# Properties

 Label 49.6.c.d Level $49$ Weight $6$ Character orbit 49.c Analytic conductor $7.859$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,6,Mod(18,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("49.18");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 49.c (of order $$3$$, degree $$2$$, not 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: $$7.85880717084$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ x^4 - x^3 - 4*x^2 - 5*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 7) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_{2} + 5 \beta_1) q^{2} + (6 \beta_{3} - 6 \beta_1 - 6) q^{3} + (9 \beta_{3} - 7 \beta_1 - 7) q^{4} + (10 \beta_{3} - 10 \beta_{2} + 4 \beta_1) q^{5} + ( - 30 \beta_{2} + 114) q^{6} + ( - 11 \beta_{2} + 1) q^{8} + ( - 36 \beta_{3} + 36 \beta_{2} + 297 \beta_1) q^{9}+O(q^{10})$$ q + (-b3 + b2 + 5*b1) * q^2 + (6*b3 - 6*b1 - 6) * q^3 + (9*b3 - 7*b1 - 7) * q^4 + (10*b3 - 10*b2 + 4*b1) * q^5 + (-30*b2 + 114) * q^6 + (-11*b2 + 1) * q^8 + (-36*b3 + 36*b2 + 297*b1) * q^9 $$q + ( - \beta_{3} + \beta_{2} + 5 \beta_1) q^{2} + (6 \beta_{3} - 6 \beta_1 - 6) q^{3} + (9 \beta_{3} - 7 \beta_1 - 7) q^{4} + (10 \beta_{3} - 10 \beta_{2} + 4 \beta_1) q^{5} + ( - 30 \beta_{2} + 114) q^{6} + ( - 11 \beta_{2} + 1) q^{8} + ( - 36 \beta_{3} + 36 \beta_{2} + 297 \beta_1) q^{9} + ( - 36 \beta_{3} + 120 \beta_1 + 120) q^{10} + ( - 124 \beta_{3} - 136 \beta_1 - 136) q^{11} + ( - 42 \beta_{3} + 42 \beta_{2} + 798 \beta_1) q^{12} + (126 \beta_{2} + 112) q^{13} + ( - 24 \beta_{2} - 816) q^{15} + (243 \beta_{3} - 243 \beta_{2} - 65 \beta_1) q^{16} + (76 \beta_{3} + 862 \beta_1 + 862) q^{17} + (441 \beta_{3} - 1989 \beta_1 - 1989) q^{18} + ( - 18 \beta_{3} + 18 \beta_{2} + 1642 \beta_1) q^{19} + ( - 56 \beta_{2} - 1232) q^{20} + (360 \beta_{2} - 1056) q^{22} + ( - 568 \beta_{3} + 568 \beta_{2} + 1328 \beta_1) q^{23} + (6 \beta_{3} - 930 \beta_1 - 930) q^{24} + ( - 180 \beta_{3} + 1709 \beta_1 + 1709) q^{25} + (392 \beta_{3} - 392 \beta_{2} - 1204 \beta_1) q^{26} + ( - 324 \beta_{2} + 3348) q^{27} + ( - 252 \beta_{2} + 3474) q^{29} + (720 \beta_{3} - 720 \beta_{2} - 3744 \beta_1) q^{30} + ( - 540 \beta_{3} + 260 \beta_1 + 260) q^{31} + ( - 1389 \beta_{3} + 3759 \beta_1 + 3759) q^{32} + ( - 816 \beta_{3} + 816 \beta_{2} - 9600 \beta_1) q^{33} + (558 \beta_{2} - 3246) q^{34} + ( - 2601 \beta_{2} + 6615) q^{36} + ( - 540 \beta_{3} + 540 \beta_{2} + 3386 \beta_1) q^{37} + (1714 \beta_{3} - 8462 \beta_1 - 8462) q^{38} + (672 \beta_{3} + 9912 \beta_1 + 9912) q^{39} + ( - 144 \beta_{3} + 144 \beta_{2} - 1536 \beta_1) q^{40} + ( - 1092 \beta_{2} + 3570) q^{41} + (4788 \beta_{2} - 3904) q^{43} + ( - 1472 \beta_{3} + 1472 \beta_{2} - 14672 \beta_1) q^{44} + ( - 2466 \beta_{3} + 3852 \beta_1 + 3852) q^{45} + (3600 \beta_{3} - 14592 \beta_1 - 14592) q^{46} + (3748 \beta_{3} - 3748 \beta_{2} - 7724 \beta_1) q^{47} + (390 \beta_{2} - 20802) q^{48} + (2429 \beta_{2} - 11065) q^{50} + (5172 \beta_{3} - 5172 \beta_{2} + 1212 \beta_1) q^{51} + (1260 \beta_{3} + 15092 \beta_1 + 15092) q^{52} + ( - 208 \beta_{3} - 4630 \beta_1 - 4630) q^{53} + ( - 4644 \beta_{3} + 4644 \beta_{2} + 21276 \beta_1) q^{54} + (3096 \beta_{2} + 17904) q^{55} + ( - 9852 \beta_{2} + 11364) q^{57} + ( - 4482 \beta_{3} + 4482 \beta_{2} + 20898 \beta_1) q^{58} + (2050 \beta_{3} - 22994 \beta_1 - 22994) q^{59} + ( - 7392 \beta_{3} + 2688 \beta_1 + 2688) q^{60} + ( - 4806 \beta_{3} + 4806 \beta_{2} + 34780 \beta_1) q^{61} + (2420 \beta_{2} - 8860) q^{62} + (1539 \beta_{2} - 36161) q^{64} + (2884 \beta_{3} - 2884 \beta_{2} + 18088 \beta_1) q^{65} + ( - 6336 \beta_{3} + 36576 \beta_1 + 36576) q^{66} + ( - 1944 \beta_{3} - 11420 \beta_1 - 11420) q^{67} + (7910 \beta_{3} - 7910 \beta_{2} + 3542 \beta_1) q^{68} + ( - 7968 \beta_{2} + 55680) q^{69} + (4200 \beta_{2} + 46608) q^{71} + ( - 2907 \beta_{3} + 2907 \beta_{2} + 5841 \beta_1) q^{72} + (5256 \beta_{3} + 6098 \beta_1 + 6098) q^{73} + (5546 \beta_{3} - 24490 \beta_1 - 24490) q^{74} + (10254 \beta_{3} - 10254 \beta_{2} - 25374 \beta_1) q^{75} + ( - 14742 \beta_{2} + 13762) q^{76} + (7224 \beta_{2} - 40152) q^{78} + ( - 14904 \beta_{3} + 14904 \beta_{2} + 33080 \beta_1) q^{79} + ( - 2752 \beta_{3} - 33760 \beta_1 - 33760) q^{80} + (11340 \beta_{3} + 24867 \beta_1 + 24867) q^{81} + ( - 7938 \beta_{3} + 7938 \beta_{2} + 33138 \beta_1) q^{82} + (15750 \beta_{2} - 66654) q^{83} + ( - 9684 \beta_{2} - 14088) q^{85} + (23056 \beta_{3} - 23056 \beta_{2} - 86552 \beta_1) q^{86} + (20844 \beta_{3} - 42012 \beta_1 - 42012) q^{87} + (2736 \beta_{3} + 18960 \beta_1 + 18960) q^{88} + ( - 22208 \beta_{3} + 22208 \beta_{2} - 31034 \beta_1) q^{89} + (13716 \beta_{2} - 53784) q^{90} + ( - 10816 \beta_{2} + 80864) q^{92} + (1560 \beta_{3} - 1560 \beta_{2} - 46920 \beta_1) q^{93} + ( - 22716 \beta_{3} + 91092 \beta_1 + 91092) q^{94} + ( - 16168 \beta_{3} - 4048 \beta_1 - 4048) q^{95} + (22554 \beta_{3} - 22554 \beta_{2} - 139230 \beta_1) q^{96} + (8820 \beta_{2} - 14798) q^{97} + (27468 \beta_{2} - 22104) q^{99}+O(q^{100})$$ q + (-b3 + b2 + 5*b1) * q^2 + (6*b3 - 6*b1 - 6) * q^3 + (9*b3 - 7*b1 - 7) * q^4 + (10*b3 - 10*b2 + 4*b1) * q^5 + (-30*b2 + 114) * q^6 + (-11*b2 + 1) * q^8 + (-36*b3 + 36*b2 + 297*b1) * q^9 + (-36*b3 + 120*b1 + 120) * q^10 + (-124*b3 - 136*b1 - 136) * q^11 + (-42*b3 + 42*b2 + 798*b1) * q^12 + (126*b2 + 112) * q^13 + (-24*b2 - 816) * q^15 + (243*b3 - 243*b2 - 65*b1) * q^16 + (76*b3 + 862*b1 + 862) * q^17 + (441*b3 - 1989*b1 - 1989) * q^18 + (-18*b3 + 18*b2 + 1642*b1) * q^19 + (-56*b2 - 1232) * q^20 + (360*b2 - 1056) * q^22 + (-568*b3 + 568*b2 + 1328*b1) * q^23 + (6*b3 - 930*b1 - 930) * q^24 + (-180*b3 + 1709*b1 + 1709) * q^25 + (392*b3 - 392*b2 - 1204*b1) * q^26 + (-324*b2 + 3348) * q^27 + (-252*b2 + 3474) * q^29 + (720*b3 - 720*b2 - 3744*b1) * q^30 + (-540*b3 + 260*b1 + 260) * q^31 + (-1389*b3 + 3759*b1 + 3759) * q^32 + (-816*b3 + 816*b2 - 9600*b1) * q^33 + (558*b2 - 3246) * q^34 + (-2601*b2 + 6615) * q^36 + (-540*b3 + 540*b2 + 3386*b1) * q^37 + (1714*b3 - 8462*b1 - 8462) * q^38 + (672*b3 + 9912*b1 + 9912) * q^39 + (-144*b3 + 144*b2 - 1536*b1) * q^40 + (-1092*b2 + 3570) * q^41 + (4788*b2 - 3904) * q^43 + (-1472*b3 + 1472*b2 - 14672*b1) * q^44 + (-2466*b3 + 3852*b1 + 3852) * q^45 + (3600*b3 - 14592*b1 - 14592) * q^46 + (3748*b3 - 3748*b2 - 7724*b1) * q^47 + (390*b2 - 20802) * q^48 + (2429*b2 - 11065) * q^50 + (5172*b3 - 5172*b2 + 1212*b1) * q^51 + (1260*b3 + 15092*b1 + 15092) * q^52 + (-208*b3 - 4630*b1 - 4630) * q^53 + (-4644*b3 + 4644*b2 + 21276*b1) * q^54 + (3096*b2 + 17904) * q^55 + (-9852*b2 + 11364) * q^57 + (-4482*b3 + 4482*b2 + 20898*b1) * q^58 + (2050*b3 - 22994*b1 - 22994) * q^59 + (-7392*b3 + 2688*b1 + 2688) * q^60 + (-4806*b3 + 4806*b2 + 34780*b1) * q^61 + (2420*b2 - 8860) * q^62 + (1539*b2 - 36161) * q^64 + (2884*b3 - 2884*b2 + 18088*b1) * q^65 + (-6336*b3 + 36576*b1 + 36576) * q^66 + (-1944*b3 - 11420*b1 - 11420) * q^67 + (7910*b3 - 7910*b2 + 3542*b1) * q^68 + (-7968*b2 + 55680) * q^69 + (4200*b2 + 46608) * q^71 + (-2907*b3 + 2907*b2 + 5841*b1) * q^72 + (5256*b3 + 6098*b1 + 6098) * q^73 + (5546*b3 - 24490*b1 - 24490) * q^74 + (10254*b3 - 10254*b2 - 25374*b1) * q^75 + (-14742*b2 + 13762) * q^76 + (7224*b2 - 40152) * q^78 + (-14904*b3 + 14904*b2 + 33080*b1) * q^79 + (-2752*b3 - 33760*b1 - 33760) * q^80 + (11340*b3 + 24867*b1 + 24867) * q^81 + (-7938*b3 + 7938*b2 + 33138*b1) * q^82 + (15750*b2 - 66654) * q^83 + (-9684*b2 - 14088) * q^85 + (23056*b3 - 23056*b2 - 86552*b1) * q^86 + (20844*b3 - 42012*b1 - 42012) * q^87 + (2736*b3 + 18960*b1 + 18960) * q^88 + (-22208*b3 + 22208*b2 - 31034*b1) * q^89 + (13716*b2 - 53784) * q^90 + (-10816*b2 + 80864) * q^92 + (1560*b3 - 1560*b2 - 46920*b1) * q^93 + (-22716*b3 + 91092*b1 + 91092) * q^94 + (-16168*b3 - 4048*b1 - 4048) * q^95 + (22554*b3 - 22554*b2 - 139230*b1) * q^96 + (8820*b2 - 14798) * q^97 + (27468*b2 - 22104) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 9 q^{2} - 6 q^{3} - 5 q^{4} - 18 q^{5} + 396 q^{6} - 18 q^{8} - 558 q^{9}+O(q^{10})$$ 4 * q - 9 * q^2 - 6 * q^3 - 5 * q^4 - 18 * q^5 + 396 * q^6 - 18 * q^8 - 558 * q^9 $$4 q - 9 q^{2} - 6 q^{3} - 5 q^{4} - 18 q^{5} + 396 q^{6} - 18 q^{8} - 558 q^{9} + 204 q^{10} - 396 q^{11} - 1554 q^{12} + 700 q^{13} - 3312 q^{15} - 113 q^{16} + 1800 q^{17} - 3537 q^{18} - 3266 q^{19} - 5040 q^{20} - 3504 q^{22} - 2088 q^{23} - 1854 q^{24} + 3238 q^{25} + 2016 q^{26} + 12744 q^{27} + 13392 q^{29} + 6768 q^{30} - 20 q^{31} + 6129 q^{32} + 20016 q^{33} - 11868 q^{34} + 21258 q^{36} - 6232 q^{37} - 15210 q^{38} + 20496 q^{39} + 3216 q^{40} + 12096 q^{41} - 6040 q^{43} + 30816 q^{44} + 5238 q^{45} - 25584 q^{46} + 11700 q^{47} - 82428 q^{48} - 39402 q^{50} - 7596 q^{51} + 31444 q^{52} - 9468 q^{53} - 37908 q^{54} + 77808 q^{55} + 25752 q^{57} - 37314 q^{58} - 43938 q^{59} - 2016 q^{60} - 64754 q^{61} - 30600 q^{62} - 141566 q^{64} - 39060 q^{65} + 66816 q^{66} - 24784 q^{67} - 14994 q^{68} + 206784 q^{69} + 194832 q^{71} - 8775 q^{72} + 17452 q^{73} - 43434 q^{74} + 40494 q^{75} + 25564 q^{76} - 146160 q^{78} - 51256 q^{79} - 70272 q^{80} + 61074 q^{81} - 58338 q^{82} - 235116 q^{83} - 75720 q^{85} + 150048 q^{86} - 63180 q^{87} + 40656 q^{88} + 84276 q^{89} - 187704 q^{90} + 301824 q^{92} + 92280 q^{93} + 159468 q^{94} - 24264 q^{95} + 255906 q^{96} - 41552 q^{97} - 33480 q^{99}+O(q^{100})$$ 4 * q - 9 * q^2 - 6 * q^3 - 5 * q^4 - 18 * q^5 + 396 * q^6 - 18 * q^8 - 558 * q^9 + 204 * q^10 - 396 * q^11 - 1554 * q^12 + 700 * q^13 - 3312 * q^15 - 113 * q^16 + 1800 * q^17 - 3537 * q^18 - 3266 * q^19 - 5040 * q^20 - 3504 * q^22 - 2088 * q^23 - 1854 * q^24 + 3238 * q^25 + 2016 * q^26 + 12744 * q^27 + 13392 * q^29 + 6768 * q^30 - 20 * q^31 + 6129 * q^32 + 20016 * q^33 - 11868 * q^34 + 21258 * q^36 - 6232 * q^37 - 15210 * q^38 + 20496 * q^39 + 3216 * q^40 + 12096 * q^41 - 6040 * q^43 + 30816 * q^44 + 5238 * q^45 - 25584 * q^46 + 11700 * q^47 - 82428 * q^48 - 39402 * q^50 - 7596 * q^51 + 31444 * q^52 - 9468 * q^53 - 37908 * q^54 + 77808 * q^55 + 25752 * q^57 - 37314 * q^58 - 43938 * q^59 - 2016 * q^60 - 64754 * q^61 - 30600 * q^62 - 141566 * q^64 - 39060 * q^65 + 66816 * q^66 - 24784 * q^67 - 14994 * q^68 + 206784 * q^69 + 194832 * q^71 - 8775 * q^72 + 17452 * q^73 - 43434 * q^74 + 40494 * q^75 + 25564 * q^76 - 146160 * q^78 - 51256 * q^79 - 70272 * q^80 + 61074 * q^81 - 58338 * q^82 - 235116 * q^83 - 75720 * q^85 + 150048 * q^86 - 63180 * q^87 + 40656 * q^88 + 84276 * q^89 - 187704 * q^90 + 301824 * q^92 + 92280 * q^93 + 159468 * q^94 - 24264 * q^95 + 255906 * q^96 - 41552 * q^97 - 33480 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20$$ (v^3 + 4*v^2 - 4*v - 25) / 20 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5$$ (-v^3 + v^2 + 9*v + 5) / 5 $$\beta_{3}$$ $$=$$ $$( 3\nu^{3} + 2\nu^{2} + 8\nu - 25 ) / 10$$ (3*v^3 + 2*v^2 + 8*v - 25) / 10
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3$$ (b3 + b2 - 2*b1 - 1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 14\beta _1 + 13 ) / 3$$ (-b3 + 2*b2 + 14*b1 + 13) / 3 $$\nu^{3}$$ $$=$$ $$( 8\beta_{3} - 4\beta_{2} - 4\beta _1 + 19 ) / 3$$ (8*b3 - 4*b2 - 4*b1 + 19) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/49\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 −1.63746 + 1.52274i 2.13746 − 0.656712i −1.63746 − 1.52274i 2.13746 + 0.656712i
−4.13746 7.16629i −12.8248 + 22.2131i −18.2371 + 31.5876i 14.3746 + 24.8975i 212.248 0 37.0241 −207.449 359.311i 118.949 206.025i
18.2 −0.362541 0.627940i 9.82475 17.0170i 15.7371 27.2575i −23.3746 40.4860i −14.2475 0 −46.0241 −71.5515 123.931i −16.9485 + 29.3557i
30.1 −4.13746 + 7.16629i −12.8248 22.2131i −18.2371 31.5876i 14.3746 24.8975i 212.248 0 37.0241 −207.449 + 359.311i 118.949 + 206.025i
30.2 −0.362541 + 0.627940i 9.82475 + 17.0170i 15.7371 + 27.2575i −23.3746 + 40.4860i −14.2475 0 −46.0241 −71.5515 + 123.931i −16.9485 29.3557i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.6.c.d 4
