LMFDB - Embedded newform 1680.4.a.bo.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,4,Mod(1,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1680.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,10,0,14,0,18,0,16,0,-76,0,30,0,-124] 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:	yes
Analytic conductor:	$99.1232088096$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1680.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} -48.5685 q^{11} -43.6569 q^{13} +15.0000 q^{15} -67.6569 q^{17} +93.2548 q^{19} +21.0000 q^{21} +104.167 q^{23} +25.0000 q^{25} +27.0000 q^{27} -58.7351 q^{29} +9.08831 q^{31} -145.706 q^{33} +35.0000 q^{35} -252.676 q^{37} -130.971 q^{39} +276.274 q^{41} +92.6375 q^{43} +45.0000 q^{45} +582.794 q^{47} +49.0000 q^{49} -202.971 q^{51} +623.019 q^{53} -242.843 q^{55} +279.765 q^{57} +524.999 q^{59} -352.794 q^{61} +63.0000 q^{63} -218.284 q^{65} +736.520 q^{67} +312.500 q^{69} +492.264 q^{71} +1164.75 q^{73} +75.0000 q^{75} -339.980 q^{77} +872.195 q^{79} +81.0000 q^{81} +529.588 q^{83} -338.284 q^{85} -176.205 q^{87} -385.216 q^{89} -305.598 q^{91} +27.2649 q^{93} +466.274 q^{95} -463.892 q^{97} -437.117 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 16 q^{11} - 76 q^{13} + 30 q^{15} - 124 q^{17} + 96 q^{19} + 42 q^{21} + 16 q^{23} + 50 q^{25} + 54 q^{27} + 188 q^{29} + 120 q^{31} + 48 q^{33} + 70 q^{35}+ \cdots + 144 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 16 * q^11 - 76 * q^13 + 30 * q^15 - 124 * q^17 + 96 * q^19 + 42 * q^21 + 16 * q^23 + 50 * q^25 + 54 * q^27 + 188 * q^29 + 120 * q^31 + 48 * q^33 + 70 * q^35 - 132 * q^37 - 228 * q^39 + 100 * q^41 + 536 * q^43 + 90 * q^45 + 928 * q^47 + 98 * q^49 - 372 * q^51 + 884 * q^53 + 80 * q^55 + 288 * q^57 - 104 * q^59 - 468 * q^61 + 126 * q^63 - 380 * q^65 + 1688 * q^67 + 48 * q^69 + 136 * q^71 + 508 * q^73 + 150 * q^75 + 112 * q^77 + 432 * q^79 + 162 * q^81 + 584 * q^83 - 620 * q^85 + 564 * q^87 - 1404 * q^89 - 532 * q^91 + 360 * q^93 + 480 * q^95 - 1188 * q^97 + 144 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−48.5685	−1.33127	−0.665635	−	0.746278i	$-0.731839\pi$
$11$	−48.5685	−1.33127	−0.665635	+	0.746278i	$0.731839\pi$
$12$	0	0
$13$	−43.6569	−0.931403	−0.465701	−	0.884942i	$-0.654198\pi$
$13$	−43.6569	−0.931403	−0.465701	+	0.884942i	$0.654198\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−67.6569	−0.965247	−0.482623	−	0.875828i	$-0.660316\pi$
$17$	−67.6569	−0.965247	−0.482623	+	0.875828i	$0.660316\pi$
$18$	0	0
$19$	93.2548	1.12601	0.563003	−	0.826455i	$-0.309646\pi$
$19$	93.2548	1.12601	0.563003	+	0.826455i	$0.309646\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	104.167	0.944357	0.472179	−	0.881503i	$-0.343468\pi$
$23$	104.167	0.944357	0.472179	+	0.881503i	$0.343468\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−58.7351	−0.376098	−0.188049	−	0.982160i	$-0.560216\pi$
$29$	−58.7351	−0.376098	−0.188049	+	0.982160i	$0.560216\pi$
$30$	0	0
$31$	9.08831	0.0526551	0.0263276	−	0.999653i	$-0.491619\pi$
$31$	9.08831	0.0526551	0.0263276	+	0.999653i	$0.491619\pi$
$32$	0	0
$33$	−145.706	−0.768609
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	−252.676	−1.12269	−0.561347	−	0.827580i	$-0.689717\pi$
$37$	−252.676	−1.12269	−0.561347	+	0.827580i	$0.689717\pi$
$38$	0	0
$39$	−130.971	−0.537745
$40$	0	0
$41$	276.274	1.05236	0.526180	−	0.850373i	$-0.323624\pi$
$41$	276.274	1.05236	0.526180	+	0.850373i	$0.323624\pi$
$42$	0	0
$43$	92.6375	0.328537	0.164268	−	0.986416i	$-0.447474\pi$
$43$	92.6375	0.328537	0.164268	+	0.986416i	$0.447474\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	582.794	1.80871	0.904354	−	0.426784i	$-0.140354\pi$
$47$	582.794	1.80871	0.904354	+	0.426784i	$0.140354\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−202.971	−0.557286
$52$	0	0
$53$	623.019	1.61468	0.807342	−	0.590083i	$-0.200905\pi$
$53$	623.019	1.61468	0.807342	+	0.590083i	$0.200905\pi$
$54$	0	0
$55$	−242.843	−0.595362
$56$	0	0
$57$	279.765	0.650100
$58$	0	0
$59$	524.999	1.15846	0.579229	−	0.815165i	$-0.303354\pi$
$59$	524.999	1.15846	0.579229	+	0.815165i	$0.303354\pi$
$60$	0	0
$61$	−352.794	−0.740502	−0.370251	−	0.928932i	$-0.620728\pi$
$61$	−352.794	−0.740502	−0.370251	+	0.928932i	$0.620728\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−218.284	−0.416536
$66$	0	0
$67$	736.520	1.34299	0.671494	−	0.741010i	$-0.265653\pi$
$67$	736.520	1.34299	0.671494	+	0.741010i	$0.265653\pi$
$68$	0	0
$69$	312.500	0.545225
$70$	0	0
$71$	492.264	0.822831	0.411415	−	0.911448i	$-0.365035\pi$
$71$	492.264	0.822831	0.411415	+	0.911448i	$0.365035\pi$
$72$	0	0
$73$	1164.75	1.86745	0.933727	−	0.357987i	$-0.116537\pi$
