LMFDB - Embedded newform 2016.3.m.c.127.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,3,Mod(127,2016)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2016.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2016.m (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$54.9320212950$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1997017344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 56*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.5
Root			$1.27733i$ of defining polynomial
Character	$\chi$	$=$	2016.127
Dual form			2016.3.m.c.127.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.57685 q^{5} -2.64575i q^{7} +O(q^{10})$	$q+4.57685 q^{5} -2.64575i q^{7} +15.7367i q^{11} +8.57685 q^{13} -28.3197 q^{17} -6.33599i q^{19} +31.0647i q^{23} -4.05242 q^{25} +0.846294 q^{29} +21.6354i q^{31} -12.1092i q^{35} -33.6637 q^{37} -66.9757 q^{41} -44.8781i q^{43} +38.4528i q^{47} -7.00000 q^{49} +14.8174 q^{53} +72.0246i q^{55} +5.80942i q^{59} -52.6015 q^{61} +39.2550 q^{65} +117.397i q^{67} -81.2543i q^{71} -47.8054 q^{73} +41.6354 q^{77} -57.4900i q^{79} -102.855i q^{83} -129.615 q^{85} +89.2955 q^{89} -22.6922i q^{91} -28.9989i q^{95} -3.44452 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 32 q^{13} + 16 q^{17} + 104 q^{25} + 80 q^{29} - 176 q^{37} - 144 q^{41} - 56 q^{49} - 48 q^{53} - 192 q^{61} + 304 q^{65} + 272 q^{73} + 112 q^{77} - 160 q^{85} + 80 q^{89} + 528 q^{97}+O(q^{100}) $ 8 * q + 32 * q^13 + 16 * q^17 + 104 * q^25 + 80 * q^29 - 176 * q^37 - 144 * q^41 - 56 * q^49 - 48 * q^53 - 192 * q^61 + 304 * q^65 + 272 * q^73 + 112 * q^77 - 160 * q^85 + 80 * q^89 + 528 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	4.57685	0.915371	0.457685	−	0.889114i	$-0.348679\pi$
$5$	4.57685	0.915371	0.457685	+	0.889114i	$0.348679\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	15.7367i	1.43061i	0.698812	+	0.715305i	$0.253712\pi$
$11$	15.7367i	1.43061i	−0.698812	+	0.715305i	$0.746288\pi$
$12$	0	0
$13$	8.57685	0.659758	0.329879	−	0.944023i	$-0.392992\pi$
$13$	8.57685	0.659758	0.329879	+	0.944023i	$0.392992\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−28.3197	−1.66587	−0.832933	−	0.553374i	$-0.813340\pi$
$17$	−28.3197	−1.66587	−0.832933	+	0.553374i	$0.813340\pi$
$18$	0	0
$19$	− 6.33599i	− 0.333473i	−0.986001	−	0.166737i	$-0.946677\pi$
$19$	− 6.33599i	− 0.333473i	0.986001	−	0.166737i	$-0.0533230\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	31.0647i	1.35064i	0.737525	+	0.675320i	$0.235995\pi$
$23$	31.0647i	1.35064i	−0.737525	+	0.675320i	$0.764005\pi$
$24$	0	0
$25$	−4.05242	−0.162097
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.846294	0.0291826	0.0145913	−	0.999894i	$-0.495355\pi$
$29$	0.846294	0.0291826	0.0145913	+	0.999894i	$0.495355\pi$
$30$	0	0
$31$	21.6354i	0.697917i	0.937138	+	0.348958i	$0.113465\pi$
$31$	21.6354i	0.697917i	−0.937138	+	0.348958i	$0.886535\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 12.1092i	− 0.345978i
$36$	0	0
$37$	−33.6637	−0.909830	−0.454915	−	0.890535i	$-0.650330\pi$
$37$	−33.6637	−0.909830	−0.454915	+	0.890535i	$0.650330\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−66.9757	−1.63355	−0.816777	−	0.576953i	$-0.804242\pi$
$41$	−66.9757	−1.63355	−0.816777	+	0.576953i	$0.804242\pi$
$42$	0	0
$43$	− 44.8781i	− 1.04368i	−0.853044	−	0.521839i	$-0.825246\pi$
$43$	− 44.8781i	− 1.04368i	0.853044	−	0.521839i	$-0.174754\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	38.4528i	0.818145i	0.912502	+	0.409073i	$0.134148\pi$
$47$	38.4528i	0.818145i	−0.912502	+	0.409073i	$0.865852\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	14.8174	0.279574	0.139787	−	0.990182i	$-0.455358\pi$
$53$	14.8174	0.279574	0.139787	+	0.990182i	$0.455358\pi$
$54$	0	0
$55$	72.0246i	1.30954i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.80942i	0.0984647i	0.998787	+	0.0492323i	$0.0156775\pi$
$59$	5.80942i	0.0984647i	−0.998787	+	0.0492323i	$0.984323\pi$
$60$	0	0
$61$	−52.6015	−0.862319	−0.431160	−	0.902276i	$-0.641895\pi$
$61$	−52.6015	−0.862319	−0.431160	+	0.902276i	$0.641895\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	39.2550	0.603923
$66$	0	0
$67$	117.397i	1.75219i	0.482138	+	0.876095i	$0.339860\pi$
$67$	117.397i	1.75219i	−0.482138	+	0.876095i	$0.660140\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 81.2543i	− 1.14443i	−0.820105	−	0.572213i	$-0.806085\pi$
$71$	− 81.2543i	− 1.14443i	0.820105	−	0.572213i	$-0.193915\pi$
$72$	0	0
$73$	−47.8054	−0.654869	−0.327435	−	0.944874i	$-0.606184\pi$
$73$	−47.8054	−0.654869	−0.327435	+	0.944874i	$0.606184\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	41.6354	0.540720
$78$	0	0
