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Properties

Label 1620.3.g.b
Level $1620$
Weight $3$
Character orbit 1620.g
Analytic conductor $44.142$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,3,Mod(161,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.g (of order \(2\), degree \(1\), 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: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} + \cdots + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + ( - \beta_{2} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + ( - \beta_{2} + 1) q^{7} + (\beta_{9} - \beta_{4} + 2 \beta_{3}) q^{11} + (\beta_{7} + 5) q^{13} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_{3}) q^{17}+ \cdots + (\beta_{7} - 2 \beta_{2} - 13 \beta_1 - 16) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{7} + 60 q^{13} + 72 q^{19} - 60 q^{25} + 24 q^{31} - 12 q^{37} - 228 q^{43} - 96 q^{49} - 120 q^{55} + 156 q^{61} + 336 q^{67} - 24 q^{73} - 240 q^{79} + 60 q^{85} + 120 q^{91} - 192 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} + \cdots + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 137 \nu^{11} + 256 \nu^{10} + 652 \nu^{9} + 1050 \nu^{8} + 36825 \nu^{7} - 137124 \nu^{6} + \cdots - 42220035 ) / 13640319 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 419 \nu^{11} - 874 \nu^{10} - 6319 \nu^{9} + 43680 \nu^{8} - 9510 \nu^{7} - 511254 \nu^{6} + \cdots - 152641665 ) / 13640319 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1417 \nu^{11} + 5975 \nu^{10} - 5086 \nu^{9} - 56283 \nu^{8} + 326775 \nu^{7} + \cdots - 189606339 ) / 31827411 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9413 \nu^{11} + 62197 \nu^{10} - 92549 \nu^{9} - 538851 \nu^{8} + 3534990 \nu^{7} + \cdots - 2794552974 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 668 \nu^{11} - 1544 \nu^{10} + 7762 \nu^{9} - 10101 \nu^{8} - 19983 \nu^{7} + 411489 \nu^{6} + \cdots + 133155495 ) / 4546773 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2083 \nu^{11} - 4211 \nu^{10} + 3016 \nu^{9} + 92022 \nu^{8} - 507729 \nu^{7} + 511659 \nu^{6} + \cdots + 269558685 ) / 13640319 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2158 \nu^{11} - 6764 \nu^{10} - 9041 \nu^{9} + 117852 \nu^{8} - 442938 \nu^{7} + 286668 \nu^{6} + \cdots + 164333367 ) / 13640319 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5314 \nu^{11} + 18710 \nu^{10} + 34784 \nu^{9} - 201489 \nu^{8} + 666678 \nu^{7} + \cdots - 720693045 ) / 31827411 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17011 \nu^{11} + 79778 \nu^{10} - 71413 \nu^{9} - 877920 \nu^{8} + 4241550 \nu^{7} + \cdots - 3659975118 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 368 \nu^{11} + 2635 \nu^{10} - 4610 \nu^{9} - 17940 \nu^{8} + 130641 \nu^{7} - 303957 \nu^{6} + \cdots - 86861079 ) / 1948617 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 997 \nu^{11} + 4316 \nu^{10} - 1075 \nu^{9} - 49668 \nu^{8} + 248970 \nu^{7} - 262962 \nu^{6} + \cdots - 86270589 ) / 4546773 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{11} - \beta_{9} + 2\beta_{4} + 6\beta_{3} - 3\beta_{2} + 6\beta _1 + 3 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - \beta_{7} + \beta_{6} + \beta_{5} + 3\beta_{4} - 4\beta_{3} + \beta_{2} - 2\beta _1 + 13 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + 6\beta_{10} - 2\beta_{7} + 2\beta_{5} - 3\beta_{4} - 31\beta_{3} - 2\beta_{2} + 9\beta _1 - 53 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{11} + 6 \beta_{10} + 11 \beta_{9} + 9 \beta_{8} + 10 \beta_{7} - 3 \beta_{6} + 8 \beta_{5} + \cdots + 217 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 51 \beta_{11} + 33 \beta_{10} - 49 \beta_{9} + 27 \beta_{8} - 5 \beta_{7} - 36 \beta_{6} - 4 \beta_{5} + \cdots + 46 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 39 \beta_{11} - 48 \beta_{10} - 160 \beta_{9} + 63 \beta_{8} + 100 \beta_{7} + 15 \beta_{6} + \cdots - 566 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 57 \beta_{11} + 222 \beta_{10} - 895 \beta_{9} - 36 \beta_{8} - 470 \beta_{7} - 114 \beta_{6} + \cdots - 7496 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2079 \beta_{11} + 222 \beta_{10} - 1045 \beta_{9} + 810 \beta_{8} + 1660 \beta_{7} + 162 \beta_{6} + \cdots - 9323 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2283 \beta_{11} - 102 \beta_{10} - 5020 \beta_{9} + 3762 \beta_{8} - 2690 \beta_{7} - 2640 \beta_{6} + \cdots + 9073 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4188 \beta_{11} - 3342 \beta_{10} - 14638 \beta_{9} + 11007 \beta_{8} + 14965 \beta_{7} + 4116 \beta_{6} + \cdots + 5653 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5418 \beta_{11} - 23511 \beta_{10} - 61057 \beta_{9} - 13419 \beta_{8} - 52835 \beta_{7} + 5472 \beta_{6} + \cdots - 241361 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.459278 − 2.96464i
