Newform orbit 900.2.r.f

• Label: 900.2.r.f
• Level: $900$
• Weight: $2$
• Character orbit: 900.r
• Analytic conductor: $7.187$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.r (of order \(6\), degree \(2\), 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 48 * q + 6 * q^6 + 4 * q^9 + 22 * q^12 - 30 * q^14 + 16 * q^18 - 4 * q^21 - 28 * q^24 - 12 * q^29 + 44 * q^33 - 6 * q^34 + 42 * q^36 - 60 * q^38 - 60 * q^41 + 18 * q^42 - 12 * q^46 + 12 * q^48 + 24 * q^49 + 18 * q^52 - 32 * q^54 - 42 * q^56 + 12 * q^57 + 18 * q^58 - 48 * q^64 + 16 * q^66 - 48 * q^68 + 36 * q^69 - 60 * q^72 + 24 * q^73 + 84 * q^74 + 6 * q^76 - 48 * q^77 - 38 * q^78 + 36 * q^82 + 50 * q^84 - 54 * q^86 - 60 * q^92 + 32 * q^93 + 18 * q^94 - 18 * q^96 + 12 * q^97

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
551.1	−1.40172	+	0.187569i	−0.398623	+	1.68556i	1.92964	−	0.525838i		0		0.242600	−	2.43745i	0.891819	+	0.514892i	−2.60618	+	1.09902i	−2.68220	−	1.34380i		0
551.2	−1.34972	−	0.422205i	0.561951	−	1.63836i	1.64349	+	1.13972i		0		−1.45020	+	1.97406i	0.811773	+	0.468677i	−1.73705	−	2.23218i	−2.36842	−	1.84135i		0
551.3	−1.34824	+	0.426910i	1.45681	+	0.936861i	1.63550	−	1.15115i		0		−2.36408	−	0.641186i	0.737635	+	0.425874i	−1.71360	+	2.25024i	1.24458	+	2.72965i		0
551.4	−1.25978	−	0.642624i	1.70656	+	0.296046i	1.17407	+	1.61912i		0		−1.95964	−	1.46963i	−3.55496	−	2.05246i	−0.438576	−	2.79422i	2.82471	+	1.01044i		0
551.5	−1.18642	−	0.769686i	−1.70656	−	0.296046i	0.815168	+	1.82634i		0		1.79683	+	1.66475i	3.55496	+	2.05246i	0.438576	−	2.79422i	2.82471	+	1.01044i		0
551.6	−1.08940	+	0.901786i	−1.35535	−	1.07843i	0.373566	−	1.96480i		0		2.44903	−	0.0473955i	3.33246	+	1.92400i	1.36487	+	2.47732i	0.673959	+	2.92332i		0
551.7	−1.04050	−	0.957789i	−0.561951	+	1.63836i	0.165280	+	1.99316i		0		2.15391	−	1.16648i	−0.811773	−	0.468677i	1.73705	−	2.23218i	−2.36842	−	1.84135i		0
551.8	−0.752973	+	1.19709i	−1.67907	+	0.425108i	−0.866065	−	1.80276i		0		0.755401	−	2.33010i	−1.87435	−	1.08216i	2.81019	+	0.320666i	2.63857	−	1.42758i		0
551.9	−0.538420	−	1.30771i	0.398623	−	1.68556i	−1.42021	+	1.40819i		0		−2.41884	+	0.386254i	−0.891819	−	0.514892i	2.60618	+	1.09902i	−2.68220	−	1.34380i		0
551.10	−0.416164	+	1.35159i	0.968388	+	1.43605i	−1.65361	−	1.12497i		0		−2.34396	+	0.711237i	1.90150	+	1.09783i	2.20868	−	1.76684i	−1.12445	+	2.78130i		0
551.11	−0.304404	−	1.38106i	−1.45681	−	0.936861i	−1.81468	+	0.840803i		0		−0.850407	+	2.29713i	−0.737635	−	0.425874i	1.71360	+	2.25024i	1.24458	+	2.72965i		0
551.12	−0.0785043	+	1.41203i	−0.427686	−	1.67842i	−1.98767	−	0.221701i		0		2.40356	−	0.472143i	−4.06955	−	2.34956i	0.469091	−	2.78926i	−2.63417	+	1.43567i		0
551.13	0.145547	+	1.40670i	−1.30449	+	1.13944i	−1.95763	+	0.409483i		0		−1.79271	−	1.66919i	2.51585	+	1.45253i	−0.860949	−	2.69421i	0.403372	−	2.97276i		0
551.14	0.236271	−	1.39434i	1.35535	+	1.07843i	−1.88835	−	0.658884i		0		1.82393	−	1.63501i	−3.33246	−	1.92400i	−1.36487	+	2.47732i	0.673959	+	2.92332i		0
