LMFDB - Embedded newform 756.4.x.a.125.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,4,Mod(125,756)] 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("756.125"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5, 3])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.x (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			125.9
Character	$\chi$	$=$	756.125
Dual form			756.4.x.a.629.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.99997 + 5.19610i) q^{5} +(-0.375477 + 18.5165i) q^{7} +(39.3810 - 22.7366i) q^{11} +(-22.7627 - 13.1421i) q^{13} +19.7249 q^{17} +27.9162i q^{19} +(60.3466 + 34.8411i) q^{23} +(44.5004 + 77.0769i) q^{25} +(119.031 - 68.7227i) q^{29} +(138.202 + 79.7908i) q^{31} +(-95.0869 - 57.4998i) q^{35} -287.829 q^{37} +(20.4288 - 35.3837i) q^{41} +(55.4158 + 95.9830i) q^{43} +(-109.666 - 189.948i) q^{47} +(-342.718 - 13.9050i) q^{49} +209.770i q^{53} +272.837i q^{55} +(-413.880 + 716.860i) q^{59} +(-594.209 + 343.066i) q^{61} +(136.575 - 78.8515i) q^{65} +(-171.449 + 296.958i) q^{67} -387.476i q^{71} -220.721i q^{73} +(406.215 + 737.734i) q^{77} +(242.301 + 419.677i) q^{79} +(354.323 + 613.706i) q^{83} +(-59.1741 + 102.493i) q^{85} +140.929 q^{89} +(251.891 - 416.550i) q^{91} +(-145.055 - 83.7476i) q^{95} +(-1304.31 + 753.041i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) $ 48 * q + 6 * q^7 + 12 * q^11 + 408 * q^23 - 600 * q^25 + 84 * q^29 + 336 * q^37 + 84 * q^43 + 318 * q^49 - 2964 * q^65 - 588 * q^67 - 2400 * q^77 + 204 * q^79 - 360 * q^85 - 1080 * q^91 - 300 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.99997	+	5.19610i	−0.268325	+	0.464753i	−0.968430	−	0.249288i	$-0.919804\pi$
$5$	−2.99997	+	5.19610i	−0.268325	+	0.464753i	0.700104	+	0.714041i	$0.253137\pi$
$6$		0			0
$7$	−0.375477	+	18.5165i	−0.0202739	+	0.999794i
$7$	−0.375477	+	18.5165i	−0.0202739	+	0.999794i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	39.3810	−	22.7366i	1.07944	−	0.623214i	0.148694	−	0.988883i	$-0.452493\pi$
$11$	39.3810	−	22.7366i	1.07944	−	0.623214i	0.930745	+	0.365669i	$0.119160\pi$
$12$		0			0
$13$	−22.7627	−	13.1421i	−0.485634	−	0.280381i	0.237127	−	0.971479i	$-0.423794\pi$
$13$	−22.7627	−	13.1421i	−0.485634	−	0.280381i	−0.722761	+	0.691098i	$0.757127\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	19.7249			0.281411			0.140706	−	0.990051i	$-0.455063\pi$
$17$	19.7249			0.281411			0.140706	+	0.990051i	$0.455063\pi$
$18$		0			0
$19$			27.9162i			0.337074i	0.985695	+	0.168537i	$0.0539043\pi$
$19$			27.9162i			0.337074i	−0.985695	+	0.168537i	$0.946096\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	60.3466	+	34.8411i	0.547093	+	0.315864i	0.747949	−	0.663757i	$-0.231039\pi$
$23$	60.3466	+	34.8411i	0.547093	+	0.315864i	−0.200856	+	0.979621i	$0.564372\pi$
$24$		0			0
$25$	44.5004	+	77.0769i	0.356003	+	0.616615i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	119.031	−	68.7227i	0.762191	−	0.440051i	−0.0678907	−	0.997693i	$-0.521627\pi$
$29$	119.031	−	68.7227i	0.762191	−	0.440051i	0.830082	+	0.557641i	$0.188294\pi$
