# Properties

 Label 121.4.c.a Level $121$ Weight $4$ Character orbit 121.c Analytic conductor $7.139$ Analytic rank $0$ Dimension $4$ CM discriminant -11 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,4,Mod(3,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(10))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("121.3");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 121.c (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: $$7.13923111069$$ 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 8 \zeta_{10}^{2} q^{3} + 8 \zeta_{10}^{3} q^{4} - 18 \zeta_{10} q^{5} + (37 \zeta_{10}^{3} - 37 \zeta_{10}^{2} + 37 \zeta_{10} - 37) q^{9} +O(q^{10})$$ q + 8*z^2 * q^3 + 8*z^3 * q^4 - 18*z * q^5 + (37*z^3 - 37*z^2 + 37*z - 37) * q^9 $$q + 8 \zeta_{10}^{2} q^{3} + 8 \zeta_{10}^{3} q^{4} - 18 \zeta_{10} q^{5} + (37 \zeta_{10}^{3} - 37 \zeta_{10}^{2} + 37 \zeta_{10} - 37) q^{9} - 64 q^{12} - 144 \zeta_{10}^{3} q^{15} - 64 \zeta_{10} q^{16} + ( - 144 \zeta_{10}^{3} + 144 \zeta_{10}^{2} - 144 \zeta_{10} + 144) q^{20} - 108 q^{23} + 199 \zeta_{10}^{2} q^{25} - 80 \zeta_{10} q^{27} + (340 \zeta_{10}^{3} - 340 \zeta_{10}^{2} + 340 \zeta_{10} - 340) q^{31} - 296 \zeta_{10}^{2} q^{36} + 434 \zeta_{10}^{3} q^{37} + 666 q^{45} - 36 \zeta_{10}^{2} q^{47} - 512 \zeta_{10}^{3} q^{48} + 343 \zeta_{10} q^{49} + ( - 738 \zeta_{10}^{3} + 738 \zeta_{10}^{2} - 738 \zeta_{10} + 738) q^{53} + 720 \zeta_{10}^{3} q^{59} + 1152 \zeta_{10} q^{60} + ( - 512 \zeta_{10}^{3} + 512 \zeta_{10}^{2} - 512 \zeta_{10} + 512) q^{64} - 416 q^{67} - 864 \zeta_{10}^{2} q^{69} - 612 \zeta_{10} q^{71} + (1592 \zeta_{10}^{3} - 1592 \zeta_{10}^{2} + 1592 \zeta_{10} - 1592) q^{75} + 1152 \zeta_{10}^{2} q^{80} + 359 \zeta_{10}^{3} q^{81} + 1674 q^{89} - 864 \zeta_{10}^{3} q^{92} - 2720 \zeta_{10} q^{93} + ( - 34 \zeta_{10}^{3} + 34 \zeta_{10}^{2} - 34 \zeta_{10} + 34) q^{97} +O(q^{100})$$ q + 8*z^2 * q^3 + 8*z^3 * q^4 - 18*z * q^5 + (37*z^3 - 37*z^2 + 37*z - 37) * q^9 - 64 * q^12 - 144*z^3 * q^15 - 64*z * q^16 + (-144*z^3 + 144*z^2 - 144*z + 144) * q^20 - 108 * q^23 + 199*z^2 * q^25 - 80*z * q^27 + (340*z^3 - 340*z^2 + 340*z - 340) * q^31 - 296*z^2 * q^36 + 434*z^3 * q^37 + 666 * q^45 - 36*z^2 * q^47 - 512*z^3 * q^48 + 343*z * q^49 + (-738*z^3 + 738*z^2 - 738*z + 738) * q^53 + 720*z^3 * q^59 + 1152*z * q^60 + (-512*z^3 + 512*z^2 - 512*z + 512) * q^64 - 416 * q^67 - 864*z^2 * q^69 - 612*z * q^71 + (1592*z^3 - 1592*z^2 + 1592*z - 1592) * q^75 + 1152*z^2 * q^80 + 359*z^3 * q^81 + 1674 * q^89 - 864*z^3 * q^92 - 2720*z * q^93 + (-34*z^3 + 34*z^2 - 34*z + 34) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{3} + 8 q^{4} - 18 q^{5} - 37 q^{9}+O(q^{10})$$ 4 * q - 8 * q^3 + 8 * q^4 - 18 * q^5 - 37 * q^9 $$4 q - 8 q^{3} + 8 q^{4} - 18 q^{5} - 37 q^{9} - 256 q^{12} - 144 q^{15} - 64 q^{16} + 144 q^{20} - 432 