# Properties

 Label 121.2.c.c Level $121$ Weight $2$ Character orbit 121.c Analytic conductor $0.966$ Analytic rank $0$ Dimension $4$ CM discriminant -11 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 121.c (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.966189864457$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{5}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q -\zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} + 3 \zeta_{10} q^{5} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q -\zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} + 3 \zeta_{10} q^{5} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{9} + 2 q^{12} -3 \zeta_{10}^{3} q^{15} -4 \zeta_{10} q^{16} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{20} -9 q^{23} + 4 \zeta_{10}^{2} q^{25} -5 \zeta_{10} q^{27} + ( 5 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{31} + 4 \zeta_{10}^{2} q^{36} -7 \zeta_{10}^{3} q^{37} + 6 q^{45} -12 \zeta_{10}^{2} q^{47} + 4 \zeta_{10}^{3} q^{48} + 7 \zeta_{10} q^{49} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{53} + 15 \zeta_{10}^{3} q^{59} + 6 \zeta_{10} q^{60} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{64} + 13 q^{67} + 9 \zeta_{10}^{2} q^{69} + 3 \zeta_{10} q^{71} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{75} -12 \zeta_{10}^{2} q^{80} -\zeta_{10}^{3} q^{81} -9 q^{89} -18 \zeta_{10}^{3} q^{92} -5 \zeta_{10} q^{93} + ( -17 + 17 \zeta_{10} - 17 \zeta_{10}^{2} + 17 \zeta_{10}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{3} + 2q^{4} + 3q^{5} + 2q^{9} + O(q^{10})$$ $$4q + q^{3} + 2q^{4} + 3q^{5} + 2q^{9} + 8q^{12} - 3q^{15} - 4q^{16} - 6q^{20} - 36q^{23} - 4q^{25} - 5q^{27} + 5q^{31} - 4q^{36} - 7q^{37} + 24q^{45} + 12q^{47} + 4q^{48} + 7q^{49} - 6q^{53} + 15q^{59} + 6q^{60} + 8q^{64} + 52q^{67} - 9q^{69} + 3q^{71} + 4q^{75} + 12q^{80} - q^{81} - 36q^{89} - 18q^{92} - 5q^{93} - 17q^{97} + O(q^{100})$$

## 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.

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 −0.309017 0.951057i −0.618034 + 1.90211i 2.42705 + 1.76336i 0 0 0 1.61803 1.17557i 0
9.1 0 0.809017 0.587785i 1.61803 + 1.17557i −0.927051 2.85317i 0 0 0 −0.618034 + 1.90211i 0
27.1 0 0.809017 + 0.587785i 1.61803 1.17557i −0.927051 + 2.85317i 0 0 0 −0.618034 1.90211i 0
81.1 0 −0.309017 + 0.951057i −0.618034 1.90211i 2.42705 1.76336i 0 0 0 1.61803 + 1.17557i 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 121.2.c.c 4
11.b odd 2 1 CM 121.2.c.c 4
11.c even 5 1 121.2.a.b 1
11.c even 5 3 inner 121.2.c.c 4
11.d odd 10 1 121.2.a.b 1
11.d odd 10 3 inner 121.2.c.c 4
33.f even 10 1 1089.2.a.g 1
33.h odd 10 1 1089.2.a.g 1
44.g even 10 1 1936.2.a.h 1
44.h odd 10 1 1936.2.a.h 1
55.h odd 10 1 3025.2.a.d 1
55.j even 10 1 3025.2.a.d 1
77.j odd 10 1 5929.2.a.e 1
77.l even 10 1 5929.2.a.e 1
88.k even 10 1 7744.2.a.n 1
88.l odd 10 1 7744.2.a.n 1
88.o even 10 1 7744.2.a.bb 1
88.p odd 10 1 7744.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.2.a.b 1 11.c even 5 1
121.2.a.b 1 11.d odd 10 1
121.2.c.c 4 1.a even 1 1 trivial
121.2.c.c 4 11.b odd 2 1 CM
121.2.c.c 4 11.c even 5 3 inner
121.2.c.c 4 11.d odd 10 3 inner
1089.2.a.g 1 33.f even 10 1
1089.2.a.g 1 33.h odd 10 1
1936.2.a.h 1 44.g even 10 1
1936.2.a.h 1 44.h odd 10 1
3025.2.a.d 1 55.h odd 10 1
3025.2.a.d 1 55.j even 10 1
5929.2.a.e 1 77.j odd 10 1
5929.2.a.e 1 77.l even 10 1
7744.2.a.n 1 88.k even 10 1
7744.2.a.n 1 88.l odd 10 1
7744.2.a.bb 1 88.o even 10 1
7744.2.a.bb 1 88.p odd 10 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$5$ $$81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 9 + T )^{4}$$
$29$ $$T^{4}$$
$31$ $$625 - 125 T + 25 T^{2} - 5 T^{3} + T^{4}$$
$37$ $$2401 + 343 T + 49 T^{2} + 7 T^{3} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$20736 - 1728 T + 144 T^{2} - 12 T^{3} + T^{4}$$
$53$ $$1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4}$$
$59$ $$50625 - 3375 T + 225 T^{2} - 15 T^{3} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -13 + T )^{4}$$
$71$ $$81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 9 + T )^{4}$$
$97$ $$83521 + 4913 T + 289 T^{2} + 17 T^{3} + T^{4}$$