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Properties

Label 270.3.g.a
Level $270$
Weight $3$
Character orbit 270.g
Analytic conductor $7.357$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + 2 \beta_{2} q^{4} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{5} + (2 \beta_{2} + 2 \beta_1 + 2) q^{7} + ( - 2 \beta_{2} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + 2 \beta_{2} q^{4} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{5} + (2 \beta_{2} + 2 \beta_1 + 2) q^{7} + ( - 2 \beta_{2} + 2) q^{8} + (3 \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{10} + ( - 5 \beta_{3} + 5 \beta_1 - 2) q^{11} + (2 \beta_{3} - 6 \beta_{2} + 6) q^{13} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{14} - 4 q^{16} + ( - 17 \beta_{2} + \beta_1 - 17) q^{17} + ( - \beta_{3} - 17 \beta_{2} - \beta_1) q^{19} + ( - 4 \beta_{3} - 6 \beta_{2} + \cdots - 2) q^{20}+ \cdots + ( - 16 \beta_{3} + 29 \beta_{2} - 29) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 12 q^{5} + 8 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 12 q^{5} + 8 q^{7} + 8 q^{8} + 16 q^{10} - 8 q^{11} + 24 q^{13} - 16 q^{16} - 68 q^{17} - 8 q^{20} + 8 q^{22} + 16 q^{23} - 16 q^{25} - 48 q^{26} - 16 q^{28} + 4 q^{31} + 16 q^{32} - 8 q^{35} - 36 q^{37} - 68 q^{38} - 16 q^{40} - 80 q^{41} - 36 q^{43} - 32 q^{46} + 48 q^{47} + 28 q^{50} + 48 q^{52} - 28 q^{53} - 36 q^{55} + 32 q^{56} - 104 q^{58} - 12 q^{61} - 4 q^{62} - 196 q^{67} + 136 q^{68} - 56 q^{70} + 200 q^{71} + 220 q^{73} + 136 q^{76} + 104 q^{77} + 48 q^{80} + 80 q^{82} + 72 q^{83} + 284 q^{85} + 72 q^{86} - 16 q^{88} + 48 q^{91} + 32 q^{92} + 32 q^{95} + 336 q^{97} - 116 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(217\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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
1.22474 − 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
−1.22474 − 1.22474i
−1.00000 + 1.00000i 0 − 2.00000i −4.22474 + 2.67423i 0 4.44949 − 4.44949i 2.00000 + 2.00000i 0 1.55051 − 6.89898i
163.2 −1.00000 + 1.00000i 0 − 2.00000i −1.77526 − 4.67423i 0 −0.449490 + 0.449490i 2.00000 + 2.00000i 0 6.44949 + 2.89898i
217.1 −1.00000 − 1.00000i 0 2.00000i −4.22474 − 2.67423i 0 4.44949 + 4.44949i 2.00000 − 2.00000i 0 1.55051 + 6.89898i
217.2 −1.00000 − 1.00000i 0 2.00000i −1.77526 + 4.67423i 0 −0.449490 − 0.449490i 2.00000 − 2.00000i 0 6.44949 − 2.89898i
\(n\): e.g. 2-40 or 990-1000
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.a ✓ 4
3.b odd 2 1 270.3.g.d yes 4
5.b even 2 1 1350.3.g.h 4
5.c odd 4 1 inner 270.3.g.a ✓ 4
5.c odd 4 1 1350.3.g.h 4
15.d odd 2 1 1350.3.g.b 4
15.e even 4 1 270.3.g.d yes 4
15.e even 4 1 1350.3.g.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.g.a ✓ 4 1.a even 1 1 trivial
270.3.g.a ✓ 4 5.c odd 4 1 inner
270.3.g.d yes 4 3.b odd 2 1
270.3.g.d yes 4 15.e even 4 1
1350.3.g.b 4 15.d odd 2 1
1350.3.g.b 4 15.e even 4 1
1350.3.g.h 4 5.b even 2 1
1350.3.g.h 4 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 \(S_{3}^{\mathrm{new}}(270, [\chi])\):

\( T_{7}^{4} - 8T_{7}^{3} + 32T_{7}^{2} + 32T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 146 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 12 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{4} - 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T - 146)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 24 T^{3} + \cdots + 3600 \) Copy content Toggle raw display
$17$ \( T^{4} + 68 T^{3} + \cdots + 330625 \) Copy content Toggle raw display
$19$ \( T^{4} + 590 T^{2} + 80089 \) Copy content Toggle raw display
$23$ \( T^{4} - 16 T^{3} + \cdots + 44521 \) Copy content Toggle raw display
$29$ \( T^{4} + 1544 T^{2} + 336400 \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T - 725)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 36 T^{3} + \cdots + 2452356 \) Copy content Toggle raw display
$41$ \( (T^{2} + 40 T - 1766)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 36 T^{3} + \cdots + 8468100 \) Copy content Toggle raw display
$47$ \( T^{4} - 48 T^{3} + \cdots + 467856 \) Copy content Toggle raw display
$53$ \( T^{4} + 28 T^{3} + \cdots + 10042561 \) Copy content Toggle raw display
$59$ \( T^{4} + 6200 T^{2} + 620944 \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T - 11607)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 196 T^{3} + \cdots + 4418404 \) Copy content Toggle raw display
$71$ \( (T^{2} - 100 T + 2350)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 220 T^{3} + \cdots + 36457444 \) Copy content Toggle raw display
$79$ \( T^{4} + 11406 T^{2} + 4995225 \) Copy content Toggle raw display
$83$ \( T^{4} - 72 T^{3} + \cdots + 19316025 \) Copy content Toggle raw display
$89$ \( T^{4} + 17900 T^{2} + 63648484 \) Copy content Toggle raw display
$97$ \( T^{4} - 336 T^{3} + \cdots + 197796096 \) Copy content Toggle raw display
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