Newform orbit 900.2.bf.f

• Label: 900.2.bf.f
• Level: $900$
• Weight: $2$
• Character orbit: 900.bf
• Analytic conductor: $7.187$
• Analytic rank: $0$
• Dimension: $192$
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(48\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 12 q^{6} + 16 q^{21} + 112 q^{26} + 124 q^{36} - 40 q^{41} + 48 q^{46} + 136 q^{56} + 20 q^{66} + 24 q^{76} - 120 q^{81} - 176 q^{86} - 304 q^{96}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−1.40283	+	0.179057i	1.67933	+	0.424102i	1.93588	−	0.502375i		0		−2.43175	−	0.294248i	0.0874592	+	0.326402i	−2.62576	+	1.05138i	2.64027	+	1.42441i		0
7.2	−1.40277	−	0.179526i	0.355417	−	1.69519i	1.93554	+	0.503668i		0		−0.802900	+	2.31416i	−0.205713	−	0.767732i	−2.62470	−	1.05401i	−2.74736	−	1.20500i		0
7.3	−1.37161	−	0.344515i	−1.72745	+	0.126167i	1.76262	+	0.945078i		0		2.41285	+	0.422080i	−0.483509	−	1.80448i	−2.09203	−	1.90353i	2.96816	−	0.435894i		0
7.4	−1.37098	+	0.347003i	1.01510	+	1.40342i	1.75918	−	0.951469i		0		−1.87867	−	1.57181i	−1.15108	−	4.29590i	−2.08164	+	1.91489i	−0.939153	+	2.84921i		0
7.5	−1.36081	+	0.384977i	−1.01510	−	1.40342i	1.70359	−	1.04776i		0		1.92163	+	1.51899i	1.15108	+	4.29590i	−1.91489	+	2.08164i	−0.939153	+	2.84921i		0
7.6	−1.33383	−	0.469996i	1.47995	−	0.899860i	1.55821	+	1.25379i		0		−2.39693	+	0.504690i	0.337895	+	1.26104i	−1.48911	−	2.40470i	1.38050	−	2.66350i		0
7.7	−1.30442	+	0.546348i	−1.67933	−	0.424102i	1.40301	−	1.42533i		0		2.42225	−	0.364290i	−0.0874592	−	0.326402i	−1.05138	+	2.62576i	2.64027	+	1.42441i		0
7.8	−1.26814	−	0.625952i	−1.37453	+	1.05388i	1.21637	+	1.58759i		0		2.40278	−	0.476087i	1.24971	+	4.66398i	−0.548769	−	2.77468i	0.778653	−	2.89719i		0
7.9	−1.26457	−	0.633147i	0.743981	+	1.56413i	1.19825	+	1.60131i		0		0.0495096	−	2.44899i	0.984552	+	3.67440i	−0.501399	−	2.78363i	−1.89298	+	2.32736i		0
7.10	−1.12507	+	0.856860i	−0.355417	+	1.69519i	0.531581	−	1.92806i		0		−1.05267	−	2.21176i	0.205713	+	0.767732i	1.05401	+	2.62470i	−2.74736	−	1.20500i		0
7.11	−1.04531	−	0.952540i	0.833803	+	1.51815i	0.185333	+	1.99139i		0		0.574517	−	2.38116i	−0.364173	−	1.35911i	1.70315	−	2.25816i	−1.60955	+	2.53167i		0
7.12	−1.01559	+	0.984163i	1.72745	−	0.126167i	0.0628479	−	1.99901i		0		−1.63021	+	1.82823i	0.483509	+	1.80448i	1.90353	+	2.09203i	2.96816	−	0.435894i		0
7.13	−0.921054	−	1.07315i	−1.43844	−	0.964826i	−0.303320	+	1.97687i		0		0.289474	+	2.43232i	−0.755079	−	2.81799i	2.40086	−	1.49529i	1.13822	+	2.77569i		0
7.14	−0.920133	+	1.07394i	−1.47995	+	0.899860i	−0.306711	−	1.97634i		0		0.395352	−	2.41737i	−0.337895	−	1.26104i	2.40470	+	1.48911i	1.38050	−	2.66350i		0
7.15	−0.785267	+	1.17616i	1.37453	−	1.05388i	−0.766712	−	1.84720i		0		0.160168	+	2.44425i	−1.24971	−	4.66398i	2.77468	+	0.548769i	0.778653	−	2.89719i		0
