Properties

Label 950.2.e.g
Level $950$
Weight $2$
Character orbit 950.e
Analytic conductor $7.586$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{6} + 2 q^{7} - q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{6} + 2 q^{7} - q^{8} + 2 \zeta_{6} q^{9} - q^{12} + 6 \zeta_{6} q^{13} + ( 2 - 2 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( 7 - 7 \zeta_{6} ) q^{17} + 2 q^{18} + ( 5 - 3 \zeta_{6} ) q^{19} + ( 2 - 2 \zeta_{6} ) q^{21} + 2 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{24} + 6 q^{26} + 5 q^{27} -2 \zeta_{6} q^{28} -10 \zeta_{6} q^{29} -2 q^{31} + \zeta_{6} q^{32} -7 \zeta_{6} q^{34} + ( 2 - 2 \zeta_{6} ) q^{36} -4 q^{37} + ( 2 - 5 \zeta_{6} ) q^{38} + 6 q^{39} + ( -2 + 2 \zeta_{6} ) q^{41} -2 \zeta_{6} q^{42} + ( -12 + 12 \zeta_{6} ) q^{43} + 2 q^{46} + \zeta_{6} q^{48} -3 q^{49} -7 \zeta_{6} q^{51} + ( 6 - 6 \zeta_{6} ) q^{52} + ( 5 - 5 \zeta_{6} ) q^{54} -2 q^{56} + ( 2 - 5 \zeta_{6} ) q^{57} -10 q^{58} + ( 1 - \zeta_{6} ) q^{59} -8 \zeta_{6} q^{61} + ( -2 + 2 \zeta_{6} ) q^{62} + 4 \zeta_{6} q^{63} + q^{64} + 8 \zeta_{6} q^{67} -7 q^{68} + 2 q^{69} + ( 12 - 12 \zeta_{6} ) q^{71} -2 \zeta_{6} q^{72} + ( -3 + 3 \zeta_{6} ) q^{73} + ( -4 + 4 \zeta_{6} ) q^{74} + ( -3 - 2 \zeta_{6} ) q^{76} + ( 6 - 6 \zeta_{6} ) q^{78} + ( 4 - 4 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 2 \zeta_{6} q^{82} + 13 q^{83} -2 q^{84} + 12 \zeta_{6} q^{86} -10 q^{87} + 13 \zeta_{6} q^{89} + 12 \zeta_{6} q^{91} + ( 2 - 2 \zeta_{6} ) q^{92} + ( -2 + 2 \zeta_{6} ) q^{93} + q^{96} + ( 15 - 15 \zeta_{6} ) q^{97} + ( -3 + 3 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{3} - q^{4} - q^{6} + 4q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + q^{2} + q^{3} - q^{4} - q^{6} + 4q^{7} - 2q^{8} + 2q^{9} - 2q^{12} + 6q^{13} + 2q^{14} - q^{16} + 7q^{17} + 4q^{18} + 7q^{19} + 2q^{21} + 2q^{23} - q^{24} + 12q^{26} + 10q^{27} - 2q^{28} - 10q^{29} - 4q^{31} + q^{32} - 7q^{34} + 2q^{36} - 8q^{37} - q^{38} + 12q^{39} - 2q^{41} - 2q^{42} - 12q^{43} + 4q^{46} + q^{48} - 6q^{49} - 7q^{51} + 6q^{52} + 5q^{54} - 4q^{56} - q^{57} - 20q^{58} + q^{59} - 8q^{61} - 2q^{62} + 4q^{63} + 2q^{64} + 8q^{67} - 14q^{68} + 4q^{69} + 12q^{71} - 2q^{72} - 3q^{73} - 4q^{74} - 8q^{76} + 6q^{78} + 4q^{79} - q^{81} + 2q^{82} + 26q^{83} - 4q^{84} + 12q^{86} - 20q^{87} + 13q^{89} + 12q^{91} + 2q^{92} - 2q^{93} + 2q^{96} + 15q^{97} - 3q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i 2.00000 −1.00000 1.00000 + 1.73205i 0
501.1 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.00000 −1.00000 1.00000 1.73205i 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
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.e.g yes 2
5.b even 2 1 950.2.e.b 2
5.c odd 4 2 950.2.j.b 4
19.c even 3 1 inner 950.2.e.g yes 2
95.i even 6 1 950.2.e.b 2
95.m odd 12 2 950.2.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.e.b 2 5.b even 2 1
950.2.e.b 2 95.i even 6 1
950.2.e.g yes 2 1.a even 1 1 trivial
950.2.e.g yes 2 19.c even 3 1 inner
950.2.j.b 4 5.c odd 4 2
950.2.j.b 4 95.m odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \)
\( T_{7} - 2 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 36 - 6 T + T^{2} \)
$17$ \( 49 - 7 T + T^{2} \)
$19$ \( 19 - 7 T + T^{2} \)
$23$ \( 4 - 2 T + T^{2} \)
$29$ \( 100 + 10 T + T^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( ( 4 + T )^{2} \)
$41$ \( 4 + 2 T + T^{2} \)
$43$ \( 144 + 12 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( 1 - T + T^{2} \)
$61$ \( 64 + 8 T + T^{2} \)
$67$ \( 64 - 8 T + T^{2} \)
$71$ \( 144 - 12 T + T^{2} \)
$73$ \( 9 + 3 T + T^{2} \)
$79$ \( 16 - 4 T + T^{2} \)
$83$ \( ( -13 + T )^{2} \)
$89$ \( 169 - 13 T + T^{2} \)
$97$ \( 225 - 15 T + T^{2} \)
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