$30$		0			0
$31$	138.202	+	79.7908i	0.800702	+	0.462286i	0.843717	−	0.536789i	$-0.180363\pi$
$31$	138.202	+	79.7908i	0.800702	+	0.462286i	−0.0430146	+	0.999074i	$0.513696\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−95.0869	−	57.4998i	−0.459218	−	0.277693i
$36$		0			0
$37$	−287.829			−1.27889			−0.639443	−	0.768839i	$-0.720835\pi$
$37$	−287.829			−1.27889			−0.639443	+	0.768839i	$0.720835\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	20.4288	−	35.3837i	0.0778157	−	0.134781i	−0.824492	−	0.565874i	$-0.808539\pi$
$41$	20.4288	−	35.3837i	0.0778157	−	0.134781i	0.902307	+	0.431093i	$0.141872\pi$
$42$		0			0
$43$	55.4158	+	95.9830i	0.196531	+	0.340402i	0.947401	−	0.320048i	$-0.103699\pi$
$43$	55.4158	+	95.9830i	0.196531	+	0.340402i	−0.750870	+	0.660450i	$0.770366\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−109.666	−	189.948i	−0.340351	−	0.589504i	0.644147	−	0.764902i	$-0.277212\pi$
$47$	−109.666	−	189.948i	−0.340351	−	0.589504i	−0.984498	+	0.175397i	$0.943879\pi$
$48$		0			0
$49$	−342.718	−	13.9050i	−0.999178	−	0.0405394i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			209.770i			0.543664i	0.962345	+	0.271832i	$0.0876295\pi$
$53$			209.770i			0.543664i	−0.962345	+	0.271832i	$0.912371\pi$
$54$		0			0
$55$			272.837i			0.668897i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−413.880	+	716.860i	−0.913263	+	1.58182i	−0.103839	+	0.994594i	$0.533113\pi$
$59$	−413.880	+	716.860i	−0.913263	+	1.58182i	−0.809425	+	0.587224i	$0.800221\pi$
$60$		0			0
$61$	−594.209	+	343.066i	−1.24722	+	0.720085i	−0.970555	−	0.240881i	$-0.922564\pi$
$61$	−594.209	+	343.066i	−1.24722	+	0.720085i	−0.276668	+	0.960965i	$0.589230\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	136.575	−	78.8515i	0.260616	−	0.150467i
$66$		0			0
$67$	−171.449	+	296.958i	−0.312624	+	0.541480i	−0.978930	−	0.204198i	$-0.934541\pi$
$67$	−171.449	+	296.958i	−0.312624	+	0.541480i	0.666306	+	0.745679i	$0.267875\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	387.476i		−	0.647675i	−0.946113	−	0.323838i	$-0.895027\pi$
$71$		−	387.476i		−	0.647675i	0.946113	−	0.323838i	$-0.104973\pi$
$72$		0			0
$73$		−	220.721i		−	0.353882i	−0.984221	−	0.176941i	$-0.943380\pi$
$73$		−	220.721i		−	0.353882i	0.984221	−	0.176941i	$-0.0566202\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	406.215	+	737.734i	0.601202	+	1.09185i
$78$		0			0
$79$	242.301	+	419.677i	0.345076	+	0.597688i	0.985367	−	0.170443i	$-0.0545200\pi$
$79$	242.301	+	419.677i	0.345076	+	0.597688i	−0.640292	+	0.768132i	$0.721187\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	354.323	+	613.706i	0.468579	+	0.811602i	0.999355	−	0.0359099i	$-0.0114329\pi$
$83$	354.323	+	613.706i	0.468579	+	0.811602i	−0.530776	+	0.847512i	$0.678100\pi$
$84$		0			0
$85$	−59.1741	+	102.493i	−0.0755098	+	0.130787i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	140.929			0.167848			0.0839240	−	0.996472i	$-0.473255\pi$
$89$	140.929			0.167848			0.0839240	+	0.996472i	$0.473255\pi$
$90$		0			0
