LMFDB - Newform orbit 825.2.n.p

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [825,2,Mod(301,825)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("825.301"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$825 = 3 \cdot 5^{2} \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	825.n (of order $5$ , degree $4$ , minimal)

Newform invariants

comment:select newform

sage:traces = [24,2,-6,-6,0,2,4,6,-6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$6$ over $\Q(\zeta_{5})$
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$ -expansion

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

\operatorname{Tr}(f)(q) =

24 q + 2 q^{2} - 6 q^{3} - 6 q^{4} + 2 q^{6} + 4 q^{7} + 6 q^{8} - 6 q^{9} + 24 q^{12} + 4 q^{13} + 2 q^{14} - 22 q^{16} + 4 q^{17} + 2 q^{18} + 8 q^{19} - 16 q^{21} - 4 q^{22} + 6 q^{24} - 38 q^{26} - 6 q^{27}+ \cdots - 120 q^{98}+O(q^{100})

24 * q + 2 * q^2 - 6 * q^3 - 6 * q^4 + 2 * q^6 + 4 * q^7 + 6 * q^8 - 6 * q^9 + 24 * q^12 + 4 * q^13 + 2 * q^14 - 22 * q^16 + 4 * q^17 + 2 * q^18 + 8 * q^19 - 16 * q^21 - 4 * q^22 + 6 * q^24 - 38 * q^26 - 6 * q^27 + 30 * q^28 - 10 * q^31 - 56 * q^32 + 10 * q^33 + 12 * q^34 - 6 * q^36 + 10 * q^37 + 4 * q^38 + 4 * q^39 + 30 * q^41 - 8 * q^42 - 64 * q^43 + 24 * q^44 + 54 * q^46 - 8 * q^47 - 2 * q^48 + 14 * q^49 + 14 * q^51 + 14 * q^52 + 26 * q^53 - 8 * q^54 + 12 * q^56 + 8 * q^57 + 20 * q^58 - 30 * q^59 + 20 * q^61 - 50 * q^62 + 4 * q^63 - 32 * q^64 + 6 * q^66 + 20 * q^67 - 62 * q^68 - 10 * q^69 - 16 * q^71 + 6 * q^72 - 12 * q^73 + 16 * q^74 - 68 * q^76 - 2 * q^77 + 32 * q^78 + 26 * q^79 - 6 * q^81 + 56 * q^82 + 48 * q^83 - 52 * q^86 + 48 * q^88 - 20 * q^89 - 20 * q^91 + 46 * q^92 - 10 * q^93 - 36 * q^94 + 14 * q^96 - 14 * q^97 - 120 * 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.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label	$a_{2}$			$a_{3}$			$a_{4}$				$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$
301.1	−1.91233	−	1.38939i	0.309017	−	0.951057i	1.10857	+	3.41183i	0	−1.91233	+	1.38939i	−0.529727	−	1.63033i	1.15951	−	3.56862i	−0.809017	−	0.587785i	0
301.2	−0.634375	−	0.460901i	0.309017	−	0.951057i	−0.428031	−	1.31734i	0	−0.634375	+	0.460901i	−0.469090	−	1.44371i	−0.820252	+	2.52448i	−0.809017	−	0.587785i	0
301.3	−0.233654	−	0.169760i	0.309017	−	0.951057i	−0.592258	−	1.82278i	0	−0.233654	+	0.169760i	1.13329	+	3.48791i	−0.349548	+	1.07580i	−0.809017	−	0.587785i	0
301.4	0.649936	+	0.472206i	0.309017	−	0.951057i	−0.418596	−	1.28830i	0	0.649936	−	0.472206i	0.157137	+	0.483617i	0.832793	−	2.56307i	−0.809017	−	0.587785i	0
301.5	1.54314	+	1.12116i	0.309017	−	0.951057i	0.506258	+	1.55810i	0	1.54314	−	1.12116i	−1.37739	−	4.23918i	0.213204	−	0.656175i	−0.809017	−	0.587785i	0
301.6	2.20531	+	1.60225i	0.309017	−	0.951057i	1.67816	+	5.16484i	0	2.20531	−	1.60225i	−0.150285	−	0.462529i	−2.88981	+	8.89393i	−0.809017	−	0.587785i	0
526.1	−0.800975	+	2.46515i	−0.809017	+	0.587785i	−3.81735	−	2.77347i	0	−0.800975	−	2.46515i	2.76550	+	2.00926i	5.70065	−	4.14176i	0.309017	−	0.951057i	0
526.2	−0.650947	+	2.00341i	−0.809017	+	0.587785i	−1.97188	−	1.43266i	0	−0.650947	−	2.00341i	−2.68296	−	1.94929i	0.745386	−	0.541555i	0.309017	−	0.951057i	0
526.3	−0.150893	+	0.464400i	−0.809017	+	0.587785i	1.42514	+	1.03542i	0	−0.150893	−	0.464400i	4.13322	+	3.00296i	−1.48598	+	1.07962i	0.309017	−	0.951057i	0
526.4	−0.0921069	+	0.283476i	−0.809017	+	0.587785i	1.54616	+	1.12335i	0	−0.0921069	−	0.283476i	−1.87778	−	1.36429i	−0.943133	+	0.685226i	0.309017	−	0.951057i	0
526.5	0.392740	−	1.20873i	−0.809017	+	0.587785i	0.311255	+	0.226140i	0	0.392740	+	1.20873i	−0.390991	−	0.284071i	2.45199	−	1.78148i	0.309017	−	0.951057i	0
526.6	0.684148	−	2.10559i	−0.809017	+	0.587785i	−2.34742	−	1.70550i	0	0.684148	+	2.10559i	1.28908	+	0.936570i	−1.61482	+	1.17323i	0.309017	−	0.951057i	0
676.1	−0.800975	−	2.46515i	−0.809017	−	0.587785i	−3.81735	+	2.77347i	0	−0.800975	+	2.46515i	2.76550	−	2.00926i	5.70065	+	4.14176i	0.309017	+	0.951057i	0
676.2	−0.650947	−	2.00341i	−0.809017	−	0.587785i	−1.97188	+	1.43266i	0	−0.650947	+	2.00341i	−2.68296	+	1.94929i	0.745386	+	0.541555i	0.309017	+	0.951057i	0
676.3	−0.150893	−	0.464400i	−0.809017	−	0.587785i	1.42514	−	1.03542i	0	−0.150893	+	0.464400i	4.13322	−	3.00296i	−1.48598	−	1.07962i	0.309017	+	0.951057i	0
676.4	−0.0921069	−	0.283476i	−0.809017	−	0.587785i	1.54616	−	1.12335i	0	−0.0921069	+	0.283476i	−1.87778	+	1.36429i	−0.943133	−	0.685226i	0.309017	+	0.951057i	0
676.5	0.392740	+	1.20873i	−0.809017	−	0.587785i	0.311255	−	0.226140i	0	0.392740	−	1.20873i	−0.390991	+	0.284071i	2.45199	+	1.78148i	0.309017	+	0.951057i	0
676.6	0.684148	+	2.10559i	−0.809017	−	0.587785i	−2.34742	+	1.70550i	0	0.684148	−	2.10559i	1.28908	−	0.936570i	−1.61482	−	1.17323i	0.309017	+	0.951057i	0
751.1	−1.91233	+	1.38939i	0.309017	+	0.951057i	1.10857	−	3.41183i	0	−1.91233	−	1.38939i	−0.529727	+	1.63033i	1.15951	+	3.56862i	−0.809017	+	0.587785i	0
751.2	−0.634375	+	0.460901i	0.309017	+	0.951057i	−0.428031	+	1.31734i	0	−0.634375	−	0.460901i	−0.469090	+	1.44371i	−0.820252	−	2.52448i	−0.809017	+	0.587785i	0

