LMFDB - Space of modular forms of level 1575, weight 4, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 47 $
Sturm bound:			$960$
Trace bound:			$11$
Distinguishing $T_p$:			$2$, $11$, $13$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(1575))$.

	Total	New	Old
Modular forms	744	143	601
Cusp forms	696	143	553
Eisenstein series	48	0	48

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

$3$	$5$	$7$	Fricke	Dim.
$+$	$+$	$+$	$+$	$16$
$+$	$+$	$-$	$-$	$10$
$+$	$-$	$+$	$-$	$14$
$+$	$-$	$-$	$+$	$18$
$-$	$+$	$+$	$-$	$19$
$-$	$+$	$-$	$+$	$22$
$-$	$-$	$+$	$+$	$23$
$-$	$-$	$-$	$-$	$21$
Plus space			$+$	$79$
Minus space			$-$	$64$

Trace form

$ 143 q - q^{2} + 569 q^{4} - 7 q^{7} + 3 q^{8} + O(q^{10}) $

$ 143 q - q^{2} + 569 q^{4} - 7 q^{7} + 3 q^{8} - 108 q^{11} - 128 q^{13} + 7 q^{14} + 2469 q^{16} - 86 q^{17} - 110 q^{19} + 256 q^{22} + 412 q^{23} - 1036 q^{26} - 35 q^{28} - 410 q^{29} + 324 q^{31} - 597 q^{32} + 362 q^{34} + 86 q^{37} + 722 q^{38} - 274 q^{41} + 240 q^{43} - 870 q^{44} + 1406 q^{46} + 604 q^{47} + 7007 q^{49} - 836 q^{52} + 394 q^{53} - 147 q^{56} - 658 q^{58} + 562 q^{59} + 296 q^{61} - 3516 q^{62} + 9031 q^{64} + 156 q^{67} - 490 q^{68} + 2248 q^{71} - 1026 q^{73} + 6316 q^{74} - 738 q^{76} + 896 q^{77} + 4260 q^{79} + 6650 q^{82} + 422 q^{83} + 4222 q^{86} + 4204 q^{88} + 1530 q^{89} + 196 q^{91} + 1368 q^{92} + 10024 q^{94} - 1270 q^{97} - 49 q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(1575))$ 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	5	7	$q$-expansion
1575.4.a.a	$1$	$92.928$	$\Q$	None	$-3$	$0$	$0$	$-7$	$-$	$+$	$+$	$q-3q^{2}+q^{4}-7q^{7}+21q^{8}+6q^{11}+\cdots$
1575.4.a.b	$1$	$92.928$	$\Q$	None	$-3$	$0$	$0$	$-7$	$-$	$+$	$+$	$q-3q^{2}+q^{4}-7q^{7}+21q^{8}+6^{2}q^{11}+\cdots$
1575.4.a.c	$1$	$92.928$	$\Q$	None	$-3$	$0$	$0$	$-7$	$+$	$+$	$+$	$q-3q^{2}+q^{4}-7q^{7}+21q^{8}+60q^{11}+\cdots$
1575.4.a.d	$1$	$92.928$	$\Q$	None	$-2$	$0$	$0$	$7$	$-$	$-$	$-$	$q-2q^{2}-4q^{4}+7q^{7}+24q^{8}+21q^{11}+\cdots$
1575.4.a.e	$1$	$92.928$	$\Q$	None	$-1$	$0$	$0$	$7$	$-$	$+$	$-$	$q-q^{2}-7q^{4}+7q^{7}+15q^{8}+8q^{11}+\cdots$
1575.4.a.f	$1$	$92.928$	$\Q$	None	$0$	$0$	$0$	$-7$	$-$	$+$	$+$	$q-8q^{4}-7q^{7}-42q^{11}-20q^{13}+\cdots$
1575.4.a.g	$1$	$92.928$	$\Q$	None	$1$	$0$	$0$	$-7$	$-$	$+$	$+$	$q+q^{2}-7q^{4}-7q^{7}-15q^{8}-12q^{11}+\cdots$
1575.4.a.h	$1$	$92.928$	$\Q$	None	$2$	$0$	$0$	$-7$	$-$	$+$	$+$	$q+2q^{2}-4q^{4}-7q^{7}-24q^{8}+21q^{11}+\cdots$
1575.4.a.i	$1$	$92.928$	$\Q$	None	$3$	$0$	$0$	$-7$	$+$	$+$	$+$	$q+3q^{2}+q^{4}-7q^{7}-21q^{8}-60q^{11}+\cdots$
1575.4.a.j	$1$	$92.928$	$\Q$	None	$3$	$0$	$0$	$7$	$-$	$-$	$-$	$q+3q^{2}+q^{4}+7q^{7}-21q^{8}+6q^{11}+\cdots$
1575.4.a.k	$1$	$92.928$	$\Q$	None	$4$	$0$	$0$	$7$	$-$	$+$	$-$	$q+4q^{2}+8q^{4}+7q^{7}-62q^{11}+62q^{13}+\cdots$