$79$	− 57.4900i	− 0.727722i	−0.931453	−	0.363861i	$-0.881458\pi$
$79$	− 57.4900i	− 0.727722i	0.931453	−	0.363861i	$-0.118542\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 102.855i	− 1.23921i	−0.784913	−	0.619606i	$-0.787292\pi$
$83$	− 102.855i	− 1.23921i	0.784913	−	0.619606i	$-0.212708\pi$
$84$	0	0
$85$	−129.615	−1.52488
$86$	0	0
$87$	0	0
$88$	0	0
$89$	89.2955	1.00332	0.501660	−	0.865065i	$-0.332723\pi$
$89$	89.2955	1.00332	0.501660	+	0.865065i	$0.332723\pi$
$90$	0	0
$91$	− 22.6922i	− 0.249365i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 28.9989i	− 0.305252i
$96$	0	0
$97$	−3.44452	−0.0355106	−0.0177553	−	0.999842i	$-0.505652\pi$
$97$	−3.44452	−0.0355106	−0.0177553	+	0.999842i	$0.505652\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−143.034	−1.41618	−0.708089	−	0.706123i	$-0.750443\pi$
$101$	−143.034	−1.41618	−0.708089	+	0.706123i	$0.750443\pi$
$102$	0	0
$103$	173.424i	1.68373i	0.539688	+	0.841865i	$0.318542\pi$
$103$	173.424i	1.68373i	−0.539688	+	0.841865i	$0.681458\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	95.5581i	0.893067i	0.894767	+	0.446533i	$0.147342\pi$
$107$	95.5581i	0.893067i	−0.894767	+	0.446533i	$0.852658\pi$
$108$	0	0
$109$	185.517	1.70199	0.850997	−	0.525170i	$-0.175998\pi$
$109$	185.517	1.70199	0.850997	+	0.525170i	$0.175998\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	103.409	0.915124	0.457562	−	0.889178i	$-0.348723\pi$
$113$	103.409	0.915124	0.457562	+	0.889178i	$0.348723\pi$
$114$	0	0
$115$	142.179i	1.23634i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	74.9269i	0.629638i
$120$	0	0
$121$	−126.644	−1.04665
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−132.969	−1.06375
$126$	0	0
$127$	− 147.970i	− 1.16512i	−0.812787	−	0.582560i	$-0.802051\pi$
$127$	− 147.970i	− 1.16512i	0.812787	−	0.582560i	$-0.197949\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	259.412i	1.98025i	0.140200	+	0.990123i	$0.455225\pi$
$131$	259.412i	1.98025i	−0.140200	+	0.990123i	$0.544775\pi$
$132$	0	0
$133$	−16.7635	−0.126041
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.07777	−0.0224654	−0.0112327	−	0.999937i	$-0.503576\pi$
$137$	−3.07777	−0.0224654	−0.0112327	+	0.999937i	$0.503576\pi$
$138$	0	0
$139$	90.3041i	0.649670i	0.945771	+	0.324835i	$0.105309\pi$
$139$	90.3041i	0.649670i	−0.945771	+	0.324835i	$0.894691\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	134.971i	0.943856i
$144$	0	0
$145$	3.87336	0.0267129
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−41.3766	−0.277695	−0.138848	−	0.990314i	$-0.544340\pi$
$149$	−41.3766	−0.277695	−0.138848	+	0.990314i	$0.544340\pi$
$150$	0	0
$151$	− 41.5182i	− 0.274955i	−0.990505	−	0.137477i	$-0.956101\pi$
$151$	− 41.5182i	− 0.274955i	0.990505	−	0.137477i	$-0.0438994\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	99.0221i	0.638853i
$156$	0	0
$157$	−26.9848	−0.171878	−0.0859389	−	0.996300i	$-0.527389\pi$
$157$	−26.9848	−0.171878	−0.0859389	+	0.996300i	$0.527389\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	82.1895	0.510494
$162$	0	0
$163$	126.684i	0.777200i	0.921407	+	0.388600i	$0.127041\pi$
$163$	126.684i	0.777200i	−0.921407	+	0.388600i	$0.872959\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 246.243i	− 1.47451i	−0.675615	−	0.737254i	$-0.736122\pi$
$167$	− 246.243i	− 1.47451i	0.675615	−	0.737254i	$-0.263878\pi$
$168$	0	0
$169$	−95.4376	−0.564720
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−74.5776	−0.431084	−0.215542	−	0.976495i	$-0.569152\pi$
$173$	−74.5776	−0.431084	−0.215542	+	0.976495i	$0.569152\pi$
$174$	0	0
$175$	10.7217i	0.0612668i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 201.073i	− 1.12331i	−0.827371	−	0.561656i	$-0.810165\pi$
$179$	− 201.073i	− 1.12331i	0.827371	−	0.561656i	$-0.189835\pi$
$180$	0	0
$181$	252.696	1.39611	0.698054	−	0.716045i	$-0.254050\pi$
$181$	252.696	1.39611	0.698054	+	0.716045i	$0.254050\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−154.074	−0.832831
$186$	0	0
$187$	− 445.659i	− 2.38320i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	332.333i	1.73996i	0.493085	+	0.869981i	$0.335869\pi$
$191$	332.333i	1.73996i	−0.493085	+	0.869981i	$0.664131\pi$
$192$	0	0
$193$	−53.2218	−0.275761	−0.137880	−	0.990449i	$-0.544029\pi$
$193$	−53.2218	−0.275761	−0.137880	+	0.990449i	$0.544029\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−276.248	−1.40227	−0.701137	−	0.713026i	$-0.747324\pi$
$197$	−276.248	−1.40227	−0.701137	+	0.713026i	$0.747324\pi$
$198$	0	0
$199$	58.5094i	0.294017i	0.989135	+	0.147008i	$0.0469644\pi$
$199$	58.5094i	0.294017i	−0.989135	+	0.147008i	$0.953036\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2.23908i	− 0.0110300i
