# Properties

 Label 11.8.a.b Level $11$ Weight $8$ Character orbit 11.a Self dual yes Analytic conductor $3.436$ Analytic rank $0$ 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] = [11,8,Mod(1,11)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(11, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("11.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 11.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: $$3.43623528033$$ Analytic rank: $$0$$ 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: $$x^{4} - x^{3} - 341x^{2} + 1417x - 1412$$ x^4 - x^3 - 341*x^2 + 1417*x - 1412 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{3} - \beta_{2} - \beta_1 - 9) q^{3} + (3 \beta_{3} - \beta_{2} + 4 \beta_1 + 152) q^{4} + (3 \beta_{3} - \beta_{2} - 5 \beta_1 + 133) q^{5} + ( - 10 \beta_{3} + 35 \beta_{2} + 10 \beta_1 + 392) q^{6} + ( - 18 \beta_{3} + 34 \beta_{2} - 18 \beta_1 + 38) q^{7} + (42 \beta_{3} - 120 \beta_{2} - 28 \beta_1 - 112) q^{8} + ( - 23 \beta_{3} + 61 \beta_{2} + 41 \beta_1 + 466) q^{9}+O(q^{10})$$ q - b2 * q^2 + (-b3 - b2 - b1 - 9) * q^3 + (3*b3 - b2 + 4*b1 + 152) * q^4 + (3*b3 - b2 - 5*b1 + 133) * q^5 + (-10*b3 + 35*b2 + 10*b1 + 392) * q^6 + (-18*b3 + 34*b2 - 18*b1 + 38) * q^7 + (42*b3 - 120*b2 - 28*b1 - 112) * q^8 + (-23*b3 + 61*b2 + 41*b1 + 466) * q^9 $$q - \beta_{2} q^{2} + ( - \beta_{3} - \beta_{2} - \beta_1 - 9) q^{3} + (3 \beta_{3} - \beta_{2} + 4 \beta_1 + 152) q^{4} + (3 \beta_{3} - \beta_{2} - 5 \beta_1 + 133) q^{5} + ( - 10 \beta_{3} + 35 \beta_{2} + 10 \beta_1 + 392) q^{6} + ( - 18 \beta_{3} + 34 \beta_{2} - 18 \beta_1 + 38) q^{7} + (42 \beta_{3} - 120 \beta_{2} - 28 \beta_1 - 112) q^{8} + ( - 23 \beta_{3} + 61 \beta_{2} + 41 \beta_1 + 466) q^{9} + (42 \beta_{3} - 103 \beta_{2} + 98 \beta_1 + 392) q^{10} - 1331 q^{11} + ( - 107 \beta_{3} - 239 \beta_{2} - 232 \beta_1 - 8648) q^{12} + (150 \beta_{3} + 250 \beta_{2} - 10 \beta_1 + 1060) q^{13} + ( - 336 \beta_{3} + 482 \beta_{2} - 28 \beta_1 - 7504) q^{14} + (11 \beta_{3} + 419 \beta_{2} + 3 \beta_1 + 1711) q^{15} + (522 \beta_{3} - 34 \beta_{2} + 696 \beta_1 + 13360) q^{16} + (348 \beta_{3} - 472 \beta_{2} + 188 \beta_1 + 13622) q^{17} + ( - 482 \beta_{3} - 680 \beta_{2} - 1002 \beta_1 - 18088) q^{18} + ( - 660 \beta_{3} + 680 \beta_{2} - 100 \beta_1 + 16936) q^{19} + (471 \beta_{3} - 2285 \beta_{2} + 16 \beta_1 + 3976) q^{20} + ( - 94 \beta_{3} - 