LMFDB - Space of modular forms of level 8450, weight 2, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 79 $
Sturm bound:			$2730$
Trace bound:			$17$
Distinguishing $T_p$:			$3$, $7$, $11$, $17$, $31$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(8450))$.

	Total	New	Old
Modular forms	1448	245	1203
Cusp forms	1281	245	1036
Eisenstein series	167	0	167

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

$2$	$5$	$13$	Fricke	Dim
$+$	$+$	$+$	$+$	$24$
$+$	$+$	$-$	$-$	$33$
$+$	$-$	$+$	$-$	$34$
$+$	$-$	$-$	$+$	$31$
$-$	$+$	$+$	$-$	$35$
$-$	$+$	$-$	$+$	$24$
$-$	$-$	$+$	$+$	$27$
$-$	$-$	$-$	$-$	$37$
Plus space			$+$	$106$
Minus space			$-$	$139$

Trace form

$ 245 q + q^{2} - 2 q^{3} + 245 q^{4} - 2 q^{6} - 8 q^{7} + q^{8} + 245 q^{9} + O(q^{10}) $

$ 245 q + q^{2} - 2 q^{3} + 245 q^{4} - 2 q^{6} - 8 q^{7} + q^{8} + 245 q^{9} + 2 q^{11} - 2 q^{12} + 245 q^{16} - 10 q^{17} + 5 q^{18} - 2 q^{19} - 16 q^{21} - 6 q^{22} + 8 q^{23} - 2 q^{24} - 8 q^{27} - 8 q^{28} - 8 q^{29} - 20 q^{31} + q^{32} + 20 q^{33} + 4 q^{34} + 245 q^{36} + 2 q^{37} - 2 q^{38} + 8 q^{41} - 12 q^{42} - 10 q^{43} + 2 q^{44} - 8 q^{46} - 2 q^{48} + 225 q^{49} + 70 q^{51} + 28 q^{53} - 2 q^{54} - 8 q^{57} + 2 q^{58} + 36 q^{59} - 16 q^{61} + 245 q^{64} + 14 q^{66} - 10 q^{68} + 72 q^{69} + 24 q^{71} + 5 q^{72} + 2 q^{73} + 8 q^{74} - 2 q^{76} + 36 q^{77} + 317 q^{81} + 22 q^{82} + 24 q^{83} - 16 q^{84} - 8 q^{86} + 28 q^{87} - 6 q^{88} + 28 q^{89} + 8 q^{92} - 32 q^{93} + 16 q^{94} - 2 q^{96} + 6 q^{97} - 7 q^{98} + 40 q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(8450))$ into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	2	5	13	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
8450.2.a.a	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$-1$	$-2$	$0$	$-5$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2q^{3}+q^{4}+2q^{6}-5q^{7}-q^{8}+\cdots$
8450.2.a.b	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$-1$	$-2$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots$
8450.2.a.c	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$1$	$-1$	$-1$	$0$	$-1$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots$
8450.2.a.d	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$1$	$-1$	$-1$	$0$	$2$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots$
8450.2.a.e	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$-1$	$-1$	$0$	$4$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+4q^{7}-q^{8}+\cdots$
8450.2.a.f	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$1$	$-1$	$0$	$0$	$-4$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-4q^{7}-q^{8}-3q^{9}+4q^{11}+\cdots$
8450.2.a.g	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$1$	$-1$	$0$	$0$	$3$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+3q^{7}-q^{8}-3q^{9}-3q^{11}+\cdots$
8450.2.a.h	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$-1$	$1$	$0$	$3$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+3q^{7}-q^{8}+\cdots$
8450.2.a.i	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$1$	$-1$	$2$	$0$	$-4$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}-4q^{7}-q^{8}+\cdots$
8450.2.a.j	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$1$	$-1$	$2$	$0$	$-1$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{7}-q^{8}+\cdots$
8450.2.a.k	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$1$	$-1$	$2$	$0$	$-1$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{7}-q^{8}+\cdots$
8450.2.a.l	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$-1$	$3$	$0$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+3q^{3}+q^{4}-3q^{6}-q^{8}+6q^{9}+\cdots$
8450.2.a.m	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$1$	$-3$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-3q^{3}+q^{4}-3q^{6}+q^{8}+6q^{9}+\cdots$
8450.2.a.n	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$1$	$-2$	$0$	$-4$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}-4q^{7}+q^{8}+\cdots$
8450.2.a.o	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$1$	$1$	$-2$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{8}+q^{9}+\cdots$
