LMFDB - Embedded newform 9.100.a.e.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9,100,Mod(1,9)] 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("9.1"); S:= CuspForms(chi, 100); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 100, names="a")

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 100 $
Character orbit:	$[\chi]$	$=$	9.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$558.609014683$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + \cdots + 77\!\cdots\!00 $ x^16 - 23468649002860563504514832190x^14 + 217899364779967801521990920766624917623329262439267904000x^12 - 1020938412603257677832087838681974125606465345076945865467053580456816687401926656000x^10 + 2560723680149202062940423557483028137055826722891150178082983594389297185903849527538033232610860847941550080000x^8 - 3371270875410797720816448891357277449921977363073090218645532111672467436485530421223936446371673490691726948609130156924240421873254400000x^6 + 2089948926077578961262357988033138536129415610819581495702947420276507911338432660567000229442463753075602798729847387416450876655567141551847186092608161054720000000x^4 - 447892986554306987388491742760325827069027555140636604993965329057379989466063410128282897560619422816547353589630248660393238082131391960519499660852340202409612175429022075905376256000000000*x^2 + 7754225595111457623740485529692504678269083037018588860565683297436066814125750099449290193702996370938300516834711757232224102201503535194746635843925419200274483940321446913689074991424666683101183528140800000000000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	multiple of $ 2^{224}\cdot 3^{268}\cdot 5^{33}\cdot 7^{18}\cdot 11^{6}\cdot 13^{2}\cdot 17^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-6.82234e13$ of defining polynomial
Character	$\chi$	$=$	9.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.22802e15 q^{2} +8.74210e29 q^{4} +6.03350e34 q^{5} -3.62729e41 q^{7} -2.95198e44 q^{8} -7.40927e49 q^{10} -9.23245e50 q^{11} -5.39424e54 q^{13} +4.45439e56 q^{14} -1.91588e59 q^{16} +4.42347e60 q^{17} -1.25825e63 q^{19} +5.27455e64 q^{20} +1.13376e66 q^{22} +4.44820e66 q^{23} +2.06259e69 q^{25} +6.62423e69 q^{26} -3.17101e71 q^{28} +4.27570e72 q^{29} -1.60336e73 q^{31} +4.22377e74 q^{32} -5.43211e75 q^{34} -2.18853e76 q^{35} +1.58686e77 q^{37} +1.54516e78 q^{38} -1.78108e79 q^{40} +4.85675e79 q^{41} +1.56419e80 q^{43} -8.07110e80 q^{44} -5.46248e81 q^{46} +9.10153e82 q^{47} -3.30496e83 q^{49} -2.53291e84 q^{50} -4.71569e84 q^{52} +6.83208e84 q^{53} -5.57040e85 q^{55} +1.07077e86 q^{56} -5.25065e87 q^{58} -2.90133e87 q^{59} -4.06602e88 q^{61} +1.96895e88 q^{62} -3.97255e89 q^{64} -3.25461e89 q^{65} +2.10745e90 q^{67} +3.86704e90 q^{68} +2.68756e91 q^{70} -5.84305e91 q^{71} +1.80379e92 q^{73} -1.94870e92 q^{74} -1.09998e93 q^{76} +3.34888e92 q^{77} -1.28939e94 q^{79} -1.15594e94 q^{80} -5.96419e94 q^{82} +3.57806e94 q^{83} +2.66890e95 q^{85} -1.92086e95 q^{86} +2.72540e95 q^{88} +3.39415e96 q^{89} +1.95664e96 q^{91} +3.88866e96 q^{92} -1.11769e98 q^{94} -7.59167e97 q^{95} +8.80491e97 q^{97} +4.05856e98 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 50\!