LMFDB - Embedded newform 672.3.g.a.463.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,3,Mod(463,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 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("672.463"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	672.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$18.3106737650$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			463.11
Character	$\chi$	$=$	672.463
Dual form			672.3.g.a.463.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +7.47825i q^{5} -2.64575i q^{7} +3.00000 q^{9} +13.5056 q^{11} +3.54193i q^{13} -12.9527i q^{15} +7.84106 q^{17} +11.1174 q^{19} +4.58258i q^{21} -22.3804i q^{23} -30.9242 q^{25} -5.19615 q^{27} +52.0985i q^{29} +15.6772i q^{31} -23.3923 q^{33} +19.7856 q^{35} +22.1574i q^{37} -6.13481i q^{39} +36.1220 q^{41} -81.8151 q^{43} +22.4347i q^{45} +66.3024i q^{47} -7.00000 q^{49} -13.5811 q^{51} -83.6865i q^{53} +100.998i q^{55} -19.2559 q^{57} -47.7464 q^{59} +102.992i q^{61} -7.93725i q^{63} -26.4874 q^{65} +10.1586 q^{67} +38.7640i q^{69} +71.0621i q^{71} +42.8065 q^{73} +53.5623 q^{75} -35.7324i q^{77} +10.3881i q^{79} +9.00000 q^{81} +103.602 q^{83} +58.6374i q^{85} -90.2373i q^{87} -60.9498 q^{89} +9.37107 q^{91} -27.1537i q^{93} +83.1387i q^{95} +52.2712 q^{97} +40.5167 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 72 q^{9} - 32 q^{11} + 16 q^{17} + 64 q^{19} - 72 q^{25} - 80 q^{41} - 32 q^{43} - 168 q^{49} - 192 q^{51} - 192 q^{65} + 32 q^{67} - 240 q^{73} + 384 q^{75} + 216 q^{81} + 320 q^{83} + 400 q^{89}+ \cdots - 96 q^{99}+O(q^{100}) $ 24 * q + 72 * q^9 - 32 * q^11 + 16 * q^17 + 64 * q^19 - 72 * q^25 - 80 * q^41 - 32 * q^43 - 168 * q^49 - 192 * q^51 - 192 * q^65 + 32 * q^67 - 240 * q^73 + 384 * q^75 + 216 * q^81 + 320 * q^83 + 400 * q^89 + 144 * q^97 - 96 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	0	0
$5$	7.47825i	1.49565i	0.663896	+	0.747825i	$0.268902\pi$
$5$	7.47825i	1.49565i	−0.663896	+	0.747825i	$0.731098\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	13.5056	1.22778	0.613889	−	0.789392i	$-0.289604\pi$
$11$	13.5056	1.22778	0.613889	+	0.789392i	$0.289604\pi$
$12$	0	0
$13$	3.54193i	0.272456i	0.990677	+	0.136228i	$0.0434980\pi$
$13$	3.54193i	0.272456i	−0.990677	+	0.136228i	$0.956502\pi$
$14$	0	0
$15$	− 12.9527i	− 0.863514i
$16$	0	0
$17$	7.84106	0.461239	0.230620	−	0.973044i	$-0.425925\pi$
$17$	7.84106	0.461239	0.230620	+	0.973044i	$0.425925\pi$
$18$	0	0
$19$	11.1174	0.585127	0.292563	−	0.956246i	$-0.405492\pi$
$19$	11.1174	0.585127	0.292563	+	0.956246i	$0.405492\pi$
$20$	0	0
$21$	4.58258i	0.218218i
$22$	0	0
$23$	− 22.3804i	− 0.973060i	−0.873664	−	0.486530i	$-0.838262\pi$
$23$	− 22.3804i	− 0.973060i	0.873664	−	0.486530i	$-0.161738\pi$
$24$	0	0
$25$	−30.9242	−1.23697
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	52.0985i	1.79650i	0.439484	+	0.898250i	$0.355161\pi$
$29$	52.0985i	1.79650i	−0.439484	+	0.898250i	$0.644839\pi$
$30$	0	0
$31$	15.6772i	0.505716i	0.967503	+	0.252858i	$0.0813705\pi$
$31$	15.6772i	0.505716i	−0.967503	+	0.252858i	$0.918630\pi$
$32$	0	0
$33$	−23.3923	−0.708858
$34$	0	0
$35$	19.7856	0.565302
$36$	0	0
$37$	22.1574i	0.598849i	0.954120	+	0.299424i	$0.0967946\pi$
$37$	22.1574i	0.598849i	−0.954120	+	0.299424i	$0.903205\pi$
$38$	0	0
$39$	− 6.13481i	− 0.157303i
$40$	0	0
$41$	36.1220	0.881023	0.440512	−	0.897747i	$-0.354797\pi$
$41$	36.1220	0.881023	0.440512	+	0.897747i	$0.354797\pi$
$42$	0	0
$43$	−81.8151	−1.90268	−0.951338	−	0.308149i	$-0.900290\pi$
$43$	−81.8151	−1.90268	−0.951338	+	0.308149i	$0.900290\pi$
$44$	0	0
$45$	22.4347i	0.498550i
$46$	0	0
$47$	66.3024i	1.41069i	0.708864	+	0.705345i	$0.249208\pi$
$47$	66.3024i	1.41069i	−0.708864	+	0.705345i	$0.750792\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	−13.5811	−0.266297
$52$	0	0
$53$	− 83.6865i	− 1.57899i	−0.613756	−	0.789496i	$-0.710342\pi$
$53$	− 83.6865i	− 1.57899i	0.613756	−	0.789496i	$-0.289658\pi$
$54$	0	0
$55$	100.998i	1.83633i
$56$	0	0
$57$	−19.2559	−0.337823
$58$	0	0
$59$	−47.7464	−0.809261	−0.404630	−	0.914480i	$-0.632600\pi$
$59$	−47.7464	−0.809261	−0.404630	+	0.914480i	$0.632600\pi$
$60$	0	0
$61$	102.992i	1.68839i	0.536039	+	0.844193i	$0.319920\pi$
$61$	102.992i	1.68839i	−0.536039	+	0.844193i	$0.680080\pi$
$62$	0	0
$63$	− 7.93725i	− 0.125988i
$64$	0	0
$65$	−26.4874	−0.407499
$66$	0	0
$67$	10.1586	0.151621	0.0758107	−	0.997122i	$-0.475846\pi$
$67$	10.1586	0.151621	0.0758107	+	0.997122i	$0.475846\pi$
$68$	0	0
$69$	38.7640i	0.561797i
$70$	0	0
$71$	71.0621i	1.00087i	0.865773	+	0.500437i	$0.166827\pi$
$71$	71.0621i	1.00087i	−0.865773	+	0.500437i	$0.833173\pi$
$72$	0	0
$73$	42.8065	0.586390	0.293195	−	0.956053i	$-0.405282\pi$
$73$	42.8065	0.586390	0.293195	+	0.956053i	$0.405282\pi$
$74$	0	0
