# Classical modular forms downloaded from the LMFDB on 07 August 2026.
# Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/230/
# Query "{'level': 230}" returned 91 forms, sorted by analytic conductor.

# Each entry in the following data list has the form:
#    [Label, Dim, $A$, Field, CM, Traces, Fricke sign, $q$-expansion]
# For more details, see the definitions at the bottom of the file.



"230.2.a.a"	2	1.836559246489449	"2.2.21.1"	[]	[-2, -1, -2, 1]	-1	"q-q^{2}-\\beta q^{3}+q^{4}-q^{5}+\\beta q^{6}+(1+\\cdots)q^{7}+\\cdots"
"230.2.a.b"	2	1.836559246489449	"2.2.13.1"	[]	[-2, 3, 2, 3]	-1	"q-q^{2}+(1+\\beta )q^{3}+q^{4}+q^{5}+(-1+\\cdots)q^{6}+\\cdots"
"230.2.a.c"	2	1.836559246489449	"2.2.5.1"	[]	[2, 1, 2, 1]	-1	"q+q^{2}+\\beta q^{3}+q^{4}+q^{5}+\\beta q^{6}+(1+\\cdots)q^{7}+\\cdots"
"230.2.a.d"	3	1.836559246489449	"3.3.1101.1"	[]	[3, 1, -3, 3]	-1	"q+q^{2}+\\beta _{1}q^{3}+q^{4}-q^{5}+\\beta _{1}q^{6}+\\cdots"
"230.2.b.a"	4	1.836559246489449	"4.0.400.1"	[]	[0, 0, 0, 0]	NULL	"q-\\beta _{3}q^{2}+\\beta _{1}q^{3}-q^{4}+(1+2\\beta _{2})q^{5}+\\cdots"
"230.2.b.b"	8	1.836559246489449	"8.0.11574317056.3"	[]	[0, 0, 0, 0]	NULL	"q-\\beta _{2}q^{2}+(\\beta _{2}-\\beta _{5})q^{3}-q^{4}+(\\beta _{1}-\\beta _{6}+\\cdots)q^{5}+\\cdots"
"230.2.e.a"	8	1.836559246489449	"8.0.110166016.2"	[]	[0, -4, -4, 0]	NULL	"q-\\beta _{3}q^{2}+(-1-\\beta _{2}-\\beta _{6}-\\beta _{7})q^{3}+\\cdots"
"230.2.e.b"	8	1.836559246489449	"8.0.110166016.2"	[]	[0, -4, 4, 0]	NULL	"q-\\beta _{3}q^{2}+(-1-\\beta _{2}-\\beta _{6}-\\beta _{7})q^{3}+\\cdots"
"230.2.e.c"	8	1.836559246489449	"8.0.16777216.1"	[]	[0, 16, 0, 0]	NULL	"q+\\zeta_{16}^{6}q^{2}+(2+2\\zeta_{16}^{4})q^{3}-\\zeta_{16}^{4}q^{4}+\\cdots"
"230.2.g.a"	10	1.836559246489449	"10.0.2357947691.1"	[]	[-1, 3, -1, -15]	NULL	"q+\\zeta_{22}^{4}q^{2}+(1-\\zeta_{22}+\\zeta_{22}^{2}+\\zeta_{22}^{4}+\\cdots)q^{3}+\\cdots"
"230.2.g.b"	20	1.836559246489449	NULL	[]	[2, -3, -2, 19]	NULL	"q+\\beta _{6}q^{2}+(\\beta _{1}-\\beta _{2}+\\beta _{5}-\\beta _{13}+\\beta _{18}+\\cdots)q^{3}+\\cdots"
"230.2.g.c"	20	1.836559246489449	NULL	[]	[2, 1, 2, -12]	NULL	"q+\\beta _{3}q^{2}+(\\beta _{5}+\\beta _{18})q^{3}+\\beta _{12}q^{4}+\\cdots"
