# Properties

 Label 26.6.a.b Level $26$ Weight $6$ Character orbit 26.a Self dual yes Analytic conductor $4.170$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,6,Mod(1,26)]

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([0]))

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

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

chi := DirichletCharacter("26.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 26.a (trivial)

## 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: yes Analytic conductor: $$4.16997931514$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2785})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 696$$ x^2 - x - 696 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{2785})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + ( - \beta + 5) q^{3} + 16 q^{4} + (\beta - 19) q^{5} + (4 \beta - 20) q^{6} + (\beta + 163) q^{7} - 64 q^{8} + ( - 9 \beta + 478) q^{9}+O(q^{10})$$ q - 4 * q^2 + (-b + 5) * q^3 + 16 * q^4 + (b - 19) * q^5 + (4*b - 20) * q^6 + (b + 163) * q^7 - 64 * q^8 + (-9*b + 478) * q^9 $$q - 4 q^{2} + ( - \beta + 5) q^{3} + 16 q^{4} + (\beta - 19) q^{5} + (4 \beta - 20) q^{6} + (\beta + 163) q^{7} - 64 q^{8} + ( - 9 \beta + 478) q^{9} + ( - 4 \beta + 76) q^{10} + (10 \beta + 38) q^{11} + ( - 16 \beta + 80) q^{12} + 169 q^{13} + ( - 4 \beta - 652) q^{14} + (23 \beta - 791) q^{15} + 256 q^{16} + ( - 11 \beta - 235) q^{17} + (36 \beta - 1912) q^{18} + (70 \beta + 586) q^{19} + (16 \beta - 304) q^{20} + ( - 159 \beta + 119) q^{21} + ( - 40 \beta - 152) q^{22} + (48 \beta - 1368) q^{23} + (64 \beta - 320) q^{24} + ( - 37 \beta - 2068) q^{25} - 676 q^{26} + ( - 271 \beta + 7439) q^{27} + (16 \beta + 2608) q^{28} + (240 \beta + 702) q^{29} + ( - 92 \beta + 3164) q^{30} + (312 \beta - 412) q^{31} - 1024 q^{32} + (2 \beta - 6770) q^{33} + (44 \beta + 940) q^{34} + (145 \beta - 2401) q^{35} + ( - 144 \beta + 7648) q^{36} + (61 \beta + 6625) q^{37} + ( - 280 \beta - 2344) q^{38} + ( - 169 \beta + 845) q^{39} + ( - 64 \beta + 1216) q^{40} + ( - 542 \beta - 5704) q^{41} + (636 \beta - 476) q^{42} + ( - 479 \beta + 7987) q^{43} + (160 \beta + 608) q^{44} + (640 \beta - 15346) q^{45} + ( - 192 \beta + 5472) q^{46} + ( - 275 \beta - 16417) q^{47} + ( - 256 \beta + 1280) q^{48} + (327 \beta + 10458) q^{49} + (148 \beta + 8272) q^{50} + (191 \beta + 6481) q^{51} + 2704 q^{52} + (142 \beta - 5464) q^{53} + (1084 \beta - 29756) q^{54} + ( - 142 \beta + 6238) q^{55} + ( - 64 \beta - 10432) q^{56} + ( - 306 \beta - 45790) q^{57} + ( - 960 \beta - 2808) q^{58} + (326 \beta + 2362) q^{59} + (368 \beta - 12656) q^{60} + (238 \beta - 272) q^{61} + ( - 1248 \beta + 1648) q^{62} + ( - 998 \beta + 71650) q^{63} + 4096 