# Properties

 Label 11.8.a.a Level $11$ Weight $8$ Character orbit 11.a Self dual yes Analytic conductor $3.436$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{15})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 15$$ x^2 - 15 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 4) q^{2} + ( - 6 \beta - 3) q^{3} + ( - 8 \beta - 52) q^{4} + (20 \beta - 235) q^{5} + (21 \beta - 348) q^{6} + (82 \beta - 614) q^{7} + ( - 148 \beta + 240) q^{8} + (36 \beta - 18) q^{9}+O(q^{10})$$ q + (b - 4) * q^2 + (-6*b - 3) * q^3 + (-8*b - 52) * q^4 + (20*b - 235) * q^5 + (21*b - 348) * q^6 + (82*b - 614) * q^7 + (-148*b + 240) * q^8 + (36*b - 18) * q^9 $$q + (\beta - 4) q^{2} + ( - 6 \beta - 3) q^{3} + ( - 8 \beta - 52) q^{4} + (20 \beta - 235) q^{5} + (21 \beta - 348) q^{6} + (82 \beta - 614) q^{7} + ( - 148 \beta + 240) q^{8} + (36 \beta - 18) q^{9} + ( - 315 \beta + 2140) q^{10} + 1331 q^{11} + (336 \beta + 3036) q^{12} + (518 \beta + 172) q^{13} + ( - 942 \beta + 7376) q^{14} + (1350 \beta - 6495) q^{15} + (1856 \beta - 3184) q^{16} + ( - 3666 \beta - 4234) q^{17} + ( - 162 \beta + 2232) q^{18} + ( - 2982 \beta - 17640) q^{19} + (840 \beta + 2620) q^{20} + (3438 \beta - 27678) q^{21} + (1331 \beta - 5324) q^{22} + (4290 \beta - 30743) q^{23} + ( - 996 \beta + 52560) q^{24} + ( - 9400 \beta + 1100) q^{25} + ( - 1900 \beta + 30392) q^{26} + (13122 \beta - 6345) q^{27} + (648 \beta - 7432) q^{28} + ( - 11468 \beta + 89520) q^{29} + ( - 11895 \beta + 106980) q^{30} + (20210 \beta - 28583) q^{31} + (8336 \beta + 93376) q^{32} + ( - 7986 \beta - 3993) q^{33} + (10430 \beta - 203024) q^{34} + ( - 31550 \beta + 242690) q^{35} + ( - 1728 \beta - 16344) q^{36} + ( - 3748 \beta - 438849) q^{37} + ( - 5712 \beta - 108360) q^{38} + ( - 2586 \beta - 186996) q^{39} + (39580 \beta - 234000) q^{40} + (68870 \beta - 141808) q^{41} + ( - 41430 \beta + 316992) q^{42} + (12760 \beta + 137742) q^{43} + ( - 10648 \beta - 69212) q^{44} + ( - 8820 \beta + 47430) q^{45} + ( - 47903 \beta + 380372) q^{46} + ( - 15252 \beta + 831256) q^{47} + (13536 \beta - 658608) q^{48} + ( - 100696 \beta - 43107) q^{49} + (38700 \beta - 568400) q^{50} + (36402 \beta + 1332462) q^{51} + ( - 28312 \beta - 257584) q^{52} + (66388 \beta + 808242) q^{53} + ( - 58833 \beta + 812700) q^{54} + (26620 \beta - 312785) q^{55} + (110552 \beta - 875520) q^{56} + (114786 \beta + 1126440) q^{57} + (135392 \beta - 1046160) q^{58} + ( - 147078 \beta - 1227065) q^{59} + ( - 18240 \beta - 310260) q^{60} + ( - 28900 \beta - 3009588) q^{61} + ( - 109423 \beta + 1326932) q^{62} + ( - 23580 \beta + 188172) q^{63} + ( - 177536 \beta + 534208) q^{64} + ( - 118290 \beta + 581180) q^{65} + (27951 \beta - 463188) q^{66} + ( - 392590 \beta - 87349) q^{67} + (224504 \beta + 1979848) q^{68} + (171588 \beta - 1452171) q^{69} + (368890 \beta - 2863760) q^{70} + (452890 \beta - 575733) q^{71} + (11304 \beta - 324000) q^{72} + ( - 195234 \beta + 442972) q^{73} + ( - 423857 \beta + 1530516) q^{74} + (21600 \beta + 3380700) q^{75} + (296184 \beta + 2348640) q^{76} + (109142 \beta - 817234) q^{77} + ( - 176652 \beta + 592824) q^{78} + (323896 \beta + 1900730) q^{79} + ( - 499840 \beta + 2975440) q^{80} + ( - 80028 \beta - 4665519) q^{81} + ( - 417288 \beta + 4699432) q^{82} + ( - 175068 \beta - 1141458) q^{83} + (42648 \beta - 210984) q^{84} + (776830 \beta - 3404210) q^{85} + (86702 \beta + 214632) q^{86} + ( - 502716 \beta + 3859920) q^{87} + ( - 196988 \beta + 319440) q^{88} + (201740 \beta - 6740985) q^{89} + (82710 \beta - 718920) q^{90} + ( - 303948 \beta + 2442952) q^{91} + (22864 \beta - 460564) q^{92} + (110868 \beta - 7189851) q^{93} + (892264 \beta - 4240144) q^{94} + (347970 \beta + 567000) q^{95} + ( - 585264 \beta - 3281088) q^{96} + (174936 \beta - 34039) q^{97} + (359677 \beta - 5869332) q^{98} + (47916 \beta - 23958) q^{99}+O(q^{100})$$ q + (b - 4) * q^2 + (-6*b - 3) * q^3 + (-8*b - 52) * q^4 + (20*b - 235) * q^5 + (21*b - 348) * q^6 + (82*b - 614) * q^7 + (-148*b + 240) * q^8 + (36*b - 18) * q^9 + (-315*b + 2140) * q^10 + 1331 * q^11 + (336*b + 3036) * q^12 + (518*b + 172) * q^13 + (-942*b + 7376) * q^14 + (1350*b - 6495) * q^15 + (1856*b - 3184) * q^16 + (-3666*b - 4234) * q^17 + (-162*b + 2232) * q^18 + (-2982*b - 17640) * q^19 + (840*b + 2620) * q^20 + (3438*b - 27678) * q^21 + (1331*b - 5324) * q^22 + (4290*b - 30743) * q^23 + (-996*b + 52560) * q^24 + (-9400*b + 1100) * q^25 + (-1900*b + 30392) * q^26 + (13122*b - 6345) * q^27 + (648*b - 7432) * q^28 + (-11468*b + 89520) * q^29 + (-11895*b + 106980) * q^30 + (20210*b - 28583) * q^31 + (8336*b + 93376) * q^32 + (-7986*b - 3993) * q^33 + (10430*b - 203024) * q^34 + (-31550*b + 242690) * q^35 + (-1728*b - 16344) * q^36 + (-3748*b - 438849) * q^37 + (-5712*b - 108360) * q^38 + (-2586*b - 186996) * q^39 + (39580*b - 234000) * q^40 + (68870*b - 141808) * q^41 + (-41430*b + 316992) * q^42 + (12760*b + 137742) * q^43 + (-10648*b - 69212) * q^44 + (-8820*b + 47430) * q^45 + (-47903*b + 380372) * q^46 + (-15252*b + 831256) * q^47 + (13536*b - 658608) * q^48 + (-100696*b - 43107) * q^49 + (38700*b - 568400) * q^50 + (36402*b + 1332462) * q^51 + (-28312*b - 257584) * q^52 + (66388*b + 808242) * q^53 + (-58833*b + 812700) * q^54 + (26620*b - 312785) * q^55 + (110552*b - 875520) * q^56 + (114786*b + 1126440) * q^57 + (135392*b - 1046160) * q^58 + (-147078*b - 1227065) * q^59 + (-18240*b - 310260) * q^60 + (-28900*b - 3009588) * q^61 + (-109423*b + 1326932) * q^62 + (-23580*b + 188172) * q^63 + (-177536*b + 534208) * q^64 + (-118290*b + 581180) * q^65 + (27951*b - 463188) * q^66 + (-392590*b - 87349) * q^67 + (224504*b + 1979848) * q^68 + (171588*b - 1452171) * q^69 + (368890*b - 2863760) * q^70 + (452890*b - 