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Properties

Label 1332.2.bt.b
Level $1332$
Weight $2$
Character orbit 1332.bt
Analytic conductor $10.636$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1332,2,Mod(145,1332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1332, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1332.145");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1332.bt (of order \(9\), degree \(6\), 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: \(10.6360735492\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 9x^{10} - 14x^{9} + 69x^{8} - 72x^{7} + 151x^{6} - 78x^{5} + 180x^{4} - 66x^{3} + 117x^{2} + 27x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 444)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{11} + \beta_{9} - 1) q^{5} + (\beta_{11} - \beta_{10} + \cdots - \beta_{2}) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots - 2 \beta_1) q^{11} + (\beta_{11} + 2 \beta_{10} + \beta_{9} + \cdots - 1) q^{13}+ \cdots + ( - 3 \beta_{11} - 4 \beta_{10} + \cdots - 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{5} - 6 q^{11} - 6 q^{13} - 12 q^{17} + 12 q^{19} - 15 q^{25} + 3 q^{29} - 36 q^{35} + 27 q^{37} - 18 q^{41} - 12 q^{43} - 9 q^{47} - 18 q^{49} + 36 q^{53} - 3 q^{55} + 6 q^{59} - 9 q^{61} - 30 q^{65}+ \cdots - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 9x^{10} - 14x^{9} + 69x^{8} - 72x^{7} + 151x^{6} - 78x^{5} + 180x^{4} - 66x^{3} + 117x^{2} + 27x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22300 \nu^{11} - 39972 \nu^{10} - 207135 \nu^{9} - 16552 \nu^{8} - 1068399 \nu^{7} + \cdots - 1430838 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 54202 \nu^{11} - 56620 \nu^{10} - 362478 \nu^{9} + 660840 \nu^{8} - 1457487 \nu^{7} + \cdots + 6547776 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 71196 \nu^{11} - 245497 \nu^{10} + 494337 \nu^{9} - 3002842 \nu^{8} + 7566829 \nu^{7} + \cdots + 2219844 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 74027 \nu^{11} + 333687 \nu^{10} + 850507 \nu^{9} + 1595916 \nu^{8} + 1206871 \nu^{7} + \cdots + 966006 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 107334 \nu^{11} + 74027 \nu^{10} - 632319 \nu^{9} + 2353183 \nu^{8} - 5810130 \nu^{7} + \cdots - 3471528 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 125253 \nu^{11} - 151332 \nu^{10} + 682852 \nu^{9} - 3373952 \nu^{8} + 7397665 \nu^{7} + \cdots - 2772270 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 144076 \nu^{11} + 324620 \nu^{10} + 1890631 \nu^{9} + 1006066 \nu^{8} + 9884477 \nu^{7} + \cdots + 1707330 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 158982 \nu^{11} + 22300 \nu^{10} - 1390866 \nu^{9} + 2432883 \nu^{8} - 10953206 \nu^{7} + \cdots + 1548036 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 200696 \nu^{11} - 199280 \nu^{10} - 1988619 \nu^{9} + 1276385 \nu^{8} - 12069988 \nu^{7} + \cdots - 1219512 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 352831 \nu^{11} + 72400 \nu^{10} + 2638417 \nu^{9} - 5007488 \nu^{8} + 18310912 \nu^{7} + \cdots + 4112292 ) / 5269059 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{6} - \beta_{5} + \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 5 \beta_{10} + 2 \beta_{8} + \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{11} - 10 \beta_{10} - 11 \beta_{9} - 7 \beta_{8} - 9 \beta_{7} - 19 \beta_{6} + \cdots + 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{11} - 31 \beta_{10} + 26 \beta_{9} - 10 \beta_{8} + 10 \beta_{7} + 25 \beta_{6} - 14 \beta_{5} + \cdots - 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 25 \beta_{11} + 182 \beta_{10} + 83 \beta_{8} + 58 \beta_{7} + 90 \beta_{6} + 83 \beta_{5} + 99 \beta_{4} + \cdots + 92 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 246 \beta_{11} - 262 \beta_{10} - 269 \beta_{9} - 154 \beta_{8} - 246 \beta_{7} - 559 \beta_{6} + \cdots + 233 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 530 \beta_{11} - 895 \beta_{10} + 866 \beta_{9} - 262 \beta_{8} + 262 \beta_{7} + 868 \beta_{6} + \cdots - 866 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 868 \beta_{11} + 5477 \beta_{10} + 2432 \beta_{8} + 1564 \beta_{7} + 2859 \beta_{6} + 2432 \beta_{5} + \cdots + 2660 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7665 \beta_{11} - 8347 \beta_{10} - 8372 \beta_{9} - 5047 \beta_{8} - 7665 \beta_{7} + \cdots + 6656 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 15458 \beta_{11} - 27715 \beta_{10} + 26024 \beta_{9} - 8347 \beta_{8} + 8347 \beta_{7} + 25741 \beta_{6} + \cdots - 26024 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1332\mathbb{Z}\right)^\times\).