$204$	0	0
$205$	−306.538	−1.49531
$206$	0	0
$207$	0	0
$208$	0	0
$209$	99.7077	0.477070
$210$	0	0
$211$	146.237i	0.693068i	0.938037	+	0.346534i	$0.112641\pi$
$211$	146.237i	0.693068i	−0.938037	+	0.346534i	$0.887359\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 205.401i	− 0.955351i
$216$	0	0
$217$	57.2420	0.263788
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−242.894	−1.09907
$222$	0	0
$223$	422.290i	1.89368i	0.321709	+	0.946839i	$0.395743\pi$
$223$	422.290i	1.89368i	−0.321709	+	0.946839i	$0.604257\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 271.103i	− 1.19429i	−0.802134	−	0.597144i	$-0.796302\pi$
$227$	− 271.103i	− 1.19429i	0.802134	−	0.597144i	$-0.203698\pi$
$228$	0	0
$229$	−400.572	−1.74922	−0.874611	−	0.484826i	$-0.838883\pi$
$229$	−400.572	−1.74922	−0.874611	+	0.484826i	$0.838883\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	257.611	1.10563	0.552813	−	0.833305i	$-0.313554\pi$
$233$	257.611	1.10563	0.552813	+	0.833305i	$0.313554\pi$
$234$	0	0
$235$	175.993i	0.748906i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 152.650i	− 0.638704i	−0.947636	−	0.319352i	$-0.896535\pi$
$239$	− 152.650i	− 0.638704i	0.947636	−	0.319352i	$-0.103465\pi$
$240$	0	0
$241$	−47.1218	−0.195526	−0.0977630	−	0.995210i	$-0.531169\pi$
$241$	−47.1218	−0.195526	−0.0977630	+	0.995210i	$0.531169\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−32.0380	−0.130767
$246$	0	0
$247$	− 54.3429i	− 0.220012i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	236.051i	0.940443i	0.882549	+	0.470221i	$0.155826\pi$
$251$	236.051i	0.940443i	−0.882549	+	0.470221i	$0.844174\pi$
$252$	0	0
$253$	−488.857	−1.93224
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27.5524	0.107208	0.0536039	−	0.998562i	$-0.482929\pi$
$257$	27.5524	0.107208	0.0536039	+	0.998562i	$0.482929\pi$
$258$	0	0
$259$	89.0658i	0.343883i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	88.2675i	0.335618i	0.985820	+	0.167809i	$0.0536691\pi$
$263$	88.2675i	0.335618i	−0.985820	+	0.167809i	$0.946331\pi$
$264$	0	0
$265$	67.8170	0.255913
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−21.9135	−0.0814629	−0.0407314	−	0.999170i	$-0.512969\pi$
$269$	−21.9135	−0.0814629	−0.0407314	+	0.999170i	$0.512969\pi$
$270$	0	0
$271$	428.897i	1.58265i	0.611399	+	0.791323i	$0.290607\pi$
$271$	428.897i	1.58265i	−0.611399	+	0.791323i	$0.709393\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 63.7717i	− 0.231897i
$276$	0	0
$277$	−457.076	−1.65009	−0.825046	−	0.565065i	$-0.808851\pi$
$277$	−457.076	−1.65009	−0.825046	+	0.565065i	$0.808851\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−95.5032	−0.339869	−0.169934	−	0.985455i	$-0.554356\pi$
$281$	−95.5032	−0.339869	−0.169934	+	0.985455i	$0.554356\pi$
$282$	0	0
$283$	131.804i	0.465737i	0.972508	+	0.232869i	$0.0748112\pi$
$283$	131.804i	0.465737i	−0.972508	+	0.232869i	$0.925189\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	177.201i	0.617426i
$288$	0	0
$289$	513.006	1.77511
$290$	0	0
$291$	0	0
$292$	0	0
$293$	325.183	1.10984	0.554920	−	0.831904i	$-0.312749\pi$
$293$	325.183	1.10984	0.554920	+	0.831904i	$0.312749\pi$
$294$	0	0
$295$	26.5888i	0.0901317i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	266.438i	0.891096i
$300$	0	0
$301$	−118.736	−0.394473
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−240.749	−0.789341
$306$	0	0
$307$	34.3658i	0.111941i	0.998432	+	0.0559704i	$0.0178252\pi$
$307$	34.3658i	0.111941i	−0.998432	+	0.0559704i	$0.982175\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	195.190i	0.627620i	0.949486	+	0.313810i	$0.101606\pi$
$311$	195.190i	0.627620i	−0.949486	+	0.313810i	$0.898394\pi$
$312$	0	0
$313$	19.8987	0.0635740	0.0317870	−	0.999495i	$-0.489880\pi$
$313$	19.8987	0.0635740	0.0317870	+	0.999495i	$0.489880\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−168.672	−0.532088	−0.266044	−	0.963961i	$-0.585717\pi$
$317$	−168.672	−0.532088	−0.266044	+	0.963961i	$0.585717\pi$
$318$	0	0
$319$	13.3179i	0.0417489i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	179.434i	0.555522i
$324$	0	0
$325$	−34.7570	−0.106945
$326$	0	0
$327$	0	0
$328$	0	0
$329$	101.737	0.309230
$330$	0	0
$331$	603.602i	1.82357i	0.410666	+	0.911786i	$0.365296\pi$
$331$	603.602i	1.82357i	−0.410666	+	0.911786i	$0.634704\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	537.308i	1.60390i
$336$	0	0
$337$	189.903	0.563511	0.281756	−	0.959486i	$-0.409083\pi$
$337$	189.903	0.563511	0.281756	+	0.959486i	$0.409083\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−340.470	−0.998447
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 81.4988i	− 0.234867i	−0.993081	−	0.117433i	$-0.962533\pi$
