Properties

Label 800.2.f.e
Level $800$
Weight $2$
Character orbit 800.f
Analytic conductor $6.388$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} -2 \zeta_{12}^{3} q^{11} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{13} + ( -2 + 4 \zeta_{12}^{2} ) q^{17} + ( -2 + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{19} -2 \zeta_{12}^{3} q^{21} + ( 3 - 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{23} + 4 q^{27} + ( -4 + 8 \zeta_{12}^{2} ) q^{29} + ( 2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{31} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{33} -2 q^{37} + ( 6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{39} + ( -2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{41} + ( -7 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{43} + ( -1 + 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{47} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{49} + ( -2 + 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{51} + ( -8 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{53} + ( 2 - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{57} + ( -2 + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{59} + ( -4 + 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{61} + ( -1 + 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{63} + ( -9 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{67} + ( 4 - 8 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{69} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{71} + ( 2 - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{73} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{77} + ( -8 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{79} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{81} + ( 3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{83} + ( -4 + 8 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{87} + ( -2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} + ( -2 + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{91} + ( 8 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{93} + ( 6 - 12 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{97} + ( 4 - 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{9} + 16q^{27} + 8q^{31} - 8q^{37} + 24q^{39} - 8q^{41} - 28q^{43} + 12q^{49} - 32q^{53} - 36q^{67} - 8q^{71} + 8q^{77} - 32q^{79} + 4q^{81} + 12q^{83} - 8q^{89} + 32q^{93} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0 −0.732051 0 0 0 2.73205i 0 −2.46410 0
49.2 0 −0.732051 0 0 0 2.73205i 0 −2.46410 0
49.3 0 2.73205 0 0 0 0.732051i 0 4.46410 0
49.4 0 2.73205 0 0 0 0.732051i 0 4.46410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.f.e 4
3.b odd 2 1 7200.2.d.n 4
4.b odd 2 1 200.2.f.e 4
5.b even 2 1 800.2.f.c 4
5.c odd 4 1 160.2.d.a 4
5.c odd 4 1 800.2.d.e 4
8.b even 2 1 800.2.f.c 4
8.d odd 2 1 200.2.f.c 4
12.b even 2 1 1800.2.d.l 4
15.d odd 2 1 7200.2.d.o 4
15.e even 4 1 1440.2.k.e 4
15.e even 4 1 7200.2.k.j 4
20.d odd 2 1 200.2.f.c 4
20.e even 4 1 40.2.d.a 4
20.e even 4 1 200.2.d.f 4
24.f even 2 1 1800.2.d.p 4
24.h odd 2 1 7200.2.d.o 4
40.e odd 2 1 200.2.f.e 4
40.f even 2 1 inner 800.2.f.e 4
40.i odd 4 1 160.2.d.a 4
40.i odd 4 1 800.2.d.e 4
40.k even 4 1 40.2.d.a 4
40.k even 4 1 200.2.d.f 4
60.h even 2 1 1800.2.d.p 4
60.l odd 4 1 360.2.k.e 4
60.l odd 4 1 1800.2.k.j 4
80.i odd 4 1 1280.2.a.d 2
80.i odd 4 1 6400.2.a.be 2
80.j even 4 1 1280.2.a.a 2
80.j even 4 1 6400.2.a.z 2
80.s even 4 1 1280.2.a.o 2
80.s even 4 1 6400.2.a.ce 2
80.t odd 4 1 1280.2.a.n 2
80.t odd 4 1 6400.2.a.cj 2
120.i odd 2 1 7200.2.d.n 4
120.m even 2 1 1800.2.d.l 4
120.q odd 4 1 360.2.k.e 4
120.q odd 4 1 1800.2.k.j 4
120.w even 4 1 1440.2.k.e 4
120.w even 4 1 7200.2.k.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.d.a 4 20.e even 4 1
40.2.d.a 4 40.k even 4 1
160.2.d.a 4 5.c odd 4 1
160.2.d.a 4 40.i odd 4 1
200.2.d.f 4 20.e even 4 1
200.2.d.f 4 40.k even 4 1
200.2.f.c 4 8.d odd 2 1
200.2.f.c 4 20.d odd 2 1
200.2.f.e 4 4.b odd 2 1
200.2.f.e 4 40.e odd 2 1
360.2.k.e 4 60.l odd 4 1
360.2.k.e 4 120.q odd 4 1
800.2.d.e 4 5.c odd 4 1
800.2.d.e 4 40.i odd 4 1
800.2.f.c 4 5.b even 2 1
800.2.f.c 4 8.b even 2 1
800.2.f.e 4 1.a even 1 1 trivial
800.2.f.e 4 40.f even 2 1 inner
1280.2.a.a 2 80.j even 4 1
1280.2.a.d 2 80.i odd 4 1
1280.2.a.n 2 80.t odd 4 1
1280.2.a.o 2 80.s even 4 1
1440.2.k.e 4 15.e even 4 1
1440.2.k.e 4 120.w even 4 1
1800.2.d.l 4 12.b even 2 1
1800.2.d.l 4 120.m even 2 1
1800.2.d.p 4 24.f even 2 1
1800.2.d.p 4 60.h even 2 1
1800.2.k.j 4 60.l odd 4 1
1800.2.k.j 4 120.q odd 4 1
6400.2.a.z 2 80.j even 4 1
6400.2.a.be 2 80.i odd 4 1
6400.2.a.ce 2 80.s even 4 1
6400.2.a.cj 2 80.t odd 4 1
7200.2.d.n 4 3.b odd 2 1
7200.2.d.n 4 120.i odd 2 1
7200.2.d.o 4 15.d odd 2 1
7200.2.d.o 4 24.h odd 2 1
7200.2.k.j 4 15.e even 4 1
7200.2.k.j 4 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -2 - 2 T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 4 + 8 T^{2} + T^{4} \)
$11$ \( ( 4 + T^{2} )^{2} \)
$13$ \( ( -12 + T^{2} )^{2} \)
$17$ \( ( 12 + T^{2} )^{2} \)
$19$ \( 16 + 56 T^{2} + T^{4} \)
$23$ \( 676 + 56 T^{2} + T^{4} \)
$29$ \( ( 48 + T^{2} )^{2} \)
$31$ \( ( -8 - 4 T + T^{2} )^{2} \)
$37$ \( ( 2 + T )^{4} \)
$41$ \( ( -8 + 4 T + T^{2} )^{2} \)
$43$ \( ( 46 + 14 T + T^{2} )^{2} \)
$47$ \( 484 + 56 T^{2} + T^{4} \)
$53$ \( ( 52 + 16 T + T^{2} )^{2} \)
$59$ \( 16 + 56 T^{2} + T^{4} \)
$61$ \( 1936 + 104 T^{2} + T^{4} \)
$67$ \( ( 78 + 18 T + T^{2} )^{2} \)
$71$ \( ( -8 + 4 T + T^{2} )^{2} \)
$73$ \( 16 + 56 T^{2} + T^{4} \)
$79$ \( ( 16 + 16 T + T^{2} )^{2} \)
$83$ \( ( 6 - 6 T + T^{2} )^{2} \)
$89$ \( ( -44 + 4 T + T^{2} )^{2} \)
$97$ \( 8464 + 248 T^{2} + T^{4} \)
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