LMFDB - Newform orbit 26.3.f.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,3,Mod(7,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([11]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("26.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 26 = 2 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	26.f (of order $12$, degree $4$, 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:	$0.708448687337$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-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 primitive root of unity $\zeta_{12}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9}+O(q^{10}) $ q + (-z^3 + z^2 + z) * q^2 + (-z^3 - z) * q^3 + 2z q^4 + (z^3 + 2z^2 - 2z - 1) * q^5 + (-2z^3 - z^2 + z - 1) q^6 + (7z^3 - 3z^2 - 4z - 4) q^7 + (2z^3 + 2) q^8 + (-6z^2 + 6) q^9	\( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9} + (2 \zeta_{12}^{2} - 4) q^{10} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{11} + ( - 4 \zeta_{12}^{2} + 2) q^{12} + ( - 2 \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 8 \zeta_{12} - 9) q^{13} + (7 \zeta_{12}^{3} - 14 \zeta_{12} - 1) q^{14} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{15} + 4 \zeta_{12}^{2} q^{16} + (13 \zeta_{12}^{2} - 6 \zeta_{12} + 13) q^{17} + ( - 6 \zeta_{12}^{3} + 6) q^{18} + ( - 17 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 12 \zeta_{12} - 12) q^{19} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{20} + (10 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 10) q^{21} + ( - 2 \zeta_{12}^{3} - 9 \zeta_{12}^{2} + \zeta_{12} + 9) q^{22} + ( - 21 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 21 \zeta_{12} + 4) q^{23} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{24} + 19 \zeta_{12}^{3} q^{25} + (15 \zeta_{12}^{3} + \zeta_{12}^{2} + 5 \zeta_{12} - 4) q^{26} + (15 \zeta_{12}^{3} - 30 \zeta_{12}) q^{27} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 8 \zeta_{12} - 14) q^{28} + ( - 16 \zeta_{12}^{3} - 27 \zeta_{12}^{2} - 16 \zeta_{12}) q^{29} + 6 \zeta_{12} q^{30} + (37 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} - 37) q^{31} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{32} + (9 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{33} + ( - 19 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 26 \zeta_{12} - 19) q^{34} + ( - 2 \zeta_{12}^{3} - 21 \zeta_{12}^{2} + \zeta_{12} + 21) q^{35} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{36} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 13 \zeta_{12} + 7) q^{37} + (7 \zeta_{12}^{3} - 34 \zeta_{12}^{2} + 17) q^{38} + ( - 15 \zeta_{12}^{3} - 14 \zeta_{12}^{2} + 21 \zeta_{12} + 4) q^{39} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{40} + ( - 26 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 13 \zeta_{12} - 13) q^{41} + (\zeta_{12}^{3} + 21 \zeta_{12}^{2} + \zeta_{12}) q^{42} + (20 \zeta_{12}^{2} - 39 \zeta_{12} + 20) q^{43} + ( - 10 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 10) q^{44} + (12 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{45} + ( - 4 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 23 \zeta_{12} + 23) q^{46} + (33 \zeta_{12}^{3} + 33) q^{47} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{48} + ( - 25 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 25 \zeta_{12} + 14) q^{49} + (19 \zeta_{12}^{3} + 19 \zeta_{12}^{2} - 19 \zeta_{12}) q^{50} + ( - 39 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{51} + (24 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 18 \zeta_{12} + 4) q^{52} + ( - 22 \zeta_{12}^{3} + 44 \zeta_{12} - 42) q^{53} + ( - 15 \zeta_{12}^{3} + 15 \zeta_{12}^{2} - 15 \zeta_{12} - 30) q^{54} + (9 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 9 \zeta_{12}) q^{55} + ( - 14 \zeta_{12}^{2} - 2 \zeta_{12} - 14) q^{56} + (22 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 7 \zeta_{12} - 22) q^{57} + ( - 