q^{23} - 199 q^{25} - 80 q^{27} - 340 q^{31} + 296 q^{36} + 434 q^{37} + 2664 q^{45} + 36 q^{47} - 512 q^{48} + 343 q^{49} + 738 q^{53} + 720 q^{59} + 1152 q^{60} + 512 q^{64} - 1664 q^{67} + 864 q^{69} - 612 q^{71} - 1592 q^{75} - 1152 q^{80} + 359 q^{81} + 6696 q^{89} - 864 q^{92} - 2720 q^{93} + 34 q^{97}+O(q^{100})$$ 4 * q - 8 * q^3 + 8 * q^4 - 18 * q^5 - 37 * q^9 - 256 * q^12 - 144 * q^15 - 64 * q^16 + 144 * q^20 - 432 * q^23 - 199 * q^25 - 80 * q^27 - 340 * q^31 + 296 * q^36 + 434 * q^37 + 2664 * q^45 + 36 * q^47 - 512 * q^48 + 343 * q^49 + 738 * q^53 + 720 * q^59 + 1152 * q^60 + 512 * q^64 - 1664 * q^67 + 864 * q^69 - 612 * q^71 - 1592 * q^75 - 1152 * q^80 + 359 * q^81 + 6696 * q^89 - 864 * q^92 - 2720 * q^93 + 34 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/121\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.809017 + 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i
0 2.47214 + 7.60845i −2.47214 + 7.60845i −14.5623 10.5801i 0 0 0 −29.9336 + 21.7481i 0
9.1 0 −6.47214 + 4.70228i 6.47214 + 4.70228i 5.56231 + 17.1190i 0 0 0 11.4336 35.1891i 0
27.1 0 −6.47214 4.70228i 6.47214 4.70228i 5.56231 17.1190i 0 0 0 11.4336 + 35.1891i 0
81.1 0 2.47214 7.60845i −2.47214 7.60845i −14.5623 + 10.5801i 0 0 0 −29.9336 21.7481i 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.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
11.c even 5 3 inner
11.d odd 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.4.c.a 4
11.b odd 2 1 CM 121.4.c.a 4
11.c even 5 1 121.4.a.a 1
11.c even 5 3 inner 121.4.c.a 4
11.d odd 10 1 121.4.a.a 1
11.d odd 10 3 inner 121.4.c.a 4
33.f even 10 1 1089.4.a.f 1
33.h odd 10 1 1089.4.a.f 1
44.g even 10 1 1936.4.a.a 1
44.h odd 10 1 1936.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.a.a 1 11.c even 5 1
121.4.a.a 1 11.d odd 10 1
121.4.c.a 4 1.a even 1 1 trivial
121.4.c.a 4 11.b odd 2 1 CM
121.4.c.a 4 11.c even 5 3 inner
121.4.c.a 4 11.d odd 10 3 inner
1089.4.a.f 1 33.f even 10 1
1089.4.a.f 1 33.h odd 10 1
1936.4.a.a 1 44.g even 10 1
1936.4.a.a 1 44.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{4}^{\mathrm{new}}(121, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$5$ $$T^{4} + 18 T^{3} + 324 T^{2} + \cdots + 104976$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$(T + 108)^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4} + 340 T^{3} + \cdots + 13363360000$$
$37$ $$T^{4} - 434 T^{3} + \cdots + 35477982736$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4} - 36 T^{3} + 1296 T^{2} + \cdots + 1679616$$
$53$ $$T^{4} - 738 T^{3} + \cdots + 296637086736$$
$59$ $$T^{4} - 720 T^{3} + \cdots + 268738560000$$
$61$ $$T^{4}$$
$67$ $$(T + 416)^{4}$$
$71$ $$T^{4} + 612 T^{3} + \cdots + 140283207936$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$(T - 1674)^{4}$$
$97$ $$T^{4} - 34 T^{3} + 1156 T^{2} + \cdots + 1336336$$