LMFDB - Embedded newform 4400.2.a.bm.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] 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("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,0,0,3,0,-3,0,-2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{3} +0.381966 q^{7} -2.61803 q^{9} -1.00000 q^{11} +5.47214 q^{13} -5.09017 q^{17} +3.76393 q^{19} +0.236068 q^{21} -0.381966 q^{23} -3.47214 q^{27} +8.32624 q^{29} +8.70820 q^{31} -0.618034 q^{33} +0.236068 q^{37} +3.38197 q^{39} -10.7082 q^{41} -6.94427 q^{43} +2.23607 q^{47} -6.85410 q^{49} -3.14590 q^{51} +3.61803 q^{53} +2.32624 q^{57} +10.7082 q^{59} -1.90983 q^{61} -1.00000 q^{63} +12.0000 q^{67} -0.236068 q^{69} -8.18034 q^{71} +11.6180 q^{73} -0.381966 q^{77} +7.38197 q^{79} +5.70820 q^{81} +13.5623 q^{83} +5.14590 q^{87} -14.3262 q^{89} +2.09017 q^{91} +5.38197 q^{93} +10.3820 q^{97} +2.61803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 3 q^{7} - 3 q^{9} - 2 q^{11} + 2 q^{13} + q^{17} + 12 q^{19} - 4 q^{21} - 3 q^{23} + 2 q^{27} + q^{29} + 4 q^{31} + q^{33} - 4 q^{37} + 9 q^{39} - 8 q^{41} + 4 q^{43} - 7 q^{49} - 13 q^{51}+ \cdots + 3 q^{99}+O(q^{100}) $ 2 * q - q^3 + 3 * q^7 - 3 * q^9 - 2 * q^11 + 2 * q^13 + q^17 + 12 * q^19 - 4 * q^21 - 3 * q^23 + 2 * q^27 + q^29 + 4 * q^31 + q^33 - 4 * q^37 + 9 * q^39 - 8 * q^41 + 4 * q^43 - 7 * q^49 - 13 * q^51 + 5 * q^53 - 11 * q^57 + 8 * q^59 - 15 * q^61 - 2 * q^63 + 24 * q^67 + 4 * q^69 + 6 * q^71 + 21 * q^73 - 3 * q^77 + 17 * q^79 - 2 * q^81 + 7 * q^83 + 17 * q^87 - 13 * q^89 - 7 * q^91 + 13 * q^93 + 23 * q^97 + 3 * 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 $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See 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
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.381966	0.144370	0.0721848	−	0.997391i	$-0.477003\pi$
$7$	0.381966	0.144370	0.0721848	+	0.997391i	$0.477003\pi$
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.47214	1.51770	0.758849	−	0.651267i	$-0.225762\pi$
$13$	5.47214	1.51770	0.758849	+	0.651267i	$0.225762\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.09017	−1.23455	−0.617274	−	0.786748i	$-0.711763\pi$
$17$	−5.09017	−1.23455	−0.617274	+	0.786748i	$0.711763\pi$
$18$	0	0
$19$	3.76393	0.863505	0.431753	−	0.901992i	$-0.357895\pi$
$19$	3.76393	0.863505	0.431753	+	0.901992i	$0.357895\pi$
$20$	0	0
$21$	0.236068	0.0515143
$22$	0	0
$23$	−0.381966	−0.0796454	−0.0398227	−	0.999207i	$-0.512679\pi$
$23$	−0.381966	−0.0796454	−0.0398227	+	0.999207i	$0.512679\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.47214	−0.668213
$28$	0	0
$29$	8.32624	1.54614	0.773072	−	0.634319i	$-0.218719\pi$
$29$	8.32624	1.54614	0.773072	+	0.634319i	$0.218719\pi$
$30$	0	0
$31$	8.70820	1.56404	0.782020	−	0.623254i	$-0.214190\pi$
$31$	8.70820	1.56404	0.782020	+	0.623254i	$0.214190\pi$
$32$	0	0
$33$	−0.618034	−0.107586
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	0	0
$39$	3.38197	0.541548
$40$	0	0
$41$	−10.7082	−1.67234	−0.836170	−	0.548470i	$-0.815210\pi$
$41$	−10.7082	−1.67234	−0.836170	+	0.548470i	$0.815210\pi$
$42$	0	0
$43$	−6.94427	−1.05899	−0.529496	−	0.848313i	$-0.677619\pi$
$43$	−6.94427	−1.05899	−0.529496	+	0.848313i	$0.677619\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.23607	0.326164	0.163082	−	0.986613i	$-0.447856\pi$
$47$	2.23607	0.326164	0.163082	+	0.986613i	$0.447856\pi$
$48$	0	0
$49$	−6.85410	−0.979157
$50$	0	0
$51$	−3.14590	−0.440514
$52$	0	0
$53$	3.61803	0.496975	0.248488	−	0.968635i	$-0.420066\pi$
$53$	3.61803	0.496975	0.248488	+	0.968635i	$0.420066\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.32624	0.308118
$58$	0	0
$59$	10.7082	1.39409	0.697045	−	0.717028i	$-0.254498\pi$
$59$	10.7082	1.39409	0.697045	+	0.717028i	$0.254498\pi$
$60$	0	0
$61$	−1.90983	−0.244529	−0.122264	−	0.992498i	$-0.539016\pi$
$61$	−1.90983	−0.244529	−0.122264	+	0.992498i	$0.539016\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−0.236068	−0.0284192
$70$	0	0
$71$	−8.18034	−0.970828	−0.485414	−	0.874284i	$-0.661331\pi$
$71$	−8.18034	−0.970828	−0.485414	+	0.874284i	$0.661331\pi$
$72$	0	0
$73$	11.6180	1.35979	0.679894	−	0.733310i	$-0.262026\pi$
$73$	11.6180	1.35979	0.679894	+	0.733310i	$0.262026\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.381966	−0.0435291
$78$	0	0
$79$	7.38197	0.830536	0.415268	−	0.909699i	$-0.363688\pi$
$79$	7.38197	0.830536	0.415268	+	0.909699i	$0.363688\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	13.5623	1.48866	0.744328	−	0.667814i	$-0.232770\pi$
