LMFDB - Newform orbit 7.7.d.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7,7,Mod(3,7)] 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("7.3"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 7 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	7.d (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.61037858534$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 primitive root of unity $\zeta_{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + 12 \zeta_{6} q^{2} + ( - 7 \zeta_{6} - 7) q^{3} + (80 \zeta_{6} - 80) q^{4} + ( - 105 \zeta_{6} + 210) q^{5} + ( - 168 \zeta_{6} + 84) q^{6} - 343 q^{7} - 192 q^{8} - 582 \zeta_{6} q^{9} + (1260 \zeta_{6} + 1260) q^{10} + \cdots + 860778 q^{99} +O(q^{100}) $ q + 12z q^2 + (-7z - 7) q^3 + (80z - 80) q^4 + (-105z + 210) q^5 + (-168z + 84) q^6 - 343 * q^7 - 192 * q^8 - 582z q^9 + (1260z + 1260) q^10 + (1479z - 1479) q^11 + (-560z + 1120) q^12 + (560z - 280) q^13 - 4116z q^14 - 2205 * q^15 + 2816z q^16 + (-1743z - 1743) q^17 + (-6984z + 6984) q^18 + (-3969z + 7938) q^19 + (16800z - 8400) q^20 + (2401z + 2401) q^21 - 17748 * q^22 + 5913z q^23 + (1344z + 1344) q^24 + (-17450z + 17450) q^25 + (3360z - 6720) q^26 + (18354z - 9177) q^27 + (-27440z + 27440) q^28 + 3978 * q^29 - 26460z q^30 + (-7399z - 7399) q^31 + (46080z - 46080) q^32 + (-10353z + 20706) q^33 + (-41832z + 20916) q^34 + (36015z - 72030) q^35 + 46560 * q^36 + 61577z q^37 + (47628z + 47628) q^38 + (-5880z + 5880) q^39 + (20160z - 40320) q^40 + (-127680z + 63840) q^41 + (57624z - 28812) q^42 - 17414 * q^43 - 118320z q^44 + (-61110z - 61110) q^45 + (70956z - 70956) q^46 + (17703z - 35406) q^47 + (-39424z + 19712) q^48 + 117649 * q^49 + 209400 * q^50 + 36603z q^51 + (-22400z - 22400) q^52 + (-60513z + 60513) q^53 + (110124z - 220248) q^54 + (310590z - 155295) q^55 + 65856 * q^56 - 83349 * q^57 + 47736z q^58 + (-124551z - 124551) q^59 + (-176400z + 176400) q^60 + (-93961z + 187922) q^61 + (-177576z + 88788) q^62 + 199626z q^63 - 372736 * q^64 + 88200z q^65 + (124236z + 124236) q^66 + (-268777z + 268777) q^67 + (-139440z + 278880) q^68 + (-82782z + 41391) q^69 + (-432180z - 432180) q^70 + 101922 * q^71 + 111744z q^72 + (183393z + 183393) q^73 + (738924z - 738924) q^74 + (122150z - 244300) q^75 + (635040z - 317520) q^76 + (-507297z + 507297) q^77 + 70560 * q^78 - 362231z q^79 + (295680z + 295680) q^80 + (231561z - 231561) q^81 + (-766080z + 1532160) q^82 + (250320z - 125160) q^83 + (192080z - 384160) q^84 - 549045 * q^85 - 208968z q^86 + (-27846z - 27846) q^87 + (-283968z + 283968) q^88 + (770511z - 1541022) q^89 + (-1466640z + 733320) q^90 + (-192080z + 96040) q^91 - 473040 * q^92 + 155379z q^93 + (-212436z - 212436) q^94 + (-1250235z + 1250235) q^95 + (-322560z + 645120) q^96 + (1748320z - 874160) q^97 + 1411788z q^98 + 860778 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{2} - 21 q^{3} - 80 q^{4} + 315 q^{5} - 686 q^{7} - 384 q^{8} - 582 q^{9} + 3780 q^{10} - 1479 q^{11} + 1680 q^{12} - 4116 q^{14} - 4410 q^{15} + 2816 q^{16} - 5229 q^{17} + 6984 q^{18} + 11907 q^{19}+ \cdots + 1721556 q^{99}+O(q^{100}) $ 2 * q + 12 * q^2 - 21 * q^3 - 80 * q^4 + 315 * q^5 - 686 * q^7 - 384 * q^8 - 582 * q^9 + 3780 * q^10 - 1479 * q^11 + 1680 * q^12 - 4116 * q^14 - 4410 * q^15 + 2816 * q^16 - 5229 * q^17 + 6984 * q^18 + 11907 * q^19 + 7203 * q^21 - 35496 * q^22 + 5913 * q^23 + 4032 * q^24 + 17450 * q^25 - 10080 * q^26 + 27440 * q^28 + 7956 * q^29 - 26460 * q^30 - 22197 * q^31 - 46080 * q^32 + 31059 * q^33 - 108045 * q^35 + 93120 * q^36 + 61577 * q^37 + 142884 * q^38 + 5880 * q^39 - 60480 * q^40 - 34828 * q^43 - 118320 * q^44 - 183330 * q^45 - 70956 * q^46 - 53109 * q^47 + 235298 * q^49 + 418800 * q^50 + 36603 * q^51 - 67200 * q^52 + 60513 * q^53 - 330372 * q^54 + 131712 * q^56 - 166698 * q^57 + 47736 * q^58 - 373653 * q^59 + 176400 * q^60 + 281883 * q^61 + 199626 * q^63 - 745472 * q^64 + 88200 * q^65 + 372708 * q^66 + 268777 * q^67 + 418320 * q^68 - 1296540 * q^70 + 203844 * q^71 + 111744 * q^72 + 550179 * q^73 - 738924 * q^74 - 366450 * q^75 + 507297 * q^77 + 141120 * q^78 - 362231 * q^79 + 887040 * q^80 - 231561 * q^81 + 2298240 * q^82 - 576240 * q^84 - 1098090 * q^85 - 208968 * q^86 - 83538 * q^87 + 283968 * q^88 - 2311533 * q^89 - 946080 * q^92 + 155379 * q^93 - 637308 * q^94 + 1250235 * q^95 + 967680 * q^96 + 1411788 * q^98 + 1721556 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$.

