LMFDB - Newform orbit 63.8.e.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,8,Mod(37,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("63.37");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$63 = 3^{2} \cdot 7$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	63.e (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:	$19.6802566055$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 4x^{7} + 362x^{6} - 1072x^{5} + 36131x^{4} - 70480x^{3} + 698554x^{2} - 663492x + 2868273$ x^8 - 4x^7 + 362x^6 - 1072x^5 + 36131x^4 - 70480x^3 + 698554x^2 - 663492*x + 2868273
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{2}\cdot 3^{3}\cdot 7^{2}$
Twist minimal:	no (minimal twist has level 21)
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{6} q^{2} + ( - \beta_{7} + 2 \beta_{6} + \cdots - 2 \beta_1) q^{4} + (2 \beta_{7} + 9 \beta_{6} + \beta_{5} + \cdots + 46) q^{5} + (14 \beta_{6} + 273 \beta_{4} + \cdots + 161) q^{7} + ( - 8 \beta_{3} - 3 \beta_{2} + \cdots + 343) q^{8}+ \cdots + ( - 35329 \beta_{7} - 449820 \beta_{6} + \cdots + 602063) q^{98}+O(q^{100})$ q - b6 * q^2 + (-b7 + 2b6 + 25b4 - 2b1) q^4 + (2b7 + 9b6 + b5 + 46b4 + 2b2 + 46) * q^5 + (14b6 + 273b4 + 7b3 - 35b1 + 161) * q^7 + (-8b3 - 3b2 - 14b1 + 343) q^8 + (-b7 + 106b6 - 14b5 - 1319b4 + 14b3 - 106b1) q^10 + (-10b7 + 83b6 + 41b5 + 1274b4 - 41b3 - 83b1) * q^11 + (15b3 - 74b2 - 203b1 - 883) q^13 + (-77b7 - 119b6 + 14b5 + 3101b4 - 91b2 + 301b1 + 5243) * q^14 + (-75b7 - 314b6 + 8b5 + 5359b4 - 75b2 + 5359) q^16 + (94b7 - 782b6 + 20b5 + 3026b4 - 20b3 + 782b1) * q^17 + (-236b7 + 731b6 + 155b5 + 17127b4 - 236b2 + 17127) q^19 + (148b3 + 37b2 + 130b1 + 21919) q^20 + (-162b3 + 235b2 + 2290b1 + 13725) q^22 + (-34b7 + 1762b6 + 204b5 - 37038b4 - 34b2 - 37038) q^23 + (482b7 + 947b6 + 741b5 + 1015b4 - 741b3 - 947b1) * q^25 + (-249b7 - 5001b6 + 622b5 + 28081b4 - 249b2 + 28081) q^26 + (161b7 - 8862b6 - 140b5 - 17101b4 + 252b3 + 315b2 + 6720b1 - 11627) q^28 + (27b3 + 496b2 - 7115b1 + 19314) q^29 + (12b7 + 3452b6 + 370b5 - 73113b4 - 370b3 - 3452b1) * q^31 + (81b7 - 9314b6 + 1640b5 + 1235b4 - 1640b3 + 9314b1) * q^32 + (712b3 + 1036b2 - 6528b1 - 122804) q^34 + (1722b7 + 12082b6 + 1036b5 + 75880b4 - 469b3 + 1162b2 - 973b1 + 44856) q^35 + (1290b7 - 16319b6 + 1223b5 + 115677b4 + 1290b2 + 115677) q^37 + (-273b7 - 38649b6 + 2198b5 - 123055b4 - 2198b3 + 38649b1) * q^38 + (-1219b7 - 5466b6 - 1792b5 - 189721b4 - 1219b2 - 189721) q^40 + (-666b3 - 112b2 + 14150b1 + 162486) q^41 + (2407b3 + 1384b2 + 53895b1 - 14283) q^43 + (2071b7 + 12294b6 + 3044b5 - 176011b4 + 2071b2 - 176011) q^44 + (164b7 + 30624b6 + 680b5 - 274108b4 - 680b3 - 30624b1) * q^46 + (-7518b7 - 12602b6 - 1652b5 + 195988b4 - 7518b2 + 195988) q^47 + (2352b7 - 16170b6 + 3430b5 - 103439b4 - 1715b3 + 1470b2 - 3969b1 + 615048) q^49 + (2374b3 + 5463b2 - 38061b1 + 138913) q^50 + (-256b7 - 13260b6 + 1316b5 + 859012b4 - 1316b3 + 13260b1) * q^52 + (2194b7 + 54619b6 - 3565b5 + 677168b4 + 3565b3 - 54619b1) * q^53 + (-7273b3 - 10064b2 - 50479b1 - 474310) q^55 + (-1561b7 + 4158b6 - 2296b5 + 63413b4 + 3360b3 - 5306b2 - 25214b1 - 629566) q^56 + (-7827b7 + 37076b6 - 3914b5 + 1106515b4 - 7827b2 + 1106515) q^58 + (8856b7 + 17311b6 - 7985b5 - 857270b4 + 7985b3 - 17311b1) * q^59 + (-14124b7 + 76820b6 + 1972b5 - 91390b4 - 14124b2 - 91390) q^61 + (-644b3 - 480b2 - 67229b1 + 533632) q^62 + (-1608b3 + 12915b2 - 64470b1 - 715847) q^64 + (-5510b7 + 21004b6 - 16094b5 - 2004608b4 - 5510b2 - 2004608) q^65 + (8976b7 + 37609b6 - 7699b5 - 1065341b4 + 7699b3 - 37609b1) * q^67 + (-1228b7 + 123824b6 - 4304b5 + 1413052b4 - 1228b2 + 1413052) q^68 + (5411b7 + 31696b6 - 8162b5 - 1665755b4 + 11704b3 + 3339b2 - 46326b1 + 135653) q^70 + (-10548b3 + 10338b2 + 215574b1 + 41400) q^71 + (6074b7 - 262693b6 - 24483b5 - 42643b4 + 24483b3 + 262693b1) * q^73 + (-27393b7 + 26611b6 - 7874b5 + 2524969b4 + 7874b3 - 26611b1) * q^74 + (13260b3 + 25752b2 - 83580b1 - 3675772) q^76 + (19684b7 - 172375b6 - 3437b5 + 3056158b4 - 5488b3 - 560b2 + 117656b1 - 539952) q^77 + (25296b7 + 425490b6 - 15864b5 + 2929975b4 + 25296b2 + 2929975) q^79 + (5353b7 + 80398b6 - 12776b5 - 1985765b4 + 12776b3 - 80398b1) * q^80 + (19590b7 - 200306b6 - 436b5 - 2158438b4 + 19590b2 - 2158438) q^82 + (-8591b3 + 11560b2 - 280143b1 + 3091288) q^83 + (-14312b3 - 17696b2 - 147588b1 - 1798976) q^85 + (33255b7 + 24133b6 - 6258b5 - 8233239b4 + 33255b2 - 8233239) q^86 + (-44209b7 + 34338b6 + 10256b5 - 3609859b4 - 10256b3 - 34338b1) * q^88 + (1276b7 + 175206b6 - 6714b5 + 1196496b4 + 1276b2 + 1196496) q^89 + (-16716b7 + 14595b6 - 4613b5 - 634193b4 - 27622b3 - 32228b2 - 509586b1 + 1590197) q^91 + (26064b3 - 29372b2 - 1264b1 - 50580) q^92 + (8132b7 - 809814b6 + 56840b5 + 1676372b4 - 56840b3 + 809814b1) * q^94 + (45446b7 + 229064b6 + 5810b5 - 2185276b4 - 5810b3 - 229064b1) * q^95 + (-8227b3 + 2590b2 - 32665b1 - 5700578) q^97 + (-35329b7 - 449820b6 - 8330b5 + 3108217b4 + 11956b3 + 10633b2 - 335699b1 + 602063) q^98
$\operatorname{Tr}(f)(q)$	$=$	$8 q - q^{2} - 101 q^{4} + 196 q^{5} + 154 q^{7} + 2694 q^{8} + 5185 q^{10} - 5210 q^{11} - 7588 q^{13} + 29932 q^{14} + 21055 q^{16} - 11436 q^{17} + 69158 q^{19} + 175982 q^{20} + 114526 q^{22} - 146220 q^{23}+ \cdots - 8665405 q^{98}+O(q^{100})$ 8 * q - q^2 - 101 * q^4 + 196 * q^5 + 154 * q^7 + 2694 * q^8 + 5185 * q^10 - 5210 * q^11 - 7588 * q^13 + 29932 * q^14 + 21055 * q^16 - 11436 * q^17 + 69158 * q^19 + 175982 * q^20 + 114526 * q^22 - 146220 * q^23 - 6230 * q^25 + 107696 * q^26 - 19201 * q^28 + 141328 * q^29 + 288618 * q^31 + 2653 * q^32 - 991992 * q^34 + 66164 * q^35 + 448902 * q^37 + 528944 * q^38 - 767361 * q^40 + 1326632 * q^41 + 1108 * q^43 - 686635 * q^44 + 1064964 * q^46 + 762180 * q^47 + 5310620 * q^49 + 1050856 * q^50 - 3423848 * q^52 - 2761920 * q^53 - 3930112 * q^55 - 5341077 * q^56 + 4451395 * q^58 + 3410898 * q^59 - 300892 * q^61 + 4132350 * q^62 - 5833102 * q^64 - 8019032 * q^65 + 4222478 * q^67 + 5770500 * q^68 + 7703801 * q^70 + 761928 * q^71 + 451674 * q^73 - 10091220 * q^74 - 29495312 * q^76 - 16516528 * q^77 + 12154822 * q^79 + 7870085 * q^80 - 8814904 * q^82 + 24175956 * q^83 - 14751000 * q^85 - 32881826 * q^86 + 14439051 * q^88 + 4955752 * q^89 + 14146174 * q^91 - 413784 * q^92 - 5960646 * q^94 + 8460784 * q^95 - 45681228 * q^97 - 8665405 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - 4x^{7} + 362x^{6} - 1072x^{5} + 36131x^{4} - 70480x^{3} + 698554x^{2} - 663492x + 2868273$ x^8 - 4*x^7 + 362*x^6 - 1072*x^5 + 36131*x^4 - 70480*x^3 + 698554*x^2 - 663492*x + 2868273 :

