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Properties

Label 800.4.d.b
Level $800$
Weight $4$
Character orbit 800.d
Analytic conductor $47.202$
Analytic rank $0$
Dimension $12$
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] = [800,4,Mod(401,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.401");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.d (of order \(2\), degree \(1\), not 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: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 10x^{9} - 32x^{8} + 56x^{7} + 160x^{6} + 448x^{5} - 2048x^{4} - 5120x^{3} - 32768x + 262144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{35} \)
Twist minimal: no (minimal twist has level 200)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{5} - 2) q^{7} + ( - \beta_{4} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{5} - 2) q^{7} + ( - \beta_{4} - 9) q^{9} + \beta_{2} q^{11} + ( - \beta_{7} - \beta_1) q^{13} + \beta_{8} q^{17} + \beta_{6} q^{19} + ( - \beta_{10} - \beta_{6} + \cdots - 4 \beta_1) q^{21}+ \cdots + ( - 5 \beta_{10} + 3 \beta_{9} + \cdots + 15 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 28 q^{7} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 28 q^{7} - 108 q^{9} - 112 q^{23} + 264 q^{31} + 116 q^{33} + 300 q^{39} - 20 q^{41} + 940 q^{47} + 228 q^{49} - 176 q^{57} + 1240 q^{63} - 628 q^{71} + 432 q^{73} - 1116 q^{79} + 1128 q^{81} - 4116 q^{87} - 424 q^{89} + 792 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 10x^{9} - 32x^{8} + 56x^{7} + 160x^{6} + 448x^{5} - 2048x^{4} - 5120x^{3} - 32768x + 262144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3 \nu^{11} + 17 \nu^{10} + 28 \nu^{9} - 14 \nu^{8} - 104 \nu^{7} + 840 \nu^{6} - 576 \nu^{5} + \cdots - 196608 ) / 131072 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5 \nu^{11} - 25 \nu^{10} - 28 \nu^{9} + 958 \nu^{8} - 1176 \nu^{7} - 392 \nu^{6} - 8384 \nu^{5} + \cdots - 2162688 ) / 131072 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 9 \nu^{10} + 40 \nu^{9} - 42 \nu^{8} + 560 \nu^{7} - 520 \nu^{6} - 5408 \nu^{5} + \cdots - 65536 ) / 16384 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{11} - 23 \nu^{10} - 116 \nu^{9} + 2 \nu^{8} + 888 \nu^{7} + 1800 \nu^{6} + 2112 \nu^{5} + \cdots + 655360 ) / 65536 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5 \nu^{11} + \nu^{10} + 20 \nu^{9} - 30 \nu^{8} + 8 \nu^{7} - 56 \nu^{6} + 1152 \nu^{5} + \cdots + 131072 ) / 32768 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31 \nu^{11} + 5 \nu^{10} - 52 \nu^{9} + 1050 \nu^{8} + 2680 \nu^{7} + 5608 \nu^{6} + 26816 \nu^{5} + \cdots - 458752 ) / 131072 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 29 \nu^{11} - 81 \nu^{10} + 388 \nu^{9} + 846 \nu^{8} - 88 \nu^{7} - 3912 \nu^{6} - 7360 \nu^{5} + \cdots - 5046272 ) / 131072 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21 \nu^{11} + 39 \nu^{10} - 44 \nu^{9} - 546 \nu^{8} + 840 \nu^{7} - 2696 \nu^{6} - 4928 \nu^{5} + \cdots - 262144 ) / 65536 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23 \nu^{11} + 115 \nu^{10} + 52 \nu^{9} + 278 \nu^{8} + 392 \nu^{7} - 8488 \nu^{6} + \cdots - 196608 ) / 65536 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11 \nu^{11} + 17 \nu^{10} + 20 \nu^{9} - 94 \nu^{8} - 440 \nu^{7} - 1016 \nu^{6} + 3200 \nu^{5} + \cdots - 622592 ) / 16384 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 49 \nu^{11} + 85 \nu^{10} + 76 \nu^{9} + 698 \nu^{8} + 1144 \nu^{7} - 5272 \nu^{6} + \cdots + 851968 ) / 65536 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{5} + \beta_{4} - 2\beta _1 + 5 ) / 64 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} + 3 \beta_{10} + \beta_{9} - 4 \beta_{8} - 4 \beta_{7} + 3 \beta_{5} + 3 \beta_{4} + \cdots + 3 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + \beta_{10} + 3 \beta_{9} - 4 \beta_{8} + 12 \beta_{7} - 8 \beta_{6} - 19 \beta_{5} + \cdots + 173 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{11} + 7 \beta_{10} - 3 \beta_{9} + 28 \beta_{8} + 12 \beta_{7} - 8 \beta_{6} + 107 \beta_{5} + \cdots + 867 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 11 \beta_{11} + 69 \beta_{10} + 7 \beta_{9} - 44 \beta_{8} + 4 \beta_{7} + 88 \beta_{6} + 265 \beta_{5} + \cdots - 447 ) / 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 89 \beta_{11} - 73 \beta_{10} - 67 \beta_{9} - 212 \beta_{8} - 132 \beta_{7} + 8 \beta_{6} + \cdots - 4021 ) / 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 85 \beta_{11} - 299 \beta_{10} + 359 \beta_{9} + 420 \beta_{8} + 852 \beta_{7} + 440 \beta_{6} + \cdots - 7423 ) / 64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 367 \beta_{11} - 625 \beta_{10} + 709 \beta_{9} - 308 \beta_{8} + 1244 \beta_{7} + 1512 \beta_{6} + \cdots + 93299 ) / 64 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 493 \beta_{11} - 19 \beta_{10} - 1521 \beta_{9} - 2972 \beta_{8} + 7444 \beta_{7} + 2040 \beta_{6} + \cdots + 212089 ) / 64 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 10087 \beta_{11} + 8679 \beta_{10} + 3789 \beta_{9} + 2412 \beta_{8} - 20356 \beta_{7} + 6632 \beta_{6} + \cdots - 142069 ) / 64 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 12171 \beta_{11} + 32949 \beta_{10} + 4839 \beta_{9} - 9532 \beta_{8} + 24820 \beta_{7} + \cdots + 2764641 ) / 64 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
2.80448 − 0.367242i
−0.412714 + 2.79815i
−1.82985 − 2.15676i
−2.74710 + 0.673381i
0.416712 − 2.79756i
2.26847 + 1.68939i
2.26847 − 1.68939i
0.416712 + 2.79756i
−2.74710 − 0.673381i
−1.82985 + 2.15676i
−0.412714 − 2.79815i
2.80448 + 0.367242i
0 − 9.63387i 0 0 0 −21.1900 0 −65.8115 0
401.2 0 − 7.69300i 0 0 0 15.6248 0 −32.1823 0
401.3 0 − 5.31349i 0 0 0 15.7169 0 −1.23319 0
401.4 0 − 5.16961i 0 0 0 −7.07059 0 0.275157 0
401.5 0 − 2.73090i 0 0 0 −31.2997 0 19.5422 0
401.6 0 − 1.26108i 0 0 0 14.2186 0 25.4097 0
401.7 0 1.26108i 0 0 0 14.2186 0 25.4097 0
401.8 0 2.73090i 0 0 0 −31.2997 0 19.5422 0
401.9 0 5.16961i 0 0 0 −7.07059 0 0.275157 0
401.10 0 5.31349i 0 0 0 15.7169 0 −1.23319 0
401.11 0 7.69300i 0 0 0 15.6248 0 −32.1823 0
401.12 0 9.63387i 0 0 0 −21.1900 0 −65.8115 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.4.d.b 12
4.b odd 2 1 200.4.d.c ✓ 12
5.b even 2 1 800.4.d.c 12
5.c odd 4 2 800.4.f.d 24
8.b even 2 1 inner 800.4.d.b 12
8.d odd 2 1 200.4.d.c ✓ 12
20.d odd 2 1 200.4.d.d yes 12
20.e even 4 2 200.4.f.d 24
40.e odd 2 1 200.4.d.d yes 12
40.f even 2 1 800.4.d.c 12
40.i odd 4 2 800.4.f.d 24
40.k even 4 2 200.4.f.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.d.c ✓ 12 4.b odd 2 1
200.4.d.c ✓ 12 8.d odd 2 1
200.4.d.d yes 12 20.d odd 2 1
200.4.d.d yes 12 40.e odd 2 1
200.4.f.d 24 20.e even 4 2
200.4.f.d 24 40.k even 4 2
800.4.d.b 12 1.a even 1 1 trivial
800.4.d.b 12 8.b even 2 1 inner
800.4.d.c 12 5.b even 2 1
800.4.d.c 12 40.f even 2 1
800.4.f.d 24 5.c odd 4 2
800.4.f.d 24 40.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{12} + 216T_{3}^{10} + 16485T_{3}^{8} + 551120T_{3}^{6} + 8086707T_{3}^{4} + 42440280T_{3}^{2} + 49154791 \) Copy content Toggle raw display
\( T_{7}^{6} + 14T_{7}^{5} - 988T_{7}^{4} - 4760T_{7}^{3} + 293344T_{7}^{2} - 370304T_{7} - 16374272 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 216 T^{10} + \cdots + 49154791 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 14 T^{5} + \cdots - 16374272)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 32\!\cdots\!75 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{6} - 17275 T^{4} + \cdots - 5006693433)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 17\!\cdots\!39 \) Copy content Toggle raw display
$23$ \( (T^{6} + 56 T^{5} + \cdots - 209281675264)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} - 132 T^{5} + \cdots + 492501925888)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 1894414654475)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 13\!\cdots\!72)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 34\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 65\!\cdots\!51 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 57\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 26\!\cdots\!19)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 21\!\cdots\!19 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 35\!\cdots\!25)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 60\!\cdots\!72)^{2} \) Copy content Toggle raw display
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