\cdots\!12 q^{4} + 19\!\cdots\!20 q^{7} + 68\!\cdots\!20 q^{10} + 29\!\cdots\!80 q^{13} + 16\!\cdots\!24 q^{16} + 10\!\cdots\!56 q^{19} + 95\!\cdots\!00 q^{22} + 95\!\cdots\!80 q^{25} + 62\!\cdots\!40 q^{28}+ \cdots + 54\!\cdots\!20 q^{97}+O(q^{100}) $ 16 * q + 5066479752027809938951985616112 * q^4 + 190417021918050518546271614944874875008320 * q^7 + 68768980212766506436066579874706216478225944453120 * q^10 + 2906085434531968835745184575729631533413760848187954080 * q^13 + 1650726374066512954764974284979209638992555040052148302033024 * q^16 + 10102390913352867605657034337137915618128037976679835509115280256 * q^19 + 9594866393479846238501012469755290641263379401995680755231448268800 * q^22 + 9552283716537234826459518662441485885155259462388349791099738613010480 * q^25 + 626795798519065058131797380926322656617877736504606812213838188629621440 * q^28 + 360500263320906470116349157746493922909306732567325527554983036842983802688 * q^31 + 15667939407309122472949275888566370569970648153886045270363377805126280232960 * q^34 + 900300930788305350093695012216549975994201106453511904692468701658936534171360 * q^37 + 25266392706772932422047146588312088589128451278313471496429808663399411296583680 * q^40 + 1077326330439836262862123926325911874578323988630505141506853398774192865684830080 * q^43 - 124404808757344647301959580495042001182336143518682412965177820183881839041199800320 * q^46 + 3571966408650746300583560211008161587430644968858153549730632290043289153515190989712 * q^49 - 31607720347139585134142146460176377170195950216716012505007184449889063471786246324640 * q^52 + 132229546764374150228079634250786081498008105681576220554537628566600545491964697804800 * q^55 + 7149063371655180714613563488023223610401398422250213937512088833217428535686671675187200 * q^58 + 23719442991336237513637862214092390888467300353866827726231700454300689497524215310799712 * q^61 + 119406672965983739007060426553963569687829386021446925454140759351489694116013691464956928 * q^64 + 2345298208407071056095407840379609764240377091211047495776067938991423512625119142957553920 * q^67 - 52091144884550837519855042363679416637195222288081899080141453092993225298148343938946662400 * q^70 - 87647546845945200844255876271724640220458439637483671237097615858424586027986835409852499360 * q^73 - 76312901680777189059950015685740132145185295782925334790205071319255296575588096221603265408 * q^76 + 28652807112741594774361584105644952451291678389388539792057712505023813092519467824982948473664 * q^79 - 114411275903482928949200267768185172681454855500223640928064473057207738633014271017458901401600 * q^82 + 445827305743749420276454807132747983955794553080863532534267963770914920274938485414320263413760 * q^85 - 1668768017430023679474068560301709226581599029633767824899764515618791478208123136565573758156800 * q^88 - 8158544031199244342577708550250656997342135988363512017194475705396945955647510364043531436208000 * q^91 + 88347702179067992595963136071715348952831094090766641883795712845975430227084093498067507102023680 * q^94 + 545605058683395782257329385012270974776896625592801475171647036100309701904096113741967793119620320 * 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 $

