LMFDB - Newform orbit 3364.2.a.r

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3364,2,Mod(1,3364)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3364, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3364.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3364 = 2^{2} \cdot 29^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3364.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:	$26.8616752400$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.3266578125.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{7} - 10x^{6} + 6x^{5} + 29x^{4} - 9x^{3} - 25x^{2} - x + 1$ x^8 - x^7 - 10x^6 + 6x^5 + 29x^4 - 9x^3 - 25*x^2 - x + 1
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots + 1) q^{3} + ( - \beta_{6} - \beta_{5} - 2 \beta_{4} + \cdots - 1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots - \beta_1) q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} + \cdots - 4) q^{99}+O(q^{100})$ q + (b6 + b5 + b4 + b3 + 1) * q^3 + (-b6 - b5 - 2b4 - b3 - b1 - 1) q^5 + (b7 + b6 + b5 + b2 - b1) * q^7 + (-b7 + b6 + b4 + b3 - b2 - b1 + 1) * q^9 + (-b7 - b6 - b5 - b4 + b2 + 1) * q^11 + (b7 + b6 + b5 + 2b2 + b1 + 1) q^13 + (2b7 + b6 - 2b4 - b3 + b2 - b1 - 3) * q^15 + (b4 + b2 - b1 + 3) * q^17 + (b5 + b3 - b2 - b1 + 1) * q^19 + (-2b5 - 3b4 - 3b3 - 3b1 + 1) * q^21 + (-b7 + 3b3 - b2) q^23 + (-2b7 + 5b4 + 3b3 - b2 + 3b1 + 3) * q^25 + (b7 + b6 - b5 - 2b4 + b3 - b2 - b1 + 1) q^27 + (b7 - b6 + b5 - 2b4 - b3 - b2 - 2b1 - 3) * q^31 + (b7 + b6 + 3b5 - b4 - 2b3 + 2b2 + b1 - 3) q^33 + (2b6 + 4b5 + 6b4 + 5b3 + 3b1 + 4) q^35 + (-b7 - b6 - 2b5 - 5b4 - b3 + b2 + b1 - 2) * q^37 + (-2b7 - 2b6 - b5 - 2b4 - 5b3 + b2 - b1) * q^39 + (b6 + 2b5 + 3b4 - b3 - b2 + 5) * q^41 + (2b6 + 3b5 + 7b4 + 2b3 - b1 + 3) * q^43 + (b7 + b6 - 3b5 - b4 - 3b3 - 2b1) q^45 + (4b7 - b6 - b5 - b4 - b3 + 2b2 + b1 - 1) * q^47 + (-2b7 - b6 + b5 + 4b4 - b3 - b2 - 2b1 + 3) q^49 + (b7 + 2b6 + 3b5 + b2) * q^51 + (-2b6 + b4 - 2b3 - 2b2 + 3b1 - 2) * q^53 + (-2b7 - 2b6 - 3b5 + b4 + 2b3 + 2b1 + 6) q^55 + (3b6 + 3b4 + 4b3 - 2b2 - 2b1 + 6) q^57 + (-2b7 + 3b6 + 2b5 - 2b2 - b1 - 2) * q^59 + (-b7 - 5b6 - 2b5 - 5b4 - 2b2 + 2) * q^61 + (2b7 + 4b6 + b5 - 4b4 + b3 - 3) q^63 + (-b7 - b6 + 4b5 + b4 + 5b3 + 2b1 - 2) q^65 + (-2b6 - 2b5 - 2b4 + 4b3 + 2b1 - 3) q^67 + (2b6 - b5 + 2b4 + 3b3 - 3b2 - 2b1 + 5) q^69 + (-b7 + b5 - 2b4 - 2b3 - 2b2 - b1 + 4) q^71 + (-b7 - 4b6 - 5b5 - 7b4 - b2 + b1 - 7) q^73 + (-3b7 - 2b6 + 4b5 + 8b4 + 6b3 - 2b2 + 7b1 + 3) q^75 + (b7 + 2b6 + 6b5 + b4 + b3 + 2b2 + 3b1 + 1) * q^77 + (2b6 - b5 - 2b4 - 2b3 - b2 + b1 + 2) q^79 + (5b7 + 2b6 - 2b5 - 3b4 + b3 - 3b1 + 2) q^81 + (4b7 + 3b6 + b5 + 5b4 + b3 + b2 - 5b1 + 1) * q^83 + (-b7 - 2b6 - 2b5 - b1 + 3) * q^85 + (-3b7 + 2b6 + b5 + 5b3 + 3b1 + 4) * q^89 + (-4b7 - 3b6 + b4 - 4b3 - 2b1 + 7) * q^91 + (b7 - 4b5 - b4 - b2 - 3b1 + 1) * q^93 + (4b7 + 2b6 - 2b5 - 3b4 - 3b3 + 2b2 - b1) * q^95 + (2b7 - 2b6 - b5 - 2b4 - 5b3 - b2) * q^97 + (-b7 - 2b6 - b4 - 9b3 - 4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 4 q^{3} - q^{5} - 2 q^{7} + 4 q^{9} + 5 q^{11} + 4 q^{13} - 17 q^{15} + 15 q^{17} + 11 q^{19} + 19 q^{21} - 3 q^{23} + 3 q^{25} + 16 q^{27} - 9 q^{31} - 16 q^{33} + 9 q^{35} - 2 q^{37} + 9 q^{39}+ \cdots - 11 q^{99}+O(q^{100})$ 8 * q + 4 * q^3 - q^5 - 2 * q^7 + 4 * q^9 + 5 * q^11 + 4 * q^13 - 17 * q^15 + 15 * q^17 + 11 * q^19 + 19 * q^21 - 3 * q^23 + 3 * q^25 + 16 * q^27 - 9 * q^31 - 16 * q^33 + 9 * q^35 - 2 * q^37 + 9 * q^39 + 38 * q^41 - 3 * q^43 + 3 * q^45 - 7 * q^47 + 12 * q^49 + 3 * q^51 - 5 * q^53 + 34 * q^55 + 34 * q^57 - 7 * q^59 + 39 * q^61 - 6 * q^63 - 21 * q^65 - 26 * q^67 + 34 * q^69 + 52 * q^71 - 34 * q^73 + 10 * q^77 + 31 * q^79 + 24 * q^81 - 17 * q^83 + 18 * q^85 + 24 * q^89 + 54 * q^91 + 6 * q^93 + 9 * q^95 + 22 * q^97 - 11 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - x^{7} - 10x^{6} + 6x^{5} + 29x^{4} - 9x^{3} - 25x^{2} - x + 1$ x^8 - x^7 - 10*x^6 + 6*x^5 + 29*x^4 - 9*x^3 - 25*x^2 - x + 1 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{7} - \nu^{6} - 8\nu^{5} + 5\nu^{4} + 14\nu^{3} - 11\nu^{2} - 4\nu + 7 ) / 3$ (v^7 - v^6 - 8v^5 + 5v^4 + 14v^3 - 11v^2 - 4*v + 7) / 3
$\beta_{3}$	$=$	$( \nu^{6} + \nu^{5} - 9\nu^{4} - 13\nu^{3} + 15\nu^{2} + 25\nu + 1 ) / 3$ (v^6 + v^5 - 9v^4 - 13v^3 + 15v^2 + 25v + 1) / 3
$\beta_{4}$	$=$	$( \nu^{7} + \nu^{6} - 9\nu^{5} - 13\nu^{4} + 15\nu^{3} + 28\nu^{2} + \nu - 6 ) / 3$ (v^7 + v^6 - 9v^5 - 13v^4 + 15v^3 + 28v^2 + v - 6) / 3
$\beta_{5}$	$=$	$( \nu^{7} - 2\nu^{6} - 9\nu^{5} + 14\nu^{4} + 24\nu^{3} - 23\nu^{2} - 17\nu + 3 ) / 3$ (v^7 - 2v^6 - 9v^5 + 14v^4 + 24v^3 - 23v^2 - 17v + 3) / 3
$\beta_{6}$	$=$	$( \nu^{7} - \nu^{6} - 11\nu^{5} + 8\nu^{4} + 35\nu^{3} - 17\nu^{2} - 31\nu + 1 ) / 3$ (v^7 - v^6 - 11v^5 + 8v^4 + 35v^3 - 17v^2 - 31*v + 1) / 3
$\beta_{7}$	$=$	$( -2\nu^{7} + 20\nu^{5} + 5\nu^{4} - 50\nu^{3} - 8\nu^{2} + 30\nu - 4 ) / 3$ (-2v^7 + 20v^5 + 5v^4 - 50v^3 - 8v^2 + 30v - 4) / 3

