# Properties

 Label 11.14.c.a Level $11$ Weight $14$ Character orbit 11.c Analytic conductor $11.795$ Analytic rank $0$ Dimension $48$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [11,14,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, 14, names="a")

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

chi := DirichletCharacter("11.3");

S:= CuspForms(chi, 14);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$14$$ 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: $$11.7954021847$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q + 59 q^{2} - 864 q^{3} - 44697 q^{4} - 52240 q^{5} - 130115 q^{6} + 390696 q^{7} + 876527 q^{8} - 3520244 q^{9}+O(q^{10})$$ 48 * q + 59 * q^2 - 864 * q^3 - 44697 * q^4 - 52240 * q^5 - 130115 * q^6 + 390696 * q^7 + 876527 * q^8 - 3520244 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q + 59 q^{2} - 864 q^{3} - 44697 q^{4} - 52240 q^{5} - 130115 q^{6} + 390696 q^{7} + 876527 q^{8} - 3520244 q^{9} - 16293072 q^{10} - 82532 q^{11} + 62972038 q^{12} - 7063968 q^{13} + 2946296 q^{14} + 81100736 q^{15} + 60644511 q^{16} - 291410392 q^{17} + 126484674 q^{18} + 365166708 q^{19} - 788032074 q^{20} - 434553632 q^{21} - 733739679 q^{22} + 4188874272 q^{23} + 1308210897 q^{24} - 5079263484 q^{25} - 7197334352 q^{26} - 3672135540 q^{27} + 15293042118 q^{28} + 9462180848 q^{29} + 14934224730 q^{30} + 3808229976 q^{31} - 37924904892 q^{32} - 29220440404 q^{33} - 42631809486 q^{34} + 81447084648 q^{35} + 20776466352 q^{36} - 92150146992 q^{37} - 85759798456 q^{38} + 122431960136 q^{39} + 156222512604 q^{40} + 224837812784 q^{41} - 100995983242 q^{42} - 323982068136 q^{43} - 117395188832 q^{44} + 101655796640 q^{45} + 504337662372 q^{46} - 131674300816 q^{47} - 508856741764 q^{48} - 364186485732 q^{49} + 292932400779 q^{50} - 46205145276 q^{51} + 971895220146 q^{52} - 374673041144 q^{53} - 171438753484 q^{54} - 609291766368 q^{55} - 55316542488 q^{56} + 581441351340 q^{57} - 172510123284 q^{58} - 866818316348 q^{59} + 450475433796 q^{60} + 396717353376 q^{61} + 2190422590172 q^{62} - 814954660544 q^{63} + 687266758335 q^{64} - 1914281234704 q^{65} + 387191529606 q^{66} - 1385875174920 q^{67} - 1442260754434 q^{68} + 2415789828408 q^{69} + 6076465410072 q^{70} + 2500983206448 q^{71} - 6853058084009 q^{72} - 6144016960080 q^{73} - 9894533340936 q^{74} + 2088242191076 q^{75} + 7382638646262 q^{76} + 8181659595976 q^{77} + 485347989556 q^{78} - 9413148440424 q^{79} - 4271997351884 q^{80} + 19193312847832 q^{81} + 24451991571525 q^{82} - 4748400435188 q^{83} - 9210465158484 q^{84} - 9425037508176 q^{85} - 2302249550185 q^{86} - 22158779783712 q^{87} - 25622627417409 q^{88} - 12037611693640 q^{89} + 8065916202334 q^{90} + 23937874042464 q^{91} + 17043827782942 q^{92} + 18075650378312 q^{93} + 10662034186230 q^{94} + 17645734105464 q^{95} + 8702791487212 q^{96} - 18472776936684 q^{97} - 52286290693988 q^{98} - 51636755872396 q^{99}+O(q^{100})$$ 48 * q + 59 * q^2 - 864 * q^3 - 44697 * q^4 - 52240 * q^5 - 130115 * q^6 + 390696 * q^7 + 876527 * q^8 - 3520244 * q^9 - 16293072 * q^10 - 82532 * q^11 + 62972038 * q^12 - 7063968 * q^13 + 2946296 * q^14 + 81100736 * q^15 + 60644511 * q^16 - 291410392 * q^17 + 126484674 * q^18 + 365166708 * q^19 - 788032074 * q^20 - 434553632 * q^21 - 733739679 * q^22 + 4188874272 * q^23 + 1308210897 * q^24 - 5079263484 * q^25 - 7197334352 * q^26 - 3672135540 * q^27 + 15293042118 * q^28 + 9462180848 * q^29 + 14934224730 * q^30 + 3808229976 * q^31 - 37924904892 * q^32 - 29220440404 * q^33 - 42631809486 * q^34 + 81447084648 * q^35 + 20776466352 * q^36 - 92150146992 * q^37 - 85759798456 * q^38 + 122431960136 * q^39 + 156222512604 * q^40 + 224837812784 * q^41 - 100995983242 * q^42 - 323982068136 * q^43 - 117395188832 * q^44 + 101655796640 * q^45 + 504337662372 * q^46 - 131674300816 * q^47 - 508856741764 * q^48 - 364186485732 * q^49 + 292932400779 * q^50 - 46205145276 * q^51 + 971895220146 * q^52 - 374673041144 * q^53 - 171438753484 * q^54 - 609291766368 * q^55 - 55316542488 * q^56 + 581441351340 * q^57 - 172510123284 * q^58 - 866818316348 * q^59 + 450475433796 * q^60 + 396717353376 * q^61 + 2190422590172 * q^62 - 814954660544 * q^63 + 687266758335 * q^64 - 1914281234704 * q^65 + 387191529606 * q^66 - 1385875174920 * q^67 - 1442260754434 * q^68 + 2415789828408 * q^69 + 6076465410072 * q^70 + 2500983206448 * q^71 - 6853058084009 * q^72 - 6144016960080 * q^73 - 9894533340936 * q^74 + 2088242191076 * q^75 + 7382638646262 * q^76 + 8181659595976 * q^77 + 485347989556 * q^78 - 9413148440424 * q^79 - 4271997351884 * q^80 + 19193312847832 * q^81 + 24451991571525 * q^82 - 4748400435188 * q^83 - 9210465158484 * q^84 - 9425037508176 * q^85 - 2302249550185 * q^86 - 22158779783712 * q^87 - 25622627417409 * q^88 - 12037611693640 * q^89 + 8065916202334 * q^90 + 23937874042464 * q^91 + 17043827782942 * q^92 + 18075650378312 * q^93 + 10662034186230 * q^94 + 17645734105464 * q^95 + 8702791487212 * q^96 - 18472776936684 * q^97 - 52286290693988 * q^98 - 51636755872396 * 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 −133.057 + 96.6717i −300.655 925.320i 5827.33 17934.7i 6362.73 + 4622.79i 129456. + 94055.7i −62415.4 + 192095.i 542063. + 1.66830e6i 524011. 380716.i −1.29350e6
3.2 −101.017 + 73.3929i 277.356 + 853.615i 2286.38 7036.76i 21001.8 + 15258.7i −90666.9 65873.3i 183952. 566147.i −30602.4 94184.4i 638103. 463609.i −3.24141e6
3.3 −97.9783 + 71.1854i 552.972 + 1701.87i 2000.91 6158.18i −26557.7 19295.3i −175328. 127383.i −163681. + 503758.i −64254.1 197754.i −1.30076e6 + 945060.i 3.97562e6
3.4 −64.6899 + 46.9999i −352.519 1084.94i −555.683 + 1710.22i −37053.3 26920.8i 73796.5 + 53616.3i 24379.1 75031.2i −246852. 759732.i 237007. 172195.i 3.66225e6
3.5 −38.1571 + 27.7228i −502.353 1546.08i −1844.05 + 5675.41i 29618.8 + 21519.3i 62030.1 + 45067.5i −28662.8 + 88215.0i −206371. 635143.i −848181. + 616240.i −1.72674e6
3.6 −16.5754 + 12.0428i 423.594 + 1303.69i −2401.75 + 7391.83i 31221.5 + 22683.7i −22721.3 16508.0i −38277.0 + 117804.i −101074. 311073.i −230340. + 167352.i −790684.
