LMFDB - Newform orbit 4212.2.a.m

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4212,2,Mod(1,4212)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4212, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4212.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4212 = 2^{2} \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4212.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,1,0,0,0,0,0,-1] 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:	$33.6329893314$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.35342001.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 12x^{3} + 15x^{2} - 15x + 3 $ x^6 - 2x^5 - 8x^4 + 12x^3 + 15x^2 - 15*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 468)
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_{4} q^{5} + (\beta_{3} - \beta_1) q^{7} + (\beta_{5} + \beta_{3} - \beta_1) q^{11} - q^{13} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{17} + (\beta_{4} + \beta_{3}) q^{19} + ( - \beta_{5} - \beta_{2} + 1) q^{23}+ \cdots + ( - 4 \beta_{5} + 2 \beta_{4} + \cdots - 2) q^{97}+O(q^{100}) $ q + b4 * q^5 + (b3 - b1) * q^7 + (b5 + b3 - b1) * q^11 - q^13 + (-b5 - b4 + b3 - b1 + 1) * q^17 + (b4 + b3) * q^19 + (-b5 - b2 + 1) * q^23 + (b4 - b3 - b1 + 1) * q^25 + (b5 - b4 + b2 + 2) * q^29 + (b5 + b4 - b2 - 2) * q^31 + (-b5 + b3 + b2 - b1 + 2) * q^35 + (b5 + 2b4 - b2 + b1 - 2) q^37 + (-b3 - 2b2 + b1 + 2) q^41 + (b5 - b3 + 2b2 + 1) q^43 + (-b4 + 3b3 + 2b2 + 2) * q^47 + (2b3 + 3b2 + b1 + 1) * q^49 + (b5 + b4 - 3b3 + b2 + 5) q^53 + (-2b5 + b4 - b3 - b2 + b1) q^55 + (b5 + 2b4 - b2) q^59 + (-2b5 + 2b4 - 2b3 - b2 + 1) q^61 - b4 * q^65 + (3b5 + b4 - 3b3 + b1 + 2) * q^67 + (b5 - b3 - b2 - 2b1 + 4) q^71 + (-b5 - 2b2 + b1) q^73 + (-b4 + b3 + b2 + 2b1 + 6) q^77 + (-2b5 - 4b4 + b3 - b2 - 1) * q^79 + (2b3 - 3b2 + b1 - 2) * q^83 + (-b4 + 4b3 + 3b2 - 2b1 - 2) q^85 + (-2b5 + b4 - 2b3 - 2b2 - b1 + 4) q^89 + (-b3 + b1) * q^91 + (-b3 + b2 - 2b1 + 8) q^95 + (-4b5 + 2b4 - 5b3 - 2b2 + 2b1 - 2) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{5} - q^{11} - 6 q^{13} + 6 q^{17} + 3 q^{19} + 7 q^{23} + 3 q^{25} + 10 q^{29} - 12 q^{31} + 13 q^{35} - 9 q^{37} + 12 q^{41} + 3 q^{43} + 17 q^{47} + 12 q^{49} + 24 q^{53} + 3 q^{55} + q^{59}+ \cdots - 12 q^{97}+O(q^{100}) $ 6 * q + q^5 - q^11 - 6 * q^13 + 6 * q^17 + 3 * q^19 + 7 * q^23 + 3 * q^25 + 10 * q^29 - 12 * q^31 + 13 * q^35 - 9 * q^37 + 12 * q^41 + 3 * q^43 + 17 * q^47 + 12 * q^49 + 24 * q^53 + 3 * q^55 + q^59 + 6 * q^61 - q^65 + 6 * q^67 + 17 * q^71 + 3 * q^73 + 41 * q^77 - 6 * q^79 - 6 * q^83 - 9 * q^85 + 21 * q^89 + 42 * q^95 - 12 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 2x^{5} - 8x^{4} + 12x^{3} + 15x^{2} - 15x + 3 $ x^6 - 2*x^5 - 8*x^4 + 12*x^3 + 15*x^2 - 15*x + 3 :

$\beta_{1}$	$=$	$ \nu^{2} - 3 $ v^2 - 3
$\beta_{2}$	$=$	$ 2\nu^{5} - 3\nu^{4} - 17\nu^{3} + 15\nu^{2} + 36\nu - 13 $ 2v^5 - 3v^4 - 17v^3 + 15v^2 + 36*v - 13
$\beta_{3}$	$=$	$ 2\nu^{5} - 3\nu^{4} - 18\nu^{3} + 17\nu^{2} + 39\nu - 17 $ 2v^5 - 3v^4 - 18v^3 + 17v^2 + 39*v - 17
$\beta_{4}$	$=$	$ 3\nu^{5} - 5\nu^{4} - 25\nu^{3} + 26\nu^{2} + 51\nu - 22 $ 3v^5 - 5v^4 - 25v^3 + 26v^2 + 51*v - 22
$\beta_{5}$	$=$	$ -3\nu^{5} + 5\nu^{4} + 26\nu^{3} - 27\nu^{2} - 57\nu + 24 $ -3v^5 + 5v^4 + 26v^3 - 27v^2 - 57*v + 24

