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Properties

Label 20.5.b.a
Level $20$
Weight $5$
Character orbit 20.b
Analytic conductor $2.067$
Analytic rank $0$
Dimension $8$
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] = [20,5,Mod(11,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.11");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 20.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: \(2.06739926168\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.246034965625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 7x^{6} - 21x^{5} + 49x^{4} - 84x^{3} + 112x^{2} - 192x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + \beta_{3} q^{3} + (\beta_{2} + \beta_1 - 2) q^{4} - \beta_{4} q^{5} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots + 6) q^{6}+ \cdots + ( - 4 \beta_{6} + 4 \beta_{5} + \cdots - 41) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + \beta_{3} q^{3} + (\beta_{2} + \beta_1 - 2) q^{4} - \beta_{4} q^{5} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots + 6) q^{6}+ \cdots + (216 \beta_{7} - 66 \beta_{6} + \cdots - 102) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{2} - 20 q^{4} + 48 q^{6} + 216 q^{8} - 328 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{2} - 20 q^{4} + 48 q^{6} + 216 q^{8} - 328 q^{9} - 50 q^{10} - 200 q^{12} + 352 q^{13} - 168 q^{14} - 272 q^{16} - 48 q^{17} + 286 q^{18} - 300 q^{20} + 16 q^{21} + 800 q^{22} + 1552 q^{24} + 1000 q^{25} - 2172 q^{26} + 40 q^{28} + 1200 q^{29} + 1400 q^{30} - 2304 q^{32} - 1120 q^{33} - 2132 q^{34} - 1044 q^{36} - 5728 q^{37} - 3360 q^{38} - 2200 q^{40} + 4896 q^{41} + 12120 q^{42} + 7920 q^{44} - 400 q^{45} + 728 q^{46} + 8640 q^{48} - 5768 q^{49} + 750 q^{50} - 12488 q^{52} + 2592 q^{53} - 17776 q^{54} + 48 q^{56} + 3840 q^{57} - 7428 q^{58} - 9800 q^{60} + 7936 q^{61} + 25680 q^{62} + 18880 q^{64} - 1200 q^{65} - 8080 q^{66} + 2712 q^{68} - 2256 q^{69} + 12000 q^{70} - 36264 q^{72} - 14448 q^{73} - 18492 q^{74} + 12000 q^{76} + 2400 q^{77} - 14480 q^{78} - 13200 q^{80} - 936 q^{81} + 27412 q^{82} + 50464 q^{84} + 11200 q^{85} - 7392 q^{86} + 18080 q^{88} + 23760 q^{89} + 19350 q^{90} - 52680 q^{92} + 11360 q^{93} - 43368 q^{94} + 2688 q^{96} - 4368 q^{97} - 21474 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 7x^{6} - 21x^{5} + 49x^{4} - 84x^{3} + 112x^{2} - 192x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 2\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 11\nu^{6} + 15\nu^{5} - 29\nu^{4} + 41\nu^{3} - 76\nu^{2} + 256 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 23\nu^{5} + 69\nu^{4} - 97\nu^{3} + 228\nu^{2} - 384\nu + 480 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 7\nu^{5} + 21\nu^{4} - 49\nu^{3} + 52\nu^{2} - 80\nu + 136 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + \nu^{6} - \nu^{5} + 7\nu^{4} - 7\nu^{3} + 2\nu^{2} + 16\nu + 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 9\nu^{6} + 37\nu^{5} - 47\nu^{4} + 131\nu^{3} - 204\nu^{2} + 224\nu - 208 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} - 3\beta_{5} + 2\beta_{4} - 2\beta_{3} - 5\beta _1 + 26 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} + \beta_{6} - \beta_{5} + 6\beta_{4} - 2\beta_{3} - 4\beta_{2} + 11\beta _1 - 14 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{7} + 5\beta_{5} - 2\beta_{4} + 5\beta_{2} - 3\beta _1 - 34 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{7} - \beta_{6} + 3\beta_{5} - 6\beta_{4} - 14\beta_{3} + 4\beta_{2} - 29\beta _1 + 142 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6\beta_{7} - 17\beta_{6} + 5\beta_{5} + 10\beta_{4} - 14\beta_{3} - 13\beta_{2} + 64\beta _1 + 160 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−1.02661 − 1.71641i
