LMFDB - Embedded newform 11.10.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [11,10,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, 10); 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, 10, names="a")

Level:	$ N $	$=$	$ 11 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	11.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3] 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:	$5.66539419780$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.2659452.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 306x - 836 $ x^3 - 306*x - 836
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$18.7255$ 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)

$f(q)$	$=$	\(q-37.4510 q^{2} +70.6582 q^{3} +890.580 q^{4} +613.897 q^{5} -2646.22 q^{6} -6611.82 q^{7} -14178.2 q^{8} -14690.4 q^{9} -22991.1 q^{10} -14641.0 q^{11} +62926.8 q^{12} -23385.3 q^{13} +247619. q^{14} +43376.9 q^{15} +75011.6 q^{16} -373567. q^{17} +550171. q^{18} -974169. q^{19} +546725. q^{20} -467179. q^{21} +548321. q^{22} +2.43507e6 q^{23} -1.00181e6 q^{24} -1.57626e6 q^{25} +875803. q^{26} -2.42876e6 q^{27} -5.88835e6 q^{28} +1.48687e6 q^{29} -1.62451e6 q^{30} -29678.2 q^{31} +4.44998e6 q^{32} -1.03451e6 q^{33} +1.39905e7 q^{34} -4.05898e6 q^{35} -1.30830e7 q^{36} +2.58734e6 q^{37} +3.64836e7 q^{38} -1.65236e6 q^{39} -8.70396e6 q^{40} +2.05042e6 q^{41} +1.74963e7 q^{42} +1.09855e7 q^{43} -1.30390e7 q^{44} -9.01841e6 q^{45} -9.11958e7 q^{46} -6.28869e7 q^{47} +5.30018e6 q^{48} +3.36250e6 q^{49} +5.90324e7 q^{50} -2.63956e7 q^{51} -2.08265e7 q^{52} +8.20335e7 q^{53} +9.09597e7 q^{54} -8.98807e6 q^{55} +9.37437e7 q^{56} -6.88330e7 q^{57} -5.56847e7 q^{58} +6.35396e7 q^{59} +3.86306e7 q^{60} -1.18264e8 q^{61} +1.11148e6 q^{62} +9.71303e7 q^{63} -2.05062e8 q^{64} -1.43562e7 q^{65} +3.87433e7 q^{66} +2.96721e8 q^{67} -3.32692e8 q^{68} +1.72058e8 q^{69} +1.52013e8 q^{70} -1.49607e8 q^{71} +2.08284e8 q^{72} +1.06570e8 q^{73} -9.68984e7 q^{74} -1.11375e8 q^{75} -8.67575e8 q^{76} +9.68036e7 q^{77} +6.18827e7 q^{78} +3.34856e8 q^{79} +4.60494e7 q^{80} +1.17539e8 q^{81} -7.67905e7 q^{82} +1.79922e8 q^{83} -4.16060e8 q^{84} -2.29332e8 q^{85} -4.11417e8 q^{86} +1.05059e8 q^{87} +2.07583e8 q^{88} -8.97889e8 q^{89} +3.37749e8 q^{90} +1.54619e8 q^{91} +2.16862e9 q^{92} -2.09701e6 q^{93} +2.35518e9 q^{94} -5.98040e8 q^{95} +3.14428e8 q^{96} -9.48620e8 q^{97} -1.25929e8 q^{98} +2.15082e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 186 q^{3} + 912 q^{4} - 1824 q^{5} - 10956 q^{6} - 7260 q^{7} - 20064 q^{8} + 14553 q^{9} - 86724 q^{10} - 43923 q^{11} - 71040 q^{12} - 93258 q^{13} + 324264 q^{14} + 540330 q^{15} + 22656 q^{16}+ \cdots - 213070473 