Newform orbit 209.2.k.c

• Label: 209.2.k.c
• Level: $209$
• Weight: $2$
• Character orbit: 209.k
• Analytic conductor: $1.669$
• Analytic rank: $0$
• Dimension: $56$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$209 = 11 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	209.k (of order $10$, degree $4$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$1.66887340224$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$14$ over $\Q(\zeta_{10})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) =$ 56 * q - 32 * q^4 - 10 * q^5 - 20 * q^6 + 20 * q^7 + 32 * q^9 - 28 * q^11 + 20 * q^16 + 10 * q^17 - 10 * q^19 - 52 * q^20 - 20 * q^23 + 40 * q^24 + 12 * q^25 + 16 * q^26 + 10 * q^28 + 20 * q^30 + 60 * q^35 - 32 * q^36 - 10 * q^38 - 40 * q^39 - 48 * q^42 - 38 * q^44 - 60 * q^45 + 6 * q^47 - 46 * q^49 - 12 * q^55 - 10 * q^57 + 74 * q^58 + 20 * q^61 - 30 * q^62 - 90 * q^63 + 48 * q^64 + 74 * q^66 - 170 * q^68 - 30 * q^73 - 20 * q^74 + 24 * q^77 + 18 * q^80 + 82 * q^81 - 12 * q^82 + 110 * q^83 + 30 * q^85 + 20 * q^92 + 82 * q^93 - 100 * q^95 + 20 * q^96 - 44 * q^99