1575.4.a.l	$1$	$92.928$	$\Q$	None	$5$	$0$	$0$	$-7$	$-$	$+$	$+$	$q+5q^{2}+17q^{4}-7q^{7}+45q^{8}-12q^{11}+\cdots$
1575.4.a.m	$2$	$92.928$	$\Q(\sqrt{17}) $	None	$-7$	$0$	$0$	$14$	$-$	$+$	$-$	$q+(-3-\beta )q^{2}+(5+7\beta )q^{4}+7q^{7}+\cdots$
1575.4.a.n	$2$	$92.928$	$\Q(\sqrt{5}) $	None	$-4$	$0$	$0$	$14$	$-$	$+$	$-$	$q+(-2-\beta )q^{2}+(1+4\beta )q^{4}+7q^{7}+\cdots$
1575.4.a.o	$2$	$92.928$	$\Q(\sqrt{17}) $	None	$-3$	$0$	$0$	$-14$	$-$	$+$	$+$	$q+(-1-\beta )q^{2}+(-3+3\beta )q^{4}-7q^{7}+\cdots$
1575.4.a.p	$2$	$92.928$	$\Q(\sqrt{57}) $	None	$-3$	$0$	$0$	$-14$	$-$	$+$	$+$	$q+(-1-\beta )q^{2}+(7+3\beta )q^{4}-7q^{7}+\cdots$
1575.4.a.q	$2$	$92.928$	$\Q(\sqrt{2}) $	None	$-2$	$0$	$0$	$14$	$-$	$+$	$-$	$q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+7q^{7}+\cdots$
1575.4.a.r	$2$	$92.928$	$\Q(\sqrt{17}) $	None	$-1$	$0$	$0$	$14$	$+$	$+$	$-$	$q-\beta q^{2}+(-4+\beta )q^{4}+7q^{7}+(-4+\cdots)q^{8}+\cdots$
1575.4.a.s	$2$	$92.928$	$\Q(\sqrt{41}) $	None	$-1$	$0$	$0$	$14$	$-$	$+$	$-$	$q-\beta q^{2}+(2+\beta )q^{4}+7q^{7}+(-10+5\beta )q^{8}+\cdots$
1575.4.a.t	$2$	$92.928$	$\Q(\sqrt{19}) $	None	$0$	$0$	$0$	$14$	$+$	$+$	$-$	$q+\beta q^{2}+11q^{4}+7q^{7}+3\beta q^{8}+10\beta q^{11}+\cdots$
1575.4.a.u	$2$	$92.928$	$\Q(\sqrt{17}) $	None	$1$	$0$	$0$	$14$	$+$	$+$	$-$	$q+\beta q^{2}+(-4+\beta )q^{4}+7q^{7}+(4-11\beta )q^{8}+\cdots$
1575.4.a.v	$2$	$92.928$	$\Q(\sqrt{41}) $	None	$1$	$0$	$0$	$-14$	$-$	$-$	$+$	$q+\beta q^{2}+(2+\beta )q^{4}-7q^{7}+(10-5\beta )q^{8}+\cdots$
1575.4.a.w	$2$	$92.928$	$\Q(\sqrt{65}) $	None	$1$	$0$	$0$	$14$	$-$	$+$	$-$	$q+\beta q^{2}+(8+\beta )q^{4}+7q^{7}+(2^{4}+\beta )q^{8}+\cdots$
1575.4.a.x	$2$	$92.928$	$\Q(\sqrt{17}) $	None	$3$	$0$	$0$	$14$	$-$	$-$	$-$	$q+(1+\beta )q^{2}+(-3+3\beta )q^{4}+7q^{7}+\cdots$
1575.4.a.y	$2$	$92.928$	$\Q(\sqrt{41}) $	None	$3$	$0$	$0$	$-14$	$-$	$+$	$+$	$q+(1+\beta )q^{2}+(3+3\beta )q^{4}-7q^{7}+(5^{2}+\cdots)q^{8}+\cdots$
1575.4.a.z	$2$	$92.928$	$\Q(\sqrt{2}) $	None	$8$	$0$	$0$	$14$	$-$	$+$	$-$	$q+(4+\beta )q^{2}+(10+8\beta )q^{4}+7q^{7}+(24+\cdots)q^{8}+\cdots$
1575.4.a.ba	$3$	$92.928$	3.3.14360.1	None	$-3$	$0$	$0$	$-21$	$-$	$+$	$+$	$q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}+\cdots$
1575.4.a.bb	$3$	$92.928$	3.3.22952.1	None	$-2$	$0$	$0$	$-21$	$+$	$+$	$+$	$q+(-1+\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots$
1575.4.a.bc	$3$	$92.928$	3.3.2292.1	None	$-1$	$0$	$0$	$-21$	$-$	$-$	$+$	$q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}-7q^{7}+\cdots$
1575.4.a.bd	$3$	$92.928$	3.3.2292.1	None	$1$	$0$	$0$	$21$	$-$	$-$	$-$	$q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+7q^{7}+\cdots$
1575.4.a.be	$3$	$92.928$	3.3.22952.1	None	$2$	$0$	$0$	$-21$	$+$	$+$	$+$	$q+(1-\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}-7q^{7}+\cdots$
