# Properties

 Label 26.4.c.b Level $26$ Weight $4$ Character orbit 26.c Analytic conductor $1.534$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,4,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, 4, names="a")

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

chi := DirichletCharacter("26.3");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$1.53404966015$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{217})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 55x^{2} + 54x + 2916$$ x^4 - x^3 + 55*x^2 + 54*x + 2916 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} + (4 \beta_{2} - 4) q^{4} + ( - \beta_{3} - 3) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 4) q^{6}+ \cdots + (3 \beta_{3} + 28 \beta_{2} + \cdots - 31) q^{9}+O(q^{10})$$ q - 2*b2 * q^2 + (b2 + b1) * q^3 + (4*b2 - 4) * q^4 + (-b3 - 3) * q^5 + (-2*b3 - 2*b2 - 2*b1 + 4) * q^6 + (b3 - 23*b2 + b1 + 22) * q^7 + 8 * q^8 + (3*b3 + 28*b2 + 3*b1 - 31) * q^9 $$q - 2 \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} + (4 \beta_{2} - 4) q^{4} + ( - \beta_{3} - 3) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 4) q^{6}+ \cdots + ( - 214 \beta_{3} + 1320) q^{99}+O(q^{100})$$ q - 2*b2 * q^2 + (b2 + b1) * q^3 + (4*b2 - 4) * q^4 + (-b3 - 3) * q^5 + (-2*b3 - 2*b2 - 2*b1 + 4) * q^6 + (b3 - 23*b2 + b1 + 22) * q^7 + 8 * q^8 + (3*b3 + 28*b2 + 3*b1 - 31) * q^9 + (8*b2 - 2*b1) * q^10 + (b2 - 7*b1) * q^11 + (4*b3 - 8) * q^12 + (5*b3 - 6*b2 + 7*b1 - 10) * q^13 + (-2*b3 - 44) * q^14 + (50*b2 - 2*b1) * q^15 - 16*b2 * q^16 + (-8*b3 - 61*b2 - 8*b1 + 69) * q^17 + (-6*b3 + 62) * q^18 + (b3 + b2 + b1 - 2) * q^19 + (4*b3 - 16*b2 + 4*b1 + 12) * q^20 + (-21*b3 - 10) * q^21 + (14*b3 - 2*b2 + 14*b1 - 12) * q^22 + (-15*b2 - 3*b1) * q^23 + (8*b2 + 8*b1) * q^24 + (7*b3 - 62) * q^25 + (-14*b3 + 22*b2 - 4*b1 + 2) * q^26 + (7*b3 - 170) * q^27 + (92*b2 - 4*b1) * q^28 + (93*b2 + 12*b1) * q^29 + (4*b3 - 100*b2 + 4*b1 + 96) * q^30 + (24*b3 + 128) * q^31 + (32*b2 - 32) * q^32 + (-13*b3 - 377*b2 - 13*b1 + 390) * q^33 + (16*b3 - 138) * q^34 + (-26*b3 + 146*b2 - 26*b1 - 120) * q^35 + (-112*b2 - 12*b1) * q^36 + (353*b2 - 4*b1) * q^37 + (-2*b3 + 4) * q^38 + (8*b3 + 97*b2 - 7*b1 - 380) * q^39 + (-8*b3 - 24) * q^40 + (-131*b2 + 20*b1) * q^41 + (62*b2 - 42*b1) * q^42 + (25*b3 - 59*b2 + 25*b1 + 34) * q^43 + (-28*b3 + 24) * q^44 + (19*b3 + 50*b2 + 19*b1 - 69) * q^45 + (6*b3 + 30*b2 + 6*b1 - 36) * q^46 + (56*b3 + 84) * q^47 + (-16*b3 - 16*b2 - 16*b1 + 32) * q^48 + (-240*b2 + 45*b1) * q^49 + (110*b2 + 14*b1) * q^50 + (-77*b3 + 570) * q^51 + (8*b3 - 20*b2 - 20*b1 + 36) * q^52 + (-b3 - 267) * q^53 + (326*b2 + 14*b1) * q^54 + (-382*b2 + 22*b1) * q^55 + (8*b3 - 184*b2 + 8*b1 + 176) * q^56 + (3*b3 - 58) * q^57 + (-24*b3 - 186*b2 - 24*b1 + 210) * q^58 + (-7*b3 - 191*b2 - 7*b1 + 198) * q^59 + (-8*b3 - 192) * q^60 + (-32*b3 + 343*b2 - 32*b1 - 311) * q^61 + (-304*b2 + 48*b1) * q^62 + (482*b2 + 38*b1) * q^63 + 64 * q^64 + (-10*b3 + 402*b2 - 27*b1 - 240) * q^65 + (26*b3 - 780) * q^66 + (85*b2 - 63*b1) * q^67 + (244*b2 + 32*b1) * q^68 + (-21*b3 - 177*b2 - 21*b1 + 198) * q^69 + (52*b3 + 240) * q^70 + (-19*b3 - 275*b2 - 19*b1 + 294) * q^71 + (24*b3 + 224*b2 + 24*b1 - 248) * q^72 + (15*b3 - 319) * q^73 + (8*b3 - 706*b2 + 8*b1 + 698) * q^74 + (-433*b2 - 69*b1) * q^75 + (-4*b2 - 4*b1) * q^76 + (155*b3 + 246) * q^77 + (14*b3 + 550*b2 + 30*b1 + 180) * q^78 + (-48*b3 + 404) * q^79 + (64*b2 - 16*b1) * q^80 + (215*b2 - 96*b1) * q^81 + (-40*b3 + 262*b2 - 40*b1 - 222) * q^82 + (-36*b3 - 864) * q^83 + (84*b3 - 124*b2 + 84*b1 + 40) * q^84 + (-37*b3 - 188*b2 - 37*b1 + 225) * q^85 + (-50*b3 - 68) * q^86 + (117*b3 + 741*b2 + 117*b1 - 858) * q^87 + (8*b2 - 56*b1) * q^88 + (451*b2 - 31*b1) * q^89 + (-38*b3 + 138) * q^90 + (-55*b3 - 155*b2 + 105*b1 - 306) * q^91 + (-12*b3 + 72) * q^92 + (-1144*b2 + 104*b1) * q^93 + (-280*b2 + 112*b1) * q^94 + (-2*b3 + 50*b2 - 2*b1 - 48) * q^95 + (32*b3 - 64) * q^96 + (-41*b3 - 419*b2 - 41*b1 + 460) * q^97 + (-90*b3 + 480*b2 - 90*b1 - 390) * q^98 + (-214*b3 + 1320) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 3 q^{3} - 8 q^{4} - 14 q^{5} + 6 q^{6} + 45 q^{7} + 32 q^{8} - 59 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 3 * q^3 - 8 * q^4 - 14 * q^5 + 6 * q^6 + 45 * q^7 + 32 * q^8 - 59 * q^9 $$4 q - 4 q^{2} + 3 q^{3} - 8 q^{4} - 14 q^{5} + 6 q^{6} + 45 q^{7} + 32 q^{8} - 59 q^{9} + 14 q^{10} - 5 q^{11} - 24 q^{12} - 35 q^{13} - 180 q^{14} + 98 q^{15} - 32 q^{16} + 130 q^{17} + 236 q^{18} - 3 q^{19} + 28 q^{20} - 82 q^{21} - 10 q^{22} - 33 q^{23} + 24 q^{24} - 234 q^{25} + 20 q^{26} - 666 q^{27} + 180 q^{28} + 198 q^{29} + 196 q^{30} + 560 q^{31} - 64 q^{32} + 767 q^{33} - 520 q^{34} - 266 q^{35} - 236 q^{36} + 702 q^{37} + 12 q^{38} - 1317 q^{39} - 112 q^{40} - 242 q^{41} + 82 q^{42} + 93 q^{43} + 40 q^{44} - 119 q^{45} - 66 q^{46} + 448 q^{47} + 