q^{99}+O(q^{100}) $ 3 * q - 186 * q^3 + 912 * q^4 - 1824 * q^5 - 10956 * q^6 - 7260 * q^7 - 20064 * q^8 + 14553 * q^9 - 86724 * q^10 - 43923 * q^11 - 71040 * q^12 - 93258 * q^13 + 324264 * q^14 + 540330 * q^15 + 22656 * q^16 + 18678 * q^17 + 1997028 * q^18 - 1027356 * q^19 - 156576 * q^20 - 1287924 * q^21 + 1674690 * q^23 - 947232 * q^24 - 1560729 * q^25 - 2162088 * q^26 - 10285434 * q^27 - 2570304 * q^28 - 2693658 * q^29 + 14272500 * q^30 - 4525302 * q^31 + 160512 * q^32 + 2723226 * q^33 + 19899408 * q^34 - 7933860 * q^35 + 20908368 * q^36 - 8820204 * q^37 + 23297712 * q^38 + 24922920 * q^39 - 5145888 * q^40 - 9771102 * q^41 - 8130360 * q^42 + 18795744 * q^43 - 13352592 * q^44 - 91915398 * q^45 - 68671020 * q^46 - 31155816 * q^47 + 76391808 * q^48 - 54229677 * q^49 + 103405236 * q^50 - 62110884 * q^51 - 86345952 * q^52 + 47500122 * q^53 - 153828180 * q^54 + 26705184 * q^55 + 113407680 * q^56 + 62103096 * q^57 - 120644760 * q^58 + 332138370 * q^59 + 290804160 * q^60 - 49031730 * q^61 - 86992620 * q^62 + 319652784 * q^63 - 421220352 * q^64 + 161689572 * q^65 + 160406796 * q^66 + 330560082 * q^67 - 382273056 * q^68 - 104664822 * q^69 - 22907160 * q^70 - 57835050 * q^71 + 302075136 * q^72 - 458816886 * q^73 - 528939708 * q^74 - 562184580 * q^75 - 1324671744 * q^76 + 106293660 * q^77 + 903869520 * q^78 - 798908748 * q^79 + 442084224 * q^80 + 1544572395 * q^81 - 376730664 * q^82 + 1239784920 * q^83 - 809016384 * q^84 - 632001744 * q^85 + 627638088 * q^86 + 505901880 * q^87 + 293757024 * q^88 - 699523368 * q^89 - 2575080288 * q^90 - 268926960 * q^91 + 3537302976 * q^92 + 614745786 * q^93 + 3327514080 * q^94 + 107303856 * q^95 + 2520807168 * q^96 - 2207436012 * q^97 - 1261919472 * q^98 - 213070473 * 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 $

$2$	−37.4510	−1.65512	−0.827559	−	0.561379i	$-0.810271\pi$
$2$	−37.4510	−1.65512	−0.827559	+	0.561379i	$0.810271\pi$
$3$	70.6582	0.503636	0.251818	−	0.967775i	$-0.418972\pi$
$3$	70.6582	0.503636	0.251818	+	0.967775i	$0.418972\pi$
$4$	890.580	1.73941
$5$	613.897	0.439269	0.219635	−	0.975582i	$-0.429513\pi$
$5$	613.897	0.439269	0.219635	+	0.975582i	$0.429513\pi$
$6$	−2646.22	−0.833577
$7$	−6611.82	−1.04083	−0.520415	−	0.853914i	$-0.674223\pi$
$7$	−6611.82	−1.04083	−0.520415	+	0.853914i	$0.674223\pi$
$8$	−14178.2	−1.22382
$9$	−14690.4	−0.746351
$10$	−22991.1	−0.727042
$11$	−14641.0	−0.301511
$11$	−14641.0	−0.301511
$12$	62926.8	0.876032
$13$	−23385.3	−0.227090	−0.113545	−	0.993533i	$-0.536221\pi$
$13$	−23385.3	−0.227090	−0.113545	+	0.993533i	$0.536221\pi$
$14$	247619.	1.72269
$15$	43376.9	0.221232
$16$	75011.6	0.286147
$17$	−373567.	−1.08480	−0.542399	−	0.840121i	$-0.682484\pi$
