LMFDB - Space of modular forms of level 8619, weight 2, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 8619 = 3 \cdot 13^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8619.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 50 $
Sturm bound:			$2184$
Trace bound:			$17$
Distinguishing $T_p$:			$2$, $5$, $7$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(8619))$.

	Total	New	Old
Modular forms	1120	414	706
Cusp forms	1065	414	651
Eisenstein series	55	0	55

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

$3$	$13$	$17$	Fricke	Dim.
$+$	$+$	$+$	$+$	$48$
$+$	$+$	$-$	$-$	$58$
$+$	$-$	$+$	$-$	$56$
$+$	$-$	$-$	$+$	$44$
$-$	$+$	$+$	$-$	$55$
$-$	$+$	$-$	$+$	$45$
$-$	$-$	$+$	$+$	$48$
$-$	$-$	$-$	$-$	$60$
Plus space			$+$	$185$
Minus space			$-$	$229$

Trace form

$ 414q - 4q^{2} + 2q^{3} + 412q^{4} - 2q^{6} - 4q^{7} - 24q^{8} + 414q^{9} + O(q^{10}) $

$ 414q - 4q^{2} + 2q^{3} + 412q^{4} - 2q^{6} - 4q^{7} - 24q^{8} + 414q^{9} - 4q^{10} + 14q^{12} - 8q^{14} - 2q^{15} + 416q^{16} - 4q^{18} + 18q^{19} + 8q^{20} + 12q^{21} + 20q^{22} + 8q^{23} - 6q^{24} + 416q^{25} + 2q^{27} - 24q^{28} - 8q^{29} + 8q^{30} - 32q^{32} - 10q^{33} - 4q^{34} - 4q^{35} + 412q^{36} + 4q^{37} - 20q^{38} + 40q^{40} + 24q^{41} + 16q^{42} + 14q^{43} - 24q^{44} - 20q^{46} - 28q^{47} + 30q^{48} + 446q^{49} - 8q^{50} + 4q^{51} - 12q^{53} - 2q^{54} + 26q^{55} + 16q^{57} + 8q^{58} + 8q^{59} + 40q^{60} + 20q^{61} + 48q^{62} - 4q^{63} + 392q^{64} - 12q^{66} - 32q^{67} - 2q^{69} + 64q^{70} - 24q^{72} - 16q^{74} + 30q^{75} + 108q^{76} + 12q^{77} + 52q^{79} + 72q^{80} + 414q^{81} + 12q^{82} + 60q^{83} + 24q^{84} - 2q^{85} + 4q^{86} + 8q^{87} + 56q^{88} + 28q^{89} - 4q^{90} + 96q^{92} + 12q^{93} + 64q^{94} + 40q^{95} + 26q^{96} + 16q^{97} - 4q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(8619))$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				A-L signs			$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	3	13	17	$q$-expansion
8619.2.a.a	$1$	$68.823$	$\Q$	None	$-2$	$-1$	$-4$	$3$	$+$	$+$	$-$	$q-2q^{2}-q^{3}+2q^{4}-4q^{5}+2q^{6}+\cdots$
8619.2.a.b	$1$	$68.823$	$\Q$	None	$-2$	$-1$	$2$	$-4$	$+$	$+$	$-$	$q-2q^{2}-q^{3}+2q^{4}+2q^{5}+2q^{6}+\cdots$
8619.2.a.c	$1$	$68.823$	$\Q$	None	$-2$	$1$	$3$	$2$	$-$	$+$	$-$	$q-2q^{2}+q^{3}+2q^{4}+3q^{5}-2q^{6}+\cdots$
8619.2.a.d	$1$	$68.823$	$\Q$	None	$-1$	$-1$	$1$	$-2$	$+$	$+$	$-$	$q-q^{2}-q^{3}-q^{4}+q^{5}+q^{6}-2q^{7}+\cdots$
8619.2.a.e	$1$	$68.823$	$\Q$	None	$-1$	$-1$	$4$	$-2$	$+$	$+$	$-$	$q-q^{2}-q^{3}-q^{4}+4q^{5}+q^{6}-2q^{7}+\cdots$
8619.2.a.f	$1$	$68.823$	$\Q$	None	$-1$	$1$	$0$	$-4$	$-$	$-$	$-$	$q-q^{2}+q^{3}-q^{4}-q^{6}-4q^{7}+3q^{8}+\cdots$
8619.2.a.g	$1$	$68.823$	$\Q$	None	$0$	$1$	$-3$	$4$	$-$	$+$	$+$	$q+q^{3}-2q^{4}-3q^{5}+4q^{7}+q^{9}+3q^{11}+\cdots$
8619.2.a.h	$1$	$68.823$	$\Q$	None	$1$	$-1$	$-1$	$2$	$+$	$+$	$-$	$q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}+2q^{7}+\cdots$
8619.2.a.i	$1$	$68.823$	$\Q$	None	$1$	$-1$	$2$	$0$	$+$	$+$	$-$	$q+q^{2}-q^{3}-q^{4}+2q^{5}-q^{6}-3q^{8}+\cdots$
8619.2.a.j	$1$	$68.823$	$\Q$	None	$1$	$1$	$0$	$2$	$-$	$+$	$-$	$q+q^{2}+q^{3}-q^{4}+q^{6}+2q^{7}-3q^{8}+\cdots$
8619.2.a.k	$1$	$68.823$	$\Q$	None	$1$	$1$	$0$	$4$	$-$	$-$	$-$	$q+q^{2}+q^{3}-q^{4}+q^{6}+4q^{7}-3q^{8}+\cdots$
8619.2.a.l	$1$	$68.823$	$\Q$	None	$2$	$-1$	$-2$	$4$	$+$	$+$	$-$	$q+2q^{2}-q^{3}+2q^{4}-2q^{5}-2q^{6}+\cdots$
8619.2.a.m	$1$	$68.823$	$\Q$	None	$2$	$-1$	$4$	$-3$	$+$	$+$	$-$	$q+2q^{2}-q^{3}+2q^{4}+4q^{5}-2q^{6}+\cdots$
8619.2.a.n	$1$	$68.823$	$\Q$	None	$2$	$1$	$-3$	$-2$	$-$	$+$	$-$	$q+2q^{2}+q^{3}+2q^{4}-3q^{5}+2q^{6}+\cdots$
8619.2.a.o	$2$	$68.823$	$\Q(\sqrt{3}) $	None	$0$	$-2$	$0$	$0$	$+$	$-$	$+$	$q-q^{3}-2q^{4}-\beta q^{5}+q^{9}-3\beta q^{11}+\cdots$
8619.2.a.p	$2$	$68.823$	$\Q(\sqrt{3}) $	None	$0$	$-2$	$0$	$0$	$+$	$-$	$-$	$q-q^{3}-2q^{4}-\beta q^{5}-2\beta q^{7}+q^{9}+\cdots$
8619.2.a.q	$2$	$68.823$	$\Q(\sqrt{17}) $	None	$1$	$-2$	$-3$	$0$	$+$	$+$	$-$	$q+\beta q^{2}-q^{3}+(2+\beta )q^{4}+(-1-\beta )q^{5}+\cdots$
8619.2.a.r	$3$	$68.823$	$\Q(\zeta_{14})^+$	None	$-2$	$3$	$-1$	$5$	$-$	$+$	$+$	$q+(-1-\beta _{2})q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots$
8619.2.a.s	$3$	$68.823$	3.3.148.1	None	$1$	$-3$	$0$	$4$	$+$	$+$	$+$	$q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots$
8619.2.a.t	$3$	$68.823$	$\Q(\zeta_{14})^+$	None	$2$	$3$	$1$	$-5$	$-$	$-$	$+$	$q+(1-\beta _{1})q^{2}+q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots$
8619.2.a.u	$3$	$68.823$	3.3.148.1	None	$3$	$3$	$6$	$2$	$-$	$+$	$+$	$q+(1+\beta _{2})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots$
8619.2.a.v	$4$	$68.823$	$\Q(\zeta_{24})^+$	None	$0$	$4$	$0$	$0$	$-$	$-$	$-$	$q-\beta _{2}q^{2}+q^{3}+(-\beta _{1}-\beta _{2})q^{5}-\beta _{2}q^{6}+\cdots$
8619.2.a.w	$5$	$68.823$	5.5.153424.1	None	$-3$	$5$	$0$	$-2$	$-$	$+$	$+$	$q+(-1-\beta _{3})q^{2}+q^{3}+(2+\beta _{3}-\beta _{4})q^{4}+\cdots$
8619.2.a.x	$5$	$68.823$	5.5.1004368.1	None	$-1$	$-5$	$-2$	$-8$	$+$	$+$	$+$	$q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots$
