LMFDB - Embedded newform 4032.3.d.n.449.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4032,3,Mod(449,4032)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4032, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4032.449"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4032.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,-64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$109.864042590$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 44x^{10} + 719x^{8} + 5356x^{6} + 17809x^{4} + 20000x^{2} + 144 $ x^12 + 44x^10 + 719x^8 + 5356x^6 + 17809x^4 + 20000*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 2016)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.9
Root			$-3.75209i$ of defining polynomial
Character	$\chi$	$=$	4032.449
Dual form			4032.3.d.n.449.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.85589i q^{5} +2.64575 q^{7} +6.89341i q^{11} -20.9186 q^{13} +3.42256i q^{17} -11.1466 q^{19} +25.1631i q^{23} +10.1321 q^{25} -0.948063i q^{29} +6.63987 q^{31} +10.2017i q^{35} -5.63459 q^{37} +30.2404i q^{41} +16.6926 q^{43} -9.40243i q^{47} +7.00000 q^{49} -44.4345i q^{53} -26.5802 q^{55} +52.2908i q^{59} -48.5469 q^{61} -80.6599i q^{65} -63.5263 q^{67} -35.6768i q^{71} +19.4279 q^{73} +18.2382i q^{77} -152.405 q^{79} -97.7442i q^{83} -13.1970 q^{85} -99.3657i q^{89} -55.3454 q^{91} -42.9803i q^{95} +133.405 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 16 q^{13} - 64 q^{19} - 124 q^{25} + 160 q^{31} - 56 q^{37} + 64 q^{43} + 84 q^{49} - 160 q^{55} - 104 q^{61} + 64 q^{67} - 64 q^{73} + 32 q^{79} + 184 q^{85} - 96 q^{97}+O(q^{100}) $ 12 * q - 16 * q^13 - 64 * q^19 - 124 * q^25 + 160 * q^31 - 56 * q^37 + 64 * q^43 + 84 * q^49 - 160 * q^55 - 104 * q^61 + 64 * q^67 - 64 * q^73 + 32 * q^79 + 184 * q^85 - 96 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.85589i	0.771179i	0.922670	+	0.385589i	$0.126002\pi$
$5$	3.85589i	0.771179i	−0.922670	+	0.385589i	$0.873998\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.89341i	0.626673i	0.949642	+	0.313337i	$0.101447\pi$
$11$	6.89341i	0.626673i	−0.949642	+	0.313337i	$0.898553\pi$
$12$	0	0
$13$	−20.9186	−1.60912	−0.804562	−	0.593869i	$-0.797600\pi$
$13$	−20.9186	−1.60912	−0.804562	+	0.593869i	$0.797600\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.42256i	0.201327i	0.994921	+	0.100664i	$0.0320966\pi$
$17$	3.42256i	0.201327i	−0.994921	+	0.100664i	$0.967903\pi$
$18$	0	0
$19$	−11.1466	−0.586666	−0.293333	−	0.956010i	$-0.594764\pi$
$19$	−11.1466	−0.586666	−0.293333	+	0.956010i	$0.594764\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	25.1631i	1.09405i	0.837117	+	0.547023i	$0.184239\pi$
$23$	25.1631i	1.09405i	−0.837117	+	0.547023i	$0.815761\pi$
$24$	0	0
$25$	10.1321	0.405283
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.948063i	− 0.0326918i	−0.999866	−	0.0163459i	$-0.994797\pi$
$29$	− 0.948063i	− 0.0326918i	0.999866	−	0.0163459i	$-0.00520330\pi$
$30$	0	0
$31$	6.63987	0.214189	0.107095	−	0.994249i	$-0.465845\pi$
$31$	6.63987	0.214189	0.107095	+	0.994249i	$0.465845\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.2017i	0.291478i
$36$	0	0
$37$	−5.63459	−0.152286	−0.0761430	−	0.997097i	$-0.524261\pi$
$37$	−5.63459	−0.152286	−0.0761430	+	0.997097i	$0.524261\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	30.2404i	0.737570i	0.929515	+	0.368785i	$0.120226\pi$
$41$	30.2404i	0.737570i	−0.929515	+	0.368785i	$0.879774\pi$
$42$	0	0
$43$	16.6926	0.388201	0.194100	−	0.980982i	$-0.437821\pi$
$43$	16.6926	0.388201	0.194100	+	0.980982i	$0.437821\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.40243i	− 0.200052i	−0.994985	−	0.100026i	$-0.968107\pi$
$47$	− 9.40243i	− 0.200052i	0.994985	−	0.100026i	$-0.0318925\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 44.4345i	− 0.838387i	−0.907897	−	0.419194i	$-0.862313\pi$
$53$	− 44.4345i	− 0.838387i	0.907897	−	0.419194i	$-0.137687\pi$
$54$	0	0
$55$	−26.5802	−0.483277
$56$	0	0
$57$	0	0
$58$	0	0
$59$	52.2908i	0.886285i	0.896451	+	0.443143i	$0.146137\pi$
$59$	52.2908i	0.886285i	−0.896451	+	0.443143i	$0.853863\pi$
$60$	0	0
$61$	−48.5469	−0.795852	−0.397926	−	0.917418i	$-0.630270\pi$
$61$	−48.5469	−0.795852	−0.397926	+	0.917418i	$0.630270\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 80.6599i	− 1.24092i
$66$	0	0
$67$	−63.5263	−0.948154	−0.474077	−	0.880483i	$-0.657218\pi$
$67$	−63.5263	−0.948154	−0.474077	+	0.880483i	$0.657218\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 35.6768i	− 0.502490i	−0.967924	−	0.251245i	$-0.919160\pi$
$71$	− 35.6768i	− 0.502490i	0.967924	−	0.251245i	$-0.0808400\pi$
$72$	0	0
