Newform orbit 197.6.b.a

• Label: 197.6.b.a
• Level: $197$
• Weight: $6$
• Character orbit: 197.b
• Analytic conductor: $31.596$
• Analytic rank: $0$
• Dimension: $82$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$197$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	197.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$31.5956125032$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) =$ 82 * q - 1272 * q^4 + 56 * q^6 - 294 * q^7 - 6644 * q^9 + 1162 * q^10 - 122 * q^15 + 13088 * q^16 + 738 * q^19 + 6534 * q^22 - 456 * q^23 - 446 * q^24 - 45670 * q^25 - 2380 * q^26 + 21660 * q^28 + 2052 * q^29 + 8068 * q^33 - 29612 * q^34 + 98540 * q^36 + 4182 * q^37 - 56242 * q^39 - 14880 * q^40 + 6074 * q^41 - 52390 * q^42 + 47302 * q^43 - 32854 * q^47 + 234940 * q^49 + 38010 * q^51 + 9186 * q^53 - 88518 * q^54 - 72930 * q^55 + 150142 * q^59 + 67874 * q^60 - 118558 * q^61 + 176982 * q^62 + 112446 * q^63 - 35834 * q^64 - 5434 * q^65 - 38102 * q^70 - 42536 * q^76 + 634854 * q^81 - 437360 * q^83 + 291628 * q^85 - 88678 * q^88 - 408654 * q^90 + 175588 * q^92 - 449962 * q^93 - 41002 * q^96 - 395422 * q^97

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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
196.1		−	10.9717i		−	14.5203i	−88.3786					75.9063i	−159.313			−163.679					618.570i	32.1600			832.823
196.2		−	10.9643i			5.73333i	−88.2161				−	30.6768i	62.8621			43.1819					616.371i	210.129			−336.350
196.3		−	10.4443i		−	19.5771i	−77.0836				−	72.4053i	−204.469			−15.2877					470.867i	−140.261			−756.224
196.4		−	10.0418i			19.2826i	−68.8383				−	85.5733i	193.633			−233.051					369.924i	−128.819			−859.313
196.5		−	9.98891i			15.6411i	−67.7783					77.2319i	156.238			51.7558					357.386i	−1.64518			771.462
196.6		−	9.94566i			30.3052i	−66.9162					67.3438i	301.406			−180.186					347.265i	−675.408			669.779
196.7		−	9.72393i		−	14.0558i	−62.5548					63.6224i	−136.678			191.537					297.113i	45.4340			618.660
196.8		−	9.63053i		−	29.9945i	−60.7471					11.8634i	−288.863			59.7609					276.850i	−656.673			114.251
196.9		−	9.60005i			25.1437i	−60.1610				−	40.5034i	241.381			177.667					270.347i	−389.205			−388.835
196.10		−	9.41411i			9.54650i	−56.6255					35.0161i	89.8718			−4.02765					231.827i	151.864			329.646
196.11		−	8.95219i		−	10.0324i	−48.1416				−	42.7481i	−89.8121			234.675					144.503i	142.351			−382.689
196.12		−	8.48038i		−	5.86620i	−39.9169				−	78.2326i	−49.7476			−57.3308					67.1380i	208.588			−663.442
196.13		−	8.43577i			1.06070i	−39.1622					26.8901i	8.94786			−197.867					60.4188i	241.875			226.839
196.14		−	8.13555i		−	22.1292i	−34.1872				−	36.5988i	−180.033			−250.448					17.7944i	−246.702			−297.751
196.15		−	8.06764i		−	16.3436i	−33.0868					45.6680i	−131.854			−4.74292					8.76825i	−24.1119			368.433
196.16		−	7.86035i		−	4.70091i	−29.7851				−	27.4832i	−36.9508			−34.7471				−	17.4099i	220.901			−216.028
196.17		−	7.19261i			21.9687i	−19.7336				−	66.0390i	158.012			−10.0615				−	88.2275i	−239.622			−474.992
196.18		−	7.05419i			12.9852i	−17.7616					97.5882i	91.6001			206.072				−	100.440i	74.3843			688.406
196.19		−	6.71821i			7.89795i	−13.1343				−	95.5906i	53.0601			108.603				−	126.743i	180.622			−642.198
196.20		−	6.64182i			17.9034i	−12.1138					27.3285i	118.911			−117.951				−	132.081i	−77.5301			181.511

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
197.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	197.6.b.a	✓	82
197.b	even	2	1	inner	197.6.b.a	✓	82

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
197.6.b.a	✓	82	1.a	even	1	1	trivial
197.6.b.a	✓	82	197.b	even	2	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{6}^{\mathrm{new}}(197, [\chi])$.