LMFDB - Embedded newform 2667.2.a.p.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2667,2,Mod(1,2667)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2667.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2667, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2667 = 3 \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2667.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,-6,-18,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$21.2961022191$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 $ x^18 - 6x^17 - 11x^16 + 123x^15 - 35x^14 - 982x^13 + 988x^12 + 3872x^11 - 5421x^10 - 7882x^9 + 13376x^8 + 7948x^7 - 15795x^6 - 3858x^5 + 8199x^4 + 1453x^3 - 1610x^2 - 380*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.981962$ of defining polynomial
Character	$\chi$	$=$	2667.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.981962 q^{2} -1.00000 q^{3} -1.03575 q^{4} +2.59441 q^{5} +0.981962 q^{6} +1.00000 q^{7} +2.98099 q^{8} +1.00000 q^{9} -2.54761 q^{10} -0.832401 q^{11} +1.03575 q^{12} -5.10508 q^{13} -0.981962 q^{14} -2.59441 q^{15} -0.855717 q^{16} +7.10660 q^{17} -0.981962 q^{18} -7.54643 q^{19} -2.68717 q^{20} -1.00000 q^{21} +0.817386 q^{22} +8.33771 q^{23} -2.98099 q^{24} +1.73097 q^{25} +5.01299 q^{26} -1.00000 q^{27} -1.03575 q^{28} -8.62369 q^{29} +2.54761 q^{30} -6.02299 q^{31} -5.12170 q^{32} +0.832401 q^{33} -6.97841 q^{34} +2.59441 q^{35} -1.03575 q^{36} -5.52425 q^{37} +7.41031 q^{38} +5.10508 q^{39} +7.73392 q^{40} -8.68580 q^{41} +0.981962 q^{42} +4.88483 q^{43} +0.862161 q^{44} +2.59441 q^{45} -8.18731 q^{46} +2.27297 q^{47} +0.855717 q^{48} +1.00000 q^{49} -1.69975 q^{50} -7.10660 q^{51} +5.28759 q^{52} +3.81937 q^{53} +0.981962 q^{54} -2.15959 q^{55} +2.98099 q^{56} +7.54643 q^{57} +8.46813 q^{58} -3.88402 q^{59} +2.68717 q^{60} -7.64885 q^{61} +5.91434 q^{62} +1.00000 q^{63} +6.74075 q^{64} -13.2447 q^{65} -0.817386 q^{66} +9.03188 q^{67} -7.36067 q^{68} -8.33771 q^{69} -2.54761 q^{70} -9.47044 q^{71} +2.98099 q^{72} -6.75232 q^{73} +5.42460 q^{74} -1.73097 q^{75} +7.81623 q^{76} -0.832401 q^{77} -5.01299 q^{78} +5.46202 q^{79} -2.22008 q^{80} +1.00000 q^{81} +8.52913 q^{82} +9.63296 q^{83} +1.03575 q^{84} +18.4374 q^{85} -4.79672 q^{86} +8.62369 q^{87} -2.48138 q^{88} +9.34318 q^{89} -2.54761 q^{90} -5.10508 q^{91} -8.63580 q^{92} +6.02299 q^{93} -2.23196 q^{94} -19.5786 q^{95} +5.12170 q^{96} -0.647975 q^{97} -0.981962 q^{98} -0.832401 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 6 q^{2} - 18 q^{3} + 22 q^{4} - 10 q^{5} + 6 q^{6} + 18 q^{7} - 21 q^{8} + 18 q^{9} - 4 q^{10} - 9 q^{11} - 22 q^{12} - 25 q^{13} - 6 q^{14} + 10 q^{15} + 34 q^{16} - 17 q^{17} - 6 q^{18} - 5 q^{19}+ \cdots - 9 q^{99}+O(q^{100}) $ 18 * q - 6 * q^2 - 18 * q^3 + 22 * q^4 - 10 * q^5 + 6 * q^6 + 18 * q^7 - 21 * q^8 + 18 * q^9 - 4 * q^10 - 9 * q^11 - 22 * q^12 - 25 * q^13 - 6 * q^14 + 10 * q^15 + 34 * q^16 - 17 * q^17 - 6 * q^18 - 5 * q^19 - 21 * q^20 - 18 * q^21 + 5 * q^22 - 14 * q^23 + 21 * q^24 + 28 * q^25 - 8 * q^26 - 18 * q^27 + 22 * q^28 - 17 * q^29 + 4 * q^30 + 5 * q^31 - 53 * q^32 + 9 * q^33 - 19 * q^34 - 10 * q^35 + 22 * q^36 - 15 * q^37 - 22 * q^38 + 25 * q^39 - q^40 - 17 * q^41 + 6 * q^42 + q^43 - 33 * q^44 - 10 * q^45 + 10 * q^46 - 31 * q^47 - 34 * q^48 + 18 * q^49 - 35 * q^50 + 17 * q^51 - 70 * q^52 - 35 * q^53 + 6 * q^54 + 4 * q^55 - 21 * q^56 + 5 * q^57 + 3 * q^58 - 46 * q^59 + 21 * q^60 - 5 * q^61 - 10 * q^62 + 18 * q^63 + 63 * q^64 - 12 * q^65 - 5 * q^66 + 6 * q^67 - 56 * q^68 + 14 * q^69 - 4 * q^70 - 22 * q^71 - 21 * q^72 - 16 * q^73 + 18 * q^74 - 28 * q^75 + 32 * q^76 - 9 * q^77 + 8 * q^78 + 46 * q^79 - 30 * q^80 + 18 * q^81 - 12 * q^82 - 46 * q^83 - 22 * q^84 + 4 * q^85 + 18 * q^86 + 17 * q^87 + 30 * q^88 - 42 * q^89 - 4 * q^90 - 25 * q^91 - 48 * q^92 - 5 * q^93 + 3 * q^94 - 2 * q^95 + 53 * q^96 - 35 * q^97 - 6 * q^98 - 9 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.981962	−0.694352	−0.347176	−	0.937800i	$-0.612859\pi$
$2$	−0.981962	−0.694352	−0.347176	+	0.937800i	$0.612859\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.03575	−0.517876
$5$	2.59441	1.16026	0.580128	−	0.814525i	$-0.303002\pi$
$5$	2.59441	1.16026	0.580128	+	0.814525i	$0.303002\pi$
$6$	0.981962	0.400884
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.98099	1.05394
$9$	1.00000	0.333333
$10$	−2.54761	−0.805626
$11$	−0.832401	−0.250978	−0.125489	−	0.992095i	$-0.540050\pi$
$11$	−0.832401	−0.250978	−0.125489	+	0.992095i	$0.540050\pi$
$12$	1.03575	0.298996
$13$	−5.10508	−1.41589	−0.707947	−	0.706265i	$-0.750379\pi$
$13$	−5.10508	−1.41589	−0.707947	+	0.706265i	$0.750379\pi$
$14$	−0.981962	−0.262440
$15$	−2.59441	−0.669874
$16$	−0.855717	−0.213929
$17$	7.10660	1.72360	0.861802	−	0.507245i	$-0.169336\pi$
$17$	7.10660	1.72360	0.861802	+	0.507245i	$0.169336\pi$
$18$	−0.981962	−0.231451
$19$	−7.54643	−1.73127	−0.865635	−	0.500675i	$-0.833085\pi$
$19$	−7.54643	−1.73127	−0.865635	+	0.500675i	$0.833085\pi$
$20$	−2.68717	−0.600868
$21$	−1.00000	−0.218218
$22$	0.817386	0.174267
$23$	8.33771	1.73853	0.869267	−	0.494344i	$-0.164592\pi$
$23$	8.33771	1.73853	0.869267	+	0.494344i	$0.164592\pi$
$24$	−2.98099	−0.608492
$25$	1.73097	0.346195
$26$	5.01299	0.983129
$27$	−1.00000	−0.192450
$28$	−1.03575	−0.195739
$29$	−8.62369	−1.60138	−0.800689	−	0.599080i	$-0.795533\pi$
$29$	−8.62369	−1.60138	−0.800689	+	0.599080i	$0.795533\pi$
$30$	2.54761	0.465128
$31$	−6.02299	−1.08176	−0.540880	−	0.841100i	$-0.681909\pi$
$31$	−6.02299	−1.08176	−0.540880	+	0.841100i	$0.681909\pi$
$32$	−5.12170	−0.905397
$33$	0.832401	0.144902
$34$	−6.97841	−1.19679
$35$	2.59441	0.438536
$36$	−1.03575	−0.172625
$37$	−5.52425	−0.908181	−0.454090	−	0.890956i	$-0.650036\pi$
$37$	−5.52425	−0.908181	−0.454090	+	0.890956i	$0.650036\pi$
$38$	7.41031	1.20211
$39$	5.10508	0.817467
$40$	7.73392	1.22284
$41$	−8.68580	−1.35649	−0.678247	−	0.734834i	$-0.737260\pi$
$41$	−8.68580	−1.35649	−0.678247	+	0.734834i	$0.737260\pi$
$42$	0.981962	0.151520
$43$	4.88483	0.744930	0.372465	−	0.928046i	$-0.378513\pi$
$43$	4.88483	0.744930	0.372465	+	0.928046i	$0.378513\pi$
$44$	0.862161	0.129976
$45$	2.59441	0.386752
$46$	−8.18731	−1.20715
$47$	2.27297	0.331546	0.165773	−	0.986164i	$-0.446988\pi$
$47$	2.27297	0.331546	0.165773	+	0.986164i	$0.446988\pi$
$48$	0.855717	0.123512
$49$	1.00000	0.142857
$50$	−1.69975	−0.240381
$51$	−7.10660	−0.995123
$52$	5.28759	0.733257
$53$	3.81937	0.524631	0.262316	−	0.964982i	$-0.415514\pi$
$53$	3.81937	0.524631	0.262316	+	0.964982i	$0.415514\pi$
$54$	0.981962	0.133628
$55$	−2.15959	−0.291199
$56$	2.98099	0.398352
$57$	7.54643	0.999549
$58$	8.46813	1.11192
$59$	−3.88402	−0.505656	−0.252828	−	0.967511i	$-0.581361\pi$
$59$	−3.88402	−0.505656	−0.252828	+	0.967511i	$0.581361\pi$
$60$	2.68717	0.346912
$61$	−7.64885	−0.979335	−0.489668	−	0.871909i	$-0.662882\pi$
$61$	−7.64885	−0.979335	−0.489668	+	0.871909i	$0.662882\pi$
$62$	5.91434	0.751122
$63$	1.00000	0.125988
$64$	6.74075	0.842594
$65$	−13.2447	−1.64280
$66$	−0.817386	−0.100613
$67$	9.03188	1.10342	0.551710	−	0.834036i	$-0.313975\pi$
$67$	9.03188	1.10342	0.551710	+	0.834036i	$0.313975\pi$
$68$	−7.36067	−0.892612
$69$	−8.33771	−1.00374
$70$	−2.54761	−0.304498
$71$	−9.47044	−1.12393	−0.561967	−	0.827160i	$-0.689955\pi$
$71$	−9.47044	−1.12393	−0.561967	+	0.827160i	$0.689955\pi$
$72$	2.98099	0.351313
$73$	−6.75232	−0.790300	−0.395150	−	0.918617i	$-0.629307\pi$
$73$	−6.75232	−0.790300	−0.395150	+	0.918617i	$0.629307\pi$
$74$	5.42460	0.630597
$75$	−1.73097	−0.199876
$76$	7.81623	0.896583
$77$	−0.832401	−0.0948609
$78$	−5.01299	−0.567610
$79$	5.46202	0.614525	0.307262	−	0.951625i	$-0.400587\pi$
$79$	5.46202	0.614525	0.307262	+	0.951625i	$0.400587\pi$
$80$	−2.22008	−0.248213
$81$	1.00000	0.111111
$82$	8.52913	0.941885
$83$	9.63296	1.05735	0.528677	−	0.848823i	$-0.322688\pi$
$83$	9.63296	1.05735	0.528677	+	0.848823i	$0.322688\pi$
$84$	1.03575	0.113010
$85$	18.4374	1.99982
$86$	−4.79672	−0.517244
$87$	8.62369	0.924556
$88$	−2.48138	−0.264516
$89$	9.34318	0.990375	0.495188	−	0.868786i	$-0.335099\pi$
$89$	9.34318	0.990375	0.495188	+	0.868786i	$0.335099\pi$
$90$	−2.54761	−0.268542
$91$	−5.10508	−0.535158
$92$	−8.63580	−0.900344
$93$	6.02299	0.624555
$94$	−2.23196	−0.230210
$95$	−19.5786	−2.00872
$96$	5.12170	0.522731
$97$	−0.647975	−0.0657918	−0.0328959	−	0.999459i	$-0.510473\pi$
$97$	−0.647975	−0.0657918	−0.0328959	+	0.999459i	$0.510473\pi$
$98$	−0.981962	−0.0991931
$99$	−0.832401	−0.0836595

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2667.2.a.p.1.8	✓	18
3.2	odd	2		8001.2.a.u.1.11		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.8	✓	18	1.1	even	1	trivial
8001.2.a.u.1.11		18	3.2	odd	2

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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2667 = 3 \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2667.a (trivial)

