LMFDB - Space of modular forms of level 4225, weight 2, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 54 $
Sturm bound:			$910$
Trace bound:			$7$
Distinguishing $T_p$:			$2$, $3$, $7$, $11$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(4225))$.

	Total	New	Old
Modular forms	496	262	234
Cusp forms	413	229	184
Eisenstein series	83	33	50

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

$5$	$13$	Fricke	Dim.
$+$	$+$	$+$	$54$
$+$	$-$	$-$	$57$
$-$	$+$	$-$	$62$
$-$	$-$	$+$	$56$
Plus space		$+$	$110$
Minus space		$-$	$119$

Trace form

$ 229 q - 3 q^{2} + 213 q^{4} + 8 q^{6} + 4 q^{7} - 3 q^{8} + 193 q^{9} + O(q^{10}) $

$ 229 q - 3 q^{2} + 213 q^{4} + 8 q^{6} + 4 q^{7} - 3 q^{8} + 193 q^{9} - 8 q^{12} + 10 q^{14} + 173 q^{16} - 11 q^{18} + 4 q^{19} + 20 q^{21} - 6 q^{22} - 2 q^{23} + 28 q^{24} + 18 q^{27} + 28 q^{28} + 6 q^{29} + 24 q^{31} + q^{32} - 20 q^{33} + 26 q^{34} + 171 q^{36} - 10 q^{37} + 36 q^{38} - 10 q^{41} + 2 q^{43} - 32 q^{44} + 4 q^{46} + 12 q^{47} + 30 q^{48} + 147 q^{49} - 58 q^{51} + 6 q^{53} + 4 q^{54} + 12 q^{56} - 4 q^{57} + 2 q^{58} - 20 q^{59} + 24 q^{61} - 6 q^{62} - 4 q^{63} + 119 q^{64} + 26 q^{66} - 16 q^{67} + 70 q^{68} - 12 q^{71} + 21 q^{72} - 14 q^{73} + 32 q^{74} + 60 q^{76} - 4 q^{77} + 22 q^{79} + 53 q^{81} + 28 q^{82} - 40 q^{83} + 64 q^{84} - 16 q^{86} + 44 q^{87} - 34 q^{88} - 2 q^{89} + 40 q^{92} + 60 q^{93} - 30 q^{94} + 36 q^{96} - 2 q^{97} - 51 q^{98} - 80 q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(4225))$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				A-L signs		$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	5	13	$q$-expansion
4225.2.a.a	$1$	$33.737$	$\Q$	None	$-2$	$-1$	$0$	$2$	$+$	$-$	$q-2q^{2}-q^{3}+2q^{4}+2q^{6}+2q^{7}+\cdots$
4225.2.a.b	$1$	$33.737$	$\Q$	None	$-2$	$1$	$0$	$-2$	$-$	$+$	$q-2q^{2}+q^{3}+2q^{4}-2q^{6}-2q^{7}+\cdots$
4225.2.a.c	$1$	$33.737$	$\Q$	None	$-2$	$1$	$0$	$2$	$-$	$-$	$q-2q^{2}+q^{3}+2q^{4}-2q^{6}+2q^{7}+\cdots$
4225.2.a.d	$1$	$33.737$	$\Q$	None	$-1$	$-2$	$0$	$-5$	$-$	$-$	$q-q^{2}-2q^{3}-q^{4}+2q^{6}-5q^{7}+3q^{8}+\cdots$
4225.2.a.e	$1$	$33.737$	$\Q$	None	$-1$	$-2$	$0$	$0$	$-$	$-$	$q-q^{2}-2q^{3}-q^{4}+2q^{6}+3q^{8}+q^{9}+\cdots$
4225.2.a.f	$1$	$33.737$	$\Q$	None	$-1$	$2$	$0$	$-5$	$+$	$-$	$q-q^{2}+2q^{3}-q^{4}-2q^{6}-5q^{7}+3q^{8}+\cdots$