$83$	13.5623	1.48866	0.744328	+	0.667814i	$0.232770\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.14590	0.551698
$88$	0	0
$89$	−14.3262	−1.51858	−0.759289	−	0.650753i	$-0.774453\pi$
$89$	−14.3262	−1.51858	−0.759289	+	0.650753i	$0.774453\pi$
$90$	0	0
$91$	2.09017	0.219109
$92$	0	0
$93$	5.38197	0.558084
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.3820	1.05413	0.527064	−	0.849825i	$-0.323293\pi$
$97$	10.3820	1.05413	0.527064	+	0.849825i	$0.323293\pi$
$98$	0	0
$99$	2.61803	0.263122
$100$	0	0
$101$	−13.6180	−1.35505	−0.677523	−	0.735502i	$-0.736946\pi$
$101$	−13.6180	−1.35505	−0.677523	+	0.735502i	$0.736946\pi$
$102$	0	0
$103$	12.3262	1.21454	0.607270	−	0.794495i	$-0.292265\pi$
$103$	12.3262	1.21454	0.607270	+	0.794495i	$0.292265\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.1803	1.27419	0.637096	−	0.770785i	$-0.280136\pi$
$107$	13.1803	1.27419	0.637096	+	0.770785i	$0.280136\pi$
$108$	0	0
$109$	3.56231	0.341207	0.170604	−	0.985340i	$-0.445428\pi$
$109$	3.56231	0.341207	0.170604	+	0.985340i	$0.445428\pi$
$110$	0	0
$111$	0.145898	0.0138480
$112$	0	0
$113$	8.23607	0.774784	0.387392	−	0.921915i	$-0.373376\pi$
$113$	8.23607	0.774784	0.387392	+	0.921915i	$0.373376\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−14.3262	−1.32446
$118$	0	0
$119$	−1.94427	−0.178231
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.61803	−0.596728
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.14590	0.190418	0.0952088	−	0.995457i	$-0.469648\pi$
$127$	2.14590	0.190418	0.0952088	+	0.995457i	$0.469648\pi$
$128$	0	0
$129$	−4.29180	−0.377872
$130$	0	0
$131$	12.5623	1.09757	0.548787	−	0.835962i	$-0.315090\pi$
$131$	12.5623	1.09757	0.548787	+	0.835962i	$0.315090\pi$
$132$	0	0
$133$	1.43769	0.124664
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.56231	−0.816963	−0.408481	−	0.912767i	$-0.633942\pi$
$137$	−9.56231	−0.816963	−0.408481	+	0.912767i	$0.633942\pi$
$138$	0	0
$139$	11.1803	0.948304	0.474152	−	0.880443i	$-0.342755\pi$
$139$	11.1803	0.948304	0.474152	+	0.880443i	$0.342755\pi$
$140$	0	0
$141$	1.38197	0.116383
$142$	0	0
$143$	−5.47214	−0.457603
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.23607	−0.349385
$148$	0	0
$149$	−7.41641	−0.607576	−0.303788	−	0.952740i	$-0.598251\pi$
$149$	−7.41641	−0.607576	−0.303788	+	0.952740i	$0.598251\pi$
$150$	0	0
$151$	−15.9443	−1.29753	−0.648763	−	0.760990i	$-0.724713\pi$
$151$	−15.9443	−1.29753	−0.648763	+	0.760990i	$0.724713\pi$
$152$	0	0
$153$	13.3262	1.07736
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.47214	−0.356915	−0.178458	−	0.983948i	$-0.557111\pi$
$157$	−4.47214	−0.356915	−0.178458	+	0.983948i	$0.557111\pi$
$158$	0	0
$159$	2.23607	0.177332
$160$	0	0
$161$	−0.145898	−0.0114984
$162$	0	0
$163$	6.38197	0.499874	0.249937	−	0.968262i	$-0.419590\pi$
$163$	6.38197	0.499874	0.249937	+	0.968262i	$0.419590\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.70820	−0.519096	−0.259548	−	0.965730i	$-0.583574\pi$
$167$	−6.70820	−0.519096	−0.259548	+	0.965730i	$0.583574\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	0	0
$171$	−9.85410	−0.753562
$172$	0	0
$173$	11.2918	0.858499	0.429250	−	0.903186i	$-0.358778\pi$
$173$	11.2918	0.858499	0.429250	+	0.903186i	$0.358778\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.61803	0.497442
$178$	0	0
$179$	−2.09017	−0.156227	−0.0781133	−	0.996944i	$-0.524890\pi$
$179$	−2.09017	−0.156227	−0.0781133	+	0.996944i	$0.524890\pi$
$180$	0	0
$181$	4.38197	0.325709	0.162854	−	0.986650i	$-0.447930\pi$
$181$	4.38197	0.325709	0.162854	+	0.986650i	$0.447930\pi$
$182$	0	0
$183$	−1.18034	−0.0872532
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.09017	0.372230
$188$	0	0
$189$	−1.32624	−0.0964696
$190$	0	0
$191$	12.0344	0.870782	0.435391	−	0.900242i	$-0.356610\pi$
$191$	12.0344	0.870782	0.435391	+	0.900242i	$0.356610\pi$
$192$	0	0
$193$	9.52786	0.685831	0.342915	−	0.939366i	$-0.388586\pi$
$193$	9.52786	0.685831	0.342915	+	0.939366i	$0.388586\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.0344	0.786171	0.393086	−	0.919502i	$-0.371408\pi$
$197$	11.0344	0.786171	0.393086	+	0.919502i	$0.371408\pi$
$198$	0	0
$199$	14.6180	1.03624	0.518122	−	0.855306i	$-0.326631\pi$
$199$	14.6180	1.03624	0.518122	+	0.855306i	$0.326631\pi$
$200$	0	0
$201$	7.41641	0.523113
$202$	0	0
$203$	3.18034	0.223216
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−3.76393	−0.260357
$210$	0	0
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	0	0
$213$	−5.05573	−0.346413
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.32624	0.225800
