LMFDB - Embedded newform 121.4.a.e.1.2

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 = [2,2,-8] 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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	\(q+4.46410 q^{2} -7.46410 q^{3} +11.9282 q^{4} -8.46410 q^{5} -33.3205 q^{6} -20.2487 q^{7} +17.5359 q^{8} +28.7128 q^{9} -37.7846 q^{10} -89.0333 q^{12} -61.5359 q^{13} -90.3923 q^{14} +63.1769 q^{15} -17.1436 q^{16} +69.3538 q^{17} +128.177 q^{18} -7.17691 q^{19} -100.962 q^{20} +151.138 q^{21} +50.3154 q^{23} -130.890 q^{24} -53.3590 q^{25} -274.703 q^{26} -12.7846 q^{27} -241.531 q^{28} +143.172 q^{29} +282.028 q^{30} -254.813 q^{31} -216.818 q^{32} +309.603 q^{34} +171.387 q^{35} +342.492 q^{36} +336.664 q^{37} -32.0385 q^{38} +459.310 q^{39} -148.426 q^{40} -178.072 q^{41} +674.697 q^{42} +55.7898 q^{43} -243.028 q^{45} +224.613 q^{46} -256.515 q^{47} +127.962 q^{48} +67.0103 q^{49} -238.200 q^{50} -517.664 q^{51} -734.013 q^{52} +213.449 q^{53} -57.0718 q^{54} -355.079 q^{56} +53.5692 q^{57} +639.133 q^{58} -797.492 q^{59} +753.587 q^{60} -168.697 q^{61} -1137.51 q^{62} -581.397 q^{63} -830.749 q^{64} +520.846 q^{65} -366.105 q^{67} +827.267 q^{68} -375.559 q^{69} +765.090 q^{70} -781.990 q^{71} +503.505 q^{72} +957.538 q^{73} +1502.90 q^{74} +398.277 q^{75} -85.6077 q^{76} +2050.41 q^{78} -585.587 q^{79} +145.105 q^{80} -679.820 q^{81} -794.931 q^{82} +656.669 q^{83} +1802.81 q^{84} -587.018 q^{85} +249.051 q^{86} -1068.65 q^{87} +72.6819 q^{89} -1084.90 q^{90} +1246.02 q^{91} +600.172 q^{92} +1901.95 q^{93} -1145.11 q^{94} +60.7461 q^{95} +1618.35 q^{96} -979.600 q^{97} +299.141 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 8 q^{3} + 10 q^{4} - 10 q^{5} - 32 q^{6} + 8 q^{7} + 42 q^{8} + 2 q^{9} - 34 q^{10} - 88 q^{12} - 130 q^{13} - 160 q^{14} + 64 q^{15} - 62 q^{16} + 14 q^{17} + 194 q^{18} + 48 q^{19} - 98 q^{20}+ \cdots - 822 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 8 * q^3 + 10 * q^4 - 10 * q^5 - 32 * q^6 + 8 * q^7 + 42 * q^8 + 2 * q^9 - 34 * q^10 - 88 * q^12 - 130 * q^13 - 160 * q^14 + 64 * q^15 - 62 * q^16 + 14 * q^17 + 194 * q^18 + 48 * q^19 - 98 * q^20 + 136 * q^21 - 128 * q^23 - 144 * q^24 - 176 * q^25 - 106 * q^26 + 16 * q^27 - 296 * q^28 + 30 * q^29 + 280 * q^30 - 184 * q^31 - 302 * q^32 + 446 * q^34 + 128 * q^35 + 394 * q^36 + 126 * q^37 - 168 * q^38 + 496 * q^39 - 186 * q^40 - 370 * q^41 + 712 * q^42 + 264 * q^43 - 202 * q^45 + 664 * q^46 + 256 * q^47 + 152 * q^48 + 522 * q^49 + 64 * q^50 - 488 * q^51 - 602 * q^52 - 162 * q^53 - 128 * q^54 + 336 * q^56 + 24 * q^57 + 918 * q^58 - 1304 * q^59 + 752 * q^60 + 300 * q^61 - 1312 * q^62 - 1336 * q^63 - 262 * q^64 + 626 * q^65 - 656 * q^67 + 934 * q^68 - 280 * q^69 + 872 * q^70 - 1176 * q^71 - 150 * q^72 + 668 * q^73 + 2022 * q^74 + 464 * q^75 - 192 * q^76 + 1960 * q^78 - 416 * q^79 + 214 * q^80 + 26 * q^81 - 322 * q^82 + 960 * q^83 + 1832 * q^84 - 502 * q^85 - 264 * q^86 - 1008 * q^87 - 1074 * q^89 - 1186 * q^90 - 688 * q^91 + 944 * q^92 + 1864 * q^93 - 2408 * q^94 - 24 * q^95 + 1664 * q^96 - 338 * q^97 - 822 * q^98

