LMFDB - Newform orbit 936.2.q.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [936,2,Mod(313,936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(936, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))

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

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

chi := DirichletCharacter("936.313");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$936 = 2^{3} \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	936.q (of order $3$ , degree $2$ , minimal)

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:	no
Analytic conductor:	$7.47399762919$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - 6 x^{11} + 29 x^{10} - 90 x^{9} + 217 x^{8} - 394 x^{7} + 555 x^{6} - 598 x^{5} + 483 x^{4} + \cdots + 3$ x^12 - 6x^11 + 29x^10 - 90x^9 + 217x^8 - 394x^7 + 555x^6 - 598x^5 + 483x^4 - 282x^3 + 112x^2 - 27*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$3^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{5} q^{3} + ( - \beta_{11} + \beta_{2}) q^{5} + (\beta_{8} - \beta_{7} + \beta_{6} + \cdots - 1) q^{7} + ( - \beta_{8} - \beta_{6} - \beta_{5} + \cdots + 2) q^{9} + ( - \beta_{10} + \beta_{9} + \cdots - 2 \beta_{4}) q^{11}+ \cdots + (\beta_{11} - 4 \beta_{10} + \beta_{9} + \cdots - 2) q^{99}+O(q^{100})$ q + b5 * q^3 + (-b11 + b2) * q^5 + (b8 - b7 + b6 + b5 + b1 - 1) * q^7 + (-b8 - b6 - b5 - b3 + b2 - 2b1 + 2) q^9 + (-b10 + b9 + b5 - 2b4) q^11 + (b10 - 1) * q^13 + (-b10 - b9 + b8 - b7 - b4 - b3 + b2 + b1) * q^15 + (b9 + b7 + b6 + b5 + b2 - 2b1 + 2) q^17 + (-b7 - b5 - b3 + b1 - 2) * q^19 + (-b9 - b6 - b4 - b3 - 2b1) q^21 + (-3b10 + b8 - b4 - b3 + 3) q^23 + (b11 + b10 - 2b9 + b8 - b7 + b6 - b5 + b1 - 1) q^25 + (2b10 + b9 + b8 + 2b7 + b6 + b4 + b3 + b2 + 2b1 - 3) q^27 + (-b11 - b8 + 3b4 - b1 + 1) q^29 + (-b11 - 2b10 - 2b9 + 3b8 - 2b7 + 2b5 - b4 - b3 + b2 + 2) q^31 + (-2b11 + b10 - 2b9 + b8 - 2b7 - b5 + 2b4 - b3 + b2 - 1) * q^33 + (b9 + b6 + b3 + 2b1 - 2) q^35 + (b9 - b7 + b6 - b5 + 2b3 + 3b2 - b1 - 4) * q^37 + (-b6 - b5) * q^39 + (2b11 + 2b10 + b9 + 3b8 - b7 - 2b6 - b5 - 2b4 - 2b3 - 2b2 - 2) q^41 + (-b11 + b10 - 2b8 - b4 - 2b1 + 2) * q^43 + (b11 - 3b10 - 3b9 + b8 + b6 + b5 + b4 - b1 + 4) * q^45 + (-2b11 - b10 - b9 + b8 - b7 + b6 + 3b4 + b1 - 1) * q^47 + (b11 - 3b10 - 3b9 + 4b8 - 3b7 + 3b5 - 2b4 - 2b3 - b2 + 3) q^49 + (-2b11 + 5b10 + b9 + 2b8 + 2b7 - b6 - b4 - 2b3 + b2 + b1 - 2) q^51 + (-b7 - b5 - 2b3 + 2b2 - 2b1 + 4) q^53 + (-b9 - 3b7 - b6 - 3b5 - b3 + b2 + b1 - 2) * q^55 + (b11 - b9 + b8 + b6 + 2b3 - 2) q^57 + (2b9 + b8 - 2b6 - 2b5 - 3b4 - 3b3) q^59 + (2b11 + 2b10 - 3b9 - b7 + b6 - 2b5 + 3b4) q^61 + (-2b11 - b10 + b9 - b8 + 2b7 - 3b5 + 2b4 + b3 + b2 + b1 + 1) * q^63 + b11 * q^65 + (b11 - b10 - 3b7 - 3b6 + 2b4 + 2b3 - b2 + 1) * q^67 + (-b11 + b10 - b9 + b7 + 3b6 + 3b5 + b4 + b3 + 1) * q^69 + (-b7 - b5 + b2 - 2b1 - 3) q^71 + (-3b9 - 2b7 - 3b6 - 2b5 - 3b3 - 3b2 - 3b1 + 3) q^73 + (b11 - 5b10 - b9 + 2b8 + b7 + 2b5 + b3 - 3b2 + b1 + 3) * q^75 + (3b8 - 2b7 - 2b6 - 2b4 - 2b3) q^77 + (-b11 + 5b10 + 2b9 + b8 + 2b5 - 2b4 + b1 - 1) * q^79 + (3b11 + 2b10 - 2b9 + b8 - b7 - 2b6 - 2b4 - 2b3 - 2b2 - b1 + 6) q^81 + (-b11 + 8b10 + 2b9 + 2b8 + b7 - b6 + b5 - 2b4 + 2b1 - 2) q^83 + (-3b11 - 3b10 - b9 - 2b7 - b6 + b5 - b4 - b3 + 3b2 + 3) * q^85 + (b11 + 2b10 + 4b9 - 2b8 + 2b7 - 4b4 + 2b2 - b1 - 3) * q^87 + (3b9 - b7 + 3b6 - b5 - 3b3 + 2b2 + 2b1 - 1) q^89 + (-b9 - b6 - b1 + 1) * q^91 + (-b11 - 4b10 - 2b9 - 3b8 + 2b7 - 2b3 + b2 - 3b1 + 5) * q^93 + (3b11 - b8 + 2b7 + 2b6 + 2b4 + 2b3 - 3b2) * q^95 + (b11 + b10 - b9 - 5b8 + b7 - b6 - 2b5 + b4 - 5b1 + 5) q^97 + (b11 - 4b10 + b9 - b8 + 2b7 - 4b4 + b3 + b2 + b1 - 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 4 q^{3} - q^{5} + 2 q^{7} + 2 q^{9} - q^{11} - 6 q^{13} - 2 q^{15} + 12 q^{17} - 14 q^{19} - 24 q^{21} + 19 q^{23} + 5 q^{25} - 7 q^{27} - 2 q^{29} + 8 q^{31} - 9 q^{33} + 2 q^{35} - 34 q^{37}+ \cdots - 39 q^{99}+O(q^{100})$ 12 * q - 4 * q^3 - q^5 + 2 * q^7 + 2 * q^9 - q^11 - 6 * q^13 - 2 * q^15 + 12 * q^17 - 14 * q^19 - 24 * q^21 + 19 * q^23 + 5 * q^25 - 7 * q^27 - 2 * q^29 + 8 * q^31 - 9 * q^33 + 2 * q^35 - 34 * q^37 - q^39 - 2 * q^41 + 15 * q^43 + 10 * q^45 - 9 * q^47 + 12 * q^49 + 4 * q^51 + 32 * q^53 - 10 * q^55 - 14 * q^57 + 5 * q^59 + 6 * q^61 + 18 * q^63 - q^65 + 8 * q^67 + 15 * q^69 - 42 * q^71 - 2 * q^73 + 10 * q^75 + 3 * q^77 + 34 * q^79 + 62 * q^81 + 44 * q^83 + 7 * q^85 - 21 * q^87 + 22 * q^89 - 4 * q^91 - 18 * q^93 + 6 * q^95 + 12 * q^97 - 39 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} - 6 x^{11} + 29 x^{10} - 90 x^{9} + 217 x^{8} - 394 x^{7} + 555 x^{6} - 598 x^{5} + 483 x^{4} + \cdots + 3$ x^12 - 6*x^11 + 29*x^10 - 90*x^9 + 217*x^8 - 394*x^7 + 555*x^6 - 598*x^5 + 483*x^4 - 282*x^3 + 112*x^2 - 27*x + 3 :

