Properties

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

Related objects

Downloads

Learn more

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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} - 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{5} - 11 q^{7} + 3 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} - 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{5} - 11 q^{7} + 3 \zeta_{6} q^{9} + 8 q^{11} + (3 \zeta_{6} - 6) q^{13} + (2 \zeta_{6} - 4) q^{15} + ( - 26 \zeta_{6} + 26) q^{17} - 19 q^{19} + (11 \zeta_{6} + 11) q^{21} - 32 \zeta_{6} q^{23} + 21 \zeta_{6} q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + (14 \zeta_{6} - 28) q^{29} + (62 \zeta_{6} - 31) q^{31} + ( - 8 \zeta_{6} - 8) q^{33} + (22 \zeta_{6} - 22) q^{35} + (54 \zeta_{6} - 27) q^{37} + 9 q^{39} + ( - 8 \zeta_{6} - 8) q^{41} + ( - 47 \zeta_{6} + 47) q^{43} + 6 q^{45} + 70 \zeta_{6} q^{47} + 72 q^{49} + (26 \zeta_{6} - 52) q^{51} + (4 \zeta_{6} - 8) q^{53} + ( - 16 \zeta_{6} + 16) q^{55} + (19 \zeta_{6} + 19) q^{57} + (62 \zeta_{6} + 62) q^{59} - 35 \zeta_{6} q^{61} - 33 \zeta_{6} q^{63} + (12 \zeta_{6} - 6) q^{65} + (9 \zeta_{6} - 18) q^{67} + (64 \zeta_{6} - 32) q^{69} + (76 \zeta_{6} + 76) q^{71} + (59 \zeta_{6} - 59) q^{73} + ( - 42 \zeta_{6} + 21) q^{75} - 88 q^{77} + (15 \zeta_{6} + 15) q^{79} + (9 \zeta_{6} - 9) q^{81} + 2 q^{83} - 52 \zeta_{6} q^{85} + 42 q^{87} + (46 \zeta_{6} - 92) q^{89} + ( - 33 \zeta_{6} + 66) q^{91} + ( - 93 \zeta_{6} + 93) q^{93} + (38 \zeta_{6} - 38) q^{95} + ( - 12 \zeta_{6} - 12) q^{97} + 24 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 2 q^{5} - 22 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + 2 q^{5} - 22 q^{7} + 3 q^{9} + 16 q^{11} - 9 q^{13} - 6 q^{15} + 26 q^{17} - 38 q^{19} + 33 q^{21} - 32 q^{23} + 21 q^{25} - 42 q^{29} - 24 q^{33} - 22 q^{35} + 18 q^{39} - 24 q^{41} + 47 q^{43} + 12 q^{45} + 70 q^{47} + 144 q^{49} - 78 q^{51} - 12 q^{53} + 16 q^{55} + 57 q^{57} + 186 q^{59} - 35 q^{61} - 33 q^{63} - 27 q^{67} + 228 q^{71} - 59 q^{73} - 176 q^{77} + 45 q^{79} - 9 q^{81} + 4 q^{83} - 52 q^{85} + 84 q^{87} - 138 q^{89} + 99 q^{91} + 93 q^{93} - 38 q^{95} - 36 q^{97} + 24 q^{99}+O(q^{100}) \) 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 1.00000 + 1.73205i 0 −11.0000 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 1.00000 1.73205i 0 −11.0000 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.b 2
4.b odd 2 1 228.3.l.c 2
12.b even 2 1 684.3.y.b 2
19.d odd 6 1 inner 912.3.be.b 2
76.f even 6 1 228.3.l.c 2
228.n odd 6 1 684.3.y.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.l.c 2 4.b odd 2 1
228.3.l.c 2 76.f even 6 1
684.3.y.b 2 12.b even 2 1
684.3.y.b 2 228.n odd 6 1
912.3.be.b 2 1.a even 1 1 trivial
912.3.be.b 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 \(S_{3}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{2} - 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{7} + 11 \) 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
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( (T + 11)^{2} \) Copy content Toggle raw display
$11$ \( (T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$17$ \( T^{2} - 26T + 676 \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 32T + 1024 \) Copy content Toggle raw display
$29$ \( T^{2} + 42T + 588 \) Copy content Toggle raw display
$31$ \( T^{2} + 2883 \) Copy content Toggle raw display
$37$ \( T^{2} + 2187 \) Copy content Toggle raw display
$41$ \( T^{2} + 24T + 192 \) Copy content Toggle raw display
$43$ \( T^{2} - 47T + 2209 \) Copy content Toggle raw display
$47$ \( T^{2} - 70T + 4900 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$59$ \( T^{2} - 186T + 11532 \) Copy content Toggle raw display
$61$ \( T^{2} + 35T + 1225 \) Copy content Toggle raw display
$67$ \( T^{2} + 27T + 243 \) Copy content Toggle raw display
$71$ \( T^{2} - 228T + 17328 \) Copy content Toggle raw display
$73$ \( T^{2} + 59T + 3481 \) Copy content Toggle raw display
$79$ \( T^{2} - 45T + 675 \) Copy content Toggle raw display
$83$ \( (T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 138T + 6348 \) Copy content Toggle raw display
$97$ \( T^{2} + 36T + 432 \) Copy content Toggle raw display
show more
show less