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Properties

Label 360.2.w.d
Level $360$
Weight $2$
Character orbit 360.w
Analytic conductor $2.875$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,2,Mod(163,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.163");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.w (of order \(4\), 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: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{12} + 28x^{8} + 80x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{15} q^{2} - \beta_{2} q^{4} + ( - \beta_{15} + \beta_{10}) q^{5} + (\beta_{11} - \beta_{6} - \beta_{2}) q^{7} + (\beta_{5} - \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{15} q^{2} - \beta_{2} q^{4} + ( - \beta_{15} + \beta_{10}) q^{5} + (\beta_{11} - \beta_{6} - \beta_{2}) q^{7} + (\beta_{5} - \beta_{4}) q^{8} + (\beta_{9} + 2 \beta_{8} + \beta_{2}) q^{10} + (\beta_{15} + \beta_{14} - \beta_1) q^{11} + ( - \beta_{7} - \beta_{3}) q^{13} + (\beta_{14} + \beta_{13} + \cdots - 2 \beta_{4}) q^{14}+ \cdots + ( - 2 \beta_{14} + 2 \beta_{12} + \cdots - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{10} - 20 q^{16} + 36 q^{22} - 24 q^{25} + 44 q^{28} + 32 q^{40} - 16 q^{43} - 32 q^{46} - 8 q^{52} - 44 q^{58} + 16 q^{67} - 44 q^{70} + 80 q^{73} - 16 q^{76} - 8 q^{82} - 28 q^{88} - 32 q^{91} - 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 5x^{12} + 28x^{8} + 80x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{14} - 4\nu^{12} + 29\nu^{10} - 36\nu^{8} + 84\nu^{6} - 256\nu^{4} + 176\nu^{2} - 384 ) / 320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{15} - 4\nu^{13} - 11\nu^{11} - 36\nu^{9} + 44\nu^{7} + 64\nu^{5} - 144\nu^{3} - 384\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} + 12\nu^{13} - 11\nu^{11} + 108\nu^{9} + 44\nu^{7} + 448\nu^{5} - 144\nu^{3} + 1152\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{12} - 9\nu^{8} - 24\nu^{4} - 96 ) / 40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{14} - 8\nu^{12} + 11\nu^{10} + 8\nu^{8} - 44\nu^{6} - 112\nu^{4} + 144\nu^{2} - 128 ) / 320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{14} - 7\nu^{10} - 12\nu^{6} - 48\nu^{2} ) / 320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{14} - 2\nu^{12} - \nu^{10} + 2\nu^{8} + 14\nu^{6} - 28\nu^{4} - 24\nu^{2} - 32 ) / 80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} + 2\nu^{13} + 9\nu^{11} - 2\nu^{9} + 24\nu^{7} + 28\nu^{5} + 136\nu^{3} - 48\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5\nu^{14} - 12\nu^{12} + 25\nu^{10} - 28\nu^{8} + 140\nu^{6} - 48\nu^{4} + 400\nu^{2} - 192 ) / 320 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5\nu^{15} + 12\nu^{13} + 25\nu^{11} + 28\nu^{9} + 140\nu^{7} + 48\nu^{5} + 400\nu^{3} + 192\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -\nu^{15} + 2\nu^{13} - 9\nu^{11} - 2\nu^{9} - 24\nu^{7} + 28\nu^{5} + 24\nu^{3} - 48\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5\nu^{15} - 12\nu^{13} + 25\nu^{11} - 28\nu^{9} + 140\nu^{7} - 48\nu^{5} + 400\nu^{3} - 192\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3\nu^{15} + 7\nu^{11} + 12\nu^{7} + 48\nu^{3} ) / 320 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{13} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{9} - \beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{12} + 2\beta_{10} + \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} + \beta_{9} + 