Properties

Label 624.4.q.b
Level $624$
Weight $4$
Character orbit 624.q
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.q (of order \(3\), 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} + 3) q^{3} - 9 q^{5} + 2 \zeta_{6} q^{7} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \zeta_{6} + 3) q^{3} - 9 q^{5} + 2 \zeta_{6} q^{7} - 9 \zeta_{6} q^{9} + ( - 30 \zeta_{6} + 30) q^{11} + (39 \zeta_{6} + 13) q^{13} + (27 \zeta_{6} - 27) q^{15} + 111 \zeta_{6} q^{17} - 46 \zeta_{6} q^{19} + 6 q^{21} + (6 \zeta_{6} - 6) q^{23} - 44 q^{25} - 27 q^{27} + ( - 105 \zeta_{6} + 105) q^{29} + 100 q^{31} - 90 \zeta_{6} q^{33} - 18 \zeta_{6} q^{35} + (17 \zeta_{6} - 17) q^{37} + ( - 39 \zeta_{6} + 156) q^{39} + ( - 231 \zeta_{6} + 231) q^{41} - 514 \zeta_{6} q^{43} + 81 \zeta_{6} q^{45} + 162 q^{47} + ( - 339 \zeta_{6} + 339) q^{49} + 333 q^{51} + 639 q^{53} + (270 \zeta_{6} - 270) q^{55} - 138 q^{57} + 600 \zeta_{6} q^{59} - 233 \zeta_{6} q^{61} + ( - 18 \zeta_{6} + 18) q^{63} + ( - 351 \zeta_{6} - 117) q^{65} + ( - 926 \zeta_{6} + 926) q^{67} + 18 \zeta_{6} q^{69} - 930 \zeta_{6} q^{71} - 253 q^{73} + (132 \zeta_{6} - 132) q^{75} + 60 q^{77} + 1324 q^{79} + (81 \zeta_{6} - 81) q^{81} - 810 q^{83} - 999 \zeta_{6} q^{85} - 315 \zeta_{6} q^{87} + (498 \zeta_{6} - 498) q^{89} + (104 \zeta_{6} - 78) q^{91} + ( - 300 \zeta_{6} + 300) q^{93} + 414 \zeta_{6} q^{95} - 1358 \zeta_{6} q^{97} - 270 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 18 q^{5} + 2 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 18 q^{5} + 2 q^{7} - 9 q^{9} + 30 q^{11} + 65 q^{13} - 27 q^{15} + 111 q^{17} - 46 q^{19} + 12 q^{21} - 6 q^{23} - 88 q^{25} - 54 q^{27} + 105 q^{29} + 200 q^{31} - 90 q^{33} - 18 q^{35} - 17 q^{37} + 273 q^{39} + 231 q^{41} - 514 q^{43} + 81 q^{45} + 324 q^{47} + 339 q^{49} + 666 q^{51} + 1278 q^{53} - 270 q^{55} - 276 q^{57} + 600 q^{59} - 233 q^{61} + 18 q^{63} - 585 q^{65} + 926 q^{67} + 18 q^{69} - 930 q^{71} - 506 q^{73} - 132 q^{75} + 120 q^{77} + 2648 q^{79} - 81 q^{81} - 1620 q^{83} - 999 q^{85} - 315 q^{87} - 498 q^{89} - 52 q^{91} + 300 q^{93} + 414 q^{95} - 1358 q^{97} - 540 q^{99}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.50000 + 2.59808i 0 −9.00000 0 1.00000 1.73205i 0 −4.50000 + 7.79423i 0
529.1 0 1.50000 2.59808i 0 −9.00000 0 1.00000 + 1.73205i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.q.b 2
4.b odd 2 1 39.4.e.a 2
12.b even 2 1 117.4.g.b 2
13.c even 3 1 inner 624.4.q.b 2
52.i odd 6 1 507.4.a.a 1
52.j odd 6 1 39.4.e.a 2
52.j odd 6 1 507.4.a.e 1
52.l even 12 2 507.4.b.c 2
156.p even 6 1 117.4.g.b 2
156.p even 6 1 1521.4.a.c 1
156.r even 6 1 1521.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.a 2 4.b odd 2 1
39.4.e.a 2 52.j odd 6 1
117.4.g.b 2 12.b even 2 1
117.4.g.b 2 156.p even 6 1
507.4.a.a 1 52.i odd 6 1
507.4.a.e 1 52.j odd 6 1
507.4.b.c 2 52.l even 12 2
624.4.q.b 2 1.a even 1 1 trivial
624.4.q.b 2 13.c even 3 1 inner
1521.4.a.c 1 156.p even 6 1
1521.4.a.j 1 156.r even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\):

\( T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} + 4 \) 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
$5$ \( (T + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$13$ \( T^{2} - 65T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} - 111T + 12321 \) Copy content Toggle raw display
$19$ \( T^{2} + 46T + 2116 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 105T + 11025 \) Copy content Toggle raw display
$31$ \( (T - 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$41$ \( T^{2} - 231T + 53361 \) Copy content Toggle raw display
$43$ \( T^{2} + 514T + 264196 \) Copy content Toggle raw display
$47$ \( (T - 162)^{2} \) Copy content Toggle raw display
$53$ \( (T - 639)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 600T + 360000 \) Copy content Toggle raw display
$61$ \( T^{2} + 233T + 54289 \) Copy content Toggle raw display
$67$ \( T^{2} - 926T + 857476 \) Copy content Toggle raw display
$71$ \( T^{2} + 930T + 864900 \) Copy content Toggle raw display
$73$ \( (T + 253)^{2} \) Copy content Toggle raw display
$79$ \( (T - 1324)^{2} \) Copy content Toggle raw display
$83$ \( (T + 810)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 498T + 248004 \) Copy content Toggle raw display
$97$ \( T^{2} + 1358 T + 1844164 \) Copy content Toggle raw display
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