$73$	1164.75	1.86745	0.933727	+	0.357987i	$0.116537\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	−339.980	−0.503173
$78$	0	0
$79$	872.195	1.24215	0.621074	−	0.783752i	$-0.286697\pi$
$79$	872.195	1.24215	0.621074	+	0.783752i	$0.286697\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	529.588	0.700359	0.350180	−	0.936683i	$-0.386121\pi$
$83$	529.588	0.700359	0.350180	+	0.936683i	$0.386121\pi$
$84$	0	0
$85$	−338.284	−0.431672
$86$	0	0
$87$	−176.205	−0.217140
$88$	0	0
$89$	−385.216	−0.458796	−0.229398	−	0.973333i	$-0.573676\pi$
$89$	−385.216	−0.458796	−0.229398	+	0.973333i	$0.573676\pi$
$90$	0	0
$91$	−305.598	−0.352037
$92$	0	0
$93$	27.2649	0.0304005
$94$	0	0
$95$	466.274	0.503565
$96$	0	0
$97$	−463.892	−0.485579	−0.242789	−	0.970079i	$-0.578062\pi$
$97$	−463.892	−0.485579	−0.242789	+	0.970079i	$0.578062\pi$
$98$	0	0
$99$	−437.117	−0.443757
$100$	0	0
$101$	−432.725	−0.426314	−0.213157	−	0.977018i	$-0.568375\pi$
$101$	−432.725	−0.426314	−0.213157	+	0.977018i	$0.568375\pi$
$102$	0	0
$103$	−512.626	−0.490393	−0.245197	−	0.969473i	$-0.578853\pi$
$103$	−512.626	−0.490393	−0.245197	+	0.969473i	$0.578853\pi$
$104$	0	0
$105$	105.000	0.0975900
$106$	0	0
$107$	−1963.09	−1.77363	−0.886817	−	0.462122i	$-0.847088\pi$
$107$	−1963.09	−1.77363	−0.886817	+	0.462122i	$0.847088\pi$
$108$	0	0
$109$	545.176	0.479068	0.239534	−	0.970888i	$-0.423005\pi$
$109$	545.176	0.479068	0.239534	+	0.970888i	$0.423005\pi$
$110$	0	0
$111$	−758.029	−0.648188
$112$	0	0
$113$	−231.823	−0.192992	−0.0964961	−	0.995333i	$-0.530764\pi$
$113$	−231.823	−0.192992	−0.0964961	+	0.995333i	$0.530764\pi$
$114$	0	0
$115$	520.833	0.422329
$116$	0	0
$117$	−392.912	−0.310468
$118$	0	0
$119$	−473.598	−0.364829
$120$	0	0
$121$	1027.90	0.772279
$122$	0	0
$123$	828.823	0.607581
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−2372.90	−1.65796	−0.828979	−	0.559280i	$-0.811078\pi$
$127$	−2372.90	−1.65796	−0.828979	+	0.559280i	$0.811078\pi$
$128$	0	0
$129$	277.913	0.189681
$130$	0	0
$131$	−1200.04	−0.800364	−0.400182	−	0.916436i	$-0.631053\pi$
$131$	−1200.04	−0.800364	−0.400182	+	0.916436i	$0.631053\pi$
$132$	0	0
$133$	652.784	0.425591
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	2781.25	1.73444	0.867221	−	0.497924i	$-0.165904\pi$
$137$	2781.25	1.73444	0.867221	+	0.497924i	$0.165904\pi$
$138$	0	0
$139$	1245.60	0.760078	0.380039	−	0.924971i	$-0.375911\pi$
$139$	1245.60	0.760078	0.380039	+	0.924971i	$0.375911\pi$
$140$	0	0
$141$	1748.38	1.04426
$142$	0	0
$143$	2120.35	1.23995
$144$	0	0
$145$	−293.675	−0.168196
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	19.4046	0.0106690	0.00533452	−	0.999986i	$-0.498302\pi$
$149$	19.4046	0.0106690	0.00533452	+	0.999986i	$0.498302\pi$
$150$	0	0
$151$	2349.80	1.26638	0.633192	−	0.773995i	$-0.281744\pi$
$151$	2349.80	1.26638	0.633192	+	0.773995i	$0.281744\pi$
$152$	0	0
$153$	−608.912	−0.321749
$154$	0	0
$155$	45.4416	0.0235481
$156$	0	0
$157$	−3898.46	−1.98172	−0.990862	−	0.134880i	$-0.956935\pi$
$157$	−3898.46	−1.98172	−0.990862	+	0.134880i	$0.956935\pi$
$158$	0	0
$159$	1869.06	0.932239
$160$	0	0
$161$	729.166	0.356934
$162$	0	0
$163$	−1527.54	−0.734024	−0.367012	−	0.930216i	$-0.619619\pi$
$163$	−1527.54	−0.734024	−0.367012	+	0.930216i	$0.619619\pi$
$164$	0	0
$165$	−728.528	−0.343732
$166$	0	0
$167$	998.518	0.462681	0.231340	−	0.972873i	$-0.425689\pi$
$167$	998.518	0.462681	0.231340	+	0.972873i	$0.425689\pi$
$168$	0	0
$169$	−291.079	−0.132489
$170$	0	0
$171$	839.294	0.375336
$172$	0	0
$173$	685.253	0.301149	0.150575	−	0.988599i	$-0.451888\pi$
$173$	685.253	0.301149	0.150575	+	0.988599i	$0.451888\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	1575.00	0.668836
$178$	0	0
$179$	−1025.58	−0.428245	−0.214122	−	0.976807i	$-0.568689\pi$
$179$	−1025.58	−0.428245	−0.214122	+	0.976807i	$0.568689\pi$
$180$	0	0
$181$	2899.40	1.19067	0.595333	−	0.803479i	$-0.297020\pi$
$181$	2899.40	1.19067	0.595333	+	0.803479i	$0.297020\pi$
$182$	0	0
$183$	−1058.38	−0.427529
$184$	0	0
$185$	−1263.38	−0.502084
$186$	0	0
$187$	3285.99	1.28500
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−1074.18	−0.406939	−0.203469	−	0.979081i	$-0.565222\pi$
$191$	−1074.18	−0.406939	−0.203469	+	0.979081i	$0.565222\pi$
$192$	0	0
$193$	898.999	0.335292	0.167646	−	0.985847i	$-0.446383\pi$
$193$	898.999	0.335292	0.167646	+	0.985847i	$0.446383\pi$
$194$	0	0
$195$	−654.853	−0.240487
$196$	0	0
$197$	3063.63	1.10799	0.553996	−	0.832519i	$-0.313102\pi$
$197$	3063.63	1.10799	0.553996	+	0.832519i	$0.313102\pi$
$198$	0	0
$199$	949.522	0.338240	0.169120	−	0.985595i	$-0.445907\pi$
$199$	949.522	0.338240	0.169120	+	0.985595i	$0.445907\pi$
$200$	0	0
