# Properties

 Label 1805.2.a.n Level $1805$ Weight $2$ Character orbit 1805.a Self dual yes Analytic conductor $14.413$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,2,Mod(1,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1805.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.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: $$14.4129975648$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ x^4 - x^3 - 5*x^2 + 2*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (\beta_{3} + \beta_1 + 1) q^{4} + q^{5} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{6} + (\beta_{2} - \beta_1 - 2) q^{7} + (\beta_{3} + 2 \beta_{2} + 3) q^{8} + (2 \beta_{2} + \beta_1 + 2) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b3 - b1) * q^3 + (b3 + b1 + 1) * q^4 + q^5 + (-b3 - 2*b2 - b1 - 3) * q^6 + (b2 - b1 - 2) * q^7 + (b3 + 2*b2 + 3) * q^8 + (2*b2 + b1 + 2) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (\beta_{3} + \beta_1 + 1) q^{4} + q^{5} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{6} + (\beta_{2} - \beta_1 - 2) q^{7} + (\beta_{3} + 2 \beta_{2} + 3) q^{8} + (2 \beta_{2} + \beta_1 + 2) q^{9} + \beta_1 q^{10} + ( - \beta_{3} - \beta_{2} - 1) q^{11} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 5) q^{12} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{13} + ( - 3 \beta_1 - 2) q^{14} + ( - \beta_{3} - \beta_1) q^{15} + (2 \beta_{2} + \beta_1) q^{16} + (2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{17} + (3 \beta_{3} + 3 \beta_1 + 5) q^{18} + (\beta_{3} + \beta_1 + 1) q^{20} + (3 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{21} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{22} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 5) q^{23} + ( - 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 2) q^{24} + q^{25} + (4 \beta_{2} + \beta_1 + 4) q^{26} + ( - 2 \beta_1 - 3) q^{27} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 5) q^{28} + ( - 4 \beta_{2} + \beta_1 - 6) q^{29} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{30} + (\beta_{3} + 2 \beta_{2} - 3 \beta_1 + 2) q^{31} + (\beta_{3} - 4 \beta_{2} + \beta_1 - 1) q^{32} + ( - \beta_{2} + 2 \beta_1 + 2) q^{33} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 4) q^{34} + (\beta_{2} - \beta_1 - 2) q^{35} + (3 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 5) q^{36} + (\beta_{3} - \beta_1 - 2) q^{37} + (\beta_{3} - 6 \beta_{2} + \beta_1 - 10) q^{39} + (\beta_{3} + 2 \beta_{2} + 3) q^{40} + (2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 9) q^{41} + (5 \beta_{3} + 6 \beta_{2} + 5 \beta_1 + 9) q^{42} + (2 \beta_{3} - \beta_{2} - 6) q^{43} + ( - \beta_{3} - 2 \beta_1 - 3) q^{44} + (2 \beta_{2} + \beta_1 + 2) q^{45} + ( - \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 1) q^{46} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 5) q^{47} + ( - \beta_{3} - 3 \beta_1 - 3) q^{48} + ( - \beta_{3} - 5 \beta_{2} + 5 \beta_1 - 1) q^{49} + \beta_1 q^{50} + (4 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{51} + (\beta_{3} + 4 \beta_{2} + \beta_1 + 9) q^{52} + ( - 2 \beta_{3} + 5 \beta_{2} - 2 \beta_1 - 2) q^{53} + ( - 2 \beta_{3} - 5 \beta_1 - 6) q^{54} + ( - \beta_{3} - \beta_{2} - 1) q^{55} + ( - 5 \beta_{3} - 6 \beta_{2} - 2 \beta_1 - 7) q^{56} + ( - 3 \beta_{3} - 5 \beta_1 - 1) q^{58} + ( - \beta_{2} + 4 \beta_1 - 4) q^{59} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 5) q^{60} + ( - 3 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 1) q^{61} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 7) q^{62} + ( - 2 \beta_{3} - 4 \beta_{2} - 5 \beta_1 - 6) q^{63} + ( - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{64} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{65} + (\beta_{3} + 4 \beta_1 + 5) q^{66} + ( - \beta_{3} + 7 \beta_{2} - \beta_1 + 4) q^{67} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{68} + (2 \beta_{3} - 4 \beta_{2} + 6 \beta_1 + 1) q^{69} + ( - 3 \beta_1 - 2) q^{70} + ( - 2 \beta_1 + 5) q^{71} + (2 \beta_{3} + 6 \beta_{2} + 5 \beta_1 + 10) q^{72} + ( - 2 \beta_{3} - 4 \beta_{2} - 9) q^{73} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 3) q^{74} + ( - \beta_{3} - \beta_1) q^{75} + (4 \beta_{3} + 5 \beta_{2} + 3) q^{77} + ( - 5 \beta_{3} + 2 \beta_{2} - 9 \beta_1 - 3) q^{78} + ( - \beta_{3} - 2 \beta_{2} + 6) q^{79} + (2 \beta_{2} + \beta_1) q^{80} + (5 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{81} + (2 \beta_{3} + 4 \beta_{2} + 7 \beta_1 - 2) q^{82} + (3 \beta_{3} + 2 \beta_1 - 2) q^{83} + (5 \beta_{3} + 4 \beta_{2} + 10 \beta_1 + 15) q^{84} + (2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{85} + ( - \beta_{3} + 4 \beta_{2} - 6 \beta_1 - 1) q^{86} + (5 \beta_{3} - 6 \beta_{2} + 9 \beta_1 - 3) q^{87} + (2 \beta_{2} - 3 \beta_1 - 4) q^{88} + ( - \beta_{3} - 3 \beta_{2} + \beta_1) q^{89} + (3 \beta_{3} + 3 \beta_1 + 5) q^{90} + ( - 4 \beta_{3} - \beta_{2} - 3 \beta_1 - 4) q^{91} + ( - 4 \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 6) q^{92} + (2 \beta_{3} + 8 \beta_{2} - \beta_1 + 7) q^{93} + (2 \beta_{3} - 4 \beta_{2} - 7 \beta_1 - 2) q^{94} + (\beta_{3} - 6 \beta_{2} + 4 \beta_1 - 5) q^{96} + (6 \beta_{2} + 3 \beta_1 + 7) q^{97} + ( - 2 \beta_{2} + 4 \beta_1 + 10) q^{98} + ( - \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b3 - b1) * q^3 + (b3 + b1 + 1) * q^4 + q^5 + (-b3 - 2*b2 - b1 - 3) * q^6 + (b2 - b1 - 2) * q^7 + (b3 + 2*b2 + 3) * q^8 + (2*b2 + b1 + 2) * q^9 + b1 * q^10 + (-b3 - b2 - 1) * q^11 + (-b3 - 2*b2 - 2*b1 - 5) * q^12 + (2*b3 - 2*b2 + 2*b1 - 1) * q^13 + (-3*b1 - 2) * q^14 + (-b3 - b1) * q^15 + (2*b2 + b1) * q^16 + (2*b3 - b2 - b1 - 1) * q^17 + (3*b3 + 3*b1 + 5) * q^18 + (b3 + b1 + 1) * q^20 + (3*b3 + 3*b2 + 2*b1 + 3) * q^21 + (-b3 - 2*b2 - b1 - 1) * q^22 + (-2*b3 - 2*b2 + b1 - 5) * q^23 + (-2*b3 + 2*b2 - 5*b1 - 2) * q^24 + q^25 + (4*b2 + b1 + 4) * q^26 + (-2*b1 - 3) * q^27 + (-3*b3 - 2*b2 - 3*b1 - 5) * q^28 + (-4*b2 + b1 - 6) * q^29 + (-b3 - 2*b2 - b1 - 3) * q^30 + (b3 + 2*b2 - 3*b1 + 2) * q^31 + (b3 - 4*b2 + b1 - 1) * q^32 + (-b2 + 2*b1 + 2) * q^33 + (-2*b3 + 4*b2 - 2*b1 - 4) * q^34 + (b2 - b1 - 2) * q^35 + (3*b3 + 2*b2 + 6*b1 + 5) * q^36 + (b3 - b1 - 2) * q^37 + (b3 - 6*b2 + b1 - 10) * q^39 + (b3 + 2*b2 + 3) * q^40 + (2*b3 + 4*b2 - 2*b1 + 9) * q^41 + (5*b3 + 6*b2 + 5*b1 + 9) * q^42 + (2*b3 - b2 - 6) * q^43 + (-b3 - 2*b1 - 3) * q^44 + (2*b2 + b1 + 2) * q^45 + (-b3 - 4*b2 - 4*b1 + 1) * q^46 + (-2*b3 + 4*b2 - 2*b1 - 5) * q^47 + (-b3 - 3*b1 - 3) * q^48 + (-b3 - 5*b2 + 5*b1 - 1) * q^49 + b1 * q^50 + (4*b3 + b2 + 3*b1 - 1) * q^51 + (b3 + 4*b2 + b1 + 9) * q^52 + (-2*b3 + 5*b2 - 2*b1 - 2) * q^53 + (-2*b3 - 5*b1 - 6) * q^54 + (-b3 - b2 - 1) * q^55 + (-5*b3 - 6*b2 - 2*b1 - 7) * q^56 + (-3*b3 - 5*b1 - 1) * q^58 + (-b2 + 4*b1 - 4) * q^59 + (-b3 - 2*b2 - 2*b1 - 5) * q^60 + (-3*b3 - 2*b2 - 4*b1 - 1) * q^61 + (-b3 + 2*b2 - b1 - 7) * q^62 + (-2*b3 - 4*b2 - 5*b1 - 6) * q^63 + (-3*b3 - 2*b2 - 2*b1 - 1) * q^64 + (2*b3 - 2*b2 + 2*b1 - 1) * q^65 + (b3 + 4*b1 + 5) * q^66 + (-b3 + 7*b2 - b1 + 4) * q^67 + (-2*b3 - 2*b2 - 4*b1) * q^68 + (2*b3 - 4*b2 + 6*b1 + 1) * q^69 + (-3*b1 - 2) * q^70 + (-2*b1 + 5) * q^71 + (2*b3 + 6*b2 + 5*b1 + 10) * q^72 + (-2*b3 - 4*b2 - 9) * q^73 + (-b3 + 2*b2 - 3*b1 - 3) * q^74 + (-b3 - b1) * q^75 + (4*b3 + 5*b2 + 3) * q^77 + (-5*b3 + 2*b2 - 9*b1 - 3) * q^78 + (-b3 - 2*b2 + 6) * q^79 + (2*b2 + b1) * q^80 + (5*b3 - 2*b2 + 2*b1) * q^81 + (2*b3 + 4*b2 + 7*b1 - 2) * q^82 + (3*b3 + 2*b1 - 2) * q^83 + (5*b3 + 4*b2 + 10*b1 + 15) * q^84 + (2*b3 - b2 - b1 - 1) * q^85 + (-b3 + 4*b2 - 6*b1 - 1) * q^86 + (5*b3 - 6*b2 + 9*b1 - 3) * q^87 + (2*b2 - 3*b1 - 4) * q^88 + (-b3 - 3*b2 + b1) * q^89 + (3*b3 + 3*b1 + 5) * q^90 + (-4*b3 - b2 - 3*b1 - 4) * q^91 + (-4*b3 + 2*b2 - 5*b1 - 6) * q^92 + (2*b3 + 8*b2 - b1 + 7) * q^93 + (2*b3 - 4*b2 - 7*b1 - 2) * q^94 + (b3 - 6*b2 + 4*b1 - 5) * q^96 + (6*b2 + 3*b1 + 7) * q^97 + (-2*b2 + 4*b1 + 10) * q^98 + (-b3 - 2*b2 - 3*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + q^{3} + 3 q^{4} + 4 q^{5} - 7 q^{6} - 11 q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10})$$ 4 * q + q^2 + q^3 + 3 * q^4 + 4 * q^5 - 7 * q^6 - 11 * q^7 + 6 * q^8 + 5 * q^9 $$4 q + q^{2} + q^{3} + 3 q^{4} + 4 q^{5} - 7 q^{6} - 11 q^{7} + 6 q^{8} + 5 q^{9} + q^{10} - 16 q^{12} - 2 q^{13} - 11 q^{14} + q^{15} - 3 q^{16} - 7 q^{17} + 17 q^{18} + 3 q^{20} + 2 q^{21} + q^{22} - 11 q^{23} - 13 q^{24} + 4 q^{25} + 9 q^{26} - 14 q^{27} - 13 q^{28} - 15 q^{29} - 7 q^{30} - q^{31} + 3 q^{32} + 12 q^{33} - 22 q^{34} - 11 q^{35} + 16 q^{36} - 11 q^{37} - 29 q^{39} + 6 q^{40} + 22 q^{41} + 19 q^{42} - 26 q^{43} - 12 q^{44} + 5 q^{45} + 10 q^{46} - 26 q^{47} - 13 q^{48} + 13 q^{49} + q^{50} - 11 q^{51} + 27 q^{52} - 16 q^{53} - 25 q^{54} - 8 q^{56} - 3 q^{58} - 10 q^{59} - 16 q^{60} + 2 q^{61} - 31 q^{62} - 17 q^{63} + 4 q^{64} - 2 q^{65} + 22 q^{66} + 3 q^{67} + 4 q^{68} + 14 q^{69} - 11 q^{70} + 18 q^{71} + 29 q^{72} - 24 q^{73} - 17 q^{74} + q^{75} - 6 q^{77} - 15 q^{78} + 30 q^{79} - 3 q^{80} - 4 q^{81} - 13 q^{82} - 12 q^{83} + 52 q^{84} - 7 q^{85} - 16 q^{86} - q^{87} - 23 q^{88} + 9 q^{89} + 17 q^{90} - 9 q^{91} - 25 q^{92} + 7 q^{93} - 11 q^{94} - 6 q^{96} + 19 q^{97} + 48 q^{98} - 9 q^{99}+O(q^{100})$$ 4 * q + q^2 + q^3 + 3 * q^4 + 4 * q^5 - 7 * q^6 - 11 * q^7 + 6 * q^8 + 5 * q^9 + q^10 - 16 * q^12 - 2 * q^13 - 11 * q^14 + q^15 - 3 * q^16 - 7 * q^17 + 17 * q^18 + 3 * q^20 + 2 * q^21 + q^22 - 11 * q^23 - 13 * q^24 + 4 * q^25 + 9 * q^26 - 14 * q^27 - 13 * q^28 - 15 * q^29 - 7 * q^30 - q^31 + 3 * q^32 + 12 * q^33 - 22 * q^34 - 11 * q^35 + 16 * q^36 - 11 * q^37 - 29 * q^39 + 6 * q^40 + 22 * q^41 + 19 * q^42 - 26 * q^43 - 12 * q^44 + 5 * q^45 + 10 * q^46 - 26 * q^47 - 13 * q^48 + 13 * q^49 + q^50 - 11 * q^51 + 27 * q^52 - 16 * q^53 - 25 * q^54 - 8 * q^56 - 3 * q^58 - 10 * q^59 - 16 * q^60 + 2 * q^61 - 31 * q^62 - 17 * q^63 + 4 * q^64 - 2 * q^65 + 22 * q^66 + 3 * q^67 + 4 * q^68 + 14 * q^69 - 11 * q^70 + 18 * q^71 + 29 * q^72 - 24 * q^73 - 17 * q^74 + q^75 - 6 * q^77 - 15 * q^78 + 30 * q^79 - 3 * q^80 - 4 * q^81 - 13 * q^82 - 12 * q^83 + 52 * q^84 - 7 * q^85 - 16 * q^86 - q^87 - 23 * q^88 + 9 * q^89 + 17 * q^90 - 9 * q^91 - 25 * q^92 + 7 * q^93 - 11 * q^94 - 6 * q^96 + 19 * q^97 + 48 * q^98 - 9 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 3\nu ) / 2$$ (v^3 - v^2 - 3*v) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 + 3$$ b3 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 4\beta _1 + 3$$ b3 + 2*b2 + 4*b1 + 3

## 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
 −1.75660 −0.820249 1.13856 2.43828
−1.75660 −0.0856374 1.08564 1.00000 0.150431 −1.86144 1.60617 −2.99267 −1.75660
1.2 −0.820249 2.32719 −1.32719 1.00000 −1.90888 −0.561717 2.72913 2.41582 −0.820249
1.3 1.13856 1.70367 −0.703671 1.00000 1.93974 −4.75660 −3.07830 −0.0975037 1.13856
1.4 2.43828 −2.94523 3.94523 1.00000 −7.18129 −3.82025 4.74301 5.67435 2.43828
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 1805.2.a.n yes 4
5.b even 2 1 9025.2.a.bh 4
19.b odd 2 1 1805.2.a.j 4
95.d odd 2 1 9025.2.a.bo 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.a.j 4 19.b odd 2 1
1805.2.a.n yes 4 1.a even 1 1 trivial
9025.2.a.bh 4 5.b even 2 1
9025.2.a.bo 4 95.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1805))$$:

 $$T_{2}^{4} - T_{2}^{3} - 5T_{2}^{2} + 2T_{2} + 4$$ T2^4 - T2^3 - 5*T2^2 + 2*T2 + 4 $$T_{3}^{4} - T_{3}^{3} - 8T_{3}^{2} + 11T_{3} + 1$$ T3^4 - T3^3 - 8*T3^2 + 11*T3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} - 5 T^{2} + 2 T + 4$$
$3$ $$T^{4} - T^{3} - 8 T^{2} + 11 T + 1$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} + 11 T^{3} + 40 T^{2} + 53 T + 19$$
$11$ $$T^{4} - 9 T^{2} - 10 T - 1$$
$13$ $$T^{4} + 2 T^{3} - 32 T^{2} + 22 T + 71$$
$17$ $$T^{4} + 7 T^{3} - 25 T^{2} - 256 T - 436$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 11 T^{3} + 2 T^{2} - 281 T - 709$$
$29$ $$T^{4} + 15 T^{3} + 49 T^{2} + \cdots - 116$$
$31$ $$T^{4} + T^{3} - 60 T^{2} + 3 T + 659$$
$37$ $$T^{4} + 11 T^{3} + 30 T^{2} + \cdots - 101$$
$41$ $$T^{4} - 22 T^{3} + 100 T^{2} + \cdots - 4061$$
$43$ $$T^{4} + 26 T^{3} + 225 T^{2} + \cdots + 379$$
$47$ $$T^{4} + 26 T^{3} + 200 T^{2} + \cdots - 1681$$
$53$ $$T^{4} + 16 T^{3} + 25 T^{2} + \cdots - 1891$$
$59$ $$T^{4} + 10 T^{3} - 41 T^{2} + \cdots + 829$$
$61$ $$T^{4} - 2 T^{3} - 131 T^{2} + \cdots - 1744$$
$67$ $$T^{4} - 3 T^{3} - 110 T^{2} + \cdots + 2389$$
$71$ $$T^{4} - 18 T^{3} + 100 T^{2} + \cdots + 19$$
$73$ $$T^{4} + 24 T^{3} + 150 T^{2} + \cdots - 2071$$
$79$ $$T^{4} - 30 T^{3} + 321 T^{2} + \cdots + 2384$$
$83$ $$T^{4} + 12 T^{3} - 5 T^{2} - 296 T + 4$$
$89$ $$T^{4} - 9 T^{3} + 13 T - 1$$
$97$ $$T^{4} - 19 T^{3} - 48 T^{2} + \cdots - 839$$