Properties

Label 930.2.i.d
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.

\(f(q)\) \(=\) \( q + q^{2} + (\zeta_{6} - 1) q^{3} + q^{4} - \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{6} + (5 \zeta_{6} - 5) q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\zeta_{6} - 1) q^{3} + q^{4} - \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{6} + (5 \zeta_{6} - 5) q^{7} + q^{8} - \zeta_{6} q^{9} - \zeta_{6} q^{10} + \zeta_{6} q^{11} + (\zeta_{6} - 1) q^{12} - 4 \zeta_{6} q^{13} + (5 \zeta_{6} - 5) q^{14} + q^{15} + q^{16} + (4 \zeta_{6} - 4) q^{17} - \zeta_{6} q^{18} + (4 \zeta_{6} - 4) q^{19} - \zeta_{6} q^{20} - 5 \zeta_{6} q^{21} + \zeta_{6} q^{22} + 2 q^{23} + (\zeta_{6} - 1) q^{24} + (\zeta_{6} - 1) q^{25} - 4 \zeta_{6} q^{26} + q^{27} + (5 \zeta_{6} - 5) q^{28} - 9 q^{29} + q^{30} + ( - \zeta_{6} - 5) q^{31} + q^{32} - q^{33} + (4 \zeta_{6} - 4) q^{34} + 5 q^{35} - \zeta_{6} q^{36} + (2 \zeta_{6} - 2) q^{37} + (4 \zeta_{6} - 4) q^{38} + 4 q^{39} - \zeta_{6} q^{40} + 2 \zeta_{6} q^{41} - 5 \zeta_{6} q^{42} + (8 \zeta_{6} - 8) q^{43} + \zeta_{6} q^{44} + (\zeta_{6} - 1) q^{45} + 2 q^{46} + 8 q^{47} + (\zeta_{6} - 1) q^{48} - 18 \zeta_{6} q^{49} + (\zeta_{6} - 1) q^{50} - 4 \zeta_{6} q^{51} - 4 \zeta_{6} q^{52} + 9 \zeta_{6} q^{53} + q^{54} + ( - \zeta_{6} + 1) q^{55} + (5 \zeta_{6} - 5) q^{56} - 4 \zeta_{6} q^{57} - 9 q^{58} + ( - 3 \zeta_{6} + 3) q^{59} + q^{60} + 6 q^{61} + ( - \zeta_{6} - 5) q^{62} + 5 q^{63} + q^{64} + (4 \zeta_{6} - 4) q^{65} - q^{66} + 10 \zeta_{6} q^{67} + (4 \zeta_{6} - 4) q^{68} + (2 \zeta_{6} - 2) q^{69} + 5 q^{70} + 4 \zeta_{6} q^{71} - \zeta_{6} q^{72} - 2 \zeta_{6} q^{73} + (2 \zeta_{6} - 2) q^{74} - \zeta_{6} q^{75} + (4 \zeta_{6} - 4) q^{76} - 5 q^{77} + 4 q^{78} + ( - 12 \zeta_{6} + 12) q^{79} - \zeta_{6} q^{80} + (\zeta_{6} - 1) q^{81} + 2 \zeta_{6} q^{82} + 9 \zeta_{6} q^{83} - 5 \zeta_{6} q^{84} + 4 q^{85} + (8 \zeta_{6} - 8) q^{86} + ( - 9 \zeta_{6} + 9) q^{87} + \zeta_{6} q^{88} - 6 q^{89} + (\zeta_{6} - 1) q^{90} + 20 q^{91} + 2 q^{92} + ( - 5 \zeta_{6} + 6) q^{93} + 8 q^{94} + 4 q^{95} + (\zeta_{6} - 1) q^{96} - q^{97} - 18 \zeta_{6} q^{98} + ( - \zeta_{6} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - q^{6} - 5 q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - q^{6} - 5 q^{7} + 2 q^{8} - q^{9} - q^{10} + q^{11} - q^{12} - 4 q^{13} - 5 q^{14} + 2 q^{15} + 2 q^{16} - 4 q^{17} - q^{18} - 4 q^{19} - q^{20} - 5 q^{21} + q^{22} + 4 q^{23} - q^{24} - q^{25} - 4 q^{26} + 2 q^{27} - 5 q^{28} - 18 q^{29} + 2 q^{30} - 11 q^{31} + 2 q^{32} - 2 q^{33} - 4 q^{34} + 10 q^{35} - q^{36} - 2 q^{37} - 4 q^{38} + 8 q^{39} - q^{40} + 2 q^{41} - 5 q^{42} - 8 q^{43} + q^{44} - q^{45} + 4 q^{46} + 16 q^{47} - q^{48} - 18 q^{49} - q^{50} - 4 q^{51} - 4 q^{52} + 9 q^{53} + 2 q^{54} + q^{55} - 5 q^{56} - 4 q^{57} - 18 q^{58} + 3 q^{59} + 2 q^{60} + 12 q^{61} - 11 q^{62} + 10 q^{63} + 2 q^{64} - 4 q^{65} - 2 q^{66} + 10 q^{67} - 4 q^{68} - 2 q^{69} + 10 q^{70} + 4 q^{71} - q^{72} - 2 q^{73} - 2 q^{74} - q^{75} - 4 q^{76} - 10 q^{77} + 8 q^{78} + 12 q^{79} - q^{80} - q^{81} + 2 q^{82} + 9 q^{83} - 5 q^{84} + 8 q^{85} - 8 q^{86} + 9 q^{87} + q^{88} - 12 q^{89} - q^{90} + 40 q^{91} + 4 q^{92} + 7 q^{93} + 16 q^{94} + 8 q^{95} - q^{96} - 2 q^{97} - 18 q^{98} + q^{99}+O(q^{100}) \) 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 −2.50000 4.33013i 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 −2.50000 + 4.33013i 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.d 2
31.c even 3 1 inner 930.2.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.d 2 1.a even 1 1 trivial
930.2.i.d 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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{2} + 5T_{7} + 25 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} + 1 \) 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
$7$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( (T - 2)^{2} \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 11T + 31 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( (T + 1)^{2} \) Copy content Toggle raw display
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