LMFDB - Newform orbit 49.7.d.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [49,7,Mod(19,49)] 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("49.19"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 49 = 7^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	49.d (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$11.2726500974$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{170})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 170x^{2} + 28900 $ x^4 + 170*x^2 + 28900
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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 basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (8 \beta_1 + 8) q^{2} + \beta_{3} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + 8 \beta_{2} q^{6} + 512 q^{8} + (1311 \beta_1 + 1311) q^{9} + 8 \beta_{3} q^{10} + 874 \beta_1 q^{11} - 49 \beta_{2} q^{13}+ \cdots - 1145814 q^{99}+O(q^{100}) $ q + (8b1 + 8) q^2 + b3 * q^3 + (b3 - b2) * q^5 + 8b2 q^6 + 512 * q^8 + (1311b1 + 1311) q^9 + 8b3 q^10 + 874b1 q^11 - 49b2 q^13 + 2040 * q^15 + (4096b1 + 4096) q^16 - 132b3 q^17 + 10488b1 q^18 + (-69b3 + 69b2) * q^19 - 6992 * q^22 + (-4738b1 - 4738) q^23 + 512b3 q^24 + 13585b1 q^25 + (392b3 - 392b2) * q^26 + 582b2 q^27 + 11146 * q^29 + (16320b1 + 16320) q^30 - 608b3 q^31 + (-874b3 + 874b2) * q^33 - 1056b2 q^34 + (-3002b1 - 3002) q^37 - 552b3 q^38 - 99960b1 q^39 + (512b3 - 512b2) * q^40 - 1274b2 q^41 + 31418 * q^43 + 1311b3 q^45 - 37904b1 q^46 + (1604b3 - 1604b2) * q^47 + 4096b2 q^48 - 108680 * q^50 + (-269280b1 - 269280) q^51 - 76406b1 q^53 + (-4656b3 + 4656b2) * q^54 + 874b2 q^55 - 140760 * q^57 + (89168b1 + 89168) q^58 + 2507b3 q^59 + (6091b3 - 6091b2) * q^61 - 4864b2 q^62 + 262144 * q^64 + (-99960b1 - 99960) q^65 - 6992b3 q^66 + 495242b1 q^67 - 4738b2 q^69 - 184406 * q^71 + (671232b1 + 671232) q^72 - 1350b3 q^73 - 24016b1 q^74 + (-13585b3 + 13585b2) * q^75 + 799680 * q^78 + (534934b1 + 534934) q^79 + 4096b3 q^80 + 231561b1 q^81 + (10192b3 - 10192b2) * q^82 + 15827b2 q^83 - 269280 * q^85 + (251344b1 + 251344) q^86 + 11146b3 q^87 + 447488b1 q^88 + (13938b3 - 13938b2) * q^89 + 10488b2 q^90 + (-1240320b1 - 1240320) q^93 + 12832b3 q^94 + 140760b1 q^95 - 18032b2 q^97 - 1145814 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{2} + 2048 q^{8} + 2622 q^{9} - 1748 q^{11} + 8160 q^{15} + 8192 q^{16} - 20976 q^{18} - 27968 q^{22} - 9476 q^{23} - 27170 q^{25} + 44584 q^{29} + 32640 q^{30} - 6004 q^{37} + 199920 q^{39}+ \cdots - 4583256 q^{99}+O(q^{100}) $ 4 * q + 16 * q^2 + 2048 * q^8 + 2622 * q^9 - 1748 * q^11 + 8160 * q^15 + 8192 * q^16 - 20976 * q^18 - 27968 * q^22 - 9476 * q^23 - 27170 * q^25 + 44584 * q^29 + 32640 * q^30 - 6004 * q^37 + 199920 * q^39 + 125672 * q^43 + 75808 * q^46 - 434720 * q^50 - 538560 * q^51 + 152812 * q^53 - 563040 * q^57 + 178336 * q^58 + 1048576 * q^64 - 199920 * q^65 - 990484 * q^67 - 737624 * q^71 + 1342464 * q^72 + 48032 * q^74 + 3198720 * q^78 + 1069868 * q^79 - 463122 * q^81 - 1077120 * q^85 + 502688 * q^86 - 894976 * q^88 - 2480640 * q^93 - 281520 * q^95 - 4583256 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} + 170x^{2} + 28900 $ x^4 + 170*x^2 + 28900 :

$\beta_{1}$	$=$	$ ( \nu^{2} ) / 170 $ (v^2) / 170
$\beta_{2}$	$=$	$ ( \nu^{3} + 340\nu ) / 85 $ (v^3 + 340*v) / 85
$\beta_{3}$	$=$	$ ( -\nu^{3} + 170\nu ) / 85 $ (-v^3 + 170*v) / 85

