LMFDB - Embedded newform 1100.3.f.e.901.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1100,3,Mod(901,1100)] 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("1100.901"); 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, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1100.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$29.9728290796$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 83x^{6} + 1611x^{4} + 7105x^{2} + 400 $ x^8 + 83x^6 + 1611x^4 + 7105*x^2 + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			901.1
Root			$-0.238817i$ of defining polynomial
Character	$\chi$	$=$	1100.901
Dual form			1100.3.f.e.901.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.14607 q^{3} -1.70661i q^{7} +8.18991 q^{9} +(-3.98865 - 10.2514i) q^{11} +16.6054i q^{13} -0.512519i q^{17} -19.5435i q^{19} +7.07571i q^{21} -11.2338 q^{23} +3.35868 q^{27} -48.1096i q^{29} +5.40252 q^{31} +(16.5372 + 42.5029i) q^{33} -0.530947 q^{37} -68.8472i q^{39} +28.0481i q^{41} -3.65348i q^{43} +3.58583 q^{47} +46.0875 q^{49} +2.12494i q^{51} -51.9269 q^{53} +81.0288i q^{57} +41.1738 q^{59} +42.3172i q^{61} -13.9770i q^{63} -73.5680 q^{67} +46.5760 q^{69} -13.3915 q^{71} +107.634i q^{73} +(-17.4951 + 6.80706i) q^{77} +15.6655i q^{79} -87.6345 q^{81} -16.3133i q^{83} +199.466i q^{87} -140.699 q^{89} +28.3389 q^{91} -22.3992 q^{93} -97.0976 q^{97} +(-32.6667 - 83.9579i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 16 q^{9} + 3 q^{11} - 28 q^{23} + 10 q^{27} + 14 q^{31} + 31 q^{33} + 24 q^{37} + 32 q^{47} + 14 q^{49} - 98 q^{53} + 128 q^{59} + 42 q^{67} - 176 q^{69} - 34 q^{71} + 136 q^{77} - 128 q^{81}+ \cdots + 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 + 16 * q^9 + 3 * q^11 - 28 * q^23 + 10 * q^27 + 14 * q^31 + 31 * q^33 + 24 * q^37 + 32 * q^47 + 14 * q^49 - 98 * q^53 + 128 * q^59 + 42 * q^67 - 176 * q^69 - 34 * q^71 + 136 * q^77 - 128 * q^81 - 146 * q^89 + 114 * q^91 + 142 * q^93 - 296 * q^97 + 8 * 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
$3$	−4.14607			−1.38202			−0.691012	−	0.722843i	$-0.742835\pi$
$3$	−4.14607			−1.38202			−0.691012	+	0.722843i	$0.742835\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	1.70661i		−	0.243801i	−0.992542	−	0.121900i	$-0.961101\pi$
$7$		−	1.70661i		−	0.243801i	0.992542	−	0.121900i	$-0.0388989\pi$
$8$		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
$13$			16.6054i			1.27734i	0.769481	+	0.638669i	$0.220515\pi$
$13$			16.6054i			1.27734i	−0.769481	+	0.638669i	$0.779485\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.512519i		−	0.0301482i	−0.999886	−	0.0150741i	$-0.995202\pi$
$17$		−	0.512519i		−	0.0301482i	0.999886	−	0.0150741i	$-0.00479842\pi$
$18$		0			0
$19$		−	19.5435i		−	1.02861i	−0.857609	−	0.514303i	$-0.828051\pi$
$19$		−	19.5435i		−	1.02861i	0.857609	−	0.514303i	$-0.171949\pi$
$20$		0			0
$21$			7.07571i			0.336939i
$22$		0			0
$23$	−11.2338			−0.488424			−0.244212	−	0.969722i	$-0.578529\pi$
$23$	−11.2338			−0.488424			−0.244212	+	0.969722i	$0.578529\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.35868			0.124395
$28$		0			0
$29$		−	48.1096i		−	1.65895i	−0.558543	−	0.829476i	$-0.688639\pi$
$29$		−	48.1096i		−	1.65895i	0.558543	−	0.829476i	$-0.311361\pi$
$30$		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
$33$	16.5372	+	42.5029i	0.501129	+	1.28797i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.530947			−0.0143499			−0.00717495	−	0.999974i	$-0.502284\pi$
$37$	−0.530947			−0.0143499			−0.00717495	+	0.999974i	$0.502284\pi$
$38$		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
$43$		−	3.65348i		−	0.0849646i	−0.999097	−	0.0424823i	$-0.986473\pi$
$43$		−	3.65348i		−	0.0849646i	0.999097	−	0.0424823i	$-0.0135266\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.58583			0.0762943			0.0381472	−	0.999272i	$-0.487854\pi$
$47$	3.58583			0.0762943			0.0381472	+	0.999272i	$0.487854\pi$
$48$		0			0
$49$	46.0875			0.940561
$50$		0			0
$51$			2.12494i			0.0416655i
$52$		0			0
$53$	−51.9269			−0.979752			−0.489876	−	0.871792i	$-0.662958\pi$
$53$	−51.9269			−0.979752			−0.489876	+	0.871792i	$0.662958\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			81.0288i			1.42156i
$58$		0			0
$59$	41.1738			0.697861			0.348931	−	0.937149i	$-0.386545\pi$
$59$	41.1738			0.697861			0.348931	+	0.937149i	$0.386545\pi$
$60$		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
$63$		−	13.9770i		−	0.221857i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−73.5680			−1.09803			−0.549015	−	0.835813i	$-0.684997\pi$