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 $

$2$	4.46410	1.57830	0.789149	−	0.614202i	$-0.210522\pi$
$2$	4.46410	1.57830	0.789149	+	0.614202i	$0.210522\pi$
$3$	−7.46410	−1.43647	−0.718234	−	0.695802i	$-0.755049\pi$
$3$	−7.46410	−1.43647	−0.718234	+	0.695802i	$0.755049\pi$
$4$	11.9282	1.49103
$5$	−8.46410	−0.757052	−0.378526	−	0.925591i	$-0.623569\pi$
$5$	−8.46410	−0.757052	−0.378526	+	0.925591i	$0.623569\pi$
$6$	−33.3205	−2.26717
$7$	−20.2487	−1.09333	−0.546664	−	0.837352i	$-0.684102\pi$
$7$	−20.2487	−1.09333	−0.546664	+	0.837352i	$0.684102\pi$
$8$	17.5359	0.774985
$9$	28.7128	1.06344
$10$	−37.7846	−1.19485
$11$	0	0
$11$	0	0
$12$	−89.0333	−2.14181
$13$	−61.5359	−1.31285	−0.656423	−	0.754393i	$-0.727931\pi$
$13$	−61.5359	−1.31285	−0.656423	+	0.754393i	$0.727931\pi$
$14$	−90.3923	−1.72560
$15$	63.1769	1.08748
$16$	−17.1436	−0.267869
$17$	69.3538	0.989457	0.494729	−	0.869048i	$-0.335267\pi$
$17$	69.3538	0.989457	0.494729	+	0.869048i	$0.335267\pi$
$18$	128.177	1.67842
$19$	−7.17691	−0.0866577	−0.0433289	−	0.999061i	$-0.513796\pi$
$19$	−7.17691	−0.0866577	−0.0433289	+	0.999061i	$0.513796\pi$
$20$	−100.962	−1.12878
$21$	151.138	1.57053
$22$	0	0
$23$	50.3154	0.456151	0.228076	−	0.973643i	$-0.426757\pi$
$23$	50.3154	0.456151	0.228076	+	0.973643i	$0.426757\pi$
$24$	−130.890	−1.11324
$25$	−53.3590	−0.426872
$26$	−274.703	−2.07206
$27$	−12.7846	−0.0911259
$28$	−241.531	−1.63018
$29$	143.172	0.916770	0.458385	−	0.888754i	$-0.348428\pi$
$29$	143.172	0.916770	0.458385	+	0.888754i	$0.348428\pi$
$30$	282.028	1.71637
$31$	−254.813	−1.47631	−0.738157	−	0.674629i	$-0.764304\pi$
$31$	−254.813	−1.47631	−0.738157	+	0.674629i	$0.764304\pi$
$32$	−216.818	−1.19776
$33$	0	0
$34$	309.603	1.56166
$35$	171.387	0.827706
$36$	342.492	1.58561
$37$	336.664	1.49587	0.747935	−	0.663771i	$-0.231045\pi$
$37$	336.664	1.49587	0.747935	+	0.663771i	$0.231045\pi$
$38$	−32.0385	−0.136772
$39$	459.310	1.88586
$40$	−148.426	−0.586704
$41$	−178.072	−0.678296	−0.339148	−	0.940733i	$-0.610139\pi$
$41$	−178.072	−0.678296	−0.339148	+	0.940733i	$0.610139\pi$
$42$	674.697	2.47876
$43$	55.7898	0.197857	0.0989286	−	0.995095i	$-0.468458\pi$
$43$	55.7898	0.197857	0.0989286	+	0.995095i	$0.468458\pi$
$44$	0	0
$45$	−243.028	−0.805078
$46$	224.613	0.719943
$47$	−256.515	−0.796098	−0.398049	−	0.917364i	$-0.630313\pi$
$47$	−256.515	−0.796098	−0.398049	+	0.917364i	$0.630313\pi$
$48$	127.962	0.384784
$49$	67.0103	0.195365
$50$	−238.200	−0.673731
$51$	−517.664	−1.42132
$52$	−734.013	−1.95749
