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Properties

Label 531.5.c.b
Level $531$
Weight $5$
Character orbit 531.c
Analytic conductor $54.889$
Analytic rank $0$
Dimension $16$
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] = [531,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.235");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(54.8894503975\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 225 x^{14} + 21073 x^{12} + 1064743 x^{10} + 31467838 x^{8} + 552171992 x^{6} + \cdots + 63591714144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: no (minimal twist has level 59)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 12) q^{4} + \beta_{8} q^{5} + ( - \beta_{6} - \beta_{4} - 5) q^{7} + (\beta_{3} - 6 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 12) q^{4} + \beta_{8} q^{5} + ( - \beta_{6} - \beta_{4} - 5) q^{7} + (\beta_{3} - 6 \beta_1) q^{8} + ( - \beta_{7} - 2 \beta_1) q^{10} + (\beta_{9} + \beta_1) q^{11} + (\beta_{13} + \beta_{5} + \cdots + 5 \beta_1) q^{13}+ \cdots + (12 \beta_{15} - 3 \beta_{13} + \cdots + 912 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 194 q^{4} - 4 q^{5} - 82 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 194 q^{4} - 4 q^{5} - 82 q^{7} - 546 q^{16} + 176 q^{17} - 632 q^{19} + 896 q^{20} - 410 q^{22} + 316 q^{25} - 2322 q^{26} + 618 q^{28} - 496 q^{29} - 2702 q^{35} - 4762 q^{41} + 5676 q^{46} + 3158 q^{49} + 9560 q^{53} + 8530 q^{59} - 1500 q^{62} + 9818 q^{64} + 9458 q^{68} - 12304 q^{71} + 13866 q^{74} - 8888 q^{76} - 25574 q^{79} + 6248 q^{80} - 18364 q^{85} - 7014 q^{86} - 2294 q^{88} - 7528 q^{94} - 49520 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 225 x^{14} + 21073 x^{12} + 1064743 x^{10} + 31467838 x^{8} + 552171992 x^{6} + \cdots + 63591714144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 38\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 705397 \nu^{14} - 133669987 \nu^{12} - 10271669799 \nu^{10} - 414173351629 \nu^{8} + \cdots - 19\!\cdots\!04 ) / 9444243757056 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 705397 \nu^{15} - 133669987 \nu^{13} - 10271669799 \nu^{11} - 414173351629 \nu^{9} + \cdots - 19\!\cdots\!04 \nu ) / 9444243757056 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 987573 \nu^{14} + 323293043 \nu^{12} + 37978153495 \nu^{10} + 2180413591869 \nu^{8} + \cdots + 27\!\cdots\!28 ) / 4722121878528 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 295283 \nu^{15} - 61851695 \nu^{13} - 5331231039 \nu^{11} - 243013326265 \nu^{9} + \cdots - 17\!\cdots\!96 \nu ) / 590265234816 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 295283 \nu^{14} + 61851695 \nu^{12} + 5331231039 \nu^{10} + 243013326265 \nu^{8} + \cdots + 17\!\cdots\!64 ) / 590265234816 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4972223 \nu^{15} - 1054014801 \nu^{13} - 92667881885 \nu^{11} - 4361975526767 \nu^{9} + \cdots - 45\!\cdots\!00 \nu ) / 9444243757056 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4972223 \nu^{14} + 1054014801 \nu^{12} + 92667881885 \nu^{10} + 4361975526767 \nu^{8} + \cdots + 45\!\cdots\!88 ) / 9444243757056 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1560493 \nu^{15} + 391732725 \nu^{13} + 39588129769 \nu^{11} + 2077157339719 \nu^{9} + \cdots + 24\!\cdots\!20 \nu ) / 2361060939264 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7193991 \nu^{14} + 1564584561 \nu^{12} + 139212555421 \nu^{10} + 6510545551871 \nu^{8} + \cdots + 52\!\cdots\!04 ) / 4722121878528 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15093379 \nu^{15} + 3262839109 \nu^{13} + 288696780641 \nu^{11} + 13435264455371 \nu^{9} + \cdots + 10\!\cdots\!84 \nu ) / 9444243757056 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16047569 \nu^{14} - 3001256927 \nu^{12} - 223843690963 \nu^{10} - 8495512685793 \nu^{8} + \cdots - 19\!\cdots\!44 ) / 9444243757056 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10086675 \nu^{15} + 2083588933 \nu^{13} + 175690600321 \nu^{11} + 7780810994891 \nu^{9} + \cdots + 55\!\cdots\!76 \nu ) / 4722121878528 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 38\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} - 2\beta_{12} + \beta_{10} + 2\beta_{8} + 2\beta_{6} + 8\beta_{4} - 58\beta_{2} + 1050 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - 2\beta_{13} + \beta_{11} + \beta_{9} - 2\beta_{7} + \beta_{5} - 63\beta_{3} + 1622\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 88\beta_{14} + 188\beta_{12} - 72\beta_{10} - 200\beta_{8} - 192\beta_{6} - 552\beta_{4} + 2903\beta_{2} - 44526 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 88 \beta_{15} + 188 \beta_{13} - 104 \beta_{11} - 120 \beta_{9} + 200 \beta_{7} + 92 \beta_{5} + \cdots - 72836 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5731 \beta_{14} - 12794 \beta_{12} + 4051 \beta_{10} + 13562 \beta_{8} + 13394 \beta_{6} + \cdots + 1991546 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5731 \beta_{15} - 12794 \beta_{13} + 7663 \beta_{11} + 9343 \beta_{9} - 13562 \beta_{7} + \cdots + 3361458 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 337122 \beta_{14} + 770172 \beta_{12} - 212450 \beta_{10} - 784632 \beta_{8} - 829048 \beta_{6} + \cdots - 91651722 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 337122 \beta_{15} + 770172 \beta_{13} - 491926 \beta_{11} - 616598 \beta_{9} + 784632 \beta_{7} + \cdots - 157925844 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 18958329 \beta_{14} - 43619178 \beta_{12} + 10786585 \beta_{10} + 41832042 \beta_{8} + \cdots + 4296332726 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 18958329 \beta_{15} - 43619178 \beta_{13} + 29424737 \beta_{11} + 37596481 \beta_{9} + \cdots + 7517254474 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1040650052 \beta_{14} + 2386144972 \beta_{12} - 535787092 \beta_{10} - 2126876872 \beta_{8} + \cdots - 204128835310 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1040650052 \beta_{15} + 2386144972 \beta_{13} - 1689536076 \beta_{11} - 2194399036 \beta_{9} + \cdots - 361518216108 \beta_1 \) 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
− 7.16407i
− 6.76646i
− 6.49637i
− 5.83876i
− 4.73769i
− 3.62558i
− 2.92448i
− 2.73020i
2.73020i
2.92448i
3.62558i
4.73769i
5.83876i
6.49637i
6.76646i
7.16407i
− 7.16407i 0 −35.3239 −2.76183 0 −74.1523 138.438i 0 19.7860i
235.2 − 6.76646i 0 −29.7850 26.7091 0 71.2878 93.2759i 0 − 180.726i
235.3 − 6.49637i 0 −26.2028 −15.1486 0 −45.0626 66.2810i 0 98.4107i
235.4 − 5.83876i 0 −18.0911 −43.4326 0 52.0443 12.2095i 0 253.593i
235.5 − 4.73769i 0 −6.44573 6.57829 0 8.91481 − 45.2652i 0 − 31.1659i
235.6 − 3.62558i 0 2.85514 42.2784 0 −22.3137 − 68.3609i 0 − 153.284i
235.7 − 2.92448i 0 7.44742 −21.5054 0 30.9748 − 68.5715i 0 62.8920i
235.8 − 2.73020i 0 8.54602 5.28252 0 −62.6931 − 67.0155i 0 − 14.4223i
235.9 2.73020i 0 8.54602 5.28252 0 −62.6931 67.0155i 0 14.4223i
235.10 2.92448i 0 7.44742 −21.5054 0 30.9748 68.5715i 0 − 62.8920i
235.11 3.62558i 0 2.85514 42.2784 0 −22.3137 68.3609i 0 153.284i
235.12 4.73769i 0 −6.44573 6.57829 0 8.91481 45.2652i 0 31.1659i
235.13 5.83876i 0 −18.0911 −43.4326 0 52.0443 − 12.2095i 0 − 253.593i
235.14 6.49637i 0 −26.2028 −15.1486 0 −45.0626 − 66.2810i 0 − 98.4107i
235.15 6.76646i 0 −29.7850 26.7091 0 71.2878 − 93.2759i 0 180.726i
235.16 7.16407i 0 −35.3239 −2.76183 0 −74.1523 − 138.438i 0 − 19.7860i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.16
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 531.5.c.b 16
3.b odd 2 1 59.5.b.b ✓ 16
59.b odd 2 1 inner 531.5.c.b 16
177.d even 2 1 59.5.b.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.5.b.b ✓ 16 3.b odd 2 1
59.5.b.b ✓ 16 177.d even 2 1
531.5.c.b 16 1.a even 1 1 trivial
531.5.c.b 16 59.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 225 T_{2}^{14} + 21073 T_{2}^{12} + 1064743 T_{2}^{10} + 31467838 T_{2}^{8} + \cdots + 63591714144 \) acting on \(S_{5}^{\mathrm{new}}(531, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 63591714144 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 2 T^{7} + \cdots + 1533430958)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 4788953563207)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 79\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 18\!\cdots\!32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 31\!\cdots\!78)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 99\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 11\!\cdots\!28)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 41\!\cdots\!97)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 11\!\cdots\!08)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 46\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 84\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 15\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 78\!\cdots\!21)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 70\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
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