# Properties

 Label 3332.1.o.g Level $3332$ Weight $1$ Character orbit 3332.o Analytic conductor $1.663$ Analytic rank $0$ Dimension $16$ Projective image $D_{10}$ CM discriminant -119 Inner twists $16$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(67,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4, 3]))

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

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

chi := DirichletCharacter("3332.67");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.o (of order $$6$$, degree $$2$$, 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: $$1.66288462209$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{60})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1$$ x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{10}$$ Projective field: Galois closure of 10.0.205346735104.4

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{60}^{4} q^{2} + ( - \zeta_{60}^{29} - \zeta_{60}^{11}) q^{3} + \zeta_{60}^{8} q^{4} + ( - \zeta_{60}^{13} + \zeta_{60}^{7}) q^{5} + ( - \zeta_{60}^{15} + \zeta_{60}^{3}) q^{6} + \zeta_{60}^{12} q^{8} + ( - \zeta_{60}^{28} + \zeta_{60}^{22} - \zeta_{60}^{10}) q^{9}+O(q^{10})$$ q + z^4 * q^2 + (-z^29 - z^11) * q^3 + z^8 * q^4 + (-z^13 + z^7) * q^5 + (-z^15 + z^3) * q^6 + z^12 * q^8 + (-z^28 + z^22 - z^10) * q^9 $$q + \zeta_{60}^{4} q^{2} + ( - \zeta_{60}^{29} - \zeta_{60}^{11}) q^{3} + \zeta_{60}^{8} q^{4} + ( - \zeta_{60}^{13} + \zeta_{60}^{7}) q^{5} + ( - \zeta_{60}^{15} + \zeta_{60}^{3}) q^{6} + \zeta_{60}^{12} q^{8} + ( - \zeta_{60}^{28} + \zeta_{60}^{22} - \zeta_{60}^{10}) q^{9} + ( - \zeta_{60}^{17} + \zeta_{60}^{11}) q^{10} + ( - \zeta_{60}^{19} + \zeta_{60}^{7}) q^{12} + (\zeta_{60}^{24} - \zeta_{60}^{18} - \zeta_{60}^{12} + \zeta_{60}^{6}) q^{15} + \zeta_{60}^{16} q^{16} - \zeta_{60}^{5} q^{17} + (\zeta_{60}^{26} - \zeta_{60}^{14} + \zeta_{60}^{2}) q^{18} + ( - \zeta_{60}^{21} + \zeta_{60}^{15}) q^{20} + ( - \zeta_{60}^{23} + \zeta_{60}^{11}) q^{24} + (\zeta_{60}^{26} - \zeta_{60}^{20} + \zeta_{60}^{14}) q^{25} + ( - \zeta_{60}^{27} + \zeta_{60}^{21} + \zeta_{60}^{9} + \zeta_{60}^{3}) q^{27} + (\zeta_{60}^{28} - \zeta_{60}^{22} - \zeta_{60}^{16} + \zeta_{60}^{10}) q^{30} + ( - \zeta_{60}^{23} - \zeta_{60}^{17}) q^{31} + \zeta_{60}^{20} q^{32} - \zeta_{60}^{9} q^{34} + ( - \zeta_{60}^{18} + \zeta_{60}^{6} - 1) q^{36} + ( - \zeta_{60}^{25} + \zeta_{60}^{19}) q^{40} + (\zeta_{60}^{21} + \zeta_{60}^{9}) q^{41} + ( - \zeta_{60}^{24} - \zeta_{60}^{6}) q^{43} + (\zeta_{60}^{29} + \zeta_{60}^{23} - \zeta_{60}^{17} - \zeta_{60}^{11} + 2 \zeta_{60}^{5}) q^{45} + ( - \zeta_{60}^{27} + \zeta_{60}^{15}) q^{48} + ( - \zeta_{60}^{24} + \zeta_{60}^{18} - 1) q^{50} + (\zeta_{60}^{16} - \zeta_{60}^{4}) q^{51} + (\zeta_{60}^{8} - \zeta_{60}^{2}) q^{53} + (\zeta_{60}^{25} - \zeta_{60}^{13} + \zeta_{60}^{7} + \zeta_{60}) q^{54} + ( - \zeta_{60}^{26} - \zeta_{60}^{20} + \zeta_{60}^{14} - \zeta_{60}^{2}) q^{60} + (\zeta_{60}^{19} - \zeta_{60}) q^{61} + ( - \zeta_{60}^{27} - \zeta_{60}^{21}) q^{62} + \zeta_{60}^{24} q^{64} + ( - \zeta_{60}^{8} - \zeta_{60}^{2}) q^{67} - \zeta_{60}^{13} q^{68} + ( - \zeta_{60}^{22} + \zeta_{60}^{10} - \zeta_{60}^{4}) q^{72} + (\zeta_{60}^{29} - \zeta_{60}^{11}) q^{73} + (\zeta_{60}^{25} - \zeta_{60}^{19} + \zeta_{60}^{13} + \zeta_{60}^{7} - \zeta_{60}) q^{75} + ( - \zeta_{60}^{29} + \zeta_{60}^{23}) q^{80} + ( - \zeta_{60}^{26} + \zeta_{60}^{20} - \zeta_{60}^{14} + \zeta_{60}^{8} - \zeta_{60}^{2}) q^{81} + (\zeta_{60}^{25} + \zeta_{60}^{13}) q^{82} + (\zeta_{60}^{18} - \zeta_{60}^{12}) q^{85} + ( - \zeta_{60}^{28} - \zeta_{60}^{10}) q^{86} + (\zeta_{60}^{27} - \zeta_{60}^{21} - \zeta_{60}^{15} + \zeta_{60}^{9} - \zeta_{60}^{3}) q^{90} + (\zeta_{60}^{28} - \zeta_{60}^{22} - \zeta_{60}^{16} - \zeta_{60}^{4}) q^{93} + (\zeta_{60}^{19} + \zeta_{60}) q^{96} + (\zeta_{60}^{27} + \zeta_{60}^{3}) q^{97} +O(q^{100})$$ q + z^4 * q^2 + (-z^29 - z^11) * q^3 + z^8 * q^4 + (-z^13 + z^7) * q^5 + (-z^15 + z^3) * q^6 + z^12 * q^8 + (-z^28 + z^22 - z^10) * q^9 + (-z^17 + z^11) * q^10 + (-z^19 + z^7) * q^12 + (z^24 - z^18 - z^12 + z^6) * q^15 + z^16 * q^16 - z^5 * q^17 + (z^26 - z^14 + z^2) * q^18 + (-z^21 + z^15) * q^20 + (-z^23 + z^11) * q^24 + (z^26 - z^20 + z^14) * q^25 + (-z^27 + z^21 + z^9 + z^3) * q^27 + (z^28 - z^22 - z^16 + z^10) * q^30 + (-z^23 - z^17) * q^31 + z^20 * q^32 - z^9 * q^34 + (-z^18 + z^6 - 1) * q^36 + (-z^25 + z^19) * q^40 + (z^21 + z^9) * q^41 + (-z^24 - z^6) * q^43 + (z^29 + z^23 - z^17 - z^11 + 2*z^5) * q^45 + (-z^27 + z^15) * q^48 + (-z^24 + z^18 - 1) * q^50 + (z^16 - z^4) * q^51 + (z^8 - z^2) * q^53 + (z^25 - z^13 + z^7 + z) * q^54 + (-z^26 - z^20 + z^14 - z^2) * q^60 + (z^19 - z) * q^61 + (-z^27 - z^21) * q^62 + z^24 * q^64 + (-z^8 - z^2) * q^67 - z^13 * q^68 + (-z^22 + z^10 - z^4) * q^72 + (z^29 - z^11) * q^73 + (z^25 - z^19 + z^13 + z^7 - z) * q^75 + (-z^29 + z^23) * q^80 + (-z^26 + z^20 - z^14 + z^8 - z^2) * q^81 + (z^25 + z^13) * q^82 + (z^18 - z^12) * q^85 + (-z^28 - z^10) * q^86 + (z^27 - z^21 - z^15 + z^9 - z^3) * q^90 + (z^28 - z^22 - z^16 - z^4) * q^93 + (z^19 + z) * q^96 + (z^27 + z^3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 2 q^{2} + 2 q^{4} - 4 q^{8} - 12 q^{9}+O(q^{10})$$ 16 * q + 2 * q^2 + 2 * q^4 - 4 * q^8 - 12 * q^9 $$16 q + 2 q^{2} + 2 q^{4} - 4 q^{8} - 12 q^{9} + 2 q^{16} - 2 q^{18} + 4 q^{25} + 10 q^{30} - 8 q^{32} - 16 q^{36} - 8 q^{50} + 4 q^{53} + 10 q^{60} - 4 q^{64} + 8 q^{72} - 8 q^{81} + 8 q^{85} - 10 q^{86}+O(q^{100})$$ 16 * q + 2 * q^2 + 2 * q^4 - 4 * q^8 - 12 * q^9 + 2 * q^16 - 2 * q^18 + 4 * q^25 + 10 * q^30 - 8 * q^32 - 16 * q^36 - 8 * q^50 + 4 * q^53 + 10 * q^60 - 4 * q^64 + 8 * q^72 - 8 * q^81 + 8 * q^85 - 10 * q^86

## Character values

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

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{60}^{10}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −0.743145 − 0.669131i 0.743145 + 0.669131i 0.406737 − 0.913545i −0.406737 + 0.913545i −0.207912 − 0.978148i 0.207912 + 0.978148i −0.994522 − 0.104528i 0.994522 + 0.104528i −0.743145 + 0.669131i 0.743145 − 0.669131i 0.406737 + 0.913545i −0.406737 − 0.913545i −0.207912 + 0.978148i 0.207912 − 0.978148i −0.994522 + 0.104528i 0.994522 − 0.104528i
−0.978148 + 0.207912i −0.951057 + 1.64728i 0.913545 0.406737i −1.40126 + 0.809017i 0.587785 1.80902i 0 −0.809017 + 0.587785i −1.30902 2.26728i 1.20243 1.08268i
67.2 −0.978148 + 0.207912i 0.951057 1.64728i 0.913545 0.406737i 1.40126 0.809017i −0.587785 + 1.80902i 0 −0.809017 + 0.587785i −1.30902 2.26728i −1.20243 + 1.08268i
67.3 −0.104528 + 0.994522i −0.587785 + 1.01807i −0.978148 0.207912i 0.535233 0.309017i −0.951057 0.690983i 0 0.309017 0.951057i −0.190983 0.330792i 0.251377 + 0.564602i
67.4 −0.104528 + 0.994522i 0.587785 1.01807i −0.978148 0.207912i −0.535233 + 0.309017i 0.951057 + 0.690983i 0 0.309017 0.951057i −0.190983 0.330792i −0.251377 0.564602i
67.5 0.669131 0.743145i −0.951057 + 1.64728i −0.104528 0.994522i 1.40126 0.809017i 0.587785 + 1.80902i 0 −0.809017 0.587785i −1.30902 2.26728i 0.336408 1.58268i
67.6 0.669131 0.743145i 0.951057 1.64728i −0.104528 0.994522i −1.40126 + 0.809017i −0.587785 1.80902i 0 −0.809017 0.587785i −1.30902 2.26728i −0.336408 + 1.58268i
67.7 0.913545 + 0.406737i −0.587785 + 1.01807i 0.669131 + 0.743145i −0.535233 + 0.309017i −0.951057 + 0.690983i 0 0.309017 + 0.951057i −0.190983 0.330792i −0.614648 + 0.0646021i
67.8 0.913545 + 0.406737i 0.587785 1.01807i 0.669131 + 0.743145i 0.535233 0.309017i 0.951057 0.690983i 0 0.309017 + 0.951057i −0.190983 0.330792i 0.614648 0.0646021i
2039.1 −0.978148 0.207912i −0.951057 1.64728i 0.913545 + 0.406737i −1.40126 0.809017i 0.587785 + 1.80902i 0 −0.809017 0.587785i −1.30902 + 2.26728i 1.20243 + 1.08268i
2039.2 −0.978148 0.207912i 0.951057 + 1.64728i 0.913545 + 0.406737i 1.40126 + 0.809017i −0.587785 1.80902i 0 −0.809017 0.587785i −1.30902 + 2.26728i −1.20243 1.08268i
2039.3 −0.104528 0.994522i −0.587785 1.01807i −0.978148 + 0.207912i 0.535233 + 0.309017i −0.951057 + 0.690983i 0 0.309017 + 0.951057i −0.190983 + 0.330792i 0.251377 0.564602i
2039.4 −0.104528 0.994522i 0.587785 + 1.01807i −0.978148 + 0.207912i −0.535233 0.309017i 0.951057 0.690983i 0 0.309017 + 0.951057i −0.190983 + 0.330792i −0.251377 + 0.564602i
2039.5 0.669131 + 0.743145i −0.951057 1.64728i −0.104528 + 0.994522i 1.40126 + 0.809017i 0.587785 1.80902i 0 −0.809017 + 0.587785i −1.30902 + 2.26728i 0.336408 + 1.58268i
2039.6 0.669131 + 0.743145i 0.951057 + 1.64728i −0.104528 + 0.994522i −1.40126 0.809017i −0.587785 + 1.80902i 0 −0.809017 + 0.587785i −1.30902 + 2.26728i −0.336408 1.58268i
2039.7 0.913545 0.406737i −0.587785 1.01807i 0.669131 0.743145i −0.535233 0.309017i −0.951057 0.690983i 0 0.309017 0.951057i −0.190983 + 0.330792i −0.614648 0.0646021i
2039.8 0.913545 0.406737i 0.587785 + 1.01807i 0.669131 0.743145i 0.535233 + 0.309017i 0.951057 + 0.690983i 0 0.309017 0.951057i −0.190983 + 0.330792i 0.614648 + 0.0646021i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 67.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
119.d odd 2 1 CM by $$\Q(\sqrt{-119})$$
4.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
17.b even 2 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
68.d odd 2 1 inner
119.h odd 6 1 inner
119.j even 6 1 inner
476.e even 2 1 inner
476.o odd 6 1 inner
476.q even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.o.g 16
4.b odd 2 1 inner 3332.1.o.g 16
7.b odd 2 1 inner 3332.1.o.g 16
7.c even 3 1 3332.1.g.i 8
7.c even 3 1 inner 3332.1.o.g 16
7.d odd 6 1 3332.1.g.i 8
7.d odd 6 1 inner 3332.1.o.g 16
17.b even 2 1 inner 3332.1.o.g 16
28.d even 2 1 inner 3332.1.o.g 16
28.f even 6 1 3332.1.g.i 8
28.f even 6 1 inner 3332.1.o.g 16
28.g odd 6 1 3332.1.g.i 8
28.g odd 6 1 inner 3332.1.o.g 16
68.d odd 2 1 inner 3332.1.o.g 16
119.d odd 2 1 CM 3332.1.o.g 16
119.h odd 6 1 3332.1.g.i 8
119.h odd 6 1 inner 3332.1.o.g 16
119.j even 6 1 3332.1.g.i 8
119.j even 6 1 inner 3332.1.o.g 16
476.e even 2 1 inner 3332.1.o.g 16
476.o odd 6 1 3332.1.g.i 8
476.o odd 6 1 inner 3332.1.o.g 16
476.q even 6 1 3332.1.g.i 8
476.q even 6 1 inner 3332.1.o.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.g.i 8 7.c even 3 1
3332.1.g.i 8 7.d odd 6 1
3332.1.g.i 8 28.f even 6 1
3332.1.g.i 8 28.g odd 6 1
3332.1.g.i 8 119.h odd 6 1
3332.1.g.i 8 119.j even 6 1
3332.1.g.i 8 476.o odd 6 1
3332.1.g.i 8 476.q even 6 1
3332.1.o.g 16 1.a even 1 1 trivial
3332.1.o.g 16 4.b odd 2 1 inner
3332.1.o.g 16 7.b odd 2 1 inner
3332.1.o.g 16 7.c even 3 1 inner
3332.1.o.g 16 7.d odd 6 1 inner
3332.1.o.g 16 17.b even 2 1 inner
3332.1.o.g 16 28.d even 2 1 inner
3332.1.o.g 16 28.f even 6 1 inner
3332.1.o.g 16 28.g odd 6 1 inner
3332.1.o.g 16 68.d odd 2 1 inner
3332.1.o.g 16 119.d odd 2 1 CM
3332.1.o.g 16 119.h odd 6 1 inner
3332.1.o.g 16 119.j even 6 1 inner
3332.1.o.g 16 476.e even 2 1 inner
3332.1.o.g 16 476.o odd 6 1 inner
3332.1.o.g 16 476.q even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$:

 $$T_{3}^{8} + 5T_{3}^{6} + 20T_{3}^{4} + 25T_{3}^{2} + 25$$ T3^8 + 5*T3^6 + 20*T3^4 + 25*T3^2 + 25 $$T_{5}^{8} - 3T_{5}^{6} + 8T_{5}^{4} - 3T_{5}^{2} + 1$$ T5^8 - 3*T5^6 + 8*T5^4 - 3*T5^2 + 1 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1)^{2}$$
$3$ $$(T^{8} + 5 T^{6} + 20 T^{4} + 25 T^{2} + \cdots + 25)^{2}$$
$5$ $$(T^{8} - 3 T^{6} + 8 T^{4} - 3 T^{2} + 1)^{2}$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$T^{16}$$
$17$ $$(T^{4} - T^{2} + 1)^{4}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$(T^{8} + 5 T^{6} + 20 T^{4} + 25 T^{2} + \cdots + 25)^{2}$$
$37$ $$T^{16}$$
$41$ $$(T^{4} + 3 T^{2} + 1)^{4}$$
$43$ $$(T^{4} + 5 T^{2} + 5)^{4}$$
$47$ $$T^{16}$$
$53$ $$(T^{4} - T^{3} + 2 T^{2} + T + 1)^{4}$$
$59$ $$T^{16}$$
$61$ $$(T^{8} - 3 T^{6} + 8 T^{4} - 3 T^{2} + 1)^{2}$$
$67$ $$(T^{8} - 5 T^{6} + 20 T^{4} - 25 T^{2} + \cdots + 25)^{2}$$
$71$ $$T^{16}$$
$73$ $$(T^{8} - 3 T^{6} + 8 T^{4} - 3 T^{2} + 1)^{2}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$T^{16}$$
$97$ $$(T^{4} + 3 T^{2} + 1)^{4}$$