# Properties

 Label 3800.1.cv.f Level $3800$ Weight $1$ Character orbit 3800.cv Analytic conductor $1.896$ Analytic rank $0$ Dimension $12$ Projective image $D_{9}$ CM discriminant -40 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(251,3800)]

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, 0, 2]))

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

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

chi := DirichletCharacter("3800.251");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3800.cv (of order $$18$$, degree $$6$$, 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 760) Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.43477921384960000.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{36} q^{2} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{17} - \zeta_{36}^{13}) q^{7} - \zeta_{36}^{3} q^{8} - \zeta_{36}^{8} q^{9} +O(q^{10})$$ q - z * q^2 + z^2 * q^4 + (-z^17 - z^13) * q^7 - z^3 * q^8 - z^8 * q^9 $$q - \zeta_{36} q^{2} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{17} - \zeta_{36}^{13}) q^{7} - \zeta_{36}^{3} q^{8} - \zeta_{36}^{8} q^{9} + (\zeta_{36}^{4} - \zeta_{36}^{2}) q^{11} - \zeta_{36}^{5} q^{13} + (\zeta_{36}^{14} - 1) q^{14} + \zeta_{36}^{4} q^{16} + \zeta_{36}^{9} q^{18} + \zeta_{36}^{10} q^{19} + ( - \zeta_{36}^{5} + \zeta_{36}^{3}) q^{22} + ( - \zeta_{36}^{15} - \zeta_{36}^{7}) q^{23} + \zeta_{36}^{6} q^{26} + ( - \zeta_{36}^{15} + \zeta_{36}) q^{28} - \zeta_{36}^{5} q^{32} - \zeta_{36}^{10} q^{36} + ( - \zeta_{36}^{13} - \zeta_{36}^{5}) q^{37} - \zeta_{36}^{11} q^{38} + ( - \zeta_{36}^{14} + \zeta_{36}^{12}) q^{41} + (\zeta_{36}^{6} - \zeta_{36}^{4}) q^{44} + (\zeta_{36}^{16} + \zeta_{36}^{8}) q^{46} + \zeta_{36}^{17} q^{47} + ( - \zeta_{36}^{16} + \cdots - \zeta_{36}^{8}) q^{49} + \cdots + ( - \zeta_{36}^{12} + \zeta_{36}^{10}) q^{99} +O(q^{100})$$ q - z * q^2 + z^2 * q^4 + (-z^17 - z^13) * q^7 - z^3 * q^8 - z^8 * q^9 + (z^4 - z^2) * q^11 - z^5 * q^13 + (z^14 - 1) * q^14 + z^4 * q^16 + z^9 * q^18 + z^10 * q^19 + (-z^5 + z^3) * q^22 + (-z^15 - z^7) * q^23 + z^6 * q^26 + (-z^15 + z) * q^28 - z^5 * q^32 - z^10 * q^36 + (-z^13 - z^5) * q^37 - z^11 * q^38 + (-z^14 + z^12) * q^41 + (z^6 - z^4) * q^44 + (z^16 + z^8) * q^46 + z^17 * q^47 + (-z^16 - z^12 - z^8) * q^49 - z^7 * q^52 + (z^13 + z^9) * q^53 + (z^16 - z^2) * q^56 - z^10 * q^59 + (-z^7 - z^3) * q^63 + z^6 * q^64 + z^11 * q^72 + (z^14 + z^6) * q^74 + z^12 * q^76 + (-z^17 + z^15 + z^3 - z) * q^77 + z^16 * q^81 + (z^15 - z^13) * q^82 + (-z^7 + z^5) * q^88 + (z^10 - 1) * q^89 + (-z^4 - 1) * q^91 + (-z^17 - z^9) * q^92 + q^94 + (z^17 + z^13 + z^9) * q^98 + (-z^12 + z^10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q+O(q^{10})$$ 12 * q $$12 q - 12 q^{14} + 6 q^{26} - 6 q^{41} + 6 q^{44} + 6 q^{49} + 6 q^{64} + 6 q^{74} - 6 q^{76} - 12 q^{89} - 12 q^{91} + 12 q^{94} + 6 q^{99}+O(q^{100})$$ 12 * q - 12 * q^14 + 6 * q^26 - 6 * q^41 + 6 * q^44 + 6 * q^49 + 6 * q^64 + 6 * q^74 - 6 * q^76 - 12 * q^89 - 12 * q^91 + 12 * q^94 + 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 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 0 −0.766044 0.642788i 0 0 1.32683 + 0.766044i 0.866025 0.500000i 0.939693 0.342020i 0
251.2 0.342020 0.939693i 0 −0.766044 0.642788i 0 0 −1.32683 0.766044i −0.866025 + 0.500000i 0.939693 0.342020i 0
651.1 −0.342020 0.939693i 0 −0.766044 + 0.642788i 0 0 1.32683 0.766044i 0.866025 + 0.500000i 0.939693 + 0.342020i 0
651.2 0.342020 + 0.939693i 0 −0.766044 + 0.642788i 0 0 −1.32683 + 0.766044i −0.866025 0.500000i 0.939693 + 0.342020i 0