$347$	− 81.4988i	− 0.234867i	0.993081	−	0.117433i	$-0.0374667\pi$
$348$	0	0
$349$	−223.673	−0.640898	−0.320449	−	0.947266i	$-0.603834\pi$
$349$	−223.673	−0.640898	−0.320449	+	0.947266i	$0.603834\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	613.342	1.73751	0.868756	−	0.495241i	$-0.164920\pi$
$353$	613.342	1.73751	0.868756	+	0.495241i	$0.164920\pi$
$354$	0	0
$355$	− 371.889i	− 1.04757i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 79.0559i	− 0.220212i	−0.993920	−	0.110106i	$-0.964881\pi$
$359$	− 79.0559i	− 0.220212i	0.993920	−	0.110106i	$-0.0351190\pi$
$360$	0	0
$361$	320.855	0.888796
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−218.798	−0.599448
$366$	0	0
$367$	0.705581i	0.00192257i	1.00000		0.000961283i	$0.000305986\pi$
$367$	0.705581i	0.00192257i	−1.00000		0.000961283i	$0.999694\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 39.2031i	− 0.105669i
$372$	0	0
$373$	250.574	0.671781	0.335890	−	0.941901i	$-0.390963\pi$
$373$	250.574	0.671781	0.335890	+	0.941901i	$0.390963\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.25854	0.0192534
$378$	0	0
$379$	− 532.859i	− 1.40596i	−0.711209	−	0.702981i	$-0.751852\pi$
$379$	− 532.859i	− 1.40596i	0.711209	−	0.702981i	$-0.248148\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 441.226i	− 1.15203i	−0.817441	−	0.576013i	$-0.804608\pi$
$383$	− 441.226i	− 1.15203i	0.817441	−	0.576013i	$-0.195392\pi$
$384$	0	0
$385$	190.559	0.494959
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−174.837	−0.449452	−0.224726	−	0.974422i	$-0.572149\pi$
$389$	−174.837	−0.449452	−0.224726	+	0.974422i	$0.572149\pi$
$390$	0	0
$391$	− 879.744i	− 2.24999i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 263.123i	− 0.666135i
$396$	0	0
$397$	129.750	0.326826	0.163413	−	0.986558i	$-0.447750\pi$
$397$	129.750	0.326826	0.163413	+	0.986558i	$0.447750\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	135.495	0.337892	0.168946	−	0.985625i	$-0.445964\pi$
$401$	135.495	0.337892	0.168946	+	0.985625i	$0.445964\pi$
$402$	0	0
$403$	185.564i	0.460456i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 529.756i	− 1.30161i
$408$	0	0
$409$	243.575	0.595538	0.297769	−	0.954638i	$-0.403758\pi$
$409$	243.575	0.595538	0.297769	+	0.954638i	$0.403758\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	15.3703	0.0372161
$414$	0	0
$415$	− 470.750i	− 1.13434i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	499.348i	1.19176i	0.803073	+	0.595881i	$0.203197\pi$
$419$	499.348i	1.19176i	−0.803073	+	0.595881i	$0.796803\pi$
$420$	0	0
$421$	537.031	1.27561	0.637804	−	0.770199i	$-0.279843\pi$
$421$	537.031	1.27561	0.637804	+	0.770199i	$0.279843\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	114.763	0.270031
$426$	0	0
$427$	139.170i	0.325926i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	559.209i	1.29747i	0.761015	+	0.648735i	$0.224702\pi$
$431$	559.209i	1.29747i	−0.761015	+	0.648735i	$0.775298\pi$
$432$	0	0
$433$	−812.706	−1.87692	−0.938459	−	0.345389i	$-0.887747\pi$
$433$	−812.706	−1.87692	−0.938459	+	0.345389i	$0.887747\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	196.826	0.450403
$438$	0	0
$439$	− 346.809i	− 0.789998i	−0.918681	−	0.394999i	$-0.870745\pi$
$439$	− 346.809i	− 0.789998i	0.918681	−	0.394999i	$-0.129255\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 369.535i	− 0.834166i	−0.908868	−	0.417083i	$-0.863052\pi$
$443$	− 369.535i	− 0.834166i	0.908868	−	0.417083i	$-0.136948\pi$
$444$	0	0
$445$	408.692	0.918409
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−315.180	−0.701961	−0.350980	−	0.936383i	$-0.614152\pi$
$449$	−315.180	−0.701961	−0.350980	+	0.936383i	$0.614152\pi$
$450$	0	0
$451$	− 1053.98i	− 2.33698i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 103.859i	− 0.228261i
$456$	0	0
$457$	781.559	1.71019	0.855097	−	0.518468i	$-0.173497\pi$
$457$	781.559	1.71019	0.855097	+	0.518468i	$0.173497\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−526.503	−1.14209	−0.571044	−	0.820919i	$-0.693461\pi$
$461$	−526.503	−1.14209	−0.571044	+	0.820919i	$0.693461\pi$
$462$	0	0
$463$	754.257i	1.62906i	0.580118	+	0.814532i	$0.303006\pi$
$463$	754.257i	1.62906i	−0.580118	+	0.814532i	$0.696994\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 480.381i	− 1.02865i	−0.857594	−	0.514326i	$-0.828042\pi$
$467$	− 480.381i	− 1.02865i	0.857594	−	0.514326i	$-0.171958\pi$
$468$	0	0
$469$	310.603	0.662266
$470$	0	0
$471$	0	0
$472$	0	0
$473$	706.234	1.49309
$474$	0	0
$475$	25.6761i	0.0540549i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 23.0908i	− 0.0482062i	−0.999709	−	0.0241031i	$-0.992327\pi$
$479$	− 23.0908i	− 0.0482062i	0.999709	−	0.0241031i	$-0.00767300\pi$
