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Properties

Label 538.4.a.b
Level $538$
Weight $4$
Character orbit 538.a
Self dual yes
Analytic conductor $31.743$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

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: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 4 x^{14} - 262 x^{13} + 870 x^{12} + 26403 x^{11} - 73750 x^{10} - 1270273 x^{9} + \cdots + 4484581281 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + ( - \beta_1 + 1) q^{3} + 4 q^{4} + (\beta_{4} + 2) q^{5} + (2 \beta_1 - 2) q^{6} + \beta_{8} q^{7} - 8 q^{8} + (\beta_{13} + \beta_{10} + \beta_{9} + \cdots + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + ( - \beta_1 + 1) q^{3} + 4 q^{4} + (\beta_{4} + 2) q^{5} + (2 \beta_1 - 2) q^{6} + \beta_{8} q^{7} - 8 q^{8} + (\beta_{13} + \beta_{10} + \beta_{9} + \cdots + 9) q^{9}+ \cdots + (\beta_{14} + 32 \beta_{13} + \cdots + 427) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 30 q^{2} + 11 q^{3} + 60 q^{4} + 29 q^{5} - 22 q^{6} + 5 q^{7} - 120 q^{8} + 142 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 30 q^{2} + 11 q^{3} + 60 q^{4} + 29 q^{5} - 22 q^{6} + 5 q^{7} - 120 q^{8} + 142 q^{9} - 58 q^{10} + 79 q^{11} + 44 q^{12} - 14 q^{13} - 10 q^{14} + 67 q^{15} + 240 q^{16} + 94 q^{17} - 284 q^{18} + 124 q^{19} + 116 q^{20} + 69 q^{21} - 158 q^{22} + 545 q^{23} - 88 q^{24} + 192 q^{25} + 28 q^{26} + 431 q^{27} + 20 q^{28} + 200 q^{29} - 134 q^{30} + 159 q^{31} - 480 q^{32} + 531 q^{33} - 188 q^{34} + 146 q^{35} + 568 q^{36} + 303 q^{37} - 248 q^{38} - 164 q^{39} - 232 q^{40} + 606 q^{41} - 138 q^{42} + 442 q^{43} + 316 q^{44} + 644 q^{45} - 1090 q^{46} + 618 q^{47} + 176 q^{48} + 286 q^{49} - 384 q^{50} - 251 q^{51} - 56 q^{52} + 916 q^{53} - 862 q^{54} + 53 q^{55} - 40 q^{56} + 190 q^{57} - 400 q^{58} + 1377 q^{59} + 268 q^{60} - 1373 q^{61} - 318 q^{62} + 1067 q^{63} + 960 q^{64} + 158 q^{65} - 1062 q^{66} + 1330 q^{67} + 376 q^{68} + 737 q^{69} - 292 q^{70} + 2291 q^{71} - 1136 q^{72} + 924 q^{73} - 606 q^{74} + 4337 q^{75} + 496 q^{76} + 4170 q^{77} + 328 q^{78} + 610 q^{79} + 464 q^{80} + 2255 q^{81} - 1212 q^{82} + 5191 q^{83} + 276 q^{84} + 1517 q^{85} - 884 q^{86} + 7295 q^{87} - 632 q^{88} + 3198 q^{89} - 1288 q^{90} + 4289 q^{91} + 2180 q^{92} + 5255 q^{93} - 1236 q^{94} + 4706 q^{95} - 352 q^{96} + 3806 q^{97} - 572 q^{98} + 7068 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 4 x^{14} - 262 x^{13} + 870 x^{12} + 26403 x^{11} - 73750 x^{10} - 1270273 x^{9} + \cdots + 4484581281 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!59 \nu^{14} + \cdots + 30\!\cdots\!69 ) / 65\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27\!\cdots\!23 \nu^{14} + \cdots - 89\!\cdots\!95 ) / 11\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 74\!\cdots\!31 \nu^{14} + \cdots + 13\!\cdots\!43 ) / 23\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!39 \nu^{14} + \cdots - 25\!\cdots\!41 ) / 23\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!46 \nu^{14} + \cdots - 61\!\cdots\!99 ) / 39\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\!\cdots\!57 \nu^{14} + \cdots - 23\!\cdots\!29 ) / 11\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!01 \nu^{14} + \cdots + 25\!\cdots\!20 ) / 11\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 89\!\cdots\!61 \nu^{14} + \cdots + 14\!\cdots\!71 ) / 78\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!67 \nu^{14} + \cdots + 22\!\cdots\!99 ) / 78\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 35\!\cdots\!63 \nu^{14} + \cdots - 65\!\cdots\!43 ) / 23\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 46\!\cdots\!21 \nu^{14} + \cdots - 95\!\cdots\!43 ) / 23\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 33\!\cdots\!38 \nu^{14} + \cdots - 63\!\cdots\!15 ) / 13\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 77\!\cdots\!53 \nu^{14} + \cdots - 16\!