# Properties

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

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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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 275) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( - \beta - 1) q^{3} + (\beta - 1) q^{4} + ( - 2 \beta - 1) q^{6} + ( - 3 \beta + 2) q^{7} + ( - 2 \beta + 1) q^{8} + (3 \beta - 1) q^{9}+O(q^{10})$$ q + b * q^2 + (-b - 1) * q^3 + (b - 1) * q^4 + (-2*b - 1) * q^6 + (-3*b + 2) * q^7 + (-2*b + 1) * q^8 + (3*b - 1) * q^9 $$q + \beta q^{2} + ( - \beta - 1) q^{3} + (\beta - 1) q^{4} + ( - 2 \beta - 1) q^{6} + ( - 3 \beta + 2) q^{7} + ( - 2 \beta + 1) q^{8} + (3 \beta - 1) q^{9} - \beta q^{12} + (2 \beta + 3) q^{13} + ( - \beta - 3) q^{14} - 3 \beta q^{16} + ( - \beta + 1) q^{17} + (2 \beta + 3) q^{18} + (6 \beta - 3) q^{19} + (4 \beta + 1) q^{21} + (5 \beta - 4) q^{23} + (3 \beta + 1) q^{24} + (5 \beta + 2) q^{26} + ( - 2 \beta + 1) q^{27} + (2 \beta - 5) q^{28} + ( - \beta + 3) q^{29} - 3 q^{31} + (\beta - 5) q^{32} - q^{34} + ( - \beta + 4) q^{36} + ( - 2 \beta - 7) q^{37} + (3 \beta + 6) q^{38} + ( - 7 \beta - 5) q^{39} + 3 q^{41} + (5 \beta + 4) q^{42} - 6 q^{43} + (\beta + 5) q^{46} + ( - 8 \beta + 1) q^{47} + (6 \beta + 3) q^{48} + ( - 3 \beta + 6) q^{49} + \beta q^{51} + (3 \beta - 1) q^{52} + ( - 7 \beta + 2) q^{53} + ( - \beta - 2) q^{54} + ( - \beta + 8) q^{56} + ( - 9 \beta - 3) q^{57} + (2 \beta - 1) q^{58} + ( - 4 \beta + 7) q^{59} + ( - 5 \beta + 8) q^{61} - 3 \beta q^{62} - 11 q^{63} + (2 \beta + 1) q^{64} - 8 q^{67} + (\beta - 2) q^{68} + ( - 6 \beta - 1) q^{69} + (10 \beta - 8) q^{71} + ( - \beta - 7) q^{72} + ( - \beta + 12) q^{73} + ( - 9 \beta - 2) q^{74} + ( - 3 \beta + 9) q^{76} + ( - 12 \beta - 7) q^{78} + ( - 3 \beta - 1) q^{79} + ( - 6 \beta + 4) q^{81} + 3 \beta q^{82} + (3 \beta - 15) q^{83} + (\beta + 3) q^{84} - 6 \beta q^{86} + ( - \beta - 2) q^{87} + (5 \beta - 15) q^{89} - 11 \beta q^{91} + ( - 4 \beta + 9) q^{92} + (3 \beta + 3) q^{93} + ( - 7 \beta - 8) q^{94} + (3 \beta + 4) q^{96} - \beta q^{97} + (3 \beta - 3) q^{98} +O(q^{100})$$ q + b * q^2 + (-b - 1) * q^3 + (b - 1) * q^4 + (-2*b - 1) * q^6 + (-3*b + 2) * q^7 + (-2*b + 1) * q^8 + (3*b - 1) * q^9 - b * q^12 + (2*b + 3) * q^13 + (-b - 3) * q^14 - 3*b * q^16 + (-b + 1) * q^17 + (2*b + 3) * q^18 + (6*b - 3) * q^19 + (4*b + 1) * q^21 + (5*b - 4) * q^23 + (3*b + 1) * q^24 + (5*b + 2) * q^26 + (-2*b + 1) * q^27 + (2*b - 5) * q^28 + (-b + 3) * q^29 - 3 * q^31 + (b - 5) * q^32 - q^34 + (-b + 4) * q^36 + (-2*b - 7) * q^37 + (3*b + 6) * q^38 + (-7*b - 5) * q^39 + 3 * q^41 + (5*b + 4) * q^42 - 6 * q^43 + (b + 5) * q^46 + (-8*b + 1) * q^47 + (6*b + 3) * q^48 + (-3*b + 6) * q^49 + b * q^51 + (3*b - 1) * q^52 + (-7*b + 2) * q^53 + (-b - 2) * q^54 + (-b + 8) * q^56 + (-9*b - 3) * q^57 + (2*b - 1) * q^58 + (-4*b + 7) * q^59 + (-5*b + 8) * q^61 - 3*b * q^62 - 11 * q^63 + (2*b + 1) * q^64 - 8 * q^67 + (b - 2) * q^68 + (-6*b - 1) * q^69 + (10*b - 8) * q^71 + (-b - 7) * q^72 + (-b + 12) * q^73 + (-9*b - 2) * q^74 + (-3*b + 9) * q^76 + (-12*b - 7) * q^78 + (-3*b - 1) * q^79 + (-6*b + 4) * q^81 + 3*b * q^82 + (3*b - 15) * q^83 + (b + 3) * q^84 - 6*b * q^86 + (-b - 2) * q^87 + (5*b - 15) * q^89 - 11*b * q^91 + (-4*b + 9) * q^92 + (3*b + 3) * q^93 + (-7*b - 8) * q^94 + (3*b + 4) * q^96 - b * q^97 + (3*b - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 3 q^{3} - q^{4} - 4 q^{6} + q^{7} + q^{9}+O(q^{10})$$ 2 * q + q^2 - 3 * q^3 - q^4 - 4 * q^6 + q^7 + q^9 $$2 q + q^{2} - 3 q^{3} - q^{4} - 4 q^{6} + q^{7} + q^{9} - q^{12} + 8 q^{13} - 7 q^{14} - 3 q^{16} + q^{17} + 8 q^{18} + 6 q^{21} - 3 q^{23} + 5 q^{24} + 9 q^{26} - 8 q^{28} + 5 q^{29} - 6 q^{31} - 9 q^{32} - 2 q^{34} + 7 q^{36} - 16 q^{37} + 15 q^{38} - 17 q^{39} + 6 q^{41} + 13 q^{42} - 12 q^{43} + 11 q^{46} - 6 q^{47} + 12 q^{48} + 9 q^{49} + q^{51} + q^{52} - 3 q^{53} - 5 q^{54} + 15 q^{56} - 15 q^{57} + 10 q^{59} + 11 q^{61} - 3 q^{62} - 22 q^{63} + 4 q^{64} - 16 q^{67} - 3 q^{68} - 8 q^{69} - 6 q^{71} - 15 q^{72} + 23 q^{73} - 13 q^{74} + 15 q^{76} - 26 q^{78} - 5 q^{79} + 2 q^{81} + 3 q^{82} - 27 q^{83} + 7 q^{84} - 6 q^{86} - 5 q^{87} - 25 q^{89} - 11 q^{91} + 14 q^{92} + 9 q^{93} - 23 q^{94} + 11 q^{96} - q^{97} - 3 q^{98}+O(q^{100})$$ 2 * q + q^2 - 3 * q^3 - q^4 - 4 * q^6 + q^7 + q^9 - q^12 + 8 * q^13 - 7 * q^14 - 3 * q^16 + q^17 + 8 * q^18 + 6 * q^21 - 3 * q^23 + 5 * q^24 + 9 * q^26 - 8 * q^28 + 5 * q^29 - 6 * q^31 - 9 * q^32 - 2 * q^34 + 7 * q^36 - 16 * q^37 + 15 * q^38 - 17 * q^39 + 6 * q^41 + 13 * q^42 - 12 * q^43 + 11 * q^46 - 6 * q^47 + 12 * q^48 + 9 * q^49 + q^51 + q^52 - 3 * q^53 - 5 * q^54 + 15 * q^56 - 15 * q^57 + 10 * q^59 + 11 * q^61 - 3 * q^62 - 22 * q^63 + 4 * q^64 - 16 * q^67 - 3 * q^68 - 8 * q^69 - 6 * q^71 - 15 * q^72 + 23 * q^73 - 13 * q^74 + 15 * q^76 - 26 * q^78 - 5 * q^79 + 2 * q^81 + 3 * q^82 - 27 * q^83 + 7 * q^84 - 6 * q^86 - 5 * q^87 - 25 * q^89 - 11 * q^91 + 14 * q^92 + 9 * q^93 - 23 * q^94 + 11 * q^96 - q^97 - 3 * q^98

