LMFDB - Newform orbit 84.6.i.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [84,6,Mod(25,84)] 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("84.25"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(84, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	84.i (of order $3$, 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:	$13.4722408643$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 - 9 \zeta_{6} q^{3} + ( - 69 \zeta_{6} + 69) q^{5} + (49 \zeta_{6} + 98) q^{7} + (81 \zeta_{6} - 81) q^{9} - 123 \zeta_{6} q^{11} - 4 q^{13} - 621 q^{15} - 1776 \zeta_{6} q^{17} + ( - 1396 \zeta_{6} + 1396) q^{19} + \cdots + 9963 q^{99} +O(q^{100}) $ q - 9z q^3 + (-69z + 69) q^5 + (49z + 98) q^7 + (81z - 81) q^9 - 123z q^11 - 4 * q^13 - 621 * q^15 - 1776z q^17 + (-1396z + 1396) q^19 + (-1323z + 441) q^21 + (-1536z + 1536) q^23 - 1636z q^25 + 729 * q^27 - 3615 * q^29 - 7295z q^31 + (1107z - 1107) q^33 + (-6762z + 10143) q^35 + (7640z - 7640) q^37 + 36z q^39 + 10032 * q^41 - 9754 * q^43 + 5589z q^45 + (-17622z + 17622) q^47 + (12005z + 7203) q^49 + (15984z - 15984) q^51 - 4197z q^53 - 8487 * q^55 - 12564 * q^57 + 20133z q^59 + (34646z - 34646) q^61 + (7938z - 11907) q^63 + (276z - 276) q^65 + 41386z q^67 - 13824 * q^69 + 30762 * q^71 + 80254z q^73 + (14724z - 14724) q^75 + (-18081z + 6027) q^77 + (-98593z + 98593) q^79 - 6561z q^81 + 73407 * q^83 - 122544 * q^85 + 32535z q^87 + (-106554z + 106554) q^89 + (-196z - 392) q^91 + (65655z - 65655) q^93 - 96324z q^95 - 161185 * q^97 + 9963 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{3} + 69 q^{5} + 245 q^{7} - 81 q^{9} - 123 q^{11} - 8 q^{13} - 1242 q^{15} - 1776 q^{17} + 1396 q^{19} - 441 q^{21} + 1536 q^{23} - 1636 q^{25} + 1458 q^{27} - 7230 q^{29} - 7295 q^{31} - 1107 q^{33}+ \cdots + 19926 q^{99}+O(q^{100}) $ 2 * q - 9 * q^3 + 69 * q^5 + 245 * q^7 - 81 * q^9 - 123 * q^11 - 8 * q^13 - 1242 * q^15 - 1776 * q^17 + 1396 * q^19 - 441 * q^21 + 1536 * q^23 - 1636 * q^25 + 1458 * q^27 - 7230 * q^29 - 7295 * q^31 - 1107 * q^33 + 13524 * q^35 - 7640 * q^37 + 36 * q^39 + 20064 * q^41 - 19508 * q^43 + 5589 * q^45 + 17622 * q^47 + 26411 * q^49 - 15984 * q^51 - 4197 * q^53 - 16974 * q^55 - 25128 * q^57 + 20133 * q^59 - 34646 * q^61 - 15876 * q^63 - 276 * q^65 + 41386 * q^67 - 27648 * q^69 + 61524 * q^71 + 80254 * q^73 - 14724 * q^75 - 6027 * q^77 + 98593 * q^79 - 6561 * q^81 + 146814 * q^83 - 245088 * q^85 + 32535 * q^87 + 106554 * q^89 - 980 * q^91 - 65655 * q^93 - 96324 * q^95 - 322370 * q^97 + 19926 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$1$	$-1 + \zeta_{6}$

