# Properties

 Label 825.2.c.f 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: 6.0.5161984.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$$ x^6 - 4*x^3 + 25*x^2 - 20*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} + \beta_{3}) q^{2} - \beta_{3} q^{3} + ( - \beta_1 - 3) q^{4} + ( - \beta_{2} + 1) q^{6} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{7} + (2 \beta_{5} - \beta_{4} - 3 \beta_{3}) q^{8} - q^{9}+O(q^{10})$$ q + (b4 + b3) * q^2 - b3 * q^3 + (-b1 - 3) * q^4 + (-b2 + 1) * q^6 + (b5 + b4 - b3) * q^7 + (2*b5 - b4 - 3*b3) * q^8 - q^9 $$q + (\beta_{4} + \beta_{3}) q^{2} - \beta_{3} q^{3} + ( - \beta_1 - 3) q^{4} + ( - \beta_{2} + 1) q^{6} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{7} + (2 \beta_{5} - \beta_{4} - 3 \beta_{3}) q^{8} - q^{9} + q^{11} + ( - \beta_{5} + 3 \beta_{3}) q^{12} + ( - \beta_{5} + 2 \beta_{3}) q^{13} + ( - 2 \beta_{2} + \beta_1 - 1) q^{14} + ( - 2 \beta_{2} + 3 \beta_1 + 5) q^{16} + (2 \beta_{5} + \beta_{4} + \beta_{3}) q^{17} + ( - \beta_{4} - \beta_{3}) q^{18} + ( - \beta_{2} + \beta_1 + 1) q^{19} + ( - \beta_{2} - \beta_1 - 1) q^{21} + (\beta_{4} + \beta_{3}) q^{22} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3}) q^{23} + (\beta_{2} - 2 \beta_1 - 3) q^{24} + (2 \beta_{2} - 2 \beta_1 - 4) q^{26} + \beta_{3} q^{27} + ( - 2 \beta_{5} - \beta_{4} + 7 \beta_{3}) q^{28} - 2 \beta_{2} q^{29} + (\beta_1 + 6) q^{31} + ( - 4 \beta_{5} + \beta_{4} + 13 \beta_{3}) q^{32} - \beta_{3} q^{33} + (3 \beta_1 - 1) q^{34} + (\beta_1 + 3) q^{36} + ( - 3 \beta_{5} + \beta_{3}) q^{37} + ( - 3 \beta_{5} + 7 \beta_{3}) q^{38} + (\beta_1 + 2) q^{39} + (\beta_{2} + 1) q^{41} + (\beta_{5} - 2 \beta_{4} + \beta_{3}) q^{42} + (3 \beta_{5} - 2 \beta_{4} + 4 \beta_{3}) q^{43} + ( - \beta_1 - 3) q^{44} + ( - \beta_{2} - 4 \beta_1 - 11) q^{46} + (\beta_{5} + 3 \beta_{3}) q^{47} + (3 \beta_{5} - 2 \beta_{4} - 5 \beta_{3}) q^{48} - 2 \beta_{2} q^{49} + ( - \beta_{2} - 2 \beta_1 + 1) q^{51} + (4 \beta_{5} - 2 \beta_{4} - 12 \beta_{3}) q^{52} + (4 \beta_{5} + 4 \beta_{4} - 2 \beta_{3}) q^{53} + (\beta_{2} - 1) q^{54} + (4 \beta_{2} - \beta_1 - 9) q^{56} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{57} + ( - 2 \beta_{5} - 2 \beta_{4} + 8 \beta_{3}) q^{58} + (2 \beta_{2} - \beta_1 + 1) q^{59} + ( - 4 \beta_{2} - \beta_1) q^{61} + ( - 2 \beta_{5} + 6 \beta_{4} + 8 \beta_{3}) q^{62} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{63} + (8 \beta_{2} - 3 \beta_1 - 15) q^{64} + ( - \beta_{2} + 1) q^{66} + ( - \beta_{5} - 6 \beta_{4} - 4 \beta_{3}) q^{67} + ( - 2 \beta_{5} + \beta_{4} + 7 \beta_{3}) q^{68} + ( - 2 \beta_{2} + \beta_1 + 1) q^{69} + (\beta_1 + 9) q^{71} + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{3}) q^{72} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{73} + (\beta_{2} - 6 \beta_1 - 7) q^{74} + (5 \beta_{2} - 4 \beta_1 - 11) q^{76} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{77} + ( - 2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3}) q^{78} + ( - \beta_{2} + 2 \beta_1 - 1) q^{79} + q^{81} + (\beta_{5} + 2 \beta_{4} - 3 \beta_{3}) q^{82} + (6 \beta_{4} + 4 \beta_{3}) q^{83} + (\beta_{2} + 2 \beta_1 + 7) q^{84} + (6 \beta_{2} + 8 \beta_1 + 10) q^{86} - 2 \beta_{4} q^{87} + (2 \beta_{5} - \beta_{4} - 3 \beta_{3}) q^{88} + (4 \beta_{2} - 2) q^{89} + (\beta_1 + 6) q^{91} + (5 \beta_{5} - 8 \beta_{4} - 13 \beta_{3}) q^{92} + (\beta_{5} - 6 \beta_{3}) q^{93} + (3 \beta_{2} + 2 \beta_1 - 1) q^{94} + ( - \beta_{2} + 4 \beta_1 + 13) q^{96} + ( - 2 \beta_{5} - 6 \beta_{4} - 11 \beta_{3}) q^{97} + ( - 2 \beta_{5} - 2 \beta_{4} + 8 \beta_{3}) q^{98} - q^{99}+O(q^{100})$$ q + (b4 + b3) * q^2 - b3 * q^3 + (-b1 - 3) * q^4 + (-b2 + 1) * q^6 + (b5 + b4 - b3) * q^7 + (2*b5 - b4 - 3*b3) * q^8 - q^9 + q^11 + (-b5 + 3*b3) * q^12 + (-b5 + 2*b3) * q^13 + (-2*b2 + b1 - 1) * q^14 + (-2*b2 + 3*b1 + 5) * q^16 + (2*b5 + b4 + b3) * q^17 + (-b4 - b3) * q^18 + (-b2 + b1 + 1) * q^19 + (-b2 - b1 - 1) * q^21 + (b4 + b3) * q^22 + (-b5 + 2*b4 + b3) * q^23 + (b2 - 2*b1 - 3) * q^24 + (2*b2 - 2*b1 - 4) * q^26 + b3 * q^27 + (-2*b5 - b4 + 7*b3) * q^28 - 2*b2 * q^29 + (b1 + 6) * q^31 + (-4*b5 + b4 + 13*b3) * q^32 - b3 * q^33 + (3*b1 - 1) * q^34 + (b1 + 3) * q^36 + (-3*b5 + b3) * q^37 + (-3*b5 + 7*b3) * q^38 + (b1 + 2) * q^39 + (b2 + 1) * q^41 + (b5 - 2*b4 + b3) * q^42 + (3*b5 - 2*b4 + 4*b3) * q^43 + (-b1 - 3) * q^44 + (-b2 - 4*b1 - 11) * q^46 + (b5 + 3*b3) * q^47 + (3*b5 - 2*b4 - 5*b3) * q^48 - 2*b2 * q^49 + (-b2 - 2*b1 + 1) * q^51 + (4*b5 - 2*b4 - 12*b3) * q^52 + (4*b5 + 4*b4 - 2*b3) * q^53 + (b2 - 1) * q^54 + (4*b2 - b1 - 9) * q^56 + (b5 - b4 - b3) * q^57 + (-2*b5 - 2*b4 + 8*b3) * q^58 + (2*b2 - b1 + 1) * q^59 + (-4*b2 - b1) * q^61 + (-2*b5 + 6*b4 + 8*b3) * q^62 + (-b5 - b4 + b3) * q^63 + (8*b2 - 3*b1 - 15) * q^64 + (-b2 + 1) * q^66 + (-b5 - 6*b4 - 4*b3) * q^67 + (-2*b5 + b4 + 7*b3) * q^68 + (-2*b2 + b1 + 1) * q^69 + (b1 + 9) * q^71 + (-2*b5 + b4 + 3*b3) * q^72 + (-2*b5 + 2*b4 - 2*b3) * q^73 + (b2 - 6*b1 - 7) * q^74 + (5*b2 - 4*b1 - 11) * q^76 + (b5 + b4 - b3) * q^77 + (-2*b5 + 2*b4 + 4*b3) * q^78 + (-b2 + 2*b1 - 1) * q^79 + q^81 + (b5 + 2*b4 - 3*b3) * q^82 + (6*b4 + 