LMFDB - Newform orbit 4000.2.a.r

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4000,2,Mod(1,4000)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4000, 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("4000.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$4000 = 2^{5} \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4000.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,18,0,0,0,12] 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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.578340050000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 21x^{6} + 129x^{4} - 220x^{2} + 80$ x^8 - 21x^6 + 129x^4 - 220*x^2 + 80
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{2}$
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{3} + \beta_{7} q^{7} + (\beta_{2} + 2) q^{9} + (\beta_{7} - \beta_{5} + \beta_1) q^{11} + (\beta_{6} + 2) q^{13} + (\beta_{6} + \beta_{3} + 1) q^{17} + (\beta_{7} + \beta_{4} + 2 \beta_1) q^{19}+ \cdots + ( - \beta_{7} + \beta_{5} + \cdots + 7 \beta_1) q^{99}+O(q^{100})$ q - b1 * q^3 + b7 * q^7 + (b2 + 2) * q^9 + (b7 - b5 + b1) * q^11 + (b6 + 2) * q^13 + (b6 + b3 + 1) * q^17 + (b7 + b4 + 2b1) q^19 + (-b6 + b3 + b2 + 2) * q^21 + (b7 - b5 + b4) * q^23 + (2b5 - b4 - 3b1) * q^27 + (b6 - b3 + 3) * q^29 + b4 * q^31 + (-b6 - b3 - b2 - 4) * q^33 + (-b6 + b2 + 2) * q^37 + (-2b7 - 3b4 - 4b1) q^39 + (b3 - b2 + 3) * q^41 + (-2b5 + b4 - b1) q^43 + (b7 - b5 - 2b4 - 2b1) * q^47 + (b6 - 4b3 - b2 + 2) q^49 + (-2b7 - 4b4 - 3b1) q^51 + (-b6 + 2b2 + 3) q^53 + (-2b6 - 4b3 - b2 - 9) * q^57 + (b7 + 2b5 + 2b4 + b1) * q^59 + (b6 + 5b3 - b2 + 6) q^61 + (-b7 + 2b5 + b4 - 4b1) * q^63 + (-b4 - 2b1) q^67 + (-2b6 - 6b3) * q^69 + (-2b7 + 2b5 + 5b4 + b1) q^71 + (6b3 - b2 - 3) q^73 + (-2b2 + 5) q^77 + (-4b5 - b4 - 3b1) * q^79 + (b6 + 9b3 + 2b2 + 12) * q^81 + (4b5 + 4b4) * q^83 + (-2b7 - 2b4 - 5b1) q^87 + (-2b6 - 5b3 - b2 - 2) * q^89 + (2b7 + 3b5 - 2b4 + 2b1) * q^91 + (-b6 - 5b3 - 1) q^93 + (2b6 + b2 - 5) q^97 + (-b7 + b5 + 5b4 + 7b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 18 q^{9} + 12 q^{13} + 18 q^{21} + 24 q^{29} - 26 q^{33} + 22 q^{37} + 18 q^{41} + 26 q^{49} + 32 q^{53} - 50 q^{57} + 22 q^{61} + 32 q^{69} - 50 q^{73} + 36 q^{77} + 60 q^{81} + 10 q^{89} + 16 q^{93}+ \cdots - 46 q^{97}+O(q^{100})$ 8 * q + 18 * q^9 + 12 * q^13 + 18 * q^21 + 24 * q^29 - 26 * q^33 + 22 * q^37 + 18 * q^41 + 26 * q^49 + 32 * q^53 - 50 * q^57 + 22 * q^61 + 32 * q^69 - 50 * q^73 + 36 * q^77 + 60 * q^81 + 10 * q^89 + 16 * q^93 - 46 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - 21x^{6} + 129x^{4} - 220x^{2} + 80$ x^8 - 21*x^6 + 129*x^4 - 220*x^2 + 80 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 5$ v^2 - 5
$\beta_{3}$	$=$	$( -\nu^{6} + 25\nu^{4} - 161\nu^{2} + 116 ) / 68$ (-v^6 + 25v^4 - 161v^2 + 116) / 68
$\beta_{4}$	$=$	$( -\nu^{7} + 25\nu^{5} - 161\nu^{3} + 116\nu ) / 68$ (-v^7 + 25v^5 - 161v^3 + 116*v) / 68
$\beta_{5}$	$=$	$( -\nu^{7} + 25\nu^{5} - 229\nu^{3} + 728\nu ) / 136$ (-v^7 + 25v^5 - 229v^3 + 728*v) / 136
$\beta_{6}$	$=$	$( 9\nu^{6} - 157\nu^{4} + 701\nu^{2} - 568 ) / 68$ (9v^6 - 157v^4 + 701*v^2 - 568) / 68
$\beta_{7}$	$=$	$( 3\nu^{7} - 58\nu^{5} + 296\nu^{3} - 263\nu ) / 34$ (3v^7 - 58v^5 + 296v^3 - 263v) / 34

