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Properties

Label 531.8.a.e
Level $531$
Weight $8$
Character orbit 531.a
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $18$
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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + \cdots + 51\!\cdots\!48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + (\beta_{2} + \beta_1 + 75) q^{4} + ( - \beta_{4} - \beta_1 - 37) q^{5} + (\beta_{7} + 5 \beta_1 + 170) q^{7} + ( - \beta_{3} - 4 \beta_{2} + \cdots - 207) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + (\beta_{2} + \beta_1 + 75) q^{4} + ( - \beta_{4} - \beta_1 - 37) q^{5} + (\beta_{7} + 5 \beta_1 + 170) q^{7} + ( - \beta_{3} - 4 \beta_{2} + \cdots - 207) q^{8}+ \cdots + (162 \beta_{17} - 424 \beta_{16} + \cdots - 2406828) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + \cdots + 51\!\cdots\!48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 202 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 318\nu + 346 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 42\!\cdots\!09 \nu^{17} + \cdots + 78\!\cdots\!20 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 63\!\cdots\!95 \nu^{17} + \cdots + 34\!\cdots\!64 ) / 51\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 59\!\cdots\!49 \nu^{17} + \cdots + 22\!\cdots\!04 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 69\!\cdots\!03 \nu^{17} + \cdots + 11\!\cdots\!40 ) / 10\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 87\!\cdots\!53 \nu^{17} + \cdots - 12\!\cdots\!04 ) / 90\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!75 \nu^{17} + \cdots - 41\!\cdots\!84 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!43 \nu^{17} + \cdots + 29\!\cdots\!28 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!61 \nu^{17} + \cdots - 26\!\cdots\!52 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 37\!\cdots\!75 \nu^{17} + \cdots - 56\!\cdots\!64 ) / 20\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 29\!\cdots\!37 \nu^{17} + \cdots + 63\!\cdots\!16 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37\!\cdots\!43 \nu^{17} + \cdots + 93\!\cdots\!96 ) / 18\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15\!\cdots\!51 \nu^{17} + \cdots + 26\!\cdots\!16 ) / 72\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 28\!\cdots\!57 \nu^{17} + \cdots - 48\!\cdots\!00 ) / 72\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 67\!\cdots\!09 \nu^{17} + \cdots - 83\!\cdots\!44 ) / 14\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 202 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 317\beta _1 - 144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} + \cdots + 64138 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6 \beta_{17} - 5 \beta_{16} - 3 \beta_{15} - 11 \beta_{14} + 9 \beta_{13} - 15 \beta_{12} + 5 \beta_{11} + \cdots - 67329 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 124 \beta_{17} - 64 \beta_{16} - 791 \beta_{15} - 647 \beta_{14} + 659 \beta_{13} + 657 \beta_{12} + \cdots + 23297602 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2474 \beta_{17} - 3415 \beta_{16} - 2716 \beta_{15} - 7840 \beta_{14} + 6558 \beta_{13} + \cdots - 31157507 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 116554 \beta_{17} - 52891 \beta_{16} - 447118 \beta_{15} - 328802 \beta_{14} + 351956 \beta_{13} + \cdots + 8984566067 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 388322 \beta_{17} - 1825573 \beta_{16} - 1726918 \beta_{15} - 4078574 \beta_{14} + 3596304 \beta_{13} + \cdots - 14615002473 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 75945018 \beta_{17} - 31120411 \beta_{16} - 221976003 \beta_{15} - 155030463 \beta_{14} + \cdots + 3585271738667 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 228023992 \beta_{17} - 908704302 \beta_{16} - 934849297 \beta_{15} - 1888558801 \beta_{14} + \cdots - 6867264837542 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 42522848694 \beta_{17} - 16097755799 \beta_{16} - 103292091881 \beta_{15} - 70929671893 \beta_{14} + \cdots + 14\!\cdots\!59 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 278185379156 \beta_{17} - 438056649436 \beta_{16} - 461373418794 \beta_{15} - 829096386854 \beta_{14} + \cdots - 32\!\cdots\!60 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 21971913318404 \beta_{17} - 7808170309630 \beta_{16} - 46346257405376 \beta_{15} + \cdots + 60\!\cdots\!04 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 195154928700636 \beta_{17} - 207010529679718 \beta_{16} - 214807477197556 \beta_{15} + \cdots - 14\!\cdots\!78 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 10\!\cdots\!12 \beta_{17} + \cdots + 25\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 11\!\cdots\!06 \beta_{17} + \cdots - 68\!\cdots\!27 \) 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
20.8794
19.9328
17.9457
15.9283
13.6971
9.20800
8.74396
6.43791
2.93322
2.80127
−6.28523
−6.66616
−10.5486
−13.7979
−16.0158
−17.7235
−20.5204
−20.9501
−21.8794 0 350.709 −336.305 0 −653.674 −4872.75 0 7358.16
1.2 −20.9328 0 310.183 149.882 0 1201.09 −3813.60 0 −3137.45
1.3 −18.9457 0 230.941 −260.561 0 254.948 −1950.29 0 4936.51
1.4 −16.9283 0 158.568 421.077 0 −62.1753 −517.469 0 −7128.13
1.5 −14.6971 0 88.0043 −457.882 0 1576.42 587.821 0 6729.53
1.6 −10.2080 0 −23.7967 397.678 0 −505.577 1549.54 0 −4059.50
1.7 −9.74396 0 −33.0553 −321.882 0 −939.921 1569.32 0 3136.41
1.8 −7.43791 0 −72.6775 −171.827 0 1529.87 1492.62 0 1278.03
1.9 −3.93322 0 −112.530 305.648 0 −1406.85 946.057 0 −1202.18
1.10 −3.80127 0 −113.550 87.8737 0 1306.06 918.197 0 −334.031
1.11 5.28523 0 −100.066 −498.961 0 341.176 −1205.38 0 −2637.13
1.12 5.66616 0 −95.8947 15.0142 0 499.350 −1268.62 0 85.0729
1.13 9.54862 0 −36.8240 −248.981 0 −1453.31 −1573.84 0 −2377.42
1.14 12.7979 0 35.7870 405.222 0 −375.715 −1180.14 0 5186.00
1.15 15.0158 0 97.4757 −6.55078 0 1466.16 −458.348 0 −98.3656
1.16 16.7235 0 151.676 298.819 0 200.095 395.941 0 4997.30
1.17 19.5204 0 253.046 49.6011 0 −119.919 2440.94 0 968.233
1.18 19.9501 0 270.005 −505.865 0 222.970 2833.01 0 −10092.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.e 18
3.b odd 2 1 177.8.a.d ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.8.a.d ✓ 18 3.b odd 2 1
531.8.a.e 18 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 24 T_{2}^{17} - 1543 T_{2}^{16} - 38223 T_{2}^{15} + 951840 T_{2}^{14} + \cdots + 85\!\cdots\!24 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(531))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 85\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 19\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 22\!\cdots\!20 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 15\!\cdots\!40 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 34\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 14\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 76\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 27\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T + 205379)^{18} \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 90\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 12\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 10\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 20\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 17\!\cdots\!84 \) Copy content Toggle raw display
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