 + 2 T + T^{2} )^{2}$$
$23$ $$484 - 52 T^{2} + T^{4}$$
$29$ $$( -48 + T^{2} )^{2}$$
$31$ $$( 46 + 14 T + T^{2} )^{2}$$
$37$ $$144 - 48 T^{2} + T^{4}$$
$41$ $$( -3 + T^{2} )^{2}$$
$43$ $$4761 - 156 T^{2} + T^{4}$$
$47$ $$2209 - 148 T^{2} + T^{4}$$
$53$ $$676 - 148 T^{2} + T^{4}$$
$59$ $$( 6 + 6 T + T^{2} )^{2}$$
$61$ $$( 97 - 20 T + T^{2} )^{2}$$
$67$ $$1089 - 228 T^{2} + T^{4}$$
$71$ $$( -18 + 6 T + T^{2} )^{2}$$
$73$ $$( -24 + T^{2} )^{2}$$
$79$ $$( -8 - 4 T + T^{2} )^{2}$$
$83$ $$( -98 + T^{2} )^{2}$$
$89$ $$( -3 + 6 T + T^{2} )^{2}$$
$97$ $$36 - 84 T^{2} + T^{4}$$
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