LMFDB - Newform orbit 3700.2.a.n

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3700,2,Mod(1,3700)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3700.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3700 = 2^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3700.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$29.5446487479$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.17268201.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 7x^{4} + 4x^{3} + 13x^{2} - 3x - 6 $ x^6 - x^5 - 7x^4 + 4x^3 + 13x^2 - 3x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{2} q^{3} - \beta_{3} q^{7} + (\beta_{5} + \beta_{3} + \beta_1 + 1) q^{9} + (\beta_1 - 1) q^{11} + ( - \beta_{4} + \beta_1 + 1) q^{13} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \cdots + 1) q^{17}+ \cdots + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots + 2) q^{99}+O(q^{100}) $ q + b2 * q^3 - b3 * q^7 + (b5 + b3 + b1 + 1) * q^9 + (b1 - 1) * q^11 + (-b4 + b1 + 1) * q^13 + (-b4 - b3 - b2 + b1 + 1) * q^17 + (-b5 + b2 + b1 + 1) * q^19 + (b5 + b4 - 2b2 + b1 + 2) q^21 + (b5 + b2 - 2) * q^23 + (-b5 + b4 + 3b2 - b1) q^27 + (b2 + 2) * q^29 + (b5 - b4 - b2 - 1) * q^31 + b4 * q^33 + q^37 + (-b5 + b3 + 3b2 - 2b1 + 1) * q^39 + (-b4 - b2 - 2b1 + 3) q^41 + (2b2 - b1 - 2) q^43 + (-b5 - b3 + b2 - 2b1 + 3) q^47 + (b5 - b4 - 2b1 + 4) q^49 + (-b5 + b4 + b2 - 2b1 - 1) q^51 + (b2 - b1) * q^53 + (b5 + b3 + b1 + 2) * q^57 + (-2b3 - b2 + 3) q^59 + (-b5 + b4 + b2 - 1) * q^61 + (-b5 + 3b4 + 4b2 - 7) * q^63 + (2b5 + b4 + b3 - b1 + 3) q^67 + (b5 + b4 + b3 + b1 + 6) * q^69 + (-b5 - 2b4 - b2 + b1 + 4) q^71 + (-b5 + 2b2 - b1 - 7) q^73 + (-b5 + b4 + 3b3 + 3b2 + b1) * q^77 + (-b4 - 3b3 - 3b2 + b1 + 4) * q^79 + (b5 - b4 - b3 - 4b2 + 2b1 + 6) * q^81 + (-b4 - b3 + b2 + 2b1 - 2) q^83 + (b5 + b3 + 2b2 + b1 + 4) q^87 + (-b5 + 2b4 + b3 + 5b2 - b1 + 2) * q^89 + (-2b5 + 3b4 - b3 - b2 - b1 - 3) * q^91 + (-2b5 + 2b2 - 3b1 - 1) q^93 + (b5 - b4 - 3b2 - 2b1 + 3) * q^97 + (b5 + b4 - b3 - b2 - b1 + 2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{3} - 2 q^{7} + 9 q^{9} - 8 q^{11} + 5 q^{13} + 2 q^{17} + 2 q^{19} + 10 q^{21} - 8 q^{23} + q^{27} + 13 q^{29} - 3 q^{31} - q^{33} + 6 q^{37} + 12 q^{39} + 22 q^{41} - 8 q^{43} + 18 q^{47} + 32 q^{49}+ \cdots + 13 q^{99}+O(q^{100}) $ 6 * q + q^3 - 2 * q^7 + 9 * q^9 - 8 * q^11 + 5 * q^13 + 2 * q^17 + 2 * q^19 + 10 * q^21 - 8 * q^23 + q^27 + 13 * q^29 - 3 * q^31 - q^33 + 6 * q^37 + 12 * q^39 + 22 * q^41 - 8 * q^43 + 18 * q^47 + 32 * q^49 - 5 * q^51 + 3 * q^53 + 15 * q^57 + 13 * q^59 - 9 * q^61 - 44 * q^63 + 27 * q^67 + 38 * q^69 + 20 * q^71 - 41 * q^73 + 3 * q^77 + 14 * q^79 + 30 * q^81 - 16 * q^83 + 29 * q^87 + 16 * q^89 - 28 * q^91 - 4 * q^93 + 23 * q^97 + 13 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} - 7x^{4} + 4x^{3} + 13x^{2} - 3x - 6 $ x^6 - x^5 - 7*x^4 + 4*x^3 + 13*x^2 - 3*x - 6 :

