# Properties

 Label 23.2.a.a Level $23$ Weight $2$ Character orbit 23.a Self dual yes Analytic conductor $0.184$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

This is the first newform (ordered by analytic conductor) with trivial character and dimension larger than $1$.

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,2,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, 2, names="a")

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

chi := DirichletCharacter("23.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$0.183655924649$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (2 \beta - 1) q^{3} + (\beta - 1) q^{4} - 2 \beta q^{5} + ( - \beta - 2) q^{6} + ( - 2 \beta + 2) q^{7} + (2 \beta - 1) q^{8} + 2 q^{9} +O(q^{10})$$ q - b * q^2 + (2*b - 1) * q^3 + (b - 1) * q^4 - 2*b * q^5 + (-b - 2) * q^6 + (-2*b + 2) * q^7 + (2*b - 1) * q^8 + 2 * q^9 $$q - \beta q^{2} + (2 \beta - 1) q^{3} + (\beta - 1) q^{4} - 2 \beta q^{5} + ( - \beta - 2) q^{6} + ( - 2 \beta + 2) q^{7} + (2 \beta - 1) q^{8} + 2 q^{9} + (2 \beta + 2) q^{10} + (2 \beta - 4) q^{11} + ( - \beta + 3) q^{12} + 3 q^{13} + 2 q^{14} + ( - 2 \beta - 4) q^{15} - 3 \beta q^{16} + (2 \beta + 2) q^{17} - 2 \beta q^{18} - 2 q^{19} - 2 q^{20} + (2 \beta - 6) q^{21} + (2 \beta - 2) q^{22} + q^{23} + 5 q^{24} + (4 \beta - 1) q^{25} - 3 \beta q^{26} + ( - 2 \beta + 1) q^{27} + (2 \beta - 4) q^{28} - 3 q^{29} + (6 \beta + 2) q^{30} + ( - 6 \beta + 3) q^{31} + ( - \beta + 5) q^{32} + ( - 6 \beta + 8) q^{33} + ( - 4 \beta - 2) q^{34} + 4 q^{35} + (2 \beta - 2) q^{36} + 2 \beta q^{37} + 2 \beta q^{38} + (6 \beta - 3) q^{39} + ( - 2 \beta - 4) q^{40} + (4 \beta - 1) q^{41} + (4 \beta - 2) q^{42} + ( - 4 \beta + 6) q^{44} - 4 \beta q^{45} - \beta q^{46} + (2 \beta - 1) q^{47} + ( - 3 \beta - 6) q^{48} + ( - 4 \beta + 1) q^{49} + ( - 3 \beta - 4) q^{50} + (6 \beta + 2) q^{51} + (3 \beta - 3) q^{52} + ( - 4 \beta - 2) q^{53} + (\beta + 2) q^{54} + (4 \beta - 4) q^{55} + (2 \beta - 6) q^{56} + ( - 4 \beta + 2) q^{57} + 3 \beta q^{58} + ( - 4 \beta + 4) q^{59} + ( - 4 \beta + 2) q^{60} + (8 \beta - 2) q^{61} + (3 \beta + 6) q^{62} + ( - 4 \beta + 4) q^{63} + (2 \beta + 1) q^{64} - 6 \beta q^{65} + ( - 2 \beta + 6) q^{66} + ( - 2 \beta - 4) q^{67} + 2 \beta q^{68} + (2 \beta - 1) q^{69} - 4 \beta q^{70} + ( - 2 \beta + 11) q^{71} + (4 \beta - 2) q^{72} + (4 \beta + 9) q^{73} + ( - 2 \beta - 2) q^{74} + (2 \beta + 9) q^{75} + ( - 2 \beta + 2) q^{76} + (8 \beta - 12) q^{77} + ( - 3 \beta - 6) q^{78} + (8 \beta - 6) q^{79} + (6 \beta + 6) q^{80} - 11 q^{81} + ( - 3 \beta - 4) q^{82} + ( - 2 \beta - 10) q^{83} + ( - 6 \beta + 8) q^{84} + ( - 8 \beta - 4) q^{85} + ( - 6 \beta + 3) q^{87} + ( - 6 \beta + 8) q^{88} + (4 \beta - 8) q^{89} + (4 \beta + 4) q^{90} + ( - 6 \beta + 6) q^{91} + (\beta - 1) q^{92} - 15 q^{93} + ( - \beta - 2) q^{94} + 4 \beta q^{95} + (9 \beta - 7) q^{96} + ( - 6 \beta + 