LMFDB - Embedded newform 4400.2.a.bn.1.1

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 = [2,0,0,0,0,0,-4,0,10,0,-2,0,8] 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:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-2.82843 q^{3} -2.00000 q^{7} +5.00000 q^{9} -1.00000 q^{11} +1.17157 q^{13} -6.82843 q^{17} +5.65685 q^{21} -2.82843 q^{23} -5.65685 q^{27} -3.65685 q^{29} +2.82843 q^{33} +7.65685 q^{37} -3.31371 q^{39} +6.00000 q^{41} -6.00000 q^{43} +2.82843 q^{47} -3.00000 q^{49} +19.3137 q^{51} -11.6569 q^{53} -1.65685 q^{59} -9.31371 q^{61} -10.0000 q^{63} +12.4853 q^{67} +8.00000 q^{69} -11.3137 q^{71} +1.17157 q^{73} +2.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +10.3431 q^{87} -13.3137 q^{89} -2.34315 q^{91} -3.65685 q^{97} -5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} - 8 q^{17} + 4 q^{29} + 4 q^{37} + 16 q^{39} + 12 q^{41} - 12 q^{43} - 6 q^{49} + 16 q^{51} - 12 q^{53} + 8 q^{59} + 4 q^{61} - 20 q^{63} + 8 q^{67} + 16 q^{69}+ \cdots - 10 q^{99}+O(q^{100}) $ 2 * q - 4 * q^7 + 10 * q^9 - 2 * q^11 + 8 * q^13 - 8 * q^17 + 4 * q^29 + 4 * q^37 + 16 * q^39 + 12 * q^41 - 12 * q^43 - 6 * q^49 + 16 * q^51 - 12 * q^53 + 8 * q^59 + 4 * q^61 - 20 * q^63 + 8 * q^67 + 16 * q^69 + 8 * q^73 + 4 * q^77 - 8 * q^79 + 2 * q^81 - 12 * q^83 + 32 * q^87 - 4 * q^89 - 16 * q^91 + 4 * q^97 - 10 * 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
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	5.00000	1.66667
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.17157	0.324936	0.162468	−	0.986714i	$-0.448055\pi$
$13$	1.17157	0.324936	0.162468	+	0.986714i	$0.448055\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	5.65685	1.23443
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	2.82843	0.492366
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.65685	1.25878	0.629390	−	0.777090i	$-0.283305\pi$
$37$	7.65685	1.25878	0.629390	+	0.777090i	$0.283305\pi$
$38$	0	0
$39$	−3.31371	−0.530618
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	19.3137	2.70446
$52$	0	0
$53$	−11.6569	−1.60119	−0.800596	−	0.599204i	$-0.795484\pi$
$53$	−11.6569	−1.60119	−0.800596	+	0.599204i	$0.795484\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.65685	−0.215704	−0.107852	−	0.994167i	$-0.534397\pi$
$59$	−1.65685	−0.215704	−0.107852	+	0.994167i	$0.534397\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	0	0
$63$	−10.0000	−1.25988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.4853	1.52532	0.762660	−	0.646800i	$-0.223893\pi$
$67$	12.4853	1.52532	0.762660	+	0.646800i	$0.223893\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	−11.3137	−1.34269	−0.671345	−	0.741145i	$-0.734283\pi$
$71$	−11.3137	−1.34269	−0.671345	+	0.741145i	$0.734283\pi$
$72$	0	0
$73$	1.17157	0.137122	0.0685611	−	0.997647i	$-0.478159\pi$
$73$	1.17157	0.137122	0.0685611	+	0.997647i	$0.478159\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.3431	1.10890
$88$	0	0
$89$	−13.3137	−1.41125	−0.705625	−	0.708585i	$-0.749334\pi$
$89$	−13.3137	−1.41125	−0.705625	+	0.708585i	$0.749334\pi$
$90$	0	0
$91$	−2.34315	−0.245628
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.65685	−0.371297	−0.185649	−	0.982616i	$-0.559439\pi$
$97$	−3.65685	−0.371297	−0.185649	+	0.982616i	$0.559439\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	9.31371	0.926749	0.463374	−	0.886163i	$-0.346639\pi$
$101$	9.31371	0.926749	0.463374	+	0.886163i	$0.346639\pi$
$102$	0	0
$103$	6.82843	0.672825	0.336412	−	0.941715i	$-0.390786\pi$
$103$	6.82843	0.672825	0.336412	+	0.941715i	$0.390786\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.65685	0.740216	0.370108	−	0.928989i	$-0.379321\pi$
$107$	7.65685	0.740216	0.370108	+	0.928989i	$0.379321\pi$
$108$	0	0
$109$	−7.65685	−0.733394	−0.366697	−	0.930341i	$-0.619511\pi$
$109$	−7.65685	−0.733394	−0.366697	+	0.930341i	$0.619511\pi$
$110$	0	0
$111$	−21.6569	−2.05558
$112$	0	0
$113$	−19.6569	−1.84916	−0.924581	−	0.380986i	$-0.875584\pi$
$113$	−19.6569	−1.84916	−0.924581	+	0.380986i	$0.875584\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.85786	0.541560
$118$	0	0
$119$	13.6569	1.25192
