Properties

Label 930.2.i.f
Level $930$
Weight $2$
Character orbit 930.i
Analytic conductor $7.426$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + ( -1 + \zeta_{6} ) q^{3} + q^{4} -\zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{6} + ( 1 - \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + q^{2} + ( -1 + \zeta_{6} ) q^{3} + q^{4} -\zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{6} + ( 1 - \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} -\zeta_{6} q^{10} -5 \zeta_{6} q^{11} + ( -1 + \zeta_{6} ) q^{12} -4 \zeta_{6} q^{13} + ( 1 - \zeta_{6} ) q^{14} + q^{15} + q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} -\zeta_{6} q^{18} + ( 2 - 2 \zeta_{6} ) q^{19} -\zeta_{6} q^{20} + \zeta_{6} q^{21} -5 \zeta_{6} q^{22} -4 q^{23} + ( -1 + \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} -4 \zeta_{6} q^{26} + q^{27} + ( 1 - \zeta_{6} ) q^{28} + 3 q^{29} + q^{30} + ( 1 + 5 \zeta_{6} ) q^{31} + q^{32} + 5 q^{33} + ( 2 - 2 \zeta_{6} ) q^{34} - q^{35} -\zeta_{6} q^{36} + ( 4 - 4 \zeta_{6} ) q^{37} + ( 2 - 2 \zeta_{6} ) q^{38} + 4 q^{39} -\zeta_{6} q^{40} -4 \zeta_{6} q^{41} + \zeta_{6} q^{42} + ( 4 - 4 \zeta_{6} ) q^{43} -5 \zeta_{6} q^{44} + ( -1 + \zeta_{6} ) q^{45} -4 q^{46} + 2 q^{47} + ( -1 + \zeta_{6} ) q^{48} + 6 \zeta_{6} q^{49} + ( -1 + \zeta_{6} ) q^{50} + 2 \zeta_{6} q^{51} -4 \zeta_{6} q^{52} -3 \zeta_{6} q^{53} + q^{54} + ( -5 + 5 \zeta_{6} ) q^{55} + ( 1 - \zeta_{6} ) q^{56} + 2 \zeta_{6} q^{57} + 3 q^{58} + ( -3 + 3 \zeta_{6} ) q^{59} + q^{60} + ( 1 + 5 \zeta_{6} ) q^{62} - q^{63} + q^{64} + ( -4 + 4 \zeta_{6} ) q^{65} + 5 q^{66} + 4 \zeta_{6} q^{67} + ( 2 - 2 \zeta_{6} ) q^{68} + ( 4 - 4 \zeta_{6} ) q^{69} - q^{70} -8 \zeta_{6} q^{71} -\zeta_{6} q^{72} -2 \zeta_{6} q^{73} + ( 4 - 4 \zeta_{6} ) q^{74} -\zeta_{6} q^{75} + ( 2 - 2 \zeta_{6} ) q^{76} -5 q^{77} + 4 q^{78} -\zeta_{6} q^{80} + ( -1 + \zeta_{6} ) q^{81} -4 \zeta_{6} q^{82} -9 \zeta_{6} q^{83} + \zeta_{6} q^{84} -2 q^{85} + ( 4 - 4 \zeta_{6} ) q^{86} + ( -3 + 3 \zeta_{6} ) q^{87} -5 \zeta_{6} q^{88} + 18 q^{89} + ( -1 + \zeta_{6} ) q^{90} -4 q^{91} -4 q^{92} + ( -6 + \zeta_{6} ) q^{93} + 2 q^{94} -2 q^{95} + ( -1 + \zeta_{6} ) q^{96} - q^{97} + 6 \zeta_{6} q^{98} + ( -5 + 5 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - q^{3} + 2q^{4} - q^{5} - q^{6} + q^{7} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - q^{3} + 2q^{4} - q^{5} - q^{6} + q^{7} + 2q^{8} - q^{9} - q^{10} - 5q^{11} - q^{12} - 4q^{13} + q^{14} + 2q^{15} + 2q^{16} + 2q^{17} - q^{18} + 2q^{19} - q^{20} + q^{21} - 5q^{22} - 8q^{23} - q^{24} - q^{25} - 4q^{26} + 2q^{27} + q^{28} + 6q^{29} + 2q^{30} + 7q^{31} + 2q^{32} + 10q^{33} + 2q^{34} - 2q^{35} - q^{36} + 4q^{37} + 2q^{38} + 8q^{39} - q^{40} - 4q^{41} + q^{42} + 4q^{43} - 5q^{44} - q^{45} - 8q^{46} + 4q^{47} - q^{48} + 6q^{49} - q^{50} + 2q^{51} - 4q^{52} - 3q^{53} + 2q^{54} - 5q^{55} + q^{56} + 2q^{57} + 6q^{58} - 3q^{59} + 2q^{60} + 7q^{62} - 2q^{63} + 2q^{64} - 4q^{65} + 10q^{66} + 4q^{67} + 2q^{68} + 4q^{69} - 2q^{70} - 8q^{71} - q^{72} - 2q^{73} + 4q^{74} - q^{75} + 2q^{76} - 10q^{77} + 8q^{78} - q^{80} - q^{81} - 4q^{82} - 9q^{83} + q^{84} - 4q^{85} + 4q^{86} - 3q^{87} - 5q^{88} + 36q^{89} - q^{90} - 8q^{91} - 8q^{92} - 11q^{93} + 4q^{94} - 4q^{95} - q^{96} - 2q^{97} + 6q^{98} - 5q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 −0.500000 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
811.1 1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i 1.00000 −0.500000 0.866025i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.i.f 2
31.c even 3 1 inner 930.2.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.f 2 1.a even 1 1 trivial
930.2.i.f 2 31.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{2} - T_{7} + 1 \)
\( T_{11}^{2} + 5 T_{11} + 25 \)
\( T_{13}^{2} + 4 T_{13} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 25 + 5 T + T^{2} \)
$13$ \( 16 + 4 T + T^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( 4 - 2 T + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( 31 - 7 T + T^{2} \)
$37$ \( 16 - 4 T + T^{2} \)
$41$ \( 16 + 4 T + T^{2} \)
$43$ \( 16 - 4 T + T^{2} \)
$47$ \( ( -2 + T )^{2} \)
$53$ \( 9 + 3 T + T^{2} \)
$59$ \( 9 + 3 T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 16 - 4 T + T^{2} \)
$71$ \( 64 + 8 T + T^{2} \)
$73$ \( 4 + 2 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 81 + 9 T + T^{2} \)
$89$ \( ( -18 + T )^{2} \)
$97$ \( ( 1 + T )^{2} \)
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