LMFDB - Embedded newform 2667.2.a.p.1.7

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			1.7
Root			$1.52505$ 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)

$f(q)$	$=$	\(q-1.52505 q^{2} -1.00000 q^{3} +0.325779 q^{4} -1.71268 q^{5} +1.52505 q^{6} +1.00000 q^{7} +2.55327 q^{8} +1.00000 q^{9} +2.61192 q^{10} -1.69211 q^{11} -0.325779 q^{12} -1.16748 q^{13} -1.52505 q^{14} +1.71268 q^{15} -4.54543 q^{16} +5.50471 q^{17} -1.52505 q^{18} -0.828520 q^{19} -0.557955 q^{20} -1.00000 q^{21} +2.58056 q^{22} -6.42297 q^{23} -2.55327 q^{24} -2.06673 q^{25} +1.78047 q^{26} -1.00000 q^{27} +0.325779 q^{28} -2.43481 q^{29} -2.61192 q^{30} -0.583990 q^{31} +1.82546 q^{32} +1.69211 q^{33} -8.39496 q^{34} -1.71268 q^{35} +0.325779 q^{36} +11.0189 q^{37} +1.26353 q^{38} +1.16748 q^{39} -4.37293 q^{40} -0.0442353 q^{41} +1.52505 q^{42} +12.9529 q^{43} -0.551255 q^{44} -1.71268 q^{45} +9.79535 q^{46} +3.58595 q^{47} +4.54543 q^{48} +1.00000 q^{49} +3.15187 q^{50} -5.50471 q^{51} -0.380341 q^{52} -12.7616 q^{53} +1.52505 q^{54} +2.89805 q^{55} +2.55327 q^{56} +0.828520 q^{57} +3.71321 q^{58} +1.61982 q^{59} +0.557955 q^{60} -5.74621 q^{61} +0.890614 q^{62} +1.00000 q^{63} +6.30693 q^{64} +1.99952 q^{65} -2.58056 q^{66} -0.393750 q^{67} +1.79332 q^{68} +6.42297 q^{69} +2.61192 q^{70} +10.0573 q^{71} +2.55327 q^{72} +7.94732 q^{73} -16.8044 q^{74} +2.06673 q^{75} -0.269914 q^{76} -1.69211 q^{77} -1.78047 q^{78} +3.92230 q^{79} +7.78485 q^{80} +1.00000 q^{81} +0.0674610 q^{82} +6.36257 q^{83} -0.325779 q^{84} -9.42780 q^{85} -19.7538 q^{86} +2.43481 q^{87} -4.32042 q^{88} -5.77339 q^{89} +2.61192 q^{90} -1.16748 q^{91} -2.09247 q^{92} +0.583990 q^{93} -5.46876 q^{94} +1.41899 q^{95} -1.82546 q^{96} +15.1754 q^{97} -1.52505 q^{98} -1.69211 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.52505	−1.07837	−0.539187	−	0.842186i	$-0.681268\pi$
$2$	−1.52505	−1.07837	−0.539187	+	0.842186i	$0.681268\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.325779	0.162890
$5$	−1.71268	−0.765933	−0.382966	−	0.923762i	$-0.625098\pi$
$5$	−1.71268	−0.765933	−0.382966	+	0.923762i	$0.625098\pi$
$6$	1.52505	0.622599
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.55327	0.902718
$9$	1.00000	0.333333
$10$	2.61192	0.825962
$11$	−1.69211	−0.510191	−0.255096	−	0.966916i	$-0.582107\pi$
$11$	−1.69211	−0.510191	−0.255096	+	0.966916i	$0.582107\pi$
$12$	−0.325779	−0.0940443
$13$	−1.16748	−0.323801	−0.161901	−	0.986807i	$-0.551762\pi$
$13$	−1.16748	−0.323801	−0.161901	+	0.986807i	$0.551762\pi$
$14$	−1.52505	−0.407587
$15$	1.71268	0.442212
$16$	−4.54543	−1.13636
$17$	5.50471	1.33509	0.667544	−	0.744570i	$-0.267345\pi$
$17$	5.50471	1.33509	0.667544	+	0.744570i	$0.267345\pi$
$18$	−1.52505	−0.359458
$19$	−0.828520	−0.190075	−0.0950377	−	0.995474i	$-0.530297\pi$
$19$	−0.828520	−0.190075	−0.0950377	+	0.995474i	$0.530297\pi$
$20$	−0.557955	−0.124762
$21$	−1.00000	−0.218218
$22$	2.58056	0.550177
$23$	−6.42297	−1.33928	−0.669641	−	0.742685i	$-0.733552\pi$
$23$	−6.42297	−1.33928	−0.669641	+	0.742685i	$0.733552\pi$
$24$	−2.55327	−0.521184
$25$	−2.06673	−0.413347
$26$	1.78047	0.349179
$27$	−1.00000	−0.192450
$28$	0.325779	0.0615664