8450.2.a.p	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$1$	$1$	$-2$	$0$	$1$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{7}+q^{8}+\cdots$
8450.2.a.q	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$1$	$0$	$0$	$-3$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-3q^{7}+q^{8}-3q^{9}+3q^{11}+\cdots$
8450.2.a.r	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$1$	$0$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}-3q^{9}+q^{16}-2q^{17}+\cdots$
8450.2.a.s	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$1$	$0$	$0$	$4$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+4q^{7}+q^{8}-3q^{9}-4q^{11}+\cdots$
8450.2.a.t	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$1$	$1$	$0$	$-4$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots$
8450.2.a.u	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$1$	$1$	$1$	$0$	$-3$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-3q^{7}+q^{8}+\cdots$
8450.2.a.v	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$1$	$1$	$1$	$0$	$-2$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots$
8450.2.a.w	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$1$	$2$	$0$	$1$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{7}+q^{8}+\cdots$
8450.2.a.x	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$1$	$2$	$0$	$5$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2q^{3}+q^{4}+2q^{6}+5q^{7}+q^{8}+\cdots$
8450.2.a.y	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓	$1$	$0$	$1$	$3$	$0$	$1$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+3q^{3}+q^{4}+3q^{6}+q^{7}+q^{8}+\cdots$
8450.2.a.z	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{3}) $	None	✓	$1$	$1$	$-2$	$-2$	$0$	$-2$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1+\beta )q^{3}+q^{4}+(1-\beta )q^{6}+\cdots$
8450.2.a.ba	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{13}) $	None	✓	$1$	$0$	$-2$	$-1$	$0$	$-5$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta q^{3}+q^{4}+\beta q^{6}+(-3+\beta )q^{7}+\cdots$
8450.2.a.bb	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{21}) $	None	✓	$1$	$0$	$-2$	$-1$	$0$	$1$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta q^{3}+q^{4}+\beta q^{6}+(1-\beta )q^{7}+\cdots$
8450.2.a.bc	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{10}) $	None	✓	$1$	$1$	$-2$	$0$	$0$	$2$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+q^{7}-q^{8}+\cdots$
8450.2.a.bd	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{13}) $	None	✓	$1$	$1$	$-2$	$1$	$0$	$-5$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+(-3+\beta )q^{7}+\cdots$
8450.2.a.be	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{21}) $	None	✓	$1$	$1$	$-2$	$1$	$0$	$1$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+(1-\beta )q^{7}+\cdots$
8450.2.a.bf	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{3}) $	None	✓	$1$	$0$	$-2$	$2$	$0$	$6$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots$
8450.2.a.bg	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{3}) $	None	✓	$1$	$0$	$2$	$-2$	$0$	$2$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots$
8450.2.a.bh	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{21}) $	None	✓	$1$	$1$	$2$	$-1$	$0$	$-1$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta q^{3}+q^{4}-\beta q^{6}+(-1+\beta )q^{7}+\cdots$
8450.2.a.bi	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{13}) $	None	✓	$1$	$1$	$2$	$-1$	$0$	$5$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta q^{3}+q^{4}-\beta q^{6}+(3-\beta )q^{7}+\cdots$
8450.2.a.bj	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{10}) $	None	✓	$1$	$0$	$2$	$0$	$0$	$-2$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}-q^{7}+q^{8}+\cdots$
8450.2.a.bk	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{21}) $	None	✓	$1$	$0$	$2$	$1$	$0$	$-1$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+(-1+\beta )q^{7}+\cdots$
8450.2.a.bl	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{13}) $	None	✓	$1$	$0$	$2$	$1$	$0$	$5$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+(3-\beta )q^{7}+\cdots$
