LMFDB - Newform orbit 1188.3.e.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1188,3,Mod(485,1188)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1188, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1188.485"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1188 = 2^{2} \cdot 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1188.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8] 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:	$32.3706554060$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 46x^{6} + 637x^{4} + 2880x^{2} + 2916 $ x^8 + 46x^6 + 637x^4 + 2880*x^2 + 2916
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2}\cdot 11 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{5} q^{5} + (\beta_{4} + 1) q^{7} - \beta_1 q^{11} + ( - \beta_{3} - 2) q^{13} + (\beta_{2} - 2 \beta_1) q^{17} + ( - 2 \beta_{7} - \beta_{4} + 3) q^{19} + ( - \beta_{6} - \beta_{5}) q^{23} + ( - \beta_{7} - \beta_{4} - 1) q^{25}+ \cdots + (6 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} - 17) q^{97}+O(q^{100}) $ q + b5 * q^5 + (b4 + 1) * q^7 - b1 * q^11 + (-b3 - 2) * q^13 + (b2 - 2b1) q^17 + (-2b7 - b4 + 3) q^19 + (-b6 - b5) * q^23 + (-b7 - b4 - 1) * q^25 + (b6 - 6b1) q^29 + (-2b7 + 2b4 - 2b3 - 2) q^31 + (b6 + 2b5 - 6b1) * q^35 + (5b7 + b4 - 5) q^37 + (b6 - 2b5 - b2 - 4b1) * q^41 + (4b7 + 4b4 - 8) * q^43 + (-b6 - 4b5 - 2b2 - 8b1) q^47 + (-5b7 - b4 + 2b3 + 12) * q^49 + (-2b6 - b5 + 2b2 - 4b1) q^53 + (b7 - b4 + b3 - 1) * q^55 + (b6 + 2b5 - 2b2 - 8b1) q^59 + (-2b7 - 8b4 + 3b3 + 4) q^61 + (-2b6 - 2b5 - b2 - 10b1) q^65 + (6b4 - 2b3 + 3) * q^67 + (8b5 - 2b2 - 8b1) q^71 + (2b7 - b4 + 6b3 + 21) * q^73 + (2b5 + b2 - b1) q^77 + (-2b7 - 12b4 - b3 - 2) * q^79 + (b6 - 8b5 + b2 - 32b1) * q^83 + (12b7 + 3b4 - b3 - 13) * q^85 + (3b6 + 2b5 - 12b1) q^89 + (b7 - 9b4 - 29) q^91 + (-b6 - 6b5 + 4b2 - 14b1) q^95 + (6b7 + 2b4 - 2b3 - 17) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7} - 20 q^{13} + 16 q^{19} - 12 q^{25} - 32 q^{31} - 20 q^{37} - 48 q^{43} + 84 q^{49} + 36 q^{61} + 16 q^{67} + 200 q^{73} - 28 q^{79} - 60 q^{85} - 228 q^{91} - 120 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 - 20 * q^13 + 16 * q^19 - 12 * q^25 - 32 * q^31 - 20 * q^37 - 48 * q^43 + 84 * q^49 + 36 * q^61 + 16 * q^67 + 200 * q^73 - 28 * q^79 - 60 * q^85 - 228 * q^91 - 120 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 46x^{6} + 637x^{4} + 2880x^{2} + 2916 $ x^8 + 46*x^6 + 637*x^4 + 2880*x^2 + 2916 :

$\beta_{1}$	$=$	$ ( \nu^{7} + 46\nu^{5} + 583\nu^{3} + 1638\nu ) / 324 $ (v^7 + 46v^5 + 583v^3 + 1638*v) / 324
$\beta_{2}$	$=$	$ ( 2\nu^{7} + 65\nu^{5} + 275\nu^{3} - 2124\nu ) / 162 $ (2v^7 + 65v^5 + 275v^3 - 2124v) / 162
$\beta_{3}$	$=$	$ ( \nu^{6} + 31\nu^{4} + 256\nu^{2} + 972 ) / 72 $ (v^6 + 31v^4 + 256v^2 + 972) / 72
$\beta_{4}$	$=$	$ ( \nu^{6} + 37\nu^{4} + 322\nu^{2} + 432 ) / 36 $ (v^6 + 37v^4 + 322v^2 + 432) / 36
$\beta_{5}$	$=$	$ ( 5\nu^{7} + 203\nu^{5} + 2240\nu^{3} + 6300\nu ) / 648 $ (5v^7 + 203v^5 + 2240v^3 + 6300v) / 648
$\beta_{6}$	$=$	$ ( \nu^{7} + 46\nu^{5} + 637\nu^{3} + 3852\nu ) / 162 $ (v^7 + 46v^5 + 637v^3 + 3852*v) / 162
$\beta_{7}$	$=$	$ ( \nu^{4} + 29\nu^{2} + 126 ) / 6 $ (v^4 + 29*v^2 + 126) / 6

