LMFDB - Newform orbit 1184.2.a.p

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1184,2,Mod(1,1184)] 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("1184.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1184, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1184 = 2^{5} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1184.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.45428759932$
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} - 20x^{6} + 116x^{4} - 221x^{2} + 128 $ x^8 - 20x^6 + 116x^4 - 221*x^2 + 128
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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_1 q^{3} + \beta_{6} q^{5} + \beta_{4} q^{7} + (\beta_{2} + 2) q^{9} + ( - \beta_{7} + \beta_1) q^{11} + (\beta_{6} + \beta_{5} - \beta_{2}) q^{13} + \beta_{7} q^{15} + 2 q^{17} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{19}+ \cdots + (\beta_{7} - 2 \beta_{4} + 2 \beta_1) q^{99}+O(q^{100}) $ q + b1 * q^3 + b6 * q^5 + b4 * q^7 + (b2 + 2) * q^9 + (-b7 + b1) * q^11 + (b6 + b5 - b2) * q^13 + b7 * q^15 + 2 * q^17 + (-b4 + b3 + b1) * q^19 + (-b6 - 2b5 + b2 + 1) q^21 - b3 * q^23 + (-b6 - b5 - b2 + 3) * q^25 + (b7 + b4 - 2b3 + 2b1) * q^27 + (b6 + b5 - b2) * q^29 + (-b7 + b3) * q^31 + (-b6 + b5 + 3) * q^33 + (2b7 - 2b4) * q^35 + q^37 + (-b7 - 3b4 + 2b3 - b1) * q^39 + (-b5 + 5) * q^41 + (b4 + b3 - b1) * q^43 + (-2b6 - b5 + b2 + 2) q^45 + (-2b7 + b4) q^47 + (-b6 + 2b5 - b2 + 4) q^49 + 2b1 q^51 + (b6 + 2b5 + b2 - 1) q^53 + (2b7 - b4 - 2b3 - 3b1) q^55 + (2b6 + 2b5 - 2b2 + 4) q^57 + (-2b7 - b4 + b3 - b1) q^59 + (-b6 + 4) * q^61 + (2b7 + 2b4 - 2b3) q^63 + (-b5 - b2 + 2) * q^65 + (-b7 + 3b4 - 2b3 + b1) * q^67 + (-b6 + 2b2) q^69 + (-2b7 + b4 - 2b1) * q^71 + (b2 + 7) * q^73 + (-b7 + b4 + 2b3 - 2b1) * q^75 + (-5b6 - 4b5 + b2 - 1) * q^77 - b3 * q^79 + (-2b6 - 3b5 + 5b2 + 7) q^81 + (b4 - 2b1) q^83 + 2b6 q^85 + (-b7 - 3b4 + 2b3 - b1) * q^87 + (2b6 - 2b5 + 4) * q^89 + (2b7 + 2b4 - 6b1) q^91 + (b5 - 3b2 - 2) q^93 + (2b7 + 2b3) * q^95 + 2 * q^97 + (b7 - 2b4 + 2b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} + 16 q^{9} + 2 q^{13} + 16 q^{17} + 2 q^{21} + 22 q^{25} + 2 q^{29} + 30 q^{33} + 8 q^{37} + 36 q^{41} + 16 q^{45} + 42 q^{49} - 2 q^{53} + 36 q^{57} + 34 q^{61} + 12 q^{65} + 2 q^{69} + 56 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) $ 8 * q - 2 * q^5 + 16 * q^9 + 2 * q^13 + 16 * q^17 + 2 * q^21 + 22 * q^25 + 2 * q^29 + 30 * q^33 + 8 * q^37 + 36 * q^41 + 16 * q^45 + 42 * q^49 - 2 * q^53 + 36 * q^57 + 34 * q^61 + 12 * q^65 + 2 * q^69 + 56 * q^73 - 14 * q^77 + 48 * q^81 - 4 * q^85 + 20 * q^89 - 12 * q^93 + 16 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 20x^{6} + 116x^{4} - 221x^{2} + 128 $ x^8 - 20*x^6 + 116*x^4 - 221*x^2 + 128 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 5 $ v^2 - 5
$\beta_{3}$	$=$	$ ( 5\nu^{7} - 88\nu^{5} + 360\nu^{3} - 153\nu ) / 44 $ (5v^7 - 88v^5 + 360v^3 - 153v) / 44
$\beta_{4}$	$=$	$ ( -\nu^{7} + 22\nu^{5} - 138\nu^{3} + 211\nu ) / 22 $ (-v^7 + 22v^5 - 138v^3 + 211*v) / 22
$\beta_{5}$	$=$	$ ( -2\nu^{6} + 33\nu^{4} - 122\nu^{2} + 81 ) / 11 $ (-2v^6 + 33v^4 - 122*v^2 + 81) / 11
$\beta_{6}$	$=$	$ ( 3\nu^{6} - 55\nu^{4} + 260\nu^{2} - 270 ) / 11 $ (3v^6 - 55v^4 + 260*v^2 - 270) / 11
$\beta_{7}$	$=$	$ ( 3\nu^{7} - 55\nu^{5} + 260\nu^{3} - 270\nu ) / 11 $ (3v^7 - 55v^5 + 260v^3 - 270v) / 11

