Properties

Label 124.4.e.c
Level $124$
Weight $4$
Character orbit 124.e
Analytic conductor $7.316$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} + 29x^{6} - 58x^{5} + 824x^{4} - 1198x^{3} + 1933x^{2} + 129x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + (\beta_{7} - \beta_{6} - 4 \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{5} + 8 \beta_{4} + \beta_1 - 8) q^{7} + ( - 3 \beta_{7} - 3 \beta_{6} - \beta_{4} - 3 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + (\beta_{7} - \beta_{6} - 4 \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{5} + 8 \beta_{4} + \beta_1 - 8) q^{7} + ( - 3 \beta_{7} - 3 \beta_{6} - \beta_{4} - 3 \beta_{3}) q^{9} + (\beta_{7} + 3 \beta_{6} - \beta_{5} - 20 \beta_{4} + 3 \beta_{3} + \beta_{2}) q^{11} + ( - 2 \beta_{7} - 4 \beta_{5} - 7 \beta_{4} + 4 \beta_{2}) q^{13} + ( - \beta_{3} - \beta_{2} - 5 \beta_1 - 38) q^{15} + (3 \beta_{7} - \beta_{6} + 2 \beta_{4} - 3 \beta_1 - 2) q^{17} + (2 \beta_{7} + 5 \beta_{6} - 4 \beta_{5} + 14 \beta_{4} - 2 \beta_1 - 14) q^{19} + ( - 4 \beta_{7} - 2 \beta_{6} - 19 \beta_{4} - 2 \beta_{3}) q^{21} + (2 \beta_{3} - 8 \beta_{2} - 6 \beta_1 - 78) q^{23} + ( - 8 \beta_{6} - 4 \beta_{5} + 12 \beta_{4} - 12) q^{25} + ( - \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 54) q^{27} + (11 \beta_{3} + 4 \beta_{2} + 19 \beta_1 - 27) q^{29} + ( - 7 \beta_{7} - 10 \beta_{6} - 9 \beta_{5} - 64 \beta_{4} - 9 \beta_{3} + \cdots + 98) q^{31}+ \cdots + (47 \beta_{7} - 29 \beta_{5} - 334 \beta_{4} - 47 \beta_1 + 334) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{5} - 32 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{5} - 32 q^{7} - 4 q^{9} - 80 q^{11} - 28 q^{13} - 304 q^{15} - 8 q^{17} - 56 q^{19} - 76 q^{21} - 624 q^{23} - 48 q^{25} - 432 q^{27} - 216 q^{29} + 528 q^{31} + 584 q^{33} + 592 q^{35} + 96 q^{37} + 128 q^{39} + 552 q^{41} - 112 q^{43} + 524 q^{45} - 304 q^{47} - 4 q^{49} - 232 q^{51} + 1316 q^{53} - 208 q^{55} + 464 q^{57} + 224 q^{59} + 2664 q^{61} - 1856 q^{63} + 504 q^{65} - 272 q^{67} + 496 q^{69} + 1120 q^{71} + 248 q^{73} - 912 q^{75} + 624 q^{77} + 824 q^{79} + 104 q^{81} + 1616 q^{83} - 2232 q^{85} + 456 q^{87} + 184 q^{89} - 6928 q^{91} - 1960 q^{93} + 544 q^{95} + 2520 q^{97} + 1336 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 29x^{6} - 58x^{5} + 824x^{4} - 1198x^{3} + 1933x^{2} + 129x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 117404 \nu^{7} + 31169 \nu^{6} + 3211838 \nu^{5} - 1568182 \nu^{4} + 88088321 \nu^{3} + 8645966 \nu^{2} + 578634 \nu + 588938652 ) / 49422123 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 42896 \nu^{7} + 27104 \nu^{6} + 1173512 \nu^{5} - 572968 \nu^{4} + 28919957 \nu^{3} + 3158984 \nu^{2} + 211416 \nu + 88248789 ) / 16474041 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 75068 \nu^{7} - 47432 \nu^{6} - 2053646 \nu^{5} + 1002694 \nu^{4} - 54728435 \nu^{3} - 5528222 \nu^{2} - 369978 \nu - 109131768 ) / 16474041 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 81647 \nu^{7} - 87009 \nu^{6} + 2364375 \nu^{5} - 4882215 \nu^{4} + 67348749 \nu^{3} - 102114519 \nu^{2} + 157428778 \nu + 10506036 ) / 10982694 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 612323 \nu^{7} + 659755 \nu^{6} - 17928125 \nu^{5} + 36812338 \nu^{4} - 510679055 \nu^{3} + 774294205 \nu^{2} - 1267988883 \nu + 3614310 ) / 16474041 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1469587 \nu^{7} + 1580537 \nu^{6} - 42949375 \nu^{5} + 88271321 \nu^{4} - 1223404357 \nu^{3} + 1854931967 \nu^{2} - 3140056428 \nu + 8658594 ) / 32948082 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4377592 \nu^{7} - 4517564 \nu^{6} + 126818713 \nu^{5} - 257729570 \nu^{4} + 3609005434 \nu^{3} - 5329786637 \nu^{2} + 8451577398 \nu + 564019506 ) / 49422123 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} - \beta_{4} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{7} + 3\beta_{6} - \beta_{5} - 57\beta_{4} + 3\beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{3} - 7\beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -168\beta_{7} - 111\beta_{6} + 43\beta_{5} + 1497\beta_{4} + 168\beta _1 - 1497 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 90\beta_{7} + 463\beta_{6} - 781\beta_{5} - 2189\beta_{4} + 463\beta_{3} + 781\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -831\beta_{3} - 406\beta_{2} - 1149\beta _1 + 10362 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5148\beta_{7} - 14365\beta_{6} + 22009\beta_{5} + 84083\beta_{4} + 5148\beta _1 - 84083 ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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
