Newform orbit 16.14.e.a

• Label: 16.14.e.a
• Level: $16$
• Weight: $14$
• Character orbit: 16.e
• Analytic conductor: $17.157$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.1569486323\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(25\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 2 q^{2} - 2 q^{3} + 360 q^{4} - 2 q^{5} - 255056 q^{6} - 1076876 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−90.1629	+	7.91484i	1174.50	−	1174.50i	8066.71	−	1427.25i	−33114.8	−	33114.8i	−96600.1	+	115192.i		−	548741.i	−716022.	+	192532.i		−	1.16456e6i	3.24783e6	+	2.72363e6i
5.2	−86.5802	−	26.3795i	913.885	−	913.885i	6800.25	+	4567.88i	15715.7	+	15715.7i	−103232.	+	55016.5i			356100.i	−468268.	−	574875.i		−	76047.5i	−946097.	−	1.77524e6i
5.3	−85.7364	−	29.0046i	−924.404	+	924.404i	6509.46	+	4973.51i	42023.8	+	42023.8i	106067.	−	52443.1i		−	291398.i	−413843.	−	615215.i		−	114722.i	−2.38408e6	−	4.82186e6i
5.4	−85.4847	+	29.7382i	−660.650	+	660.650i	6423.28	−	5084.32i	−6982.48	−	6982.48i	36829.0	−	76122.0i			294350.i	−397894.	+	625648.i			721406.i	804541.	+	389249.i
5.5	−81.6959	−	38.9586i	−1316.54	+	1316.54i	5156.45	+	6365.52i	−43811.8	−	43811.8i	158847.	−	56265.6i		−	156812.i	−173269.	−	720925.i		−	1.87225e6i	1.87240e6	+	5.28609e6i
5.6	−71.4691	+	55.5353i	293.446	−	293.446i	2023.65	−	7938.12i	22716.6	+	22716.6i	−4675.68	+	37268.9i		−	343566.i	296217.	+	679714.i			1.42210e6i	−2.88511e6	−	361960.i
5.7	−56.2421	+	70.9142i	1621.89	−	1621.89i	−1865.65	−	7976.73i	8037.57	+	8037.57i	23796.5	+	206234.i			516249.i	670591.	+	316327.i		−	3.66675e6i	−1.02203e6	+	117928.i
5.8	−55.8189	−	71.2478i	375.766	−	375.766i	−1960.51	+	7953.95i	−11477.0	−	11477.0i	−47747.4	−	5797.69i			9991.15i	676135.	−	304298.i			1.31192e6i	−177079.	+	1.45835e6i
5.9	−40.5224	+	80.9316i	−1639.97	+	1639.97i	−4907.86	−	6559.10i	9558.16	+	9558.16i	−66269.9	−	199181.i		−	133901.i	729717.	−	131411.i		−	3.78468e6i	−1.16088e6	+	386238.i
5.10	−30.3106	+	85.2834i	−26.4760	+	26.4760i	−6354.53	−	5169.99i	−38066.2	−	38066.2i	−1455.46	−	3060.47i			23704.0i	633524.	−	385231.i			1.59292e6i	4.40022e6	−	2.09260e6i
5.11	−21.6798	−	87.8748i	−1056.56	+	1056.56i	−7251.97	+	3810.23i	11923.3	+	11923.3i	115752.	+	69939.2i			170872.i	492044.	+	554660.i		−	638332.i	789265.	−	1.30626e6i
5.12	−18.7534	−	88.5455i	1610.25	−	1610.25i	−7488.62	+	3321.06i	39780.2	+	39780.2i	−172778.	−	112383.i		−	439933.i	434502.	+	600803.i		−	3.59146e6i	2.77634e6	−	4.26837e6i
5.13	−0.844839	+	90.5057i	381.223	−	381.223i	−8190.57	−	152.926i	29472.6	+	29472.6i	34180.8	+	34824.9i		−	167372.i	20760.3	−	741165.i			1.30366e6i	−2.69234e6	+	2.64254e6i
5.14	20.0123	−	88.2695i	983.557	−	983.557i	−7391.01	−	3532.96i	−35371.2	−	35371.2i	−67134.8	−	106501.i			334186.i	−459764.	+	581698.i		−	340445.i	−3.83006e6	+	2.41434e6i
5.15	28.6247	−	85.8640i	−336.873	+	336.873i	−6553.25	−	4915.66i	−6822.00	−	6822.00i	19282.4	+	38568.2i		−	509047.i	−609663.	+	421979.i			1.36736e6i	−781042.	+	390487.i
5.16	34.4647	+	83.6910i	−787.355	+	787.355i	−5816.37	+	5768.77i	13376.5	+	13376.5i	−93030.5	−	38758.6i			547213.i	−683254.	−	287958.i			354466.i	−658474.	+	1.58051e6i
5.17	37.4414	+	82.4023i	1317.13	−	1317.13i	−5388.28	+	6170.52i	−24836.8	−	24836.8i	157850.	+	59219.5i		−	73212.9i	−710210.	−	212974.i		−	1.87536e6i	1.11668e6	−	2.97653e6i
5.18	54.8815	−	71.9724i	242.259	−	242.259i	−2168.05	−	7899.90i	42421.3	+	42421.3i	−4140.43	−	30731.5i			421955.i	−687560.	−	277518.i			1.47694e6i	5.38130e6	−	725019.i
5.19	58.4516	+	69.1043i	−1009.54	+	1009.54i	−1358.81	+	8078.52i	−19923.8	−	19923.8i	−128773.	−	10754.3i		−	484985.i	−637685.	+	378303.i		−	444038.i	212241.	−	2.54140e6i
5.20	64.2863	−	63.7124i	−1618.47	+	1618.47i	73.4667	−	8191.67i	−23191.8	−	23191.8i	−928.945	+	207162.i			280152.i	−517188.	−	531293.i		−	3.64459e6i	−2.96853e6	−	13311.3i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	16.14.e.a	✓	50
4.b	odd	2	1		64.14.e.a		50
8.b	even	2	1		128.14.e.b		50
8.d	odd	2	1		128.14.e.a		50
16.e	even	4	1	inner	16.14.e.a	✓	50
16.e	even	4	1		128.14.e.b		50
16.f	odd	4	1		64.14.e.a		50
16.f	odd	4	1		128.14.e.a		50

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
16.14.e.a	✓	50	1.a	even	1	1	trivial
16.14.e.a	✓	50	16.e	even	4	1	inner
64.14.e.a		50	4.b	odd	2	1
64.14.e.a		50	16.f	odd	4	1
128.14.e.a		50	8.d	odd	2	1
128.14.e.a		50	16.f	odd	4	1
128.14.e.b		50	8.b	even	2	1
128.14.e.b		50	16.e	even	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(16, [\chi])\).