$201$	2209.56	0.775375
$202$	0	0
$203$	−411.145	−0.142151
$204$	0	0
$205$	1381.37	0.470630
$206$	0	0
$207$	937.499	0.314786
$208$	0	0
$209$	−4529.25	−1.49902
$210$	0	0
$211$	−2306.64	−0.752587	−0.376294	−	0.926500i	$-0.622802\pi$
$211$	−2306.64	−0.752587	−0.376294	+	0.926500i	$0.622802\pi$
$212$	0	0
$213$	1476.79	0.475062
$214$	0	0
$215$	463.188	0.146926
$216$	0	0
$217$	63.6182	0.0199018
$218$	0	0
$219$	3494.26	1.07817
$220$	0	0
$221$	2953.69	0.899033
$222$	0	0
$223$	3227.61	0.969222	0.484611	−	0.874730i	$-0.338961\pi$
$223$	3227.61	0.969222	0.484611	+	0.874730i	$0.338961\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	637.820	0.186492	0.0932458	−	0.995643i	$-0.470276\pi$
$227$	637.820	0.186492	0.0932458	+	0.995643i	$0.470276\pi$
$228$	0	0
$229$	544.774	0.157204	0.0786019	−	0.996906i	$-0.474954\pi$
$229$	544.774	0.157204	0.0786019	+	0.996906i	$0.474954\pi$
$230$	0	0
$231$	−1019.94	−0.290507
$232$	0	0
$233$	−5748.54	−1.61631	−0.808154	−	0.588972i	$-0.799533\pi$
$233$	−5748.54	−1.61631	−0.808154	+	0.588972i	$0.799533\pi$
$234$	0	0
$235$	2913.97	0.808878
$236$	0	0
$237$	2616.59	0.717154
$238$	0	0
$239$	2678.10	0.724820	0.362410	−	0.932019i	$-0.381954\pi$
$239$	2678.10	0.724820	0.362410	+	0.932019i	$0.381954\pi$
$240$	0	0
$241$	−2202.16	−0.588604	−0.294302	−	0.955713i	$-0.595087\pi$
$241$	−2202.16	−0.588604	−0.294302	+	0.955713i	$0.595087\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	−4071.21	−1.04877
$248$	0	0
$249$	1588.76	0.404353
$250$	0	0
$251$	5716.90	1.43764	0.718820	−	0.695196i	$-0.244683\pi$
$251$	5716.90	1.43764	0.718820	+	0.695196i	$0.244683\pi$
$252$	0	0
$253$	−5059.22	−1.25719
$254$	0	0
$255$	−1014.85	−0.249226
$256$	0	0
$257$	4724.29	1.14666	0.573332	−	0.819323i	$-0.305650\pi$
$257$	4724.29	1.14666	0.573332	+	0.819323i	$0.305650\pi$
$258$	0	0
$259$	−1768.73	−0.424339
$260$	0	0
$261$	−528.616	−0.125366
$262$	0	0
$263$	−5975.36	−1.40097	−0.700487	−	0.713665i	$-0.747034\pi$
$263$	−5975.36	−1.40097	−0.700487	+	0.713665i	$0.747034\pi$
$264$	0	0
$265$	3115.10	0.722109
$266$	0	0
$267$	−1155.65	−0.264886
$268$	0	0
$269$	4486.11	1.01681	0.508407	−	0.861117i	$-0.330234\pi$
$269$	4486.11	1.01681	0.508407	+	0.861117i	$0.330234\pi$
$270$	0	0
$271$	−3827.68	−0.857989	−0.428994	−	0.903307i	$-0.641132\pi$
$271$	−3827.68	−0.857989	−0.428994	+	0.903307i	$0.641132\pi$
$272$	0	0
$273$	−916.794	−0.203249
$274$	0	0
$275$	−1214.21	−0.266254
$276$	0	0
$277$	−3420.54	−0.741950	−0.370975	−	0.928643i	$-0.620976\pi$
$277$	−3420.54	−0.741950	−0.370975	+	0.928643i	$0.620976\pi$
$278$	0	0
$279$	81.7948	0.0175517
$280$	0	0
$281$	5235.92	1.11156	0.555781	−	0.831329i	$-0.312419\pi$
$281$	5235.92	1.11156	0.555781	+	0.831329i	$0.312419\pi$
$282$	0	0
$283$	6985.88	1.46738	0.733688	−	0.679486i	$-0.237797\pi$
$283$	6985.88	1.46738	0.733688	+	0.679486i	$0.237797\pi$
$284$	0	0
$285$	1398.82	0.290734
$286$	0	0
$287$	1933.92	0.397755
$288$	0	0
$289$	−335.550	−0.0682984
$290$	0	0
$291$	−1391.68	−0.280349
$292$	0	0
$293$	7399.70	1.47541	0.737705	−	0.675123i	$-0.235910\pi$
$293$	7399.70	1.47541	0.737705	+	0.675123i	$0.235910\pi$
$294$	0	0
$295$	2625.00	0.518078
$296$	0	0
$297$	−1311.35	−0.256203
$298$	0	0
$299$	−4547.58	−0.879577
$300$	0	0
$301$	648.463	0.124175
$302$	0	0
$303$	−1298.17	−0.246133
$304$	0	0
$305$	−1763.97	−0.331163
$306$	0	0
$307$	−2668.64	−0.496116	−0.248058	−	0.968745i	$-0.579792\pi$
$307$	−2668.64	−0.496116	−0.248058	+	0.968745i	$0.579792\pi$
$308$	0	0
$309$	−1537.88	−0.283129
$310$	0	0
$311$	−6189.25	−1.12849	−0.564244	−	0.825608i	$-0.690832\pi$
$311$	−6189.25	−1.12849	−0.564244	+	0.825608i	$0.690832\pi$
$312$	0	0
$313$	−2921.59	−0.527598	−0.263799	−	0.964578i	$-0.584976\pi$
$313$	−2921.59	−0.527598	−0.263799	+	0.964578i	$0.584976\pi$
$314$	0	0
$315$	315.000	0.0563436
$316$	0	0
$317$	9825.56	1.74088	0.870439	−	0.492276i	$-0.163835\pi$
$317$	9825.56	1.74088	0.870439	+	0.492276i	$0.163835\pi$
$318$	0	0
$319$	2852.68	0.500687
$320$	0	0
$321$	−5889.26	−1.02401
$322$	0	0
$323$	−6309.33	−1.08687
$324$	0	0
$325$	−1091.42	−0.186281
$326$	0	0
$327$	1635.53	0.276590
$328$	0	0
$329$	4079.56	0.683627
$330$	0	0
$331$	9258.17	1.53739	0.768693	−	0.639618i	$-0.220907\pi$
$331$	9258.17	1.53739	0.768693	+	0.639618i	$0.220907\pi$
$332$	0	0
$333$	−2274.09	−0.374232
$334$	0	0
$335$	3682.60	0.600603
$336$	0	0
$337$	−3693.98	−0.597103	−0.298552	−	0.954394i	$-0.596503\pi$
$337$	−3693.98	−0.597103	−0.298552	+	0.954394i	$0.596503\pi$
$338$	0	0
$339$	−695.470	−0.111424
$340$	0	0
$341$	−441.406	−0.0700982
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	1562.50	0.243832
$346$	0	0
$347$	−3832.83	−0.592960	−0.296480	−	0.955039i	$-0.595813\pi$
