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Properties

Label 5.22.b.a
Level $5$
Weight $22$
Character orbit 5.b
Analytic conductor $13.974$
Analytic rank $0$
Dimension $10$
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] = [5,22,Mod(4,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.4");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 5.b (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: \(13.9738672144\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 3780655 x^{8} + 4653816871660 x^{6} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{10}\cdot 5^{19}\cdot 7^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} - 6 \beta_1) q^{3} + (\beta_{2} - 927372) q^{4} + ( - \beta_{5} + 2 \beta_{4} + \cdots - 2517597) q^{5}+ \cdots + (\beta_{7} + 16 \beta_{5} + \cdots - 4679690553) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} - 6 \beta_1) q^{3} + (\beta_{2} - 927372) q^{4} + ( - \beta_{5} + 2 \beta_{4} + \cdots - 2517597) q^{5}+ \cdots + (138692384055 \beta_{9} + \cdots - 22\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 9273720 q^{4} - 25175970 q^{5} + 190183320 q^{6} - 46796905530 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 9273720 q^{4} - 25175970 q^{5} + 190183320 q^{6} - 46796905530 q^{9} + 42469996280 q^{10} + 150626450520 q^{11} + 1196972791560 q^{14} + 28918735560 q^{15} + 13236859984160 q^{16} - 111339219544600 q^{19} + 210425436114840 q^{20} + 153512457036120 q^{21} - 925398618703200 q^{24} + 233060326559650 q^{25} - 621077824626480 q^{26} - 26\!\cdots\!00 q^{29}+ \cdots - 22\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 3780655 x^{8} + 4653816871660 x^{6} + \cdots + 14\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} + 3024524 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10621 \nu^{8} - 22795614323 \nu^{6} + \cdots + 18\!\cdots\!20 ) / 56\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2433333971 \nu^{9} + \cdots - 17\!\cdots\!04 \nu ) / 10\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 985090675420483 \nu^{9} + \cdots + 17\!\cdots\!16 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35\!\cdots\!89 \nu^{9} + \cdots + 75\!\cdots\!72 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 985090675420483 \nu^{9} + \cdots + 23\!\cdots\!84 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 52\!\cdots\!99 \nu^{9} + \cdots - 46\!\cdots\!48 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 428771300096059 \nu^{9} + \cdots + 83\!\cdots\!88 ) / 26\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 3024524 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} - 2\beta_{6} + 46\beta_{5} - 4793\beta_{4} - 11\beta_{2} - 5273588\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 255 \beta_{9} + 11 \beta_{7} + 255 \beta_{6} + 29756 \beta_{5} - 1020 \beta_{4} - 6486 \beta_{3} + \cdots + 3988574788564 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 149720 \beta_{9} - 2443727 \beta_{8} + 3924990 \beta_{6} - 122464266 \beta_{5} + \cdots + 8430242099108 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 669466305 \beta_{9} - 24560565 \beta_{7} - 669466305 \beta_{6} - 78051060420 \beta_{5} + \cdots - 63\!\cdots\!24 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 415182465960 \beta_{9} + 5027251450389 \beta_{8} - 7601221412298 \beta_{6} + \cdots - 14\!\cdots\!96 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 14\!\cdots\!35 \beta_{9} + 51129798555159 \beta_{7} + \cdots + 11\!\cdots\!04 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 91\!\cdots\!00 \beta_{9} + \cdots + 27\!\cdots\!44 \beta_1 ) / 8 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
− 1378.95i
− 934.563i
− 867.225i
− 441.894i
− 241.650i
241.650i
441.894i
867.225i
934.563i
1378.95i
− 2757.90i 90721.2i −5.50886e6 −2.14352e7 − 4.16791e6i 2.50200e8 − 2.49542e8i 9.40915e9i 2.23002e9 −1.14947e10 + 5.91160e10i
4.2 − 1869.13i − 163448.i −1.39648e6 −1.04751e7 + 1.91601e7i −3.05505e8 1.05403e9i − 1.30964e9i −1.62549e10 3.58126e10 + 1.95794e10i
4.3 − 1734.45i − 21936.5i −911162. 2.08547e7 − 6.47443e6i −3.80478e7 − 8.17780e8i − 2.05704e9i 9.97914e9 −1.12296e10 − 3.61714e10i
4.4 − 883.789i 198986.i 1.31607e6 1.03034e7 + 1.92530e7i 1.75862e8 4.00513e8i − 3.01657e9i −2.91352e10 1.70156e10 − 9.10600e9i
4.5 − 483.300i 26035.4i 1.86357e6 −1.18358e7 − 1.83508e7i 1.25829e7 7.88368e8i − 1.91422e9i 9.78251e9 −8.86894e9 + 5.72022e9i
4.6 483.300i − 26035.4i 1.86357e6 −1.18358e7 + 1.83508e7i 1.25829e7 − 7.88368e8i 1.91422e9i 9.78251e9 −8.86894e9 − 5.72022e9i
4.7 883.789i − 198986.i 1.31607e6 1.03034e7 − 1.92530e7i 1.75862e8 − 4.00513e8i 3.01657e9i −2.91352e10 1.70156e10 + 9.10600e9i
4.8 1734.45i 21936.5i −911162. 2.08547e7 + 6.47443e6i −3.80478e7 8.17780e8i 2.05704e9i 9.97914e9 −1.12296e10 + 3.61714e10i
4.9 1869.13i 163448.i −1.39648e6 −1.04751e7 − 1.91601e7i −3.05505e8 − 1.05403e9i 1.30964e9i −1.62549e10 3.58126e10 − 1.95794e10i
4.10 2757.90i − 90721.2i −5.50886e6 −2.14352e7 + 4.16791e6i 2.50200e8 2.49542e8i − 9.40915e9i 2.23002e9 −1.14947e10 − 5.91160e10i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.22.b.a ✓ 10
3.b odd 2 1 45.22.b.b 10
4.b odd 2 1 80.22.c.a 10
5.b even 2 1 inner 5.22.b.a ✓ 10
5.c odd 4 2 25.22.a.f 10
15.d odd 2 1 45.22.b.b 10
20.d odd 2 1 80.22.c.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.22.b.a ✓ 10 1.a even 1 1 trivial
5.22.b.a ✓ 10 5.b even 2 1 inner
25.22.a.f 10 5.c odd 4 2
45.22.b.b 10 3.b odd 2 1
45.22.b.b 10 15.d odd 2 1
80.22.c.a 10 4.b odd 2 1
80.22.c.a 10 20.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{22}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 46\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots - 21\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 83\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 12\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 50\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 69\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 63\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 37\!\cdots\!24 \) Copy content Toggle raw display
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