# Properties

 Label 11.10.a.a Level $11$ Weight $10$ Character orbit 11.a Self dual yes Analytic conductor $5.665$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [11,10,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, 10, names="a")

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

chi := DirichletCharacter("11.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$10$$ 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: $$5.66539419780$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.2659452.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 306x - 836$$ x^3 - 306*x - 836 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + ( - \beta_{2} + 4 \beta_1 - 62) q^{3} + (8 \beta_{2} + 12 \beta_1 + 304) q^{4} + ( - 3 \beta_{2} + 34 \beta_1 - 608) q^{5} + ( - 44 \beta_{2} + 47 \beta_1 - 3652) q^{6} + (22 \beta_{2} - 122 \beta_1 - 2420) q^{7} + ( - 200 \beta_1 - 6688) q^{8} + (315 \beta_{2} - 666 \beta_1 + 4851) q^{9}+O(q^{10})$$ q - b1 * q^2 + (-b2 + 4*b1 - 62) * q^3 + (8*b2 + 12*b1 + 304) * q^4 + (-3*b2 + 34*b1 - 608) * q^5 + (-44*b2 + 47*b1 - 3652) * q^6 + (22*b2 - 122*b1 - 2420) * q^7 + (-200*b1 - 6688) * q^8 + (315*b2 - 666*b1 + 4851) * q^9 $$q - \beta_1 q^{2} + ( - \beta_{2} + 4 \beta_1 - 62) q^{3} + (8 \beta_{2} + 12 \beta_1 + 304) q^{4} + ( - 3 \beta_{2} + 34 \beta_1 - 608) q^{5} + ( - 44 \beta_{2} + 47 \beta_1 - 3652) q^{6} + (22 \beta_{2} - 122 \beta_1 - 2420) q^{7} + ( - 200 \beta_1 - 6688) q^{8} + (315 \beta_{2} - 666 \beta_1 + 4851) q^{9} + ( - 308 \beta_{2} + 299 \beta_1 - 28908) q^{10} - 14641 q^{11} + ( - 392 \beta_{2} + 2492 \beta_1 - 23680) q^{12} + ( - 726 \beta_{2} + 538 \beta_1 - 31086) q^{13} + (1240 \beta_{2} + 3158 \beta_1 + 108088) q^{14} + (2335 \beta_{2} - 4720 \beta_1 + 180110) q^{15} + ( - 2496 \beta_{2} + 2944 \beta_1 + 7552) q^{16} + ( - 2156 \beta_{2} - 9154 \beta_1 + 6226) q^{17} + (9108 \beta_{2} - 7254 \beta_1 + 665676) q^{18} + ( - 7876 \beta_{2} - 13262 \beta_1 - 342452) q^{19} + ( - 4552 \beta_{2} + 18076 \beta_1 - 52192) q^{20} + ( - 4642 \beta_{2} + 1114 \beta_1 - 429308) q^{21} + 14641 \beta_1 q^{22} + (23639 \beta_{2} + 39292 \beta_1 + 558230) q^{23} + ( - 2112 \beta_{2} - 17352 \beta_1 - 315744) q^{24} + (15047 \beta_{2} - 35086 \beta_1 - 520243) q^{25} + ( - 13016 \beta_{2} + 48588 \beta_1 - 720696) q^{26} + ( - 38727 \beta_{2} + 44424 \beta_1 - 3428478) q^{27} + ( - 21648 \beta_{2} - 124440 \beta_1 - 856768) q^{28} + (15422 \beta_{2} + 56616 \beta_1 - 897886) q^{29} + (65780 \beta_{2} - 200525 \beta_1 + 4757500) q^{30} + (4231 \beta_{2} + 37548 \beta_1 - 1508434) q^{31} + ( - 53504 \beta_{2} + 141888 \beta_1 + 53504) q^{32} + (14641 \beta_{2} - 58564 \beta_1 + 907742) q^{33} + (47360 \beta_{2} + 174770 \beta_1 + 6633136) q^{34} + ( - 50490 \beta_{2} - 14650 \beta_1 - 2644620) q^{35} + (6048 \beta_{2} - 538200 \beta_1 + 6969456) q^{36} + ( - 73373 \beta_{2} + 181182 \beta_1 - 2940068) q^{37} + (11584 \beta_{2} + 761504 \beta_1 + 7765904) q^{38} + (110660 \beta_{2} - 316610 \beta_1 + 8307640) q^{39} + ( - 41536 \beta_{2} - 167592 \beta_1 - 1715296) q^{40} + ( - 13046 \beta_{2} + 147690 \beta_1 - 3257034) q^{41} + ( - 64616 \beta_{2} + 569126 \beta_1 - 2710120) q^{42} + (409750 \beta_{2} - 61556 \beta_1 + 6265248) q^{43} + ( - 117128 \beta_{2} - 175692 \beta_1 - 4450864) q^{44} + ( - 508536 \beta_{2} + 810108 \beta_1 - 30638466) q^{45} + ( - 30668 \beta_{2} - 1809821 \beta_1 - 22890340) q^{46} + ( - 45640 \beta_{2} - 1380980 \beta_1 - 10385272) q^{47} + (314176 \beta_{2} - 682240 \beta_1 + 25463936) q^{48} + (61036 \beta_{2} + 544512 \beta_1 - 18076559) q^{49} + (461252 \beta_{2} + 444724 \beta_1 + 34468412) q^{50} + ( - 242990 \beta_{2} - 40738 \beta_1 - 20703628) q^{51} + ( - 173184 \beta_{2} + 291712 \beta_1 - 28781984) q^{52} + (450204 \beta_{2} + 1561532 \beta_1 + 15833374) q^{53} + ( - 820116 \beta_{2} + 4173381 \beta_1 - 51276060) q^{54} + (43923 \beta_{2} - 497794 \beta_1 + 8901728) q^{55} + (100864 \beta_{2} + 1447536 \beta_1 + 37802560) q^{56} + (365376 \beta_{2} - 2557974 \beta_1 + 20701032) q^{57} + ( - 267864 \beta_{2} - 290432 \beta_1 - 40214920) q^{58} + ( - 264531 \beta_{2} + \cdots + 110712790) q^{59}+ \cdots + ( - 4611915 \beta_{2} + \cdots - 71023491) q^{99}+O(q^{100})$$ q - b1 * q^2 + (-b2 + 4*b1 - 62) * q^3 + (8*b2 + 12*b1 + 304) * q^4 + (-3*b2 + 34*b1 - 608) * q^5 + (-44*b2 + 47*b1 - 3652) * q^6 + (22*b2 - 122*b1 - 2420) * q^7 + (-200*b1 - 6688) * q^8 + (315*b2 - 666*b1 + 4851) * q^9 + (-308*b2 + 299*b1 - 28908) * q^10 - 14641 * q^11 + (-392*b2 + 2492*b1 - 23680) * q^12 + (-726*b2 + 538*b1 - 31086) * q^13 + (1240*b2 + 3158*b1 + 108088) * q^14 + (2335*b2 - 4720*b1 + 180110) * q^15 + (-2496*b2 + 2944*b1 + 7552) * q^16 + (-2156*b2 - 9154*b1 + 6226) * q^17 + (9108*b2 - 7254*b1 + 665676) * q^18 + (-7876*b2 - 13262*b1 - 342452) * q^19 + (-4552*b2 + 18076*b1 - 52192) * q^20 + (-4642*b2 + 1114*b1 - 429308) * q^21 + 14641*b1 * q^22 + (23639*b2 + 39292*b1 + 558230) * q^23 + (-2112*b2 - 17352*b1 - 315744) * q^24 + (15047*b2 - 35086*b1 - 520243) * q^25 + (-13016*b2 + 48588*b1 - 720696) * q^26 + (-38727*b2 + 44424*b1 - 3428478) * q^27 + (-21648*b2 - 124440*b1 - 856768) * q^28 + (15422*b2 + 56616*b1 - 897886) * q^29 + (65780*b2 - 200525*b1 + 4757500) * q^30 + (4231*b2 + 37548*b1 - 1508434) * q^31 + (-53504*b2 + 141888*b1 + 53504) * q^32 + (14641*b2 - 58564*b1 + 907742) * q^33 + (47360*b2 + 174770*b1 + 6633136) * q^34 + (-50490*b2 - 14650*b1 - 2644620) * q^35 + (6048*b2 - 538200*b1 + 6969456) * q^36 + (-73373*b2 + 181182*b1 - 2940068) * q^37 + (11584*b2 + 761504*b1 + 7765904) * q^38 + (110660*b2 - 316610*b1 + 8307640) * q^39 + (-41536*b2 - 167592*b1 - 1715296) * q^40 + (-13046*b2 + 147690*b1 - 3257034) * q^41 + (-64616*b2 + 569126*b1 - 2710120) * q^42 + (409750*b2 - 61556*b1 + 6265248) * q^43 + (-117128*b2 - 175692*b1 - 4450864) * q^44 + (-508536*b2 + 810108*b1 - 30638466) * q^45 + (-30668*b2 - 1809821*b1 - 22890340) * q^46 + (-45640*b2 - 1380980*b1 - 10385272) * q^47 + (314176*b2 - 682240*b1 + 25463936) * q^48 + (61036*b2 + 544512*b1 - 18076559) * q^49 + (461252*b2 + 444724*b1 + 34468412) * q^50 + (-242990*b2 - 40738*b1 - 20703628) * q^51 + (-173184*b2 + 291712*b1 - 28781984) * q^52 + (450204*b2 + 1561532*b1 + 15833374) * q^53 + (-820116*b2 + 4173381*b1 - 51276060) * q^54 + (43923*b2 - 497794*b1 + 8901728) * q^55 + (100864*b2 + 1447536*b1 + 37802560) * q^56 + (365376*b2 - 2557974*b1 + 20701032) * q^57 + (-267864*b2 - 290432*b1 - 40214920) * q^58 + (-264531*b2 - 1138488*b1 + 110712790) * q^59 + (1198040*b2 - 2105300*b1 + 96934720) * q^60 + (-1536766*b2 - 2017864*b1 - 16343910) * q^61 + (-249612*b2 + 918235*b1 - 28997540) * q^62 + (402732*b2 - 435924*b1 + 106550928) * q^63 + (-499200*b2 - 1497856*b1 - 140406784) * q^64 + (924704*b2 - 2245802*b1 + 53896524) * q^65 + (644204*b2 - 688127*b1 + 53468932) * q^66 + (1419385*b2 + 4330920*b1 + 110186694) * q^67 + (274032*b2 - 5606408*b1 - 127424352) * q^68 + (-649585*b2 + 5823166*b1 - 34888274) * q^69 + (-488680*b2 + 4486590*b1 - 7635720) * q^70 + (-2085075*b2 - 2525368*b1 - 19278350) * q^71 + (-285120*b2 + 3003408*b1 + 100691712) * q^72 + (2551054*b2 + 5761370*b1 - 152938962) * q^73 + (-2329932*b2 + 3187193*b1 - 176313236) * q^74 + (-2182160*b2 + 3028880*b1 - 187394860) * q^75 + (-1920512*b2 - 10496080*b1 - 441557248) * q^76 + (-322102*b2 + 1786202*b1 + 35431220) * q^77 + (3860800*b2 - 8160100*b1 + 301289840) * q^78 + (2692646*b2 + 14819100*b1 - 266302916) * q^79 + (3172928*b2 - 4157824*b1 + 147361408) * q^80 + (2164968*b2 - 11600172*b1 + 514857465) * q^81 + (-1338072*b2 + 1915272*b1 - 125576888) * q^82 + (-2076162*b2 - 5280024*b1 + 413261640) * q^83 + (-2951696*b2 - 2557432*b1 - 269672128) * q^84 + (-861058*b2 - 104166*b1 - 210667248) * q^85 + (5409448*b2 - 19048326*b1 + 209212696) * q^86 + (2201496*b2 - 2705436*b1 + 168633960) * q^87 + (2928200*b1 + 97919008) * q^88 + (-10415009*b2 - 12980682*b1 - 233174456) * q^89 + (-12583296*b2 + 37698858*b1 - 858360096) * q^90 + (-129880*b2 + 6581620*b1 - 89642320) * q^91 + (2007384*b2 + 25502732*b1 + 1179100992) * q^92 + (2834759*b2 - 6825362*b1 + 204915262) * q^93 + (10500160*b2 + 28463152*b1 + 1109171360) * q^94 + (4164952*b2 - 18830446*b1 + 35767952) * q^95 + (10309376*b2 - 18760640*b1 + 840269056) * q^96 + (3441005*b2 - 7257674*b1 - 735812004) * q^97 + (-3623664*b2 + 9528227*b1 - 420639824) * q^98 + (-4611915*b2 + 9750906*b1 - 71023491) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 186 q^{3} + 912 q^{4} - 1824 q^{5} - 10956 q^{6} - 7260 q^{7} - 20064 