$75$	53.5623	0.714164
$76$	0	0
$77$	− 35.7324i	− 0.464057i
$78$	0	0
$79$	10.3881i	0.131494i	0.997836	+	0.0657472i	$0.0209431\pi$
$79$	10.3881i	0.131494i	−0.997836	+	0.0657472i	$0.979057\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	103.602	1.24822	0.624111	−	0.781335i	$-0.285461\pi$
$83$	103.602	1.24822	0.624111	+	0.781335i	$0.285461\pi$
$84$	0	0
$85$	58.6374i	0.689852i
$86$	0	0
$87$	− 90.2373i	− 1.03721i
$88$	0	0
$89$	−60.9498	−0.684830	−0.342415	−	0.939549i	$-0.611245\pi$
$89$	−60.9498	−0.684830	−0.342415	+	0.939549i	$0.611245\pi$
$90$	0	0
$91$	9.37107	0.102979
$92$	0	0
$93$	− 27.1537i	− 0.291975i
$94$	0	0
$95$	83.1387i	0.875145i
$96$	0	0
$97$	52.2712	0.538878	0.269439	−	0.963017i	$-0.413162\pi$
$97$	52.2712	0.538878	0.269439	+	0.963017i	$0.413162\pi$
$98$	0	0
$99$	40.5167	0.409260
$100$	0	0
$101$	− 22.2697i	− 0.220492i	−0.993904	−	0.110246i	$-0.964836\pi$
$101$	− 22.2697i	− 0.220492i	0.993904	−	0.110246i	$-0.0351639\pi$
$102$	0	0
$103$	133.270i	1.29388i	0.762539	+	0.646942i	$0.223952\pi$
$103$	133.270i	1.29388i	−0.762539	+	0.646942i	$0.776048\pi$
$104$	0	0
$105$	−34.2696	−0.326378
$106$	0	0
$107$	−157.452	−1.47151	−0.735755	−	0.677248i	$-0.763173\pi$
$107$	−157.452	−1.47151	−0.735755	+	0.677248i	$0.763173\pi$
$108$	0	0
$109$	134.357i	1.23264i	0.787498	+	0.616318i	$0.211376\pi$
$109$	134.357i	1.23264i	−0.787498	+	0.616318i	$0.788624\pi$
$110$	0	0
$111$	− 38.3778i	− 0.345746i
$112$	0	0
$113$	179.708	1.59034	0.795169	−	0.606388i	$-0.207382\pi$
$113$	179.708	1.59034	0.795169	+	0.606388i	$0.207382\pi$
$114$	0	0
$115$	167.366	1.45536
$116$	0	0
$117$	10.6258i	0.0908188i
$118$	0	0
$119$	− 20.7455i	− 0.174332i
$120$	0	0
$121$	61.4003	0.507440
$122$	0	0
$123$	−62.5651	−0.508659
$124$	0	0
$125$	− 44.3027i	− 0.354422i
$126$	0	0
$127$	34.9032i	0.274828i	0.990514	+	0.137414i	$0.0438791\pi$
$127$	34.9032i	0.274828i	−0.990514	+	0.137414i	$0.956121\pi$
$128$	0	0
$129$	141.708	1.09851
$130$	0	0
$131$	−127.352	−0.972150	−0.486075	−	0.873917i	$-0.661572\pi$
$131$	−127.352	−0.972150	−0.486075	+	0.873917i	$0.661572\pi$
$132$	0	0
$133$	− 29.4139i	− 0.221157i
$134$	0	0
$135$	− 38.8581i	− 0.287838i
$136$	0	0
$137$	−155.552	−1.13542	−0.567708	−	0.823230i	$-0.692170\pi$
$137$	−155.552	−1.13542	−0.567708	+	0.823230i	$0.692170\pi$
$138$	0	0
$139$	50.8834	0.366067	0.183034	−	0.983107i	$-0.441408\pi$
$139$	50.8834	0.366067	0.183034	+	0.983107i	$0.441408\pi$
$140$	0	0
$141$	− 114.839i	− 0.814462i
$142$	0	0
$143$	47.8358i	0.334516i
$144$	0	0
$145$	−389.606	−2.68694
$146$	0	0
$147$	12.1244	0.0824786
$148$	0	0
$149$	− 253.074i	− 1.69848i	−0.528007	−	0.849240i	$-0.677060\pi$
$149$	− 253.074i	− 1.69848i	0.528007	−	0.849240i	$-0.322940\pi$
$150$	0	0
$151$	− 279.766i	− 1.85275i	−0.376598	−	0.926377i	$-0.622906\pi$
$151$	− 279.766i	− 1.85275i	0.376598	−	0.926377i	$-0.377094\pi$
$152$	0	0
$153$	23.5232	0.153746
$154$	0	0
$155$	−117.238	−0.756374
$156$	0	0
$157$	− 73.6080i	− 0.468841i	−0.972135	−	0.234420i	$-0.924681\pi$
$157$	− 73.6080i	− 0.468841i	0.972135	−	0.234420i	$-0.0753192\pi$
$158$	0	0
$159$	144.949i	0.911631i
$160$	0	0
$161$	−59.2129	−0.367782
$162$	0	0
$163$	−171.461	−1.05191	−0.525953	−	0.850513i	$-0.676291\pi$
$163$	−171.461	−1.05191	−0.525953	+	0.850513i	$0.676291\pi$
$164$	0	0
$165$	− 174.934i	− 1.06020i
$166$	0	0
$167$	− 181.048i	− 1.08412i	−0.840339	−	0.542061i	$-0.817644\pi$
$167$	− 181.048i	− 1.08412i	0.840339	−	0.542061i	$-0.182356\pi$
$168$	0	0
$169$	156.455	0.925768
$170$	0	0
$171$	33.3522	0.195042
$172$	0	0
$173$	− 288.442i	− 1.66729i	−0.552298	−	0.833647i	$-0.686249\pi$
$173$	− 288.442i	− 1.66729i	0.552298	−	0.833647i	$-0.313751\pi$
$174$	0	0
$175$	81.8178i	0.467530i
$176$	0	0
$177$	82.6992	0.467227
$178$	0	0
$179$	33.2048	0.185502	0.0927510	−	0.995689i	$-0.470434\pi$
$179$	33.2048	0.185502	0.0927510	+	0.995689i	$0.470434\pi$
$180$	0	0
$181$	196.790i	1.08724i	0.839332	+	0.543619i	$0.182946\pi$
$181$	196.790i	1.08724i	−0.839332	+	0.543619i	$0.817054\pi$
$182$	0	0
$183$	− 178.387i	− 0.974790i
$184$	0	0
$185$	−165.699	−0.895668
$186$	0	0
$187$	105.898	0.566300
$188$	0	0
$189$	13.7477i	0.0727393i
$190$	0	0
$191$	− 123.996i	− 0.649191i	−0.945853	−	0.324596i	$-0.894772\pi$
$191$	− 123.996i	− 0.649191i	0.945853	−	0.324596i	$-0.105228\pi$
$192$	0	0
$193$	68.7028	0.355973	0.177986	−	0.984033i	$-0.443042\pi$
$193$	68.7028	0.355973	0.177986	+	0.984033i	$0.443042\pi$
$194$	0	0
$195$	45.8776	0.235270
$196$	0	0
$197$	174.809i	0.887357i	0.896186	+	0.443678i	$0.146327\pi$
$197$	174.809i	0.887357i	−0.896186	+	0.443678i	$0.853673\pi$
$198$	0	0
$199$	131.589i	0.661251i	0.943762	+	0.330625i	$0.107260\pi$
$199$	131.589i	0.661251i	−0.943762	+	0.330625i	$0.892740\pi$
$200$	0	0
$201$	−17.5953	−0.0875386
$202$	0	0
$203$	137.840	0.679014
$204$	0	0