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

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} $

3.1

0.500000	+	0.866025i
0.500000	−	0.866025i

6.00000

10.3923i

−10.5000

−

6.06218i

−40.0000

69.2820i

157.500

−

90.9327i

−

145.492i

−343.000

−192.000

−291.000

−

504.027i

1890.00

1091.19i

5.1

6.00000

−

10.3923i

−10.5000

6.06218i

−40.0000

−

69.2820i

157.500

90.9327i

145.492i

−343.000

−192.000

−291.000

504.027i

1890.00

−

1091.19i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
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	7.7.d.a	✓	2
3.b	odd	2	1		63.7.m.a		2
4.b	odd	2	1		112.7.s.a		2
7.b	odd	2	1		49.7.d.b		2
7.c	even	3	1		49.7.b.a		2
7.c	even	3	1		49.7.d.b		2
7.d	odd	6	1	inner	7.7.d.a	✓	2
7.d	odd	6	1		49.7.b.a		2
21.g	even	6	1		63.7.m.a		2
21.g	even	6	1		441.7.d.a		2
21.h	odd	6	1		441.7.d.a		2
28.f	even	6	1		112.7.s.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.7.d.a	✓	2	1.a	even	1	1	trivial
7.7.d.a	✓	2	7.d	odd	6	1	inner
49.7.b.a		2	7.c	even	3	1
49.7.b.a		2	7.d	odd	6	1
49.7.d.b		2	7.b	odd	2	1
49.7.d.b		2	7.c	even	3	1
63.7.m.a		2	3.b	odd	2	1
63.7.m.a		2	21.g	even	6	1
112.7.s.a		2	4.b	odd	2	1
112.7.s.a		2	28.f	even	6	1
441.7.d.a		2	21.g	even	6	1
441.7.d.a		2	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{2} - 12T_{2} + 144 $ T2^2 - 12*T2 + 144 acting on $S_{7}^{\mathrm{new}}(7, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} - 12T + 144 $ T^2 - 12*T + 144
$3$	$ T^{2} + 21T + 147 $ T^2 + 21*T + 147
$5$	$ T^{2} - 315T + 33075 $ T^2 - 315*T + 33075
$7$	$ (T + 343)^{2} $ (T + 343)^2
$11$	$ T^{2} + 1479 T + 2187441 $ T^2 + 1479*T + 2187441
$13$	$ T^{2} + 235200 $ T^2 + 235200
$17$	$ T^{2} + 5229 T + 9114147 $ T^2 + 5229*T + 9114147
$19$	$ T^{2} - 11907 T + 47258883 $ T^2 - 11907*T + 47258883
$23$	$ T^{2} - 5913 T + 34963569 $ T^2 - 5913*T + 34963569
$29$	$ (T - 3978)^{2} $ (T - 3978)^2
$31$	$ T^{2} + 22197 T + 164235603 $ T^2 + 22197*T + 164235603
$37$	$ T^{2} + \cdots + 3791726929 $ T^2 - 61577*T + 3791726929
$41$	$ T^{2} + 12226636800 $ T^2 + 12226636800
$43$	$ (T + 17414)^{2} $ (T + 17414)^2
$47$	$ T^{2} + 53109 T + 940188627 $ T^2 + 53109*T + 940188627
$53$	$ T^{2} + \cdots + 3661823169 $ T^2 - 60513*T + 3661823169
$59$	$ T^{2} + \cdots + 46538854803 $ T^2 + 373653*T + 46538854803
$61$	$ T^{2} + \cdots + 26486008563 $ T^2 - 281883*T + 26486008563
$67$	$ T^{2} + \cdots + 72241075729 $ T^2 - 268777*T + 72241075729
$71$	$ (T - 101922)^{2} $ (T - 101922)^2
$73$	$ T^{2} + \cdots + 100898977347 $ T^2 - 550179*T + 100898977347
$79$	$ T^{2} + \cdots + 131211297361 $ T^2 + 362231*T + 131211297361
$83$	$ T^{2} + 46995076800 $ T^2 + 46995076800
$89$	$ T^{2} + \cdots + 1781061603363 $ T^2 + 2311533*T + 1781061603363
$97$	$ T^{2} + 2292467116800 $ T^2 + 2292467116800
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Elliptic curves over $\Q$
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Belyi maps

