Properties

Label 930.2.d.i
Level $930$
Weight $2$
Character orbit 930.d
Analytic conductor $7.426$
Analytic rank $0$
Dimension $6$
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.11669056.1
Defining polynomial: \(x^{6} - 2 x^{5} + 7 x^{4} + 8 x^{3} - x^{2} + 54 x + 58\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 7 x^{4} + 8 x^{3} - x^{2} + 54 x + 58\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - \nu^{4} + 6 \nu^{3} + 14 \nu^{2} + 13 \nu + 42 \)\()/25\)
\(\beta_{3}\)\(=\)\((\)\( 4 \nu^{5} - 14 \nu^{4} + 49 \nu^{3} - 29 \nu^{2} + 27 \nu + 213 \)\()/25\)
\(\beta_{4}\)\(=\)\((\)\( -6 \nu^{5} + 16 \nu^{4} - 61 \nu^{3} + 26 \nu^{2} - 53 \nu - 272 \)\()/25\)
\(\beta_{5}\)\(=\)\((\)\( -6 \nu^{5} + 21 \nu^{4} - 61 \nu^{3} + 31 \nu^{2} - 3 \nu - 232 \)\()/25\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 2 \beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{5} + \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 3 \beta_{1} - 8\)
\(\nu^{4}\)\(=\)\(5 \beta_{5} - 6 \beta_{4} - \beta_{3} - 2 \beta_{2} - 10 \beta_{1} - 7\)
\(\nu^{5}\)\(=\)\(-7 \beta_{5} - 26 \beta_{4} - 39 \beta_{3} - 17 \beta_{2} - 5 \beta_{1} + 13\)

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
−1.23545 0.0526623i
0.630356 + 2.20530i
1.60509 2.15264i
−1.23545 + 0.0526623i
0.630356 2.20530i
1.60509 + 2.15264i
1.00000i 1.00000i −1.00000 −2.23545 0.0526623i 1.00000 2.00000i 1.00000i −1.00000 −0.0526623 + 2.23545i
559.2 1.00000i 1.00000i −1.00000 −0.369644 + 2.20530i 1.00000 2.00000i 1.00000i −1.00000 2.20530 + 0.369644i
559.3 1.00000i 1.00000i −1.00000 0.605092 2.15264i 1.00000 2.00000i 1.00000i −1.00000 −2.15264 0.605092i
559.4 1.00000i 1.00000i −1.00000 −2.23545 + 0.0526623i 1.00000 2.00000i 1.00000i −1.00000 −0.0526623 2.23545i
559.5 1.00000i 1.00000i −1.00000 −0.369644 2.20530i 1.00000 2.00000i 1.00000i −1.00000 2.20530 0.369644i
559.6 1.00000i 1.00000i −1.00000 0.605092 + 2.15264i 1.00000 2.00000i 1.00000i −1.00000 −2.15264 + 0.605092i
\(n\): 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.i 6
3.b odd 2 1 2790.2.d.j 6
5.b even 2 1 inner 930.2.d.i 6
5.c odd 4 1 4650.2.a.ci 3
5.c odd 4 1 4650.2.a.cp 3
15.d odd 2 1 2790.2.d.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.i 6 1.a even 1 1 trivial
930.2.d.i 6 5.b even 2 1 inner
2790.2.d.j 6 3.b odd 2 1
2790.2.d.j 6 15.d odd 2 1
4650.2.a.ci 3 5.c odd 4 1
4650.2.a.cp 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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{11}^{3} - 8 T_{11}^{2} + 13 T_{11} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{3} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( 125 + 100 T + 60 T^{2} + 36 T^{3} + 12 T^{4} + 4 T^{5} + T^{6} \)
$7$ \( ( 4 + T^{2} )^{3} \)
$11$ \( ( 8 + 13 T - 8 T^{2} + T^{3} )^{2} \)
$13$ \( 196 + 345 T^{2} + 50 T^{4} + T^{6} \)
$17$ \( 16 + 1209 T^{2} + 74 T^{4} + T^{6} \)
$19$ \( ( -4 - 35 T - 2 T^{2} + T^{3} )^{2} \)
$23$ \( 256 + 960 T^{2} + 68 T^{4} + T^{6} \)
$29$ \( ( -16 - 32 T - 2 T^{2} + T^{3} )^{2} \)
$31$ \( ( 1 + T )^{6} \)
$37$ \( 16384 + 5632 T^{2} + 164 T^{4} + T^{6} \)
$41$ \( ( -2 + T )^{6} \)
$43$ \( 256 + 5776 T^{2} + 152 T^{4} + T^{6} \)
$47$ \( 16 + 1209 T^{2} + 74 T^{4} + T^{6} \)
$53$ \( 64 + 1776 T^{2} + 188 T^{4} + T^{6} \)
$59$ \( ( 280 - 76 T - 2 T^{2} + T^{3} )^{2} \)
$61$ \( ( 1372 - 91 T - 14 T^{2} + T^{3} )^{2} \)
$67$ \( 425104 + 21089 T^{2} + 286 T^{4} + T^{6} \)
$71$ \( ( 20 - 79 T + 8 T^{2} + T^{3} )^{2} \)
$73$ \( 153664 + 18992 T^{2} + 268 T^{4} + T^{6} \)
$79$ \( ( 28 - 7 T - 6 T^{2} + T^{3} )^{2} \)
$83$ \( 59536 + 32513 T^{2} + 370 T^{4} + T^{6} \)
$89$ \( ( 320 - 144 T - 2 T^{2} + T^{3} )^{2} \)
$97$ \( 55696 + 6185 T^{2} + 150 T^{4} + T^{6} \)
show more
show less