$67$	−73.5680			−1.09803			−0.549015	+	0.835813i	$0.684997\pi$
$68$		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
$73$			107.634i			1.47443i	0.675657	+	0.737216i	$0.263860\pi$
$73$			107.634i			1.47443i	−0.675657	+	0.737216i	$0.736140\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−17.4951	+	6.80706i	−0.227209	+	0.0884034i
$78$		0			0
$79$			15.6655i			0.198298i	0.995073	+	0.0991489i	$0.0316120\pi$
$79$			15.6655i			0.198298i	−0.995073	+	0.0991489i	$0.968388\pi$
$80$		0			0
$81$	−87.6345			−1.08191
$82$		0			0
$83$		−	16.3133i		−	0.196546i	−0.995160	−	0.0982728i	$-0.968668\pi$
$83$		−	16.3133i		−	0.196546i	0.995160	−	0.0982728i	$-0.0313318\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			199.466i			2.29271i
$88$		0			0
$89$	−140.699			−1.58088			−0.790442	−	0.612536i	$-0.790149\pi$
$89$	−140.699			−1.58088			−0.790442	+	0.612536i	$0.790149\pi$
$90$		0			0
$91$	28.3389			0.311416
$92$		0			0
$93$	−22.3992			−0.240852
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−97.0976			−1.00101			−0.500503	−	0.865735i	$-0.666852\pi$
$97$	−97.0976			−1.00101			−0.500503	+	0.865735i	$0.666852\pi$
$98$		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
$103$	121.975			1.18422			0.592112	−	0.805856i	$-0.298294\pi$
$103$	121.975			1.18422			0.592112	+	0.805856i	$0.298294\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			109.905i			1.02715i	0.858046	+	0.513573i	$0.171678\pi$
$107$			109.905i			1.02715i	−0.858046	+	0.513573i	$0.828322\pi$
$108$		0			0
$109$			127.789i			1.17238i	0.810174	+	0.586189i	$0.199372\pi$
$109$			127.789i			1.17238i	−0.810174	+	0.586189i	$0.800628\pi$
$110$		0			0
$111$	2.20134			0.0198319
$112$		0			0
$113$	−100.189			−0.886631			−0.443316	−	0.896366i	$-0.646198\pi$
$113$	−100.189			−0.886631			−0.443316	+	0.896366i	$0.646198\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			135.997i			1.16237i
$118$		0			0
$119$	−0.874669			−0.00735016
$120$		0			0
$121$	−89.1813	+	81.7783i	−0.737035	+	0.675854i
$122$		0			0
$123$		−	116.290i		−	0.945444i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			159.105i			1.25279i	0.779505	+	0.626396i	$0.215471\pi$
$127$			159.105i			1.25279i	−0.779505	+	0.626396i	$0.784529\pi$
$128$		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
$133$	−33.3531			−0.250775
$134$		0			0
$135$		0			0
$136$		0			0
$137$	29.0495			0.212040			0.106020	−	0.994364i	$-0.466189\pi$
$137$	29.0495			0.212040			0.106020	+	0.994364i	$0.466189\pi$
$138$		0			0
$139$		−	62.4675i		−	0.449406i	−0.974427	−	0.224703i	$-0.927859\pi$
$139$		−	62.4675i		−	0.449406i	0.974427	−	0.224703i	$-0.0721412\pi$
$140$		0			0
$141$	−14.8671			−0.105441
$142$		0			0
$143$	170.228	−	66.2332i	1.19041	−	0.463169i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−191.082			−1.29988
$148$		0			0
$149$			48.4621i			0.325249i	0.986688	+	0.162624i	$0.0519959\pi$
$149$			48.4621i			0.325249i	−0.986688	+	0.162624i	$0.948004\pi$
$150$		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
$153$		−	4.19749i		−	0.0274346i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	215.729			1.37407			0.687036	−	0.726624i	$-0.258912\pi$
$157$	215.729			1.37407			0.687036	+	0.726624i	$0.258912\pi$
$158$		0			0
$159$	215.293			1.35404
$160$		0			0
$161$			19.1716i			0.119078i
$162$		0			0
$163$	−107.030			−0.656629			−0.328314	−	0.944569i	$-0.606481\pi$
$163$	−107.030			−0.656629			−0.328314	+	0.944569i	$0.606481\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			7.69504i			0.0460781i	0.999735	+	0.0230391i	$0.00733421\pi$
$167$			7.69504i			0.0460781i	−0.999735	+	0.0230391i	$0.992666\pi$
$168$		0			0
$169$	−106.739			−0.631593
$170$		0			0
$171$		−	160.060i		−	0.936021i
$172$		0			0
$173$			83.0910i			0.480295i	0.970736	+	0.240147i	$0.0771958\pi$
$173$			83.0910i			0.480295i	−0.970736	+	0.240147i	$0.922804\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−170.710			−0.964461
$178$		0			0
$179$	304.889			1.70329			0.851645	−	0.524119i	$-0.175605\pi$
$179$	304.889			1.70329			0.851645	+	0.524119i	$0.175605\pi$
$180$		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
$183$		−	175.450i		−	0.958745i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−5.25403	+	2.04426i	−0.0280964	+	0.0109319i
$188$		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
$193$			278.288i			1.44191i	0.692983	+	0.720954i	$0.256296\pi$
$193$			278.288i			1.44191i	−0.692983	+	0.720954i	$0.743704\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	157.345i		−	0.798704i	−0.916798	−	0.399352i	$-0.869235\pi$