"230.2.g.d"	30	1.836559246489449	NULL	[]	[-3, -1, 3, 8]	NULL	NULL
"230.2.j.a"	120	1.836559246489449	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"230.2.l.a"	240	1.836559246489449	NULL	[]	[0, -8, 0, 0]	NULL	NULL
"230.3.c.a"	24	6.267046080286828	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"230.3.d.a"	16	6.267046080286828	NULL	[]	[0, 0, 0, 0]	NULL	"q-\\beta _{6}q^{2}-\\beta _{5}q^{3}+2q^{4}+\\beta _{2}q^{5}-\\beta _{10}q^{6}+\\cdots"
"230.3.f.a"	20	6.267046080286828	NULL	[]	[-20, 0, 4, 8]	NULL	"q+(-1-\\beta _{8})q^{2}+\\beta _{1}q^{3}+2\\beta _{8}q^{4}+\\cdots"
"230.3.f.b"	24	6.267046080286828	NULL	[]	[24, 0, 4, 8]	NULL	NULL
"230.3.h.a"	160	6.267046080286828	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"230.3.i.a"	240	6.267046080286828	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"230.3.k.a"	240	6.267046080286828	NULL	[]	[-24, 0, -4, -74]	NULL	NULL
"230.3.k.b"	240	6.267046080286828	NULL	[]	[24, 8, -4, 50]	NULL	NULL
"230.4.a.a"	1	13.570439301320341	"1.1.1.1"	[]	[-2, -5, -5, 12]	-1	"q-2q^{2}-5q^{3}+4q^{4}-5q^{5}+10q^{6}+\\cdots"
"230.4.a.b"	1	13.570439301320341	"1.1.1.1"	[]	[-2, 4, -5, 3]	-1	"q-2q^{2}+4q^{3}+4q^{4}-5q^{5}-8q^{6}+\\cdots"
"230.4.a.c"	1	13.570439301320341	"1.1.1.1"	[]	[-2, 7, 5, 20]	1	"q-2q^{2}+7q^{3}+4q^{4}+5q^{5}-14q^{6}+\\cdots"
"230.4.a.d"	1	13.570439301320341	"1.1.1.1"	[]	[2, -1, 5, -32]	-1	"q+2q^{2}-q^{3}+4q^{4}+5q^{5}-2q^{6}+\\cdots"
"230.4.a.e"	1	13.570439301320341	"1.1.1.1"	[]	[2, 1, -5, -18]	-1	"q+2q^{2}+q^{3}+4q^{4}-5q^{5}+2q^{6}+\\cdots"
"230.4.a.f"	2	13.570439301320341	"2.2.73.1"	[]	[-4, -3, 10, -17]	-1	"q-2q^{2}+(-1-\\beta )q^{3}+4q^{4}+5q^{5}+\\cdots"
"230.4.a.g"	3	13.570439301320341	"3.3.318165.1"	[]	[-6, -1, 15, 7]	1	"q-2q^{2}-\\beta _{1}q^{3}+4q^{4}+5q^{5}+2\\beta _{1}q^{6}+\\cdots"
"230.4.a.h"	4	13.570439301320341	NULL	[]	[-8, -4, -20, -1]	1	"q-2q^{2}+(-1+\\beta _{1})q^{3}+4q^{4}-5q^{5}+\\cdots"
"230.4.a.i"	4	13.570439301320341	NULL	[]	[8, 4, -20, 26]	1	"q+2q^{2}+(1-\\beta _{1})q^{3}+4q^{4}-5q^{5}+\\cdots"
"230.4.a.j"	4	13.570439301320341	NULL	[]	[8, 14, 20, 8]	1	"q+2q^{2}+(4-\\beta _{1})q^{3}+4q^{4}+5q^{5}+\\cdots"
