# Properties

 Label 124.4.a.b Level $124$ Weight $4$ Character orbit 124.a Self dual yes Analytic conductor $7.316$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 124.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.31623684071$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.4000044.1 Defining polynomial: $$x^{4} - x^{3} - 21x^{2} + 16x + 62$$ x^4 - x^3 - 21*x^2 + 16*x + 62 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_{3} + \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1 + 4) q^{7} + (2 \beta_{3} - 2 \beta_{2} + 15) q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b3 + b1 + 1) * q^5 + (b2 + b1 + 4) * q^7 + (2*b3 - 2*b2 + 15) * q^9 $$q + \beta_1 q^{3} + ( - \beta_{3} + \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1 + 4) q^{7} + (2 \beta_{3} - 2 \beta_{2} + 15) q^{9} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 25) q^{11} + (2 \beta_{3} + 4 \beta_{2} + \beta_1 + 12) q^{13} + ( - 3 \beta_{3} - 7 \beta_{2} - 6 \beta_1 + 45) q^{15} + (\beta_{3} + 11 \beta_{2} + 4 \beta_1 + 3) q^{17} + ( - 8 \beta_{3} - 3 \beta_{2} - 9 \beta_1 + 48) q^{19} + (3 \beta_{3} - 5 \beta_{2} + 35) q^{21} + (3 \beta_{3} + 11 \beta_{2} - 4 \beta_1 + 43) q^{23} + ( - 13 \beta_{3} - 18 \beta_{2} - 3 \beta_1 + 30) q^{25} + (8 \beta_{3} + 16 \beta_{2} + 10 \beta_1 + 8) q^{27} + (8 \beta_{3} - 20 \beta_{2} - 5 \beta_1 + 58) q^{29} + 31 q^{31} + ( - 2 \beta_{3} + 14 \beta_{2} + 36 \beta_1 - 122) q^{33} + ( - 2 \beta_{3} - 5 \beta_{2} - 9 \beta_1 + 58) q^{35} + (15 \beta_{3} + 7 \beta_{2} + 7 \beta_1 - 71) q^{37} + (16 \beta_{3} - 4 \beta_{2} + 10 \beta_1 + 8) q^{39} + ( - 17 \beta_{3} + 18 \beta_{2} + \beta_1 - 123) q^{41} + ( - 16 \beta_{3} + 6 \beta_{2} + 3 \beta_1 - 80) q^{43} + ( - 7 \beta_{3} + 18 \beta_{2} + 25 \beta_1 - 221) q^{45} + ( - 4 \beta_{3} - 40 \beta_{2} - 22 \beta_1 + 108) q^{47} + (3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 240) q^{49} + (24 \beta_{3} - 36 \beta_{2} - 34 \beta_1 + 88) q^{51} + (3 \beta_{3} - 35 \beta_{2} - 43 \beta_1 - 59) q^{53} + ( - 12 \beta_{3} + 30 \beta_{2} + 48 \beta_1 - 228) q^{55} + ( - 61 \beta_{3} - 13 \beta_{2} + 4 \beta_1 - 333) q^{57} + (26 \beta_{3} + 45 \beta_{2} - 25 \beta_1 - 6) q^{59} + (2 \beta_{3} + 20 \beta_{2} - 73 \beta_1 - 220) q^{61} + (10 \beta_{3} + 3 \beta_{2} + 49 \beta_1 - 82) q^{63} + (23 \beta_{3} + 23 \beta_{2} - 26 \beta_1 - 125) q^{65} + ( - 4 \beta_{3} + 16 \beta_{2} - 32 \beta_1 + 24) q^{67} + (18 \beta_{3} - 10 \beta_{2} + 20 \beta_1 - 254) q^{69} + (6 \beta_{3} - 15 \beta_{2} - 35 \beta_1 + 386) q^{71} + (54 \beta_{3} - 58 \beta_{2} - 40 \beta_1 - 192) q^{73} + ( - 89 \beta_{3} - 5 \beta_{2} + 11 \beta_1 + 39) q^{75} + ( - 4 \beta_{3} + 34 \beta_{2} + 42 \beta_1 - 76) q^{77} + ( - 15 \beta_{3} - 85 \beta_{2} + 68 \beta_1 + 129) q^{79} + (22 \beta_{3} + 26 \beta_{2} - 121) q^{81} + (41 \beta_{3} + 11 \beta_{2} + 57 \beta_1 + 329) q^{83} + (49 \beta_{3} + 5 \beta_{2} - 100 \beta_1 + 173) q^{85} + (10 \beta_{3} + 110 \beta_{2} + 194 \beta_1 - 94) q^{87} + ( - 39 \beta_{3} - 19 \beta_{2} + 68 \beta_1 + 19) q^{89} + (13 \beta_{3} + 37 \beta_{2} + 4 \beta_1 + 253) q^{91} + 31 \beta_1 q^{93} + ( - 108 \beta_{3} - 31 \beta_{2} + 139 \beta_1 + 488) q^{95} + ( - 27 \beta_{3} - 110 \beta_{2} - 29 \beta_1 + 91) q^{97} + (49 \beta_{3} - 97 \beta_{2} - 111 \beta_1 + 745) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b3 + b1 + 1) * q^5 + (b2 + b1 + 4) * q^7 + (2*b3 - 2*b2 + 15) * q^9 + (b3 - b2 - 3*b1 + 25) * q^11 + (2*b3 + 4*b2 + b1 + 12) * q^13 + (-3*b3 - 7*b2 - 6*b1 + 45) * q^15 + (b3 + 11*b2 + 4*b1 + 3) * q^17 + (-8*b3 - 3*b2 - 9*b1 + 48) * q^19 + (3*b3 - 5*b2 + 35) * q^21 + (3*b3 + 11*b2 - 4*b1 + 43) * q^23 + (-13*b3 - 18*b2 - 3*b1 + 30) * q^25 + (8*b3 + 16*b2 + 10*b1 + 8) * q^27 + (8*b3 - 20*b2 - 5*b1 + 58) * q^29 + 31 * q^31 + (-2*b3 + 14*b2 + 36*b1 - 122) * q^33 + (-2*b3 - 5*b2 - 9*b1 + 58) * q^35 + (15*b3 + 7*b2 + 7*b1 - 71) * q^37 + (16*b3 - 4*b2 + 10*b1 + 8) * q^39 + (-17*b3 + 18*b2 + b1 - 123) * q^41 + (-16*b3 + 6*b2 + 3*b1 - 80) * q^43 + (-7*b3 + 18*b2 + 25*b1 - 221) * q^45 + (-4*b3 - 40*b2 - 22*b1 + 108) * q^47 + (3*b3 + 6*b2 - 3*b1 - 240) * q^49 + (24*b3 - 36*b2 - 34*b1 + 88) * q^51 + (3*b3 - 35*b2 - 43*b1 - 59) * q^53 + (-12*b3 + 30*b2 + 48*b1 - 228) * q^55 + (-61*b3 - 13*b2 + 4*b1 - 333) * q^57 + (26*b3 + 45*b2 - 25*b1 - 6) * q^59 + (2*b3 + 20*b2 - 73*b1 - 220) * q^61 + (10*b3 + 3*b2 + 49*b1 - 82) * q^63 + (23*b3 + 23*b2 - 26*b1 - 125) * q^65 + (-4*b3 + 16*b2 - 32*b1 + 24) * q^67 + (18*b3 - 10*b2 + 20*b1 - 254) * q^69 + (6*b3 - 15*b2 - 35*b1 + 386) * q^71 + (54*b3 - 58*b2 - 40*b1 - 192) * q^73 + (-89*b3 - 5*b2 + 11*b1 + 39) * q^75 + (-4*b3 + 34*b2 + 42*b1 - 76) * q^77 + (-15*b3 - 85*b2 + 68*b1 + 129) * q^79 + (22*b3 + 26*b2 - 121) * q^81 + (41*b3 + 11*b2 + 57*b1 + 329) * q^83 + (49*b3 + 5*b2 - 100*b1 + 173) * q^85 + (10*b3 + 110*b2 + 194*b1 - 94) * q^87 + (-39*b3 - 19*b2 + 68*b1 + 19) * q^89 + (13*b3 + 37*b2 + 4*b1 + 253) * q^91 + 31*b1 * q^93 + (-108*b3 - 31*b2 + 139*b1 + 488) * q^95 + (-27*b3 - 110*b2 - 29*b1 + 91) * q^97 + (49*b3 - 97*b2 - 111*b1 + 745) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 6 q^{5} + 16 q^{7} + 64 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 6 * q^5 + 16 * q^7 + 64 * q^9 $$4 q + 2 q^{3} + 6 q^{5} + 16 q^{7} + 64 q^{9} + 96 q^{11} + 42 q^{13} + 182 q^{15} - 2 q^{17} + 180 q^{19} + 150 q^{21} + 142 q^{23} + 150 q^{25} + 20 q^{27} + 262 q^{29} + 124 q^{31} - 444 q^{33} + 224 q^{35} - 284 q^{37} + 60 q^{39} - 526 q^{41} - 326 q^{43} - 870 q^{45} + 468 q^{47} - 978 q^{49} + 356 q^{51} - 252 q^{53} - 876 q^{55} - 1298 q^{57} - 164 q^{59} - 1066 q^{61} - 236 q^{63} - 598 q^{65} - 956 q^{69} + 1504 q^{71} - 732 q^{73} + 188 q^{75} - 288 q^{77} + 822 q^{79} - 536 q^{81} + 1408 q^{83} + 482 q^{85} - 208 q^{87} + 250 q^{89} + 946 q^{91} + 62 q^{93} + 2292 q^{95} + 526 q^{97} + 2952 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 + 6 * q^5 + 16 * q^7 + 64 * q^9 + 96 * q^11 + 42 * q^13 + 182 * q^15 - 2 * q^17 + 180 * q^19 + 150 * q^21 + 142 * q^23 + 150 * q^25 + 20 * q^27 + 262 * q^29 + 124 * q^31 - 444 * q^33 + 224 * q^35 - 284 * q^37 + 60 * q^39 - 526 * q^41 - 326 * q^43 - 870 * q^45 + 468 * q^47 - 978 * q^49 + 356 * q^51 - 252 * q^53 - 876 * q^55 - 1298 * q^57 - 164 * q^59 - 1066 * q^61 - 236 * q^63 - 598 * q^65 - 956 * q^69 + 1504 * q^71 - 732 * q^73 + 188 * q^75 - 288 * q^77 + 822 * q^79 - 536 * q^81 + 1408 * q^83 + 482 * q^85 - 208 * q^87 + 250 * q^89 + 946 * q^91 + 62 * q^93 + 2292 * q^95 + 526 * q^97 + 2952 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 21x^{2} + 16x + 62$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 2\nu^{2} - 16\nu + 20 ) / 3$$ (v^3 - 2*v^2 - 16*v + 20) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 16\nu - 43 ) / 3$$ (v^3 + 4*v^2 - 16*v - 43) / 3
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - \beta_{2} + 21 ) / 2$$ (b3 - b2 + 21) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 8\beta _1 + 1$$ b3 + 2*b2 + 8*b1 + 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.13619 −1.48616 2.37430 4.24805
0 −8.27238 0 −14.2221 0 −10.5389 0 41.4323 0
1.2 0 −2.97232 0 2.58408 0 13.0539 0 −18.1653 0
1.3 0 4.74860 0 20.7669 0 3.45568 0 −4.45080 0
1.4 0 8.49610 0 −3.12892 0 10.0292 0 45.1838 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$31$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.4.a.b 4
3.b odd 2 1 1116.4.a.f 4
4.b odd 2 1 496.4.a.f 4
8.b even 2 1 1984.4.a.m 4
8.d odd 2 1 1984.4.a.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.a.b 4 1.a even 1 1 trivial
496.4.a.f 4 4.b odd 2 1
1116.4.a.f 4 3.b odd 2 1
1984.4.a.m 4 8.b even 2 1
1984.4.a.o 4 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 2T_{3}^{3} - 84T_{3}^{2} + 128T_{3} + 992$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(124))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2 T^{3} - 84 T^{2} + 128 T + 992$$
$5$ $$T^{4} - 6 T^{3} - 307 T^{2} + \cdots + 2388$$
$7$ $$T^{4} - 16 T^{3} - 69 T^{2} + \cdots - 4768$$
$11$ $$T^{4} - 96 T^{3} + 2364 T^{2} + \cdots + 16416$$
$13$ $$T^{4} - 42 T^{3} - 1536 T^{2} + \cdots + 441648$$
$17$ $$T^{4} + 2 T^{3} - 12864 T^{2} + \cdots + 12601872$$
$19$ $$T^{4} - 180 T^{3} + \cdots - 110971856$$
$23$ $$T^{4} - 142 T^{3} - 8700 T^{2} + \cdots - 7610112$$
$29$ $$T^{4} - 262 T^{3} + \cdots - 189481680$$
$31$ $$(T - 31)^{4}$$
$37$ $$T^{4} + 284 T^{3} + \cdots - 250857520$$
$41$ $$T^{4} + 526 T^{3} + \cdots + 949605156$$
$43$ $$T^{4} + 326 T^{3} + \cdots + 551576736$$
$47$ $$T^{4} - 468 T^{3} + \cdots + 1256256000$$
$53$ $$T^{4} + 252 T^{3} + \cdots - 7248886080$$
$59$ $$T^{4} + 164 T^{3} + \cdots + 42747696$$
$61$ $$T^{4} + 1066 T^{3} + \cdots + 37784393520$$
$67$ $$T^{4} - 135984 T^{2} + \cdots + 2730691584$$
$71$ $$T^{4} - 1504 T^{3} + \cdots + 9642905280$$
$73$ $$T^{4} + 732 T^{3} + \cdots + 231605048496$$
$79$ $$T^{4} - 822 T^{3} + \cdots + 4306806912$$
$83$ $$T^{4} - 1408 T^{3} + \cdots - 747812064$$
$89$ $$T^{4} - 250 T^{3} + \cdots + 25296843024$$
$97$ $$T^{4} - 526 T^{3} + \cdots + 180198106052$$