$347$	−3832.83	−0.592960	−0.296480	+	0.955039i	$0.595813\pi$
$348$	0	0
$349$	8325.22	1.27690	0.638451	−	0.769662i	$-0.279575\pi$
$349$	8325.22	1.27690	0.638451	+	0.769662i	$0.279575\pi$
$350$	0	0
$351$	−1178.74	−0.179248
$352$	0	0
$353$	−8991.52	−1.35572	−0.677862	−	0.735189i	$-0.737093\pi$
$353$	−8991.52	−1.35572	−0.677862	+	0.735189i	$0.737093\pi$
$354$	0	0
$355$	2461.32	0.367981
$356$	0	0
$357$	−1420.79	−0.210634
$358$	0	0
$359$	12893.8	1.89557	0.947783	−	0.318917i	$-0.103319\pi$
$359$	12893.8	1.89557	0.947783	+	0.318917i	$0.103319\pi$
$360$	0	0
$361$	1837.46	0.267891
$362$	0	0
$363$	3083.71	0.445875
$364$	0	0
$365$	5823.77	0.835150
$366$	0	0
$367$	7480.17	1.06393	0.531964	−	0.846767i	$-0.321454\pi$
$367$	7480.17	1.06393	0.531964	+	0.846767i	$0.321454\pi$
$368$	0	0
$369$	2486.47	0.350787
$370$	0	0
$371$	4361.14	0.610293
$372$	0	0
$373$	−3523.32	−0.489090	−0.244545	−	0.969638i	$-0.578639\pi$
$373$	−3523.32	−0.489090	−0.244545	+	0.969638i	$0.578639\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	2564.19	0.350298
$378$	0	0
$379$	−13515.4	−1.83177	−0.915886	−	0.401438i	$-0.868510\pi$
$379$	−13515.4	−1.83177	−0.915886	+	0.401438i	$0.868510\pi$
$380$	0	0
$381$	−7118.69	−0.957222
$382$	0	0
$383$	657.182	0.0876774	0.0438387	−	0.999039i	$-0.486041\pi$
$383$	657.182	0.0876774	0.0438387	+	0.999039i	$0.486041\pi$
$384$	0	0
$385$	−1699.90	−0.225026
$386$	0	0
$387$	833.738	0.109512
$388$	0	0
$389$	−9741.87	−1.26975	−0.634875	−	0.772615i	$-0.718948\pi$
$389$	−9741.87	−1.26975	−0.634875	+	0.772615i	$0.718948\pi$
$390$	0	0
$391$	−7047.58	−0.911538
$392$	0	0
$393$	−3600.11	−0.462091
$394$	0	0
$395$	4360.98	0.555505
$396$	0	0
$397$	4407.42	0.557184	0.278592	−	0.960410i	$-0.410132\pi$
$397$	4407.42	0.557184	0.278592	+	0.960410i	$0.410132\pi$
$398$	0	0
$399$	1958.35	0.245715
$400$	0	0
$401$	−11569.5	−1.44078	−0.720391	−	0.693568i	$-0.756037\pi$
$401$	−11569.5	−1.44078	−0.720391	+	0.693568i	$0.756037\pi$
$402$	0	0
$403$	−396.767	−0.0490431
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	12272.1	1.49461
$408$	0	0
$409$	−3083.03	−0.372729	−0.186364	−	0.982481i	$-0.559670\pi$
$409$	−3083.03	−0.372729	−0.186364	+	0.982481i	$0.559670\pi$
$410$	0	0
$411$	8343.76	1.00138
$412$	0	0
$413$	3674.99	0.437856
$414$	0	0
$415$	2647.94	0.313210
$416$	0	0
$417$	3736.81	0.438831
$418$	0	0
$419$	5415.21	0.631385	0.315692	−	0.948862i	$-0.397763\pi$
$419$	5415.21	0.631385	0.315692	+	0.948862i	$0.397763\pi$
$420$	0	0
$421$	4188.34	0.484863	0.242432	−	0.970168i	$-0.422055\pi$
$421$	4188.34	0.484863	0.242432	+	0.970168i	$0.422055\pi$
$422$	0	0
$423$	5245.15	0.602902
$424$	0	0
$425$	−1691.42	−0.193049
$426$	0	0
$427$	−2469.56	−0.279884
$428$	0	0
$429$	6361.05	0.715884
$430$	0	0
$431$	9108.41	1.01795	0.508975	−	0.860781i	$-0.330024\pi$
$431$	9108.41	1.01795	0.508975	+	0.860781i	$0.330024\pi$
$432$	0	0
$433$	−16847.0	−1.86979	−0.934893	−	0.354930i	$-0.884505\pi$
$433$	−16847.0	−1.86979	−0.934893	+	0.354930i	$0.884505\pi$
$434$	0	0
$435$	−881.026	−0.0971080
$436$	0	0
$437$	9714.03	1.06335
$438$	0	0
$439$	−8434.14	−0.916946	−0.458473	−	0.888708i	$-0.651603\pi$
$439$	−8434.14	−0.916946	−0.458473	+	0.888708i	$0.651603\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	4298.49	0.461010	0.230505	−	0.973071i	$-0.425962\pi$
$443$	4298.49	0.461010	0.230505	+	0.973071i	$0.425962\pi$
$444$	0	0
$445$	−1926.08	−0.205180
$446$	0	0
$447$	58.2139	0.00615978
$448$	0	0
$449$	10545.0	1.10835	0.554173	−	0.832402i	$-0.313035\pi$
$449$	10545.0	1.10835	0.554173	+	0.832402i	$0.313035\pi$
$450$	0	0
$451$	−13418.2	−1.40098
$452$	0	0
$453$	7049.40	0.731147
$454$	0	0
$455$	−1527.99	−0.157436
$456$	0	0
$457$	11952.4	1.22344	0.611719	−	0.791075i	$-0.290478\pi$
$457$	11952.4	1.22344	0.611719	+	0.791075i	$0.290478\pi$
$458$	0	0
$459$	−1826.74	−0.185762
$460$	0	0
$461$	−17200.9	−1.73780	−0.868900	−	0.494988i	$-0.835173\pi$
$461$	−17200.9	−1.73780	−0.868900	+	0.494988i	$0.835173\pi$
$462$	0	0
$463$	10368.7	1.04076	0.520381	−	0.853934i	$-0.325790\pi$
$463$	10368.7	1.04076	0.520381	+	0.853934i	$0.325790\pi$
$464$	0	0
$465$	136.325	0.0135955
$466$	0	0
$467$	16879.5	1.67257	0.836284	−	0.548296i	$-0.184723\pi$
$467$	16879.5	1.67257	0.836284	+	0.548296i	$0.184723\pi$
$468$	0	0
$469$	5155.64	0.507602
$470$	0	0
$471$	−11695.4	−1.14415
$472$	0	0
$473$	−4499.27	−0.437371
$474$	0	0
$475$	2331.37	0.225201
$476$	0	0
$477$	5607.17	0.538228
$478$	0	0
$479$	−7329.12	−0.699115	−0.349558	−	0.936915i	$-0.613668\pi$
$479$	−7329.12	−0.699115	−0.349558	+	0.936915i	$0.613668\pi$
$480$	0	0
$481$	11031.0	1.04568
$482$	0	0
$483$	2187.50	0.206076
$484$	0	0
$485$	−2319.46	−0.217157
$486$	0	0
