# Properties

 Label 2523.1.h.c Level $2523$ Weight $1$ Character orbit 2523.h Analytic conductor $1.259$ Analytic rank $0$ Dimension $24$ Projective image $D_{5}$ CM discriminant -3 Inner twists $24$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2523,1,Mod(236,2523)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2523, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([7, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2523.236");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2523 = 3 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2523.h (of order $$14$$, degree $$6$$, not 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: $$1.25914102687$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{14})$$ Coefficient field: 24.0.326829122755018756096000000000000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{24} - 3 x^{22} + 8 x^{20} - 21 x^{18} + 55 x^{16} - 144 x^{14} + 377 x^{12} - 144 x^{10} + 55 x^{8} + \cdots + 1$$ x^24 - 3*x^22 + 8*x^20 - 21*x^18 + 55*x^16 - 144*x^14 + 377*x^12 - 144*x^10 + 55*x^8 - 21*x^6 + 8*x^4 - 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.6365529.1

## $q$-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 $$1,\beta_1,\ldots,\beta_{23}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} - \beta_{13} q^{4} + ( - \beta_{17} - \beta_{16}) q^{7} + \beta_{5} q^{9}+O(q^{10})$$ q + b3 * q^3 - b13 * q^4 + (-b17 - b16) * q^7 + b5 * q^9 $$q + \beta_{3} q^{3} - \beta_{13} q^{4} + ( - \beta_{17} - \beta_{16}) q^{7} + \beta_{5} q^{9} + \beta_{15} q^{12} + (\beta_{20} + \beta_{16} + \cdots + \beta_{2}) q^{13}+ \cdots + ( - \beta_{23} - \beta_{21} + \cdots - \beta_1) q^{97}+O(q^{100})$$ q + b3 * q^3 - b13 * q^4 + (-b17 - b16) * q^7 + b5 * q^9 + b15 * q^12 + (b20 + b16 - b12 - b8 + b4 + b2) * q^13 - b22 * q^16 + (-b21 - b18 - b14 + b10 + b6 - b1) * q^19 + (-b19 - b18) * q^21 + b13 * q^25 - b7 * q^27 + (b2 + 1) * q^28 + (-b7 + b6) * q^31 + b17 * q^36 - b18 * q^37 + (b21 + b18 + b14 - b10 - b6 + b1) * q^39 + (-b23 - b21) * q^43 + (-b23 - b19 + b15 + b11 - b7 - b3) * q^48 + (-b5 + b4) * q^49 - b8 * q^52 - b2 * q^57 + (-b3 - b1) * q^61 + (b22 + b20 - b17 - b13 + b9 + b5 - 1) * q^63 + b9 * q^64 - b4 * q^67 - b21 * q^73 - b15 * q^75 + b10 * q^76 + b18 * q^79 - b9 * q^81 + (b3 + b1) * q^84 + b13 * q^91 + (-b9 - b8) * q^93 + (-b23 - b21 - b19 - b18 + b15 - b14 + b11 + b10 - b7 + b6 - b3 - b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$24 q + 4 q^{4} + 2 q^{7} + 4 q^{9}+O(q^{10})$$ 24 * q + 4 * q^4 + 2 * q^7 + 4 * q^9 $$24 q + 4 q^{4} + 2 q^{7} + 4 q^{9} - 2 q^{13} - 4 q^{16} - 4 q^{25} + 