Newform orbit 213.2.l.a

• Label: 213.2.l.a
• Level: $213$
• Weight: $2$
• Character orbit: 213.l
• Analytic conductor: $1.701$
• Analytic rank: $0$
• Dimension: $132$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 213 = 3 \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	213.l (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.70081356305\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 132 * q - 10 * q^3 + 10 * q^4 - 15 * q^6 - 14 * q^7 - 14 * q^9 + 2 * q^10 + 20 * q^12 - 14 * q^13 + 22 * q^15 - 30 * q^16 + 8 * q^18 - 14 * q^19 - 28 * q^21 - 14 * q^22 + 8 * q^24 - 84 * q^25 - 7 * q^27 - 42 * q^28 - 5 * q^30 + 14 * q^31 - 7 * q^33 - 42 * q^34 - 43 * q^36 + 22 * q^37 - 35 * q^39 + 64 * q^40 - 28 * q^42 + 16 * q^43 - 50 * q^45 + 55 * q^48 + 12 * q^49 - 98 * q^51 - 168 * q^52 - 38 * q^54 + 70 * q^55 + 48 * q^57 + 86 * q^58 + 116 * q^60 + 14 * q^61 + 105 * q^63 + 2 * q^64 - 42 * q^67 - 28 * q^69 + 266 * q^72 - 62 * q^73 + 18 * q^75 - 120 * q^76 + 21 * q^78 - 74 * q^79 - 22 * q^81 - 182 * q^82 + 42 * q^84 - 2 * q^87 + 15 * q^90 - 22 * q^91 + 14 * q^94 - 210 * q^96 - 42 * q^97 + 49 * q^99