$487$	17209.5	1.60131	0.800655	−	0.599125i	$-0.204485\pi$
$487$	17209.5	1.60131	0.800655	+	0.599125i	$0.204485\pi$
$488$	0	0
$489$	−4582.61	−0.423789
$490$	0	0
$491$	−11392.5	−1.04712	−0.523560	−	0.851989i	$-0.675396\pi$
$491$	−11392.5	−1.04712	−0.523560	+	0.851989i	$0.675396\pi$
$492$	0	0
$493$	3973.83	0.363027
$494$	0	0
$495$	−2185.58	−0.198454
$496$	0	0
$497$	3445.85	0.311001
$498$	0	0
$499$	−19079.4	−1.71164	−0.855822	−	0.517271i	$-0.826948\pi$
$499$	−19079.4	−1.71164	−0.855822	+	0.517271i	$0.826948\pi$
$500$	0	0
$501$	2995.55	0.267129
$502$	0	0
$503$	13499.4	1.19663	0.598317	−	0.801259i	$-0.295836\pi$
$503$	13499.4	1.19663	0.598317	+	0.801259i	$0.295836\pi$
$504$	0	0
$505$	−2163.62	−0.190654
$506$	0	0
$507$	−873.237	−0.0764928
$508$	0	0
$509$	−4328.85	−0.376960	−0.188480	−	0.982077i	$-0.560356\pi$
$509$	−4328.85	−0.376960	−0.188480	+	0.982077i	$0.560356\pi$
$510$	0	0
$511$	8153.27	0.705831
$512$	0	0
$513$	2517.88	0.216700
$514$	0	0
$515$	−2563.13	−0.219311
$516$	0	0
$517$	−28305.5	−2.40788
$518$	0	0
$519$	2055.76	0.173869
$520$	0	0
$521$	19395.7	1.63098	0.815490	−	0.578771i	$-0.196468\pi$
$521$	19395.7	1.63098	0.815490	+	0.578771i	$0.196468\pi$
$522$	0	0
$523$	−20413.8	−1.70675	−0.853377	−	0.521294i	$-0.825450\pi$
$523$	−20413.8	−1.70675	−0.853377	+	0.521294i	$0.825450\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	−614.887	−0.0508252
$528$	0	0
$529$	−1316.34	−0.108189
$530$	0	0
$531$	4724.99	0.386153
$532$	0	0
$533$	−12061.3	−0.980171
$534$	0	0
$535$	−9815.43	−0.793193
$536$	0	0
$537$	−3076.75	−0.247247
$538$	0	0
$539$	−2379.86	−0.190181
$540$	0	0
$541$	−4919.18	−0.390928	−0.195464	−	0.980711i	$-0.562621\pi$
$541$	−4919.18	−0.390928	−0.195464	+	0.980711i	$0.562621\pi$
$542$	0	0
$543$	8698.19	0.687431
$544$	0	0
$545$	2725.88	0.214246
$546$	0	0
$547$	−15334.2	−1.19862	−0.599308	−	0.800518i	$-0.704558\pi$
$547$	−15334.2	−1.19862	−0.599308	+	0.800518i	$0.704558\pi$
$548$	0	0
$549$	−3175.15	−0.246834
$550$	0	0
$551$	−5477.33	−0.423488
$552$	0	0
$553$	6105.37	0.469487
$554$	0	0
$555$	−3790.14	−0.289879
$556$	0	0
$557$	−8613.78	−0.655256	−0.327628	−	0.944807i	$-0.606249\pi$
$557$	−8613.78	−0.655256	−0.327628	+	0.944807i	$0.606249\pi$
$558$	0	0
$559$	−4044.26	−0.306000
$560$	0	0
$561$	9857.98	0.741897
$562$	0	0
$563$	2320.81	0.173731	0.0868654	−	0.996220i	$-0.472315\pi$
$563$	2320.81	0.173731	0.0868654	+	0.996220i	$0.472315\pi$
$564$	0	0
$565$	−1159.12	−0.0863087
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	−1736.04	−0.127906	−0.0639529	−	0.997953i	$-0.520371\pi$
$569$	−1736.04	−0.127906	−0.0639529	+	0.997953i	$0.520371\pi$
$570$	0	0
$571$	−23897.8	−1.75148	−0.875738	−	0.482786i	$-0.839625\pi$
$571$	−23897.8	−1.75148	−0.875738	+	0.482786i	$0.839625\pi$
$572$	0	0
$573$	−3222.55	−0.234946
$574$	0	0
$575$	2604.16	0.188871
$576$	0	0
$577$	8029.26	0.579311	0.289655	−	0.957131i	$-0.406459\pi$
$577$	8029.26	0.579311	0.289655	+	0.957131i	$0.406459\pi$
$578$	0	0
$579$	2697.00	0.193581
$580$	0	0
$581$	3707.12	0.264711
$582$	0	0
$583$	−30259.1	−2.14958
$584$	0	0
$585$	−1964.56	−0.138845
$586$	0	0
$587$	8015.14	0.563578	0.281789	−	0.959476i	$-0.409072\pi$
$587$	8015.14	0.563578	0.281789	+	0.959476i	$0.409072\pi$
$588$	0	0
$589$	847.529	0.0592900
$590$	0	0
$591$	9190.88	0.639700
$592$	0	0
$593$	12820.3	0.887801	0.443901	−	0.896076i	$-0.353594\pi$
$593$	12820.3	0.887801	0.443901	+	0.896076i	$0.353594\pi$
$594$	0	0
$595$	−2367.99	−0.163157
$596$	0	0
$597$	2848.57	0.195283
$598$	0	0
$599$	15330.5	1.04572	0.522860	−	0.852418i	$-0.324865\pi$
$599$	15330.5	1.04572	0.522860	+	0.852418i	$0.324865\pi$
$600$	0	0
$601$	107.658	0.00730694	0.00365347	−	0.999993i	$-0.498837\pi$
$601$	107.658	0.00730694	0.00365347	+	0.999993i	$0.498837\pi$
$602$	0	0
$603$	6628.68	0.447663
$604$	0	0
$605$	5139.52	0.345374
$606$	0	0
$607$	8213.06	0.549189	0.274595	−	0.961560i	$-0.411456\pi$
$607$	8213.06	0.549189	0.274595	+	0.961560i	$0.411456\pi$
$608$	0	0
$609$	−1233.44	−0.0820712
$610$	0	0
$611$	−25443.0	−1.68463
$612$	0	0
$613$	1242.60	0.0818732	0.0409366	−	0.999162i	$-0.486966\pi$
$613$	1242.60	0.0818732	0.0409366	+	0.999162i	$0.486966\pi$
$614$	0	0
$615$	4144.11	0.271718
$616$	0	0
$617$	−13170.6	−0.859367	−0.429683	−	0.902980i	$-0.641375\pi$
$617$	−13170.6	−0.859367	−0.429683	+	0.902980i	$0.641375\pi$
$618$	0	0
$619$	−19774.1	−1.28399	−0.641993	−	0.766711i	$-0.721892\pi$
$619$	−19774.1	−1.28399	−0.641993	+	0.766711i	$0.721892\pi$
$620$	0	0
$621$	2812.50	0.181742
$622$	0	0
$623$	−2696.51	−0.173409
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−13587.8	−0.865459
$628$	0	0
$629$	17095.3	1.08368
$630$	0	0
$631$	14308.8	0.902735	0.451367	−	0.892338i	$-0.350936\pi$
