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Properties

Label 531.3.c.d
Level $531$
Weight $3$
Character orbit 531.c
Analytic conductor $14.469$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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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: \( x^{20} - 264 x^{18} + 33518 x^{16} - 2330346 x^{14} + 94572949 x^{12} - 2154660388 x^{10} + \cdots + 545424869874889 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} + (\beta_{2} - 2) q^{4} + \beta_{12} q^{5} - \beta_{4} q^{7} + (\beta_{13} + 2 \beta_{11}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} + (\beta_{2} - 2) q^{4} + \beta_{12} q^{5} - \beta_{4} q^{7} + (\beta_{13} + 2 \beta_{11}) q^{8} + \beta_1 q^{10} + (\beta_{18} + \beta_{11}) q^{11} - \beta_{6} q^{13} + ( - \beta_{13} - \beta_{11} - \beta_{10}) q^{14} + (\beta_{5} - \beta_{4} - \beta_{2} + 5) q^{16} + \beta_{15} q^{17} + (\beta_{3} - \beta_{2} + 3) q^{19} + ( - \beta_{14} - 4 \beta_{12}) q^{20} + (\beta_{5} + \beta_{3} - 2 \beta_{2} + 7) q^{22} + ( - \beta_{19} - \beta_{11}) q^{23} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots + 8) q^{25}+ \cdots + ( - 3 \beta_{19} - 4 \beta_{18} + \cdots + \beta_{10}) q^{98}+O(q^{100}) \) 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 \( 20 q - 40 q^{4} - 8 q^{7} + 88 q^{16} + 60 q^{19} + 136 q^{22} + 148 q^{25} - 136 q^{28} - 84 q^{46} - 100 q^{49} + 36 q^{64} - 552 q^{76} + 252 q^{79} + 180 q^{85} - 788 q^{88} + 584 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 264 x^{18} + 33518 x^{16} - 2330346 x^{14} + 94572949 x^{12} - 2154660388 x^{10} + \cdots + 545424869874889 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 63\!\cdots\!13 \nu^{18} + \cdots - 59\!\cdots\!23 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 84\!\cdots\!12 \nu^{18} + \cdots + 86\!\cdots\!65 ) / 23\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 32\!\cdots\!49 \nu^{18} + \cdots - 40\!\cdots\!53 ) / 95\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 40\!\cdots\!23 \nu^{18} + \cdots + 89\!\cdots\!28 ) / 47\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18\!\cdots\!21 \nu^{18} + \cdots + 68\!\cdots\!94 ) / 15\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10\!\cdots\!54 \nu^{18} + \cdots - 24\!\cdots\!54 ) / 95\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 49\!\cdots\!63 \nu^{18} + \cdots - 12\!\cdots\!43 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 50\!\cdots\!73 \nu^{18} + \cdots + 12\!\cdots\!71 ) / 19\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 77\!\cdots\!55 \nu^{18} + \cdots + 16\!\cdots\!21 ) / 19\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 92\!\cdots\!43 \nu^{19} + \cdots + 87\!\cdots\!99 \nu ) / 22\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 42\!\cdots\!23 \nu^{19} + \cdots - 88\!\cdots\!51 \nu ) / 44\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 42\!\cdots\!23 \nu^{19} + \cdots + 35\!\cdots\!45 \nu ) / 44\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!11 \nu^{19} + \cdots - 51\!\cdots\!89 \nu ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 53\!\cdots\!95 \nu^{19} + \cdots - 47\!\cdots\!69 \nu ) / 49\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 55\!\cdots\!01 \nu^{19} + \cdots + 21\!\cdots\!91 \nu ) / 44\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 38\!\cdots\!17 \nu^{19} + \cdots - 13\!\cdots\!65 \nu ) / 49\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 79\!\cdots\!26 \nu^{19} + \cdots - 17\!\cdots\!42 \nu ) / 74\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 12\!