4225.2.a.g	$1$	$33.737$	$\Q$	None	$-1$	$2$	$0$	$-4$	$+$	$+$	$q-q^{2}+2q^{3}-q^{4}-2q^{6}-4q^{7}+3q^{8}+\cdots$
4225.2.a.h	$1$	$33.737$	$\Q$	None	$-1$	$2$	$0$	$0$	$-$	$-$	$q-q^{2}+2q^{3}-q^{4}-2q^{6}+3q^{8}+q^{9}+\cdots$
4225.2.a.i	$1$	$33.737$	$\Q$	None	$0$	$-1$	$0$	$-4$	$+$	$+$	$q-q^{3}-2q^{4}-4q^{7}-2q^{9}+6q^{11}+\cdots$
4225.2.a.j	$1$	$33.737$	$\Q$	None	$0$	$1$	$0$	$4$	$-$	$+$	$q+q^{3}-2q^{4}+4q^{7}-2q^{9}+6q^{11}+\cdots$
4225.2.a.k	$1$	$33.737$	$\Q$	None	$1$	$-2$	$0$	$0$	$-$	$-$	$q+q^{2}-2q^{3}-q^{4}-2q^{6}-3q^{8}+q^{9}+\cdots$
4225.2.a.l	$1$	$33.737$	$\Q$	None	$1$	$-2$	$0$	$5$	$-$	$-$	$q+q^{2}-2q^{3}-q^{4}-2q^{6}+5q^{7}-3q^{8}+\cdots$
4225.2.a.m	$1$	$33.737$	$\Q$	None	$1$	$2$	$0$	$0$	$-$	$-$	$q+q^{2}+2q^{3}-q^{4}+2q^{6}-3q^{8}+q^{9}+\cdots$
4225.2.a.n	$1$	$33.737$	$\Q$	None	$1$	$2$	$0$	$5$	$+$	$-$	$q+q^{2}+2q^{3}-q^{4}+2q^{6}+5q^{7}-3q^{8}+\cdots$
4225.2.a.o	$1$	$33.737$	$\Q$	None	$2$	$-1$	$0$	$-2$	$+$	$-$	$q+2q^{2}-q^{3}+2q^{4}-2q^{6}-2q^{7}+\cdots$
4225.2.a.p	$1$	$33.737$	$\Q$	None	$2$	$-1$	$0$	$2$	$+$	$+$	$q+2q^{2}-q^{3}+2q^{4}-2q^{6}+2q^{7}+\cdots$
4225.2.a.q	$1$	$33.737$	$\Q$	None	$2$	$1$	$0$	$-2$	$-$	$-$	$q+2q^{2}+q^{3}+2q^{4}+2q^{6}-2q^{7}+\cdots$
4225.2.a.r	$2$	$33.737$	$\Q(\sqrt{2}) $	None	$-2$	$0$	$0$	$4$	$+$	$+$	$q+(-1+\beta )q^{2}-\beta q^{3}+(1-2\beta )q^{4}+\cdots$
4225.2.a.s	$2$	$33.737$	$\Q(\sqrt{2}) $	None	$-2$	$0$	$0$	$-2$	$+$	$+$	$q+(-1+\beta )q^{2}+2\beta q^{3}+(1-2\beta )q^{4}+\cdots$
4225.2.a.t	$2$	$33.737$	$\Q(\sqrt{13}) $	None	$-1$	$-2$	$0$	$-2$	$+$	$+$	$q-\beta q^{2}-q^{3}+(1+\beta )q^{4}+\beta q^{6}-q^{7}+\cdots$
4225.2.a.u	$2$	$33.737$	$\Q(\sqrt{5}) $	None	$-1$	$0$	$0$	$-4$	$+$	$+$	$q-\beta q^{2}+(-1+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots$
4225.2.a.v	$2$	$33.737$	$\Q(\sqrt{3}) $	None	$0$	$-4$	$0$	$0$	$+$	$-$	$q+\beta q^{2}-2q^{3}+q^{4}-2\beta q^{6}-\beta q^{8}+\cdots$
4225.2.a.w	$2$	$33.737$	$\Q(\sqrt{3}) $	None	$0$	$-2$	$0$	$4$	$+$	$+$	$q+\beta q^{2}+(-1+\beta )q^{3}+q^{4}+(3-\beta )q^{6}+\cdots$
4225.2.a.x	$2$	$33.737$	$\Q(\sqrt{13}) $	None	$1$	$-2$	$0$	$2$	$+$	$+$	$q+\beta q^{2}-q^{3}+(1+\beta )q^{4}-\beta q^{6}+q^{7}+\cdots$
4225.2.a.y	$2$	$33.737$	$\Q(\sqrt{5}) $	None	$1$	$0$	$0$	$4$	$+$	$+$	$q+\beta q^{2}+(-1+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots$
4225.2.a.z	$2$	$33.737$	$\Q(\sqrt{2}) $	None	$2$	$0$	$0$	$2$	$-$	$+$	$q+(1+\beta )q^{2}+2\beta q^{3}+(1+2\beta )q^{4}+(4+\cdots)q^{6}+\cdots$
4225.2.a.ba	$3$	$33.737$	3.3.148.1	None	$-3$	$4$	$0$	$-2$	$-$	$+$	$q+(-1-\beta _{2})q^{2}+(1+\beta _{1})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots$
4225.2.a.bb	$3$	$33.737$	$\Q(\zeta_{14})^+$	None	$-2$	$2$	$0$	$-3$	$+$	$-$	$q+(-1-\beta _{2})q^{2}+(\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots$
4225.2.a.bc	$3$	$33.737$	3.3.564.1	None	$-1$	$2$	$0$	$-2$	$+$	$-$	$q-\beta _{1}q^{2}+(1-\beta _{1})q^{3}+(2+\beta _{2})q^{4}+\cdots$
4225.2.a.bd	$3$	$33.737$	$\Q(\zeta_{14})^+$	None	$-1$	$5$	$0$	$-5$	$+$	$+$	$q-\beta _{1}q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots$
4225.2.a.be	$3$	$33.737$	3.3.564.1	None	$1$	$2$	$0$	$2$	$+$	$-$	$q+\beta _{1}q^{2}+(1-\beta _{1})q^{3}+(2+\beta _{2})q^{4}+\cdots$
4225.2.a.bf	$3$	$33.737$	$\Q(\zeta_{14})^+$	None	$1$	$5$	$0$	$5$	$+$	$-$	$q+\beta _{1}q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots$
4225.2.a.bg	$3$	$33.737$	$\Q(\zeta_{14})^+$	None	$2$	$2$	$0$	$3$	$+$	$+$	$q+(1-\beta _{1})q^{2}+(1+\beta _{2})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots$
4225.2.a.bh	$3$	$33.737$	3.3.148.1	None	$3$	$-4$	$0$	$2$	$-$	$+$	$q+(1+\beta _{2})q^{2}+(-1-\beta _{1})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots$
4225.2.a.bi	$4$	$33.737$	4.4.4752.1	None	$-2$	$2$	$0$	$-10$	$+$	$-$	$q-\beta _{1}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots$
4225.2.a.bj	$4$	$33.737$	$\Q(\sqrt{3}, \sqrt{7})$	None	$0$	$-4$	$0$	$0$	$-$	$-$	$q+\beta _{3}q^{2}-q^{3}-\beta _{2}q^{4}-\beta _{3}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots$
4225.2.a.bk	$4$	$33.737$	$\Q(\sqrt{3}, \sqrt{7})$	None	$0$	$4$	$0$	$0$	$-$	$-$	$q+\beta _{3}q^{2}+q^{3}-\beta _{2}q^{4}+\beta _{3}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots$
4225.2.a.bl	$4$	$33.737$	4.4.4752.1	None	$2$	$2$	$0$	$10$	$+$	$-$	$q+\beta _{1}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots$
4225.2.a.bm	$5$	$33.737$	5.5.1068321.1	None	$0$	$-3$	$0$	$-2$	$+$	$+$	$q+\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots$
4225.2.a.bn	$5$	$33.737$	5.5.1068321.1	None	$0$	$-3$	$0$	$2$	$+$	$+$	$q-\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots$
4225.2.a.bo	$5$	$33.737$	5.5.1068321.1	None	$0$	$3$	$0$	$2$	$-$	$+$	$q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots$