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 5$ b2 + 5
$\nu^{3}$	$=$	$-2\beta_{5} + \beta_{4} + 9\beta_1$ -2b5 + b4 + 9b1
$\nu^{4}$	$=$	$\beta_{6} + 9\beta_{3} + 11\beta_{2} + 48$ b6 + 9b3 + 11b2 + 48
$\nu^{5}$	$=$	$2\beta_{7} - 22\beta_{5} + 23\beta_{4} + 94\beta_1$ 2b7 - 22b5 + 23b4 + 94b1
$\nu^{6}$	$=$	$25\beta_{6} + 157\beta_{3} + 114\beta_{2} + 511$ 25b6 + 157b3 + 114*b2 + 511
$\nu^{7}$	$=$	$50\beta_{7} - 228\beta_{5} + 346\beta_{4} + 1017\beta_1$ 50b7 - 228b5 + 346b4 + 1017b1

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

3.33481
2.72684
1.39511
0.705030
−0.705030
−1.39511
−2.72684
−3.33481

−3.33481

−1.78602

8.12097

1.2

−2.72684

−2.84526

4.43565

1.3

−1.39511

4.73991

−1.05368

1.4

−0.705030

−2.69218

−2.50293

1.5

0.705030

2.69218

−2.50293

1.6

1.39511

−4.73991

−1.05368

1.7

2.72684

2.84526

4.43565

1.8

3.33481

1.78602

8.12097

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$+1$

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4000.2.a.r	yes	8
4.b	odd	2	1	inner	4000.2.a.r	yes	8
5.b	even	2	1		4000.2.a.q	✓	8
5.c	odd	4	2		4000.2.c.h		16
8.b	even	2	1		8000.2.a.ca		8
8.d	odd	2	1		8000.2.a.ca		8
20.d	odd	2	1		4000.2.a.q	✓	8
20.e	even	4	2		4000.2.c.h		16
40.e	odd	2	1		8000.2.a.cb		8
40.f	even	2	1		8000.2.a.cb		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4000.2.a.q	✓	8	5.b	even	2	1
4000.2.a.q	✓	8	20.d	odd	2	1
4000.2.a.r	yes	8	1.a	even	1	1	trivial
4000.2.a.r	yes	8	4.b	odd	2	1	inner
4000.2.c.h		16	5.c	odd	4	2
4000.2.c.h		16	20.e	even	4	2
8000.2.a.ca		8	8.b	even	2	1
8000.2.a.ca		8	8.d	odd	2	1
8000.2.a.cb		8	40.e	odd	2	1
8000.2.a.cb		8	40.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4000))$ :

T_{3}^{8} - 21T_{3}^{6} + 129T_{3}^{4} - 220T_{3}^{2} + 80

T3^8 - 21*T3^6 + 129*T3^4 - 220*T3^2 + 80

T_{7}^{8} - 41T_{7}^{6} + 524T_{7}^{4} - 2605T_{7}^{2} + 4205

T7^8 - 41*T7^6 + 524*T7^4 - 2605*T7^2 + 4205

T_{11}^{8} - 49T_{11}^{6} + 684T_{11}^{4} - 2445T_{11}^{2} + 5

T11^8 - 49*T11^6 + 684*T11^4 - 2445*T11^2 + 5

T_{13}^{4} - 6T_{13}^{3} - 17T_{13}^{2} + 78T_{13} + 149

T13^4 - 6*T13^3 - 17*T13^2 + 78*T13 + 149

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$T^{8} - 21 T^{6} + \cdots + 80$ T^8 - 21T^6 + 129T^4 - 220*T^2 + 80
$5$	$T^{8}$ T^8
$7$	$T^{8} - 41 T^{6} + \cdots + 4205$ T^8 - 41T^6 + 524T^4 - 2605*T^2 + 4205
$11$	$T^{8} - 49 T^{6} + \cdots + 5$ T^8 - 49T^6 + 684T^4 - 2445*T^2 + 5
$13$	$(T^{4} - 6 T^{3} + \cdots + 149)^{2}$ (T^4 - 6T^3 - 17T^2 + 78*T + 149)^2
$17$	$(T^{4} - 33 T^{2} + \cdots + 176)^{2}$ (T^4 - 33T^2 + 20T + 176)^2
$19$	$T^{8} - 80 T^{6} + \cdots + 3125$ T^8 - 80T^6 + 1645T^4 - 5500*T^2 + 3125
$23$	$T^{8} - 91 T^{6} + \cdots + 20480$ T^8 - 91T^6 + 2329T^4 - 15680*T^2 + 20480
$29$	$(T^{4} - 12 T^{3} + \cdots + 20)^{2}$ (T^4 - 12T^3 + 21T^2 + 70*T + 20)^2
$31$	$T^{8} - 29 T^{6} + \cdots + 80$ T^8 - 29T^6 + 209T^4 - 460*T^2 + 80
$37$	$(T^{4} - 11 T^{3} + \cdots - 320)^{2}$ (T^4 - 11T^3 - 11T^2 + 280*T - 320)^2
$41$	$(T^{4} - 9 T^{3} - 6 T^{2} + \cdots + 95)^{2}$ (T^4 - 9T^3 - 6T^2 + 95*T + 95)^2
$43$	$T^{8} - 234 T^{6} + \cdots + 3872000$ T^8 - 234T^6 + 17109T^4 - 473200*T^2 + 3872000
$47$	$T^{8} - 176 T^{6} + \cdots + 17405$ T^8 - 176T^6 + 8069T^4 - 51460*T^2 + 17405
$53$	$(T^{4} - 16 T^{3} + \cdots + 2501)^{2}$ (T^4 - 16T^3 - 59T^2 + 1084*T + 2501)^2
$59$	$T^{8} - 196 T^{6} + \cdots + 85805$ T^8 - 196T^6 + 11809T^4 - 202760*T^2 + 85805
$61$	$(T^{4} - 11 T^{3} + \cdots - 2620)^{2}$ (T^4 - 11T^3 - 61T^2 + 980*T - 2620)^2
$67$	$T^{8} - 81 T^{6} + \cdots + 80$ T^8 - 81T^6 + 369T^4 - 380*T^2 + 80
$71$	$T^{8} - 694 T^{6} + \cdots + 849686480$ T^8 - 694T^6 + 178669T^4 - 20224700*T^2 + 849686480
$73$	$(T^{4} + 25 T^{3} + \cdots - 4124)^{2}$ (T^4 + 25T^3 + 123T^2 - 670*T - 4124)^2
$79$	$T^{8} - 546 T^{6} + \cdots + 3494480$ T^8 - 546T^6 + 78069T^4 - 1154440*T^2 + 3494480
$83$	$T^{8} - 576 T^{6} + \cdots + 8192000$ T^8 - 576T^6 + 90624T^4 - 2867200*T^2 + 8192000
$89$	$(T^{4} - 5 T^{3} + \cdots - 2000)^{2}$ (T^4 - 5T^3 - 245T^2 + 1900*T - 2000)^2
$97$	$(T^{4} + 23 T^{3} + \cdots - 244)^{2}$ (T^4 + 23T^3 + 19T^2 - 1418*T - 244)^2
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