↑

Properties

Label 549.2.y.b
Level $549$
Weight $2$
Character orbit 549.y
Analytic conductor $4.384$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [549,2,Mod(64,549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(549, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("549.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 549 = 3^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 549.y (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: \(4.38378707097\)
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: \( x^{16} + 17x^{14} + 111x^{12} + 361x^{10} + 624x^{8} + 558x^{6} + 229x^{4} + 34x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 61)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_{15} + \beta_{9} + \beta_{5} + \cdots - 1) q^{4}+ \cdots + (\beta_{15} + \beta_{13} + \cdots - 2 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + (\beta_{15} + \beta_{9} + \beta_{5} + \cdots - 1) q^{4}+ \cdots + (4 \beta_{15} - 4 \beta_{14} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 5 q^{2} + 3 q^{4} + 10 q^{7} + 5 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 5 q^{2} + 3 q^{4} + 10 q^{7} + 5 q^{8} - 5 q^{10} - 12 q^{13} + 18 q^{14} + 19 q^{16} + 3 q^{19} + 13 q^{20} + 19 q^{22} + 15 q^{23} - 2 q^{25} - 10 q^{26} + 35 q^{28} - 15 q^{31} - 14 q^{34} - 10 q^{35} - 5 q^{37} + 15 q^{38} - 12 q^{41} - 25 q^{43} + 50 q^{44} + 27 q^{46} - 6 q^{47} - 30 q^{49} - 46 q^{52} + 20 q^{53} + 20 q^{55} + 28 q^{56} - 41 q^{58} - 5 q^{59} - 53 q^{61} - 16 q^{62} + 17 q^{64} - 20 q^{65} - 55 q^{67} - 80 q^{68} - 17 q^{70} + 50 q^{71} - 11 q^{73} - 24 q^{74} - 19 q^{76} - 63 q^{77} + 40 q^{79} + 49 q^{80} - 31 q^{83} + 55 q^{85} - 35 q^{86} + 27 q^{88} - 60 q^{89} - 15 q^{91} + 5 q^{92} + 65 q^{94} - 48 q^{95} + 45 q^{97} - 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 17x^{14} + 111x^{12} + 361x^{10} + 624x^{8} + 558x^{6} + 229x^{4} + 34x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15 \nu^{15} - 9 \nu^{14} + 226 \nu^{13} - 162 \nu^{12} + 1231 \nu^{11} - 1117 \nu^{10} + 3082 \nu^{9} + \cdots - 123 ) / 88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9 \nu^{15} - 3 \nu^{14} + 162 \nu^{13} - 54 \nu^{12} + 1117 \nu^{11} - 387 \nu^{10} + 3750 \nu^{9} + \cdots - 41 ) / 88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9 \nu^{15} - 29 \nu^{14} + 162 \nu^{13} - 434 \nu^{12} + 1117 \nu^{11} - 2333 \nu^{10} + 3750 \nu^{9} + \cdots - 15 ) / 88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9 \nu^{15} - 3 \nu^{14} - 162 \nu^{13} - 54 \nu^{12} - 1117 \nu^{11} - 387 \nu^{10} - 3750 \nu^{9} + \cdots - 41 ) / 88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9 \nu^{15} + 29 \nu^{14} + 162 \nu^{13} + 434 \nu^{12} + 1117 \nu^{11} + 2333 \nu^{10} + 3750 \nu^{9} + \cdots + 15 ) / 88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41 \nu^{15} - 9 \nu^{14} - 650 \nu^{13} - 118 \nu^{12} - 3837 \nu^{11} - 501 \nu^{10} - 10850 \nu^{9} + \cdots - 123 ) / 88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18 \nu^{15} + 21 \nu^{14} - 302 \nu^{13} + 334 \nu^{12} - 1926 \nu^{11} + 1983 \nu^{10} - 6004 \nu^{9} + \cdots + 133 ) / 44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25 \nu^{15} + 5 \nu^{14} - 406 \nu^{13} + 68 \nu^{12} - 2477 \nu^{11} + 293 \nu^{10} - 7300 \nu^{9} + \cdots - 49 ) / 44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25 \nu^{15} - 13 \nu^{14} + 406 \nu^{13} - 212 \nu^{12} + 2477 \nu^{11} - 1303 \nu^{10} + 7300 \nu^{9} + \cdots - 31 ) / 44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 41 \nu^{15} + 9 \nu^{14} - 694 \nu^{13} + 162 \nu^{12} - 4497 \nu^{11} + 1117 \nu^{10} - 14414 \nu^{9} + \cdots + 79 ) / 88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25 \nu^{15} + 5 \nu^{14} + 406 \nu^{13} + 68 \nu^{12} + 2477 \nu^{11} + 293 \nu^{10} + 7300 \nu^{9} + \cdots - 49 ) / 44 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 43 \nu^{15} - 15 \nu^{14} + 730 \nu^{13} - 226 \nu^{12} + 4755 \nu^{11} - 1231 \nu^{10} + 15394 \nu^{9} + \cdots - 117 ) / 88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 41 \nu^{15} + 9 \nu^{14} + 694 \nu^{13} + 162 \nu^{12} + 4497 \nu^{11} + 1117 \nu^{10} + 14414 \nu^{9} + \cdots + 79 ) / 88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 43 \nu^{15} - 15 \nu^{14} - 730 \nu^{13} - 226 \nu^{12} - 4755 \nu^{11} - 1231 \nu^{10} - 15394 \nu^{9} + \cdots - 117 ) / 88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 18 \nu^{15} - 21 \nu^{14} - 302 \nu^{13} - 334 \nu^{12} - 1926 \nu^{11} - 1983 \nu^{10} - 6004 \nu^{9} + \cdots - 133 ) / 44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \cdots + 4 \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - 2 \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{10} + 5 \beta_{9} - \beta_{7} + 5 \beta_{6} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} + 7 \beta_{14} + 9 \beta_{13} + 5 \beta_{12} - 6 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{15} + 17 \beta_{14} - 10 \beta_{13} + 17 \beta_{12} + \beta_{11} - 10 \beta_{10} + \cdots - 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 22 \beta_{15} - 45 \beta_{14} - 63 \beta_{13} - 27 \beta_{12} + 34 \beta_{11} - 25 \beta_{10} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 76 \beta_{15} - 119 \beta_{14} + 76 \beta_{13} - 119 \beta_{12} - 13 \beta_{11} + 76 \beta_{10} + \cdots + 70 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 174 \beta_{15} + 286 \beta_{14} + 414 \beta_{13} + 162 \beta_{12} - 201 \beta_{11} + 140 \beta_{10} + \cdots + 53 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 527 \beta_{15} + 790 \beta_{14} - 528 \beta_{13} + 790 \beta_{12} + 112 \beta_{11} - 528 \beta_{10} + \cdots - 419 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1227 \beta_{15} - 1819 \beta_{14} - 2670 \beta_{13} - 1019 \beta_{12} + 1231 \beta_{11} - 846 \beta_{10} + \cdots - 339 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3513 \beta_{15} - 5144 \beta_{14} + 3529 \beta_{13} - 5144 \beta_{12} - 827 \beta_{11} + 3529 \beta_{10} + \cdots + 2619 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8237 \beta_{15} + 11597 \beta_{14} + 17125 \beta_{13} + 6511 \beta_{12} - 7700 \beta_{11} + 5299 \beta_{10} + \cdots + 2158 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 22979 \beta_{15} + 33222 \beta_{14} - 23138 \beta_{13} + 33222 \beta_{12} + 5694 \beta_{11} + \cdots - 16639 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 54035 \beta_{15} - 74084 \beta_{14} - 109689 \beta_{13} - 41772 \beta_{12} + 48722 \beta_{11} + \cdots - 13755 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/549\mathbb{Z}\right)^\times\).

\(n\) \(245\) \(307\)
\(\chi(n)\) \(1\) \(1 + \beta_{1} + \beta_{10} + \beta_{13}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
− 1.60228i
0.196205i
0.776536i
2.53165i
1.60228i
− 0.196205i
− 0.776536i
− 2.53165i
− 1.46081i
− 0.475317i
1.25523i
1.85647i
1.46081i
0.475317i
− 1.25523i
− 1.85647i
−1.52385 − 0.495130i 0 0.458946 + 0.333444i 1.72728 + 1.25494i 0 −1.05248 + 0.341973i 1.34932 + 1.85718i 0 −2.01077 − 2.76758i
64.2 0.186602 + 0.0606307i 0 −1.58689 − 1.15294i −2.93210 − 2.13030i 0 −0.222647 + 0.0723423i −0.456866 − 0.628822i 0 −0.417975 − 0.575293i
64.3 0.738530 + 0.239963i 0 −1.13019 − 0.821131i 0.515969 + 0.374873i 0 1.31473 − 0.427183i −1.55051 − 2.13410i 0 0.291103 + 0.400668i
64.4 2.40774 + 0.782322i 0 3.56715 + 2.59169i −0.429180 − 0.311818i 0 3.57843 − 1.16270i 3.58511 + 4.93448i 0 −0.789413 − 1.08653i
163.1 −1.52385 + 0.495130i 0 0.458946 − 0.333444i 1.72728 − 1.25494i 0 −1.05248 − 0.341973i 1.34932 − 1.85718i 0 −2.01077 + 2.76758i
163.2 0.186602 − 0.0606307i 0 −1.58689 + 1.15294i −2.93210 + 2.13030i 0 −0.222647 − 0.0723423i −0.456866 + 0.628822i 0 −0.417975 + 0.575293i