\cdots\!79 ) / 23\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{10} + \beta_{9} + \beta _1 + 35 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5 \beta_{14} - 4 \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + 6 \beta_{8} + 2 \beta_{7} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + 78 \beta_{13} + 14 \beta_{12} - 13 \beta_{11} + 82 \beta_{10} + 76 \beta_{9} + 3 \beta_{8} + \cdots + 2231 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 657 \beta_{14} - 384 \beta_{13} + 174 \beta_{12} - 87 \beta_{11} + 99 \beta_{10} + 261 \beta_{9} + \cdots + 5022 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1035 \beta_{14} + 5650 \beta_{13} + 2049 \beta_{12} - 2109 \beta_{11} + 6415 \beta_{10} + 6247 \beta_{9} + \cdots + 166394 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 65999 \beta_{14} - 29860 \beta_{13} + 21013 \beta_{12} - 19640 \beta_{11} + 10354 \beta_{10} + \cdots + 613310 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 188902 \beta_{14} + 404268 \beta_{13} + 228647 \beta_{12} - 251206 \beta_{11} + 510418 \beta_{10} + \cdots + 13263902 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6141441 \beta_{14} - 2187783 \beta_{13} + 2264826 \beta_{12} - 2483997 \beta_{11} + 1108809 \beta_{10} + \cdots + 65112696 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 24708771 \beta_{14} + 28948252 \beta_{13} + 23234007 \beta_{12} - 26733849 \beta_{11} + \cdots + 1100849126 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 556272746 \beta_{14} - 156042928 \beta_{13} + 229968751 \beta_{12} - 267363050 \beta_{11} + \cdots + 6514778954 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2801306548 \beta_{14} + 2086626876 \beta_{13} + 2264386787 \beta_{12} - 2688759937 \beta_{11} + \cdots + 93932831150 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 49916523660 \beta_{14} - 10909115355 \beta_{13} + 22516418811 \beta_{12} - 26809897983 \beta_{11} + \cdots + 632425202838 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 294380867034 \beta_{14} + 151928981542 \beta_{13} + 216046861956 \beta_{12} - 261677336742 \beta_{11} + \cdots + 8175515534072 \) 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
9.60732
9.56722
7.81309
4.53450
3.08712
2.53854
2.25606
2.11620
−1.46980
−1.89711
−4.52243
−6.13461
−6.99488
−8.02621
−8.47502
−2.00000 −8.60732 4.00000 0.977680 17.2146 26.8253 −8.00000 47.0859 −1.95536
1.2 −2.00000 −8.56722 4.00000 −3.91913 17.1344 −8.79649 −8.00000 46.3972 7.83826
1.3 −2.00000 −6.81309 4.00000 10.9766 13.6262 −4.36588 −8.00000 19.4181 −21.9532
1.4 −2.00000 −3.53450 4.00000 −5.15607 7.06899 −9.22028 −8.00000 −14.5073 10.3121
1.5 −2.00000 −2.08712 4.00000 12.0240 4.17425 −1.17722 −8.00000 −22.6439 −24.0480
1.6 −2.00000 −1.53854 4.00000 16.1523 3.07708 −33.2509 −8.00000 −24.6329 −32.3046
1.7 −2.00000 −1.25606 4.00000 −6.40351 2.51212 −19.2637 −8.00000 −25.4223 12.8070
1.8 −2.00000 −1.11620 4.00000 −11.2565 2.23239 26.6107 −8.00000 −25.7541 22.5130
1.9 −2.00000 2.46980 4.00000 18.7148 −4.93959 20.4098 −8.00000 −20.9001 −37.4297
1.10 −2.00000 2.89711 4.00000 −18.0320 −5.79421 −16.8014 −8.00000 −18.6068 36.0640
1.11 −2.00000 5.52243 4.00000 −2.50340 −11.0449 8.99122 −8.00000 3.49718 5.00680
1.12 −2.00000 7.13461 4.00000 −2.07069 −14.2692 19.9491 −8.00000 23.9026 4.14137
1.13 −2.00000 7.99488 4.00000 8.61560 −15.9898 25.7038 −8.00000 36.9181 −17.2312
1.14 −2.00000 9.02621 4.00000 21.4254 −18.0524 −11.0193 −8.00000 54.4724 −42.8507
1.15 −2.00000 9.47502 4.00000 −10.5451 −18.9500 −19.5948 −8.00000 62.7759 21.0902
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.b ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.4.a.b ✓ 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} - 11 T_{3}^{14} - 213 T_{3}^{13} + 2445 T_{3}^{12} + 16316 T_{3}^{11} - 198170 T_{3}^{10} + \cdots - 1541023936 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(538))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots - 1541023936 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 10337508482932 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 65\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 41\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots - 11\!\cdots\!72 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 22\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 14\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 12\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 12\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 30\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 39\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 29\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 14\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 56\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 95\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 26\!\cdots\!62 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 74\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 10\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 33\!\cdots\!27 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 20\!\cdots\!68 \) Copy content Toggle raw display
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