Properties

Label 930.2.d.a
Level $930$
Weight $2$
Character orbit 930.d
Analytic conductor $7.426$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} + (2 i - 1) q^{5} - q^{6} + i q^{7} - i q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + i q^{3} - q^{4} + (2 i - 1) q^{5} - q^{6} + i q^{7} - i q^{8} - q^{9} + ( - i - 2) q^{10} - 3 q^{11} - i q^{12} + 4 i q^{13} - q^{14} + ( - i - 2) q^{15} + q^{16} - i q^{18} + q^{19} + ( - 2 i + 1) q^{20} - q^{21} - 3 i q^{22} - 5 i q^{23} + q^{24} + ( - 4 i - 3) q^{25} - 4 q^{26} - i q^{27} - i q^{28} - 2 q^{29} + ( - 2 i + 1) q^{30} - q^{31} + i q^{32} - 3 i q^{33} + ( - i - 2) q^{35} + q^{36} + 4 i q^{37} + i q^{38} - 4 q^{39} + (i + 2) q^{40} - 10 q^{41} - i q^{42} + 5 i q^{43} + 3 q^{44} + ( - 2 i + 1) q^{45} + 5 q^{46} - 8 i q^{47} + i q^{48} + 6 q^{49} + ( - 3 i + 4) q^{50} - 4 i q^{52} - 5 i q^{53} + q^{54} + ( - 6 i + 3) q^{55} + q^{56} + i q^{57} - 2 i q^{58} + 6 q^{59} + (i + 2) q^{60} - 2 q^{61} - i q^{62} - i q^{63} - q^{64} + ( - 4 i - 8) q^{65} + 3 q^{66} + 2 i q^{67} + 5 q^{69} + ( - 2 i + 1) q^{70} - 5 q^{71} + i q^{72} + 7 i q^{73} - 4 q^{74} + ( - 3 i + 4) q^{75} - q^{76} - 3 i q^{77} - 4 i q^{78} - 3 q^{79} + (2 i - 1) q^{80} + q^{81} - 10 i q^{82} + 2 i q^{83} + q^{84} - 5 q^{86} - 2 i q^{87} + 3 i q^{88} - q^{89} + (i + 2) q^{90} - 4 q^{91} + 5 i q^{92} - i q^{93} + 8 q^{94} + (2 i - 1) q^{95} - q^{96} - 10 i q^{97} + 6 i q^{98} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{9} - 4 q^{10} - 6 q^{11} - 2 q^{14} - 4 q^{15} + 2 q^{16} + 2 q^{19} + 2 q^{20} - 2 q^{21} + 2 q^{24} - 6 q^{25} - 8 q^{26} - 4 q^{29} + 2 q^{30} - 2 q^{31} - 4 q^{35} + 2 q^{36} - 8 q^{39} + 4 q^{40} - 20 q^{41} + 6 q^{44} + 2 q^{45} + 10 q^{46} + 12 q^{49} + 8 q^{50} + 2 q^{54} + 6 q^{55} + 2 q^{56} + 12 q^{59} + 4 q^{60} - 4 q^{61} - 2 q^{64} - 16 q^{65} + 6 q^{66} + 10 q^{69} + 2 q^{70} - 10 q^{71} - 8 q^{74} + 8 q^{75} - 2 q^{76} - 6 q^{79} - 2 q^{80} + 2 q^{81} + 2 q^{84} - 10 q^{86} - 2 q^{89} + 4 q^{90} - 8 q^{91} + 16 q^{94} - 2 q^{95} - 2 q^{96} + 6 q^{99}+O(q^{100}) \) 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
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 −1.00000 2.00000i −1.00000 1.00000i 1.00000i −1.00000 −2.00000 + 1.00000i
559.2 1.00000i 1.00000i −1.00000 −1.00000 + 2.00000i −1.00000 1.00000i 1.00000i −1.00000 −2.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
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.a 2
3.b odd 2 1 2790.2.d.f 2
5.b even 2 1 inner 930.2.d.a 2
5.c odd 4 1 4650.2.a.p 1
5.c odd 4 1 4650.2.a.bc 1
15.d odd 2 1 2790.2.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.a 2 1.a even 1 1 trivial
930.2.d.a 2 5.b even 2 1 inner
2790.2.d.f 2 3.b odd 2 1
2790.2.d.f 2 15.d odd 2 1
4650.2.a.p 1 5.c odd 4 1
4650.2.a.bc 1 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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 25 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 25 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 25 \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T + 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 49 \) Copy content Toggle raw display
$79$ \( (T + 3)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4 \) Copy content Toggle raw display
$89$ \( (T + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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