$53$	213.449	0.553197	0.276598	−	0.960986i	$-0.410793\pi$
$53$	213.449	0.553197	0.276598	+	0.960986i	$0.410793\pi$
$54$	−57.0718	−0.143824
$55$	0	0
$56$	−355.079	−0.847312
$57$	53.5692	0.124481
$58$	639.133	1.44694
$59$	−797.492	−1.75974	−0.879870	−	0.475215i	$-0.842370\pi$
$59$	−797.492	−1.75974	−0.879870	+	0.475215i	$0.842370\pi$
$60$	753.587	1.62146
$61$	−168.697	−0.354090	−0.177045	−	0.984203i	$-0.556654\pi$
$61$	−168.697	−0.354090	−0.177045	+	0.984203i	$0.556654\pi$
$62$	−1137.51	−2.33006
$63$	−581.397	−1.16269
$64$	−830.749	−1.62256
$65$	520.846	0.993892
$66$	0	0
$67$	−366.105	−0.667565	−0.333783	−	0.942650i	$-0.608325\pi$
$67$	−366.105	−0.667565	−0.333783	+	0.942650i	$0.608325\pi$
$68$	827.267	1.47531
$69$	−375.559	−0.655246
$70$	765.090	1.30637
$71$	−781.990	−1.30711	−0.653557	−	0.756877i	$-0.726724\pi$
$71$	−781.990	−1.30711	−0.653557	+	0.756877i	$0.726724\pi$
$72$	503.505	0.824148
$73$	957.538	1.53522	0.767612	−	0.640915i	$-0.221445\pi$
$73$	957.538	1.53522	0.767612	+	0.640915i	$0.221445\pi$
$74$	1502.90	2.36093
$75$	398.277	0.613187
$76$	−85.6077	−0.129209
$77$	0	0
$78$	2050.41	2.97645
$79$	−585.587	−0.833971	−0.416985	−	0.908913i	$-0.636913\pi$
$79$	−585.587	−0.833971	−0.416985	+	0.908913i	$0.636913\pi$
$80$	145.105	0.202791
$81$	−679.820	−0.932538
$82$	−794.931	−1.07055
$83$	656.669	0.868419	0.434210	−	0.900812i	$-0.357028\pi$
$83$	656.669	0.868419	0.434210	+	0.900812i	$0.357028\pi$
$84$	1802.81	2.34170
$85$	−587.018	−0.749071
$86$	249.051	0.312278
$87$	−1068.65	−1.31691
$88$	0	0
$89$	72.6819	0.0865648	0.0432824	−	0.999063i	$-0.486218\pi$
$89$	72.6819	0.0865648	0.0432824	+	0.999063i	$0.486218\pi$
$90$	−1084.90	−1.27065
$91$	1246.02	1.43537
$92$	600.172	0.680133
$93$	1901.95	2.12068
$94$	−1145.11	−1.25648
$95$	60.7461	0.0656044
$96$	1618.35	1.72054
$97$	−979.600	−1.02539	−0.512697	−	0.858569i	$-0.671354\pi$
$97$	−979.600	−1.02539	−0.512697	+	0.858569i	$0.671354\pi$
$98$	299.141	0.308345
$99$	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.e.1.2	yes	2
3.2	odd	2		1089.4.a.k.1.1		2
4.3	odd	2		1936.4.a.y.1.2		2
11.2	odd	10		121.4.c.g.81.2		8
11.3	even	5		121.4.c.d.9.2		8
11.4	even	5		121.4.c.d.27.2		8
11.5	even	5		121.4.c.d.3.1		8
11.6	odd	10		121.4.c.g.3.2		8
11.7	odd	10		121.4.c.g.27.1		8
11.8	odd	10		121.4.c.g.9.1		8
11.9	even	5		121.4.c.d.81.1		8
11.10	odd	2		121.4.a.b.1.1	✓	2
33.32	even	2		1089.4.a.x.1.2		2
44.43	even	2		1936.4.a.z.1.2		2

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

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Elliptic curves over $\Q$ over number fields
