Properties

Label 930.2.i.n
Level $930$
Weight $2$
Character orbit 930.i
Analytic conductor $7.426$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.3636603.4
Defining polynomial: \(x^{6} - x^{5} + 3 x^{4} + 4 x^{3} + 12 x^{2} - 16 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} -\beta_{1} q^{3} + q^{4} + ( -1 - \beta_{1} ) q^{5} + \beta_{1} q^{6} + ( \beta_{1} - \beta_{4} ) q^{7} - q^{8} + ( -1 - \beta_{1} ) q^{9} +O(q^{10})\) \( q - q^{2} -\beta_{1} q^{3} + q^{4} + ( -1 - \beta_{1} ) q^{5} + \beta_{1} q^{6} + ( \beta_{1} - \beta_{4} ) q^{7} - q^{8} + ( -1 - \beta_{1} ) q^{9} + ( 1 + \beta_{1} ) q^{10} + ( 2 + 2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{11} -\beta_{1} q^{12} -2 \beta_{2} q^{13} + ( -\beta_{1} + \beta_{4} ) q^{14} - q^{15} + q^{16} + ( 4 \beta_{1} + \beta_{4} ) q^{17} + ( 1 + \beta_{1} ) q^{18} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{19} + ( -1 - \beta_{1} ) q^{20} + ( 1 + \beta_{1} + \beta_{2} ) q^{21} + ( -2 - 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{22} + ( 2 + \beta_{2} + \beta_{4} ) q^{23} + \beta_{1} q^{24} + \beta_{1} q^{25} + 2 \beta_{2} q^{26} - q^{27} + ( \beta_{1} - \beta_{4} ) q^{28} + ( 4 + \beta_{2} + \beta_{4} + \beta_{5} ) q^{29} + q^{30} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{31} - q^{32} + ( 2 + \beta_{5} ) q^{33} + ( -4 \beta_{1} - \beta_{4} ) q^{34} + ( 1 + \beta_{2} + \beta_{4} ) q^{35} + ( -1 - \beta_{1} ) q^{36} + ( -6 \beta_{1} - \beta_{4} ) q^{37} + ( -\beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{38} + ( -2 \beta_{2} - 2 \beta_{4} ) q^{39} + ( 1 + \beta_{1} ) q^{40} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{41} + ( -1 - \beta_{1} - \beta_{2} ) q^{42} + ( \beta_{1} + \beta_{3} + \beta_{4} ) q^{43} + ( 2 + 2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{44} + \beta_{1} q^{45} + ( -2 - \beta_{2} - \beta_{4} ) q^{46} + ( 3 \beta_{2} + 3 \beta_{4} ) q^{47} -\beta_{1} q^{48} + ( -1 - \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{5} ) q^{49} -\beta_{1} q^{50} + ( 4 + 4 \beta_{1} - \beta_{2} ) q^{51} -2 \beta_{2} q^{52} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{53} + q^{54} + ( -2 \beta_{1} + \beta_{3} ) q^{55} + ( -\beta_{1} + \beta_{4} ) q^{56} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{57} + ( -4 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{58} + ( 3 \beta_{1} - 3 \beta_{4} ) q^{59} - q^{60} + ( -3 + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{61} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} ) q^{62} + ( 1 + \beta_{2} + \beta_{4} ) q^{63} + q^{64} -2 \beta_{4} q^{65} + ( -2 - \beta_{5} ) q^{66} + ( -6 - 6 \beta_{1} + \beta_{2} ) q^{67} + ( 4 \beta_{1} + \beta_{4} ) q^{68} + ( -2 \beta_{1} + \beta_{4} ) q^{69} + ( -1 - \beta_{2} - \beta_{4} ) q^{70} + ( 4 + 4 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 1 + \beta_{1} ) q^{72} + ( 2 + 2 \beta_{1} ) q^{73} + ( 6 \beta_{1} + \beta_{4} ) q^{74} + ( 1 + \beta_{1} ) q^{75} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{76} + ( 2 - 5 \beta_{2} - 5 \beta_{4} - \beta_{5} ) q^{77} + ( 2 \beta_{2} + 2 \beta_{4} ) q^{78} + ( \beta_{1} + \beta_{3} + \beta_{4} ) q^{79} + ( -1 - \beta_{1} ) q^{80} + \beta_{1} q^{81} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{82} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{83} + ( 1 + \beta_{1} + \beta_{2} ) q^{84} + ( 4 - \beta_{2} - \beta_{4} ) q^{85} + ( -\beta_{1} - \beta_{3} - \beta_{4} ) q^{86} + ( -4 \beta_{1} + \beta_{3} + \beta_{4} ) q^{87} + ( -2 - 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{88} + ( -10 + 2 \beta_{2} + 2 \beta_{4} ) q^{89} -\beta_{1} q^{90} + ( 14 + 4 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{91} + ( 2 + \beta_{2} + \beta_{4} ) q^{92} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{93} + ( -3 \beta_{2} - 3 \beta_{4} ) q^{94} + ( 1 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{95} + \beta_{1} q^{96} + ( 8 - 5 \beta_{2} - 5 \beta_{4} - \beta_{5} ) q^{97} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{98} + ( -2 \beta_{1} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + 3q^{3} + 6q^{4} - 3q^{5} - 3q^{6} - 4q^{7} - 6q^{8} - 3q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + 3q^{3} + 6q^{4} - 3q^{5} - 3q^{6} - 4q^{7} - 6q^{8} - 3q^{9} + 3q^{10} + 5q^{11} + 3q^{12} - 2q^{13} + 4q^{14} - 6q^{15} + 6q^{16} - 11q^{17} + 3q^{18} - 2q^{19} - 3q^{20} + 4q^{21} - 5q^{22} + 14q^{23} - 3q^{24} - 3q^{25} + 2q^{26} - 6q^{27} - 4q^{28} + 24q^{29} + 6q^{30} + 8q^{31} - 6q^{32} + 10q^{33} + 11q^{34} + 8q^{35} - 3q^{36} + 17q^{37} + 2q^{38} - 4q^{39} + 3q^{40} - 12q^{41} - 4q^{42} - 3q^{43} + 5q^{44} - 3q^{45} - 14q^{46} + 6q^{47} + 3q^{48} - 5q^{49} + 3q^{50} + 11q^{51} - 2q^{52} + 2q^{53} + 6q^{54} + 5q^{55} + 4q^{56} + 2q^{57} - 24q^{58} - 12q^{59} - 6q^{60} - 16q^{61} - 8q^{62} + 8q^{63} + 6q^{64} - 2q^{65} - 10q^{66} - 17q^{67} - 11q^{68} + 7q^{69} - 8q^{70} + 10q^{71} + 3q^{72} + 6q^{73} - 17q^{74} + 3q^{75} - 2q^{76} + 4q^{77} + 4q^{78} - 3q^{79} - 3q^{80} - 3q^{81} + 12q^{82} + 4q^{84} + 22q^{85} + 3q^{86} + 12q^{87} - 5q^{88} - 56q^{89} + 3q^{90} + 88q^{91} + 14q^{92} + q^{93} - 6q^{94} + 4q^{95} - 3q^{96} + 40q^{97} + 5q^{98} + 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 3 x^{4} + 4 x^{3} + 12 x^{2} - 16 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} + 3 \nu^{4} + 7 \nu^{3} - 2 \nu^{2} + 8 \nu + 32 \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 5 \nu^{4} + \nu^{3} - 22 \nu^{2} - 24 \nu + 32 \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} - 13 \nu^{4} - 9 \nu^{3} - 82 \nu^{2} + 40 \nu - 224 \)\()/64\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{5} + 3 \nu^{4} + 7 \nu^{3} + 38 \nu^{2} + 120 \nu - 32 \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 3 \nu^{3} - 4 \nu^{2} + 4 \nu + 8 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_{1} + 3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} - 2 \beta_{2} - \beta_{1} - 2\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 36 \beta_{1} - 3\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 10 \beta_{2} + 3 \beta_{1} - 19\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-14 \beta_{5} + 20 \beta_{4} + \beta_{3} + 58 \beta_{2} - 75 \beta_{1} + 36\)\()/6\)

