# 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 $2$

# Related objects

## 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: $$x^{8} - x^{7} + 29x^{6} - 58x^{5} + 824x^{4} - 1198x^{3} + 1933x^{2} + 129x + 9$$ x^8 - x^7 + 29*x^6 - 58*x^5 + 824*x^4 - 1198*x^3 + 1933*x^2 + 129*x + 9 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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{3} + (\beta_{7} - \beta_{6} - 4 \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{5} + 8 \beta_{4} + \beta_1 - 8) q^{7} + ( - 3 \beta_{7} - 3 \beta_{6} - \beta_{4} - 3 \beta_{3}) q^{9}+O(q^{10})$$ q - b6 * q^3 + (b7 - b6 - 4*b4 - b3) * q^5 + (-b7 - b6 - b5 + 8*b4 + b1 - 8) * q^7 + (-3*b7 - 3*b6 - b4 - 3*b3) * q^9 $$q - \beta_{6} q^{3} + (\beta_{7} - \beta_{6} - 4 \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{5} + 8 \beta_{4} + \beta_1 - 8) q^{7} + ( - 3 \beta_{7} - 3 \beta_{6} - \beta_{4} - 3 \beta_{3}) q^{9} + (\beta_{7} + 3 \beta_{6} - \beta_{5} - 20 \beta_{4} + 3 \beta_{3} + \beta_{2}) q^{11} + ( - 2 \beta_{7} - 4 \beta_{5} - 7 \beta_{4} + 4 \beta_{2}) q^{13} + ( - \beta_{3} - \beta_{2} - 5 \beta_1 - 38) q^{15} + (3 \beta_{7} - \beta_{6} + 2 \beta_{4} - 3 \beta_1 - 2) q^{17} + (2 \beta_{7} + 5 \beta_{6} - 4 \beta_{5} + 14 \beta_{4} - 2 \beta_1 - 14) q^{19} + ( - 4 \beta_{7} - 2 \beta_{6} - 19 \beta_{4} - 2 \beta_{3}) q^{21} + (2 \beta_{3} - 8 \beta_{2} - 6 \beta_1 - 78) q^{23} + ( - 8 \beta_{6} - 4 \beta_{5} + 12 \beta_{4} - 12) q^{25} + ( - \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 54) q^{27} + (11 \beta_{3} + 4 \beta_{2} + 19 \beta_1 - 27) q^{29} + ( - 7 \beta_{7} - 10 \beta_{6} - 9 \beta_{5} - 64 \beta_{4} - 9 \beta_{3} + \cdots + 98) q^{31}+ \cdots + (47 \beta_{7} - 29 \beta_{5} - 334 \beta_{4} - 47 \beta_1 + 334) q^{99}+O(q^{100})$$ q - b6 * q^3 + (b7 - b6 - 4*b4 - b3) * q^5 + (-b7 - b6 - b5 + 8*b4 + b1 - 8) * q^7 + (-3*b7 - 3*b6 - b4 - 3*b3) * q^9 + (b7 + 3*b6 - b5 - 20*b4 + 3*b3 + b2) * q^11 + (-2*b7 - 4*b5 - 7*b4 + 4*b2) * q^13 + (-b3 - b2 - 5*b1 - 38) * q^15 + (3*b7 - b6 + 2*b4 - 3*b1 - 2) * q^17 + (2*b7 + 5*b6 - 4*b5 + 14*b4 - 2*b1 - 14) * q^19 + (-4*b7 - 2*b6 - 19*b4 - 2*b3) * q^21 + (2*b3 - 8*b2 - 6*b1 - 78) * q^23 + (-8*b6 - 4*b5 + 12*b4 - 12) * q^25 + (-b3 + 3*b2 - 3*b1 - 54) * q^27 + (11*b3 + 4*b2 + 19*b1 - 27) * q^29 + (-7*b7 - 10*b6 - 9*b5 - 64*b4 - 9*b3 + 9*b2 + 8*b1 + 98) * q^31 + (-6*b3 - 2*b2 + 4*b1 + 73) * q^33 + (-b3 + 12*b2 - 22*b1 + 74) * q^35 + (-11*b7 - 23*b6 + 6*b5 - 24*b4 + 11*b1 + 24) * q^37 + (-23*b3 - 2*b2 - 8*b1 + 16) * q^39 + (11*b7 - 27*b6 + 18*b5 + 138*b4 - 27*b3 - 18*b2) * q^41 + (-8*b7 + 3*b6 - 8*b5 + 28*b4 + 8*b1 - 28) * q^43 + (17*b7 + 43*b6 + 6*b5 - 131*b4 - 17*b1 + 131) * q^45 + (50*b3 + 8*b2 + 42*b1 - 