# Properties

 Label 121.4.a.b Level $121$ Weight $4$ Character orbit 121.a Self dual yes Analytic conductor $7.139$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,4,Mod(1,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("121.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 121.a (trivial)

## 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: yes Analytic conductor: $$7.13923111069$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + (\beta - 4) q^{3} + ( - 2 \beta + 5) q^{4} + (\beta - 5) q^{5} + ( - 5 \beta + 16) q^{6} + ( - 7 \beta - 4) q^{7} + ( - \beta - 21) q^{8} + ( - 8 \beta + 1) q^{9}+O(q^{10})$$ q + (b - 1) * q^2 + (b - 4) * q^3 + (-2*b + 5) * q^4 + (b - 5) * q^5 + (-5*b + 16) * q^6 + (-7*b - 4) * q^7 + (-b - 21) * q^8 + (-8*b + 1) * q^9 $$q + (\beta - 1) q^{2} + (\beta - 4) q^{3} + ( - 2 \beta + 5) q^{4} + (\beta - 5) q^{5} + ( - 5 \beta + 16) q^{6} + ( - 7 \beta - 4) q^{7} + ( - \beta - 21) q^{8} + ( - 8 \beta + 1) q^{9} + ( - 6 \beta + 17) q^{10} + (13 \beta - 44) q^{12} + (\beta + 65) q^{13} + (3 \beta - 80) q^{14} + ( - 9 \beta + 32) q^{15} + ( - 4 \beta - 31) q^{16} + (18 \beta - 7) q^{17} + (9 \beta - 97) q^{18} + ( - 9 \beta - 24) q^{19} + (15 \beta - 49) q^{20} + (24 \beta - 68) q^{21} + ( - 33 \beta - 64) q^{23} + ( - 17 \beta + 72) q^{24} + ( - 10 \beta - 88) q^{25} + (64 \beta - 53) q^{26} + (6 \beta + 8) q^{27} + ( - 27 \beta + 148) q^{28} + (37 \beta - 15) q^{29} + (41 \beta - 140) q^{30} + (47 \beta - 92) q^{31} + ( - 19 \beta + 151) q^{32} + ( - 25 \beta + 223) q^{34} + (31 \beta - 64) q^{35} + ( - 42 \beta + 197) q^{36} + ( - 79 \beta + 63) q^{37} + ( - 15 \beta - 84) q^{38} + (61 \beta - 248) q^{39} + ( - 16 \beta + 93) q^{40} + (2 \beta + 185) q^{41} + ( - 92 \beta + 356) q^{42} + ( - 22 \beta - 132) q^{43} + (41 \beta - 101) q^{45} + ( - 31 \beta - 332) q^{46} + (111 \beta + 128) q^{47} + ( - 15 \beta + 76) q^{48} + (56 \beta + 261) q^{49} + ( - 78 \beta - 32) q^{50} + ( - 79 \beta + 244) q^{51} + ( - 125 \beta + 301) q^{52} + ( - 85 \beta - 81) q^{53} + (2 \beta + 64) q^{54} + (151 \beta + 168) q^{56} + (12 \beta - 12) q^{57} + ( - 52 \beta + 459) q^{58} + (42 \beta - 652) q^{59} + ( - 109 \beta + 376) q^{60} + ( - 92 \beta - 150) q^{61} + ( - 139 \beta + 656) q^{62} + (25 \beta + 668) q^{63} + (202 \beta - 131) q^{64} + (60 \beta - 313) q^{65} + (11 \beta - 328) q^{67} + (104 \beta - 467) q^{68} + (68 \beta - 140) q^{69} + ( - 95 \beta + 436) q^{70} + (56 \beta - 588) q^{71} + (167 \beta + 75) q^{72} + (180 \beta - 334) q^{73} + (142 \beta - 1011) q^{74} + ( - 48 \beta + 232) q^{75} + (3 \beta + 96) q^{76} + ( - 309 \beta + 980) q^{78} + ( - 109 \beta + 208) q^{79} + ( - 11 \beta + 107) q^{80} + (200 \beta + 13) q^{81} + (183 \beta - 161) q^{82} + (51 \beta - 480) q^{83} + (256 \beta - 916) q^{84} + ( - 97 \beta + 251) q^{85} + ( - 110 \beta - 132) q^{86} + ( - 163 \beta + 504) q^{87} + ( - 176 \beta - 537) q^{89} + ( - 142 \beta + 593) q^{90} + ( - 459 \beta - 344) q^{91} + ( - 37 \beta + 472) q^{92} + ( - 280 \beta + 932) q^{93} + (17 \beta + 1204) q^{94} + (21 \beta + 12) q^{95} + (227 \beta - 832) q^{96} + (234 \beta - 169) q^{97} + (205 \beta + 411) q^{98}+O(q^{100})$$ q + (b - 1) * q^2 + (b - 4) * q^3 + (-2*b + 5) * q^4 + (b - 5) * q^5 + (-5*b + 16) * q^6 + (-7*b - 4) * q^7 + (-b - 21) * q^8 + (-8*b + 1) * q^9 + (-6*b + 17) * q^10 + (13*b - 44) * q^12 + (b + 65) * q^13 + (3*b - 80) * q^14 + (-9*b + 32) * q^15 + (-4*b - 31) * q^16 + (18*b - 7) * q^17 + (9*b - 97) * q^18 + (-9*b - 24) * q^19 + (15*b - 49) * q^20 + (24*b - 68) * q^21 + (-33*b - 64) * q^23 + (-17*b + 72) * q^24 + (-10*b - 88) * q^25 + (64*b - 53) * q^26 + (6*b + 8) * q^27 + (-27*b + 148) * q^28 + (37*b - 15) * q^29 + (41*b - 140) * q^30 + (47*b - 92) * q^31 + (-19*b + 151) * q^32 + (-25*b + 223) * q^34 + (31*b - 64) * q^35 + (-42*b + 197) * q^36 + (-79*b + 63) * q^37 + (-15*b - 84) * q^38 + (61*b - 248) * q^39 + (-16*b + 93) * q^40 + (2*b + 185) * q^41 + (-92*b + 356) * q^42 + (-22*b - 132) * q^43 + (41*b - 101) * q^45 + (-31*b - 332) * q^46 + (111*b + 128) * q^47 + (-15*b + 76) * q^48 + (56*b + 261) * q^49 + (-78*b - 32) * q^50 + (-79*b + 244) * q^51 + (-125*b + 301) * q^52 + (-85*b - 81) * q^53 + (2*b + 64) * q^54 + (151*b + 168) * q^56 + (12*b - 12) * q^57 + (-52*b + 459) * q^58 + (42*b - 652) * q^59 + (-109*b + 376) * q^60 + (-92*b - 150) * q^61 + (-139*b + 656) * q^62 + (25*b + 668) * q^63 + (202*b - 131) * q^64 + (60*b - 313) * q^65 + (11*b - 328) * q^67 + (104*b - 467) * q^68 + (68*b - 140) * q^69 + (-95*b + 436) * q^70 + (56*b - 588) * q^71 + (167*b + 75) * q^72 + (180*b - 334) * q^73 + (142*b - 1011) * q^74 + (-48*b + 232) * q^75 + (3*b + 96) * q^76 + (-309*b + 980) * q^78 + (-109*b + 208) * q^79 + (-11*b + 107) * q^80 + (200*b + 13) * q^81 + (183*b - 161) * q^82 + (51*b - 480) * q^83 + (256*b - 916) * q^84 + (-97*b + 251) * q^85 + (-110*b - 132) * q^86 + (-163*b + 504) * q^87 + (-176*b - 537) * q^89 + (-142*b + 593) * q^90 + (-459*b - 344) * q^91 + (-37*b + 472) * q^92 + (-280*b + 932) * q^93 + (17*b + 1204) * q^94 + (21*b + 12) * q^95 + (227*b - 832) * q^96 + (234*b - 169) * q^97 + (205*b + 411) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 8 q^{3} + 10 q^{4} - 10 q^{5} + 32 q^{6} - 8 q^{7} - 42 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 8 * q^3 + 10 * q^4 - 10 * q^5 + 32 * q^6 - 8 * q^7 - 42 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 8 q^{3} + 10 q^{4} - 10 q^{5} + 32 q^{6} - 8 q^{7} - 42 q^{8} + 2 q^{9} + 34 q^{10} - 88 q^{12} + 130 q^{13} - 160 q^{14} + 64 q^{15} - 62 q^{16} - 14 q^{17} - 194 q^{18} - 48 q^{19} - 98 q^{20} - 136 q^{21} - 128 q^{23} + 144 q^{24} - 176 q^{25} - 106 q^{26} + 16 q^{27} + 296 q^{28} - 30 q^{29} - 280 q^{30} - 184 q^{31} + 302 q^{32} + 446 q^{34} - 128 q^{35} + 394 q^{36} + 126 q^{37} - 168 q^{38} - 496 q^{39} + 186 q^{40} + 370 q^{41} + 712 q^{42} - 264 q^{43} - 202 q^{45} - 664 q^{46} + 256 q^{47} + 152 q^{48} + 522 q^{49} - 64 q^{50} + 488 q^{51} + 602 q^{52} - 162 q^{53} + 128 q^{54} + 336 q^{56} - 24 q^{57} + 918 q^{58} - 1304 q^{59} + 752 q^{60} - 300 q^{61} + 1312 q^{62} + 1336 q^{63} - 262 q^{64} - 626 q^{65} - 656 q^{67} - 934 q^{68} - 280 q^{69} + 872 q^{70} - 1176 q^{71} + 150 