q^{8} + 14553 q^{9}+O(q^{10})$$ 3 * q - 186 * q^3 + 912 * q^4 - 1824 * q^5 - 10956 * q^6 - 7260 * q^7 - 20064 * q^8 + 14553 * q^9 $$3 q - 186 q^{3} + 912 q^{4} - 1824 q^{5} - 10956 q^{6} - 7260 q^{7} - 20064 q^{8} + 14553 q^{9} - 86724 q^{10} - 43923 q^{11} - 71040 q^{12} - 93258 q^{13} + 324264 q^{14} + 540330 q^{15} + 22656 q^{16} + 18678 q^{17} + 1997028 q^{18} - 1027356 q^{19} - 156576 q^{20} - 1287924 q^{21} + 1674690 q^{23} - 947232 q^{24} - 1560729 q^{25} - 2162088 q^{26} - 10285434 q^{27} - 2570304 q^{28} - 2693658 q^{29} + 14272500 q^{30} - 4525302 q^{31} + 160512 q^{32} + 2723226 q^{33} + 19899408 q^{34} - 7933860 q^{35} + 20908368 q^{36} - 8820204 q^{37} + 23297712 q^{38} + 24922920 q^{39} - 5145888 q^{40} - 9771102 q^{41} - 8130360 q^{42} + 18795744 q^{43} - 13352592 q^{44} - 91915398 q^{45} - 68671020 q^{46} - 31155816 q^{47} + 76391808 q^{48} - 54229677 q^{49} + 103405236 q^{50} - 62110884 q^{51} - 86345952 q^{52} + 47500122 q^{53} - 153828180 q^{54} + 26705184 q^{55} + 113407680 q^{56} + 62103096 q^{57} - 120644760 q^{58} + 332138370 q^{59} + 290804160 q^{60} - 49031730 q^{61} - 86992620 q^{62} + 319652784 q^{63} - 421220352 q^{64} + 161689572 q^{65} + 160406796 q^{66} + 330560082 q^{67} - 382273056 q^{68} - 104664822 q^{69} - 22907160 q^{70} - 57835050 q^{71} + 302075136 q^{72} - 458816886 q^{73} - 528939708 q^{74} - 562184580 q^{75} - 1324671744 q^{76} + 106293660 q^{77} + 903869520 q^{78} - 798908748 q^{79} + 442084224 q^{80} + 1544572395 q^{81} - 376730664 q^{82} + 1239784920 q^{83} - 809016384 q^{84} - 632001744 q^{85} + 627638088 q^{86} + 505901880 q^{87} + 293757024 q^{88} - 699523368 q^{89} - 2575080288 q^{90} - 268926960 q^{91} + 3537302976 q^{92} + 614745786 q^{93} + 3327514080 q^{94} + 107303856 q^{95} + 2520807168 q^{96} - 2207436012 q^{97} - 1261919472 q^{98} - 213070473 q^{99}+O(q^{100})$$ 3 * q - 186 * q^3 + 912 * q^4 - 1824 * q^5 - 10956 * q^6 - 7260 * q^7 - 20064 * q^8 + 14553 * q^9 - 86724 * q^10 - 43923 * q^11 - 71040 * q^12 - 93258 * q^13 + 324264 * q^14 + 540330 * q^15 + 22656 * q^16 + 18678 * q^17 + 1997028 * q^18 - 1027356 * q^19 - 156576 * q^20 - 1287924 * q^21 + 1674690 * q^23 - 947232 * q^24 - 1560729 * q^25 - 2162088 * q^26 - 10285434 * q^27 - 2570304 * q^28 - 2693658 * q^29 + 14272500 * q^30 - 4525302 * q^31 + 160512 * q^32 + 2723226 * q^33 + 19899408 * q^34 - 7933860 * q^35 + 20908368 * q^36 - 8820204 * q^37 + 23297712 * q^38 + 24922920 * q^39 - 5145888 * q^40 - 9771102 * q^41 - 8130360 * q^42 + 18795744 * q^43 - 13352592 * q^44 - 91915398 * q^45 - 68671020 * q^46 - 31155816 * q^47 + 76391808 * q^48 - 54229677 * q^49 + 103405236 * q^50 - 62110884 * q^51 - 86345952 * q^52 + 47500122 * q^53 - 153828180 * q^54 + 26705184 * q^55 + 113407680 * q^56 + 62103096 * q^57 - 120644760 * q^58 + 332138370 * q^59 + 290804160 * q^60 - 49031730 * q^61 - 86992620 * q^62 + 319652784 * q^63 - 421220352 * q^64 + 161689572 * q^65 + 160406796 * q^66 + 330560082 * q^67 - 382273056 * q^68 - 104664822 * q^69 - 22907160 * q^70 - 57835050 * q^71 + 302075136 * q^72 - 458816886 * q^73 - 528939708 * q^74 - 562184580 * q^75 - 1324671744 * q^76 + 106293660 * q^77 + 903869520 * q^78 - 798908748 * q^79 + 442084224 * q^80 + 1544572395 * q^81 - 376730664 * q^82 + 1239784920 * q^83 - 809016384 * q^84 - 632001744 * q^85 + 627638088 * q^86 + 505901880 * q^87 + 293757024 * q^88 - 699523368 * q^89 - 2575080288 * q^90 - 268926960 * q^91 + 3537302976 * q^92 + 614745786 * q^93 + 3327514080 * q^94 + 107303856 * q^95 + 2520807168 * q^96 - 2207436012 * q^97 - 1261919472 * q^98 - 213070473 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 306x - 836$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 6\nu - 204 ) / 2$$ (v^2 - 6*v - 204) / 2
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$2\beta_{2} + 3\beta _1 + 204$$ 2*b2 + 3*b1 + 204

## 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
 18.7255 −2.80408 −15.9214
−37.4510 70.6582 890.580 613.897 −2646.22 −6611.82 −14178.2 −14690.4 −22991.1
1.2 5.60816 5.22371 −480.549 −529.708 29.2954 −3708.24 −5566.37 −19655.7 −2970.69
1.3 31.8429 −261.882 501.969 −1908.19 −8339.07 3060.06 −319.425 48899.1 −60762.2
 $$n$$: 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 11.10.a.a 3
3.b odd 2 1 99.10.a.b 3
4.b odd 2 1 176.10.a.g 3
5.b even 2 1 275.10.a.a 3
11.b odd 2 1 121.10.a.b 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.10.a.a 3 1.a even 1 1 trivial
99.10.a.b 3 3.b odd 2 1
121.10.a.b 3 11.b odd 2 1
176.10.a.g 3 4.b odd 2 1
275.10.a.a 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 1224T_{2} + 6688$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(11))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 1224T + 6688$$
$3$ $$T^{3} + 186 T^{2} + \cdots + 96660$$
$5$ $$T^{3} + 1824 T^{2} + \cdots - 620517350$$
$7$ $$T^{3} + \cdots - 75027235360$$
$11$ $$(T + 14641)^{3}$$
$13$ $$T^{3} + \cdots - 73087940648800$$
$17$ $$T^{3} + \cdots + 13\!\cdots\!12$$
$19$ $$T^{3} + \cdots - 20\!\cdots\!76$$
$23$ $$T^{3} + \cdots + 44\!\cdots\!32$$
$29$ $$T^{3} + \cdots - 61\!\cdots\!96$$
$31$ $$T^{3} + \cdots + 14\!\cdots\!36$$
$37$ $$T^{3} + \cdots + 95\!\cdots\!26$$
$41$ $$T^{3} + \cdots - 53\!\cdots\!12$$
$43$ $$T^{3} + \cdots + 12\!\cdots\!88$$
$47$ $$T^{3} + \cdots + 27\!\cdots\!48$$
$53$ $$T^{3} + \cdots - 34\!\cdots\!76$$
$59$ $$T^{3} + \cdots - 11\!\cdots\!88$$
$61$ $$T^{3} + \cdots - 99\!\cdots\!00$$
$67$ $$T^{3} + \cdots + 92\!\cdots\!16$$
$71$ $$T^{3} + \cdots - 24\!\cdots\!84$$
$73$ $$T^{3} + \cdots - 66\!\cdots\!08$$
$79$ $$T^{3} + \cdots - 10\!\cdots\!48$$
$83$ $$T^{3} + \cdots - 48\!\cdots\!84$$
$89$ $$T^{3} + \cdots - 39\!\cdots\!30$$
$97$ $$T^{3} + \cdots + 24\!\cdots\!50$$