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Properties

Label 531.3.c.c
Level $531$
Weight $3$
Character orbit 531.c
Analytic conductor $14.469$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [531,3,Mod(235,531)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.235");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 531.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(14.4687020375\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + \cdots + 570861 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11}\cdot 3 \)
Twist minimal: no (minimal twist has level 177)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{6} q^{5} - \beta_{12} q^{7} + (\beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{6} q^{5} - \beta_{12} q^{7} + (\beta_{3} - \beta_1) q^{8} + (\beta_{5} + \beta_{3} - \beta_1) q^{10} + (\beta_{9} - \beta_{5}) q^{11} + (\beta_{7} - \beta_1) q^{13} + ( - \beta_{19} - \beta_{10} + \cdots - 2 \beta_1) q^{14}+ \cdots + (\beta_{19} + 2 \beta_{17} + \cdots + 27 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + \cdots + 570861 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 9\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24659 \nu^{19} - 525507 \nu^{17} + 8932411 \nu^{15} + 381955669 \nu^{13} + \cdots - 66292360677 \nu ) / 1272772800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 224699 \nu^{19} - 12416727 \nu^{17} - 286167029 \nu^{15} - 3569199791 \nu^{13} + \cdots - 47542367097 \nu ) / 2545545600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 224699 \nu^{18} + 12416727 \nu^{16} + 286167029 \nu^{14} + 3569199791 \nu^{12} + \cdots + 27178002297 ) / 2545545600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10915 \nu^{19} - 672471 \nu^{17} - 17577133 \nu^{15} - 253866655 \nu^{13} + \cdots - 16365365145 \nu ) / 101821824 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 161281 \nu^{18} - 9446113 \nu^{16} - 229662151 \nu^{14} - 2993102729 \nu^{12} + \cdots - 24709031943 ) / 848515200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 533161 \nu^{19} - 29389353 \nu^{17} - 671121631 \nu^{15} - 8215396849 \nu^{13} + \cdots - 204166271583 \nu ) / 2545545600 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 243091 \nu^{19} - 14075643 \nu^{17} - 341817061 \nu^{15} - 4524039619 \nu^{13} + \cdots - 69057391773 \nu ) / 848515200 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16387 \nu^{18} + 876991 \nu^{16} + 19289197 \nu^{14} + 224839863 \nu^{12} + 1491695530 \nu^{10} + \cdots + 69554801 ) / 56567680 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 455827 \nu^{18} + 27087171 \nu^{16} + 671585917 \nu^{14} + 8996141443 \nu^{12} + \cdots + 127003783581 ) / 1272772800 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 238493 \nu^{18} + 13660914 \nu^{16} + 327904553 \nu^{14} + 4285329662 \nu^{12} + \cdots + 69406113204 ) / 636386400 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 80081 \nu^{18} - 4602113 \nu^{16} - 110442251 \nu^{14} - 1435697629 \nu^{12} + \cdots - 25003644843 ) / 212128800 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 101093 \nu^{19} - 5880439 \nu^{17} - 142869803 \nu^{15} - 1879285587 \nu^{13} + \cdots - 31926757929 \nu ) / 212128800 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 101093 \nu^{18} - 5880439 \nu^{16} - 142869803 \nu^{14} - 1879285587 \nu^{12} + \cdots - 31714629129 ) / 212128800 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 480783 \nu^{19} + 26683459 \nu^{17} + 613547793 \nu^{15} + 7555234747 \nu^{13} + \cdots - 45147198051 \nu ) / 848515200 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 59621 \nu^{18} + 3377817 \nu^{16} + 79756187 \nu^{14} + 1017726833 \nu^{12} + 7607793830 \nu^{10} + \cdots + 9860581287 ) / 101821824 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 90587 \nu^{19} + 5241276 \nu^{17} + 126656027 \nu^{15} + 1657491608 \nu^{13} + \cdots + 27510621786 \nu ) / 106064400 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{16} - \beta_{13} + \beta_{8} + \beta_{6} - 13\beta_{2} + 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + \beta_{10} + \beta_{9} - \beta_{4} - 15\beta_{3} + 94\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22 \beta_{16} - 6 \beta_{14} + 18 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 22 \beta_{8} - 16 \beta_{6} + \cdots - 597 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{19} + 26 \beta_{15} - 20 \beta_{10} - 24 \beta_{9} - 2 \beta_{7} - 2 \beta_{5} + \cdots - 1049 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{18} - 380 \beta_{16} + 158 \beta_{14} - 266 \beta_{13} + 38 \beta_{12} + 48 \beta_{11} + \cdots + 6756 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 72 \beta_{19} + 2 \beta_{17} - 490 \beta_{15} + 304 \beta_{10} + 400 \beta_{9} + 46 \beta_{7} + \cdots + 12173 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 104 \beta_{18} + 5965 \beta_{16} - 2944 \beta_{14} + 