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Properties

Label 531.8.a.b
Level $531$
Weight $8$
Character orbit 531.a
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $16$
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] = [531,8,Mod(1,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.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: \(165.876448532\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + \cdots - 23\!\cdots\!32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 61) q^{4} + ( - \beta_{4} - \beta_1 + 5) q^{5} + ( - \beta_{9} + \beta_{2} + 5 \beta_1 - 148) q^{7} + (\beta_{3} + 42 \beta_1 - 67) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 61) q^{4} + ( - \beta_{4} - \beta_1 + 5) q^{5} + ( - \beta_{9} + \beta_{2} + 5 \beta_1 - 148) q^{7} + (\beta_{3} + 42 \beta_1 - 67) q^{8} + (\beta_{14} - \beta_{13} - \beta_{10} + \cdots - 204) q^{10}+ \cdots + (487 \beta_{15} + 31 \beta_{14} + \cdots - 221055) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{2} + 974 q^{4} + 68 q^{5} - 2343 q^{7} - 819 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{2} + 974 q^{4} + 68 q^{5} - 2343 q^{7} - 819 q^{8} - 3479 q^{10} - 898 q^{11} - 8172 q^{13} + 13315 q^{14} + 3138 q^{16} + 44985 q^{17} - 40137 q^{19} - 130657 q^{20} + 109394 q^{22} + 2833 q^{23} + 285746 q^{25} + 129420 q^{26} + 112890 q^{28} - 144375 q^{29} - 141759 q^{31} + 36224 q^{32} - 341332 q^{34} + 78859 q^{35} - 297971 q^{37} - 329075 q^{38} - 203048 q^{40} - 659077 q^{41} - 1431608 q^{43} - 254916 q^{44} + 873113 q^{46} + 1574073 q^{47} + 1893545 q^{49} - 302533 q^{50} - 4972548 q^{52} - 587736 q^{53} - 4624036 q^{55} + 5798506 q^{56} - 6991380 q^{58} - 3286064 q^{59} - 6117131 q^{61} + 11570258 q^{62} - 19063011 q^{64} + 5335514 q^{65} - 16518710 q^{67} + 17284669 q^{68} - 39189486 q^{70} + 10882582 q^{71} - 21097441 q^{73} + 16717030 q^{74} - 40864952 q^{76} + 3404601 q^{77} - 3784458 q^{79} + 27466195 q^{80} - 24990117 q^{82} + 1951425 q^{83} - 23238675 q^{85} + 35910572 q^{86} - 27843055 q^{88} - 10499443 q^{89} + 699217 q^{91} + 20062766 q^{92} - 59358988 q^{94} + 29236333 q^{95} - 25158976 q^{97} - 2120460 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + \cdots - 23\!\cdots\!32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 298\nu + 67 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 72\!\cdots\!81 \nu^{15} + \cdots + 98\!\cdots\!76 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!49 \nu^{15} + \cdots + 18\!\cdots\!64 ) / 80\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36\!\cdots\!57 \nu^{15} + \cdots + 12\!\cdots\!48 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 50\!\cdots\!83 \nu^{15} + \cdots + 10\!\cdots\!52 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 50\!\cdots\!91 \nu^{15} + \cdots - 48\!\cdots\!64 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!16 \nu^{15} + \cdots + 10\!\cdots\!64 ) / 25\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 72\!\cdots\!59 \nu^{15} + \cdots + 79\!\cdots\!16 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 77\!\cdots\!87 \nu^{15} + \cdots - 87\!\cdots\!08 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 42\!\cdots\!39 \nu^{15} + \cdots - 20\!\cdots\!96 ) / 80\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 63\!\cdots\!26 \nu^{15} + \cdots - 41\!\cdots\!04 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13\!\cdots\!11 \nu^{15} + \cdots - 22\!\cdots\!24 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 69\!\cdots\!33 \nu^{15} + \cdots + 11\!\cdots\!92 ) / 10\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 189 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 298\beta _1 - 67 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - 4 \beta_{14} + 4 \beta_{13} + 3 \beta_{12} - 3 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + \cdots + 56410 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{15} + 2 \beta_{14} + 20 \beta_{13} + 16 \beta_{12} + 20 \beta_{11} - 6 \beta_{10} + \cdots - 29763 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 322 \beta_{15} - 2056 \beta_{14} + 2752 \beta_{13} + 1738 \beta_{12} - 1718 \beta_{11} + \cdots + 18428719 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3715 \beta_{15} + 1540 \beta_{14} + 14628 \beta_{13} + 11923 \beta_{12} + 7041 \beta_{11} + \cdots - 12727306 