Properties

Label 50.6.b.b
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: \( 2 \)
Twist minimal: no (minimal twist has level 10)
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.

\(f(q)\) \(=\) \( q + 4 i q^{2} -6 i q^{3} -16 q^{4} + 24 q^{6} -118 i q^{7} -64 i q^{8} + 207 q^{9} +O(q^{10})\) \( q + 4 i q^{2} -6 i q^{3} -16 q^{4} + 24 q^{6} -118 i q^{7} -64 i q^{8} + 207 q^{9} + 192 q^{11} + 96 i q^{12} -1106 i q^{13} + 472 q^{14} + 256 q^{16} + 762 i q^{17} + 828 i q^{18} + 2740 q^{19} -708 q^{21} + 768 i q^{22} -1566 i q^{23} -384 q^{24} + 4424 q^{26} -2700 i q^{27} + 1888 i q^{28} -5910 q^{29} -6868 q^{31} + 1024 i q^{32} -1152 i q^{33} -3048 q^{34} -3312 q^{36} -5518 i q^{37} + 10960 i q^{38} -6636 q^{39} -378 q^{41} -2832 i q^{42} + 2434 i q^{43} -3072 q^{44} + 6264 q^{46} + 13122 i q^{47} -1536 i q^{48} + 2883 q^{49} + 4572 q^{51} + 17696 i q^{52} + 9174 i q^{53} + 10800 q^{54} -7552 q^{56} -16440 i q^{57} -23640 i q^{58} + 34980 q^{59} -9838 q^{61} -27472 i q^{62} -24426 i q^{63} -4096 q^{64} + 4608 q^{66} + 33722 i q^{67} -12192 i q^{68} -9396 q^{69} + 70212 q^{71} -13248 i q^{72} -21986 i q^{73} + 22072 q^{74} -43840 q^{76} -22656 i q^{77} -26544 i q^{78} -4520 q^{79} + 34101 q^{81} -1512 i q^{82} + 109074 i q^{83} + 11328 q^{84} -9736 q^{86} + 35460 i q^{87} -12288 i q^{88} -38490 q^{89} -130508 q^{91} + 25056 i q^{92} + 41208 i q^{93} -52488 q^{94} + 6144 q^{96} -1918 i q^{97} + 11532 i q^{98} + 39744 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + 48q^{6} + 414q^{9} + O(q^{10}) \) \( 2q - 32q^{4} + 48q^{6} + 414q^{9} + 384q^{11} + 944q^{14} + 512q^{16} + 5480q^{19} - 1416q^{21} - 768q^{24} + 8848q^{26} - 11820q^{29} - 13736q^{31} - 6096q^{34} - 6624q^{36} - 13272q^{39} - 756q^{41} - 6144q^{44} + 12528q^{46} + 5766q^{49} + 9144q^{51} + 21600q^{54} - 15104q^{56} + 69960q^{59} - 19676q^{61} - 8192q^{64} + 9216q^{66} - 18792q^{69} + 140424q^{71} + 44144q^{74} - 87680q^{76} - 9040q^{79} + 68202q^{81} + 22656q^{84} - 19472q^{86} - 76980q^{89} - 261016q^{91} - 104976q^{94} + 12288q^{96} + 79488q^{99} + O(q^{100}) \)

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 6.00000i −16.0000 0 24.0000 118.000i 64.0000i 207.000 0
49.2 4.00000i 6.00000i −16.0000 0 24.0000 118.000i 64.0000i 207.000 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.b 2
3.b odd 2 1 450.6.c.f 2
4.b odd 2 1 400.6.c.i 2
5.b even 2 1 inner 50.6.b.b 2
5.c odd 4 1 10.6.a.c 1
5.c odd 4 1 50.6.a.b 1
15.d odd 2 1 450.6.c.f 2
15.e even 4 1 90.6.a.b 1
15.e even 4 1 450.6.a.u 1
20.d odd 2 1 400.6.c.i 2
20.e even 4 1 80.6.a.c 1
20.e even 4 1 400.6.a.i 1
35.f even 4 1 490.6.a.k 1
40.i odd 4 1 320.6.a.f 1
40.k even 4 1 320.6.a.k 1
60.l odd 4 1 720.6.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.c 1 5.c odd 4 1
50.6.a.b 1 5.c odd 4 1
50.6.b.b 2 1.a even 1 1 trivial
50.6.b.b 2 5.b even 2 1 inner
80.6.a.c 1 20.e even 4 1
90.6.a.b 1 15.e even 4 1
320.6.a.f 1 40.i odd 4 1
320.6.a.k 1 40.k even 4 1
400.6.a.i 1 20.e even 4 1
400.6.c.i 2 4.b odd 2 1
400.6.c.i 2 20.d odd 2 1
450.6.a.u 1 15.e even 4 1
450.6.c.f 2 3.b odd 2 1
450.6.c.f 2 15.d odd 2 1
490.6.a.k 1 35.f even 4 1
720.6.a.v 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 36 \) acting on \(S_{6}^{\mathrm{new}}(50, [\chi])\).