# Properties

 Label 49.16.c.b Level $49$ Weight $16$ Character orbit 49.c Analytic conductor $69.920$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,16,Mod(18,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("49.18");

S:= CuspForms(chi, 16);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$16$$ Character orbit: $$[\chi]$$ $$=$$ 49.c (of order $$3$$, degree $$2$$, not 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: $$69.9198174990$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 216 \zeta_{6} q^{2} + (3348 \zeta_{6} - 3348) q^{3} + (13888 \zeta_{6} - 13888) q^{4} + 52110 \zeta_{6} q^{5} + 723168 q^{6} - 4078080 q^{8} + 3139803 \zeta_{6} q^{9} +O(q^{10})$$ q - 216*z * q^2 + (3348*z - 3348) * q^3 + (13888*z - 13888) * q^4 + 52110*z * q^5 + 723168 * q^6 - 4078080 * q^8 + 3139803*z * q^9 $$q - 216 \zeta_{6} q^{2} + (3348 \zeta_{6} - 3348) q^{3} + (13888 \zeta_{6} - 13888) q^{4} + 52110 \zeta_{6} q^{5} + 723168 q^{6} - 4078080 q^{8} + 3139803 \zeta_{6} q^{9} + ( - 11255760 \zeta_{6} + 11255760) q^{10} + (20586852 \zeta_{6} - 20586852) q^{11} - 46497024 \zeta_{6} q^{12} + 190073338 q^{13} - 174464280 q^{15} + 1335947264 \zeta_{6} q^{16} + ( - 1646527986 \zeta_{6} + 1646527986) q^{17} + ( - 678197448 \zeta_{6} + 678197448) q^{18} + 1563257180 \zeta_{6} q^{19} - 723703680 q^{20} + 4446760032 q^{22} - 9451116072 \zeta_{6} q^{23} + ( - 13653411840 \zeta_{6} + 13653411840) q^{24} + ( - 27802126025 \zeta_{6} + 27802126025) q^{25} - 41055841008 \zeta_{6} q^{26} - 58552201080 q^{27} - 36902568330 q^{29} + 37684284480 \zeta_{6} q^{30} + ( - 71588483552 \zeta_{6} + 71588483552) q^{31} + ( - 154934083584 \zeta_{6} + 154934083584) q^{32} - 68924780496 \zeta_{6} q^{33} - 355650044976 q^{34} - 43605584064 q^{36} + 1033652081554 \zeta_{6} q^{37} + ( - 337663550880 \zeta_{6} + 337663550880) q^{38} + (636365535624 \zeta_{6} - 636365535624) q^{39} - 212508748800 \zeta_{6} q^{40} - 1641974018202 q^{41} - 492403109308 q^{43} - 285910200576 \zeta_{6} q^{44} + (163615134330 \zeta_{6} - 163615134330) q^{45} + (2041441071552 \zeta_{6} - 2041441071552) q^{46} - 3410684952624 \zeta_{6} q^{47} - 4472751439872 q^{48} - 6005259221400 q^{50} + 5512575697128 \zeta_{6} q^{51} + (2639738518144 \zeta_{6} - 2639738518144) q^{52} + (6797151655902 \zeta_{6} - 6797151655902) q^{53} + 12647275433280 \zeta_{6} q^{54} - 1072780857720 q^{55} - 5233785038640 q^{57} + 7970954759280 \zeta_{6} q^{58} + ( - 9858856815540 \zeta_{6} + 9858856815540) q^{59} + ( - 2422959920640 \zeta_{6} + 2422959920640) q^{60} + 4931842626902 \zeta_{6} q^{61} - 15463112447232 q^{62} + 10310557892608 q^{64} + 9904721643180 \zeta_{6} q^{65} + (14887752587136 \zeta_{6} - 14887752587136) q^{66} + ( - 28837826625364 \zeta_{6} + 28837826625364) q^{67} + 22866980669568 \zeta_{6} q^{68} + 31642336609056 q^{69} + 125050114914552 q^{71} - 12804367818240 \zeta_{6} q^{72} + (82171455513478 \zeta_{6} - 82171455513478) q^{73} + ( - 223268849615664 \zeta_{6} + 223268849615664) q^{74} + 93081517931700 \zeta_{6} q^{75} - 21710515715840 q^{76} + 137454955694784 q^{78} + 25413078694480 \zeta_{6} q^{79} + (69616211927040 \zeta_{6} - 69616211927040) q^{80} + ( - 150980027970519 \zeta_{6} + 