Newform orbit 113.2.i.a

• Label: 113.2.i.a
• Level: $113$
• Weight: $2$
• Character orbit: 113.i
• Analytic conductor: $0.902$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	113.i (of order \(56\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.902309542840\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(9\) over \(\Q(\zeta_{56})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{56}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 28 q^{2} - 24 q^{3} + 16 q^{4} - 24 q^{5} - 20 q^{6} - 12 q^{7} - 28 q^{8} - 40 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
9.1	−2.09762	−	1.67279i	0.250458	+	0.0140654i	1.15672	+	5.06790i	−0.181281	−	3.22800i	−0.501836	−	0.448468i	−2.82655	−	1.36120i	3.72303	−	7.73096i	−2.91861	−	0.328848i	−5.01953	+	7.07436i
9.2	−1.72874	−	1.37862i	0.800598	+	0.0449606i	0.642891	+	2.81669i	0.220921	+	3.93387i	−1.32204	−	1.18145i	3.01720	+	1.45301i	0.853007	−	1.77129i	−2.34220	−	0.263903i	5.04140	−	7.10518i
9.3	−1.45586	−	1.16101i	−3.08023	−	0.172982i	0.326546	+	1.43069i	0.0298052	+	0.530731i	4.28355	+	3.82802i	0.199681	+	0.0961612i	−0.430242	+	0.893406i	6.47674	+	0.729753i	0.572793	−	0.807276i
9.4	−0.746348	−	0.595193i	2.38913	+	0.134171i	−0.242261	−	1.06141i	−0.0508015	−	0.904605i	−1.70326	−	1.52213i	−0.744186	−	0.358381i	−1.27932	+	2.65653i	2.70880	+	0.305208i	−0.500499	+	0.705387i
9.5	0.413535	+	0.329783i	−2.06378	−	0.115899i	−0.382788	−	1.67710i	−0.155155	−	2.76280i	−0.815224	−	0.728529i	0.425022	+	0.204680i	0.853773	−	1.77288i	1.26462	+	0.142488i	0.846962	−	1.19368i
9.6	0.453389	+	0.361566i	0.657112	+	0.0369026i	−0.370210	−	1.62200i	0.0497726	+	0.886284i	0.284585	+	0.254321i	3.50853	+	1.68962i	0.921833	−	1.91421i	−2.55070	−	0.287395i	−0.297884	+	0.419828i
9.7	1.26594	+	1.00956i	1.18834	+	0.0667358i	0.138368	+	0.606232i	0.0726647	+	1.29391i	1.43700	+	1.28418i	−4.25336	−	2.04831i	0.968234	−	2.01056i	−1.57344	−	0.177284i	−1.21429	+	1.71138i
9.8	1.60094	+	1.27671i	−2.83589	−	0.159260i	0.487987	+	2.13801i	0.213928	+	3.80935i	−4.33676	−	3.87557i	1.40937	+	0.678715i	−0.171465	+	0.356050i	5.03577	+	0.567395i	−4.52093	+	6.37166i
9.9	2.02914	+	1.61819i	−0.454102	−	0.0255018i	1.05384	+	4.61719i	−0.172294	−	3.06798i	−0.880170	−	0.786568i	−0.557246	−	0.268356i	−3.08090	+	6.39755i	−2.77558	−	0.312733i	4.61495	−	6.50416i
11.1	−2.68046	+	0.611798i	−1.07689	+	1.94849i	5.00864	−	2.41203i	2.39325	−	1.32270i	1.69449	−	5.88169i	1.92645	+	2.41569i	−7.65067	+	6.10121i	−1.04082	−	1.65645i	−5.60578	+	5.00963i
11.2	−2.02615	+	0.462456i	0.465405	−	0.842086i	2.08949	−	1.00624i	−3.07068	+	1.69711i	−0.553553	+	1.92142i	1.56686	+	1.96478i	−0.518582	+	0.413556i	1.10359	+	1.75635i	5.43683	−	4.85865i
11.3	−1.80431	+	0.411822i	0.333572	−	0.603553i	1.28400	−	0.618341i	0.962549	−	0.531982i	−0.353311	+	1.22637i	−2.04811	−	2.56825i	0.831804	−	0.663342i	1.34309	+	2.13752i	−1.51765	+	1.35626i
11.4	−1.05417	+	0.240608i	−1.25122	+	2.26391i	−0.748550	+	0.360483i	0.0415455	−	0.0229614i	0.774285	−	2.68760i	−2.62482	−	3.29142i	2.39313	−	1.90846i	−1.96364	−	3.12511i	−0.0382714	+	0.0342014i
11.5	−0.563377	+	0.128587i	1.63111	−	2.95127i	−1.50108	+	0.722881i	0.736041	−	0.406795i	−0.539434	+	1.87242i	0.689118	+	0.864127i	1.65631	−	1.32086i	−4.45338	−	7.08751i	−0.362360	+	0.323825i
11.6	0.100068	−	0.0228398i	−0.560665	+	1.01445i	−1.79245	+	0.863196i	−1.92116	+	1.06179i	−0.0329348	+	0.114319i	1.45361	+	1.82277i	−0.320147	+	0.255309i	0.881338	+	1.40264i	−0.167996	+	0.150130i
11.7	1.32968	−	0.303490i	0.556428	−	1.00678i	−0.125999	+	0.0606780i	0.819825	−	0.453101i	0.434322	−	1.50757i	−0.897119	−	1.12495i	−2.28176	+	1.81964i	0.892101	+	1.41977i	0.952592	−	0.851288i
11.8	1.57736	−	0.360021i	−1.47266	+	2.66459i	0.556497	−	0.267995i	1.61626	−	0.893273i	−1.36361	+	4.73319i	0.685395	+	0.859459i	−1.74857	+	1.39444i	−3.33518	−	5.30791i	2.22781	−	1.99090i
11.9	2.54434	−	0.580729i	−0.546666	+	0.989117i	4.33448	−	2.08738i	−2.98726	+	1.65100i	−0.816495	+	2.83412i	−2.77584	−	3.48080i	5.73540	−	4.57383i	0.916586	+	1.45874i	−6.64182	+	5.93550i
13.1	−1.13768	−	2.36241i	2.28120	+	1.61860i	−3.03969	+	3.81165i	1.14558	+	1.61455i	1.22852	−	7.23057i	0.706923	−	3.09723i	7.35017	+	1.67763i	1.59318	+	4.55306i	2.51092	−	4.54317i
13.2	−1.03880	−	2.15708i	−0.761509	−	0.540319i	−2.32693	+	2.91787i	−1.58731	−	2.23711i	−0.374461	+	2.20392i	−0.707914	+	3.10157i	4.04299	+	0.922786i	−0.702886	−	2.00873i	−3.17673	+	5.74786i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
113.i	even	56	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	113.2.i.a	✓	216
113.i	even	56	1	inner	113.2.i.a	✓	216

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
113.2.i.a	✓	216	1.a	even	1	1	trivial
113.2.i.a	✓	216	113.i	even	56	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(113, [\chi])\).