$\beta_{1}$	$=$	$- 3 \nu^{10} + 15 \nu^{9} - 71 \nu^{8} + 194 \nu^{7} - 433 \nu^{6} + 683 \nu^{5} - 832 \nu^{4} + \cdots - 24$ -3v^10 + 15v^9 - 71v^8 + 194v^7 - 433v^6 + 683v^5 - 832v^4 + 722v^3 - 429v^2 + 154v - 24
$\beta_{2}$	$=$	$5 \nu^{10} - 25 \nu^{9} + 119 \nu^{8} - 326 \nu^{7} + 735 \nu^{6} - 1169 \nu^{5} + 1455 \nu^{4} + \cdots + 58$ 5v^10 - 25v^9 + 119v^8 - 326v^7 + 735v^6 - 1169v^5 + 1455v^4 - 1292v^3 + 812v^2 - 314v + 58
$\beta_{3}$	$=$	$- 6 \nu^{10} + 30 \nu^{9} - 142 \nu^{8} + 388 \nu^{7} - 867 \nu^{6} + 1369 \nu^{5} - 1678 \nu^{4} + \cdots - 59$ -6v^10 + 30v^9 - 142v^8 + 388v^7 - 867v^6 + 1369v^5 - 1678v^4 + 1467v^3 - 896v^2 + 335v - 59
$\beta_{4}$	$=$	$( - 50 \nu^{11} + 314 \nu^{10} - 1514 \nu^{9} + 4796 \nu^{8} - 11585 \nu^{7} + 21207 \nu^{6} + \cdots + 742 ) / 13$ (-50v^11 + 314v^10 - 1514v^9 + 4796v^8 - 11585v^7 + 21207v^6 - 29724v^5 + 31704v^4 - 24782v^3 + 13513v^2 - 4596*v + 742) / 13
$\beta_{5}$	$=$	$( 118 \nu^{11} - 610 \nu^{10} + 2896 \nu^{9} - 8119 \nu^{8} + 18407 \nu^{7} - 29963 \nu^{6} + 37814 \nu^{5} + \cdots - 149 ) / 13$ (118v^11 - 610v^10 + 2896v^9 - 8119v^8 + 18407v^7 - 29963v^6 + 37814v^5 - 34731v^4 + 22657v^3 - 9565v^2 + 2187*v - 149) / 13
$\beta_{6}$	$=$	$( 131 \nu^{11} - 688 \nu^{10} + 3273 \nu^{9} - 9289 \nu^{8} + 21228 \nu^{7} - 35085 \nu^{6} + 45029 \nu^{5} + \cdots - 461 ) / 13$ (131v^11 - 688v^10 + 3273v^9 - 9289v^8 + 21228v^7 - 35085v^6 + 45029v^5 - 42505v^4 + 28936v^3 - 13231v^2 + 3643*v - 461) / 13
$\beta_{7}$	$=$	$( - 118 \nu^{11} + 688 \nu^{10} - 3286 \nu^{9} + 9965 \nu^{8} - 23451 \nu^{7} + 41234 \nu^{6} + \cdots + 942 ) / 13$ (-118v^11 + 688v^10 - 3286v^9 + 9965v^8 - 23451v^7 + 41234v^6 - 55611v^5 + 56545v^4 - 41728v^3 + 21226v^2 - 6555*v + 942) / 13
$\beta_{8}$	$=$	$( - 135 \nu^{11} + 762 \nu^{10} - 3638 \nu^{9} + 10825 \nu^{8} - 25293 \nu^{7} + 43761 \nu^{6} + \cdots + 836 ) / 13$ (-135v^11 + 762v^10 - 3638v^9 + 10825v^8 - 25293v^7 + 43761v^6 - 58342v^5 + 58267v^4 - 42214v^3 + 20889v^2 - 6229*v + 836) / 13
$\beta_{9}$	$=$	$( - 131 \nu^{11} + 753 \nu^{10} - 3598 \nu^{9} + 10823 \nu^{8} - 25414 \nu^{7} + 44393 \nu^{6} + \cdots + 981 ) / 13$ (-131v^11 + 753v^10 - 3598v^9 + 10823v^8 - 25414v^7 + 44393v^6 - 59667v^5 + 60263v^4 - 44289v^3 + 22331v^2 - 6906*v + 981) / 13
$\beta_{10}$	$=$	$( - 196 \nu^{11} + 1078 \nu^{10} - 5132 \nu^{9} + 15009 \nu^{8} - 34722 \nu^{7} + 59031 \nu^{6} + \cdots + 968 ) / 13$ (-196v^11 + 1078v^10 - 5132v^9 + 15009v^8 - 34722v^7 + 59031v^6 - 77425v^5 + 75616v^4 - 53389v^3 + 25594v^2 - 7387*v + 968) / 13
$\beta_{11}$	$=$	$( - 193 \nu^{11} + 1094 \nu^{10} - 5232 \nu^{9} + 15625 \nu^{8} - 36649 \nu^{7} + 63756 \nu^{6} + \cdots + 1496 ) / 13$ (-193v^11 + 1094v^10 - 5232v^9 + 15625v^8 - 36649v^7 + 63756v^6 - 85650v^5 + 86486v^4 - 63782v^3 + 32506v^2 - 10199*v + 1496) / 13