4225.2.a.bp	$5$	$33.737$	5.5.1068321.1	None	$0$	$3$	$0$	$-2$	$-$	$+$	$q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots$
4225.2.a.bq	$6$	$33.737$	6.6.199374400.1	None	$0$	$0$	$0$	$0$	$-$	$+$	$q+\beta _{1}q^{2}+(-\beta _{1}+\beta _{5})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots$
4225.2.a.br	$6$	$33.737$	6.6.199374400.1	None	$0$	$0$	$0$	$0$	$-$	$+$	$q+\beta _{1}q^{2}+(\beta _{1}-\beta _{5})q^{3}+(1+\beta _{2})q^{4}+\cdots$
4225.2.a.bs	$9$	$33.737$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	$-3$	$-7$	$0$	$-7$	$+$	$+$	$q+(-\beta _{1}-\beta _{6})q^{2}+(-1+\beta _{4})q^{3}+(2+\cdots)q^{4}+\cdots$
4225.2.a.bt	$9$	$33.737$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	$3$	$-7$	$0$	$7$	$+$	$-$	$q+(\beta _{1}+\beta _{6})q^{2}+(-1+\beta _{4})q^{3}+(2-\beta _{3}+\cdots)q^{4}+\cdots$
4225.2.a.bu	$10$	$33.737$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	$0$	$-6$	$0$	$0$	$-$	$-$	$q+\beta _{1}q^{2}+(-1-\beta _{7})q^{3}+(1+\beta _{2})q^{4}+\cdots$
4225.2.a.bv	$10$	$33.737$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	$0$	$6$	$0$	$0$	$+$	$-$	$q+\beta _{1}q^{2}+(1+\beta _{7})q^{3}+(1+\beta _{2})q^{4}+\cdots$
4225.2.a.bw	$12$	$33.737$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	$-7$	$-1$	$0$	$-12$	$-$	$-$	$q+(-1+\beta _{1})q^{2}+\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$
4225.2.a.bx	$12$	$33.737$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	$-7$	$1$	$0$	$-12$	$+$	$+$	$q+(-1+\beta _{1})q^{2}-\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$
4225.2.a.by	$12$	$33.737$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	$7$	$-1$	$0$	$12$	$-$	$+$	$q+(1-\beta _{1})q^{2}+\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$
4225.2.a.bz	$12$	$33.737$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	$7$	$1$	$0$	$12$	$+$	$-$	$q+(1-\beta _{1})q^{2}-\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$
4225.2.a.ca	$18$	$33.737$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	None	$0$	$0$	$0$	$0$	$-$	$-$	$q+\beta _{1}q^{2}-\beta _{14}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{6}+\cdots$
4225.2.a.cb	$18$	$33.737$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	None	$0$	$0$	$0$	$0$	$-$	$+$	$q+\beta _{1}q^{2}+\beta _{14}q^{3}+(1+\beta _{2})q^{4}+(1+\cdots)q^{6}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(4225))$ into lower level spaces

$ S_{2}^{\mathrm{old}}(\Gamma_0(4225)) \cong $ $S_{2}^{\mathrm{new}}(\Gamma_0(65))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(169))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(325))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(845))$$^{\oplus 2}$

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Level:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 54 \)
Sturm bound:			\(910\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(3\), \(7\), \(11\)

\(5\)	\(13\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(54\)
\(+\)	\(-\)	\(-\)	\(57\)
\(-\)	\(+\)	\(-\)	\(62\)
\(-\)	\(-\)	\(+\)	\(56\)
Plus space		\(+\)	\(110\)
Minus space		\(-\)	\(119\)

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Defining parameters

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Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4225))\) 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	4225.2.a

Level	$4225$
Weight	$2$
Character orbit	4225.a
Rep. character	$\chi_{4225}(1,\cdot)$
Character field	$\Q$
Dimension	$229$
Newform subspaces	$54$
Sturm bound	$910$
Trace bound	$7$

Label	Dim.	\(A\)	Field	CM	Traces				A-L signs		$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	5	13	$q$-expansion
4225.2.a.a	\(1\)	\(33.737\)	\(\Q\)	None	\(-2\)	\(-1\)	\(0\)	\(2\)	\(+\)	\(-\)	\(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+2q^{7}+\cdots\)
4225.2.a.b	\(1\)	\(33.737\)	\(\Q\)	None	\(-2\)	\(1\)	\(0\)	\(-2\)	\(-\)	\(+\)	\(q-2q^{2}+q^{3}+2q^{4}-2q^{6}-2q^{7}+\cdots\)
4225.2.a.c	\(1\)	\(33.737\)	\(\Q\)	None	\(-2\)	\(1\)	\(0\)	\(2\)	\(-\)	\(-\)	\(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+2q^{7}+\cdots\)
4225.2.a.d	\(1\)	\(33.737\)	\(\Q\)	None	\(-1\)	\(-2\)	\(0\)	\(-5\)	\(-\)	\(-\)	\(q-q^{2}-2q^{3}-q^{4}+2q^{6}-5q^{7}+3q^{8}+\cdots\)
4225.2.a.e	\(1\)	\(33.737\)	\(\Q\)	None	\(-1\)	\(-2\)	\(0\)	\(0\)	\(-\)	\(-\)	\(q-q^{2}-2q^{3}-q^{4}+2q^{6}+3q^{8}+q^{9}+\cdots\)
4225.2.a.f	\(1\)	\(33.737\)	\(\Q\)	None	\(-1\)	\(2\)	\(0\)	\(-5\)	\(+\)	\(-\)	\(q-q^{2}+2q^{3}-q^{4}-2q^{6}-5q^{7}+3q^{8}+\cdots\)
4225.2.a.g	\(1\)	\(33.737\)	\(\Q\)	None	\(-1\)	\(2\)	\(0\)	\(-4\)	\(+\)	\(+\)	\(q-q^{2}+2q^{3}-q^{4}-2q^{6}-4q^{7}+3q^{8}+\cdots\)
4225.2.a.h	\(1\)	\(33.737\)	\(\Q\)	None	\(-1\)	\(2\)	\(0\)	\(0\)	\(-\)	\(-\)	\(q-q^{2}+2q^{3}-q^{4}-2q^{6}+3q^{8}+q^{9}+\cdots\)
4225.2.a.i	\(1\)	\(33.737\)	\(\Q\)	None	\(0\)	\(-1\)	\(0\)	\(-4\)	\(+\)	\(+\)	\(q-q^{3}-2q^{4}-4q^{7}-2q^{9}+6q^{11}+\cdots\)
4225.2.a.j	\(1\)	\(33.737\)	\(\Q\)	None	\(0\)	\(1\)	\(0\)	\(4\)	\(-\)	\(+\)	\(q+q^{3}-2q^{4}+4q^{7}-2q^{9}+6q^{11}+\cdots\)
4225.2.a.k	\(1\)	\(33.737\)	\(\Q\)	None	\(1\)	\(-2\)	\(0\)	\(0\)	\(-\)	\(-\)	\(q+q^{2}-2q^{3}-q^{4}-2q^{6}-3q^{8}+q^{9}+\cdots\)
4225.2.a.l	\(1\)	\(33.737\)	\(\Q\)	None	\(1\)	\(-2\)	\(0\)	\(5\)	\(-\)	\(-\)	\(q+q^{2}-2q^{3}-q^{4}-2q^{6}+5q^{7}-3q^{8}+\cdots\)
4225.2.a.m	\(1\)	\(33.737\)	\(\Q\)	None	\(1\)	\(2\)	\(0\)	\(0\)	\(-\)	\(-\)	\(q+q^{2}+2q^{3}-q^{4}+2q^{6}-3q^{8}+q^{9}+\cdots\)
4225.2.a.n	\(1\)	\(33.737\)	\(\Q\)	None	\(1\)	\(2\)	\(0\)	\(5\)	\(+\)	\(-\)	\(q+q^{2}+2q^{3}-q^{4}+2q^{6}+5q^{7}-3q^{8}+\cdots\)
4225.2.a.o	\(1\)	\(33.737\)	\(\Q\)	None	\(2\)	\(-1\)	\(0\)	\(-2\)	\(+\)	\(-\)	\(q+2q^{2}-q^{3}+2q^{4}-2q^{6}-2q^{7}+\cdots\)
