Properties

 Label 108.4.b.b Level $108$ Weight $4$ Character orbit 108.b Analytic conductor $6.372$ Analytic rank $0$ Dimension $12$ Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(107,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("108.107");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 108.b (of order $$2$$, degree $$1$$, 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: $$6.37220628062$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + \cdots + 9496$$ x^12 + 3*x^10 - 12*x^9 + 73*x^8 - 12*x^7 + 589*x^6 + 84*x^5 + 2452*x^4 + 852*x^3 + 6854*x^2 - 888*x + 9496 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{14}\cdot 3^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - \beta_{6} q^{4} + \beta_{5} q^{5} - \beta_{4} q^{7} + (\beta_{9} - \beta_1) q^{8}+O(q^{10})$$ q - b1 * q^2 - b6 * q^4 + b5 * q^5 - b4 * q^7 + (b9 - b1) * q^8 $$q - \beta_1 q^{2} - \beta_{6} q^{4} + \beta_{5} q^{5} - \beta_{4} q^{7} + (\beta_{9} - \beta_1) q^{8} + (\beta_{6} - \beta_{4} + \beta_{3} + 4) q^{10} + (\beta_{9} + \beta_{2} - \beta_1) q^{11} + (\beta_{10} - \beta_{6} - 6) q^{13} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots - \beta_1) q^{14}+ \cdots + (22 \beta_{9} - 38 \beta_{8} + \cdots - 24 \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 - b6 * q^4 + b5 * q^5 - b4 * q^7 + (b9 - b1) * q^8 + (b6 - b4 + b3 + 4) * q^10 + (b9 + b2 - b1) * q^11 + (b10 - b6 - 6) * q^13 + (b9 + b8 + b7 + 2*b5 + b2 - b1) * q^14 + (b11 - b10 - b6 + 2*b4 + 8) * q^16 + (3*b9 + 2*b8 + b7 + 2*b5 - b2 - 6*b1) * q^17 + (-2*b11 - b10 - b6 - 4*b3) * q^19 + (-b8 - 3*b7 + 6*b5 + b2 - 7*b1) * q^20 + (-b11 - b10 - 3*b6 + 5*b4 + b3 + 4) * q^22 + (b9 - 4*b8 - b7 + b2 - 18*b1) * q^23 + (4*b11 + 3*b10 + 17*b6 - 24) * q^25 + (2*b9 - 2*b8 + 6*b7 + 4*b5 - 2*b2 + 6*b1) * q^26 + (-b6 + 4*b4 + 4*b3 - 24) * q^28 + (-3*b9 + 2*b8 - b7 + 2*b5 + b2 - 10*b1) * q^29 + (-6*b11 - b10 + 11*b6 - 7*b4 - 4*b3 + 8) * q^31 + (-b9 + 8*b8 - 8*b7 - 8*b5 - 11*b1) * q^32 + (6*b11 - 2*b10 - 2*b6 + 2*b4 + 2*b3 - 32) * q^34 + (-2*b9 + 8*b8 + 3*b7 - 2*b2 + 37*b1) * q^35 + (5*b10 - 5*b6 - 20) * q^37 + (-2*b9 - 6*b8 + 10*b7 - 20*b5 + 2*b2 + 16*b1) * q^38 + (-5*b11 - 3*b10 - 3*b6 - 6*b4 + 4*b3 + 64) * q^40 + (-3*b9 - 18*b8 - b7 - 8*b5 + b2 + 70*b1) * q^41 + (4*b11 - 2*b10 - 26*b6 - 6*b4 - 8*b3 - 16) * q^43 + (-4*b9 - 13*b8 - 15*b7 - 10*b5 - 3*b2 - 3*b1) * q^44 + (-2*b11 - 2*b10 - 20*b6 + 4*b4 + 136) * q^46 + (-6*b9 + 8*b8 - 4*b7 - 6*b2 + 34*b1) * q^47 + (-4*b11 + 3*b10 - 23*b6 + 16) * q^49 + (-10*b9 + 26*b8 + 18*b7 + 12*b5 - 6*b2 + 32*b1) * q^50 + (12*b11 + 4*b10 + 14*b6 + 8*b3 + 80) * q^52 + (-9*b9 + 14*b8 - 3*b7 - 19*b5 + 