## 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
 −0.618034 1.61803
−0.618034 −0.381966 −1.61803 0 0.236068 3.85410 2.23607 −2.85410 0
1.2 1.61803 −2.61803 0.618034 0 −4.23607 −2.85410 −2.23607 3.85410 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

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 3025.2.a.m 2
5.b even 2 1 3025.2.a.i 2
11.b odd 2 1 275.2.a.d 2
33.d even 2 1 2475.2.a.s 2
44.c even 2 1 4400.2.a.bv 2
55.d odd 2 1 275.2.a.g yes 2
55.e even 4 2 275.2.b.e 4
165.d even 2 1 2475.2.a.n 2
165.l odd 4 2 2475.2.c.p 4
220.g even 2 1 4400.2.a.bg 2
220.i odd 4 2 4400.2.b.x 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.d 2 11.b odd 2 1
275.2.a.g yes 2 55.d odd 2 1
275.2.b.e 4 55.e even 4 2
2475.2.a.n 2 165.d even 2 1
2475.2.a.s 2 33.d even 2 1
2475.2.c.p 4 165.l odd 4 2
3025.2.a.i 2 5.b even 2 1
3025.2.a.m 2 1.a even 1 1 trivial
4400.2.a.bg 2 220.g even 2 1
4400.2.a.bv 2 44.c even 2 1
4400.2.b.x 4 220.i odd 4 2

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} - 1$$ T2^2 - T2 - 1 $$T_{3}^{2} + 3T_{3} + 1$$ T3^2 + 3*T3 + 1 $$T_{19}^{2} - 45$$ T19^2 - 45

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 1$$
$3$ $$T^{2} + 3T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T - 11$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 8T + 11$$
$17$ $$T^{2} - T - 1$$
$19$ $$T^{2} - 45$$
$23$ $$T^{2} + 3T - 29$$
$29$ $$T^{2} - 5T + 5$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 16T + 59$$
$41$ $$(T - 3)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 6T - 71$$
$53$ $$T^{2} + 3T - 59$$
$59$ $$T^{2} - 10T + 5$$
$61$ $$T^{2} - 11T - 1$$
$67$ $$(T + 8)^{2}$$
$71$ $$T^{2} + 6T - 116$$
$73$ $$T^{2} - 23T + 131$$
$79$ $$T^{2} + 5T - 5$$
$83$ $$T^{2} + 27T + 171$$
$89$ $$T^{2} + 25T + 125$$
$97$ $$T^{2} + T - 1$$
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