# Properties

 Label 49.4.e.a Level $49$ Weight $4$ Character orbit 49.e Analytic conductor $2.891$ Analytic rank $0$ Dimension $78$ 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(8,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([12]))

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

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

chi := DirichletCharacter("49.8");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 49.e (of order $$7$$, degree $$6$$, 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: $$78$$ Relative dimension: $$13$$ over $$\Q(\zeta_{7})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10})$$ 78 * q - 5 * q^2 - 5 * q^3 - 53 * q^4 - 23 * q^5 + 19 * q^6 - 31 * q^8 - 174 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9} + 9 q^{10} - 103 q^{11} + 364 q^{12} - 35 q^{13} + 161 q^{14} - 245 q^{15} - 205 q^{16} - 285 q^{17} + 16 q^{18} + 628 q^{19} + 553 q^{20} - 21 q^{21} - 605 q^{22} + 149 q^{23} + 653 q^{24} - 370 q^{25} - 511 q^{26} - 65 q^{27} + 70 q^{28} - 187 q^{29} + 84 q^{30} + 1276 q^{31} + 1399 q^{32} - 23 q^{33} - 765 q^{34} - 805 q^{35} - 1691 q^{36} - 1531 q^{37} - 1041 q^{38} - 1351 q^{39} - 1759 q^{40} - 301 q^{41} + 3395 q^{42} - 257 q^{43} - 883 q^{44} + 3105 q^{45} + 1593 q^{46} + 733 q^{47} - 1948 q^{48} + 1288 q^{49} + 6148 q^{50} + 1197 q^{51} - 1099 q^{52} - 285 q^{53} + 660 q^{54} + 2641 q^{55} - 1988 q^{56} - 2352 q^{57} + 1173 q^{58} - 3603 q^{59} - 175 q^{60} - 2613 q^{61} - 1927 q^{62} - 3066 q^{63} + 1589 q^{64} - 371 q^{65} - 2175 q^{66} + 352 q^{67} + 6076 q^{68} + 5549 q^{69} - 6293 q^{70} - 2623 q^{71} + 6220 q^{72} + 2039 q^{73} - 2411 q^{74} - 3903 q^{75} + 4130 q^{76} + 1029 q^{77} - 3759 q^{78} + 44 q^{79} - 1608 q^{80} + 1394 q^{81} - 10920 q^{82} - 553 q^{83} - 7798 q^{84} + 497 q^{85} - 2985 q^{86} - 4273 q^{87} - 2197 q^{88} - 3957 q^{89} - 2958 q^{90} + 14119 q^{91} - 9136 q^{92} + 6272 q^{93} + 14912 q^{94} + 5866 q^{95} + 21882 q^{96} - 1540 q^{97} - 2303 q^{98} + 10768 q^{99}+O(q^{100})$$ 78 * q - 5 * q^2 - 5 * q^3 - 53 * q^4 - 23 * q^5 + 19 * q^6 - 31 * q^8 - 174 * q^9 + 9 * q^10 - 103 * q^11 + 364 * q^12 - 35 * q^13 + 161 * q^14 - 245 * q^15 - 205 * q^16 - 285 * q^17 + 16 * q^18 + 628 * q^19 + 553 * q^20 - 21 * q^21 - 605 * q^22 + 149 * q^23 + 653 * q^24 - 370 * q^25 - 511 * q^26 - 65 * q^27 + 70 * q^28 - 187 * q^29 + 84 * q^30 + 1276 * q^31 + 1399 * q^32 - 23 * q^33 - 765 * q^34 - 805 * q^35 - 1691 * q^36 - 1531 * q^37 - 1041 * q^38 - 1351 * q^39 - 1759 * q^40 - 301 * q^41 + 3395 * q^42 - 257 * q^43 - 883 * q^44 + 3105 * q^45 + 1593 * q^46 + 733 * q^47 - 1948 * q^48 + 1288 * q^49 + 6148 * q^50 + 1197 * q^51 - 1099 * q^52 - 285 * q^53 + 660 * q^54 + 2641 * q^55 - 1988 * q^56 - 2352 * q^57 + 1173 * q^58 - 3603 * q^59 - 175 * q^60 - 2613 * q^61 - 1927 * q^62 - 3066 * q^63 + 1589 * q^64 - 371 * q^65 - 2175 * q^66 + 352 * q^67 + 6076 * q^68 + 5549 * q^69 - 6293 * q^70 - 2623 * q^71 + 6220 * q^72 + 2039 * q^73 - 2411 * q^74 - 3903 * q^75 + 4130 * q^76 + 1029 * q^77 - 3759 * q^78 + 44 * q^79 - 1608 * q^80 + 1394 * q^81 - 10920 * q^82 - 553 * q^83 - 7798 * q^84 + 497 * q^85 - 2985 * q^86 - 4273 * q^87 - 2197 * q^88 - 3957 * q^89 - 2958 * q^90 + 14119 * q^91 - 9136 * q^92 + 6272 * q^93 + 14912 * q^94 + 5866 * q^95 + 21882 * q^96 - 1540 * q^97 - 2303 * q^98 + 10768 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1 −3.44244 4.31668i 3.65176 1.75859i −5.00318 + 21.9203i −12.6967 + 6.11441i −20.1622 9.70961i −15.1671 + 10.6282i 72.0505 34.6977i −6.59154 + 8.26553i 70.1015 + 33.7591i
8.2 −2.77071 3.47436i −6.15987 + 2.96643i −2.61419 + 11.4535i 1.98451 0.955692i 27.3737 + 13.1825i 18.5202 0.0393741i 15.0065 7.22673i 12.3100 15.4362i −8.81894 4.24698i