$197$		−	157.345i		−	0.798704i	0.916798	−	0.399352i	$-0.130765\pi$
$198$		0			0
$199$	−339.116			−1.70410			−0.852050	−	0.523460i	$-0.824641\pi$
$199$	−339.116			−1.70410			−0.852050	+	0.523460i	$0.824641\pi$
$200$		0			0
$201$	305.018			1.51750
$202$		0			0
$203$	−82.1042			−0.404454
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−92.0035			−0.444461
$208$		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
$213$	55.5220			0.260667
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	9.21998i		−	0.0424884i
$218$		0			0
$219$		−	446.256i		−	2.03770i
$220$		0			0
$221$	8.51059			0.0385095
$222$		0			0
$223$	−247.418			−1.10950			−0.554749	−	0.832017i	$-0.687186\pi$
$223$	−247.418			−1.10950			−0.554749	+	0.832017i	$0.687186\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	339.323i		−	1.49481i	−0.664366	−	0.747407i	$-0.731298\pi$
$227$		−	339.323i		−	1.49481i	0.664366	−	0.747407i	$-0.268702\pi$
$228$		0			0
$229$	152.661			0.666642			0.333321	−	0.942813i	$-0.391831\pi$
$229$	152.661			0.666642			0.333321	+	0.942813i	$0.391831\pi$
$230$		0			0
$231$	72.5358	−	28.2226i	0.314008	−	0.122176i
$232$		0			0
$233$			397.264i			1.70499i	0.522731	+	0.852497i	$0.324913\pi$
$233$			397.264i			1.70499i	−0.522731	+	0.852497i	$0.675087\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	64.9504i		−	0.274052i
$238$		0			0
$239$		−	99.6838i		−	0.417087i	−0.978013	−	0.208544i	$-0.933128\pi$
$239$		−	99.6838i		−	0.417087i	0.978013	−	0.208544i	$-0.0668723\pi$
$240$		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
$243$	333.111			1.37083
$244$		0			0
$245$		0			0
$246$		0			0
$247$	324.528			1.31388
$248$		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
$253$	44.8076	+	115.161i	0.177105	+	0.455183i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.4659			−0.0446146			−0.0223073	−	0.999751i	$-0.507101\pi$
$257$	−11.4659			−0.0446146			−0.0223073	+	0.999751i	$0.507101\pi$
$258$		0			0
$259$			0.906117i			0.00349852i
$260$		0			0
$261$		−	394.013i		−	1.50963i
$262$		0			0
$263$		−	164.627i		−	0.625960i	−0.949760	−	0.312980i	$-0.898673\pi$
$263$		−	164.627i		−	0.625960i	0.949760	−	0.312980i	$-0.101327\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	583.347			2.18482
$268$		0			0
$269$	361.059			1.34223			0.671113	−	0.741355i	$-0.265817\pi$
$269$	361.059			1.34223			0.671113	+	0.741355i	$0.265817\pi$
$270$		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
$273$	−117.495			−0.430385
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	390.963i		−	1.41142i	−0.708502	−	0.705708i	$-0.750629\pi$
$277$		−	390.963i		−	1.41142i	0.708502	−	0.705708i	$-0.249371\pi$
$278$		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
$283$			197.972i			0.699549i	0.936834	+	0.349775i	$0.113742\pi$
$283$			197.972i			0.699549i	−0.936834	+	0.349775i	$0.886258\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	47.8672			0.166785
$288$		0			0
$289$	288.737			0.999091
$290$		0			0
$291$	402.574			1.38341
$292$		0			0
$293$		−	127.860i		−	0.436384i	−0.975906	−	0.218192i	$-0.929984\pi$
$293$		−	127.860i		−	0.436384i	0.975906	−	0.218192i	$-0.0700159\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−13.3966	−	34.4311i	−0.0451064	−	0.115929i
$298$		0			0
$299$		−	186.541i		−	0.623883i
$300$		0			0
$301$	−6.23505			−0.0207145
$302$		0			0
$303$		−	700.438i		−	2.31168i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	559.530i		−	1.82257i	−0.411772	−	0.911287i	$-0.635090\pi$
$307$		−	559.530i		−	1.82257i	0.411772	−	0.911287i	$-0.364910\pi$
$308$		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
$313$	−490.115			−1.56586			−0.782931	−	0.622108i	$-0.786276\pi$
$313$	−490.115			−1.56586			−0.782931	+	0.622108i	$0.786276\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−178.470			−0.562997			−0.281499	−	0.959562i	$-0.590832\pi$
$317$	−178.470			−0.562997			−0.281499	+	0.959562i	$0.590832\pi$
$318$		0			0
$319$	−493.189	+	191.892i	−1.54605	+	0.601544i
$320$		0			0
$321$		−	455.672i		−	1.41954i
$322$		0			0
$323$	−10.0164			−0.0310106
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	529.823i		−	1.62025i
$328$		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
$333$	−4.34841			−0.0130583
$334$		0			0
$335$		0			0
$336$		0			0
$337$			391.858i			1.16278i	0.813624	+	0.581392i	$0.197492\pi$
$337$			391.858i			1.16278i	−0.813624	+	0.581392i	$0.802508\pi$
$338$		0			0
$339$	415.392			1.22535
$340$		0			0
$341$	−21.5488	−	55.3832i	−0.0631929	−	0.162414i
$342$		0			0