$631$	14308.8	0.902735	0.451367	+	0.892338i	$0.350936\pi$
$632$	0	0
$633$	−6919.93	−0.434507
$634$	0	0
$635$	−11864.5	−0.741461
$636$	0	0
$637$	−2139.19	−0.133058
$638$	0	0
$639$	4430.38	0.274277
$640$	0	0
$641$	11537.5	0.710925	0.355463	−	0.934691i	$-0.384323\pi$
$641$	11537.5	0.710925	0.355463	+	0.934691i	$0.384323\pi$
$642$	0	0
$643$	−19603.0	−1.20228	−0.601139	−	0.799144i	$-0.705286\pi$
$643$	−19603.0	−1.20228	−0.601139	+	0.799144i	$0.705286\pi$
$644$	0	0
$645$	1389.56	0.0848279
$646$	0	0
$647$	−21650.0	−1.31553	−0.657765	−	0.753223i	$-0.728498\pi$
$647$	−21650.0	−1.31553	−0.657765	+	0.753223i	$0.728498\pi$
$648$	0	0
$649$	−25498.4	−1.54222
$650$	0	0
$651$	190.855	0.0114903
$652$	0	0
$653$	−2927.33	−0.175429	−0.0877145	−	0.996146i	$-0.527956\pi$
$653$	−2927.33	−0.175429	−0.0877145	+	0.996146i	$0.527956\pi$
$654$	0	0
$655$	−6000.18	−0.357934
$656$	0	0
$657$	10482.8	0.622484
$658$	0	0
$659$	4778.76	0.282480	0.141240	−	0.989975i	$-0.454891\pi$
$659$	4778.76	0.282480	0.141240	+	0.989975i	$0.454891\pi$
$660$	0	0
$661$	−31510.3	−1.85417	−0.927086	−	0.374849i	$-0.877695\pi$
$661$	−31510.3	−1.85417	−0.927086	+	0.374849i	$0.877695\pi$
$662$	0	0
$663$	8861.06	0.519057
$664$	0	0
$665$	3263.92	0.190330
$666$	0	0
$667$	−6118.23	−0.355170
$668$	0	0
$669$	9682.82	0.559581
$670$	0	0
$671$	17134.7	0.985808
$672$	0	0
$673$	−8992.15	−0.515040	−0.257520	−	0.966273i	$-0.582905\pi$
$673$	−8992.15	−0.515040	−0.257520	+	0.966273i	$0.582905\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	19340.8	1.09797	0.548985	−	0.835832i	$-0.315014\pi$
$677$	19340.8	1.09797	0.548985	+	0.835832i	$0.315014\pi$
$678$	0	0
$679$	−3247.25	−0.183531
$680$	0	0
$681$	1913.46	0.107671
$682$	0	0
$683$	4255.14	0.238387	0.119194	−	0.992871i	$-0.461969\pi$
$683$	4255.14	0.238387	0.119194	+	0.992871i	$0.461969\pi$
$684$	0	0
$685$	13906.3	0.775666
$686$	0	0
$687$	1634.32	0.0907616
$688$	0	0
$689$	−27199.1	−1.50392
$690$	0	0
$691$	17505.5	0.963733	0.481867	−	0.876245i	$-0.339959\pi$
$691$	17505.5	0.963733	0.481867	+	0.876245i	$0.339959\pi$
$692$	0	0
$693$	−3059.82	−0.167724
$694$	0	0
$695$	6228.02	0.339917
$696$	0	0
$697$	−18691.8	−1.01579
$698$	0	0
$699$	−17245.6	−0.933175
$700$	0	0
$701$	−3240.77	−0.174611	−0.0873054	−	0.996182i	$-0.527826\pi$
$701$	−3240.77	−0.174611	−0.0873054	+	0.996182i	$0.527826\pi$
$702$	0	0
$703$	−23563.3	−1.26416
$704$	0	0
$705$	8741.91	0.467006
$706$	0	0
$707$	−3029.07	−0.161132
$708$	0	0
$709$	19949.3	1.05672	0.528358	−	0.849022i	$-0.322808\pi$
$709$	19949.3	1.05672	0.528358	+	0.849022i	$0.322808\pi$
$710$	0	0
$711$	7849.76	0.414049
$712$	0	0
$713$	946.698	0.0497253
$714$	0	0
$715$	10601.7	0.554522
$716$	0	0
$717$	8034.30	0.418475
$718$	0	0
$719$	11259.4	0.584011	0.292006	−	0.956417i	$-0.405677\pi$
$719$	11259.4	0.584011	0.292006	+	0.956417i	$0.405677\pi$
$720$	0	0
$721$	−3588.38	−0.185351
$722$	0	0
$723$	−6606.47	−0.339830
$724$	0	0
$725$	−1468.38	−0.0752195
$726$	0	0
$727$	12228.5	0.623840	0.311920	−	0.950108i	$-0.399028\pi$
$727$	12228.5	0.623840	0.311920	+	0.950108i	$0.399028\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−6267.56	−0.317119
$732$	0	0
$733$	−26635.1	−1.34214	−0.671072	−	0.741392i	$-0.734166\pi$
$733$	−26635.1	−1.34214	−0.671072	+	0.741392i	$0.734166\pi$
$734$	0	0
$735$	735.000	0.0368856
$736$	0	0
$737$	−35771.7	−1.78788
$738$	0	0
$739$	6074.00	0.302349	0.151174	−	0.988507i	$-0.451694\pi$
$739$	6074.00	0.302349	0.151174	+	0.988507i	$0.451694\pi$
$740$	0	0
$741$	−12213.6	−0.605505
$742$	0	0
$743$	4016.87	0.198337	0.0991686	−	0.995071i	$-0.468382\pi$
$743$	4016.87	0.198337	0.0991686	+	0.995071i	$0.468382\pi$
$744$	0	0
$745$	97.0231	0.00477134
$746$	0	0
$747$	4766.29	0.233453
$748$	0	0
$749$	−13741.6	−0.670370
$750$	0	0
$751$	23913.2	1.16192	0.580962	−	0.813931i	$-0.302676\pi$
$751$	23913.2	1.16192	0.580962	+	0.813931i	$0.302676\pi$
$752$	0	0
$753$	17150.7	0.830022
$754$	0	0
$755$	11749.0	0.566344
$756$	0	0
$757$	31044.9	1.49055	0.745275	−	0.666758i	$-0.232318\pi$
$757$	31044.9	1.49055	0.745275	+	0.666758i	$0.232318\pi$
$758$	0	0
$759$	−15177.6	−0.725842
$760$	0	0
$761$	14011.8	0.667446	0.333723	−	0.942671i	$-0.391695\pi$
$761$	14011.8	0.667446	0.333723	+	0.942671i	$0.391695\pi$
$762$	0	0
$763$	3816.23	0.181071
$764$	0	0
$765$	−3044.56	−0.143891
$766$	0	0
$767$	−22919.8	−1.07899
$768$	0	0
$769$	3342.49	0.156740	0.0783701	−	0.996924i	$-0.475028\pi$
$769$	3342.49	0.156740	0.0783701	+	0.996924i	$0.475028\pi$
$770$	0	0
$771$	14172.9	0.662027
$772$	0	0
$773$	−21074.6	−0.980594	−0.490297	−	0.871555i	$-0.663112\pi$
$773$	−21074.6	−0.980594	−0.490297	+	0.871555i	$0.663112\pi$
$774$	0	0
$775$	227.208	0.0105310
