# Properties

 Label 26.6.c.a Level $26$ Weight $6$ Character orbit 26.c Analytic conductor $4.170$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 314x^{4} + 24649x^{2} + 245388$$ x^6 + 314*x^4 + 24649*x^2 + 245388 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 4 \beta_1 q^{2} + \beta_{3} q^{3} + (16 \beta_1 - 16) q^{4} + (\beta_{4} - \beta_{3} + \beta_{2}) q^{5} - 4 \beta_{2} q^{6} + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - 68) q^{7}+ \cdots + ( - 3 \beta_{5} + 3 \beta_{4} + \cdots - 72) q^{9}+O(q^{10})$$ q - 4*b1 * q^2 + b3 * q^3 + (16*b1 - 16) * q^4 + (b4 - b3 + b2) * q^5 - 4*b2 * q^6 + (-2*b5 + 2*b4 - b2 + 68*b1 - 68) * q^7 + 64 * q^8 + (-3*b5 + 3*b4 - 9*b2 + 72*b1 - 72) * q^9 $$q - 4 \beta_1 q^{2} + \beta_{3} q^{3} + (16 \beta_1 - 16) q^{4} + (\beta_{4} - \beta_{3} + \beta_{2}) q^{5} - 4 \beta_{2} q^{6} + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - 68) q^{7}+ \cdots + (330 \beta_{4} - 3906 \beta_{3} + \cdots + 29790) q^{99}+O(q^{100})$$ q - 4*b1 * q^2 + b3 * q^3 + (16*b1 - 16) * q^4 + (b4 - b3 + b2) * q^5 - 4*b2 * q^6 + (-2*b5 + 2*b4 - b2 + 68*b1 - 68) * q^7 + 64 * q^8 + (-3*b5 + 3*b4 - 9*b2 + 72*b1 - 72) * q^9 + (-4*b5 + 4*b3) * q^10 + (-2*b5 + 23*b3 + 168*b1) * q^11 + (-16*b3 + 16*b2) * q^12 + (b5 + 9*b4 + 8*b3 - 15*b2 - 144*b1 - 113) * q^13 + (-8*b4 - 4*b3 + 4*b2 + 272) * q^14 + (12*b5 - 16*b3 - 402*b1) * q^15 - 256*b1 * q^16 + (18*b5 - 18*b4 - 24*b2 + 69*b1 - 69) * q^17 + (-12*b4 - 36*b3 + 36*b2 + 288) * q^18 + (14*b5 - 14*b4 + 73*b2 + 350*b1 - 350) * q^19 + (16*b5 - 16*b4 - 16*b2) * q^20 + (15*b4 - 127*b3 + 127*b2 + 141) * q^21 + (8*b5 - 8*b4 - 92*b2 - 672*b1 + 672) * q^22 + (-28*b5 + 121*b3 - 1704*b1) * q^23 + 64*b3 * q^24 + (5*b4 + 145*b3 - 145*b2 + 511) * q^25 + (-40*b5 + 4*b4 - 60*b3 + 28*b2 + 1028*b1 - 576) * q^26 + (15*b3 - 15*b2 + 2574) * q^27 + (32*b5 + 16*b3 - 1088*b1) * q^28 + (-16*b5 - 266*b3 + 1923*b1) * q^29 + (-48*b5 + 48*b4 + 64*b2 + 1608*b1 - 1608) * q^30 + (-2*b4 + 26*b3 - 26*b2 - 2524) * q^31 + (1024*b1 - 1024) * q^32 + (-87*b5 + 87*b4 + 11*b2 + 7419*b1 - 7419) * q^33 + (72*b4 - 96*b3 + 96*b2 + 276) * q^34 + (22*b5 - 22*b4 - 310*b2 - 6066*b1 + 6066) * q^35 + (48*b5 + 144*b3 - 1152*b1) * q^36 + (22*b5 + 386*b3 - 1295*b1) * q^37 + (56*b4 + 292*b3 - 292*b2 + 1400) * q^38 + (66*b5 - 30*b4 - 473*b3 - 106*b2 + 1650*b1 + 2292) * q^39 + (64*b4 - 64*b3 + 64*b2) * q^40 + (-198*b5 + 264*b3 - 3081*b1) * q^41 + (-60*b5 + 508*b3 - 564*b1) * q^42 + (106*b5 - 106*b4 + 743*b2 + 4160*b1 - 4160) * q^43 + (32*b4 - 368*b3 + 368*b2 - 2688) * q^44 + (-87*b5 + 87*b4 - 315*b2 - 6084*b1 + 6084) * q^45 + (112*b5 - 112*b4 - 484*b2 + 6816*b1 - 6816) * q^46 + (-268*b4 - 674*b3 + 674*b2 + 2538) * q^47 - 256*b2 * q^48 + (135*b5 - 171*b3 - 372*b1) * q^49 + (-20*b5 - 580*b3 - 2044*b1) * q^50 + (-234*b4 + 165*b3 - 165*b2 + 9126) * q^51 + (144*b5 - 160*b4 + 112*b3 + 128*b2 - 1808*b1 + 4112) * q^52 + (33*b4 + 51*b3 - 51*b2 + 18756) * q^53 + (-60*b3 - 10296*b1) * q^54 + (410*b5 - 794*b3 - 15714*b1) * q^55 + (-128*b5 + 128*b4 - 64*b2 + 4352*b1 - 4352) * q^56 + (93*b4 + 657*b3 - 657*b2 - 21777) * q^57 + (64*b5 - 64*b4 + 1064*b2 - 7692*b1 + 7692) * q^58 + (160*b5 - 160*b4 + 623*b2 + 30714*b1 - 30714) * q^59 + (-192*b4 + 256*b3 - 256*b2 + 6432) * q^60 + (-200*b5 + 200*b4 + 1226*b2 - 3679*b1 + 3679) * q^61 + (8*b5 - 104*b3 + 10096*b1) * q^62 + (30*b5 + 666*b3 - 24786*b1) * q^63 + 4096 * q^64 + (-211*b5 + 220*b4 + 1419*b3 - 1034*b2 + 6048*b1 + 23076) * q^65 + (-348*b4 + 44*b3 - 44*b2 + 29676) * q^66 + (530*b5 - 359*b3 - 26546*b1) * q^67 + (-288*b5 + 384*b3 - 1104*b1) * q^68 + (-615*b5 + 615*b4 - 2093*b2 + 40551*b1 - 40551) * q^69 + (88*b4 - 1240*b3 + 1240*b2 - 24264) * q^70 + (-274*b5 + 274*b4 - 239*b2 - 1002*b1 + 1002) * q^71 + (-192*b5 + 192*b4 - 576*b2 + 4608*b1 - 4608) * q^72 + (531*b4 + 2613*b3 - 2613*b2 + 4112) * q^73 + (-88*b5 + 88*b4 - 1544*b2 + 5180*b1 - 5180) * q^74 + (-390*b5 - 919*b3 + 45240*b1) * q^75 + (-224*b5 - 1168*b3 - 5600*b1) * q^76 + (695*b4 - 3119*b3 + 3119*b2 - 20595) * q^77 + (-144*b5 + 264*b4 - 424*b3 + 2316*b2 - 15768*b1 + 6600) * q^78 + (234*b4 + 120*b3 - 120*b2 + 31430) * q^79 + (-256*b5 + 256*b3) * q^80 + (-774*b5 + 252*b3 + 22221*b1) * q^81 + (792*b5 - 792*b4 - 1056*b2 + 12324*b1 - 12324) * q^82 + (-588*b4 + 2928*b3 - 2928*b2 - 14544) * q^83 + (240*b5 - 240*b4 - 2032*b2 + 2256*b1 - 2256) * q^84 + (87*b5 - 87*b4 + 2637*b2 + 67860*b1 - 67860) * q^85 + (424*b4 + 2972*b3 - 2972*b2 + 16640) * q^86 + (654*b5 - 654*b4 + 4717*b2 - 82398*b1 + 82398) * q^87 + (-128*b5 + 1472*b3 + 10752*b1) * q^88 + (9*b5 - 1299*b3 - 111135*b1) * q^89 + (-348*b4 - 1260*b3 + 1260*b2 - 24336) * q^90 + (64*b5 - 464*b4 + 928*b3 - 557*b2 - 65662*b1 + 80674) * q^91 + (448*b4 - 1936*b3 + 1936*b2 + 27264) * q^92 + (-96*b5 - 2708*b3 + 8364*b1) * q^93 + (1072*b5 + 2696*b3 - 10152*b1) * q^94 + (1464*b5 - 1464*b4 + 288*b2 + 15930*b1 - 15930) * q^95 + (-1024*b3 + 1024*b2) * q^96 + (1863*b5 - 1863*b4 - 1917*b2 + 35831*b1 - 35831) * q^97 + (-540*b5 + 540*b4 + 684*b2 + 1488*b1 - 1488) * q^98 + (330*b4 - 3906*b3 + 3906*b2 + 29790) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 12 q^{2} - 48 q^{4} + 2 q^{5} - 202 q^{7} + 384 q^{8} - 213 q^{9}+O(q^{10})$$ 6 * q - 12 * q^2 - 48 * q^4 + 2 * q^5 - 202 * q^7 + 384 * q^8 - 213 * q^9 $$6 q - 12 q^{2} - 48 q^{4} + 2 q^{5} - 202 q^{7} + 384 q^{8} - 213 q^{9} - 4 q^{10} + 502 q^{11} - 1091 q^{13} + 1616 q^{14} - 1194 q^{15} - 768 q^{16} - 225 q^{17} + 1704 q^{18} - 1064 q^{19} - 16 q^{20} + 876 q^{21} + 2008 q^{22} - 5140 q^{23} + 3076 q^{25} - 404 q^{26} + 15444 q^{27} - 3232 q^{28} + 5753 q^{29} - 4776 q^{30} - 15148 q^{31} - 3072 q^{32} - 22170 q^{33} + 1800 q^{34} + 18176 q^{35} - 3408 q^{36} - 3863 q^{37} + 8512 q^{38} + 18708 q^{39} + 128 q^{40} - 9441 q^{41} - 1752 q^{42} - 12586 q^{43} - 16064 q^{44} + 18339 q^{45} - 20560 q^{46} + 14692 q^{47} - 981 q^{49} - 6152 q^{50} + 54288 q^{51} + 19072 q^{52} + 112602 q^{53} - 30888 q^{54} - 46732 q^{55} - 12928 q^{56} - 130476 q^{57} + 23012 q^{58} - 92302 q^{59} + 38208 q^{60} + 11237 q^{61} + 30296 q^{62} - 74328 q^{63} + 24576 q^{64} + 156829 q^{65} + 177360 q^{66} - 79108 q^{67} - 3600 q^{68} - 121038 q^{69} - 145408 q^{70} + 3280 q^{71} - 13632 q^{72} + 25734 q^{73} - 15452 q^{74} + 135330 q^{75} - 17024 q^{76} - 122180 q^{77} - 7320 q^{78} + 189048 q^{79} - 256 q^{80} + 65889 q^{81} - 37764 q^{82} - 88440 q^{83} - 7008 q^{84} - 203667 q^{85} + 100688 q^{86} + 246540 q^{87} + 32128 q^{88} - 333396 q^{89} - 146712 q^{90} + 286194 q^{91} + 164480 q^{92} + 24996 q^{93} - 29384 q^{94} - 49254 q^{95} - 109356 q^{97} - 3924 q^{98} + 179400 q^{99}+O(q^{100})$$ 6 * q - 12 * q^2 - 48 * q^4 + 2 * q^5 - 202 * q^7 + 384 * q^8 - 213 * q^9 - 4 * q^10 + 502 * q^11 - 1091 * q^13 + 1616 * q^14 - 1194 * q^15 - 768 * q^16 - 225 * q^17 + 1704 * q^18 - 1064 * q^19 - 16 * q^20 + 876 * q^21 + 2008 * q^22 - 5140 * q^23 + 3076 * q^25 - 404 * q^26 + 15444 * q^27 - 3232 * q^28 + 5753 * q^29 - 4776 * q^30 - 15148 * q^31 - 3072 * q^32 - 22170 * q^33 + 1800 * q^34 + 18176 * q^35 - 3408 * q^36 - 3863 * q^37 + 8512 * q^38 + 18708 * q^39 + 128 * q^40 - 9441 * q^41 - 1752 * q^42 - 12586 * q^43 - 16064 * q^44 + 18339 * q^45 - 20560 * q^46 + 14692 * q^47 - 981 * q^49 - 6152 * q^50 + 54288 * q^51 + 19072 * q^52 + 112602 * q^53 - 30888 * q^54 - 46732 * q^55 - 12928 * q^56 - 130476 * q^57 + 23012 * q^58 - 92302 * q^59 + 38208 * q^60 + 11237 * q^61 + 30296 * q^62 - 74328 * q^63 + 24576 * q^64 + 156829 * q^65 + 177360 * q^66 - 79108 * q^67 - 3600 * q^68 - 121038 * q^69 - 145408 * q^70 + 3280 * q^71 - 13632 * q^72 + 25734 * q^73 - 15452 * q^74 + 135330 * q^75 - 17024 * q^76 - 122180 * q^77 - 7320 * q^78 + 189048 * q^79 - 256 * q^80 + 65889 * q^81 - 37764 * q^82 - 88440 * q^83 - 7008 * q^84 - 203667 * q^85 + 100688 * q^86 + 246540 * q^87 + 32128 * q^88 - 333396 * q^89 - 146712 * q^90 + 286194 * q^91 + 164480 * q^92 + 24996 * q^93 - 29384 * q^94 - 49254 * q^95 - 109356 * q^97 - 3924 * q^98 + 179400 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 314x^{4} + 24649x^{2} + 245388$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 157\nu + 286 ) / 572$$ (v^3 + 157*v + 286) / 572 $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 157\nu^{2} + 858\nu ) / 572$$ (v^4 + 157*v^2 + 858*v) / 572 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} - 157\nu^{2} + 858\nu ) / 572$$ (-v^4 - 157*v^2 + 858*v) / 572 $$\beta_{4}$$ $$=$$ $$( 3\nu^{4} + 757\nu^{2} + 30030 ) / 286$$ (3*v^4 + 757*v^2 + 30030) / 286 $$\beta_{5}$$ $$=$$ $$( \nu^{5} + 3\nu^{4} + 262\nu^{3} + 757\nu^{2} + 13911\nu + 30030 ) / 572$$ (v^5 + 3*v^4 + 262*v^3 + 757*v^2 + 13911*v + 30030) / 572
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{4} + 3\beta_{3} - 3\beta_{2} - 105$$ b4 + 3*b3 - 3*b2 - 105 $$\nu^{3}$$ $$=$$ $$( -157\beta_{3} - 157\beta_{2} + 1716\beta _1 - 858 ) / 3$$ (-157*b3 - 157*b2 + 1716*b1 - 858) / 3 $$\nu^{4}$$ $$=$$ $$-157\beta_{4} - 757\beta_{3} + 757\beta_{2} + 16485$$ -157*b4 - 757*b3 + 757*b2 + 16485 $$\nu^{5}$$ $$=$$ $$( 1716\beta_{5} - 858\beta_{4} + 27223\beta_{3} + 27223\beta_{2} - 449592\beta _1 + 224796 ) / 3$$ (1716*b5 - 858*b4 + 27223*b3 + 27223*b2 - 449592*b1 + 224796) / 3

## Character values

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

 $$n$$ $$15$$ $$\chi(n)$$ $$-1 + \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 − 13.8811i 3.40712i 10.4740i 13.8811i − 3.40712i − 10.4740i
−2.00000 3.46410i −12.0214 20.8217i −8.00000 + 13.8564i 8.48514 −48.0857 + 83.2869i −61.5791 + 106.658i 64.0000 −167.529 + 290.169i −16.9703 29.3934i