8450.2.a.bm	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{3}) $	None	✓	$1$	$1$	$2$	$2$	$0$	$-6$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots$
8450.2.a.bn	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	$1$	$1$	$-3$	$-3$	$0$	$4$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(1+\cdots)q^{6}+\cdots$
8450.2.a.bo	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	$1$	$1$	$-3$	$-1$	$0$	$-5$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots$
8450.2.a.bp	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	$1$	$1$	$-3$	$-1$	$0$	$4$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(2-\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.bq	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.2808.1	None	✓	$1$	$1$	$-3$	$0$	$0$	$-3$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots$
8450.2.a.br	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.2808.1	None	✓	$1$	$0$	$-3$	$0$	$0$	$-3$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots$
8450.2.a.bs	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.940.1	None	✓	$1$	$0$	$-3$	$0$	$0$	$2$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.bt	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.148.1	None	✓	$1$	$0$	$-3$	$0$	$0$	$5$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}+(2-\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.bu	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.148.1	None	✓	$1$	$0$	$-3$	$0$	$0$	$5$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+(2-\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.bv	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	$1$	$0$	$-3$	$1$	$0$	$-2$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}-2\beta _{1}q^{7}+\cdots$
8450.2.a.bw	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	$1$	$1$	$-3$	$1$	$0$	$4$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(2-\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.bx	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	$1$	$1$	$3$	$-3$	$0$	$-4$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(-1+\cdots)q^{6}+\cdots$
8450.2.a.by	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	$1$	$1$	$3$	$-1$	$0$	$-4$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-2+\cdots)q^{7}+\cdots$
8450.2.a.bz	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	$1$	$1$	$3$	$-1$	$0$	$5$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.ca	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.148.1	None	✓	$1$	$1$	$3$	$0$	$0$	$-5$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}+(-2+\cdots)q^{7}+\cdots$
8450.2.a.cb	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.148.1	None	✓	$1$	$1$	$3$	$0$	$0$	$-5$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+(-2+\cdots)q^{7}+\cdots$
8450.2.a.cc	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.940.1	None	✓	$1$	$1$	$3$	$0$	$0$	$-2$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots$
8450.2.a.cd	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.2808.1	None	✓	$1$	$1$	$3$	$0$	$0$	$3$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.ce	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.2808.1	None	✓	$1$	$0$	$3$	$0$	$0$	$3$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.cf	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	$1$	$1$	$3$	$1$	$0$	$-4$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-2+\cdots)q^{7}+\cdots$
8450.2.a.cg	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	$1$	$0$	$3$	$1$	$0$	$2$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+2\beta _{1}q^{7}+\cdots$
8450.2.a.ch	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	4.4.4752.1	None	✓	$1$	$0$	$-4$	$-2$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(\beta _{1}-\beta _{3})q^{7}+\cdots$
8450.2.a.ci	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	4.4.4752.1	None	✓	$1$	$0$	$-4$	$-2$	$0$	$0$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+(-1-\beta _{2}+\beta _{3})q^{3}+q^{4}+(1+\cdots)q^{6}+\cdots$
