# Properties

 Label 3332.1.be.b Level $3332$ Weight $1$ Character orbit 3332.be Analytic conductor $1.663$ Analytic rank $0$ Dimension $6$ Projective image $D_{7}$ CM discriminant -68 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(407,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([7, 10, 7]))

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

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

chi := DirichletCharacter("3332.407");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.be (of order $$14$$, 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: $$1.66288462209$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{7}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{7} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{14}^{4} q^{2} + ( - \zeta_{14}^{3} + 1) q^{3} - \zeta_{14} q^{4} + (\zeta_{14}^{4} + 1) q^{6} - \zeta_{14}^{5} q^{7} - \zeta_{14}^{5} q^{8} + (\zeta_{14}^{6} - \zeta_{14}^{3} + 1) q^{9} +O(q^{10})$$ q + z^4 * q^2 + (-z^3 + 1) * q^3 - z * q^4 + (z^4 + 1) * q^6 - z^5 * q^7 - z^5 * q^8 + (z^6 - z^3 + 1) * q^9 $$q + \zeta_{14}^{4} q^{2} + ( - \zeta_{14}^{3} + 1) q^{3} - \zeta_{14} q^{4} + (\zeta_{14}^{4} + 1) q^{6} - \zeta_{14}^{5} q^{7} - \zeta_{14}^{5} q^{8} + (\zeta_{14}^{6} - \zeta_{14}^{3} + 1) q^{9} + ( - \zeta_{14} + 1) q^{11} + (\zeta_{14}^{4} - \zeta_{14}) q^{12} + ( - \zeta_{14} + 1) q^{13} + \zeta_{14}^{2} q^{14} + \zeta_{14}^{2} q^{16} + \zeta_{14}^{6} q^{17} + (\zeta_{14}^{4} - \zeta_{14}^{3} + 1) q^{18} + ( - \zeta_{14}^{5} - \zeta_{14}) q^{21} + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{22} + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{23} + ( - \zeta_{14}^{5} - \zeta_{14}) q^{24} - \zeta_{14}^{3} q^{25} + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{26} + (\zeta_{14}^{6} - \zeta_{14}^{3} + \cdots + 1) q^{27} + \cdots + (\zeta_{14}^{6} + \zeta_{14}^{4} + \cdots + 2) q^{99} +O(q^{100})$$ q + z^4 * q^2 + (-z^3 + 1) * q^3 - z * q^4 + (z^4 + 1) * q^6 - z^5 * q^7 - z^5 * q^8 + (z^6 - z^3 + 1) * q^9 + (-z + 1) * q^11 + (z^4 - z) * q^12 + (-z + 1) * q^13 + z^2 * q^14 + z^2 * q^16 + z^6 * q^17 + (z^4 - z^3 + 1) * q^18 + (-z^5 - z) * q^21 + (-z^5 + z^4) * q^22 + (-z^5 + z^4) * q^23 + (-z^5 - z) * q^24 - z^3 * q^25 + (-z^5 + z^4) * q^26 + (z^6 - z^3 + z^2 + 1) * q^27 + z^6 * q^28 + (-z^5 + z^2) * q^31 + z^6 * q^32 + (z^4 - z^3 - z + 1) * q^33 - z^3 * q^34 + (z^4 - z + 1) * q^36 + (z^4 - z^3 - z + 1) * q^39 + (-z^5 + z^2) * q^42 + (z^2 - z) * q^44 + (z^2 - z) * q^46 + (-z^5 + z^2) * q^48 - z^3 * q^49 + q^50 + (z^6 + z^2) * q^51 + (z^2 - z) * q^52 + (z^6 - z^3) * q^53 + (z^6 + z^4 - z^3 + 1) * q^54 - z^3 * q^56 + (z^6 + z^2) * q^62 + (-z^5 + z^4 - z) * q^63 - z^3 * q^64 + (-z^5 + z^4 - z + 1) * q^66 + q^68 + (-z^5 + z^4 - z + 1) * q^69 + (z^6 - z^3) * q^71 + (-z^5 + z^4 - z) * q^72 + (z^6 - z^3) * q^75 + (z^6 - z^5) * q^77 + (-z^5 + z^4 - z + 1) * q^78 + (z^4 - z^3) * q^79 + (z^6 - z^5 - z^3 + z^2 + 1) * q^81 + (z^6 + z^2) * q^84 + (z^6 - z^5) * q^88 + (-z^5 - z) * q^89 + (z^6 - z^5) * q^91 + (z^6 - z^5) * q^92 + (-2*z^5 + z^2 - z) * q^93 + (z^6 + z^2) * q^96 + q^98 + (z^6 + z^4 - z^3 - z + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{2} + 5 q^{3} - q^{4} + 5 q^{6} - q^{7} - q^{8} + 4 q^{9}+O(q^{10})$$ 6 * q - q^2 + 5 * q^3 - q^4 + 5 * q^6 - q^7 - q^8 + 4 * q^9 $$6 q - q^{2} + 5 q^{3} - q^{4} + 5 q^{6} - q^{7} - q^{8} + 