"230.4.b.a"	14	13.570439301320341	NULL	[]	[0, 0, -6, 0]	NULL	"q-2\\beta _{8}q^{2}+(\\beta _{1}-\\beta _{8})q^{3}-4q^{4}+(-\\beta _{7}+\\cdots)q^{5}+\\cdots"
"230.4.b.b"	18	13.570439301320341	NULL	[]	[0, 0, -6, 0]	NULL	"q-2\\beta _{5}q^{2}+\\beta _{1}q^{3}-4q^{4}+(-\\beta _{5}-\\beta _{12}+\\cdots)q^{5}+\\cdots"
"230.4.e.a"	72	13.570439301320341	NULL	[]	[0, -16, 0, 0]	NULL	NULL
"230.4.g.a"	50	13.570439301320341	NULL	[]	[-10, -5, 25, -19]	NULL	NULL
"230.4.g.b"	60	13.570439301320341	NULL	[]	[12, -3, -30, 100]	NULL	NULL
"230.4.g.c"	60	13.570439301320341	NULL	[]	[12, 5, 30, -3]	NULL	NULL
"230.4.g.d"	70	13.570439301320341	NULL	[]	[-14, 3, -35, -78]	NULL	NULL
"230.4.j.a"	360	13.570439301320341	NULL	[]	[0, 0, 32, 0]	NULL	NULL
"230.4.l.a"	720	13.570439301320341	NULL	[]	[0, 16, 0, 0]	NULL	NULL
"230.5.c.a"	48	23.775091509284195	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"230.5.d.a"	32	23.775091509284195	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"230.5.f.a"	44	23.775091509284195	NULL	[]	[-88, 0, 24, -80]	NULL	NULL
"230.5.f.b"	44	23.775091509284195	NULL	[]	[88, 0, 24, -80]	NULL	NULL
"230.6.a.a"	1	36.88827855698555	"1.1.1.1"	[]	[-4, 8, -25, 199]	1	"q-4q^{2}+8q^{3}+2^{4}q^{4}-5^{2}q^{5}-2^{5}q^{6}+\\cdots"
"230.6.a.b"	2	36.88827855698555	"2.2.8.1"	[]	[-8, 6, -50, -164]	1	"q-4q^{2}+(3+6\\beta )q^{3}+2^{4}q^{4}-5^{2}q^{5}+\\cdots"
"230.6.a.c"	3	36.88827855698555	"3.3.27980.1"	[]	[-12, -26, 75, 1]	1	"q-4q^{2}+(-9+\\beta _{2})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\\cdots"
"230.6.a.d"	3	36.88827855698555	"3.3.27980.1"	[]	[12, -34, 75, -121]	1	"q+4q^{2}+(-11-\\beta _{2})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\\cdots"
"230.6.a.e"	3	36.88827855698555	NULL	[]	[12, 6, -75, 5]	1	"q+4q^{2}+(2-\\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\\cdots"
"230.6.a.f"	5	36.88827855698555	NULL	[]	[-20, 1, 125, 102]	-1	"q-4q^{2}+\\beta _{1}q^{3}+2^{4}q^{4}+5^{2}q^{5}-4\\beta _{1}q^{6}+\\cdots"
"230.6.a.g"	5	36.88827855698555	NULL	[]	[-20, 5, -125, 130]	-1	"q-4q^{2}+(1+\\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\\cdots"
