LMFDB - Newform orbit 8464.2.a.ca

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-4,0,0,0,0,0,8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.5780865024.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 15x^{4} - 8x^{2} + 1 $ x^8 - 8x^6 + 15x^4 - 8*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2116)
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_{5} + \beta_{2}) q^{3} - \beta_{7} q^{5} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{7} + \beta_{6} - \beta_{3}) q^{11} + ( - 2 \beta_{2} - 1) q^{13}+ \cdots + ( - 3 \beta_{7} - 3 \beta_{6} + \cdots + \beta_{3}) q^{99}+O(q^{100}) $ q + (b5 + b2) * q^3 - b7 * q^5 + (b6 + b4 - b3) * q^7 + (-b5 - b2 + b1 + 1) * q^9 + (-b7 + b6 - b3) * q^11 + (-2b2 - 1) q^13 + (b7 + b6 - 4b4 - 3b3) * q^15 + (-b7 - 2b4 - b3) q^17 + (b7 + b6 - 2b4 - 3b3) * q^19 + (b7 + 2b6 - 3b4 - 4b3) q^21 + (-b5 + 1) * q^25 + (2b5 - 2b1 - 6) * q^27 + (-b2 - b1 + 2) * q^29 + (2b1 - 2) q^31 + (b7 + 2b6 - 5b4 - 6b3) q^33 + (3b5 - b2 + 2b1 + 2) * q^35 + (-b7 + 2b6 - 4b4 - 3b3) q^37 + (b5 + b2 - 4) * q^39 + (-2b5 - 2b2 + 5) * q^41 + (-b7 - 2b6 + 3b4 + 4b3) q^43 + (-2b7 - 2b6 + 9b4 + 2b3) * q^45 + (-2b5 + 2b2 - b1 + 2) * q^47 + (3b5 + 5b2 + b1 + 7) * q^49 + (-b7 - b4 - 2b3) q^51 + (-2b4 + 5b3) * q^53 + (2b5 + b1 + 8) q^55 + (-3b7 + 5b4) * q^57 + (-3b5 - b2 - 8) q^59 + (-b7 + 5b4 + 3b3) * q^61 + (-4b7 - 3b6 + 3b4 + b3) q^63 + (b7 + 2b6 + 6b4 + 4b3) q^65 + (b7 + 4b6 + b4 + 2b3) * q^67 + (-4b5 + 2b2 + b1 + 6) * q^71 + (-3b5 - 2b2 - 2b1 - 2) q^73 + (b5 + b2 - b1 - 2) * q^75 + (7b5 + 3b2 + 3b1 + 12) q^77 + (b7 + b4 + 6b3) q^79 + (-9b5 - 5b2 + b1 + 5) * q^81 + (b7 + 4b6 + 3b4 + 2b3) q^83 + (3b2 - b1 + 7) q^85 + (2b2 + b1) q^87 + (3b7 - 2b6 + 5b4 + 3b3) * q^89 + (-4b7 - 3b6 - 3b4 + 3b3) * q^91 + (4b5 - 2b1 - 4) * q^93 + (6b5 + 4b2 + b1 - 2) * q^95 + (-b7 - b4 + 5b3) q^97 + (-3b7 - 3b6 + 8b4 + b3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 8 q^{9} + 8 q^{25} - 40 q^{27} + 24 q^{29} - 24 q^{31} + 12 q^{35} - 36 q^{39} + 48 q^{41} + 12 q^{47} + 32 q^{49} + 60 q^{55} - 60 q^{59} + 36 q^{71} - 16 q^{75} + 72 q^{77} + 56 q^{81}+ \cdots - 36 q^{95}+O(q^{100}) $ 8 * q - 4 * q^3 + 8 * q^9 + 8 * q^25 - 40 * q^27 + 24 * q^29 - 24 * q^31 + 12 * q^35 - 36 * q^39 + 48 * q^41 + 12 * q^47 + 32 * q^49 + 60 * q^55 - 60 * q^59 + 36 * q^71 - 16 * q^75 + 72 * q^77 + 56 * q^81 + 48 * q^85 - 12 * q^87 - 24 * q^93 - 36 * q^95

