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Properties

Label 220.3.p.b
Level $220$
Weight $3$
Character orbit 220.p
Analytic conductor $5.995$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [220,3,Mod(41,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 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("220.41");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 220.p (of order \(10\), degree \(4\), 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 33 x^{14} - 111 x^{13} + 735 x^{12} - 1436 x^{11} + 10633 x^{10} - 25103 x^{9} + \cdots + 75625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{3}+ \cdots + ( - \beta_{14} + \beta_{12} - \beta_{11} + \cdots - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{3}+ \cdots + (8 \beta_{15} - 9 \beta_{14} + \cdots - 56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 9 q^{3} + 20 q^{5} + 10 q^{7} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 9 q^{3} + 20 q^{5} + 10 q^{7} - 19 q^{9} + 23 q^{11} - 5 q^{13} + 15 q^{15} + 25 q^{17} + 30 q^{19} - 168 q^{23} - 20 q^{25} - 225 q^{27} - 105 q^{29} + 40 q^{31} + 106 q^{33} - 16 q^{37} + 115 q^{39} + 255 q^{41} - 30 q^{45} + 102 q^{47} + 12 q^{49} + 380 q^{51} + 297 q^{53} + 55 q^{55} - 150 q^{57} - 22 q^{59} - 30 q^{61} - 320 q^{63} - 68 q^{67} - 137 q^{69} - 240 q^{71} + 85 q^{73} - 30 q^{75} + 226 q^{77} - 215 q^{79} - 463 q^{81} - 610 q^{83} + 25 q^{85} - 160 q^{89} + 161 q^{91} - 253 q^{93} + 90 q^{95} + 179 q^{97} + 150 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 33 x^{14} - 111 x^{13} + 735 x^{12} - 1436 x^{11} + 10633 x^{10} - 25103 x^{9} + \cdots + 75625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 51\!\cdots\!71 \nu^{15} + \cdots - 32\!\cdots\!25 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 96\!\cdots\!54 \nu^{15} + \cdots - 19\!\cdots\!25 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!41 \nu^{15} + \cdots + 35\!\cdots\!75 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22\!\cdots\!43 \nu^{15} + \cdots + 12\!\cdots\!75 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25\!\cdots\!05 \nu^{15} + \cdots + 13\!\cdots\!75 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 46\!\cdots\!59 \nu^{15} + \cdots + 91\!\cdots\!00 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!29 \nu^{15} + \cdots - 27\!\cdots\!50 ) / 16\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\!\cdots\!48 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 16\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 71\!\cdots\!86 \nu^{15} + \cdots + 11\!\cdots\!25 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!09 \nu^{15} + \cdots - 13\!\cdots\!75 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!71 \nu^{15} + \cdots - 29\!\cdots\!25 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 71\!\cdots\!94 \nu^{15} + \cdots - 87\!\cdots\!25 ) / 16\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 15\!\cdots\!57 \nu^{15} + \cdots + 25\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21\!\cdots\!82 \nu^{15} + \cdots + 78\!\cdots\!75 ) / 33\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} - \beta_{11} - 11\beta_{7} + 2\beta_{5} + \beta_{4} + \beta_{3} - \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_{9} - 2 \beta_{8} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + 23 \beta_{14} + \beta_{13} + 4 \beta_{12} + 21 \beta_{11} + \beta_{10} - 42 \beta_{9} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 26 \beta_{15} - 36 \beta_{14} - 2 \beta_{13} - 29 \beta_{12} - 14 \beta_{11} + 26 \beta_{10} + \cdots - 435 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7 \beta_{15} - 428 \beta_{14} + 20 \beta_{13} - 508 \beta_{12} + 428 \beta_{11} - 17 \beta_{10} + \cdots + 884 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 70 \beta_{15} + 700 \beta_{14} - 70 \beta_{13} + 1129 \beta_{12} + 1129 \beta_{11} - 1030 \beta_{10} + \cdots + 8558 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1088 \beta_{15} - 8854 \beta_{14} - 379 \beta_{13} + 8854 \beta_{12} - 11434 \beta_{11} - 165 \beta_{10} + \cdots - 117935 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2036 \beta_{15} - 32622 \beta_{14} + 13091 \beta_{13} - 24926 \beta_{12} - 20159 \beta_{11} + \cdots - 32622 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 