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{3} + \beta_{2} ) / 6 $ (b3 + b2) / 6
$\nu^{2}$	$=$	$ 170\beta_1 $ 170*b1
$\nu^{3}$	$=$	$ ( -170\beta_{3} + 85\beta_{2} ) / 3 $ (-170b3 + 85b2) / 3

Display $\beta_i$ in terms of $\nu^j$

Character values

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

$n$	$3$
$\chi(n)$	$1 + \beta_{1}$

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} $

19.1

−6.51920	+	11.2916i
6.51920	−	11.2916i
−6.51920	−	11.2916i
6.51920	+	11.2916i

4.00000

−

6.92820i

−39.1152

22.5832i

−39.1152

−

22.5832i

361.331i

512.000

655.500

−

1135.36i

−312.922

180.665i

19.2

4.00000

−

6.92820i

39.1152

−

22.5832i

39.1152

22.5832i

−

361.331i

512.000

655.500

−

1135.36i

312.922

−

180.665i

31.1

4.00000

6.92820i

−39.1152

−

22.5832i

−39.1152

22.5832i

−

361.331i

512.000

655.500

1135.36i

−312.922

−

180.665i

31.2

4.00000

6.92820i

39.1152

22.5832i

39.1152

−

22.5832i

361.331i

512.000

655.500

1135.36i

312.922

180.665i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	49.7.d.d		4
7.b	odd	2	1	inner	49.7.d.d		4
7.c	even	3	1		7.7.b.b	✓	2
7.c	even	3	1	inner	49.7.d.d		4
7.d	odd	6	1		7.7.b.b	✓	2
7.d	odd	6	1	inner	49.7.d.d		4
21.g	even	6	1		63.7.d.d		2
21.h	odd	6	1		63.7.d.d		2
28.f	even	6	1		112.7.c.b		2
28.g	odd	6	1		112.7.c.b		2
35.i	odd	6	1		175.7.d.e		2
35.j	even	6	1		175.7.d.e		2
35.k	even	12	2		175.7.c.c		4
35.l	odd	12	2		175.7.c.c		4
56.j	odd	6	1		448.7.c.d		2
56.k	odd	6	1		448.7.c.c		2
56.m	even	6	1		448.7.c.c		2
56.p	even	6	1		448.7.c.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.7.b.b	✓	2	7.c	even	3	1
7.7.b.b	✓	2	7.d	odd	6	1
49.7.d.d		4	1.a	even	1	1	trivial
49.7.d.d		4	7.b	odd	2	1	inner
49.7.d.d		4	7.c	even	3	1	inner
49.7.d.d		4	7.d	odd	6	1	inner
63.7.d.d		2	21.g	even	6	1
63.7.d.d		2	21.h	odd	6	1
112.7.c.b		2	28.f	even	6	1
112.7.c.b		2	28.g	odd	6	1
175.7.c.c		4	35.k	even	12	2
175.7.c.c		4	35.l	odd	12	2
175.7.d.e		2	35.i	odd	6	1
175.7.d.e		2	35.j	even	6	1
448.7.c.c		2	56.k	odd	6	1
448.7.c.c		2	56.m	even	6	1
448.7.c.d		2	56.j	odd	6	1
448.7.c.d		2	56.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{2} - 8T_{2} + 64 $ T2^2 - 8*T2 + 64 acting on $S_{7}^{\mathrm{new}}(49, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - 8 T + 64)^{2} $ (T^2 - 8*T + 64)^2
$3$	$ T^{4} - 2040 T^{2} + 4161600 $ T^4 - 2040*T^2 + 4161600
$5$	$ T^{4} - 2040 T^{2} + 4161600 $ T^4 - 2040*T^2 + 4161600
$7$	$ T^{4} $ T^4
$11$	$ (T^{2} + 874 T + 763876)^{2} $ (T^2 + 874*T + 763876)^2
$13$	$ (T^{2} + 4898040)^{2} $ (T^2 + 4898040)^2
$17$	$ T^{4} + \cdots + 12\!\cdots\!00 $ T^4 - 35544960*T^2 + 1263444181401600
$19$	$ T^{4} + \cdots + 94331490753600 $ T^4 - 9712440*T^2 + 94331490753600
$23$	$ (T^{2} + 4738 T + 22448644)^{2} $ (T^2 + 4738*T + 22448644)^2
$29$	$ (T - 11146)^{4} $ (T - 11146)^4
$31$	$ T^{4} + \cdots + 56\!\cdots\!00 $ T^4 - 754114560*T^2 + 568688769603993600
$37$	$ (T^{2} + 3002 T + 9012004)^{2} $ (T^2 + 3002*T + 9012004)^2
$41$	$ (T^{2} + 3311075040)^{2} $ (T^2 + 3311075040)^2
$43$	$ (T - 31418)^{4} $ (T - 31418)^4
$47$	$ T^{4} + \cdots + 27\!\cdots\!00 $ T^4 - 5248544640*T^2 + 27547220838072729600
$53$	$ (T^{2} - 76406 T + 5837876836)^{2} $ (T^2 - 76406*T + 5837876836)^2