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−16.9706	−1.53018
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.34315	0.385392	0.192696	−	0.981259i	$-0.438277\pi$
$127$	4.34315	0.385392	0.192696	+	0.981259i	$0.438277\pi$
$128$	0	0
$129$	16.9706	1.49417
$130$	0	0
$131$	11.3137	0.988483	0.494242	−	0.869325i	$-0.335446\pi$
$131$	11.3137	0.988483	0.494242	+	0.869325i	$0.335446\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.9706	0.937278	0.468639	−	0.883390i	$-0.344744\pi$
$137$	10.9706	0.937278	0.468639	+	0.883390i	$0.344744\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−1.17157	−0.0979718
$144$	0	0
$145$	0	0
$146$	0	0
$147$	8.48528	0.699854
$148$	0	0
$149$	0.343146	0.0281116	0.0140558	−	0.999901i	$-0.495526\pi$
$149$	0.343146	0.0281116	0.0140558	+	0.999901i	$0.495526\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	−34.1421	−2.76023
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	32.9706	2.61474
$160$	0	0
$161$	5.65685	0.445823
$162$	0	0
$163$	16.4853	1.29123	0.645613	−	0.763664i	$-0.276602\pi$
$163$	16.4853	1.29123	0.645613	+	0.763664i	$0.276602\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.9706	−1.77752	−0.888758	−	0.458377i	$-0.848431\pi$
$167$	−22.9706	−1.77752	−0.888758	+	0.458377i	$0.848431\pi$
$168$	0	0
$169$	−11.6274	−0.894417
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.1421	1.68344	0.841718	−	0.539918i	$-0.181545\pi$
$173$	22.1421	1.68344	0.841718	+	0.539918i	$0.181545\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.68629	0.352243
$178$	0	0
$179$	−9.65685	−0.721787	−0.360894	−	0.932607i	$-0.617528\pi$
$179$	−9.65685	−0.721787	−0.360894	+	0.932607i	$0.617528\pi$
$180$	0	0
$181$	21.3137	1.58424	0.792118	−	0.610368i	$-0.208979\pi$
$181$	21.3137	1.58424	0.792118	+	0.610368i	$0.208979\pi$
$182$	0	0
$183$	26.3431	1.94734
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.82843	0.499344
$188$	0	0
$189$	11.3137	0.822951
$190$	0	0
$191$	−3.31371	−0.239772	−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371	−0.239772	−0.119886	+	0.992788i	$0.538253\pi$
$192$	0	0
$193$	1.17157	0.0843317	0.0421658	−	0.999111i	$-0.486574\pi$
$193$	1.17157	0.0843317	0.0421658	+	0.999111i	$0.486574\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.8284	0.771493	0.385747	−	0.922605i	$-0.373944\pi$
$197$	10.8284	0.771493	0.385747	+	0.922605i	$0.373944\pi$
$198$	0	0
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	0	0
$201$	−35.3137	−2.49084
$202$	0	0
$203$	7.31371	0.513322
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−14.1421	−0.982946
$208$	0	0
$209$	0	0
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	32.0000	2.19260
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−3.31371	−0.223920
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−10.8284	−0.725125	−0.362563	−	0.931959i	$-0.618098\pi$
$223$	−10.8284	−0.725125	−0.362563	+	0.931959i	$0.618098\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.3137	1.68013	0.840065	−	0.542486i	$-0.182517\pi$
$227$	25.3137	1.68013	0.840065	+	0.542486i	$0.182517\pi$
$228$	0	0
$229$	1.31371	0.0868123	0.0434062	−	0.999058i	$-0.486179\pi$
$229$	1.31371	0.0868123	0.0434062	+	0.999058i	$0.486179\pi$
$230$	0	0
$231$	−5.65685	−0.372194
$232$	0	0
$233$	6.14214	0.402385	0.201192	−	0.979552i	$-0.435518\pi$
$233$	6.14214	0.402385	0.201192	+	0.979552i	$0.435518\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	11.3137	0.734904
$238$	0	0
$239$	23.3137	1.50804	0.754019	−	0.656852i	$-0.228113\pi$
$239$	23.3137	1.50804	0.754019	+	0.656852i	$0.228113\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	14.1421	0.907218
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	16.9706	1.07547
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	2.82843	0.177822
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.31371	0.580973	0.290487	−	0.956879i	$-0.406183\pi$
$257$	9.31371	0.580973	0.290487	+	0.956879i	$0.406183\pi$
$258$	0	0
$259$	−15.3137	−0.951548
$260$	0	0
$261$	−18.2843	−1.13177
$262$	0	0
$263$	−10.9706	−0.676474	−0.338237	−	0.941061i	$-0.609831\pi$
$263$	−10.9706	−0.676474	−0.338237	+	0.941061i	$0.609831\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	37.6569	2.30456
