# Properties

 Label 475.3.c.f Level $475$ Weight $3$ Character orbit 475.c Analytic conductor $12.943$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 475.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.9428125571$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 42 x^{6} + 771 x^{4} - 7098 x^{2} + 28561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{3} q^{3} + ( -1 + 2 \beta_{4} ) q^{4} + ( -1 - 3 \beta_{4} ) q^{6} + \beta_{2} q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} + ( 2 - 4 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{3} q^{3} + ( -1 + 2 \beta_{4} ) q^{4} + ( -1 - 3 \beta_{4} ) q^{6} + \beta_{2} q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} + ( 2 - 4 \beta_{4} ) q^{9} + ( -2 + 3 \beta_{4} ) q^{11} + ( 2 \beta_{1} + \beta_{3} ) q^{12} + ( -5 \beta_{1} + 4 \beta_{3} ) q^{13} -\beta_{5} q^{14} + ( -11 + 4 \beta_{4} ) q^{16} + ( 3 \beta_{2} - \beta_{6} ) q^{17} + ( 6 \beta_{1} - 4 \beta_{3} ) q^{18} + ( -6 - 7 \beta_{4} + \beta_{5} ) q^{19} -\beta_{7} q^{21} + ( -5 \beta_{1} + 3 \beta_{3} ) q^{22} + 3 \beta_{2} q^{23} + ( -15 - 11 \beta_{4} ) q^{24} + ( 21 - 22 \beta_{4} ) q^{26} + ( -4 \beta_{1} + 7 \beta_{3} ) q^{27} + ( -\beta_{2} + 2 \beta_{6} ) q^{28} + ( -2 \beta_{5} - \beta_{7} ) q^{29} + ( -\beta_{5} + \beta_{7} ) q^{31} + ( -11 \beta_{1} + 12 \beta_{3} ) q^{32} + ( 3 \beta_{1} + \beta_{3} ) q^{33} + ( -4 \beta_{5} + \beta_{7} ) q^{34} + ( -18 + 8 \beta_{4} ) q^{36} + ( 12 \beta_{1} - 12 \beta_{3} ) q^{37} + ( \beta_{1} + 5 \beta_{2} - 7 \beta_{3} - 2 \beta_{6} ) q^{38} + ( -23 - \beta_{4} ) q^{39} + ( -\beta_{5} + 3 \beta_{7} ) q^{41} + ( -\beta_{2} - 3 \beta_{6} ) q^{42} + ( -2 \beta_{2} - 3 \beta_{6} ) q^{43} + ( 14 - 7 \beta_{4} ) q^{44} -3 \beta_{5} q^{46} + ( -2 \beta_{2} - 2 \beta_{6} ) q^{47} + ( 4 \beta_{1} - 7 \beta_{3} ) q^{48} + ( -2 + 2 \beta_{4} ) q^{49} + ( \beta_{5} - 2 \beta_{7} ) q^{51} + ( 23 \beta_{1} - 6 \beta_{3} ) q^{52} + ( -19 \beta_{1} + 2 \beta_{3} ) q^{53} + ( 13 - 29 \beta_{4} ) q^{54} + ( -\beta_{5} - 2 \beta_{7} ) q^{56} + ( -7 \beta_{1} + \beta_{2} - 13 \beta_{3} + 3 \beta_{6} ) q^{57} + ( -11 \beta_{2} + \beta_{6} ) q^{58} + ( -3 \beta_{5} + 4 \beta_{7} ) q^{59} + ( 22 + \beta_{4} ) q^{61} + ( -4 \beta_{2} + 5 \beta_{6} ) q^{62} + ( 2 \beta_{2} - 4 \beta_{6} ) q^{63} + ( -1 - 42 \beta_{4} ) q^{64} + ( -16 + 3 \beta_{4} ) q^{66} + ( -8 \beta_{1} - 19 \beta_{3} ) q^{67} + ( -7 \beta_{2} + 7 \beta_{6} ) q^{68} -3 \beta_{7} q^{69} + ( 5 \beta_{5} - \beta_{7} ) q^{71} + ( -2 \beta_{1} - 8 \beta_{3} ) q^{72} + ( -7 \beta_{2} - 5 \beta_{6} ) q^{73} + ( -48 + 60 \beta_{4} ) q^{74} + ( -22 - 5 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} ) q^{76} + ( -2 \beta_{2} + 3 \beta_{6} ) q^{77} + ( -22 \beta_{1} - \beta_{3} ) q^{78} + ( 4 \beta_{5} - 6 \beta_{7} ) q^{79} + ( -27 - 52 \beta_{4} ) q^{81} + ( -2 \beta_{2} + 11 \beta_{6} ) q^{82} + ( 12 \beta_{2} + 4 \beta_{6} ) q^{83} + ( -2 \beta_{5} - \beta_{7} ) q^{84} + ( -\beta_{5} + 3 \beta_{7} ) q^{86} + ( -9 \beta_{2} - 10 \beta_{6} ) q^{87} + ( \beta_{1} + 5 \beta_{3} ) q^{88} + ( \beta_{5} + 3 \beta_{7} ) q^{89} + ( 5 \beta_{5} - 4 \beta_{7} ) q^{91} + ( -3 \beta_{2} + 6 \beta_{6} ) q^{92} + ( 6 \beta_{2} + \beta_{6} ) q^{93} + 2 \beta_{7} q^{94} + ( -73 - 15 \beta_{4} ) q^{96} -10 \beta_{3} q^{97} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{98} + ( -28 + 14 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} - 8q^{6} + 16q^{9} + O(q^{10})$$ $$8q - 8q^{4} - 8q^{6} + 16q^{9} - 16q^{11} - 88q^{16} - 48q^{19} - 120q^{24} + 168q^{26} - 144q^{36} - 184q^{39} + 112q^{44} - 16q^{49} + 104q^{54} + 176q^{61} - 8q^{64} - 128q^{66} - 384q^{74} - 176q^{76} - 216q^{81} - 584q^{96} - 224q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 42 x^{6} + 771 x^{4} - 7098 x^{2} + 28561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} + 42 \nu^{5} - 771 \nu^{3} + 4901 \nu$$$$)/2197$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 42 \nu^{5} - 771 \nu^{3} + 9295 \nu$$$$)/2197$$ $$\beta_{3}$$ $$=$$ $$($$$$-10 \nu^{7} + 251 \nu^{5} - 2809 \nu^{3} + 10985 \nu$$$$)/4394$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 42 \nu^{4} - 602 \nu^{2} + 3549$$$$)/338$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} - 42 \nu^{4} + 940 \nu^{2} - 7098$$$$)/169$$ $$\beta_{6}$$ $$=$$ $$($$$$-34 \nu^{7} + 1259 \nu^{5} - 16919 \nu^{3} + 93457 \nu$$$$)/4394$$ $$\beta_{7}$$ $$=$$ $$($$$$23 \nu^{6} - 628 \nu^{4} + 7424 \nu^{2} - 32955$$$$)/338$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{4} + 21$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} - \beta_{3} + 4 \beta_{2} - 16 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{7} + 19 \beta_{5} + 84 \beta_{4} + 111$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$58 \beta_{6} - 110 \beta_{3} + 7 \beta_{2} - 443 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$42 \beta_{7} + 98 \beta_{5} + 824 \beta_{4} - 441$$ $$\nu^{7}$$ $$=$$ $$($$$$894 \beta_{6} - 3078 \beta_{3} - 973 \beta_{2} - 3229 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 −3.32308 + 1.39897i 3.32308 + 1.39897i −3.52946 + 0.736813i 3.52946 + 0.736813i −3.52946 − 0.736813i 3.52946 − 0.736813i −3.32308 − 1.39897i 3.32308 − 1.39897i