3*b2 - 62*b1) * q^53 + (-4*b11 + 2*b10 + 26*b6 - 15*b4 + 8*b3 + 16) * q^55 + (-3*b9 - 12*b8 - 20*b7 + 8*b5 - 4*b2 + 15*b1) * q^56 + (-6*b11 + 2*b10 - 10*b6 - 6*b4 + 2*b3 - 80) * q^58 + (-6*b9 - 32*b8 + 8*b7 - 6*b2 - 114*b1) * q^59 + (8*b11 - 4*b10 + 44*b6 + 28) * q^61 + (-11*b9 - 31*b8 + 17*b7 - 6*b5 + 9*b2 + 25*b1) * q^62 + (-9*b11 - 7*b10 - 19*b6 - 2*b4 - 16*b3 - 264) * q^64 + (9*b9 + 14*b8 + 3*b7 - 18*b5 - 3*b2 - 50*b1) * q^65 + 22*b4 * q^67 + (4*b9 + 46*b8 - 22*b7 - 4*b5 + 2*b2 + 6*b1) * q^68 + (5*b11 + 5*b10 + 41*b6 - 7*b4 + b3 - 276) * q^70 + (7*b9 + 28*b8 - 17*b7 + 7*b2 + 88*b1) * q^71 + (-4*b11 - 9*b10 - 11*b6 + 5) * q^73 + (10*b9 - 10*b8 + 30*b7 + 20*b5 - 10*b2 + 20*b1) * q^74 + (4*b11 + 12*b10 - 8*b6 + 32*b4 - 8*b3 + 48) * q^76 + (12*b9 - 48*b8 + 4*b7 + 21*b5 - 4*b2 + 200*b1) * q^77 + (10*b11 + 3*b10 - 9*b6 + 34*b4 + 12*b3 - 8) * q^79 + (b9 - 36*b8 - 28*b7 + 16*b5 + 12*b2 - 73*b1) * q^80 + (-6*b11 + 2*b10 + 60*b6 + 4*b4 - 8*b3 + 520) * q^82 + (13*b9 + 24*b8 + 26*b7 + 13*b2 + 109*b1) * q^83 + (-8*b11 - 13*b10 - 27*b6 - 30) * q^85 + (34*b9 + 58*b8 + 26*b7 - 28*b5 + 10*b2 - 6*b1) * q^86 + (-13*b11 - 11*b10 - 7*b6 - 22*b4 - 28*b3 + 416) * q^88 + (-12*b9 + 48*b8 - 4*b7 + 70*b5 + 4*b2 - 200*b1) * q^89 + (14*b11 + 3*b10 - 21*b6 + 38*b4 + 12*b3 - 16) * q^91 + (12*b9 - 16*b8 - 16*b7 - 16*b5 - 148*b1) * q^92 + (2*b11 + 2*b10 + 46*b6 - 34*b4 - 10*b3 - 264) * q^94 + (11*b9 - 76*b8 - 31*b7 + 11*b2 - 346*b1) * q^95 + (-16*b11 - 12*b10 - 68*b6 + 11) * q^97 + (22*b9 - 38*b8 + 18*b7 + 12*b5 - 6*b2 - 24*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{4}+O(q^{10})$$ 12 * q + 6 * q^4 $$12 q + 6 q^{4} + 42 q^{10} - 72 q^{13} + 114 q^{16} + 66 q^{22} - 384 q^{25} - 282 q^{28} - 324 q^{34} - 240 q^{37} + 774 q^{40} + 1752 q^{46} + 288 q^{49} + 924 q^{52} - 948 q^{58} + 144 q^{61} - 3066 q^{64} - 3558 q^{70} + 156 q^{73} + 576 q^{76} + 5832 q^{82} - 168 q^{85} + 5022 q^{88} - 3444 q^{94} + 516 q^{97}+O(q^{100})$$ 12 * q + 6 * q^4 + 42 * q^10 - 72 * q^13 + 114 * q^16 + 66 * q^22 - 384 * q^25 - 282 * q^28 - 324 * q^34 - 240 * q^37 + 774 * q^40 + 1752 * q^46 + 288 * q^49 + 924 * q^52 - 948 * q^58 + 144 * q^61 - 3066 * q^64 - 3558 * q^70 + 156 * q^73 + 576 * q^76 + 5832 * q^82 - 168 * q^85 + 5022 * q^88 - 3444 * q^94 + 516 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + \cdots + 9496$$ :

