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Properties

Label 1444.2.e.h
Level $1444$
Weight $2$
Character orbit 1444.e
Analytic conductor $11.530$
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] = [1444,2,Mod(429,1444)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1444, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1444.429");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1444 = 2^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1444.e (of order \(3\), degree \(2\), not 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: \(11.5303980519\)
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: \( x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} + \cdots + 4161 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 76)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{11} - \beta_{5}) q^{3} + \beta_{9} q^{5} + ( - \beta_{11} - \beta_{8} - \beta_{5} + \cdots - 1) q^{7} + ( - 2 \beta_{11} - \beta_{10} + \cdots - 2) q^{9} + ( - \beta_{11} - \beta_{8} - \beta_{7} + \cdots - 1) q^{11}+ \cdots + (8 \beta_{11} + \beta_{10} + 7 \beta_{8} + \cdots + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} + 3 q^{5} - 6 q^{7} - 9 q^{9} - 6 q^{11} + 12 q^{13} + 6 q^{17} + 21 q^{21} - 9 q^{25} - 18 q^{27} + 21 q^{29} + 12 q^{31} + 9 q^{33} - 3 q^{35} + 12 q^{37} + 60 q^{39} + 36 q^{41} + 18 q^{43}+ \cdots + 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} + \cdots + 4161 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{10} + 5 \nu^{9} + 9 \nu^{8} - 66 \nu^{7} + 49 \nu^{6} + 105 \nu^{5} - 553 \nu^{4} + \cdots - 5578 ) / 1853 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22 \nu^{10} - 110 \nu^{9} - 89 \nu^{8} + 1016 \nu^{7} + 230 \nu^{6} - 4708 \nu^{5} - 1786 \nu^{4} + \cdots - 13970 ) / 1853 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 995 \nu^{11} + 7831 \nu^{10} + 46995 \nu^{9} - 326814 \nu^{8} + 54349 \nu^{7} + 1703408 \nu^{6} + \cdots + 10936061 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3014 \nu^{11} + 25876 \nu^{10} - 250822 \nu^{9} - 84243 \nu^{8} + 2839567 \nu^{7} + \cdots - 91337490 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3014 \nu^{11} + 59030 \nu^{10} - 173708 \nu^{9} - 679911 \nu^{8} + 2764229 \nu^{7} + \cdots - 108083119 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5549 \nu^{11} + 28161 \nu^{10} + 42520 \nu^{9} - 88867 \nu^{8} - 1402242 \nu^{7} + \cdots + 13272986 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5549 \nu^{11} + 32878 \nu^{10} + 18935 \nu^{9} - 645473 \nu^{8} + 965692 \nu^{7} + \cdots - 41137609 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 24220 \nu^{11} - 81323 \nu^{10} - 427002 \nu^{9} + 1543220 \nu^{8} + 2521526 \nu^{7} + \cdots - 9775054 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24441 \nu^{11} + 84897 \nu^{10} + 198319 \nu^{9} - 340479 \nu^{8} - 2357880 \nu^{7} + \cdots + 137176039 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24441 \nu^{11} + 183954 \nu^{10} - 296966 \nu^{9} - 1231992 \nu^{8} + 4179882 \nu^{7} + \cdots - 114419307 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1506 \nu^{11} + 8283 \nu^{10} + 6301 \nu^{9} - 90477 \nu^{8} - 1422 \nu^{7} + 485184 \nu^{6} + \cdots - 1259576 ) / 514153 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{11} + 4\beta_{8} + 3\beta_{5} - 3\beta_{4} + 2\beta_{3} - 2\beta_{2} + \beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{11} + \beta_{10} - \beta_{9} + 4 \beta_{8} - \beta_{7} + \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + \cdots + 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4 \beta_{11} + \beta_{10} + 7 \beta_{8} + \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 8 \beta_{3} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 21 \beta_{11} + 8 \beta_{10} - 2 \beta_{9} + 38 \beta_{8} + \beta_{7} + 17 \beta_{6} + 18 \beta_{5} + \cdots + 39 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 21 \beta_{11} + 25 \beta_{10} + 5 \beta_{9} + 85 \beta_{8} + 29 \beta_{7} + 64 \beta_{6} + 27 \beta_{5} + \cdots + 57 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{11} + 15 \beta_{10} + 10 \beta_{9} + 54 \beta_{8} + 21 \beta_{7} + 57 \beta_{6} + 24 \beta_{5} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 270 \beta_{11} + 119 \beta_{10} + 133 \beta_{9} + 155 \beta_{8} + 211 \beta_{7} + 470 \beta_{6} + \cdots - 207 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1089 \beta_{11} + 166 \beta_{10} + 506 \beta_{9} - 50 \beta_{8} + 470 \beta_{7} + 1204 \beta_{6} + \cdots - 1071 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1477 \beta_{11} + 86 \beta_{10} + 542 \beta_{9} - 627 \beta_{8} + 306 \beta_{7} + 936 \beta_{6} + \cdots - 1329 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 14004 \beta_{11} - 130 \beta_{10} + 5005 \beta_{9} - 8080 \beta_{8} + 1003 \beta_{7} + 6620 \beta_{6} + \cdots - 12339 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 43776 \beta_{11} - 2189 \beta_{10} + 13871 \beta_{9} - 30497 \beta_{8} - 2023 \beta_{7} + \cdots - 36225 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times\).