$268$	0	0
$269$	17.3137	1.05564	0.527818	−	0.849358i	$-0.323010\pi$
$269$	17.3137	1.05564	0.527818	+	0.849358i	$0.323010\pi$
$270$	0	0
$271$	−7.31371	−0.444276	−0.222138	−	0.975015i	$-0.571304\pi$
$271$	−7.31371	−0.444276	−0.222138	+	0.975015i	$0.571304\pi$
$272$	0	0
$273$	6.62742	0.401110
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.82843	−0.410280	−0.205140	−	0.978733i	$-0.565765\pi$
$277$	−6.82843	−0.410280	−0.205140	+	0.978733i	$0.565765\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.3137	1.03285	0.516425	−	0.856333i	$-0.327263\pi$
$281$	17.3137	1.03285	0.516425	+	0.856333i	$0.327263\pi$
$282$	0	0
$283$	32.6274	1.93950	0.969749	−	0.244103i	$-0.0784935\pi$
$283$	32.6274	1.93950	0.969749	+	0.244103i	$0.0784935\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	29.6274	1.74279
$290$	0	0
$291$	10.3431	0.606326
$292$	0	0
$293$	9.17157	0.535809	0.267905	−	0.963445i	$-0.413669\pi$
$293$	9.17157	0.535809	0.267905	+	0.963445i	$0.413669\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.65685	0.328244
$298$	0	0
$299$	−3.31371	−0.191637
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	−26.3431	−1.51337
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−16.3431	−0.932753	−0.466376	−	0.884586i	$-0.654441\pi$
$307$	−16.3431	−0.932753	−0.466376	+	0.884586i	$0.654441\pi$
$308$	0	0
$309$	−19.3137	−1.09872
$310$	0	0
$311$	−4.68629	−0.265735	−0.132868	−	0.991134i	$-0.542419\pi$
$311$	−4.68629	−0.265735	−0.132868	+	0.991134i	$0.542419\pi$
$312$	0	0
$313$	1.31371	0.0742552	0.0371276	−	0.999311i	$-0.488179\pi$
$313$	1.31371	0.0742552	0.0371276	+	0.999311i	$0.488179\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.31371	0.0737852	0.0368926	−	0.999319i	$-0.488254\pi$
$317$	1.31371	0.0737852	0.0368926	+	0.999319i	$0.488254\pi$
$318$	0	0
$319$	3.65685	0.204745
$320$	0	0
$321$	−21.6569	−1.20877
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	21.6569	1.19763
$328$	0	0
$329$	−5.65685	−0.311872
$330$	0	0
$331$	7.31371	0.401998	0.200999	−	0.979591i	$-0.435581\pi$
$331$	7.31371	0.401998	0.200999	+	0.979591i	$0.435581\pi$
$332$	0	0
$333$	38.2843	2.09797
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.4853	1.11590	0.557952	−	0.829873i	$-0.311587\pi$
$337$	20.4853	1.11590	0.557952	+	0.829873i	$0.311587\pi$
$338$	0	0
$339$	55.5980	3.01967
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.9706	0.588931	0.294465	−	0.955662i	$-0.404858\pi$
$347$	10.9706	0.588931	0.294465	+	0.955662i	$0.404858\pi$
$348$	0	0
$349$	26.9706	1.44370	0.721851	−	0.692049i	$-0.243292\pi$
$349$	26.9706	1.44370	0.721851	+	0.692049i	$0.243292\pi$
$350$	0	0
$351$	−6.62742	−0.353745
$352$	0	0
$353$	−21.3137	−1.13441	−0.567207	−	0.823575i	$-0.691976\pi$
$353$	−21.3137	−1.13441	−0.567207	+	0.823575i	$0.691976\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−38.6274	−2.04438
$358$	0	0
$359$	−0.686292	−0.0362211	−0.0181105	−	0.999836i	$-0.505765\pi$
$359$	−0.686292	−0.0362211	−0.0181105	+	0.999836i	$0.505765\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−2.82843	−0.148454
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8.48528	−0.442928	−0.221464	−	0.975169i	$-0.571084\pi$
$367$	−8.48528	−0.442928	−0.221464	+	0.975169i	$0.571084\pi$
$368$	0	0
$369$	30.0000	1.56174
$370$	0	0
$371$	23.3137	1.21039
$372$	0	0
$373$	−35.7990	−1.85360	−0.926801	−	0.375554i	$-0.877453\pi$
$373$	−35.7990	−1.85360	−0.926801	+	0.375554i	$0.877453\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.28427	−0.220651
$378$	0	0
$379$	−33.6569	−1.72884	−0.864418	−	0.502773i	$-0.832313\pi$
$379$	−33.6569	−1.72884	−0.864418	+	0.502773i	$0.832313\pi$
$380$	0	0
$381$	−12.2843	−0.629342
$382$	0	0
$383$	−5.85786	−0.299323	−0.149661	−	0.988737i	$-0.547818\pi$
$383$	−5.85786	−0.299323	−0.149661	+	0.988737i	$0.547818\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−30.0000	−1.52499
$388$	0	0
$389$	20.6274	1.04585	0.522926	−	0.852378i	$-0.324840\pi$
$389$	20.6274	1.04585	0.522926	+	0.852378i	$0.324840\pi$
$390$	0	0
$391$	19.3137	0.976736
$392$	0	0
$393$	−32.0000	−1.61419
