Properties

Label 720.6.o.c
Level $720$
Weight $6$
Character orbit 720.o
Analytic conductor $115.476$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,6,Mod(719,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.719");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 720.o (of order \(2\), degree \(1\), 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: \(115.476350265\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 370x^{6} - 1096x^{5} + 36273x^{4} - 70724x^{3} + 465952x^{2} - 430772x + 1070929 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 5 \beta_{4} + 5 \beta_1) q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \beta_{4} + 5 \beta_1) q^{5} - \beta_{2} q^{7} - 7 \beta_{3} q^{11} - 11 \beta_{5} q^{13} - 44 \beta_1 q^{17} - 26 \beta_{6} q^{19} - 11 \beta_{7} q^{23} + (50 \beta_{5} + 3025) q^{25} + 3058 \beta_{4} q^{29} - 99 \beta_{6} q^{31} + (5 \beta_{7} - 205 \beta_{3}) q^{35} - 397 \beta_{5} q^{37} + 9251 \beta_{4} q^{41} + 54 \beta_{2} q^{43} + 44 \beta_{7} q^{47} + 11483 q^{49} - 672 \beta_1 q^{53} + (35 \beta_{6} - 105 \beta_{2}) q^{55} + 187 \beta_{3} q^{59} + 4510 q^{61} + (6765 \beta_{4} + 110 \beta_1) q^{65} + 88 \beta_{2} q^{67} + 924 \beta_{3} q^{71} - 2134 \beta_{5} q^{73} + 4830 \beta_1 q^{77} + 1679 \beta_{6} q^{79} - 122 \beta_{7} q^{83} + ( - 220 \beta_{5} - 27060) q^{85} - 12563 \beta_{4} q^{89} - 451 \beta_{6} q^{91} + ( - 390 \beta_{7} - 260 \beta_{3}) q^{95} - 5082 \beta_{5} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24200 q^{25} + 91864 q^{49} + 36080 q^{61} - 216480 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 370x^{6} - 1096x^{5} + 36273x^{4} - 70724x^{3} + 465952x^{2} - 430772x + 1070929 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} + 667\nu^{4} - 1324\nu^{3} + 56279\nu^{2} - 55618\nu + 345414 ) / 14803 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 3\nu^{5} + 277\nu^{4} - 549\nu^{3} + 3901\nu^{2} - 3627\nu - 1150184 ) / 6893 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{6} + 9\nu^{5} - 1197\nu^{4} + 2379\nu^{3} - 120771\nu^{2} + 119583\nu - 755886 ) / 7991 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 702 \nu^{7} - 2457 \nu^{6} + 250651 \nu^{5} - 620485 \nu^{4} + 22976045 \nu^{3} - 33844811 \nu^{2} + 171147235 \nu - 79953440 ) / 108364853 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2226 \nu^{7} + 7791 \nu^{6} - 827585 \nu^{5} + 2049485 \nu^{4} - 82855295 \nu^{3} + 122237353 \nu^{2} - 1418157135 \nu + 688773806 ) / 125626511 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15696 \nu^{7} + 54936 \nu^{6} - 5968704 \nu^{5} + 14784420 \nu^{4} - 624864240 \nu^{3} + 922539408 \nu^{2} - 10764578064 \nu + 5229023970 ) / 232717963 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16 \nu^{7} + 56 \nu^{6} - 5892 \nu^{5} + 14590 \nu^{4} - 557352 \nu^{3} + 821466 \nu^{2} - 4167556 \nu + 1947352 ) / 15721 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - 6\beta_{5} - 6\beta_{4} + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 6\beta_{5} - 6\beta_{4} + 2\beta_{3} - 6\beta_{2} + 12\beta _1 - 1086 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{7} - 214\beta_{6} + 891\beta_{5} + 1623\beta_{4} + 3\beta_{3} - 9\beta_{2} + 18\beta _1 - 1632 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{7} - 143\beta_{6} + 596\beta_{5} + 1084\beta_{4} - 284\beta_{3} + 364\beta_{2} - 1180\beta _1 + 61410 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2700 \beta_{7} + 37950 \beta_{6} - 155781 \beta_{5} - 455169 \beta_{4} - 2135 \beta_{3} + 2745 \beta_{2} - 8880 \beta _1 + 463296 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 8145 \beta_{7} + 114923 \beta_{6} - 471816 \beta_{5} - 1373640 \beta_{4} + 223444 \beta_{3} - 193068 \beta_{2} + 917010 \beta _1 - 32477334 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 656880 \beta_{7} - 6718599 \beta_{6} + 27562674 \beta_{5} + 110492052 \beta_{4} + 789530 \beta_{3} - 685356 \beta_{2} + 3240636 \beta _1 - 115294110 ) / 12 \) Copy content Toggle raw display

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
719.1
0.500000 13.9112i
0.500000 3.18740i
0.500000 + 13.9112i
0.500000 + 3.18740i
0.500000 + 12.4970i
0.500000 + 1.77318i
0.500000 12.4970i
0.500000 1.77318i
0 0 0 −55.4527 7.07107i 0 −168.196 0 0 0
719.2 0 0 0 −55.4527 7.07107i 0 168.196 0 0 0
719.3 0 0 0 −55.4527 + 7.07107i 0 −168.196 0 0 0
719.4 0 0 0 −55.4527 + 7.07107i 0 168.196 0 0 0
719.5 0 0 0 55.4527 7.07107i 0 −168.196 0 0 0
719.6 0 0 0 55.4527 7.07107i 0 168.196 0 0 0
719.7 0 0 0 55.4527 + 7.07107i 0 −168.196 0 0 0
719.8 0 0 0 55.4527 + 7.07107i 0 168.196 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 719.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.6.o.c 8
3.b odd 2 1 inner 720.6.o.c 8
4.b odd 2 1 inner 720.6.o.c 8
5.b even 2 1 inner 720.6.o.c 8
12.b even 2 1 inner 720.6.o.c 8
15.d odd 2 1 inner 720.6.o.c 8
20.d odd 2 1 inner 720.6.o.c 8
60.h even 2 1 inner 720.6.o.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.6.o.c 8 1.a even 1 1 trivial
720.6.o.c 8 3.b odd 2 1 inner
720.6.o.c 8 4.b odd 2 1 inner
720.6.o.c 8 5.b even 2 1 inner
720.6.o.c 8 12.b even 2 1 inner
720.6.o.c 8 15.d odd 2 1 inner
720.6.o.c 8 20.d odd 2 1 inner
720.6.o.c 8 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 28290 \) acting on \(S_{6}^{\mathrm{new}}(720, [\chi])\). 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
$5$ \( (T^{4} - 6050 T^{2} + 9765625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 28290)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 101430)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 29766)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 238128)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2798640)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 6846180)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 18702728)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 40576140)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 38771814)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 171162002)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 82493640)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 109538880)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 55544832)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 72385830)^{4} \) Copy content Toggle raw display
$61$ \( (T - 4510)^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} - 219077760)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1767316320)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 1120273176)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 11670829740)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 842136720)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 315657938)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 6353374104)^{4} \) Copy content Toggle raw display
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