$73$	19.4279	0.266135	0.133068	−	0.991107i	$-0.457517\pi$
$73$	19.4279	0.266135	0.133068	+	0.991107i	$0.457517\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.2382i	0.236860i
$78$	0	0
$79$	−152.405	−1.92918	−0.964588	−	0.263760i	$-0.915037\pi$
$79$	−152.405	−1.92918	−0.964588	+	0.263760i	$0.915037\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 97.7442i	− 1.17764i	−0.808264	−	0.588821i	$-0.799592\pi$
$83$	− 97.7442i	− 1.17764i	0.808264	−	0.588821i	$-0.200408\pi$
$84$	0	0
$85$	−13.1970	−0.155259
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 99.3657i	− 1.11647i	−0.829684	−	0.558234i	$-0.811479\pi$
$89$	− 99.3657i	− 1.11647i	0.829684	−	0.558234i	$-0.188521\pi$
$90$	0	0
$91$	−55.3454	−0.608192
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 42.9803i	− 0.452424i
$96$	0	0
$97$	133.405	1.37531	0.687657	−	0.726036i	$-0.258639\pi$
$97$	133.405	1.37531	0.687657	+	0.726036i	$0.258639\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	161.774i	1.60172i	0.598852	+	0.800860i	$0.295624\pi$
$101$	161.774i	1.60172i	−0.598852	+	0.800860i	$0.704376\pi$
$102$	0	0
$103$	88.2106	0.856414	0.428207	−	0.903681i	$-0.359145\pi$
$103$	88.2106	0.856414	0.428207	+	0.903681i	$0.359145\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 128.321i	− 1.19926i	−0.800276	−	0.599632i	$-0.795314\pi$
$107$	− 128.321i	− 1.19926i	0.800276	−	0.599632i	$-0.204686\pi$
$108$	0	0
$109$	−158.797	−1.45686	−0.728429	−	0.685122i	$-0.759749\pi$
$109$	−158.797	−1.45686	−0.728429	+	0.685122i	$0.759749\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 9.63061i	− 0.0852266i	−0.999092	−	0.0426133i	$-0.986432\pi$
$113$	− 9.63061i	− 0.0852266i	0.999092	−	0.0426133i	$-0.0135684\pi$
$114$	0	0
$115$	−97.0261	−0.843705
$116$	0	0
$117$	0	0
$118$	0	0
$119$	9.05525i	0.0760946i
$120$	0	0
$121$	73.4809	0.607280
$122$	0	0
$123$	0	0
$124$	0	0
$125$	135.466i	1.08372i
$126$	0	0
$127$	−187.202	−1.47403	−0.737017	−	0.675875i	$-0.763766\pi$
$127$	−187.202	−1.47403	−0.737017	+	0.675875i	$0.763766\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 103.874i	− 0.792928i	−0.918050	−	0.396464i	$-0.870237\pi$
$131$	− 103.874i	− 0.792928i	0.918050	−	0.396464i	$-0.129763\pi$
$132$	0	0
$133$	−29.4913	−0.221739
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 198.199i	− 1.44671i	−0.690476	−	0.723356i	$-0.742599\pi$
$137$	− 198.199i	− 1.44671i	0.690476	−	0.723356i	$-0.257401\pi$
$138$	0	0
$139$	59.7079	0.429554	0.214777	−	0.976663i	$-0.431098\pi$
$139$	59.7079	0.429554	0.214777	+	0.976663i	$0.431098\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 144.200i	− 1.00839i
$144$	0	0
$145$	3.65563	0.0252112
$146$	0	0
$147$	0	0
$148$	0	0
$149$	56.1968i	0.377160i	0.982058	+	0.188580i	$0.0603884\pi$
$149$	56.1968i	0.377160i	−0.982058	+	0.188580i	$0.939612\pi$
$150$	0	0
$151$	8.98303	0.0594903	0.0297451	−	0.999558i	$-0.490530\pi$
$151$	8.98303	0.0594903	0.0297451	+	0.999558i	$0.490530\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	25.6026i	0.165178i
$156$	0	0
$157$	13.3517	0.0850424	0.0425212	−	0.999096i	$-0.486461\pi$
$157$	13.3517	0.0850424	0.0425212	+	0.999096i	$0.486461\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	66.5752i	0.413511i
$162$	0	0
$163$	−136.711	−0.838715	−0.419358	−	0.907821i	$-0.637745\pi$
$163$	−136.711	−0.838715	−0.419358	+	0.907821i	$0.637745\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 45.5297i	− 0.272633i	−0.990665	−	0.136316i	$-0.956474\pi$
$167$	− 45.5297i	− 0.272633i	0.990665	−	0.136316i	$-0.0435264\pi$
$168$	0	0
$169$	268.588	1.58928
$170$	0	0
$171$	0	0
$172$	0	0
$173$	38.8939i	0.224820i	0.993662	+	0.112410i	$0.0358570\pi$
$173$	38.8939i	0.224820i	−0.993662	+	0.112410i	$0.964143\pi$
$174$	0	0
$175$	26.8070	0.153183
$176$	0	0
$177$	0	0
$178$	0	0
$179$	68.6751i	0.383660i	0.981428	+	0.191830i	$0.0614422\pi$
$179$	68.6751i	0.383660i	−0.981428	+	0.191830i	$0.938558\pi$
$180$	0	0
$181$	104.985	0.580026	0.290013	−	0.957023i	$-0.406340\pi$
$181$	104.985	0.580026	0.290013	+	0.957023i	$0.406340\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 21.7264i	− 0.117440i
$186$	0	0
$187$	−23.5931	−0.126166
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 43.1970i	− 0.226162i	−0.993586	−	0.113081i	$-0.963928\pi$
$191$	− 43.1970i	− 0.226162i	0.993586	−	0.113081i	$-0.0360720\pi$
$192$	0	0
$193$	−92.0404	−0.476893	−0.238447	−	0.971156i	$-0.576638\pi$
$193$	−92.0404	−0.476893	−0.238447	+	0.971156i	$0.576638\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	86.2740i	0.437939i	0.975732	+	0.218969i	$0.0702695\pi$
$197$	86.2740i	0.437939i	−0.975732	+	0.218969i	$0.929730\pi$
$198$	0	0
