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Properties

Label 572.2.n.a
Level $572$
Weight $2$
Character orbit 572.n
Analytic conductor $4.567$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [572,2,Mod(53,572)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(572, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("572.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.n (of order \(5\), 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: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 22 x^{18} - 72 x^{17} + 236 x^{16} - 556 x^{15} + 1232 x^{14} - 1981 x^{13} + \cdots + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( - \beta_{10} - \beta_{9}) q^{5} + ( - \beta_{15} + \beta_{11} - \beta_{7} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{14} - \beta_{13} - \beta_{8} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + ( - \beta_{10} - \beta_{9}) q^{5} + ( - \beta_{15} + \beta_{11} - \beta_{7} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{19} + \beta_{17} + \beta_{16} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{3} + q^{5} + 3 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + q^{3} + q^{5} + 3 q^{7} + 12 q^{9} - q^{11} - 5 q^{13} + 14 q^{15} - 3 q^{17} - 7 q^{19} - 32 q^{21} - 30 q^{23} - 12 q^{25} + 13 q^{27} - 14 q^{29} + 11 q^{31} - 10 q^{33} - 10 q^{35} + 45 q^{37} - 4 q^{39} + 9 q^{41} - 8 q^{43} - 34 q^{45} + 39 q^{47} + 30 q^{49} - 55 q^{51} + 36 q^{53} + 11 q^{55} - 31 q^{57} - 31 q^{59} - 7 q^{61} + 14 q^{63} - 14 q^{65} + 26 q^{67} + 32 q^{69} + 33 q^{71} - 44 q^{73} + 37 q^{75} - 73 q^{77} + 21 q^{79} + 16 q^{81} - 25 q^{83} + 15 q^{85} + 32 q^{87} - 2 q^{89} + 3 q^{91} + 33 q^{93} - 37 q^{95} + 52 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 22 x^{18} - 72 x^{17} + 236 x^{16} - 556 x^{15} + 1232 x^{14} - 1981 x^{13} + \cdots + 400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 72\!\cdots\!07 \nu^{19} + \cdots + 18\!\cdots\!20 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 92\!\cdots\!14 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31\!\cdots\!65 \nu^{19} + \cdots - 47\!\cdots\!00 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21\!\cdots\!59 \nu^{19} + \cdots + 62\!\cdots\!00 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 50\!\cdots\!27 \nu^{19} + \cdots + 41\!\cdots\!00 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\!\cdots\!29 \nu^{19} + \cdots + 92\!\cdots\!20 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!45 \nu^{19} + \cdots + 15\!\cdots\!00 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\!\cdots\!39 \nu^{19} + \cdots - 42\!\cdots\!40 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17\!\cdots\!29 \nu^{19} + \cdots + 60\!\cdots\!60 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!01 \nu^{19} + \cdots - 51\!\cdots\!50 ) / 27\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!02 \nu^{19} + \cdots - 18\!\cdots\!80 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 61\!\cdots\!76 \nu^{19} + \cdots + 14\!\cdots\!50 ) / 54\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 28\!\cdots\!51 \nu^{19} + \cdots - 40\!\cdots\!00 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 23\!\cdots\!45 \nu^{19} + \cdots - 38\!\cdots\!00 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 49\!\cdots\!95 \nu^{19} + \cdots - 48\!\cdots\!20 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 58\!\cdots\!99 \nu^{19} + \cdots - 27\!\cdots\!80 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 71\!\cdots\!25 \nu^{19} + \cdots - 52\!\cdots\!40 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 10\!\cdots\!79 \nu^{19} + \cdots + 67\!