Properties

Label 800.6.a.t
Level $800$
Weight $6$
Character orbit 800.a
Self dual yes
Analytic conductor $128.307$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,6,Mod(1,800)]
 
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, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.a (trivial)

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: yes
Analytic conductor: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 226x^{3} + 455x^{2} + 9816x + 4656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} - 22) q^{7} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 121) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} - 22) q^{7} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 121) q^{9} + ( - \beta_{4} + \beta_{3} - 7 \beta_1) q^{11} + (\beta_{4} + 2 \beta_{2} + 5 \beta_1 + 57) q^{13} + ( - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 172) q^{17} + (2 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} - 17 \beta_1 + 896) q^{19} + ( - 5 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} - 59 \beta_1 - 93) q^{21} + (8 \beta_{3} - 2 \beta_{2} - 22 \beta_1 + 788) q^{23} + (5 \beta_{4} - 13 \beta_{3} + 14 \beta_{2} + 73 \beta_1 - 988) q^{27} + ( - 5 \beta_{4} - 6 \beta_{3} - 20 \beta_{2} + 89 \beta_1 - 635) q^{29} + (6 \beta_{4} + 2 \beta_{3} - 24 \beta_{2} + 26 \beta_1 + 1656) q^{31} + (2 \beta_{4} - 14 \beta_{3} + 66 \beta_{2} + 148 \beta_1 - 2669) q^{33} + (3 \beta_{4} + 8 \beta_{3} + 22 \beta_{2} + 359 \beta_1 + 717) q^{37} + ( - 12 \beta_{4} + 20 \beta_{3} - 29 \beta_{2} - 10 \beta_1 + 1766) q^{39} + (10 \beta_{4} - 22 \beta_{3} + 58 \beta_{2} - 236 \beta_1 - 2339) q^{41} + (5 \beta_{4} - 5 \beta_{3} - 44 \beta_{2} - 58 \beta_1 + 72) q^{43} + ( - 30 \beta_{4} + 22 \beta_{3} - 61 \beta_{2} - 326 \beta_1 + 1878) q^{47} + (20 \beta_{4} + 24 \beta_{3} - 56 \beta_{2} + 652 \beta_1 + 5385) q^{49} + ( - 11 \beta_{4} + 11 \beta_{3} + 120 \beta_{2} - 187 \beta_1 + 560) q^{51} + ( - 22 \beta_{4} + 78 \beta_{3} - 18 \beta_{2} + 184 \beta_1 + 6142) q^{53} + (16 \beta_{4} - 59 \beta_{3} + 19 \beta_{2} + 2033 \beta_1 - 5486) q^{57} + (15 \beta_{4} - 95 \beta_{3} + 110 \beta_{2} - 230 \beta_1 - 7940) q^{59} + (45 \beta_{4} + 10 \beta_{3} - 88 \beta_{2} - 1229 \beta_1 + 6729) q^{61} + ( - 30 \beta_{4} - 50 \beta_{3} + 194 \beta_{2} + 572 \beta_1 - 17468) q^{63} + (35 \beta_{4} + 53 \beta_{3} + 74 \beta_{2} + 597 \beta_1 + 13916) q^{67} + (10 \beta_{4} - 66 \beta_{3} + 158 \beta_{2} + 2120 \beta_1 - 7734) q^{69} + (2 \beta_{4} - 90 \beta_{3} - 221 \beta_{2} + 850 \beta_1 + 10374) q^{71} + ( - 111 \beta_{3} - 177 \beta_{2} + 1677 \beta_1 - 8032) q^{73} + ( - 3 \beta_{4} + 48 \beta_{3} - 190 \beta_{2} + 3729 \beta_1 + 12175) q^{77} + (38 \beta_{4} - 118 \beta_{3} + 25 \beta_{2} + 2436 \beta_1 + 330) q^{79} + ( - 80 \beta_{4} - 19 \beta_{3} - 165 \beta_{2} - 2823 \beta_1 - 3616) q^{81} + (25 \beta_{4} - 17 \beta_{3} - 128 \beta_{2} - 1383 \beta_1 - 13528) q^{83} + (110 \beta_{4} - 22 \beta_{3} - 163 \beta_{2} - 1166 \beta_1 + 33530) q^{87} + ( - 130 \beta_{4} + 31 \beta_{3} + 261 \beta_{2} + 2537 \beta_1 - 20246) q^{89} + (28 \beta_{4} - 20 \beta_{3} + 226 \beta_{2} - 1260 \beta_1 + 36796) q^{91} + (108 \beta_{4} - 108 \beta_{3} - 420 \beta_{2} + 2896 \beta_1 + 12510) q^{93} + ( - 90 \beta_{4} + 232 \beta_{3} - 76 \beta_{2} + 4006 \beta_1 + 5124) q^{97} + ( - 91 \beta_{4} + 363 \beta_{3} + 34 \beta_{2} - 5896 \beta_1 + 47844) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{3} - 110 q^{7} + 606 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{3} - 110 q^{7} + 606 q^{9} + 5 q^{11} + 280 q^{13} - 865 q^{17} + 4485 q^{19} - 418 q^{21} + 3946 q^{23} - 4987 q^{27} - 3252 q^{29} + 8250 q^{31} - 13465 q^{33} + 3210 q^{37} + 8800 q^{39} - 11415 q^{41} + 428 q^{43} + 9672 q^{47} + 26225 q^{49} + 2965 q^{51} + 30370 q^{53} - 29345 q^{57} - 39280 q^{59} + 34854 q^{61} - 87812 q^{63} + 68877 q^{67} - 40658 q^{69} + 51200 q^{71} - 41615 