\(n\) \(667\) \(1037\) \(1297\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−0.130479 + 0.225997i
0.896524 − 1.55282i
0.735449 − 1.27384i
−0.561801 + 0.973068i
0.621301 + 1.07612i
−1.56099 − 2.70372i
−0.130479 − 0.225997i
0.896524 + 1.55282i
0.735449 + 1.27384i
−0.561801 − 0.973068i
0.621301 − 1.07612i
−1.56099 + 2.70372i
0 0 0 −1.68491 − 0.613258i 0 2.82666 + 1.02882i 0 0 0
145.2 0 0 0 0.245221 + 0.0892531i 0 −2.06061 − 0.750002i 0 0 0
181.1 0 0 0 −0.860729 + 0.722237i 0 1.42570 − 1.19631i 0 0 0
181.2 0 0 0 1.12677 − 0.945475i 0 −1.25205 + 1.05060i 0 0 0
793.1 0 0 0 −0.542127 + 3.07456i 0 −0.803091 + 4.55455i 0 0 0
793.2 0 0 0 0.215775 − 1.22372i 0 −0.136602 + 0.774709i 0 0 0
937.1 0 0 0 −1.68491 + 0.613258i 0 2.82666 − 1.02882i 0 0 0
937.2 0 0 0 0.245221 − 0.0892531i 0 −2.06061 + 0.750002i 0 0 0
1045.1 0 0 0 −0.860729 − 0.722237i 0 1.42570 + 1.19631i 0 0 0
1045.2 0 0 0 1.12677 + 0.945475i 0 −1.25205 − 1.05060i 0 0 0
1117.1 0 0 0 −0.542127 − 3.07456i 0 −0.803091 − 4.55455i 0 0 0
1117.2 0 0 0 0.215775 + 1.22372i 0 −0.136602 − 0.774709i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1332.2.bt.b 12
3.b odd 2 1 444.2.u.a ✓ 12
37.f even 9 1 inner 1332.2.bt.b 12
111.p odd 18 1 444.2.u.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.u.a ✓ 12 3.b odd 2 1
444.2.u.a ✓ 12 111.p odd 18 1
1332.2.bt.b 12 1.a even 1 1 trivial
1332.2.bt.b 12 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 3 T_{5}^{11} + 12 T_{5}^{10} + 20 T_{5}^{9} + 9 T_{5}^{8} + 21 T_{5}^{7} + 70 T_{5}^{6} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(1332, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{12} + 9 T^{10} + \cdots + 5329 \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + \cdots + 110889 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{12} + 12 T^{11} + \cdots + 59049 \) Copy content Toggle raw display
$19$ \( T^{12} - 12 T^{11} + \cdots + 494209 \) Copy content Toggle raw display
$23$ \( T^{12} + 63 T^{10} + \cdots + 239121 \) Copy content Toggle raw display
$29$ \( T^{12} - 3 T^{11} + \cdots + 106929 \) Copy content Toggle raw display
$31$ \( (T^{6} - 42 T^{4} + \cdots + 109)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( T^{12} + 18 T^{11} + \cdots + 2152089 \) Copy content Toggle raw display
$43$ \( (T^{6} + 6 T^{5} + \cdots + 3743)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 231983361 \) Copy content Toggle raw display
$53$ \( T^{12} - 36 T^{11} + \cdots + 6765201 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 184932801 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 417425761 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 143736121 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 231331178961 \) Copy content Toggle raw display
$73$ \( (T^{6} + 21 T^{5} + \cdots + 258371)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 336020786929 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1659992049 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 8131891329 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 159693769 \) Copy content Toggle raw display
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