Properties

Label 531.8.a.d
Level $531$
Weight $8$
Character orbit 531.a
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
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: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + \cdots - 58\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + (\beta_{2} - 3 \beta_1 + 69) q^{4} + ( - \beta_{3} + 3 \beta_1 + 63) q^{5} + (\beta_{11} - \beta_{3} - 2 \beta_{2} - 6 \beta_1 - 141) q^{7} + (\beta_{9} + \beta_{8} + \beta_{3} + 3 \beta_{2} - 86 \beta_1 + 401) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{2} + (\beta_{2} - 3 \beta_1 + 69) q^{4} + ( - \beta_{3} + 3 \beta_1 + 63) q^{5} + (\beta_{11} - \beta_{3} - 2 \beta_{2} - 6 \beta_1 - 141) q^{7} + (\beta_{9} + \beta_{8} + \beta_{3} + 3 \beta_{2} - 86 \beta_1 + 401) q^{8} + ( - \beta_{16} + 2 \beta_{14} - \beta_{10} + 2 \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + \cdots - 361) q^{10}+ \cdots + ( - 9124 \beta_{16} - 7652 \beta_{15} + 9435 \beta_{14} + \cdots + 5557208) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 32 q^{2} + 1166 q^{4} + 1072 q^{5} - 2407 q^{7} + 6645 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 32 q^{2} + 1166 q^{4} + 1072 q^{5} - 2407 q^{7} + 6645 q^{8} - 6391 q^{10} + 8888 q^{11} - 12702 q^{13} + 17555 q^{14} + 139226 q^{16} + 36167 q^{17} - 71037 q^{19} + 274883 q^{20} - 325182 q^{22} + 269995 q^{23} + 97329 q^{25} + 336906 q^{26} - 901362 q^{28} + 543825 q^{29} - 633109 q^{31} + 837062 q^{32} - 529288 q^{34} + 287621 q^{35} - 867607 q^{37} + 1727169 q^{38} - 815662 q^{40} + 1428939 q^{41} - 477060 q^{43} + 1667926 q^{44} + 5305549 q^{46} + 1217849 q^{47} + 4350738 q^{49} - 4561369 q^{50} + 4175994 q^{52} + 3487068 q^{53} - 960484 q^{55} + 5363196 q^{56} - 3082906 q^{58} + 3491443 q^{59} + 998917 q^{61} + 5742614 q^{62} + 17531621 q^{64} + 6075816 q^{65} - 356026 q^{67} + 16149231 q^{68} - 548798 q^{70} + 12879428 q^{71} - 6176157 q^{73} + 5971906 q^{74} - 17624580 q^{76} - 239687 q^{77} - 18886490 q^{79} + 70463349 q^{80} - 19351611 q^{82} + 22824893 q^{83} - 7973079 q^{85} + 27502196 q^{86} - 62527651 q^{88} + 30609647 q^{89} - 36301521 q^{91} + 41388548 q^{92} + 1010176 q^{94} + 29303629 q^{95} - 26249806 q^{97} + 93110852 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + \cdots - 58\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 193 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59\!\cdots\!07 \nu^{16} + \cdots + 21\!\cdots\!32 ) / 86\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 48\!\cdots\!11 \nu^{16} + \cdots + 12\!\cdots\!20 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 35\!\cdots\!73 \nu^{16} + \cdots - 84\!\cdots\!00 ) / 86\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 45\!\cdots\!93 \nu^{16} + \cdots + 11\!\cdots\!36 ) / 86\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!07 \nu^{16} + \cdots - 31\!\cdots\!48 ) / 21\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 70\!\cdots\!35 \nu^{16} + \cdots + 16\!\cdots\!92 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 56\!\cdots\!87 \nu^{16} + \cdots - 13\!\cdots\!76 ) / 86\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 31\!\cdots\!73 \nu^{16} + \cdots - 77\!\cdots\!04 ) / 43\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 66\!\cdots\!61 \nu^{16} + \cdots - 15\!\cdots\!32 ) / 86\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 31\!\cdots\!53 \nu^{16} + \cdots + 75\!\cdots\!48 ) / 21\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 66\!\cdots\!77 \nu^{16} + \cdots - 15\!\cdots\!20 ) / 43\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!89 \nu^{16} + \cdots - 25\!