LMFDB - Newform orbit 390.2.x.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(49,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 5]))

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

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

chi := DirichletCharacter("390.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$390 = 2 \cdot 3 \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	390.x (of order $6$ , 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:	$3.11416567883$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197$ x^12 - 2x^11 - 8x^10 + 34x^9 + 8x^8 - 134x^7 + 98x^6 + 154x^5 + 104x^4 + 190x^3 - 1196x^2 - 338*x + 2197
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$ -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.

$f(q)$	$=$	$q + ( - \beta_{6} - 1) q^{2} + \beta_{4} q^{3} + \beta_{6} q^{4} + (\beta_{8} - \beta_{6} + \cdots + \beta_{3}) q^{5} + ( - \beta_{7} - \beta_{4}) q^{6} + ( - \beta_{9} - \beta_{7} + \cdots + \beta_{2}) q^{7}+ \cdots + (\beta_{11} - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{99}+O(q^{100})$ q + (-b6 - 1) * q^2 + b4 * q^3 + b6 * q^4 + (b8 - b6 + b4 + b3) * q^5 + (-b7 - b4) * q^6 + (-b9 - b7 + b4 + b2) * q^7 + q^8 - b6 * q^9 + (-b11 + b10 - b7 + b6 - b4 + b2) * q^10 + (b9 - b8 + b5 - b2) * q^11 + b7 * q^12 + (b11 - b5 - 2b1) q^13 + (-b10 - b4 - b3 - b2) * q^14 + (-b9 + b8 - b7 - b6 + b3 + b2 - b1 + 1) * q^15 + (-b6 - 1) * q^16 + (b11 - 2b9 + 3b8 + b7 - 2b5 + 3b4 + 2b3 + b2 - 3b1 + 4) * q^17 - q^18 + (-2b11 - b9 + 2b8 - b7 - b4 + b2 - 2b1) q^19 + (b11 - b10 - b8 + b7 - b3 - b2) * q^20 + (-b11 + b10 + b9 - b6 + b5 - b4 + b1 - 1) * q^21 + (b11 - b9 + b8 - b7 - b6 - b5 + b3 - b1) * q^22 + (2b10 + 2b9 - b7 + 3b5 + b3 - b2 - b1 + 1) q^23 + b4 * q^24 + (-b11 + b8 - b7 - 2b6 - b4 + 3b2 + b1 - 1) * q^25 + (b11 + b5 - b2 + b1) * q^26 - b7 * q^27 + (b10 + b9 + b7 + b3) * q^28 + (-2b10 - 3b9 + b8 + b7 + b6 + 2b4 - b3 - b1 + 2) q^29 + (b7 - b3 - b2 - 1) * q^30 + (3b11 - 2b10 - b9 - b8 - b7 + b6 - 3b5 + 2b4 - b3 - b2 - 2b1 + 1) q^31 + b6 * q^32 + (-b10 - b6 - b5 + b4) * q^33 + (-b11 + b10 + b9 - b8 - b7 - b6 + b5 - b4 - 2b3 + 2b1 - 2) * q^34 + (-b9 - 3b7 + 2b5 - b4 - b2 + 2) * q^35 + (b6 + 1) * q^36 + (-b10 - 2b9 - b8 - 3b6 + 2b5 - 2b3 - 2b2 + b1 - 4) q^37 + (2b11 + b10 + 2b9 - 2b8 + 2b7 + 2b6 - b4 - b3 - 3b2) * q^38 + (b11 - 2b5) q^39 + (b8 - b6 + b4 + b3) * q^40 + (-b10 + b9 - 2b8 + b6 + b5 - 2b4 - b2 + b1 - 2) * q^41 + (-b8 + b6 - 1) * q^42 + (-b11 + 2b9 - b8 - 2b7 - b6 + b5 - 3b4 - b3 + b2 + b1 - 4) q^43 + (-b11 + b7 + b6 - b3 + b2 + b1) * q^44 + (-b11 + b10 + b8 - b7 + b3 + b2) * q^45 + (b11 - b9 - b8 - 2b7 - b6 - 2b4 - 2b2 + b1 - 2) q^46 + (b11 - b10 + b9 - b8 - 3b7 - b6 + b5 - 5b4 + 2b1 + 1) q^47 + (-b7 - b4) * q^48 + (-4b11 + 3b10 + 3b9 + b7 + 3b6 + 2b5 - 2b4 + b3 + 2b2 + 2b1 + 1) * q^49 + (-b11 + b10 + b9 - b8 + b7 + 2b6 + 3b5 - b4 - 2b2 - 1) q^50 + (2b10 - b9 + 2b8 - b5 + 2b4 + 2b3 + 2b2 - 2b1 + 3) * q^51 + (-2b11 + b2 + b1) q^52 + (b11 + 2b10 + 5b9 - b8 + b7 + 5b6 - b5 - 2b4 + b3 - 3b2 + 2) q^53 - b4 * q^54 + (2b11 - b10 - 3b9 + 2b8 - 2b7 - 2b6 - 3b5 + 2b4 + 2b3 + b2 - 3b1 - 1) q^55 + (-b9 - b7 + b4 + b2) * q^56 + (-b11 + b10 - b9 + b6 - b5 - b4 - b1 - 1) * q^57 + (2b9 + b7 - b6 - b5 - b3 - 2b2) * q^58 + (3b11 - 2b10 - b8 + 6b7 - 3b6 - b5 + 7b4 - b3 - b2 + b1 - 4) q^59 + (b9 - b8 + b6 + b1) * q^60 + (b11 - b8 + 3b7 - 2b5 - b4 - 2b3 - b2 - b1) q^61 + (2b8 - b7 - 2b6 + b5 - 2b4 + b3 + b2 + b1 + 1) q^62 + (-b10 - b9 - b7 - b3) * q^63 + q^64 + (2b11 + b10 - 3b9 + 3b7 + 4b6 - 2b5 + 2b4 + b3 - 2b1 + 3) q^65 + (b10 + b8 - b7 - b4 + b3 + b2 - b1) * q^66 + (-4b11 + 2b10 + 2b9 - 2b8 + 2b7 + 2b6 + 4b5 - 2b4 - 2b3 + 4b1 - 2) * q^67 + (-b10 + b9 - 2b8 + b6 + b5 - 2b4 - b2 + b1 - 2) * q^68 + (-b11 - 2b10 - 2b9 + b8 - 3b6 - 3b5 + 3b4 - b3 + 3b2 + b1) * q^69 + (-b10 + 2b7 - 2b6 - 2b5 - b4 - b3 - b2 - 2) q^70 + (-2b10 - 4b9 + 4b8 + 6b7 - 2b6 - 2b5 + 8b4 - 4b1) * q^71 - b6 * q^72 + (b11 - b10 + 3b9 - 5b8 + b7 - b6 + 3b5 + 3b4 + 6b1) q^73 + (-2b10 + b9 + 2b7 + 3b6 - 3b5 + b4 - b3 - b2) * q^74 + (-b9 - 2b7 + b5 - b4 + 2b1 - 1) * q^75 + (-b10 - b9 - b7 - 2b6 + 2b4 + b3 + 2b2 + 2b1) * q^76 + (-2b11 + b10 + 2b9 + 2b7 + 2b6 - b4 - b3 + b2 + 2b1) q^77 + (-b11 + b5 + 2b2 + b1) q^78 + (-b11 + 4b10 - b9 + 3b8 - 3b7 + b6 - b5 - 4b4 + 3b3 + 3b2 - 4b1 + 3) q^79 + (-b11 + b10 - b7 + b6 - b4 + b2) * q^80 + (-b6 - 1) * q^81 + (b11 - 2b9 + 3b8 + b7 - 2b5 + 3b4 + 2b3 + b2 - 3b1 + 4) * q^82 + (-2b10 + 2b9 + 2b5 - 2b4 - 2b3 - 2b2 + 6) * q^83 + (b11 - b10 - b9 + b8 - b5 + b4 - b1 + 2) * q^84 + (-b10 - b9 + 2b8 + 2b7 + b6 + b5 + 2b3 + 3b2 + 3b1 + 4) q^85 + (b11 + b10 + b7 + 3b6 + 2b5 - b4 + 2b3 - 2b2 - b1 + 2) * q^86 + (-3b11 + 3b10 + 2b9 - b8 + b7 + 2b6 + 2b5 - b4 + b3 + b2 + b1) q^87 + (b9 - b8 + b5 - b2) * q^88 + (-5b10 - b9 - 4b8 + b7 - 2b6 - 2b5 - b3 + 4b1 - 2) q^89 + (-b8 + b6 - b4 - b3) * q^90 + (-b10 + 3b9 + 5b7 - 2b5 + 4b4 - b3 + 2b1) q^91 + (-b11 - 2b10 - b9 + b8 + 3b7 + b6 - 3b5 + 2b4 - b3 + 3b2 + 1) q^92 + (2b11 + b10 + 2b9 - b8 + b7 - 2b6 - b5 + b3 - b2 - 2) q^93 + (-2b11 + b10 - b9 + 2b8 + 4b7 - 2b6 + b5 + 2b4 + 2b3 + b2 - b1 - 1) * q^94 + (-2b11 - b9 - 2b8 + b7 + 2b5 - 7b4 - b2 + 2b1 - 6) q^95 + b7 * q^96 + (2b11 - 4b10 - 2b9 - 2b8 + 8b6 - 4b5 + 4b4 - 2b1) * q^97 + (2b11 - b9 - 2b8 - b7 - b6 + b4 - b2 - 2b1) q^98 + (b11 - b7 - b6 + b3 - b2 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 6 q^{2} - 6 q^{4} - 2 q^{5} - 2 q^{7} + 12 q^{8} + 6 q^{9} - 2 q^{10} + 6 q^{11} - 8 q^{13} + 4 q^{14} + 6 q^{15} - 6 q^{16} + 18 q^{17} - 12 q^{18} - 6 q^{19} + 4 q^{20} - 6 q^{22} + 6 q^{23} - 10 q^{25}+ \cdots + 8 q^{98}+O(q^{100})$ 12 * q - 6 * q^2 - 6 * q^4 - 2 * q^5 - 2 * q^7 + 12 * q^8 + 6 * q^9 - 2 * q^10 + 6 * q^11 - 8 * q^13 + 4 * q^14 + 6 * q^15 - 6 * q^16 + 18 * q^17 - 12 * q^18 - 6 * q^19 + 4 * q^20 - 6 * q^22 + 6 * q^23 - 10 * q^25 - 2 * q^26 - 2 * q^28 + 14 * q^29 - 6 * q^30 - 6 * q^32 + 6 * q^33 + 26 * q^35 + 6 * q^36 - 12 * q^37 - 2 * q^39 - 2 * q^40 - 18 * q^41 - 12 * q^42 - 36 * q^43 - 4 * q^45 - 6 * q^46 + 16 * q^47 + 8 * q^49 - 10 * q^50 + 16 * q^51 + 10 * q^52 - 28 * q^55 - 2 * q^56 - 8 * q^57 + 14 * q^58 - 36 * q^59 + 10 * q^61 + 6 * q^62 + 2 * q^63 + 12 * q^64 + 6 * q^65 - 12 * q^66 + 4 * q^67 - 18 * q^68 + 16 * q^69 - 4 * q^70 - 12 * q^71 + 6 * q^72 + 28 * q^73 - 12 * q^74 - 8 * q^75 + 6 * q^76 - 2 * q^78 + 4 * q^79 - 2 * q^80 - 6 * q^81 + 18 * q^82 + 72 * q^83 + 12 * q^84 + 18 * q^85 + 6 * q^87 + 6 * q^88 + 18 * q^89 + 2 * q^90 + 2 * q^91 - 16 * q^93 - 8 * q^94 - 42 * q^95 - 48 * q^97 + 8 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197$ x^12 - 2*x^11 - 8*x^10 + 34*x^9 + 8*x^8 - 134*x^7 + 98*x^6 + 154*x^5 + 104*x^4 + 190*x^3 - 1196*x^2 - 338*x + 2197 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( - 203419 \nu^{11} - 163110633 \nu^{10} + 591783880 \nu^{9} + 97338749 \nu^{8} + \cdots + 81183629852 ) / 63907274600$ (-203419v^11 - 163110633v^10 + 591783880v^9 + 97338749v^8 - 4513282461v^7 + 6489146722v^6 + 4655211661v^5 - 5740233327v^4 - 8165110694v^3 - 33307875431v^2 + 64277898945*v + 81183629852) / 63907274600
$\beta_{3}$	$=$	$( 60968787 \nu^{11} - 2097063441 \nu^{10} + 2300362460 \nu^{9} + 17147379373 \nu^{8} + \cdots + 1269272187404 ) / 830794569800$ (60968787v^11 - 2097063441v^10 + 2300362460v^9 + 17147379373v^8 - 55879543947v^7 - 372787906v^6 + 135156205247v^5 - 272345430479v^4 + 414518069862v^3 - 554153957987v^2 - 833086363785*v + 1269272187404) / 830794569800
