LMFDB - Embedded newform 2667.2.a.m.1.14

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 = [14,-5,-14,15] 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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 5 x^{13} - 9 x^{12} + 76 x^{11} - 12 x^{10} - 414 x^{9} + 331 x^{8} + 959 x^{7} - 1067 x^{6} + \cdots + 44 $ x^14 - 5x^13 - 9x^12 + 76x^11 - 12x^10 - 414x^9 + 331x^8 + 959x^7 - 1067x^6 - 875x^5 + 1134x^4 + 301x^3 - 418x^2 - 42*x + 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-2.33388$ 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)

$f(q)$	$=$	\(q+2.33388 q^{2} -1.00000 q^{3} +3.44698 q^{4} -0.335557 q^{5} -2.33388 q^{6} -1.00000 q^{7} +3.37706 q^{8} +1.00000 q^{9} -0.783148 q^{10} -5.16521 q^{11} -3.44698 q^{12} -5.99956 q^{13} -2.33388 q^{14} +0.335557 q^{15} +0.987696 q^{16} +7.35302 q^{17} +2.33388 q^{18} +5.24027 q^{19} -1.15666 q^{20} +1.00000 q^{21} -12.0550 q^{22} +1.66716 q^{23} -3.37706 q^{24} -4.88740 q^{25} -14.0022 q^{26} -1.00000 q^{27} -3.44698 q^{28} +1.61282 q^{29} +0.783148 q^{30} -6.38054 q^{31} -4.44897 q^{32} +5.16521 q^{33} +17.1610 q^{34} +0.335557 q^{35} +3.44698 q^{36} -11.1071 q^{37} +12.2302 q^{38} +5.99956 q^{39} -1.13320 q^{40} -3.12952 q^{41} +2.33388 q^{42} -8.78359 q^{43} -17.8044 q^{44} -0.335557 q^{45} +3.89095 q^{46} -11.1208 q^{47} -0.987696 q^{48} +1.00000 q^{49} -11.4066 q^{50} -7.35302 q^{51} -20.6803 q^{52} -4.86128 q^{53} -2.33388 q^{54} +1.73322 q^{55} -3.37706 q^{56} -5.24027 q^{57} +3.76411 q^{58} +14.4216 q^{59} +1.15666 q^{60} -9.80157 q^{61} -14.8914 q^{62} -1.00000 q^{63} -12.3587 q^{64} +2.01319 q^{65} +12.0550 q^{66} -4.47288 q^{67} +25.3457 q^{68} -1.66716 q^{69} +0.783148 q^{70} -0.675275 q^{71} +3.37706 q^{72} +13.2597 q^{73} -25.9226 q^{74} +4.88740 q^{75} +18.0631 q^{76} +5.16521 q^{77} +14.0022 q^{78} +10.6397 q^{79} -0.331428 q^{80} +1.00000 q^{81} -7.30391 q^{82} +10.1444 q^{83} +3.44698 q^{84} -2.46736 q^{85} -20.4998 q^{86} -1.61282 q^{87} -17.4432 q^{88} -15.6055 q^{89} -0.783148 q^{90} +5.99956 q^{91} +5.74667 q^{92} +6.38054 q^{93} -25.9547 q^{94} -1.75841 q^{95} +4.44897 q^{96} +13.0787 q^{97} +2.33388 q^{98} -5.16521 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 5 q^{2} - 14 q^{3} + 15 q^{4} - 4 q^{5} + 5 q^{6} - 14 q^{7} - 12 q^{8} + 14 q^{9} + 4 q^{10} - 3 q^{11} - 15 q^{12} - 13 q^{13} + 5 q^{14} + 4 q^{15} + 13 q^{16} - 5 q^{17} - 5 q^{18} + 21 q^{19}+ \cdots - 3 q^{99}+O(q^{100}) $ 14 * q - 5 * q^2 - 14 * q^3 + 15 * q^4 - 4 * q^5 + 5 * q^6 - 14 * q^7 - 12 * q^8 + 14 * q^9 + 4 * q^10 - 3 * q^11 - 15 * q^12 - 13 * q^13 + 5 * q^14 + 4 * q^15 + 13 * q^16 - 5 * q^17 - 5 * q^18 + 21 * q^19 - 3 * q^20 + 14 * q^21 - 3 * q^22 - 10 * q^23 + 12 * q^24 + 4 * q^25 - 6 * q^26 - 14 * q^27 - 15 * q^28 - 15 * q^29 - 4 * q^30 + 33 * q^31 - 29 * q^32 + 3 * q^33 + 28 * q^34 + 4 * q^35 + 15 * q^36 - 29 * q^37 - 15 * q^38 + 13 * q^39 + 3 * q^40 - q^41 - 5 * q^42 - 25 * q^43 - 26 * q^44 - 4 * q^45 - 4 * q^46 - 9 * q^47 - 13 * q^48 + 14 * q^49 - 28 * q^50 + 5 * q^51 - 13 * q^52 - 35 * q^53 + 5 * q^54 + 14 * q^55 + 12 * q^56 - 21 * q^57 - 23 * q^58 + 10 * q^59 + 3 * q^60 + q^61 - 43 * q^62 - 14 * q^63 - 2 * q^64 - 24 * q^65 + 3 * q^66 - 38 * q^67 - 2 * q^68 + 10 * q^69 - 4 * q^70 - 10 * q^71 - 12 * q^72 + 8 * q^73 - 25 * q^74 - 4 * q^75 + 26 * q^76 + 3 * q^77 + 6 * q^78 + 26 * q^79 - 48 * q^80 + 14 * q^81 + 6 * q^82 - 30 * q^83 + 15 * q^84 - 32 * q^85 - 50 * q^86 + 15 * q^87 - 29 * q^88 + 4 * q^89 + 4 * q^90 + 13 * q^91 - 32 * q^92 - 33 * q^93 - 7 * q^94 - 32 * q^95 + 29 * q^96 + 15 * q^97 - 5 * q^98 - 3 * 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 $

$2$	2.33388	1.65030	0.825150	−	0.564914i	$-0.191091\pi$
$2$	2.33388	1.65030	0.825150	+	0.564914i	$0.191091\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.44698	1.72349
$5$	−0.335557	−0.150066	−0.0750328	−	0.997181i	$-0.523906\pi$
$5$	−0.335557	−0.150066	−0.0750328	+	0.997181i	$0.523906\pi$
$6$	−2.33388	−0.952801
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.37706	1.19397
$9$	1.00000	0.333333
$10$	−0.783148	−0.247653
$11$	−5.16521	−1.55737	−0.778684	−	0.627416i	$-0.784113\pi$
$11$	−5.16521	−1.55737	−0.778684	+	0.627416i	$0.784113\pi$
$12$	−3.44698	−0.995057
$13$	−5.99956	−1.66398	−0.831989	−	0.554793i	$-0.812798\pi$
$13$	−5.99956	−1.66398	−0.831989	+	0.554793i	$0.812798\pi$
$14$	−2.33388	−0.623755
$15$	0.335557	0.0866404
$16$	0.987696	0.246924
$17$	7.35302	1.78337	0.891685	−	0.452656i	$-0.149524\pi$
$17$	7.35302	1.78337	0.891685	+	0.452656i	$0.149524\pi$
$18$	2.33388	0.550100
$19$	5.24027	1.20220	0.601101	−	0.799173i	$-0.294729\pi$
$19$	5.24027	1.20220	0.601101	+	0.799173i	$0.294729\pi$
$20$	−1.15666	−0.258636
$21$	1.00000	0.218218
$22$	−12.0550	−2.57013
$23$	1.66716	0.347627	0.173814	−	0.984779i	$-0.444391\pi$
$23$	1.66716	0.347627	0.173814	+	0.984779i	$0.444391\pi$
$24$	−3.37706	−0.689340
$25$	−4.88740	−0.977480
$26$	−14.0022	−2.74606
$27$	−1.00000	−0.192450
$28$	−3.44698	−0.651417
$29$	1.61282	0.299492	0.149746	−	0.988724i	$-0.452154\pi$
$29$	1.61282	0.299492	0.149746	+	0.988724i	$0.452154\pi$
$30$	0.783148	0.142983
$31$	−6.38054	−1.14598	−0.572989	−	0.819563i	$-0.694216\pi$
$31$	−6.38054	−1.14598	−0.572989	+	0.819563i	$0.694216\pi$
$32$	−4.44897	−0.786474
$33$	5.16521	0.899147
$34$	17.1610	2.94309
$35$	0.335557	0.0567195
$36$	3.44698	0.574496
$37$	−11.1071	−1.82600	−0.913000	−	0.407960i	$-0.866240\pi$
$37$	−11.1071	−1.82600	−0.913000	+	0.407960i	$0.866240\pi$
$38$	12.2302	1.98399
$39$	5.99956	0.960698
$40$	−1.13320	−0.179174
$41$	−3.12952	−0.488749	−0.244374	−	0.969681i	$-0.578583\pi$
$41$	−3.12952	−0.488749	−0.244374	+	0.969681i	$0.578583\pi$
$42$	2.33388	0.360125
$43$	−8.78359	−1.33948	−0.669742	−	0.742594i	$-0.733595\pi$