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

$\nu$	$=$	$ ( -\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 3 $ (-b5 - b4 - b3 + b2 + b1 + 1) / 3
$\nu^{2}$	$=$	$ \beta _1 + 3 $ b1 + 3
$\nu^{3}$	$=$	$ -\beta_{5} - \beta_{4} - 2\beta_{3} + 2\beta_{2} + 3\beta _1 + 3 $ -b5 - b4 - 2b3 + 2b2 + 3*b1 + 3
$\nu^{4}$	$=$	$ \beta_{5} - \beta_{4} + 3\beta_{2} + 8\beta _1 + 17 $ b5 - b4 + 3b2 + 8b1 + 17
$\nu^{5}$	$=$	$ -\beta_{5} - 4\beta_{4} - 11\beta_{3} + 16\beta_{2} + 24\beta _1 + 29 $ -b5 - 4b4 - 11b3 + 16b2 + 24b1 + 29

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

0.335792
2.02803
−2.07603
2.92333
0.439707
−1.65083

−2.94030

0.188839

1.2

−1.82377

−1.37569

1.3

−0.697259

2.19680

1.4

0.0520874

−5.04236

1.5

3.18901

4.63253

1.6

3.22022

−0.600114

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ +1 $
$13$	$ +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	4212.2.a.m		6
3.b	odd	2	1		4212.2.a.k		6
9.c	even	3	2		468.2.i.c	✓	12
9.d	odd	6	2		1404.2.i.c		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
468.2.i.c	✓	12	9.c	even	3	2
1404.2.i.c		12	9.d	odd	6	2
4212.2.a.k		6	3.b	odd	2	1
4212.2.a.m		6	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4212))$:

$ T_{5}^{6} - T_{5}^{5} - 16T_{5}^{4} + 5T_{5}^{3} + 65T_{5}^{2} + 35T_{5} - 2 $

T5^6 - T5^5 - 16*T5^4 + 5*T5^3 + 65*T5^2 + 35*T5 - 2

$ T_{11}^{6} + T_{11}^{5} - 37T_{11}^{4} + 10T_{11}^{3} + 335T_{11}^{2} - 473T_{11} + 172 $

T11^6 + T11^5 - 37*T11^4 + 10*T11^3 + 335*T11^2 - 473*T11 + 172

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - T^{5} - 16 T^{4} + \cdots - 2 $ T^6 - T^5 - 16T^4 + 5T^3 + 65T^2 + 35T - 2
$7$	$ T^{6} - 27 T^{4} + \cdots - 8 $ T^6 - 27T^4 + 7T^3 + 81T^2 + 27T - 8
$11$	$ T^{6} + T^{5} + \cdots + 172 $ T^6 + T^5 - 37T^4 + 10T^3 + 335T^2 - 473T + 172
$13$	$ (T + 1)^{6} $ (T + 1)^6
$17$	$ T^{6} - 6 T^{5} + \cdots + 1917 $ T^6 - 6T^5 - 42T^4 + 243T^3 + 369T^2 - 2268*T + 1917
$19$	$ T^{6} - 3 T^{5} + \cdots + 216 $ T^6 - 3T^5 - 33T^4 + 117T^3 + 72T^2 - 459*T + 216
$23$	$ T^{6} - 7 T^{5} + \cdots + 151 $ T^6 - 7T^5 - 16T^4 + 101T^3 + 17T^2 - 265*T + 151
$29$	$ T^{6} - 10 T^{5} + \cdots - 146 $ T^6 - 10T^5 - 7T^4 + 158T^3 + 236T^2 - 73*T - 146
$31$	$ T^{6} + 12 T^{5} + \cdots + 5686 $ T^6 + 12T^5 + 3T^4 - 368T^3 - 768T^2 + 2481*T + 5686
$37$	$ T^{6} + 9 T^{5} + \cdots - 8046 $ T^6 + 9T^5 - 69T^4 - 666T^3 + 855T^2 + 9045*T - 8046
$41$	$ T^{6} - 12 T^{5} + \cdots + 1350 $ T^6 - 12T^5 - 69T^4 + 1467T^3 - 5949T^2 + 5265*T + 1350
$43$	$ T^{6} - 3 T^{5} + \cdots - 5237 $ T^6 - 3T^5 - 123T^4 + 367T^3 + 3774T^2 - 11361*T - 5237
$47$	$ T^{6} - 17 T^{5} + \cdots - 24062 $ T^6 - 17T^5 - 43T^4 + 1375T^3 - 16T^2 - 19727*T - 24062
$53$	$ T^{6} - 24 T^{5} + \cdots + 63477 $ T^6 - 24T^5 + 75T^4 + 1683T^3 - 10350T^2 - 5724*T + 63477
$59$	$ T^{6} - T^{5} + \cdots + 3466 $ T^6 - T^5 - 82T^4 - 181T^3 + 983T^2 + 3839T + 3466
$61$	$ T^{6} - 6 T^{5} + \cdots - 102089 $ T^6 - 6T^5 - 132T^4 + 586T^3 + 6099T^2 - 14133*T - 102089
$67$	$ T^{6} - 6 T^{5} + \cdots - 448652 $ T^6 - 6T^5 - 339T^4 + 2089T^3 + 29103T^2 - 151383*T - 448652
$71$	$ T^{6} - 17 T^{5} + \cdots + 268 $ T^6 - 17T^5 - 70T^4 + 1786T^3 - 4615T^2 - 1097*T + 268
$73$	$ T^{6} - 3 T^{5} + \cdots - 15206 $ T^6 - 3T^5 - 117T^4 + 140T^3 + 3633T^2 + 675*T - 15206
$79$	$ T^{6} + 6 T^{5} + \cdots - 156017 $ T^6 + 6T^5 - 312T^4 - 1918T^3 + 12903T^2 + 53211*T - 156017
$83$	$ T^{6} + 6 T^{5} + \cdots - 138186 $ T^6 + 6T^5 - 327T^4 - 2619T^3 + 11889T^2 + 75843*T - 138186
$89$	$ T^{6} - 21 T^{5} + \cdots + 37584 $ T^6 - 21T^5 - 21T^4 + 2637T^3 - 14850T^2 + 12231*T + 37584
$97$	$ T^{6} + 12 T^{5} + \cdots + 298300 $ T^6 + 12T^5 - 372T^4 - 4747T^3 + 6762T^2 + 151395*T + 298300
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Level:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{5} + (\beta_{3} - \beta_1) q^{7} + (\beta_{5} + \beta_{3} - \beta_1) q^{11} - q^{13} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{17} + (\beta_{4} + \beta_{3}) q^{19} + ( - \beta_{5} - \beta_{2} + 1) q^{23}+ \cdots + ( - 4 \beta_{5} + 2 \beta_{4} + \cdots - 2) q^{97}+O(q^{100}) \) q + b4 * q^5 + (b3 - b1) * q^7 + (b5 + b3 - b1) * q^11 - q^13 + (-b5 - b4 + b3 - b1 + 1) * q^17 + (b4 + b3) * q^19 + (-b5 - b2 + 1) * q^23 + (b4 - b3 - b1 + 1) * q^25 + (b5 - b4 + b2 + 2) * q^29 + (b5 + b4 - b2 - 2) * q^31 + (-b5 + b3 + b2 - b1 + 2) * q^35 + (b5 + 2b4 - b2 + b1 - 2) q^37 + (-b3 - 2b2 + b1 + 2) q^41 + (b5 - b3 + 2b2 + 1) q^43 + (-b4 + 3b3 + 2b2 + 2) * q^47 + (2b3 + 3b2 + b1 + 1) * q^49 + (b5 + b4 - 3b3 + b2 + 5) q^53 + (-2b5 + b4 - b3 - b2 + b1) q^55 + (b5 + 2b4 - b2) q^59 + (-2b5 + 2b4 - 2b3 - b2 + 1) q^61 - b4 * q^65 + (3b5 + b4 - 3b3 + b1 + 2) * q^67 + (b5 - b3 - b2 - 2b1 + 4) q^71 + (-b5 - 2b2 + b1) q^73 + (-b4 + b3 + b2 + 2b1 + 6) q^77 + (-2b5 - 4b4 + b3 - b2 - 1) * q^79 + (2b3 - 3b2 + b1 - 2) * q^83 + (-b4 + 4b3 + 3b2 - 2b1 - 2) q^85 + (-2b5 + b4 - 2b3 - 2b2 - b1 + 4) q^89 + (-b3 + b1) * q^91 + (-b3 + b2 - 2b1 + 8) q^95 + (-4b5 + 2b4 - 5b3 - 2b2 + 2b1 - 2) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{5} - q^{11} - 6 q^{13} + 6 q^{17} + 3 q^{19} + 7 q^{23} + 3 q^{25} + 10 q^{29} - 12 q^{31} + 13 q^{35} - 9 q^{37} + 12 q^{41} + 3 q^{43} + 17 q^{47} + 12 q^{49} + 24 q^{53} + 3 q^{55} + q^{59}+ \cdots - 12 q^{97}+O(q^{100}) \) 6 * q + q^5 - q^11 - 6 * q^13 + 6 * q^17 + 3 * q^19 + 7 * q^23 + 3 * q^25 + 10 * q^29 - 12 * q^31 + 13 * q^35 - 9 * q^37 + 12 * q^41 + 3 * q^43 + 17 * q^47 + 12 * q^49 + 24 * q^53 + 3 * q^55 + q^59 + 6 * q^61 - q^65 + 6 * q^67 + 17 * q^71 + 3 * q^73 + 41 * q^77 - 6 * q^79 - 6 * q^83 - 9 * q^85 + 21 * q^89 + 42 * q^95 - 12 * q^97

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