Properties

Label 930.2.z.a
Level $930$
Weight $2$
Character orbit 930.z
Analytic conductor $7.426$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.z (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{20}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.587785 0.809017i
−0.587785 + 0.809017i
0.951057 0.309017i
−0.951057 + 0.309017i
0.951057 + 0.309017i
−0.951057 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
−0.587785 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i 2.23607 −1.00000 −3.07768 1.00000i 0.951057 0.309017i −0.309017 0.951057i −1.31433 1.80902i
109.2 0.587785 + 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 2.23607 −1.00000 3.07768 + 1.00000i −0.951057 + 0.309017i −0.309017 0.951057i 1.31433 + 1.80902i
349.1 −0.951057 0.309017i 0.951057 0.309017i 0.809017 + 0.587785i −2.23607 −1.00000 −0.726543 + 1.00000i −0.587785 0.809017i 0.809017 0.587785i 2.12663 + 0.690983i
349.2 0.951057 + 0.309017i −0.951057 + 0.309017i 0.809017 + 0.587785i −2.23607 −1.00000 0.726543 1.00000i 0.587785 + 0.809017i 0.809017 0.587785i −2.12663 0.690983i
469.1 −0.951057 + 0.309017i 0.951057 + 0.309017i 0.809017 0.587785i −2.23607 −1.00000 −0.726543 1.00000i −0.587785 + 0.809017i 0.809017 + 0.587785i 2.12663 0.690983i
469.2 0.951057 0.309017i −0.951057 0.309017i 0.809017 0.587785i −2.23607 −1.00000 0.726543 + 1.00000i 0.587785 0.809017i 0.809017 + 0.587785i −2.12663 + 0.690983i
529.1 −0.587785 + 0.809017i 0.587785 + 0.809017i −0.309017 0.951057i 2.23607 −1.00000 −3.07768 + 1.00000i 0.951057 + 0.309017i −0.309017 + 0.951057i −1.31433 + 1.80902i
529.2 0.587785 0.809017i −0.587785 0.809017i −0.309017 0.951057i 2.23607 −1.00000 3.07768 1.00000i −0.951057 0.309017i −0.309017 + 0.951057i 1.31433 1.80902i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.d even 5 1 inner
155.n even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.z.a 8
5.b even 2 1 inner 930.2.z.a 8
31.d even 5 1 inner 930.2.z.a 8
155.n even 10 1 inner 930.2.z.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.z.a 8 1.a even 1 1 trivial
930.2.z.a 8 5.b even 2 1 inner
930.2.z.a 8 31.d even 5 1 inner
930.2.z.a 8 155.n even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 16 T_{7}^{6} + 96 T_{7}^{4} + 64 T_{7}^{2} + 256 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$3$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$5$ \( ( -5 + T^{2} )^{4} \)
$7$ \( 256 + 64 T^{2} + 96 T^{4} - 16 T^{6} + T^{8} \)
$11$ \( ( 121 - 66 T + 16 T^{2} - T^{3} + T^{4} )^{2} \)
$13$ \( 1 - 76 T^{2} + 2206 T^{4} + 29 T^{6} + T^{8} \)
$17$ \( 6561 + 729 T^{2} + 486 T^{4} - 36 T^{6} + T^{8} \)
$19$ \( ( 400 + 200 T + 60 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$23$ \( 625 - 1375 T^{2} + 1150 T^{4} + 20 T^{6} + T^{8} \)
$29$ \( ( 1 - 11 T + 46 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$31$ \( ( 961 + 341 T + 91 T^{2} + 11 T^{3} + T^{4} )^{2} \)
$37$ \( ( 2025 + 135 T^{2} + T^{4} )^{2} \)
$41$ \( ( 1936 + 704 T + 136 T^{2} + 14 T^{3} + T^{4} )^{2} \)
$43$ \( 65536 - 4096 T^{2} + 256 T^{4} - 16 T^{6} + T^{8} \)
$47$ \( 390625 - 62500 T^{2} + 3750 T^{4} + 25 T^{6} + T^{8} \)
$53$ \( 3748096 - 147136 T^{2} + 2656 T^{4} - 16 T^{6} + T^{8} \)
$59$ \( ( 961 - 31 T + 76 T^{2} + 14 T^{3} + T^{4} )^{2} \)
$61$ \( ( 20 - 10 T + T^{2} )^{4} \)
$67$ \( ( 64 + T^{2} )^{4} \)
$71$ \( ( 256 - 64 T + 16 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$73$ \( 3748096 + 240064 T^{2} + 48016 T^{4} - 356 T^{6} + T^{8} \)
$79$ \( ( 961 - 372 T + 94 T^{2} - 13 T^{3} + T^{4} )^{2} \)
$83$ \( 160000 - 32000 T^{2} + 2400 T^{4} + 20 T^{6} + T^{8} \)
$89$ \( ( 1296 - 864 T + 216 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$97$ \( 65536 + 45056 T^{2} + 82176 T^{4} - 464 T^{6} + T^{8} \)
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