$2$	−1.22802e15	−1.54249	−0.771243	−	0.636541i	$-0.780364\pi$
$2$	−1.22802e15	−1.54249	−0.771243	+	0.636541i	$0.780364\pi$
$3$	0	0
$3$	0	0
$4$	8.74210e29	1.37926
$5$	6.03350e34	1.51899	0.759494	−	0.650514i	$-0.225447\pi$
$5$	6.03350e34	1.51899	0.759494	+	0.650514i	$0.225447\pi$
$6$	0	0
$7$	−3.62729e41	−0.533616	−0.266808	−	0.963750i	$-0.585969\pi$
$7$	−3.62729e41	−0.533616	−0.266808	+	0.963750i	$0.585969\pi$
$8$	−2.95198e44	−0.585003
$9$	0	0
$10$	−7.40927e49	−2.34302
$11$	−9.23245e50	−0.260843	−0.130421	−	0.991459i	$-0.541633\pi$
$11$	−9.23245e50	−0.260843	−0.130421	+	0.991459i	$0.541633\pi$
$12$	0	0
$13$	−5.39424e54	−0.390602	−0.195301	−	0.980743i	$-0.562568\pi$
$13$	−5.39424e54	−0.390602	−0.195301	+	0.980743i	$0.562568\pi$
$14$	4.45439e56	0.823095
$15$	0	0
$16$	−1.91588e59	−0.476901
$17$	4.42347e60	0.547698	0.273849	−	0.961773i	$-0.411703\pi$
$17$	4.42347e60	0.547698	0.273849	+	0.961773i	$0.411703\pi$
$18$	0	0
$19$	−1.25825e63	−0.633089	−0.316544	−	0.948578i	$-0.602523\pi$
$19$	−1.25825e63	−0.633089	−0.316544	+	0.948578i	$0.602523\pi$
$20$	5.27455e64	2.09508
$21$	0	0
$22$	1.13376e66	0.402346
$23$	4.44820e66	0.174846	0.0874231	−	0.996171i	$-0.472137\pi$
$23$	4.44820e66	0.174846	0.0874231	+	0.996171i	$0.472137\pi$
$24$	0	0
$25$	2.06259e69	1.30732
$26$	6.62423e69	0.602497
$27$	0	0
$28$	−3.17101e71	−0.735996
$29$	4.27570e72	1.74705	0.873527	−	0.486776i	$-0.161827\pi$
$29$	4.27570e72	1.74705	0.873527	+	0.486776i	$0.161827\pi$
$30$	0	0
$31$	−1.60336e73	−0.241338	−0.120669	−	0.992693i	$-0.538504\pi$
$31$	−1.60336e73	−0.241338	−0.120669	+	0.992693i	$0.538504\pi$
$32$	4.22377e74	1.32062
$33$	0	0
$34$	−5.43211e75	−0.844816
$35$	−2.18853e76	−0.810557
$36$	0	0
$37$	1.58686e77	0.375451	0.187726	−	0.982222i	$-0.439888\pi$
$37$	1.58686e77	0.375451	0.187726	+	0.982222i	$0.439888\pi$
$38$	1.54516e78	0.976530
$39$	0	0
$40$	−1.78108e79	−0.888613
$41$	4.85675e79	0.713748	0.356874	−	0.934153i	$-0.383843\pi$
$41$	4.85675e79	0.713748	0.356874	+	0.934153i	$0.383843\pi$
$42$	0	0
$43$	1.56419e80	0.217571	0.108786	−	0.994065i	$-0.465304\pi$
$43$	1.56419e80	0.217571	0.108786	+	0.994065i	$0.465304\pi$
$44$	−8.07110e80	−0.359770
$45$	0	0
$46$	−5.46248e81	−0.269698
$47$	9.10153e82	1.54978	0.774891	−	0.632095i	$-0.217805\pi$
$47$	9.10153e82	1.54978	0.774891	+	0.632095i	$0.217805\pi$
$48$	0	0
$49$	−3.30496e83	−0.715254
$50$	−2.53291e84	−2.01653
$51$	0	0
$52$	−4.71569e84	−0.538741
$53$	6.83208e84	0.304017	0.152008	−	0.988379i	$-0.451426\pi$
$53$	6.83208e84	0.304017	0.152008	+	0.988379i	$0.451426\pi$
$54$	0	0
$55$	−5.57040e85	−0.396217
$56$	1.07077e86	0.312167
$57$	0	0
$58$	−5.25065e87	−2.69480
$59$	−2.90133e87	−0.638885	−0.319442	−	0.947606i	$-0.603496\pi$
$59$	−2.90133e87	−0.638885	−0.319442	+	0.947606i	$0.603496\pi$
$60$	0	0
$61$	−4.06602e88	−1.71926	−0.859631	−	0.510915i	$-0.829307\pi$
$61$	−4.06602e88	−1.71926	−0.859631	+	0.510915i	$0.829307\pi$
$62$	1.96895e88	0.372261
$63$	0	0
$64$	−3.97255e89	−1.56013
