Properties

Label 800.2.ba.b
Level $800$
Weight $2$
Character orbit 800.ba
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.ba (of order \(8\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + ( 1 - \zeta_{8}^{3} ) q^{3} + 2 q^{4} + ( 1 + \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( -1 + \zeta_{8}^{2} ) q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( 1 - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + ( 1 - \zeta_{8}^{3} ) q^{3} + 2 q^{4} + ( 1 + \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( -1 + \zeta_{8}^{2} ) q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( 1 - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{9} + ( -2 + 2 \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{11} + ( 2 - 2 \zeta_{8}^{3} ) q^{12} + ( -\zeta_{8} - \zeta_{8}^{2} ) q^{13} + 2 \zeta_{8}^{3} q^{14} + 4 q^{16} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{17} + ( -1 + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{18} + ( 2 + 3 \zeta_{8} - 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{19} + ( -1 + \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{21} + ( 3 - 3 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{22} + ( -3 + 4 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{23} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{24} + ( -1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{26} + ( 1 - \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{27} + ( -2 + 2 \zeta_{8}^{2} ) q^{28} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{29} -4 q^{31} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{33} + 4 q^{34} + ( 2 - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{36} + ( \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{37} + ( 5 - \zeta_{8} + \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{38} + ( -1 - 2 \zeta_{8} - \zeta_{8}^{2} ) q^{39} + ( -3 + 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{41} + ( 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{42} + ( -3 - 3 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{43} + ( -4 + 4 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{44} + ( 4 - 6 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{46} + ( -6 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{47} + ( 4 - 4 \zeta_{8}^{3} ) q^{48} + 5 \zeta_{8}^{2} q^{49} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{51} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{52} + ( -4 - 4 \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{53} + ( -4 + 4 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{54} + 4 \zeta_{8}^{3} q^{56} + ( 5 - 5 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{57} + ( -3 - \zeta_{8} - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{58} + ( 4 + 4 \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{59} + ( 1 + \zeta_{8}^{3} ) q^{61} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{62} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{63} + 8 q^{64} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{66} + ( -3 + 2 \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{67} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{68} + ( 1 + \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{69} + ( -3 + 4 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{71} + ( -2 + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{72} + ( -7 - 7 \zeta_{8}^{2} ) q^{73} + ( -1 + \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{74} + ( 4 + 6 \zeta_{8} - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{76} + ( 3 - \zeta_{8} - \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{77} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{78} + 6 \zeta_{8}^{2} q^{79} + ( 5 \zeta_{8} + 3 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{81} + ( -4 + 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{82} + ( 1 - 4 \zeta_{8} - 4 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{83} + ( -2 + 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{84} + ( -7 - 7 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{86} + ( -1 - 4 \zeta_{8} - \zeta_{8}^{2} ) q^{87} + ( 6 - 6 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{88} + ( -3 + 8 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{89} + ( 1 + \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{91} + ( -6 + 8 \zeta_{8} - 6 \zeta_{8}^{2} ) q^{92} + ( -4 + 4 \zeta_{8}^{3} ) q^{93} + ( -8 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{94} + ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{96} + ( 6 \zeta_{8} - 10 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{97} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{98} + ( -5 + 2 \zeta_{8} + 2 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 8q^{4} + 4q^{6} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 8q^{4} + 4q^{6} - 4q^{7} + 4q^{9} - 8q^{11} + 8q^{12} + 16q^{16} - 4q^{18} + 8q^{19} - 4q^{21} + 12q^{22} - 12q^{23} + 8q^{24} - 4q^{26} + 4q^{27} - 8q^{28} + 4q^{29} - 16q^{31} + 16q^{34} + 8q^{36} + 20q^{38} - 4q^{39} - 12q^{41} - 12q^{43} - 16q^{44} + 16q^{46} - 24q^{47} + 16q^{48} + 8q^{51} - 16q^{53} - 16q^{54} + 20q^{57} - 12q^{58} + 16q^{59} + 4q^{61} + 32q^{64} - 12q^{67} + 4q^{69} - 12q^{71} - 8q^{72} - 28q^{73} - 4q^{74} + 16q^{76} + 12q^{77} - 8q^{78} - 16q^{82} + 4q^{83} - 8q^{84} - 28q^{86} - 4q^{87} + 24q^{88} - 12q^{89} + 4q^{91} - 24q^{92} - 16q^{93} - 32q^{94} + 16q^{96} - 20q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(\zeta_{8}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−1.41421 0.292893 + 0.707107i 2.00000 0 −0.414214 1.00000i −1.00000 + 1.00000i −2.82843 1.70711 1.70711i 0
349.1 −1.41421 0.292893 0.707107i 2.00000 0 −0.414214 + 1.00000i −1.00000 1.00000i −2.82843 1.70711 + 1.70711i 0
549.1 1.41421 1.70711 0.707107i 2.00000 0 2.41421 1.00000i −1.00000 + 1.00000i 2.82843 0.292893 0.292893i 0
749.1 1.41421 1.70711 + 0.707107i 2.00000 0 2.41421 + 1.00000i −1.00000 1.00000i 2.82843 0.292893 + 0.292893i 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
160.z even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.ba.b 4
5.b even 2 1 800.2.ba.a 4
5.c odd 4 1 32.2.g.a 4
5.c odd 4 1 800.2.y.a 4
15.e even 4 1 288.2.v.a 4
20.e even 4 1 128.2.g.a 4
32.g even 8 1 800.2.ba.a 4
40.i odd 4 1 256.2.g.b 4
40.k even 4 1 256.2.g.a 4
60.l odd 4 1 1152.2.v.a 4
80.i odd 4 1 512.2.g.a 4
80.j even 4 1 512.2.g.b 4
80.s even 4 1 512.2.g.c 4
80.t odd 4 1 512.2.g.d 4
160.u even 8 1 512.2.g.b 4
160.u even 8 1 512.2.g.c 4
160.v odd 8 1 32.2.g.a 4
160.v odd 8 1 256.2.g.b 4
160.z even 8 1 inner 800.2.ba.b 4
160.ba even 8 1 128.2.g.a 4
160.ba even 8 1 256.2.g.a 4
160.bb odd 8 1 512.2.g.a 4
160.bb odd 8 1 512.2.g.d 4
160.bb odd 8 1 800.2.y.a 4
320.bi odd 16 2 4096.2.a.e 4
320.bj even 16 2 4096.2.a.f 4
480.br even 8 1 288.2.v.a 4
480.ca odd 8 1 1152.2.v.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.a 4 5.c odd 4 1
32.2.g.a 4 160.v odd 8 1
128.2.g.a 4 20.e even 4 1
128.2.g.a 4 160.ba even 8 1
256.2.g.a 4 40.k even 4 1
256.2.g.a 4 160.ba even 8 1
256.2.g.b 4 40.i odd 4 1
256.2.g.b 4 160.v odd 8 1
288.2.v.a 4 15.e even 4 1
288.2.v.a 4 480.br even 8 1
512.2.g.a 4 80.i odd 4 1
512.2.g.a 4 160.bb odd 8 1
512.2.g.b 4 80.j even 4 1
512.2.g.b 4 160.u even 8 1
512.2.g.c 4 80.s even 4 1
512.2.g.c 4 160.u even 8 1
512.2.g.d 4 80.t odd 4 1
512.2.g.d 4 160.bb odd 8 1
800.2.y.a 4 5.c odd 4 1
800.2.y.a 4 160.bb odd 8 1
800.2.ba.a 4 5.b even 2 1
800.2.ba.a 4 32.g even 8 1
800.2.ba.b 4 1.a even 1 1 trivial
800.2.ba.b 4 160.z even 8 1 inner
1152.2.v.a 4 60.l odd 4 1
1152.2.v.a 4 480.ca odd 8 1
4096.2.a.e 4 320.bi odd 16 2
4096.2.a.f 4 320.bj even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4 T_{3}^{3} + 6 T_{3}^{2} - 4 T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{2} \)
$3$ \( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 2 + 2 T + T^{2} )^{2} \)
$11$ \( 2 - 4 T + 18 T^{2} + 8 T^{3} + T^{4} \)
$13$ \( 2 - 4 T + 2 T^{2} + T^{4} \)
$17$ \( ( -8 + T^{2} )^{2} \)
$19$ \( 578 - 68 T + 18 T^{2} - 8 T^{3} + T^{4} \)
$23$ \( 4 + 24 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$29$ \( 98 - 28 T + 6 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( ( 4 + T )^{4} \)
$37$ \( 2 - 4 T + 2 T^{2} + T^{4} \)
$41$ \( 4 + 24 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$43$ \( 1922 + 124 T + 38 T^{2} + 12 T^{3} + T^{4} \)
$47$ \( ( 4 + 12 T + T^{2} )^{2} \)
$53$ \( 98 + 84 T + 82 T^{2} + 16 T^{3} + T^{4} \)
$59$ \( 1058 - 460 T + 114 T^{2} - 16 T^{3} + T^{4} \)
$61$ \( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} \)
$67$ \( 578 + 340 T + 86 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( 4 + 24 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$73$ \( ( 98 + 14 T + T^{2} )^{2} \)
$79$ \( ( 36 + T^{2} )^{2} \)
$83$ \( 1058 - 276 T + 22 T^{2} - 4 T^{3} + T^{4} \)
$89$ \( 2116 - 552 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$97$ \( 784 + 344 T^{2} + T^{4} \)
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