# Properties

 Label 1856.4.a.z Level $1856$ Weight $4$ Character orbit 1856.a Self dual yes Analytic conductor $109.508$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.507544971$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Defining polynomial: $$x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198$$ x^5 - x^4 - 34*x^3 + 74*x^2 + 94*x - 198 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 232) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{3} + ( - \beta_{3} - 2) q^{5} + (\beta_{4} + \beta_{2} + 6) q^{7} + (\beta_{4} - \beta_{3} - 3 \beta_1 + 6) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^3 + (-b3 - 2) * q^5 + (b4 + b2 + 6) * q^7 + (b4 - b3 - 3*b1 + 6) * q^9 $$q + (\beta_1 - 1) q^{3} + ( - \beta_{3} - 2) q^{5} + (\beta_{4} + \beta_{2} + 6) q^{7} + (\beta_{4} - \beta_{3} - 3 \beta_1 + 6) q^{9} + (2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 8) q^{11} + (2 \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_1 - 7) q^{13} + ( - \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + 18) q^{15} + (\beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 17) q^{17} + (2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 5 \beta_1 - 31) q^{19} + (\beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 11 \beta_1 - 17) q^{21} + (5 \beta_{4} - 8 \beta_{3} + 9 \beta_{2} + 6 \beta_1 + 64) q^{23} + (5 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 6 \beta_1 + 27) q^{25} + ( - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 8 \beta_1 - 68) q^{27} - 29 q^{29} + ( - 3 \beta_{4} - 3 \beta_{2} + 13 \beta_1 + 85) q^{31} + (\beta_{4} + 5 \beta_{3} - 7 \beta_{2} + 6 \beta_1 + 20) q^{33} + ( - 7 \beta_{4} - 8 \beta_{3} - 5 \beta_{2} - 26 \beta_1 - 112) q^{35} + (12 \beta_{4} - 14 \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{37} + (11 \beta_{4} - 20 \beta_{3} + 13 \beta_{2} - 9 \beta_1 + 135) q^{39} + ( - 5 \beta_{4} - 2 \beta_{3} + 18 \beta_{2} + 17 \beta_1 + 15) q^{41} + (22 \beta_{4} - 30 \beta_{3} + 27 \beta_{2} - 14 \beta_1 - 74) q^{43} + (6 \beta_{4} - 16 \beta_{3} + 15 \beta_{2} + 9 \beta_1 + 25) q^{45} + ( - 11 \beta_{4} - 6 \beta_{3} - 24 \beta_{2} - 18 \beta_1 + 144) q^{47} + (12 \beta_{4} + 8 \beta_{3} + 24 \beta_{2} + 36 \beta_1 - 35) q^{49} + ( - 9 \beta_{4} + 24 \beta_{3} - 23 \beta_{2} + 40 \beta_1 - 126) q^{51} + ( - 6 \beta_{4} + 15 \beta_{3} + 3 \beta_{2} + 9 \beta_1 + 21) q^{53} + (\beta_{4} - 11 \beta_{2} - 11 \beta_1 + 149) q^{55} + ( - 2 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} - 7 \beta_1 - 113) q^{57} + ( - 33 \beta_{4} + 8 \beta_{3} - 7 \beta_{2} + 12 \beta_1 - 174) q^{59} + ( - 17 \beta_{4} - 4 \beta_{3} + 20 \beta_{2} + 27 \beta_1 + 125) q^{61} + ( - 12 \beta_{4} - 8 \beta_{3} - 2 \beta_{2} - 42 \beta_1 + 130) q^{63} + ( - 9 \beta_{4} - 11 \beta_{3} - 22 \beta_{2} + 29 \beta_1 - 143) q^{65} + (28 \beta_{4} - 18 \beta_{2} + 50 \beta_1 - 30) q^{67} + ( - \beta_{4} + 58 \beta_{3} - 42 \beta_{2} + 93 \beta_1 + 193) q^{69} + ( - 8 \beta_{4} + 4 \beta_{3} + 36 \beta_{2} - 2 \beta_1 + 210) q^{71} + (30 \beta_{4} - 2 \beta_{3} + 26 \beta_{2} + 72 \beta_1 - 242) q^{73} + (22 \beta_{4} - 48 \beta_{3} + 36 \beta_{2} + 34 \beta_1 + 78) q^{75} + ( - 19 \beta_{4} + 4 \beta_{3} + 8 \beta_{2} + \beta_1 + 185) q^{77} + ( - 5 \beta_{4} + 10 \beta_{3} - 2 \beta_{2} + 42) q^{79} + ( - 34 \beta_{4} + 16 \beta_{3} + 9 \beta_{2} + 15 \beta_1 - 358) q^{81} + ( - 33 \beta_{4} - 40 \beta_{3} + 19 \beta_{2} - 46 \beta_1 - 68) q^{83} + (13 \beta_{4} - 14 \beta_{3} + 8 \beta_{2} - 31 \beta_1 + 341) q^{85} + ( - 29 \beta_1 + 29) q^{87} + ( - 13 \beta_{4} - 26 \beta_{3} - 36 \beta_{2} + 71 \beta_1 - 367) q^{89} + ( - 19 \beta_{4} + 56 \beta_{3} + 5 \beta_{2} + 96 \beta_1 + 214) q^{91} + (10 \beta_{4} - 25 \beta_{3} - 6 \beta_{2} + 44 \beta_1 + 364) q^{93} + (16 \beta_{4} + 6 \beta_{3} + 29 \beta_{2} + 47 \beta_1 + 189) q^{95} + (15 \beta_{4} - 6 \beta_{3} - 24 \beta_{2} + 27 \beta_1 - 515) q^{97} + ( - 34 \beta_{4} - 14 \beta_{3} + 13 \beta_{2} + 3 \beta_1 + 313) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^3 + (-b3 - 2) * q^5 + (b4 + b2 + 6) * q^7 + (b4 - b3 - 3*b1 + 6) * q^9 + (2*b4 - 2*b3 + b2 - 8) * q^11 + (2*b4 + b3 - b2 + 5*b1 - 7) * q^13 + (-b4 + 2*b3 - 6*b2 + 18) * q^15 + (b4 - 4*b3 + 2*b2 - 5*b1 + 17) * q^17 + (2*b4 - 2*b3 - b2 - 5*b1 - 31) * q^19 + (b4 + 4*b3 + 2*b2 + 11*b1 - 17) * q^21 + (5*b4 - 8*b3 + 9*b2 + 6*b1 + 64) * q^23 + (5*b4 + 3*b3 - 3*b2 + 6*b1 + 27) * q^25 + (-2*b4 + 2*b3 - 3*b2 - 8*b1 - 68) * q^27 - 29 * q^29 + (-3*b4 - 3*b2 + 13*b1 + 85) * q^31 + (b4 + 5*b3 - 7*b2 + 6*b1 + 20) * q^33 + (-7*b4 - 8*b3 - 5*b2 - 26*b1 - 112) * q^35 + (12*b4 - 14*b3 + 2*b2 - 6*b1) * q^37 + (11*b4 - 20*b3 + 13*b2 - 9*b1 + 135) * q^39 + (-5*b4 - 2*b3 + 18*b2 + 17*b1 + 15) * q^41 + (22*b4 - 30*b3 + 27*b2 - 14*b1 - 74) * q^43 + (6*b4 - 16*b3 + 15*b2 + 9*b1 + 25) * q^45 + (-11*b4 - 6*b3 - 24*b2 - 18*b1 + 144) * q^47 + (12*b4 + 8*b3 + 24*b2 + 36*b1 - 35) * q^49 + (-9*b4 + 24*b3 - 23*b2 + 40*b1 - 126) * q^51 + (-6*b4 + 15*b3 + 3*b2 + 9*b1 + 21) * q^53 + (b4 - 11*b2 - 11*b1 + 149) * q^55 + (-2*b4 - 4*b3 - 5*b2 - 7*b1 - 113) * q^57 + (-33*b4 + 8*b3 - 7*b2 + 12*b1 - 174) * q^59 + (-17*b4 - 4*b3 + 20*b2 + 27*b1 + 125) * q^61 + (-12*b4 - 8*b3 - 2*b2 - 42*b1 + 130) * q^63 + (-9*b4 - 11*b3 - 22*b2 + 29*b1 - 143) * q^65 + (28*b4 - 18*b2 + 50*b1 - 30) * q^67 + (-b4 + 58*b3 - 42*b2 + 93*b1 + 193) * q^69 + (-8*b4 + 4*b3 + 36*b2 - 2*b1 + 210) * q^71 + (30*b4 - 2*b3 + 26*b2 + 72*b1 - 242) * q^73 + (22*b4 - 48*b3 + 36*b2 + 34*b1 + 78) * q^75 + (-19*b4 + 4*b3 + 8*b2 + b1 + 185) * q^77 + (-5*b4 + 10*b3 - 2*b2 + 42) * q^79 + (-34*b4 + 16*b3 + 9*b2 + 15*b1 - 358) * q^81 + (-33*b4 - 40*b3 + 19*b2 - 46*b1 - 68) * q^83 + (13*b4 - 14*b3 + 8*b2 - 31*b1 + 341) * q^85 + (-29*b1 + 29) * q^87 + (-13*b4 - 26*b3 - 36*b2 + 71*b1 - 367) * q^89 + (-19*b4 + 56*b3 + 5*b2 + 96*b1 + 214) * q^91 + (10*b4 - 25*b3 - 6*b2 + 44*b1 + 364) * q^93 + (16*b4 + 6*b3 + 29*b2 + 47*b1 + 189) * q^95 + (15*b4 - 6*b3 - 24*b2 + 27*b1 - 515) * q^97 + (-34*b4 - 14*b3 + 13*b2 + 3*b1 + 313) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9}+O(q^{10})$$ 5 * q - 4 * q^3 - 10 * q^5 + 32 * q^7 + 29 * q^9 $$5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9} - 36 q^{11} - 26 q^{13} + 88 q^{15} + 82 q^{17} - 156 q^{19} - 72 q^{21} + 336 q^{23} + 151 q^{25} - 352 q^{27} - 145 q^{29} + 432 q^{31} + 108 q^{33} - 600 q^{35} + 18 q^{37} + 688 q^{39} + 82 q^{41} - 340 q^{43} + 146 q^{45} + 680 q^{47} - 115 q^{49} - 608 q^{51} + 102 q^{53} + 736 q^{55} - 576 q^{57} - 924 q^{59} + 618 q^{61} + 584 q^{63} - 704 q^{65} - 44 q^{67} + 1056 q^{69} + 1032 q^{71} - 1078 q^{73} + 468 q^{75} + 888 q^{77} + 200 q^{79} - 1843 q^{81} - 452 q^{83} + 1700 q^{85} + 116 q^{87} - 1790 q^{89} + 1128 q^{91} + 1884 q^{93} + 1024 q^{95} - 2518 q^{97} + 1500 q^{99}+O(q^{100})$$ 5 * q - 4 * q^3 - 10 * q^5 + 32 * q^7 + 29 * q^9 - 36 * q^11 - 26 * q^13 + 88 * q^15 + 82 * q^17 - 156 * q^19 - 72 * q^21 + 336 * q^23 + 151 * q^25 - 352 * q^27 - 145 * q^29 + 432 * q^31 + 108 * q^33 - 600 * q^35 + 18 * q^37 + 688 * q^39 + 82 * q^41 - 340 * q^43 + 146 * q^45 + 680 * q^47 - 115 * q^49 - 608 * q^51 + 102 * q^53 + 736 * q^55 - 576 * q^57 - 924 * q^59 + 618 * q^61 + 584 * q^63 - 704 * q^65 - 44 * q^67 + 1056 * q^69 + 1032 * q^71 - 1078 * q^73 + 468 * q^75 + 888 * q^77 + 200 * q^79 - 1843 * q^81 - 452 * q^83 + 1700 * q^85 + 116 * q^87 - 1790 * q^89 + 1128 * q^91 + 1884 * q^93 + 1024 * q^95 - 2518 * q^97 + 1500 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{4} - 5\nu^{3} + 23\nu^{2} + 64\nu - 90 ) / 19$$ (-v^4 - 5*v^3 + 23*v^2 + 64*v - 90) / 19 $$\beta_{2}$$ $$=$$ $$( -\nu^{4} - 5\nu^{3} + 23\nu^{2} + 140\nu - 109 ) / 19$$ (-v^4 - 5*v^3 + 23*v^2 + 140*v - 109) / 19 $$\beta_{3}$$ $$=$$ $$( -5\nu^{4} - 6\nu^{3} + 134\nu^{2} - 60\nu - 203 ) / 19$$ (-5*v^4 - 6*v^3 + 134*v^2 - 60*v - 203) / 19 $$\beta_{4}$$ $$=$$ $$( -9\nu^{4} - 7\nu^{3} + 283\nu^{2} - 184\nu - 829 ) / 19$$ (-9*v^4 - 7*v^3 + 283*v^2 - 184*v - 829) / 19
