# Properties

 Label 26.6.c.b Level $26$ Weight $6$ Character orbit 26.c Analytic conductor $4.170$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,6,Mod(3,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("26.3");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 26.c (of order $$3$$, degree $$2$$, 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: $$4.16997931514$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 715x^{6} - 6060x^{5} + 441025x^{4} - 2166450x^{3} + 59373900x^{2} + 212706000x + 4928040000$$ x^8 + 715*x^6 - 6060*x^5 + 441025*x^4 - 2166450*x^3 + 59373900*x^2 + 212706000*x + 4928040000 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (4 \beta_{2} + 4) q^{2} - \beta_1 q^{3} + 16 \beta_{2} q^{4} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \cdots - 3) q^{5}+ \cdots + ( - \beta_{7} - \beta_{4} + \cdots + 115 \beta_{2}) q^{9}+O(q^{10})$$ q + (4*b2 + 4) * q^2 - b1 * q^3 + 16*b2 * q^4 + (-b6 - b4 + b3 + b1 - 3) * q^5 + 4*b3 * q^6 + (b7 - b3 - 32*b2) * q^7 - 64 * q^8 + (-b7 - b4 + 7*b3 + 115*b2) * q^9 $$q + (4 \beta_{2} + 4) q^{2} - \beta_1 q^{3} + 16 \beta_{2} q^{4} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \cdots - 3) q^{5}+ \cdots + ( - 206 \beta_{6} + 248 \beta_{5} + \cdots - 66872) q^{99}+O(q^{100})$$ q + (4*b2 + 4) * q^2 - b1 * q^3 + 16*b2 * q^4 + (-b6 - b4 + b3 + b1 - 3) * q^5 + 4*b3 * q^6 + (b7 - b3 - 32*b2) * q^7 - 64 * q^8 + (-b7 - b4 + 7*b3 + 115*b2) * q^9 + (-4*b6 - 12*b2 + 4*b1 - 12) * q^10 + (-b7 - b5 + 112*b2 - 7*b1 + 112) * q^11 + (16*b3 + 16*b1) * q^12 + (b5 + 5*b4 - 15*b3 + 69*b2 - 24*b1 + 207) * q^13 + (-4*b5 - 4*b3 - 4*b1 + 128) * q^14 + (5*b7 + 10*b6 + 5*b5 - 200*b2 + 16*b1 - 200) * q^15 + (-256*b2 - 256) * q^16 + (-4*b7 - 10*b4 - 40*b3 + 127*b2) * q^17 + (-4*b6 + 4*b5 - 4*b4 + 28*b3 + 28*b1 - 460) * q^18 + (-14*b4 - 7*b3 - 1012*b2) * q^19 + (16*b4 - 16*b3 - 48*b2) * q^20 + (35*b6 - 5*b5 + 35*b4 - 161*b3 - 161*b1 + 290) * q^21 + (-4*b7 + 28*b3 + 448*b2) * q^22 + (-4*b7 - 16*b6 - 4*b5 + 76*b2 + 87*b1 + 76) * q^23 + 64*b1 * q^24 + (21*b6 - b5 + 21*b4 + 143*b3 + 143*b1 + 2474) * q^25 + (4*b7 + 20*b6 + 4*b5 + 20*b4 + 36*b3 + 828*b2 - 60*b1 + 552) * q^26 + (-30*b6 + 15*b5 - 30*b4 + 49*b3 + 49*b1 - 2280) * q^27 + (-16*b7 - 16*b5 + 512*b2 - 16*b1 + 512) * q^28 + (5*b7 - 46*b6 + 5*b5 - 1031*b2 - 6*b1 - 1031) * q^29 + (20*b7 - 40*b4 - 64*b3 - 800*b2) * q^30 + (-4*b6 - 19*b5 - 4*b4 - 234*b3 - 234*b1 - 3120) * q^31 - 1024*b2 * q^32 + (-11*b7 - 41*b4 + 283*b3 + 2438*b2) * q^33 + (-40*b6 + 16*b5 - 40*b4 - 160*b3 - 160*b1 - 508) * q^34 + (30*b7 - 2*b4 + 602*b3 - 600*b2) * q^35 + (16*b7 - 16*b6 + 16*b5 - 1840*b2 + 112*b1 - 1840) * q^36 + (15*b7 - 20*b6 + 15*b5 + 4057*b2 + 446*b1 + 4057) * q^37 + (-56*b6 - 56*b4 - 28*b3 - 28*b1 + 4048) * q^38 + (-20*b7 - 74*b6 - 31*b5 - 64*b4 + 102*b3 + 8660*b2 - 151*b1 + 4648) * q^39 + (64*b6 + 64*b4 - 64*b3 - 64*b1 + 192) * q^40 + (-14*b7 - 10*b6 - 14*b5 - 1507*b2 + 8*b1 - 1507) * q^41 + (-20*b7 + 140*b6 - 20*b5 + 1160*b2 - 644*b1 + 1160) * q^42 + (-17*b7 + 180*b4 - 73*b3 - 1196*b2) * q^43 + (16*b5 + 112*b3 + 112*b1 - 1792) * q^44 + (-4*b7 + 53*b4 - 619*b3 - 6239*b2) * q^45 + (-16*b7 + 64*b4 - 348*b3 + 304*b2) * q^46 + (-90*b6 + 19*b5 - 90*b4 + 34*b3 + 34*b1 - 5392) * q^47 - 256*b3 * q^48 + (3*b7 + 231*b6 + 3*b5 - 23775*b2 + 507*b1 - 23775) * q^49 + (-4*b7 + 84*b6 - 4*b5 + 9896*b2 + 572*b1 + 9896) * q^50 + (14*b6 + 16*b5 + 14*b4 + 395*b3 + 395*b1 + 16172) * q^51 + (16*b7 + 80*b6 + 384*b3 + 2208*b2 + 144*b1 - 1104) * q^52 + (91*b6 - 46*b5 + 91*b4 - 563*b3 - 563*b1 - 3731) * q^53 + (60*b7 - 120*b6 + 60*b5 - 9120*b2 + 196*b1 - 9120) * q^54 + (-22*b6 - 840*b2 + 778*b1 - 840) * q^55 + (-64*b7 + 64*b3 + 2048*b2) * q^56 + (161*b6 + 49*b5 + 161*b4 - 977*b3 - 977*b1 + 4718) * q^57 + (20*b7 + 184*b4 + 24*b3 - 4124*b2) * q^58 + (95*b7 - 210*b4 + 367*b3 + 1636*b2) * q^59 + (-160*b6 - 80*b5 - 160*b4 - 256*b3 - 256*b1 + 3200) * q^60 + (-143*b7 + 54*b4 + 122*b3 - 1673*b2) * q^61 + (-76*b7 - 16*b6 - 76*b5 - 12480*b2 - 936*b1 - 12480) * q^62 + (-78*b7 - 54*b6 - 78*b5 + 43992*b2 - 1994*b1 + 43992) * q^63 + 4096 * q^64 + (65*b7 + 13*b6 + 155*b5 - 148*b4 - 986*b3 + 24943*b2 + 37*b1 - 6317) * q^65 + (-164*b6 + 44*b5 - 164*b4 + 1132*b3 + 1132*b1 - 9752) * q^66 + (40*b7 - 110*b6 + 40*b5 + 6096*b2 + 87*b1 + 6096) * q^67 + (64*b7 - 160*b6 + 64*b5 - 2032*b2 - 640*b1 - 2032) * q^68 + (135*b7 - 225*b4 - 141*b3 - 28890*b2) * q^69 + (-8*b6 - 120*b5 - 8*b4 + 2408*b3 + 2408*b1 + 2400) * q^70 + (-12*b7 - 250*b4 + 1489*b3 + 4644*b2) * q^71 + (64*b7 + 64*b4 - 448*b3 - 7360*b2) * q^72 + (255*b6 - 39*b5 + 255*b4 + 1755*b3 + 1755*b1 - 10555) * q^73 + (60*b7 + 80*b4 - 1784*b3 + 16228*b2) * q^74 + (55*b7 - 340*b6 + 55*b5 - 54580*b2 - 1721*b1 - 54580) * q^75 + (-224*b6 + 16192*b2 - 112*b1 + 16192) * q^76 + (-21*b6 - 113*b5 - 21*b4 - 1553*b3 - 1553*b1 + 44882) * q^77 + (-124*b7 - 256*b6 - 44*b5 + 40*b4 + 1012*b3 + 18592*b2 + 408*b1 - 16048) * q^78 + (-246*b6 - 96*b5 - 246*b4 + 312*b3 + 312*b1 - 66160) * q^79 + (256*b6 + 768*b2 - 256*b1 + 768) * q^80 + (-14*b7 + 14*b6 - 14*b5 + 16163*b2 + 2932*b1 + 16163) * q^81 + (-56*b7 + 40*b4 - 32*b3 - 6028*b2) * q^82 + (-480*b6 + 150*b5 - 480*b4 + 528*b3 + 528*b1 + 17400) * q^83 + (-80*b7 - 560*b4 + 2576*b3 + 4640*b2) * q^84 + (-380*b7 + 727*b4 - 83*b3 - 42697*b2) * q^85 + (720*b6 + 68*b5 + 720*b4 - 292*b3 - 292*b1 + 4784) * q^86 + (198*b7 - 342*b4 - 1875*b3 + 9756*b2) * q^87 + (64*b7 + 64*b5 - 7168*b2 + 448*b1 - 7168) * q^88 + (241*b7 + 365*b6 + 241*b5 + 38540*b2 - 3277*b1 + 38540) * q^89 + (212*b6 + 16*b5 + 212*b4 - 2476*b3 - 2476*b1 + 24956) * q^90 + (57*b7 + 480*b6 - 304*b5 + 1106*b4 - 3757*b3 + 35432*b2 + 672*b1 + 9748) * q^91 + (256*b6 + 64*b5 + 256*b4 - 1392*b3 - 1392*b1 - 1216) * q^92 + (-294*b7 + 924*b6 - 294*b5 + 83112*b2 - 812*b1 + 83112) * q^93 + (76*b7 - 360*b6 + 76*b5 - 21568*b2 + 136*b1 - 21568) * q^94 + (-91*b7 - 550*b4 + 3392*b3 - 71024*b2) * q^95 + (-1024*b3 - 1024*b1) * q^96 + (119*b7 + 581*b4 - 2557*b3 - 51684*b2) * q^97 + (12*b7 - 924*b4 - 2028*b3 - 95100*b2) * q^98 + (-206*b6 + 248*b5 - 206*b4 + 4306*b3 + 4306*b1 - 66872) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 16 q^{2} - 64 q^{4} - 24 q^{5} + 126 q^{7} - 512 q^{8} - 458 q^{9}+O(q^{10})$$ 8 * q + 16 * q^2 - 64 * q^4 - 24 * q^5 + 126 * q^7 - 512 * q^8 - 458 * q^9 $$8 q + 16 q^{2} - 64 q^{4} - 24 q^{5} + 126 q^{7} - 512 q^{8} - 458 q^{9} - 48 q^{10} + 446 q^{11} + 1384 q^{13} + 1008 q^{14} - 790 q^{15} - 1024 q^{16} - 500 q^{17} - 3664 q^{18} + 4048 q^{19} + 192 q^{20} + 2300 q^{21} - 1784 q^{22} + 296 q^{23} + 19788 q^{25} + 1112 q^{26} - 18180 q^{27} + 2016 q^{28} - 4114 q^{29} + 3160 q^{30} - 25036 q^{31} + 4096 q^{32} - 9730 q^{33} - 4000 q^{34} + 2340 q^{35} - 7328 q^{36} + 16258 q^{37} + 32384 q^{38} + 2460 q^{39} + 1536 q^{40} - 6056 q^{41} + 4600 q^{42} + 4818 q^{43} - 14272 q^{44} + 24964 q^{45} - 1184 q^{46} - 43060 q^{47} - 95094 q^{49} + 39576 q^{50} + 129440 q^{51} - 17696 q^{52} - 30032 q^{53} - 36360 q^{54} - 3360 q^{55} - 8064 q^{56} + 37940 q^{57} + 16456 q^{58} - 6734 q^{59} + 25280 q^{60} + 6978 q^{61} - 50072 q^{62} + 175812 q^{63} + 32768 q^{64} - 149818 q^{65} - 77840 q^{66} + 24464 q^{67} - 8000 q^{68} + 115290 q^{69} + 18720 q^{70} - 18552 q^{71} + 29312 q^{72} - 84596 q^{73} - 65032 q^{74} - 218210 q^{75} + 64768 q^{76} + 358604 q^{77} - 202680 q^{78} - 529664 q^{79} + 3072 q^{80} + 64624 q^{81} + 24224 q^{82} + 139800 q^{83} - 18400 q^{84} + 171548 q^{85} + 38544 q^{86} - 39420 q^{87} - 28544 q^{88} + 154642 q^{89} + 199712 q^{90} - 65074 q^{91} - 9472 q^{92} + 331860 q^{93} - 86120 q^{94} + 284278 q^{95} + 206498 q^{97} + 380376 q^{98} - 533984 q^{99}+O(q^{100})$$ 8 * q + 16 * q^2 - 64 * q^4 - 24 * q^5 + 126 * q^7 - 