$65$	−3.25461e89	−0.593319
$66$	0	0
$67$	2.10745e90	0.857144	0.428572	−	0.903508i	$-0.359017\pi$
$67$	2.10745e90	0.857144	0.428572	+	0.903508i	$0.359017\pi$
$68$	3.86704e90	0.755418
$69$	0	0
$70$	2.68756e91	1.25027
$71$	−5.84305e91	−1.34695	−0.673474	−	0.739210i	$-0.735199\pi$
$71$	−5.84305e91	−1.34695	−0.673474	+	0.739210i	$0.735199\pi$
$72$	0	0
$73$	1.80379e92	1.05124	0.525622	−	0.850718i	$-0.323833\pi$
$73$	1.80379e92	1.05124	0.525622	+	0.850718i	$0.323833\pi$
$74$	−1.94870e92	−0.579128
$75$	0	0
$76$	−1.09998e93	−0.873195
$77$	3.34888e92	0.139190
$78$	0	0
$79$	−1.28939e94	−1.50607	−0.753034	−	0.657982i	$-0.771410\pi$
$79$	−1.28939e94	−1.50607	−0.753034	+	0.657982i	$0.771410\pi$
$80$	−1.15594e94	−0.724407
$81$	0	0
$82$	−5.96419e94	−1.10095
$83$	3.57806e94	0.362478	0.181239	−	0.983439i	$-0.441989\pi$
$83$	3.57806e94	0.362478	0.181239	+	0.983439i	$0.441989\pi$
$84$	0	0
$85$	2.66890e95	0.831947
$86$	−1.92086e95	−0.335600
$87$	0	0
$88$	2.72540e95	0.152594
$89$	3.39415e96	1.08624	0.543120	−	0.839655i	$-0.317243\pi$
$89$	3.39415e96	1.08624	0.543120	+	0.839655i	$0.317243\pi$
$90$	0	0
$91$	1.95664e96	0.208431
$92$	3.88866e96	0.241159
$93$	0	0
$94$	−1.11769e98	−2.39052
$95$	−7.59167e97	−0.961655
$96$	0	0
$97$	8.80491e97	0.397671	0.198836	−	0.980033i	$-0.436284\pi$
$97$	8.80491e97	0.397671	0.198836	+	0.980033i	$0.436284\pi$
$98$	4.05856e98	1.10327
$99$	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	9.100.a.e.1.3	✓	16
3.2	odd	2	inner	9.100.a.e.1.14	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.100.a.e.1.3	✓	16	1.1	even	1	trivial
9.100.a.e.1.14	yes	16	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 100 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

\(f(q)\)	\(=\)	\(q-1.22802e15 q^{2} +8.74210e29 q^{4} +6.03350e34 q^{5} -3.62729e41 q^{7} -2.95198e44 q^{8} -7.40927e49 q^{10} -9.23245e50 q^{11} -5.39424e54 q^{13} +4.45439e56 q^{14} -1.91588e59 q^{16} +4.42347e60 q^{17} -1.25825e63 q^{19} +5.27455e64 q^{20} +1.13376e66 q^{22} +4.44820e66 q^{23} +2.06259e69 q^{25} +6.62423e69 q^{26} -3.17101e71 q^{28} +4.27570e72 q^{29} -1.60336e73 q^{31} +4.22377e74 q^{32} -5.43211e75 q^{34} -2.18853e76 q^{35} +1.58686e77 q^{37} +1.54516e78 q^{38} -1.78108e79 q^{40} +4.85675e79 q^{41} +1.56419e80 q^{43} -8.07110e80 q^{44} -5.46248e81 q^{46} +9.10153e82 q^{47} -3.30496e83 q^{49} -2.53291e84 q^{50} -4.71569e84 q^{52} +6.83208e84 q^{53} -5.57040e85 q^{55} +1.07077e86 q^{56} -5.25065e87 q^{58} -2.90133e87 q^{59} -4.06602e88 q^{61} +1.96895e88 q^{62} -3.97255e89 q^{64} -3.25461e89 q^{65} +2.10745e90 q^{67} +3.86704e90 q^{68} +2.68756e91 q^{70} -5.84305e91 q^{71} +1.80379e92 q^{73} -1.94870e92 q^{74} -1.09998e93 q^{76} +3.34888e92 q^{77} -1.28939e94 q^{79} -1.15594e94 q^{80} -5.96419e94 q^{82} +3.57806e94 q^{83} +2.66890e95 q^{85} -1.92086e95 q^{86} +2.72540e95 q^{88} +3.39415e96 q^{89} +1.95664e96 q^{91} +3.88866e96 q^{92} -1.11769e98 q^{94} -7.59167e97 q^{95} +8.80491e97 q^{97} +4.05856e98 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 50\!\cdots\!12 q^{4} + 19\!\cdots\!20 q^{7} + 68\!\cdots\!20 q^{10} + 29\!\cdots\!80 q^{13} + 16\!