$199$	−3.93405	−0.0197691	−0.00988456	−	0.999951i	$-0.503146\pi$
$199$	−3.93405	−0.0197691	−0.00988456	+	0.999951i	$0.503146\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2.50834i	− 0.0123563i
$204$	0	0
$205$	−116.604	−0.568798
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 76.8384i	− 0.367648i
$210$	0	0
$211$	94.1420	0.446171	0.223085	−	0.974799i	$-0.428387\pi$
$211$	94.1420	0.446171	0.223085	+	0.974799i	$0.428387\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	64.3650i	0.299372i
$216$	0	0
$217$	17.5675	0.0809560
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 71.5953i	− 0.323961i
$222$	0	0
$223$	149.630	0.670986	0.335493	−	0.942043i	$-0.391097\pi$
$223$	149.630	0.670986	0.335493	+	0.942043i	$0.391097\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 24.5021i	− 0.107939i	−0.998543	−	0.0539693i	$-0.982813\pi$
$227$	− 24.5021i	− 0.107939i	0.998543	−	0.0539693i	$-0.0171873\pi$
$228$	0	0
$229$	−335.268	−1.46405	−0.732027	−	0.681276i	$-0.761425\pi$
$229$	−335.268	−1.46405	−0.732027	+	0.681276i	$0.761425\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 60.9364i	− 0.261530i	−0.991413	−	0.130765i	$-0.958257\pi$
$233$	− 60.9364i	− 0.261530i	0.991413	−	0.130765i	$-0.0417433\pi$
$234$	0	0
$235$	36.2548	0.154276
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 319.472i	− 1.33670i	−0.743846	−	0.668351i	$-0.767000\pi$
$239$	− 319.472i	− 1.33670i	0.743846	−	0.668351i	$-0.233000\pi$
$240$	0	0
$241$	274.230	1.13789	0.568943	−	0.822377i	$-0.307353\pi$
$241$	274.230	1.13789	0.568943	+	0.822377i	$0.307353\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	26.9913i	0.110168i
$246$	0	0
$247$	233.172	0.944017
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 455.650i	− 1.81534i	−0.419687	−	0.907669i	$-0.637860\pi$
$251$	− 455.650i	− 1.81534i	0.419687	−	0.907669i	$-0.362140\pi$
$252$	0	0
$253$	−173.459	−0.685610
$254$	0	0
$255$	0	0
$256$	0	0
$257$	86.2818i	0.335727i	0.985810	+	0.167863i	$0.0536868\pi$
$257$	86.2818i	0.335727i	−0.985810	+	0.167863i	$0.946313\pi$
$258$	0	0
$259$	−14.9077	−0.0575587
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 290.234i	− 1.10355i	−0.833992	−	0.551776i	$-0.813950\pi$
$263$	− 290.234i	− 1.10355i	0.833992	−	0.551776i	$-0.186050\pi$
$264$	0	0
$265$	171.335	0.646546
$266$	0	0
$267$	0	0
$268$	0	0
$269$	309.859i	1.15189i	0.817488	+	0.575945i	$0.195366\pi$
$269$	309.859i	1.15189i	−0.817488	+	0.575945i	$0.804634\pi$
$270$	0	0
$271$	−475.422	−1.75432	−0.877162	−	0.480195i	$-0.840566\pi$
$271$	−475.422	−1.75432	−0.877162	+	0.480195i	$0.840566\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	69.8446i	0.253980i
$276$	0	0
$277$	−399.021	−1.44051	−0.720254	−	0.693710i	$-0.755975\pi$
$277$	−399.021	−1.44051	−0.720254	+	0.693710i	$0.755975\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 57.9182i	− 0.206115i	−0.994675	−	0.103057i	$-0.967137\pi$
$281$	− 57.9182i	− 0.206115i	0.994675	−	0.103057i	$-0.0328625\pi$
$282$	0	0
$283$	187.466	0.662423	0.331211	−	0.943557i	$-0.392543\pi$
$283$	187.466	0.662423	0.331211	+	0.943557i	$0.392543\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	80.0085i	0.278775i
$288$	0	0
$289$	277.286	0.959467
$290$	0	0
$291$	0	0
$292$	0	0
$293$	59.4001i	0.202731i	0.994849	+	0.101365i	$0.0323211\pi$
$293$	59.4001i	0.202731i	−0.994849	+	0.101365i	$0.967679\pi$
$294$	0	0
$295$	−201.628	−0.683485
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 526.376i	− 1.76046i
$300$	0	0
$301$	44.1646	0.146726
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 187.192i	− 0.613744i
$306$	0	0
$307$	−230.365	−0.750374	−0.375187	−	0.926949i	$-0.622421\pi$
$307$	−230.365	−0.750374	−0.375187	+	0.926949i	$0.622421\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 285.635i	− 0.918440i	−0.888323	−	0.459220i	$-0.848129\pi$
$311$	− 285.635i	− 0.918440i	0.888323	−	0.459220i	$-0.151871\pi$
$312$	0	0
$313$	−158.225	−0.505513	−0.252756	−	0.967530i	$-0.581337\pi$
$313$	−158.225	−0.505513	−0.252756	+	0.967530i	$0.581337\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 398.409i	− 1.25681i	−0.777887	−	0.628405i	$-0.783708\pi$
$317$	− 398.409i	− 1.25681i	0.777887	−	0.628405i	$-0.216292\pi$
$318$	0	0
$319$	6.53538	0.0204871
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 38.1501i	− 0.118112i
$324$	0	0
$325$	−211.949	−0.652151
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 24.8765i	− 0.0756124i
$330$	0	0
$331$	−625.718	−1.89039	−0.945193	−	0.326511i	$-0.894127\pi$
$331$	−625.718	−1.89039	−0.945193	+	0.326511i	$0.894127\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 244.951i	− 0.731196i