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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
18.1	−2.04958	−	1.48911i	2.98595	+	0.970193i	1.36531	+	4.20198i	−2.80019	+	2.03445i	−4.67522	−	6.43489i	0.130591	−	0.0424315i	1.89316	−	5.82655i	5.54755	+	4.03053i	8.76873
18.2	−1.65927	−	1.20553i	−1.03335	−	0.335757i	0.681836	+	2.09848i	−2.27071	+	1.64977i	1.30985	+	1.80285i	2.26837	−	0.737037i	0.130856	−	0.402733i	−1.47196	−	1.06944i	5.75655
18.3	−1.50855	−	1.09603i	−1.74193	−	0.565986i	0.456422	+	1.40472i	0.620262	−	0.450647i	2.00745	+	2.76302i	−2.46234	+	0.800063i	−0.301354	+	0.927471i	0.286918	+	0.208458i	−1.42962
18.4	−1.47842	−	1.07413i	1.96879	+	0.639697i	0.413920	+	1.27391i	3.04034	−	2.20894i	−2.22357	−	3.06047i	0.571977	−	0.185846i	−0.373003	+	1.14799i	1.03986	+	0.755500i	−6.86759
18.5	−0.899298	−	0.653378i	1.75604	+	0.570573i	−0.236200	−	0.726950i	−0.523027	+	0.380001i	−1.20641	−	1.66048i	3.53379	−	1.14820i	−0.949561	+	2.92245i	0.331086	+	0.240548i	0.718642
18.6	−0.465575	−	0.338260i	−1.32976	−	0.432066i	−0.515694	−	1.58714i	0.698529	−	0.507511i	0.472954	+	0.650965i	1.64346	−	0.533991i	−0.652440	+	2.00800i	−0.845462	−	0.614264i	−0.496888
18.7	−0.300425	−	0.218271i	−3.03656	−	0.986639i	−0.575421	−	1.77096i	−0.574232	+	0.417204i	0.696903	+	0.959205i	−2.06780	+	0.671870i	−0.443184	+	1.36398i	5.82021	+	4.22863i	0.263577
18.8	0.300425	+	0.218271i	3.03656	+	0.986639i	−0.575421	−	1.77096i	−0.574232	+	0.417204i	0.696903	+	0.959205i	−2.06780	+	0.671870i	0.443184	−	1.36398i	5.82021	+	4.22863i	−0.263577
18.9	0.465575	+	0.338260i	1.32976	+	0.432066i	−0.515694	−	1.58714i	0.698529	−	0.507511i	0.472954	+	0.650965i	1.64346	−	0.533991i	0.652440	−	2.00800i	−0.845462	−	0.614264i	0.496888
18.10	0.899298	+	0.653378i	−1.75604	−	0.570573i	−0.236200	−	0.726950i	−0.523027	+	0.380001i	−1.20641	−	1.66048i	3.53379	−	1.14820i	0.949561	−	2.92245i	0.331086	+	0.240548i	−0.718642
18.11	1.47842	+	1.07413i	−1.96879	−	0.639697i	0.413920	+	1.27391i	3.04034	−	2.20894i	−2.22357	−	3.06047i	0.571977	−	0.185846i	0.373003	−	1.14799i	1.03986	+	0.755500i	6.86759
18.12	1.50855	+	1.09603i	1.74193	+	0.565986i	0.456422	+	1.40472i	0.620262	−	0.450647i	2.00745	+	2.76302i	−2.46234	+	0.800063i	0.301354	−	0.927471i	0.286918	+	0.208458i	1.42962
18.13	1.65927	+	1.20553i	1.03335	+	0.335757i	0.681836	+	2.09848i	−2.27071	+	1.64977i	1.30985	+	1.80285i	2.26837	−	0.737037i	−0.130856	+	0.402733i	−1.47196	−	1.06944i	−5.75655
18.14	2.04958	+	1.48911i	−2.98595	−	0.970193i	1.36531	+	4.20198i	−2.80019	+	2.03445i	−4.67522	−	6.43489i	0.130591	−	0.0424315i	−1.89316	+	5.82655i	5.54755	+	4.03053i	−8.76873
94.1	−0.805271	−	2.47837i	−0.729604	+	1.00421i	−3.87582	+	2.81595i	0.147707	−	0.454595i	3.07634	+	0.999563i	2.60156	+	3.58074i	5.88358	+	4.27467i	0.450928	+	1.38781i	−1.24560
94.2	−0.787101	−	2.42245i	1.24680	−	1.71608i	−3.63069	+	2.63785i	0.405883	−	1.24918i	−5.13846	−	1.66959i	−1.04090	−	1.43268i	5.12646	+	3.72459i	−0.463350	−	1.42604i	−3.34554
94.3	−0.713349	−	2.19546i	−0.368052	+	0.506580i	−2.69316	+	1.95669i	−1.07334	+	3.30339i	1.37473	+	0.446676i	−1.86145	−	2.56207i	2.48186	+	1.80318i	0.805890	+	2.48027i	8.01814
94.4	−0.548706	−	1.68874i	−1.65185	+	2.27357i	−0.932742	+	0.677676i	−0.00109312	+	0.00336429i	4.74586	+	1.54202i	−0.981625	−	1.35109i	−1.21684	−	0.884085i	−1.51348	−	4.65802i	0.00628122
94.5	−0.497006	−	1.52963i	1.67150	−	2.30062i	−0.474709	+	0.344896i	−0.600188	+	1.84719i	−4.34983	−	1.41335i	0.354186	+	0.487495i	−1.83886	−	1.33601i	−1.57189	−	4.83779i	3.12380
94.6	−0.366018	−	1.12649i	0.922652	−	1.26992i	0.483026	−	0.350939i	1.00302	−	3.08698i	−1.76826	−	0.574543i	1.99434	+	2.74498i	−2.48862	−	1.80809i	0.165637	+	0.509779i	−3.84457

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.d	odd	10	1	inner
19.b	odd	2	1	inner
209.k	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	209.2.k.c	✓	56
11.d	odd	10	1	inner	209.2.k.c	✓	56
19.b	odd	2	1	inner	209.2.k.c	✓	56
209.k	even	10	1	inner	209.2.k.c	✓	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
209.2.k.c	✓	56	1.a	even	1	1	trivial
209.2.k.c	✓	56	11.d	odd	10	1	inner
209.2.k.c	✓	56	19.b	odd	2	1	inner
209.2.k.c	✓	56	209.k	even	10	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^56 + 30*T2^54 + 476*T2^52 + 5344*T2^50 + 49318*T2^48 + 390884*T2^46 + 2670740*T2^44 + 15817791*T2^42 + 83239940*T2^40 + 392992104*T2^38 + 1659448958*T2^36 + 6256470720*T2^34 + 21324935609*T2^32 + 65204905784*T2^30 + 176552104275*T2^28 + 420853603440*T2^26 + 886177918350*T2^24 + 1578947537135*T2^22 + 2260115919690*T2^20 + 2557181454830*T2^18 + 2577606898837*T2^16 + 2130138998865*T2^14 + 1132550463627*T2^12 + 125795922858*T2^10 + 139107023541*T2^8 + 12576988422*T2^6 + 2061302175*T2^4 + 273217536*T2^2 + 24413481