# Properties

 Label 475.2.g.b Level $475$ Weight $2$ Character orbit 475.g Analytic conductor $3.793$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.g (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 35 x^{8} + 223 x^{4} + 289$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{7} q^{2} + \beta_{5} q^{3} + ( -\beta_{6} - \beta_{8} ) q^{4} + ( -2 + \beta_{2} + 2 \beta_{4} ) q^{6} + ( 1 + \beta_{2} + \beta_{6} - \beta_{8} ) q^{7} + \beta_{3} q^{8} + ( 2 \beta_{6} - \beta_{8} - \beta_{10} ) q^{9} +O(q^{10})$$ $$q -\beta_{7} q^{2} + \beta_{5} q^{3} + ( -\beta_{6} - \beta_{8} ) q^{4} + ( -2 + \beta_{2} + 2 \beta_{4} ) q^{6} + ( 1 + \beta_{2} + \beta_{6} - \beta_{8} ) q^{7} + \beta_{3} q^{8} + ( 2 \beta_{6} - \beta_{8} - \beta_{10} ) q^{9} + ( -1 + \beta_{2} - \beta_{4} ) q^{11} + ( \beta_{7} + \beta_{9} ) q^{12} -\beta_{5} q^{13} + ( \beta_{3} + \beta_{9} ) q^{14} + ( 2 + \beta_{2} + \beta_{4} ) q^{16} + ( -2 + \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{8} + \beta_{10} ) q^{17} + ( \beta_{3} - \beta_{5} ) q^{18} + ( -\beta_{3} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{19} + ( \beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{7} + \beta_{9} + 2 \beta_{11} ) q^{21} + ( 3 \beta_{7} + \beta_{9} + \beta_{11} ) q^{22} + ( 1 + \beta_{2} - \beta_{6} + \beta_{8} ) q^{23} + ( -\beta_{6} + 2 \beta_{8} + 3 \beta_{10} ) q^{24} + ( 2 - \beta_{2} - 2 \beta_{4} ) q^{26} + 2 \beta_{9} q^{27} + ( -2 + \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{8} - \beta_{10} ) q^{28} + ( 2 \beta_{1} - \beta_{5} - 2 \beta_{7} + \beta_{11} ) q^{29} + ( -\beta_{1} - \beta_{3} - 2 \beta_{5} - \beta_{7} + \beta_{9} - 2 \beta_{11} ) q^{31} + ( -2 \beta_{7} - \beta_{9} - \beta_{11} ) q^{32} -2 \beta_{3} q^{33} + ( -2 \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{7} + \beta_{9} - \beta_{11} ) q^{34} + ( -2 + \beta_{4} ) q^{36} + ( -2 \beta_{7} - \beta_{11} ) q^{37} + ( -3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{8} + \beta_{10} ) q^{38} + ( -5 \beta_{6} + \beta_{8} + \beta_{10} ) q^{39} + ( -3 \beta_{1} + \beta_{3} - \beta_{5} - 3 \beta_{7} - \beta_{9} - \beta_{11} ) q^{41} + ( -1 - 2 \beta_{2} + 3 \beta_{4} - \beta_{6} + 2 \beta_{8} + 3 \beta_{10} ) q^{42} + ( 1 - \beta_{2} - 4 \beta_{4} - \beta_{6} - \beta_{8} + 4 \beta_{10} ) q^{43} + ( 5 \beta_{6} + 3 \beta_{8} - \beta_{10} ) q^{44} + ( -\beta_{3} + \beta_{9} ) q^{46} + ( 3 - \beta_{2} + 2 \beta_{4} + 3 \beta_{6} + \beta_{8} + 2 \beta_{10} ) q^{47} + ( 2 \beta_{1} + 3 \beta_{5} ) q^{48} + ( \beta_{6} - 2 \beta_{8} - 2 \beta_{10} ) q^{49} + ( 2 \beta_{1} - \beta_{5} + 2 \beta_{7} - \beta_{11} ) q^{51} + ( -\beta_{7} - \beta_{9} ) q^{52} + ( -2 \beta_{1} - 3 \beta_{5} ) q^{53} + ( 6 \beta_{8} - 2 \beta_{10} ) q^{54} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{7} - \beta_{9} - \beta_{11} ) q^{56} + ( 1 + 2 \beta_{2} - 3 \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{10} + \beta_{11} ) q^{57} + ( 