$\beta_{4}$	$=$	$( - 120243408 \nu^{11} + 454030419 \nu^{10} + 2051209685 \nu^{9} - 7036618932 \nu^{8} + \cdots + 618192439839 ) / 830794569800$ (-120243408v^11 + 454030419v^10 + 2051209685v^9 - 7036618932v^8 + 2513656023v^7 + 29967292429v^6 - 18436259198v^5 + 35262141211v^4 + 8309942667v^3 + 28687118858v^2 + 98318215515*v + 618192439839) / 830794569800
$\beta_{5}$	$=$	$( 16426431 \nu^{11} + 83789417 \nu^{10} - 226795620 \nu^{9} + 267354099 \nu^{8} + \cdots + 20321135952 ) / 63907274600$ (16426431v^11 + 83789417v^10 - 226795620v^9 + 267354099v^8 + 1065744289v^7 - 511723478v^6 + 4136894311v^5 + 1601173623v^4 + 3964105106v^3 - 3499453881v^2 + 44426935995*v + 20321135952) / 63907274600
$\beta_{6}$	$=$	$( 4036 \nu^{11} - 37023 \nu^{10} + 57555 \nu^{9} + 233294 \nu^{8} - 817691 \nu^{7} + \cdots + 11867687 ) / 11796200$ (4036v^11 - 37023v^10 + 57555v^9 + 233294v^8 - 817691v^7 + 35557v^6 + 1844066v^5 - 940887v^4 + 77961v^3 - 7481036v^2 - 4439955*v + 11867687) / 11796200
$\beta_{7}$	$=$	$( - 20638 \nu^{11} + 28887 \nu^{10} - 18833 \nu^{9} - 91862 \nu^{8} - 291763 \nu^{7} + \cdots + 21764327 ) / 47610004$ (-20638v^11 + 28887v^10 - 18833v^9 - 91862v^8 - 291763v^7 - 1390803v^6 + 3193024v^5 - 2792113v^4 - 7615595v^3 - 12957312v^2 + 2476201*v + 21764327) / 47610004
$\beta_{8}$	$=$	$( 5665538 \nu^{11} - 7145579 \nu^{10} - 9226575 \nu^{9} + 110201552 \nu^{8} - 247897203 \nu^{7} + \cdots - 40106339 ) / 12781454920$ (5665538v^11 - 7145579v^10 - 9226575v^9 + 110201552v^8 - 247897203v^7 + 32851611v^6 + 1496635908v^5 - 1228577871v^4 - 1963928377v^3 + 3156976042v^2 + 8732837145*v - 40106339) / 12781454920
$\beta_{9}$	$=$	$( - 680246389 \nu^{11} - 845264698 \nu^{10} + 11881898255 \nu^{9} - 20291622981 \nu^{8} + \cdots + 1108736562037 ) / 830794569800$ (-680246389v^11 - 845264698v^10 + 11881898255v^9 - 20291622981v^8 - 56431077566v^7 + 138431742257v^6 - 46377269759v^5 - 85505629812v^4 - 58320816489v^3 - 633066289511v^2 + 550784663520*v + 1108736562037) / 830794569800
$\beta_{10}$	$=$	$( - 1202455216 \nu^{11} + 4870077513 \nu^{10} - 1280213355 \nu^{9} - 35058222714 \nu^{8} + \cdots - 1182431604497 ) / 830794569800$ (-1202455216v^11 + 4870077513v^10 - 1280213355v^9 - 35058222714v^8 + 75140023171v^7 - 22116401667v^6 - 93428783546v^5 + 206023103197v^4 - 574822497041v^3 + 859741244016v^2 + 783010043855*v - 1182431604497) / 830794569800
$\beta_{11}$	$=$	$( - 78524401 \nu^{11} + 161812093 \nu^{10} + 556127095 \nu^{9} - 2253760279 \nu^{8} + \cdots + 16572567908 ) / 31953637300$ (-78524401v^11 + 161812093v^10 + 556127095v^9 - 2253760279v^8 - 695335944v^7 + 7168384238v^6 - 1904719281v^5 - 3795941083v^4 - 26424465301v^3 - 15605355549v^2 + 99935137380*v + 16572567908) / 31953637300

