# Properties

 Label 23.4.a.b Level $23$ Weight $4$ Character orbit 23.a Self dual yes Analytic conductor $1.357$ Analytic rank $0$ 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] = [23,4,Mod(1,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("23.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 23.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: $$1.35704393013$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.334189.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 16x^{2} - 5x + 4$$ x^4 - 2*x^3 - 16*x^2 - 5*x + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 1) q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 6) q^{4}+ \cdots + (2 \beta_{3} + \beta_{2} + 5 \beta_1 - 10) q^{9}+O(q^{10})$$ q + (b3 + 1) * q^2 + (b2 + b1 + 1) * q^3 + (-2*b3 - 2*b2 - 3*b1 + 6) * q^4 + (-2*b3 - 2*b2 + 2*b1 + 2) * q^5 + (b3 + 3*b2 + b1 - 5) * q^6 + (-2*b3 + 4*b2 - 4*b1 + 4) * q^7 + (5*b3 - b2 + 4*b1 - 15) * q^8 + (2*b3 + b2 + 5*b1 - 10) * q^9 $$q + (\beta_{3} + 1) q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 6) q^{4}+ \cdots + ( - 48 \beta_{3} - 46 \beta_{2} + \cdots - 340) q^{99}+O(q^{100})$$ q + (b3 + 1) * q^2 + (b2 + b1 + 1) * q^3 + (-2*b3 - 2*b2 - 3*b1 + 6) * q^4 + (-2*b3 - 2*b2 + 2*b1 + 2) * q^5 + (b3 + 3*b2 + b1 - 5) * q^6 + (-2*b3 + 4*b2 - 4*b1 + 4) * q^7 + (5*b3 - b2 + 4*b1 - 15) * q^8 + (2*b3 + b2 + 5*b1 - 10) * q^9 + (4*b3 - 6*b2 + 4*b1 - 16) * q^10 + (2*b3 - 4*b2 - 12*b1 + 10) * q^11 + (-6*b3 + b2 - 8*b1 - 16) * q^12 + (-10*b3 - 11*b2 - 11*b1 + 31) * q^13 + (18*b3 + 24*b2 + 10*b1 - 38) * q^14 + (-6*b3 + 2*b2 + 18*b1 - 10) * q^15 + (-19*b3 - 2*b2 + 8*b1 + 3) * q^16 + (10*b3 - 6*b2 - 2*b1 + 32) * q^17 + (-20*b3 - 5*b2 - 5*b1 + 6) * q^18 + (24*b3 + 24*b2 + 32*b1 + 14) * q^19 + (-22*b3 - 20*b2 - 34*b1 + 46) * q^20 + (6*b3 - 8*b2 - 28*b1 + 64) * q^21 + (12*b3 - 8*b2 - 10*b1 + 68) * q^22 - 23 * q^23 + (3*b3 + 11*b1 - 51) * q^24 + (16*b3 + 28*b2 + 8*b1 + 43) * q^25 + (61*b3 - 13*b2 + 19*b1 - 33) * q^26 + (4*b3 - 29*b2 - 9*b1 - 25) * q^27 + (-62*b3 + 18*b2 + 2*b1 + 34) * q^28 + (-26*b3 + 23*b2 + 35*b1 - 31) * q^29 + (-8*b3 + 2*b2 + 20*b1 - 116) * q^30 + (12*b3 + 9*b2 - 19*b1 - 35) * q^31 + (10*b3 + 30*b2 + 23*b1 - 122) * q^32 + (-6*b3 + 6*b2 - 54*b1 - 82) * q^33 + (-2*b3 - 42*b2 - 36*b1 + 194) * q^34 + (-84*b3 - 48*b2 - 12*b1 - 212) * q^35 + (50*b3 + 17*b2 + 15*b1 - 144) * q^36 + (10*b3 - 54*b2 + 30*b1 + 46) * q^37 + (-66*b3 + 16*b2 - 48*b1 + 174) * q^38 + (-32*b3 + 11*b2 - 13*b1 - 85) * q^39 + (94*b3 + 46*b2 + 14*b1 + 