# Properties

 Label 121.4.a.c 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: $$1$$ Twist minimal: no (minimal twist has level 11) 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + (4 \beta - 1) q^{3} + ( - 2 \beta - 4) q^{4} + ( - 8 \beta + 1) q^{5} + ( - 5 \beta + 13) q^{6} + ( - 4 \beta - 10) q^{7} + ( - 10 \beta + 6) q^{8} + ( - 8 \beta + 22) q^{9}+O(q^{10})$$ q + (b - 1) * q^2 + (4*b - 1) * q^3 + (-2*b - 4) * q^4 + (-8*b + 1) * q^5 + (-5*b + 13) * q^6 + (-4*b - 10) * q^7 + (-10*b + 6) * q^8 + (-8*b + 22) * q^9 $$q + (\beta - 1) q^{2} + (4 \beta - 1) q^{3} + ( - 2 \beta - 4) q^{4} + ( - 8 \beta + 1) q^{5} + ( - 5 \beta + 13) q^{6} + ( - 4 \beta - 10) q^{7} + ( - 10 \beta + 6) q^{8} + ( - 8 \beta + 22) q^{9} + (9 \beta - 25) q^{10} + ( - 14 \beta - 20) q^{12} + ( - 20 \beta - 40) q^{13} + ( - 6 \beta - 2) q^{14} + (12 \beta - 97) q^{15} + (32 \beta - 4) q^{16} + (12 \beta + 62) q^{17} + (30 \beta - 46) q^{18} + (60 \beta - 36) q^{19} + (30 \beta + 44) q^{20} + ( - 36 \beta - 38) q^{21} + (36 \beta - 49) q^{23} + (34 \beta - 126) q^{24} + ( - 16 \beta + 68) q^{25} + ( - 20 \beta - 20) q^{26} + ( - 12 \beta - 91) q^{27} + (36 \beta + 64) q^{28} + ( - 56 \beta - 72) q^{29} + ( - 109 \beta + 133) q^{30} + ( - 28 \beta - 17) q^{31} + (44 \beta + 52) q^{32} + (50 \beta - 26) q^{34} + (76 \beta + 86) q^{35} + ( - 12 \beta - 40) q^{36} + (8 \beta + 27) q^{37} + ( - 96 \beta + 216) q^{38} + ( - 140 \beta - 200) q^{39} + ( - 58 \beta + 246) q^{40} + ( - 4 \beta - 268) q^{41} + ( - 2 \beta - 70) q^{42} + ( - 16 \beta + 30) q^{43} + ( - 184 \beta + 214) q^{45} + ( - 85 \beta + 157) q^{46} + (120 \beta - 136) q^{47} + ( - 48 \beta + 388) q^{48} + (80 \beta - 195) q^{49} + (84 \beta - 116) q^{50} + (236 \beta + 82) q^{51} + (160 \beta + 280) q^{52} + (56 \beta - 246) q^{53} + ( - 79 \beta + 55) q^{54} + (76 \beta + 60) q^{56} + ( - 204 \beta + 756) q^{57} + ( - 16 \beta - 96) q^{58} + (132 \beta + 317) q^{59} + (146 \beta + 316) q^{60} + (184 \beta - 420) q^{61} + (11 \beta - 67) q^{62} + ( - 8 \beta - 124) q^{63} + ( - 248 \beta + 112) q^{64} + (300 \beta + 440) q^{65} + (20 \beta + 377) q^{67} + ( - 172 \beta - 320) q^{68} + ( - 232 \beta + 481) q^{69} + (10 \beta + 142) q^{70} + ( - 76 \beta - 339) q^{71} + ( - 268 \beta + 372) q^{72} + ( - 468 \beta + 200) q^{73} + (19 \beta - 3) q^{74} + (288 \beta - 260) q^{75} + ( - 168 \beta - 216) q^{76} + ( - 60 \beta - 220) q^{78} + (656 \beta - 158) q^{79} + (64 \beta - 772) q^{80} + ( - 136 \beta - 647) q^{81} + ( - 264 \beta + 256) q^{82} + (120 \beta - 234) q^{83} + (220 \beta + 368) q^{84} + ( - 484 \beta - 226) q^{85} + (46 \beta - 78) q^{86} + ( - 232 \beta - 600) q^{87} + (328 \beta - 921) q^{89} + (398 \beta - 766) q^{90} + (360 \beta + 640) q^{91} + ( - 46 \beta - 20) q^{92} + ( - 40 \beta - 319) q^{93} + ( - 256 \beta + 496) q^{94} + (348 \beta - 1476) q^{95} + (164 \beta + 476) q^{96} + ( - 144 \beta + 1097) q^{97} + ( - 275 \beta + 435) q^{98}+O(q^{100})$$ q + (b - 1) * q^2 + (4*b - 1) * q^3 + (-2*b - 4) * q^4 + (-8*b + 1) * q^5 + (-5*b + 13) * q^6 + (-4*b - 10) * q^7 + (-10*b + 6) * q^8 + (-8*b + 22) * q^9 + (9*b - 25) * q^10 + (-14*b - 20) * q^12 + (-20*b - 40) * q^13 + (-6*b - 2) * q^14 + (12*b - 97) * q^15 + (32*b - 4) * q^16 + (12*b + 62) * q^17 + (30*b - 46) * q^18 + (60*b - 36) * q^19 + (30*b + 44) * q^20 + (-36*b - 38) * q^21 + (36*b - 49) * q^23 + (34*b - 126) * q^24 + (-16*b + 68) * q^25 + (-20*b - 20) * q^26 + (-12*b - 91) * q^27 + (36*b + 64) * q^28 + (-56*b - 72) * q^29 + (-109*b + 133) * q^30 + (-28*b - 17) * q^31 + (44*b + 52) * q^32 + (50*b - 26) * q^34 + (76*b + 86) * q^35 + (-12*b - 40) * q^36 + (8*b + 27) * q^37 + (-96*b + 216) * q^38 + (-140*b - 200) * q^39 + (-58*b + 246) * q^40 + (-4*b - 268) * q^41 + (-2*b - 70) * q^42 + (-16*b + 30) * q^43 + (-184*b + 214) * q^45 + (-85*b + 157) * q^46 + (120*b - 136) * q^47 + (-48*b + 388) * q^48 + (80*b - 195) * q^49 + (84*b - 116) * q^50 + (236*b + 82) * q^51 + (160*b + 280) * q^52 + (56*b - 246) * q^53 + (-79*b + 55) * q^54 + (76*b + 60) * q^56 + (-204*b + 756) * q^57 + (-16*b - 96) * q^58 + (132*b + 317) * q^59 + (146*b + 316) * q^60 + (184*b - 420) * q^61 + (11*b - 67) * q^62 + (-8*b - 124) * q^63 + (-248*b + 112) * q^64 + (300*b + 440) * q^65 + (20*b + 377) * q^67 + (-172*b - 320) * q^68 + (-232*b + 481) * q^69 + (10*b + 142) * q^70 + (-76*b - 339) * q^71 + (-268*b + 372) * q^72 + (-468*b + 200) * q^73 + (19*b - 3) * q^74 + (288*b - 260) * q^75 + (-168*b - 216) * q^76 + (-60*b - 220) * q^78 + (656*b - 158) * q^79 + (64*b - 772) * q^80 + (-136*b - 647) * q^81 + (-264*b + 256) * q^82 + (120*b - 234) * q^83 + (220*b + 368) * q^84 + (-484*b - 226) * q^85 + (46*b - 78) * q^86 + (-232*b - 600) * q^87 + (328*b - 921) * q^89 + (398*b - 766) * q^90 + (360*b + 640) * q^91 + (-46*b - 20) * q^92 + (-40*b - 319) * q^93 + (-256*b + 496) * q^94 + (348*b - 1476) * q^95 + (164*b + 476) * q^96 + (-144*b + 1097) * q^97 + (-275*b + 435) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} - 8 q^{4} + 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 - 8 * q^4 + 2 * q^5 + 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9 $$2 q - 2 q^{2} - 2 q^{3} - 8 q^{4} + 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9} - 50 q^{10} - 40 q^{12} - 80 q^{13} - 4 q^{14} - 194 q^{15} - 8 q^{16} + 124 q^{17} - 92 q^{18} - 72 q^{19} + 88 q^{20} - 76 q^{21} - 98 q^{23} - 252 q^{24} + 136 q^{25} - 40 q^{26} - 182 q^{27} + 128 q^{28} - 144 q^{29} + 266 q^{30} - 34 q^{31} + 104 q^{32} - 52 q^{34} + 172 q^{35} - 80 q^{36} + 54 q^{37} + 432 q^{38} - 400 q^{39} + 492 q^{40} - 536 q^{41} - 140 q^{42} + 60 q^{43} + 428 q^{45} + 314 q^{46} - 272 q^{47} + 776 q^{48} - 390 q^{49} - 232 q^{50} + 164 q^{51} + 560 q^{52} - 492 q^{53} + 110 q^{54} + 120 q^{56} + 1512 q^{57} - 192 q^{58} + 634 q^{59} + 632 q^{60} - 840 q^{61} - 134 q^{62} - 248 q^{63} + 224 q^{64} + 880 q^{65} + 754 q^{67} - 640 q^{68} + 962 q^{69} + 284 q^{70} - 678 q^{71} + 744 q^{72} + 400 q^{73} - 6 q^{74} - 520 q^{75} - 432 q^{76} - 440 q^{78} - 316 q^{79} - 1544 q^{80} - 1294 q^{81} + 512 q^{82} - 468 q^{83} + 736 q^{84} - 452 q^{85} - 156 q^{86} - 1200 q^{87} - 1842 q^{89} - 1532 q^{90} + 1280 q^{91} - 40 q^{92} - 638 q^{93} + 992 