# Properties

 Label 49.2.g.a Level $49$ Weight $2$ Character orbit 49.g Analytic conductor $0.391$ Analytic rank $0$ Dimension $48$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,2,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, 2, names="a")

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

chi := DirichletCharacter("49.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$0.391266969904$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$4$$ over $$\Q(\zeta_{21})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 13 q^{2} - 14 q^{3} - 9 q^{4} - 14 q^{5} - 14 q^{7} - 20 q^{8} + 6 q^{9}+O(q^{10})$$ 48 * q - 13 * q^2 - 14 * q^3 - 9 * q^4 - 14 * q^5 - 14 * q^7 - 20 * q^8 + 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 13 q^{2} - 14 q^{3} - 9 q^{4} - 14 q^{5} - 14 q^{7} - 20 q^{8} + 6 q^{9} - 14 q^{10} - 3 q^{11} + 21 q^{12} - 14 q^{13} + 21 q^{14} - 12 q^{15} - 3 q^{16} - 7 q^{17} + 2 q^{18} + 21 q^{19} + 14 q^{20} - 14 q^{21} - 20 q^{22} + 15 q^{23} + 28 q^{24} - 4 q^{25} + 7 q^{27} + 28 q^{28} + 12 q^{29} + 11 q^{30} + 35 q^{31} + 45 q^{32} - 14 q^{33} + 70 q^{34} - 12 q^{36} + 15 q^{37} - 28 q^{38} - 7 q^{39} - 42 q^{40} - 42 q^{41} + 28 q^{42} - 30 q^{43} - 50 q^{44} + 7 q^{45} - 78 q^{46} + 21 q^{47} - 84 q^{48} - 70 q^{49} + 40 q^{50} - 52 q^{51} - 70 q^{52} + 11 q^{53} - 77 q^{54} - 7 q^{55} - 28 q^{56} - 12 q^{57} + 16 q^{58} - 28 q^{59} + 56 q^{60} + 7 q^{61} - 28 q^{62} + 35 q^{63} - 32 q^{64} + 14 q^{65} + 154 q^{66} + 11 q^{67} + 77 q^{68} + 70 q^{69} + 70 q^{70} + 19 q^{71} + 170 q^{72} + 7 q^{73} + 34 q^{74} + 112 q^{75} + 119 q^{76} + 7 q^{77} + 28 q^{78} + 15 q^{79} + 70 q^{80} + 64 q^{81} - 14 q^{82} - 84 q^{84} - 26 q^{85} - 33 q^{86} - 112 q^{87} - 77 q^{88} - 14 q^{89} - 182 q^{90} + 84 q^{91} - 38 q^{92} - 80 q^{93} + 14 q^{94} - 61 q^{95} - 70 q^{96} - 161 q^{98} - 16 q^{99}+O(q^{100})$$ 48 * q - 13 * q^2 - 14 * q^3 - 9 * q^4 - 14 * q^5 - 14 * q^7 - 20 * q^8 + 6 * q^9 - 14 * q^10 - 3 * q^11 + 21 * q^12 - 14 * q^13 + 21 * q^14 - 12 * q^15 - 3 * q^16 - 7 * q^17 + 2 * q^18 + 21 * q^19 + 14 * q^20 - 14 * q^21 - 20 * q^22 + 15 * q^23 + 28 * q^24 - 4 * q^25 + 7 * q^27 + 28 * q^28 + 12 * q^29 + 11 * q^30 + 35 * q^31 + 45 * q^32 - 14 * q^33 + 70 * q^34 - 12 * q^36 + 15 * q^37 - 28 * q^38 - 7 * q^39 - 42 * q^40 - 42 * q^41 + 28 * q^42 - 30 * q^43 - 50 * q^44 + 7 * q^45 - 78 * q^46 + 21 * q^47 - 84 * q^48 - 70 * q^49 + 40 * q^50 - 52 * q^51 - 70 * q^52 + 11 * q^53 - 77 * q^54 - 7 * q^55 - 28 * q^56 - 12 * q^57 + 16 * q^58 - 28 * q^59 + 56 * q^60 + 7 * q^61 - 28 * q^62 + 35 * q^63 - 32 * q^64 + 14 * q^65 + 154 * q^66 + 11 * q^67 + 77 * q^68 + 70 * q^69 + 70 * q^70 + 19 * q^71 + 170 * q^72 + 7 * q^73 + 34 * q^74 + 112 * q^75 + 119 * q^76 + 7 * q^77 + 28 * q^78 + 15 * q^79 + 70 * q^80 + 64 * q^81 - 14 * q^82 - 84 * q^84 - 26 * q^85 - 33 * q^86 - 112 * q^87 - 77 * q^88 - 14 * q^89 - 182 * q^90 + 84 * q^91 - 38 * q^92 - 80 * q^93 + 14 * q^94 - 61 * q^95 - 70 * q^96 - 161 * q^98 - 16 * 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 −2.50546 0.772833i −0.539916 + 1.37568i 4.02760 + 2.74597i 2.04183 0.307757i 2.41591 3.02946i 0.189789 + 2.63894i −4.69930 5.89274i 0.598158 + 0.555009i −5.35358 0.806922i
2.2 −1.16438 0.359164i 0.671120 1.70999i −0.425690 0.290231i −0.478452 + 0.0721150i −1.39561 + 1.75004i 1.90196 1.83918i 1.91089 + 2.39618i −0.274497 0.254696i 0.583002 + 0.0878735i
2.3 0.681025 + 0.210069i −0.953694 + 2.42997i −1.23281 0.840516i 2.65874 0.400741i −1.15995 + 1.45453i −0.518691 2.59441i −1.55172 1.94579i −2.79608 2.59438i 1.89485 + 0.285603i
2.4 1.16258 + 0.358609i 0.190201 0.484624i −0.429482 0.292816i −2.56601 + 0.386764i 0.394914 0.495207i −0.339612 + 2.62386i −1.91142 2.39684i 2.00047 + 1.85617i −3.12189 0.470550i
4.1 −2.09732 1.42993i −1.40598 1.30456i 1.62338 + 4.13631i −2.38971 + 0.737129i 1.08336 + 4.74653i −1.40100 2.24437i 1.38019 6.04701i 0.0507148 + 0.676742i 6.06605 + 1.87113i
4.2 −0.968018 0.659983i 0.630335 + 0.584866i −0.229202 0.583997i 3.03447 0.936009i −0.224174 0.982171i −2.50722 + 0.844891i −0.684966 + 3.00103i −0.168936 2.25429i −3.55517 1.09662i
4.3 0.174575 + 0.119023i 1.58372 + 1.46948i −0.714372 1.82019i −3.75050 + 1.15688i 0.101576 + 0.445032i 1.85388 1.88763i 0.185965 0.814768i 0.124614 + 1.66286i −0.792437 0.244435i
4.4 1.52543 + 1.04002i −1.14975 1.06682i 0.514606 + 1.31120i −1.32986 + 0.410207i −0.644358 2.82312i −1.73764 + 1.99515i 0.242977 1.06455i −0.0403518 0.538458i −2.45523 0.757337i
9.1 −1.73170 1.60678i 2.78061 + 0.419109i 0.267571 + 3.57049i −0.567937 1.44708i −4.14175 5.19359i −2.62161 + 0.356599i 2.32789 2.91908i 4.68940 + 1.44649i −1.34164 + 3.41845i
9.2 −1.02480 0.950878i −1.96135 0.295625i −0.00340854 0.0454838i −1.11243 2.83442i 1.72889 + 2.16796i 2.44418 + 1.01290i −1.78303 + 2.23585i 0.892763 + 0.275381i −1.55517 + 3.96251i
9.3 0.168237 + 0.156102i 0.223157 + 0.0336355i −0.145524 1.94188i 0.711168 + 1.81203i 0.0322928 + 0.0404939i −2.04196 + 1.68237i 0.564834 0.708279i −2.81805 0.869254i −0.163215 + 0.415865i
9.4 1.51353 + 1.40435i −2.00916 0.302832i 0.169112 + 2.25665i −0.0830372 0.211575i −2.61564 3.27991i 1.07894 2.41576i −0.338532 + 0.424506i 1.07830 + 0.332612i 0.171447 0.436839i