1575.4.a.bf	$4$	$92.928$	4.4.26729725.1	None	$-6$	$0$	$0$	$-28$	$-$	$-$	$+$	$q+(-1-\beta _{1})q^{2}+(3+3\beta _{1}+\beta _{2})q^{4}+\cdots$
1575.4.a.bg	$4$	$92.928$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	$-4$	$0$	$0$	$28$	$-$	$-$	$-$	$q+(-1+\beta _{1})q^{2}+(9+\beta _{2}-\beta _{3})q^{4}+\cdots$
1575.4.a.bh	$4$	$92.928$	4.4.3030748.1	None	$0$	$0$	$0$	$-28$	$+$	$-$	$+$	$q+\beta _{1}q^{2}+(3+\beta _{3})q^{4}-7q^{7}+(3\beta _{1}+\cdots)q^{8}+\cdots$
1575.4.a.bi	$4$	$92.928$	4.4.3030748.1	None	$0$	$0$	$0$	$28$	$+$	$+$	$-$	$q+\beta _{1}q^{2}+(3+\beta _{3})q^{4}+7q^{7}+(3\beta _{1}+\cdots)q^{8}+\cdots$
1575.4.a.bj	$4$	$92.928$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	$0$	$0$	$0$	$-28$	$-$	$-$	$+$	$q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}-7q^{7}+(2+4\beta _{1}+\cdots)q^{8}+\cdots$
1575.4.a.bk	$4$	$92.928$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	$0$	$0$	$0$	$28$	$-$	$+$	$-$	$q-\beta _{1}q^{2}+(4+\beta _{2})q^{4}+7q^{7}+(-2+\cdots)q^{8}+\cdots$
1575.4.a.bl	$4$	$92.928$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	$4$	$0$	$0$	$-28$	$-$	$+$	$+$	$q+(1-\beta _{1})q^{2}+(9+\beta _{2}-\beta _{3})q^{4}-7q^{7}+\cdots$
1575.4.a.bm	$4$	$92.928$	4.4.26729725.1	None	$6$	$0$	$0$	$28$	$-$	$+$	$-$	$q+(1+\beta _{1})q^{2}+(3+3\beta _{1}+\beta _{2})q^{4}+7q^{7}+\cdots$
1575.4.a.bn	$5$	$92.928$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	$-4$	$0$	$0$	$35$	$-$	$-$	$-$	$q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots$
1575.4.a.bo	$5$	$92.928$	5.5.78066700.1	None	$-1$	$0$	$0$	$35$	$-$	$-$	$-$	$q+\beta _{1}q^{2}+(5-\beta _{1}-\beta _{3})q^{4}+7q^{7}+\cdots$
1575.4.a.bp	$5$	$92.928$	5.5.78066700.1	None	$1$	$0$	$0$	$-35$	$-$	$-$	$+$	$q-\beta _{1}q^{2}+(5-\beta _{1}-\beta _{3})q^{4}-7q^{7}+\cdots$
1575.4.a.bq	$5$	$92.928$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	$4$	$0$	$0$	$-35$	$-$	$-$	$+$	$q+(1-\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots$
1575.4.a.br	$8$	$92.928$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$0$	$0$	$0$	$-56$	$+$	$+$	$+$	$q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}-7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots$
1575.4.a.bs	$8$	$92.928$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$0$	$0$	$0$	$56$	$+$	$-$	$-$	$q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots$
1575.4.a.bt	$10$	$92.928$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	$0$	$0$	$0$	$-70$	$+$	$-$	$+$	$q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}-7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots$
1575.4.a.bu	$10$	$92.928$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	$0$	$0$	$0$	$70$	$+$	$-$	$-$	$q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(1575))$ into lower level spaces

