# Properties

 Label 55.5.i.a Level $55$ Weight $5$ Character orbit 55.i Analytic conductor $5.685$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,5,Mod(6,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 9]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("55.6");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 55.i (of order $$10$$, degree $$4$$, minimal)

## 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: no Analytic conductor: $$5.68534796961$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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 primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{2} + (4 \zeta_{10}^{3} + 9 \zeta_{10}^{2} + \cdots - 4) q^{3}+ \cdots + ( - 104 \zeta_{10}^{3} + 65 \zeta_{10} - 65) q^{9}+O(q^{10})$$ q + (-z^3 + 2*z^2 + 2) * q^2 + (4*z^3 + 9*z^2 - 9*z - 4) * q^3 + (4*z^2 - 13*z + 4) * q^4 + (-10*z^3 + 5*z^2 - 10*z) * q^5 + (21*z^3 - 17*z^2 + 13*z - 34) * q^6 + (-28*z^3 - 16*z^2 - 14*z + 44) * q^7 + (-25*z^3 - 21*z^2 - 21*z - 25) * q^8 + (-104*z^3 + 65*z - 65) * q^9 $$q + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{2} + (4 \zeta_{10}^{3} + 9 \zeta_{10}^{2} + \cdots - 4) q^{3}+ \cdots + (429 \zeta_{10}^{3} - 4147 \zeta_{10}^{2} + \cdots - 16588) q^{99}+O(q^{100})$$ q + (-z^3 + 2*z^2 + 2) * q^2 + (4*z^3 + 9*z^2 - 9*z - 4) * q^3 + (4*z^2 - 13*z + 4) * q^4 + (-10*z^3 + 5*z^2 - 10*z) * q^5 + (21*z^3 - 17*z^2 + 13*z - 34) * q^6 + (-28*z^3 - 16*z^2 - 14*z + 44) * q^7 + (-25*z^3 - 21*z^2 - 21*z - 25) * q^8 + (-104*z^3 + 65*z - 65) * q^9 + (-20*z^3 - 10*z^2 - 10*z + 5) * q^10 + (77*z^3 - 99*z^2 - 33*z - 33) * q^11 + (-153*z^3 + 153*z^2 - 16) * q^12 + (36*z^3 - 72*z^2 + 18*z - 54) * q^13 + (-146*z^3 + 74*z^2 - 74*z + 146) * q^14 + (-25*z^2 + 175*z - 25) * q^15 + (-152*z^3 + 137*z^2 - 152*z) * q^16 + (229*z^3 + 14*z^2 - 257*z + 28) * q^17 + (-78*z^3 - 65*z^2 - 39*z + 143) * q^18 + (-165*z^3 + 202*z^2 + 202*z - 165) * q^19 + (5*z^3 + 110*z - 110) * q^20 + (358*z^3 + 534*z^2 - 176*z + 88) * q^21 + (-44*z^3 - 99*z^2 - 154*z - 154) * q^22 + (416*z^3 - 416*z^2 - 278) * q^23 + (-48*z^3 + 96*z^2 + 361*z + 457) * q^24 + (125*z^3 - 125*z^2 + 125*z - 125) * q^25 + (-90*z^2 - 90*z - 90) * q^26 + (884*z^3 - 1637*z^2 + 884*z) * q^27 + (340*z^3 - 6*z^2 - 328*z - 12) * q^28 + (-1560*z^3 + 662*z^2 - 780*z + 898) * q^29 + (150*z^3 + 125*z^2 + 125*z + 150) * q^30 + (26*z^3 + 288*z - 288) * q^31 + (554*z^3 - 216*z^2 + 770*z - 385) * q^32 + (-1562*z^3 + 2112*z^2 - 1595*z + 1551) * q^33 + (201*z^3 - 201*z^2 - 673) * q^34 + (-10*z^3 + 20*z^2 - 380*z - 360) * q^35 + (1196*z^3 - 2457*z^2 + 2457*z - 1196) * q^36 + (198*z^2 - 614*z + 198) * q^37 + (441*z^3 - 128*z^2 + 