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.31371	0.467442	0.233721	−	0.972304i	$-0.424910\pi$
$397$	9.31371	0.467442	0.233721	+	0.972304i	$0.424910\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.31371	−0.265354	−0.132677	−	0.991159i	$-0.542357\pi$
$401$	−5.31371	−0.265354	−0.132677	+	0.991159i	$0.542357\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.65685	−0.379536
$408$	0	0
$409$	1.02944	0.0509024	0.0254512	−	0.999676i	$-0.491898\pi$
$409$	1.02944	0.0509024	0.0254512	+	0.999676i	$0.491898\pi$
$410$	0	0
$411$	−31.0294	−1.53057
$412$	0	0
$413$	3.31371	0.163057
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−11.3137	−0.554035
$418$	0	0
$419$	25.6569	1.25342	0.626710	−	0.779253i	$-0.284401\pi$
$419$	25.6569	1.25342	0.626710	+	0.779253i	$0.284401\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	14.1421	0.687614
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.6274	0.901444
$428$	0	0
$429$	3.31371	0.159987
$430$	0	0
$431$	11.3137	0.544962	0.272481	−	0.962161i	$-0.412156\pi$
$431$	11.3137	0.544962	0.272481	+	0.962161i	$0.412156\pi$
$432$	0	0
$433$	7.65685	0.367965	0.183982	−	0.982930i	$-0.441101\pi$
$433$	7.65685	0.367965	0.183982	+	0.982930i	$0.441101\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	0	0
$443$	−26.8284	−1.27466	−0.637329	−	0.770592i	$-0.719961\pi$
$443$	−26.8284	−1.27466	−0.637329	+	0.770592i	$0.719961\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.970563	−0.0459060
$448$	0	0
$449$	28.6274	1.35101	0.675506	−	0.737355i	$-0.263925\pi$
$449$	28.6274	1.35101	0.675506	+	0.737355i	$0.263925\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	−33.9411	−1.59469
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.485281	−0.0227005	−0.0113503	−	0.999936i	$-0.503613\pi$
$457$	−0.485281	−0.0227005	−0.0113503	+	0.999936i	$0.503613\pi$
$458$	0	0
$459$	38.6274	1.80297
$460$	0	0
$461$	12.6274	0.588117	0.294059	−	0.955787i	$-0.404994\pi$
$461$	12.6274	0.588117	0.294059	+	0.955787i	$0.404994\pi$
$462$	0	0
$463$	−6.14214	−0.285449	−0.142725	−	0.989762i	$-0.545586\pi$
$463$	−6.14214	−0.285449	−0.142725	+	0.989762i	$0.545586\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.8284	−0.686178	−0.343089	−	0.939303i	$-0.611473\pi$
$467$	−14.8284	−0.686178	−0.343089	+	0.939303i	$0.611473\pi$
$468$	0	0
$469$	−24.9706	−1.15303
$470$	0	0
$471$	−39.5980	−1.82458
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−58.2843	−2.66865
$478$	0	0
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	8.97056	0.409022
$482$	0	0
$483$	−16.0000	−0.728025
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−24.4853	−1.10953	−0.554767	−	0.832006i	$-0.687193\pi$
$487$	−24.4853	−1.10953	−0.554767	+	0.832006i	$0.687193\pi$
$488$	0	0
$489$	−46.6274	−2.10856
$490$	0	0
$491$	0.686292	0.0309719	0.0154860	−	0.999880i	$-0.495070\pi$
$491$	0.686292	0.0309719	0.0154860	+	0.999880i	$0.495070\pi$
$492$	0	0
$493$	24.9706	1.12462
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.6274	1.01498
$498$	0	0
$499$	−9.65685	−0.432300	−0.216150	−	0.976360i	$-0.569350\pi$
$499$	−9.65685	−0.432300	−0.216150	+	0.976360i	$0.569350\pi$
$500$	0	0
$501$	64.9706	2.90267
$502$	0	0
$503$	16.6274	0.741380	0.370690	−	0.928757i	$-0.379121\pi$
$503$	16.6274	0.741380	0.370690	+	0.928757i	$0.379121\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	32.8873	1.46058
$508$	0	0
$509$	−13.3137	−0.590120	−0.295060	−	0.955479i	$-0.595340\pi$
$509$	−13.3137	−0.590120	−0.295060	+	0.955479i	$0.595340\pi$
$510$	0	0
$511$	−2.34315	−0.103655
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−2.82843	−0.124394
$518$	0	0
$519$	−62.6274	−2.74904
$520$	0	0
$521$	25.3137	1.10901	0.554507	−	0.832179i	$-0.312907\pi$
$521$	25.3137	1.10901	0.554507	+	0.832179i	$0.312907\pi$
$522$	0	0
$523$	−41.5980	−1.81895	−0.909476	−	0.415756i	$-0.863517\pi$
$523$	−41.5980	−1.81895	−0.909476	+	0.415756i	$0.863517\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	−8.28427	−0.359507
$532$	0	0
$533$	7.02944	0.304479
$534$	0	0
$535$	0	0
$536$	0	0
$537$	27.3137	1.17867
$538$	0	0
$539$	3.00000	0.129219
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	0	0
$543$	−60.2843	−2.58705