32 \zeta_{12}^{3} - 43 \zeta_{12}^{2} - 11 \zeta_{12} + 11) q^{58} + ( - 13 \zeta_{12}^{3} + 23 \zeta_{12}^{2} - 10 \zeta_{12} - 10) q^{59} + (6 \zeta_{12}^{3} + 6) q^{60} + ( - 12 \zeta_{12}^{3} - 39 \zeta_{12}^{2} + 6 \zeta_{12} + 39) q^{61} + (64 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 64 \zeta_{12} - 20) q^{62} + (24 \zeta_{12}^{3} + 24 \zeta_{12}^{2} + 18 \zeta_{12} - 42) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( - 7 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 2 \zeta_{12} - 25) q^{65} + (9 \zeta_{12}^{3} - 18 \zeta_{12} - 3) q^{66} + (23 \zeta_{12}^{3} - 23 \zeta_{12}^{2} + 10 \zeta_{12} + 33) q^{67} + (26 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 26 \zeta_{12}) q^{68} + ( - 21 \zeta_{12}^{2} - 6 \zeta_{12} - 21) q^{69} + ( - 22 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 22) q^{70} + (25 \zeta_{12}^{3} + 29 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{71} + ( - 12 \zeta_{12}^{2} + 12 \zeta_{12} + 12) q^{72} + ( - 7 \zeta_{12}^{3} - 54 \zeta_{12}^{2} - 54 \zeta_{12} - 7) q^{73} + ( - 14 \zeta_{12}^{3} + 19 \zeta_{12}^{2} + 7 \zeta_{12} - 19) q^{74} + ( - 19 \zeta_{12}^{2} + 38) q^{75} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 24 \zeta_{12} + 34) q^{76} + (15 \zeta_{12}^{3} + 64 \zeta_{12}^{2} - 32) q^{77} + (2 \zeta_{12}^{3} - 25 \zeta_{12}^{2} + 5 \zeta_{12} + 35) q^{78} + ( - 36 \zeta_{12}^{3} + 72 \zeta_{12} + 48) q^{79} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 8) q^{80} - 9 \zeta_{12}^{2} q^{81} + ( - 13 \zeta_{12}^{2} + 39 \zeta_{12} - 13) q^{82} + ( - 25 \zeta_{12}^{3} + 46 \zeta_{12}^{2} - 46 \zeta_{12} + 25) q^{83} + (2 \zeta_{12}^{3} + 22 \zeta_{12}^{2} + 20 \zeta_{12} - 20) q^{84} + ( - 12 \zeta_{12}^{3} + 45 \zeta_{12}^{2} - 33 \zeta_{12} - 33) q^{85} + ( - 59 \zeta_{12}^{3} + 40 \zeta_{12}^{2} + 40 \zeta_{12} - 59) q^{86} + (54 \zeta_{12}^{3} + 48 \zeta_{12}^{2} - 27 \zeta_{12} - 48) q^{87} + ( - 18 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 18 \zeta_{12} + 4) q^{88} + ( - 56 \zeta_{12}^{3} - 56 \zeta_{12}^{2} + 43 \zeta_{12} + 13) q^{89} + (24 \zeta_{12}^{2} - 12) q^{90} + ( - 37 \zeta_{12}^{3} - 25 \zeta_{12}^{2} - 86 \zeta_{12} + 22) q^{91} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 42) q^{92} + (17 \zeta_{12}^{3} - 17 \zeta_{12}^{2} + 47 \zeta_{12} + 64) q^{93} + 66 \zeta_{12}^{2} q^{94} + ( - 7 \zeta_{12}^{2} + 51 \zeta_{12} - 7) q^{95} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{96} + (69 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 71 \zeta_{12} + 71) q^{97} + ( - 14 \zeta_{12}^{3} - 18 \zeta_{12}^{2} + 32 \zeta_{12} + 32) q^{98} + ( - 24 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12} - 24) q^{99} +O(q^{100}) \) q + (-z^3 + z^2 + z) * q^2 + (-z^3 - z) * q^3 + 2z q^4 + (z^3 + 2z^2 - 2z - 1) * q^5 + (-2z^3 - z^2 + z - 1) q^6 + (7z^3 - 3z^2 - 4z - 4) q^7 + (2z^3 + 2) q^8 + (-6z^2 + 6) q^9 + (2z^2 - 4) q^10 + (-5z^3 - 5z^2 + 4z + 1) q^11 + (-4z^2 + 2) q^12 + (-2z^3 + 12z^2 + 8z - 9) q^13 + (7z^3 - 14z - 1) * q^14 + (-3z^3 + 3z^2 + 3z) q^15 + 4z^2 q^16 + (13z^2 - 6z + 13) * q^17 + (-6z^3 + 6) q^18 + (-17z^3 - 5z^2 + 12z - 12) q^19 + (4z^3 - 2z^2 - 2z - 2) q^20 + (10z^3 + z^2 + z + 10) q^21 + (-2z^3 - 9z^2 + z + 9) * q^22 + (-21z^3 - 2z^2 + 21z + 4) q^23 + (-2z^3 - 2z^2 - 2z + 4) q^24 + 19z^3 q^25 + (15z^3 + z^2 + 5z - 4) * q^26 + (15z^3 - 30z) * q^27 + (-6z^3 + 6z^2 - 8z - 14) q^28 + (-16z^3 - 27z^2 - 16z) q^29 + 6z q^30 + (37z^3 + 10z^2 - 10z - 37) q^31 + (4z^2 + 4z - 4) * q^32 + (9z^3 - 3z^2 - 6z - 6) q^33 + (-19z^3 + 26z^2 + 26z - 19) q^34 + (-2z^3 - 21z^2 + z + 21) * q^35 + (-12z^3 + 12z) * q^36 + (6z^3 + 6z^2 - 13z + 7) q^37 + (7z^3 - 34z^2 + 17) * q^38 + (-15z^3 - 14z^2 + 21z + 4) q^39 + (4z^3 - 8z) * q^40 + (-26z^3 + 26z^2 + 13z - 13) q^41 + (z^3 + 21z^2 + z) q^42 + (20z^2 - 39z + 20) * q^43 + (-10z^3 - 2z^2 + 2z + 10) q^44 + (12z^3 + 6z^2 - 6z + 6) q^45 + (-4z^3 - 19z^2 + 23z + 23) q^46 + (33z^3 + 33) q^47 + (-8z^3 + 4z) * q^48 + (-25z^3 - 7z^2 + 25z + 14) q^49 + (19z^3 + 19z^2 - 19z) q^50 + (-39z^3 + 12z^2 - 6) * q^51 + (24z^3 + 12z^2 - 18z + 4) q^52 + (-22z^3 + 44z - 42) * q^53 + (-15z^3 + 15z^2 - 15z - 30) q^54 + (9z^3 + 3z^2 + 9z) q^55 + (-14z^2 - 2z - 14) * q^56 + (22z^3 - 7z^2 + 7z - 22) q^57 + (-32z^3 - 43z^2 - 11z + 11) q^58 + (-13z^3 + 23z^2 - 10z - 10) q^59 + (6z^3 + 6) q^60 + (-12z^3 - 39z^2 + 6z + 39) q^61 + (64z^3 + 10z^2 - 64z - 20) q^62 + (24z^3 + 24z^2 + 18z - 42) q^63 + 8z^3 q^64 + (-7z^3 - 10z^2 + 2z - 25) q^65 + (9z^3 - 18z - 3) * q^66 + (23z^3 - 23z^2 + 10z + 33) q^67 + (26z^3 - 12z^2 + 26z) q^68 + (-21z^2 - 6z - 21) * q^69 + (-22z^3 - 2z^2 + 2z + 22) q^70 + (25z^3 + 29z^2 + 4z - 4) q^71 + (-12z^2 + 12z + 12) * q^72 + (-7z^3 - 54z^2 - 54z - 7) q^73 + (-14z^3 + 19z^2 + 7z - 19) q^74 + (-19z^2 + 38) q^75 + (-10z^3 - 10z^2 - 24z + 34) q^76 + (15z^3 + 64z^2 - 32) * q^77 + (2z^3 - 25z^2 + 5z + 35) q^78 + (-36z^3 + 72z + 48) * q^79 + (-4z^3 + 4z^2 - 4z - 8) q^80 - 9z^2 q^81 + (-13z^2 + 39z - 13) * q^82 + (-25z^3 + 46z^2 - 46z + 25) q^83 + (2z^3 + 22z^2 + 20z - 20) q^84 + (-12z^3 + 45z^2 - 33z - 33) q^85 + (-59z^3 + 40z^2 + 40z - 59) q^86 + (54z^3 + 48z^2 - 27z - 48) q^87 + (-18z^3 - 2z^2 + 18z + 4) q^88 + (-56z^3 - 56z^2 + 43z + 13) q^89 + (24z^2 - 12) q^90 + (-37z^3 - 25z^2 - 86z + 22) q^91 + (-4z^3 + 8z + 42) * q^92 + (17z^3 - 17z^2 + 47z + 64) q^93 + 66z^2 q^94 + (-7z^2 + 51z - 7) * q^95 + (-4z^3 - 8z^2 + 8z + 4) q^96 + (69z^3 - 2z^2 - 71z + 71) q^97 + (-14z^3 - 18z^2 + 32z + 32) q^98 + (-24z^3 - 6z^2 - 6z - 24) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 6 q^{6} - 22 q^{7} + 8 q^{8} + 12 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 6 * q^6 - 22 * q^7 + 8 * q^8 + 12 * q^9	$ 4 q + 2 q^{2} - 6 q^{6} - 22 q^{7} + 8 q^{8} + 12 q^{9} - 12 q^{10} - 6 q^{11} - 12 q^{13} - 4 q^{14} + 6 q^{15} + 8 q^{16} + 78 q^{17} + 24 q^{18} - 58 q^{19} - 12 q^{20} + 42 q^{21} + 18 q^{22} + 12 q^{23} + 12 q^{24} - 14 q^{26} - 44 q^{28} - 54 q^{29} - 128 q^{31} - 8 q^{32} - 30 q^{33} - 24 q^{34} + 42 q^{35} + 40 q^{37} - 12 q^{39} + 42 q^{42} + 120 q^{43} + 36 q^{44} + 36 q^{45} + 54 q^{46} + 132 q^{47} + 42 q^{49} + 38 q^{50} + 40 q^{52} - 168 q^{53} - 90 q^{54} + 6 q^{55} - 84 q^{56} - 102 q^{57} - 42 q^{58} + 6 q^{59} + 24 q^{60} + 78 q^{61} - 60 q^{62} - 120 q^{63} - 120 q^{65} - 12 q^{66} + 86 q^{67} - 24 q^{68} - 126 q^{69} + 84 q^{70} + 42 q^{71} + 24 q^{72} - 136 q^{73} - 38 q^{74} + 114 q^{75} + 116 q^{76} + 90 q^{78} + 192 q^{79} - 24 q^{80} - 18 q^{81} - 78 q^{82} + 192 q^{83} - 36 q^{84} - 42 q^{85} - 156 q^{86} - 96 q^{87} + 12 q^{88} - 60 q^{89} + 38 q^{91} + 168 q^{92} + 222 q^{93} + 132 q^{94} - 42 q^{95} + 280 q^{97} + 92 q^{98} - 108 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 - 6 * q^6 - 22 * q^7 + 8 * q^8 + 12 * q^9 - 12 * q^10 - 6 * q^11 - 12 * q^13 - 4 * q^14 + 6 * q^15 + 8 * q^16 + 78 * q^17 + 24 * q^18 - 58 * q^19 - 12 * q^20 + 42 * q^21 + 18 * q^22 + 12 * q^23 + 12 * q^24 - 14 * q^26 - 44 * q^28 - 54 * q^29 - 128 * q^31 - 8 * q^32 - 30 * q^33 - 24 * q^34 + 42 * q^35 + 40 * q^37 - 12 * q^39 + 42 * q^42 + 120 * q^43 + 36 * q^44 + 36 * q^45 + 54 * q^46 + 132 * q^47 + 42 * q^49 + 38 * q^50 + 40 * q^52 - 168 * q^53 - 90 * q^54 + 6 * q^55 - 84 * q^56 - 102 * q^57 - 42 * q^58 + 6 * q^59 + 24 * q^60 + 78 * q^61 - 60 * q^62 - 120 * q^63 - 120 * q^65 - 12 * q^66 + 86 * q^67 - 24 * q^68 - 126 * q^69 + 84 * q^70 + 42 * q^71 + 24 * q^72 - 136 * q^73 - 38 * q^74 + 114 * q^75 + 116 * q^76 + 90 * q^78 + 192 * q^79 - 24 * q^80 - 18 * q^81 - 78 * q^82 + 192 * q^83 - 36 * q^84 - 42 * q^85 - 156 * q^86 - 96 * q^87 + 12 * q^88 - 60 * q^89 + 38 * q^91 + 168 * q^92 + 222 * q^93 + 132 * q^94 - 42 * q^95 + 280 * q^97 + 92 * q^98 - 108 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$15$
$\chi(n)$	$\zeta_{12}$

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} $

7.1

0.866025	−	0.500000i
−0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i

1.36603

−

0.366025i

−0.866025

1.50000i

1.73205

−

1.00000i

−1.73205

−

1.73205i

−0.633975

2.36603i

−8.96410

−

2.40192i

2.00000

−

2.00000i

3.00000

5.19615i

−3.00000

−

1.73205i

11.1

−0.366025

1.36603i

0.866025

1.50000i

−1.73205

−

1.00000i

1.73205

1.73205i

−2.36603

0.633975i

−2.03590

−

7.59808i

2.00000

−

2.00000i

3.00000

−

5.19615i

−3.00000

1.73205i

15.1

1.36603

0.366025i

−0.866025

−

1.50000i

1.73205

1.00000i

−1.73205

1.73205i

−0.633975

−

2.36603i

−8.96410

2.40192i

2.00000

2.00000i

3.00000

−

5.19615i

−3.00000

1.73205i

19.1

−0.366025

−

1.36603i

0.866025

−

1.50000i

−1.73205

1.00000i

1.73205

−

1.73205i

−2.36603

−

0.633975i

−2.03590

7.59808i

2.00000

2.00000i

3.00000

5.19615i

−3.00000

−

1.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.f	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	26.3.f.a	✓	4
3.b	odd	2	1		234.3.bb.b		4
4.b	odd	2	1		208.3.bd.c		4
13.b	even	2	1		338.3.f.d		4
13.c	even	3	1		338.3.d.d		4
13.c	even	3	1		338.3.f.f		4
13.d	odd	4	1		338.3.f.c		4
13.d	odd	4	1		338.3.f.f		4
13.e	even	6	1		338.3.d.e		4
13.e	even	6	1		338.3.f.c		4
13.f	odd	12	1	inner	26.3.f.a	✓	4
13.f	odd	12	1		338.3.d.d		4
13.f	odd	12	1		338.3.d.e		4
13.f	odd	12	1		338.3.f.d		4
39.k	even	12	1		234.3.bb.b		4
52.l	even	12	1		208.3.bd.c		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.3.f.a	✓	4	1.a	even	1	1	trivial
26.3.f.a	✓	4	13.f	odd	12	1	inner
208.3.bd.c		4	4.b	odd	2	1
208.3.bd.c		4	52.l	even	12	1
234.3.bb.b		4	3.b	odd	2	1
234.3.bb.b		4	39.k	even	12	1
338.3.d.d		4	13.c	even	3	1
338.3.d.d		4	13.f	odd	12	1
338.3.d.e		4	13.e	even	6	1
338.3.d.e		4	13.f	odd	12	1
338.3.f.c		4	13.d	odd	4	1
338.3.f.c		4	13.e	even	6	1
338.3.f.d		4	13.b	even	2	1
338.3.f.d		4	13.f	odd	12	1
338.3.f.f		4	13.c	even	3	1
338.3.f.f		4	13.d	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{4} + 3T_{3}^{2} + 9 $ T3^4 + 3*T3^2 + 9 acting on $S_{3}^{\mathrm{new}}(26, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 $ T^4 - 2T^3 + 2T^2 - 4*T + 4
$3$	$ T^{4} + 3T^{2} + 9 $ T^4 + 3*T^2 + 9
$5$	$ T^{4} + 36 $ T^4 + 36
$7$	$ T^{4} + 22 T^{3} + 221 T^{2} + \cdots + 5329 $ T^4 + 22T^3 + 221T^2 + 1460*T + 5329
$11$	$ T^{4} + 6 T^{3} + 45 T^{2} + \cdots + 1521 $ T^4 + 6T^3 + 45T^2 + 468*T + 1521
$13$	$ T^{4} + 12 T^{3} + 182 T^{2} + \cdots + 28561 $ T^4 + 12T^3 + 182T^2 + 2028*T + 28561
$17$	$ T^{4} - 78 T^{3} + 2499 T^{2} + \cdots + 221841 $ T^4 - 78T^3 + 2499T^2 - 36738*T + 221841
$19$	$ T^{4} + 58 T^{3} + 1325 T^{2} + \cdots + 167281 $ T^4 + 58T^3 + 1325T^2 + 17996*T + 167281
$23$	$ T^{4} - 12 T^{3} - 381 T^{2} + \cdots + 184041 $ T^4 - 12T^3 - 381T^2 + 5148*T + 184041
$29$	$ T^{4} + 54 T^{3} + 2955 T^{2} + \cdots + 1521 $ T^4 + 54T^3 + 2955T^2 - 2106*T + 1521
$31$	$ T^{4} + 128 T^{3} + 8192 T^{2} + \cdots + 3602404 $ T^4 + 128T^3 + 8192T^2 + 242944*T + 3602404
$37$	$ T^{4} - 40 T^{3} + 401 T^{2} + \cdots + 11449 $ T^4 - 40T^3 + 401T^2 + 214*T + 11449
$41$	$ T^{4} + 1521 T^{2} - 39546 T + 257049 $ T^4 + 1521T^2 - 39546T + 257049
$43$	$ T^{4} - 120 T^{3} + 4479 T^{2} + \cdots + 103041 $ T^4 - 120T^3 + 4479T^2 + 38520*T + 103041
$47$	$ (T^{2} - 66 T + 2178)^{2} $ (T^2 - 66*T + 2178)^2
$53$	$ (T^{2} + 84 T + 312)^{2} $ (T^2 + 84*T + 312)^2
$59$	$ T^{4} - 6 T^{3} + 1305 T^{2} + \cdots + 84681 $ T^4 - 6T^3 + 1305T^2 + 20952*T + 84681
$61$	$ T^{4} - 78 T^{3} + 4671 T^{2} + \cdots + 1996569 $ T^4 - 78T^3 + 4671T^2 - 110214*T + 1996569
$67$	$ T^{4} - 86 T^{3} + 4985 T^{2} + \cdots + 2399401 $ T^4 - 86T^3 + 4985T^2 - 173488*T + 2399401
$71$	$ T^{4} - 42 T^{3} + 3357 T^{2} + \cdots + 154449 $ T^4 - 42T^3 + 3357T^2 - 42444*T + 154449
$73$	$ T^{4} + 136 T^{3} + 9248 T^{2} + \cdots + 4251844 $ T^4 + 136T^3 + 9248T^2 - 280432*T + 4251844
$79$	$ (T^{2} - 96 T - 1584)^{2} $ (T^2 - 96*T - 1584)^2
$83$	$ T^{4} - 192 T^{3} + 18432 T^{2} + \cdots + 2056356 $ T^4 - 192T^3 + 18432T^2 - 275328*T + 2056356
$89$	$ T^{4} + 60 T^{3} + 5661 T^{2} + \cdots + 21594609 $ T^4 + 60T^3 + 5661T^2 + 641286*T + 21594609
$97$	$ T^{4} - 280 T^{3} + \cdots + 20043529 $ T^4 - 280T^3 + 24089T^2 - 599918*T + 20043529
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 26 = 2 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	26.f (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9}+O(q^{10}) \) q + (-z^3 + z^2 + z) * q^2 + (-z^3 - z) * q^3 + 2z q^4 + (z^3 + 2z^2 - 2z - 1) * q^5 + (-2z^3 - z^2 + z - 1) q^6 + (7z^3 - 3z^2 - 4z - 4) q^7 + (2z^3 + 2) q^8 + (-6z^2 + 6) q^9	\( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9} + (2 \zeta_{12}^{2} - 4) q^{10} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{11} + ( - 4 \zeta_{12}^{2} + 2) q^{12} + ( - 2 \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 8 \zeta_{12} - 9) q^{13} + (7 \zeta_{12}^{3} - 14 \zeta_{12} - 1) q^{14} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{15} + 4 \zeta_{12}^{2} q^{16} + (13 \zeta_{12}^{2} - 6 \zeta_{12} + 13) q^{17} + ( - 6 \zeta_{12}^{3} + 6) q^{18} + ( - 17 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 12 \zeta_{12} - 12) q^{19} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{20} + (10 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 10) q^{21} + ( - 2 \zeta_{12}^{3} - 9 \zeta_{12}^{2} + \zeta_{12} + 9) q^{22} + ( - 21 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 21 \zeta_{12} + 4) q^{23} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{24} + 19 \zeta_{12}^{3} q^{25} + (15 \zeta_{12}^{3} + \zeta_{12}^{2} + 5 \zeta_{12} - 4) q^{26} + (15 \zeta_{12}^{3} - 30 \zeta_{12}) q^{27} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 8 \zeta_{12} - 14) q^{28} + ( - 16 \zeta_{12}^{3} - 27 \zeta_{12}^{2} - 16 \zeta_{12}) q^{29} + 6 \zeta_{12} q^{30} + (37 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} - 37) q^{31} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{32} + (9 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{33} + ( - 19 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 26 \zeta_{12} - 19) q^{34} + ( - 2 \zeta_{12}^{3} - 21 \zeta_{12}^{2} + \zeta_{12} + 21) q^{35} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{36} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 13 \zeta_{12} + 7) q^{37} + (7 \zeta_{12}^{3} - 34 \zeta_{12}^{2} + 17) q^{38} + ( - 15 \zeta_{12}^{3} - 14 \zeta_{12}^{2} + 21 \zeta_{12} + 4) q^{39} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{40} + ( - 26 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 13 \zeta_{12} - 13) q^{41} + (\zeta_{12}^{3} + 21 \zeta_{12}^{2} + \zeta_{12}) q^{42} + (20 \zeta_{12}^{2} - 39 \zeta_{12} + 20) q^{43} + ( - 10 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 10) q^{44} + (12 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{45} + ( - 4 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 23 \zeta_{12} + 23) q^{46} + (33 \zeta_{12}^{3} + 33) q^{47} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{48} + ( - 25 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 25 \zeta_{12} + 14) q^{49} + (19 \zeta_{12}^{3} + 19 \zeta_{12}^{2} - 19 \zeta_{12}) q^{50} + ( - 39 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{51} + (24 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 18 \zeta_{12} + 4) q^{52} + ( - 22 \zeta_{12}^{3} + 44 \zeta_{12} - 42) q^{53} + ( - 15 \zeta_{12}^{3} + 15 \zeta_{12}^{2} - 15 \zeta_{12} - 30) q^{54} + (9 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 9 \zeta_{12}) q^{55} + ( - 14 \zeta_{12}^{2} - 2 \zeta_{12} - 14) q^{56} + (22 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 7 \zeta_{12} - 22) q^{57} + ( - 32 \zeta_{12}^{3} - 43 \zeta_{12}^{2} - 11 \zeta_{12} + 11) q^{58} + ( - 13 \zeta_{12}^{3} + 23 \zeta_{12}^{2} - 10 \zeta_{12} - 10) q^{59} + (6 \zeta_{12}^{3} + 6) q^{60} + ( - 12 \zeta_{12}^{3} - 39 \zeta_{12}^{2} + 6 \zeta_{12} + 39) q^{61} + (64 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 64 \zeta_{12} - 20) q^{62} + (24 \zeta_{12}^{3} + 24 \zeta_{12}^{2} + 18 \zeta_{12} - 42) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( - 7 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 2 \zeta_{12} - 25) q^{65} + (9 \zeta_{12}^{3} - 18 \zeta_{12} - 3) q^{66} + (23 \zeta_{12}^{3} - 23 \zeta_{12}^{2} + 10 \zeta_{12} + 33) q^{67} + (26 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 26 \zeta_{12}) q^{68} + ( - 21 \zeta_{12}^{2} - 6 \zeta_{12} - 21) q^{69} + ( - 22 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 22) q^{70} + (25 \zeta_{12}^{3} + 29 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{71} + ( - 12 \zeta_{12}^{2} + 12 \zeta_{12} + 12) q^{72} + ( - 7 \zeta_{12}^{3} - 54 \zeta_{12}^{2} - 54 \zeta_{12} - 7) q^{73} + ( - 14 \zeta_{12}^{3} + 19 \zeta_{12}^{2} + 7 \zeta_{12} - 19) q^{74} + ( - 19 \zeta_{12}^{2} + 38) q^{75} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 24 \zeta_{12} + 34) q^{76} + (15 \zeta_{12}^{3} + 64 \zeta_{12}^{2} - 32) q^{77} + (2 \zeta_{12}^{3} - 25 \zeta_{12}^{2} + 5 \zeta_{12} + 35) q^{78} + ( - 36 \zeta_{12}^{3} + 72 \zeta_{12} + 48) q^{79} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 8) q^{80} - 9 \zeta_{12}^{2} q^{81} + ( - 13 \zeta_{12}^{2} + 39 \zeta_{12} - 13) q^{82} + ( - 25 \zeta_{12}^{3} + 46 \zeta_{12}^{2} - 46 \zeta_{12} + 25) q^{83} + (2 \zeta_{12}^{3} + 22 \zeta_{12}^{2} + 20 \zeta_{12} - 20) q^{84} + ( - 12 \zeta_{12}^{3} + 45 \zeta_{12}^{2} - 33 \zeta_{12} - 33) q^{85} + ( - 59 \zeta_{12}^{3} + 40 \zeta_{12}^{2} + 40 \zeta_{12} - 59) q^{86} + (54 \zeta_{12}^{3} + 48 \zeta_{12}^{2} - 27 \zeta_{12} - 48) q^{87} + ( - 18 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 18 \zeta_{12} + 4) q^{88} + ( - 56 \zeta_{12}^{3} - 56 \zeta_{12}^{2} + 43 \zeta_{12} + 13) q^{89} + (24 \zeta_{12}^{2} - 12) q^{90} + ( - 37 \zeta_{12}^{3} - 25 \zeta_{12}^{2} - 86 \zeta_{12} + 22) q^{91} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 42) q^{92} + (17 \zeta_{12}^{3} - 17 \zeta_{12}^{2} + 47 \zeta_{12} + 64) q^{93} + 66 \zeta_{12}^{2} q^{94} + ( - 7 \zeta_{12}^{2} + 51 \zeta_{12} - 7) q^{95} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{96} + (69 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 71 \zeta_{12} + 71) q^{97} + ( - 14 \zeta_{12}^{3} - 18 \zeta_{12}^{2} + 32 \zeta_{12} + 32) q^{98} + ( - 24 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12} - 24) q^{99} +O(q^{100}) \) q + (-z^3 + z^2 + z) * q^2 + (-z^3 - z) * q^3 + 2z q^4 + (z^3 + 2z^2 - 2z - 1) * q^5 + (-2z^3 - z^2 + z - 1) q^6 + (7z^3 - 3z^2 - 4z - 4) q^7 + (2z^3 + 2) q^8 + (-6z^2 + 6) q^9 + (2z^2 - 4) q^10 + (-5z^3 - 5z^2 + 4z + 1) q^11 + (-4z^2 + 2) q^12 + (-2z^3 + 12z^2 + 8z - 9) q^13 + (7z^3 - 14z - 1) * q^14 + (-3z^3 + 3z^2 + 3z) q^15 + 4z^2 q^16 + (13z^2 - 6z + 13) * q^17 + (-6z^3 + 6) q^18 + (-17z^3 - 5z^2 + 12z - 12) q^19 + (4z^3 - 2z^2 - 2z - 2) q^20 + (10z^3 + z^2 + z + 10) q^21 + (-2z^3 - 9z^2 + z + 9) * q^22 + (-21z^3 - 2z^2 + 21z + 4) q^23 + (-2z^3 - 2z^2 - 2z + 4) q^24 + 19z^3 q^25 + (15z^3 + z^2 + 5z - 4) * q^26 + (15z^3 - 30z) * q^27 + (-6z^3 + 6z^2 - 8z - 14) q^28 + (-16z^3 - 27z^2 - 16z) q^29 + 6z q^30 + (37z^3 + 10z^2 - 10z - 37) q^31 + (4z^2 + 4z - 4) * q^32 + (9z^3 - 3z^2 - 6z - 6) q^33 + (-19z^3 + 26z^2 + 26z - 19) q^34 + (-2z^3 - 21z^2 + z + 21) * q^35 + (-12z^3 + 12z) * q^36 + (6z^3 + 6z^2 - 13z + 7) q^37 + (7z^3 - 34z^2 + 17) * q^38 + (-15z^3 - 14z^2 + 21z + 4) q^39 + (4z^3 - 8z) * q^40 + (-26z^3 + 26z^2 + 13z - 13) q^41 + (z^3 + 21z^2 + z) q^42 + (20z^2 - 39z + 20) * q^43 + (-10z^3 - 2z^2 + 2z + 10) q^44 + (12z^3 + 6z^2 - 6z + 6) q^45 + (-4z^3 - 19z^2 + 23z + 23) q^46 + (33z^3 + 33) q^47 + (-8z^3 + 4z) * q^48 + (-25z^3 - 7z^2 + 25z + 14) q^49 + (19z^3 + 19z^2 - 19z) q^50 + (-39z^3 + 12z^2 - 6) * q^51 + (24z^3 + 12z^2 - 18z + 4) q^52 + (-22z^3 + 44z - 42) * q^53 + (-15z^3 + 15z^2 - 15z - 30) q^54 + (9z^3 + 3z^2 + 9z) q^55 + (-14z^2 - 2z - 14) * q^56 + (22z^3 - 7z^2 + 7z - 22) q^57 + (-32z^3 - 43z^2 - 11z + 11) q^58 + (-13z^3 + 23z^2 - 10z - 10) q^59 + (6z^3 + 6) q^60 + (-12z^3 - 39z^2 + 6z + 39) q^61 + (64z^3 + 10z^2 - 64z - 20) q^62 + (24z^3 + 24z^2 + 18z - 42) q^63 + 8z^3 q^64 + (-7z^3 - 10z^2 + 2z - 25) q^65 + (9z^3 - 18z - 3) * q^66 + (23z^3 - 23z^2 + 10z + 33) q^67 + (26z^3 - 12z^2 + 26z) q^68 + (-21z^2 - 6z - 21) * q^69 + (-22z^3 - 2z^2 + 2z + 22) q^70 + (25z^3 + 29z^2 + 4z - 4) q^71 + (-12z^2 + 12z + 12) * q^72 + (-7z^3 - 54z^2 - 54z - 7) q^73 + (-14z^3 + 19z^2 + 7z - 19) q^74 + (-19z^2 + 38) q^75 + (-10z^3 - 10z^2 - 24z + 34) q^76 + (15z^3 + 64z^2 - 32) * q^77 + (2z^3 - 25z^2 + 5z + 35) q^78 + (-36z^3 + 72z + 48) * q^79 + (-4z^3 + 4z^2 - 4z - 8) q^80 - 9z^2 q^81 + (-13z^2 + 39z - 13) * q^82 + (-25z^3 + 46z^2 - 46z + 25) q^83 + (2z^3 + 22z^2 + 20z - 20) q^84 + (-12z^3 + 45z^2 - 33z - 33) q^85 + (-59z^3 + 40z^2 + 40z - 59) q^86 + (54z^3 + 48z^2 - 27z - 48) q^87 + (-18z^3 - 2z^2 + 18z + 4) q^88 + (-56z^3 - 56z^2 + 43z + 13) q^89 + (24z^2 - 12) q^90 + (-37z^3 - 25z^2 - 86z + 22) q^91 + (-4z^3 + 8z + 42) * q^92 + (17z^3 - 17z^2 + 47z + 64) q^93 + 66z^2 q^94 + (-7z^2 + 51z - 7) * q^95 + (-4z^3 - 8z^2 + 8z + 4) q^96 + (69z^3 - 2z^2 - 71z + 71) q^97 + (-14z^3 - 18z^2 + 32z + 32) q^98 + (-24z^3 - 6z^2 - 6z - 24) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 6 q^{6} - 22 q^{7} + 8 q^{8} + 12 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 6 * q^6 - 22 * q^7 + 8 * q^8 + 12 * q^9	\( 4 q + 2 q^{2} - 6 q^{6} - 22 q^{7} + 8 q^{8} + 12 q^{9} - 12 q^{10} - 6 q^{11} - 12 q^{13} - 4 q^{14} + 6 q^{15} + 8 q^{16} + 78 q^{17} + 24 q^{18} - 58 q^{19} - 12 q^{20} + 42 q^{21} + 18 q^{22} + 12 q^{23} + 12 q^{24} - 14 q^{26} - 44 q^{28} - 54 q^{29} - 128 q^{31} - 8 q^{32} - 30 q^{33} - 24 q^{34} + 42 q^{35} + 40 q^{37} - 12 q^{39} + 42 q^{42} + 120 q^{43} + 36 q^{44} + 36 q^{45} + 54 q^{46} + 132 q^{47} + 42 q^{49} + 38 q^{50} + 40 q^{52} - 168 q^{53} - 90 q^{54} + 6 q^{55} - 84 q^{56} - 102 q^{57} - 42 q^{58} + 6 q^{59} + 24 q^{60} + 78 q^{61} - 60 q^{62} - 120 q^{63} - 120 q^{65} - 12 q^{66} + 86 q^{67} - 24 q^{68} - 126 q^{69} + 84 q^{70} + 42 q^{71} + 24 q^{72} - 136 q^{73} - 38 q^{74} + 114 q^{75} + 116 q^{76} + 90 q^{78} + 192 q^{79} - 24 q^{80} - 18 q^{81} - 78 q^{82} + 192 q^{83} - 36 q^{84} - 42 q^{85} - 156 q^{86} - 96 q^{87} + 12 q^{88} - 60 q^{89} + 38 q^{91} + 168 q^{92} + 222 q^{93} + 132 q^{94} - 42 q^{95} + 280 q^{97} + 92 q^{98} - 108 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 6 * q^6 - 22 * q^7 + 8 * q^8 + 12 * q^9 - 12 * q^10 - 6 * q^11 - 12 * q^13 - 4 * q^14 + 6 * q^15 + 8 * q^16 + 78 * q^17 + 24 * q^18 - 58 * q^19 - 12 * q^20 + 42 * q^21 + 18 * q^22 + 12 * q^23 + 12 * q^24 - 14 * q^26 - 44 * q^28 - 54 * q^29 - 128 * q^31 - 8 * q^32 - 30 * q^33 - 24 * q^34 + 42 * q^35 + 40 * q^37 - 12 * q^39 + 42 * q^42 + 120 * q^43 + 36 * q^44 + 36 * q^45 + 54 * q^46 + 132 * q^47 + 42 * q^49 + 38 * q^50 + 40 * q^52 - 168 * q^53 - 90 * q^54 + 6 * q^55 - 84 * q^56 - 102 * q^57 - 42 * q^58 + 6 * q^59 + 24 * q^60 + 78 * q^61 - 60 * q^62 - 120 * q^63 - 120 * q^65 - 12 * q^66 + 86 * q^67 - 24 * q^68 - 126 * q^69 + 84 * q^70 + 42 * q^71 + 24 * q^72 - 136 * q^73 - 38 * q^74 + 114 * q^75 + 116 * q^76 + 90 * q^78 + 192 * q^79 - 24 * q^80 - 18 * q^81 - 78 * q^82 + 192 * q^83 - 36 * q^84 - 42 * q^85 - 156 * q^86 - 96 * q^87 + 12 * q^88 - 60 * q^89 + 38 * q^91 + 168 * q^92 + 222 * q^93 + 132 * q^94 - 42 * q^95 + 280 * q^97 + 92 * q^98 - 108 * q^99

