Properties

Label 930.2.i.l
Level $930$
Weight $2$
Character orbit 930.i
Analytic conductor $7.426$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} + ( 1 + \beta_{1} ) q^{5} + \beta_{1} q^{6} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + q^{8} + ( -1 - \beta_{1} ) q^{9} +O(q^{10})\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} + ( 1 + \beta_{1} ) q^{5} + \beta_{1} q^{6} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + q^{8} + ( -1 - \beta_{1} ) q^{9} + ( 1 + \beta_{1} ) q^{10} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{11} + \beta_{1} q^{12} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{14} - q^{15} + q^{16} + ( -\beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{17} + ( -1 - \beta_{1} ) q^{18} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( 1 + \beta_{1} ) q^{20} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{21} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{22} + ( 1 - \beta_{3} ) q^{23} + \beta_{1} q^{24} + \beta_{1} q^{25} + q^{27} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{28} + 3 q^{29} - q^{30} + ( 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} + q^{32} + ( -2 - \beta_{3} ) q^{33} + ( -\beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{34} + ( 2 - \beta_{3} ) q^{35} + ( -1 - \beta_{1} ) q^{36} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{37} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{38} + ( 1 + \beta_{1} ) q^{40} + ( 1 + \beta_{1} + \beta_{2} ) q^{41} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{42} + ( 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{43} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{44} -\beta_{1} q^{45} + ( 1 - \beta_{3} ) q^{46} + ( 3 - \beta_{3} ) q^{47} + \beta_{1} q^{48} + ( -2 - 2 \beta_{1} - 4 \beta_{2} ) q^{49} + \beta_{1} q^{50} + ( 1 + \beta_{1} + 3 \beta_{2} ) q^{51} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{53} + q^{54} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{55} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{56} + ( 1 + \beta_{1} - \beta_{2} ) q^{57} + 3 q^{58} + ( -4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{59} - q^{60} + ( -3 - 5 \beta_{3} ) q^{61} + ( 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{62} + ( -2 + \beta_{3} ) q^{63} + q^{64} + ( -2 - \beta_{3} ) q^{66} + ( -3 - 3 \beta_{1} + \beta_{2} ) q^{67} + ( -\beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{68} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{69} + ( 2 - \beta_{3} ) q^{70} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -1 - \beta_{1} ) q^{72} + ( 6 + 6 \beta_{1} + 4 \beta_{2} ) q^{73} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{74} + ( -1 - \beta_{1} ) q^{75} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{76} - q^{77} + ( -6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{79} + ( 1 + \beta_{1} ) q^{80} + \beta_{1} q^{81} + ( 1 + \beta_{1} + \beta_{2} ) q^{82} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{83} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{84} + ( 1 - 3 \beta_{3} ) q^{85} + ( 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{86} + 3 \beta_{1} q^{87} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{88} + ( 2 + 4 \beta_{3} ) q^{89} -\beta_{1} q^{90} + ( 1 - \beta_{3} ) q^{92} + ( -4 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{93} + ( 3 - \beta_{3} ) q^{94} + ( 1 + \beta_{3} ) q^{95} + \beta_{1} q^{96} + ( -3 + 2 \beta_{3} ) q^{97} + ( -2 - 2 \beta_{1} - 4 \beta_{2} ) q^{98} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 2q^{3} + 4q^{4} + 2q^{5} - 2q^{6} + 4q^{7} + 4q^{8} - 2q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 2q^{3} + 4q^{4} + 2q^{5} - 2q^{6} + 4q^{7} + 4q^{8} - 2q^{9} + 2q^{10} + 4q^{11} - 2q^{12} + 4q^{14} - 4q^{15} + 4q^{16} + 2q^{17} - 2q^{18} + 2q^{19} + 2q^{20} + 4q^{21} + 4q^{22} + 4q^{23} - 2q^{24} - 2q^{25} + 4q^{27} + 4q^{28} + 12q^{29} - 4q^{30} - 8q^{31} + 4q^{32} - 8q^{33} + 2q^{34} + 8q^{35} - 2q^{36} - 2q^{37} + 2q^{38} + 2q^{40} + 2q^{41} + 4q^{42} - 12q^{43} + 4q^{44} + 2q^{45} + 4q^{46} + 12q^{47} - 2q^{48} - 4q^{49} - 2q^{50} + 2q^{51} + 6q^{53} + 4q^{54} - 4q^{55} + 4q^{56} + 2q^{57} + 12q^{58} + 8q^{59} - 4q^{60} - 12q^{61} - 8q^{62} - 8q^{63} + 4q^{64} - 8q^{66} - 6q^{67} + 2q^{68} - 2q^{69} + 8q^{70} - 4q^{71} - 2q^{72} + 12q^{73} - 2q^{74} - 2q^{75} + 2q^{76} - 4q^{77} + 12q^{79} + 2q^{80} - 2q^{81} + 2q^{82} + 4q^{83} + 4q^{84} + 4q^{85} - 12q^{86} - 6q^{87} + 4q^{88} + 8q^{89} + 2q^{90} + 4q^{92} - 8q^{93} + 12q^{94} + 4q^{95} - 2q^{96} - 12q^{97} - 4q^{98} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + 2 \nu^{2} - 2 \nu - 1 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 2 \nu^{2} + 6 \nu - 1 \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 3 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2\)