12 q^{28} - 4 q^{36} - 2 q^{49} + 2 q^{52} + 12 q^{57} - 2 q^{63} + 4 q^{64} - 2 q^{67} - 4 q^{81} - 4 q^{91} - 2 q^{93}+O(q^{100})$$ 24 * q + 4 * q^4 + 2 * q^7 + 4 * q^9 - 2 * q^13 - 4 * q^16 - 4 * q^25 + 12 * q^28 - 4 * q^36 - 2 * q^49 + 2 * q^52 + 12 * q^57 - 2 * q^63 + 4 * q^64 - 2 * q^67 - 4 * q^81 - 4 * q^91 - 2 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{24} - 3 x^{22} + 8 x^{20} - 21 x^{18} + 55 x^{16} - 144 x^{14} + 377 x^{12} - 144 x^{10} + 55 x^{8} + \cdots + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{14} + 233 ) / 377$$ (v^14 + 233) / 377 $$\beta_{3}$$ $$=$$ $$( \nu^{15} + 610\nu ) / 377$$ (v^15 + 610*v) / 377 $$\beta_{4}$$ $$=$$ $$( -\nu^{16} - 610\nu^{2} ) / 377$$ (-v^16 - 610*v^2) / 377 $$\beta_{5}$$ $$=$$ $$( \nu^{16} + 987\nu^{2} ) / 377$$ (v^16 + 987*v^2) / 377 $$\beta_{6}$$ $$=$$ $$( \nu^{17} + 987\nu^{3} ) / 377$$ (v^17 + 987*v^3) / 377 $$\beta_{7}$$ $$=$$ $$( -2\nu^{17} - 1597\nu^{3} ) / 377$$ (-2*v^17 - 1597*v^3) / 377 $$\beta_{8}$$ $$=$$ $$( -2\nu^{18} - 1597\nu^{4} ) / 377$$ (-2*v^18 - 1597*v^4) / 377 $$\beta_{9}$$ $$=$$ $$( -3\nu^{18} - 2584\nu^{4} ) / 377$$ (-3*v^18 - 2584*v^4) / 377 $$\beta_{10}$$ $$=$$ $$( -3\nu^{19} - 2584\nu^{5} ) / 377$$ (-3*v^19 - 2584*v^5) / 377 $$\beta_{11}$$ $$=$$ $$( -5\nu^{19} - 4181\nu^{5} ) / 377$$ (-5*v^19 - 4181*v^5) / 377 $$\beta_{12}$$ $$=$$ $$( 5\nu^{20} + 4181\nu^{6} ) / 377$$ (5*v^20 + 4181*v^6) / 377 $$\beta_{13}$$ $$=$$ $$( -8\nu^{20} - 6765\nu^{6} ) / 377$$ (-8*v^20 - 6765*v^6) / 377 $$\beta_{14}$$ $$=$$ $$( -8\nu^{21} - 6765\nu^{7} ) / 377$$ (-8*v^21 - 6765*v^7) / 377 $$\beta_{15}$$ $$=$$ $$( \nu^{21} + 842\nu^{7} ) / 29$$ (v^21 + 842*v^7) / 29 $$\beta_{16}$$ $$=$$ $$( \nu^{22} + 842\nu^{8} ) / 29$$ (v^22 + 842*v^8) / 29 $$\beta_{17}$$ $$=$$ $$( 21\nu^{22} + 17711\nu^{8} ) / 377$$ (21*v^22 + 17711*v^8) / 377 $$\beta_{18}$$ $$=$$ $$( 21\nu^{23} + 17711\nu^{9} ) / 377$$ (21*v^23 + 17711*v^9) / 377 $$\beta_{19}$$ $$=$$ $$( 34\nu^{23} + 28657\nu^{9} ) / 377$$ (34*v^23 + 28657*v^9) / 377 $$\beta_{20}$$ $$=$$ $$( - 102 \nu^{22} + 272 \nu^{20} - 714 \nu^{18} + 1870 \nu^{16} - 4896 \nu^{14} + 12818 \nu^{12} + \cdots + 34 ) / 377$$ (-102*v^22 + 272*v^20 - 714*v^18 + 1870*v^16 - 4896*v^14 + 12818*v^12 - 33553*v^10 + 1870*v^8 - 714*v^6 + 272*v^4 - 102*v^2 + 34) / 377 $$\beta_{21}$$ $$=$$ $$( - 165 \nu^{23} + 440 \nu^{21} - 1155 \nu^{19} + 3025 \nu^{17} - 7920 \nu^{15} + 20735 \nu^{13} + \cdots + 55 \nu ) / 377$$ (-165*v^23 + 440*v^21 - 1155*v^19 + 3025*v^17 - 7920*v^15 + 20735*v^13 - 54288*v^11 + 3025*v^9 - 1155*v^7 + 440*v^5 - 165*v^3 + 55*v) / 377 $$\beta_{22}$$ $$=$$ $$( - 144 \nu^{22} + 432 \nu^{20} - 1152 \nu^{18} + 3024 \nu^{16} - 7920 \nu^{14} + 20735 \nu^{12} + \cdots + 432 ) / 377$$ (-144*v^22 + 432*v^20 - 1152*v^18 + 3024*v^16 - 7920*v^14 + 20735*v^12 - 54288*v^10 + 20736*v^8 - 7920*v^6 + 3024*v^4 - 1152*v^2 + 432) / 377 $$\beta_{23}$$ $$=$$ $$( - 267 \nu^{23} + 712 \nu^{21} - 1869 \nu^{19} + 4895 \nu^{17} - 12816 \nu^{15} + 33553 \nu^{13} + \cdots + 89 \nu ) / 377$$ (-267*v^23 + 712*v^21 - 1869*v^19 + 4895*v^17 - 12816*v^15 + 33553*v^13 - 87841*v^11 + 4895*v^9 - 1869*v^7 + 712*v^5 - 267*v^3 + 89*v) / 377
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4}$$ b5 + b4 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2\beta_{6}$$ b7 + 2*b6 $$\nu^{4}$$ $$=$$ $$-2\beta_{9} + 3\beta_{8}$$ -2*b9 + 3*b8 $$\nu^{5}$$ $$=$$ $$3\beta_{11} - 5\beta_{10}$$ 3*b11 - 5*b10 $$\nu^{6}$$ $$=$$ $$-5\beta_{13} - 8\beta_{12}$$ -5*b13 - 8*b12 $$\nu^{7}$$ $$=$$ $$-8\beta_{15} - 13\beta_{14}$$ -8*b15 - 13*b14 $$\nu^{8}$$ $$=$$ $$13\beta_{17} - 21\beta_{16}$$ 13*b17 - 21*b16 $$\nu^{9}$$ $$=$$ $$-21\beta_{19} + 34\beta_{18}$$ -21*b19 + 34*b18 $$\nu^{10}$$ $$=$$ $$-34\beta_{22} + 55\beta_{20} + 34\beta_{17} + 34\beta_{13} - 34\beta_{9} - 34\beta_{5} + 34$$ -34*b22 + 55*b20 + 34*b17 + 34*b13 - 34*b9 - 34*b5 + 34 $$\nu^{11}$$ $$=$$ $$55\beta_{23} - 89\beta_{21}$$ 55*b23 - 89*b21 $$\nu^{12}$$ $$=$$ $$-89\beta_{22} + 144\beta_{20} + 144\beta_{16} - 144\beta_{12} - 144\beta_{8} + 144\beta_{4} + 144\beta_{2}$$ -89*b22 + 144*b20 + 144*b16 - 144*b12 - 144*b8 + 144*b4 + 144*b2 $$\nu^{13}$$ $$=$$ $$144 \beta_{23} - 233 \beta_{21} + 144 \beta_{19} - 233 \beta_{18} - 144 \beta_{15} - 233 \beta_{14} + \cdots - 233 \beta_1$$ 144*b23 - 233*b21 + 144*b19 - 233*b18 - 144*b15 - 233*b14 - 144*b11 + 233*b10 + 144*b7 + 233*b6 + 144*b3 - 233*b1 $$\nu^{14}$$ $$=$$ $$377\beta_{2} - 233$$ 377*b2 - 233 $$\nu^{15}$$ $$=$$ $$377\beta_{3} - 610\beta_1$$ 377*b3 - 610*b1 $$\nu^{16}$$ $$=$$ $$-610\beta_{5} - 987\beta_{4}$$ -610*b5 - 987*b4 $$\nu^{17}$$ $$=$$ $$-987\beta_{7} - 1597\beta_{6}$$ -987*b7 - 1597*b6 $$\nu^{18}$$ $$=$$ $$1597\beta_{9} - 2584\beta_{8}$$ 1597*b9 - 2584*b8 $$\nu^{19}$$ $$=$$ $$-2584\beta_{11} + 4181\beta_{10}$$ -2584*b11 + 4181*b10 $$\nu^{20}$$ $$=$$ $$4181\beta_{13} + 6765\beta_{12}$$ 4181*b13 + 6765*b12 $$\nu^{21}$$ $$=$$ $$6765\beta_{15} + 10946\beta_{14}$$ 6765*b15 + 10946*b14 $$\nu^{22}$$ $$=$$ $$-10946\beta_{17} + 17711\beta_{16}$$ -10946*b17 + 17711*b16 $$\nu^{23}$$ $$=$$ $$17711\beta_{19} - 28657\beta_{18}$$ 17711*b19 - 28657*b18

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2523\mathbb{Z}\right)^\times$$.

