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Properties

Label 360.4.s.b
Level $360$
Weight $4$
Character orbit 360.s
Analytic conductor $21.241$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,4,Mod(17,360)]
 
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([0, 0, 2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.s (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: \(21.2406876021\)
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: \( x^{16} + 209x^{12} + 9456x^{8} + 5504x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{27}\cdot 5^{4} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{13} - 2 \beta_{6} - \beta_{5}) q^{5} + (\beta_{3} - 4 \beta_{2} - 4) q^{7} + ( - \beta_{15} - \beta_{13} - 4 \beta_{5}) q^{11} + (\beta_{10} + \beta_{4} - 15 \beta_{2} + 14) q^{13} + (\beta_{15} + \beta_{14} + \cdots - 3 \beta_{5}) q^{17}+ \cdots + (6 \beta_{11} - 21 \beta_{9} + \cdots - 80) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{7} + 232 q^{13} - 384 q^{25} + 240 q^{31} + 624 q^{37} + 1600 q^{43} + 1264 q^{55} - 4416 q^{61} + 2640 q^{67} + 2536 q^{73} + 1800 q^{85} - 4192 q^{91} - 1496 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 209x^{12} + 9456x^{8} + 5504x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 619\nu^{12} + 128747\nu^{8} + 5749352\nu^{4} + 1035124 ) / 90660 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -195\nu^{14} - 40783\nu^{10} - 1849900\nu^{6} - 1379200\nu^{2} ) / 290112 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 956 \nu^{14} + 76 \nu^{12} + 200073 \nu^{10} + 15368 \nu^{8} + 9093933 \nu^{6} + 622268 \nu^{4} + \cdots - 1527344 ) / 181320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 956 \nu^{14} - 76 \nu^{12} + 200073 \nu^{10} - 15368 \nu^{8} + 9093933 \nu^{6} - 622268 \nu^{4} + \cdots + 1527344 ) / 181320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6983 \nu^{15} + 2102 \nu^{13} + 1458559 \nu^{11} + 438566 \nu^{9} + 65848504 \nu^{7} + \cdots + 7110912 \nu ) / 2901120 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6983 \nu^{15} - 2102 \nu^{13} + 1458559 \nu^{11} - 438566 \nu^{9} + 65848504 \nu^{7} + \cdots - 7110912 \nu ) / 2901120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4717 \nu^{15} + 8711 \nu^{13} + 988421 \nu^{11} + 1820703 \nu^{9} + 45145496 \nu^{7} + \cdots + 44467456 \nu ) / 1450560 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4717 \nu^{15} + 8711 \nu^{13} - 988421 \nu^{11} + 1820703 \nu^{9} - 45145496 \nu^{7} + \cdots + 44467456 \nu ) / 1450560 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -21411\nu^{14} - 4468303\nu^{10} - 201136588\nu^{6} - 62855296\nu^{2} ) / 1450560 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12921 \nu^{14} + 4904 \nu^{12} + 2698253 \nu^{10} + 1026632 \nu^{8} + 121682708 \nu^{6} + \cdots + 11885904 ) / 362640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12921 \nu^{14} - 4904 \nu^{12} + 2698253 \nu^{10} - 1026632 \nu^{8} + 121682708 \nu^{6} + \cdots - 11885904 ) / 362640 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 78143 \nu^{15} + 3924 \nu^{13} + 16328199 \nu^{11} + 817332 \nu^{9} + 738139464 \nu^{7} + \cdots - 20092416 \nu ) / 2901120 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 78143 \nu^{15} - 3924 \nu^{13} + 16328199 \nu^{11} - 817332 \nu^{9} + 738139464 \nu^{7} + \cdots + 20092416 \nu ) / 2901120 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 110673 \nu^{15} + 8784 \nu^{13} + 23137529 \nu^{11} + 1820752 \nu^{9} + 1047929384 \nu^{7} + \cdots - 75727616 \nu ) / 2901120 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 110673 \nu^{15} + 8784 \nu^{13} - 23137529 \nu^{11} + 1820752 \nu^{9} - 1047929384 \nu^{7} + \cdots - 75727616 \nu ) / 2901120 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} + 4\beta_{13} - 4\beta_{12} + \beta_{8} + \beta_{7} + 5\beta_{6} - 5\beta_{5} ) / 40 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} + 6\beta_{9} - 12\beta_{4} - 12\beta_{3} - 214\beta_{2} ) / 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{15} - 3\beta_{14} + 6\beta_{13} + 6\beta_{12} - 4\beta_{8} + 4\beta_{7} - 25\beta_{6} - 25\beta_{5} ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{11} - 9\beta_{10} + 68\beta_{4} - 68\beta_{3} + 44\beta _1 - 1058 ) / 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 13 \beta_{15} - 13 \beta_{14} - 24 \beta_{13} + 24 \beta_{12} - 19 \beta_{8} - 19 \beta_{7} + \cdots + 167 \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -129\beta_{11} - 