LMFDB - Embedded newform 2116.4.a.i.1.13

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			1.13
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)

$f(q)$	$=$	$q-1.94926 q^{3} +3.12092 q^{5} +11.3326 q^{7} -23.2004 q^{9} -49.4800 q^{11} +91.4256 q^{13} -6.08349 q^{15} +77.1371 q^{17} -135.362 q^{19} -22.0902 q^{21} -115.260 q^{25} +97.8535 q^{27} -33.1089 q^{29} +239.353 q^{31} +96.4493 q^{33} +35.3682 q^{35} +35.4026 q^{37} -178.212 q^{39} -343.521 q^{41} +276.966 q^{43} -72.4067 q^{45} +217.775 q^{47} -214.572 q^{49} -150.360 q^{51} -55.2680 q^{53} -154.423 q^{55} +263.855 q^{57} -505.111 q^{59} -13.1966 q^{61} -262.921 q^{63} +285.332 q^{65} +329.223 q^{67} +1082.77 q^{71} -1198.37 q^{73} +224.671 q^{75} -560.738 q^{77} -277.151 q^{79} +435.669 q^{81} +691.879 q^{83} +240.739 q^{85} +64.5378 q^{87} -1475.16 q^{89} +1036.09 q^{91} -466.561 q^{93} -422.454 q^{95} +1243.35 q^{97} +1147.96 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.94926	−0.375135	−0.187567	−	0.982252i	$-0.560060\pi$
$3$	−1.94926	−0.375135	−0.187567	+	0.982252i	$0.560060\pi$
$4$	0	0
$5$	3.12092	0.279144	0.139572	−	0.990212i	$-0.455427\pi$
$5$	3.12092	0.279144	0.139572	+	0.990212i	$0.455427\pi$
$6$	0	0
$7$	11.3326	0.611904	0.305952	−	0.952047i	$-0.401025\pi$
$7$	11.3326	0.611904	0.305952	+	0.952047i	$0.401025\pi$
$8$	0	0
$9$	−23.2004	−0.859274
$10$	0	0
$11$	−49.4800	−1.35625	−0.678127	−	0.734945i	$-0.737208\pi$
$11$	−49.4800	−1.35625	−0.678127	+	0.734945i	$0.737208\pi$
$12$	0	0
$13$	91.4256	1.95053	0.975265	−	0.221039i	$-0.0709447\pi$
$13$	91.4256	1.95053	0.975265	+	0.221039i	$0.0709447\pi$
$14$	0	0
$15$	−6.08349	−0.104717
$16$	0	0
$17$	77.1371	1.10050	0.550250	−	0.835000i	$-0.314532\pi$
$17$	77.1371	1.10050	0.550250	+	0.835000i	$0.314532\pi$
$18$	0	0
$19$	−135.362	−1.63443	−0.817215	−	0.576333i	$-0.804483\pi$
$19$	−135.362	−1.63443	−0.817215	+	0.576333i	$0.804483\pi$
$20$	0	0
$21$	−22.0902	−0.229546
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−115.260	−0.922079
$26$	0	0
$27$	97.8535	0.697478
$28$	0	0
$29$	−33.1089	−0.212006	−0.106003	−	0.994366i	$-0.533805\pi$
$29$	−33.1089	−0.212006	−0.106003	+	0.994366i	$0.533805\pi$
$30$	0	0
$31$	239.353	1.38674	0.693372	−	0.720580i	$-0.256124\pi$
$31$	239.353	1.38674	0.693372	+	0.720580i	$0.256124\pi$
$32$	0	0
$33$	96.4493	0.508778
$34$	0	0
$35$	35.3682	0.170809
$36$	0	0
$37$	35.4026	0.157301	0.0786507	−	0.996902i	$-0.474939\pi$
$37$	35.4026	0.157301	0.0786507	+	0.996902i	$0.474939\pi$
$38$	0	0
$39$	−178.212	−0.731712
$40$	0	0
$41$	−343.521	−1.30851	−0.654256	−	0.756273i	$-0.727018\pi$
$41$	−343.521	−1.30851	−0.654256	+	0.756273i	$0.727018\pi$
$42$	0	0
$43$	276.966	0.982254	0.491127	−	0.871088i	$-0.336585\pi$
$43$	276.966	0.982254	0.491127	+	0.871088i	$0.336585\pi$
$44$	0	0
$45$	−72.4067	−0.239861
$46$	0	0
$47$	217.775	0.675868	0.337934	−	0.941170i	$-0.390272\pi$
$47$	217.775	0.675868	0.337934	+	0.941170i	$0.390272\pi$
$48$	0	0
$49$	−214.572	−0.625574
$50$	0	0
$51$	−150.360	−0.412836
$52$	0	0
$53$	−55.2680	−0.143239	−0.0716193	−	0.997432i	$-0.522817\pi$
$53$	−55.2680	−0.143239	−0.0716193	+	0.997432i	$0.522817\pi$
$54$	0	0
$55$	−154.423	−0.378590
$56$	0	0
$57$	263.855	0.613132
$58$	0	0
$59$	−505.111	−1.11457	−0.557287	−	0.830320i	$-0.688158\pi$
$59$	−505.111	−1.11457	−0.557287	+	0.830320i	$0.688158\pi$
$60$	0	0
$61$	−13.1966	−0.0276991	−0.0138496	−	0.999904i	$-0.504409\pi$
$61$	−13.1966	−0.0276991	−0.0138496	+	0.999904i	$0.504409\pi$
$62$	0	0
$63$	−262.921	−0.525793
$64$	0	0
$65$	285.332	0.544479
$66$	0	0
$67$	329.223	0.600313	0.300156	−	0.953890i	$-0.402961\pi$
$67$	329.223	0.600313	0.300156	+	0.953890i	$0.402961\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1082.77	1.80988	0.904940	−	0.425540i	$-0.139916\pi$
$71$	1082.77	1.80988	0.904940	+	0.425540i	$0.139916\pi$
$72$	0	0
$73$	−1198.37	−1.92136	−0.960679	−	0.277662i	$-0.910440\pi$
$73$	−1198.37	−1.92136	−0.960679	+	0.277662i	$0.910440\pi$
$74$	0	0
$75$	224.671	0.345904
$76$	0	0
$77$	−560.738	−0.829896
$78$	0	0
$79$	−277.151	−0.394708	−0.197354	−	0.980332i	$-0.563235\pi$
$79$	−277.151	−0.394708	−0.197354	+	0.980332i	$0.563235\pi$
$80$	0	0
$81$	435.669	0.597625
$82$	0	0
$83$	691.879	0.914984	0.457492	−	0.889214i	$-0.348748\pi$
$83$	691.879	0.914984	0.457492	+	0.889214i	$0.348748\pi$
$84$	0	0
$85$	240.739	0.307198
$86$	0	0
$87$	64.5378	0.0795308
$88$	0	0
$89$	−1475.16	−1.75692	−0.878461	−	0.477813i	$-0.841429\pi$
$89$	−1475.16	−1.75692	−0.878461	+	0.477813i	$0.841429\pi$
$90$	0	0
$91$	1036.09	1.19354
$92$	0	0
$93$	−466.561	−0.520216
$94$	0	0
$95$	−422.454	−0.456241
$96$	0	0
$97$	1243.35	1.30148	0.650740	−	0.759301i	$-0.274459\pi$
$97$	1243.35	1.30148	0.650740	+	0.759301i	$0.274459\pi$
$98$	0	0
$99$	1147.96	1.16539

