Properties

Label 930.2.i.m
Level $930$
Weight $2$
Character orbit 930.i
Analytic conductor $7.426$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
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.

\(f(q)\) \(=\) \( q + q^{2} + ( 1 + \beta_{2} ) q^{3} + q^{4} + \beta_{2} q^{5} + ( 1 + \beta_{2} ) q^{6} + ( 1 + \beta_{1} + \beta_{2} ) q^{7} + q^{8} + \beta_{2} q^{9} +O(q^{10})\) \( q + q^{2} + ( 1 + \beta_{2} ) q^{3} + q^{4} + \beta_{2} q^{5} + ( 1 + \beta_{2} ) q^{6} + ( 1 + \beta_{1} + \beta_{2} ) q^{7} + q^{8} + \beta_{2} q^{9} + \beta_{2} q^{10} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + ( 1 + \beta_{2} ) q^{12} + 2 \beta_{2} q^{13} + ( 1 + \beta_{1} + \beta_{2} ) q^{14} - q^{15} + q^{16} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{17} + \beta_{2} q^{18} + ( 6 + 6 \beta_{2} ) q^{19} + \beta_{2} q^{20} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{21} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{22} + ( 2 + 2 \beta_{3} ) q^{23} + ( 1 + \beta_{2} ) q^{24} + ( -1 - \beta_{2} ) q^{25} + 2 \beta_{2} q^{26} - q^{27} + ( 1 + \beta_{1} + \beta_{2} ) q^{28} + ( 3 - \beta_{3} ) q^{29} - q^{30} + ( -1 + 5 \beta_{2} ) q^{31} + q^{32} + ( 1 - \beta_{3} ) q^{33} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{34} + ( -1 + \beta_{3} ) q^{35} + \beta_{2} q^{36} + ( -4 - 2 \beta_{1} - 4 \beta_{2} ) q^{37} + ( 6 + 6 \beta_{2} ) q^{38} -2 q^{39} + \beta_{2} q^{40} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{41} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{42} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{43} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{44} + ( -1 - \beta_{2} ) q^{45} + ( 2 + 2 \beta_{3} ) q^{46} + ( -2 - 4 \beta_{3} ) q^{47} + ( 1 + \beta_{2} ) q^{48} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{49} + ( -1 - \beta_{2} ) q^{50} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{51} + 2 \beta_{2} q^{52} + ( -2 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{53} - q^{54} + ( 1 + \beta_{1} + \beta_{2} ) q^{55} + ( 1 + \beta_{1} + \beta_{2} ) q^{56} + 6 \beta_{2} q^{57} + ( 3 - \beta_{3} ) q^{58} + ( 3 + \beta_{1} + 3 \beta_{2} ) q^{59} - q^{60} -6 q^{61} + ( -1 + 5 \beta_{2} ) q^{62} + ( -1 + \beta_{3} ) q^{63} + q^{64} + ( -2 - 2 \beta_{2} ) q^{65} + ( 1 - \beta_{3} ) q^{66} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{68} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{69} + ( -1 + \beta_{3} ) q^{70} -12 \beta_{2} q^{71} + \beta_{2} q^{72} + ( -4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{73} + ( -4 - 2 \beta_{1} - 4 \beta_{2} ) q^{74} -\beta_{2} q^{75} + ( 6 + 6 \beta_{2} ) q^{76} + ( 9 - 2 \beta_{3} ) q^{77} -2 q^{78} + \beta_{2} q^{80} + ( -1 - \beta_{2} ) q^{81} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{82} + ( 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{83} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{84} + ( -2 - 2 \beta_{3} ) q^{85} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{86} + ( 3 + \beta_{1} + 3 \beta_{2} ) q^{87} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{88} + 12 q^{89} + ( -1 - \beta_{2} ) q^{90} + ( -2 + 2 \beta_{3} ) q^{91} + ( 2 + 2 \beta_{3} ) q^{92} + ( -6 - \beta_{2} ) q^{93} + ( -2 - 4 \beta_{3} ) q^{94} -6 q^{95} + ( 1 + \beta_{2} ) q^{96} + ( -3 + 3 \beta_{3} ) q^{97} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{98} + ( 1 + \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 2q^{3} + 4q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + 4q^{8} - 2q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 2q^{3} + 4q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + 4q^{8} - 2q^{9} - 2q^{10} + 2q^{11} + 2q^{12} - 4q^{13} + 2q^{14} - 4q^{15} + 4q^{16} + 4q^{17} - 2q^{18} + 12q^{19} - 2q^{20} - 2q^{21} + 2q^{22} + 8q^{23} + 2q^{24} - 2q^{25} - 4q^{26} - 4q^{27} + 2q^{28} + 12q^{29} - 4q^{30} - 14q^{31} + 4q^{32} + 4q^{33} + 4q^{34} - 4q^{35} - 2q^{36} - 8q^{37} + 12q^{38} - 8q^{39} - 2q^{40} - 2q^{42} - 8q^{43} + 2q^{44} - 2q^{45} + 8q^{46} - 8q^{47} + 2q^{48} - 4q^{49} - 2q^{50} - 4q^{51} - 4q^{52} + 14q^{53} - 4q^{54} + 2q^{55} + 2q^{56} - 12q^{57} + 12q^{58} + 6q^{59} - 4q^{60} - 24q^{61} - 14q^{62} - 4q^{63} + 4q^{64} - 4q^{65} + 4q^{66} + 4q^{68} + 4q^{69} - 4q^{70} + 24q^{71} - 2q^{72} + 4q^{73} - 8q^{74} + 2q^{75} + 12q^{76} + 36q^{77} - 8q^{78} - 2q^{80} - 2q^{81} + 6q^{83} - 2q^{84} - 8q^{85} - 8q^{86} + 6q^{87} + 2q^{88} + 48q^{89} - 2q^{90} - 8q^{91} + 8q^{92} - 22q^{93} - 8q^{94} - 24q^{95} + 2q^{96} - 12q^{97} - 4q^{98} + 2q^{99} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
1.00000 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.914214 1.58346i 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 1.91421 + 3.31552i 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 −0.914214 + 1.58346i 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 1.91421 3.31552i 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.m 4
31.c even 3 1 inner 930.2.i.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.m 4 1.a even 1 1 trivial
930.2.i.m 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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{4} - 2 T_{7}^{3} + 11 T_{7}^{2} + 14 T_{7} + 49 \)
\( T_{11}^{4} - 2 T_{11}^{3} + 11 T_{11}^{2} + 14 T_{11} + 49 \)
\( T_{13}^{2} + 2 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( 49 + 14 T + 11 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( 49 + 14 T + 11 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( ( 4 + 2 T + T^{2} )^{2} \)
$17$ \( 784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( ( 36 - 6 T + T^{2} )^{2} \)
$23$ \( ( -28 - 4 T + T^{2} )^{2} \)
$29$ \( ( 1 - 6 T + T^{2} )^{2} \)
$31$ \( ( 31 + 7 T + T^{2} )^{2} \)
$37$ \( 256 - 128 T + 80 T^{2} + 8 T^{3} + T^{4} \)
$41$ \( 16384 + 128 T^{2} + T^{4} \)
$43$ \( 256 - 128 T + 80 T^{2} + 8 T^{3} + T^{4} \)
$47$ \( ( -124 + 4 T + T^{2} )^{2} \)
$53$ \( 289 - 238 T + 179 T^{2} - 14 T^{3} + T^{4} \)
$59$ \( 1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4} \)
$61$ \( ( 6 + T )^{4} \)
$67$ \( T^{4} \)
$71$ \( ( 144 - 12 T + T^{2} )^{2} \)
$73$ \( 15376 + 496 T + 140 T^{2} - 4 T^{3} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( 529 + 138 T + 59 T^{2} - 6 T^{3} + T^{4} \)
$89$ \( ( -12 + T )^{4} \)
$97$ \( ( -63 + 6 T + T^{2} )^{2} \)
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