4\beta_{8} - 2\beta_{7} - \beta_{6} + \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -4\beta_{15} + \beta_{14} - \beta_{13} + 4\beta_{12} - 2\beta_{10} + 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -\beta_{11} + 3\beta_{9} + 2\beta_{8} + 2\beta_{7} - 4\beta_{6} + \beta_{3} + \beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4\beta_{14} + 2\beta_{12} - 6\beta_{10} + 3\beta_{5} - 9\beta_{4} - 12\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3\beta_{11} - 9\beta_{9} - 8\beta_{8} + 6\beta_{7} - 3\beta_{6} + 3\beta_{3} - 12\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -4\beta_{15} - 9\beta_{14} - 15\beta_{13} - 12\beta_{12} + 18\beta_{10} - 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -15\beta_{11} - 3\beta_{9} - 18\beta_{8} - 18\beta_{7} - 4\beta_{6} + 15\beta_{3} + 15\beta_{2} - 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -36\beta_{14} + 30\beta_{12} + 6\beta_{10} - 11\beta_{5} + 17\beta_{4} + 12\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -11\beta_{11} + 17\beta_{9} - 104\beta_{8} - 6\beta_{7} + 11\beta_{6} - 11\beta_{3} + 12\beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 132\beta_{15} + \beta_{14} + 23\beta_{13} + 12\beta_{12} - 34\beta_{10} + 11\beta_{4} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(\beta_{8}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−0.512386 + 1.31813i
0.788026 + 1.17431i
1.17431 + 0.788026i
−1.31813 + 0.512386i
1.31813 − 0.512386i
−1.17431 − 0.788026i
−0.788026 − 1.17431i
0.512386 − 1.31813i
−0.512386 − 1.31813i
0.788026 − 1.17431i
1.17431 − 0.788026i
−1.31813 − 0.512386i
1.31813 + 0.512386i
−1.17431 + 0.788026i
−0.788026 + 1.17431i
0.512386 + 1.31813i
−1.31813 − 0.512386i 0 1.47492 + 1.35078i 1.83051 + 1.28422i 0 2.94984 + 2.94984i −1.25201 − 2.53623i 0 −1.75483 − 2.63069i
163.2 −1.17431 + 0.788026i 0 0.758030 − 1.85078i 0.386289 − 2.20245i 0 1.51606 + 1.51606i 0.568298 + 2.77075i 0 1.28196 + 2.89077i
163.3 −0.788026 + 1.17431i 0 −0.758030 − 1.85078i −0.386289 + 2.20245i 0 −1.51606 − 1.51606i 2.77075 + 0.568298i 0 −2.28196 − 2.18921i
163.4 −0.512386 − 1.31813i 0 −1.47492 + 1.35078i 1.83051 + 1.28422i 0 −2.94984 − 2.94984i 2.53623 + 1.25201i 0 0.754834 − 3.07087i
163.5 0.512386 + 1.31813i 0 −1.47492 + 1.35078i −1.83051 − 1.28422i 0 −2.94984 − 2.94984i −2.53623 − 1.25201i 0 0.754834 − 3.07087i
163.6 0.788026 − 1.17431i 0 −0.758030 − 1.85078i 0.386289 − 2.20245i 0 −1.51606 − 1.51606i −2.77075 − 0.568298i 0 −2.28196 − 2.18921i
163.7 1.17431 − 0.788026i 0 0.758030 − 1.85078i −0.386289 + 2.20245i 0 1.51606 + 1.51606i −0.568298 − 2.77075i 0 1.28196 + 2.89077i
163.8 1.31813 + 0.512386i 0 1.47492 + 1.35078i −1.83051 − 1.28422i 0 2.94984 + 2.94984i 1.25201 + 2.53623i 0 −1.75483 − 2.63069i
307.1 −1.31813 + 0.512386i 0 1.47492 − 1.35078i 1.83051 − 1.28422i 0 2.94984 − 2.94984i −1.25201 + 2.53623i 0 −1.75483 + 2.63069i
307.2 −1.17431 − 0.788026i 0 0.758030 + 1.85078i 0.386289 + 2.20245i 0 1.51606 − 1.51606i 0.568298 − 2.77075i 0 1.28196 − 2.89077i
307.3 −0.788026 − 1.17431i 0 −0.758030 + 1.85078i −0.386289 − 2.20245i 0 −1.51606 + 1.51606i 2.77075 − 0.568298i 0 −2.28196 + 2.18921i
307.4 −0.512386 + 1.31813i 0 −1.47492 − 1.35078i 1.83051 − 1.28422i 0 −2.94984 + 2.94984i 2.53623 − 1.25201i 0 0.754834 + 3.07087i
307.5 0.512386 − 1.31813i 0 −1.47492 − 1.35078i −1.83051 + 1.28422i 0 −2.94984 + 2.94984i −2.53623 + 1.25201i 0 0.754834 + 3.07087i