$205$	270.129i	1.31770i
$206$	0	0
$207$	− 67.1412i	− 0.324353i
$208$	0	0
$209$	150.147	0.718406
$210$	0	0
$211$	21.4133	0.101485	0.0507424	−	0.998712i	$-0.483841\pi$
$211$	21.4133	0.101485	0.0507424	+	0.998712i	$0.483841\pi$
$212$	0	0
$213$	− 123.083i	− 0.577855i
$214$	0	0
$215$	− 611.834i	− 2.84574i
$216$	0	0
$217$	41.4779	0.191143
$218$	0	0
$219$	−74.1430	−0.338552
$220$	0	0
$221$	27.7725i	0.125667i
$222$	0	0
$223$	16.9401i	0.0759648i	0.999278	+	0.0379824i	$0.0120931\pi$
$223$	16.9401i	0.0759648i	−0.999278	+	0.0379824i	$0.987907\pi$
$224$	0	0
$225$	−92.7726	−0.412323
$226$	0	0
$227$	413.555	1.82183	0.910914	−	0.412596i	$-0.135378\pi$
$227$	413.555	1.82183	0.910914	+	0.412596i	$0.135378\pi$
$228$	0	0
$229$	− 36.0491i	− 0.157420i	−0.996898	−	0.0787099i	$-0.974920\pi$
$229$	− 36.0491i	− 0.157420i	0.996898	−	0.0787099i	$-0.0250801\pi$
$230$	0	0
$231$	61.8903i	0.267923i
$232$	0	0
$233$	303.232	1.30143	0.650713	−	0.759324i	$-0.274470\pi$
$233$	303.232	1.30143	0.650713	+	0.759324i	$0.274470\pi$
$234$	0	0
$235$	−495.826	−2.10990
$236$	0	0
$237$	− 17.9926i	− 0.0759183i
$238$	0	0
$239$	47.4692i	0.198616i	0.995057	+	0.0993078i	$0.0316629\pi$
$239$	47.4692i	0.198616i	−0.995057	+	0.0993078i	$0.968337\pi$
$240$	0	0
$241$	−273.047	−1.13298	−0.566488	−	0.824070i	$-0.691698\pi$
$241$	−273.047	−1.13298	−0.566488	+	0.824070i	$0.691698\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	− 52.3477i	− 0.213664i
$246$	0	0
$247$	39.3771i	0.159421i
$248$	0	0
$249$	−179.445	−0.720662
$250$	0	0
$251$	384.102	1.53029	0.765144	−	0.643859i	$-0.222668\pi$
$251$	384.102	1.53029	0.765144	+	0.643859i	$0.222668\pi$
$252$	0	0
$253$	− 302.260i	− 1.19470i
$254$	0	0
$255$	− 101.563i	− 0.398286i
$256$	0	0
$257$	308.955	1.20216	0.601080	−	0.799189i	$-0.294737\pi$
$257$	308.955	1.20216	0.601080	+	0.799189i	$0.294737\pi$
$258$	0	0
$259$	58.6230	0.226344
$260$	0	0
$261$	156.296i	0.598834i
$262$	0	0
$263$	− 18.7484i	− 0.0712865i	−0.999365	−	0.0356433i	$-0.988652\pi$
$263$	− 18.7484i	− 0.0712865i	0.999365	−	0.0356433i	$-0.0113480\pi$
$264$	0	0
$265$	625.829	2.36162
$266$	0	0
$267$	105.568	0.395387
$268$	0	0
$269$	344.672i	1.28131i	0.767829	+	0.640655i	$0.221337\pi$
$269$	344.672i	1.28131i	−0.767829	+	0.640655i	$0.778663\pi$
$270$	0	0
$271$	− 54.1955i	− 0.199983i	−0.994988	−	0.0999917i	$-0.968118\pi$
$271$	− 54.1955i	− 0.199983i	0.994988	−	0.0999917i	$-0.0318816\pi$
$272$	0	0
$273$	−16.2312	−0.0594548
$274$	0	0
$275$	−417.649	−1.51872
$276$	0	0
$277$	72.5083i	0.261763i	0.991398	+	0.130881i	$0.0417807\pi$
$277$	72.5083i	0.261763i	−0.991398	+	0.130881i	$0.958219\pi$
$278$	0	0
$279$	47.0316i	0.168572i
$280$	0	0
$281$	−5.38105	−0.0191497	−0.00957483	−	0.999954i	$-0.503048\pi$
$281$	−5.38105	−0.0191497	−0.00957483	+	0.999954i	$0.503048\pi$
$282$	0	0
$283$	545.863	1.92885	0.964423	−	0.264365i	$-0.0851625\pi$
$283$	545.863	1.92885	0.964423	+	0.264365i	$0.0851625\pi$
$284$	0	0
$285$	− 144.001i	− 0.505265i
$286$	0	0
$287$	− 95.5697i	− 0.332995i
$288$	0	0
$289$	−227.518	−0.787258
$290$	0	0
$291$	−90.5363	−0.311121
$292$	0	0
$293$	− 425.941i	− 1.45373i	−0.686783	−	0.726863i	$-0.740978\pi$
$293$	− 425.941i	− 1.45373i	0.686783	−	0.726863i	$-0.259022\pi$
$294$	0	0
$295$	− 357.059i	− 1.21037i
$296$	0	0
$297$	−70.1770	−0.236286
$298$	0	0
$299$	79.2698	0.265116
$300$	0	0
$301$	216.462i	0.719144i
$302$	0	0
$303$	38.5722i	0.127301i
$304$	0	0
$305$	−770.197	−2.52524
$306$	0	0
$307$	−444.426	−1.44764	−0.723822	−	0.689987i	$-0.757616\pi$
$307$	−444.426	−1.44764	−0.723822	+	0.689987i	$0.757616\pi$
$308$	0	0
$309$	− 230.830i	− 0.747024i
$310$	0	0
$311$	17.4956i	0.0562559i	0.999604	+	0.0281279i	$0.00895458\pi$
$311$	17.4956i	0.0562559i	−0.999604	+	0.0281279i	$0.991045\pi$
$312$	0	0
$313$	−382.054	−1.22062	−0.610311	−	0.792162i	$-0.708955\pi$
$313$	−382.054	−1.22062	−0.610311	+	0.792162i	$0.708955\pi$
$314$	0	0
$315$	59.3568	0.188434
$316$	0	0
$317$	390.680i	1.23243i	0.787579	+	0.616214i	$0.211334\pi$
$317$	390.680i	1.23243i	−0.787579	+	0.616214i	$0.788666\pi$
$318$	0	0
$319$	703.620i	2.20571i
$320$	0	0
$321$	272.714	0.849577
$322$	0	0
$323$	87.1723	0.269883
$324$	0	0
$325$	− 109.531i	− 0.337020i
$326$	0	0
$327$	− 232.714i	− 0.711662i
$328$	0	0
$329$	175.420	0.533191
$330$	0	0
$331$	283.248	0.855735	0.427867	−	0.903842i	$-0.359265\pi$
$331$	283.248	0.855735	0.427867	+	0.903842i	$0.359265\pi$
$332$	0	0
$333$	66.4722i	0.199616i
$334$	0	0
$335$	75.9688i	0.226772i
$336$	0	0
$337$	−119.881	−0.355730	−0.177865	−	0.984055i	$-0.556919\pi$
$337$	−119.881	−0.355730	−0.177865	+	0.984055i	$0.556919\pi$
$338$	0	0
$339$	−311.264	−0.918182
$340$	0	0
$341$	211.729i	0.620907i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	−289.887	−0.840251
$346$	0	0
$347$	−95.3186	−0.274693	−0.137347	−	0.990523i	$-0.543857\pi$