$59$	$ T^{4} + \cdots + 16\!\cdots\!00 $ T^4 - 12821499960*T^2 + 164390861224280001600
$61$	$ T^{4} + \cdots + 57\!\cdots\!00 $ T^4 - 75684573240*T^2 + 5728154626520924097600
$67$	$ (T^{2} + 495242 T + 245264638564)^{2} $ (T^2 + 495242*T + 245264638564)^2
$71$	$ (T + 184406)^{4} $ (T + 184406)^4
$73$	$ T^{4} + \cdots + 13\!\cdots\!00 $ T^4 - 3717900000*T^2 + 13822780410000000000
$79$	$ (T^{2} - 534934 T + 286154384356)^{2} $ (T^2 - 534934*T + 286154384356)^2
$83$	$ (T^{2} + 511007615160)^{2} $ (T^2 + 511007615160)^2
$89$	$ T^{4} + \cdots + 15\!\cdots\!00 $ T^4 - 396306401760*T^2 + 157058764075958531097600
$97$	$ (T^{2} + 663312168960)^{2} $ (T^2 + 663312168960)^2
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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:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	49.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (8 \beta_1 + 8) q^{2} + \beta_{3} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + 8 \beta_{2} q^{6} + 512 q^{8} + (1311 \beta_1 + 1311) q^{9} + 8 \beta_{3} q^{10} + 874 \beta_1 q^{11} - 49 \beta_{2} q^{13}+ \cdots - 1145814 q^{99}+O(q^{100}) \) q + (8b1 + 8) q^2 + b3 * q^3 + (b3 - b2) * q^5 + 8b2 q^6 + 512 * q^8 + (1311b1 + 1311) q^9 + 8b3 q^10 + 874b1 q^11 - 49b2 q^13 + 2040 * q^15 + (4096b1 + 4096) q^16 - 132b3 q^17 + 10488b1 q^18 + (-69b3 + 69b2) * q^19 - 6992 * q^22 + (-4738b1 - 4738) q^23 + 512b3 q^24 + 13585b1 q^25 + (392b3 - 392b2) * q^26 + 582b2 q^27 + 11146 * q^29 + (16320b1 + 16320) q^30 - 608b3 q^31 + (-874b3 + 874b2) * q^33 - 1056b2 q^34 + (-3002b1 - 3002) q^37 - 552b3 q^38 - 99960b1 q^39 + (512b3 - 512b2) * q^40 - 1274b2 q^41 + 31418 * q^43 + 1311b3 q^45 - 37904b1 q^46 + (1604b3 - 1604b2) * q^47 + 4096b2 q^48 - 108680 * q^50 + (-269280b1 - 269280) q^51 - 76406b1 q^53 + (-4656b3 + 4656b2) * q^54 + 874b2 q^55 - 140760 * q^57 + (89168b1 + 89168) q^58 + 2507b3 q^59 + (6091b3 - 6091b2) * q^61 - 4864b2 q^62 + 262144 * q^64 + (-99960b1 - 99960) q^65 - 6992b3 q^66 + 495242b1 q^67 - 4738b2 q^69 - 184406 * q^71 + (671232b1 + 671232) q^72 - 1350b3 q^73 - 24016b1 q^74 + (-13585b3 + 13585b2) * q^75 + 799680 * q^78 + (534934b1 + 534934) q^79 + 4096b3 q^80 + 231561b1 q^81 + (10192b3 - 10192b2) * q^82 + 15827b2 q^83 - 269280 * q^85 + (251344b1 + 251344) q^86 + 11146b3 q^87 + 447488b1 q^88 + (13938b3 - 13938b2) * q^89 + 10488b2 q^90 + (-1240320b1 - 1240320) q^93 + 12832b3 q^94 + 140760b1 q^95 - 18032b2 q^97 - 1145814 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{2} + 2048 q^{8} + 2622 q^{9} - 1748 q^{11} + 8160 q^{15} + 8192 q^{16} - 20976 q^{18} - 27968 q^{22} - 9476 q^{23} - 27170 q^{25} + 44584 q^{29} + 32640 q^{30} - 6004 q^{37} + 199920 q^{39}+ \cdots - 4583256 q^{99}+O(q^{100}) \) 4 * q + 16 * q^2 + 2048 * q^8 + 2622 * q^9 - 1748 * q^11 + 8160 * q^15 + 8192 * q^16 - 20976 * q^18 - 27968 * q^22 - 9476 * q^23 - 27170 * q^25 + 44584 * q^29 + 32640 * q^30 - 6004 * q^37 + 199920 * q^39 + 125672 * q^43 + 75808 * q^46 - 434720 * q^50 - 538560 * q^51 + 152812 * q^53 - 563040 * q^57 + 178336 * q^58 + 1048576 * q^64 - 199920 * q^65 - 990484 * q^67 - 737624 * q^71 + 1342464 * q^72 + 48032 * q^74 + 3198720 * q^78 + 1069868 * q^79 - 463122 * q^81 - 1077120 * q^85 + 502688 * q^86 - 894976 * q^88 - 2480640 * q^93 - 281520 * q^95 - 4583256 * q^99