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} $

25.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−4.50000

7.79423i

34.5000

59.7558i

122.500

−

42.4352i

−40.5000

−

70.1481i

37.1

−4.50000

−

7.79423i

34.5000

−

59.7558i

122.500

42.4352i

−40.5000

70.1481i

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	84.6.i.a	✓	2
3.b	odd	2	1		252.6.k.a		2
4.b	odd	2	1		336.6.q.d		2
7.b	odd	2	1		588.6.i.d		2
7.c	even	3	1	inner	84.6.i.a	✓	2
7.c	even	3	1		588.6.a.d		1
7.d	odd	6	1		588.6.a.c		1
7.d	odd	6	1		588.6.i.d		2
21.h	odd	6	1		252.6.k.a		2
28.g	odd	6	1		336.6.q.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.6.i.a	✓	2	1.a	even	1	1	trivial
84.6.i.a	✓	2	7.c	even	3	1	inner
252.6.k.a		2	3.b	odd	2	1
252.6.k.a		2	21.h	odd	6	1
336.6.q.d		2	4.b	odd	2	1
336.6.q.d		2	28.g	odd	6	1
588.6.a.c		1	7.d	odd	6	1
588.6.a.d		1	7.c	even	3	1
588.6.i.d		2	7.b	odd	2	1
588.6.i.d		2	7.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{2} - 69T_{5} + 4761 $ T5^2 - 69*T5 + 4761 acting on $S_{6}^{\mathrm{new}}(84, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 9T + 81 $ T^2 + 9*T + 81
$5$	$ T^{2} - 69T + 4761 $ T^2 - 69*T + 4761
$7$	$ T^{2} - 245T + 16807 $ T^2 - 245*T + 16807
$11$	$ T^{2} + 123T + 15129 $ T^2 + 123*T + 15129
$13$	$ (T + 4)^{2} $ (T + 4)^2
$17$	$ T^{2} + 1776 T + 3154176 $ T^2 + 1776*T + 3154176
$19$	$ T^{2} - 1396 T + 1948816 $ T^2 - 1396*T + 1948816
$23$	$ T^{2} - 1536 T + 2359296 $ T^2 - 1536*T + 2359296
$29$	$ (T + 3615)^{2} $ (T + 3615)^2
$31$	$ T^{2} + 7295 T + 53217025 $ T^2 + 7295*T + 53217025
$37$	$ T^{2} + 7640 T + 58369600 $ T^2 + 7640*T + 58369600
$41$	$ (T - 10032)^{2} $ (T - 10032)^2
$43$	$ (T + 9754)^{2} $ (T + 9754)^2
$47$	$ T^{2} - 17622 T + 310534884 $ T^2 - 17622*T + 310534884
$53$	$ T^{2} + 4197 T + 17614809 $ T^2 + 4197*T + 17614809
$59$	$ T^{2} - 20133 T + 405337689 $ T^2 - 20133*T + 405337689
$61$	$ T^{2} + \cdots + 1200345316 $ T^2 + 34646*T + 1200345316
$67$	$ T^{2} + \cdots + 1712800996 $ T^2 - 41386*T + 1712800996
$71$	$ (T - 30762)^{2} $ (T - 30762)^2
$73$	$ T^{2} + \cdots + 6440704516 $ T^2 - 80254*T + 6440704516
$79$	$ T^{2} + \cdots + 9720579649 $ T^2 - 98593*T + 9720579649
$83$	$ (T - 73407)^{2} $ (T - 73407)^2
$89$	$ T^{2} + \cdots + 11353754916 $ T^2 - 106554*T + 11353754916
$97$	$ (T + 161185)^{2} $ (T + 161185)^2
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Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	84.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - 9 \zeta_{6} q^{3} + ( - 69 \zeta_{6} + 69) q^{5} + (49 \zeta_{6} + 98) q^{7} + (81 \zeta_{6} - 81) q^{9} - 123 \zeta_{6} q^{11} - 4 q^{13} - 621 q^{15} - 1776 \zeta_{6} q^{17} + ( - 1396 \zeta_{6} + 1396) q^{19} + \cdots + 9963 q^{99} +O(q^{100}) \) q - 9z q^3 + (-69z + 69) q^5 + (49z + 98) q^7 + (81z - 81) q^9 - 123z q^11 - 4 * q^13 - 621 * q^15 - 1776z q^17 + (-1396z + 1396) q^19 + (-1323z + 441) q^21 + (-1536z + 1536) q^23 - 1636z q^25 + 729 * q^27 - 3615 * q^29 - 7295z q^31 + (1107z - 1107) q^33 + (-6762z + 10143) q^35 + (7640z - 7640) q^37 + 36z q^39 + 10032 * q^41 - 9754 * q^43 + 5589z q^45 + (-17622z + 17622) q^47 + (12005z + 7203) q^49 + (15984z - 15984) q^51 - 4197z q^53 - 8487 * q^55 - 12564 * q^57 + 20133z q^59 + (34646z - 34646) q^61 + (7938z - 11907) q^63 + (276z - 276) q^65 + 41386z q^67 - 13824 * q^69 + 30762 * q^71 + 80254z q^73 + (14724z - 14724) q^75 + (-18081z + 6027) q^77 + (-98593z + 98593) q^79 - 6561z q^81 + 73407 * q^83 - 122544 * q^85 + 32535z q^87 + (-106554z + 106554) q^89 + (-196z - 392) q^91 + (65655z - 65655) q^93 - 96324z q^95 - 161185 * q^97 + 9963 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{3} + 69 q^{5} + 245 q^{7} - 81 q^{9} - 123 q^{11} - 8 q^{13} - 1242 q^{15} - 1776 q^{17} + 1396 q^{19} - 441 q^{21} + 1536 q^{23} - 1636 q^{25} + 1458 q^{27} - 7230 q^{29} - 7295 q^{31} - 1107 q^{33}+ \cdots + 19926 q^{99}+O(q^{100}) \) 2 * q - 9 * q^3 + 69 * q^5 + 245 * q^7 - 81 * q^9 - 123 * q^11 - 8 * q^13 - 1242 * q^15 - 1776 * q^17 + 1396 * q^19 - 441 * q^21 + 1536 * q^23 - 1636 * q^25 + 1458 * q^27 - 7230 * q^29 - 7295 * q^31 - 1107 * q^33 + 13524 * q^35 - 7640 * q^37 + 36 * q^39 + 20064 * q^41 - 19508 * q^43 + 5589 * q^45 + 17622 * q^47 + 26411 * q^49 - 15984 * q^51 - 4197 * q^53 - 16974 * q^55 - 25128 * q^57 + 20133 * q^59 - 34646 * q^61 - 15876 * q^63 - 276 * q^65 + 41386 * q^67 - 27648 * q^69 + 61524 * q^71 + 80254 * q^73 - 14724 * q^75 - 6027 * q^77 + 98593 * q^79 - 6561 * q^81 + 146814 * q^83 - 245088 * q^85 + 32535 * q^87 + 106554 * q^89 - 980 * q^91 - 65655 * q^93 - 96324 * q^95 - 322370 * q^97 + 19926 * q^99

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