22) * q^40 + (-18*b3 - 7*b2 + 21*b1 - 49) * q^41 + (66*b3 - 16*b2 - 26*b1 + 210) * q^42 + (28*b3 + 14*b2 + 70*b1 - 24) * q^43 + (18*b3 - 14*b2 + 52*b1 + 194) * q^44 + (52*b3 + 48*b2 + 40*b1 + 36) * q^45 + (-23*b3 - 23) * q^46 + (36*b3 + 13*b2 - 71*b1 - 119) * q^47 + (-23*b3 - 25*b2 + 55*b1 + 105) * q^48 + (32*b3 + 8*b2 - 92*b1 + 365) * q^49 + (15*b3 + 72*b2 - 20*b1 + 103) * q^50 + (-2*b3 + 56*b2 + 32*b1 - 116) * q^51 + (-168*b3 - 105*b2 - 108*b1 + 558) * q^52 + (-120*b3 - 26*b2 + 14*b1 - 118) * q^53 + (-57*b3 - 115*b2 - 41*b1 + 181) * q^54 + (-40*b3 - 108*b2 - 120*b1 - 176) * q^55 + (92*b3 + 2*b2 + 124*b1 - 560) * q^56 + (72*b3 + 70*b2 + 158*b1 + 270) * q^57 + (35*b3 + 109*b2 + 101*b1 - 519) * q^58 + (28*b3 + 60*b2 + 36*b1 - 304) * q^59 + (-62*b3 - 12*b2 - 118*b1 - 170) * q^60 + (36*b3 - 26*b2 + 14*b1 + 206) * q^61 + (-43*b3 + 31*b2 - 27*b1 + 95) * q^62 + (44*b3 - 52*b2 + 20*b1 - 248) * q^63 + (7*b3 + 93*b2 - 64*b1 - 189) * q^64 + (-78*b3 - 66*b2 - 134*b1 + 374) * q^65 + (-4*b3 + 90*b2 + 24*b1 - 136) * q^66 + (110*b3 + 84*b2 + 92*b1 + 110) * q^67 + (114*b3 - 80*b2 - 20*b1 + 158) * q^68 + (-23*b2 - 23*b1 - 23) * q^69 + (4*b3 - 12*b2 + 204*b1 - 1052) * q^70 + (68*b3 - 111*b2 - 143*b1 + 33) * q^71 + (-132*b3 - 7*b2 - 93*b1 + 358) * q^72 + (-66*b3 - b2 + 127*b1 + 281) * q^73 + (-68*b3 - 266*b2 - 84*b1 + 416) * q^74 + (72*b3 + 55*b2 + 35*b1 + 355) * q^75 + (244*b3 + 52*b2 - 42*b1 - 828) * q^76 + (36*b3 + 208*b2 + 132*b1 + 84) * q^77 + (35*b3 + 121*b2 + 107*b1 - 543) * q^78 + (-132*b3 - 50*b2 - 98*b1 - 214) * q^79 + (-52*b3 + 142*b2 + 36*b1 + 632) * q^80 + (-108*b3 - 24*b2 - 156*b1 - 203) * q^81 + (-23*b3 - 13*b2 + 47*b1 - 269) * q^82 + (-22*b3 + 6*b2 - 146*b1 + 96) * q^83 + (-26*b3 - 106*b2 + 10*b1 + 662) * q^84 + (44*b3 - 96*b2 + 28*b1 + 60) * q^85 + (-164*b3 - 70*b2 - 70*b1 + 200) * q^86 + (20*b3 - 71*b2 + 133*b1 + 517) * q^87 + (-22*b3 - 80*b2 + 12*b1 - 98) * q^88 + (160*b3 + 82*b2 + 82*b1 + 604) * q^89 + (-112*b3 + 48*b2 - 108*b1 + 432) * q^90 + (-350*b3 + 56*b2 - 116) * q^91 + (46*b3 + 46*b2 + 69*b1 - 138) * q^92 + (30*b3 - 39*b2 - 167*b1 - 19) * q^93 + (-143*b3 + 51*b2 - 95*b1 + 355) * q^94 + (68*b3 + 36*b2 + 540*b1 - 484) * q^95 + (70*b3 - 109*b2 - 44*b1 + 284) * q^96 + (282*b3 + 150*b2 - 38*b1 + 388) * q^97 + (369*b3 + 60*b2 - 88*b1 + 833) * q^98 + (-48*b3 - 46*b2 - 94*b1 - 340) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9 $$4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9} - 70 q^{10} + 8 q^{11} - 67 q^{12} + 111 q^{13} - 144 q^{14} + 10 q^{15} + 64 q^{16} + 98 q^{17} + 49 q^{18} + 96 q^{19} + 140 q^{20} + 180 q^{21} + 220 q^{22} - 92 q^{23} - 188 q^{24} + 184 q^{25} - 229 q^{26} - 155 q^{27} + 282 q^{28} + 21 q^{29} - 406 q^{30} - 193 q^{31} - 432 q^{32} - 418 q^{33} + 666 q^{34} - 752 q^{35} - 629 q^{36} + 170 q^{37} + 748 q^{38} - 291 q^{39} - 26 q^{40} - 125 q^{41} + 640 q^{42} + 2 q^{43} + 830 q^{44} + 168 q^{45} - 46 q^{46} - 677 q^{47} + 551 q^{48} + 1220 q^{49} + 414 q^{50} - 340 q^{51} + 2247 q^{52} - 230 q^{53} + 641 q^{54} - 972 q^{55} - 2174 q^{56} + 1322 q^{57} - 1835 q^{58} - 1140 q^{59} - 804 q^{60} + 754 q^{61} + 443 q^{62} - 1092 q^{63} - 805 q^{64} + 1318 q^{65} - 398 q^{66} + 488 q^{67} + 284 q^{68} - 161 q^{69} - 3820 q^{70} - 401 q^{71} + 1503 q^{72} + 1509 q^{73} + 1366 q^{74} + 1401 q^{75} - 3832 q^{76} + 736 q^{77} - 1907 q^{78} - 838 q^{79} + 2846 q^{80} - 932 q^{81} - 949 q^{82} + 142 q^{83} + 2614 q^{84} + 112 q^{85} + 918 q^{86} + 2223 q^{87} - 404 q^{88} + 2342 q^{89} + 1784 q^{90} + 292 q^{91} - 460 q^{92} - 509 q^{93} + 1567 q^{94} - 956 q^{95} + 799 q^{96} + 1062 q^{97} + 2478 q^{98} - 1498 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9 - 70 * q^10 + 8 * q^11 - 67 * q^12 + 111 * q^13 - 144 * q^14 + 10 * q^15 + 64 * q^16 + 98 * q^17 + 49 * q^18 + 96 * q^19 + 140 * q^20 + 180 * q^21 + 220 * q^22 - 92 * q^23 - 188 * q^24 + 184 * q^25 - 229 * q^26 - 155 * q^27 + 282 * q^28 + 21 * q^29 - 406 * q^30 - 193 * q^31 - 432 * q^32 - 418 * q^33 + 666 * q^34 - 752 * q^35 - 629 * q^36 + 170 * q^37 + 748 * q^38 - 291 * q^39 - 26 * q^40 - 125 * q^41 + 640 * q^42 + 2 * q^43 + 830 * q^44 + 168 * q^45 - 46 * q^46 - 677 * q^47 + 551 * q^48 + 1220 * q^49 + 414 * q^50 - 340 * q^51 + 2247 * q^52 - 230 * q^53 + 641 * q^54 - 972 * q^55 - 2174 * q^56 + 1322 * q^57 - 1835 * q^58 - 1140 * q^59 - 804 * q^60 + 754 * q^61 + 443 * q^62 - 1092 * q^63 - 805 * q^64 + 1318 * q^65 - 398 * q^66 + 488 * q^67 + 284 * q^68 - 161 * q^69 - 3820 * q^70 - 401 * q^71 + 1503 * q^72 + 1509 * q^73 + 1366 * q^74 + 1401 * q^75 - 3832 * q^76 + 736 * q^77 - 1907 * q^78 - 838 * q^79 + 2846 * q^80 - 932 * q^81 - 949 * q^82 + 142 * q^83 + 2614 * q^84 + 112 * q^85 + 918 * q^86 + 2223 * q^87 - 404 * q^88 + 2342 * q^89 + 1784 * q^90 + 292 * q^91 - 460 * q^92 - 509 * q^93 + 1567 * q^94 - 956 * q^95 + 799 * q^96 + 1062 * q^97 + 2478 * q^98 - 1498 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 16x^{2} - 5x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 20\nu - 10 ) / 3$$ (v^3 - v^2 - 20*v - 10) / 3 $$\beta_{3}$$ $$=$$ $$( -2\nu^{3} + 5\nu^{2} + 28\nu - 1 ) / 3$$ (-2*v^3 + 5*v^2 + 28*v - 1) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 4\beta _1 + 7$$ b3 + 2*b2 + 4*b1 + 7 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5\beta_{2} + 24\beta _1 + 17$$ b3 + 5*b2 + 24*b1 + 17

