Properties

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

Related objects

Downloads

Learn more

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.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} + 3) q^{3} + 7 q^{5} - 10 \zeta_{6} q^{7} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \zeta_{6} + 3) q^{3} + 7 q^{5} - 10 \zeta_{6} q^{7} - 9 \zeta_{6} q^{9} + (22 \zeta_{6} - 22) q^{11} + ( - 13 \zeta_{6} - 39) q^{13} + ( - 21 \zeta_{6} + 21) q^{15} - 37 \zeta_{6} q^{17} + 30 \zeta_{6} q^{19} - 30 q^{21} + (162 \zeta_{6} - 162) q^{23} - 76 q^{25} - 27 q^{27} + ( - 113 \zeta_{6} + 113) q^{29} - 196 q^{31} + 66 \zeta_{6} q^{33} - 70 \zeta_{6} q^{35} + (13 \zeta_{6} - 13) q^{37} + (117 \zeta_{6} - 156) q^{39} + (285 \zeta_{6} - 285) q^{41} - 246 \zeta_{6} q^{43} - 63 \zeta_{6} q^{45} + 462 q^{47} + ( - 243 \zeta_{6} + 243) q^{49} - 111 q^{51} - 537 q^{53} + (154 \zeta_{6} - 154) q^{55} + 90 q^{57} + 576 \zeta_{6} q^{59} + 635 \zeta_{6} q^{61} + (90 \zeta_{6} - 90) q^{63} + ( - 91 \zeta_{6} - 273) q^{65} + ( - 202 \zeta_{6} + 202) q^{67} + 486 \zeta_{6} q^{69} - 1086 \zeta_{6} q^{71} - 805 q^{73} + (228 \zeta_{6} - 228) q^{75} + 220 q^{77} - 884 q^{79} + (81 \zeta_{6} - 81) q^{81} - 518 q^{83} - 259 \zeta_{6} q^{85} - 339 \zeta_{6} q^{87} + (194 \zeta_{6} - 194) q^{89} + (520 \zeta_{6} - 130) q^{91} + (588 \zeta_{6} - 588) q^{93} + 210 \zeta_{6} q^{95} + 1202 \zeta_{6} q^{97} + 198 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 14 q^{5} - 10 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + 14 q^{5} - 10 q^{7} - 9 q^{9} - 22 q^{11} - 91 q^{13} + 21 q^{15} - 37 q^{17} + 30 q^{19} - 60 q^{21} - 162 q^{23} - 152 q^{25} - 54 q^{27} + 113 q^{29} - 392 q^{31} + 66 q^{33} - 70 q^{35} - 13 q^{37} - 195 q^{39} - 285 q^{41} - 246 q^{43} - 63 q^{45} + 924 q^{47} + 243 q^{49} - 222 q^{51} - 1074 q^{53} - 154 q^{55} + 180 q^{57} + 576 q^{59} + 635 q^{61} - 90 q^{63} - 637 q^{65} + 202 q^{67} + 486 q^{69} - 1086 q^{71} - 1610 q^{73} - 228 q^{75} + 440 q^{77} - 1768 q^{79} - 81 q^{81} - 1036 q^{83} - 259 q^{85} - 339 q^{87} - 194 q^{89} + 260 q^{91} - 588 q^{93} + 210 q^{95} + 1202 q^{97} + 396 q^{99}+O(q^{100}) \) 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 7.00000 0 −5.00000 + 8.66025i 0 −4.50000 + 7.79423i 0
529.1 0 1.50000 2.59808i 0 7.00000 0 −5.00000 8.66025i 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.c 2
4.b odd 2 1 39.4.e.b 2
12.b even 2 1 117.4.g.a 2
13.c even 3 1 inner 624.4.q.c 2
52.i odd 6 1 507.4.a.d 1
52.j odd 6 1 39.4.e.b 2
52.j odd 6 1 507.4.a.b 1
52.l even 12 2 507.4.b.d 2
156.p even 6 1 117.4.g.a 2
156.p even 6 1 1521.4.a.h 1
156.r even 6 1 1521.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.b 2 4.b odd 2 1
39.4.e.b 2 52.j odd 6 1
117.4.g.a 2 12.b even 2 1
117.4.g.a 2 156.p even 6 1
507.4.a.b 1 52.j odd 6 1
507.4.a.d 1 52.i odd 6 1
507.4.b.d 2 52.l even 12 2
624.4.q.c 2 1.a even 1 1 trivial
624.4.q.c 2 13.c even 3 1 inner
1521.4.a.e 1 156.r even 6 1
1521.4.a.h 1 156.p even 6 1

Hecke kernels

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

\( T_{5} - 7 \) Copy content Toggle raw display
\( T_{7}^{2} + 10T_{7} + 100 \) 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
$5$ \( (T - 7)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$11$ \( T^{2} + 22T + 484 \) Copy content Toggle raw display
$13$ \( T^{2} + 91T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} + 37T + 1369 \) Copy content Toggle raw display
$19$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$23$ \( T^{2} + 162T + 26244 \) Copy content Toggle raw display
$29$ \( T^{2} - 113T + 12769 \) Copy content Toggle raw display
$31$ \( (T + 196)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$41$ \( T^{2} + 285T + 81225 \) Copy content Toggle raw display
$43$ \( T^{2} + 246T + 60516 \) Copy content Toggle raw display
$47$ \( (T - 462)^{2} \) Copy content Toggle raw display
$53$ \( (T + 537)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 576T + 331776 \) Copy content Toggle raw display
$61$ \( T^{2} - 635T + 403225 \) Copy content Toggle raw display
$67$ \( T^{2} - 202T + 40804 \) Copy content Toggle raw display
$71$ \( T^{2} + 1086 T + 1179396 \) Copy content Toggle raw display
$73$ \( (T + 805)^{2} \) Copy content Toggle raw display
$79$ \( (T + 884)^{2} \) Copy content Toggle raw display
$83$ \( (T + 518)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 194T + 37636 \) Copy content Toggle raw display
$97$ \( T^{2} - 1202 T + 1444804 \) Copy content Toggle raw display
show more
show less