8619.2.a.y	$6$	$68.823$	6.6.15187408.1	None	$-4$	$6$	$-4$	$-2$	$-$	$+$	$-$	$q+(-1+\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$
8619.2.a.z	$6$	$68.823$	6.6.83831632.1	None	$-2$	$-6$	$0$	$-2$	$+$	$+$	$-$	$q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots$
8619.2.a.ba	$6$	$68.823$	6.6.485125.1	None	$-2$	$6$	$4$	$5$	$-$	$+$	$-$	$q-\beta _{1}q^{2}+q^{3}+(\beta _{3}+\beta _{4})q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots$
8619.2.a.bb	$6$	$68.823$	6.6.31337472.1	None	$0$	$-6$	$0$	$0$	$+$	$-$	$-$	$q+\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{5})q^{5}+\cdots$
8619.2.a.bc	$6$	$68.823$	6.6.485125.1	None	$2$	$6$	$-4$	$-5$	$-$	$+$	$-$	$q+\beta _{1}q^{2}+q^{3}+(\beta _{3}+\beta _{4})q^{4}+(-2+\cdots)q^{5}+\cdots$
8619.2.a.bd	$8$	$68.823$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$0$	$-8$	$-1$	$-2$	$+$	$+$	$+$	$q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(\beta _{1}+\beta _{7})q^{5}+\cdots$
8619.2.a.be	$8$	$68.823$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$0$	$-8$	$1$	$2$	$+$	$+$	$+$	$q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$
8619.2.a.bf	$8$	$68.823$	8.8.$\cdots$.1	None	$0$	$-8$	$0$	$0$	$+$	$-$	$+$	$q+(\beta _{1}+\beta _{7})q^{2}-q^{3}+(2+\beta _{5})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$
8619.2.a.bg	$8$	$68.823$	8.8.$\cdots$.1	None	$0$	$8$	$0$	$0$	$-$	$-$	$+$	$q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$
8619.2.a.bh	$9$	$68.823$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	$-1$	$-9$	$-5$	$3$	$+$	$+$	$-$	$q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots$
8619.2.a.bi	$9$	$68.823$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	$1$	$-9$	$5$	$-3$	$+$	$+$	$-$	$q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+(1+\cdots)q^{5}+\cdots$
8619.2.a.bj	$10$	$68.823$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	$0$	$10$	$0$	$0$	$-$	$-$	$-$	$q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots$
8619.2.a.bk	$11$	$68.823$	$\mathbb{Q}[x]/(x^{11} - \cdots)$	None	$0$	$11$	$0$	$-3$	$-$	$+$	$+$	$q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots$
8619.2.a.bl	$11$	$68.823$	$\mathbb{Q}[x]/(x^{11} - \cdots)$	None	$0$	$11$	$0$	$3$	$-$	$+$	$+$	$q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots$
8619.2.a.bm	$12$	$68.823$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	$0$	$-12$	$0$	$0$	$+$	$-$	$-$	$q+\beta _{7}q^{2}-q^{3}+(1-\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$