$218$	0	0
$219$	7.18034	0.485202
$220$	0	0
$221$	−27.8541	−1.87367
$222$	0	0
$223$	25.6525	1.71782	0.858908	−	0.512129i	$-0.171143\pi$
$223$	25.6525	1.71782	0.858908	+	0.512129i	$0.171143\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.8541	−0.919529	−0.459765	−	0.888041i	$-0.652066\pi$
$227$	−13.8541	−0.919529	−0.459765	+	0.888041i	$0.652066\pi$
$228$	0	0
$229$	12.6180	0.833823	0.416912	−	0.908947i	$-0.363112\pi$
$229$	12.6180	0.833823	0.416912	+	0.908947i	$0.363112\pi$
$230$	0	0
$231$	−0.236068	−0.0155321
$232$	0	0
$233$	−16.5623	−1.08503	−0.542516	−	0.840045i	$-0.682528\pi$
$233$	−16.5623	−1.08503	−0.542516	+	0.840045i	$0.682528\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.56231	0.296354
$238$	0	0
$239$	22.3820	1.44777	0.723885	−	0.689921i	$-0.242355\pi$
$239$	22.3820	1.44777	0.723885	+	0.689921i	$0.242355\pi$
$240$	0	0
$241$	0.145898	0.00939812	0.00469906	−	0.999989i	$-0.498504\pi$
$241$	0.145898	0.00939812	0.00469906	+	0.999989i	$0.498504\pi$
$242$	0	0
$243$	13.9443	0.894525
$244$	0	0
$245$	0	0
$246$	0	0
$247$	20.5967	1.31054
$248$	0	0
$249$	8.38197	0.531186
$250$	0	0
$251$	−18.5066	−1.16812	−0.584062	−	0.811709i	$-0.698538\pi$
$251$	−18.5066	−1.16812	−0.584062	+	0.811709i	$0.698538\pi$
$252$	0	0
$253$	0.381966	0.0240140
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.05573	0.315368	0.157684	−	0.987490i	$-0.449597\pi$
$257$	5.05573	0.315368	0.157684	+	0.987490i	$0.449597\pi$
$258$	0	0
$259$	0.0901699	0.00560289
$260$	0	0
$261$	−21.7984	−1.34929
$262$	0	0
$263$	15.4721	0.954053	0.477026	−	0.878889i	$-0.341715\pi$
$263$	15.4721	0.954053	0.477026	+	0.878889i	$0.341715\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.85410	−0.541862
$268$	0	0
$269$	−2.61803	−0.159624	−0.0798122	−	0.996810i	$-0.525432\pi$
$269$	−2.61803	−0.159624	−0.0798122	+	0.996810i	$0.525432\pi$
$270$	0	0
$271$	−12.7082	−0.771968	−0.385984	−	0.922505i	$-0.626138\pi$
$271$	−12.7082	−0.771968	−0.385984	+	0.922505i	$0.626138\pi$
$272$	0	0
$273$	1.29180	0.0781831
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−26.1246	−1.56968	−0.784838	−	0.619701i	$-0.787254\pi$
$277$	−26.1246	−1.56968	−0.784838	+	0.619701i	$0.787254\pi$
$278$	0	0
$279$	−22.7984	−1.36490
$280$	0	0
$281$	4.23607	0.252703	0.126351	−	0.991986i	$-0.459673\pi$
$281$	4.23607	0.252703	0.126351	+	0.991986i	$0.459673\pi$
$282$	0	0
$283$	−27.4508	−1.63178	−0.815892	−	0.578205i	$-0.803754\pi$
$283$	−27.4508	−1.63178	−0.815892	+	0.578205i	$0.803754\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.09017	−0.241435
$288$	0	0
$289$	8.90983	0.524108
$290$	0	0
$291$	6.41641	0.376136
$292$	0	0
$293$	−11.5279	−0.673465	−0.336733	−	0.941600i	$-0.609322\pi$
$293$	−11.5279	−0.673465	−0.336733	+	0.941600i	$0.609322\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.47214	0.201474
$298$	0	0
$299$	−2.09017	−0.120878
$300$	0	0
$301$	−2.65248	−0.152886
$302$	0	0
$303$	−8.41641	−0.483510
$304$	0	0
$305$	0	0
$306$	0	0
$307$	23.8541	1.36143	0.680713	−	0.732550i	$-0.261670\pi$
$307$	23.8541	1.36143	0.680713	+	0.732550i	$0.261670\pi$
$308$	0	0
$309$	7.61803	0.433375
$310$	0	0
$311$	−5.94427	−0.337069	−0.168534	−	0.985696i	$-0.553903\pi$
$311$	−5.94427	−0.337069	−0.168534	+	0.985696i	$0.553903\pi$
$312$	0	0
$313$	−10.6525	−0.602114	−0.301057	−	0.953606i	$-0.597339\pi$
$313$	−10.6525	−0.602114	−0.301057	+	0.953606i	$0.597339\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.38197	0.189950	0.0949751	−	0.995480i	$-0.469723\pi$
$317$	3.38197	0.189950	0.0949751	+	0.995480i	$0.469723\pi$
$318$	0	0
$319$	−8.32624	−0.466180
$320$	0	0
$321$	8.14590	0.454660
$322$	0	0
$323$	−19.1591	−1.06604
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.20163	0.121750
$328$	0	0
$329$	0.854102	0.0470882
$330$	0	0
$331$	−13.3607	−0.734369	−0.367185	−	0.930148i	$-0.619678\pi$
$331$	−13.3607	−0.734369	−0.367185	+	0.930148i	$0.619678\pi$
$332$	0	0
$333$	−0.618034	−0.0338681
$334$	0	0
$335$	0	0
$336$	0	0
$337$	35.8885	1.95497	0.977487	−	0.210997i	$-0.0676709\pi$
$337$	35.8885	1.95497	0.977487	+	0.210997i	$0.0676709\pi$
$338$	0	0
$339$	5.09017	0.276460
$340$	0	0
$341$	−8.70820	−0.471576
$342$	0	0
$343$	−5.29180	−0.285730
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.61803	−0.194226	−0.0971131	−	0.995273i	$-0.530961\pi$
$347$	−3.61803	−0.194226	−0.0971131	+	0.995273i	$0.530961\pi$
$348$	0	0
$349$	−8.12461	−0.434900	−0.217450	−	0.976071i	$-0.569774\pi$
$349$	−8.12461	−0.434900	−0.217450	+	0.976071i	$0.569774\pi$
$350$	0	0
$351$	−19.0000	−1.01414
