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Properties

Label 180.3.o.b
Level $180$
Weight $3$
Character orbit 180.o
Analytic conductor $4.905$
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] = [180,3,Mod(41,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.41");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.o (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: \(4.90464475849\)
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: \( x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} + \cdots + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{8} q^{5} + ( - \beta_{9} - \beta_{8} + \cdots + \beta_1) q^{7}+ \cdots + (\beta_{11} - \beta_{10} + \beta_{8} + \cdots + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{8} q^{5} + ( - \beta_{9} - \beta_{8} + \cdots + \beta_1) q^{7}+ \cdots + (7 \beta_{11} - 2 \beta_{10} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} - 6 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{3} - 6 q^{7} + 26 q^{9} + 48 q^{11} - 30 q^{13} + 10 q^{15} + 72 q^{19} - 128 q^{21} - 78 q^{23} + 30 q^{25} - 106 q^{27} + 150 q^{29} - 12 q^{31} + 96 q^{33} - 12 q^{37} + 40 q^{39} + 90 q^{41} + 114 q^{43} - 20 q^{45} + 12 q^{47} + 48 q^{49} - 144 q^{51} - 120 q^{55} - 158 q^{57} + 48 q^{59} - 78 q^{61} - 212 q^{63} - 150 q^{65} - 168 q^{67} - 150 q^{69} - 24 q^{73} - 10 q^{75} - 258 q^{77} + 120 q^{79} + 434 q^{81} + 114 q^{83} - 30 q^{85} - 330 q^{87} + 120 q^{91} + 82 q^{93} + 120 q^{95} + 96 q^{97} - 120 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 242 \nu^{11} - 32960 \nu^{10} + 98305 \nu^{9} + 191607 \nu^{8} - 1678629 \nu^{7} + \cdots + 1811977614 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 239 \nu^{11} + 7615 \nu^{10} - 11402 \nu^{9} - 41574 \nu^{8} + 169188 \nu^{7} + \cdots - 109358748 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32 \nu^{11} - 298 \nu^{10} - 388 \nu^{9} + 2047 \nu^{8} - 9822 \nu^{7} + 68334 \nu^{6} + \cdots + 42456231 ) / 3536379 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1264 \nu^{11} - 7514 \nu^{10} + 7570 \nu^{9} + 69045 \nu^{8} - 447600 \nu^{7} + 644742 \nu^{6} + \cdots + 255564072 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 631 \nu^{11} - 1051 \nu^{10} + 15968 \nu^{9} - 34098 \nu^{8} - 73320 \nu^{7} + \cdots + 377145963 ) / 31827411 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 653 \nu^{11} + 1682 \nu^{10} - 15724 \nu^{9} + 45705 \nu^{8} + 66441 \nu^{7} - 1137465 \nu^{6} + \cdots - 426924270 ) / 31827411 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4012 \nu^{11} - 10310 \nu^{10} + 3856 \nu^{9} + 127275 \nu^{8} - 811137 \nu^{7} + \cdots + 459460269 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 554 \nu^{11} + 2386 \nu^{10} - 755 \nu^{9} - 32880 \nu^{8} + 119670 \nu^{7} - 115038 \nu^{6} + \cdots - 74637936 ) / 10609137 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 9665 \nu^{11} + 29059 \nu^{10} + 12094 \nu^{9} - 278598 \nu^{8} + 1431609 \nu^{7} + \cdots - 638142543 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14237 \nu^{11} + 71170 \nu^{10} - 45263 \nu^{9} - 637482 \nu^{8} + 3402366 \nu^{7} + \cdots - 2067718833 ) / 95482233 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} - 2\beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} - \beta_{9} + 5\beta_{8} + \beta_{6} + 3\beta_{5} + \beta_{4} - 9\beta_{3} - \beta_{2} + 5\beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{11} + 6 \beta_{10} + 8 \beta_{9} + 20 \beta_{8} + 3 \beta_{7} - 2 \beta_{6} - 9 \beta_{5} + \cdots + 46 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{11} + 27 \beta_{10} - 16 \beta_{9} + 5 \beta_{8} - 21 \beta_{7} - 2 \beta_{6} + 66 \beta_{5} + \cdots - 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{11} + 15 \beta_{10} + 20 \beta_{9} + 83 \beta_{8} - 78 \beta_{7} - 137 \beta_{6} + \cdots + 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 140 \beta_{11} - 78 \beta_{10} - 412 \beta_{9} - 37 \beta_{8} - 342 \beta_{7} - 173 \beta_{6} + \cdots - 1172 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 178 \beta_{11} + 