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	49.7.d.d

Level	$49$
Weight	$7$
Character orbit	49.d
Analytic conductor	$11.273$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(11.2726500974\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{170})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} + 170x^{2} + 28900 \) x^4 + 170*x^2 + 28900
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{2}\cdot 3 \)
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{2} ) / 170 \) (v^2) / 170
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + 340\nu ) / 85 \) (v^3 + 340*v) / 85
\(\beta_{3}\)	\(=\)	\( ( -\nu^{3} + 170\nu ) / 85 \) (-v^3 + 170*v) / 85

\(\nu\)	\(=\)	\( ( \beta_{3} + \beta_{2} ) / 6 \) (b3 + b2) / 6
\(\nu^{2}\)	\(=\)	\( 170\beta_1 \) 170*b1
\(\nu^{3}\)	\(=\)	\( ( -170\beta_{3} + 85\beta_{2} ) / 3 \) (-170b3 + 85b2) / 3

$p$	$F_p(T)$
$2$	\( (T^{2} - 8 T + 64)^{2} \) (T^2 - 8*T + 64)^2
$3$	\( T^{4} - 2040 T^{2} + 4161600 \) T^4 - 2040*T^2 + 4161600
$5$	\( T^{4} - 2040 T^{2} + 4161600 \) T^4 - 2040*T^2 + 4161600
$7$	\( T^{4} \) T^4
$11$	\( (T^{2} + 874 T + 763876)^{2} \) (T^2 + 874*T + 763876)^2
$13$	\( (T^{2} + 4898040)^{2} \) (T^2 + 4898040)^2
$17$	\( T^{4} + \cdots + 12\!\cdots\!00 \) T^4 - 35544960*T^2 + 1263444181401600
$19$	\( T^{4} + \cdots + 94331490753600 \) T^4 - 9712440*T^2 + 94331490753600
$23$	\( (T^{2} + 4738 T + 22448644)^{2} \) (T^2 + 4738*T + 22448644)^2
$29$	\( (T - 11146)^{4} \) (T - 11146)^4
$31$	\( T^{4} + \cdots + 56\!\cdots\!00 \) T^4 - 754114560*T^2 + 568688769603993600
$37$	\( (T^{2} + 3002 T + 9012004)^{2} \) (T^2 + 3002*T + 9012004)^2
$41$	\( (T^{2} + 3311075040)^{2} \) (T^2 + 3311075040)^2
$43$	\( (T - 31418)^{4} \) (T - 31418)^4
$47$	\( T^{4} + \cdots + 27\!\cdots\!00 \) T^4 - 5248544640*T^2 + 27547220838072729600
$53$	\( (T^{2} - 76406 T + 5837876836)^{2} \) (T^2 - 76406*T + 5837876836)^2
$59$	\( T^{4} + \cdots + 16\!\cdots\!00 \) T^4 - 12821499960*T^2 + 164390861224280001600
$61$	\( T^{4} + \cdots + 57\!\cdots\!00 \) T^4 - 75684573240*T^2 + 5728154626520924097600
$67$	\( (T^{2} + 495242 T + 245264638564)^{2} \) (T^2 + 495242*T + 245264638564)^2
$71$	\( (T + 184406)^{4} \) (T + 184406)^4
$73$	\( T^{4} + \cdots + 13\!\cdots\!00 \) T^4 - 3717900000*T^2 + 13822780410000000000
$79$	\( (T^{2} - 534934 T + 286154384356)^{2} \) (T^2 - 534934*T + 286154384356)^2
$83$	\( (T^{2} + 511007615160)^{2} \) (T^2 + 511007615160)^2
$89$	\( T^{4} + \cdots + 15\!\cdots\!00 \) T^4 - 396306401760*T^2 + 157058764075958531097600
$97$	\( (T^{2} + 663312168960)^{2} \) (T^2 + 663312168960)^2
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\(n\)	\(3\)
\(\chi(n)\)	\(1 + \beta_{1}\)

\(n\):		e.g. 2-40 or 80-90
Significant digits:
Format: