Newform orbit 37.11.g.a

• Label: 37.11.g.a
• Level: $37$
• Weight: $11$
• Character orbit: 37.g
• Analytic conductor: $23.508$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.5082183489\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 6 q^{2} - 6 q^{3} - 3768 q^{4} + 2922 q^{5} - 2052 q^{6} - 2 q^{7} + 20880 q^{8} + 1052964 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
8.1	−57.5818	−	15.4290i	155.272	+	89.6465i	2190.80	+	1264.86i	1994.02	−	534.297i	−7557.70	−	7557.70i	1378.57	−	2387.76i	−63470.2	−	63470.2i	−13451.5	−	23298.7i	−123063.
8.2	−54.2077	−	14.5249i	−356.538	−	205.848i	1840.69	+	1062.72i	1975.98	−	529.461i	16337.2	+	16337.2i	14659.3	−	25390.7i	−43708.4	−	43708.4i	55221.9	+	95647.1i	−114803.
8.3	−52.9068	−	14.1763i	−219.093	−	126.493i	1711.35	+	988.050i	−2671.29	+	715.769i	9798.29	+	9798.29i	−11032.3	+	19108.5i	−36875.3	−	36875.3i	2476.56	+	4289.53i	151476.
8.4	−49.5335	−	13.2725i	39.3340	+	22.7095i	1390.60	+	802.863i	−4045.30	+	1083.93i	−1646.94	−	1646.94i	4712.15	−	8161.69i	−21094.0	−	21094.0i	−28493.1	−	49351.4i	214764.
8.5	−43.3980	−	11.6284i	393.244	+	227.040i	861.352	+	497.302i	−4035.49	+	1081.31i	−14425.9	−	14425.9i	7702.44	−	13341.0i	933.920	+	933.920i	73569.6	+	127426.i	187706.
8.6	−41.0206	−	10.9914i	−196.274	−	113.319i	675.070	+	389.752i	3858.05	−	1033.76i	6805.74	+	6805.74i	−8257.62	+	14302.6i	7342.02	+	7342.02i	−3842.19	−	6654.87i	−169622.
8.7	−40.1395	−	10.7554i	294.444	+	169.997i	608.694	+	351.429i	1914.73	−	513.049i	−9990.46	−	9990.46i	−9723.20	+	16841.1i	9436.46	+	9436.46i	28273.6	+	48971.4i	−82374.2
8.8	−33.6907	−	9.02740i	−34.0392	−	19.6525i	166.761	+	96.2797i	3757.77	−	1006.89i	969.394	+	969.394i	8651.45	−	14984.7i	20506.1	+	20506.1i	−28752.1	−	49800.0i	−135692.
8.9	−27.9992	−	7.50237i	−35.7432	−	20.6364i	−159.139	−	91.8788i	−1410.80	+	378.024i	845.961	+	845.961i	7821.46	−	13547.2i	24755.2	+	24755.2i	−28672.8	−	49662.7i	42337.5
8.10	−24.0184	−	6.43570i	−350.545	−	202.387i	−351.346	−	202.850i	−3310.95	+	887.167i	7117.01	+	7117.01i	2258.75	−	3912.27i	25137.9	+	25137.9i	52396.6	+	90753.7i	85233.2
8.11	−20.4348	−	5.47548i	129.456	+	74.7417i	−499.211	−	288.219i	−3560.99	+	954.164i	−2236.17	−	2236.17i	−12767.8	+	22114.5i	23941.4	+	23941.4i	−18351.9	−	31786.4i	77992.5
8.12	−19.8168	−	5.30990i	278.697	+	160.906i	−522.299	−	301.550i	4955.27	−	1327.76i	−4668.48	−	4668.48i	8535.82	−	14784.5i	23604.2	+	23604.2i	22256.7	+	38549.7i	−105248.
8.13	−9.38107	−	2.51365i	−275.634	−	159.137i	−805.124	−	464.839i	2126.03	−	569.669i	2185.73	+	2185.73i	−7724.56	+	13379.3i	13416.7	+	13416.7i	21125.0	+	36589.5i	−21376.4
8.14	−2.23188	−	0.598031i	276.550	+	159.666i	−882.186	−	509.331i	−1565.42	+	419.454i	−521.741	−	521.741i	11439.9	−	19814.4i	3337.40	+	3337.40i	21462.0	+	37173.3i	3744.69
8.15	0.259656	+	0.0695747i	270.672	+	156.273i	−886.747	−	511.964i	−656.767	+	175.980i	59.4092	+	59.4092i	−1021.79	+	1769.78i	−389.274	−	389.274i	19317.8	+	33459.4i	−182.778
8.16	2.54900	+	0.683004i	30.7968	+	17.7805i	−880.779	−	508.518i	2807.41	−	752.242i	66.3569	+	66.3569i	−7543.00	+	13064.9i	−3808.57	−	3808.57i	−28892.2	−	50042.8i	7669.88
8.17	8.41925	+	2.25593i	−149.333	−	86.2176i	−821.016	−	474.014i	−1530.72	+	410.155i	−1062.77	−	1062.77i	8137.72	−	14094.9i	−12154.2	−	12154.2i	−14657.6	−	25387.6i	−13812.8
8.18	12.1045	+	3.24340i	−326.253	−	188.362i	−750.810	−	433.480i	4604.88	−	1233.87i	−3338.20	−	3338.20i	11511.3	−	19938.2i	−16756.0	−	16756.0i	41436.3	+	71769.7i	59741.8
8.19	15.3889	+	4.12345i	−135.779	−	78.3918i	−666.994	−	385.089i	−5456.89	+	1462.17i	−1766.24	−	1766.24i	−1683.82	+	2916.47i	−20212.3	−	20212.3i	−17233.9	−	29850.1i	−90004.9
8.20	25.9310	+	6.94820i	385.296	+	222.451i	−262.669	−	151.652i	3458.18	−	926.615i	8445.50	+	8445.50i	−8871.91	+	15366.6i	−25196.0	−	25196.0i	69444.4	+	120281.i	96112.4

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.g	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	37.11.g.a	✓	120
37.g	odd	12	1	inner	37.11.g.a	✓	120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
37.11.g.a	✓	120	1.a	even	1	1	trivial
37.11.g.a	✓	120	37.g	odd	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(37, [\chi])\).