# Properties

 Label 1232.4.a.s Level $1232$ Weight $4$ Character orbit 1232.a Self dual yes Analytic conductor $72.690$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1232.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.6903531271$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.522072.1 Defining polynomial: $$x^{4} - x^{3} - 12x^{2} + 5x + 1$$ x^4 - x^3 - 12*x^2 + 5*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 77) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 4) q^{3} + (2 \beta_{2} - \beta_1 + 4) q^{5} + 7 q^{7} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 18) q^{9}+O(q^{10})$$ q + (-b2 - 4) * q^3 + (2*b2 - b1 + 4) * q^5 + 7 * q^7 + (3*b3 + 4*b2 + 3*b1 + 18) * q^9 $$q + ( - \beta_{2} - 4) q^{3} + (2 \beta_{2} - \beta_1 + 4) q^{5} + 7 q^{7} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 18) q^{9} + 11 q^{11} + ( - 3 \beta_{2} - 10 \beta_1 + 18) q^{13} + ( - 6 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 70) q^{15} + ( - 2 \beta_{3} - 5 \beta_{2} + 11 \beta_1 - 6) q^{17} + ( - 4 \beta_{3} + 4 \beta_{2} + 11 \beta_1 - 66) q^{19} + ( - 7 \beta_{2} - 28) q^{21} + (2 \beta_{3} - 18 \beta_{2} + 4 \beta_1 - 14) q^{23} + (14 \beta_{3} - 4 \beta_{2} - 12 \beta_1 + 17) q^{25} + ( - 18 \beta_{3} - 12 \beta_{2} - 36 \beta_1 - 86) q^{27} + ( - 6 \beta_{3} + 14 \beta_{2} - 2 \beta_1 - 88) q^{29} + (16 \beta_{3} + 9 \beta_{2} + 9 \beta_1 + 6) q^{31} + ( - 11 \beta_{2} - 44) q^{33} + (14 \beta_{2} - 7 \beta_1 + 28) q^{35} + ( - 23 \beta_{3} - 16 \beta_{2} - 9 \beta_1 + 29) q^{37} + (9 \beta_{3} + 2 \beta_{2} + 89 \beta_1 + 55) q^{39} + ( - 14 \beta_{3} - 23 \beta_{2} + 37 \beta_1 - 10) q^{41} + ( - 37 \beta_{3} - 2 \beta_{2} + 37 \beta_1 - 103) q^{43} + (18 \beta_{3} + 42 \beta_{2} + 17 \beta_1 + 210) q^{45} + ( - 8 \beta_{3} + 39 \beta_{2} + 7 \beta_1 + 18) q^{47} + 49 q^{49} + (19 \beta_{3} - 6 \beta_{2} - 73 \beta_1 + 121) q^{51} + (7 \beta_{3} - 8 \beta_{2} + 3 \beta_1 + 147) q^{53} + (22 \beta_{2} - 11 \beta_1 + 44) q^{55} + ( - 4 \beta_{3} + 64 \beta_{2} - 100 \beta_1 + 96) q^{57} + (74 \beta_{3} - 3 \beta_{2} - 50 \beta_1 + 46) q^{59} + (48 \beta_{3} + 29 \beta_{2} + 72 \beta_1 - 86) q^{61} + (21 \beta_{3} + 28 \beta_{2} + 21 \beta_1 + 126) q^{63} + (2 \beta_{3} + 54 \beta_{2} - 144 \beta_1 + 90) q^{65} + ( - 2 \beta_{3} + 62 \beta_{2} - 122 \beta_1 - 242) q^{67} + (50 \beta_{3} - 4 \beta_{2} + 22 \beta_1 + 566) q^{69} + ( - 2 \beta_{3} + 26 \beta_{2} - 94 \beta_1 - 386) q^{71} + ( - 78 \beta_{3} - 47 \beta_{2} + 139 \beta_1 + 310) q^{73} + ( - 16 \beta_{3} - 63 \beta_{2} + 108 \beta_1 + 124) q^{75} + 77 q^{77} + (71 \beta_{3} + 6 \beta_{2} + 43 \beta_1 + 277) q^{79} + ( - 9 \beta_{3} + 140 \beta_{2} + 243 \beta_1 + 314) q^{81} + (34 \beta_{3} + 76 \beta_{2} + 9 \beta_1 - 96) q^{83} + ( - 52 \beta_{3} - 2 \beta_{2} + 134 \beta_1 - 476) q^{85} + ( - 30 \beta_{3} + 122 \beta_{2} - 26 \beta_1 - 58) q^{87} + ( - 74 \beta_{3} - 48 \beta_{2} - 30 \beta_1 - 740) q^{89} + ( - 