# Properties

 Label 121.4.c.f Level $121$ Weight $4$ Character orbit 121.c Analytic conductor $7.139$ Analytic rank $0$ Dimension $8$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,4,Mod(3,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([8]))

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

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

chi := DirichletCharacter("121.3");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## 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: no Analytic conductor: $$7.13923111069$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.324000000.3 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81$$ x^8 + 3*x^6 + 9*x^4 + 27*x^2 + 81 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

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 a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{7} - \beta_{2}) q^{2} + ( - \beta_{6} + 4 \beta_1) q^{3} + (2 \beta_{7} + 2 \beta_{5} + \cdots + 2 \beta_1) q^{4}+ \cdots + ( - 8 \beta_{7} + 22 \beta_{2}) q^{9}+O(q^{10})$$ q + (b7 - b2) * q^2 + (-b6 + 4*b1) * q^3 + (2*b7 + 2*b5 - 4*b4 + 2*b3 + 2*b1) * q^4 + (-b6 - b4 - 8*b3 - b2 - 1) * q^5 + (-13*b6 - 13*b4 - 5*b3 - 13*b2 - 13) * q^6 + (4*b7 + 4*b5 - 10*b4 + 4*b3 + 4*b1) * q^7 + (6*b6 - 10*b1) * q^8 + (-8*b7 + 22*b2) * q^9 $$q + (\beta_{7} - \beta_{2}) q^{2} + ( - \beta_{6} + 4 \beta_1) q^{3} + (2 \beta_{7} + 2 \beta_{5} + \cdots + 2 \beta_1) q^{4}+ \cdots + ( - 275 \beta_{5} + 435) q^{98}+O(q^{100})$$ q + (b7 - b2) * q^2 + (-b6 + 4*b1) * q^3 + (2*b7 + 2*b5 - 4*b4 + 2*b3 + 2*b1) * q^4 + (-b6 - b4 - 8*b3 - b2 - 1) * q^5 + (-13*b6 - 13*b4 - 5*b3 - 13*b2 - 13) * q^6 + (4*b7 + 4*b5 - 10*b4 + 4*b3 + 4*b1) * q^7 + (6*b6 - 10*b1) * q^8 + (-8*b7 + 22*b2) * q^9 + (9*b5 - 25) * q^10 + (-14*b5 - 20) * q^12 + (-20*b7 - 40*b2) * q^13 + (-2*b6 - 6*b1) * q^14 + (-12*b7 - 12*b5 - 97*b4 - 12*b3 - 12*b1) * q^15 + (4*b6 + 4*b4 + 32*b3 + 4*b2 + 4) * q^16 + (-62*b6 - 62*b4 + 12*b3 - 62*b2 - 62) * q^17 + (-30*b7 - 30*b5 - 46*b4 - 30*b3 - 30*b1) * q^18 + (-36*b6 + 60*b1) * q^19 + (30*b7 + 44*b2) * q^20 + (-36*b5 - 38) * q^21 + (36*b5 - 49) * q^23 + (34*b7 - 126*b2) * q^24 + (68*b6 - 16*b1) * q^25 + (20*b7 + 20*b5 - 20*b4 + 20*b3 + 20*b1) * q^26 + (91*b6 + 91*b4 - 12*b3 + 91*b2 + 91) * q^27 + (-64*b6 - 64*b4 + 36*b3 - 64*b2 - 64) * q^28 + (56*b7 + 56*b5 - 72*b4 + 56*b3 + 56*b1) * q^29 + (133*b6 - 109*b1) * q^30 + (-28*b7 - 17*b2) * q^31 + (44*b5 + 52) * q^32 + (50*b5 - 26) * q^34 + (76*b7 + 86*b2) * q^35 + (-40*b6 - 12*b1) * q^36 + (-8*b7 - 8*b5 + 27*b4 - 8*b3 - 8*b1) * q^37 + (-216*b6 - 216*b4 - 96*b3 - 216*b2 - 216) * q^38 + (200*b6 + 200*b4 - 140*b3 + 200*b2 + 200) * q^39 + (58*b7 + 58*b5 + 246*b4 + 58*b3 + 58*b1) * q^40 + (-268*b6 - 4*b1) * q^41 + (-2*b7 - 70*b2) * q^42 + (-16*b5 + 30) * q^43 + (-184*b5 + 214) * q^45 + (-85*b7 + 157*b2) * q^46 + (-136*b6 + 120*b1) * q^47 + (48*b7 + 48*b5 + 388*b4 + 48*b3 + 48*b1) * q^48 + (195*b6 + 195*b4 + 80*b3 + 195*b2 + 195) * q^49 + (116*b6 + 116*b4 + 84*b3 + 116*b2 + 116) * q^50 + (-236*b7 - 236*b5 + 82*b4 - 236*b3 - 236*b1) * q^51 + (280*b6 + 160*b1) * q^52 + (56*b7 - 246*b2) * q^53 + (-79*b5 + 55) * q^54 + (76*b5 + 60) * q^56 + (-204*b7 + 756*b2) * q^57 + (-96*b6 - 16*b1) * q^58 + (-132*b7 - 132*b5 + 317*b4 - 132*b3 - 132*b1) * q^59 + (-316*b6 - 316*b4 + 146*b3 - 316*b2 - 316) * q^60 + (420*b6 + 420*b4 + 184*b3 + 420*b2 + 420) * q^61 + (-11*b7 - 11*b5 - 67*b4 - 11*b3 - 11*b1) * q^62 + (-124*b6 - 8*b1) * q^63 + (-248*b7 + 112*b2) * q^64 + (300*b5 + 440) * q^65 + (20*b5 + 377) * q^67 + (-172*b7 - 320*b2) * q^68 + (481*b6 - 232*b1) * q^69 + (-10*b7 - 10*b5 + 142*b4 - 10*b3 - 10*b1) * q^70 + (339*b6 + 339*b4 - 76*b3 + 339*b2 + 339) * q^71 + (-372*b6 - 372*b4 - 268*b3 - 372*b2 - 372) * q^72 + (468*b7 + 468*b5 + 200*b4 + 468*b3 + 468*b1) * q^73 + (-3*b6 + 19*b1) * q^74 + (288*b7 - 260*b2) * q^75 + (-168*b5 - 216) * q^76 + (-60*b5 - 220) * q^78 + (656*b7 - 158*b2) * q^79 + (-772*b6 + 64*b1) * q^80 + (136*b7 + 136*b5 - 647*b4 + 136*b3 + 136*b1) * q^81 + (-256*b6 - 256*b4 - 264*b3 - 256*b2 - 256) * q^82 + (234*b6 + 234*b4 + 120*b3 + 234*b2 + 234) * q^83 + (-220*b7 - 220*b5 + 368*b4 - 220*b3 - 220*b1) * q^84 + (-226*b6 - 484*b1) * q^85 + (46*b7 - 78*b2) * q^86 + (-232*b5 - 600) * q^87 + (328*b5 - 921) * q^89 + (398*b7 - 766*b2) * q^90 + (640*b6 + 360*b1) * q^91 + (46*b7 + 46*b5 - 20*b4 + 46*b3 + 46*b1) * q^92 + (319*b6 + 319*b4 - 40*b3 + 319*b2 + 319) * q^93 + (-496*b6 - 496*b4 - 256*b3 - 496*b2 - 496) * q^94 + (-348*b7 - 348*b5 - 1476*b4 - 348*b3 - 348*b1) * q^95 + (476*b6 + 164*b1) * q^96 + (-144*b7 + 1097*b2) * q^97 + (-275*b5 + 435) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} - 26 q^{6} + 20 q^{7} - 12 q^{8} - 44 q^{9}+O(q^{10})$$ 8 * 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 $$8 q + 2 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} - 26 q^{6} + 20 q^{7} - 12 q^{8} - 44 q^{9} - 200 q^{10} - 160 q^{12} + 80 q^{13} + 4 q^{14} + 194 q^{15} + 8 q^{16} - 124 q^{17} + 92 q^{18} + 72 q^{19} - 88 q^{20} - 304 q^{21} - 392 q^{23} + 252 q^{24} - 136 q^{25} + 40 q^{26} + 182 q^{27} - 128 q^{28} + 144 q^{29} - 266 q^{30} + 34 q^{31} + 416 q^{32} - 208 q^{34} - 172 q^{35} + 80 q^{36} - 54 q^{37} - 432 q^{38} + 400 q^{39} - 492 q^{40} + 536 q^{41} + 140 q^{42} + 240 q^{43} + 1712 q^{45} - 314 q^{46} + 272 q^{47} - 776 q^{48} + 390 q^{49} + 232 q^{50} - 164 q^{51} - 560 q^{52} + 492 q^{53} + 440 q^{54} + 480 q^{56} - 1512 q^{57} + 192 q^{58} - 634 q^{59} - 632 q^{60} + 840 q^{61} + 134 