Newform orbit 525.2.w.a

• Label: 525.2.w.a
• Level: $525$
• Weight: $2$
• Character orbit: 525.w
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $304$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.w (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(304\)
Relative dimension:	\(76\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 304 q - 84 q^{4} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
104.1	−0.840823	+	2.58779i	−0.654037	+	1.60382i	−4.37163	−	3.17618i	1.62287	+	1.53827i	−3.60041	−	3.04104i	−0.317962	+	2.62658i	7.49243	−	5.44357i	−2.14447	−	2.09792i	−5.34527	+	2.90624i
104.2	−0.840823	+	2.58779i	0.654037	−	1.60382i	−4.37163	−	3.17618i	−1.62287	−	1.53827i	3.60041	+	3.04104i	0.317962	+	2.62658i	7.49243	−	5.44357i	−2.14447	−	2.09792i	5.34527	−	2.90624i
104.3	−0.825889	+	2.54182i	−1.65709	−	0.504029i	−4.16074	−	3.02296i	−1.82317	+	1.29462i	2.64973	−	3.79577i	2.62130	−	0.358847i	6.79573	−	4.93738i	2.49191	+	1.67044i	−1.78496	−	5.70340i
104.4	−0.825889	+	2.54182i	1.65709	+	0.504029i	−4.16074	−	3.02296i	1.82317	−	1.29462i	−2.64973	+	3.79577i	−2.62130	−	0.358847i	6.79573	−	4.93738i	2.49191	+	1.67044i	1.78496	+	5.70340i
104.5	−0.780529	+	2.40222i	−0.282161	+	1.70891i	−3.54341	−	2.57444i	−0.607661	−	2.15192i	−3.88495	−	2.01167i	1.54353	−	2.14884i	4.86320	−	3.53332i	−2.84077	−	0.964377i	5.64368	+	0.219898i
104.6	−0.780529	+	2.40222i	0.282161	−	1.70891i	−3.54341	−	2.57444i	0.607661	+	2.15192i	3.88495	+	2.01167i	−1.54353	−	2.14884i	4.86320	−	3.53332i	−2.84077	−	0.964377i	−5.64368	−	0.219898i
104.7	−0.742792	+	2.28608i	−1.73121	−	0.0538885i	−3.05638	−	2.22059i	2.20023	+	0.398729i	1.40912	−	3.91766i	−0.0744001	−	2.64471i	3.45740	−	2.51195i	2.99419	+	0.186585i	−2.54584	+	4.73373i
104.8	−0.742792	+	2.28608i	1.73121	+	0.0538885i	−3.05638	−	2.22059i	−2.20023	−	0.398729i	−1.40912	+	3.91766i	0.0744001	−	2.64471i	3.45740	−	2.51195i	2.99419	+	0.186585i	2.54584	−	4.73373i
104.9	−0.715634	+	2.20250i	−1.15693	−	1.28900i	−2.72082	−	1.97679i	1.79617	−	1.33182i	3.66695	−	1.62568i	−1.56332	+	2.13449i	2.55388	−	1.85551i	−0.323024	+	2.98256i	1.64793	+	4.90917i
104.10	−0.715634	+	2.20250i	1.15693	+	1.28900i	−2.72082	−	1.97679i	−1.79617	+	1.33182i	−3.66695	+	1.62568i	1.56332	+	2.13449i	2.55388	−	1.85551i	−0.323024	+	2.98256i	−1.64793	−	4.90917i
104.11	−0.677246	+	2.08435i	−1.04597	+	1.38056i	−2.26781	−	1.64766i	−2.02988	+	0.937853i	−2.16919	−	3.11514i	−2.59371	−	0.522162i	1.42406	−	1.03464i	−0.811893	−	2.88805i	−0.580080	−	4.86614i