$352$	0	0
$353$	19.3607	1.03047	0.515233	−	0.857050i	$-0.327706\pi$
$353$	19.3607	1.03047	0.515233	+	0.857050i	$0.327706\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.20163	−0.0635968
$358$	0	0
$359$	−31.8885	−1.68301	−0.841506	−	0.540247i	$-0.818331\pi$
$359$	−31.8885	−1.68301	−0.841506	+	0.540247i	$0.818331\pi$
$360$	0	0
$361$	−4.83282	−0.254359
$362$	0	0
$363$	0.618034	0.0324384
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−35.9787	−1.87807	−0.939037	−	0.343817i	$-0.888280\pi$
$367$	−35.9787	−1.87807	−0.939037	+	0.343817i	$0.888280\pi$
$368$	0	0
$369$	28.0344	1.45941
$370$	0	0
$371$	1.38197	0.0717481
$372$	0	0
$373$	14.4164	0.746453	0.373227	−	0.927740i	$-0.378251\pi$
$373$	14.4164	0.746453	0.373227	+	0.927740i	$0.378251\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	45.5623	2.34658
$378$	0	0
$379$	−29.4721	−1.51388	−0.756941	−	0.653483i	$-0.773307\pi$
$379$	−29.4721	−1.51388	−0.756941	+	0.653483i	$0.773307\pi$
$380$	0	0
$381$	1.32624	0.0679452
$382$	0	0
$383$	18.9443	0.968007	0.484004	−	0.875066i	$-0.339182\pi$
$383$	18.9443	0.968007	0.484004	+	0.875066i	$0.339182\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	18.1803	0.924159
$388$	0	0
$389$	32.9443	1.67034	0.835170	−	0.549991i	$-0.185369\pi$
$389$	32.9443	1.67034	0.835170	+	0.549991i	$0.185369\pi$
$390$	0	0
$391$	1.94427	0.0983261
$392$	0	0
$393$	7.76393	0.391639
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−24.7984	−1.24459	−0.622297	−	0.782781i	$-0.713801\pi$
$397$	−24.7984	−1.24459	−0.622297	+	0.782781i	$0.713801\pi$
$398$	0	0
$399$	0.888544	0.0444828
$400$	0	0
$401$	0.472136	0.0235773	0.0117887	−	0.999931i	$-0.496247\pi$
$401$	0.472136	0.0235773	0.0117887	+	0.999931i	$0.496247\pi$
$402$	0	0
$403$	47.6525	2.37374
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.236068	−0.0117015
$408$	0	0
$409$	−20.5279	−1.01504	−0.507519	−	0.861641i	$-0.669437\pi$
$409$	−20.5279	−1.01504	−0.507519	+	0.861641i	$0.669437\pi$
$410$	0	0
$411$	−5.90983	−0.291510
$412$	0	0
$413$	4.09017	0.201264
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.90983	0.338376
$418$	0	0
$419$	12.5279	0.612026	0.306013	−	0.952027i	$-0.401005\pi$
$419$	12.5279	0.612026	0.306013	+	0.952027i	$0.401005\pi$
$420$	0	0
$421$	4.67376	0.227785	0.113893	−	0.993493i	$-0.463668\pi$
$421$	4.67376	0.227785	0.113893	+	0.993493i	$0.463668\pi$
$422$	0	0
$423$	−5.85410	−0.284636
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.729490	−0.0353025
$428$	0	0
$429$	−3.38197	−0.163283
$430$	0	0
$431$	5.76393	0.277639	0.138819	−	0.990318i	$-0.455669\pi$
$431$	5.76393	0.277639	0.138819	+	0.990318i	$0.455669\pi$
$432$	0	0
$433$	10.8885	0.523270	0.261635	−	0.965167i	$-0.415738\pi$
$433$	10.8885	0.523270	0.261635	+	0.965167i	$0.415738\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.43769	−0.0687742
$438$	0	0
$439$	−23.9230	−1.14178	−0.570891	−	0.821026i	$-0.693402\pi$
$439$	−23.9230	−1.14178	−0.570891	+	0.821026i	$0.693402\pi$
$440$	0	0
$441$	17.9443	0.854489
$442$	0	0
$443$	24.2918	1.15414	0.577069	−	0.816695i	$-0.304196\pi$
$443$	24.2918	1.15414	0.577069	+	0.816695i	$0.304196\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4.58359	−0.216796
$448$	0	0
$449$	7.72949	0.364777	0.182389	−	0.983227i	$-0.441617\pi$
$449$	7.72949	0.364777	0.182389	+	0.983227i	$0.441617\pi$
$450$	0	0
$451$	10.7082	0.504230
$452$	0	0
$453$	−9.85410	−0.462986
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.1459	0.895607	0.447804	−	0.894132i	$-0.352206\pi$
$457$	19.1459	0.895607	0.447804	+	0.894132i	$0.352206\pi$
$458$	0	0
$459$	17.6738	0.824941
$460$	0	0
$461$	6.88854	0.320831	0.160416	−	0.987050i	$-0.448717\pi$
$461$	6.88854	0.320831	0.160416	+	0.987050i	$0.448717\pi$
$462$	0	0
$463$	−36.2361	−1.68403	−0.842016	−	0.539452i	$-0.818631\pi$
$463$	−36.2361	−1.68403	−0.842016	+	0.539452i	$0.818631\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−34.7082	−1.60610	−0.803052	−	0.595909i	$-0.796792\pi$
$467$	−34.7082	−1.60610	−0.803052	+	0.595909i	$0.796792\pi$
$468$	0	0
$469$	4.58359	0.211651
$470$	0	0
$471$	−2.76393	−0.127355
$472$	0	0
$473$	6.94427	0.319298
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−9.47214	−0.433699
$478$	0	0
$479$	4.41641	0.201791	0.100895	−	0.994897i	$-0.467829\pi$
$479$	4.41641	0.201791	0.100895	+	0.994897i	$0.467829\pi$
$480$	0	0
$481$	1.29180	0.0589008
$482$	0	0
$483$	−0.0901699	−0.00410287
$484$	0	0
$485$	0	0
$486$	0	0
$487$	25.1803	1.14103	0.570515	−	0.821287i	$-0.306744\pi$
$487$	25.1803	1.14103	0.570515	+	0.821287i	$0.306744\pi$
