# Properties

 Label 475.2.e.d Level $475$ Weight $2$ Character orbit 475.e 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.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.3518667.1 Defining polynomial: $$x^{6} - x^{5} + 7 x^{4} - 8 x^{3} + 43 x^{2} - 42 x + 49$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{3} + ( -3 + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{4} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{6} + ( -1 + \beta_{4} ) q^{7} + ( 2 - \beta_{2} + \beta_{4} ) q^{8} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{3} + ( -3 + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{4} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{6} + ( -1 + \beta_{4} ) q^{7} + ( 2 - \beta_{2} + \beta_{4} ) q^{8} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + ( -2 + \beta_{4} ) q^{11} + ( 1 + 2 \beta_{2} - \beta_{4} ) q^{12} + ( -5 - 5 \beta_{3} ) q^{13} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{14} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{16} + ( -\beta_{1} - 2 \beta_{5} ) q^{17} + ( -5 + \beta_{4} ) q^{18} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{19} + ( 2 \beta_{1} - 3 \beta_{3} - \beta_{5} ) q^{21} + ( -3 \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{22} + ( 3 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{23} -7 \beta_{3} q^{24} -5 \beta_{2} q^{26} + ( 3 - \beta_{2} ) q^{27} + ( 4 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{28} + ( 1 - \beta_{1} - \beta_{4} + \beta_{5} ) q^{29} -\beta_{2} q^{31} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{32} + ( 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{33} + ( 9 + 4 \beta_{1} - 3 \beta_{2} + 10 \beta_{3} + \beta_{4} - \beta_{5} ) q^{34} + ( -4 \beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{36} + ( 3 - 2 \beta_{2} - 5 \beta_{4} ) q^{37} + ( 7 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + 3 \beta_{5} ) q^{38} -5 \beta_{4} q^{39} + ( -3 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{41} + ( -8 + 5 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{42} + ( -2 \beta_{1} - \beta_{5} ) q^{43} + ( 7 - 3 \beta_{1} + 4 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{44} + ( -9 + 4 \beta_{2} - \beta_{4} ) q^{46} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{47} + ( 2 + \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{48} + ( -2 + \beta_{2} - 2 \beta_{4} ) q^{49} + ( -1 - \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{51} + ( 10 \beta_{3} + 5 \beta_{5} ) q^{52} + ( 2 + 3 \beta_{1} + \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{53} + ( 3 \beta_{1} + 4 \beta_{3} + \beta_{5} ) q^{54} + ( 5 + \beta_{2} - \beta_{4} ) q^{56} + ( -2 + 2 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{57} + ( 3 - 2 \beta_{4} ) q^{58} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{59} + ( 5 + 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{61} + ( 4 \beta_{3} + \beta_{5} ) q^{62} + ( 4 - 3 \beta_{1} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{63} + ( -5 + 2 \beta_{2} + 5 \beta_{4} ) q^{64} + ( -11 + 6 \beta_{1} - \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} ) q^{66} + ( -8 + 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{67} + ( -14 + 7 \beta_{2} + \beta_{4} ) q^{68} + ( 1 - 5 \beta_{2} - 3 \beta_{4} ) q^{69} + ( 2 \beta_{1} - 