LMFDB - Newform orbit 8208.2.a.ce

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8208,2,Mod(1,8208)] 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("8208.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8208 = 2^{4} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-2,0,-2,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$65.5412099791$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 18x^{4} + 17x^{3} + 72x^{2} + 29x - 3 $ x^6 - 2x^5 - 18x^4 + 17x^3 + 72x^2 + 29*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4104)
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_1 q^{5} - \beta_{4} q^{7} + (\beta_{3} + \beta_1 - 1) q^{11} - \beta_{5} q^{13} + (\beta_{4} - \beta_{3} - \beta_1 - 1) q^{17} - q^{19} + (\beta_{5} + \beta_{4} + \beta_1) q^{23} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{25}+ \cdots + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 1) q^{97}+O(q^{100}) $ q - b1 * q^5 - b4 * q^7 + (b3 + b1 - 1) * q^11 - b5 * q^13 + (b4 - b3 - b1 - 1) * q^17 - q^19 + (b5 + b4 + b1) * q^23 + (-b3 + b2 + b1 + 1) * q^25 + (b5 + b3 + 1) * q^29 + (-b3 + b1 - 1) * q^31 - b2 * q^35 + (b4 - b2 + 2) * q^37 + (-b2 - 1) * q^41 + (-2b4 + b2 - 1) q^43 + (b2 + b1 - 2) * q^47 + (2b5 + b4 + b3 - b2 - b1 + 4) q^49 + (b5 + 2b4 - b1 - 2) q^53 + (-b5 + b4 - b2 + b1 - 4) * q^55 + (-b5 + 2b4 - 2b3 - b1 - 3) * q^59 + (b5 - b4 + b3 - b2 + 2b1 + 3) q^61 + (-b4 + b3 + b2 + b1) * q^65 + (2b5 + b3 - b2 + b1 + 1) q^67 + (b5 - 2b3 + b2 - b1 + 2) q^71 + (-b5 + b4 - 2b3 + b2 + b1 + 2) q^73 + (-3b5 - 2b3 + 2b2 + b1 - 6) q^77 + (2b4 - b3 - 5) q^79 + (-b5 + b4 - b2 + b1 - 6) * q^83 + (b5 - b4 + 2b2 + b1 + 4) q^85 + (-b4 + 2b3 - 3) q^89 + (-b5 + b4 + 2b3 + b1 - 4) q^91 + b1 * q^95 + (-b5 - b4 - b3 + 2b1 + 1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} - 2 q^{7} - 3 q^{11} + q^{13} - 7 q^{17} - 6 q^{19} + 3 q^{23} + 10 q^{25} + 6 q^{29} - 5 q^{31} - 3 q^{35} + 11 q^{37} - 9 q^{41} - 7 q^{43} - 7 q^{47} + 20 q^{49} - 11 q^{53} - 22 q^{55}+ \cdots + 8 q^{97}+O(q^{100}) $ 6 * q - 2 * q^5 - 2 * q^7 - 3 * q^11 + q^13 - 7 * q^17 - 6 * q^19 + 3 * q^23 + 10 * q^25 + 6 * q^29 - 5 * q^31 - 3 * q^35 + 11 * q^37 - 9 * q^41 - 7 * q^43 - 7 * q^47 + 20 * q^49 - 11 * q^53 - 22 * q^55 - 17 * q^59 + 17 * q^61 + 4 * q^65 + 4 * q^67 + 10 * q^71 + 18 * q^73 - 27 * q^77 - 27 * q^79 - 34 * q^83 + 29 * q^85 - 18 * q^89 - 17 * q^91 + 2 * q^95 + 8 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 2x^{5} - 18x^{4} + 17x^{3} + 72x^{2} + 29x - 3 $ x^6 - 2*x^5 - 18*x^4 + 17*x^3 + 72*x^2 + 29*x - 3 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -\nu^{5} + 5\nu^{4} + 11\nu^{3} - 50\nu^{2} - 22\nu - 3 ) / 12 $ (-v^5 + 5v^4 + 11v^3 - 50v^2 - 22v - 3) / 12
$\beta_{3}$	$=$	$ ( -\nu^{5} + 5\nu^{4} + 11\nu^{3} - 62\nu^{2} - 10\nu + 69 ) / 12 $ (-v^5 + 5v^4 + 11v^3 - 62v^2 - 10v + 69) / 12
$\beta_{4}$	$=$	$ ( \nu^{5} - \nu^{4} - 23\nu^{3} + 6\nu^{2} + 122\nu + 51 ) / 12 $ (v^5 - v^4 - 23v^3 + 6v^2 + 122*v + 51) / 12
$\beta_{5}$	$=$	$ ( 5\nu^{5} - 13\nu^{4} - 79\nu^{3} + 130\nu^{2} + 242\nu + 3 ) / 12 $ (5v^5 - 13v^4 - 79v^3 + 130v^2 + 242*v + 3) / 12

