LMFDB - Newform orbit 23.2.c.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,2,Mod(2,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("23.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	23.c (of order $11$, degree $10$, minimal)

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:	no
Analytic conductor:	$0.183655924649$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-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 primitive root of unity $\zeta_{22}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \cdots - \zeta_{22}) q^{2}+ \cdots + ( - \zeta_{22}^{9} + \zeta_{22}^{7} + \cdots - \zeta_{22}) q^{9}+O(q^{10}) $ q + (-z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - z) * q^2 + (z^5 - z^4 + z - 1) * q^3 + (z^9 - 2z^8 + 2z^7 - z^6 - z^4 - 1) * q^4 + (z^9 + z^7 - 2z^6 + z^5 + z^3 + z - 1) q^5 + (z^9 - z^7 + 2z^4 - 2z^3 + 1) * q^6 + (-z^7 + z^6 - z^5 - z^4 - z^3 + z^2 - z) * q^7 + (z^9 - z^8 - z^6 + 2z^5 - 2z^4 + 3z^3 - 2z^2 + 2z - 1) q^8 + (-z^9 + z^7 + z^6 + z^4 + z^3 - z) * q^9	$ q + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \cdots - \zeta_{22}) q^{2}+ \cdots + ( - 4 \zeta_{22}^{8} + \zeta_{22}^{7} + \cdots - 4) q^{99}+O(q^{100}) $ q + (-z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - z) * q^2 + (z^5 - z^4 + z - 1) * q^3 + (z^9 - 2z^8 + 2z^7 - z^6 - z^4 - 1) * q^4 + (z^9 + z^7 - 2z^6 + z^5 + z^3 + z - 1) q^5 + (z^9 - z^7 + 2z^4 - 2z^3 + 1) * q^6 + (-z^7 + z^6 - z^5 - z^4 - z^3 + z^2 - z) * q^7 + (z^9 - z^8 - z^6 + 2z^5 - 2z^4 + 3z^3 - 2z^2 + 2z - 1) q^8 + (-z^9 + z^7 + z^6 + z^4 + z^3 - z) * q^9 + (-2z^9 + 2z^8 + 2z^6 - 2z^5 - z^3 + 1) * q^10 + (-3z^9 + 2z^8 - 2z^7 + 3z^6 - 3z^5 + 3z^4 - 3z^3 + 2z^2 - 2z + 3) q^11 + (-2z^9 + 3z^8 - z^7 - z^6 + z^3 + z^2 - 3z + 2) q^12 + (-2z^8 - 2z^6 + z^3 - z^2 + z - 1) * q^13 + (z^9 - z^8 - z^6 + 3z^5 - z^4 - z^2 + z) q^14 + (2z^9 - 2z^8 + z^5 - 2z^4 + 2z^3 - 2z^2 + z) q^15 + (z^8 + z^7 + 2z^6 - 2z^5 + 2z^4 - 2z^3 + 2z^2 + z + 1) q^16 + (3z^9 + z^7 + z^6 + z^4 - z^3 - z - 1) q^17 + (2z^9 - 3z^8 + z^7 - z^6 + z^5 - 3z^4 + 2z^3 + 3z - 3) q^18 + (2z^9 + 2z^8 - 2z^7 - 2z^6 - 2z^2) q^19 + (-z^9 - z^8 - z^7 + z^4 - 3z^3 + 3z^2 - z) * q^20 + (-2z^9 + z^8 + 2z^7 - 2z^6 - z^5 + 2z^4) * q^21 + (z^9 - 4z^8 + 3z^7 - 4z^6 + 4z^5 - 3z^4 + 4z^3 - z^2 - 3) * q^22 + (-z^9 + 3z^7 - z^6 + 2z^5 - 3z^4 - z^3 - z^2 - 2) q^23 + (-2z^9 + 2z^8 - 2z^7 + 3z^6 - 3z^5 + 2z^4 - 2z^3 + 2z^2 - 2) * q^24 + (-z^9 - z^8 - 2z^7 + 2z^6 + z^5 + z^4) * q^25 + (z^9 + z^8 + z^7 - z^5 - z^4 + z + 1) * q^26 + (-z^4 + z^3 - 3z^2 + z - 1) q^27 + (z^9 + 3z^8 - 2z^7 + 3z^6 - 2z^5 + 3z^4 + z^3 - 2z + 2) * q^28 + (-2z^9 - 2z^7 + z^6 - 2z^5 - 2z^4 + 2z^3 + 2z^2 - z + 2) * q^29 + (3z^8 - 3z^7 + 2z^6 - 3z^5 + 4z^4 - 3z^3 + 2z^2 - 3z + 3) * q^30 + (-z^9 + z^8 + 2z^6 + 2z^4 - 2z^3 + 2z^2 + 2) * q^31 + (2z^9 - 4z^8 + z^7 - 2z^6 + 3z^5 - 2z^4 + z^3 - 4z^2 + 2z) q^32 + (2z^9 - z^8 + z^7 - z^6 + 2z^5 - z^3 + 1) * q^33 + (-z^9 + 2z^7 - 4z^6 + 4z^5 - 4z^4 + 4z^3 - 2z^2 + 1) * q^34 + (z^9 - 4z^8 + 2z^7 - 2z^6 + 2z^3 - 2z^2 + 4z - 1) * q^35 + (z^9 - 2z^8 - z^7 - 2z^6 + z^5 - 2z^3 - 2z^2 + 2z + 2) q^36 + (-z^9 + 4z^8 - 2z^7 + z^6 - 3z^5 + z^4 - 2z^3 + 4z^2 - z) q^37 + (-6z^9 + 6z^8 + 4z^6 - 8z^5 + 6z^4 - 4z^3 + 6z^2 - 8z + 4) * q^38 + (z^8 - 2z^7 + 2z^6 + 2z^2 - 2z + 1) * q^39 + (-2z^9 + z^7 - z^6 + 4z^5 - z^4 + z^3 - 4z^2 + z - 1) q^40 + (3z^9 - z^8 + z^6 - z^4 + 3z^3) * q^41 + (-z^9 - 3z^8 + 3z^7 + z^6 - 4z^4 + 3z^3 - z^2 + 3z - 4) q^42 + (-z^9 + z^8 - z^7 + 3z^4 - z^3 + z^2 - 3z) * q^43 + (2z^9 + z^8 - 2z^7 + 2z^6 - z^5 - 2z^4 - 4z^2 + 4z - 4) * q^44 + (-z^9 + 3z^8 + 2z^7 + z^6 - z^5 - 2z^4 - 3z^3 + z^2) * q^45 + (2z^9 - 3z^8 - 3z^7 + z^6 + 2z^5 + z^4 + z^3 - 3z^2 + 5z - 4) * q^46 + (-z^9 - 2z^8 - z^7 - z^6 + z^5 + z^4 + 2z^3 + z^2 - 2) * q^47 + (z^9 + z^7 - z^6 - z^4 + 2z^2 - 4z + 2) * q^48 + (-z^9 - z^8 - z^7 + z^5 + 2z^4 - 2z^3 + 2z^2 - 2z - 1) * q^49 + (4z^9 - 4z^6 + z^4 + z^2 + 1) * q^50 + (-3z^8 + 3z^7 - 4z^6 + 3z^5 - 3z^4 + z - 1) q^51 + (z^9 + 3z^7 - z^6 + 2z^5 - 2z^2 + z - 3) q^52 + (z^8 + 2z^7 - 6z^6 + z^5 - 2z^4 + z^3 - 6z^2 + 2z + 1) q^53 + (z^9 - z^8 + z^6 - 3z^5 + 4z^4 - 3z^3 + 4z^2 - 3z + 1) q^54 + (3z^9 + 3z^8 + z^7 - 3z^5 + z^3 + 3z^2 + 3z) q^55 + (-z^9 - z^7 - z^5 - z^3 + 4z^2 - 4z + 1) * q^56 + (4z^9 - 8z^8 + 2z^7 + 4z^5 - 4z^4 - 2z^2 + 8z - 4) q^57 + (z^9 - z^8 - 2z^7 + 5z^6 - z^5 + z^4 - 5z^3 + 2z^2 + z - 1) * q^58 + (3z^8 + 3z^7 + 3z^6 + 2z^3 - 2) * q^59 + (-3z^8 + 6z^7 - 3z^6 + z^5 - 3z^4 + 6z^3 - 3z^2) * q^60 + (2z^9 - 2z^8 - 3z^6 + 5z^5 - 6z^4 + 4z^3 - 6z^2 + 5z - 3) * q^61 + (-z^8 - 3z^6 + 3z^5 - z^4 + 3z^3 - 3z^2 - 1) * q^62 + (-z^9 - 5z^7 + 2z^6 - z^5 + 2z^4 - 2z^3 + z^2 - 2z + 5) q^63 + (-4z^9 + 4z^8 - 3z^7 - z^6 - 3z^5 + 4z^4 - 4z^3 - 5z + 5) q^64 + (-6z^9 + 4z^8 - 4z^7 + 6z^6 + 4z^4 - 5z^3 + 4z^2 - 5z + 4) * q^65 + (-3z^9 + 5z^8 - 3z^7 - z^5 + z^4 + 3z^3 - 3z^2 - z + 1) q^66 + (-z^9 + 3z^8 - 2z^7 + 2z^6 - 3z^5 + z^4 + 7z^2 - 6z + 7) * q^67 + (-6z^9 + 4z^8 - 9z^7 + 11z^6 - 11z^5 + 9z^4 - 4z^3 + 6z^2 + 3) * q^68 + (-2z^9 + 3z^8 - 2z^7 + 2z^6 - 5z^5 + 2z^4 + 3z^3 - 2z^2 - 3z + 4) q^69 + (2z^9 + z^7 - z^6 + z^5 - z^4 - 2z^2 + 3) * q^70 + (z^9 - z^8 + 3z^7 - 3z^6 + z^5 - z^4 - 3z^2 + 3z - 3) * q^71 + (z^9 + 3z^8 + z^7 - 2z^5 + z^3 - z^2 + 2) * q^72 + (-4z^9 + 3z^8 - 3z^7 + 4z^6 + 4z^4 - 2z^3 - z^2 - 2z + 4) q^73 + (-2z^9 - 3z^8 + 4z^7 + 4z^5 - 3z^4 - 2z^3) * q^74 + (-2z^9 + 2z^7 + z^6 - 2z^5 + z^4 - z^3 + 2z^2 - z - 2) * q^75 + (-6z^8 + 6z^7 + 8z^5 - 10z^4 + 8z^3 + 6z - 6) * q^76 + (z^9 - z^8 - z^6 + z^5 - 4z^4 - z^3 - 4z^2 + z - 1) * q^77 + (z^8 - z^7 - z^3 + z^2) * q^78 + (2z^9 - 8z^8 + 6z^7 - 8z^6 + 2z^5 + 5z^3 - 2z^2 + 2z - 5) * q^79 + (6z^9 - z^8 - 2z^7 - 2z^6 + 2z^3 + 2z^2 + z - 6) q^80 + (2z^9 + 5z^8 - 4z^7 + z^6 - 3z^5 + 3z^4 - z^3 + 4z^2 - 5z - 2) q^81 + (-z^9 + 2z^8 - z^7 + 2z^6 - z^5 - 3z^3 + 2z^2 - 2z + 3) q^82 + (3z^9 - 3z^8 - z^7 - 2z^6 + 4z^5 - 2z^4 - z^3 - 3z^2 + 3z) q^83 + (3z^9 - 3z^8 - 3z^6 + 5z^5 - 6z^3 + 5z - 3) * q^84 + (-z^8 - z^7 - z^6 - 4z^5 + 2z^4 - 4z^3 - z^2 - z - 1) q^85 + (3z^7 - 3z^6 - 5z^4 + 5z^3 + 3z - 3) q^86 + (-3z^9 + 7z^8 - 2z^7 - z^6 - 2z^5 + 7z^4 - 3z^3) * q^87 + (5z^9 - 3z^8 + 3z^7 - 5z^6 + 2z^4 - z^3 + 5z^2 - z + 2) * q^88 + (11z^9 - 8z^8 + 11z^7 + 3z^5 - 7z^4 + 4z^3 - 4z^2 + 7z - 3) * q^89 + (-z^9 - 5z^8 - z^7 + z^6 + 5z^5 + z^4 - 3z^2 + z - 3) q^90 + (5z^9 - 4z^8 + 3z^7 - z^6 + z^5 - 