LMFDB - Embedded newform 1100.3.e.c.549.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1100,3,Mod(549,1100)] 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("1100.549"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1100.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$29.9728290796$
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} + 174 x^{14} + 10969 x^{12} + 318076 x^{10} + 4442560 x^{8} + 28982576 x^{6} + 77210944 x^{4} + \cdots + 26790976 $ x^16 + 174x^14 + 10969x^12 + 318076x^10 + 4442560x^8 + 28982576x^6 + 77210944x^4 + 82074624*x^2 + 26790976
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			549.14
Root			$-0.761183i$ of defining polynomial
Character	$\chi$	$=$	1100.549
Dual form			1100.3.e.c.549.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.14607i q^{3} +1.70661 q^{7} -8.18991 q^{9} +(-3.98865 - 10.2514i) q^{11} +16.6054 q^{13} +0.512519 q^{17} +19.5435i q^{19} +7.07571i q^{21} +11.2338i q^{23} +3.35868i q^{27} +48.1096i q^{29} +5.40252 q^{31} +(42.5029 - 16.5372i) q^{33} -0.530947i q^{37} +68.8472i q^{39} +28.0481i q^{41} -3.65348 q^{43} +3.58583i q^{47} -46.0875 q^{49} +2.12494i q^{51} +51.9269i q^{53} -81.0288 q^{57} -41.1738 q^{59} +42.3172i q^{61} -13.9770 q^{63} -73.5680i q^{67} -46.5760 q^{69} -13.3915 q^{71} +107.634 q^{73} +(-6.80706 - 17.4951i) q^{77} -15.6655i q^{79} -87.6345 q^{81} -16.3133 q^{83} -199.466 q^{87} +140.699 q^{89} +28.3389 q^{91} +22.3992i q^{93} -97.0976i q^{97} +(32.6667 + 83.9579i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{9} + 6 q^{11} + 28 q^{31} - 28 q^{49} - 256 q^{59} + 352 q^{69} - 68 q^{71} - 256 q^{81} + 292 q^{89} + 228 q^{91} - 16 q^{99}+O(q^{100}) $ 16 * q - 32 * q^9 + 6 * q^11 + 28 * q^31 - 28 * q^49 - 256 * q^59 + 352 * q^69 - 68 * q^71 - 256 * q^81 + 292 * q^89 + 228 * q^91 - 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

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
$3$			4.14607i			1.38202i	0.722843	+	0.691012i	$0.242835\pi$
$3$			4.14607i			1.38202i	−0.722843	+	0.691012i	$0.757165\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.70661			0.243801			0.121900	−	0.992542i	$-0.461101\pi$
$7$	1.70661			0.243801			0.121900	+	0.992542i	$0.461101\pi$
$8$		0			0
$9$	−8.18991			−0.909990
$10$		0			0
$11$	−3.98865	−	10.2514i	−0.362605	−	0.931943i
$11$	−3.98865	−	10.2514i	−0.362605	−	0.931943i
$12$		0			0
$13$	16.6054			1.27734			0.638669	−	0.769481i	$-0.279485\pi$
$13$	16.6054			1.27734			0.638669	+	0.769481i	$0.279485\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.512519			0.0301482			0.0150741	−	0.999886i	$-0.495202\pi$
$17$	0.512519			0.0301482			0.0150741	+	0.999886i	$0.495202\pi$
$18$		0			0
$19$			19.5435i			1.02861i	0.857609	+	0.514303i	$0.171949\pi$
$19$			19.5435i			1.02861i	−0.857609	+	0.514303i	$0.828051\pi$
$20$		0			0
$21$			7.07571i			0.336939i
$22$		0			0
$23$			11.2338i			0.488424i	0.969722	+	0.244212i	$0.0785293\pi$
$23$			11.2338i			0.488424i	−0.969722	+	0.244212i	$0.921471\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			3.35868i			0.124395i
$28$		0			0
$29$			48.1096i			1.65895i	0.558543	+	0.829476i	$0.311361\pi$
$29$			48.1096i			1.65895i	−0.558543	+	0.829476i	$0.688639\pi$
$30$		0			0
$31$	5.40252			0.174275			0.0871374	−	0.996196i	$-0.472228\pi$
$31$	5.40252			0.174275			0.0871374	+	0.996196i	$0.472228\pi$
$32$		0			0
$33$	42.5029	−	16.5372i	1.28797	−	0.501129i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	0.530947i		−	0.0143499i	−0.999974	−	0.00717495i	$-0.997716\pi$
$37$		−	0.530947i		−	0.0143499i	0.999974	−	0.00717495i	$-0.00228388\pi$
$38$		0			0
$39$			68.8472i			1.76531i
$40$		0			0
$41$			28.0481i			0.684101i	0.939682	+	0.342051i	$0.111121\pi$
$41$			28.0481i			0.684101i	−0.939682	+	0.342051i	$0.888879\pi$
$42$		0			0
$43$	−3.65348			−0.0849646			−0.0424823	−	0.999097i	$-0.513527\pi$
$43$	−3.65348			−0.0849646			−0.0424823	+	0.999097i	$0.513527\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.58583i			0.0762943i	0.999272	+	0.0381472i	$0.0121456\pi$
$47$			3.58583i			0.0762943i	−0.999272	+	0.0381472i	$0.987854\pi$
$48$		0			0
$49$	−46.0875			−0.940561
$50$		0			0
$51$			2.12494i			0.0416655i
$52$		0			0
$53$			51.9269i			0.979752i	0.871792	+	0.489876i	$0.162958\pi$
$53$			51.9269i			0.979752i	−0.871792	+	0.489876i	$0.837042\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−81.0288			−1.42156
$58$		0			0
$59$	−41.1738			−0.697861			−0.348931	−	0.937149i	$-0.613455\pi$
$59$	−41.1738			−0.697861			−0.348931	+	0.937149i	$0.613455\pi$
$60$		0			0
$61$			42.3172i			0.693725i	0.937916	+	0.346863i	$0.112753\pi$
$61$			42.3172i			0.693725i	−0.937916	+	0.346863i	$0.887247\pi$
$62$		0			0
$63$	−13.9770			−0.221857
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	73.5680i		−	1.09803i	−0.835813	−	0.549015i	$-0.815003\pi$
$67$		−	73.5680i		−	1.09803i	0.835813	−	0.549015i	$-0.184997\pi$
$68$		0			0
$69$	−46.5760			−0.675014
$70$		0			0
$71$	−13.3915			−0.188612			−0.0943062	−	0.995543i	$-0.530063\pi$
$71$	−13.3915			−0.188612			−0.0943062	+	0.995543i	$0.530063\pi$
$72$		0			0
$73$	107.634			1.47443			0.737216	−	0.675657i	$-0.236140\pi$