$347$	−95.3186	−0.274693	−0.137347	+	0.990523i	$0.543857\pi$
$348$	0	0
$349$	− 42.8609i	− 0.122811i	−0.998113	−	0.0614053i	$-0.980442\pi$
$349$	− 42.8609i	− 0.122811i	0.998113	−	0.0614053i	$-0.0195582\pi$
$350$	0	0
$351$	− 18.4044i	− 0.0524342i
$352$	0	0
$353$	324.090	0.918101	0.459050	−	0.888410i	$-0.348190\pi$
$353$	324.090	0.918101	0.459050	+	0.888410i	$0.348190\pi$
$354$	0	0
$355$	−531.420	−1.49696
$356$	0	0
$357$	35.9323i	0.100651i
$358$	0	0
$359$	− 352.212i	− 0.981092i	−0.871415	−	0.490546i	$-0.836797\pi$
$359$	− 352.212i	− 0.981092i	0.871415	−	0.490546i	$-0.163203\pi$
$360$	0	0
$361$	−237.403	−0.657627
$362$	0	0
$363$	−106.348	−0.292971
$364$	0	0
$365$	320.117i	0.877034i
$366$	0	0
$367$	402.579i	1.09695i	0.836168	+	0.548473i	$0.184791\pi$
$367$	402.579i	1.09695i	−0.836168	+	0.548473i	$0.815209\pi$
$368$	0	0
$369$	108.366	0.293674
$370$	0	0
$371$	−221.414	−0.596803
$372$	0	0
$373$	− 349.786i	− 0.937765i	−0.883261	−	0.468882i	$-0.844657\pi$
$373$	− 349.786i	− 0.937765i	0.883261	−	0.468882i	$-0.155343\pi$
$374$	0	0
$375$	76.7346i	0.204625i
$376$	0	0
$377$	−184.529	−0.489468
$378$	0	0
$379$	97.5057	0.257271	0.128635	−	0.991692i	$-0.458940\pi$
$379$	97.5057	0.257271	0.128635	+	0.991692i	$0.458940\pi$
$380$	0	0
$381$	− 60.4540i	− 0.158672i
$382$	0	0
$383$	3.49274i	0.00911944i	0.999990	+	0.00455972i	$0.00145141\pi$
$383$	3.49274i	0.00911944i	−0.999990	+	0.00455972i	$0.998549\pi$
$384$	0	0
$385$	267.216	0.694066
$386$	0	0
$387$	−245.445	−0.634225
$388$	0	0
$389$	− 1.14711i	− 0.00294887i	−0.999999	−	0.00147444i	$-0.999531\pi$
$389$	− 1.14711i	− 0.00294887i	0.999999	−	0.00147444i	$-0.000469328\pi$
$390$	0	0
$391$	− 175.486i	− 0.448813i
$392$	0	0
$393$	220.580	0.561271
$394$	0	0
$395$	−77.6844	−0.196669
$396$	0	0
$397$	− 683.504i	− 1.72167i	−0.508882	−	0.860836i	$-0.669941\pi$
$397$	− 683.504i	− 1.72167i	0.508882	−	0.860836i	$-0.330059\pi$
$398$	0	0
$399$	50.9464i	0.127685i
$400$	0	0
$401$	308.774	0.770011	0.385005	−	0.922914i	$-0.374200\pi$
$401$	308.774	0.770011	0.385005	+	0.922914i	$0.374200\pi$
$402$	0	0
$403$	−55.5275	−0.137785
$404$	0	0
$405$	67.3042i	0.166183i
$406$	0	0
$407$	299.248i	0.735254i
$408$	0	0
$409$	357.161	0.873253	0.436627	−	0.899643i	$-0.356173\pi$
$409$	357.161	0.873253	0.436627	+	0.899643i	$0.356173\pi$
$410$	0	0
$411$	269.424	0.655533
$412$	0	0
$413$	126.325i	0.305872i
$414$	0	0
$415$	774.765i	1.86690i
$416$	0	0
$417$	−88.1326	−0.211349
$418$	0	0
$419$	309.516	0.738702	0.369351	−	0.929290i	$-0.379580\pi$
$419$	309.516	0.738702	0.369351	+	0.929290i	$0.379580\pi$
$420$	0	0
$421$	108.620i	0.258005i	0.991644	+	0.129002i	$0.0411775\pi$
$421$	108.620i	0.258005i	−0.991644	+	0.129002i	$0.958823\pi$
$422$	0	0
$423$	198.907i	0.470230i
$424$	0	0
$425$	−242.479	−0.570538
$426$	0	0
$427$	272.490	0.638150
$428$	0	0
$429$	− 82.8540i	− 0.193133i
$430$	0	0
$431$	116.061i	0.269283i	0.990894	+	0.134641i	$0.0429882\pi$
$431$	116.061i	0.269283i	−0.990894	+	0.134641i	$0.957012\pi$
$432$	0	0
$433$	363.996	0.840637	0.420318	−	0.907377i	$-0.361918\pi$
$433$	363.996	0.840637	0.420318	+	0.907377i	$0.361918\pi$
$434$	0	0
$435$	674.817	1.55130
$436$	0	0
$437$	− 248.812i	− 0.569364i
$438$	0	0
$439$	264.500i	0.602507i	0.953544	+	0.301253i	$0.0974050\pi$
$439$	264.500i	0.602507i	−0.953544	+	0.301253i	$0.902595\pi$
$440$	0	0
$441$	−21.0000	−0.0476190
$442$	0	0
$443$	−361.826	−0.816763	−0.408382	−	0.912811i	$-0.633907\pi$
$443$	−361.826	−0.816763	−0.408382	+	0.912811i	$0.633907\pi$
$444$	0	0
$445$	− 455.798i	− 1.02427i
$446$	0	0
$447$	438.336i	0.980618i
$448$	0	0
$449$	58.1925	0.129605	0.0648023	−	0.997898i	$-0.479358\pi$
$449$	58.1925	0.129605	0.0648023	+	0.997898i	$0.479358\pi$
$450$	0	0
$451$	487.847	1.08170
$452$	0	0
$453$	484.569i	1.06969i
$454$	0	0
$455$	70.0792i	0.154020i
$456$	0	0
$457$	282.240	0.617594	0.308797	−	0.951128i	$-0.400074\pi$
$457$	282.240	0.617594	0.308797	+	0.951128i	$0.400074\pi$
$458$	0	0
$459$	−40.7434	−0.0887655
$460$	0	0
$461$	263.726i	0.572073i	0.958219	+	0.286036i	$0.0923378\pi$
$461$	263.726i	0.572073i	−0.958219	+	0.286036i	$0.907662\pi$
$462$	0	0
$463$	203.652i	0.439853i	0.975516	+	0.219927i	$0.0705818\pi$
$463$	203.652i	0.439853i	−0.975516	+	0.219927i	$0.929418\pi$
$464$	0	0
$465$	203.062	0.436692
$466$	0	0
$467$	−100.538	−0.215285	−0.107643	−	0.994190i	$-0.534330\pi$
$467$	−100.538	−0.215285	−0.107643	+	0.994190i	$0.534330\pi$
$468$	0	0
$469$	− 26.8772i	− 0.0573075i
$470$	0	0
$471$	127.493i	0.270685i
$472$	0	0
$473$	−1104.96	−2.33607
$474$	0	0
$475$	−343.797	−0.723783
$476$	0	0
$477$	− 251.060i	− 0.526330i
$478$	0	0
$479$	491.688i	1.02649i	0.858242	+	0.513245i	$0.171557\pi$
$479$	491.688i	1.02649i	−0.858242	+	0.513245i	$0.828443\pi$
$480$	0	0
$481$	−78.4800	−0.163160
$482$	0	0
$483$	102.560	0.212339
