# Properties

 Label 825.2.c.g Level $825$ Weight $2$ Character orbit 825.c Analytic conductor $6.588$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(199,825)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{4} + \beta_{2} - 1) q^{4} + \beta_{2} q^{6} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{8} - q^{9}+O(q^{10})$$ q - b1 * q^2 + b3 * q^3 + (b4 + b2 - 1) * q^4 + b2 * q^6 + (b5 + b1) * q^7 + (-b5 - 2*b3 + 2*b1) * q^8 - q^9 $$q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{4} + \beta_{2} - 1) q^{4} + \beta_{2} q^{6} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{8} - q^{9} + q^{11} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{12} + ( - \beta_{5} + \beta_1) q^{13} + ( - \beta_{4} + \beta_{2} + 4) q^{14} + ( - 4 \beta_{2} + 3) q^{16} + ( - \beta_{5} + 2 \beta_{3} + 3 \beta_1) q^{17} + \beta_1 q^{18} + (2 \beta_{4} - 2) q^{19} + (\beta_{4} - \beta_{2}) q^{21} - \beta_1 q^{22} + (2 \beta_{5} + 2 \beta_1) q^{23} + ( - \beta_{4} - 2 \beta_{2} + 2) q^{24} + ( - \beta_{4} - 3 \beta_{2} + 2) q^{26} - \beta_{3} q^{27} + (\beta_{5} - 4 \beta_{3} - 3 \beta_1) q^{28} + (2 \beta_{2} + 4) q^{29} + (2 \beta_{4} + 2 \beta_{2} + 4) q^{31} + (2 \beta_{5} + 8 \beta_{3} - 3 \beta_1) q^{32} + \beta_{3} q^{33} + ( - 3 \beta_{4} - 3 \beta_{2} + 8) q^{34} + ( - \beta_{4} - \beta_{2} + 1) q^{36} + 2 \beta_{3} q^{37} + (2 \beta_{3} + 6 \beta_1) q^{38} + ( - \beta_{4} - \beta_{2}) q^{39} + (2 \beta_{2} - 4) q^{41} + (\beta_{5} + 4 \beta_{3} + \beta_1) q^{42} + (3 \beta_{5} - \beta_1) q^{43} + (\beta_{4} + \beta_{2} - 1) q^{44} + ( - 2 \beta_{4} + 2 \beta_{2} + 8) q^{46} + ( - 2 \beta_{5} + 4 \beta_{3} + 2 \beta_1) q^{47} + (3 \beta_{3} - 4 \beta_1) q^{48} + ( - 4 \beta_{2} - 5) q^{49} + ( - \beta_{4} - 3 \beta_{2} - 2) q^{51} + (\beta_{5} + 8 \beta_{3} - 5 \beta_1) q^{52} + ( - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{53} - \beta_{2} q^{54} + (\beta_{4} + 3 \beta_{2}) q^{56} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{57} + ( - 2 \beta_{5} - 6 \beta_{3} - 2 \beta_1) q^{58} + ( - 2 \beta_{4} + 2 \beta_{2} - 4) q^{59} + ( - 2 \beta_{4} + 2 \beta_{2} - 2) q^{61} + ( - 2 \beta_{5} - 4 \beta_{3} + 2 \beta_1) q^{62} + ( - \beta_{5} - \beta_1) q^{63} + (3 \beta_{4} + 7 \beta_{2} - 1) q^{64} + \beta_{2} q^{66} + (2 \beta_{5} - 2 \beta_1) q^{67} + (\beta_{5} + 10 \beta_{3} - 11 \beta_1) q^{68} + (2 \beta_{4} - 2 \beta_{2}) q^{69} + (2 \beta_{4} + 2 \beta_{2} + 4) q^{71} + (\beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{72} + (\beta_{5} - 4 \beta_{3} + 3 \beta_1) q^{73} + 2 \beta_{2} q^{74} + ( - 2 \beta_{4} - 4 \beta_{2} + 14) q^{76} + (\beta_{5} + \beta_1) q^{77} + (\beta_{5} + 2 \beta_{3} - 3 \beta_1) q^{78} + ( - 2 \beta_{4} - 4 \beta_{2} - 6) q^{79} + q^{81} + ( - 2 \beta_{5} - 6 \beta_{3} + 6 \beta_1) q^{82} + ( - 3 \beta_{5} + 2 \beta_{3} + 3 \beta_1) q^{83} + (\beta_{4} + 3 \beta_{2} + 4) q^{84} + (\beta_{4} + 7 \beta_{2}) q^{86} + (4 \beta_{3} + 2 \beta_1) q^{87} + ( - \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{88} + ( - 4 \beta_{2} + 2) q^{89} + (2 \beta_{4} + 2 \beta_{2} + 4) q^{91} + (2 \beta_{5} - 8 \beta_{3} - 6 \beta_1) q^{92} + ( - 2 \beta_{5} + 4 \beta_{3} + 2 \beta_1) q^{93} + ( - 2 \beta_{4} - 2 \beta_{2} + 4) q^{94} + (2 \beta_{4} + 3 \beta_{2} - 8) q^{96} + ( - 2 \beta_{5} - 6 \beta_{3} + 2 \beta_1) q^{97} + (4 \beta_{5} + 12 \beta_{3} + \beta_1) q^{98} - q^{99}+O(q^{100})$$ q - b1 * q^2 + b3 * q^3 + (b4 + b2 - 1) * q^4 + b2 * q^6 + (b5 + b1) * q^7 + (-b5 - 2*b3 + 2*b1) * q^8 - q^9 + q^11 + (-b5 - b3 + b1) * q^12 + (-b5 + b1) * q^13 + (-b4 + b2 + 4) * q^14 + (-4*b2 + 3) * q^16 + (-b5 + 2*b3 + 3*b1) * q^17 + b1 * q^18 + (2*b4 - 2) * q^19 + (b4 - b2) * q^21 - b1 * q^22 + (2*b5 + 2*b1) * q^23 + (-b4 - 2*b2 + 2) * q^24 + (-b4 - 3*b2 + 2) * q^26 - b3 * q^27 + (b5 - 4*b3 - 3*b1) * q^28 + (2*b2 + 4) * q^29 + (2*b4 + 2*b2 + 4) * q^31 + (2*b5 + 8*b3 - 3*b1) * q^32 + b3 * q^33 + (-3*b4 - 3*b2 + 8) * q^34 + (-b4 - b2 + 1) * q^36 + 2*b3 * q^37 + (2*b3 + 6*b1) * q^38 + (-b4 - b2) * q^39 + (2*b2 - 4) * q^41 + (b5 + 4*b3 + b1) * q^42 + (3*b5 - b1) * q^43 + (b4 + b2 - 1) * q^44 + (-2*b4 + 2*b2 + 8) * q^46 + (-2*b5 + 4*b3 + 2*b1) * q^47 + (3*b3 - 4*b1) * q^48 + (-4*b2 - 5) * q^49 + (-b4 - 3*b2 - 2) * q^51 + (b5 + 8*b3 - 5*b1) * q^52 + (-2*b5 - 2*b3 - 2*b1) * q^53 - b2 * q^54 + (b4 + 3*b2) * q^56 + (-2*b5 - 2*b3) * q^57 + (-2*b5 - 6*b3 - 2*b1) * q^58 + (-2*b4 + 2*b2 - 4) * q^59 + (-2*b4 + 2*b2 - 2) * q^61 + (-2*b5 - 4*b3 + 2*b1) * q^62 + (-b5 - b1) * q^63 + (3*b4 + 7*b2 - 1) * q^64 + b2 * q^66 + (2*b5 - 2*b1) * q^67 + (b5 + 10*b3 - 11*b1) * q^68 + (2*b4 - 2*b2) * q^69 + (2*b4 + 2*b2 + 4) * q^71 + (b5 + 2*b3 - 2*b1) * q^72 + (b5 - 4*b3 + 3*b1) * q^73 + 2*b2 * q^74 + (-2*b4 - 4*b2 + 14) * q^76 + (b5 + b1) * q^77 + (b5 + 2*b3 - 3*b1) * q^78 + (-2*b4 - 4*b2 - 6) * q^79 + q^81 + (-2*b5 - 6*b3 + 6*b1) * q^82 + (-3*b5 + 2*b3 + 3*b1) * q^83 + (b4 + 3*b2 + 4) * q^84 + (b4 + 7*b2) * q^86 + (4*b3 + 2*b1) * q^87 + (-b5 - 2*b3 + 2*b1) * q^88 + (-4*b2 + 2) * q^89 + (2*b4 + 2*b2 + 4) * q^91 + (2*b5 - 8*b3 - 6*b1) * q^92 + (-2*b5 + 4*b3 + 2*b1) * q^93 + (-2*b4 - 2*b2 + 4) * q^94 + (2*b4 + 3*b2 - 8) * q^96 + (-2*b5 - 6*b3 + 2*b1) * q^97 + (4*b5 + 12*b3 + b1) * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 10 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10})$$ 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 $$6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 6 q^{11} + 24 q^{14} + 26 q^{16} - 16 q^{19} + 18 q^{24} + 20 q^{26} + 20 q^{29} + 16 q^{31} + 60 q^{34} + 10 q^{36} + 4 q^{39} - 28 q^{41} - 10 q^{44} + 48 q^{46} - 22 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} - 12 q^{61} - 26 q^{64} - 2 q^{66} + 16 q^{71} - 4 q^{74} + 96 q^{76} - 24 q^{79} + 6 q^{81} + 16 q^{84} - 16 q^{86} + 20 q^{89} + 16 q^{91} + 32 q^{94} - 58 q^{96} - 6 q^{99}+O(q^{100})$$ 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 + 6 * q^11 + 24 * q^14 + 26 * q^16 - 16 * q^19 + 18 * q^24 + 20 * q^26 + 20 * q^29 + 16 * q^31 + 60 * q^34 + 10 * q^36 + 4 * q^39 - 28 * q^41 - 10 * q^44 + 48 * q^46 - 22 * q^49 - 4 * q^51 + 2 * q^54 - 8 * q^56 - 24 * q^59 - 12 * q^61 - 26 * q^64 - 2 * q^66 + 16 * q^71 - 4 * q^74 + 96 * q^76 - 24 * q^79 + 6 * q^81 + 16 * q^84 - 16 * q^86 + 20 * q^89 + 16 * q^91 + 32 * q^94 - 58 * q^96 - 6 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23$$ (-3*v^5 + v^4 + 11*v^3 - 26*v^2 + 6*v - 1) / 23 $$\beta_{2}$$ $$=$$ $$( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23$$ (-4*v^5 + 9*v^4 - 16*v^3 - 4*v^2 + 8*v - 9) / 23 $$\beta_{3}$$ $$=$$ $$( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23$$ (7*v^5 - 10*v^4 + 5*v^3 + 30*v^2 + 32*v - 13) / 23 $$\beta_{4}$$ $$=$$ $$( -6\nu^{5} + 25\nu^{4} - 24\nu^{3} - 6\nu^{2} + 12\nu + 67 ) / 23$$ (-6*v^5 + 25*v^4 - 24*v^3 - 6*v^2 + 12*v + 67) / 23 $$\beta_{5}$$ $$=$$ $$( -25\nu^{5} + 39\nu^{4} - 31\nu^{3} - 48\nu^{2} - 134\nu + 53 ) / 23$$ (-25*v^5 + 39*v^4 - 31*v^3 - 48*v^2 - 134*v + 53) / 23
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 2$$ (b3 + b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 4\beta_{3} + \beta_1 ) / 2$$ (b5 + 4*b3 + b1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{5} + \beta_{4} + 4\beta_{3} - 3\beta_{2} + 3\beta _1 - 4 ) / 2$$ (b5 + b4 + 4*b3 - 3*b2 + 3*b1 - 4) / 2 $$\nu^{4}$$ $$=$$ $$2\beta_{4} - 3\beta_{2} - 7$$ 2*b4 - 3*b2 - 7 $$\nu^{5}$$ $$=$$ $$( -5\beta_{5} + 5\beta_{4} - 18\beta_{3} - 11\beta_{2} - 11\beta _1 - 18 ) / 2$$ (-5*b5 + 5*b4 - 18*b3 - 11*b2 - 11*b1 - 18) / 2

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −0.854638 + 0.854638i 1.45161 + 1.45161i 0.403032 − 0.403032i 0.403032 + 0.403032i 1.45161 − 1.45161i −0.854638 − 0.854638i
2.70928i 1.00000i −5.34017 0 −2.70928 1.07838i 9.04945i −1.00000 0
199.2 1.90321i 1.00000i −1.62222 0 1.90321 4.42864i 0.719004i −1.00000 0