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

$\nu$	$=$	$ ( 4\beta_{6} - 4\beta_{5} + \beta_{2} - 2\beta_1 ) / 33 $ (4b6 - 4b5 + b2 - 2*b1) / 33
$\nu^{2}$	$=$	$ ( \beta_{7} - \beta_{4} + 2\beta_{3} - 36 ) / 3 $ (b7 - b4 + 2*b3 - 36) / 3
$\nu^{3}$	$=$	$ ( -65\beta_{6} + 164\beta_{5} - 41\beta_{2} - 116\beta_1 ) / 33 $ (-65b6 + 164b5 - 41b2 - 116b1) / 33
$\nu^{4}$	$=$	$ ( -11\beta_{7} + 29\beta_{4} - 58\beta_{3} + 666 ) / 3 $ (-11b7 + 29b4 - 58*b3 + 666) / 3
$\nu^{5}$	$=$	$ ( 1345\beta_{6} - 4612\beta_{5} + 955\beta_{2} + 5020\beta_1 ) / 33 $ (1345b6 - 4612b5 + 955b2 + 5020b1) / 33
$\nu^{6}$	$=$	$ ( 85\beta_{7} - 643\beta_{4} + 1502\beta_{3} - 14346 ) / 3 $ (85b7 - 643b4 + 1502*b3 - 14346) / 3
$\nu^{7}$	$=$	$ ( -30527\beta_{6} + 123092\beta_{5} - 21665\beta_{2} - 149324\beta_1 ) / 33 $ (-30527b6 + 123092b5 - 21665b2 - 149324b1) / 33

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1188\mathbb{Z}\right)^\times$.

$n$	$353$	$541$	$595$
$\chi(n)$	$-1$	$1$	$1$

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} $

485.1

	−	1.18638i
		4.98184i
	−	3.69664i
		2.47158i
	−	2.47158i
		3.69664i
	−	4.98184i
		1.18638i

