Properties

Label 930.2.d.h
Level $930$
Weight $2$
Character orbit 930.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.3534400.1
Defining polynomial: \(x^{6} - 2 x^{5} - 3 x^{4} + 16 x^{3} + x^{2} - 12 x + 40\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 3 x^{4} + 16 x^{3} + x^{2} - 12 x + 40\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{5} - 70 \nu^{4} + 183 \nu^{3} + 120 \nu^{2} - 966 \nu + 240 \)\()/445\)
\(\beta_{3}\)\(=\)\((\)\( 13 \nu^{5} - 5 \nu^{4} - 184 \nu^{3} + 390 \nu^{2} + 643 \nu - 1000 \)\()/445\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{5} + 8 \nu^{4} - 26 \nu^{3} - \nu^{2} + 57 \nu - 180 \)\()/89\)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{5} + 24 \nu^{4} + 11 \nu^{3} - 92 \nu^{2} - 7 \nu - 6 \)\()/178\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(4 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_{1} - 4\)
\(\nu^{4}\)\(=\)\(14 \beta_{5} - 14 \beta_{4} + 9 \beta_{3} - 3 \beta_{2} - 10 \beta_{1} - 6\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} - 32 \beta_{4} + 6 \beta_{3} - 17 \beta_{2} - 25 \beta_{1} - 42\)

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
0.627553 + 1.14620i
−1.81837 + 0.301352i
2.19082 1.44755i
0.627553 1.14620i
−1.81837 0.301352i
2.19082 + 1.44755i
1.00000i 1.00000i −1.00000 −2.14620 + 0.627553i −1.00000 0.255105i 1.00000i −1.00000 0.627553 + 2.14620i
559.2 1.00000i 1.00000i −1.00000 −1.30135 1.81837i −1.00000 4.63675i 1.00000i −1.00000 −1.81837 + 1.30135i
559.3 1.00000i 1.00000i −1.00000 0.447553 + 2.19082i −1.00000 3.38164i 1.00000i −1.00000 2.19082 0.447553i
559.4 1.00000i 1.00000i −1.00000 −2.14620 0.627553i −1.00000 0.255105i 1.00000i −1.00000 0.627553 2.14620i
559.5 1.00000i 1.00000i −1.00000 −1.30135 + 1.81837i −1.00000 4.63675i 1.00000i −1.00000 −1.81837 1.30135i
559.6 1.00000i 1.00000i −1.00000 0.447553 2.19082i −1.00000 3.38164i 1.00000i −1.00000 2.19082 + 0.447553i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.d.h 6
3.b odd 2 1 2790.2.d.k 6
5.b even 2 1 inner 930.2.d.h 6
5.c odd 4 1 4650.2.a.ck 3
5.c odd 4 1 4650.2.a.cn 3
15.d odd 2 1 2790.2.d.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.h 6 1.a even 1 1 trivial
930.2.d.h 6 5.b even 2 1 inner
2790.2.d.k 6 3.b odd 2 1
2790.2.d.k 6 15.d odd 2 1
4650.2.a.ck 3 5.c odd 4 1
4650.2.a.cn 3 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{6} + 33 T_{7}^{4} + 248 T_{7}^{2} + 16 \)
\( T_{11}^{3} - T_{11}^{2} - 35 T_{11} + 67 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{3} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( 125 + 150 T + 100 T^{2} + 50 T^{3} + 20 T^{4} + 6 T^{5} + T^{6} \)
$7$ \( 16 + 248 T^{2} + 33 T^{4} + T^{6} \)
$11$ \( ( 67 - 35 T - T^{2} + T^{3} )^{2} \)
$13$ \( 1600 + 721 T^{2} + 62 T^{4} + T^{6} \)
$17$ \( 400 + 281 T^{2} + 42 T^{4} + T^{6} \)
$19$ \( ( -85 - 45 T + T^{2} + T^{3} )^{2} \)
$23$ \( 1600 + 480 T^{2} + 41 T^{4} + T^{6} \)
$29$ \( ( -16 - 16 T + 6 T^{2} + T^{3} )^{2} \)
$31$ \( ( -1 + T )^{6} \)
$37$ \( T^{6} \)
$41$ \( ( 472 - 100 T - 6 T^{2} + T^{3} )^{2} \)
$43$ \( 16 + 248 T^{2} + 33 T^{4} + T^{6} \)
$47$ \( 1600 + 905 T^{2} + 106 T^{4} + T^{6} \)
$53$ \( 8464 + 6504 T^{2} + 161 T^{4} + T^{6} \)
$59$ \( ( 40 - 44 T + 2 T^{2} + T^{3} )^{2} \)
$61$ \( ( -10 + 65 T - 16 T^{2} + T^{3} )^{2} \)
$67$ \( 11236 + 3449 T^{2} + 114 T^{4} + T^{6} \)
$71$ \( ( 43 - 75 T - 9 T^{2} + T^{3} )^{2} \)
$73$ \( 26896 + 13528 T^{2} + 233 T^{4} + T^{6} \)
$79$ \( ( -1175 - 141 T + 11 T^{2} + T^{3} )^{2} \)
$83$ \( 62500 + 7625 T^{2} + 166 T^{4} + T^{6} \)
$89$ \( ( 736 + 264 T + 29 T^{2} + T^{3} )^{2} \)
$97$ \( 100 + 2721 T^{2} + 138 T^{4} + T^{6} \)
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