$73$	107.634			1.47443			0.737216	+	0.675657i	$0.236140\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.80706	−	17.4951i	−0.0884034	−	0.227209i
$78$		0			0
$79$		−	15.6655i		−	0.198298i	−0.995073	−	0.0991489i	$-0.968388\pi$
$79$		−	15.6655i		−	0.198298i	0.995073	−	0.0991489i	$-0.0316120\pi$
$80$		0			0
$81$	−87.6345			−1.08191
$82$		0			0
$83$	−16.3133			−0.196546			−0.0982728	−	0.995160i	$-0.531332\pi$
$83$	−16.3133			−0.196546			−0.0982728	+	0.995160i	$0.531332\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−199.466			−2.29271
$88$		0			0
$89$	140.699			1.58088			0.790442	−	0.612536i	$-0.209851\pi$
$89$	140.699			1.58088			0.790442	+	0.612536i	$0.209851\pi$
$90$		0			0
$91$	28.3389			0.311416
$92$		0			0
$93$			22.3992i			0.240852i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	97.0976i		−	1.00101i	−0.865735	−	0.500503i	$-0.833148\pi$
$97$		−	97.0976i		−	1.00101i	0.865735	−	0.500503i	$-0.166852\pi$
$98$		0			0
$99$	32.6667	+	83.9579i	0.329967	+	0.848059i
$100$		0			0
$101$			168.940i			1.67268i	0.548214	+	0.836338i	$0.315308\pi$
$101$			168.940i			1.67268i	−0.548214	+	0.836338i	$0.684692\pi$
$102$		0			0
$103$		−	121.975i		−	1.18422i	−0.805856	−	0.592112i	$-0.798294\pi$
$103$		−	121.975i		−	1.18422i	0.805856	−	0.592112i	$-0.201706\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−109.905			−1.02715			−0.513573	−	0.858046i	$-0.671678\pi$
$107$	−109.905			−1.02715			−0.513573	+	0.858046i	$0.671678\pi$
$108$		0			0
$109$		−	127.789i		−	1.17238i	−0.810174	−	0.586189i	$-0.800628\pi$
$109$		−	127.789i		−	1.17238i	0.810174	−	0.586189i	$-0.199372\pi$
$110$		0			0
$111$	2.20134			0.0198319
$112$		0			0
$113$			100.189i			0.886631i	0.896366	+	0.443316i	$0.146198\pi$
$113$			100.189i			0.886631i	−0.896366	+	0.443316i	$0.853802\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−135.997			−1.16237
$118$		0			0
$119$	0.874669			0.00735016
$120$		0			0
$121$	−89.1813	+	81.7783i	−0.737035	+	0.675854i
$122$		0			0
$123$	−116.290			−0.945444
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−159.105			−1.25279			−0.626396	−	0.779505i	$-0.715471\pi$
$127$	−159.105			−1.25279			−0.626396	+	0.779505i	$0.715471\pi$
$128$		0			0
$129$		−	15.1476i		−	0.117423i
$130$		0			0
$131$			103.534i			0.790335i	0.918609	+	0.395167i	$0.129313\pi$
$131$			103.534i			0.790335i	−0.918609	+	0.395167i	$0.870687\pi$
$132$		0			0
$133$			33.3531i			0.250775i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			29.0495i			0.212040i	0.994364	+	0.106020i	$0.0338107\pi$
$137$			29.0495i			0.212040i	−0.994364	+	0.106020i	$0.966189\pi$
$138$		0			0
$139$			62.4675i			0.449406i	0.974427	+	0.224703i	$0.0721412\pi$
$139$			62.4675i			0.449406i	−0.974427	+	0.224703i	$0.927859\pi$
$140$		0			0
$141$	−14.8671			−0.105441
$142$		0			0
$143$	−66.2332	−	170.228i	−0.463169	−	1.19041i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	191.082i		−	1.29988i
$148$		0			0
$149$		−	48.4621i		−	0.325249i	−0.986688	−	0.162624i	$-0.948004\pi$
$149$		−	48.4621i		−	0.325249i	0.986688	−	0.162624i	$-0.0519959\pi$
$150$		0			0
$151$			134.512i			0.890811i	0.895329	+	0.445405i	$0.146940\pi$
$151$			134.512i			0.890811i	−0.895329	+	0.445405i	$0.853060\pi$
$152$		0			0
$153$	−4.19749			−0.0274346
$154$		0			0
$155$		0			0
$156$		0			0
$157$			215.729i			1.37407i	0.726624	+	0.687036i	$0.241088\pi$
$157$			215.729i			1.37407i	−0.726624	+	0.687036i	$0.758912\pi$
$158$		0			0
$159$	−215.293			−1.35404
$160$		0			0
$161$			19.1716i			0.119078i
$162$		0			0
$163$			107.030i			0.656629i	0.944569	+	0.328314i	$0.106481\pi$
$163$			107.030i			0.656629i	−0.944569	+	0.328314i	$0.893519\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.69504			−0.0460781			−0.0230391	−	0.999735i	$-0.507334\pi$
$167$	−7.69504			−0.0460781			−0.0230391	+	0.999735i	$0.507334\pi$
$168$		0			0
$169$	106.739			0.631593
$170$		0			0
$171$		−	160.060i		−	0.936021i
$172$		0			0
$173$	83.0910			0.480295			0.240147	−	0.970736i	$-0.422804\pi$
$173$	83.0910			0.480295			0.240147	+	0.970736i	$0.422804\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	170.710i		−	0.964461i
$178$		0			0
$179$	−304.889			−1.70329			−0.851645	−	0.524119i	$-0.824395\pi$
$179$	−304.889			−1.70329			−0.851645	+	0.524119i	$0.824395\pi$
$180$		0			0
$181$	−19.7418			−0.109071			−0.0545353	−	0.998512i	$-0.517368\pi$
$181$	−19.7418			−0.109071			−0.0545353	+	0.998512i	$0.517368\pi$
$182$		0			0
$183$	−175.450			−0.958745
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.04426	−	5.25403i	−0.0109319	−	0.0280964i
$188$		0			0
$189$			5.73194i			0.0303277i
$190$		0			0
$191$	−159.618			−0.835699			−0.417849	−	0.908516i	$-0.637216\pi$
$191$	−159.618			−0.835699			−0.417849	+	0.908516i	$0.637216\pi$
$192$		0			0
$193$	278.288			1.44191			0.720954	−	0.692983i	$-0.243704\pi$
$193$	278.288			1.44191			0.720954	+	0.692983i	$0.243704\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	157.345			0.798704			0.399352	−	0.916798i	$-0.369235\pi$
$197$	157.345			0.798704			0.399352	+	0.916798i	$0.369235\pi$
$198$		0			0
$199$	339.116			1.70410			0.852050	−	0.523460i	$-0.175359\pi$