$336$	0	0
$337$	−62.6776	−0.185987	−0.0929934	−	0.995667i	$-0.529644\pi$
$337$	−62.6776	−0.185987	−0.0929934	+	0.995667i	$0.529644\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	45.7713i	0.134227i
$342$	0	0
$343$	18.5203	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 470.894i	− 1.35704i	−0.734580	−	0.678522i	$-0.762621\pi$
$347$	− 470.894i	− 1.35704i	0.734580	−	0.678522i	$-0.237379\pi$
$348$	0	0
$349$	−99.3869	−0.284776	−0.142388	−	0.989811i	$-0.545478\pi$
$349$	−99.3869	−0.284776	−0.142388	+	0.989811i	$0.545478\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	606.481i	1.71808i	0.511911	+	0.859039i	$0.328938\pi$
$353$	606.481i	1.71808i	−0.511911	+	0.859039i	$0.671062\pi$
$354$	0	0
$355$	137.566	0.387510
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 572.425i	− 1.59450i	−0.603651	−	0.797249i	$-0.706288\pi$
$359$	− 572.425i	− 1.59450i	0.603651	−	0.797249i	$-0.293712\pi$
$360$	0	0
$361$	−236.752	−0.655823
$362$	0	0
$363$	0	0
$364$	0	0
$365$	74.9118i	0.205238i
$366$	0	0
$367$	−384.530	−1.04777	−0.523883	−	0.851790i	$-0.675517\pi$
$367$	−384.530	−1.04777	−0.523883	+	0.851790i	$0.675517\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 117.563i	− 0.316881i
$372$	0	0
$373$	579.800	1.55442	0.777212	−	0.629239i	$-0.216633\pi$
$373$	579.800	1.55442	0.777212	+	0.629239i	$0.216633\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.8322i	0.0526052i
$378$	0	0
$379$	−128.966	−0.340280	−0.170140	−	0.985420i	$-0.554422\pi$
$379$	−128.966	−0.340280	−0.170140	+	0.985420i	$0.554422\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	525.916i	1.37315i	0.727059	+	0.686575i	$0.240887\pi$
$383$	525.916i	1.37315i	−0.727059	+	0.686575i	$0.759113\pi$
$384$	0	0
$385$	−70.3247	−0.182662
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 134.120i	− 0.344781i	−0.985029	−	0.172391i	$-0.944851\pi$
$389$	− 134.120i	− 0.344781i	0.985029	−	0.172391i	$-0.0551491\pi$
$390$	0	0
$391$	−86.1222	−0.220261
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 587.657i	− 1.48774i
$396$	0	0
$397$	275.606	0.694222	0.347111	−	0.937824i	$-0.387163\pi$
$397$	275.606	0.694222	0.347111	+	0.937824i	$0.387163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 387.895i	− 0.967320i	−0.875256	−	0.483660i	$-0.839307\pi$
$401$	− 387.895i	− 0.967320i	0.875256	−	0.483660i	$-0.160693\pi$
$402$	0	0
$403$	−138.897	−0.344657
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 38.8415i	− 0.0954336i
$408$	0	0
$409$	−261.605	−0.639622	−0.319811	−	0.947481i	$-0.603619\pi$
$409$	−261.605	−0.639622	−0.319811	+	0.947481i	$0.603619\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	138.349i	0.334984i
$414$	0	0
$415$	376.891	0.908172
$416$	0	0
$417$	0	0
$418$	0	0
$419$	345.798i	0.825293i	0.910891	+	0.412646i	$0.135395\pi$
$419$	345.798i	0.825293i	−0.910891	+	0.412646i	$0.864605\pi$
$420$	0	0
$421$	484.947	1.15189	0.575947	−	0.817487i	$-0.304633\pi$
$421$	484.947	1.15189	0.575947	+	0.817487i	$0.304633\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	34.6777i	0.0815946i
$426$	0	0
$427$	−128.443	−0.300804
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 727.249i	− 1.68735i	−0.536851	−	0.843677i	$-0.680386\pi$
$431$	− 727.249i	− 1.68735i	0.536851	−	0.843677i	$-0.319614\pi$
$432$	0	0
$433$	−211.148	−0.487640	−0.243820	−	0.969821i	$-0.578401\pi$
$433$	−211.148	−0.487640	−0.243820	+	0.969821i	$0.578401\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 280.484i	− 0.641840i
$438$	0	0
$439$	−759.909	−1.73100	−0.865500	−	0.500908i	$-0.832999\pi$
$439$	−759.909	−1.73100	−0.865500	+	0.500908i	$0.832999\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	121.346i	0.273919i	0.990577	+	0.136959i	$0.0437330\pi$
$443$	121.346i	0.273919i	−0.990577	+	0.136959i	$0.956267\pi$
$444$	0	0
$445$	383.143	0.860996
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 409.519i	− 0.912068i	−0.889962	−	0.456034i	$-0.849270\pi$
$449$	− 409.519i	− 0.912068i	0.889962	−	0.456034i	$-0.150730\pi$
$450$	0	0
$451$	−208.459	−0.462215
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 213.406i	− 0.469024i
$456$	0	0
$457$	375.001	0.820570	0.410285	−	0.911957i	$-0.365429\pi$
$457$	375.001	0.820570	0.410285	+	0.911957i	$0.365429\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 713.233i	− 1.54714i	−0.633709	−	0.773571i	$-0.718468\pi$
$461$	− 713.233i	− 1.54714i	0.633709	−	0.773571i	$-0.281532\pi$
$462$	0	0
$463$	−642.413	−1.38750	−0.693751	−	0.720215i	$-0.744043\pi$
$463$	−642.413	−1.38750	−0.693751	+	0.720215i	$0.744043\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 251.943i	− 0.539492i	−0.962932	−	0.269746i	$-0.913060\pi$