922 \beta_{2} + 18 \beta_1 + 25570) q^{21} + 1331 \beta_{2} q^{22} + ( - 789 \beta_{3} - 3029 \beta_{2} + 211 \beta_1 - 2201) q^{23} + (606 \beta_{3} + 8568 \beta_{2} + 2068 \beta_1 + 35728) q^{24} + (765 \beta_{3} + 2153 \beta_{2} - 2211 \beta_1 + 5080) q^{25} + (1200 \beta_{3} - 2620 \beta_{2} + 340 \beta_1 - 77840) q^{26} + (1121 \beta_{3} - 4567 \beta_{2} + 81 \beta_1 - 30731) q^{27} + ( - 3510 \beta_{3} + 8394 \beta_{2} - 1920 \beta_1 - 119440) q^{28} + (66 \beta_{3} - 1662 \beta_{2} + 3778 \beta_1 + 59464) q^{29} + ( - 1114 \beta_{3} - 1477 \beta_{2} - 1630 \beta_1 - 118104) q^{30} + ( - 1533 \beta_{3} - 3389 \beta_{2} - 709 \beta_1 + 47287) q^{31} + (1512 \beta_{3} - 14564 \beta_{2} - 1848 \beta_1 - 44352) q^{32} + (1331 \beta_{3} + 1331 \beta_{2} + 1331 \beta_1 + 11979) q^{33} + (5940 \beta_{3} - 21250 \beta_{2} + 2040 \beta_1 + 102144) q^{34} + ( - 1386 \beta_{3} + 12698 \beta_{2} - 6042 \beta_1 + 37014) q^{35} + ( - 1282 \beta_{3} + 29894 \beta_{2} + 7644 \beta_1 + 213856) q^{36} + ( - 3555 \beta_{3} + 14753 \beta_{2} + 4485 \beta_1 + 33095) q^{37} + ( - 10620 \beta_{3} - 6276 \beta_{2} - 6600 \beta_1 - 147840) q^{38} + (6500 \beta_{3} - 1000 \beta_{2} - 3020 \beta_1 - 253420) q^{39} + (7602 \beta_{3} + 576 \beta_{2} + 140 \beta_1 + 562352) q^{40} + (7374 \beta_{3} + 8162 \beta_{2} + 1582 \beta_1 + 72856) q^{41} + (1544 \beta_{3} - 25522 \beta_{2} + 2684 \beta_1 + 262416) q^{42} + ( - 12282 \beta_{3} + 8126 \beta_{2} - 1082 \beta_1 + 175962) q^{43} + ( - 3993 \beta_{3} + 1331 \beta_{2} - 5324 \beta_1 - 202312) q^{44} + ( - 3752 \beta_{3} - 13808 \beta_{2} + 4984 \beta_1 - 485442) q^{45} + ( - 1170 \beta_{3} + 6475 \beta_{2} + 2850 \beta_1 + 880488) q^{46} + (6312 \beta_{3} - 10160 \beta_{2} + 6712 \beta_1 - 430592) q^{47} + ( - 4130 \beta_{3} - 33398 \beta_{2} - 28680 \beta_1 - 1441840) q^{48} + (12180 \beta_{3} - 34860 \beta_{2} - 1820 \beta_1 + 34525) q^{49} + (3486 \beta_{3} + 18082 \beta_{2} + 28462 \beta_1 - 521864) q^{50} + ( - 8926 \beta_{3} + 8342 \beta_{2} - 12718 \beta_1 - 515094) q^{51} + (4260 \beta_{3} + 22860 \beta_{2} + 16600 \beta_1 + 511680) q^{52} + ( - 2220 \beta_{3} + 50316 \beta_{2} - 7580 \beta_1 + 272770) q^{53} + (28274 \beta_{3} + 10457 \beta_{2} + 26102 \beta_1 + 1211448) q^{54} + ( - 3993 \beta_{3} + 1331 \beta_{2} + 6655 \beta_1 - 177023) q^{55} + ( - 27804 \beta_{3} + 138648 \beta_{2} - 31192 \beta_1 - 1085728) q^{56} + ( - 30816 \beta_{3} - 69276 \beta_{2} - 22656 \beta_1 + 385896) q^{57} + (5844 \beta_{3} - 114876 \beta_{2} - 45716 \beta_1 + 250096) q^{58} + (25797 \beta_{3} - 4979 \beta_{2} + 36901 \beta_1 - 1157219) q^{59} + ( - 11459 \beta_{3} + 100297 \beta_{2} + 19432 \beta_1 + 348216) q^{60} + (11646 \beta_{3} - 56962 \beta_{2} - 3522 \beta_1 + 76772) q^{61} + ( - 9762 \beta_{3} - 20821 \beta_{2} + 11218 \beta_1 + 1074472) q^{62} + (1276 \beta_{3} - 74012 \beta_{2} + 22396 \beta_1 + 126324) q^{63} + ( - 3468 \beta_{3} + 40356 \beta_{2} + 7136 \beta_1 + 2386656) q^{64} + ( - 480 \beta_{3} + 26300 \beta_{2} - 50080 \beta_1 + 1119020) q^{65} + (13310 \beta_{3} - 46585 \beta_{2} - 13310 \beta_1 - 521752) q^{66} + (23241 \beta_{3} - 36015 \beta_{2} - 3239 \beta_1 - 842871) q^{67} + (96426 \beta_{3} - 168758 \beta_{2} + 79896 \beta_1 + 3759504) q^{68} + ( - 56447 \beta_{3} + 50629 \beta_{2} + 25849 \beta_1 + 1795197) q^{69} + ( - 56112 \beta_{3} + 78290 \beta_{2} + 22708 \beta_1 - 3139472) q^{70} + (27849 \beta_{3} + 178977 \beta_{2} + 12817 \beta_1 - 850931) q^{71} + ( - 44652 \beta_{3} - 187272 \beta_{2} - 108592 \beta_1 - 6411328) q^{72} + ( - 28806 \beta_{3} + 128398 \beta_{2} - 15206 \beta_1 + 2862756) q^{73} + ( - 90474 \beta_{3} - 34917 \beta_{2} - 150242 \beta_1 - 4182920) q^{74} + (66016 \beta_{3} + 117856 \beta_{2} + 27984 \beta_1 + 811436) q^{75} + ( - 34752 \beta_{3} + 284984 \beta_{2} + 45344 \beta_1 + 553792) q^{76} + (23958 \beta_{3} - 45254 \beta_{2} + 23958 \beta_1 - 50578) q^{77} + (87500 \beta_{3} + 210200 \beta_{2} + 98280 \beta_1 + 85120) q^{78} + ( - 44706 \beta_{3} - 153738 \beta_{2} - 32818 \beta_1 + 133366) q^{79} + (36810 \beta_{3} - 370082 \beta_{2} + 54504 \beta_1 - 1103760) q^{80} + (71464 \beta_{3} + 111952 \beta_{2} - 12968 \beta_1 - 218799) q^{81} + (71376 \beta_{3} - 182704 \beta_{2} + 4196 \beta_1 - 2786896) q^{82} + ( - 84258 \beta_{3} + 153630 \beta_{2} + 75102 \beta_1 - 1036422) q^{83} + (108670 \beta_{3} - 227570 \beta_{2} + 74560 \beta_1 + 3636432) q^{84} + (68322 \beta_{3} - 162202 \beta_{2} - 14414 \beta_1 + 2574622) q^{85} + ( - 184044 \beta_{3} + 6978 \beta_{2} - 115612 \beta_1 - 1526896) q^{86} + ( - 115464 \beta_{3} - 265968 \beta_{2} - 133384 \beta_1 - 4229536) q^{87} + ( - 55902 \beta_{3} + 159720 \beta_{2} + 37268 \beta_1 + 149072) q^{88} + ( - 78267 \beta_{3} - 141351 \beta_{2} + 34469 \beta_1 + 4574915) q^{89} + ( - 7352 \beta_{3} + 450634 \beta_{2} - 44560 \beta_1 + 3797248) q^{90} + (84600 \beta_{3} + 168240 \beta_{2} - 74040 \beta_1 - 301880) q^{91} + (66357 \beta_{3} - 510991 \beta_{2} - 102168 \beta_1 - 1625352) q^{92} + ( - 123967 \beta_{3} + 38885 \beta_{2} + 553 \beta_1 + 3312077) q^{93} + (112536 \beta_{3} + 244408 \beta_{2} - 2832 \beta_1 + 2115456) q^{94} + ( - 2232 \beta_{3} + 116724 \beta_{2} - 99240 \beta_1 - 1440552) q^{95} + ( - 31064 \beta_{3} + 766948 \beta_{2} + 237368 \beta_1 + 6615616) q^{96} + (149319 \beta_{3} - 125589 \beta_{2} + 234279 \beta_1 + 2914901) q^{97} + (262920 \beta_{3} - 202245 \beta_{2} + 262360 \beta_1 + 9180640) q^{98} + (30613 \beta_{3} - 81191 \beta_{2} - 54571 \beta_1 - 620246) q^{99}+O(q^{100})$$ q - b2 * q^2 + (-b3 - b2 - b1 - 9) * q^3 + (3*b3 - b2 + 4*b1 + 152) * q^4 + (3*b3 - b2 - 5*b1 + 133) * q^5 + (-10*b3 + 35*b2 + 10*b1 + 392) * q^6 + (-18*b3 + 34*b2 - 18*b1 + 38) * q^7 + (42*b3 - 120*b2 - 28*b1 - 112) * q^8 + (-23*b3 + 61*b2 + 41*b1 + 466) * q^9 + (42*b3 - 103*b2 + 98*b1 + 392) * q^10 - 1331 * q^11 + (-107*b3 - 239*b2 - 232*b1 - 8648) * q^12 + (150*b3 + 250*b2 - 10*b1 + 1060) * q^13 + (-336*b3 + 482*b2 - 28*b1 - 7504) * q^14 + (11*b3 + 419*b2 + 3*b1 + 1711) * q^15 + (522*b3 - 34*b2 + 696*b1 + 13360) * q^16 + (348*b3 - 472*b2 + 188*b1 + 13622) * q^17 + (-482*b3 - 680*b2 - 1002*b1 - 18088) * q^18 + (-660*b3 + 680*b2 - 100*b1 + 16936) * q^19 + (471*b3 - 2285*b2 + 16*b1 + 3976) * q^20 + (-94*b3 - 922*b2 + 18*b1 + 25570) * q^21 + 1331*b2 * q^22 + (-789*b3 - 3029*b2 + 211*b1 - 2201) * q^23 + (606*b3 + 8568*b2 + 2068*b1 + 35728) * q^24 + (765*b3 + 2153*b2 - 2211*b1 + 5080) * q^25 + (1200*b3 - 2620*b2 + 340*b1 - 77840) * q^26 + (1121*b3 - 4567*b2 + 81*b1 - 30731) * q^27 + (-3510*b3 + 8394*b2 - 1920*b1 - 119440) * q^28 + (66*b3 - 1662*b2 + 3778*b1 + 59464) * q^29 + (-1114*b3 - 1477*b2 - 1630*b1 - 118104) * q^30 + (-1533*b3 - 3389*b2 - 709*b1 + 47287) * q^31 + (1512*b3 - 14564*b2 - 1848*b1 - 44352) * q^32 + (1331*b3 + 1331*b2 + 1331*b1 + 11979) * q^33 + (5940*b3 - 21250*b2 + 2040*b1 + 102144) * q^34 + (-1386*b3 + 12698*b2 - 6042*b1 + 37014) * q^35 + (-1282*b3 + 29894*b2 + 7644*b1 + 213856) * q^36 + (-3555*b3 + 14753*b2 + 4485*b1 + 33095) * q^37 + (-10620*b3 - 6276*b2 - 6600*b1 - 147840) * q^38 + (6500*b3 - 1000*b2 - 3020*b1 - 253420) * q^39 + (7602*b3 + 576*b2 + 140*b1 + 