8619.2.a.bn	$16$	$68.823$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	$0$	$16$	$0$	$0$	$-$	$-$	$+$	$q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{14}q^{5}+\cdots$
8619.2.a.bo	$20$	$68.823$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None	$0$	$20$	$0$	$0$	$-$	$-$	$-$	$q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{15}q^{5}+\cdots$
8619.2.a.bp	$21$	$68.823$		None	$-9$	$21$	$-26$	$5$	$-$	$-$	$+$
8619.2.a.bq	$21$	$68.823$		None	$9$	$21$	$26$	$-5$	$-$	$+$	$+$
8619.2.a.br	$22$	$68.823$		None	$0$	$-22$	$0$	$0$	$+$	$-$	$+$
8619.2.a.bs	$24$	$68.823$		None	$-11$	$24$	$-23$	$-10$	$-$	$+$	$-$
8619.2.a.bt	$24$	$68.823$		None	$-7$	$-24$	$13$	$12$	$+$	$-$	$+$
8619.2.a.bu	$24$	$68.823$		None	$-1$	$-24$	$-11$	$-2$	$+$	$-$	$-$
8619.2.a.bv	$24$	$68.823$		None	$1$	$-24$	$11$	$2$	$+$	$+$	$-$
8619.2.a.bw	$24$	$68.823$		None	$7$	$-24$	$-13$	$-12$	$+$	$+$	$+$
8619.2.a.bx	$24$	$68.823$		None	$11$	$24$	$23$	$10$	$-$	$-$	$-$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(8619))$ into lower level spaces

$ S_{2}^{\mathrm{old}}(\Gamma_0(8619)) \cong $ $S_{2}^{\mathrm{new}}(\Gamma_0(17))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(39))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(51))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(169))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(221))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(507))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(663))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(2873))$$^{\oplus 2}$

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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8619.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 50 \)
Sturm bound:			\(2184\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

\(3\)	\(13\)	\(17\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(48\)
\(+\)	\(+\)	\(-\)	\(-\)	\(58\)
\(+\)	\(-\)	\(+\)	\(-\)	\(56\)
\(+\)	\(-\)	\(-\)	\(+\)	\(44\)
\(-\)	\(+\)	\(+\)	\(-\)	\(55\)
\(-\)	\(+\)	\(-\)	\(+\)	\(45\)
\(-\)	\(-\)	\(+\)	\(+\)	\(48\)
\(-\)	\(-\)	\(-\)	\(-\)	\(60\)
Plus space			\(+\)	\(185\)
Minus space			\(-\)	\(229\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8619))\) 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	8619.2.a

Level	$8619$
Weight	$2$
Character orbit	8619.a
Rep. character	$\chi_{8619}(1,\cdot)$
Character field	$\Q$
Dimension	$414$
Newform subspaces	$50$
Sturm bound	$2184$
Trace bound	$17$

Label	Dim.	\(A\)	Field	CM	Traces				A-L signs			$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	3	13	17	$q$-expansion
8619.2.a.a	\(1\)	\(68.823\)	\(\Q\)	None	\(-2\)	\(-1\)	\(-4\)	\(3\)	\(+\)	\(+\)	\(-\)	\(q-2q^{2}-q^{3}+2q^{4}-4q^{5}+2q^{6}+\cdots\)
8619.2.a.b	\(1\)	\(68.823\)	\(\Q\)	None	\(-2\)	\(-1\)	\(2\)	\(-4\)	\(+\)	\(+\)	\(-\)	\(q-2q^{2}-q^{3}+2q^{4}+2q^{5}+2q^{6}+\cdots\)
8619.2.a.c	\(1\)	\(68.823\)	\(\Q\)	None	\(-2\)	\(1\)	\(3\)	\(2\)	\(-\)	\(+\)	\(-\)	\(q-2q^{2}+q^{3}+2q^{4}+3q^{5}-2q^{6}+\cdots\)
8619.2.a.d	\(1\)	\(68.823\)	\(\Q\)	None	\(-1\)	\(-1\)	\(1\)	\(-2\)	\(+\)	\(+\)	\(-\)	\(q-q^{2}-q^{3}-q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
8619.2.a.e	\(1\)	\(68.823\)	\(\Q\)	None	\(-1\)	\(-1\)	\(4\)	\(-2\)	\(+\)	\(+\)	\(-\)	\(q-q^{2}-q^{3}-q^{4}+4q^{5}+q^{6}-2q^{7}+\cdots\)
8619.2.a.f	\(1\)	\(68.823\)	\(\Q\)	None	\(-1\)	\(1\)	\(0\)	\(-4\)	\(-\)	\(-\)	\(-\)	\(q-q^{2}+q^{3}-q^{4}-q^{6}-4q^{7}+3q^{8}+\cdots\)
8619.2.a.g	\(1\)	\(68.823\)	\(\Q\)	None	\(0\)	\(1\)	\(-3\)	\(4\)	\(-\)	\(+\)	\(+\)	\(q+q^{3}-2q^{4}-3q^{5}+4q^{7}+q^{9}+3q^{11}+\cdots\)
8619.2.a.h	\(1\)	\(68.823\)	\(\Q\)	None	\(1\)	\(-1\)	\(-1\)	\(2\)	\(+\)	\(+\)	\(-\)	\(q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
8619.2.a.i	\(1\)	\(68.823\)	\(\Q\)	None	\(1\)	\(-1\)	\(2\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q+q^{2}-q^{3}-q^{4}+2q^{5}-q^{6}-3q^{8}+\cdots\)
8619.2.a.j	\(1\)	\(68.823\)	\(\Q\)	None	\(1\)	\(1\)	\(0\)	\(2\)	\(-\)	\(+\)	\(-\)	\(q+q^{2}+q^{3}-q^{4}+q^{6}+2q^{7}-3q^{8}+\cdots\)
8619.2.a.k	\(1\)	\(68.823\)	\(\Q\)	None	\(1\)	\(1\)	\(0\)	\(4\)	\(-\)	\(-\)	\(-\)	\(q+q^{2}+q^{3}-q^{4}+q^{6}+4q^{7}-3q^{8}+\cdots\)
8619.2.a.l	\(1\)	\(68.823\)	\(\Q\)	None	\(2\)	\(-1\)	\(-2\)	\(4\)	\(+\)	\(+\)	\(-\)	\(q+2q^{2}-q^{3}+2q^{4}-2q^{5}-2q^{6}+\cdots\)
8619.2.a.m	\(1\)	\(68.823\)	\(\Q\)	None	\(2\)	\(-1\)	\(4\)	\(-3\)	\(+\)	\(+\)	\(-\)	\(q+2q^{2}-q^{3}+2q^{4}+4q^{5}-2q^{6}+\cdots\)
8619.2.a.n	\(1\)	\(68.823\)	\(\Q\)	None	\(2\)	\(1\)	\(-3\)	\(-2\)	\(-\)	\(+\)	\(-\)	\(q+2q^{2}+q^{3}+2q^{4}-3q^{5}+2q^{6}+\cdots\)
8619.2.a.o	\(2\)	\(68.823\)	\(\Q(\sqrt{3}) \)	None	\(0\)	\(-2\)	\(0\)	\(0\)	\(+\)	\(-\)	\(+\)	\(q-q^{3}-2q^{4}-\beta q^{5}+q^{9}-3\beta q^{11}+\cdots\)
8619.2.a.p	\(2\)	\(68.823\)	\(\Q(\sqrt{3}) \)	None	\(0\)	\(-2\)	\(0\)	\(0\)	\(+\)	\(-\)	\(-\)	\(q-q^{3}-2q^{4}-\beta q^{5}-2\beta q^{7}+q^{9}+\cdots\)
8619.2.a.q	\(2\)	\(68.823\)	\(\Q(\sqrt{17}) \)	None	\(1\)	\(-2\)	\(-3\)	\(0\)	\(+\)	\(+\)	\(-\)	\(q+\beta q^{2}-q^{3}+(2+\beta )q^{4}+(-1-\beta )q^{5}+\cdots\)