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.1.13		30
23.4	even	11		92.4.e.a.85.3	yes	60
23.6	even	11		92.4.e.a.13.3	✓	60
23.22	odd	2		2116.4.a.j.1.13		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.4.e.a.13.3	✓	60	23.6	even	11
92.4.e.a.85.3	yes	60	23.4	even	11
2116.4.a.i.1.13		30	1.1	even	1	trivial
2116.4.a.j.1.13		30	23.22	odd	2

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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)

\(f(q)\)	\(=\)	\(q-1.94926 q^{3} +3.12092 q^{5} +11.3326 q^{7} -23.2004 q^{9} -49.4800 q^{11} +91.4256 q^{13} -6.08349 q^{15} +77.1371 q^{17} -135.362 q^{19} -22.0902 q^{21} -115.260 q^{25} +97.8535 q^{27} -33.1089 q^{29} +239.353 q^{31} +96.4493 q^{33} +35.3682 q^{35} +35.4026 q^{37} -178.212 q^{39} -343.521 q^{41} +276.966 q^{43} -72.4067 q^{45} +217.775 q^{47} -214.572 q^{49} -150.360 q^{51} -55.2680 q^{53} -154.423 q^{55} +263.855 q^{57} -505.111 q^{59} -13.1966 q^{61} -262.921 q^{63} +285.332 q^{65} +329.223 q^{67} +1082.77 q^{71} -1198.37 q^{73} +224.671 q^{75} -560.738 q^{77} -277.151 q^{79} +435.669 q^{81} +691.879 q^{83} +240.739 q^{85} +64.5378 q^{87} -1475.16 q^{89} +1036.09 q^{91} -466.561 q^{93} -422.454 q^{95} +1243.35 q^{97} +1147.96 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{3} - 50 q^{5} + 2 q^{7} + 272 q^{9} - 92 q^{11} - 28 q^{13} + 64 q^{15} - 316 q^{17} - 222 q^{19} - 512 q^{21} + 786 q^{25} - 212 q^{27} + 218 q^{29} + 350 q^{31} - 576 q^{33} + 150 q^{35} - 314 q^{37}+ \cdots - 930 q^{99}+O(q^{100}) \) 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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