# Properties

 Label 100.4.e.f Level $100$ Weight $4$ Character orbit 100.e Analytic conductor $5.900$ Analytic rank $0$ Dimension $24$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 100.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.90019100057$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 36 q^{6}+O(q^{10})$$ 24 * q + 36 * q^6 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 36 q^{6} - 676 q^{16} + 512 q^{21} + 2072 q^{26} - 4600 q^{36} - 392 q^{41} + 5016 q^{46} - 8224 q^{56} + 1088 q^{61} + 11140 q^{66} - 6700 q^{76} - 2424 q^{81} + 5216 q^{86} + 796 q^{96}+O(q^{100})$$ 24 * q + 36 * q^6 - 676 * q^16 + 512 * q^21 + 2072 * q^26 - 4600 * q^36 - 392 * q^41 + 5016 * q^46 - 8224 * q^56 + 1088 * q^61 + 11140 * q^66 - 6700 * q^76 - 2424 * q^81 + 5216 * q^86 + 796 * q^96

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −2.66990 0.933623i −3.69377 3.69377i 6.25670 + 4.98535i 0 6.41339 + 13.3106i 19.3489 19.3489i −12.0503 19.1518i 0.287847i 0
7.2 −2.38148 + 1.52596i −0.243692 0.243692i 3.34287 7.26809i 0 0.952213 + 0.208482i −9.53656 + 9.53656i 3.12986 + 22.4099i 26.8812i 0
7.3 −2.21943 1.75332i 6.14790 + 6.14790i 1.85174 + 7.78274i 0 −2.86560 24.4241i 16.4522 16.4522i 9.53582 20.5199i 48.5934i 0
7.4 −1.75332 2.21943i −6.14790 6.14790i −1.85174 + 7.78274i 0 −2.86560 + 24.4241i −16.4522 + 16.4522i 20.5199 9.53582i 48.5934i 0
7.5 −1.52596 + 2.38148i −0.243692 0.243692i −3.34287 7.26809i 0 0.952213 0.208482i −9.53656 + 9.53656i 22.4099 + 3.12986i 26.8812i 0
7.6 −0.933623 2.66990i 3.69377 + 3.69377i −6.25670 + 4.98535i 0 6.41339 13.3106i −19.3489 + 19.3489i 19.1518 + 12.0503i 0.287847i 0
7.7 0.933623 + 2.66990i −3.69377 3.69377i −6.25670 + 4.98535i 0 6.41339 13.3106i 19.3489 19.3489i −19.1518 12.0503i 0.287847i 0
7.8 1.52596 2.38148i 0.243692 + 0.243692i −3.34287 7.26809i 0 0.952213 0.208482i 9.53656 9.53656i −22.4099 3.12986i 26.8812i 0
7.9 1.75332 + 2.21943i 6.14790 + 6.14790i −1.85174 + 7.78274i 0 −2.86560 + 24.4241i 16.4522 16.4522i −20.5199 + 9.53582i 48.5934i 0
7.10 2.21943 + 1.75332i −6.14790 6.14790i 1.85174 + 7.78274i 0 −2.86560 24.4241i −16.4522 + 16.4522i −9.53582 + 20.5199i 48.5934i 0
7.11 2.38148 1.52596i 0.243692 + 0.243692i 3.34287 7.26809i 0 0.952213 + 0.208482i 9.53656 9.53656i −3.12986 22.4099i 26.8812i 0
7.12 2.66990 + 0.933623i 3.69377 + 3.69377i 6.25670 + 4.98535i 0 6.41339 + 13.3106i −19.3489 + 19.3489i 12.0503 + 19.1518i 0.287847i 0
43.1 −2.66990 + 0.933623i −3.69377 + 3.69377i 6.25670 4.98535i 0 6.41339 13.3106i 19.3489 + 19.3489i −12.0503 + 19.1518i 0.287847i 0
43.2 −2.38148 1.52596i −0.243692 + 0.243692i 3.34287 + 7.26809i 0 0.952213 0.208482i −9.53656 9.53656i 3.12986 22.4099i 26.8812i 0
43.3 −2.21943 + 1.75332i 6.14790 6.14790i 1.85174 7.78274i 0 −2.86560 + 24.4241i 16.4522 + 16.4522i 9.53582 + 20.5199i 48.5934i 0
43.4 −1.75332 + 2.21943i −6.14790 + 6.14790i −1.85174 7.78274i 0 −2.86560 24.4241i −16.4522 16.4522i 20.5199 + 9.53582i 48.5934i 0
43.5 −1.52596 2.38148i −0.243692 + 0.243692i −3.34287 + 7.26809i 0 0.952213 + 0.208482i −9.53656 9.53656i 22.4099 3.12986i 26.8812i 0
43.6 −0.933623 + 2.66990i 3.69377 3.69377i −6.25670 4.98535i 0 6.41339 + 13.3106i −19.3489 19.3489i 19.1518 12.0503i 0.287847i 0
43.7 0.933623 2.66990i −3.69377 + 3.69377i −6.25670 4.98535i 0 6.41339 + 13.3106i 19.3489 + 19.3489i −19.1518 + 12.0503i 0.287847i 0
43.8 1.52596 + 2.38148i 0.243692 0.243692i −3.34287 + 7.26809i 0 0.952213 + 0.208482i 9.53656 + 9.53656i −22.4099 + 3.12986i 26.8812i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.4.e.f 24
4.b odd 2 1 inner 100.4.e.f 24
5.b even 2 1 inner 100.4.e.f 24
5.c odd 4 2 inner 100.4.e.f 24
20.d odd 2 1 inner 100.4.e.f 24
20.e even 4 2 inner 100.4.e.f 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.e.f 24 1.a even 1 1 trivial
100.4.e.f 24 4.b odd 2 1 inner
100.4.e.f 24 5.b even 2 1 inner
100.4.e.f 24 5.c odd 4 2 inner
100.4.e.f 24 20.d odd 2 1 inner
100.4.e.f 24 20.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 6459T_{3}^{8} + 4255155T_{3}^{4} + 60025$$ acting on $$S_{4}^{\mathrm{new}}(100, [\chi])$$.