LMFDB - Embedded newform 2116.4.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2116,4,Mod(1,2116)] 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("2116.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2116, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2116 = 2^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2116.a (trivial)

Newform invariants

comment:select newform

sage:traces = [30,0,-2,0,-50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$124.848041572$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 92)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	2116.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.19219 q^{3} +7.93087 q^{5} +7.58084 q^{7} +57.4963 q^{9} +47.6376 q^{11} +15.5483 q^{13} -72.9020 q^{15} -65.8744 q^{17} +28.2468 q^{19} -69.6845 q^{21} -62.1014 q^{25} -280.328 q^{27} +268.580 q^{29} -264.303 q^{31} -437.894 q^{33} +60.1226 q^{35} -63.2075 q^{37} -142.923 q^{39} -483.018 q^{41} -205.068 q^{43} +455.996 q^{45} +123.059 q^{47} -285.531 q^{49} +605.530 q^{51} +150.981 q^{53} +377.808 q^{55} -259.650 q^{57} -366.411 q^{59} -710.493 q^{61} +435.870 q^{63} +123.312 q^{65} +809.367 q^{67} -340.468 q^{71} -592.304 q^{73} +570.847 q^{75} +361.133 q^{77} +1030.78 q^{79} +1024.43 q^{81} +1144.88 q^{83} -522.441 q^{85} -2468.84 q^{87} +403.284 q^{89} +117.869 q^{91} +2429.53 q^{93} +224.022 q^{95} -1489.06 q^{97} +2738.99 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 2 q^{3} - 50 q^{5} + 2 q^{7} + 272 q^{9} - 92 q^{11} - 28 q^{13} + 64 q^{15} - 316 q^{17} - 222 q^{19} - 512 q^{21} + 786 q^{25} - 212 q^{27} + 218 q^{29} + 350 q^{31} - 576 q^{33} + 150 q^{35} - 314 q^{37}+ \cdots - 930 q^{99}+O(q^{100}) $ 30 * q - 2 * q^3 - 50 * q^5 + 2 * q^7 + 272 * q^9 - 92 * q^11 - 28 * q^13 + 64 * q^15 - 316 * q^17 - 222 * q^19 - 512 * q^21 + 786 * q^25 - 212 * q^27 + 218 * q^29 + 350 * q^31 - 576 * q^33 + 150 * q^35 - 314 * q^37 - 144 * q^39 - 90 * q^41 - 224 * q^43 - 1406 * q^45 - 734 * q^47 + 1002 * q^49 - 1126 * q^51 - 1866 * q^53 - 306 * q^55 - 1246 * q^57 - 1070 * q^59 - 1656 * q^61 - 1026 * q^63 - 2524 * q^65 + 46 * q^67 - 1064 * q^71 - 2570 * q^73 + 3138 * q^75 + 3628 * q^77 - 410 * q^79 + 4590 * q^81 - 2264 * q^83 - 2322 * q^85 + 1584 * q^87 - 5736 * q^89 - 2350 * q^91 + 2022 * q^93 + 3182 * q^95 - 3144 * q^97 - 930 * 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 $