512 * q^8 - 458 * q^9 - 48 * q^10 + 446 * q^11 + 1384 * q^13 + 1008 * q^14 - 790 * q^15 - 1024 * q^16 - 500 * q^17 - 3664 * q^18 + 4048 * q^19 + 192 * q^20 + 2300 * q^21 - 1784 * q^22 + 296 * q^23 + 19788 * q^25 + 1112 * q^26 - 18180 * q^27 + 2016 * q^28 - 4114 * q^29 + 3160 * q^30 - 25036 * q^31 + 4096 * q^32 - 9730 * q^33 - 4000 * q^34 + 2340 * q^35 - 7328 * q^36 + 16258 * q^37 + 32384 * q^38 + 2460 * q^39 + 1536 * q^40 - 6056 * q^41 + 4600 * q^42 + 4818 * q^43 - 14272 * q^44 + 24964 * q^45 - 1184 * q^46 - 43060 * q^47 - 95094 * q^49 + 39576 * q^50 + 129440 * q^51 - 17696 * q^52 - 30032 * q^53 - 36360 * q^54 - 3360 * q^55 - 8064 * q^56 + 37940 * q^57 + 16456 * q^58 - 6734 * q^59 + 25280 * q^60 + 6978 * q^61 - 50072 * q^62 + 175812 * q^63 + 32768 * q^64 - 149818 * q^65 - 77840 * q^66 + 24464 * q^67 - 8000 * q^68 + 115290 * q^69 + 18720 * q^70 - 18552 * q^71 + 29312 * q^72 - 84596 * q^73 - 65032 * q^74 - 218210 * q^75 + 64768 * q^76 + 358604 * q^77 - 202680 * q^78 - 529664 * q^79 + 3072 * q^80 + 64624 * q^81 + 24224 * q^82 + 139800 * q^83 - 18400 * q^84 + 171548 * q^85 + 38544 * q^86 - 39420 * q^87 - 28544 * q^88 + 154642 * q^89 + 199712 * q^90 - 65074 * q^91 - 9472 * q^92 + 331860 * q^93 - 86120 * q^94 + 284278 * q^95 + 206498 * q^97 + 380376 * q^98 - 533984 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 715x^{6} - 6060x^{5} + 441025x^{4} - 2166450x^{3} + 59373900x^{2} + 212706000x + 4928040000$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 134431 \nu^{7} + 2686860 \nu^{6} - 82919485 \nu^{5} + 2471955960 \nu^{4} + \cdots - 10075464126000 ) / 144937028106000$$ (-134431*v^7 + 2686860*v^6 - 82919485*v^5 + 2471955960*v^4 - 67428617575*v^3 + 1436218471050*v^2 - 7182376326900*v - 10075464126000) / 144937028106000 $$\beta_{3}$$ $$=$$ $$( - 14927 \nu^{7} - 73326 \nu^{6} - 9207245 \nu^{5} + 45228810 \nu^{4} - 6361002395 \nu^{3} + \cdots - 3680451918000 ) / 805205711700$$ (-14927*v^7 - 73326*v^6 - 9207245*v^5 + 45228810*v^4 - 6361002395*v^3 - 4440646800*v^2 - 908088023700*v - 3680451918000) / 805205711700 $$\beta_{4}$$ $$=$$ $$( - 15871907 \nu^{7} - 422731980 \nu^{6} - 16547051825 \nu^{5} - 164565014880 \nu^{4} + \cdots - 62\!\cdots\!00 ) / 72468514053000$$ (-15871907*v^7 - 422731980*v^6 - 16547051825*v^5 - 164565014880*v^4 - 5719029885275*v^3 - 63829290396750*v^2 - 1068135940059300*v - 6289672377222000) / 72468514053000 $$\beta_{5}$$ $$=$$ $$( - 530299 \nu^{7} - 13559550 \nu^{6} - 327098065 \nu^{5} + 1606805970 \nu^{4} + \cdots + 681813571046400 ) / 2415617135100$$ (-530299*v^7 - 13559550*v^6 - 327098065*v^5 + 1606805970*v^4 - 31748537635*v^3 - 157759131600*v^2 - 3655013544000*v + 681813571046400) / 2415617135100 $$\beta_{6}$$ $$=$$ $$( - 16040413 \nu^{7} + 101650500 \nu^{6} - 3137047375 \nu^{5} + 261259344480 \nu^{4} + \cdots + 51\!