Properties

Label 800.3.b.h
Level $800$
Weight $3$
Character orbit 800.b
Analytic conductor $21.798$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 800.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.7984211488\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
Defining polynomial: \(x^{6} + 9 x^{4} + 14 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{5} q^{7} + ( -3 - \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{5} q^{7} + ( -3 - \beta_{2} + \beta_{4} ) q^{9} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{11} + ( -1 + 2 \beta_{2} + \beta_{4} ) q^{13} + ( -12 - 2 \beta_{2} - 2 \beta_{4} ) q^{17} + ( 6 \beta_{1} + 3 \beta_{3} + 2 \beta_{5} ) q^{19} + ( 2 - \beta_{2} - \beta_{4} ) q^{21} + ( -2 \beta_{1} + 4 \beta_{3} + 3 \beta_{5} ) q^{23} + ( -4 \beta_{1} + 8 \beta_{3} + 2 \beta_{5} ) q^{27} + ( -6 - 4 \beta_{2} ) q^{29} + ( 4 \beta_{1} + 4 \beta_{5} ) q^{31} + ( -22 - 6 \beta_{2} ) q^{33} + ( -37 + 4 \beta_{2} + 3 \beta_{4} ) q^{37} + ( 4 \beta_{1} - 10 \beta_{3} - 4 \beta_{5} ) q^{39} + ( -10 - 3 \beta_{2} - \beta_{4} ) q^{41} + ( -\beta_{1} - 6 \beta_{5} ) q^{43} + ( 8 \beta_{1} + 12 \beta_{3} - \beta_{5} ) q^{47} + ( 13 - \beta_{2} - 7 \beta_{4} ) q^{49} + ( -12 \beta_{1} + 8 \beta_{3} + 4 \beta_{5} ) q^{51} + ( -11 + 6 \beta_{2} - 5 \beta_{4} ) q^{53} + ( -70 + 2 \beta_{2} + 8 \beta_{4} ) q^{57} + ( -6 \beta_{1} + 13 \beta_{3} - 2 \beta_{5} ) q^{59} + ( -20 + 3 \beta_{2} + 7 \beta_{4} ) q^{61} + ( 2 \beta_{1} + 4 \beta_{3} - 7 \beta_{5} ) q^{63} + ( 13 \beta_{1} + 16 \beta_{3} + 2 \beta_{5} ) q^{67} + ( 26 + 13 \beta_{2} + \beta_{4} ) q^{69} + ( -10 \beta_{3} - 8 \beta_{5} ) q^{71} + ( -10 - 6 \beta_{2} - 4 \beta_{4} ) q^{73} + ( -62 - 2 \beta_{2} - 12 \beta_{4} ) q^{77} + ( -12 \beta_{1} - 20 \beta_{3} + 4 \beta_{5} ) q^{79} + ( 33 + 13 \beta_{2} + 7 \beta_{4} ) q^{81} + ( 3 \beta_{1} + 16 \beta_{3} + 14 \beta_{5} ) q^{83} + ( -26 \beta_{1} + 24 \beta_{3} + 8 \beta_{5} ) q^{87} + ( -22 + 12 \beta_{2} + 16 \beta_{4} ) q^{89} + ( 8 \beta_{1} + 10 \beta_{3} ) q^{91} + ( -56 + 8 \beta_{4} ) q^{93} + ( -32 - 2 \beta_{2} - 6 \beta_{4} ) q^{97} + ( -34 \beta_{1} + 27 \beta_{3} - 6 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 18 q^{9} + O(q^{10}) \) \( 6 q - 18 q^{9} - 80 q^{17} + 8 q^{21} - 44 q^{29} - 144 q^{33} - 208 q^{37} - 68 q^{41} + 62 q^{49} - 64 q^{53} - 400 q^{57} - 100 q^{61} + 184 q^{69} - 80 q^{73} - 400 q^{77} + 238 q^{81} - 76 q^{89} - 320 q^{93} - 208 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 9 x^{4} + 14 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{4} + 20 \nu^{2} - 9 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( 4 \nu^{5} + 40 \nu^{3} + 76 \nu \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( 4 \nu^{4} + 40 \nu^{2} + 51 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( -16 \nu^{5} - 140 \nu^{3} - 194 \nu \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} - \beta_{2} - 12\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} + 4 \beta_{3} - 11 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-5 \beta_{4} + 10 \beta_{2} + 69\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-10 \beta_{5} - 35 \beta_{3} + 72 \beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
2.65109i
1.37720i
0.273891i
0.273891i
1.37720i
2.65109i
0 5.30219i 0 0 0 0.206625i 0 −19.1132 0