551.15	0.295692	+	1.38296i	0.531460	−	1.64850i	−1.82513	+	0.817857i		0		2.43695	+	0.247538i	3.61144	+	2.08506i	−1.67074	−	2.28224i	−2.43510	−	1.75222i		0
551.16	0.660227	−	1.25064i	1.67907	−	0.425108i	−1.12820	−	1.65141i		0		0.576911	−	2.38058i	1.87435	+	1.08216i	−2.81019	+	0.320666i	2.63857	−	1.42758i		0
551.17	0.896034	+	1.09413i	−1.60382	−	0.654027i	−0.394247	+	1.96076i		0		−0.721488	−	2.34082i	−1.04714	−	0.604565i	−2.49858	+	1.32555i	2.14450	+	2.09789i		0
551.18	0.962433	−	1.03621i	−0.968388	−	1.43605i	−0.147445	−	1.99456i		0		−2.42005	−	0.378648i	−1.90150	−	1.09783i	−2.20868	−	1.76684i	−1.12445	+	2.78130i		0
551.19	0.981980	+	1.01770i	1.63013	−	0.585384i	−0.0714305	+	1.99872i		0		2.19650	+	1.08415i	1.44910	+	0.836639i	−2.10425	+	1.89001i	2.31465	−	1.90850i		0
551.20	1.18360	−	0.774003i	0.427686	+	1.67842i	0.801838	−	1.83223i		0		1.80531	+	1.65555i	4.06955	+	2.34956i	−0.469091	−	2.78926i	−2.63417	+	1.43567i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
9.d	odd	6	1	inner
36.h	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.r.f		48
4.b	odd	2	1	inner	900.2.r.f		48
5.b	even	2	1		180.2.q.a	✓	48
5.c	odd	4	1		900.2.o.b		48
5.c	odd	4	1		900.2.o.c		48
9.d	odd	6	1	inner	900.2.r.f		48
15.d	odd	2	1		540.2.q.a		48
20.d	odd	2	1		180.2.q.a	✓	48
20.e	even	4	1		900.2.o.b		48
20.e	even	4	1		900.2.o.c		48
36.h	even	6	1	inner	900.2.r.f		48
45.h	odd	6	1		180.2.q.a	✓	48
45.h	odd	6	1		1620.2.e.b		48
45.j	even	6	1		540.2.q.a		48
45.j	even	6	1		1620.2.e.b		48
45.l	even	12	1		900.2.o.b		48
45.l	even	12	1		900.2.o.c		48
60.h	even	2	1		540.2.q.a		48
180.n	even	6	1		180.2.q.a	✓	48
180.n	even	6	1		1620.2.e.b		48
180.p	odd	6	1		540.2.q.a		48
180.p	odd	6	1		1620.2.e.b		48
180.v	odd	12	1		900.2.o.b		48
180.v	odd	12	1		900.2.o.c		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
180.2.q.a	✓	48	5.b	even	2	1
180.2.q.a	✓	48	20.d	odd	2	1
180.2.q.a	✓	48	45.h	odd	6	1
180.2.q.a	✓	48	180.n	even	6	1
540.2.q.a		48	15.d	odd	2	1
540.2.q.a		48	45.j	even	6	1
540.2.q.a		48	60.h	even	2	1
540.2.q.a		48	180.p	odd	6	1
900.2.o.b		48	5.c	odd	4	1
900.2.o.b		48	20.e	even	4	1
900.2.o.b		48	45.l	even	12	1
900.2.o.b		48	180.v	odd	12	1
900.2.o.c		48	5.c	odd	4	1
900.2.o.c		48	20.e	even	4	1
900.2.o.c		48	45.l	even	12	1
900.2.o.c		48	180.v	odd	12	1
900.2.r.f		48	1.a	even	1	1	trivial
900.2.r.f		48	4.b	odd	2	1	inner
900.2.r.f		48	9.d	odd	6	1	inner
900.2.r.f		48	36.h	even	6	1	inner
1620.2.e.b		48	45.h	odd	6	1
1620.2.e.b		48	45.j	even	6	1
1620.2.e.b		48	180.n	even	6	1
1620.2.e.b		48	180.p	odd	6	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 - 32*T13^21 + 4740*T13^20 - 2976*T13^19 + 146552*T13^18 - 204360*T13^17 + 3297000*T13^16 - 5536448*T13^15 + 46859520*T13^14 - 101875392*T13^13 + 475241392*T13^12 - 821526240*T13^11 + 2358339552*T13^10 - 3010704448*T13^9 + 7141580160*T13^8 - 6982958976*T13^7 + 11961078656*T13^6 - 4631235840*T13^5 + 7202693376*T13^4 + 179287040*T13^3 + 4116879360*T13^2 + 563601408*T13 + 82591744