LMFDB - Newform orbit 3800.2.a.bd

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.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:	$30.3431527681$
Analytic rank:	$0$
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} - 12x^{4} + 16x^{3} + 33x^{2} - 4x - 7 $ x^6 - 2x^5 - 12x^4 + 16x^3 + 33x^2 - 4*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
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_1 q^{3} - \beta_{3} q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{11} + \beta_{4} q^{13} + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{17} + q^{19} + ( - \beta_{4} - \beta_{3} + 1) q^{21}+ \cdots + (\beta_{5} - 3 \beta_{4} + \cdots - 6 \beta_1) q^{99}+O(q^{100}) $ q + b1 * q^3 - b3 * q^7 + (b2 + b1 + 1) * q^9 + (-b5 - b3 - b1) * q^11 + b4 * q^13 + (-b5 + b4 - b2) * q^17 + q^19 + (-b4 - b3 + 1) * q^21 + (-b3 + b2) * q^23 + (2b3 + 3b1 + 3) * q^27 + (-b5 + 2b1) q^29 + (b4 + b3 - b1 + 2) * q^31 + (-2b4 - b3 - b2 + b1 - 4) q^33 + (b5 - b4 - 3b3 + b2 - b1 - 1) q^37 + (b5 + b4 + 3b3 + b1 + 3) q^39 + (b5 - 2b3 + b2 - 2b1 + 2) * q^41 + (2b5 + b4 + b2) q^43 + (b3 - b2 + 3b1 + 3) q^47 + (b5 - b4 - b3 + b1 - 1) * q^49 + (b5 + b3 + b2 - b1 + 3) * q^51 + (-b5 + b1 - 2) * q^53 + b1 * q^57 + (-b5 + b4 - b3 - b2 - 2b1) q^59 + (b4 + b3 - 2b2 - b1 + 4) q^61 + (-b5 - 2b4 - b3 - 2) q^63 + (2b5 - b4 + b3 + b2 + 2b1 + 4) * q^67 + (-b4 + b3 - b2 + 4b1) q^69 + (-b4 + 2b3 + b2 + 2b1 + 4) * q^71 + (-b4 + b3 + 2b1 - 4) q^73 + (-2b4 + b3 + b2 + 3b1 + 2) * q^77 + (-b5 - b4 + b3 - b2 + b1 + 4) * q^79 + (2b4 + 2b3 + 3b1 + 7) q^81 + (-b5 - b2 - 2b1 - 2) q^83 + (-b4 + 2b2 + 4b1 + 7) * q^87 + (-b5 + b4 - 2b3 - 2b2 - 2) * q^89 + (2b5 + 2b3 - b2 - 3b1 + 3) q^91 + (b5 + 2b4 + 4b3 - b2 + 2b1 - 2) q^93 + (-2b5 + b4 - b3 - 2b2 - b1 - 2) * q^97 + (b5 - 3b4 - 6b3 + 2b2 - 6b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 2 q^{7} + 10 q^{9} + 3 q^{11} - 3 q^{13} - 2 q^{17} + 6 q^{19} + 11 q^{21} + 4 q^{23} + 20 q^{27} + 7 q^{29} + 5 q^{31} - 16 q^{33} + 8 q^{39} + 11 q^{41} - 7 q^{43} + 20 q^{47} - 2 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 + 2 * q^7 + 10 * q^9 + 3 * q^11 - 3 * q^13 - 2 * q^17 + 6 * q^19 + 11 * q^21 + 4 * q^23 + 20 * q^27 + 7 * q^29 + 5 * q^31 - 16 * q^33 + 8 * q^39 + 11 * q^41 - 7 * q^43 + 20 * q^47 - 2 * q^49 + 13 * q^51 - 7 * q^53 + 2 * q^57 - 4 * q^59 + 13 * q^61 - q^63 + 25 * q^67 + 7 * q^69 + 29 * q^71 - 19 * q^73 + 24 * q^77 + 28 * q^79 + 38 * q^81 - 15 * q^83 + 57 * q^87 - 12 * q^89 - 27 * q^93 - 13 * q^97 + 10 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 2x^{5} - 12x^{4} + 16x^{3} + 33x^{2} - 4x - 7 $ x^6 - 2*x^5 - 12*x^4 + 16*x^3 + 33*x^2 - 4*x - 7 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - \nu - 4 $ v^2 - v - 4
$\beta_{3}$	$=$	$ ( \nu^{3} - 9\nu - 3 ) / 2 $ (v^3 - 9*v - 3) / 2
$\beta_{4}$	$=$	$ ( \nu^{4} - \nu^{3} - 9\nu^{2} + 6\nu + 5 ) / 2 $ (v^4 - v^3 - 9v^2 + 6v + 5) / 2
$\beta_{5}$	$=$	$ ( \nu^{5} - 2\nu^{4} - 11\nu^{3} + 15\nu^{2} + 24\nu - 2 ) / 2 $ (v^5 - 2v^4 - 11v^3 + 15v^2 + 24v - 2) / 2

