Newform orbit 416.3.be.a

• Label: 416.3.be.a
• Level: $416$
• Weight: $3$
• Character orbit: 416.be
• Analytic conductor: $11.335$
• Analytic rank: $0$
• Dimension: $440$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	416.be (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.3351789974\)
Analytic rank:	\(0\)
Dimension:	\(440\)
Relative dimension:	\(110\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 440 * q - 8 * q^3 - 8 * q^4 - 8 * q^9 - 8 * q^10 + 40 * q^12 - 4 * q^13 - 40 * q^14 - 8 * q^16 + 104 * q^22 - 8 * q^23 - 8 * q^25 - 104 * q^26 + 184 * q^27 - 8 * q^29 + 24 * q^30 - 8 * q^35 - 272 * q^36 - 288 * q^38 - 196 * q^39 + 352 * q^40 - 208 * q^42 - 8 * q^43 - 112 * q^48 + 64 * q^51 - 428 * q^52 - 8 * q^53 + 248 * q^55 - 368 * q^56 - 8 * q^61 + 296 * q^62 - 8 * q^64 - 8 * q^65 + 856 * q^66 - 8 * q^69 - 8 * q^74 + 576 * q^75 - 8 * q^77 + 484 * q^78 - 16 * q^79 + 1032 * q^82 - 8 * q^87 - 568 * q^88 - 440 * q^90 - 196 * q^91 - 944 * q^92 - 792 * q^94

