Properties

 Label 3025.2.a.l Level $3025$ Weight $2$ Character orbit 3025.a Self dual yes Analytic conductor $24.155$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

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
 −1.73205 1.73205
−1.73205 1.00000 1.00000 0 −1.73205 1.73205 1.73205 −2.00000 0
1.2 1.73205 1.00000 1.00000 0 1.73205 −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
$$5$$ $$1$$
$$11$$ $$1$$

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.l 2
5.b even 2 1 605.2.a.e 2
11.b odd 2 1 inner 3025.2.a.l 2
15.d odd 2 1 5445.2.a.u 2
20.d odd 2 1 9680.2.a.bu 2
55.d odd 2 1 605.2.a.e 2
55.h odd 10 4 605.2.g.i 8
55.j even 10 4 605.2.g.i 8
165.d even 2 1 5445.2.a.u 2
220.g even 2 1 9680.2.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.e 2 5.b even 2 1
605.2.a.e 2 55.d odd 2 1
605.2.g.i 8 55.h odd 10 4
605.2.g.i 8 55.j even 10 4
3025.2.a.l 2 1.a even 1 1 trivial
3025.2.a.l 2 11.b odd 2 1 inner
5445.2.a.u 2 15.d odd 2 1
5445.2.a.u 2 165.d even 2 1
9680.2.a.bu 2 20.d odd 2 1
9680.2.a.bu 2 220.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3025))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{3} - 1$$ T3 - 1 $$T_{19}^{2} - 12$$ T19^2 - 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 3$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 12$$
$17$ $$T^{2} - 48$$
$19$ $$T^{2} - 12$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} - 147$$
$43$ $$T^{2} - 75$$
$47$ $$(T + 9)^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$(T + 12)^{2}$$
$61$ $$T^{2} - 75$$
$67$ $$(T - 5)^{2}$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2} - 108$$
$83$ $$T^{2} - 12$$
$89$ $$(T - 3)^{2}$$
$97$ $$(T - 10)^{2}$$
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