LMFDB - Newform search results

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Level	Weight	Analytic conductor	$Nk^2$	Dim.

Bad $p$	Char.	Primitive character	Character order	Coefficient field

Self-twist	CM/RM discriminant	Inner twist count	Is self-dual	Analytic rank

Coefficient ring index	Coefficient ring gens.		Projective image	Projective image type
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Results (39 matches)

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Label	Dim.	$A$	Field	CM	Traces				A-L signs			$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	2	5	7	$q$-expansion
9800.2.a.g	$1$	$78.253$	$\Q$	None	$0$	$-2$	$0$	$0$	$+$	$+$	$-$	$q-2q^{3}+q^{9}-q^{11}-3q^{13}-2q^{17}+\cdots$
9800.2.a.i	$1$	$78.253$	$\Q$	None	$0$	$-2$	$0$	$0$	$+$	$+$	$-$	$q-2q^{3}+q^{9}+4q^{11}-2q^{13}+2q^{19}+\cdots$
9800.2.a.j	$1$	$78.253$	$\Q$	None	$0$	$-2$	$0$	$0$	$-$	$+$	$+$	$q-2q^{3}+q^{9}+4q^{11}+2q^{13}+3q^{17}+\cdots$
9800.2.a.l	$1$	$78.253$	$\Q$	None	$0$	$-2$	$0$	$0$	$+$	$+$	$-$	$q-2q^{3}+q^{9}+5q^{11}-8q^{17}+2q^{19}+\cdots$
9800.2.a.m	$1$	$78.253$	$\Q$	None	$0$	$-1$	$0$	$0$	$+$	$+$	$-$	$q-q^{3}-2q^{9}-5q^{11}-7q^{13}+3q^{17}+\cdots$
9800.2.a.p	$1$	$78.253$	$\Q$	None	$0$	$-1$	$0$	$0$	$-$	$-$	$-$	$q-q^{3}-2q^{9}-q^{11}+q^{13}+3q^{17}+\cdots$
9800.2.a.q	$1$	$78.253$	$\Q$	None	$0$	$-1$	$0$	$0$	$-$	$-$	$-$	$q-q^{3}-2q^{9}-q^{11}+6q^{13}-7q^{17}+\cdots$
9800.2.a.u	$1$	$78.253$	$\Q$	None	$0$	$0$	$0$	$0$	$+$	$+$	$-$	$q-3q^{9}-4q^{11}+2q^{13}-6q^{17}-8q^{19}+\cdots$
9800.2.a.y	$1$	$78.253$	$\Q$	None	$0$	$1$	$0$	$0$	$+$	$+$	$-$	$q+q^{3}-2q^{9}-5q^{11}+7q^{13}-3q^{17}+\cdots$
9800.2.a.z	$1$	$78.253$	$\Q$	None	$0$	$1$	$0$	$0$	$-$	$+$	$+$	$q+q^{3}-2q^{9}-2q^{11}-4q^{13}+6q^{19}+\cdots$
9800.2.a.ba	$1$	$78.253$	$\Q$	None	$0$	$1$	$0$	$0$	$+$	$+$	$-$	$q+q^{3}-2q^{9}-q^{11}-6q^{13}+7q^{17}+\cdots$
9800.2.a.bc	$1$	$78.253$	$\Q$	None	$0$	$1$	$0$	$0$	$+$	$+$	$-$	$q+q^{3}-2q^{9}+2q^{11}+4q^{17}+2q^{19}+\cdots$
9800.2.a.be	$1$	$78.253$	$\Q$	None	$0$	$1$	$0$	$0$	$-$	$+$	$+$	$q+q^{3}-2q^{9}+3q^{11}+6q^{13}+5q^{17}+\cdots$
9800.2.a.bf	$1$	$78.253$	$\Q$	None	$0$	$2$	$0$	$0$	$-$	$-$	$-$	$q+2q^{3}+q^{9}-4q^{11}+4q^{13}+4q^{19}+\cdots$
9800.2.a.bm	$1$	$78.253$	$\Q$	None	$0$	$2$	$0$	$0$	$+$	$-$	$+$	$q+2q^{3}+q^{9}+4q^{11}-2q^{13}-3q^{17}+\cdots$
9800.2.a.bn	$1$	$78.253$	$\Q$	None	$0$	$2$	$0$	$0$	$+$	$+$	$-$	$q+2q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{19}+\cdots$
9800.2.a.bo	$1$	$78.253$	$\Q$	None	$0$	$2$	$0$	$0$	$-$	$-$	$-$	$q+2q^{3}+q^{9}+5q^{11}+8q^{17}+2q^{19}+\cdots$
9800.2.a.bp	$1$	$78.253$	$\Q$	None	$0$	$3$	$0$	$0$	$+$	$+$	$-$	$q+3q^{3}+6q^{9}-q^{11}+2q^{13}+3q^{17}+\cdots$
9800.2.a.br	$2$	$78.253$	$\Q(\sqrt{2}) $	None	$0$	$-2$	$0$	$0$	$+$	$+$	$-$	$q+(-1+\beta )q^{3}-2\beta q^{9}+(-2+2\beta )q^{11}+\cdots$
9800.2.a.bt	$2$	$78.253$	$\Q(\sqrt{17}) $	None	$0$	$-1$	$0$	$0$	$-$	$-$	$-$	$q-\beta q^{3}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots$
9800.2.a.bu	$2$	$78.253$	$\Q(\sqrt{33}) $	None	$0$	$-1$	$0$	$0$	$+$	$+$	$-$	$q-\beta q^{3}+(5+\beta )q^{9}+(4-\beta )q^{11}+(2+\cdots)q^{13}+\cdots$
9800.2.a.bx	$2$	$78.253$	$\Q(\sqrt{17}) $	None	$0$	$1$	$0$	$0$	$+$	$+$	$-$	$q+\beta q^{3}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots$
