LMFDB - Embedded newform 1680.3.n.c.1471.4

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			1471.4
Root			$2.51939 + 1.56062i$ of defining polynomial
Character	$\chi$	$=$	1680.1471
Dual form			1680.3.n.c.1471.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -2.23607 q^{5} +2.64575i q^{7} -3.00000 q^{9} +5.10452i q^{11} -12.4178 q^{13} +3.87298i q^{15} -26.0758 q^{17} -15.5836i q^{19} +4.58258 q^{21} -18.3400i q^{23} +5.00000 q^{25} +5.19615i q^{27} +46.5279 q^{29} +2.63506i q^{31} +8.84128 q^{33} -5.91608i q^{35} +28.7266 q^{37} +21.5082i q^{39} +61.8324 q^{41} +67.6436i q^{43} +6.70820 q^{45} -0.353999i q^{47} -7.00000 q^{49} +45.1646i q^{51} -39.4015 q^{53} -11.4140i q^{55} -26.9916 q^{57} -23.4528i q^{59} +4.09977 q^{61} -7.93725i q^{63} +27.7669 q^{65} +88.1856i q^{67} -31.7657 q^{69} +99.4186i q^{71} +96.6327 q^{73} -8.66025i q^{75} -13.5053 q^{77} -81.1594i q^{79} +9.00000 q^{81} -97.3455i q^{83} +58.3073 q^{85} -80.5887i q^{87} +94.0310 q^{89} -32.8543i q^{91} +4.56406 q^{93} +34.8460i q^{95} +3.09617 q^{97} -15.3135i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	5.10452i	0.464047i	0.972710	+	0.232023i	$0.0745346\pi$
$11$	5.10452i	0.464047i	−0.972710	+	0.232023i	$0.925465\pi$
$12$	0	0
$13$	−12.4178	−0.955212	−0.477606	−	0.878574i	$-0.658495\pi$
$13$	−12.4178	−0.955212	−0.477606	+	0.878574i	$0.658495\pi$
$14$	0	0
$15$	3.87298i	0.258199i
$16$	0	0
$17$	−26.0758	−1.53387	−0.766936	−	0.641724i	$-0.778220\pi$
$17$	−26.0758	−1.53387	−0.766936	+	0.641724i	$0.778220\pi$
$18$	0	0
$19$	− 15.5836i	− 0.820190i	−0.912043	−	0.410095i	$-0.865496\pi$
$19$	− 15.5836i	− 0.820190i	0.912043	−	0.410095i	$-0.134504\pi$
$20$	0	0
$21$	4.58258	0.218218
$22$	0	0
$23$	− 18.3400i	− 0.797389i	−0.917084	−	0.398695i	$-0.869463\pi$
$23$	− 18.3400i	− 0.797389i	0.917084	−	0.398695i	$-0.130537\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	46.5279	1.60441	0.802206	−	0.597048i	$-0.203660\pi$
$29$	46.5279	1.60441	0.802206	+	0.597048i	$0.203660\pi$
$30$	0	0
$31$	2.63506i	0.0850019i	0.999096	+	0.0425010i	$0.0135326\pi$
$31$	2.63506i	0.0850019i	−0.999096	+	0.0425010i	$0.986467\pi$
$32$	0	0
$33$	8.84128	0.267918
$34$	0	0
$35$	− 5.91608i	− 0.169031i
$36$	0	0
$37$	28.7266	0.776393	0.388197	−	0.921577i	$-0.373098\pi$
$37$	28.7266	0.776393	0.388197	+	0.921577i	$0.373098\pi$
$38$	0	0
$39$	21.5082i	0.551492i
$40$	0	0
$41$	61.8324	1.50811	0.754054	−	0.656813i	$-0.228096\pi$
$41$	61.8324	1.50811	0.754054	+	0.656813i	$0.228096\pi$
$42$	0	0
$43$	67.6436i	1.57311i	0.617522	+	0.786553i	$0.288136\pi$
$43$	67.6436i	1.57311i	−0.617522	+	0.786553i	$0.711864\pi$
$44$	0	0
$45$	6.70820	0.149071
$46$	0	0
$47$	− 0.353999i	− 0.00753189i	−0.999993	−	0.00376595i	$-0.998801\pi$
$47$	− 0.353999i	− 0.00753189i	0.999993	−	0.00376595i	$-0.00119874\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	45.1646i	0.885581i
$52$	0	0
$53$	−39.4015	−0.743424	−0.371712	−	0.928348i	$-0.621229\pi$
$53$	−39.4015	−0.743424	−0.371712	+	0.928348i	$0.621229\pi$
$54$	0	0
$55$	− 11.4140i	− 0.207528i
$56$	0	0
$57$	−26.9916	−0.473537
$58$	0	0
$59$	− 23.4528i	− 0.397505i	−0.980050	−	0.198752i	$-0.936311\pi$
$59$	− 23.4528i	− 0.397505i	0.980050	−	0.198752i	$-0.0636889\pi$
$60$	0	0
$61$	4.09977	0.0672093	0.0336046	−	0.999435i	$-0.489301\pi$
$61$	4.09977	0.0672093	0.0336046	+	0.999435i	$0.489301\pi$
$62$	0	0
$63$	− 7.93725i	− 0.125988i
$64$	0	0
$65$	27.7669	0.427184
$66$	0	0
$67$	88.1856i	1.31620i	0.752929	+	0.658101i	$0.228640\pi$
$67$	88.1856i	1.31620i	−0.752929	+	0.658101i	$0.771360\pi$
$68$	0	0
$69$	−31.7657	−0.460373
$70$	0	0
$71$	99.4186i	1.40026i	0.714015	+	0.700131i	$0.246875\pi$
$71$	99.4186i	1.40026i	−0.714015	+	0.700131i	$0.753125\pi$
$72$	0	0
$73$	96.6327	1.32374	0.661868	−	0.749620i	$-0.269764\pi$
$73$	96.6327	1.32374	0.661868	+	0.749620i	$0.269764\pi$
$74$	0	0
$75$	− 8.66025i	− 0.115470i
$76$	0	0
$77$	−13.5053	−0.175393
$78$	0	0
$79$	− 81.1594i	− 1.02733i	−0.857989	−	0.513667i	$-0.828287\pi$
$79$	− 81.1594i	− 1.02733i	0.857989	−	0.513667i	$-0.171713\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 97.3455i	− 1.17284i	−0.810008	−	0.586419i	$-0.800537\pi$
$83$	− 97.3455i	− 1.17284i	0.810008	−	0.586419i	$-0.199463\pi$
$84$	0	0
$85$	58.3073	0.685968
$86$	0	0
$87$	− 80.5887i	− 0.926307i
$88$	0	0
$89$	94.0310	1.05653	0.528264	−	0.849080i	$-0.322843\pi$
$89$	94.0310	1.05653	0.528264	+	0.849080i	$0.322843\pi$
$90$	0	0
$91$	− 32.8543i	− 0.361036i
$92$	0	0
$93$	4.56406	0.0490759
$94$	0	0
$95$	34.8460i	0.366800i
$96$	0	0
$97$	3.09617	0.0319192	0.0159596	−	0.999873i	$-0.494920\pi$
$97$	3.09617	0.0319192	0.0159596	+	0.999873i	$0.494920\pi$
$98$	0	0
$99$	− 15.3135i	− 0.154682i
$100$	0	0
$101$	−96.7821	−0.958239	−0.479120	−	0.877750i	$-0.659044\pi$
$101$	−96.7821	−0.958239	−0.479120	+	0.877750i	$0.659044\pi$
$102$	0	0
$103$	173.284i	1.68237i	0.540745	+	0.841186i	$0.318142\pi$
$103$	173.284i	1.68237i	−0.540745	+	0.841186i	$0.681858\pi$
$104$	0	0
$105$	−10.2470	−0.0975900
$106$	0	0
$107$	111.618i	1.04316i	0.853202	+	0.521581i	$0.174658\pi$
$107$	111.618i	1.04316i	−0.853202	+	0.521581i	$0.825342\pi$
$108$	0	0
$109$	138.733	1.27278	0.636388	−	0.771369i	$-0.280428\pi$
$109$	138.733	1.27278	0.636388	+	0.771369i	$0.280428\pi$
$110$	0	0
$111$	− 49.7558i	− 0.448251i
$112$	0	0
$113$	215.262	1.90497	0.952487	−	0.304579i	$-0.0985158\pi$
$113$	215.262	1.90497	0.952487	+	0.304579i	$0.0985158\pi$
$114$	0	0
$115$	41.0094i	0.356603i
$116$	0	0
$117$	37.2533	0.318404
$118$	0	0
$119$	− 68.9901i	− 0.579749i
$120$	0	0
$121$	94.9439	0.784660
$122$	0	0
$123$	− 107.097i	− 0.870706i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	− 216.668i	− 1.70604i	−0.521874	−	0.853022i	$-0.674767\pi$
$127$	− 216.668i	− 1.70604i	0.521874	−	0.853022i	$-0.325233\pi$
$128$	0	0
$129$	117.162	0.908234
$130$	0	0
$131$	− 140.526i	− 1.07272i	−0.843989	−	0.536361i	$-0.819799\pi$
$131$	− 140.526i	− 1.07272i	0.843989	−	0.536361i	$-0.180201\pi$
$132$	0	0
$133$	41.2304	0.310003
$134$	0	0
$135$	− 11.6190i	− 0.0860663i
$136$	0	0
$137$	58.1404	0.424382	0.212191	−	0.977228i	$-0.431940\pi$
$137$	58.1404	0.424382	0.212191	+	0.977228i	$0.431940\pi$
$138$	0	0
$139$	223.817i	1.61020i	0.593141	+	0.805099i	$0.297888\pi$
$139$	223.817i	1.61020i	−0.593141	+	0.805099i	$0.702112\pi$
$140$	0	0
$141$	−0.613144	−0.00434854
$142$	0	0
$143$	− 63.3866i	− 0.443263i
$144$	0	0
$145$	−104.040	−0.717514
$146$	0	0
$147$	12.1244i	0.0824786i
$148$	0	0
$149$	105.629	0.708923	0.354461	−	0.935071i	$-0.384664\pi$
$149$	105.629	0.708923	0.354461	+	0.935071i	$0.384664\pi$
$150$	0	0
$151$	73.6111i	0.487491i	0.969839	+	0.243745i	$0.0783762\pi$
$151$	73.6111i	0.487491i	−0.969839	+	0.243745i	$0.921624\pi$
$152$	0	0
$153$	78.2274	0.511290
$154$	0	0
$155$	− 5.89217i	− 0.0380140i
$156$	0	0
$157$	205.320	1.30777	0.653885	−	0.756594i	$-0.273138\pi$
$157$	205.320	1.30777	0.653885	+	0.756594i	$0.273138\pi$
$158$	0	0
$159$	68.2454i	0.429216i
$160$	0	0
$161$	48.5230	0.301385
$162$	0	0
$163$	204.188i	1.25269i	0.779547	+	0.626344i	$0.215450\pi$
$163$	204.188i	1.25269i	−0.779547	+	0.626344i	$0.784550\pi$
$164$	0	0
$165$	−19.7697	−0.119816
$166$	0	0
$167$	20.8370i	0.124773i	0.998052	+	0.0623863i	$0.0198711\pi$
$167$	20.8370i	0.124773i	−0.998052	+	0.0623863i	$0.980129\pi$
$168$	0	0
$169$	−14.7993	−0.0875701
$170$	0	0
$171$	46.7508i	0.273397i
$172$	0	0
$173$	−129.149	−0.746529	−0.373264	−	0.927725i	$-0.621762\pi$
$173$	−129.149	−0.746529	−0.373264	+	0.927725i	$0.621762\pi$
$174$	0	0
$175$	13.2288i	0.0755929i
$176$	0	0
$177$	−40.6214	−0.229499
$178$	0	0
$179$	− 335.588i	− 1.87480i	−0.348261	−	0.937398i	$-0.613228\pi$
$179$	− 335.588i	− 1.87480i	0.348261	−	0.937398i	$-0.386772\pi$
$180$	0	0
$181$	−150.609	−0.832094	−0.416047	−	0.909343i	$-0.636585\pi$
$181$	−150.609	−0.832094	−0.416047	+	0.909343i	$0.636585\pi$
$182$	0	0
$183$	− 7.10100i	− 0.0388033i
$184$	0	0
$185$	−64.2345	−0.347214
$186$	0	0
$187$	− 133.104i	− 0.711788i
$188$	0	0
$189$	−13.7477	−0.0727393
$190$	0	0
$191$	− 247.834i	− 1.29756i	−0.760977	−	0.648779i	$-0.775280\pi$
$191$	− 247.834i	− 1.29756i	0.760977	−	0.648779i	$-0.224720\pi$
$192$	0	0
$193$	198.054	1.02619	0.513094	−	0.858332i	$-0.328499\pi$
$193$	198.054	1.02619	0.513094	+	0.858332i	$0.328499\pi$
$194$	0	0
$195$	− 48.0938i	− 0.246635i
$196$	0	0
$197$	72.3188	0.367101	0.183550	−	0.983010i	$-0.441241\pi$
$197$	72.3188	0.367101	0.183550	+	0.983010i	$0.441241\pi$
$198$	0	0
$199$	− 218.281i	− 1.09689i	−0.836187	−	0.548445i	$-0.815220\pi$
$199$	− 218.281i	− 1.09689i	0.836187	−	0.548445i	$-0.184780\pi$
$200$	0	0
$201$	152.742	0.759910
$202$	0	0
$203$	123.101i	0.606410i
$204$	0	0
$205$	−138.261	−0.674446
$206$	0	0
$207$	55.0199i	0.265796i
$208$	0	0
$209$	79.5468	0.380607
$210$	0	0
$211$	267.621i	1.26834i	0.773192	+	0.634172i	$0.218659\pi$
$211$	267.621i	1.26834i	−0.773192	+	0.634172i	$0.781341\pi$
$212$	0	0
$213$	172.198	0.808441
$214$	0	0
$215$	− 151.256i	− 0.703515i
$216$	0	0
$217$	−6.97171	−0.0321277
$218$	0	0
$219$	− 167.373i	− 0.764259i
$220$	0	0
$221$	323.803	1.46517
$222$	0	0
$223$	137.858i	0.618198i	0.951030	+	0.309099i	$0.100027\pi$
