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Properties

Label 538.4.a.a
Level $538$
Weight $4$
Character orbit 538.a
Self dual yes
Analytic conductor $31.743$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,4,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 538.a (trivial)

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: yes
Analytic conductor: \(31.7430275831\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 2 x^{12} - 189 x^{11} + 344 x^{10} + 12502 x^{9} - 15678 x^{8} - 385197 x^{7} + 263029 x^{6} + \cdots + 3268107 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta_1 - 1) q^{3} + 4 q^{4} + (\beta_{2} - 3) q^{5} + ( - 2 \beta_1 - 2) q^{6} + (\beta_{12} + \beta_{10} - 2 \beta_{9} + \cdots - 7) q^{7}+ \cdots + ( - \beta_{12} - 2 \beta_{10} + \cdots + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \beta_1 - 1) q^{3} + 4 q^{4} + (\beta_{2} - 3) q^{5} + ( - 2 \beta_1 - 2) q^{6} + (\beta_{12} + \beta_{10} - 2 \beta_{9} + \cdots - 7) q^{7}+ \cdots + (28 \beta_{12} + 22 \beta_{11} + \cdots + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 26 q^{2} - 15 q^{3} + 52 q^{4} - 41 q^{5} - 30 q^{6} - 60 q^{7} + 104 q^{8} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 26 q^{2} - 15 q^{3} + 52 q^{4} - 41 q^{5} - 30 q^{6} - 60 q^{7} + 104 q^{8} + 48 q^{9} - 82 q^{10} - 110 q^{11} - 60 q^{12} - 140 q^{13} - 120 q^{14} - 113 q^{15} + 208 q^{16} - 214 q^{17} + 96 q^{18} - 120 q^{19} - 164 q^{20} - 353 q^{21} - 220 q^{22} - 459 q^{23} - 120 q^{24} - 90 q^{25} - 280 q^{26} - 465 q^{27} - 240 q^{28} - 707 q^{29} - 226 q^{30} - 556 q^{31} + 416 q^{32} - 807 q^{33} - 428 q^{34} - 570 q^{35} + 192 q^{36} - 915 q^{37} - 240 q^{38} - 734 q^{39} - 328 q^{40} - 991 q^{41} - 706 q^{42} - 201 q^{43} - 440 q^{44} - 1154 q^{45} - 918 q^{46} - 686 q^{47} - 240 q^{48} - 407 q^{49} - 180 q^{50} - 251 q^{51} - 560 q^{52} - 2068 q^{53} - 930 q^{54} - 459 q^{55} - 480 q^{56} - 1982 q^{57} - 1414 q^{58} - 949 q^{59} - 452 q^{60} - 1731 q^{61} - 1112 q^{62} - 1856 q^{63} + 832 q^{64} - 2258 q^{65} - 1614 q^{66} - 1047 q^{67} - 856 q^{68} + 587 q^{69} - 1140 q^{70} - 2152 q^{71} + 384 q^{72} - 149 q^{73} - 1830 q^{74} + 441 q^{75} - 480 q^{76} + 886 q^{77} - 1468 q^{78} - 1608 q^{79} - 656 q^{80} + 1633 q^{81} - 1982 q^{82} + 2073 q^{83} - 1412 q^{84} - 465 q^{85} - 402 q^{86} + 4403 q^{87} - 880 q^{88} + 815 q^{89} - 2308 q^{90} + 2051 q^{91} - 1836 q^{92} - 2113 q^{93} - 1372 q^{94} - 158 q^{95} - 480 q^{96} - 1011 q^{97} - 814 q^{98} + 767 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 2 x^{12} - 189 x^{11} + 344 x^{10} + 12502 x^{9} - 15678 x^{8} - 385197 x^{7} + 263029 x^{6} + \cdots + 3268107 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 47\!\cdots\!53 \nu^{12} + \cdots + 41\!\cdots\!94 ) / 14\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 53\!\cdots\!21 \nu^{12} + \cdots - 13\!\cdots\!61 ) / 49\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 56\!\cdots\!95 \nu^{12} + \cdots - 63\!\cdots\!52 ) / 49\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18\!\cdots\!73 \nu^{12} + \cdots - 17\!\cdots\!46 ) / 14\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!22 \nu^{12} + \cdots + 39\!\cdots\!05 ) / 14\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!45 \nu^{12} + \cdots - 20\!\cdots\!61 ) / 14\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 82\!\cdots\!28 \nu^{12} + \cdots + 72\!\cdots\!55 ) / 49\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 40\!\cdots\!44 \nu^{12} + \cdots + 37\!\cdots\!19 ) / 14\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 45\!\cdots\!35 \nu^{12} + \cdots + 43\!\cdots\!25 ) / 14\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 57\!\cdots\!38 \nu^{12} + \cdots + 42\!\cdots\!99 ) / 14\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 91\!\cdots\!38 \nu^{12} + \cdots + 42\!\cdots\!50 ) / 14\!