$2$	0	0
$2$	0	0
$3$	−9.19219	−1.76904	−0.884519	−	0.466505i	$-0.845513\pi$
$3$	−9.19219	−1.76904	−0.884519	+	0.466505i	$0.845513\pi$
$4$	0	0
$5$	7.93087	0.709358	0.354679	−	0.934988i	$-0.384590\pi$
$5$	7.93087	0.709358	0.354679	+	0.934988i	$0.384590\pi$
$6$	0	0
$7$	7.58084	0.409327	0.204663	−	0.978832i	$-0.434390\pi$
$7$	7.58084	0.409327	0.204663	+	0.978832i	$0.434390\pi$
$8$	0	0
$9$	57.4963	2.12949
$10$	0	0
$11$	47.6376	1.30575	0.652877	−	0.757464i	$-0.273562\pi$
$11$	47.6376	1.30575	0.652877	+	0.757464i	$0.273562\pi$
$12$	0	0
$13$	15.5483	0.331717	0.165859	−	0.986150i	$-0.446960\pi$
$13$	15.5483	0.331717	0.165859	+	0.986150i	$0.446960\pi$
$14$	0	0
$15$	−72.9020	−1.25488
$16$	0	0
$17$	−65.8744	−0.939818	−0.469909	−	0.882715i	$-0.655713\pi$
$17$	−65.8744	−0.939818	−0.469909	+	0.882715i	$0.655713\pi$
$18$	0	0
$19$	28.2468	0.341067	0.170533	−	0.985352i	$-0.445451\pi$
$19$	28.2468	0.341067	0.170533	+	0.985352i	$0.445451\pi$
$20$	0	0
$21$	−69.6845	−0.724115
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−62.1014	−0.496811
$26$	0	0
$27$	−280.328	−1.99812
$28$	0	0
$29$	268.580	1.71979	0.859897	−	0.510467i	$-0.170527\pi$
$29$	268.580	1.71979	0.859897	+	0.510467i	$0.170527\pi$
$30$	0	0
$31$	−264.303	−1.53130	−0.765650	−	0.643257i	$-0.777583\pi$
$31$	−264.303	−1.53130	−0.765650	+	0.643257i	$0.777583\pi$
$32$	0	0
$33$	−437.894	−2.30993
$34$	0	0
$35$	60.1226	0.290359
$36$	0	0
$37$	−63.2075	−0.280845	−0.140422	−	0.990092i	$-0.544846\pi$
$37$	−63.2075	−0.280845	−0.140422	+	0.990092i	$0.544846\pi$
$38$	0	0
$39$	−142.923	−0.586820
$40$	0	0
$41$	−483.018	−1.83987	−0.919936	−	0.392068i	$-0.871760\pi$
$41$	−483.018	−1.83987	−0.919936	+	0.392068i	$0.871760\pi$
$42$	0	0
$43$	−205.068	−0.727269	−0.363634	−	0.931542i	$-0.618464\pi$
$43$	−205.068	−0.727269	−0.363634	+	0.931542i	$0.618464\pi$
$44$	0	0
$45$	455.996	1.51057
$46$	0	0
$47$	123.059	0.381914	0.190957	−	0.981598i	$-0.438841\pi$
$47$	123.059	0.381914	0.190957	+	0.981598i	$0.438841\pi$
$48$	0	0
$49$	−285.531	−0.832452
$50$	0	0
$51$	605.530	1.66257
$52$	0	0
$53$	150.981	0.391299	0.195650	−	0.980674i	$-0.437319\pi$
$53$	150.981	0.391299	0.195650	+	0.980674i	$0.437319\pi$
$54$	0	0
$55$	377.808	0.926247
$56$	0	0
$57$	−259.650	−0.603360
$58$	0	0
$59$	−366.411	−0.808520	−0.404260	−	0.914644i	$-0.632471\pi$
$59$	−366.411	−0.808520	−0.404260	+	0.914644i	$0.632471\pi$
$60$	0	0
$61$	−710.493	−1.49130	−0.745650	−	0.666338i	$-0.767861\pi$
$61$	−710.493	−1.49130	−0.745650	+	0.666338i	$0.767861\pi$
$62$	0	0
$63$	435.870	0.871659
$64$	0	0
$65$	123.312	0.235306
$66$	0	0
$67$	809.367	1.47582	0.737910	−	0.674899i	$-0.235813\pi$
$67$	809.367	1.47582	0.737910	+	0.674899i	$0.235813\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−340.468	−0.569100	−0.284550	−	0.958661i	$-0.591844\pi$
$71$	−340.468	−0.569100	−0.284550	+	0.958661i	$0.591844\pi$
$72$	0	0
$73$	−592.304	−0.949643	−0.474821	−	0.880082i	$-0.657487\pi$
$73$	−592.304	−0.949643	−0.474821	+	0.880082i	$0.657487\pi$
$74$	0	0
$75$	570.847	0.878877
$76$	0	0
$77$	361.133	0.534480
$78$	0	0
$79$	1030.78	1.46800	0.734001	−	0.679148i	$-0.237651\pi$
$79$	1030.78	1.46800	0.734001	+	0.679148i	$0.237651\pi$
$80$	0	0
$81$	1024.43	1.40525
$82$	0	0
$83$	1144.88	1.51405	0.757027	−	0.653384i	$-0.226651\pi$
$83$	1144.88	1.51405	0.757027	+	0.653384i	$0.226651\pi$
$84$	0	0
$85$	−522.441	−0.666667
$86$	0	0
$87$	−2468.84	−3.04238
$88$	0	0
$89$	403.284	0.480315	0.240158	−	0.970734i	$-0.422801\pi$
$89$	403.284	0.480315	0.240158	+	0.970734i	$0.422801\pi$
$90$	0	0
$91$	117.869	0.135781
$92$	0	0
$93$	2429.53	2.70893
$94$	0	0
$95$	224.022	0.241938
$96$	0	0
$97$	−1489.06	−1.55867	−0.779334	−	0.626608i	$-0.784443\pi$
$97$	−1489.06	−1.55867	−0.779334	+	0.626608i	$0.784443\pi$
$98$	0	0
$99$	2738.99	2.78059