\cdots\!24 q^{16} + 10\!\cdots\!56 q^{19} + 95\!\cdots\!00 q^{22} + 95\!\cdots\!80 q^{25} + 62\!\cdots\!40 q^{28}+ \cdots + 54\!\cdots\!20 q^{97}+O(q^{100}) \) 16 * q + 5066479752027809938951985616112 * q^4 + 190417021918050518546271614944874875008320 * q^7 + 68768980212766506436066579874706216478225944453120 * q^10 + 2906085434531968835745184575729631533413760848187954080 * q^13 + 1650726374066512954764974284979209638992555040052148302033024 * q^16 + 10102390913352867605657034337137915618128037976679835509115280256 * q^19 + 9594866393479846238501012469755290641263379401995680755231448268800 * q^22 + 9552283716537234826459518662441485885155259462388349791099738613010480 * q^25 + 626795798519065058131797380926322656617877736504606812213838188629621440 * q^28 + 360500263320906470116349157746493922909306732567325527554983036842983802688 * q^31 + 15667939407309122472949275888566370569970648153886045270363377805126280232960 * q^34 + 900300930788305350093695012216549975994201106453511904692468701658936534171360 * q^37 + 25266392706772932422047146588312088589128451278313471496429808663399411296583680 * q^40 + 1077326330439836262862123926325911874578323988630505141506853398774192865684830080 * q^43 - 124404808757344647301959580495042001182336143518682412965177820183881839041199800320 * q^46 + 3571966408650746300583560211008161587430644968858153549730632290043289153515190989712 * q^49 - 31607720347139585134142146460176377170195950216716012505007184449889063471786246324640 * q^52 + 132229546764374150228079634250786081498008105681576220554537628566600545491964697804800 * q^55 + 7149063371655180714613563488023223610401398422250213937512088833217428535686671675187200 * q^58 + 23719442991336237513637862214092390888467300353866827726231700454300689497524215310799712 * q^61 + 119406672965983739007060426553963569687829386021446925454140759351489694116013691464956928 * q^64 + 2345298208407071056095407840379609764240377091211047495776067938991423512625119142957553920 * q^67 - 52091144884550837519855042363679416637195222288081899080141453092993225298148343938946662400 * q^70 - 87647546845945200844255876271724640220458439637483671237097615858424586027986835409852499360 * q^73 - 76312901680777189059950015685740132145185295782925334790205071319255296575588096221603265408 * q^76 + 28652807112741594774361584105644952451291678389388539792057712505023813092519467824982948473664 * q^79 - 114411275903482928949200267768185172681454855500223640928064473057207738633014271017458901401600 * q^82 + 445827305743749420276454807132747983955794553080863532534267963770914920274938485414320263413760 * q^85 - 1668768017430023679474068560301709226581599029633767824899764515618791478208123136565573758156800 * q^88 - 8158544031199244342577708550250656997342135988363512017194475705396945955647510364043531436208000 * q^91 + 88347702179067992595963136071715348952831094090766641883795712845975430227084093498067507102023680 * q^94 + 545605058683395782257329385012270974776896625592801475171647036100309701904096113741967793119620320 * q^97

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