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−34.0000	−1.45374	−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000	−1.45374	−0.726868	+	0.686778i	$0.759025\pi$
$548$	0	0
$549$	−46.5685	−1.98750
$550$	0	0
$551$	0	0
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.85786	−0.417691	−0.208846	−	0.977949i	$-0.566971\pi$
$557$	−9.85786	−0.417691	−0.208846	+	0.977949i	$0.566971\pi$
$558$	0	0
$559$	−7.02944	−0.297314
$560$	0	0
$561$	−19.3137	−0.815425
$562$	0	0
$563$	0.343146	0.0144619	0.00723093	−	0.999974i	$-0.497698\pi$
$563$	0.343146	0.0144619	0.00723093	+	0.999974i	$0.497698\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	31.6569	1.32712	0.663562	−	0.748121i	$-0.269044\pi$
$569$	31.6569	1.32712	0.663562	+	0.748121i	$0.269044\pi$
$570$	0	0
$571$	21.9411	0.918208	0.459104	−	0.888383i	$-0.348171\pi$
$571$	21.9411	0.918208	0.459104	+	0.888383i	$0.348171\pi$
$572$	0	0
$573$	9.37258	0.391545
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26.9706	1.12280	0.561400	−	0.827545i	$-0.310263\pi$
$577$	26.9706	1.12280	0.561400	+	0.827545i	$0.310263\pi$
$578$	0	0
$579$	−3.31371	−0.137713
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	11.6569	0.482778
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.14214	−0.0884154	−0.0442077	−	0.999022i	$-0.514076\pi$
$587$	−2.14214	−0.0884154	−0.0442077	+	0.999022i	$0.514076\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−30.6274	−1.25984
$592$	0	0
$593$	−3.51472	−0.144332	−0.0721661	−	0.997393i	$-0.522991\pi$
$593$	−3.51472	−0.144332	−0.0721661	+	0.997393i	$0.522991\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29.2548	1.19732
$598$	0	0
$599$	5.65685	0.231133	0.115566	−	0.993300i	$-0.463132\pi$
$599$	5.65685	0.231133	0.115566	+	0.993300i	$0.463132\pi$
$600$	0	0
$601$	23.9411	0.976579	0.488289	−	0.872682i	$-0.337621\pi$
$601$	23.9411	0.976579	0.488289	+	0.872682i	$0.337621\pi$
$602$	0	0
$603$	62.4264	2.54220
$604$	0	0
$605$	0	0
$606$	0	0
$607$	38.2843	1.55391	0.776955	−	0.629556i	$-0.216763\pi$
$607$	38.2843	1.55391	0.776955	+	0.629556i	$0.216763\pi$
$608$	0	0
$609$	−20.6863	−0.838251
$610$	0	0
$611$	3.31371	0.134058
$612$	0	0
$613$	25.4558	1.02815	0.514076	−	0.857745i	$-0.328135\pi$
$613$	25.4558	1.02815	0.514076	+	0.857745i	$0.328135\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.343146	−0.0138145	−0.00690726	−	0.999976i	$-0.502199\pi$
$617$	−0.343146	−0.0138145	−0.00690726	+	0.999976i	$0.502199\pi$
$618$	0	0
$619$	14.3431	0.576500	0.288250	−	0.957555i	$-0.406927\pi$
$619$	14.3431	0.576500	0.288250	+	0.957555i	$0.406927\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	0	0
$623$	26.6274	1.06680
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−52.2843	−2.08471
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	−45.2548	−1.79872
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.51472	−0.139258
$638$	0	0
$639$	−56.5685	−2.23782
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	−1.45584	−0.0574129	−0.0287064	−	0.999588i	$-0.509139\pi$
$643$	−1.45584	−0.0574129	−0.0287064	+	0.999588i	$0.509139\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.1127	1.06591	0.532955	−	0.846144i	$-0.321081\pi$
$647$	27.1127	1.06591	0.532955	+	0.846144i	$0.321081\pi$
$648$	0	0
$649$	1.65685	0.0650372
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11.6569	−0.456168	−0.228084	−	0.973641i	$-0.573246\pi$
$653$	−11.6569	−0.456168	−0.228084	+	0.973641i	$0.573246\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.85786	0.228537
$658$	0	0
$659$	−45.9411	−1.78961	−0.894806	−	0.446455i	$-0.852686\pi$
$659$	−45.9411	−1.78961	−0.894806	+	0.446455i	$0.852686\pi$
$660$	0	0
$661$	44.6274	1.73581	0.867903	−	0.496734i	$-0.165468\pi$
$661$	44.6274	1.73581	0.867903	+	0.496734i	$0.165468\pi$
$662$	0	0
$663$	22.6274	0.878776
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.3431	0.400488
$668$	0	0
$669$	30.6274	1.18412
$670$	0	0
$671$	9.31371	0.359552
$672$	0	0
$673$	12.4853	0.481272	0.240636	−	0.970615i	$-0.422644\pi$
$673$	12.4853	0.481272	0.240636	+	0.970615i	$0.422644\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.8284	−0.877368	−0.438684	−	0.898641i	$-0.644555\pi$