3.2 −2.00000 3.46410i 2.95065 + 5.11068i −8.00000 + 13.8564i 69.7863 11.8026 20.4427i 44.6383 77.3157i 64.0000 104.087 180.284i −139.573 241.747i
3.3 −2.00000 3.46410i 9.07077 + 15.7110i −8.00000 + 13.8564i −77.2714 36.2831 62.8442i −84.0591 + 145.595i 64.0000 −43.0579 + 74.5784i 154.543 + 267.676i
9.1 −2.00000 + 3.46410i −12.0214 + 20.8217i −8.00000 13.8564i 8.48514 −48.0857 83.2869i −61.5791 106.658i 64.0000 −167.529 290.169i −16.9703 + 29.3934i
9.2 −2.00000 + 3.46410i 2.95065 5.11068i −8.00000 13.8564i 69.7863 11.8026 + 20.4427i 44.6383 + 77.3157i 64.0000 104.087 + 180.284i −139.573 + 241.747i
9.3 −2.00000 + 3.46410i 9.07077 15.7110i −8.00000 13.8564i −77.2714 36.2831 + 62.8442i −84.0591 145.595i 64.0000 −43.0579 74.5784i 154.543 267.676i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.3 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.a 6
3.b odd 2 1 234.6.h.d 6
4.b odd 2 1 208.6.i.a 6
13.c even 3 1 inner 26.6.c.a 6
13.c even 3 1 338.6.a.i 3
13.e even 6 1 338.6.a.h 3
13.f odd 12 2 338.6.b.d 6
39.i odd 6 1 234.6.h.d 6
52.j odd 6 1 208.6.i.a 6

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 471T_{3}^{4} - 5148T_{3}^{3} + 221841T_{3}^{2} - 1212354T_{3} + 6625476$$ acting on $$S_{6}^{\mathrm{new}}(26, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 4 T + 16)^{3}$$
$3$ $$T^{6} + 471 T^{4} + \cdots + 6625476$$
$5$ $$(T^{3} - T^{2} + \cdots + 45756)^{2}$$
$7$ $$T^{6} + \cdots + 3416893098256$$
$11$ $$T^{6} + \cdots + 14\!\cdots\!56$$
$13$ $$T^{6} + \cdots + 51\!\cdots\!57$$
$17$ $$T^{6} + \cdots + 45\!\cdots\!21$$
$19$ $$T^{6} + \cdots + 63\!\cdots\!00$$
$23$ $$T^{6} + \cdots + 77\!\cdots\!00$$
$29$ $$T^{6} + \cdots + 20\!\cdots\!25$$
$31$ $$(T^{3} + 7574 T^{2} + \cdots + 15208306048)^{2}$$
$37$ $$T^{6} + \cdots + 16\!\cdots\!41$$
$41$ $$T^{6} + \cdots + 19\!\cdots\!69$$
$43$ $$T^{6} + \cdots + 83\!\cdots\!04$$
$47$ $$(T^{3} + \cdots + 3064802787936)^{2}$$
$53$ $$(T^{3} + \cdots - 6503035859892)^{2}$$
$59$ $$T^{6} + \cdots + 40\!\cdots\!00$$
$61$ $$T^{6} + \cdots + 11\!\cdots\!41$$
$67$ $$T^{6} + \cdots + 37\!\cdots\!36$$
$71$ $$T^{6} + \cdots + 51\!\cdots\!44$$
$73$ $$(T^{3} + \cdots + 80467235311348)^{2}$$
$79$ $$(T^{3} + \cdots - 24544367880128)^{2}$$
$83$ $$(T^{3} + \cdots - 232191799909056)^{2}$$
$89$ $$T^{6} + \cdots + 16\!\cdots\!76$$
$97$ $$T^{6} + \cdots + 91\!\cdots\!00$$