$488$	0	0
$489$	3.94427	0.178366
$490$	0	0
$491$	7.65248	0.345351	0.172676	−	0.984979i	$-0.444759\pi$
$491$	7.65248	0.345351	0.172676	+	0.984979i	$0.444759\pi$
$492$	0	0
$493$	−42.3820	−1.90879
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.12461	−0.140158
$498$	0	0
$499$	−40.5623	−1.81582	−0.907909	−	0.419167i	$-0.862322\pi$
$499$	−40.5623	−1.81582	−0.907909	+	0.419167i	$0.862322\pi$
$500$	0	0
$501$	−4.14590	−0.185225
$502$	0	0
$503$	28.1803	1.25650	0.628250	−	0.778012i	$-0.283772\pi$
$503$	28.1803	1.25650	0.628250	+	0.778012i	$0.283772\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	10.4721	0.465084
$508$	0	0
$509$	26.8541	1.19029	0.595144	−	0.803619i	$-0.297095\pi$
$509$	26.8541	1.19029	0.595144	+	0.803619i	$0.297095\pi$
$510$	0	0
$511$	4.43769	0.196312
$512$	0	0
$513$	−13.0689	−0.577005
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−2.23607	−0.0983422
$518$	0	0
$519$	6.97871	0.306332
$520$	0	0
$521$	2.76393	0.121090	0.0605450	−	0.998165i	$-0.480716\pi$
$521$	2.76393	0.121090	0.0605450	+	0.998165i	$0.480716\pi$
$522$	0	0
$523$	17.4164	0.761566	0.380783	−	0.924664i	$-0.375654\pi$
$523$	17.4164	0.761566	0.380783	+	0.924664i	$0.375654\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−44.3262	−1.93088
$528$	0	0
$529$	−22.8541	−0.993657
$530$	0	0
$531$	−28.0344	−1.21659
$532$	0	0
$533$	−58.5967	−2.53811
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.29180	−0.0557451
$538$	0	0
$539$	6.85410	0.295227
$540$	0	0
$541$	40.4508	1.73912	0.869559	−	0.493829i	$-0.164403\pi$
$541$	40.4508	1.73912	0.869559	+	0.493829i	$0.164403\pi$
$542$	0	0
$543$	2.70820	0.116220
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.67376	0.114322	0.0571609	−	0.998365i	$-0.481795\pi$
$547$	2.67376	0.114322	0.0571609	+	0.998365i	$0.481795\pi$
$548$	0	0
$549$	5.00000	0.213395
$550$	0	0
$551$	31.3394	1.33510
$552$	0	0
$553$	2.81966	0.119904
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.47214	−0.316605	−0.158302	−	0.987391i	$-0.550602\pi$
$557$	−7.47214	−0.316605	−0.158302	+	0.987391i	$0.550602\pi$
$558$	0	0
$559$	−38.0000	−1.60723
$560$	0	0
$561$	3.14590	0.132820
$562$	0	0
$563$	1.03444	0.0435965	0.0217983	−	0.999762i	$-0.493061\pi$
$563$	1.03444	0.0435965	0.0217983	+	0.999762i	$0.493061\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.18034	0.0915657
$568$	0	0
$569$	−20.2016	−0.846896	−0.423448	−	0.905920i	$-0.639180\pi$
$569$	−20.2016	−0.846896	−0.423448	+	0.905920i	$0.639180\pi$
$570$	0	0
$571$	34.1591	1.42951	0.714756	−	0.699374i	$-0.246538\pi$
$571$	34.1591	1.42951	0.714756	+	0.699374i	$0.246538\pi$
$572$	0	0
$573$	7.43769	0.310714
$574$	0	0
$575$	0	0
$576$	0	0
$577$	29.7426	1.23820	0.619101	−	0.785311i	$-0.287497\pi$
$577$	29.7426	1.23820	0.619101	+	0.785311i	$0.287497\pi$
$578$	0	0
$579$	5.88854	0.244720
$580$	0	0
$581$	5.18034	0.214917
$582$	0	0
$583$	−3.61803	−0.149844
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.20163	−0.297243	−0.148621	−	0.988894i	$-0.547484\pi$
$587$	−7.20163	−0.297243	−0.148621	+	0.988894i	$0.547484\pi$
$588$	0	0
$589$	32.7771	1.35056
$590$	0	0
$591$	6.81966	0.280523
$592$	0	0
$593$	−16.3475	−0.671312	−0.335656	−	0.941985i	$-0.608958\pi$
$593$	−16.3475	−0.671312	−0.335656	+	0.941985i	$0.608958\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.03444	0.369755
$598$	0	0
$599$	34.3951	1.40535	0.702673	−	0.711513i	$-0.251990\pi$
$599$	34.3951	1.40535	0.702673	+	0.711513i	$0.251990\pi$
$600$	0	0
$601$	4.67376	0.190647	0.0953234	−	0.995446i	$-0.469611\pi$
$601$	4.67376	0.190647	0.0953234	+	0.995446i	$0.469611\pi$
$602$	0	0
$603$	−31.4164	−1.27938
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.83282	−0.358513	−0.179256	−	0.983802i	$-0.557369\pi$
$607$	−8.83282	−0.358513	−0.179256	+	0.983802i	$0.557369\pi$
$608$	0	0
$609$	1.96556	0.0796484
$610$	0	0
$611$	12.2361	0.495018
$612$	0	0
$613$	46.3262	1.87110	0.935550	−	0.353195i	$-0.114905\pi$
$613$	46.3262	1.87110	0.935550	+	0.353195i	$0.114905\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.2361	−0.532864	−0.266432	−	0.963854i	$-0.585845\pi$
$617$	−13.2361	−0.532864	−0.266432	+	0.963854i	$0.585845\pi$
$618$	0	0
$619$	−40.7082	−1.63620	−0.818100	−	0.575075i	$-0.804973\pi$
$619$	−40.7082	−1.63620	−0.818100	+	0.575075i	$0.804973\pi$
$620$	0	0
$621$	1.32624	0.0532201
$622$	0	0
$623$	−5.47214	−0.219236
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.32624	−0.0929010
$628$	0	0
$629$	−1.20163	−0.0479120
$630$	0	0
$631$	−38.4508	−1.53070	−0.765352	−	0.643612i	$-0.777435\pi$