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

$\nu$	$=$	$( \beta_{10} - \beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{3} - \beta _1 + 2 ) / 3$ (b10 - b9 - 2b8 + b7 + b6 - b5 + 2b4 + b3 - b1 + 2) / 3
$\nu^{2}$	$=$	$( \beta_{10} - \beta_{9} - 2\beta_{8} + 4\beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{4} + 4\beta_{3} - \beta _1 - 4 ) / 3$ (b10 - b9 - 2b8 + 4b7 + b6 + 2b5 + 2b4 + 4*b3 - b1 - 4) / 3
$\nu^{3}$	$=$	$( 2 \beta_{11} - 9 \beta_{10} + 6 \beta_{9} + 6 \beta_{8} + 3 \beta_{7} - 6 \beta_{6} + 6 \beta_{5} + \cdots - 6 ) / 3$ (2b11 - 9b10 + 6b9 + 6b8 + 3b7 - 6b6 + 6b5 - 13b4 - 2b3 - b2 + 3b1 - 6) / 3
$\nu^{4}$	$=$	$( 4 \beta_{11} - 19 \beta_{10} + 7 \beta_{9} + 14 \beta_{8} - 22 \beta_{7} - 19 \beta_{6} - 14 \beta_{5} + \cdots + 28 ) / 3$ (4b11 - 19b10 + 7b9 + 14b8 - 22b7 - 19b6 - 14b5 - 28b4 - 38b3 - 5b2 + 4*b1 + 28) / 3
$\nu^{5}$	$=$	$( - 11 \beta_{11} + 54 \beta_{10} - 51 \beta_{9} - 15 \beta_{8} - 54 \beta_{7} + 21 \beta_{6} - 51 \beta_{5} + \cdots + 57 ) / 3$ (-11b11 + 54b10 - 51b9 - 15b8 - 54b7 + 21b6 - 51b5 + 67b4 - 34b3 - 2b2 - 15*b1 + 57) / 3
$\nu^{6}$	$=$	$( - 43 \beta_{11} + 210 \beta_{10} - 102 \beta_{9} - 81 \beta_{8} + 90 \beta_{7} + 180 \beta_{6} + \cdots - 186 ) / 3$ (-43b11 + 210b10 - 102b9 - 81b8 + 90b7 + 180b6 + 78b5 + 272b4 + 256b3 + 41b2 - 15*b1 - 186) / 3
$\nu^{7}$	$=$	$( 35 \beta_{11} - 205 \beta_{10} + 325 \beta_{9} + 29 \beta_{8} + 506 \beta_{7} + 53 \beta_{6} + \cdots - 614 ) / 3$ (35b11 - 205b10 + 325b9 + 29b8 + 506b7 + 53b6 + 460b5 - 231b4 + 546b3 + 77b2 + 130*b1 - 614) / 3
$\nu^{8}$	$=$	$( 350 \beta_{11} - 1845 \beta_{10} + 1134 \beta_{9} + 528 \beta_{8} - 102 \beta_{7} - 1332 \beta_{6} + \cdots + 891 ) / 3$ (350b11 - 1845b10 + 1134b9 + 528b8 - 102b7 - 1332b6 - 210b5 - 2260b4 - 1367b3 - 220b2 + 186*b1 + 891) / 3
