# Properties

 Label 49.4.g.a Level $49$ Weight $4$ Character orbit 49.g Analytic conductor $2.891$ Analytic rank $0$ Dimension $156$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,4,Mod(2,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([26]))

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

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

chi := DirichletCharacter("49.2");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 49.g (of order $$21$$, degree $$12$$, 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: $$2.89109359028$$ Analytic rank: $$0$$ Dimension: $$156$$ Relative dimension: $$13$$ over $$\Q(\zeta_{21})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10})$$ 156 * q - 13 * q^2 - 7 * q^3 + 35 * q^4 - 7 * q^5 - 70 * q^6 - 42 * q^7 + 16 * q^8 + 198 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9} + 85 q^{11} - 427 q^{12} + 14 q^{13} - 203 q^{14} + 230 q^{15} + 127 q^{16} + 189 q^{17} - 16 q^{18} - 490 q^{19} - 406 q^{20} + 231 q^{21} + 590 q^{22} - 377 q^{23} - 812 q^{24} + 142 q^{25} + 490 q^{26} + 14 q^{27} - 280 q^{28} + 172 q^{29} - 39 q^{30} - 1162 q^{31} - 1099 q^{32} - 49 q^{33} + 882 q^{34} + 847 q^{35} + 812 q^{36} - 377 q^{37} - 1596 q^{38} - 770 q^{39} - 1694 q^{40} - 938 q^{41} - 3290 q^{42} - 10 q^{43} + 4354 q^{44} + 1176 q^{45} + 3408 q^{46} + 1379 q^{47} + 9604 q^{48} + 2576 q^{49} - 6184 q^{50} + 2271 q^{51} + 6874 q^{52} + 1905 q^{53} + 1617 q^{54} - 1834 q^{55} + 1736 q^{56} + 2673 q^{57} - 4716 q^{58} + 63 q^{59} - 10850 q^{60} - 2625 q^{61} - 4298 q^{62} - 1806 q^{63} - 2756 q^{64} - 196 q^{65} + 2100 q^{66} - 202 q^{67} - 5635 q^{68} - 3248 q^{69} + 6650 q^{70} + 2608 q^{71} - 6424 q^{72} - 2471 q^{73} + 2582 q^{74} + 3612 q^{75} - 4529 q^{76} - 1281 q^{77} + 5040 q^{78} - 608 q^{79} - 12348 q^{80} - 17375 q^{81} - 3234 q^{82} - 9842 q^{83} - 11564 q^{84} - 512 q^{85} - 2301 q^{86} + 2254 q^{87} + 4033 q^{88} + 4977 q^{89} + 18312 q^{90} - 8932 q^{91} + 7714 q^{92} + 14164 q^{93} + 2506 q^{94} + 11504 q^{95} + 12642 q^{96} + 11116 q^{97} + 29771 q^{98} - 10804 q^{99}+O(q^{100})$$ 156 * q - 13 * q^2 - 7 * q^3 + 35 * q^4 - 7 * q^5 - 70 * q^6 - 42 * q^7 + 16 * q^8 + 198 * q^9 + 85 * q^11 - 427 * q^12 + 14 * q^13 - 203 * q^14 + 230 * q^15 + 127 * q^16 + 189 * q^17 - 16 * q^18 - 490 * q^19 - 406 * q^20 + 231 * q^21 + 590 * q^22 - 377 * q^23 - 812 * q^24 + 142 * q^25 + 490 * q^26 + 14 * q^27 - 280 * q^28 + 172 * q^29 - 39 * q^30 - 1162 * q^31 - 1099 * q^32 - 49 * q^33 + 882 * q^34 + 847 * q^35 + 812 * q^36 - 377 * q^37 - 1596 * q^38 - 770 * q^39 - 1694 * q^40 - 938 * q^41 - 3290 * q^42 - 10 * q^43 + 4354 * q^44 + 1176 * q^45 + 3408 * q^46 + 1379 * q^47 + 9604 * q^48 + 2576 * q^49 - 6184 * q^50 + 2271 * q^51 + 6874 * q^52 + 1905 * q^53 + 1617 * q^54 - 1834 * q^55 + 1736 * q^56 + 2673 * q^57 - 4716 * q^58 + 63 * q^59 - 10850 * q^60 - 2625 * q^61 - 4298 * q^62 - 1806 * q^63 - 2756 * q^64 - 196 * q^65 + 2100 * q^66 - 202 * q^67 - 5635 * q^68 - 3248 * q^69 + 6650 * q^70 + 2608 * q^71 - 6424 * q^72 - 2471 * q^73 + 2582 * q^74 + 3612 * q^75 - 4529 * q^76 - 1281 * q^77 + 5040 * q^78 - 608 * q^79 - 12348 * q^80 - 17375 * q^81 - 3234 * q^82 - 9842 * q^83 - 11564 * q^84 - 512 * q^85 - 2301 * q^86 + 2254 * q^87 + 4033 * q^88 + 4977 * q^89 + 18312 * q^90 - 8932 * q^91 + 7714 * q^92 + 14164 * q^93 + 2506 * q^94 + 11504 * q^95 + 12642 * q^96 + 11116 * q^97 + 29771 * q^98 - 10804 * 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 −4.61656 1.42402i 0.231546 0.589970i 12.6749 + 8.64158i −16.6086 + 2.50334i −1.90907 + 2.39390i 18.4370 + 1.75414i −22.1109 27.7262i 19.4979 + 18.0915i 80.2392 + 12.0941i
2.2 −4.56991 1.40963i 2.47210 6.29880i 12.2871 + 8.37722i 16.6312 2.50675i −20.1763 + 25.3002i −5.68057 17.6276i −20.4882 25.6914i −13.7713 12.7779i −79.5368 11.9882i
