# Properties

 Label 3800.1.cq.b Level $3800$ Weight $1$ Character orbit 3800.cq Analytic conductor $1.896$ Analytic rank $0$ Dimension $12$ Projective image $D_{9}$ CM discriminant -8 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(99,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 9, 2]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3800.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3800.cq (of order $$18$$, degree $$6$$, 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: $$1.89644704801$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{18})$$ Coefficient field: $$\Q(\zeta_{36})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{6} + 1$$ x^12 - x^6 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.69564674215936.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{36} q^{2} + ( - \zeta_{36}^{5} + \zeta_{36}^{3}) q^{3} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{6} + \zeta_{36}^{4}) q^{6} + \zeta_{36}^{3} q^{8} + (\zeta_{36}^{10} + \cdots + \zeta_{36}^{6}) q^{9}+O(q^{10})$$ q + z * q^2 + (-z^5 + z^3) * q^3 + z^2 * q^4 + (-z^6 + z^4) * q^6 + z^3 * q^8 + (z^10 - z^8 + z^6) * q^9 $$q + \zeta_{36} q^{2} + ( - \zeta_{36}^{5} + \zeta_{36}^{3}) q^{3} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{6} + \zeta_{36}^{4}) q^{6} + \zeta_{36}^{3} q^{8} + (\zeta_{36}^{10} + \cdots + \zeta_{36}^{6}) q^{9}+ \cdots + ( - \zeta_{36}^{16} + \zeta_{36}^{14} + \cdots - 1) q^{99}+O(q^{100})$$ q + z * q^2 + (-z^5 + z^3) * q^3 + z^2 * q^4 + (-z^6 + z^4) * q^6 + z^3 * q^8 + (z^10 - z^8 + z^6) * q^9 + (z^16 + z^8) * q^11 + (-z^7 + z^5) * q^12 + z^4 * q^16 - z * q^17 + (z^11 - z^9 + z^7) * q^18 + z^2 * q^19 + (z^17 + z^9) * q^22 + (-z^8 + z^6) * q^24 + (-z^15 + z^13 - z^11 + z^9) * q^27 + z^5 * q^32 + (-z^13 + z^11 + z^3 - z) * q^33 - z^2 * q^34 + (z^12 - z^10 + z^8) * q^36 + z^3 * q^38 + (-z^6 - z^2) * q^41 - z^7 * q^43 + (z^10 - 1) * q^44 + (-z^9 + z^7) * q^48 - z^12 * q^49 + (z^6 - z^4) * q^51 + (-z^16 + z^14 - z^12 + z^10) * q^54 + (-z^7 + z^5) * q^57 + (z^14 + z^6) * q^59 + z^6 * q^64 + (-z^14 + z^12 + z^4 - z^2) * q^66 + (-z^15 + z) * q^67 - z^3 * q^68 + (z^13 - z^11 + z^9) * q^72 + (-z^17 - z^9) * q^73 + z^4 * q^76 + (z^16 - z^14 + z^12 - z^2 - 1) * q^81 + (-z^7 - z^3) * q^82 + (-z^17 - z^13) * q^83 - z^8 * q^86 + (z^11 - z) * q^88 - z^14 * q^89 + (-z^10 + z^8) * q^96 + (z^17 - z^3) * q^97 - z^13 * q^98 + (-z^16 + z^14 - z^8 + z^6 - z^4 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 6 q^{6} + 6 q^{9}+O(q^{10})$$ 12 * q - 6 * q^6 + 6 * q^9 $$12 q - 6 q^{6} + 6 q^{9} + 6 q^{24} - 6 q^{36} - 6 q^{41} - 12 q^{44} + 6 q^{49} + 6 q^{51} + 6 q^{54} + 6 q^{59} + 6 q^{64} - 6 q^{66} + 6 q^{81} - 6 q^{99}+O(q^{100})$$ 12 * q - 6 * q^6 + 6 * q^9 + 6 * q^24 - 6 * q^36 - 6 * q^41 - 12 * q^44 + 6 * q^49 + 6 * q^51 + 6 * q^54 + 6 * q^59 + 6 * q^64 - 6 * q^66 + 6 * q^81 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$-\zeta_{36}^{2}$$ $$-1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 −0.342020 + 0.939693i 0.342020 − 0.939693i −0.342020 − 0.939693i 0.342020 + 0.939693i −0.642788 − 0.766044i 0.642788 + 0.766044i −0.642788 + 0.766044i 0.642788 − 0.766044i −0.984808 − 0.173648i 0.984808 + 0.173648i −0.984808 + 0.173648i 0.984808 − 0.173648i
