# 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 $2$

# 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.250722553392.1 Defining polynomial: $$x^{6} + 46x^{4} - 24x^{3} + 2116x^{2} - 552x + 144$$ x^6 + 46*x^4 - 24*x^3 + 2116*x^2 - 552*x + 144 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.

 $$f(q)$$ $$=$$ $$q + (3 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{3} + ( - \beta_{3} + 2 \beta_1) q^{5} + ( - 15 \beta_{3} - \beta_{2} - \beta_1 + 15) q^{7} + ( - \beta_{5} + \beta_{4} - 13 \beta_{3} + 6 \beta_1) q^{9}+O(q^{10})$$ q + (3*b3 - b2 - b1 - 3) * q^3 + (-b3 + 2*b1) * q^5 + (-15*b3 - b2 - b1 + 15) * q^7 + (-b5 + b4 - 13*b3 + 6*b1) * q^9 $$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})$$ q + (3*b3 - b2 - b1 - 3) * q^3 + (-b3 + 2*b1) * q^5 + (-15*b3 - b2 - b1 + 15) * q^7 + (-b5 + b4 - 13*b3 + 6*b1) * q^9 + (-b5 + b4 + 20*b3 - 5*b1) * q^11 + (b5 - b4 + 38*b3 + 4*b1) * q^13 + (-2*b4 + 7*b2 + 65) * q^15 + (-3*b5 - 42*b3 - 2*b2 - 2*b1 + 42) * q^17 + (-6*b5 - 23*b3 + 3*b2 + 3*b1 + 23) * q^19 + (-b5 + b4 + 14*b3 - 12*b1) * q^21 + (6*b4 - 4*b2 + 14) * q^23 + (4*b5 - 4*b2 - 4*b1) * q^25 + (-9*b4 + 19*b2 + 156) * q^27 + (b4 - 18*b2 + 55) * q^29 + (-5*b5 + 6*b4 + 43*b3 + 5*b2 + 6*b1 - 134) * q^31 + (2*b4 - 20*b2 - 203) * q^33 + (-2*b4 - 29*b2 + 47) * q^35 + (6*b5 - 41*b3 + 58*b2 + 58*b1 + 41) * q^37 + (-b4 - 41*b2 - 2) * q^39 + (b5 - b4 - 236*b3 + 14*b1) * q^41 + (2*b5 - 35*b3 + 47*b2 + 47*b1 + 35) * q^43 + (13*b5 + 409*b3 - 62*b2 - 62*b1 - 409) * q^45 + (4*b4 + 32*b2 - 96) * q^47 + (-b5 + b4 + 87*b3 - 30*b1) * q^49 + (7*b5 - 7*b4 + 100*b3 - 81*b1) * q^51 + (b5 - b4 - 422*b3 - 20*b1) * q^53 + (-9*b5 - 306*b3 + 15*b2 + 15*b1 + 306) * q^55 + (21*b5 - 21*b4 + 234*b3 - 122*b1) * q^57 + (13*b5 + 334*b3 - b2 - b1 - 334) * q^59 + (-5*b4 + 14*b2 - 415) * q^61 + (9*b4 - 62*b2 + 3) * q^63 + (7*b5 + 186*b3 + 102*b2 + 102*b1 - 186) * q^65 + (-21*b5 + 21*b4 + 240*b3 + 15*b1) * q^67 + (-22*b5 - 154*b3 + 88*b2 + 88*b1 + 154) * q^69 + (21*b5 - 21*b4 - 356*b3 + 115*b1) * q^71 + (29*b5 - 29*b4 - 56*b3 + 6*b1) * q^73 + (-16*b5 + 16*b4 - 172*b3 + 72*b1) * q^75 + (20*b4 + 70*b2 + 157) * q^77 + (2*b5 - 225*b3 - 49*b2 - 49*b1 + 225) * q^79 + (19*b5 + 814*b3 - 186*b2 - 186*b1 - 814) * q^81 + (-10*b5 + 10*b4 - 3*b3 - 149*b1) * q^83 + (-b4 - 172*b2 + 10) * q^85 + (-21*b5 - 405*b3 + 14*b2 + 14*b1 + 405) * q^87 + (33*b4 + 54*b2 - 105) * q^89 + (-19*b4 - 113*b2 + 682) * q^91 + (2*b5 - 21*b4 - 259*b3 + 184*b2 + 134*b1 + 376) * q^93 + (12*b4 - 229*b2 - 353) * q^95 + (13*b4 + 74*b2 - 353) * q^97 + (-53*b5 - 713*b3 + 158*b2 + 158*b1 + 713) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 9 q^{3} - 3 q^{5} + 45 q^{7} - 38 q^{9}+O(q^{10})$$ 6 * q - 9 * q^3 - 3 * q^5 + 45 * q^7 - 38 * q^9 $$6 q - 9 q^{3} - 3 q^{5} + 45 q^{7} - 38 q^{9} + 61 q^{11} + 113 q^{13} + 386 q^{15} + 123 q^{17} + 63 q^{19} + 43 q^{21} + 96 q^{23} + 4 q^{25} + 918 q^{27} + 332 q^{29} - 668 q^{31} - 1214 q^{33} + 278 q^{35} + 129 q^{37} - 14 q^{39} - 709 q^{41} + 107 q^{43} - 1214 q^{45} - 568 q^{47} + 262 q^{49} + 293 q^{51} - 1267 q^{53} + 909 q^{55} + 681 q^{57} - 989 q^{59} - 2500 q^{61} + 36 q^{63} - 551 q^{65} + 741 q^{67} + 440 q^{69} - 1089 q^{71} - 197 q^{73} - 500 q^{75} + 982 q^{77} + 677 q^{79} - 2423 q^{81} + q^{83} + 58 q^{85} + 1194 q^{87} - 564 q^{89} + 4054 q^{91} + 1439 q^{93} - 2094 q^{95} - 2092 q^{97} + 2086 q^{99}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 46x^{4} - 24x^{3} + 2116x^{2} - 552x + 144$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 12 ) / 46$$ (v^3 - 12) / 46 $$\beta_{3}$$ $$=$$ $$( \nu^{5} + 46\nu^{3} - 12\nu^{2} + 2116\nu ) / 552$$ (v^5 + 46*v^3 - 12*v^2 + 2116*v) / 552 $$\beta_{4}$$ $$=$$ $$( \nu^{4} + 46\nu^{2} - 12\nu + 1426 ) / 46$$ (v^4 + 46*v^2 - 12*v + 1426) / 46 $$\beta_{5}$$ $$=$$ $$( -31\nu^{5} - 1426\nu^{3} + 924\nu^{2} - 65596\nu + 17112 ) / 552$$ (-31*v^5 - 1426*v^3 + 924*v^2 - 65596*v + 17112) / 552
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 31\beta_{3} - 31$$ b5 + 31*b3 - 31 $$\nu^{3}$$ $$=$$ $$46\beta_{2} + 12$$ 46*b2 + 12 $$\nu^{4}$$ $$=$$ $$-46\beta_{5} + 46\beta_{4} - 1426\beta_{3} + 12\beta_1$$ -46*b5 + 46*b4 - 1426*b3 + 12*b1 $$\nu^{5}$$ $$=$$ $$12\beta_{5} + 924\beta_{3} - 2116\beta_{2} - 2116\beta _1 - 924$$ 12*b5 + 924*b3 - 2116*b2 - 2116*b1 - 924

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

## Hecke characteristic polynomials

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