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	2116.4.a.i.1.2		30
23.4	even	11		92.4.e.a.85.1	yes	60
23.6	even	11		92.4.e.a.13.1	✓	60
23.22	odd	2		2116.4.a.j.1.2		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.4.e.a.13.1	✓	60	23.6	even	11
92.4.e.a.85.1	yes	60	23.4	even	11
2116.4.a.i.1.2		30	1.1	even	1	trivial
2116.4.a.j.1.2		30	23.22	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:	\( N \)	\(=\)	\( 2116 = 2^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2116.a (trivial)

\(f(q)\)	\(=\)	\(q-9.19219 q^{3} +7.93087 q^{5} +7.58084 q^{7} +57.4963 q^{9} +47.6376 q^{11} +15.5483 q^{13} -72.9020 q^{15} -65.8744 q^{17} +28.2468 q^{19} -69.6845 q^{21} -62.1014 q^{25} -280.328 q^{27} +268.580 q^{29} -264.303 q^{31} -437.894 q^{33} +60.1226 q^{35} -63.2075 q^{37} -142.923 q^{39} -483.018 q^{41} -205.068 q^{43} +455.996 q^{45} +123.059 q^{47} -285.531 q^{49} +605.530 q^{51} +150.981 q^{53} +377.808 q^{55} -259.650 q^{57} -366.411 q^{59} -710.493 q^{61} +435.870 q^{63} +123.312 q^{65} +809.367 q^{67} -340.468 q^{71} -592.304 q^{73} +570.847 q^{75} +361.133 q^{77} +1030.78 q^{79} +1024.43 q^{81} +1144.88 q^{83} -522.441 q^{85} -2468.84 q^{87} +403.284 q^{89} +117.869 q^{91} +2429.53 q^{93} +224.022 q^{95} -1489.06 q^{97} +2738.99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{3} - 50 q^{5} + 2 q^{7} + 272 q^{9} - 92 q^{11} - 28 q^{13} + 64 q^{15} - 316 q^{17} - 222 q^{19} - 512 q^{21} + 786 q^{25} - 212 q^{27} + 218 q^{29} + 350 q^{31} - 576 q^{33} + 150 q^{35} - 314 q^{37}+ \cdots - 930 q^{99}+O(q^{100}) \) 30 * q - 2 * q^3 - 50 * q^5 + 2 * q^7 + 272 * q^9 - 92 * q^11 - 28 * q^13 + 64 * q^15 - 316 * q^17 - 222 * q^19 - 512 * q^21 + 786 * q^25 - 212 * q^27 + 218 * q^29 + 350 * q^31 - 576 * q^33 + 150 * q^35 - 314 * q^37 - 144 * q^39 - 90 * q^41 - 224 * q^43 - 1406 * q^45 - 734 * q^47 + 1002 * q^49 - 1126 * q^51 - 1866 * q^53 - 306 * q^55 - 1246 * q^57 - 1070 * q^59 - 1656 * q^61 - 1026 * q^63 - 2524 * q^65 + 46 * q^67 - 1064 * q^71 - 2570 * q^73 + 3138 * q^75 + 3628 * q^77 - 410 * q^79 + 4590 * q^81 - 2264 * q^83 - 2322 * q^85 + 1584 * q^87 - 5736 * q^89 - 2350 * q^91 + 2022 * q^93 + 3182 * q^95 - 3144 * q^97 - 930 * q^99

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