−0.342020 + 0.939693i 1.85083 0.326352i −0.766044 0.642788i 0 −0.326352 + 1.85083i 0 0.866025 0.500000i 2.37939 0.866025i 0
99.2 0.342020 0.939693i −1.85083 + 0.326352i −0.766044 0.642788i 0 −0.326352 + 1.85083i 0 −0.866025 + 0.500000i 2.37939 0.866025i 0
499.1 −0.342020 0.939693i 1.85083 + 0.326352i −0.766044 + 0.642788i 0 −0.326352 1.85083i 0 0.866025 + 0.500000i 2.37939 + 0.866025i 0
499.2 0.342020 + 0.939693i −1.85083 0.326352i −0.766044 + 0.642788i 0 −0.326352 1.85083i 0 −0.866025 0.500000i 2.37939 + 0.866025i 0
899.1 −0.642788 0.766044i 0.524005 1.43969i −0.173648 + 0.984808i 0 −1.43969 + 0.524005i 0 0.866025 0.500000i −1.03209 0.866025i 0
899.2 0.642788 + 0.766044i −0.524005 + 1.43969i −0.173648 + 0.984808i 0 −1.43969 + 0.524005i 0 −0.866025 + 0.500000i −1.03209 0.866025i 0
1099.1 −0.642788 + 0.766044i 0.524005 + 1.43969i −0.173648 0.984808i 0 −1.43969 0.524005i 0 0.866025 + 0.500000i −1.03209 + 0.866025i 0
1099.2 0.642788 0.766044i −0.524005 1.43969i −0.173648 0.984808i 0 −1.43969 0.524005i 0 −0.866025 0.500000i −1.03209 + 0.866025i 0
1499.1 −0.984808 0.173648i −0.223238 + 0.266044i 0.939693 + 0.342020i 0 0.266044 0.223238i 0 −0.866025 0.500000i 0.152704 + 0.866025i 0
1499.2 0.984808 + 0.173648i 0.223238 0.266044i 0.939693 + 0.342020i 0 0.266044 0.223238i 0 0.866025 + 0.500000i 0.152704 + 0.866025i 0
2099.1 −0.984808 + 0.173648i −0.223238 0.266044i 0.939693 0.342020i 0 0.266044 + 0.223238i 0 −0.866025 + 0.500000i 0.152704 0.866025i 0
2099.2 0.984808 0.173648i 0.223238 + 0.266044i 0.939693 0.342020i 0 0.266044 + 0.223238i 0 0.866025 0.500000i 0.152704 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 99.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
5.b even 2 1 inner
19.e even 9 1 inner
40.e odd 2 1 inner
95.p even 18 1 inner
152.u odd 18 1 inner
760.bz odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.cq.b 12
5.b even 2 1 inner 3800.1.cq.b 12
5.c odd 4 1 152.1.u.a 6
5.c odd 4 1 3800.1.cv.c 6
8.d odd 2 1 CM 3800.1.cq.b 12
15.e even 4 1 1368.1.eh.a 6
19.e even 9 1 inner 3800.1.cq.b 12
20.e even 4 1 608.1.bg.a 6
40.e odd 2 1 inner 3800.1.cq.b 12
40.i odd 4 1 608.1.bg.a 6
40.k even 4 1 152.1.u.a 6
40.k even 4 1 3800.1.cv.c 6
95.g even 4 1 2888.1.u.e 6
95.l even 12 1 2888.1.u.a 6
95.l even 12 1 2888.1.u.f 6
95.m odd 12 1 2888.1.u.b 6
95.m odd 12 1 2888.1.u.g 6
95.p even 18 1 inner 3800.1.cq.b 12
95.q odd 36 1 152.1.u.a 6
95.q odd 36 1 2888.1.f.d 3
95.q odd 36 2 2888.1.k.b 6
95.q odd 36 1 2888.1.u.b 6
95.q odd 36 1 2888.1.u.g 6
95.q odd 36 1 3800.1.cv.c 6
95.r even 36 1 2888.1.f.c 3
95.r even 36 2 2888.1.k.c 6
95.r even 36 1 2888.1.u.a 6
95.r even 36 1 2888.1.u.e 6
95.r even 36 1 2888.1.u.f 6
120.q odd 4 1 1368.1.eh.a 6
152.u odd 18 1 inner 3800.1.cq.b 12
285.bi even 36 1 1368.1.eh.a 6
380.bj even 36 1 608.1.bg.a 6
760.y odd 4 1 2888.1.u.e 6
760.bu odd 12 1 2888.1.u.a 6
