Newform orbit 900.2.o.b

• Label: 900.2.o.b
• Level: $900$
• Weight: $2$
• Character orbit: 900.o
• Analytic conductor: $7.187$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{6} - 12 q^{8} - 4 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
299.1	−1.39434	−	0.236271i	−1.07843	+	1.35535i	1.88835	+	0.658884i		0		1.82393	−	1.63501i	−1.92400	+	3.33246i	−2.47732	−	1.36487i	−0.673959	−	2.92332i		0
299.2	−1.38296	+	0.295692i	−1.64850	−	0.531460i	1.82513	−	0.817857i		0		2.43695	+	0.247538i	−2.08506	+	3.61144i	−2.28224	+	1.67074i	2.43510	+	1.75222i		0
299.3	−1.38106	+	0.304404i	0.936861	−	1.45681i	1.81468	−	0.840803i		0		−0.850407	+	2.29713i	−0.425874	+	0.737635i	−2.25024	+	1.71360i	−1.24458	−	2.72965i		0
299.4	−1.35159	−	0.416164i	1.43605	−	0.968388i	1.65361	+	1.12497i		0		−2.34396	+	0.711237i	−1.09783	+	1.90150i	−1.76684	−	2.20868i	1.12445	−	2.78130i		0
299.5	−1.19709	−	0.752973i	0.425108	+	1.67907i	0.866065	+	1.80276i		0		0.755401	−	2.33010i	1.08216	−	1.87435i	0.320666	−	2.81019i	−2.63857	+	1.42758i		0
299.6	−1.09413	+	0.896034i	−0.654027	+	1.60382i	0.394247	−	1.96076i		0		−0.721488	−	2.34082i	0.604565	−	1.04714i	1.32555	+	2.49858i	−2.14450	−	2.09789i		0
299.7	−0.957789	+	1.04050i	−1.63836	−	0.561951i	−0.165280	−	1.99316i		0		2.15391	−	1.16648i	−0.468677	+	0.811773i	2.23218	+	1.73705i	2.36842	+	1.84135i		0
299.8	−0.774003	−	1.18360i	−1.67842	+	0.427686i	−0.801838	+	1.83223i		0		1.80531	+	1.65555i	2.34956	−	4.06955i	2.78926	−	0.469091i	2.63417	−	1.43567i		0
299.9	−0.577305	−	1.29101i	1.13944	+	1.30449i	−1.33344	+	1.49062i		0		1.02631	−	2.22411i	−1.45253	+	2.51585i	2.69421	+	0.860949i	−0.403372	+	2.97276i		0
299.10	−0.422205	+	1.34972i	1.63836	+	0.561951i	−1.64349	−	1.13972i		0		−1.45020	+	1.97406i	0.468677	−	0.811773i	2.23218	−	1.73705i	2.36842	+	1.84135i		0
299.11	−0.228922	+	1.39556i	0.654027	−	1.60382i	−1.89519	−	0.638951i		0		2.08851	+	1.27989i	−0.604565	+	1.04714i	1.32555	−	2.49858i	−2.14450	−	2.09789i		0
299.12	−0.187569	−	1.40172i	1.68556	+	0.398623i	−1.92964	+	0.525838i		0		0.242600	−	2.43745i	−0.514892	+	0.891819i	1.09902	+	2.60618i	2.68220	+	1.34380i		0
299.13	0.341569	−	1.37234i	−0.585384	−	1.63013i	−1.76666	−	0.937501i		0		−2.43705	+	0.246546i	−0.836639	+	1.44910i	−1.89001	+	2.10425i	−2.31465	+	1.90850i		0
299.14	0.426910	+	1.34824i	−0.936861	+	1.45681i	−1.63550	+	1.15115i		0		−2.36408	−	0.641186i	0.425874	−	0.737635i	−2.25024	−	1.71360i	−1.24458	−	2.72965i		0
299.15	0.435401	+	1.34552i	1.64850	+	0.531460i	−1.62085	+	1.17168i		0		0.00266887	+	2.44949i	2.08506	−	3.61144i	−2.28224	−	1.67074i	2.43510	+	1.75222i		0
299.16	0.642624	−	1.25978i	0.296046	−	1.70656i	−1.17407	−	1.61912i		0		−1.95964	−	1.46963i	2.05246	−	3.55496i	−2.79422	+	0.438576i	−2.82471	−	1.01044i		0
299.17	0.769686	−	1.18642i	−0.296046	+	1.70656i	−0.815168	−	1.82634i		0		1.79683	+	1.66475i	−2.05246	+	3.55496i	−2.79422	−	0.438576i	−2.82471	−	1.01044i		0
299.18	0.901786	+	1.08940i	1.07843	−	1.35535i	−0.373566	+	1.96480i		0		2.44903	−	0.0473955i	1.92400	−	3.33246i	−2.47732	+	1.36487i	−0.673959	−	2.92332i		0
299.19	1.01770	−	0.981980i	0.585384	+	1.63013i	0.0714305	−	1.99872i		0		2.19650	+	1.08415i	0.836639	−	1.44910i	−1.89001	−	2.10425i	−2.31465	+	1.90850i		0
299.20	1.03621	+	0.962433i	−1.43605	+	0.968388i	0.147445	+	1.99456i		0		−2.42005	−	0.378648i	1.09783	−	1.90150i	−1.76684	+	2.20868i	1.12445	−	2.78130i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
45.h	odd	6	1	inner
180.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.o.b		48
4.b	odd	2	1	inner	900.2.o.b		48
5.b	even	2	1		900.2.o.c		48
5.c	odd	4	1		180.2.q.a	✓	48
5.c	odd	4	1		900.2.r.f		48
9.d	odd	6	1		900.2.o.c		48
15.e	even	4	1		540.2.q.a		48
20.d	odd	2	1		900.2.o.c		48
20.e	even	4	1		180.2.q.a	✓	48
20.e	even	4	1		900.2.r.f		48
36.h	even	6	1		900.2.o.c		48
45.h	odd	6	1	inner	900.2.o.b		48
45.k	odd	12	1		540.2.q.a		48
45.k	odd	12	1		1620.2.e.b		48
45.l	even	12	1		180.2.q.a	✓	48
45.l	even	12	1		900.2.r.f		48
45.l	even	12	1		1620.2.e.b		48
60.l	odd	4	1		540.2.q.a		48
180.n	even	6	1	inner	900.2.o.b		48
180.v	odd	12	1		180.2.q.a	✓	48
180.v	odd	12	1		900.2.r.f		48
180.v	odd	12	1		1620.2.e.b		48
180.x	even	12	1		540.2.q.a		48
180.x	even	12	1		1620.2.e.b		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
180.2.q.a	✓	48	5.c	odd	4	1
180.2.q.a	✓	48	20.e	even	4	1
180.2.q.a	✓	48	45.l	even	12	1
180.2.q.a	✓	48	180.v	odd	12	1
540.2.q.a		48	15.e	even	4	1
540.2.q.a		48	45.k	odd	12	1
540.2.q.a		48	60.l	odd	4	1
540.2.q.a		48	180.x	even	12	1
900.2.o.b		48	1.a	even	1	1	trivial
900.2.o.b		48	4.b	odd	2	1	inner
900.2.o.b		48	45.h	odd	6	1	inner
900.2.o.b		48	180.n	even	6	1	inner
900.2.o.c		48	5.b	even	2	1
900.2.o.c		48	9.d	odd	6	1
900.2.o.c		48	20.d	odd	2	1
900.2.o.c		48	36.h	even	6	1
900.2.r.f		48	5.c	odd	4	1
900.2.r.f		48	20.e	even	4	1
900.2.r.f		48	45.l	even	12	1
900.2.r.f		48	180.v	odd	12	1
1620.2.e.b		48	45.k	odd	12	1
1620.2.e.b		48	45.l	even	12	1
1620.2.e.b		48	180.v	odd	12	1
1620.2.e.b		48	180.x	even	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