"230.6.a.h"	6	36.88827855698555	NULL	[]	[24, 11, 150, 366]	-1	"q+4q^{2}+(2-\\beta _{1})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\\cdots"
"230.6.a.i"	6	36.88827855698555	NULL	[]	[24, 15, -150, 106]	-1	"q+4q^{2}+(2+\\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\\cdots"
"230.6.b.a"	26	36.88827855698555	NULL	[]	[0, 0, -30, 0]	NULL	NULL
"230.6.b.b"	30	36.88827855698555	NULL	[]	[0, 0, -30, 0]	NULL	NULL
"230.7.c.a"	72	52.91243923256573	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"230.7.d.a"	48	52.91243923256573	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"230.8.a.a"	5	71.848555861348	NULL	[]	[40, -103, 625, -1404]	-1	"q+8q^{2}+(-21+\\beta _{1})q^{3}+2^{6}q^{4}+5^{3}q^{5}+\\cdots"
"230.8.a.b"	5	71.848555861348	NULL	[]	[40, -35, -625, -480]	-1	"q+8q^{2}+(-7-\\beta _{1})q^{3}+2^{6}q^{4}-5^{3}q^{5}+\\cdots"
"230.8.a.c"	6	71.848555861348	NULL	[]	[-48, -33, 750, -672]	-1	"q-8q^{2}+(-5-\\beta _{1})q^{3}+2^{6}q^{4}+5^{3}q^{5}+\\cdots"
"230.8.a.d"	6	71.848555861348	NULL	[]	[-48, 35, -750, -292]	-1	"q-8q^{2}+(6-\\beta _{1})q^{3}+2^{6}q^{4}-5^{3}q^{5}+\\cdots"
"230.8.a.e"	8	71.848555861348	NULL	[]	[-64, 8, -1000, -363]	1	"q-8q^{2}+(1-\\beta _{1})q^{3}+2^{6}q^{4}-5^{3}q^{5}+\\cdots"
"230.8.a.f"	8	71.848555861348	NULL	[]	[-64, 48, 1000, 771]	1	"q-8q^{2}+(6+\\beta _{1})q^{3}+2^{6}q^{4}+5^{3}q^{5}+\\cdots"
"230.8.a.g"	8	71.848555861348	NULL	[]	[64, -8, -1000, 963]	1	"q+8q^{2}+(-1+\\beta _{1})q^{3}+2^{6}q^{4}-5^{3}q^{5}+\\cdots"
"230.8.a.h"	8	71.848555861348	NULL	[]	[64, 32, 1000, 1269]	1	"q+8q^{2}+(4-\\beta _{1})q^{3}+2^{6}q^{4}+5^{3}q^{5}+\\cdots"
"230.8.b.a"	36	71.848555861348	NULL	[]	[0, 0, 84, 0]	NULL	NULL
"230.8.b.b"	40	71.848555861348	NULL	[]	[0, 0, 84, 0]	NULL	NULL
"230.9.c.a"	96	93.69708031405926	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"230.9.d.a"	64	93.69708031405926	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"230.10.a.a"	7	118.45824231769082	NULL	[]	[-112, -22, -4375, 4897]	1	"q-2^{4}q^{2}+(-3-\\beta _{1})q^{3}+2^{8}q^{4}-5^{4}q^{5}+\\cdots"
"230.10.a.b"	7	118.45824231769082	NULL	[]	[-112, 28, 4375, -5795]	1	"q-2^{4}q^{2}+(4-\\beta _{1})q^{3}+2^{8}q^{4}+5^{4}q^{5}+\\cdots"
