LMFDB - Embedded newform 1680.3.n.c.1471.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,3,Mod(1471,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 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("1680.1471"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1680.n (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,0,0,48,0,0,0,-96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$45.7766844125$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7 x^{15} + 6 x^{14} + 115 x^{13} - 535 x^{12} + 605 x^{11} + 2453 x^{10} - 10997 x^{9} + \cdots + 92164 $ x^16 - 7x^15 + 6x^14 + 115x^13 - 535x^12 + 605x^11 + 2453x^10 - 10997x^9 + 23723x^8 - 49514x^7 + 98500x^6 - 121640x^5 + 87351x^4 - 81896x^3 + 155185x^2 - 191418*x + 92164
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{36} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.14
Root			$2.52830 + 0.173365i$ of defining polynomial
Character	$\chi$	$=$	1680.1471
Dual form			1680.3.n.c.1471.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +2.23607 q^{5} -2.64575i q^{7} -3.00000 q^{9} +8.79995i q^{11} -9.38422 q^{13} +3.87298i q^{15} +1.25490 q^{17} +17.2362i q^{19} +4.58258 q^{21} +4.11360i q^{23} +5.00000 q^{25} -5.19615i q^{27} +10.3954 q^{29} -44.7612i q^{31} -15.2420 q^{33} -5.91608i q^{35} -45.8730 q^{37} -16.2539i q^{39} -47.7876 q^{41} +14.7389i q^{43} -6.70820 q^{45} +32.1352i q^{47} -7.00000 q^{49} +2.17355i q^{51} -15.9421 q^{53} +19.6773i q^{55} -29.8539 q^{57} +34.8978i q^{59} -44.2374 q^{61} +7.93725i q^{63} -20.9837 q^{65} +93.7730i q^{67} -7.12496 q^{69} -2.89105i q^{71} -17.8725 q^{73} +8.66025i q^{75} +23.2825 q^{77} -43.7268i q^{79} +9.00000 q^{81} -34.8743i q^{83} +2.80604 q^{85} +18.0054i q^{87} +104.018 q^{89} +24.8283i q^{91} +77.5287 q^{93} +38.5413i q^{95} -110.722 q^{97} -26.3998i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} + 48 q^{13} - 96 q^{17} + 80 q^{25} + 112 q^{29} + 48 q^{33} - 224 q^{37} + 160 q^{41} - 112 q^{49} - 96 q^{53} + 80 q^{61} - 80 q^{65} + 48 q^{69} - 48 q^{73} + 144 q^{81} - 160 q^{89}+ \cdots + 592 q^{97}+O(q^{100}) $ 16 * q - 48 * q^9 + 48 * q^13 - 96 * q^17 + 80 * q^25 + 112 * q^29 + 48 * q^33 - 224 * q^37 + 160 * q^41 - 112 * q^49 - 96 * q^53 + 80 * q^61 - 80 * q^65 + 48 * q^69 - 48 * q^73 + 144 * q^81 - 160 * q^89 + 592 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	8.79995i	0.799995i	0.916516	+	0.399998i	$0.130989\pi$
$11$	8.79995i	0.799995i	−0.916516	+	0.399998i	$0.869011\pi$
$12$	0	0
$13$	−9.38422	−0.721863	−0.360931	−	0.932592i	$-0.617541\pi$
$13$	−9.38422	−0.721863	−0.360931	+	0.932592i	$0.617541\pi$
$14$	0	0
$15$	3.87298i	0.258199i
$16$	0	0
$17$	1.25490	0.0738176	0.0369088	−	0.999319i	$-0.488249\pi$
$17$	1.25490	0.0738176	0.0369088	+	0.999319i	$0.488249\pi$
$18$	0	0
$19$	17.2362i	0.907167i	0.891214	+	0.453583i	$0.149855\pi$
$19$	17.2362i	0.907167i	−0.891214	+	0.453583i	$0.850145\pi$
$20$	0	0
$21$	4.58258	0.218218
$22$	0	0
$23$	4.11360i	0.178852i	0.995993	+	0.0894260i	$0.0285033\pi$
$23$	4.11360i	0.178852i	−0.995993	+	0.0894260i	$0.971497\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	10.3954	0.358463	0.179231	−	0.983807i	$-0.442639\pi$
$29$	10.3954	0.358463	0.179231	+	0.983807i	$0.442639\pi$
$30$	0	0
$31$	− 44.7612i	− 1.44391i	−0.691940	−	0.721955i	$-0.743244\pi$
$31$	− 44.7612i	− 1.44391i	0.691940	−	0.721955i	$-0.256756\pi$
$32$	0	0
$33$	−15.2420	−0.461877
$34$	0	0
$35$	− 5.91608i	− 0.169031i
$36$	0	0
$37$	−45.8730	−1.23981	−0.619905	−	0.784677i	$-0.712829\pi$
$37$	−45.8730	−1.23981	−0.619905	+	0.784677i	$0.712829\pi$
$38$	0	0
$39$	− 16.2539i	− 0.416768i
$40$	0	0
$41$	−47.7876	−1.16555	−0.582776	−	0.812633i	$-0.698033\pi$
$41$	−47.7876	−1.16555	−0.582776	+	0.812633i	$0.698033\pi$
$42$	0	0
$43$	14.7389i	0.342765i	0.985205	+	0.171382i	$0.0548234\pi$
$43$	14.7389i	0.342765i	−0.985205	+	0.171382i	$0.945177\pi$
$44$	0	0
$45$	−6.70820	−0.149071
$46$	0	0
$47$	32.1352i	0.683727i	0.939750	+	0.341863i	$0.111058\pi$
$47$	32.1352i	0.683727i	−0.939750	+	0.341863i	$0.888942\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	2.17355i	0.0426186i
$52$	0	0
$53$	−15.9421	−0.300794	−0.150397	−	0.988626i	$-0.548055\pi$
$53$	−15.9421	−0.300794	−0.150397	+	0.988626i	$0.548055\pi$
$54$	0	0
$55$	19.6773i	0.357769i
$56$	0	0
$57$	−29.8539	−0.523753
$58$	0	0
$59$	34.8978i	0.591489i	0.955267	+	0.295744i	$0.0955677\pi$
$59$	34.8978i	0.591489i	−0.955267	+	0.295744i	$0.904432\pi$
$60$	0	0
$61$	−44.2374	−0.725204	−0.362602	−	0.931944i	$-0.618112\pi$
$61$	−44.2374	−0.725204	−0.362602	+	0.931944i	$0.618112\pi$
$62$	0	0
$63$	7.93725i	0.125988i
$64$	0	0
$65$	−20.9837	−0.322827
$66$	0	0
$67$	93.7730i	1.39960i	0.714340	+	0.699798i	$0.246727\pi$
$67$	93.7730i	1.39960i	−0.714340	+	0.699798i	$0.753273\pi$