9800.2.a.by	$2$	$78.253$	$\Q(\sqrt{17}) $	None	$0$	$1$	$0$	$0$	$+$	$+$	$-$	$q+\beta q^{3}+(1+\beta )q^{9}-\beta q^{11}+(2-3\beta )q^{13}+\cdots$
9800.2.a.cc	$3$	$78.253$	$\Q(\zeta_{18})^+$	None	$0$	$-3$	$0$	$0$	$+$	$-$	$+$	$q+(-1+\beta _{1})q^{3}+(-2\beta _{1}+\beta _{2})q^{9}+\cdots$
9800.2.a.ce	$3$	$78.253$	3.3.1944.1	None	$0$	$0$	$0$	$0$	$-$	$+$	$+$	$q+\beta _{1}q^{3}+(3+\beta _{1}+\beta _{2})q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots$
9800.2.a.cg	$3$	$78.253$	3.3.568.1	None	$0$	$1$	$0$	$0$	$-$	$-$	$-$	$q+\beta _{1}q^{3}+(1+2\beta _{1}+\beta _{2})q^{9}+(2-\beta _{2})q^{11}+\cdots$
9800.2.a.ci	$3$	$78.253$	$\Q(\zeta_{18})^+$	None	$0$	$3$	$0$	$0$	$-$	$+$	$+$	$q+(1-\beta _{1})q^{3}+(-2\beta _{1}+\beta _{2})q^{9}+(-2+\cdots)q^{11}+\cdots$
9800.2.a.ck	$4$	$78.253$	4.4.43449.1	None	$0$	$-3$	$0$	$0$	$+$	$+$	$-$	$q+(-1+\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{9}+\cdots$
9800.2.a.cl	$4$	$78.253$	4.4.16448.2	None	$0$	$-2$	$0$	$0$	$-$	$+$	$+$	$q-\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2})q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots$
9800.2.a.cp	$4$	$78.253$	4.4.13448.1	None	$0$	$0$	$0$	$0$	$-$	$-$	$-$	$q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(2+\beta _{2})q^{11}+\cdots$
9800.2.a.cq	$4$	$78.253$	$\Q(\sqrt{7}, \sqrt{15})$	None	$0$	$0$	$0$	$0$	$-$	$-$	$-$	$q+\beta _{2}q^{3}+(2-\beta _{3})q^{9}+q^{11}+(\beta _{1}+\beta _{2}+\cdots)q^{13}+\cdots$
9800.2.a.cr	$4$	$78.253$	$\Q(\sqrt{7}, \sqrt{15})$	None	$0$	$0$	$0$	$0$	$+$	$+$	$-$	$q+\beta _{2}q^{3}+(2-\beta _{3})q^{9}+q^{11}+(\beta _{1}+\beta _{2}+\cdots)q^{13}+\cdots$
9800.2.a.cs	$4$	$78.253$	4.4.16448.2	None	$0$	$2$	$0$	$0$	$-$	$+$	$+$	$q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2})q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots$
9800.2.a.cu	$4$	$78.253$	4.4.43449.1	None	$0$	$3$	$0$	$0$	$-$	$-$	$-$	$q+(1-\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots$
9800.2.a.cx	$6$	$78.253$	6.6.239575536.1	None	$0$	$1$	$0$	$0$	$+$	$-$	$+$	$q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(\beta _{1}+\beta _{5})q^{11}+\cdots$
9800.2.a.cy	$6$	$78.253$	6.6.239575536.1	None	$0$	$1$	$0$	$0$	$-$	$-$	$-$	$q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(\beta _{1}+\beta _{5})q^{11}+\cdots$
9800.2.a.cz	$8$	$78.253$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$0$	$0$	$0$	$0$	$-$	$+$	$+$	$q+\beta _{1}q^{3}+(2+\beta _{2})q^{9}+(1+\beta _{3}+\beta _{6}+\cdots)q^{11}+\cdots$
9800.2.a.da	$8$	$78.253$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$0$	$0$	$0$	$0$	$+$	$-$	$+$	$q+\beta _{1}q^{3}+(2+\beta _{2})q^{9}+(1+\beta _{3}+\beta _{6}+\cdots)q^{11}+\cdots$
9800.2.a.dc	$10$	$78.253$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	$0$	$0$	$0$	$0$	$+$	$-$	$+$	$q+\beta _{7}q^{3}+(1-\beta _{4}-\beta _{8})q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots$

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Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Label	Dim.	\(A\)	Field	CM	Traces				A-L signs			$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	2	5	7	$q$-expansion
