# Properties

 Label 11.10.c.a Level $11$ Weight $10$ Character orbit 11.c Analytic conductor $5.665$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [11,10,Mod(3,11)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(11, base_ring=CyclotomicField(10))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("11.3");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 11.c (of order $$5$$, degree $$4$$, 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: $$5.66539419780$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9}+O(q^{10})$$ 32 * q - 21 * q^2 + 69 * q^3 - 1625 * q^4 + 225 * q^5 + 8173 * q^6 - 10675 * q^7 + 22863 * q^8 - 93587 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9} - 16272 q^{10} - 44637 q^{11} + 339622 q^{12} + 164099 q^{13} - 125832 q^{14} - 547009 q^{15} - 1428705 q^{16} + 1430145 q^{17} + 1483170 q^{18} + 2175116 q^{19} - 79194 q^{20} - 3427202 q^{21} - 4280751 q^{22} - 4045452 q^{23} + 15199185 q^{24} - 1591199 q^{25} - 11064048 q^{26} - 377496 q^{27} + 3790710 q^{28} + 2621889 q^{29} + 18259050 q^{30} + 3571591 q^{31} - 56548860 q^{32} + 10586762 q^{33} + 8515154 q^{34} + 3015333 q^{35} + 3379584 q^{36} + 20945529 q^{37} + 55333848 q^{38} + 16458893 q^{39} - 21310356 q^{40} - 53829591 q^{41} - 22644634 q^{42} - 4383434 q^{43} - 21997776 q^{44} - 165146020 q^{45} - 2025868 q^{46} + 78754497 q^{47} + 281712140 q^{48} + 227397397 q^{49} - 74287941 q^{50} - 117901326 q^{51} - 157487230 q^{52} - 30528081 q^{53} - 337412524 q^{54} - 248082383 q^{55} - 327187608 q^{56} + 326803764 q^{57} + 279888572 q^{58} + 254767908 q^{59} + 1309524756 q^{60} - 10620465 q^{61} + 686572332 q^{62} - 172608476 q^{63} - 1338445825 q^{64} - 869006394 q^{65} - 2484759594 q^{66} - 946203354 q^{67} + 2053775934 q^{68} + 1552022058 q^{69} - 10909448 q^{70} + 1396168719 q^{71} + 2446779799 q^{72} + 1566720997 q^{73} - 604530504 q^{74} - 2015724064 q^{75} - 4233309866 q^{76} - 3256040955 q^{77} - 4298015660 q^{78} + 2890358835 q^{79} + 5092570836 q^{80} + 1937450890 q^{81} + 4014664309 q^{82} - 480997020 q^{83} + 4617031020 q^{84} - 3775782881 q^{85} - 3721184697 q^{86} - 4032861402 q^{87} - 8695160705 q^{88} - 1559422158 q^{89} + 5473366174 q^{90} + 5038455823 q^{91} + 6839439726 q^{92} + 9679941995 q^{93} + 1219591670 q^{94} + 2592967899 q^{95} - 11940342116 q^{96} - 3010541778 q^{97} - 7371061380 q^{98} - 9888905941 q^{99}+O(q^{100})$$ 32 * q - 21 * q^2 + 69 * q^3 - 1625 * q^4 + 225 * q^5 + 8173 * q^6 - 10675 * q^7 + 22863 * q^8 - 93587 * q^9 - 16272 * q^10 - 44637 * q^11 + 339622 * q^12 + 164099 * q^13 - 125832 * q^14 - 547009 * q^15 - 1428705 * q^16 + 1430145 * q^17 + 1483170 * q^18 + 2175116 * q^19 - 79194 * q^20 - 3427202 * q^21 - 4280751 * q^22 - 4045452 * q^23 + 15199185 * q^24 - 1591199 * q^25 - 11064048 * q^26 - 377496 * q^27 + 3790710 * q^28 + 2621889 * q^29 + 18259050 * q^30 + 3571591 * q^31 - 56548860 * q^32 + 10586762 * q^33 + 8515154 * q^34 + 3015333 * q^35 + 3379584 * q^36 + 20945529 * q^37 + 55333848 * q^38 + 16458893 * q^39 - 21310356 * q^40 - 53829591 * q^41 - 22644634 * q^42 - 4383434 * q^43 - 21997776 * q^44 - 165146020 * q^45 - 2025868 * q^46 + 78754497 * q^47 + 281712140 * q^48 + 227397397 * q^49 - 74287941 * q^50 - 117901326 * q^51 - 157487230 * q^52 - 30528081 * q^53 - 337412524 * q^54 - 248082383 * q^55 - 327187608 * q^56 + 326803764 * q^57 + 279888572 * q^58 + 254767908 * q^59 + 1309524756 * q^60 - 10620465 * q^61 + 686572332 * q^62 - 172608476 * q^63 - 1338445825 * q^64 - 869006394 * q^65 - 2484759594 * q^66 - 946203354 * q^67 + 2053775934 * q^68 + 1552022058 * q^69 - 10909448 * q^70 + 1396168719 * q^71 + 2446779799 * q^72 + 1566720997 * q^73 - 604530504 * q^74 - 2015724064 * q^75 - 4233309866 * q^76 - 3256040955 * q^77 - 4298015660 * q^78 + 2890358835 * q^79 + 5092570836 * q^80 + 1937450890 * q^81 + 4014664309 * q^82 - 480997020 * q^83 + 4617031020 * q^84 - 3775782881 * q^85 - 3721184697 * q^86 - 4032861402 * q^87 - 8695160705 * q^88 - 1559422158 * q^89 + 5473366174 * q^90 + 5038455823 * q^91 + 6839439726 * q^92 + 9679941995 * q^93 + 1219591670 * q^94 + 2592967899 * q^95 - 11940342116 * q^96 - 3010541778 * q^97 - 7371061380 * q^98 - 9888905941 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −33.4202 + 24.2812i −18.4920 56.9126i 369.117 1136.03i −314.303 228.354i 1999.91 + 1453.02i −1491.31 + 4589.77i 8712.23 + 26813.5i 13026.8 9464.52i 16048.8