q^{62} + 248 q^{63} - 224 q^{64} + 3520 q^{65} + 3016 q^{67} + 640 q^{68} - 962 q^{69} - 284 q^{70} + 678 q^{71} - 744 q^{72} - 400 q^{73} + 6 q^{74} + 520 q^{75} - 1728 q^{76} - 1760 q^{78} + 316 q^{79} + 1544 q^{80} + 1294 q^{81} - 512 q^{82} + 468 q^{83} - 736 q^{84} + 452 q^{85} + 156 q^{86} - 4800 q^{87} - 7368 q^{89} + 1532 q^{90} - 1280 q^{91} + 40 q^{92} + 638 q^{93} - 992 q^{94} + 2952 q^{95} - 952 q^{96} - 2194 q^{97} + 3480 q^{98}+O(q^{100})$$ 8 * 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 - 200 * q^10 - 160 * 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 - 304 * q^21 - 392 * 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 + 416 * q^32 - 208 * 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 + 240 * q^43 + 1712 * 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 + 440 * q^54 + 480 * 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 + 3520 * q^65 + 3016 * 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 - 1728 * q^76 - 1760 * 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 - 4800 * q^87 - 7368 * 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 + 3480 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3 $$\beta_{4}$$ $$=$$ $$( \nu^{4} ) / 9$$ (v^4) / 9 $$\beta_{5}$$ $$=$$ $$( \nu^{5} ) / 9$$ (v^5) / 9 $$\beta_{6}$$ $$=$$ $$( \nu^{6} ) / 27$$ (v^6) / 27 $$\beta_{7}$$ $$=$$ $$( \nu^{7} ) / 27$$ (v^7) / 27
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3 $$\nu^{4}$$ $$=$$ $$9\beta_{4}$$ 9*b4 $$\nu^{5}$$ $$=$$ $$9\beta_{5}$$ 9*b5 $$\nu^{6}$$ $$=$$ $$27\beta_{6}$$ 27*b6 $$\nu^{7}$$ $$=$$ $$27\beta_{7}$$ 27*b7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/121\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.535233 + 1.64728i −0.535233 − 1.64728i 1.40126 − 1.01807i −1.40126 + 1.01807i 1.40126 + 1.01807i −1.40126 − 1.01807i 0.535233 − 1.64728i −0.535233 + 1.64728i
−0.592242 + 0.430289i 1.83192 + 5.63806i −2.30653 + 7.09878i 10.4011 + 7.55681i −3.51093 2.55084i −5.23110 + 16.0997i −3.49823 10.7664i −6.58831 + 4.78668i −9.41154
3.2 2.21028 1.60586i −2.44995 7.54017i −0.165602 + 0.509670i −12.0191 8.73238i −17.5235 12.7316i −0.949237 + 2.92145i 7.20643 + 22.1791i −29.0084 + 21.0759i −40.5885
9.1 −0.844250 + 2.59833i 6.41405 4.66008i 0.433551 + 0.314993i 4.59088 + 14.1293i 6.69339 + 20.6001i 2.48514 + 1.80556i −18.8667 + 13.7075i 11.0802 34.1015i −40.5885
9.2 0.226216 0.696222i −4.79602 + 3.48451i 6.03859 + 4.38729i −3.97285 12.2272i 1.34106 + 4.12734i 13.6952 + 9.95015i 9.15848 6.65403i 2.51651 7.74502i −9.41154
27.1 −0.844250 2.59833i 6.41405 + 4.66008i 0.433551 0.314993i 4.59088 14.1293i 6.69339 20.6001i 2.48514 1.80556i −18.8667 13.7075i 11.0802 + 34.1015i −40.5885