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta _1 + 1 ) / 4$$ (b2 - b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{4} - 2\beta_{3} + \beta _1 + 27 ) / 2$$ (b4 - 2*b3 + b1 + 27) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{4} + 4\beta_{3} + 10\beta_{2} - 21\beta _1 - 43 ) / 2$$ (-b4 + 4*b3 + 10*b2 - 21*b1 - 43) / 2 $$\nu^{4}$$ $$=$$ $$14\beta_{4} - 33\beta_{3} - 9\beta_{2} + 29\beta _1 + 344$$ 14*b4 - 33*b3 - 9*b2 + 29*b1 + 344

## 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
 4.30242 −1.69859 1.52813 2.92646 −6.05843
0 −7.82921 0 7.04208 0 14.1473 0 34.2965 0
1.2 0 −7.11424 0 −16.3850 0 5.74609 0 23.6124 0
1.3 0 1.01127 0 −0.397400 0 −14.4223 0 −25.9773 0
1.4 0 4.03215 0 −15.2584 0 33.3511 0 −10.7418 0
1.5 0 5.90002 0 14.9988 0 −6.82211 0 7.81028 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$29$$ $$1$$

## 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 1856.4.a.z 5
4.b odd 2 1 1856.4.a.ba 5
8.b even 2 1 232.4.a.d 5
8.d odd 2 1 464.4.a.m 5
24.h odd 2 1 2088.4.a.f 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.a.d 5 8.b even 2 1
464.4.a.m 5 8.d odd 2 1
1856.4.a.z 5 1.a even 1 1 trivial
1856.4.a.ba 5 4.b odd 2 1
2088.4.a.f 5 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1856))$$:

 $$T_{3}^{5} + 4T_{3}^{4} - 74T_{3}^{3} - 128T_{3}^{2} + 1525T_{3} - 1340$$ T3^5 + 4*T3^4 - 74*T3^3 - 128*T3^2 + 1525*T3 - 1340 $$T_{5}^{5} + 10T_{5}^{4} - 338T_{5}^{3} - 2304T_{5}^{2} + 25545T_{5} + 10494$$ T5^5 + 10*T5^4 - 338*T5^3 - 2304*T5^2 + 25545*T5 + 10494 $$T_{7}^{5} - 32T_{7}^{4} - 288T_{7}^{3} + 7872T_{7}^{2} + 15680T_{7} - 266752$$ T7^5 - 32*T7^4 - 288*T7^3 + 7872*T7^2 + 15680*T7 - 266752

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} + 4 T^{4} - 74 T^{3} + \cdots - 1340$$
$5$ $$T^{5} + 10 T^{4} - 338 T^{3} + \cdots + 10494$$
$7$ $$T^{5} - 32 T^{4} - 288 T^{3} + \cdots - 266752$$
$11$ $$T^{5} + 36 T^{4} - 1346 T^{3} + \cdots - 1741860$$
$13$ $$T^{5} + 26 T^{4} + \cdots - 105404410$$
$17$ $$T^{5} - 82 T^{4} - 2776 T^{3} + \cdots - 49184$$
$19$ $$T^{5} + 156 T^{4} + 4216 T^{3} + \cdots + 1820736$$
$23$ $$T^{5} - 336 T^{4} + \cdots + 7489438848$$
$29$ $$(T + 29)^{5}$$
$31$ $$T^{5} - 432 T^{4} + \cdots + 445071048$$
$37$ $$T^{5} - 18 T^{4} + \cdots + 15294686720$$
$41$ $$T^{5} - 82 T^{4} + \cdots - 731491061376$$
$43$ $$T^{5} + 340 T^{4} + \cdots + 571309913052$$
$47$ $$T^{5} - 680 T^{4} + \cdots + 2559413417896$$
$53$ $$T^{5} - 102 T^{4} + \cdots - 103910584482$$
$59$ $$T^{5} + 924 T^{4} + \cdots - 16799541984192$$
$61$ $$T^{5} - 618 T^{4} + \cdots - 2366067286944$$
$67$ $$T^{5} + 44 T^{4} + \cdots - 29804817076224$$
$71$ $$T^{5} - 1032 T^{4} + \cdots - 50004302698368$$
$73$ $$T^{5} + 1078 T^{4} + \cdots - 4755790305792$$
$79$ $$T^{5} - 200 T^{4} + \cdots - 13404287016$$
$83$ $$T^{5} + \cdots + 181825403631808$$
$89$ $$T^{5} + 1790 T^{4} + \cdots - 79946879886976$$
$97$ $$T^{5} + 2518 T^{4} + \cdots - 41073987679360$$