$343$		−	162.277i		−	0.473111i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	515.467i		−	1.48550i	−0.669571	−	0.742748i	$-0.733522\pi$
$347$		−	515.467i		−	1.48550i	0.669571	−	0.742748i	$-0.266478\pi$
$348$		0			0
$349$			413.637i			1.18521i	0.805495	+	0.592603i	$0.201900\pi$
$349$			413.637i			1.18521i	−0.805495	+	0.592603i	$0.798100\pi$
$350$		0			0
$351$			55.7722i			0.158895i
$352$		0			0
$353$	672.441			1.90493			0.952466	−	0.304646i	$-0.0985381\pi$
$353$	672.441			1.90493			0.952466	+	0.304646i	$0.0985381\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	3.62644			0.0101581
$358$		0			0
$359$			655.797i			1.82673i	0.407138	+	0.913367i	$0.366527\pi$
$359$			655.797i			1.82673i	−0.407138	+	0.913367i	$0.633473\pi$
$360$		0			0
$361$	−20.9485			−0.0580291
$362$		0			0
$363$	369.752	−	339.059i	1.01860	−	0.934047i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−642.013			−1.74935			−0.874677	−	0.484706i	$-0.838927\pi$
$367$	−642.013			−1.74935			−0.874677	+	0.484706i	$0.838927\pi$
$368$		0			0
$369$			229.712i			0.622526i
$370$		0			0
$371$			88.6187i			0.238865i
$372$		0			0
$373$			444.369i			1.19134i	0.803230	+	0.595669i	$0.203113\pi$
$373$			444.369i			1.19134i	−0.803230	+	0.595669i	$0.796887\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	798.879			2.11904
$378$		0			0
$379$	93.9534			0.247898			0.123949	−	0.992289i	$-0.460444\pi$
$379$	93.9534			0.247898			0.123949	+	0.992289i	$0.460444\pi$
$380$		0			0
$381$		−	659.659i		−	1.73139i
$382$		0			0
$383$	81.6845			0.213275			0.106638	−	0.994298i	$-0.465992\pi$
$383$	81.6845			0.213275			0.106638	+	0.994298i	$0.465992\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	29.9217i		−	0.0773170i
$388$		0			0
$389$	−648.091			−1.66604			−0.833022	−	0.553240i	$-0.813391\pi$
$389$	−648.091			−1.66604			−0.833022	+	0.553240i	$0.813391\pi$
$390$		0			0
$391$			5.75752i			0.0147251i
$392$		0			0
$393$		−	429.259i		−	1.09226i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−121.217			−0.305334			−0.152667	−	0.988278i	$-0.548786\pi$
$397$	−121.217			−0.305334			−0.152667	+	0.988278i	$0.548786\pi$
$398$		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
$403$			89.7110i			0.222608i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.11776	+	5.44293i	0.00520335	+	0.0133733i
$408$		0			0
$409$			434.331i			1.06193i	0.847392	+	0.530967i	$0.178171\pi$
$409$			434.331i			1.06193i	−0.847392	+	0.530967i	$0.821829\pi$
$410$		0			0
$411$	−120.441			−0.293044
$412$		0			0
$413$		−	70.2675i		−	0.170139i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			258.995i			0.621090i
$418$		0			0
$419$	−521.101			−1.24368			−0.621839	−	0.783145i	$-0.713614\pi$
$419$	−521.101			−1.24368			−0.621839	+	0.783145i	$0.713614\pi$
$420$		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
$423$	29.3677			0.0694271
$424$		0			0
$425$		0			0
$426$		0			0
$427$	72.2189			0.169131
$428$		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
$433$	−434.851			−1.00428			−0.502138	−	0.864788i	$-0.667453\pi$
$433$	−434.851			−1.00428			−0.502138	+	0.864788i	$0.667453\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			219.547i			0.502396i
$438$		0			0
$439$			762.031i			1.73583i	0.496709	+	0.867917i	$0.334542\pi$
$439$			762.031i			1.73583i	−0.496709	+	0.867917i	$0.665458\pi$
$440$		0			0
$441$	377.453			0.855902
$442$		0			0
$443$	−156.276			−0.352768			−0.176384	−	0.984321i	$-0.556440\pi$
$443$	−156.276			−0.352768			−0.176384	+	0.984321i	$0.556440\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	200.927i		−	0.449502i
$448$		0			0
$449$	545.999			1.21603			0.608017	−	0.793924i	$-0.291965\pi$
$449$	545.999			1.21603			0.608017	+	0.793924i	$0.291965\pi$
$450$		0			0
$451$	287.532	−	111.874i	0.637543	−	0.248058i
$452$		0			0
$453$		−	557.698i		−	1.23112i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			310.676i			0.679816i	0.940459	+	0.339908i	$0.110396\pi$
$457$			310.676i			0.679816i	−0.940459	+	0.339908i	$0.889604\pi$
$458$		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
$463$	573.469			1.23859			0.619297	−	0.785157i	$-0.287418\pi$
$463$	573.469			1.23859			0.619297	+	0.785157i	$0.287418\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	341.159			0.730533			0.365267	−	0.930903i	$-0.380978\pi$
$467$	341.159			0.730533			0.365267	+	0.930903i	$0.380978\pi$
$468$		0			0
$469$			125.552i			0.267701i
$470$		0			0
$471$	−894.429			−1.89900
$472$		0			0
$473$	−37.4532	+	14.5725i	−0.0791822	+	0.0308086i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−425.277			−0.891565
$478$		0			0