8619.2.a.r	\(3\)	\(68.823\)	\(\Q(\zeta_{14})^+\)	None	\(-2\)	\(3\)	\(-1\)	\(5\)	\(-\)	\(+\)	\(+\)	\(q+(-1-\beta _{2})q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
8619.2.a.s	\(3\)	\(68.823\)	3.3.148.1	None	\(1\)	\(-3\)	\(0\)	\(4\)	\(+\)	\(+\)	\(+\)	\(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
8619.2.a.t	\(3\)	\(68.823\)	\(\Q(\zeta_{14})^+\)	None	\(2\)	\(3\)	\(1\)	\(-5\)	\(-\)	\(-\)	\(+\)	\(q+(1-\beta _{1})q^{2}+q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
8619.2.a.u	\(3\)	\(68.823\)	3.3.148.1	None	\(3\)	\(3\)	\(6\)	\(2\)	\(-\)	\(+\)	\(+\)	\(q+(1+\beta _{2})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
8619.2.a.v	\(4\)	\(68.823\)	\(\Q(\zeta_{24})^+\)	None	\(0\)	\(4\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q-\beta _{2}q^{2}+q^{3}+(-\beta _{1}-\beta _{2})q^{5}-\beta _{2}q^{6}+\cdots\)
8619.2.a.w	\(5\)	\(68.823\)	5.5.153424.1	None	\(-3\)	\(5\)	\(0\)	\(-2\)	\(-\)	\(+\)	\(+\)	\(q+(-1-\beta _{3})q^{2}+q^{3}+(2+\beta _{3}-\beta _{4})q^{4}+\cdots\)
8619.2.a.x	\(5\)	\(68.823\)	5.5.1004368.1	None	\(-1\)	\(-5\)	\(-2\)	\(-8\)	\(+\)	\(+\)	\(+\)	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots\)
8619.2.a.y	\(6\)	\(68.823\)	6.6.15187408.1	None	\(-4\)	\(6\)	\(-4\)	\(-2\)	\(-\)	\(+\)	\(-\)	\(q+(-1+\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
8619.2.a.z	\(6\)	\(68.823\)	6.6.83831632.1	None	\(-2\)	\(-6\)	\(0\)	\(-2\)	\(+\)	\(+\)	\(-\)	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
8619.2.a.ba	\(6\)	\(68.823\)	6.6.485125.1	None	\(-2\)	\(6\)	\(4\)	\(5\)	\(-\)	\(+\)	\(-\)	\(q-\beta _{1}q^{2}+q^{3}+(\beta _{3}+\beta _{4})q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots\)
8619.2.a.bb	\(6\)	\(68.823\)	6.6.31337472.1	None	\(0\)	\(-6\)	\(0\)	\(0\)	\(+\)	\(-\)	\(-\)	\(q+\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{5})q^{5}+\cdots\)
8619.2.a.bc	\(6\)	\(68.823\)	6.6.485125.1	None	\(2\)	\(6\)	\(-4\)	\(-5\)	\(-\)	\(+\)	\(-\)	\(q+\beta _{1}q^{2}+q^{3}+(\beta _{3}+\beta _{4})q^{4}+(-2+\cdots)q^{5}+\cdots\)
8619.2.a.bd	\(8\)	\(68.823\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(0\)	\(-8\)	\(-1\)	\(-2\)	\(+\)	\(+\)	\(+\)	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(\beta _{1}+\beta _{7})q^{5}+\cdots\)
8619.2.a.be	\(8\)	\(68.823\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(0\)	\(-8\)	\(1\)	\(2\)	\(+\)	\(+\)	\(+\)	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
8619.2.a.bf	\(8\)	\(68.823\)	8.8.\(\cdots\).1	None	\(0\)	\(-8\)	\(0\)	\(0\)	\(+\)	\(-\)	\(+\)	\(q+(\beta _{1}+\beta _{7})q^{2}-q^{3}+(2+\beta _{5})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