$91$	251.891	−	416.550i	0.290169	−	0.479850i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−145.055	−	83.7476i	−0.156656	−	0.0904455i
$96$		0			0
$97$	−1304.31	+	753.041i	−1.36528	+	0.788245i	−0.990321	−	0.138796i	$-0.955677\pi$
$97$	−1304.31	+	753.041i	−1.36528	+	0.788245i	−0.374960	+	0.927041i	$0.622343\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	923.516	+	1599.58i	0.909834	+	1.57588i	0.814293	+	0.580454i	$0.197125\pi$
$101$	923.516	+	1599.58i	0.909834	+	1.57588i	0.0955414	+	0.995425i	$0.469542\pi$
$102$		0			0
$103$	−696.481	−	402.114i	−0.666275	−	0.384674i	0.128389	−	0.991724i	$-0.459019\pi$
$103$	−696.481	−	402.114i	−0.666275	−	0.384674i	−0.794664	+	0.607050i	$0.792353\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			282.997i			0.255685i	0.991794	+	0.127843i	$0.0408053\pi$
$107$			282.997i			0.255685i	−0.991794	+	0.127843i	$0.959195\pi$
$108$		0			0
$109$	1826.43			1.60496			0.802478	−	0.596681i	$-0.203514\pi$
$109$	1826.43			1.60496			0.802478	+	0.596681i	$0.203514\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	815.811	+	471.008i	0.679159	+	0.392113i	0.799538	−	0.600615i	$-0.205078\pi$
$113$	815.811	+	471.008i	0.679159	+	0.392113i	−0.120379	+	0.992728i	$0.538411\pi$
$114$		0			0
$115$	−362.076	+	209.044i	−0.293598	+	0.169509i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.40625	+	365.235i	−0.00570529	+	0.281353i
$120$		0			0
$121$	368.410	−	638.105i	0.276792	−	0.479418i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1283.99			−0.918749
$126$		0			0
$127$	320.551			0.223971			0.111985	−	0.993710i	$-0.464279\pi$
$127$	320.551			0.223971			0.111985	+	0.993710i	$0.464279\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−306.907	+	531.578i	−0.204691	+	0.354536i	−0.950034	−	0.312145i	$-0.898952\pi$
$131$	−306.907	+	531.578i	−0.204691	+	0.354536i	0.745343	+	0.666681i	$0.232286\pi$
$132$		0			0
$133$	−516.908	−	10.4819i	−0.337005	−	0.00683379i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1515.86	+	875.182i	−0.945318	+	0.545780i	−0.891623	−	0.452778i	$-0.850433\pi$
$137$	−1515.86	+	875.182i	−0.945318	+	0.545780i	−0.0536948	+	0.998557i	$0.517100\pi$
$138$		0			0
$139$	−1120.72	−	647.048i	−0.683872	−	0.394834i	0.117440	−	0.993080i	$-0.462531\pi$
$139$	−1120.72	−	647.048i	−0.683872	−	0.394834i	−0.801312	+	0.598246i	$0.795865\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1195.23			−0.698950
$144$		0			0
$145$			824.664i			0.472308i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2499.23	+	1442.93i	1.37413	+	0.793352i	0.991445	−	0.130528i	$-0.0416674\pi$
$149$	2499.23	+	1442.93i	1.37413	+	0.793352i	0.382681	+	0.923880i	$0.375001\pi$
$150$		0			0
$151$	1540.63	+	2668.45i	0.830297	+	1.43812i	0.897803	+	0.440398i	$0.145163\pi$
$151$	1540.63	+	2668.45i	0.830297	+	1.43812i	−0.0675056	+	0.997719i	$0.521504\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−829.202	+	478.740i	−0.429697	+	0.248086i
$156$		0			0
$157$	−1090.13	−	629.388i	−0.554153	−	0.319941i	0.196642	−	0.980475i	$-0.436996\pi$
