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Properties

Label 220.3.p.a
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: \( x^{16} - 2 x^{15} + 5 x^{14} + 36 x^{13} + 396 x^{12} + 1918 x^{11} + 8573 x^{10} + 28624 x^{9} + \cdots + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 5\cdot 11 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{14} - \beta_{7}) q^{3} + (2 \beta_{7} + \beta_{6} - 2) q^{5} + (\beta_{15} + \beta_{14} + \beta_{12} + \cdots - 1) q^{7}+ \cdots + (\beta_{15} - \beta_{14} - 2 \beta_{13} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{14} - \beta_{7}) q^{3} + (2 \beta_{7} + \beta_{6} - 2) q^{5} + (\beta_{15} + \beta_{14} + \beta_{12} + \cdots - 1) q^{7}+ \cdots + (16 \beta_{15} + 29 \beta_{14} + \cdots - 26) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{3} - 20 q^{5} - 10 q^{7} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{3} - 20 q^{5} - 10 q^{7} - 19 q^{9} - 37 q^{11} + 35 q^{13} + 15 q^{15} + 55 q^{17} + 30 q^{19} + 72 q^{23} - 20 q^{25} - 25 q^{27} - 145 q^{29} - 80 q^{31} - 94 q^{33} + 44 q^{37} + 175 q^{39} + 35 q^{41} + 30 q^{45} + 2 q^{47} + 172 q^{49} - 460 q^{51} + 47 q^{53} + 85 q^{55} + 10 q^{57} + 98 q^{59} + 330 q^{61} + 90 q^{63} + 152 q^{67} - 317 q^{69} - 20 q^{71} - 495 q^{73} - 30 q^{75} - 204 q^{77} + 25 q^{79} + 57 q^{81} + 190 q^{83} - 25 q^{85} + 160 q^{89} - 119 q^{91} + 617 q^{93} - 90 q^{95} - 521 q^{97} - 70 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 5 x^{14} + 36 x^{13} + 396 x^{12} + 1918 x^{11} + 8573 x^{10} + 28624 x^{9} + \cdots + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 20\!\cdots\!48 \nu^{15} + \cdots - 19\!\cdots\!07 ) / 31\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\!\cdots\!94 \nu^{15} + \cdots + 45\!\cdots\!93 ) / 62\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 71\!\cdots\!97 \nu^{15} + \cdots - 20\!\cdots\!92 ) / 31\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\!\cdots\!64 \nu^{15} + \cdots + 10\!\cdots\!73 ) / 62\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21\!\cdots\!87 \nu^{15} + \cdots - 23\!\cdots\!74 ) / 56\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 26\!\cdots\!55 \nu^{15} + \cdots - 83\!\cdots\!54 ) / 62\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 44\!\cdots\!03 \nu^{15} + \cdots + 74\!\cdots\!31 ) / 62\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!73 \nu^{15} + \cdots + 30\!\cdots\!02 ) / 31\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 79\!\cdots\!85 \nu^{15} + \cdots + 12\!\cdots\!91 ) / 62\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 98\!\cdots\!87 \nu^{15} + \cdots - 40\!\cdots\!23 ) / 56\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 59\!\cdots\!08 \nu^{15} + \cdots - 30\!\cdots\!82 ) / 31\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!19 \nu^{15} + \cdots - 28\!\cdots\!48 ) / 56\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 70\!\cdots\!02 \nu^{15} + \cdots + 28\!\cdots\!03 ) / 31\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 15\!\cdots\!61 \nu^{15} + \cdots + 28\!\cdots\!44 ) / 56\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 94\!\cdots\!97 \nu^{15} + \cdots - 28\!\cdots\!48 ) / 31\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 6 \beta_{15} - 9 \beta_{14} - 10 \beta_{13} - 27 \beta_{12} - 20 \beta_{11} + 40 \beta_{10} + \cdots + 36 ) / 55 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4 \beta_{15} + 11 \beta_{14} - 2 \beta_{12} - 5 \beta_{10} + 4 \beta_{9} + 6 \beta_{8} - 51 \beta_{7} + \cdots + 16 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5 \beta_{15} + 13 \beta_{14} - 3 \beta_{13} + 3 \beta_{12} + \beta_{11} - 14 \beta_{10} + 4 \beta_{9} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 161 \beta_{15} + 309 \beta_{14} - 210 \beta_{13} + 247 \beta_{12} - 60 \beta_{11} - 665 \beta_{10} + \cdots + 109 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 761 \beta_{15} + 489 \beta_{14} - 1250 \beta_{13} + 1862 \beta_{12} - 760 \beta_{11} - 3655 \beta_{10} + \cdots - 1811 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 351 \beta_{15} - 1076 \beta_{14} - 737 \beta_{13} + 1882 \beta_{12} - 906 \beta_{11} - 2414 \beta_{10} + \cdots - 4646 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 9221 \beta_{15} - 65079 \beta_{14} + 7690 \beta_{13} + 27778 \beta_{12} - 13660 \beta_{11} + \cdots - 161234 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 156956 \beta_{15} - 440159 \beta_{14} + 