$480$	0	0
$481$	−288.729	−0.600267
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−15.7651	−0.0325053
$486$	0	0
$487$	− 517.867i	− 1.06338i	−0.846939	−	0.531691i	$-0.821557\pi$
$487$	− 517.867i	− 1.06338i	0.846939	−	0.531691i	$-0.178443\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 385.683i	− 0.785505i	−0.919644	−	0.392753i	$-0.871523\pi$
$491$	− 385.683i	− 0.785505i	0.919644	−	0.392753i	$-0.128477\pi$
$492$	0	0
$493$	−23.9668	−0.0486142
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−214.979	−0.432552
$498$	0	0
$499$	726.518i	1.45595i	0.685605	+	0.727974i	$0.259538\pi$
$499$	726.518i	1.45595i	−0.685605	+	0.727974i	$0.740462\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 253.383i	− 0.503743i	−0.967761	−	0.251872i	$-0.918954\pi$
$503$	− 253.383i	− 0.503743i	0.967761	−	0.251872i	$-0.0810461\pi$
$504$	0	0
$505$	−654.646	−1.29633
$506$	0	0
$507$	0	0
$508$	0	0
$509$	338.344	0.664723	0.332361	−	0.943152i	$-0.392155\pi$
$509$	338.344	0.664723	0.332361	+	0.943152i	$0.392155\pi$
$510$	0	0
$511$	126.481i	0.247517i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	793.737i	1.54124i
$516$	0	0
$517$	−605.121	−1.17045
$518$	0	0
$519$	0	0
$520$	0	0
$521$	333.599	0.640305	0.320152	−	0.947366i	$-0.396266\pi$
$521$	333.599	0.640305	0.320152	+	0.947366i	$0.396266\pi$
$522$	0	0
$523$	− 38.7942i	− 0.0741763i	−0.999312	−	0.0370882i	$-0.988192\pi$
$523$	− 38.7942i	− 0.0741763i	0.999312	−	0.0370882i	$-0.0118082\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 612.709i	− 1.16264i
$528$	0	0
$529$	−436.017	−0.824229
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−574.441	−1.07775
$534$	0	0
$535$	437.355i	0.817487i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 110.157i	− 0.204373i
$540$	0	0
$541$	543.111	1.00390	0.501951	−	0.864896i	$-0.332616\pi$
$541$	543.111	1.00390	0.501951	+	0.864896i	$0.332616\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	849.086	1.55796
$546$	0	0
$547$	− 262.532i	− 0.479949i	−0.970779	−	0.239974i	$-0.922861\pi$
$547$	− 262.532i	− 0.479949i	0.970779	−	0.239974i	$-0.0771391\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 5.36212i	− 0.00973161i
$552$	0	0
$553$	−152.104	−0.275053
$554$	0	0
$555$	0	0
$556$	0	0
$557$	652.276	1.17105	0.585526	−	0.810654i	$-0.300888\pi$
$557$	652.276	1.17105	0.585526	+	0.810654i	$0.300888\pi$
$558$	0	0
$559$	− 384.913i	− 0.688574i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 993.435i	− 1.76454i	−0.470746	−	0.882269i	$-0.656015\pi$
$563$	− 993.435i	− 1.76454i	0.470746	−	0.882269i	$-0.343985\pi$
$564$	0	0
$565$	473.288	0.837678
$566$	0	0
$567$	0	0
$568$	0	0
$569$	450.590	0.791898	0.395949	−	0.918272i	$-0.370416\pi$
$569$	450.590	0.791898	0.395949	+	0.918272i	$0.370416\pi$
$570$	0	0
$571$	− 373.493i	− 0.654103i	−0.945007	−	0.327051i	$-0.893945\pi$
$571$	− 373.493i	− 0.654103i	0.945007	−	0.327051i	$-0.106055\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 125.887i	− 0.218934i
$576$	0	0
$577$	370.687	0.642439	0.321219	−	0.947005i	$-0.395907\pi$
$577$	370.687	0.642439	0.321219	+	0.947005i	$0.395907\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−272.128	−0.468378
$582$	0	0
$583$	233.177i	0.399961i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	182.976i	0.311714i	0.987780	+	0.155857i	$0.0498139\pi$
$587$	182.976i	0.311714i	−0.987780	+	0.155857i	$0.950186\pi$
$588$	0	0
$589$	137.082	0.232737
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−333.716	−0.562759	−0.281379	−	0.959597i	$-0.590792\pi$
$593$	−333.716	−0.562759	−0.281379	+	0.959597i	$0.590792\pi$
$594$	0	0
$595$	342.930i	0.576352i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	219.331i	0.366163i	0.983098	+	0.183081i	$0.0586072\pi$
$599$	219.331i	0.366163i	−0.983098	+	0.183081i	$0.941393\pi$
$600$	0	0
$601$	268.193	0.446245	0.223122	−	0.974790i	$-0.428375\pi$
$601$	268.193	0.446245	0.223122	+	0.974790i	$0.428375\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−579.631	−0.958068
$606$	0	0
$607$	− 585.050i	− 0.963839i	−0.876216	−	0.481919i	$-0.839940\pi$
$607$	− 585.050i	− 0.963839i	0.876216	−	0.481919i	$-0.160060\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	329.804i	0.539778i
$612$	0	0
$613$	29.3420	0.0478662	0.0239331	−	0.999714i	$-0.492381\pi$
$613$	29.3420	0.0478662	0.0239331	+	0.999714i	$0.492381\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	413.079	0.669497	0.334748	−	0.942308i	$-0.391349\pi$
$617$	413.079	0.669497	0.334748	+	0.942308i	$0.391349\pi$
$618$	0	0
$619$	225.596i	0.364453i	0.983257	+	0.182226i	$0.0583304\pi$
$619$	225.596i	0.364453i	−0.983257	+	0.182226i	$0.941670\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 236.254i	− 0.379219i
