LMFDB - Newform orbit 676.3.g.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [676,3,Mod(437,676)] 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("676.437"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(676, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 676 = 2^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	676.g (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$18.4196658708$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	6.0.20819026944.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 42x^{4} + 441x^{2} + 36 $ x^6 + 42x^4 + 441x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{3} q^{3} + (\beta_{5} - \beta_{2} + 1) q^{5} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 5) q^{9} + ( - 2 \beta_{3} + 3 \beta_{2} + \cdots + 3) q^{11}+ \cdots + ( - 6 \beta_{4} - 21 \beta_{3} + \cdots + 3) q^{99}+O(q^{100}) $ q + b3 * q^3 + (b5 - b2 + 1) * q^5 + (-b4 + b3 - b2 - b1 - 1) * q^7 + (b5 - b4 + b3 + 5) * q^9 + (-2b3 + 3b2 + 2b1 + 3) q^11 + (-b5 + 4b3 + 4b2 + 4b1 - 4) q^15 + (b5 + b4 - 2b2 + 5b1) * q^17 + (b3 - 13b2 + b1 + 13) q^19 + (-b4 + 3b3 + 10b2 - 3b1 + 10) q^21 + (-b5 - b4 + 20b2) q^23 + (-b5 - b4 - 27b2 - 7b1) * q^25 + (3b3 + 6) q^27 + (-b5 + b4 + 6b3 + 10) q^29 + (-4b5 - 5b3 - b2 - 5b1 + 1) q^31 + (4b4 + b3 - 28b2 - b1 - 28) * q^33 + (-4b5 + 4b4 + b3 + 40) * q^35 + (-b4 + 9b2 + 9) q^37 + (4b5 - 3b3 + 11b2 - 3b1 - 11) * q^41 + (-4b5 - 4b4 + 14b2 - b1) q^43 + (-3b3 - 51b2 - 3b1 + 51) q^45 + (5b4 - 7b3 + 19b2 + 7b1 + 19) * q^47 + (5b5 + 5b4 + 15b2 - 3b1) * q^49 + (4b5 + 4b4 - 62b2 + 13b1) * q^51 + (b5 - b4 - 4b3 + 38) q^53 + (5b5 - 5b4 - 16b3 + 22) q^55 + (2b5 + 14b3 - 14b2 + 14b1 + 14) * q^57 + (-2b4 - 3b3 - 31b2 + 3b1 - 31) * q^59 + (2b5 - 2b4 - 8b3 + 34) q^61 + (4b4 + 7b3 + 47b2 - 7b1 + 47) * q^63 + (-2b5 - 8b3 + 13b2 - 8b1 - 13) * q^67 + (b5 + b4 - 8b2 - 26b1) * q^69 + (-b5 + 5b3 - 35b2 + 5b1 + 35) q^71 + (-6b4 + 7b3 + 37b2 - 7b1 + 37) * q^73 + (-6b5 - 6b4 + 90b2 + 14b1) * q^75 + (-b5 - b4 - 46b2 + 6b1) * q^77 + (-5b5 + 5b4 - 12b3 - 10) q^79 + (-6b5 + 6b4 - 3) * q^81 + (8b5 - 6b3 + 13b2 - 6b1 - 13) * q^83 + (-7b4 - 13b3 - 32b2 + 13b1 - 32) * q^85 + (7b5 - 7b4 + 10b3 + 92) q^87 + (7b3 - 39b2 - 7b1 - 39) q^89 + (-6b5 - 16b3 + 54b2 - 16b1 - 54) * q^93 + (12b5 + 12b4 - 18b2 + 8b1) * q^95 + (6b5 - 8b3 - 59b2 - 8b1 + 59) * q^97 + (-6b4 - 21b3 + 3b2 + 21b1 + 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{5} - 6 q^{7} + 30 q^{9} + 18 q^{11} - 24 q^{15} + 78 q^{19} + 60 q^{21} + 36 q^{27} + 60 q^{29} + 6 q^{31} - 168 q^{33} + 240 q^{35} + 54 q^{37} - 66 q^{41} + 306 q^{45} + 114 q^{47} + 228 q^{53}+ \cdots + 18 q^{99}+O(q^{100}) $ 6 * q + 6 * q^5 - 6 * q^7 + 30 * q^9 + 18 * q^11 - 24 * q^15 + 78 * q^19 + 60 * q^21 + 36 * q^27 + 60 * q^29 + 6 * q^31 - 168 * q^33 + 240 * q^35 + 54 * q^37 - 66 * q^41 + 306 * q^45 + 114 * q^47 + 228 * q^53 + 132 * q^55 + 84 * q^57 - 186 * q^59 + 204 * q^61 + 282 * q^63 - 78 * q^67 + 210 * q^71 + 222 * q^73 - 60 * q^79 - 18 * q^81 - 78 * q^83 - 192 * q^85 + 552 * q^87 - 234 * q^89 - 324 * q^93 + 354 * q^97 + 18 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 42x^{4} + 441x^{2} + 36 $ x^6 + 42*x^4 + 441*x^2 + 36 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{3} + 21\nu ) / 6 $ (v^3 + 21*v) / 6
$\beta_{3}$	$=$	$ ( \nu^{4} + 21\nu^{2} ) / 6 $ (v^4 + 21*v^2) / 6
$\beta_{4}$	$=$	$ ( \nu^{5} + \nu^{4} + 35\nu^{3} + 27\nu^{2} + 288\nu + 84 ) / 12 $ (v^5 + v^4 + 35v^3 + 27v^2 + 288*v + 84) / 12
$\beta_{5}$	$=$	$ ( \nu^{5} - \nu^{4} + 35\nu^{3} - 27\nu^{2} + 288\nu - 84 ) / 12 $ (v^5 - v^4 + 35v^3 - 27v^2 + 288*v - 84) / 12

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{5} + \beta_{4} - \beta_{3} - 14 $ -b5 + b4 - b3 - 14
$\nu^{3}$	$=$	$ 6\beta_{2} - 21\beta_1 $ 6b2 - 21b1
$\nu^{4}$	$=$	$ 21\beta_{5} - 21\beta_{4} + 27\beta_{3} + 294 $ 21b5 - 21b4 + 27*b3 + 294
$\nu^{5}$	$=$	$ 6\beta_{5} + 6\beta_{4} - 210\beta_{2} + 447\beta_1 $ 6b5 + 6b4 - 210b2 + 447b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/676\mathbb{Z}\right)^\times$.

$n$	$339$	$509$
$\chi(n)$	$1$	$-\beta_{2}$

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} $

437.1

	−	4.43242i
	−	0.286838i
		4.71926i
		4.43242i
		0.286838i
	−	4.71926i

−4.43242

6.03938

6.03938i

−0.393041

0.393041i

10.6463

437.2

−0.286838

−5.81544

−

5.81544i

−8.10228

8.10228i

−8.91772

437.3

4.71926

2.77606

2.77606i

5.49532

−

5.49532i

13.2714

577.1

−4.43242

6.03938

−

6.03938i

−0.393041

−

0.393041i

10.6463

577.2

−0.286838

−5.81544

5.81544i

−8.10228

−

8.10228i

−8.91772

577.3

4.71926

2.77606

−

2.77606i

5.49532

5.49532i

13.2714

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.d	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	676.3.g.b		6
13.b	even	2	1		52.3.g.a	✓	6
13.d	odd	4	1		52.3.g.a	✓	6
13.d	odd	4	1	inner	676.3.g.b		6
39.d	odd	2	1		468.3.m.c		6
39.f	even	4	1		468.3.m.c		6
52.b	odd	2	1		208.3.t.d		6
52.f	even	4	1		208.3.t.d		6
65.d	even	2	1		1300.3.t.a		6
65.f	even	4	1		1300.3.k.a		6
65.g	odd	4	1		1300.3.t.a		6
65.h	odd	4	1		1300.3.k.a		6
65.h	odd	4	1		1300.3.k.b		6
65.k	even	4	1		1300.3.k.b		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
52.3.g.a	✓	6	13.b	even	2	1
52.3.g.a	✓	6	13.d	odd	4	1
208.3.t.d		6	52.b	odd	2	1
208.3.t.d		6	52.f	even	4	1
468.3.m.c		6	39.d	odd	2	1
468.3.m.c		6	39.f	even	4	1
676.3.g.b		6	1.a	even	1	1	trivial
676.3.g.b		6	13.d	odd	4	1	inner
1300.3.k.a		6	65.f	even	4	1
1300.3.k.a		6	65.h	odd	4	1
1300.3.k.b		6	65.h	odd	4	1
1300.3.k.b		6	65.k	even	4	1
1300.3.t.a		6	65.d	even	2	1
1300.3.t.a		6	65.g	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(676, [\chi])$:

$ T_{3}^{3} - 21T_{3} - 6 $

T3^3 - 21*T3 - 6

$ T_{5}^{6} - 6T_{5}^{5} + 18T_{5}^{4} + 24T_{5}^{3} + 4761T_{5}^{2} - 26910T_{5} + 76050 $

T5^6 - 6*T5^5 + 18*T5^4 + 24*T5^3 + 4761*T5^2 - 26910*T5 + 76050

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ (T^{3} - 21 T - 6)^{2} $ (T^3 - 21*T - 6)^2
$5$	$ T^{6} - 6 T^{5} + \cdots + 76050 $ T^6 - 6T^5 + 18T^4 + 24T^3 + 4761T^2 - 26910*T + 76050
$7$	$ T^{6} + 6 T^{5} + \cdots + 2450 $ T^6 + 6T^5 + 18T^4 - 452T^3 + 7569T^2 + 6090*T + 2450
$11$	$ T^{6} - 18 T^{5} + \cdots + 596232 $ T^6 - 18T^5 + 162T^4 + 960T^3 + 12996T^2 - 124488*T + 596232
$13$	$ T^{6} $ T^6
$17$	$ T^{6} + 1122 T^{4} + \cdots + 14745600 $ T^6 + 1122T^4 + 248769T^2 + 14745600
$19$	$ T^{6} - 78 T^{5} + \cdots + 29799200 $ T^6 - 78T^5 + 3042T^4 - 68096T^3 + 944784T^2 - 7503840*T + 29799200
$23$	$ T^{6} + 1500 T^{4} + \cdots + 20358144 $ T^6 + 1500T^4 + 561060T^2 + 20358144
$29$	$ (T^{3} - 30 T^{2} + \cdots + 20316)^{2} $ (T^3 - 30T^2 - 750T + 20316)^2
$31$	$ T^{6} - 6 T^{5} + \cdots + 56180000 $ T^6 - 6T^5 + 18T^4 - 16T^3 + 3111696T^2 - 18698400*T + 56180000
$37$	$ T^{6} - 54 T^{5} + \cdots + 873842 $ T^6 - 54T^5 + 1458T^4 - 20872T^3 + 168921T^2 - 543342*T + 873842
$41$	$ T^{6} + \cdots + 1839089952 $ T^6 + 66T^5 + 2178T^4 - 14592T^3 + 1299600T^2 + 69138720*T + 1839089952
$43$	$ T^{6} + \cdots + 4493289024 $ T^6 + 5238T^4 + 8647857T^2 + 4493289024
$47$	$ T^{6} + \cdots + 7858818450 $ T^6 - 114T^5 + 6498T^4 - 19692T^3 + 859329T^2 - 116217990*T + 7858818450
$53$	$ (T^{3} - 114 T^{2} + \cdots - 38076)^{2} $ (T^3 - 114T^2 + 3750T - 38076)^2
$59$	$ T^{6} + \cdots + 1542123648 $ T^6 + 186T^5 + 17298T^4 + 864048T^3 + 24443136T^2 + 274569984*T + 1542123648
$61$	$ (T^{3} - 102 T^{2} + \cdots - 2712)^{2} $ (T^3 - 102T^2 + 1140T - 2712)^2
$67$	$ T^{6} + 78 T^{5} + \cdots + 30952712 $ T^6 + 78T^5 + 3042T^4 - 131888T^3 + 2528100T^2 - 12510120*T + 30952712
$71$	$ T^{6} + \cdots + 2561418738 $ T^6 - 210T^5 + 22050T^4 - 1210476T^3 + 37271025T^2 - 436959270*T + 2561418738
$73$	$ T^{6} + \cdots + 2377740800 $ T^6 - 222T^5 + 24642T^4 - 1059968T^3 + 19927296T^2 + 307837440*T + 2377740800
$79$	$ (T^{3} + 30 T^{2} + \cdots - 37452)^{2} $ (T^3 + 30T^2 - 5034T - 37452)^2
$83$	$ T^{6} + \cdots + 73972426248 $ T^6 + 78T^5 + 3042T^4 - 118464T^3 + 41602500T^2 + 2480902200*T + 73972426248
$89$	$ T^{6} + \cdots + 2347495200 $ T^6 + 234T^5 + 27378T^4 + 1585392T^3 + 49956624T^2 + 484299360*T + 2347495200
$97$	$ T^{6} + \cdots + 52696863368 $ T^6 - 354T^5 + 62658T^4 - 4753840T^3 + 205807716T^2 - 4657342824*T + 52696863368