 $$\beta_{1}$$ $$=$$ $$( - 33253968445 \nu^{11} - 2668198080286 \nu^{10} + 792575592373 \nu^{9} + \cdots - 56\!\cdots\!84 ) / 22\!\cdots\!40$$ (-33253968445*v^11 - 2668198080286*v^10 + 792575592373*v^9 + 1425147997482*v^8 + 24644484803959*v^7 - 193768555419338*v^6 - 61078234561525*v^5 - 887616443102162*v^4 - 85890745896528*v^3 - 2267866287233700*v^2 - 2389458067685830*v - 5634039587496684) / 2246281435673440 $$\beta_{2}$$ $$=$$ $$( 16345777051 \nu^{11} - 1455335526278 \nu^{10} - 1946811804847 \nu^{9} + \cdots - 28\!\cdots\!40 ) / 112314071783672$$ (16345777051*v^11 - 1455335526278*v^10 - 1946811804847*v^9 - 1614515226786*v^8 + 21894737772791*v^7 - 72858396390586*v^6 - 162793265999057*v^5 - 670564599311566*v^4 - 710732231788632*v^3 - 1759491359890068*v^2 - 2792533544858270*v - 2835784389936540) / 112314071783672 $$\beta_{3}$$ $$=$$ $$( 2029314856 \nu^{11} - 20657394999 \nu^{10} - 72348050754 \nu^{9} - 44845772052 \nu^{8} + \cdots - 67923831391169 ) / 6381481351345$$ (2029314856*v^11 - 20657394999*v^10 - 72348050754*v^9 - 44845772052*v^8 + 509091530022*v^7 - 18858252443*v^6 - 5180674803238*v^5 - 15414104133564*v^4 - 24647091348478*v^3 - 3230408502820*v^2 - 21152396498056*v - 67923831391169) / 6381481351345 $$\beta_{4}$$ $$=$$ $$( - 2849350 \nu^{11} + 6585588 \nu^{10} - 7900308 \nu^{9} + 8937888 \nu^{8} - 274261062 \nu^{7} + \cdots - 3774567143 ) / 3046053151$$ (-2849350*v^11 + 6585588*v^10 - 7900308*v^9 + 8937888*v^8 - 274261062*v^7 + 514828262*v^6 - 1348583642*v^5 + 577023966*v^4 - 3610618898*v^3 - 4417697498*v^2 - 20418808208*v - 3774567143) / 3046053151 $$\beta_{5}$$ $$=$$ $$( - 454920367261 \nu^{11} - 699408466494 \nu^{10} - 909094946747 \nu^{9} + \cdots + 873663409442004 ) / 449256287134688$$ (-454920367261*v^11 - 699408466494*v^10 - 909094946747*v^9 + 6055960107450*v^8 - 16604234843593*v^7 - 44620628131914*v^6 - 261918488231045*v^5 - 346379732140290*v^4 - 372084361441392*v^3 - 4408588811940*v^2 + 876761148010650*v + 873663409442004) / 449256287134688 $$\beta_{6}$$ $$=$$ $$( - 23425387202 \nu^{11} - 19598582637 \nu^{10} - 48064283082 \nu^{9} + \cdots - 38283473678982 ) / 12762962702690$$ (-23425387202*v^11 - 19598582637*v^10 - 48064283082*v^9 + 315709019049*v^8 - 1232017295874*v^7 - 1577989182089*v^6 - 13160711736994*v^5 - 9640316846217*v^4 - 29399170369744*v^3 - 39045109376800*v^2 - 83437255959808*v - 38283473678982) / 12762962702690 $$\beta_{7}$$ $$=$$ $$( - 1144977542195 \nu^{11} + 4352732408014 \nu^{10} - 29629791457 \nu^{9} + \cdots + 19\!\cdots\!76 ) / 561570358918360$$ (-1144977542195*v^11 + 4352732408014*v^10 - 29629791457*v^9 + 26260811026362*v^8 - 128693237482831*v^7 + 247175465899922*v^6 - 505821450280655*v^5 + 2483737463807558*v^4 - 214719710377848*v^3 + 7168378187690340*v^2 + 893868308969470*v + 19027629960424476) / 561570358918360 $$\beta_{8}$$ $$=$$ $$( 159260070855 \nu^{11} + 117861110662 \nu^{10} + 438728620059 \nu^{9} + \cdots - 385589858796132 ) / 70196294864795$$ (159260070855*v^11 + 117861110662*v^10 + 438728620059*v^9 - 2345594945694*v^8 + 10405960829647*v^7 + 5462455990466*v^6 + 93693140746805*v^5 + 30241542078854*v^4 + 390731460563916*v^3 + 122017097284380*v^2 + 719519253923230*v - 385589858796132) / 70196294864795 $$\beta_{9}$$ $$=$$ $$( 716000535355 \nu^{11} - 22806431518 \nu^{10} + 1186916251229 \nu^{9} + \cdots - 24\!