LMFDB - Embedded newform 121.4.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [121,4,Mod(1,121)] 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("121.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 121 = 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	121.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.13923111069$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{26}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 26 $ x^2 - 26
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.09902$ of defining polynomial
Character	$\chi$	$=$	121.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.09902 q^{2} -5.00000 q^{3} +18.0000 q^{4} +5.00000 q^{5} +25.4951 q^{6} -20.3961 q^{7} -50.9902 q^{8} -2.00000 q^{9} -25.4951 q^{10} -90.0000 q^{12} -61.1882 q^{13} +104.000 q^{14} -25.0000 q^{15} +116.000 q^{16} +20.3961 q^{17} +10.1980 q^{18} +101.980 q^{19} +90.0000 q^{20} +101.980 q^{21} +35.0000 q^{23} +254.951 q^{24} -100.000 q^{25} +312.000 q^{26} +145.000 q^{27} -367.129 q^{28} +203.961 q^{29} +127.475 q^{30} +15.0000 q^{31} -183.565 q^{32} -104.000 q^{34} -101.980 q^{35} -36.0000 q^{36} -265.000 q^{37} -520.000 q^{38} +305.941 q^{39} -254.951 q^{40} +101.980 q^{41} -520.000 q^{42} +448.714 q^{43} -10.0000 q^{45} -178.466 q^{46} +380.000 q^{47} -580.000 q^{48} +73.0000 q^{49} +509.902 q^{50} -101.980 q^{51} -1101.39 q^{52} +510.000 q^{53} -739.358 q^{54} +1040.00 q^{56} -509.902 q^{57} -1040.00 q^{58} +21.0000 q^{59} -450.000 q^{60} -203.961 q^{61} -76.4853 q^{62} +40.7922 q^{63} +8.00000 q^{64} -305.941 q^{65} +585.000 q^{67} +367.129 q^{68} -175.000 q^{69} +520.000 q^{70} +313.000 q^{71} +101.980 q^{72} -469.110 q^{73} +1351.24 q^{74} +500.000 q^{75} +1835.65 q^{76} -1560.00 q^{78} +611.882 q^{79} +580.000 q^{80} -671.000 q^{81} -520.000 q^{82} -652.674 q^{83} +1835.65 q^{84} +101.980 q^{85} -2288.00 q^{86} -1019.80 q^{87} -185.000 q^{89} +50.9902 q^{90} +1248.00 q^{91} +630.000 q^{92} -75.0000 q^{93} -1937.63 q^{94} +509.902 q^{95} +917.824 q^{96} +785.000 q^{97} -372.228 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{3} + 36 q^{4} + 10 q^{5} - 4 q^{9} - 180 q^{12} + 208 q^{14} - 50 q^{15} + 232 q^{16} + 180 q^{20} + 70 q^{23} - 200 q^{25} + 624 q^{26} + 290 q^{27} + 30 q^{31} - 208 q^{34} - 72 q^{36} - 530 q^{37}+ \cdots + 1570 q^{97}+O(q^{100}) $ 2 * q - 10 * q^3 + 36 * q^4 + 10 * q^5 - 4 * q^9 - 180 * q^12 + 208 * q^14 - 50 * q^15 + 232 * q^16 + 180 * q^20 + 70 * q^23 - 200 * q^25 + 624 * q^26 + 290 * q^27 + 30 * q^31 - 208 * q^34 - 72 * q^36 - 530 * q^37 - 1040 * q^38 - 1040 * q^42 - 20 * q^45 + 760 * q^47 - 1160 * q^48 + 146 * q^49 + 1020 * q^53 + 2080 * q^56 - 2080 * q^58 + 42 * q^59 - 900 * q^60 + 16 * q^64 + 1170 * q^67 - 350 * q^69 + 1040 * q^70 + 626 * q^71 + 1000 * q^75 - 3120 * q^78 + 1160 * q^80 - 1342 * q^81 - 1040 * q^82 - 4576 * q^86 - 370 * q^89 + 2496 * q^91 + 1260 * q^92 - 150 * q^93 + 1570 * q^97