$223$	137.858i	0.618198i	−0.951030	+	0.309099i	$0.899973\pi$
$224$	0	0
$225$	−15.0000	−0.0666667
$226$	0	0
$227$	− 171.639i	− 0.756118i	−0.925782	−	0.378059i	$-0.876592\pi$
$227$	− 171.639i	− 0.756118i	0.925782	−	0.378059i	$-0.123408\pi$
$228$	0	0
$229$	−26.6275	−0.116277	−0.0581386	−	0.998309i	$-0.518517\pi$
$229$	−26.6275	−0.116277	−0.0581386	+	0.998309i	$0.518517\pi$
$230$	0	0
$231$	23.3918i	0.101263i
$232$	0	0
$233$	−114.142	−0.489881	−0.244941	−	0.969538i	$-0.578768\pi$
$233$	−114.142	−0.489881	−0.244941	+	0.969538i	$0.578768\pi$
$234$	0	0
$235$	0.791566i	0.00336836i
$236$	0	0
$237$	−140.572	−0.593132
$238$	0	0
$239$	− 99.6449i	− 0.416924i	−0.978030	−	0.208462i	$-0.933154\pi$
$239$	− 99.6449i	− 0.416924i	0.978030	−	0.208462i	$-0.0668458\pi$
$240$	0	0
$241$	87.0912	0.361374	0.180687	−	0.983541i	$-0.442168\pi$
$241$	87.0912	0.361374	0.180687	+	0.983541i	$0.442168\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	15.6525	0.0638877
$246$	0	0
$247$	193.513i	0.783455i
$248$	0	0
$249$	−168.607	−0.677138
$250$	0	0
$251$	401.400i	1.59920i	0.600531	+	0.799602i	$0.294956\pi$
$251$	401.400i	1.59920i	−0.600531	+	0.799602i	$0.705044\pi$
$252$	0	0
$253$	93.6166	0.370026
$254$	0	0
$255$	− 100.991i	− 0.396044i
$256$	0	0
$257$	80.8278	0.314505	0.157252	−	0.987558i	$-0.449736\pi$
$257$	80.8278	0.314505	0.157252	+	0.987558i	$0.449736\pi$
$258$	0	0
$259$	76.0033i	0.293449i
$260$	0	0
$261$	−139.584	−0.534804
$262$	0	0
$263$	− 27.9109i	− 0.106125i	−0.998591	−	0.0530625i	$-0.983102\pi$
$263$	− 27.9109i	− 0.106125i	0.998591	−	0.0530625i	$-0.0168983\pi$
$264$	0	0
$265$	88.1044	0.332469
$266$	0	0
$267$	− 162.866i	− 0.609987i
$268$	0	0
$269$	−40.8922	−0.152016	−0.0760078	−	0.997107i	$-0.524217\pi$
$269$	−40.8922	−0.152016	−0.0760078	+	0.997107i	$0.524217\pi$
$270$	0	0
$271$	38.7754i	0.143083i	0.997438	+	0.0715413i	$0.0227918\pi$
$271$	38.7754i	0.143083i	−0.997438	+	0.0715413i	$0.977208\pi$
$272$	0	0
$273$	−56.9053	−0.208444
$274$	0	0
$275$	25.5226i	0.0928094i
$276$	0	0
$277$	−386.534	−1.39543	−0.697715	−	0.716375i	$-0.745800\pi$
$277$	−386.534	−1.39543	−0.697715	+	0.716375i	$0.745800\pi$
$278$	0	0
$279$	− 7.90518i	− 0.0283340i
$280$	0	0
$281$	357.365	1.27176	0.635881	−	0.771787i	$-0.280637\pi$
$281$	357.365	1.27176	0.635881	+	0.771787i	$0.280637\pi$
$282$	0	0
$283$	− 247.517i	− 0.874619i	−0.899311	−	0.437309i	$-0.855931\pi$
$283$	− 247.517i	− 0.874619i	0.899311	−	0.437309i	$-0.144069\pi$
$284$	0	0
$285$	60.3551	0.211772
$286$	0	0
$287$	163.593i	0.570011i
$288$	0	0
$289$	390.948	1.35276
$290$	0	0
$291$	− 5.36272i	− 0.0184286i
$292$	0	0
$293$	−87.7105	−0.299353	−0.149677	−	0.988735i	$-0.547823\pi$
$293$	−87.7105	−0.299353	−0.149677	+	0.988735i	$0.547823\pi$
$294$	0	0
$295$	52.4420i	0.177770i
$296$	0	0
$297$	−26.5238	−0.0893059
$298$	0	0
$299$	227.741i	0.761676i
$300$	0	0
$301$	−178.968	−0.594578
$302$	0	0
$303$	167.632i	0.553240i
$304$	0	0
$305$	−9.16736	−0.0300569
$306$	0	0
$307$	241.887i	0.787906i	0.919131	+	0.393953i	$0.128893\pi$
$307$	241.887i	0.787906i	−0.919131	+	0.393953i	$0.871107\pi$
$308$	0	0
$309$	300.137	0.971318
$310$	0	0
$311$	− 277.512i	− 0.892320i	−0.894953	−	0.446160i	$-0.852791\pi$
$311$	− 277.512i	− 0.892320i	0.894953	−	0.446160i	$-0.147209\pi$
$312$	0	0
$313$	−46.5328	−0.148667	−0.0743335	−	0.997233i	$-0.523683\pi$
$313$	−46.5328	−0.148667	−0.0743335	+	0.997233i	$0.523683\pi$
$314$	0	0
$315$	17.7482i	0.0563436i
$316$	0	0
$317$	−411.665	−1.29863	−0.649313	−	0.760521i	$-0.724944\pi$
$317$	−411.665	−1.29863	−0.649313	+	0.760521i	$0.724944\pi$
$318$	0	0
$319$	237.503i	0.744522i
$320$	0	0
$321$	193.329	0.602270
$322$	0	0
$323$	406.355i	1.25807i
$324$	0	0
$325$	−62.0888	−0.191042
$326$	0	0
$327$	− 240.292i	− 0.734838i
$328$	0	0
$329$	0.936593	0.00284679
$330$	0	0
$331$	− 106.573i	− 0.321972i	−0.986957	−	0.160986i	$-0.948533\pi$
$331$	− 106.573i	− 0.321972i	0.986957	−	0.160986i	$-0.0514674\pi$
$332$	0	0
$333$	−86.1797	−0.258798
$334$	0	0
$335$	− 197.189i	− 0.588624i
$336$	0	0
$337$	−472.309	−1.40151	−0.700756	−	0.713401i	$-0.747154\pi$
$337$	−472.309	−1.40151	−0.700756	+	0.713401i	$0.747154\pi$
$338$	0	0
$339$	− 372.845i	− 1.09984i
$340$	0	0
$341$	−13.4507	−0.0394449
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	71.0303	0.205885
$346$	0	0
$347$	382.728i	1.10296i	0.834188	+	0.551481i	$0.185937\pi$
$347$	382.728i	1.10296i	−0.834188	+	0.551481i	$0.814063\pi$
$348$	0	0
$349$	267.858	0.767501	0.383751	−	0.923437i	$-0.374632\pi$
$349$	267.858	0.767501	0.383751	+	0.923437i	$0.374632\pi$
$350$	0	0
$351$	− 64.5246i	− 0.183831i
$352$	0	0
