# Properties

 Label 55.6.a.b Level $55$ Weight $6$ Character orbit 55.a Self dual yes Analytic conductor $8.821$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,6,Mod(1,55)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("55.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 55.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$8.82111008971$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 33x^{2} - 8x + 116$$ x^4 - 2*x^3 - 33*x^2 - 8*x + 116 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{2} + ( - \beta_{3} + \beta_1) q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 15) q^{4} - 25 q^{5} + (5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 43) q^{6} + (8 \beta_{3} + 5 \beta_{2} + 9 \beta_1 - 28) q^{7} + ( - 3 \beta_{3} - 5 \beta_{2} + \cdots - 33) q^{8}+ \cdots + ( - 6 \beta_{3} - 9 \beta_{2} + \cdots + 7) q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^2 + (-b3 + b1) * q^3 + (b3 - b2 + b1 + 15) * q^4 - 25 * q^5 + (5*b3 + 5*b2 + 5*b1 - 43) * q^6 + (8*b3 + 5*b2 + 9*b1 - 28) * q^7 + (-3*b3 - 5*b2 + 5*b1 - 33) * q^8 + (-6*b3 - 9*b2 + 9*b1 + 7) * q^9 $$q + ( - \beta_1 - 1) q^{2} + ( - \beta_{3} + \beta_1) q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 15) q^{4} - 25 q^{5} + (5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 43) q^{6} + (8 \beta_{3} + 5 \beta_{2} + 9 \beta_1 - 28) q^{7} + ( - 3 \beta_{3} - 5 \beta_{2} + \cdots - 33) q^{8}+ \cdots + (726 \beta_{3} + 1089 \beta_{2} + \cdots - 847) q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^2 + (-b3 + b1) * q^3 + (b3 - b2 + b1 + 15) * q^4 - 25 * q^5 + (5*b3 + 5*b2 + 5*b1 - 43) * q^6 + (8*b3 + 5*b2 + 9*b1 - 28) * q^7 + (-3*b3 - 5*b2 + 5*b1 - 33) * q^8 + (-6*b3 - 9*b2 + 9*b1 + 7) * q^9 + (25*b1 + 25) * q^10 - 121 * q^11 + (-23*b3 - 5*b2 + 21*b1 - 197) * q^12 + (25*b3 - 15*b2 + 10*b1 + 200) * q^13 + (-77*b3 - 13*b2 + 23*b1 - 405) * q^14 + (25*b3 - 25*b1) * q^15 + (b3 + 39*b2 - 19*b1 - 673) * q^16 + (17*b3 + 50*b2 - 27*b1 - 960) * q^17 + (63*b3 + 15*b2 - 40*b1 - 412) * q^18 + (-30*b3 + 5*b2 - 65*b1 - 826) * q^19 + (-25*b3 + 25*b2 - 25*b1 - 375) * q^20 + (38*b3 + 7*b2 - 337*b1 - 1104) * q^21 + (121*b1 + 121) * q^22 + (-9*b3 - 145*b2 - 306*b1 - 640) * q^23 + (-23*b3 - 57*b2 + 117*b1 + 671) * q^24 + 625 * q^25 + (-100*b3 - 120*b2 - 430*b1 - 750) * q^26 + (134*b3 - 105*b2 + 43*b1 + 1332) * q^27 + (235*b3 + 145*b2 + 411*b1 + 461) * q^28 + (162*b3 + 273*b2 + 277*b1 - 1490) * q^29 + (-125*b3 - 125*b2 - 125*b1 + 1075) * q^30 + (36*b3 + 19*b2 - 219*b1 + 1224) * q^31 + (-47*b3 + 215*b2 + 781*b1 + 2639) * q^32 + (121*b3 - 121*b1) * q^33 + (-275*b3 + 5*b2 + 1225*b1 + 2201) * q^34 + (-200*b3 - 125*b2 - 225*b1 + 700) * q^35 + (-206*b3 + 26*b2 - 86*b1 + 1854) * q^36 + (-280*b3 - 205*b2 + 263*b1 + 2608) * q^37 + (225*b3 + 65*b2 + 1011*b1 + 