LMFDB - Space of modular forms of level 690, weight 4, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	690.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 20 $
Sturm bound:			$576$
Trace bound:			$7$
Distinguishing $T_p$:			$7$, $11$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(690))$.

	Total	New	Old
Modular forms	440	44	396
Cusp forms	424	44	380
Eisenstein series	16	0	16

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$2$	$3$	$5$	$23$	Fricke	Total			Cusp			Eisenstein
$2$	$3$	$5$	$23$	Fricke	All	New	Old	All	New	Old	All	New	Old
$+$	$+$	$+$	$+$	$+$	$31$	$3$	$28$	$30$	$3$	$27$	$1$	$0$	$1$
$+$	$+$	$+$	$-$	$-$	$26$	$3$	$23$	$25$	$3$	$22$	$1$	$0$	$1$
$+$	$+$	$-$	$+$	$-$	$25$	$2$	$23$	$24$	$2$	$22$	$1$	$0$	$1$
$+$	$+$	$-$	$-$	$+$	$28$	$4$	$24$	$27$	$4$	$23$	$1$	$0$	$1$
$+$	$-$	$+$	$+$	$-$	$27$	$3$	$24$	$26$	$3$	$23$	$1$	$0$	$1$
$+$	$-$	$+$	$-$	$+$	$27$	$2$	$25$	$26$	$2$	$24$	$1$	$0$	$1$
$+$	$-$	$-$	$+$	$+$	$27$	$3$	$24$	$26$	$3$	$23$	$1$	$0$	$1$
$+$	$-$	$-$	$-$	$-$	$29$	$2$	$27$	$28$	$2$	$26$	$1$	$0$	$1$
$-$	$+$	$+$	$+$	$-$	$27$	$3$	$24$	$26$	$3$	$23$	$1$	$0$	$1$
$-$	$+$	$+$	$-$	$+$	$29$	$3$	$26$	$28$	$3$	$25$	$1$	$0$	$1$
$-$	$+$	$-$	$+$	$+$	$29$	$4$	$25$	$28$	$4$	$24$	$1$	$0$	$1$
$-$	$+$	$-$	$-$	$-$	$25$	$2$	$23$	$24$	$2$	$22$	$1$	$0$	$1$
$-$	$-$	$+$	$+$	$+$	$26$	$3$	$23$	$25$	$3$	$22$	$1$	$0$	$1$
$-$	$-$	$+$	$-$	$-$	$29$	$2$	$27$	$28$	$2$	$26$	$1$	$0$	$1$
$-$	$-$	$-$	$+$	$-$	$28$	$1$	$27$	$27$	$1$	$26$	$1$	$0$	$1$
$-$	$-$	$-$	$-$	$+$	$27$	$4$	$23$	$26$	$4$	$22$	$1$	$0$	$1$
Plus space				$+$	$224$	$26$	$198$	$216$	$26$	$190$	$8$	$0$	$8$
Minus space				$-$	$216$	$18$	$198$	$208$	$18$	$190$	$8$	$0$	$8$

Trace form

$ 44 q - 12 q^{3} + 176 q^{4} - 56 q^{7} + 396 q^{9} - 48 q^{12} + 160 q^{14} + 704 q^{16} + 16 q^{17} - 152 q^{19} - 168 q^{21} - 224 q^{22} + 1100 q^{25} - 108 q^{27} - 224 q^{28} - 336 q^{29} + 408 q^{31}+ \cdots + 1024 q^{98}+O(q^{100}) $

44 * q - 12 * q^3 + 176 * q^4 - 56 * q^7 + 396 * q^9 - 48 * q^12 + 160 * q^14 + 704 * q^16 + 16 * q^17 - 152 * q^19 - 168 * q^21 - 224 * q^22 + 1100 * q^25 - 108 * q^27 - 224 * q^28 - 336 * q^29 + 408 * q^31 + 384 * q^33 + 80 * q^35 + 1584 * q^36 + 472 * q^37 - 992 * q^38 + 168 * q^39 + 576 * q^41 + 392 * q^43 + 1824 * q^47 - 192 * q^48 + 1268 * q^49 - 480 * q^51 + 1744 * q^53 + 640 * q^56 - 840 * q^57 + 464 * q^58 + 704 * q^59 - 1368 * q^61 - 1408 * q^62 - 504 * q^63 + 2816 * q^64 - 1352 * q^67 + 64 * q^68 + 560 * q^70 - 512 * q^71 - 952 * q^73 + 3328 * q^74 - 300 * q^75 - 608 * q^76 + 960 * q^77 + 624 * q^78 + 3176 * q^79 + 3564 * q^81 + 3840 * q^82 + 3664 * q^83 - 672 * q^84 - 440 * q^85 + 3904 * q^86 - 1392 * q^87 - 896 * q^88 + 1328 * q^89 + 3184 * q^91 + 576 * q^93 - 1632 * q^94 - 80 * q^95 + 1544 * q^97 + 1024 * q^98