163.3 0.738530 − 0.239963i 0 −1.13019 + 0.821131i 0.515969 − 0.374873i 0 1.31473 + 0.427183i −1.55051 + 2.13410i 0 0.291103 − 0.400668i
163.4 2.40774 − 0.782322i 0 3.56715 − 2.59169i −0.429180 + 0.311818i 0 3.57843 + 1.16270i 3.58511 − 4.93448i 0 −0.789413 + 1.08653i
235.1 −0.858642 + 1.18182i 0 −0.0413973 − 0.127408i 0.701732 + 2.15971i 0 2.88318 + 3.96835i −2.59251 − 0.842356i 0 −3.15492 − 1.02510i
235.2 −0.279384 + 0.384540i 0 0.548219 + 1.68724i −0.00765597 − 0.0235626i 0 −2.52527 − 3.47573i −1.70608 − 0.554340i 0 0.0111997 + 0.00363901i
235.3 0.737805 − 1.01550i 0 0.131147 + 0.403630i −0.620635 − 1.91012i 0 1.40505 + 1.93389i 2.89423 + 0.940394i 0 −2.39763 − 0.779038i
235.4 1.09120 − 1.50191i 0 −0.446986 − 1.37568i 1.04459 + 3.21492i 0 −0.380995 − 0.524395i 0.977305 + 0.317546i 0 5.96841 + 1.93925i
271.1 −0.858642 − 1.18182i 0 −0.0413973 + 0.127408i 0.701732 − 2.15971i 0 2.88318 − 3.96835i −2.59251 + 0.842356i 0 −3.15492 + 1.02510i
271.2 −0.279384 − 0.384540i 0 0.548219 − 1.68724i −0.00765597 + 0.0235626i 0 −2.52527 + 3.47573i −1.70608 + 0.554340i 0 0.0111997 − 0.00363901i
271.3 0.737805 + 1.01550i 0 0.131147 − 0.403630i −0.620635 + 1.91012i 0 1.40505 − 1.93389i 2.89423 − 0.940394i 0 −2.39763 + 0.779038i
271.4 1.09120 + 1.50191i 0 −0.446986 + 1.37568i 1.04459 − 3.21492i 0 −0.380995 + 0.524395i 0.977305 − 0.317546i 0 5.96841 − 1.93925i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 549.2.y.b 16
3.b odd 2 1 61.2.g.a ✓ 16
12.b even 2 1 976.2.bd.b 16
61.g even 10 1 inner 549.2.y.b 16
183.l odd 10 1 61.2.g.a ✓ 16
183.r even 20 2 3721.2.a.k 16
732.y even 10 1 976.2.bd.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.2.g.a ✓ 16 3.b odd 2 1
61.2.g.a ✓ 16 183.l odd 10 1
549.2.y.b 16 1.a even 1 1 trivial
549.2.y.b 16 61.g even 10 1 inner
976.2.bd.b 16 12.b even 2 1
976.2.bd.b 16 732.y even 10 1
3721.2.a.k 16 183.r even 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 5 T_{2}^{15} + 7 T_{2}^{14} + 5 T_{2}^{13} - 19 T_{2}^{12} + 46 T_{2}^{10} + 10 T_{2}^{9} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(549, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 5 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 11 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{16} - 10 T^{15} + \cdots + 1936 \) Copy content Toggle raw display
$11$ \( T^{16} + 87 T^{14} + \cdots + 1936 \) Copy content Toggle raw display
$13$ \( (T^{8} + 6 T^{7} + \cdots + 10261)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 25 T^{14} + \cdots + 4879681 \) Copy content Toggle raw display
$19$ \( T^{16} - 3 T^{15} + \cdots + 7311616 \) Copy content Toggle raw display
$23$ \( T^{16} - 15 T^{15} + \cdots + 30976 \) Copy content Toggle raw display
$29$ \( T^{16} + 185 T^{14} + \cdots + 383161 \) Copy content Toggle raw display
$31$ \( T^{16} + 15 T^{15} + \cdots + 1008016 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 46787420416 \) Copy content Toggle raw display
$41$ \( T^{16} + 12 T^{15} + \cdots + 1437601 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 591267856 \) Copy content Toggle raw display
$47$ \( (T^{8} + 3 T^{7} + \cdots - 3524)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 162537001 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 138529862416 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 191707312997281 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 395837272336 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 121101216016 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 687101735056 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 2515192996096 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 5678526736 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 394856641 \) Copy content Toggle raw display
show more
show less