351.2 0 2.75441i 0 0 0 3.84997i 0 1.41325 0
351.3 0 0.547781i 0 0 0 10.0566i 0 8.69994 0
351.4 0 0.547781i 0 0 0 10.0566i 0 8.69994 0
351.5 0 2.75441i 0 0 0 3.84997i 0 1.41325 0
351.6 0 5.30219i 0 0 0 0.206625i 0 −19.1132 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 351.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.3.b.h 6
4.b odd 2 1 inner 800.3.b.h 6
5.b even 2 1 800.3.b.i 6
5.c odd 4 1 160.3.h.a 6
5.c odd 4 1 160.3.h.b yes 6
8.b even 2 1 1600.3.b.v 6
8.d odd 2 1 1600.3.b.v 6
15.e even 4 1 1440.3.j.a 6
15.e even 4 1 1440.3.j.b 6
20.d odd 2 1 800.3.b.i 6
20.e even 4 1 160.3.h.a 6
20.e even 4 1 160.3.h.b yes 6
40.e odd 2 1 1600.3.b.w 6
40.f even 2 1 1600.3.b.w 6
40.i odd 4 1 320.3.h.f 6
40.i odd 4 1 320.3.h.g 6
40.k even 4 1 320.3.h.f 6
40.k even 4 1 320.3.h.g 6
60.l odd 4 1 1440.3.j.a 6
60.l odd 4 1 1440.3.j.b 6
80.i odd 4 1 1280.3.e.f 6
80.i odd 4 1 1280.3.e.g 6
80.j even 4 1 1280.3.e.h 6
80.j even 4 1 1280.3.e.i 6
80.s even 4 1 1280.3.e.f 6
80.s even 4 1 1280.3.e.g 6
80.t odd 4 1 1280.3.e.h 6
80.t odd 4 1 1280.3.e.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.h.a 6 5.c odd 4 1
160.3.h.a 6 20.e even 4 1
160.3.h.b yes 6 5.c odd 4 1
160.3.h.b yes 6 20.e even 4 1
320.3.h.f 6 40.i odd 4 1
320.3.h.f 6 40.k even 4 1
320.3.h.g 6 40.i odd 4 1
320.3.h.g 6 40.k even 4 1
800.3.b.h 6 1.a even 1 1 trivial
800.3.b.h 6 4.b odd 2 1 inner
800.3.b.i 6 5.b even 2 1
800.3.b.i 6 20.d odd 2 1
1280.3.e.f 6 80.i odd 4 1
1280.3.e.f 6 80.s even 4 1
1280.3.e.g 6 80.i odd 4 1
1280.3.e.g 6 80.s even 4 1
1280.3.e.h 6 80.j even 4 1
1280.3.e.h 6 80.t odd 4 1
1280.3.e.i 6 80.j even 4 1
1280.3.e.i 6 80.t odd 4 1
1440.3.j.a 6 15.e even 4 1
1440.3.j.a 6 60.l odd 4 1
1440.3.j.b 6 15.e even 4 1
1440.3.j.b 6 60.l odd 4 1
1600.3.b.v 6 8.b even 2 1
1600.3.b.v 6 8.d odd 2 1
1600.3.b.w 6 40.e odd 2 1
1600.3.b.w 6 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{6} + 36 T_{3}^{4} + 224 T_{3}^{2} + 64 \)
\( T_{13}^{3} - 208 T_{13} + 832 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 64 + 224 T^{2} + 36 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 64 + 1504 T^{2} + 116 T^{4} + T^{6} \)
$11$ \( 2560000 + 86784 T^{2} + 560 T^{4} + T^{6} \)
$13$ \( ( 832 - 208 T + T^{3} )^{2} \)
$17$ \( ( -2560 + 256 T + 40 T^{2} + T^{3} )^{2} \)
$19$ \( 11505664 + 533248 T^{2} + 1712 T^{4} + T^{6} \)
$23$ \( 83905600 + 645984 T^{2} + 1460 T^{4} + T^{6} \)
$29$ \( ( 2120 - 948 T + 22 T^{2} + T^{3} )^{2} \)
$31$ \( 419430400 + 1900544 T^{2} + 2560 T^{4} + T^{6} \)
$37$ \( ( 20800 + 2704 T + 104 T^{2} + T^{3} )^{2} \)
$41$ \( ( -5000 - 100 T + 34 T^{2} + T^{3} )^{2} \)
$43$ \( 39438400 + 2459744 T^{2} + 4260 T^{4} + T^{6} \)
$47$ \( 493550656 + 17317088 T^{2} + 8308 T^{4} + T^{6} \)
$53$ \( ( -224320 - 5968 T + 32 T^{2} + T^{3} )^{2} \)
$59$ \( 25416011776 + 41968384 T^{2} + 12464 T^{4} + T^{6} \)
$61$ \( ( -81544 - 1732 T + 50 T^{2} + T^{3} )^{2} \)
$67$ \( 613057600 + 30359904 T^{2} + 14180 T^{4} + T^{6} \)
$71$ \( 14231535616 + 19738624 T^{2} + 8384 T^{4} + T^{6} \)
$73$ \( ( -55808 - 1408 T + 40 T^{2} + T^{3} )^{2} \)
$79$ \( 37060870144 + 159514624 T^{2} + 25856 T^{4} + T^{6} \)
$83$ \( 233675560000 + 145948000 T^{2} + 24164 T^{4} + T^{6} \)
$89$ \( ( -155000 - 13940 T + 38 T^{2} + T^{3} )^{2} \)
$97$ \( ( 6656 + 1664 T + 104 T^{2} + T^{3} )^{2} \)
show more
show less