\cdots\!11 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} - 2\beta_{10} + 3\beta_{9} + \beta_{8} + 3\beta_{4} - \beta_{3} + \beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + 6 \beta_{11} + \beta_{10} + 3 \beta_{9} - 12 \beta_{8} - 10 \beta_{6} - 5 \beta_{5} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 109 \beta_{12} + 8 \beta_{11} - 192 \beta_{10} + 287 \beta_{9} + 100 \beta_{8} - 19 \beta_{7} + \cdots + 2032 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 229 \beta_{12} + 698 \beta_{11} + 168 \beta_{10} - 2 \beta_{9} - 1330 \beta_{8} + 37 \beta_{7} + \cdots - 2790 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9854 \beta_{12} + 227 \beta_{11} - 16967 \beta_{10} + 24674 \beta_{9} + 9980 \beta_{8} + \cdots + 157129 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 31818 \beta_{12} + 64223 \beta_{11} + 28548 \beta_{10} - 30152 \beta_{9} - 128516 \beta_{8} + \cdots - 436709 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 877117 \beta_{12} - 43898 \beta_{11} - 1494624 \beta_{10} + 2115418 \beta_{9} + 994671 \beta_{8} + \cdots + 13287352 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3752735 \beta_{12} + 5652287 \beta_{11} + 3922224 \beta_{10} - 5209932 \beta_{9} - 12132859 \beta_{8} + \cdots - 54095631 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 78775342 \beta_{12} - 9809672 \beta_{11} - 132611759 \beta_{10} + 183974105 \beta_{9} + \cdots + 1169043035 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 410951602 \beta_{12} + 496129850 \beta_{11} + 474987377 \beta_{10} - 670004857 \beta_{9} + \cdots - 6051696225 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 7157571350 \beta_{12} - 1384329918 \beta_{11} - 11876658728 \beta_{10} + 16271702057 \beta_{9} + \cdots + 105205327319 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
8.65137
7.80002
5.46835
4.40455
3.76771
1.49210
−0.0526328
−1.68110
−4.24178
−4.40749
−4.59428
−4.79800
−9.80881
2.00000 −9.65137 4.00000 −9.25595 −19.3027 −10.9930 8.00000 66.1490 −18.5119
1.2 2.00000 −8.80002 4.00000 −2.39986 −17.6000 −8.30098 8.00000 50.4404 −4.79973
1.3 2.00000 −6.46835 4.00000 13.2925 −12.9367 −3.82128 8.00000 14.8395 26.5850
1.4 2.00000 −5.40455 4.00000 1.92533 −10.8091 8.45594 8.00000 2.20911 3.85067
1.5 2.00000 −4.76771 4.00000 −21.2458 −9.53542 31.3938 8.00000 −4.26894 −42.4916
1.6 2.00000 −2.49210 4.00000 13.9170 −4.98420 0.532507 8.00000 −20.7894 27.8341
1.7 2.00000 −0.947367 4.00000 0.467492 −1.89473 −20.1380 8.00000 −26.1025 0.934985
1.8 2.00000 0.681098 4.00000 −10.5046 1.36220 15.6888 8.00000 −26.5361 −21.0091
1.9 2.00000 3.24178 4.00000 8.55862 6.48356 −26.1411 8.00000 −16.4909 17.1172
1.10 2.00000 3.40749 4.00000 −1.51765 6.81499 −24.7241 8.00000 −15.3890 −3.03530
1.11 2.00000 3.59428 4.00000 −11.0192 7.18855 9.57416 8.00000 −14.0812 −22.0383
1.12 2.00000 3.79800 4.00000 −7.07981 7.59600 −4.46441 8.00000 −12.5752 −14.1596
1.13 2.00000 8.80881 4.00000 −16.1381 17.6176 −27.0625 8.00000 50.5952 −32.2763
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(269\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 538.4.a.a ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.4.a.a ✓ 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{13} + 15 T_{3}^{12} - 87 T_{3}^{11} - 2005 T_{3}^{10} - 178 T_{3}^{9} + 83808 T_{3}^{8} + \cdots + 30236464 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(538))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{13} \) Copy content Toggle raw display
$3$ \( T^{13} + 15 T^{12} + \cdots + 30236464 \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots - 13498695011 \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots - 11643086778944 \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots + 35\!\cdots\!40 \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots + 23\!\cdots\!42 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots - 92\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots - 16\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots - 16\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 42\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots - 61\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots + 52\!\cdots\!23 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 39\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 78\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 22\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 66\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots + 98\!\cdots\!77 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 20\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 53\!\cdots\!46 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 30\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 82\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots + 29\!\cdots\!85 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 17\!\cdots\!12 \) Copy content Toggle raw display
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