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Properties

Label 5.21.c.a
Level $5$
Weight $21$
Character orbit 5.c
Analytic conductor $12.676$
Analytic rank $0$
Dimension $18$
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] = [5,21,Mod(2,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 21, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.2");
 
S:= CuspForms(chi, 21);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 21 \)
Character orbit: \([\chi]\) \(=\) 5.c (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: \(12.6756882551\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 7326009 x^{16} + 21914129354136 x^{14} + \cdots + 79\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{16}\cdot 5^{32} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{5} - 1636 \beta_{4} + \cdots + 1636) q^{3}+ \cdots + (2 \beta_{17} - \beta_{16} + \beta_{15} + \cdots + 35) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{5} - 1636 \beta_{4} + \cdots + 1636) q^{3}+ \cdots + (79672849856 \beta_{17} + \cdots + 420033902864) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{2} + 29448 q^{3} - 7302140 q^{5} + 19792536 q^{6} - 585532752 q^{7} + 930113700 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{2} + 29448 q^{3} - 7302140 q^{5} + 19792536 q^{6} - 585532752 q^{7} + 930113700 q^{8} - 17138778090 q^{10} - 6506343064 q^{11} + 140311795848 q^{12} - 369354655602 q^{13} + 998626204320 q^{15} - 6345876020232 q^{16} + 6508314764998 q^{17} - 11265991705902 q^{18} + 38356516779780 q^{20} - 97402974719064 q^{21} + 161054386758096 q^{22} - 81655963656152 q^{23} + 33913283845350 q^{25} + 234104101103636 q^{26} - 511935279143400 q^{27} + 772213614545352 q^{28} - 877127470680480 q^{30} + 17\!\cdots\!36 q^{31}+ \cdots - 27\!\cdots\!98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 7326009 x^{16} + 21914129354136 x^{14} + \cdots + 79\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 40\!\cdots\!31 \nu^{16} + \cdots - 18\!\cdots\!36 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 40\!\cdots\!31 \nu^{16} + \cdots - 18\!\cdots\!36 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 62\!\cdots\!67 \nu^{16} + \cdots + 25\!\cdots\!48 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32\!\cdots\!49 \nu^{17} + \cdots - 10\!\cdots\!44 \nu ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30\!\cdots\!29 \nu^{17} + \cdots - 21\!\cdots\!20 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!29 \nu^{17} + \cdots - 21\!\cdots\!20 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 64\!\cdots\!99 \nu^{17} + \cdots - 27\!\cdots\!44 \nu ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 94\!\cdots\!85 \nu^{17} + \cdots - 79\!\cdots\!56 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31\!\cdots\!33 \nu^{17} + \cdots - 27\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 82\!\cdots\!71 \nu^{17} + \cdots + 82\!\cdots\!76 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!97 \nu^{17} + \cdots + 14\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!27 \nu^{17} + \cdots + 10\!\cdots\!96 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 23\!\cdots\!53 \nu^{17} + \cdots - 43\!\cdots\!92 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 92\!\cdots\!29 \nu^{17} + \cdots + 97\!\cdots\!12 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10\!\cdots\!39 \nu^{17} + \cdots + 56\!\cdots\!68 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 71\!\cdots\!01 \nu^{17} + \cdots + 46\!\cdots\!56 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 38\!