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Properties

Label 220.3.e.b
Level $220$
Weight $3$
Character orbit 220.e
Analytic conductor $5.995$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [220,3,Mod(109,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("220.109");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 220.e (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: \(5.99456581593\)
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: \( x^{8} + 87x^{6} + 2505x^{4} + 26424x^{2} + 62500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{6} + \beta_{4} + \beta_1) q^{5} - \beta_{2} q^{7} + ( - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{6} + \beta_{4} + \beta_1) q^{5} - \beta_{2} q^{7} + ( - \beta_1 + 2) q^{9} + (\beta_{5} - \beta_1 - 6) q^{11} + \beta_{3} q^{13} + (2 \beta_{6} + 2 \beta_{4} - 3 \beta_1 - 5) q^{15} - \beta_{2} q^{17} + ( - \beta_{7} + \beta_{5}) q^{19} + ( - \beta_{7} - \beta_{5}) q^{21} + (4 \beta_{6} - 2 \beta_{4}) q^{23} + ( - 2 \beta_{6} + 8 \beta_{4} + \cdots - 5) q^{25}+ \cdots + ( - \beta_{7} + 4 \beta_{5} + 3 \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{5} + 20 q^{9} - 44 q^{11} - 28 q^{15} - 32 q^{25} - 52 q^{31} - 92 q^{45} + 284 q^{49} - 60 q^{55} + 384 q^{59} + 200 q^{69} - 60 q^{71} - 464 q^{75} - 336 q^{81} + 380 q^{89} - 32 q^{91} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 87x^{6} + 2505x^{4} + 26424x^{2} + 62500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{6} + 324\nu^{4} + 5524\nu^{2} + 17351 ) / 959 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{6} - 167\nu^{4} - 1917\nu^{2} + 2166 ) / 548 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -79\nu^{6} - 5311\nu^{4} - 89389\nu^{2} - 204906 ) / 3836 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -143\nu^{7} - 8691\nu^{5} - 115215\nu^{3} + 124618\nu ) / 239750 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -143\nu^{7} - 8691\nu^{5} - 115215\nu^{3} + 604118\nu ) / 239750 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 309\nu^{7} + 22133\nu^{5} + 418295\nu^{3} + 1430766\nu ) / 479500 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2237\nu^{7} + 149369\nu^{5} + 2719435\nu^{3} + 11563738\nu ) / 239750 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 5\beta _1 - 41 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 8\beta_{6} - 14\beta_{5} + 21\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -51\beta_{3} - 11\beta_{2} - 213\beta _1 + 1173 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -101\beta_{7} + 1094\beta_{6} + 907\beta_{5} - 1305\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1100\beta_{3} - 196\beta_{2} + 4331\beta _1 - 18827 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4527\beta_{7} - 53598\beta_{6} - 31693\beta_{5} + 41249\beta_{4} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/220\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(111\) \(177\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
− 1.80726i
4.92199i
6.28668i
− 4.47052i
− 6.28668i
4.47052i
1.80726i
− 4.92199i
0 − 3.11473i 0 2.70156 − 4.20732i 0 −12.3180 0 −0.701562 0
109.2 0 − 3.11473i 0 2.70156 − 4.20732i 0 12.3180 0 −0.701562 0
109.3 0 − 1.81616i 0 −3.70156 + 3.36131i 0 −4.15538 0 5.70156 0
109.4 0 − 1.81616i 0 −3.70156 + 3.36131i 0 4.15538 0 5.70156 0
109.5 0 1.81616i 0 −3.70156 − 3.36131i 0 −4.15538 0 5.70156 0
109.6 0 1.81616i 0 −3.70156 − 3.36131i 0 4.15538 0 5.70156 0
109.7 0 3.11473i 0 2.70156 + 4.20732i 0 −12.3180 0 −0.701562 0
109.8 0 3.11473i 0 2.70156 + 4.20732i 0 12.3180 0 −0.701562 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 220.3.e.b ✓ 8
3.b odd 2 1 1980.3.p.b 8
4.b odd 2 1 880.3.i.h 8
5.b even 2 1 inner 220.3.e.b ✓ 8
5.c odd 4 2 1100.3.f.d 8
11.b odd 2 1 inner 220.3.e.b ✓ 8
15.d odd 2 1 1980.3.p.b 8
20.d odd 2 1 880.3.i.h 8
33.d even 2 1 1980.3.p.b 8
44.c even 2 1 880.3.i.h 8
55.d odd 2 1 inner 220.3.e.b ✓ 8
55.e even 4 2 1100.3.f.d 8
165.d even 2 1 1980.3.p.b 8
220.g even 2 1 880.3.i.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.e.b ✓ 8 1.a even 1 1 trivial
220.3.e.b ✓ 8 5.b even 2 1 inner
220.3.e.b ✓ 8 11.b odd 2 1 inner
220.3.e.b ✓ 8 55.d odd 2 1 inner
880.3.i.h 8 4.b odd 2 1
880.3.i.h 8 20.d odd 2 1
880.3.i.h 8 44.c even 2 1
880.3.i.h 8 220.g even 2 1
1100.3.f.d 8 5.c odd 4 2
1100.3.f.d 8 55.e even 4 2
1980.3.p.b 8 3.b odd 2 1
1980.3.p.b 8 15.d odd 2 1
1980.3.p.b 8 33.d even 2 1
1980.3.p.b 8 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 13T_{3}^{2} + 32 \) acting on \(S_{3}^{\mathrm{new}}(220, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 13 T^{2} + 32)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 169 T^{2} + 2620)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 22 T^{3} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 636 T^{2} + 41920)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 169 T^{2} + 2620)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1465 T^{2} + 524000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 596 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2881 T^{2} + 524000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 13 T - 214)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2197 T^{2} + 800)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 5860 T^{2} + 8384000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 6496 T^{2} + 10071280)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 1300 T^{2} + 320000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 6629 T^{2} + 10580000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 96 T + 2140)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 5201 T^{2} + 20960)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 3988 T^{2} + 236672)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 15 T - 11106)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 16416 T^{2} + 5543920)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 22084 T^{2} + 85852160)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 8236 T^{2} + 4192000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 95 T - 3166)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 20852 T^{2} + 108339200)^{2} \) Copy content Toggle raw display
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