q^{94} - 2952 q^{95} + 952 q^{96} + 2194 q^{97} + 870 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 - 8 * q^4 + 2 * q^5 + 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9 - 50 * q^10 - 40 * q^12 - 80 * q^13 - 4 * q^14 - 194 * q^15 - 8 * q^16 + 124 * q^17 - 92 * q^18 - 72 * q^19 + 88 * q^20 - 76 * q^21 - 98 * q^23 - 252 * q^24 + 136 * q^25 - 40 * q^26 - 182 * q^27 + 128 * q^28 - 144 * q^29 + 266 * q^30 - 34 * q^31 + 104 * q^32 - 52 * q^34 + 172 * q^35 - 80 * q^36 + 54 * q^37 + 432 * q^38 - 400 * q^39 + 492 * q^40 - 536 * q^41 - 140 * q^42 + 60 * q^43 + 428 * q^45 + 314 * q^46 - 272 * q^47 + 776 * q^48 - 390 * q^49 - 232 * q^50 + 164 * q^51 + 560 * q^52 - 492 * q^53 + 110 * q^54 + 120 * q^56 + 1512 * q^57 - 192 * q^58 + 634 * q^59 + 632 * q^60 - 840 * q^61 - 134 * q^62 - 248 * q^63 + 224 * q^64 + 880 * q^65 + 754 * q^67 - 640 * q^68 + 962 * q^69 + 284 * q^70 - 678 * q^71 + 744 * q^72 + 400 * q^73 - 6 * q^74 - 520 * q^75 - 432 * q^76 - 440 * q^78 - 316 * q^79 - 1544 * q^80 - 1294 * q^81 + 512 * q^82 - 468 * q^83 + 736 * q^84 - 452 * q^85 - 156 * q^86 - 1200 * q^87 - 1842 * q^89 - 1532 * q^90 + 1280 * q^91 - 40 * q^92 - 638 * q^93 + 992 * q^94 - 2952 * q^95 + 952 * q^96 + 2194 * q^97 + 870 * 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
−2.73205 −7.92820 −0.535898 14.8564 21.6603 −3.07180 23.3205 35.8564 −40.5885
1.2 0.732051 5.92820 −7.46410 −12.8564 4.33975 −16.9282 −11.3205 8.14359 −9.41154
 $$n$$: 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.c 2
3.b odd 2 1 1089.4.a.v 2
4.b odd 2 1 1936.4.a.w 2
11.b odd 2 1 11.4.a.a 2
11.c even 5 4 121.4.c.f 8
11.d odd 10 4 121.4.c.c 8
33.d even 2 1 99.4.a.c 2
44.c even 2 1 176.4.a.i 2
55.d odd 2 1 275.4.a.b 2
55.e even 4 2 275.4.b.c 4
77.b even 2 1 539.4.a.e 2
88.b odd 2 1 704.4.a.p 2
88.g even 2 1 704.4.a.n 2
132.d odd 2 1 1584.4.a.bc 2
143.d odd 2 1 1859.4.a.a 2
165.d even 2 1 2475.4.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 11.b odd 2 1
99.4.a.c 2 33.d even 2 1
121.4.a.c 2 1.a even 1 1 trivial
121.4.c.c 8 11.d odd 10 4
121.4.c.f 8 11.c even 5 4
176.4.a.i 2 44.c even 2 1
275.4.a.b 2 55.d odd 2 1
275.4.b.c 4 55.e even 4 2
539.4.a.e 2 77.b even 2 1
704.4.a.n 2 88.g even 2 1
704.4.a.p 2 88.b odd 2 1
1089.4.a.v 2 3.b odd 2 1
1584.4.a.bc 2 132.d odd 2 1
1859.4.a.a 2 143.d odd 2 1
1936.4.a.w 2 4.b odd 2 1
2475.4.a.q 2 165.d even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 2$$
$3$ $$T^{2} + 2T - 47$$
$5$ $$T^{2} - 2T - 191$$
$7$ $$T^{2} + 20T + 52$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 80T + 400$$
$17$ $$T^{2} - 124T + 3412$$
$19$ $$T^{2} + 72T - 9504$$
$23$ $$T^{2} + 98T - 1487$$
$29$ $$T^{2} + 144T - 4224$$
$31$ $$T^{2} + 34T - 2063$$
$37$ $$T^{2} - 54T + 537$$
$41$ $$T^{2} + 536T + 71776$$
$43$ $$T^{2} - 60T + 132$$
$47$ $$T^{2} + 272T - 24704$$
$53$ $$T^{2} + 492T + 51108$$
$59$ $$T^{2} - 634T + 48217$$
$61$ $$T^{2} + 840T + 74832$$
$67$ $$T^{2} - 754T + 140929$$
$71$ $$T^{2} + 678T + 97593$$
$73$ $$T^{2} - 400T - 617072$$
$79$ $$T^{2} + 316 T - 1266044$$
$83$ $$T^{2} + 468T + 11556$$
$89$ $$T^{2} + 1842 T + 525489$$
$97$ $$T^{2} - 2194 T + 1141201$$