38) * q^47 + (41*b7 + 9*b6 + 28*b5 - b4 + 9*b3 - 28*b2) * q^49 + (-9*b7 + 17*b6 + 3*b5 - 58*b4 + 17*b3 - 3*b2) * q^51 + (-12*b7 + 40*b6 + 2*b5 + 329*b4 + 40*b3 - 2*b2) * q^53 + (-28*b7 - b6 + 10*b5 + 52*b4 + 28*b1 - 52) * q^55 + (-b7 + 37*b6 + 6*b5 + 116*b4 + 37*b3 - 6*b2) * q^57 + (61*b7 + 35*b6 - 5*b5 - 56*b4 - 61*b1 + 56) * q^59 + (-21*b3 + 10*b2 - 31*b1 + 333) * q^61 + (-22*b3 - 23*b2 + 29*b1 - 232) * q^63 + (27*b7 + 13*b6 + 46*b5 - 126*b4 - 27*b1 + 126) * q^65 + (-27*b7 + 3*b6 + 15*b5 - 68*b4 + 3*b3 - 15*b2) * q^67 + (-42*b7 + 100*b6 + 14*b5 - 124*b4 + 42*b1 + 124) * q^69 + (-4*b7 - 85*b6 - 26*b5 + 280*b4 - 85*b3 + 26*b2) * q^71 + (63*b7 + 89*b6 - 14*b5 + 62*b4 + 89*b3 + 14*b2) * q^73 + (-36*b7 - 16*b6 + 4*b5 - 228*b4 - 16*b3 - 4*b2) * q^75 + (-b3 + 16*b2 + b1 + 78) * q^77 + (33*b7 - 119*b6 - 13*b5 - 206*b4 - 33*b1 + 206) * q^79 + (-75*b7 - 3*b6 - 26*b4 + 75*b1 + 26) * q^81 + (-32*b7 - 57*b6 - 32*b5 + 404*b4 - 57*b3 + 32*b2) * q^83 + (40*b3 - 10*b2 - 4*b1 - 279) * q^85 + (17*b7 - 116*b6 - 23*b5 - 114*b4 - 17*b1 + 114) * q^87 + (17*b3 + 12*b2 + 65*b1 + 23) * q^89 + (15*b3 - 57*b2 + 113*b1 - 866) * q^91 + (13*b7 - 149*b6 - 8*b5 + 52*b4 - 145*b3 - 2*b2 - 56*b1 - 271) * q^93 + (65*b3 + 39*b2 - 19*b1 + 68) * q^95 + (37*b3 - 42*b2 - 89*b1 + 315) * q^97 + (47*b7 - 29*b5 - 334*b4 - 47*b1 + 334) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 16 q^{5} - 32 q^{7} - 4 q^{9}+O(q^{10})$$ 8 * q - 16 * q^5 - 32 * q^7 - 4 * q^9 $$8 q - 16 q^{5} - 32 q^{7} - 4 q^{9} - 80 q^{11} - 28 q^{13} - 304 q^{15} - 8 q^{17} - 56 q^{19} - 76 q^{21} - 624 q^{23} - 48 q^{25} - 432 q^{27} - 216 q^{29} + 528 q^{31} + 584 q^{33} + 592 q^{35} + 96 q^{37} + 128 q^{39} + 552 q^{41} - 112 q^{43} + 524 q^{45} - 304 q^{47} - 4 q^{49} - 232 q^{51} + 1316 q^{53} - 208 q^{55} + 464 q^{57} + 224 q^{59} + 2664 q^{61} - 1856 q^{63} + 504 q^{65} - 272 q^{67} + 496 q^{69} + 1120 q^{71} + 248 q^{73} - 912 q^{75} + 624 q^{77} + 824 q^{79} + 104 q^{81} + 1616 q^{83} - 2232 q^{85} + 456 q^{87} + 184 q^{89} - 6928 q^{91} - 1960 q^{93} + 544 q^{95} + 2520 q^{97} + 1336 q^{99}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 29x^{6} - 58x^{5} + 824x^{4} - 1198x^{3} + 1933x^{2} + 129x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( 117404 \nu^{7} + 31169 \nu^{6} + 3211838 \nu^{5} - 1568182 \nu^{4} + 88088321 \nu^{3} + 8645966 \nu^{2} + 578634 \nu + 588938652 ) / 49422123$$ (117404*v^7 + 31169*v^6 + 3211838*v^5 - 1568182*v^4 + 88088321*v^3 + 8645966*v^2 + 578634*v + 588938652) / 49422123 $$\beta_{2}$$ $$=$$ $$( 42896 \nu^{7} + 27104 \nu^{6} + 1173512 \nu^{5} - 572968 \nu^{4} + 28919957 \nu^{3} + 3158984 \nu^{2} + 211416 \nu + 88248789 ) / 16474041$$ (42896*v^7 + 27104*v^6 + 1173512*v^5 - 572968*v^4 + 