Properties

Label 624.4.bv.a
Level $624$
Weight $4$
Character orbit 624.bv
Analytic conductor $36.817$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.bv (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q + (3 \zeta_{6} - 3) q^{3} + ( - 6 \zeta_{6} + 3) q^{5} + (6 \zeta_{6} - 12) q^{7} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (3 \zeta_{6} - 3) q^{3} + ( - 6 \zeta_{6} + 3) q^{5} + (6 \zeta_{6} - 12) q^{7} - 9 \zeta_{6} q^{9} + ( - 30 \zeta_{6} - 30) q^{11} + ( - 39 \zeta_{6} - 13) q^{13} + (9 \zeta_{6} + 9) q^{15} + 117 \zeta_{6} q^{17} + ( - 14 \zeta_{6} + 28) q^{19} + ( - 36 \zeta_{6} + 18) q^{21} + ( - 18 \zeta_{6} + 18) q^{23} + 98 q^{25} + 27 q^{27} + ( - 99 \zeta_{6} + 99) q^{29} + (224 \zeta_{6} - 112) q^{31} + ( - 90 \zeta_{6} + 180) q^{33} + 54 \zeta_{6} q^{35} + (65 \zeta_{6} + 65) q^{37} + ( - 39 \zeta_{6} + 156) q^{39} + ( - 21 \zeta_{6} - 21) q^{41} - 82 \zeta_{6} q^{43} + (27 \zeta_{6} - 54) q^{45} + ( - 84 \zeta_{6} + 42) q^{47} + (235 \zeta_{6} - 235) q^{49} - 351 q^{51} - 261 q^{53} + (270 \zeta_{6} - 270) q^{55} + (84 \zeta_{6} - 42) q^{57} + ( - 456 \zeta_{6} + 912) q^{59} + 719 \zeta_{6} q^{61} + (54 \zeta_{6} + 54) q^{63} + (195 \zeta_{6} - 273) q^{65} + (406 \zeta_{6} + 406) q^{67} + 54 \zeta_{6} q^{69} + ( - 270 \zeta_{6} + 540) q^{71} + ( - 790 \zeta_{6} + 395) q^{73} + (294 \zeta_{6} - 294) q^{75} + 540 q^{77} + 440 q^{79} + (81 \zeta_{6} - 81) q^{81} + (1380 \zeta_{6} - 690) q^{83} + ( - 351 \zeta_{6} + 702) q^{85} + 297 \zeta_{6} q^{87} + (876 \zeta_{6} + 876) q^{89} + (156 \zeta_{6} + 390) q^{91} + ( - 336 \zeta_{6} - 336) q^{93} - 126 \zeta_{6} q^{95} + (668 \zeta_{6} - 1336) q^{97} + (540 \zeta_{6} - 270) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 18 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - 18 q^{7} - 9 q^{9} - 90 q^{11} - 65 q^{13} + 27 q^{15} + 117 q^{17} + 42 q^{19} + 18 q^{23} + 196 q^{25} + 54 q^{27} + 99 q^{29} + 270 q^{33} + 54 q^{35} + 195 q^{37} + 273 q^{39} - 63 q^{41} - 82 q^{43} - 81 q^{45} - 235 q^{49} - 702 q^{51} - 522 q^{53} - 270 q^{55} + 1368 q^{59} + 719 q^{61} + 162 q^{63} - 351 q^{65} + 1218 q^{67} + 54 q^{69} + 810 q^{71} - 294 q^{75} + 1080 q^{77} + 880 q^{79} - 81 q^{81} + 1053 q^{85} + 297 q^{87} + 2628 q^{89} + 936 q^{91} - 1008 q^{93} - 126 q^{95} - 2004 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.50000 2.59808i 0 5.19615i 0 −9.00000 5.19615i 0 −4.50000 + 7.79423i 0
433.1 0 −1.50000 + 2.59808i 0 5.19615i 0 −9.00000 + 5.19615i 0 −4.50000 7.79423i 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
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.bv.a 2
4.b odd 2 1 39.4.j.a 2
12.b even 2 1 117.4.q.b 2
13.e even 6 1 inner 624.4.bv.a 2
52.i odd 6 1 39.4.j.a 2
52.i odd 6 1 507.4.b.a 2
52.j odd 6 1 507.4.b.a 2
52.l even 12 2 507.4.a.g 2
156.r even 6 1 117.4.q.b 2
156.v odd 12 2 1521.4.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.a 2 4.b odd 2 1
39.4.j.a 2 52.i odd 6 1
117.4.q.b 2 12.b even 2 1
117.4.q.b 2 156.r even 6 1
507.4.a.g 2 52.l even 12 2
507.4.b.a 2 52.i odd 6 1
507.4.b.a 2 52.j odd 6 1
624.4.bv.a 2 1.a even 1 1 trivial
624.4.bv.a 2 13.e even 6 1 inner
1521.4.a.m 2 156.v odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 27 \) acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 27 \) Copy content Toggle raw display
$7$ \( T^{2} + 18T + 108 \) Copy content Toggle raw display
$11$ \( T^{2} + 90T + 2700 \) Copy content Toggle raw display
$13$ \( T^{2} + 65T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} - 117T + 13689 \) Copy content Toggle raw display
$19$ \( T^{2} - 42T + 588 \) Copy content Toggle raw display
$23$ \( T^{2} - 18T + 324 \) Copy content Toggle raw display
$29$ \( T^{2} - 99T + 9801 \) Copy content Toggle raw display
$31$ \( T^{2} + 37632 \) Copy content Toggle raw display
$37$ \( T^{2} - 195T + 12675 \) Copy content Toggle raw display
$41$ \( T^{2} + 63T + 1323 \) Copy content Toggle raw display
$43$ \( T^{2} + 82T + 6724 \) Copy content Toggle raw display
$47$ \( T^{2} + 5292 \) Copy content Toggle raw display
$53$ \( (T + 261)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 1368 T + 623808 \) Copy content Toggle raw display
$61$ \( T^{2} - 719T + 516961 \) Copy content Toggle raw display
$67$ \( T^{2} - 1218 T + 494508 \) Copy content Toggle raw display
$71$ \( T^{2} - 810T + 218700 \) Copy content Toggle raw display
$73$ \( T^{2} + 468075 \) Copy content Toggle raw display
$79$ \( (T - 440)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1428300 \) Copy content Toggle raw display
$89$ \( T^{2} - 2628 T + 2302128 \) Copy content Toggle raw display
$97$ \( T^{2} + 2004 T + 1338672 \) Copy content Toggle raw display
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