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$-\beta_{11} + 2\beta_{10} + 2\beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{5} - 3\beta_{4} + \beta_{3} + 2$ -b11 + 2b10 + 2b9 + b8 - 2b7 + b5 - 3b4 + b3 + 2
$\nu^{3}$	$=$	$- 2 \beta_{11} + \beta_{10} + 3 \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + 4 \beta_{4} + \cdots - 5$ -2b11 + b10 + 3b9 + b8 + 2b7 - b6 + b5 + 4b4 + b3 - b2 + 3*b1 - 5
$\nu^{4}$	$=$	$-7\beta_{11} + 7\beta_{10} + 12\beta_{9} + \beta_{8} + 2\beta_{7} + 9\beta_{5} - 6\beta_{4} + \beta_{2} - 3$ -7b11 + 7b10 + 12b9 + b8 + 2b7 + 9b5 - 6b4 + b2 - 3
$\nu^{5}$	$=$	$- 4 \beta_{11} - 5 \beta_{10} - 8 \beta_{9} + 4 \beta_{8} + 29 \beta_{7} - 12 \beta_{6} - 7 \beta_{5} + \cdots - 42$ -4b11 - 5b10 - 8b9 + 4b8 + 29b7 - 12b6 - 7b5 + 47b4 - b3 + 12*b2 - 42
$\nu^{6}$	$=$	$- 6 \beta_{11} + 8 \beta_{10} - 2 \beta_{9} - 10 \beta_{8} - 24 \beta_{6} + 18 \beta_{5} - 20 \beta_{4} + \cdots - 7$ -6b11 + 8b10 - 2b9 - 10b8 - 24b6 + 18b5 - 20b4 + 6b3 + 38b2 - 32b1 - 7
$\nu^{7}$	$=$	$50 \beta_{11} - 70 \beta_{10} - 162 \beta_{9} - 50 \beta_{8} + 54 \beta_{7} - 36 \beta_{6} - 90 \beta_{5} + \cdots - 104$ 50b11 - 70b10 - 162b9 - 50b8 + 54b7 - 36b6 - 90b5 + 202b4 - 8b3 + 76b2 - 11*b1 - 104
$\nu^{8}$	$=$	$81 \beta_{11} + 80 \beta_{10} - 90 \beta_{9} - 155 \beta_{8} - 382 \beta_{7} + 114 \beta_{6} + \cdots + 388$ 81b11 + 80b10 - 90b9 - 155b8 - 382b7 + 114b6 + 113b5 - 491b4 + 73b3 + 64b2 - 112*b1 + 388
$\nu^{9}$	$=$	$404 \beta_{11} - 223 \beta_{10} - 627 \beta_{9} - 213 \beta_{8} - 540 \beta_{7} + 459 \beta_{6} + \cdots + 249$ 404b11 - 223b10 - 627b9 - 213b8 - 540b7 + 459b6 - 365b5 + 328b4 + 7b3 - 231b2 + 163*b1 + 249
$\nu^{10}$	$=$	$327 \beta_{11} + 649 \beta_{10} + 866 \beta_{9} - 57 \beta_{8} - 2616 \beta_{7} + 1480 \beta_{6} + \cdots + 2981$ 327b11 + 649b10 + 866b9 - 57b8 - 2616b7 + 1480b6 + 1181b5 - 3528b4 + 248b3 - 1421b2 - 202*b1 + 2981
$\nu^{11}$	$=$	$1120 \beta_{11} - 2139 \beta_{10} - 614 \beta_{9} + 1672 \beta_{8} + 195 \beta_{7} + 1784 \beta_{6} + \cdots - 288$ 1120b11 - 2139b10 - 614b9 + 1672b8 + 195b7 + 1784b6 - 1755b5 + 3931b4 - 1427b3 - 3608b2 + 1392*b1 - 288