$\beta_{1}$	$=$	$( \nu^{6} - 3\nu^{5} + 334\nu^{4} - 663\nu^{3} + 29038\nu^{2} - 28707\nu + 287577 ) / 6912$ (v^6 - 3v^5 + 334v^4 - 663v^3 + 29038v^2 - 28707*v + 287577) / 6912
$\beta_{2}$	$=$	$( \nu^{6} - 3\nu^{5} + 238\nu^{4} - 471\nu^{3} + 1486\nu^{2} - 1251\nu - 797319 ) / 6912$ (v^6 - 3v^5 + 238v^4 - 471v^3 + 1486v^2 - 1251*v - 797319) / 6912
$\beta_{3}$	$=$	$( -\nu^{6} + 3\nu^{5} - 670\nu^{4} + 1335\nu^{3} - 89182\nu^{2} + 88515\nu - 851625 ) / 6912$ (-v^6 + 3v^5 - 670v^4 + 1335v^3 - 89182v^2 + 88515*v - 851625) / 6912
$\beta_{4}$	$=$	$( -8\nu^{7} + 28\nu^{6} - 2882\nu^{5} + 7135\nu^{4} - 272324\nu^{3} + 401365\nu^{2} - 3032736\nu - 444825 ) / 3789072$ (-8v^7 + 28v^6 - 2882v^5 + 7135v^4 - 272324v^3 + 401365v^2 - 3032736*v - 444825) / 3789072
$\beta_{5}$	$=$	$( 477 \nu^{7} - 6055 \nu^{6} + 191574 \nu^{5} - 3380155 \nu^{4} + 25170582 \nu^{3} + \cdots - 4290009840 ) / 60625152$ (477v^7 - 6055v^6 + 191574v^5 - 3380155v^4 + 25170582v^3 - 419640535v^2 + 1508080917*v - 4290009840) / 60625152
$\beta_{6}$	$=$	$( - 541 \nu^{7} + 6279 \nu^{6} - 214630 \nu^{5} + 1963707 \nu^{4} - 24402118 \nu^{3} + \cdots + 1509692976 ) / 60625152$ (-541v^7 + 6279v^6 - 214630v^5 + 1963707v^4 - 24402118v^3 + 159089943v^2 - 633490725*v + 1509692976) / 60625152
$\beta_{7}$	$=$	$( 2357 \nu^{7} - 8876 \nu^{6} + 804941 \nu^{5} - 2136137 \nu^{4} + 71969297 \nu^{3} + \cdots + 119000043 ) / 8660736$ (2357v^7 - 8876v^6 + 804941v^5 - 2136137v^4 + 71969297v^3 - 106459397v^2 + 796868436*v + 119000043) / 8660736