−

6.47276i

2.36935

485.2

−

5.30468i

−0.564603

485.3

−

4.78414i

11.8134

485.4

−

3.61605i

−9.61820

485.5

3.61605i

−9.61820

485.6

4.78414i

11.8134

485.7

5.30468i

−0.564603

485.8

6.47276i

2.36935

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1188.3.e.c	✓	8
3.b	odd	2	1	inner	1188.3.e.c	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1188.3.e.c	✓	8	1.a	even	1	1	trivial
1188.3.e.c	✓	8	3.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} + 106T_{5}^{6} + 3997T_{5}^{4} + 63360T_{5}^{2} + 352836 $ T5^8 + 106*T5^6 + 3997*T5^4 + 63360*T5^2 + 352836 acting on $S_{3}^{\mathrm{new}}(1188, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ T^{8} + 106 T^{6} + \cdots + 352836 $ T^8 + 106T^6 + 3997T^4 + 63360*T^2 + 352836
$7$	$ (T^{4} - 4 T^{3} + \cdots + 152)^{2} $ (T^4 - 4T^3 - 111T^2 + 208*T + 152)^2
$11$	$ (T^{2} + 11)^{4} $ (T^2 + 11)^4
$13$	$ (T^{4} + 10 T^{3} + \cdots - 17817)^{2} $ (T^4 + 10T^3 - 304T^2 - 4846*T - 17817)^2
$17$	$ T^{8} + \cdots + 24158484900 $ T^8 + 2206T^6 + 1568653T^4 + 404388864*T^2 + 24158484900
$19$	$ (T^{4} - 8 T^{3} + \cdots + 28488)^{2} $ (T^4 - 8T^3 - 715T^2 + 8072*T + 28488)^2
$23$	$ T^{8} + \cdots + 228078835776 $ T^8 + 2978T^6 + 3203865T^4 + 1454254560*T^2 + 228078835776
$29$	$ T^{8} + \cdots + 4115479104 $ T^8 + 4160T^6 + 4648788T^4 + 746472672*T^2 + 4115479104
$31$	$ (T^{4} + 16 T^{3} + \cdots + 1136160)^{2} $ (T^4 + 16T^3 - 2236T^2 - 12592*T + 1136160)^2
$37$	$ (T^{4} + 10 T^{3} + \cdots + 1429356)^{2} $ (T^4 + 10T^3 - 4807T^2 - 65212*T + 1429356)^2
$41$	$ T^{8} + \cdots + 4115479104 $ T^8 + 5414T^6 + 2194281T^4 + 249052320*T^2 + 4115479104
$43$	$ (T^{4} + 24 T^{3} + \cdots + 37888)^{2} $ (T^4 + 24T^3 - 3248T^2 - 66048*T + 37888)^2
$47$	$ T^{8} + \cdots + 228999940969536 $ T^8 + 17360T^6 + 107827380T^4 + 275933119968*T^2 + 228999940969536
$53$	$ T^{8} + \cdots + 6264598514724 $ T^8 + 18730T^6 + 100945069T^4 + 133602164784*T^2 + 6264598514724
$59$	$ T^{8} + \cdots + 2504040397056 $ T^8 + 12744T^6 + 33641892T^4 + 26054436672*T^2 + 2504040397056
$61$	$ (T^{4} - 18 T^{3} + \cdots - 1229609)^{2} $ (T^4 - 18T^3 - 7568T^2 - 211626*T - 1229609)^2
$67$	$ (T^{4} - 8 T^{3} + \cdots + 932385)^{2} $ (T^4 - 8T^3 - 4306T^2 + 94904*T + 932385)^2
$71$	$ T^{8} + \cdots + 15706700596224 $ T^8 + 16760T^6 + 78597072T^4 + 116636433408*T^2 + 15706700596224
$73$	$ (T^{4} - 100 T^{3} + \cdots - 24227568)^{2} $ (T^4 - 100T^3 - 7967T^2 + 1014984*T - 24227568)^2
$79$	$ (T^{4} + 14 T^{3} + \cdots + 25297691)^{2} $ (T^4 + 14T^3 - 16332T^2 - 344138*T + 25297691)^2
$83$	$ T^{8} + \cdots + 11\!\cdots\!00 $ T^8 + 55030T^6 + 992169337T^4 + 5950800462816*T^2 + 1190490132110400
$89$	$ T^{8} + \cdots + 669865914805824 $ T^8 + 30928T^6 + 280911316T^4 + 792944035488*T^2 + 669865914805824
$97$	$ (T^{4} + 60 T^{3} + \cdots + 15077077)^{2} $ (T^4 + 60T^3 - 7478T^2 - 208572*T + 15077077)^2
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Level:	\( N \)	\(=\)	\( 1188 = 2^{2} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1188.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{5} + (\beta_{4} + 1) q^{7} - \beta_1 q^{11} + ( - \beta_{3} - 2) q^{13} + (\beta_{2} - 2 \beta_1) q^{17} + ( - 2 \beta_{7} - \beta_{4} + 3) q^{19} + ( - \beta_{6} - \beta_{5}) q^{23} + ( - \beta_{7} - \beta_{4} - 1) q^{25}+ \cdots + (6 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} - 17) q^{97}+O(q^{100}) \) q + b5 * q^5 + (b4 + 1) * q^7 - b1 * q^11 + (-b3 - 2) * q^13 + (b2 - 2b1) q^17 + (-2b7 - b4 + 3) q^19 + (-b6 - b5) * q^23 + (-b7 - b4 - 1) * q^25 + (b6 - 6b1) q^29 + (-2b7 + 2b4 - 2b3 - 2) q^31 + (b6 + 2b5 - 6b1) * q^35 + (5b7 + b4 - 5) q^37 + (b6 - 2b5 - b2 - 4b1) * q^41 + (4b7 + 4b4 - 8) * q^43 + (-b6 - 4b5 - 2b2 - 8b1) q^47 + (-5b7 - b4 + 2b3 + 12) * q^49 + (-2b6 - b5 + 2b2 - 4b1) q^53 + (b7 - b4 + b3 - 1) * q^55 + (b6 + 2b5 - 2b2 - 8b1) q^59 + (-2b7 - 8b4 + 3b3 + 4) q^61 + (-2b6 - 2b5 - b2 - 10b1) q^65 + (6b4 - 2b3 + 3) * q^67 + (8b5 - 2b2 - 8b1) q^71 + (2b7 - b4 + 6b3 + 21) * q^73 + (2b5 + b2 - b1) q^77 + (-2b7 - 12b4 - b3 - 2) * q^79 + (b6 - 8b5 + b2 - 32b1) * q^83 + (12b7 + 3b4 - b3 - 13) * q^85 + (3b6 + 2b5 - 12b1) q^89 + (b7 - 9b4 - 29) q^91 + (-b6 - 6b5 + 4b2 - 14b1) q^95 + (6b7 + 2b4 - 2b3 - 17) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7} - 20 q^{13} + 16 q^{19} - 12 q^{25} - 32 q^{31} - 20 q^{37} - 48 q^{43} + 84 q^{49} + 36 q^{61} + 16 q^{67} + 200 q^{73} - 28 q^{79} - 60 q^{85} - 228 q^{91} - 120 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 - 20 * q^13 + 16 * q^19 - 12 * q^25 - 32 * q^31 - 20 * q^37 - 48 * q^43 + 84 * q^49 + 36 * q^61 + 16 * q^67 + 200 * q^73 - 28 * q^79 - 60 * q^85 - 228 * q^91 - 120 * q^97

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