Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [270,3,Mod(269,270)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.269");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 270.b (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: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{8}^{3} - \zeta_{8}) q^{2} + 2 q^{4} + ( - 4 \zeta_{8}^{3} + 3 \zeta_{8}) q^{5} - 11 \zeta_{8}^{2} q^{7} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{8}^{3} - \zeta_{8}) q^{2} + 2 q^{4} + ( - 4 \zeta_{8}^{3} + 3 \zeta_{8}) q^{5} - 11 \zeta_{8}^{2} q^{7} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{8} + (\zeta_{8}^{2} - 7) q^{10} + (5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{11} - 15 \zeta_{8}^{2} q^{13} + (11 \zeta_{8}^{3} + 11 \zeta_{8}) q^{14} + 4 q^{16} + (16 \zeta_{8}^{3} - 16 \zeta_{8}) q^{17} - 3 q^{19} + ( - 8 \zeta_{8}^{3} + 6 \zeta_{8}) q^{20} - 10 \zeta_{8}^{2} q^{22} + ( - 13 \zeta_{8}^{3} + 13 \zeta_{8}) q^{23} + ( - 7 \zeta_{8}^{2} + 24) q^{25} + (15 \zeta_{8}^{3} + 15 \zeta_{8}) q^{26} - 22 \zeta_{8}^{2} q^{28} + ( - 9 \zeta_{8}^{3} - 9 \zeta_{8}) q^{29} - 8 q^{31} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{32} + 32 q^{34} + ( - 33 \zeta_{8}^{3} - 44 \zeta_{8}) q^{35} - 65 \zeta_{8}^{2} q^{37} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{38} + (2 \zeta_{8}^{2} - 14) q^{40} + ( - 55 \zeta_{8}^{3} - 55 \zeta_{8}) q^{41} + 32 \zeta_{8}^{2} q^{43} + (10 \zeta_{8}^{3} + 10 \zeta_{8}) q^{44} - 26 q^{46} + ( - 40 \zeta_{8}^{3} + 40 \zeta_{8}) q^{47} - 72 q^{49} + (31 \zeta_{8}^{3} - 17 \zeta_{8}) q^{50} - 30 \zeta_{8}^{2} q^{52} + (9 \zeta_{8}^{3} - 9 \zeta_{8}) q^{53} + (35 \zeta_{8}^{2} + 5) q^{55} + (22 \zeta_{8}^{3} + 22 \zeta_{8}) q^{56} + 18 \zeta_{8}^{2} q^{58} + (56 \zeta_{8}^{3} + 56 \zeta_{8}) q^{59} + 95 q^{61} + ( - 8 \zeta_{8}^{3} + 8 \zeta_{8}) q^{62} + 8 q^{64} + ( - 45 \zeta_{8}^{3} - 60 \zeta_{8}) q^{65} - 19 \zeta_{8}^{2} q^{67} + (32 \zeta_{8}^{3} - 32 \zeta_{8}) q^{68} + (77 \zeta_{8}^{2} + 11) q^{70} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{71} + 119 \zeta_{8}^{2} q^{73} + (65 \zeta_{8}^{3} + 65 \zeta_{8}) q^{74} - 6 q^{76} + ( - 55 \zeta_{8}^{3} + 55 \zeta_{8}) q^{77} + 99 q^{79} + ( - 16 \zeta_{8}^{3} + 12 \zeta_{8}) q^{80} + 110 \zeta_{8}^{2} q^{82} + (77 \zeta_{8}^{3} - 77 \zeta_{8}) q^{83} + (16 \zeta_{8}^{2} - 112) q^{85} + ( - 32 \zeta_{8}^{3} - 32 \zeta_{8}) q^{86} - 20 \zeta_{8}^{2} q^{88} + (64 \zeta_{8}^{3} + 64 \zeta_{8}) q^{89} - 165 q^{91} + ( - 26 \zeta_{8}^{3} + 26 \zeta_{8}) q^{92} - 80 q^{94} + (12 \zeta_{8}^{3} - 9 \zeta_{8}) q^{95} + 95 \zeta_{8}^{2} q^{97} + ( - 72 \zeta_{8}^{3} + 72 \zeta_{8}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 28 q^{10} + 16 q^{16} - 12 q^{19} + 96 q^{25} - 32 q^{31} + 128 q^{34} - 56 q^{40} - 104 q^{46} - 288 q^{49} + 20 q^{55} + 380 q^{61} + 32 q^{64} + 44 q^{70} - 24 q^{76} + 396 q^{79} - 448 q^{85} - 660 q^{91} - 320 q^{94}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
−1.41421 0 2.00000 4.94975 0.707107i 0 11.0000i −2.82843 0 −7.00000 + 1.00000i
269.2 −1.41421 0 2.00000 4.94975 + 0.707107i 0 11.0000i −2.82843 0 −7.00000 1.00000i
269.3 1.41421 0 2.00000 −4.94975 0.707107i 0 11.0000i 2.82843 0 −7.00000 1.00000i
269.4 1.41421 0 2.00000 −4.94975 + 0.707107i 0 11.0000i 2.82843 0 −7.00000 + 1.00000i
\(n\): 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
5.b even 2 1 inner
15.d 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.b.b 4
3.b odd 2 1 inner 270.3.b.b 4
4.b odd 2 1 2160.3.c.j 4
5.b even 2 1 inner 270.3.b.b 4
5.c odd 4 1 1350.3.d.b 2
5.c odd 4 1 1350.3.d.j 2
9.c even 3 2 810.3.j.b 8
9.d odd 6 2 810.3.j.b 8
12.b even 2 1 2160.3.c.j 4
15.d odd 2 1 inner 270.3.b.b 4
15.e even 4 1 1350.3.d.b 2
15.e even 4 1 1350.3.d.j 2
20.d odd 2 1 2160.3.c.j 4
45.h odd 6 2 810.3.j.b 8
45.j even 6 2 810.3.j.b 8
60.h even 2 1 2160.3.c.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.b 4 1.a even 1 1 trivial
270.3.b.b 4 3.b odd 2 1 inner
270.3.b.b 4 5.b even 2 1 inner
270.3.b.b 4 15.d odd 2 1 inner
810.3.j.b 8 9.c even 3 2
810.3.j.b 8 9.d odd 6 2
810.3.j.b 8 45.h odd 6 2
810.3.j.b 8 45.j even 6 2
1350.3.d.b 2 5.c odd 4 1
1350.3.d.b 2 15.e even 4 1
1350.3.d.j 2 5.c odd 4 1
1350.3.d.j 2 15.e even 4 1
2160.3.c.j 4 4.b odd 2 1
2160.3.c.j 4 12.b even 2 1
2160.3.c.j 4 20.d odd 2 1
2160.3.c.j 4 60.h even 2 1

Hecke kernels

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

\( T_{7}^{2} + 121 \) Copy content Toggle raw display
\( T_{17}^{2} - 512 \) 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
$5$ \( T^{4} - 48T^{2} + 625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 225)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 512)^{2} \) Copy content Toggle raw display
$19$ \( (T + 3)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 338)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4225)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 6050)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1024)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 3200)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 6272)^{2} \) Copy content Toggle raw display
$61$ \( (T - 95)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 361)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 14161)^{2} \) Copy content Toggle raw display
$79$ \( (T - 99)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 11858)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 8192)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 9025)^{2} \) Copy content Toggle raw display
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