562352) * q^40 + (7374*b3 + 8162*b2 + 1582*b1 + 72856) * q^41 + (1544*b3 - 25522*b2 + 2684*b1 + 262416) * q^42 + (-12282*b3 + 8126*b2 - 1082*b1 + 175962) * q^43 + (-3993*b3 + 1331*b2 - 5324*b1 - 202312) * q^44 + (-3752*b3 - 13808*b2 + 4984*b1 - 485442) * q^45 + (-1170*b3 + 6475*b2 + 2850*b1 + 880488) * q^46 + (6312*b3 - 10160*b2 + 6712*b1 - 430592) * q^47 + (-4130*b3 - 33398*b2 - 28680*b1 - 1441840) * q^48 + (12180*b3 - 34860*b2 - 1820*b1 + 34525) * q^49 + (3486*b3 + 18082*b2 + 28462*b1 - 521864) * q^50 + (-8926*b3 + 8342*b2 - 12718*b1 - 515094) * q^51 + (4260*b3 + 22860*b2 + 16600*b1 + 511680) * q^52 + (-2220*b3 + 50316*b2 - 7580*b1 + 272770) * q^53 + (28274*b3 + 10457*b2 + 26102*b1 + 1211448) * q^54 + (-3993*b3 + 1331*b2 + 6655*b1 - 177023) * q^55 + (-27804*b3 + 138648*b2 - 31192*b1 - 1085728) * q^56 + (-30816*b3 - 69276*b2 - 22656*b1 + 385896) * q^57 + (5844*b3 - 114876*b2 - 45716*b1 + 250096) * q^58 + (25797*b3 - 4979*b2 + 36901*b1 - 1157219) * q^59 + (-11459*b3 + 100297*b2 + 19432*b1 + 348216) * q^60 + (11646*b3 - 56962*b2 - 3522*b1 + 76772) * q^61 + (-9762*b3 - 20821*b2 + 11218*b1 + 1074472) * q^62 + (1276*b3 - 74012*b2 + 22396*b1 + 126324) * q^63 + (-3468*b3 + 40356*b2 + 7136*b1 + 2386656) * q^64 + (-480*b3 + 26300*b2 - 50080*b1 + 1119020) * q^65 + (13310*b3 - 46585*b2 - 13310*b1 - 521752) * q^66 + (23241*b3 - 36015*b2 - 3239*b1 - 842871) * q^67 + (96426*b3 - 168758*b2 + 79896*b1 + 3759504) * q^68 + (-56447*b3 + 50629*b2 + 25849*b1 + 1795197) * q^69 + (-56112*b3 + 78290*b2 + 22708*b1 - 3139472) * q^70 + (27849*b3 + 178977*b2 + 12817*b1 - 850931) * q^71 + (-44652*b3 - 187272*b2 - 108592*b1 - 6411328) * q^72 + (-28806*b3 + 128398*b2 - 15206*b1 + 2862756) * q^73 + (-90474*b3 - 34917*b2 - 150242*b1 - 4182920) * q^74 + (66016*b3 + 117856*b2 + 27984*b1 + 811436) * q^75 + (-34752*b3 + 284984*b2 + 45344*b1 + 553792) * q^76 + (23958*b3 - 45254*b2 + 23958*b1 - 50578) * q^77 + (87500*b3 + 210200*b2 + 98280*b1 + 85120) * q^78 + (-44706*b3 - 153738*b2 - 32818*b1 + 133366) * q^79 + (36810*b3 - 370082*b2 + 54504*b1 - 1103760) * q^80 + (71464*b3 + 111952*b2 - 12968*b1 - 218799) * q^81 + (71376*b3 - 182704*b2 + 4196*b1 - 2786896) * q^82 + (-84258*b3 + 153630*b2 + 75102*b1 - 1036422) * q^83 + (108670*b3 - 227570*b2 + 74560*b1 + 3636432) * q^84 + (68322*b3 - 162202*b2 - 14414*b1 + 2574622) * q^85 + (-184044*b3 + 6978*b2 - 115612*b1 - 1526896) * q^86 + (-115464*b3 - 265968*b2 - 133384*b1 - 4229536) * q^87 + (-55902*b3 + 159720*b2 + 37268*b1 + 149072) * q^88 + (-78267*b3 - 141351*b2 + 34469*b1 + 4574915) * q^89 + (-7352*b3 + 450634*b2 - 44560*b1 + 3797248) * q^90 + (84600*b3 + 168240*b2 - 74040*b1 - 301880) * q^91 + (66357*b3 - 510991*b2 - 102168*b1 - 1625352) * q^92 + (-123967*b3 + 38885*b2 + 553*b1 + 3312077) * q^93 + (112536*b3 + 244408*b2 - 2832*b1 + 2115456) * q^94 + (-2232*b3 + 116724*b2 - 99240*b1 - 1440552) * q^95 + (-31064*b3 + 766948*b2 + 237368*b1 + 6615616) * q^96 + (149319*b3 - 125589*b2 + 234279*b1 + 2914901) * q^97 + (262920*b3 - 202245*b2 + 262360*b1 + 9180640) * q^98 + (30613*b3 - 81191*b2 - 54571*b1 - 620246) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 35 q^{3} + 604 q^{4} + 537 q^{5} + 1558 q^{6} + 170 q^{7} - 420 q^{8} + 1823 q^{9}+O(q^{10})$$ 4 * q - 35 * q^3 + 604 * q^4 + 537 * q^5 + 1558 * q^6 + 170 * q^7 - 420 * q^8 + 1823 * q^9 $$4 q - 35 q^{3} + 604 q^{4} + 537 q^{5} + 1558 q^{6} + 170 q^{7} - 420 q^{8} + 1823 q^{9} + 1470 q^{10} - 5324 q^{11} - 34360 q^{12} + 4250 q^{13} - 29988 q^{14} + 6841 q^{15} + 52744 q^{16} + 54300 q^{17} - 71350 q^{18} + 67844 q^{19} + 15888 q^{20} + 102262 q^{21} - 9015 q^{23} + 140844 q^{24} + 22531 q^{25} - 311700 q^{26} - 123005 q^{27} - 475840 q^{28} + 234078 q^{29} - 470786 q^{30} + 189857 q^{31} - 175560 q^{32} + 46585 q^{33} + 406536 q^{34} + 154098 q^{35} + 847780 q^{36} + 127895 q^{37} - 584760 q^{38} - 1010660 q^{39} + 2249268 q^{40} + 289842 q^{41} + 1046980 q^{42} + 704930 q^{43} - 803924 q^{44} - 1946752 q^{45} + 3519102 q^{46} - 1729080 q^{47} - 5738680 q^{48} + 139920 q^{49} - 2115918 q^{50} - 2047658 q^{51} + 2030120 q^{52} + 1098660 q^{53} + 4819690 q^{54} - 714747 q^{55} - 4311720 q^{56} + 1566240 q^{57} + 1046100 q^{58} - 4665777 q^{59} + 1373432 q^{60} + 310610 q^{61} + 4286670 q^{62} + 482900 q^{63} + 9539488 q^{64} + 4526160 q^{65} - 2073698 q^{66} - 3368245 q^{67} + 14958120 q^{68} + 7154939 q^{69} - 12580596 q^{70} - 3416541 q^{71} - 25536720 q^{72} + 11466230 q^{73} - 16581438 q^{74} + 3217760 q^{75} + 2169824 q^{76} - 226270 q^{77} + 242200 q^{78} + 566282 q^{79} - 4469544 q^{80} - 862228 q^{81} - 11151780 q^{82} - 4220790 q^{83} + 14471168 q^{84} + 10312902 q^{85} - 5991972 q^{86} - 16784760 q^{87} + 559020 q^{88} + 18265191 q^{89} + 15233552 q^{90} - 1133480 q^{91} - 6399240 q^{92} + 13247755 q^{93} + 