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Properties

Label 100.5.b.d
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: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 10x^{6} - 12x^{5} - 32x^{4} - 192x^{3} + 2560x^{2} - 12288x + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{6} - \beta_{3} - 6) q^{6} + (\beta_{4} + \beta_{2} + 2 \beta_1) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_1 + 3) q^{8} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots - 17) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{6} - \beta_{3} - 6) q^{6} + (\beta_{4} + \beta_{2} + 2 \beta_1) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_1 + 3) q^{8} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots - 17) q^{9}+ \cdots + ( - 144 \beta_{7} + 18 \beta_{6} + \cdots + 432) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} - 11 q^{4} - 51 q^{6} + 27 q^{8} - 160 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} - 11 q^{4} - 51 q^{6} + 27 q^{8} - 160 q^{9} + 145 q^{12} - 176 q^{13} + 246 q^{14} + 193 q^{16} + 24 q^{17} + 682 q^{18} + 448 q^{21} + 1025 q^{22} - 449 q^{24} - 588 q^{26} - 2810 q^{28} - 1392 q^{29} - 1773 q^{32} + 2600 q^{33} + 2917 q^{34} + 4722 q^{36} - 256 q^{37} + 5505 q^{38} - 3384 q^{41} + 2790 q^{42} - 7155 q^{44} - 7726 q^{46} - 16035 q^{48} + 1000 q^{49} - 3236 q^{52} + 144 q^{53} + 15767 q^{54} + 12234 q^{56} - 6120 q^{57} + 20904 q^{58} + 1696 q^{61} + 11190 q^{62} - 12071 q^{64} - 20575 q^{66} - 31371 q^{68} - 8688 q^{69} - 23298 q^{72} - 9336 q^{73} + 22782 q^{74} + 21765 q^{76} + 5280 q^{77} + 53380 q^{78} - 264 q^{81} + 20869 q^{82} - 32978 q^{84} - 31656 q^{86} - 47245 q^{88} + 12168 q^{89} - 11190 q^{92} + 14960 q^{93} + 30396 q^{94} + 40839 q^{96} - 14736 q^{97} + 48177 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 10x^{6} - 12x^{5} - 32x^{4} - 192x^{3} + 2560x^{2} - 12288x + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - \nu^{6} + 18\nu^{5} + 116\nu^{4} - 80\nu^{3} + 256\nu^{2} - 3072\nu - 4096 ) / 4096 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} + 11\nu^{6} + 18\nu^{5} + 84\nu^{4} + 720\nu^{3} - 384\nu^{2} + 2048\nu + 2048 ) / 2048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 11\nu^{6} - 18\nu^{5} - 84\nu^{4} + 1328\nu^{3} + 384\nu^{2} + 4096 ) / 2048 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 29\nu^{6} + 86\nu^{5} - 228\nu^{4} + 144\nu^{3} - 768\nu^{2} - 13312\nu + 45056 ) / 4096 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 10\nu^{5} - 12\nu^{4} - 32\nu^{3} - 192\nu^{2} + 2048\nu - 10752 ) / 512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{7} - 4\beta_{6} - \beta_{5} + 3\beta_{4} + 20\beta_{3} - 2\beta_{2} + 7\beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18\beta_{7} + 20\beta_{6} + 3\beta_{5} - \beta_{4} + 60\beta_{3} + 6\beta_{2} + 39\beta _1 + 207 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -54\beta_{7} + 36\beta_{6} - 33\beta_{5} + 43\beta_{4} - 84\beta_{3} + 6\beta_{2} + 227\beta _1 - 1597 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 146\beta_{7} - 140\beta_{6} - 109\beta_{5} + 207\beta_{4} - 612\beta_{3} + 126\beta_{2} - 1705\beta _1 + 3879 \) 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
3.73903 + 1.42115i
3.73903 − 1.42115i
1.70077 + 3.62041i
1.70077 − 3.62041i
−0.450692 + 3.97453i
−0.450692 − 3.97453i
−3.48911 + 1.95604i
−3.48911 − 1.95604i
−3.73903 − 1.42115i 6.97928i 11.9607 + 10.6274i 0 9.91857 − 26.0957i 83.5849i −29.6183 − 56.7341i 32.2896 0
51.2 −3.73903 + 1.42115i − 6.97928i 11.9607 − 10.6274i 0 9.91857 + 26.0957i − 83.5849i −29.6183 + 56.7341i 32.2896 0
51.3 −1.70077 − 3.62041i − 15.8668i −10.2147 + 12.3150i 0 −57.4443 + 26.9858i 34.2627i 61.9583 + 16.0365i −170.755 0
51.4 −1.70077 + 3.62041i 15.8668i −10.2147 − 12.3150i 0 −57.4443 − 26.9858i − 34.2627i 61.9583 − 16.0365i −170.755 0
51.5 0.450692 − 3.97453i 8.38124i −15.5938 − 3.58258i 0 33.3115 + 3.77736i − 30.6984i −21.2670 + 60.3632i 10.7548 0
51.6 0.450692 + 3.97453i − 8.38124i −15.5938 + 3.58258i 0 33.3115 − 3.77736i 30.6984i −21.2670 − 60.3632i 10.7548 0
51.7 3.48911 − 1.95604i − 5.76967i 8.34780 − 13.6497i 0 −11.2857 − 20.1310i 1.11471i 2.42703 − 63.9540i 47.7109 0
51.8 3.48911 + 1.95604i 5.76967i 8.34780 + 13.6497i 0 −11.2857 + 20.1310i − 1.11471i 2.42703 + 63.9540i 47.7109 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.d ✓ 8
4.b odd 2 1 inner 100.5.b.d ✓ 8
5.b even 2 1 100.5.b.f yes 8
5.c odd 4 2 100.5.d.b 16
20.d odd 2 1 100.5.b.f yes 8
20.e even 4 2 100.5.d.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.5.b.d ✓ 8 1.a even 1 1 trivial
100.5.b.d ✓ 8 4.b odd 2 1 inner
100.5.b.f yes 8 5.b even 2 1
100.5.b.f yes 8 20.d odd 2 1
100.5.d.b 16 5.c odd 4 2
100.5.d.b 16 20.e even 4 2

Hecke kernels

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

\( T_{3}^{8} + 404T_{3}^{6} + 45710T_{3}^{4} + 1972260T_{3}^{2} + 28676025 \) Copy content Toggle raw display
\( T_{13}^{4} + 88T_{13}^{3} - 81904T_{13}^{2} - 5492992T_{13} + 921948160 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( T^{8} + 404 T^{6} + \cdots + 28676025 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 9604000000 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{4} + 88 T^{3} + \cdots + 921948160)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 12 T^{3} + \cdots + 2671438105)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 87\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + 696 T^{3} + \cdots - 708907790336)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + 128 T^{3} + \cdots + 3293908240)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1692 T^{3} + \cdots + 967612358881)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 6308970663920)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 7239157791376)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 45\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 85263437429065)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 25569083993929)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 26\!\cdots\!20)^{2} \) Copy content Toggle raw display
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