↑

Properties

Label 1428.2.w.c
Level $1428$
Weight $2$
Character orbit 1428.w
Analytic conductor $11.403$
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] = [1428,2,Mod(421,1428)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1428, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1428.421");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1428.w (of order \(4\), degree \(2\), 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: \(11.4026374086\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 64 x^{18} + 1546 x^{16} + 17508 x^{14} + 94093 x^{12} + 217608 x^{10} + 224200 x^{8} + 94444 x^{6} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{3} + (\beta_{10} + \beta_{7}) q^{5} + \beta_{12} q^{7} - \beta_{9} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{3} + (\beta_{10} + \beta_{7}) q^{5} + \beta_{12} q^{7} - \beta_{9} q^{9} + ( - \beta_{16} - \beta_{14} + \beta_{10} + \cdots + 1) q^{11}+ \cdots + (\beta_{18} - \beta_{7} + \cdots - \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{5} + 8 q^{11} - 4 q^{13} - 20 q^{21} + 4 q^{29} + 8 q^{31} - 12 q^{33} - 12 q^{35} + 16 q^{37} - 4 q^{39} + 4 q^{45} - 32 q^{47} + 20 q^{51} - 28 q^{55} - 28 q^{57} - 48 q^{61} - 12 q^{65} + 24 q^{67} - 36 q^{69} + 44 q^{71} + 28 q^{73} + 8 q^{75} - 8 q^{79} - 20 q^{81} + 28 q^{85} - 96 q^{89} + 4 q^{91} - 28 q^{95} + 16 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 64 x^{18} + 1546 x^{16} + 17508 x^{14} + 94093 x^{12} + 217608 x^{10} + 224200 x^{8} + 94444 x^{6} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11649382 \nu^{18} - 727687772 \nu^{16} - 16892307065 \nu^{14} - 177965379541 \nu^{12} + \cdots - 228419158 ) / 3124870180 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1530734983 \nu^{19} - 1530288051 \nu^{18} - 88954612225 \nu^{17} - 96408510069 \nu^{16} + \cdots + 309457353364 ) / 399983383040 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1530734983 \nu^{19} - 1530288051 \nu^{18} + 88954612225 \nu^{17} - 96408510069 \nu^{16} + \cdots + 309457353364 ) / 399983383040 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 169726895 \nu^{19} - 7311168085 \nu^{18} - 13519044697 \nu^{17} - 469746078531 \nu^{16} + \cdots + 1564895636556 ) / 399983383040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 169726895 \nu^{19} + 7311168085 \nu^{18} - 13519044697 \nu^{17} + 469746078531 \nu^{16} + \cdots - 1564895636556 ) / 399983383040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2804699055 \nu^{19} - 182442765 \nu^{18} - 178191879961 \nu^{17} - 11705927755 \nu^{16} + \cdots - 60221294740 ) / 399983383040 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2804699055 \nu^{19} + 182442765 \nu^{18} - 178191879961 \nu^{17} + 11705927755 \nu^{16} + \cdots + 60221294740 ) / 399983383040 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 114209579 \nu^{19} - 7286114292 \nu^{17} - 175112633590 \nu^{15} - 1965796695002 \nu^{13} + \cdots - 73999217528 \nu ) / 6249740360 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15055323685 \nu^{19} + 2804699055 \nu^{18} + 963358273075 \nu^{17} + 178191879961 \nu^{16} + \cdots + 492592240604 ) / 399983383040 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 122256237 \nu^{18} - 7806641579 \nu^{16} - 187884193215 \nu^{14} - 2113794240541 \nu^{12} + \cdots - 5170477716 ) / 858333440 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15055323685 \nu^{19} - 2804699055 \nu^{18} + 963358273075 \nu^{17} - 178191879961 \nu^{16} + \cdots - 492592240604 ) / 399983383040 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 35075984534 \nu^{19} + 5907941043 \nu^{18} + 2245117090746 \nu^{17} + 378932756629 \nu^{16} + \cdots + 939247639276 ) / 199991691520 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 35075984534 \nu^{19} - 5907941043 \nu^{18} + 2245117090746 \nu^{17} - 378932756629 \nu^{16} + \cdots - 939247639276 ) / 199991691520 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 