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 $2$

Related objects

Downloads

Learn more

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: \( x^{6} - x^{5} + 3x^{4} + 4x^{3} + 12x^{2} - 16x + 64 \) Copy content Toggle raw display
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.

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

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

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} + 7\nu^{3} - 2\nu^{2} + 8\nu + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 5\nu^{4} + \nu^{3} - 22\nu^{2} - 24\nu + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - 13\nu^{4} - 9\nu^{3} - 82\nu^{2} + 40\nu - 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{5} + 3\nu^{4} + 7\nu^{3} + 38\nu^{2} + 120\nu - 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 3\nu^{3} - 4\nu^{2} + 4\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{4} + \beta_{3} - 2\beta_{2} + 3\beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} - 2\beta_{2} - \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{5} - 2\beta_{4} + 2\beta_{3} - 4\beta_{2} + 36\beta _1 - 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{5} + 2\beta_{4} - 3\beta_{3} + 10\beta_{2} + 3\beta _1 - 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -14\beta_{5} + 20\beta_{4} + \beta_{3} + 58\beta_{2} - 75\beta _1 + 36 ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{1}\)

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 4 T^{5} + 21 T^{4} + 4 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{6} - 5 T^{5} + 54 T^{4} + \cdots + 16641 \) Copy content Toggle raw display
$13$ \( T^{6} + 2 T^{5} + 44 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( T^{6} + 11 T^{5} + 91 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( T^{6} + 2 T^{5} + 64 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$23$ \( (T^{3} - 7 T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 12 T^{2} + 9 T + 146)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 8 T^{5} - 28 T^{4} + \cdots + 29791 \) Copy content Toggle raw display
$37$ \( T^{6} - 17 T^{5} + 203 T^{4} + \cdots + 15376 \) Copy content Toggle raw display
$41$ \( T^{6} + 12 T^{5} + 192 T^{4} + \cdots + 419904 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + 45 T^{4} - 140 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( (T^{3} - 3 T^{2} - 90 T + 108)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} - 2 T^{5} + 59 T^{4} + \cdots + 8464 \) Copy content Toggle raw display
$59$ \( T^{6} + 12 T^{5} + 189 T^{4} + \cdots + 104976 \) Copy content Toggle raw display
$61$ \( (T^{3} + 8 T^{2} - 40 T - 248)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 17 T^{5} + 203 T^{4} + \cdots + 15376 \) Copy content Toggle raw display
$71$ \( T^{6} - 10 T^{5} + 108 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 4)^{3} \) Copy content Toggle raw display
$79$ \( T^{6} + 3 T^{5} + 45 T^{4} - 140 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( T^{6} + 177 T^{4} - 424 T^{3} + \cdots + 44944 \) Copy content Toggle raw display
$89$ \( (T^{3} + 28 T^{2} + 220 T + 432)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 20 T^{2} - 119 T + 2766)^{2} \) Copy content Toggle raw display
show more
show less