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Properties

Label 180.2.r.a
Level $180$
Weight $2$
Character orbit 180.r
Analytic conductor $1.437$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [180,2,Mod(49,180)]
 
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, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 180.r (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: \(1.43730723638\)
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: \( x^{12} - 4x^{10} + 10x^{8} - 6x^{6} + 90x^{4} - 324x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{9} q^{5} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_{3}) q^{7}+ \cdots + (\beta_{5} + \beta_{4} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{9} q^{5} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_{3}) q^{7}+ \cdots + (3 \beta_{10} + 3 \beta_{9} - \beta_{6} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{5} + 8 q^{9} + 2 q^{11} - 5 q^{15} + 10 q^{21} - 3 q^{25} - 18 q^{29} + 6 q^{31} - 34 q^{35} - 42 q^{39} + 14 q^{41} - 31 q^{45} + 16 q^{51} - 6 q^{55} - 34 q^{59} + 6 q^{61} + 15 q^{65} + 14 q^{69} + 41 q^{75} - 6 q^{79} - 8 q^{81} - 12 q^{85} + 112 q^{89} + 12 q^{91} + 36 q^{95} + 82 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{10} + 10x^{8} - 6x^{6} + 90x^{4} - 324x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} + 31\nu^{8} + 17\nu^{6} - 21\nu^{4} + 45\nu^{2} + 2997 ) / 1296 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} + 31\nu^{9} + 17\nu^{7} - 21\nu^{5} + 45\nu^{3} + 2997\nu ) / 1296 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} - 3 \nu^{10} + 13 \nu^{9} + 39 \nu^{8} - 19 \nu^{7} - 57 \nu^{6} + 69 \nu^{5} + \cdots + 2673 ) / 972 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4 \nu^{11} + 15 \nu^{10} - 52 \nu^{9} - 249 \nu^{8} + 76 \nu^{7} + 177 \nu^{6} - 276 \nu^{5} + \cdots - 19683 ) / 3888 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 24 \nu^{10} - 13 \nu^{9} - 42 \nu^{8} + 19 \nu^{7} + 24 \nu^{6} - 69 \nu^{5} - 90 \nu^{4} + \cdots - 3402 ) / 972 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{11} + 24 \nu^{10} + 13 \nu^{9} - 42 \nu^{8} - 19 \nu^{7} + 24 \nu^{6} + 69 \nu^{5} + \cdots - 3402 ) / 972 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} + 4\nu^{9} - 10\nu^{7} + 6\nu^{5} - 90\nu^{3} + 324\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5 \nu^{11} + 15 \nu^{10} - 11 \nu^{9} - 33 \nu^{8} + 11 \nu^{7} + 33 \nu^{6} + 9 \nu^{5} + 27 \nu^{4} + \cdots - 2619 ) / 864 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5 \nu^{11} + 15 \nu^{10} + 11 \nu^{9} - 33 \nu^{8} - 11 \nu^{7} + 33 \nu^{6} - 9 \nu^{5} + \cdots - 2619 ) / 864 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 37\nu^{11} - 139\nu^{9} + 91\nu^{7} - 375\nu^{5} + 3519\nu^{3} - 12393\nu ) / 3888 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{10} + 3\beta_{9} - 3\beta_{7} - \beta_{6} - 2\beta_{5} - 3\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{11} - \beta_{10} + \beta_{9} - 8\beta_{8} + 5\beta_{7} - 5\beta_{6} - 5\beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{10} + 9\beta_{9} - 3\beta_{7} - 10\beta_{6} - 3\beta_{5} - 10\beta_{4} + 14\beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -11\beta_{11} + \beta_{10} - \beta_{9} - 28\beta_{8} - 2\beta_{7} + 2\beta_{6} + 4\beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -3\beta_{10} - 3\beta_{9} + 6\beta_{6} - 4\beta_{5} + 2\beta_{4} + 29\beta_{2} - 90 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2\beta_{11} - 7\beta_{10} + 7\beta_{9} + 16\beta_{8} + 2\beta_{7} - 2\beta_{6} + 39\beta_{3} - 98\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -3\beta_{10} - 3\beta_{9} + 12\beta_{7} + 37\beta_{6} - 88\beta_{5} - 63\beta_{4} - 51\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4\beta_{11} - 134\beta_{10} + 134\beta_{9} + 143\beta_{8} - 32\beta_{7} + 32\beta_{6} + 86\beta_{3} - 158\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-1\) \(1\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−1.62372 − 0.602950i
−1.54493 + 0.783067i
−0.690466 − 1.58848i
0.690466 + 1.58848i
1.54493 − 0.783067i
1.62372 + 0.602950i
−1.62372 + 0.602950i
−1.54493 − 0.783067i
−0.690466 + 1.58848i
0.690466 − 1.58848i
1.54493 + 0.783067i
1.62372 − 0.602950i
0 −1.62372 − 0.602950i 0 0.194047 + 2.22763i 0 −3.26460 + 1.88482i 0 2.27290 + 1.95804i 0
