↑

Properties

Label 828.2.q.a
Level $828$
Weight $2$
Character orbit 828.q
Analytic conductor $6.612$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [828,2,Mod(73,828)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(828, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("828.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 828.q (of order \(11\), degree \(10\), 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: \(6.61161328736\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 9 x^{19} + 51 x^{18} - 200 x^{17} + 633 x^{16} - 1688 x^{15} + 3957 x^{14} - 8161 x^{13} + \cdots + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 92)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{19} - \beta_{18} + \cdots - \beta_{2}) q^{5}+ \cdots + (\beta_{16} + \beta_{13} + \beta_{11} + \cdots + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{19} - \beta_{18} + \cdots - \beta_{2}) q^{5}+ \cdots + (3 \beta_{19} - 3 \beta_{18} + \cdots + 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{5} + 2 q^{7} + 2 q^{11} + 6 q^{13} + 9 q^{17} - 11 q^{19} + 22 q^{23} - 16 q^{25} + q^{29} - 13 q^{31} - 14 q^{35} + 34 q^{37} - 28 q^{41} + 44 q^{43} - 26 q^{47} + 60 q^{49} - 14 q^{53} + 26 q^{55} + 10 q^{59} - 56 q^{61} + 87 q^{65} - 44 q^{67} + 37 q^{71} - 12 q^{73} + 47 q^{77} - 6 q^{79} + 25 q^{83} + 8 q^{85} - 10 q^{89} + 26 q^{91} - 29 q^{95} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 9 x^{19} + 51 x^{18} - 200 x^{17} + 633 x^{16} - 1688 x^{15} + 3957 x^{14} - 8161 x^{13} + \cdots + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 26\!\cdots\!24 \nu^{19} + \cdots + 24\!\cdots\!76 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 58\!\cdots\!64 \nu^{19} + \cdots + 73\!\cdots\!27 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 95\!\cdots\!62 \nu^{19} + \cdots - 48\!\cdots\!81 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!69 \nu^{19} + \cdots + 45\!\cdots\!85 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!65 \nu^{19} + \cdots - 35\!\cdots\!17 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28\!\cdots\!41 \nu^{19} + \cdots + 87\!\cdots\!95 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!51 \nu^{19} + \cdots - 28\!\cdots\!69 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 45\!\cdots\!44 \nu^{19} + \cdots + 12\!\cdots\!17 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 86\!\cdots\!65 \nu^{19} + \cdots + 50\!\cdots\!07 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 91\!\cdots\!89 \nu^{19} + \cdots - 46\!\cdots\!37 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12\!\cdots\!80 \nu^{19} + \cdots + 40\!\cdots\!07 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!63 \nu^{19} + \cdots + 59\!\cdots\!43 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16\!\cdots\!55 \nu^{19} + \cdots - 62\!\cdots\!30 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 18\!\cdots\!81 \nu^{19} + \cdots - 37\!\cdots\!88 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 20\!\cdots\!15 \nu^{19} + \cdots - 37\!\cdots\!16 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 22\!\cdots\!50 \nu^{19} + \cdots - 97\!\cdots\!34 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 31\!\cdots\!77 \nu^{19} + \cdots + 90\!\cdots\!66 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 39\!\cdots\!49 \nu^{19} + \cdots - 10\!\cdots\!67 ) / 20\!\cdots\!31 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} - \beta_{15} - 4\beta_{12} - \beta_{11} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - 5 \beta_{17} - 2 \beta_{15} - 2 \beta_{14} - 7 \beta_{12} - \beta_{11} - 7 \beta_{10} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{19} - 2 \beta_{18} - 7 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} - 11 \beta_{14} - 3 \beta_{12} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 11 \beta_{18} + 11 \beta_{16} + 12 \beta_{15} + 23 \beta_{13} + 24 \beta_{12} + 24 \beta_{11} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5 \beta_{19} + 18 \beta_{17} + 18 \beta_{16} + 77 \beta_{15} + 98 \beta_{14} + 93 \beta_{13} + \cdots - 45 \beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 114 \beta_{18} + 19 \beta_{17} - 19 \beta_{16} + 148 \beta_{15} + 227 \beta_{14} - \beta_{13} + \cdots - 132 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 82 \beta_{19} + 342 \beta_{18} - 211 \beta_{17} - 131 \beta_{16} - 102 \beta_{15} - 102 \beta_{14} + \cdots - 305 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 67 \beta_{19} - 86 \beta_{18} - 830 \beta_{17} - 80 \beta_{16} - 1124 \beta_{15} - 2005 \beta_{14} + \cdots + 977 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1761 \beta_{19} - 4000 \beta_{18} + 1035 \beta_{16} - 1908 \beta_{15} - 3683 \beta_{14} + 615 \beta_{13} + \cdots + 6479 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6957 \beta_{19} - 11824 \beta_{18} + 6957 \beta_{17} + 3994 \beta_{16} + 3237 \beta_{15} + \cdots + 7927 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 939 \beta_{19} + 15349 \beta_{17} + 6292 \beta_{16} + 21527 \beta_{15} + 64536 \beta_{14} + \cdots - 41065 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 69465 \beta_{19} + 94268 \beta_{18} - 22527 \beta_{17} + 3016 \beta_{16} + 37992 \beta_{15} + \cdots - 197987 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 219027 \beta_{19} + 260035 \beta_{18} - 180672 \beta_{17} + 10817 \beta_{15} - 353220 \beta_{14} + \cdots - 227454 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 10817 \beta_{19} + 10817 \beta_{18} - 190828 \beta_{17} - 10817 \beta_{16} + 10817 \beta_{15} + \cdots + 1038335 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 2107484 \beta_{19} - 1840420 \beta_{18} + 1359709 \beta_{17} - 267064 \beta_{16} + 292387 \beta_{15} + \cdots + 4815018 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 6205628 \beta_{19} - 4662471 \beta_{18} + 5660522 \beta_{17} - 1517834 \beta_{16} - 194412 \beta_{15} + \cdots + 4662471 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 1673060 \beta_{18} + 3385306 \beta_{17} - 3385306 \beta_{16} - 8227854 \beta_{15} + 25425140 \beta_{14} + \cdots - 25983466 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 50776689 \beta_{19} + 35015101 \beta_{18} - 39163529 \beta_{17} + 4148428 \beta_{16} - 32082975 \beta_{15} + \cdots - 104318793 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/828\mathbb{Z}\right)^\times\).

\(n\) \(415\) \(461\) \(649\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
0.291382 − 2.02660i
−0.250875 + 1.74487i
−0.420431 + 0.123450i
2.26168 − 0.664090i
−0.115644 − 0.0743196i
1.53106 + 0.983952i
−0.420431 − 0.123450i
2.26168 + 0.664090i
0.291382 + 2.02660i
−0.250875 − 1.74487i
−0.858865 − 1.88065i
1.20400 + 2.63640i
−0.967148 + 1.11615i
1.82483 − 2.10597i
−0.967148 − 1.11615i
1.82483 + 2.10597i
−0.858865 + 1.88065i
1.20400 − 2.63640i
−0.115644 + 0.0743196i
1.53106 − 0.983952i
0 0 0 −3.22799 − 2.07450i 0 0.267813 + 1.86268i 0 0 0
73.2 0 0 0 0.630070 + 0.404921i 0 −0.0283674 − 0.197300i 0 0 0
289.1 0 0 0 −0.781296 + 1.71080i 0 4.72847 + 1.38840i 0 0 0
289.2 0 0 0 0.105327 − 0.230633i 0 −3.93129 − 1.15433i 0 0 0
325.1 0 0 0 −1.77577 + 2.04934i 0 −0.600040 + 0.385622i 0 0 0
325.2 0 0 0 2.72780 − 3.14805i 0 1.70185 − 1.09371i 0 0 0
361.1 0 0 0 −0.781296 − 1.71080i 0 4.72847 − 1.38840i 0 0 0
361.2 0 0 0 0.105327 + 0.230633i 0 −3.93129 + 1.15433i 0 0 0
397.1 0 0 0 −3.22799 + 2.07450i 0 0.267813 − 1.86268i 0 0 0
397.2 0 0 0 0.630070 − 0.404921i 0 −0.0283674 + 0.197300i 0 0 0
469.1 0 0 0 0.187926 + 1.30705i 0 1.18148 − 2.58708i 0 0 0
469.2 0 0 0 0.556197 + 3.86843i 0 −1.06324 + 2.32817i 0 0 0
541.1 0 0 0 −0.926812 + 0.272136i 0 −1.14874 − 1.32572i 0 0 0
541.2 0 0 0 1.50454 − 0.441774i 0 −0.107929 − 0.124557i 0 0 0
577.1 0 0 0 −0.926812 − 0.272136i 0 −1.14874 + 1.32572i 0 0 0
577.2 0 0 0 1.50454 + 0.441774i 0 −0.107929 + 0.124557i 0 0 0
685.1 0 0 0 0.187926 − 1.30705i 0 1.18148 + 2.58708i 0 0 0
685.2 0 0 0 0.556197 − 3.86843i 0 −1.06324 − 2.32817i 0 0 0
721.1 0 0 0 −1.77577 − 2.04934i 0 −0.600040 − 0.385622i 0 0 0
721.2 0 0 0 2.72780 + 3.14805i 0 1.70185 + 1.09371i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 828.2.q.a 20
3.b odd 2 1 92.2.e.a ✓ 20
12.b even 2 1 368.2.m.d 20
23.c even 11 1 inner 828.2.q.a 20
69.g even 22 1 2116.2.a.i 10
69.h odd 22 1 92.2.e.a ✓ 20
69.h odd 22 1 2116.2.a.j 10
276.j odd 22 1 8464.2.a.cd 10
276.o even 22 1 368.2.m.d 20
276.o even 22 1 8464.2.a.ce 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.2.e.a ✓ 20 3.b odd 2 1
92.2.e.a ✓ 20 69.h odd 22 1
368.2.m.d 20 12.b even 2 1
368.2.m.d 20 276.o even 22 1
828.2.q.a 20 1.a even 1 1 trivial
828.2.q.a 20 23.c even 11 1 inner
2116.2.a.i 10 69.g even 22 1
2116.2.a.j 10 69.h odd 22 1
8464.2.a.cd 10 276.j odd 22 1
8464.2.a.ce 10 276.o even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 2 T_{5}^{19} + 15 T_{5}^{18} + 63 T_{5}^{17} + 324 T_{5}^{16} + 637 T_{5}^{15} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(828, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + 2 T^{19} + \cdots + 14641 \) Copy content Toggle raw display
$7$ \( T^{20} - 2 T^{19} + \cdots + 529 \) Copy content Toggle raw display
$11$ \( T^{20} - 2 T^{19} + \cdots + 25715041 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 14988860041 \) Copy content Toggle raw display
$17$ \( T^{20} - 9 T^{19} + \cdots + 139129 \) Copy content Toggle raw display
$19$ \( T^{20} + 11 T^{19} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 10805664116809 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 2079845655889 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 1866439595329 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 65806215082609 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 150230641573321 \) Copy content Toggle raw display
$47$ \( (T^{10} + 13 T^{9} + \cdots - 3761152)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 130889227506489 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 6274789532209 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 791079807551881 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 14858951082361 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 59472089161 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 12\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 1214604572281 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 19165912096321 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 153528281766409 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
show more
show less