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Properties

Label 360.4.s.c
Level $360$
Weight $4$
Character orbit 360.s
Analytic conductor $21.241$
Analytic rank $0$
Dimension $16$
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] = [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: \( x^{16} + 112 x^{14} + 4660 x^{12} + 89172 x^{10} + 793554 x^{8} + 2902488 x^{6} + 3754000 x^{4} + \cdots + 5041 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{25} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{5}) q^{5} + (\beta_{11} + \beta_{4} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{5}) q^{5} + (\beta_{11} + \beta_{4} + 1) q^{7} + ( - \beta_{10} - \beta_{9} + \beta_{5}) q^{11} + (\beta_{13} + \beta_{12} + 2 \beta_{4} + \cdots - 2) q^{13}+ \cdots + (22 \beta_{14} - 25 \beta_{13} + \cdots - 307) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{7} - 40 q^{13} - 160 q^{25} + 464 q^{31} + 32 q^{37} - 416 q^{43} + 2592 q^{61} + 368 q^{67} - 3352 q^{73} - 2040 q^{85} + 7552 q^{91} - 4536 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 112 x^{14} + 4660 x^{12} + 89172 x^{10} + 793554 x^{8} + 2902488 x^{6} + 3754000 x^{4} + \cdots + 5041 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17141 \nu^{14} + 1906123 \nu^{12} + 78410859 \nu^{10} + 1471495137 \nu^{8} + 12627930303 \nu^{6} + \cdots + 3388470139 ) / 2050319880 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 49094289 \nu^{15} + 585572216 \nu^{14} + 5478355401 \nu^{13} + 66038984908 \nu^{12} + \cdots + 253689535223116 ) / 14401300385700 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 174345536 \nu^{15} + 585572216 \nu^{14} + 19553568349 \nu^{13} + 66038984908 \nu^{12} + \cdots + 253689535223116 ) / 14401300385700 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8937593 \nu^{15} + 1001276950 \nu^{13} + 41678761980 \nu^{11} + 798196064790 \nu^{9} + \cdots + 6046422091360 \nu ) / 576052015428 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 673 \nu^{15} + 75305 \nu^{13} + 3128299 \nu^{11} + 59690771 \nu^{9} + 528140087 \nu^{7} + \cdots + 255368149 \nu ) / 16520280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 55170575501 \nu^{15} - 4978795125 \nu^{14} - 6167569337665 \nu^{13} + \cdots - 46\!\cdots\!27 ) / 403236410799600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 55170575501 \nu^{15} + 4978795125 \nu^{14} - 6167569337665 \nu^{13} + \cdots + 46\!\cdots\!27 ) / 403236410799600 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 61935321665 \nu^{15} - 13443794975 \nu^{14} - 6945792721525 \nu^{13} + \cdots - 11\!\cdots\!81 ) / 403236410799600 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1177157873 \nu^{15} + 243428345 \nu^{14} + 131834598445 \nu^{13} + 27125857711 \nu^{12} + \cdots + 22138398492979 ) / 5679386067600 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1177157873 \nu^{15} - 243428345 \nu^{14} + 131834598445 \nu^{13} - 27125857711 \nu^{12} + \cdots - 22138398492979 ) / 5679386067600 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3636012727 \nu^{15} - 351556216 \nu^{14} - 407290291028 \nu^{13} + \cdots - 170317809720956 ) / 14401300385700 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3636012727 \nu^{15} - 351556216 \nu^{14} + 407290291028 \nu^{13} - 39965660588 \nu^{12} + \cdots - 170317809720956 ) / 14401300385700 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 77480757 \nu^{15} - 4311448 \nu^{14} - 8680567788 \nu^{13} - 485893274 \nu^{12} + \cdots + 577787424502 ) / 202835216700 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 77480757 \nu^{15} - 4311448 \nu^{14} + 8680567788 \nu^{13} - 485893274 \nu^{12} + \cdots + 780622641202 ) / 202835216700 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 200684106149 \nu^{15} + 8818412645 \nu^{14} + 22473618548785 \nu^{13} + \cdots - 47\!\cdots\!45 ) / 403236410799600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - 2\beta_{10} - 3\beta_{9} - \beta_{8} - \beta_{7} + 16\beta_{4} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + 4 \beta_{14} + 4 \beta_{13} - \beta_{9} + \beta_{8} + 3 \beta_{7} - 2 \beta_{6} + \cdots - 112 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 31 \beta_{15} + 12 \beta_{14} - 12 \beta_{13} + 46 \beta_{10} + 77 \beta_{9} + 31 \beta_{8} + \cdots - 12 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 58 \beta_{15} - 128 \beta_{14} - 128 \beta_{13} - 5 \beta_{12} - 5 \beta_{11} + 16 \beta_{10} + \cdots + 3224 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 999 \beta_{15} - 610 \beta_{14} + 610 \beta_{13} + 