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 $

$2$	−5.09902	−1.80278	−0.901388	−	0.433013i	$-0.857451\pi$
$2$	−5.09902	−1.80278	−0.901388	+	0.433013i	$0.857451\pi$
$3$	−5.00000	−0.962250	−0.481125	−	0.876652i	$-0.659772\pi$
$3$	−5.00000	−0.962250	−0.481125	+	0.876652i	$0.659772\pi$
$4$	18.0000	2.25000
$5$	5.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	5.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	25.4951	1.73472
$7$	−20.3961	−1.10128	−0.550642	−	0.834741i	$-0.685617\pi$
$7$	−20.3961	−1.10128	−0.550642	+	0.834741i	$0.685617\pi$
$8$	−50.9902	−2.25347
$9$	−2.00000	−0.0740741
$10$	−25.4951	−0.806226
$11$	0	0
$11$	0	0
$12$	−90.0000	−2.16506
$13$	−61.1882	−1.30543	−0.652714	−	0.757604i	$-0.726370\pi$
$13$	−61.1882	−1.30543	−0.652714	+	0.757604i	$0.726370\pi$
$14$	104.000	1.98537
$15$	−25.0000	−0.430331
$16$	116.000	1.81250
$17$	20.3961	0.290987	0.145493	−	0.989359i	$-0.453523\pi$
$17$	20.3961	0.290987	0.145493	+	0.989359i	$0.453523\pi$
$18$	10.1980	0.133539
$19$	101.980	1.23136	0.615682	−	0.787995i	$-0.288881\pi$
$19$	101.980	1.23136	0.615682	+	0.787995i	$0.288881\pi$
$20$	90.0000	1.00623
$21$	101.980	1.05971
$22$	0	0
$23$	35.0000	0.317305	0.158652	−	0.987335i	$-0.449285\pi$
$23$	35.0000	0.317305	0.158652	+	0.987335i	$0.449285\pi$
$24$	254.951	2.16840
$25$	−100.000	−0.800000
$26$	312.000	2.35339
$27$	145.000	1.03353
$28$	−367.129	−2.47789
$29$	203.961	1.30602	0.653010	−	0.757349i	$-0.273506\pi$
$29$	203.961	1.30602	0.653010	+	0.757349i	$0.273506\pi$
$30$	127.475	0.775791
$31$	15.0000	0.0869058	0.0434529	−	0.999055i	$-0.486164\pi$
$31$	15.0000	0.0869058	0.0434529	+	0.999055i	$0.486164\pi$
$32$	−183.565	−1.01406
$33$	0	0
$34$	−104.000	−0.524584
$35$	−101.980	−0.492509
$36$	−36.0000	−0.166667
$37$	−265.000	−1.17745	−0.588726	−	0.808333i	$-0.700370\pi$
$37$	−265.000	−1.17745	−0.588726	+	0.808333i	$0.700370\pi$
$38$	−520.000	−2.21987
$39$	305.941	1.25615
$40$	−254.951	−1.00778
$41$	101.980	0.388455	0.194228	−	0.980956i	$-0.437780\pi$
$41$	101.980	0.388455	0.194228	+	0.980956i	$0.437780\pi$
$42$	−520.000	−1.91042
$43$	448.714	1.59135	0.795677	−	0.605721i	$-0.207115\pi$
$43$	448.714	1.59135	0.795677	+	0.605721i	$0.207115\pi$
$44$	0	0
$45$	−10.0000	−0.0331269
$46$	−178.466	−0.572029
$47$	380.000	1.17933	0.589667	−	0.807646i	$-0.299259\pi$
$47$	380.000	1.17933	0.589667	+	0.807646i	$0.299259\pi$
$48$	−580.000	−1.74408
$49$	73.0000	0.212828
$50$	509.902	1.44222
$51$	−101.980	−0.280002