8.3 −2.45388 3.07706i 2.79026 1.34372i −1.66664 + 7.30203i 16.4856 7.93907i −10.9817 5.28849i −6.70370 17.2644i −1.80909 + 0.871210i −10.8543 + 13.6108i −64.8827 31.2459i
8.4 −1.61233 2.02180i −3.25484 + 1.56745i 0.292112 1.27982i −3.53374 + 1.70176i 8.41693 + 4.05338i −10.4442 + 15.2944i −21.6976 + 10.4490i −8.69714 + 10.9059i 9.13817 + 4.40071i
8.5 −1.46127 1.83238i 7.38411 3.55600i 0.557879 2.44423i −0.458224 + 0.220669i −17.3061 8.33419i 12.0280 + 14.0828i −22.1867 + 10.6846i 25.0457 31.4064i 1.07394 + 0.517181i
8.6 −0.802750 1.00662i −1.70862 + 0.822828i 1.41130 6.18330i −15.9369 + 7.67483i 2.19987 + 1.05940i 0.116992 18.5199i −16.6372 + 8.01205i −14.5919 + 18.2976i 20.5190 + 9.88143i
8.7 0.153998 + 0.193108i −8.98937 + 4.32905i 1.76659 7.73995i 14.2902 6.88179i −2.22032 1.06925i −13.6848 12.4791i 3.54697 1.70813i 45.2339 56.7215i 3.52959 + 1.69976i
8.8 0.521845 + 0.654373i 5.78023 2.78361i 1.62429 7.11646i −1.28950 + 0.620991i 4.83790 + 2.32981i −17.3557 6.46381i 11.5371 5.55599i 8.82835 11.0704i −1.07928 0.519753i
8.9 0.777062 + 0.974405i −0.189548 + 0.0912813i 1.43453 6.28508i 7.66912 3.69325i −0.236235 0.113765i 18.3768 + 2.30053i 16.2220 7.81211i −16.8066 + 21.0748i 9.55811 + 4.60294i
8.10 1.88794 + 2.36740i −5.83533 + 2.81015i −0.260101 + 1.13958i −12.6471 + 6.09051i −17.6695 8.50917i −3.94107 + 18.0961i 18.6363 8.97477i 9.31994 11.6868i −38.2955 18.4422i
8.11 2.59695 + 3.25648i 2.96773 1.42918i −2.08030 + 9.11440i 9.52410 4.58657i 12.3612 + 5.95282i −16.3066 + 8.78031i −5.06162 + 2.43755i −10.0694 + 12.6266i 39.6697 + 19.1039i
8.12 2.64790 + 3.32037i 6.62695 3.19137i −2.23327 + 9.78459i −17.1881 + 8.27737i 28.1441 + 13.5535i 16.3388 8.72030i −7.79127 + 3.75208i 16.8974 21.1887i −72.9964 35.1532i
8.13 3.18020 + 3.98784i −4.68696 + 2.25712i −4.00905 + 17.5648i 4.76744 2.29588i −23.9065 11.5128i 7.38357 16.9848i −46.0311 + 22.1674i 0.0387732 0.0486201i 24.3170 + 11.7104i
15.1 −1.16701 + 5.11301i −1.24065 + 1.55573i −17.5732 8.46282i 0.849213 1.06488i −6.50660 8.15902i −18.2375 + 3.22369i 37.6195 47.1733i 5.12699 + 22.4628i 4.45370 + 5.58476i
15.2 −0.892344 + 3.90961i 5.02886 6.30599i −7.28105 3.50637i 8.76329 10.9888i 20.1665 + 25.2880i 3.97541 18.0886i 0.203423 0.255085i −8.46803 37.1008i 35.1422 + 44.0669i
15.3 −0.846844 + 3.71026i 3.04811 3.82222i −5.84116 2.81296i −12.1071 + 15.1818i 11.6002 + 14.5461i 13.1709 + 13.0203i −3.59906 + 4.51308i 0.689741 + 3.02195i −46.0758 57.7772i
15.4 −0.764068 + 3.34760i −5.99643 + 7.51929i −3.41489 1.64452i 0.968529 1.21450i −20.5899 25.8189i 18.5012 + 0.839908i −9.01256 + 11.3014i −14.5744 63.8547i 3.32563 + 4.17021i
15.5 −0.424669 + 1.86059i 0.113640 0.142500i 3.92628 + 1.89080i 11.1399 13.9690i 0.216876 + 0.271954i 1.18252 + 18.4825i −14.7045 + 18.4389i 6.00067 + 26.2907i 21.2598 + 26.6590i
15.6 −0.379589 + 1.66309i −2.24247 + 2.81197i 4.58597 + 2.20849i −4.66393 + 5.84838i −3.82533 4.79682i −14.0409 12.0769i −13.9224 + 17.4581i 3.12958 + 13.7116i −7.95600 9.97651i
15.7 0.0613960 0.268993i 2.25069 2.82228i 7.13916 + 3.43804i −2.36861 + 2.97014i −0.620990 0.778697i 9.47466 15.9132i 2.73935 3.43503i 3.10843 + 13.6189i 0.653525 + 0.819495i
See all 78 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.13 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.e even 7 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.4.e.a 78
49.e even 7 1 inner 49.4.e.a 78
49.e even 7 1 2401.4.a.d 39
49.f odd 14 1 2401.4.a.c 39

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.e.a 78 1.a even 1 1 trivial
49.4.e.a 78 49.e even 7 1 inner
2401.4.a.c 39 49.f odd 14 1
2401.4.a.d 39 49.e even 7 1

## Hecke kernels

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