$484$	0	0
$485$	390.897i	0.805973i
$486$	0	0
$487$	− 242.746i	− 0.498451i	−0.968446	−	0.249225i	$-0.919824\pi$
$487$	− 242.746i	− 0.498451i	0.968446	−	0.249225i	$-0.0801760\pi$
$488$	0	0
$489$	296.979	0.607319
$490$	0	0
$491$	525.167	1.06959	0.534793	−	0.844983i	$-0.320390\pi$
$491$	525.167	1.06959	0.534793	+	0.844983i	$0.320390\pi$
$492$	0	0
$493$	408.508i	0.828616i
$494$	0	0
$495$	302.994i	0.612109i
$496$	0	0
$497$	188.013	0.378295
$498$	0	0
$499$	231.230	0.463387	0.231694	−	0.972789i	$-0.425573\pi$
$499$	231.230	0.463387	0.231694	+	0.972789i	$0.425573\pi$
$500$	0	0
$501$	313.585i	0.625918i
$502$	0	0
$503$	− 180.388i	− 0.358624i	−0.983792	−	0.179312i	$-0.942613\pi$
$503$	− 180.388i	− 0.358624i	0.983792	−	0.179312i	$-0.0573872\pi$
$504$	0	0
$505$	166.538	0.329779
$506$	0	0
$507$	−270.988	−0.534492
$508$	0	0
$509$	− 379.548i	− 0.745674i	−0.927897	−	0.372837i	$-0.878385\pi$
$509$	− 379.548i	− 0.745674i	0.927897	−	0.372837i	$-0.121615\pi$
$510$	0	0
$511$	− 113.255i	− 0.221634i
$512$	0	0
$513$	−57.7677	−0.112608
$514$	0	0
$515$	−996.626	−1.93520
$516$	0	0
$517$	895.452i	1.73201i
$518$	0	0
$519$	499.596i	0.962613i
$520$	0	0
$521$	576.665	1.10684	0.553421	−	0.832902i	$-0.313322\pi$
$521$	576.665	1.10684	0.553421	+	0.832902i	$0.313322\pi$
$522$	0	0
$523$	−738.590	−1.41222	−0.706109	−	0.708103i	$-0.749551\pi$
$523$	−738.590	−1.41222	−0.706109	+	0.708103i	$0.749551\pi$
$524$	0	0
$525$	− 141.713i	− 0.269929i
$526$	0	0
$527$	122.926i	0.233256i
$528$	0	0
$529$	28.1183	0.0531536
$530$	0	0
$531$	−143.239	−0.269754
$532$	0	0
$533$	127.941i	0.240040i
$534$	0	0
$535$	− 1177.46i	− 2.20086i
$536$	0	0
$537$	−57.5125	−0.107100
$538$	0	0
$539$	−94.5390	−0.175397
$540$	0	0
$541$	− 119.151i	− 0.220243i	−0.993918	−	0.110121i	$-0.964876\pi$
$541$	− 119.151i	− 0.220243i	0.993918	−	0.110121i	$-0.0351240\pi$
$542$	0	0
$543$	− 340.850i	− 0.627717i
$544$	0	0
$545$	−1004.76	−1.84359
$546$	0	0
$547$	316.005	0.577706	0.288853	−	0.957374i	$-0.406726\pi$
$547$	316.005	0.577706	0.288853	+	0.957374i	$0.406726\pi$
$548$	0	0
$549$	308.975i	0.562796i
$550$	0	0
$551$	579.201i	1.05118i
$552$	0	0
$553$	27.4842	0.0497002
$554$	0	0
$555$	286.998	0.517114
$556$	0	0
$557$	611.086i	1.09710i	0.836117	+	0.548551i	$0.184820\pi$
$557$	611.086i	1.09710i	−0.836117	+	0.548551i	$0.815180\pi$
$558$	0	0
$559$	− 289.783i	− 0.518396i
$560$	0	0
$561$	−183.421	−0.326953
$562$	0	0
$563$	263.337	0.467739	0.233869	−	0.972268i	$-0.424861\pi$
$563$	263.337	0.467739	0.233869	+	0.972268i	$0.424861\pi$
$564$	0	0
$565$	1343.90i	2.37859i
$566$	0	0
$567$	− 23.8118i	− 0.0419961i
$568$	0	0
$569$	−263.864	−0.463733	−0.231867	−	0.972748i	$-0.574483\pi$
$569$	−263.864	−0.463733	−0.231867	+	0.972748i	$0.574483\pi$
$570$	0	0
$571$	−198.806	−0.348172	−0.174086	−	0.984730i	$-0.555697\pi$
$571$	−198.806	−0.348172	−0.174086	+	0.984730i	$0.555697\pi$
$572$	0	0
$573$	214.767i	0.374811i
$574$	0	0
$575$	692.096i	1.20364i
$576$	0	0
$577$	−419.509	−0.727051	−0.363526	−	0.931584i	$-0.618427\pi$
$577$	−419.509	−0.727051	−0.363526	+	0.931584i	$0.618427\pi$
$578$	0	0
$579$	−118.997	−0.205521
$580$	0	0
$581$	− 274.106i	− 0.471784i
$582$	0	0
$583$	− 1130.23i	− 1.93865i
$584$	0	0
$585$	−79.4623	−0.135833
$586$	0	0
$587$	536.991	0.914806	0.457403	−	0.889260i	$-0.348780\pi$
$587$	536.991	0.914806	0.457403	+	0.889260i	$0.348780\pi$
$588$	0	0
$589$	174.290i	0.295908i
$590$	0	0
$591$	− 302.779i	− 0.512316i
$592$	0	0
$593$	201.757	0.340231	0.170115	−	0.985424i	$-0.445586\pi$
$593$	201.757	0.340231	0.170115	+	0.985424i	$0.445586\pi$
$594$	0	0
$595$	155.140	0.260740
$596$	0	0
$597$	− 227.919i	− 0.381773i
$598$	0	0
$599$	− 487.011i	− 0.813040i	−0.913642	−	0.406520i	$-0.866742\pi$
$599$	− 487.011i	− 0.813040i	0.913642	−	0.406520i	$-0.133258\pi$
$600$	0	0
$601$	−671.112	−1.11666	−0.558330	−	0.829619i	$-0.688558\pi$
$601$	−671.112	−1.11666	−0.558330	+	0.829619i	$0.688558\pi$
$602$	0	0
$603$	30.4759	0.0505405
$604$	0	0
$605$	459.167i	0.758953i
$606$	0	0
$607$	− 463.776i	− 0.764046i	−0.924153	−	0.382023i	$-0.875228\pi$
$607$	− 463.776i	− 0.764046i	0.924153	−	0.382023i	$-0.124772\pi$
$608$	0	0
$609$	−238.745	−0.392029
$610$	0	0
$611$	−234.839	−0.384351
$612$	0	0
$613$	− 704.945i	− 1.14999i	−0.818156	−	0.574996i	$-0.805003\pi$
$613$	− 704.945i	− 1.14999i	0.818156	−	0.574996i	$-0.194997\pi$
$614$	0	0
$615$	− 467.877i	− 0.760776i
$616$	0	0
$617$	662.653	1.07399	0.536996	−	0.843585i	$-0.319559\pi$
$617$	662.653	1.07399	0.536996	+	0.843585i	$0.319559\pi$
$618$	0	0
$619$	−325.613	−0.526031	−0.263015	−	0.964792i	$-0.584717\pi$
$619$	−325.613	−0.526031	−0.263015	+	0.964792i	$0.584717\pi$
$620$	0	0
$621$	116.292i	0.187266i
$622$	0	0
$623$	161.258i	0.258841i
$624$	0	0
$625$	−441.798	−0.706878
$626$	0	0
$627$	−260.062	−0.414772
$628$	0	0