$199$	339.116			1.70410			0.852050	+	0.523460i	$0.175359\pi$
$200$		0			0
$201$	305.018			1.51750
$202$		0			0
$203$			82.1042i			0.404454i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	92.0035i		−	0.444461i
$208$		0			0
$209$	200.348	−	77.9523i	0.958602	−	0.372977i
$210$		0			0
$211$		−	156.642i		−	0.742381i	−0.928557	−	0.371191i	$-0.878950\pi$
$211$		−	156.642i		−	0.742381i	0.928557	−	0.371191i	$-0.121050\pi$
$212$		0			0
$213$		−	55.5220i		−	0.260667i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	9.21998			0.0424884
$218$		0			0
$219$			446.256i			2.03770i
$220$		0			0
$221$	8.51059			0.0385095
$222$		0			0
$223$			247.418i			1.10950i	0.832017	+	0.554749i	$0.187186\pi$
$223$			247.418i			1.10950i	−0.832017	+	0.554749i	$0.812814\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	339.323			1.49481			0.747407	−	0.664366i	$-0.231298\pi$
$227$	339.323			1.49481			0.747407	+	0.664366i	$0.231298\pi$
$228$		0			0
$229$	−152.661			−0.666642			−0.333321	−	0.942813i	$-0.608169\pi$
$229$	−152.661			−0.666642			−0.333321	+	0.942813i	$0.608169\pi$
$230$		0			0
$231$	72.5358	−	28.2226i	0.314008	−	0.122176i
$232$		0			0
$233$	397.264			1.70499			0.852497	−	0.522731i	$-0.175087\pi$
$233$	397.264			1.70499			0.852497	+	0.522731i	$0.175087\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	64.9504			0.274052
$238$		0			0
$239$			99.6838i			0.417087i	0.978013	+	0.208544i	$0.0668723\pi$
$239$			99.6838i			0.417087i	−0.978013	+	0.208544i	$0.933128\pi$
$240$		0			0
$241$		−	247.891i		−	1.02859i	−0.857613	−	0.514296i	$-0.828053\pi$
$241$		−	247.891i		−	1.02859i	0.857613	−	0.514296i	$-0.171947\pi$
$242$		0			0
$243$		−	333.111i		−	1.37083i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			324.528i			1.31388i
$248$		0			0
$249$		−	67.6360i		−	0.271631i
$250$		0			0
$251$	−117.545			−0.468309			−0.234154	−	0.972199i	$-0.575232\pi$
$251$	−117.545			−0.468309			−0.234154	+	0.972199i	$0.575232\pi$
$252$		0			0
$253$	115.161	−	44.8076i	0.455183	−	0.177105i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	11.4659i		−	0.0446146i	−0.999751	−	0.0223073i	$-0.992899\pi$
$257$		−	11.4659i		−	0.0446146i	0.999751	−	0.0223073i	$-0.00710122\pi$
$258$		0			0
$259$		−	0.906117i		−	0.00349852i
$260$		0			0
$261$		−	394.013i		−	1.50963i
$262$		0			0
$263$	−164.627			−0.625960			−0.312980	−	0.949760i	$-0.601327\pi$
$263$	−164.627			−0.625960			−0.312980	+	0.949760i	$0.601327\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			583.347i			2.18482i
$268$		0			0
$269$	−361.059			−1.34223			−0.671113	−	0.741355i	$-0.734183\pi$
$269$	−361.059			−1.34223			−0.671113	+	0.741355i	$0.734183\pi$
$270$		0			0
$271$			155.764i			0.574776i	0.957814	+	0.287388i	$0.0927869\pi$
$271$			155.764i			0.574776i	−0.957814	+	0.287388i	$0.907213\pi$
$272$		0			0
$273$			117.495i			0.430385i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	390.963			1.41142			0.705708	−	0.708502i	$-0.250629\pi$
$277$	390.963			1.41142			0.705708	+	0.708502i	$0.250629\pi$
$278$		0			0
$279$	−44.2462			−0.158588
$280$		0			0
$281$			82.7795i			0.294589i	0.989093	+	0.147295i	$0.0470565\pi$
$281$			82.7795i			0.294589i	−0.989093	+	0.147295i	$0.952943\pi$
$282$		0			0
$283$	197.972			0.699549			0.349775	−	0.936834i	$-0.386258\pi$
$283$	197.972			0.699549			0.349775	+	0.936834i	$0.386258\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			47.8672i			0.166785i
$288$		0			0
$289$	−288.737			−0.999091
$290$		0			0
$291$	402.574			1.38341
$292$		0			0
$293$	−127.860			−0.436384			−0.218192	−	0.975906i	$-0.570016\pi$
$293$	−127.860			−0.436384			−0.218192	+	0.975906i	$0.570016\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	34.4311	−	13.3966i	0.115929	−	0.0451064i
$298$		0			0
$299$			186.541i			0.623883i
$300$		0			0
$301$	−6.23505			−0.0207145
$302$		0			0
$303$	−700.438			−2.31168
$304$		0			0
$305$		0			0
$306$		0			0
$307$	559.530			1.82257			0.911287	−	0.411772i	$-0.135090\pi$
$307$	559.530			1.82257			0.911287	+	0.411772i	$0.135090\pi$
$308$		0			0
$309$	505.717			1.63662
$310$		0			0
$311$	208.493			0.670395			0.335197	−	0.942148i	$-0.391197\pi$
$311$	208.493			0.670395			0.335197	+	0.942148i	$0.391197\pi$
$312$		0			0
$313$			490.115i			1.56586i	0.622108	+	0.782931i	$0.286276\pi$
$313$			490.115i			1.56586i	−0.622108	+	0.782931i	$0.713724\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	178.470i		−	0.562997i	−0.959562	−	0.281499i	$-0.909168\pi$
$317$		−	178.470i		−	0.562997i	0.959562	−	0.281499i	$-0.0908315\pi$
$318$		0			0
$319$	493.189	−	191.892i	1.54605	−	0.601544i
$320$		0			0
$321$		−	455.672i		−	1.41954i
$322$		0			0
$323$			10.0164i			0.0310106i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	529.823			1.62025
$328$		0			0
$329$			6.11961i			0.0186006i
$330$		0			0
$331$	−424.264			−1.28177			−0.640883	−	0.767639i	$-0.721431\pi$
$331$	−424.264			−1.28177			−0.640883	+	0.767639i	$0.721431\pi$
$332$		0			0
$333$			4.34841i			0.0130583i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−391.858			−1.16278			−0.581392	−	0.813624i	$-0.697492\pi$
$337$	−391.858			−1.16278			−0.581392	+	0.813624i	$0.697492\pi$
$338$		0			0
$339$	−415.392			−1.22535