11.1 −1.73170 + 1.60678i 2.78061 0.419109i 0.267571 3.57049i −0.567937 + 1.44708i −4.14175 + 5.19359i −2.62161 0.356599i 2.32789 + 2.91908i 4.68940 1.44649i −1.34164 3.41845i
11.2 −1.02480 + 0.950878i −1.96135 + 0.295625i −0.00340854 + 0.0454838i −1.11243 + 2.83442i 1.72889 2.16796i 2.44418 1.01290i −1.78303 2.23585i 0.892763 0.275381i −1.55517 3.96251i
11.3 0.168237 0.156102i 0.223157 0.0336355i −0.145524 + 1.94188i 0.711168 1.81203i 0.0322928 0.0404939i −2.04196 1.68237i 0.564834 + 0.708279i −2.81805 + 0.869254i −0.163215 0.415865i
11.4 1.51353 1.40435i −2.00916 + 0.302832i 0.169112 2.25665i −0.0830372 + 0.211575i −2.61564 + 3.27991i 1.07894 + 2.41576i −0.338532 0.424506i 1.07830 0.332612i 0.171447 + 0.436839i
16.1 −0.887059 2.26019i −0.0483362 0.645002i −2.85548 + 2.64950i 1.29033 0.879733i −1.41495 + 0.681404i −2.25101 + 1.39030i 4.14619 + 1.99670i 2.55280 0.384773i −3.13296 2.13602i
16.2 −0.354207 0.902506i 0.218016 + 2.90922i 0.777050 0.720997i −0.605902 + 0.413097i 2.54837 1.22723i −0.310399 2.62748i −2.67297 1.28723i −5.44954 + 0.821385i 0.587437 + 0.400508i
16.3 0.0341744 + 0.0870750i −0.0823774 1.09925i 1.45969 1.35439i −2.09852 + 1.43075i 0.0929020 0.0447392i −0.301609 + 2.62850i 0.336373 + 0.161989i 1.76493 0.266020i −0.196298 0.133834i
16.4 0.940144 + 2.39545i −0.173202 2.31121i −3.38820 + 3.14379i −0.392378 + 0.267519i 5.37356 2.58777i −1.60453 2.10369i −6.07919 2.92758i −2.34522 + 0.353485i −1.00972 0.688415i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.4 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 49.2.g.a 48
3.b odd 2 1 441.2.bb.d 48
4.b odd 2 1 784.2.bg.c 48
7.b odd 2 1 343.2.g.g 48
7.c even 3 1 343.2.e.d 48
7.c even 3 1 343.2.g.i 48
7.d odd 6 1 343.2.e.c 48
7.d odd 6 1 343.2.g.h 48
49.e even 7 1 343.2.g.i 48
49.f odd 14 1 343.2.g.h 48
49.g even 21 1 inner 49.2.g.a 48
49.g even 21 1 343.2.e.d 48
49.g even 21 1 2401.2.a.h 24
49.h odd 42 1 343.2.e.c 48
49.h odd 42 1 343.2.g.g 48
49.h odd 42 1 2401.2.a.i 24
147.n odd 42 1 441.2.bb.d 48
196.o odd 42 1 784.2.bg.c 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.g.a 48 1.a even 1 1 trivial
49.2.g.a 48 49.g even 21 1 inner
343.2.e.c 48 7.d odd 6 1
343.2.e.c 48 49.h odd 42 1
343.2.e.d 48 7.c even 3 1
343.2.e.d 48 49.g even 21 1
343.2.g.g 48 7.b odd 2 1
343.2.g.g 48 49.h odd 42 1
343.2.g.h 48 7.d odd 6 1
343.2.g.h 48 49.f odd 14 1
343.2.g.i 48 7.c even 3 1
343.2.g.i 48 49.e even 7 1
441.2.bb.d 48 3.b odd 2 1
441.2.bb.d 48 147.n odd 42 1
784.2.bg.c 48 4.b odd 2 1
784.2.bg.c 48 196.o odd 42 1
2401.2.a.h 24 49.g even 21 1
2401.2.a.i 24 49.h odd 42 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(49, [\chi])$$.