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Properties

Label 1421.2.b.l
Level $1421$
Weight $2$
Character orbit 1421.b
Analytic conductor $11.347$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1421,2,Mod(1275,1421)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1421.1275"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1421, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1421 = 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1421.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,-20,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.3467421272\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 10 x^{19} + 61 x^{18} - 256 x^{17} + 871 x^{16} - 2474 x^{15} + 5887 x^{14} - 11788 x^{13} + \cdots + 784 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{16} q^{2} + \beta_{13} q^{3} + ( - \beta_{8} - 1) q^{4} + \beta_{2} q^{5} - \beta_{11} q^{6} + (\beta_{17} + 2 \beta_{16} + \beta_{15}) q^{8} + ( - \beta_{9} + \beta_{8} + \beta_{4} - 1) q^{9}+ \cdots + (\beta_{16} + \beta_{15} + \cdots + \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{4} - 12 q^{9} + 36 q^{16} + 48 q^{22} + 56 q^{23} + 84 q^{25} + 12 q^{29} + 24 q^{30} - 68 q^{36} - 40 q^{51} - 32 q^{53} + 128 q^{57} - 64 q^{58} - 196 q^{64} + 104 q^{65} - 96 q^{71} - 40 q^{74}+ \cdots - 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 10 x^{19} + 61 x^{18} - 256 x^{17} + 871 x^{16} - 2474 x^{15} + 5887 x^{14} - 11788 x^{13} + \cdots + 784 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 54\!\cdots\!13 \nu^{19} + \cdots + 31\!\cdots\!28 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!95 \nu^{19} + \cdots - 10\!\cdots\!72 ) / 28\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\!\cdots\!79 \nu^{19} + \cdots + 34\!\cdots\!80 ) / 54\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25\!\cdots\!17 \nu^{19} + \cdots - 11\!\cdots\!28 ) / 54\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 48\!\cdots\!21 \nu^{19} + \cdots - 12\!\cdots\!04 ) / 75\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 47\!\cdots\!87 \nu^{19} + \cdots + 21\!\cdots\!88 ) / 71\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!46 \nu^{19} + \cdots + 16\!\cdots\!44 ) / 40\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 51\!\cdots\!31 \nu^{19} + \cdots + 20\!\cdots\!24 ) / 54\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 26\!\cdots\!27 \nu^{19} + \cdots + 14\!\cdots\!92 ) / 27\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 39\!\cdots\!07 \nu^{19} + \cdots + 27\!\cdots\!52 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 38\!\cdots\!93 \nu^{19} + \cdots - 43\!\cdots\!28 ) / 28\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 71\!\cdots\!14 \nu^{19} + \cdots + 29\!\cdots\!16 ) / 37\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 39\!\cdots\!63 \nu^{19} + \cdots - 14\!\cdots\!28 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 45\!\cdots\!99 \nu^{19} + \cdots - 22\!\cdots\!84 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 52\!\cdots\!89 \nu^{19} + \cdots - 18\!\cdots\!24 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 28\!\cdots\!59 \nu^{19} + \cdots - 12\!\cdots\!36 ) / 75\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 35\!\cdots\!43 \nu^{19} + \cdots - 15\!\cdots\!48 ) / 75\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 11\!\cdots\!34 \nu^{19} + \cdots - 49\!\cdots\!44 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 12\!\cdots\!21 \nu^{19} + \cdots - 53\!\cdots\!16 ) / 20\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{16} - \beta_{15} + 2\beta_{13} - 2\beta_{12} + \beta_{11} + \beta_{10} - \beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{19} - 2 \beta_{18} - \beta_{16} + \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{11} - \beta_{9} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{18} + 2 \beta_{17} - 7 \beta_{16} + 5 \beta_{15} + 8 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \cdots - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 7 \beta_{19} + 14 \beta_{18} + 2 \beta_{17} - \beta_{16} + 2 \beta_{15} - 3 \beta_{14} - 12 \beta_{13} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 36 \beta_{19} + 104 \beta_{18} - 50 \beta_{17} + 145 \beta_{16} + \beta_{15} - 96 \beta_{14} + \cdots + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 73 \beta_{19} + 162 \beta_{18} - 68 \beta_{17} + 181 \beta_{16} + 30 \beta_{15} - 97 \beta_{14} + \cdots + 166 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 300 \beta_{19} + 324 \beta_{18} + 134 \beta_{17} - 365 \beta_{16} + 267 \beta_{15} + 308 \beta_{14} + \cdots + 1710 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 973 \beta_{19} - 2198 \beta_{18} + 754 \beta_{17} - 1841 \beta_{16} - 266 \beta_{15} + 1133 \beta_{14} + \cdots + 2932 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 10672 \beta_{19} - 19620 \beta_{18} + 6394 \beta_{17} - 14141 \beta_{16} - 9485 \beta_{15} + 2992 \beta_{14} + \cdots + 3026 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 8227 \beta_{19} - 13502 \beta_{18} + 6752 \beta_{17} - 14045 \beta_{16} - 12488 \beta_{15} + \cdots - 45100 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 48680 \beta_{19} + 87620 \beta_{18} - 32522 \beta_{17} + 75197 \beta_{16} + 51913 \beta_{15} + \cdots - 367706 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 223891 \beta_{19} + 402206 \beta_{18} - 152638 \beta_{17} + 342927 \beta_{16} + 218794 \beta_{15} + \cdots - 114394 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1414732 \beta_{19} + 2651192 \beta_{18} - 927962 \beta_{17} + 2124373 \beta_{16} + 989485 \beta_{15} + \cdots + 2836810 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 156305 \beta_{19} + 406650 \beta_{18} - 238532 \beta_{17} + 603077 \beta_{16} - 107834 \beta_{15} + \cdots + 7097566 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 15443436 \beta_{19} - 28929164 \beta_{18} + 10530470 \beta_{17} - 23901305 \beta_{16} - 11772489 \beta_{15} + \cdots + 33437846 ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 35798445 \beta_{19} - 67873102 \beta_{18} + 27159762 \beta_{17} - 62035001 \beta_{16} - 27569658 \beta_{15} + \cdots - 12012956 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 138645336 \beta_{19} - 258554148 \beta_{18} + 98875882 \beta_{17} - 223883633 \beta_{16} + \cdots - 552213174 ) / 4 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 151182349 \beta_{19} + 291435194 \beta_{18} - 119993808 \beta_{17} + 277025763 \beta_{16} + \cdots - 1110336372 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 3222172448 \beta_{19} + 6011272996 \beta_{18} - 2260251146 \beta_{17} + 5132364745 \beta_{16} + \cdots - 3001185810 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1421\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1277\)
\(\chi(n)\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1275.1
0.823114 − 1.06873i
0.176886 + 0.0447042i
2.24023 − 2.77589i
−1.24023 − 2.46921i
0.711468 − 0.241861i
0.288532 + 2.85463i
2.26418 − 0.142127i
−1.26418 + 0.370207i
1.34271 − 1.93018i
−0.342705 + 0.680469i
−0.342705 − 0.680469i
1.34271 + 1.93018i
−1.26418 − 0.370207i
2.26418 + 0.142127i
0.288532 − 2.85463i
0.711468 + 0.241861i
−1.24023 + 2.46921i
2.24023 + 2.77589i
0.176886 − 0.0447042i
0.823114 + 1.06873i
− 2.77959i − 1.11343i −5.72610 3.58884 −3.09488 0 10.3570i 1.76027 − 9.97548i
1275.2 − 2.77959i 1.11343i −5.72610 −3.58884 3.09488 0 10.3570i 1.76027 9.97548i
1275.3 − 1.87376i − 0.306676i −1.51098 −1.29728 −0.574637 0 − 0.916302i 2.90595 2.43078i
1275.4 − 1.87376i 0.306676i −1.51098 1.29728 0.574637 0 − 0.916302i 2.90595 − 2.43078i
1275.5 − 1.52717i − 3.09649i −0.332235 −3.78449 −4.72885 0 − 2.54695i −6.58825 5.77954i
1275.6 − 1.52717i 3.09649i −0.332235 3.78449 4.72885 0 − 2.54695i −6.58825 − 5.77954i
1275.7 − 1.10638i − 0.512335i 0.775929 −3.07838 −0.566835 0 − 3.07123i 2.73751 3.40585i
1275.8 − 1.10638i 0.512335i 0.775929 3.07838 0.566835 0 − 3.07123i 2.73751 − 3.40585i
1275.9 − 0.454544i − 2.61065i 1.79339 2.76379 −1.18665 0 − 1.72426i −3.81548 − 1.25627i
1275.10 − 0.454544i 2.61065i 1.79339 −2.76379 1.18665 0 − 1.72426i −3.81548 1.25627i
1275.11 0.454544i − 2.61065i 1.79339 −2.76379 1.18665 0 1.72426i −3.81548 − 1.25627i
1275.12 0.454544i 2.61065i 1.79339 2.76379 −1.18665 0 1.72426i −3.81548 1.25627i
1275.13 1.10638i − 0.512335i 0.775929 3.07838 0.566835 0 3.07123i 2.73751 3.40585i
1275.14 1.10638i 0.512335i 0.775929 −3.07838 −0.566835 0 3.07123i 2.73751 − 3.40585i
1275.15 1.52717i − 3.09649i −0.332235 3.78449 4.72885 0 2.54695i −6.58825 5.77954i
1275.16 1.52717i 3.09649i −0.332235 −3.78449 −4.72885 0 2.54695i −6.58825 − 5.77954i
1275.17 1.87376i − 0.306676i −1.51098 1.29728 0.574637 0 0.916302i 2.90595 2.43078i
1275.18 1.87376i 0.306676i −1.51098 −1.29728 −0.574637 0 0.916302i 2.90595 − 2.43078i
1275.19 2.77959i − 1.11343i −5.72610 −3.58884 3.09488 0 − 10.3570i 1.76027 − 9.97548i
1275.20 2.77959i 1.11343i −5.72610 3.58884 −3.09488 0 − 10.3570i 1.76027 9.97548i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1275.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
29.b even 2 1 inner
203.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1421.2.b.l ✓ 20
7.b odd 2 1 inner 1421.2.b.l ✓ 20
29.b even 2 1 inner 1421.2.b.l ✓ 20
203.c odd 2 1 inner 1421.2.b.l ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1421.2.b.l ✓ 20 1.a even 1 1 trivial
1421.2.b.l ✓ 20 7.b odd 2 1 inner
1421.2.b.l ✓ 20 29.b even 2 1 inner
1421.2.b.l ✓ 20 203.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1421, [\chi])\):

\( T_{2}^{10} + 15T_{2}^{8} + 73T_{2}^{6} + 143T_{2}^{4} + 104T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{3}^{10} + 18T_{3}^{8} + 92T_{3}^{6} + 112T_{3}^{4} + 31T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{5}^{10} - 46T_{5}^{8} + 797T_{5}^{6} - 6342T_{5}^{4} + 21980T_{5}^{2} - 22472 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 15 T^{8} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{10} + 18 T^{8} + \cdots + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} - 46 T^{8} + \cdots - 22472)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( (T^{10} + 40 T^{8} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} - 118 T^{8} + \cdots - 3931208)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + 68 T^{8} + \cdots + 21632)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + 104 T^{8} + \cdots + 1059968)^{2} \) Copy content Toggle raw display
$23$ \( (T^{5} - 14 T^{4} + \cdots - 4192)^{4} \) Copy content Toggle raw display
$29$ \( (T^{10} - 6 T^{9} + \cdots + 20511149)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 96 T^{8} + \cdots + 1352)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 200 T^{8} + \cdots + 39337984)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 258 T^{8} + \cdots + 2097152)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 232 T^{8} + \cdots + 7139584)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + 282 T^{8} + \cdots + 7242818)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + 8 T^{4} + \cdots - 638)^{4} \) Copy content Toggle raw display
$59$ \( (T^{10} - 160 T^{8} + \cdots - 36992)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 252 T^{8} + \cdots + 23011328)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 155 T^{3} + \cdots + 3896)^{4} \) Copy content Toggle raw display
$71$ \( (T^{5} + 24 T^{4} + \cdots - 20776)^{4} \) Copy content Toggle raw display
$73$ \( (T^{10} + 446 T^{8} + \cdots + 298070528)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 476 T^{8} + \cdots + 1324377664)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} - 392 T^{8} + \cdots - 5068928)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 574 T^{8} + \cdots + 3986173472)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 554 T^{8} + \cdots + 415180928)^{2} \) Copy content Toggle raw display
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