$340$		0			0
$341$	−21.5488	−	55.3832i	−0.0631929	−	0.162414i
$342$		0			0
$343$	−162.277			−0.473111
$344$		0			0
$345$		0			0
$346$		0			0
$347$	515.467			1.48550			0.742748	−	0.669571i	$-0.233522\pi$
$347$	515.467			1.48550			0.742748	+	0.669571i	$0.233522\pi$
$348$		0			0
$349$		−	413.637i		−	1.18521i	−0.805495	−	0.592603i	$-0.798100\pi$
$349$		−	413.637i		−	1.18521i	0.805495	−	0.592603i	$-0.201900\pi$
$350$		0			0
$351$			55.7722i			0.158895i
$352$		0			0
$353$		−	672.441i		−	1.90493i	−0.304646	−	0.952466i	$-0.598538\pi$
$353$		−	672.441i		−	1.90493i	0.304646	−	0.952466i	$-0.401462\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			3.62644i			0.0101581i
$358$		0			0
$359$		−	655.797i		−	1.82673i	−0.407138	−	0.913367i	$-0.633473\pi$
$359$		−	655.797i		−	1.82673i	0.407138	−	0.913367i	$-0.366527\pi$
$360$		0			0
$361$	−20.9485			−0.0580291
$362$		0			0
$363$	−339.059	−	369.752i	−0.934047	−	1.01860i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	642.013i		−	1.74935i	−0.484706	−	0.874677i	$-0.661073\pi$
$367$		−	642.013i		−	1.74935i	0.484706	−	0.874677i	$-0.338927\pi$
$368$		0			0
$369$		−	229.712i		−	0.622526i
$370$		0			0
$371$			88.6187i			0.238865i
$372$		0			0
$373$	444.369			1.19134			0.595669	−	0.803230i	$-0.296887\pi$
$373$	444.369			1.19134			0.595669	+	0.803230i	$0.296887\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			798.879i			2.11904i
$378$		0			0
$379$	−93.9534			−0.247898			−0.123949	−	0.992289i	$-0.539556\pi$
$379$	−93.9534			−0.247898			−0.123949	+	0.992289i	$0.539556\pi$
$380$		0			0
$381$		−	659.659i		−	1.73139i
$382$		0			0
$383$		−	81.6845i		−	0.213275i	−0.994298	−	0.106638i	$-0.965992\pi$
$383$		−	81.6845i		−	0.213275i	0.994298	−	0.106638i	$-0.0340085\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	29.9217			0.0773170
$388$		0			0
$389$	648.091			1.66604			0.833022	−	0.553240i	$-0.186609\pi$
$389$	648.091			1.66604			0.833022	+	0.553240i	$0.186609\pi$
$390$		0			0
$391$			5.75752i			0.0147251i
$392$		0			0
$393$	−429.259			−1.09226
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	121.217i		−	0.305334i	−0.988278	−	0.152667i	$-0.951214\pi$
$397$		−	121.217i		−	0.305334i	0.988278	−	0.152667i	$-0.0487861\pi$
$398$		0			0
$399$	−138.284			−0.346577
$400$		0			0
$401$	−46.6180			−0.116254			−0.0581272	−	0.998309i	$-0.518513\pi$
$401$	−46.6180			−0.116254			−0.0581272	+	0.998309i	$0.518513\pi$
$402$		0			0
$403$	89.7110			0.222608
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.44293	+	2.11776i	−0.0133733	+	0.00520335i
$408$		0			0
$409$		−	434.331i		−	1.06193i	−0.847392	−	0.530967i	$-0.821829\pi$
$409$		−	434.331i		−	1.06193i	0.847392	−	0.530967i	$-0.178171\pi$
$410$		0			0
$411$	−120.441			−0.293044
$412$		0			0
$413$	−70.2675			−0.170139
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−258.995			−0.621090
$418$		0			0
$419$	521.101			1.24368			0.621839	−	0.783145i	$-0.286386\pi$
$419$	521.101			1.24368			0.621839	+	0.783145i	$0.286386\pi$
$420$		0			0
$421$	678.997			1.61282			0.806410	−	0.591357i	$-0.201408\pi$
$421$	678.997			1.61282			0.806410	+	0.591357i	$0.201408\pi$
$422$		0			0
$423$		−	29.3677i		−	0.0694271i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			72.2189i			0.169131i
$428$		0			0
$429$	705.778	−	274.608i	1.64517	−	0.640111i
$430$		0			0
$431$			731.778i			1.69786i	0.528505	+	0.848930i	$0.322753\pi$
$431$			731.778i			1.69786i	−0.528505	+	0.848930i	$0.677247\pi$
$432$		0			0
$433$			434.851i			1.00428i	0.864788	+	0.502138i	$0.167453\pi$
$433$			434.851i			1.00428i	−0.864788	+	0.502138i	$0.832547\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−219.547			−0.502396
$438$		0			0
$439$		−	762.031i		−	1.73583i	−0.496709	−	0.867917i	$-0.665458\pi$
$439$		−	762.031i		−	1.73583i	0.496709	−	0.867917i	$-0.334542\pi$
$440$		0			0
$441$	377.453			0.855902
$442$		0			0
$443$			156.276i			0.352768i	0.984321	+	0.176384i	$0.0564400\pi$
$443$			156.276i			0.352768i	−0.984321	+	0.176384i	$0.943560\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	200.927			0.449502
$448$		0			0
$449$	−545.999			−1.21603			−0.608017	−	0.793924i	$-0.708035\pi$
$449$	−545.999			−1.21603			−0.608017	+	0.793924i	$0.708035\pi$
$450$		0			0
$451$	287.532	−	111.874i	0.637543	−	0.248058i
$452$		0			0
$453$	−557.698			−1.23112
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−310.676			−0.679816			−0.339908	−	0.940459i	$-0.610396\pi$
$457$	−310.676			−0.679816			−0.339908	+	0.940459i	$0.610396\pi$
$458$		0			0
$459$			1.72139i			0.00375030i
$460$		0			0
$461$		−	527.875i		−	1.14507i	−0.819882	−	0.572533i	$-0.805961\pi$
$461$		−	527.875i		−	1.14507i	0.819882	−	0.572533i	$-0.194039\pi$
$462$		0			0
$463$		−	573.469i		−	1.23859i	−0.785157	−	0.619297i	$-0.787418\pi$
$463$		−	573.469i		−	1.23859i	0.785157	−	0.619297i	$-0.212582\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			341.159i			0.730533i	0.930903	+	0.365267i	$0.119022\pi$
$467$			341.159i			0.730533i	−0.930903	+	0.365267i	$0.880978\pi$
$468$		0			0
$469$		−	125.552i		−	0.267701i
$470$		0			0
$471$	−894.429			−1.89900
$472$		0			0
