# Properties

 Label 11.12.c.a Level $11$ Weight $12$ Character orbit 11.c Analytic conductor $8.452$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [11,12,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, 12, names="a")

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

chi := DirichletCharacter("11.3");

S:= CuspForms(chi, 12);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$12$$ 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: $$8.45177498616$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$10$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 11 q^{2} - 276 q^{3} - 6137 q^{4} + 6038 q^{5} - 30371 q^{6} + 55440 q^{7} - 347809 q^{8} - 240812 q^{9}+O(q^{10})$$ 40 * q + 11 * q^2 - 276 * q^3 - 6137 * q^4 + 6038 * q^5 - 30371 * q^6 + 55440 * q^7 - 347809 * q^8 - 240812 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 11 q^{2} - 276 q^{3} - 6137 q^{4} + 6038 q^{5} - 30371 q^{6} + 55440 q^{7} - 347809 q^{8} - 240812 q^{9} + 782320 q^{10} - 288860 q^{11} - 5871146 q^{12} - 794178 q^{13} + 9181256 q^{14} - 9707296 q^{15} - 10033697 q^{16} + 11803154 q^{17} - 2661582 q^{18} - 33326568 q^{19} + 34173582 q^{20} + 72722452 q^{21} + 77957825 q^{22} + 75694752 q^{23} - 297034287 q^{24} - 118409216 q^{25} + 605949712 q^{26} + 348141600 q^{27} - 546845442 q^{28} - 400113538 q^{29} - 828772494 q^{30} + 514005960 q^{31} - 84440796 q^{32} + 1211751794 q^{33} - 899802542 q^{34} - 1820315508 q^{35} + 971066424 q^{36} + 1207982934 q^{37} + 1815036992 q^{38} - 2825057488 q^{39} - 1263004028 q^{40} - 3269102782 q^{41} + 4736681342 q^{42} + 2749122728 q^{43} + 5968984648 q^{44} - 1896048208 q^{45} - 1240862964 q^{46} - 2081937184 q^{47} - 4090087516 q^{48} - 2514603052 q^{49} + 7217881803 q^{50} - 1172394432 q^{51} + 23226253050 q^{52} - 7132071266 q^{53} - 34172203516 q^{54} - 9211291408 q^{55} + 33017775912 q^{56} + 30518561370 q^{57} - 34218647932 q^{58} - 12866186432 q^{59} - 55228105332 q^{60} + 2702502990 q^{61} - 18405787180 q^{62} + 33243939784 q^{63} + 28267347551 q^{64} + 28056414980 q^{65} + 64510531470 q^{66} - 549542392 q^{67} - 17332191778 q^{68} - 33459566280 q^{69} - 90854725304 q^{70} - 37002489528 q^{71} + 37881288247 q^{72} - 41936764254 q^{73} + 118297253448 q^{74} + 72805696880 q^{75} + 136670345702 q^{76} + 85397422330 q^{77} + 113064629380 q^{78} - 140458240296 q^{79} - 257276697596 q^{80} + 28633292464 q^{81} - 94584428059 q^{82} - 238929577568 q^{83} - 339093796404 q^{84} + 48160626338 q^{85} + 176827923479 q^{86} + 448810202016 q^{87} + 742233436703 q^{88} + 311499953960 q^{89} - 913432848482 q^{90} - 410566597652 q^{91} + 77316479782 q^{92} - 480470187862 q^{93} - 508753213738 q^{94} - 524637979272 q^{95} + 324858127396 q^{96} + 474017921070 q^{97} + 1187521497580 q^{98} + 1382319662900 q^{99}+O(q^{100})$$ 40 * q + 11 * q^2 - 276 * q^3 - 6137 * q^4 + 6038 * q^5 - 30371 * q^6 + 55440 * q^7 - 347809 * q^8 - 240812 * q^9 + 782320 * q^10 - 288860 * q^11 - 