$479$			298.366i			0.622893i	0.950264	+	0.311446i	$0.100813\pi$
$479$			298.366i			0.622893i	−0.950264	+	0.311446i	$0.899187\pi$
$480$		0			0
$481$		−	8.81658i		−	0.0183297i
$482$		0			0
$483$		−	79.4868i		−	0.164569i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	459.677			0.943896			0.471948	−	0.881626i	$-0.343551\pi$
$487$	459.677			0.943896			0.471948	+	0.881626i	$0.343551\pi$
$488$		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
$493$	−24.6571			−0.0500144
$494$		0			0
$495$		0			0
$496$		0			0
$497$			22.8540i			0.0459839i
$498$		0			0
$499$	155.577			0.311777			0.155889	−	0.987775i	$-0.450176\pi$
$499$	155.577			0.311777			0.155889	+	0.987775i	$0.450176\pi$
$500$		0			0
$501$		−	31.9042i		−	0.0636810i
$502$		0			0
$503$		−	908.189i		−	1.80554i	−0.430119	−	0.902772i	$-0.641528\pi$
$503$		−	908.189i		−	1.80554i	0.430119	−	0.902772i	$-0.358472\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	442.549			0.872877
$508$		0			0
$509$	99.5660			0.195611			0.0978055	−	0.995206i	$-0.468818\pi$
$509$	99.5660			0.195611			0.0978055	+	0.995206i	$0.468818\pi$
$510$		0			0
$511$	183.688			0.359468
$512$		0			0
$513$		−	65.6403i		−	0.127954i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−14.3026	−	36.7597i	−0.0276647	−	0.0711019i
$518$		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
$523$			788.239i			1.50715i	0.657363	+	0.753574i	$0.271672\pi$
$523$			788.239i			1.50715i	−0.657363	+	0.753574i	$0.728328\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	2.76890i		−	0.00525407i
$528$		0			0
$529$	−402.803			−0.761442
$530$		0			0
$531$	337.210			0.635047
$532$		0			0
$533$	−465.751			−0.873829
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1264.09			−2.35399
$538$		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
$543$	81.8509			0.150738
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	318.855i		−	0.582916i	−0.956584	−	0.291458i	$-0.905860\pi$
$547$		−	318.855i		−	0.582916i	0.956584	−	0.291458i	$-0.0941403\pi$
$548$		0			0
$549$			346.574i			0.631283i
$550$		0			0
$551$	−940.230			−1.70641
$552$		0			0
$553$	26.7349			0.0483452
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	643.510i		−	1.15531i	−0.816279	−	0.577657i	$-0.803967\pi$
$557$		−	643.510i		−	1.15531i	0.816279	−	0.577657i	$-0.196033\pi$
$558$		0			0
$559$	60.6675			0.108529
$560$		0			0
$561$	21.7836	−	8.47566i	0.0388299	−	0.0151081i
$562$		0			0
$563$		−	156.186i		−	0.277417i	−0.990333	−	0.138709i	$-0.955705\pi$
$563$		−	156.186i		−	0.277417i	0.990333	−	0.138709i	$-0.0442951\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			149.558i			0.263770i
$568$		0			0
$569$		−	810.820i		−	1.42499i	−0.701676	−	0.712496i	$-0.747565\pi$
$569$		−	810.820i		−	1.42499i	0.701676	−	0.712496i	$-0.252435\pi$
$570$		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
$573$	661.790			1.15496
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−122.633			−0.212536			−0.106268	−	0.994338i	$-0.533890\pi$
$577$	−122.633			−0.212536			−0.106268	+	0.994338i	$0.533890\pi$
$578$		0			0
$579$		−	1153.80i		−	1.99275i
$580$		0			0
$581$	−27.8404			−0.0479180
$582$		0			0
$583$	207.118	+	532.322i	0.355263	+	0.913073i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−959.478			−1.63455			−0.817273	−	0.576251i	$-0.804515\pi$
$587$	−959.478			−1.63455			−0.817273	+	0.576251i	$0.804515\pi$
$588$		0			0
$589$		−	105.584i		−	0.179260i
$590$		0			0
$591$			652.362i			1.10383i
$592$		0			0
$593$		−	277.185i		−	0.467428i	−0.972305	−	0.233714i	$-0.924912\pi$
$593$		−	277.185i		−	0.467428i	0.972305	−	0.233714i	$-0.0750880\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1406.00			2.35511
$598$		0			0
$599$	281.461			0.469885			0.234943	−	0.972009i	$-0.424510\pi$
$599$	281.461			0.469885			0.234943	+	0.972009i	$0.424510\pi$
$600$		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
$603$	−602.516			−0.999197
$604$		0			0
$605$		0			0
$606$		0			0
$607$			558.941i			0.920825i	0.887705	+	0.460413i	$0.152299\pi$
$607$			558.941i			0.920825i	−0.887705	+	0.460413i	$0.847701\pi$
$608$		0			0
$609$	340.410			0.558965
$610$		0			0
$611$			59.5442i			0.0974537i
$612$		0			0
$613$			1035.10i			1.68857i	0.535892	+	0.844286i	$0.319975\pi$
$613$			1035.10i			1.68857i	−0.535892	+	0.844286i	$0.680025\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−291.717			−0.472800			−0.236400	−	0.971656i	$-0.575968\pi$
$617$	−291.717			−0.472800			−0.236400	+	0.971656i	$0.575968\pi$
$618$		0			0
$619$	269.541			0.435446			0.217723	−	0.976011i	$-0.430137\pi$