−1.02661 + 1.71641i
−0.641015 − 1.89449i
−0.641015 + 1.89449i
1.21760 − 1.58665i
1.21760 + 1.58665i
1.95003 − 0.444269i
1.95003 + 0.444269i
−2.05323 − 3.43282i 15.5779i −7.56853 + 14.0967i 11.1803 53.4761 − 31.9849i 37.6230i 63.9314 − 2.96235i −161.671 −22.9558 − 38.3801i
11.2 −2.05323 + 3.43282i − 15.5779i −7.56853 − 14.0967i 11.1803 53.4761 + 31.9849i − 37.6230i 63.9314 + 2.96235i −161.671 −22.9558 + 38.3801i
11.3 −1.28203 − 3.78898i − 8.14153i −12.7128 + 9.71518i −11.1803 −30.8481 + 10.4377i − 63.6032i 53.1089 + 35.7134i 14.7155 14.3335 + 42.3621i
11.4 −1.28203 + 3.78898i 8.14153i −12.7128 − 9.71518i −11.1803 −30.8481 − 10.4377i 63.6032i 53.1089 − 35.7134i 14.7155 14.3335 − 42.3621i
11.5 2.43519 − 3.17330i − 3.20523i −4.13968 − 15.4552i 11.1803 −10.1712 − 7.80536i 30.6227i −59.1249 − 24.4999i 70.7265 27.2263 − 35.4786i
11.6 2.43519 + 3.17330i 3.20523i −4.13968 + 15.4552i 11.1803 −10.1712 + 7.80536i − 30.6227i −59.1249 + 24.4999i 70.7265 27.2263 + 35.4786i
11.7 3.90006 − 0.888538i 12.9912i 14.4210 − 6.93071i −11.1803 11.5432 + 50.6665i − 78.0345i 50.0846 − 39.8438i −87.7712 −43.6040 + 9.93415i
11.8 3.90006 + 0.888538i − 12.9912i 14.4210 + 6.93071i −11.1803 11.5432 − 50.6665i 78.0345i 50.0846 + 39.8438i −87.7712 −43.6040 − 9.93415i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.5.b.a ✓ 8
3.b odd 2 1 180.5.c.a 8
4.b odd 2 1 inner 20.5.b.a ✓ 8
5.b even 2 1 100.5.b.c 8
5.c odd 4 2 100.5.d.c 16
8.b even 2 1 320.5.b.d 8
8.d odd 2 1 320.5.b.d 8
12.b even 2 1 180.5.c.a 8
20.d odd 2 1 100.5.b.c 8
20.e even 4 2 100.5.d.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.5.b.a ✓ 8 1.a even 1 1 trivial
20.5.b.a ✓ 8 4.b odd 2 1 inner
100.5.b.c 8 5.b even 2 1
100.5.b.c 8 20.d odd 2 1
100.5.d.c 16 5.c odd 4 2
100.5.d.c 16 20.e even 4 2
180.5.c.a 8 3.b odd 2 1
180.5.c.a 8 12.b even 2 1
320.5.b.d 8 8.b even 2 1
320.5.b.d 8 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(20, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( T^{8} + 488 T^{6} + \cdots + 27889920 \) Copy content Toggle raw display
$5$ \( (T^{2} - 125)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 32698357408000 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 204994355200000 \) Copy content Toggle raw display
$13$ \( (T^{4} - 176 T^{3} + \cdots - 8015600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 24 T^{3} + \cdots + 688291600)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( (T^{4} - 600 T^{3} + \cdots - 98595968624)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 6304245167600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 1586334915856)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 43\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( (T^{4} - 1296 T^{3} + \cdots + 486559363600)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 17262940540816)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 39753982895600)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 26555598339856)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
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