Newform orbit 76.4.f.a

• Label: 76.4.f.a
• Level: $76$
• Weight: $4$
• Character orbit: 76.f
• Analytic conductor: $4.484$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	76.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.48414516044\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 3 q^{2} + 5 q^{4} - 2 q^{5} + 21 q^{6} - 228 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
27.1	−2.82835	−	0.0213793i	0.662470	−	1.14743i	7.99909	+	0.120936i	0.369055	−	0.639222i	−1.89823	+	3.23117i		−	4.50637i	−22.6216	−	0.513064i	12.6223	+	21.8624i	−1.05748	+	1.80005i
27.2	−2.76333	−	0.603342i	−2.62655	+	4.54932i	7.27196	+	3.33446i	6.06129	−	10.4985i	10.0028	−	10.9866i			11.2918i	−18.0830	−	13.6017i	−0.297553	−	0.515377i	−23.0835	+	25.3537i
27.3	−2.68912	+	0.876719i	4.88508	−	8.46120i	6.46273	−	4.71521i	6.65041	−	11.5188i	−5.71846	+	27.0360i			4.05088i	−13.2451	+	18.3457i	−34.2280	−	59.2846i	−7.78495	+	36.8061i
27.4	−2.49901	−	1.32474i	3.47531	−	6.01941i	4.49010	+	6.62110i	−9.47803	+	16.4164i	−16.6590	+	10.4387i			33.3441i	−2.44954	−	22.4944i	−10.6555	−	18.4559i	45.4333	−	28.4689i
27.5	−2.46181	+	1.39265i	−4.46255	+	7.72937i	4.12104	−	6.85689i	−6.17128	+	10.6890i	0.221653	−	25.2430i			29.2047i	−0.595974	+	22.6196i	−26.3288	−	45.6028i	0.306525	−	34.9087i
27.6	−2.45296	+	1.40819i	0.676811	−	1.17227i	4.03401	−	6.90846i	−5.49309	+	9.51431i	−0.00941024	+	3.82861i		−	11.1169i	−0.166844	+	22.6268i	12.5839	+	21.7959i	0.0763748	−	31.0735i
27.7	−2.39677	−	1.50183i	−3.47531	+	6.01941i	3.48899	+	7.19909i	−9.47803	+	16.4164i	17.3697	−	9.20780i		−	33.3441i	2.44954	−	22.4944i	−10.6555	−	18.4559i	47.3714	−	25.1119i
27.8	−1.90417	−	2.09144i	2.62655	−	4.54932i	−0.748247	+	7.96493i	6.06129	−	10.4985i	−14.5160	+	3.16942i		−	11.2918i	18.0830	−	13.6017i	−0.297553	−	0.515377i	−33.4987	+	7.31407i
27.9	−1.80915	+	2.17416i	−2.57005	+	4.45145i	−1.45393	−	7.86677i	7.41606	−	12.8450i	−5.02855	−	13.6410i		−	20.1038i	19.7340	+	11.0711i	0.289720	+	0.501809i	14.5103	+	39.3623i
27.10	−1.43269	−	2.43873i	−0.662470	+	1.14743i	−3.89481	+	6.98788i	0.369055	−	0.639222i	3.74739	−	0.0283263i			4.50637i	22.6216	−	0.513064i	12.6223	+	21.8624i	−2.08763	+	0.0157803i
27.11	−1.06422	+	2.62058i	1.31663	−	2.28047i	−5.73488	−	5.57774i	1.30962	−	2.26832i	4.57498	+	5.87726i			35.2043i	20.7201	−	9.09277i	10.0330	+	17.3776i	4.55060	+	5.84595i
27.12	−1.03532	+	2.63213i	3.69266	−	6.39588i	−5.85623	−	5.45019i	−3.68797	+	6.38776i	13.0117	+	16.3413i		−	17.6395i	20.4087	−	9.77170i	−13.7715	−	23.8529i	−12.9952	−	16.3206i
27.13	−0.585298	−	2.76721i	−4.88508	+	8.46120i	−7.31485	+	3.23928i	6.65041	−	11.5188i	26.2731	+	8.56569i		−	4.05088i	13.2451	+	18.3457i	−34.2280	−	59.2846i	−35.7675	−	11.6611i
27.14	−0.0248348	−	2.82832i	4.46255	−	7.72937i	−7.99877	+	0.140481i	−6.17128	+	10.6890i	−21.9719	−	12.4296i		−	29.2047i	0.595974	+	22.6196i	−26.3288	−	45.6028i	30.3851	+	17.1889i
27.15	−0.00695190	−	2.82842i	−0.676811	+	1.17227i	−7.99990	+	0.0393258i	−5.49309	+	9.51431i	3.32038	+	1.90615i			11.1169i	0.166844	+	22.6268i	12.5839	+	21.7959i	26.9486	+	15.4706i
27.16	0.147992	+	2.82455i	−2.98979	+	5.17847i	−7.95620	+	0.836024i	−1.90510	+	3.29973i	−15.0693	−	7.67845i		−	2.74987i	−3.53885	−	22.3490i	−4.37772	−	7.58243i	−9.60221	−	4.89273i
27.17	0.750206	+	2.72712i	2.20715	−	3.82289i	−6.87438	+	4.09180i	9.79476	−	16.9650i	12.0813	+	3.15120i		−	17.1729i	−16.3160	−	15.6776i	3.75702	+	6.50734i	53.6137	+	13.9842i
27.18	0.978299	−	2.65385i	2.57005	−	4.45145i	−6.08586	−	5.19252i	7.41606	−	12.8450i	−9.29922	−	11.1754i			20.1038i	−19.7340	+	11.0711i	0.289720	+	0.501809i	−26.8336	−	32.2474i
27.19	1.37769	+	2.47022i	1.48658	−	2.57483i	−4.20394	+	6.80639i	−8.90564	+	15.4250i	8.40843	+	0.124854i		−	5.78146i	−22.6050	−	1.00755i	9.08017	+	15.7273i	−50.3724	−	0.747961i
27.20	1.73738	−	2.23193i	−1.31663	+	2.28047i	−1.96303	−	7.75542i	1.30962	−	2.26832i	2.80237	+	6.90067i		−	35.2043i	−20.7201	−	9.09277i	10.0330	+	17.3776i	−2.78744	−	6.86391i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
19.d	odd	6	1	inner
76.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	76.4.f.a	✓	56
4.b	odd	2	1	inner	76.4.f.a	✓	56
19.d	odd	6	1	inner	76.4.f.a	✓	56
76.f	even	6	1	inner	76.4.f.a	✓	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
76.4.f.a	✓	56	1.a	even	1	1	trivial
76.4.f.a	✓	56	4.b	odd	2	1	inner
76.4.f.a	✓	56	19.d	odd	6	1	inner
76.4.f.a	✓	56	76.f	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(76, [\chi])\).