8 - 3 \beta_{2} - 2 \beta_{4} - 8 \beta_{6} - 3 \beta_{8} + 2 \beta_{10} ) q^{58} + ( -3 \beta_{1} + \beta_{3} - \beta_{5} + 3 \beta_{7} + \beta_{9} + \beta_{11} ) q^{59} + ( -1 - 5 \beta_{2} - 3 \beta_{4} ) q^{61} + ( 1 - 4 \beta_{2} - 5 \beta_{4} + \beta_{6} + 4 \beta_{8} - 5 \beta_{10} ) q^{62} + ( -6 - 2 \beta_{2} + 3 \beta_{4} + 6 \beta_{6} - 2 \beta_{8} - 3 \beta_{10} ) q^{63} + ( -6 \beta_{8} + \beta_{10} ) q^{64} + ( -6 \beta_{2} - 2 \beta_{4} ) q^{66} + ( -2 \beta_{7} + 3 \beta_{11} ) q^{67} + ( -4 + 4 \beta_{2} + \beta_{4} + 4 \beta_{6} + 4 \beta_{8} - \beta_{10} ) q^{68} + ( \beta_{1} - \beta_{3} + 2 \beta_{5} - \beta_{7} - \beta_{9} - 2 \beta_{11} ) q^{69} + ( \beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{7} + \beta_{9} + 2 \beta_{11} ) q^{71} + ( \beta_{7} - 2 \beta_{9} - 3 \beta_{11} ) q^{72} + ( -4 + \beta_{2} + 5 \beta_{4} + 4 \beta_{6} + \beta_{8} - 5 \beta_{10} ) q^{73} + ( -4 \beta_{6} - \beta_{8} - 2 \beta_{10} ) q^{74} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{9} + \beta_{11} ) q^{76} + ( 3 - \beta_{2} - 3 \beta_{4} + 3 \beta_{6} + \beta_{8} - 3 \beta_{10} ) q^{77} + ( -3 \beta_{1} - \beta_{3} + \beta_{5} ) q^{78} + ( -3 \beta_{1} - \beta_{3} - \beta_{5} + 3 \beta_{7} - \beta_{9} + \beta_{11} ) q^{79} + ( 4 - \beta_{2} + 3 \beta_{4} ) q^{81} + ( -7 + 5 \beta_{2} - \beta_{4} - 7 \beta_{6} - 5 \beta_{8} - \beta_{10} ) q^{82} + ( 4 - 3 \beta_{4} - 4 \beta_{6} + 3 \beta_{10} ) q^{83} + ( 2 \beta_{1} - \beta_{5} - 2 \beta_{7} + \beta_{11} ) q^{84} + ( 2 \beta_{1} + \beta_{3} + 4 \beta_{5} + 2 \beta_{7} - \beta_{9} + 4 \beta_{11} ) q^{86} + ( -9 + \beta_{2} + 5 \beta_{4} - 9 \beta_{6} - \beta_{8} + 5 \beta_{10} ) q^{87} + ( \beta_{1} - \beta_{3} - 3 \beta_{5} ) q^{88} + ( 4 \beta_{1} + 3 \beta_{5} - 4 \beta_{7} - 3 \beta_{11} ) q^{89} + ( -\beta_{1} + \beta_{3} - 2 \beta_{5} - \beta_{7} - \beta_{9} - 2 \beta_{11} ) q^{91} + ( 2 - \beta_{2} - \beta_{4} + 2 \beta_{6} + \beta_{8} - \beta_{10} ) q^{92} + ( 7 + \beta_{2} + 3 \beta_{4} - 7 \beta_{6} + \beta_{8} - 3 \beta_{10} ) q^{93} + ( 6 \beta_{1} - \beta_{3} + 2 \beta_{5} - 6 \beta_{7} - \beta_{9} - 2 \beta_{11} ) q^{94} + ( 2 + 5 \beta_{2} ) q^{96} + ( 4 \beta_{7} - 2 \beta_{9} + \beta_{11} ) q^{97} + ( -3 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{98} + ( 5 \beta_{6} - \beta_{8} - 3 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 16 q^{6} + 12 q^{7} + O(q^{10})$$ $$12 q - 16 q^{6} + 12 q^{7} - 16 q^{11} + 28 q^{16} - 20 q^{17} + 12 q^{23} + 16 q^{26} - 20 q^{28} - 20 q^{36} - 4 q^{38} - 4 q^{43} + 44 q^{47} + 88 q^{58} - 24 q^{61} - 8 q^{62} - 60 q^{63} - 8 q^{66} - 44 q^{68} - 28 q^{73} - 20 q^{76} + 24 q^{77} + 60 q^{81} - 88 q^{82} + 36 q^{83} - 88 q^{87} + 20 q^{92} + 96 q^{93} + 24 q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 35 x^{8} + 223 x^{4} + 289$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{8} + 