−0.0334318 + 0.0579056i
−2.74884 + 4.76113i
2.44925 4.24222i
0.833026 1.44284i
−0.0334318 0.0579056i
−2.74884 4.76113i
2.44925 + 4.24222i
0.833026 + 1.44284i
0 −3.31198 + 5.73652i 0 7.27011 + 12.5922i 0 −4.03211 + 6.98382i 0 −8.43843 14.6158i 0
5.2 0 −0.961703 + 1.66572i 0 −4.12668 7.14762i 0 2.48358 4.30169i 0 11.6503 + 20.1788i 0
5.3 0 0.269527 0.466835i 0 −7.15729 12.3968i 0 −17.6457 + 30.5633i 0 13.3547 + 23.1310i 0
5.4 0 4.00416 6.93540i 0 −3.98613 6.90419i 0 3.19423 5.53257i 0 −18.5665 32.1582i 0
25.1 0 −3.31198 5.73652i 0 7.27011 12.5922i 0 −4.03211 6.98382i 0 −8.43843 + 14.6158i 0
25.2 0 −0.961703 1.66572i 0 −4.12668 + 7.14762i 0 2.48358 + 4.30169i 0 11.6503 20.1788i 0
25.3 0 0.269527 + 0.466835i 0 −7.15729 + 12.3968i 0 −17.6457 30.5633i 0 13.3547 23.1310i 0
25.4 0 4.00416 + 6.93540i 0 −3.98613 + 6.90419i 0 3.19423 + 5.53257i 0 −18.5665 + 32.1582i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.4
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.c 8
3.b odd 2 1 1116.4.i.e 8
4.b odd 2 1 496.4.i.e 8
31.c even 3 1 inner 124.4.e.c 8
93.h odd 6 1 1116.4.i.e 8
124.i odd 6 1 496.4.i.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.e.c 8 1.a even 1 1 trivial
124.4.e.c 8 31.c even 3 1 inner
496.4.i.e 8 4.b odd 2 1
496.4.i.e 8 124.i odd 6 1
1116.4.i.e 8 3.b odd 2 1
1116.4.i.e 8 93.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 56T_{3}^{6} + 144T_{3}^{5} + 3081T_{3}^{4} + 4032T_{3}^{3} + 8264T_{3}^{2} - 3960T_{3} + 3025 \) acting on \(S_{4}^{\mathrm{new}}(124, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 56 T^{6} + 144 T^{5} + \cdots + 3025 \) Copy content Toggle raw display
$5$ \( T^{8} + 16 T^{7} + \cdots + 187553025 \) Copy content Toggle raw display
$7$ \( T^{8} + 32 T^{7} + 1200 T^{6} + \cdots + 81558961 \) Copy content Toggle raw display
$11$ \( T^{8} + 80 T^{7} + \cdots + 4788778401 \) Copy content Toggle raw display
$13$ \( T^{8} + 28 T^{7} + \cdots + 2366893017841 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + \cdots + 12137208561 \) Copy content Toggle raw display
$19$ \( T^{8} + 56 T^{7} + \cdots + 77787196287025 \) Copy content Toggle raw display
$23$ \( (T^{4} + 312 T^{3} + 4368 T^{2} + \cdots - 279217152)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 108 T^{3} - 53612 T^{2} + \cdots + 638715456)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 528 T^{7} + \cdots + 78\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{8} - 96 T^{7} + \cdots + 44\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{8} - 552 T^{7} + \cdots + 20\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( T^{8} + 112 T^{7} + \cdots + 1137223889649 \) Copy content Toggle raw display
$47$ \( (T^{4} + 152 T^{3} + \cdots + 13923790848)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 1316 T^{7} + \cdots + 74\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{8} - 224 T^{7} + \cdots + 49\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( (T^{4} - 1332 T^{3} + 487812 T^{2} + \cdots + 725031296)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 272 T^{7} + \cdots + 29\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{8} - 1120 T^{7} + \cdots + 69\!\cdots\!29 \) Copy content Toggle raw display
$73$ \( T^{8} - 248 T^{7} + \cdots + 35\!\cdots\!49 \) Copy content Toggle raw display
$79$ \( T^{8} - 824 T^{7} + \cdots + 32\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( T^{8} - 1616 T^{7} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{4} - 92 T^{3} - 648972 T^{2} + \cdots + 75650082624)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1260 T^{3} + \cdots - 319459930240)^{2} \) Copy content Toggle raw display
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