Newform orbit 13.14.c.a

• Label: 13.14.c.a
• Level: $13$
• Weight: $14$
• Character orbit: 13.c
• Analytic conductor: $13.940$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	13.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.9400207637\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 65 q^{2} + 728 q^{3} - 53249 q^{4} + 41860 q^{5} + 143910 q^{6} - 173992 q^{7} + 2616978 q^{8} - 5231838 q^{9}+O(q^{10}) \)

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	−82.8702	−	143.535i	451.483	+	781.992i	−9638.95	+	16695.1i	−67212.0			74829.1	−	129608.i	−137812.	+	238698.i	1.83738e6			389487.	−	674611.i	5.56987e6	+	9.64731e6i
3.2	−82.3546	−	142.642i	−255.684	−	442.858i	−9468.55	+	16400.0i	48546.3			−42113.5	+	72942.8i	216096.	−	374290.i	1.76982e6			666413.	−	1.15426e6i	−3.99801e6	−	6.92476e6i
3.3	−59.2686	−	102.656i	1072.90	+	1858.33i	−2929.54	+	5074.11i	45885.5			127179.	−	220281.i	−177681.	+	307752.i	−276538.			−1.50509e6	+	2.60689e6i	−2.71957e6	−	4.71043e6i
3.4	−56.7104	−	98.2253i	−906.506	−	1570.12i	−2336.14	+	4046.32i	−10940.5			−102817.	+	178084.i	−89772.2	+	155490.i	−399209.			−846346.	+	1.46591e6i	620440.	+	1.07463e6i
3.5	−45.4515	−	78.7243i	108.764	+	188.384i	−35.6760	+	61.7927i	−2568.18			9886.95	−	17124.7i	77358.1	−	133988.i	−738191.			773502.	−	1.33975e6i	116728.	+	202178.i
3.6	−16.9887	−	29.4253i	797.844	+	1381.91i	3518.77	−	6094.69i	−28205.3			27108.7	−	46953.6i	177172.	−	306871.i	−517460.			−475948.	+	824367.i	479171.	+	829948.i
3.7	−3.81407	−	6.60616i	−187.264	−	324.351i	4066.91	−	7044.09i	42407.7			−1428.48	+	2474.20i	−178632.	+	309400.i	−124536.			727026.	−	1.25925e6i	−161746.	−	280152.i
3.8	8.80659	+	15.2535i	−449.698	−	778.901i	3940.89	−	6825.82i	−51640.9			7920.62	−	13718.9i	−75519.2	+	130803.i	283110.			392704.	−	680184.i	−454780.	−	787702.i
3.9	11.2286	+	19.4486i	−1142.04	−	1978.08i	3843.84	−	6657.72i	17999.0			25647.2	−	44422.2i	281521.	−	487609.i	356614.			−1.81136e6	+	3.13737e6i	202104.	+	350055.i
3.10	37.0467	+	64.1667i	958.983	+	1661.01i	1351.09	−	2340.15i	−22075.6			−71054.3	+	123070.i	−250997.	+	434740.i	807186.			−1.04213e6	+	1.80503e6i	−817827.	−	1.41652e6i
3.11	40.7956	+	70.6601i	482.143	+	835.097i	767.437	−	1329.24i	41512.4			−39338.7	+	68136.6i	158394.	−	274346.i	793627.			332237.	−	575452.i	1.69352e6	+	2.93327e6i
3.12	60.7949	+	105.300i	−355.726	−	616.135i	−3296.05	+	5708.93i	−32981.4			43252.7	−	74915.8i	47112.8	−	81601.8i	194532.			544080.	−	942374.i	−2.00510e6	−	3.47294e6i
3.13	71.5224	+	123.880i	−865.569	−	1499.21i	−6134.91	+	10626.0i	41380.4			123815.	−	214454.i	−187839.	+	325347.i	−583312.			−701257.	+	1.21461e6i	2.95963e6	+	5.12623e6i
3.14	84.7632	+	146.814i	654.369	+	1133.40i	−10273.6	+	17794.4i	−1177.50			−110933.	+	192141.i	53602.7	−	92842.5i	−2.09453e6			−59235.2	+	102598.i	−99808.8	−	172874.i
9.1	−82.8702	+	143.535i	451.483	−	781.992i	−9638.95	−	16695.1i	−67212.0			74829.1	+	129608.i	−137812.	−	238698.i	1.83738e6			389487.	+	674611.i	5.56987e6	−	9.64731e6i
9.2	−82.3546	+	142.642i	−255.684	+	442.858i	−9468.55	−	16400.0i	48546.3			−42113.5	−	72942.8i	216096.	+	374290.i	1.76982e6			666413.	+	1.15426e6i	−3.99801e6	+	6.92476e6i
9.3	−59.2686	+	102.656i	1072.90	−	1858.33i	−2929.54	−	5074.11i	45885.5			127179.	+	220281.i	−177681.	−	307752.i	−276538.			−1.50509e6	−	2.60689e6i	−2.71957e6	+	4.71043e6i
9.4	−56.7104	+	98.2253i	−906.506	+	1570.12i	−2336.14	−	4046.32i	−10940.5			−102817.	−	178084.i	−89772.2	−	155490.i	−399209.			−846346.	−	1.46591e6i	620440.	−	1.07463e6i
9.5	−45.4515	+	78.7243i	108.764	−	188.384i	−35.6760	−	61.7927i	−2568.18			9886.95	+	17124.7i	77358.1	+	133988.i	−738191.			773502.	+	1.33975e6i	116728.	−	202178.i
9.6	−16.9887	+	29.4253i	797.844	−	1381.91i	3518.77	+	6094.69i	−28205.3			27108.7	+	46953.6i	177172.	+	306871.i	−517460.			−475948.	−	824367.i	479171.	−	829948.i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	13.14.c.a	✓	28
13.c	even	3	1	inner	13.14.c.a	✓	28
13.c	even	3	1		169.14.a.e		14
13.e	even	6	1		169.14.a.c		14

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.14.c.a	✓	28	1.a	even	1	1	trivial
13.14.c.a	✓	28	13.c	even	3	1	inner
169.14.a.c		14	13.e	even	6	1
169.14.a.e		14	13.c	even	3	1

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(13, [\chi])\).