\cdots\!20 ) / 21\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + 2\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} + \beta_{17} - \beta_{16} - \beta_{15} - \beta_{10} - \beta_{9} - \beta_{6} - 5 \beta_{5} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{19} - 2 \beta_{18} + 2 \beta_{17} + 2 \beta_{16} + 9 \beta_{14} + \beta_{13} - 2 \beta_{12} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{19} - 11 \beta_{17} + 10 \beta_{16} + 10 \beta_{15} + 10 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} + \cdots - 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13 \beta_{19} + 28 \beta_{18} - 11 \beta_{17} - 28 \beta_{16} - 5 \beta_{15} - 75 \beta_{14} - 47 \beta_{13} + \cdots - 47 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 110 \beta_{19} - 87 \beta_{18} + 110 \beta_{17} - 30 \beta_{15} - 26 \beta_{14} - 87 \beta_{12} + \cdots + 355 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 215 \beta_{18} - 37 \beta_{17} + 294 \beta_{16} + 215 \beta_{15} + 406 \beta_{14} + 325 \beta_{13} + \cdots - 285 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 946 \beta_{19} + 1062 \beta_{18} - 736 \beta_{17} - 736 \beta_{16} - 965 \beta_{14} + 59 \beta_{13} + \cdots - 1984 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1291 \beta_{19} + 1348 \beta_{17} - 1911 \beta_{16} - 1911 \beta_{15} - 1911 \beta_{14} - 976 \beta_{13} + \cdots + 7959 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5134 \beta_{19} - 9369 \beta_{18} + 1348 \beta_{17} + 9369 \beta_{16} + 3153 \beta_{15} + 11869 \beta_{14} + \cdots + 5967 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 18559 \beta_{19} + 16737 \beta_{18} - 18559 \beta_{17} + 8761 \beta_{15} - 8320 \beta_{14} + \cdots - 78862 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 52771 \beta_{18} + 29282 \beta_{17} - 81721 \beta_{16} - 52771 \beta_{15} - 91550 \beta_{14} + \cdots + 80364 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 169447 \beta_{19} - 227679 \beta_{18} + 145927 \beta_{17} + 145927 \beta_{16} + 281283 \beta_{14} + \cdots + 508358 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 363035 \beta_{19} - 491764 \beta_{17} + 450730 \beta_{16} + 450730 \beta_{15} + 450730 \beta_{14} + \cdots - 1667114 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 950228 \beta_{19} + 2011007 \beta_{18} - 491764 \beta_{17} - 2011007 \beta_{16} - 740416 \beta_{15} + \cdots - 1492754 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 4959022 \beta_{19} - 3869709 \beta_{18} + 4959022 \beta_{17} - 2288193 \beta_{15} + 1217201 \beta_{14} + \cdots + 16831149 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 11058567 \beta_{18} - 4207543 \beta_{17} + 17654843 \beta_{16} + 11058567 \beta_{15} + \cdots - 18074266 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 42613209 \beta_{19} + 53417028 \beta_{18} - 33352998 \beta_{17} - 33352998 \beta_{16} - 53619230 \beta_{14} + \cdots - 111204725 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(353\) \(365\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.911154 + 2.80424i
−0.601689 + 1.85181i
0.170304 − 0.524141i
0.465638 − 1.43309i
0.758867 − 2.33555i
−1.11876 + 0.812829i
−0.384533 + 0.279379i
0.695167 − 0.505068i
1.06177 − 0.771421i
1.86439 − 1.35456i
−0.911154 − 2.80424i
−0.601689 − 1.85181i
0.170304 + 0.524141i
0.465638 + 1.43309i
0.758867 + 2.33555i
−1.11876 − 0.812829i
−0.384533 − 0.279379i
0.695167 + 0.505068i
1.06177 + 0.771421i
1.86439 + 1.35456i
0 −2.38543 + 1.73312i 0 −1.16829 − 3.59562i 0 2.62868 + 1.90985i 0 1.75954 − 5.41530i 0
53.2 0 −1.57524 + 1.14448i 0 −0.125488 − 0.386211i 0 −1.28300 − 0.932153i 0 0.244502 − 0.752499i 0
53.3 0 0.445861 − 0.323937i 0 0.710005 + 2.18517i 0 3.17229 + 2.30481i 0 −0.833194 + 2.56431i 0