48 q^{48} - 435 q^{49} + 234 q^{50} + 2126 q^{51} + 100 q^{52} - 1070 q^{53} + 666 q^{54} - 742 q^{55} + 360 q^{56} - 226 q^{57} + 396 q^{58} + 389 q^{59} - 784 q^{60} - 654 q^{61} - 560 q^{62} + 1002 q^{63} + 256 q^{64} - 203 q^{65} - 3068 q^{66} + 107 q^{67} + 520 q^{68} + 375 q^{69} + 1064 q^{70} + 569 q^{71} - 472 q^{72} - 1246 q^{73} + 1404 q^{74} - 935 q^{75} - 12 q^{76} + 1294 q^{77} + 1878 q^{78} + 1520 q^{79} + 112 q^{80} + 334 q^{81} - 484 q^{82} - 3528 q^{83} + 164 q^{84} + 413 q^{85} - 372 q^{86} - 1599 q^{87} - 40 q^{88} + 871 q^{89} + 476 q^{90} - 1539 q^{91} + 264 q^{92} - 2184 q^{93} - 448 q^{94} - 98 q^{95} - 192 q^{96} + 879 q^{97} - 870 q^{98} + 4852 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 3 * q^3 - 8 * q^4 - 14 * q^5 + 6 * q^6 + 45 * q^7 + 32 * q^8 - 59 * q^9 + 14 * q^10 - 5 * q^11 - 24 * q^12 - 35 * q^13 - 180 * q^14 + 98 * q^15 - 32 * q^16 + 130 * q^17 + 236 * q^18 - 3 * q^19 + 28 * q^20 - 82 * q^21 - 10 * q^22 - 33 * q^23 + 24 * q^24 - 234 * q^25 + 20 * q^26 - 666 * q^27 + 180 * q^28 + 198 * q^29 + 196 * q^30 + 560 * q^31 - 64 * q^32 + 767 * q^33 - 520 * q^34 - 266 * q^35 - 236 * q^36 + 702 * q^37 + 12 * q^38 - 1317 * q^39 - 112 * q^40 - 242 * q^41 + 82 * q^42 + 93 * q^43 + 40 * q^44 - 119 * q^45 - 66 * q^46 + 448 * q^47 + 48 * q^48 - 435 * q^49 + 234 * q^50 + 2126 * q^51 + 100 * q^52 - 1070 * q^53 + 666 * q^54 - 742 * q^55 + 360 * q^56 - 226 * q^57 + 396 * q^58 + 389 * q^59 - 784 * q^60 - 654 * q^61 - 560 * q^62 + 1002 * q^63 + 256 * q^64 - 203 * q^65 - 3068 * q^66 + 107 * q^67 + 520 * q^68 + 375 * q^69 + 1064 * q^70 + 569 * q^71 - 472 * q^72 - 1246 * q^73 + 1404 * q^74 - 935 * q^75 - 12 * q^76 + 1294 * q^77 + 1878 * q^78 + 1520 * q^79 + 112 * q^80 + 334 * q^81 - 484 * q^82 - 3528 * q^83 + 164 * q^84 + 413 * q^85 - 372 * q^86 - 1599 * q^87 - 40 * q^88 + 871 * q^89 + 476 * q^90 - 1539 * q^91 + 264 * q^92 - 2184 * q^93 - 448 * q^94 - 98 * q^95 - 192 * q^96 + 879 * q^97 - 870 * q^98 + 4852 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 55x^{2} + 54x + 2916$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 55\nu^{2} - 55\nu + 2916 ) / 2970$$ (-v^3 + 55*v^2 - 55*v + 2916) / 2970 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 109 ) / 55$$ (v^3 + 109) / 55
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 54\beta_{2} + \beta _1 - 55$$ b3 + 54*b2 + b1 - 55 $$\nu^{3}$$ $$=$$ $$55\beta_{3} - 109$$ 55*b3 - 109

