LMFDB - Embedded newform 110.3.d.a.21.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [110,3,Mod(21,110)] 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("110.21"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 110 = 2 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	110.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$2.99728290796$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4956160000.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 19x^{4} - 30x^{2} + 25 $ x^8 - 4x^6 + 19x^4 - 30*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			21.4
Root			$1.09132 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	110.21
Dual form			110.3.d.a.21.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} +4.76766 q^{3} -2.00000 q^{4} -2.23607 q^{5} -6.74249i q^{6} -9.73055i q^{7} +2.82843i q^{8} +13.7306 q^{9} +3.16228i q^{10} +(10.0795 - 4.40491i) q^{11} -9.53532 q^{12} +16.6335i q^{13} -13.7611 q^{14} -10.6608 q^{15} +4.00000 q^{16} +12.8351i q^{17} -19.4180i q^{18} +4.69221i q^{19} +4.47214 q^{20} -46.3920i q^{21} +(-6.22949 - 14.2546i) q^{22} -32.4557 q^{23} +13.4850i q^{24} +5.00000 q^{25} +23.5233 q^{26} +22.5538 q^{27} +19.4611i q^{28} +29.4791i q^{29} +15.0767i q^{30} -23.4043 q^{31} -5.65685i q^{32} +(48.0557 - 21.0011i) q^{33} +18.1516 q^{34} +21.7582i q^{35} -27.4611 q^{36} -11.1821 q^{37} +6.63579 q^{38} +79.3026i q^{39} -6.32456i q^{40} +69.0178i q^{41} -65.6081 q^{42} -65.2097i q^{43} +(-20.1590 + 8.80982i) q^{44} -30.7025 q^{45} +45.8992i q^{46} +45.7097 q^{47} +19.0706 q^{48} -45.6836 q^{49} -7.07107i q^{50} +61.1934i q^{51} -33.2669i q^{52} -1.00392 q^{53} -31.8958i q^{54} +(-22.5385 + 9.84968i) q^{55} +27.5222 q^{56} +22.3709i q^{57} +41.6898 q^{58} +94.0820 q^{59} +21.3216 q^{60} -113.061i q^{61} +33.0987i q^{62} -133.606i q^{63} -8.00000 q^{64} -37.1935i q^{65} +(-29.7001 - 67.9611i) q^{66} -13.0292 q^{67} -25.6702i q^{68} -154.738 q^{69} +30.7707 q^{70} +36.0057 q^{71} +38.8359i q^{72} -83.2194i q^{73} +15.8139i q^{74} +23.8383 q^{75} -9.38442i q^{76} +(-42.8622 - 98.0793i) q^{77} +112.151 q^{78} -0.104581i q^{79} -8.94427 q^{80} -16.0465 q^{81} +97.6059 q^{82} +41.9202i q^{83} +92.7839i q^{84} -28.7002i q^{85} -92.2204 q^{86} +140.546i q^{87} +(12.4590 + 28.5092i) q^{88} -145.516 q^{89} +43.4199i q^{90} +161.853 q^{91} +64.9113 q^{92} -111.584 q^{93} -64.6433i q^{94} -10.4921i q^{95} -26.9700i q^{96} +89.1978 q^{97} +64.6064i q^{98} +(138.398 - 60.4820i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} - 16 q^{4} + 40 q^{9} + 16 q^{12} - 32 q^{14} - 40 q^{15} + 32 q^{16} - 136 q^{23} + 40 q^{25} + 80 q^{26} + 64 q^{27} - 64 q^{31} + 88 q^{33} + 112 q^{34} - 80 q^{36} - 48 q^{37} - 208 q^{42}+ \cdots + 704 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 - 16 * q^4 + 40 * q^9 + 16 * q^12 - 32 * q^14 - 40 * q^15 + 32 * q^16 - 136 * q^23 + 40 * q^25 + 80 * q^26 + 64 * q^27 - 64 * q^31 + 88 * q^33 + 112 * q^34 - 80 * q^36 - 48 * q^37 - 208 * q^42 + 152 * q^47 - 32 * q^48 - 232 * q^49 + 352 * q^53 + 64 * q^56 - 64 * q^58 + 80 * q^59 + 80 * q^60 - 64 * q^64 - 176 * q^66 + 24 * q^67 + 112 * q^69 - 256 * q^71 - 40 * q^75 - 440 * q^77 + 224 * q^78 + 88 * q^81 + 368 * q^82 + 32 * q^86 - 368 * q^89 + 240 * q^91 + 272 * q^92 - 272 * q^93 - 32 * q^97 + 704 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/110\mathbb{Z}\right)^\times$.