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{7} + \beta_{6} + \beta_{4} + 3$ b7 + b6 + b4 + 3
$\nu^{3}$	$=$	$\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 4\beta _1 + 2$ b7 + b6 - b5 + b4 - b3 + b2 + 4*b1 + 2
$\nu^{4}$	$=$	$7\beta_{7} + 8\beta_{6} - 2\beta_{5} + 6\beta_{4} + 2\beta_{2} + 2\beta _1 + 16$ 7b7 + 8b6 - 2b5 + 6b4 + 2b2 + 2b1 + 16
$\nu^{5}$	$=$	$12\beta_{7} + 12\beta_{6} - 9\beta_{5} + 11\beta_{4} - 7\beta_{3} + 10\beta_{2} + 21\beta _1 + 22$ 12b7 + 12b6 - 9b5 + 11b4 - 7b3 + 10b2 + 21*b1 + 22
$\nu^{6}$	$=$	$49\beta_{7} + 58\beta_{6} - 22\beta_{5} + 41\beta_{4} - 3\beta_{3} + 21\beta_{2} + 24\beta _1 + 102$ 49b7 + 58b6 - 22b5 + 41b4 - 3b3 + 21b2 + 24*b1 + 102
$\nu^{7}$	$=$	$107\beta_{7} + 111\beta_{6} - 70\beta_{5} + 96\beta_{4} - 45\beta_{3} + 80\beta_{2} + 130\beta _1 + 196$ 107b7 + 111b6 - 70b5 + 96b4 - 45b3 + 80b2 + 130*b1 + 196

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

−0.944572
1.58001
0.178891
2.77166
−1.72594
−0.241749
1.51688
−2.13519

−2.54007

2.86661

−4.47762

3.45197

1.2

−0.729419

0.767443

−2.02171

−2.46795

1.3

−0.498509

1.93765

0.267484

−2.75149

1.4

−0.243314

−3.14638

−0.923802

−2.94080

1.5

0.271575

0.836327

3.75591

−2.92625

1.6

2.21825

−0.358466

3.12857

1.92063

1.7

2.27574

−4.41066

−3.82663

2.17901

1.8

3.24575

0.507473

2.09779

7.53487

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$29$	$+1$

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	3364.2.a.r	yes	8
29.b	even	2	1		3364.2.a.q	✓	8
29.c	odd	4	2		3364.2.c.k		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3364.2.a.q	✓	8	29.b	even	2	1
3364.2.a.r	yes	8	1.a	even	1	1	trivial
3364.2.c.k		16	29.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 $S_{2}^{\mathrm{new}}(\Gamma_0(3364))$ :