$17$	−373567.	−1.08480	−0.542399	+	0.840121i	$0.682484\pi$
$18$	550171.	1.23530
$19$	−974169.	−1.71492	−0.857458	−	0.514553i	$-0.827958\pi$
$19$	−974169.	−1.71492	−0.857458	+	0.514553i	$0.827958\pi$
$20$	546725.	0.764071
$21$	−467179.	−0.524199
$22$	548321.	0.499037
$23$	2.43507e6	1.81441	0.907206	−	0.420686i	$-0.138211\pi$
$23$	2.43507e6	1.81441	0.907206	+	0.420686i	$0.138211\pi$
$24$	−1.00181e6	−0.616358
$25$	−1.57626e6	−0.807043
$26$	875803.	0.375860
$27$	−2.42876e6	−0.879525
$28$	−5.88835e6	−1.81043
$29$	1.48687e6	0.390374	0.195187	−	0.980766i	$-0.437469\pi$
$29$	1.48687e6	0.390374	0.195187	+	0.980766i	$0.437469\pi$
$30$	−1.62451e6	−0.366165
$31$	−29678.2	−0.00577178	−0.00288589	−	0.999996i	$-0.500919\pi$
$31$	−29678.2	−0.00577178	−0.00288589	+	0.999996i	$0.500919\pi$
$32$	4.44998e6	0.750211
$33$	−1.03451e6	−0.151852
$34$	1.39905e7	1.79547
$35$	−4.05898e6	−0.457204
$36$	−1.30830e7	−1.29821
$37$	2.58734e6	0.226958	0.113479	−	0.993540i	$-0.463801\pi$
$37$	2.58734e6	0.226958	0.113479	+	0.993540i	$0.463801\pi$
$38$	3.64836e7	2.83839
$39$	−1.65236e6	−0.114371
$40$	−8.70396e6	−0.537585
$41$	2.05042e6	0.113323	0.0566613	−	0.998393i	$-0.481954\pi$
$41$	2.05042e6	0.113323	0.0566613	+	0.998393i	$0.481954\pi$
$42$	1.74963e7	0.867611
$43$	1.09855e7	0.490016	0.245008	−	0.969521i	$-0.421209\pi$
$43$	1.09855e7	0.490016	0.245008	+	0.969521i	$0.421209\pi$
$44$	−1.30390e7	−0.524453
$45$	−9.01841e6	−0.327849
$46$	−9.11958e7	−3.00307
$47$	−6.28869e7	−1.87984	−0.939918	−	0.341399i	$-0.889099\pi$
$47$	−6.28869e7	−1.87984	−0.939918	+	0.341399i	$0.889099\pi$
$48$	5.30018e6	0.144114
$49$	3.36250e6	0.0833258
$50$	5.90324e7	1.33575
$51$	−2.63956e7	−0.546343
$52$	−2.08265e7	−0.395003
$53$	8.20335e7	1.42807	0.714036	−	0.700109i	$-0.246865\pi$
$53$	8.20335e7	1.42807	0.714036	+	0.700109i	$0.246865\pi$
$54$	9.09597e7	1.45572
$55$	−8.98807e6	−0.132445
$56$	9.37437e7	1.27378
$57$	−6.88330e7	−0.863694
$58$	−5.56847e7	−0.646115
$59$	6.35396e7	0.682669	0.341335	−	0.939942i	$-0.389121\pi$
$59$	6.35396e7	0.682669	0.341335	+	0.939942i	$0.389121\pi$
$60$	3.86306e7	0.384814
$61$	−1.18264e8	−1.09363	−0.546814	−	0.837254i	$-0.684159\pi$
$61$	−1.18264e8	−1.09363	−0.546814	+	0.837254i	$0.684159\pi$
$62$	1.11148e6	0.00955297
$63$	9.71303e7	0.776824
$64$	−2.05062e8	−1.52783
$65$	−1.43562e7	−0.0997536
$66$	3.87433e7	0.251333
$67$	2.96721e8	1.79892	0.899459	−	0.437005i	$-0.143961\pi$
$67$	2.96721e8	1.79892	0.899459	+	0.437005i	$0.143961\pi$
$68$	−3.32692e8	−1.88691
$69$	1.72058e8	0.913804
$70$	1.52013e8	0.756727
$71$	−1.49607e8	−0.698696	−0.349348	−	0.936993i	$-0.613597\pi$
$71$	−1.49607e8	−0.698696	−0.349348	+	0.936993i	$0.613597\pi$
$72$	2.08284e8	0.913396
