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Properties

Label 720.3.bh.o
Level $720$
Weight $3$
Character orbit 720.bh
Analytic conductor $19.619$
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] = [720,3,Mod(433,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.433");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 720.bh (of order \(4\), degree \(2\), not 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: \(19.6185790339\)
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: \( x^{8} - 4x^{7} + 8x^{6} + 269x^{4} - 1116x^{3} + 2312x^{2} + 680x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 5 \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} + 1) q^{5} + (\beta_{5} - \beta_{2} - \beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + 1) q^{5} + (\beta_{5} - \beta_{2} - \beta_1 - 1) q^{7} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots + 4) q^{11}+ \cdots + (6 \beta_{7} + 12 \beta_{5} + \cdots - 35) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{5} - 4 q^{7} + 32 q^{11} - 4 q^{13} - 52 q^{17} - 40 q^{23} - 84 q^{25} - 96 q^{31} + 24 q^{35} - 60 q^{37} + 152 q^{41} + 88 q^{43} - 16 q^{47} - 108 q^{53} - 116 q^{55} + 264 q^{61} - 164 q^{65} + 216 q^{67} - 240 q^{71} - 208 q^{73} - 168 q^{77} + 336 q^{83} - 12 q^{85} - 592 q^{91} - 128 q^{95} - 208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} + 269x^{4} - 1116x^{3} + 2312x^{2} + 680x + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 833 \nu^{7} + 10788 \nu^{6} - 59834 \nu^{5} - 980 \nu^{4} - 18043 \nu^{3} + 2993192 \nu^{2} + \cdots + 9290260 ) / 1916820 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2465 \nu^{7} + 10455 \nu^{6} - 21142 \nu^{5} + 2900 \nu^{4} - 663785 \nu^{3} + 2966245 \nu^{2} + \cdots - 838660 ) / 958410 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6953 \nu^{7} - 12966 \nu^{6} - 11750 \nu^{5} - 8180 \nu^{4} + 1740137 \nu^{3} - 3756574 \nu^{2} + \cdots + 9543760 ) / 1916820 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3096 \nu^{7} - 17905 \nu^{6} + 34598 \nu^{5} + 15150 \nu^{4} + 876904 \nu^{3} - 4737945 \nu^{2} + \cdots + 2940830 ) / 319470 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24497 \nu^{7} - 89580 \nu^{6} + 219802 \nu^{5} - 28820 \nu^{4} + 6354293 \nu^{3} - 25482760 \nu^{2} + \cdots + 8140660 ) / 1916820 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4775 \nu^{7} + 20987 \nu^{6} - 31148 \nu^{5} + 24410 \nu^{4} - 1286695 \nu^{3} + 5631403 \nu^{2} + \cdots + 227650 ) / 319470 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10999 \nu^{7} - 39674 \nu^{6} + 93162 \nu^{5} - 12940 \nu^{4} + 2991231 \nu^{3} - 11289026 \nu^{2} + \cdots + 5043540 ) / 638940 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{5} + 3\beta_{3} + 2\beta_{2} - 3\beta _1 + 5 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} - 5\beta_{6} + 7\beta_{5} + 5\beta_{4} + \beta_{3} + 104\beta_{2} - \beta _1 - 5 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 41\beta_{7} - 51\beta_{5} + 7\beta_{3} + 28\beta_{2} - 17\beta _1 - 35 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{7} + 85\beta_{6} + 3\beta_{5} + 85\beta_{4} + 99\beta_{3} - 34\beta_{2} + 31\beta _1 - 1505 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 279\beta_{7} - 29\beta_{5} - 837\beta_{3} + 642\beta_{2} + 587\beta _1 - 195 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 297 \beta_{7} + 1395 \beta_{6} - 1493 \beta_{5} - 1395 \beta_{4} + 301 \beta_{3} - 22396 \beta_{2} + \cdots + 895 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8609\beta_{7} + 13599\beta_{5} + 200\beta_{4} + 457\beta_{3} + 8228\beta_{2} + 4533\beta _1 - 8685 ) / 10 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
