# Properties

 Label 2475.2.a.bj Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

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## 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: $$4$$ Coefficient field: 4.4.48704.2 Defining polynomial: $$x^{4} - 2 x^{3} - 6 x^{2} + 4 x + 6$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 495) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{1} - \beta_{3} ) q^{7} + ( 2 - 2 \beta_{1} - \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{1} - \beta_{3} ) q^{7} + ( 2 - 2 \beta_{1} - \beta_{3} ) q^{8} + q^{11} + ( -1 - \beta_{1} + \beta_{3} ) q^{13} + ( 1 + \beta_{2} - 2 \beta_{3} ) q^{14} + ( 3 - 2 \beta_{1} - 2 \beta_{3} ) q^{16} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{17} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{19} + ( 1 - \beta_{1} ) q^{22} + ( -2 - 2 \beta_{2} ) q^{23} + ( 3 + 4 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{26} + ( 3 - 5 \beta_{1} - 3 \beta_{3} ) q^{28} + ( 2 + 2 \beta_{2} + 2 \beta_{3} ) q^{29} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{31} + ( 3 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{32} + ( 3 - 4 \beta_{1} - \beta_{2} ) q^{34} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{37} + ( 4 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{38} + ( 2 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -3 + \beta_{1} + \beta_{3} ) q^{43} + ( 2 + \beta_{2} ) q^{44} + ( -6 + 6 \beta_{1} + 2 \beta_{3} ) q^{46} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( 7 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{49} + ( -3 - 3 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{52} + ( 4 - 2 \beta_{2} ) q^{53} + ( 13 - 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{56} + ( 8 - 2 \beta_{1} + 2 \beta_{3} ) q^{58} + ( 8 - 4 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{61} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{62} + ( 2 - 6 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{64} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 11 + \beta_{1} - \beta_{3} ) q^{68} + ( 4 - 2 \beta_{2} ) q^{71} + ( -1 - \beta_{1} + \beta_{3} ) q^{73} + ( -2 - 2 \beta_{2} + 2 \beta_{3} ) q^{74} + ( -6 - 4 \beta_{1} + 2 \beta_{3} ) q^{76} + ( -1 - \beta_{1} - \beta_{3} ) q^{77} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{79} + ( 8 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{82} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{83} + ( -5 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{86} + ( 2 - 2 \beta_{1} - \beta_{3} ) q^{88} + ( 2 + 4 \beta_{1} ) q^{89} + ( -8 + 6 \beta_{1} - 2 \beta_{3} ) q^{91} + ( -18 + 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{92} + ( -10 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{94} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{97} + ( 19 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} + 8q^{4} - 4q^{7} + 6q^{8} + O(q^{10})$$ $$4q + 2q^{2} + 8q^{4} - 4q^{7} + 6q^{8} + 4q^{11} - 8q^{13} + 8q^{14} + 12q^{16} + 4q^{17} + 4q^{19} + 2q^{22} - 8q^{23} + 16q^{26} + 8q^{28} + 4q^{29} + 14q^{32} + 4q^{34} - 8q^{37} + 20q^{38} + 4q^{41} - 12q^{43} + 8q^{44} - 16q^{46} + 20q^{49} - 20q^{52} + 16q^{53} + 48q^{56} + 24q^{58} + 24q^{59} + 8q^{61} - 20q^{62} + 48q^{68} + 16q^{71} - 8q^{73} - 12q^{74} - 36q^{76} - 4q^{77} + 12q^{79} + 40q^{82} + 8q^{83} - 16q^{86} + 6q^{88} + 16q^{89} - 16q^{91} - 72q^{92} - 40q^{94} - 8q^{97} + 62q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 6 x^{2} + 4 x + 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3 \nu^{2} - 3 \nu + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3 \beta_{2} + 9 \beta_{1} + 4$$

## 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
 3.28632 1.26270 −0.852061 −1.69696
−2.28632 0 3.22727 0 0 −2.51962 −2.80595 0 0
1.2 −0.262696 0 −1.93099 0 0 −0.704647 1.03266 0 0
1.3 1.85206 0 1.43013 0 0 −4.90749 −1.05543 0 0
1.4 2.69696 0 5.27358 0 0 4.13176 8.82872 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.bj 4
3.b odd 2 1 2475.2.a.bf 4
5.b even 2 1 495.2.a.f 4
5.c odd 4 2 2475.2.c.t 8
15.d odd 2 1 495.2.a.g yes 4
15.e even 4 2 2475.2.c.s 8
20.d odd 2 1 7920.2.a.cm 4
55.d odd 2 1 5445.2.a.bs 4
60.h even 2 1 7920.2.a.cn 4
165.d even 2 1 5445.2.a.bh 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.a.f 4 5.b even 2 1
495.2.a.g yes 4 15.d odd 2 1
2475.2.a.bf 4 3.b odd 2 1
2475.2.a.bj 4 1.a even 1 1 trivial
2475.2.c.s 8 15.e even 4 2
2475.2.c.t 8 5.c odd 4 2
5445.2.a.bh 4 165.d even 2 1
5445.2.a.bs 4 55.d odd 2 1
7920.2.a.cm 4 20.d odd 2 1
7920.2.a.cn 4 60.h even 2 1

## Hecke kernels

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

 $$T_{2}^{4} - 2 T_{2}^{3} - 6 T_{2}^{2} + 10 T_{2} + 3$$ $$T_{7}^{4} + 4 T_{7}^{3} - 16 T_{7}^{2} - 64 T_{7} - 36$$ $$T_{29}^{4} - 4 T_{29}^{3} - 88 T_{29}^{2} + 240 T_{29} - 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 10 T - 6 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$-36 - 64 T - 16 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$( -1 + T )^{4}$$
$13$ $$-292 - 168 T - 8 T^{2} + 8 T^{3} + T^{4}$$
$17$ $$-324 + 240 T - 40 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$288 + 192 T - 56 T^{2} - 4 T^{3} + T^{4}$$
$23$ $$-192 - 256 T - 32 T^{2} + 8 T^{3} + T^{4}$$
$29$ $$-144 + 240 T - 88 T^{2} - 4 T^{3} + T^{4}$$
$31$ $$144 + 192 T - 88 T^{2} + T^{4}$$
$37$ $$976 - 224 T - 56 T^{2} + 8 T^{3} + T^{4}$$
$41$ $$2160 + 432 T - 120 T^{2} - 4 T^{3} + T^{4}$$
$43$ $$-36 + 32 T^{2} + 12 T^{3} + T^{4}$$
$47$ $$-576 + 576 T - 128 T^{2} + T^{4}$$
$53$ $$-240 + 128 T + 40 T^{2} - 16 T^{3} + T^{4}$$
$59$ $$-8496 + 1376 T + 88 T^{2} - 24 T^{3} + T^{4}$$
$61$ $$-2224 + 1056 T - 104 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$6208 + 576 T - 224 T^{2} + T^{4}$$
$71$ $$-240 + 128 T + 40 T^{2} - 16 T^{3} + T^{4}$$
$73$ $$-292 - 168 T - 8 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$160 + 192 T - 8 T^{2} - 12 T^{3} + T^{4}$$
$83$ $$1836 + 360 T - 72 T^{2} - 8 T^{3} + T^{4}$$
$89$ $$720 + 512 T - 24 T^{2} - 16 T^{3} + T^{4}$$
$97$ $$-2864 - 1184 T - 104 T^{2} + 8 T^{3} + T^{4}$$
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