$353$	598.049	1.69419	0.847095	−	0.531441i	$-0.178349\pi$
$353$	598.049	1.69419	0.847095	+	0.531441i	$0.178349\pi$
$354$	0	0
$355$	− 222.307i	− 0.626216i
$356$	0	0
$357$	−119.494	−0.334718
$358$	0	0
$359$	− 105.553i	− 0.294019i	−0.989135	−	0.147010i	$-0.953035\pi$
$359$	− 105.553i	− 0.294019i	0.989135	−	0.147010i	$-0.0469648\pi$
$360$	0	0
$361$	118.151	0.327288
$362$	0	0
$363$	− 164.448i	− 0.453024i
$364$	0	0
$365$	−216.077	−0.591993
$366$	0	0
$367$	254.558i	0.693618i	0.937936	+	0.346809i	$0.112735\pi$
$367$	254.558i	0.693618i	−0.937936	+	0.346809i	$0.887265\pi$
$368$	0	0
$369$	−185.497	−0.502702
$370$	0	0
$371$	− 104.247i	− 0.280988i
$372$	0	0
$373$	90.0152	0.241328	0.120664	−	0.992693i	$-0.461498\pi$
$373$	90.0152	0.241328	0.120664	+	0.992693i	$0.461498\pi$
$374$	0	0
$375$	19.3649i	0.0516398i
$376$	0	0
$377$	−577.772	−1.53255
$378$	0	0
$379$	649.951i	1.71491i	0.514558	+	0.857456i	$0.327956\pi$
$379$	649.951i	1.71491i	−0.514558	+	0.857456i	$0.672044\pi$
$380$	0	0
$381$	−375.279	−0.984985
$382$	0	0
$383$	− 6.65993i	− 0.0173889i	−0.999962	−	0.00869443i	$-0.997232\pi$
$383$	− 6.65993i	− 0.0173889i	0.999962	−	0.00869443i	$-0.00276756\pi$
$384$	0	0
$385$	30.1987	0.0784382
$386$	0	0
$387$	− 202.931i	− 0.524369i
$388$	0	0
$389$	25.3910	0.0652724	0.0326362	−	0.999467i	$-0.489610\pi$
$389$	25.3910	0.0652724	0.0326362	+	0.999467i	$0.489610\pi$
$390$	0	0
$391$	478.229i	1.22309i
$392$	0	0
$393$	−243.399	−0.619336
$394$	0	0
$395$	181.478i	0.459438i
$396$	0	0
$397$	−591.020	−1.48872	−0.744358	−	0.667781i	$-0.767244\pi$
$397$	−591.020	−1.48872	−0.744358	+	0.667781i	$0.767244\pi$
$398$	0	0
$399$	− 71.4131i	− 0.178980i
$400$	0	0
$401$	−322.719	−0.804785	−0.402393	−	0.915467i	$-0.631821\pi$
$401$	−322.719	−0.804785	−0.402393	+	0.915467i	$0.631821\pi$
$402$	0	0
$403$	− 32.7215i	− 0.0811949i
$404$	0	0
$405$	−20.1246	−0.0496904
$406$	0	0
$407$	146.635i	0.360283i
$408$	0	0
$409$	338.317	0.827181	0.413590	−	0.910463i	$-0.364275\pi$
$409$	338.317	0.827181	0.413590	+	0.910463i	$0.364275\pi$
$410$	0	0
$411$	− 100.702i	− 0.245017i
$412$	0	0
$413$	62.0502	0.150243
$414$	0	0
$415$	217.671i	0.524509i
$416$	0	0
$417$	387.663	0.929648
$418$	0	0
$419$	− 166.464i	− 0.397288i	−0.980072	−	0.198644i	$-0.936346\pi$
$419$	− 166.464i	− 0.397288i	0.980072	−	0.198644i	$-0.0636538\pi$
$420$	0	0
$421$	463.057	1.09990	0.549949	−	0.835199i	$-0.314647\pi$
$421$	463.057	1.09990	0.549949	+	0.835199i	$0.314647\pi$
$422$	0	0
$423$	1.06200i	0.00251063i
$424$	0	0
$425$	−130.379	−0.306774
$426$	0	0
$427$	10.8470i	0.0254027i
$428$	0	0
$429$	−109.789	−0.255918
$430$	0	0
$431$	− 117.540i	− 0.272714i	−0.990660	−	0.136357i	$-0.956461\pi$
$431$	− 117.540i	− 0.272714i	0.990660	−	0.136357i	$-0.0435395\pi$
$432$	0	0
$433$	117.959	0.272423	0.136211	−	0.990680i	$-0.456507\pi$
$433$	117.959	0.272423	0.136211	+	0.990680i	$0.456507\pi$
$434$	0	0
$435$	180.202i	0.414257i
$436$	0	0
$437$	−285.803	−0.654011
$438$	0	0
$439$	52.7908i	0.120252i	0.998191	+	0.0601262i	$0.0191503\pi$
$439$	52.7908i	0.120252i	−0.998191	+	0.0601262i	$0.980850\pi$
$440$	0	0
$441$	21.0000	0.0476190
$442$	0	0
$443$	− 315.351i	− 0.711853i	−0.934514	−	0.355927i	$-0.884165\pi$
$443$	− 315.351i	− 0.711853i	0.934514	−	0.355927i	$-0.115835\pi$
$444$	0	0
$445$	−210.260	−0.472494
$446$	0	0
$447$	− 182.956i	− 0.409297i
$448$	0	0
$449$	476.573	1.06141	0.530705	−	0.847557i	$-0.321927\pi$
$449$	476.573	1.06141	0.530705	+	0.847557i	$0.321927\pi$
$450$	0	0
$451$	315.624i	0.699833i
$452$	0	0
$453$	127.498	0.281453
$454$	0	0
$455$	73.4644i	0.161460i
$456$	0	0
$457$	59.3708	0.129914	0.0649571	−	0.997888i	$-0.479309\pi$
$457$	59.3708	0.129914	0.0649571	+	0.997888i	$0.479309\pi$
$458$	0	0
$459$	− 135.494i	− 0.295194i
$460$	0	0
$461$	−209.996	−0.455522	−0.227761	−	0.973717i	$-0.573140\pi$
$461$	−209.996	−0.455522	−0.227761	+	0.973717i	$0.573140\pi$
$462$	0	0
$463$	405.442i	0.875684i	0.899052	+	0.437842i	$0.144257\pi$
$463$	405.442i	0.875684i	−0.899052	+	0.437842i	$0.855743\pi$
$464$	0	0
$465$	−10.2055	−0.0219474
$466$	0	0
$467$	− 774.009i	− 1.65741i	−0.559689	−	0.828703i	$-0.689079\pi$
$467$	− 774.009i	− 1.65741i	0.559689	−	0.828703i	$-0.310921\pi$
$468$	0	0
$469$	−233.317	−0.497478
$470$	0	0
$471$	− 355.625i	− 0.755042i
$472$	0	0
$473$	−345.288	−0.729995
$474$	0	0
$475$	− 77.9180i	− 0.164038i
$476$	0	0
$477$	118.204	0.247808
$478$	0	0
$479$	− 113.617i	− 0.237196i	−0.992942	−	0.118598i	$-0.962160\pi$
$479$	− 113.617i	− 0.237196i	0.992942	−	0.118598i	$-0.0378399\pi$
$480$	0	0
$481$	−356.719	−0.741620
$482$	0	0
$483$	− 84.0442i	− 0.174005i