$68$	0	0
$69$	−7.12496	−0.103260
$70$	0	0
$71$	− 2.89105i	− 0.0407191i	−0.999793	−	0.0203595i	$-0.993519\pi$
$71$	− 2.89105i	− 0.0407191i	0.999793	−	0.0203595i	$-0.00648109\pi$
$72$	0	0
$73$	−17.8725	−0.244829	−0.122414	−	0.992479i	$-0.539064\pi$
$73$	−17.8725	−0.244829	−0.122414	+	0.992479i	$0.539064\pi$
$74$	0	0
$75$	8.66025i	0.115470i
$76$	0	0
$77$	23.2825	0.302370
$78$	0	0
$79$	− 43.7268i	− 0.553504i	−0.960941	−	0.276752i	$-0.910742\pi$
$79$	− 43.7268i	− 0.553504i	0.960941	−	0.276752i	$-0.0892581\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 34.8743i	− 0.420173i	−0.977683	−	0.210086i	$-0.932626\pi$
$83$	− 34.8743i	− 0.420173i	0.977683	−	0.210086i	$-0.0673745\pi$
$84$	0	0
$85$	2.80604	0.0330122
$86$	0	0
$87$	18.0054i	0.206959i
$88$	0	0
$89$	104.018	1.16875	0.584373	−	0.811485i	$-0.301340\pi$
$89$	104.018	1.16875	0.584373	+	0.811485i	$0.301340\pi$
$90$	0	0
$91$	24.8283i	0.272839i
$92$	0	0
$93$	77.5287	0.833641
$94$	0	0
$95$	38.5413i	0.405697i
$96$	0	0
$97$	−110.722	−1.14146	−0.570731	−	0.821137i	$-0.693340\pi$
$97$	−110.722	−1.14146	−0.570731	+	0.821137i	$0.693340\pi$
$98$	0	0
$99$	− 26.3998i	− 0.266665i
$100$	0	0
$101$	−128.652	−1.27379	−0.636893	−	0.770952i	$-0.719781\pi$
$101$	−128.652	−1.27379	−0.636893	+	0.770952i	$0.719781\pi$
$102$	0	0
$103$	− 9.30677i	− 0.0903570i	−0.998979	−	0.0451785i	$-0.985614\pi$
$103$	− 9.30677i	− 0.0903570i	0.998979	−	0.0451785i	$-0.0143857\pi$
$104$	0	0
$105$	10.2470	0.0975900
$106$	0	0
$107$	− 1.12107i	− 0.0104773i	−0.999986	−	0.00523867i	$-0.998332\pi$
$107$	− 1.12107i	− 0.0104773i	0.999986	−	0.00523867i	$-0.00166753\pi$
$108$	0	0
$109$	−154.428	−1.41677	−0.708384	−	0.705827i	$-0.750575\pi$
$109$	−154.428	−1.41677	−0.708384	+	0.705827i	$0.750575\pi$
$110$	0	0
$111$	− 79.4543i	− 0.715805i
$112$	0	0
$113$	−118.217	−1.04617	−0.523085	−	0.852280i	$-0.675219\pi$
$113$	−118.217	−1.04617	−0.523085	+	0.852280i	$0.675219\pi$
$114$	0	0
$115$	9.19828i	0.0799851i
$116$	0	0
$117$	28.1527	0.240621
$118$	0	0
$119$	− 3.32015i	− 0.0279004i
$120$	0	0
$121$	43.5609	0.360008
$122$	0	0
$123$	− 82.7706i	− 0.672932i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	62.0975i	0.488957i	0.969655	+	0.244478i	$0.0786167\pi$
$127$	62.0975i	0.488957i	−0.969655	+	0.244478i	$0.921383\pi$
$128$	0	0
$129$	−25.5285	−0.197895
$130$	0	0
$131$	54.4183i	0.415407i	0.978192	+	0.207704i	$0.0665989\pi$
$131$	54.4183i	0.415407i	−0.978192	+	0.207704i	$0.933401\pi$
$132$	0	0
$133$	45.6026	0.342877
$134$	0	0
$135$	− 11.6190i	− 0.0860663i
$136$	0	0
$137$	−119.071	−0.869131	−0.434565	−	0.900640i	$-0.643098\pi$
$137$	−119.071	−0.869131	−0.434565	+	0.900640i	$0.643098\pi$
$138$	0	0
$139$	197.091i	1.41792i	0.705250	+	0.708959i	$0.250835\pi$
$139$	197.091i	1.41792i	−0.705250	+	0.708959i	$0.749165\pi$
$140$	0	0
$141$	−55.6597	−0.394750
$142$	0	0
$143$	− 82.5806i	− 0.577487i
$144$	0	0
$145$	23.2449	0.160309
$146$	0	0
$147$	− 12.1244i	− 0.0824786i
$148$	0	0
$149$	−48.8826	−0.328071	−0.164036	−	0.986454i	$-0.552451\pi$
$149$	−48.8826	−0.328071	−0.164036	+	0.986454i	$0.552451\pi$
$150$	0	0
$151$	1.21455i	0.00804336i	0.999992	+	0.00402168i	$0.00128014\pi$
$151$	1.21455i	0.00804336i	−0.999992	+	0.00402168i	$0.998720\pi$
$152$	0	0
$153$	−3.76470	−0.0246059
$154$	0	0
$155$	− 100.089i	− 0.645736i
$156$	0	0
$157$	−102.408	−0.652281	−0.326140	−	0.945321i	$-0.605748\pi$
$157$	−102.408	−0.652281	−0.326140	+	0.945321i	$0.605748\pi$
$158$	0	0
$159$	− 27.6125i	− 0.173664i
$160$	0	0
$161$	10.8836	0.0675997
$162$	0	0
$163$	− 135.591i	− 0.831849i	−0.909399	−	0.415925i	$-0.863458\pi$
$163$	− 135.591i	− 0.831849i	0.909399	−	0.415925i	$-0.136542\pi$
$164$	0	0
$165$	−34.0820	−0.206558
$166$	0	0
$167$	− 208.183i	− 1.24661i	−0.781981	−	0.623303i	$-0.785790\pi$
$167$	− 208.183i	− 1.24661i	0.781981	−	0.623303i	$-0.214210\pi$
$168$	0	0
$169$	−80.9365	−0.478914
$170$	0	0
$171$	− 51.7085i	− 0.302389i
$172$	0	0
$173$	−188.537	−1.08981	−0.544904	−	0.838499i	$-0.683434\pi$
$173$	−188.537	−1.08981	−0.544904	+	0.838499i	$0.683434\pi$
$174$	0	0
$175$	− 13.2288i	− 0.0755929i
$176$	0	0
$177$	−60.4448	−0.341496
$178$	0	0
$179$	79.3279i	0.443172i	0.975141	+	0.221586i	$0.0711234\pi$
$179$	79.3279i	0.443172i	−0.975141	+	0.221586i	$0.928877\pi$
$180$	0	0
$181$	157.255	0.868814	0.434407	−	0.900717i	$-0.356958\pi$
$181$	157.255	0.868814	0.434407	+	0.900717i	$0.356958\pi$
$182$	0	0
$183$	− 76.6215i	− 0.418697i
$184$	0	0
$185$	−102.575	−0.554460
$186$	0	0
$187$	11.0430i	0.0590537i
$188$	0	0
$189$	−13.7477	−0.0727393
$190$	0	0
$191$	− 207.530i	− 1.08654i	−0.839557	−	0.543271i	$-0.817186\pi$
$191$	− 207.530i	− 1.08654i	0.839557	−	0.543271i	$-0.182814\pi$
$192$	0	0
$193$	−25.2788	−0.130978	−0.0654891	−	0.997853i	$-0.520861\pi$