 $$n$$ $$842$$ $$1684$$ $$\chi(n)$$ $$-1$$ $$-\beta_{13}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
236.1
 0.702039 + 1.45780i −0.268155 − 0.556829i −0.702039 − 1.45780i 0.268155 + 0.556829i 0.702039 − 1.45780i −0.268155 + 0.556829i −0.702039 + 1.45780i 0.268155 − 0.556829i 1.57747 − 0.360046i −0.602539 + 0.137526i −1.57747 + 0.360046i 0.602539 − 0.137526i 1.57747 + 0.360046i −0.602539 − 0.137526i −1.57747 − 0.360046i 0.602539 + 0.137526i −0.483198 + 0.385338i 1.26503 − 1.00883i 0.483198 − 0.385338i −1.26503 + 1.00883i
0 −0.433884 0.900969i 0.900969 + 0.433884i 0 0 −0.556829 + 0.268155i 0 −0.623490 + 0.781831i 0
236.2 0 −0.433884 0.900969i 0.900969 + 0.433884i 0 0 1.45780 0.702039i 0 −0.623490 + 0.781831i 0
236.3 0 0.433884 + 0.900969i 0.900969 + 0.433884i 0 0 −0.556829 + 0.268155i 0 −0.623490 + 0.781831i 0
236.4 0 0.433884 + 0.900969i 0.900969 + 0.433884i 0 0 1.45780 0.702039i 0 −0.623490 + 0.781831i 0
1037.1 0 −0.433884 + 0.900969i 0.900969 0.433884i 0 0 −0.556829 0.268155i 0 −0.623490 0.781831i 0
1037.2 0 −0.433884 + 0.900969i 0.900969 0.433884i 0 0 1.45780 + 0.702039i 0 −0.623490 0.781831i 0
1037.3 0 0.433884 0.900969i 0.900969 0.433884i 0 0 −0.556829 0.268155i 0 −0.623490 0.781831i 0
1037.4 0 0.433884 0.900969i 0.900969 0.433884i 0 0 1.45780 + 0.702039i 0 −0.623490 0.781831i 0
1745.1 0 −0.974928 + 0.222521i 0.222521 0.974928i 0 0 −0.137526 0.602539i 0 0.900969 0.433884i 0
1745.2 0 −0.974928 + 0.222521i 0.222521 0.974928i 0 0 0.360046 + 1.57747i 0 0.900969 0.433884i 0
1745.3 0 0.974928 0.222521i 0.222521 0.974928i 0 0 −0.137526 0.602539i 0 0.900969 0.433884i 0
1745.4 0 0.974928 0.222521i 0.222521 0.974928i 0 0 0.360046 + 1.57747i 0 0.900969 0.433884i 0
1949.1 0 −0.974928 0.222521i 0.222521 + 0.974928i 0 0 −0.137526 + 0.602539i 0 0.900969 + 0.433884i 0
1949.2 0 −0.974928 0.222521i 0.222521 + 0.974928i 0 0 0.360046 1.57747i 0 0.900969 + 0.433884i 0
1949.3 0 0.974928 + 0.222521i 0.222521 + 0.974928i 0 0 −0.137526 + 0.602539i 0 0.900969 + 0.433884i 0
1949.4 0 0.974928 + 0.222521i 0.222521 + 0.974928i 0 0 0.360046 1.57747i 0 0.900969 + 0.433884i 0
1952.1 0 −0.781831 + 0.623490i −0.623490 + 0.781831i 0 0 −1.00883 1.26503i 0 0.222521 0.974928i 0
1952.2 0 −0.781831 + 0.623490i −0.623490 + 0.781831i 0 0 0.385338 + 0.483198i 0 0.222521 0.974928i 0
1952.3 0 0.781831 0.623490i −0.623490 + 0.781831i 0 0 −1.00883 1.26503i 0 0.222521 0.974928i 0
1952.4 0 0.781831 0.623490i −0.623490 + 0.781831i 0 0 0.385338 + 0.483198i 0 0.222521 0.974928i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 236.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