−2.88489 + 2.88489i
−0.137883 + 0.137883i
2.36263 − 2.36263i
2.66015 − 2.66015i
−2.88489 − 2.88489i
−0.137883 − 0.137883i
2.36263 + 2.36263i
2.66015 + 2.66015i
0 0 0 −1.88489 − 4.63111i 0 −6.02356 + 6.02356i 0 0 0
433.2 0 0 0 0.862117 + 4.92511i 0 −7.33876 + 7.33876i 0 0 0
433.3 0 0 0 3.36263 − 3.70037i 0 8.78825 − 8.78825i 0 0 0
433.4 0 0 0 3.66015 + 3.40637i 0 2.57407 − 2.57407i 0 0 0
577.1 0 0 0 −1.88489 + 4.63111i 0 −6.02356 − 6.02356i 0 0 0
577.2 0 0 0 0.862117 − 4.92511i 0 −7.33876 − 7.33876i 0 0 0
577.3 0 0 0 3.36263 + 3.70037i 0 8.78825 + 8.78825i 0 0 0
577.4 0 0 0 3.66015 − 3.40637i 0 2.57407 + 2.57407i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.4
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 720.3.bh.o 8
3.b odd 2 1 240.3.bg.e 8
4.b odd 2 1 360.3.v.f 8
5.c odd 4 1 inner 720.3.bh.o 8
12.b even 2 1 120.3.u.b ✓ 8
15.d odd 2 1 1200.3.bg.q 8
15.e even 4 1 240.3.bg.e 8
15.e even 4 1 1200.3.bg.q 8
20.d odd 2 1 1800.3.v.t 8
20.e even 4 1 360.3.v.f 8
20.e even 4 1 1800.3.v.t 8
24.f even 2 1 960.3.bg.l 8
24.h odd 2 1 960.3.bg.k 8
60.h even 2 1 600.3.u.h 8
60.l odd 4 1 120.3.u.b ✓ 8
60.l odd 4 1 600.3.u.h 8
120.q odd 4 1 960.3.bg.l 8
120.w even 4 1 960.3.bg.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.u.b ✓ 8 12.b even 2 1
120.3.u.b ✓ 8 60.l odd 4 1
240.3.bg.e 8 3.b odd 2 1
240.3.bg.e 8 15.e even 4 1
360.3.v.f 8 4.b odd 2 1
360.3.v.f 8 20.e even 4 1
600.3.u.h 8 60.h even 2 1
600.3.u.h 8 60.l odd 4 1
720.3.bh.o 8 1.a even 1 1 trivial
720.3.bh.o 8 5.c odd 4 1 inner
960.3.bg.k 8 24.h odd 2 1
960.3.bg.k 8 120.w even 4 1
960.3.bg.l 8 24.f even 2 1
960.3.bg.l 8 120.q odd 4 1
1200.3.bg.q 8 15.d odd 2 1
1200.3.bg.q 8 15.e even 4 1
1800.3.v.t 8 20.d odd 2 1
1800.3.v.t 8 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{8} + 4T_{7}^{7} + 8T_{7}^{6} + 120T_{7}^{5} + 20900T_{7}^{4} + 120000T_{7}^{3} + 320000T_{7}^{2} - 3200000T_{7} + 16000000 \) Copy content Toggle raw display
\( T_{11}^{4} - 16T_{11}^{3} - 358T_{11}^{2} + 7696T_{11} - 32288 \) 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
$5$ \( T^{8} - 12 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 16000000 \) Copy content Toggle raw display
$11$ \( (T^{4} - 16 T^{3} + \cdots - 32288)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 53824 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 2570084416 \) Copy content Toggle raw display
$19$ \( T^{8} + 888 T^{6} + \cdots + 82591744 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 3064729600 \) Copy content Toggle raw display
$29$ \( T^{8} + 4500 T^{6} + \cdots + 619810816 \) Copy content Toggle raw display
$31$ \( (T^{4} + 48 T^{3} + \cdots - 1815680)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 16133718222400 \) Copy content Toggle raw display
$41$ \( (T^{4} - 76 T^{3} + \cdots - 63872)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 949580087296 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 287296000000 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 5939943840000 \) Copy content Toggle raw display
$59$ \( T^{8} + 2348 T^{6} + \cdots + 861184 \) Copy content Toggle raw display
$61$ \( (T^{4} - 132 T^{3} + \cdots - 13929728)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 3555366682624 \) Copy content Toggle raw display
$71$ \( (T^{4} + 120 T^{3} + \cdots - 4371968)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 1421016580096 \) Copy content Toggle raw display
$83$ \( T^{8} - 336 T^{7} + \cdots + 2166784 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 693289369600 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
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