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 54326 \beta_{15} - 837682 \beta_{14} + 1420036 \beta_{13} + 734014 \beta_{12} - 710438 \beta_{11} + \cdots + 6300524291 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1081644 \beta_{15} + 1020800 \beta_{14} + 7884360 \beta_{13} + 6213940 \beta_{12} + 1058156 \beta_{11} + \cdots - 4947659831 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9178535 \beta_{15} - 317171524 \beta_{14} + 657958388 \beta_{13} + 275901219 \beta_{12} + \cdots + 2218326673922 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 192171720 \beta_{15} + 611574026 \beta_{14} + 3800835556 \beta_{13} + 2776451832 \beta_{12} + \cdots - 1752912931667 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 14484661278 \beta_{15} - 116815310624 \beta_{14} + 289230246224 \beta_{13} + 97765146802 \beta_{12} + \cdots + 796832833506959 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 21120806365 \beta_{15} + 332609278476 \beta_{14} + 1734027120372 \beta_{13} + 1141712395035 \beta_{12} + \cdots - 570094464406250 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 9321682989434 \beta_{15} - 42615689960586 \beta_{14} + 123413715766356 \beta_{13} + \cdots + 29\!\cdots\!43 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 43827714804276 \beta_{15} + 167484351779752 \beta_{14} + 765851434829912 \beta_{13} + \cdots - 16\!\cdots\!91 \) 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
−19.6562
−18.7189
−17.9442
−14.0604
−13.0039
−7.02227
−3.09726
1.05882
1.97136
4.55626
7.00808
14.7989
15.0467
16.2952
19.0314
19.7363
−19.6562 0 258.365 436.761 0 956.841 −2562.48 0 −8585.06
1.2 −18.7189 0 222.398 −443.832 0 −1695.79 −1767.03 0 8308.06
1.3 −17.9442 0 193.993 −1.22320 0 −719.259 −1184.20 0 21.9494
1.4 −14.0604 0 69.6949 −153.219 0 −215.221 819.793 0 2154.32
1.5 −13.0039 0 41.1009 −167.303 0 887.373 1130.03 0 2175.59
1.6 −7.02227 0 −78.6878 266.773 0 665.758 1451.42 0 −1873.35
1.7 −3.09726 0 −118.407 156.435 0 11.3597 763.187 0 −484.520
1.8 1.05882 0 −126.879 −151.597 0 −1574.54 −269.870 0 −160.513
1.9 1.97136 0 −124.114 339.775 0 364.700 −497.007 0 669.820
1.10 4.55626 0 −107.241 540.445 0 −1238.10 −1071.82 0 2462.40
1.11 7.00808 0 −78.8868 −449.079 0 −271.717 −1449.88 0 −3147.18
1.12 14.7989 0 91.0084 296.536 0 −1410.76 −547.437 0 4388.42
1.13 15.0467 0 98.4026 −159.890 0 980.332 −445.343 0 −2405.81
1.14 16.2952 0 137.534 −495.569 0 565.301 155.363 0 −8075.41
1.15 19.0314 0 234.195 −37.6075 0 1158.54 2021.05 0 −715.724
1.16 19.7363 0 261.522 90.5952 0 −807.818 2635.23 0 1788.01
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(59\) \(1\)

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 531.8.a.b 16
3.b odd 2 1 177.8.a.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.8.a.a ✓ 16 3.b odd 2 1
531.8.a.b 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 6 T_{2}^{15} - 1493 T_{2}^{14} + 8791 T_{2}^{13} + 890490 T_{2}^{12} + \cdots - 23\!\cdots\!32 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(531))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots - 23\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 23\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots - 51\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots - 41\!\cdots\!40 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 23\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 16\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 67\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 10\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots - 50\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 10\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 82\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 21\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( (T + 205379)^{16} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots - 45\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 35\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 74\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots - 69\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 49\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 91\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 38\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 16\!\cdots\!80 \) Copy content Toggle raw display
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