$193$	−25.2788	−0.130978	−0.0654891	+	0.997853i	$0.520861\pi$
$194$	0	0
$195$	− 36.3449i	− 0.186384i
$196$	0	0
$197$	−198.866	−1.00947	−0.504735	−	0.863274i	$-0.668410\pi$
$197$	−198.866	−1.00947	−0.504735	+	0.863274i	$0.668410\pi$
$198$	0	0
$199$	75.2401i	0.378091i	0.981968	+	0.189045i	$0.0605393\pi$
$199$	75.2401i	0.378091i	−0.981968	+	0.189045i	$0.939461\pi$
$200$	0	0
$201$	−162.420	−0.808058
$202$	0	0
$203$	− 27.5037i	− 0.135486i
$204$	0	0
$205$	−106.856	−0.521251
$206$	0	0
$207$	− 12.3408i	− 0.0596174i
$208$	0	0
$209$	−151.677	−0.725729
$210$	0	0
$211$	76.3853i	0.362016i	0.983482	+	0.181008i	$0.0579359\pi$
$211$	76.3853i	0.362016i	−0.983482	+	0.181008i	$0.942064\pi$
$212$	0	0
$213$	5.00745	0.0235092
$214$	0	0
$215$	32.9572i	0.153289i
$216$	0	0
$217$	−118.427	−0.545746
$218$	0	0
$219$	− 30.9561i	− 0.141352i
$220$	0	0
$221$	−11.7762	−0.0532861
$222$	0	0
$223$	112.706i	0.505409i	0.967544	+	0.252704i	$0.0813200\pi$
$223$	112.706i	0.505409i	−0.967544	+	0.252704i	$0.918680\pi$
$224$	0	0
$225$	−15.0000	−0.0666667
$226$	0	0
$227$	− 363.020i	− 1.59921i	−0.600527	−	0.799604i	$-0.705043\pi$
$227$	− 363.020i	− 1.59921i	0.600527	−	0.799604i	$-0.294957\pi$
$228$	0	0
$229$	331.342	1.44691	0.723453	−	0.690374i	$-0.242554\pi$
$229$	331.342	1.44691	0.723453	+	0.690374i	$0.242554\pi$
$230$	0	0
$231$	40.3264i	0.174573i
$232$	0	0
$233$	217.379	0.932957	0.466478	−	0.884533i	$-0.345523\pi$
$233$	217.379	0.932957	0.466478	+	0.884533i	$0.345523\pi$
$234$	0	0
$235$	71.8564i	0.305772i
$236$	0	0
$237$	75.7371	0.319566
$238$	0	0
$239$	240.882i	1.00787i	0.863740	+	0.503937i	$0.168116\pi$
$239$	240.882i	1.00787i	−0.863740	+	0.503937i	$0.831884\pi$
$240$	0	0
$241$	−179.376	−0.744297	−0.372148	−	0.928173i	$-0.621379\pi$
$241$	−179.376	−0.744297	−0.372148	+	0.928173i	$0.621379\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−15.6525	−0.0638877
$246$	0	0
$247$	− 161.748i	− 0.654850i
$248$	0	0
$249$	60.4041	0.242587
$250$	0	0
$251$	− 111.351i	− 0.443629i	−0.975089	−	0.221814i	$-0.928802\pi$
$251$	− 111.351i	− 0.443629i	0.975089	−	0.221814i	$-0.0711979\pi$
$252$	0	0
$253$	−36.1994	−0.143081
$254$	0	0
$255$	4.86020i	0.0190596i
$256$	0	0
$257$	−341.799	−1.32996	−0.664978	−	0.746863i	$-0.731559\pi$
$257$	−341.799	−1.32996	−0.664978	+	0.746863i	$0.731559\pi$
$258$	0	0
$259$	121.368i	0.468604i
$260$	0	0
$261$	−31.1863	−0.119488
$262$	0	0
$263$	92.5794i	0.352013i	0.984389	+	0.176006i	$0.0563180\pi$
$263$	92.5794i	0.352013i	−0.984389	+	0.176006i	$0.943682\pi$
$264$	0	0
$265$	−35.6476	−0.134519
$266$	0	0
$267$	180.165i	0.674776i
$268$	0	0
$269$	168.803	0.627522	0.313761	−	0.949502i	$-0.398411\pi$
$269$	168.803	0.627522	0.313761	+	0.949502i	$0.398411\pi$
$270$	0	0
$271$	65.5407i	0.241847i	0.992662	+	0.120924i	$0.0385857\pi$
$271$	65.5407i	0.241847i	−0.992662	+	0.120924i	$0.961414\pi$
$272$	0	0
$273$	−43.0039	−0.157523
$274$	0	0
$275$	43.9997i	0.159999i
$276$	0	0
$277$	−124.917	−0.450964	−0.225482	−	0.974247i	$-0.572396\pi$
$277$	−124.917	−0.450964	−0.225482	+	0.974247i	$0.572396\pi$
$278$	0	0
$279$	134.284i	0.481303i
$280$	0	0
$281$	135.699	0.482913	0.241457	−	0.970412i	$-0.422375\pi$
$281$	135.699	0.482913	0.241457	+	0.970412i	$0.422375\pi$
$282$	0	0
$283$	198.538i	0.701549i	0.936460	+	0.350775i	$0.114082\pi$
$283$	198.538i	0.701549i	−0.936460	+	0.350775i	$0.885918\pi$
$284$	0	0
$285$	−66.7554	−0.234230
$286$	0	0
$287$	126.434i	0.440537i
$288$	0	0
$289$	−287.425	−0.994551
$290$	0	0
$291$	− 191.776i	− 0.659024i
$292$	0	0
$293$	79.6878	0.271972	0.135986	−	0.990711i	$-0.456580\pi$
$293$	79.6878	0.271972	0.135986	+	0.990711i	$0.456580\pi$
$294$	0	0
$295$	78.0339i	0.264522i
$296$	0	0
$297$	45.7259	0.153959
$298$	0	0
$299$	− 38.6029i	− 0.129107i
$300$	0	0
$301$	38.9954	0.129553
$302$	0	0
$303$	− 222.833i	− 0.735421i
$304$	0	0
$305$	−98.9179	−0.324321
$306$	0	0
$307$	322.160i	1.04938i	0.851293	+	0.524690i	$0.175819\pi$
$307$	322.160i	1.04938i	−0.851293	+	0.524690i	$0.824181\pi$
$308$	0	0
$309$	16.1198	0.0521676
$310$	0	0
$311$	− 168.522i	− 0.541870i	−0.962598	−	0.270935i	$-0.912667\pi$
$311$	− 168.522i	− 0.541870i	0.962598	−	0.270935i	$-0.0873328\pi$
$312$	0	0
$313$	2.78345	0.00889282	0.00444641	−	0.999990i	$-0.498585\pi$
$313$	2.78345	0.00889282	0.00444641	+	0.999990i	$0.498585\pi$
$314$	0	0
$315$	17.7482i	0.0563436i
$316$	0	0
$317$	304.673	0.961115	0.480557	−	0.876963i	$-0.340434\pi$
$317$	304.673	0.961115	0.480557	+	0.876963i	$0.340434\pi$
$318$	0	0
$319$	91.4792i	0.286769i
$320$	0	0
$321$	1.94176	0.00604909
$322$	0	0
$323$	21.6296i	0.0669648i
$324$	0	0
$325$	−46.9211	−0.144373
$326$	0	0
$327$	− 267.477i	− 0.817971i
$328$	0	0
$329$	85.0216	0.258424
$330$	0	0
$331$	432.011i	1.30517i	0.757716	+	0.652584i	$0.226315\pi$
$331$	432.011i	1.30517i	−0.757716	+	0.652584i	$0.773685\pi$
