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Properties

Label 180.5.l.b
Level $180$
Weight $5$
Character orbit 180.l
Analytic conductor $18.607$
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] = [180,5,Mod(37,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.37");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 180.l (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: \(18.6065933551\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{6} + 28x^{5} + 97x^{4} - 168x^{3} + 288x^{2} + 864x + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 60)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{6} + \beta_{2} + \cdots - 1) q^{5}+ \cdots + (\beta_{6} - 2 \beta_{5} + \beta_{2} + \cdots - 18) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{6} + \beta_{2} + \cdots - 1) q^{5}+ \cdots + ( - 66 \beta_{7} - 116 \beta_{6} + \cdots - 1985) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{5} - 140 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{5} - 140 q^{7} - 288 q^{11} + 300 q^{13} + 1020 q^{17} - 1320 q^{23} - 2036 q^{25} + 1472 q^{31} - 1416 q^{35} - 300 q^{37} + 3480 q^{41} - 6360 q^{43} - 4800 q^{47} - 3900 q^{53} + 11172 q^{55} - 11544 q^{61} + 16380 q^{65} - 920 q^{67} + 3600 q^{71} + 2960 q^{73} - 19800 q^{77} - 12720 q^{83} + 1396 q^{85} + 32400 q^{91} + 37200 q^{95} - 15600 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} + 28x^{5} + 97x^{4} - 168x^{3} + 288x^{2} + 864x + 1296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 343\nu^{7} - 2023\nu^{6} + 4880\nu^{5} + 5404\nu^{4} + 17563\nu^{3} - 175455\nu^{2} + 180684\nu + 145584 ) / 486540 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 679 \nu^{7} - 7639 \nu^{6} + 27410 \nu^{5} - 33368 \nu^{4} - 80621 \nu^{3} - 299175 \nu^{2} + \cdots - 1025208 ) / 162180 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5111 \nu^{7} + 31556 \nu^{6} - 53080 \nu^{5} - 326228 \nu^{4} + 795289 \nu^{3} + 1209960 \nu^{2} + \cdots - 8360928 ) / 973080 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6509 \nu^{7} - 25394 \nu^{6} + 70870 \nu^{5} + 88532 \nu^{4} + 602549 \nu^{3} - 204570 \nu^{2} + \cdots - 1979208 ) / 486540 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15607 \nu^{7} + 67372 \nu^{6} - 81200 \nu^{5} - 547276 \nu^{4} - 1091767 \nu^{3} + \cdots - 7861536 ) / 973080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8593 \nu^{7} + 32818 \nu^{6} - 36410 \nu^{5} - 414964 \nu^{4} - 394633 \nu^{3} + 1248210 \nu^{2} + \cdots - 11926764 ) / 486540 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23551 \nu^{7} + 132916 \nu^{6} - 384800 \nu^{5} - 167788 \nu^{4} - 1461151 \nu^{3} + \cdots - 4002048 ) / 973080 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + 2\beta_{6} - \beta_{5} - 2\beta_{3} - 2\beta_{2} - 10\beta _1 + 10 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} + \beta_{6} - 2\beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} - 87\beta _1 - 1 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -30\beta_{7} - 5\beta_{6} - 33\beta_{4} + 14\beta_{3} - 22\beta_{2} - 294\beta _1 - 311 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -32\beta_{7} - 56\beta_{6} + 53\beta_{5} - 53\beta_{4} + 25\beta_{3} - 21\beta_{2} + 21\beta _1 - 1216 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 77\beta_{7} - 574\beta_{6} + 721\beta_{5} + 350\beta_{3} + 126\beta_{2} + 6526\beta _1 - 6078 ) / 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1061 \beta_{7} - 727 \beta_{6} + 1140 \beta_{5} + 1140 \beta_{4} + 413 \beta_{3} + 549 \beta_{2} + \cdots + 727 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11422\beta_{7} + 1581\beta_{6} + 14761\beta_{4} - 1582\beta_{3} + 6502\beta_{2} + 129590\beta _1 + 134511 ) / 20 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−1.84806 − 1.84806i
3.17086 + 3.17086i
1.62332 + 1.62332i
−0.946115 − 0.946115i
−1.84806 + 1.84806i
3.17086 − 3.17086i
1.62332 − 1.62332i
−0.946115 + 0.946115i
0 0 0 −11.5321 + 22.1813i 0 −9.71439 − 9.71439i 0 0 0
37.2 0 0 0 −11.0239 − 22.4382i 0 21.1834 + 21.1834i 0 0 0
37.3 0 0 0 −4.94005 − 24.5071i 0 −52.2300 − 52.2300i 0 0 0
37.4 0 0 0 21.4961 + 12.7640i 0 −29.2390 − 29.2390i 0 0 0
73.1 0 0 0 −11.5321 − 22.1813i 0 −9.71439 + 9.71439i 0 0 0
73.2 0 0 0 −11.0239 + 22.4382i 0 21.1834 − 21.1834i 0 0 0
73.3 0 0 0 −4.94005 + 24.5071i 0 −52.2300 + 52.2300i 0 0 0
73.4 0 0 0 21.4961 − 12.7640i 0 −29.2390 + 29.2390i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.5.l.b 8
3.b odd 2 1 60.5.k.a ✓ 8
5.b even 2 1 900.5.l.k 8
5.c odd 4 1 inner 180.5.l.b 8
5.c odd 4 1 900.5.l.k 8
12.b even 2 1 240.5.bg.d 8
15.d odd 2 1 300.5.k.d 8
15.e even 4 1 60.5.k.a ✓ 8
15.e even 4 1 300.5.k.d 8
60.l odd 4 1 240.5.bg.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.k.a ✓ 8 3.b odd 2 1
60.5.k.a ✓ 8 15.e even 4 1
180.5.l.b 8 1.a even 1 1 trivial
180.5.l.b 8 5.c odd 4 1 inner
240.5.bg.d 8 12.b even 2 1
240.5.bg.d 8 60.l odd 4 1
300.5.k.d 8 15.d odd 2 1
300.5.k.d 8 15.e even 4 1
900.5.l.k 8 5.b even 2 1
900.5.l.k 8 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 140 T_{7}^{7} + 9800 T_{7}^{6} + 245480 T_{7}^{5} + 3113188 T_{7}^{4} + 69856960 T_{7}^{3} + \cdots + 1580189787136 \) acting on \(S_{5}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 1580189787136 \) Copy content Toggle raw display
$11$ \( (T^{4} + 144 T^{3} + \cdots + 311125216)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 69\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 1071194996864)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 3221711600000)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 162789583002624)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 301327179200000)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 66\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 71\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
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