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Properties

Label 100.5.b.c
Level $100$
Weight $5$
Character orbit 100.b
Analytic conductor $10.337$
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] = [100,5,Mod(51,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, 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("100.51");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 100.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: \(10.3369963084\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + \beta_{3} q^{3} + (\beta_{2} + \beta_1 - 2) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 6) q^{6}+ \cdots + (\beta_{7} + 4 \beta_{5} - 4 \beta_{4} + \cdots - 41) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + \beta_{3} q^{3} + (\beta_{2} + \beta_1 - 2) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 6) q^{6}+ \cdots + (216 \beta_{6} + 252 \beta_{5} + \cdots - 102) q^{99}+O(q^{100}) \) 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 \( 8 q - 6 q^{2} - 20 q^{4} + 48 q^{6} - 216 q^{8} - 328 q^{9} + 200 q^{12} - 352 q^{13} - 168 q^{14} - 272 q^{16} + 48 q^{17} - 286 q^{18} + 16 q^{21} - 800 q^{22} + 1552 q^{24} - 2172 q^{26} - 40 q^{28} + 1200 q^{29} + 2304 q^{32} + 1120 q^{33} - 2132 q^{34} - 1044 q^{36} + 5728 q^{37} + 3360 q^{38} + 4896 q^{41} - 12120 q^{42} + 7920 q^{44} + 728 q^{46} - 8640 q^{48} - 5768 q^{49} + 12488 q^{52} - 2592 q^{53} - 17776 q^{54} + 48 q^{56} - 3840 q^{57} + 7428 q^{58} + 7936 q^{61} - 25680 q^{62} + 18880 q^{64} - 8080 q^{66} - 2712 q^{68} - 2256 q^{69} + 36264 q^{72} + 14448 q^{73} - 18492 q^{74} + 12000 q^{76} - 2400 q^{77} + 14480 q^{78} - 936 q^{81} - 27412 q^{82} + 50464 q^{84} - 7392 q^{86} - 18080 q^{88} + 23760 q^{89} + 52680 q^{92} - 11360 q^{93} - 43368 q^{94} + 2688 q^{96} + 4368 q^{97} + 21474 q^{98}+O(q^{100}) \) 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
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + \nu^{6} - \nu^{5} + 7\nu^{4} - 7\nu^{3} + 2\nu^{2} + 16\nu + 24 ) / 4 \) 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
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 9\nu^{6} + 37\nu^{5} - 47\nu^{4} + 131\nu^{3} - 204\nu^{2} + 224\nu - 208 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 23\nu^{5} + 69\nu^{4} - 97\nu^{3} + 228\nu^{2} - 416\nu + 496 ) / 16 \) 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
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 3\beta_{5} + \beta_{4} + 2\beta_{3} - 4\beta _1 + 26 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} + 2\beta_{3} - 4\beta_{2} + 14\beta _1 - 14 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} + 5\beta_{5} + 5\beta_{2} - 4\beta _1 - 34 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{7} + 2\beta_{6} + 3\beta_{5} - \beta_{4} + 14\beta_{3} + 4\beta_{2} - 32\beta _1 + 142 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5\beta_{7} + 6\beta_{6} + 5\beta_{5} - 17\beta_{4} + 14\beta_{3} - 13\beta_{2} + 69\beta _1 + 160 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
1.95003 + 0.444269i
1.95003 − 0.444269i
1.21760 + 1.58665i
1.21760 − 1.58665i
−0.641015 + 1.89449i
−0.641015 − 1.89449i
−1.02661 + 1.71641i
−1.02661 − 1.71641i
−3.90006 − 0.888538i 12.9912i 14.4210 + 6.93071i 0 11.5432 − 50.6665i − 78.0345i −50.0846 − 39.8438i −87.7712 0
51.2 −3.90006 + 0.888538i − 12.9912i 14.4210 − 6.93071i 0 11.5432 + 50.6665i 78.0345i −50.0846 + 39.8438i −87.7712 0
51.3 −2.43519 − 3.17330i − 3.20523i −4.13968 + 15.4552i 0 −10.1712 + 7.80536i 30.6227i 59.1249 − 24.4999i 70.7265 0
51.4 −2.43519 + 3.17330i 3.20523i −4.13968 − 15.4552i 0 −10.1712 − 7.80536i − 30.6227i 59.1249 + 24.4999i 70.7265 0
51.5 1.28203 − 3.78898i − 8.14153i −12.7128 − 9.71518i 0 −30.8481 − 10.4377i − 63.6032i −53.1089 + 35.7134i 14.7155 0
51.6 1.28203 + 3.78898i 8.14153i −12.7128 + 9.71518i 0 −30.8481 + 10.4377i 63.6032i −53.1089 − 35.7134i 14.7155 0
51.7 2.05323 − 3.43282i 15.5779i −7.56853 − 14.0967i 0 53.4761 + 31.9849i 37.6230i −63.9314 − 2.96235i −161.671 0
51.8 2.05323 + 3.43282i − 15.5779i −7.56853 + 14.0967i 0 53.4761 − 31.9849i − 37.6230i −63.9314 + 2.96235i −161.671 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.8
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 100.5.b.c 8
4.b odd 2 1 inner 100.5.b.c 8
5.b even 2 1 20.5.b.a ✓ 8
5.c odd 4 2 100.5.d.c 16
15.d odd 2 1 180.5.c.a 8
20.d odd 2 1 20.5.b.a ✓ 8
20.e even 4 2 100.5.d.c 16
40.e odd 2 1 320.5.b.d 8
40.f even 2 1 320.5.b.d 8
60.h even 2 1 180.5.c.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.5.b.a ✓ 8 5.b even 2 1
20.5.b.a ✓ 8 20.d odd 2 1
100.5.b.c 8 1.a even 1 1 trivial
100.5.b.c 8 4.b odd 2 1 inner
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 15.d odd 2 1
180.5.c.a 8 60.h even 2 1
320.5.b.d 8 40.e odd 2 1
320.5.b.d 8 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(100, [\chi])\):

\( T_{3}^{8} + 488T_{3}^{6} + 73136T_{3}^{4} + 3415680T_{3}^{2} + 27889920 \) Copy content Toggle raw display
\( T_{13}^{4} + 176T_{13}^{3} - 36616T_{13}^{2} - 2659520T_{13} - 8015600 \) Copy content Toggle raw display

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
$5$ \( T^{8} \) 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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