21 \beta_{2} - 70 \beta_1 + 126) q^{91} + ( - 59 \beta_{3} - 104 \beta_{2} - 99 \beta_1 - 289) q^{93} + (2 \beta_{3} - 196 \beta_{2} + 218 \beta_1 - 214) q^{95} + (104 \beta_{3} + 154 \beta_{2} - 72 \beta_1 - 102) q^{97} + (33 \beta_{3} + 44 \beta_{2} + 33 \beta_1 + 198) q^{99}+O(q^{100})$$ q + (-b2 - 4) * q^3 + (2*b2 - b1 + 4) * q^5 + 7 * q^7 + (3*b3 + 4*b2 + 3*b1 + 18) * q^9 + 11 * q^11 + (-3*b2 - 10*b1 + 18) * q^13 + (-6*b3 - 2*b2 + 2*b1 - 70) * q^15 + (-2*b3 - 5*b2 + 11*b1 - 6) * q^17 + (-4*b3 + 4*b2 + 11*b1 - 66) * q^19 + (-7*b2 - 28) * q^21 + (2*b3 - 18*b2 + 4*b1 - 14) * q^23 + (14*b3 - 4*b2 - 12*b1 + 17) * q^25 + (-18*b3 - 12*b2 - 36*b1 - 86) * q^27 + (-6*b3 + 14*b2 - 2*b1 - 88) * q^29 + (16*b3 + 9*b2 + 9*b1 + 6) * q^31 + (-11*b2 - 44) * q^33 + (14*b2 - 7*b1 + 28) * q^35 + (-23*b3 - 16*b2 - 9*b1 + 29) * q^37 + (9*b3 + 2*b2 + 89*b1 + 55) * q^39 + (-14*b3 - 23*b2 + 37*b1 - 10) * q^41 + (-37*b3 - 2*b2 + 37*b1 - 103) * q^43 + (18*b3 + 42*b2 + 17*b1 + 210) * q^45 + (-8*b3 + 39*b2 + 7*b1 + 18) * q^47 + 49 * q^49 + (19*b3 - 6*b2 - 73*b1 + 121) * q^51 + (7*b3 - 8*b2 + 3*b1 + 147) * q^53 + (22*b2 - 11*b1 + 44) * q^55 + (-4*b3 + 64*b2 - 100*b1 + 96) * q^57 + (74*b3 - 3*b2 - 50*b1 + 46) * q^59 + (48*b3 + 29*b2 + 72*b1 - 86) * q^61 + (21*b3 + 28*b2 + 21*b1 + 126) * q^63 + (2*b3 + 54*b2 - 144*b1 + 90) * q^65 + (-2*b3 + 62*b2 - 122*b1 - 242) * q^67 + (50*b3 - 4*b2 + 22*b1 + 566) * q^69 + (-2*b3 + 26*b2 - 94*b1 - 386) * q^71 + (-78*b3 - 47*b2 + 139*b1 + 310) * q^73 + (-16*b3 - 63*b2 + 108*b1 + 124) * q^75 + 77 * q^77 + (71*b3 + 6*b2 + 43*b1 + 277) * q^79 + (-9*b3 + 140*b2 + 243*b1 + 314) * q^81 + (34*b3 + 76*b2 + 9*b1 - 96) * q^83 + (-52*b3 - 2*b2 + 134*b1 - 476) * q^85 + (-30*b3 + 122*b2 - 26*b1 - 58) * q^87 + (-74*b3 - 48*b2 - 30*b1 - 740) * q^89 + (-21*b2 - 70*b1 + 126) * q^91 + (-59*b3 - 104*b2 - 99*b1 - 289) * q^93 + (2*b3 - 196*b2 + 218*b1 - 214) * q^95 + (104*b3 + 154*b2 - 72*b1 - 102) * q^97 + (33*b3 + 44*b2 + 33*b1 + 198) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9}+O(q^{10})$$ 4 * q - 14 * q^3 + 10 * q^5 + 28 * q^7 + 76 * q^9 $$4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + 44 q^{11} + 58 q^{13} - 284 q^{15} + 4 q^{17} - 258 q^{19} - 98 q^{21} - 8 q^{23} + 80 q^{25} - 428 q^{27} - 396 q^{29} + 56 q^{31} - 154 q^{33} + 70 q^{35} + 84 q^{37} + 412 q^{39} + 52 q^{41} - 408 q^{43} + 826 q^{45} - 8 q^{47} + 196 q^{49} + 388 q^{51} + 624 q^{53} + 110 q^{55} + 48 q^{57} + 238 q^{59} - 162 q^{61} + 532 q^{63} - 32 q^{65} - 1340 q^{67} + 2416 q^{69} - 1788 q^{71} + 1456 q^{73} + 806 q^{75} + 308 q^{77} + 1324 q^{79} + 1444 q^{81} - 450 q^{83} - 1736 q^{85} - 588 q^{87} - 3072 q^{89} + 406 q^{91} - 1264 q^{93} - 24 q^{95} - 652 q^{97} + 836 q^{99}+O(q^{100})$$ 4 * q - 14 * q^3 + 10 * q^5 + 28 * q^7 + 76 * q^9 + 44 * q^11 + 58 * q^13 - 284 * q^15 + 4 * q^17 - 258 * q^19 - 98 * q^21 - 8 * q^23 + 80 * q^25 - 428 * q^27 - 396 * q^29 + 56 * q^31 - 154 * q^33 + 70 * q^35 + 84 * q^37 + 412 * q^39 + 