8 \beta_{3} - 3 \beta_{5} ) q^{71} + ( 6 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{72} + ( \beta_{1} + 7 \beta_{3} + 2 \beta_{5} ) q^{73} + ( 8 \beta_{1} + 3 \beta_{3} - 8 \beta_{5} ) q^{74} + ( -1 - 2 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{76} + ( 6 + \beta_{2} - 3 \beta_{4} ) q^{77} + ( 5 \beta_{1} - 5 \beta_{3} - 10 \beta_{5} ) q^{78} + ( 2 \beta_{1} - 8 \beta_{3} + 2 \beta_{5} ) q^{79} + ( \beta_{1} - 7 \beta_{3} + \beta_{5} ) q^{81} + ( 13 - 4 \beta_{2} + 9 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{82} + ( 2 - 3 \beta_{2} ) q^{83} + ( -11 - \beta_{2} + 6 \beta_{4} ) q^{84} + ( 12 + 2 \beta_{1} - 3 \beta_{2} + 11 \beta_{3} - \beta_{4} + \beta_{5} ) q^{86} + ( -4 - \beta_{2} + \beta_{4} ) q^{87} + ( 3 + 2 \beta_{2} - 2 \beta_{4} ) q^{88} + ( 2 + 4 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{89} + ( 5 - 5 \beta_{1} - 5 \beta_{4} + 5 \beta_{5} ) q^{91} + ( -6 \beta_{1} - 15 \beta_{3} - 2 \beta_{5} ) q^{92} + ( \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{93} + 7 q^{94} + ( 5 + 3 \beta_{2} - \beta_{4} ) q^{96} + ( \beta_{1} - 3 \beta_{3} + 3 \beta_{5} ) q^{97} + ( -6 \beta_{3} - 5 \beta_{5} ) q^{98} + ( 5 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{2} + q^{3} - 7 q^{4} + 6 q^{6} - 4 q^{7} + 12 q^{8} - 4 q^{9} + O(q^{10})$$ $$6 q + q^{2} + q^{3} - 7 q^{4} + 6 q^{6} - 4 q^{7} + 12 q^{8} - 4 q^{9} - 10 q^{11} + 8 q^{12} - 15 q^{13} - 7 q^{14} - 3 q^{16} + q^{17} - 28 q^{18} + 12 q^{21} - 8 q^{22} + 4 q^{23} + 21 q^{24} - 10 q^{26} + 16 q^{27} + 11 q^{28} + 2 q^{29} - 2 q^{31} - 6 q^{32} + 11 q^{33} + 25 q^{34} - 3 q^{36} + 4 q^{37} + 19 q^{38} - 10 q^{39} + 2 q^{41} - 23 q^{42} - q^{43} + 18 q^{44} - 48 q^{46} + 6 q^{47} + q^{48} - 14 q^{49} + 6 q^{51} - 35 q^{52} + 11 q^{53} - 10 q^{54} + 30 q^{56} + 19 q^{57} + 14 q^{58} - 6 q^{59} + 9 q^{61} - 13 q^{62} + 9 q^{63} - 16 q^{64} - 29 q^{66} - 20 q^{67} - 68 q^{68} - 10 q^{69} + 29 q^{71} + 11 q^{72} - 22 q^{73} + 7 q^{74} - 19 q^{76} + 32 q^{77} + 30 q^{78} + 24 q^{79} + 21 q^{81} + 31 q^{82} + 6 q^{83} - 56 q^{84} + 32 q^{86} - 24 q^{87} + 18 q^{88} + 14 q^{89} + 10 q^{91} + 41 q^{92} + 6 q^{93} + 42 q^{94} + 34 q^{96} + 7 q^{97} + 23 q^{98} + 13 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 7 x^{4} - 8 x^{3} + 43 x^{2} - 42 x + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 7 \nu^{4} - 49 \nu^{3} + 43 \nu^{2} - 42 \nu + 294$$$$)/259$$ $$\beta_{3}$$ $$=$$ $$($$$$-6 \nu^{5} + 5 \nu^{4} - 35 \nu^{3} - \nu^{2} - 215 \nu - 49$$$$)/259$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{5} + 35 \nu^{4} + 14 \nu^{3} + 215 \nu^{2} - 210 \nu + 952$$$$)/259$$ $$\beta_{5}$$ $$=$$ $$($$$$18 \nu^{5} + 22 \nu^{4} + 105 \nu^{3} + 3 \nu^{2} + 608 \nu + 147$$$$)/259$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} - 4 \beta_{3} + \beta_{2} - 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} - 5 \beta_{2} + 2$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{5} + 21 \beta_{3} + \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$6 \beta_{5} - 6 \beta_{4} - 25 \beta_{3} + 29 \beta_{2} - 35 \beta_{1} - 19$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 −1.25351 − 2.17114i 0.610938 + 1.05818i 1.14257 + 1.97899i −1.25351 + 2.17114i 0.610938 − 1.05818i 1.14257 − 1.97899i