$624$	0	0
$625$	−507.267	−0.811628
$626$	0	0
$627$	0	0
$628$	0	0
$629$	953.346	1.51565
$630$	0	0
$631$	− 914.619i	− 1.44948i	−0.689025	−	0.724738i	$-0.741961\pi$
$631$	− 914.619i	− 1.44948i	0.689025	−	0.724738i	$-0.258039\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 677.239i	− 1.06652i
$636$	0	0
$637$	−60.0380	−0.0942511
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1025.61	1.60002	0.800009	−	0.599987i	$-0.204828\pi$
$641$	1025.61	1.60002	0.800009	+	0.599987i	$0.204828\pi$
$642$	0	0
$643$	− 864.377i	− 1.34429i	−0.740421	−	0.672144i	$-0.765374\pi$
$643$	− 864.377i	− 1.34429i	0.740421	−	0.672144i	$-0.234626\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	689.240i	1.06529i	0.846340	+	0.532643i	$0.178801\pi$
$647$	689.240i	1.06529i	−0.846340	+	0.532643i	$0.821199\pi$
$648$	0	0
$649$	−91.4211	−0.140865
$650$	0	0
$651$	0	0
$652$	0	0
$653$	565.975	0.866730	0.433365	−	0.901218i	$-0.357326\pi$
$653$	565.975	0.866730	0.433365	+	0.901218i	$0.357326\pi$
$654$	0	0
$655$	1187.29i	1.81266i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 289.064i	− 0.438640i	−0.975653	−	0.219320i	$-0.929616\pi$
$659$	− 289.064i	− 0.438640i	0.975653	−	0.219320i	$-0.0703838\pi$
$660$	0	0
$661$	283.787	0.429329	0.214665	−	0.976688i	$-0.431134\pi$
$661$	283.787	0.429329	0.214665	+	0.976688i	$0.431134\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−76.7239	−0.115374
$666$	0	0
$667$	26.2899i	0.0394151i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 827.774i	− 1.23364i
$672$	0	0
$673$	987.489	1.46729	0.733647	−	0.679531i	$-0.237817\pi$
$673$	987.489	1.46729	0.733647	+	0.679531i	$0.237817\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−787.301	−1.16293	−0.581463	−	0.813573i	$-0.697519\pi$
$677$	−787.301	−1.16293	−0.581463	+	0.813573i	$0.697519\pi$
$678$	0	0
$679$	9.11336i	0.0134217i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	193.168i	0.282823i	0.989951	+	0.141411i	$0.0451640\pi$
$683$	193.168i	0.282823i	−0.989951	+	0.141411i	$0.954836\pi$
$684$	0	0
$685$	−14.0865	−0.0205642
$686$	0	0
$687$	0	0
$688$	0	0
$689$	127.087	0.184451
$690$	0	0
$691$	− 967.147i	− 1.39963i	−0.714322	−	0.699817i	$-0.753265\pi$
$691$	− 967.147i	− 1.39963i	0.714322	−	0.699817i	$-0.246735\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	413.308i	0.594688i
$696$	0	0
$697$	1896.73	2.72128
$698$	0	0
$699$	0	0
$700$	0	0
$701$	246.322	0.351386	0.175693	−	0.984445i	$-0.443783\pi$
$701$	246.322	0.351386	0.175693	+	0.984445i	$0.443783\pi$
$702$	0	0
$703$	213.293i	0.303404i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	378.432i	0.535265i
$708$	0	0
$709$	1191.27	1.68021	0.840103	−	0.542428i	$-0.182495\pi$
$709$	1191.27	1.68021	0.840103	+	0.542428i	$0.182495\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−672.098	−0.942635
$714$	0	0
$715$	617.744i	0.863978i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	434.163i	0.603843i	0.953333	+	0.301922i	$0.0976281\pi$
$719$	434.163i	0.603843i	−0.953333	+	0.301922i	$0.902372\pi$
$720$	0	0
$721$	458.838	0.636390
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.42954	−0.00473040
$726$	0	0
$727$	− 931.815i	− 1.28173i	−0.767655	−	0.640863i	$-0.778577\pi$
$727$	− 931.815i	− 1.28173i	0.767655	−	0.640863i	$-0.221423\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1270.94i	1.73863i
$732$	0	0
$733$	−1426.30	−1.94584	−0.972918	−	0.231150i	$-0.925751\pi$
$733$	−1426.30	−1.94584	−0.972918	+	0.231150i	$0.925751\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1847.44	−2.50670
$738$	0	0
$739$	− 216.382i	− 0.292804i	−0.989225	−	0.146402i	$-0.953231\pi$
$739$	− 216.382i	− 0.292804i	0.989225	−	0.146402i	$-0.0467693\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	81.5074i	0.109700i	0.998495	+	0.0548502i	$0.0174681\pi$
$743$	81.5074i	0.109700i	−0.998495	+	0.0548502i	$0.982532\pi$
$744$	0	0
$745$	−189.375	−0.254194
$746$	0	0
$747$	0	0
$748$	0	0
$749$	252.823	0.337547
$750$	0	0
$751$	142.603i	0.189884i	0.995483	+	0.0949422i	$0.0302666\pi$
$751$	142.603i	0.189884i	−0.995483	+	0.0949422i	$0.969733\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 190.023i	− 0.251686i
$756$	0	0
$757$	−372.503	−0.492078	−0.246039	−	0.969260i	$-0.579129\pi$
$757$	−372.503	−0.492078	−0.246039	+	0.969260i	$0.579129\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−663.881	−0.872380	−0.436190	−	0.899855i	$-0.643672\pi$
$761$	−663.881	−0.872380	−0.436190	+	0.899855i	$0.643672\pi$
$762$	0	0
$763$	− 490.833i	− 0.643294i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	49.8265i	0.0649628i
$768$	0	0
$769$	732.653	0.952734	0.476367	−	0.879247i	$-0.341953\pi$
