# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,6,Mod(25,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("26.25");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 26.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: $$4.16997931514$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{2} + 4 q^{3} - 16 q^{4} + 34 \beta q^{5} + 8 \beta q^{6} + 41 \beta q^{7} - 32 \beta q^{8} - 227 q^{9} +O(q^{10})$$ q + 2*b * q^2 + 4 * q^3 - 16 * q^4 + 34*b * q^5 + 8*b * q^6 + 41*b * q^7 - 32*b * q^8 - 227 * q^9 $$q + 2 \beta q^{2} + 4 q^{3} - 16 q^{4} + 34 \beta q^{5} + 8 \beta q^{6} + 41 \beta q^{7} - 32 \beta q^{8} - 227 q^{9} - 272 q^{10} + 195 \beta q^{11} - 64 q^{12} + ( - 169 \beta + 507) q^{13} - 328 q^{14} + 136 \beta q^{15} + 256 q^{16} + 1738 q^{17} - 454 \beta q^{18} - 537 \beta q^{19} - 544 \beta q^{20} + 164 \beta q^{21} - 1560 q^{22} + 2104 q^{23} - 128 \beta q^{24} - 1499 q^{25} + (1014 \beta + 1352) q^{26} - 1880 q^{27} - 656 \beta q^{28} - 1690 q^{29} - 1088 q^{30} - 715 \beta q^{31} + 512 \beta q^{32} + 780 \beta q^{33} + 3476 \beta q^{34} - 5576 q^{35} + 3632 q^{36} + 4426 \beta q^{37} + 4296 q^{38} + ( - 676 \beta + 2028) q^{39} + 4352 q^{40} + 3380 \beta q^{41} - 1312 q^{42} - 16916 q^{43} - 3120 \beta q^{44} - 7718 \beta q^{45} + 4208 \beta q^{46} - 12579 \beta q^{47} + 1024 q^{48} + 10083 q^{49} - 2998 \beta q^{50} + 6952 q^{51} + (2704 \beta - 8112) q^{52} + 38214 q^{53} - 3760 \beta q^{54} - 26520 q^{55} + 5248 q^{56} - 2148 \beta q^{57} - 3380 \beta q^{58} + 10643 \beta q^{59} - 2176 \beta q^{60} - 5458 q^{61} + 5720 q^{62} - 9307 \beta q^{63} - 4096 q^{64} + (17238 \beta + 22984) q^{65} - 6240 q^{66} + 22271 \beta q^{67} - 27808 q^{68} + 8416 q^{69} - 11152 \beta q^{70} - 8895 \beta q^{71} + 7264 \beta q^{72} + 15532 \beta q^{73} - 35408 q^{74} - 5996 q^{75} + 8592 \beta q^{76} - 31980 q^{77} + (4056 \beta + 5408) q^{78} - 45360 q^{79} + 8704 \beta q^{80} + 47641 q^{81} - 27040 q^{82} - 62273 \beta q^{83} - 2624 \beta q^{84} + 59092 \beta q^{85} - 33832 \beta q^{86} - 6760 q^{87} + 24960 q^{88} - 9372 \beta q^{89} + 61744 q^{90} + (20787 \beta + 27716) q^{91} - 33664 q^{92} - 2860 \beta q^{93} + 100632 q^{94} + 73032 q^{95} + 2048 \beta q^{96} - 60744 \beta q^{97} + 20166 \beta q^{98} - 44265 \beta q^{99} +O(q^{100})$$ q + 2*b * q^2 + 4 * q^3 - 16 * q^4 + 34*b * q^5 + 8*b * q^6 + 41*b * q^7 - 32*b * q^8 - 227 * q^9 - 272 * q^10 + 195*b * q^11 - 64 * q^12 + (-169*b + 507) * q^13 - 328 * q^14 + 136*b * q^15 + 256 * q^16 + 1738 * q^17 - 454*b * q^18 - 537*b * q^19 - 544*b * q^20 + 164*b * q^21 - 1560 * q^22 + 2104 * q^23 - 128*b * q^24 - 1499 * q^25 + (1014*b + 1352) * q^26 - 1880 * q^27 - 656*b * q^28 - 1690 * q^29 - 1088 * q^30 - 715*b * q^31 + 512*b * q^32 + 780*b * q^33 + 3476*b * q^34 - 5576 * q^35 + 3632 * q^36 + 4426*b * q^37 + 4296 * q^38 + (-676*b + 2028) * q^39 + 4352 * q^40 + 3380*b * q^41 - 1312 * q^42 - 16916 * q^43 - 3120*b * q^44 - 7718*b * q^45 + 4208*b * q^46 - 12579*b * q^47 + 1024 * q^48 + 10083 * q^49 - 2998*b * q^50 + 6952 * q^51 + (2704*b - 8112) * q^52 + 38214 * q^53 - 3760*b * q^54 - 26520 * q^55 + 5248 * q^56 - 2148*b * q^57 - 3380*b * q^58 + 10643*b * q^59 - 2176*b * q^60 - 5458 * q^61 + 5720 * q^62 - 9307*b * q^63 - 4096 * q^64 + (17238*b + 22984) * q^65 - 6240 * q^66 + 22271*b * q^67 - 27808 * q^68 + 8416 * q^69 - 11152*b * q^70 - 8895*b * q^71 + 7264*b * q^72 + 15532*b * q^73 - 35408 * q^74 - 5996 * q^75 + 8592*b * q^76 - 31980 * q^77 + (4056*b + 5408) * q^78 - 45360 * q^79 + 8704*b * q^80 + 47641 * q^81 - 27040 * q^82 - 62273*b * q^83 - 2624*b * q^84 + 59092*b * q^85 - 33832*b * q^86 - 6760 * q^87 + 24960 * q^88 - 9372*b * q^89 + 61744 * q^90 + (20787*b + 27716) * q^91 - 33664 * q^92 - 2860*b * q^93 + 100632 * q^94 + 73032 * q^95 + 2048*b * q^96 - 60744*b * q^97 + 20166*b * q^98 - 44265*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{3} - 32 q^{4} - 454 q^{9}+O(q^{10})$$ 2 * q + 8 * q^3 - 32 * q^4 - 454 * q^9 $$2 q + 8 q^{3} - 32 q^{4} - 454 q^{9} - 544 q^{10} - 128 q^{12} + 1014 q^{13} - 656 q^{14} + 512 q^{16} + 3476 q^{17} - 3120 q^{22} + 4208 q^{23} - 2998 q^{25} + 2704 q^{26} - 3760 q^{27} - 3380 q^{29} - 2176 q^{30} - 11152 q^{35} + 7264 q^{36} + 8592 q^{38} + 4056 q^{39} + 8704 q^{40} - 2624 q^{42} - 33832 q^{43} + 2048 q^{48} + 20166 q^{49} + 13904 q^{51} - 16224 q^{52} + 76428 q^{53} - 53040 q^{55} + 10496 q^{56} - 10916 q^{61} + 11440 q^{62} - 8192 q^{64} + 45968 q^{65} - 12480 q^{66} - 55616 q^{68} + 16832 q^{69} - 70816 q^{74} - 11992 q^{75} - 63960 q^{77} + 10816 q^{78} - 90720 q^{79} + 95282 q^{81} - 54080 q^{82} - 13520 q^{87} + 49920 q^{88} + 123488 q^{90} + 55432 q^{91} - 67328 q^{92} + 201264 q^{94} + 146064 q^{95}+O(q^{100})$$ 2 * q + 8 * q^3 - 32 * q^4 - 454 * q^9 - 544 * q^10 - 128 * q^12 + 1014 * q^13 - 656 * q^14 + 512 * q^16 + 3476 * q^17 - 3120 * q^22 + 4208 * q^23 - 2998 * q^25 + 2704 * q^26 - 3760 * q^27 - 3380 * q^29 - 2176 * q^30 - 11152 * q^35 + 7264 * q^36 + 8592 * q^38 + 4056 * q^39 + 8704 * q^40 - 2624 * q^42 - 33832 * q^43 + 2048 * q^48 + 20166 * q^49 + 13904 * q^51 - 16224 * q^52 + 76428 * q^53 - 53040 * q^55 + 10496 * q^56 - 10916 * q^61 + 11440 * q^62 - 8192 * q^64 + 45968 * q^65 - 12480 * q^66 - 55616 * q^68 + 16832 * q^69 - 70816 * q^74 - 11992 * q^75 - 63960 * q^77 + 10816 * q^78 - 90720 * q^79 + 95282 * q^81 - 54080 * q^82 - 13520 * q^87 + 49920 * q^88 + 123488 * q^90 + 55432 * q^91 - 67328 * q^92 + 201264 * q^94 + 146064 * q^95

## Character values

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

 $$n$$ $$15$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 − 1.00000i 1.00000i
4.00000i 4.00000 −16.0000 68.0000i 16.0000i 82.0000i 64.0000i −227.000 −272.000
25.2 4.00000i 4.00000 −16.0000 68.0000i 16.0000i 82.0000i 64.0000i −227.000 −272.000
 $$n$$: e.g. 2-40 or 990-1000 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.6.b.b 2
3.b odd 2 1 234.6.b.a 2
4.b odd 2 1 208.6.f.a 2
13.b even 2 1 inner 26.6.b.b 2
13.d odd 4 1 338.6.a.b 1
13.d odd 4 1 338.6.a.e 1
39.d odd 2 1 234.6.b.a 2
52.b odd 2 1 208.6.f.a 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$(T - 4)^{2}$$
$5$ $$T^{2} + 4624$$
$7$ $$T^{2} + 6724$$
$11$ $$T^{2} + 152100$$
$13$ $$T^{2} - 1014 T + 371293$$
$17$ $$(T - 1738)^{2}$$
$19$ $$T^{2} + 1153476$$
$23$ $$(T - 2104)^{2}$$
$29$ $$(T + 1690)^{2}$$
$31$ $$T^{2} + 2044900$$
$37$ $$T^{2} + 78357904$$
$41$ $$T^{2} + 45697600$$
$43$ $$(T + 16916)^{2}$$
$47$ $$T^{2} + 632924964$$
$53$ $$(T - 38214)^{2}$$
$59$ $$T^{2} + 453093796$$
$61$ $$(T + 5458)^{2}$$
$67$ $$T^{2} + 1983989764$$
$71$ $$T^{2} + 316484100$$
$73$ $$T^{2} + 964972096$$
$79$ $$(T + 45360)^{2}$$
$83$ $$T^{2} + 15511706116$$
$89$ $$T^{2} + 351337536$$
$97$ $$T^{2} + 14759334144$$