# Properties

 Label 825.2.n.c Level $825$ Weight $2$ Character orbit 825.n Analytic conductor $6.588$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
301.1
 0.809017 − 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 + 0.587785i
−0.309017 0.224514i 0.309017 0.951057i −0.572949 1.76336i 0 −0.309017 + 0.224514i 0.309017 + 0.951057i −0.454915 + 1.40008i −0.809017 0.587785i 0
526.1 0.809017 2.48990i −0.809017 + 0.587785i −3.92705 2.85317i 0 0.809017 + 2.48990i −0.809017 0.587785i −6.04508 + 4.39201i 0.309017 0.951057i 0
676.1 0.809017 + 2.48990i −0.809017 0.587785i −3.92705 + 2.85317i 0 0.809017 2.48990i −0.809017 + 0.587785i −6.04508 4.39201i 0.309017 + 0.951057i 0
751.1 −0.309017 + 0.224514i 0.309017 + 0.951057i −0.572949 + 1.76336i 0 −0.309017 0.224514i 0.309017 0.951057i −0.454915 1.40008i −0.809017 + 0.587785i 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.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.n.c 4
5.b even 2 1 33.2.e.b 4
5.c odd 4 2 825.2.bx.d 8
11.c even 5 1 inner 825.2.n.c 4
11.c even 5 1 9075.2.a.cb 2
11.d odd 10 1 9075.2.a.u 2
15.d odd 2 1 99.2.f.a 4
20.d odd 2 1 528.2.y.b 4
45.h odd 6 2 891.2.n.b 8
45.j even 6 2 891.2.n.c 8
55.d odd 2 1 363.2.e.f 4
55.h odd 10 1 363.2.a.i 2
55.h odd 10 2 363.2.e.b 4
55.h odd 10 1 363.2.e.f 4
55.j even 10 1 33.2.e.b 4
55.j even 10 1 363.2.a.d 2
55.j even 10 2 363.2.e.k 4
55.k odd 20 2 825.2.bx.d 8
165.o odd 10 1 99.2.f.a 4
165.o odd 10 1 1089.2.a.t 2
165.r even 10 1 1089.2.a.l 2
220.n odd 10 1 528.2.y.b 4
220.n odd 10 1 5808.2.a.cj 2
220.o even 10 1 5808.2.a.ci 2
495.bl even 30 2 891.2.n.c 8
495.bp odd 30 2 891.2.n.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.b 4 5.b even 2 1
33.2.e.b 4 55.j even 10 1
99.2.f.a 4 15.d odd 2 1
99.2.f.a 4 165.o odd 10 1
363.2.a.d 2 55.j even 10 1
363.2.a.i 2 55.h odd 10 1
363.2.e.b 4 55.h odd 10 2
363.2.e.f 4 55.d odd 2 1
363.2.e.f 4 55.h odd 10 1
363.2.e.k 4 55.j even 10 2
528.2.y.b 4 20.d odd 2 1
528.2.y.b 4 220.n odd 10 1
825.2.n.c 4 1.a even 1 1 trivial
825.2.n.c 4 11.c even 5 1 inner
825.2.bx.d 8 5.c odd 4 2
825.2.bx.d 8 55.k odd 20 2
891.2.n.b 8 45.h odd 6 2
891.2.n.b 8 495.bp odd 30 2
891.2.n.c 8 45.j even 6 2
891.2.n.c 8 495.bl even 30 2
1089.2.a.l 2 165.r even 10 1
1089.2.a.t 2 165.o odd 10 1
5808.2.a.ci 2 220.o even 10 1
5808.2.a.cj 2 220.n odd 10 1
9075.2.a.u 2 11.d odd 10 1
9075.2.a.cb 2 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{3} + 6 T_{2}^{2} + 4 T_{2} + 1$$ $$T_{13}^{4} + 7 T_{13}^{3} + 19 T_{13}^{2} + 3 T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 6 T^{2} - T^{3} + T^{4}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$11$ $$121 + 121 T + 51 T^{2} + 11 T^{3} + T^{4}$$
$13$ $$1 + 3 T + 19 T^{2} + 7 T^{3} + T^{4}$$
$17$ $$81 - 27 T + 54 T^{2} + 12 T^{3} + T^{4}$$
$19$ $$25 + 25 T + 40 T^{2} + 10 T^{3} + T^{4}$$
$23$ $$( -1 - 4 T + T^{2} )^{2}$$
$29$ $$1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4}$$
$31$ $$961 + 403 T + 94 T^{2} + 12 T^{3} + T^{4}$$
$37$ $$121 - 11 T + 31 T^{2} + 9 T^{3} + T^{4}$$
$41$ $$1 + 7 T + 19 T^{2} + 3 T^{3} + T^{4}$$
$43$ $$( -45 + T^{2} )^{2}$$
$47$ $$121 + 88 T + 114 T^{2} + 17 T^{3} + T^{4}$$
$53$ $$1 - T + 6 T^{2} + 4 T^{3} + T^{4}$$
$59$ $$5041 + 781 T + 76 T^{2} + 6 T^{3} + T^{4}$$
$61$ $$9801 + 2376 T + 306 T^{2} + 21 T^{3} + T^{4}$$
$67$ $$( -9 - 3 T + T^{2} )^{2}$$
$71$ $$3025 - 1100 T + 190 T^{2} - 15 T^{3} + T^{4}$$
$73$ $$1936 + 704 T + 136 T^{2} + 14 T^{3} + T^{4}$$
$79$ $$14641 + 1331 T + 121 T^{2} + 11 T^{3} + T^{4}$$
$83$ $$121 + 77 T + 69 T^{2} + 13 T^{3} + T^{4}$$
$89$ $$( 31 + 12 T + T^{2} )^{2}$$
$97$ $$81 - 108 T + 54 T^{2} + 3 T^{3} + T^{4}$$