Newform orbit 900.2.z.b

• Label: 900.2.z.b
• Level: $900$
• Weight: $2$
• Character orbit: 900.z
• Analytic conductor: $7.187$
• Analytic rank: $0$
• Dimension: $224$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.z (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(56\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 224 * q + 8 * q^4 - 12 * q^10 - 8 * q^16 - 8 * q^25 + 92 * q^34 + 40 * q^37 + 36 * q^40 - 40 * q^46 + 368 * q^49 - 100 * q^52 - 120 * q^58 + 48 * q^61 - 16 * q^64 - 40 * q^70 + 24 * q^76 - 120 * q^85 - 120 * q^88 - 4 * q^94 + 200 * 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
179.1	−1.41421		0.000842959i		0		2.00000	−	0.00238425i	−1.84360	−	1.26536i		0		−3.63587			−2.82842	+	0.00505775i		0		2.60831	+	1.78793i
179.2	−1.41421	−	0.00291448i		0		1.99998	+	0.00824339i	−1.16580	+	1.90812i		0		−0.904297			−2.82837	−	0.0174868i		0		1.65425	−	2.69508i
179.3	−1.40055	+	0.196087i		0		1.92310	−	0.549261i	0.280394	−	2.21842i		0		3.98153			−2.58570	+	1.14636i		0		0.0422965	+	3.16199i
179.4	−1.37756	−	0.319872i		0		1.79536	+	0.881287i	1.39186	+	1.75006i		0		−0.321783			−2.19133	−	1.78832i		0		−1.35758	−	2.85604i
179.5	−1.35544	−	0.403479i		0		1.67441	+	1.09378i	2.20262	+	0.385325i		0		−3.23604			−1.82824	−	2.15813i		0		−2.83004	−	1.41099i
179.6	−1.33373	−	0.470284i		0		1.55767	+	1.25446i	−2.20262	−	0.385325i		0		3.23604			−1.48755	−	2.40566i		0		2.75648	+	1.54977i
179.7	−1.31270	+	0.526124i		0		1.44639	−	1.38129i	−0.701604	+	2.12315i		0		3.43527			−1.17195	+	2.57421i		0		−0.196039	−	3.15620i
179.8	−1.30704	+	0.540055i		0		1.41668	−	1.41174i	−0.0446450	−	2.23562i		0		1.57369			−1.08924	+	2.61028i		0		1.26571	+	2.89793i
179.9	−1.30249	−	0.550930i		0		1.39295	+	1.43516i	−1.39186	−	1.75006i		0		0.321783			−1.02363	−	2.63670i		0		0.848721	+	3.04626i
179.10	−1.29792	+	0.561608i		0		1.36919	−	1.45785i	2.23219	−	0.131662i		0		−3.32757			−0.958364	+	2.66112i		0		−2.82326	+	1.42450i
179.11	−1.20607	+	0.738508i		0		0.909211	−	1.78139i	−2.22983	+	0.166952i		0		1.74766			0.218995	+	2.81994i		0		2.56603	−	1.84810i
179.12	−1.14583	−	0.828894i		0		0.625869	+	1.89955i	1.16580	−	1.90812i		0		0.904297			0.857384	−	2.69535i		0		−2.91744	+	1.22006i
179.13	−1.14363	−	0.831936i		0		0.615766	+	1.90285i	1.84360	+	1.26536i		0		3.63587			0.878841	−	2.68843i		0		−1.05570	−	2.98086i
179.14	−1.08008	+	0.912918i		0		0.333163	−	1.97206i	1.78494	+	1.34684i		0		3.03823			1.44048	+	2.43414i		0		−3.15744	+	0.174809i
179.15	−1.01781	−	0.981862i		0		0.0718926	+	1.99871i	−0.280394	+	2.21842i		0		−3.98153			1.88928	−	2.10490i		0		2.46357	−	1.98263i
179.16	−0.926472	+	1.06848i		0		−0.283300	−	1.97983i	0.286438	+	2.21765i		0		−3.53478			2.37788	+	1.53156i		0		−2.63489	−	1.74853i
179.17	−0.891829	+	1.09756i		0		−0.409281	−	1.95767i	−2.20167	+	0.390696i		0		−2.21362			2.51368	+	1.29670i		0		1.53470	−	2.76490i
179.18	−0.857437	+	1.12463i		0		−0.529603	−	1.92861i	1.40263	−	1.74145i		0		−1.33837			2.62308	+	1.05805i		0		0.755824	+	3.07062i
179.19	−0.772651	+	1.18449i		0		−0.806021	−	1.83039i	2.06636	−	0.854496i		0		4.63434			2.79085	+	0.459531i		0		−0.584433	+	3.10780i
179.20	−0.752753	−	1.19723i		0		−0.866727	+	1.80244i	0.701604	−	2.12315i		0		−3.43527			2.81037	−	0.319117i		0		−3.07003	+	0.758221i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner
25.e	even	10	1	inner
75.h	odd	10	1	inner
100.h	odd	10	1	inner
300.r	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.z.b	✓	224
3.b	odd	2	1	inner	900.2.z.b	✓	224
4.b	odd	2	1	inner	900.2.z.b	✓	224
12.b	even	2	1	inner	900.2.z.b	✓	224
25.e	even	10	1	inner	900.2.z.b	✓	224
75.h	odd	10	1	inner	900.2.z.b	✓	224
100.h	odd	10	1	inner	900.2.z.b	✓	224
300.r	even	10	1	inner	900.2.z.b	✓	224

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
900.2.z.b	✓	224	1.a	even	1	1	trivial
900.2.z.b	✓	224	3.b	odd	2	1	inner
900.2.z.b	✓	224	4.b	odd	2	1	inner
900.2.z.b	✓	224	12.b	even	2	1	inner
900.2.z.b	✓	224	25.e	even	10	1	inner
900.2.z.b	✓	224	75.h	odd	10	1	inner
900.2.z.b	✓	224	100.h	odd	10	1	inner
900.2.z.b	✓	224	300.r	even	10	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^56 - 242*T7^54 + 27611*T7^52 - 1976506*T7^50 + 99645909*T7^48 - 3764565188*T7^46 + 110707971134*T7^44 - 2598514615264*T7^42 + 49520478026456*T7^40 - 775339561781240*T7^38 + 10053831449006180*T7^36 - 108524216940649300*T7^34 + 977854813607297655*T7^32 - 7360199638180538230*T7^30 + 46225834071497963645*T7^28 - 241547509830847885690*T7^26 + 1045220855524344497260*T7^24 - 3720829771534486619650*T7^22 + 10802194924466139729125*T7^20 - 25290052797007943922250*T7^18 + 47067717145077319770425*T7^16 - 68363065541553388044900*T7^14 + 75641559479234771236500*T7^12 - 61718796082679531351800*T7^10 + 35478756910779392103400*T7^8 - 13425984640565960606000*T7^6 + 2999599638469274296000*T7^4 - 327595716060300448000*T7^2 + 13236713416590962000