307.6 0.788026 + 1.17431i 0 −0.758030 + 1.85078i 0.386289 + 2.20245i 0 −1.51606 + 1.51606i −2.77075 + 0.568298i 0 −2.28196 + 2.18921i
307.7 1.17431 + 0.788026i 0 0.758030 + 1.85078i −0.386289 − 2.20245i 0 1.51606 − 1.51606i −0.568298 + 2.77075i 0 1.28196 − 2.89077i
307.8 1.31813 − 0.512386i 0 1.47492 − 1.35078i −1.83051 + 1.28422i 0 2.94984 − 2.94984i 1.25201 − 2.53623i 0 −1.75483 + 2.63069i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
8.d odd 2 1 inner
15.e even 4 1 inner
24.f even 2 1 inner
40.k even 4 1 inner
120.q odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.w.d ✓ 16
3.b odd 2 1 inner 360.2.w.d ✓ 16
4.b odd 2 1 1440.2.bi.d 16
5.c odd 4 1 inner 360.2.w.d ✓ 16
8.b even 2 1 1440.2.bi.d 16
8.d odd 2 1 inner 360.2.w.d ✓ 16
12.b even 2 1 1440.2.bi.d 16
15.e even 4 1 inner 360.2.w.d ✓ 16
20.e even 4 1 1440.2.bi.d 16
24.f even 2 1 inner 360.2.w.d ✓ 16
24.h odd 2 1 1440.2.bi.d 16
40.i odd 4 1 1440.2.bi.d 16
40.k even 4 1 inner 360.2.w.d ✓ 16
60.l odd 4 1 1440.2.bi.d 16
120.q odd 4 1 inner 360.2.w.d ✓ 16
120.w even 4 1 1440.2.bi.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.w.d ✓ 16 1.a even 1 1 trivial
360.2.w.d ✓ 16 3.b odd 2 1 inner
360.2.w.d ✓ 16 5.c odd 4 1 inner
360.2.w.d ✓ 16 8.d odd 2 1 inner
360.2.w.d ✓ 16 15.e even 4 1 inner
360.2.w.d ✓ 16 24.f even 2 1 inner
360.2.w.d ✓ 16 40.k even 4 1 inner
360.2.w.d ✓ 16 120.q odd 4 1 inner
1440.2.bi.d 16 4.b odd 2 1
1440.2.bi.d 16 8.b even 2 1
1440.2.bi.d 16 12.b even 2 1
1440.2.bi.d 16 20.e even 4 1
1440.2.bi.d 16 24.h odd 2 1
1440.2.bi.d 16 40.i odd 4 1
1440.2.bi.d 16 60.l odd 4 1
1440.2.bi.d 16 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 324T_{7}^{4} + 6400 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 5 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 6 T^{6} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 324 T^{4} + 6400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 18 T^{2} + 40)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 804 T^{4} + 102400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1796 T^{4} + 25600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{8} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2880 T^{4} + 16384)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 58 T^{2} + 800)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 76 T^{2} + 1280)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 804 T^{4} + 102400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 92 T^{2} + 640)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} + \cdots + 6400)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 11520 T^{4} + 262144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 420 T^{4} + 16384)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 18 T^{2} + 40)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 220 T^{2} + 8000)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + \cdots + 6400)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{2} - 10 T + 50)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 76 T^{2} + 1280)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 21136 T^{4} + 104857600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 348 T^{2} + 2560)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + \cdots + 2500)^{4} \) Copy content Toggle raw display
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