$467$	− 251.943i	− 0.539492i	0.962932	−	0.269746i	$-0.0869397\pi$
$468$	0	0
$469$	−168.075	−0.358369
$470$	0	0
$471$	0	0
$472$	0	0
$473$	115.069i	0.243275i
$474$	0	0
$475$	−112.939	−0.237766
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 174.575i	− 0.364458i	−0.983256	−	0.182229i	$-0.941669\pi$
$479$	− 174.575i	− 0.364458i	0.983256	−	0.182229i	$-0.0583312\pi$
$480$	0	0
$481$	117.868	0.245047
$482$	0	0
$483$	0	0
$484$	0	0
$485$	514.397i	1.06061i
$486$	0	0
$487$	169.624	0.348304	0.174152	−	0.984719i	$-0.444282\pi$
$487$	169.624	0.348304	0.174152	+	0.984719i	$0.444282\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	56.9134i	0.115913i	0.998319	+	0.0579566i	$0.0184585\pi$
$491$	56.9134i	0.115913i	−0.998319	+	0.0579566i	$0.981541\pi$
$492$	0	0
$493$	3.24481	0.00658176
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 94.3919i	− 0.189923i
$498$	0	0
$499$	−21.0923	−0.0422691	−0.0211345	−	0.999777i	$-0.506728\pi$
$499$	−21.0923	−0.0422691	−0.0211345	+	0.999777i	$0.506728\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	420.228i	0.835444i	0.908575	+	0.417722i	$0.137171\pi$
$503$	420.228i	0.835444i	−0.908575	+	0.417722i	$0.862829\pi$
$504$	0	0
$505$	−623.782	−1.23521
$506$	0	0
$507$	0	0
$508$	0	0
$509$	242.312i	0.476055i	0.971258	+	0.238028i	$0.0765009\pi$
$509$	242.312i	0.476055i	−0.971258	+	0.238028i	$0.923499\pi$
$510$	0	0
$511$	51.4013	0.100590
$512$	0	0
$513$	0	0
$514$	0	0
$515$	340.131i	0.660448i
$516$	0	0
$517$	64.8148	0.125367
$518$	0	0
$519$	0	0
$520$	0	0
$521$	52.9968i	0.101721i	0.998706	+	0.0508607i	$0.0161965\pi$
$521$	52.9968i	0.101721i	−0.998706	+	0.0508607i	$0.983804\pi$
$522$	0	0
$523$	136.510	0.261013	0.130506	−	0.991447i	$-0.458340\pi$
$523$	136.510	0.261013	0.130506	+	0.991447i	$0.458340\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.7254i	0.0431222i
$528$	0	0
$529$	−104.180	−0.196938
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 632.586i	− 1.18684i
$534$	0	0
$535$	494.793	0.924847
$536$	0	0
$537$	0	0
$538$	0	0
$539$	48.2538i	0.0895248i
$540$	0	0
$541$	305.615	0.564908	0.282454	−	0.959281i	$-0.408852\pi$
$541$	305.615	0.564908	0.282454	+	0.959281i	$0.408852\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 612.306i	− 1.12350i
$546$	0	0
$547$	102.020	0.186508	0.0932542	−	0.995642i	$-0.470273\pi$
$547$	102.020	0.186508	0.0932542	+	0.995642i	$0.470273\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.5677i	0.0191792i
$552$	0	0
$553$	−403.226	−0.729160
$554$	0	0
$555$	0	0
$556$	0	0
$557$	403.429i	0.724289i	0.932122	+	0.362145i	$0.117955\pi$
$557$	403.429i	0.724289i	−0.932122	+	0.362145i	$0.882045\pi$
$558$	0	0
$559$	−349.187	−0.624663
$560$	0	0
$561$	0	0
$562$	0	0
$563$	813.539i	1.44501i	0.691367	+	0.722503i	$0.257009\pi$
$563$	813.539i	1.44501i	−0.691367	+	0.722503i	$0.742991\pi$
$564$	0	0
$565$	37.1346	0.0657250
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1045.50i	1.83743i	0.394923	+	0.918714i	$0.370771\pi$
$569$	1045.50i	1.83743i	−0.394923	+	0.918714i	$0.629229\pi$
$570$	0	0
$571$	−939.318	−1.64504	−0.822521	−	0.568735i	$-0.807433\pi$
$571$	−939.318	−1.64504	−0.822521	+	0.568735i	$0.807433\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	254.954i	0.443399i
$576$	0	0
$577$	−456.546	−0.791241	−0.395620	−	0.918414i	$-0.629470\pi$
$577$	−456.546	−0.791241	−0.395620	+	0.918414i	$0.629470\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 258.607i	− 0.445107i
$582$	0	0
$583$	306.305	0.525395
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 555.407i	− 0.946178i	−0.881015	−	0.473089i	$-0.843139\pi$
$587$	− 555.407i	− 0.946178i	0.881015	−	0.473089i	$-0.156861\pi$
$588$	0	0
$589$	−74.0123	−0.125658
$590$	0	0
$591$	0	0
$592$	0	0
$593$	506.971i	0.854925i	0.904033	+	0.427463i	$0.140592\pi$
$593$	506.971i	0.854925i	−0.904033	+	0.427463i	$0.859408\pi$
$594$	0	0
$595$	−34.9161	−0.0586825
$596$	0	0
$597$	0	0
$598$	0	0
$599$	730.835i	1.22009i	0.792366	+	0.610046i	$0.208849\pi$
$599$	730.835i	1.22009i	−0.792366	+	0.610046i	$0.791151\pi$
$600$	0	0
$601$	−196.232	−0.326510	−0.163255	−	0.986584i	$-0.552199\pi$
$601$	−196.232	−0.326510	−0.163255	+	0.986584i	$0.552199\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	283.335i	0.468322i
$606$	0	0
$607$	−655.779	−1.08036	−0.540180	−	0.841549i	$-0.681644\pi$
$607$	−655.779	−1.08036	−0.540180	+	0.841549i	$0.681644\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	196.686i	0.321908i
$612$	0	0
$613$	341.221	0.556641	0.278320	−	0.960488i	$-0.410222\pi$
$613$	341.221	0.556641	0.278320	+	0.960488i	$0.410222\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	246.010i	0.398720i	0.979926	+	0.199360i	$0.0638863\pi$