9800.2.a.g	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(-2\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q-2q^{3}+q^{9}-q^{11}-3q^{13}-2q^{17}+\cdots\)
9800.2.a.i	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(-2\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q-2q^{3}+q^{9}+4q^{11}-2q^{13}+2q^{19}+\cdots\)
9800.2.a.j	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(-2\)	\(0\)	\(0\)	\(-\)	\(+\)	\(+\)	\(q-2q^{3}+q^{9}+4q^{11}+2q^{13}+3q^{17}+\cdots\)
9800.2.a.l	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(-2\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q-2q^{3}+q^{9}+5q^{11}-8q^{17}+2q^{19}+\cdots\)
9800.2.a.m	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q-q^{3}-2q^{9}-5q^{11}-7q^{13}+3q^{17}+\cdots\)
9800.2.a.p	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q-q^{3}-2q^{9}-q^{11}+q^{13}+3q^{17}+\cdots\)
9800.2.a.q	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q-q^{3}-2q^{9}-q^{11}+6q^{13}-7q^{17}+\cdots\)
9800.2.a.u	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q-3q^{9}-4q^{11}+2q^{13}-6q^{17}-8q^{19}+\cdots\)
9800.2.a.y	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q+q^{3}-2q^{9}-5q^{11}+7q^{13}-3q^{17}+\cdots\)
9800.2.a.z	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(0\)	\(-\)	\(+\)	\(+\)	\(q+q^{3}-2q^{9}-2q^{11}-4q^{13}+6q^{19}+\cdots\)
9800.2.a.ba	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q+q^{3}-2q^{9}-q^{11}-6q^{13}+7q^{17}+\cdots\)
9800.2.a.bc	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q+q^{3}-2q^{9}+2q^{11}+4q^{17}+2q^{19}+\cdots\)
9800.2.a.be	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(0\)	\(-\)	\(+\)	\(+\)	\(q+q^{3}-2q^{9}+3q^{11}+6q^{13}+5q^{17}+\cdots\)
9800.2.a.bf	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(2\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q+2q^{3}+q^{9}-4q^{11}+4q^{13}+4q^{19}+\cdots\)
9800.2.a.bm	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(2\)	\(0\)	\(0\)	\(+\)	\(-\)	\(+\)	\(q+2q^{3}+q^{9}+4q^{11}-2q^{13}-3q^{17}+\cdots\)
9800.2.a.bn	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(2\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q+2q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{19}+\cdots\)
9800.2.a.bo	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(2\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q+2q^{3}+q^{9}+5q^{11}+8q^{17}+2q^{19}+\cdots\)
9800.2.a.bp	\(1\)	\(78.253\)	\(\Q\)	None	\(0\)	\(3\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q+3q^{3}+6q^{9}-q^{11}+2q^{13}+3q^{17}+\cdots\)
9800.2.a.br	\(2\)	\(78.253\)	\(\Q(\sqrt{2}) \)	None	\(0\)	\(-2\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q+(-1+\beta )q^{3}-2\beta q^{9}+(-2+2\beta )q^{11}+\cdots\)
9800.2.a.bt	\(2\)	\(78.253\)	\(\Q(\sqrt{17}) \)	None	\(0\)	\(-1\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q-\beta q^{3}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots\)
9800.2.a.bu	\(2\)	\(78.253\)	\(\Q(\sqrt{33}) \)	None	\(0\)	\(-1\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q-\beta q^{3}+(5+\beta )q^{9}+(4-\beta )q^{11}+(2+\cdots)q^{13}+\cdots\)
9800.2.a.bx	\(2\)	\(78.253\)	\(\Q(\sqrt{17}) \)	None	\(0\)	\(1\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q+\beta q^{3}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots\)