$619$	269.541			0.435446			0.217723	+	0.976011i	$0.430137\pi$
$620$		0			0
$621$	−37.7306			−0.0607578
$622$		0			0
$623$			240.117i			0.385421i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	830.656	−	323.196i	1.32481	−	0.515464i
$628$		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
$633$			649.451i			1.02599i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			765.301i			1.20141i
$638$		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
$643$	−511.939			−0.796172			−0.398086	−	0.917348i	$-0.630325\pi$
$643$	−511.939			−0.796172			−0.398086	+	0.917348i	$0.630325\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−521.728			−0.806381			−0.403190	−	0.915116i	$-0.632099\pi$
$647$	−521.728			−0.806381			−0.403190	+	0.915116i	$0.632099\pi$
$648$		0			0
$649$	−164.228	−	422.088i	−0.253048	−	0.650367i
$650$		0			0
$651$			38.2267i			0.0587199i
$652$		0			0
$653$	−314.135			−0.481065			−0.240532	−	0.970641i	$-0.577322\pi$
$653$	−314.135			−0.481065			−0.240532	+	0.970641i	$0.577322\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			881.509i			1.34172i
$658$		0			0
$659$		−	691.038i		−	1.04862i	−0.851528	−	0.524308i	$-0.824324\pi$
$659$		−	691.038i		−	1.04862i	0.851528	−	0.524308i	$-0.175676\pi$
$660$		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
$663$	−35.2855			−0.0532210
$664$		0			0
$665$		0			0
$666$		0			0
$667$			540.451i			0.810272i
$668$		0			0
$669$	1025.81			1.53335
$670$		0			0
$671$	433.810	−	168.789i	0.646512	−	0.251548i
$672$		0			0
$673$			888.140i			1.31967i	0.751409	+	0.659837i	$0.229375\pi$
$673$			888.140i			1.31967i	−0.751409	+	0.659837i	$0.770625\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	666.114i		−	0.983920i	−0.870618	−	0.491960i	$-0.836281\pi$
$677$		−	666.114i		−	0.983920i	0.870618	−	0.491960i	$-0.163719\pi$
$678$		0			0
$679$			165.707i			0.244046i
$680$		0			0
$681$			1406.86i			2.06587i
$682$		0			0
$683$	−727.901			−1.06574			−0.532871	−	0.846197i	$-0.678887\pi$
$683$	−727.901			−1.06574			−0.532871	+	0.846197i	$0.678887\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−632.944			−0.921316
$688$		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
$693$	−143.283	+	55.7493i	−0.206758	+	0.0804463i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	14.3752			0.0206244
$698$		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
$703$			10.3766i			0.0147604i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	288.315			0.407800
$708$		0			0
$709$	62.5477			0.0882197			0.0441098	−	0.999027i	$-0.485955\pi$
$709$	62.5477			0.0882197			0.0441098	+	0.999027i	$0.485955\pi$
$710$		0			0
$711$			128.299i			0.180449i
$712$		0			0
$713$	−60.6906			−0.0851200
$714$		0			0
$715$		0			0
$716$		0			0
$717$			413.296i			0.576424i
$718$		0			0
$719$	958.637			1.33329			0.666646	−	0.745374i	$-0.267729\pi$
$719$	958.637			1.33329			0.666646	+	0.745374i	$0.267729\pi$
$720$		0			0
$721$		−	208.163i		−	0.288715i
$722$		0			0
$723$			1027.77i			1.42154i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	678.507			0.933297			0.466648	−	0.884443i	$-0.345461\pi$
$727$	678.507			0.933297			0.466648	+	0.884443i	$0.345461\pi$
$728$		0			0
$729$	−592.391			−0.812608
$730$		0			0
$731$	−1.87248			−0.00256153
$732$		0			0
$733$			925.664i			1.26284i	0.775440	+	0.631422i	$0.217528\pi$
$733$			925.664i			1.26284i	−0.775440	+	0.631422i	$0.782472\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	293.437	+	754.173i	0.398151	+	1.02330i
$738$		0			0
$739$			771.476i			1.04395i	0.852962	+	0.521973i	$0.174804\pi$
$739$			771.476i			1.04395i	−0.852962	+	0.521973i	$0.825196\pi$
$740$		0			0
$741$	−1345.52			−1.81581
$742$		0			0
$743$		−	816.220i		−	1.09855i	−0.835643	−	0.549274i	$-0.814905\pi$
$743$		−	816.220i		−	1.09855i	0.835643	−	0.549274i	$-0.185095\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	133.604i		−	0.178855i
$748$		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
$753$	487.352			0.647214
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1457.18			−1.92495			−0.962473	−	0.271379i	$-0.912520\pi$
$757$	−1457.18			−1.92495			−0.962473	+	0.271379i	$0.912520\pi$
$758$		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
$763$	218.086			0.285827
$764$		0			0
$765$		0			0
$766$		0			0
$767$			683.708i			0.891405i
$768$		0			0
$769$		−	654.531i		−	0.851146i	−0.904924	−	0.425573i	$-0.860073\pi$
$769$		−	654.531i		−	0.851146i	0.904924	−	0.425573i	$-0.139927\pi$
$770$		0			0
$771$	47.5386			0.0616584
$772$		0			0
$773$	682.354			0.882735			0.441367	−	0.897326i	$-0.354494\pi$
