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Properties

Label 320.4.o.b
Level $320$
Weight $4$
Character orbit 320.o
Analytic conductor $18.881$
Analytic rank $0$
Dimension $12$
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] = [320,4,Mod(223,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.223");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.o (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: \(18.8806112018\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 130x^{10} + 5747x^{8} + 104630x^{6} + 725953x^{4} + 1488340x^{2} + 28224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{12} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + \beta_{8} q^{5} + ( - \beta_{9} + \beta_{2} + 1) q^{7} + ( - \beta_{11} + 14 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + \beta_{8} q^{5} + ( - \beta_{9} + \beta_{2} + 1) q^{7} + ( - \beta_{11} + 14 \beta_{2}) q^{9} + (\beta_{8} + \beta_{7} + \cdots + \beta_{3}) q^{11}+ \cdots + ( - 3 \beta_{8} + 3 \beta_{7} + \cdots - 101 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{7} - 88 q^{15} - 84 q^{17} - 464 q^{23} + 468 q^{25} - 672 q^{33} + 992 q^{39} - 448 q^{41} - 448 q^{47} + 2296 q^{55} + 2272 q^{57} + 192 q^{63} + 148 q^{65} - 140 q^{73} - 800 q^{79} - 5820 q^{81} - 8064 q^{87} + 4568 q^{95} + 6244 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 130x^{10} + 5747x^{8} + 104630x^{6} + 725953x^{4} + 1488340x^{2} + 28224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2760\nu^{10} - 321378\nu^{8} - 11832856\nu^{6} - 158572136\nu^{4} - 422738464\nu^{2} + 2978463243 ) / 75578573 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 91961 \nu^{11} + 12186770 \nu^{9} + 555495619 \nu^{7} + 10615839334 \nu^{5} + \cdots + 185076465980 \nu ) / 25394400528 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 91961 \nu^{11} - 231840 \nu^{10} + 12186770 \nu^{9} - 26995752 \nu^{8} + 555495619 \nu^{7} + \cdots - 22798893264 ) / 25394400528 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 91961 \nu^{11} - 231840 \nu^{10} - 12186770 \nu^{9} - 26995752 \nu^{8} - 555495619 \nu^{7} + \cdots - 22798893264 ) / 25394400528 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 85514643 \nu^{11} + 38699080 \nu^{10} - 11072549126 \nu^{9} + 5196236584 \nu^{8} + \cdots + 29013585644064 ) / 2920356060720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 85514643 \nu^{11} - 38699080 \nu^{10} - 11072549126 \nu^{9} - 5196236584 \nu^{8} + \cdots - 29013585644064 ) / 2920356060720 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 84812853 \nu^{11} + 35979440 \nu^{10} - 10972758646 \nu^{9} + 4465519016 \nu^{8} + \cdots + 8241418232376 ) / 1460178030360 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 84812853 \nu^{11} + 35979440 \nu^{10} + 10972758646 \nu^{9} + 4465519016 \nu^{8} + \cdots + 8241418232376 ) / 1460178030360 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3765841 \nu^{11} - 1459248 \nu^{10} + 485982498 \nu^{9} - 188318480 \nu^{8} + \cdots + 780597694800 ) / 63486001320 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3765841 \nu^{11} - 1459248 \nu^{10} - 485982498 \nu^{9} - 188318480 \nu^{8} + \cdots + 780597694800 ) / 63486001320 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1350053 \nu^{11} + 176425178 \nu^{9} + 7884082983 \nu^{7} + 146885950958 \nu^{5} + \cdots + 2438122920268 \nu ) / 8464800176 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} - 2\beta_{3} + \beta _1 - 43 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{11} + \beta_{8} - \beta_{7} + 7\beta_{6} + 7\beta_{5} - 37\beta_{4} + 37\beta_{3} - 125\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15 \beta_{10} + 15 \beta_{9} + 28 \beta_{8} + 28 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + 140 \beta_{4} + \cdots + 1774 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 250 \beta_{11} - 75 \beta_{10} + 75 \beta_{9} - 118 \beta_{8} + 118 \beta_{7} - 481 \beta_{6} + \cdots + 8042 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1155 \beta_{10} - 1155 \beta_{9} - 2606 \beta_{8} - 2606 \beta_{7} - 668 \beta_{6} + 668 \beta_{5} + \cdots - 88099 