3z^4 + 4z^3 - 5z^2 - 3) * q^91 + (4z^8 - z^7 - 2z^6 + z^5 + 7z^4 - 5z^3 + 8z^2 - 6z + 8) * q^92 + (z^9 - 2z^8 + 3z^7 - z^6 + z^5 - 3z^4 + 2z^3 - z^2 - 1) * q^93 + (4z^9 + 2z^7 - 2z^6 - 4z^4 + 5z) q^94 + (-4z^9 - 4z^7 - 2z^5 + 4z^4 + 2z^3 - 2z^2 - 4z + 2) q^95 + (-4z^9 + z^8 - z^7 + 4z^6 - 4z^4 + 5z^2 - 4) * q^96 + (-5z^9 + 2z^8 - 7z^7 + 6z^6 - 7z^5 + 2z^4 - 5z^3) q^97 + (3z^9 + 2z^7 - 6z^6 + 4z^5 - 6z^4 + 6z^3 - 4z^2 + 6z - 2) * q^98 + (-4z^8 + z^7 + z^6 + 4z^5 + 2z^4 + 4z^3 + z^2 + z - 4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 7 q^{2} - 7 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} - 5 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10}) $ 10 * q - 7 * q^2 - 7 * q^3 - 3 * q^4 - 3 * q^5 + 6 * q^6 - 5 * q^7 + 4 * q^8 - 2 * q^9	\( 10 q - 7 q^{2} - 7 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} - 5 q^{7} + 4 q^{8} - 2 q^{9} + q^{10} + 7 q^{11} + 12 q^{12} - 3 q^{13} + 9 q^{14} + 12 q^{15} + q^{16} - 10 q^{17} - 14 q^{18} + 2 q^{19} - 9 q^{20} - 2 q^{21} - 6 q^{22} - 12 q^{23} - 38 q^{24} - 4 q^{25} + 12 q^{26} - 4 q^{27} + 7 q^{28} + 14 q^{29} + 7 q^{30} + 10 q^{31} + 21 q^{32} + 16 q^{33} + 29 q^{34} + 7 q^{35} + 27 q^{36} - 19 q^{37} - 8 q^{38} + q^{39} + q^{40} + 7 q^{41} - 25 q^{42} - 11 q^{43} - 34 q^{44} - 6 q^{45} - 29 q^{46} - 18 q^{47} + 18 q^{48} - 18 q^{49} + 16 q^{50} + 7 q^{51} - 20 q^{52} + 29 q^{53} - 6 q^{54} - q^{55} - 2 q^{56} - 8 q^{57} - 23 q^{58} - 21 q^{59} + 25 q^{60} + 3 q^{61} + 4 q^{62} + 34 q^{63} + 24 q^{64} + 2 q^{65} + 2 q^{66} + 45 q^{67} - 30 q^{68} + 26 q^{69} + 38 q^{70} - 14 q^{71} + 19 q^{72} + 19 q^{73} + 10 q^{74} - 28 q^{75} - 16 q^{76} + 2 q^{77} - 4 q^{78} - 15 q^{79} - 52 q^{80} - 44 q^{81} + 16 q^{82} + 18 q^{83} - 17 q^{84} - 19 q^{85} - 11 q^{86} - 23 q^{87} + 27 q^{88} + 25 q^{89} - 20 q^{90} - 4 q^{91} + 52 q^{92} + 4 q^{93} + 17 q^{94} + 6 q^{95} - 51 q^{96} - 34 q^{97} + 17 q^{98} - 30 q^{99}+O(q^{100}) \) 10 * q - 7 * q^2 - 7 * q^3 - 3 * q^4 - 3 * q^5 + 6 * q^6 - 5 * q^7 + 4 * q^8 - 2 * q^9 + q^10 + 7 * q^11 + 12 * q^12 - 3 * q^13 + 9 * q^14 + 12 * q^15 + q^16 - 10 * q^17 - 14 * q^18 + 2 * q^19 - 9 * q^20 - 2 * q^21 - 6 * q^22 - 12 * q^23 - 38 * q^24 - 4 * q^25 + 12 * q^26 - 4 * q^27 + 7 * q^28 + 14 * q^29 + 7 * q^30 + 10 * q^31 + 21 * q^32 + 16 * q^33 + 29 * q^34 + 7 * q^35 + 27 * q^36 - 19 * q^37 - 8 * q^38 + q^39 + q^40 + 7 * q^41 - 25 * q^42 - 11 * q^43 - 34 * q^44 - 6 * q^45 - 29 * q^46 - 18 * q^47 + 18 * q^48 - 18 * q^49 + 16 * q^50 + 7 * q^51 - 20 * q^52 + 29 * q^53 - 6 * q^54 - q^55 - 2 * q^56 - 8 * q^57 - 23 * q^58 - 21 * q^59 + 25 * q^60 + 3 * q^61 + 4 * q^62 + 34 * q^63 + 24 * q^64 + 2 * q^65 + 2 * q^66 + 45 * q^67 - 30 * q^68 + 26 * q^69 + 38 * q^70 - 14 * q^71 + 19 * q^72 + 19 * q^73 + 10 * q^74 - 28 * q^75 - 16 * q^76 + 2 * q^77 - 4 * q^78 - 15 * q^79 - 52 * q^80 - 44 * q^81 + 16 * q^82 + 18 * q^83 - 17 * q^84 - 19 * q^85 - 11 * q^86 - 23 * q^87 + 27 * q^88 + 25 * q^89 - 20 * q^90 - 4 * q^91 + 52 * q^92 + 4 * q^93 + 17 * q^94 + 6 * q^95 - 51 * q^96 - 34 * q^97 + 17 * q^98 - 30 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/23\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-\zeta_{22}$