−1.25351 2.17114i 0.610938 + 1.05818i −2.14257 + 3.71104i 0 1.53163 2.65287i 0.221876 5.72889 0.753509 1.30512i 0
26.2 0.610938 + 1.05818i 1.14257 + 1.97899i 0.253509 0.439091i 0 −1.39608 + 2.41808i 1.28514 3.06327 −1.11094 + 1.92420i 0
26.3 1.14257 + 1.97899i −1.25351 2.17114i −1.61094 + 2.79023i 0 2.86445 4.96137i −3.50702 −2.79216 −1.64257 + 2.84502i 0
201.1 −1.25351 + 2.17114i 0.610938 1.05818i −2.14257 3.71104i 0 1.53163 + 2.65287i 0.221876 5.72889 0.753509 + 1.30512i 0
201.2 0.610938 1.05818i 1.14257 1.97899i 0.253509 + 0.439091i 0 −1.39608 2.41808i 1.28514 3.06327 −1.11094 1.92420i 0
201.3 1.14257 1.97899i −1.25351 + 2.17114i −1.61094 2.79023i 0 2.86445 + 4.96137i −3.50702 −2.79216 −1.64257 2.84502i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 201.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.e.d 6
5.b even 2 1 95.2.e.b 6
5.c odd 4 2 475.2.j.b 12
15.d odd 2 1 855.2.k.g 6
19.c even 3 1 inner 475.2.e.d 6
19.c even 3 1 9025.2.a.z 3
19.d odd 6 1 9025.2.a.ba 3
20.d odd 2 1 1520.2.q.j 6
95.h odd 6 1 1805.2.a.g 3
95.i even 6 1 95.2.e.b 6
95.i even 6 1 1805.2.a.h 3
95.m odd 12 2 475.2.j.b 12
285.n odd 6 1 855.2.k.g 6
380.p odd 6 1 1520.2.q.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.b 6 5.b even 2 1
95.2.e.b 6 95.i even 6 1
475.2.e.d 6 1.a even 1 1 trivial
475.2.e.d 6 19.c even 3 1 inner
475.2.j.b 12 5.c odd 4 2
475.2.j.b 12 95.m odd 12 2
855.2.k.g 6 15.d odd 2 1
855.2.k.g 6 285.n odd 6 1
1520.2.q.j 6 20.d odd 2 1
1520.2.q.j 6 380.p odd 6 1
1805.2.a.g 3 95.h odd 6 1
1805.2.a.h 3 95.i even 6 1
9025.2.a.z 3 19.c even 3 1
9025.2.a.ba 3 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - T_{2}^{5} + 7 T_{2}^{4} - 8 T_{2}^{3} + 43 T_{2}^{2} - 42 T_{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$49 - 42 T + 43 T^{2} - 8 T^{3} + 7 T^{4} - T^{5} + T^{6}$$
$3$ $$49 - 42 T + 43 T^{2} - 8 T^{3} + 7 T^{4} - T^{5} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$( 1 - 5 T + 2 T^{2} + T^{3} )^{2}$$
$11$ $$( -1 + 2 T + 5 T^{2} + T^{3} )^{2}$$
$13$ $$( 25 + 5 T + T^{2} )^{3}$$
$17$ $$49 - 308 T + 1943 T^{2} + 30 T^{3} + 45 T^{4} - T^{5} + T^{6}$$
$19$ $$6859 + 133 T^{3} + T^{6}$$
$23$ $$2401 - 1911 T + 1717 T^{2} + 58 T^{3} + 55 T^{4} - 4 T^{5} + T^{6}$$
$29$ $$1 + 5 T + 23 T^{2} + 12 T^{3} + 9 T^{4} - 2 T^{5} + T^{6}$$
$31$ $$( -7 - 6 T + T^{2} + T^{3} )^{2}$$
$37$ $$( 227 - 119 T - 2 T^{2} + T^{3} )^{2}$$
$41$ $$1369 - 1591 T + 1923 T^{2} + 12 T^{3} + 47 T^{4} - 2 T^{5} + T^{6}$$
$43$ $$14641 + 5324 T + 2057 T^{2} + 198 T^{3} + 45 T^{4} + T^{5} + T^{6}$$
$47$ $$2401 - 343 T + 343 T^{2} - 56 T^{3} + 43 T^{4} - 6 T^{5} + T^{6}$$
$53$ $$96721 - 13062 T + 5185 T^{2} - 160 T^{3} + 163 T^{4} - 11 T^{5} + T^{6}$$
$59$ $$2401 + 343 T + 343 T^{2} + 56 T^{3} + 43 T^{4} + 6 T^{5} + T^{6}$$
$61$ $$2401 - 2401 T + 2842 T^{2} + 343 T^{3} + 130 T^{4} - 9 T^{5} + T^{6}$$
$67$ $$7744 + 9504 T + 9904 T^{2} + 1984 T^{3} + 292 T^{4} + 20 T^{5} + T^{6}$$
$71$ $$218089 - 110212 T + 42153 T^{2} - 5910 T^{3} + 605 T^{4} - 29 T^{5} + T^{6}$$
$73$ $$5929 + 9009 T + 11995 T^{2} + 2420 T^{3} + 367 T^{4} + 22 T^{5} + T^{6}$$
$79$ $$61504 + 28768 T + 19408 T^{2} - 3280 T^{3} + 460 T^{4} - 24 T^{5} + T^{6}$$
$83$ $$( -77 - 54 T - 3 T^{2} + T^{3} )^{2}$$
$89$ $$3136 - 2016 T + 2080 T^{2} + 392 T^{3} + 232 T^{4} - 14 T^{5} + T^{6}$$
$97$ $$14641 - 7986 T + 5203 T^{2} + 220 T^{3} + 115 T^{4} - 7 T^{5} + T^{6}$$