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Properties

Label 360.4.k.b
Level $360$
Weight $4$
Character orbit 360.k
Analytic conductor $21.241$
Analytic rank $0$
Dimension $8$
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] = [360,4,Mod(181,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.181");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.k (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: \(21.2406876021\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.55839580416.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 7x^{6} - 6x^{5} + 18x^{4} - 24x^{3} + 112x^{2} - 128x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 120)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_{7} - \beta_{6} - 3 \beta_{2} - 3) q^{4} + 5 \beta_{2} q^{5} + ( - 2 \beta_{6} + \beta_{4} + \cdots - 10) q^{7}+ \cdots + ( - \beta_{7} - 3 \beta_{6} + \beta_{5} + \cdots - 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + (\beta_{7} - \beta_{6} - 3 \beta_{2} - 3) q^{4} + 5 \beta_{2} q^{5} + ( - 2 \beta_{6} + \beta_{4} + \cdots - 10) q^{7}+ \cdots + (58 \beta_{7} - 58 \beta_{6} + \cdots + 212) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 24 q^{4} - 80 q^{7} - 44 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 24 q^{4} - 80 q^{7} - 44 q^{8} + 10 q^{10} - 76 q^{14} + 88 q^{16} + 216 q^{17} + 100 q^{20} + 272 q^{22} + 32 q^{23} - 200 q^{25} + 264 q^{26} + 432 q^{28} - 136 q^{31} + 8 q^{32} + 84 q^{34} - 680 q^{38} - 100 q^{40} - 176 q^{41} - 256 q^{44} + 880 q^{46} + 848 q^{47} - 1320 q^{49} + 50 q^{50} + 1728 q^{52} - 40 q^{55} + 872 q^{56} - 36 q^{58} - 1988 q^{62} - 1728 q^{64} - 120 q^{65} - 1344 q^{68} + 20 q^{70} - 3088 q^{71} - 496 q^{73} + 2800 q^{74} + 1312 q^{76} - 2056 q^{79} + 960 q^{80} - 124 q^{82} - 2632 q^{86} - 3728 q^{88} - 3472 q^{89} - 1632 q^{92} + 3136 q^{94} - 320 q^{95} - 4256 q^{97} + 2022 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 7x^{6} - 6x^{5} + 18x^{4} - 24x^{3} + 112x^{2} - 128x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 22\nu^{6} - 15\nu^{5} + 18\nu^{4} - 234\nu^{3} + 384\nu^{2} - 592\nu + 1024 ) / 576 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -29\nu^{7} - 10\nu^{6} - 3\nu^{5} + 18\nu^{4} - 306\nu^{3} + 624\nu^{2} - 1040\nu - 1408 ) / 2880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -17\nu^{7} + 50\nu^{6} - 39\nu^{5} + 54\nu^{4} - 18\nu^{3} + 552\nu^{2} - 2000\nu + 1856 ) / 720 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -17\nu^{7} + 50\nu^{6} - 39\nu^{5} + 54\nu^{4} - 18\nu^{3} + 552\nu^{2} - 560\nu + 1856 ) / 720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\nu^{7} - 20\nu^{6} + 21\nu^{5} - 36\nu^{4} + 522\nu^{3} + 492\nu^{2} + 1520\nu + 1936 ) / 720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 79\nu^{7} - 190\nu^{6} + 393\nu^{5} - 378\nu^{4} + 846\nu^{3} - 2184\nu^{2} + 6160\nu - 8032 ) / 1440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 199\nu^{7} + 230\nu^{6} + 393\nu^{5} + 1602\nu^{4} + 1926\nu^{3} + 2496\nu^{2} + 6640\nu + 23168 ) / 2880 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 3\beta_{2} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + 2\beta_{3} - \beta_{2} - 3\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} - 2\beta_{5} - 4\beta_{4} - 2\beta_{3} + 13\beta_{2} - 6\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{7} + 10\beta_{6} - 5\beta_{5} + 6\beta_{4} + 8\beta_{3} + 19\beta_{2} + 3\beta _1 + 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{7} + 9\beta_{6} - 8\beta_{5} + 32\beta_{4} + 2\beta_{3} - 65\beta_{2} + 12\beta _1 - 61 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2\beta_{7} - 6\beta_{6} + 13\beta_{5} - 50\beta_{4} + 12\beta_{3} - 95\beta_{2} + 45\beta _1 - 113 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