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
0.715814 + 1.86751i
−1.55951 1.25217i
1.34370 1.48137i
0.715814 1.86751i
−1.55951 + 1.25217i
1.34370 + 1.48137i
−1.00000 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i −2.25941 3.91341i −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 −0.695350 1.20438i −1.00000 −0.500000 + 0.866025i 0.500000 0.866025i
211.3 −1.00000 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i 0.954758 + 1.65369i −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 −2.25941 + 3.91341i −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 −0.695350 + 1.20438i −1.00000 −0.500000 0.866025i 0.500000 + 0.866025i
811.3 −1.00000 0.500000 0.866025i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i 0.954758 1.65369i −1.00000 −0.500000 0.866025i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 811.3
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.n 6
31.c even 3 1 inner 930.2.i.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.n 6 1.a even 1 1 trivial
930.2.i.n 6 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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{6} + 4 T_{7}^{5} + 21 T_{7}^{4} + 4 T_{7}^{3} + 73 T_{7}^{2} + 60 T_{7} + 144 \)
\( T_{11}^{6} - 5 T_{11}^{5} + 54 T_{11}^{4} - 113 T_{11}^{3} + 1486 T_{11}^{2} - 3741 T_{11} + 16641 \)
\( T_{13}^{6} + 2 T_{13}^{5} + 44 T_{13}^{4} - 16 T_{13}^{3} + 1664 T_{13}^{2} + 1280 T_{13} + 1024 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( ( 1 - T + T^{2} )^{3} \)
$5$ \( ( 1 + T + T^{2} )^{3} \)
$7$ \( 144 + 60 T + 73 T^{2} + 4 T^{3} + 21 T^{4} + 4 T^{5} + T^{6} \)
$11$ \( 16641 - 3741 T + 1486 T^{2} - 113 T^{3} + 54 T^{4} - 5 T^{5} + T^{6} \)
$13$ \( 1024 + 1280 T + 1664 T^{2} - 16 T^{3} + 44 T^{4} + 2 T^{5} + T^{6} \)
$17$ \( 144 + 360 T + 768 T^{2} + 306 T^{3} + 91 T^{4} + 11 T^{5} + T^{6} \)
$19$ \( 20736 + 8640 T + 3888 T^{2} + 168 T^{3} + 64 T^{4} + 2 T^{5} + T^{6} \)
$23$ \( ( 12 + 6 T - 7 T^{2} + T^{3} )^{2} \)
$29$ \( ( 146 + 9 T - 12 T^{2} + T^{3} )^{2} \)
$31$ \( 29791 - 7688 T - 868 T^{2} + 508 T^{3} - 28 T^{4} - 8 T^{5} + T^{6} \)
$37$ \( 15376 - 10664 T + 5288 T^{2} - 1214 T^{3} + 203 T^{4} - 17 T^{5} + T^{6} \)
$41$ \( 419904 + 31104 T + 10080 T^{2} + 720 T^{3} + 192 T^{4} + 12 T^{5} + T^{6} \)
$43$ \( 256 - 576 T + 1248 T^{2} - 140 T^{3} + 45 T^{4} + 3 T^{5} + T^{6} \)
$47$ \( ( 108 - 90 T - 3 T^{2} + T^{3} )^{2} \)
$53$ \( 8464 - 5060 T + 3209 T^{2} - 74 T^{3} + 59 T^{4} - 2 T^{5} + T^{6} \)
$59$ \( 104976 + 14580 T + 5913 T^{2} + 108 T^{3} + 189 T^{4} + 12 T^{5} + T^{6} \)
$61$ \( ( -248 - 40 T + 8 T^{2} + T^{3} )^{2} \)
$67$ \( 15376 + 10664 T + 5288 T^{2} + 1214 T^{3} + 203 T^{4} + 17 T^{5} + T^{6} \)
$71$ \( 9216 - 768 T + 1024 T^{2} - 112 T^{3} + 108 T^{4} - 10 T^{5} + T^{6} \)
$73$ \( ( 4 - 2 T + T^{2} )^{3} \)
$79$ \( 256 - 576 T + 1248 T^{2} - 140 T^{3} + 45 T^{4} + 3 T^{5} + T^{6} \)
$83$ \( 44944 - 37524 T + 31329 T^{2} - 424 T^{3} + 177 T^{4} + T^{6} \)
$89$ \( ( 432 + 220 T + 28 T^{2} + T^{3} )^{2} \)
$97$ \( ( 2766 - 119 T - 20 T^{2} + T^{3} )^{2} \)
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