Properties

Label 912.3.be.a
Level $912$
Weight $3$
Character orbit 912.be
Analytic conductor $24.850$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.be (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} - 1) q^{3} + (6 \zeta_{6} - 6) q^{5} + 5 q^{7} + 3 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} - 1) q^{3} + (6 \zeta_{6} - 6) q^{5} + 5 q^{7} + 3 \zeta_{6} q^{9} + (11 \zeta_{6} - 22) q^{13} + ( - 6 \zeta_{6} + 12) q^{15} + (6 \zeta_{6} - 6) q^{17} + (16 \zeta_{6} + 5) q^{19} + ( - 5 \zeta_{6} - 5) q^{21} - 24 \zeta_{6} q^{23} - 11 \zeta_{6} q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + ( - 18 \zeta_{6} + 36) q^{29} + ( - 34 \zeta_{6} + 17) q^{31} + (30 \zeta_{6} - 30) q^{35} + (70 \zeta_{6} - 35) q^{37} + 33 q^{39} + ( - 24 \zeta_{6} - 24) q^{41} + (25 \zeta_{6} - 25) q^{43} - 18 q^{45} - 42 \zeta_{6} q^{47} - 24 q^{49} + ( - 6 \zeta_{6} + 12) q^{51} + ( - 36 \zeta_{6} + 72) q^{53} + ( - 37 \zeta_{6} + 11) q^{57} + ( - 42 \zeta_{6} - 42) q^{59} - 43 \zeta_{6} q^{61} + 15 \zeta_{6} q^{63} + ( - 132 \zeta_{6} + 66) q^{65} + (33 \zeta_{6} - 66) q^{67} + (48 \zeta_{6} - 24) q^{69} + ( - 36 \zeta_{6} - 36) q^{71} + (11 \zeta_{6} - 11) q^{73} + (22 \zeta_{6} - 11) q^{75} + ( - \zeta_{6} - 1) q^{79} + (9 \zeta_{6} - 9) q^{81} - 126 q^{83} - 36 \zeta_{6} q^{85} - 54 q^{87} + (6 \zeta_{6} - 12) q^{89} + (55 \zeta_{6} - 110) q^{91} + (51 \zeta_{6} - 51) q^{93} + (30 \zeta_{6} - 126) q^{95} + ( - 76 \zeta_{6} - 76) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 6 q^{5} + 10 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - 6 q^{5} + 10 q^{7} + 3 q^{9} - 33 q^{13} + 18 q^{15} - 6 q^{17} + 26 q^{19} - 15 q^{21} - 24 q^{23} - 11 q^{25} + 54 q^{29} - 30 q^{35} + 66 q^{39} - 72 q^{41} - 25 q^{43} - 36 q^{45} - 42 q^{47} - 48 q^{49} + 18 q^{51} + 108 q^{53} - 15 q^{57} - 126 q^{59} - 43 q^{61} + 15 q^{63} - 99 q^{67} - 108 q^{71} - 11 q^{73} - 3 q^{79} - 9 q^{81} - 252 q^{83} - 36 q^{85} - 108 q^{87} - 18 q^{89} - 165 q^{91} - 51 q^{93} - 222 q^{95} - 228 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(\zeta_{6}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.50000 + 0.866025i 0 −3.00000 5.19615i 0 5.00000 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 −3.00000 + 5.19615i 0 5.00000 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.be.a 2
4.b odd 2 1 228.3.l.a 2
12.b even 2 1 684.3.y.e 2
19.d odd 6 1 inner 912.3.be.a 2
76.f even 6 1 228.3.l.a 2
228.n odd 6 1 684.3.y.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.l.a 2 4.b odd 2 1
228.3.l.a 2 76.f even 6 1
684.3.y.e 2 12.b even 2 1
684.3.y.e 2 228.n odd 6 1
912.3.be.a 2 1.a even 1 1 trivial
912.3.be.a 2 19.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{2} + 6T_{5} + 36 \) Copy content Toggle raw display
\( T_{7} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$7$ \( (T - 5)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 33T + 363 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 26T + 361 \) Copy content Toggle raw display
$23$ \( T^{2} + 24T + 576 \) Copy content Toggle raw display
$29$ \( T^{2} - 54T + 972 \) Copy content Toggle raw display
$31$ \( T^{2} + 867 \) Copy content Toggle raw display
$37$ \( T^{2} + 3675 \) Copy content Toggle raw display
$41$ \( T^{2} + 72T + 1728 \) Copy content Toggle raw display
$43$ \( T^{2} + 25T + 625 \) Copy content Toggle raw display
$47$ \( T^{2} + 42T + 1764 \) Copy content Toggle raw display
$53$ \( T^{2} - 108T + 3888 \) Copy content Toggle raw display
$59$ \( T^{2} + 126T + 5292 \) Copy content Toggle raw display
$61$ \( T^{2} + 43T + 1849 \) Copy content Toggle raw display
$67$ \( T^{2} + 99T + 3267 \) Copy content Toggle raw display
$71$ \( T^{2} + 108T + 3888 \) Copy content Toggle raw display
$73$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$79$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$83$ \( (T + 126)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 108 \) Copy content Toggle raw display
$97$ \( T^{2} + 228T + 17328 \) Copy content Toggle raw display
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