50 \beta_{12} - 50 \beta_{11} - 1462 \beta_{10} + \cdots + 610 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2425 \beta_{15} + 4288 \beta_{14} + 4288 \beta_{13} + 345 \beta_{12} + 345 \beta_{11} - 944 \beta_{10} + \cdots - 106684 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 34203 \beta_{15} + 25830 \beta_{14} - 25830 \beta_{13} - 4130 \beta_{12} + 4130 \beta_{11} + \cdots - 25830 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 46834 \beta_{15} - 75364 \beta_{14} - 75364 \beta_{13} - 9215 \beta_{12} - 9215 \beta_{11} + \cdots + 1868908 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1214793 \beta_{15} - 1035828 \beta_{14} + 1035828 \beta_{13} + 235980 \beta_{12} - 235980 \beta_{11} + \cdots + 1035828 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3517349 \beta_{15} + 5441460 \beta_{14} + 5441460 \beta_{13} + 892000 \beta_{12} + 892000 \beta_{11} + \cdots - 134612872 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 43982407 \beta_{15} + 40628236 \beta_{14} - 40628236 \beta_{13} - 11663080 \beta_{12} + \cdots - 40628236 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 130684910 \beta_{15} - 199173168 \beta_{14} - 199173168 \beta_{13} - 40842655 \beta_{12} + \cdots + 4915781312 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1608564739 \beta_{15} - 1577494750 \beta_{14} + 1577494750 \beta_{13} + 535165670 \beta_{12} + \cdots + 1577494750 ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 4838638449 \beta_{15} + 7344658952 \beta_{14} + 7344658952 \beta_{13} + 1801821065 \beta_{12} + \cdots - 180861429868 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 59168568639 \beta_{15} + 60946663190 \beta_{14} - 60946663190 \beta_{13} - 23499155050 \beta_{12} + \cdots - 60946663190 ) / 16 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(1\) \(-1\)

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
1.89891i
6.18105i
1.61396i
5.89610i
− 3.89610i
0.386037i
− 4.18105i
0.101086i
− 1.89891i
− 6.18105i
− 1.61396i
− 5.89610i
3.89610i
− 0.386037i
4.18105i
− 0.101086i
0 0 0 −11.1044 − 1.30106i 0 −9.61894 − 9.61894i 0 0 0
17.2 0 0 0 −8.43406 − 7.33939i 0 −16.3809 − 16.3809i 0 0 0
17.3 0 0 0 −5.33480 + 9.82547i 0 5.07144 + 5.07144i 0 0 0
17.4 0 0 0 −2.66447 + 10.8582i 0 24.9284 + 24.9284i 0 0 0
17.5 0 0 0 2.66447 − 10.8582i 0 24.9284 + 24.9284i 0 0 0
17.6 0 0 0 5.33480 − 9.82547i 0 5.07144 + 5.07144i 0 0 0
17.7 0 0 0 8.43406 + 7.33939i 0 −16.3809 − 16.3809i 0 0 0
17.8 0 0 0 11.1044 + 1.30106i 0 −9.61894 − 9.61894i 0 0 0
233.1 0 0 0 −11.1044 + 1.30106i 0 −9.61894 + 9.61894i 0 0 0
233.2 0 0 0 −8.43406 + 7.33939i 0 −16.3809 + 16.3809i 0 0 0
233.3 0 0 0 −5.33480 − 9.82547i 0 5.07144 − 5.07144i 0 0 0
233.4 0 0 0 −2.66447 − 10.8582i 0 24.9284 − 24.9284i 0 0 0
233.5 0 0 0 2.66447 + 10.8582i 0 24.9284 − 24.9284i 0 0 0
233.6 0 0 0 5.33480 + 9.82547i 0 5.07144 − 5.07144i 0 0 0
233.7 0 0 0 8.43406 − 7.33939i 0 −16.3809 + 16.3809i 0 0 0
233.8 0 0 0 11.1044 − 1.30106i 0 −9.61894 + 9.61894i 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.c ✓ 16
3.b odd 2 1 inner 360.4.s.c ✓ 16
4.b odd 2 1 720.4.w.f 16
5.c odd 4 1 inner 360.4.s.c ✓ 16
12.b even 2 1 720.4.w.f 16
15.e even 4 1 inner 360.4.s.c ✓ 16
20.e even 4 1 720.4.w.f 16
60.l odd 4 1 720.4.w.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.s.c ✓ 16 1.a even 1 1 trivial
360.4.s.c ✓ 16 3.b odd 2 1 inner
360.4.s.c ✓ 16 5.c odd 4 1 inner
360.4.s.c ✓ 16 15.e even 4 1 inner
720.4.w.f 16 4.b odd 2 1
720.4.w.f 16 12.b even 2 1
720.4.w.f 16 20.e even 4 1
720.4.w.f 16 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 8 T_{7}^{7} + 32 T_{7}^{6} + 13696 T_{7}^{5} + 824704 T_{7}^{4} + 6351360 T_{7}^{3} + \cdots + 6348902400 \) 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
$7$ \( (T^{8} - 8 T^{7} + \cdots + 6348902400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 928 T^{6} + \cdots + 323424256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 20 T^{7} + \cdots + 19992828816)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 13589544960000 \) Copy content Toggle raw display
$19$ \( (T^{8} + 8720 T^{6} + \cdots + 7845707776)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 116 T^{3} + \cdots + 234386944)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 64\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 33\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 648 T^{3} + \cdots - 36718565376)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 19\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
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