188675 \beta_{13} - 69737 \beta_{12} + 27195 \beta_{11} + \cdots - 704074 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 234191 \beta_{15} - 391873 \beta_{14} + 304280 \beta_{13} - 328055 \beta_{12} + 138103 \beta_{11} + \cdots - 208192 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5634159 \beta_{15} - 3453881 \beta_{14} + 7994360 \beta_{13} - 13099428 \beta_{12} + 5518490 \beta_{11} + \cdots + 14119984 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 13100599 \beta_{15} + 34857219 \beta_{14} + 23740595 \beta_{13} - 67652613 \beta_{12} + \cdots + 168643124 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 13784035 \beta_{15} + 89961654 \beta_{14} - 9909370 \beta_{13} - 38289636 \beta_{12} + 17192042 \beta_{11} + \cdots + 222936140 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1102387369 \beta_{15} + 3067324896 \beta_{14} - 1288973000 \beta_{13} + 507271828 \beta_{12} + \cdots + 4786709691 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 8090887604 \beta_{15} + 13630360086 \beta_{14} - 10529842940 \beta_{13} + 11390473828 \beta_{12} + \cdots + 7108847761 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 7775756752 \beta_{15} + 4771780752 \beta_{14} - 11113567366 \beta_{13} + 18127633774 \beta_{12} + \cdots - 19436453841 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(111\) \(177\)
\(\chi(n)\) \(1 - \beta_{4} - \beta_{6} - \beta_{7}\) \(1\) \(1\)

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
4.74372 + 3.44651i
−1.13609 − 0.825419i
−1.95255 − 1.41861i
−0.0370387 − 0.0269102i
−0.584772 + 1.79974i
0.968143 − 2.97964i
−0.737287 + 2.26913i
−0.264119 + 0.812875i
−0.584772 − 1.79974i
0.968143 + 2.97964i
−0.737287 − 2.26913i
−0.264119 − 0.812875i
4.74372 − 3.44651i
−1.13609 + 0.825419i
−1.95255 + 1.41861i
−0.0370387 + 0.0269102i
0 −1.21659 − 3.74427i 0 −1.80902 − 1.31433i 0 −6.54602 − 2.12693i 0 −5.25831 + 3.82039i 0
41.2 0 0.147899 + 0.455187i 0 −1.80902 − 1.31433i 0 −3.05737 − 0.993400i 0 7.09583 − 5.15542i 0
41.3 0 0.901587 + 2.77480i 0 −1.80902 − 1.31433i 0 11.1733 + 3.63044i 0 0.394502 − 0.286622i 0
41.4 0 1.59415 + 4.90629i 0 −1.80902 − 1.31433i 0 −9.66012 − 3.13876i 0 −14.2492 + 10.3527i 0
61.1 0 −3.86726 − 2.80973i 0 −0.690983 + 2.12663i 0 5.14876 + 7.08667i 0 4.27996 + 13.1723i 0
61.2 0 −1.69407 − 1.23081i 0 −0.690983 + 2.12663i 0 −0.687326 − 0.946023i 0 −1.42619 − 4.38936i 0
61.3 0 0.323084 + 0.234734i 0 −0.690983 + 2.12663i 0 −4.97040 − 6.84117i 0 −2.73187 − 8.40783i 0
61.4 0 3.31119 + 2.40572i 0 −0.690983 + 2.12663i 0 3.59914 + 4.95378i 0 2.39532 + 7.37205i 0
101.1 0 −3.86726 + 2.80973i 0 −0.690983 − 2.12663i 0 5.14876 − 7.08667i 0 4.27996 − 13.1723i 0
101.2 0 −1.69407 + 1.23081i 0 −0.690983 − 2.12663i 0 −0.687326 + 0.946023i 0 −1.42619 + 4.38936i 0
101.3 0 0.323084 − 0.234734i 0 −0.690983 − 2.12663i 0 −4.97040 + 6.84117i 0 −2.73187 + 8.40783i 0
101.4 0 3.31119 − 2.40572i 0 −0.690983 − 2.12663i 0 3.59914 − 4.95378i 0 2.39532 − 7.37205i 0
161.1 0 −1.21659 + 3.74427i 0 −1.80902 + 1.31433i 0 −6.54602 + 2.12693i 0 −5.25831 − 3.82039i 0
161.2 0 0.147899 − 0.455187i 0 −1.80902 + 1.31433i 0 −3.05737 + 0.993400i 0 7.09583 + 5.15542i 0
161.3 0 0.901587 − 2.77480i 0 −1.80902 + 1.31433i 0 11.1733 − 3.63044i 0 0.394502 + 0.286622i 0
161.4 0 1.59415 − 4.90629i 0 −1.80902 + 1.31433i 0 −9.66012 + 3.13876i 0 −14.2492 − 10.3527i 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.a ✓ 16
11.c even 5 1 2420.3.f.c 16
11.d odd 10 1 inner 220.3.p.a ✓ 16
11.d odd 10 1 2420.3.f.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.p.a ✓ 16 1.a even 1 1 trivial
220.3.p.a ✓ 16 11.d odd 10 1 inner
2420.3.f.c 16 11.c even 5 1
2420.3.f.c 16 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + T_{3}^{15} + 28 T_{3}^{14} + 72 T_{3}^{13} + 611 T_{3}^{12} + 76 T_{3}^{11} + 11155 T_{3}^{10} + \cdots + 215296 \) 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
$3$ \( T^{16} + T^{15} + \cdots + 215296 \) Copy content Toggle raw display
$5$ \( (T^{4} + 5 T^{3} + 15 T^{2} + \cdots + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 1961064144400 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{8} - 36 T^{7} + \cdots - 19587964)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 97\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 2025401077036)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 107140783101329)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
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