Properties

 Label 11.13.d.a Level $11$ Weight $13$ Character orbit 11.d Analytic conductor $10.054$ Analytic rank $0$ Dimension $44$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [11,13,Mod(2,11)]

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([1]))

N = Newforms(chi, 13, names="a")

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

chi := DirichletCharacter("11.2");

S:= CuspForms(chi, 13);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$13$$ Character orbit: $$[\chi]$$ $$=$$ 11.d (of order $$10$$, 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: $$10.0539319900$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$11$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 5 q^{2} - 1083 q^{3} + 16247 q^{4} - 723 q^{5} + 147195 q^{6} - 219605 q^{7} + 921595 q^{8} - 3695582 q^{9}+O(q^{10})$$ 44 * q - 5 * q^2 - 1083 * q^3 + 16247 * q^4 - 723 * q^5 + 147195 * q^6 - 219605 * q^7 + 921595 * q^8 - 3695582 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 5 q^{2} - 1083 q^{3} + 16247 q^{4} - 723 q^{5} + 147195 q^{6} - 219605 q^{7} + 921595 q^{8} - 3695582 q^{9} + 2565608 q^{11} + 7500402 q^{12} - 6670805 q^{13} - 249422 q^{14} + 31344279 q^{15} - 182249929 q^{16} - 7887605 q^{17} + 296147970 q^{18} - 59195705 q^{19} - 119854528 q^{20} + 615523885 q^{22} - 407959208 q^{23} + 217767665 q^{24} - 1102003838 q^{25} - 1289107598 q^{26} + 945406779 q^{27} + 5018997220 q^{28} - 541321205 q^{29} - 7192403540 q^{30} + 1789230381 q^{31} + 8868147617 q^{33} + 4668584258 q^{34} - 3628416965 q^{35} - 18485968210 q^{36} - 7637278083 q^{37} - 1555712420 q^{38} + 24275524235 q^{39} + 17352998460 q^{40} - 20307362045 q^{41} - 51939651970 q^{42} + 73504485716 q^{44} + 33796434396 q^{45} - 10698599530 q^{46} - 13177100523 q^{47} - 48055484488 q^{48} - 11538977128 q^{49} + 8631835695 q^{50} + 103295029335 q^{51} - 124762325930 q^{52} + 80932774677 q^{53} - 87742505883 q^{55} - 162644024476 q^{56} + 156654446895 q^{57} + 68439789940 q^{58} + 153452464089 q^{59} + 211654410344 q^{60} - 130004870645 q^{61} - 478111392680 q^{62} + 56162082000 q^{63} + 88053755815 q^{64} - 547107987930 q^{66} + 172356405472 q^{67} + 550428178040 q^{68} + 147718471202 q^{69} + 656066411580 q^{70} - 276485879307 q^{71} - 1923582671435 q^{72} + 401204208595 q^{73} + 668106311350 q^{74} + 545281450749 q^{75} - 431450680685 q^{77} + 349062971320 q^{78} + 454959862075 q^{79} + 732789378812 q^{80} + 681177766216 q^{81} + 455720922385 q^{82} - 1768304661305 q^{83} - 2152859747190 q^{84} - 2586009108005 q^{85} - 686286163053 q^{86} + 2647326267355 q^{88} + 587790027856 q^{89} + 6704158241200 q^{90} - 245650237373 q^{91} + 3006033812992 q^{92} + 794315205079 q^{93} - 1053727188680 q^{94} - 7101305012165 q^{95} - 13699998107060 q^{96} + 647608267617 q^{97} + 6944343091354 q^{99}+O(q^{100})$$ 44 * q - 5 * q^2 - 1083 * q^3 + 16247 * q^4 - 723 * q^5 + 147195 * q^6 - 219605 * q^7 + 921595 * q^8 - 3695582 * q^9 + 2565608 * q^11 + 7500402 * q^12 - 6670805 * q^13 - 249422 * q^14 + 31344279 * q^15 - 182249929 * q^16 - 7887605 * q^17 + 296147970 * q^18 - 59195705 * q^19 - 119854528 * q^20 + 615523885 * q^22 - 407959208 * q^23 + 217767665 * q^24 - 1102003838 * q^25 - 1289107598 * q^26 + 945406779 * q^27 + 5018997220 * q^28 - 541321205 * q^29 - 7192403540 * q^30 + 1789230381 * q^31 + 8868147617 * q^33 + 4668584258 * q^34 - 3628416965 * q^35 - 18485968210 * q^36 - 7637278083 * q^37 - 1555712420 * q^38 + 24275524235 * q^39 + 17352998460 * q^40 - 20307362045 * q^41 - 51939651970 * q^42 + 73504485716 * q^44 + 33796434396 * q^45 - 10698599530 * q^46 - 13177100523 * q^47 - 48055484488 * q^48 - 11538977128 * q^49 + 8631835695 * q^50 + 103295029335 * q^51 - 124762325930 * q^52 + 80932774677 * q^53 - 87742505883 * q^55 - 162644024476 * q^56 + 156654446895 * q^57 + 68439789940 * q^58 + 153452464089 * q^59 + 211654410344 * q^60 - 130004870645 * q^61 - 478111392680 * q^62 + 56162082000 * q^63 + 88053755815 * q^64 - 547107987930 * q^66 + 172356405472 * q^67 + 550428178040 * q^68 + 147718471202 * q^69 + 656066411580 * q^70 - 276485879307 * q^71 - 1923582671435 * q^72 + 401204208595 * q^73 + 668106311350 * q^74 + 545281450749 * q^75 - 431450680685 * q^77 + 349062971320 * q^78 + 454959862075 * q^79 + 732789378812 * q^80 + 681177766216 * q^81 + 455720922385 * q^82 - 1768304661305 * q^83 - 2152859747190 * q^84 - 2586009108005 * q^85 - 686286163053 * q^86 + 2647326267355 * q^88 + 587790027856 * q^89 + 6704158241200 * q^90 - 245650237373 * q^91 + 3006033812992 * q^92 + 794315205079 * q^93 - 1053727188680 * q^94 - 7101305012165 * q^95 - 13699998107060 * q^96 + 647608267617 * q^97 + 6944343091354 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −102.224 33.2144i −276.412 + 200.825i 6032.72 + 4383.03i −6925.60 21314.8i 34926.2 11348.2i 128352. 