3911) * q^38 + (-280*b3 - 10*b2 + 170*b1 - 5240) * q^39 + (75*b3 + 125*b2 - 125*b1 + 825) * q^40 + (-108*b3 + 348*b2 - 1348*b1 - 4990) * q^41 + (81*b3 - 475*b2 + 963*b1 + 16499) * q^42 + (-622*b3 - 100*b2 - 466*b1 - 2738) * q^43 + (-121*b3 + 121*b2 - 121*b1 - 1815) * q^44 + (150*b3 + 225*b2 - 225*b1 - 175) * q^45 + (940*b3 - 560*b2 - 330*b1 + 14598) * q^46 + (-667*b3 + 205*b2 + 910*b1 + 1068) * q^47 + (985*b3 + 255*b2 - 1627*b1 + 263) * q^48 + (570*b3 - 365*b2 + 1885*b1 + 5615) * q^49 + (-625*b1 - 625) * q^50 + (1402*b3 + 403*b2 - 2403*b1 - 3030) * q^51 + (710*b3 + 210*b2 + 90*b1 + 14310) * q^52 + (-1060*b3 - 5*b2 - 381*b1 - 5056) * q^53 + (-427*b3 - 703*b2 - 2737*b1 - 3817) * q^54 + 3025 * q^55 + (63*b3 + 177*b2 - 1357*b1 - 6967) * q^56 + (1056*b3 + 225*b2 - 121*b1 + 3580) * q^57 + (-2341*b3 + 175*b2 + 2591*b1 - 11465) * q^58 + (1334*b3 - 524*b2 - 2966*b1 - 16728) * q^59 + (575*b3 + 125*b2 - 525*b1 + 4925) * q^60 + (-2112*b3 - 523*b2 + 2483*b1 - 10962) * q^61 + (-73*b3 - 325*b2 - 1271*b1 + 8761) * q^62 + (626*b3 + 650*b2 - 2028*b1 - 15322) * q^63 + (-1391*b3 + 151*b2 - 291*b1 - 16673) * q^64 + (-625*b3 + 375*b2 - 250*b1 - 5000) * q^65 + (-605*b3 - 605*b2 - 605*b1 + 5203) * q^66 + (39*b3 + 965*b2 + 4580*b1 - 3576) * q^67 + (-139*b3 + 735*b2 + 73*b1 - 27001) * q^68 + (976*b3 + 914*b2 + 5246*b1 - 16080) * q^69 + (1925*b3 + 325*b2 - 575*b1 + 10125) * q^70 + (-1968*b3 - 1467*b2 + 4367*b1 - 12112) * q^71 + (-798*b3 + 310*b2 + 638*b1 + 15930) * q^72 + (-2031*b3 - 3055*b2 + 1022*b1 + 396) * q^73 + (2237*b3 + 973*b2 - 2643*b1 - 14071) * q^74 + (-625*b3 + 625*b1) * q^75 + (-1661*b3 + 81*b2 - 2501*b1 - 24595) * q^76 + (-968*b3 - 605*b2 - 1089*b1 + 3388) * q^77 + (1550*b3 + 1270*b2 + 6570*b1 - 1750) * q^78 + (2688*b3 + 1222*b2 + 3498*b1 - 48900) * q^79 + (-25*b3 - 975*b2 + 475*b1 + 16825) * q^80 + (-408*b3 + 2088*b2 - 288*b1 - 31055) * q^81 + (604*b3 - 220*b2 + 7966*b1 + 67670) * q^82 + (2802*b3 - 2590*b2 + 2880*b1 + 16382) * q^83 + (-765*b3 - 535*b2 - 9445*b1 - 26187) * q^84 + (-425*b3 - 1250*b2 + 675*b1 + 24000) * q^85 + (4598*b3 + 1822*b2 + 5148*b1 + 25940) * q^86 + (2344*b3 + 355*b2 - 12953*b1 - 12614) * q^87 + (363*b3 + 605*b2 - 605*b1 + 3993) * q^88 + (-2454*b3 - 4311*b2 - 29*b1 - 40868) * q^89 + (-1575*b3 - 375*b2 + 1000*b1 + 10300) * q^90 + (3290*b3 + 2020*b2 + 9150*b1 + 20820) * q^91 + (-2782*b3 - 570*b2 - 13426*b1 + 17682) * q^92 + (-162*b3 + 1315*b2 + 1485*b1 - 16242) * q^93 + (2272*b3 + 3988*b2 + 3702*b1 - 40722) * q^94 + (750*b3 - 125*b2 + 1625*b1 + 20650) * q^95 + (-4567*b3 - 3233*b2 - 7147*b1 + 50407) * q^96 + (706*b3 + 2500*b2 - 7374*b1 + 47582) * q^97 + (-3845*b3 - 1125*b2 - 11020*b1 - 94400) * q^98 + (726*b3 + 1089*b2 - 1089*b1 - 847) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 5 q^{2} + 61 q^{4} - 100 q^{5} - 157 q^{6} - 90 