## 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
 −0.743529 5.22031 −2.83969 0.362907
−5.07751 1.55870 17.7811 10.0635 −7.91434 24.3381 −49.6639 −24.5704 −51.0976
1.2 −0.0323756 6.42170 −7.99895 14.1026 −0.207906 −14.0109 0.517976 14.2382 −0.456580
1.3 2.86845 3.43737 0.228032 −17.9704 9.85995 32.7301 −22.2935 −15.1845 −51.5473
1.4 4.24143 −4.41777 9.98977 7.80430 −18.7377 −27.0572 8.43948 −7.48328 33.1014
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$23$$ $$+1$$

## 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 23.4.a.b 4
3.b odd 2 1 207.4.a.e 4
4.b odd 2 1 368.4.a.l 4
5.b even 2 1 575.4.a.i 4
5.c odd 4 2 575.4.b.g 8
7.b odd 2 1 1127.4.a.c 4
8.b even 2 1 1472.4.a.y 4
8.d odd 2 1 1472.4.a.bf 4
23.b odd 2 1 529.4.a.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.4.a.b 4 1.a even 1 1 trivial
207.4.a.e 4 3.b odd 2 1
368.4.a.l 4 4.b odd 2 1
529.4.a.g 4 23.b odd 2 1
575.4.a.i 4 5.b even 2 1
575.4.b.g 8 5.c odd 4 2
1127.4.a.c 4 7.b odd 2 1
1472.4.a.y 4 8.b even 2 1
1472.4.a.bf 4 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2T_{2}^{3} - 24T_{2}^{2} + 61T_{2} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(23))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + \cdots + 2$$
$3$ $$T^{4} - 7 T^{3} + \cdots - 152$$
$5$ $$T^{4} - 14 T^{3} + \cdots - 19904$$
$7$ $$T^{4} - 16 T^{3} + \cdots + 301984$$
$11$ $$T^{4} - 8 T^{3} + \cdots - 81440$$
$13$ $$T^{4} - 111 T^{3} + \cdots + 1322658$$
$17$ $$T^{4} - 98 T^{3} + \cdots - 855280$$
$19$ $$T^{4} - 96 T^{3} + \cdots + 66996944$$
$23$ $$(T + 23)^{4}$$
$29$ $$T^{4} - 21 T^{3} + \cdots + 325399050$$
$31$ $$T^{4} + 193 T^{3} + \cdots - 58104720$$
$37$ $$T^{4} + \cdots + 2389345472$$
$41$ $$T^{4} + 125 T^{3} + \cdots + 29467114$$
$43$ $$T^{4} - 2 T^{3} + \cdots + 78004224$$
$47$ $$T^{4} + \cdots - 3169103456$$
$53$ $$T^{4} + \cdots + 7631805536$$
$59$ $$T^{4} + \cdots + 1146071296$$
$61$ $$T^{4} - 754 T^{3} + \cdots - 621762112$$
$67$ $$T^{4} + \cdots - 1826338144$$
$71$ $$T^{4} + \cdots - 5581505296$$
$73$ $$T^{4} + \cdots - 14695752674$$
$79$ $$T^{4} + \cdots - 61908677856$$
$83$ $$T^{4} + \cdots + 7015211408$$
$89$ $$T^{4} + \cdots - 213195182848$$
$97$ $$T^{4} + \cdots + 60054540368$$