$\nu^{9}$	$=$	$( 84 \beta_{11} - 319 \beta_{10} - 1424 \beta_{9} + 209 \beta_{8} - 3892 \beta_{7} - 1861 \beta_{6} + \cdots + 5650 ) / 3$ (84b11 - 319b10 - 1424b9 + 209b8 - 3892b7 - 1861b6 - 3767b5 - 488b4 - 5572b3 - 843b2 - 971*b1 + 5650) / 3
$\nu^{10}$	$=$	$- 842 \beta_{11} + 4603 \beta_{10} - 3412 \beta_{9} - 1185 \beta_{8} - 1102 \beta_{7} + 2727 \beta_{6} + \cdots - 426$ -842b11 + 4603b10 - 3412b9 - 1185b8 - 1102b7 + 2727b6 - 666b5 + 5519b4 + 1549b3 + 248b2 - 799*b1 - 426
$\nu^{11}$	$=$	$( - 3098 \beta_{11} + 16465 \beta_{10} + 650 \beta_{9} - 4619 \beta_{8} + 25891 \beta_{7} + 22648 \beta_{6} + \cdots - 44647 ) / 3$ (-3098b11 + 16465b10 + 650b9 - 4619b8 + 25891b7 + 22648b6 + 26975b5 + 20079b4 + 46950b3 + 7159b2 + 5495*b1 - 44647) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$ .

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$1$	$-1 + \beta_{10}$	$1$	$1$

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}

313.1

0.500000	−	2.70881i
0.500000	−	1.30710i
0.500000	+	1.70293i
0.500000	−	0.272034i
0.500000	+	0.667413i
0.500000	+	0.185550i
0.500000	+	2.70881i
0.500000	+	1.30710i
0.500000	−	1.70293i
0.500000	+	0.272034i
0.500000	−	0.667413i
0.500000	−	0.185550i

−1.71033

−

0.273454i

−0.558529

−

0.967401i

0.697927

−

1.20884i

2.85045

0.935391i

313.2

−1.69504

0.356160i

0.348249

0.603184i

−0.902727

1.56357i

2.74630

−

1.20741i

313.3

−0.793714

−

1.53949i

0.671409

1.16292i

2.29976

−

3.98331i

−1.74004

2.44382i

313.4

0.0876778

1.72983i

−0.391618

−

0.678302i

−0.823587

1.42649i

−2.98463

0.303335i

313.5

0.409577

−

1.68293i

−1.87385

−

3.24560i

0.175717

−

0.304350i

−2.66449

−

1.37858i

313.6

1.70182

−

0.322173i

1.30434

2.25918i

−0.447094

0.774390i

2.79241

−

1.09656i

625.1