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$ .

$n$	$131$	$157$	$301$
$\chi(n)$	$1$	$-1$	$-\beta_{6}$

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}

49.1

−2.39378	+	0.0429626i
1.75374	+	1.62986i
2.00607	−	1.30680i
−0.330925	+	1.46916i
−1.44229	+	0.433312i
1.40719	−	0.536449i
−2.39378	−	0.0429626i
1.75374	−	1.62986i
2.00607	+	1.30680i
−0.330925	−	1.46916i
−1.44229	−	0.433312i
1.40719	+	0.536449i

−0.500000

−

0.866025i

−0.866025

0.500000i

−0.500000

0.866025i

−2.10012

−

0.767774i

0.866025

0.500000i

−0.823063

1.42559i

1.00000

0.500000

−

0.866025i

0.385150

2.20265i

49.2

−0.500000

−

0.866025i

−0.866025

0.500000i

−0.500000

0.866025i

−1.40066

1.74303i

0.866025

0.500000i

0.763837

−

1.32301i

1.00000

0.500000

−

0.866025i

2.20984

0.341491i

49.3

−0.500000

−

0.866025i

−0.866025

0.500000i

−0.500000

0.866025i

1.26873

−

1.84128i

0.866025

0.500000i

−2.17283

3.76344i

1.00000

0.500000

−

0.866025i

−2.22896

−

0.178114i

49.4

−0.500000

−

0.866025i

0.866025

−

0.500000i

−0.500000

0.866025i

−0.571769

−

2.16173i

−0.866025

−

0.500000i

−0.603137

1.04466i

1.00000

0.500000

−

0.866025i

−1.58623

1.57603i

49.5

−0.500000

−

0.866025i

0.866025

−

0.500000i

−0.500000

0.866025i

−0.230377

2.22417i

−0.866025

−

0.500000i

0.432713

−

0.749482i

1.00000

0.500000

−

0.866025i

2.04138

−

0.912572i

49.6

−0.500000

−

0.866025i

0.866025

−

0.500000i

−0.500000

0.866025i

2.03420

−

0.928463i

−0.866025

−

0.500000i

1.40247

−

2.42916i

1.00000

0.500000

−

0.866025i

−1.82117

−

1.29743i

199.1

−0.500000

0.866025i

−0.866025

−

0.500000i

−0.500000

−

0.866025i

−2.10012

0.767774i

0.866025

−

0.500000i

−0.823063

−

1.42559i

1.00000

0.500000

0.866025i

0.385150

−

2.20265i

199.2

−0.500000

0.866025i

−0.866025

−

0.500000i

−0.500000

−

0.866025i

−1.40066

−

1.74303i

0.866025

−

0.500000i

0.763837

1.32301i

1.00000

0.500000

0.866025i

2.20984

−

0.341491i

199.3

−0.500000

0.866025i

−0.866025

−

0.500000i

−0.500000

−

0.866025i

1.26873

1.84128i

0.866025

−

0.500000i

−2.17283

−

3.76344i

1.00000

0.500000

0.866025i

−2.22896

0.178114i

199.4

−0.500000

0.866025i

0.866025

0.500000i

−0.500000

−

0.866025i

−0.571769

2.16173i

−0.866025

0.500000i

−0.603137

−

1.04466i

1.00000

0.500000

0.866025i

−1.58623

−

1.57603i

199.5

−0.500000

0.866025i

0.866025

0.500000i

−0.500000

−

0.866025i

−0.230377

−

2.22417i

−0.866025

0.500000i

0.432713

0.749482i

1.00000

0.500000

0.866025i

2.04138

0.912572i

199.6

−0.500000

0.866025i

0.866025

0.500000i

−0.500000

−

0.866025i

2.03420

0.928463i

