LMFDB - Embedded newform 7942.2.a.bi.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7942,2,Mod(1,7942)] 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("7942.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7942, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,-3,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$63.4171892853$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.14510 q^{3} +1.00000 q^{4} -1.60147 q^{5} +2.14510 q^{6} +2.89167 q^{7} +1.00000 q^{8} +1.60147 q^{9} -1.60147 q^{10} -1.00000 q^{11} +2.14510 q^{12} -4.89167 q^{13} +2.89167 q^{14} -3.43531 q^{15} +1.00000 q^{16} -5.74657 q^{17} +1.60147 q^{18} -1.60147 q^{20} +6.20293 q^{21} -1.00000 q^{22} -7.74657 q^{23} +2.14510 q^{24} -2.43531 q^{25} -4.89167 q^{26} -3.00000 q^{27} +2.89167 q^{28} -5.34803 q^{29} -3.43531 q^{30} +7.43531 q^{31} +1.00000 q^{32} -2.14510 q^{33} -5.74657 q^{34} -4.63091 q^{35} +1.60147 q^{36} +7.49314 q^{37} -10.4931 q^{39} -1.60147 q^{40} +2.05783 q^{41} +6.20293 q^{42} -10.6887 q^{43} -1.00000 q^{44} -2.56469 q^{45} -7.74657 q^{46} -7.08727 q^{47} +2.14510 q^{48} +1.36176 q^{49} -2.43531 q^{50} -12.3270 q^{51} -4.89167 q^{52} -8.32698 q^{53} -3.00000 q^{54} +1.60147 q^{55} +2.89167 q^{56} -5.34803 q^{58} +2.54364 q^{59} -3.43531 q^{60} +13.4931 q^{61} +7.43531 q^{62} +4.63091 q^{63} +1.00000 q^{64} +7.83384 q^{65} -2.14510 q^{66} +10.3848 q^{67} -5.74657 q^{68} -16.6172 q^{69} -4.63091 q^{70} -14.9789 q^{71} +1.60147 q^{72} -3.74657 q^{73} +7.49314 q^{74} -5.22399 q^{75} -2.89167 q^{77} -10.4931 q^{78} +8.69607 q^{79} -1.60147 q^{80} -11.2397 q^{81} +2.05783 q^{82} -6.10833 q^{83} +6.20293 q^{84} +9.20293 q^{85} -10.6887 q^{86} -11.4721 q^{87} -1.00000 q^{88} -3.20293 q^{89} -2.56469 q^{90} -14.1451 q^{91} -7.74657 q^{92} +15.9495 q^{93} -7.08727 q^{94} +2.14510 q^{96} +6.98627 q^{97} +1.36176 q^{98} -1.60147 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{11} - 6 q^{14} + 9 q^{15} + 3 q^{16} - 9 q^{17} + 3 q^{18} - 3 q^{20} + 15 q^{21} - 3 q^{22} - 15 q^{23} + 12 q^{25} - 9 q^{27}+ \cdots - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 6 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^11 - 6 * q^14 + 9 * q^15 + 3 * q^16 - 9 * q^17 + 3 * q^18 - 3 * q^20 + 15 * q^21 - 3 * q^22 - 15 * q^23 + 12 * q^25 - 9 * q^27 - 6 * q^28 - 6 * q^29 + 9 * q^30 + 3 * q^31 + 3 * q^32 - 9 * q^34 + 3 * q^36 + 6 * q^37 - 15 * q^39 - 3 * q^40 + 9 * q^41 + 15 * q^42 - 21 * q^43 - 3 * q^44 - 27 * q^45 - 15 * q^46 - 12 * q^47 + 27 * q^49 + 12 * q^50 - 3 * q^51 + 9 * q^53 - 9 * q^54 + 3 * q^55 - 6 * q^56 - 6 * q^58 + 3 * q^59 + 9 * q^60 + 24 * q^61 + 3 * q^62 + 3 * q^64 + 6 * q^65 - 9 * q^68 - 3 * q^69 - 21 * q^71 + 3 * q^72 - 3 * q^73 + 6 * q^74 - 36 * q^75 + 6 * q^77 - 15 * q^78 + 6 * q^79 - 3 * q^80 - 9 * q^81 + 9 * q^82 - 33 * q^83 + 15 * q^84 + 24 * q^85 - 21 * q^86 + 6 * q^87 - 3 * q^88 - 6 * q^89 - 27 * q^90 - 36 * q^91 - 15 * q^92 + 36 * q^93 - 12 * q^94 - 12 * q^97 + 27 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.14510	1.23848	0.619238	−	0.785204i	$-0.287442\pi$
$3$	2.14510	1.23848	0.619238	+	0.785204i	$0.287442\pi$
$4$	1.00000	0.500000
$5$	−1.60147	−0.716197	−0.358099	−	0.933684i	$-0.616575\pi$
$5$	−1.60147	−0.716197	−0.358099	+	0.933684i	$0.616575\pi$
$6$	2.14510	0.875735
$7$	2.89167	1.09295	0.546474	−	0.837476i	$-0.315970\pi$
$7$	2.89167	1.09295	0.546474	+	0.837476i	$0.315970\pi$
$8$	1.00000	0.353553
$9$	1.60147	0.533822
$10$	−1.60147	−0.506428
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.14510	0.619238
$13$	−4.89167	−1.35671	−0.678353	−	0.734736i	$-0.737306\pi$
$13$	−4.89167	−1.35671	−0.678353	+	0.734736i	$0.737306\pi$
$14$	2.89167	0.772832
$15$	−3.43531	−0.886993
$16$	1.00000	0.250000
$17$	−5.74657	−1.39375	−0.696874	−	0.717194i	$-0.745426\pi$
$17$	−5.74657	−1.39375	−0.696874	+	0.717194i	$0.745426\pi$
$18$	1.60147	0.377469
$19$	0	0
$19$	0	0
$20$	−1.60147	−0.358099
$21$	6.20293	1.35359
$22$	−1.00000	−0.213201
$23$	−7.74657	−1.61527	−0.807636	−	0.589682i	$-0.799253\pi$
$23$	−7.74657	−1.61527	−0.807636	+	0.589682i	$0.799253\pi$
$24$	2.14510	0.437867
$25$	−2.43531	−0.487062
$26$	−4.89167	−0.959336
$27$	−3.00000	−0.577350
$28$	2.89167	0.546474
$29$	−5.34803	−0.993105	−0.496552	−	0.868007i	$-0.665401\pi$
$29$	−5.34803	−0.993105	−0.496552	+	0.868007i	$0.665401\pi$
$30$	−3.43531	−0.627199
$31$	7.43531	1.33542	0.667710	−	0.744421i	$-0.267274\pi$
$31$	7.43531	1.33542	0.667710	+	0.744421i	$0.267274\pi$
$32$	1.00000	0.176777
$33$	−2.14510	−0.373414
$34$	−5.74657	−0.985528
$35$	−4.63091	−0.782767
$36$	1.60147	0.266911
$37$	7.49314	1.23186	0.615932	−	0.787799i	$-0.288780\pi$
$37$	7.49314	1.23186	0.615932	+	0.787799i	$0.288780\pi$
$38$	0	0
$39$	−10.4931	−1.68025
$40$	−1.60147	−0.253214
$41$	2.05783	0.321379	0.160689	−	0.987005i	$-0.448628\pi$
$41$	2.05783	0.321379	0.160689	+	0.987005i	$0.448628\pi$
$42$	6.20293	0.957133
$43$	−10.6887	−1.63002	−0.815009	−	0.579449i	$-0.803268\pi$
$43$	−10.6887	−1.63002	−0.815009	+	0.579449i	$0.803268\pi$
$44$	−1.00000	−0.150756
$45$	−2.56469	−0.382322
$46$	−7.74657	−1.14217
$47$	−7.08727	−1.03379	−0.516893	−	0.856050i	$-0.672911\pi$
$47$	−7.08727	−1.03379	−0.516893	+	0.856050i	$0.672911\pi$
$48$	2.14510	0.309619
$49$	1.36176	0.194537
