Properties

Label 124.5.c.a
Level $124$
Weight $5$
Character orbit 124.c
Analytic conductor $12.818$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 124.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.8178754224\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \( x^{10} + 478x^{8} + 69668x^{6} + 4198200x^{4} + 101304000x^{2} + 622080000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} - 1) q^{5} - \beta_{4} q^{7} + (\beta_{2} - 15) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} - 1) q^{5} - \beta_{4} q^{7} + (\beta_{2} - 15) q^{9} - \beta_{6} q^{11} + (\beta_{7} + 2 \beta_1) q^{13} + ( - \beta_{8} - \beta_1) q^{15} + (\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_1) q^{17} + ( - \beta_{9} + \beta_{4} + \beta_{3} - \beta_{2} - 31) q^{19} + ( - \beta_{8} + \beta_{7} + 2 \beta_{5} - 4 \beta_1) q^{21} + ( - 3 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} - 11 \beta_1) q^{23} + ( - \beta_{9} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 75) q^{25} + ( - 3 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} + 5 \beta_{5} - 26 \beta_1) q^{27} + ( - 3 \beta_{8} - \beta_{7} - 3 \beta_{6} - 4 \beta_{5} + 14 \beta_1) q^{29} + (\beta_{9} - 4 \beta_{8} + 2 \beta_{7} - 5 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - 4 \beta_{3} + \cdots + 75) q^{31}+ \cdots + ( - 19 \beta_{8} - 12 \beta_{7} - 66 \beta_{6} + 6 \beta_{5} - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9} - 310 q^{19} - 756 q^{25} + 734 q^{31} - 372 q^{33} - 666 q^{35} - 1924 q^{39} - 210 q^{41} + 650 q^{45} - 1992 q^{47} + 188 q^{49} + 1352 q^{51} + 5610 q^{59} + 3478 q^{63} - 5420 q^{67} + 10160 q^{69} - 1734 q^{71} + 11598 q^{81} - 13244 q^{87} + 3088 q^{93} + 4302 q^{95} + 2942 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 478x^{8} + 69668x^{6} + 4198200x^{4} + 101304000x^{2} + 622080000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{8} - 4358\nu^{6} - 406348\nu^{4} - 11667000\nu^{2} - 67392000 ) / 475200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -13\nu^{8} - 5314\nu^{6} - 534884\nu^{4} - 16456200\nu^{2} - 77284800 ) / 475200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53\nu^{9} + 21014\nu^{7} + 1951444\nu^{5} + 53082840\nu^{3} + 235656000\nu ) / 3564000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -29\nu^{9} - 12002\nu^{7} - 1266292\nu^{5} - 46705320\nu^{3} - 503136000\nu ) / 2376000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37\nu^{9} + 15256\nu^{7} + 1578176\nu^{5} + 52379160\nu^{3} + 314010000\nu ) / 1782000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\nu^{9} + 4358\nu^{7} + 406348\nu^{5} + 11667000\nu^{3} + 67392000\nu ) / 475200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\nu^{8} + 5314\nu^{6} + 539284\nu^{4} + 17785000\nu^{2} + 125966400 ) / 17600 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{8} - 2\beta_{7} - 3\beta_{6} + 5\beta_{5} - 188\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{9} + 108\beta_{4} - 302\beta_{2} + 17928 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 1042\beta_{8} + 576\beta_{7} + 858\beta_{6} - 1722\beta_{5} + 46276\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1336\beta_{9} - 38976\beta_{4} + 3432\beta_{3} + 84564\beta_{2} - 4408344 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -305452\beta_{8} - 156872\beta_{7} - 237660\beta_{6} + 499436\beta_{5} - 12388224\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 381536\beta_{9} + 11451984\beta_{4} - 1402896\beta_{3} - 23407256\beta_{2} + 1179926928 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 85747400\beta_{8} + 42993248\beta_{7} + 65643336\beta_{6} - 139558712\beta_{5} + 3391793104\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
16.6119i
8.69414i
8.43572i
6.79929i
3.01087i
3.01087i
6.79929i
8.43572i
8.69414i
16.6119i
0 16.6119i 0 −1.78751 0 −31.2283 0 −194.955 0
61.2 0 8.69414i 0 32.3102 0 39.7108 0 5.41184 0
61.3 0 8.43572i 0 −14.7147 0 70.0514 0 9.83861 0
61.4 0 6.79929i 0 −34.9624 0 −12.5932 0 34.7697 0
61.5 0 3.01087i 0 16.1544 0 −66.9407 0 71.9347 0
61.6 0 3.01087i 0 16.1544 0 −66.9407 0 71.9347 0
61.7 0 6.79929i 0 −34.9624 0 −12.5932 0 34.7697 0
61.8 0 8.43572i 0 −14.7147 0 70.0514 0 9.83861 0
61.9 0 8.69414i 0 32.3102 0 39.7108 0 5.41184 0
61.10 0 16.6119i 0 −1.78751 0 −31.2283 0 −194.955 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.5.c.a 10
3.b odd 2 1 1116.5.h.b 10
4.b odd 2 1 496.5.e.c 10
31.b odd 2 1 inner 124.5.c.a 10
93.c even 2 1 1116.5.h.b 10
124.d even 2 1 496.5.e.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.5.c.a 10 1.a even 1 1 trivial
124.5.c.a 10 31.b odd 2 1 inner
496.5.e.c 10 4.b odd 2 1
496.5.e.c 10 124.d even 2 1
1116.5.h.b 10 3.b odd 2 1
1116.5.h.b 10 93.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(124, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 478 T^{8} + \cdots + 622080000 \) Copy content Toggle raw display
$5$ \( (T^{5} + 3 T^{4} - 1369 T^{3} + \cdots + 479988)^{2} \) Copy content Toggle raw display
$7$ \( (T^{5} + T^{4} - 6049 T^{3} + \cdots + 73231744)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 61994 T^{8} + \cdots + 71\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( T^{10} + 112342 T^{8} + \cdots + 67\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{10} + 272828 T^{8} + \cdots + 17\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( (T^{5} + 155 T^{4} + \cdots - 1981857577500)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 1404700 T^{8} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{10} + 6900214 T^{8} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{10} - 734 T^{9} + \cdots + 67\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{10} + 12067770 T^{8} + \cdots + 25\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( (T^{5} + 105 T^{4} + \cdots - 21406740928848)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 25360238 T^{8} + \cdots + 36\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( (T^{5} + 996 T^{4} + \cdots - 118001846116800)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 32516362 T^{8} + \cdots + 32\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( (T^{5} - 2805 T^{4} + \cdots + 43\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 75017078 T^{8} + \cdots + 82\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( (T^{5} + 2710 T^{4} + \cdots + 18\!\cdots\!88)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 867 T^{4} + \cdots + 19\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 108829832 T^{8} + \cdots + 33\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{10} + 184367764 T^{8} + \cdots + 25\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{10} + 283021210 T^{8} + \cdots + 17\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{10} + 266101028 T^{8} + \cdots + 31\!\cdots\!08 \) Copy content Toggle raw display
$97$ \( (T^{5} - 1471 T^{4} + \cdots - 17\!\cdots\!08)^{2} \) Copy content Toggle raw display
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