T_{3}^{8} - 4T_{3}^{7} - 6T_{3}^{6} + 32T_{3}^{5} - T_{3}^{4} - 41T_{3}^{3} - 14T_{3}^{2} + 3T_{3} + 1

T_{5}^{8} + T_{5}^{7} - 21T_{5}^{6} + 7T_{5}^{5} + 109T_{5}^{4} - 156T_{5}^{3} + 51T_{5}^{2} + 18T_{5} - 9

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$T^{8} - 4 T^{7} + \cdots + 1$ T^8 - 4T^7 - 6T^6 + 32T^5 - T^4 - 41T^3 - 14T^2 + 3T + 1
$5$	$T^{8} + T^{7} - 21 T^{6} + \cdots - 9$ T^8 + T^7 - 21T^6 + 7T^5 + 109T^4 - 156T^3 + 51T^2 + 18T - 9
$7$	$T^{8} + 2 T^{7} + \cdots + 211$ T^8 + 2T^7 - 32T^6 - 46T^5 + 315T^4 + 289T^3 - 887T^2 - 578*T + 211
$11$	$T^{8} - 5 T^{7} + \cdots + 5931$ T^8 - 5T^7 - 43T^6 + 155T^5 + 739T^4 - 1230T^3 - 5262T^2 - 135*T + 5931
$13$	$T^{8} - 4 T^{7} + \cdots + 20851$ T^8 - 4T^7 - 65T^6 + 239T^5 + 1239T^4 - 3716T^3 - 8180T^2 + 13966*T + 20851
$17$	$T^{8} - 15 T^{7} + \cdots + 81$ T^8 - 15T^7 + 63T^6 + 45T^5 - 891T^4 + 1890T^3 - 1053T^2 + 81
$19$	$T^{8} - 11 T^{7} + \cdots - 549$ T^8 - 11T^7 - T^6 + 308T^5 - 851T^4 - 144T^3 + 1821T^2 + 117T - 549
$23$	$T^{8} + 3 T^{7} + \cdots - 31599$ T^8 + 3T^7 - 79T^6 - 276T^5 + 1234T^4 + 5487T^3 - 2541T^2 - 31059*T - 31599
$29$	$T^{8}$ T^8
$31$	$T^{8} + 9 T^{7} + \cdots - 134009$ T^8 + 9T^7 - 65T^6 - 604T^5 + 944T^4 + 12961T^3 + 5715T^2 - 90161*T - 134009
$37$	$T^{8} + 2 T^{7} + \cdots + 16651$ T^8 + 2T^7 - 124T^6 - 244T^5 + 4219T^4 + 7363T^3 - 24406T^2 - 6401*T + 16651
$41$	$T^{8} - 38 T^{7} + \cdots - 1341909$ T^8 - 38T^7 + 548T^6 - 3301T^5 - 185T^4 + 114339T^3 - 642072T^2 + 1509642*T - 1341909
$43$	$T^{8} + 3 T^{7} + \cdots - 8009$ T^8 + 3T^7 - 135T^6 - 372T^5 + 4589T^4 + 11517T^3 - 3465T^2 - 18708*T - 8009
$47$	$T^{8} + 7 T^{7} + \cdots + 8745291$ T^8 + 7T^7 - 274T^6 - 1714T^5 + 24304T^4 + 128733T^3 - 790296T^2 - 2745441*T + 8745291
$53$	$T^{8} + 5 T^{7} + \cdots + 5291775$ T^8 + 5T^7 - 300T^6 - 1240T^5 + 28930T^4 + 90300T^3 - 950475T^2 - 1959075*T + 5291775
$59$	$T^{8} + 7 T^{7} + \cdots - 36189$ T^8 + 7T^7 - 280T^6 - 1423T^5 + 22759T^4 + 60663T^3 - 406695T^2 + 437508*T - 36189
$61$	$T^{8} - 39 T^{7} + \cdots + 15536221$ T^8 - 39T^7 + 415T^6 + 1499T^5 - 50536T^4 + 205894T^3 + 924645T^2 - 8264039*T + 15536221
$67$	$T^{8} + 26 T^{7} + \cdots - 4409$ T^8 + 26T^7 + 94T^6 - 2758T^5 - 33336T^4 - 149186T^3 - 288494T^2 - 189842*T - 4409
$71$	$T^{8} - 52 T^{7} + \cdots - 5139$ T^8 - 52T^7 + 1055T^6 - 10337T^5 + 47104T^4 - 60048T^3 - 131370T^2 + 84807*T - 5139
$73$	$T^{8} + 34 T^{7} + \cdots - 744749$ T^8 + 34T^7 + 280T^6 - 2039T^5 - 38196T^4 - 124279T^3 + 225760T^2 + 807959*T - 744749
$79$	$T^{8} - 31 T^{7} + \cdots + 1438231$ T^8 - 31T^7 + 219T^6 + 2048T^5 - 33331T^4 + 109636T^3 + 253831T^2 - 1698093*T + 1438231
$83$	$T^{8} + 17 T^{7} + \cdots - 6356619$ T^8 + 17T^7 - 294T^6 - 7129T^5 - 8006T^4 + 659853T^3 + 4915794T^2 + 9409374*T - 6356619
$89$	$T^{8} - 24 T^{7} + \cdots + 1672641$ T^8 - 24T^7 - 25T^6 + 3879T^5 - 16196T^4 - 116376T^3 + 350745T^2 + 1868616*T + 1672641
$97$	$T^{8} - 22 T^{7} + \cdots + 343171$ T^8 - 22T^7 - 25T^6 + 2333T^5 - 1721T^4 - 82523T^3 + 34550T^2 + 964897*T + 343171
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
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