2.3 −3.74530 1.15527i −3.45495 + 8.80308i 6.08268 + 4.14710i −0.875096 + 0.131899i 23.1098 28.9787i −6.85447 17.2051i 1.55937 + 1.95538i −45.7650 42.4637i 3.42987 + 0.516970i
2.4 −3.15708 0.973830i −0.0268992 + 0.0685381i 2.40889 + 1.64236i −0.551282 + 0.0830924i 0.151667 0.190185i −15.0526 + 10.7898i 10.4737 + 13.1336i 19.7884 + 18.3610i 1.82136 + 0.274526i
2.5 −1.76903 0.545674i 3.43379 8.74916i −3.77820 2.57593i −8.01721 + 1.20840i −10.8487 + 13.6038i −0.295568 + 18.5179i 14.5122 + 18.1977i −44.9645 41.7210i 14.8421 + 2.23708i
2.6 −1.42883 0.440736i −1.06746 + 2.71984i −4.76260 3.24708i 14.5837 2.19813i 2.72395 3.41573i 18.2694 + 3.03821i 12.8321 + 16.0909i 13.5343 + 12.5580i −21.8064 3.28678i
2.7 0.0991526 + 0.0305845i −0.212463 + 0.541346i −6.60101 4.50050i −19.4416 + 2.93036i −0.0376231 + 0.0471778i 0.208913 18.5191i −1.03442 1.29712i 19.5445 + 18.1346i −2.01731 0.304061i
2.8 0.724682 + 0.223535i 1.81654 4.62846i −6.13471 4.18258i 7.40499 1.11612i 2.35103 2.94810i −12.1170 14.0064i −7.29347 9.14572i 1.66955 + 1.54912i 5.61575 + 0.846439i
2.9 0.871702 + 0.268884i −2.65215 + 6.75756i −5.92235 4.03779i −5.10865 + 0.770006i −4.12888 + 5.17746i −7.37832 + 16.9871i −8.62695 10.8178i −18.8383 17.4794i −4.66026 0.702422i
2.10 2.83978 + 0.875955i 2.09482 5.33750i 0.687126 + 0.468475i 2.48561 0.374645i 10.6242 13.3224i 17.0480 + 7.23649i −13.2822 16.6554i −4.30828 3.99750i 7.38675 + 1.11337i
2.11 3.77876 + 1.16559i −0.769406 + 1.96041i 6.31048 + 4.30242i 13.4588 2.02859i −5.19244 + 6.51112i −18.2161 + 3.34267i −0.893508 1.12042i 16.5412 + 15.3480i 53.2221 + 8.02194i
2.12 4.09539 + 1.26326i −2.23891 + 5.70464i 8.56651 + 5.84055i −5.73194 + 0.863951i −16.3757 + 20.5344i 16.2381 8.90649i 6.32787 + 7.93490i −7.73783 7.17966i −24.5659 3.70272i
2.13 5.05101 + 1.55803i 2.29854 5.85659i 16.4753 + 11.2327i −16.6099 + 2.50354i 20.7347 26.0005i −18.4107 2.01144i 39.3507 + 49.3442i −9.22394 8.55857i −87.7972 13.2333i
4.1 −4.51874 3.08083i 0.164357 + 0.152501i 8.00480 + 20.3959i 12.0019 3.70210i −0.272857 1.19547i 1.40336 + 18.4670i 16.9288 74.1698i −2.01396 26.8744i −65.6390 20.2470i
4.2 −3.34475 2.28041i 5.73197 + 5.31849i 3.06435 + 7.80784i −15.4193 + 4.75622i −7.04365 30.8603i −18.5181 0.281065i 0.349197 1.52993i 2.55141 + 34.0463i 62.4198 + 19.2540i
4.3 −2.89337 1.97267i −3.27221 3.03617i 1.55746 + 3.96835i −9.83041 + 3.03228i 3.47838 + 15.2398i 18.2590 + 3.09998i −2.91198 + 12.7582i −0.528663 7.05451i 34.4248 + 10.6186i
4.4 −2.79294 1.90420i 1.97416 + 1.83175i 1.25183 + 3.18962i 6.58232 2.03038i −2.02569 8.87514i 3.93477 18.0974i −3.44015 + 15.0723i −1.47573 19.6922i −22.2503 6.86330i
4.5 −2.38820 1.62825i −6.73991 6.25373i 0.129585 + 0.330178i 10.8842 3.35733i 5.91365 + 25.9094i −18.4252 + 1.87423i −4.91735 + 21.5443i 4.29963 + 57.3746i −31.4602 9.70418i
4.6 −0.609565 0.415594i 5.72670 + 5.31360i −2.72388 6.94033i 7.13618 2.20122i −1.28249 5.61896i 12.7167 + 13.4643i −2.53731 + 11.1167i 2.54302 + 33.9342i −5.26478 1.62397i
4.7 −0.0954438 0.0650725i −1.38664 1.28661i −2.91785 7.43457i −13.2250 + 4.07936i 0.0486229 + 0.213031i −12.7050 + 13.4753i −0.410932 + 1.80041i −1.75032 23.3564i 1.52769 + 0.471231i
See next 80 embeddings (of 156 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.13 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.4.g.a 156
49.g even 21 1 inner 49.4.g.a 156
49.g even 21 1 2401.4.a.g 78
49.h odd 42 1 2401.4.a.f 78

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.g.a 156 1.a even 1 1 trivial
49.4.g.a 156 49.g even 21 1 inner
2401.4.a.f 78 49.h odd 42 1
2401.4.a.g 78 49.g even 21 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(49, [\chi])$$.