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 17227 \beta_{11} + 7035 \beta_{10} - 7035 \beta_{9} + 8241 \beta_{8} - 8241 \beta_{7} + \cdots - 506417 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 73380 \beta_{10} + 73380 \beta_{9} + 184816 \beta_{8} + 184816 \beta_{7} + 53608 \beta_{6} + \cdots + 4786231 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1107345 \beta_{11} - 496620 \beta_{10} + 496620 \beta_{9} - 514281 \beta_{8} + \cdots + 31446647 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 4454475 \beta_{10} - 4454475 \beta_{9} - 11956302 \beta_{8} - 11956302 \beta_{7} - 3608106 \beta_{6} + \cdots - 272789440 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 68933106 \beta_{11} + 31975185 \beta_{10} - 31975185 \beta_{9} + 31123786 \beta_{8} + \cdots - 1931282234 \beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
7.75639i
4.95079i
1.86165i
0.138353i
− 2.95079i
− 5.75639i
− 7.75639i
− 4.95079i
− 1.86165i
− 0.138353i
2.95079i
5.75639i
0 −6.75639 − 6.75639i 0 11.1678 − 0.529594i 0 −6.57815 − 6.57815i 0 64.2977i 0
223.2 0 −3.95079 − 3.95079i 0 −9.54934 + 5.81465i 0 19.9724 + 19.9724i 0 4.21745i 0
223.3 0 −0.861647 − 0.861647i 0 −5.48549 − 9.74215i 0 −11.3943 − 11.3943i 0 − 25.5151i 0
223.4 0 0.861647 + 0.861647i 0 5.48549 + 9.74215i 0 −11.3943 − 11.3943i 0 − 25.5151i 0
223.5 0 3.95079 + 3.95079i 0 9.54934 − 5.81465i 0 19.9724 + 19.9724i 0 4.21745i 0
223.6 0 6.75639 + 6.75639i 0 −11.1678 + 0.529594i 0 −6.57815 − 6.57815i 0 64.2977i 0
287.1 0 −6.75639 + 6.75639i 0 11.1678 + 0.529594i 0 −6.57815 + 6.57815i 0 − 64.2977i 0
287.2 0 −3.95079 + 3.95079i 0 −9.54934 − 5.81465i 0 19.9724 − 19.9724i 0 − 4.21745i 0
287.3 0 −0.861647 + 0.861647i 0 −5.48549 + 9.74215i 0 −11.3943 + 11.3943i 0 25.5151i 0
287.4 0 0.861647 − 0.861647i 0 5.48549 − 9.74215i 0 −11.3943 + 11.3943i 0 25.5151i 0
287.5 0 3.95079 − 3.95079i 0 9.54934 + 5.81465i 0 19.9724 − 19.9724i 0 − 4.21745i 0
287.6 0 6.75639 − 6.75639i 0 −11.1678 − 0.529594i 0 −6.57815 + 6.57815i 0 − 64.2977i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
20.e even 4 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.4.o.b yes 12
4.b odd 2 1 320.4.o.a ✓ 12
5.c odd 4 1 320.4.o.a ✓ 12
8.b even 2 1 inner 320.4.o.b yes 12
8.d odd 2 1 320.4.o.a ✓ 12
20.e even 4 1 inner 320.4.o.b yes 12
40.i odd 4 1 320.4.o.a ✓ 12
40.k even 4 1 inner 320.4.o.b yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.4.o.a ✓ 12 4.b odd 2 1
320.4.o.a ✓ 12 5.c odd 4 1
320.4.o.a ✓ 12 8.d odd 2 1
320.4.o.a ✓ 12 40.i odd 4 1
320.4.o.b yes 12 1.a even 1 1 trivial
320.4.o.b yes 12 8.b even 2 1 inner
320.4.o.b yes 12 20.e even 4 1 inner
320.4.o.b yes 12 40.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{12} + 9312T_{3}^{8} + 8143488T_{3}^{4} + 17909824 \) Copy content Toggle raw display
\( T_{7}^{6} - 4T_{7}^{5} + 8T_{7}^{4} + 8260T_{7}^{3} + 322624T_{7}^{2} + 3401184T_{7} + 17928072 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 9312 T^{8} + \cdots + 17909824 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 3814697265625 \) Copy content Toggle raw display
$7$ \( (T^{6} - 4 T^{5} + \cdots + 17928072)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 1988 T^{4} + \cdots - 74373376)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{6} + 42 T^{5} + \cdots + 5415906888)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 18268 T^{4} + \cdots + 4041907776)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 3797117611848)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 33164606712384)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 1640100280896)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{3} + 112 T^{2} + \cdots + 9853704)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 74138349389832)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 41\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 27\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 200 T^{2} + \cdots + 228096000)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 90\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 77\!\cdots\!08)^{2} \) Copy content Toggle raw display
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