# Properties

 Label 2475.2.a.n Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ 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: $$0$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta - 1) q^{4} + ( - 3 \beta + 2) q^{7} + (2 \beta - 1) q^{8} +O(q^{10})$$ q - b * q^2 + (b - 1) * q^4 + (-3*b + 2) * q^7 + (2*b - 1) * q^8 $$q - \beta q^{2} + (\beta - 1) q^{4} + ( - 3 \beta + 2) q^{7} + (2 \beta - 1) q^{8} - q^{11} + (2 \beta + 3) q^{13} + (\beta + 3) q^{14} - 3 \beta q^{16} + (\beta - 1) q^{17} + ( - 6 \beta + 3) q^{19} + \beta q^{22} + (5 \beta - 4) q^{23} + ( - 5 \beta - 2) q^{26} + (2 \beta - 5) q^{28} + ( - \beta + 3) q^{29} - 3 q^{31} + ( - \beta + 5) q^{32} - q^{34} + (2 \beta + 7) q^{37} + (3 \beta + 6) q^{38} + 3 q^{41} - 6 q^{43} + ( - \beta + 1) q^{44} + ( - \beta - 5) q^{46} + ( - 8 \beta + 1) q^{47} + ( - 3 \beta + 6) q^{49} + (3 \beta - 1) q^{52} + ( - 7 \beta + 2) q^{53} + (\beta - 8) q^{56} + ( - 2 \beta + 1) q^{58} + (4 \beta - 7) q^{59} + (5 \beta - 8) q^{61} + 3 \beta q^{62} + (2 \beta + 1) q^{64} + 8 q^{67} + ( - \beta + 2) q^{68} + ( - 10 \beta + 8) q^{71} + ( - \beta + 12) q^{73} + ( - 9 \beta - 2) q^{74} + (3 \beta - 9) q^{76} + (3 \beta - 2) q^{77} + (3 \beta + 1) q^{79} - 3 \beta q^{82} + ( - 3 \beta + 15) q^{83} + 6 \beta q^{86} + ( - 2 \beta + 1) q^{88} + ( - 5 \beta + 15) q^{89} - 11 \beta q^{91} + ( - 4 \beta + 9) q^{92} + (7 \beta + 8) q^{94} + \beta q^{97} + ( - 3 \beta + 3) q^{98} +O(q^{100})$$ q - b * q^2 + (b - 1) * q^4 + (-3*b + 2) * q^7 + (2*b - 1) * q^8 - q^11 + (2*b + 3) * q^13 + (b + 3) * q^14 - 3*b * q^16 + (b - 1) * q^17 + (-6*b + 3) * q^19 + b * q^22 + (5*b - 4) * q^23 + (-5*b - 2) * q^26 + (2*b - 5) * q^28 + (-b + 3) * q^29 - 3 * q^31 + (-b + 5) * q^32 - q^34 + (2*b + 7) * q^37 + (3*b + 6) * q^38 + 3 * q^41 - 6 * q^43 + (-b + 1) * q^44 + (-b - 5) * q^46 + (-8*b + 1) * q^47 + (-3*b + 6) * q^49 + (3*b - 1) * q^52 + (-7*b + 2) * q^53 + (b - 8) * q^56 + (-2*b + 1) * q^58 + (4*b - 7) * q^59 + (5*b - 8) * q^61 + 3*b * q^62 + (2*b + 1) * q^64 + 8 * q^67 + (-b + 2) * q^68 + (-10*b + 8) * q^71 + (-b + 12) * q^73 + (-9*b - 2) * q^74 + (3*b - 9) * q^76 + (3*b - 2) * q^77 + (3*b + 1) * q^79 - 3*b * q^82 + (-3*b + 15) * q^83 + 6*b * q^86 + (-2*b + 1) * q^88 + (-5*b + 15) * q^89 - 11*b * q^91 + (-4*b + 9) * q^92 + (7*b + 8) * q^94 + b * q^97 + (-3*b + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + q^{7}+O(q^{10})$$ 2 * q - q^2 - q^4 + q^7 $$2 q - q^{2} - q^{4} + q^{7} - 2 q^{11} + 8 q^{13} + 7 q^{14} - 3 q^{16} - q^{17} + q^{22} - 3 q^{23} - 9 q^{26} - 8 q^{28} + 5 q^{29} - 6 q^{31} + 9 q^{32} - 2 q^{34} + 16 q^{37} + 15 q^{38} + 6 q^{41} - 12 q^{43} + q^{44} - 11 q^{46} - 6 q^{47} + 9 q^{49} + q^{52} - 3 q^{53} - 15 q^{56} - 10 q^{59} - 11 q^{61} + 3 q^{62} + 4 q^{64} + 16 q^{67} + 3 q^{68} + 6 q^{71} + 23 q^{73} - 13 q^{74} - 15 q^{76} - q^{77} + 5 q^{79} - 3 q^{82} + 27 q^{83} + 6 q^{86} + 25 q^{89} - 11 q^{91} + 14 q^{92} + 23 q^{94} + q^{97} + 3 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 + q^7 - 2 * q^11 + 8 * q^13 + 7 * q^14 - 3 * q^16 - q^17 + q^22 - 3 * q^23 - 9 * q^26 - 8 * q^28 + 5 * q^29 - 6 * q^31 + 9 * q^32 - 2 * q^34 + 16 * q^37 + 15 * q^38 + 6 * q^41 - 12 * q^43 + q^44 - 11 * q^46 - 6 * q^47 + 9 * q^49 + q^52 - 3 * q^53 - 15 * q^56 - 10 * q^59 - 11 * q^61 + 3 * q^62 + 4 * q^64 + 16 * q^67 + 3 * q^68 + 6 * q^71 + 23 * q^73 - 13 * q^74 - 15 * q^76 - q^77 + 5 * q^79 - 3 * q^82 + 27 * q^83 + 6 * q^86 + 25 * q^89 - 11 * q^91 + 14 * q^92 + 23 * q^94 + 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
 1.61803 −0.618034
−1.61803 0 0.618034 0 0 −2.85410 2.23607 0 0
1.2 0.618034 0 −1.61803 0 0 3.85410 −2.23607 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.n 2
3.b odd 2 1 275.2.a.g yes 2
5.b even 2 1 2475.2.a.s 2
5.c odd 4 2 2475.2.c.p 4
12.b even 2 1 4400.2.a.bg 2
15.d odd 2 1 275.2.a.d 2
15.e even 4 2 275.2.b.e 4
33.d even 2 1 3025.2.a.i 2
60.h even 2 1 4400.2.a.bv 2
60.l odd 4 2 4400.2.b.x 4
165.d even 2 1 3025.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.d 2 15.d odd 2 1
275.2.a.g yes 2 3.b odd 2 1
275.2.b.e 4 15.e even 4 2
2475.2.a.n 2 1.a even 1 1 trivial
2475.2.a.s 2 5.b even 2 1
2475.2.c.p 4 5.c odd 4 2
3025.2.a.i 2 33.d even 2 1
3025.2.a.m 2 165.d even 2 1
4400.2.a.bg 2 12.b even 2 1
4400.2.a.bv 2 60.h even 2 1
4400.2.b.x 4 60.l odd 4 2

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} - 1$$ T2^2 + T2 - 1 $$T_{7}^{2} - T_{7} - 11$$ T7^2 - T7 - 11 $$T_{29}^{2} - 5T_{29} + 5$$ T29^2 - 5*T29 + 5

## Hecke characteristic polynomials

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