$73$	1.06570e8	0.439222	0.219611	−	0.975588i	$-0.429521\pi$
$73$	1.06570e8	0.439222	0.219611	+	0.975588i	$0.429521\pi$
$74$	−9.68984e7	−0.375642
$75$	−1.11375e8	−0.406456
$76$	−8.67575e8	−2.98295
$77$	9.68036e7	0.313822
$78$	6.18827e7	0.189297
$79$	3.34856e8	0.967243	0.483622	−	0.875277i	$-0.339321\pi$
$79$	3.34856e8	0.967243	0.483622	+	0.875277i	$0.339321\pi$
$80$	4.60494e7	0.125695
$81$	1.17539e8	0.303390
$82$	−7.67905e7	−0.187562
$83$	1.79922e8	0.416133	0.208066	−	0.978115i	$-0.433283\pi$
$83$	1.79922e8	0.416133	0.208066	+	0.978115i	$0.433283\pi$
$84$	−4.16060e8	−0.911800
$85$	−2.29332e8	−0.476518
$86$	−4.11417e8	−0.811034
$87$	1.05059e8	0.196606
$88$	2.07583e8	0.368995
$89$	−8.97889e8	−1.51694	−0.758469	−	0.651709i	$-0.774052\pi$
$89$	−8.97889e8	−1.51694	−0.758469	+	0.651709i	$0.774052\pi$
$90$	3.37749e8	0.542628
$91$	1.54619e8	0.236362
$92$	2.16862e9	3.15601
$93$	−2.09701e6	−0.00290688
$94$	2.35518e9	3.11135
$95$	−5.98040e8	−0.753310
$96$	3.14428e8	0.377833
$97$	−9.48620e8	−1.08798	−0.543988	−	0.839093i	$-0.683086\pi$
$97$	−9.48620e8	−1.08798	−0.543988	+	0.839093i	$0.683086\pi$
$98$	−1.25929e8	−0.137914
$99$	2.15082e8	0.225033

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	11.10.a.a.1.1	✓	3
3.2	odd	2		99.10.a.b.1.3		3
4.3	odd	2		176.10.a.g.1.1		3
5.4	even	2		275.10.a.a.1.3		3
11.10	odd	2		121.10.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.a.a.1.1	✓	3	1.1	even	1	trivial
99.10.a.b.1.3		3	3.2	odd	2
121.10.a.b.1.3		3	11.10	odd	2
176.10.a.g.1.1		3	4.3	odd	2
275.10.a.a.1.3		3	5.4	even	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	11.a (trivial)

\(f(q)\)	\(=\)	\(q-37.4510 q^{2} +70.6582 q^{3} +890.580 q^{4} +613.897 q^{5} -2646.22 q^{6} -6611.82 q^{7} -14178.2 q^{8} -14690.4 q^{9} -22991.1 q^{10} -14641.0 q^{11} +62926.8 q^{12} -23385.3 q^{13} +247619. q^{14} +43376.9 q^{15} +75011.6 q^{16} -373567. q^{17} +550171. q^{18} -974169. q^{19} +546725. q^{20} -467179. q^{21} +548321. q^{22} +2.43507e6 q^{23} -1.00181e6 q^{24} -1.57626e6 q^{25} +875803. q^{26} -2.42876e6 q^{27} -5.88835e6 q^{28} +1.48687e6 q^{29} -1.62451e6 q^{30} -29678.2 q^{31} +4.44998e6 q^{32} -1.03451e6 q^{33} +1.39905e7 q^{34} -4.05898e6 q^{35} -1.30830e7 q^{36} +2.58734e6 q^{37} +3.64836e7 q^{38} -1.65236e6 q^{39} -8.70396e6 q^{40} +2.05042e6 q^{41} +1.74963e7 q^{42} +1.09855e7 q^{43} -1.30390e7 q^{44} -9.01841e6 q^{45} -9.11958e7 q^{46} -6.28869e7 q^{47} +5.30018e6 q^{48} +3.36250e6 q^{49} +5.90324e7 q^{50} -2.63956e7 q^{51} -2.08265e7 q^{52} +8.20335e7 q^{53} +9.09597e7 q^{54} -8.98807e6 q^{55} +9.37437e7 q^{56} -6.88330e7 q^{57} -5.56847e7 q^{58} +6.35396e7 q^{59} +3.86306e7 q^{60} -1.18264e8 q^{61} +1.11148e6 q^{62} +9.71303e7 q^{63} -2.05062e8 