$677$	−22.8284	−0.877368	−0.438684	+	0.898641i	$0.644555\pi$
$678$	0	0
$679$	7.31371	0.280674
$680$	0	0
$681$	−71.5980	−2.74364
$682$	0	0
$683$	−7.79899	−0.298420	−0.149210	−	0.988806i	$-0.547673\pi$
$683$	−7.79899	−0.298420	−0.149210	+	0.988806i	$0.547673\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3.71573	−0.141764
$688$	0	0
$689$	−13.6569	−0.520285
$690$	0	0
$691$	39.3137	1.49556	0.747782	−	0.663944i	$-0.231119\pi$
$691$	39.3137	1.49556	0.747782	+	0.663944i	$0.231119\pi$
$692$	0	0
$693$	10.0000	0.379869
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−40.9706	−1.55187
$698$	0	0
$699$	−17.3726	−0.657091
$700$	0	0
$701$	−12.6274	−0.476931	−0.238465	−	0.971151i	$-0.576644\pi$
$701$	−12.6274	−0.476931	−0.238465	+	0.971151i	$0.576644\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18.6274	−0.700556
$708$	0	0
$709$	24.6274	0.924902	0.462451	−	0.886645i	$-0.346970\pi$
$709$	24.6274	0.924902	0.462451	+	0.886645i	$0.346970\pi$
$710$	0	0
$711$	−20.0000	−0.750059
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−65.9411	−2.46262
$718$	0	0
$719$	−18.3431	−0.684084	−0.342042	−	0.939685i	$-0.611118\pi$
$719$	−18.3431	−0.684084	−0.342042	+	0.939685i	$0.611118\pi$
$720$	0	0
$721$	−13.6569	−0.508608
$722$	0	0
$723$	−16.9706	−0.631142
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−19.5147	−0.723761	−0.361880	−	0.932225i	$-0.617865\pi$
$727$	−19.5147	−0.723761	−0.361880	+	0.932225i	$0.617865\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	40.9706	1.51535
$732$	0	0
$733$	17.4558	0.644746	0.322373	−	0.946613i	$-0.395519\pi$
$733$	17.4558	0.644746	0.322373	+	0.946613i	$0.395519\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.4853	−0.459901
$738$	0	0
$739$	−29.9411	−1.10140	−0.550701	−	0.834703i	$-0.685640\pi$
$739$	−29.9411	−1.10140	−0.550701	+	0.834703i	$0.685640\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−49.5980	−1.81957	−0.909787	−	0.415076i	$-0.863755\pi$
$743$	−49.5980	−1.81957	−0.909787	+	0.415076i	$0.863755\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−30.0000	−1.09764
$748$	0	0
$749$	−15.3137	−0.559551
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	33.9411	1.23688
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−13.3137	−0.483895	−0.241947	−	0.970289i	$-0.577786\pi$
$757$	−13.3137	−0.483895	−0.241947	+	0.970289i	$0.577786\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	15.3137	0.554393
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.94113	−0.0700900
$768$	0	0
$769$	−18.9706	−0.684096	−0.342048	−	0.939682i	$-0.611121\pi$
$769$	−18.9706	−0.684096	−0.342048	+	0.939682i	$0.611121\pi$
$770$	0	0
$771$	−26.3431	−0.948725
$772$	0	0
$773$	−26.2843	−0.945380	−0.472690	−	0.881229i	$-0.656717\pi$
$773$	−26.2843	−0.945380	−0.472690	+	0.881229i	$0.656717\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	43.3137	1.55387
$778$	0	0
$779$	0	0
$780$	0	0
$781$	11.3137	0.404836
$782$	0	0
$783$	20.6863	0.739268
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−14.9706	−0.533643	−0.266821	−	0.963746i	$-0.585973\pi$
$787$	−14.9706	−0.533643	−0.266821	+	0.963746i	$0.585973\pi$
$788$	0	0
$789$	31.0294	1.10468
$790$	0	0
$791$	39.3137	1.39783
$792$	0	0
$793$	−10.9117	−0.387485
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32.6274	−1.15572	−0.577861	−	0.816135i	$-0.696113\pi$
$797$	−32.6274	−1.15572	−0.577861	+	0.816135i	$0.696113\pi$
$798$	0	0
$799$	−19.3137	−0.683270
$800$	0	0
$801$	−66.5685	−2.35208
$802$	0	0
$803$	−1.17157	−0.0413439
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−48.9706	−1.72385
$808$	0	0
$809$	−10.9706	−0.385704	−0.192852	−	0.981228i	$-0.561774\pi$
$809$	−10.9706	−0.385704	−0.192852	+	0.981228i	$0.561774\pi$
$810$	0	0
$811$	−53.9411	−1.89413	−0.947065	−	0.321043i	$-0.895967\pi$
$811$	−53.9411	−1.89413	−0.947065	+	0.321043i	$0.895967\pi$
$812$	0	0
$813$	20.6863	0.725500
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−11.7157	−0.409381
$820$	0	0
$821$	−41.3137	−1.44186	−0.720929	−	0.693009i	$-0.756285\pi$
$821$	−41.3137	−1.44186	−0.720929	+	0.693009i	$0.756285\pi$
$822$	0	0
$823$	19.5147	0.680240	0.340120	−	0.940382i	$-0.389532\pi$