$629$	173.738i	0.276212i
$630$	0	0
$631$	− 368.582i	− 0.584124i	−0.956399	−	0.292062i	$-0.905659\pi$
$631$	− 368.582i	− 0.584124i	0.956399	−	0.292062i	$-0.0943413\pi$
$632$	0	0
$633$	−37.0889	−0.0585922
$634$	0	0
$635$	−261.015	−0.411046
$636$	0	0
$637$	− 24.7935i	− 0.0389223i
$638$	0	0
$639$	213.186i	0.333625i
$640$	0	0
$641$	444.594	0.693595	0.346797	−	0.937940i	$-0.387269\pi$
$641$	444.594	0.693595	0.346797	+	0.937940i	$0.387269\pi$
$642$	0	0
$643$	1180.78	1.83636	0.918181	−	0.396162i	$-0.129658\pi$
$643$	1180.78	1.83636	0.918181	+	0.396162i	$0.129658\pi$
$644$	0	0
$645$	1059.73i	1.64299i
$646$	0	0
$647$	− 112.967i	− 0.174601i	−0.996182	−	0.0873005i	$-0.972176\pi$
$647$	− 112.967i	− 0.174601i	0.996182	−	0.0873005i	$-0.0278240\pi$
$648$	0	0
$649$	−644.842	−0.993593
$650$	0	0
$651$	−71.8419	−0.110356
$652$	0	0
$653$	− 364.690i	− 0.558484i	−0.960221	−	0.279242i	$-0.909917\pi$
$653$	− 364.690i	− 0.558484i	0.960221	−	0.279242i	$-0.0900831\pi$
$654$	0	0
$655$	− 952.368i	− 1.45400i
$656$	0	0
$657$	128.419	0.195463
$658$	0	0
$659$	−12.4844	−0.0189444	−0.00947220	−	0.999955i	$-0.503015\pi$
$659$	−12.4844	−0.0189444	−0.00947220	+	0.999955i	$0.503015\pi$
$660$	0	0
$661$	− 215.021i	− 0.325296i	−0.986684	−	0.162648i	$-0.947996\pi$
$661$	− 215.021i	− 0.325296i	0.986684	−	0.162648i	$-0.0520036\pi$
$662$	0	0
$663$	− 48.1034i	− 0.0725542i
$664$	0	0
$665$	219.964	0.330774
$666$	0	0
$667$	1165.99	1.74810
$668$	0	0
$669$	− 29.3412i	− 0.0438583i
$670$	0	0
$671$	1390.96i	2.07297i
$672$	0	0
$673$	−107.429	−0.159627	−0.0798135	−	0.996810i	$-0.525432\pi$
$673$	−107.429	−0.159627	−0.0798135	+	0.996810i	$0.525432\pi$
$674$	0	0
$675$	160.687	0.238055
$676$	0	0
$677$	916.328i	1.35351i	0.736207	+	0.676756i	$0.236615\pi$
$677$	916.328i	1.35351i	−0.736207	+	0.676756i	$0.763385\pi$
$678$	0	0
$679$	− 138.296i	− 0.203677i
$680$	0	0
$681$	−716.298	−1.05183
$682$	0	0
$683$	−497.699	−0.728696	−0.364348	−	0.931263i	$-0.618708\pi$
$683$	−497.699	−0.728696	−0.364348	+	0.931263i	$0.618708\pi$
$684$	0	0
$685$	− 1163.26i	− 1.69819i
$686$	0	0
$687$	62.4389i	0.0908863i
$688$	0	0
$689$	296.412	0.430206
$690$	0	0
$691$	−663.410	−0.960073	−0.480036	−	0.877249i	$-0.659377\pi$
$691$	−663.410	−0.960073	−0.480036	+	0.877249i	$0.659377\pi$
$692$	0	0
$693$	− 107.197i	− 0.154686i
$694$	0	0
$695$	380.518i	0.547508i
$696$	0	0
$697$	283.235	0.406362
$698$	0	0
$699$	−525.214	−0.751379
$700$	0	0
$701$	− 1295.94i	− 1.84871i	−0.381537	−	0.924354i	$-0.624605\pi$
$701$	− 1295.94i	− 1.84871i	0.381537	−	0.924354i	$-0.375395\pi$
$702$	0	0
$703$	246.333i	0.350402i
$704$	0	0
$705$	858.796	1.21815
$706$	0	0
$707$	−58.9201	−0.0833382
$708$	0	0
$709$	− 871.601i	− 1.22934i	−0.788785	−	0.614669i	$-0.789290\pi$
$709$	− 871.601i	− 1.22934i	0.788785	−	0.614669i	$-0.210710\pi$
$710$	0	0
$711$	31.1642i	0.0438314i
$712$	0	0
$713$	350.861	0.492092
$714$	0	0
$715$	−357.728	−0.500319
$716$	0	0
$717$	− 82.2190i	− 0.114671i
$718$	0	0
$719$	1027.81i	1.42950i	0.699378	+	0.714752i	$0.253460\pi$
$719$	1027.81i	1.42950i	−0.699378	+	0.714752i	$0.746540\pi$
$720$	0	0
$721$	352.599	0.489042
$722$	0	0
$723$	472.931	0.654123
$724$	0	0
$725$	− 1611.11i	− 2.22221i
$726$	0	0
$727$	− 460.819i	− 0.633864i	−0.948448	−	0.316932i	$-0.897347\pi$
$727$	− 460.819i	− 0.633864i	0.948448	−	0.316932i	$-0.102653\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	−641.517	−0.877589
$732$	0	0
$733$	− 1332.20i	− 1.81747i	−0.417376	−	0.908734i	$-0.637050\pi$
$733$	− 1332.20i	− 1.81747i	0.417376	−	0.908734i	$-0.362950\pi$
$734$	0	0
$735$	90.6690i	0.123359i
$736$	0	0
$737$	137.198	0.186157
$738$	0	0
$739$	22.2967	0.0301715	0.0150857	−	0.999886i	$-0.495198\pi$
$739$	22.2967	0.0301715	0.0150857	+	0.999886i	$0.495198\pi$
$740$	0	0
$741$	− 68.2031i	− 0.0920420i
$742$	0	0
$743$	− 1408.39i	− 1.89555i	−0.318941	−	0.947775i	$-0.603327\pi$
$743$	− 1408.39i	− 1.89555i	0.318941	−	0.947775i	$-0.396673\pi$
$744$	0	0
$745$	1892.55	2.54033
$746$	0	0
$747$	310.807	0.416074
$748$	0	0
$749$	416.578i	0.556179i
$750$	0	0
$751$	− 408.216i	− 0.543563i	−0.962359	−	0.271782i	$-0.912387\pi$
$751$	− 408.216i	− 0.543563i	0.962359	−	0.271782i	$-0.0876129\pi$
$752$	0	0
$753$	−665.285	−0.883512
$754$	0	0
$755$	2092.16	2.77107
$756$	0	0
$757$	364.759i	0.481849i	0.970544	+	0.240924i	$0.0774506\pi$
$757$	364.759i	0.481849i	−0.970544	+	0.240924i	$0.922549\pi$
$758$	0	0
$759$	523.529i	0.689762i
$760$	0	0
$761$	1245.82	1.63709	0.818543	−	0.574446i	$-0.194782\pi$
$761$	1245.82	1.63709	0.818543	+	0.574446i	$0.194782\pi$
$762$	0	0
$763$	355.476	0.465892
$764$	0	0
$765$	175.912i	0.229951i
$766$	0	0
$767$	− 169.114i	− 0.220488i
$768$	0	0
$769$	32.0583	0.0416883	0.0208442	−	0.999783i	$-0.493365\pi$
$769$	32.0583	0.0416883	0.0208442	+	0.999783i	$0.493365\pi$
$770$	0	0