8464656 q^{94} - 5662968 q^{95} + 26225096 q^{96} + 11425325 q^{97} + 36460200 q^{98} - 2426413 q^{99}+O(q^{100})$$ 4 * q - 35 * q^3 + 604 * q^4 + 537 * q^5 + 1558 * q^6 + 170 * q^7 - 420 * q^8 + 1823 * q^9 + 1470 * q^10 - 5324 * q^11 - 34360 * q^12 + 4250 * q^13 - 29988 * q^14 + 6841 * q^15 + 52744 * q^16 + 54300 * q^17 - 71350 * q^18 + 67844 * q^19 + 15888 * q^20 + 102262 * q^21 - 9015 * q^23 + 140844 * q^24 + 22531 * q^25 - 311700 * q^26 - 123005 * q^27 - 475840 * q^28 + 234078 * q^29 - 470786 * q^30 + 189857 * q^31 - 175560 * q^32 + 46585 * q^33 + 406536 * q^34 + 154098 * q^35 + 847780 * q^36 + 127895 * q^37 - 584760 * q^38 - 1010660 * q^39 + 2249268 * q^40 + 289842 * q^41 + 1046980 * q^42 + 704930 * q^43 - 803924 * q^44 - 1946752 * q^45 + 3519102 * q^46 - 1729080 * q^47 - 5738680 * q^48 + 139920 * q^49 - 2115918 * q^50 - 2047658 * q^51 + 2030120 * q^52 + 1098660 * q^53 + 4819690 * q^54 - 714747 * q^55 - 4311720 * q^56 + 1566240 * q^57 + 1046100 * q^58 - 4665777 * q^59 + 1373432 * q^60 + 310610 * q^61 + 4286670 * q^62 + 482900 * q^63 + 9539488 * q^64 + 4526160 * q^65 - 2073698 * q^66 - 3368245 * q^67 + 14958120 * q^68 + 7154939 * q^69 - 12580596 * q^70 - 3416541 * q^71 - 25536720 * q^72 + 11466230 * q^73 - 16581438 * q^74 + 3217760 * q^75 + 2169824 * q^76 - 226270 * q^77 + 242200 * q^78 + 566282 * q^79 - 4469544 * q^80 - 862228 * q^81 - 11151780 * q^82 - 4220790 * q^83 + 14471168 * q^84 + 10312902 * q^85 - 5991972 * q^86 - 16784760 * q^87 + 559020 * q^88 + 18265191 * q^89 + 15233552 * q^90 - 1133480 * q^91 - 6399240 * q^92 + 13247755 * q^93 + 8464656 * q^94 - 5662968 * q^95 + 26225096 * q^96 + 11425325 * q^97 + 36460200 * q^98 - 2426413 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 341x^{2} + 1417x - 1412$$ :

 $$\beta_{1}$$ $$=$$ $$3\nu - 1$$ 3*v - 1 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 321\nu + 204 ) / 28$$ (v^3 + 4*v^2 - 321*v + 204) / 28 $$\beta_{3}$$ $$=$$ $$( -9\nu^{3} - 8\nu^{2} + 3029\nu - 6652 ) / 28$$ (-9*v^3 - 8*v^2 + 3029*v - 6652) / 28
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 3$$ (b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( 3\beta_{3} + 27\beta_{2} - 5\beta _1 + 511 ) / 3$$ (3*b3 + 27*b2 - 5*b1 + 511) / 3 $$\nu^{3}$$ $$=$$ $$( -12\beta_{3} - 24\beta_{2} + 341\beta _1 - 2335 ) / 3$$ (-12*b3 - 24*b2 + 341*b1 - 2335) / 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.