q^{7} - 135 q^{8} + 22 q^{9}+O(q^{10})$$ 4 * q - 5 * q^2 + 61 * q^4 - 100 * q^5 - 157 * q^6 - 90 * q^7 - 135 * q^8 + 22 * q^9 $$4 q - 5 q^{2} + 61 q^{4} - 100 q^{5} - 157 q^{6} - 90 q^{7} - 135 q^{8} + 22 q^{9} + 125 q^{10} - 484 q^{11} - 795 q^{12} + 820 q^{13} - 1687 q^{14} - 2671 q^{16} - 3800 q^{17} - 1610 q^{18} - 3394 q^{19} - 1525 q^{20} - 4708 q^{21} + 605 q^{22} - 3020 q^{23} + 2721 q^{24} + 2500 q^{25} - 3650 q^{26} + 5400 q^{27} + 2635 q^{28} - 5248 q^{29} + 3925 q^{30} + 4732 q^{31} + 11505 q^{32} + 9759 q^{34} + 2250 q^{35} + 7150 q^{36} + 10210 q^{37} + 16945 q^{38} - 21080 q^{39} + 3375 q^{40} - 21068 q^{41} + 66565 q^{42} - 12140 q^{43} - 7381 q^{44} - 550 q^{45} + 58442 q^{46} + 4720 q^{47} + 665 q^{48} + 24550 q^{49} - 3125 q^{50} - 12718 q^{51} + 58250 q^{52} - 21670 q^{53} - 19135 q^{54} + 12100 q^{55} - 28985 q^{56} + 15480 q^{57} - 45435 q^{58} - 69068 q^{59} + 19875 q^{60} - 44000 q^{61} + 33375 q^{62} - 62040 q^{63} - 68223 q^{64} - 20500 q^{65} + 18997 q^{66} - 8720 q^{67} - 107335 q^{68} - 57184 q^{69} + 42175 q^{70} - 47516 q^{71} + 63870 q^{72} - 2480 q^{73} - 55717 q^{74} - 102461 q^{76} + 10890 q^{77} + 2390 q^{78} - 188192 q^{79} + 66775 q^{80} - 122828 q^{81} + 279030 q^{82} + 68620 q^{83} - 115493 q^{84} + 95000 q^{85} + 115328 q^{86} - 60710 q^{87} + 16335 q^{88} - 170266 q^{89} + 40250 q^{90} + 97740 q^{91} + 53950 q^{92} - 62330 q^{93} - 152926 q^{94} + 84850 q^{95} + 186681 q^{96} + 186160 q^{97} - 393590 q^{98} - 2662 q^{99}+O(q^{100})$$ 4 * q - 5 * q^2 + 61 * q^4 - 100 * q^5 - 157 * q^6 - 90 * q^7 - 135 * q^8 + 22 * q^9 + 125 * q^10 - 484 * q^11 - 795 * q^12 + 820 * q^13 - 1687 * q^14 - 2671 * q^16 - 3800 * q^17 - 1610 * q^18 - 3394 * q^19 - 1525 * q^20 - 4708 * q^21 + 605 * q^22 - 3020 * q^23 + 2721 * q^24 + 2500 * q^25 - 3650 * q^26 + 5400 * q^27 + 2635 * q^28 - 5248 * q^29 + 3925 * q^30 + 4732 * q^31 + 11505 * q^32 + 9759 * q^34 + 2250 * q^35 + 7150 * q^36 + 10210 * q^37 + 16945 * q^38 - 21080 * q^39 + 3375 * q^40 - 21068 * q^41 + 66565 * q^42 - 12140 * q^43 - 7381 * q^44 - 550 * q^45 + 58442 * q^46 + 4720 * q^47 + 665 * q^48 + 24550 * q^49 - 3125 * q^50 - 12718 * q^51 + 58250 * q^52 - 21670 * q^53 - 19135 * q^54 + 12100 * q^55 - 28985 * q^56 + 15480 * q^57 - 45435 * q^58 - 69068 * q^59 + 19875 * q^60 - 44000 * q^61 + 33375 * q^62 - 62040 * q^63 - 68223 * q^64 - 20500 * q^65 + 18997 * q^66 - 8720 * q^67 - 107335 * q^68 - 57184 * q^69 + 42175 * q^70 - 47516 * q^71 + 63870 * q^72 - 2480 * q^73 - 55717 * q^74 - 102461 * q^76 + 10890 * q^77 + 2390 * q^78 - 188192 * q^79 + 66775 * q^80 - 122828 * q^81 + 279030 * q^82 + 68620 * q^83 - 115493 * q^84 + 95000 * q^85 + 115328 * q^86 - 60710 * q^87 + 16335 * q^88 - 170266 * q^89 + 40250 * q^90 + 97740 * q^91 + 53950 * q^92 - 62330 * q^93 - 152926 * q^94 + 