5871146 * q^12 - 794178 * q^13 + 9181256 * q^14 - 9707296 * q^15 - 10033697 * q^16 + 11803154 * q^17 - 2661582 * q^18 - 33326568 * q^19 + 34173582 * q^20 + 72722452 * q^21 + 77957825 * q^22 + 75694752 * q^23 - 297034287 * q^24 - 118409216 * q^25 + 605949712 * q^26 + 348141600 * q^27 - 546845442 * q^28 - 400113538 * q^29 - 828772494 * q^30 + 514005960 * q^31 - 84440796 * q^32 + 1211751794 * q^33 - 899802542 * q^34 - 1820315508 * q^35 + 971066424 * q^36 + 1207982934 * q^37 + 1815036992 * q^38 - 2825057488 * q^39 - 1263004028 * q^40 - 3269102782 * q^41 + 4736681342 * q^42 + 2749122728 * q^43 + 5968984648 * q^44 - 1896048208 * q^45 - 1240862964 * q^46 - 2081937184 * q^47 - 4090087516 * q^48 - 2514603052 * q^49 + 7217881803 * q^50 - 1172394432 * q^51 + 23226253050 * q^52 - 7132071266 * q^53 - 34172203516 * q^54 - 9211291408 * q^55 + 33017775912 * q^56 + 30518561370 * q^57 - 34218647932 * q^58 - 12866186432 * q^59 - 55228105332 * q^60 + 2702502990 * q^61 - 18405787180 * q^62 + 33243939784 * q^63 + 28267347551 * q^64 + 28056414980 * q^65 + 64510531470 * q^66 - 549542392 * q^67 - 17332191778 * q^68 - 33459566280 * q^69 - 90854725304 * q^70 - 37002489528 * q^71 + 37881288247 * q^72 - 41936764254 * q^73 + 118297253448 * q^74 + 72805696880 * q^75 + 136670345702 * q^76 + 85397422330 * q^77 + 113064629380 * q^78 - 140458240296 * q^79 - 257276697596 * q^80 + 28633292464 * q^81 - 94584428059 * q^82 - 238929577568 * q^83 - 339093796404 * q^84 + 48160626338 * q^85 + 176827923479 * q^86 + 448810202016 * q^87 + 742233436703 * q^88 + 311499953960 * q^89 - 913432848482 * q^90 - 410566597652 * q^91 + 77316479782 * q^92 - 480470187862 * q^93 - 508753213738 * q^94 - 524637979272 * q^95 + 324858127396 * q^96 + 474017921070 * q^97 + 1187521497580 * q^98 + 1382319662900 * 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 −66.2723 + 48.1496i 155.409 + 478.301i 1440.76 4434.21i −3908.19 2839.47i −33329.3 24215.2i 15024.9 46241.9i 66180.4 + 203682.i −61304.7 + 44540.5i 395724.
3.2 −55.1898 + 40.0977i −247.284 761.061i 805.219 2478.21i −3143.04 2283.56i 44164.4 + 32087.3i 15890.9 48907.1i 11757.6 + 36186.1i −374750. + 272272.i 265029.
3.3 −49.0498 + 35.6368i −8.11730 24.9825i 503.038 1548.19i 10386.7 + 7546.38i 1288.45 + 936.111i −13532.7 + 41649.2i −7871.40 24225.7i 142757. 103719.i −778394.
3.4 −32.2868 + 23.4578i −12.6213 38.8444i −140.693 + 433.009i −5225.04 3796.22i 1318.70 + 958.095i −10421.9 + 32075.4i −30871.8 95013.6i 141965. 103144.i 257751.
3.5 −6.97513 + 5.06773i 187.776 + 577.916i −609.896 + 1877.07i −462.160 335.779i −4238.49 3079.44i 5696.63 17532.4i −10714.8 32976.7i −155412. + 112914.i 4925.26
3.6 4.91043 3.56764i −108.526 334.010i −621.482 + 1912.73i 6091.05 + 4425.40i −1724.54 1252.95i 20002.0 61559.7i 7613.44 + 23431.7i 43530.5 31626.8i 45697.9
3.7 21.2132 15.4123i −143.663 442.150i −420.407 + 1293.88i −5027.95 3653.02i −9862.09 7165.22i −13499.1 + 41546.0i 27617.8 + 84998.8i −31542.8 + 22917.2i −162960.