35926215473 \nu^{19} - 6428424549 \nu^{18} + 2296353919207 \nu^{17} - 414710540755 \nu^{16} + \cdots - 1536345547380 ) / 199991691520 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 90524427205 \nu^{19} - 23405155667 \nu^{18} + 5795121225555 \nu^{17} - 1497775080469 \nu^{16} + \cdots + 392974246164 ) / 399983383040 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 54306721719 \nu^{19} - 5907941043 \nu^{18} + 3473681856369 \nu^{17} - 378932756629 \nu^{16} + \cdots - 939247639276 ) / 199991691520 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 146217297289 \nu^{19} + 37843294043 \nu^{18} + 9348752567855 \nu^{17} + 2422126545933 \nu^{16} + \cdots + 2317875361292 ) / 399983383040 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 94527810695 \nu^{19} + 6039923834017 \nu^{17} + 145512806639645 \nu^{15} + \cdots + 3006324479548 \nu ) / 199991691520 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{18} + \beta_{16} + \beta_{13} - \beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3 \beta_{19} - 2 \beta_{18} - \beta_{17} - 2 \beta_{16} - \beta_{14} + 3 \beta_{12} + 5 \beta_{10} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 18 \beta_{18} - 18 \beta_{16} - \beta_{15} - 2 \beta_{14} - 15 \beta_{13} + 5 \beta_{12} + 19 \beta_{11} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 69 \beta_{19} + 52 \beta_{18} + 17 \beta_{17} + 52 \beta_{16} + 18 \beta_{14} - 17 \beta_{13} + \cdots - 52 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 323 \beta_{18} + 323 \beta_{16} + 37 \beta_{15} + 49 \beta_{14} + 237 \beta_{13} - 144 \beta_{12} + \cdots - 1493 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1437 \beta_{19} - 1148 \beta_{18} - 291 \beta_{17} - 1148 \beta_{16} - 321 \beta_{14} + 536 \beta_{13} + \cdots + 1148 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6041 \beta_{18} - 6041 \beta_{16} - 915 \beta_{15} - 1065 \beta_{14} - 4061 \beta_{13} + 3292 \beta_{12} + \cdots + 27145 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 29125 \beta_{19} + 23944 \beta_{18} + 5321 \beta_{17} + 23944 \beta_{16} + 5901 \beta_{14} + \cdots - 23944 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 116075 \beta_{18} + 116075 \beta_{16} + 20065 \beta_{15} + 22047 \beta_{14} + 73963 \beta_{13} + \cdots - 513741 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 583455 \beta_{19} - 486824 \beta_{18} - 101761 \beta_{17} - 486824 \beta_{16} - 110945 \beta_{14} + \cdots + 486824 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2264111 \beta_{18} - 2264111 \beta_{16} - 419541 \beta_{15} - 444729 \beta_{14} - 1399841 \beta_{13} + \cdots + 9945553 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 11623985 \beta_{19} + 9776700 \beta_{18} + 1994773 \beta_{17} + 9776700 \beta_{16} + 2116623 \beta_{14} + \cdots - 9776700 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 44517921 \beta_{18} + 44517921 \beta_{16} + 8588941 \beta_{15} + 8841465 \beta_{14} + \cdots - 194913391 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 230966147 \beta_{19} - 195139396 \beta_{18} - 39598669 \beta_{17} - 195139396 \beta_{16} + \cdots + 195139396 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 879002855 \beta_{18} - 879002855 \beta_{16} - 174111937 \beta_{15} - 174307685 \beta_{14} + \cdots + 3844771525 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 4583340301 \beta_{19} + 3882839392 \beta_{18} + 790965361 \beta_{17} + 3882839392 \beta_{16} + \cdots - 3882839392 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 17393528509 \beta_{18} + 17393528509 \beta_{16} + 3512919321 \beta_{15} + 3419589501 \beta_{14} + \cdots - 76097062743 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 90895952075 \beta_{19} - 77137859128 \beta_{18} - 15845972181 \beta_{17} - 77137859128 \beta_{16} + \cdots + 77137859128 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1428\mathbb{Z}\right)^\times\).

\(n\) \(409\) \(715\) \(953\) \(1261\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
421.1
− 3.23509i
− 1.21232i
0.253312i
0.319114i
4.46077i
4.45281i
1.20790i
0.819490i
0.101303i
− 3.16728i
3.23509i
1.21232i
− 0.253312i
− 0.319114i
− 4.46077i
− 4.45281i
− 1.20790i
− 0.819490i
− 0.101303i
3.16728i
0 −0.707107 − 0.707107i 0 −2.99466 − 2.99466i 0 0.707107 − 0.707107i 0 1.00000i 0
421.2 0 −0.707107 − 0.707107i 0 −1.56434 − 1.56434i 0 0.707107 − 0.707107i 0 1.00000i 0
421.3 0 −0.707107 − 0.707107i 0 −0.527988 − 0.527988i 0 0.707107 − 0.707107i 0 1.00000i 0
421.4 0 −0.707107 − 0.707107i 0 −0.481459 − 0.481459i 0 0.707107 − 0.707107i 0 1.00000i 0
421.5 0 −0.707107 − 0.707107i 0 2.44713 + 2.44713i 0 0.707107 − 0.707107i 0 1.00000i 0
421.6 0 0.707107 + 0.707107i 0 −2.44150 − 2.44150i 0 −0.707107 + 0.707107i 0 1.00000i 0
421.7 0 0.707107 + 0.707107i 0 −0.147007 − 0.147007i 0 −0.707107 + 0.707107i 0 1.00000i 0
421.8 0 0.707107 + 0.707107i 0 0.127640 + 0.127640i 0 −0.707107 + 0.707107i 0 1.00000i 0
421.9 0 0.707107 + 0.707107i 0 0.635475 + 0.635475i 0 −0.707107 + 0.707107i 0 1.00000i 0
421.10 0 0.707107 + 0.707107i 0 2.94671 + 2.94671i 0 −0.707107 + 0.707107i 0 1.00000i 0
1177.1 0 −0.707107 + 0.707107i 0 −2.99466 + 2.99466i 0 0.707107 + 0.707107i 0 − 1.00000i 0
1177.2 0 −0.707107 + 0.707107i 0 −1.56434 + 1.56434i 0 0.707107 + 0.707107i 0 − 1.00000i 0
1177.3 0 −0.707107 + 0.707107i 0 −0.527988 + 0.527988i 0 0.707107 + 0.707107i 0 − 1.00000i 0
1177.4 0 −0.707107 + 0.707107i 0 −0.481459 + 0.481459i 0 0.707107 + 0.707107i 0 − 1.00000i 0
1177.5 0 −0.707107 + 0.707107i 0 2.44713 − 2.44713i 0 0.707107 + 0.707107i 0 − 1.00000i 0
1177.6 0 0.707107 − 0.707107i 0 −2.44150 + 2.44150i 0 −0.707107 − 0.707107i 0 − 1.00000i 0
1177.7 0 0.707107 − 0.707107i 0 −0.147007 + 0.147007i 0 −0.707107 − 0.707107i 0 − 1.00000i 0
1177.8 0 0.707107 − 0.707107i 0 0.127640 − 0.127640i 0 −0.707107 − 0.707107i 0 − 1.00000i 0
1177.9 0 0.707107 − 0.707107i 0 0.635475 − 0.635475i 0 −0.707107 − 0.707107i 0 − 1.00000i 0
1177.10 0 0.707107 − 0.707107i 0 2.94671 − 2.94671i 0 −0.707107 − 0.707107i 0 − 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1428.2.w.c ✓ 20
3.b odd 2 1 4284.2.z.d 20
17.c even 4 1 inner 1428.2.w.c ✓ 20
51.f odd 4 1 4284.2.z.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1428.2.w.c ✓ 20 1.a even 1 1 trivial
1428.2.w.c ✓ 20 17.c even 4 1 inner
4284.2.z.d 20 3.b odd 2 1
4284.2.z.d 20 51.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 4 T_{5}^{19} + 8 T_{5}^{18} + 4 T_{5}^{17} + 451 T_{5}^{16} + 1780 T_{5}^{15} + 3520 T_{5}^{14} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(1428, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} + 4 T^{19} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 1004549161984 \) Copy content Toggle raw display
$13$ \( (T^{10} + 2 T^{9} + \cdots + 61472)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 2015993900449 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 156999797824 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 6921108640000 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 9519514624 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 4161798144 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 654364509184 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 180699807744 \) Copy content Toggle raw display
$47$ \( (T^{10} + 16 T^{9} + \cdots - 7483904)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 58\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( (T^{10} - 12 T^{9} + \cdots - 10418176)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 4720503619584 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 54842437402624 \) Copy content Toggle raw display
$89$ \( (T^{10} + 48 T^{9} + \cdots + 101617664)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
show more
show less