Properties

 Label 363.4.a.r Level $363$ Weight $4$ Character orbit 363.a Self dual yes Analytic conductor $21.418$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,4,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("363.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 363.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: $$21.4176933321$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 7x^{2} + 8x + 1$$ x^4 - 2*x^3 - 7*x^2 + 8*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_1) q^{2} - 3 q^{3} + 2 \beta_{2} q^{4} + ( - 4 \beta_{2} + 4) q^{5} + ( - 3 \beta_{3} + 3 \beta_1) q^{6} + ( - 4 \beta_{3} - 13 \beta_1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{8} + 9 q^{9}+O(q^{10})$$ q + (b3 - b1) * q^2 - 3 * q^3 + 2*b2 * q^4 + (-4*b2 + 4) * q^5 + (-3*b3 + 3*b1) * q^6 + (-4*b3 - 13*b1) * q^7 + (-2*b3 - 2*b1) * q^8 + 9 * q^9 $$q + (\beta_{3} - \beta_1) q^{2} - 3 q^{3} + 2 \beta_{2} q^{4} + ( - 4 \beta_{2} + 4) q^{5} + ( - 3 \beta_{3} + 3 \beta_1) q^{6} + ( - 4 \beta_{3} - 13 \beta_1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{8} + 9 q^{9} + ( - 8 \beta_{3} + 16 \beta_1) q^{10} - 6 \beta_{2} q^{12} + (24 \beta_{3} - 8 \beta_1) q^{13} + (9 \beta_{2} + 19) q^{14} + (12 \beta_{2} - 12) q^{15} + ( - 16 \beta_{2} - 4) q^{16} + (8 \beta_{3} - 40 \beta_1) q^{17} + (9 \beta_{3} - 9 \beta_1) q^{18} + (20 \beta_{3} - 39 \beta_1) q^{19} + (8 \beta_{2} - 120) q^{20} + (12 \beta_{3} + 39 \beta_1) q^{21} + (24 \beta_{2} + 34) q^{23} + (6 \beta_{3} + 6 \beta_1) q^{24} + ( - 32 \beta_{2} + 131) q^{25} + (32 \beta_{2} + 144) q^{26} - 27 q^{27} + (78 \beta_{3} + 40 \beta_1) q^{28} + (104 \beta_{3} + 22 \beta_1) q^{29} + (24 \beta_{3} - 48 \beta_1) q^{30} + ( - 28 \beta_{2} - 51) q^{31} + ( - 36 \beta_{3} + 100 \beta_1) q^{32} + (48 \beta_{2} + 160) q^{34} + ( - 172 \beta_{3} - 132 \beta_1) q^{35} + 18 \beta_{2} q^{36} + ( - 72 \beta_{2} - 43) q^{37} + (59 \beta_{2} + 217) q^{38} + ( - 72 \beta_{3} + 24 \beta_1) q^{39} + ( - 32 \beta_{3} - 48 \beta_1) q^{40} + (52 \beta_{3} - 186 \beta_1) q^{41} + ( - 27 \beta_{2} - 57) q^{42} + (40 \beta_{3} + 54 \beta_1) q^{43} + ( - 36 \beta_{2} + 36) q^{45} + (106 \beta_{3} - 154 \beta_1) q^{46} + ( - 28 \beta_{2} + 166) q^{47} + (48 \beta_{2} + 12) q^{48} + ( - 104 \beta_{2} + 244) q^{49} + (35 \beta_{3} + 29 \beta_1) q^{50} + ( - 24 \beta_{3} + 120 \beta_1) q^{51} + (48 \beta_{3} - 240 \beta_1) q^{52} + (24 \beta_{2} + 630) q^{53} + ( - 27 \beta_{3} + 27 \beta_1) q^{54} + ( - 34 \beta_{2} + 118) q^{56} + ( - 60 \beta_{3} + 117 \beta_1) q^{57} + (82 \beta_{2} + 454) q^{58} + ( - 24 \beta_{2} + 90) q^{59} + ( - 24 \beta_{2} + 360) q^{60} + ( - 208 \beta_{3} + 85 \beta_1) q^{61} + ( - 135 \beta_{3} + 191 \beta_1) q^{62} + ( - 36 \beta_{3} - 117 \beta_1) q^{63} + ( - 8 \beta_{2} - 448) q^{64} + 448 \beta_1 q^{65} + ( - 60 \beta_{2} + 657) q^{67} + (240 \beta_{3} - 80 \beta_1) q^{68} + ( - 72 \beta_{2} - 102) q^{69} + ( - 40 \beta_{2} - 464) q^{70} + (20 \beta_{2} + 128) q^{71} + ( - 18 \beta_{3} - 18 \beta_1) q^{72} + (16 \beta_{3} + 365 \beta_1) q^{73} + ( - 259 \beta_{3} + 403 \beta_1) q^{74} + (96 \beta_{2} - 393) q^{75} + (234 \beta_{3} - 200 \beta_1) q^{76} + ( - 96 \beta_{2} - 432) q^{78} + (276 \beta_{3} - 113 \beta_1) q^{79} + ( - 48 \beta_{2} + 944) q^{80} + 81 q^{81} + (238 \beta_{2} + 818) q^{82} + (68 \beta_{3} + 126 \beta_1) q^{83} + ( - 234 \beta_{3} - 120 \beta_1) q^{84} - 448 \beta_{3} q^{85} + ( - 14 \beta_{2} + 38) q^{86} + ( - 312 \beta_{3} - 66 \beta_1) q^{87} + ( - 76 \beta_{2} - 136) q^{89} + ( - 72 \beta_{3} + 144 \beta_1) q^{90} + (280 \beta_{2} - 168) q^{91} + (68 \beta_{2} + 720) q^{92} + (84 \beta_{2} + 153) q^{93} + (82 \beta_{3} - 26 \beta_1) q^{94} + ( - 388 \beta_{3} + 244 \beta_1) q^{95} + (108 \beta_{3} - 300 \beta_1) q^{96} + (208 \beta_{2} + 5) q^{97} + ( - 68 \beta_{3} + 276 \beta_1) q^{98}+O(q^{100})$$ q + (b3 - b1) * q^2 - 3 * q^3 + 2*b2 * q^4 + (-4*b2 + 4) * q^5 + (-3*b3 + 3*b1) * q^6 + (-4*b3 - 13*b1) * q^7 + (-2*b3 - 2*b1) * q^8 + 9 * q^9 + (-8*b3 + 16*b1) * q^10 - 6*b2 * q^12 + (24*b3 - 8*b1) * q^13 + (9*b2 + 19) * q^14 + (12*b2 - 12) * q^15 + (-16*b2 - 4) * q^16 + (8*b3 - 40*b1) * q^17 + (9*b3 - 9*b1) * q^18 + (20*b3 - 39*b1) * q^19 + (8*b2 - 120) * q^20 + (12*b3 + 39*b1) * q^21 + (24*b2 + 34) * q^23 + (6*b3 + 6*b1) * q^24 + (-32*b2 + 131) * q^25 + (32*b2 + 144) * q^26 - 27 * q^27 + (78*b3 + 40*b1) * q^28 + (104*b3 + 22*b1) * q^29 + (24*b3 - 48*b1) * q^30 + (-28*b2 - 51) * q^31 + (-36*b3 + 100*b1) * q^32 + (48*b2 + 160) * q^34 + (-172*b3 - 132*b1) * q^35 + 18*b2 * q^36 + (-72*b2 - 43) * q^37 + (59*b2 + 217) * q^38 + (-72*b3 + 24*b1) * q^39 + (-32*b3 - 48*b1) * q^40 + (52*b3 - 186*b1) * q^41 + (-27*b2 - 57) * q^42 + (40*b3 + 54*b1) * q^43 + (-36*b2 + 36) * q^45 + (106*b3 - 154*b1) * q^46 + (-28*b2 + 166) * q^47 + (48*b2 + 12) * q^48 + (-104*b2 + 244) * q^49 + (35*b3 + 29*b1) * q^50 + (-24*b3 + 120*b1) * q^51 + (48*b3 - 240*b1) * q^52 + (24*b2 + 630) * q^53 + (-27*b3 + 27*b1) * q^54 + (-34*b2 + 118) * q^56 + (-60*b3 + 117*b1) * q^57 + (82*b2 + 454) * q^58 + (-24*b2 + 90) * q^59 + (-24*b2 + 360) * q^60 + (-208*b3 + 85*b1) * q^61 + (-135*b3 + 191*b1) * q^62 + (-36*b3 - 117*b1) * q^63 + (-8*b2 - 448) * q^64 + 448*b1 * q^65 + (-60*b2 + 657) * q^67 + (240*b3 - 80*b1) * q^68 + (-72*b2 - 102) * q^69 + (-40*b2 - 464) * q^70 + (20*b2 + 128) * q^71 + (-18*b3 - 18*b1) * q^72 + (16*b3 + 365*b1) * q^73 + (-259*b3 + 403*b1) * q^74 + (96*b2 - 393) * q^75 + (234*b3 - 200*b1) * q^76 + (-96*b2 - 432) * q^78 + (276*b3 - 113*b1) * q^79 + (-48*b2 + 944) * q^80 + 81 * q^81 + (238*b2 + 818) * q^82 + (68*b3 + 126*b1) * q^83 + (-234*b3 - 120*b1) * q^84 - 448*b3 * q^85 + (-14*b2 + 38) * q^86 + (-312*b3 - 66*b1) * q^87 + (-76*b2 - 136) * q^89 + (-72*b3 + 144*b1) * q^90 + (280*b2 - 168) * q^91 + (68*b2 + 720) * q^92 + (84*b2 + 153) * q^93 + (82*b3 - 26*b1) * q^94 + (-388*b3 + 244*b1) * q^95 + (108*b3 - 300*b1) * q^96 + (208*b2 + 5) * q^97 + (-68*b3 + 276*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} + 16 q^{5} + 36 q^{9}+O(q^{10})$$ 4 * q - 12 * q^3 + 16 * q^5 + 36 * q^9 $$4 q - 12 q^{3} + 16 q^{5} + 36 q^{9} + 76 q^{14} - 48 q^{15} - 16 q^{16} - 480 q^{20} + 136 q^{23} + 524 q^{25} + 576 q^{26} - 108 q^{27} - 204 q^{31} + 640 q^{34} - 172 q^{37} + 868 q^{38} - 228 q^{42} + 144 q^{45} + 664 q^{47} + 48 q^{48} + 976 q^{49} + 2520 q^{53} + 472 q^{56} + 1816 q^{58} + 360 q^{59} + 1440 q^{60} - 1792 q^{64} + 2628 q^{67} - 408 q^{69} - 1856 q^{70} + 512 q^{71} - 1572 q^{75} - 1728 q^{78} + 3776 q^{80} + 324 q^{81} + 3272 q^{82} + 152 q^{86} - 544 q^{89} - 672 q^{91} + 2880 q^{92} + 612 q^{93} + 20 q^{97}+O(q^{100})$$ 4 * q - 12 * q^3 + 16 * q^5 + 36 * q^9 + 76 * q^14 - 48 * q^15 - 16 * q^16 - 480 * q^20 + 136 * q^23 + 524 * q^25 + 576 * q^26 - 108 * q^27 - 204 * q^31 + 640 * q^34 - 172 * q^37 + 868 * q^38 - 228 * q^42 + 144 * q^45 + 664 * q^47 + 48 * q^48 + 976 * q^49 + 2520 * q^53 + 472 * q^56 + 1816 * q^58 + 360 * q^59 + 1440 * q^60 - 1792 * q^64 + 2628 * q^67 - 408 * q^69 - 1856 * q^70 + 512 * q^71 - 1572 * q^75 - 1728 * q^78 + 3776 * q^80 + 324 * q^81 + 3272 * q^82 + 152 * q^86 - 544 * q^89 - 672 * q^91 + 2880 * q^92 + 612 * q^93 + 20 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 7x^{2} + 8x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{3} - 3\nu^{2} - 19\nu + 10 ) / 7$$ (2*v^3 - 3*v^2 - 19*v + 10) / 7 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4 $$\beta_{3}$$ $$=$$ $$( 4\nu^{3} - 6\nu^{2} - 24\nu + 13 ) / 7$$ (4*v^3 - 6*v^2 - 24*v + 13) / 7
 $$\nu$$ $$=$$ $$( \beta_{3} - 2\beta _1 + 1 ) / 2$$ (b3 - 2*b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 2\beta_{2} - 2\beta _1 + 9 ) / 2$$ (b3 + 2*b2 - 2*b1 + 9) / 2 $$\nu^{3}$$ $$=$$ $$( 11\beta_{3} + 3\beta_{2} - 15\beta _1 + 13 ) / 2$$ (11*b3 + 3*b2 - 15*b1 + 13) / 2

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
 −2.35008 1.11402 −0.114017 3.35008
−3.96812 −3.00000 7.74597 −11.4919 11.9044 −13.5724 1.00803 9.00000 45.6014
1.2 −0.504017 −3.00000 −7.74597 19.4919 1.51205 31.4609 7.93624 9.00000 −9.82427
1.3 0.504017 −3.00000 −7.74597 19.4919 −1.51205 −31.4609 −7.93624 9.00000 9.82427
1.4 3.96812 −3.00000 7.74597 −11.4919 −11.9044 13.5724 −1.00803 9.00000 −45.6014
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$11$$ $$1$$

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.4.a.r 4
3.b odd 2 1 1089.4.a.bb 4
11.b odd 2 1 inner 363.4.a.r 4
33.d even 2 1 1089.4.a.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.a.r 4 1.a even 1 1 trivial
363.4.a.r 4 11.b odd 2 1 inner
1089.4.a.bb 4 3.b odd 2 1
1089.4.a.bb 4 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(363))$$:

 $$T_{2}^{4} - 16T_{2}^{2} + 4$$ T2^4 - 16*T2^2 + 4 $$T_{5}^{2} - 8T_{5} - 224$$ T5^2 - 8*T5 - 224 $$T_{7}^{4} - 1174T_{7}^{2} + 182329$$ T7^4 - 1174*T7^2 + 182329

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 16T^{2} + 4$$
$3$ $$(T + 3)^{4}$$
$5$ $$(T^{2} - 8 T - 224)^{2}$$
$7$ $$T^{4} - 1174 T^{2} + 182329$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 6144 T^{2} + \cdots + 7225344$$
$17$ $$T^{4} - 10240 T^{2} + \cdots + 20070400$$
$19$ $$T^{4} - 13126 T^{2} + \cdots + 6568969$$
$23$ $$(T^{2} - 68 T - 7484)^{2}$$
$29$ $$T^{4} - 111064 T^{2} + \cdots + 2769706384$$
$31$ $$(T^{2} + 102 T - 9159)^{2}$$
$37$ $$(T^{2} + 86 T - 75911)^{2}$$
$41$ $$T^{4} - 234616 T^{2} + \cdots + 8148311824$$
$43$ $$T^{4} - 33496 T^{2} + \cdots + 559504$$
$47$ $$(T^{2} - 332 T + 15796)^{2}$$
$53$ $$(T^{2} - 1260 T + 388260)^{2}$$
$59$ $$(T^{2} - 180 T - 540)^{2}$$
$61$ $$T^{4} - 475990 T^{2} + \cdots + 37886676025$$
$67$ $$(T^{2} - 1314 T + 377649)^{2}$$
$71$ $$(T^{2} - 256 T + 10384)^{2}$$
$73$ $$T^{4} - 801910 T^{2} + \cdots + 158718576025$$
$79$ $$T^{4} - 838374 T^{2} + \cdots + 117356260329$$
$83$ $$T^{4} - 141496 T^{2} + \cdots + 600642064$$
$89$ $$(T^{2} + 272 T - 68144)^{2}$$
$97$ $$(T^{2} - 10 T - 648935)^{2}$$
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