$484$	0	0
$485$	−6.92324	−0.0142747
$486$	0	0
$487$	801.556i	1.64590i	0.568110	+	0.822952i	$0.307675\pi$
$487$	801.556i	1.64590i	−0.568110	+	0.822952i	$0.692325\pi$
$488$	0	0
$489$	353.664	0.723240
$490$	0	0
$491$	− 588.018i	− 1.19759i	−0.800901	−	0.598796i	$-0.795646\pi$
$491$	− 588.018i	− 1.19759i	0.800901	−	0.598796i	$-0.204354\pi$
$492$	0	0
$493$	−1213.25	−2.46096
$494$	0	0
$495$	34.2421i	0.0691760i
$496$	0	0
$497$	−263.037	−0.529249
$498$	0	0
$499$	− 23.7739i	− 0.0476432i	−0.999716	−	0.0238216i	$-0.992417\pi$
$499$	− 23.7739i	− 0.0476432i	0.999716	−	0.0238216i	$-0.00758336\pi$
$500$	0	0
$501$	36.0908	0.0720375
$502$	0	0
$503$	371.403i	0.738376i	0.929355	+	0.369188i	$0.120364\pi$
$503$	371.403i	0.738376i	−0.929355	+	0.369188i	$0.879636\pi$
$504$	0	0
$505$	216.411	0.428538
$506$	0	0
$507$	25.6332i	0.0505586i
$508$	0	0
$509$	165.052	0.324268	0.162134	−	0.986769i	$-0.448162\pi$
$509$	165.052	0.324268	0.162134	+	0.986769i	$0.448162\pi$
$510$	0	0
$511$	255.666i	0.500325i
$512$	0	0
$513$	80.9748	0.157846
$514$	0	0
$515$	− 387.476i	− 0.752380i
$516$	0	0
$517$	1.80699	0.00349515
$518$	0	0
$519$	223.693i	0.431008i
$520$	0	0
$521$	47.0690	0.0903435	0.0451717	−	0.998979i	$-0.485616\pi$
$521$	47.0690	0.0903435	0.0451717	+	0.998979i	$0.485616\pi$
$522$	0	0
$523$	667.863i	1.27699i	0.769628	+	0.638493i	$0.220442\pi$
$523$	667.863i	1.27699i	−0.769628	+	0.638493i	$0.779558\pi$
$524$	0	0
$525$	22.9129	0.0436436
$526$	0	0
$527$	− 68.7113i	− 0.130382i
$528$	0	0
$529$	192.646	0.364170
$530$	0	0
$531$	70.3583i	0.132502i
$532$	0	0
$533$	−767.820	−1.44056
$534$	0	0
$535$	− 249.586i	− 0.466516i
$536$	0	0
$537$	−581.256	−1.08241
$538$	0	0
$539$	− 35.7316i	− 0.0662924i
$540$	0	0
$541$	−714.702	−1.32108	−0.660538	−	0.750793i	$-0.729672\pi$
$541$	−714.702	−1.32108	−0.660538	+	0.750793i	$0.729672\pi$
$542$	0	0
$543$	260.862i	0.480410i
$544$	0	0
$545$	−310.216	−0.569203
$546$	0	0
$547$	667.532i	1.22035i	0.792266	+	0.610175i	$0.208901\pi$
$547$	667.532i	1.22035i	−0.792266	+	0.610175i	$0.791099\pi$
$548$	0	0
$549$	−12.2993	−0.0224031
$550$	0	0
$551$	− 725.073i	− 1.31592i
$552$	0	0
$553$	214.728	0.388296
$554$	0	0
$555$	111.257i	0.200464i
$556$	0	0
$557$	−594.544	−1.06740	−0.533702	−	0.845673i	$-0.679199\pi$
$557$	−594.544	−1.06740	−0.533702	+	0.845673i	$0.679199\pi$
$558$	0	0
$559$	− 839.982i	− 1.50265i
$560$	0	0
$561$	−230.544	−0.410951
$562$	0	0
$563$	− 1077.37i	− 1.91363i	−0.290705	−	0.956813i	$-0.593890\pi$
$563$	− 1077.37i	− 1.91363i	0.290705	−	0.956813i	$-0.406110\pi$
$564$	0	0
$565$	−481.341	−0.851930
$566$	0	0
$567$	23.8118i	0.0419961i
$568$	0	0
$569$	213.690	0.375553	0.187777	−	0.982212i	$-0.439872\pi$
$569$	213.690	0.375553	0.187777	+	0.982212i	$0.439872\pi$
$570$	0	0
$571$	665.176i	1.16493i	0.812855	+	0.582466i	$0.197912\pi$
$571$	665.176i	1.16493i	−0.812855	+	0.582466i	$0.802088\pi$
$572$	0	0
$573$	−429.260	−0.749146
$574$	0	0
$575$	− 91.6998i	− 0.159478i
$576$	0	0
$577$	−671.641	−1.16402	−0.582011	−	0.813181i	$-0.697734\pi$
$577$	−671.641	−1.16402	−0.582011	+	0.813181i	$0.697734\pi$
$578$	0	0
$579$	− 343.040i	− 0.592470i
$580$	0	0
$581$	257.552	0.443291
$582$	0	0
$583$	− 201.126i	− 0.344984i
$584$	0	0
$585$	−83.3008	−0.142395
$586$	0	0
$587$	156.431i	0.266492i	0.991083	+	0.133246i	$0.0425401\pi$
$587$	156.431i	0.266492i	−0.991083	+	0.133246i	$0.957460\pi$
$588$	0	0
$589$	41.0637	0.0697177
$590$	0	0
$591$	− 125.260i	− 0.211946i
$592$	0	0
$593$	338.402	0.570662	0.285331	−	0.958429i	$-0.407897\pi$
$593$	338.402	0.570662	0.285331	+	0.958429i	$0.407897\pi$
$594$	0	0
$595$	154.267i	0.259272i
$596$	0	0
$597$	−378.074	−0.633289
$598$	0	0
$599$	629.125i	1.05029i	0.851012	+	0.525146i	$0.175989\pi$
$599$	629.125i	1.05029i	−0.851012	+	0.525146i	$0.824011\pi$
$600$	0	0
$601$	−552.824	−0.919840	−0.459920	−	0.887960i	$-0.652122\pi$
$601$	−552.824	−0.919840	−0.459920	+	0.887960i	$0.652122\pi$
$602$	0	0
$603$	− 264.557i	− 0.438734i
$604$	0	0
$605$	−212.301	−0.350911
$606$	0	0
$607$	− 123.936i	− 0.204178i	−0.994775	−	0.102089i	$-0.967447\pi$
$607$	− 123.936i	− 0.204178i	0.994775	−	0.102089i	$-0.0325527\pi$
$608$	0	0
$609$	213.218	0.350111
$610$	0	0
$611$	4.39587i	0.00719455i
$612$	0	0
$613$	1164.11	1.89903	0.949515	−	0.313721i	$-0.101576\pi$
$613$	1164.11	1.89903	0.949515	+	0.313721i	$0.101576\pi$
$614$	0	0
$615$	239.476i	0.389392i
$616$	0	0
$617$	323.657	0.524566	0.262283	−	0.964991i	$-0.415525\pi$
$617$	323.657	0.524566	0.262283	+	0.964991i	$0.415525\pi$
$618$	0	0
$619$	− 382.614i	− 0.618116i	−0.951043	−	0.309058i	$-0.899986\pi$