3.8 44.3233 32.2028i 122.608 + 377.349i 294.671 906.903i 3995.51 + 2902.91i 17586.1 + 12777.0i −12513.3 + 38512.0i 18528.6 + 57025.2i 15955.6 11592.4i 270576.
3.9 56.7091 41.2016i 40.8374 + 125.685i 885.488 2725.25i −8463.91 6149.39i 7494.26 + 5444.90i 23115.4 71141.9i −17707.8 54499.0i 129186. 93859.1i −733345.
3.10 66.9203 48.6204i −196.291 604.123i 1481.51 4559.62i 7813.25 + 5676.66i −42508.6 30884.3i −7835.02 + 24113.7i −70198.2 216048.i −183119. + 133044.i 798867.
4.1 −66.2723 48.1496i 155.409 478.301i 1440.76 + 4434.21i −3908.19 + 2839.47i −33329.3 + 24215.2i 15024.9 + 46241.9i 66180.4 203682.i −61304.7 44540.5i 395724.
4.2 −55.1898 40.0977i −247.284 + 761.061i 805.219 + 2478.21i −3143.04 + 2283.56i 44164.4 32087.3i 15890.9 + 48907.1i 11757.6 36186.1i −374750. 272272.i 265029.
4.3 −49.0498 35.6368i −8.11730 + 24.9825i 503.038 + 1548.19i 10386.7 7546.38i 1288.45 936.111i −13532.7 41649.2i −7871.40 + 24225.7i 142757. + 103719.i −778394.
4.4 −32.2868 23.4578i −12.6213 + 38.8444i −140.693 433.009i −5225.04 + 3796.22i 1318.70 958.095i −10421.9 32075.4i −30871.8 + 95013.6i 141965. + 103144.i 257751.
4.5 −6.97513 5.06773i 187.776 577.916i −609.896 1877.07i −462.160 + 335.779i −4238.49 + 3079.44i 5696.63 + 17532.4i −10714.8 + 32976.7i −155412. 112914.i 4925.26
4.6 4.91043 + 3.56764i −108.526 + 334.010i −621.482 1912.73i 6091.05 4425.40i −1724.54 + 1252.95i 20002.0 + 61559.7i 7613.44 23431.7i 43530.5 + 31626.8i 45697.9
4.7 21.2132 + 15.4123i −143.663 + 442.150i −420.407 1293.88i −5027.95 + 3653.02i −9862.09 + 7165.22i −13499.1 41546.0i 27617.8 84998.8i −31542.8 22917.2i −162960.
4.8 44.3233 + 32.2028i 122.608 377.349i 294.671 + 906.903i 3995.51 2902.91i 17586.1 12777.0i −12513.3 38512.0i 18528.6 57025.2i 15955.6 + 11592.4i 270576.
4.9 56.7091 + 41.2016i 40.8374 125.685i 885.488 + 2725.25i −8463.91 + 6149.39i 7494.26 5444.90i 23115.4 + 71141.9i −17707.8 + 54499.0i 129186. + 93859.1i −733345.
4.10 66.9203 + 48.6204i −196.291 + 604.123i 1481.51 + 4559.62i 7813.25 5676.66i −42508.6 + 30884.3i −7835.02 24113.7i −70198.2 + 216048.i −183119. 133044.i 798867.
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 9.10 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 11.12.c.a 40
3.b odd 2 1 99.12.f.a 40
11.c even 5 1 inner 11.12.c.a 40
11.c even 5 1 121.12.a.j 20
11.d odd 10 1 121.12.a.h 20
33.h odd 10 1 99.12.f.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.12.c.a 40 1.a even 1 1 trivial
11.12.c.a 40 11.c even 5 1 inner
99.12.f.a 40 3.b odd 2 1
99.12.f.a 40 33.h odd 10 1
121.12.a.h 20 11.d odd 10 1
121.12.a.j 20 11.c even 5 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{12}^{\mathrm{new}}(11, [\chi])$$.