$n$	$67$	$101$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	1.41421i		−	0.707107i
$2$		−	1.41421i		−	0.707107i
$3$	4.76766			1.58922			0.794610	−	0.607120i	$-0.207675\pi$
$3$	4.76766			1.58922			0.794610	+	0.607120i	$0.207675\pi$
$4$	−2.00000			−0.500000
$5$	−2.23607			−0.447214
$5$	−2.23607			−0.447214
$6$		−	6.74249i		−	1.12375i
$7$		−	9.73055i		−	1.39008i	−0.718972	−	0.695039i	$-0.755387\pi$
$7$		−	9.73055i		−	1.39008i	0.718972	−	0.695039i	$-0.244613\pi$
$8$			2.82843i			0.353553i
$9$	13.7306			1.52562
$10$			3.16228i			0.316228i
$11$	10.0795	−	4.40491i	0.916320	−	0.400447i
$11$	10.0795	−	4.40491i	0.916320	−	0.400447i
$12$	−9.53532			−0.794610
$13$			16.6335i			1.27950i	0.768585	+	0.639748i	$0.220961\pi$
$13$			16.6335i			1.27950i	−0.768585	+	0.639748i	$0.779039\pi$
$14$	−13.7611			−0.982934
$15$	−10.6608			−0.710721
$16$	4.00000			0.250000
$17$			12.8351i			0.755007i	0.926008	+	0.377503i	$0.123217\pi$
$17$			12.8351i			0.755007i	−0.926008	+	0.377503i	$0.876783\pi$
$18$		−	19.4180i		−	1.07878i
$19$			4.69221i			0.246958i	0.992347	+	0.123479i	$0.0394052\pi$
$19$			4.69221i			0.246958i	−0.992347	+	0.123479i	$0.960595\pi$
$20$	4.47214			0.223607
$21$		−	46.3920i		−	2.20914i
$22$	−6.22949	−	14.2546i	−0.283158	−	0.647936i
$23$	−32.4557			−1.41112			−0.705558	−	0.708652i	$-0.749304\pi$
$23$	−32.4557			−1.41112			−0.705558	+	0.708652i	$0.749304\pi$
$24$			13.4850i			0.561874i
$25$	5.00000			0.200000
$26$	23.5233			0.904741
$27$	22.5538			0.835325
$28$			19.4611i			0.695039i
$29$			29.4791i			1.01652i	0.861203	+	0.508260i	$0.169711\pi$
$29$			29.4791i			1.01652i	−0.861203	+	0.508260i	$0.830289\pi$
$30$			15.0767i			0.502555i
$31$	−23.4043			−0.754979			−0.377489	−	0.926014i	$-0.623213\pi$
$31$	−23.4043			−0.754979			−0.377489	+	0.926014i	$0.623213\pi$
$32$		−	5.65685i		−	0.176777i
$33$	48.0557	−	21.0011i	1.45623	−	0.636398i
$34$	18.1516			0.533870
$35$			21.7582i			0.621662i
$36$	−27.4611			−0.762810
$37$	−11.1821			−0.302219			−0.151109	−	0.988517i	$-0.548285\pi$
$37$	−11.1821			−0.302219			−0.151109	+	0.988517i	$0.548285\pi$
$38$	6.63579			0.174626
$39$			79.3026i			2.03340i
$40$		−	6.32456i		−	0.158114i
$41$			69.0178i			1.68336i	0.539976	+	0.841681i	$0.318433\pi$
$41$			69.0178i			1.68336i	−0.539976	+	0.841681i	$0.681567\pi$
$42$	−65.6081			−1.56210
$43$		−	65.2097i		−	1.51650i	−0.651961	−	0.758252i	$-0.726054\pi$
$43$		−	65.2097i		−	1.51650i	0.651961	−	0.758252i	$-0.273946\pi$
$44$	−20.1590	+	8.80982i	−0.458160	+	0.200223i
$45$	−30.7025			−0.682278
$46$			45.8992i			0.997810i
$47$	45.7097			0.972547			0.486274	−	0.873807i	$-0.338356\pi$
$47$	45.7097			0.972547			0.486274	+	0.873807i	$0.338356\pi$
$48$	19.0706			0.397305
$49$	−45.6836			−0.932319
$50$		−	7.07107i		−	0.141421i
$51$			61.1934i			1.19987i
$52$		−	33.2669i		−	0.639748i
$53$	−1.00392			−0.0189419			−0.00947094	−	0.999955i	$-0.503015\pi$
$53$	−1.00392			−0.0189419			−0.00947094	+	0.999955i	$0.503015\pi$
$54$		−	31.8958i		−	0.590664i
$55$	−22.5385	+	9.84968i	−0.409791	+	0.179085i
$56$	27.5222			0.491467
$57$			22.3709i			0.392471i
$58$	41.6898			0.718789
$59$	94.0820			1.59461			0.797305	−	0.603576i	$-0.206258\pi$
$59$	94.0820			1.59461			0.797305	+	0.603576i	$0.206258\pi$
$60$	21.3216			0.355360
$61$		−	113.061i		−	1.85346i	−0.375723	−	0.926732i	$-0.622606\pi$
$61$		−	113.061i		−	1.85346i	0.375723	−	0.926732i	$-0.377394\pi$
$62$			33.0987i			0.533851i
$63$		−	133.606i		−	2.12073i
$64$	−8.00000			−0.125000
$65$		−	37.1935i		−	0.572208i
$66$	−29.7001	−	67.9611i	−0.450001	−	1.02971i
$67$	−13.0292			−0.194466			−0.0972329	−	0.995262i	$-0.530999\pi$
$67$	−13.0292			−0.194466			−0.0972329	+	0.995262i	$0.530999\pi$
$68$		−	25.6702i		−	0.377503i
$69$	−154.738			−2.24257
$70$	30.7707			0.439582
$71$	36.0057			0.507122			0.253561	−	0.967319i	$-0.418398\pi$
$71$	36.0057			0.507122			0.253561	+	0.967319i	$0.418398\pi$
$72$			38.8359i			0.539388i
$73$		−	83.2194i		−	1.13999i	−0.821648	−	0.569996i	$-0.806945\pi$
$73$		−	83.2194i		−	1.13999i	0.821648	−	0.569996i	$-0.193055\pi$
$74$			15.8139i			0.213701i
$75$	23.8383			0.317844
$76$		−	9.38442i		−	0.123479i
$77$	−42.8622	−	98.0793i	−0.556652	−	1.27376i
$78$	112.151			1.43783
$79$		−	0.104581i		−	0.00132381i	−1.00000		0.000661904i	$-0.999789\pi$
$79$		−	0.104581i		−	0.00132381i	1.00000		0.000661904i	$-0.000210690\pi$
$80$	−8.94427			−0.111803
$81$	−16.0465			−0.198105
$82$	97.6059			1.19032
$83$			41.9202i			0.505062i	0.967589	+	0.252531i	$0.0812630\pi$
$83$			41.9202i			0.505062i	−0.967589	+	0.252531i	$0.918737\pi$
$84$			92.7839i			1.10457i
$85$		−	28.7002i		−	0.337649i
$86$	−92.2204			−1.07233
$87$			140.546i			1.61548i
$88$	12.4590	+	28.5092i	0.141579	+	0.323968i
$89$	−145.516			−1.63501			−0.817505	−	0.575921i	$-0.804643\pi$
$89$	−145.516			−1.63501			−0.817505	+	0.575921i	$0.804643\pi$
$90$			43.4199i			0.482443i
$91$	161.853			1.77860
$92$	64.9113			0.705558
$93$	−111.584			−1.19983
$94$		−	64.6433i		−	0.687695i
$95$		−	10.4921i		−	0.110443i
$96$		−	26.9700i		−	0.280937i
$97$	89.1978			0.919565			0.459782	−	0.888032i	$-0.347927\pi$
$97$	89.1978			0.919565			0.459782	+	0.888032i	$0.347927\pi$
$98$			64.6064i			0.659249i
$99$	138.398	−	60.4820i	1.39796	−	0.610929i