150980027970519) q^{81} + 354666387931632 \zeta_{6} q^{82} + 281736730890468 q^{83} + 85800573350460 q^{85} + 106359071610528 \zeta_{6} q^{86} + ( - 123549798768840 \zeta_{6} + 123549798768840) q^{87} + ( - 83954829404160 \zeta_{6} + 83954829404160) q^{88} + 715618564776810 \zeta_{6} q^{89} + 35340869015280 q^{90} + 131257100007936 q^{92} + 239678242932096 \zeta_{6} q^{93} + (736707949766784 \zeta_{6} - 736707949766784) q^{94} + (81461331649800 \zeta_{6} - 81461331649800) q^{95} + 518719311839232 \zeta_{6} q^{96} - 612786136081826 q^{97} - 64638659670156 q^{99} +O(q^{100})$$ q - 216*z * q^2 + (3348*z - 3348) * q^3 + (13888*z - 13888) * q^4 + 52110*z * q^5 + 723168 * q^6 - 4078080 * q^8 + 3139803*z * q^9 + (-11255760*z + 11255760) * q^10 + (20586852*z - 20586852) * q^11 - 46497024*z * q^12 + 190073338 * q^13 - 174464280 * q^15 + 1335947264*z * q^16 + (-1646527986*z + 1646527986) * q^17 + (-678197448*z + 678197448) * q^18 + 1563257180*z * q^19 - 723703680 * q^20 + 4446760032 * q^22 - 9451116072*z * q^23 + (-13653411840*z + 13653411840) * q^24 + (-27802126025*z + 27802126025) * q^25 - 41055841008*z * q^26 - 58552201080 * q^27 - 36902568330 * q^29 + 37684284480*z * q^30 + (-71588483552*z + 71588483552) * q^31 + (-154934083584*z + 154934083584) * q^32 - 68924780496*z * q^33 - 355650044976 * q^34 - 43605584064 * q^36 + 1033652081554*z * q^37 + (-337663550880*z + 337663550880) * q^38 + (636365535624*z - 636365535624) * q^39 - 212508748800*z * q^40 - 1641974018202 * q^41 - 492403109308 * q^43 - 285910200576*z * q^44 + (163615134330*z - 163615134330) * q^45 + (2041441071552*z - 2041441071552) * q^46 - 3410684952624*z * q^47 - 4472751439872 * q^48 - 6005259221400 * q^50 + 5512575697128*z * q^51 + (2639738518144*z - 2639738518144) * q^52 + (6797151655902*z - 6797151655902) * q^53 + 12647275433280*z * q^54 - 1072780857720 * q^55 - 5233785038640 * q^57 + 7970954759280*z * q^58 + (-9858856815540*z + 9858856815540) * q^59 + (-2422959920640*z + 2422959920640) * q^60 + 4931842626902*z * q^61 - 15463112447232 * q^62 + 10310557892608 * q^64 + 9904721643180*z * q^65 + (14887752587136*z - 14887752587136) * q^66 + (-28837826625364*z + 28837826625364) * q^67 + 22866980669568*z * q^68 + 31642336609056 * q^69 + 125050114914552 * q^71 - 12804367818240*z * q^72 + (82171455513478*z - 82171455513478) * q^73 + (-223268849615664*z + 223268849615664) * q^74 + 93081517931700*z * q^75 - 21710515715840 * q^76 + 137454955694784 * q^78 + 25413078694480*z * q^79 + (69616211927040*z - 69616211927040) * q^80 + (-150980027970519*z + 150980027970519) * q^81 + 354666387931632*z * q^82 + 281736730890468 * q^83 + 85800573350460 * q^85 + 106359071610528*z * q^86 + (-123549798768840*z + 123549798768840) * q^87 + (-83954829404160*z + 83954829404160) * q^88 + 715618564776810*z * q^89 + 35340869015280 * q^90 + 131257100007936 * q^92 + 239678242932096*z * q^93 + (736707949766784*z - 736707949766784) * q^94 + (81461331649800*z - 81461331649800) * q^95 + 518719311839232*z * q^96 - 612786136081826 * q^97 - 64638659670156 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 216 q^{2} - 3348 q^{3} - 13888 q^{4} + 52110 q^{5} + 1446336 q^{6} - 8156160 q^{8} + 3139803 