LMFDB - Embedded newform 11.8.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [11,8,Mod(1,11)] 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("11.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(11, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	11.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.43623528033$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{15}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 15 $ x^2 - 15
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.87298$ of defining polynomial
Character	$\chi$	$=$	11.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-11.7460 q^{2} +43.4758 q^{3} +9.96773 q^{4} -389.919 q^{5} -510.665 q^{6} -1249.17 q^{7} +1386.40 q^{8} -296.855 q^{9} +4579.98 q^{10} +1331.00 q^{11} +433.355 q^{12} -3840.41 q^{13} +14672.7 q^{14} -16952.1 q^{15} -17560.5 q^{16} +24162.7 q^{17} +3486.85 q^{18} +5458.47 q^{19} -3886.61 q^{20} -54308.6 q^{21} -15633.9 q^{22} -63973.2 q^{23} +60275.0 q^{24} +73912.1 q^{25} +45109.3 q^{26} -107988. q^{27} -12451.4 q^{28} +178351. q^{29} +199118. q^{30} -185129. q^{31} +28805.6 q^{32} +57866.3 q^{33} -283814. q^{34} +487075. q^{35} -2958.97 q^{36} -409817. q^{37} -64115.0 q^{38} -166965. q^{39} -540585. q^{40} -675273. q^{41} +637907. q^{42} +38903.5 q^{43} +13267.1 q^{44} +115749. q^{45} +751427. q^{46} +949397. q^{47} -763457. q^{48} +736881. q^{49} -868169. q^{50} +1.05049e6 q^{51} -38280.2 q^{52} +294003. q^{53} +1.26842e6 q^{54} -518983. q^{55} -1.73185e6 q^{56} +237311. q^{57} -2.09490e6 q^{58} -87803.7 q^{59} -168974. q^{60} -2.78573e6 q^{61} +2.17452e6 q^{62} +370822. q^{63} +1.90940e6 q^{64} +1.49745e6 q^{65} -679696. q^{66} +2.95364e6 q^{67} +240847. q^{68} -2.78129e6 q^{69} -5.72117e6 q^{70} -4.08380e6 q^{71} -411560. q^{72} +1.95525e6 q^{73} +4.81370e6 q^{74} +3.21339e6 q^{75} +54408.6 q^{76} -1.66264e6 q^{77} +1.96116e6 q^{78} -608158. q^{79} +6.84718e6 q^{80} -4.04562e6 q^{81} +7.93173e6 q^{82} +214613. q^{83} -541334. q^{84} -9.42151e6 q^{85} -456959. q^{86} +7.75394e6 q^{87} +1.84530e6 q^{88} -8.30366e6 q^{89} -1.35959e6 q^{90} +4.79732e6 q^{91} -637668. q^{92} -8.04863e6 q^{93} -1.11516e7 q^{94} -2.12836e6 q^{95} +1.25235e6 q^{96} -1.38909e6 q^{97} -8.65538e6 q^{98} -395114. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} - 6 q^{3} - 104 q^{4} - 470 q^{5} - 696 q^{6} - 1228 q^{7} + 480 q^{8} - 36 q^{9} + 4280 q^{10} + 2662 q^{11} + 6072 q^{12} + 344 q^{13} + 14752 q^{14} - 12990 q^{15} - 6368 q^{16} - 8468 q^{17}+ \cdots - 47916 q^{99}+O(q^{100}) $ 2 * q - 8 * q^2 - 6 * q^3 - 104 * q^4 - 470 * q^5 - 696 * q^6 - 1228 * q^7 + 480 * q^8 - 36 * q^9 + 4280 * q^10 + 2662 * q^11 + 6072 * q^12 + 344 * q^13 + 14752 * q^14 - 12990 * q^15 - 6368 * q^16 - 8468 * q^17 + 4464 * q^18 - 35280 * q^19 + 5240 * q^20 - 55356 * q^21 - 10648 * q^22 - 61486 * q^23 + 105120 * q^24 + 2200 * q^25 + 60784 * q^26 - 12690 * q^27 - 14864 * q^28 + 179040 * q^29 + 213960 * q^30 - 57166 * q^31 + 186752 * q^32 - 7986 * q^33 - 406048 * q^34 + 485380 * q^35 - 32688 * q^36 - 877698 * q^37 - 216720 * q^38 - 373992 * q^39 - 468000 * q^40 - 283616 * q^41 + 633984 * q^42 + 275484 * q^43 - 138424 * q^44 + 94860 * q^45 + 760744 * q^46 + 1662512 * q^47 - 1317216 * q^48 - 86214 * q^49 - 1136800 * q^50 + 2664924 * q^51 - 515168 * q^52 + 1616484 * q^53 + 1625400 * q^54 - 625570 * q^55 - 1751040 * q^56 + 2252880 * q^57 - 2092320 * q^58 - 2454130 * q^59 - 620520 * q^60 - 6019176 * q^61 + 2653864 * q^62 + 376344 * q^63 + 1068416 * q^64 + 1162360 * q^65 - 926376 * q^66 - 174698 * q^67 + 3959696 * q^68 - 2904342 * q^69 - 5727520 * q^70 - 1151466 * q^71 - 648000 * q^72 + 885944 * q^73 + 3061032 * q^74 + 6761400 * q^75 + 4697280 * q^76 - 1634468 * q^77 + 1185648 * q^78 + 3801460 * q^79 + 5950880 * q^80 - 9331038 * q^81 + 9398864 * q^82 - 2282916 * q^83 - 421968 * q^84 - 6808420 * q^85 + 429264 * q^86 + 7719840 * q^87 + 638880 * q^88 - 13481970 * q^89 - 1437840 * q^90 + 4885904 * q^91 - 921128 * q^92 - 14379702 * q^93 - 8480288 * q^94 + 1134000 * q^95 - 6562176 * q^96 - 68078 * q^97 - 11738664 * q^98 - 47916 * 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 $

$2$	−11.7460	−1.03821	−0.519103	−	0.854712i	$-0.673734\pi$
$2$	−11.7460	−1.03821	−0.519103	+	0.854712i	$0.673734\pi$
$3$	43.4758	0.929658	0.464829	−	0.885400i	$-0.346116\pi$
$3$	43.4758	0.929658	0.464829	+	0.885400i	$0.346116\pi$
$4$	9.96773	0.0778729
$5$	−389.919	−1.39502	−0.697509	−	0.716576i	$-0.745708\pi$
$5$	−389.919	−1.39502	−0.697509	+	0.716576i	$0.745708\pi$
$6$	−510.665	−0.965177
$7$	−1249.17	−1.37651	−0.688253	−	0.725471i	$-0.741622\pi$
$7$	−1249.17	−1.37651	−0.688253	+	0.725471i	$0.741622\pi$
$8$	1386.40	0.957358
$9$	−296.855	−0.135736
$10$	4579.98	1.44832
$11$	1331.00	0.301511
$11$	1331.00	0.301511
$12$	433.355	0.0723952
$13$	−3840.41	−0.484815	−0.242407	−	0.970175i	$-0.577937\pi$
$13$	−3840.41	−0.484815	−0.242407	+	0.970175i	$0.577937\pi$
$14$	14672.7	1.42910
$15$	−16952.1	−1.29689
$16$	−17560.5	−1.07181
$17$	24162.7	1.19282	0.596409	−	0.802680i	$-0.296593\pi$
$17$	24162.7	1.19282	0.596409	+	0.802680i	$0.296593\pi$
$18$	3486.85	0.140922
$19$	5458.47	0.182572	0.0912859	−	0.995825i	$-0.470902\pi$
