# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 475.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.9428125571$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + \beta_{3} q^{3} + 9 q^{4} + 13 q^{6} + \beta_{1} q^{7} + 5 \beta_{3} q^{8} + 4 q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} + \beta_{3} q^{3} + 9 q^{4} + 13 q^{6} + \beta_{1} q^{7} + 5 \beta_{3} q^{8} + 4 q^{9} -10 q^{11} + 9 \beta_{3} q^{12} -\beta_{3} q^{13} + \beta_{2} q^{14} + 29 q^{16} -3 \beta_{1} q^{17} + 4 \beta_{3} q^{18} + ( 6 - \beta_{2} ) q^{19} + \beta_{2} q^{21} -10 \beta_{3} q^{22} + 7 \beta_{1} q^{23} + 65 q^{24} -13 q^{26} -5 \beta_{3} q^{27} + 9 \beta_{1} q^{28} -\beta_{2} q^{29} -2 \beta_{2} q^{31} + 9 \beta_{3} q^{32} -10 \beta_{3} q^{33} -3 \beta_{2} q^{34} + 36 q^{36} -6 \beta_{3} q^{37} + ( -13 \beta_{1} + 6 \beta_{3} ) q^{38} -13 q^{39} + 2 \beta_{2} q^{41} + 13 \beta_{1} q^{42} -4 \beta_{1} q^{43} -90 q^{44} + 7 \beta_{2} q^{46} -2 \beta_{1} q^{47} + 29 \beta_{3} q^{48} + 24 q^{49} -3 \beta_{2} q^{51} -9 \beta_{3} q^{52} + 21 \beta_{3} q^{53} -65 q^{54} + 5 \beta_{2} q^{56} + ( -13 \beta_{1} + 6 \beta_{3} ) q^{57} -13 \beta_{1} q^{58} -\beta_{2} q^{59} -40 q^{61} -26 \beta_{1} q^{62} + 4 \beta_{1} q^{63} + q^{64} -130 q^{66} + 11 \beta_{3} q^{67} -27 \beta_{1} q^{68} + 7 \beta_{2} q^{69} + 6 \beta_{2} q^{71} + 20 \beta_{3} q^{72} + 21 \beta_{1} q^{73} -78 q^{74} + ( 54 - 9 \beta_{2} ) q^{76} -10 \beta_{1} q^{77} -13 \beta_{3} q^{78} + 2 \beta_{2} q^{79} -101 q^{81} + 26 \beta_{1} q^{82} -8 \beta_{1} q^{83} + 9 \beta_{2} q^{84} -4 \beta_{2} q^{86} -13 \beta_{1} q^{87} -50 \beta_{3} q^{88} -\beta_{2} q^{91} + 63 \beta_{1} q^{92} -26 \beta_{1} q^{93} -2 \beta_{2} q^{94} + 117 q^{96} + 34 \beta_{3} q^{97} + 24 \beta_{3} q^{98} -40 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 36 q^{4} + 52 q^{6} + 16 q^{9} + O(q^{10})$$ $$4 q + 36 q^{4} + 52 q^{6} + 16 q^{9} - 40 q^{11} + 116 q^{16} + 24 q^{19} + 260 q^{24} - 52 q^{26} + 144 q^{36} - 52 q^{39} - 360 q^{44} + 96 q^{49} - 260 q^{54} - 160 q^{61} + 4 q^{64} - 520 q^{66} - 312 q^{74} + 216 q^{76} - 404 q^{81} + 468 q^{96} - 160 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$5 \nu^{3} + 20 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{3} + 50 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{2} + 5 \beta_{1}$$$$)/5$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
474.1
 2.30278i − 2.30278i − 1.30278i 1.30278i
−3.60555 −3.60555 9.00000 0 13.0000 5.00000i −18.0278 4.00000 0
474.2 −3.60555 −3.60555 9.00000 0 13.0000 5.00000i −18.0278 4.00000 0
474.3 3.60555 3.60555 9.00000 0 13.0000 5.00000i 18.0278 4.00000 0
474.4 3.60555 3.60555 9.00000 0 13.0000 5.00000i 18.0278 4.00000 0
 $$n$$: e.g. 2-40 or 990-1000 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.d.b 4
5.b even 2 1 inner 475.3.d.b 4
5.c odd 4 1 19.3.b.b 2
5.c odd 4 1 475.3.c.b 2
15.e even 4 1 171.3.c.b 2
19.b odd 2 1 inner 475.3.d.b 4
20.e even 4 1 304.3.e.d 2
40.i odd 4 1 1216.3.e.g 2
40.k even 4 1 1216.3.e.h 2
60.l odd 4 1 2736.3.o.d 2
95.d odd 2 1 inner 475.3.d.b 4
95.g even 4 1 19.3.b.b 2
95.g even 4 1 475.3.c.b 2
95.l even 12 2 361.3.d.b 4
95.m odd 12 2 361.3.d.b 4
95.q odd 36 6 361.3.f.d 12
95.r even 36 6 361.3.f.d 12
285.j odd 4 1 171.3.c.b 2
380.j odd 4 1 304.3.e.d 2
760.t even 4 1 1216.3.e.g 2
760.y odd 4 1 1216.3.e.h 2
1140.w even 4 1 2736.3.o.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 5.c odd 4 1
19.3.b.b 2 95.g even 4 1
171.3.c.b 2 15.e even 4 1
171.3.c.b 2 285.j odd 4 1
304.3.e.d 2 20.e even 4 1
304.3.e.d 2 380.j odd 4 1
361.3.d.b 4 95.l even 12 2
361.3.d.b 4 95.m odd 12 2
361.3.f.d 12 95.q odd 36 6
361.3.f.d 12 95.r even 36 6
475.3.c.b 2 5.c odd 4 1
475.3.c.b 2 95.g even 4 1
475.3.d.b 4 1.a even 1 1 trivial
475.3.d.b 4 5.b even 2 1 inner
475.3.d.b 4 19.b odd 2 1 inner
475.3.d.b 4 95.d odd 2 1 inner
1216.3.e.g 2 40.i odd 4 1
1216.3.e.g 2 760.t even 4 1
1216.3.e.h 2 40.k even 4 1
1216.3.e.h 2 760.y odd 4 1
2736.3.o.d 2 60.l odd 4 1
2736.3.o.d 2 1140.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 13$$ acting on $$S_{3}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -13 + T^{2} )^{2}$$
$3$ $$( -13 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 25 + T^{2} )^{2}$$
$11$ $$( 10 + T )^{4}$$
$13$ $$( -13 + T^{2} )^{2}$$
$17$ $$( 225 + T^{2} )^{2}$$
$19$ $$( 361 - 12 T + T^{2} )^{2}$$
$23$ $$( 1225 + T^{2} )^{2}$$
$29$ $$( 325 + T^{2} )^{2}$$
$31$ $$( 1300 + T^{2} )^{2}$$
$37$ $$( -468 + T^{2} )^{2}$$
$41$ $$( 1300 + T^{2} )^{2}$$
$43$ $$( 400 + T^{2} )^{2}$$
$47$ $$( 100 + T^{2} )^{2}$$
$53$ $$( -5733 + T^{2} )^{2}$$
$59$ $$( 325 + T^{2} )^{2}$$
$61$ $$( 40 + T )^{4}$$
$67$ $$( -1573 + T^{2} )^{2}$$
$71$ $$( 11700 + T^{2} )^{2}$$
$73$ $$( 11025 + T^{2} )^{2}$$
$79$ $$( 1300 + T^{2} )^{2}$$
$83$ $$( 1600 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$( -15028 + T^{2} )^{2}$$