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Properties

Label 531.8.a.c
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: \( x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 68) q^{4} + (\beta_{4} - 2 \beta_1 + 19) q^{5} + (\beta_{10} - 2 \beta_1 + 185) q^{7} + ( - \beta_{5} + 2 \beta_{4} + \cdots - 128) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 68) q^{4} + (\beta_{4} - 2 \beta_1 + 19) q^{5} + (\beta_{10} - 2 \beta_1 + 185) q^{7} + ( - \beta_{5} + 2 \beta_{4} + \cdots - 128) q^{8}+ \cdots + ( - 3846 \beta_{16} + 449 \beta_{15} + \cdots - 1559125) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 2 q^{2} + 1166 q^{4} + 318 q^{5} + 3145 q^{7} - 2355 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - 2 q^{2} + 1166 q^{4} + 318 q^{5} + 3145 q^{7} - 2355 q^{8} + 6521 q^{10} + 1764 q^{11} + 18192 q^{13} + 7827 q^{14} + 139226 q^{16} + 15507 q^{17} + 52083 q^{19} - 721 q^{20} - 234434 q^{22} - 63823 q^{23} + 202153 q^{25} + 367956 q^{26} + 182306 q^{28} + 502955 q^{29} + 347531 q^{31} + 243908 q^{32} - 330872 q^{34} - 92641 q^{35} + 447615 q^{37} - 775669 q^{38} + 2203270 q^{40} - 940335 q^{41} + 478562 q^{43} + 596924 q^{44} - 3078663 q^{46} - 703121 q^{47} + 1895082 q^{49} + 876967 q^{50} + 6278296 q^{52} + 1005974 q^{53} + 5212846 q^{55} - 3425294 q^{56} + 6710166 q^{58} + 3491443 q^{59} + 11510749 q^{61} - 5996234 q^{62} + 29496941 q^{64} - 11094180 q^{65} + 14007144 q^{67} - 19688159 q^{68} + 30909708 q^{70} - 5229074 q^{71} + 5452211 q^{73} - 12819662 q^{74} + 41929340 q^{76} - 9930777 q^{77} + 15275654 q^{79} - 36576105 q^{80} + 32025935 q^{82} - 7826609 q^{83} + 11836945 q^{85} - 51649136 q^{86} + 30223741 q^{88} + 6436185 q^{89} + 11633535 q^{91} - 43357972 q^{92} - 4494252 q^{94} - 23741055 q^{95} + 26377540 q^{97} - 26517816 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + \cdots + 24\!\cdots\!16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 196 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25\!\cdots\!23 \nu^{16} + \cdots - 18\!\cdots\!68 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!41 \nu^{16} + \cdots - 85\!\cdots\!48 ) / 10\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30\!\cdots\!41 \nu^{16} + \cdots - 15\!\cdots\!40 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!95 \nu^{16} + \cdots + 62\!\cdots\!00 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!01 \nu^{16} + \cdots - 16\!\cdots\!76 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22\!\cdots\!77 \nu^{16} + \cdots + 11\!\cdots\!44 ) / 41\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 55\!\cdots\!51 \nu^{16} + \cdots - 31\!\cdots\!96 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 72\!\cdots\!87 \nu^{16} + \cdots + 45\!\cdots\!88 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!01 \nu^{16} + \cdots - 74\!\cdots\!20 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12\!\cdots\!33 \nu^{16} + \cdots - 91\!\cdots\!52 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12\!\cdots\!47 \nu^{16} + \cdots - 72\!\cdots\!96 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37\!\cdots\!15 \nu^{16} + \cdots - 26\!\cdots\!04 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 43\!\cdots\!78 \nu^{16} + \cdots + 30\!\cdots\!64 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 10\!\cdots\!61 \nu^{16} + \cdots - 65\!\cdots\!52 ) / 41\!\cdots\!44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 196 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 2\beta_{4} + \beta_{3} + 343\beta _1 + 128 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{16} + \beta_{15} + 2 \beta_{14} - 5 \beta_{13} - 2 \beta_{12} - \beta_{11} + 6 \beta_{10} + \cdots + 67005 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{16} - 9 \beta_{15} - 30 \beta_{14} - 52 \beta_{13} - 11 \beta_{12} - 14 \beta_{11} + \cdots + 49443 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4428 \beta_{16} + 620 \beta_{15} + 1572 \beta_{14} - 3762 \beta_{13} - 2026 \beta_{12} - 518 \beta_{11} + \cdots + 27441788 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9070 \beta_{16} - 4023 \beta_{15} - 20770 \beta_{14} - 39287 \beta_{13} - 7888 \beta_{12} + \cdots + 16725561 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2479010 \beta_{16} + 290925 \beta_{15} + 963786 \beta_{14} - 2165060 \beta_{13} - 1340093 \beta_{12} + \cdots + 12112167141 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4315240 \beta_{16} - 408182 \beta_{15} - 10904788 \beta_{14} - 22246216 \beta_{13} - 3753658 \beta_{12} + \cdots + 5128297790 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1271346030 \beta_{16} + 122290725 \beta_{15} + 543072090 \beta_{14} - 1138061973 \beta_{13} + \cdots + 5534138272313 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1772327478 \beta_{16} + 788518275 \beta_{15} - 5296131630 \beta_{14} - 11335515344 \beta_{13} + \cdots + 1353066234647 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 630160762572 \beta_{16} + 47674381828 \beta_{15} + 293440025772 \beta_{14} - 574597910894 \beta_{13} + \cdots + 25\!\cdots\!52 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 638016799126 \beta_{16} + 872848948757 \beta_{15} - 2522693475330 \beta_{14} - 5490034897119 \beta_{13} + \cdots + 217406968471469 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 307896248891050 \beta_{16} + 17087373773281 \beta_{15} + 154652656782170 \beta_{14} + \cdots + 12\!\cdots\!65 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 186458253689432 \beta_{16} + 647735166648634 \beta_{15} + \cdots - 77\!\cdots\!26 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 14\!\cdots\!22 \beta_{16} + \cdots + 56\!\cdots\!25 \) 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
21.8139
21.5848
16.5905
15.1891
11.5967
9.14085
7.49302
3.56813
−3.62331
−3.88629
−5.66556
−8.14464
−12.3679
−13.8304
−14.3045
−20.9857
−22.1687
−21.8139 0 347.844 −424.937 0 1192.89 −4795.65 0 9269.52
1.2 −21.5848 0 337.903 113.334 0 −647.523 −4530.72 0 −2446.30
1.3 −16.5905 0 147.246 −174.998 0 1111.07 −319.307 0 2903.31
1.4 −15.1891 0 102.710 418.807 0 −814.653 384.131 0 −6361.32
1.5 −11.5967 0 6.48306 401.104 0 213.573 1409.19 0 −4651.48
1.6 −9.14085 0 −44.4448 27.9722 0 −1415.76 1576.29 0 −255.690
1.7 −7.49302 0 −71.8546 −400.807 0 −272.200 1497.52 0 3003.25
1.8 −3.56813 0 −115.268 210.232 0 1544.27 868.013 0 −750.136
1.9 3.62331 0 −114.872 −300.817 0 1353.78 −880.000 0 −1089.96
1.10 3.88629 0 −112.897 26.2639 0 −644.460 −936.194 0 102.069
1.11 5.66556 0 −95.9014 −117.894 0 1197.66 −1268.53 0 −667.937
1.12 8.14464 0 −61.6648 54.3103 0 −413.400 −1544.75 0 442.338
1.13 12.3679 0 24.9655 477.980 0 1344.57 −1274.32 0 5911.62
1.14 13.8304 0 63.2803 149.469 0 14.2725 −895.100 0 2067.22
1.15 14.3045 0 76.6177 −348.885 0 −1316.66 −734.996 0 −4990.62
1.16 20.9857 0 312.401 465.631 0 286.579 3869.79 0 9771.60
1.17 22.1687 0 363.451 −258.765 0 411.003 5219.64 0 −5736.49
\(n\): 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.c 17
3.b odd 2 1 177.8.a.c ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.8.a.c ✓ 17 3.b odd 2 1
531.8.a.c 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} + 2 T_{2}^{16} - 1669 T_{2}^{15} - 2385 T_{2}^{14} + 1108684 T_{2}^{13} + \cdots - 24\!\cdots\!16 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(531))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + \cdots - 24\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{17} \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{17} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots + 47\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{17} + \cdots + 46\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 10\!\cdots\!08 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 41\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots - 98\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 52\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 52\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots + 44\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 68\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T - 205379)^{17} \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 61\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 99\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 46\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 22\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 45\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 44\!\cdots\!88 \) Copy content Toggle raw display
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