4225.2.a.p	\(1\)	\(33.737\)	\(\Q\)	None	\(2\)	\(-1\)	\(0\)	\(2\)	\(+\)	\(+\)	\(q+2q^{2}-q^{3}+2q^{4}-2q^{6}+2q^{7}+\cdots\)
4225.2.a.q	\(1\)	\(33.737\)	\(\Q\)	None	\(2\)	\(1\)	\(0\)	\(-2\)	\(-\)	\(-\)	\(q+2q^{2}+q^{3}+2q^{4}+2q^{6}-2q^{7}+\cdots\)
4225.2.a.r	\(2\)	\(33.737\)	\(\Q(\sqrt{2}) \)	None	\(-2\)	\(0\)	\(0\)	\(4\)	\(+\)	\(+\)	\(q+(-1+\beta )q^{2}-\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
4225.2.a.s	\(2\)	\(33.737\)	\(\Q(\sqrt{2}) \)	None	\(-2\)	\(0\)	\(0\)	\(-2\)	\(+\)	\(+\)	\(q+(-1+\beta )q^{2}+2\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
4225.2.a.t	\(2\)	\(33.737\)	\(\Q(\sqrt{13}) \)	None	\(-1\)	\(-2\)	\(0\)	\(-2\)	\(+\)	\(+\)	\(q-\beta q^{2}-q^{3}+(1+\beta )q^{4}+\beta q^{6}-q^{7}+\cdots\)
4225.2.a.u	\(2\)	\(33.737\)	\(\Q(\sqrt{5}) \)	None	\(-1\)	\(0\)	\(0\)	\(-4\)	\(+\)	\(+\)	\(q-\beta q^{2}+(-1+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
4225.2.a.v	\(2\)	\(33.737\)	\(\Q(\sqrt{3}) \)	None	\(0\)	\(-4\)	\(0\)	\(0\)	\(+\)	\(-\)	\(q+\beta q^{2}-2q^{3}+q^{4}-2\beta q^{6}-\beta q^{8}+\cdots\)
4225.2.a.w	\(2\)	\(33.737\)	\(\Q(\sqrt{3}) \)	None	\(0\)	\(-2\)	\(0\)	\(4\)	\(+\)	\(+\)	\(q+\beta q^{2}+(-1+\beta )q^{3}+q^{4}+(3-\beta )q^{6}+\cdots\)
4225.2.a.x	\(2\)	\(33.737\)	\(\Q(\sqrt{13}) \)	None	\(1\)	\(-2\)	\(0\)	\(2\)	\(+\)	\(+\)	\(q+\beta q^{2}-q^{3}+(1+\beta )q^{4}-\beta q^{6}+q^{7}+\cdots\)
4225.2.a.y	\(2\)	\(33.737\)	\(\Q(\sqrt{5}) \)	None	\(1\)	\(0\)	\(0\)	\(4\)	\(+\)	\(+\)	\(q+\beta q^{2}+(-1+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
4225.2.a.z	\(2\)	\(33.737\)	\(\Q(\sqrt{2}) \)	None	\(2\)	\(0\)	\(0\)	\(2\)	\(-\)	\(+\)	\(q+(1+\beta )q^{2}+2\beta q^{3}+(1+2\beta )q^{4}+(4+\cdots)q^{6}+\cdots\)
4225.2.a.ba	\(3\)	\(33.737\)	3.3.148.1	None	\(-3\)	\(4\)	\(0\)	\(-2\)	\(-\)	\(+\)	\(q+(-1-\beta _{2})q^{2}+(1+\beta _{1})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
4225.2.a.bb	\(3\)	\(33.737\)	\(\Q(\zeta_{14})^+\)	None	\(-2\)	\(2\)	\(0\)	\(-3\)	\(+\)	\(-\)	\(q+(-1-\beta _{2})q^{2}+(\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
4225.2.a.bc	\(3\)	\(33.737\)	3.3.564.1	None	\(-1\)	\(2\)	\(0\)	\(-2\)	\(+\)	\(-\)	\(q-\beta _{1}q^{2}+(1-\beta _{1})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
4225.2.a.bd	\(3\)	\(33.737\)	\(\Q(\zeta_{14})^+\)	None	\(-1\)	\(5\)	\(0\)	\(-5\)	\(+\)	\(+\)	\(q-\beta _{1}q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
4225.2.a.be	\(3\)	\(33.737\)	3.3.564.1	None	\(1\)	\(2\)	\(0\)	\(2\)	\(+\)	\(-\)	\(q+\beta _{1}q^{2}+(1-\beta _{1})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
4225.2.a.bf	\(3\)	\(33.737\)	\(\Q(\zeta_{14})^+\)	None	\(1\)	\(5\)	\(0\)	\(5\)	\(+\)	\(-\)	\(q+\beta _{1}q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
4225.2.a.bg	\(3\)	\(33.737\)	\(\Q(\zeta_{14})^+\)	None	\(2\)	\(2\)	\(0\)	\(3\)	\(+\)	\(+\)	\(q+(1-\beta _{1})q^{2}+(1+\beta _{2})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)
4225.2.a.bh	\(3\)	\(33.737\)	3.3.148.1	None	\(3\)	\(-4\)	\(0\)	\(2\)	\(-\)	\(+\)	\(q+(1+\beta _{2})q^{2}+(-1-\beta _{1})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
4225.2.a.bi	\(4\)	\(33.737\)	4.4.4752.1	None	\(-2\)	\(2\)	\(0\)	\(-10\)	\(+\)	\(-\)	\(q-\beta _{1}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
4225.2.a.bj	\(4\)	\(33.737\)	\(\Q(\sqrt{3}, \sqrt{7})\)	None	\(0\)	\(-4\)	\(0\)	\(0\)	\(-\)	\(-\)	\(q+\beta _{3}q^{2}-q^{3}-\beta _{2}q^{4}-\beta _{3}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
