For a newform $f \in S_k^{\rm new}(\Gamma_1(N))$, its **trace form** $\mathrm{Tr}(f)$ is the sum of its distinct conjugates under $\mathrm{Aut}(\C)$ (equivalently, the sum under all embeddings of the coefficient field into $\C$). The trace form is a modular form $\mathrm{Tr}(f) \in S_k^{\rm new}(\Gamma_1(N))$ whose $q$-expansion has integral coefficients $a_n(\mathrm{Tr}(f)) \in \Z$.

The coefficient $a_1$ is equal to the dimension of the newform.

For $p$ prime, the coefficient $a_p$ is the trace of Frobenius in the direct sum of the $\ell$-adic Galois representations attached to the conjugates of $f$ (for any prime $\ell$). When $f$ has weight $k=2$, the coefficient $a_p(f)$ is the trace of Frobenius acting on the modular abelian variety associated to $f$.

For a newspace $S_k^{\rm new}(N,\chi)$, its trace form is the sum of the trace forms $\mathrm{Tr}(f)$ over all newforms $f\in S_k^{\rm new}(N,k)$; it is also a modular form in $S_k^{\rm new}(\Gamma_1(N))$.

The graphical plot displayed in the properties box on the home page of each newform or newspace is computed using the trace form.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2019-03-08 15:44:50

**Referred to by:**

- cmf.picture_description
- portrait.cmf
- rcs.cande.cmf
- rcs.rigor.cmf
- rcs.source.cmf
- lmfdb/classical_modular_forms/main.py (line 853)
- lmfdb/classical_modular_forms/templates/cmf_newform_common.html (lines 203-205)
- lmfdb/classical_modular_forms/templates/cmf_newform_common.html (line 242)
- lmfdb/classical_modular_forms/templates/cmf_trace_search_results.html (line 8)

**History:**(expand/hide all)

- 2019-03-08 15:44:50 by Andrew Sutherland (Reviewed)