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	26.3.f.a

Level	$26$
Weight	$3$
Character orbit	26.f
Analytic conductor	$0.708$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.708448687337\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{12})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} - x^{2} + 1 \) x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$p$	$F_p(T)$
$2$	\( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) T^4 - 2T^3 + 2T^2 - 4*T + 4
$3$	\( T^{4} + 3T^{2} + 9 \) T^4 + 3*T^2 + 9
$5$	\( T^{4} + 36 \) T^4 + 36
$7$	\( T^{4} + 22 T^{3} + 221 T^{2} + \cdots + 5329 \) T^4 + 22T^3 + 221T^2 + 1460*T + 5329
$11$	\( T^{4} + 6 T^{3} + 45 T^{2} + \cdots + 1521 \) T^4 + 6T^3 + 45T^2 + 468*T + 1521
$13$	\( T^{4} + 12 T^{3} + 182 T^{2} + \cdots + 28561 \) T^4 + 12T^3 + 182T^2 + 2028*T + 28561
$17$	\( T^{4} - 78 T^{3} + 2499 T^{2} + \cdots + 221841 \) T^4 - 78T^3 + 2499T^2 - 36738*T + 221841
$19$	\( T^{4} + 58 T^{3} + 1325 T^{2} + \cdots + 167281 \) T^4 + 58T^3 + 1325T^2 + 17996*T + 167281
$23$	\( T^{4} - 12 T^{3} - 381 T^{2} + \cdots + 184041 \) T^4 - 12T^3 - 381T^2 + 5148*T + 184041
$29$	\( T^{4} + 54 T^{3} + 2955 T^{2} + \cdots + 1521 \) T^4 + 54T^3 + 2955T^2 - 2106*T + 1521
$31$	\( T^{4} + 128 T^{3} + 8192 T^{2} + \cdots + 3602404 \) T^4 + 128T^3 + 8192T^2 + 242944*T + 3602404
$37$	\( T^{4} - 40 T^{3} + 401 T^{2} + \cdots + 11449 \) T^4 - 40T^3 + 401T^2 + 214*T + 11449
$41$	\( T^{4} + 1521 T^{2} - 39546 T + 257049 \) T^4 + 1521T^2 - 39546T + 257049
$43$	\( T^{4} - 120 T^{3} + 4479 T^{2} + \cdots + 103041 \) T^4 - 120T^3 + 4479T^2 + 38520*T + 103041
$47$	\( (T^{2} - 66 T + 2178)^{2} \) (T^2 - 66*T + 2178)^2
$53$	\( (T^{2} + 84 T + 312)^{2} \) (T^2 + 84*T + 312)^2
$59$	\( T^{4} - 6 T^{3} + 1305 T^{2} + \cdots + 84681 \) T^4 - 6T^3 + 1305T^2 + 20952*T + 84681
$61$	\( T^{4} - 78 T^{3} + 4671 T^{2} + \cdots + 1996569 \) T^4 - 78T^3 + 4671T^2 - 110214*T + 1996569
$67$	\( T^{4} - 86 T^{3} + 4985 T^{2} + \cdots + 2399401 \) T^4 - 86T^3 + 4985T^2 - 173488*T + 2399401
$71$	\( T^{4} - 42 T^{3} + 3357 T^{2} + \cdots + 154449 \) T^4 - 42T^3 + 3357T^2 - 42444*T + 154449
$73$	\( T^{4} + 136 T^{3} + 9248 T^{2} + \cdots + 4251844 \) T^4 + 136T^3 + 9248T^2 - 280432*T + 4251844
$79$	\( (T^{2} - 96 T - 1584)^{2} \) (T^2 - 96*T - 1584)^2
$83$	\( T^{4} - 192 T^{3} + 18432 T^{2} + \cdots + 2056356 \) T^4 - 192T^3 + 18432T^2 - 275328*T + 2056356
$89$	\( T^{4} + 60 T^{3} + 5661 T^{2} + \cdots + 21594609 \) T^4 + 60T^3 + 5661T^2 + 641286*T + 21594609
$97$	\( T^{4} - 280 T^{3} + \cdots + 20043529 \) T^4 - 280T^3 + 24089T^2 - 599918*T + 20043529
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\(n\)	\(15\)
\(\chi(n)\)	\(\zeta_{12}\)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
Format: