# Properties

 Label 2475.2.a.z Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.7629745003$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 825) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{2} + 3) q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{7} + ( - 2 \beta_{2} + \beta_1 - 3) q^{8}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b2 + 3) * q^4 + (-b2 - b1 - 1) * q^7 + (-2*b2 + b1 - 3) * q^8 $$q + (\beta_1 - 1) q^{2} + (\beta_{2} + 3) q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{7} + ( - 2 \beta_{2} + \beta_1 - 3) q^{8} - q^{11} + ( - \beta_{2} - 2) q^{13} + (\beta_{2} - 2 \beta_1 - 1) q^{14} + (3 \beta_{2} - 2 \beta_1 + 5) q^{16} + (2 \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{2} + \beta_1 - 1) q^{19} + ( - \beta_1 + 1) q^{22} + (\beta_{2} - 2 \beta_1 + 1) q^{23} + (2 \beta_{2} - 2 \beta_1 + 4) q^{26} + ( - 2 \beta_{2} - \beta_1 - 7) q^{28} - 2 \beta_1 q^{29} + (\beta_{2} + 6) q^{31} + ( - 4 \beta_{2} + \beta_1 - 13) q^{32} + ( - 3 \beta_{2} + 1) q^{34} + (3 \beta_{2} + 1) q^{37} + (3 \beta_{2} + 7) q^{38} + ( - \beta_1 - 1) q^{41} + (3 \beta_{2} - 2 \beta_1 - 4) q^{43} + ( - \beta_{2} - 3) q^{44} + ( - 4 \beta_{2} - \beta_1 - 11) q^{46} + (\beta_{2} - 3) q^{47} + 2 \beta_1 q^{49} + ( - 4 \beta_{2} + 2 \beta_1 - 12) q^{52} + ( - 4 \beta_{2} - 4 \beta_1 - 2) q^{53} + (\beta_{2} - 4 \beta_1 + 9) q^{56} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{58} + ( - \beta_{2} + 2 \beta_1 + 1) q^{59} + ( - \beta_{2} - 4 \beta_1) q^{61} + ( - 2 \beta_{2} + 6 \beta_1 - 8) q^{62} + (3 \beta_{2} - 8 \beta_1 + 15) q^{64} + (\beta_{2} + 6 \beta_1 - 4) q^{67} + (2 \beta_{2} - \beta_1 + 7) q^{68} + ( - \beta_{2} - 9) q^{71} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{73} + ( - 6 \beta_{2} + \beta_1 - 7) q^{74} + ( - 4 \beta_{2} + 5 \beta_1 - 11) q^{76} + (\beta_{2} + \beta_1 + 1) q^{77} + ( - 2 \beta_{2} + \beta_1 + 1) q^{79} + ( - \beta_{2} - 2 \beta_1 - 3) q^{82} + ( - 6 \beta_1 + 4) q^{83} + ( - 8 \beta_{2} - 6 \beta_1 - 10) q^{86} + (2 \beta_{2} - \beta_1 + 3) q^{88} + (4 \beta_1 - 2) q^{89} + (\beta_{2} + 6) q^{91} + (5 \beta_{2} - 8 \beta_1 + 13) q^{92} + ( - 2 \beta_{2} - 3 \beta_1 + 1) q^{94} + (2 \beta_{2} + 6 \beta_1 - 11) q^{97} + (2 \beta_{2} + 2 \beta_1 + 8) q^{98}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b2 + 3) * q^4 + (-b2 - b1 - 1) * q^7 + (-2*b2 + b1 - 3) * q^8 - q^11 + (-b2 - 2) * q^13 + (b2 - 2*b1 - 1) * q^14 + (3*b2 - 2*b1 + 5) * q^16 + (2*b2 + b1 - 1) * q^17 + (-b2 + b1 - 1) * q^19 + (-b1 + 1) * q^22 + (b2 - 2*b1 + 1) * q^23 + (2*b2 - 2*b1 + 4) * q^26 + (-2*b2 - b1 - 7) * q^28 - 2*b1 * q^29 + (b2 + 6) * q^31 + (-4*b2 + b1 - 13) * q^32 + (-3*b2 + 1) * q^34 + (3*b2 + 1) * q^37 + (3*b2 + 7) * q^38 + (-b1 - 1) * q^41 + (3*b2 - 2*b1 - 4) * q^43 + (-b2 - 3) * q^44 + (-4*b2 - b1 - 11) * q^46 + (b2 - 3) * q^47 + 2*b1 * q^49 + (-4*b2 + 2*b1 - 12) * q^52 + (-4*b2 - 4*b1 - 2) * q^53 + (b2 - 4*b1 + 9) * q^56 + (-2*b2 - 2*b1 - 8) * q^58 + (-b2 + 2*b1 + 1) * q^59 + (-b2 - 4*b1) * q^61 + (-2*b2 + 6*b1 - 8) * q^62 + (3*b2 - 8*b1 + 15) * q^64 + (b2 + 6*b1 - 4) * q^67 + (2*b2 - b1 + 7) * q^68 + (-b2 - 9) * q^71 + (-2*b2 + 2*b1 + 2) * q^73 + (-6*b2 + b1 - 7) * q^74 + (-4*b2 + 5*b1 - 11) * q^76 + (b2 + b1 + 1) * q^77 + (-2*b2 + b1 + 1) * q^79 + (-b2 - 2*b1 - 3) * q^82 + (-6*b1 + 4) * q^83 + (-8*b2 - 6*b1 - 10) * q^86 + (2*b2 - b1 + 3) * q^88 + (4*b1 - 2) * q^89 + (b2 + 6) * q^91 + (5*b2 - 8*b1 + 13) * q^92 + (-2*b2 - 3*b1 + 1) * q^94 + (2*b2 + 6*b1 - 11) * q^97 + (2*b2 + 2*b1 + 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 8 q^{4} - 3 q^{7} - 6 q^{8}+O(q^{10})$$ 3 * q - 2 * q^2 + 8 * q^4 - 3 * q^7 - 6 * q^8 $$3 q - 2 q^{2} + 8 q^{4} - 3 q^{7} - 6 q^{8} - 3 q^{11} - 5 q^{13} - 6 q^{14} + 10 q^{16} - 4 q^{17} - q^{19} + 2 q^{22} + 8 q^{26} - 20 q^{28} - 2 q^{29} + 17 q^{31} - 34 q^{32} + 6 q^{34} + 18 q^{38} - 4 q^{41} - 17 q^{43} - 8 q^{44} - 30 q^{46} - 10 q^{47} + 2 q^{49} - 30 q^{52} - 6 q^{53} + 22 q^{56} - 24 q^{58} + 6 q^{59} - 3 q^{61} - 16 q^{62} + 34 q^{64} - 7 q^{67} + 18 q^{68} - 26 q^{71} + 10 q^{73} - 14 q^{74} - 24 q^{76} + 3 q^{77} + 6 q^{79} - 10 q^{82} + 6 q^{83} - 28 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 26 q^{92} + 2 q^{94} - 29 q^{97} + 24 q^{98}+O(q^{100})$$ 3 * q - 2 * q^2 + 8 * q^4 - 3 * q^7 - 6 * q^8 - 3 * q^11 - 5 * q^13 - 6 * q^14 + 10 * q^16 - 4 * q^17 - q^19 + 2 * q^22 + 8 * q^26 - 20 * q^28 - 2 * q^29 + 17 * q^31 - 34 * q^32 + 6 * q^34 + 18 * q^38 - 4 * q^41 - 17 * q^43 - 8 * q^44 - 30 * q^46 - 10 * q^47 + 2 * q^49 - 30 * q^52 - 6 * q^53 + 22 * q^56 - 24 * q^58 + 6 * q^59 - 3 * q^61 - 16 * q^62 + 34 * q^64 - 7 * q^67 + 18 * q^68 - 26 * q^71 + 10 * q^73 - 14 * q^74 - 24 * q^76 + 3 * q^77 + 6 * q^79 - 10 * q^82 + 6 * q^83 - 28 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 26 * q^92 + 2 * q^94 - 29 * q^97 + 24 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 4$$ v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 4$$ b2 + 2*b1 + 4