\cdots\!72 ) / 102103701621520$$ (716000535355*v^11 - 22806431518*v^10 + 1186916251229*v^9 - 10373131256454*v^8 + 49370641188207*v^7 + 1347814842646*v^6 + 379075967096835*v^5 - 39097470243746*v^4 + 1094945871786096*v^3 - 524652329852580*v^2 + 2831947063781450*v - 2432191580688972) / 102103701621520 $$\beta_{10}$$ $$=$$ $$( - 114943514282 \nu^{11} + 101958949683 \nu^{10} + 310110651558 \nu^{9} + \cdots + 11\!\cdots\!98 ) / 12762962702690$$ (-114943514282*v^11 + 101958949683*v^10 + 310110651558*v^9 + 1675099805049*v^8 - 9122517442074*v^7 + 1428113010271*v^6 - 26263858992874*v^5 + 63325710114663*v^4 + 4591871758616*v^3 + 245754664595480*v^2 - 29530710450688*v + 1169684312559898) / 12762962702690 $$\beta_{11}$$ $$=$$ $$( 29835469322 \nu^{11} + 31182551757 \nu^{10} + 48231870066 \nu^{9} - 289609225533 \nu^{8} + \cdots + 58683428040738 ) / 2552592540538$$ (29835469322*v^11 + 31182551757*v^10 + 48231870066*v^9 - 289609225533*v^8 + 1601507488746*v^7 + 2210589724721*v^6 + 14942761872730*v^5 + 20689356220701*v^4 + 49348940923840*v^3 + 69766706628088*v^2 + 108997215408544*v + 58683428040738) / 2552592540538
 $$\nu$$ $$=$$ $$( - 2 \beta_{11} - \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} - 9 \beta_{6} + 8 \beta_{5} + \cdots - 4 ) / 72$$ (-2*b11 - b10 + 3*b9 - 2*b8 + 5*b7 - 9*b6 + 8*b5 - b2 + 14*b1 - 4) / 72 $$\nu^{2}$$ $$=$$ $$( - \beta_{10} - 6 \beta_{9} + 6 \beta_{8} - 2 \beta_{7} - 11 \beta_{6} + 6 \beta_{5} - 2 \beta_{4} + \cdots - 24 ) / 36$$ (-b10 - 6*b9 + 6*b8 - 2*b7 - 11*b6 + 6*b5 - 2*b4 - 2*b3 - 2*b1 - 24) / 36 $$\nu^{3}$$ $$=$$ $$( 13 \beta_{11} - \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 13 \beta_{7} + 30 \beta_{6} + 46 \beta_{5} + \cdots + 224 ) / 72$$ (13*b11 - b10 - 3*b9 - 7*b8 - 13*b7 + 30*b6 + 46*b5 + 48*b4 - 6*b3 - 5*b2 + 92*b1 + 224) / 72 $$\nu^{4}$$ $$=$$ $$( - 2 \beta_{11} + 3 \beta_{10} + 18 \beta_{9} - 32 \beta_{8} + 26 \beta_{7} - 61 \beta_{6} + 44 \beta_{5} + \cdots - 850 ) / 36$$ (-2*b11 + 3*b10 + 18*b9 - 32*b8 + 26*b7 - 61*b6 + 44*b5 + 14*b4 - 28*b3 + 26*b2 - 40*b1 - 850) / 36 $$\nu^{5}$$ $$=$$ $$( 119 \beta_{11} + 23 \beta_{10} - 129 \beta_{9} - 149 \beta_{8} - 387 \beta_{7} + 392 \beta_{6} + \cdots - 572 ) / 72$$ (119*b11 + 23*b10 - 129*b9 - 149*b8 - 387*b7 + 392*b6 + 