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Properties

Label 8046.2.a.q
Level $8046$
Weight $2$
Character orbit 8046.a
Self dual yes
Analytic conductor $64.248$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8046,2,Mod(1,8046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8046, 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("8046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8046.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: \(64.2476334663\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + \cdots - 7083 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - \beta_1 q^{5} - \beta_{11} q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - \beta_1 q^{5} - \beta_{11} q^{7} - q^{8} + \beta_1 q^{10} + (\beta_{12} - \beta_{9}) q^{11} + ( - \beta_{11} + \beta_{4}) q^{13} + \beta_{11} q^{14} + q^{16} + (\beta_{3} - 1) q^{17} + (\beta_{12} - \beta_{11} - \beta_{9} + \cdots + 1) q^{19}+ \cdots + (\beta_{9} + \beta_{8} - \beta_{6} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} - 4 q^{14} + 14 q^{16} - 9 q^{17} + 14 q^{19} - 2 q^{20} + 2 q^{22} - 30 q^{23} + 18 q^{25} - 4 q^{26} + 4 q^{28} - 6 q^{29} + 11 q^{31} - 14 q^{32} + 9 q^{34} + 18 q^{35} + 13 q^{37} - 14 q^{38} + 2 q^{40} + 2 q^{41} + 12 q^{43} - 2 q^{44} + 30 q^{46} - 21 q^{47} + 32 q^{49} - 18 q^{50} + 4 q^{52} - 22 q^{53} - 7 q^{55} - 4 q^{56} + 6 q^{58} - 14 q^{59} + 31 q^{61} - 11 q^{62} + 14 q^{64} + 24 q^{67} - 9 q^{68} - 18 q^{70} - 28 q^{71} + 24 q^{73} - 13 q^{74} + 14 q^{76} - 16 q^{77} + 65 q^{79} - 2 q^{80} - 2 q^{82} + 15 q^{83} - 19 q^{85} - 12 q^{86} + 2 q^{88} + 11 q^{89} + 68 q^{91} - 30 q^{92} + 21 q^{94} - 8 q^{95} + 23 q^{97} - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + \cdots - 7083 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6885747582 \nu^{13} + 33914318185 \nu^{12} + 202430140259 \nu^{11} - 1106373672733 \nu^{10} + \cdots + 19597736159910 ) / 837106466514 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11004140892 \nu^{13} + 49436090470 \nu^{12} + 340112131925 \nu^{11} + \cdots + 34191931183770 ) / 837106466514 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19118330939 \nu^{13} + 86554061461 \nu^{12} + 595251271041 \nu^{11} + \cdots + 53337950263866 ) / 837106466514 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23639242991 \nu^{13} + 109271241761 \nu^{12} + 703785403742 \nu^{11} + \cdots + 49744016665776 ) / 837106466514 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25609954388 \nu^{13} + 109910586859 \nu^{12} + 842563500174 \nu^{11} + \cdots + 105376014149448 ) / 837106466514 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31764401845 \nu^{13} - 134861372299 \nu^{12} - 1026991737247 \nu^{11} + \cdots - 101151393317133 ) / 837106466514 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37435189105 \nu^{13} - 162933719250 \nu^{12} - 1191384212828 \nu^{11} + \cdots - 113370111470034 ) / 837106466514 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37752739446 \nu^{13} - 161274827528 \nu^{12} - 1202197738381 \nu^{11} + \cdots - 109196500860438 ) / 837106466514 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 38350226953 \nu^{13} + 161631948769 \nu^{12} + 1243644226300 \nu^{11} + \cdots + 123665771390928 ) / 837106466514 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 38926641911 \nu^{13} + 173061237061 \nu^{12} + 1207022732481 \nu^{11} + \cdots + 106272852513750 ) / 837106466514 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 48334159531 \nu^{13} + 220483599101 \nu^{12} + 1494088890444 \nu^{11} + \cdots + 150684116159928 ) / 837106466514 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 102355805795 \nu^{13} + 456007662555 \nu^{12} + 3186087569260 \nu^{11} + \cdots + 295489137970695 ) / 837106466514 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} - \beta_{10} - \beta_{8} + \beta_{6} + \beta_{5} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} + 2\beta_{11} - 2\beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + 9\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{13} - 9 \beta_{12} - 2 \beta_{11} - 15 \beta_{10} - 4 \beta_{9} - 18 \beta_{8} - 2 \beta_{7} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 24 \beta_{13} - 3 \beta_{12} + 51 \beta_{11} - 45 \beta_{10} - 14 \beta_{9} - 29 \beta_{8} + \cdots + 70 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 147 \beta_{13} - 92 \beta_{12} - 37 \beta_{11} - 251 \beta_{10} - 91 \beta_{9} - 330 \beta_{8} + \cdots + 1249 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 495 \beta_{13} - 61 \beta_{12} + 968 \beta_{11} - 869 \beta_{10} - 225 \beta_{9} - 659 \beta_{8} + \cdots + 1468 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2920 \beta_{13} - 1131 \beta_{12} - 437 \beta_{11} - 4441 \beta_{10} - 1716 \beta_{9} - 6010 \beta_{8} + \cdots + 20427 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 9734 \beta_{13} - 1052 \beta_{12} + 17068 \beta_{11} - 16349 \beta_{10} - 4100 \beta_{9} + \cdots + 30197 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 54851 \beta_{13} - 16214 \beta_{12} - 2601 \beta_{11} - 80429 \beta_{10} - 31101 \beta_{9} + \cdots + 348706 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 187187 \beta_{13} - 18966 \beta_{12} + 295104 \beta_{11} - 305918 \beta_{10} - 78259 \beta_{9} + \cdots + 613563 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1012139 \beta_{13} - 256618 \beta_{12} + 47933 \beta_{11} - 1470566 \beta_{10} - 559263 \beta_{9} + \cdots + 6080293 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3556677 \beta_{13} - 363417 \beta_{12} + 5088500 \beta_{11} - 5713758 \beta_{10} - 1507447 \beta_{9} + \cdots + 12308542 \) Copy content Toggle raw display

