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Properties

Label 270.3.g.e
Level $270$
Weight $3$
Character orbit 270.g
Analytic conductor $7.357$
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] = [270,3,Mod(163,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([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("270.163");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 270.g (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: \(7.35696713773\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.122645643264.38
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 24x^{6} + 164x^{4} + 336x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + 2 \beta_1 q^{4} + (\beta_{3} - \beta_1 + 2) q^{5} + ( - \beta_{7} - 2 \beta_{5} - \beta_1 - 1) q^{7} + ( - 2 \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + 2 \beta_1 q^{4} + (\beta_{3} - \beta_1 + 2) q^{5} + ( - \beta_{7} - 2 \beta_{5} - \beta_1 - 1) q^{7} + ( - 2 \beta_1 + 2) q^{8} + ( - \beta_{5} - \beta_{3} - \beta_{2} + \cdots - 2) q^{10}+ \cdots + ( - 4 \beta_{6} - 4 \beta_{5} + \cdots + 20) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 12 q^{5} - 8 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 12 q^{5} - 8 q^{7} + 16 q^{8} - 16 q^{10} - 40 q^{11} - 24 q^{13} - 32 q^{16} + 8 q^{17} + 8 q^{20} + 40 q^{22} - 16 q^{23} + 16 q^{25} + 48 q^{26} + 16 q^{28} - 64 q^{31} + 32 q^{32} - 124 q^{35} - 24 q^{37} + 8 q^{38} + 16 q^{40} + 56 q^{41} - 72 q^{43} + 32 q^{46} + 80 q^{50} - 48 q^{52} - 80 q^{53} - 96 q^{55} - 32 q^{56} - 88 q^{58} + 216 q^{61} + 64 q^{62} + 156 q^{65} + 136 q^{67} - 16 q^{68} + 152 q^{70} - 56 q^{71} + 368 q^{73} - 16 q^{76} - 320 q^{77} - 48 q^{80} - 56 q^{82} + 168 q^{83} - 56 q^{85} + 144 q^{86} - 80 q^{88} - 312 q^{91} - 32 q^{92} + 4 q^{95} - 312 q^{97} + 128 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 24x^{6} + 164x^{4} + 336x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 20\nu^{3} + 74\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{6} + 2\nu^{5} + 50\nu^{4} + 40\nu^{3} + 102\nu^{2} + 268\nu - 180 ) / 80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{6} + 4\nu^{5} - 50\nu^{4} + 80\nu^{3} - 102\nu^{2} + 416\nu + 180 ) / 80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 20\nu^{5} - 5\nu^{4} + 94\nu^{3} - 60\nu^{2} + 80\nu - 50 ) / 40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 20\nu^{5} + 5\nu^{4} + 94\nu^{3} + 60\nu^{2} + 80\nu + 50 ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 6\nu^{6} - 72\nu^{5} + 120\nu^{4} - 462\nu^{3} + 564\nu^{2} - 648\nu + 480 ) / 80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} + 6\nu^{6} + 72\nu^{5} + 120\nu^{4} + 462\nu^{3} + 564\nu^{2} + 648\nu + 480 ) / 80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 3\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - 2\beta_{2} - 2\beta _1 - 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} + 3\beta_{5} + 3\beta_{4} - 8\beta_{3} - 8\beta_{2} + 48\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{7} - 4\beta_{6} + 12\beta_{5} - 12\beta_{4} - 8\beta_{3} + 8\beta_{2} + 8\beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 40\beta_{7} - 40\beta_{6} - 60\beta_{5} - 60\beta_{4} + 86\beta_{3} + 86\beta_{2} - 618\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 166\beta_{7} + 166\beta_{6} - 532\beta_{5} + 532\beta_{4} + 292\beta_{3} - 292\beta_{2} - 292\beta _1 - 1976 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -612\beta_{7} + 612\beta_{6} + 978\beta_{5} + 978\beta_{4} - 1048\beta_{3} - 1048\beta_{2} + 8088\beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
2.59703i
3.72938i
− 1.72938i
− 0.597032i
− 2.59703i