575733) * q^71 + (11304*b - 324000) * q^72 + (-195234*b + 442972) * q^73 + (-423857*b + 1530516) * q^74 + (21600*b + 3380700) * q^75 + (296184*b + 2348640) * q^76 + (109142*b - 817234) * q^77 + (-176652*b + 592824) * q^78 + (323896*b + 1900730) * q^79 + (-499840*b + 2975440) * q^80 + (-80028*b - 4665519) * q^81 + (-417288*b + 4699432) * q^82 + (-175068*b - 1141458) * q^83 + (42648*b - 210984) * q^84 + (776830*b - 3404210) * q^85 + (86702*b + 214632) * q^86 + (-502716*b + 3859920) * q^87 + (-196988*b + 319440) * q^88 + (201740*b - 6740985) * q^89 + (82710*b - 718920) * q^90 + (-303948*b + 2442952) * q^91 + (22864*b - 460564) * q^92 + (110868*b - 7189851) * q^93 + (892264*b - 4240144) * q^94 + (347970*b + 567000) * q^95 + (-585264*b - 3281088) * q^96 + (174936*b - 34039) * q^97 + (359677*b - 5869332) * q^98 + (47916*b - 23958) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{2} - 6 q^{3} - 104 q^{4} - 470 q^{5} - 696 q^{6} - 1228 q^{7} + 480 q^{8} - 36 q^{9}+O(q^{10})$$ 2 * q - 8 * q^2 - 6 * q^3 - 104 * q^4 - 470 * q^5 - 696 * q^6 - 1228 * q^7 + 480 * q^8 - 36 * q^9 $$2 q - 8 q^{2} - 6 q^{3} - 104 q^{4} - 470 q^{5} - 696 q^{6} - 1228 q^{7} + 480 q^{8} - 36 q^{9} + 4280 q^{10} + 2662 q^{11} + 6072 q^{12} + 344 q^{13} + 14752 q^{14} - 12990 q^{15} - 6368 q^{16} - 8468 q^{17} + 4464 q^{18} - 35280 q^{19} + 5240 q^{20} - 55356 q^{21} - 10648 q^{22} - 61486 q^{23} + 105120 q^{24} + 2200 q^{25} + 60784 q^{26} - 12690 q^{27} - 14864 q^{28} + 179040 q^{29} + 213960 q^{30} - 57166 q^{31} + 186752 q^{32} - 7986 q^{33} - 406048 q^{34} + 485380 q^{35} - 32688 q^{36} - 877698 q^{37} - 216720 q^{38} - 373992 q^{39} - 468000 q^{40} - 283616 q^{41} + 633984 q^{42} + 275484 q^{43} - 138424 q^{44} + 94860 q^{45} + 760744 q^{46} + 1662512 q^{47} - 1317216 q^{48} - 86214 q^{49} - 1136800 q^{50} + 2664924 q^{51} - 515168 q^{52} + 1616484 q^{53} + 1625400 q^{54} - 625570 q^{55} - 1751040 q^{56} + 2252880 q^{57} - 2092320 q^{58} - 2454130 q^{59} - 620520 q^{60} - 6019176 q^{61} + 2653864 q^{62} + 376344 q^{63} + 1068416 q^{64} + 1162360 q^{65} - 926376 q^{66} - 174698 q^{67} + 3959696 q^{68} - 2904342 q^{69} - 5727520 q^{70} - 1151466 q^{71} - 648000 q^{72} + 885944 q^{73} + 3061032 q^{74} + 6761400 q^{75} + 4697280 q^{76} - 1634468 q^{77} + 1185648 q^{78} + 3801460 q^{79} + 5950880 q^{80} - 9331038 q^{81} + 9398864 q^{82} - 2282916 q^{83} - 421968 q^{84} - 6808420 q^{85} + 429264 q^{86} + 7719840 q^{87} + 638880 q^{88} - 13481970 q^{89} - 1437840 q^{90} + 4885904 q^{91} - 921128 q^{92} - 14379702 q^{93} - 8480288 q^{94} + 1134000 q^{95} - 6562176 q^{96} - 68078 q^{97} - 11738664 q^{98} - 47916 q^{99}+O(q^{100})$$ 2 * q - 8 * q^2 - 6 * q^3 - 104 * q^4 - 470 * q^5 - 696 * q^6 - 1228 * q^7 + 480 * q^8 - 36 * q^9 + 4280 * q^10 + 2662 * q^11 + 6072 * q^12 + 344 * q^13 + 14752 * q^14 - 12990 * q^15 - 6368 * q^16 - 8468 * q^17 + 4464 * q^18 - 35280 * q^19 + 5240 * q^20 - 55356 * q^21 - 10648 * q^22 - 61486 * q^23 + 105120 * q^24 + 2200 * q^25 + 60784 * q^26 - 12690 * q^27 - 14864 * q^28 + 179040 * q^29 + 213960 * q^30 - 57166 * q^31 + 186752 * q^32 - 7986 * q^33 - 406048 * q^34 + 485380 * q^35 - 32688 * q^36 - 877698 * q^37 - 216720 * q^38 - 373992 * q^39 - 468000 * q^40 - 283616 * q^41 + 633984 * q^42 + 275484 * q^43 - 138424 * q^44 + 94860 * q^45 + 760744 * q^46 + 1662512 * q^47 - 1317216 * q^48 - 86214 * q^49 - 1136800 * q^50 + 2664924 * q^51 - 515168 * q^52 + 1616484 * q^53 + 1625400 * q^54 - 625570 * q^55 - 1751040 * q^56 + 2252880 * q^57 - 2092320 * q^58 - 2454130 * q^59 - 620520 * q^60 - 6019176 * q^61 + 2653864 * q^62 + 376344 * q^63 + 1068416 * q^64 + 1162360 * q^65 - 926376 * q^66 - 174698 * q^67 + 3959696 * q^68 - 2904342 * q^69 - 5727520 * q^70 - 1151466 * q^71 - 648000 * q^72 + 885944 * q^73 + 3061032 * q^74 + 6761400 * q^75 + 4697280 * q^76 - 1634468 * q^77 + 1185648 * q^78 + 3801460 * q^79 + 5950880 * q^80 - 9331038 * q^81 + 9398864 * q^82 - 2282916 * q^83 - 421968 * q^84 - 6808420 * q^85 + 429264 * q^86 + 7719840 * q^87 + 638880 * q^88 - 13481970 * q^89 - 1437840 * q^90 + 4885904 * q^91 - 921128 * q^92 - 14379702 * q^93 - 8480288 * q^94 + 1134000 * q^95 - 6562176 * q^96 - 68078 * q^97 - 11738664 * q^98 - 47916 * q^99

## 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
 −3.87298 3.87298
−11.7460 43.4758 9.96773 −389.919 −510.665 −1249.17 1386.40 −296.855 4579.98
1.2 3.74597 −49.4758 −113.968 −80.0807 −185.335 21.1693 −906.403 260.855 −299.980
 $$n$$: 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.a 2
3.b odd 2 1 99.8.a.c 2
4.b odd 2 1 176.8.a.d 2
5.b even 2 1 275.8.a.a 2
7.b odd 2 1 539.8.a.a 2
11.b odd 2 1 121.8.a.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 8T_{2} - 44$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(11))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 8T - 44$$
$3$ $$T^{2} + 6T - 2151$$
$5$ $$T^{2} + 470T + 31225$$
$7$ $$T^{2} + 1228T - 26444$$
$11$ $$(T - 1331)^{2}$$
$13$ $$T^{2} - 344 T - 16069856$$
$17$ $$T^{2} + 8468 T - 788446604$$
$19$ $$T^{2} + 35280 T - 222369840$$
$23$ $$T^{2} + 61486 T - 159113951$$
$29$ $$T^{2} - 179040 T + 122928960$$
$31$ $$T^{2} + 57166 T - 23689658111$$
$37$ $$T^{2} + 877698 T + 191745594561$$
$41$ $$T^{2} + 283616 T - 264475105136$$
$43$ $$T^{2} - 275484 T + 9203802564$$
$47$ $$T^{2} - 1662512 T + 677029127296$$
$53$ $$T^{2} - 1616484 T + 388813137924$$
$59$ $$T^{2} + 2454130 T + 207772229185$$
$61$ $$T^{2} + 6019176 T + 9007507329744$$
$67$ $$T^{2} + 174698 T - 9239984638199$$
$71$ $$T^{2} + 1151466 T - 11975092638711$$
$73$ $$T^{2} - 885944 T - 2090754692576$$
$79$ $$T^{2} - 3801460 T - 2681742596060$$
$83$ $$T^{2} + 2282916 T - 536001911676$$
$89$ $$T^{2} + 13481970 T + 42998937114225$$
$97$ $$T^{2} + 68078 T - 1834997592239$$