Newform orbit 197.2.d.a

• Label: 197.2.d.a
• Level: $197$
• Weight: $2$
• Character orbit: 197.d
• Analytic conductor: $1.573$
• Analytic rank: $0$
• Dimension: $90$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$197$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	197.d (of order $7$, degree $6$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$1.57305291982$
Analytic rank:	$0$
Dimension:	$90$
Relative dimension:	$15$ over $\Q(\zeta_{7})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) =$ 90 * q - 2 * q^2 - 5 * q^3 - 26 * q^4 + q^5 - 10 * q^6 - 11 * q^7 - 16 * q^9 + 9 * q^10 - 7 * q^11 + 15 * q^12 + 9 * q^13 - 40 * q^14 - 16 * q^15 - 10 * q^16 + 13 * q^17 + 15 * q^18 - 20 * q^19 + 26 * q^20 + 13 * q^21 - 17 * q^22 + 25 * q^23 - 17 * q^24 + 16 * q^25 - 2 * q^26 - 41 * q^27 - 50 * q^28 - 33 * q^29 - 72 * q^30 - 15 * q^31 + 46 * q^32 - 32 * q^33 + 31 * q^34 + 45 * q^35 - 64 * q^36 - 27 * q^37 + 43 * q^38 + 14 * q^39 + 31 * q^40 + 35 * q^41 + 14 * q^42 + 16 * q^43 + 49 * q^44 - 8 * q^45 + 17 * q^46 - 4 * q^47 + 41 * q^48 - 48 * q^49 - 52 * q^50 + 51 * q^51 - 2 * q^52 + 41 * q^53 + 14 * q^54 + 53 * q^55 + 52 * q^56 - 104 * q^57 - 79 * q^58 - 17 * q^59 + 11 * q^60 + 21 * q^61 + 23 * q^62 + 48 * q^63 + 22 * q^64 + 5 * q^65 - 94 * q^66 - 5 * q^67 - 200 * q^68 + 54 * q^69 + 64 * q^70 - 94 * q^71 + 75 * q^72 + 31 * q^73 + 18 * q^74 + 36 * q^75 - 80 * q^76 + 22 * q^77 - 45 * q^78 - 25 * q^79 + 131 * q^80 + 6 * q^81 + 51 * q^82 - 130 * q^83 + 108 * q^84 - 129 * q^85 + 32 * q^86 + 98 * q^87 + 54 * q^88 + 17 * q^89 - 42 * q^90 - 119 * q^91 + 31 * q^92 + 60 * q^93 - 111 * q^94 - 77 * q^95 - 98 * q^96 + 49 * q^97 + 45 * q^98 + 25 * 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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
36.1	−2.45864	+	1.18402i	0.124979	+	0.0601869i	3.39601	−	4.25846i	−0.578930	−	0.725956i	−0.378541			−0.626559	−	0.301735i	−2.09300	+	9.17002i	−1.85847	−	2.33045i	2.28292	+	1.09940i
36.2	−2.08071	+	1.00202i	2.19341	+	1.05629i	2.07835	−	2.60617i	2.34290	+	2.93790i	−5.62228			−0.184771	−	0.0889812i	−0.685233	+	3.00220i	1.82483	+	2.28827i	−7.81874	−	3.76531i
36.3	−1.81302	+	0.873105i	−1.68155	−	0.809790i	1.27775	−	1.60225i	1.13035	+	1.41742i	3.75571			−2.93980	−	1.41573i	−0.0221004	+	0.0968284i	0.301366	+	0.377901i	−3.28690	−	1.58289i
36.4	−1.70417	+	0.820685i	0.577242	+	0.277985i	0.983693	−	1.23351i	−1.63883	−	2.05503i	−1.21186			1.92605	+	0.927536i	0.177735	−	0.778708i	−1.61454	−	2.02456i	4.47938	+	2.15715i
36.5	−0.948843	+	0.456939i	−1.49781	−	0.721309i	−0.555469	+	0.696537i	0.812989	+	1.01946i	1.75078			−0.337287	−	0.162429i	0.677469	−	2.96818i	−0.147309	−	0.184720i	−1.23723	−	0.595818i
36.6	−0.593780	+	0.285950i	2.71133	+	1.30571i	−0.976172	+	1.22408i	−0.811676	−	1.01781i	−1.98330			1.33888	+	0.644768i	0.522910	−	2.29102i	3.77595	+	4.73489i	0.773000	+	0.372257i
