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: \( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \) Copy content Toggle raw display
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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{5} - 70\nu^{4} + 183\nu^{3} + 120\nu^{2} - 966\nu + 240 ) / 445 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{5} - 5\nu^{4} - 184\nu^{3} + 390\nu^{2} + 643\nu - 1000 ) / 445 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{5} + 8\nu^{4} - 26\nu^{3} - \nu^{2} + 57\nu - 180 ) / 89 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{5} + 24\nu^{4} + 11\nu^{3} - 92\nu^{2} - 7\nu - 6 ) / 178 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{5} - 4\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 14\beta_{5} - 14\beta_{4} + 9\beta_{3} - 3\beta_{2} - 10\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - 32\beta_{4} + 6\beta_{3} - 17\beta_{2} - 25\beta _1 - 42 \) 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
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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

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