# Properties

 Label 475.2.b.d Level $475$ Weight $2$ Character orbit 475.b 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4 \nu^{5} + 9 \nu^{4} - 16 \nu^{3} - 4 \nu^{2} + 8 \nu - 9$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-7 \nu^{5} + 10 \nu^{4} - 5 \nu^{3} - 7 \nu^{2} - 32 \nu + 13$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$-6 \nu^{5} + 25 \nu^{4} - 24 \nu^{3} - 6 \nu^{2} + 12 \nu + 67$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$21 \nu^{5} - 30 \nu^{4} + 15 \nu^{3} + 67 \nu^{2} + 96 \nu - 39$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 3 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{4} + \beta_{3} + 4 \beta_{2} - 3 \beta_{1} - 4$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{3} - 3 \beta_{1} - 7$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{5} + 6 \beta_{4} + 5 \beta_{3} - 17 \beta_{2} - 11 \beta_{1} - 18$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
324.1
 −0.854638 − 0.854638i 0.403032 − 0.403032i 1.45161 + 1.45161i 1.45161 − 1.45161i 0.403032 + 0.403032i −0.854638 + 0.854638i
2.17009i 1.70928i −2.70928 0 −3.70928 1.07838i 1.53919i 0.0783777 0
324.2 1.48119i 0.806063i −0.193937 0 −1.19394 3.35026i 2.67513i 2.35026 0
324.3 0.311108i 2.90321i 1.90321 0 0.903212 4.42864i 1.21432i −5.42864 0
324.4 0.311108i 2.90321i 1.90321 0 0.903212 4.42864i 1.21432i −5.42864 0
324.5 1.48119i 0.806063i −0.193937 0 −1.19394 3.35026i 2.67513i 2.35026 0
324.6 2.17009i 1.70928i −2.70928 0 −3.70928 1.07838i 1.53919i 0.0783777 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 324.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b.d 6
5.b even 2 1 inner 475.2.b.d 6
5.c odd 4 1 95.2.a.a 3
5.c odd 4 1 475.2.a.f 3
15.e even 4 1 855.2.a.i 3
15.e even 4 1 4275.2.a.bk 3
20.e even 4 1 1520.2.a.p 3
20.e even 4 1 7600.2.a.bx 3
35.f even 4 1 4655.2.a.u 3
40.i odd 4 1 6080.2.a.bo 3
40.k even 4 1 6080.2.a.by 3
95.g even 4 1 1805.2.a.f 3
95.g even 4 1 9025.2.a.bb 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.a 3 5.c odd 4 1
475.2.a.f 3 5.c odd 4 1
475.2.b.d 6 1.a even 1 1 trivial
475.2.b.d 6 5.b even 2 1 inner
855.2.a.i 3 15.e even 4 1
1520.2.a.p 3 20.e even 4 1
1805.2.a.f 3 95.g even 4 1
4275.2.a.bk 3 15.e even 4 1
4655.2.a.u 3 35.f even 4 1
6080.2.a.bo 3 40.i odd 4 1
6080.2.a.by 3 40.k even 4 1
7600.2.a.bx 3 20.e even 4 1
9025.2.a.bb 3 95.g even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 7 T_{2}^{4} + 11 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 11 T^{2} + 7 T^{4} + T^{6}$$
$3$ $$16 + 32 T^{2} + 12 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$256 + 256 T^{2} + 32 T^{4} + T^{6}$$
$11$ $$( -16 + 8 T + 8 T^{2} + T^{3} )^{2}$$
$13$ $$16 + 80 T^{2} + 40 T^{4} + T^{6}$$
$17$ $$10816 + 1712 T^{2} + 76 T^{4} + T^{6}$$
$19$ $$( -1 + T )^{6}$$
$23$ $$256 + 192 T^{2} + 32 T^{4} + T^{6}$$
$29$ $$( 40 + 12 T - 10 T^{2} + T^{3} )^{2}$$
$31$ $$( 64 - 48 T - 4 T^{2} + T^{3} )^{2}$$
$37$ $$59536 + 5616 T^{2} + 152 T^{4} + T^{6}$$
$41$ $$( -104 - 36 T + 2 T^{2} + T^{3} )^{2}$$
$43$ $$350464 + 25472 T^{2} + 304 T^{4} + T^{6}$$
$47$ $$256 + 256 T^{2} + 32 T^{4} + T^{6}$$
$53$ $$8464 + 2832 T^{2} + 104 T^{4} + T^{6}$$
$59$ $$( -160 + 112 T - 20 T^{2} + T^{3} )^{2}$$
$61$ $$( 232 - 84 T + 2 T^{2} + T^{3} )^{2}$$
$67$ $$13456 + 5312 T^{2} + 156 T^{4} + T^{6}$$
$71$ $$( -64 - 80 T + 4 T^{2} + T^{3} )^{2}$$
$73$ $$64 + 432 T^{2} + 44 T^{4} + T^{6}$$
$79$ $$( 160 - 192 T + T^{3} )^{2}$$
$83$ $$1149184 + 38976 T^{2} + 368 T^{4} + T^{6}$$
$89$ $$( -680 - 132 T + 2 T^{2} + T^{3} )^{2}$$
$97$ $$3055504 + 73520 T^{2} + 520 T^{4} + T^{6}$$