\(f(q)\)	\(=\)	\(q-0.981962 q^{2} -1.00000 q^{3} -1.03575 q^{4} +2.59441 q^{5} +0.981962 q^{6} +1.00000 q^{7} +2.98099 q^{8} +1.00000 q^{9} -2.54761 q^{10} -0.832401 q^{11} +1.03575 q^{12} -5.10508 q^{13} -0.981962 q^{14} -2.59441 q^{15} -0.855717 q^{16} +7.10660 q^{17} -0.981962 q^{18} -7.54643 q^{19} -2.68717 q^{20} -1.00000 q^{21} +0.817386 q^{22} +8.33771 q^{23} -2.98099 q^{24} +1.73097 q^{25} +5.01299 q^{26} -1.00000 q^{27} -1.03575 q^{28} -8.62369 q^{29} +2.54761 q^{30} -6.02299 q^{31} -5.12170 q^{32} +0.832401 q^{33} -6.97841 q^{34} +2.59441 q^{35} -1.03575 q^{36} -5.52425 q^{37} +7.41031 q^{38} +5.10508 q^{39} +7.73392 q^{40} -8.68580 q^{41} +0.981962 q^{42} +4.88483 q^{43} +0.862161 q^{44} +2.59441 q^{45} -8.18731 q^{46} +2.27297 q^{47} +0.855717 q^{48} +1.00000 q^{49} -1.69975 q^{50} -7.10660 q^{51} +5.28759 q^{52} +3.81937 q^{53} +0.981962 q^{54} -2.15959 q^{55} +2.98099 q^{56} +7.54643 q^{57} +8.46813 q^{58} -3.88402 q^{59} +2.68717 q^{60} -7.64885 q^{61} +5.91434 q^{62} +1.00000 q^{63} +6.74075 q^{64} -13.2447 q^{65} -0.817386 q^{66} +9.03188 q^{67} -7.36067 q^{68} -8.33771 q^{69} -2.54761 q^{70} -9.47044 q^{71} +2.98099 q^{72} -6.75232 q^{73} +5.42460 q^{74} -1.73097 q^{75} +7.81623 q^{76} -0.832401 q^{77} -5.01299 q^{78} +5.46202 q^{79} -2.22008 q^{80} +1.00000 q^{81} +8.52913 q^{82} +9.63296 q^{83} +1.03575 q^{84} +18.4374 q^{85} -4.79672 q^{86} +8.62369 q^{87} -2.48138 q^{88} +9.34318 q^{89} -2.54761 q^{90} -5.10508 q^{91} -8.63580 q^{92} +6.02299 q^{93} -2.23196 q^{94} -19.5786 q^{95} +5.12170 q^{96} -0.647975 q^{97} -0.981962 q^{98} -0.832401 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 6 q^{2} - 18 q^{3} + 22 q^{4} - 10 q^{5} + 6 q^{6} + 18 q^{7} - 21 q^{8} + 18 q^{9} - 4 q^{10} - 9 q^{11} - 22 q^{12} - 25 q^{13} - 6 q^{14} + 10 q^{15} + 34 q^{16} - 17 q^{17} - 6 q^{18} - 5 q^{19}+ \cdots - 9 q^{99}+O(q^{100}) \) 18 * q - 6 * q^2 - 18 * q^3 + 22 * q^4 - 10 * q^5 + 6 * q^6 + 18 * q^7 - 21 * q^8 + 18 * q^9 - 4 * q^10 - 9 * q^11 - 22 * q^12 - 25 * q^13 - 6 * q^14 + 10 * q^15 + 34 * q^16 - 17 * q^17 - 6 * q^18 - 5 * q^19 - 21 * q^20 - 18 * q^21 + 5 * q^22 - 14 * q^23 + 21 * q^24 + 28 * q^25 - 8 * q^26 - 18 * q^27 + 22 * q^28 - 17 * q^29 + 4 * q^30 + 5 * q^31 - 53 * q^32 + 9 * q^33 - 19 * q^34 - 10 * q^35 + 22 * q^36 - 15 * q^37 - 22 * q^38 + 25 * q^39 - q^40 - 17 * q^41 + 6 * q^42 + q^43 - 33 * q^44 - 10 * q^45 + 10 * q^46 - 31 * q^47 - 34 * q^48 + 18 * q^49 - 35 * q^50 + 17 * q^51 - 70 * q^52 - 35 * q^53 + 6 * q^54 + 4 * q^55 - 21 * q^56 + 5 * q^57 + 3 * q^58 - 46 * q^59 + 21 * q^60 - 5 * q^61 - 10 * q^62 + 18 * q^63 + 63 * q^64 - 12 * q^65 - 5 * q^66 + 6 * q^67 - 56 * q^68 + 14 * q^69 - 4 * q^70 - 22 * q^71 - 21 * q^72 - 16 * q^73 + 18 * q^74 - 28 * q^75 + 32 * q^76 - 9 * q^77 + 8 * q^78 + 46 * q^79 - 30 * q^80 + 18 * q^81 - 12 * q^82 - 46 * q^83 - 22 * q^84 + 4 * q^85 + 18 * q^86 + 17 * q^87 + 30 * q^88 - 42 * q^89 - 4 * q^90 - 25 * q^91 - 48 * q^92 - 5 * q^93 + 3 * q^94 - 2 * q^95 + 53 * q^96 - 35 * q^97 - 6 * q^98 - 9 * q^99

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