$771$	−535.126	−0.694067
$772$	0	0
$773$	− 1257.99i	− 1.62742i	−0.581272	−	0.813709i	$-0.697445\pi$
$773$	− 1257.99i	− 1.62742i	0.581272	−	0.813709i	$-0.302555\pi$
$774$	0	0
$775$	− 484.805i	− 0.625554i
$776$	0	0
$777$	−101.538	−0.130680
$778$	0	0
$779$	401.582	0.515510
$780$	0	0
$781$	959.734i	1.22885i
$782$	0	0
$783$	− 270.712i	− 0.345737i
$784$	0	0
$785$	550.459	0.701221
$786$	0	0
$787$	−633.258	−0.804648	−0.402324	−	0.915497i	$-0.631797\pi$
$787$	−633.258	−0.804648	−0.402324	+	0.915497i	$0.631797\pi$
$788$	0	0
$789$	32.4731i	0.0411573i
$790$	0	0
$791$	− 475.463i	− 0.601091i
$792$	0	0
$793$	−364.789	−0.460012
$794$	0	0
$795$	−1083.97	−1.36348
$796$	0	0
$797$	− 552.494i	− 0.693218i	−0.938010	−	0.346609i	$-0.887333\pi$
$797$	− 552.494i	− 0.693218i	0.938010	−	0.346609i	$-0.112667\pi$
$798$	0	0
$799$	519.882i	0.650665i
$800$	0	0
$801$	−182.850	−0.228277
$802$	0	0
$803$	578.125	0.719957
$804$	0	0
$805$	− 442.809i	− 0.550073i
$806$	0	0
$807$	− 596.990i	− 0.739765i
$808$	0	0
$809$	−184.780	−0.228406	−0.114203	−	0.993457i	$-0.536431\pi$
$809$	−184.780	−0.228406	−0.114203	+	0.993457i	$0.536431\pi$
$810$	0	0
$811$	−781.297	−0.963375	−0.481688	−	0.876343i	$-0.659976\pi$
$811$	−781.297	−0.963375	−0.481688	+	0.876343i	$0.659976\pi$
$812$	0	0
$813$	93.8694i	0.115461i
$814$	0	0
$815$	− 1282.23i	− 1.57328i
$816$	0	0
$817$	−909.572	−1.11331
$818$	0	0
$819$	28.1132	0.0343263
$820$	0	0
$821$	− 1272.12i	− 1.54947i	−0.632283	−	0.774737i	$-0.717882\pi$
$821$	− 1272.12i	− 1.54947i	0.632283	−	0.774737i	$-0.282118\pi$
$822$	0	0
$823$	− 92.4404i	− 0.112321i	−0.998422	−	0.0561606i	$-0.982114\pi$
$823$	− 92.4404i	− 0.112321i	0.998422	−	0.0561606i	$-0.0178859\pi$
$824$	0	0
$825$	723.389	0.876835
$826$	0	0
$827$	−479.059	−0.579273	−0.289637	−	0.957137i	$-0.593534\pi$
$827$	−479.059	−0.579273	−0.289637	+	0.957137i	$0.593534\pi$
$828$	0	0
$829$	− 1205.72i	− 1.45443i	−0.686409	−	0.727215i	$-0.740814\pi$
$829$	− 1205.72i	− 1.45443i	0.686409	−	0.727215i	$-0.259186\pi$
$830$	0	0
$831$	− 125.588i	− 0.151129i
$832$	0	0
$833$	−54.8875	−0.0658913
$834$	0	0
$835$	1353.92	1.62147
$836$	0	0
$837$	− 81.4610i	− 0.0973250i
$838$	0	0
$839$	1231.72i	1.46808i	0.679107	+	0.734039i	$0.262367\pi$
$839$	1231.72i	1.46808i	−0.679107	+	0.734039i	$0.737633\pi$
$840$	0	0
$841$	−1873.26	−2.22742
$842$	0	0
$843$	9.32026	0.0110561
$844$	0	0
$845$	1170.01i	1.38462i
$846$	0	0
$847$	− 162.450i	− 0.191794i
$848$	0	0
$849$	−945.463	−1.11362
$850$	0	0
$851$	495.891	0.582716
$852$	0	0
$853$	− 1181.27i	− 1.38484i	−0.721493	−	0.692421i	$-0.756544\pi$
$853$	− 1181.27i	− 1.38484i	0.721493	−	0.692421i	$-0.243456\pi$
$854$	0	0
$855$	249.416i	0.291715i
$856$	0	0
$857$	−1499.85	−1.75012	−0.875060	−	0.484014i	$-0.839178\pi$
$857$	−1499.85	−1.75012	−0.875060	+	0.484014i	$0.839178\pi$
$858$	0	0
$859$	−789.796	−0.919437	−0.459718	−	0.888065i	$-0.652050\pi$
$859$	−789.796	−0.919437	−0.459718	+	0.888065i	$0.652050\pi$
$860$	0	0
$861$	165.532i	0.192255i
$862$	0	0
$863$	59.4710i	0.0689119i	0.999406	+	0.0344560i	$0.0109698\pi$
$863$	59.4710i	0.0689119i	−0.999406	+	0.0344560i	$0.989030\pi$
$864$	0	0
$865$	2157.04	2.49369
$866$	0	0
$867$	394.072	0.454524
$868$	0	0
$869$	140.297i	0.161446i
$870$	0	0
$871$	35.9812i	0.0413102i
$872$	0	0
$873$	156.813	0.179626
$874$	0	0
$875$	−117.214	−0.133959
$876$	0	0
$877$	514.929i	0.587148i	0.955936	+	0.293574i	$0.0948448\pi$
$877$	514.929i	0.587148i	−0.955936	+	0.293574i	$0.905155\pi$
$878$	0	0
$879$	737.752i	0.839309i
$880$	0	0
$881$	−1362.15	−1.54614	−0.773071	−	0.634319i	$-0.781281\pi$
$881$	−1362.15	−1.54614	−0.773071	+	0.634319i	$0.781281\pi$
$882$	0	0
$883$	−1150.48	−1.30292	−0.651459	−	0.758684i	$-0.725843\pi$
$883$	−1150.48	−1.30292	−0.651459	+	0.758684i	$0.725843\pi$
$884$	0	0
$885$	618.445i	0.698808i
$886$	0	0
$887$	− 321.470i	− 0.362424i	−0.983444	−	0.181212i	$-0.941998\pi$
$887$	− 321.470i	− 0.362424i	0.983444	−	0.181212i	$-0.0580021\pi$
$888$	0	0
$889$	92.3451	0.103875
$890$	0	0
$891$	121.550	0.136420
$892$	0	0
$893$	737.111i	0.825432i
$894$	0	0
$895$	248.314i	0.277446i
$896$	0	0
$897$	−137.299	−0.153065
$898$	0	0
$899$	−816.758	−0.908519
$900$	0	0
$901$	− 656.191i	− 0.728292i
$902$	0	0
$903$	− 374.924i	− 0.415198i
$904$	0	0
$905$	−1471.64	−1.62613
$906$	0	0
$907$	−116.706	−0.128673	−0.0643365	−	0.997928i	$-0.520493\pi$
$907$	−116.706	−0.128673	−0.0643365	+	0.997928i	$0.520493\pi$
$908$	0	0
$909$	− 66.8091i	− 0.0734973i
$910$	0	0
$911$	− 1122.30i	− 1.23195i	−0.787767	−	0.615973i	$-0.788763\pi$
$911$	− 1122.30i	− 1.23195i	0.787767	−	0.615973i	$-0.211237\pi$
$912$	0	0
$913$	1399.21	1.53254
$914$	0	0
$915$	1334.02	1.45795
$916$	0	0
$917$	336.941i	0.367438i
$918$	0	0
$919$	1476.99i	1.60717i	0.595187	+	0.803587i	$0.297078\pi$
$919$	1476.99i	1.60717i	−0.595187	+	0.803587i	$0.702922\pi$