$43$	−8.78359	−1.33948	−0.669742	+	0.742594i	$0.733595\pi$
$44$	−17.8044	−2.68411
$45$	−0.335557	−0.0500219
$46$	3.89095	0.573689
$47$	−11.1208	−1.62214	−0.811071	−	0.584948i	$-0.801115\pi$
$47$	−11.1208	−1.62214	−0.811071	+	0.584948i	$0.801115\pi$
$48$	−0.987696	−0.142562
$49$	1.00000	0.142857
$50$	−11.4066	−1.61314
$51$	−7.35302	−1.02963
$52$	−20.6803	−2.86785
$53$	−4.86128	−0.667748	−0.333874	−	0.942618i	$-0.608356\pi$
$53$	−4.86128	−0.667748	−0.333874	+	0.942618i	$0.608356\pi$
$54$	−2.33388	−0.317600
$55$	1.73322	0.233708
$56$	−3.37706	−0.451279
$57$	−5.24027	−0.694091
$58$	3.76411	0.494252
$59$	14.4216	1.87754	0.938768	−	0.344551i	$-0.111969\pi$
$59$	14.4216	1.87754	0.938768	+	0.344551i	$0.111969\pi$
$60$	1.15666	0.149324
$61$	−9.80157	−1.25496	−0.627481	−	0.778632i	$-0.715914\pi$
$61$	−9.80157	−1.25496	−0.627481	+	0.778632i	$0.715914\pi$
$62$	−14.8914	−1.89121
$63$	−1.00000	−0.125988
$64$	−12.3587	−1.54484
$65$	2.01319	0.249706
$66$	12.0550	1.48386
$67$	−4.47288	−0.546450	−0.273225	−	0.961950i	$-0.588090\pi$
$67$	−4.47288	−0.546450	−0.273225	+	0.961950i	$0.588090\pi$
$68$	25.3457	3.07362
$69$	−1.66716	−0.200703
$70$	0.783148	0.0936041
$71$	−0.675275	−0.0801404	−0.0400702	−	0.999197i	$-0.512758\pi$
$71$	−0.675275	−0.0801404	−0.0400702	+	0.999197i	$0.512758\pi$
$72$	3.37706	0.397991
$73$	13.2597	1.55193	0.775966	−	0.630775i	$-0.217263\pi$
$73$	13.2597	1.55193	0.775966	+	0.630775i	$0.217263\pi$
$74$	−25.9226	−3.01345
$75$	4.88740	0.564349
$76$	18.0631	2.07198
$77$	5.16521	0.588630
$78$	14.0022	1.58544
$79$	10.6397	1.19707	0.598533	−	0.801098i	$-0.295751\pi$
$79$	10.6397	1.19707	0.598533	+	0.801098i	$0.295751\pi$
$80$	−0.331428	−0.0370548
$81$	1.00000	0.111111
$82$	−7.30391	−0.806582
$83$	10.1444	1.11349	0.556746	−	0.830682i	$-0.312050\pi$
$83$	10.1444	1.11349	0.556746	+	0.830682i	$0.312050\pi$
$84$	3.44698	0.376096
$85$	−2.46736	−0.267623
$86$	−20.4998	−2.21055
$87$	−1.61282	−0.172912
$88$	−17.4432	−1.85946
$89$	−15.6055	−1.65417	−0.827087	−	0.562073i	$-0.810004\pi$
$89$	−15.6055	−1.65417	−0.827087	+	0.562073i	$0.810004\pi$
$90$	−0.783148	−0.0825511
$91$	5.99956	0.628924
$92$	5.74667	0.599132
$93$	6.38054	0.661631
$94$	−25.9547	−2.67702
$95$	−1.75841	−0.180409
$96$	4.44897	0.454071
$97$	13.0787	1.32794	0.663968	−	0.747761i	$-0.268871\pi$
$97$	13.0787	1.32794	0.663968	+	0.747761i	$0.268871\pi$
$98$	2.33388	0.235757
$99$	−5.16521	−0.519123

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.m.1.14	✓	14
3.2	odd	2		8001.2.a.p.1.1		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.m.1.14	✓	14	1.1	even	1	trivial
8001.2.a.p.1.1		14	3.2	odd	2