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	676.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{3} + (\beta_{5} - \beta_{2} + 1) q^{5} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 5) q^{9} + ( - 2 \beta_{3} + 3 \beta_{2} + \cdots + 3) q^{11}+ \cdots + ( - 6 \beta_{4} - 21 \beta_{3} + \cdots + 3) q^{99}+O(q^{100}) \) q + b3 * q^3 + (b5 - b2 + 1) * q^5 + (-b4 + b3 - b2 - b1 - 1) * q^7 + (b5 - b4 + b3 + 5) * q^9 + (-2b3 + 3b2 + 2b1 + 3) q^11 + (-b5 + 4b3 + 4b2 + 4b1 - 4) q^15 + (b5 + b4 - 2b2 + 5b1) * q^17 + (b3 - 13b2 + b1 + 13) q^19 + (-b4 + 3b3 + 10b2 - 3b1 + 10) q^21 + (-b5 - b4 + 20b2) q^23 + (-b5 - b4 - 27b2 - 7b1) * q^25 + (3b3 + 6) q^27 + (-b5 + b4 + 6b3 + 10) q^29 + (-4b5 - 5b3 - b2 - 5b1 + 1) q^31 + (4b4 + b3 - 28b2 - b1 - 28) * q^33 + (-4b5 + 4b4 + b3 + 40) * q^35 + (-b4 + 9b2 + 9) q^37 + (4b5 - 3b3 + 11b2 - 3b1 - 11) * q^41 + (-4b5 - 4b4 + 14b2 - b1) q^43 + (-3b3 - 51b2 - 3b1 + 51) q^45 + (5b4 - 7b3 + 19b2 + 7b1 + 19) * q^47 + (5b5 + 5b4 + 15b2 - 3b1) * q^49 + (4b5 + 4b4 - 62b2 + 13b1) * q^51 + (b5 - b4 - 4b3 + 38) q^53 + (5b5 - 5b4 - 16b3 + 22) q^55 + (2b5 + 14b3 - 14b2 + 14b1 + 14) * q^57 + (-2b4 - 3b3 - 31b2 + 3b1 - 31) * q^59 + (2b5 - 2b4 - 8b3 + 34) q^61 + (4b4 + 7b3 + 47b2 - 7b1 + 47) * q^63 + (-2b5 - 8b3 + 13b2 - 8b1 - 13) * q^67 + (b5 + b4 - 8b2 - 26b1) * q^69 + (-b5 + 5b3 - 35b2 + 5b1 + 35) q^71 + (-6b4 + 7b3 + 37b2 - 7b1 + 37) * q^73 + (-6b5 - 6b4 + 90b2 + 14b1) * q^75 + (-b5 - b4 - 46b2 + 6b1) * q^77 + (-5b5 + 5b4 - 12b3 - 10) q^79 + (-6b5 + 6b4 - 3) * q^81 + (8b5 - 6b3 + 13b2 - 6b1 - 13) * q^83 + (-7b4 - 13b3 - 32b2 + 13b1 - 32) * q^85 + (7b5 - 7b4 + 10b3 + 92) q^87 + (7b3 - 39b2 - 7b1 - 39) q^89 + (-6b5 - 16b3 + 54b2 - 16b1 - 54) * q^93 + (12b5 + 12b4 - 18b2 + 8b1) * q^95 + (6b5 - 8b3 - 59b2 - 8b1 + 59) * q^97 + (-6b4 - 21b3 + 3b2 + 21b1 + 3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{5} - 6 q^{7} + 30 q^{9} + 18 q^{11} - 24 q^{15} + 78 q^{19} + 60 q^{21} + 36 q^{27} + 60 q^{29} + 6 q^{31} - 168 q^{33} + 240 q^{35} + 54 q^{37} - 66 q^{41} + 306 q^{45} + 114 q^{47} + 228 q^{53}+ \cdots + 18 q^{99}+O(q^{100}) \) 6 * q + 6 * q^5 - 6 * q^7 + 30 * q^9 + 18 * q^11 - 24 * q^15 + 78 * q^19 + 60 * q^21 + 36 * q^27 + 60 * q^29 + 6 * q^31 - 168 * q^33 + 240 * q^35 + 54 * q^37 - 66 * q^41 + 306 * q^45 + 114 * q^47 + 228 * q^53 + 132 * q^55 + 84 * q^57 - 186 * q^59 + 204 * q^61 + 282 * q^63 - 78 * q^67 + 210 * q^71 + 222 * q^73 - 60 * q^79 - 18 * q^81 - 78 * q^83 - 192 * q^85 + 552 * q^87 - 234 * q^89 - 324 * q^93 + 354 * q^97 + 18 * q^99

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