Properties

 Label 23.25.b.a Level $23$ Weight $25$ Character orbit 23.b Self dual yes Analytic conductor $83.942$ Analytic rank $0$ Dimension $1$ CM discriminant -23 Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,25,Mod(22,23)]

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([1]))

N = Newforms(chi, 25, names="a")

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

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 25);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$25$$ Character orbit: $$[\chi]$$ $$=$$ 23.b (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$83.9424450193$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 1951 q^{2} - 1062686 q^{3} - 12970815 q^{4} + 2073300386 q^{6} + 58038408481 q^{8} + 846871998115 q^{9}+O(q^{10})$$ q - 1951 * q^2 - 1062686 * q^3 - 12970815 * q^4 + 2073300386 * q^6 + 58038408481 * q^8 + 846871998115 * q^9 $$q - 1951 q^{2} - 1062686 q^{3} - 12970815 q^{4} + 2073300386 q^{6} + 58038408481 q^{8} + 846871998115 q^{9} + 13783903509090 q^{12} + 25363320370274 q^{13} + 104381230004609 q^{16} - 16\!\cdots\!65 q^{18}+ \cdots - 37\!\cdots\!51 q^{98}+O(q^{100})$$ q - 1951 * q^2 - 1062686 * q^3 - 12970815 * q^4 + 2073300386 * q^6 + 58038408481 * q^8 + 846871998115 * q^9 + 13783903509090 * q^12 + 25363320370274 * q^13 + 104381230004609 * q^16 - 1652247268322365 * q^18 + 21914624432020321 * q^23 - 61676604155039966 * q^24 + 59604644775390625 * q^25 - 49483838042404574 * q^26 - 599825101783988924 * q^27 - 647932355939762206 * q^29 + 1237087799571624194 * q^31 - 1177370695120961055 * q^32 - 10984620016230013725 * q^36 - 26953245471004995964 * q^39 + 12576527614080568514 * q^41 - 42755432266871646271 * q^46 + 192261621286365409154 * q^47 - 110924471788677919774 * q^48 + 191581231380566414401 * q^49 - 116288661956787109375 * q^50 - 328982936308555553310 * q^52 + 1170258773580562390724 * q^54 + 1264116026438476063906 * q^58 - 3163397976515704019038 * q^59 - 2413558296964238802494 * q^62 + 545823784047988829761 * q^64 - 23288364579165946842206 * q^69 + 2861536906024780963394 * q^71 + 49151102957719032013315 * q^72 + 3932686548449574639554 * q^73 - 63341021537780761718750 * q^75 + 52585781913930747125764 * q^78 + 398244072228062297956549 * q^81 - 24536805375071189170814 * q^82 + 688548643604202139645316 * q^87 - 284250539302215659931615 * q^92 - 1314635885375571028225084 * q^93 - 375102423129698913259454 * q^94 + 1251175354515313619693730 * q^96 - 373774982423485074496351 * q^98

Character values

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

 $$n$$ $$5$$ $$\chi(n)$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0
−1951.00 −1.06269e6 −1.29708e7 0 2.07330e9 0 5.80384e10 8.46872e11 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by $$\Q(\sqrt{-23})$$

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.25.b.a 1
23.b odd 2 1 CM 23.25.b.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.25.b.a 1 1.a even 1 1 trivial
23.25.b.a 1 23.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1951$$ acting on $$S_{25}^{\mathrm{new}}(23, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1951$$
$3$ $$T + 1062686$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 25363320370274$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 21\!\cdots\!21$$
$29$ $$T + 64\!\cdots\!06$$
$31$ $$T - 12\!\cdots\!94$$
$37$ $$T$$
$41$ $$T - 12\!\cdots\!14$$
$43$ $$T$$
$47$ $$T - 19\!\cdots\!54$$
$53$ $$T$$
$59$ $$T + 31\!\cdots\!38$$
$61$ $$T$$
$67$ $$T$$
$71$ $$T - 28\!\cdots\!94$$
$73$ $$T - 39\!\cdots\!54$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$