Newform orbit 104.3.x.a

• Label: 104.3.x.a
• Level: $104$
• Weight: $3$
• Character orbit: 104.x
• Analytic conductor: $2.834$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 104 = 2^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	104.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.83379474935\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 4 q^{2} - 6 q^{4} - 6 q^{6} - 8 q^{7} + 14 q^{8} + 128 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
37.1	−1.99704	−	0.108761i	−2.91455	+	1.68271i	3.97634	+	0.434401i	3.32349	−	3.32349i	6.00348	−	3.04346i	0.104412	+	0.389671i	−7.89367	−	1.29999i	1.16305	−	2.01446i	−6.99861	+	6.27568i
37.2	−1.97604	−	0.308635i	2.12272	−	1.22555i	3.80949	+	1.21975i	−6.26471	+	6.26471i	−4.57283	+	1.76660i	1.47970	+	5.52230i	−7.15125	−	3.58602i	−1.49605	+	2.59123i	14.3128	−	10.4458i
37.3	−1.96585	+	0.368002i	2.86455	−	1.65385i	3.72915	−	1.44687i	2.51902	−	2.51902i	−5.02266	+	4.30538i	−1.41367	−	5.27589i	−6.79851	+	4.21667i	0.970422	−	1.68082i	−4.02502	+	5.87902i
37.4	−1.73744	+	0.990605i	−3.44798	+	1.99069i	2.03740	−	3.44224i	−4.11868	+	4.11868i	4.01867	−	6.87429i	−1.19292	−	4.45205i	−0.129975	+	7.99894i	3.42569	−	5.93347i	3.07598	−	11.2360i
37.5	−1.65217	−	1.12709i	−1.28958	+	0.744538i	1.45935	+	3.72428i	0.276490	−	0.276490i	2.96976	+	0.223360i	−2.37559	−	8.86584i	1.78649	−	7.79798i	−3.39133	+	5.87395i	−0.768437	+	0.145181i
37.6	−1.47936	−	1.34592i	3.57445	−	2.06371i	0.377021	+	3.98219i	5.28854	−	5.28854i	−8.06548	−	1.75793i	2.65665	+	9.91475i	4.80194	−	6.39854i	4.01779	−	6.95901i	−14.9416	+	0.705733i
37.7	−1.42761	+	1.40069i	−0.612804	+	0.353803i	0.0761534	−	3.99928i	1.03191	−	1.03191i	0.379280	−	1.36344i	2.66670	+	9.95228i	5.49301	+	5.81608i	−4.24965	+	7.36061i	−0.0277847	+	2.91855i
37.8	−1.08444	−	1.68047i	−0.153386	+	0.0885577i	−1.64799	+	3.64474i	−2.18154	+	2.18154i	0.315157	+	0.161727i	0.791672	+	2.95456i	7.91203	−	1.18310i	−4.48432	+	7.76706i	6.03176	+	1.30027i
37.9	−0.950881	+	1.75950i	4.87568	−	2.81498i	−2.19165	−	3.34614i	−0.709122	+	0.709122i	0.316745	+	11.2555i	1.00876	+	3.76475i	7.97152	−	0.674417i	11.3482	−	19.6556i	−0.573406	−	1.92199i
37.10	−0.702252	+	1.87266i	−0.561288	+	0.324060i	−3.01368	−	2.63015i	6.07540	−	6.07540i	−0.212687	−	1.27867i	−2.40837	−	8.98816i	7.04174	−	3.79657i	−4.28997	+	7.43045i	7.11068	+	15.6436i
37.11	−0.684744	−	1.87913i	−4.39268	+	2.53611i	−3.06225	+	2.57345i	4.23770	−	4.23770i	7.77355	+	6.51782i	1.53484	+	5.72808i	6.93270	+	3.99221i	8.36376	−	14.4865i	−10.8649	−	5.06144i
37.12	−0.495189	−	1.93773i	4.05656	−	2.34206i	−3.50958	+	1.91908i	−2.23435	+	2.23435i	−6.54703	−	6.70075i	−3.40412	−	12.7043i	5.45656	+	5.85029i	6.47046	−	11.2072i	5.43598	+	3.22313i
37.13	−0.328160	+	1.97289i	0.561288	−	0.324060i	−3.78462	−	1.29485i	−6.07540	+	6.07540i	0.455143	+	1.21371i	−2.40837	−	8.98816i	3.79657	−	7.04174i	−4.28997	+	7.43045i	−9.99242	−	13.9798i
37.14	−0.0562608	+	1.99921i	−4.87568	+	2.81498i	−3.99367	−	0.224954i	0.709122	−	0.709122i	−5.35342	−	9.90588i	1.00876	+	3.76475i	0.674417	−	7.97152i	11.3482	−	19.6556i	1.37779	+	1.45758i
37.15	0.361928	−	1.96698i	−1.99448	+	1.15151i	−3.73802	−	1.42381i	−3.45975	+	3.45975i	1.54314	+	4.33986i	0.417969	+	1.55988i	−4.15350	+	6.83729i	−1.84804	+	3.20090i	5.55308	+	8.05744i
37.16	0.536005	+	1.92684i	0.612804	−	0.353803i	−3.42540	+	2.06559i	−1.03191	+	1.03191i	1.01019	+	0.991133i	2.66670	+	9.95228i	−5.81608	−	5.49301i	−4.24965	+	7.36061i	−2.54143	−	1.43521i
37.17	0.670051	−	1.88442i	1.99448	−	1.15151i	−3.10206	−	2.52531i	3.45975	−	3.45975i	−0.833528	−	4.53000i	0.417969	+	1.55988i	−6.83729	+	4.15350i	−1.84804	+	3.20090i	−4.20141	−	8.83783i
37.18	1.00937	+	1.72661i	3.44798	−	1.99069i	−1.96236	+	3.48556i	4.11868	−	4.11868i	6.91741	+	3.94397i	−1.19292	−	4.45205i	−7.99894	+	0.129975i	3.42569	−	5.93347i	11.2686	+	2.95410i
37.19	1.39771	−	1.43053i	−4.05656	+	2.34206i	−0.0928144	−	3.99892i	2.23435	−	2.23435i	−2.31952	+	9.07653i	−3.40412	−	12.7043i	−5.85029	−	5.45656i	6.47046	−	11.2072i	−0.0733246	−	6.31926i
37.20	1.51848	+	1.30162i	−2.86455	+	1.65385i	0.611546	+	3.95298i	−2.51902	+	2.51902i	−6.50244	−	1.21724i	−1.41367	−	5.27589i	−4.21667	+	6.79851i	0.970422	−	1.68082i	−7.10389	+	0.546255i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
13.f	odd	12	1	inner
104.x	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	104.3.x.a	✓	104
4.b	odd	2	1		416.3.bv.a		104
8.b	even	2	1	inner	104.3.x.a	✓	104
8.d	odd	2	1		416.3.bv.a		104
13.f	odd	12	1	inner	104.3.x.a	✓	104
52.l	even	12	1		416.3.bv.a		104
104.u	even	12	1		416.3.bv.a		104
104.x	odd	12	1	inner	104.3.x.a	✓	104

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.3.x.a	✓	104	1.a	even	1	1	trivial
104.3.x.a	✓	104	8.b	even	2	1	inner
104.3.x.a	✓	104	13.f	odd	12	1	inner
104.3.x.a	✓	104	104.x	odd	12	1	inner
416.3.bv.a		104	4.b	odd	2	1
416.3.bv.a		104	8.d	odd	2	1
416.3.bv.a		104	52.l	even	12	1
416.3.bv.a		104	104.u	even	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(104, [\chi])\).