28919957*v^3 + 3158984*v^2 + 211416*v + 88248789) / 16474041 $$\beta_{3}$$ $$=$$ $$( - 75068 \nu^{7} - 47432 \nu^{6} - 2053646 \nu^{5} + 1002694 \nu^{4} - 54728435 \nu^{3} - 5528222 \nu^{2} - 369978 \nu - 109131768 ) / 16474041$$ (-75068*v^7 - 47432*v^6 - 2053646*v^5 + 1002694*v^4 - 54728435*v^3 - 5528222*v^2 - 369978*v - 109131768) / 16474041 $$\beta_{4}$$ $$=$$ $$( 81647 \nu^{7} - 87009 \nu^{6} + 2364375 \nu^{5} - 4882215 \nu^{4} + 67348749 \nu^{3} - 102114519 \nu^{2} + 157428778 \nu + 10506036 ) / 10982694$$ (81647*v^7 - 87009*v^6 + 2364375*v^5 - 4882215*v^4 + 67348749*v^3 - 102114519*v^2 + 157428778*v + 10506036) / 10982694 $$\beta_{5}$$ $$=$$ $$( - 612323 \nu^{7} + 659755 \nu^{6} - 17928125 \nu^{5} + 36812338 \nu^{4} - 510679055 \nu^{3} + 774294205 \nu^{2} - 1267988883 \nu + 3614310 ) / 16474041$$ (-612323*v^7 + 659755*v^6 - 17928125*v^5 + 36812338*v^4 - 510679055*v^3 + 774294205*v^2 - 1267988883*v + 3614310) / 16474041 $$\beta_{6}$$ $$=$$ $$( - 1469587 \nu^{7} + 1580537 \nu^{6} - 42949375 \nu^{5} + 88271321 \nu^{4} - 1223404357 \nu^{3} + 1854931967 \nu^{2} - 3140056428 \nu + 8658594 ) / 32948082$$ (-1469587*v^7 + 1580537*v^6 - 42949375*v^5 + 88271321*v^4 - 1223404357*v^3 + 1854931967*v^2 - 3140056428*v + 8658594) / 32948082 $$\beta_{7}$$ $$=$$ $$( 4377592 \nu^{7} - 4517564 \nu^{6} + 126818713 \nu^{5} - 257729570 \nu^{4} + 3609005434 \nu^{3} - 5329786637 \nu^{2} + 8451577398 \nu + 564019506 ) / 49422123$$ (4377592*v^7 - 4517564*v^6 + 126818713*v^5 - 257729570*v^4 + 3609005434*v^3 - 5329786637*v^2 + 8451577398*v + 564019506) / 49422123
 $$\nu$$ $$=$$ $$( -\beta_{6} + \beta_{5} - \beta_{4} + 1 ) / 4$$ (-b6 + b5 - b4 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( 6\beta_{7} + 3\beta_{6} - \beta_{5} - 57\beta_{4} + 3\beta_{3} + \beta_{2} ) / 4$$ (6*b7 + 3*b6 - b5 - 57*b4 + 3*b3 + b2) / 4 $$\nu^{3}$$ $$=$$ $$-4\beta_{3} - 7\beta_{2} + 11$$ -4*b3 - 7*b2 + 11 $$\nu^{4}$$ $$=$$ $$( -168\beta_{7} - 111\beta_{6} + 43\beta_{5} + 1497\beta_{4} + 168\beta _1 - 1497 ) / 4$$ (-168*b7 - 111*b6 + 43*b5 + 1497*b4 + 168*b1 - 1497) / 4 $$\nu^{5}$$ $$=$$ $$( 90\beta_{7} + 463\beta_{6} - 781\beta_{5} - 2189\beta_{4} + 463\beta_{3} + 781\beta_{2} ) / 4$$ (90*b7 + 463*b6 - 781*b5 - 2189*b4 + 463*b3 + 781*b2) / 4 $$\nu^{6}$$ $$=$$ $$-831\beta_{3} - 406\beta_{2} - 1149\beta _1 + 10362$$ -831*b3 - 406*b2 - 1149*b1 + 10362 $$\nu^{7}$$ $$=$$ $$( -5148\beta_{7} - 14365\beta_{6} + 22009\beta_{5} + 84083\beta_{4} + 5148\beta _1 - 84083 ) / 4$$ (-5148*b7 - 14365*b6 + 22009*b5 + 84083*b4 + 5148*b1 - 84083) / 4

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{4}^{\mathrm{new}}(124, [\chi])$$.

## Hecke characteristic polynomials

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