14) q^{97} + (3 \beta + 4) q^{98} + (4 \beta - 8) q^{99} +O(q^{100})$$ q - b * q^2 + (2*b - 1) * q^3 + (b - 1) * q^4 - 2*b * q^5 + (-b - 2) * q^6 + (-2*b + 2) * q^7 + (2*b - 1) * q^8 + 2 * q^9 + (2*b + 2) * q^10 + (2*b - 4) * q^11 + (-b + 3) * q^12 + 3 * q^13 + 2 * q^14 + (-2*b - 4) * q^15 - 3*b * q^16 + (2*b + 2) * q^17 - 2*b * q^18 - 2 * q^19 - 2 * q^20 + (2*b - 6) * q^21 + (2*b - 2) * q^22 + q^23 + 5 * q^24 + (4*b - 1) * q^25 - 3*b * q^26 + (-2*b + 1) * q^27 + (2*b - 4) * q^28 - 3 * q^29 + (6*b + 2) * q^30 + (-6*b + 3) * q^31 + (-b + 5) * q^32 + (-6*b + 8) * q^33 + (-4*b - 2) * q^34 + 4 * q^35 + (2*b - 2) * q^36 + 2*b * q^37 + 2*b * q^38 + (6*b - 3) * q^39 + (-2*b - 4) * q^40 + (4*b - 1) * q^41 + (4*b - 2) * q^42 + (-4*b + 6) * q^44 - 4*b * q^45 - b * q^46 + (2*b - 1) * q^47 + (-3*b - 6) * q^48 + (-4*b + 1) * q^49 + (-3*b - 4) * q^50 + (6*b + 2) * q^51 + (3*b - 3) * q^52 + (-4*b - 2) * q^53 + (b + 2) * q^54 + (4*b - 4) * q^55 + (2*b - 6) * q^56 + (-4*b + 2) * q^57 + 3*b * q^58 + (-4*b + 4) * q^59 + (-4*b + 2) * q^60 + (8*b - 2) * q^61 + (3*b + 6) * q^62 + (-4*b + 4) * q^63 + (2*b + 1) * q^64 - 6*b * q^65 + (-2*b + 6) * q^66 + (-2*b - 4) * q^67 + 2*b * q^68 + (2*b - 1) * q^69 - 4*b * q^70 + (-2*b + 11) * q^71 + (4*b - 2) * q^72 + (4*b + 9) * q^73 + (-2*b - 2) * q^74 + (2*b + 9) * q^75 + (-2*b + 2) * q^76 + (8*b - 12) * q^77 + (-3*b - 6) * q^78 + (8*b - 6) * q^79 + (6*b + 6) * q^80 - 11 * q^81 + (-3*b - 4) * q^82 + (-2*b - 10) * q^83 + (-6*b + 8) * q^84 + (-8*b - 4) * q^85 + (-6*b + 3) * q^87 + (-6*b + 8) * q^88 + (4*b - 8) * q^89 + (4*b + 4) * q^90 + (-6*b + 6) * q^91 + (b - 1) * q^92 - 15 * q^93 + (-b - 2) * q^94 + 4*b * q^95 + (9*b - 7) * q^96 + (-6*b + 14) * q^97 + (3*b + 4) * q^98 + (4*b - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 2 q^{5} - 5 q^{6} + 2 q^{7} + 4 q^{9}+O(q^{10})$$ 2 * q - q^2 - q^4 - 2 * q^5 - 5 * q^6 + 2 * q^7 + 4 * q^9 $$2 q - q^{2} - q^{4} - 2 q^{5} - 5 q^{6} + 2 q^{7} + 4 q^{9} + 6 q^{10} - 6 q^{11} + 5 q^{12} + 6 q^{13} + 4 q^{14} - 10 q^{15} - 3 q^{16} + 6 q^{17} - 2 q^{18} - 4 q^{19} - 4 q^{20} - 10 q^{21} - 2 q^{22} + 2 q^{23} + 10 q^{24} + 2 q^{25} - 3 q^{26} - 6 q^{28} - 6 q^{29} + 10 q^{30} + 9 q^{32} + 10 q^{33} - 8 q^{34} + 8 q^{35} - 2 q^{36} + 2 q^{37} + 2 q^{38} - 10 q^{40} + 2 q^{41} + 8 q^{44} - 4 q^{45} - q^{46} - 15 q^{48} - 2 q^{49} - 11 q^{50} + 10 q^{51} - 3 q^{52} - 8 q^{53} + 5 q^{54} - 4 q^{55} - 10 q^{56} + 3 q^{58} + 4 q^{59} + 4 q^{61} + 15 q^{62} + 4 q^{63} + 4 q^{64} - 6 q^{65} + 10 q^{66} - 10 q^{67} + 2 q^{68} - 4 q^{70} + 20 q^{71} + 22 q^{73} - 6 q^{74} + 20 q^{75} + 2 q^{76} - 16 q^{77} - 15 q^{78} - 4 q^{79} + 18 q^{80} - 22 q^{81} - 11 q^{82} - 22 q^{83} + 10 q^{84} - 16 q^{85} + 10 q^{88} - 12 q^{89} + 12 q^{90} + 6 q^{91} - q^{92} - 30 q^{93} - 5 q^{94} + 4 q^{95} - 5 q^{96} + 22 q^{97} + 11 q^{98} - 12 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^4 - 2 * q^5 - 5 * q^6 + 2 * q^7 + 4 * q^9 + 6 * q^10 - 6 * q^11 + 5 * q^12 + 6 * q^13 + 4 * q^14 - 10 * q^15 - 3 * q^16 + 6 * q^17 - 2 * q^18 - 4 * q^19 - 4 * q^20 - 10 * q^21 - 2 * q^22 + 2 * q^23 + 10 * q^24 + 2 * q^25 - 3 * q^26 - 6 * q^28 - 6 * q^29 + 10 * q^30 + 9 * q^32 + 10 * q^33 - 8 * q^34 + 8 * q^35 - 2 * q^36 + 2 * q^37 + 2 * q^38 - 10 * q^40 + 2 * q^41 + 8 * q^44 - 4 * q^45 - q^46 - 15 * q^48 - 2 * q^49 - 11 * q^50 + 10 * q^51 - 3 * q^52 - 8 * q^53 + 5 * q^54 - 4 * q^55 - 10 * q^56 + 3 * q^58 + 4 * q^59 + 4 * q^61 + 15 * q^62 + 4 * q^63 + 4 * q^64 - 6 * q^65 + 10 * q^66 - 10 * q^67 + 2 * q^68 - 4 * q^70 + 20 * q^71 + 22 * q^73 - 6 * q^74 + 20 * q^75 + 2 * q^76 - 16 * q^77 - 15 * q^78 - 4 * q^79 + 18 * q^80 - 22 * q^81 - 11 * q^82 - 22 * q^83 + 10 * q^84 - 16 * q^85 + 10 * q^88 - 12 * q^89 + 12 * q^90 + 6 * q^91 - q^92 - 30 * q^93 - 5 * q^94 + 4 * q^95 - 5 * q^96 + 22 * q^97 + 11 * q^98 - 12 * 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
 1.61803 −0.618034
−1.61803 2.23607 0.618034 −3.23607 −3.61803 −1.23607 2.23607 2.00000 5.23607
1.2 0.618034 −2.23607 −1.61803 1.23607 −1.38197 3.23607 −2.23607 2.00000 0.763932
 $$n$$: 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 23.2.a.a 2
3.b odd 2 1 207.2.a.d 2
4.b odd 2 1 368.2.a.h 2
5.b even 2 1 575.2.a.f 2
5.c odd 4 2 575.2.b.d 4
7.b odd 2 1 1127.2.a.c 2
8.b even 2 1 1472.2.a.t 2
8.d odd 2 1 1472.2.a.s 2
11.b odd 2 1 2783.2.a.c 2
12.b even 2 1 3312.2.a.ba 2
13.b even 2 1 3887.2.a.i 2
15.d odd 2 1 5175.2.a.be 2
17.b even 2 1 6647.2.a.b 2
19.b odd 2 1 8303.2.a.e 2
20.d odd 2 1 9200.2.a.bt 2
23.b odd 2 1 529.2.a.a 2
23.c even 11 10 529.2.c.o 20
23.d odd 22 10 529.2.c.n 20
69.c even 2 1 4761.2.a.w 2
92.b even 2 1 8464.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.a.a 2 1.a even 1 1 trivial
207.2.a.d 2 3.b odd 2 1
368.2.a.h 2 4.b odd 2 1
529.2.a.a 2 23.b odd 2 1
529.2.c.n 20 23.d odd 22 10
529.2.c.o 20 23.c even 11 10
575.2.a.f 2 5.b even 2 1
575.2.b.d 4 5.c odd 4 2
1127.2.a.c 2 7.b odd 2 1
1472.2.a.s 2 8.d odd 2 1
1472.2.a.t 2 8.b even 2 1
2783.2.a.c 2 11.b odd 2 1
3312.2.a.ba 2 12.b even 2 1
3887.2.a.i 2 13.b even 2 1
4761.2.a.w 2 69.c even 2 1
5175.2.a.be 2 15.d odd 2 1
6647.2.a.b 2 17.b even 2 1
8303.2.a.e 2 19.b odd 2 1
8464.2.a.bb 2 92.b even 2 1
9200.2.a.bt 2 20.d odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(23))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$T^{2} - 5$$
$5$ $$T^{2} + 2T - 4$$
$7$ $$T^{2} - 2T - 4$$
$11$ $$T^{2} + 6T + 4$$
$13$ $$(T - 3)^{2}$$
$17$ $$T^{2} - 6T + 4$$
$19$ $$(T + 2)^{2}$$
$23$ $$(T - 1)^{2}$$
$29$ $$(T + 3)^{2}$$
$31$ $$T^{2} - 45$$
$37$ $$T^{2} - 2T - 4$$
$41$ $$T^{2} - 2T - 19$$
$43$ $$T^{2}$$
$47$ $$T^{2} - 5$$
$53$ $$T^{2} + 8T - 4$$
$59$ $$T^{2} - 4T - 16$$
$61$ $$T^{2} - 4T - 76$$
$67$ $$T^{2} + 10T + 20$$
$71$ $$T^{2} - 20T + 95$$
$73$ $$T^{2} - 22T + 101$$
$79$ $$T^{2} + 4T - 76$$
$83$ $$T^{2} + 22T + 116$$
$89$ $$T^{2} + 12T + 16$$
$97$ $$T^{2} - 22T + 76$$