− 3.72938i
1.72938i
0.597032i
−1.00000 + 1.00000i 0 − 2.00000i −4.07625 + 2.89555i 0 1.68717 − 1.68717i 2.00000 + 2.00000i 0 1.18070 − 6.97180i
163.2 −1.00000 + 1.00000i 0 − 2.00000i 1.97347 + 4.59406i 0 −6.46083 + 6.46083i 2.00000 + 2.00000i 0 −6.56753 − 2.62059i
163.3 −1.00000 + 1.00000i 0 − 2.00000i 3.47602 − 3.59406i 0 6.91032 − 6.91032i 2.00000 + 2.00000i 0 0.118044 + 7.07008i
163.4 −1.00000 + 1.00000i 0 − 2.00000i 4.62676 − 1.89555i 0 −6.13666 + 6.13666i 2.00000 + 2.00000i 0 −2.73121 + 6.52231i
217.1 −1.00000 − 1.00000i 0 2.00000i −4.07625 − 2.89555i 0 1.68717 + 1.68717i 2.00000 − 2.00000i 0 1.18070 + 6.97180i
217.2 −1.00000 − 1.00000i 0 2.00000i 1.97347 − 4.59406i 0 −6.46083 − 6.46083i 2.00000 − 2.00000i 0 −6.56753 + 2.62059i
217.3 −1.00000 − 1.00000i 0 2.00000i 3.47602 + 3.59406i 0 6.91032 + 6.91032i 2.00000 − 2.00000i 0 0.118044 − 7.07008i
217.4 −1.00000 − 1.00000i 0 2.00000i 4.62676 + 1.89555i 0 −6.13666 − 6.13666i 2.00000 − 2.00000i 0 −2.73121 − 6.52231i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.4
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 270.3.g.e ✓ 8
3.b odd 2 1 270.3.g.f yes 8
5.b even 2 1 1350.3.g.r 8
5.c odd 4 1 inner 270.3.g.e ✓ 8
5.c odd 4 1 1350.3.g.r 8
15.d odd 2 1 1350.3.g.o 8
15.e even 4 1 270.3.g.f yes 8
15.e even 4 1 1350.3.g.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.g.e ✓ 8 1.a even 1 1 trivial
270.3.g.e ✓ 8 5.c odd 4 1 inner
270.3.g.f yes 8 3.b odd 2 1
270.3.g.f yes 8 15.e even 4 1
1350.3.g.o 8 15.d odd 2 1
1350.3.g.o 8 15.e even 4 1
1350.3.g.r 8 5.b even 2 1
1350.3.g.r 8 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(270, [\chi])\):

\( T_{7}^{8} + 8T_{7}^{7} + 32T_{7}^{6} - 136T_{7}^{5} + 9298T_{7}^{4} + 73672T_{7}^{3} + 301088T_{7}^{2} - 1434824T_{7} + 3418801 \) Copy content Toggle raw display
\( T_{11}^{4} + 20T_{11}^{3} - 84T_{11}^{2} - 2920T_{11} - 11750 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{4} \) 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
$7$ \( T^{8} + 8 T^{7} + \cdots + 3418801 \) Copy content Toggle raw display
$11$ \( (T^{4} + 20 T^{3} + \cdots - 11750)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 24 T^{7} + \cdots + 553896225 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 2457779776 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 55806030289 \) Copy content Toggle raw display
$23$ \( T^{8} + 16 T^{7} + \cdots + 58400164 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 53999534884 \) Copy content Toggle raw display
$31$ \( (T^{4} + 32 T^{3} + \cdots - 1029500)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 1342597277025 \) Copy content Toggle raw display
$41$ \( (T^{4} - 28 T^{3} + \cdots + 44890)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 72 T^{7} + \cdots + 431475984 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 75488661504 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 10620755102500 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 14890769628736 \) Copy content Toggle raw display
$61$ \( (T^{4} - 108 T^{3} + \cdots + 2786625)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 264446106716401 \) Copy content Toggle raw display
$71$ \( (T^{4} + 28 T^{3} + \cdots - 1934390)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 79929589912225 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 385814724455889 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 339174175889124 \) Copy content Toggle raw display
$89$ \( T^{8} + 28624 T^{6} + \cdots + 577921600 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 26203905050625 \) Copy content Toggle raw display
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