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

$\nu$	$=$	$( 2\beta_{6} + 2\beta_{5} - \beta_{4} - \beta_{3} - \beta _1 + 10 ) / 21$ (2b6 + 2b5 - b4 - b3 - b1 + 10) / 21
$\nu^{2}$	$=$	$( 2\beta_{6} + 2\beta_{5} - \beta_{4} + 3\beta_{3} - 14\beta_{2} + 17\beta _1 - 1861 ) / 21$ (2b6 + 2b5 - b4 + 3b3 - 14b2 + 17*b1 - 1861) / 21
$\nu^{3}$	$=$	$( 14\beta_{7} - 994\beta_{6} - 308\beta_{5} + 4858\beta_{4} + 160\beta_{3} - 14\beta_{2} + 524\beta _1 - 367 ) / 21$ (14b7 - 994b6 - 308b5 + 4858b4 + 160b3 - 14b2 + 524*b1 - 367) / 21
$\nu^{4}$	$=$	$( 28 \beta_{7} - 1990 \beta_{6} - 618 \beta_{5} + 9717 \beta_{4} - 827 \beta_{3} + 2478 \beta_{2} + \cdots + 298912 ) / 21$ (28b7 - 1990b6 - 618b5 + 9717b4 - 827b3 + 2478b2 - 2605*b1 + 298912) / 21
$\nu^{5}$	$=$	$( - 6482 \beta_{7} + 176108 \beta_{6} + 52026 \beta_{5} - 1385973 \beta_{4} - 28863 \beta_{3} + \cdots + 46809 ) / 21$ (-6482b7 + 176108b6 + 52026b5 - 1385973b4 - 28863b3 + 2954b2 - 97099*b1 + 46809) / 21
$\nu^{6}$	$=$	$( - 19516 \beta_{7} + 533300 \beta_{6} + 157624 \beta_{5} - 4182212 \beta_{4} + 179888 \beta_{3} + \cdots - 51651831 ) / 21$ (-19516b7 + 533300b6 + 157624b5 - 4182212b4 + 179888b3 - 421540b2 + 548984*b1 - 51651831) / 21
$\nu^{7}$	$=$	$( 1815240 \beta_{7} - 30172774 \beta_{6} - 8915230 \beta_{5} + 318339239 \beta_{4} + 5373047 \beta_{3} + \cdots - 16885466 ) / 21$ (1815240b7 - 30172774b6 - 8915230b5 + 318339239b4 + 5373047b3 - 555324b2 + 17972795*b1 - 16885466) / 21

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$ .

$n$	$10$	$29$
$\chi(n)$	$-1 - \beta_{4}$	$1$

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}

37.1

0.500000	+	2.45385i
0.500000	−	12.0633i
0.500000	+	13.6728i
0.500000	−	4.06331i
0.500000	−	2.45385i
0.500000	+	12.0633i
0.500000	−	13.6728i
0.500000	+	4.06331i

−8.70345

15.0748i

−87.5001

−

151.555i

−37.1845

64.4054i

−789.501

−

447.472i

818.128

−647.267

−

1121.10i

37.2

−2.81424

4.87440i

48.1601

83.4158i

251.471

−

435.561i

854.822

−

304.667i

−1262.58

1415.40

2451.54i

37.3

3.40109

−

5.89086i

40.8652

70.7806i

6.28050

−

10.8781i

−894.339

−

153.953i

1426.62

−42.7211

−

73.9951i

37.4

7.61660

−

13.1923i

−52.0252

−

90.1103i

−122.567

212.292i

906.017

−

51.7322i

364.830

1867.09

3233.89i

46.1