760.bu odd 12 1 2888.1.u.f 6
760.bw even 12 1 2888.1.u.b 6
760.bw even 12 1 2888.1.u.g 6
760.bz odd 18 1 inner 3800.1.cq.b 12
760.cn odd 36 1 2888.1.f.c 3
760.cn odd 36 2 2888.1.k.c 6
760.cn odd 36 1 2888.1.u.a 6
760.cn odd 36 1 2888.1.u.e 6
760.cn odd 36 1 2888.1.u.f 6
760.cp even 36 1 152.1.u.a 6
760.cp even 36 1 2888.1.f.d 3
760.cp even 36 2 2888.1.k.b 6
760.cp even 36 1 2888.1.u.b 6
760.cp even 36 1 2888.1.u.g 6
760.cp even 36 1 3800.1.cv.c 6
760.cq odd 36 1 608.1.bg.a 6
2280.fl odd 36 1 1368.1.eh.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a 6 5.c odd 4 1
152.1.u.a 6 40.k even 4 1
152.1.u.a 6 95.q odd 36 1
152.1.u.a 6 760.cp even 36 1
608.1.bg.a 6 20.e even 4 1
608.1.bg.a 6 40.i odd 4 1
608.1.bg.a 6 380.bj even 36 1
608.1.bg.a 6 760.cq odd 36 1
1368.1.eh.a 6 15.e even 4 1
1368.1.eh.a 6 120.q odd 4 1
1368.1.eh.a 6 285.bi even 36 1
1368.1.eh.a 6 2280.fl odd 36 1
2888.1.f.c 3 95.r even 36 1
2888.1.f.c 3 760.cn odd 36 1
2888.1.f.d 3 95.q odd 36 1
2888.1.f.d 3 760.cp even 36 1
2888.1.k.b 6 95.q odd 36 2
2888.1.k.b 6 760.cp even 36 2
2888.1.k.c 6 95.r even 36 2
2888.1.k.c 6 760.cn odd 36 2
2888.1.u.a 6 95.l even 12 1
2888.1.u.a 6 95.r even 36 1
2888.1.u.a 6 760.bu odd 12 1
2888.1.u.a 6 760.cn odd 36 1
2888.1.u.b 6 95.m odd 12 1
2888.1.u.b 6 95.q odd 36 1
2888.1.u.b 6 760.bw even 12 1
2888.1.u.b 6 760.cp even 36 1
2888.1.u.e 6 95.g even 4 1
2888.1.u.e 6 95.r even 36 1
2888.1.u.e 6 760.y odd 4 1
2888.1.u.e 6 760.cn odd 36 1
2888.1.u.f 6 95.l even 12 1
2888.1.u.f 6 95.r even 36 1
2888.1.u.f 6 760.bu odd 12 1
2888.1.u.f 6 760.cn odd 36 1
2888.1.u.g 6 95.m odd 12 1
2888.1.u.g 6 95.q odd 36 1
2888.1.u.g 6 760.bw even 12 1
2888.1.u.g 6 760.cp even 36 1
3800.1.cq.b 12 1.a even 1 1 trivial
3800.1.cq.b 12 5.b even 2 1 inner
3800.1.cq.b 12 8.d odd 2 1 CM
3800.1.cq.b 12 19.e even 9 1 inner
3800.1.cq.b 12 40.e odd 2 1 inner
3800.1.cq.b 12 95.p even 18 1 inner
3800.1.cq.b 12 152.u odd 18 1 inner
3800.1.cq.b 12 760.bz odd 18 1 inner
3800.1.cv.c 6 5.c odd 4 1
3800.1.cv.c 6 40.k even 4 1
3800.1.cv.c 6 95.q odd 36 1
3800.1.cv.c 6 760.cp even 36 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - 3T_{3}^{10} - 6T_{3}^{8} + 8T_{3}^{6} + 69T_{3}^{4} + 3T_{3}^{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3800, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - T^{6} + 1$$
$3$ $$T^{12} - 3 T^{10} + \cdots + 1$$
$5$ $$T^{12}$$
$7$ $$T^{12}$$
$11$ $$(T^{6} + 3 T^{4} + 2 T^{3} + \cdots + 1)^{2}$$
$13$ $$T^{12}$$
$17$ $$T^{12} - T^{6} + 1$$
$19$ $$(T^{6} - T^{3} + 1)^{2}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12}$$
$37$ $$T^{12}$$
$41$ $$(T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 1)^{2}$$
$43$ $$T^{12} - T^{6} + 1$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$(T^{6} - 3 T^{5} + 6 T^{4} + \cdots + 1)^{2}$$
$61$ $$T^{12}$$
$67$ $$T^{12} - 3 T^{10} + \cdots + 1$$
$71$ $$T^{12}$$
$73$ $$T^{12} + 6 T^{10} + \cdots + 1$$
$79$ $$T^{12}$$
$83$ $$T^{12} - 6 T^{10} + \cdots + 1$$
$89$ $$(T^{6} + T^{3} + 1)^{2}$$
$97$ $$T^{12} - 3 T^{10} + \cdots + 1$$