Properties

Label 800.6.c.i
Level 800
Weight 6
Character orbit 800.c
Analytic conductor 128.307
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

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

Self dual: no
Analytic conductor: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Defining polynomial: \(x^{4} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + 7 \beta_{2} q^{7} + 203 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + 7 \beta_{2} q^{7} + 203 q^{9} -57 \beta_{3} q^{11} + 73 \beta_{1} q^{13} -351 \beta_{1} q^{17} -216 \beta_{3} q^{19} + 280 q^{21} + 647 \beta_{2} q^{23} -446 \beta_{2} q^{27} + 4010 q^{29} + 361 \beta_{3} q^{31} + 2280 \beta_{1} q^{33} -7389 \beta_{1} q^{37} + 73 \beta_{3} q^{39} -4350 q^{41} -1965 \beta_{2} q^{43} + 951 \beta_{2} q^{47} + 14847 q^{49} -351 \beta_{3} q^{51} + 9077 \beta_{1} q^{53} + 8640 \beta_{1} q^{57} + 1558 \beta_{3} q^{59} -42130 q^{61} + 1421 \beta_{2} q^{63} + 2559 \beta_{2} q^{67} + 25880 q^{69} + 3593 \beta_{3} q^{71} -13133 \beta_{1} q^{73} -15960 \beta_{1} q^{77} + 686 \beta_{3} q^{79} + 31489 q^{81} + 15615 \beta_{2} q^{83} -4010 \beta_{2} q^{87} -30570 q^{89} -511 \beta_{3} q^{91} -14440 \beta_{1} q^{93} + 33441 \beta_{1} q^{97} -11571 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 812q^{9} + O(q^{10}) \) \( 4q + 812q^{9} + 1120q^{21} + 16040q^{29} - 17400q^{41} + 59388q^{49} - 168520q^{61} + 103520q^{69} + 125956q^{81} - 122280q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{2} \)\(/5\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} + 10 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{3} + 20 \nu \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2}\)\()/8\)
\(\nu^{2}\)\(=\)\(5 \beta_{1}\)\(/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{3} + 10 \beta_{2}\)\()/8\)

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
1.58114 + 1.58114i
−1.58114 + 1.58114i
1.58114 1.58114i
−1.58114 1.58114i
0 6.32456i 0 0 0 44.2719i 0 203.000 0
449.2 0 6.32456i 0 0 0 44.2719i 0 203.000 0
449.3 0 6.32456i 0 0 0 44.2719i 0 203.000 0
449.4 0 6.32456i 0 0 0 44.2719i 0 203.000 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 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.i 4
4.b odd 2 1 inner 800.6.c.i 4
5.b even 2 1 inner 800.6.c.i 4
5.c odd 4 1 160.6.a.d 2
5.c odd 4 1 800.6.a.h 2
20.d odd 2 1 inner 800.6.c.i 4
20.e even 4 1 160.6.a.d 2
20.e even 4 1 800.6.a.h 2
40.i odd 4 1 320.6.a.s 2
40.k even 4 1 320.6.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.d 2 5.c odd 4 1
160.6.a.d 2 20.e even 4 1
320.6.a.s 2 40.i odd 4 1
320.6.a.s 2 40.k even 4 1
800.6.a.h 2 5.c odd 4 1
800.6.a.h 2 20.e even 4 1
800.6.c.i 4 1.a even 1 1 trivial
800.6.c.i 4 4.b odd 2 1 inner
800.6.c.i 4 5.b even 2 1 inner
800.6.c.i 4 20.d odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} + 40 \)
\( T_{11}^{2} - 519840 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 446 T^{2} + 59049 T^{4} )^{2} \)
$5$ 1
$7$ \( ( 1 - 31654 T^{2} + 282475249 T^{4} )^{2} \)
$11$ \( ( 1 - 197738 T^{2} + 25937424601 T^{4} )^{2} \)
$13$ \( ( 1 - 721270 T^{2} + 137858491849 T^{4} )^{2} \)
$17$ \( ( 1 - 2346910 T^{2} + 2015993900449 T^{4} )^{2} \)
$19$ \( ( 1 - 2512762 T^{2} + 6131066257801 T^{4} )^{2} \)
$23$ \( ( 1 + 3871674 T^{2} + 41426511213649 T^{4} )^{2} \)
$29$ \( ( 1 - 4010 T + 20511149 T^{2} )^{4} \)
$31$ \( ( 1 + 36406942 T^{2} + 819628286980801 T^{4} )^{2} \)
$37$ \( ( 1 + 79701370 T^{2} + 4808584372417849 T^{4} )^{2} \)
$41$ \( ( 1 + 4350 T + 115856201 T^{2} )^{4} \)
$43$ \( ( 1 - 139567886 T^{2} + 21611482313284249 T^{4} )^{2} \)
$47$ \( ( 1 - 422513974 T^{2} + 52599132235830049 T^{4} )^{2} \)
$53$ \( ( 1 - 506823270 T^{2} + 174887470365513049 T^{4} )^{2} \)
$59$ \( ( 1 + 1041470358 T^{2} + 511116753300641401 T^{4} )^{2} \)
$61$ \( ( 1 + 42130 T + 844596301 T^{2} )^{4} \)
$67$ \( ( 1 - 2438310974 T^{2} + 1822837804551761449 T^{4} )^{2} \)
$71$ \( ( 1 + 1542914862 T^{2} + 3255243551009881201 T^{4} )^{2} \)
$73$ \( ( 1 - 3456240430 T^{2} + 4297625829703557649 T^{4} )^{2} \)
$79$ \( ( 1 + 6078817438 T^{2} + 9468276082626847201 T^{4} )^{2} \)
$83$ \( ( 1 + 1875047714 T^{2} + 15516041187205853449 T^{4} )^{2} \)
$89$ \( ( 1 + 30570 T + 5584059449 T^{2} )^{4} \)
$97$ \( ( 1 - 12701478590 T^{2} + 73742412689492826049 T^{4} )^{2} \)
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