## Character values

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

 $$n$$ $$15$$ $$\chi(n)$$ $$-1 + \beta_{2}$$

## 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
 −3.43273 − 5.94566i 3.93273 + 6.81169i −3.43273 + 5.94566i 3.93273 − 6.81169i
−1.00000 1.73205i −2.93273 5.07964i −2.00000 + 3.46410i −10.8655 −5.86546 + 10.1593i 14.9327 25.8642i 8.00000 −3.70181 + 6.41172i 10.8655 + 18.8195i
3.2 −1.00000 1.73205i 4.43273 + 7.67771i −2.00000 + 3.46410i 3.86546 8.86546 15.3554i 7.56727 13.1069i 8.00000 −25.7982 + 44.6838i −3.86546 6.69517i
9.1 −1.00000 + 1.73205i −2.93273 + 5.07964i −2.00000 3.46410i −10.8655 −5.86546 10.1593i 14.9327 + 25.8642i 8.00000 −3.70181 6.41172i 10.8655 18.8195i
9.2 −1.00000 + 1.73205i 4.43273 7.67771i −2.00000 3.46410i 3.86546 8.86546 + 15.3554i 7.56727 + 13.1069i 8.00000 −25.7982 44.6838i −3.86546 + 6.69517i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.4.c.b 4
3.b odd 2 1 234.4.h.h 4
4.b odd 2 1 208.4.i.d 4
13.b even 2 1 338.4.c.j 4
13.c even 3 1 inner 26.4.c.b 4
13.c even 3 1 338.4.a.h 2
13.d odd 4 2 338.4.e.f 8
13.e even 6 1 338.4.a.g 2
13.e even 6 1 338.4.c.j 4
13.f odd 12 2 338.4.b.e 4
13.f odd 12 2 338.4.e.f 8
39.i odd 6 1 234.4.h.h 4
52.j odd 6 1 208.4.i.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.c.b 4 1.a even 1 1 trivial
26.4.c.b 4 13.c even 3 1 inner
208.4.i.d 4 4.b odd 2 1
208.4.i.d 4 52.j odd 6 1
234.4.h.h 4 3.b odd 2 1
234.4.h.h 4 39.i odd 6 1
338.4.a.g 2 13.e even 6 1
338.4.a.h 2 13.c even 3 1
338.4.b.e 4 13.f odd 12 2
338.4.c.j 4 13.b even 2 1
338.4.c.j 4 13.e even 6 1
338.4.e.f 8 13.d odd 4 2
338.4.e.f 8 13.f odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 3T_{3}^{3} + 61T_{3}^{2} + 156T_{3} + 2704$$ acting on $$S_{4}^{\mathrm{new}}(26, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 4)^{2}$$
$3$ $$T^{4} - 3 T^{3} + \cdots + 2704$$
$5$ $$(T^{2} + 7 T - 42)^{2}$$
$7$ $$T^{4} - 45 T^{3} + \cdots + 204304$$
$11$ $$T^{4} + 5 T^{3} + \cdots + 7033104$$
$13$ $$T^{4} + 35 T^{3} + \cdots + 4826809$$
$17$ $$T^{4} - 130 T^{3} + \cdots + 567009$$
$19$ $$T^{4} + 3 T^{3} + \cdots + 2704$$
$23$ $$T^{4} + 33 T^{3} + \cdots + 46656$$
$29$ $$T^{4} - 198 T^{3} + \cdots + 3956121$$
$31$ $$(T^{2} - 280 T - 11648)^{2}$$
$37$ $$T^{4} + \cdots + 14965362889$$
$41$ $$T^{4} + 242 T^{3} + \cdots + 49829481$$
$43$ $$T^{4} + \cdots + 1007681536$$
$47$ $$(T^{2} - 224 T - 157584)^{2}$$
$53$ $$(T^{2} + 535 T + 71502)^{2}$$
$59$ $$T^{4} + \cdots + 1237069584$$
$61$ $$T^{4} + \cdots + 2639596129$$
$67$ $$T^{4} + \cdots + 45137551936$$
$71$ $$T^{4} + \cdots + 3764558736$$
$73$ $$(T^{2} + 623 T + 84826)^{2}$$
$79$ $$(T^{2} - 760 T + 19408)^{2}$$
$83$ $$(T^{2} + 1764 T + 707616)^{2}$$
$89$ $$T^{4} + \cdots + 18913400676$$
$97$ $$T^{4} + \cdots + 10397065156$$