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{3} + \beta_{2} + \beta _1 + 6 $ -b3 + b2 + b1 + 6
$\nu^{3}$	$=$	$ \beta_{5} - 3\beta_{4} + \beta_{3} + \beta_{2} + 13\beta _1 + 7 $ b5 - 3b4 + b3 + b2 + 13b1 + 7
$\nu^{4}$	$=$	$ 3\beta_{5} - 6\beta_{4} - 8\beta_{3} + 17\beta_{2} + 25\beta _1 + 75 $ 3b5 - 6b4 - 8b3 + 17b2 + 25*b1 + 75
$\nu^{5}$	$=$	$ 26\beta_{5} - 63\beta_{4} + 21\beta_{3} + 34\beta_{2} + 196\beta _1 + 149 $ 26b5 - 63b4 + 21b3 + 34b2 + 196*b1 + 149

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

4.27383
2.93526
0.0851277
−0.611152
−1.40017
−3.28289

−4.27383

1.76802

1.2

−2.93526

−1.89965

1.3

−0.0851277

−5.11790

1.4

0.611152

1.35784

1.5

1.40017

4.51232

1.6

3.28289

−2.62063

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ +1 $
$19$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8208.2.a.ce		6
3.b	odd	2	1		8208.2.a.cf		6
4.b	odd	2	1		4104.2.a.s	✓	6
12.b	even	2	1		4104.2.a.t	yes	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4104.2.a.s	✓	6	4.b	odd	2	1
4104.2.a.t	yes	6	12.b	even	2	1
8208.2.a.ce		6	1.a	even	1	1	trivial
8208.2.a.cf		6	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8208))$:

$ T_{5}^{6} + 2T_{5}^{5} - 18T_{5}^{4} - 17T_{5}^{3} + 72T_{5}^{2} - 29T_{5} - 3 $

T5^6 + 2*T5^5 - 18*T5^4 - 17*T5^3 + 72*T5^2 - 29*T5 - 3

$ T_{7}^{6} + 2T_{7}^{5} - 29T_{7}^{4} - 41T_{7}^{3} + 165T_{7}^{2} + 116T_{7} - 276 $

T7^6 + 2*T7^5 - 29*T7^4 - 41*T7^3 + 165*T7^2 + 116*T7 - 276

$ T_{11}^{6} + 3T_{11}^{5} - 45T_{11}^{4} - 88T_{11}^{3} + 601T_{11}^{2} + 572T_{11} - 2608 $

T11^6 + 3*T11^5 - 45*T11^4 - 88*T11^3 + 601*T11^2 + 572*T11 - 2608

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + 2 T^{5} + \cdots - 3 $ T^6 + 2T^5 - 18T^4 - 17T^3 + 72T^2 - 29*T - 3
$7$	$ T^{6} + 2 T^{5} + \cdots - 276 $ T^6 + 2T^5 - 29T^4 - 41T^3 + 165T^2 + 116*T - 276
$11$	$ T^{6} + 3 T^{5} + \cdots - 2608 $ T^6 + 3T^5 - 45T^4 - 88T^3 + 601T^2 + 572*T - 2608
$13$	$ T^{6} - T^{5} + \cdots - 684 $ T^6 - T^5 - 47T^4 - 14T^3 + 454T^2 + 303T - 684
$17$	$ T^{6} + 7 T^{5} + \cdots + 384 $ T^6 + 7T^5 - 31T^4 - 203T^3 + 274T^2 + 1040*T + 384
$19$	$ (T + 1)^{6} $ (T + 1)^6
$23$	$ T^{6} - 3 T^{5} + \cdots - 864 $ T^6 - 3T^5 - 78T^4 + 277T^3 + 578T^2 - 640*T - 864
$29$	$ T^{6} - 6 T^{5} + \cdots + 456 $ T^6 - 6T^5 - 59T^4 + 335T^3 - 126T^2 - 844*T + 456
$31$	$ T^{6} + 5 T^{5} + \cdots - 6256 $ T^6 + 5T^5 - 71T^4 - 342T^3 + 1145T^2 + 4120*T - 6256
$37$	$ T^{6} - 11 T^{5} + \cdots + 11696 $ T^6 - 11T^5 - 56T^4 + 765T^3 - 76T^2 - 9268*T + 11696
$41$	$ T^{6} + 9 T^{5} + \cdots + 192 $ T^6 + 9T^5 - 67T^4 - 424T^3 + 1720T^2 + 1433*T + 192
$43$	$ T^{6} + 7 T^{5} + \cdots + 4364 $ T^6 + 7T^5 - 153T^4 - 632T^3 + 5238T^2 - 8969*T + 4364
$47$	$ T^{6} + 7 T^{5} + \cdots + 11906 $ T^6 + 7T^5 - 96T^4 - 758T^3 - 147T^2 + 7519*T + 11906
$53$	$ T^{6} + 11 T^{5} + \cdots + 86508 $ T^6 + 11T^5 - 109T^4 - 1464T^3 - 405T^2 + 34452*T + 86508
$59$	$ T^{6} + 17 T^{5} + \cdots + 282816 $ T^6 + 17T^5 - 99T^4 - 2810T^3 - 7492T^2 + 67635*T + 282816
$61$	$ T^{6} - 17 T^{5} + \cdots - 55112 $ T^6 - 17T^5 - 110T^4 + 3773T^3 - 26222T^2 + 68620*T - 55112
$67$	$ T^{6} - 4 T^{5} + \cdots - 111744 $ T^6 - 4T^5 - 245T^4 + 727T^3 + 14944T^2 - 17508*T - 111744
$71$	$ T^{6} - 10 T^{5} + \cdots + 69984 $ T^6 - 10T^5 - 225T^4 + 2439T^3 + 8586T^2 - 112104*T + 69984
$73$	$ T^{6} - 18 T^{5} + \cdots - 183972 $ T^6 - 18T^5 - 104T^4 + 2286T^3 + 5075T^2 - 76996*T - 183972
$79$	$ T^{6} + 27 T^{5} + \cdots - 496 $ T^6 + 27T^5 + 199T^4 - 102T^3 - 4188T^2 - 2893*T - 496
$83$	$ T^{6} + 34 T^{5} + \cdots - 16844 $ T^6 + 34T^5 + 288T^4 - 1162T^3 - 21197T^2 - 48288*T - 16844
$89$	$ T^{6} + 18 T^{5} + \cdots + 18306 $ T^6 + 18T^5 - 14T^4 - 987T^3 - 1935T^2 + 8559*T + 18306
$97$	$ T^{6} - 8 T^{5} + \cdots + 57168 $ T^6 - 8T^5 - 203T^4 + 1407T^3 + 7967T^2 - 52908*T + 57168
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Level:	\( N \)	\(=\)	\( 8208 = 2^{4} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8208.a (trivial)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{5} - \beta_{4} q^{7} + (\beta_{3} + \beta_1 - 1) q^{11} - \beta_{5} q^{13} + (\beta_{4} - \beta_{3} - \beta_1 - 1) q^{17} - q^{19} + (\beta_{5} + \beta_{4} + \beta_1) q^{23} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{25}+ \cdots + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 1) q^{97}+O(q^{100}) \) q - b1 * q^5 - b4 * q^7 + (b3 + b1 - 1) * q^11 - b5 * q^13 + (b4 - b3 - b1 - 1) * q^17 - q^19 + (b5 + b4 + b1) * q^23 + (-b3 + b2 + b1 + 1) * q^25 + (b5 + b3 + 1) * q^29 + (-b3 + b1 - 1) * q^31 - b2 * q^35 + (b4 - b2 + 2) * q^37 + (-b2 - 1) * q^41 + (-2b4 + b2 - 1) q^43 + (b2 + b1 - 2) * q^47 + (2b5 + b4 + b3 - b2 - b1 + 4) q^49 + (b5 + 2b4 - b1 - 2) q^53 + (-b5 + b4 - b2 + b1 - 4) * q^55 + (-b5 + 2b4 - 2b3 - b1 - 3) * q^59 + (b5 - b4 + b3 - b2 + 2b1 + 3) q^61 + (-b4 + b3 + b2 + b1) * q^65 + (2b5 + b3 - b2 + b1 + 1) q^67 + (b5 - 2b3 + b2 - b1 + 2) q^71 + (-b5 + b4 - 2b3 + b2 + b1 + 2) q^73 + (-3b5 - 2b3 + 2b2 + b1 - 6) q^77 + (2b4 - b3 - 5) q^79 + (-b5 + b4 - b2 + b1 - 6) * q^83 + (b5 - b4 + 2b2 + b1 + 4) q^85 + (-b4 + 2b3 - 3) q^89 + (-b5 + b4 + 2b3 + b1 - 4) q^91 + b1 * q^95 + (-b5 - b4 - b3 + 2b1 + 1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 2 q^{7} - 3 q^{11} + q^{13} - 7 q^{17} - 6 q^{19} + 3 q^{23} + 10 q^{25} + 6 q^{29} - 5 q^{31} - 3 q^{35} + 11 q^{37} - 9 q^{41} - 7 q^{43} - 7 q^{47} + 20 q^{49} - 11 q^{53} - 22 q^{55}+ \cdots + 8 q^{97}+O(q^{100}) \) 6 * q - 2 * q^5 - 2 * q^7 - 3 * q^11 + q^13 - 7 * q^17 - 6 * q^19 + 3 * q^23 + 10 * q^25 + 6 * q^29 - 5 * q^31 - 3 * q^35 + 11 * q^37 - 9 * q^41 - 7 * q^43 - 7 * q^47 + 20 * q^49 - 11 * q^53 - 22 * q^55 - 17 * q^59 + 17 * q^61 + 4 * q^65 + 4 * q^67 + 10 * q^71 + 18 * q^73 - 27 * q^77 - 27 * q^79 - 34 * q^83 + 29 * q^85 - 18 * q^89 - 17 * q^91 + 2 * q^95 + 8 * q^97

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