# Properties

 Label 121.2.c.b Level $121$ Weight $2$ Character orbit 121.c Analytic conductor $0.966$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + 2 \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} -2 \zeta_{10} q^{6} + 2 \zeta_{10}^{3} q^{7} -3 \zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + 2 \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} -2 \zeta_{10} q^{6} + 2 \zeta_{10}^{3} q^{7} -3 \zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} + q^{10} -2 q^{12} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{13} -2 \zeta_{10}^{2} q^{14} -2 \zeta_{10}^{3} q^{15} + \zeta_{10} q^{16} + 5 \zeta_{10} q^{17} -\zeta_{10}^{3} q^{18} + 6 \zeta_{10}^{2} q^{19} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{20} -4 q^{21} + 2 q^{23} + ( 6 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{24} -4 \zeta_{10}^{2} q^{25} -\zeta_{10}^{3} q^{26} + 4 \zeta_{10} q^{27} -2 \zeta_{10} q^{28} -9 \zeta_{10}^{3} q^{29} + 2 \zeta_{10}^{2} q^{30} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{31} + 5 q^{32} -5 q^{34} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{35} -\zeta_{10}^{2} q^{36} + 3 \zeta_{10}^{3} q^{37} -6 \zeta_{10} q^{38} -2 \zeta_{10} q^{39} + 3 \zeta_{10}^{3} q^{40} -5 \zeta_{10}^{2} q^{41} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{42} + q^{45} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{46} + 2 \zeta_{10}^{2} q^{47} + 2 \zeta_{10}^{3} q^{48} + 3 \zeta_{10} q^{49} + 4 \zeta_{10} q^{50} + 10 \zeta_{10}^{3} q^{51} -\zeta_{10}^{2} q^{52} + ( -9 + 9 \zeta_{10} - 9 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{53} -4 q^{54} + 6 q^{56} + ( -12 + 12 \zeta_{10} - 12 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{57} + 9 \zeta_{10}^{2} q^{58} -8 \zeta_{10}^{3} q^{59} + 2 \zeta_{10} q^{60} -6 \zeta_{10} q^{61} + 2 \zeta_{10}^{3} q^{62} -2 \zeta_{10}^{2} q^{63} + ( -7 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{64} + q^{65} + 2 q^{67} + ( -5 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{68} + 4 \zeta_{10}^{2} q^{69} + 2 \zeta_{10}^{3} q^{70} -12 \zeta_{10} q^{71} + 3 \zeta_{10} q^{72} + 2 \zeta_{10}^{3} q^{73} -3 \zeta_{10}^{2} q^{74} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{75} -6 q^{76} + 2 q^{78} + ( 10 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{79} -\zeta_{10}^{2} q^{80} + 11 \zeta_{10}^{3} q^{81} + 5 \zeta_{10} q^{82} -6 \zeta_{10} q^{83} -4 \zeta_{10}^{3} q^{84} -5 \zeta_{10}^{2} q^{85} + 18 q^{87} -9 q^{89} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{90} -2 \zeta_{10}^{2} q^{91} + 2 \zeta_{10}^{3} q^{92} + 4 \zeta_{10} q^{93} -2 \zeta_{10} q^{94} -6 \zeta_{10}^{3} q^{95} + 10 \zeta_{10}^{2} q^{96} + ( 13 - 13 \zeta_{10} + 13 \zeta_{10}^{2} - 13 \zeta_{10}^{3} ) q^{97} -3 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} + 2q^{7} + 3q^{8} - q^{9} + O(q^{10})$$ $$4q - q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} + 2q^{7} + 3q^{8} - q^{9} + 4q^{10} - 8q^{12} - q^{13} + 2q^{14} - 2q^{15} + q^{16} + 5q^{17} - q^{18} - 6q^{19} + q^{20} - 16q^{21} + 8q^{23} + 6q^{24} + 4q^{25} - q^{26} + 4q^{27} - 2q^{28} - 9q^{29} - 2q^{30} + 2q^{31} + 20q^{32} - 20q^{34} + 2q^{35} + q^{36} + 3q^{37} - 6q^{38} - 2q^{39} + 3q^{40} + 5q^{41} + 4q^{42} + 4q^{45} - 2q^{46} - 2q^{47} + 2q^{48} + 3q^{49} + 4q^{50} + 10q^{51} + q^{52} - 9q^{53} - 16q^{54} + 24q^{56} - 12q^{57} - 9q^{58} - 8q^{59} + 2q^{60} - 6q^{61} + 2q^{62} + 2q^{63} - 7q^{64} + 4q^{65} + 8q^{67} - 5q^{68} - 4q^{69} + 2q^{70} - 12q^{71} + 3q^{72} + 2q^{73} + 3q^{74} + 8q^{75} - 24q^{76} + 8q^{78} + 10q^{79} + q^{80} + 11q^{81} + 5q^{82} - 6q^{83} - 4q^{84} + 5q^{85} + 72q^{87} - 36q^{89} - q^{90} + 2q^{91} + 2q^{92} + 4q^{93} - 2q^{94} - 6q^{95} - 10q^{96} + 13q^{97} - 12q^{98} + O(q^{100})$$

## 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
−0.809017 + 0.587785i 0.618034 + 1.90211i −0.309017 + 0.951057i −0.809017 0.587785i −1.61803 1.17557i −0.618034 + 1.90211i −0.927051 2.85317i −0.809017 + 0.587785i 1.00000
9.1 0.309017 0.951057i −1.61803 + 1.17557i 0.809017 + 0.587785i 0.309017 + 0.951057i 0.618034 + 1.90211i 1.61803 + 1.17557i 2.42705 1.76336i 0.309017 0.951057i 1.00000
27.1 0.309017 + 0.951057i −1.61803 1.17557i 0.809017 0.587785i 0.309017 0.951057i 0.618034 1.90211i 1.61803 1.17557i 2.42705 + 1.76336i 0.309017 + 0.951057i 1.00000
81.1 −0.809017 0.587785i 0.618034 1.90211i −0.309017 0.951057i −0.809017 + 0.587785i −1.61803 + 1.17557i −0.618034 1.90211i −0.927051 + 2.85317i −0.809017 0.587785i 1.00000
 $$n$$: 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 121.2.c.b 4
11.b odd 2 1 121.2.c.d 4
11.c even 5 1 121.2.a.c yes 1
11.c even 5 3 inner 121.2.c.b 4
11.d odd 10 1 121.2.a.a 1
11.d odd 10 3 121.2.c.d 4
33.f even 10 1 1089.2.a.i 1
33.h odd 10 1 1089.2.a.c 1
44.g even 10 1 1936.2.a.a 1
44.h odd 10 1 1936.2.a.b 1
55.h odd 10 1 3025.2.a.e 1
55.j even 10 1 3025.2.a.b 1
77.j odd 10 1 5929.2.a.g 1
77.l even 10 1 5929.2.a.a 1
88.k even 10 1 7744.2.a.be 1
88.l odd 10 1 7744.2.a.bf 1
88.o even 10 1 7744.2.a.c 1
88.p odd 10 1 7744.2.a.f 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(121, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$3$ $$16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$7$ $$16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$17$ $$625 - 125 T + 25 T^{2} - 5 T^{3} + T^{4}$$
$19$ $$1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$( -2 + T )^{4}$$
$29$ $$6561 + 729 T + 81 T^{2} + 9 T^{3} + T^{4}$$
$31$ $$16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$37$ $$81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4}$$
$41$ $$625 - 125 T + 25 T^{2} - 5 T^{3} + T^{4}$$
$43$ $$T^{4}$$
$47$ $$16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$53$ $$6561 + 729 T + 81 T^{2} + 9 T^{3} + T^{4}$$
$59$ $$4096 + 512 T + 64 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4}$$
$67$ $$( -2 + T )^{4}$$
$71$ $$20736 + 1728 T + 144 T^{2} + 12 T^{3} + T^{4}$$
$73$ $$16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$79$ $$10000 - 1000 T + 100 T^{2} - 10 T^{3} + T^{4}$$
$83$ $$1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4}$$
$89$ $$( 9 + T )^{4}$$
$97$ $$28561 - 2197 T + 169 T^{2} - 13 T^{3} + T^{4}$$