$617$	246.010i	0.398720i	−0.979926	+	0.199360i	$0.936114\pi$
$618$	0	0
$619$	−760.921	−1.22928	−0.614638	−	0.788810i	$-0.710698\pi$
$619$	−760.921	−1.22928	−0.614638	+	0.788810i	$0.710698\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 262.897i	− 0.421985i
$624$	0	0
$625$	−269.039	−0.430462
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 19.2847i	− 0.0306594i
$630$	0	0
$631$	70.2909	0.111396	0.0556980	−	0.998448i	$-0.482262\pi$
$631$	70.2909	0.111396	0.0556980	+	0.998448i	$0.482262\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 721.832i	− 1.13674i
$636$	0	0
$637$	−146.430	−0.229875
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1116.18i	1.74130i	0.491900	+	0.870652i	$0.336303\pi$
$641$	1116.18i	1.74130i	−0.491900	+	0.870652i	$0.663697\pi$
$642$	0	0
$643$	513.359	0.798381	0.399191	−	0.916868i	$-0.369291\pi$
$643$	513.359	0.798381	0.399191	+	0.916868i	$0.369291\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 249.620i	− 0.385812i	−0.981217	−	0.192906i	$-0.938209\pi$
$647$	− 249.620i	− 0.385812i	0.981217	−	0.192906i	$-0.0617912\pi$
$648$	0	0
$649$	−360.462	−0.555412
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 916.213i	− 1.40308i	−0.712629	−	0.701541i	$-0.752496\pi$
$653$	− 916.213i	− 1.40308i	0.712629	−	0.701541i	$-0.247504\pi$
$654$	0	0
$655$	400.525	0.611489
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1080.96i	− 1.64030i	−0.572149	−	0.820149i	$-0.693890\pi$
$659$	− 1080.96i	− 1.64030i	0.572149	−	0.820149i	$-0.306110\pi$
$660$	0	0
$661$	449.943	0.680701	0.340350	−	0.940299i	$-0.389454\pi$
$661$	449.943	0.680701	0.340350	+	0.940299i	$0.389454\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 113.715i	− 0.171000i
$666$	0	0
$667$	23.8562	0.0357664
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 334.654i	− 0.498739i
$672$	0	0
$673$	−614.341	−0.912840	−0.456420	−	0.889764i	$-0.650869\pi$
$673$	−614.341	−0.912840	−0.456420	+	0.889764i	$0.650869\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 211.920i	− 0.313029i	−0.987676	−	0.156514i	$-0.949974\pi$
$677$	− 211.920i	− 0.313029i	0.987676	−	0.156514i	$-0.0500258\pi$
$678$	0	0
$679$	352.957	0.519820
$680$	0	0
$681$	0	0
$682$	0	0
$683$	816.178i	1.19499i	0.801873	+	0.597495i	$0.203837\pi$
$683$	816.178i	1.19499i	−0.801873	+	0.597495i	$0.796163\pi$
$684$	0	0
$685$	764.236	1.11567
$686$	0	0
$687$	0	0
$688$	0	0
$689$	929.508i	1.34907i
$690$	0	0
$691$	1070.12	1.54866	0.774330	−	0.632782i	$-0.218087\pi$
$691$	1070.12	1.54866	0.774330	+	0.632782i	$0.218087\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	230.227i	0.331263i
$696$	0	0
$697$	−103.500	−0.148493
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 600.000i	− 0.855920i	−0.903798	−	0.427960i	$-0.859232\pi$
$701$	− 600.000i	− 0.855920i	0.903798	−	0.427960i	$-0.140768\pi$
$702$	0	0
$703$	62.8067	0.0893410
$704$	0	0
$705$	0	0
$706$	0	0
$707$	428.013i	0.605393i
$708$	0	0
$709$	911.547	1.28568	0.642840	−	0.766001i	$-0.277756\pi$
$709$	911.547	1.28568	0.642840	+	0.766001i	$0.277756\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	167.080i	0.234333i
$714$	0	0
$715$	556.022	0.777653
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 515.213i	− 0.716569i	−0.933612	−	0.358284i	$-0.883362\pi$
$719$	− 515.213i	− 0.716569i	0.933612	−	0.358284i	$-0.116638\pi$
$720$	0	0
$721$	233.383	0.323694
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 9.60586i	− 0.0132495i
$726$	0	0
$727$	−1223.59	−1.68307	−0.841534	−	0.540204i	$-0.818347\pi$
$727$	−1223.59	−1.68307	−0.841534	+	0.540204i	$0.818347\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	57.1316i	0.0781554i
$732$	0	0
$733$	376.404	0.513512	0.256756	−	0.966476i	$-0.417346\pi$
$733$	376.404	0.513512	0.256756	+	0.966476i	$0.417346\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 437.913i	− 0.594183i
$738$	0	0
$739$	−504.166	−0.682228	−0.341114	−	0.940022i	$-0.610804\pi$
$739$	−504.166	−0.682228	−0.341114	+	0.940022i	$0.610804\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1134.40i	1.52679i	0.645934	+	0.763393i	$0.276468\pi$
$743$	1134.40i	1.52679i	−0.645934	+	0.763393i	$0.723532\pi$
$744$	0	0
$745$	−216.689	−0.290857
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 339.506i	− 0.453279i
$750$	0	0
$751$	−949.876	−1.26481	−0.632407	−	0.774636i	$-0.717933\pi$
$751$	−949.876	−1.26481	−0.632407	+	0.774636i	$0.717933\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	34.6376i	0.0458777i
$756$	0	0
$757$	−750.370	−0.991241	−0.495621	−	0.868539i	$-0.665059\pi$
