# Properties

 Label 26.6.a.c 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: $$\Q(\sqrt{849})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 212$$ x^2 - x - 212 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 $$\beta = \frac{1}{2}(1 + \sqrt{849})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + ( - \beta + 5) q^{3} + 16 q^{4} + (3 \beta + 35) q^{5} + ( - 4 \beta + 20) q^{6} + (9 \beta + 73) q^{7} + 64 q^{8} + ( - 9 \beta - 6) q^{9}+O(q^{10})$$ q + 4 * q^2 + (-b + 5) * q^3 + 16 * q^4 + (3*b + 35) * q^5 + (-4*b + 20) * q^6 + (9*b + 73) * q^7 + 64 * q^8 + (-9*b - 6) * q^9 $$q + 4 q^{2} + ( - \beta + 5) q^{3} + 16 q^{4} + (3 \beta + 35) q^{5} + ( - 4 \beta + 20) q^{6} + (9 \beta + 73) q^{7} + 64 q^{8} + ( - 9 \beta - 6) q^{9} + (12 \beta + 140) q^{10} + ( - 24 \beta - 98) q^{11} + ( - 16 \beta + 80) q^{12} - 169 q^{13} + (36 \beta + 292) q^{14} + ( - 23 \beta - 461) q^{15} + 256 q^{16} + (129 \beta - 159) q^{17} + ( - 36 \beta - 24) q^{18} + ( - 60 \beta - 1218) q^{19} + (48 \beta + 560) q^{20} + ( - 37 \beta - 1543) q^{21} + ( - 96 \beta - 392) q^{22} + ( - 108 \beta - 1468) q^{23} + ( - 64 \beta + 320) q^{24} + (219 \beta + 8) q^{25} - 676 q^{26} + (213 \beta + 663) q^{27} + (144 \beta + 1168) q^{28} + ( - 264 \beta + 1082) q^{29} + ( - 92 \beta - 1844) q^{30} + ( - 378 \beta + 1588) q^{31} + 1024 q^{32} + (2 \beta + 4598) q^{33} + (516 \beta - 636) q^{34} + (561 \beta + 8279) q^{35} + ( - 144 \beta - 96) q^{36} + ( - 177 \beta + 8991) q^{37} + ( - 240 \beta - 4872) q^{38} + (169 \beta - 845) q^{39} + (192 \beta + 2240) q^{40} + ( - 462 \beta + 6048) q^{41} + ( - 148 \beta - 6172) q^{42} + ( - 219 \beta - 1925) q^{43} + ( - 384 \beta - 1568) q^{44} + ( - 360 \beta - 5934) q^{45} + ( - 432 \beta - 5872) q^{46} + (405 \beta - 12947) q^{47} + ( - 256 \beta + 1280) q^{48} + (1395 \beta + 5694) q^{49} + (876 \beta + 32) q^{50} + (675 \beta - 28143) q^{51} - 2704 q^{52} + ( - 798 \beta - 1908) q^{53} + (852 \beta + 2652) q^{54} + ( - 1206 \beta - 18694) q^{55} + (576 \beta + 4672) q^{56} + (978 \beta + 6630) q^{57} + ( - 1056 \beta + 4328) q^{58} + ( - 456 \beta - 11482) q^{59} + ( - 368 \beta - 7376) q^{60} + ( - 402 \beta + 48616) q^{61} + ( - 1512 \beta + 6352) q^{62} + ( - 792 \beta - 17610) q^{63} + 4096 q^{64} + ( - 507 \beta - 5915) q^{65} + (8 \beta + 18392) q^{66} + ( - 276 \beta + 36358) q^{67} + (2064 \beta - 2544) q^{68} + (1036 \beta + 15556) q^{69} + (2244 \beta + 33116) q^{70} + (3219 \beta - 19949) q^{71} + ( - 576 \beta - 384) q^{72} + ( - 3084 \beta + 25118) q^{73} + ( - 