1051.1 −0.642788 0.766044i 0 −0.173648 + 0.984808i 0 0 0.300767 + 0.173648i 0.866025 0.500000i −0.766044 0.642788i 0
1051.2 0.642788 + 0.766044i 0 −0.173648 + 0.984808i 0 0 −0.300767 0.173648i −0.866025 + 0.500000i −0.766044 0.642788i 0
1251.1 −0.642788 + 0.766044i 0 −0.173648 0.984808i 0 0 0.300767 0.173648i 0.866025 + 0.500000i −0.766044 + 0.642788i 0
1251.2 0.642788 0.766044i 0 −0.173648 0.984808i 0 0 −0.300767 + 0.173648i −0.866025 0.500000i −0.766044 + 0.642788i 0
1651.1 −0.984808 0.173648i 0 0.939693 + 0.342020i 0 0 1.62760 0.939693i −0.866025 0.500000i −0.173648 0.984808i 0
1651.2 0.984808 + 0.173648i 0 0.939693 + 0.342020i 0 0 −1.62760 + 0.939693i 0.866025 + 0.500000i −0.173648 0.984808i 0
2251.1 −0.984808 + 0.173648i 0 0.939693 0.342020i 0 0 1.62760 + 0.939693i −0.866025 + 0.500000i −0.173648 + 0.984808i 0
2251.2 0.984808 0.173648i 0 0.939693 0.342020i 0 0 −1.62760 0.939693i 0.866025 0.500000i −0.173648 + 0.984808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
5.b even 2 1 inner
8.d odd 2 1 inner
19.e even 9 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.cv.f 12
5.b even 2 1 inner 3800.1.cv.f 12
5.c odd 4 1 760.1.bz.a 6
5.c odd 4 1 760.1.bz.b yes 6
8.d odd 2 1 inner 3800.1.cv.f 12
19.e even 9 1 inner 3800.1.cv.f 12
20.e even 4 1 3040.1.dv.a 6
20.e even 4 1 3040.1.dv.b 6
40.e odd 2 1 CM 3800.1.cv.f 12
40.i odd 4 1 3040.1.dv.a 6
40.i odd 4 1 3040.1.dv.b 6
40.k even 4 1 760.1.bz.a 6
40.k even 4 1 760.1.bz.b yes 6
95.p even 18 1 inner 3800.1.cv.f 12
95.q odd 36 1 760.1.bz.a 6
95.q odd 36 1 760.1.bz.b yes 6
152.u odd 18 1 inner 3800.1.cv.f 12
380.bj even 36 1 3040.1.dv.a 6
380.bj even 36 1 3040.1.dv.b 6
760.bz odd 18 1 inner 3800.1.cv.f 12
760.cp even 36 1 760.1.bz.a 6
760.cp even 36 1 760.1.bz.b yes 6
760.cq odd 36 1 3040.1.dv.a 6
760.cq odd 36 1 3040.1.dv.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.1.bz.a 6 5.c odd 4 1
760.1.bz.a 6 40.k even 4 1
760.1.bz.a 6 95.q odd 36 1
760.1.bz.a 6 760.cp even 36 1
760.1.bz.b yes 6 5.c odd 4 1
760.1.bz.b yes 6 40.k even 4 1
760.1.bz.b yes 6 95.q odd 36 1
760.1.bz.b yes 6 760.cp even 36 1
3040.1.dv.a 6 20.e even 4 1
3040.1.dv.a 6 40.i odd 4 1
3040.1.dv.a 6 380.bj even 36 1
3040.1.dv.a 6 760.cq odd 36 1
3040.1.dv.b 6 20.e even 4 1
3040.1.dv.b 6 40.i odd 4 1
3040.1.dv.b 6 380.bj even 36 1
3040.1.dv.b 6 760.cq odd 36 1
3800.1.cv.f 12 1.a even 1 1 trivial
3800.1.cv.f 12 5.b even 2 1 inner
3800.1.cv.f 12 8.d odd 2 1 inner
3800.1.cv.f 12 19.e even 9 1 inner
3800.1.cv.f 12 40.e odd 2 1 CM
3800.1.cv.f 12 95.p even 18 1 inner
3800.1.cv.f 12 152.u odd 18 1 inner
3800.1.cv.f 12 760.bz odd 18 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{1}^{\mathrm{new}}(3800, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - T^{6} + 1$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$T^{12} - 6 T^{10} + \cdots + 1$$
$11$ $$(T^{6} + 3 T^{4} + 2 T^{3} + \cdots + 1)^{2}$$
$13$ $$T^{12} - T^{6} + 1$$
$17$ $$T^{12}$$
$19$ $$(T^{6} - T^{3} + 1)^{2}$$
$23$ $$T^{12} - 3 T^{10} + \cdots + 1$$
$29$ $$T^{12}$$
$31$ $$T^{12}$$
$37$ $$(T^{6} + 6 T^{4} + 9 T^{2} + 1)^{2}$$
$41$ $$(T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 1)^{2}$$
$43$ $$T^{12}$$
$47$ $$T^{12} - T^{6} + 1$$
$53$ $$T^{12} + 6 T^{10} + \cdots + 1$$
$59$ $$(T^{6} + T^{3} + 1)^{2}$$
$61$ $$T^{12}$$
$67$ $$T^{12}$$
$71$ $$T^{12}$$
$73$ $$T^{12}$$
$79$ $$T^{12}$$
$83$ $$T^{12}$$
$89$ $$(T^{6} + 6 T^{5} + 15 T^{4} + \cdots + 1)^{2}$$
$97$ $$T^{12}$$