53.4 0 1.21906 − 0.885697i 0 −1.09133 − 3.35878i 0 −3.27072 − 2.37632i 0 −0.225410 + 0.693741i 0
53.5 0 1.98674 − 1.44345i 0 0.248052 + 0.763424i 0 0.0617615 + 0.0448724i 0 0.936532 − 2.88235i 0
157.1 0 −0.427329 − 1.31518i 0 −1.65997 − 1.20604i 0 0.287529 − 0.884923i 0 0.879951 − 0.639322i 0
157.2 0 −0.146878 − 0.452045i 0 3.01285 + 2.18896i 0 0.736420 − 2.26647i 0 2.24428 − 1.63056i 0
157.3 0 0.265530 + 0.817217i 0 −1.45473 − 1.05692i 0 0.157953 − 0.486129i 0 1.82971 − 1.32936i 0
157.4 0 0.405560 + 1.24819i 0 1.75793 + 1.27721i 0 −1.12323 + 3.45694i 0 1.03356 − 0.750928i 0
157.5 0 0.712135 + 2.19173i 0 0.270974 + 0.196874i 0 0.132310 − 0.407207i 0 −1.86947 + 1.35825i 0
313.1 0 −2.38543 − 1.73312i 0 −1.16829 + 3.59562i 0 2.62868 − 1.90985i 0 1.75954 + 5.41530i 0
313.2 0 −1.57524 − 1.14448i 0 −0.125488 + 0.386211i 0 −1.28300 + 0.932153i 0 0.244502 + 0.752499i 0
313.3 0 0.445861 + 0.323937i 0 0.710005 − 2.18517i 0 3.17229 − 2.30481i 0 −0.833194 − 2.56431i 0
313.4 0 1.21906 + 0.885697i 0 −1.09133 + 3.35878i 0 −3.27072 + 2.37632i 0 −0.225410 − 0.693741i 0
313.5 0 1.98674 + 1.44345i 0 0.248052 − 0.763424i 0 0.0617615 − 0.0448724i 0 0.936532 + 2.88235i 0
521.1 0 −0.427329 + 1.31518i 0 −1.65997 + 1.20604i 0 0.287529 + 0.884923i 0 0.879951 + 0.639322i 0
521.2 0 −0.146878 + 0.452045i 0 3.01285 − 2.18896i 0 0.736420 + 2.26647i 0 2.24428 + 1.63056i 0
521.3 0 0.265530 − 0.817217i 0 −1.45473 + 1.05692i 0 0.157953 + 0.486129i 0 1.82971 + 1.32936i 0
521.4 0 0.405560 − 1.24819i 0 1.75793 − 1.27721i 0 −1.12323 − 3.45694i 0 1.03356 + 0.750928i 0
521.5 0 0.712135 − 2.19173i 0 0.270974 − 0.196874i 0 0.132310 + 0.407207i 0 −1.86947 − 1.35825i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.n.a ✓ 20
11.c even 5 1 inner 572.2.n.a ✓ 20
11.c even 5 1 6292.2.a.w 10
11.d odd 10 1 6292.2.a.x 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.n.a ✓ 20 1.a even 1 1 trivial
572.2.n.a ✓ 20 11.c even 5 1 inner
6292.2.a.w 10 11.c even 5 1
6292.2.a.x 10 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - T_{3}^{19} + 2 T_{3}^{18} + 2 T_{3}^{17} + 46 T_{3}^{16} - 129 T_{3}^{15} + 422 T_{3}^{14} + \cdots + 400 \) acting on \(S_{2}^{\mathrm{new}}(572, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - T^{19} + \cdots + 400 \) Copy content Toggle raw display
$5$ \( T^{20} - T^{19} + \cdots + 10000 \) Copy content Toggle raw display
$7$ \( T^{20} - 3 T^{19} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 25937424601 \) Copy content Toggle raw display
$13$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 5606265625 \) Copy content Toggle raw display
$19$ \( T^{20} + 7 T^{19} + \cdots + 87025 \) Copy content Toggle raw display
$23$ \( (T^{10} + 15 T^{9} + \cdots + 2384644)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 7055832001 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 87658037041 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 1362906814096 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 453666314709136 \) Copy content Toggle raw display
$43$ \( (T^{10} + 4 T^{9} + \cdots + 28117204)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 63\!\cdots\!41 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 28\!\cdots\!81 \) Copy content Toggle raw display
$59$ \( T^{20} + 31 T^{19} + \cdots + 844561 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 287193860772025 \) Copy content Toggle raw display
$67$ \( (T^{10} - 13 T^{9} + \cdots + 1577216)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 57\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 9965038995025 \) Copy content Toggle raw display
$89$ \( (T^{10} + T^{9} + \cdots + 723464500)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
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