Properties

Label 50.6.b.c
Level $50$
Weight $6$
Character orbit 50.b
Analytic conductor $8.019$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.01919099065\)
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: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 4 i q^{2} -11 i q^{3} -16 q^{4} + 44 q^{6} + 142 i q^{7} -64 i q^{8} + 122 q^{9} +O(q^{10})\) \( q + 4 i q^{2} -11 i q^{3} -16 q^{4} + 44 q^{6} + 142 i q^{7} -64 i q^{8} + 122 q^{9} + 777 q^{11} + 176 i q^{12} + 884 i q^{13} -568 q^{14} + 256 q^{16} + 27 i q^{17} + 488 i q^{18} -1145 q^{19} + 1562 q^{21} + 3108 i q^{22} + 1854 i q^{23} -704 q^{24} -3536 q^{26} -4015 i q^{27} -2272 i q^{28} + 4920 q^{29} + 1802 q^{31} + 1024 i q^{32} -8547 i q^{33} -108 q^{34} -1952 q^{36} -13178 i q^{37} -4580 i q^{38} + 9724 q^{39} -15123 q^{41} + 6248 i q^{42} + 7844 i q^{43} -12432 q^{44} -7416 q^{46} + 6732 i q^{47} -2816 i q^{48} -3357 q^{49} + 297 q^{51} -14144 i q^{52} + 3414 i q^{53} + 16060 q^{54} + 9088 q^{56} + 12595 i q^{57} + 19680 i q^{58} -33960 q^{59} + 47402 q^{61} + 7208 i q^{62} + 17324 i q^{63} -4096 q^{64} + 34188 q^{66} + 13177 i q^{67} -432 i q^{68} + 20394 q^{69} -7548 q^{71} -7808 i q^{72} -59821 i q^{73} + 52712 q^{74} + 18320 q^{76} + 110334 i q^{77} + 38896 i q^{78} -75830 q^{79} -14519 q^{81} -60492 i q^{82} + 46299 i q^{83} -24992 q^{84} -31376 q^{86} -54120 i q^{87} -49728 i q^{88} + 30585 q^{89} -125528 q^{91} -29664 i q^{92} -19822 i q^{93} -26928 q^{94} + 11264 q^{96} -104018 i q^{97} -13428 i q^{98} + 94794 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + 88q^{6} + 244q^{9} + O(q^{10}) \) \( 2q - 32q^{4} + 88q^{6} + 244q^{9} + 1554q^{11} - 1136q^{14} + 512q^{16} - 2290q^{19} + 3124q^{21} - 1408q^{24} - 7072q^{26} + 9840q^{29} + 3604q^{31} - 216q^{34} - 3904q^{36} + 19448q^{39} - 30246q^{41} - 24864q^{44} - 14832q^{46} - 6714q^{49} + 594q^{51} + 32120q^{54} + 18176q^{56} - 67920q^{59} + 94804q^{61} - 8192q^{64} + 68376q^{66} + 40788q^{69} - 15096q^{71} + 105424q^{74} + 36640q^{76} - 151660q^{79} - 29038q^{81} - 49984q^{84} - 62752q^{86} + 61170q^{89} - 251056q^{91} - 53856q^{94} + 22528q^{96} + 189588q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
4.00000i 11.0000i −16.0000 0 44.0000 142.000i 64.0000i 122.000 0
49.2 4.00000i 11.0000i −16.0000 0 44.0000 142.000i 64.0000i 122.000 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 50.6.b.c 2
3.b odd 2 1 450.6.c.a 2
4.b odd 2 1 400.6.c.g 2
5.b even 2 1 inner 50.6.b.c 2
5.c odd 4 1 50.6.a.a 1
5.c odd 4 1 50.6.a.f yes 1
15.d odd 2 1 450.6.c.a 2
15.e even 4 1 450.6.a.j 1
15.e even 4 1 450.6.a.n 1
20.d odd 2 1 400.6.c.g 2
20.e even 4 1 400.6.a.e 1
20.e even 4 1 400.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.6.a.a 1 5.c odd 4 1
50.6.a.f yes 1 5.c odd 4 1
50.6.b.c 2 1.a even 1 1 trivial
50.6.b.c 2 5.b even 2 1 inner
400.6.a.e 1 20.e even 4 1
400.6.a.j 1 20.e even 4 1
400.6.c.g 2 4.b odd 2 1
400.6.c.g 2 20.d odd 2 1
450.6.a.j 1 15.e even 4 1
450.6.a.n 1 15.e even 4 1
450.6.c.a 2 3.b odd 2 1
450.6.c.a 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 121 \) acting on \(S_{6}^{\mathrm{new}}(50, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( 121 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 20164 + T^{2} \)
$11$ \( ( -777 + T )^{2} \)
$13$ \( 781456 + T^{2} \)
$17$ \( 729 + T^{2} \)
$19$ \( ( 1145 + T )^{2} \)
$23$ \( 3437316 + T^{2} \)
$29$ \( ( -4920 + T )^{2} \)
$31$ \( ( -1802 + T )^{2} \)
$37$ \( 173659684 + T^{2} \)
$41$ \( ( 15123 + T )^{2} \)
$43$ \( 61528336 + T^{2} \)
$47$ \( 45319824 + T^{2} \)
$53$ \( 11655396 + T^{2} \)
$59$ \( ( 33960 + T )^{2} \)
$61$ \( ( -47402 + T )^{2} \)
$67$ \( 173633329 + T^{2} \)
$71$ \( ( 7548 + T )^{2} \)
$73$ \( 3578552041 + T^{2} \)
$79$ \( ( 75830 + T )^{2} \)
$83$ \( 2143597401 + T^{2} \)
$89$ \( ( -30585 + T )^{2} \)
$97$ \( 10819744324 + T^{2} \)
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