129\beta_{10} - 604\beta_{9} + 768\beta_{4} + 768\beta_{3} + 11636\beta_{2} ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 719 \beta_{15} + 719 \beta_{14} - 2048 \beta_{13} - 2048 \beta_{12} + 357 \beta_{8} + \cdots + 12005 \beta_{5} ) / 20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1713\beta_{11} + 1713\beta_{10} - 8816\beta_{4} + 8816\beta_{3} - 7868\beta _1 + 126066 ) / 20 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8127 \beta_{15} + 8127 \beta_{14} + 9988 \beta_{13} - 9988 \beta_{12} + 17457 \beta_{8} + \cdots - 160005 \beta_{5} ) / 20 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4373\beta_{11} + 4373\beta_{10} + 19868\beta_{9} - 20496\beta_{4} - 20496\beta_{3} - 294228\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 93407 \beta_{15} - 93407 \beta_{14} + 314864 \beta_{13} + 314864 \beta_{12} - 2821 \beta_{8} + \cdots - 2046885 \beta_{5} ) / 20 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 272697\beta_{11} - 272697\beta_{10} + 1202064\beta_{4} - 1202064\beta_{3} + 1230732\beta _1 - 16427314 ) / 20 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1086799 \beta_{15} - 1086799 \beta_{14} - 963556 \beta_{13} + 963556 \beta_{12} + \cdots + 25563125 \beta_{5} ) / 20 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 3352681 \beta_{11} - 3352681 \beta_{10} - 15067596 \beta_{9} + 14189712 \beta_{4} + \cdots + 198019684 \beta_{2} ) / 20 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2551251 \beta_{15} + 2551251 \beta_{14} - 9301296 \beta_{13} - 9301296 \beta_{12} + \cdots + 62911073 \beta_{5} ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−2.00890 + 2.00890i
2.44635 − 2.44635i
−0.605632 + 0.605632i
0.335980 − 0.335980i
−0.335980 + 0.335980i
0.605632 − 0.605632i
−2.44635 + 2.44635i
2.00890 − 2.00890i
−2.00890 − 2.00890i
2.44635 + 2.44635i
−0.605632 − 0.605632i
0.335980 + 0.335980i
−0.335980 − 0.335980i
0.605632 + 0.605632i
−2.44635 − 2.44635i
2.00890 + 2.00890i
0 0 0 −10.4827 + 3.88762i 0 7.82474 + 7.82474i 0 0 0
17.2 0 0 0 −7.08469 − 8.64912i 0 3.01926 + 3.01926i 0 0 0
17.3 0 0 0 −5.12357 + 9.93725i 0 −14.2458 − 14.2458i 0 0 0
17.4 0 0 0 −3.95851 − 10.4561i 0 −12.5982 − 12.5982i 0 0 0
17.5 0 0 0 3.95851 + 10.4561i 0 −12.5982 − 12.5982i 0 0 0
17.6 0 0 0 5.12357 − 9.93725i 0 −14.2458 − 14.2458i 0 0 0
17.7 0 0 0 7.08469 + 8.64912i 0 3.01926 + 3.01926i 0 0 0
17.8 0 0 0 10.4827 − 3.88762i 0 7.82474 + 7.82474i 0 0 0
233.1 0 0 0 −10.4827 − 3.88762i 0 7.82474 − 7.82474i 0 0 0
233.2 0 0 0 −7.08469 + 8.64912i 0 3.01926 − 3.01926i 0 0 0
233.3 0 0 0 −5.12357 − 9.93725i 0 −14.2458 + 14.2458i 0 0 0
233.4 0 0 0 −3.95851 + 10.4561i 0 −12.5982 + 12.5982i 0 0 0
233.5 0 0 0 3.95851 − 10.4561i 0 −12.5982 + 12.5982i 0 0 0
233.6 0 0 0 5.12357 + 9.93725i 0 −14.2458 + 14.2458i 0 0 0
233.7 0 0 0 7.08469 − 8.64912i 0 3.01926 − 3.01926i 0 0 0
233.8 0 0 0 10.4827 + 3.88762i 0 7.82474 − 7.82474i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
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
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.4.s.b ✓ 16
3.b odd 2 1 inner 360.4.s.b ✓ 16
4.b odd 2 1 720.4.w.g 16
5.c odd 4 1 inner 360.4.s.b ✓ 16
12.b even 2 1 720.4.w.g 16
15.e even 4 1 inner 360.4.s.b ✓ 16
20.e even 4 1 720.4.w.g 16
60.l odd 4 1 720.4.w.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.s.b ✓ 16 1.a even 1 1 trivial
360.4.s.b ✓ 16 3.b odd 2 1 inner
360.4.s.b ✓ 16 5.c odd 4 1 inner
360.4.s.b ✓ 16 15.e even 4 1 inner
720.4.w.g 16 4.b odd 2 1
720.4.w.g 16 12.b even 2 1
720.4.w.g 16 20.e even 4 1
720.4.w.g 16 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 32 T_{7}^{7} + 512 T_{7}^{6} - 384 T_{7}^{5} - 2944 T_{7}^{4} + 380928 T_{7}^{3} + \cdots + 287641600 \) acting on \(S_{4}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + 32 T^{7} + \cdots + 287641600)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 1189460709376)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 6963666543376)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 5200443080704)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 60 T^{3} + \cdots - 20146688)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 31\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 38\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 1104 T^{3} + \cdots - 263401472)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 20\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
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