$29$	−2.43481	−0.452133	−0.226066	−	0.974112i	$-0.572587\pi$
$29$	−2.43481	−0.452133	−0.226066	+	0.974112i	$0.572587\pi$
$30$	−2.61192	−0.476869
$31$	−0.583990	−0.104888	−0.0524438	−	0.998624i	$-0.516701\pi$
$31$	−0.583990	−0.104888	−0.0524438	+	0.998624i	$0.516701\pi$
$32$	1.82546	0.322699
$33$	1.69211	0.294559
$34$	−8.39496	−1.43972
$35$	−1.71268	−0.289495
$36$	0.325779	0.0542965
$37$	11.0189	1.81150	0.905749	−	0.423814i	$-0.139309\pi$
$37$	11.0189	1.81150	0.905749	+	0.423814i	$0.139309\pi$
$38$	1.26353	0.204972
$39$	1.16748	0.186947
$40$	−4.37293	−0.691421
$41$	−0.0442353	−0.00690839	−0.00345419	−	0.999994i	$-0.501100\pi$
$41$	−0.0442353	−0.00690839	−0.00345419	+	0.999994i	$0.501100\pi$
$42$	1.52505	0.235320
$43$	12.9529	1.97529	0.987647	−	0.156695i	$-0.0500839\pi$
$43$	12.9529	1.97529	0.987647	+	0.156695i	$0.0500839\pi$
$44$	−0.551255	−0.0831048
$45$	−1.71268	−0.255311
$46$	9.79535	1.44425
$47$	3.58595	0.523065	0.261533	−	0.965195i	$-0.415772\pi$
$47$	3.58595	0.523065	0.261533	+	0.965195i	$0.415772\pi$
$48$	4.54543	0.656076
$49$	1.00000	0.142857
$50$	3.15187	0.445742
$51$	−5.50471	−0.770814
$52$	−0.380341	−0.0527439
$53$	−12.7616	−1.75295	−0.876473	−	0.481451i	$-0.840110\pi$
$53$	−12.7616	−1.75295	−0.876473	+	0.481451i	$0.840110\pi$
$54$	1.52505	0.207533
$55$	2.89805	0.390772
$56$	2.55327	0.341195
$57$	0.828520	0.109740
$58$	3.71321	0.487568
$59$	1.61982	0.210883	0.105442	−	0.994426i	$-0.466374\pi$
$59$	1.61982	0.210883	0.105442	+	0.994426i	$0.466374\pi$
$60$	0.557955	0.0720316
$61$	−5.74621	−0.735727	−0.367864	−	0.929880i	$-0.619911\pi$
$61$	−5.74621	−0.735727	−0.367864	+	0.929880i	$0.619911\pi$
$62$	0.890614	0.113108
$63$	1.00000	0.125988
$64$	6.30693	0.788366
$65$	1.99952	0.248010
$66$	−2.58056	−0.317645
$67$	−0.393750	−0.0481042	−0.0240521	−	0.999711i	$-0.507657\pi$
$67$	−0.393750	−0.0481042	−0.0240521	+	0.999711i	$0.507657\pi$
$68$	1.79332	0.217472
$69$	6.42297	0.773234
$70$	2.61192	0.312184
$71$	10.0573	1.19358	0.596792	−	0.802396i	$-0.296442\pi$
$71$	10.0573	1.19358	0.596792	+	0.802396i	$0.296442\pi$
$72$	2.55327	0.300906
$73$	7.94732	0.930163	0.465081	−	0.885268i	$-0.346025\pi$
$73$	7.94732	0.930163	0.465081	+	0.885268i	$0.346025\pi$
$74$	−16.8044	−1.95347
$75$	2.06673	0.238646
$76$	−0.269914	−0.0309613
$77$	−1.69211	−0.192834
$78$	−1.78047	−0.201599
$79$	3.92230	0.441293	0.220647	−	0.975354i	$-0.429183\pi$
$79$	3.92230	0.441293	0.220647	+	0.975354i	$0.429183\pi$
$80$	7.78485	0.870373
$81$	1.00000	0.111111
$82$	0.0674610	0.00744982
$83$	6.36257	0.698382	0.349191	−	0.937051i	$-0.386456\pi$
$83$	6.36257	0.698382	0.349191	+	0.937051i	$0.386456\pi$
$84$	−0.325779	−0.0355454
$85$	−9.42780	−1.02259
$86$	−19.7538	−2.13010
$87$	2.43481	0.261039
$88$	−4.32042	−0.460559
$89$	−5.77339	−0.611978	−0.305989	−	0.952035i	$-0.598987\pi$
$89$	−5.77339	−0.611978	−0.305989	+	0.952035i	$0.598987\pi$
$90$	2.61192	0.275321
$91$	−1.16748	−0.122385
$92$	−2.09247	−0.218155
$93$	0.583990	0.0605569
$94$	−5.46876	−0.564060
$95$	1.41899	0.145585
$96$	−1.82546	−0.186310
$97$	15.1754	1.54083	0.770415	−	0.637542i	$-0.220049\pi$
$97$	15.1754	1.54083	0.770415	+	0.637542i	$0.220049\pi$
$98$	−1.52505	−0.154053
$99$	−1.69211	−0.170064

