Newform orbit 169.4.i.a

• Label: 169.4.i.a
• Level: $169$
• Weight: $4$
• Character orbit: 169.i
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $1080$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.i (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.97132279097\)
Analytic rank:	\(0\)
Dimension:	\(1080\)
Relative dimension:	\(45\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1080 q - 27 q^{2} - 19 q^{3} + 155 q^{4} - 30 q^{5} - 56 q^{6} + 9 q^{7} + 4 q^{8} + 364 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−4.36857	+	3.28276i	0.666615	−	3.26530i	6.08208	−	20.9978i	11.6758	+	6.12792i	7.80705	+	16.4530i	−30.8182	−	5.00845i	26.8589	+	70.8210i	14.6217	+	6.22971i	−71.1229	+	11.5586i
3.2	−4.25141	+	3.19473i	−1.68365	+	8.24708i	5.64247	−	19.4801i	−11.4022	−	5.98435i	−19.1893	−	40.4405i	−33.3973	−	5.42759i	23.1588	+	61.0648i	−40.3401	−	17.1873i	67.5939	−	10.9851i
3.3	−4.12652	+	3.10088i	0.990669	−	4.85262i	5.18696	−	17.9075i	−10.2304	−	5.36931i	10.9593	+	23.0963i	3.16043	+	0.513619i	19.4817	+	51.3691i	2.27298	+	0.968427i	58.8654	−	9.56655i
3.4	−4.11168	+	3.08973i	−0.342888	+	1.67958i	5.13377	−	17.7238i	−10.9675	−	5.75617i	−3.77959	−	7.96533i	28.5869	+	4.64582i	19.0630	+	50.2651i	22.1360	+	9.43128i	62.8797	−	10.2189i
3.5	−3.78216	+	2.84211i	−1.59681	+	7.82171i	4.00142	−	13.8145i	15.7312	+	8.25639i	−16.1907	−	34.1213i	15.0706	+	2.44921i	10.7072	+	28.2326i	−33.7899	−	14.3965i	−82.9636	+	13.4829i
3.6	−3.76951	+	2.83260i	1.50425	−	7.36828i	3.95981	−	13.6708i	13.5333	+	7.10282i	15.2011	+	32.0357i	27.2448	+	4.42771i	10.4213	+	27.4788i	−27.1894	−	11.5843i	−71.1333	+	11.5603i
3.7	−3.59815	+	2.70384i	−1.15347	+	5.65006i	3.41023	−	11.7735i	1.58751	+	0.833188i	−11.1265	−	23.4486i	7.26213	+	1.18021i	6.79493	+	17.9167i	−5.75321	−	2.45122i	−7.96489	+	1.29442i
3.8	−3.30946	+	2.48690i	−0.286302	+	1.40240i	2.54213	−	8.77646i	3.60691	+	1.89305i	−2.54012	−	5.35320i	−2.89951	−	0.471216i	1.66938	+	4.40180i	22.9547	+	9.78008i	−16.6447	+	2.70503i
3.9	−3.14814	+	2.36567i	1.68249	−	8.24137i	2.08864	−	7.21081i	−7.14936	−	3.75227i	14.1997	+	29.9252i	−10.7212	−	1.74236i	−0.688137	−	1.81447i	−40.2500	−	17.1489i	31.3839	−	5.10037i
3.10	−3.11755	+	2.34268i	0.204779	−	1.00307i	2.00519	−	6.92271i	4.77311	+	2.50512i	1.71148	+	3.60686i	−12.1144	−	1.96879i	−1.09622	−	2.89048i	23.8752	+	10.1723i	−20.7491	+	3.37206i
3.11	−2.50470	+	1.88216i	0.456815	−	2.23763i	0.505246	−	1.74431i	−17.9736	−	9.43330i	3.06739	+	6.46438i	−16.3193	−	2.65214i	−6.87038	−	18.1157i	20.0411	+	8.53873i	62.7734	−	10.2017i
3.12	−2.34304	+	1.76068i	−1.17092	+	5.73554i	0.164101	−	0.566542i	−14.8440	−	7.79075i	−7.35495	−	15.5002i	−0.299423	−	0.0486610i	−7.70132	−	20.3067i	−6.68597	−	2.84863i	48.4972	−	7.88156i
3.13	−2.15092	+	1.61631i	1.23779	−	6.06311i	−0.211743	+	0.731021i	−2.40618	−	1.26286i	7.13748	+	15.0419i	23.6338	+	3.84087i	−8.35871	−	22.0401i	−10.3897	−	4.42663i	7.21667	−	1.17282i
3.14	−2.01266	+	1.51241i	−1.46355	+	7.16893i	−0.462348	+	1.59621i	12.1937	+	6.39972i	−7.89677	−	16.6421i	−29.4267	−	4.78231i	−8.62553	−	22.7437i	−24.4121	−	10.4010i	−34.2206	+	5.56140i
3.15	−1.96807	+	1.47891i	−1.90327	+	9.32284i	−0.539605	+	1.86293i	−2.56285	−	1.34509i	−10.0419	−	21.1628i	15.5737	+	2.53097i	−8.67688	−	22.8791i	−58.4535	−	24.9047i	7.03313	−	1.14300i
3.16	−1.80785	+	1.35851i	1.80580	−	8.84539i	−0.802965	+	2.77216i	13.2860	+	6.97301i	8.75196	+	18.4444i	−29.1390	−	4.73555i	−8.72958	−	23.0180i	−50.1405	−	21.3629i	−33.4920	+	5.44298i
3.17	−1.80419	+	1.35576i	0.148573	−	0.727759i	−0.808720	+	2.79203i	17.9465	+	9.41905i	0.718615	+	1.51445i	10.7406	+	1.74552i	−8.72846	−	23.0150i	24.3319	+	10.3668i	−45.1489	+	7.33742i
3.18	−1.36129	+	1.02295i	−0.443383	+	2.17184i	−1.41904	+	4.89909i	−2.67571	−	1.40432i	−1.61809	−	3.41006i	33.0738	+	5.37501i	−7.91035	−	20.8579i	20.3192	+	8.65719i	5.07897	−	0.825413i
3.19	−0.917761	+	0.689653i	−1.14631	+	5.61498i	−1.85907	+	6.41827i	0.345192	+	0.181171i	−2.82035	−	5.94377i	−16.0749	−	2.61243i	−5.97689	−	15.7598i	−5.37458	−	2.28989i	−0.441748	+	0.0717911i
3.20	−0.869369	+	0.653288i	1.46258	−	7.16421i	−1.89672	+	6.54825i	0.457687	+	0.240213i	3.40877	+	7.18382i	7.99891	+	1.29995i	−5.71391	−	15.0664i	−24.3473	−	10.3734i	−0.554827	+	0.0901682i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
169.i	even	39	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	169.4.i.a	✓	1080
169.i	even	39	1	inner	169.4.i.a	✓	1080

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
169.4.i.a	✓	1080	1.a	even	1	1	trivial
169.4.i.a	✓	1080	169.i	even	39	1	inner

Hecke kernels

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