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	110.3.d.a.21.4	✓	8
3.2	odd	2		990.3.b.b.901.7		8
4.3	odd	2		880.3.j.c.241.2		8
5.2	odd	4		550.3.c.b.549.10		16
5.3	odd	4		550.3.c.b.549.7		16
5.4	even	2		550.3.d.f.351.5		8
11.10	odd	2	inner	110.3.d.a.21.8	yes	8
33.32	even	2		990.3.b.b.901.4		8
44.43	even	2		880.3.j.c.241.1		8
55.32	even	4		550.3.c.b.549.2		16
55.43	even	4		550.3.c.b.549.15		16
55.54	odd	2		550.3.d.f.351.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.3.d.a.21.4	✓	8	1.1	even	1	trivial
110.3.d.a.21.8	yes	8	11.10	odd	2	inner
550.3.c.b.549.2		16	55.32	even	4
550.3.c.b.549.7		16	5.3	odd	4
550.3.c.b.549.10		16	5.2	odd	4
550.3.c.b.549.15		16	55.43	even	4
550.3.d.f.351.1		8	55.54	odd	2
550.3.d.f.351.5		8	5.4	even	2
880.3.j.c.241.1		8	44.43	even	2
880.3.j.c.241.2		8	4.3	odd	2
990.3.b.b.901.4		8	33.32	even	2
990.3.b.b.901.7		8	3.2	odd	2

Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 110 = 2 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	110.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +4.76766 q^{3} -2.00000 q^{4} -2.23607 q^{5} -6.74249i q^{6} -9.73055i q^{7} +2.82843i q^{8} +13.7306 q^{9} +3.16228i q^{10} +(10.0795 - 4.40491i) q^{11} -9.53532 q^{12} +16.6335i q^{13} -13.7611 q^{14} -10.6608 q^{15} +4.00000 q^{16} +12.8351i q^{17} -19.4180i q^{18} +4.69221i q^{19} +4.47214 q^{20} -46.3920i q^{21} +(-6.22949 - 14.2546i) q^{22} -32.4557 q^{23} +13.4850i q^{24} +5.00000 q^{25} +23.5233 q^{26} +22.5538 q^{27} +19.4611i q^{28} +29.4791i q^{29} +15.0767i q^{30} -23.4043 q^{31} -5.65685i q^{32} +(48.0557 - 21.0011i) q^{33} +18.1516 q^{34} +21.7582i q^{35} -27.4611 q^{36} -11.1821 q^{37} +6.63579 q^{38} +79.3026i q^{39} -6.32456i q^{40} +69.0178i q^{41} -65.6081 q^{42} -65.2097i q^{43} +(-20.1590 + 8.80982i) q^{44} -30.7025 q^{45} +45.8992i q^{46} +45.7097 q^{47} +19.0706 q^{48} -45.6836 q^{49} -7.07107i q^{50} +61.1934i q^{51} -33.2669i q^{52} -1.00392 q^{53} -31.8958i q^{54} +(-22.5385 + 9.84968i) q^{55} +27.5222 q^{56} +22.3709i q^{57} +41.6898 q^{58} +94.0820 q^{59} +21.3216 q^{60} -113.061i q^{61} +33.0987i q^{62} -133.606i q^{63} -8.00000 q^{64} -37.1935i q^{65} +(-29.7001 - 67.9611i) q^{66} -13.0292 q^{67} -25.6702i q^{68} -154.738 q^{69} +30.7707 q^{70} +36.0057 q^{71} +38.8359i q^{72} -83.2194i q^{73} +15.8139i q^{74} +23.8383 q^{75} -9.38442i q^{76} +(-42.8622 - 98.0793i) q^{77} +112.151 q^{78} -0.104581i q^{79} -8.94427 q^{80} -16.0465 q^{81} +97.6059 q^{82} +41.9202i q^{83} +92.7839i q^{84} -28.7002i q^{85} -92.2204 q^{86} +140.546i q^{87} +(12.4590 + 28.5092i) q^{88} -145.516 q^{89} +43.4199i q^{90} +161.853 q^{91} +64.9113 q^{92} -111.584 q^{93} -64.6433i q^{94} -10.4921i q^{95} -26.9700i q^{96} +89.1978 q^{97} +64.6064i q^{98} +(138.398 - 60.4820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} - 16 q^{4} + 40 q^{9} + 16 q^{12} - 32 q^{14} - 40 q^{15} + 32 q^{16} - 136 q^{23} + 40 q^{25} + 80 q^{26} + 64 q^{27} - 64 q^{31} + 88 q^{33} + 112 q^{34} - 80 q^{36} - 48 q^{37} - 208 q^{42}+ \cdots + 704 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 - 16 * q^4 + 40 * q^9 + 16 * q^12 - 32 * q^14 - 40 * q^15 + 32 * q^16 - 136 * q^23 + 40 * q^25 + 80 * q^26 + 64 * q^27 - 64 * q^31 + 88 * q^33 + 112 * q^34 - 80 * q^36 - 48 * q^37 - 208 * q^42 + 152 * q^47 - 32 * q^48 - 232 * q^49 + 352 * q^53 + 64 * q^56 - 64 * q^58 + 80 * q^59 + 80 * q^60 - 64 * q^64 - 176 * q^66 + 24 * q^67 + 112 * q^69 - 256 * q^71 - 40 * q^75 - 440 * q^77 + 224 * q^78 + 88 * q^81 + 368 * q^82 + 32 * q^86 - 368 * q^89 + 240 * q^91 + 272 * q^92 - 272 * q^93 - 32 * q^97 + 704 * q^99

Introduction

L-functions

Modular forms

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