$50$	−2.43531	−0.344405
$51$	−12.3270	−1.72612
$52$	−4.89167	−0.678353
$53$	−8.32698	−1.14380	−0.571899	−	0.820324i	$-0.693793\pi$
$53$	−8.32698	−1.14380	−0.571899	+	0.820324i	$0.693793\pi$
$54$	−3.00000	−0.408248
$55$	1.60147	0.215942
$56$	2.89167	0.386416
$57$	0	0
$58$	−5.34803	−0.702231
$59$	2.54364	0.331153	0.165577	−	0.986197i	$-0.447051\pi$
$59$	2.54364	0.331153	0.165577	+	0.986197i	$0.447051\pi$
$60$	−3.43531	−0.443496
$61$	13.4931	1.72762	0.863810	−	0.503818i	$-0.168072\pi$
$61$	13.4931	1.72762	0.863810	+	0.503818i	$0.168072\pi$
$62$	7.43531	0.944285
$63$	4.63091	0.583440
$64$	1.00000	0.125000
$65$	7.83384	0.971669
$66$	−2.14510	−0.264044
$67$	10.3848	1.26871	0.634353	−	0.773043i	$-0.281267\pi$
$67$	10.3848	1.26871	0.634353	+	0.773043i	$0.281267\pi$
$68$	−5.74657	−0.696874
$69$	−16.6172	−2.00047
$70$	−4.63091	−0.553500
$71$	−14.9789	−1.77767	−0.888837	−	0.458224i	$-0.848486\pi$
$71$	−14.9789	−1.77767	−0.888837	+	0.458224i	$0.848486\pi$
$72$	1.60147	0.188735
$73$	−3.74657	−0.438503	−0.219251	−	0.975668i	$-0.570361\pi$
$73$	−3.74657	−0.438503	−0.219251	+	0.975668i	$0.570361\pi$
$74$	7.49314	0.871059
$75$	−5.22399	−0.603214
$76$	0	0
$77$	−2.89167	−0.329536
$78$	−10.4931	−1.18811
$79$	8.69607	0.978384	0.489192	−	0.872176i	$-0.337292\pi$
$79$	8.69607	0.978384	0.489192	+	0.872176i	$0.337292\pi$
$80$	−1.60147	−0.179049
$81$	−11.2397	−1.24886
$82$	2.05783	0.227249
$83$	−6.10833	−0.670476	−0.335238	−	0.942133i	$-0.608817\pi$
$83$	−6.10833	−0.670476	−0.335238	+	0.942133i	$0.608817\pi$
$84$	6.20293	0.676795
$85$	9.20293	0.998198
$86$	−10.6887	−1.15260
$87$	−11.4721	−1.22994
$88$	−1.00000	−0.106600
$89$	−3.20293	−0.339510	−0.169755	−	0.985486i	$-0.554298\pi$
$89$	−3.20293	−0.339510	−0.169755	+	0.985486i	$0.554298\pi$
$90$	−2.56469	−0.270342
$91$	−14.1451	−1.48281
$92$	−7.74657	−0.807636
$93$	15.9495	1.65389
$94$	−7.08727	−0.730997
$95$	0	0
$96$	2.14510	0.218934
$97$	6.98627	0.709349	0.354674	−	0.934990i	$-0.384592\pi$
$97$	6.98627	0.709349	0.354674	+	0.934990i	$0.384592\pi$
$98$	1.36176	0.137559
$99$	−1.60147	−0.160953

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	7942.2.a.bi.1.3		3
19.18	odd	2		418.2.a.g.1.1	✓	3
57.56	even	2		3762.2.a.bg.1.2		3
76.75	even	2		3344.2.a.q.1.3		3
209.208	even	2		4598.2.a.bo.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.g.1.1	✓	3	19.18	odd	2
3344.2.a.q.1.3		3	76.75	even	2
3762.2.a.bg.1.2		3	57.56	even	2
4598.2.a.bo.1.1		3	209.208	even	2
7942.2.a.bi.1.3		3	1.1	even	1	trivial