$157$	−1090.13	−	629.388i	−0.554153	−	0.319941i	−0.750795	+	0.660535i	$0.770330\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−667.793	+	1104.32i	−0.326891	+	0.540576i
$162$		0			0
$163$	−2906.94			−1.39687			−0.698434	−	0.715675i	$-0.746119\pi$
$163$	−2906.94			−1.39687			−0.698434	+	0.715675i	$0.746119\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	286.775	−	496.709i	0.132882	−	0.230159i	−0.791904	−	0.610645i	$-0.790910\pi$
$167$	286.775	−	496.709i	0.132882	−	0.230159i	0.924786	+	0.380487i	$0.124244\pi$
$168$		0			0
$169$	−753.072	−	1304.36i	−0.342773	−	0.593700i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1931.13	+	3344.82i	0.848679	+	1.46995i	0.882388	+	0.470523i	$0.155935\pi$
$173$	1931.13	+	3344.82i	0.848679	+	1.46995i	−0.0337092	+	0.999432i	$0.510732\pi$
$174$		0			0
$175$	−1443.90	+	795.048i	−0.623706	+	0.343429i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2248.26i		−	0.938787i	−0.882989	−	0.469394i	$-0.844473\pi$
$179$		−	2248.26i		−	0.938787i	0.882989	−	0.469394i	$-0.155527\pi$
$180$		0			0
$181$			166.407i			0.0683368i	0.999416	+	0.0341684i	$0.0108783\pi$
$181$			166.407i			0.0683368i	−0.999416	+	0.0341684i	$0.989122\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	863.478	−	1495.59i	0.343157	−	0.594366i
$186$		0			0
$187$	776.787	−	448.478i	0.303766	−	0.175379i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1758.88	−	1015.49i	0.666324	−	0.384703i	−0.128358	−	0.991728i	$-0.540971\pi$
$191$	1758.88	−	1015.49i	0.666324	−	0.384703i	0.794682	+	0.607025i	$0.207637\pi$
$192$		0			0
$193$	251.118	−	434.949i	0.0936573	−	0.162219i	−0.815390	−	0.578912i	$-0.803478\pi$
$193$	251.118	−	434.949i	0.0936573	−	0.162219i	0.909047	+	0.416693i	$0.136811\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3949.11i		−	1.42823i	−0.700026	−	0.714117i	$-0.746828\pi$
$197$		−	3949.11i		−	1.42823i	0.700026	−	0.714117i	$-0.253172\pi$
$198$		0			0
$199$			629.709i			0.224316i	0.993690	+	0.112158i	$0.0357763\pi$
$199$			629.709i			0.224316i	−0.993690	+	0.112158i	$0.964224\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1227.81	+	2229.84i	0.424508	+	0.770956i
$204$		0			0
$205$	122.572	+	212.300i	0.0417599	+	0.0723302i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	634.720	+	1099.37i	0.210069	+	0.363851i
$210$		0			0
$211$	−1494.95	+	2589.33i	−0.487757	+	0.844820i	−0.999901	−	0.0140797i	$-0.995518\pi$
$211$	−1494.95	+	2589.33i	−0.487757	+	0.844820i	0.512144	+	0.858900i	$0.328851\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−664.983			−0.210937
$216$		0			0
$217$	−1529.33	+	2529.05i	−0.478424	+	0.791165i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−448.992	−	259.226i	−0.136663	−	0.0789023i
$222$		0			0
$223$	2829.72	−	1633.74i	0.849740	−	0.490598i	−0.0108230	−	0.999941i	$-0.503445\pi$
$223$	2829.72	−	1633.74i	0.849740	−	0.490598i	0.860563	+	0.509344i	$0.170112\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2836.86	−	4913.59i	−0.829468	−	1.43668i	−0.898457	−	0.439063i	$-0.855311\pi$
$227$	−2836.86	−	4913.59i	−0.829468	−	1.43668i	0.0689889	−	0.997617i	$-0.478023\pi$
