Properties

 Label 475.2.a.j Level $475$ Weight $2$ Character orbit 475.a Self dual yes Analytic conductor $3.793$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.66064384.1 Defining polynomial: $$x^{6} - 9 x^{4} + 13 x^{2} - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} + ( -\beta_{1} - \beta_{5} ) q^{3} + ( 1 - \beta_{3} ) q^{4} + \beta_{2} q^{6} + ( -\beta_{4} - \beta_{5} ) q^{7} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{8} + ( 3 + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{2} + ( -\beta_{1} - \beta_{5} ) q^{3} + ( 1 - \beta_{3} ) q^{4} + \beta_{2} q^{6} + ( -\beta_{4} - \beta_{5} ) q^{7} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{8} + ( 3 + 2 \beta_{3} ) q^{9} + ( -\beta_{2} - \beta_{3} ) q^{11} + ( -\beta_{1} + \beta_{5} ) q^{12} + ( -\beta_{1} + \beta_{5} ) q^{13} + ( 2 - 2 \beta_{3} ) q^{14} + ( -1 - \beta_{2} - \beta_{3} ) q^{16} + ( -2 \beta_{1} + \beta_{4} + \beta_{5} ) q^{17} + ( -2 \beta_{1} + \beta_{4} + 2 \beta_{5} ) q^{18} - q^{19} + ( 4 + 2 \beta_{3} ) q^{21} + ( 4 \beta_{1} - 2 \beta_{4} ) q^{22} + 2 \beta_{1} q^{23} + ( 2 - \beta_{2} + 2 \beta_{3} ) q^{24} + ( 2 + \beta_{2} + 2 \beta_{3} ) q^{26} -4 \beta_{5} q^{27} + ( 2 \beta_{1} - 4 \beta_{4} ) q^{28} + 6 q^{29} -2 \beta_{2} q^{31} + ( 2 \beta_{1} + \beta_{4} + 2 \beta_{5} ) q^{32} + ( 2 \beta_{1} + 2 \beta_{4} ) q^{33} + ( 2 \beta_{2} + 4 \beta_{3} ) q^{34} + ( -5 + 2 \beta_{2} + \beta_{3} ) q^{36} + ( \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{37} + \beta_{4} q^{38} + ( -2 + 2 \beta_{2} - 2 \beta_{3} ) q^{39} + ( 2 + 2 \beta_{2} ) q^{41} + ( -2 \beta_{1} + 2 \beta_{5} ) q^{42} + ( \beta_{4} + \beta_{5} ) q^{43} + ( 2 - 2 \beta_{2} - 4 \beta_{3} ) q^{44} + ( -2 - 2 \beta_{2} - 2 \beta_{3} ) q^{46} + ( -4 \beta_{1} + 3 \beta_{4} - \beta_{5} ) q^{47} + ( 3 \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{48} + ( -1 - \beta_{2} - \beta_{3} ) q^{49} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{51} + ( -3 \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{52} + ( -\beta_{1} + 4 \beta_{4} + \beta_{5} ) q^{53} + ( -4 - 4 \beta_{3} ) q^{54} + ( 6 - 2 \beta_{2} - 2 \beta_{3} ) q^{56} + ( \beta_{1} + \beta_{5} ) q^{57} -6 \beta_{4} q^{58} + ( 4 + 2 \beta_{3} ) q^{59} + ( -2 + 3 \beta_{2} + \beta_{3} ) q^{61} + ( 6 \beta_{1} + 2 \beta_{5} ) q^{62} + ( -4 \beta_{1} + 3 \beta_{4} - 5 \beta_{5} ) q^{63} + ( -1 + 3 \beta_{3} ) q^{64} + ( -8 - 2 \beta_{2} ) q^{66} + ( 3 \beta_{1} + 2 \beta_{4} + 5 \beta_{5} ) q^{67} + ( -6 \beta_{1} + 6 \beta_{4} ) q^{68} + ( -4 - 2 \beta_{2} ) q^{69} + ( 8 - 2 \beta_{3} ) q^{71} + ( -3 \beta_{1} + 5 \beta_{4} - 5 \beta_{5} ) q^{72} + ( 2 \beta_{1} - \beta_{4} + 3 \beta_{5} ) q^{73} + ( -6 - \beta_{2} + 2 \beta_{3} ) q^{74} + ( -1 + \beta_{3} ) q^{76} + ( 4 \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{77} + ( -4 \beta_{1} - 2 \beta_{4} - 4 \beta_{5} ) q^{78} + ( -4 - 2 \beta_{2} ) q^{79} + ( 7 - 4 \beta_{2} + 2 \beta_{3} ) q^{81} + ( -6 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{82} + ( -2 \beta_{1} - 4 \beta_{4} ) q^{83} + ( -4 + 2 \beta_{2} ) q^{84} + ( -2 + 2 \beta_{3} ) q^{86} + ( -6 \beta_{1} - 6 \beta_{5} ) q^{87} + ( 2 \beta_{1} - 6 \beta_{4} - 2 \beta_{5} ) q^{88} + ( 2 + 2 \beta_{2} - 6 \beta_{3} ) q^{89} + ( -4 + 2 \beta_{2} ) q^{91} + ( 4 \beta_{1} - 2 \beta_{4} ) q^{92} + ( 4 \beta_{1} + 4 \beta_{4} - 4 \beta_{5} ) q^{93} + ( -6 + 4 \beta_{2} + 6 \beta_{3} ) q^{94} + ( -12 - \beta_{2} - 4 \beta_{3} ) q^{96} + ( -\beta_{1} + \beta_{5} ) q^{97} + ( 4 \beta_{1} - \beta_{4} ) q^{98} + ( -4 - \beta_{2} + 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 8 q^{4} + 14 q^{9} + O(q^{10})$$ $$6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51} - 16 q^{54} + 40 q^{56} + 20 q^{59} - 14 q^{61} - 12 q^{64} - 48 q^{66} - 24 q^{69} + 52 q^{71} - 40 q^{74} - 8 q^{76} - 24 q^{79} + 38 q^{81} - 24 q^{84} - 16 q^{86} + 24 q^{89} - 24 q^{91} - 48 q^{94} - 64 q^{96} - 30 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 9 x^{4} + 13 x^{2} - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{4} - 8 \nu^{2} + 5$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 8 \nu^{3} + 7 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$-\nu^{5} + 9 \nu^{3} - 13 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2 \beta_{4} + 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{3} + 8 \beta_{2} + 19$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{5} + 18 \beta_{4} + 41 \beta_{1}$$