20939 \beta_{15} + 264164 \beta_{14} + 1408 \beta_{13} + 187196 \beta_{12} + 153936 \beta_{11} + \cdots + 2855465 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 300662 \beta_{15} + 510217 \beta_{14} - 486450 \beta_{13} + 898472 \beta_{12} - 510217 \beta_{11} + \cdots - 5278521 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 145030 \beta_{15} - 4401810 \beta_{14} + 145030 \beta_{13} - 6254636 \beta_{12} - 6254636 \beta_{11} + \cdots - 32490852 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 11023592 \beta_{15} + 12761006 \beta_{14} + 7114891 \beta_{13} - 12761006 \beta_{12} + \cdots + 274091275 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5789229 \beta_{15} + 151171833 \beta_{14} - 22736369 \beta_{13} + 122452944 \beta_{12} + \cdots + 151171833 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 172714796 \beta_{15} - 633792586 \beta_{14} - 44279332 \beta_{13} - 317518854 \beta_{12} + \cdots - 7508982935 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(111\) \(177\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
1.27755 − 3.93190i
0.777205 − 2.39199i
−0.295420 + 0.909211i
−1.56835 + 4.82689i
3.61325 − 2.62518i
0.884441 − 0.642584i
0.214694 − 0.155984i
−3.40337 + 2.47269i
3.61325 + 2.62518i
0.884441 + 0.642584i
0.214694 + 0.155984i
−3.40337 − 2.47269i
1.27755 + 3.93190i
0.777205 + 2.39199i
−0.295420 − 0.909211i
−1.56835 − 4.82689i
0 −1.08657 − 3.34411i 0 1.80902 + 1.31433i 0 −6.54766 − 2.12746i 0 −2.72130 + 1.97714i 0
41.2 0 −0.586222 − 1.80421i 0 1.80902 + 1.31433i 0 6.31172 + 2.05080i 0 4.36965 − 3.17474i 0
41.3 0 0.486403 + 1.49700i 0 1.80902 + 1.31433i 0 −2.66579 − 0.866168i 0 5.27674 − 3.83378i 0
41.4 0 1.75934 + 5.41468i 0 1.80902 + 1.31433i 0 10.9919 + 3.57148i 0 −18.9423 + 13.7624i 0
61.1 0 −2.30423 − 1.67412i 0 0.690983 − 2.12663i 0 1.86008 + 2.56018i 0 −0.274352 − 0.844368i 0
61.2 0 0.424576 + 0.308473i 0 0.690983 − 2.12663i 0 5.21482 + 7.17758i 0 −2.69604 − 8.29757i 0
61.3 0 1.09432 + 0.795072i 0 0.690983 − 2.12663i 0 −7.53649 − 10.3731i 0 −2.21575 − 6.81938i 0
61.4 0 4.71238 + 3.42375i 0 0.690983 − 2.12663i 0 −2.62857 − 3.61792i 0 7.70337 + 23.7085i 0
101.1 0 −2.30423 + 1.67412i 0 0.690983 + 2.12663i 0 1.86008 − 2.56018i 0 −0.274352 + 0.844368i 0
101.2 0 0.424576 − 0.308473i 0 0.690983 + 2.12663i 0 5.21482 − 7.17758i 0 −2.69604 + 8.29757i 0
101.3 0 1.09432 − 0.795072i 0 0.690983 + 2.12663i 0 −7.53649 + 10.3731i 0 −2.21575 + 6.81938i 0
101.4 0 4.71238 − 3.42375i 0 0.690983 + 2.12663i 0 −2.62857 + 3.61792i 0 7.70337 − 23.7085i 0
161.1 0 −1.08657 + 3.34411i 0 1.80902 − 1.31433i 0 −6.54766 + 2.12746i 0 −2.72130 − 1.97714i 0
161.2 0 −0.586222 + 1.80421i 0 1.80902 − 1.31433i 0 6.31172 − 2.05080i 0 4.36965 + 3.17474i 0
161.3 0 0.486403 − 1.49700i 0 1.80902 − 1.31433i 0 −2.66579 + 0.866168i 0 5.27674 + 3.83378i 0
161.4 0 1.75934 − 5.41468i 0 1.80902 − 1.31433i 0 10.9919 − 3.57148i 0 −18.9423 − 13.7624i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 220.3.p.b ✓ 16
11.c even 5 1 2420.3.f.a 16
11.d odd 10 1 inner 220.3.p.b ✓ 16
11.d odd 10 1 2420.3.f.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.p.b ✓ 16 1.a even 1 1 trivial
220.3.p.b ✓ 16 11.d odd 10 1 inner
2420.3.f.a 16 11.c even 5 1
2420.3.f.a 16 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 9 T_{3}^{15} + 68 T_{3}^{14} - 258 T_{3}^{13} + 1071 T_{3}^{12} - 1994 T_{3}^{11} + \cdots + 495616 \) acting on \(S_{3}^{\mathrm{new}}(220, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 9 T^{15} + \cdots + 495616 \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{3} + 15 T^{2} + \cdots + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 5677831152400 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 586694626368400 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 33\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{8} + 84 T^{7} + \cdots + 107673335396)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 85\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 67\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 70950001317124)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 58\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 367433895690751)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
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