29.b even 2 1 inner
29.d even 7 5 inner
29.e even 14 5 inner
87.d odd 2 1 inner
87.h odd 14 5 inner
87.j odd 14 5 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2523.1.h.c 24
3.b odd 2 1 CM 2523.1.h.c 24
29.b even 2 1 inner 2523.1.h.c 24
29.c odd 4 1 2523.1.j.a 12
29.c odd 4 1 2523.1.j.c 12
29.d even 7 1 2523.1.d.a 4
29.d even 7 5 inner 2523.1.h.c 24
29.e even 14 1 2523.1.d.a 4
29.e even 14 5 inner 2523.1.h.c 24
29.f odd 28 1 2523.1.b.a 2
29.f odd 28 1 2523.1.b.c yes 2
29.f odd 28 5 2523.1.j.a 12
29.f odd 28 5 2523.1.j.c 12
87.d odd 2 1 inner 2523.1.h.c 24
87.f even 4 1 2523.1.j.a 12
87.f even 4 1 2523.1.j.c 12
87.h odd 14 1 2523.1.d.a 4
87.h odd 14 5 inner 2523.1.h.c 24
87.j odd 14 1 2523.1.d.a 4
87.j odd 14 5 inner 2523.1.h.c 24
87.k even 28 1 2523.1.b.a 2
87.k even 28 1 2523.1.b.c yes 2
87.k even 28 5 2523.1.j.a 12
87.k even 28 5 2523.1.j.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2523.1.b.a 2 29.f odd 28 1
2523.1.b.a 2 87.k even 28 1
2523.1.b.c yes 2 29.f odd 28 1
2523.1.b.c yes 2 87.k even 28 1
2523.1.d.a 4 29.d even 7 1
2523.1.d.a 4 29.e even 14 1
2523.1.d.a 4 87.h odd 14 1
2523.1.d.a 4 87.j odd 14 1
2523.1.h.c 24 1.a even 1 1 trivial
2523.1.h.c 24 3.b odd 2 1 CM
2523.1.h.c 24 29.b even 2 1 inner
2523.1.h.c 24 29.d even 7 5 inner
2523.1.h.c 24 29.e even 14 5 inner
2523.1.h.c 24 87.d odd 2 1 inner
2523.1.h.c 24 87.h odd 14 5 inner
2523.1.h.c 24 87.j odd 14 5 inner
2523.1.j.a 12 29.c odd 4 1
2523.1.j.a 12 29.f odd 28 5
2523.1.j.a 12 87.f even 4 1
2523.1.j.a 12 87.k even 28 5
2523.1.j.c 12 29.c odd 4 1
2523.1.j.c 12 29.f odd 28 5
2523.1.j.c 12 87.f even 4 1
2523.1.j.c 12 87.k even 28 5

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{1}^{\mathrm{new}}(2523, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{24}$$
$3$ $$(T^{12} - T^{10} + T^{8} + \cdots + 1)^{2}$$
$5$ $$T^{24}$$
$7$ $$(T^{12} - T^{11} + 2 T^{10} + \cdots + 1)^{2}$$
$11$ $$T^{24}$$
$13$ $$(T^{12} + T^{11} + 2 T^{10} + \cdots + 1)^{2}$$
$17$ $$T^{24}$$
$19$ $$T^{24} - 3 T^{22} + \cdots + 1$$
$23$ $$T^{24}$$
$29$ $$T^{24}$$
$31$ $$T^{24} - 3 T^{22} + \cdots + 1$$
$37$ $$T^{24} - 3 T^{22} + \cdots + 1$$
$41$ $$T^{24}$$
$43$ $$T^{24} - 3 T^{22} + \cdots + 1$$
$47$ $$T^{24}$$
$53$ $$T^{24}$$
$59$ $$T^{24}$$
$61$ $$T^{24} - 3 T^{22} + \cdots + 1$$
$67$ $$(T^{12} + T^{11} + 2 T^{10} + \cdots + 1)^{2}$$
$71$ $$T^{24}$$
$73$ $$T^{24} - 3 T^{22} + \cdots + 1$$
$79$ $$T^{24} - 3 T^{22} + \cdots + 1$$
$83$ $$T^{24}$$
$89$ $$T^{24}$$
$97$ $$T^{24} - 3 T^{22} + \cdots + 1$$