\cdots\!20 ) / 54\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21\!\cdots\!51 \nu^{16} + \cdots + 52\!\cdots\!68 ) / 86\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 37\!\cdots\!17 \nu^{16} + \cdots - 91\!\cdots\!84 ) / 86\!\cdots\!08 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 193 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} - \beta_{3} + 3\beta_{2} + 336\beta _1 + 253 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} - \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} + 6 \beta_{11} + 3 \beta_{10} - 20 \beta_{9} + 7 \beta_{7} + 4 \beta_{6} - 10 \beta_{5} + 9 \beta_{4} - 42 \beta_{3} + 452 \beta_{2} + 1094 \beta _1 + 64899 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 148 \beta_{16} - 49 \beta_{15} - 132 \beta_{14} + \beta_{13} - 10 \beta_{12} - 147 \beta_{11} + 51 \beta_{10} - 822 \beta_{9} - 433 \beta_{8} - 120 \beta_{7} + 67 \beta_{6} - 36 \beta_{5} + 201 \beta_{4} - 671 \beta_{3} + 2687 \beta_{2} + \cdots + 234684 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 766 \beta_{16} - 1912 \beta_{15} + 1532 \beta_{14} + 5300 \beta_{13} + 5146 \beta_{12} + 3504 \beta_{11} + 1244 \beta_{10} - 17931 \beta_{9} - 685 \beta_{8} + 3860 \beta_{7} + 2368 \beta_{6} - 7870 \beta_{5} + \cdots + 25899709 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 114021 \beta_{16} - 57379 \beta_{15} - 98702 \beta_{14} + 27464 \beta_{13} + 12614 \beta_{12} - 117618 \beta_{11} + 21245 \beta_{10} - 532076 \beta_{9} - 170026 \beta_{8} - 94623 \beta_{7} + \cdots + 168642875 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 635630 \beta_{16} - 1725907 \beta_{15} + 716880 \beta_{14} + 3990095 \beta_{13} + 3826244 \beta_{12} + 1480107 \beta_{11} + 273189 \beta_{10} - 12300698 \beta_{9} - 644885 \beta_{8} + \cdots + 11512527046 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 66441380 \beta_{16} - 46530816 \beta_{15} - 54081184 \beta_{14} + 34859488 \beta_{13} + 23733736 \beta_{12} - 69574116 \beta_{11} + 2638544 \beta_{10} - 323608777 \beta_{9} + \cdots + 111819940985 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 501465529 \beta_{16} - 1237972625 \beta_{15} + 233253826 \beta_{14} + 2511200180 \beta_{13} + 2389824196 \beta_{12} + 503051806 \beta_{11} - 39641013 \beta_{10} + \cdots + 5535112643107 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 35736675116 \beta_{16} - 32314549633 \beta_{15} - 26860437140 \beta_{14} + 29741506937 \beta_{13} + 22882238054 \beta_{12} - 37289224731 \beta_{11} + \cdots + 71002791721804 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 361105418406 \beta_{16} - 804046573904 \beta_{15} + 30850932748 \beta_{14} + 1473767444980 \beta_{13} + 1394674156634 \beta_{12} + 115674091168 \beta_{11} + \cdots + 28\!\cdots\!65 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 18880479302309 \beta_{16} - 20707039000419 \beta_{15} - 12995109669646 \beta_{14} + 21435326874856 \beta_{13} + 17532114461750 \beta_{12} + \cdots + 43\!\cdots\!91 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 241898320413574 \beta_{16} - 496953590726531 \beta_{15} - 31391010068768 \beta_{14} + 841704157680503 \beta_{13} + 792723427066324 \beta_{12} + \cdots + 14\!\cdots\!94 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 10\!\cdots\!48 \beta_{16} + \cdots + 26\!\cdots\!45 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 15\!\cdots\!53 \beta_{16} + \cdots + 81\!\cdots\!79 \) Copy content Toggle raw display

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
24.0278
18.2619
17.5255
15.6681
12.7630
7.01814
4.11298
2.41303
2.39686
−4.01497
−4.85375
−10.1391
−11.3335
−15.8998
−15.9892
−19.6388
−20.3182