$776$	0	0
$777$	−5306.20	−0.244992
$778$	0	0
$779$	25763.9	1.18496
$780$	0	0
$781$	−23908.5	−1.09541
$782$	0	0
$783$	−1585.85	−0.0723800
$784$	0	0
$785$	−19492.3	−0.886254
$786$	0	0
$787$	−21394.8	−0.969048	−0.484524	−	0.874778i	$-0.661007\pi$
$787$	−21394.8	−0.969048	−0.484524	+	0.874778i	$0.661007\pi$
$788$	0	0
$789$	−17926.1	−0.808853
$790$	0	0
$791$	−1622.76	−0.0729442
$792$	0	0
$793$	15401.9	0.689706
$794$	0	0
$795$	9345.29	0.416910
$796$	0	0
$797$	20645.0	0.917547	0.458773	−	0.888553i	$-0.348289\pi$
$797$	20645.0	0.917547	0.458773	+	0.888553i	$0.348289\pi$
$798$	0	0
$799$	−39430.0	−1.74585
$800$	0	0
$801$	−3466.95	−0.152932
$802$	0	0
$803$	−56570.4	−2.48608
$804$	0	0
$805$	3645.83	0.159626
$806$	0	0
$807$	13458.3	0.587058
$808$	0	0
$809$	−15939.0	−0.692688	−0.346344	−	0.938108i	$-0.612577\pi$
$809$	−15939.0	−0.692688	−0.346344	+	0.938108i	$0.612577\pi$
$810$	0	0
$811$	−22829.2	−0.988460	−0.494230	−	0.869331i	$-0.664550\pi$
$811$	−22829.2	−0.988460	−0.494230	+	0.869331i	$0.664550\pi$
$812$	0	0
$813$	−11483.0	−0.495360
$814$	0	0
$815$	−7637.69	−0.328266
$816$	0	0
$817$	8638.90	0.369935
$818$	0	0
$819$	−2750.38	−0.117346
$820$	0	0
$821$	−5700.22	−0.242313	−0.121157	−	0.992633i	$-0.538660\pi$
$821$	−5700.22	−0.242313	−0.121157	+	0.992633i	$0.538660\pi$
$822$	0	0
$823$	32438.0	1.37390	0.686948	−	0.726707i	$-0.258950\pi$
$823$	32438.0	1.37390	0.686948	+	0.726707i	$0.258950\pi$
$824$	0	0
$825$	−3642.64	−0.153722
$826$	0	0
$827$	12762.6	0.536638	0.268319	−	0.963330i	$-0.413532\pi$
$827$	12762.6	0.536638	0.268319	+	0.963330i	$0.413532\pi$
$828$	0	0
$829$	−30766.7	−1.28899	−0.644494	−	0.764609i	$-0.722932\pi$
$829$	−30766.7	−1.28899	−0.644494	+	0.764609i	$0.722932\pi$
$830$	0	0
$831$	−10261.6	−0.428365
$832$	0	0
$833$	−3315.19	−0.137892
$834$	0	0
$835$	4992.59	0.206917
$836$	0	0
$837$	245.384	0.0101335
$838$	0	0
$839$	9779.71	0.402423	0.201212	−	0.979548i	$-0.435512\pi$
$839$	9779.71	0.402423	0.201212	+	0.979548i	$0.435512\pi$
$840$	0	0
$841$	−20939.2	−0.858551
$842$	0	0
$843$	15707.8	0.641760
$844$	0	0
$845$	−1455.40	−0.0592510
$846$	0	0
$847$	7195.32	0.291894
$848$	0	0
$849$	20957.6	0.847190
$850$	0	0
$851$	−26320.4	−1.06023
$852$	0	0
$853$	24201.5	0.971445	0.485723	−	0.874113i	$-0.338556\pi$
$853$	24201.5	0.971445	0.485723	+	0.874113i	$0.338556\pi$
$854$	0	0
$855$	4196.47	0.167855
$856$	0	0
$857$	21036.7	0.838507	0.419254	−	0.907869i	$-0.362292\pi$
$857$	21036.7	0.838507	0.419254	+	0.907869i	$0.362292\pi$
$858$	0	0
$859$	6179.19	0.245438	0.122719	−	0.992441i	$-0.460839\pi$
$859$	6179.19	0.245438	0.122719	+	0.992441i	$0.460839\pi$
$860$	0	0
$861$	5801.76	0.229644
$862$	0	0
$863$	−50256.2	−1.98232	−0.991160	−	0.132671i	$-0.957644\pi$
$863$	−50256.2	−1.98232	−0.991160	+	0.132671i	$0.957644\pi$
$864$	0	0
$865$	3426.27	0.134678
$866$	0	0
$867$	−1006.65	−0.0394321
$868$	0	0
$869$	−42361.2	−1.65363
$870$	0	0
$871$	−32154.1	−1.25086
$872$	0	0
$873$	−4175.03	−0.161860
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−9175.95	−0.353306	−0.176653	−	0.984273i	$-0.556527\pi$
$877$	−9175.95	−0.353306	−0.176653	+	0.984273i	$0.556527\pi$
$878$	0	0
$879$	22199.1	0.851828
$880$	0	0
$881$	−26172.7	−1.00089	−0.500444	−	0.865769i	$-0.666830\pi$
$881$	−26172.7	−1.00089	−0.500444	+	0.865769i	$0.666830\pi$
$882$	0	0
$883$	−18615.5	−0.709471	−0.354736	−	0.934967i	$-0.615429\pi$
$883$	−18615.5	−0.709471	−0.354736	+	0.934967i	$0.615429\pi$
$884$	0	0
$885$	7874.99	0.299113
$886$	0	0
$887$	12837.8	0.485964	0.242982	−	0.970031i	$-0.421874\pi$
$887$	12837.8	0.485964	0.242982	+	0.970031i	$0.421874\pi$
$888$	0	0
$889$	−16610.3	−0.626649
$890$	0	0
$891$	−3934.05	−0.147919
$892$	0	0
$893$	54348.4	2.03662
$894$	0	0
$895$	−5127.92	−0.191517
$896$	0	0
$897$	−13642.7	−0.507824
$898$	0	0
$899$	−533.803	−0.0198035
$900$	0	0
$901$	−42151.5	−1.55857
$902$	0	0
$903$	1945.39	0.0716926
$904$	0	0
$905$	14497.0	0.532482
$906$	0	0
$907$	26766.1	0.979885	0.489942	−	0.871755i	$-0.337018\pi$
$907$	26766.1	0.979885	0.489942	+	0.871755i	$0.337018\pi$
$908$	0	0
$909$	−3894.52	−0.142105
$910$	0	0
$911$	−5022.67	−0.182666	−0.0913328	−	0.995820i	$-0.529113\pi$
$911$	−5022.67	−0.182666	−0.0913328	+	0.995820i	$0.529113\pi$
$912$	0	0
$913$	−25721.3	−0.932367
$914$	0	0
$915$	−5291.91	−0.191197
$916$	0	0
$917$	−8400.26	−0.302509
$918$	0	0
$919$	−9541.79	−0.342497	−0.171248	−	0.985228i	$-0.554780\pi$
$919$	−9541.79	−0.342497	−0.171248	+	0.985228i	$0.554780\pi$
$920$	0	0
$921$	−8005.93	−0.286432
$922$	0	0
$923$	−21490.7	−0.766387
$924$	0	0
$925$	−6316.90	−0.224539
$926$	0	0
$927$	−4613.63	−0.163464
$928$	0	0
$929$	−25479.6	−0.899846	−0.449923	−	0.893067i	$-0.648549\pi$