$228$		0			0
$229$	4528.48	+	2614.52i	1.30677	+	0.754465i	0.981556	−	0.191175i	$-0.0612298\pi$
$229$	4528.48	+	2614.52i	1.30677	+	0.754465i	0.325215	+	0.945640i	$0.394563\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	3433.84i		−	0.965487i	−0.875762	−	0.482743i	$-0.839640\pi$
$233$		−	3433.84i		−	0.965487i	0.875762	−	0.482743i	$-0.160360\pi$
$234$		0			0
$235$	1315.98			0.365299
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−469.677	−	271.168i	−0.127117	−	0.0733908i	0.435093	−	0.900385i	$-0.356715\pi$
$239$	−469.677	−	271.168i	−0.127117	−	0.0733908i	−0.562210	+	0.826995i	$0.690049\pi$
$240$		0			0
$241$	−3110.33	+	1795.75i	−0.831343	+	0.479976i	−0.854312	−	0.519760i	$-0.826021\pi$
$241$	−3110.33	+	1795.75i	−0.831343	+	0.479976i	0.0229693	+	0.999736i	$0.492688\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1100.40	−	1739.08i	0.286946	−	0.453493i
$246$		0			0
$247$	366.876	−	635.448i	0.0945092	−	0.163695i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4334.39			−1.08998			−0.544988	−	0.838444i	$-0.683466\pi$
$251$	−4334.39			−1.08998			−0.544988	+	0.838444i	$0.683466\pi$
$252$		0			0
$253$	3168.68			0.787404
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3282.88	−	5686.12i	0.796812	−	1.38012i	−0.124870	−	0.992173i	$-0.539852\pi$
$257$	3282.88	−	5686.12i	0.796812	−	1.38012i	0.921682	−	0.387946i	$-0.126815\pi$
$258$		0			0
$259$	108.073	−	5329.57i	0.0259279	−	1.27862i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6274.79	+	3622.75i	−1.47118	+	0.849386i	−0.999476	−	0.0323711i	$-0.989694\pi$
$263$	−6274.79	+	3622.75i	−1.47118	+	0.849386i	−0.471704	+	0.881757i	$0.656361\pi$
$264$		0			0
$265$	−1089.99	−	629.305i	−0.252669	−	0.145879i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3455.87			0.783302			0.391651	−	0.920114i	$-0.371904\pi$
$269$	3455.87			0.783302			0.391651	+	0.920114i	$0.371904\pi$
$270$		0			0
$271$			2471.30i			0.553951i	0.960877	+	0.276975i	$0.0893321\pi$
$271$			2471.30i			0.553951i	−0.960877	+	0.276975i	$0.910668\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3504.94	+	2023.58i	0.768567	+	0.443732i
$276$		0			0
$277$	2256.85	+	3908.97i	0.489533	+	0.847897i	0.999927	−	0.0120440i	$-0.00383381\pi$
$277$	2256.85	+	3908.97i	0.489533	+	0.847897i	−0.510394	+	0.859941i	$0.670500\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2090.48	+	1206.94i	−0.443799	+	0.256228i	−0.705208	−	0.709001i	$-0.749146\pi$
$281$	−2090.48	+	1206.94i	−0.443799	+	0.256228i	0.261409	+	0.965228i	$0.415813\pi$
$282$		0			0
$283$	3893.62	+	2247.98i	0.817851	+	0.472187i	0.849675	−	0.527307i	$-0.176798\pi$
$283$	3893.62	+	2247.98i	0.817851	+	0.472187i	−0.0318238	+	0.999493i	$0.510132\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	647.511	+	391.555i	0.133175	+	0.0805322i
$288$		0			0
$289$	−4523.93			−0.920808
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1569.64	+	2718.69i	−0.312966	+	0.542073i	−0.979003	−	0.203845i	$-0.934656\pi$
$293$	−1569.64	+	2718.69i	−0.312966	+	0.542073i	0.666037	+	0.745919i	$0.267989\pi$