52 * q^41 - 408 * q^43 + 826 * q^45 - 8 * q^47 + 196 * q^49 + 388 * q^51 + 624 * q^53 + 110 * q^55 + 48 * q^57 + 238 * q^59 - 162 * q^61 + 532 * q^63 - 32 * q^65 - 1340 * q^67 + 2416 * q^69 - 1788 * q^71 + 1456 * q^73 + 806 * q^75 + 308 * q^77 + 1324 * q^79 + 1444 * q^81 - 450 * q^83 - 1736 * q^85 - 588 * q^87 - 3072 * q^89 + 406 * q^91 - 1264 * q^93 - 24 * q^95 - 652 * q^97 + 836 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 12x^{2} + 5x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} - 12\nu - 3$$ v^3 - 12*v - 3 $$\beta_{3}$$ $$=$$ $$-2\nu^{3} + 2\nu^{2} + 24\nu - 7$$ -2*v^3 + 2*v^2 + 24*v - 7
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 2\beta_{2} + 13 ) / 2$$ (b3 + 2*b2 + 13) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{2} + 6\beta _1 + 3$$ b2 + 6*b1 + 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.79597 −3.20317 −0.148103 0.555307
0 −10.1459 0 8.69995 0 7.00000 0 75.9402 0
1.2 0 −6.57251 0 15.5514 0 7.00000 0 16.1978 0
1.3 0 −2.77399 0 1.84418 0 7.00000 0 −19.3050 0
1.4 0 5.49244 0 −16.0955 0 7.00000 0 3.16692 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 1232.4.a.s 4
4.b odd 2 1 77.4.a.d 4
12.b even 2 1 693.4.a.l 4
20.d odd 2 1 1925.4.a.p 4
28.d even 2 1 539.4.a.g 4
44.c even 2 1 847.4.a.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.a.d 4 4.b odd 2 1
539.4.a.g 4 28.d even 2 1
693.4.a.l 4 12.b even 2 1
847.4.a.d 4 44.c even 2 1
1232.4.a.s 4 1.a even 1 1 trivial
1925.4.a.p 4 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1232))$$:

 $$T_{3}^{4} + 14T_{3}^{3} + 6T_{3}^{2} - 436T_{3} - 1016$$ T3^4 + 14*T3^3 + 6*T3^2 - 436*T3 - 1016 $$T_{5}^{4} - 10T_{5}^{3} - 240T_{5}^{2} + 2648T_{5} - 4016$$ T5^4 - 10*T5^3 - 240*T5^2 + 2648*T5 - 4016

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 14 T^{3} + 6 T^{2} + \cdots - 1016$$
$5$ $$T^{4} - 10 T^{3} - 240 T^{2} + \cdots - 4016$$
$7$ $$(T - 7)^{4}$$
$11$ $$(T - 11)^{4}$$
$13$ $$T^{4} - 58 T^{3} - 4926 T^{2} + \cdots - 4947656$$
$17$ $$T^{4} - 4 T^{3} - 6186 T^{2} + \cdots - 2705024$$
$19$ $$T^{4} + 258 T^{3} + \cdots - 14423904$$
$23$ $$T^{4} + 8 T^{3} - 22392 T^{2} + \cdots - 17449856$$
$29$ $$T^{4} + 396 T^{3} + \cdots + 22336464$$
$31$ $$T^{4} - 56 T^{3} - 31890 T^{2} + \cdots - 11250248$$
$37$ $$T^{4} - 84 T^{3} - 63516 T^{2} + \cdots + 11157312$$
$41$ $$T^{4} - 52 T^{3} + \cdots - 659233664$$
$43$ $$T^{4} + 408 T^{3} + \cdots + 1210397376$$
$47$ $$T^{4} + 8 T^{3} - 121650 T^{2} + \cdots - 318931592$$
$53$ $$T^{4} - 624 T^{3} + \cdots + 403923072$$
$59$ $$T^{4} - 238 T^{3} + \cdots + 17599820728$$
$61$ $$T^{4} + 162 T^{3} + \cdots + 6668930664$$
$67$ $$T^{4} + 1340 T^{3} + \cdots - 140865466496$$
$71$ $$T^{4} + 1788 T^{3} + \cdots - 72982082688$$
$73$ $$T^{4} - 1456 T^{3} + \cdots - 322052228384$$
$79$ $$T^{4} - 1324 T^{3} + \cdots - 59537293568$$
$83$ $$T^{4} + 450 T^{3} + \cdots + 21951092064$$
$89$ $$T^{4} + 3072 T^{3} + \cdots - 109303561968$$
$97$ $$T^{4} + 652 T^{3} + \cdots + 868634650768$$