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
51.1	−1.99787	+	0.0923444i	−3.11870	−	1.29181i	3.98295	−	0.368984i	0.586174	+	1.41515i	6.35003	+	2.29286i	8.09949	−	8.09949i	−7.92332	+	1.10498i	1.69355	+	1.69355i	−1.30178	−	2.77315i
51.2	−1.99616	−	0.123854i	−0.885286	−	0.366698i	3.96932	+	0.494465i	−0.454051	−	1.09618i	1.72176	+	0.841634i	−8.45119	+	8.45119i	−7.86216	−	1.47865i	−5.71470	−	5.71470i	0.770593	+	2.24438i
51.3	−1.99291	+	0.168199i	2.73329	+	1.13217i	3.94342	−	0.670412i	−2.31992	−	5.60078i	−5.63765	−	1.79657i	7.44559	−	7.44559i	−7.74613	+	1.99935i	−0.174878	−	0.174878i	5.56544	+	10.7717i
51.4	−1.98733	+	0.224787i	3.22684	+	1.33660i	3.89894	−	0.893452i	−0.176830	−	0.426906i	−6.71325	−	1.93091i	−0.188699	+	0.188699i	−7.54764	+	2.65201i	2.26205	+	2.26205i	0.447383	+	0.808653i
51.5	−1.97986	+	0.283121i	−0.499130	−	0.206746i	3.83968	−	1.12108i	1.15470	+	2.78770i	1.04674	+	0.268014i	2.72764	−	2.72764i	−7.28463	+	3.30668i	−6.15757	−	6.15757i	−3.07541	−	5.19234i
51.6	−1.96792	+	0.356756i	−4.95804	−	2.05369i	3.74545	−	1.40414i	2.61067	+	6.30273i	10.4897	+	2.27269i	−8.59359	+	8.59359i	−6.86983	+	4.09945i	14.0006	+	14.0006i	−7.38615	−	11.4719i
51.7	−1.96333	−	0.381226i	−1.28388	−	0.531799i	3.70933	+	1.49694i	3.53737	+	8.53997i	2.31794	+	1.53354i	1.84272	−	1.84272i	−6.71197	−	4.35309i	−4.99843	−	4.99843i	−3.68937	−	18.1153i
51.8	−1.95091	−	0.440378i	0.255951	+	0.106019i	3.61214	+	1.71828i	−2.72741	−	6.58455i	−0.452651	−	0.319548i	−2.25164	+	2.25164i	−6.29028	−	4.94292i	−6.30969	−	6.30969i	2.42126	+	14.0470i
51.9	−1.92993	+	0.524772i	−3.77448	−	1.56344i	3.44923	−	2.02554i	−2.28619	−	5.51936i	8.10491	+	1.03658i	−3.65748	+	3.65748i	−5.59381	+	5.71920i	5.43837	+	5.43837i	7.30859	+	9.45223i
51.10	−1.92545	+	0.540980i	4.91460	+	2.03570i	3.41468	−	2.08325i	2.19173	+	5.29131i	−10.5641	−	1.26092i	−1.13789	+	1.13789i	−5.44779	+	5.85847i	13.6453	+	13.6453i	−7.08255	−	9.00245i
51.11	−1.90708	−	0.602529i	2.65847	+	1.10117i	3.27392	+	2.29814i	2.74000	+	6.61494i	−4.40643	−	3.70183i	−4.57539	+	4.57539i	−4.85893	−	6.35537i	−0.509089	−	0.509089i	−1.23971	−	14.2662i
51.12	−1.90033	−	0.623488i	−4.38843	−	1.81775i	3.22252	+	2.36967i	−1.71600	−	4.14279i	7.20613	+	6.19046i	1.73486	−	1.73486i	−4.64640	−	6.51237i	9.59016	+	9.59016i	0.677988	+	8.94258i
51.13	−1.89304	−	0.645295i	4.68963	+	1.94251i	3.16719	+	2.44314i	−0.683091	−	1.64913i	−7.62416	−	6.70344i	3.12122	−	3.12122i	−4.41906	−	6.66872i	11.8553	+	11.8553i	0.228943	+	3.56266i
51.14	−1.87191	−	0.704229i	−3.85446	−	1.59657i	3.00812	+	2.63651i	0.599273	+	1.44677i	6.09086	+	5.70306i	0.783280	−	0.783280i	−3.77423	−	7.05373i	5.94386	+	5.94386i	−0.102927	−	3.13026i
51.15	−1.83810	+	0.788281i	1.32174	+	0.547481i	2.75723	−	2.89788i	1.20868	+	2.91800i	−2.86105	+	0.0355745i	−1.19470	+	1.19470i	−2.78371	+	7.50006i	−4.91671	−	4.91671i	−4.52187	−	4.41080i
51.16	−1.81111	+	0.848459i	3.26014	+	1.35040i	2.56023	−	3.07330i	−3.69360	−	8.91714i	−7.05023	+	0.320385i	−5.21443	+	5.21443i	−2.02929	+	7.73835i	2.44101	+	2.44101i	14.2553	+	13.0160i
51.17	−1.79828	+	0.875320i	−4.22104	−	1.74841i	2.46763	−	3.14814i	−2.88910	−	6.97489i	9.12104	−	0.550624i	2.87965	−	2.87965i	−1.68186	+	7.82121i	8.39629	+	8.39629i	11.3007	+	10.0139i
51.18	−1.76306	−	0.944258i	2.02772	+	0.839910i	2.21675	+	3.32956i	0.464706	+	1.12190i	−2.78190	−	3.39550i	6.62794	−	6.62794i	−0.764303	−	7.96341i	−2.95776	−	2.95776i	0.240058	−	2.41678i
51.19	−1.66437	+	1.10899i	−0.157380	−	0.0651888i	1.54028	−	3.69155i	−0.933238	−	2.25303i	0.334232	−	0.0660338i	−4.44018	+	4.44018i	1.53028	+	7.85228i	−6.34344	−	6.34344i	4.05185	+	2.71494i
51.20	−1.65663	+	1.12052i	−2.80483	−	1.16180i	1.48887	−	3.71258i	1.24533	+	3.00650i	5.94839	−	1.21819i	−1.43368	+	1.43368i	1.69352	+	7.81870i	0.153325	+	0.153325i	−5.43191	−	3.58525i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner
32.h	odd	8	1	inner
416.be	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	416.3.be.a	✓	440
13.b	even	2	1	inner	416.3.be.a	✓	440
32.h	odd	8	1	inner	416.3.be.a	✓	440
416.be	odd	8	1	inner	416.3.be.a	✓	440

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
416.3.be.a	✓	440	1.a	even	1	1	trivial
416.3.be.a	✓	440	13.b	even	2	1	inner
416.3.be.a	✓	440	32.h	odd	8	1	inner
416.3.be.a	✓	440	416.be	odd	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(416, [\chi])\).