q^{72} - 668 q^{73} - 2022 q^{74} + 464 q^{75} + 192 q^{76} + 1960 q^{78} + 416 q^{79} + 214 q^{80} + 26 q^{81} - 322 q^{82} - 960 q^{83} - 1832 q^{84} + 502 q^{85} - 264 q^{86} + 1008 q^{87} - 1074 q^{89} + 1186 q^{90} - 688 q^{91} + 944 q^{92} + 1864 q^{93} + 2408 q^{94} + 24 q^{95} - 1664 q^{96} - 338 q^{97} + 822 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 8 * q^3 + 10 * q^4 - 10 * q^5 + 32 * q^6 - 8 * q^7 - 42 * q^8 + 2 * q^9 + 34 * q^10 - 88 * q^12 + 130 * q^13 - 160 * q^14 + 64 * q^15 - 62 * q^16 - 14 * q^17 - 194 * q^18 - 48 * q^19 - 98 * q^20 - 136 * q^21 - 128 * q^23 + 144 * q^24 - 176 * q^25 - 106 * q^26 + 16 * q^27 + 296 * q^28 - 30 * q^29 - 280 * q^30 - 184 * q^31 + 302 * q^32 + 446 * q^34 - 128 * q^35 + 394 * q^36 + 126 * q^37 - 168 * q^38 - 496 * q^39 + 186 * q^40 + 370 * q^41 + 712 * q^42 - 264 * q^43 - 202 * q^45 - 664 * q^46 + 256 * q^47 + 152 * q^48 + 522 * q^49 - 64 * q^50 + 488 * q^51 + 602 * q^52 - 162 * q^53 + 128 * q^54 + 336 * q^56 - 24 * q^57 + 918 * q^58 - 1304 * q^59 + 752 * q^60 - 300 * q^61 + 1312 * q^62 + 1336 * q^63 - 262 * q^64 - 626 * q^65 - 656 * q^67 - 934 * q^68 - 280 * q^69 + 872 * q^70 - 1176 * q^71 + 150 * q^72 - 668 * q^73 - 2022 * q^74 + 464 * q^75 + 192 * q^76 + 1960 * q^78 + 416 * q^79 + 214 * q^80 + 26 * q^81 - 322 * q^82 - 960 * q^83 - 1832 * q^84 + 502 * q^85 - 264 * q^86 + 1008 * q^87 - 1074 * q^89 + 1186 * q^90 - 688 * q^91 + 944 * q^92 + 1864 * q^93 + 2408 * q^94 + 24 * q^95 - 1664 * q^96 - 338 * q^97 + 822 * q^98

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−4.46410 −7.46410 11.9282 −8.46410 33.3205 20.2487 −17.5359 28.7128 37.7846
1.2 2.46410 −0.535898 −1.92820 −1.53590 −1.32051 −28.2487 −24.4641 −26.7128 −3.78461
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.4.a.b 2
3.b odd 2 1 1089.4.a.x 2
4.b odd 2 1 1936.4.a.z 2
11.b odd 2 1 121.4.a.e yes 2
11.c even 5 4 121.4.c.g 8
11.d odd 10 4 121.4.c.d 8
33.d even 2 1 1089.4.a.k 2
44.c even 2 1 1936.4.a.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.a.b 2 1.a even 1 1 trivial
121.4.a.e yes 2 11.b odd 2 1
121.4.c.d 8 11.d odd 10 4
121.4.c.g 8 11.c even 5 4
1089.4.a.k 2 33.d even 2 1
1089.4.a.x 2 3.b odd 2 1
1936.4.a.y 2 44.c even 2 1
1936.4.a.z 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2T_{2} - 11$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(121))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 11$$
$3$ $$T^{2} + 8T + 4$$
$5$ $$T^{2} + 10T + 13$$
$7$ $$T^{2} + 8T - 572$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 130T + 4213$$
$17$ $$T^{2} + 14T - 3839$$
$19$ $$T^{2} + 48T - 396$$
$23$ $$T^{2} + 128T - 8972$$
$29$ $$T^{2} + 30T - 16203$$
$31$ $$T^{2} + 184T - 18044$$
$37$ $$T^{2} - 126T - 70923$$
$41$ $$T^{2} - 370T + 34177$$
$43$ $$T^{2} + 264T + 11616$$
$47$ $$T^{2} - 256T - 131468$$
$53$ $$T^{2} + 162T - 80139$$
$59$ $$T^{2} + 1304 T + 403936$$
$61$ $$T^{2} + 300T - 79068$$
$67$ $$T^{2} + 656T + 106132$$
$71$ $$T^{2} + 1176 T + 308112$$
$73$ $$T^{2} + 668T - 277244$$
$79$ $$T^{2} - 416T - 99308$$
$83$ $$T^{2} + 960T + 199188$$
$89$ $$T^{2} + 1074T - 83343$$
$97$ $$T^{2} + 338T - 628511$$