9800.2.a.by	\(2\)	\(78.253\)	\(\Q(\sqrt{17}) \)	None	\(0\)	\(1\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q+\beta q^{3}+(1+\beta )q^{9}-\beta q^{11}+(2-3\beta )q^{13}+\cdots\)
9800.2.a.cc	\(3\)	\(78.253\)	\(\Q(\zeta_{18})^+\)	None	\(0\)	\(-3\)	\(0\)	\(0\)	\(+\)	\(-\)	\(+\)	\(q+(-1+\beta _{1})q^{3}+(-2\beta _{1}+\beta _{2})q^{9}+\cdots\)
9800.2.a.ce	\(3\)	\(78.253\)	3.3.1944.1	None	\(0\)	\(0\)	\(0\)	\(0\)	\(-\)	\(+\)	\(+\)	\(q+\beta _{1}q^{3}+(3+\beta _{1}+\beta _{2})q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
9800.2.a.cg	\(3\)	\(78.253\)	3.3.568.1	None	\(0\)	\(1\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q+\beta _{1}q^{3}+(1+2\beta _{1}+\beta _{2})q^{9}+(2-\beta _{2})q^{11}+\cdots\)
9800.2.a.ci	\(3\)	\(78.253\)	\(\Q(\zeta_{18})^+\)	None	\(0\)	\(3\)	\(0\)	\(0\)	\(-\)	\(+\)	\(+\)	\(q+(1-\beta _{1})q^{3}+(-2\beta _{1}+\beta _{2})q^{9}+(-2+\cdots)q^{11}+\cdots\)
9800.2.a.ck	\(4\)	\(78.253\)	4.4.43449.1	None	\(0\)	\(-3\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q+(-1+\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{9}+\cdots\)
9800.2.a.cl	\(4\)	\(78.253\)	4.4.16448.2	None	\(0\)	\(-2\)	\(0\)	\(0\)	\(-\)	\(+\)	\(+\)	\(q-\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2})q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
9800.2.a.cp	\(4\)	\(78.253\)	4.4.13448.1	None	\(0\)	\(0\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(2+\beta _{2})q^{11}+\cdots\)
9800.2.a.cq	\(4\)	\(78.253\)	\(\Q(\sqrt{7}, \sqrt{15})\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q+\beta _{2}q^{3}+(2-\beta _{3})q^{9}+q^{11}+(\beta _{1}+\beta _{2}+\cdots)q^{13}+\cdots\)
9800.2.a.cr	\(4\)	\(78.253\)	\(\Q(\sqrt{7}, \sqrt{15})\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q+\beta _{2}q^{3}+(2-\beta _{3})q^{9}+q^{11}+(\beta _{1}+\beta _{2}+\cdots)q^{13}+\cdots\)
9800.2.a.cs	\(4\)	\(78.253\)	4.4.16448.2	None	\(0\)	\(2\)	\(0\)	\(0\)	\(-\)	\(+\)	\(+\)	\(q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2})q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
9800.2.a.cu	\(4\)	\(78.253\)	4.4.43449.1	None	\(0\)	\(3\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q+(1-\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
9800.2.a.cx	\(6\)	\(78.253\)	6.6.239575536.1	None	\(0\)	\(1\)	\(0\)	\(0\)	\(+\)	\(-\)	\(+\)	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(\beta _{1}+\beta _{5})q^{11}+\cdots\)
9800.2.a.cy	\(6\)	\(78.253\)	6.6.239575536.1	None	\(0\)	\(1\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(\beta _{1}+\beta _{5})q^{11}+\cdots\)
9800.2.a.cz	\(8\)	\(78.253\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(-\)	\(+\)	\(+\)	\(q+\beta _{1}q^{3}+(2+\beta _{2})q^{9}+(1+\beta _{3}+\beta _{6}+\cdots)q^{11}+\cdots\)
9800.2.a.da	\(8\)	\(78.253\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(+\)	\(-\)	\(+\)	\(q+\beta _{1}q^{3}+(2+\beta _{2})q^{9}+(1+\beta _{3}+\beta _{6}+\cdots)q^{11}+\cdots\)
9800.2.a.dc	\(10\)	\(78.253\)	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(+\)	\(-\)	\(+\)	\(q+\beta _{7}q^{3}+(1-\beta _{4}-\beta _{8})q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.