$332$	0	0
$333$	137.619	0.413270
$334$	0	0
$335$	209.683i	0.625919i
$336$	0	0
$337$	−594.368	−1.76370	−0.881851	−	0.471528i	$-0.843703\pi$
$337$	−594.368	−1.76370	−0.881851	+	0.471528i	$0.843703\pi$
$338$	0	0
$339$	− 204.758i	− 0.604007i
$340$	0	0
$341$	393.896	1.15512
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	−15.9319	−0.0461794
$346$	0	0
$347$	− 216.608i	− 0.624231i	−0.950044	−	0.312115i	$-0.898962\pi$
$347$	− 216.608i	− 0.624231i	0.950044	−	0.312115i	$-0.101038\pi$
$348$	0	0
$349$	454.046	1.30099	0.650496	−	0.759510i	$-0.274561\pi$
$349$	454.046	1.30099	0.650496	+	0.759510i	$0.274561\pi$
$350$	0	0
$351$	48.7618i	0.138923i
$352$	0	0
$353$	8.40161	0.0238006	0.0119003	−	0.999929i	$-0.496212\pi$
$353$	8.40161	0.0238006	0.0119003	+	0.999929i	$0.496212\pi$
$354$	0	0
$355$	− 6.46459i	− 0.0182101i
$356$	0	0
$357$	5.75067	0.0161083
$358$	0	0
$359$	− 173.200i	− 0.482451i	−0.970469	−	0.241226i	$-0.922451\pi$
$359$	− 173.200i	− 0.482451i	0.970469	−	0.241226i	$-0.0775494\pi$
$360$	0	0
$361$	63.9143	0.177048
$362$	0	0
$363$	75.4498i	0.207851i
$364$	0	0
$365$	−39.9641	−0.109491
$366$	0	0
$367$	380.268i	1.03615i	0.855335	+	0.518076i	$0.173352\pi$
$367$	380.268i	1.03615i	−0.855335	+	0.518076i	$0.826648\pi$
$368$	0	0
$369$	143.363	0.388517
$370$	0	0
$371$	42.1788i	0.113690i
$372$	0	0
$373$	215.367	0.577391	0.288696	−	0.957421i	$-0.406778\pi$
$373$	215.367	0.577391	0.288696	+	0.957421i	$0.406778\pi$
$374$	0	0
$375$	19.3649i	0.0516398i
$376$	0	0
$377$	−97.5529	−0.258761
$378$	0	0
$379$	322.776i	0.851652i	0.904805	+	0.425826i	$0.140016\pi$
$379$	322.776i	0.851652i	−0.904805	+	0.425826i	$0.859984\pi$
$380$	0	0
$381$	−107.556	−0.282299
$382$	0	0
$383$	− 637.804i	− 1.66528i	−0.553811	−	0.832642i	$-0.686827\pi$
$383$	− 637.804i	− 1.66528i	0.553811	−	0.832642i	$-0.313173\pi$
$384$	0	0
$385$	52.0612	0.135224
$386$	0	0
$387$	− 44.2167i	− 0.114255i
$388$	0	0
$389$	581.065	1.49374	0.746870	−	0.664970i	$-0.231556\pi$
$389$	581.065	1.49374	0.746870	+	0.664970i	$0.231556\pi$
$390$	0	0
$391$	5.16215i	0.0132024i
$392$	0	0
$393$	−94.2553	−0.239835
$394$	0	0
$395$	− 97.7762i	− 0.247535i
$396$	0	0
$397$	370.929	0.934329	0.467165	−	0.884170i	$-0.345276\pi$
$397$	370.929	0.934329	0.467165	+	0.884170i	$0.345276\pi$
$398$	0	0
$399$	78.9861i	0.197960i
$400$	0	0
$401$	−75.4668	−0.188197	−0.0940983	−	0.995563i	$-0.529997\pi$
$401$	−75.4668	−0.188197	−0.0940983	+	0.995563i	$0.529997\pi$
$402$	0	0
$403$	420.049i	1.04230i
$404$	0	0
$405$	20.1246	0.0496904
$406$	0	0
$407$	− 403.680i	− 0.991842i
$408$	0	0
$409$	598.394	1.46307	0.731533	−	0.681806i	$-0.238805\pi$
$409$	598.394	1.46307	0.731533	+	0.681806i	$0.238805\pi$
$410$	0	0
$411$	− 206.237i	− 0.501793i
$412$	0	0
$413$	92.3310	0.223562
$414$	0	0
$415$	− 77.9813i	− 0.187907i
$416$	0	0
$417$	−341.371	−0.818635
$418$	0	0
$419$	− 422.299i	− 1.00787i	−0.863740	−	0.503937i	$-0.831884\pi$
$419$	− 422.299i	− 1.00787i	0.863740	−	0.503937i	$-0.168116\pi$
$420$	0	0
$421$	−42.6375	−0.101277	−0.0506383	−	0.998717i	$-0.516126\pi$
$421$	−42.6375	−0.101277	−0.0506383	+	0.998717i	$0.516126\pi$
$422$	0	0
$423$	− 96.4055i	− 0.227909i
$424$	0	0
$425$	6.27449	0.0147635
$426$	0	0
$427$	117.041i	0.274101i
$428$	0	0
$429$	143.034	0.333412
$430$	0	0
$431$	796.932i	1.84903i	0.381145	+	0.924515i	$0.375530\pi$
$431$	796.932i	1.84903i	−0.381145	+	0.924515i	$0.624470\pi$
$432$	0	0
$433$	489.393	1.13024	0.565119	−	0.825009i	$-0.308830\pi$
$433$	489.393	1.13024	0.565119	+	0.825009i	$0.308830\pi$
$434$	0	0
$435$	40.2613i	0.0925547i
$436$	0	0
$437$	−70.9027	−0.162249
$438$	0	0
$439$	514.652i	1.17233i	0.810193	+	0.586164i	$0.199363\pi$
$439$	514.652i	1.17233i	−0.810193	+	0.586164i	$0.800637\pi$
$440$	0	0
$441$	21.0000	0.0476190
$442$	0	0
$443$	− 450.951i	− 1.01795i	−0.860782	−	0.508974i	$-0.830025\pi$
$443$	− 450.951i	− 1.01795i	0.860782	−	0.508974i	$-0.169975\pi$
$444$	0	0
$445$	232.592	0.522679
$446$	0	0
$447$	− 84.6671i	− 0.189412i
$448$	0	0
$449$	−255.396	−0.568812	−0.284406	−	0.958704i	$-0.591796\pi$
$449$	−255.396	−0.568812	−0.284406	+	0.958704i	$0.591796\pi$
$450$	0	0
$451$	− 420.528i	− 0.932436i
$452$	0	0
$453$	−2.10366	−0.00464384
$454$	0	0
$455$	55.5178i	0.122017i
$456$	0	0
$457$	−354.776	−0.776314	−0.388157	−	0.921593i	$-0.626888\pi$
$457$	−354.776	−0.776314	−0.388157	+	0.921593i	$0.626888\pi$
$458$	0	0
$459$	− 6.52064i	− 0.0142062i
$460$	0	0
$461$	771.084	1.67263	0.836316	−	0.548247i	$-0.184705\pi$
$461$	771.084	1.67263	0.836316	+	0.548247i	$0.184705\pi$
$462$	0	0
$463$	167.817i	0.362455i	0.983441	+	0.181227i	$0.0580070\pi$
$463$	167.817i	0.362455i	−0.983441	+	0.181227i	$0.941993\pi$
$464$	0	0
$465$	173.359	0.372816
$466$	0	0
$467$	91.5618i	0.196064i	0.995183	+	0.0980319i	$0.0312547\pi$