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
4.30228
3.13855
2.95487
2.34343
2.04034
1.69938
0.316300
−0.541630
−1.46245
−1.53522
−2.39851
−2.42130
−2.45968
−3.97637
−1.00000 0 1.00000 −4.30228 0 −2.58072 −1.00000 0 4.30228
1.2 −1.00000 0 1.00000 −3.13855 0 −2.45592 −1.00000 0 3.13855
1.3 −1.00000 0 1.00000 −2.95487 0 2.46629 −1.00000 0 2.95487
1.4 −1.00000 0 1.00000 −2.34343 0 0.305937 −1.00000 0 2.34343
1.5 −1.00000 0 1.00000 −2.04034 0 2.45177 −1.00000 0 2.04034
1.6 −1.00000 0 1.00000 −1.69938 0 −0.423602 −1.00000 0 1.69938
1.7 −1.00000 0 1.00000 −0.316300 0 −2.35786 −1.00000 0 0.316300
1.8 −1.00000 0 1.00000 0.541630 0 3.83971 −1.00000 0 −0.541630
1.9 −1.00000 0 1.00000 1.46245 0 −0.313566 −1.00000 0 −1.46245
1.10 −1.00000 0 1.00000 1.53522 0 4.84041 −1.00000 0 −1.53522
1.11 −1.00000 0 1.00000 2.39851 0 2.84704 −1.00000 0 −2.39851
1.12 −1.00000 0 1.00000 2.42130 0 −4.62365 −1.00000 0 −2.42130
1.13 −1.00000 0 1.00000 2.45968 0 −3.97455 −1.00000 0 −2.45968
1.14 −1.00000 0 1.00000 3.97637 0 3.97870 −1.00000 0 −3.97637
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(149\) \(1\)

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 8046.2.a.q ✓ 14
3.b odd 2 1 8046.2.a.r yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.q ✓ 14 1.a even 1 1 trivial
8046.2.a.r yes 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\( T_{5}^{14} + 2 T_{5}^{13} - 42 T_{5}^{12} - 70 T_{5}^{11} + 680 T_{5}^{10} + 882 T_{5}^{9} - 5559 T_{5}^{8} + \cdots - 7083 \) Copy content Toggle raw display
\( T_{11}^{14} + 2 T_{11}^{13} - 93 T_{11}^{12} - 150 T_{11}^{11} + 3294 T_{11}^{10} + 3815 T_{11}^{9} + \cdots - 331776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots - 7083 \) Copy content Toggle raw display
$7$ \( T^{14} - 4 T^{13} + \cdots - 14207 \) Copy content Toggle raw display
$11$ \( T^{14} + 2 T^{13} + \cdots - 331776 \) Copy content Toggle raw display
$13$ \( T^{14} - 4 T^{13} + \cdots + 35947449 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 149673024 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 1993837504 \) Copy content Toggle raw display
$23$ \( T^{14} + 30 T^{13} + \cdots + 2953827 \) Copy content Toggle raw display
$29$ \( T^{14} + 6 T^{13} + \cdots - 592587 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 325771295 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 3105338304 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 155961639 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 303883523 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 202855104 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 79069521057 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 993589824 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 561724628544 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 100336092480 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 1347360351345 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 3927013138821 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 5603301056 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 9100099410624 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 5482645053867 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 56680087660608 \) Copy content Toggle raw display
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