$631$	−38.4508	−1.53070	−0.765352	+	0.643612i	$0.777435\pi$
$632$	0	0
$633$	0.618034	0.0245646
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−37.5066	−1.48606
$638$	0	0
$639$	21.4164	0.847220
$640$	0	0
$641$	−10.4164	−0.411423	−0.205712	−	0.978613i	$-0.565951\pi$
$641$	−10.4164	−0.411423	−0.205712	+	0.978613i	$0.565951\pi$
$642$	0	0
$643$	−25.4164	−1.00233	−0.501163	−	0.865353i	$-0.667094\pi$
$643$	−25.4164	−1.00233	−0.501163	+	0.865353i	$0.667094\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.4164	−0.645396	−0.322698	−	0.946502i	$-0.604590\pi$
$647$	−16.4164	−0.645396	−0.322698	+	0.946502i	$0.604590\pi$
$648$	0	0
$649$	−10.7082	−0.420334
$650$	0	0
$651$	2.05573	0.0805703
$652$	0	0
$653$	−47.3820	−1.85420	−0.927100	−	0.374815i	$-0.877706\pi$
$653$	−47.3820	−1.85420	−0.927100	+	0.374815i	$0.877706\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−30.4164	−1.18666
$658$	0	0
$659$	19.7426	0.769064	0.384532	−	0.923112i	$-0.374363\pi$
$659$	19.7426	0.769064	0.384532	+	0.923112i	$0.374363\pi$
$660$	0	0
$661$	26.5967	1.03449	0.517247	−	0.855836i	$-0.326957\pi$
$661$	26.5967	1.03449	0.517247	+	0.855836i	$0.326957\pi$
$662$	0	0
$663$	−17.2148	−0.668567
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.18034	−0.123143
$668$	0	0
$669$	15.8541	0.612955
$670$	0	0
$671$	1.90983	0.0737282
$672$	0	0
$673$	18.5967	0.716852	0.358426	−	0.933558i	$-0.383314\pi$
$673$	18.5967	0.716852	0.358426	+	0.933558i	$0.383314\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.59675	−0.0613680	−0.0306840	−	0.999529i	$-0.509769\pi$
$677$	−1.59675	−0.0613680	−0.0306840	+	0.999529i	$0.509769\pi$
$678$	0	0
$679$	3.96556	0.152184
$680$	0	0
$681$	−8.56231	−0.328108
$682$	0	0
$683$	1.18034	0.0451645	0.0225822	−	0.999745i	$-0.492811\pi$
$683$	1.18034	0.0451645	0.0225822	+	0.999745i	$0.492811\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.79837	0.297527
$688$	0	0
$689$	19.7984	0.754258
$690$	0	0
$691$	−40.6312	−1.54568	−0.772842	−	0.634599i	$-0.781165\pi$
$691$	−40.6312	−1.54568	−0.772842	+	0.634599i	$0.781165\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	0	0
$696$	0	0
$697$	54.5066	2.06458
$698$	0	0
$699$	−10.2361	−0.387164
$700$	0	0
$701$	−37.5279	−1.41741	−0.708704	−	0.705506i	$-0.750720\pi$
$701$	−37.5279	−1.41741	−0.708704	+	0.705506i	$0.750720\pi$
$702$	0	0
$703$	0.888544	0.0335121
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.20163	−0.195627
$708$	0	0
$709$	−24.3607	−0.914885	−0.457442	−	0.889239i	$-0.651234\pi$
$709$	−24.3607	−0.914885	−0.457442	+	0.889239i	$0.651234\pi$
$710$	0	0
$711$	−19.3262	−0.724791
$712$	0	0
$713$	−3.32624	−0.124569
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.8328	0.516596
$718$	0	0
$719$	−49.8328	−1.85845	−0.929225	−	0.369514i	$-0.879524\pi$
$719$	−49.8328	−1.85845	−0.929225	+	0.369514i	$0.879524\pi$
$720$	0	0
$721$	4.70820	0.175343
$722$	0	0
$723$	0.0901699	0.00335346
$724$	0	0
$725$	0	0
$726$	0	0
$727$	13.0902	0.485488	0.242744	−	0.970090i	$-0.421953\pi$
$727$	13.0902	0.485488	0.242744	+	0.970090i	$0.421953\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	35.3475	1.30738
$732$	0	0
$733$	4.12461	0.152346	0.0761730	−	0.997095i	$-0.475730\pi$
$733$	4.12461	0.152346	0.0761730	+	0.997095i	$0.475730\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.0000	−0.442026
$738$	0	0
$739$	2.85410	0.104990	0.0524949	−	0.998621i	$-0.483283\pi$
$739$	2.85410	0.104990	0.0524949	+	0.998621i	$0.483283\pi$
$740$	0	0
$741$	12.7295	0.467630
$742$	0	0
$743$	47.2148	1.73214	0.866071	−	0.499921i	$-0.166638\pi$
$743$	47.2148	1.73214	0.866071	+	0.499921i	$0.166638\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−35.5066	−1.29912
$748$	0	0
$749$	5.03444	0.183955
$750$	0	0
$751$	0.0344419	0.00125680	0.000628401	−	1.00000i	$-0.499800\pi$
$751$	0.0344419	0.00125680	0.000628401		1.00000i	$0.499800\pi$
$752$	0	0
$753$	−11.4377	−0.416813
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.88854	0.177677	0.0888386	−	0.996046i	$-0.471684\pi$
$757$	4.88854	0.177677	0.0888386	+	0.996046i	$0.471684\pi$
$758$	0	0
$759$	0.236068	0.00856872
$760$	0	0
$761$	8.00000	0.290000	0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000	0.290000	0.145000	+	0.989432i	$0.453682\pi$
$762$	0	0
$763$	1.36068	0.0492599
$764$	0	0
$765$	0	0
$766$	0	0
$767$	58.5967	2.11581
$768$	0	0
$769$	−41.6525	−1.50203	−0.751013	−	0.660287i	$-0.770435\pi$
$769$	−41.6525	−1.50203	−0.751013	+	0.660287i	$0.770435\pi$
$770$	0	0
$771$	3.12461	0.112530
$772$	0	0