$473$	14.5725	+	37.4532i	0.0308086	+	0.0791822i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	425.277i		−	0.891565i
$478$		0			0
$479$		−	298.366i		−	0.622893i	−0.950264	−	0.311446i	$-0.899187\pi$
$479$		−	298.366i		−	0.622893i	0.950264	−	0.311446i	$-0.100813\pi$
$480$		0			0
$481$		−	8.81658i		−	0.0183297i
$482$		0			0
$483$	−79.4868			−0.164569
$484$		0			0
$485$		0			0
$486$		0			0
$487$			459.677i			0.943896i	0.881626	+	0.471948i	$0.156449\pi$
$487$			459.677i			0.943896i	−0.881626	+	0.471948i	$0.843551\pi$
$488$		0			0
$489$	−443.756			−0.907476
$490$		0			0
$491$			761.166i			1.55024i	0.631816	+	0.775119i	$0.282310\pi$
$491$			761.166i			1.55024i	−0.631816	+	0.775119i	$0.717690\pi$
$492$		0			0
$493$			24.6571i			0.0500144i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−22.8540			−0.0459839
$498$		0			0
$499$	−155.577			−0.311777			−0.155889	−	0.987775i	$-0.549824\pi$
$499$	−155.577			−0.311777			−0.155889	+	0.987775i	$0.549824\pi$
$500$		0			0
$501$		−	31.9042i		−	0.0636810i
$502$		0			0
$503$	−908.189			−1.80554			−0.902772	−	0.430119i	$-0.858472\pi$
$503$	−908.189			−1.80554			−0.902772	+	0.430119i	$0.858472\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			442.549i			0.872877i
$508$		0			0
$509$	−99.5660			−0.195611			−0.0978055	−	0.995206i	$-0.531182\pi$
$509$	−99.5660			−0.195611			−0.0978055	+	0.995206i	$0.531182\pi$
$510$		0			0
$511$	183.688			0.359468
$512$		0			0
$513$	−65.6403			−0.127954
$514$		0			0
$515$		0			0
$516$		0			0
$517$	36.7597	−	14.3026i	0.0711019	−	0.0276647i
$518$		0			0
$519$			344.501i			0.663779i
$520$		0			0
$521$	−285.253			−0.547511			−0.273755	−	0.961799i	$-0.588266\pi$
$521$	−285.253			−0.547511			−0.273755	+	0.961799i	$0.588266\pi$
$522$		0			0
$523$	788.239			1.50715			0.753574	−	0.657363i	$-0.228328\pi$
$523$	788.239			1.50715			0.753574	+	0.657363i	$0.228328\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.76890			0.00525407
$528$		0			0
$529$	402.803			0.761442
$530$		0			0
$531$	337.210			0.635047
$532$		0			0
$533$			465.751i			0.873829i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	1264.09i		−	2.35399i
$538$		0			0
$539$	183.827	+	472.460i	0.341052	+	0.876549i
$540$		0			0
$541$		−	656.926i		−	1.21428i	−0.794595	−	0.607140i	$-0.792317\pi$
$541$		−	656.926i		−	1.21428i	0.794595	−	0.607140i	$-0.207683\pi$
$542$		0			0
$543$		−	81.8509i		−	0.150738i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	318.855			0.582916			0.291458	−	0.956584i	$-0.405860\pi$
$547$	318.855			0.582916			0.291458	+	0.956584i	$0.405860\pi$
$548$		0			0
$549$		−	346.574i		−	0.631283i
$550$		0			0
$551$	−940.230			−1.70641
$552$		0			0
$553$		−	26.7349i		−	0.0483452i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	643.510			1.15531			0.577657	−	0.816279i	$-0.303967\pi$
$557$	643.510			1.15531			0.577657	+	0.816279i	$0.303967\pi$
$558$		0			0
$559$	−60.6675			−0.108529
$560$		0			0
$561$	21.7836	−	8.47566i	0.0388299	−	0.0151081i
$562$		0			0
$563$	−156.186			−0.277417			−0.138709	−	0.990333i	$-0.544295\pi$
$563$	−156.186			−0.277417			−0.138709	+	0.990333i	$0.544295\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−149.558			−0.263770
$568$		0			0
$569$			810.820i			1.42499i	0.701676	+	0.712496i	$0.252435\pi$
$569$			810.820i			1.42499i	−0.701676	+	0.712496i	$0.747565\pi$
$570$		0			0
$571$			293.650i			0.514273i	0.966375	+	0.257136i	$0.0827789\pi$
$571$			293.650i			0.514273i	−0.966375	+	0.257136i	$0.917221\pi$
$572$		0			0
$573$		−	661.790i		−	1.15496i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	122.633i		−	0.212536i	−0.994338	−	0.106268i	$-0.966110\pi$
$577$		−	122.633i		−	0.212536i	0.994338	−	0.106268i	$-0.0338902\pi$
$578$		0			0
$579$			1153.80i			1.99275i
$580$		0			0
$581$	−27.8404			−0.0479180
$582$		0			0
$583$	532.322	−	207.118i	0.913073	−	0.355263i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	959.478i		−	1.63455i	−0.576251	−	0.817273i	$-0.695485\pi$
$587$		−	959.478i		−	1.63455i	0.576251	−	0.817273i	$-0.304515\pi$
$588$		0			0
$589$			105.584i			0.179260i
$590$		0			0
$591$			652.362i			1.10383i
$592$		0			0
$593$	−277.185			−0.467428			−0.233714	−	0.972305i	$-0.575088\pi$
$593$	−277.185			−0.467428			−0.233714	+	0.972305i	$0.575088\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			1406.00i			2.35511i
$598$		0			0
$599$	−281.461			−0.469885			−0.234943	−	0.972009i	$-0.575490\pi$
$599$	−281.461			−0.469885			−0.234943	+	0.972009i	$0.575490\pi$
$600$		0			0
$601$		−	626.332i		−	1.04215i	−0.853511	−	0.521075i	$-0.825531\pi$
$601$		−	626.332i		−	1.04215i	0.853511	−	0.521075i	$-0.174469\pi$
$602$		0			0
$603$			602.516i			0.999197i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−558.941			−0.920825			−0.460413	−	0.887705i	$-0.652299\pi$
$607$	−558.941			−0.920825			−0.460413	+	0.887705i	$0.652299\pi$
$608$		0			0
$609$	−340.410			−0.558965
$610$		0			0
$611$			59.5442i			0.0974537i
$612$		0			0
$613$	1035.10			1.68857			0.844286	−	0.535892i	$-0.180025\pi$
$613$	1035.10			1.68857			0.844286	+	0.535892i	$0.180025\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	291.717i		−	0.472800i	−0.971656	−	0.236400i	$-0.924032\pi$