$52$	−1101.39	−2.93721
$53$	510.000	1.32177	0.660886	−	0.750487i	$-0.270181\pi$
$53$	510.000	1.32177	0.660886	+	0.750487i	$0.270181\pi$
$54$	−739.358	−1.86322
$55$	0	0
$56$	1040.00	2.48171
$57$	−509.902	−1.18488
$58$	−1040.00	−2.35446
$59$	21.0000	0.0463384	0.0231692	−	0.999732i	$-0.492624\pi$
$59$	21.0000	0.0463384	0.0231692	+	0.999732i	$0.492624\pi$
$60$	−450.000	−0.968246
$61$	−203.961	−0.428107	−0.214053	−	0.976822i	$-0.568667\pi$
$61$	−203.961	−0.428107	−0.214053	+	0.976822i	$0.568667\pi$
$62$	−76.4853	−0.156672
$63$	40.7922	0.0815766
$64$	8.00000	0.0156250
$65$	−305.941	−0.583805
$66$	0	0
$67$	585.000	1.06670	0.533352	−	0.845894i	$-0.320932\pi$
$67$	585.000	1.06670	0.533352	+	0.845894i	$0.320932\pi$
$68$	367.129	0.654720
$69$	−175.000	−0.305326
$70$	520.000	0.887884
$71$	313.000	0.523187	0.261593	−	0.965178i	$-0.415752\pi$
$71$	313.000	0.523187	0.261593	+	0.965178i	$0.415752\pi$
$72$	101.980	0.166924
$73$	−469.110	−0.752125	−0.376063	−	0.926594i	$-0.622722\pi$
$73$	−469.110	−0.752125	−0.376063	+	0.926594i	$0.622722\pi$
$74$	1351.24	2.12268
$75$	500.000	0.769800
$76$	1835.65	2.77057
$77$	0	0
$78$	−1560.00	−2.26455
$79$	611.882	0.871420	0.435710	−	0.900087i	$-0.356497\pi$
$79$	611.882	0.871420	0.435710	+	0.900087i	$0.356497\pi$
$80$	580.000	0.810575
$81$	−671.000	−0.920439
$82$	−520.000	−0.700297
$83$	−652.674	−0.863137	−0.431568	−	0.902080i	$-0.642040\pi$
$83$	−652.674	−0.863137	−0.431568	+	0.902080i	$0.642040\pi$
$84$	1835.65	2.38435
$85$	101.980	0.130133
$86$	−2288.00	−2.86885
$87$	−1019.80	−1.25672
$88$	0	0
$89$	−185.000	−0.220337	−0.110168	−	0.993913i	$-0.535139\pi$
$89$	−185.000	−0.220337	−0.110168	+	0.993913i	$0.535139\pi$
$90$	50.9902	0.0597204
$91$	1248.00	1.43765
$92$	630.000	0.713935
$93$	−75.0000	−0.0836251
$94$	−1937.63	−2.12607
$95$	509.902	0.550682
$96$	917.824	0.975781
$97$	785.000	0.821698	0.410849	−	0.911703i	$-0.365232\pi$
$97$	785.000	0.821698	0.410849	+	0.911703i	$0.365232\pi$
$98$	−372.228	−0.383681
$99$	0	0

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	121.4.a.d.1.1	✓	2
3.2	odd	2		1089.4.a.r.1.2		2
4.3	odd	2		1936.4.a.ba.1.2		2
11.2	odd	10		121.4.c.e.81.1		8
11.3	even	5		121.4.c.e.9.1		8
11.4	even	5		121.4.c.e.27.1		8
11.5	even	5		121.4.c.e.3.2		8
11.6	odd	10		121.4.c.e.3.1		8
11.7	odd	10		121.4.c.e.27.2		8
11.8	odd	10		121.4.c.e.9.2		8
11.9	even	5		121.4.c.e.81.2		8
11.10	odd	2	inner	121.4.a.d.1.2	yes	2
33.32	even	2		1089.4.a.r.1.1		2
44.43	even	2		1936.4.a.ba.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.4.a.d.1.1	✓	2	1.1	even	1	trivial
121.4.a.d.1.2	yes	2	11.10	odd	2	inner
121.4.c.e.3.1		8	11.6	odd	10
121.4.c.e.3.2		8	11.5	even	5
121.4.c.e.9.1		8	11.3	even	5