$773$	682.354			0.882735			0.441367	+	0.897326i	$0.354494\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	3.75683i		−	0.00483504i
$778$		0			0
$779$	548.159			0.703670
$780$		0			0
$781$	53.4140	+	137.281i	0.0683917	+	0.175776i
$782$		0			0
$783$		−	161.585i		−	0.206366i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			651.105i			0.827326i	0.910430	+	0.413663i	$0.135751\pi$
$787$			651.105i			0.827326i	−0.910430	+	0.413663i	$0.864249\pi$
$788$		0			0
$789$			682.557i			0.865091i
$790$		0			0
$791$			170.984i			0.216162i
$792$		0			0
$793$	−702.694			−0.886122
$794$		0			0
$795$		0			0
$796$		0			0
$797$	288.164			0.361561			0.180780	−	0.983524i	$-0.442138\pi$
$797$	288.164			0.361561			0.180780	+	0.983524i	$0.442138\pi$
$798$		0			0
$799$		−	1.83781i		−	0.00230014i
$800$		0			0
$801$	−1152.31			−1.43859
$802$		0			0
$803$	1103.39	−	429.313i	1.37409	−	0.534636i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1496.98			−1.85499
$808$		0			0
$809$			37.5190i			0.0463771i	0.999731	+	0.0231885i	$0.00738180\pi$
$809$			37.5190i			0.0463771i	−0.999731	+	0.0231885i	$0.992618\pi$
$810$		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
$813$		−	645.810i		−	0.794354i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−71.4018			−0.0873951
$818$		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
$823$	−1296.62			−1.57548			−0.787738	−	0.616010i	$-0.788748\pi$
$823$	−1296.62			−1.57548			−0.787738	+	0.616010i	$0.788748\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	48.1443i		−	0.0582156i	−0.999576	−	0.0291078i	$-0.990733\pi$
$827$		−	48.1443i		−	0.0582156i	0.999576	−	0.0291078i	$-0.00926661\pi$
$828$		0			0
$829$	−8.95955			−0.0108077			−0.00540383	−	0.999985i	$-0.501720\pi$
$829$	−8.95955			−0.0108077			−0.00540383	+	0.999985i	$0.501720\pi$
$830$		0			0
$831$			1620.96i			1.95061i
$832$		0			0
$833$		−	23.6207i		−	0.0283562i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	18.1453			0.0216790
$838$		0			0
$839$	−1504.58			−1.79331			−0.896653	−	0.442734i	$-0.854009\pi$
$839$	−1504.58			−1.79331			−0.896653	+	0.442734i	$0.854009\pi$
$840$		0			0
$841$	−1473.53			−1.75212
$842$		0			0
$843$		−	343.210i		−	0.407129i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	139.563	+	152.197i	0.164774	+	0.179690i
$848$		0			0
$849$		−	820.808i		−	0.966794i
$850$		0			0
$851$	5.96452			0.00700884
$852$		0			0
$853$		−	419.571i		−	0.491877i	−0.969285	−	0.245938i	$-0.920904\pi$
$853$		−	419.571i		−	0.491877i	0.969285	−	0.245938i	$-0.0790961\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			285.389i			0.333010i	0.986041	+	0.166505i	$0.0532481\pi$
$857$			285.389i			0.333010i	−0.986041	+	0.166505i	$0.946752\pi$
$858$		0			0
$859$	−575.894			−0.670423			−0.335212	−	0.942143i	$-0.608808\pi$
$859$	−575.894			−0.670423			−0.335212	+	0.942143i	$0.608808\pi$
$860$		0			0
$861$	−198.461			−0.230500
$862$		0			0
$863$	1065.79			1.23499			0.617493	−	0.786576i	$-0.288148\pi$
$863$	1065.79			1.23499			0.617493	+	0.786576i	$0.288148\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1197.13			−1.38077
$868$		0			0
$869$	160.593	−	62.4843i	0.184802	−	0.0719037i
$870$		0			0
$871$		−	1221.63i		−	1.40256i
$872$		0			0
$873$	−795.221			−0.910906
$874$		0			0
$875$		0			0
$876$		0			0
$877$			120.742i			0.137676i	0.997628	+	0.0688382i	$0.0219292\pi$
$877$			120.742i			0.137676i	−0.997628	+	0.0688382i	$0.978071\pi$
$878$		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
$883$	306.072			0.346627			0.173314	−	0.984867i	$-0.444553\pi$
$883$	306.072			0.346627			0.173314	+	0.984867i	$0.444553\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1125.95i			1.26939i	0.772764	+	0.634694i	$0.218874\pi$
$887$			1125.95i			1.26939i	−0.772764	+	0.634694i	$0.781126\pi$
$888$		0			0
$889$	271.529			0.305432
$890$		0			0
$891$	349.544	+	898.374i	0.392305	+	1.00828i
$892$		0			0
$893$		−	70.0797i		−	0.0784767i
$894$		0			0
$895$		0			0
$896$		0			0
$897$			773.412i			0.862221i
$898$		0			0
$899$		−	259.913i		−	0.289113i
$900$		0			0
$901$			26.6135i			0.0295378i
$902$		0			0
$903$	25.8510			0.0286279
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−874.007			−0.963624			−0.481812	−	0.876275i	$-0.660021\pi$
$907$	−874.007			−0.963624			−0.481812	+	0.876275i	$0.660021\pi$
$908$		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
$913$	−167.234	+	65.0680i	−0.183169	+	0.0712684i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	176.692			0.192684
$918$		0			0
$919$			1421.78i			1.54710i	0.633736	+	0.773550i	$0.281521\pi$