7.16	−0.778572	+	1.18060i	−0.743981	−	1.56413i	−0.787651	−	1.83837i		0		2.42586	+	0.339438i	−0.984552	−	3.67440i	2.78363	+	0.501399i	−1.89298	+	2.32736i		0
7.17	−0.693094	−	1.23273i	−0.895996	+	1.48229i	−1.03924	+	1.70879i		0		2.44827	+	0.0771521i	−0.218614	−	0.815879i	2.82677	+	0.0967471i	−1.39438	−	2.65626i		0
7.18	−0.617949	−	1.27206i	0.560959	−	1.63870i	−1.23628	+	1.57214i		0		−2.43117	+	0.299055i	0.987896	+	3.68688i	2.76381	+	0.601123i	−2.37065	−	1.83848i		0
7.19	−0.547152	−	1.30408i	1.71963	−	0.207028i	−1.40125	+	1.42706i		0		−1.21088	−	2.12926i	−0.671010	−	2.50424i	2.62770	+	1.04652i	2.91428	−	0.712025i		0
7.20	−0.428992	+	1.34758i	−0.833803	−	1.51815i	−1.63193	−	1.15620i		0		2.40352	−	0.472340i	0.364173	+	1.35911i	2.25816	−	1.70315i	−1.60955	+	2.53167i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
9.c	even	3	1	inner
20.d	odd	2	1	inner
20.e	even	4	2	inner
36.f	odd	6	1	inner
45.j	even	6	1	inner
45.k	odd	12	2	inner
180.p	odd	6	1	inner
180.x	even	12	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.bf.f	✓	192
4.b	odd	2	1	inner	900.2.bf.f	✓	192
5.b	even	2	1	inner	900.2.bf.f	✓	192
5.c	odd	4	2	inner	900.2.bf.f	✓	192
9.c	even	3	1	inner	900.2.bf.f	✓	192
20.d	odd	2	1	inner	900.2.bf.f	✓	192
20.e	even	4	2	inner	900.2.bf.f	✓	192
36.f	odd	6	1	inner	900.2.bf.f	✓	192
45.j	even	6	1	inner	900.2.bf.f	✓	192
45.k	odd	12	2	inner	900.2.bf.f	✓	192
180.p	odd	6	1	inner	900.2.bf.f	✓	192
180.x	even	12	2	inner	900.2.bf.f	✓	192

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
900.2.bf.f	✓	192	1.a	even	1	1	trivial
900.2.bf.f	✓	192	4.b	odd	2	1	inner
900.2.bf.f	✓	192	5.b	even	2	1	inner
900.2.bf.f	✓	192	5.c	odd	4	2	inner
900.2.bf.f	✓	192	9.c	even	3	1	inner
900.2.bf.f	✓	192	20.d	odd	2	1	inner
900.2.bf.f	✓	192	20.e	even	4	2	inner
900.2.bf.f	✓	192	36.f	odd	6	1	inner
900.2.bf.f	✓	192	45.j	even	6	1	inner
900.2.bf.f	✓	192	45.k	odd	12	2	inner
900.2.bf.f	✓	192	180.p	odd	6	1	inner
900.2.bf.f	✓	192	180.x	even	12	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^96 - 1494*T7^92 + 1388463*T7^88 - 803005622*T7^84 + 338679670527*T7^80 - 101344508912682*T7^76 + 22829751617848386*T7^72 - 3713925781340444736*T7^68 + 448899872325217188579*T7^64 - 36999832702706158701974*T7^60 + 2221009428648941292199086*T7^56 - 87165622127256232406726388*T7^52 + 2398988713393586699301476518*T7^48 - 35611590272921750339128694628*T7^44 + 369916625460446127949328384472*T7^40 - 2106808037362273172540895428724*T7^36 + 8534132996844607877712087344097*T7^32 - 20802190707648943012592699615064*T7^28 + 35390377273503616654266566023536*T7^24 - 26844649743556114697251458131040*T7^20 + 14546771900451636991602499608192*T7^16 - 4140040958299242594325209583872*T7^12 + 813386351076120162482772623616*T7^8 - 10574100996620934976089329664*T7^4 + 129116401871455302786809856