Character values

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

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

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
−0.309017 + 0.535233i
0.809017 1.40126i
−0.309017 0.535233i
0.809017 + 1.40126i
1.00000 −0.500000 0.866025i 1.00000 0.500000 0.866025i −0.500000 0.866025i −0.118034 0.204441i 1.00000 −0.500000 + 0.866025i 0.500000 0.866025i
211.2 1.00000 −0.500000 0.866025i 1.00000 0.500000 0.866025i −0.500000 0.866025i 2.11803 + 3.66854i 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 −0.118034 + 0.204441i 1.00000 −0.500000 0.866025i 0.500000 + 0.866025i
811.2 1.00000 −0.500000 + 0.866025i 1.00000 0.500000 + 0.866025i −0.500000 + 0.866025i 2.11803 3.66854i 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.l 4
31.c even 3 1 inner 930.2.i.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.l 4 1.a even 1 1 trivial
930.2.i.l 4 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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{4} - 4 T_{7}^{3} + 17 T_{7}^{2} + 4 T_{7} + 1 \)
\( T_{11}^{4} - 4 T_{11}^{3} + 17 T_{11}^{2} + 4 T_{11} + 1 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 1 + 4 T + 17 T^{2} - 4 T^{3} + T^{4} \)
$11$ \( 1 + 4 T + 17 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 1936 + 88 T + 48 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 16 + 8 T + 8 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( ( -4 - 2 T + T^{2} )^{2} \)
$29$ \( ( -3 + T )^{4} \)
$31$ \( 961 + 248 T + 33 T^{2} + 8 T^{3} + T^{4} \)
$37$ \( 16 - 8 T + 8 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( 16 + 8 T + 8 T^{2} - 2 T^{3} + T^{4} \)
$43$ \( 256 + 192 T + 128 T^{2} + 12 T^{3} + T^{4} \)
$47$ \( ( 4 - 6 T + T^{2} )^{2} \)
$53$ \( 121 + 66 T + 47 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( 121 - 88 T + 53 T^{2} - 8 T^{3} + T^{4} \)
$61$ \( ( -116 + 6 T + T^{2} )^{2} \)
$67$ \( 16 + 24 T + 32 T^{2} + 6 T^{3} + T^{4} \)
$71$ \( 256 - 64 T + 32 T^{2} + 4 T^{3} + T^{4} \)
$73$ \( 1936 + 528 T + 188 T^{2} - 12 T^{3} + T^{4} \)
$79$ \( 256 - 192 T + 128 T^{2} - 12 T^{3} + T^{4} \)
$83$ \( 1681 + 164 T + 57 T^{2} - 4 T^{3} + T^{4} \)
$89$ \( ( -76 - 4 T + T^{2} )^{2} \)
$97$ \( ( -11 + 6 T + T^{2} )^{2} \)
show more
show less