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Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.14510 q^{3} +1.00000 q^{4} -1.60147 q^{5} +2.14510 q^{6} +2.89167 q^{7} +1.00000 q^{8} +1.60147 q^{9} -1.60147 q^{10} -1.00000 q^{11} +2.14510 q^{12} -4.89167 q^{13} +2.89167 q^{14} -3.43531 q^{15} +1.00000 q^{16} -5.74657 q^{17} +1.60147 q^{18} -1.60147 q^{20} +6.20293 q^{21} -1.00000 q^{22} -7.74657 q^{23} +2.14510 q^{24} -2.43531 q^{25} -4.89167 q^{26} -3.00000 q^{27} +2.89167 q^{28} -5.34803 q^{29} -3.43531 q^{30} +7.43531 q^{31} +1.00000 q^{32} -2.14510 q^{33} -5.74657 q^{34} -4.63091 q^{35} +1.60147 q^{36} +7.49314 q^{37} -10.4931 q^{39} -1.60147 q^{40} +2.05783 q^{41} +6.20293 q^{42} -10.6887 q^{43} -1.00000 q^{44} -2.56469 q^{45} -7.74657 q^{46} -7.08727 q^{47} +2.14510 q^{48} +1.36176 q^{49} -2.43531 q^{50} -12.3270 q^{51} -4.89167 q^{52} -8.32698 q^{53} -3.00000 q^{54} +1.60147 q^{55} +2.89167 q^{56} -5.34803 q^{58} +2.54364 q^{59} -3.43531 q^{60} +13.4931 q^{61} +7.43531 q^{62} +4.63091 q^{63} +1.00000 q^{64} +7.83384 q^{65} -2.14510 q^{66} +10.3848 q^{67} -5.74657 q^{68} -16.6172 q^{69} -4.63091 q^{70} -14.9789 q^{71} +1.60147 q^{72} -3.74657 q^{73} +7.49314 q^{74} -5.22399 q^{75} -2.89167 q^{77} -10.4931 q^{78} +8.69607 q^{79} -1.60147 q^{80} -11.2397 q^{81} +2.05783 q^{82} -6.10833 q^{83} +6.20293 q^{84} +9.20293 q^{85} -10.6887 q^{86} -11.4721 q^{87} -1.00000 q^{88} -3.20293 q^{89} -2.56469 q^{90} -14.1451 q^{91} -7.74657 q^{92} +15.9495 q^{93} -7.08727 q^{94} +2.14510 q^{96} +6.98627 q^{97} +1.36176 q^{98} -1.60147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{11} - 6 q^{14} + 9 q^{15} + 3 q^{16} - 9 q^{17} + 3 q^{18} - 3 q^{20} + 15 q^{21} - 3 q^{22} - 15 q^{23} + 12 q^{25} - 9 q^{27}+ \cdots - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 6 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^11 - 6 * q^14 + 9 * q^15 + 3 * q^16 - 9 * q^17 + 3 * q^18 - 3 * q^20 + 15 * q^21 - 3 * q^22 - 15 * q^23 + 12 * q^25 - 9 * q^27 - 6 * q^28 - 6 * q^29 + 9 * q^30 + 3 * q^31 + 3 * q^32 - 9 * q^34 + 3 * q^36 + 6 * q^37 - 15 * q^39 - 3 * q^40 + 9 * q^41 + 15 * q^42 - 21 * q^43 - 3 * q^44 - 27 * q^45 - 15 * q^46 - 12 * q^47 + 27 * q^49 + 12 * q^50 - 3 * q^51 + 9 * q^53 - 9 * q^54 + 3 * q^55 - 6 * q^56 - 6 * q^58 + 3 * q^59 + 9 * q^60 + 24 * q^61 + 3 * q^62 + 3 * q^64 + 6 * q^65 - 9 * q^68 - 3 * q^69 - 21 * q^71 + 3 * q^72 - 3 * q^73 + 6 * q^74 - 36 * q^75 + 6 * q^77 - 15 * q^78 + 6 * q^79 - 3 * q^80 - 9 * q^81 + 9 * q^82 - 33 * q^83 + 15 * q^84 + 24 * q^85 - 21 * q^86 + 6 * q^87 - 3 * q^88 - 6 * q^89 - 27 * q^90 - 36 * q^91 - 15 * q^92 + 36 * q^93 - 12 * q^94 - 12 * q^97 + 27 * q^98 - 3 * q^99

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