Newform orbit 798.2.p.c

• Label: 798.2.p.c
• Level: $798$
• Weight: $2$
• Character orbit: 798.p
• Analytic conductor: $6.372$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	798.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.37206208130\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(25\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 50 q^{2} - 3 q^{3} + 50 q^{4} + 3 q^{6} - 50 q^{8} - 3 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
107.1	−1.00000			−1.73197	−	0.0167509i	1.00000				−	3.64338i	1.73197	+	0.0167509i	2.23585	−	1.41457i	−1.00000			2.99944	+	0.0580241i			3.64338i
107.2	−1.00000			−1.72597	−	0.145042i	1.00000					0.610954i	1.72597	+	0.145042i	−2.63918	+	0.186412i	−1.00000			2.95793	+	0.500674i		−	0.610954i
107.3	−1.00000			−1.69615	+	0.350829i	1.00000					4.07740i	1.69615	−	0.350829i	−1.01122	−	2.44488i	−1.00000			2.75384	−	1.19012i		−	4.07740i
107.4	−1.00000			−1.66624	+	0.472919i	1.00000				−	1.80606i	1.66624	−	0.472919i	−2.02215	+	1.70614i	−1.00000			2.55270	−	1.57599i			1.80606i
107.5	−1.00000			−1.41815	−	0.994415i	1.00000					2.50019i	1.41815	+	0.994415i	2.58513	+	0.563124i	−1.00000			1.02228	+	2.82045i		−	2.50019i
107.6	−1.00000			−1.35878	−	1.07411i	1.00000				−	1.73455i	1.35878	+	1.07411i	−0.675179	−	2.55815i	−1.00000			0.692588	+	2.91896i			1.73455i
107.7	−1.00000			−1.32098	+	1.12028i	1.00000					0.0905266i	1.32098	−	1.12028i	2.19691	−	1.47431i	−1.00000			0.489951	−	2.95972i		−	0.0905266i
107.8	−1.00000			−1.11989	+	1.32131i	1.00000					3.26298i	1.11989	−	1.32131i	0.0400578	+	2.64545i	−1.00000			−0.491712	−	2.95943i		−	3.26298i
107.9	−1.00000			−0.915186	−	1.47052i	1.00000				−	2.04210i	0.915186	+	1.47052i	0.305109	+	2.62810i	−1.00000			−1.32487	+	2.69160i			2.04210i
107.10	−1.00000			−0.412154	+	1.68230i	1.00000				−	1.95673i	0.412154	−	1.68230i	1.98940	+	1.74422i	−1.00000			−2.66026	−	1.38673i			1.95673i
107.11	−1.00000			−0.306658	+	1.70469i	1.00000					2.52686i	0.306658	−	1.70469i	−2.32425	−	1.26406i	−1.00000			−2.81192	−	1.04551i		−	2.52686i
107.12	−1.00000			−0.165673	+	1.72411i	1.00000				−	1.16878i	0.165673	−	1.72411i	−1.14346	−	2.38590i	−1.00000			−2.94510	−	0.571278i			1.16878i
107.13	−1.00000			−0.0922912	−	1.72959i	1.00000				−	3.23749i	0.0922912	+	1.72959i	2.52933	+	0.776191i	−1.00000			−2.98296	+	0.319252i			3.23749i
107.14	−1.00000			−0.0759161	−	1.73039i	1.00000					0.138945i	0.0759161	+	1.73039i	−2.56896	−	0.632805i	−1.00000			−2.98847	+	0.262728i		−	0.138945i
107.15	−1.00000			0.320237	−	1.70219i	1.00000					2.51691i	−0.320237	+	1.70219i	−0.461738	+	2.60515i	−1.00000			−2.79490	−	1.09021i		−	2.51691i
107.16	−1.00000			0.377439	−	1.69043i	1.00000					2.98905i	−0.377439	+	1.69043i	2.60274	−	0.475099i	−1.00000			−2.71508	−	1.27606i		−	2.98905i
107.17	−1.00000			0.722655	−	1.57409i	1.00000				−	3.34851i	−0.722655	+	1.57409i	−2.58671	−	0.555829i	−1.00000			−1.95554	−	2.27505i			3.34851i
107.18	−1.00000			0.828880	+	1.52084i	1.00000					1.59780i	−0.828880	−	1.52084i	2.18393	−	1.49348i	−1.00000			−1.62592	+	2.52119i		−	1.59780i
107.19	−1.00000			0.872368	+	1.49632i	1.00000				−	2.75270i	−0.872368	−	1.49632i	−1.41302	+	2.23683i	−1.00000			−1.47795	+	2.61068i			2.75270i
107.20	−1.00000			1.16908	+	1.27799i	1.00000					3.13526i	−1.16908	−	1.27799i	−0.151832	+	2.64139i	−1.00000			−0.266493	+	2.98814i		−	3.13526i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
399.bm	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	798.2.p.c	✓	50
3.b	odd	2	1		798.2.p.d	yes	50
7.c	even	3	1		798.2.bh.d	yes	50
19.d	odd	6	1		798.2.bh.c	yes	50
21.h	odd	6	1		798.2.bh.c	yes	50
57.f	even	6	1		798.2.bh.d	yes	50
133.j	odd	6	1		798.2.p.d	yes	50
399.bm	even	6	1	inner	798.2.p.c	✓	50