$617$		−	291.717i		−	0.472800i	0.971656	−	0.236400i	$-0.0759675\pi$
$618$		0			0
$619$	−269.541			−0.435446			−0.217723	−	0.976011i	$-0.569863\pi$
$619$	−269.541			−0.435446			−0.217723	+	0.976011i	$0.569863\pi$
$620$		0			0
$621$	−37.7306			−0.0607578
$622$		0			0
$623$	240.117			0.385421
$624$		0			0
$625$		0			0
$626$		0			0
$627$	323.196	+	830.656i	0.515464	+	1.32481i
$628$		0			0
$629$		−	0.272120i		−	0.000432624i
$630$		0			0
$631$	−172.219			−0.272930			−0.136465	−	0.990645i	$-0.543574\pi$
$631$	−172.219			−0.272930			−0.136465	+	0.990645i	$0.543574\pi$
$632$		0			0
$633$	649.451			1.02599
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−765.301			−1.20141
$638$		0			0
$639$	109.675			0.171635
$640$		0			0
$641$	377.939			0.589608			0.294804	−	0.955558i	$-0.404746\pi$
$641$	377.939			0.589608			0.294804	+	0.955558i	$0.404746\pi$
$642$		0			0
$643$			511.939i			0.796172i	0.917348	+	0.398086i	$0.130325\pi$
$643$			511.939i			0.796172i	−0.917348	+	0.398086i	$0.869675\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	521.728i		−	0.806381i	−0.915116	−	0.403190i	$-0.867901\pi$
$647$		−	521.728i		−	0.806381i	0.915116	−	0.403190i	$-0.132099\pi$
$648$		0			0
$649$	164.228	+	422.088i	0.253048	+	0.650367i
$650$		0			0
$651$			38.2267i			0.0587199i
$652$		0			0
$653$			314.135i			0.481065i	0.970641	+	0.240532i	$0.0773221\pi$
$653$			314.135i			0.481065i	−0.970641	+	0.240532i	$0.922678\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−881.509			−1.34172
$658$		0			0
$659$			691.038i			1.04862i	0.851528	+	0.524308i	$0.175676\pi$
$659$			691.038i			1.04862i	−0.851528	+	0.524308i	$0.824324\pi$
$660$		0			0
$661$	1127.61			1.70591			0.852957	−	0.521981i	$-0.174807\pi$
$661$	1127.61			1.70591			0.852957	+	0.521981i	$0.174807\pi$
$662$		0			0
$663$			35.2855i			0.0532210i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−540.451			−0.810272
$668$		0			0
$669$	−1025.81			−1.53335
$670$		0			0
$671$	433.810	−	168.789i	0.646512	−	0.251548i
$672$		0			0
$673$	888.140			1.31967			0.659837	−	0.751409i	$-0.270625\pi$
$673$	888.140			1.31967			0.659837	+	0.751409i	$0.270625\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	666.114			0.983920			0.491960	−	0.870618i	$-0.336281\pi$
$677$	666.114			0.983920			0.491960	+	0.870618i	$0.336281\pi$
$678$		0			0
$679$		−	165.707i		−	0.244046i
$680$		0			0
$681$			1406.86i			2.06587i
$682$		0			0
$683$			727.901i			1.06574i	0.846197	+	0.532871i	$0.178887\pi$
$683$			727.901i			1.06574i	−0.846197	+	0.532871i	$0.821113\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	632.944i		−	0.921316i
$688$		0			0
$689$			862.266i			1.25148i
$690$		0			0
$691$	−610.238			−0.883123			−0.441561	−	0.897231i	$-0.645575\pi$
$691$	−610.238			−0.883123			−0.441561	+	0.897231i	$0.645575\pi$
$692$		0			0
$693$	55.7493	+	143.283i	0.0804463	+	0.206758i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			14.3752i			0.0206244i
$698$		0			0
$699$			1647.08i			2.35634i
$700$		0			0
$701$		−	669.548i		−	0.955133i	−0.878596	−	0.477566i	$-0.841519\pi$
$701$		−	669.548i		−	0.955133i	0.878596	−	0.477566i	$-0.158481\pi$
$702$		0			0
$703$	10.3766			0.0147604
$704$		0			0
$705$		0			0
$706$		0			0
$707$			288.315i			0.407800i
$708$		0			0
$709$	−62.5477			−0.0882197			−0.0441098	−	0.999027i	$-0.514045\pi$
$709$	−62.5477			−0.0882197			−0.0441098	+	0.999027i	$0.514045\pi$
$710$		0			0
$711$			128.299i			0.180449i
$712$		0			0
$713$			60.6906i			0.0851200i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−413.296			−0.576424
$718$		0			0
$719$	−958.637			−1.33329			−0.666646	−	0.745374i	$-0.732271\pi$
$719$	−958.637			−1.33329			−0.666646	+	0.745374i	$0.732271\pi$
$720$		0			0
$721$		−	208.163i		−	0.288715i
$722$		0			0
$723$	1027.77			1.42154
$724$		0			0
$725$		0			0
$726$		0			0
$727$			678.507i			0.933297i	0.884443	+	0.466648i	$0.154539\pi$
$727$			678.507i			0.933297i	−0.884443	+	0.466648i	$0.845461\pi$
$728$		0			0
$729$	592.391			0.812608
$730$		0			0
$731$	−1.87248			−0.00256153
$732$		0			0
$733$	925.664			1.26284			0.631422	−	0.775440i	$-0.282472\pi$
$733$	925.664			1.26284			0.631422	+	0.775440i	$0.282472\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−754.173	+	293.437i	−1.02330	+	0.398151i
$738$		0			0
$739$		−	771.476i		−	1.04395i	−0.852962	−	0.521973i	$-0.825196\pi$
$739$		−	771.476i		−	1.04395i	0.852962	−	0.521973i	$-0.174804\pi$
$740$		0			0
$741$	−1345.52			−1.81581
$742$		0			0
$743$	−816.220			−1.09855			−0.549274	−	0.835643i	$-0.685095\pi$
$743$	−816.220			−1.09855			−0.549274	+	0.835643i	$0.685095\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	133.604			0.178855
$748$		0			0
$749$	−187.564			−0.250419
$750$		0			0
$751$	704.308			0.937827			0.468914	−	0.883244i	$-0.344646\pi$
$751$	704.308			0.937827			0.468914	+	0.883244i	$0.344646\pi$
$752$		0			0
$753$		−	487.352i		−	0.647214i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	1457.18i		−	1.92495i	−0.271379	−	0.962473i	$-0.587480\pi$
$757$		−	1457.18i		−	1.92495i	0.271379	−	0.962473i	$-0.412520\pi$
$758$		0			0
$759$	185.775	+	477.467i	0.244763	+	0.629074i
$760$		0			0
$761$			1121.73i			1.47403i	0.675878	+	0.737014i	$0.263765\pi$