$467$	91.5618i	0.196064i	−0.995183	+	0.0980319i	$0.968745\pi$
$468$	0	0
$469$	248.100	0.528998
$470$	0	0
$471$	− 177.376i	− 0.376594i
$472$	0	0
$473$	−129.701	−0.274210
$474$	0	0
$475$	86.1809i	0.181433i
$476$	0	0
$477$	47.8263	0.100265
$478$	0	0
$479$	− 457.015i	− 0.954102i	−0.878876	−	0.477051i	$-0.841706\pi$
$479$	− 457.015i	− 0.954102i	0.878876	−	0.477051i	$-0.158294\pi$
$480$	0	0
$481$	430.482	0.894973
$482$	0	0
$483$	18.8509i	0.0390287i
$484$	0	0
$485$	−247.582	−0.510478
$486$	0	0
$487$	144.211i	0.296122i	0.988978	+	0.148061i	$0.0473032\pi$
$487$	144.211i	0.296122i	−0.988978	+	0.148061i	$0.952697\pi$
$488$	0	0
$489$	234.851	0.480268
$490$	0	0
$491$	285.763i	0.582002i	0.956723	+	0.291001i	$0.0939883\pi$
$491$	285.763i	0.582002i	−0.956723	+	0.291001i	$0.906012\pi$
$492$	0	0
$493$	13.0452	0.0264608
$494$	0	0
$495$	− 59.0318i	− 0.119256i
$496$	0	0
$497$	−7.64901	−0.0153904
$498$	0	0
$499$	576.622i	1.15556i	0.816194	+	0.577778i	$0.196080\pi$
$499$	576.622i	1.15556i	−0.816194	+	0.577778i	$0.803920\pi$
$500$	0	0
$501$	360.584	0.719728
$502$	0	0
$503$	574.430i	1.14201i	0.820948	+	0.571004i	$0.193446\pi$
$503$	574.430i	1.14201i	−0.820948	+	0.571004i	$0.806554\pi$
$504$	0	0
$505$	−287.676	−0.569655
$506$	0	0
$507$	− 140.186i	− 0.276501i
$508$	0	0
$509$	525.130	1.03169	0.515845	−	0.856682i	$-0.327478\pi$
$509$	525.130	1.03169	0.515845	+	0.856682i	$0.327478\pi$
$510$	0	0
$511$	47.2862i	0.0925365i
$512$	0	0
$513$	89.5618	0.174584
$514$	0	0
$515$	− 20.8106i	− 0.0404089i
$516$	0	0
$517$	−282.788	−0.546978
$518$	0	0
$519$	− 326.555i	− 0.629201i
$520$	0	0
$521$	−403.662	−0.774783	−0.387392	−	0.921915i	$-0.626624\pi$
$521$	−403.662	−0.774783	−0.387392	+	0.921915i	$0.626624\pi$
$522$	0	0
$523$	65.2733i	0.124806i	0.998051	+	0.0624028i	$0.0198763\pi$
$523$	65.2733i	0.124806i	−0.998051	+	0.0624028i	$0.980124\pi$
$524$	0	0
$525$	22.9129	0.0436436
$526$	0	0
$527$	− 56.1707i	− 0.106586i
$528$	0	0
$529$	512.078	0.968012
$530$	0	0
$531$	− 104.694i	− 0.197163i
$532$	0	0
$533$	448.449	0.841368
$534$	0	0
$535$	− 2.50680i	− 0.00468561i
$536$	0	0
$537$	−137.400	−0.255866
$538$	0	0
$539$	− 61.5996i	− 0.114285i
$540$	0	0
$541$	−41.6239	−0.0769388	−0.0384694	−	0.999260i	$-0.512248\pi$
$541$	−41.6239	−0.0769388	−0.0384694	+	0.999260i	$0.512248\pi$
$542$	0	0
$543$	272.374i	0.501610i
$544$	0	0
$545$	−345.311	−0.633598
$546$	0	0
$547$	932.793i	1.70529i	0.522492	+	0.852644i	$0.325002\pi$
$547$	932.793i	1.70529i	−0.522492	+	0.852644i	$0.674998\pi$
$548$	0	0
$549$	132.712	0.241735
$550$	0	0
$551$	179.177i	0.325186i
$552$	0	0
$553$	−115.690	−0.209205
$554$	0	0
$555$	− 177.665i	− 0.320118i
$556$	0	0
$557$	−439.314	−0.788715	−0.394358	−	0.918957i	$-0.629033\pi$
$557$	−439.314	−0.788715	−0.394358	+	0.918957i	$0.629033\pi$
$558$	0	0
$559$	− 138.313i	− 0.247429i
$560$	0	0
$561$	−19.1271	−0.0340947
$562$	0	0
$563$	− 954.938i	− 1.69616i	−0.529869	−	0.848080i	$-0.677759\pi$
$563$	− 954.938i	− 1.69616i	0.529869	−	0.848080i	$-0.322241\pi$
$564$	0	0
$565$	−264.342	−0.467862
$566$	0	0
$567$	− 23.8118i	− 0.0419961i
$568$	0	0
$569$	930.534	1.63539	0.817693	−	0.575655i	$-0.195253\pi$
$569$	930.534	1.63539	0.817693	+	0.575655i	$0.195253\pi$
$570$	0	0
$571$	194.324i	0.340322i	0.985416	+	0.170161i	$0.0544288\pi$
$571$	194.324i	0.340322i	−0.985416	+	0.170161i	$0.945571\pi$
$572$	0	0
$573$	359.452	0.627316
$574$	0	0
$575$	20.5680i	0.0357704i
$576$	0	0
$577$	−276.445	−0.479107	−0.239554	−	0.970883i	$-0.577001\pi$
$577$	−276.445	−0.479107	−0.239554	+	0.970883i	$0.577001\pi$
$578$	0	0
$579$	− 43.7841i	− 0.0756203i
$580$	0	0
$581$	−92.2688	−0.158810
$582$	0	0
$583$	− 140.290i	− 0.240634i
$584$	0	0
$585$	62.9512	0.107609
$586$	0	0
$587$	518.590i	0.883458i	0.897149	+	0.441729i	$0.145635\pi$
$587$	518.590i	0.883458i	−0.897149	+	0.441729i	$0.854365\pi$
$588$	0	0
$589$	771.512	1.30987
$590$	0	0
$591$	− 344.445i	− 0.582818i
$592$	0	0
$593$	427.203	0.720410	0.360205	−	0.932873i	$-0.382707\pi$
$593$	427.203	0.720410	0.360205	+	0.932873i	$0.382707\pi$
$594$	0	0
$595$	− 7.42408i	− 0.0124774i
$596$	0	0
$597$	−130.320	−0.218291
$598$	0	0
$599$	217.842i	0.363676i	0.983329	+	0.181838i	$0.0582046\pi$
$599$	217.842i	0.363676i	−0.983329	+	0.181838i	$0.941795\pi$
$600$	0	0
$601$	−402.101	−0.669053	−0.334527	−	0.942386i	$-0.608576\pi$
$601$	−402.101	−0.669053	−0.334527	+	0.942386i	$0.608576\pi$
$602$	0	0
$603$	− 281.319i	− 0.466532i
$604$	0	0
$605$	97.4052	0.161000
$606$	0	0
$607$	− 467.555i	− 0.770272i	−0.922860	−	0.385136i	$-0.874155\pi$
$607$	− 467.555i	− 0.770272i	0.922860	−	0.385136i	$-0.125845\pi$
$608$	0	0
$609$	47.6378	0.0782230
$610$	0	0
$611$	− 301.563i	− 0.493557i
$612$	0	0
$613$	483.990	0.789543	0.394772	−	0.918779i	$-0.370824\pi$
$613$	483.990	0.789543	0.394772	+	0.918779i	$0.370824\pi$
