Newform orbit 52.5.i.a

• Label: 52.5.i.a
• Level: $52$
• Weight: $5$
• Character orbit: 52.i
• Analytic conductor: $5.375$
• Analytic rank: $0$
• Dimension: $52$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	52.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.37523808036\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(26\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 52 * q - 3 * q^2 - q^4 + 96 * q^6 + 592 * q^9 - 61 * q^10 + 164 * q^12 - 240 * q^13 + 244 * q^14 + 135 * q^16 - 302 * q^17 + 1665 * q^20 + 954 * q^22 + 882 * q^24 - 3668 * q^25 - 909 * q^26 - 4908 * q^28 - 422 * q^29 - 150 * q^30 - 2523 * q^32 - 6 * q^33 + 4207 * q^36 + 2094 * q^37 - 8432 * q^38 + 4434 * q^40 - 2706 * q^41 + 3226 * q^42 + 8604 * q^45 - 8568 * q^46 - 5890 * q^48 - 1060 * q^49 + 2616 * q^50 - 5410 * q^52 + 880 * q^53 + 24006 * q^54 + 9736 * q^56 + 15393 * q^58 + 978 * q^61 - 15710 * q^62 + 13802 * q^64 + 9300 * q^65 - 25676 * q^66 + 4279 * q^68 + 1818 * q^69 + 20181 * q^72 + 5553 * q^74 - 3714 * q^76 + 2588 * q^77 + 16526 * q^78 - 40149 * q^80 - 12090 * q^81 - 3947 * q^82 - 36036 * q^84 + 12060 * q^85 + 2552 * q^88 - 1158 * q^89 - 68582 * q^90 + 1260 * q^92 - 50940 * q^93 + 6342 * q^94 - 12390 * q^97 + 19395 * q^98

