# Properties

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

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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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} - 13 q^{3} - 16 q^{4} - 51 i q^{5} - 52 i q^{6} - 105 i q^{7} - 64 i q^{8} - 74 q^{9} +O(q^{10})$$ q + 4*i * q^2 - 13 * q^3 - 16 * q^4 - 51*i * q^5 - 52*i * q^6 - 105*i * q^7 - 64*i * q^8 - 74 * q^9 $$q + 4 i q^{2} - 13 q^{3} - 16 q^{4} - 51 i q^{5} - 52 i q^{6} - 105 i q^{7} - 64 i q^{8} - 74 q^{9} + 204 q^{10} - 120 i q^{11} + 208 q^{12} + ( - 117 i - 598) q^{13} + 420 q^{14} + 663 i q^{15} + 256 q^{16} - 1101 q^{17} - 296 i q^{18} + 1170 i q^{19} + 816 i q^{20} + 1365 i q^{21} + 480 q^{22} + 1050 q^{23} + 832 i q^{24} + 524 q^{25} + ( - 2392 i + 468) q^{26} + 4121 q^{27} + 1680 i q^{28} - 4104 q^{29} - 2652 q^{30} - 9624 i q^{31} + 1024 i q^{32} + 1560 i q^{33} - 4404 i q^{34} - 5355 q^{35} + 1184 q^{36} - 8709 i q^{37} - 4680 q^{38} + (1521 i + 7774) q^{39} - 3264 q^{40} + 9480 i q^{41} - 5460 q^{42} + 9995 q^{43} + 1920 i q^{44} + 3774 i q^{45} + 4200 i q^{46} + 2943 i q^{47} - 3328 q^{48} + 5782 q^{49} + 2096 i q^{50} + 14313 q^{51} + (1872 i + 9568) q^{52} - 750 q^{53} + 16484 i q^{54} - 6120 q^{55} - 6720 q^{56} - 15210 i q^{57} - 16416 i q^{58} + 40938 i q^{59} - 10608 i q^{60} - 57920 q^{61} + 38496 q^{62} + 7770 i q^{63} - 4096 q^{64} + (30498 i - 5967) q^{65} - 6240 q^{66} - 22812 i q^{67} + 17616 q^{68} - 13650 q^{69} - 21420 i q^{70} - 63741 i q^{71} + 4736 i q^{72} - 58866 i q^{73} + 34836 q^{74} - 6812 q^{75} - 18720 i q^{76} - 12600 q^{77} + (31096 i - 6084) q^{78} + 63202 q^{79} - 13056 i q^{80} - 35591 q^{81} - 37920 q^{82} - 55458 i q^{83} - 21840 i q^{84} + 56151 i q^{85} + 39980 i q^{86} + 53352 q^{87} - 7680 q^{88} + 104778 i q^{89} - 15096 q^{90} + (62790 i - 12285) q^{91} - 16800 q^{92} + 125112 i q^{93} - 11772 q^{94} + 59670 q^{95} - 13312 i q^{96} - 160452 i q^{97} + 23128 i q^{98} + 8880 i q^{99} +O(q^{100})$$ q + 4*i * q^2 - 13 * q^3 - 16 * q^4 - 51*i * q^5 - 52*i * q^6 - 105*i * q^7 - 64*i * q^8 - 74 * q^9 + 204 * q^10 - 120*i * q^11 + 208 * q^12 + (-117*i - 598) * q^13 + 420 * q^14 + 663*i * q^15 + 256 * q^16 - 1101 * q^17 - 296*i * q^18 + 1170*i * q^19 + 816*i * q^20 + 1365*i * q^21 + 480 * q^22 + 1050 * q^23 + 832*i * q^24 + 524 * q^25 + (-2392*i + 468) * q^26 + 4121 * q^27 + 1680*i * q^28 - 4104 * q^29 - 2652 * q^30 - 9624*i * q^31 + 1024*i * q^32 + 1560*i * q^33 - 4404*i * q^34 - 5355 * q^35 + 1184 * q^36 - 8709*i * q^37 - 4680 * q^38 + (1521*i + 7774) * q^39 - 3264 * q^40 + 9480*i * q^41 - 5460 * q^42 + 9995 * q^43 + 1920*i * q^44 + 3774*i * q^45 + 4200*i * q^46 + 2943*i * q^47 - 3328 * q^48 + 5782 * q^49 + 2096*i * q^50 + 14313 * q^51 + (1872*i + 9568) * q^52 - 750 * q^53 + 16484*i * q^54 - 6120 * q^55 - 6720 * q^56 - 15210*i * q^57 - 16416*i * q^58 + 40938*i * q^59 - 10608*i * q^60 - 57920 * q^61 + 38496 * q^62 + 7770*i * q^63 - 4096 * q^64 + (30498*i - 5967) * q^65 - 6240 * q^66 - 22812*i * q^67 + 17616 * q^68 - 13650 * q^69 - 21420*i * q^70 - 63741*i * q^71 + 4736*i * q^72 - 58866*i * q^73 + 34836 * q^74 - 6812 * q^75 - 18720*i * q^76 - 12600 * q^77 + (31096*i - 6084) * q^78 + 63202 * q^79 - 13056*i * q^80 - 35591 * q^81 - 37920 * q^82 - 55458*i * q^83 - 21840*i * q^84 + 56151*i * q^85 + 39980*i * q^86 + 53352 * q^87 - 7680 * q^88 + 104778*i * q^89 - 15096 * q^90 + (62790*i - 12285) * q^91 - 16800 * q^92 + 125112*i * q^93 - 11772 * q^94 + 59670 * q^95 - 13312*i * q^96 - 160452*i * q^97 + 23128*i * q^98 + 8880*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 26 q^{3} - 32 q^{4} - 148 q^{9}+O(q^{10})$$ 2 * q - 26 * q^3 - 32 * q^4 - 148 * q^9 $$2 q - 26 q^{3} - 32 q^{4} - 148 q^{9} + 408 q^{10} + 416 q^{12} - 1196 q^{13} + 840 q^{14} + 512 q^{16} - 2202 q^{17} + 960 q^{22} + 2100 q^{23} + 1048 q^{25} + 936 q^{26} + 8242 q^{27} - 8208 q^{29} - 5304 q^{30} - 10710 q^{35} + 2368 q^{36} - 9360 q^{38} + 15548 q^{39} - 6528 q^{40} - 10920 q^{42} + 19990 q^{43} - 6656 q^{48} + 11564 q^{49} + 28626 q^{51} + 19136 q^{52} - 1500 q^{53} - 12240 q^{55} - 13440 q^{56} - 115840 q^{61} + 76992 q^{62} - 8192 q^{64} - 11934 q^{65} - 12480 q^{66} + 35232 q^{68} - 27300 q^{69} + 69672 q^{74} - 13624 q^{75} - 25200 q^{77} - 12168 q^{78} + 126404 q^{79} - 71182 q^{81} - 75840 q^{82} + 106704 q^{87} - 15360 q^{88} - 30192 q^{90} - 24570 q^{91} - 33600 q^{92} - 23544 q^{94} + 119340 q^{95}+O(q^{100})$$ 2 * q - 26 * q^3 - 32 * q^4 - 148 * q^9 + 408 * q^10 + 416 * q^12 - 1196 * q^13 + 840 * q^14 + 512 * q^16 - 2202 * q^17 + 960 * q^22 + 2100 * q^23 + 1048 * q^25 + 936 * q^26 + 8242 * q^27 - 8208 * q^29 - 5304 * q^30 - 10710 * q^35 + 2368 * q^36 - 9360 * q^38 + 15548 * q^39 - 6528 * q^40 - 10920 * q^42 + 19990 * q^43 - 6656 * q^48 + 11564 * q^49 + 28626 * q^51 + 19136 * q^52 - 1500 * q^53 - 12240 * q^55 - 13440 * q^56 - 115840 * q^61 + 76992 * q^62 - 8192 * q^64 - 11934 * q^65 - 12480 * q^66 + 35232 * q^68 - 27300 * q^69 + 69672 * q^74 - 13624 * q^75 - 25200 * q^77 - 12168 * q^78 + 126404 * q^79 - 71182 * q^81 - 75840 * q^82 + 106704 * q^87 - 15360 * q^88 - 30192 * q^90 - 24570 * q^91 - 33600 * q^92 - 23544 * q^94 + 119340 * 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 −13.0000 −16.0000 51.0000i 52.0000i 105.000i 64.0000i −74.0000 204.000
25.2 4.00000i −13.0000 −16.0000 51.0000i 52.0000i 105.000i 64.0000i −74.0000 204.000
 $$n$$: 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.a 2
3.b odd 2 1 234.6.b.b 2
4.b odd 2 1 208.6.f.b 2
13.b even 2 1 inner 26.6.b.a 2
13.d odd 4 1 338.6.a.a 1
13.d odd 4 1 338.6.a.c 1
39.d odd 2 1 234.6.b.b 2
52.b odd 2 1 208.6.f.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 13$$ acting on $$S_{6}^{\mathrm{new}}(26, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$(T + 13)^{2}$$
$5$ $$T^{2} + 2601$$
$7$ $$T^{2} + 11025$$
$11$ $$T^{2} + 14400$$
$13$ $$T^{2} + 1196 T + 371293$$
$17$ $$(T + 1101)^{2}$$
$19$ $$T^{2} + 1368900$$
$23$ $$(T - 1050)^{2}$$
$29$ $$(T + 4104)^{2}$$
$31$ $$T^{2} + 92621376$$
$37$ $$T^{2} + 75846681$$
$41$ $$T^{2} + 89870400$$
$43$ $$(T - 9995)^{2}$$
$47$ $$T^{2} + 8661249$$
$53$ $$(T + 750)^{2}$$
$59$ $$T^{2} + 1675919844$$
$61$ $$(T + 57920)^{2}$$
$67$ $$T^{2} + 520387344$$
$71$ $$T^{2} + 4062915081$$
$73$ $$T^{2} + 3465205956$$
$79$ $$(T - 63202)^{2}$$
$83$ $$T^{2} + 3075589764$$
$89$ $$T^{2} + 10978429284$$
$97$ $$T^{2} + 25744844304$$
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