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
 −1.30397 2.68667 0.285442 −0.285442 −2.68667 1.30397
−2.41987 0.537080 3.85577 0 −1.29966 −3.18676 −4.49073 −2.71155 0
1.2 −1.82254 −2.31446 1.32164 0 4.21819 −1.45033 1.23634 2.35673 0
1.3 −0.906968 3.21789 −1.17741 0 −2.91852 2.59637 2.88181 7.35482 0
1.4 0.906968 −3.21789 −1.17741 0 −2.91852 −2.59637 −2.88181 7.35482 0
1.5 1.82254 2.31446 1.32164 0 4.21819 1.45033 −1.23634 2.35673 0
1.6 2.41987 −0.537080 3.85577 0 −1.29966 3.18676 4.49073 −2.71155 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$19$$ $$1$$

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.a.j 6
3.b odd 2 1 4275.2.a.br 6
4.b odd 2 1 7600.2.a.ck 6
5.b even 2 1 inner 475.2.a.j 6
5.c odd 4 2 95.2.b.b 6
15.d odd 2 1 4275.2.a.br 6
15.e even 4 2 855.2.c.d 6
19.b odd 2 1 9025.2.a.bx 6
20.d odd 2 1 7600.2.a.ck 6
20.e even 4 2 1520.2.d.h 6
95.d odd 2 1 9025.2.a.bx 6
95.g even 4 2 1805.2.b.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.b 6 5.c odd 4 2
475.2.a.j 6 1.a even 1 1 trivial
475.2.a.j 6 5.b even 2 1 inner
855.2.c.d 6 15.e even 4 2
1520.2.d.h 6 20.e even 4 2
1805.2.b.e 6 95.g even 4 2
4275.2.a.br 6 3.b odd 2 1
4275.2.a.br 6 15.d odd 2 1
7600.2.a.ck 6 4.b odd 2 1
7600.2.a.ck 6 20.d odd 2 1
9025.2.a.bx 6 19.b odd 2 1
9025.2.a.bx 6 95.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 10 T_{2}^{4} + 27 T_{2}^{2} - 16$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(475))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-16 + 27 T^{2} - 10 T^{4} + T^{6}$$
$3$ $$-16 + 60 T^{2} - 16 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$-144 + 104 T^{2} - 19 T^{4} + T^{6}$$
$11$ $$( 12 - 16 T - T^{2} + T^{3} )^{2}$$
$13$ $$-576 + 236 T^{2} - 28 T^{4} + T^{6}$$
$17$ $$-5184 + 1008 T^{2} - 59 T^{4} + T^{6}$$
$19$ $$( 1 + T )^{6}$$
$23$ $$-64 + 208 T^{2} - 36 T^{4} + T^{6}$$
$29$ $$( -6 + T )^{6}$$
$31$ $$( 128 - 56 T + T^{3} )^{2}$$
$37$ $$-1296 + 764 T^{2} - 56 T^{4} + T^{6}$$
$41$ $$( -24 - 44 T - 6 T^{2} + T^{3} )^{2}$$
$43$ $$-144 + 104 T^{2} - 19 T^{4} + T^{6}$$
$47$ $$-85264 + 7464 T^{2} - 187 T^{4} + T^{6}$$
$53$ $$-64 + 2476 T^{2} - 156 T^{4} + T^{6}$$
$59$ $$( 48 + 8 T - 10 T^{2} + T^{3} )^{2}$$
$61$ $$( -776 - 104 T + 7 T^{2} + T^{3} )^{2}$$
$67$ $$-484416 + 28556 T^{2} - 340 T^{4} + T^{6}$$
$71$ $$( -432 + 200 T - 26 T^{2} + T^{3} )^{2}$$
$73$ $$-5184 + 1616 T^{2} - 131 T^{4} + T^{6}$$
$79$ $$( -32 - 8 T + 12 T^{2} + T^{3} )^{2}$$
$83$ $$-141376 + 11728 T^{2} - 228 T^{4} + T^{6}$$
$89$ $$( 3456 - 284 T - 12 T^{2} + T^{3} )^{2}$$
$97$ $$-576 + 236 T^{2} - 28 T^{4} + T^{6}$$