176662.i −212330. 292248.i −128151. + 394409.i 2.40891e6i
2.2 −92.8176 30.1583i 691.311 502.267i 4391.85 + 3190.87i 1083.89 + 3335.88i −79313.3 + 25770.5i −69230.1 + 95287.1i −76445.3 105218.i 61414.8 189015.i 342317.i
2.3 −79.4376 25.8108i −641.631 + 466.172i 2330.39 + 1693.13i 6734.78 + 20727.5i 63001.9 20470.5i −24839.2 + 34188.3i 59673.9 + 82134.1i 30149.4 92790.4i 1.82037e6i
2.4 −28.7412 9.33860i −796.380 + 578.604i −2574.88 1870.76i −7517.68 23137.0i 28292.3 9192.72i −91007.2 + 125261.i 129293. + 177956.i 135214. 416146.i 735192.i
2.5 −21.1209 6.86260i 669.622 486.509i −2914.74 2117.68i −1648.51 5073.61i −17481.7 + 5680.16i 14639.8 20149.9i 100496. + 138321.i 47478.3 146123.i 118472.i
2.6 −8.21322 2.66864i −83.6942 + 60.8074i −3253.40 2363.73i 3579.35 + 11016.1i 849.672 276.075i 61235.8 84283.8i 41204.4 + 56713.0i −160917. + 495252.i 100030.i
2.7 56.5486 + 18.3737i −199.859 + 145.206i −453.588 329.551i 1735.17 + 5340.30i −13969.7 + 4539.04i −101553. + 139776.i −162745. 224000.i −145365. + 447389.i 333868.i
2.8 59.3643 + 19.2886i −999.365 + 726.081i −161.661 117.454i 875.230 + 2693.68i −73331.7 + 23826.9i 110980. 152751.i −157610. 216932.i 307312. 945809.i 176791.i
2.9 73.1050 + 23.7533i 491.809 357.320i 1466.40 + 1065.40i −9397.34 28922.0i 44441.2 14439.8i 26666.1 36702.7i −103169. 142000.i −50026.1 + 153965.i 2.33757e6i
2.10 75.1176 + 24.4072i 1006.77 731.464i 1733.20 + 1259.25i 7624.07 + 23464.5i 93479.4 30373.3i 11948.6 16445.8i −90698.2 124835.i 314329. 967407.i 1.94868e6i
2.11 116.921 + 37.9900i −222.926 + 161.965i 8913.56 + 6476.08i 698.010 + 2148.25i −32217.8 + 10468.2i 9655.01 13289.0i 500175. + 688432.i −140761. + 433218.i 277694.i
6.1 −102.224 + 33.2144i −276.412 200.825i 6032.72 4383.03i −6925.60 + 21314.8i 34926.2 + 11348.2i 128352. + 176662.i −212330. + 292248.i −128151. 394409.i 2.40891e6i
6.2 −92.8176 + 30.1583i 691.311 + 502.267i 4391.85 3190.87i 1083.89 3335.88i −79313.3 25770.5i −69230.1 95287.1i −76445.3 + 105218.i 61414.8 + 189015.i 342317.i
6.3 −79.4376 + 25.8108i −641.631 466.172i 2330.39 1693.13i 6734.78 20727.5i 63001.9 + 20470.5i −24839.2 34188.3i 59673.9 82134.1i 30149.4 + 92790.4i 1.82037e6i
6.4 −28.7412 + 9.33860i −796.380 578.604i −2574.88 + 1870.76i −7517.68 + 23137.0i 28292.3 + 9192.72i −91007.2 125261.i 129293. 177956.i 135214. + 416146.i 735192.i
6.5 −21.1209 + 6.86260i 669.622 + 486.509i −2914.74 + 2117.68i −1648.51 + 5073.61i −17481.7 5680.16i 14639.8 + 20149.9i 100496. 138321.i 47478.3 + 146123.i 118472.i
6.6 −8.21322 + 2.66864i −83.6942 60.8074i −3253.40 + 2363.73i 3579.35 11016.1i 849.672 + 276.075i 61235.8 + 84283.8i 41204.4 56713.0i −160917. 495252.i 100030.i
6.7 56.5486 18.3737i −199.859 145.206i −453.588 + 329.551i 1735.17 5340.30i −13969.7 4539.04i −101553. 139776.i −162745. + 224000.i −145365. 447389.i 333868.i
6.8 59.3643 19.2886i −999.365 726.081i −161.661 + 117.454i 875.230 2693.68i −73331.7 23826.9i 110980. + 152751.i −157610. + 216932.i 307312. + 945809.i 176791.i
6.9 73.1050 23.7533i 491.809 + 357.320i 1466.40 1065.40i −9397.34 + 28922.0i 44441.2 + 14439.8i 26666.1 + 36702.7i −103169. + 142000.i −50026.1 153965.i 2.33757e6i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.11 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.d odd 10 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.13.d.a 44
11.d odd 10 1 inner 11.13.d.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.13.d.a 44 1.a even 1 1 trivial
11.13.d.a 44 11.d odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{13}^{\mathrm{new}}(11, [\chi])$$.