Newform orbit 825.2.cz.a

• Label: 825.2.cz.a
• Level: $825$
• Weight: $2$
• Character orbit: 825.cz
• Analytic conductor: $6.588$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.cz (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(60\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 8 q^{5}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
142.1	−2.45422	−	1.25049i	0.156434	−	0.987688i	3.28392	+	4.51993i	0.131481	−	2.23220i	−1.61902	+	2.22839i	0.387829	−	0.387829i	−1.54557	−	9.75837i	−0.951057	−	0.309017i	−3.11403	+	5.31390i
142.2	−2.40823	−	1.22706i	−0.156434	+	0.987688i	3.11835	+	4.29205i	−1.80592	−	1.31858i	1.58868	−	2.18663i	3.33198	−	3.33198i	−1.39751	−	8.82356i	−0.951057	−	0.309017i	2.73110	+	5.39142i
142.3	−2.39692	−	1.22129i	−0.156434	+	0.987688i	3.07810	+	4.23664i	1.20624	−	1.88281i	1.58122	−	2.17636i	−2.89121	+	2.89121i	−1.36213	−	8.60012i	−0.951057	−	0.309017i	−5.19072	+	3.03979i
142.4	−2.30557	−	1.17475i	0.156434	−	0.987688i	2.76005	+	3.79888i	−0.876008	+	2.05733i	−1.52095	+	2.09341i	1.22660	−	1.22660i	−1.09118	−	6.88945i	−0.951057	−	0.309017i	4.43654	−	3.71423i
142.5	−2.29231	−	1.16799i	−0.156434	+	0.987688i	2.71491	+	3.73675i	1.51453	+	1.64505i	1.51221	−	2.08137i	0.352110	−	0.352110i	−1.05400	−	6.65468i	−0.951057	−	0.309017i	−1.55036	−	5.53992i
142.6	−2.22999	−	1.13624i	−0.156434	+	0.987688i	2.50627	+	3.44958i	−1.77557	+	1.35917i	1.47110	−	2.02479i	−0.297823	+	0.297823i	−0.886371	−	5.59633i	−0.951057	−	0.309017i	5.50386	−	1.01346i
142.7	−2.00096	−	1.01954i	0.156434	−	0.987688i	1.78882	+	2.46210i	0.509920	+	2.17715i	−1.32001	+	1.81684i	−3.14238	+	3.14238i	−0.366530	−	2.31418i	−0.951057	−	0.309017i	1.19936	−	4.87628i
142.8	−1.98883	−	1.01336i	0.156434	−	0.987688i	1.75297	+	2.41276i	−1.76001	−	1.37927i	−1.31200	+	1.80582i	−1.12467	+	1.12467i	−0.343013	−	2.16570i	−0.951057	−	0.309017i	2.10266	+	4.52665i
142.9	−1.96626	−	1.00186i	−0.156434	+	0.987688i	1.68690	+	2.32182i	2.23126	+	0.146482i	1.29712	−	1.78533i	−0.173996	+	0.173996i	−0.300313	−	1.89610i	−0.951057	−	0.309017i	−4.24050	−	2.52344i
142.10	−1.89870	−	0.967438i	−0.156434	+	0.987688i	1.49357	+	2.05572i	−1.42641	−	1.72202i	1.25255	−	1.72399i	−3.16117	+	3.16117i	−0.180349	−	1.13868i	−0.951057	−	0.309017i	1.04238	+	4.64957i
142.11	−1.65864	−	0.845118i	−0.156434	+	0.987688i	0.861285	+	1.18546i	0.261131	+	2.22077i	1.09418	−	1.50601i	1.97026	−	1.97026i	0.155707	+	0.983096i	−0.951057	−	0.309017i	1.44369	−	3.90414i
142.12	−1.65535	−	0.843445i	0.156434	−	0.987688i	0.853226	+	1.17437i	−2.13062	−	0.678586i	−1.09202	+	1.50303i	3.15735	−	3.15735i	0.159384	+	1.00631i	−0.951057	−	0.309017i	2.95457	+	2.92036i
142.13	−1.58747	−	0.808856i	0.156434	−	0.987688i	0.690238	+	0.950032i	1.87394	−	1.21997i	−1.04723	+	1.44139i	−1.40203	+	1.40203i	0.230132	+	1.45300i	−0.951057	−	0.309017i	−3.96161	+	0.420921i
142.14	−1.58057	−	0.805343i	0.156434	−	0.987688i	0.674066	+	0.927773i	2.19015	−	0.450832i	−1.04268	+	1.43513i	3.37989	−	3.37989i	0.236768	+	1.49489i	−0.951057	−	0.309017i	−3.82477	−	1.05125i
142.15	−1.49489	−	0.761685i	0.156434	−	0.987688i	0.478963	+	0.659236i	0.564835	−	2.16355i	−0.986159	+	1.35733i	−0.965903	+	0.965903i	0.311050	+	1.96389i	−0.951057	−	0.309017i	−2.49231	+	2.80405i
142.16	−1.44115	−	0.734301i	−0.156434	+	0.987688i	0.362134	+	0.498434i	0.255005	−	2.22148i	0.950705	−	1.30853i	0.650324	−	0.650324i	0.350158	+	2.21081i	−0.951057	−	0.309017i	−1.99873	+	3.01423i
142.17	−1.15602	−	0.589023i	−0.156434	+	0.987688i	−0.186129	−	0.256185i	−1.91670	+	1.15164i	0.762613	−	1.04965i	−2.05829	+	2.05829i	0.470198	+	2.96871i	−0.951057	−	0.309017i	2.89409	−	0.202338i
142.18	−1.15286	−	0.587412i	0.156434	−	0.987688i	−0.191536	−	0.263627i	1.57650	+	1.58576i	−0.760527	+	1.04678i	−1.06566	+	1.06566i	0.470773	+	2.97235i	−0.951057	−	0.309017i	−0.885994	−	2.75422i
142.19	−1.13108	−	0.576312i	−0.156434	+	0.987688i	−0.228374	−	0.314330i	−1.84508	+	1.26320i	0.746156	−	1.02700i	2.95742	−	2.95742i	0.474323	+	2.99476i	−0.951057	−	0.309017i	2.81492	−	0.365429i
142.20	−1.10854	−	0.564831i	−0.156434	+	0.987688i	−0.265737	−	0.365756i	2.23603	−	0.0121541i	0.731291	−	1.00654i	−1.46196	+	1.46196i	0.477246	+	3.01321i	−0.951057	−	0.309017i	−2.48561	−	1.24951i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner
25.f	odd	20	1	inner
275.bo	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	825.2.cz.a	✓	480
11.b	odd	2	1	inner	825.2.cz.a	✓	480
25.f	odd	20	1	inner	825.2.cz.a	✓	480
275.bo	even	20	1	inner	825.2.cz.a	✓	480

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
825.2.cz.a	✓	480	1.a	even	1	1	trivial
825.2.cz.a	✓	480	11.b	odd	2	1	inner
825.2.cz.a	✓	480	25.f	odd	20	1	inner
825.2.cz.a	✓	480	275.bo	even	20	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(825, [\chi])\).