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(690))$ into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				A-L signs				Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	2	3	5	23	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
690.4.a.a	$690$	$4$	690.a	1.a	$1$	$1$	$1$	$40.711$	$\Q$	None	✓	✓			690.4.a.a	$1$	$1$	$-2$	$-3$	$5$	$-19$	$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots$
690.4.a.b	$690$	$4$	690.a	1.a	$1$	$1$	$1$	$40.711$	$\Q$	None	✓	✓			690.4.a.b	$1$	$0$	$-2$	$-3$	$5$	$-18$	$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots$
690.4.a.c	$690$	$4$	690.a	1.a	$1$	$1$	$1$	$40.711$	$\Q$	None	✓	✓			690.4.a.c	$1$	$1$	$-2$	$-3$	$5$	$16$	$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots$
690.4.a.d	$690$	$4$	690.a	1.a	$1$	$1$	$1$	$40.711$	$\Q$	None	✓	✓			690.4.a.d	$1$	$0$	$-2$	$3$	$-5$	$-16$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots$
690.4.a.e	$690$	$4$	690.a	1.a	$1$	$1$	$1$	$40.711$	$\Q$	None	✓	✓			690.4.a.e	$1$	$0$	$-2$	$3$	$-5$	$-5$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots$
690.4.a.f	$690$	$4$	690.a	1.a	$1$	$1$	$1$	$40.711$	$\Q$	None	✓	✓			690.4.a.f	$1$	$1$	$2$	$3$	$-5$	$-20$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots$
690.4.a.g	$690$	$4$	690.a	1.a	$1$	$1$	$1$	$40.711$	$\Q$	None	✓	✓			690.4.a.g	$1$	$1$	$2$	$3$	$-5$	$-5$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots$
690.4.a.h	$690$	$4$	690.a	1.a	$1$	$1$	$1$	$40.711$	$\Q$	None	✓	✓	✓	✓	690.4.a.h	$1$	$1$	$2$	$3$	$5$	$-7$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots$
690.4.a.i	$690$	$4$	690.a	1.a	$1$	$2$	$2$	$40.711$	$\Q(\sqrt{6}) $	None	✓	✓	✓	✓	690.4.a.i	$1$	$1$	$-4$	$6$	$10$	$-26$	$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots$
690.4.a.j	$690$	$4$	690.a	1.a	$1$	$2$	$2$	$40.711$	$\Q(\sqrt{14}) $	None	✓	✓	✓	✓	690.4.a.j	$1$	$1$	$4$	$-6$	$10$	$-2$	$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots$
690.4.a.k	$690$	$4$	690.a	1.a	$1$	$3$	$3$	$40.711$	3.3.471057.3	None	✓	✓	✓	✓	690.4.a.k	$1$	$1$	$-6$	$-9$	$-15$	$-35$	$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q-2q^{2}-3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots$
690.4.a.l	$690$	$4$	690.a	1.a	$1$	$3$	$3$	$40.711$	3.3.617756.1	None	✓	✓	✓	✓	690.4.a.l	$1$	$0$	$-6$	$-9$	$-15$	$30$	$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots$
690.4.a.m	$690$	$4$	690.a	1.a	$1$	$3$	$3$	$40.711$	3.3.931848.1	None	✓	✓	✓		690.4.a.m	$1$	$0$	$-6$	$-9$	$15$	$6$	$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots$
690.4.a.n	$690$	$4$	690.a	1.a	$1$	$3$	$3$	$40.711$	3.3.207308.1	None	✓	✓	✓	✓	690.4.a.n	$1$	$1$	$-6$	$9$	$-15$	$2$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots$
690.4.a.o	$690$	$4$	690.a	1.a	$1$	$3$	$3$	$40.711$	3.3.460593.1	None	✓	✓	✓	✓	690.4.a.o	$1$	$0$	$-6$	$9$	$15$	$-3$	$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots$
690.4.a.p	$690$	$4$	690.a	1.a	$1$	$3$	$3$	$40.711$	3.3.162793.1	None	✓	✓	✓	✓	690.4.a.p	$1$	$0$	$6$	$-9$	$-15$	$3$	$-$	$+$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+2q^{2}-3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots$
690.4.a.q	$690$	$4$	690.a	1.a	$1$	$3$	$3$	$40.711$	3.3.396732.1	None	✓	✓	✓	✓	690.4.a.q	$1$	$1$	$6$	$-9$	$-15$	$12$	$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots$
690.4.a.r	$690$	$4$	690.a	1.a	$1$	$3$	$3$	$40.711$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓	✓	✓	✓	690.4.a.r	$1$	$0$	$6$	$9$	$-15$	$-2$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots$
690.4.a.s	$690$	$4$	690.a	1.a	$1$	$4$	$4$	$40.711$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	✓	✓	✓	690.4.a.s	$1$	$0$	$8$	$-12$	$20$	$7$	$-$	$+$	$-$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	$q+2q^{2}-3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots$
690.4.a.t	$690$	$4$	690.a	1.a	$1$	$4$	$4$	$40.711$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	✓	✓	✓	690.4.a.t	$1$	$0$	$8$	$12$	$20$	$26$	$-$	$-$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q+2q^{2}+3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(690))$ into lower level spaces