q^{64} + (169 \beta - 3211) q^{65} + ( - 8 \beta + 27080) q^{66} + ( - 918 \beta + 19862) q^{67} + ( - 176 \beta - 3760) q^{68} + (1560 \beta - 40248) q^{69} + ( - 580 \beta + 9604) q^{70} + (707 \beta + 36169) q^{71} + (576 \beta - 30592) q^{72} + ( - 624 \beta + 25322) q^{73} + ( - 244 \beta - 26500) q^{74} + (1920 \beta + 15412) q^{75} + (1120 \beta + 9376) q^{76} + (1678 \beta + 13154) q^{77} + (676 \beta - 3380) q^{78} + (1680 \beta + 39176) q^{79} + (256 \beta - 4864) q^{80} + ( - 6336 \beta + 109657) q^{81} + (2168 \beta + 22816) q^{82} + ( - 2460 \beta - 37620) q^{83} + ( - 2544 \beta + 1904) q^{84} + ( - 37 \beta - 3191) q^{85} + (1916 \beta - 31948) q^{86} + (258 \beta - 163530) q^{87} + ( - 640 \beta - 2432) q^{88} + ( - 896 \beta + 20234) q^{89} + ( - 2560 \beta + 61384) q^{90} + (169 \beta + 27547) q^{91} + (768 \beta - 21888) q^{92} + (1660 \beta - 219212) q^{93} + (1100 \beta + 65668) q^{94} + ( - 674 \beta + 37586) q^{95} + (1024 \beta - 5120) q^{96} + ( - 1580 \beta - 74306) q^{97} + ( - 1308 \beta - 41832) q^{98} + (4348 \beta - 44476) q^{99}+O(q^{100})$$ q - 4 * q^2 + (-b + 5) * q^3 + 16 * q^4 + (b - 19) * q^5 + (4*b - 20) * q^6 + (b + 163) * q^7 - 64 * q^8 + (-9*b + 478) * q^9 + (-4*b + 76) * q^10 + (10*b + 38) * q^11 + (-16*b + 80) * q^12 + 169 * q^13 + (-4*b - 652) * q^14 + (23*b - 791) * q^15 + 256 * q^16 + (-11*b - 235) * q^17 + (36*b - 1912) * q^18 + (70*b + 586) * q^19 + (16*b - 304) * q^20 + (-159*b + 119) * q^21 + (-40*b - 152) * q^22 + (48*b - 1368) * q^23 + (64*b - 320) * q^24 + (-37*b - 2068) * q^25 - 676 * q^26 + (-271*b + 7439) * q^27 + (16*b + 2608) * q^28 + (240*b + 702) * q^29 + (-92*b + 3164) * q^30 + (312*b - 412) * q^31 - 1024 * q^32 + (2*b - 6770) * q^33 + (44*b + 940) * q^34 + (145*b - 2401) * q^35 + (-144*b + 7648) * q^36 + (61*b + 6625) * q^37 + (-280*b - 2344) * q^38 + (-169*b + 845) * q^39 + (-64*b + 1216) * q^40 + (-542*b - 5704) * q^41 + (636*b - 476) * q^42 + (-479*b + 7987) * q^43 + (160*b + 608) * q^44 + (640*b - 15346) * q^45 + (-192*b + 5472) * q^46 + (-275*b - 16417) * q^47 + (-256*b + 1280) * q^48 + (327*b + 10458) * q^49 + (148*b + 8272) * q^50 + (191*b + 6481) * q^51 + 2704 * q^52 + (142*b - 5464) * q^53 + (1084*b - 29756) * q^54 + (-142*b + 6238) * q^55 + (-64*b - 10432) * q^56 + (-306*b - 45790) * q^57 + (-960*b - 2808) * q^58 + (326*b + 2362) * q^59 + (368*b - 12656) * q^60 + (238*b - 272) * q^61 + (-1248*b + 1648) * q^62 + (-998*b + 71650) * q^63 + 4096 * q^64 + (169*b - 3211) * q^65 + (-8*b + 27080) * q^66 + (-918*b + 19862) * q^67 + (-176*b - 3760) * q^68 + (1560*b - 40248) * q^69 + (-580*b + 9604) * q^70 + (707*b + 36169) * q^71 + (576*b - 30592) * q^72 + (-624*b + 25322) * q^73 + (-244*b - 26500) * q^74 + (1920*b + 15412) * q^75 + (1120*b + 9376) * q^76 + (1678*b + 13154) * q^77 + (676*b - 3380) * q^78 + (1680*b + 39176) * q^79 + (256*b - 4864) * q^80 + (-6336*b + 109657) * q^81 + (2168*b + 22816) * q^82 + (-2460*b - 37620) * q^83 + (-2544*b + 1904) * q^84 + (-37*b - 3191) * q^85 + (1916*b - 31948) * q^86 + (258*b - 163530) * q^87 + (-640*b - 2432) * q^88 + (-896*b + 20234) * q^89 + (-2560*b + 61384) * q^90 + (169*b + 27547) * q^91 + (768*b - 21888) * q^92 + (1660*b - 219212) * q^93 + (1100*b + 65668) * q^94 + (-674*b + 37586) * q^95 + (1024*b - 5120) * q^96 + (-1580*b - 74306) * q^97 + (-1308*b - 41832) * q^98 + (4348*b - 44476) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{2} + 9 q^{3} + 32 q^{4} - 37 q^{5} - 36 q^{6} + 327 q^{7} - 128 q^{8} + 947 q^{9}+O(q^{10})$$ 2 * q - 8 * q^2 + 9 * q^3 + 32 * q^4 - 37 * q^5 - 36 * q^6 + 327 * q^7 - 128 * q^8 + 947 * q^9 $$2 q - 8 q^{2} + 9 q^{3} + 32 q^{4} - 37 q^{5} - 36 q^{6} + 327 q^{7} - 128 q^{8} + 947 q^{9} + 148 q^{10} + 86 q^{11} + 144 q^{12} + 338 q^{13} - 1308 q^{14} - 1559 q^{15} + 512 q^{16} - 481 q^{17} - 3788 q^{18} + 1242 q^{19} - 592 q^{20} + 79 q^{21} - 344 q^{22} - 2688 q^{23} - 576 q^{24} - 4173 q^{25} - 1352 q^{26} + 14607 q^{27} + 5232 q^{28} + 1644 q^{29} + 6236 q^{30} - 512 q^{31} - 2048 q^{32} - 13538 q^{33} + 1924 q^{34} - 4657 q^{35} + 15152 q^{36} + 13311 q^{37} - 4968 q^{38} + 1521 q^{39} + 2368 q^{40} - 11950 q^{41} - 316 q^{42} + 15495 q^{43} + 1376 q^{44} - 30052 q^{45} + 10752 q^{46} - 33109 q^{47} + 2304 q^{48} + 21243 q^{49} + 16692 q^{50} + 13153 q^{51} + 5408 q^{52} - 10786 q^{53} - 58428 q^{54} + 12334 q^{55} - 20928 q^{56} - 91886 q^{57} - 6576 q^{58} + 5050 q^{59} - 24944 q^{60} - 306 q^{61} + 2048 q^{62} + 142302 q^{63} + 8192 q^{64} - 6253 q^{65} + 54152 q^{66} + 38806 q^{67} - 7696 q^{68} - 78936 q^{69} + 18628 q^{70} + 73045 q^{71} - 60608 q^{72} + 50020 q^{73} - 53244 q^{74} + 32744 q^{75} + 19872 q^{76} + 27986 q^{77} - 6084 q^{78} + 80032 q^{79} - 9472 q^{80} + 212978 q^{81} + 47800 q^{82} - 77700 q^{83} + 1264 q^{84} - 6419 q^{85} - 61980 q^{86} - 326802 q^{87} - 5504 q^{88} + 39572 q^{89} + 120208 q^{90} + 55263 q^{91} - 43008 q^{92} - 436764 q^{93} + 132436 q^{94} + 74498 q^{95} - 9216 q^{96} - 150192 q^{97} - 84972 q^{98} - 84604 q^{99}+O(q^{100})$$ 2 * q - 8 * q^2 + 9 * q^3 + 32 * q^4 - 37 * q^5 - 36 * q^6 + 327 * q^7 - 128 * q^8 + 947 * q^9 + 148 * q^10 + 86 * q^11 + 144 * q^12 + 338 * q^13 - 1308 * q^14 - 1559 * q^15 + 512 * q^16 - 481 * q^17 - 3788 * q^18 + 1242 * q^19 - 592 * q^20 + 79 * q^21 - 344 * q^22 - 2688 * q^23 - 576 * q^24 - 4173 * q^25 - 1352 * q^26 + 14607 * q^27 + 5232 * q^28 + 1644 * q^29 + 6236 * q^30 - 512 * q^31 - 2048 * q^32 - 13538 * q^33 + 1924 * q^34 - 4657 * q^35 + 15152 * q^36 + 13311 * q^37 - 4968 * q^38 + 1521 * q^39 + 2368 * q^40 - 11950 * q^41 - 316 * q^42 + 15495 * q^43 + 1376 * q^44 - 30052 * q^45 + 10752 * q^46 - 33109 * q^47 + 2304 * q^48 + 21243 * q^49 + 16692 * q^50 + 13153 * q^51 + 5408 * q^52 - 10786 * q^53 - 58428 * q^54 + 12334 * q^55 - 20928 * q^56 - 91886 * q^57 - 6576 * q^58 + 5050 * q^59 - 24944 * q^60 - 306 * q^61 + 2048 * q^62 + 142302 * q^63 + 8192 * q^64 - 6253 * q^65 + 54152 * q^66 + 38806 * q^67 - 7696 * q^68 - 78936 * q^69 + 18628 * q^70 + 73045 * q^71 - 60608 * q^72 + 50020 * q^73 - 53244 * q^74 + 32744 * q^75 + 19872 * q^76 + 27986 * q^77 - 6084 * q^78 + 80032 * q^79 - 9472 * q^80 + 212978 * q^81 + 47800 * q^82 - 77700 * q^83 + 1264 * q^84 - 6419 * q^85 - 61980 * q^86 - 326802 * q^87 - 5504 * q^88 + 39572 * q^89 + 120208 * q^90 + 55263 * q^91 - 43008 * q^92 - 436764 * q^93 + 132436 * q^94 + 74498 * q^95 - 9216 * q^96 - 150192 * q^97 - 84972 * q^98 - 84604 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 26.8865 −25.8865
−4.00000 −21.8865 16.0000 7.88655 87.5462 189.887 −64.0000 236.021 −31.5462
1.2 −4.00000 30.8865 16.0000 −44.8865 −123.546 137.113 −64.0000 710.979 179.546
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.6.a.b 2
3.b odd 2 1 234.6.a.m 2
4.b odd 2 1 208.6.a.f 2
5.b even 2 1 650.6.a.e 2
5.c odd 4 2 650.6.b.d 4
8.b even 2 1 832.6.a.l 2
8.d odd 2 1 832.6.a.n 2
13.b even 2 1 338.6.a.g 2
13.d odd 4 2 338.6.b.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.a.b 2 1.a even 1 1 trivial
208.6.a.f 2 4.b odd 2 1
234.6.a.m 2 3.b odd 2 1
338.6.a.g 2 13.b even 2 1
338.6.b.c 4 13.d odd 4 2
650.6.a.e 2 5.b even 2 1
650.6.b.d 4 5.c odd 4 2
832.6.a.l 2 8.b even 2 1
832.6.a.n 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 9T_{3} - 676$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(26))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 4)^{2}$$
$3$ $$T^{2} - 9T - 676$$
$5$ $$T^{2} + 37T - 354$$
$7$ $$T^{2} - 327T + 26036$$
$11$ $$T^{2} - 86T - 67776$$
$13$ $$(T - 169)^{2}$$
$17$ $$T^{2} + 481T - 26406$$
$19$ $$T^{2} - 1242 T - 3025984$$
$23$ $$T^{2} + 2688 T + 202176$$
$29$ $$T^{2} - 1644 T - 39428316$$
$31$ $$T^{2} + 512 T - 67710224$$
$37$ $$T^{2} - 13311 T + 41704934$$
$41$ $$T^{2} + 11950 T - 168832560$$
$43$ $$T^{2} - 15495 T - 99724540$$
$47$ $$T^{2} + 33109 T + 221397564$$
$53$ $$T^{2} + 10786 T + 15045264$$
$59$ $$T^{2} - 5050 T - 67619040$$
$61$ $$T^{2} + 306 T - 39414976$$
$67$ $$T^{2} - 38806 T - 210270176$$
$71$ $$T^{2} - 73045 T + 985873140$$
$73$ $$T^{2} - 50020 T + 354397060$$
$79$ $$T^{2} - 80032 T - 363815744$$
$83$ $$T^{2} + \cdots - 2704104000$$
$89$ $$T^{2} - 39572 T - 167474844$$
$97$ $$T^{2} + \cdots + 3901290716$$