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} $

2.1

−0.841254	−	0.540641i
0.142315	+	0.989821i
−0.415415	−	0.909632i
−0.415415	+	0.909632i
0.142315	−	0.989821i
0.959493	−	0.281733i
−0.841254	+	0.540641i
0.654861	+	0.755750i
0.654861	−	0.755750i
0.959493	+	0.281733i

−2.11435

−

1.35881i

−0.226900

−

1.57812i

1.79329

3.92676i

1.41899

0.416652i

−1.66463

3.64502i

−0.804632

0.928595i

0.828708

−

5.76379i

0.439490

−

0.129046i

−2.43409

−

2.80909i

3.1

−0.313607

−

2.18119i

−1.04408

2.28621i

−2.74024

0.804606i

0.809721

−

0.934468i

5.31408

1.56036i

−1.99611

1.28282i

0.783524

1.71568i

−2.17208

−

2.50672i

−2.29218

−

1.47310i

4.1

0.198939

0.435615i

−2.11435

0.620830i

1.15954

−

1.33818i

−2.18251

−

1.40261i

−0.691070

−

0.797537i

0.483568

3.36329i

1.73259

0.508735i

1.56130

−

1.00339i

0.176814

−

1.22977i

6.1

0.198939

−

0.435615i

−2.11435

−

0.620830i

1.15954

1.33818i

−2.18251

1.40261i

−0.691070

0.797537i

0.483568

−

3.36329i

1.73259

−

0.508735i

1.56130

1.00339i

0.176814

1.22977i

8.1

−0.313607

2.18119i

−1.04408

−

2.28621i

−2.74024

−

0.804606i

0.809721

0.934468i

5.31408

−

1.56036i

−1.99611

−

1.28282i

0.783524

−

1.71568i

−2.17208

2.50672i

−2.29218

1.47310i

9.1

−0.226900

0.0666238i

−0.313607

−

0.361922i

−1.63546

1.05105i

−0.215370

−

1.49793i

0.0952700

0.0612263i

−1.05773

2.31611i

0.610783

−

0.704881i

0.394306

−

2.74246i

0.148666

0.325532i

12.1

−2.11435

1.35881i

−0.226900

1.57812i

1.79329

−

3.92676i

1.41899

−

0.416652i

−1.66463

−

3.64502i

−0.804632

−

0.928595i

0.828708

5.76379i

0.439490

0.129046i

−2.43409

2.80909i

13.1

−1.04408

−

1.20493i

0.198939

0.127850i

−0.0771283

0.536439i

−1.33083

2.91411i

−0.0536570

−

0.373193i

0.874908

0.256896i

−1.95561

1.25679i

−1.22301

−

2.67803i

4.90079

−

1.43900i

16.1

−1.04408

1.20493i

0.198939

−

0.127850i

−0.0771283

−

0.536439i

−1.33083

−

2.91411i

−0.0536570

0.373193i

0.874908

−

0.256896i

−1.95561

−

1.25679i

−1.22301

2.67803i

4.90079

1.43900i

18.1

−0.226900

−

0.0666238i

−0.313607

0.361922i

−1.63546

−

1.05105i

−0.215370

1.49793i

0.0952700

−

0.0612263i

−1.05773

−

2.31611i

0.610783

0.704881i

0.394306

2.74246i

0.148666

−

0.325532i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	23.2.c.a	✓	10
3.b	odd	2	1		207.2.i.c		10
4.b	odd	2	1		368.2.m.c		10
5.b	even	2	1		575.2.k.b		10
5.c	odd	4	2		575.2.p.b		20
23.b	odd	2	1		529.2.c.a		10
23.c	even	11	1	inner	23.2.c.a	✓	10
23.c	even	11	1		529.2.a.i		5
23.c	even	11	2		529.2.c.b		10
23.c	even	11	2		529.2.c.d		10
23.c	even	11	2		529.2.c.g		10
23.c	even	11	2		529.2.c.i		10
23.d	odd	22	1		529.2.a.j		5
23.d	odd	22	1		529.2.c.a		10
23.d	odd	22	2		529.2.c.c		10
23.d	odd	22	2		529.2.c.e		10
23.d	odd	22	2		529.2.c.f		10
23.d	odd	22	2		529.2.c.h		10
69.g	even	22	1		4761.2.a.bn		5
69.h	odd	22	1		207.2.i.c		10
69.h	odd	22	1		4761.2.a.bo		5
92.g	odd	22	1		368.2.m.c		10
92.g	odd	22	1		8464.2.a.bs		5
92.h	even	22	1		8464.2.a.bt		5
115.j	even	22	1		575.2.k.b		10
115.k	odd	44	2		575.2.p.b		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
23.2.c.a	✓	10	1.a	even	1	1	trivial
23.2.c.a	✓	10	23.c	even	11	1	inner
207.2.i.c		10	3.b	odd	2	1
207.2.i.c		10	69.h	odd	22	1
368.2.m.c		10	4.b	odd	2	1
368.2.m.c		10	92.g	odd	22	1
529.2.a.i		5	23.c	even	11	1
529.2.a.j		5	23.d	odd	22	1
529.2.c.a		10	23.b	odd	2	1
529.2.c.a		10	23.d	odd	22	1
529.2.c.b		10	23.c	even	11	2
529.2.c.c		10	23.d	odd	22	2
529.2.c.d		10	23.c	even	11	2
529.2.c.e		10	23.d	odd	22	2
529.2.c.f		10	23.d	odd	22	2
529.2.c.g		10	23.c	even	11	2
529.2.c.h		10	23.d	odd	22	2
529.2.c.i		10	23.c	even	11	2
575.2.k.b		10	5.b	even	2	1
575.2.k.b		10	115.j	even	22	1
575.2.p.b		20	5.c	odd	4	2
575.2.p.b		20	115.k	odd	44	2
4761.2.a.bn		5	69.g	even	22	1
4761.2.a.bo		5	69.h	odd	22	1
8464.2.a.bs		5	92.g	odd	22	1
8464.2.a.bt		5	92.h	even	22	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} + 7 T^{9} + \cdots + 1 $ T^10 + 7T^9 + 27T^8 + 68T^7 + 124T^6 + 142T^5 + 103T^4 + 28T^3 + 20T^2 + 8*T + 1