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Classical	Maass
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	690.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(576\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

\(2\)	\(3\)	\(5\)	\(23\)	Fricke	Total			Cusp			Eisenstein
\(2\)	\(3\)	\(5\)	\(23\)	Fricke	All	New	Old	All	New	Old	All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(31\)	\(3\)	\(28\)	\(30\)	\(3\)	\(27\)	\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(26\)	\(3\)	\(23\)	\(25\)	\(3\)	\(22\)	\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(25\)	\(2\)	\(23\)	\(24\)	\(2\)	\(22\)	\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(28\)	\(4\)	\(24\)	\(27\)	\(4\)	\(23\)	\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(27\)	\(3\)	\(24\)	\(26\)	\(3\)	\(23\)	\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(27\)	\(2\)	\(25\)	\(26\)	\(2\)	\(24\)	\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(27\)	\(3\)	\(24\)	\(26\)	\(3\)	\(23\)	\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(29\)	\(2\)	\(27\)	\(28\)	\(2\)	\(26\)	\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(27\)	\(3\)	\(24\)	\(26\)	\(3\)	\(23\)	\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(29\)	\(3\)	\(26\)	\(28\)	\(3\)	\(25\)	\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(29\)	\(4\)	\(25\)	\(28\)	\(4\)	\(24\)	\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(25\)	\(2\)	\(23\)	\(24\)	\(2\)	\(22\)	\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(26\)	\(3\)	\(23\)	\(25\)	\(3\)	\(22\)	\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(29\)	\(2\)	\(27\)	\(28\)	\(2\)	\(26\)	\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(28\)	\(1\)	\(27\)	\(27\)	\(1\)	\(26\)	\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(27\)	\(4\)	\(23\)	\(26\)	\(4\)	\(22\)	\(1\)	\(0\)	\(1\)
Plus space				\(+\)	\(224\)	\(26\)	\(198\)	\(216\)	\(26\)	\(190\)	\(8\)	\(0\)	\(8\)
Minus space				\(-\)	\(216\)	\(18\)	\(198\)	\(208\)	\(18\)	\(190\)	\(8\)	\(0\)	\(8\)

Introduction

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(690))\) into newform subspaces

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(690))\) into lower level spaces

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

Label	690.4.a

Level	$690$
Weight	$4$
Character orbit	690.a
Rep. character	$\chi_{690}(1,\cdot)$
Character field	$\Q$
Dimension	$44$
Newform subspaces	$20$
Sturm bound	$576$
Trace bound	$7$