# Properties

 Label 1815.4.a.bc Level $1815$ Weight $4$ Character orbit 1815.a Self dual yes Analytic conductor $107.088$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.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: $$107.088466660$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 52x^{5} + 37x^{4} + 765x^{3} - 296x^{2} - 2962x + 1692$$ x^7 - x^6 - 52*x^5 + 37*x^4 + 765*x^3 - 296*x^2 - 2962*x + 1692 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 7) q^{4} + 5 q^{5} + 3 \beta_1 q^{6} + (\beta_{3} - \beta_1 + 2) q^{7} + ( - \beta_{4} + \beta_{3} - 7 \beta_1 - 4) q^{8} + 9 q^{9}+O(q^{10})$$ q - b1 * q^2 - 3 * q^3 + (b2 + 7) * q^4 + 5 * q^5 + 3*b1 * q^6 + (b3 - b1 + 2) * q^7 + (-b4 + b3 - 7*b1 - 4) * q^8 + 9 * q^9 $$q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 7) q^{4} + 5 q^{5} + 3 \beta_1 q^{6} + (\beta_{3} - \beta_1 + 2) q^{7} + ( - \beta_{4} + \beta_{3} - 7 \beta_1 - 4) q^{8} + 9 q^{9} - 5 \beta_1 q^{10} + ( - 3 \beta_{2} - 21) q^{12} + ( - \beta_{6} + \beta_{3} - \beta_{2} - 13) q^{13} + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots + 11) q^{14}+ \cdots + (7 \beta_{6} + 19 \beta_{5} + \cdots - 735) q^{98}+O(q^{100})$$ q - b1 * q^2 - 3 * q^3 + (b2 + 7) * q^4 + 5 * q^5 + 3*b1 * q^6 + (b3 - b1 + 2) * q^7 + (-b4 + b3 - 7*b1 - 4) * q^8 + 9 * q^9 - 5*b1 * q^10 + (-3*b2 - 21) * q^12 + (-b6 + b3 - b2 - 13) * q^13 + (b6 + b5 - b4 - 2*b3 + 4*b2 - 4*b1 + 11) * q^14 - 15 * q^15 + (b6 + 2*b5 - b4 + 8*b2 + 3*b1 + 48) * q^16 + (3*b5 - b3 + 4*b2 - 5*b1 + 12) * q^17 - 9*b1 * q^18 + (b6 + b5 + 2*b4 - 5*b2 + b1 - 38) * q^19 + (5*b2 + 35) * q^20 + (-3*b3 + 3*b1 - 6) * q^21 + (b6 - b5 + 2*b3 + b2 - 21*b1 - 33) * q^23 + (3*b4 - 3*b3 + 21*b1 + 12) * q^24 + 25 * q^25 + (5*b5 - 12*b3 + 2*b2 + 24*b1 + 9) * q^26 - 27 * q^27 + (-2*b6 - 5*b5 - 4*b4 + 12*b3 + 6*b2 - 36*b1 + 31) * q^28 + (-b6 + 2*b5 - 4*b4 - b3 - 13*b2 - 6*b1 + 5) * q^29 + 15*b1 * q^30 + (b6 + 4*b4 + 3*b3 + b2 - 4*b1 + 114) * q^31 + (-b6 - 3*b5 - 4*b4 + 13*b3 + 6*b2 - 62*b1 - 49) * q^32 + (-4*b6 - b5 - 9*b4 + 9*b3 + 5*b2 - 45*b1 + 66) * q^34 + (5*b3 - 5*b1 + 10) * q^35 + (9*b2 + 63) * q^36 + (b6 - 5*b5 + 4*b4 + 4*b3 - 11*b2 + 5*b1 - 48) * q^37 + (-6*b5 + 3*b4 + b3 - 11*b2 + 70*b1 - 9) * q^38 + (3*b6 - 3*b3 + 3*b2 + 39) * q^39 + (-5*b4 + 5*b3 - 35*b1 - 20) * q^40 + (b6 + 3*b5 - 8*b4 + 5*b3 + 5*b2 + 8*b1 - 20) * q^41 + (-3*b6 - 3*b5 + 3*b4 + 6*b3 - 12*b2 + 12*b1 - 33) * q^42 + (3*b6 + 7*b5 + 2*b4 + b3 + 5*b2 - 26*b1 - 110) * q^43 + 45 * q^45 + (4*b6 - 2*b5 - b4 + 5*b3 + 27*b2 + 17*b1 + 293) * q^46 + (-2*b6 - 10*b5 - 8*b4 + 7*b3 + 14*b2 - 29*b1 + 91) * q^47 + (-3*b6 - 6*b5 + 3*b4 - 24*b2 - 9*b1 - 144) * q^48 + (-2*b6 - 2*b4 + 9*b3 + 12*b2 + 45*b1 + 109) * q^49 - 25*b1 * q^50 + (-9*b5 + 3*b3 - 12*b2 + 15*b1 - 36) * q^51 + (-9*b6 - 12*b5 + 23*b3 - 47*b2 - 6*b1 - 211) * q^52 + (-b6 + 5*b5 + 2*b4 - 19*b2 - 45*b1 + 59) * q^53 + 27*b1 * q^54 + (7*b6 + 16*b5 - 17*b3 + 57*b2 - 52*b1 + 405) * q^56 + (-3*b6 - 3*b5 - 6*b4 + 15*b2 - 3*b1 + 114) * q^57 + (-4*b6 + 7*b5 + 10*b4 - 10*b3 + 28*b2 + 108*b1 + 169) * q^58 + (-b6 - 3*b5 + 6*b4 + 5*b3 - 3*b2 - 58*b1 - 120) * q^59 + (-15*b2 - 105) * q^60 + (-3*b6 - 2*b5 + 6*b4 - 15*b3 - 13*b2 - 42*b1 + 14) * q^61 + (4*b6 - 5*b5 - 4*b4 - 4*b3 - 10*b2 - 137*b1 + 23) * q^62 + (9*b3 - 9*b1 + 18) * q^63 + (7*b6 + 5*b5 - 5*b4 - 24*b3 + 57*b2 - 37*b1 + 488) * q^64 + (-5*b6 + 5*b3 - 5*b2 - 65) * q^65 + (-3*b5 + 14*b4 - 15*b3 + 2*b2 - 19*b1 - 5) * q^67 + (6*b6 + 10*b5 - 12*b4 - 24*b3 + 89*b2 - 54*b1 + 585) * q^68 + (-3*b6 + 3*b5 - 6*b3 - 3*b2 + 63*b1 + 99) * q^69 + (5*b6 + 5*b5 - 5*b4 - 10*b3 + 20*b2 - 20*b1 + 55) * q^70 + (-10*b6 - 3*b5 + 6*b4 + 2*b3 + 46*b1 - 93) * q^71 + (-9*b4 + 9*b3 - 63*b1 - 36) * q^72 + (-3*b6 + 10*b5 + 12*b4 + 4*b3 - 19*b2 + 139*b1 + 23) * q^73 + (10*b6 - 4*b5 + 17*b4 - 23*b3 - 21*b2 + 124*b1 - 73) * q^74 - 75 * q^75 + (-b6 - 10*b5 + 6*b4 - 25*b3 - 51*b2 + 90*b1 - 721) * q^76 + (-15*b5 + 36*b3 - 6*b2 - 72*b1 - 27) * q^78 + (2*b6 + 3*b5 + 28*b4 - 4*b3 - 38*b2 + 28*b1 + 72) * q^79 + (5*b6 + 10*b5 - 5*b4 + 40*b2 + 15*b1 + 240) * q^80 + 81 * q^81 + (3*b6 + 9*b5 - 16*b4 + 23*b3 + 59*b2 - 30*b1 - 142) * q^82 + (3*b6 + 18*b4 + b3 + 25*b2 - 68*b1 + 177) * q^83 + (6*b6 + 15*b5 + 12*b4 - 36*b3 - 18*b2 + 108*b1 - 93) * q^84 + (15*b5 - 5*b3 + 20*b2 - 25*b1 + 60) * q^85 + (-3*b6 - 13*b5 - 20*b4 + 33*b3 + 27*b2 + 44*b1 + 340) * q^86 + (3*b6 - 6*b5 + 12*b4 + 3*b3 + 39*b2 + 18*b1 - 15) * q^87 + (-2*b6 - 16*b5 + 10*b4 - 4*b3 - 16*b2 - 26*b1 + 298) * q^89 - 45*b1 * q^90 + (b6 - 14*b5 + 6*b4 + 13*b3 + 27*b2 + 310*b1 + 409) * q^91 + (3*b6 - 2*b5 - 28*b4 + 37*b3 - 2*b2 - 368*b1 - 154) * q^92 + (-3*b6 - 12*b4 - 9*b3 - 3*b2 + 12*b1 - 342) * q^93 + (15*b6 + 23*b5 - b4 - 12*b3 + 86*b2 - 189*b1 + 383) * q^94 + (5*b6 + 5*b5 + 10*b4 - 25*b2 + 5*b1 - 190) * q^95 + (3*b6 + 9*b5 + 12*b4 - 39*b3 - 18*b2 + 186*b1 + 147) * q^96 + (-7*b6 - 10*b5 + 6*b4 + 43*b2 - 45*b1 + 347) * q^97 + (7*b6 + 19*b5 - 21*b4 - 20*b3 - 8*b2 - 211*b1 - 735) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - q^{2} - 21 q^{3} + 49 q^{4} + 35 q^{5} + 3 q^{6} + 16 q^{7} - 30 q^{8} + 63 q^{9}+O(q^{10})$$ 7 * q - q^2 - 21 * q^3 + 49 * q^4 + 35 * q^5 + 3 * q^6 + 16 * q^7 - 30 * q^8 + 63 * q^9 $$7 q - q^{2} - 21 q^{3} + 49 q^{4} + 35 q^{5} + 3 q^{6} + 16 q^{7} - 30 q^{8} + 63 q^{9} - 5 q^{10} - 147 q^{12} - 90 q^{13} + 68 q^{14} - 105 q^{15} + 337 q^{16} + 67 q^{17} - 9 q^{18} - 270 q^{19} + 245 q^{20} - 48 q^{21} - 241 q^{23} + 90 q^{24} + 175 q^{25} + 36 q^{26} - 189 q^{27} + 236 q^{28} + 26 q^{29} + 15 q^{30} + 797 q^{31} - 351 q^{32} + 457 q^{34} + 80 q^{35} + 441 q^{36} - 310 q^{37} + 22 q^{38} + 270 q^{39} - 150 q^{40} - 108 q^{41} - 204 q^{42} - 812 q^{43} + 315 q^{45} + 2099 q^{46} + 671 q^{47} - 1011 q^{48} + 835 q^{49} - 25 q^{50} - 201 q^{51} - 1396 q^{52} + 347 q^{53} + 27 q^{54} + 2698 q^{56} + 810 q^{57} + 1212 q^{58} - 888 q^{59} - 735 q^{60} - q^{61} + 43 q^{62} + 144 q^{63} + 3316 q^{64} - 450 q^{65} - 118 q^{67} + 3975 q^{68} + 723 q^{69} + 340 q^{70} - 622 q^{71} - 270 q^{72} + 252 q^{73} - 458 q^{74} - 525 q^{75} - 5016 q^{76} - 108 q^{78} + 459 q^{79} + 1685 q^{80} + 567 q^{81} - 944 q^{82} + 1144 q^{83} - 708 q^{84} + 335 q^{85} + 2596 q^{86} - 78 q^{87} + 2072 q^{89} - 45 q^{90} + 3244 q^{91} - 1267 q^{92} - 2391 q^{93} + 2419 q^{94} - 1350 q^{95} + 1053 q^{96} + 2388 q^{97} - 5417 q^{98}+O(q^{100})$$ 7 * q - q^2 - 21 * q^3 + 49 * q^4 + 35 * q^5 + 3 * q^6 + 16 * q^7 - 30 * q^8 + 63 * q^9 - 5 * q^10 - 147 * q^12 - 90 * q^13 + 68 * q^14 - 105 * q^15 + 337 * q^16 + 67 * q^17 - 9 * q^18 - 270 * q^19 + 245 * q^20 - 48 * q^21 - 241 * q^23 + 90 * q^24 + 175 * q^25 + 36 * q^26 - 189 * q^27 + 236 * q^28 + 26 * q^29 + 15 * q^30 + 797 * q^31 - 351 * q^32 + 457 * q^34 + 80 * q^35 + 441 * q^36 - 310 * q^37 + 22 * q^38 + 270 * q^39 - 150 * q^40 - 108 * q^41 - 204 * q^42 - 812 * q^43 + 315 * q^45 + 2099 * q^46 + 671 * q^47 - 1011 * q^48 + 835 * q^49 - 25 * q^50 - 201 * q^51 - 1396 * q^52 + 347 * q^53 + 27 * q^54 + 2698 * q^56 + 810 * q^57 + 1212 * q^58 - 888 * q^59 - 735 * q^60 - q^61 + 43 * q^62 + 144 * q^63 + 3316 * q^64 - 450 * q^65 - 118 * q^67 + 3975 * q^68 + 723 * q^69 + 340 * q^70 - 622 * q^71 - 270 * q^72 + 252 * q^73 - 458 * q^74 - 525 * q^75 - 5016 * q^76 - 108 * q^78 + 459 * q^79 + 1685 * q^80 + 567 * q^81 - 944 * q^82 + 1144 * q^83 - 708 * q^84 + 335 * q^85 + 2596 * q^86 - 78 * q^87 + 2072 * q^89 - 45 * q^90 + 3244 * q^91 - 1267 * q^92 - 2391 * q^93 + 2419 * q^94 - 1350 * q^95 + 1053 * q^96 + 2388 * q^97 - 5417 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 52x^{5} + 37x^{4} + 765x^{3} - 296x^{2} - 2962x + 1692$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 15$$ v^2 - 15 $$\beta_{3}$$ $$=$$ $$( -\nu^{6} - 9\nu^{5} + 56\nu^{4} + 335\nu^{3} - 893\nu^{2} - 2336\nu + 3384 ) / 94$$ (-v^6 - 9*v^5 + 56*v^4 + 335*v^3 - 893*v^2 - 2336*v + 3384) / 94 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} - 9\nu^{5} + 56\nu^{4} + 429\nu^{3} - 893\nu^{2} - 4498\nu + 3008 ) / 94$$ (-v^6 - 9*v^5 + 56*v^4 + 429*v^3 - 893*v^2 - 4498*v + 3008) / 94 $$\beta_{5}$$ $$=$$ $$( -4\nu^{6} + 11\nu^{5} + 177\nu^{4} - 399\nu^{3} - 1786\nu^{2} + 2312\nu + 1551 ) / 47$$ (-4*v^6 + 11*v^5 + 177*v^4 - 399*v^3 - 1786*v^2 + 2312*v + 1551) / 47 $$\beta_{6}$$ $$=$$ $$( 15\nu^{6} - 53\nu^{5} - 558\nu^{4} + 2025\nu^{3} + 3243\nu^{2} - 14028\nu + 9588 ) / 94$$ (15*v^6 - 53*v^5 - 558*v^4 + 2025*v^3 + 3243*v^2 - 14028*v + 9588) / 94
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 15$$ b2 + 15 $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + 23\beta _1 + 4$$ b4 - b3 + 23*b1 + 4 $$\nu^{4}$$ $$=$$ $$\beta_{6} + 2\beta_{5} - \beta_{4} + 32\beta_{2} + 3\beta _1 + 344$$ b6 + 2*b5 - b4 + 32*b2 + 3*b1 + 344 $$\nu^{5}$$ $$=$$ $$\beta_{6} + 3\beta_{5} + 36\beta_{4} - 45\beta_{3} - 6\beta_{2} + 606\beta _1 + 177$$ b6 + 3*b5 + 36*b4 - 45*b3 - 6*b2 + 606*b1 + 177 $$\nu^{6}$$ $$=$$ $$47\beta_{6} + 85\beta_{5} - 45\beta_{4} - 24\beta_{3} + 953\beta_{2} + 83\beta _1 + 9000$$ 47*b6 + 85*b5 - 45*b4 - 24*b3 + 953*b2 + 83*b1 + 9000

## 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
 5.46553 4.56895 2.12247 0.589700 −2.97676 −3.36428 −5.40561
−5.46553 −3.00000 21.8720 5.00000 16.3966 −24.1442 −75.8177 9.00000 −27.3276
1.2 −4.56895 −3.00000 12.8753 5.00000 13.7069 33.6877 −22.2751 9.00000 −22.8448
1.3 −2.12247 −3.00000 −3.49513 5.00000 6.36740 −18.5956 24.3980 9.00000 −10.6123
1.4 −0.589700 −3.00000 −7.65225 5.00000 1.76910 20.2476 9.23013 9.00000 −2.94850
1.5 2.97676 −3.00000 0.861071 5.00000 −8.93027 −1.47808 −21.2508 9.00000 14.8838
1.6 3.36428 −3.00000 3.31835 5.00000 −10.0928 −16.1008 −15.7504 9.00000 16.8214
1.7 5.40561 −3.00000 21.2207 5.00000 −16.2168 22.3833 71.4658 9.00000 27.0281
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

## 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 1815.4.a.bc 7
11.b odd 2 1 1815.4.a.bd yes 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.4.a.bc 7 1.a even 1 1 trivial
1815.4.a.bd yes 7 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{7} + T_{2}^{6} - 52T_{2}^{5} - 37T_{2}^{4} + 765T_{2}^{3} + 296T_{2}^{2} - 2962T_{2} - 1692$$ T2^7 + T2^6 - 52*T2^5 - 37*T2^4 + 765*T2^3 + 296*T2^2 - 2962*T2 - 1692 $$T_{7}^{7} - 16T_{7}^{6} - 1490T_{7}^{5} + 14184T_{7}^{4} + 722529T_{7}^{3} - 2670992T_{7}^{2} - 115840784T_{7} - 163131144$$ T7^7 - 16*T7^6 - 1490*T7^5 + 14184*T7^4 + 722529*T7^3 - 2670992*T7^2 - 115840784*T7 - 163131144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} + T^{6} + \cdots - 1692$$
$3$ $$(T + 3)^{7}$$
$5$ $$(T - 5)^{7}$$
$7$ $$T^{7} - 16 T^{6} + \cdots - 163131144$$
$11$ $$T^{7}$$
$13$ $$T^{7} + \cdots + 609973453952$$
$17$ $$T^{7} + \cdots + 78973270549092$$
$19$ $$T^{7} + \cdots + 26372327985216$$
$23$ $$T^{7} + \cdots + 100765569878688$$
$29$ $$T^{7} + \cdots - 90\!\cdots\!12$$
$31$ $$T^{7} + \cdots + 130509871517520$$
$37$ $$T^{7} + \cdots + 478375739919752$$
$41$ $$T^{7} + \cdots - 26\!\cdots\!08$$
$43$ $$T^{7} + \cdots - 19\!\cdots\!28$$
$47$ $$T^{7} + \cdots + 22\!\cdots\!72$$
$53$ $$T^{7} + \cdots + 17\!\cdots\!24$$
$59$ $$T^{7} + \cdots + 950579470970880$$
$61$ $$T^{7} + \cdots + 29\!\cdots\!52$$
$67$ $$T^{7} + \cdots - 39\!\cdots\!68$$
$71$ $$T^{7} + \cdots + 18\!\cdots\!20$$
$73$ $$T^{7} + \cdots + 55\!\cdots\!44$$
$79$ $$T^{7} + \cdots - 38\!\cdots\!56$$
$83$ $$T^{7} + \cdots - 20\!\cdots\!16$$
$89$ $$T^{7} + \cdots + 30\!\cdots\!12$$
$97$ $$T^{7} + \cdots - 45\!\cdots\!60$$