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Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 47 \)
Sturm bound:			\(960\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(11\), \(13\)

\(3\)	\(5\)	\(7\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(16\)
\(+\)	\(+\)	\(-\)	\(-\)	\(10\)
\(+\)	\(-\)	\(+\)	\(-\)	\(14\)
\(+\)	\(-\)	\(-\)	\(+\)	\(18\)
\(-\)	\(+\)	\(+\)	\(-\)	\(19\)
\(-\)	\(+\)	\(-\)	\(+\)	\(22\)
\(-\)	\(-\)	\(+\)	\(+\)	\(23\)
\(-\)	\(-\)	\(-\)	\(-\)	\(21\)
Plus space			\(+\)	\(79\)
Minus space			\(-\)	\(64\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\) into newform subspaces

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1575))\) 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	1575.4.a

Level	$1575$
Weight	$4$
Character orbit	1575.a
Rep. character	$\chi_{1575}(1,\cdot)$
Character field	$\Q$
Dimension	$143$
Newform subspaces	$47$
Sturm bound	$960$
Trace bound	$11$

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	5	7	$q$-expansion
1575.4.a.a	$1$	$92.928$	\(\Q\)	None	\(-3\)	\(0\)	\(0\)	\(-7\)	$-$	$+$	$+$	\(q-3q^{2}+q^{4}-7q^{7}+21q^{8}+6q^{11}+\cdots\)
1575.4.a.b	$1$	$92.928$	\(\Q\)	None	\(-3\)	\(0\)	\(0\)	\(-7\)	$-$	$+$	$+$	\(q-3q^{2}+q^{4}-7q^{7}+21q^{8}+6^{2}q^{11}+\cdots\)
1575.4.a.c	$1$	$92.928$	\(\Q\)	None	\(-3\)	\(0\)	\(0\)	\(-7\)	$+$	$+$	$+$	\(q-3q^{2}+q^{4}-7q^{7}+21q^{8}+60q^{11}+\cdots\)
1575.4.a.d	$1$	$92.928$	\(\Q\)	None	\(-2\)	\(0\)	\(0\)	\(7\)	$-$	$-$	$-$	\(q-2q^{2}-4q^{4}+7q^{7}+24q^{8}+21q^{11}+\cdots\)
1575.4.a.e	$1$	$92.928$	\(\Q\)	None	\(-1\)	\(0\)	\(0\)	\(7\)	$-$	$+$	$-$	\(q-q^{2}-7q^{4}+7q^{7}+15q^{8}+8q^{11}+\cdots\)
1575.4.a.f	$1$	$92.928$	\(\Q\)	None	\(0\)	\(0\)	\(0\)	\(-7\)	$-$	$+$	$+$	\(q-8q^{4}-7q^{7}-42q^{11}-20q^{13}+\cdots\)
1575.4.a.g	$1$	$92.928$	\(\Q\)	None	\(1\)	\(0\)	\(0\)	\(-7\)	$-$	$+$	$+$	\(q+q^{2}-7q^{4}-7q^{7}-15q^{8}-12q^{11}+\cdots\)
1575.4.a.h	$1$	$92.928$	\(\Q\)	None	\(2\)	\(0\)	\(0\)	\(-7\)	$-$	$+$	$+$	\(q+2q^{2}-4q^{4}-7q^{7}-24q^{8}+21q^{11}+\cdots\)
1575.4.a.i	$1$	$92.928$	\(\Q\)	None	\(3\)	\(0\)	\(0\)	\(-7\)	$+$	$+$	$+$	\(q+3q^{2}+q^{4}-7q^{7}-21q^{8}-60q^{11}+\cdots\)
1575.4.a.j	$1$	$92.928$	\(\Q\)	None	\(3\)	\(0\)	\(0\)	\(7\)	$-$	$-$	$-$	\(q+3q^{2}+q^{4}+7q^{7}-21q^{8}+6q^{11}+\cdots\)
1575.4.a.k	$1$	$92.928$	\(\Q\)	None	\(4\)	\(0\)	\(0\)	\(7\)	$-$	$+$	$-$	\(q+4q^{2}+8q^{4}+7q^{7}-62q^{11}+62q^{13}+\cdots\)
1575.4.a.l	$1$	$92.928$	\(\Q\)	None	\(5\)	\(0\)	\(0\)	\(-7\)	$-$	$+$	$+$	\(q+5q^{2}+17q^{4}-7q^{7}+45q^{8}-12q^{11}+\cdots\)