$19$	5458.47	0.182572	0.0912859	+	0.995825i	$0.470902\pi$
$20$	−3886.61	−0.108634
$21$	−54308.6	−1.27968
$22$	−15633.9	−0.313031
$23$	−63973.2	−1.09635	−0.548177	−	0.836362i	$-0.684678\pi$
$23$	−63973.2	−1.09635	−0.548177	+	0.836362i	$0.684678\pi$
$24$	60275.0	0.890016
$25$	73912.1	0.946075
$26$	45109.3	0.503338
$27$	−107988.	−1.05585
$28$	−12451.4	−0.107193
$29$	178351.	1.35794	0.678972	−	0.734164i	$-0.262426\pi$
$29$	178351.	1.35794	0.678972	+	0.734164i	$0.262426\pi$
$30$	199118.	1.34644
$31$	−185129.	−1.11611	−0.558057	−	0.829803i	$-0.688453\pi$
$31$	−185129.	−1.11611	−0.558057	+	0.829803i	$0.688453\pi$
$32$	28805.6	0.155400
$33$	57866.3	0.280302
$34$	−283814.	−1.23839
$35$	487075.	1.92025
$36$	−2958.97	−0.0105702
$37$	−409817.	−1.33010	−0.665050	−	0.746799i	$-0.731590\pi$
$37$	−409817.	−1.33010	−0.665050	+	0.746799i	$0.731590\pi$
$38$	−64115.0	−0.189547
$39$	−166965.	−0.450712
$40$	−540585.	−1.33553
$41$	−675273.	−1.53016	−0.765078	−	0.643938i	$-0.777300\pi$
$41$	−675273.	−1.53016	−0.765078	+	0.643938i	$0.777300\pi$
$42$	637907.	1.32857
$43$	38903.5	0.0746189	0.0373094	−	0.999304i	$-0.488121\pi$
$43$	38903.5	0.0746189	0.0373094	+	0.999304i	$0.488121\pi$
$44$	13267.1	0.0234796
$45$	115749.	0.189354
$46$	751427.	1.13824
$47$	949397.	1.33385	0.666923	−	0.745127i	$-0.267611\pi$
$47$	949397.	1.33385	0.666923	+	0.745127i	$0.267611\pi$
$48$	−763457.	−0.996416
$49$	736881.	0.894769
$50$	−868169.	−0.982221
$51$	1.05049e6	1.10891
$52$	−38280.2	−0.0377539
$53$	294003.	0.271260	0.135630	−	0.990760i	$-0.456694\pi$
$53$	294003.	0.271260	0.135630	+	0.990760i	$0.456694\pi$
$54$	1.26842e6	1.09619
$55$	−518983.	−0.420614
$56$	−1.73185e6	−1.31781
$57$	237311.	0.169729
$58$	−2.09490e6	−1.40983
$59$	−87803.7	−0.0556584	−0.0278292	−	0.999613i	$-0.508859\pi$
$59$	−87803.7	−0.0556584	−0.0278292	+	0.999613i	$0.508859\pi$
$60$	−168974.	−0.100993
$61$	−2.78573e6	−1.57139	−0.785696	−	0.618613i	$-0.787695\pi$
$61$	−2.78573e6	−1.57139	−0.785696	+	0.618613i	$0.787695\pi$
$62$	2.17452e6	1.15876
$63$	370822.	0.186842
$64$	1.90940e6	0.910471
$65$	1.49745e6	0.676325
$66$	−679696.	−0.291012
$67$	2.95364e6	1.19976	0.599882	−	0.800089i	$-0.295214\pi$
$67$	2.95364e6	1.19976	0.599882	+	0.800089i	$0.295214\pi$
$68$	240847.	0.0928883
$69$	−2.78129e6	−1.01923
$70$	−5.72117e6	−1.99362
$71$	−4.08380e6	−1.35413	−0.677065	−	0.735923i	$-0.736749\pi$
$71$	−4.08380e6	−1.35413	−0.677065	+	0.735923i	$0.736749\pi$
$72$	−411560.	−0.129948
$73$	1.95525e6	0.588263	0.294132	−	0.955765i	$-0.404970\pi$
$73$	1.95525e6	0.588263	0.294132	+	0.955765i	$0.404970\pi$