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−2.17375	+	1.73351i	−0.833794	+	1.51815i	1.27510	−	5.58657i		−	2.20653i	−0.819269	−	4.74547i	2.00764	+	1.60104i	4.49993	+	9.34420i	−1.60957	−	2.53165i	3.82504	+	4.79645i
23.2	−1.95866	+	1.56198i	1.56487	−	0.742422i	0.951534	−	4.16894i		−	1.25570i	−1.90540	+	3.89845i	−1.63347	−	1.30265i	2.47413	+	5.13758i	1.89762	−	2.32358i	1.96138	+	2.45949i
23.3	−1.74810	+	1.39407i	1.11251	+	1.32753i	0.667405	−	2.92409i			3.11868i	−3.79544	−	0.769744i	−0.583477	−	0.465307i	0.969434	+	2.01305i	−0.524652	+	2.95377i	−4.34764	−	5.45177i
23.4	−1.44834	+	1.15502i	−1.73136	+	0.0489550i	0.318597	−	1.39586i			0.149249i	2.45106	−	2.07065i	−2.05988	−	1.64270i	−0.456734	−	0.948417i	2.99521	−	0.169517i	−0.172385	−	0.216164i
23.5	−1.43340	+	1.14310i	0.715776	−	1.57723i	0.302917	−	1.32717i			1.45768i	0.776937	+	3.07900i	1.88203	+	1.50087i	−0.508072	−	1.05502i	−1.97533	−	2.25789i	−1.66627	−	2.08944i
23.6	−1.11358	+	0.888049i	−0.0493701	+	1.73135i	0.00638361	−	0.0279684i		−	3.18442i	−1.48254	−	1.97183i	−3.37622	−	2.69244i	−1.21825	−	2.52972i	−2.99513	−	0.170954i	2.82792	+	3.54610i
23.7	−1.01153	+	0.806668i	1.70091	+	0.326966i	−0.0725631	+	0.317920i		−	3.12921i	−1.98427	+	1.04133i	1.87483	+	1.49512i	−1.30577	−	2.71146i	2.78619	+	1.11228i	2.52424	+	3.16529i
23.8	−0.782614	+	0.624114i	−0.908953	+	1.47438i	−0.222075	+	0.972975i			2.40824i	−0.208824	−	1.72116i	2.42488	+	1.93378i	−1.30208	−	2.70381i	−1.34761	−	2.68029i	−1.50302	−	1.88472i
23.9	−0.569704	+	0.454324i	−0.628011	−	1.61419i	−0.326889	+	1.43220i		−	2.87043i	1.09114	+	0.634289i	0.816025	+	0.650759i	−1.09677	−	2.27748i	−2.21120	+	2.02746i	1.30411	+	1.63530i
23.10	−0.524364	+	0.418166i	−0.528285	−	1.64952i	−0.344947	+	1.51131i			3.23325i	0.966787	+	0.644037i	−3.33086	−	2.65627i	−1.03310	−	2.14526i	−2.44183	+	1.74283i	−1.35204	−	1.69540i
23.11	−0.208082	+	0.165940i	1.72906	+	0.101725i	−0.429280	+	1.88080i			0.782738i	−0.376667	+	0.265753i	0.755976	+	0.602871i	−0.453728	−	0.942175i	2.97930	+	0.351777i	−0.129888	−	0.162874i
23.12	0.208082	−	0.165940i	−1.60197	−	0.658560i	−0.429280	+	1.88080i		−	0.782738i	−0.442622	+	0.128796i	0.755976	+	0.602871i	0.453728	+	0.942175i	2.13260	+	2.10998i	−0.129888	−	0.162874i
23.13	0.524364	−	0.418166i	1.19167	−	1.25695i	−0.344947	+	1.51131i		−	3.23325i	0.0992530	−	1.15742i	−3.33086	−	2.65627i	1.03310	+	2.14526i	−0.159854	−	2.99574i	−1.35204	−	1.69540i
23.14	0.569704	−	0.454324i	1.26619	−	1.18185i	−0.326889	+	1.43220i			2.87043i	0.184410	−	1.24856i	0.816025	+	0.650759i	1.09677	+	2.27748i	0.206465	−	2.99289i	1.30411	+	1.63530i
23.15	0.782614	−	0.624114i	0.179227	+	1.72275i	−0.222075	+	0.972975i		−	2.40824i	1.21546	+	1.23639i	2.42488	+	1.93378i	1.30208	+	2.70381i	−2.93576	+	0.617528i	−1.50302	−	1.88472i
23.16	1.01153	−	0.806668i	−1.67433	−	0.443411i	−0.0725631	+	0.317920i			3.12921i	−2.05132	+	0.902106i	1.87483	+	1.49512i	1.30577	+	2.71146i	2.60677	+	1.48483i	2.52424	+	3.16529i
23.17	1.11358	−	0.888049i	−0.706722	+	1.58131i	0.00638361	−	0.0279684i			3.18442i	0.617291	+	2.38852i	−3.37622	−	2.69244i	1.21825	+	2.52972i	−2.00109	−	2.23510i	2.82792	+	3.54610i
23.18	1.43340	−	1.14310i	0.0394435	−	1.73160i	0.302917	−	1.32717i		−	1.45768i	−1.92285	−	2.52716i	1.88203	+	1.50087i	0.508072	+	1.05502i	−2.99689	−	0.136601i	−1.66627	−	2.08944i
23.19	1.44834	−	1.15502i	1.53866	+	0.795315i	0.318597	−	1.39586i		−	0.149249i	3.14711	−	0.625286i	−2.05988	−	1.64270i	0.456734	+	0.948417i	1.73495	+	2.44744i	−0.172385	−	0.216164i
23.20	1.74810	−	1.39407i	−1.57833	+	0.713361i	0.667405	−	2.92409i		−	3.11868i	−1.76461	+	3.44732i	−0.583477	−	0.465307i	−0.969434	−	2.01305i	1.98223	−	2.25183i	−4.34764	−	5.45177i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
71.f	odd	14	1	inner
213.l	even	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	213.2.l.a	✓	132
3.b	odd	2	1	inner	213.2.l.a	✓	132
71.f	odd	14	1	inner	213.2.l.a	✓	132
213.l	even	14	1	inner	213.2.l.a	✓	132

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
213.2.l.a	✓	132	1.a	even	1	1	trivial
213.2.l.a	✓	132	3.b	odd	2	1	inner
213.2.l.a	✓	132	71.f	odd	14	1	inner
213.2.l.a	✓	132	213.l	even	14	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(213, [\chi])\).