4225.2.a.bk	\(4\)	\(33.737\)	\(\Q(\sqrt{3}, \sqrt{7})\)	None	\(0\)	\(4\)	\(0\)	\(0\)	\(-\)	\(-\)	\(q+\beta _{3}q^{2}+q^{3}-\beta _{2}q^{4}+\beta _{3}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
4225.2.a.bl	\(4\)	\(33.737\)	4.4.4752.1	None	\(2\)	\(2\)	\(0\)	\(10\)	\(+\)	\(-\)	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)
4225.2.a.bm	\(5\)	\(33.737\)	5.5.1068321.1	None	\(0\)	\(-3\)	\(0\)	\(-2\)	\(+\)	\(+\)	\(q+\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bn	\(5\)	\(33.737\)	5.5.1068321.1	None	\(0\)	\(-3\)	\(0\)	\(2\)	\(+\)	\(+\)	\(q-\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bo	\(5\)	\(33.737\)	5.5.1068321.1	None	\(0\)	\(3\)	\(0\)	\(2\)	\(-\)	\(+\)	\(q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bp	\(5\)	\(33.737\)	5.5.1068321.1	None	\(0\)	\(3\)	\(0\)	\(-2\)	\(-\)	\(+\)	\(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bq	\(6\)	\(33.737\)	6.6.199374400.1	None	\(0\)	\(0\)	\(0\)	\(0\)	\(-\)	\(+\)	\(q+\beta _{1}q^{2}+(-\beta _{1}+\beta _{5})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.br	\(6\)	\(33.737\)	6.6.199374400.1	None	\(0\)	\(0\)	\(0\)	\(0\)	\(-\)	\(+\)	\(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{5})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bs	\(9\)	\(33.737\)	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	\(-3\)	\(-7\)	\(0\)	\(-7\)	\(+\)	\(+\)	\(q+(-\beta _{1}-\beta _{6})q^{2}+(-1+\beta _{4})q^{3}+(2+\cdots)q^{4}+\cdots\)
4225.2.a.bt	\(9\)	\(33.737\)	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	\(3\)	\(-7\)	\(0\)	\(7\)	\(+\)	\(-\)	\(q+(\beta _{1}+\beta _{6})q^{2}+(-1+\beta _{4})q^{3}+(2-\beta _{3}+\cdots)q^{4}+\cdots\)
4225.2.a.bu	\(10\)	\(33.737\)	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	\(0\)	\(-6\)	\(0\)	\(0\)	\(-\)	\(-\)	\(q+\beta _{1}q^{2}+(-1-\beta _{7})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bv	\(10\)	\(33.737\)	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	\(0\)	\(6\)	\(0\)	\(0\)	\(+\)	\(-\)	\(q+\beta _{1}q^{2}+(1+\beta _{7})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bw	\(12\)	\(33.737\)	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	\(-7\)	\(-1\)	\(0\)	\(-12\)	\(-\)	\(-\)	\(q+(-1+\beta _{1})q^{2}+\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.bx	\(12\)	\(33.737\)	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	\(-7\)	\(1\)	\(0\)	\(-12\)	\(+\)	\(+\)	\(q+(-1+\beta _{1})q^{2}-\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.by	\(12\)	\(33.737\)	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	\(7\)	\(-1\)	\(0\)	\(12\)	\(-\)	\(+\)	\(q+(1-\beta _{1})q^{2}+\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.bz	\(12\)	\(33.737\)	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	\(7\)	\(1\)	\(0\)	\(12\)	\(+\)	\(-\)	\(q+(1-\beta _{1})q^{2}-\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.ca	\(18\)	\(33.737\)	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(-\)	\(-\)	\(q+\beta _{1}q^{2}-\beta _{14}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{6}+\cdots\)
4225.2.a.cb	\(18\)	\(33.737\)	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(-\)	\(+\)	\(q+\beta _{1}q^{2}+\beta _{14}q^{3}+(1+\beta _{2})q^{4}+(1+\cdots)q^{6}+\cdots\)