Newform orbit 475.2.bi.a

• Label: 475.2.bi.a
• Level: $475$
• Weight: $2$
• Character orbit: 475.bi
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $2304$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.bi (of order \(180\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(2304\)
Relative dimension:	\(48\) over \(\Q(\zeta_{180})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{180}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2304 q - 48 q^{2} - 48 q^{3} - 60 q^{4} - 48 q^{5} - 36 q^{6} - 30 q^{7} - 72 q^{8} - 60 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−2.62434	−	0.903631i	−2.47135	−	0.570556i	4.49457	+	3.51154i	1.68444	−	1.47060i	5.97008	+	3.73052i	0.0634865	−	0.236935i	−5.59878	−	8.62136i	3.08565	+	1.50497i	−5.74942	+	2.33723i
2.2	−2.56561	−	0.883409i	0.702914	+	0.162281i	4.22590	+	3.30163i	−1.98966	+	1.02042i	−1.66004	−	1.03731i	0.919986	−	3.43344i	−4.96961	−	7.65253i	−2.22863	−	1.08697i	6.00613	−	0.860317i
2.3	−2.37287	−	0.817046i	3.28556	+	0.758531i	3.38695	+	2.64617i	−0.200815	−	2.22703i	−7.17646	−	4.48435i	0.308540	−	1.15149i	−3.14110	−	4.83687i	7.52313	+	3.66928i	−1.34308	+	5.44854i
2.4	−2.35508	−	0.810920i	0.00625138	+	0.00144324i	3.31280	+	2.58824i	−1.87745	−	1.21457i	−0.0135522	−	0.00846832i	−1.26696	+	4.72836i	−2.98990	−	4.60405i	−2.69635	−	1.31510i	3.43662	+	4.38288i
2.5	−2.28668	−	0.787366i	0.766057	+	0.176858i	3.03292	+	2.36958i	2.20863	+	0.349212i	−1.61247	−	1.00759i	0.287418	−	1.07266i	−2.43524	−	3.74994i	−2.14082	−	1.04415i	−4.77547	−	2.53753i
2.6	−2.23217	−	0.768598i	−2.11065	−	0.487282i	2.81582	+	2.19996i	0.259366	+	2.22097i	4.33681	+	2.70994i	−0.637349	+	2.37862i	−2.02295	−	3.11507i	1.52102	+	0.741853i	1.12809	−	5.15694i
2.7	−2.15383	−	0.741624i	2.53352	+	0.584910i	2.51296	+	1.96334i	−0.432784	+	2.19379i	−5.02300	−	3.13872i	−0.361347	+	1.34856i	−1.47513	−	2.27150i	3.38023	+	1.64865i	2.55911	−	4.40408i
2.8	−2.11303	−	0.727576i	−2.93221	−	0.676953i	2.35952	+	1.84346i	−1.89401	+	1.18859i	5.70331	+	3.56382i	0.732295	−	2.73296i	−1.21018	−	1.86352i	5.44318	+	2.65482i	4.86688	−	1.13350i
2.9	−1.79315	−	0.617429i	−0.497419	−	0.114838i	1.25813	+	0.982958i	1.00275	+	1.99862i	0.821041	+	0.513043i	0.500493	−	1.86786i	0.416679	+	0.641630i	−2.46214	−	1.20087i	−0.564078	−	4.20295i
2.10	−1.68202	−	0.579164i	−1.00903	−	0.232952i	0.917723	+	0.717004i	2.04123	−	0.912897i	1.56228	+	0.976222i	−0.783209	+	2.92298i	0.809393	+	1.24636i	−1.73251	−	0.845003i	−3.96210	+	0.353300i
2.11	−1.65294	−	0.569153i	1.76866	+	0.408326i	0.832254	+	0.650228i	0.672217	−	2.13263i	−2.69108	−	1.68157i	−0.575171	+	2.14657i	0.898671	+	1.38383i	0.265032	+	0.129265i	−2.32493	+	3.14252i
2.12	−1.63974	−	0.564606i	−0.536741	−	0.123916i	0.793930	+	0.620286i	−0.505939	−	2.17808i	0.810149	+	0.506237i	1.26758	−	4.73067i	0.937428	+	1.44351i	−2.42365	−	1.18209i	−0.400151	+	3.85713i
2.13	−1.53158	−	0.527365i	1.56218	+	0.360658i	0.491602	+	0.384081i	−2.11036	−	0.739181i	−2.20241	−	1.37622i	−0.0254288	+	0.0949017i	1.21407	+	1.86950i	−0.386047	−	0.188288i	2.84236	+	2.24504i
2.14	−1.31509	−	0.452821i	2.79569	+	0.645436i	−0.0516128	−	0.0403243i	1.94765	+	1.09847i	−3.38431	−	2.11475i	1.08161	−	4.03663i	1.56465	+	2.40935i	4.70291	+	2.29376i	−2.06393	−	2.32653i
2.15	−1.18387	−	0.407639i	−3.17084	−	0.732046i	−0.340643	−	0.266139i	−0.821350	−	2.07976i	3.45545	+	2.15921i	−0.594087	+	2.21716i	1.65866	+	2.55411i	6.82196	+	3.32729i	0.124582	+	2.79698i
2.16	−1.15799	−	0.398726i	−2.70353	−	0.624158i	−0.394073	−	0.307884i	2.13344	−	0.669642i	2.88178	+	1.80073i	0.726074	−	2.70974i	1.66762	+	2.56791i	4.22309	+	2.05974i	−2.73750	−	0.0752243i
2.17	−1.08643	−	0.374089i	−1.56658	−	0.361674i	−0.535624	−	0.418476i	−1.77819	+	1.35575i	1.56669	+	0.978977i	−0.847061	+	3.16128i	1.67699	+	2.58234i	−0.373010	−	0.181929i	2.43905	−	0.807730i
2.18	−0.967384	−	0.333097i	−0.527689	−	0.121827i	−0.751143	−	0.586857i	−2.21092	+	0.334424i	0.469898	+	0.293625i	0.292070	−	1.09002i	1.64563	+	2.53405i	−2.43277	−	1.18654i	2.25020	+	0.412934i
2.19	−0.865537	−	0.298028i	2.22234	+	0.513069i	−0.915688	−	0.715414i	2.21532	−	0.303886i	−1.77061	−	1.10640i	−0.390977	+	1.45915i	1.57648	+	2.42757i	1.97919	+	0.965317i	−2.00801	−	0.397204i
2.20	−0.507373	−	0.174703i	−2.17799	−	0.502828i	−1.34912	−	1.05404i	1.04478	+	1.97698i	1.01721	+	0.635621i	0.724035	−	2.70213i	1.08488	+	1.67056i	1.79441	+	0.875194i	−0.184708	−	1.18559i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.f	odd	18	1	inner
25.f	odd	20	1	inner
475.bi	even	180	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	475.2.bi.a	✓	2304
19.f	odd	18	1	inner	475.2.bi.a	✓	2304
25.f	odd	20	1	inner	475.2.bi.a	✓	2304
475.bi	even	180	1	inner	475.2.bi.a	✓	2304

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
475.2.bi.a	✓	2304	1.a	even	1	1	trivial
475.2.bi.a	✓	2304	19.f	odd	18	1	inner
475.2.bi.a	✓	2304	25.f	odd	20	1	inner
475.2.bi.a	✓	2304	475.bi	even	180	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(475, [\chi])\).