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

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

$\nu$	$=$	$ ( \beta_{4} + \beta_{3} + 2\beta_{2} ) / 3 $ (b4 + b3 + 2*b2) / 3
$\nu^{2}$	$=$	$ ( \beta_{5} + \beta_{4} + \beta_{2} + 7 ) / 3 $ (b5 + b4 + b2 + 7) / 3
$\nu^{3}$	$=$	$ ( \beta_{5} + 4\beta_{4} + 3\beta_{3} + 7\beta_{2} + 3\beta _1 + 4 ) / 3 $ (b5 + 4b4 + 3b3 + 7b2 + 3b1 + 4) / 3
$\nu^{4}$	$=$	$ 2\beta_{5} + 2\beta_{4} + 3\beta_{2} + \beta _1 + 9 $ 2b5 + 2b4 + 3*b2 + b1 + 9
$\nu^{5}$	$=$	$ ( 10\beta_{5} + 19\beta_{4} + 12\beta_{3} + 34\beta_{2} + 21\beta _1 + 31 ) / 3 $ (10b5 + 19b4 + 12b3 + 34b2 + 21*b1 + 31) / 3

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

−1.25357
1.54959
−0.739309
−1.90711
0.890614
2.45978

−3.17857

−3.29374

7.10328

1.2

−1.31241

−2.90449

−1.27757

1.3

−0.247976

1.40772

−2.93851

1.4

0.257851

4.98593

−2.93351

1.5

2.62860

−4.66591

3.90956

1.6

2.85250

2.47048

5.13675

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$5$	$ -1 $
$37$	$ -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	3700.2.a.n	yes	6
5.b	even	2	1		3700.2.a.m	✓	6
5.c	odd	4	2		3700.2.d.k		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3700.2.a.m	✓	6	5.b	even	2	1
3700.2.a.n	yes	6	1.a	even	1	1	trivial
3700.2.d.k		12	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(3700))$:

$ T_{3}^{6} - T_{3}^{5} - 13T_{3}^{4} + 11T_{3}^{3} + 32T_{3}^{2} - T_{3} - 2 $

$ T_{7}^{6} + 2T_{7}^{5} - 35T_{7}^{4} - 66T_{7}^{3} + 294T_{7}^{2} + 351T_{7} - 774 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - T^{5} - 13 T^{4} + \cdots - 2 $ T^6 - T^5 - 13T^4 + 11T^3 + 32*T^2 - T - 2
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + 2 T^{5} + \cdots - 774 $ T^6 + 2T^5 - 35T^4 - 66T^3 + 294T^2 + 351*T - 774
$11$	$ T^{6} + 8 T^{5} + \cdots - 18 $ T^6 + 8T^5 + 9T^4 - 43T^3 - 41T^2 + 93*T - 18
$13$	$ T^{6} - 5 T^{5} + \cdots - 428 $ T^6 - 5T^5 - 37T^4 + 175T^3 + 122T^2 - 629*T - 428
$17$	$ T^{6} - 2 T^{5} + \cdots - 288 $ T^6 - 2T^5 - 48T^4 + 163T^3 + 88T^2 - 417*T - 288
$19$	$ T^{6} - 2 T^{5} + \cdots + 127 $ T^6 - 2T^5 - 64T^4 + 112T^3 + 341T^2 - 677*T + 127
$23$	$ T^{6} + 8 T^{5} + \cdots - 1557 $ T^6 + 8T^5 - 36T^4 - 301T^3 + 367T^2 + 2364*T - 1557
$29$	$ T^{6} - 13 T^{5} + \cdots - 72 $ T^6 - 13T^5 + 57T^4 - 85T^3 - 26T^2 + 147*T - 72
$31$	$ T^{6} + 3 T^{5} + \cdots - 11024 $ T^6 + 3T^5 - 69T^4 - 124T^3 + 1551T^2 + 1191*T - 11024
$37$	$ (T - 1)^{6} $ (T - 1)^6
$41$	$ T^{6} - 22 T^{5} + \cdots + 23031 $ T^6 - 22T^5 + 51T^4 + 1607T^3 - 10484T^2 + 12042*T + 23031
$43$	$ T^{6} + 8 T^{5} + \cdots - 1557 $ T^6 + 8T^5 - 44T^4 - 237T^3 + 507T^2 + 1044*T - 1557
$47$	$ T^{6} - 18 T^{5} + \cdots + 7614 $ T^6 - 18T^5 + 33T^4 + 900T^3 - 5202T^2 + 6021*T + 7614
$53$	$ T^{6} - 3 T^{5} + \cdots + 81 $ T^6 - 3T^5 - 27T^4 + 72T^3 + 144T^2 - 270*T + 81
$59$	$ T^{6} - 13 T^{5} + \cdots - 2151 $ T^6 - 13T^5 - 69T^4 + 1067T^3 - 1340T^2 - 5991*T - 2151
$61$	$ T^{6} + 9 T^{5} + \cdots - 7346 $ T^6 + 9T^5 - 39T^4 - 388T^3 + 639T^2 + 4245*T - 7346
$67$	$ T^{6} - 27 T^{5} + \cdots - 131876 $ T^6 - 27T^5 + 144T^4 + 2045T^3 - 27513T^2 + 109845*T - 131876
$71$	$ T^{6} - 20 T^{5} + \cdots - 89118 $ T^6 - 20T^5 - 51T^4 + 2242T^3 - 830T^2 - 51621*T - 89118
$73$	$ T^{6} + 41 T^{5} + \cdots - 288 $ T^6 + 41T^5 + 628T^4 + 4560T^3 + 15816T^2 + 21024*T - 288
$79$	$ T^{6} - 14 T^{5} + \cdots + 107917 $ T^6 - 14T^5 - 211T^4 + 2809T^3 + 7790T^2 - 105350*T + 107917
$83$	$ T^{6} + 16 T^{5} + \cdots + 1296 $ T^6 + 16T^5 - 24T^4 - 551T^3 + 136T^2 + 4365*T + 1296
$89$	$ T^{6} - 16 T^{5} + \cdots + 130716 $ T^6 - 16T^5 - 216T^4 + 3983T^3 - 1526T^2 - 108075*T + 130716
$97$	$ T^{6} - 23 T^{5} + \cdots + 282706 $ T^6 - 23T^5 - 37T^4 + 3220T^3 - 10033T^2 - 72443*T + 282706
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Level:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3700.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{3} - \beta_{3} q^{7} + (\beta_{5} + \beta_{3} + \beta_1 + 1) q^{9} + (\beta_1 - 1) q^{11} + ( - \beta_{4} + \beta_1 + 1) q^{13} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \cdots + 1) q^{17}+ \cdots + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots + 2) q^{99}+O(q^{100}) \) q + b2 * q^3 - b3 * q^7 + (b5 + b3 + b1 + 1) * q^9 + (b1 - 1) * q^11 + (-b4 + b1 + 1) * q^13 + (-b4 - b3 - b2 + b1 + 1) * q^17 + (-b5 + b2 + b1 + 1) * q^19 + (b5 + b4 - 2b2 + b1 + 2) q^21 + (b5 + b2 - 2) * q^23 + (-b5 + b4 + 3b2 - b1) q^27 + (b2 + 2) * q^29 + (b5 - b4 - b2 - 1) * q^31 + b4 * q^33 + q^37 + (-b5 + b3 + 3b2 - 2b1 + 1) * q^39 + (-b4 - b2 - 2b1 + 3) q^41 + (2b2 - b1 - 2) q^43 + (-b5 - b3 + b2 - 2b1 + 3) q^47 + (b5 - b4 - 2b1 + 4) q^49 + (-b5 + b4 + b2 - 2b1 - 1) q^51 + (b2 - b1) * q^53 + (b5 + b3 + b1 + 2) * q^57 + (-2b3 - b2 + 3) q^59 + (-b5 + b4 + b2 - 1) * q^61 + (-b5 + 3b4 + 4b2 - 7) * q^63 + (2b5 + b4 + b3 - b1 + 3) q^67 + (b5 + b4 + b3 + b1 + 6) * q^69 + (-b5 - 2b4 - b2 + b1 + 4) q^71 + (-b5 + 2b2 - b1 - 7) q^73 + (-b5 + b4 + 3b3 + 3b2 + b1) * q^77 + (-b4 - 3b3 - 3b2 + b1 + 4) * q^79 + (b5 - b4 - b3 - 4b2 + 2b1 + 6) * q^81 + (-b4 - b3 + b2 + 2b1 - 2) q^83 + (b5 + b3 + 2b2 + b1 + 4) q^87 + (-b5 + 2b4 + b3 + 5b2 - b1 + 2) * q^89 + (-2b5 + 3b4 - b3 - b2 - b1 - 3) * q^91 + (-2b5 + 2b2 - 3b1 - 1) q^93 + (b5 - b4 - 3b2 - 2b1 + 3) * q^97 + (b5 + b4 - b3 - b2 - b1 + 2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{3} - 2 q^{7} + 9 q^{9} - 8 q^{11} + 5 q^{13} + 2 q^{17} + 2 q^{19} + 10 q^{21} - 8 q^{23} + q^{27} + 13 q^{29} - 3 q^{31} - q^{33} + 6 q^{37} + 12 q^{39} + 22 q^{41} - 8 q^{43} + 18 q^{47} + 32 q^{49}+ \cdots + 13 q^{99}+O(q^{100}) \) 6 * q + q^3 - 2 * q^7 + 9 * q^9 - 8 * q^11 + 5 * q^13 + 2 * q^17 + 2 * q^19 + 10 * q^21 - 8 * q^23 + q^27 + 13 * q^29 - 3 * q^31 - q^33 + 6 * q^37 + 12 * q^39 + 22 * q^41 - 8 * q^43 + 18 * q^47 + 32 * q^49 - 5 * q^51 + 3 * q^53 + 15 * q^57 + 13 * q^59 - 9 * q^61 - 44 * q^63 + 27 * q^67 + 38 * q^69 + 20 * q^71 - 41 * q^73 + 3 * q^77 + 14 * q^79 + 30 * q^81 - 16 * q^83 + 29 * q^87 + 16 * q^89 - 28 * q^91 - 4 * q^93 + 23 * q^97 + 13 * q^99

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