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Properties

Label 220.3.j.a
Level $220$
Weight $3$
Character orbit 220.j
Analytic conductor $5.995$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [220,3,Mod(133,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.133");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 220.j (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: \(5.99456581593\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 2 x^{18} - 6 x^{17} + 1579 x^{16} - 3420 x^{15} + 3700 x^{14} - 2060 x^{13} + \cdots + 30250000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{13} q^{5} + ( - \beta_{11} - \beta_{3}) q^{7} + (\beta_{19} + \beta_{14} + \cdots - 5 \beta_{4}) q^{9} + \beta_{5} q^{11} + (\beta_{19} - \beta_{17} + \beta_{15} + \cdots - 2) q^{13}+ \cdots + (\beta_{18} + \beta_{17} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3} + 8 q^{5} - 28 q^{13} + 28 q^{15} + 24 q^{17} + 106 q^{23} - 6 q^{25} - 50 q^{27} + 88 q^{31} - 22 q^{33} - 252 q^{35} + 2 q^{37} + 72 q^{41} + 168 q^{43} + 38 q^{45} + 108 q^{47} + 112 q^{51}+ \cdots - 542 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 2 x^{18} - 6 x^{17} + 1579 x^{16} - 3420 x^{15} + 3700 x^{14} - 2060 x^{13} + \cdots + 30250000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 32\!\cdots\!37 \nu^{19} + \cdots + 42\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 62\!\cdots\!89 \nu^{19} + \cdots + 34\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 70\!\cdots\!77 \nu^{19} + \cdots + 79\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21\!\cdots\!19 \nu^{19} + \cdots - 18\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 44\!\cdots\!31 \nu^{19} + \cdots - 84\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 57\!\cdots\!84 \nu^{19} + \cdots + 34\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 68\!\cdots\!14 \nu^{19} + \cdots + 27\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 53\!\cdots\!19 \nu^{19} + \cdots + 59\!\cdots\!00 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 72\!\cdots\!67 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 81\!\cdots\!43 \nu^{19} + \cdots + 19\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 17\!\cdots\!43 \nu^{19} + \cdots - 74\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!03 \nu^{19} + \cdots + 91\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\!\cdots\!58 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!58 \nu^{19} + \cdots - 31\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 30\!\cdots\!87 \nu^{19} + \cdots + 18\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 16\!\cdots\!32 \nu^{19} + \cdots - 68\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 22\!\cdots\!57 \nu^{19} + \cdots - 24\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 27\!\cdots\!21 \nu^{19} + \cdots + 18\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} + \beta_{14} + \beta_{9} - 14\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{18} + \beta_{17} - 2 \beta_{16} - \beta_{14} - \beta_{13} - \beta_{12} + 3 \beta_{11} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{16} - 27 \beta_{15} - 9 \beta_{14} + 9 \beta_{13} - 4 \beta_{12} - 36 \beta_{11} - \beta_{9} + \cdots - 298 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{19} - 60 \beta_{18} - 55 \beta_{17} - 60 \beta_{16} + 2 \beta_{15} + 116 \beta_{14} + \cdots + 169 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 747 \beta_{19} + 72 \beta_{18} - 188 \beta_{17} - 955 \beta_{14} + 300 \beta_{13} + 182 \beta_{11} + \cdots + 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6 \beta_{19} - 1616 \beta_{18} - 1431 \beta_{17} + 1616 \beta_{16} - 6 \beta_{15} + 1269 \beta_{14} + \cdots - 2866 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2584 \beta_{17} - 1774 \beta_{16} + 21287 \beta_{15} + 15343 \beta_{14} - 17723 \beta_{13} + \cdots + 217984 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2492 \beta_{19} + 43774 \beta_{18} + 70687 \beta_{17} + 43774 \beta_{16} + 2492 \beta_{15} + \cdots - 186047 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 619903 \beta_{19} - 34848 \beta_{18} + 233368 \beta_{17} + 844977 \beta_{14} - 343718 \beta_{13} + \cdots - 99634 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 156100 \beta_{19} + 1211874 \beta_{18} + 1334263 \beta_{17} - 1211874 \beta_{16} - 156100 \beta_{15} + \cdots + 2440032 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4471976 \beta_{17} + 485882 \beta_{16} - 18329047 \beta_{15} - 17880293 \beta_{14} + 20372933 \beta_{13} + \cdots - 185787214 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 6840950 \beta_{19} - 34269220 \beta_{18} - 73377355 \beta_{17} - 34269220 \beta_{16} + \cdots + 171470333 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 547844419 \beta_{19} - 420760 \beta_{18} - 237274100 \beta_{17} - 773187675 \beta_{14} + \cdots + 97126056 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 259203170 \beta_{19} - 986501648 \beta_{18} - 1186294759 \beta_{17} + 986501648 \beta_{16} + \cdots - 1920985746 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 5450652880 \beta_{17} + 393924322 \beta_{16} + 16505328583 \beta_{15} + 18848913327 \beta_{14} + \cdots + 168146200616 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 9085296292 \beta_{19} + 28814269710 \beta_{18} + 72872825015 \beta_{17} + 28814269710 \beta_{16} + \cdots - 156761348423 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 500260484703 \beta_{19} + 21271059072 \beta_{18} + 228663351792 \beta_{17} + 725481699249 \beta_{14} + \cdots - 87567387082 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 303708550636 \beta_{19} + 851590872914 \beta_{18} + 1056536463919 \beta_{17} - 851590872914 \beta_{16} + \cdots + 1619938852776 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(111\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1
3.92142 + 3.92142i
3.02805 + 3.02805i
2.66982 + 2.66982i
1.52486 + 1.52486i
0.522070 + 0.522070i
0.176450 + 0.176450i
−1.36101 − 1.36101i
−2.08033 − 2.08033i
−3.44912 − 3.44912i
−3.95222 − 3.95222i
3.92142 − 3.92142i
3.02805 − 3.02805i
2.66982 − 2.66982i
1.52486 − 1.52486i
0.522070 − 0.522070i
0.176450 − 0.176450i
−1.36101 + 1.36101i
−2.08033 + 2.08033i
−3.44912 + 3.44912i
−3.95222 + 3.95222i
0 −3.92142 − 3.92142i 0 3.38645 + 3.67858i 0 −3.90761 + 3.90761i 0 21.7551i 0
133.2 0 −3.02805 − 3.02805i 0 −1.23337 − 4.84549i 0 3.53136 − 3.53136i 0 9.33821i 0
133.3 0 −2.66982 − 2.66982i 0 −4.91667 + 0.909014i 0 0.0733527 − 0.0733527i 0 5.25586i 0
133.4 0 −1.52486 − 1.52486i 0 4.49108 − 2.19777i 0 0.0256967 − 0.0256967i 0 − 4.34959i 0
133.5 0 −0.522070 − 0.522070i 0 −4.02463 + 2.96687i 0 3.40356 − 3.40356i 0 − 8.45489i 0
133.6 0 −0.176450 − 0.176450i 0 0.718279 + 4.94814i 0 −8.33978 + 8.33978i 0 − 8.93773i 0
133.7 0 1.36101 + 1.36101i 0 4.24143 + 2.64770i 0 4.16351 − 4.16351i 0 − 5.29528i 0
133.8 0 2.08033 + 2.08033i 0 −0.0655940 − 4.99957i 0 7.13039 − 7.13039i 0 − 0.344475i 0
133.9 0 3.44912 + 3.44912i 0 −3.20203 + 3.84018i 0 1.11033 − 1.11033i 0 14.7928i 0
133.10 0 3.95222 + 3.95222i 0 4.60507 − 1.94766i 0 −7.19082 + 7.19082i 0 22.2400i 0
177.1 0 −3.92142 + 3.92142i 0 3.38645 − 3.67858i 0 −3.90761 − 3.90761i 0 − 21.7551i 0
177.2 0 −3.02805 + 3.02805i 0 −1.23337 + 4.84549i 0 3.53136 + 3.53136i 0 − 9.33821i 0
177.3 0 −2.66982 + 2.66982i 0 −4.91667 − 0.909014i 0 0.0733527 + 0.0733527i 0 − 5.25586i 0
177.4 0 −1.52486 + 1.52486i 0 4.49108 + 2.19777i 0 0.0256967 + 0.0256967i 0 4.34959i 0
177.5 0 −0.522070 + 0.522070i 0 −4.02463 − 2.96687i 0 3.40356 + 3.40356i 0 8.45489i 0
177.6 0 −0.176450 + 0.176450i 0 0.718279 − 4.94814i 0 −8.33978 − 8.33978i 0 8.93773i 0
177.7 0 1.36101 − 1.36101i 0 4.24143 − 2.64770i 0 4.16351 + 4.16351i 0 5.29528i 0
177.8 0 2.08033 − 2.08033i 0 −0.0655940 + 4.99957i 0 7.13039 + 7.13039i 0 0.344475i 0
177.9 0 3.44912 − 3.44912i 0 −3.20203 − 3.84018i 0 1.11033 + 1.11033i 0 − 14.7928i 0
177.10 0 3.95222 − 3.95222i 0 4.60507 + 1.94766i 0 −7.19082 − 7.19082i 0 − 22.2400i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.10
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 220.3.j.a ✓ 20
3.b odd 2 1 1980.3.s.a 20
5.b even 2 1 1100.3.j.b 20
5.c odd 4 1 inner 220.3.j.a ✓ 20
5.c odd 4 1 1100.3.j.b 20
15.e even 4 1 1980.3.s.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.j.a ✓ 20 1.a even 1 1 trivial
220.3.j.a ✓ 20 5.c odd 4 1 inner
1100.3.j.b 20 5.b even 2 1
1100.3.j.b 20 5.c odd 4 1
1980.3.s.a 20 3.b odd 2 1
1980.3.s.a 20 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(220, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 2 T^{19} + \cdots + 30250000 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 95367431640625 \) Copy content Toggle raw display
$7$ \( T^{20} - 656 T^{17} + \cdots + 31360000 \) Copy content Toggle raw display
$11$ \( (T^{2} - 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 1641573808000)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 97\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 26\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
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