Properties

Label 624.4.a.r
Level $624$
Weight $4$
Character orbit 624.a
Self dual yes
Analytic conductor $36.817$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + (\beta + 12) q^{5} - \beta q^{7} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + (\beta + 12) q^{5} - \beta q^{7} + 9 q^{9} + ( - 6 \beta + 22) q^{11} - 13 q^{13} + (3 \beta + 36) q^{15} + ( - 2 \beta + 82) q^{17} + (\beta - 24) q^{19} - 3 \beta q^{21} + (24 \beta - 4) q^{23} + (24 \beta + 75) q^{25} + 27 q^{27} + (12 \beta + 202) q^{29} + ( - 13 \beta - 20) q^{31} + ( - 18 \beta + 66) q^{33} + ( - 12 \beta - 56) q^{35} + ( - 14 \beta - 50) q^{37} - 39 q^{39} + ( - 47 \beta + 100) q^{41} + (26 \beta + 308) q^{43} + (9 \beta + 108) q^{45} + (16 \beta + 162) q^{47} - 287 q^{49} + ( - 6 \beta + 246) q^{51} + (60 \beta - 82) q^{53} + ( - 50 \beta - 72) q^{55} + (3 \beta - 72) q^{57} + (20 \beta - 70) q^{59} + ( - 68 \beta + 314) q^{61} - 9 \beta q^{63} + ( - 13 \beta - 156) q^{65} + ( - 85 \beta + 236) q^{67} + (72 \beta - 12) q^{69} + ( - 42 \beta - 214) q^{71} + ( - 38 \beta - 450) q^{73} + (72 \beta + 225) q^{75} + ( - 22 \beta + 336) q^{77} + ( - 44 \beta + 216) q^{79} + 81 q^{81} + (32 \beta + 694) q^{83} + (58 \beta + 872) q^{85} + (36 \beta + 606) q^{87} + (95 \beta + 480) q^{89} + 13 \beta q^{91} + ( - 39 \beta - 60) q^{93} + ( - 12 \beta - 232) q^{95} + (110 \beta - 266) q^{97} + ( - 54 \beta + 198) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 24 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 24 q^{5} + 18 q^{9} + 44 q^{11} - 26 q^{13} + 72 q^{15} + 164 q^{17} - 48 q^{19} - 8 q^{23} + 150 q^{25} + 54 q^{27} + 404 q^{29} - 40 q^{31} + 132 q^{33} - 112 q^{35} - 100 q^{37} - 78 q^{39} + 200 q^{41} + 616 q^{43} + 216 q^{45} + 324 q^{47} - 574 q^{49} + 492 q^{51} - 164 q^{53} - 144 q^{55} - 144 q^{57} - 140 q^{59} + 628 q^{61} - 312 q^{65} + 472 q^{67} - 24 q^{69} - 428 q^{71} - 900 q^{73} + 450 q^{75} + 672 q^{77} + 432 q^{79} + 162 q^{81} + 1388 q^{83} + 1744 q^{85} + 1212 q^{87} + 960 q^{89} - 120 q^{93} - 464 q^{95} - 532 q^{97} + 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.74166
3.74166
0 3.00000 0 4.51669 0 7.48331 0 9.00000 0
1.2 0 3.00000 0 19.4833 0 −7.48331 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.a.r 2
3.b odd 2 1 1872.4.a.t 2
4.b odd 2 1 39.4.a.b 2
8.b even 2 1 2496.4.a.s 2
8.d odd 2 1 2496.4.a.bc 2
12.b even 2 1 117.4.a.c 2
20.d odd 2 1 975.4.a.j 2
28.d even 2 1 1911.4.a.h 2
52.b odd 2 1 507.4.a.f 2
52.f even 4 2 507.4.b.f 4
156.h even 2 1 1521.4.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.b 2 4.b odd 2 1
117.4.a.c 2 12.b even 2 1
507.4.a.f 2 52.b odd 2 1
507.4.b.f 4 52.f even 4 2
624.4.a.r 2 1.a even 1 1 trivial
975.4.a.j 2 20.d odd 2 1
1521.4.a.s 2 156.h even 2 1
1872.4.a.t 2 3.b odd 2 1
1911.4.a.h 2 28.d even 2 1
2496.4.a.s 2 8.b even 2 1
2496.4.a.bc 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(624))\):

\( T_{5}^{2} - 24T_{5} + 88 \) Copy content Toggle raw display
\( T_{7}^{2} - 56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 24T + 88 \) Copy content Toggle raw display
$7$ \( T^{2} - 56 \) Copy content Toggle raw display
$11$ \( T^{2} - 44T - 1532 \) Copy content Toggle raw display
$13$ \( (T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 164T + 6500 \) Copy content Toggle raw display
$19$ \( T^{2} + 48T + 520 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T - 32240 \) Copy content Toggle raw display
$29$ \( T^{2} - 404T + 32740 \) Copy content Toggle raw display
$31$ \( T^{2} + 40T - 9064 \) Copy content Toggle raw display
$37$ \( T^{2} + 100T - 8476 \) Copy content Toggle raw display
$41$ \( T^{2} - 200T - 113704 \) Copy content Toggle raw display
$43$ \( T^{2} - 616T + 57008 \) Copy content Toggle raw display
$47$ \( T^{2} - 324T + 11908 \) Copy content Toggle raw display
$53$ \( T^{2} + 164T - 194876 \) Copy content Toggle raw display
$59$ \( T^{2} + 140T - 17500 \) Copy content Toggle raw display
$61$ \( T^{2} - 628T - 160348 \) Copy content Toggle raw display
$67$ \( T^{2} - 472T - 348904 \) Copy content Toggle raw display
$71$ \( T^{2} + 428T - 52988 \) Copy content Toggle raw display
$73$ \( T^{2} + 900T + 121636 \) Copy content Toggle raw display
$79$ \( T^{2} - 432T - 61760 \) Copy content Toggle raw display
$83$ \( T^{2} - 1388 T + 424292 \) Copy content Toggle raw display
$89$ \( T^{2} - 960T - 275000 \) Copy content Toggle raw display
$97$ \( T^{2} + 532T - 606844 \) Copy content Toggle raw display
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