3.7 19.9401 14.4873i 214.083 + 658.880i −2343.74 + 7213.30i −38433.9 27923.9i 13814.3 + 10036.6i 86353.5 265769.i 120161. + 369817.i 901543. 655009.i −1.17092e6
3.8 55.3587 40.2205i −140.614 432.765i −1084.57 + 3337.95i 7579.93 + 5507.14i −25190.2 18301.8i −99964.7 + 307660.i 247435. + 761526.i 1.12232e6 815414.i 641115.
3.9 80.7641 58.6786i −728.875 2243.25i 548.202 1687.19i −12915.0 9383.32i −190497. 138404.i 94448.4 290682.i 197989. + 609348.i −3.21106e6 + 2.33297e6i −1.59367e6
3.10 96.0816 69.8074i 738.641 + 2273.30i 1827.14 5623.35i −7207.48 5236.54i 229663. + 166860.i −816.456 + 2512.79i 83648.5 + 257444.i −3.33248e6 + 2.42119e6i −1.05806e6
3.11 112.273 81.5711i 22.6986 + 69.8590i 3419.92 10525.4i 39045.2 + 28368.0i 8246.91 + 5991.73i 128438. 395292.i −123297. 379468.i 1.28547e6 933948.i 6.69773e6
3.12 137.025 99.5545i −84.9208 261.359i 6333.29 19491.9i −37394.7 27168.8i −37655.8 27358.5i −156206. + 480753.i −643923. 1.98179e6i 1.22874e6 892730.i −7.82879e6
4.1 −133.057 96.6717i −300.655 + 925.320i 5827.33 + 17934.7i 6362.73 4622.79i 129456. 94055.7i −62415.4 192095.i 542063. 1.66830e6i 524011. + 380716.i −1.29350e6
4.2 −101.017 73.3929i 277.356 853.615i 2286.38 + 7036.76i 21001.8 15258.7i −90666.9 + 65873.3i 183952. + 566147.i −30602.4 + 94184.4i 638103. + 463609.i −3.24141e6
4.3 −97.9783 71.1854i 552.972 1701.87i 2000.91 + 6158.18i −26557.7 + 19295.3i −175328. + 127383.i −163681. 503758.i −64254.1 + 197754.i −1.30076e6 945060.i 3.97562e6
4.4 −64.6899 46.9999i −352.519 + 1084.94i −555.683 1710.22i −37053.3 + 26920.8i 73796.5 53616.3i 24379.1 + 75031.2i −246852. + 759732.i 237007. + 172195.i 3.66225e6
4.5 −38.1571 27.7228i −502.353 + 1546.08i −1844.05 5675.41i 29618.8 21519.3i 62030.1 45067.5i −28662.8 88215.0i −206371. + 635143.i −848181. 616240.i −1.72674e6
4.6 −16.5754 12.0428i 423.594 1303.69i −2401.75 7391.83i 31221.5 22683.7i −22721.3 + 16508.0i −38277.0 117804.i −101074. + 311073.i −230340. 167352.i −790684.
4.7 19.9401 + 14.4873i 214.083 658.880i −2343.74 7213.30i −38433.9 + 27923.9i 13814.3 10036.6i 86353.5 + 265769.i 120161. 369817.i 901543. + 655009.i −1.17092e6
4.8 55.3587 + 40.2205i −140.614 + 432.765i −1084.57 3337.95i 7579.93 5507.14i −25190.2 + 18301.8i −99964.7 307660.i 247435. 761526.i 1.12232e6 + 815414.i 641115.
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 9.12 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 11.14.c.a 48
11.c even 5 1 inner 11.14.c.a 48
11.c even 5 1 121.14.a.h 24
11.d odd 10 1 121.14.a.i 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.14.c.a 48 1.a even 1 1 trivial
11.14.c.a 48 11.c even 5 1 inner
121.14.a.h 24 11.c even 5 1
121.14.a.i 24 11.d odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{14}^{\mathrm{new}}(11, [\chi])$$.