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
798.2.p.c	✓	50	1.a	even	1	1	trivial
798.2.p.c	✓	50	399.bm	even	6	1	inner
798.2.p.d	yes	50	3.b	odd	2	1
798.2.p.d	yes	50	133.j	odd	6	1
798.2.bh.c	yes	50	19.d	odd	6	1
798.2.bh.c	yes	50	21.h	odd	6	1
798.2.bh.d	yes	50	7.c	even	3	1
798.2.bh.d	yes	50	57.f	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\):

T5^50 + 154*T5^48 + 11091*T5^46 + 496680*T5^44 + 15511693*T5^42 + 359151444*T5^40 + 6397835009*T5^38 + 89814995056*T5^36 + 1009611897357*T5^34 + 9183552140468*T5^32 + 68034852087246*T5^30 + 411844896855372*T5^28 + 2038084442686399*T5^26 + 8227306111056804*T5^24 + 26960426841416374*T5^22 + 71155707270222422*T5^20 + 149527257827129961*T5^18 + 246184784758870924*T5^16 + 310514690407908184*T5^14 + 290682307253004504*T5^12 + 192855062829105175*T5^10 + 84488925003176052*T5^8 + 21718065850595154*T5^6 + 2589843479457384*T5^4 + 59211058136049*T5^2 + 322885837872

T11^50 - 3*T11^49 - 133*T11^48 + 408*T11^47 + 10271*T11^46 - 29751*T11^45 - 539483*T11^44 + 1454952*T11^43 + 21385223*T11^42 - 52190625*T11^41 - 661754523*T11^40 + 1438224690*T11^39 + 16465642419*T11^38 - 31140966207*T11^37 - 332882310077*T11^36 + 538147880478*T11^35 + 5531276437011*T11^34 - 7459472018181*T11^33 - 75775440266947*T11^32 + 83298397348638*T11^31 + 861104239593504*T11^30 - 744668714934480*T11^29 - 8127712811157113*T11^28 + 5261223864438933*T11^27 + 63904421719982721*T11^26 - 28218984571997928*T11^25 - 417754354656537013*T11^24 + 103751110874196915*T11^23 + 2267164107706422754*T11^22 - 144477213955828929*T11^21 - 10137605696303106914*T11^20 - 1116308336227640193*T11^19 + 37065274624026597399*T11^18 + 10230565222603428402*T11^17 - 108949827103979173634*T11^16 - 46748321020090855182*T11^15 + 253488464034029717203*T11^14 + 145148291330668317357*T11^13 - 451468247006968072237*T11^12 - 314429155838030044158*T11^11 + 604619393326248692362*T11^10 + 490512221476801913802*T11^9 - 563599349871093180453*T11^8 - 508321574347716638937*T11^7 + 364805372308215045015*T11^6 + 351596069989467355332*T11^5 - 120730589447951952315*T11^4 - 112403549262077886519*T11^3 + 34854526430021769282*T11^2 + 16368468382490754411*T11 + 1871940019044751803