441*z) * q^38 + (-450*z^3 + 540*z^2 - 630*z + 1080) * q^39 + (710*z^3 - 270*z^2 + 355*z - 440) * q^40 + (1304*z^3 + 153*z^2 + 153*z + 1304) * q^41 + (1520*z^3 + 1250*z - 1250) * q^42 + (529*z^3 + 809*z^2 - 280*z + 140) * q^43 + (66*z^3 + 1298*z^2 - 1100*z + 957) * q^44 + (1365*z^3 - 1365*z^2 + 130) * q^45 + (278*z^3 - 556*z^2 - 416*z - 972) * q^46 + (-1412*z^3 + 1248*z^2 - 1248*z + 1412) * q^47 + (-1173*z^2 + 3209*z - 1173) * q^48 + (-976*z^3 + 149*z^2 - 976*z) * q^49 + (250*z^3 - 125*z^2 - 250) * q^50 + (-6206*z^3 + 5472*z^2 - 3103*z + 734) * q^51 + (396*z^3 + 18*z^2 + 18*z + 396) * q^52 + (-2350*z^3 + 80*z - 80) * q^53 + (-622*z^3 + 884*z^2 - 1506*z + 753) * q^54 + (220*z^3 + 1100*z^2 + 165*z - 440) * q^55 + (1504*z^3 - 1504*z^2 - 3362) * q^56 + (4111*z^3 - 8222*z^2 + 5448*z - 2774) * q^57 + (-3474*z^3 + 1016*z^2 - 1016*z + 3474) * q^58 + (-1615*z^2 - 3038*z - 1615) * q^59 + (925*z^3 - 2375*z^2 + 925*z) * q^60 + (-3138*z^3 + 2012*z^2 - 886*z + 4024) * q^61 + (628*z^3 - 288*z^2 + 314*z - 340) * q^62 + (-4160*z^3 + 494*z^2 + 494*z - 4160) * q^63 + (2071*z^3 + 3324*z - 3324) * q^64 + (450*z^3 + 450*z^2) * q^65 + (-2046*z^3 + 1507*z^2 + 1067*z + 2519) * q^66 + (2099*z^3 - 2099*z^2 - 6487) * q^67 + (-3215*z^3 + 6430*z^2 - 4313*z + 2117) * q^68 + (-6520*z^3 + 6650*z^2 - 6650*z + 6520) * q^69 + (-1100*z^2 - 350*z - 1100) * q^70 + (392*z^3 - 4234*z^2 + 392*z) * q^71 + (3419*z^3 - 559*z^2 - 2301*z - 1118) * q^72 + (1178*z^3 - 3783*z^2 + 589*z + 2605) * q^73 + (-416*z^3 - 218*z^2 - 218*z - 416) * q^74 + (-500*z^3 - 1625*z + 1625) * q^75 + (475*z^3 - 5431*z^2 + 5906*z - 2953) * q^76 + (7656*z^3 - 4796*z^2 + 2596*z - 4422) * q^77 + (-1530*z^3 + 1530*z^2 + 1890) * q^78 + (904*z^3 - 1808*z^2 + 2960*z + 1152) * q^79 + (1595*z^3 - 2205*z^2 + 2205*z - 1595) * q^80 + (12480*z^2 - 12536*z + 12480) * q^81 + (1763*z^3 + 2761*z^2 + 1763*z) * q^82 + (8331*z^3 - 1919*z^2 - 4493*z - 3838) * q^83 + (-8732*z^3 + 7294*z^2 - 4366*z + 1438) * q^84 + (-1355*z^3 + 2360*z^2 + 2360*z - 1355) * q^85 + (2256*z^3 + 1867*z - 1867) * q^86 + (13732*z^3 - 4424*z^2 + 18156*z - 9078) * q^87 + (2959*z^3 + 2574*z^2 + 4730*z - 1320) * q^88 + (-4087*z^3 + 4087*z^2 + 12248) * q^89 + (-130*z^3 + 260*z^2 - 1365*z - 1105) * q^90 + (3960*z^3 - 2700*z^2 + 2700*z - 3960) * q^91 + (4296*z^2 - 3458*z + 4296) * q^92 + (2254*z^3 - 6102*z^2 + 2254*z) * q^93 + (-2988*z^3 + 1576*z^2 - 164*z + 3152) * q^94 + (1280*z^3 - 3485*z^2 + 640*z + 2205) * q^95 + (1268*z^3 - 5681*z^2 - 5681*z + 1268) * q^96 + (-7882*z^3 - 9309*z + 9309) * q^97 + (-2630*z^3 - 976*z^2 - 1654*z + 827) * q^98 + (429*z^3 - 4147*z^2 + 16445*z - 16588) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 5 q^{2} - 