−0.866025

0.500000i

1.40247

2.42916i

1.00000

0.500000

0.866025i

−1.82117

1.29743i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
65.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	390.2.x.a	✓	12
3.b	odd	2	1		1170.2.bj.d		12
5.b	even	2	1		390.2.x.b	yes	12
5.c	odd	4	1		1950.2.bc.i		12
5.c	odd	4	1		1950.2.bc.j		12
13.e	even	6	1		390.2.x.b	yes	12
15.d	odd	2	1		1170.2.bj.c		12
39.h	odd	6	1		1170.2.bj.c		12
65.l	even	6	1	inner	390.2.x.a	✓	12
65.r	odd	12	1		1950.2.bc.i		12
65.r	odd	12	1		1950.2.bc.j		12
195.y	odd	6	1		1170.2.bj.d		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
390.2.x.a	✓	12	1.a	even	1	1	trivial
390.2.x.a	✓	12	65.l	even	6	1	inner
390.2.x.b	yes	12	5.b	even	2	1
390.2.x.b	yes	12	13.e	even	6	1
1170.2.bj.c		12	15.d	odd	2	1
1170.2.bj.c		12	39.h	odd	6	1
1170.2.bj.d		12	3.b	odd	2	1
1170.2.bj.d		12	195.y	odd	6	1
1950.2.bc.i		12	5.c	odd	4	1
1950.2.bc.i		12	65.r	odd	12	1
1950.2.bc.j		12	5.c	odd	4	1
1950.2.bc.j		12	65.r	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{12} + 2 T_{7}^{11} + 19 T_{7}^{10} - 6 T_{7}^{9} + 205 T_{7}^{8} + 20 T_{7}^{7} + 708 T_{7}^{6} + \cdots + 1024$ T7^12 + 2*T7^11 + 19*T7^10 - 6*T7^9 + 205*T7^8 + 20*T7^7 + 708*T7^6 - 112*T7^5 + 1648*T7^4 - 64*T7^3 + 1664*T7^2 - 512*T7 + 1024 acting on $S_{2}^{\mathrm{new}}(390, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} + T + 1)^{6}$ (T^2 + T + 1)^6
$3$	$(T^{4} - T^{2} + 1)^{3}$ (T^4 - T^2 + 1)^3
$5$	$T^{12} + 2 T^{11} + \cdots + 15625$ T^12 + 2T^11 + 7T^10 + 6T^9 + 15T^8 + 32T^7 + 114T^6 + 160T^5 + 375T^4 + 750T^3 + 4375T^2 + 6250*T + 15625
$7$	$T^{12} + 2 T^{11} + \cdots + 1024$ T^12 + 2T^11 + 19T^10 - 6T^9 + 205T^8 + 20T^7 + 708T^6 - 112T^5 + 1648T^4 - 64T^3 + 1664T^2 - 512*T + 1024
$11$	$T^{12} - 6 T^{11} + \cdots + 16$ T^12 - 6T^11 - 2T^10 + 84T^9 + 87T^8 - 444T^7 - 330T^6 + 1446T^5 + 2585T^4 + 1176T^3 - 4T^2 - 96*T + 16
$13$	$T^{12} + 8 T^{11} + \cdots + 4826809$ T^12 + 8T^11 + 36T^10 + 40T^9 - 232T^8 - 2040T^7 - 7838T^6 - 26520T^5 - 39208T^4 + 87880T^3 + 1028196T^2 + 2970344*T + 4826809
$17$	$T^{12} - 18 T^{11} + \cdots + 65536$ T^12 - 18T^11 + 95T^10 + 234T^9 - 2695T^8 - 4632T^7 + 75744T^6 - 85248T^5 - 300288T^4 + 307200T^3 + 1269760T^2 + 491520*T + 65536
$19$	$T^{12} + 6 T^{11} + \cdots + 1982464$ T^12 + 6T^11 - 53T^10 - 390T^9 + 2701T^8 + 14568T^7 - 43876T^6 - 254496T^5 + 672400T^4 + 2581632T^3 + 837120T^2 - 3649536*T + 1982464
$23$	$T^{12} + \cdots + 190660864$ T^12 - 6T^11 - 93T^10 + 630T^9 + 7441T^8 - 53748T^7 - 129352T^6 + 1617792T^5 + 1290352T^4 - 44643456T^3 + 171279232T^2 - 284334336*T + 190660864
$29$	$T^{12} - 14 T^{11} + \cdots + 21904$ T^12 - 14T^11 + 190T^10 - 960T^9 + 6703T^8 - 15668T^7 + 148926T^6 - 196610T^5 + 1737745T^4 + 1644412T^3 + 11075036T^2 - 490768*T + 21904