$614$	0	0
$615$	− 185.081i	− 0.300944i
$616$	0	0
$617$	−36.8122	−0.0596632	−0.0298316	−	0.999555i	$-0.509497\pi$
$617$	−36.8122	−0.0596632	−0.0298316	+	0.999555i	$0.509497\pi$
$618$	0	0
$619$	− 1044.96i	− 1.68814i	−0.536234	−	0.844069i	$-0.680154\pi$
$619$	− 1044.96i	− 1.68814i	0.536234	−	0.844069i	$-0.319846\pi$
$620$	0	0
$621$	21.3749	0.0344201
$622$	0	0
$623$	− 275.207i	− 0.441745i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 262.713i	− 0.419000i
$628$	0	0
$629$	−57.5659	−0.0915198
$630$	0	0
$631$	727.116i	1.15232i	0.817336	+	0.576161i	$0.195450\pi$
$631$	727.116i	1.15232i	−0.817336	+	0.576161i	$0.804550\pi$
$632$	0	0
$633$	−132.303	−0.209010
$634$	0	0
$635$	138.854i	0.218668i
$636$	0	0
$637$	65.6895	0.103123
$638$	0	0
$639$	8.67316i	0.0135730i
$640$	0	0
$641$	−905.378	−1.41245	−0.706223	−	0.707989i	$-0.749603\pi$
$641$	−905.378	−1.41245	−0.706223	+	0.707989i	$0.749603\pi$
$642$	0	0
$643$	413.042i	0.642366i	0.947017	+	0.321183i	$0.104081\pi$
$643$	413.042i	0.642366i	−0.947017	+	0.321183i	$0.895919\pi$
$644$	0	0
$645$	−57.0835	−0.0885015
$646$	0	0
$647$	− 596.729i	− 0.922301i	−0.887322	−	0.461150i	$-0.847437\pi$
$647$	− 596.729i	− 0.922301i	0.887322	−	0.461150i	$-0.152563\pi$
$648$	0	0
$649$	−307.099	−0.473188
$650$	0	0
$651$	− 205.122i	− 0.315087i
$652$	0	0
$653$	1026.13	1.57141	0.785704	−	0.618602i	$-0.212301\pi$
$653$	1026.13	1.57141	0.785704	+	0.618602i	$0.212301\pi$
$654$	0	0
$655$	121.683i	0.185776i
$656$	0	0
$657$	53.6175	0.0816096
$658$	0	0
$659$	− 396.535i	− 0.601723i	−0.953668	−	0.300861i	$-0.902726\pi$
$659$	− 396.535i	− 0.601723i	0.953668	−	0.300861i	$-0.0972741\pi$
$660$	0	0
$661$	718.231	1.08658	0.543291	−	0.839544i	$-0.317178\pi$
$661$	718.231	1.08658	0.543291	+	0.839544i	$0.317178\pi$
$662$	0	0
$663$	− 20.3970i	− 0.0307648i
$664$	0	0
$665$	101.971	0.153339
$666$	0	0
$667$	42.7626i	0.0641118i
$668$	0	0
$669$	−195.213	−0.291798
$670$	0	0
$671$	− 389.287i	− 0.580159i
$672$	0	0
$673$	554.302	0.823629	0.411814	−	0.911268i	$-0.364895\pi$
$673$	554.302	0.823629	0.411814	+	0.911268i	$0.364895\pi$
$674$	0	0
$675$	− 25.9808i	− 0.0384900i
$676$	0	0
$677$	−129.171	−0.190799	−0.0953994	−	0.995439i	$-0.530413\pi$
$677$	−129.171	−0.190799	−0.0953994	+	0.995439i	$0.530413\pi$
$678$	0	0
$679$	292.942i	0.431432i
$680$	0	0
$681$	628.770	0.923304
$682$	0	0
$683$	443.991i	0.650060i	0.945704	+	0.325030i	$0.105374\pi$
$683$	443.991i	0.650060i	−0.945704	+	0.325030i	$0.894626\pi$
$684$	0	0
$685$	−266.251	−0.388687
$686$	0	0
$687$	573.900i	0.835372i
$688$	0	0
$689$	149.604	0.217132
$690$	0	0
$691$	− 306.527i	− 0.443600i	−0.975092	−	0.221800i	$-0.928807\pi$
$691$	− 306.527i	− 0.443600i	0.975092	−	0.221800i	$-0.0711932\pi$
$692$	0	0
$693$	−69.8474	−0.100790
$694$	0	0
$695$	440.708i	0.634112i
$696$	0	0
$697$	−59.9686	−0.0860382
$698$	0	0
$699$	376.511i	0.538643i
$700$	0	0
$701$	1004.44	1.43287	0.716436	−	0.697653i	$-0.245772\pi$
$701$	1004.44	1.43287	0.716436	+	0.697653i	$0.245772\pi$
$702$	0	0
$703$	− 790.675i	− 1.12471i
$704$	0	0
$705$	−124.459	−0.176537
$706$	0	0
$707$	340.382i	0.481446i
$708$	0	0
$709$	125.899	0.177572	0.0887862	−	0.996051i	$-0.471701\pi$
$709$	125.899	0.177572	0.0887862	+	0.996051i	$0.471701\pi$
$710$	0	0
$711$	131.180i	0.184501i
$712$	0	0
$713$	184.130	0.258246
$714$	0	0
$715$	− 184.656i	− 0.258260i
$716$	0	0
$717$	−417.220	−0.581896
$718$	0	0
$719$	− 1117.32i	− 1.55399i	−0.629506	−	0.776995i	$-0.716743\pi$
$719$	− 1117.32i	− 1.55399i	0.629506	−	0.776995i	$-0.283257\pi$
$720$	0	0
$721$	−24.6234	−0.0341517
$722$	0	0
$723$	− 310.688i	− 0.429720i
$724$	0	0
$725$	51.9771	0.0716926
$726$	0	0
$727$	380.280i	0.523082i	0.965192	+	0.261541i	$0.0842306\pi$
$727$	380.280i	0.523082i	−0.965192	+	0.261541i	$0.915769\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	18.4958i	0.0253021i
$732$	0	0
$733$	−31.7992	−0.0433822	−0.0216911	−	0.999765i	$-0.506905\pi$
$733$	−31.7992	−0.0433822	−0.0216911	+	0.999765i	$0.506905\pi$
$734$	0	0
$735$	− 27.1109i	− 0.0368856i
$736$	0	0
$737$	−825.197	−1.11967
$738$	0	0
$739$	180.660i	0.244465i	0.992501	+	0.122233i	$0.0390054\pi$
$739$	180.660i	0.244465i	−0.992501	+	0.122233i	$0.960995\pi$
$740$	0	0
$741$	280.156	0.378078
$742$	0	0
$743$	1232.99i	1.65948i	0.558152	+	0.829738i	$0.311510\pi$
$743$	1232.99i	1.65948i	−0.558152	+	0.829738i	$0.688490\pi$
$744$	0	0
$745$	−109.305	−0.146718
$746$	0	0
$747$	104.623i	0.140058i
$748$	0	0
$749$	−2.96608	−0.00396006
$750$	0	0
$751$	− 233.664i	− 0.311137i	−0.987825	−	0.155569i	$-0.950279\pi$
$751$	− 233.664i	− 0.311137i	0.987825	−	0.155569i	$-0.0497210\pi$
$752$	0	0
$753$	192.865	0.256129
$754$	0	0
$755$	2.71581i	0.00359710i
$756$	0	0
$757$	191.053	0.252381	0.126191	−	0.992006i	$-0.459725\pi$