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.1.7	✓	18
3.2	odd	2		8001.2.a.u.1.12		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.7	✓	18	1.1	even	1	trivial
8001.2.a.u.1.12		18	3.2	odd	2

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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

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

\(f(q)\)	\(=\)	\(q-1.52505 q^{2} -1.00000 q^{3} +0.325779 q^{4} -1.71268 q^{5} +1.52505 q^{6} +1.00000 q^{7} +2.55327 q^{8} +1.00000 q^{9} +2.61192 q^{10} -1.69211 q^{11} -0.325779 q^{12} -1.16748 q^{13} -1.52505 q^{14} +1.71268 q^{15} -4.54543 q^{16} +5.50471 q^{17} -1.52505 q^{18} -0.828520 q^{19} -0.557955 q^{20} -1.00000 q^{21} +2.58056 q^{22} -6.42297 q^{23} -2.55327 q^{24} -2.06673 q^{25} +1.78047 q^{26} -1.00000 q^{27} +0.325779 q^{28} -2.43481 q^{29} -2.61192 q^{30} -0.583990 q^{31} +1.82546 q^{32} +1.69211 q^{33} -8.39496 q^{34} -1.71268 q^{35} +0.325779 q^{36} +11.0189 q^{37} +1.26353 q^{38} +1.16748 q^{39} -4.37293 q^{40} -0.0442353 q^{41} +1.52505 q^{42} +12.9529 q^{43} -0.551255 q^{44} -1.71268 q^{45} +9.79535 q^{46} +3.58595 q^{47} +4.54543 q^{48} +1.00000 q^{49} +3.15187 q^{50} -5.50471 q^{51} -0.380341 q^{52} -12.7616 q^{53} +1.52505 q^{54} +2.89805 q^{55} +2.55327 q^{56} +0.828520 q^{57} +3.71321 q^{58} +1.61982 q^{59} +0.557955 q^{60} -5.74621 q^{61} +0.890614 q^{62} +1.00000 q^{63} +6.30693 q^{64} +1.99952 q^{65} -2.58056 q^{66} -0.393750 q^{67} +1.79332 q^{68} +6.42297 q^{69} +2.61192 q^{70} +10.0573 q^{71} +2.55327 q^{72} +7.94732 q^{73} -16.8044 q^{74} +2.06673 q^{75} -0.269914 q^{76} -1.69211 q^{77} -1.78047 q^{78} +3.92230 q^{79} +7.78485 q^{80} +1.00000 q^{81} +0.0674610 q^{82} +6.36257 q^{83} -0.325779 q^{84} -9.42780 q^{85} -19.7538 q^{86} +2.43481 q^{87} -4.32042 q^{88} -5.77339 q^{89} +2.61192 q^{90} -1.16748 q^{91} -2.09247 q^{92} +0.583990 q^{93} -5.46876 q^{94} +1.41899 q^{95} -1.82546 q^{96} +15.1754 q^{97} -1.52505 q^{98} -1.69211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 6 q^{2} - 18 q^{3} + 22 q^{4} - 10 q^{5} + 6 q^{6} + 18 q^{7} - 21 q^{8} + 18 q^{9} - 4 q^{10} - 9 q^{11} - 22 q^{12} - 25 q^{13} - 6 q^{14} + 10 q^{15} + 34 q^{16} - 17 q^{17} - 6 q^{18} - 5 q^{19}+ \cdots - 9 q^{99}+O(q^{100}) \) 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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