$619$	− 382.614i	− 0.618116i	0.951043	−	0.309058i	$-0.100014\pi$
$620$	0	0
$621$	95.2972	0.153458
$622$	0	0
$623$	248.783i	0.399330i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 137.779i	− 0.219743i
$628$	0	0
$629$	−749.068	−1.19089
$630$	0	0
$631$	− 98.2864i	− 0.155763i	−0.996963	−	0.0778815i	$-0.975184\pi$
$631$	− 98.2864i	− 0.155763i	0.996963	−	0.0778815i	$-0.0248156\pi$
$632$	0	0
$633$	463.533	0.732279
$634$	0	0
$635$	484.484i	0.762966i
$636$	0	0
$637$	86.9243	0.136459
$638$	0	0
$639$	− 298.256i	− 0.466754i
$640$	0	0
$641$	−1010.53	−1.57649	−0.788244	−	0.615363i	$-0.789009\pi$
$641$	−1010.53	−1.57649	−0.788244	+	0.615363i	$0.789009\pi$
$642$	0	0
$643$	− 448.858i	− 0.698069i	−0.937110	−	0.349034i	$-0.886510\pi$
$643$	− 448.858i	− 0.698069i	0.937110	−	0.349034i	$-0.113490\pi$
$644$	0	0
$645$	−261.982	−0.406174
$646$	0	0
$647$	− 405.975i	− 0.627472i	−0.949510	−	0.313736i	$-0.898419\pi$
$647$	− 405.975i	− 0.627472i	0.949510	−	0.313736i	$-0.101581\pi$
$648$	0	0
$649$	119.715	0.184461
$650$	0	0
$651$	12.0754i	0.0185489i
$652$	0	0
$653$	196.084	0.300282	0.150141	−	0.988665i	$-0.452027\pi$
$653$	196.084	0.300282	0.150141	+	0.988665i	$0.452027\pi$
$654$	0	0
$655$	314.227i	0.479735i
$656$	0	0
$657$	−289.898	−0.441245
$658$	0	0
$659$	920.381i	1.39663i	0.715790	+	0.698316i	$0.246067\pi$
$659$	920.381i	1.39663i	−0.715790	+	0.698316i	$0.753933\pi$
$660$	0	0
$661$	1019.52	1.54239	0.771197	−	0.636597i	$-0.219658\pi$
$661$	1019.52	1.54239	0.771197	+	0.636597i	$0.219658\pi$
$662$	0	0
$663$	− 560.843i	− 0.845918i
$664$	0	0
$665$	−92.1939	−0.138637
$666$	0	0
$667$	− 853.320i	− 1.27934i
$668$	0	0
$669$	238.777	0.356917
$670$	0	0
$671$	20.9273i	0.0311883i
$672$	0	0
$673$	688.377	1.02285	0.511424	−	0.859328i	$-0.329118\pi$
$673$	688.377	1.02285	0.511424	+	0.859328i	$0.329118\pi$
$674$	0	0
$675$	25.9808i	0.0384900i
$676$	0	0
$677$	−433.825	−0.640805	−0.320402	−	0.947282i	$-0.603818\pi$
$677$	−433.825	−0.640805	−0.320402	+	0.947282i	$0.603818\pi$
$678$	0	0
$679$	8.19169i	0.0120643i
$680$	0	0
$681$	−297.287	−0.436545
$682$	0	0
$683$	918.373i	1.34462i	0.740272	+	0.672308i	$0.234697\pi$
$683$	918.373i	1.34462i	−0.740272	+	0.672308i	$0.765303\pi$
$684$	0	0
$685$	−130.006	−0.189790
$686$	0	0
$687$	46.1202i	0.0671327i
$688$	0	0
$689$	489.278	0.710128
$690$	0	0
$691$	897.634i	1.29904i	0.760346	+	0.649518i	$0.225029\pi$
$691$	897.634i	1.29904i	−0.760346	+	0.649518i	$0.774971\pi$
$692$	0	0
$693$	40.5158	0.0584644
$694$	0	0
$695$	− 500.471i	− 0.720102i
$696$	0	0
$697$	−1612.33	−2.31324
$698$	0	0
$699$	197.700i	0.282833i
$700$	0	0
$701$	−971.305	−1.38560	−0.692799	−	0.721130i	$-0.743623\pi$
$701$	−971.305	−1.38560	−0.692799	+	0.721130i	$0.743623\pi$
$702$	0	0
$703$	− 447.663i	− 0.636790i
$704$	0	0
$705$	1.37103	0.00194473
$706$	0	0
$707$	− 256.061i	− 0.362180i
$708$	0	0
$709$	368.861	0.520255	0.260128	−	0.965574i	$-0.416235\pi$
$709$	368.861	0.520255	0.260128	+	0.965574i	$0.416235\pi$
$710$	0	0
$711$	243.478i	0.342445i
$712$	0	0
$713$	48.3269	0.0677796
$714$	0	0
$715$	141.737i	0.198233i
$716$	0	0
$717$	−172.590	−0.240711
$718$	0	0
$719$	81.4585i	0.113294i	0.998394	+	0.0566471i	$0.0180410\pi$
$719$	81.4585i	0.113294i	−0.998394	+	0.0566471i	$0.981959\pi$
$720$	0	0
$721$	−458.467	−0.635877
$722$	0	0
$723$	− 150.846i	− 0.208640i
$724$	0	0
$725$	232.640	0.320882
$726$	0	0
$727$	286.135i	0.393583i	0.980445	+	0.196792i	$0.0630522\pi$
$727$	286.135i	0.393583i	−0.980445	+	0.196792i	$0.936948\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 1763.86i	− 2.41294i
$732$	0	0
$733$	−597.050	−0.814529	−0.407265	−	0.913310i	$-0.633517\pi$
$733$	−597.050	−0.814529	−0.407265	+	0.913310i	$0.633517\pi$
$734$	0	0
$735$	− 27.1109i	− 0.0368856i
$736$	0	0
$737$	−450.145	−0.610780
$738$	0	0
$739$	1115.03i	1.50883i	0.656395	+	0.754417i	$0.272080\pi$
$739$	1115.03i	1.50883i	−0.656395	+	0.754417i	$0.727920\pi$
$740$	0	0
$741$	335.175	0.452328
$742$	0	0
$743$	− 661.612i	− 0.890460i	−0.895416	−	0.445230i	$-0.853122\pi$
$743$	− 661.612i	− 0.890460i	0.895416	−	0.445230i	$-0.146878\pi$
$744$	0	0
$745$	−236.195	−0.317040
$746$	0	0
$747$	292.036i	0.390946i
$748$	0	0
$749$	−295.314	−0.394278
$750$	0	0
$751$	− 176.622i	− 0.235182i	−0.993062	−	0.117591i	$-0.962483\pi$
$751$	− 176.622i	− 0.235182i	0.993062	−	0.117591i	$-0.0375172\pi$
$752$	0	0
$753$	695.245	0.923301
$754$	0	0
$755$	− 164.599i	− 0.218013i
$756$	0	0
$757$	−1255.24	−1.65818	−0.829088	−	0.559119i	$-0.811140\pi$
$757$	−1255.24	−1.65818	−0.829088	+	0.559119i	$0.811140\pi$
$758$	0	0
$759$	− 162.149i	− 0.213635i
$760$	0	0
$761$	565.354	0.742909	0.371455	−	0.928451i	$-0.378859\pi$
$761$	565.354	0.742909	0.371455	+	0.928451i	$0.378859\pi$
$762$	0	0
$763$	367.052i	0.481064i
$764$	0	0
$765$	−174.922	−0.228656
$766$	0	0
$767$	291.231i	0.379701i
$768$	0	0
$769$	700.628	0.911090	0.455545	−	0.890213i	$-0.349444\pi$
$769$	700.628	0.911090	0.455545	+	0.890213i	$0.349444\pi$
$770$	0	0
$771$	− 139.998i	− 0.181579i
$772$	0	0
$773$	1204.71	1.55849	0.779243	−	0.626722i	$-0.215604\pi$
$773$	1204.71	1.55849	0.779243	+	0.626722i	$0.215604\pi$
$774$	0	0
$775$	13.1753i	0.0170004i
$776$	0	0
$777$	131.642	0.169423
$778$	0	0
$779$	− 963.572i	− 1.23693i
$780$	0	0
$781$	−507.484	−0.649787
$782$	0	0
$783$	241.766i	0.308769i
$784$	0	0
$785$	−459.109	−0.584853
$786$	0	0
$787$	− 1360.26i	− 1.72841i	−0.503141	−	0.864205i	$-0.667822\pi$
$787$	− 1360.26i	− 1.72841i	0.503141	−	0.864205i	$-0.332178\pi$
$788$	0	0
$789$	−48.3430	−0.0612713
$790$	0	0
$791$	569.530i	0.720013i
$792$	0	0
$793$	−50.9099	−0.0641991
$794$	0	0
$795$	− 152.601i	− 0.191951i
$796$	0	0
$797$	−848.626	−1.06478	−0.532388	−	0.846501i	$-0.678705\pi$
$797$	−848.626	−1.06478	−0.532388	+	0.846501i	$0.678705\pi$
$798$	0	0
$799$	9.23081i	0.0115530i
$800$	0	0
$801$	−282.093	−0.352176
$802$	0	0
$803$	493.263i	0.614276i
$804$	0	0
$805$	−108.501	−0.134783
$806$	0	0
$807$	70.8274i	0.0877663i
$808$	0	0
$809$	1247.56	1.54211	0.771053	−	0.636771i	$-0.219730\pi$
$809$	1247.56	1.54211	0.771053	+	0.636771i	$0.219730\pi$
$810$	0	0
$811$	1358.85i	1.67552i	0.546036	+	0.837762i	$0.316136\pi$
$811$	1358.85i	1.67552i	−0.546036	+	0.837762i	$0.683864\pi$
$812$	0	0
$813$	67.1609	0.0826088
$814$	0	0
$815$	− 456.579i	− 0.560219i
$816$	0	0
$817$	1054.13	1.29025
$818$	0	0
$819$	98.5629i	0.120345i
$820$	0	0
$821$	1170.58	1.42580	0.712900	−	0.701266i	$-0.247381\pi$
$821$	1170.58	1.42580	0.712900	+	0.701266i	$0.247381\pi$
$822$	0	0
$823$	− 604.336i	− 0.734308i	−0.930160	−	0.367154i	$-0.880332\pi$
$823$	− 604.336i	− 0.734308i	0.930160	−	0.367154i	$-0.119668\pi$
$824$	0	0
$825$	44.2064	0.0535835
$826$	0	0
$827$	271.951i	0.328841i	0.986390	+	0.164420i	$0.0525753\pi$
$827$	271.951i	0.328841i	−0.986390	+	0.164420i	$0.947425\pi$
$828$	0	0
$829$	−1131.33	−1.36469	−0.682347	−	0.731029i	$-0.739041\pi$
$829$	−1131.33	−1.36469	−0.682347	+	0.731029i	$0.739041\pi$
$830$	0	0
$831$	669.497i	0.805652i
$832$	0	0
$833$	182.531	0.219124
$834$	0	0
$835$	− 46.5930i	− 0.0558000i
$836$	0	0
$837$	−13.6922	−0.0163586
$838$	0	0
$839$	1076.26i	1.28279i	0.767210	+	0.641396i	$0.221644\pi$
$839$	1076.26i	1.28279i	−0.767210	+	0.641396i	$0.778356\pi$
$840$	0	0
$841$	1323.85	1.57414
$842$	0	0
$843$	− 618.974i	− 0.734252i
$844$	0	0
$845$	33.0923	0.0391625
$846$	0	0
$847$	251.198i	0.296574i
$848$	0	0
$849$	−428.712	−0.504961
$850$	0	0
$851$	− 526.844i	− 0.619088i
$852$	0	0
$853$	255.300	0.299296	0.149648	−	0.988739i	$-0.452186\pi$
$853$	255.300	0.299296	0.149648	+	0.988739i	$0.452186\pi$
$854$	0	0
$855$	− 104.538i	− 0.122267i
$856$	0	0
$857$	−589.026	−0.687312	−0.343656	−	0.939096i	$-0.611665\pi$
$857$	−589.026	−0.687312	−0.343656	+	0.939096i	$0.611665\pi$
$858$	0	0
$859$	− 966.822i	− 1.12552i	−0.826620	−	0.562760i	$-0.809740\pi$
$859$	− 966.822i	− 1.12552i	0.826620	−	0.562760i	$-0.190260\pi$
$860$	0	0
$861$	283.352	0.329096
$862$	0	0
$863$	− 934.859i	− 1.08327i	−0.840615	−	0.541633i	$-0.817806\pi$
$863$	− 934.859i	− 1.08327i	0.840615	−	0.541633i	$-0.182194\pi$
$864$	0	0
$865$	288.787	0.333858
$866$	0	0
$867$	− 677.142i	− 0.781017i
$868$	0	0
$869$	414.280	0.476731
$870$	0	0
$871$	− 1095.07i	− 1.25725i
$872$	0	0
$873$	−9.28850	−0.0106397
$874$	0	0
$875$	− 29.5804i	− 0.0338062i
$876$	0	0
$877$	−675.488	−0.770225	−0.385113	−	0.922870i	$-0.625837\pi$
$877$	−675.488	−0.770225	−0.385113	+	0.922870i	$0.625837\pi$
$878$	0	0
$879$	151.919i	0.172832i
$880$	0	0
$881$	895.222	1.01614	0.508071	−	0.861315i	$-0.330359\pi$
$881$	895.222	1.01614	0.508071	+	0.861315i	$0.330359\pi$
$882$	0	0
$883$	778.770i	0.881959i	0.897517	+	0.440979i	$0.145369\pi$
$883$	778.770i	0.881959i	−0.897517	+	0.440979i	$0.854631\pi$
$884$	0	0
$885$	90.8322	0.102635
$886$	0	0
$887$	1121.57i	1.26445i	0.774783	+	0.632227i	$0.217859\pi$
$887$	1121.57i	1.26445i	−0.774783	+	0.632227i	$0.782141\pi$
$888$	0	0
$889$	573.249	0.644824
$890$	0	0
$891$	45.9406i	0.0515608i
$892$	0	0
$893$	−5.51658	−0.00617758
$894$	0	0
$895$	750.398i	0.838434i
$896$	0	0
$897$	394.459	0.439754
$898$	0	0
$899$	122.604i	0.136378i
$900$	0	0
$901$	1027.43	1.14032
$902$	0	0
$903$	309.982i	0.343280i
$904$	0	0
$905$	336.772	0.372124
$906$	0	0
