# Properties

 Label 1911.4.a.h Level $1911$ Weight $4$ Character orbit 1911.a Self dual yes Analytic conductor $112.753$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1911,4,Mod(1,1911)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1911, 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("1911.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1911.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: $$112.752650021$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 14$$ x^2 - 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) 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 $$\beta = \sqrt{14}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + 3 q^{3} + (2 \beta + 7) q^{4} + (2 \beta - 12) q^{5} + (3 \beta + 3) q^{6} + (\beta + 27) q^{8} + 9 q^{9}+O(q^{10})$$ q + (b + 1) * q^2 + 3 * q^3 + (2*b + 7) * q^4 + (2*b - 12) * q^5 + (3*b + 3) * q^6 + (b + 27) * q^8 + 9 * q^9 $$q + (\beta + 1) q^{2} + 3 q^{3} + (2 \beta + 7) q^{4} + (2 \beta - 12) q^{5} + (3 \beta + 3) q^{6} + (\beta + 27) q^{8} + 9 q^{9} + ( - 10 \beta + 16) q^{10} + ( - 12 \beta - 22) q^{11} + (6 \beta + 21) q^{12} + 13 q^{13} + (6 \beta - 36) q^{15} + (12 \beta - 15) q^{16} + ( - 4 \beta - 82) q^{17} + (9 \beta + 9) q^{18} + ( - 2 \beta - 24) q^{19} + ( - 10 \beta - 28) q^{20} + ( - 34 \beta - 190) q^{22} + (48 \beta + 4) q^{23} + (3 \beta + 81) q^{24} + ( - 48 \beta + 75) q^{25} + (13 \beta + 13) q^{26} + 27 q^{27} + ( - 24 \beta + 202) q^{29} + ( - 30 \beta + 48) q^{30} + (26 \beta - 20) q^{31} + ( - 11 \beta - 63) q^{32} + ( - 36 \beta - 66) q^{33} + ( - 86 \beta - 138) q^{34} + (18 \beta + 63) q^{36} + (28 \beta - 50) q^{37} + ( - 26 \beta - 52) q^{38} + 39 q^{39} + (42 \beta - 296) q^{40} + ( - 94 \beta - 100) q^{41} + (52 \beta - 308) q^{43} + ( - 128 \beta - 490) q^{44} + (18 \beta - 108) q^{45} + (52 \beta + 676) q^{46} + ( - 32 \beta + 162) q^{47} + (36 \beta - 45) q^{48} + (27 \beta - 597) q^{50} + ( - 12 \beta - 246) q^{51} + (26 \beta + 91) q^{52} + ( - 120 \beta - 82) q^{53} + (27 \beta + 27) q^{54} + (100 \beta - 72) q^{55} + ( - 6 \beta - 72) q^{57} + (178 \beta - 134) q^{58} + ( - 40 \beta - 70) q^{59} + ( - 30 \beta - 84) q^{60} + ( - 136 \beta - 314) q^{61} + (6 \beta + 344) q^{62} + ( - 170 \beta - 97) q^{64} + (26 \beta - 156) q^{65} + ( - 102 \beta - 570) q^{66} + ( - 170 \beta - 236) q^{67} + ( - 192 \beta - 686) q^{68} + (144 \beta + 12) q^{69} + ( - 84 \beta + 214) q^{71} + (9 \beta + 243) q^{72} + ( - 76 \beta + 450) q^{73} + ( - 22 \beta + 342) q^{74} + ( - 144 \beta + 225) q^{75} + ( - 62 \beta - 224) q^{76} + (39 \beta + 39) q^{78} + ( - 88 \beta - 216) q^{79} + ( - 174 \beta + 516) q^{80} + 81 q^{81} + ( - 194 \beta - 1416) q^{82} + ( - 64 \beta + 694) q^{83} + ( - 116 \beta + 872) q^{85} + ( - 256 \beta + 420) q^{86} + ( - 72 \beta + 606) q^{87} + ( - 346 \beta - 762) q^{88} + (190 \beta - 480) q^{89} + ( - 90 \beta + 144) q^{90} + (344 \beta + 1372) q^{92} + (78 \beta - 60) q^{93} + (130 \beta - 286) q^{94} + ( - 24 \beta + 232) q^{95} + ( - 33 \beta - 189) q^{96} + (220 \beta + 266) q^{97} + ( - 108 \beta - 198) q^{99}+O(q^{100})$$ q + (b + 1) * q^2 + 3 * q^3 + (2*b + 7) * q^4 + (2*b - 12) * q^5 + (3*b + 3) * q^6 + (b + 27) * q^8 + 9 * q^9 + (-10*b + 16) * q^10 + (-12*b - 22) * q^11 + (6*b + 21) * q^12 + 13 * q^13 + (6*b - 36) * q^15 + (12*b - 15) * q^16 + (-4*b - 82) * q^17 + (9*b + 9) * q^18 + (-2*b - 24) * q^19 + (-10*b - 28) * q^20 + (-34*b - 190) * q^22 + (48*b + 4) * q^23 + (3*b + 81) * q^24 + (-48*b + 75) * q^25 + (13*b + 13) * q^26 + 27 * q^27 + (-24*b + 202) * q^29 + (-30*b + 48) * q^30 + (26*b - 20) * q^31 + (-11*b - 63) * q^32 + (-36*b - 66) * q^33 + (-86*b - 138) * q^34 + (18*b + 63) * q^36 + (28*b - 50) * q^37 + (-26*b - 52) * q^38 + 39 * q^39 + (42*b - 296) * q^40 + (-94*b - 100) * q^41 + (52*b - 308) * q^43 + (-128*b - 490) * q^44 + (18*b - 108) * q^45 + (52*b + 676) * q^46 + (-32*b + 162) * q^47 + (36*b - 45) * q^48 + (27*b - 597) * q^50 + (-12*b - 246) * q^51 + (26*b + 91) * q^52 + (-120*b - 82) * q^53 + (27*b + 27) * q^54 + (100*b - 72) * q^55 + (-6*b - 72) * q^57 + (178*b - 134) * q^58 + (-40*b - 70) * q^59 + (-30*b - 84) * q^60 + (-136*b - 314) * q^61 + (6*b + 344) * q^62 + (-170*b - 97) * q^64 + (26*b - 156) * q^65 + (-102*b - 570) * q^66 + (-170*b - 236) * q^67 + (-192*b - 686) * q^68 + (144*b + 12) * q^69 + (-84*b + 214) * q^71 + (9*b + 243) * q^72 + (-76*b + 450) * q^73 + (-22*b + 342) * q^74 + (-144*b + 225) * q^75 + (-62*b - 224) * q^76 + (39*b + 39) * q^78 + (-88*b - 216) * q^79 + (-174*b + 516) * q^80 + 81 * q^81 + (-194*b - 1416) * q^82 + (-64*b + 694) * q^83 + (-116*b + 872) * q^85 + (-256*b + 420) * q^86 + (-72*b + 606) * q^87 + (-346*b - 762) * q^88 + (190*b - 480) * q^89 + (-90*b + 144) * q^90 + (344*b + 1372) * q^92 + (78*b - 60) * q^93 + (130*b - 286) * q^94 + (-24*b + 232) * q^95 + (-33*b - 189) * q^96 + (220*b + 266) * q^97 + (-108*b - 198) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 6 q^{3} + 14 q^{4} - 24 q^{5} + 6 q^{6} + 54 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 6 * q^3 + 14 * q^4 - 24 * q^5 + 6 * q^6 + 54 * q^8 + 18 * q^9 $$2 q + 2 q^{2} + 6 q^{3} + 14 q^{4} - 24 q^{5} + 6 q^{6} + 54 q^{8} + 18 q^{9} + 32 q^{10} - 44 q^{11} + 42 q^{12} + 26 q^{13} - 72 q^{15} - 30 q^{16} - 164 q^{17} + 18 q^{18} - 48 q^{19} - 56 q^{20} - 380 q^{22} + 8 q^{23} + 162 q^{24} + 150 q^{25} + 26 q^{26} + 54 q^{27} + 404 q^{29} + 96 q^{30} - 40 q^{31} - 126 q^{32} - 132 q^{33} - 276 q^{34} + 126 q^{36} - 100 q^{37} - 104 q^{38} + 78 q^{39} - 592 q^{40} - 200 q^{41} - 616 q^{43} - 980 q^{44} - 216 q^{45} + 1352 q^{46} + 324 q^{47} - 90 q^{48} - 1194 q^{50} - 492 q^{51} + 182 q^{52} - 164 q^{53} + 54 q^{54} - 144 q^{55} - 144 q^{57} - 268 q^{58} - 140 q^{59} - 168 q^{60} - 628 q^{61} + 688 q^{62} - 194 q^{64} - 312 q^{65} - 1140 q^{66} - 472 q^{67} - 1372 q^{68} + 24 q^{69} + 428 q^{71} + 486 q^{72} + 900 q^{73} + 684 q^{74} + 450 q^{75} - 448 q^{76} + 78 q^{78} - 432 q^{79} + 1032 q^{80} + 162 q^{81} - 2832 q^{82} + 1388 q^{83} + 1744 q^{85} + 840 q^{86} + 1212 q^{87} - 1524 q^{88} - 960 q^{89} + 288 q^{90} + 2744 q^{92} - 120 q^{93} - 572 q^{94} + 464 q^{95} - 378 q^{96} + 532 q^{97} - 396 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 6 * q^3 + 14 * q^4 - 24 * q^5 + 6 * q^6 + 54 * q^8 + 18 * q^9 + 32 * q^10 - 44 * q^11 + 42 * q^12 + 26 * q^13 - 72 * q^15 - 30 * q^16 - 164 * q^17 + 18 * q^18 - 48 * q^19 - 56 * q^20 - 380 * q^22 + 8 * q^23 + 162 * q^24 + 150 * q^25 + 26 * q^26 + 54 * q^27 + 404 * q^29 + 96 * q^30 - 40 * q^31 - 126 * q^32 - 132 * q^33 - 276 * q^34 + 126 * q^36 - 100 * q^37 - 104 * q^38 + 78 * q^39 - 592 * q^40 - 200 * q^41 - 616 * q^43 - 980 * q^44 - 216 * q^45 + 1352 * q^46 + 324 * q^47 - 90 * q^48 - 1194 * q^50 - 492 * q^51 + 182 * q^52 - 164 * q^53 + 54 * q^54 - 144 * q^55 - 144 * q^57 - 268 * q^58 - 140 * q^59 - 168 * q^60 - 628 * q^61 + 688 * q^62 - 194 * q^64 - 312 * q^65 - 1140 * q^66 - 472 * q^67 - 1372 * q^68 + 24 * q^69 + 428 * q^71 + 486 * q^72 + 900 * q^73 + 684 * q^74 + 450 * q^75 - 448 * q^76 + 78 * q^78 - 432 * q^79 + 1032 * q^80 + 162 * q^81 - 2832 * q^82 + 1388 * q^83 + 1744 * q^85 + 840 * q^86 + 1212 * q^87 - 1524 * q^88 - 960 * q^89 + 288 * q^90 + 2744 * q^92 - 120 * q^93 - 572 * q^94 + 464 * q^95 - 378 * q^96 + 532 * q^97 - 396 * q^99

## 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
 −3.74166 3.74166
−2.74166 3.00000 −0.483315 −19.4833 −8.22497 0 23.2583 9.00000 53.4166
1.2 4.74166 3.00000 14.4833 −4.51669 14.2250 0 30.7417 9.00000 −21.4166
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 1911.4.a.h 2
7.b odd 2 1 39.4.a.b 2
21.c even 2 1 117.4.a.c 2
28.d even 2 1 624.4.a.r 2
35.c odd 2 1 975.4.a.j 2
56.e even 2 1 2496.4.a.s 2
56.h odd 2 1 2496.4.a.bc 2
84.h odd 2 1 1872.4.a.t 2
91.b odd 2 1 507.4.a.f 2
91.i even 4 2 507.4.b.f 4
273.g even 2 1 1521.4.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.b 2 7.b odd 2 1
117.4.a.c 2 21.c even 2 1
507.4.a.f 2 91.b odd 2 1
507.4.b.f 4 91.i even 4 2
624.4.a.r 2 28.d even 2 1
975.4.a.j 2 35.c odd 2 1
1521.4.a.s 2 273.g even 2 1
1872.4.a.t 2 84.h odd 2 1
1911.4.a.h 2 1.a even 1 1 trivial
2496.4.a.s 2 56.e even 2 1
2496.4.a.bc 2 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1911))$$:

 $$T_{2}^{2} - 2T_{2} - 13$$ T2^2 - 2*T2 - 13 $$T_{5}^{2} + 24T_{5} + 88$$ T5^2 + 24*T5 + 88

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 13$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} + 24T + 88$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 44T - 1532$$
$13$ $$(T - 13)^{2}$$
$17$ $$T^{2} + 164T + 6500$$
$19$ $$T^{2} + 48T + 520$$
$23$ $$T^{2} - 8T - 32240$$
$29$ $$T^{2} - 404T + 32740$$
$31$ $$T^{2} + 40T - 9064$$
$37$ $$T^{2} + 100T - 8476$$
$41$ $$T^{2} + 200T - 113704$$
$43$ $$T^{2} + 616T + 57008$$
$47$ $$T^{2} - 324T + 11908$$
$53$ $$T^{2} + 164T - 194876$$
$59$ $$T^{2} + 140T - 17500$$
$61$ $$T^{2} + 628T - 160348$$
$67$ $$T^{2} + 472T - 348904$$
$71$ $$T^{2} - 428T - 52988$$
$73$ $$T^{2} - 900T + 121636$$
$79$ $$T^{2} + 432T - 61760$$
$83$ $$T^{2} - 1388 T + 424292$$
$89$ $$T^{2} + 960T - 275000$$
$97$ $$T^{2} - 532T - 606844$$