# Properties

 Label 121.2.c.e 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: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{2} - \zeta_{10}^{2} q^{3} - 2 \zeta_{10}^{3} q^{4} - \zeta_{10} q^{5} - 2 \zeta_{10} q^{6} + 2 \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{9} +O(q^{10})$$ q + (-2*z^3 + 2*z^2 - 2*z + 2) * q^2 - z^2 * q^3 - 2*z^3 * q^4 - z * q^5 - 2*z * q^6 + 2*z^3 * q^7 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^9 $$q + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{2} - \zeta_{10}^{2} q^{3} - 2 \zeta_{10}^{3} q^{4} - \zeta_{10} q^{5} - 2 \zeta_{10} q^{6} + 2 \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{9} - 2 q^{10} - 2 q^{12} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{13} + 4 \zeta_{10}^{2} q^{14} + \zeta_{10}^{3} q^{15} + 4 \zeta_{10} q^{16} + 2 \zeta_{10} q^{17} - 4 \zeta_{10}^{3} q^{18} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{20} + 2 q^{21} - q^{23} - 4 \zeta_{10}^{2} q^{25} + 8 \zeta_{10}^{3} q^{26} - 5 \zeta_{10} q^{27} + 4 \zeta_{10} q^{28} + 2 \zeta_{10}^{2} q^{30} + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 7) q^{31} + 8 q^{32} + 4 q^{34} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{35} - 4 \zeta_{10}^{2} q^{36} - 3 \zeta_{10}^{3} q^{37} + 4 \zeta_{10} q^{39} - 8 \zeta_{10}^{2} q^{41} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{42} - 6 q^{43} - 2 q^{45} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{46} + 8 \zeta_{10}^{2} q^{47} - 4 \zeta_{10}^{3} q^{48} + 3 \zeta_{10} q^{49} - 8 \zeta_{10} q^{50} - 2 \zeta_{10}^{3} q^{51} + 8 \zeta_{10}^{2} q^{52} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{53} - 10 q^{54} - 5 \zeta_{10}^{3} q^{59} + 2 \zeta_{10} q^{60} - 12 \zeta_{10} q^{61} + 14 \zeta_{10}^{3} q^{62} + 4 \zeta_{10}^{2} q^{63} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{64} + 4 q^{65} - 7 q^{67} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{68} + \zeta_{10}^{2} q^{69} - 4 \zeta_{10}^{3} q^{70} + 3 \zeta_{10} q^{71} - 4 \zeta_{10}^{3} q^{73} - 6 \zeta_{10}^{2} q^{74} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{75} + 8 q^{78} + ( - 10 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 10 \zeta_{10} + 10) q^{79} - 4 \zeta_{10}^{2} q^{80} - \zeta_{10}^{3} q^{81} - 16 \zeta_{10} q^{82} + 6 \zeta_{10} q^{83} - 4 \zeta_{10}^{3} q^{84} - 2 \zeta_{10}^{2} q^{85} + (12 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 12 \zeta_{10} - 12) q^{86} + 15 q^{89} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{90} - 8 \zeta_{10}^{2} q^{91} + 2 \zeta_{10}^{3} q^{92} + 7 \zeta_{10} q^{93} + 16 \zeta_{10} q^{94} - 8 \zeta_{10}^{2} q^{96} + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 7) q^{97} + 6 q^{98} +O(q^{100})$$ q + (-2*z^3 + 2*z^2 - 2*z + 2) * q^2 - z^2 * q^3 - 2*z^3 * q^4 - z * q^5 - 2*z * q^6 + 2*z^3 * q^7 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^9 - 2 * q^10 - 2 * q^12 + (4*z^3 - 4*z^2 + 4*z - 4) * q^13 + 4*z^2 * q^14 + z^3 * q^15 + 4*z * q^16 + 2*z * q^17 - 4*z^3 * q^18 + (2*z^3 - 2*z^2 + 2*z - 2) * q^20 + 2 * q^21 - q^23 - 4*z^2 * q^25 + 8*z^3 * q^26 - 5*z * q^27 + 4*z * q^28 + 2*z^2 * q^30 + (7*z^3 - 7*z^2 + 7*z - 7) * q^31 + 8 * q^32 + 4 * q^34 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^35 - 4*z^2 * q^36 - 3*z^3 * q^37 + 4*z * q^39 - 8*z^2 * q^41 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^42 - 6 * q^43 - 2 * q^45 + (2*z^3 - 2*z^2 + 2*z - 2) * q^46 + 8*z^2 * q^47 - 4*z^3 * q^48 + 3*z * q^49 - 8*z * q^50 - 2*z^3 * q^51 + 8*z^2 * q^52 + (-6*z^3 + 6*z^2 - 6*z + 6) * q^53 - 10 * q^54 - 5*z^3 * q^59 + 2*z * q^60 - 12*z * q^61 + 14*z^3 * q^62 + 4*z^2 * q^63 + (-8*z^3 + 8*z^2 - 8*z + 8) * q^64 + 4 * q^65 - 7 * q^67 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^68 + z^2 * q^69 - 4*z^3 * q^70 + 3*z * q^71 - 4*z^3 * q^73 - 6*z^2 * q^74 + (4*z^3 - 4*z^2 + 4*z - 4) * q^75 + 8 * q^78 + (-10*z^3 + 10*z^2 - 10*z + 10) * q^79 - 4*z^2 * q^80 - z^3 * q^81 - 16*z * q^82 + 6*z * q^83 - 4*z^3 * q^84 - 2*z^2 * q^85 + (12*z^3 - 12*z^2 + 12*z - 12) * q^86 + 15 * q^89 + (4*z^3 - 4*z^2 + 4*z - 4) * q^90 - 8*z^2 * q^91 + 2*z^3 * q^92 + 7*z * q^93 + 16*z * q^94 - 8*z^2 * q^96 + (-7*z^3 + 7*z^2 - 7*z + 7) * q^97 + 6 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + q^{3} - 2 q^{4} - q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 + q^3 - 2 * q^4 - q^5 - 2 * q^6 + 2 * q^7 + 2 * q^9 $$4 q + 2 q^{2} + q^{3} - 2 q^{4} - q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{9} - 8 q^{10} - 8 q^{12} - 4 q^{13} - 4 q^{14} + q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} - 2 q^{20} + 8 q^{21} - 4 q^{23} + 4 q^{25} + 8 q^{26} - 5 q^{27} + 4 q^{28} - 2 q^{30} - 7 q^{31} + 32 q^{32} + 16 q^{34} + 2 q^{35} + 4 q^{36} - 3 q^{37} + 4 q^{39} + 8 q^{41} + 4 q^{42} - 24 q^{43} - 8 q^{45} - 2 q^{46} - 8 q^{47} - 4 q^{48} + 3 q^{49} - 8 q^{50} - 2 q^{51} - 8 q^{52} + 6 q^{53} - 40 q^{54} - 5 q^{59} + 2 q^{60} - 12 q^{61} + 14 q^{62} - 4 q^{63} + 8 q^{64} + 16 q^{65} - 28 q^{67} + 4 q^{68} - q^{69} - 4 q^{70} + 3 q^{71} - 4 q^{73} + 6 q^{74} - 4 q^{75} + 32 q^{78} + 10 q^{79} + 4 q^{80} - q^{81} - 16 q^{82} + 6 q^{83} - 4 q^{84} + 2 q^{85} - 12 q^{86} + 60 q^{89} - 4 q^{90} + 8 q^{91} + 2 q^{92} + 7 q^{93} + 16 q^{94} + 8 q^{96} + 7 q^{97} + 24 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 + q^3 - 2 * q^4 - q^5 - 2 * q^6 + 2 * q^7 + 2 * q^9 - 8 * q^10 - 8 * q^12 - 4 * q^13 - 4 * q^14 + q^15 + 4 * q^16 + 2 * q^17 - 4 * q^18 - 2 * q^20 + 8 * q^21 - 4 * q^23 + 4 * q^25 + 8 * q^26 - 5 * q^27 + 4 * q^28 - 2 * q^30 - 7 * q^31 + 32 * q^32 + 16 * q^34 + 2 * q^35 + 4 * q^36 - 3 * q^37 + 4 * q^39 + 8 * q^41 + 4 * q^42 - 24 * q^43 - 8 * q^45 - 2 * q^46 - 8 * q^47 - 4 * q^48 + 3 * q^49 - 8 * q^50 - 2 * q^51 - 8 * q^52 + 6 * q^53 - 40 * q^54 - 5 * q^59 + 2 * q^60 - 12 * q^61 + 14 * q^62 - 4 * q^63 + 8 * q^64 + 16 * q^65 - 28 * q^67 + 4 * q^68 - q^69 - 4 * q^70 + 3 * q^71 - 4 * q^73 + 6 * q^74 - 4 * q^75 + 32 * q^78 + 10 * q^79 + 4 * q^80 - q^81 - 16 * q^82 + 6 * q^83 - 4 * q^84 + 2 * q^85 - 12 * q^86 + 60 * q^89 - 4 * q^90 + 8 * q^91 + 2 * q^92 + 7 * q^93 + 16 * q^94 + 8 * q^96 + 7 * q^97 + 24 * q^98

## 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
1.61803 1.17557i −0.309017 0.951057i 0.618034 1.90211i −0.809017 0.587785i −1.61803 1.17557i −0.618034 + 1.90211i 0 1.61803 1.17557i −2.00000
9.1 −0.618034 + 1.90211i 0.809017 0.587785i −1.61803 1.17557i 0.309017 + 0.951057i 0.618034 + 1.90211i 1.61803 + 1.17557i 0 −0.618034 + 1.90211i −2.00000
27.1 −0.618034 1.90211i 0.809017 + 0.587785i −1.61803 + 1.17557i 0.309017 0.951057i 0.618034 1.90211i 1.61803 1.17557i 0 −0.618034 1.90211i −2.00000
81.1 1.61803 + 1.17557i −0.309017 + 0.951057i 0.618034 + 1.90211i −0.809017 + 0.587785i −1.61803 + 1.17557i −0.618034 1.90211i 0 1.61803 + 1.17557i −2.00000
 $$n$$: 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.e 4
11.b odd 2 1 121.2.c.a 4
11.c even 5 1 11.2.a.a 1
11.c even 5 3 inner 121.2.c.e 4
11.d odd 10 1 121.2.a.d 1
11.d odd 10 3 121.2.c.a 4
33.f even 10 1 1089.2.a.b 1
33.h odd 10 1 99.2.a.d 1
44.g even 10 1 1936.2.a.i 1
44.h odd 10 1 176.2.a.b 1
55.h odd 10 1 3025.2.a.a 1
55.j even 10 1 275.2.a.b 1
55.k odd 20 2 275.2.b.a 2
77.j odd 10 1 539.2.a.a 1
77.l even 10 1 5929.2.a.h 1
77.m even 15 2 539.2.e.h 2
77.p odd 30 2 539.2.e.g 2
88.k even 10 1 7744.2.a.k 1
88.l odd 10 1 704.2.a.c 1
88.o even 10 1 704.2.a.h 1
88.p odd 10 1 7744.2.a.x 1
99.m even 15 2 891.2.e.k 2
99.n odd 30 2 891.2.e.b 2
132.o even 10 1 1584.2.a.g 1
143.n even 10 1 1859.2.a.b 1
165.o odd 10 1 2475.2.a.a 1
165.v even 20 2 2475.2.c.a 2
176.v odd 20 2 2816.2.c.f 2
176.w even 20 2 2816.2.c.j 2
187.j even 10 1 3179.2.a.a 1
209.m odd 10 1 3971.2.a.b 1
220.n odd 10 1 4400.2.a.i 1
220.v even 20 2 4400.2.b.h 2
231.u even 10 1 4851.2.a.t 1
253.f odd 10 1 5819.2.a.a 1
264.t odd 10 1 6336.2.a.br 1
264.w even 10 1 6336.2.a.bu 1
308.t even 10 1 8624.2.a.j 1
319.k even 10 1 9251.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 11.c even 5 1
99.2.a.d 1 33.h odd 10 1
121.2.a.d 1 11.d odd 10 1
121.2.c.a 4 11.b odd 2 1
121.2.c.a 4 11.d odd 10 3
121.2.c.e 4 1.a even 1 1 trivial
121.2.c.e 4 11.c even 5 3 inner
176.2.a.b 1 44.h odd 10 1
275.2.a.b 1 55.j even 10 1
275.2.b.a 2 55.k odd 20 2
539.2.a.a 1 77.j odd 10 1
539.2.e.g 2 77.p odd 30 2
539.2.e.h 2 77.m even 15 2
704.2.a.c 1 88.l odd 10 1
704.2.a.h 1 88.o even 10 1
891.2.e.b 2 99.n odd 30 2
891.2.e.k 2 99.m even 15 2
1089.2.a.b 1 33.f even 10 1
1584.2.a.g 1 132.o even 10 1
1859.2.a.b 1 143.n even 10 1
1936.2.a.i 1 44.g even 10 1
2475.2.a.a 1 165.o odd 10 1
2475.2.c.a 2 165.v even 20 2
2816.2.c.f 2 176.v odd 20 2
2816.2.c.j 2 176.w even 20 2
3025.2.a.a 1 55.h odd 10 1
3179.2.a.a 1 187.j even 10 1
3971.2.a.b 1 209.m odd 10 1
4400.2.a.i 1 220.n odd 10 1
4400.2.b.h 2 220.v even 20 2
4851.2.a.t 1 231.u even 10 1
5819.2.a.a 1 253.f odd 10 1
5929.2.a.h 1 77.l even 10 1
6336.2.a.br 1 264.t odd 10 1
6336.2.a.bu 1 264.w even 10 1
7744.2.a.k 1 88.k even 10 1
7744.2.a.x 1 88.p odd 10 1
8624.2.a.j 1 308.t even 10 1
9251.2.a.d 1 319.k even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2T_{2}^{3} + 4T_{2}^{2} - 8T_{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(121, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$3$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$5$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$7$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$17$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$19$ $$T^{4}$$
$23$ $$(T + 1)^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4} + 7 T^{3} + 49 T^{2} + \cdots + 2401$$
$37$ $$T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81$$
$41$ $$T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$43$ $$(T + 6)^{4}$$
$47$ $$T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$53$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$59$ $$T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625$$
$61$ $$T^{4} + 12 T^{3} + 144 T^{2} + \cdots + 20736$$
$67$ $$(T + 7)^{4}$$
$71$ $$T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81$$
$73$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$79$ $$T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000$$
$83$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$89$ $$(T - 15)^{4}$$
$97$ $$T^{4} - 7 T^{3} + 49 T^{2} + \cdots + 2401$$