Properties

Label 51.2.d.b
Level 51
Weight 2
Character orbit 51.d
Analytic conductor 0.407
Analytic rank 0
Dimension 2
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 51 = 3 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 51.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.407237050309\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \( + i q^{3} \) \(- q^{4}\) \( + i q^{6} \) \( -4 i q^{7} \) \( -3 q^{8} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \( + i q^{3} \) \(- q^{4}\) \( + i q^{6} \) \( -4 i q^{7} \) \( -3 q^{8} \) \(- q^{9}\) \( + 4 i q^{11} \) \( -i q^{12} \) \( + 2 q^{13} \) \( -4 i q^{14} \) \(- q^{16}\) \( + ( 1 + 4 i ) q^{17} \) \(- q^{18}\) \( -4 q^{19} \) \( + 4 q^{21} \) \( + 4 i q^{22} \) \( -4 i q^{23} \) \( -3 i q^{24} \) \( + 5 q^{25} \) \( + 2 q^{26} \) \( -i q^{27} \) \( + 4 i q^{28} \) \( -4 i q^{31} \) \( + 5 q^{32} \) \( -4 q^{33} \) \( + ( 1 + 4 i ) q^{34} \) \(+ q^{36}\) \( + 8 i q^{37} \) \( -4 q^{38} \) \( + 2 i q^{39} \) \( -8 i q^{41} \) \( + 4 q^{42} \) \( -4 q^{43} \) \( -4 i q^{44} \) \( -4 i q^{46} \) \( -8 q^{47} \) \( -i q^{48} \) \( -9 q^{49} \) \( + 5 q^{50} \) \( + ( -4 + i ) q^{51} \) \( -2 q^{52} \) \( + 6 q^{53} \) \( -i q^{54} \) \( + 12 i q^{56} \) \( -4 i q^{57} \) \( -12 q^{59} \) \( -8 i q^{61} \) \( -4 i q^{62} \) \( + 4 i q^{63} \) \( + 7 q^{64} \) \( -4 q^{66} \) \( + 12 q^{67} \) \( + ( -1 - 4 i ) q^{68} \) \( + 4 q^{69} \) \( + 12 i q^{71} \) \( + 3 q^{72} \) \( + 8 i q^{74} \) \( + 5 i q^{75} \) \( + 4 q^{76} \) \( + 16 q^{77} \) \( + 2 i q^{78} \) \( -4 i q^{79} \) \(+ q^{81}\) \( -8 i q^{82} \) \( + 12 q^{83} \) \( -4 q^{84} \) \( -4 q^{86} \) \( -12 i q^{88} \) \( -10 q^{89} \) \( -8 i q^{91} \) \( + 4 i q^{92} \) \( + 4 q^{93} \) \( -8 q^{94} \) \( + 5 i q^{96} \) \( + 16 i q^{97} \) \( -9 q^{98} \) \( -4 i q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut 2q^{18} \) \(\mathstrut -\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 8q^{21} \) \(\mathstrut +\mathstrut 10q^{25} \) \(\mathstrut +\mathstrut 4q^{26} \) \(\mathstrut +\mathstrut 10q^{32} \) \(\mathstrut -\mathstrut 8q^{33} \) \(\mathstrut +\mathstrut 2q^{34} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 8q^{38} \) \(\mathstrut +\mathstrut 8q^{42} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut -\mathstrut 16q^{47} \) \(\mathstrut -\mathstrut 18q^{49} \) \(\mathstrut +\mathstrut 10q^{50} \) \(\mathstrut -\mathstrut 8q^{51} \) \(\mathstrut -\mathstrut 4q^{52} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut -\mathstrut 24q^{59} \) \(\mathstrut +\mathstrut 14q^{64} \) \(\mathstrut -\mathstrut 8q^{66} \) \(\mathstrut +\mathstrut 24q^{67} \) \(\mathstrut -\mathstrut 2q^{68} \) \(\mathstrut +\mathstrut 8q^{69} \) \(\mathstrut +\mathstrut 6q^{72} \) \(\mathstrut +\mathstrut 8q^{76} \) \(\mathstrut +\mathstrut 32q^{77} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 24q^{83} \) \(\mathstrut -\mathstrut 8q^{84} \) \(\mathstrut -\mathstrut 8q^{86} \) \(\mathstrut -\mathstrut 20q^{89} \) \(\mathstrut +\mathstrut 8q^{93} \) \(\mathstrut -\mathstrut 16q^{94} \) \(\mathstrut -\mathstrut 18q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/51\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
1.00000i
1.00000i
1.00000 1.00000i −1.00000 0 1.00000i 4.00000i −3.00000 −1.00000 0
16.2 1.00000 1.00000i −1.00000 0 1.00000i 4.00000i −3.00000 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
17.b Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(51, [\chi])\).