q^{9}+O(q^{10})$$ 2 * q - 216 * q^2 - 3348 * q^3 - 13888 * q^4 + 52110 * q^5 + 1446336 * q^6 - 8156160 * q^8 + 3139803 * q^9 $$2 q - 216 q^{2} - 3348 q^{3} - 13888 q^{4} + 52110 q^{5} + 1446336 q^{6} - 8156160 q^{8} + 3139803 q^{9} + 11255760 q^{10} - 20586852 q^{11} - 46497024 q^{12} + 380146676 q^{13} - 348928560 q^{15} + 1335947264 q^{16} + 1646527986 q^{17} + 678197448 q^{18} + 1563257180 q^{19} - 1447407360 q^{20} + 8893520064 q^{22} - 9451116072 q^{23} + 13653411840 q^{24} + 27802126025 q^{25} - 41055841008 q^{26} - 117104402160 q^{27} - 73805136660 q^{29} + 37684284480 q^{30} + 71588483552 q^{31} + 154934083584 q^{32} - 68924780496 q^{33} - 711300089952 q^{34} - 87211168128 q^{36} + 1033652081554 q^{37} + 337663550880 q^{38} - 636365535624 q^{39} - 212508748800 q^{40} - 3283948036404 q^{41} - 984806218616 q^{43} - 285910200576 q^{44} - 163615134330 q^{45} - 2041441071552 q^{46} - 3410684952624 q^{47} - 8945502879744 q^{48} - 12010518442800 q^{50} + 5512575697128 q^{51} - 2639738518144 q^{52} - 6797151655902 q^{53} + 12647275433280 q^{54} - 2145561715440 q^{55} - 10467570077280 q^{57} + 7970954759280 q^{58} + 9858856815540 q^{59} + 2422959920640 q^{60} + 4931842626902 q^{61} - 30926224894464 q^{62} + 20621115785216 q^{64} + 9904721643180 q^{65} - 14887752587136 q^{66} + 28837826625364 q^{67} + 22866980669568 q^{68} + 63284673218112 q^{69} + 250100229829104 q^{71} - 12804367818240 q^{72} - 82171455513478 q^{73} + 223268849615664 q^{74} + 93081517931700 q^{75} - 43421031431680 q^{76} + 274909911389568 q^{78} + 25413078694480 q^{79} - 69616211927040 q^{80} + 150980027970519 q^{81} + 354666387931632 q^{82} + 563473461780936 q^{83} + 171601146700920 q^{85} + 106359071610528 q^{86} + 123549798768840 q^{87} + 83954829404160 q^{88} + 715618564776810 q^{89} + 70681738030560 q^{90} + 262514200015872 q^{92} + 239678242932096 q^{93} - 736707949766784 q^{94} - 81461331649800 q^{95} + 518719311839232 q^{96} - 12\!\cdots\!52 q^{97}+ \cdots - 129277319340312 q^{99}+O(q^{100})$$ 2 * q - 216 * q^2 - 3348 * q^3 - 13888 * q^4 + 52110 * q^5 + 1446336 * q^6 - 8156160 * q^8 + 3139803 * q^9 + 11255760 * q^10 - 20586852 * q^11 - 46497024 * q^12 + 380146676 * q^13 - 348928560 * q^15 + 1335947264 * q^16 + 1646527986 * q^17 + 678197448 * q^18 + 1563257180 * q^19 - 1447407360 * q^20 + 8893520064 * q^22 - 9451116072 * q^23 + 13653411840 * q^24 + 27802126025 * q^25 - 41055841008 * q^26 - 117104402160 * q^27 - 73805136660 * q^29 + 37684284480 * q^30 + 71588483552 * q^31 + 154934083584 * q^32 - 68924780496 * q^33 - 711300089952 * q^34 - 87211168128 * q^36 + 1033652081554 * q^37 + 337663550880 * q^38 - 636365535624 * q^39 - 212508748800 * q^40 - 3283948036404 * q^41 - 984806218616 * q^43 - 285910200576 * q^44 - 163615134330 * q^45 - 2041441071552 * q^46 - 3410684952624 * q^47 - 8945502879744 * q^48 - 12010518442800 * q^50 + 5512575697128 * q^51 - 2639738518144 * q^52 - 6797151655902 * q^53 + 12647275433280 * q^54 - 2145561715440 * q^55 - 10467570077280 * q^57 + 7970954759280 * q^58 + 9858856815540 * q^59 + 2422959920640 * q^60 + 4931842626902 * q^61 - 30926224894464 * q^62 + 20621115785216 * q^64 + 9904721643180 * q^65 - 14887752587136 * q^66 + 