4*b3) * q^83 + (b2 + 2*b1 + 7) * q^84 + (6*b2 + 8*b1 + 10) * q^86 - 2*b4 * q^87 + (2*b5 - b4 - 3*b3) * q^88 + (4*b2 - 2) * q^89 + (b1 + 6) * q^91 + (5*b5 - 8*b4 - 13*b3) * q^92 + (b5 - 6*b3) * q^93 + (3*b2 + 2*b1 - 1) * q^94 + (-b2 + 4*b1 + 13) * q^96 + (-2*b5 - 6*b4 - 11*b3) * q^97 + (-2*b5 - 2*b4 + 8*b3) * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 16 q^{4} + 4 q^{6} - 6 q^{9}+O(q^{10})$$ 6 * q - 16 * q^4 + 4 * q^6 - 6 * q^9 $$6 q - 16 q^{4} + 4 q^{6} - 6 q^{9} + 6 q^{11} - 12 q^{14} + 20 q^{16} + 2 q^{19} - 6 q^{21} - 12 q^{24} - 16 q^{26} - 4 q^{29} + 34 q^{31} - 12 q^{34} + 16 q^{36} + 10 q^{39} + 8 q^{41} - 16 q^{44} - 60 q^{46} - 4 q^{49} + 8 q^{51} - 4 q^{54} - 44 q^{56} + 12 q^{59} - 6 q^{61} - 68 q^{64} + 4 q^{66} + 52 q^{71} - 28 q^{74} - 48 q^{76} - 12 q^{79} + 6 q^{81} + 40 q^{84} + 56 q^{86} - 4 q^{89} + 34 q^{91} - 4 q^{94} + 68 q^{96} - 6 q^{99}+O(q^{100})$$ 6 * q - 16 * q^4 + 4 * q^6 - 6 * q^9 + 6 * q^11 - 12 * q^14 + 20 * q^16 + 2 * q^19 - 6 * q^21 - 12 * q^24 - 16 * q^26 - 4 * q^29 + 34 * q^31 - 12 * q^34 + 16 * q^36 + 10 * q^39 + 8 * q^41 - 16 * q^44 - 60 * q^46 - 4 * q^49 + 8 * q^51 - 4 * q^54 - 44 * q^56 + 12 * q^59 - 6 * q^61 - 68 * q^64 + 4 * q^66 + 52 * q^71 - 28 * q^74 - 48 * q^76 - 12 * q^79 + 6 * q^81 + 40 * q^84 + 56 * q^86 - 4 * q^89 + 34 * q^91 - 4 * q^94 + 68 * q^96 - 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{5} + 25\nu^{4} + 10\nu^{3} - 4\nu^{2} + 323 ) / 121$$ (2*v^5 + 25*v^4 + 10*v^3 - 4*v^2 + 323) / 121 $$\beta_{2}$$ $$=$$ $$( -7\nu^{5} - 27\nu^{4} - 35\nu^{3} + 14\nu^{2} - 223 ) / 121$$ (-7*v^5 - 27*v^4 - 35*v^3 + 14*v^2 - 223) / 121 $$\beta_{3}$$ $$=$$ $$( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242$$ (-25*v^5 - 10*v^4 - 4*v^3 + 50*v^2 - 605*v + 258) / 242 $$\beta_{4}$$ $$=$$ $$( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242$$ (-65*v^5 - 26*v^4 + 38*v^3 + 372*v^2 - 1331*v + 574) / 242 $$\beta_{5}$$ $$=$$ $$( 75\nu^{5} + 30\nu^{4} + 12\nu^{3} - 392\nu^{2} + 1815\nu - 774 ) / 242$$ (75*v^5 + 30*v^4 + 12*v^3 - 392*v^2 + 1815*v - 774) / 242
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 ) / 2$$ (b5 + b4 + b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{5} - 3\beta_{3}$$ -b5 - 3*b3 $$\nu^{3}$$ $$=$$ $$( 5\beta_{5} + 5\beta_{4} + 4\beta_{3} - 5\beta_{2} - 5\beta _1 + 4 ) / 2$$ (5*b5 + 5*b4 + 4*b3 - 5*b2 - 5*b1 + 4) / 2 $$\nu^{4}$$ $$=$$ $$2\beta_{2} + 7\beta _1 - 15$$ 2*b2 + 7*b1 - 15 $$\nu^{5}$$ $$=$$ $$( -29\beta_{5} - 25\beta_{4} - 32\beta_{3} - 25\beta_{2} - 29\beta _1 + 32 ) / 2$$ (-29*b5 - 25*b4 - 32*b3 - 25*b2 - 29*b1 + 32) / 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