$31$	$T^{12} + \cdots + 177209344$ T^12 + 198T^10 + 15433T^8 + 592640T^6 + 11339776T^4 + 93061120*T^2 + 177209344
$37$	$T^{12} + \cdots + 227195329$ T^12 + 12T^11 + 209T^10 + 1028T^9 + 14658T^8 + 60444T^7 + 734609T^6 + 1319436T^5 + 11678146T^4 + 32084244T^3 + 129327441T^2 + 175510012*T + 227195329
$41$	$T^{12} + 18 T^{11} + \cdots + 65536$ T^12 + 18T^11 + 95T^10 - 234T^9 - 2695T^8 + 4632T^7 + 75744T^6 + 85248T^5 - 300288T^4 - 307200T^3 + 1269760T^2 - 491520*T + 65536
$43$	$T^{12} + \cdots + 349241344$ T^12 + 36T^11 + 535T^10 + 3708T^9 + 6253T^8 - 57864T^7 + 54524T^6 + 6267168T^5 + 51560464T^4 + 211680000T^3 + 494054400T^2 + 627916800*T + 349241344
$47$	$(T^{6} - 8 T^{5} + \cdots + 5956)^{2}$ (T^6 - 8T^5 - 98T^4 + 1280T^3 - 4283T^2 + 2352*T + 5956)^2
$53$	$T^{12} + \cdots + 2473271824$ T^12 + 508T^10 + 87438T^8 + 5778420T^6 + 127872521T^4 + 989681000*T^2 + 2473271824
$59$	$T^{12} + \cdots + 4983230464$ T^12 + 36T^11 + 459T^10 + 972T^9 - 26427T^8 - 77796T^7 + 3235868T^6 + 37281744T^5 + 157498704T^4 + 129534336T^3 - 820266240T^2 - 752228352*T + 4983230464
$61$	$T^{12} - 10 T^{11} + \cdots + 89718784$ T^12 - 10T^11 + 135T^10 - 762T^9 + 7561T^8 - 39396T^7 + 256760T^6 - 835296T^5 + 3388992T^4 - 7106048T^3 + 26851328T^2 - 41828352*T + 89718784
$67$	$T^{12} - 4 T^{11} + \cdots + 83759104$ T^12 - 4T^11 + 164T^10 + 784T^9 + 18784T^8 + 35776T^7 + 394560T^6 - 320768T^5 + 6100480T^4 - 2632704T^3 + 25142272T^2 + 2928640*T + 83759104
$71$	$T^{12} + 12 T^{11} + \cdots + 4194304$ T^12 + 12T^11 - 172T^10 - 2640T^9 + 36080T^8 + 505152T^7 + 1150080T^6 - 6873600T^5 - 703488T^4 + 35782656T^3 + 56295424T^2 + 25165824*T + 4194304
$73$	$(T^{6} - 14 T^{5} + \cdots - 230528)^{2}$ (T^6 - 14T^5 - 263T^4 + 3068T^3 + 18280T^2 - 114240*T - 230528)^2
$79$	$(T^{6} - 2 T^{5} + \cdots - 29312)^{2}$ (T^6 - 2T^5 - 179T^4 + 788T^3 + 3892T^2 - 12576*T - 29312)^2
$83$	$(T^{6} - 36 T^{5} + \cdots - 6912)^{2}$ (T^6 - 36T^5 + 360T^4 + 144T^3 - 16560T^2 + 51840*T - 6912)^2
$89$	$T^{12} + \cdots + 1341001056256$ T^12 - 18T^11 - 309T^10 + 7506T^9 + 96433T^8 - 1612140T^7 - 15114712T^6 + 191431776T^5 + 2363129536T^4 - 1837575168T^3 - 61874871296T^2 + 39576354816*T + 1341001056256
$97$	$T^{12} + \cdots + 415519473664$ T^12 + 48T^11 + 1640T^10 + 32704T^9 + 522672T^8 + 5437824T^7 + 53039744T^6 + 348175872T^5 + 3300888832T^4 + 14104406016T^3 + 88030961664T^2 - 141504348160*T + 415519473664
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.6
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	390.2.x.a

Level	$390$
Weight	$2$
Character orbit	390.x
Analytic conductor	$3.114$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$