$761$			1121.73i			1.47403i	−0.675878	+	0.737014i	$0.736235\pi$
$762$		0			0
$763$		−	218.086i		−	0.285827i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−683.708			−0.891405
$768$		0			0
$769$			654.531i			0.851146i	0.904924	+	0.425573i	$0.139927\pi$
$769$			654.531i			0.851146i	−0.904924	+	0.425573i	$0.860073\pi$
$770$		0			0
$771$	47.5386			0.0616584
$772$		0			0
$773$		−	682.354i		−	0.882735i	−0.897326	−	0.441367i	$-0.854494\pi$
$773$		−	682.354i		−	0.882735i	0.897326	−	0.441367i	$-0.145506\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	3.75683			0.00483504
$778$		0			0
$779$	−548.159			−0.703670
$780$		0			0
$781$	53.4140	+	137.281i	0.0683917	+	0.175776i
$782$		0			0
$783$	−161.585			−0.206366
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−651.105			−0.827326			−0.413663	−	0.910430i	$-0.635751\pi$
$787$	−651.105			−0.827326			−0.413663	+	0.910430i	$0.635751\pi$
$788$		0			0
$789$		−	682.557i		−	0.865091i
$790$		0			0
$791$			170.984i			0.216162i
$792$		0			0
$793$			702.694i			0.886122i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			288.164i			0.361561i	0.983524	+	0.180780i	$0.0578623\pi$
$797$			288.164i			0.361561i	−0.983524	+	0.180780i	$0.942138\pi$
$798$		0			0
$799$			1.83781i			0.00230014i
$800$		0			0
$801$	−1152.31			−1.43859
$802$		0			0
$803$	−429.313	−	1103.39i	−0.534636	−	1.37409i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	1496.98i		−	1.85499i
$808$		0			0
$809$		−	37.5190i		−	0.0463771i	−0.999731	−	0.0231885i	$-0.992618\pi$
$809$		−	37.5190i		−	0.0463771i	0.999731	−	0.0231885i	$-0.00738180\pi$
$810$		0			0
$811$		−	879.835i		−	1.08488i	−0.840096	−	0.542438i	$-0.817501\pi$
$811$		−	879.835i		−	1.08488i	0.840096	−	0.542438i	$-0.182499\pi$
$812$		0			0
$813$	−645.810			−0.794354
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	71.4018i		−	0.0873951i
$818$		0			0
$819$	−232.093			−0.283386
$820$		0			0
$821$			901.103i			1.09757i	0.835964	+	0.548784i	$0.184909\pi$
$821$			901.103i			1.09757i	−0.835964	+	0.548784i	$0.815091\pi$
$822$		0			0
$823$			1296.62i			1.57548i	0.616010	+	0.787738i	$0.288748\pi$
$823$			1296.62i			1.57548i	−0.616010	+	0.787738i	$0.711252\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	48.1443			0.0582156			0.0291078	−	0.999576i	$-0.490733\pi$
$827$	48.1443			0.0582156			0.0291078	+	0.999576i	$0.490733\pi$
$828$		0			0
$829$	8.95955			0.0108077			0.00540383	−	0.999985i	$-0.498280\pi$
$829$	8.95955			0.0108077			0.00540383	+	0.999985i	$0.498280\pi$
$830$		0			0
$831$			1620.96i			1.95061i
$832$		0			0
$833$	−23.6207			−0.0283562
$834$		0			0
$835$		0			0
$836$		0			0
$837$			18.1453i			0.0216790i
$838$		0			0
$839$	1504.58			1.79331			0.896653	−	0.442734i	$-0.145991\pi$
$839$	1504.58			1.79331			0.896653	+	0.442734i	$0.145991\pi$
$840$		0			0
$841$	−1473.53			−1.75212
$842$		0			0
$843$	−343.210			−0.407129
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−152.197	+	139.563i	−0.179690	+	0.164774i
$848$		0			0
$849$			820.808i			0.966794i
$850$		0			0
$851$	5.96452			0.00700884
$852$		0			0
$853$	−419.571			−0.491877			−0.245938	−	0.969285i	$-0.579096\pi$
$853$	−419.571			−0.491877			−0.245938	+	0.969285i	$0.579096\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−285.389			−0.333010			−0.166505	−	0.986041i	$-0.553248\pi$
$857$	−285.389			−0.333010			−0.166505	+	0.986041i	$0.553248\pi$
$858$		0			0
$859$	575.894			0.670423			0.335212	−	0.942143i	$-0.391192\pi$
$859$	575.894			0.670423			0.335212	+	0.942143i	$0.391192\pi$
$860$		0			0
$861$	−198.461			−0.230500
$862$		0			0
$863$		−	1065.79i		−	1.23499i	−0.786576	−	0.617493i	$-0.788148\pi$
$863$		−	1065.79i		−	1.23499i	0.786576	−	0.617493i	$-0.211852\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	1197.13i		−	1.38077i
$868$		0			0
$869$	−160.593	+	62.4843i	−0.184802	+	0.0719037i
$870$		0			0
$871$		−	1221.63i		−	1.40256i
$872$		0			0
$873$			795.221i			0.910906i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−120.742			−0.137676			−0.0688382	−	0.997628i	$-0.521929\pi$
$877$	−120.742			−0.137676			−0.0688382	+	0.997628i	$0.521929\pi$
$878$		0			0
$879$		−	530.119i		−	0.603093i
$880$		0			0
$881$	1206.22			1.36914			0.684572	−	0.728946i	$-0.259989\pi$
$881$	1206.22			1.36914			0.684572	+	0.728946i	$0.259989\pi$
$882$		0			0
$883$		−	306.072i		−	0.346627i	−0.984867	−	0.173314i	$-0.944553\pi$
$883$		−	306.072i		−	0.346627i	0.984867	−	0.173314i	$-0.0554475\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1125.95			−1.26939			−0.634694	−	0.772764i	$-0.718874\pi$
$887$	−1125.95			−1.26939			−0.634694	+	0.772764i	$0.718874\pi$
$888$		0			0
$889$	−271.529			−0.305432
$890$		0			0
$891$	349.544	+	898.374i	0.392305	+	1.00828i
$892$		0			0
$893$	−70.0797			−0.0784767
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−773.412			−0.862221
$898$		0			0
$899$			259.913i			0.289113i
$900$		0			0
$901$			26.6135i			0.0295378i
$902$		0			0
$903$		−	25.8510i		−	0.0286279i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	874.007i		−	0.963624i	−0.876275	−	0.481812i	$-0.839979\pi$
$907$		−	874.007i		−	0.963624i	0.876275	−	0.481812i	$-0.160021\pi$