\(n\) \(723\) \(1085\)
\(\chi(n)\) \(1\) \(-1 - \beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
429.1
2.75227 − 0.342020i
2.26253 + 0.984808i
2.25236 − 0.642788i
−1.75227 − 0.342020i
−1.26253 + 0.984808i
−1.25236 − 0.642788i
2.75227 + 0.342020i
2.26253 − 0.984808i
2.25236 + 0.642788i
−1.75227 + 0.342020i
−1.26253 − 0.984808i
−1.25236 + 0.642788i
0 −1.34598 + 2.33131i 0 0.641102 − 1.11042i 0 −3.34467 0 −2.12333 − 3.67771i 0
429.2 0 −0.544440 + 0.942998i 0 1.60017 − 2.77158i 0 −0.556791 0 0.907170 + 1.57126i 0
429.3 0 −0.243160 + 0.421165i 0 −1.39668 + 2.41913i 0 −3.36570 0 1.38175 + 2.39326i 0
429.4 0 0.906288 − 1.56974i 0 −0.141102 + 0.244396i 0 1.15987 0 −0.142716 − 0.247192i 0
429.5 0 1.21809 − 2.10979i 0 −1.10017 + 1.90556i 0 2.96827 0 −1.46748 − 2.54175i 0
429.6 0 1.50920 − 2.61402i 0 1.89668 − 3.28515i 0 0.139023 0 −3.05539 − 5.29210i 0
653.1 0 −1.34598 − 2.33131i 0 0.641102 + 1.11042i 0 −3.34467 0 −2.12333 + 3.67771i 0
653.2 0 −0.544440 − 0.942998i 0 1.60017 + 2.77158i 0 −0.556791 0 0.907170 − 1.57126i 0
653.3 0 −0.243160 − 0.421165i 0 −1.39668 − 2.41913i 0 −3.36570 0 1.38175 − 2.39326i 0
653.4 0 0.906288 + 1.56974i 0 −0.141102 − 0.244396i 0 1.15987 0 −0.142716 + 0.247192i 0
653.5 0 1.21809 + 2.10979i 0 −1.10017 − 1.90556i 0 2.96827 0 −1.46748 + 2.54175i 0
653.6 0 1.50920 + 2.61402i 0 1.89668 + 3.28515i 0 0.139023 0 −3.05539 + 5.29210i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 429.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.2.e.h 12
19.b odd 2 1 1444.2.e.g 12
19.c even 3 1 1444.2.a.g 6
19.c even 3 1 inner 1444.2.e.h 12
19.d odd 6 1 1444.2.a.h 6
19.d odd 6 1 1444.2.e.g 12
19.f odd 18 2 76.2.i.a ✓ 12
57.j even 18 2 684.2.bo.c 12
76.f even 6 1 5776.2.a.bw 6
76.g odd 6 1 5776.2.a.by 6
76.k even 18 2 304.2.u.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.i.a ✓ 12 19.f odd 18 2
304.2.u.e 12 76.k even 18 2
684.2.bo.c 12 57.j even 18 2
1444.2.a.g 6 19.c even 3 1
1444.2.a.h 6 19.d odd 6 1
1444.2.e.g 12 19.b odd 2 1
1444.2.e.g 12 19.d odd 6 1
1444.2.e.h 12 1.a even 1 1 trivial
1444.2.e.h 12 19.c even 3 1 inner
5776.2.a.bw 6 76.f even 6 1
5776.2.a.by 6 76.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 3 T_{3}^{11} + 18 T_{3}^{10} - 27 T_{3}^{9} + 147 T_{3}^{8} - 192 T_{3}^{7} + 709 T_{3}^{6} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(1444, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 3 T^{11} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( (T^{6} + 3 T^{5} - 12 T^{4} + \cdots - 3)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 3 T^{5} - 24 T^{4} + \cdots - 27)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 12 T^{11} + \cdots + 130321 \) Copy content Toggle raw display
$17$ \( T^{12} - 6 T^{11} + \cdots + 8982009 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 151585344 \) Copy content Toggle raw display
$29$ \( T^{12} - 21 T^{11} + \cdots + 263169 \) Copy content Toggle raw display
$31$ \( (T^{6} - 6 T^{5} + \cdots - 361)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 6 T^{5} + \cdots + 11096)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 360278361 \) Copy content Toggle raw display
$43$ \( T^{12} - 18 T^{11} + \cdots + 1203409 \) Copy content Toggle raw display
$47$ \( T^{12} + 30 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$53$ \( T^{12} - 18 T^{11} + \cdots + 95004009 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 12621848409 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1418049649 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 6080256576 \) Copy content Toggle raw display
$71$ \( T^{12} + 12 T^{11} + \cdots + 16842816 \) Copy content Toggle raw display
$73$ \( T^{12} - 24 T^{11} + \cdots + 36864 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1548343801 \) Copy content Toggle raw display
$83$ \( (T^{6} + 3 T^{5} + \cdots - 109269)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 7695324729 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 16983563041 \) Copy content Toggle raw display
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