\cdots\!00 ) / 72468514053000$$ (-16040413*v^7 + 101650500*v^6 - 3137047375*v^5 + 261259344480*v^4 - 2550989888125*v^3 + 54335665308750*v^2 + 848184626139300*v + 5102143546500000) / 72468514053000 $$\beta_{7}$$ $$=$$ $$( - 488757 \nu^{7} + 23819015 \nu^{6} - 113780565 \nu^{5} + 17653868945 \nu^{4} + \cdots + 60207766398000 ) / 2013014279250$$ (-488757*v^7 + 23819015*v^6 - 113780565*v^5 + 17653868945*v^4 - 287725671375*v^3 + 6823507643825*v^2 - 21933468816300*v + 60207766398000) / 2013014279250
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{4} + 7\beta_{3} + 358\beta_{2}$$ -b7 - b4 + 7*b3 + 358*b2 $$\nu^{3}$$ $$=$$ $$30\beta_{6} - 15\beta_{5} + 30\beta_{4} - 535\beta_{3} - 535\beta _1 + 2280$$ 30*b6 - 15*b5 + 30*b4 - 535*b3 - 535*b1 + 2280 $$\nu^{4}$$ $$=$$ $$715\beta_{7} - 715\beta_{6} + 715\beta_{5} - 185770\beta_{2} + 8035\beta _1 - 185770$$ 715*b7 - 715*b6 + 715*b5 - 185770*b2 + 8035*b1 - 185770 $$\nu^{5}$$ $$=$$ $$-13755\beta_{7} - 24480\beta_{4} + 333535\beta_{3} + 2714940\beta_{2}$$ -13755*b7 - 24480*b4 + 333535*b3 + 2714940*b2 $$\nu^{6}$$ $$=$$ $$531925\beta_{6} - 486475\beta_{5} + 531925\beta_{4} - 6874675\beta_{3} - 6874675\beta _1 + 114602350$$ 531925*b6 - 486475*b5 + 531925*b4 - 6874675*b3 - 6874675*b1 + 114602350 $$\nu^{7}$$ $$=$$ $$10948275 \beta_{7} - 17563650 \beta_{6} + 10948275 \beta_{5} - 2344009200 \beta_{2} + 225266575 \beta _1 - 2344009200$$ 10948275*b7 - 17563650*b6 + 10948275*b5 - 2344009200*b2 + 225266575*b1 - 2344009200

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/26\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$\chi(n)$$ $$\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 9.50363 + 16.4608i 8.18663 + 14.1797i −4.17141 − 7.22509i −13.5189 − 23.4153i 9.50363 − 16.4608i 8.18663 − 14.1797i −4.17141 + 7.22509i −13.5189 + 23.4153i
2.00000 + 3.46410i −9.50363 16.4608i −8.00000 + 13.8564i 84.1993 38.0145 65.8431i 108.756 188.370i −64.0000 −59.1379 + 102.430i 168.399 + 291.675i
3.2 2.00000 + 3.46410i −8.18663 14.1797i −8.00000 + 13.8564i −101.283 32.7465 56.7187i −37.1663 + 64.3739i −64.0000 −12.5419 + 21.7232i −202.565 350.854i
3.3 2.00000 + 3.46410i 4.17141 + 7.22509i −8.00000 + 13.8564i 52.7082 −16.6856 + 28.9003i −121.202 + 209.927i −64.0000 86.6987 150.167i 105.416 + 182.587i