58 \nu^{4} + 417$$$$)/190$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{9} + 58 \nu^{5} + 607 \nu$$$$)/190$$ $$\beta_{4}$$ $$=$$ $$($$$$-7 \nu^{8} - 216 \nu^{4} - 639$$$$)/190$$ $$\beta_{5}$$ $$=$$ $$($$$$-7 \nu^{9} - 216 \nu^{5} - 829 \nu$$$$)/190$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{10} + 332 \nu^{6} + 2993 \nu^{2}$$$$)/3230$$ $$\beta_{7}$$ $$=$$ $$($$$$9 \nu^{11} + 332 \nu^{7} + 2993 \nu^{3}$$$$)/3230$$ $$\beta_{8}$$ $$=$$ $$($$$$-27 \nu^{10} - 996 \nu^{6} - 5749 \nu^{2}$$$$)/3230$$ $$\beta_{9}$$ $$=$$ $$($$$$-18 \nu^{11} - 664 \nu^{7} - 4371 \nu^{3}$$$$)/1615$$ $$\beta_{10}$$ $$=$$ $$($$$$-32 \nu^{10} - 1001 \nu^{6} - 3464 \nu^{2}$$$$)/1615$$ $$\beta_{11}$$ $$=$$ $$($$$$-73 \nu^{11} - 2334 \nu^{7} - 9921 \nu^{3}$$$$)/3230$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} + 3 \beta_{6}$$ $$\nu^{3}$$ $$=$$ $$\beta_{9} + 4 \beta_{7}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 7 \beta_{2} - 12$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 7 \beta_{3} - 18 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$9 \beta_{10} - 40 \beta_{8} - 56 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$9 \beta_{11} - 40 \beta_{9} - 87 \beta_{7}$$ $$\nu^{8}$$ $$=$$ $$-58 \beta_{4} - 216 \beta_{2} + 279$$ $$\nu^{9}$$ $$=$$ $$-58 \beta_{5} - 216 \beta_{3} + 437 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-332 \beta_{10} + 1143 \beta_{8} + 1427 \beta_{6}$$ $$\nu^{11}$$ $$=$$ $$-332 \beta_{11} + 1143 \beta_{9} + 2238 \beta_{7}$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$\beta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 −1.61467 + 1.61467i −1.10924 + 1.10924i −0.813901 + 0.813901i 0.813901 − 0.813901i 1.10924 − 1.10924i 1.61467 − 1.61467i −1.61467 − 1.61467i −1.10924 − 1.10924i −0.813901 − 0.813901i 0.813901 + 0.813901i 1.10924 + 1.10924i 1.61467 + 1.61467i
−1.61467 1.61467i 1.11233 1.11233i 3.21432i 0 −3.59210 −1.21432 + 1.21432i 1.96073 1.96073i 0.525428i 0
18.2 −1.10924 1.10924i −1.29790 + 1.29790i 0.460811i 0 2.87936 1.53919 1.53919i −1.70732 + 1.70732i 0.369102i 0
18.3 −0.813901 0.813901i 2.01945 2.01945i 0.675131i 0 −3.28726 2.67513 2.67513i −2.17729 + 2.17729i 5.15633i 0
18.4 0.813901 + 0.813901i −2.01945 + 2.01945i 0.675131i 0 −3.28726 2.67513 2.67513i 2.17729 2.17729i 5.15633i 0
18.5 1.10924 + 1.10924i 1.29790 1.29790i 0.460811i 0 2.87936 1.53919 1.53919i 1.70732 1.70732i 0.369102i 0
18.6 1.61467 + 1.61467i −1.11233 + 1.11233i 3.21432i 0 −3.59210 −1.21432 + 1.21432i −1.96073 + 1.96073i 0.525428i 0
132.1 −1.61467 + 1.61467i 1.11233 + 1.11233i 3.21432i 0 −3.59210 −1.21432 1.21432i 1.96073 + 1.96073i 0.525428i 0
132.2 −1.10924 + 1.10924i −1.29790 1.29790i 0.460811i 0 2.87936 1.53919 + 1.53919i −1.70732 1.70732i 0.369102i 0
132.3 −0.813901 + 0.813901i 2.01945 + 2.01945i 0.675131i 0 −3.28726 2.67513 + 2.67513i −2.17729 2.17729i 5.15633i 0