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 8x^{6} + 15x^{4} - 8x^{2} + 1 $ x^8 - 8*x^6 + 15*x^4 - 8*x^2 + 1 :

$\beta_{1}$	$=$	$ \nu^{4} - 7\nu^{2} + 5 $ v^4 - 7*v^2 + 5
$\beta_{2}$	$=$	$ \nu^{6} - 7\nu^{4} + 9\nu^{2} - 3 $ v^6 - 7v^4 + 9v^2 - 3
$\beta_{3}$	$=$	$ \nu^{7} - 7\nu^{5} + 8\nu^{3} $ v^7 - 7v^5 + 8v^3
$\beta_{4}$	$=$	$ \nu^{7} - 8\nu^{5} + 15\nu^{3} - 7\nu $ v^7 - 8v^5 + 15v^3 - 7*v
$\beta_{5}$	$=$	$ \nu^{6} - 8\nu^{4} + 14\nu^{2} - 4 $ v^6 - 8v^4 + 14v^2 - 4
$\beta_{6}$	$=$	$ -\nu^{7} + 8\nu^{5} - 14\nu^{3} + 3\nu $ -v^7 + 8v^5 - 14v^3 + 3*v
$\beta_{7}$	$=$	$ -\nu^{7} + 8\nu^{5} - 15\nu^{3} + 9\nu $ -v^7 + 8v^5 - 15v^3 + 9*v

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

$\nu$	$=$	$ ( \beta_{7} + \beta_{4} ) / 2 $ (b7 + b4) / 2
$\nu^{2}$	$=$	$ ( -\beta_{5} + \beta_{2} - \beta _1 + 4 ) / 2 $ (-b5 + b2 - b1 + 4) / 2
$\nu^{3}$	$=$	$ 2\beta_{7} + \beta_{6} + 3\beta_{4} $ 2b7 + b6 + 3b4
$\nu^{4}$	$=$	$ ( -7\beta_{5} + 7\beta_{2} - 5\beta _1 + 18 ) / 2 $ (-7b5 + 7b2 - 5*b1 + 18) / 2
$\nu^{5}$	$=$	$ ( 21\beta_{7} + 14\beta_{6} + 33\beta_{4} + 2\beta_{3} ) / 2 $ (21b7 + 14b6 + 33b4 + 2b3) / 2
$\nu^{6}$	$=$	$ -20\beta_{5} + 21\beta_{2} - 13\beta _1 + 48 $ -20b5 + 21b2 - 13*b1 + 48
$\nu^{7}$	$=$	$ ( 115\beta_{7} + 82\beta_{6} + 183\beta_{4} + 16\beta_{3} ) / 2 $ (115b7 + 82b6 + 183b4 + 16b3) / 2

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

0.424402
−0.424402
1.29177
−1.29177
2.35625
−2.35625
0.774132
−0.774132

−3.33225

−2.78066

−2.13881

8.10387

1.2

−3.33225

2.78066

2.13881

8.10387

1.3

−1.09479

−2.06590

−1.07969

−1.80144

1.4

−1.09479

2.06590

1.07969

−1.80144

1.5

0.600196

−2.78066

4.17439

−2.63977

1.6

0.600196

2.78066

−4.17439

−2.63977

1.7

1.82684

−2.06590

−4.56446

0.337339

1.8

1.82684

2.06590

4.56446

0.337339

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$23$	$ +1 $

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8464.2.a.ca		8
4.b	odd	2	1		2116.2.a.h	✓	8
23.b	odd	2	1	inner	8464.2.a.ca		8
92.b	even	2	1		2116.2.a.h	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2116.2.a.h	✓	8	4.b	odd	2	1
2116.2.a.h	✓	8	92.b	even	2	1
8464.2.a.ca		8	1.a	even	1	1	trivial
8464.2.a.ca		8	23.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8464))$:

$ T_{3}^{4} + 2T_{3}^{3} - 6T_{3}^{2} - 4T_{3} + 4 $

T3^4 + 2*T3^3 - 6*T3^2 - 4*T3 + 4

$ T_{5}^{4} - 12T_{5}^{2} + 33 $

T5^4 - 12*T5^2 + 33

$ T_{7}^{8} - 44T_{7}^{6} + 588T_{7}^{4} - 2288T_{7}^{2} + 1936 $

T7^8 - 44*T7^6 + 588*T7^4 - 2288*T7^2 + 1936

$ T_{13}^{4} - 30T_{13}^{2} + 24T_{13} + 69 $

T13^4 - 30*T13^2 + 24*T13 + 69

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ (T^{4} + 2 T^{3} - 6 T^{2} + \cdots + 4)^{2} $ (T^4 + 2T^3 - 6T^2 - 4*T + 4)^2
$5$	$ (T^{4} - 12 T^{2} + 33)^{2} $ (T^4 - 12*T^2 + 33)^2
$7$	$ T^{8} - 44 T^{6} + \cdots + 1936 $ T^8 - 44T^6 + 588T^4 - 2288*T^2 + 1936
$11$	$ T^{8} - 60 T^{6} + \cdots + 1296 $ T^8 - 60T^6 + 972T^4 - 4752*T^2 + 1296
$13$	$ (T^{4} - 30 T^{2} + \cdots + 69)^{2} $ (T^4 - 30T^2 + 24T + 69)^2
$17$	$ T^{8} - 48 T^{6} + \cdots + 144 $ T^8 - 48T^6 + 408T^4 - 576*T^2 + 144
$19$	$ T^{8} - 68 T^{6} + \cdots + 35344 $ T^8 - 68T^6 + 1548T^4 - 13328*T^2 + 35344
$23$	$ T^{8} $ T^8
$29$	$ (T^{4} - 12 T^{3} + 30 T^{2} + \cdots - 3)^{2} $ (T^4 - 12T^3 + 30T^2 - 12*T - 3)^2
$31$	$ (T^{4} + 12 T^{3} + \cdots + 144)^{2} $ (T^4 + 12T^3 - 24T^2 - 432*T + 144)^2
$37$	$ T^{8} - 176 T^{6} + \cdots + 1430416 $ T^8 - 176T^6 + 9432T^4 - 199232*T^2 + 1430416
$41$	$ (T^{4} - 24 T^{3} + \cdots + 429)^{2} $ (T^4 - 24T^3 + 186T^2 - 528*T + 429)^2
$43$	$ T^{8} - 128 T^{6} + \cdots + 30976 $ T^8 - 128T^6 + 4464T^4 - 28160*T^2 + 30976
$47$	$ (T^{4} - 6 T^{3} + \cdots - 156)^{2} $ (T^4 - 6T^3 - 78T^2 - 204*T - 156)^2
$53$	$ (T^{4} - 156 T^{2} + 4761)^{2} $ (T^4 - 156*T^2 + 4761)^2
$59$	$ (T^{4} + 30 T^{3} + \cdots + 852)^{2} $ (T^4 + 30T^3 + 294T^2 + 1020*T + 852)^2
$61$	$ T^{8} - 176 T^{6} + \cdots + 121801 $ T^8 - 176T^6 + 7674T^4 - 103184*T^2 + 121801
$67$	$ T^{8} - 368 T^{6} + \cdots + 2408704 $ T^8 - 368T^6 + 33264T^4 - 537344*T^2 + 2408704
$71$	$ (T^{4} - 18 T^{3} + \cdots - 876)^{2} $ (T^4 - 18T^3 - 90T^2 + 2100*T - 876)^2
$73$	$ (T^{4} - 78 T^{2} + \cdots + 321)^{2} $ (T^4 - 78T^2 + 192T + 321)^2
$79$	$ T^{8} - 272 T^{6} + \cdots + 565504 $ T^8 - 272T^6 + 18672T^4 - 265472*T^2 + 565504
$83$	$ T^{8} - 432 T^{6} + \cdots + 389376 $ T^8 - 432T^6 + 50160T^4 - 744192*T^2 + 389376
$89$	$ T^{8} - 456 T^{6} + \cdots + 471969 $ T^8 - 456T^6 + 67314T^4 - 3181896*T^2 + 471969