104.12	−0.677246	+	2.08435i	1.04597	−	1.38056i	−2.26781	−	1.64766i	2.02988	−	0.937853i	2.16919	+	3.11514i	2.59371	−	0.522162i	1.42406	−	1.03464i	−0.811893	−	2.88805i	0.580080	+	4.86614i
104.13	−0.605685	+	1.86411i	−0.698093	−	1.58514i	−1.49001	−	1.08256i	−1.90667	−	1.16816i	3.37770	−	0.341225i	−2.25871	−	1.37777i	−0.250932	+	0.182313i	−2.02533	+	2.21315i	3.33242	−	2.84671i
104.14	−0.605685	+	1.86411i	0.698093	+	1.58514i	−1.49001	−	1.08256i	1.90667	+	1.16816i	−3.37770	+	0.341225i	2.25871	−	1.37777i	−0.250932	+	0.182313i	−2.02533	+	2.21315i	−3.33242	+	2.84671i
104.15	−0.588826	+	1.81222i	−1.26929	−	1.17851i	−1.31939	−	0.958593i	−0.430450	−	2.19425i	2.88312	−	1.60630i	2.48611	−	0.905127i	−0.569061	+	0.413447i	0.222220	+	2.99176i	4.22991	+	0.511957i
104.16	−0.588826	+	1.81222i	1.26929	+	1.17851i	−1.31939	−	0.958593i	0.430450	+	2.19425i	−2.88312	+	1.60630i	−2.48611	−	0.905127i	−0.569061	+	0.413447i	0.222220	+	2.99176i	−4.22991	−	0.511957i
104.17	−0.534897	+	1.64624i	−1.66454	+	0.478872i	−0.805968	−	0.585570i	−2.04228	−	0.910555i	0.102015	−	2.99638i	0.846893	+	2.50655i	−1.40566	+	1.02127i	2.54136	−	1.59420i	2.59140	−	2.87503i
104.18	−0.534897	+	1.64624i	1.66454	−	0.478872i	−0.805968	−	0.585570i	2.04228	+	0.910555i	−0.102015	+	2.99638i	−0.846893	+	2.50655i	−1.40566	+	1.02127i	2.54136	−	1.59420i	−2.59140	+	2.87503i
104.19	−0.507173	+	1.56092i	−0.407162	−	1.68351i	−0.561203	−	0.407738i	0.134898	+	2.23200i	2.83433	+	0.218287i	1.99984	+	1.73224i	−1.73452	+	1.26020i	−2.66844	+	1.37092i	−3.55238	−	0.921442i
104.20	−0.507173	+	1.56092i	0.407162	+	1.68351i	−0.561203	−	0.407738i	−0.134898	−	2.23200i	−2.83433	−	0.218287i	−1.99984	+	1.73224i	−1.73452	+	1.26020i	−2.66844	+	1.37092i	3.55238	+	0.921442i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
21.c	even	2	1	inner
25.e	even	10	1	inner
75.h	odd	10	1	inner
175.m	odd	10	1	inner
525.w	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	525.2.w.a	✓	304
3.b	odd	2	1	inner	525.2.w.a	✓	304
7.b	odd	2	1	inner	525.2.w.a	✓	304
21.c	even	2	1	inner	525.2.w.a	✓	304
25.e	even	10	1	inner	525.2.w.a	✓	304
75.h	odd	10	1	inner	525.2.w.a	✓	304
175.m	odd	10	1	inner	525.2.w.a	✓	304
525.w	even	10	1	inner	525.2.w.a	✓	304

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
525.2.w.a	✓	304	1.a	even	1	1	trivial
525.2.w.a	✓	304	3.b	odd	2	1	inner
525.2.w.a	✓	304	7.b	odd	2	1	inner
525.2.w.a	✓	304	21.c	even	2	1	inner
525.2.w.a	✓	304	25.e	even	10	1	inner
525.2.w.a	✓	304	75.h	odd	10	1	inner
525.2.w.a	✓	304	175.m	odd	10	1	inner
525.2.w.a	✓	304	525.w	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(525, [\chi])\).