$773$	−48.6312	−1.74914	−0.874571	−	0.484897i	$-0.838857\pi$
$773$	−48.6312	−1.74914	−0.874571	+	0.484897i	$0.838857\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.0557281	0.00199923
$778$	0	0
$779$	−40.3050	−1.44407
$780$	0	0
$781$	8.18034	0.292716
$782$	0	0
$783$	−28.9098	−1.03315
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.76393	0.0628774	0.0314387	−	0.999506i	$-0.489991\pi$
$787$	1.76393	0.0628774	0.0314387	+	0.999506i	$0.489991\pi$
$788$	0	0
$789$	9.56231	0.340427
$790$	0	0
$791$	3.14590	0.111855
$792$	0	0
$793$	−10.4508	−0.371121
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−35.5623	−1.25968	−0.629841	−	0.776724i	$-0.716880\pi$
$797$	−35.5623	−1.25968	−0.629841	+	0.776724i	$0.716880\pi$
$798$	0	0
$799$	−11.3820	−0.402665
$800$	0	0
$801$	37.5066	1.32523
$802$	0	0
$803$	−11.6180	−0.409992
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.61803	−0.0569575
$808$	0	0
$809$	−50.2492	−1.76667	−0.883334	−	0.468744i	$-0.844707\pi$
$809$	−50.2492	−1.76667	−0.883334	+	0.468744i	$0.844707\pi$
$810$	0	0
$811$	13.9443	0.489650	0.244825	−	0.969567i	$-0.421270\pi$
$811$	13.9443	0.489650	0.244825	+	0.969567i	$0.421270\pi$
$812$	0	0
$813$	−7.85410	−0.275455
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−26.1378	−0.914445
$818$	0	0
$819$	−5.47214	−0.191212
$820$	0	0
$821$	−52.7214	−1.83999	−0.919994	−	0.391932i	$-0.871807\pi$
$821$	−52.7214	−1.83999	−0.919994	+	0.391932i	$0.871807\pi$
$822$	0	0
$823$	−14.8328	−0.517039	−0.258520	−	0.966006i	$-0.583235\pi$
$823$	−14.8328	−0.517039	−0.258520	+	0.966006i	$0.583235\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.23607	−0.321170	−0.160585	−	0.987022i	$-0.551338\pi$
$827$	−9.23607	−0.321170	−0.160585	+	0.987022i	$0.551338\pi$
$828$	0	0
$829$	−23.1591	−0.804347	−0.402174	−	0.915563i	$-0.631745\pi$
$829$	−23.1591	−0.804347	−0.402174	+	0.915563i	$0.631745\pi$
$830$	0	0
$831$	−16.1459	−0.560095
$832$	0	0
$833$	34.8885	1.20882
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−30.2361	−1.04511
$838$	0	0
$839$	8.85410	0.305678	0.152839	−	0.988251i	$-0.451158\pi$
$839$	8.85410	0.305678	0.152839	+	0.988251i	$0.451158\pi$
$840$	0	0
$841$	40.3262	1.39056
$842$	0	0
$843$	2.61803	0.0901699
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.381966	0.0131245
$848$	0	0
$849$	−16.9656	−0.582256
$850$	0	0
$851$	−0.0901699	−0.00309099
$852$	0	0
$853$	−11.5623	−0.395886	−0.197943	−	0.980214i	$-0.563426\pi$
$853$	−11.5623	−0.395886	−0.197943	+	0.980214i	$0.563426\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.8328	−0.404201	−0.202101	−	0.979365i	$-0.564777\pi$
$857$	−11.8328	−0.404201	−0.202101	+	0.979365i	$0.564777\pi$
$858$	0	0
$859$	21.7082	0.740674	0.370337	−	0.928897i	$-0.379242\pi$
$859$	21.7082	0.740674	0.370337	+	0.928897i	$0.379242\pi$
$860$	0	0
$861$	−2.52786	−0.0861494
$862$	0	0
$863$	−52.4164	−1.78428	−0.892138	−	0.451764i	$-0.850795\pi$
$863$	−52.4164	−1.78428	−0.892138	+	0.451764i	$0.850795\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.50658	0.187013
$868$	0	0
$869$	−7.38197	−0.250416
$870$	0	0
$871$	65.6656	2.22500
$872$	0	0
$873$	−27.1803	−0.919915
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−45.8885	−1.54955	−0.774773	−	0.632239i	$-0.782136\pi$
$877$	−45.8885	−1.54955	−0.774773	+	0.632239i	$0.782136\pi$
$878$	0	0
$879$	−7.12461	−0.240307
$880$	0	0
$881$	14.4508	0.486861	0.243431	−	0.969918i	$-0.421727\pi$
$881$	14.4508	0.486861	0.243431	+	0.969918i	$0.421727\pi$
$882$	0	0
$883$	25.8328	0.869343	0.434672	−	0.900589i	$-0.356864\pi$
$883$	25.8328	0.869343	0.434672	+	0.900589i	$0.356864\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.0000	−0.570804	−0.285402	−	0.958408i	$-0.592127\pi$
$887$	−17.0000	−0.570804	−0.285402	+	0.958408i	$0.592127\pi$
$888$	0	0
$889$	0.819660	0.0274905
$890$	0	0
$891$	−5.70820	−0.191232
$892$	0	0
$893$	8.41641	0.281644
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.29180	−0.0431318
$898$	0	0
$899$	72.5066	2.41823
$900$	0	0
$901$	−18.4164	−0.613540
$902$	0	0
$903$	−1.63932	−0.0545532
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−23.7771	−0.789505	−0.394753	−	0.918787i	$-0.629170\pi$
$907$	−23.7771	−0.789505	−0.394753	+	0.918787i	$0.629170\pi$
$908$	0	0
$909$	35.6525	1.18252
$910$	0	0
$911$	13.0557	0.432556	0.216278	−	0.976332i	$-0.430608\pi$
$911$	13.0557	0.432556	0.216278	+	0.976332i	$0.430608\pi$
$912$	0	0
$913$	−13.5623	−0.448847
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.79837	0.158456
$918$	0	0
$919$	52.3607	1.72722	0.863610	−	0.504161i	$-0.168198\pi$