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	825.2.n.p		24
5.b	even	2	1		825.2.n.o		24
5.c	odd	4	2		165.2.s.a	✓	48
11.c	even	5	1	inner	825.2.n.p		24
11.c	even	5	1		9075.2.a.dy		12
11.d	odd	10	1		9075.2.a.ea		12
15.e	even	4	2		495.2.ba.c		48
55.h	odd	10	1		9075.2.a.dx		12
55.j	even	10	1		825.2.n.o		24
55.j	even	10	1		9075.2.a.dz		12
55.k	odd	20	2		165.2.s.a	✓	48
55.k	odd	20	2		1815.2.c.j		24
55.l	even	20	2		1815.2.c.k		24
165.v	even	20	2		495.2.ba.c		48

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
165.2.s.a	✓	48	5.c	odd	4	2
165.2.s.a	✓	48	55.k	odd	20	2
495.2.ba.c		48	15.e	even	4	2
495.2.ba.c		48	165.v	even	20	2
825.2.n.o		24	5.b	even	2	1
825.2.n.o		24	55.j	even	10	1
825.2.n.p		24	1.a	even	1	1	trivial
825.2.n.p		24	11.c	even	5	1	inner
1815.2.c.j		24	55.k	odd	20	2
1815.2.c.k		24	55.l	even	20	2
9075.2.a.dx		12	55.h	odd	10	1
9075.2.a.dy		12	11.c	even	5	1
9075.2.a.dz		12	55.j	even	10	1
9075.2.a.ea		12	11.d	odd	10	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(825, [\chi])$ :

T_{2}^{24} - 2 T_{2}^{23} + 11 T_{2}^{22} - 24 T_{2}^{21} + 95 T_{2}^{20} - 116 T_{2}^{19} + 556 T_{2}^{18} + \cdots + 25

T2^24 - 2*T2^23 + 11*T2^22 - 24*T2^21 + 95*T2^20 - 116*T2^19 + 556*T2^18 - 664*T2^17 + 3074*T2^16 - 3928*T2^15 + 11247*T2^14 - 11892*T2^13 + 28569*T2^12 - 2156*T2^11 + 20101*T2^10 + 17346*T2^9 + 19092*T2^8 - 2050*T2^7 + 6734*T2^6 + 5936*T2^5 + 6351*T2^4 + 3020*T2^3 + 1015*T2^2 + 200*T2 + 25

T_{13}^{24} - 4 T_{13}^{23} + 35 T_{13}^{22} - 204 T_{13}^{21} + 1406 T_{13}^{20} - 5008 T_{13}^{19} + \cdots + 445252201

T13^24 - 4*T13^23 + 35*T13^22 - 204*T13^21 + 1406*T13^20 - 5008*T13^19 + 37765*T13^18 - 172500*T13^17 + 912187*T13^16 - 3656790*T13^15 + 16867121*T13^14 - 53107856*T13^13 + 179026864*T13^12 - 403634158*T13^11 + 1055676078*T13^10 - 1648620226*T13^9 + 3942142454*T13^8 - 937403730*T13^7 + 10556380266*T13^6 + 4304737420*T13^5 + 6137639649*T13^4 - 81003326*T13^3 - 456822124*T13^2 - 5064240*T13 + 445252201

Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 301.6
Significant digits:
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