Properties

Label 930.2.j.c
Level $930$
Weight $2$
Character orbit 930.j
Analytic conductor $7.426$
Analytic rank $0$
Dimension $8$
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.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1698758656.6
Defining polynomial: \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 18\nu^{5} + 28\nu^{4} + 89\nu^{3} + 74\nu^{2} + 104\nu - 16 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 18\nu^{5} - 28\nu^{4} + 89\nu^{3} - 74\nu^{2} + 104\nu + 16 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} - 10\nu^{5} - 20\nu^{4} + 15\nu^{3} - 2\nu^{2} + 120\nu + 80 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} - 10\nu^{5} - 20\nu^{4} + 15\nu^{3} - 2\nu^{2} + 184\nu + 80 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 10\nu^{5} - 20\nu^{4} - 15\nu^{3} - 2\nu^{2} - 120\nu + 80 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 45\nu^{2} + 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} + 46\nu^{5} + 179\nu^{3} + 168\nu ) / 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{2} - 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} - 2\beta_{5} - 5\beta_{4} + 7\beta_{3} - 4\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{6} + 4\beta_{5} + 4\beta_{3} - 22\beta_{2} + 22\beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -52\beta_{7} + 22\beta_{5} + 37\beta_{4} - 59\beta_{3} + 56\beta_{2} + 56\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 89\beta_{6} - 56\beta_{5} - 56\beta_{3} + 218\beta_{2} - 218\beta _1 - 325 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 580\beta_{7} - 218\beta_{5} - 325\beta_{4} + 543\beta_{3} - 620\beta_{2} - 620\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(\beta_{7}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
497.1
2.16053i
3.16053i
1.69230i
0.692297i
2.16053i
3.16053i
1.69230i
0.692297i
−0.707107 + 0.707107i −1.70711 0.292893i 1.00000i −2.23483 + 0.0743018i 1.41421 1.00000i 0.218591 + 0.218591i 0.707107 + 0.707107i 2.82843 + 1.00000i 1.52773 1.63280i
497.2 −0.707107 + 0.707107i −1.70711 0.292893i 1.00000i 1.52773 + 1.63280i 1.41421 1.00000i −1.33991 1.33991i 0.707107 + 0.707107i 2.82843 + 1.00000i −2.23483 0.0743018i
497.3 0.707107 0.707107i −0.292893 1.70711i 1.00000i −0.489528 + 2.18183i −1.41421 1.00000i −0.474719 0.474719i −0.707107 0.707107i −2.82843 + 1.00000i 1.19663 + 1.88893i
497.4 0.707107 0.707107i −0.292893 1.70711i 1.00000i 1.19663 1.88893i −1.41421 1.00000i 3.59604 + 3.59604i −0.707107 0.707107i −2.82843 + 1.00000i −0.489528 2.18183i
683.1 −0.707107 0.707107i −1.70711 + 0.292893i 1.00000i −2.23483 0.0743018i 1.41421 + 1.00000i 0.218591 0.218591i 0.707107 0.707107i 2.82843 1.00000i 1.52773 + 1.63280i
683.2 −0.707107 0.707107i −1.70711 + 0.292893i 1.00000i 1.52773 1.63280i 1.41421 + 1.00000i −1.33991 + 1.33991i 0.707107 0.707107i 2.82843 1.00000i −2.23483 + 0.0743018i
683.3 0.707107 + 0.707107i −0.292893 + 1.70711i 1.00000i −0.489528 2.18183i −1.41421 + 1.00000i −0.474719 + 0.474719i −0.707107 + 0.707107i −2.82843 1.00000i 1.19663 1.88893i
683.4 0.707107 + 0.707107i −0.292893 + 1.70711i 1.00000i 1.19663 + 1.88893i −1.41421 + 1.00000i 3.59604 3.59604i −0.707107 + 0.707107i −2.82843 1.00000i −0.489528 + 2.18183i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 683.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.j.c 8
3.b odd 2 1 930.2.j.f yes 8
5.c odd 4 1 930.2.j.f yes 8
15.e even 4 1 inner 930.2.j.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.j.c 8 1.a even 1 1 trivial
930.2.j.c 8 15.e even 4 1 inner
930.2.j.f yes 8 3.b odd 2 1
930.2.j.f yes 8 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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{8} - 4T_{7}^{7} + 8T_{7}^{6} + 48T_{7}^{5} + 117T_{7}^{4} + 52T_{7}^{3} + 8T_{7}^{2} - 8T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{8} + 22T_{11}^{6} + 97T_{11}^{4} + 144T_{11}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{8} - 4T_{17}^{7} + 8T_{17}^{6} + 168T_{17}^{5} + 3540T_{17}^{4} - 7424T_{17}^{3} + 15488T_{17}^{2} + 339328T_{17} + 3717184 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 4 T^{3} + 8 T^{2} + 12 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{6} + 16 T^{5} + 2 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + 8 T^{6} + 48 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{8} + 22 T^{6} + 97 T^{4} + 144 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + 8 T^{6} + 24 T^{5} + \cdots + 3136 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + 8 T^{6} + \cdots + 3717184 \) Copy content Toggle raw display
$19$ \( T^{8} + 118 T^{6} + 4417 T^{4} + \cdots + 107584 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + 72 T^{6} + \cdots + 37636 \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} - 62 T^{2} + 8 T + 248)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T + 8)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + 196 T^{6} + 11140 T^{4} + \cdots + 1048576 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 85264 \) Copy content Toggle raw display
$47$ \( T^{8} - 28 T^{7} + 392 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} + 72 T^{6} + \cdots + 15376 \) Copy content Toggle raw display
$59$ \( (T^{4} - 12 T^{3} - 30 T^{2} + 512 T - 1024)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 28 T^{3} + 186 T^{2} + 720 T - 7792)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 2408704 \) Copy content Toggle raw display
$71$ \( T^{8} + 286 T^{6} + 22561 T^{4} + \cdots + 583696 \) Copy content Toggle raw display
$73$ \( T^{8} + 36 T^{7} + 648 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$79$ \( T^{8} + 278 T^{6} + 20289 T^{4} + \cdots + 602176 \) Copy content Toggle raw display
$83$ \( T^{8} + 12 T^{7} + 72 T^{6} + \cdots + 18800896 \) Copy content Toggle raw display
$89$ \( (T^{4} - 18 T^{3} + 113 T^{2} - 288 T + 248)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 4129024 \) Copy content Toggle raw display
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