\cdots\!45 \nu^{19} + \cdots + 28\!\cdots\!93 \nu ) / 49\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 18\!\cdots\!27 \nu^{19} + \cdots - 39\!\cdots\!53 \nu ) / 44\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{12} - \beta_{11} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 9 \beta_{19} - 15 \beta_{18} + \beta_{17} - \beta_{16} + \beta_{15} + \beta_{14} + \cdots - 9 \beta_{10} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{9} - 16\beta_{7} + 48\beta_{6} + 6\beta_{5} - 22\beta_{4} - 2\beta_{3} + 43\beta_{2} + 240\beta _1 + 276 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 706 \beta_{19} - 1202 \beta_{18} - 18 \beta_{17} + 38 \beta_{16} - 10 \beta_{15} + \cdots - 822 \beta_{10} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 308 \beta_{9} - 144 \beta_{8} - 1530 \beta_{7} + 4020 \beta_{6} - 7248 \beta_{5} + 6350 \beta_{4} + \cdots - 119537 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 27888 \beta_{19} - 47989 \beta_{18} - 9456 \beta_{17} + 13393 \beta_{16} - 15582 \beta_{15} + \cdots - 35269 \beta_{10} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5712 \beta_{9} - 5192 \beta_{8} - 4120 \beta_{7} + 18440 \beta_{6} - 1023751 \beta_{5} + 975535 \beta_{4} + \cdots - 16323311 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2150087 \beta_{19} + 3687178 \beta_{18} - 1001279 \beta_{17} + 1397338 \beta_{16} + \cdots + 2287450 \beta_{10} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1751638 \beta_{9} + 463134 \beta_{8} + 14768262 \beta_{7} - 36497702 \beta_{6} - 71381370 \beta_{5} + \cdots - 1120820994 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 524199144 \beta_{19} + 902264467 \beta_{18} - 43905520 \beta_{17} + 60369661 \beta_{16} + \cdots + 596709963 \beta_{10} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 262550160 \beta_{9} + 88207932 \beta_{8} + 2048358416 \beta_{7} - 5086050292 \beta_{6} + \cdots - 7231549091 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 51149022836 \beta_{19} + 88108625410 \beta_{18} + 3073621042 \beta_{17} - 4579648614 \beta_{16} + \cdots + 58729746700 \beta_{10} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 18696523084 \beta_{9} + 6541005536 \beta_{8} + 143965920788 \beta_{7} - 357710695504 \beta_{6} + \cdots + 8849228947211 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2234014445207 \beta_{19} + 3848935277133 \beta_{18} + 795270193659 \beta_{17} + \cdots + 2571711232065 \beta_{10} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 161520206480 \beta_{9} + 59468825312 \beta_{8} + 1194069512128 \beta_{7} - 2971542060064 \beta_{6} + \cdots + 12\!\cdots\!08 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 151125512945714 \beta_{19} - 260504923467616 \beta_{18} + 78993642325928 \beta_{17} + \cdots - 173624245874198 \beta_{10} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 143238477678224 \beta_{9} - 50278648697576 \beta_{8} + \cdots + 88\!\cdots\!51 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 39\!\cdots\!52 \beta_{19} + \cdots - 45\!\cdots\!15 \beta_{10} \) 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
−8.96460 − 3.66991i
8.96460 − 3.66991i
−0.561432 − 3.11779i
0.561432 − 3.11779i
−5.18142 − 2.15227i
5.18142 − 2.15227i
−5.47552 − 1.46909i
5.47552 − 1.46909i
−4.94897 − 0.143626i
4.94897 − 0.143626i
−4.94897 + 0.143626i
4.94897 + 0.143626i
−5.47552 + 1.46909i
5.47552 + 1.46909i
−5.18142 + 2.15227i
5.18142 + 2.15227i
−0.561432 + 3.11779i
0.561432 + 3.11779i
−8.96460 + 3.66991i
8.96460 + 3.66991i
− 3.66991i 0 −9.46822 −8.96460 0 4.32189 20.0678i 0 32.8992i