$929$	−25479.6	−0.899846	−0.449923	+	0.893067i	$0.648549\pi$
$930$	0	0
$931$	4569.49	0.160858
$932$	0	0
$933$	−18567.7	−0.651533
$934$	0	0
$935$	16430.0	0.574671
$936$	0	0
$937$	−33608.3	−1.17176	−0.585878	−	0.810399i	$-0.699250\pi$
$937$	−33608.3	−1.17176	−0.585878	+	0.810399i	$0.699250\pi$
$938$	0	0
$939$	−8764.77	−0.304609
$940$	0	0
$941$	−19173.6	−0.664232	−0.332116	−	0.943239i	$-0.607762\pi$
$941$	−19173.6	−0.664232	−0.332116	+	0.943239i	$0.607762\pi$
$942$	0	0
$943$	28778.5	0.993804
$944$	0	0
$945$	945.000	0.0325300
$946$	0	0
$947$	979.315	0.0336045	0.0168023	−	0.999859i	$-0.494651\pi$
$947$	979.315	0.0336045	0.0168023	+	0.999859i	$0.494651\pi$
$948$	0	0
$949$	−50849.5	−1.73935
$950$	0	0
$951$	29476.7	1.00510
$952$	0	0
$953$	3048.61	0.103624	0.0518122	−	0.998657i	$-0.483500\pi$
$953$	3048.61	0.103624	0.0518122	+	0.998657i	$0.483500\pi$
$954$	0	0
$955$	−5370.92	−0.181989
$956$	0	0
$957$	8558.03	0.289072
$958$	0	0
$959$	19468.8	0.655557
$960$	0	0
$961$	−29708.4	−0.997227
$962$	0	0
$963$	−17667.8	−0.591211
$964$	0	0
$965$	4495.00	0.149947
$966$	0	0
$967$	14467.9	0.481133	0.240567	−	0.970633i	$-0.422667\pi$
$967$	14467.9	0.481133	0.240567	+	0.970633i	$0.422667\pi$
$968$	0	0
$969$	−18928.0	−0.627507
$970$	0	0
$971$	−12952.8	−0.428090	−0.214045	−	0.976824i	$-0.568664\pi$
$971$	−12952.8	−0.428090	−0.214045	+	0.976824i	$0.568664\pi$
$972$	0	0
$973$	8719.23	0.287282
$974$	0	0
$975$	−3274.26	−0.107549
$976$	0	0
$977$	47244.2	1.54706	0.773529	−	0.633760i	$-0.218489\pi$
$977$	47244.2	1.54706	0.773529	+	0.633760i	$0.218489\pi$
$978$	0	0
$979$	18709.4	0.610781
$980$	0	0
$981$	4906.58	0.159689
$982$	0	0
$983$	1536.84	0.0498651	0.0249326	−	0.999689i	$-0.492063\pi$
$983$	1536.84	0.0498651	0.0249326	+	0.999689i	$0.492063\pi$
$984$	0	0
$985$	15318.1	0.495509
$986$	0	0
$987$	12238.7	0.394692
$988$	0	0
$989$	9649.73	0.310256
$990$	0	0
$991$	−3785.22	−0.121334	−0.0606668	−	0.998158i	$-0.519323\pi$
$991$	−3785.22	−0.121334	−0.0606668	+	0.998158i	$0.519323\pi$
$992$	0	0
$993$	27774.5	0.887610
$994$	0	0
$995$	4747.61	0.151266
$996$	0	0
$997$	−25894.9	−0.822566	−0.411283	−	0.911508i	$-0.634919\pi$
$997$	−25894.9	−0.822566	−0.411283	+	0.911508i	$0.634919\pi$
$998$	0	0
$999$	−6822.26	−0.216063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.4.a.bo.1.1		2
4.3	odd	2		105.4.a.e.1.1	✓	2
12.11	even	2		315.4.a.k.1.2		2
20.3	even	4		525.4.d.l.274.4		4
20.7	even	4		525.4.d.l.274.1		4
20.19	odd	2		525.4.a.l.1.2		2
28.27	even	2		735.4.a.o.1.1		2
60.59	even	2		1575.4.a.q.1.1		2
84.83	odd	2		2205.4.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.e.1.1	✓	2	4.3	odd	2
315.4.a.k.1.2		2	12.11	even	2
525.4.a.l.1.2		2	20.19	odd	2
525.4.d.l.274.1		4	20.7	even	4
525.4.d.l.274.4		4	20.3	even	4
735.4.a.o.1.1		2	28.27	even	2
1575.4.a.q.1.1		2	60.59	even	2
1680.4.a.bo.1.1		2	1.1	even	1	trivial
2205.4.a.bb.1.2		2	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1680.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} -48.5685 q^{11} -43.6569 q^{13} +15.0000 q^{15} -67.6569 q^{17} +93.2548 q^{19} +21.0000 q^{21} +104.167 q^{23} +25.0000 q^{25} +27.0000 q^{27} -58.7351 q^{29} +9.08831 q^{31} -145.706 q^{33} +35.0000 q^{35} -252.676 q^{37} -130.971 q^{39} +276.274 q^{41} +92.6375 q^{43} +45.0000 q^{45} +582.794 q^{47} +49.0000 q^{49} -202.971 q^{51} +623.019 q^{53} -242.843 q^{55} +279.765 q^{57} +524.999 q^{59} -352.794 q^{61} +63.0000 q^{63} -218.284 q^{65} +736.520 q^{67} +312.500 q^{69} +492.264 q^{71} +1164.75 q^{73} +75.0000 q^{75} -339.980 q^{77} +872.195 q^{79} +81.0000 q^{81} +529.588 q^{83} -338.284 q^{85} -176.205 q^{87} -385.216 q^{89} -305.598 q^{91} +27.2649 q^{93} +466.274 q^{95} -463.892 q^{97} -437.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 16 q^{11} - 76 q^{13} + 30 q^{15} - 124 q^{17} + 96 q^{19} + 42 q^{21} + 16 q^{23} + 50 q^{25} + 54 q^{27} + 188 q^{29} + 120 q^{31} + 48 q^{33} + 70 q^{35}+ \cdots + 144 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 16 * q^11 - 76 * q^13 + 30 * q^15 - 124 * q^17 + 96 * q^19 + 42 * q^21 + 16 * q^23 + 50 * q^25 + 54 * q^27 + 188 * q^29 + 120 * q^31 + 48 * q^33 + 70 * q^35 - 132 * q^37 - 228 * q^39 + 100 * q^41 + 536 * q^43 + 90 * q^45 + 928 * q^47 + 98 * q^49 - 372 * q^51 + 884 * q^53 + 80 * q^55 + 288 * q^57 - 104 * q^59 - 468 * q^61 + 126 * q^63 - 380 * q^65 + 1688 * q^67 + 48 * q^69 + 136 * q^71 + 508 * q^73 + 150 * q^75 + 112 * q^77 + 432 * q^79 + 162 * q^81 + 584 * q^83 - 620 * q^85 + 564 * q^87 - 1404 * q^89 - 532 * q^91 + 360 * q^93 + 480 * q^95 - 1188 * q^97 + 144 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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