0.694547 + 1.87553i
0.694547 − 1.87553i
1.61974 + 1.17321i
1.61974 − 1.17321i
−1.53189 + 1.28581i
−1.53189 − 1.28581i
0.217599 + 1.98813i
0.217599 − 1.98813i
−2.57007 − 1.18098i 0 5.21057 + 6.07042i − 5.00000i 0 1.05849 −6.22250 − 21.7550i 0 −5.90490 + 12.8504i
181.2 −2.57007 + 1.18098i 0 5.21057 − 6.07042i 5.00000i 0 1.05849 −6.22250 + 21.7550i 0 −5.90490 − 12.8504i
181.3 −0.446531 − 2.79296i 0 −7.60122 + 2.49428i 5.00000i 0 −3.73490 10.3606 + 20.1161i 0 13.9648 − 2.23265i
181.4 −0.446531 + 2.79296i 0 −7.60122 − 2.49428i − 5.00000i 0 −3.73490 10.3606 − 20.1161i 0 13.9648 + 2.23265i
181.5 0.246076 − 2.81770i 0 −7.87889 − 1.38674i − 5.00000i 0 −19.1119 −5.84623 + 21.8591i 0 −14.0885 − 1.23038i
181.6 0.246076 + 2.81770i 0 −7.87889 + 1.38674i 5.00000i 0 −19.1119 −5.84623 − 21.8591i 0 −14.0885 + 1.23038i
181.7 1.77053 − 2.20573i 0 −1.73045 − 7.81060i 5.00000i 0 −18.2117 −20.2919 − 10.0120i 0 11.0286 + 8.85264i
181.8 1.77053 + 2.20573i 0 −1.73045 + 7.81060i − 5.00000i 0 −18.2117 −20.2919 + 10.0120i 0 11.0286 − 8.85264i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.4.k.b 8
3.b odd 2 1 120.4.k.b ✓ 8
4.b odd 2 1 1440.4.k.b 8
8.b even 2 1 inner 360.4.k.b 8
8.d odd 2 1 1440.4.k.b 8
12.b even 2 1 480.4.k.b 8
24.f even 2 1 480.4.k.b 8
24.h odd 2 1 120.4.k.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.k.b ✓ 8 3.b odd 2 1
120.4.k.b ✓ 8 24.h odd 2 1
360.4.k.b 8 1.a even 1 1 trivial
360.4.k.b 8 8.b even 2 1 inner
480.4.k.b 8 12.b even 2 1
480.4.k.b 8 24.f even 2 1
1440.4.k.b 8 4.b odd 2 1
1440.4.k.b 8 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 40T_{7}^{3} + 444T_{7}^{2} + 784T_{7} - 1376 \) acting on \(S_{4}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 40 T^{3} + \cdots - 1376)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 62236278784 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 38353305600 \) Copy content Toggle raw display
$17$ \( (T^{4} - 108 T^{3} + \cdots - 61632)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 31548801716224 \) Copy content Toggle raw display
$23$ \( (T^{4} - 16 T^{3} + \cdots + 7277888)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( (T^{4} + 68 T^{3} + \cdots - 140627840)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 82\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{4} + 88 T^{3} + \cdots - 3143107120)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 70\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( (T^{4} - 424 T^{3} + \cdots - 10676481472)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 90\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{4} + 1544 T^{3} + \cdots - 11209493248)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 248 T^{3} + \cdots + 112584267280)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1028 T^{3} + \cdots - 443106379136)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + 1736 T^{3} + \cdots - 68323863856)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 1040188142960)^{2} \) Copy content Toggle raw display
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