3739 \beta_{13} - 470 \beta_{12} - 794 \beta_{11} + \cdots - 79374 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1680 \beta_{19} - 104 \beta_{17} + 8115 \beta_{15} - 4209 \beta_{10} - 5825 \beta_{9} + \cdots - 145290 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2984 \beta_{18} - 88960 \beta_{16} + 47820 \beta_{14} - 51676 \beta_{13} + 4256 \beta_{12} + \cdots + 958629 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 32288 \beta_{19} + 2984 \beta_{17} - 125504 \beta_{15} + 55932 \beta_{10} + 79636 \beta_{9} + \cdots + 1772449 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 65740 \beta_{18} + 1286096 \beta_{16} - 724052 \beta_{14} + 709288 \beta_{13} - 19600 \beta_{12} + \cdots - 11827082 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 556644 \beta_{19} - 65740 \beta_{17} + 1862340 \beta_{15} - 728888 \beta_{10} - 1053640 \beta_{9} + \cdots - 22001437 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1250416 \beta_{18} - 18219809 \beta_{16} + 10525640 \beta_{14} - 9701529 \beta_{13} - 286768 \beta_{12} + \cdots + 148356592 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 8966204 \beta_{19} + 1250416 \beta_{17} - 26899245 \beta_{15} + 9414761 \beta_{10} + \cdots + 276904842 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 21697080 \beta_{18} + 254556070 \beta_{16} - 149178658 \beta_{14} + 132391686 \beta_{13} + \cdots - 1885103977 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 138008538 \beta_{19} - 21697080 \beta_{17} + 381406514 \beta_{15} - 121233592 \beta_{10} + \cdots - 3523643537 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\) \(119\) \(415\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
− 3.65759i
− 3.38526i
− 3.07256i
− 2.96307i
− 2.72775i
− 1.86235i
− 1.61360i
− 1.39995i
− 1.08628i
− 0.537675i
0.537675i
1.08628i
1.39995i
1.61360i
1.86235i
2.72775i
2.96307i
3.07256i
3.38526i
3.65759i
− 3.65759i 0 −9.37798 −0.395869 0 −6.20847 19.6705i 0 1.44793i
235.2 − 3.38526i 0 −7.45997 −9.27487 0 −9.96030 11.7129i 0 31.3978i
235.3 − 3.07256i 0 −5.44061 6.03179 0 −9.30133 4.42637i 0 − 18.5330i
235.4 − 2.96307i 0 −4.77980 −3.25089 0 11.8224 2.31061i 0 9.63262i
235.5 − 2.72775i 0 −3.44061 −5.71516 0 8.69147 − 1.52587i 0 15.5895i
235.6 − 1.86235i 0 0.531642 9.04540 0 1.29987 − 8.43952i 0 − 16.8457i
235.7 − 1.61360i 0 1.39629 6.36659 0 6.66564 − 8.70746i 0 − 10.2731i
235.8 − 1.39995i 0 2.04015 −0.273484 0 −10.8938 − 8.45589i 0 0.382863i
235.9 − 1.08628i 0 2.82000 −3.33671 0 −1.22637 − 7.40842i 0 3.62460i
235.10 − 0.537675i 0 3.71091 0.803210 0 5.11098 − 4.14596i 0 − 0.431866i
235.11 0.537675i 0 3.71091 0.803210 0 5.11098 4.14596i 0 0.431866i
235.12 1.08628i 0 2.82000 −3.33671 0 −1.22637 7.40842i 0 − 3.62460i
235.13 1.39995i 0 2.04015 −0.273484 0 −10.8938 8.45589i 0 − 0.382863i
235.14 1.61360i 0 1.39629 6.36659 0 6.66564 8.70746i 0 10.2731i
235.15 1.86235i 0 0.531642 9.04540 0 1.29987 8.43952i 0 16.8457i
235.16 2.72775i 0 −3.44061 −5.71516 0 8.69147 1.52587i 0 − 15.5895i
235.17 2.96307i 0 −4.77980 −3.25089 0 11.8224 − 2.31061i 0 − 9.63262i
235.18 3.07256i 0 −5.44061 6.03179 0 −9.30133 − 4.42637i 0 18.5330i
235.19 3.38526i 0 −7.45997 −9.27487 0 −9.96030 − 11.7129i 0 − 31.3978i
235.20 3.65759i 0 −9.37798 −0.395869 0 −6.20847 − 19.6705i 0 − 1.44793i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.3.c.c 20
3.b odd 2 1 177.3.c.a ✓ 20
59.b odd 2 1 inner 531.3.c.c 20
177.d even 2 1 177.3.c.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.3.c.a ✓ 20 3.b odd 2 1
177.3.c.a ✓ 20 177.d even 2 1
531.3.c.c 20 1.a even 1 1 trivial
531.3.c.c 20 59.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 60 T_{2}^{18} + 1522 T_{2}^{16} + 21286 T_{2}^{14} + 179593 T_{2}^{12} + 941588 T_{2}^{10} + \cdots + 570861 \) acting on \(S_{3}^{\mathrm{new}}(531, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 60 T^{18} + \cdots + 570861 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} - 150 T^{8} + \cdots + 17368)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 4 T^{9} + \cdots - 34966256)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 74\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( (T^{10} + 8 T^{9} + \cdots - 139536176)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 726592156400)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 14765383626976)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 35\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 103416495474416)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 49\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 26\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 71\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 242741808356634)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 37\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 60\!\cdots\!36 \) Copy content Toggle raw display
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