q^{64} -1.43562e7 q^{65} +3.87433e7 q^{66} +2.96721e8 q^{67} -3.32692e8 q^{68} +1.72058e8 q^{69} +1.52013e8 q^{70} -1.49607e8 q^{71} +2.08284e8 q^{72} +1.06570e8 q^{73} -9.68984e7 q^{74} -1.11375e8 q^{75} -8.67575e8 q^{76} +9.68036e7 q^{77} +6.18827e7 q^{78} +3.34856e8 q^{79} +4.60494e7 q^{80} +1.17539e8 q^{81} -7.67905e7 q^{82} +1.79922e8 q^{83} -4.16060e8 q^{84} -2.29332e8 q^{85} -4.11417e8 q^{86} +1.05059e8 q^{87} +2.07583e8 q^{88} -8.97889e8 q^{89} +3.37749e8 q^{90} +1.54619e8 q^{91} +2.16862e9 q^{92} -2.09701e6 q^{93} +2.35518e9 q^{94} -5.98040e8 q^{95} +3.14428e8 q^{96} -9.48620e8 q^{97} -1.25929e8 q^{98} +2.15082e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 186 q^{3} + 912 q^{4} - 1824 q^{5} - 10956 q^{6} - 7260 q^{7} - 20064 q^{8} + 14553 q^{9} - 86724 q^{10} - 43923 q^{11} - 71040 q^{12} - 93258 q^{13} + 324264 q^{14} + 540330 q^{15} + 22656 q^{16}+ \cdots - 213070473 q^{99}+O(q^{100}) \) 3 * q - 186 * q^3 + 912 * q^4 - 1824 * q^5 - 10956 * q^6 - 7260 * q^7 - 20064 * q^8 + 14553 * q^9 - 86724 * q^10 - 43923 * q^11 - 71040 * q^12 - 93258 * q^13 + 324264 * q^14 + 540330 * q^15 + 22656 * q^16 + 18678 * q^17 + 1997028 * q^18 - 1027356 * q^19 - 156576 * q^20 - 1287924 * q^21 + 1674690 * q^23 - 947232 * q^24 - 1560729 * q^25 - 2162088 * q^26 - 10285434 * q^27 - 2570304 * q^28 - 2693658 * q^29 + 14272500 * q^30 - 4525302 * q^31 + 160512 * q^32 + 2723226 * q^33 + 19899408 * q^34 - 7933860 * q^35 + 20908368 * q^36 - 8820204 * q^37 + 23297712 * q^38 + 24922920 * q^39 - 5145888 * q^40 - 9771102 * q^41 - 8130360 * q^42 + 18795744 * q^43 - 13352592 * q^44 - 91915398 * q^45 - 68671020 * q^46 - 31155816 * q^47 + 76391808 * q^48 - 54229677 * q^49 + 103405236 * q^50 - 62110884 * q^51 - 86345952 * q^52 + 47500122 * q^53 - 153828180 * q^54 + 26705184 * q^55 + 113407680 * q^56 + 62103096 * q^57 - 120644760 * q^58 + 332138370 * q^59 + 290804160 * q^60 - 49031730 * q^61 - 86992620 * q^62 + 319652784 * q^63 - 421220352 * q^64 + 161689572 * q^65 + 160406796 * q^66 + 330560082 * q^67 - 382273056 * q^68 - 104664822 * q^69 - 22907160 * q^70 - 57835050 * q^71 + 302075136 * q^72 - 458816886 * q^73 - 528939708 * q^74 - 562184580 * q^75 - 1324671744 * q^76 + 106293660 * q^77 + 903869520 * q^78 - 798908748 * q^79 + 442084224 * q^80 + 1544572395 * q^81 - 376730664 * q^82 + 1239784920 * q^83 - 809016384 * q^84 - 632001744 * q^85 + 627638088 * q^86 + 505901880 * q^87 + 293757024 * q^88 - 699523368 * q^89 - 2575080288 * q^90 - 268926960 * q^91 + 3537302976 * q^92 + 614745786 * q^93 + 3327514080 * q^94 + 107303856 * q^95 + 2520807168 * q^96 - 2207436012 * q^97 - 1261919472 * q^98 - 213070473 * q^99

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