$757$	−750.370	−0.991241	−0.495621	+	0.868539i	$0.665059\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1100.48i	1.44610i	0.690794	+	0.723052i	$0.257261\pi$
$761$	1100.48i	1.44610i	−0.690794	+	0.723052i	$0.742739\pi$
$762$	0	0
$763$	−420.139	−0.550640
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1093.85i	− 1.42614i
$768$	0	0
$769$	−525.202	−0.682968	−0.341484	−	0.939888i	$-0.610929\pi$
$769$	−525.202	−0.682968	−0.341484	+	0.939888i	$0.610929\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	211.562i	0.273689i	0.990593	+	0.136845i	$0.0436961\pi$
$773$	211.562i	0.273689i	−0.990593	+	0.136845i	$0.956304\pi$
$774$	0	0
$775$	67.2758	0.0868074
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 337.079i	− 0.432707i
$780$	0	0
$781$	245.935	0.314897
$782$	0	0
$783$	0	0
$784$	0	0
$785$	51.4826i	0.0655829i
$786$	0	0
$787$	−1219.20	−1.54917	−0.774585	−	0.632470i	$-0.782041\pi$
$787$	−1219.20	−1.54917	−0.774585	+	0.632470i	$0.782041\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 25.4802i	− 0.0322126i
$792$	0	0
$793$	1015.53	1.28062
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1096.10i	− 1.37528i	−0.726053	−	0.687639i	$-0.758647\pi$
$797$	− 1096.10i	− 1.37528i	0.726053	−	0.687639i	$-0.241353\pi$
$798$	0	0
$799$	32.1804	0.0402759
$800$	0	0
$801$	0	0
$802$	0	0
$803$	133.924i	0.166780i
$804$	0	0
$805$	−256.707	−0.318891
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 213.728i	− 0.264188i	−0.991237	−	0.132094i	$-0.957830\pi$
$809$	− 213.728i	− 0.264188i	0.991237	−	0.132094i	$-0.0421701\pi$
$810$	0	0
$811$	263.712	0.325169	0.162585	−	0.986695i	$-0.448017\pi$
$811$	263.712	0.325169	0.162585	+	0.986695i	$0.448017\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 527.141i	− 0.646799i
$816$	0	0
$817$	−186.067	−0.227744
$818$	0	0
$819$	0	0
$820$	0	0
$821$	401.805i	0.489409i	0.969598	+	0.244705i	$0.0786910\pi$
$821$	401.805i	0.489409i	−0.969598	+	0.244705i	$0.921309\pi$
$822$	0	0
$823$	682.384	0.829142	0.414571	−	0.910017i	$-0.363932\pi$
$823$	682.384	0.829142	0.414571	+	0.910017i	$0.363932\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 964.623i	− 1.16641i	−0.812324	−	0.583206i	$-0.801798\pi$
$827$	− 964.623i	− 1.16641i	0.812324	−	0.583206i	$-0.198202\pi$
$828$	0	0
$829$	379.477	0.457753	0.228877	−	0.973455i	$-0.426495\pi$
$829$	379.477	0.457753	0.228877	+	0.973455i	$0.426495\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	23.9580i	0.0287610i
$834$	0	0
$835$	175.558	0.210249
$836$	0	0
$837$	0	0
$838$	0	0
$839$	616.642i	0.734973i	0.930029	+	0.367487i	$0.119782\pi$
$839$	616.642i	0.734973i	−0.930029	+	0.367487i	$0.880218\pi$
$840$	0	0
$841$	840.101	0.998931
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1035.65i	1.22562i
$846$	0	0
$847$	194.412	0.229530
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 141.783i	− 0.166608i
$852$	0	0
$853$	−411.601	−0.482533	−0.241267	−	0.970459i	$-0.577563\pi$
$853$	−411.601	−0.482533	−0.241267	+	0.970459i	$0.577563\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 564.267i	− 0.658421i	−0.944256	−	0.329211i	$-0.893217\pi$
$857$	− 564.267i	− 0.658421i	0.944256	−	0.329211i	$-0.106783\pi$
$858$	0	0
$859$	1196.28	1.39265	0.696324	−	0.717728i	$-0.254818\pi$
$859$	1196.28	1.39265	0.696324	+	0.717728i	$0.254818\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1171.34i	− 1.35729i	−0.734468	−	0.678644i	$-0.762568\pi$
$863$	− 1171.34i	− 1.35729i	0.734468	−	0.678644i	$-0.237432\pi$
$864$	0	0
$865$	−149.971	−0.173377
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1050.59i	− 1.20896i
$870$	0	0
$871$	1328.88	1.52570
$872$	0	0
$873$	0	0
$874$	0	0
$875$	358.408i	0.409609i
$876$	0	0
$877$	239.563	0.273162	0.136581	−	0.990629i	$-0.456389\pi$
$877$	239.563	0.273162	0.136581	+	0.990629i	$0.456389\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1004.79i	− 1.14051i	−0.821468	−	0.570255i	$-0.806844\pi$
$881$	− 1004.79i	− 1.14051i	0.821468	−	0.570255i	$-0.193156\pi$
$882$	0	0
$883$	1469.24	1.66391	0.831957	−	0.554841i	$-0.187221\pi$
$883$	1469.24	1.66391	0.831957	+	0.554841i	$0.187221\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1327.12i	1.49619i	0.663591	+	0.748096i	$0.269032\pi$
$887$	1327.12i	1.49619i	−0.663591	+	0.748096i	$0.730968\pi$
$888$	0	0
$889$	−495.290	−0.557132
$890$	0	0
$891$	0	0
$892$	0	0
$893$	104.806i	0.117363i
$894$	0	0
$895$	−264.804	−0.295870
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 6.29502i	− 0.00700224i
$900$	0	0
$901$	152.080	0.168790
$902$	0	0
$903$	0	0
$904$	0	0
$905$	404.810i	0.447304i
$906$	0	0
$907$	−1238.98	−1.36602	−0.683010	−	0.730409i	$-0.739329\pi$
