Properties

Label 800.6.c.k
Level $800$
Weight $6$
Character orbit 800.c
Analytic conductor $128.307$
Analytic rank $0$
Dimension $6$
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] = [800,6,Mod(449,800)]
 
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, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.449");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.c (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: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.6140289600.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 32x^{4} + 116x^{3} + 256x^{2} + 2778x + 7605 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 160)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 2 \beta_1) q^{3} + (\beta_{5} + \beta_{3} - \beta_1) q^{7} + (2 \beta_{4} - 6 \beta_{2} - 157) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 2 \beta_1) q^{3} + (\beta_{5} + \beta_{3} - \beta_1) q^{7} + (2 \beta_{4} - 6 \beta_{2} - 157) q^{9} + ( - 3 \beta_{4} + 131) q^{11} + ( - 2 \beta_{5} - 20 \beta_{3} - 53 \beta_1) q^{13} + (6 \beta_{5} + 60 \beta_{3} - 211 \beta_1) q^{17} + ( - \beta_{4} + 25 \beta_{2} + 1072) q^{19} + (4 \beta_{4} + 36 \beta_{2} - 260) q^{21} + ( - 7 \beta_{5} - 43 \beta_{3} - 1009 \beta_1) q^{23} + (20 \beta_{5} - 182 \beta_{3} + 2368 \beta_1) q^{27} + (52 \beta_{4} + 20 \beta_{2} - 118) q^{29} + ( - 42 \beta_{4} + 135 \beta_{2} - 1061) q^{31} + (6 \beta_{5} + 380 \beta_{3} - 526 \beta_1) q^{33} + (76 \beta_{5} - 200 \beta_{3} + 2019 \beta_1) q^{37} + ( - 44 \beta_{4} - 55 \beta_{2} + 7083) q^{39} + ( - 22 \beta_{4} + 34 \beta_{2} + 4154) q^{41} + (14 \beta_{5} + 221 \beta_{3} - 4460 \beta_1) q^{43} + ( - 25 \beta_{5} + 77 \beta_{3} + 6093 \beta_1) q^{47} + ( - 74 \beta_{4} + 158 \beta_{2} + 1311) q^{49} + (132 \beta_{4} - 205 \beta_{2} - 23839) q^{51} + (114 \beta_{5} + 1140 \beta_{3} - 3869 \beta_1) q^{53} + ( - 98 \beta_{5} + 1580 \beta_{3} - 12082 \beta_1) q^{57} + (185 \beta_{4} - 35 \beta_{2} + 11730) q^{59} + ( - 32 \beta_{4} + 992 \beta_{2} - 7726) q^{61} + (91 \beta_{5} + 263 \beta_{3} - 13555 \beta_1) q^{63} + ( - 82 \beta_{5} + 2345 \beta_{3} - 2404 \beta_1) q^{67} + ( - 100 \beta_{4} - 1124 \beta_{2} + 8604) q^{69} + (56 \beta_{4} - 515 \beta_{2} - 29517) q^{71} + ( - 150 \beta_{5} + 2340 \beta_{3} + 10295 \beta_1) q^{73} + ( - 106 \beta_{5} + 860 \beta_{3} + 22426 \beta_1) q^{77} + ( - 164 \beta_{4} + 350 \beta_{2} - 30922) q^{79} + (162 \beta_{4} + 2458 \beta_{2} + 51337) q^{81} + ( - 376 \beta_{5} - 2923 \beta_{3} - 4194 \beta_1) q^{83} + ( - 184 \beta_{5} - 4094 \beta_{3} - 3068 \beta_1) q^{87} + ( - 36 \beta_{4} + 300 \beta_{2} - 57474) q^{89} + ( - 106 \beta_{4} - 1055 \beta_{2} + 35217) q^{91} + ( - 456 \beta_{5} + 4720 \beta_{3} - 54764 \beta_1) q^{93} + ( - 430 \beta_{5} - 1420 \beta_{3} - 28155 \beta_1) q^{97} + (43 \beta_{4} - 1800 \beta_{2} - 117731) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 934 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 934 q^{9} + 792 q^{11} + 6384 q^{19} - 1640 q^{21} - 852 q^{29} - 6552 q^{31} + 42696 q^{39} + 24900 q^{41} + 7698 q^{49} - 142888 q^{51} + 70080 q^{59} - 48276 q^{61} + 54072 q^{69} - 176184 q^{71} - 185904 q^{79} + 302782 q^{81} - 345372 q^{89} + 213624 q^{91} - 702872 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 32x^{4} + 116x^{3} + 256x^{2} + 2778x + 7605 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7\nu^{5} - 275\nu^{4} + 2174\nu^{3} - 5515\nu^{2} + 8137\nu + 106320 ) / 58125 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{5} + 256\nu^{3} + 1440\nu^{2} - 72\nu + 25205 ) / 625 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -173\nu^{5} + 685\nu^{4} - 6166\nu^{3} + 2645\nu^{2} - 27323\nu - 212280 ) / 11625 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -16\nu^{5} + 200\nu^{4} - 112\nu^{3} - 680\nu^{2} + 20544\nu + 12965 ) / 625 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1061\nu^{5} - 1825\nu^{4} + 32802\nu^{3} + 89655\nu^{2} + 693051\nu + 1899360 ) / 19375 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{3} - 2\beta_{2} + 5\beta _1 + 10 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} - 2\beta_{4} + 22\beta_{3} + 8\beta_{2} + 79\beta _1 - 318 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{5} + 4\beta_{4} + 2\beta_{3} + 28\beta_{2} + 395\beta _1 - 1408 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -115\beta_{5} + 106\beta_{4} - 454\beta_{3} + 204\beta_{2} - 2959\beta _1 - 2030 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -211\beta_{5} + 104\beta_{4} - 4070\beta_{3} - 750\beta_{2} - 39455\beta _1 + 46622 ) / 32 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−2.90341 + 0.978064i
3.05894 4.88658i
0.844467 + 4.86464i
0.844467 4.86464i
3.05894 + 4.88658i
−2.90341 0.978064i
0 29.2272i 0 0 0 44.5253i 0 −611.232 0
449.2 0 18.4715i 0 0 0 121.899i 0 −98.1968 0
449.3 0 0.755735i 0 0 0 172.424i 0 242.429 0
449.4 0 0.755735i 0 0 0 172.424i 0 242.429 0
449.5 0 18.4715i 0 0 0 121.899i 0 −98.1968 0
449.6 0 29.2272i 0 0 0 44.5253i 0 −611.232 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.6.c.k 6
4.b odd 2 1 800.6.c.j 6
5.b even 2 1 inner 800.6.c.k 6
5.c odd 4 1 160.6.a.f 3
5.c odd 4 1 800.6.a.o 3
20.d odd 2 1 800.6.c.j 6
20.e even 4 1 160.6.a.g yes 3
20.e even 4 1 800.6.a.n 3
40.i odd 4 1 320.6.a.y 3
40.k even 4 1 320.6.a.x 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.f 3 5.c odd 4 1
160.6.a.g yes 3 20.e even 4 1
320.6.a.x 3 40.k even 4 1
320.6.a.y 3 40.i odd 4 1
800.6.a.n 3 20.e even 4 1
800.6.a.o 3 5.c odd 4 1
800.6.c.j 6 4.b odd 2 1
800.6.c.j 6 20.d odd 2 1
800.6.c.k 6 1.a even 1 1 trivial
800.6.c.k 6 5.b even 2 1 inner