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + \beta _1 + 4 $ b2 + b1 + 4
$\nu^{3}$	$=$	$ 2\beta_{3} + 9\beta _1 + 3 $ 2b3 + 9b1 + 3
$\nu^{4}$	$=$	$ 2\beta_{4} + 2\beta_{3} + 9\beta_{2} + 12\beta _1 + 34 $ 2b4 + 2b3 + 9b2 + 12b1 + 34
$\nu^{5}$	$=$	$ 2\beta_{5} + 4\beta_{4} + 26\beta_{3} + 3\beta_{2} + 84\beta _1 + 43 $ 2b5 + 4b4 + 26b3 + 3b2 + 84*b1 + 43

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

−2.79951
−1.22174
−0.471016
0.486697
2.70452
3.30105

−2.79951

−0.127550

4.83726

1.2

−1.22174

−3.08602

−1.50735

1.3

−0.471016

−0.567324

−2.77814

1.4

0.486697

3.63249

−2.76313

1.5

2.70452

3.77934

4.31442

1.6

3.30105

−1.63094

7.89694

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$5$	$ +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	3800.2.a.bd	yes	6
4.b	odd	2	1		7600.2.a.ci		6
5.b	even	2	1		3800.2.a.bb	✓	6
5.c	odd	4	2		3800.2.d.p		12
20.d	odd	2	1		7600.2.a.cm		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3800.2.a.bb	✓	6	5.b	even	2	1
3800.2.a.bd	yes	6	1.a	even	1	1	trivial
3800.2.d.p		12	5.c	odd	4	2
7600.2.a.ci		6	4.b	odd	2	1
7600.2.a.cm		6	20.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 $S_{2}^{\mathrm{new}}(\Gamma_0(3800))$:

$ T_{3}^{6} - 2T_{3}^{5} - 12T_{3}^{4} + 16T_{3}^{3} + 33T_{3}^{2} - 4T_{3} - 7 $

$ T_{7}^{6} - 2T_{7}^{5} - 18T_{7}^{4} + 16T_{7}^{3} + 87T_{7}^{2} + 50T_{7} + 5 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - 2 T^{5} + \cdots - 7 $ T^6 - 2T^5 - 12T^4 + 16T^3 + 33T^2 - 4*T - 7
$5$	$ T^{6} $ T^6
$7$	$ T^{6} - 2 T^{5} + \cdots + 5 $ T^6 - 2T^5 - 18T^4 + 16T^3 + 87T^2 + 50*T + 5
$11$	$ T^{6} - 3 T^{5} + \cdots - 5400 $ T^6 - 3T^5 - 52T^4 + 115T^3 + 906T^2 - 1080*T - 5400
$13$	$ T^{6} + 3 T^{5} + \cdots + 37 $ T^6 + 3T^5 - 43T^4 - 61T^3 + 491T^2 - 303*T + 37
$17$	$ T^{6} + 2 T^{5} + \cdots - 745 $ T^6 + 2T^5 - 62T^4 - 182T^3 + 303T^2 + 624*T - 745
$19$	$ (T - 1)^{6} $ (T - 1)^6
$23$	$ T^{6} - 4 T^{5} + \cdots - 587 $ T^6 - 4T^5 - 42T^4 + 114T^3 + 469T^2 - 714*T - 587
$29$	$ T^{6} - 7 T^{5} + \cdots - 14717 $ T^6 - 7T^5 - 73T^4 + 335T^3 + 1957T^2 - 3169*T - 14717
$31$	$ T^{6} - 5 T^{5} + \cdots - 296 $ T^6 - 5T^5 - 70T^4 + 609T^3 - 1528T^2 + 1228*T - 296
$37$	$ T^{6} - 208 T^{4} + \cdots - 111157 $ T^6 - 208T^4 + 189T^3 + 11924T^2 - 19221T - 111157
$41$	$ T^{6} - 11 T^{5} + \cdots - 3584 $ T^6 - 11T^5 - 84T^4 + 639T^3 + 3136T^2 - 960*T - 3584
$43$	$ T^{6} + 7 T^{5} + \cdots - 79576 $ T^6 + 7T^5 - 215T^4 - 975T^3 + 12686T^2 + 18112*T - 79576
$47$	$ T^{6} - 20 T^{5} + \cdots - 90017 $ T^6 - 20T^5 + 12T^4 + 2337T^3 - 21020T^2 + 72555*T - 90017
$53$	$ T^{6} + 7 T^{5} + \cdots - 1187 $ T^6 + 7T^5 - 32T^4 - 181T^3 + 358T^2 + 714*T - 1187
$59$	$ T^{6} + 4 T^{5} + \cdots - 11944 $ T^6 + 4T^5 - 93T^4 - 355T^3 + 1926T^2 + 4496*T - 11944
$61$	$ T^{6} - 13 T^{5} + \cdots - 3880 $ T^6 - 13T^5 - 120T^4 + 1547T^3 + 3438T^2 - 35240*T - 3880
$67$	$ T^{6} - 25 T^{5} + \cdots + 14207 $ T^6 - 25T^5 + 103T^4 + 1603T^3 - 14043T^2 + 26469*T + 14207
$71$	$ T^{6} - 29 T^{5} + \cdots - 39960 $ T^6 - 29T^5 + 177T^4 + 1769T^3 - 22974T^2 + 71064*T - 39960
$73$	$ T^{6} + 19 T^{5} + \cdots + 1723 $ T^6 + 19T^5 + 67T^4 - 377T^3 - 1207T^2 + 2797*T + 1723
$79$	$ T^{6} - 28 T^{5} + \cdots + 15040 $ T^6 - 28T^5 + 185T^4 + 689T^3 - 6260T^2 - 13216*T + 15040
$83$	$ T^{6} + 15 T^{5} + \cdots - 200 $ T^6 + 15T^5 - 2T^4 - 627T^3 - 2100T^2 - 1292*T - 200
$89$	$ T^{6} + 12 T^{5} + \cdots - 274808 $ T^6 + 12T^5 - 217T^4 - 2179T^3 + 14514T^2 + 88364*T - 274808
$97$	$ T^{6} + 13 T^{5} + \cdots + 1784 $ T^6 + 13T^5 - 134T^4 - 2517T^3 - 8524T^2 + 3556*T + 1784
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Level:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} - \beta_{3} q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{11} + \beta_{4} q^{13} + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{17} + q^{19} + ( - \beta_{4} - \beta_{3} + 1) q^{21}+ \cdots + (\beta_{5} - 3 \beta_{4} + \cdots - 6 \beta_1) q^{99}+O(q^{100}) \) q + b1 * q^3 - b3 * q^7 + (b2 + b1 + 1) * q^9 + (-b5 - b3 - b1) * q^11 + b4 * q^13 + (-b5 + b4 - b2) * q^17 + q^19 + (-b4 - b3 + 1) * q^21 + (-b3 + b2) * q^23 + (2b3 + 3b1 + 3) * q^27 + (-b5 + 2b1) q^29 + (b4 + b3 - b1 + 2) * q^31 + (-2b4 - b3 - b2 + b1 - 4) q^33 + (b5 - b4 - 3b3 + b2 - b1 - 1) q^37 + (b5 + b4 + 3b3 + b1 + 3) q^39 + (b5 - 2b3 + b2 - 2b1 + 2) * q^41 + (2b5 + b4 + b2) q^43 + (b3 - b2 + 3b1 + 3) q^47 + (b5 - b4 - b3 + b1 - 1) * q^49 + (b5 + b3 + b2 - b1 + 3) * q^51 + (-b5 + b1 - 2) * q^53 + b1 * q^57 + (-b5 + b4 - b3 - b2 - 2b1) q^59 + (b4 + b3 - 2b2 - b1 + 4) q^61 + (-b5 - 2b4 - b3 - 2) q^63 + (2b5 - b4 + b3 + b2 + 2b1 + 4) * q^67 + (-b4 + b3 - b2 + 4b1) q^69 + (-b4 + 2b3 + b2 + 2b1 + 4) * q^71 + (-b4 + b3 + 2b1 - 4) q^73 + (-2b4 + b3 + b2 + 3b1 + 2) * q^77 + (-b5 - b4 + b3 - b2 + b1 + 4) * q^79 + (2b4 + 2b3 + 3b1 + 7) q^81 + (-b5 - b2 - 2b1 - 2) q^83 + (-b4 + 2b2 + 4b1 + 7) * q^87 + (-b5 + b4 - 2b3 - 2b2 - 2) * q^89 + (2b5 + 2b3 - b2 - 3b1 + 3) q^91 + (b5 + 2b4 + 4b3 - b2 + 2b1 - 2) q^93 + (-2b5 + b4 - b3 - 2b2 - b1 - 2) * q^97 + (b5 - 3b4 - 6b3 + 2b2 - 6b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 2 q^{7} + 10 q^{9} + 3 q^{11} - 3 q^{13} - 2 q^{17} + 6 q^{19} + 11 q^{21} + 4 q^{23} + 20 q^{27} + 7 q^{29} + 5 q^{31} - 16 q^{33} + 8 q^{39} + 11 q^{41} - 7 q^{43} + 20 q^{47} - 2 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 + 2 * q^7 + 10 * q^9 + 3 * q^11 - 3 * q^13 - 2 * q^17 + 6 * q^19 + 11 * q^21 + 4 * q^23 + 20 * q^27 + 7 * q^29 + 5 * q^31 - 16 * q^33 + 8 * q^39 + 11 * q^41 - 7 * q^43 + 20 * q^47 - 2 * q^49 + 13 * q^51 - 7 * q^53 + 2 * q^57 - 4 * q^59 + 13 * q^61 - q^63 + 25 * q^67 + 7 * q^69 + 29 * q^71 - 19 * q^73 + 24 * q^77 + 28 * q^79 + 38 * q^81 - 15 * q^83 + 57 * q^87 - 12 * q^89 - 27 * q^93 - 13 * q^97 + 10 * q^99

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