30 q^{3} - q^{4} - 25 q^{5} - 85 q^{6} + 150 q^{7} - 125 q^{8} - 299 q^{9}+O(q^{10})$$ 4 * q + 5 * q^2 - 30 * q^3 - q^4 - 25 * q^5 - 85 * q^6 + 150 * q^7 - 125 * q^8 - 299 * q^9 $$4 q + 5 q^{2} - 30 q^{3} - q^{4} - 25 q^{5} - 85 q^{6} + 150 q^{7} - 125 q^{8} - 299 q^{9} + 11 q^{11} - 370 q^{12} - 90 q^{13} + 290 q^{14} + 100 q^{15} - 441 q^{16} + 70 q^{17} + 520 q^{18} - 825 q^{19} - 325 q^{20} - 715 q^{22} - 280 q^{23} + 2045 q^{24} - 125 q^{25} - 360 q^{26} + 3405 q^{27} - 30 q^{28} + 590 q^{29} + 750 q^{30} - 838 q^{31} + 935 q^{33} - 2290 q^{34} - 1850 q^{35} + 1326 q^{36} - 20 q^{37} + 1010 q^{38} + 2700 q^{39} - 425 q^{40} + 6520 q^{41} - 2230 q^{42} + 1496 q^{44} + 3250 q^{45} - 3470 q^{46} + 1740 q^{47} - 310 q^{48} - 2101 q^{49} - 625 q^{50} - 11845 q^{51} + 1980 q^{52} - 2590 q^{53} - 2475 q^{55} - 10440 q^{56} + 6685 q^{57} + 8390 q^{58} - 7883 q^{59} + 4225 q^{60} + 10060 q^{61} - 130 q^{62} - 20800 q^{63} - 7901 q^{64} + 7590 q^{66} - 21750 q^{67} - 5490 q^{68} + 6260 q^{69} - 3650 q^{70} + 5018 q^{71} - 2795 q^{72} + 15970 q^{73} - 2080 q^{74} + 4375 q^{75} - 2640 q^{77} + 4500 q^{78} + 10280 q^{79} - 375 q^{80} + 24904 q^{81} + 765 q^{82} - 9595 q^{83} - 14640 q^{84} - 6775 q^{85} - 3345 q^{86} - 165 q^{88} + 40818 q^{89} - 6175 q^{90} - 6480 q^{91} + 9430 q^{92} + 10610 q^{93} + 7880 q^{94} + 14225 q^{95} + 6340 q^{96} + 20045 q^{97} - 45331 q^{99}+O(q^{100})$$ 4 * q + 5 * q^2 - 30 * q^3 - q^4 - 25 * q^5 - 85 * q^6 + 150 * q^7 - 125 * q^8 - 299 * q^9 + 11 * q^11 - 370 * q^12 - 90 * q^13 + 290 * q^14 + 100 * q^15 - 441 * q^16 + 70 * q^17 + 520 * q^18 - 825 * q^19 - 325 * q^20 - 715 * q^22 - 280 * q^23 + 2045 * q^24 - 125 * q^25 - 360 * q^26 + 3405 * q^27 - 30 * q^28 + 590 * q^29 + 750 * q^30 - 838 * q^31 + 935 * q^33 - 2290 * q^34 - 1850 * q^35 + 1326 * q^36 - 20 * q^37 + 1010 * q^38 + 2700 * q^39 - 425 * q^40 + 6520 * q^41 - 2230 * q^42 + 1496 * q^44 + 3250 * q^45 - 3470 * q^46 + 1740 * q^47 - 310 * q^48 - 2101 * q^49 - 625 * q^50 - 11845 * q^51 + 1980 * q^52 - 2590 * q^53 - 2475 * q^55 - 10440 * q^56 + 6685 * q^57 + 8390 * q^58 - 7883 * q^59 + 4225 * q^60 + 10060 * q^61 - 130 * q^62 - 20800 * q^63 - 7901 * q^64 + 7590 * q^66 - 21750 * q^67 - 5490 * q^68 + 6260 * q^69 - 3650 * q^70 + 5018 * q^71 - 2795 * q^72 + 15970 * q^73 - 2080 * q^74 + 4375 * q^75 - 2640 * q^77 + 4500 * q^78 + 10280 * q^79 - 375 * q^80 + 24904 * q^81 + 765 * q^82 - 9595 * q^83 - 14640 * q^84 - 6775 * q^85 - 3345 * q^86 - 165 * q^88 + 40818 * q^89 - 6175 * q^90 - 6480 * q^91 + 9430 * q^92 + 10610 * q^93 + 7880 * q^94 + 14225 * q^95 + 6340 * q^96 + 20045 * q^97 - 45331 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$$.