"230.10.a.c"	7	118.45824231769082	NULL	[]	[112, -198, -4375, -10112]	1	"q+2^{4}q^{2}+(-28-\\beta _{1})q^{3}+2^{8}q^{4}+\\cdots"
"230.10.a.d"	7	118.45824231769082	NULL	[]	[112, -148, 4375, -7970]	1	"q+2^{4}q^{2}+(-21-\\beta _{1})q^{3}+2^{8}q^{4}+\\cdots"
"230.10.a.e"	9	118.45824231769082	NULL	[]	[-144, -103, -5625, 881]	-1	"q-2^{4}q^{2}+(-11-\\beta _{1})q^{3}+2^{8}q^{4}+\\cdots"
"230.10.a.f"	9	118.45824231769082	NULL	[]	[-144, 271, 5625, 7825]	-1	"q-2^{4}q^{2}+(30+\\beta _{1})q^{3}+2^{8}q^{4}+5^{4}q^{5}+\\cdots"
"230.10.a.g"	10	118.45824231769082	NULL	[]	[160, -117, -6250, 3508]	-1	"q+2^{4}q^{2}+(-12+\\beta _{1})q^{3}+2^{8}q^{4}+\\cdots"
"230.10.a.h"	10	118.45824231769082	NULL	[]	[160, 257, 6250, 7222]	-1	"q+2^{4}q^{2}+(26-\\beta _{1})q^{3}+2^{8}q^{4}+5^{4}q^{5}+\\cdots"
"230.10.b.a"	48	118.45824231769082	NULL	[]	[0, 0, 420, 0]	NULL	NULL
"230.10.b.b"	52	118.45824231769082	NULL	[]	[0, 0, 420, 0]	NULL	NULL
"230.11.d.a"	80	146.13216811495653	NULL	[]	[0, 0, 0, 0]	NULL	NULL
"230.12.a.a"	8	176.71893152870544	NULL	[]	[256, -805, 25000, -76141]	-1	"q+2^{5}q^{2}+(-101+\\beta _{1})q^{3}+2^{10}q^{4}+\\cdots"
"230.12.a.b"	8	176.71893152870544	NULL	[]	[256, -233, -25000, 32317]	-1	"q+2^{5}q^{2}+(-29-\\beta _{1})q^{3}+2^{10}q^{4}+\\cdots"
"230.12.a.c"	9	176.71893152870544	NULL	[]	[-288, -759, 28125, -20503]	-1	"q-2^{5}q^{2}+(-84-\\beta _{1})q^{3}+2^{10}q^{4}+\\cdots"
"230.12.a.d"	9	176.71893152870544	NULL	[]	[-288, -187, -28125, -24905]	-1	"q-2^{5}q^{2}+(-21+\\beta _{1})q^{3}+2^{10}q^{4}+\\cdots"
"230.12.a.e"	11	176.71893152870544	NULL	[]	[-352, -430, -34375, 14718]	1	"q-2^{5}q^{2}+(-39-\\beta _{1})q^{3}+2^{10}q^{4}+\\cdots"
"230.12.a.f"	11	176.71893152870544	NULL	[]	[-352, -30, 34375, 7102]	1	"q-2^{5}q^{2}+(-3+\\beta _{1})q^{3}+2^{10}q^{4}+\\cdots"
"230.12.a.g"	11	176.71893152870544	NULL	[]	[352, 10, -34375, 59922]	1	"q+2^{5}q^{2}+(1-\\beta _{1})q^{3}+2^{10}q^{4}-5^{5}q^{5}+\\cdots"
"230.12.a.h"	11	176.71893152870544	NULL	[]	[352, 410, 34375, 97938]	1	"q+2^{5}q^{2}+(37+\\beta _{1})q^{3}+2^{10}q^{4}+5^{5}q^{5}+\\cdots"
"230.13.d.a"	96	210.21857797354568	NULL	[]	[0, 0, 0, 0]	NULL	NULL


