# Properties

 Label 825.3.b.a Level $825$ Weight $3$ Character orbit 825.b Analytic conductor $22.480$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 825.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.4796218097$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.39744.5 Defining polynomial: $$x^{4} - 2 x^{3} + 12 x^{2} + 4 x + 22$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 33) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + \beta_{1} q^{3} + ( -5 - 2 \beta_{1} ) q^{4} + ( \beta_{2} - \beta_{3} ) q^{6} + \beta_{2} q^{7} + ( -2 \beta_{2} + 3 \beta_{3} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} + \beta_{1} q^{3} + ( -5 - 2 \beta_{1} ) q^{4} + ( \beta_{2} - \beta_{3} ) q^{6} + \beta_{2} q^{7} + ( -2 \beta_{2} + 3 \beta_{3} ) q^{8} + 3 q^{9} + ( 5 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{11} + ( -6 - 5 \beta_{1} ) q^{12} + ( -\beta_{2} + 6 \beta_{3} ) q^{13} + ( 3 - 7 \beta_{1} ) q^{14} + ( 1 + 12 \beta_{1} ) q^{16} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{17} -3 \beta_{3} q^{18} + ( 4 \beta_{2} + 2 \beta_{3} ) q^{19} + ( -\beta_{2} - 2 \beta_{3} ) q^{21} + ( 3 + 16 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{22} + ( 23 - 5 \beta_{1} ) q^{23} + ( -\beta_{2} + 7 \beta_{3} ) q^{24} + ( 51 + 19 \beta_{1} ) q^{26} + 3 \beta_{1} q^{27} + ( -3 \beta_{2} + 4 \beta_{3} ) q^{28} + ( -4 \beta_{2} + 8 \beta_{3} ) q^{29} + ( 20 - 18 \beta_{1} ) q^{31} + ( 4 \beta_{2} - \beta_{3} ) q^{32} + ( 3 + 5 \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{33} + ( 12 + 18 \beta_{1} ) q^{34} + ( -15 - 6 \beta_{1} ) q^{36} + ( 16 - 2 \beta_{1} ) q^{37} + ( 30 - 24 \beta_{1} ) q^{38} + ( -5 \beta_{2} + 8 \beta_{3} ) q^{39} + ( -6 \beta_{2} - 4 \beta_{3} ) q^{41} + ( -21 + 3 \beta_{1} ) q^{42} + ( -12 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -31 - 15 \beta_{1} + 8 \beta_{2} - 15 \beta_{3} ) q^{44} + ( -5 \beta_{2} - 18 \beta_{3} ) q^{46} + ( -25 + 3 \beta_{1} ) q^{47} + ( 36 + \beta_{1} ) q^{48} + ( 25 + 10 \beta_{1} ) q^{49} + 6 \beta_{3} q^{51} + ( 15 \beta_{2} - 46 \beta_{3} ) q^{52} + ( -7 + 11 \beta_{1} ) q^{53} + ( 3 \beta_{2} - 3 \beta_{3} ) q^{54} + ( 39 + \beta_{1} ) q^{56} + ( -6 \beta_{2} - 6 \beta_{3} ) q^{57} + ( 60 + 44 \beta_{1} ) q^{58} + ( 10 - 42 \beta_{1} ) q^{59} + ( -11 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -18 \beta_{2} - 2 \beta_{3} ) q^{62} + 3 \beta_{2} q^{63} + ( 7 + 18 \beta_{1} ) q^{64} + ( 48 + 3 \beta_{1} + 5 \beta_{2} - 8 \beta_{3} ) q^{66} + 34 q^{67} + ( 10 \beta_{2} - 22 \beta_{3} ) q^{68} + ( -15 + 23 \beta_{1} ) q^{69} + ( -71 + \beta_{1} ) q^{71} + ( -6 \beta_{2} + 9 \beta_{3} ) q^{72} + ( 10 \beta_{2} - 4 \beta_{3} ) q^{73} + ( -2 \beta_{2} - 14 \beta_{3} ) q^{74} + ( -8 \beta_{2} + 2 \beta_{3} ) q^{76} + ( 45 - 13 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{77} + ( 57 + 51 \beta_{1} ) q^{78} + ( 5 \beta_{2} - 24 \beta_{3} ) q^{79} + 9 q^{81} + ( -54 + 34 \beta_{1} ) q^{82} + ( 6 \beta_{2} - 16 \beta_{3} ) q^{83} + ( -\beta_{2} + 10 \beta_{3} ) q^{84} + ( -54 + 80 \beta_{1} ) q^{86} + ( -4 \beta_{2} + 16 \beta_{3} ) q^{87} + ( -99 - 22 \beta_{1} - 11 \beta_{2} + 22 \beta_{3} ) q^{88} + ( 76 + 18 \beta_{1} ) q^{89} + ( 6 + 32 \beta_{1} ) q^{91} + ( -85 - 21 \beta_{1} ) q^{92} + ( -54 + 20 \beta_{1} ) q^{93} + ( 3 \beta_{2} + 22 \beta_{3} ) q^{94} + ( -3 \beta_{2} - 9 \beta_{3} ) q^{96} + ( 94 + 42 \beta_{1} ) q^{97} + ( 10 \beta_{2} - 35 \beta_{3} ) q^{98} + ( 15 + 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 