3.4 2.00000 + 3.46410i 13.5189 + 23.4153i −8.00000 + 13.8564i −47.6248 −54.0754 + 93.6614i 112.612 195.050i −64.0000 −244.019 + 422.653i −95.2496 164.977i
9.1 2.00000 3.46410i −9.50363 + 16.4608i −8.00000 13.8564i 84.1993 38.0145 + 65.8431i 108.756 + 188.370i −64.0000 −59.1379 102.430i 168.399 291.675i
9.2 2.00000 3.46410i −8.18663 + 14.1797i −8.00000 13.8564i −101.283 32.7465 + 56.7187i −37.1663 64.3739i −64.0000 −12.5419 21.7232i −202.565 + 350.854i
9.3 2.00000 3.46410i 4.17141 7.22509i −8.00000 13.8564i 52.7082 −16.6856 28.9003i −121.202 209.927i −64.0000 86.6987 + 150.167i 105.416 182.587i
9.4 2.00000 3.46410i 13.5189 23.4153i −8.00000 13.8564i −47.6248 −54.0754 93.6614i 112.612 + 195.050i −64.0000 −244.019 422.653i −95.2496 + 164.977i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.6.c.b 8
3.b odd 2 1 234.6.h.f 8
4.b odd 2 1 208.6.i.c 8
13.c even 3 1 inner 26.6.c.b 8
13.c even 3 1 338.6.a.j 4
13.e even 6 1 338.6.a.k 4
13.f odd 12 2 338.6.b.e 8
39.i odd 6 1 234.6.h.f 8
52.j odd 6 1 208.6.i.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.c.b 8 1.a even 1 1 trivial
26.6.c.b 8 13.c even 3 1 inner
208.6.i.c 8 4.b odd 2 1
208.6.i.c 8 52.j odd 6 1
234.6.h.f 8 3.b odd 2 1
234.6.h.f 8 39.i odd 6 1
338.6.a.j 4 13.c even 3 1
338.6.a.k 4 13.e even 6 1
338.6.b.e 8 13.f odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 715T_{3}^{6} + 6060T_{3}^{5} + 441025T_{3}^{4} + 2166450T_{3}^{3} + 59373900T_{3}^{2} - 212706000T_{3} + 4928040000$$ acting on $$S_{6}^{\mathrm{new}}(26, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 4 T + 16)^{4}$$
$3$ $$T^{8} + \cdots + 4928040000$$
$5$ $$(T^{4} + 12 T^{3} + \cdots + 21407004)^{2}$$
$7$ $$T^{8} + \cdots + 77\!\cdots\!84$$
$11$ $$T^{8} + \cdots + 79\!\cdots\!64$$
$13$ $$T^{8} + \cdots + 19\!\cdots\!01$$
$17$ $$T^{8} + \cdots + 13\!\cdots\!09$$
$19$ $$T^{8} + \cdots + 59\!\cdots\!00$$
$23$ $$T^{8} + \cdots + 13\!\cdots\!76$$
$29$ $$T^{8} + \cdots + 23\!\cdots\!49$$
$31$ $$(T^{4} + \cdots - 12\!\cdots\!20)^{2}$$
$37$ $$T^{8} + \cdots + 51\!\cdots\!41$$
$41$ $$T^{8} + \cdots + 29\!\cdots\!69$$
$43$ $$T^{8} + \cdots + 37\!\cdots\!24$$
$47$ $$(T^{4} + \cdots - 20\!\cdots\!68)^{2}$$
$53$ $$(T^{4} + \cdots - 27\!\cdots\!44)^{2}$$
$59$ $$T^{8} + \cdots + 11\!\cdots\!96$$
$61$ $$T^{8} + \cdots + 36\!\cdots\!09$$
$67$ $$T^{8} + \cdots + 14\!\cdots\!96$$
$71$ $$T^{8} + \cdots + 12\!\cdots\!44$$
$73$ $$(T^{4} + \cdots + 14\!\cdots\!24)^{2}$$
$79$ $$(T^{4} + \cdots + 11\!\cdots\!20)^{2}$$
$83$ $$(T^{4} + \cdots + 11\!\cdots\!00)^{2}$$
$89$ $$T^{8} + \cdots + 24\!\cdots\!36$$
$97$ $$T^{8} + \cdots + 67\!\cdots\!64$$