86*b5 - 100*b4 - 130*b3 - 7*b2 - 1704*b1 - 572) / 72 $$\nu^{6}$$ $$=$$ $$( 167 \beta_{11} + 172 \beta_{10} + 297 \beta_{9} - 633 \beta_{8} - 183 \beta_{7} + 627 \beta_{6} + \cdots - 5210 ) / 36$$ (167*b11 + 172*b10 + 297*b9 - 633*b8 - 183*b7 + 627*b6 - 6*b5 + 384*b4 + 204*b3 + 57*b2 - 552*b1 - 5210) / 36 $$\nu^{7}$$ $$=$$ $$( - 75 \beta_{11} + 683 \beta_{10} + 525 \beta_{9} + 543 \beta_{8} - 249 \beta_{7} + 2386 \beta_{6} + \cdots - 35724 ) / 72$$ (-75*b11 + 683*b10 + 525*b9 + 543*b8 - 249*b7 + 2386*b6 - 2682*b5 - 2996*b4 + 154*b3 + 1239*b2 - 16092*b1 - 35724) / 72 $$\nu^{8}$$ $$=$$ $$( 1272 \beta_{11} + 529 \beta_{10} + 72 \beta_{9} - 1448 \beta_{8} - 3032 \beta_{7} + 11519 \beta_{6} + \cdots + 30702 ) / 36$$ (1272*b11 + 529*b10 + 72*b9 - 1448*b8 - 3032*b7 + 11519*b6 - 5200*b5 - 1918*b4 + 2228*b3 - 1528*b2 - 6176*b1 + 30702) / 36 $$\nu^{9}$$ $$=$$ $$( - 7085 \beta_{11} + 5021 \beta_{10} + 15741 \beta_{9} + 1455 \beta_{8} + 20007 \beta_{7} + \cdots - 156028 ) / 72$$ (-7085*b11 + 5021*b10 + 15741*b9 + 1455*b8 + 20007*b7 - 7938*b6 - 32010*b5 - 3564*b4 + 17118*b3 + 2295*b2 + 64596*b1 - 156028) / 72 $$\nu^{10}$$ $$=$$ $$( - 12715 \beta_{11} - 7950 \beta_{10} - 17487 \beta_{9} + 56011 \beta_{8} + 8305 \beta_{7} + \cdots + 415114 ) / 36$$ (-12715*b11 - 7950*b10 - 17487*b9 + 56011*b8 + 8305*b7 - 461*b6 - 44218*b5 - 46676*b4 + 1624*b3 - 6211*b2 - 40964*b1 + 415114) / 36 $$\nu^{11}$$ $$=$$ $$( - 22857 \beta_{11} - 49075 \beta_{10} + 2019 \beta_{9} + 26941 \beta_{8} + 102273 \beta_{7} + \cdots + 3750276 ) / 72$$ (-22857*b11 - 49075*b10 + 2019*b9 + 26941*b8 + 102273*b7 - 60590*b6 - 49390*b5 + 130636*b4 + 96910*b3 - 115975*b2 + 1521828*b1 + 3750276) / 72

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$-1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 2.61836 + 1.60260i 2.61836 − 1.60260i −1.29835 − 1.36719i −1.29835 + 1.36719i −0.453986 + 2.07664i −0.453986 − 2.07664i −2.18604 + 2.07664i −2.18604 − 2.07664i 0.433705 − 1.36719i 0.433705 + 1.36719i 0.886307 + 1.60260i 0.886307 − 1.60260i
−2.72048 0.773954i 0 6.80199 + 4.21105i 20.8488i 0 13.9048i −15.2455 16.7205i 0 16.1360 56.7186i
107.2 −2.72048 + 0.773954i 0 6.80199 4.21105i 20.8488i 0 13.9048i −15.2455 + 16.7205i 0 16.1360 + 56.7186i
107.3 −2.27435 1.68146i 0 2.34537 + 7.64848i 5.83890i 0 8.83113i 7.52641 21.3390i 0 −9.81789 + 13.2797i
107.4 −2.27435 + 1.68146i 0 2.34537 7.64848i 5.83890i 0 8.83113i 7.52641 + 21.3390i 0 −9.81789 13.2797i
107.5 −0.419903 2.79708i 0 −7.64736 + 2.34901i 1.49508i 0 26.1852i 9.78152 + 20.4040i 0 4.18188 0.627790i
107.6 −0.419903 + 2.79708i 0 −7.64736 2.34901i 1.49508i 0 26.1852i 9.78152 20.4040i 0 4.18188 + 0.627790i
