LMFDB - Newform orbit 98.14.c.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [98,14,Mod(67,98)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("98.67"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 98 = 2 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	98.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-64,-1026] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$105.086310373$
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 14)
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 - 64 \zeta_{6} q^{2} + (1026 \zeta_{6} - 1026) q^{3} + (4096 \zeta_{6} - 4096) q^{4} + 4320 \zeta_{6} q^{5} + 65664 q^{6} + 262144 q^{8} + 541647 \zeta_{6} q^{9} + ( - 276480 \zeta_{6} + 276480) q^{10} + \cdots + 4759621182864 q^{99} +O(q^{100}) $ q - 64z q^2 + (1026z - 1026) q^3 + (4096z - 4096) q^4 + 4320z q^5 + 65664 * q^6 + 262144 * q^8 + 541647z q^9 + (-276480z + 276480) q^10 + (-8787312z + 8787312) q^11 - 4202496z q^12 + 20420932 * q^13 - 4432320 * q^15 - 16777216z q^16 + (-1719462z + 1719462) q^17 + (-34665408z + 34665408) q^18 - 109702942z q^19 - 17694720 * q^20 - 562387968 * q^22 + 646760160z q^23 + (268959744z - 268959744) q^24 + (-1202040725z + 1202040725) q^25 - 1306939648z q^26 - 2191505220 * q^27 + 728867274 * q^29 + 283668480z q^30 + (-1028049116z + 1028049116) q^31 + (1073741824z - 1073741824) q^32 + 9015782112z q^33 - 110045568 * q^34 - 2218586112 * q^36 - 14229390962z q^37 + (7020988288z - 7020988288) q^38 + (20951876232z - 20951876232) q^39 + 1132462080z q^40 - 44544458406 * q^41 - 54689828968 * q^43 + 35992829952z q^44 + (2339915040z - 2339915040) q^45 + (-41392650240z + 41392650240) q^46 + 47868325716z q^47 + 17213423616 * q^48 - 76930606400 * q^50 + 1764168012z q^51 + (83644137472z - 83644137472) q^52 + (-169986882858z + 169986882858) q^53 + 140256334080z q^54 + 37961187840 * q^55 + 112555218492 * q^57 - 46647505536z q^58 + (300765540198z - 300765540198) q^59 + (-18154782720z + 18154782720) q^60 + 369996272360z q^61 - 65795143424 * q^62 + 68719476736 * q^64 + 88218426240z q^65 + (-577010055168z + 577010055168) q^66 + (-787010801908z + 787010801908) q^67 + 7042916352z q^68 - 663575924160 * q^69 + 559441472256 * q^71 + 141989511168z q^72 + (-121137579650z + 121137579650) q^73 + (910681021568z - 910681021568) q^74 + 1233293783850z q^75 + 449343250432 * q^76 + 1340920078848 * q^78 - 290426785064z q^79 + (-72477573120z + 72477573120) q^80 + (-1384924085739z + 1384924085739) q^81 + 2850845337984z