 0.432320 + 0.432320i 1.32001 − 1.32001i −1.75233 − 1.75233i −1.75233 + 1.75233i 1.32001 + 1.32001i 0.432320 − 0.432320i
2.76156i 1.00000i −5.62620 0 2.76156 1.86464i 10.0140i −1.00000 0
199.2 2.12489i 1.00000i −2.51514 0 −2.12489 3.64002i 1.09461i −1.00000 0
199.3 1.36333i 1.00000i 0.141336 0 1.36333 2.50466i 2.91934i −1.00000 0
199.4 1.36333i 1.00000i 0.141336 0 1.36333 2.50466i 2.91934i −1.00000 0
199.5 2.12489i 1.00000i −2.51514 0 −2.12489 3.64002i 1.09461i −1.00000 0
199.6 2.76156i 1.00000i −5.62620 0 2.76156 1.86464i 10.0140i −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.f 6
3.b odd 2 1 2475.2.c.q 6
5.b even 2 1 inner 825.2.c.f 6
5.c odd 4 1 825.2.a.i 3
5.c odd 4 1 825.2.a.m yes 3
15.d odd 2 1 2475.2.c.q 6
15.e even 4 1 2475.2.a.z 3
15.e even 4 1 2475.2.a.bd 3
55.e even 4 1 9075.2.a.cd 3
55.e even 4 1 9075.2.a.cj 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.i 3 5.c odd 4 1
825.2.a.m yes 3 5.c odd 4 1
825.2.c.f 6 1.a even 1 1 trivial
825.2.c.f 6 5.b even 2 1 inner
2475.2.a.z 3 15.e even 4 1
2475.2.a.bd 3 15.e even 4 1
2475.2.c.q 6 3.b odd 2 1
2475.2.c.q 6 15.d odd 2 1
9075.2.a.cd 3 55.e even 4 1
9075.2.a.cj 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 $$S_{2}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{6} + 14T_{2}^{4} + 57T_{2}^{2} + 64$$ T2^6 + 14*T2^4 + 57*T2^2 + 64 $$T_{7}^{6} + 23T_{7}^{4} + 151T_{7}^{2} + 289$$ T7^6 + 23*T7^4 + 151*T7^2 + 289

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 14 T^{4} + \cdots + 64$$
$3$ $$(T^{2} + 1)^{3}$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 23 T^{4} + \cdots + 289$$
$11$ $$(T - 1)^{6}$$
$13$ $$T^{6} + 25 T^{4} + \cdots + 64$$
$17$ $$T^{6} + 66 T^{4} + \cdots + 484$$
$19$ $$(T^{3} - T^{2} - 19 T - 25)^{2}$$
$23$ $$T^{6} + 86 T^{4} + \cdots + 3364$$
$29$ $$(T^{3} + 2 T^{2} - 24 T + 16)^{2}$$
$31$ $$(T^{3} - 17 T^{2} + \cdots - 136)^{2}$$
$37$ $$T^{6} + 150 T^{4} + \cdots + 1156$$
$41$ $$(T^{3} - 4 T^{2} - T + 2)^{2}$$
$43$ $$T^{6} + 353 T^{4} + \cdots + 1210000$$
$47$ $$T^{6} + 50 T^{4} + \cdots + 64$$
$53$ $$T^{6} + 332 T^{4} + \cdots + 678976$$
$59$ $$(T^{3} - 6 T^{2} + \cdots + 136)^{2}$$
$61$ $$(T^{3} + 3 T^{2} + \cdots + 244)^{2}$$
$67$ $$T^{6} + 433 T^{4} + \cdots + 2521744$$
$71$ $$(T^{3} - 26 T^{2} + \cdots - 580)^{2}$$
$73$ $$T^{6} + 188 T^{4} + \cdots + 222784$$
$79$ $$(T^{3} + 6 T^{2} + \cdots - 212)^{2}$$
$83$ $$T^{6} + 468 T^{4} + \cdots + 1763584$$
$89$ $$(T^{3} + 2 T^{2} + \cdots - 328)^{2}$$
$97$ $$T^{6} + 691 T^{4} + \cdots + 4635409$$