7.b odd 2 1 49.6.c.e 4
7.c even 3 1 49.6.a.f 2
7.c even 3 1 inner 49.6.c.d 4
7.d odd 6 1 7.6.a.b 2
7.d odd 6 1 49.6.c.e 4
21.g even 6 1 63.6.a.f 2
21.h odd 6 1 441.6.a.l 2
28.f even 6 1 112.6.a.h 2
28.g odd 6 1 784.6.a.v 2
35.i odd 6 1 175.6.a.c 2
35.k even 12 2 175.6.b.c 4
56.j odd 6 1 448.6.a.w 2
56.m even 6 1 448.6.a.u 2
77.i even 6 1 847.6.a.c 2
84.j odd 6 1 1008.6.a.bq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 7.d odd 6 1
49.6.a.f 2 7.c even 3 1
49.6.c.d 4 1.a even 1 1 trivial
49.6.c.d 4 7.c even 3 1 inner
49.6.c.e 4 7.b odd 2 1
49.6.c.e 4 7.d odd 6 1
63.6.a.f 2 21.g even 6 1
112.6.a.h 2 28.f even 6 1
175.6.a.c 2 35.i odd 6 1
175.6.b.c 4 35.k even 12 2
441.6.a.l 2 21.h odd 6 1
448.6.a.u 2 56.m even 6 1
448.6.a.w 2 56.j odd 6 1
784.6.a.v 2 28.g odd 6 1
847.6.a.c 2 77.i even 6 1
1008.6.a.bq 2 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(49, [\chi])$$:

 $$T_{2}^{4} + 9T_{2}^{3} + 75T_{2}^{2} + 54T_{2} + 36$$ T2^4 + 9*T2^3 + 75*T2^2 + 54*T2 + 36 $$T_{3}^{4} + 6T_{3}^{3} + 540T_{3}^{2} - 3024T_{3} + 254016$$ T3^4 + 6*T3^3 + 540*T3^2 - 3024*T3 + 254016

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 9 T^{3} + 75 T^{2} + 54 T + 36$$
$3$ $$T^{4} + 6 T^{3} + 540 T^{2} + \cdots + 254016$$
$5$ $$T^{4} + 18 T^{3} + 1668 T^{2} + \cdots + 1806336$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 396 T^{3} + \cdots + 32365449216$$
$13$ $$(T^{2} - 350 T - 195608)^{2}$$
$17$ $$T^{4} - 1800 T^{3} + \cdots + 529535646864$$
$19$ $$T^{4} + 3266 T^{3} + \cdots + 7086627333184$$
$23$ $$T^{4} + 2088 T^{3} + \cdots + 12302247591936$$
$29$ $$(T^{2} - 6696 T + 10304172)^{2}$$
$31$ $$T^{4} + 20 T^{3} + \cdots + 17265687040000$$
$37$ $$T^{4} + 6232 T^{3} + \cdots + 30848648872336$$
$41$ $$(T^{2} - 6048 T - 7848036)^{2}$$
$43$ $$(T^{2} + 3020 T - 324400352)^{2}$$
$47$ $$T^{4} - 11700 T^{3} + \cdots + 27\!\cdots\!24$$
$53$ $$T^{4} + \cdots + 474989071531536$$
$59$ $$T^{4} + 43938 T^{3} + \cdots + 17\!\cdots\!96$$
$61$ $$T^{4} + 64754 T^{3} + \cdots + 51\!\cdots\!56$$
$67$ $$T^{4} + 24784 T^{3} + \cdots + 99\!\cdots\!76$$
$71$ $$(T^{2} - 97416 T + 2121099264)^{2}$$
$73$ $$T^{4} - 17452 T^{3} + \cdots + 10\!\cdots\!44$$
$79$ $$T^{4} + 51256 T^{3} + \cdots + 62\!\cdots\!36$$
$83$ $$(T^{2} + 117558 T - 79919784)^{2}$$
$89$ $$T^{4} - 84276 T^{3} + \cdots + 27\!\cdots\!24$$
$97$ $$(T^{2} + 20776 T - 1000631156)^{2}$$