q^82 + 3965105603046 * q^83 + 7428075840 * q^85 + 3500149053952z q^86 + (747817823124z - 747817823124) q^87 + (-2303541116928z + 2303541116928) q^88 - 6025919250630z q^89 + 149754562560 * q^90 - 2649129615360 * q^92 + 1054778393016z q^93 + (-3063572845824z + 3063572845824) q^94 + (-473916709440z + 473916709440) q^95 - 1101659111424z q^96 - 11302818199190 * q^97 + 4759621182864 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 64 q^{2} - 1026 q^{3} - 4096 q^{4} + 4320 q^{5} + 131328 q^{6} + 524288 q^{8} + 541647 q^{9} + 276480 q^{10} + 8787312 q^{11} - 4202496 q^{12} + 40841864 q^{13} - 8864640 q^{15} - 16777216 q^{16}+ \cdots + 9519242365728 q^{99}+O(q^{100}) $ 2 * q - 64 * q^2 - 1026 * q^3 - 4096 * q^4 + 4320 * q^5 + 131328 * q^6 + 524288 * q^8 + 541647 * q^9 + 276480 * q^10 + 8787312 * q^11 - 4202496 * q^12 + 40841864 * q^13 - 8864640 * q^15 - 16777216 * q^16 + 1719462 * q^17 + 34665408 * q^18 - 109702942 * q^19 - 35389440 * q^20 - 1124775936 * q^22 + 646760160 * q^23 - 268959744 * q^24 + 1202040725 * q^25 - 1306939648 * q^26 - 4383010440 * q^27 + 1457734548 * q^29 + 283668480 * q^30 + 1028049116 * q^31 - 1073741824 * q^32 + 9015782112 * q^33 - 220091136 * q^34 - 4437172224 * q^36 - 14229390962 * q^37 - 7020988288 * q^38 - 20951876232 * q^39 + 1132462080 * q^40 - 89088916812 * q^41 - 109379657936 * q^43 + 35992829952 * q^44 - 2339915040 * q^45 + 41392650240 * q^46 + 47868325716 * q^47 + 34426847232 * q^48 - 153861212800 * q^50 + 1764168012 * q^51 - 83644137472 * q^52 + 169986882858 * q^53 + 140256334080 * q^54 + 75922375680 * q^55 + 225110436984 * q^57 - 46647505536 * q^58 - 300765540198 * q^59 + 18154782720 * q^60 + 369996272360 * q^61 - 131590286848 * q^62 + 137438953472 * q^64 + 88218426240 * q^65 + 577010055168 * q^66 + 787010801908 * q^67 + 7042916352 * q^68 - 1327151848320 * q^69 + 1118882944512 * q^71 + 141989511168 * q^72 + 121137579650 * q^73 - 910681021568 * q^74 + 1233293783850 * q^75 + 898686500864 * q^76 + 2681840157696 * q^78 - 290426785064 * q^79 + 72477573120 * q^80 + 1384924085739 * q^81 + 2850845337984 * q^82 + 7930211206092 * q^83 + 14856151680 * q^85 + 3500149053952 * q^86 - 747817823124 * q^87 + 2303541116928 * q^88 - 6025919250630 * q^89 + 299509125120 * q^90 - 5298259230720 * q^92 + 1054778393016 * q^93 + 3063572845824 * q^94 + 473916709440 * q^95 - 1101659111424 * q^96 - 22605636398380 * q^97 + 9519242365728 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/98\mathbb{Z}\right)^\times$.

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

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} $

67.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−32.0000

−

55.4256i

−513.000

888.542i

−2048.00

3547.24i

2160.00

3741.23i

65664.0

262144.

270824.

469080.i

138240.

−

239439.i

79.1

−32.0000

55.4256i

−513.000

−

888.542i

−2048.00

−

3547.24i

2160.00

−

3741.23i

65664.0

262144.

270824.

−

469080.i

138240.

239439.i

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	98.14.c.b		2
7.b	odd	2	1		98.14.c.c		2
7.c	even	3	1		98.14.a.d		1
7.c	even	3	1	inner	98.14.c.b		2
7.d	odd	6	1		14.14.a.b	✓	1
7.d	odd	6	1		98.14.c.c		2
21.g	even	6	1		126.14.a.a		1
28.f	even	6	1		112.14.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.14.a.b	✓	1	7.d	odd	6	1
98.14.a.d		1	7.c	even	3	1
98.14.c.b		2	1.a	even	1	1	trivial
98.14.c.b		2	7.c	even	3	1	inner
98.14.c.c		2	7.b	odd	2	1
98.14.c.c		2	7.d	odd	6	1
112.14.a.b		1	28.f	even	6	1
126.14.a.a		1	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{2} + 1026T_{3} + 1052676 $ T3^2 + 1026*T3 + 1052676 acting on $S_{14}^{\mathrm{new}}(98, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 64T + 4096 $ T^2 + 64*T + 4096
$3$	$ T^{2} + 1026 T + 1052676 $ T^2 + 1026*T + 1052676
$5$	$ T^{2} - 4320 T + 18662400 $ T^2 - 4320*T + 18662400
$7$	$ T^{2} $ T^2
$11$	$ T^{2} + \cdots + 77216852185344 $ T^2 - 8787312*T + 77216852185344
$13$	$ (T - 20420932)^{2} $ (T - 20420932)^2
$17$	$ T^{2} + \cdots + 2956549569444 $ T^2 - 1719462*T + 2956549569444
$19$	$ T^{2} + \cdots + 12\!\cdots\!64 $ T^2 + 109702942*T + 12034735483455364
$23$	$ T^{2} + \cdots + 41\!\cdots\!00 $ T^2 - 646760160*T + 418298704563225600
$29$	$ (T - 728867274)^{2} $ (T - 728867274)^2
$31$	$ T^{2} + \cdots + 10\!\cdots\!56 $ T^2 - 1028049116*T + 1056884984908381456
$37$	$ T^{2} + \cdots + 20\!\cdots\!44 $ T^2 + 14229390962*T + 202475567149447285444
$41$	$ (T + 44544458406)^{2} $ (T + 44544458406)^2
$43$	$ (T + 54689828968)^{2} $ (T + 54689828968)^2
$47$	$ T^{2} + \cdots + 22\!\cdots\!56 $ T^2 - 47868325716*T + 2291376606853066912656
$53$	$ T^{2} + \cdots + 28\!\cdots\!64 $ T^2 - 169986882858*T + 28895540343779414248164
$59$	$ T^{2} + \cdots + 90\!\cdots\!04 $ T^2 + 300765540198*T + 90459910170594753879204
$61$	$ T^{2} + \cdots + 13\!\cdots\!00 $ T^2 - 369996272360*T + 136897241560295299969600
$67$	$ T^{2} + \cdots + 61\!\cdots\!64 $ T^2 - 787010801908*T + 619386002319873216440464
$71$	$ (T - 559441472256)^{2} $ (T - 559441472256)^2
$73$	$ T^{2} + \cdots + 14\!\cdots\!00 $ T^2 - 121137579650*T + 14674313203460094122500
$79$	$ T^{2} + \cdots + 84\!\cdots\!96 $ T^2 + 290426785064*T + 84347717482610853484096
$83$	$ (T - 3965105603046)^{2} $ (T - 3965105603046)^2
$89$	$ T^{2} + \cdots + 36\!\cdots\!00 $ T^2 + 6025919250630*T + 36311702815113220755396900
$97$	$ (T + 11302818199190)^{2} $ (T + 11302818199190)^2
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	98.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - 64 \zeta_{6} q^{2} + (1026 \zeta_{6} - 1026) q^{3} + (4096 \zeta_{6} - 4096) q^{4} + 4320 \zeta_{6} q^{5} + 65664 q^{6} + 262144 q^{8} + 541647 \zeta_{6} q^{9} + ( - 276480 \zeta_{6} + 276480) q^{10} + \cdots + 4759621182864 q^{99} +O(q^{100}) \) q - 64z q^2 + (1026z - 1026) q^3 + (4096z - 4096) q^4 + 4320z q^5 + 65664 * q^6 + 262144 * q^8 + 541647z q^9 + (-276480z + 276480) q^10 + (-8787312z + 8787312) q^11 - 4202496z q^12 + 20420932 * q^13 - 4432320 * q^15 - 16777216z q^16 + (-1719462z + 1719462) q^17 + (-34665408z + 34665408) q^18 - 109702942z q^19 - 17694720 * q^20 - 562387968 * q^22 + 646760160z q^23 + (268959744z - 268959744) q^24 + (-1202040725z + 1202040725) q^25 - 1306939648z q^26 - 2191505220 * q^27 + 728867274 * q^29 + 283668480z q^30 + (-1028049116z + 1028049116) q^31 + (1073741824z - 1073741824) q^32 + 9015782112z q^33 - 110045568 * q^34 - 2218586112 * q^36 - 14229390962z q^37 + (7020988288z - 7020988288) q^38 + (20951876232z - 20951876232) q^39 + 1132462080z q^40 - 44544458406 * q^41 - 54689828968 * q^43 + 35992829952z q^44 + (2339915040z - 2339915040) q^45 + (-41392650240z + 41392650240) q^46 + 47868325716z q^47 + 17213423616 * q^48 - 76930606400 * q^50 + 1764168012z q^51 + (83644137472z - 83644137472) q^52 + (-169986882858z + 169986882858) q^53 + 140256334080z q^54 + 37961187840 * q^55 + 112555218492 * q^57 - 46647505536z q^58 + (300765540198z - 300765540198) q^59 + (-18154782720z + 18154782720) q^60 + 369996272360z q^61 - 65795143424 * q^62 + 68719476736 * q^64 + 88218426240z q^65 + (-577010055168z + 577010055168) q^66 + (-787010801908z + 787010801908) q^67 + 7042916352z q^68 - 663575924160 * q^69 + 559441472256 * q^71 + 141989511168z q^72 + (-121137579650z + 121137579650) q^73 + (910681021568z - 910681021568) q^74 + 1233293783850z q^75 + 449343250432 * q^76 + 1340920078848 * q^78 - 290426785064z q^79 + (-72477573120z + 72477573120) q^80 + (-1384924085739z + 1384924085739) q^81 + 2850845337984z q^82 + 3965105603046 * q^83 + 7428075840 * q^85 + 3500149053952z q^86 + (747817823124z - 747817823124) q^87 + (-2303541116928z + 2303541116928) q^88 - 6025919250630z q^89 + 149754562560 * q^90 - 2649129615360 * q^92 + 1054778393016z q^93 + (-3063572845824z + 3063572845824) q^94 + (-473916709440z + 473916709440) q^95 - 1101659111424z q^96 - 11302818199190 * q^97 + 4759621182864 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 64 q^{2} - 1026 q^{3} - 4096 q^{4} + 4320 q^{5} + 131328 q^{6} + 524288 q^{8} + 541647 q^{9} + 276480 q^{10} + 8787312 q^{11} - 4202496 q^{12} + 40841864 q^{13} - 8864640 q^{15} - 16777216 q^{16}+ \cdots + 9519242365728 q^{99}+O(q^{100}) \) 2 * q - 64 * q^2 - 1026 * q^3 - 4096 * q^4 + 4320 * q^5 + 131328 * q^6 + 524288 * q^8 + 541647 * q^9 + 276480 * q^10 + 8787312 * q^11 - 4202496 * q^12 + 40841864 * q^13 - 8864640 * q^15 - 16777216 * q^16 + 1719462 * q^17 + 34665408 * q^18 - 109702942 * q^19 - 35389440 * q^20 - 1124775936 * q^22 + 646760160 * q^23 - 268959744 * q^24 + 1202040725 * q^25 - 1306939648 * q^26 - 4383010440 * q^27 + 1457734548 * q^29 + 283668480 * q^30 + 1028049116 * q^31 - 1073741824 * q^32 + 9015782112 * q^33 - 220091136 * q^34 - 4437172224 * q^36 - 14229390962 * q^37 - 7020988288 * q^38 - 20951876232 * q^39 + 1132462080 * q^40 - 89088916812 * q^41 - 109379657936 * q^43 + 35992829952 * q^44 - 2339915040 * q^45 + 41392650240 * q^46 + 47868325716 * q^47 + 34426847232 * q^48 - 153861212800 * q^50 + 1764168012 * q^51 - 83644137472 * q^52 + 169986882858 * q^53 + 140256334080 * q^54 + 75922375680 * q^55 + 225110436984 * q^57 - 46647505536 * q^58 - 300765540198 * q^59 + 18154782720 * q^60 + 369996272360 * q^61 - 131590286848 * q^62 + 137438953472 * q^64 + 88218426240 * q^65 + 577010055168 * q^66 + 787010801908 * q^67 + 7042916352 * q^68 - 1327151848320 * q^69 + 1118882944512 * q^71 + 141989511168 * q^72 + 121137579650 * q^73 - 910681021568 * q^74 + 1233293783850 * q^75 + 898686500864 * q^76 + 2681840157696 * q^78 - 290426785064 * q^79 + 72477573120 * q^80 + 1384924085739 * q^81 + 2850845337984 * q^82 + 7930211206092 * q^83 + 14856151680 * q^85 + 3500149053952 * q^86 - 747817823124 * q^87 + 2303541116928 * q^88 - 6025919250630 * q^89 + 299509125120 * q^90 - 5298259230720 * q^92 + 1054778393016 * q^93 + 3063572845824 * q^94 + 473916709440 * q^95 - 1101659111424 * q^96 - 22605636398380 * q^97 + 9519242365728 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more