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Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.33388 q^{2} -1.00000 q^{3} +3.44698 q^{4} -0.335557 q^{5} -2.33388 q^{6} -1.00000 q^{7} +3.37706 q^{8} +1.00000 q^{9} -0.783148 q^{10} -5.16521 q^{11} -3.44698 q^{12} -5.99956 q^{13} -2.33388 q^{14} +0.335557 q^{15} +0.987696 q^{16} +7.35302 q^{17} +2.33388 q^{18} +5.24027 q^{19} -1.15666 q^{20} +1.00000 q^{21} -12.0550 q^{22} +1.66716 q^{23} -3.37706 q^{24} -4.88740 q^{25} -14.0022 q^{26} -1.00000 q^{27} -3.44698 q^{28} +1.61282 q^{29} +0.783148 q^{30} -6.38054 q^{31} -4.44897 q^{32} +5.16521 q^{33} +17.1610 q^{34} +0.335557 q^{35} +3.44698 q^{36} -11.1071 q^{37} +12.2302 q^{38} +5.99956 q^{39} -1.13320 q^{40} -3.12952 q^{41} +2.33388 q^{42} -8.78359 q^{43} -17.8044 q^{44} -0.335557 q^{45} +3.89095 q^{46} -11.1208 q^{47} -0.987696 q^{48} +1.00000 q^{49} -11.4066 q^{50} -7.35302 q^{51} -20.6803 q^{52} -4.86128 q^{53} -2.33388 q^{54} +1.73322 q^{55} -3.37706 q^{56} -5.24027 q^{57} +3.76411 q^{58} +14.4216 q^{59} +1.15666 q^{60} -9.80157 q^{61} -14.8914 q^{62} -1.00000 q^{63} -12.3587 q^{64} +2.01319 q^{65} +12.0550 q^{66} -4.47288 q^{67} +25.3457 q^{68} -1.66716 q^{69} +0.783148 q^{70} -0.675275 q^{71} +3.37706 q^{72} +13.2597 q^{73} -25.9226 q^{74} +4.88740 q^{75} +18.0631 q^{76} +5.16521 q^{77} +14.0022 q^{78} +10.6397 q^{79} -0.331428 q^{80} +1.00000 q^{81} -7.30391 q^{82} +10.1444 q^{83} +3.44698 q^{84} -2.46736 q^{85} -20.4998 q^{86} -1.61282 q^{87} -17.4432 q^{88} -15.6055 q^{89} -0.783148 q^{90} +5.99956 q^{91} +5.74667 q^{92} +6.38054 q^{93} -25.9547 q^{94} -1.75841 q^{95} +4.44897 q^{96} +13.0787 q^{97} +2.33388 q^{98} -5.16521 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 5 q^{2} - 14 q^{3} + 15 q^{4} - 4 q^{5} + 5 q^{6} - 14 q^{7} - 12 q^{8} + 14 q^{9} + 4 q^{10} - 3 q^{11} - 15 q^{12} - 13 q^{13} + 5 q^{14} + 4 q^{15} + 13 q^{16} - 5 q^{17} - 5 q^{18} + 21 q^{19}+ \cdots - 3 q^{99}+O(q^{100}) \) 14 * q - 5 * q^2 - 14 * q^3 + 15 * q^4 - 4 * q^5 + 5 * q^6 - 14 * q^7 - 12 * q^8 + 14 * q^9 + 4 * q^10 - 3 * q^11 - 15 * q^12 - 13 * q^13 + 5 * q^14 + 4 * q^15 + 13 * q^16 - 5 * q^17 - 5 * q^18 + 21 * q^19 - 3 * q^20 + 14 * q^21 - 3 * q^22 - 10 * q^23 + 12 * q^24 + 4 * q^25 - 6 * q^26 - 14 * q^27 - 15 * q^28 - 15 * q^29 - 4 * q^30 + 33 * q^31 - 29 * q^32 + 3 * q^33 + 28 * q^34 + 4 * q^35 + 15 * q^36 - 29 * q^37 - 15 * q^38 + 13 * q^39 + 3 * q^40 - q^41 - 5 * q^42 - 25 * q^43 - 26 * q^44 - 4 * q^45 - 4 * q^46 - 9 * q^47 - 13 * q^48 + 14 * q^49 - 28 * q^50 + 5 * q^51 - 13 * q^52 - 35 * q^53 + 5 * q^54 + 14 * q^55 + 12 * q^56 - 21 * q^57 - 23 * q^58 + 10 * q^59 + 3 * q^60 + q^61 - 43 * q^62 - 14 * q^63 - 2 * q^64 - 24 * q^65 + 3 * q^66 - 38 * q^67 - 2 * q^68 + 10 * q^69 - 4 * q^70 - 10 * q^71 - 12 * q^72 + 8 * q^73 - 25 * q^74 - 4 * q^75 + 26 * q^76 + 3 * q^77 + 6 * q^78 + 26 * q^79 - 48 * q^80 + 14 * q^81 + 6 * q^82 - 30 * q^83 + 15 * q^84 - 32 * q^85 - 50 * q^86 + 15 * q^87 - 29 * q^88 + 4 * q^89 + 4 * q^90 + 13 * q^91 - 32 * q^92 - 33 * q^93 - 7 * q^94 - 32 * q^95 + 29 * q^96 + 15 * q^97 - 5 * q^98 - 3 * q^99

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