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	7.d (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + 12 \zeta_{6} q^{2} + ( - 7 \zeta_{6} - 7) q^{3} + (80 \zeta_{6} - 80) q^{4} + ( - 105 \zeta_{6} + 210) q^{5} + ( - 168 \zeta_{6} + 84) q^{6} - 343 q^{7} - 192 q^{8} - 582 \zeta_{6} q^{9} + (1260 \zeta_{6} + 1260) q^{10} + \cdots + 860778 q^{99} +O(q^{100}) \) q + 12z q^2 + (-7z - 7) q^3 + (80z - 80) q^4 + (-105z + 210) q^5 + (-168z + 84) q^6 - 343 * q^7 - 192 * q^8 - 582z q^9 + (1260z + 1260) q^10 + (1479z - 1479) q^11 + (-560z + 1120) q^12 + (560z - 280) q^13 - 4116z q^14 - 2205 * q^15 + 2816z q^16 + (-1743z - 1743) q^17 + (-6984z + 6984) q^18 + (-3969z + 7938) q^19 + (16800z - 8400) q^20 + (2401z + 2401) q^21 - 17748 * q^22 + 5913z q^23 + (1344z + 1344) q^24 + (-17450z + 17450) q^25 + (3360z - 6720) q^26 + (18354z - 9177) q^27 + (-27440z + 27440) q^28 + 3978 * q^29 - 26460z q^30 + (-7399z - 7399) q^31 + (46080z - 46080) q^32 + (-10353z + 20706) q^33 + (-41832z + 20916) q^34 + (36015z - 72030) q^35 + 46560 * q^36 + 61577z q^37 + (47628z + 47628) q^38 + (-5880z + 5880) q^39 + (20160z - 40320) q^40 + (-127680z + 63840) q^41 + (57624z - 28812) q^42 - 17414 * q^43 - 118320z q^44 + (-61110z - 61110) q^45 + (70956z - 70956) q^46 + (17703z - 35406) q^47 + (-39424z + 19712) q^48 + 117649 * q^49 + 209400 * q^50 + 36603z q^51 + (-22400z - 22400) q^52 + (-60513z + 60513) q^53 + (110124z - 220248) q^54 + (310590z - 155295) q^55 + 65856 * q^56 - 83349 * q^57 + 47736z q^58 + (-124551z - 124551) q^59 + (-176400z + 176400) q^60 + (-93961z + 187922) q^61 + (-177576z + 88788) q^62 + 199626z q^63 - 372736 * q^64 + 88200z q^65 + (124236z + 124236) q^66 + (-268777z + 268777) q^67 + (-139440z + 278880) q^68 + (-82782z + 41391) q^69 + (-432180z - 432180) q^70 + 101922 * q^71 + 111744z q^72 + (183393z + 183393) q^73 + (738924z - 738924) q^74 + (122150z - 244300) q^75 + (635040z - 317520) q^76 + (-507297z + 507297) q^77 + 70560 * q^78 - 362231z q^79 + (295680z + 295680) q^80 + (231561z - 231561) q^81 + (-766080z + 1532160) q^82 + (250320z - 125160) q^83 + (192080z - 384160) q^84 - 549045 * q^85 - 208968z q^86 + (-27846z - 27846) q^87 + (-283968z + 283968) q^88 + (770511z - 1541022) q^89 + (-1466640z + 733320) q^90 + (-192080z + 96040) q^91 - 473040 * q^92 + 155379z q^93 + (-212436z - 212436) q^94 + (-1250235z + 1250235) q^95 + (-322560z + 645120) q^96 + (1748320z - 874160) q^97 + 1411788z q^98 + 860778 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{2} - 21 q^{3} - 80 q^{4} + 315 q^{5} - 686 q^{7} - 384 q^{8} - 582 q^{9} + 3780 q^{10} - 1479 q^{11} + 1680 q^{12} - 4116 q^{14} - 4410 q^{15} + 2816 q^{16} - 5229 q^{17} + 6984 q^{18} + 11907 q^{19}+ \cdots + 1721556 q^{99}+O(q^{100}) \) 2 * q + 12 * q^2 - 21 * q^3 - 80 * q^4 + 315 * q^5 - 686 * q^7 - 384 * q^8 - 582 * q^9 + 3780 * q^10 - 1479 * q^11 + 1680 * q^12 - 4116 * q^14 - 4410 * q^15 + 2816 * q^16 - 5229 * q^17 + 6984 * q^18 + 11907 * q^19 + 7203 * q^21 - 35496 * q^22 + 5913 * q^23 + 4032 * q^24 + 17450 * q^25 - 10080 * q^26 + 27440 * q^28 + 7956 * q^29 - 26460 * q^30 - 22197 * q^31 - 46080 * q^32 + 31059 * q^33 - 108045 * q^35 + 93120 * q^36 + 61577 * q^37 + 142884 * q^38 + 5880 * q^39 - 60480 * q^40 - 34828 * q^43 - 118320 * q^44 - 183330 * q^45 - 70956 * q^46 - 53109 * q^47 + 235298 * q^49 + 418800 * q^50 + 36603 * q^51 - 67200 * q^52 + 60513 * q^53 - 330372 * q^54 + 131712 * q^56 - 166698 * q^57 + 47736 * q^58 - 373653 * q^59 + 176400 * q^60 + 281883 * q^61 + 199626 * q^63 - 745472 * q^64 + 88200 * q^65 + 372708 * q^66 + 268777 * q^67 + 418320 * q^68 - 1296540 * q^70 + 203844 * q^71 + 111744 * q^72 + 550179 * q^73 - 738924 * q^74 - 366450 * q^75 + 507297 * q^77 + 141120 * q^78 - 362231 * q^79 + 887040 * q^80 - 231561 * q^81 + 2298240 * q^82 - 576240 * q^84 - 1098090 * q^85 - 208968 * q^86 - 83538 * q^87 + 283968 * q^88 - 2311533 * q^89 - 946080 * q^92 + 155379 * q^93 - 637308 * q^94 + 1250235 * q^95 + 967680 * q^96 + 1411788 * q^98 + 1721556 * q^99

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