\cdots\!81 \nu^{17} + \cdots - 26\!\cdots\!36 ) / 29\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} + 2\beta_{5} + \beta_{3} - 31\beta_{2} - 31\beta _1 - 1627995 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{17} - \beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} + 4 \beta_{11} - 2 \beta_{10} + \cdots + 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 302 \beta_{17} - 253 \beta_{16} - 253 \beta_{15} - 935 \beta_{14} + 935 \beta_{13} + \cdots + 4374290096441 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5340724 \beta_{17} + 2252047 \beta_{16} - 2252047 \beta_{15} - 2063509 \beta_{14} - 2063509 \beta_{13} + \cdots - 29695127 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 752221990 \beta_{17} + 880920065 \beta_{16} + 880920065 \beta_{15} + 2878617875 \beta_{14} + \cdots - 69\!\cdots\!61 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11639024494588 \beta_{17} - 4319612645399 \beta_{16} + 4319612645399 \beta_{15} + 3821660837909 \beta_{14} + \cdots + 57937332066375 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 15\!\cdots\!18 \beta_{17} + \cdots + 11\!\cdots\!69 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 23\!\cdots\!36 \beta_{17} + \cdots - 11\!\cdots\!63 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 31\!\cdots\!10 \beta_{17} + \cdots - 21\!\cdots\!49 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 47\!\cdots\!80 \beta_{17} + \cdots + 23\!\cdots\!35 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 64\!\cdots\!10 \beta_{17} + \cdots + 38\!\cdots\!25 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 94\!\cdots\!52 \beta_{17} + \cdots - 48\!\cdots\!75 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 13\!\cdots\!82 \beta_{17} + \cdots - 70\!\cdots\!61 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 18\!\cdots\!04 \beta_{17} + \cdots + 99\!\cdots\!07 ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 28\!\cdots\!10 \beta_{17} + \cdots + 13\!\cdots\!61 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 36\!\cdots\!08 \beta_{17} + \cdots - 20\!\cdots\!95 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
1286.49i
1196.41i
670.386i
439.799i
− 3.77310i
− 569.040i
− 782.816i
− 836.109i
− 1400.36i
− 1286.49i
− 1196.41i
− 670.386i
− 439.799i
3.77310i
569.040i
782.816i
836.109i
1400.36i
−1286.49 − 1286.49i 59231.4 − 59231.4i 2.26156e6i 2.84523e6 − 9.34195e6i −1.52402e8 −2.38026e8 − 2.38026e8i 1.56049e9 − 1.56049e9i − 3.52992e9i −1.56787e10 + 8.35799e9i
2.2 −1196.41 − 1196.41i −49580.3 + 49580.3i 1.81424e6i −9.20993e6 + 3.24723e6i 1.18637e8 2.14321e8 + 2.14321e8i 9.16051e8 − 9.16051e8i − 1.42964e9i 1.49039e10 + 7.13386e9i
2.3 −670.386 − 670.386i 17254.5 − 17254.5i − 149742.i 4.96780e6 + 8.40764e6i −2.31343e7 9.51724e7 + 9.51724e7i −8.03335e8 + 8.03335e8i 2.89135e9i 2.30602e9 − 8.96670e9i
2.4 −439.799 − 439.799i −48235.4 + 48235.4i − 661730.i 5.04058e6 − 8.36421e6i 4.24277e7 −2.17438e8 − 2.17438e8i −7.52190e8 + 7.52190e8i − 1.16652e9i −5.89541e9 + 1.46173e9i
2.5 3.77310 + 3.77310i 38858.9 − 38858.9i − 1.04855e6i −8.90462e6 − 4.00939e6i 293237. 4.46902e7 + 4.46902e7i 7.91266e6 − 7.91266e6i 4.66753e8i −1.84702e7 − 4.87259e7i
2.6 569.040 + 569.040i −56179.4 + 56179.4i − 400963.i −4.35767e6 + 8.73946e6i −6.39367e7 −1.35277e8 − 1.35277e8i 8.24846e8 − 8.24846e8i − 2.82548e9i −7.45279e9 + 2.49341e9i
2.7 782.816 + 782.816i −12801.5 + 12801.5i 177026.i 7.95503e6 − 5.66435e6i −2.00424e7 3.15497e8 + 3.15497e8i 6.82263e8 − 6.82263e8i 3.15903e9i 1.06615e10 + 1.79318e9i
