Properties

Label 800.4.c.f
Level $800$
Weight $4$
Character orbit 800.c
Analytic conductor $47.202$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{3} + 6 i q^{7} + 23 q^{9} +O(q^{10})\) \( q + 2 i q^{3} + 6 i q^{7} + 23 q^{9} + 60 q^{11} -50 i q^{13} -30 i q^{17} -40 q^{19} -12 q^{21} -178 i q^{23} + 100 i q^{27} -166 q^{29} + 20 q^{31} + 120 i q^{33} + 10 i q^{37} + 100 q^{39} -250 q^{41} -142 i q^{43} + 214 i q^{47} + 307 q^{49} + 60 q^{51} -490 i q^{53} -80 i q^{57} + 800 q^{59} + 250 q^{61} + 138 i q^{63} -774 i q^{67} + 356 q^{69} + 100 q^{71} + 230 i q^{73} + 360 i q^{77} + 1320 q^{79} + 421 q^{81} -982 i q^{83} -332 i q^{87} -874 q^{89} + 300 q^{91} + 40 i q^{93} -310 i q^{97} + 1380 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 46q^{9} + O(q^{10}) \) \( 2q + 46q^{9} + 120q^{11} - 80q^{19} - 24q^{21} - 332q^{29} + 40q^{31} + 200q^{39} - 500q^{41} + 614q^{49} + 120q^{51} + 1600q^{59} + 500q^{61} + 712q^{69} + 200q^{71} + 2640q^{79} + 842q^{81} - 1748q^{89} + 600q^{91} + 2760q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 2.00000i 0 0 0 6.00000i 0 23.0000 0
449.2 0 2.00000i 0 0 0 6.00000i 0 23.0000 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
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 800.4.c.f 2
4.b odd 2 1 800.4.c.e 2
5.b even 2 1 inner 800.4.c.f 2
5.c odd 4 1 160.4.a.a 1
5.c odd 4 1 800.4.a.h 1
15.e even 4 1 1440.4.a.o 1
20.d odd 2 1 800.4.c.e 2
20.e even 4 1 160.4.a.b yes 1
20.e even 4 1 800.4.a.d 1
40.i odd 4 1 320.4.a.i 1
40.i odd 4 1 1600.4.a.r 1
40.k even 4 1 320.4.a.f 1
40.k even 4 1 1600.4.a.bj 1
60.l odd 4 1 1440.4.a.n 1
80.i odd 4 1 1280.4.d.f 2
80.j even 4 1 1280.4.d.k 2
80.s even 4 1 1280.4.d.k 2
80.t odd 4 1 1280.4.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 5.c odd 4 1
160.4.a.b yes 1 20.e even 4 1
320.4.a.f 1 40.k even 4 1
320.4.a.i 1 40.i odd 4 1
800.4.a.d 1 20.e even 4 1
800.4.a.h 1 5.c odd 4 1
800.4.c.e 2 4.b odd 2 1
800.4.c.e 2 20.d odd 2 1
800.4.c.f 2 1.a even 1 1 trivial
800.4.c.f 2 5.b even 2 1 inner
1280.4.d.f 2 80.i odd 4 1
1280.4.d.f 2 80.t odd 4 1
1280.4.d.k 2 80.j even 4 1
1280.4.d.k 2 80.s even 4 1
1440.4.a.n 1 60.l odd 4 1
1440.4.a.o 1 15.e even 4 1
1600.4.a.r 1 40.i odd 4 1
1600.4.a.bj 1 40.k even 4 1

Hecke kernels

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

\( T_{3}^{2} + 4 \)
\( T_{11} - 60 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 36 + T^{2} \)
$11$ \( ( -60 + T )^{2} \)
$13$ \( 2500 + T^{2} \)
$17$ \( 900 + T^{2} \)
$19$ \( ( 40 + T )^{2} \)
$23$ \( 31684 + T^{2} \)
$29$ \( ( 166 + T )^{2} \)
$31$ \( ( -20 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( 250 + T )^{2} \)
$43$ \( 20164 + T^{2} \)
$47$ \( 45796 + T^{2} \)
$53$ \( 240100 + T^{2} \)
$59$ \( ( -800 + T )^{2} \)
$61$ \( ( -250 + T )^{2} \)
$67$ \( 599076 + T^{2} \)
$71$ \( ( -100 + T )^{2} \)
$73$ \( 52900 + T^{2} \)
$79$ \( ( -1320 + T )^{2} \)
$83$ \( 964324 + T^{2} \)
$89$ \( ( 874 + T )^{2} \)
$97$ \( 96100 + T^{2} \)
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