Properties

Label 124.4.e.b
Level $124$
Weight $4$
Character orbit 124.e
Analytic conductor $7.316$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 124.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.31623684071\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.250722553392.1
Defining polynomial: \( x^{6} + 46x^{4} - 24x^{3} + 2116x^{2} - 552x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{3} + ( - \beta_{3} + 2 \beta_1) q^{5} + ( - 15 \beta_{3} - \beta_{2} - \beta_1 + 15) q^{7} + ( - \beta_{5} + \beta_{4} - 13 \beta_{3} + 6 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{3} + ( - \beta_{3} + 2 \beta_1) q^{5} + ( - 15 \beta_{3} - \beta_{2} - \beta_1 + 15) q^{7} + ( - \beta_{5} + \beta_{4} - 13 \beta_{3} + 6 \beta_1) q^{9} + ( - \beta_{5} + \beta_{4} + 20 \beta_{3} - 5 \beta_1) q^{11} + (\beta_{5} - \beta_{4} + 38 \beta_{3} + 4 \beta_1) q^{13} + ( - 2 \beta_{4} + 7 \beta_{2} + 65) q^{15} + ( - 3 \beta_{5} - 42 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 42) q^{17} + ( - 6 \beta_{5} - 23 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 23) q^{19} + ( - \beta_{5} + \beta_{4} + 14 \beta_{3} - 12 \beta_1) q^{21} + (6 \beta_{4} - 4 \beta_{2} + 14) q^{23} + (4 \beta_{5} - 4 \beta_{2} - 4 \beta_1) q^{25} + ( - 9 \beta_{4} + 19 \beta_{2} + 156) q^{27} + (\beta_{4} - 18 \beta_{2} + 55) q^{29} + ( - 5 \beta_{5} + 6 \beta_{4} + 43 \beta_{3} + 5 \beta_{2} + 6 \beta_1 - 134) q^{31} + (2 \beta_{4} - 20 \beta_{2} - 203) q^{33} + ( - 2 \beta_{4} - 29 \beta_{2} + 47) q^{35} + (6 \beta_{5} - 41 \beta_{3} + 58 \beta_{2} + 58 \beta_1 + 41) q^{37} + ( - \beta_{4} - 41 \beta_{2} - 2) q^{39} + (\beta_{5} - \beta_{4} - 236 \beta_{3} + 14 \beta_1) q^{41} + (2 \beta_{5} - 35 \beta_{3} + 47 \beta_{2} + 47 \beta_1 + 35) q^{43} + (13 \beta_{5} + 409 \beta_{3} - 62 \beta_{2} - 62 \beta_1 - 409) q^{45} + (4 \beta_{4} + 32 \beta_{2} - 96) q^{47} + ( - \beta_{5} + \beta_{4} + 87 \beta_{3} - 30 \beta_1) q^{49} + (7 \beta_{5} - 7 \beta_{4} + 100 \beta_{3} - 81 \beta_1) q^{51} + (\beta_{5} - \beta_{4} - 422 \beta_{3} - 20 \beta_1) q^{53} + ( - 9 \beta_{5} - 306 \beta_{3} + 15 \beta_{2} + 15 \beta_1 + 306) q^{55} + (21 \beta_{5} - 21 \beta_{4} + 234 \beta_{3} - 122 \beta_1) q^{57} + (13 \beta_{5} + 334 \beta_{3} - \beta_{2} - \beta_1 - 334) q^{59} + ( - 5 \beta_{4} + 14 \beta_{2} - 415) q^{61} + (9 \beta_{4} - 62 \beta_{2} + 3) q^{63} + (7 \beta_{5} + 186 \beta_{3} + 102 \beta_{2} + 102 \beta_1 - 186) q^{65} + ( - 21 \beta_{5} + 21 \beta_{4} + 240 \beta_{3} + 15 \beta_1) q^{67} + ( - 22 \beta_{5} - 154 \beta_{3} + 88 \beta_{2} + 88 \beta_1 + 154) q^{69} + (21 \beta_{5} - 21 \beta_{4} - 356 \beta_{3} + 115 \beta_1) q^{71} + (29 \beta_{5} - 29 \beta_{4} - 56 \beta_{3} + 6 \beta_1) q^{73} + ( - 16 \beta_{5} + 16 \beta_{4} - 172 \beta_{3} + 72 \beta_1) q^{75} + (20 \beta_{4} + 70 \beta_{2} + 157) q^{77} + (2 \beta_{5} - 225 \beta_{3} - 49 \beta_{2} - 49 \beta_1 + 225) q^{79} + (19 \beta_{5} + 814 \beta_{3} - 186 \beta_{2} - 186 \beta_1 - 814) q^{81} + ( - 10 \beta_{5} + 10 \beta_{4} - 3 \beta_{3} - 149 \beta_1) q^{83} + ( - \beta_{4} - 172 \beta_{2} + 10) q^{85} + ( - 21 \beta_{5} - 405 \beta_{3} + 14 \beta_{2} + 14 \beta_1 + 405) q^{87} + (33 \beta_{4} + 54 \beta_{2} - 105) q^{89} + ( - 19 \beta_{4} - 113 \beta_{2} + 682) q^{91} + (2 \beta_{5} - 21 \beta_{4} - 259 \beta_{3} + 184 \beta_{2} + 134 \beta_1 + 376) q^{93} + (12 \beta_{4} - 229 \beta_{2} - 353) q^{95} + (13 \beta_{4} + 74 \beta_{2} - 353) q^{97} + ( - 53 \beta_{5} - 713 \beta_{3} + 158 \beta_{2} + 158 \beta_1 + 713) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} - 3 q^{5} + 45 q^{7} - 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 9 q^{3} - 3 q^{5} + 45 q^{7} - 38 q^{9} + 61 q^{11} + 113 q^{13} + 386 q^{15} + 123 q^{17} + 63 q^{19} + 43 q^{21} + 96 q^{23} + 4 q^{25} + 918 q^{27} + 332 q^{29} - 668 q^{31} - 1214 q^{33} + 278 q^{35} + 129 q^{37} - 14 q^{39} - 709 q^{41} + 107 q^{43} - 1214 q^{45} - 568 q^{47} + 262 q^{49} + 293 q^{51} - 1267 q^{53} + 909 q^{55} + 681 q^{57} - 989 q^{59} - 2500 q^{61} + 36 q^{63} - 551 q^{65} + 741 q^{67} + 440 q^{69} - 1089 q^{71} - 197 q^{73} - 500 q^{75} + 982 q^{77} + 677 q^{79} - 2423 q^{81} + q^{83} + 