$74$	4.81370e6	1.38092
$75$	3.21339e6	0.879526
$76$	54408.6	0.0142174
$77$	−1.66264e6	−0.415032
$78$	1.96116e6	0.467932
$79$	−608158.	−0.138778	−0.0693891	−	0.997590i	$-0.522105\pi$
$79$	−608158.	−0.138778	−0.0693891	+	0.997590i	$0.522105\pi$
$80$	6.84718e6	1.49519
$81$	−4.04562e6	−0.845840
$82$	7.93173e6	1.58862
$83$	214613.	0.0411986	0.0205993	−	0.999788i	$-0.493443\pi$
$83$	214613.	0.0411986	0.0205993	+	0.999788i	$0.493443\pi$
$84$	−541334.	−0.0996524
$85$	−9.42151e6	−1.66400
$86$	−456959.	−0.0774698
$87$	7.75394e6	1.26242
$88$	1.84530e6	0.288654
$89$	−8.30366e6	−1.24855	−0.624273	−	0.781206i	$-0.714605\pi$
$89$	−8.30366e6	−1.24855	−0.624273	+	0.781206i	$0.714605\pi$
$90$	−1.35959e6	−0.196589
$91$	4.79732e6	0.667351
$92$	−637668.	−0.0853763
$93$	−8.04863e6	−1.03760
$94$	−1.11516e7	−1.38481
$95$	−2.12836e6	−0.254691
$96$	1.25235e6	0.144469
$97$	−1.38909e6	−0.154536	−0.0772678	−	0.997010i	$-0.524620\pi$
$97$	−1.38909e6	−0.154536	−0.0772678	+	0.997010i	$0.524620\pi$
$98$	−8.65538e6	−0.928955
$99$	−395114.	−0.0409260

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	11.8.a.a.1.1	✓	2
3.2	odd	2		99.8.a.c.1.2		2
4.3	odd	2		176.8.a.d.1.1		2
5.4	even	2		275.8.a.a.1.2		2
7.6	odd	2		539.8.a.a.1.1		2
11.10	odd	2		121.8.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.8.a.a.1.1	✓	2	1.1	even	1	trivial
99.8.a.c.1.2		2	3.2	odd	2
121.8.a.b.1.2		2	11.10	odd	2
176.8.a.d.1.1		2	4.3	odd	2
275.8.a.a.1.2		2	5.4	even	2
539.8.a.a.1.1		2	7.6	odd	2

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

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	11.a (trivial)

\(f(q)\)	\(=\)	\(q-11.7460 q^{2} +43.4758 q^{3} +9.96773 q^{4} -389.919 q^{5} -510.665 q^{6} -1249.17 q^{7} +1386.40 q^{8} -296.855 q^{9} +4579.98 q^{10} +1331.00 q^{11} +433.355 q^{12} -3840.41 q^{13} +14672.7 q^{14} -16952.1 q^{15} -17560.5 q^{16} +24162.7 q^{17} +3486.85 q^{18} +5458.47 q^{19} -3886.61 q^{20} -54308.6 q^{21} -15633.9 q^{22} -63973.2 q^{23} +60275.0 q^{24} +73912.1 q^{25} +45109.3 q^{26} -107988. q^{27} -12451.4 q^{28} +178351. q^{29} +199118. q^{30} -185129. q^{31} +28805.6 q^{32} +57866.3 q^{33} -283814. q^{34} +487075. q^{35} -2958.97 q^{36} -409817. q^{37} -64115.0 q^{38} -166965. q^{39} -540585. q^{40} -675273. q^{41} +637907. q^{42} +38903.5 q^{43} +13267.1 q^{44} +115749. q^{45} +751427. q^{46} +949397. q^{47} -763457. q^{48} +736881. q^{49} -868169. q^{50} +1.05049e6 q^{51} -38280.2 q^{52} +294003. q^{53} +1.26842e6 q^{54} -518983. q^{55} -1.73185e6 q^{56} +237311. q^{57} -2.09490e6 q^{58} -87803.7 q^{59} -168974. q^{60} -2.78573e6 