Newform orbit 169.4.h.a

• Label: 169.4.h.a
• Level: $169$
• Weight: $4$
• Character orbit: 169.h
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $528$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.h (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.97132279097\)
Analytic rank:	\(0\)
Dimension:	\(528\)
Relative dimension:	\(44\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 528 q - 13 q^{2} - 11 q^{3} + 157 q^{4} - 13 q^{5} - 13 q^{6} - 13 q^{7} - 13 q^{8} - 339 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
12.1	−4.54493	+	3.13714i	4.40875	+	2.31389i	7.97791	−	21.0360i	−4.48511	−	5.06264i	−27.2965	+	3.31439i	2.07596	+	8.42250i	19.1609	+	77.7389i	−1.25477	−	1.81784i	36.2667	+	8.93894i
12.2	−4.26969	+	2.94716i	−2.73220	−	1.43397i	6.70772	−	17.6868i	6.73172	+	7.59854i	15.8918	−	1.92961i	3.54523	+	14.3836i	13.5532	+	54.9874i	−9.92910	−	14.3848i	−51.1365	−	12.6040i
12.3	−4.22925	+	2.91924i	−8.43671	−	4.42793i	6.52775	−	17.2123i	−3.39733	−	3.83479i	48.6071	−	5.90198i	0.351475	+	1.42599i	12.8006	+	51.9342i	36.2338	+	52.4938i	25.5628	+	6.30067i
12.4	−3.99097	+	2.75477i	−3.64670	−	1.91393i	5.50225	−	14.5082i	6.25474	+	7.06014i	19.8263	−	2.40735i	−8.42904	−	34.1979i	8.72323	+	35.3915i	−5.70250	−	8.26149i	−44.4115	−	10.9465i
12.5	−3.98439	+	2.75022i	8.05272	+	4.22640i	5.47477	−	14.4358i	14.3051	+	16.1471i	−43.7087	+	5.30720i	−2.97876	−	12.0853i	8.61907	+	34.9689i	31.6462	+	45.8474i	−101.405	−	24.9941i
12.6	−3.82658	+	2.64130i	−0.515339	−	0.270471i	4.82943	−	12.7341i	−9.07725	−	10.2461i	2.68638	−	0.326186i	−2.31988	−	9.41213i	6.25264	+	25.3679i	−15.1453	−	21.9418i	61.7978	+	15.2318i
12.7	−3.51754	+	2.42799i	1.95844	+	1.02787i	3.64116	−	9.60096i	9.47471	+	10.6947i	−9.38456	+	1.13949i	5.03943	+	20.4458i	2.32009	+	9.41299i	−12.5588	−	18.1945i	−59.2943	−	14.6147i
12.8	−3.43216	+	2.36905i	6.52784	+	3.42607i	3.33048	−	8.78175i	−6.38443	−	7.20653i	−30.5211	+	3.70593i	−3.10428	−	12.5945i	1.38935	+	5.63683i	15.5369	+	22.5091i	38.9850	+	9.60894i
12.9	−3.03941	+	2.09795i	−4.12592	−	2.16545i	1.99976	−	5.27293i	−11.0346	−	12.4555i	17.0834	−	2.07430i	−0.976064	−	3.96005i	−2.08637	−	8.46473i	−3.00371	−	4.35163i	59.6696	+	14.7072i
12.10	−2.90590	+	2.00580i	−3.44169	−	1.80634i	1.58418	−	4.17715i	4.49093	+	5.06921i	13.6243	−	1.65430i	2.43551	+	9.88126i	−2.98502	−	12.1107i	−6.75537	−	9.78685i	−23.2180	−	5.72272i
12.11	−2.89276	+	1.99673i	5.07271	+	2.66236i	1.54431	−	4.07201i	−3.05174	−	3.44470i	−19.9902	+	2.42725i	6.84874	+	27.7864i	−3.06611	−	12.4397i	3.30644	+	4.79021i	15.7061	+	3.87121i
12.12	−2.45236	+	1.69274i	3.10130	+	1.62769i	0.311859	−	0.822303i	5.37501	+	6.06714i	−10.3608	+	1.25802i	−5.13062	−	20.8157i	−5.07782	−	20.6015i	−8.36906	−	12.1247i	−23.4516	−	5.78030i
12.13	−2.28978	+	1.58052i	−7.48539	−	3.92863i	−0.0917941	+	0.242041i	0.259627	+	0.293058i	23.3492	−	2.83510i	6.07172	+	24.6339i	−5.49913	−	22.3108i	25.2591	+	36.5941i	−1.05767	−	0.260693i
12.14	−2.12313	+	1.46549i	−7.37628	−	3.87137i	−0.476815	+	1.25726i	13.1976	+	14.8970i	21.3343	−	2.59045i	−5.27042	−	21.3829i	−5.76925	−	23.4068i	24.0842	+	34.8920i	−49.8518	−	12.2874i
12.15	−1.58142	+	1.09157i	1.40125	+	0.735435i	−1.52749	+	4.02767i	−12.1968	−	13.7674i	−3.01875	+	0.366542i	7.63975	+	30.9957i	−5.65977	−	22.9626i	−13.9151	−	20.1595i	34.3163	+	8.45821i
12.16	−1.21218	+	0.836708i	0.815974	+	0.428256i	−2.06754	+	5.45165i	1.46999	+	1.65927i	−1.34743	+	0.163608i	−3.10032	−	12.5785i	−4.87513	−	19.7792i	−14.8553	−	21.5217i	−3.17022	−	0.781389i
12.17	−1.08949	+	0.752017i	7.70099	+	4.04179i	−2.21539	+	5.84151i	1.12991	+	1.27541i	−11.4296	+	1.38781i	−1.44954	−	5.88103i	−4.51377	−	18.3131i	27.6315	+	40.0311i	−2.19016	−	0.539825i
12.18	−1.07690	+	0.743328i	−1.42990	−	0.750468i	−2.22967	+	5.87916i	−5.16749	−	5.83289i	2.09770	−	0.254706i	−1.49760	−	6.07599i	−4.47422	−	18.1526i	−13.8563	−	20.0744i	9.90060	+	2.44028i
12.19	−0.997032	+	0.688202i	−6.71687	−	3.52528i	−2.31639	+	6.10781i	−8.17932	−	9.23255i	9.12304	−	1.10774i	−7.34833	−	29.8134i	−4.21331	−	17.0941i	17.3509	+	25.1372i	14.5089	+	3.57612i
12.20	−0.954593	+	0.658908i	7.15527	+	3.75538i	−2.35975	+	6.22215i	7.67220	+	8.66012i	−9.30483	+	1.12981i	5.78913	+	23.4874i	−4.06792	−	16.5042i	21.7573	+	31.5209i	−13.0301	−	3.21162i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
169.h	even	26	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	169.4.h.a	✓	528
169.h	even	26	1	inner	169.4.h.a	✓	528

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
169.4.h.a	✓	528	1.a	even	1	1	trivial
169.4.h.a	✓	528	169.h	even	26	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(169, [\chi])\).