−22.0278 0 357.226 492.460 0 −1209.81 −5049.35 0 −10847.8
1.2 −16.2619 0 136.450 30.6671 0 −1356.62 −137.421 0 −498.706
1.3 −15.5255 0 113.041 141.845 0 105.285 232.240 0 −2202.21
1.4 −13.6681 0 58.8158 207.404 0 882.793 945.614 0 −2834.81
1.5 −10.7630 0 −12.1577 −451.863 0 −504.672 1508.52 0 4863.40
1.6 −5.01814 0 −102.818 77.5687 0 −216.829 1158.28 0 −389.251
1.7 −2.11298 0 −123.535 537.118 0 1259.15 531.490 0 −1134.92
1.8 −0.413031 0 −127.829 −231.152 0 302.150 105.665 0 95.4729
1.9 −0.396855 0 −127.843 −247.233 0 −652.929 101.532 0 98.1157
1.10 6.01497 0 −91.8201 385.807 0 847.649 −1322.21 0 2320.62
1.11 6.85375 0 −81.0262 190.727 0 −799.288 −1432.61 0 1307.20
1.12 12.1391 0 19.3578 −236.334 0 1426.66 −1318.82 0 −2868.89
1.13 13.3335 0 49.7814 −152.093 0 −1328.47 −1042.93 0 −2027.93
1.14 17.8998 0 192.403 −255.355 0 −1072.82 1152.80 0 −4570.81
1.15 17.9892 0 195.612 98.9095 0 159.201 1216.28 0 1779.30
1.16 21.6388 0 340.238 399.107 0 −1780.57 4592.58 0 8636.21
1.17 22.3182 0 370.104 84.4166 0 1532.13 5403.34 0 1884.03
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
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.d 17
3.b odd 2 1 177.8.a.b 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.8.a.b 17 3.b odd 2 1
531.8.a.d 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 32 T_{2}^{16} - 1159 T_{2}^{15} + 43065 T_{2}^{14} + 456514 T_{2}^{13} - 22317501 T_{2}^{12} - 60303860 T_{2}^{11} + 5698856604 T_{2}^{10} - 4350215008 T_{2}^{9} + \cdots + 14\!\cdots\!16 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(531))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} - 32 T^{16} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{17} \) Copy content Toggle raw display
$5$ \( T^{17} - 1072 T^{16} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{17} + 2407 T^{16} + \cdots + 24\!\cdots\!92 \) Copy content Toggle raw display
$11$ \( T^{17} - 8888 T^{16} + \cdots - 63\!\cdots\!60 \) Copy content Toggle raw display
$13$ \( T^{17} + 12702 T^{16} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{17} - 36167 T^{16} + \cdots - 39\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{17} + 71037 T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{17} - 269995 T^{16} + \cdots + 15\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{17} - 543825 T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{17} + 633109 T^{16} + \cdots + 83\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{17} + 867607 T^{16} + \cdots - 36\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{17} - 1428939 T^{16} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{17} + 477060 T^{16} + \cdots + 42\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{17} - 1217849 T^{16} + \cdots - 19\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{17} - 3487068 T^{16} + \cdots - 92\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T - 205379)^{17} \) Copy content Toggle raw display
$61$ \( T^{17} - 998917 T^{16} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{17} + 356026 T^{16} + \cdots + 41\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{17} - 12879428 T^{16} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{17} + 6176157 T^{16} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{17} + 18886490 T^{16} + \cdots - 80\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{17} - 22824893 T^{16} + \cdots - 56\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{17} - 30609647 T^{16} + \cdots + 36\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{17} + 26249806 T^{16} + \cdots + 80\!\cdots\!88 \) Copy content Toggle raw display
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