## 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
 −1.76156 −0.363328 3.12489
−2.76156 0 5.62620 0 0 −1.86464 −10.0140 0 0
1.2 −1.36333 0 −0.141336 0 0 2.50466 2.91934 0 0
1.3 2.12489 0 2.51514 0 0 −3.64002 1.09461 0 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## 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 2475.2.a.z 3
3.b odd 2 1 825.2.a.m yes 3
5.b even 2 1 2475.2.a.bd 3
5.c odd 4 2 2475.2.c.q 6
15.d odd 2 1 825.2.a.i 3
15.e even 4 2 825.2.c.f 6
33.d even 2 1 9075.2.a.cd 3
165.d even 2 1 9075.2.a.cj 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.i 3 15.d odd 2 1
825.2.a.m yes 3 3.b odd 2 1
825.2.c.f 6 15.e even 4 2
2475.2.a.z 3 1.a even 1 1 trivial
2475.2.a.bd 3 5.b even 2 1
2475.2.c.q 6 5.c odd 4 2
9075.2.a.cd 3 33.d even 2 1
9075.2.a.cj 3 165.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2475))$$:

 $$T_{2}^{3} + 2T_{2}^{2} - 5T_{2} - 8$$ T2^3 + 2*T2^2 - 5*T2 - 8 $$T_{7}^{3} + 3T_{7}^{2} - 7T_{7} - 17$$ T7^3 + 3*T7^2 - 7*T7 - 17 $$T_{29}^{3} + 2T_{29}^{2} - 24T_{29} + 16$$ T29^3 + 2*T29^2 - 24*T29 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} - 5 T - 8$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 3 T^{2} - 7 T - 17$$
$11$ $$(T + 1)^{3}$$
$13$ $$T^{3} + 5T^{2} - 8$$
$17$ $$T^{3} + 4 T^{2} - 25 T + 22$$
$19$ $$T^{3} + T^{2} - 19 T + 25$$
$23$ $$T^{3} - 43T - 58$$
$29$ $$T^{3} + 2 T^{2} - 24 T + 16$$
$31$ $$T^{3} - 17 T^{2} + 88 T - 136$$
$37$ $$T^{3} - 75T - 34$$
$41$ $$T^{3} + 4T^{2} - T - 2$$
$43$ $$T^{3} + 17 T^{2} - 32 T - 1100$$
$47$ $$T^{3} + 10 T^{2} + 25 T + 8$$
$53$ $$T^{3} + 6 T^{2} - 148 T - 824$$
$59$ $$T^{3} - 6 T^{2} - 31 T + 136$$
$61$ $$T^{3} + 3 T^{2} - 88 T + 244$$
$67$ $$T^{3} + 7 T^{2} - 192 T - 1588$$
$71$ $$T^{3} + 26 T^{2} + 217 T + 580$$
$73$ $$T^{3} - 10 T^{2} - 44 T + 472$$
$79$ $$T^{3} - 6 T^{2} - 37 T + 212$$
$83$ $$T^{3} - 6 T^{2} - 216 T + 1328$$
$89$ $$T^{3} + 2 T^{2} - 100 T - 328$$
$97$ $$T^{3} + 29 T^{2} + 75 T - 2153$$