49.2 0 −1.54493 + 0.783067i 0 0.801701 − 2.08741i 0 1.96151 − 1.13248i 0 1.77361 − 2.41957i 0
49.3 0 −0.690466 − 1.58848i 0 −1.87175 − 1.22333i 0 1.11574 − 0.644175i 0 −2.04651 + 2.19358i 0
49.4 0 0.690466 + 1.58848i 0 1.99531 + 1.00932i 0 −1.11574 + 0.644175i 0 −2.04651 + 2.19358i 0
49.5 0 1.54493 − 0.783067i 0 1.40690 − 1.73800i 0 −1.96151 + 1.13248i 0 1.77361 − 2.41957i 0
49.6 0 1.62372 + 0.602950i 0 −2.02621 + 0.945767i 0 3.26460 − 1.88482i 0 2.27290 + 1.95804i 0
169.1 0 −1.62372 + 0.602950i 0 0.194047 − 2.22763i 0 −3.26460 − 1.88482i 0 2.27290 − 1.95804i 0
169.2 0 −1.54493 − 0.783067i 0 0.801701 + 2.08741i 0 1.96151 + 1.13248i 0 1.77361 + 2.41957i 0
169.3 0 −0.690466 + 1.58848i 0 −1.87175 + 1.22333i 0 1.11574 + 0.644175i 0 −2.04651 − 2.19358i 0
169.4 0 0.690466 − 1.58848i 0 1.99531 − 1.00932i 0 −1.11574 − 0.644175i 0 −2.04651 − 2.19358i 0
169.5 0 1.54493 + 0.783067i 0 1.40690 + 1.73800i 0 −1.96151 − 1.13248i 0 1.77361 + 2.41957i 0
169.6 0 1.62372 − 0.602950i 0 −2.02621 − 0.945767i 0 3.26460 + 1.88482i 0 2.27290 − 1.95804i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.2.r.a ✓ 12
3.b odd 2 1 540.2.r.a 12
4.b odd 2 1 720.2.by.e 12
5.b even 2 1 inner 180.2.r.a ✓ 12
5.c odd 4 2 900.2.i.f 12
9.c even 3 1 inner 180.2.r.a ✓ 12
9.c even 3 1 1620.2.d.c 6
9.d odd 6 1 540.2.r.a 12
9.d odd 6 1 1620.2.d.d 6
12.b even 2 1 2160.2.by.e 12
15.d odd 2 1 540.2.r.a 12
15.e even 4 2 2700.2.i.f 12
20.d odd 2 1 720.2.by.e 12
36.f odd 6 1 720.2.by.e 12
36.h even 6 1 2160.2.by.e 12
45.h odd 6 1 540.2.r.a 12
45.h odd 6 1 1620.2.d.d 6
45.j even 6 1 inner 180.2.r.a ✓ 12
45.j even 6 1 1620.2.d.c 6
45.k odd 12 2 900.2.i.f 12
45.k odd 12 2 8100.2.a.bc 6
45.l even 12 2 2700.2.i.f 12
45.l even 12 2 8100.2.a.bd 6
60.h even 2 1 2160.2.by.e 12
180.n even 6 1 2160.2.by.e 12
180.p odd 6 1 720.2.by.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.r.a ✓ 12 1.a even 1 1 trivial
180.2.r.a ✓ 12 5.b even 2 1 inner
180.2.r.a ✓ 12 9.c even 3 1 inner
180.2.r.a ✓ 12 45.j even 6 1 inner
540.2.r.a 12 3.b odd 2 1
540.2.r.a 12 9.d odd 6 1
540.2.r.a 12 15.d odd 2 1
540.2.r.a 12 45.h odd 6 1
720.2.by.e 12 4.b odd 2 1
720.2.by.e 12 20.d odd 2 1
720.2.by.e 12 36.f odd 6 1
720.2.by.e 12 180.p odd 6 1
900.2.i.f 12 5.c odd 4 2
900.2.i.f 12 45.k odd 12 2
1620.2.d.c 6 9.c even 3 1
1620.2.d.c 6 45.j even 6 1
1620.2.d.d 6 9.d odd 6 1
1620.2.d.d 6 45.h odd 6 1
2160.2.by.e 12 12.b even 2 1
2160.2.by.e 12 36.h even 6 1
2160.2.by.e 12 60.h even 2 1
2160.2.by.e 12 180.n even 6 1
2700.2.i.f 12 15.e even 4 2
2700.2.i.f 12 45.l even 12 2
8100.2.a.bc 6 45.k odd 12 2
8100.2.a.bd 6 45.l even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(180, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} - T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} - 21 T^{10} + \cdots + 14641 \) Copy content Toggle raw display
$11$ \( (T^{6} - T^{5} + 23 T^{4} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 39 T^{10} + \cdots + 104976 \) Copy content Toggle raw display
$17$ \( (T^{6} + 36 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 42 T + 72)^{4} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 815730721 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 9)^{6} \) Copy content Toggle raw display
$31$ \( (T^{6} - 3 T^{5} + \cdots + 128164)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 60 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 7 T^{5} + \cdots + 9409)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 8318169616 \) Copy content Toggle raw display
$47$ \( T^{12} - 105 T^{10} + \cdots + 923521 \) Copy content Toggle raw display
$53$ \( (T^{6} + 264 T^{4} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 17 T^{5} + \cdots + 7744)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 3 T^{5} + \cdots + 961)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 214358881 \) Copy content Toggle raw display
$71$ \( (T^{3} - 42 T - 72)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + 384 T^{4} + \cdots + 369664)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 3 T^{5} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 3637263079921 \) Copy content Toggle raw display
$89$ \( (T^{3} - 28 T^{2} + \cdots - 662)^{4} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 3102044416 \) Copy content Toggle raw display
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