Hecke kernels

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

\( T_{3}^{6} + 1196T_{3}^{4} + 292144T_{3}^{2} + 166464 \) Copy content Toggle raw display
\( T_{11}^{3} - 396T_{11}^{2} - 155280T_{11} + 59934400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 1196 T^{4} + 292144 T^{2} + \cdots + 166464 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 46572 T^{4} + \cdots + 875811479104 \) Copy content Toggle raw display
$11$ \( (T^{3} - 396 T^{2} - 155280 T + 59934400)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 833072769000000 \) Copy content Toggle raw display
$17$ \( T^{6} + 6031884 T^{4} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{3} - 3192 T^{2} + 1901760 T - 126323200)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 16700748 T^{4} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{3} + 426 T^{2} + \cdots - 159249002312)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 3276 T^{2} + \cdots - 229217617600)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 368496204 T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{3} - 12450 T^{2} + \cdots - 1203781400)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 294952236 T^{4} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{6} + 487597932 T^{4} + \cdots + 26\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{6} + 2165424204 T^{4} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{3} - 35040 T^{2} + \cdots - 887454720000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 24138 T^{2} + \cdots - 47677792189640)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 6873664524 T^{4} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{3} + 88092 T^{2} + \cdots + 11664981864000)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 9143837100 T^{4} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{3} + 92952 T^{2} + \cdots + 1808931596800)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 15962385612 T^{4} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{3} + 172686 T^{2} + \cdots + 175016035497384)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 20166336300 T^{4} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
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