$908$		0			0
$909$		−	1383.61i		−	1.52212i
$910$		0			0
$911$	−234.753			−0.257687			−0.128844	−	0.991665i	$-0.541127\pi$
$911$	−234.753			−0.257687			−0.128844	+	0.991665i	$0.541127\pi$
$912$		0			0
$913$	65.0680	+	167.234i	0.0712684	+	0.183169i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			176.692i			0.192684i
$918$		0			0
$919$		−	1421.78i		−	1.54710i	−0.633736	−	0.773550i	$-0.718479\pi$
$919$		−	1421.78i		−	1.54710i	0.633736	−	0.773550i	$-0.281521\pi$
$920$		0			0
$921$			2319.85i			2.51884i
$922$		0			0
$923$	−222.371			−0.240922
$924$		0			0
$925$		0			0
$926$		0			0
$927$			998.964i			1.07763i
$928$		0			0
$929$	455.451			0.490259			0.245130	−	0.969490i	$-0.421169\pi$
$929$	455.451			0.490259			0.245130	+	0.969490i	$0.421169\pi$
$930$		0			0
$931$		−	900.711i		−	0.967466i
$932$		0			0
$933$			864.426i			0.926502i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1512.60			−1.61430			−0.807152	−	0.590343i	$-0.798992\pi$
$937$	−1512.60			−1.61430			−0.807152	+	0.590343i	$0.798992\pi$
$938$		0			0
$939$	−2032.05			−2.16406
$940$		0			0
$941$		−	419.883i		−	0.446209i	−0.974795	−	0.223105i	$-0.928381\pi$
$941$		−	419.883i		−	0.446209i	0.974795	−	0.223105i	$-0.0716191\pi$
$942$		0			0
$943$	−315.086			−0.334132
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1110.89i			1.17307i	0.809925	+	0.586533i	$0.199508\pi$
$947$			1110.89i			1.17307i	−0.809925	+	0.586533i	$0.800492\pi$
$948$		0			0
$949$	1787.30			1.88335
$950$		0			0
$951$	739.950			0.778076
$952$		0			0
$953$	−810.140			−0.850095			−0.425047	−	0.905171i	$-0.639743\pi$
$953$	−810.140			−0.850095			−0.425047	+	0.905171i	$0.639743\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	795.600	+	2044.80i	0.831348	+	2.13668i
$958$		0			0
$959$			49.5760i			0.0516955i
$960$		0			0
$961$	−931.813			−0.969628
$962$		0			0
$963$	900.109			0.934692
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1345.29			−1.39120			−0.695601	−	0.718428i	$-0.744862\pi$
$967$	−1345.29			−1.39120			−0.695601	+	0.718428i	$0.744862\pi$
$968$		0			0
$969$	−41.5288			−0.0428574
$970$		0			0
$971$	560.812			0.577561			0.288781	−	0.957395i	$-0.406750\pi$
$971$	560.812			0.577561			0.288781	+	0.957395i	$0.406750\pi$
$972$		0			0
$973$			106.607i			0.109566i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1447.74i		−	1.48182i	−0.671603	−	0.740912i	$-0.734394\pi$
$977$		−	1447.74i		−	1.48182i	0.671603	−	0.740912i	$-0.265606\pi$
$978$		0			0
$979$	−561.199	−	1442.36i	−0.573236	−	1.47329i
$980$		0			0
$981$			1046.58i			1.06685i
$982$		0			0
$983$		−	543.853i		−	0.553259i	−0.960977	−	0.276629i	$-0.910783\pi$
$983$		−	543.853i		−	0.553259i	0.960977	−	0.276629i	$-0.0892174\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−25.3723			−0.0257065
$988$		0			0
$989$		−	41.0423i		−	0.0414988i
$990$		0			0
$991$	−398.772			−0.402394			−0.201197	−	0.979551i	$-0.564483\pi$
$991$	−398.772			−0.402394			−0.201197	+	0.979551i	$0.564483\pi$
$992$		0			0
$993$		−	1759.03i		−	1.77143i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	264.448			0.265244			0.132622	−	0.991167i	$-0.457660\pi$
$997$	264.448			0.265244			0.132622	+	0.991167i	$0.457660\pi$
$998$		0			0
$999$	1.78328			0.00178506

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.3.e.c.549.14		16
5.2	odd	4		1100.3.f.c.901.8	yes	8
5.3	odd	4		1100.3.f.e.901.1	yes	8
5.4	even	2	inner	1100.3.e.c.549.3		16
11.10	odd	2	inner	1100.3.e.c.549.13		16
55.32	even	4		1100.3.f.c.901.7	✓	8
55.43	even	4		1100.3.f.e.901.2	yes	8
55.54	odd	2	inner	1100.3.e.c.549.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.3.e.c.549.3		16	5.4	even	2	inner
1100.3.e.c.549.4		16	55.54	odd	2	inner
1100.3.e.c.549.13		16	11.10	odd	2	inner
1100.3.e.c.549.14		16	1.1	even	1	trivial
1100.3.f.c.901.7	✓	8	55.32	even	4
1100.3.f.c.901.8	yes	8	5.2	odd	4
1100.3.f.e.901.1	yes	8	5.3	odd	4
1100.3.f.e.901.2	yes	8	55.43	even	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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1100.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.14607i q^{3} +1.70661 q^{7} -8.18991 q^{9} +(-3.98865 - 10.2514i) q^{11} +16.6054 q^{13} +0.512519 q^{17} +19.5435i q^{19} +7.07571i q^{21} +11.2338i q^{23} +3.35868i q^{27} +48.1096i q^{29} +5.40252 q^{31} +(42.5029 - 16.5372i) q^{33} -0.530947i q^{37} +68.8472i q^{39} +28.0481i q^{41} -3.65348 q^{43} +3.58583i q^{47} -46.0875 q^{49} +2.12494i q^{51} +51.9269i q^{53} -81.0288 q^{57} -41.1738 q^{59} +42.3172i q^{61} -13.9770 q^{63} -73.5680i q^{67} -46.5760 q^{69} -13.3915 q^{71} +107.634 q^{73} +(-6.80706 - 17.4951i) q^{77} -15.6655i q^{79} -87.6345 q^{81} -16.3133 q^{83} -199.466 q^{87} +140.699 q^{89} +28.3389 q^{91} +22.3992i q^{93} -97.0976i q^{97} +(32.6667 + 83.9579i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9} + 6 q^{11} + 28 q^{31} - 28 q^{49} - 256 q^{59} + 352 q^{69} - 68 q^{71} - 256 q^{81} + 292 q^{89} + 228 q^{91} - 16 q^{99}+O(q^{100}) \) 16 * q - 32 * q^9 + 6 * q^11 + 28 * q^31 - 28 * q^49 - 256 * q^59 + 352 * q^69 - 68 * q^71 - 256 * q^81 + 292 * q^89 + 228 * q^91 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more