$907$	−1238.98	−1.36602	−0.683010	+	0.730409i	$0.739329\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 302.136i	− 0.331653i	−0.986155	−	0.165826i	$-0.946971\pi$
$911$	− 302.136i	− 0.331653i	0.986155	−	0.165826i	$-0.0530292\pi$
$912$	0	0
$913$	673.791	0.737996
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 274.824i	− 0.299699i
$918$	0	0
$919$	1023.81	1.11404	0.557022	−	0.830498i	$-0.311944\pi$
$919$	1023.81	1.11404	0.557022	+	0.830498i	$0.311944\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	746.309i	0.808569i
$924$	0	0
$925$	−57.0901	−0.0617190
$926$	0	0
$927$	0	0
$928$	0	0
$929$	330.204i	0.355441i	0.984081	+	0.177720i	$0.0568722\pi$
$929$	330.204i	0.355441i	−0.984081	+	0.177720i	$0.943128\pi$
$930$	0	0
$931$	−78.0265	−0.0838094
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 90.9726i	− 0.0972969i
$936$	0	0
$937$	−320.286	−0.341821	−0.170910	−	0.985287i	$-0.554671\pi$
$937$	−320.286	−0.341821	−0.170910	+	0.985287i	$0.554671\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1473.27i	1.56565i	0.622244	+	0.782823i	$0.286221\pi$
$941$	1473.27i	1.56565i	−0.622244	+	0.782823i	$0.713779\pi$
$942$	0	0
$943$	−760.940	−0.806936
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1058.79i	1.11805i	0.829152	+	0.559023i	$0.188824\pi$
$947$	1058.79i	1.11805i	−0.829152	+	0.559023i	$0.811176\pi$
$948$	0	0
$949$	−406.404	−0.428245
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1683.17i	− 1.76619i	−0.469199	−	0.883093i	$-0.655457\pi$
$953$	− 1683.17i	− 1.76619i	0.469199	−	0.883093i	$-0.344543\pi$
$954$	0	0
$955$	166.563	0.174412
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 524.387i	− 0.546806i
$960$	0	0
$961$	−916.912	−0.954123
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 354.898i	− 0.367770i
$966$	0	0
$967$	764.881	0.790983	0.395492	−	0.918470i	$-0.370574\pi$
$967$	764.881	0.790983	0.395492	+	0.918470i	$0.370574\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 631.310i	− 0.650165i	−0.945686	−	0.325082i	$-0.894608\pi$
$971$	− 631.310i	− 0.650165i	0.945686	−	0.325082i	$-0.105392\pi$
$972$	0	0
$973$	157.972	0.162356
$974$	0	0
$975$	0	0
$976$	0	0
$977$	367.900i	0.376560i	0.982115	+	0.188280i	$0.0602913\pi$
$977$	367.900i	0.376560i	−0.982115	+	0.188280i	$0.939709\pi$
$978$	0	0
$979$	684.968	0.699661
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1678.27i	1.70729i	0.520852	+	0.853647i	$0.325614\pi$
$983$	1678.27i	1.70729i	−0.520852	+	0.853647i	$0.674386\pi$
$984$	0	0
$985$	−332.663	−0.337729
$986$	0	0
$987$	0	0
$988$	0	0
$989$	420.038i	0.424710i
$990$	0	0
$991$	1079.56	1.08936	0.544681	−	0.838644i	$-0.316651\pi$
$991$	1079.56	1.08936	0.544681	+	0.838644i	$0.316651\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 15.1693i	− 0.0152455i
$996$	0	0
$997$	677.368	0.679406	0.339703	−	0.940533i	$-0.389673\pi$
$997$	677.368	0.679406	0.339703	+	0.940533i	$0.389673\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.3.d.n.449.9		12
3.2	odd	2	inner	4032.3.d.n.449.4		12
4.3	odd	2		4032.3.d.o.449.9		12
8.3	odd	2		2016.3.d.e.449.4	✓	12
8.5	even	2		2016.3.d.f.449.4	yes	12
12.11	even	2		4032.3.d.o.449.4		12
24.5	odd	2		2016.3.d.f.449.9	yes	12
24.11	even	2		2016.3.d.e.449.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.3.d.e.449.4	✓	12	8.3	odd	2
2016.3.d.e.449.9	yes	12	24.11	even	2
2016.3.d.f.449.4	yes	12	8.5	even	2
2016.3.d.f.449.9	yes	12	24.5	odd	2
4032.3.d.n.449.4		12	3.2	odd	2	inner
4032.3.d.n.449.9		12	1.1	even	1	trivial
4032.3.d.o.449.4		12	12.11	even	2
4032.3.d.o.449.9		12	4.3	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4032.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.85589i q^{5} +2.64575 q^{7} +6.89341i q^{11} -20.9186 q^{13} +3.42256i q^{17} -11.1466 q^{19} +25.1631i q^{23} +10.1321 q^{25} -0.948063i q^{29} +6.63987 q^{31} +10.2017i q^{35} -5.63459 q^{37} +30.2404i q^{41} +16.6926 q^{43} -9.40243i q^{47} +7.00000 q^{49} -44.4345i q^{53} -26.5802 q^{55} +52.2908i q^{59} -48.5469 q^{61} -80.6599i q^{65} -63.5263 q^{67} -35.6768i q^{71} +19.4279 q^{73} +18.2382i q^{77} -152.405 q^{79} -97.7442i q^{83} -13.1970 q^{85} -99.3657i q^{89} -55.3454 q^{91} -42.9803i q^{95} +133.405 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{13} - 64 q^{19} - 124 q^{25} + 160 q^{31} - 56 q^{37} + 64 q^{43} + 84 q^{49} - 160 q^{55} - 104 q^{61} + 64 q^{67} - 64 q^{73} + 32 q^{79} + 184 q^{85} - 96 q^{97}+O(q^{100}) \) 12 * q - 16 * q^13 - 64 * q^19 - 124 * q^25 + 160 * q^31 - 56 * q^37 + 64 * q^43 + 84 * q^49 - 160 * q^55 - 104 * q^61 + 64 * q^67 - 64 * q^73 + 32 * q^79 + 184 * q^85 - 96 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more