# Label --
#    The **label** of a newform $f\in S_k^{\rm new}(N,\chi)$ has the format \( N.k.a.x \), where

#    -  \( N\) is the level;

#    - \(k\) is the weight;

#    - \(N.a\) is the label of the Galois orbit of the Dirichlet character $\chi$;

#    - \(x\) is the label of the Galois orbit of the newform $f$.

#    For each embedding of the coefficient field of $f$ into the complex numbers, the corresponding modular form over $\C$ has a label of the form \(N.k.a.x.n.i\), where

#    - \(n\) determines the Conrey label \(N.n\) of the Dirichlet character \(\chi\);

#    - \(i\) is an integer ranging from 1 to the relative dimension of the newform that distinguishes embeddings with the same character $\chi$.


# Dim --
#    The **dimension** of a space of modular forms is its dimension as a complex vector space; for spaces of newforms $S_k^{\rm new}(N,\chi)$ this is the same as the dimension of the $\Q$-vector space spanned by its eigenforms.

#    The **dimension** of a newform refers to the dimension of its newform subspace, equivalently, the cardinality of its newform orbit.  This is equal to the degree of its coefficient field (as an extension of $\Q$).

#    The **relative dimension** of $S_k^{\rm new}(N,\chi)$  is its dimension as a $\Q(\chi)$-vector space, where $\Q(\chi)$ is the field generated by the values of $\chi$, and similarly for newform subspaces.


#$A$ (analytic_conductor) --
#    The **analytic conductor** of a newform $f \in S_k^{\mathrm{new}}(N,\chi)$ is the positive real number
#    \[
#    N\left(\frac{\exp(\psi(k/2))}{2\pi}\right)^2,
#    \]
#    where $\psi(x):=\Gamma'(x)/\Gamma(x)$ is the logarithmic derivative of the Gamma function.


#Field (nf_label) --
#    The **coefficient field** of a modular form is the subfield of $\C$ generated by the coefficients $a_n$ of its $q$-expansion $\sum a_nq^n$.  The space of cusp forms $S_k^\mathrm{new}(N,\chi)$ has a basis of modular forms that are simultaneous eigenforms for all Hecke operators and with algebraic Fourier coefficients.  For such eigenforms the coefficient field will be a number field, and Galois conjugate eigenforms will share the same coefficient field.  Moreover, if $m$ is the smallest positive integer such that the values of the character $\chi$ are contained in the cyclotomic field $\Q(\zeta_m)$, the coefficient field will contain $\Q(\zeta_m)$
#    For eigenforms, the coefficient field is also known as the **Hecke field**.


#CM (cm_discs) --
#    A newform $f$ admits a **self-twist** by a primitive
#     Dirichlet character $\chi$ if the equality
#    \[
#    a_p(f) = \chi(p)a_p(f)
#    \]
#    holds for all but finitely many primes $p$.

#    For non-trivial $\chi$ this can hold only when $\chi$ has order $2$ and $a_p=0$ for all primes $p$ not dividing the level of $f$ for which $\chi(p)=-1$.
#    The character $\chi$ is then the Kronecker character of a quadratic field $K$ and may be identified by the discriminant $D$ of $K$.

#    If $D$ is negative, the modular form $f$ is said to have complex multiplication (CM) by $K$, and if $D$ is positive, $f$ is said to have real multiplication (RM) by $K$.  The latter can occur only when $f$ is a modular form of weight $1$ whose projective image is dihedral.

#    It is possible for a modular form to have multiple non-trivial self twists; this occurs precisely when $f$ is a modular form of weight one whose projective image is isomorphic to $D_2:=C_2\times C_2$; in this case $f$ admits three non-trivial self twists, two of which are CM and one of which is RM.



#Traces (trace_display) --
#    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.


#Fricke sign (fricke_eigenval) --
#    The **Fricke involution** is the Atkin-Lehner involution $w_N$ on the space $S_k(\Gamma_0(N))$ (induced by the corresponding involution on the modular curve $X_0(N)$).

#    For a newform $f \in S_k^{\textup{new}}(\Gamma_0(N))$, the sign of the functional equation satisfied by the L-function attached to $f$ is $i^{-k}$ times the eigenvalue of $\omega_N$ on $f$.  So, for example when $k=2$, the signs swap, and the analytic rank of $f$ is even when $w_N f = -f$ and odd when $w_N f = +f$.


#$q$-expansion (qexp_display) --
#    The **$q$-expansion** of a modular form $f(z)$ is its Fourier expansion at the cusp $z=i\infty$, expressed as a power series $\sum_{n=0}^{\infty} a_n q^n$ in the variable $q=e^{2\pi iz}$.

#    For cusp forms, the constant coefficient $a_0$ of the $q$-expansion is zero.

#    For newforms, we have $a_1=1$ and the coefficients $a_n$ are algebraic integers in a number field $K \subseteq \C$.

#    Accordingly, we define the **$q$-expansion** of a newform orbit $[f]$ to be the $q$-expansion of any newform $f$ in the orbit, but with coefficients $a_n \in K$ (without an embedding into $\C$).  Each embedding $K \hookrightarrow \C$ then gives rise to an embedded newform whose $q$-expansion has $a_n \in \C$, as above.