522 \beta_{10} - 322 \beta_{9} + 968 \beta_{8} + 600 \beta_{7} + 304 \beta_{6} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 280 \beta_{11} + 1500 \beta_{10} - 2851 \beta_{9} - 646 \beta_{8} - 996 \beta_{7} + 1510 \beta_{6} + \cdots - 98 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2488 \beta_{11} + 5043 \beta_{10} - 3598 \beta_{9} + 2924 \beta_{8} - 513 \beta_{7} - 389 \beta_{6} + \cdots + 5191 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7589 \beta_{11} - 19899 \beta_{10} - 23272 \beta_{9} - 2200 \beta_{8} - 25518 \beta_{7} - 12926 \beta_{6} + \cdots - 20057 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
−2.99781 + 0.114662i
−2.85525 + 0.920635i
0.459278 + 2.96464i
0.841761 − 2.87949i
2.65605 + 1.39478i
2.89597 − 0.783177i
−2.99781 − 0.114662i
−2.85525 − 0.920635i
0.459278 − 2.96464i
0.841761 + 2.87949i
2.65605 − 1.39478i
2.89597 + 0.783177i
0 −2.99781 + 0.114662i 0 1.93649 + 1.11803i 0 −0.801399 − 1.38806i 0 8.97371 − 0.687471i 0
41.2 0 −2.85525 + 0.920635i 0 −1.93649 − 1.11803i 0 0.594587 + 1.02985i 0 7.30486 − 5.25728i 0
41.3 0 0.459278 + 2.96464i 0 −1.93649 − 1.11803i 0 4.13490 + 7.16186i 0 −8.57813 + 2.72318i 0
41.4 0 0.841761 − 2.87949i 0 1.93649 + 1.11803i 0 −5.98742 − 10.3705i 0 −7.58288 − 4.84768i 0
41.5 0 2.65605 + 1.39478i 0 1.93649 + 1.11803i 0 1.41583 + 2.45229i 0 5.10917 + 7.40921i 0
41.6 0 2.89597 − 0.783177i 0 −1.93649 − 1.11803i 0 −2.35650 − 4.08158i 0 7.77327 − 4.53611i 0
101.1 0 −2.99781 − 0.114662i 0 1.93649 − 1.11803i 0 −0.801399 + 1.38806i 0 8.97371 + 0.687471i 0
101.2 0 −2.85525 − 0.920635i 0 −1.93649 + 1.11803i 0 0.594587 − 1.02985i 0 7.30486 + 5.25728i 0
101.3 0 0.459278 − 2.96464i 0 −1.93649 + 1.11803i 0 4.13490 − 7.16186i 0 −8.57813 − 2.72318i 0
101.4 0 0.841761 + 2.87949i 0 1.93649 − 1.11803i 0 −5.98742 + 10.3705i 0 −7.58288 + 4.84768i 0
101.5 0 2.65605 − 1.39478i 0 1.93649 − 1.11803i 0 1.41583 − 2.45229i 0 5.10917 − 7.40921i 0
101.6 0 2.89597 + 0.783177i 0 −1.93649 + 1.11803i 0 −2.35650 + 4.08158i 0 7.77327 + 4.53611i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.o.b ✓ 12
3.b odd 2 1 540.3.o.b 12
4.b odd 2 1 720.3.bs.b 12
5.b even 2 1 900.3.p.c 12
5.c odd 4 2 900.3.u.c 24
9.c even 3 1 540.3.o.b 12
9.c even 3 1 1620.3.g.b 12
9.d odd 6 1 inner 180.3.o.b ✓ 12
9.d odd 6 1 1620.3.g.b 12
12.b even 2 1 2160.3.bs.b 12
15.d odd 2 1 2700.3.p.c 12
15.e even 4 2 2700.3.u.c 24
36.f odd 6 1 2160.3.bs.b 12
36.h even 6 1 720.3.bs.b 12
45.h odd 6 1 900.3.p.c 12
45.j even 6 1 2700.3.p.c 12
45.k odd 12 2 2700.3.u.c 24
45.l even 12 2 900.3.u.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.b ✓ 12 1.a even 1 1 trivial
180.3.o.b ✓ 12 9.d odd 6 1 inner
540.3.o.b 12 3.b odd 2 1
540.3.o.b 12 9.c even 3 1
720.3.bs.b 12 4.b odd 2 1
720.3.bs.b 12 36.h even 6 1
900.3.p.c 12 5.b even 2 1
900.3.p.c 12 45.h odd 6 1
900.3.u.c 24 5.c odd 4 2
900.3.u.c 24 45.l even 12 2
1620.3.g.b 12 9.c even 3 1
1620.3.g.b 12 9.d odd 6 1
2160.3.bs.b 12 12.b even 2 1
2160.3.bs.b 12 36.f odd 6 1
2700.3.p.c 12 15.d odd 2 1
2700.3.p.c 12 45.j even 6 1
2700.3.u.c 24 15.e even 4 2
2700.3.u.c 24 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 6 T_{7}^{11} + 141 T_{7}^{10} - 50 T_{7}^{9} + 11340 T_{7}^{8} + 14346 T_{7}^{7} + \cdots + 6345361 \) acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + \cdots + 6345361 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 416649744 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 15716943091600 \) 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
$31$ \( T^{12} + \cdots + 177690926165776 \) 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
$43$ \( T^{12} + \cdots + 67\!\cdots\!96 \) 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
$61$ \( T^{12} + \cdots + 11\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 50\!\cdots\!41 \) 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
$79$ \( T^{12} + \cdots + 86\!\cdots\!00 \) 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
$97$ \( T^{12} + \cdots + 60\!\cdots\!16 \) Copy content Toggle raw display
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