2.65605 + 1.39478i
−2.85525 − 0.920635i
−2.99781 + 0.114662i
2.89597 + 0.783177i
0.841761 − 2.87949i
0.459278 + 2.96464i
2.65605 − 1.39478i
−2.85525 + 0.920635i
−2.99781 − 0.114662i
2.89597 − 0.783177i
0.841761 + 2.87949i
0 0 0 − 2.23607i 0 −8.26980 0 0 0
161.2 0 0 0 − 2.23607i 0 −2.83166 0 0 0
161.3 0 0 0 − 2.23607i 0 −1.18917 0 0 0
161.4 0 0 0 − 2.23607i 0 1.60280 0 0 0
161.5 0 0 0 − 2.23607i 0 4.71301 0 0 0
161.6 0 0 0 − 2.23607i 0 11.9748 0 0 0
161.7 0 0 0 2.23607i 0 −8.26980 0 0 0
161.8 0 0 0 2.23607i 0 −2.83166 0 0 0
161.9 0 0 0 2.23607i 0 −1.18917 0 0 0
161.10 0 0 0 2.23607i 0 1.60280 0 0 0
161.11 0 0 0 2.23607i 0 4.71301 0 0 0
161.12 0 0 0 2.23607i 0 11.9748 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.3.g.b 12
3.b odd 2 1 inner 1620.3.g.b 12
9.c even 3 1 180.3.o.b ✓ 12
9.c even 3 1 540.3.o.b 12
9.d odd 6 1 180.3.o.b ✓ 12
9.d odd 6 1 540.3.o.b 12
36.f odd 6 1 720.3.bs.b 12
36.f odd 6 1 2160.3.bs.b 12
36.h even 6 1 720.3.bs.b 12
36.h even 6 1 2160.3.bs.b 12
45.h odd 6 1 900.3.p.c 12
45.h odd 6 1 2700.3.p.c 12
45.j even 6 1 900.3.p.c 12
45.j even 6 1 2700.3.p.c 12
45.k odd 12 2 900.3.u.c 24
45.k odd 12 2 2700.3.u.c 24
45.l even 12 2 900.3.u.c 24
45.l even 12 2 2700.3.u.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.b ✓ 12 9.c even 3 1
180.3.o.b ✓ 12 9.d odd 6 1
540.3.o.b 12 9.c even 3 1
540.3.o.b 12 9.d odd 6 1
720.3.bs.b 12 36.f odd 6 1
720.3.bs.b 12 36.h even 6 1
900.3.p.c 12 45.h odd 6 1
900.3.p.c 12 45.j even 6 1
900.3.u.c 24 45.k odd 12 2
900.3.u.c 24 45.l even 12 2
1620.3.g.b 12 1.a even 1 1 trivial
1620.3.g.b 12 3.b odd 2 1 inner
2160.3.bs.b 12 36.f odd 6 1
2160.3.bs.b 12 36.h even 6 1
2700.3.p.c 12 45.h odd 6 1
2700.3.p.c 12 45.j even 6 1
2700.3.u.c 24 45.k odd 12 2
2700.3.u.c 24 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 6T_{7}^{5} - 105T_{7}^{4} + 290T_{7}^{3} + 1425T_{7}^{2} - 996T_{7} - 2519 \) acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\). 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
$5$ \( (T^{2} + 5)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 6 T^{5} + \cdots - 2519)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 416649744 \) Copy content Toggle raw display
$13$ \( (T^{6} - 30 T^{5} + \cdots - 3964460)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 191070131587344 \) Copy content Toggle raw display
$19$ \( (T^{6} - 36 T^{5} + \cdots - 19580204)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 13626529936569 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 3840796442025 \) Copy content Toggle raw display
$31$ \( (T^{6} - 12 T^{5} + \cdots - 13330076)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 6 T^{5} + \cdots - 2144769884)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{6} + 114 T^{5} + \cdots + 25912310836)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 47\!\cdots\!49 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 29\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{6} - 78 T^{5} + \cdots - 1087778171)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 168 T^{5} + \cdots + 71107458829)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 28\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{6} + 12 T^{5} + \cdots - 2761132736)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 120 T^{5} + \cdots + 92748115600)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 15\!\cdots\!29 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13\!\cdots\!29 \) Copy content Toggle raw display
$97$ \( (T^{6} + 96 T^{5} + \cdots - 776285253104)^{2} \) Copy content Toggle raw display
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