107.7 0.419903 2.79708i 0 −7.64736 2.34901i 1.49508i 0 26.1852i −9.78152 + 20.4040i 0 4.18188 + 0.627790i
107.8 0.419903 + 2.79708i 0 −7.64736 + 2.34901i 1.49508i 0 26.1852i −9.78152 20.4040i 0 4.18188 0.627790i
107.9 2.27435 1.68146i 0 2.34537 7.64848i 5.83890i 0 8.83113i −7.52641 21.3390i 0 −9.81789 13.2797i
107.10 2.27435 + 1.68146i 0 2.34537 + 7.64848i 5.83890i 0 8.83113i −7.52641 + 21.3390i 0 −9.81789 + 13.2797i
107.11 2.72048 0.773954i 0 6.80199 4.21105i 20.8488i 0 13.9048i 15.2455 16.7205i 0 16.1360 + 56.7186i
107.12 2.72048 + 0.773954i 0 6.80199 + 4.21105i 20.8488i 0 13.9048i 15.2455 + 16.7205i 0 16.1360 56.7186i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 107.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.b.b 12
3.b odd 2 1 inner 108.4.b.b 12
4.b odd 2 1 inner 108.4.b.b 12
8.b even 2 1 1728.4.c.j 12
8.d odd 2 1 1728.4.c.j 12
12.b even 2 1 inner 108.4.b.b 12
24.f even 2 1 1728.4.c.j 12
24.h odd 2 1 1728.4.c.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.b.b 12 1.a even 1 1 trivial
108.4.b.b 12 3.b odd 2 1 inner
108.4.b.b 12 4.b odd 2 1 inner
108.4.b.b 12 12.b even 2 1 inner
1728.4.c.j 12 8.b even 2 1
1728.4.c.j 12 8.d odd 2 1
1728.4.c.j 12 24.f even 2 1
1728.4.c.j 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 471T_{5}^{4} + 15867T_{5}^{2} + 33125$$ acting on $$S_{4}^{\mathrm{new}}(108, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 3 T^{10} + \cdots + 262144$$
$3$ $$T^{12}$$
$5$ $$(T^{6} + 471 T^{4} + \cdots + 33125)^{2}$$
$7$ $$(T^{6} + 957 T^{4} + \cdots + 10338975)^{2}$$
$11$ $$(T^{6} - 4929 T^{4} + \cdots - 2112318675)^{2}$$
$13$ $$(T^{3} + 18 T^{2} + \cdots - 70760)^{4}$$
$17$ $$(T^{6} + 17532 T^{4} + \cdots + 42531205952)^{2}$$
$19$ $$(T^{6} + \cdots + 1093873434624)^{2}$$
$23$ $$(T^{6} - 20688 T^{4} + \cdots - 15885545472)^{2}$$
$29$ $$(T^{6} + 28380 T^{4} + \cdots + 251748219200)^{2}$$
$31$ $$(T^{6} + \cdots + 37183780589679)^{2}$$
$37$ $$(T^{3} + 60 T^{2} + \cdots - 8168000)^{4}$$
$41$ $$(T^{6} + 223920 T^{4} + \cdots + 855375564800)^{2}$$
$43$ $$(T^{6} + \cdots + 41\!\cdots\!00)^{2}$$
$47$ $$(T^{6} + \cdots - 62306621744832)^{2}$$
$53$ $$(T^{6} + \cdots + 820610118691973)^{2}$$
$59$ $$(T^{6} + \cdots - 98\!\cdots\!00)^{2}$$
$61$ $$(T^{3} - 36 T^{2} + \cdots - 68226752)^{4}$$
$67$ $$(T^{6} + \cdots + 11\!\cdots\!00)^{2}$$
$71$ $$(T^{6} + \cdots - 467529795115200)^{2}$$
$73$ $$(T^{3} - 39 T^{2} + \cdots + 14711755)^{4}$$
$79$ $$(T^{6} + \cdots + 49\!\cdots\!84)^{2}$$
$83$ $$(T^{6} + \cdots - 39\!\cdots\!43)^{2}$$
$89$ $$(T^{6} + \cdots + 19\!\cdots\!00)^{2}$$
$97$ $$(T^{3} - 129 T^{2} + \cdots - 298155155)^{4}$$