132.4 0.813901 0.813901i −2.01945 2.01945i 0.675131i 0 −3.28726 2.67513 + 2.67513i 2.17729 + 2.17729i 5.15633i 0
132.5 1.10924 1.10924i 1.29790 + 1.29790i 0.460811i 0 2.87936 1.53919 + 1.53919i 1.70732 + 1.70732i 0.369102i 0
132.6 1.61467 1.61467i −1.11233 1.11233i 3.21432i 0 −3.59210 −1.21432 1.21432i −1.96073 1.96073i 0.525428i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 132.6 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.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.g.b 12
5.b even 2 1 95.2.g.b 12
5.c odd 4 1 95.2.g.b 12
5.c odd 4 1 inner 475.2.g.b 12
15.d odd 2 1 855.2.p.f 12
15.e even 4 1 855.2.p.f 12
19.b odd 2 1 inner 475.2.g.b 12
95.d odd 2 1 95.2.g.b 12
95.g even 4 1 95.2.g.b 12
95.g even 4 1 inner 475.2.g.b 12
285.b even 2 1 855.2.p.f 12
285.j odd 4 1 855.2.p.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.g.b 12 5.b even 2 1
95.2.g.b 12 5.c odd 4 1
95.2.g.b 12 95.d odd 2 1
95.2.g.b 12 95.g even 4 1
475.2.g.b 12 1.a even 1 1 trivial
475.2.g.b 12 5.c odd 4 1 inner
475.2.g.b 12 19.b odd 2 1 inner
475.2.g.b 12 95.g even 4 1 inner
855.2.p.f 12 15.d odd 2 1
855.2.p.f 12 15.e even 4 1
855.2.p.f 12 285.b even 2 1
855.2.p.f 12 285.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 35 T_{2}^{8} + 223 T_{2}^{4} + 289$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$289 + 223 T^{4} + 35 T^{8} + T^{12}$$
$3$ $$4624 + 1232 T^{4} + 84 T^{8} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$( 200 - 40 T + 4 T^{2} - 8 T^{3} + 18 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$11$ $$( -20 - 4 T + 4 T^{2} + T^{3} )^{4}$$
$13$ $$4624 + 1232 T^{4} + 84 T^{8} + T^{12}$$
$17$ $$( 200 - 120 T + 36 T^{2} + 80 T^{3} + 50 T^{4} + 10 T^{5} + T^{6} )^{2}$$
$19$ $$47045881 - 9122470 T^{2} + 903583 T^{4} - 58228 T^{6} + 2503 T^{8} - 70 T^{10} + T^{12}$$
$23$ $$( 200 - 40 T + 4 T^{2} - 8 T^{3} + 18 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$29$ $$( -13600 + 4448 T^{2} - 132 T^{4} + T^{6} )^{2}$$
$31$ $$( 13600 + 3728 T^{2} + 148 T^{4} + T^{6} )^{2}$$
$37$ $$4624 + 49200 T^{4} + 692 T^{8} + T^{12}$$
$41$ $$( 54400 + 6432 T^{2} + 176 T^{4} + T^{6} )^{2}$$
$43$ $$( 57800 - 33320 T + 9604 T^{2} - 536 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$47$ $$( 33800 + 9880 T + 1444 T^{2} - 1096 T^{3} + 242 T^{4} - 22 T^{5} + T^{6} )^{2}$$
$53$ $$4624 + 8604592 T^{4} + 7124 T^{8} + T^{12}$$
$59$ $$( -54400 + 6432 T^{2} - 176 T^{4} + T^{6} )^{2}$$
$61$ $$( -460 - 88 T + 6 T^{2} + T^{3} )^{4}$$
$67$ $$3270467344 + 104485936 T^{4} + 21428 T^{8} + T^{12}$$
$71$ $$( 13600 + 2832 T^{2} + 140 T^{4} + T^{6} )^{2}$$
$73$ $$( 897800 + 163480 T + 14884 T^{2} - 368 T^{3} + 98 T^{4} + 14 T^{5} + T^{6} )^{2}$$
$79$ $$( -54400 + 6688 T^{2} - 200 T^{4} + T^{6} )^{2}$$
$83$ $$( 33800 - 1560 T + 36 T^{2} - 152 T^{3} + 162 T^{4} - 18 T^{5} + T^{6} )^{2}$$
$89$ $$( -340000 + 20768 T^{2} - 348 T^{4} + T^{6} )^{2}$$
$97$ $$1806250000 + 110070000 T^{4} + 31956 T^{8} + T^{12}$$