$3$	$ T^{10} + 7 T^{9} + \cdots + 1 $ T^10 + 7T^9 + 27T^8 + 68T^7 + 113T^6 + 131T^5 + 103T^4 + 17T^3 - 2T^2 - 3*T + 1
$5$	$ T^{10} + 3 T^{9} + \cdots + 529 $ T^10 + 3T^9 + 9T^8 - 6T^7 - 18T^6 - 32T^5 + 124T^4 - 233T^3 + 489T^2 - 667*T + 529
$7$	$ T^{10} + 5 T^{9} + \cdots + 529 $ T^10 + 5T^9 + 25T^8 + 81T^7 + 207T^6 + 298T^5 + 170T^4 - 448T^3 - 425T^2 - 46*T + 529
$11$	$ T^{10} - 7 T^{9} + \cdots + 529 $ T^10 - 7T^9 + 16T^8 + 31T^7 - 74T^6 - 813T^5 + 3282T^4 - 6111T^3 + 11317T^2 - 2208*T + 529
$13$	$ T^{10} + 3 T^{9} + \cdots + 1 $ T^10 + 3T^9 + 9T^8 + 71T^7 + 125T^6 + 166T^5 + 1290T^4 - 684T^3 + 159T^2 - 18*T + 1
$17$	$ T^{10} + 10 T^{9} + \cdots + 529 $ T^10 + 10T^9 + 56T^8 + 65T^7 - 274T^6 - 2003T^5 + 10880T^4 - 20197T^3 + 14521T^2 - 1541*T + 529
$19$	$ T^{10} - 2 T^{9} + \cdots + 541696 $ T^10 - 2T^9 - 40T^8 - 184T^7 + 544T^6 + 8416T^5 + 52864T^4 + 180096T^3 + 450816T^2 + 635904*T + 541696
$23$	$ T^{10} + 12 T^{9} + \cdots + 6436343 $ T^10 + 12T^9 - 10T^8 - 527T^7 - 32T^6 + 14103T^5 - 736T^4 - 278783T^3 - 121670T^2 + 3358092*T + 6436343
$29$	$ T^{10} - 14 T^{9} + \cdots + 4932841 $ T^10 - 14T^9 + 86T^8 + 6T^7 - 2394T^6 + 13100T^5 - 833T^4 - 184633T^3 + 1146117T^2 - 1896734*T + 4932841
$31$	$ T^{10} - 10 T^{9} + \cdots + 17161 $ T^10 - 10T^9 + 56T^8 - 164T^7 + 375T^6 - 505T^5 + 56T^4 + 3862T^3 - 1066T^2 + 2751*T + 17161
$37$	$ T^{10} + 19 T^{9} + \cdots + 529 $ T^10 + 19T^9 + 174T^8 + 1117T^7 + 6956T^6 + 38345T^5 + 148800T^4 + 346183T^3 + 370815T^2 + 4462*T + 529
$41$	$ T^{10} - 7 T^{9} + \cdots + 1849 $ T^10 - 7T^9 + 16T^8 + 9T^7 + 1235T^6 + 4181T^5 + 15173T^4 + 27978T^3 + 23747T^2 + 9331*T + 1849
$43$	$ T^{10} + 11 T^{9} + \cdots + 64009 $ T^10 + 11T^9 + 66T^8 + 341T^7 + 1826T^6 + 6875T^5 + 20449T^4 + 55297T^3 + 106843T^2 + 114103*T + 64009
$47$	$ (T^{5} + 9 T^{4} - 5 T^{3} + \cdots + 1)^{2} $ (T^5 + 9T^4 - 5T^3 - 97T^2 - 106T + 1)^2
$53$	$ T^{10} + \cdots + 517426009 $ T^10 - 29T^9 + 478T^8 - 5579T^7 + 55454T^6 - 448645T^5 + 2974063T^4 - 16104853T^3 + 83193185T^2 - 312339057*T + 517426009
$59$	$ T^{10} + 21 T^{9} + \cdots + 4489 $ T^10 + 21T^9 + 265T^8 + 1649T^7 + 3433T^6 - 22111T^5 - 98174T^4 - 33991T^3 + 1176352T^2 + 20569*T + 4489
$61$	$ T^{10} - 3 T^{9} + \cdots + 529 $ T^10 - 3T^9 + 218T^8 + 281T^7 + 7066T^6 + 46573T^5 + 215097T^4 + 359141T^3 + 236219T^2 - 22103*T + 529
$67$	$ T^{10} + \cdots + 1113757129 $ T^10 - 45T^9 + 1090T^8 - 17513T^7 + 203501T^6 - 1799216T^5 + 12627902T^4 - 70242899T^3 + 291205025T^2 - 798415652*T + 1113757129
$71$	$ T^{10} + 14 T^{9} + \cdots + 529 $ T^10 + 14T^9 + 130T^8 + 566T^7 + 785T^6 - 197T^5 + 1686T^4 + 1560T^3 + 1204T^2 + 851*T + 529
$73$	$ T^{10} - 19 T^{9} + \cdots + 982081 $ T^10 - 19T^9 + 251T^8 - 2415T^7 + 15063T^6 - 73963T^5 + 379118T^4 - 1529101T^3 + 3411556T^2 - 2470563*T + 982081
$79$	$ T^{10} + \cdots + 517426009 $ T^10 + 15T^9 + 115T^8 + 2715T^7 + 26304T^6 + 111816T^5 + 2966242T^4 + 21480574T^3 + 54168013T^2 - 24589507*T + 517426009
$83$	$ T^{10} - 18 T^{9} + \cdots + 529 $ T^10 - 18T^9 + 192T^8 - 321T^7 + 1455T^6 - 2914T^5 + 224T^4 - 17T^3 + 3804T^2 + 2346*T + 529
$89$	$ T^{10} + \cdots + 78310985281 $ T^10 - 25T^9 + 262T^8 - 5615T^7 + 111566T^6 - 1048455T^5 + 11505750T^4 - 150274617T^3 + 1491272689T^2 - 10362512230*T + 78310985281
$97$	$ T^{10} + 34 T^{9} + \cdots + 2374681 $ T^10 + 34T^9 + 430T^8 + 4423T^7 + 52812T^6 + 183261T^5 + 1071940T^4 + 3733313T^3 + 17156261T^2 + 10511161*T + 2374681
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	23.c (of order \(11\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \cdots - \zeta_{22}) q^{2}+ \cdots + ( - \zeta_{22}^{9} + \zeta_{22}^{7} + \cdots - \zeta_{22}) q^{9}+O(q^{10}) \) q + (-z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - z) * q^2 + (z^5 - z^4 + z - 1) * q^3 + (z^9 - 2z^8 + 2z^7 - z^6 - z^4 - 1) * q^4 + (z^9 + z^7 - 2z^6 + z^5 + z^3 + z - 1) q^5 + (z^9 - z^7 + 2z^4 - 2z^3 + 1) * q^6 + (-z^7 + z^6 - z^5 - z^4 - z^3 + z^2 - z) * q^7 + (z^9 - z^8 - z^6 + 2z^5 - 2z^4 + 3z^3 - 2z^2 + 2z - 1) q^8 + (-z^9 + z^7 + z^6 + z^4 + z^3 - z) * q^9	\( q + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \cdots - \zeta_{22}) q^{2}+ \cdots + ( - 4 \zeta_{22}^{8} + \zeta_{22}^{7} + \cdots - 4) q^{99}+O(q^{100}) \) q + (-z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - z) * q^2 + (z^5 - z^4 + z - 1) * q^3 + (z^9 - 2z^8 + 2z^7 - z^6 - z^4 - 1) * q^4 + (z^9 + z^7 - 2z^6 + z^5 + z^3 + z - 1) q^5 + (z^9 - z^7 + 2z^4 - 2z^3 + 1) * q^6 + (-z^7 + z^6 - z^5 - z^4 - z^3 + z^2 - z) * q^7 + (z^9 - z^8 - z^6 + 2z^5 - 2z^4 + 3z^3 - 2z^2 + 2z - 1) q^8 + (-z^9 + z^7 + z^6 + z^4 + z^3 - z) * q^9 + (-2z^9 + 2z^8 + 2z^6 - 2z^5 - z^3 + 1) * q^10 + (-3z^9 + 2z^8 - 2z^7 + 3z^6 - 3z^5 + 3z^4 - 3z^3 + 2z^2 - 2z + 3) q^11 + (-2z^9 + 3z^8 - z^7 - z^6 + z^3 + z^2 - 3z + 2) q^12 + (-2z^8 - 2z^6 + z^3 - z^2 + z - 1) * q^13 + (z^9 - z^8 - z^6 + 3z^5 - z^4 - z^2 + z) q^14 + (2z^9 - 2z^8 + z^5 - 2z^4 + 2z^3 - 2z^2 + z) q^15 + (z^8 + z^7 + 2z^6 - 2z^5 + 2z^4 - 2z^3 + 2z^2 + z + 1) q^16 + (3z^9 + z^7 + z^6 + z^4 - z^3 - z - 1) q^17 + (2z^9 - 3z^8 + z^7 - z^6 + z^5 - 3z^4 + 2z^3 + 3z - 3) q^18 + (2z^9 + 2z^8 - 2z^7 - 2z^6 - 2z^2) q^19 + (-z^9 - z^8 - z^7 + z^4 - 3z^3 + 3z^2 - z) * q^20 + (-2z^9 + z^8 + 2z^7 - 2z^6 - z^5 + 2z^4) * q^21 + (z^9 - 4z^8 + 3z^7 - 4z^6 + 4z^5 - 3z^4 + 4z^3 - z^2 - 3) * q^22 + (-z^9 + 3z^7 - z^6 + 2z^5 - 3z^4 - z^3 - z^2 - 2) q^23 + (-2z^9 + 2z^8 - 2z^7 + 3z^6 - 3z^5 + 2z^4 - 2z^3 + 2z^2 - 2) * q^24 + (-z^9 - z^8 - 2z^7 + 2z^6 + z^5 + z^4) * q^25 + (z^9 + z^8 + z^7 - z^5 - z^4 + z + 1) * q^26 + (-z^4 + z^3 - 3z^2 + z - 1) q^27 + (z^9 + 3z^8 - 2z^7 + 3z^6 - 2z^5 + 3z^4 + z^3 - 2z + 2) * q^28 + (-2z^9 - 2z^7 + z^6 - 2z^5 - 2z^4 + 2z^3 + 2z^2 - z + 2) * q^29 + (3z^8 - 3z^7 + 2z^6 - 3z^5 + 4z^4 - 3z^3 + 2z^2 - 3z + 3) * q^30 + (-z^9 + z^8 + 2z^6 + 2z^4 - 2z^3 + 2z^2 + 2) * q^31 + (2z^9 - 4z^8 + z^7 - 2z^6 + 3z^5 - 2z^4 + z^3 - 4z^2 + 2z) q^32 + (2z^9 - z^8 + z^7 - z^6 + 2z^5 - z^3 + 1) * q^33 + (-z^9 + 2z^7 - 4z^6 + 4z^5 - 4z^4 + 4z^3 - 2z^2 + 1) * q^34 + (z^9 - 4z^8 + 2z^7 - 2z^6 + 2z^3 - 2z^2 + 4z - 1) * q^35 + (z^9 - 2z^8 - z^7 - 2z^6 + z^5 - 2z^3 - 2z^2 + 2z + 2) q^36 + (-z^9 + 4z^8 - 2z^7 + z^6 - 3z^5 + z^4 - 2z^3 + 4z^2 - z) q^37 + (-6z^9 + 6z^8 + 4z^6 - 8z^5 + 6z^4 - 4z^3 + 6z^2 - 8z + 4) * q^38 + (z^8 - 2z^7 + 2z^6 + 2z^2 - 2z + 1) * q^39 + (-2z^9 + z^7 - z^6 + 4z^5 - z^4 + z^3 - 4z^2 + z - 1) q^40 + (3z^9 - z^8 + z^6 - z^4 + 3z^3) * q^41 + (-z^9 - 3z^8 + 3z^7 + z^6 - 4z^4 + 3z^3 - z^2 + 3z - 4) q^42 + (-z^9 + z^8 - z^7 + 3z^4 - z^3 + z^2 - 3z) * q^43 + (2z^9 + z^8 - 2z^7 + 2z^6 - z^5 - 2z^4 - 4z^2 + 4z - 4) * q^44 + (-z^9 + 3z^8 + 2z^7 + z^6 - z^5 - 2z^4 - 3z^3 + z^2) * q^45 + (2z^9 - 3z^8 - 3z^7 + z^6 + 2z^5 + z^4 + z^3 - 3z^2 + 5z - 4) * q^46 + (-z^9 - 2z^8 - z^7 - z^6 + z^5 + z^4 + 2z^3 + z^2 - 2) * q^47 + (z^9 + z^7 - z^6 - z^4 + 2z^2 - 4z + 2) * q^48 + (-z^9 - z^8 - z^7 + z^5 + 2z^4 - 2z^3 + 2z^2 - 2z - 1) * q^49 + (4z^9 - 4z^6 + z^4 + z^2 + 1) * q^50 + (-3z^8 + 3z^7 - 4z^6 + 3z^5 - 3z^4 + z - 1) q^51 + (z^9 + 3z^7 - z^6 + 2z^5 - 2z^2 + z - 3) q^52 + (z^8 + 2z^7 - 6z^6 + z^5 - 2z^4 + z^3 - 6z^2 + 2z + 1) q^53 + (z^9 - z^8 + z^6 - 3z^5 + 4z^4 - 3z^3 + 4z^2 - 3z + 1) q^54 + (3z^9 + 3z^8 + z^7 - 3z^5 + z^3 + 3z^2 + 3z) q^55 + (-z^9 - z^7 - z^5 - z^3 + 4z^2 - 4z + 1) * q^56 + (4z^9 - 8z^8 + 2z^7 + 4z^5 - 4z^4 - 2z^2 + 8z - 4) q^57 + (z^9 - z^8 - 2z^7 + 5z^6 - z^5 + z^4 - 5z^3 + 2z^2 + z - 1) * q^58 + (3z^8 + 3z^7 + 3z^6 + 2z^3 - 2) * q^59 + (-3z^8 + 6z^7 - 3z^6 + z^5 - 3z^4 + 6z^3 - 3z^2) * q^60 + (2z^9 - 2z^8 - 3z^6 + 5z^5 - 6z^4 + 4z^3 - 6z^2 + 5z - 3) * q^61 + (-z^8 - 3z^6 + 3z^5 - z^4 + 3z^3 - 3z^2 - 1) * q^62 + (-z^9 - 5z^7 + 2z^6 - z^5 + 2z^4 - 2z^3 + z^2 - 2z + 5) q^63 + (-4z^9 + 4z^8 - 3z^7 - z^6 - 3z^5 + 4z^4 - 4z^3 - 5z + 5) q^64 + (-6z^9 + 4z^8 - 4z^7 + 6z^6 + 4z^4 - 5z^3 + 4z^2 - 5z + 4) * q^65 + (-3z^9 + 5z^8 - 3z^7 - z^5 + z^4 + 3z^3 - 3z^2 - z + 1) q^66 + (-z^9 + 3z^8 - 2z^7 + 2z^6 - 3z^5 + z^4 + 7z^2 - 6z + 7) * q^67 + (-6z^9 + 4z^8 - 9z^7 + 11z^6 - 11z^5 + 9z^4 - 4z^3 + 6z^2 + 3) * q^68 + (-2z^9 + 3z^8 - 2z^7 + 2z^6 - 5z^5 + 2z^4 + 3z^3 - 2z^2 - 3z + 4) q^69 + (2z^9 + z^7 - z^6 + z^5 - z^4 - 2z^2 + 3) * q^70 + (z^9 - z^8 + 3z^7 - 3z^6 + z^5 - z^4 - 3z^2 + 3z - 3) * q^71 + (z^9 + 3z^8 + z^7 - 2z^5 + z^3 - z^2 + 2) * q^72 + (-4z^9 + 3z^8 - 3z^7 + 4z^6 + 4z^4 - 2z^3 - z^2 - 2z + 4) q^73 + (-2z^9 - 3z^8 + 4z^7 + 4z^5 - 3z^4 - 2z^3) * q^74 + (-2z^9 + 2z^7 + z^6 - 2z^5 + z^4 - z^3 + 2z^2 - z - 2) * q^75 + (-6z^8 + 6z^7 + 8z^5 - 10z^4 + 8z^3 + 6z - 6) * q^76 + (z^9 - z^8 - z^6 + z^5 - 4z^4 - z^3 - 4z^2 + z - 1) * q^77 + (z^8 - z^7 - z^3 + z^2) * q^78 + (2z^9 - 8z^8 + 6z^7 - 8z^6 + 2z^5 + 5z^3 - 2z^2 + 2z - 5) * q^79 + (6z^9 - z^8 - 2z^7 - 2z^6 + 2z^3 + 2z^2 + z - 6) q^80 + (2z^9 + 5z^8 - 4z^7 + z^6 - 3z^5 + 3z^4 - z^3 + 4z^2 - 5z - 2) q^81 + (-z^9 + 2z^8 - z^7 + 2z^6 - z^5 - 3z^3 + 2z^2 - 2z + 3) q^82 + (3z^9 - 3z^8 - z^7 - 2z^6 + 4z^5 - 2z^4 - z^3 - 3z^2 + 3z) q^83 + (3z^9 - 3z^8 - 3z^6 + 5z^5 - 6z^3 + 5z - 3) * q^84 + (-z^8 - z^7 - z^6 - 4z^5 + 2z^4 - 4z^3 - z^2 - z - 1) q^85 + (3z^7 - 3z^6 - 5z^4 + 5z^3 + 3z - 3) q^86 + (-3z^9 + 7z^8 - 2z^7 - z^6 - 2z^5 + 7z^4 - 3z^3) * q^87 + (5z^9 - 3z^8 + 3z^7 - 5z^6 + 2z^4 - z^3 + 5z^2 - z + 2) * q^88 + (11z^9 - 8z^8 + 11z^7 + 3z^5 - 7z^4 + 4z^3 - 4z^2 + 7z - 3) * q^89 + (-z^9 - 5z^8 - z^7 + z^6 + 5z^5 + z^4 - 3z^2 + z - 3) q^90 + (5z^9 - 4z^8 + 3z^7 - z^6 + z^5 - 3z^4 + 4z^3 - 5z^2 - 3) * q^91 + (4z^8 - z^7 - 2z^6 + z^5 + 7z^4 - 5z^3 + 8z^2 - 6z + 8) * q^92 + (z^9 - 2z^8 + 3z^7 - z^6 + z^5 - 3z^4 + 2z^3 - z^2 - 1) * q^93 + (4z^9 + 2z^7 - 2z^6 - 4z^4 + 5z) q^94 + (-4z^9 - 4z^7 - 2z^5 + 4z^4 + 2z^3 - 2z^2 - 4z + 2) q^95 + (-4z^9 + z^8 - z^7 + 4z^6 - 4z^4 + 5z^2 - 4) * q^96 + (-5z^9 + 2z^8 - 7z^7 + 6z^6 - 7z^5 + 2z^4 - 5z^3) q^97 + (3z^9 + 2z^7 - 6z^6 + 4z^5 - 6z^4 + 6z^3 - 4z^2 + 6z - 2) * q^98 + (-4z^8 + z^7 + z^6 + 4z^5 + 2z^4 + 4z^3 + z^2 + z - 4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 7 q^{2} - 7 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} - 5 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10}) \) 10 * q - 7 * q^2 - 7 * q^3 - 3 * q^4 - 3 * q^5 + 6 * q^6 - 5 * q^7 + 4 * q^8 - 2 * q^9	\( 10 q - 7 q^{2} - 7 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} - 5 q^{7} + 4 q^{8} - 2 q^{9} + q^{10} + 7 q^{11} + 12 q^{12} - 3 q^{13} + 9 q^{14} + 12 q^{15} + q^{16} - 10 q^{17} - 14 q^{18} + 2 q^{19} - 9 q^{20} - 2 q^{21} - 6 q^{22} - 12 q^{23} - 38 q^{24} - 4 q^{25} + 12 q^{26} - 4 q^{27} + 7 q^{28} + 14 q^{29} + 7 q^{30} + 10 q^{31} + 21 q^{32} + 16 q^{33} + 29 q^{34} + 7 q^{35} + 27 q^{36} - 19 q^{37} - 8 q^{38} + q^{39} + q^{40} + 7 q^{41} - 25 q^{42} - 11 q^{43} - 34 q^{44} - 6 q^{45} - 29 q^{46} - 18 q^{47} + 18 q^{48} - 18 q^{49} + 16 q^{50} + 7 q^{51} - 20 q^{52} + 29 q^{53} - 6 q^{54} - q^{55} - 2 q^{56} - 8 q^{57} - 23 q^{58} - 21 q^{59} + 25 q^{60} + 3 q^{61} + 4 q^{62} + 34 q^{63} + 24 q^{64} + 2 q^{65} + 2 q^{66} + 45 q^{67} - 30 q^{68} + 26 q^{69} + 38 q^{70} - 14 q^{71} + 19 q^{72} + 19 q^{73} + 10 q^{74} - 28 q^{75} - 16 q^{76} + 2 q^{77} - 4 q^{78} - 15 q^{79} - 52 q^{80} - 44 q^{81} + 16 q^{82} + 18 q^{83} - 17 q^{84} - 19 q^{85} - 11 q^{86} - 23 q^{87} + 27 q^{88} + 25 q^{89} - 20 q^{90} - 4 q^{91} + 52 q^{92} + 4 q^{93} + 17 q^{94} + 6 q^{95} - 51 q^{96} - 34 q^{97} + 17 q^{98} - 30 q^{99}+O(q^{100}) \) 10 * q - 7 * q^2 - 7 * q^3 - 3 * q^4 - 3 * q^5 + 6 * q^6 - 5 * q^7 + 4 * q^8 - 2 * q^9 + q^10 + 7 * q^11 + 12 * q^12 - 3 * q^13 + 9 * q^14 + 12 * q^15 + q^16 - 10 * q^17 - 14 * q^18 + 2 * q^19 - 9 * q^20 - 2 * q^21 - 6 * q^22 - 12 * q^23 - 38 * q^24 - 4 * q^25 + 12 * q^26 - 4 * q^27 + 7 * q^28 + 14 * q^29 + 7 * q^30 + 10 * q^31 + 21 * q^32 + 16 * q^33 + 29 * q^34 + 7 * q^35 + 27 * q^36 - 19 * q^37 - 8 * q^38 + q^39 + q^40 + 7 * q^41 - 25 * q^42 - 11 * q^43 - 34 * q^44 - 6 * q^45 - 29 * q^46 - 18 * q^47 + 18 * q^48 - 18 * q^49 + 16 * q^50 + 7 * q^51 - 20 * q^52 + 29 * q^53 - 6 * q^54 - q^55 - 2 * q^56 - 8 * q^57 - 23 * q^58 - 21 * q^59 + 25 * q^60 + 3 * q^61 + 4 * q^62 + 34 * q^63 + 24 * q^64 + 2 * q^65 + 2 * q^66 + 45 * q^67 - 30 * q^68 + 26 * q^69 + 38 * q^70 - 14 * q^71 + 19 * q^72 + 19 * q^73 + 10 * q^74 - 28 * q^75 - 16 * q^76 + 2 * q^77 - 4 * q^78 - 15 * q^79 - 52 * q^80 - 44 * q^81 + 16 * q^82 + 18 * q^83 - 17 * q^84 - 19 * q^85 - 11 * q^86 - 23 * q^87 + 27 * q^88 + 25 * q^89 - 20 * q^90 - 4 * q^91 + 52 * q^92 + 4 * q^93 + 17 * q^94 + 6 * q^95 - 51 * q^96 - 34 * q^97 + 17 * q^98 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	23.2.c.a

Level	$23$
Weight	$2$
Character orbit	23.c
Analytic conductor	$0.184$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.183655924649\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} + 7 T^{9} + \cdots + 1 \) T^10 + 7T^9 + 27T^8 + 68T^7 + 124T^6 + 142T^5 + 103T^4 + 28T^3 + 20T^2 + 8*T + 1
$3$	\( T^{10} + 7 T^{9} + \cdots + 1 \) T^10 + 7T^9 + 27T^8 + 68T^7 + 113T^6 + 131T^5 + 103T^4 + 17T^3 - 2T^2 - 3*T + 1
$5$	\( T^{10} + 3 T^{9} + \cdots + 529 \) T^10 + 3T^9 + 9T^8 - 6T^7 - 18T^6 - 32T^5 + 124T^4 - 233T^3 + 489T^2 - 667*T + 529
$7$	\( T^{10} + 5 T^{9} + \cdots + 529 \) T^10 + 5T^9 + 25T^8 + 81T^7 + 207T^6 + 298T^5 + 170T^4 - 448T^3 - 425T^2 - 46*T + 529
$11$	\( T^{10} - 7 T^{9} + \cdots + 529 \) T^10 - 7T^9 + 16T^8 + 31T^7 - 74T^6 - 813T^5 + 3282T^4 - 6111T^3 + 11317T^2 - 2208*T + 529
$13$	\( T^{10} + 3 T^{9} + \cdots + 1 \) T^10 + 3T^9 + 9T^8 + 71T^7 + 125T^6 + 166T^5 + 1290T^4 - 684T^3 + 159T^2 - 18*T + 1
$17$	\( T^{10} + 10 T^{9} + \cdots + 529 \) T^10 + 10T^9 + 56T^8 + 65T^7 - 274T^6 - 2003T^5 + 10880T^4 - 20197T^3 + 14521T^2 - 1541*T + 529
$19$	\( T^{10} - 2 T^{9} + \cdots + 541696 \) T^10 - 2T^9 - 40T^8 - 184T^7 + 544T^6 + 8416T^5 + 52864T^4 + 180096T^3 + 450816T^2 + 635904*T + 541696
$23$	\( T^{10} + 12 T^{9} + \cdots + 6436343 \) T^10 + 12T^9 - 10T^8 - 527T^7 - 32T^6 + 14103T^5 - 736T^4 - 278783T^3 - 121670T^2 + 3358092*T + 6436343
$29$	\( T^{10} - 14 T^{9} + \cdots + 4932841 \) T^10 - 14T^9 + 86T^8 + 6T^7 - 2394T^6 + 13100T^5 - 833T^4 - 184633T^3 + 1146117T^2 - 1896734*T + 4932841
$31$	\( T^{10} - 10 T^{9} + \cdots + 17161 \) T^10 - 10T^9 + 56T^8 - 164T^7 + 375T^6 - 505T^5 + 56T^4 + 3862T^3 - 1066T^2 + 2751*T + 17161
$37$	\( T^{10} + 19 T^{9} + \cdots + 529 \) T^10 + 19T^9 + 174T^8 + 1117T^7 + 6956T^6 + 38345T^5 + 148800T^4 + 346183T^3 + 370815T^2 + 4462*T + 529
$41$	\( T^{10} - 7 T^{9} + \cdots + 1849 \) T^10 - 7T^9 + 16T^8 + 9T^7 + 1235T^6 + 4181T^5 + 15173T^4 + 27978T^3 + 23747T^2 + 9331*T + 1849
$43$	\( T^{10} + 11 T^{9} + \cdots + 64009 \) T^10 + 11T^9 + 66T^8 + 341T^7 + 1826T^6 + 6875T^5 + 20449T^4 + 55297T^3 + 106843T^2 + 114103*T + 64009
$47$	\( (T^{5} + 9 T^{4} - 5 T^{3} + \cdots + 1)^{2} \) (T^5 + 9T^4 - 5T^3 - 97T^2 - 106T + 1)^2
$53$	\( T^{10} + \cdots + 517426009 \) T^10 - 29T^9 + 478T^8 - 5579T^7 + 55454T^6 - 448645T^5 + 2974063T^4 - 16104853T^3 + 83193185T^2 - 312339057*T + 517426009
$59$	\( T^{10} + 21 T^{9} + \cdots + 4489 \) T^10 + 21T^9 + 265T^8 + 1649T^7 + 3433T^6 - 22111T^5 - 98174T^4 - 33991T^3 + 1176352T^2 + 20569*T + 4489
$61$	\( T^{10} - 3 T^{9} + \cdots + 529 \) T^10 - 3T^9 + 218T^8 + 281T^7 + 7066T^6 + 46573T^5 + 215097T^4 + 359141T^3 + 236219T^2 - 22103*T + 529
$67$	\( T^{10} + \cdots + 1113757129 \) T^10 - 45T^9 + 1090T^8 - 17513T^7 + 203501T^6 - 1799216T^5 + 12627902T^4 - 70242899T^3 + 291205025T^2 - 798415652*T + 1113757129
$71$	\( T^{10} + 14 T^{9} + \cdots + 529 \) T^10 + 14T^9 + 130T^8 + 566T^7 + 785T^6 - 197T^5 + 1686T^4 + 1560T^3 + 1204T^2 + 851*T + 529
$73$	\( T^{10} - 19 T^{9} + \cdots + 982081 \) T^10 - 19T^9 + 251T^8 - 2415T^7 + 15063T^6 - 73963T^5 + 379118T^4 - 1529101T^3 + 3411556T^2 - 2470563*T + 982081
$79$	\( T^{10} + \cdots + 517426009 \) T^10 + 15T^9 + 115T^8 + 2715T^7 + 26304T^6 + 111816T^5 + 2966242T^4 + 21480574T^3 + 54168013T^2 - 24589507*T + 517426009
$83$	\( T^{10} - 18 T^{9} + \cdots + 529 \) T^10 - 18T^9 + 192T^8 - 321T^7 + 1455T^6 - 2914T^5 + 224T^4 - 17T^3 + 3804T^2 + 2346*T + 529
$89$	\( T^{10} + \cdots + 78310985281 \) T^10 - 25T^9 + 262T^8 - 5615T^7 + 111566T^6 - 1048455T^5 + 11505750T^4 - 150274617T^3 + 1491272689T^2 - 10362512230*T + 78310985281
$97$	\( T^{10} + 34 T^{9} + \cdots + 2374681 \) T^10 + 34T^9 + 430T^8 + 4423T^7 + 52812T^6 + 183261T^5 + 1071940T^4 + 3733313T^3 + 17156261T^2 + 10511161*T + 2374681
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\(n\)	\(5\)
\(\chi(n)\)	\(-\zeta_{22}\)