# Properties

 Label 726.2.e.k Level $726$ Weight $2$ Character orbit 726.e Analytic conductor $5.797$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [726,2,Mod(487,726)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(726, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 8]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("726.487");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$726 = 2 \cdot 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 726.e (of order $$5$$, degree $$4$$, 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: $$5.79713918674$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 66) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} + \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} + \zeta_{10} q^{6} - 2 \zeta_{10}^{3} q^{7} - \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10})$$ q + (-z^3 + z^2 - z + 1) * q^2 + z^2 * q^3 - z^3 * q^4 + z * q^6 - 2*z^3 * q^7 - z^2 * q^8 + (z^3 - z^2 + z - 1) * q^9 $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} + \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} + \zeta_{10} q^{6} - 2 \zeta_{10}^{3} q^{7} - \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} + q^{12} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{13} - 2 \zeta_{10}^{2} q^{14} - \zeta_{10} q^{16} + 6 \zeta_{10} q^{17} + \zeta_{10}^{3} q^{18} - 4 \zeta_{10}^{2} q^{19} + 2 q^{21} + 6 q^{23} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{24} - 5 \zeta_{10}^{2} q^{25} - 4 \zeta_{10}^{3} q^{26} - \zeta_{10} q^{27} - 2 \zeta_{10} q^{28} - 6 \zeta_{10}^{3} q^{29} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 8) q^{31} - q^{32} + 6 q^{34} + \zeta_{10}^{2} q^{36} + 10 \zeta_{10}^{3} q^{37} - 4 \zeta_{10} q^{38} + 4 \zeta_{10} q^{39} + 6 \zeta_{10}^{2} q^{41} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{42} + 8 q^{43} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{46} - 6 \zeta_{10}^{2} q^{47} - \zeta_{10}^{3} q^{48} + 3 \zeta_{10} q^{49} - 5 \zeta_{10} q^{50} + 6 \zeta_{10}^{3} q^{51} - 4 \zeta_{10}^{2} q^{52} - q^{54} - 2 q^{56} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{57} - 6 \zeta_{10}^{2} q^{58} - 8 \zeta_{10} q^{61} + 8 \zeta_{10}^{3} q^{62} + 2 \zeta_{10}^{2} q^{63} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{64} - 4 q^{67} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{68} + 6 \zeta_{10}^{2} q^{69} - 6 \zeta_{10} q^{71} + \zeta_{10} q^{72} - 2 \zeta_{10}^{3} q^{73} + 10 \zeta_{10}^{2} q^{74} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{75} - 4 q^{76} + 4 q^{78} + (14 \zeta_{10}^{3} - 14 \zeta_{10}^{2} + 14 \zeta_{10} - 14) q^{79} - \zeta_{10}^{3} q^{81} + 6 \zeta_{10} q^{82} + 12 \zeta_{10} q^{83} - 2 \zeta_{10}^{3} q^{84} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{86} + 6 q^{87} - 6 q^{89} - 8 \zeta_{10}^{2} q^{91} - 6 \zeta_{10}^{3} q^{92} - 8 \zeta_{10} q^{93} - 6 \zeta_{10} q^{94} - \zeta_{10}^{2} q^{96} + (14 \zeta_{10}^{3} - 14 \zeta_{10}^{2} + 14 \zeta_{10} - 14) q^{97} + 3 q^{98} +O(q^{100})$$ q + (-z^3 + z^2 - z + 1) * q^2 + z^2 * q^3 - z^3 * q^4 + z * q^6 - 2*z^3 * q^7 - z^2 * q^8 + (z^3 - z^2 + z - 1) * q^9 + q^12 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^13 - 2*z^2 * q^14 - z * q^16 + 6*z * q^17 + z^3 * q^18 - 4*z^2 * q^19 + 2 * q^21 + 6 * q^23 + (-z^3 + z^2 - z + 1) * q^24 - 5*z^2 * q^25 - 4*z^3 * q^26 - z * q^27 - 2*z * q^28 - 6*z^3 * q^29 + (8*z^3 - 8*z^2 + 8*z - 8) * q^31 - q^32 + 6 * q^34 + z^2 * q^36 + 10*z^3 * q^37 - 4*z * q^38 + 4*z * q^39 + 6*z^2 * q^41 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^42 + 8 * q^43 + (-6*z^3 + 6*z^2 - 6*z + 6) * q^46 - 6*z^2 * q^47 - z^3 * q^48 + 3*z * q^49 - 5*z * q^50 + 6*z^3 * q^51 - 4*z^2 * q^52 - q^54 - 2 * q^56 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^57 - 6*z^2 * q^58 - 8*z * q^61 + 8*z^3 * q^62 + 2*z^2 * q^63 + (z^3 - z^2 + z - 1) * q^64 - 4 * q^67 + (-6*z^3 + 6*z^2 - 6*z + 6) * q^68 + 6*z^2 * q^69 - 6*z * q^71 + z * q^72 - 2*z^3 * q^73 + 10*z^2 * q^74 + (-5*z^3 + 5*z^2 - 5*z + 5) * q^75 - 4 * q^76 + 4 * q^78 + (14*z^3 - 14*z^2 + 14*z - 14) * q^79 - z^3 * q^81 + 6*z * q^82 + 12*z * q^83 - 2*z^3 * q^84 + (-8*z^3 + 8*z^2 - 8*z + 8) * q^86 + 6 * q^87 - 6 * q^89 - 8*z^2 * q^91 - 6*z^3 * q^92 - 8*z * q^93 - 6*z * q^94 - z^2 * q^96 + (14*z^3 - 14*z^2 + 14*z - 14) * q^97 + 3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{7} + q^{8} - q^{9}+O(q^{10})$$ 4 * q + q^2 - q^3 - q^4 + q^6 - 2 * q^7 + q^8 - q^9 $$4 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{7} + q^{8} - q^{9} + 4 q^{12} + 4 q^{13} + 2 q^{14} - q^{16} + 6 q^{17} + q^{18} + 4 q^{19} + 8 q^{21} + 24 q^{23} + q^{24} + 5 q^{25} - 4 q^{26} - q^{27} - 2 q^{28} - 6 q^{29} - 8 q^{31} - 4 q^{32} + 24 q^{34} - q^{36} + 10 q^{37} - 4 q^{38} + 4 q^{39} - 6 q^{41} + 2 q^{42} + 32 q^{43} + 6 q^{46} + 6 q^{47} - q^{48} + 3 q^{49} - 5 q^{50} + 6 q^{51} + 4 q^{52} - 4 q^{54} - 8 q^{56} + 4 q^{57} + 6 q^{58} - 8 q^{61} + 8 q^{62} - 2 q^{63} - q^{64} - 16 q^{67} + 6 q^{68} - 6 q^{69} - 6 q^{71} + q^{72} - 2 q^{73} - 10 q^{74} + 5 q^{75} - 16 q^{76} + 16 q^{78} - 14 q^{79} - q^{81} + 6 q^{82} + 12 q^{83} - 2 q^{84} + 8 q^{86} + 24 q^{87} - 24 q^{89} + 8 q^{91} - 6 q^{92} - 8 q^{93} - 6 q^{94} + q^{96} - 14 q^{97} + 12 q^{98}+O(q^{100})$$ 4 * q + q^2 - q^3 - q^4 + q^6 - 2 * q^7 + q^8 - q^9 + 4 * q^12 + 4 * q^13 + 2 * q^14 - q^16 + 6 * q^17 + q^18 + 4 * q^19 + 8 * q^21 + 24 * q^23 + q^24 + 5 * q^25 - 4 * q^26 - q^27 - 2 * q^28 - 6 * q^29 - 8 * q^31 - 4 * q^32 + 24 * q^34 - q^36 + 10 * q^37 - 4 * q^38 + 4 * q^39 - 6 * q^41 + 2 * q^42 + 32 * q^43 + 6 * q^46 + 6 * q^47 - q^48 + 3 * q^49 - 5 * q^50 + 6 * q^51 + 4 * q^52 - 4 * q^54 - 8 * q^56 + 4 * q^57 + 6 * q^58 - 8 * q^61 + 8 * q^62 - 2 * q^63 - q^64 - 16 * q^67 + 6 * q^68 - 6 * q^69 - 6 * q^71 + q^72 - 2 * q^73 - 10 * q^74 + 5 * q^75 - 16 * q^76 + 16 * q^78 - 14 * q^79 - q^81 + 6 * q^82 + 12 * q^83 - 2 * q^84 + 8 * q^86 + 24 * q^87 - 24 * q^89 + 8 * q^91 - 6 * q^92 - 8 * q^93 - 6 * q^94 + q^96 - 14 * q^97 + 12 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/726\mathbb{Z}\right)^\times$$.

 $$n$$ $$485$$ $$607$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
487.1
 0.809017 + 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i
0.809017 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i 0 0.809017 + 0.587785i 0.618034 1.90211i −0.309017 0.951057i −0.809017 + 0.587785i 0
493.1 −0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0 −0.309017 0.951057i −1.61803 1.17557i 0.809017 0.587785i 0.309017 0.951057i 0
511.1 −0.309017 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0 −0.309017 + 0.951057i −1.61803 + 1.17557i 0.809017 + 0.587785i 0.309017 + 0.951057i 0
565.1 0.809017 + 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i 0 0.809017 0.587785i 0.618034 + 1.90211i −0.309017 + 0.951057i −0.809017 0.587785i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.2.e.k 4
11.b odd 2 1 726.2.e.b 4
11.c even 5 1 66.2.a.a 1
11.c even 5 3 inner 726.2.e.k 4
11.d odd 10 1 726.2.a.i 1
11.d odd 10 3 726.2.e.b 4
33.f even 10 1 2178.2.a.b 1
33.h odd 10 1 198.2.a.e 1
44.g even 10 1 5808.2.a.l 1
44.h odd 10 1 528.2.a.d 1
55.j even 10 1 1650.2.a.m 1
55.k odd 20 2 1650.2.c.d 2
77.j odd 10 1 3234.2.a.d 1
88.l odd 10 1 2112.2.a.v 1
88.o even 10 1 2112.2.a.i 1
99.m even 15 2 1782.2.e.s 2
99.n odd 30 2 1782.2.e.f 2
132.o even 10 1 1584.2.a.h 1
165.o odd 10 1 4950.2.a.g 1
165.v even 20 2 4950.2.c.r 2
231.u even 10 1 9702.2.a.bu 1
264.t odd 10 1 6336.2.a.bj 1
264.w even 10 1 6336.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.a 1 11.c even 5 1
198.2.a.e 1 33.h odd 10 1
528.2.a.d 1 44.h odd 10 1
726.2.a.i 1 11.d odd 10 1
726.2.e.b 4 11.b odd 2 1
726.2.e.b 4 11.d odd 10 3
726.2.e.k 4 1.a even 1 1 trivial
726.2.e.k 4 11.c even 5 3 inner
1584.2.a.h 1 132.o even 10 1
1650.2.a.m 1 55.j even 10 1
1650.2.c.d 2 55.k odd 20 2
1782.2.e.f 2 99.n odd 30 2
1782.2.e.s 2 99.m even 15 2
2112.2.a.i 1 88.o even 10 1
2112.2.a.v 1 88.l odd 10 1
2178.2.a.b 1 33.f even 10 1
3234.2.a.d 1 77.j odd 10 1
4950.2.a.g 1 165.o odd 10 1
4950.2.c.r 2 165.v even 20 2
5808.2.a.l 1 44.g even 10 1
6336.2.a.bf 1 264.w even 10 1
6336.2.a.bj 1 264.t odd 10 1
9702.2.a.bu 1 231.u even 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(726, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{4} + 2T_{7}^{3} + 4T_{7}^{2} + 8T_{7} + 16$$ T7^4 + 2*T7^3 + 4*T7^2 + 8*T7 + 16 $$T_{13}^{4} - 4T_{13}^{3} + 16T_{13}^{2} - 64T_{13} + 256$$ T13^4 - 4*T13^3 + 16*T13^2 - 64*T13 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$3$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256$$
$17$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$19$ $$T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256$$
$23$ $$(T - 6)^{4}$$
$29$ $$T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$31$ $$T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$37$ $$T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000$$
$41$ $$T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$43$ $$(T - 8)^{4}$$
$47$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$67$ $$(T + 4)^{4}$$
$71$ $$T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$73$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$79$ $$T^{4} + 14 T^{3} + 196 T^{2} + \cdots + 38416$$
$83$ $$T^{4} - 12 T^{3} + 144 T^{2} + \cdots + 20736$$
$89$ $$(T + 6)^{4}$$
$97$ $$T^{4} + 14 T^{3} + 196 T^{2} + \cdots + 38416$$