$920$	0	0
$921$	769.769	0.835797
$922$	0	0
$923$	−251.697	−0.272695
$924$	0	0
$925$	− 685.200i	− 0.740757i
$926$	0	0
$927$	399.810i	0.431295i
$928$	0	0
$929$	−413.368	−0.444960	−0.222480	−	0.974937i	$-0.571415\pi$
$929$	−413.368	−0.444960	−0.222480	+	0.974937i	$0.571415\pi$
$930$	0	0
$931$	−77.8219	−0.0835895
$932$	0	0
$933$	− 30.3032i	− 0.0324793i
$934$	0	0
$935$	791.932i	0.846986i
$936$	0	0
$937$	−51.5540	−0.0550203	−0.0275102	−	0.999622i	$-0.508758\pi$
$937$	−51.5540	−0.0550203	−0.0275102	+	0.999622i	$0.508758\pi$
$938$	0	0
$939$	661.738	0.704726
$940$	0	0
$941$	247.999i	0.263548i	0.991280	+	0.131774i	$0.0420674\pi$
$941$	247.999i	0.263548i	−0.991280	+	0.131774i	$0.957933\pi$
$942$	0	0
$943$	− 808.423i	− 0.857289i
$944$	0	0
$945$	−102.809	−0.108793
$946$	0	0
$947$	−835.213	−0.881957	−0.440978	−	0.897518i	$-0.645368\pi$
$947$	−835.213	−0.881957	−0.440978	+	0.897518i	$0.645368\pi$
$948$	0	0
$949$	151.618i	0.159766i
$950$	0	0
$951$	− 676.677i	− 0.711542i
$952$	0	0
$953$	1791.85	1.88022	0.940112	−	0.340867i	$-0.110721\pi$
$953$	1791.85	1.88022	0.940112	+	0.340867i	$0.110721\pi$
$954$	0	0
$955$	927.270	0.970963
$956$	0	0
$957$	− 1218.71i	− 1.27346i
$958$	0	0
$959$	411.552i	0.429147i
$960$	0	0
$961$	715.226	0.744252
$962$	0	0
$963$	−472.355	−0.490503
$964$	0	0
$965$	513.776i	0.532411i
$966$	0	0
$967$	− 605.335i	− 0.625993i	−0.949754	−	0.312997i	$-0.898667\pi$
$967$	− 605.335i	− 0.625993i	0.949754	−	0.312997i	$-0.101333\pi$
$968$	0	0
$969$	−150.987	−0.155817
$970$	0	0
$971$	−113.638	−0.117032	−0.0585158	−	0.998286i	$-0.518637\pi$
$971$	−113.638	−0.117032	−0.0585158	+	0.998286i	$0.518637\pi$
$972$	0	0
$973$	− 134.625i	− 0.138360i
$974$	0	0
$975$	189.714i	0.194578i
$976$	0	0
$977$	551.794	0.564784	0.282392	−	0.959299i	$-0.408872\pi$
$977$	551.794	0.564784	0.282392	+	0.959299i	$0.408872\pi$
$978$	0	0
$979$	−823.162	−0.840819
$980$	0	0
$981$	403.072i	0.410878i
$982$	0	0
$983$	− 377.431i	− 0.383958i	−0.981399	−	0.191979i	$-0.938509\pi$
$983$	− 377.431i	− 0.383958i	0.981399	−	0.191979i	$-0.0614906\pi$
$984$	0	0
$985$	−1307.27	−1.32718
$986$	0	0
$987$	−303.836	−0.307838
$988$	0	0
$989$	1831.05i	1.85142i
$990$	0	0
$991$	179.491i	0.181121i	0.995891	+	0.0905607i	$0.0288659\pi$
$991$	179.491i	0.181121i	−0.995891	+	0.0905607i	$0.971134\pi$
$992$	0	0
$993$	−490.600	−0.494059
$994$	0	0
$995$	−984.055	−0.989000
$996$	0	0
$997$	1542.02i	1.54666i	0.634005	+	0.773329i	$0.281410\pi$
$997$	1542.02i	1.54666i	−0.634005	+	0.773329i	$0.718590\pi$
$998$	0	0
$999$	− 115.133i	− 0.115249i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.3.g.a.463.11		24
3.2	odd	2		2016.3.g.d.1135.3		24
4.3	odd	2		168.3.g.a.43.2	yes	24
8.3	odd	2	inner	672.3.g.a.463.2		24
8.5	even	2		168.3.g.a.43.1	✓	24
12.11	even	2		504.3.g.d.379.23		24
24.5	odd	2		504.3.g.d.379.24		24
24.11	even	2		2016.3.g.d.1135.22		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.g.a.43.1	✓	24	8.5	even	2
168.3.g.a.43.2	yes	24	4.3	odd	2
504.3.g.d.379.23		24	12.11	even	2
504.3.g.d.379.24		24	24.5	odd	2
672.3.g.a.463.2		24	8.3	odd	2	inner
672.3.g.a.463.11		24	1.1	even	1	trivial
2016.3.g.d.1135.3		24	3.2	odd	2
2016.3.g.d.1135.22		24	24.11	even	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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	672.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +7.47825i q^{5} -2.64575i q^{7} +3.00000 q^{9} +13.5056 q^{11} +3.54193i q^{13} -12.9527i q^{15} +7.84106 q^{17} +11.1174 q^{19} +4.58258i q^{21} -22.3804i q^{23} -30.9242 q^{25} -5.19615 q^{27} +52.0985i q^{29} +15.6772i q^{31} -23.3923 q^{33} +19.7856 q^{35} +22.1574i q^{37} -6.13481i q^{39} +36.1220 q^{41} -81.8151 q^{43} +22.4347i q^{45} +66.3024i q^{47} -7.00000 q^{49} -13.5811 q^{51} -83.6865i q^{53} +100.998i q^{55} -19.2559 q^{57} -47.7464 q^{59} +102.992i q^{61} -7.93725i q^{63} -26.4874 q^{65} +10.1586 q^{67} +38.7640i q^{69} +71.0621i q^{71} +42.8065 q^{73} +53.5623 q^{75} -35.7324i q^{77} +10.3881i q^{79} +9.00000 q^{81} +103.602 q^{83} +58.6374i q^{85} -90.2373i q^{87} -60.9498 q^{89} +9.37107 q^{91} -27.1537i q^{93} +83.1387i q^{95} +52.2712 q^{97} +40.5167 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 72 q^{9} - 32 q^{11} + 16 q^{17} + 64 q^{19} - 72 q^{25} - 80 q^{41} - 32 q^{43} - 168 q^{49} - 192 q^{51} - 192 q^{65} + 32 q^{67} - 240 q^{73} + 384 q^{75} + 216 q^{81} + 320 q^{83} + 400 q^{89}+ \cdots - 96 q^{99}+O(q^{100}) \) 24 * q + 72 * q^9 - 32 * q^11 + 16 * q^17 + 64 * q^19 - 72 * q^25 - 80 * q^41 - 32 * q^43 - 168 * q^49 - 192 * q^51 - 192 * q^65 + 32 * q^67 - 240 * q^73 + 384 * q^75 + 216 * q^81 + 320 * q^83 + 400 * q^89 + 144 * q^97 - 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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