$919$	52.3607	1.72722	0.863610	+	0.504161i	$0.168198\pi$
$920$	0	0
$921$	14.7426	0.485787
$922$	0	0
$923$	−44.7639	−1.47342
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−32.2705	−1.05990
$928$	0	0
$929$	−36.8197	−1.20801	−0.604007	−	0.796979i	$-0.706430\pi$
$929$	−36.8197	−1.20801	−0.604007	+	0.796979i	$0.706430\pi$
$930$	0	0
$931$	−25.7984	−0.845508
$932$	0	0
$933$	−3.67376	−0.120274
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.583592	−0.0190651	−0.00953256	−	0.999955i	$-0.503034\pi$
$937$	−0.583592	−0.0190651	−0.00953256	+	0.999955i	$0.503034\pi$
$938$	0	0
$939$	−6.58359	−0.214847
$940$	0	0
$941$	−8.81966	−0.287513	−0.143756	−	0.989613i	$-0.545918\pi$
$941$	−8.81966	−0.287513	−0.143756	+	0.989613i	$0.545918\pi$
$942$	0	0
$943$	4.09017	0.133194
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.12461	−0.264014	−0.132007	−	0.991249i	$-0.542142\pi$
$947$	−8.12461	−0.264014	−0.132007	+	0.991249i	$0.542142\pi$
$948$	0	0
$949$	63.5755	2.06375
$950$	0	0
$951$	2.09017	0.0677784
$952$	0	0
$953$	−0.111456	−0.00361042	−0.00180521	−	0.999998i	$-0.500575\pi$
$953$	−0.111456	−0.00361042	−0.00180521	+	0.999998i	$0.500575\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−5.14590	−0.166343
$958$	0	0
$959$	−3.65248	−0.117945
$960$	0	0
$961$	44.8328	1.44622
$962$	0	0
$963$	−34.5066	−1.11196
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.72949	0.0877745	0.0438872	−	0.999036i	$-0.486026\pi$
$967$	2.72949	0.0877745	0.0438872	+	0.999036i	$0.486026\pi$
$968$	0	0
$969$	−11.8409	−0.380386
$970$	0	0
$971$	0.381966	0.0122579	0.00612894	−	0.999981i	$-0.498049\pi$
$971$	0.381966	0.0122579	0.00612894	+	0.999981i	$0.498049\pi$
$972$	0	0
$973$	4.27051	0.136906
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.5410	1.07307	0.536536	−	0.843877i	$-0.319733\pi$
$977$	33.5410	1.07307	0.536536	+	0.843877i	$0.319733\pi$
$978$	0	0
$979$	14.3262	0.457869
$980$	0	0
$981$	−9.32624	−0.297764
$982$	0	0
$983$	40.9443	1.30592	0.652960	−	0.757393i	$-0.273527\pi$
$983$	40.9443	1.30592	0.652960	+	0.757393i	$0.273527\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.527864	0.0168021
$988$	0	0
$989$	2.65248	0.0843438
$990$	0	0
$991$	49.0902	1.55940	0.779700	−	0.626153i	$-0.215371\pi$
$991$	49.0902	1.55940	0.779700	+	0.626153i	$0.215371\pi$
$992$	0	0
$993$	−8.25735	−0.262039
$994$	0	0
$995$	0	0
$996$	0	0
$997$	56.0344	1.77463	0.887314	−	0.461165i	$-0.152568\pi$
$997$	56.0344	1.77463	0.887314	+	0.461165i	$0.152568\pi$
$998$	0	0
$999$	−0.819660	−0.0259329

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bm.1.2		2
4.3	odd	2		2200.2.a.q.1.1	yes	2
5.2	odd	4		4400.2.b.z.4049.2		4
5.3	odd	4		4400.2.b.z.4049.3		4
5.4	even	2		4400.2.a.bo.1.1		2
20.3	even	4		2200.2.b.k.1849.2		4
20.7	even	4		2200.2.b.k.1849.3		4
20.19	odd	2		2200.2.a.p.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.p.1.2	✓	2	20.19	odd	2
2200.2.a.q.1.1	yes	2	4.3	odd	2
2200.2.b.k.1849.2		4	20.3	even	4
2200.2.b.k.1849.3		4	20.7	even	4
4400.2.a.bm.1.2		2	1.1	even	1	trivial
4400.2.a.bo.1.1		2	5.4	even	2
4400.2.b.z.4049.2		4	5.2	odd	4
4400.2.b.z.4049.3		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{3} +0.381966 q^{7} -2.61803 q^{9} -1.00000 q^{11} +5.47214 q^{13} -5.09017 q^{17} +3.76393 q^{19} +0.236068 q^{21} -0.381966 q^{23} -3.47214 q^{27} +8.32624 q^{29} +8.70820 q^{31} -0.618034 q^{33} +0.236068 q^{37} +3.38197 q^{39} -10.7082 q^{41} -6.94427 q^{43} +2.23607 q^{47} -6.85410 q^{49} -3.14590 q^{51} +3.61803 q^{53} +2.32624 q^{57} +10.7082 q^{59} -1.90983 q^{61} -1.00000 q^{63} +12.0000 q^{67} -0.236068 q^{69} -8.18034 q^{71} +11.6180 q^{73} -0.381966 q^{77} +7.38197 q^{79} +5.70820 q^{81} +13.5623 q^{83} +5.14590 q^{87} -14.3262 q^{89} +2.09017 q^{91} +5.38197 q^{93} +10.3820 q^{97} +2.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 3 q^{7} - 3 q^{9} - 2 q^{11} + 2 q^{13} + q^{17} + 12 q^{19} - 4 q^{21} - 3 q^{23} + 2 q^{27} + q^{29} + 4 q^{31} + q^{33} - 4 q^{37} + 9 q^{39} - 8 q^{41} + 4 q^{43} - 7 q^{49} - 13 q^{51}+ \cdots + 3 q^{99}+O(q^{100}) \) 2 * q - q^3 + 3 * q^7 - 3 * q^9 - 2 * q^11 + 2 * q^13 + q^17 + 12 * q^19 - 4 * q^21 - 3 * q^23 + 2 * q^27 + q^29 + 4 * q^31 + q^33 - 4 * q^37 + 9 * q^39 - 8 * q^41 + 4 * q^43 - 7 * q^49 - 13 * q^51 + 5 * q^53 - 11 * q^57 + 8 * q^59 - 15 * q^61 - 2 * q^63 + 24 * q^67 + 4 * q^69 + 6 * q^71 + 21 * q^73 - 3 * q^77 + 17 * q^79 - 2 * q^81 + 7 * q^83 + 17 * q^87 - 13 * q^89 - 7 * q^91 + 13 * q^93 + 23 * q^97 + 3 * q^99

Introduction

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