36.7	−0.448644	+	0.216056i	−2.62086	−	1.26214i	−1.09238	+	1.36980i	−1.50240	−	1.88395i	1.44852			3.49725	+	1.68419i	0.415748	−	1.82151i	3.40543	+	4.27028i	1.08108	+	0.520620i
36.8	−0.120109	+	0.0578412i	−0.0851283	−	0.0409956i	−1.23590	+	1.54977i	−1.69772	−	2.12888i	0.0125959			−3.18358	−	1.53313i	0.118130	−	0.517563i	−1.86490	−	2.33851i	0.327048	+	0.157498i
36.9	−0.0351696	+	0.0169368i	1.43946	+	0.693208i	−1.24603	+	1.56247i	1.82344	+	2.28652i	−0.0623661			−1.43652	−	0.691792i	0.0347315	−	0.152169i	−0.278956	−	0.349800i	−0.102856	−	0.0495329i
36.10	0.857171	−	0.412792i	−2.30589	−	1.11046i	−0.682634	+	0.855996i	2.52569	+	3.16712i	−2.43493			0.566338	+	0.272734i	−0.655194	+	2.87059i	2.21355	+	2.77571i	3.47231	+	1.67218i
36.11	0.887644	−	0.427467i	0.442784	+	0.213234i	−0.641795	+	0.804785i	0.198374	+	0.248753i	0.484186			3.85822	+	1.85802i	−0.664127	+	2.90973i	−1.71988	−	2.15666i	0.282420	+	0.136006i
36.12	1.52637	−	0.735059i	−2.33495	−	1.12445i	0.542502	−	0.680276i	−1.58350	−	1.98564i	−4.39053			−1.94366	−	0.936020i	−0.425949	+	1.86621i	2.31713	+	2.90559i	−3.87657	−	1.86686i
36.13	1.66827	−	0.803398i	2.04773	+	0.986133i	0.890709	−	1.11691i	−1.79031	−	2.24498i	4.20843			0.174909	+	0.0842317i	−0.235439	+	1.03153i	1.35026	+	1.69317i	−4.79034	−	2.30690i
36.14	1.85821	−	0.894867i	0.813278	+	0.391654i	1.40518	−	1.76204i	1.13765	+	1.42657i	1.86172			−3.23536	−	1.55807i	0.116449	−	0.510195i	−1.36244	−	1.70845i	3.39059	+	1.63282i
36.15	2.28193	−	1.09892i	−1.44751	−	0.697085i	2.75260	−	3.45165i	0.532937	+	0.668281i	−4.06916			1.25930	+	0.606448i	1.36097	−	5.96281i	−0.261106	−	0.327417i	1.95051	+	0.939317i
104.1	−2.45864	−	1.18402i	0.124979	−	0.0601869i	3.39601	+	4.25846i	−0.578930	+	0.725956i	−0.378541			−0.626559	+	0.301735i	−2.09300	−	9.17002i	−1.85847	+	2.33045i	2.28292	−	1.09940i
104.2	−2.08071	−	1.00202i	2.19341	−	1.05629i	2.07835	+	2.60617i	2.34290	−	2.93790i	−5.62228			−0.184771	+	0.0889812i	−0.685233	−	3.00220i	1.82483	−	2.28827i	−7.81874	+	3.76531i
104.3	−1.81302	−	0.873105i	−1.68155	+	0.809790i	1.27775	+	1.60225i	1.13035	−	1.41742i	3.75571			−2.93980	+	1.41573i	−0.0221004	−	0.0968284i	0.301366	−	0.377901i	−3.28690	+	1.58289i
104.4	−1.70417	−	0.820685i	0.577242	−	0.277985i	0.983693	+	1.23351i	−1.63883	+	2.05503i	−1.21186			1.92605	−	0.927536i	0.177735	+	0.778708i	−1.61454	+	2.02456i	4.47938	−	2.15715i
104.5	−0.948843	−	0.456939i	−1.49781	+	0.721309i	−0.555469	−	0.696537i	0.812989	−	1.01946i	1.75078			−0.337287	+	0.162429i	0.677469	+	2.96818i	−0.147309	+	0.184720i	−1.23723	+	0.595818i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
197.d	even	7	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	197.2.d.a	✓	90
197.d	even	7	1	inner	197.2.d.a	✓	90

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
197.2.d.a	✓	90	1.a	even	1	1	trivial
197.2.d.a	✓	90	197.d	even	7	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(197, [\chi])$.