$823$	19.5147	0.680240	0.340120	+	0.940382i	$0.389532\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.2843	0.774900	0.387450	−	0.921891i	$-0.373356\pi$
$827$	22.2843	0.774900	0.387450	+	0.921891i	$0.373356\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	19.3137	0.669985
$832$	0	0
$833$	20.4853	0.709773
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26.3431	−0.909466	−0.454733	−	0.890628i	$-0.650265\pi$
$839$	−26.3431	−0.909466	−0.454733	+	0.890628i	$0.650265\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	−48.9706	−1.68664
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	−92.2843	−3.16719
$850$	0	0
$851$	−21.6569	−0.742387
$852$	0	0
$853$	15.5147	0.531214	0.265607	−	0.964081i	$-0.414428\pi$
$853$	15.5147	0.531214	0.265607	+	0.964081i	$0.414428\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.7696	−0.846112	−0.423056	−	0.906104i	$-0.639043\pi$
$857$	−24.7696	−0.846112	−0.423056	+	0.906104i	$0.639043\pi$
$858$	0	0
$859$	−24.2843	−0.828569	−0.414284	−	0.910148i	$-0.635968\pi$
$859$	−24.2843	−0.828569	−0.414284	+	0.910148i	$0.635968\pi$
$860$	0	0
$861$	33.9411	1.15671
$862$	0	0
$863$	−9.17157	−0.312204	−0.156102	−	0.987741i	$-0.549893\pi$
$863$	−9.17157	−0.312204	−0.156102	+	0.987741i	$0.549893\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−83.7990	−2.84596
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	14.6274	0.495631
$872$	0	0
$873$	−18.2843	−0.618829
$874$	0	0
$875$	0	0
$876$	0	0
$877$	49.4558	1.67001	0.835003	−	0.550246i	$-0.185466\pi$
$877$	49.4558	1.67001	0.835003	+	0.550246i	$0.185466\pi$
$878$	0	0
$879$	−25.9411	−0.874972
$880$	0	0
$881$	−7.37258	−0.248389	−0.124194	−	0.992258i	$-0.539635\pi$
$881$	−7.37258	−0.248389	−0.124194	+	0.992258i	$0.539635\pi$
$882$	0	0
$883$	37.1716	1.25092	0.625462	−	0.780255i	$-0.284911\pi$
$883$	37.1716	1.25092	0.625462	+	0.780255i	$0.284911\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.2843	1.28546	0.642730	−	0.766093i	$-0.277802\pi$
$887$	38.2843	1.28546	0.642730	+	0.766093i	$0.277802\pi$
$888$	0	0
$889$	−8.68629	−0.291329
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.37258	0.312941
$898$	0	0
$899$	0	0
$900$	0	0
$901$	79.5980	2.65179
$902$	0	0
$903$	−33.9411	−1.12949
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−27.5147	−0.913611	−0.456806	−	0.889567i	$-0.651007\pi$
$907$	−27.5147	−0.913611	−0.456806	+	0.889567i	$0.651007\pi$
$908$	0	0
$909$	46.5685	1.54458
$910$	0	0
$911$	9.94113	0.329364	0.164682	−	0.986347i	$-0.447340\pi$
$911$	9.94113	0.329364	0.164682	+	0.986347i	$0.447340\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−22.6274	−0.747223
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	46.2254	1.52318
$922$	0	0
$923$	−13.2548	−0.436288
$924$	0	0
$925$	0	0
$926$	0	0
$927$	34.1421	1.12137
$928$	0	0
$929$	5.31371	0.174337	0.0871686	−	0.996194i	$-0.472218\pi$
$929$	5.31371	0.174337	0.0871686	+	0.996194i	$0.472218\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	13.2548	0.433944
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.45584	0.0475604	0.0237802	−	0.999717i	$-0.492430\pi$
$937$	1.45584	0.0475604	0.0237802	+	0.999717i	$0.492430\pi$
$938$	0	0
$939$	−3.71573	−0.121258
$940$	0	0
$941$	−6.68629	−0.217967	−0.108983	−	0.994044i	$-0.534760\pi$
$941$	−6.68629	−0.217967	−0.108983	+	0.994044i	$0.534760\pi$
$942$	0	0
$943$	−16.9706	−0.552638
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.1716	1.33790	0.668948	−	0.743309i	$-0.266745\pi$
$947$	41.1716	1.33790	0.668948	+	0.743309i	$0.266745\pi$
$948$	0	0
$949$	1.37258	0.0445559
$950$	0	0
$951$	−3.71573	−0.120491
$952$	0	0
$953$	−53.1716	−1.72240	−0.861198	−	0.508269i	$-0.830285\pi$
$953$	−53.1716	−1.72240	−0.861198	+	0.508269i	$0.830285\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−10.3431	−0.334346
$958$	0	0
$959$	−21.9411	−0.708516
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	38.2843	1.23369
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−14.9706	−0.481421	−0.240710	−	0.970597i	$-0.577380\pi$
$967$	−14.9706	−0.481421	−0.240710	+	0.970597i	$0.577380\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.68629	0.278756	0.139378	−	0.990239i	$-0.455490\pi$