$769$	732.653	0.952734	0.476367	+	0.879247i	$0.341953\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	928.973	1.20178	0.600888	−	0.799333i	$-0.294814\pi$
$773$	928.973	1.20178	0.600888	+	0.799333i	$0.294814\pi$
$774$	0	0
$775$	− 87.6758i	− 0.113130i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	424.358i	0.544747i
$780$	0	0
$781$	1278.67	1.63723
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−123.506	−0.157332
$786$	0	0
$787$	− 470.202i	− 0.597461i	−0.954338	−	0.298730i	$-0.903437\pi$
$787$	− 470.202i	− 0.597461i	0.954338	−	0.298730i	$-0.0965631\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 273.595i	− 0.345884i
$792$	0	0
$793$	−451.155	−0.568922
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−832.694	−1.04479	−0.522393	−	0.852705i	$-0.674960\pi$
$797$	−832.694	−1.04479	−0.522393	+	0.852705i	$0.674960\pi$
$798$	0	0
$799$	− 1088.97i	− 1.36292i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 752.300i	− 0.936862i
$804$	0	0
$805$	376.169	0.467291
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−976.293	−1.20679	−0.603395	−	0.797442i	$-0.706186\pi$
$809$	−976.293	−1.20679	−0.603395	+	0.797442i	$0.706186\pi$
$810$	0	0
$811$	564.470i	0.696018i	0.937491	+	0.348009i	$0.113142\pi$
$811$	564.470i	0.696018i	−0.937491	+	0.348009i	$0.886858\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	579.812i	0.711426i
$816$	0	0
$817$	−284.347	−0.348038
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−779.033	−0.948883	−0.474441	−	0.880287i	$-0.657350\pi$
$821$	−779.033	−0.948883	−0.474441	+	0.880287i	$0.657350\pi$
$822$	0	0
$823$	363.394i	0.441548i	0.975325	+	0.220774i	$0.0708582\pi$
$823$	363.394i	0.441548i	−0.975325	+	0.220774i	$0.929142\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	620.477i	0.750275i	0.926969	+	0.375137i	$0.122404\pi$
$827$	620.477i	0.750275i	−0.926969	+	0.375137i	$0.877596\pi$
$828$	0	0
$829$	737.842	0.890038	0.445019	−	0.895521i	$-0.353197\pi$
$829$	737.842	0.890038	0.445019	+	0.895521i	$0.353197\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	198.238	0.237981
$834$	0	0
$835$	− 1127.02i	− 1.34972i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 897.093i	− 1.06924i	−0.845092	−	0.534620i	$-0.820455\pi$
$839$	− 897.093i	− 1.06924i	0.845092	−	0.534620i	$-0.179545\pi$
$840$	0	0
$841$	−840.284	−0.999148
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−436.804	−0.516928
$846$	0	0
$847$	335.069i	0.395595i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1045.75i	− 1.22885i
$852$	0	0
$853$	−1470.98	−1.72448	−0.862239	−	0.506501i	$-0.830939\pi$
$853$	−1470.98	−1.72448	−0.862239	+	0.506501i	$0.830939\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−851.862	−0.994005	−0.497002	−	0.867749i	$-0.665566\pi$
$857$	−851.862	−0.994005	−0.497002	+	0.867749i	$0.665566\pi$
$858$	0	0
$859$	140.400i	0.163446i	0.996655	+	0.0817229i	$0.0260422\pi$
$859$	140.400i	0.163446i	−0.996655	+	0.0817229i	$0.973958\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 32.1489i	− 0.0372525i	−0.999827	−	0.0186263i	$-0.994071\pi$
$863$	− 32.1489i	− 0.0372525i	0.999827	−	0.0186263i	$-0.00592927\pi$
$864$	0	0
$865$	−341.330	−0.394602
$866$	0	0
$867$	0	0
$868$	0	0
$869$	904.704	1.04109
$870$	0	0
$871$	1006.89i	1.15602i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	351.802i	0.402059i
$876$	0	0
$877$	−1039.93	−1.18578	−0.592888	−	0.805285i	$-0.702012\pi$
$877$	−1039.93	−1.18578	−0.592888	+	0.805285i	$0.702012\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	416.344	0.472581	0.236291	−	0.971682i	$-0.424068\pi$
$881$	416.344	0.472581	0.236291	+	0.971682i	$0.424068\pi$
$882$	0	0
$883$	231.281i	0.261926i	0.991387	+	0.130963i	$0.0418069\pi$
$883$	231.281i	0.261926i	−0.991387	+	0.130963i	$0.958193\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 6.35223i	− 0.00716148i	−0.999994	−	0.00358074i	$-0.998860\pi$
$887$	− 6.35223i	− 0.00716148i	0.999994	−	0.00358074i	$-0.00113979\pi$
$888$	0	0
$889$	−391.493	−0.440374
$890$	0	0
$891$	0	0
$892$	0	0
$893$	243.637	0.272830
$894$	0	0
$895$	− 920.280i	− 1.02825i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	18.3099i	0.0203670i
$900$	0	0
$901$	−419.624	−0.465732
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1156.55	1.27796
$906$	0	0
$907$	201.716i	0.222399i	0.993798	+	0.111200i	$0.0354693\pi$
$907$	201.716i	0.222399i	−0.993798	+	0.111200i	$0.964531\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	484.675i	0.532025i	0.963970	+	0.266013i	$0.0857063\pi$
$911$	484.675i	0.532025i	−0.963970	+	0.266013i	$0.914294\pi$
$912$	0	0
$913$	1618.59	1.77283
$914$	0	0
$915$	0	0
$916$	0	0
$917$	686.340	0.748463