28837826625364 * q^67 + 22866980669568 * q^68 + 63284673218112 * q^69 + 250100229829104 * q^71 - 12804367818240 * q^72 - 82171455513478 * q^73 + 223268849615664 * q^74 + 93081517931700 * q^75 - 43421031431680 * q^76 + 274909911389568 * q^78 + 25413078694480 * q^79 - 69616211927040 * q^80 + 150980027970519 * q^81 + 354666387931632 * q^82 + 563473461780936 * q^83 + 171601146700920 * q^85 + 106359071610528 * q^86 + 123549798768840 * q^87 + 83954829404160 * q^88 + 715618564776810 * q^89 + 70681738030560 * q^90 + 262514200015872 * q^92 + 239678242932096 * q^93 - 736707949766784 * q^94 - 81461331649800 * q^95 + 518719311839232 * q^96 - 1225572272163652 * q^97 - 129277319340312 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/49\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 0.5 + 0.866025i 0.5 − 0.866025i
−108.000 187.061i −1674.00 + 2899.45i −6944.00 + 12027.4i 26055.0 + 45128.6i 723168. 0 −4.07808e6 1.56990e6 + 2.71915e6i 5.62788e6 9.74777e6i
30.1 −108.000 + 187.061i −1674.00 2899.45i −6944.00 12027.4i 26055.0 45128.6i 723168. 0 −4.07808e6 1.56990e6 2.71915e6i 5.62788e6 + 9.74777e6i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.16.c.b 2
7.b odd 2 1 49.16.c.c 2
7.c even 3 1 49.16.a.a 1
7.c even 3 1 inner 49.16.c.b 2
7.d odd 6 1 1.16.a.a 1
7.d odd 6 1 49.16.c.c 2
21.g even 6 1 9.16.a.a 1
28.f even 6 1 16.16.a.d 1
35.i odd 6 1 25.16.a.a 1
35.k even 12 2 25.16.b.a 2
56.j odd 6 1 64.16.a.i 1
56.m even 6 1 64.16.a.c 1
77.i even 6 1 121.16.a.a 1
84.j odd 6 1 144.16.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.16.a.a 1 7.d odd 6 1
9.16.a.a 1 21.g even 6 1
16.16.a.d 1 28.f even 6 1
25.16.a.a 1 35.i odd 6 1
25.16.b.a 2 35.k even 12 2
49.16.a.a 1 7.c even 3 1
49.16.c.b 2 1.a even 1 1 trivial
49.16.c.b 2 7.c even 3 1 inner
49.16.c.c 2 7.b odd 2 1
49.16.c.c 2 7.d odd 6 1
64.16.a.c 1 56.m even 6 1
64.16.a.i 1 56.j odd 6 1
121.16.a.a 1 77.i even 6 1
144.16.a.f 1 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{16}^{\mathrm{new}}(49, [\chi])$$:

 $$T_{2}^{2} + 216T_{2} + 46656$$ T2^2 + 216*T2 + 46656 $$T_{3}^{2} + 3348T_{3} + 11209104$$ T3^2 + 3348*T3 + 11209104

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 216T + 46656$$
$3$ $$T^{2} + 3348 T + 11209104$$
$5$ $$T^{2} - 52110 T + 2715452100$$
$7$ $$T^{2}$$
$11$ $$T^{2} + \cdots + 423818475269904$$
$13$ $$(T - 190073338)^{2}$$
$17$ $$T^{2} - 1646527986 T + 27\!\cdots\!96$$
$19$ $$T^{2} - 1563257180 T + 24\!\cdots\!00$$
$23$ $$T^{2} + 9451116072 T + 89\!\cdots\!84$$
$29$ $$(T + 36902568330)^{2}$$
$31$ $$T^{2} - 71588483552 T + 51\!\cdots\!04$$
$37$ $$T^{2} - 1033652081554 T + 10\!\cdots\!16$$
$41$ $$(T + 1641974018202)^{2}$$
$43$ $$(T + 492403109308)^{2}$$
$47$ $$T^{2} + 3410684952624 T + 11\!\cdots\!76$$
$53$ $$T^{2} + 6797151655902 T + 46\!\cdots\!04$$
$59$ $$T^{2} - 9858856815540 T + 97\!\cdots\!00$$
$61$ $$T^{2} - 4931842626902 T + 24\!\cdots\!04$$
$67$ $$T^{2} - 28837826625364 T + 83\!\cdots\!96$$
$71$ $$(T - 125050114914552)^{2}$$
$73$ $$T^{2} + 82171455513478 T + 67\!\cdots\!84$$
$79$ $$T^{2} - 25413078694480 T + 64\!\cdots\!00$$
$83$ $$(T - 281736730890468)^{2}$$
$89$ $$T^{2} - 715618564776810 T + 51\!\cdots\!00$$
$97$ $$(T + 612786136081826)^{2}$$