$757$	191.053	0.252381	0.126191	+	0.992006i	$0.459725\pi$
$758$	0	0
$759$	− 62.6993i	− 0.0826077i
$760$	0	0
$761$	494.084	0.649256	0.324628	−	0.945842i	$-0.394761\pi$
$761$	494.084	0.649256	0.324628	+	0.945842i	$0.394761\pi$
$762$	0	0
$763$	408.577i	0.535488i
$764$	0	0
$765$	−8.41811	−0.0110041
$766$	0	0
$767$	− 327.489i	− 0.426974i
$768$	0	0
$769$	−26.7946	−0.0348434	−0.0174217	−	0.999848i	$-0.505546\pi$
$769$	−26.7946	−0.0348434	−0.0174217	+	0.999848i	$0.505546\pi$
$770$	0	0
$771$	− 592.013i	− 0.767850i
$772$	0	0
$773$	258.308	0.334164	0.167082	−	0.985943i	$-0.446566\pi$
$773$	258.308	0.334164	0.167082	+	0.985943i	$0.446566\pi$
$774$	0	0
$775$	− 223.806i	− 0.288782i
$776$	0	0
$777$	−210.216	−0.270549
$778$	0	0
$779$	− 823.676i	− 1.05735i
$780$	0	0
$781$	25.4411	0.0325751
$782$	0	0
$783$	− 54.0162i	− 0.0689862i
$784$	0	0
$785$	−228.991	−0.291709
$786$	0	0
$787$	− 1051.45i	− 1.33602i	−0.744151	−	0.668012i	$-0.767146\pi$
$787$	− 1051.45i	− 1.33602i	0.744151	−	0.668012i	$-0.232854\pi$
$788$	0	0
$789$	−160.352	−0.203235
$790$	0	0
$791$	312.774i	0.395415i
$792$	0	0
$793$	415.134	0.523498
$794$	0	0
$795$	− 61.7435i	− 0.0776648i
$796$	0	0
$797$	−646.015	−0.810558	−0.405279	−	0.914193i	$-0.632826\pi$
$797$	−646.015	−0.810558	−0.405279	+	0.914193i	$0.632826\pi$
$798$	0	0
$799$	40.3264i	0.0504710i
$800$	0	0
$801$	−312.055	−0.389582
$802$	0	0
$803$	− 157.277i	− 0.195862i
$804$	0	0
$805$	24.3364	0.0302315
$806$	0	0
$807$	292.376i	0.362300i
$808$	0	0
$809$	981.931	1.21376	0.606880	−	0.794794i	$-0.292421\pi$
$809$	981.931	1.21376	0.606880	+	0.794794i	$0.292421\pi$
$810$	0	0
$811$	− 24.1176i	− 0.0297381i	−0.999889	−	0.0148691i	$-0.995267\pi$
$811$	− 24.1176i	− 0.0297381i	0.999889	−	0.0148691i	$-0.00473314\pi$
$812$	0	0
$813$	−113.520	−0.139631
$814$	0	0
$815$	− 303.192i	− 0.372014i
$816$	0	0
$817$	−254.042	−0.310945
$818$	0	0
$819$	− 74.4849i	− 0.0909462i
$820$	0	0
$821$	465.783	0.567336	0.283668	−	0.958923i	$-0.408449\pi$
$821$	465.783	0.567336	0.283668	+	0.958923i	$0.408449\pi$
$822$	0	0
$823$	720.915i	0.875960i	0.898985	+	0.437980i	$0.144306\pi$
$823$	720.915i	0.875960i	−0.898985	+	0.437980i	$0.855694\pi$
$824$	0	0
$825$	−76.2098	−0.0923755
$826$	0	0
$827$	90.1914i	0.109059i	0.998512	+	0.0545293i	$0.0173658\pi$
$827$	90.1914i	0.109059i	−0.998512	+	0.0545293i	$0.982634\pi$
$828$	0	0
$829$	1037.66	1.25170	0.625850	−	0.779943i	$-0.284752\pi$
$829$	1037.66	1.25170	0.625850	+	0.779943i	$0.284752\pi$
$830$	0	0
$831$	− 216.363i	− 0.260364i
$832$	0	0
$833$	−8.78429	−0.0105454
$834$	0	0
$835$	− 465.512i	− 0.557499i
$836$	0	0
$837$	−232.586	−0.277880
$838$	0	0
$839$	1230.91i	1.46712i	0.679625	+	0.733559i	$0.262142\pi$
$839$	1230.91i	1.46712i	−0.679625	+	0.733559i	$0.737858\pi$
$840$	0	0
$841$	−732.935	−0.871504
$842$	0	0
$843$	235.037i	0.278810i
$844$	0	0
$845$	−180.979	−0.214177
$846$	0	0
$847$	− 115.251i	− 0.136070i
$848$	0	0
$849$	−343.879	−0.405040
$850$	0	0
$851$	− 188.703i	− 0.221743i
$852$	0	0
$853$	−354.911	−0.416074	−0.208037	−	0.978121i	$-0.566707\pi$
$853$	−354.911	−0.416074	−0.208037	+	0.978121i	$0.566707\pi$
$854$	0	0
$855$	− 115.624i	− 0.135232i
$856$	0	0
$857$	−1706.61	−1.99138	−0.995689	−	0.0927500i	$-0.970434\pi$
$857$	−1706.61	−1.99138	−0.995689	+	0.0927500i	$0.970434\pi$
$858$	0	0
$859$	646.956i	0.753150i	0.926386	+	0.376575i	$0.122898\pi$
$859$	646.956i	0.753150i	−0.926386	+	0.376575i	$0.877102\pi$
$860$	0	0
$861$	−218.990	−0.254344
$862$	0	0
$863$	664.063i	0.769482i	0.923025	+	0.384741i	$0.125709\pi$
$863$	664.063i	0.769482i	−0.923025	+	0.384741i	$0.874291\pi$
$864$	0	0
$865$	−421.581	−0.487377
$866$	0	0
$867$	− 497.835i	− 0.574204i
$868$	0	0
$869$	384.794	0.442801
$870$	0	0
$871$	− 879.986i	− 1.01032i
$872$	0	0
$873$	332.166	0.380487
$874$	0	0
$875$	− 29.5804i	− 0.0338062i
$876$	0	0
$877$	−634.315	−0.723278	−0.361639	−	0.932318i	$-0.617783\pi$
$877$	−634.315	−0.723278	−0.361639	+	0.932318i	$0.617783\pi$
$878$	0	0
$879$	138.023i	0.157023i
$880$	0	0
$881$	−638.388	−0.724618	−0.362309	−	0.932058i	$-0.618011\pi$
$881$	−638.388	−0.724618	−0.362309	+	0.932058i	$0.618011\pi$
$882$	0	0
$883$	1427.58i	1.61673i	0.588679	+	0.808367i	$0.299648\pi$
$883$	1427.58i	1.61673i	−0.588679	+	0.808367i	$0.700352\pi$
$884$	0	0
$885$	−135.159	−0.152722
$886$	0	0
$887$	− 509.818i	− 0.574767i	−0.957816	−	0.287383i	$-0.907215\pi$
$887$	− 509.818i	− 0.574767i	0.957816	−	0.287383i	$-0.0927854\pi$
$888$	0	0
$889$	164.295	0.184808
$890$	0	0
$891$	79.1995i	0.0888883i
$892$	0	0
$893$	−553.887	−0.620254
$894$	0	0
$895$	177.383i	0.198193i
$896$	0	0
$897$	66.8622	0.0745398
$898$	0	0
$899$	− 465.311i	− 0.517588i
$900$	0	0
$901$	−20.0057	−0.0222039
$902$	0	0
$903$	67.5421i	0.0747974i
$904$	0	0
$905$	351.634	0.388545
$906$	0	0
$907$	1238.64i	1.36564i	0.730586	+	0.682821i	$0.239247\pi$
