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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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:
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.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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{2} - T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 5T_{11} + 25 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

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