$97$	$ T^{8} - 272 T^{6} + \cdots + 1590121 $ T^8 - 272T^6 + 22170T^4 - 590192*T^2 + 1590121
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Level:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\( q + (\beta_{5} + \beta_{2}) q^{3} - \beta_{7} q^{5} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{7} + \beta_{6} - \beta_{3}) q^{11} + ( - 2 \beta_{2} - 1) q^{13}+ \cdots + ( - 3 \beta_{7} - 3 \beta_{6} + \cdots + \beta_{3}) q^{99}+O(q^{100}) \) q + (b5 + b2) * q^3 - b7 * q^5 + (b6 + b4 - b3) * q^7 + (-b5 - b2 + b1 + 1) * q^9 + (-b7 + b6 - b3) * q^11 + (-2b2 - 1) q^13 + (b7 + b6 - 4b4 - 3b3) * q^15 + (-b7 - 2b4 - b3) q^17 + (b7 + b6 - 2b4 - 3b3) * q^19 + (b7 + 2b6 - 3b4 - 4b3) q^21 + (-b5 + 1) * q^25 + (2b5 - 2b1 - 6) * q^27 + (-b2 - b1 + 2) * q^29 + (2b1 - 2) q^31 + (b7 + 2b6 - 5b4 - 6b3) q^33 + (3b5 - b2 + 2b1 + 2) * q^35 + (-b7 + 2b6 - 4b4 - 3b3) q^37 + (b5 + b2 - 4) * q^39 + (-2b5 - 2b2 + 5) * q^41 + (-b7 - 2b6 + 3b4 + 4b3) q^43 + (-2b7 - 2b6 + 9b4 + 2b3) * q^45 + (-2b5 + 2b2 - b1 + 2) * q^47 + (3b5 + 5b2 + b1 + 7) * q^49 + (-b7 - b4 - 2b3) q^51 + (-2b4 + 5b3) * q^53 + (2b5 + b1 + 8) q^55 + (-3b7 + 5b4) * q^57 + (-3b5 - b2 - 8) q^59 + (-b7 + 5b4 + 3b3) * q^61 + (-4b7 - 3b6 + 3b4 + b3) q^63 + (b7 + 2b6 + 6b4 + 4b3) q^65 + (b7 + 4b6 + b4 + 2b3) * q^67 + (-4b5 + 2b2 + b1 + 6) * q^71 + (-3b5 - 2b2 - 2b1 - 2) q^73 + (b5 + b2 - b1 - 2) * q^75 + (7b5 + 3b2 + 3b1 + 12) q^77 + (b7 + b4 + 6b3) q^79 + (-9b5 - 5b2 + b1 + 5) * q^81 + (b7 + 4b6 + 3b4 + 2b3) q^83 + (3b2 - b1 + 7) q^85 + (2b2 + b1) q^87 + (3b7 - 2b6 + 5b4 + 3b3) * q^89 + (-4b7 - 3b6 - 3b4 + 3b3) * q^91 + (4b5 - 2b1 - 4) * q^93 + (6b5 + 4b2 + b1 - 2) * q^95 + (-b7 - b4 + 5b3) q^97 + (-3b7 - 3b6 + 8b4 + b3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 8 q^{9} + 8 q^{25} - 40 q^{27} + 24 q^{29} - 24 q^{31} + 12 q^{35} - 36 q^{39} + 48 q^{41} + 12 q^{47} + 32 q^{49} + 60 q^{55} - 60 q^{59} + 36 q^{71} - 16 q^{75} + 72 q^{77} + 56 q^{81}+ \cdots - 36 q^{95}+O(q^{100}) \) 8 * q - 4 * q^3 + 8 * q^9 + 8 * q^25 - 40 * q^27 + 24 * q^29 - 24 * q^31 + 12 * q^35 - 36 * q^39 + 48 * q^41 + 12 * q^47 + 32 * q^49 + 60 * q^55 - 60 * q^59 + 36 * q^71 - 16 * q^75 + 72 * q^77 + 56 * q^81 + 48 * q^85 - 12 * q^87 - 24 * q^93 - 36 * q^95

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