$907$	1238.64i	1.36564i	−0.730586	+	0.682821i	$0.760753\pi$
$908$	0	0
$909$	385.957	0.424595
$910$	0	0
$911$	1396.21i	1.53261i	0.642478	+	0.766304i	$0.277906\pi$
$911$	1396.21i	1.53261i	−0.642478	+	0.766304i	$0.722094\pi$
$912$	0	0
$913$	306.892	0.336136
$914$	0	0
$915$	− 171.331i	− 0.187247i
$916$	0	0
$917$	143.977	0.157009
$918$	0	0
$919$	− 251.692i	− 0.273876i	−0.990580	−	0.136938i	$-0.956274\pi$
$919$	− 251.692i	− 0.273876i	0.990580	−	0.136938i	$-0.0437261\pi$
$920$	0	0
$921$	−557.997	−0.605860
$922$	0	0
$923$	27.1303i	0.0293936i
$924$	0	0
$925$	−229.365	−0.247962
$926$	0	0
$927$	27.9203i	0.0301190i
$928$	0	0
$929$	374.998	0.403657	0.201829	−	0.979421i	$-0.435312\pi$
$929$	374.998	0.403657	0.201829	+	0.979421i	$0.435312\pi$
$930$	0	0
$931$	− 120.653i	− 0.129595i
$932$	0	0
$933$	291.888	0.312849
$934$	0	0
$935$	24.6930i	0.0264096i
$936$	0	0
$937$	−530.104	−0.565746	−0.282873	−	0.959157i	$-0.591287\pi$
$937$	−530.104	−0.565746	−0.282873	+	0.959157i	$0.591287\pi$
$938$	0	0
$939$	4.82108i	0.00513427i
$940$	0	0
$941$	991.428	1.05359	0.526795	−	0.849992i	$-0.323394\pi$
$941$	991.428	1.05359	0.526795	+	0.849992i	$0.323394\pi$
$942$	0	0
$943$	− 196.579i	− 0.208461i
$944$	0	0
$945$	−30.7409	−0.0325300
$946$	0	0
$947$	− 1520.18i	− 1.60526i	−0.596476	−	0.802631i	$-0.703433\pi$
$947$	− 1520.18i	− 1.60526i	0.596476	−	0.802631i	$-0.296567\pi$
$948$	0	0
$949$	167.719	0.176733
$950$	0	0
$951$	527.710i	0.554900i
$952$	0	0
$953$	−76.4275	−0.0801967	−0.0400984	−	0.999196i	$-0.512767\pi$
$953$	−76.4275	−0.0801967	−0.0400984	+	0.999196i	$0.512767\pi$
$954$	0	0
$955$	− 464.050i	− 0.485917i
$956$	0	0
$957$	−158.447	−0.165566
$958$	0	0
$959$	315.032i	0.328501i
$960$	0	0
$961$	−1042.56	−1.08487
$962$	0	0
$963$	3.36322i	0.00349244i
$964$	0	0
$965$	−56.5251	−0.0585752
$966$	0	0
$967$	111.839i	0.115655i	0.998327	+	0.0578276i	$0.0184174\pi$
$967$	111.839i	0.115655i	−0.998327	+	0.0578276i	$0.981583\pi$
$968$	0	0
$969$	−37.4636	−0.0386622
$970$	0	0
$971$	612.000i	0.630278i	0.949045	+	0.315139i	$0.102051\pi$
$971$	612.000i	0.630278i	−0.949045	+	0.315139i	$0.897949\pi$
$972$	0	0
$973$	521.453	0.535923
$974$	0	0
$975$	− 81.2697i	− 0.0833535i
$976$	0	0
$977$	−384.848	−0.393907	−0.196954	−	0.980413i	$-0.563105\pi$
$977$	−384.848	−0.393907	−0.196954	+	0.980413i	$0.563105\pi$
$978$	0	0
$979$	915.357i	0.934992i
$980$	0	0
$981$	463.283	0.472256
$982$	0	0
$983$	− 75.5987i	− 0.0769061i	−0.999260	−	0.0384531i	$-0.987757\pi$
$983$	− 75.5987i	− 0.0769061i	0.999260	−	0.0384531i	$-0.0122430\pi$
$984$	0	0
$985$	−444.677	−0.451449
$986$	0	0
$987$	147.262i	0.149201i
$988$	0	0
$989$	−60.6299	−0.0613042
$990$	0	0
$991$	784.734i	0.791860i	0.918281	+	0.395930i	$0.129578\pi$
$991$	784.734i	0.791860i	−0.918281	+	0.395930i	$0.870422\pi$
$992$	0	0
$993$	−748.265	−0.753539
$994$	0	0
$995$	168.242i	0.169087i
$996$	0	0
$997$	−195.160	−0.195747	−0.0978737	−	0.995199i	$-0.531204\pi$
$997$	−195.160	−0.195747	−0.0978737	+	0.995199i	$0.531204\pi$
$998$	0	0
$999$	238.363i	0.238602i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.n.c.1471.14	yes	16
4.3	odd	2	inner	1680.3.n.c.1471.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.3.n.c.1471.7	✓	16	4.3	odd	2	inner
1680.3.n.c.1471.14	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1680.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +2.23607 q^{5} -2.64575i q^{7} -3.00000 q^{9} +8.79995i q^{11} -9.38422 q^{13} +3.87298i q^{15} +1.25490 q^{17} +17.2362i q^{19} +4.58258 q^{21} +4.11360i q^{23} +5.00000 q^{25} -5.19615i q^{27} +10.3954 q^{29} -44.7612i q^{31} -15.2420 q^{33} -5.91608i q^{35} -45.8730 q^{37} -16.2539i q^{39} -47.7876 q^{41} +14.7389i q^{43} -6.70820 q^{45} +32.1352i q^{47} -7.00000 q^{49} +2.17355i q^{51} -15.9421 q^{53} +19.6773i q^{55} -29.8539 q^{57} +34.8978i q^{59} -44.2374 q^{61} +7.93725i q^{63} -20.9837 q^{65} +93.7730i q^{67} -7.12496 q^{69} -2.89105i q^{71} -17.8725 q^{73} +8.66025i q^{75} +23.2825 q^{77} -43.7268i q^{79} +9.00000 q^{81} -34.8743i q^{83} +2.80604 q^{85} +18.0054i q^{87} +104.018 q^{89} +24.8283i q^{91} +77.5287 q^{93} +38.5413i q^{95} -110.722 q^{97} -26.3998i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} + 48 q^{13} - 96 q^{17} + 80 q^{25} + 112 q^{29} + 48 q^{33} - 224 q^{37} + 160 q^{41} - 112 q^{49} - 96 q^{53} + 80 q^{61} - 80 q^{65} + 48 q^{69} - 48 q^{73} + 144 q^{81} - 160 q^{89}+ \cdots + 592 q^{97}+O(q^{100}) \) 16 * q - 48 * q^9 + 48 * q^13 - 96 * q^17 + 80 * q^25 + 112 * q^29 + 48 * q^33 - 224 * q^37 + 160 * q^41 - 112 * q^49 - 96 * q^53 + 80 * q^61 - 80 * q^65 + 48 * q^69 - 48 * q^73 + 144 * q^81 - 160 * q^89 + 592 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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