20q^{4} + 12q^{9} + O(q^{10})$$ $$4q - 20q^{4} + 12q^{9} + 20q^{11} - 24q^{12} + 12q^{14} + 4q^{16} + 12q^{22} + 92q^{23} + 204q^{26} + 80q^{31} + 12q^{33} + 48q^{34} - 60q^{36} + 64q^{37} + 120q^{38} - 84q^{42} - 124q^{44} - 100q^{47} + 144q^{48} + 100q^{49} - 28q^{53} + 156q^{56} + 240q^{58} + 40q^{59} + 28q^{64} + 192q^{66} + 136q^{67} - 60q^{69} - 284q^{71} + 180q^{77} + 228q^{78} + 36q^{81} - 216q^{82} - 216q^{86} - 396q^{88} + 304q^{89} + 24q^{91} - 340q^{92} - 216q^{93} + 376q^{97} + 60q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} + 12 x^{2} + 4 x + 22$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} + 2 \nu + 23$$$$)/13$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} + 28 \nu + 10$$$$)/13$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{3} + 10 \nu^{2} - 32 \nu + 9$$$$)/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 2 \beta_{1} - 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{3} - 2 \beta_{2} + 12 \beta_{1} - 19$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 −0.366025 − 1.29224i 1.36603 + 3.21405i 1.36603 − 3.21405i −0.366025 + 1.29224i
3.53045i 1.73205 −8.46410 0 6.11492i 2.58447i 15.7603i 3.00000 0
76.2 2.35285i −1.73205 −1.53590 0 4.07525i 6.42810i 5.79766i 3.00000 0
76.3 2.35285i −1.73205 −1.53590 0 4.07525i 6.42810i 5.79766i 3.00000 0
76.4 3.53045i 1.73205 −8.46410 0 6.11492i 2.58447i 15.7603i 3.00000 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
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.3.b.a 4
5.b even 2 1 33.3.c.a 4
5.c odd 4 2 825.3.h.a 8
11.b odd 2 1 inner 825.3.b.a 4
15.d odd 2 1 99.3.c.b 4
20.d odd 2 1 528.3.j.c 4
40.e odd 2 1 2112.3.j.d 4
40.f even 2 1 2112.3.j.a 4
55.d odd 2 1 33.3.c.a 4
55.e even 4 2 825.3.h.a 8
55.h odd 10 4 363.3.g.e 16
55.j even 10 4 363.3.g.e 16
60.h even 2 1 1584.3.j.f 4
165.d even 2 1 99.3.c.b 4
220.g even 2 1 528.3.j.c 4
440.c even 2 1 2112.3.j.d 4
440.o odd 2 1 2112.3.j.a 4
660.g odd 2 1 1584.3.j.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.c.a 4 5.b even 2 1
33.3.c.a 4 55.d odd 2 1
99.3.c.b 4 15.d odd 2 1
99.3.c.b 4 165.d even 2 1
363.3.g.e 16 55.h odd 10 4
363.3.g.e 16 55.j even 10 4
528.3.j.c 4 20.d odd 2 1
528.3.j.c 4 220.g even 2 1
825.3.b.a 4 1.a even 1 1 trivial
825.3.b.a 4 11.b odd 2 1 inner
825.3.h.a 8 5.c odd 4 2
825.3.h.a 8 55.e even 4 2
1584.3.j.f 4 60.h even 2 1
1584.3.j.f 4 660.g odd 2 1
2112.3.j.a 4 40.f even 2 1
2112.3.j.a 4 440.o odd 2 1
2112.3.j.d 4 40.e odd 2 1
2112.3.j.d 4 440.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{4} + 18 T_{2}^{2} + 69$$ $$T_{23}^{2} - 46 T_{23} + 454$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$69 + 18 T^{2} + T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$276 + 48 T^{2} + T^{4}$$
$11$ $$14641 - 2420 T + 330 T^{2} - 20 T^{3} + T^{4}$$
$13$ $$33396 + 624 T^{2} + T^{4}$$
$17$ $$9936 + 216 T^{2} + T^{4}$$
$19$ $$9936 + 936 T^{2} + T^{4}$$
$23$ $$( 454 - 46 T + T^{2} )^{2}$$
$29$ $$70656 + 1536 T^{2} + T^{4}$$
$31$ $$( -572 - 40 T + T^{2} )^{2}$$
$37$ $$( 244 - 32 T + T^{2} )^{2}$$
$41$ $$4416 + 2304 T^{2} + T^{4}$$
$43$ $$3843024 + 7272 T^{2} + T^{4}$$
$47$ $$( 598 + 50 T + T^{2} )^{2}$$
$53$ $$( -314 + 14 T + T^{2} )^{2}$$
$59$ $$( -5192 - 20 T + T^{2} )^{2}$$
$61$ $$2596884 + 6144 T^{2} + T^{4}$$
$67$ $$( -34 + T )^{4}$$
$71$ $$( 5038 + 142 T + T^{2} )^{2}$$
$73$ $$4809024 + 4608 T^{2} + T^{4}$$
$79$ $$5643924 + 10128 T^{2} + T^{4}$$
$83$ $$4416 + 5184 T^{2} + T^{4}$$
$89$ $$( 4804 - 152 T + T^{2} )^{2}$$
$97$ $$( 3544 - 188 T + T^{2} )^{2}$$