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Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+4.46410 q^{2} -7.46410 q^{3} +11.9282 q^{4} -8.46410 q^{5} -33.3205 q^{6} -20.2487 q^{7} +17.5359 q^{8} +28.7128 q^{9} -37.7846 q^{10} -89.0333 q^{12} -61.5359 q^{13} -90.3923 q^{14} +63.1769 q^{15} -17.1436 q^{16} +69.3538 q^{17} +128.177 q^{18} -7.17691 q^{19} -100.962 q^{20} +151.138 q^{21} +50.3154 q^{23} -130.890 q^{24} -53.3590 q^{25} -274.703 q^{26} -12.7846 q^{27} -241.531 q^{28} +143.172 q^{29} +282.028 q^{30} -254.813 q^{31} -216.818 q^{32} +309.603 q^{34} +171.387 q^{35} +342.492 q^{36} +336.664 q^{37} -32.0385 q^{38} +459.310 q^{39} -148.426 q^{40} -178.072 q^{41} +674.697 q^{42} +55.7898 q^{43} -243.028 q^{45} +224.613 q^{46} -256.515 q^{47} +127.962 q^{48} +67.0103 q^{49} -238.200 q^{50} -517.664 q^{51} -734.013 q^{52} +213.449 q^{53} -57.0718 q^{54} -355.079 q^{56} +53.5692 q^{57} +639.133 q^{58} -797.492 q^{59} +753.587 q^{60} -168.697 q^{61} -1137.51 q^{62} -581.397 q^{63} -830.749 q^{64} +520.846 q^{65} -366.105 q^{67} +827.267 q^{68} -375.559 q^{69} +765.090 q^{70} -781.990 q^{71} +503.505 q^{72} +957.538 q^{73} +1502.90 q^{74} +398.277 q^{75} -85.6077 q^{76} +2050.41 q^{78} -585.587 q^{79} +145.105 q^{80} -679.820 q^{81} -794.931 q^{82} +656.669 q^{83} +1802.81 q^{84} -587.018 q^{85} +249.051 q^{86} -1068.65 q^{87} +72.6819 q^{89} -1084.90 q^{90} +1246.02 q^{91} +600.172 q^{92} +1901.95 q^{93} -1145.11 q^{94} +60.7461 q^{95} +1618.35 q^{96} -979.600 q^{97} +299.141 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 8 q^{3} + 10 q^{4} - 10 q^{5} - 32 q^{6} + 8 q^{7} + 42 q^{8} + 2 q^{9} - 34 q^{10} - 88 q^{12} - 130 q^{13} - 160 q^{14} + 64 q^{15} - 62 q^{16} + 14 q^{17} + 194 q^{18} + 48 q^{19} - 98 q^{20}+ \cdots - 822 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 8 * q^3 + 10 * q^4 - 10 * q^5 - 32 * q^6 + 8 * q^7 + 42 * q^8 + 2 * q^9 - 34 * q^10 - 88 * q^12 - 130 * q^13 - 160 * q^14 + 64 * q^15 - 62 * q^16 + 14 * q^17 + 194 * q^18 + 48 * q^19 - 98 * q^20 + 136 * q^21 - 128 * q^23 - 144 * q^24 - 176 * q^25 - 106 * q^26 + 16 * q^27 - 296 * q^28 + 30 * q^29 + 280 * q^30 - 184 * q^31 - 302 * q^32 + 446 * q^34 + 128 * q^35 + 394 * q^36 + 126 * q^37 - 168 * q^38 + 496 * q^39 - 186 * q^40 - 370 * q^41 + 712 * q^42 + 264 * q^43 - 202 * q^45 + 664 * q^46 + 256 * q^47 + 152 * q^48 + 522 * q^49 + 64 * q^50 - 488 * q^51 - 602 * q^52 - 162 * q^53 - 128 * q^54 + 336 * q^56 + 24 * q^57 + 918 * q^58 - 1304 * q^59 + 752 * q^60 + 300 * q^61 - 1312 * q^62 - 1336 * q^63 - 262 * q^64 + 626 * q^65 - 656 * q^67 + 934 * q^68 - 280 * q^69 + 872 * q^70 - 1176 * q^71 - 150 * q^72 + 668 * q^73 + 2022 * q^74 + 464 * q^75 - 192 * q^76 + 1960 * q^78 - 416 * q^79 + 214 * q^80 + 26 * q^81 - 322 * q^82 + 960 * q^83 + 1832 * q^84 - 502 * q^85 - 264 * q^86 - 1008 * q^87 - 1074 * q^89 - 1186 * q^90 - 688 * q^91 + 944 * q^92 + 1864 * q^93 - 2408 * q^94 - 24 * q^95 + 1664 * q^96 - 338 * q^97 - 822 * q^98

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