2.8 836.109 + 836.109i 68486.6 − 68486.6i 349580.i 7.10218e6 + 6.70272e6i 1.14524e8 −2.53016e8 − 2.53016e8i 5.84437e8 − 5.84437e8i − 5.89403e9i 3.33994e8 + 1.15424e10i
2.9 1400.36 + 1400.36i −2310.64 + 2310.64i 2.87341e6i −9.08967e6 − 3.57006e6i −6.47145e6 −1.18690e8 − 1.18690e8i −2.55542e9 + 2.55542e9i 3.47611e9i −7.72941e9 − 1.77281e10i
3.1 −1286.49 + 1286.49i 59231.4 + 59231.4i − 2.26156e6i 2.84523e6 + 9.34195e6i −1.52402e8 −2.38026e8 + 2.38026e8i 1.56049e9 + 1.56049e9i 3.52992e9i −1.56787e10 − 8.35799e9i
3.2 −1196.41 + 1196.41i −49580.3 − 49580.3i − 1.81424e6i −9.20993e6 − 3.24723e6i 1.18637e8 2.14321e8 − 2.14321e8i 9.16051e8 + 9.16051e8i 1.42964e9i 1.49039e10 − 7.13386e9i
3.3 −670.386 + 670.386i 17254.5 + 17254.5i 149742.i 4.96780e6 − 8.40764e6i −2.31343e7 9.51724e7 − 9.51724e7i −8.03335e8 − 8.03335e8i − 2.89135e9i 2.30602e9 + 8.96670e9i
3.4 −439.799 + 439.799i −48235.4 − 48235.4i 661730.i 5.04058e6 + 8.36421e6i 4.24277e7 −2.17438e8 + 2.17438e8i −7.52190e8 − 7.52190e8i 1.16652e9i −5.89541e9 − 1.46173e9i
3.5 3.77310 − 3.77310i 38858.9 + 38858.9i 1.04855e6i −8.90462e6 + 4.00939e6i 293237. 4.46902e7 − 4.46902e7i 7.91266e6 + 7.91266e6i − 4.66753e8i −1.84702e7 + 4.87259e7i
3.6 569.040 − 569.040i −56179.4 − 56179.4i 400963.i −4.35767e6 − 8.73946e6i −6.39367e7 −1.35277e8 + 1.35277e8i 8.24846e8 + 8.24846e8i 2.82548e9i −7.45279e9 − 2.49341e9i
3.7 782.816 − 782.816i −12801.5 − 12801.5i − 177026.i 7.95503e6 + 5.66435e6i −2.00424e7 3.15497e8 − 3.15497e8i 6.82263e8 + 6.82263e8i − 3.15903e9i 1.06615e10 − 1.79318e9i
3.8 836.109 − 836.109i 68486.6 + 68486.6i − 349580.i 7.10218e6 − 6.70272e6i 1.14524e8 −2.53016e8 + 2.53016e8i 5.84437e8 + 5.84437e8i 5.89403e9i 3.33994e8 − 1.15424e10i
3.9 1400.36 − 1400.36i −2310.64 − 2310.64i − 2.87341e6i −9.08967e6 + 3.57006e6i −6.47145e6 −1.18690e8 + 1.18690e8i −2.55542e9 − 2.55542e9i − 3.47611e9i −7.72941e9 + 1.77281e10i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.9
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.21.c.a ✓ 18
3.b odd 2 1 45.21.g.a 18
5.b even 2 1 25.21.c.b 18
5.c odd 4 1 inner 5.21.c.a ✓ 18
5.c odd 4 1 25.21.c.b 18
15.e even 4 1 45.21.g.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.21.c.a ✓ 18 1.a even 1 1 trivial
5.21.c.a ✓ 18 5.c odd 4 1 inner
25.21.c.b 18 5.b even 2 1
25.21.c.b 18 5.c odd 4 1
45.21.g.a 18 3.b odd 2 1
45.21.g.a 18 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{21}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 40\!\cdots\!12 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 59\!\cdots\!12 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 18\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( (T^{9} + \cdots + 55\!\cdots\!88)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 10\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 15\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 50\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 13\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 50\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots + 20\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 54\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 79\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 24\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 48\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 51\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 81\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 61\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 44\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
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