121.4.c.e.9.2		8	11.8	odd	10
121.4.c.e.27.1		8	11.4	even	5
121.4.c.e.27.2		8	11.7	odd	10
121.4.c.e.81.1		8	11.2	odd	10
121.4.c.e.81.2		8	11.9	even	5
1089.4.a.r.1.1		2	33.32	even	2
1089.4.a.r.1.2		2	3.2	odd	2
1936.4.a.ba.1.1		2	44.43	even	2
1936.4.a.ba.1.2		2	4.3	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	121.a (trivial)

\(f(q)\)	\(=\)	\(q-5.09902 q^{2} -5.00000 q^{3} +18.0000 q^{4} +5.00000 q^{5} +25.4951 q^{6} -20.3961 q^{7} -50.9902 q^{8} -2.00000 q^{9} -25.4951 q^{10} -90.0000 q^{12} -61.1882 q^{13} +104.000 q^{14} -25.0000 q^{15} +116.000 q^{16} +20.3961 q^{17} +10.1980 q^{18} +101.980 q^{19} +90.0000 q^{20} +101.980 q^{21} +35.0000 q^{23} +254.951 q^{24} -100.000 q^{25} +312.000 q^{26} +145.000 q^{27} -367.129 q^{28} +203.961 q^{29} +127.475 q^{30} +15.0000 q^{31} -183.565 q^{32} -104.000 q^{34} -101.980 q^{35} -36.0000 q^{36} -265.000 q^{37} -520.000 q^{38} +305.941 q^{39} -254.951 q^{40} +101.980 q^{41} -520.000 q^{42} +448.714 q^{43} -10.0000 q^{45} -178.466 q^{46} +380.000 q^{47} -580.000 q^{48} +73.0000 q^{49} +509.902 q^{50} -101.980 q^{51} -1101.39 q^{52} +510.000 q^{53} -739.358 q^{54} +1040.00 q^{56} -509.902 q^{57} -1040.00 q^{58} +21.0000 q^{59} -450.000 q^{60} -203.961 q^{61} -76.4853 q^{62} +40.7922 q^{63} +8.00000 q^{64} -305.941 q^{65} +585.000 q^{67} +367.129 q^{68} -175.000 q^{69} +520.000 q^{70} +313.000 q^{71} +101.980 q^{72} -469.110 q^{73} +1351.24 q^{74} +500.000 q^{75} +1835.65 q^{76} -1560.00 q^{78} +611.882 q^{79} +580.000 q^{80} -671.000 q^{81} -520.000 q^{82} -652.674 q^{83} +1835.65 q^{84} +101.980 q^{85} -2288.00 q^{86} -1019.80 q^{87} -185.000 q^{89} +50.9902 q^{90} +1248.00 q^{91} +630.000 q^{92} -75.0000 q^{93} -1937.63 q^{94} +509.902 q^{95} +917.824 q^{96} +785.000 q^{97} -372.228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{3} + 36 q^{4} + 10 q^{5} - 4 q^{9} - 180 q^{12} + 208 q^{14} - 50 q^{15} + 232 q^{16} + 180 q^{20} + 70 q^{23} - 200 q^{25} + 624 q^{26} + 290 q^{27} + 30 q^{31} - 208 q^{34} - 72 q^{36} - 530 q^{37}+ \cdots + 1570 q^{97}+O(q^{100}) \) 2 * q - 10 * q^3 + 36 * q^4 + 10 * q^5 - 4 * q^9 - 180 * q^12 + 208 * q^14 - 50 * q^15 + 232 * q^16 + 180 * q^20 + 70 * q^23 - 200 * q^25 + 624 * q^26 + 290 * q^27 + 30 * q^31 - 208 * q^34 - 72 * q^36 - 530 * q^37 - 1040 * q^38 - 1040 * q^42 - 20 * q^45 + 760 * q^47 - 1160 * q^48 + 146 * q^49 + 1020 * q^53 + 2080 * q^56 - 2080 * q^58 + 42 * q^59 - 900 * q^60 + 16 * q^64 + 1170 * q^67 - 350 * q^69 + 1040 * q^70 + 626 * q^71 + 1000 * q^75 - 3120 * q^78 + 1160 * q^80 - 1342 * q^81 - 1040 * q^82 - 4576 * q^86 - 370 * q^89 + 2496 * q^91 + 1260 * q^92 - 150 * q^93 + 1570 * q^97

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