Properties

Label 270.3.d.a
Level $270$
Weight $3$
Character orbit 270.d
Analytic conductor $7.357$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [270,3,Mod(161,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 270.d (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: \(7.35696713773\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 2 q^{4} + \beta_{2} q^{5} - q^{7} + 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 2 q^{4} + \beta_{2} q^{5} - q^{7} + 2 \beta_1 q^{8} + \beta_{3} q^{10} + 6 \beta_{2} q^{11} + (6 \beta_{3} + 5) q^{13} + \beta_1 q^{14} + 4 q^{16} - 6 \beta_1 q^{17} + (6 \beta_{3} - 7) q^{19} - 2 \beta_{2} q^{20} + 6 \beta_{3} q^{22} + (6 \beta_{2} + 12 \beta_1) q^{23} - 5 q^{25} + ( - 12 \beta_{2} - 5 \beta_1) q^{26} + 2 q^{28} + (6 \beta_{2} + 6 \beta_1) q^{29} + (12 \beta_{3} + 2) q^{31} - 4 \beta_1 q^{32} - 12 q^{34} - \beta_{2} q^{35} + (6 \beta_{3} + 5) q^{37} + ( - 12 \beta_{2} + 7 \beta_1) q^{38} - 2 \beta_{3} q^{40} + ( - 6 \beta_{2} - 30 \beta_1) q^{41} + ( - 12 \beta_{3} + 38) q^{43} - 12 \beta_{2} q^{44} + (6 \beta_{3} + 24) q^{46} + (12 \beta_{2} - 30 \beta_1) q^{47} - 48 q^{49} + 5 \beta_1 q^{50} + ( - 12 \beta_{3} - 10) q^{52} + ( - 6 \beta_{2} + 54 \beta_1) q^{53} - 30 q^{55} - 2 \beta_1 q^{56} + (6 \beta_{3} + 12) q^{58} + 6 \beta_1 q^{59} + ( - 24 \beta_{3} - 25) q^{61} + ( - 24 \beta_{2} - 2 \beta_1) q^{62} - 8 q^{64} + (5 \beta_{2} + 30 \beta_1) q^{65} + 71 q^{67} + 12 \beta_1 q^{68} - \beta_{3} q^{70} + (30 \beta_{2} - 42 \beta_1) q^{71} + (6 \beta_{3} + 41) q^{73} + ( - 12 \beta_{2} - 5 \beta_1) q^{74} + ( - 12 \beta_{3} + 14) q^{76} - 6 \beta_{2} q^{77} + ( - 30 \beta_{3} - 19) q^{79} + 4 \beta_{2} q^{80} + ( - 6 \beta_{3} - 60) q^{82} + (42 \beta_{2} + 42 \beta_1) q^{83} + 6 \beta_{3} q^{85} + (24 \beta_{2} - 38 \beta_1) q^{86} - 12 \beta_{3} q^{88} + (24 \beta_{2} - 66 \beta_1) q^{89} + ( - 6 \beta_{3} - 5) q^{91} + ( - 12 \beta_{2} - 24 \beta_1) q^{92} + (12 \beta_{3} - 60) q^{94} + ( - 7 \beta_{2} + 30 \beta_1) q^{95} + ( - 30 \beta_{3} + 5) q^{97} + 48 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 4 q^{7} + 20 q^{13} + 16 q^{16} - 28 q^{19} - 20 q^{25} + 8 q^{28} + 8 q^{31} - 48 q^{34} + 20 q^{37} + 152 q^{43} + 96 q^{46} - 192 q^{49} - 40 q^{52} - 120 q^{55} + 48 q^{58} - 100 q^{61} - 32 q^{64} + 284 q^{67} + 164 q^{73} + 56 q^{76} - 76 q^{79} - 240 q^{82} - 20 q^{91} - 240 q^{94} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 7\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 7\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(217\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−1.58114 + 0.707107i
1.58114 + 0.707107i
1.58114 0.707107i
−1.58114 0.707107i
1.41421i 0 −2.00000 2.23607i 0 −1.00000 2.82843i 0 −3.16228
161.2 1.41421i 0 −2.00000 2.23607i 0 −1.00000 2.82843i 0 3.16228
161.3 1.41421i 0 −2.00000 2.23607i 0 −1.00000 2.82843i 0 3.16228
161.4 1.41421i 0 −2.00000 2.23607i 0 −1.00000 2.82843i 0 −3.16228
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.3.d.a 4
3.b odd 2 1 inner 270.3.d.a 4
4.b odd 2 1 2160.3.l.f 4
5.b even 2 1 1350.3.d.m 4
5.c odd 4 2 1350.3.b.h 8
9.c even 3 2 810.3.h.b 8
9.d odd 6 2 810.3.h.b 8
12.b even 2 1 2160.3.l.f 4
15.d odd 2 1 1350.3.d.m 4
15.e even 4 2 1350.3.b.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.d.a 4 1.a even 1 1 trivial
270.3.d.a 4 3.b odd 2 1 inner
810.3.h.b 8 9.c even 3 2
810.3.h.b 8 9.d odd 6 2
1350.3.b.h 8 5.c odd 4 2
1350.3.b.h 8 15.e even 4 2
1350.3.d.m 4 5.b even 2 1
1350.3.d.m 4 15.d odd 2 1
2160.3.l.f 4 4.b odd 2 1
2160.3.l.f 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 1 \) acting on \(S_{3}^{\mathrm{new}}(270, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 10 T - 335)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 14 T - 311)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 936 T^{2} + 11664 \) Copy content Toggle raw display
$29$ \( T^{4} + 504 T^{2} + 11664 \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 1436)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T - 335)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 3960 T^{2} + 2624400 \) Copy content Toggle raw display
$43$ \( (T^{2} - 76 T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 5040 T^{2} + 1166400 \) Copy content Toggle raw display
$53$ \( T^{4} + 12024 T^{2} + 31945104 \) Copy content Toggle raw display
$59$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 50 T - 5135)^{2} \) Copy content Toggle raw display
$67$ \( (T - 71)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 16056 T^{2} + 944784 \) Copy content Toggle raw display
$73$ \( (T^{2} - 82 T + 1321)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 38 T - 8639)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 24696 T^{2} + 28005264 \) Copy content Toggle raw display
$89$ \( T^{4} + 23184 T^{2} + 34012224 \) Copy content Toggle raw display
$97$ \( (T^{2} - 10 T - 8975)^{2} \) Copy content Toggle raw display
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