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$1$$ $$\zeta_{10}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 0.809017 − 0.587785i −0.309017 + 0.951057i 0.809017 + 0.587785i −0.309017 − 0.951057i
2.92705 0.951057i −9.73607 7.07367i −5.28115 + 3.83698i −3.45492 + 10.6331i −35.2254 11.4454i 36.3820 + 50.0755i −40.7533 + 56.0921i 19.7239 + 60.7038i 34.4095i
41.1 −0.427051 0.587785i −5.26393 16.2007i 4.78115 14.7149i −9.04508 6.57164i −7.27458 + 10.0126i 38.6180 + 12.5478i −21.7467 + 7.06593i −169.224 + 122.948i 8.12299i
46.1 2.92705 + 0.951057i −9.73607 + 7.07367i −5.28115 3.83698i −3.45492 10.6331i −35.2254 + 11.4454i 36.3820 50.0755i −40.7533 56.0921i 19.7239 60.7038i 34.4095i
51.1 −0.427051 + 0.587785i −5.26393 + 16.2007i 4.78115 + 14.7149i −9.04508 + 6.57164i −7.27458 10.0126i 38.6180 12.5478i −21.7467 7.06593i −169.224 122.948i 8.12299i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.5.i.a 4
11.d odd 10 1 inner 55.5.i.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.5.i.a 4 1.a even 1 1 trivial
55.5.i.a 4 11.d odd 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 5T_{2}^{3} + 5T_{2}^{2} + 5T_{2} + 5$$ acting on $$S_{5}^{\mathrm{new}}(55, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 5 T^{3} + \cdots + 5$$
$3$ $$T^{4} + 30 T^{3} + \cdots + 42025$$
$5$ $$T^{4} + 25 T^{3} + \cdots + 15625$$
$7$ $$T^{4} - 150 T^{3} + \cdots + 6316880$$
$11$ $$T^{4} - 11 T^{3} + \cdots + 214358881$$
$13$ $$T^{4} + 90 T^{3} + 13122000$$
$17$ $$T^{4} + \cdots + 15374067005$$
$19$ $$T^{4} + \cdots + 11002271405$$
$23$ $$(T^{2} + 140 T - 211420)^{2}$$
$29$ $$T^{4} + \cdots + 1262209670480$$
$31$ $$T^{4} + \cdots + 8056139536$$
$37$ $$T^{4} + \cdots + 46751088400$$
$41$ $$T^{4} + \cdots + 17606617786805$$
$43$ $$T^{4} + \cdots + 1360442338205$$
$47$ $$T^{4} + \cdots + 3011196678400$$
$53$ $$T^{4} + \cdots + 32536756810000$$
$59$ $$T^{4} + \cdots + 132885308152921$$
$61$ $$T^{4} + \cdots + 1324868697680$$
$67$ $$(T^{2} + 10875 T + 24059155)^{2}$$
$71$ $$T^{4} + \cdots + 259640499396496$$
$73$ $$T^{4} + \cdots + 317879444778005$$
$79$ $$T^{4} + \cdots + 138787779399680$$
$83$ $$T^{4} + \cdots + 12\!\cdots\!05$$
$89$ $$(T^{2} - 20409 T + 83252359)^{2}$$
$97$ $$T^{4} + \cdots + 95\!\cdots\!25$$