1575.4.a.m	$2$	$92.928$	\(\Q(\sqrt{17}) \)	None	\(-7\)	\(0\)	\(0\)	\(14\)	$-$	$+$	$-$	\(q+(-3-\beta )q^{2}+(5+7\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.n	$2$	$92.928$	\(\Q(\sqrt{5}) \)	None	\(-4\)	\(0\)	\(0\)	\(14\)	$-$	$+$	$-$	\(q+(-2-\beta )q^{2}+(1+4\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.o	$2$	$92.928$	\(\Q(\sqrt{17}) \)	None	\(-3\)	\(0\)	\(0\)	\(-14\)	$-$	$+$	$+$	\(q+(-1-\beta )q^{2}+(-3+3\beta )q^{4}-7q^{7}+\cdots\)
1575.4.a.p	$2$	$92.928$	\(\Q(\sqrt{57}) \)	None	\(-3\)	\(0\)	\(0\)	\(-14\)	$-$	$+$	$+$	\(q+(-1-\beta )q^{2}+(7+3\beta )q^{4}-7q^{7}+\cdots\)
1575.4.a.q	$2$	$92.928$	\(\Q(\sqrt{2}) \)	None	\(-2\)	\(0\)	\(0\)	\(14\)	$-$	$+$	$-$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.r	$2$	$92.928$	\(\Q(\sqrt{17}) \)	None	\(-1\)	\(0\)	\(0\)	\(14\)	$+$	$+$	$-$	\(q-\beta q^{2}+(-4+\beta )q^{4}+7q^{7}+(-4+\cdots)q^{8}+\cdots\)
1575.4.a.s	$2$	$92.928$	\(\Q(\sqrt{41}) \)	None	\(-1\)	\(0\)	\(0\)	\(14\)	$-$	$+$	$-$	\(q-\beta q^{2}+(2+\beta )q^{4}+7q^{7}+(-10+5\beta )q^{8}+\cdots\)
1575.4.a.t	$2$	$92.928$	\(\Q(\sqrt{19}) \)	None	\(0\)	\(0\)	\(0\)	\(14\)	$+$	$+$	$-$	\(q+\beta q^{2}+11q^{4}+7q^{7}+3\beta q^{8}+10\beta q^{11}+\cdots\)
1575.4.a.u	$2$	$92.928$	\(\Q(\sqrt{17}) \)	None	\(1\)	\(0\)	\(0\)	\(14\)	$+$	$+$	$-$	\(q+\beta q^{2}+(-4+\beta )q^{4}+7q^{7}+(4-11\beta )q^{8}+\cdots\)
1575.4.a.v	$2$	$92.928$	\(\Q(\sqrt{41}) \)	None	\(1\)	\(0\)	\(0\)	\(-14\)	$-$	$-$	$+$	\(q+\beta q^{2}+(2+\beta )q^{4}-7q^{7}+(10-5\beta )q^{8}+\cdots\)
1575.4.a.w	$2$	$92.928$	\(\Q(\sqrt{65}) \)	None	\(1\)	\(0\)	\(0\)	\(14\)	$-$	$+$	$-$	\(q+\beta q^{2}+(8+\beta )q^{4}+7q^{7}+(2^{4}+\beta )q^{8}+\cdots\)
1575.4.a.x	$2$	$92.928$	\(\Q(\sqrt{17}) \)	None	\(3\)	\(0\)	\(0\)	\(14\)	$-$	$-$	$-$	\(q+(1+\beta )q^{2}+(-3+3\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.y	$2$	$92.928$	\(\Q(\sqrt{41}) \)	None	\(3\)	\(0\)	\(0\)	\(-14\)	$-$	$+$	$+$	\(q+(1+\beta )q^{2}+(3+3\beta )q^{4}-7q^{7}+(5^{2}+\cdots)q^{8}+\cdots\)
1575.4.a.z	$2$	$92.928$	\(\Q(\sqrt{2}) \)	None	\(8\)	\(0\)	\(0\)	\(14\)	$-$	$+$	$-$	\(q+(4+\beta )q^{2}+(10+8\beta )q^{4}+7q^{7}+(24+\cdots)q^{8}+\cdots\)
1575.4.a.ba	$3$	$92.928$	3.3.14360.1	None	\(-3\)	\(0\)	\(0\)	\(-21\)	$-$	$+$	$+$	\(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1575.4.a.bb	$3$	$92.928$	3.3.22952.1	None	\(-2\)	\(0\)	\(0\)	\(-21\)	$+$	$+$	$+$	\(q+(-1+\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
1575.4.a.bc	$3$	$92.928$	3.3.2292.1	None	\(-1\)	\(0\)	\(0\)	\(-21\)	$-$	$-$	$+$	\(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}-7q^{7}+\cdots\)
1575.4.a.bd	$3$	$92.928$	3.3.2292.1	None	\(1\)	\(0\)	\(0\)	\(21\)	$-$	$-$	$-$	\(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+7q^{7}+\cdots\)
1575.4.a.be	$3$	$92.928$	3.3.22952.1	None	\(2\)	\(0\)	\(0\)	\(-21\)	$+$	$+$	$+$	\(q+(1-\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}-7q^{7}+\cdots\)