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 5 $ b2 + 5
$\nu^{3}$	$=$	$ \beta_{7} + \beta_{4} - 2\beta_{3} + 8\beta_1 $ b7 + b4 - 2b3 + 8b1
$\nu^{4}$	$=$	$ -2\beta_{6} - 3\beta_{5} + 14\beta_{2} + 43 $ -2b6 - 3b5 + 14*b2 + 43
$\nu^{5}$	$=$	$ 15\beta_{7} + 20\beta_{4} - 28\beta_{3} + 79\beta_1 $ 15b7 + 20b4 - 28b3 + 79b1
$\nu^{6}$	$=$	$ -33\beta_{6} - 55\beta_{5} + 170\beta_{2} + 445 $ -33b6 - 55b5 + 170*b2 + 445
$\nu^{7}$	$=$	$ 192\beta_{7} + 280\beta_{4} - 340\beta_{3} + 845\beta_1 $ 192b7 + 280b4 - 340b3 + 845b1

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

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} $

1.1

−3.39115
−2.36494
−1.34313
−1.05032
1.05032
1.34313
2.36494
3.39115

−3.39115

0.805051

−1.94652

8.49990

1.2

−2.36494

−1.03927

5.11652

2.59295

1.3

−1.34313

3.42359

−1.69567

−1.19601

1.4

−1.05032

−4.18937

−4.01957

−1.89684

1.5

1.05032

−4.18937

4.01957

−1.89684

1.6

1.34313

3.42359

1.69567

−1.19601

1.7

2.36494

−1.03927

−5.11652

2.59295

1.8

3.39115

0.805051

1.94652

8.49990

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$37$	$ -1 $

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1184.2.a.p	✓	8
4.b	odd	2	1	inner	1184.2.a.p	✓	8
8.b	even	2	1		2368.2.a.bj		8
8.d	odd	2	1		2368.2.a.bj		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1184.2.a.p	✓	8	1.a	even	1	1	trivial
1184.2.a.p	✓	8	4.b	odd	2	1	inner
2368.2.a.bj		8	8.b	even	2	1
2368.2.a.bj		8	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1184))$:

$ T_{3}^{8} - 20T_{3}^{6} + 116T_{3}^{4} - 221T_{3}^{2} + 128 $

T3^8 - 20*T3^6 + 116*T3^4 - 221*T3^2 + 128

$ T_{5}^{4} + T_{5}^{3} - 15T_{5}^{2} - 4T_{5} + 12 $

T5^4 + T5^3 - 15*T5^2 - 4*T5 + 12

$ T_{7}^{8} - 49T_{7}^{6} + 716T_{7}^{4} - 3280T_{7}^{2} + 4608 $

T7^8 - 49*T7^6 + 716*T7^4 - 3280*T7^2 + 4608

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} - 20 T^{6} + \cdots + 128 $ T^8 - 20T^6 + 116T^4 - 221*T^2 + 128
$5$	$ (T^{4} + T^{3} - 15 T^{2} + \cdots + 12)^{2} $ (T^4 + T^3 - 15T^2 - 4T + 12)^2
$7$	$ T^{8} - 49 T^{6} + \cdots + 4608 $ T^8 - 49T^6 + 716T^4 - 3280*T^2 + 4608
$11$	$ T^{8} - 64 T^{6} + \cdots + 3200 $ T^8 - 64T^6 + 1280T^4 - 7869*T^2 + 3200
$13$	$ (T^{4} - T^{3} - 37 T^{2} + \cdots + 152)^{2} $ (T^4 - T^3 - 37T^2 + 30T + 152)^2
$17$	$ (T - 2)^{8} $ (T - 2)^8
$19$	$ T^{8} - 96 T^{6} + \cdots + 2048 $ T^8 - 96T^6 + 2512T^4 - 10112*T^2 + 2048
$23$	$ T^{8} - 79 T^{6} + \cdots + 7200 $ T^8 - 79T^6 + 1541T^4 - 9436*T^2 + 7200
$29$	$ (T^{4} - T^{3} - 37 T^{2} + \cdots + 152)^{2} $ (T^4 - T^3 - 37T^2 + 30T + 152)^2
$31$	$ T^{8} - 111 T^{6} + \cdots + 41472 $ T^8 - 111T^6 + 3449T^4 - 25384*T^2 + 41472
$37$	$ (T - 1)^{8} $ (T - 1)^8
$41$	$ (T^{4} - 18 T^{3} + \cdots - 678)^{2} $ (T^4 - 18T^3 + 86T^2 + 31*T - 678)^2
$43$	$ T^{8} - 196 T^{6} + \cdots + 4333568 $ T^8 - 196T^6 + 13808T^4 - 411264*T^2 + 4333568
$47$	$ T^{8} - 233 T^{6} + \cdots + 4608 $ T^8 - 233T^6 + 11980T^4 - 114640*T^2 + 4608
$53$	$ (T^{4} + T^{3} - 218 T^{2} + \cdots + 8856)^{2} $ (T^4 + T^3 - 218T^2 - 292T + 8856)^2
$59$	$ T^{8} - 328 T^{6} + \cdots + 11674112 $ T^8 - 328T^6 + 29696T^4 - 1022784*T^2 + 11674112
$61$	$ (T^{4} - 17 T^{3} + \cdots + 76)^{2} $ (T^4 - 17T^3 + 93T^2 - 180*T + 76)^2
$67$	$ T^{8} - 539 T^{6} + \cdots + 259555328 $ T^8 - 539T^6 + 105113T^4 - 8735552*T^2 + 259555328
$71$	$ T^{8} - 349 T^{6} + \cdots + 30732800 $ T^8 - 349T^6 + 43000T^4 - 2117584*T^2 + 30732800
$73$	$ (T^{4} - 28 T^{3} + \cdots + 1210)^{2} $ (T^4 - 28T^3 + 260T^2 - 957*T + 1210)^2
$79$	$ T^{8} - 79 T^{6} + \cdots + 7200 $ T^8 - 79T^6 + 1541T^4 - 9436*T^2 + 7200
$83$	$ T^{8} - 125 T^{6} + \cdots + 8192 $ T^8 - 125T^6 + 2832T^4 - 11008*T^2 + 8192
$89$	$ (T^{4} - 10 T^{3} + \cdots + 1920)^{2} $ (T^4 - 10T^3 - 224T^2 + 1024*T + 1920)^2
$97$	$ (T - 2)^{8} $ (T - 2)^8
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Level:	\( N \)	\(=\)	\( 1184 = 2^{5} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1184.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + \beta_{6} q^{5} + \beta_{4} q^{7} + (\beta_{2} + 2) q^{9} + ( - \beta_{7} + \beta_1) q^{11} + (\beta_{6} + \beta_{5} - \beta_{2}) q^{13} + \beta_{7} q^{15} + 2 q^{17} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{19}+ \cdots + (\beta_{7} - 2 \beta_{4} + 2 \beta_1) q^{99}+O(q^{100}) \) q + b1 * q^3 + b6 * q^5 + b4 * q^7 + (b2 + 2) * q^9 + (-b7 + b1) * q^11 + (b6 + b5 - b2) * q^13 + b7 * q^15 + 2 * q^17 + (-b4 + b3 + b1) * q^19 + (-b6 - 2b5 + b2 + 1) q^21 - b3 * q^23 + (-b6 - b5 - b2 + 3) * q^25 + (b7 + b4 - 2b3 + 2b1) * q^27 + (b6 + b5 - b2) * q^29 + (-b7 + b3) * q^31 + (-b6 + b5 + 3) * q^33 + (2b7 - 2b4) * q^35 + q^37 + (-b7 - 3b4 + 2b3 - b1) * q^39 + (-b5 + 5) * q^41 + (b4 + b3 - b1) * q^43 + (-2b6 - b5 + b2 + 2) q^45 + (-2b7 + b4) q^47 + (-b6 + 2b5 - b2 + 4) q^49 + 2b1 q^51 + (b6 + 2b5 + b2 - 1) q^53 + (2b7 - b4 - 2b3 - 3b1) q^55 + (2b6 + 2b5 - 2b2 + 4) q^57 + (-2b7 - b4 + b3 - b1) q^59 + (-b6 + 4) * q^61 + (2b7 + 2b4 - 2b3) q^63 + (-b5 - b2 + 2) * q^65 + (-b7 + 3b4 - 2b3 + b1) * q^67 + (-b6 + 2b2) q^69 + (-2b7 + b4 - 2b1) * q^71 + (b2 + 7) * q^73 + (-b7 + b4 + 2b3 - 2b1) * q^75 + (-5b6 - 4b5 + b2 - 1) * q^77 - b3 * q^79 + (-2b6 - 3b5 + 5b2 + 7) q^81 + (b4 - 2b1) q^83 + 2b6 q^85 + (-b7 - 3b4 + 2b3 - b1) * q^87 + (2b6 - 2b5 + 4) * q^89 + (2b7 + 2b4 - 6b1) q^91 + (b5 - 3b2 - 2) q^93 + (2b7 + 2b3) * q^95 + 2 * q^97 + (b7 - 2b4 + 2b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} + 16 q^{9} + 2 q^{13} + 16 q^{17} + 2 q^{21} + 22 q^{25} + 2 q^{29} + 30 q^{33} + 8 q^{37} + 36 q^{41} + 16 q^{45} + 42 q^{49} - 2 q^{53} + 36 q^{57} + 34 q^{61} + 12 q^{65} + 2 q^{69} + 56 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) \) 8 * q - 2 * q^5 + 16 * q^9 + 2 * q^13 + 16 * q^17 + 2 * q^21 + 22 * q^25 + 2 * q^29 + 30 * q^33 + 8 * q^37 + 36 * q^41 + 16 * q^45 + 42 * q^49 - 2 * q^53 + 36 * q^57 + 34 * q^61 + 12 * q^65 + 2 * q^69 + 56 * q^73 - 14 * q^77 + 48 * q^81 - 4 * q^85 + 20 * q^89 - 12 * q^93 + 16 * q^97

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