4 q^{9} + 5 q^{11} - 2 q^{12} + 5 q^{13} - q^{14} - q^{16} - q^{17} + 4 q^{18} - 2 q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} - q^{25} - 2 q^{26} + 3 q^{27} - q^{28} - 2 q^{31} - q^{32} + 3 q^{33} - q^{34} + 4 q^{36} + 3 q^{39} - 2 q^{42} - 2 q^{44} - 2 q^{46} - 2 q^{48} - q^{49} + 6 q^{50} - 2 q^{51} - 2 q^{52} - 2 q^{53} + 3 q^{54} - q^{56} - 2 q^{62} - 3 q^{63} - q^{64} + 3 q^{66} + 6 q^{68} + 3 q^{69} - 2 q^{71} - 3 q^{72} - 2 q^{75} - 2 q^{77} + 3 q^{78} - 2 q^{79} + 2 q^{81} - 2 q^{84} - 2 q^{88} - 2 q^{89} - 2 q^{91} - 2 q^{92} - 4 q^{93} - 2 q^{96} + 6 q^{98} + 8 q^{99}+O(q^{100})$$ 6 * q - q^2 + 5 * q^3 - q^4 + 5 * q^6 - q^7 - q^8 + 4 * q^9 + 5 * q^11 - 2 * q^12 + 5 * q^13 - q^14 - q^16 - q^17 + 4 * q^18 - 2 * q^21 - 2 * q^22 - 2 * q^23 - 2 * q^24 - q^25 - 2 * q^26 + 3 * q^27 - q^28 - 2 * q^31 - q^32 + 3 * q^33 - q^34 + 4 * q^36 + 3 * q^39 - 2 * q^42 - 2 * q^44 - 2 * q^46 - 2 * q^48 - q^49 + 6 * q^50 - 2 * q^51 - 2 * q^52 - 2 * q^53 + 3 * q^54 - q^56 - 2 * q^62 - 3 * q^63 - q^64 + 3 * q^66 + 6 * q^68 + 3 * q^69 - 2 * q^71 - 3 * q^72 - 2 * q^75 - 2 * q^77 + 3 * q^78 - 2 * q^79 + 2 * q^81 - 2 * q^84 - 2 * q^88 - 2 * q^89 - 2 * q^91 - 2 * q^92 - 4 * q^93 - 2 * q^96 + 6 * q^98 + 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{14}^{3}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
407.1
 0.900969 + 0.433884i 0.900969 − 0.433884i −0.623490 + 0.781831i 0.222521 − 0.974928i 0.222521 + 0.974928i −0.623490 − 0.781831i
−0.222521 + 0.974928i 0.777479 0.974928i −0.900969 0.433884i 0 0.777479 + 0.974928i 0.623490 0.781831i 0.623490 0.781831i −0.123490 0.541044i 0
1359.1 −0.222521 0.974928i 0.777479 + 0.974928i −0.900969 + 0.433884i 0 0.777479 0.974928i 0.623490 + 0.781831i 0.623490 + 0.781831i −0.123490 + 0.541044i 0
1835.1 −0.900969 + 0.433884i 0.0990311 0.433884i 0.623490 0.781831i 0 0.0990311 + 0.433884i −0.222521 + 0.974928i −0.222521 + 0.974928i 0.722521 + 0.347948i 0
2311.1 0.623490 + 0.781831i 1.62349 0.781831i −0.222521 + 0.974928i 0 1.62349 + 0.781831i −0.900969 + 0.433884i −0.900969 + 0.433884i 1.40097 1.75676i 0
2787.1 0.623490 0.781831i 1.62349 + 0.781831i −0.222521 0.974928i 0 1.62349 0.781831i −0.900969 0.433884i −0.900969 0.433884i 1.40097 + 1.75676i 0
3263.1 −0.900969 0.433884i 0.0990311 + 0.433884i 0.623490 + 0.781831i 0 0.0990311 0.433884i −0.222521 0.974928i −0.222521 0.974928i 0.722521 0.347948i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 407.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$
49.e even 7 1 inner
3332.be odd 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.be.b yes 6
4.b odd 2 1 3332.1.be.a 6
17.b even 2 1 3332.1.be.a 6
49.e even 7 1 inner 3332.1.be.b yes 6
68.d odd 2 1 CM 3332.1.be.b yes 6
196.k odd 14 1 3332.1.be.a 6
833.r even 14 1 3332.1.be.a 6
3332.be odd 14 1 inner 3332.1.be.b yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.be.a 6 4.b odd 2 1
3332.1.be.a 6 17.b even 2 1
3332.1.be.a 6 196.k odd 14 1
3332.1.be.a 6 833.r even 14 1
3332.1.be.b yes 6 1.a even 1 1 trivial
3332.1.be.b yes 6 49.e even 7 1 inner
3332.1.be.b yes 6 68.d odd 2 1 CM
3332.1.be.b yes 6 3332.be odd 14 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 5T_{3}^{5} + 11T_{3}^{4} - 13T_{3}^{3} + 9T_{3}^{2} - 3T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$.

## Hecke characteristic polynomials

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