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−3.99628	−	0.172396i	−12.9573	+	7.48092i	15.9406	+	1.37789i			28.0008i	53.0709	−	27.6621i	−33.5372	+	58.0882i	−63.4654	−	8.25451i	71.4283	−	123.717i	4.82722	−	111.899i
23.2	−3.93752	+	0.704243i	9.57461	−	5.52790i	15.0081	−	5.54594i		−	20.6402i	−33.8072	+	28.5091i	0.499659	−	0.865435i	−55.1889	+	32.4066i	20.6154	−	35.7070i	14.5357	+	81.2710i
23.3	−3.89412	+	0.914238i	−2.36372	+	1.36470i	14.3283	−	7.12031i			7.63478i	7.95696	−	7.47529i	28.3503	−	49.1041i	−49.2866	+	40.8268i	−36.7752	+	63.6965i	−6.98000	−	29.7307i
23.4	−3.65004	−	1.63623i	10.1376	−	5.85294i	10.6455	+	11.9446i			40.5931i	−46.5793	+	4.77603i	−26.4101	+	45.7437i	−19.3124	−	61.0166i	28.0138	−	48.5213i	66.4196	−	148.166i
23.5	−3.50457	−	1.92822i	−4.16271	+	2.40334i	8.56396	+	13.5151i		−	43.7800i	19.2226	−	0.396065i	−27.5011	+	47.6332i	−3.95290	−	63.8778i	−28.9479	+	50.1392i	−84.4173	+	153.430i
23.6	−3.23747	−	2.34921i	−3.63166	+	2.09674i	4.96240	+	15.2110i			7.96541i	16.6831	+	1.74341i	35.6880	−	61.8134i	19.6682	−	60.9029i	−31.7074	+	54.9188i	18.7124	−	25.7878i
23.7	−2.73881	+	2.91529i	2.36372	−	1.36470i	−0.997804	−	15.9689i			7.63478i	−2.49531	+	10.6286i	−28.3503	+	49.1041i	49.2866	+	40.8268i	−36.7752	+	63.6965i	−22.2576	−	20.9102i
23.8	−2.57865	+	3.05787i	−9.57461	+	5.52790i	−2.70112	−	15.7704i		−	20.6402i	7.78597	−	43.5324i	−0.499659	+	0.865435i	55.1889	+	32.4066i	20.6154	−	35.7070i	63.1149	+	53.2238i
23.9	−1.84884	+	3.54708i	12.9573	−	7.48092i	−9.16356	−	13.1160i			28.0008i	2.57936	+	59.7917i	33.5372	−	58.0882i	63.4654	−	8.25451i	71.4283	−	123.717i	−99.3211	−	51.7691i
23.10	−1.80419	−	3.57000i	11.6983	−	6.75402i	−9.48982	+	12.8819i		−	26.8049i	−45.2178	−	29.5775i	5.90736	−	10.2318i	63.1098	+	10.6373i	50.7336	−	87.8731i	−95.6934	+	48.3610i
23.11	−1.10537	−	3.84424i	−15.4197	+	8.90255i	−13.5563	+	8.49861i		−	7.00386i	51.2680	+	49.4363i	18.9216	−	32.7731i	47.6554	+	42.7195i	118.011	−	204.401i	−26.9245	+	7.74186i
23.12	−1.02671	−	3.86599i	0.384527	−	0.222007i	−13.8917	+	7.93852i			20.9340i	−1.25307	−	1.25864i	−11.6885	+	20.2451i	44.9530	+	45.5546i	−40.4014	+	69.9773i	80.9306	−	21.4932i
23.13	−0.408003	+	3.97914i	−10.1376	+	5.85294i	−15.6671	−	3.24700i			40.5931i	−19.1535	−	42.7269i	26.4101	−	45.7437i	19.3124	−	61.0166i	28.0138	−	48.5213i	−161.526	−	16.5621i
23.14	−0.0823987	+	3.99915i	4.16271	−	2.40334i	−15.9864	−	0.659050i		−	43.7800i	9.26832	+	16.8453i	27.5011	−	47.6332i	3.95290	−	63.8778i	−28.9479	+	50.1392i	175.083	+	3.60742i
23.15	0.415743	+	3.97834i	3.63166	−	2.09674i	−15.6543	+	3.30793i			7.96541i	9.85137	+	13.5763i	−35.6880	+	61.8134i	−19.6682	−	60.9029i	−31.7074	+	54.9188i	−31.6891	+	3.31156i
23.16	1.20110	−	3.81541i	−4.08378	+	2.35777i	−13.1147	−	9.16535i		−	26.8316i	4.09085	+	18.4132i	−20.7689	+	35.9729i	−50.7217	+	39.0296i	−29.3818	+	50.8908i	−102.374	−	32.2273i
23.17	1.65700	−	3.64065i	9.17215	−	5.29554i	−10.5087	−	12.0651i			8.58415i	−4.08095	−	42.1673i	32.7370	−	56.7022i	−61.3379	+	18.2665i	15.5856	−	26.9950i	31.2519	+	14.2240i
23.18	2.18962	+	3.34747i	−11.6983	+	6.75402i	−6.41114	+	14.6594i		−	26.8049i	−48.2237	−	24.3710i	−5.90736	+	10.2318i	−63.1098	+	10.6373i	50.7336	−	87.8731i	89.7286	−	58.6925i
23.19	2.47042	−	3.14596i	−7.98357	+	4.60932i	−3.79408	−	15.5436i			41.7731i	−5.22203	+	36.5029i	−5.71487	+	9.89844i	−58.2726	−	26.4633i	1.99158	−	3.44951i	131.416	+	103.197i
23.20	2.77652	+	2.87940i	15.4197	−	8.90255i	−0.581861	+	15.9894i		−	7.00386i	68.4471	+	19.6813i	−18.9216	+	32.7731i	−47.6554	+	42.7195i	118.011	−	204.401i	20.1669	−	19.4464i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
13.e	even	6	1	inner
52.i	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	52.5.i.a	✓	52
4.b	odd	2	1	inner	52.5.i.a	✓	52
13.e	even	6	1	inner	52.5.i.a	✓	52
52.i	odd	6	1	inner	52.5.i.a	✓	52

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
52.5.i.a	✓	52	1.a	even	1	1	trivial
52.5.i.a	✓	52	4.b	odd	2	1	inner
52.5.i.a	✓	52	13.e	even	6	1	inner
52.5.i.a	✓	52	52.i	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(52, [\chi])\).