2.79793i 1.15894i −3.82843 0 3.24264 −6.64617 0.480049i 7.65685 0
151.2 2.79793i 1.15894i −3.82843 0 3.24264 6.64617 0.480049i 7.65685 0
151.3 1.47363i 3.55765i 1.82843 0 −5.24264 −7.05893 8.58892i −3.65685 0
151.4 1.47363i 3.55765i 1.82843 0 −5.24264 7.05893 8.58892i −3.65685 0
151.5 1.47363i 3.55765i 1.82843 0 −5.24264 −7.05893 8.58892i −3.65685 0
151.6 1.47363i 3.55765i 1.82843 0 −5.24264 7.05893 8.58892i −3.65685 0
151.7 2.79793i 1.15894i −3.82843 0 3.24264 −6.64617 0.480049i 7.65685 0
151.8 2.79793i 1.15894i −3.82843 0 3.24264 6.64617 0.480049i 7.65685 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.8 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
19.b odd 2 1 inner
95.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.3.c.f 8
5.b even 2 1 inner 475.3.c.f 8
5.c odd 4 2 95.3.d.d 8
15.e even 4 2 855.3.g.g 8
19.b odd 2 1 inner 475.3.c.f 8
95.d odd 2 1 inner 475.3.c.f 8
95.g even 4 2 95.3.d.d 8
285.j odd 4 2 855.3.g.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.3.d.d 8 5.c odd 4 2
95.3.d.d 8 95.g even 4 2
475.3.c.f 8 1.a even 1 1 trivial
475.3.c.f 8 5.b even 2 1 inner
475.3.c.f 8 19.b odd 2 1 inner
475.3.c.f 8 95.d odd 2 1 inner
855.3.g.g 8 15.e even 4 2
855.3.g.g 8 285.j odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(475, [\chi])$$:

 $$T_{2}^{4} + 10 T_{2}^{2} + 17$$ $$T_{7}^{4} - 94 T_{7}^{2} + 2201$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 17 + 10 T^{2} + T^{4} )^{2}$$
$3$ $$( 17 + 14 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 2201 - 94 T^{2} + T^{4} )^{2}$$
$11$ $$( -14 + 4 T + T^{2} )^{4}$$
$13$ $$( 16337 + 394 T^{2} + T^{4} )^{2}$$
$17$ $$( 107849 - 986 T^{2} + T^{4} )^{2}$$
$19$ $$( 130321 + 8664 T + 474 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$23$ $$( 178281 - 846 T^{2} + T^{4} )^{2}$$
$29$ $$( 1833433 + 2978 T^{2} + T^{4} )^{2}$$
$31$ $$( 149668 + 908 T^{2} + T^{4} )^{2}$$
$37$ $$( 1410048 + 2880 T^{2} + T^{4} )^{2}$$
$41$ $$( 7333732 + 5956 T^{2} + T^{4} )^{2}$$
$43$ $$( 431396 - 2164 T^{2} + T^{4} )^{2}$$
$47$ $$( 35216 - 1192 T^{2} + T^{4} )^{2}$$
$53$ $$( 1342337 + 3514 T^{2} + T^{4} )^{2}$$
$59$ $$( 35957737 + 12294 T^{2} + T^{4} )^{2}$$
$61$ $$( 482 - 44 T + T^{2} )^{4}$$
$67$ $$( 833 + 6302 T^{2} + T^{4} )^{2}$$
$71$ $$( 7333732 + 10860 T^{2} + T^{4} )^{2}$$
$73$ $$( 2201 - 9866 T^{2} + T^{4} )^{2}$$
$79$ $$( 173016208 + 26440 T^{2} + T^{4} )^{2}$$
$83$ $$( 27609344 - 17312 T^{2} + T^{4} )^{2}$$
$89$ $$( 149668 + 7372 T^{2} + T^{4} )^{2}$$
$97$ $$( 170000 + 1400 T^{2} + T^{4} )^{2}$$