199.3 0.193937i 1.00000i 1.96239 0 −0.193937 3.35026i 0.768452i −1.00000 0
199.4 0.193937i 1.00000i 1.96239 0 −0.193937 3.35026i 0.768452i −1.00000 0
199.5 1.90321i 1.00000i −1.62222 0 1.90321 4.42864i 0.719004i −1.00000 0
199.6 2.70928i 1.00000i −5.34017 0 −2.70928 1.07838i 9.04945i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.6 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.c.g 6
3.b odd 2 1 2475.2.c.r 6
5.b even 2 1 inner 825.2.c.g 6
5.c odd 4 1 165.2.a.c 3
5.c odd 4 1 825.2.a.k 3
15.d odd 2 1 2475.2.c.r 6
15.e even 4 1 495.2.a.e 3
15.e even 4 1 2475.2.a.bb 3
20.e even 4 1 2640.2.a.be 3
35.f even 4 1 8085.2.a.bk 3
55.e even 4 1 1815.2.a.m 3
55.e even 4 1 9075.2.a.cf 3
60.l odd 4 1 7920.2.a.cj 3
165.l odd 4 1 5445.2.a.z 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 5.c odd 4 1
495.2.a.e 3 15.e even 4 1
825.2.a.k 3 5.c odd 4 1
825.2.c.g 6 1.a even 1 1 trivial
825.2.c.g 6 5.b even 2 1 inner
1815.2.a.m 3 55.e even 4 1
2475.2.a.bb 3 15.e even 4 1
2475.2.c.r 6 3.b odd 2 1
2475.2.c.r 6 15.d odd 2 1
2640.2.a.be 3 20.e even 4 1
5445.2.a.z 3 165.l odd 4 1
7920.2.a.cj 3 60.l odd 4 1
8085.2.a.bk 3 35.f even 4 1
9075.2.a.cf 3 55.e even 4 1

## Hecke kernels

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

 $$T_{2}^{6} + 11T_{2}^{4} + 27T_{2}^{2} + 1$$ T2^6 + 11*T2^4 + 27*T2^2 + 1 $$T_{7}^{6} + 32T_{7}^{4} + 256T_{7}^{2} + 256$$ T7^6 + 32*T7^4 + 256*T7^2 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 11 T^{4} + 27 T^{2} + 1$$
$3$ $$(T^{2} + 1)^{3}$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256$$
$11$ $$(T - 1)^{6}$$
$13$ $$T^{6} + 28 T^{4} + 176 T^{2} + \cdots + 64$$
$17$ $$T^{6} + 108 T^{4} + 3440 T^{2} + \cdots + 33856$$
$19$ $$(T^{3} + 8 T^{2} - 16 T - 160)^{2}$$
$23$ $$T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 16384$$
$29$ $$(T^{3} - 10 T^{2} + 12 T + 40)^{2}$$
$31$ $$(T^{3} - 8 T^{2} - 32 T + 128)^{2}$$
$37$ $$(T^{2} + 4)^{3}$$
$41$ $$(T^{3} + 14 T^{2} + 44 T + 8)^{2}$$
$43$ $$T^{6} + 176 T^{4} + 9600 T^{2} + \cdots + 160000$$
$47$ $$T^{6} + 128 T^{4} + 3072 T^{2} + \cdots + 16384$$
$53$ $$T^{6} + 140 T^{4} + 2608 T^{2} + \cdots + 64$$
$59$ $$(T^{3} + 12 T^{2} - 16 T - 320)^{2}$$
$61$ $$(T^{3} + 6 T^{2} - 52 T - 248)^{2}$$
$67$ $$T^{6} + 112 T^{4} + 2816 T^{2} + \cdots + 4096$$
$71$ $$(T^{3} - 8 T^{2} - 32 T + 128)^{2}$$
$73$ $$T^{6} + 188 T^{4} + 9648 T^{2} + \cdots + 118336$$
$79$ $$(T^{3} + 12 T^{2} - 64 T - 800)^{2}$$
$83$ $$T^{6} + 240 T^{4} + 14400 T^{2} + \cdots + 256$$
$89$ $$(T^{3} - 10 T^{2} - 52 T + 200)^{2}$$
$97$ $$T^{6} + 268 T^{4} + 11312 T^{2} + \cdots + 64$$