8619.2.a.bg	\(8\)	\(68.823\)	8.8.\(\cdots\).1	None	\(0\)	\(8\)	\(0\)	\(0\)	\(-\)	\(-\)	\(+\)	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
8619.2.a.bh	\(9\)	\(68.823\)	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	\(-1\)	\(-9\)	\(-5\)	\(3\)	\(+\)	\(+\)	\(-\)	\(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots\)
8619.2.a.bi	\(9\)	\(68.823\)	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	\(1\)	\(-9\)	\(5\)	\(-3\)	\(+\)	\(+\)	\(-\)	\(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+(1+\cdots)q^{5}+\cdots\)
8619.2.a.bj	\(10\)	\(68.823\)	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	\(0\)	\(10\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots\)
8619.2.a.bk	\(11\)	\(68.823\)	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	\(0\)	\(11\)	\(0\)	\(-3\)	\(-\)	\(+\)	\(+\)	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots\)
8619.2.a.bl	\(11\)	\(68.823\)	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	\(0\)	\(11\)	\(0\)	\(3\)	\(-\)	\(+\)	\(+\)	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots\)
8619.2.a.bm	\(12\)	\(68.823\)	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	\(0\)	\(-12\)	\(0\)	\(0\)	\(+\)	\(-\)	\(-\)	\(q+\beta _{7}q^{2}-q^{3}+(1-\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
8619.2.a.bn	\(16\)	\(68.823\)	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	\(0\)	\(16\)	\(0\)	\(0\)	\(-\)	\(-\)	\(+\)	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{14}q^{5}+\cdots\)
8619.2.a.bo	\(20\)	\(68.823\)	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	\(0\)	\(20\)	\(0\)	\(0\)	\(-\)	\(-\)	\(-\)	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{15}q^{5}+\cdots\)
8619.2.a.bp	\(21\)	\(68.823\)		None	\(-9\)	\(21\)	\(-26\)	\(5\)	\(-\)	\(-\)	\(+\)
8619.2.a.bq	\(21\)	\(68.823\)		None	\(9\)	\(21\)	\(26\)	\(-5\)	\(-\)	\(+\)	\(+\)
8619.2.a.br	\(22\)	\(68.823\)		None	\(0\)	\(-22\)	\(0\)	\(0\)	\(+\)	\(-\)	\(+\)
8619.2.a.bs	\(24\)	\(68.823\)		None	\(-11\)	\(24\)	\(-23\)	\(-10\)	\(-\)	\(+\)	\(-\)
8619.2.a.bt	\(24\)	\(68.823\)		None	\(-7\)	\(-24\)	\(13\)	\(12\)	\(+\)	\(-\)	\(+\)
8619.2.a.bu	\(24\)	\(68.823\)		None	\(-1\)	\(-24\)	\(-11\)	\(-2\)	\(+\)	\(-\)	\(-\)
8619.2.a.bv	\(24\)	\(68.823\)		None	\(1\)	\(-24\)	\(11\)	\(2\)	\(+\)	\(+\)	\(-\)
8619.2.a.bw	\(24\)	\(68.823\)		None	\(7\)	\(-24\)	\(-13\)	\(-12\)	\(+\)	\(+\)	\(+\)
8619.2.a.bx	\(24\)	\(68.823\)		None	\(11\)	\(24\)	\(23\)	\(10\)	\(-\)	\(-\)	\(-\)