$907$	− 441.365i	− 0.486621i	−0.969948	−	0.243310i	$-0.921767\pi$
$907$	− 441.365i	− 0.486621i	0.969948	−	0.243310i	$-0.0782334\pi$
$908$	0	0
$909$	290.346	0.319413
$910$	0	0
$911$	411.453i	0.451650i	0.974168	+	0.225825i	$0.0725077\pi$
$911$	411.453i	0.451650i	−0.974168	+	0.225825i	$0.927492\pi$
$912$	0	0
$913$	496.902	0.544252
$914$	0	0
$915$	15.8783i	0.0173534i
$916$	0	0
$917$	371.798	0.405450
$918$	0	0
$919$	910.962i	0.991254i	0.868536	+	0.495627i	$0.165062\pi$
$919$	910.962i	0.991254i	−0.868536	+	0.495627i	$0.834938\pi$
$920$	0	0
$921$	418.961	0.454898
$922$	0	0
$923$	− 1234.56i	− 1.33755i
$924$	0	0
$925$	143.633	0.155279
$926$	0	0
$927$	− 519.853i	− 0.560791i
$928$	0	0
$929$	762.270	0.820528	0.410264	−	0.911967i	$-0.365437\pi$
$929$	762.270	0.820528	0.410264	+	0.911967i	$0.365437\pi$
$930$	0	0
$931$	109.085i	0.117170i
$932$	0	0
$933$	−480.664	−0.515181
$934$	0	0
$935$	297.631i	0.318321i
$936$	0	0
$937$	834.729	0.890853	0.445427	−	0.895318i	$-0.353052\pi$
$937$	834.729	0.890853	0.445427	+	0.895318i	$0.353052\pi$
$938$	0	0
$939$	80.5971i	0.0858329i
$940$	0	0
$941$	−276.685	−0.294033	−0.147017	−	0.989134i	$-0.546967\pi$
$941$	−276.685	−0.294033	−0.147017	+	0.989134i	$0.546967\pi$
$942$	0	0
$943$	− 1134.00i	− 1.20255i
$944$	0	0
$945$	30.7409	0.0325300
$946$	0	0
$947$	− 268.037i	− 0.283038i	−0.989936	−	0.141519i	$-0.954801\pi$
$947$	− 268.037i	− 0.283038i	0.989936	−	0.141519i	$-0.0451987\pi$
$948$	0	0
$949$	−1199.96	−1.26445
$950$	0	0
$951$	713.024i	0.749762i
$952$	0	0
$953$	952.185	0.999145	0.499572	−	0.866272i	$-0.333490\pi$
$953$	952.185	0.999145	0.499572	+	0.866272i	$0.333490\pi$
$954$	0	0
$955$	554.173i	0.580286i
$956$	0	0
$957$	411.366	0.429850
$958$	0	0
$959$	153.825i	0.160401i
$960$	0	0
$961$	954.056	0.992775
$962$	0	0
$963$	− 334.855i	− 0.347721i
$964$	0	0
$965$	−442.863	−0.458925
$966$	0	0
$967$	284.006i	0.293698i	0.989159	+	0.146849i	$0.0469131\pi$
$967$	284.006i	0.293698i	−0.989159	+	0.146849i	$0.953087\pi$
$968$	0	0
$969$	703.828	0.726345
$970$	0	0
$971$	− 388.559i	− 0.400164i	−0.979779	−	0.200082i	$-0.935879\pi$
$971$	− 388.559i	− 0.400164i	0.979779	−	0.200082i	$-0.0641208\pi$
$972$	0	0
$973$	−592.165	−0.608597
$974$	0	0
$975$	107.541i	0.110298i
$976$	0	0
$977$	−304.974	−0.312153	−0.156077	−	0.987745i	$-0.549885\pi$
$977$	−304.974	−0.312153	−0.156077	+	0.987745i	$0.549885\pi$
$978$	0	0
$979$	479.983i	0.490279i
$980$	0	0
$981$	−416.198	−0.424259
$982$	0	0
$983$	770.648i	0.783976i	0.919970	+	0.391988i	$0.128212\pi$
$983$	770.648i	0.783976i	−0.919970	+	0.391988i	$0.871788\pi$
$984$	0	0
$985$	−161.710	−0.164172
$986$	0	0
$987$	− 1.62223i	− 0.00164359i
$988$	0	0
$989$	1240.58	1.25438
$990$	0	0
$991$	− 1234.76i	− 1.24597i	−0.782232	−	0.622987i	$-0.785919\pi$
$991$	− 1234.76i	− 1.24597i	0.782232	−	0.622987i	$-0.214081\pi$
$992$	0	0
$993$	−184.589	−0.185891
$994$	0	0
$995$	488.091i	0.490544i
$996$	0	0
$997$	−1515.67	−1.52023	−0.760117	−	0.649786i	$-0.774858\pi$
$997$	−1515.67	−1.52023	−0.760117	+	0.649786i	$0.774858\pi$
$998$	0	0
$999$	149.268i	0.149417i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.n.c.1471.4	✓	16
4.3	odd	2	inner	1680.3.n.c.1471.9	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.3.n.c.1471.4	✓	16	1.1	even	1	trivial
1680.3.n.c.1471.9	yes	16	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 $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -2.23607 q^{5} +2.64575i q^{7} -3.00000 q^{9} +5.10452i q^{11} -12.4178 q^{13} +3.87298i q^{15} -26.0758 q^{17} -15.5836i q^{19} +4.58258 q^{21} -18.3400i q^{23} +5.00000 q^{25} +5.19615i q^{27} +46.5279 q^{29} +2.63506i q^{31} +8.84128 q^{33} -5.91608i q^{35} +28.7266 q^{37} +21.5082i q^{39} +61.8324 q^{41} +67.6436i q^{43} +6.70820 q^{45} -0.353999i q^{47} -7.00000 q^{49} +45.1646i q^{51} -39.4015 q^{53} -11.4140i q^{55} -26.9916 q^{57} -23.4528i q^{59} +4.09977 q^{61} -7.93725i q^{63} +27.7669 q^{65} +88.1856i q^{67} -31.7657 q^{69} +99.4186i q^{71} +96.6327 q^{73} -8.66025i q^{75} -13.5053 q^{77} -81.1594i q^{79} +9.00000 q^{81} -97.3455i q^{83} +58.3073 q^{85} -80.5887i q^{87} +94.0310 q^{89} -32.8543i q^{91} +4.56406 q^{93} +34.8460i q^{95} +3.09617 q^{97} -15.3135i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} + 48 q^{13} - 96 q^{17} + 80 q^{25} + 112 q^{29} + 48 q^{33} - 224 q^{37} + 160 q^{41} - 112 q^{49} - 96 q^{53} + 80 q^{61} - 80 q^{65} + 48 q^{69} - 48 q^{73} + 144 q^{81} - 160 q^{89}+ \cdots + 592 q^{97}+O(q^{100}) \) 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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