Properties

Label 550.6.b.g
Level $550$
Weight $6$
Character orbit 550.b
Analytic conductor $88.211$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 550.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(88.2111008971\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
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.

\(f(q)\) \(=\) \( q - 4 i q^{2} + 21 i q^{3} - 16 q^{4} + 84 q^{6} + 98 i q^{7} + 64 i q^{8} - 198 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 4 i q^{2} + 21 i q^{3} - 16 q^{4} + 84 q^{6} + 98 i q^{7} + 64 i q^{8} - 198 q^{9} + 121 q^{11} - 336 i q^{12} - 824 i q^{13} + 392 q^{14} + 256 q^{16} + 978 i q^{17} + 792 i q^{18} + 2140 q^{19} - 2058 q^{21} - 484 i q^{22} - 3699 i q^{23} - 1344 q^{24} - 3296 q^{26} + 945 i q^{27} - 1568 i q^{28} - 3480 q^{29} - 7813 q^{31} - 1024 i q^{32} + 2541 i q^{33} + 3912 q^{34} + 3168 q^{36} - 13597 i q^{37} - 8560 i q^{38} + 17304 q^{39} + 6492 q^{41} + 8232 i q^{42} - 14234 i q^{43} - 1936 q^{44} - 14796 q^{46} - 20352 i q^{47} + 5376 i q^{48} + 7203 q^{49} - 20538 q^{51} + 13184 i q^{52} + 366 i q^{53} + 3780 q^{54} - 6272 q^{56} + 44940 i q^{57} + 13920 i q^{58} - 9825 q^{59} + 26132 q^{61} + 31252 i q^{62} - 19404 i q^{63} - 4096 q^{64} + 10164 q^{66} + 17093 i q^{67} - 15648 i q^{68} + 77679 q^{69} - 23583 q^{71} - 12672 i q^{72} + 35176 i q^{73} - 54388 q^{74} - 34240 q^{76} + 11858 i q^{77} - 69216 i q^{78} + 42490 q^{79} - 67959 q^{81} - 25968 i q^{82} - 22674 i q^{83} + 32928 q^{84} - 56936 q^{86} - 73080 i q^{87} + 7744 i q^{88} + 17145 q^{89} + 80752 q^{91} + 59184 i q^{92} - 164073 i q^{93} - 81408 q^{94} + 21504 q^{96} - 30727 i q^{97} - 28812 i q^{98} - 23958 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} + 168 q^{6} - 396 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} + 168 q^{6} - 396 q^{9} + 242 q^{11} + 784 q^{14} + 512 q^{16} + 4280 q^{19} - 4116 q^{21} - 2688 q^{24} - 6592 q^{26} - 6960 q^{29} - 15626 q^{31} + 7824 q^{34} + 6336 q^{36} + 34608 q^{39} + 12984 q^{41} - 3872 q^{44} - 29592 q^{46} + 14406 q^{49} - 41076 q^{51} + 7560 q^{54} - 12544 q^{56} - 19650 q^{59} + 52264 q^{61} - 8192 q^{64} + 20328 q^{66} + 155358 q^{69} - 47166 q^{71} - 108776 q^{74} - 68480 q^{76} + 84980 q^{79} - 135918 q^{81} + 65856 q^{84} - 113872 q^{86} + 34290 q^{89} + 161504 q^{91} - 162816 q^{94} + 43008 q^{96} - 47916 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/550\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
4.00000i 21.0000i −16.0000 0 84.0000 98.0000i 64.0000i −198.000 0
199.2 4.00000i 21.0000i −16.0000 0 84.0000 98.0000i 64.0000i −198.000 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
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 550.6.b.g 2
5.b even 2 1 inner 550.6.b.g 2
5.c odd 4 1 22.6.a.a 1
5.c odd 4 1 550.6.a.g 1
15.e even 4 1 198.6.a.d 1
20.e even 4 1 176.6.a.d 1
35.f even 4 1 1078.6.a.b 1
40.i odd 4 1 704.6.a.i 1
40.k even 4 1 704.6.a.b 1
55.e even 4 1 242.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.a 1 5.c odd 4 1
176.6.a.d 1 20.e even 4 1
198.6.a.d 1 15.e even 4 1
242.6.a.c 1 55.e even 4 1
550.6.a.g 1 5.c odd 4 1
550.6.b.g 2 1.a even 1 1 trivial
550.6.b.g 2 5.b even 2 1 inner
704.6.a.b 1 40.k even 4 1
704.6.a.i 1 40.i odd 4 1
1078.6.a.b 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(550, [\chi])\):

\( T_{3}^{2} + 441 \) Copy content Toggle raw display
\( T_{7}^{2} + 9604 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 441 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 9604 \) Copy content Toggle raw display
$11$ \( (T - 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 678976 \) Copy content Toggle raw display
$17$ \( T^{2} + 956484 \) Copy content Toggle raw display
$19$ \( (T - 2140)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 13682601 \) Copy content Toggle raw display
$29$ \( (T + 3480)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7813)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 184878409 \) Copy content Toggle raw display
$41$ \( (T - 6492)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 202606756 \) Copy content Toggle raw display
$47$ \( T^{2} + 414203904 \) Copy content Toggle raw display
$53$ \( T^{2} + 133956 \) Copy content Toggle raw display
$59$ \( (T + 9825)^{2} \) Copy content Toggle raw display
$61$ \( (T - 26132)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 292170649 \) Copy content Toggle raw display
$71$ \( (T + 23583)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1237350976 \) Copy content Toggle raw display
$79$ \( (T - 42490)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 514110276 \) Copy content Toggle raw display
$89$ \( (T - 17145)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 944148529 \) Copy content Toggle raw display
show more
show less