3.2 −21.4033 + 15.5504i 76.8224 + 236.435i 58.0694 178.719i 1489.97 + 1082.53i −5320.92 3865.87i −720.731 + 2218.18i −2649.49 8154.29i −34076.0 + 24757.6i −48724.0
3.3 −14.4842 + 10.5234i −4.17901 12.8617i −59.1658 + 182.094i −591.288 429.596i 195.879 + 142.314i 2103.13 6472.78i −3891.91 11978.1i 15775.9 11461.9i 13085.2
3.4 −7.91186 + 5.74830i −77.1605 237.476i −128.662 + 395.981i 690.317 + 501.544i 1975.57 + 1435.33i −2608.39 + 8027.81i −2805.56 8634.62i −34517.1 + 25078.1i −8344.72
3.5 7.79135 5.66074i 51.4829 + 158.448i −129.556 + 398.731i −1985.48 1442.54i 1298.06 + 943.093i −2299.51 + 7077.16i 2771.43 + 8529.59i −6531.46 + 4745.38i −23635.4
3.6 12.1558 8.83167i 13.4151 + 41.2873i −88.4528 + 272.230i 1436.78 + 1043.88i 527.706 + 383.401i 741.893 2283.31i 3706.29 + 11406.8i 14399.2 10461.6i 26684.4
3.7 24.5942 17.8687i −45.7664 140.854i 127.366 391.991i −601.873 437.286i −3642.47 2646.41i 430.583 1325.20i 937.875 + 2886.48i −1821.51 + 1323.41i −22616.3
3.8 35.8136 26.0201i 63.0538 + 194.060i 447.352 1376.81i 376.543 + 273.575i 7307.64 + 5309.31i −442.773 + 1362.72i −12799.4 39392.6i −17759.5 + 12903.0i 20603.8
4.1 −33.4202 24.2812i −18.4920 + 56.9126i 369.117 + 1136.03i −314.303 + 228.354i 1999.91 1453.02i −1491.31 4589.77i 8712.23 26813.5i 13026.8 + 9464.52i 16048.8
4.2 −21.4033 15.5504i 76.8224 236.435i 58.0694 + 178.719i 1489.97 1082.53i −5320.92 + 3865.87i −720.731 2218.18i −2649.49 + 8154.29i −34076.0 24757.6i −48724.0
4.3 −14.4842 10.5234i −4.17901 + 12.8617i −59.1658 182.094i −591.288 + 429.596i 195.879 142.314i 2103.13 + 6472.78i −3891.91 + 11978.1i 15775.9 + 11461.9i 13085.2
4.4 −7.91186 5.74830i −77.1605 + 237.476i −128.662 395.981i 690.317 501.544i 1975.57 1435.33i −2608.39 8027.81i −2805.56 + 8634.62i −34517.1 25078.1i −8344.72
4.5 7.79135 + 5.66074i 51.4829 158.448i −129.556 398.731i −1985.48 + 1442.54i 1298.06 943.093i −2299.51 7077.16i 2771.43 8529.59i −6531.46 4745.38i −23635.4
4.6 12.1558 + 8.83167i 13.4151 41.2873i −88.4528 272.230i 1436.78 1043.88i 527.706 383.401i 741.893 + 2283.31i 3706.29 11406.8i 14399.2 + 10461.6i 26684.4
4.7 24.5942 + 17.8687i −45.7664 + 140.854i 127.366 + 391.991i −601.873 + 437.286i −3642.47 + 2646.41i 430.583 + 1325.20i 937.875 2886.48i −1821.51 1323.41i −22616.3
4.8 35.8136 + 26.0201i 63.0538 194.060i 447.352 + 1376.81i 376.543 273.575i 7307.64 5309.31i −442.773 1362.72i −12799.4 + 39392.6i −17759.5 12903.0i 20603.8
5.1 −13.1726 40.5409i 121.812 + 88.5013i −1055.84 + 767.109i −602.641 + 1854.74i 1983.36 6104.15i −3576.09 + 2598.18i 27350.5 + 19871.3i 923.206 + 2841.34i 83131.1
5.2 −10.8459 33.3802i −181.934 132.183i −582.388 + 423.130i 311.221 957.840i −2439.05 + 7506.62i 7268.43 5280.82i 5902.49 + 4288.41i 9545.25 + 29377.3i −35348.4
5.3 −6.86703 21.1346i 42.7536 + 31.0623i 14.7033 10.6826i 597.948 1840.30i 362.897 1116.88i −7706.91 + 5599.40i −9531.54 6925.07i −5219.38 16063.6i −43000.0
5.4 −4.28817 13.1976i 55.2813 + 40.1642i 258.428 187.759i −237.559 + 731.131i 293.017 901.813i 9125.19 6629.84i −9334.16 6781.66i −4639.52 14279.0i 10667.9
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 9.8 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 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.10.c.a 32
3.b odd 2 1 99.10.f.a 32
11.c even 5 1 inner 11.10.c.a 32
11.c even 5 1 121.10.a.h 16
11.d odd 10 1 121.10.a.i 16
33.h odd 10 1 99.10.f.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.10.c.a 32 1.a even 1 1 trivial
11.10.c.a 32 11.c even 5 1 inner
99.10.f.a 32 3.b odd 2 1
99.10.f.a 32 33.h odd 10 1
121.10.a.h 16 11.c even 5 1
121.10.a.i 16 11.d odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{10}^{\mathrm{new}}(11, [\chi])$$.