$918$	0	0
$919$	650.067i	0.707363i	0.935366	+	0.353681i	$0.115070\pi$
$919$	650.067i	0.707363i	−0.935366	+	0.353681i	$0.884930\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 696.906i	− 0.755044i
$924$	0	0
$925$	136.419	0.147480
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−522.394	−0.562318	−0.281159	−	0.959661i	$-0.590719\pi$
$929$	−522.394	−0.562318	−0.281159	+	0.959661i	$0.590719\pi$
$930$	0	0
$931$	44.3520i	0.0476391i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 2039.72i	− 2.18151i
$936$	0	0
$937$	1579.17	1.68535	0.842676	−	0.538422i	$-0.180979\pi$
$937$	1579.17	1.68535	0.842676	+	0.538422i	$0.180979\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1080.68	1.14844	0.574218	−	0.818703i	$-0.305306\pi$
$941$	1080.68	1.14844	0.574218	+	0.818703i	$0.305306\pi$
$942$	0	0
$943$	− 2080.58i	− 2.20634i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 538.713i	− 0.568863i	−0.958696	−	0.284431i	$-0.908195\pi$
$947$	− 538.713i	− 0.568863i	0.958696	−	0.284431i	$-0.0918048\pi$
$948$	0	0
$949$	−410.020	−0.432055
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1099.01	1.15322	0.576608	−	0.817021i	$-0.304376\pi$
$953$	1099.01	1.15322	0.576608	+	0.817021i	$0.304376\pi$
$954$	0	0
$955$	1521.04i	1.59271i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.14300i	0.00849114i
$960$	0	0
$961$	492.908	0.512912
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−243.588	−0.252423
$966$	0	0
$967$	− 847.122i	− 0.876031i	−0.898967	−	0.438016i	$-0.855681\pi$
$967$	− 847.122i	− 0.876031i	0.898967	−	0.438016i	$-0.144319\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1059.01i	− 1.09064i	−0.838227	−	0.545322i	$-0.816408\pi$
$971$	− 1059.01i	− 1.09064i	0.838227	−	0.545322i	$-0.183592\pi$
$972$	0	0
$973$	238.922	0.245552
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1223.46	1.25226	0.626131	−	0.779718i	$-0.284638\pi$
$977$	1223.46	1.25226	0.626131	+	0.779718i	$0.284638\pi$
$978$	0	0
$979$	1405.22i	1.43536i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	256.802i	0.261244i	0.991432	+	0.130622i	$0.0416974\pi$
$983$	256.802i	0.261244i	−0.991432	+	0.130622i	$0.958303\pi$
$984$	0	0
$985$	−1264.35	−1.28360
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1394.13	1.40963
$990$	0	0
$991$	988.781i	0.997761i	0.866671	+	0.498881i	$0.166255\pi$
$991$	988.781i	0.997761i	−0.866671	+	0.498881i	$0.833745\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	267.789i	0.269134i
$996$	0	0
$997$	−581.918	−0.583669	−0.291835	−	0.956469i	$-0.594266\pi$
$997$	−581.918	−0.583669	−0.291835	+	0.956469i	$0.594266\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.3.m.c.127.5		8
3.2	odd	2		224.3.d.b.127.5	yes	8
4.3	odd	2	inner	2016.3.m.c.127.6		8
12.11	even	2		224.3.d.b.127.4	✓	8
21.20	even	2		1568.3.d.n.1471.4		8
24.5	odd	2		448.3.d.e.127.4		8
24.11	even	2		448.3.d.e.127.5		8
48.5	odd	4		1792.3.g.d.127.5		8
48.11	even	4		1792.3.g.f.127.3		8
48.29	odd	4		1792.3.g.f.127.4		8
48.35	even	4		1792.3.g.d.127.6		8
84.83	odd	2		1568.3.d.n.1471.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.d.b.127.4	✓	8	12.11	even	2
224.3.d.b.127.5	yes	8	3.2	odd	2
448.3.d.e.127.4		8	24.5	odd	2
448.3.d.e.127.5		8	24.11	even	2
1568.3.d.n.1471.4		8	21.20	even	2
1568.3.d.n.1471.5		8	84.83	odd	2
1792.3.g.d.127.5		8	48.5	odd	4
1792.3.g.d.127.6		8	48.35	even	4
1792.3.g.f.127.3		8	48.11	even	4
1792.3.g.f.127.4		8	48.29	odd	4
2016.3.m.c.127.5		8	1.1	even	1	trivial
2016.3.m.c.127.6		8	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2016.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.57685 q^{5} -2.64575i q^{7} +O(q^{10})\)	\(q+4.57685 q^{5} -2.64575i q^{7} +15.7367i q^{11} +8.57685 q^{13} -28.3197 q^{17} -6.33599i q^{19} +31.0647i q^{23} -4.05242 q^{25} +0.846294 q^{29} +21.6354i q^{31} -12.1092i q^{35} -33.6637 q^{37} -66.9757 q^{41} -44.8781i q^{43} +38.4528i q^{47} -7.00000 q^{49} +14.8174 q^{53} +72.0246i q^{55} +5.80942i q^{59} -52.6015 q^{61} +39.2550 q^{65} +117.397i q^{67} -81.2543i q^{71} -47.8054 q^{73} +41.6354 q^{77} -57.4900i q^{79} -102.855i q^{83} -129.615 q^{85} +89.2955 q^{89} -22.6922i q^{91} -28.9989i q^{95} -3.44452 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 32 q^{13} + 16 q^{17} + 104 q^{25} + 80 q^{29} - 176 q^{37} - 144 q^{41} - 56 q^{49} - 48 q^{53} - 192 q^{61} + 304 q^{65} + 272 q^{73} + 112 q^{77} - 160 q^{85} + 80 q^{89} + 528 q^{97}+O(q^{100}) \) 8 * q + 32 * q^13 + 16 * q^17 + 104 * q^25 + 80 * q^29 - 176 * q^37 - 144 * q^41 - 56 * q^49 - 48 * q^53 - 192 * q^61 + 304 * q^65 + 272 * q^73 + 112 * q^77 - 160 * q^85 + 80 * q^89 + 528 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more