$971$	8.68629	0.278756	0.139378	+	0.990239i	$0.455490\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32.3431	−1.03475	−0.517374	−	0.855759i	$-0.673091\pi$
$977$	−32.3431	−1.03475	−0.517374	+	0.855759i	$0.673091\pi$
$978$	0	0
$979$	13.3137	0.425508
$980$	0	0
$981$	−38.2843	−1.22232
$982$	0	0
$983$	−21.8579	−0.697158	−0.348579	−	0.937279i	$-0.613336\pi$
$983$	−21.8579	−0.697158	−0.348579	+	0.937279i	$0.613336\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16.0000	0.509286
$988$	0	0
$989$	16.9706	0.539633
$990$	0	0
$991$	57.9411	1.84056	0.920280	−	0.391260i	$-0.127961\pi$
$991$	57.9411	1.84056	0.920280	+	0.391260i	$0.127961\pi$
$992$	0	0
$993$	−20.6863	−0.656460
$994$	0	0
$995$	0	0
$996$	0	0
$997$	41.4558	1.31292	0.656460	−	0.754361i	$-0.272053\pi$
$997$	41.4558	1.31292	0.656460	+	0.754361i	$0.272053\pi$
$998$	0	0
$999$	−43.3137	−1.37039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bn.1.1		2
4.3	odd	2		275.2.a.c.1.1		2
5.2	odd	4		4400.2.b.q.4049.3		4
5.3	odd	4		4400.2.b.q.4049.2		4
5.4	even	2		880.2.a.m.1.2		2
12.11	even	2		2475.2.a.x.1.2		2
15.14	odd	2		7920.2.a.ch.1.2		2
20.3	even	4		275.2.b.d.199.4		4
20.7	even	4		275.2.b.d.199.1		4
20.19	odd	2		55.2.a.b.1.2	✓	2
40.19	odd	2		3520.2.a.bn.1.2		2
40.29	even	2		3520.2.a.bo.1.1		2
44.43	even	2		3025.2.a.o.1.2		2
55.54	odd	2		9680.2.a.bn.1.2		2
60.23	odd	4		2475.2.c.l.199.1		4
60.47	odd	4		2475.2.c.l.199.4		4
60.59	even	2		495.2.a.b.1.1		2
140.139	even	2		2695.2.a.f.1.2		2
220.19	even	10		605.2.g.l.251.1		8
220.39	even	10		605.2.g.l.366.2		8
220.59	odd	10		605.2.g.f.511.2		8
220.79	even	10		605.2.g.l.81.2		8
220.119	odd	10		605.2.g.f.81.1		8
220.139	even	10		605.2.g.l.511.1		8
220.159	odd	10		605.2.g.f.366.1		8
220.179	odd	10		605.2.g.f.251.2		8
220.219	even	2		605.2.a.d.1.1		2
260.259	odd	2		9295.2.a.g.1.1		2
660.659	odd	2		5445.2.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.a.b.1.2	✓	2	20.19	odd	2
275.2.a.c.1.1		2	4.3	odd	2
275.2.b.d.199.1		4	20.7	even	4
275.2.b.d.199.4		4	20.3	even	4
495.2.a.b.1.1		2	60.59	even	2
605.2.a.d.1.1		2	220.219	even	2
605.2.g.f.81.1		8	220.119	odd	10
605.2.g.f.251.2		8	220.179	odd	10
605.2.g.f.366.1		8	220.159	odd	10
605.2.g.f.511.2		8	220.59	odd	10
605.2.g.l.81.2		8	220.79	even	10
605.2.g.l.251.1		8	220.19	even	10
605.2.g.l.366.2		8	220.39	even	10
605.2.g.l.511.1		8	220.139	even	10
880.2.a.m.1.2		2	5.4	even	2
2475.2.a.x.1.2		2	12.11	even	2
2475.2.c.l.199.1		4	60.23	odd	4
2475.2.c.l.199.4		4	60.47	odd	4
2695.2.a.f.1.2		2	140.139	even	2
3025.2.a.o.1.2		2	44.43	even	2
3520.2.a.bn.1.2		2	40.19	odd	2
3520.2.a.bo.1.1		2	40.29	even	2
4400.2.a.bn.1.1		2	1.1	even	1	trivial
4400.2.b.q.4049.2		4	5.3	odd	4
4400.2.b.q.4049.3		4	5.2	odd	4
5445.2.a.y.1.2		2	660.659	odd	2
7920.2.a.ch.1.2		2	15.14	odd	2
9295.2.a.g.1.1		2	260.259	odd	2
9680.2.a.bn.1.2		2	55.54	odd	2

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)

\(f(q)\)	\(=\)	\(q-2.82843 q^{3} -2.00000 q^{7} +5.00000 q^{9} -1.00000 q^{11} +1.17157 q^{13} -6.82843 q^{17} +5.65685 q^{21} -2.82843 q^{23} -5.65685 q^{27} -3.65685 q^{29} +2.82843 q^{33} +7.65685 q^{37} -3.31371 q^{39} +6.00000 q^{41} -6.00000 q^{43} +2.82843 q^{47} -3.00000 q^{49} +19.3137 q^{51} -11.6569 q^{53} -1.65685 q^{59} -9.31371 q^{61} -10.0000 q^{63} +12.4853 q^{67} +8.00000 q^{69} -11.3137 q^{71} +1.17157 q^{73} +2.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +10.3431 q^{87} -13.3137 q^{89} -2.34315 q^{91} -3.65685 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} - 8 q^{17} + 4 q^{29} + 4 q^{37} + 16 q^{39} + 12 q^{41} - 12 q^{43} - 6 q^{49} + 16 q^{51} - 12 q^{53} + 8 q^{59} + 4 q^{61} - 20 q^{63} + 8 q^{67} + 16 q^{69}+ \cdots - 10 q^{99}+O(q^{100}) \) 2 * q - 4 * q^7 + 10 * q^9 - 2 * q^11 + 8 * q^13 - 8 * q^17 + 4 * q^29 + 4 * q^37 + 16 * q^39 + 12 * q^41 - 12 * q^43 - 6 * q^49 + 16 * q^51 - 12 * q^53 + 8 * q^59 + 4 * q^61 - 20 * q^63 + 8 * q^67 + 16 * q^69 + 8 * q^73 + 4 * q^77 - 8 * q^79 + 2 * q^81 - 12 * q^83 + 32 * q^87 - 4 * q^89 - 16 * q^91 + 4 * q^97 - 10 * q^99

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