8450.2.a.cj	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	$\Q(\sqrt{3}, \sqrt{11})$	None	✓	$2$	$1$	$-4$	$0$	$0$	$-2$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+\beta _{2}q^{7}+\cdots$
8450.2.a.ck	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	4.4.4752.1	None	✓	$1$	$1$	$-4$	$2$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(\beta _{1}-\beta _{3})q^{7}+\cdots$
8450.2.a.cl	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	4.4.4752.1	None	✓	$1$	$1$	$4$	$-2$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.cm	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	4.4.4752.1	None	✓	$1$	$1$	$4$	$-2$	$0$	$0$	$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+(-1-\beta _{2}+\beta _{3})q^{3}+q^{4}+(-1+\cdots)q^{6}+\cdots$
8450.2.a.cn	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	$\Q(\sqrt{3}, \sqrt{11})$	None	✓	$2$	$0$	$4$	$0$	$0$	$2$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}-\beta _{2}q^{7}+\cdots$
8450.2.a.co	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	4.4.4752.1	None	✓	$1$	$0$	$4$	$2$	$0$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.cp	$8450$	$2$	8450.a	1.a	$1$	$6$	$6$	$67.474$	6.6.20439713.1	None	✓	$1$	$0$	$-6$	$2$	$0$	$-3$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots$
8450.2.a.cq	$8450$	$2$	8450.a	1.a	$1$	$6$	$6$	$67.474$	6.6.20439713.1	None	✓	$1$	$0$	$6$	$2$	$0$	$3$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-\beta _{2}+\cdots)q^{7}+\cdots$
8450.2.a.cr	$8450$	$2$	8450.a	1.a	$1$	$8$	$8$	$67.474$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$2$	$1$	$-8$	$0$	$0$	$-10$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots$
8450.2.a.cs	$8450$	$2$	8450.a	1.a	$1$	$8$	$8$	$67.474$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$2$	$0$	$8$	$0$	$0$	$10$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{7})q^{7}+\cdots$
8450.2.a.ct	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$1$	$-9$	$-7$	$0$	$-1$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1-\beta _{4})q^{3}+q^{4}+(1+\beta _{4}+\cdots)q^{6}+\cdots$
8450.2.a.cu	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$0$	$-9$	$0$	$0$	$4$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}-\beta _{8}q^{7}+\cdots$
8450.2.a.cv	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$0$	$-9$	$0$	$0$	$4$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}-\beta _{8}q^{7}+\cdots$
8450.2.a.cw	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$0$	$-9$	$7$	$0$	$-1$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(1+\beta _{4})q^{3}+q^{4}+(-1-\beta _{4}+\cdots)q^{6}+\cdots$
8450.2.a.cx	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$1$	$9$	$-7$	$0$	$1$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-1-\beta _{4})q^{3}+q^{4}+(-1-\beta _{4}+\cdots)q^{6}+\cdots$
8450.2.a.cy	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$0$	$9$	$0$	$0$	$-4$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+\beta _{8}q^{7}+\cdots$
8450.2.a.cz	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$0$	$9$	$0$	$0$	$-4$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+\beta _{8}q^{7}+\cdots$
8450.2.a.da	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$0$	$9$	$7$	$0$	$1$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(1+\beta _{4})q^{3}+q^{4}+(1+\beta _{4})q^{6}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(8450))$ into lower level spaces

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 79 \)
Sturm bound:			\(2730\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(17\), \(31\)

\(2\)	\(5\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(24\)
\(+\)	\(+\)	\(-\)	$-$	\(33\)
\(+\)	\(-\)	\(+\)	$-$	\(34\)
\(+\)	\(-\)	\(-\)	$+$	\(31\)
\(-\)	\(+\)	\(+\)	$-$	\(35\)
\(-\)	\(+\)	\(-\)	$+$	\(24\)
\(-\)	\(-\)	\(+\)	$+$	\(27\)
\(-\)	\(-\)	\(-\)	$-$	\(37\)
Plus space			\(+\)	\(106\)
Minus space			\(-\)	\(139\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8450))\) 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	8450.2.a

Level	$8450$
Weight	$2$
Character orbit	8450.a
Rep. character	$\chi_{8450}(1,\cdot)$
Character field	$\Q$
Dimension	$245$
Newform subspaces	$79$
Sturm bound	$2730$
Trace bound	$17$