708 \beta + 35964) q^{74} + (868 \beta - 46388) q^{75} + ( - 960 \beta - 19488) q^{76} + ( - 2850 \beta - 52946) q^{77} + (676 \beta - 3380) q^{78} + (1056 \beta + 25484) q^{79} + (768 \beta + 8960) q^{80} + (2376 \beta - 40383) q^{81} + ( - 1848 \beta + 24192) q^{82} + ( - 1134 \beta - 18312) q^{83} + ( - 592 \beta - 24688) q^{84} + (4425 \beta + 76479) q^{85} + ( - 876 \beta - 7700) q^{86} + ( - 2138 \beta + 61378) q^{87} + ( - 1536 \beta - 6272) q^{88} + (2820 \beta - 53946) q^{89} + ( - 1440 \beta - 23736) q^{90} + ( - 1521 \beta - 12337) q^{91} + ( - 1728 \beta - 23488) q^{92} + ( - 3100 \beta + 88076) q^{93} + (1620 \beta - 51788) q^{94} + ( - 5934 \beta - 80790) q^{95} + ( - 1024 \beta + 5120) q^{96} + (5724 \beta + 49334) q^{97} + (5580 \beta + 22776) q^{98} + (1242 \beta + 46380) q^{99}+O(q^{100})$$ q + 4 * q^2 + (-b + 5) * q^3 + 16 * q^4 + (3*b + 35) * q^5 + (-4*b + 20) * q^6 + (9*b + 73) * q^7 + 64 * q^8 + (-9*b - 6) * q^9 + (12*b + 140) * q^10 + (-24*b - 98) * q^11 + (-16*b + 80) * q^12 - 169 * q^13 + (36*b + 292) * q^14 + (-23*b - 461) * q^15 + 256 * q^16 + (129*b - 159) * q^17 + (-36*b - 24) * q^18 + (-60*b - 1218) * q^19 + (48*b + 560) * q^20 + (-37*b - 1543) * q^21 + (-96*b - 392) * q^22 + (-108*b - 1468) * q^23 + (-64*b + 320) * q^24 + (219*b + 8) * q^25 - 676 * q^26 + (213*b + 663) * q^27 + (144*b + 1168) * q^28 + (-264*b + 1082) * q^29 + (-92*b - 1844) * q^30 + (-378*b + 1588) * q^31 + 1024 * q^32 + (2*b + 4598) * q^33 + (516*b - 636) * q^34 + (561*b + 8279) * q^35 + (-144*b - 96) * q^36 + (-177*b + 8991) * q^37 + (-240*b - 4872) * q^38 + (169*b - 845) * q^39 + (192*b + 2240) * q^40 + (-462*b + 6048) * q^41 + (-148*b - 6172) * q^42 + (-219*b - 1925) * q^43 + (-384*b - 1568) * q^44 + (-360*b - 5934) * q^45 + (-432*b - 5872) * q^46 + (405*b - 12947) * q^47 + (-256*b + 1280) * q^48 + (1395*b + 5694) * q^49 + (876*b + 32) * q^50 + (675*b - 28143) * q^51 - 2704 * q^52 + (-798*b - 1908) * q^53 + (852*b + 2652) * q^54 + (-1206*b - 18694) * q^55 + (576*b + 4672) * q^56 + (978*b + 6630) * q^57 + (-1056*b + 4328) * q^58 + (-456*b - 11482) * q^59 + (-368*b - 7376) * q^60 + (-402*b + 48616) * q^61 + (-1512*b + 6352) * q^62 + (-792*b - 17610) * q^63 + 4096 * q^64 + (-507*b - 5915) * q^65 + (8*b + 18392) * q^66 + (-276*b + 36358) * q^67 + (2064*b - 2544) * q^68 + (1036*b + 15556) * q^69 + (2244*b + 33116) * q^70 + (3219*b - 19949) * q^71 + (-576*b - 384) * q^72 + (-3084*b + 25118) * q^73 + (-708*b + 35964) * q^74 + (868*b - 46388) * q^75 + (-960*b - 19488) * q^76 + (-2850*b - 52946) * q^77 + (676*b - 3380) * q^78 + (1056*b + 25484) * q^79 + (768*b + 8960) * q^80 + (2376*b - 40383) * q^81 + (-1848*b + 24192) * q^82 + (-1134*b - 18312) * q^83 + (-592*b - 24688) * q^84 + (4425*b + 76479) * q^85 + (-876*b - 7700) * q^86 + (-2138*b + 61378) * q^87 + (-1536*b - 6272) * q^88 + (2820*b - 53946) * q^89 + (-1440*b - 23736) * q^90 + (-1521*b - 12337) * q^91 + (-1728*b - 23488) * q^92 + (-3100*b + 88076) * q^93 + (1620*b - 51788) * q^94 + (-5934*b - 80790) * q^95 + (-1024*b + 5120) * q^96 + (5724*b + 49334) * q^97 + (5580*b + 22776) * q^98 + (1242*b + 46380) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{2} + 9 q^{3} + 32 q^{4} + 73 q^{5} + 36 q^{6} + 155 q^{7} + 128 q^{8} - 21 q^{9}+O(q^{10})$$ 2 * q + 8 * q^2 + 9 * q^3 + 32 * q^4 + 73 * q^5 + 36 * q^6 + 155 * q^7 + 128 * q^8 - 21 * q^9 $$2 q + 8 q^{2} + 9 q^{3} + 32 q^{4} + 73 q^{5} + 36 q^{6} + 155 q^{7} + 128 q^{8} - 21 q^{9} + 292 q^{10} - 220 q^{11} + 144 q^{12} - 338 q^{13} + 620 q^{14} - 945 q^{15} + 512 q^{16} - 189 q^{17} - 84 q^{18} - 2496 q^{19} + 1168 q^{20} - 3123 q^{21} - 880 q^{22} - 3044 q^{23} + 576 q^{24} + 235 q^{25} - 1352 q^{26} + 1539 q^{27} + 2480 q^{28} + 1900 q^{29} - 3780 q^{30} + 2798 q^{31} + 2048 q^{32} + 9198 q^{33} - 756 q^{34} + 17119 q^{35} - 336 q^{36} + 17805 q^{37} - 9984 q^{38} - 1521 q^{39} + 4672 q^{40} + 11634 q^{41} - 12492 q^{42} - 4069 q^{43} - 3520 q^{44} - 12228 q^{45} - 12176 q^{46} - 25489 q^{47} + 2304 q^{48} + 12783 q^{49} + 940 q^{50} - 55611 q^{51} - 5408 q^{52} - 4614 q^{53} + 6156 q^{54} - 38594 q^{55} + 9920 q^{56} + 14238 q^{57} + 7600 q^{58} - 23420 q^{59} - 15120 q^{60} + 96830 q^{61} + 11192 q^{62} - 36012 q^{63} + 8192 q^{64} - 12337 q^{65} + 36792 q^{66} + 72440 q^{67} - 3024 q^{68} + 32148 q^{69} + 68476 q^{70} - 36679 q^{71} - 1344 q^{72} + 47152 q^{73} + 71220 q^{74} - 91908 q^{75} - 39936 q^{76} - 108742 q^{77} - 6084 q^{78} + 52024 q^{79} + 18688 q^{80} - 78390 q^{81} + 46536 q^{82} - 37758 q^{83} - 49968 q^{84} + 157383 q^{85} - 16276 q^{86} + 120618 q^{87} - 14080 q^{88} - 105072 q^{89} - 48912 q^{90} - 26195 q^{91} - 48704 q^{92} + 173052 q^{93} - 101956 q^{94} - 167514 q^{95} + 9216 q^{96} + 104392 q^{97} + 51132 q^{98} + 94002 q^{99}+O(q^{100})$$ 2 * q + 8 * q^2 + 9 * q^3 + 32 * q^4 + 73 * q^5 + 36 * q^6 + 155 * q^7 + 128 * q^8 - 21 * q^9 + 292 * q^10 - 220 * q^11 + 144 * q^12 - 338 * q^13 + 620 * q^14 - 945 * q^15 + 512 * q^16 - 189 * q^17 - 84 * q^18 - 2496 * q^19 + 1168 * q^20 - 3123 * q^21 - 880 * q^22 - 3044 * q^23 + 576 * q^24 + 235 * q^25 - 1352 * q^26 + 1539 * q^27 + 2480 * q^28 + 1900 * q^29 - 3780 * q^30 + 2798 * q^31 + 2048 * q^32 + 9198 * q^33 - 756 * q^34 + 17119 * q^35 - 336 * q^36 + 17805 * q^37 - 9984 * q^38 - 1521 * q^39 + 4672 * q^40 + 11634 * q^41 - 12492 * q^42 - 4069 * q^43 - 3520 * q^44 - 12228 * q^45 - 12176 * q^46 - 25489 * q^47 + 2304 * q^48 + 12783 * q^49 + 940 * q^50 - 55611 * q^51 - 5408 * q^52 - 4614 * q^53 + 6156 * q^54 - 38594 * q^55 + 9920 * q^56 + 14238 * q^57 + 7600 * q^58 - 23420 * q^59 - 15120 * q^60 + 96830 * q^61 + 11192 * q^62 - 36012 * q^63 + 8192 * q^64 - 12337 * q^65 + 36792 * q^66 + 72440 * q^67 - 3024 * q^68 + 32148 * q^69 + 68476 * q^70 - 36679 * q^71 - 1344 * q^72 + 47152 * q^73 + 71220 * q^74 - 91908 * q^75 - 39936 * q^76 - 108742 * q^77 - 6084 * q^78 + 52024 * q^79 + 18688 * q^80 - 78390 * q^81 + 46536 * q^82 - 37758 * q^83 - 49968 * q^84 + 157383 * q^85 - 16276 * q^86 + 120618 * q^87 - 14080 * q^88 - 105072 * q^89 - 48912 * q^90 - 26195 * q^91 - 48704 * q^92 + 173052 * q^93 - 101956 * q^94 - 167514 * q^95 + 9216 * q^96 + 104392 * q^97 + 51132 * q^98 + 94002 * 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
 15.0688 −14.0688
4.00000 −10.0688 16.0000 80.2064 −40.2752 208.619 64.0000 −141.619 320.826
1.2 4.00000 19.0688 16.0000 −7.20641 76.2752 −53.6192 64.0000 120.619 −28.8256
 $$n$$: 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.c 2
3.b odd 2 1 234.6.a.h 2
4.b odd 2 1 208.6.a.g 2
5.b even 2 1 650.6.a.b 2
5.c odd 4 2 650.6.b.h 4
8.b even 2 1 832.6.a.k 2
8.d odd 2 1 832.6.a.m 2
13.b even 2 1 338.6.a.f 2
13.d odd 4 2 338.6.b.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 9T_{3} - 192$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(26))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 4)^{2}$$
$3$ $$T^{2} - 9T - 192$$
$5$ $$T^{2} - 73T - 578$$
$7$ $$T^{2} - 155T - 11186$$
$11$ $$T^{2} + 220T - 110156$$
$13$ $$(T + 169)^{2}$$
$17$ $$T^{2} + 189 T - 3523122$$
$19$ $$T^{2} + 2496 T + 793404$$
$23$ $$T^{2} + 3044 T - 159200$$
$29$ $$T^{2} - 1900 T - 13890476$$
$31$ $$T^{2} - 2798 T - 28369928$$
$37$ $$T^{2} - 17805 T + 72604926$$
$41$ $$T^{2} - 11634 T - 11466000$$
$43$ $$T^{2} + 4069 T - 6040532$$
$47$ $$T^{2} + 25489 T + 127607974$$
$53$ $$T^{2} + 4614 T - 129839400$$
$59$ $$T^{2} + 23420 T + 92989684$$
$61$ $$T^{2} + \cdots + 2309711776$$
$67$ $$T^{2} + \cdots + 1295720044$$
$71$ $$T^{2} + \cdots - 1862988962$$
$73$ $$T^{2} + \cdots - 1462893860$$
$79$ $$T^{2} - 52024 T + 439936528$$
$83$ $$T^{2} + 37758 T + 83472480$$
$89$ $$T^{2} + \cdots + 1072134396$$
$97$ $$T^{2} + \cdots - 4229773940$$