27.2 0.226216 + 0.696222i −4.79602 3.48451i 6.03859 4.38729i −3.97285 + 12.2272i 1.34106 4.12734i 13.6952 9.95015i 9.15848 + 6.65403i 2.51651 + 7.74502i −9.41154
81.1 −0.592242 0.430289i 1.83192 5.63806i −2.30653 7.09878i 10.4011 7.55681i −3.51093 + 2.55084i −5.23110 16.0997i −3.49823 + 10.7664i −6.58831 4.78668i −9.41154
81.2 2.21028 + 1.60586i −2.44995 + 7.54017i −0.165602 0.509670i −12.0191 + 8.73238i −17.5235 + 12.7316i −0.949237 2.92145i 7.20643 22.1791i −29.0084 21.0759i −40.5885
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.2 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 121.4.c.f 8
11.b odd 2 1 121.4.c.c 8
11.c even 5 1 121.4.a.c 2
11.c even 5 3 inner 121.4.c.f 8
11.d odd 10 1 11.4.a.a 2
11.d odd 10 3 121.4.c.c 8
33.f even 10 1 99.4.a.c 2
33.h odd 10 1 1089.4.a.v 2
44.g even 10 1 176.4.a.i 2
44.h odd 10 1 1936.4.a.w 2
55.h odd 10 1 275.4.a.b 2
55.l even 20 2 275.4.b.c 4
77.l even 10 1 539.4.a.e 2
88.k even 10 1 704.4.a.n 2
88.p odd 10 1 704.4.a.p 2
132.n odd 10 1 1584.4.a.bc 2
143.l odd 10 1 1859.4.a.a 2
165.r even 10 1 2475.4.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 11.d odd 10 1
99.4.a.c 2 33.f even 10 1
121.4.a.c 2 11.c even 5 1
121.4.c.c 8 11.b odd 2 1
121.4.c.c 8 11.d odd 10 3
121.4.c.f 8 1.a even 1 1 trivial
121.4.c.f 8 11.c even 5 3 inner
176.4.a.i 2 44.g even 10 1
275.4.a.b 2 55.h odd 10 1
275.4.b.c 4 55.l even 20 2
539.4.a.e 2 77.l even 10 1
704.4.a.n 2 88.k even 10 1
704.4.a.p 2 88.p odd 10 1
1089.4.a.v 2 33.h odd 10 1
1584.4.a.bc 2 132.n odd 10 1
1859.4.a.a 2 143.l odd 10 1
1936.4.a.w 2 44.h odd 10 1
2475.4.a.q 2 165.r even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 2T_{2}^{7} + 6T_{2}^{6} - 16T_{2}^{5} + 44T_{2}^{4} + 32T_{2}^{3} + 24T_{2}^{2} + 16T_{2} + 16$$ acting on $$S_{4}^{\mathrm{new}}(121, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 2 T^{7} + \cdots + 16$$
$3$ $$T^{8} - 2 T^{7} + \cdots + 4879681$$
$5$ $$T^{8} + \cdots + 1330863361$$
$7$ $$T^{8} - 20 T^{7} + \cdots + 7311616$$
$11$ $$T^{8}$$
$13$ $$T^{8} + \cdots + 25600000000$$
$17$ $$T^{8} + \cdots + 135530203361536$$
$19$ $$T^{8} + \cdots + 81\!\cdots\!56$$
$23$ $$(T^{2} + 98 T - 1487)^{4}$$
$29$ $$T^{8} + \cdots + 318343244414976$$
$31$ $$T^{8} + \cdots + 18113272128961$$
$37$ $$T^{8} + \cdots + 83156680161$$
$41$ $$T^{8} + \cdots + 26\!\cdots\!76$$
$43$ $$(T^{2} - 60 T + 132)^{4}$$
$47$ $$T^{8} + \cdots + 37\!\cdots\!56$$
$53$ $$T^{8} + \cdots + 68\!\cdots\!96$$
$59$ $$T^{8} + \cdots + 54\!\cdots\!21$$
$61$ $$T^{8} + \cdots + 31\!\cdots\!76$$
$67$ $$(T^{2} - 754 T + 140929)^{4}$$
$71$ $$T^{8} + \cdots + 90\!\cdots\!01$$
$73$ $$T^{8} + \cdots + 14\!\cdots\!56$$
$79$ $$T^{8} + \cdots + 25\!\cdots\!96$$
$83$ $$T^{8} + \cdots + 17\!\cdots\!96$$
$89$ $$(T^{2} + 1842 T + 525489)^{4}$$
$97$ $$T^{8} + \cdots + 16\!\cdots\!01$$