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 $2$

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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: 6.0.11669056.1
Defining polynomial: \( x^{6} - 2x^{5} + 7x^{4} + 8x^{3} - x^{2} + 54x + 58 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{3} q^{3} - q^{4} + (\beta_1 - 1) q^{5} + q^{6} + 2 \beta_{3} q^{7} - \beta_{3} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{3} q^{3} - q^{4} + (\beta_1 - 1) q^{5} + q^{6} + 2 \beta_{3} q^{7} - \beta_{3} q^{8} - q^{9} + (\beta_{5} - \beta_{3}) q^{10} + ( - \beta_{2} + \beta_1 + 2) q^{11} + \beta_{3} q^{12} + (\beta_{5} + \beta_{4} - 4 \beta_{3}) q^{13} - 2 q^{14} + ( - \beta_{5} + \beta_{3}) q^{15} + q^{16} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{17} - \beta_{3} q^{18} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 2) q^{19} + ( - \beta_1 + 1) q^{20} + 2 q^{21} + (\beta_{5} + \beta_{4} + 2 \beta_{3}) q^{22} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{23} - q^{24} + (\beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{25} + (\beta_{2} - \beta_1 + 4) q^{26} + \beta_{3} q^{27} - 2 \beta_{3} q^{28} + (2 \beta_{2} - 2 \beta_1 + 2) q^{29} + (\beta_1 - 1) q^{30} - q^{31} + \beta_{3} q^{32} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3}) q^{33} + (\beta_{5} - \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 2) q^{34} + (2 \beta_{5} - 2 \beta_{3}) q^{35} + q^{36} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{37} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{38} + ( - \beta_{2} + \beta_1 - 4) q^{39} + ( - \beta_{5} + \beta_{3}) q^{40} + 2 q^{41} + 2 \beta_{3} q^{42} + (2 \beta_{2} + 2 \beta_1) q^{43} + (\beta_{2} - \beta_1 - 2) q^{44} + ( - \beta_1 + 1) q^{45} + (2 \beta_{2} - 2 \beta_1 + 2) q^{46} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{47} - \beta_{3} q^{48} + 3 q^{49} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{2} - 1) q^{50} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 2) q^{51} + ( - \beta_{5} - \beta_{4} + 4 \beta_{3}) q^{52} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{53} - q^{54} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{55} + 2 q^{56} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{57} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{58} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 2) q^{59} + (\beta_{5} - \beta_{3}) q^{60} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 6) q^{61} - \beta_{3} q^{62} - 2 \beta_{3} q^{63} - q^{64} + ( - 4 \beta_{5} - 2 \beta_{4} + 7 \beta_{3} + \beta_{2} - 1) q^{65} + ( - \beta_{2} + \beta_1 + 2) q^{66} + (\beta_{5} + \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{67} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{68} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{69} + ( - 2 \beta_1 + 2) q^{70} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_1 - 2) q^{71} + \beta_{3} q^{72} + (4 \beta_{5} + 4 \beta_{4} - 2 \beta_{3}) q^{73} + (2 \beta_{5} - 2 \beta_{4} + 2) q^{74} + (2 \beta_{5} + 2 \beta_{4} - \beta_{2} + 1) q^{75} + (\beta_{5} - \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 2) q^{76} + (2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3}) q^{77} + (\beta_{5} + \beta_{4} - 4 \beta_{3}) q^{78} + (\beta_{5} - \beta_{4} + 2) q^{79} + (\beta_1 - 1) q^{80} + q^{81} + 2 \beta_{3} q^{82} + ( - 4 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{83} - 2 q^{84} + (2 \beta_{5} + 5 \beta_{4} - 7 \beta_{3} - 2 \beta_1 - 3) q^{85} + (2 \beta_{5} - 2 \beta_{4}) q^{86} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{87} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3}) q^{88} + (2 \beta_{5} - 2 \beta_{4} - 4 \beta_{2} + 4 \beta_1 - 2) q^{89} + ( - \beta_{5} + \beta_{3}) q^{90} + (2 \beta_{2} - 2 \beta_1 + 8) q^{91} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{92} + \beta_{3} q^{93} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 2) q^{94} + (2 \beta_{5} + 3 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 7) q^{95} + q^{96} + ( - \beta_{5} - \beta_{4} - 6 \beta_{3}) q^{97} + 3 \beta_{3} q^{98} + (\beta_{2} - \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9} + 16 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} + 4 q^{20} + 12 q^{21} - 6 q^{24} - 8 q^{25} + 20 q^{26} + 4 q^{29} - 4 q^{30} - 6 q^{31} - 4 q^{34} + 6 q^{36} - 20 q^{39} + 12 q^{41} - 16 q^{44} + 4 q^{45} + 4 q^{46} + 18 q^{49} - 8 q^{50} + 4 q^{51} - 6 q^{54} + 6 q^{55} + 12 q^{56} + 4 q^{59} + 28 q^{61} - 6 q^{64} - 8 q^{65} + 16 q^{66} - 4 q^{69} + 8 q^{70} - 16 q^{71} + 12 q^{74} + 8 q^{75} - 4 q^{76} + 12 q^{79} - 4 q^{80} + 6 q^{81} - 12 q^{84} - 22 q^{85} + 4 q^{89} + 40 q^{91} + 4 q^{94} - 28 q^{95} + 6 q^{96} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} + 6\nu^{3} + 14\nu^{2} + 13\nu + 42 ) / 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{5} - 14\nu^{4} + 49\nu^{3} - 29\nu^{2} + 27\nu + 213 ) / 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{5} + 16\nu^{4} - 61\nu^{3} + 26\nu^{2} - 53\nu - 272 ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 21\nu^{4} - 61\nu^{3} + 31\nu^{2} - 3\nu - 232 ) / 25 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 2\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + \beta_{4} + 4\beta_{3} + 2\beta_{2} - 3\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{5} - 6\beta_{4} - \beta_{3} - 2\beta_{2} - 10\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{5} - 26\beta_{4} - 39\beta_{3} - 17\beta_{2} - 5\beta _1 + 13 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{3} - 8T_{11}^{2} + 13T_{11} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

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