T7^48 + 96*T7^46 + 5319*T7^44 + 199112*T7^42 + 5590407*T7^40 + 121687632*T7^38 + 2114888258*T7^36 + 29605899444*T7^34 + 337285177761*T7^32 + 3126602088404*T7^30 + 23683715619477*T7^28 + 146174345487120*T7^26 + 736757632470817*T7^24 + 3018513885868788*T7^22 + 10051535648023638*T7^20 + 26992477029956632*T7^18 + 58430010384411459*T7^16 + 100971127532302332*T7^14 + 139552157903210843*T7^12 + 152277535518285108*T7^10 + 130696321187995329*T7^8 + 85467569720545384*T7^6 + 41565265782323376*T7^4 + 13451810346046080*T7^2 + 2552640623984896

T13^24 - 84*T13^22 + 4740*T13^20 + 3672*T13^19 - 146040*T13^18 - 170520*T13^17 + 3244776*T13^16 + 5819328*T13^15 - 35963616*T13^14 - 65404800*T13^13 + 301451504*T13^12 + 392675040*T13^11 - 1393037856*T13^10 - 1515840192*T13^9 + 4662985344*T13^8 + 3642742656*T13^7 - 7644889728*T13^6 - 6567878400*T13^5 + 7844225280*T13^4 + 8929999872*T13^3 + 1588525056*T13^2 - 790437888*T13 + 82591744