58 q^{85} + 1194 q^{87} - 564 q^{89} + 4054 q^{91} + 1439 q^{93} - 2094 q^{95} - 2092 q^{97} + 2086 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 46x^{4} - 24x^{3} + 2116x^{2} - 552x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 12 ) / 46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 46\nu^{3} - 12\nu^{2} + 2116\nu ) / 552 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 46\nu^{2} - 12\nu + 1426 ) / 46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -31\nu^{5} - 1426\nu^{3} + 924\nu^{2} - 65596\nu + 17112 ) / 552 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 31\beta_{3} - 31 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 46\beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -46\beta_{5} + 46\beta_{4} - 1426\beta_{3} + 12\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{5} + 924\beta_{3} - 2116\beta_{2} - 2116\beta _1 - 924 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−3.45459 5.98353i
0.130629 + 0.226255i
3.32396 + 5.75727i
−3.45459 + 5.98353i
0.130629 0.226255i
3.32396 5.75727i
0 −4.95459 + 8.58160i 0 −7.40918 12.8331i 0 4.04541 7.00685i 0 −35.5960 61.6540i 0
5.2 0 −1.36937 + 2.37182i 0 −0.238743 0.413515i 0 7.63063 13.2166i 0 9.74964 + 16.8869i 0
5.3 0 1.82396 3.15920i 0 6.14793 + 10.6485i 0 10.8240 18.7477i 0 6.84632 + 11.8582i 0
25.1 0 −4.95459 8.58160i 0 −7.40918 + 12.8331i 0 4.04541 + 7.00685i 0 −35.5960 + 61.6540i 0
25.2 0 −1.36937 2.37182i 0 −0.238743 + 0.413515i 0 7.63063 + 13.2166i 0 9.74964 16.8869i 0
25.3 0 1.82396 + 3.15920i 0 6.14793 10.6485i 0 10.8240 + 18.7477i 0 6.84632 11.8582i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.4.e.b 6
3.b odd 2 1 1116.4.i.d 6
4.b odd 2 1 496.4.i.b 6
31.c even 3 1 inner 124.4.e.b 6
93.h odd 6 1 1116.4.i.d 6
124.i odd 6 1 496.4.i.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.e.b 6 1.a even 1 1 trivial
124.4.e.b 6 31.c even 3 1 inner
496.4.i.b 6 4.b odd 2 1
496.4.i.b 6 124.i odd 6 1
1116.4.i.d 6 3.b odd 2 1
1116.4.i.d 6 93.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 9T_{3}^{5} + 100T_{3}^{4} + 27T_{3}^{3} + 1252T_{3}^{2} + 1881T_{3} + 9801 \) acting on \(S_{4}^{\mathrm{new}}(124, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 9 T^{5} + 100 T^{4} + \cdots + 9801 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + 190 T^{4} + \cdots + 7569 \) Copy content Toggle raw display
$7$ \( T^{6} - 45 T^{5} + 1396 T^{4} + \cdots + 7144929 \) Copy content Toggle raw display
$11$ \( T^{6} - 61 T^{5} + \cdots + 2460060801 \) Copy content Toggle raw display
$13$ \( T^{6} - 113 T^{5} + \cdots + 292512609 \) Copy content Toggle raw display
$17$ \( T^{6} - 123 T^{5} + \cdots + 139818409929 \) Copy content Toggle raw display
$19$ \( T^{6} - 63 T^{5} + \cdots + 3749757272041 \) Copy content Toggle raw display
$23$ \( (T^{3} - 48 T^{2} - 26224 T - 882816)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 166 T^{2} - 7072 T + 1260064)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 668 T^{5} + \cdots + 26439622160671 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 935157255676281 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 116486668824201 \) Copy content Toggle raw display
$43$ \( T^{6} - 107 T^{5} + \cdots + 70307168493969 \) Copy content Toggle raw display
$47$ \( (T^{3} + 284 T^{2} - 26896 T + 407104)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 1267 T^{5} + \cdots + 43\!\cdots\!89 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 362920445321841 \) Copy content Toggle raw display
$61$ \( (T^{3} + 1250 T^{2} + 491664 T + 59458752)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 144877789637361 \) Copy content Toggle raw display
$71$ \( T^{6} + 1089 T^{5} + \cdots + 28\!\cdots\!89 \) Copy content Toggle raw display
$73$ \( T^{6} + 197 T^{5} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 533707532656321 \) Copy content Toggle raw display
$83$ \( T^{6} - T^{5} + \cdots + 66\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{3} + 282 T^{2} - 811584 T - 230488416)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 1046 T^{2} + 28240 T - 1795008)^{2} \) Copy content Toggle raw display
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