$919$			1421.78i			1.54710i	−0.633736	+	0.773550i	$0.718479\pi$
$920$		0			0
$921$			2319.85i			2.51884i
$922$		0			0
$923$		−	222.371i		−	0.240922i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	998.964			1.07763
$928$		0			0
$929$	−455.451			−0.490259			−0.245130	−	0.969490i	$-0.578831\pi$
$929$	−455.451			−0.490259			−0.245130	+	0.969490i	$0.578831\pi$
$930$		0			0
$931$		−	900.711i		−	0.967466i
$932$		0			0
$933$	−864.426			−0.926502
$934$		0			0
$935$		0			0
$936$		0			0
$937$			1512.60i			1.61430i	0.590343	+	0.807152i	$0.298992\pi$
$937$			1512.60i			1.61430i	−0.590343	+	0.807152i	$0.701008\pi$
$938$		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
$943$		−	315.086i		−	0.334132i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1110.89			1.17307			0.586533	−	0.809925i	$-0.300492\pi$
$947$	1110.89			1.17307			0.586533	+	0.809925i	$0.300492\pi$
$948$		0			0
$949$	−1787.30			−1.88335
$950$		0			0
$951$	739.950			0.778076
$952$		0			0
$953$		−	810.140i		−	0.850095i	−0.905171	−	0.425047i	$-0.860257\pi$
$953$		−	810.140i		−	0.850095i	0.905171	−	0.425047i	$-0.139743\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	2044.80	−	795.600i	2.13668	−	0.831348i
$958$		0			0
$959$		−	49.5760i		−	0.0516955i
$960$		0			0
$961$	−931.813			−0.969628
$962$		0			0
$963$			900.109i			0.934692i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1345.29i			1.39120i	0.718428	+	0.695601i	$0.244862\pi$
$967$			1345.29i			1.39120i	−0.718428	+	0.695601i	$0.755138\pi$
$968$		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
$973$	−106.607			−0.109566
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1447.74			−1.48182			−0.740912	−	0.671603i	$-0.765606\pi$
$977$	−1447.74			−1.48182			−0.740912	+	0.671603i	$0.765606\pi$
$978$		0			0
$979$	561.199	+	1442.36i	0.573236	+	1.47329i
$980$		0			0
$981$			1046.58i			1.06685i
$982$		0			0
$983$	543.853			0.553259			0.276629	−	0.960977i	$-0.410783\pi$
$983$	543.853			0.553259			0.276629	+	0.960977i	$0.410783\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			25.3723i			0.0257065i
$988$		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
$993$	1759.03			1.77143
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	264.448i		−	0.265244i	−0.991167	−	0.132622i	$-0.957660\pi$
$997$		−	264.448i		−	0.265244i	0.991167	−	0.132622i	$-0.0423396\pi$
$998$		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.f.e.901.1	yes	8
5.2	odd	4		1100.3.e.c.549.14		16
5.3	odd	4		1100.3.e.c.549.3		16
5.4	even	2		1100.3.f.c.901.8	yes	8
11.10	odd	2	inner	1100.3.f.e.901.2	yes	8
55.32	even	4		1100.3.e.c.549.13		16
55.43	even	4		1100.3.e.c.549.4		16
55.54	odd	2		1100.3.f.c.901.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.3.e.c.549.3		16	5.3	odd	4
1100.3.e.c.549.4		16	55.43	even	4
1100.3.e.c.549.13		16	55.32	even	4
1100.3.e.c.549.14		16	5.2	odd	4
1100.3.f.c.901.7	✓	8	55.54	odd	2
1100.3.f.c.901.8	yes	8	5.4	even	2
1100.3.f.e.901.1	yes	8	1.1	even	1	trivial
1100.3.f.e.901.2	yes	8	11.10	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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1100.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.14607 q^{3} -1.70661i q^{7} +8.18991 q^{9} +(-3.98865 - 10.2514i) q^{11} +16.6054i q^{13} -0.512519i q^{17} -19.5435i q^{19} +7.07571i q^{21} -11.2338 q^{23} +3.35868 q^{27} -48.1096i q^{29} +5.40252 q^{31} +(16.5372 + 42.5029i) q^{33} -0.530947 q^{37} -68.8472i q^{39} +28.0481i q^{41} -3.65348i q^{43} +3.58583 q^{47} +46.0875 q^{49} +2.12494i q^{51} -51.9269 q^{53} +81.0288i q^{57} +41.1738 q^{59} +42.3172i q^{61} -13.9770i q^{63} -73.5680 q^{67} +46.5760 q^{69} -13.3915 q^{71} +107.634i q^{73} +(-17.4951 + 6.80706i) q^{77} +15.6655i q^{79} -87.6345 q^{81} -16.3133i q^{83} +199.466i q^{87} -140.699 q^{89} +28.3389 q^{91} -22.3992 q^{93} -97.0976 q^{97} +(-32.6667 - 83.9579i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 16 q^{9} + 3 q^{11} - 28 q^{23} + 10 q^{27} + 14 q^{31} + 31 q^{33} + 24 q^{37} + 32 q^{47} + 14 q^{49} - 98 q^{53} + 128 q^{59} + 42 q^{67} - 176 q^{69} - 34 q^{71} + 136 q^{77} - 128 q^{81}+ \cdots + 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 + 16 * q^9 + 3 * q^11 - 28 * q^23 + 10 * q^27 + 14 * q^31 + 31 * q^33 + 24 * q^37 + 32 * q^47 + 14 * q^49 - 98 * q^53 + 128 * q^59 + 42 * q^67 - 176 * q^69 - 34 * q^71 + 136 * q^77 - 128 * q^81 - 146 * q^89 + 114 * q^91 + 142 * q^93 - 296 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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