1575.4.a.bf	$4$	$92.928$	4.4.26729725.1	None	\(-6\)	\(0\)	\(0\)	\(-28\)	$-$	$-$	$+$	\(q+(-1-\beta _{1})q^{2}+(3+3\beta _{1}+\beta _{2})q^{4}+\cdots\)
1575.4.a.bg	$4$	$92.928$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	\(-4\)	\(0\)	\(0\)	\(28\)	$-$	$-$	$-$	\(q+(-1+\beta _{1})q^{2}+(9+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1575.4.a.bh	$4$	$92.928$	4.4.3030748.1	None	\(0\)	\(0\)	\(0\)	\(-28\)	$+$	$-$	$+$	\(q+\beta _{1}q^{2}+(3+\beta _{3})q^{4}-7q^{7}+(3\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bi	$4$	$92.928$	4.4.3030748.1	None	\(0\)	\(0\)	\(0\)	\(28\)	$+$	$+$	$-$	\(q+\beta _{1}q^{2}+(3+\beta _{3})q^{4}+7q^{7}+(3\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bj	$4$	$92.928$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(-28\)	$-$	$-$	$+$	\(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}-7q^{7}+(2+4\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bk	$4$	$92.928$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(28\)	$-$	$+$	$-$	\(q-\beta _{1}q^{2}+(4+\beta _{2})q^{4}+7q^{7}+(-2+\cdots)q^{8}+\cdots\)
1575.4.a.bl	$4$	$92.928$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	\(4\)	\(0\)	\(0\)	\(-28\)	$-$	$+$	$+$	\(q+(1-\beta _{1})q^{2}+(9+\beta _{2}-\beta _{3})q^{4}-7q^{7}+\cdots\)
1575.4.a.bm	$4$	$92.928$	4.4.26729725.1	None	\(6\)	\(0\)	\(0\)	\(28\)	$-$	$+$	$-$	\(q+(1+\beta _{1})q^{2}+(3+3\beta _{1}+\beta _{2})q^{4}+7q^{7}+\cdots\)
1575.4.a.bn	$5$	$92.928$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	\(-4\)	\(0\)	\(0\)	\(35\)	$-$	$-$	$-$	\(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1575.4.a.bo	$5$	$92.928$	5.5.78066700.1	None	\(-1\)	\(0\)	\(0\)	\(35\)	$-$	$-$	$-$	\(q+\beta _{1}q^{2}+(5-\beta _{1}-\beta _{3})q^{4}+7q^{7}+\cdots\)
1575.4.a.bp	$5$	$92.928$	5.5.78066700.1	None	\(1\)	\(0\)	\(0\)	\(-35\)	$-$	$-$	$+$	\(q-\beta _{1}q^{2}+(5-\beta _{1}-\beta _{3})q^{4}-7q^{7}+\cdots\)
1575.4.a.bq	$5$	$92.928$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	\(4\)	\(0\)	\(0\)	\(-35\)	$-$	$-$	$+$	\(q+(1-\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1575.4.a.br	$8$	$92.928$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(-56\)	$+$	$+$	$+$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}-7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bs	$8$	$92.928$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(56\)	$+$	$-$	$-$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bt	$10$	$92.928$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(-70\)	$+$	$-$	$+$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}-7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bu	$10$	$92.928$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(70\)	$+$	$-$	$-$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)