235.2 − 3.66991i 0 −9.46822 8.96460 0 4.32189 20.0678i 0 − 32.8992i
235.3 − 3.11779i 0 −5.72063 −0.561432 0 −0.0649187 5.36456i 0 1.75043i
235.4 − 3.11779i 0 −5.72063 0.561432 0 −0.0649187 5.36456i 0 − 1.75043i
235.5 − 2.15227i 0 −0.632287 −5.18142 0 −10.7194 − 7.24824i 0 11.1518i
235.6 − 2.15227i 0 −0.632287 5.18142 0 −10.7194 − 7.24824i 0 − 11.1518i
235.7 − 1.46909i 0 1.84176 −5.47552 0 8.41407 − 8.58210i 0 8.04406i
235.8 − 1.46909i 0 1.84176 5.47552 0 8.41407 − 8.58210i 0 − 8.04406i
235.9 − 0.143626i 0 3.97937 −4.94897 0 −3.95168 − 1.14604i 0 0.710799i
235.10 − 0.143626i 0 3.97937 4.94897 0 −3.95168 − 1.14604i 0 − 0.710799i
235.11 0.143626i 0 3.97937 −4.94897 0 −3.95168 1.14604i 0 − 0.710799i
235.12 0.143626i 0 3.97937 4.94897 0 −3.95168 1.14604i 0 0.710799i
235.13 1.46909i 0 1.84176 −5.47552 0 8.41407 8.58210i 0 − 8.04406i
235.14 1.46909i 0 1.84176 5.47552 0 8.41407 8.58210i 0 8.04406i
235.15 2.15227i 0 −0.632287 −5.18142 0 −10.7194 7.24824i 0 − 11.1518i
235.16 2.15227i 0 −0.632287 5.18142 0 −10.7194 7.24824i 0 11.1518i
235.17 3.11779i 0 −5.72063 −0.561432 0 −0.0649187 − 5.36456i 0 − 1.75043i
235.18 3.11779i 0 −5.72063 0.561432 0 −0.0649187 − 5.36456i 0 1.75043i
235.19 3.66991i 0 −9.46822 −8.96460 0 4.32189 − 20.0678i 0 − 32.8992i
235.20 3.66991i 0 −9.46822 8.96460 0 4.32189 − 20.0678i 0 32.8992i
\(n\): 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
3.b odd 2 1 inner
59.b odd 2 1 inner
177.d even 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.d ✓ 20
3.b odd 2 1 inner 531.3.c.d ✓ 20
59.b odd 2 1 inner 531.3.c.d ✓ 20
177.d even 2 1 inner 531.3.c.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
531.3.c.d ✓ 20 1.a even 1 1 trivial
531.3.c.d ✓ 20 3.b odd 2 1 inner
531.3.c.d ✓ 20 59.b odd 2 1 inner
531.3.c.d ✓ 20 177.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 30T_{2}^{8} + 299T_{2}^{6} + 1127T_{2}^{4} + 1332T_{2}^{2} + 27 \) acting on \(S_{3}^{\mathrm{new}}(531, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 30 T^{8} + \cdots + 27)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} - 162 T^{8} + \cdots - 499384)^{2} \) Copy content Toggle raw display
$7$ \( (T^{5} + 2 T^{4} + \cdots + 100)^{4} \) Copy content Toggle raw display
$11$ \( (T^{10} + 581 T^{8} + \cdots + 955867500)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 967 T^{8} + \cdots + 16992414522)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 151063660000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - 15 T^{4} + \cdots - 1205188)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + 1555 T^{8} + \cdots + 4958731008)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} - 5099 T^{8} + \cdots - 74406218464)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 69784856505000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 36\!\cdots\!98)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 53\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 17\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 41\!\cdots\!52)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 26\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 11\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 40\!\cdots\!34)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 95\!\cdots\!28)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 63 T^{4} + \cdots + 497986900)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 16\!\cdots\!52)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 40\!\cdots\!08)^{2} \) Copy content Toggle raw display
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