Properties

Label 850.2.b.i
Level $850$
Weight $2$
Character orbit 850.b
Analytic conductor $6.787$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 850.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.78728417181\)
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: no (minimal twist has level 170)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/850\mathbb{Z}\right)^\times\).

\(n\) \(477\) \(751\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
1.00000i
1.00000i
1.00000 1.00000i 1.00000 0 1.00000i 2.00000i 1.00000 2.00000 0
101.2 1.00000 1.00000i 1.00000 0 1.00000i 2.00000i 1.00000 2.00000 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
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 850.2.b.i 2
5.b even 2 1 850.2.b.c 2
5.c odd 4 1 170.2.d.a 2
5.c odd 4 1 170.2.d.b yes 2
15.e even 4 1 1530.2.f.b 2
15.e even 4 1 1530.2.f.e 2
17.b even 2 1 inner 850.2.b.i 2
20.e even 4 1 1360.2.o.a 2
20.e even 4 1 1360.2.o.b 2
85.c even 2 1 850.2.b.c 2
85.g odd 4 1 170.2.d.a 2
85.g odd 4 1 170.2.d.b yes 2
255.o even 4 1 1530.2.f.b 2
255.o even 4 1 1530.2.f.e 2
340.r even 4 1 1360.2.o.a 2
340.r even 4 1 1360.2.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.d.a 2 5.c odd 4 1
170.2.d.a 2 85.g odd 4 1
170.2.d.b yes 2 5.c odd 4 1
170.2.d.b yes 2 85.g odd 4 1
850.2.b.c 2 5.b even 2 1
850.2.b.c 2 85.c even 2 1
850.2.b.i 2 1.a even 1 1 trivial
850.2.b.i 2 17.b even 2 1 inner
1360.2.o.a 2 20.e even 4 1
1360.2.o.a 2 340.r even 4 1
1360.2.o.b 2 20.e even 4 1
1360.2.o.b 2 340.r even 4 1
1530.2.f.b 2 15.e even 4 1
1530.2.f.b 2 255.o even 4 1
1530.2.f.e 2 15.e even 4 1
1530.2.f.e 2 255.o even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(850, [\chi])\):

\( T_{3}^{2} + 1 \)
\( T_{7}^{2} + 4 \)
\( T_{13} + 1 \)