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
 16.6649 −19.8969 1.64802 2.58394
−21.2011 −77.4315 321.485 −138.468 1641.63 −91.3115 −4102.09 3808.63 2935.68
1.2 −10.6261 12.2971 −15.0863 512.130 −130.670 973.904 1520.45 −2035.78 −5441.94
1.3 11.0598 59.6211 −5.68000 −60.1766 659.400 698.069 −1478.48 1367.68 −665.544
1.4 20.7673 −29.4867 303.281 223.515 −612.360 −1410.66 3640.12 −1317.53 4641.80
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$1$$

## 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 11.8.a.b 4
3.b odd 2 1 99.8.a.g 4
4.b odd 2 1 176.8.a.j 4
5.b even 2 1 275.8.a.b 4
7.b odd 2 1 539.8.a.b 4
11.b odd 2 1 121.8.a.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.8.a.b 4 1.a even 1 1 trivial
99.8.a.g 4 3.b odd 2 1
121.8.a.c 4 11.b odd 2 1
176.8.a.j 4 4.b odd 2 1
275.8.a.b 4 5.b even 2 1
539.8.a.b 4 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 558T_{2}^{2} + 140T_{2} + 51744$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(11))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 558 T^{2} + 140 T + 51744$$
$3$ $$T^{4} + 35 T^{3} - 4673 T^{2} + \cdots + 1673964$$
$5$ $$T^{4} - 537 T^{3} + \cdots + 953818350$$
$7$ $$T^{4} - 170 T^{3} + \cdots + 87571440704$$
$11$ $$(T + 1331)^{4}$$
$13$ $$T^{4} + \cdots - 518474088880000$$
$17$ $$T^{4} - 54300 T^{3} + \cdots - 58\!\cdots\!56$$
$19$ $$T^{4} - 67844 T^{3} + \cdots - 31\!\cdots\!84$$
$23$ $$T^{4} + 9015 T^{3} + \cdots + 10\!\cdots\!44$$
$29$ $$T^{4} - 234078 T^{3} + \cdots - 41\!\cdots\!64$$
$31$ $$T^{4} - 189857 T^{3} + \cdots - 61\!\cdots\!36$$
$37$ $$T^{4} - 127895 T^{3} + \cdots + 41\!\cdots\!94$$
$41$ $$T^{4} - 289842 T^{3} + \cdots - 66\!\cdots\!84$$
$43$ $$T^{4} - 704930 T^{3} + \cdots + 28\!\cdots\!64$$
$47$ $$T^{4} + 1729080 T^{3} + \cdots - 74\!\cdots\!04$$
$53$ $$T^{4} - 1098660 T^{3} + \cdots + 18\!\cdots\!84$$
$59$ $$T^{4} + 4665777 T^{3} + \cdots + 29\!\cdots\!64$$
$61$ $$T^{4} - 310610 T^{3} + \cdots + 10\!\cdots\!84$$
$67$ $$T^{4} + 3368245 T^{3} + \cdots - 84\!\cdots\!64$$
$71$ $$T^{4} + 3416541 T^{3} + \cdots + 59\!\cdots\!44$$
$73$ $$T^{4} - 11466230 T^{3} + \cdots - 60\!\cdots\!76$$
$79$ $$T^{4} - 566282 T^{3} + \cdots + 25\!\cdots\!16$$
$83$ $$T^{4} + 4220790 T^{3} + \cdots + 70\!\cdots\!96$$
$89$ $$T^{4} - 18265191 T^{3} + \cdots - 24\!\cdots\!34$$
$97$ $$T^{4} - 11425325 T^{3} + \cdots + 46\!\cdots\!46$$