q^{61} +2.17452e6 q^{62} +370822. q^{63} +1.90940e6 q^{64} +1.49745e6 q^{65} -679696. q^{66} +2.95364e6 q^{67} +240847. q^{68} -2.78129e6 q^{69} -5.72117e6 q^{70} -4.08380e6 q^{71} -411560. q^{72} +1.95525e6 q^{73} +4.81370e6 q^{74} +3.21339e6 q^{75} +54408.6 q^{76} -1.66264e6 q^{77} +1.96116e6 q^{78} -608158. q^{79} +6.84718e6 q^{80} -4.04562e6 q^{81} +7.93173e6 q^{82} +214613. q^{83} -541334. q^{84} -9.42151e6 q^{85} -456959. q^{86} +7.75394e6 q^{87} +1.84530e6 q^{88} -8.30366e6 q^{89} -1.35959e6 q^{90} +4.79732e6 q^{91} -637668. q^{92} -8.04863e6 q^{93} -1.11516e7 q^{94} -2.12836e6 q^{95} +1.25235e6 q^{96} -1.38909e6 q^{97} -8.65538e6 q^{98} -395114. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} - 6 q^{3} - 104 q^{4} - 470 q^{5} - 696 q^{6} - 1228 q^{7} + 480 q^{8} - 36 q^{9} + 4280 q^{10} + 2662 q^{11} + 6072 q^{12} + 344 q^{13} + 14752 q^{14} - 12990 q^{15} - 6368 q^{16} - 8468 q^{17}+ \cdots - 47916 q^{99}+O(q^{100}) \) 2 * q - 8 * q^2 - 6 * q^3 - 104 * q^4 - 470 * q^5 - 696 * q^6 - 1228 * q^7 + 480 * q^8 - 36 * q^9 + 4280 * q^10 + 2662 * q^11 + 6072 * q^12 + 344 * q^13 + 14752 * q^14 - 12990 * q^15 - 6368 * q^16 - 8468 * q^17 + 4464 * q^18 - 35280 * q^19 + 5240 * q^20 - 55356 * q^21 - 10648 * q^22 - 61486 * q^23 + 105120 * q^24 + 2200 * q^25 + 60784 * q^26 - 12690 * q^27 - 14864 * q^28 + 179040 * q^29 + 213960 * q^30 - 57166 * q^31 + 186752 * q^32 - 7986 * q^33 - 406048 * q^34 + 485380 * q^35 - 32688 * q^36 - 877698 * q^37 - 216720 * q^38 - 373992 * q^39 - 468000 * q^40 - 283616 * q^41 + 633984 * q^42 + 275484 * q^43 - 138424 * q^44 + 94860 * q^45 + 760744 * q^46 + 1662512 * q^47 - 1317216 * q^48 - 86214 * q^49 - 1136800 * q^50 + 2664924 * q^51 - 515168 * q^52 + 1616484 * q^53 + 1625400 * q^54 - 625570 * q^55 - 1751040 * q^56 + 2252880 * q^57 - 2092320 * q^58 - 2454130 * q^59 - 620520 * q^60 - 6019176 * q^61 + 2653864 * q^62 + 376344 * q^63 + 1068416 * q^64 + 1162360 * q^65 - 926376 * q^66 - 174698 * q^67 + 3959696 * q^68 - 2904342 * q^69 - 5727520 * q^70 - 1151466 * q^71 - 648000 * q^72 + 885944 * q^73 + 3061032 * q^74 + 6761400 * q^75 + 4697280 * q^76 - 1634468 * q^77 + 1185648 * q^78 + 3801460 * q^79 + 5950880 * q^80 - 9331038 * q^81 + 9398864 * q^82 - 2282916 * q^83 - 421968 * q^84 - 6808420 * q^85 + 429264 * q^86 + 7719840 * q^87 + 638880 * q^88 - 13481970 * q^89 - 1437840 * q^90 + 4885904 * q^91 - 921128 * q^92 - 14379702 * q^93 - 8480288 * q^94 + 1134000 * q^95 - 6562176 * q^96 - 68078 * q^97 - 11738664 * q^98 - 47916 * q^99

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