84850 * q^95 + 186681 * q^96 + 186160 * q^97 - 393590 * q^98 - 2662 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 33x^{2} - 8x + 116$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 4\nu^{2} - 17\nu + 22 ) / 4$$ (v^3 - 4*v^2 - 17*v + 22) / 4 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 41\nu + 38 ) / 4$$ (-v^3 + 41*v + 38) / 4 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 8\nu^{2} + 17\nu - 90 ) / 4$$ (-v^3 + 8*v^2 + 17*v - 90) / 4
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + 2\beta _1 + 2 ) / 6$$ (b3 + b2 + 2*b1 + 2) / 6 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 + 17$$ b3 + b1 + 17 $$\nu^{3}$$ $$=$$ $$( 41\beta_{3} + 17\beta_{2} + 82\beta _1 + 310 ) / 6$$ (41*b3 + 17*b2 + 82*b1 + 310) / 6

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 6.71109 −2.50110 1.74666 −3.95665
−8.50380 −13.0311 40.3146 −25.0000 110.814 217.433 −70.7058 −73.1914 212.595
1.2 −6.96278 22.6701 16.4803 −25.0000 −157.847 −169.118 108.060 270.932 174.070
1.3 2.64192 6.66534 −25.0202 −25.0000 17.6093 −12.8802 −150.643 −198.573 −66.0481
1.4 7.82466 −16.3044 29.2253 −25.0000 −127.576 −125.436 −21.7111 22.8321 −195.616
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$11$$ $$+1$$

## 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 55.6.a.b 4
3.b odd 2 1 495.6.a.g 4
4.b odd 2 1 880.6.a.n 4
5.b even 2 1 275.6.a.d 4
5.c odd 4 2 275.6.b.d 8
11.b odd 2 1 605.6.a.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.6.a.b 4 1.a even 1 1 trivial
275.6.a.d 4 5.b even 2 1
275.6.b.d 8 5.c odd 4 2
495.6.a.g 4 3.b odd 2 1
605.6.a.c 4 11.b odd 2 1
880.6.a.n 4 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 5T_{2}^{3} - 82T_{2}^{2} - 300T_{2} + 1224$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(55))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 5 T^{3} + \cdots + 1224$$
$3$ $$T^{4} - 497 T^{2} + \cdots + 32104$$
$5$ $$(T + 25)^{4}$$
$7$ $$T^{4} + 90 T^{3} + \cdots - 59409676$$
$11$ $$(T + 121)^{4}$$
$13$ $$T^{4} + \cdots - 4887860000$$
$17$ $$T^{4} + \cdots - 500025407256$$
$19$ $$T^{4} + \cdots + 123062234816$$
$23$ $$T^{4} + \cdots - 26010252218016$$
$29$ $$T^{4} + \cdots + 563529236434956$$
$31$ $$T^{4} + \cdots - 2285702081856$$
$37$ $$T^{4} + \cdots + 21956626606324$$
$41$ $$T^{4} + \cdots + 17\!\cdots\!96$$
$43$ $$T^{4} + \cdots + 559080671259904$$
$47$ $$T^{4} + \cdots + 11\!\cdots\!16$$
$53$ $$T^{4} + \cdots + 21\!\cdots\!44$$
$59$ $$T^{4} + \cdots - 60\!\cdots\!76$$
$61$ $$T^{4} + \cdots - 81\!\cdots\!16$$
$67$ $$T^{4} + \cdots - 10\!\cdots\!64$$
$71$ $$T^{4} + \cdots - 30\!\cdots\!76$$
$73$ $$T^{4} + \cdots + 48\!\cdots\!44$$
$79$ $$T^{4} + \cdots - 14\!\cdots\!44$$
$83$ $$T^{4} + \cdots - 77\!\cdots\!04$$
$89$ $$T^{4} + \cdots + 56\!\cdots\!36$$
$97$ $$T^{4} + \cdots - 54\!\cdots\!24$$