q^{73} + 57050 q^{77} - 550 q^{79} - 15219 q^{81} - 66223 q^{83} + 168860 q^{87} - 103829 q^{89} + 185280 q^{91} + 59870 q^{93} + 21150 q^{97} + 244390 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 226x^{3} + 455x^{2} + 9816x + 4656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{4} - 12\nu^{3} + 272\nu^{2} + 1098\nu - 96 ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{4} - 12\nu^{3} + 356\nu^{2} + 1140\nu - 7782 ) / 21 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{4} + 180\nu^{3} + 568\nu^{2} - 23904\nu - 67356 ) / 105 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} - \beta _1 + 365 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{4} - 10\beta_{3} + 11\beta_{2} + 553\beta _1 + 105 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -15\beta_{4} + 166\beta_{3} - 211\beta_{2} - 697\beta _1 + 50231 ) / 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−13.4365
−5.94994
−0.488029
10.4009
11.4736
0 −27.8730 0 0 0 −108.742 0 533.902 0
1.2 0 −12.8999 0 0 0 121.874 0 −76.5929 0
1.3 0 −1.97606 0 0 0 −48.9425 0 −239.095 0
1.4 0 19.8018 0 0 0 160.939 0 149.111 0
1.5 0 21.9471 0 0 0 −235.128 0 238.676 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.6.a.t yes 5
4.b odd 2 1 800.6.a.u yes 5
5.b even 2 1 800.6.a.v yes 5
5.c odd 4 2 800.6.c.o 10
20.d odd 2 1 800.6.a.s 5
20.e even 4 2 800.6.c.n 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.6.a.s 5 20.d odd 2 1
800.6.a.t yes 5 1.a even 1 1 trivial
800.6.a.u yes 5 4.b odd 2 1
800.6.a.v yes 5 5.b even 2 1
800.6.c.n 10 20.e even 4 2
800.6.c.o 10 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3}^{5} + T_{3}^{4} - 910T_{3}^{3} + 914T_{3}^{2} + 161613T_{3} + 308781 \) Copy content Toggle raw display
\( T_{11}^{5} - 5T_{11}^{4} - 581526T_{11}^{3} + 34767310T_{11}^{2} + 82493492725T_{11} - 11275947155625 \) Copy content Toggle raw display
\( T_{13}^{5} - 280T_{13}^{4} - 581616T_{13}^{3} + 34060800T_{13}^{2} + 49678848000T_{13} - 5363712000000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + T^{4} - 910 T^{3} + \cdots + 308781 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 110 T^{4} + \cdots + 24544747872 \) Copy content Toggle raw display
$11$ \( T^{5} - 5 T^{4} + \cdots - 11275947155625 \) Copy content Toggle raw display
$13$ \( T^{5} - 280 T^{4} + \cdots - 5363712000000 \) Copy content Toggle raw display
$17$ \( T^{5} + 865 T^{4} + \cdots - 10944985381875 \) Copy content Toggle raw display
$19$ \( T^{5} - 4485 T^{4} + \cdots + 13\!\cdots\!75 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots + 377296470599904 \) Copy content Toggle raw display
$29$ \( T^{5} + 3252 T^{4} + \cdots + 75\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{5} - 8250 T^{4} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{5} - 3210 T^{4} + \cdots - 69\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{5} + 11415 T^{4} + \cdots - 82\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{5} - 428 T^{4} + \cdots - 66\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{5} - 9672 T^{4} + \cdots + 44\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{5} - 30370 T^{4} + \cdots - 66\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{5} + 39280 T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{5} - 34854 T^{4} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{5} - 68877 T^{4} + \cdots - 12\!\cdots\!77 \) Copy content Toggle raw display
$71$ \( T^{5} - 51200 T^{4} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{5} + 41615 T^{4} + \cdots + 28\!\cdots\!75 \) Copy content Toggle raw display
$79$ \( T^{5} + 550 T^{4} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{5} + 66223 T^{4} + \cdots + 32\!\cdots\!75 \) Copy content Toggle raw display
$89$ \( T^{5} + 103829 T^{4} + \cdots + 16\!\cdots\!69 \) Copy content Toggle raw display
$97$ \( T^{5} - 21150 T^{4} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
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