Properties

Label 930.2.i.i
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.

\(f(q)\) \(=\) \( q + q^{2} + ( 1 - \zeta_{6} ) q^{3} + q^{4} + \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{6} + ( 3 - 3 \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + q^{2} + ( 1 - \zeta_{6} ) q^{3} + q^{4} + \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{6} + ( 3 - 3 \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} + \zeta_{6} q^{10} + 3 \zeta_{6} q^{11} + ( 1 - \zeta_{6} ) q^{12} + 2 \zeta_{6} q^{13} + ( 3 - 3 \zeta_{6} ) q^{14} + q^{15} + q^{16} + ( 4 - 4 \zeta_{6} ) q^{17} -\zeta_{6} q^{18} + \zeta_{6} q^{20} -3 \zeta_{6} q^{21} + 3 \zeta_{6} q^{22} -4 q^{23} + ( 1 - \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} + 2 \zeta_{6} q^{26} - q^{27} + ( 3 - 3 \zeta_{6} ) q^{28} -5 q^{29} + q^{30} + ( 5 + \zeta_{6} ) q^{31} + q^{32} + 3 q^{33} + ( 4 - 4 \zeta_{6} ) q^{34} + 3 q^{35} -\zeta_{6} q^{36} + ( 6 - 6 \zeta_{6} ) q^{37} + 2 q^{39} + \zeta_{6} q^{40} + 2 \zeta_{6} q^{41} -3 \zeta_{6} q^{42} + ( 8 - 8 \zeta_{6} ) q^{43} + 3 \zeta_{6} q^{44} + ( 1 - \zeta_{6} ) q^{45} -4 q^{46} -8 q^{47} + ( 1 - \zeta_{6} ) q^{48} -2 \zeta_{6} q^{49} + ( -1 + \zeta_{6} ) q^{50} -4 \zeta_{6} q^{51} + 2 \zeta_{6} q^{52} + 3 \zeta_{6} q^{53} - q^{54} + ( -3 + 3 \zeta_{6} ) q^{55} + ( 3 - 3 \zeta_{6} ) q^{56} -5 q^{58} + ( -3 + 3 \zeta_{6} ) q^{59} + q^{60} -8 q^{61} + ( 5 + \zeta_{6} ) q^{62} -3 q^{63} + q^{64} + ( -2 + 2 \zeta_{6} ) q^{65} + 3 q^{66} + 10 \zeta_{6} q^{67} + ( 4 - 4 \zeta_{6} ) q^{68} + ( -4 + 4 \zeta_{6} ) q^{69} + 3 q^{70} + 8 \zeta_{6} q^{71} -\zeta_{6} q^{72} -14 \zeta_{6} q^{73} + ( 6 - 6 \zeta_{6} ) q^{74} + \zeta_{6} q^{75} + 9 q^{77} + 2 q^{78} + ( -8 + 8 \zeta_{6} ) q^{79} + \zeta_{6} q^{80} + ( -1 + \zeta_{6} ) q^{81} + 2 \zeta_{6} q^{82} + 9 \zeta_{6} q^{83} -3 \zeta_{6} q^{84} + 4 q^{85} + ( 8 - 8 \zeta_{6} ) q^{86} + ( -5 + 5 \zeta_{6} ) q^{87} + 3 \zeta_{6} q^{88} -8 q^{89} + ( 1 - \zeta_{6} ) q^{90} + 6 q^{91} -4 q^{92} + ( 6 - 5 \zeta_{6} ) q^{93} -8 q^{94} + ( 1 - \zeta_{6} ) q^{96} -7 q^{97} -2 \zeta_{6} q^{98} + ( 3 - 3 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + q^{6} + 3q^{7} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + q^{6} + 3q^{7} + 2q^{8} - q^{9} + q^{10} + 3q^{11} + q^{12} + 2q^{13} + 3q^{14} + 2q^{15} + 2q^{16} + 4q^{17} - q^{18} + q^{20} - 3q^{21} + 3q^{22} - 8q^{23} + q^{24} - q^{25} + 2q^{26} - 2q^{27} + 3q^{28} - 10q^{29} + 2q^{30} + 11q^{31} + 2q^{32} + 6q^{33} + 4q^{34} + 6q^{35} - q^{36} + 6q^{37} + 4q^{39} + q^{40} + 2q^{41} - 3q^{42} + 8q^{43} + 3q^{44} + q^{45} - 8q^{46} - 16q^{47} + q^{48} - 2q^{49} - q^{50} - 4q^{51} + 2q^{52} + 3q^{53} - 2q^{54} - 3q^{55} + 3q^{56} - 10q^{58} - 3q^{59} + 2q^{60} - 16q^{61} + 11q^{62} - 6q^{63} + 2q^{64} - 2q^{65} + 6q^{66} + 10q^{67} + 4q^{68} - 4q^{69} + 6q^{70} + 8q^{71} - q^{72} - 14q^{73} + 6q^{74} + q^{75} + 18q^{77} + 4q^{78} - 8q^{79} + q^{80} - q^{81} + 2q^{82} + 9q^{83} - 3q^{84} + 8q^{85} + 8q^{86} - 5q^{87} + 3q^{88} - 16q^{89} + q^{90} + 12q^{91} - 8q^{92} + 7q^{93} - 16q^{94} + q^{96} - 14q^{97} - 2q^{98} + 3q^{99} + O(q^{100}) \)

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 1.50000 + 2.59808i 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 1.50000 2.59808i 1.00000 −0.500000 0.866025i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.i 2
31.c even 3 1 inner 930.2.i.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.i 2 1.a even 1 1 trivial
930.2.i.i 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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{2} - 3 T_{7} + 9 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)
\( T_{13}^{2} - 2 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 9 - 3 T + T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( 4 - 2 T + T^{2} \)
$17$ \( 16 - 4 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( 31 - 11 T + T^{2} \)
$37$ \( 36 - 6 T + T^{2} \)
$41$ \( 4 - 2 T + T^{2} \)
$43$ \( 64 - 8 T + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( 9 - 3 T + T^{2} \)
$59$ \( 9 + 3 T + T^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( 100 - 10 T + T^{2} \)
$71$ \( 64 - 8 T + T^{2} \)
$73$ \( 196 + 14 T + T^{2} \)
$79$ \( 64 + 8 T + T^{2} \)
$83$ \( 81 - 9 T + T^{2} \)
$89$ \( ( 8 + T )^{2} \)
$97$ \( ( 7 + T )^{2} \)
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