$294$		0			0
$295$	−2483.25	−	4301.12i	−0.490103	−	0.848884i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−915.768	−	1586.16i	−0.177125	−	0.306789i
$300$		0			0
$301$	−1798.07	+	990.065i	−0.344316	+	0.189589i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	4116.75i		−	0.772868i
$306$		0			0
$307$		−	10294.0i		−	1.91371i	−0.290560	−	0.956857i	$-0.593842\pi$
$307$		−	10294.0i		−	1.91371i	0.290560	−	0.956857i	$-0.406158\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3940.59	−	6825.30i	0.718489	−	1.24446i	−0.243109	−	0.969999i	$-0.578167\pi$
$311$	3940.59	−	6825.30i	0.718489	−	1.24446i	0.961598	−	0.274461i	$-0.0884994\pi$
$312$		0			0
$313$	7096.84	−	4097.36i	1.28159	−	0.739925i	0.304449	−	0.952529i	$-0.401528\pi$
$313$	7096.84	−	4097.36i	1.28159	−	0.739925i	0.977139	+	0.212604i	$0.0681943\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7590.80	+	4382.55i	−1.34493	+	0.776494i	−0.987526	−	0.157457i	$-0.949670\pi$
$317$	−7590.80	+	4382.55i	−1.34493	+	0.776494i	−0.357401	+	0.933951i	$0.616337\pi$
$318$		0			0
$319$	3125.05	−	5412.74i	0.548493	−	0.950017i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			550.644i			0.0948564i
$324$		0			0
$325$		−	2339.31i		−	0.399266i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3558.33	−	1959.31i	0.596283	−	0.328329i
$330$		0			0
$331$	−3431.76	−	5943.98i	−0.569869	−	0.987041i	−0.996578	−	0.0826527i	$-0.973661\pi$
$331$	−3431.76	−	5943.98i	−0.569869	−	0.987041i	0.426710	−	0.904389i	$-0.359673\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1028.68	−	1781.73i	−0.167770	−	0.290586i
$336$		0			0
$337$	2323.81	−	4024.96i	0.375627	−	0.650605i	−0.614794	−	0.788688i	$-0.710761\pi$
$337$	2323.81	−	4024.96i	0.375627	−	0.650605i	0.990421	+	0.138083i	$0.0440941\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7256.70			1.15241
$342$		0			0
$343$	386.154	−	6340.70i	0.0607882	−	0.998151i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8295.04	−	4789.14i	−1.28329	−	0.740907i	−0.305840	−	0.952083i	$-0.598937\pi$
$347$	−8295.04	−	4789.14i	−1.28329	−	0.740907i	−0.977448	+	0.211176i	$0.932271\pi$
$348$		0			0
$349$	−755.168	+	435.997i	−0.115826	+	0.0668721i	−0.556794	−	0.830651i	$-0.687969\pi$
$349$	−755.168	+	435.997i	−0.115826	+	0.0668721i	0.440968	+	0.897523i	$0.354635\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4148.63	+	7185.64i	0.625523	+	1.08344i	0.988440	+	0.151615i	$0.0484475\pi$
$353$	4148.63	+	7185.64i	0.625523	+	1.08344i	−0.362917	+	0.931821i	$0.618219\pi$
$354$		0			0
$355$	2013.36	+	1162.42i	0.301009	+	0.173788i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			976.696i			0.143588i	0.997419	+	0.0717939i	$0.0228724\pi$
$359$			976.696i			0.143588i	−0.997419	+	0.0717939i	$0.977128\pi$
$360$		0			0
$361$	6079.69			0.886381
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1146.89	+	662.155i	0.164468	+	0.0949556i
$366$		0			0
$367$	6001.51	−	3464.97i	0.853614	−	0.492834i	−0.00825490	−	0.999966i	$-0.502628\pi$
$367$	6001.51	−	3464.97i	0.853614	−	0.492834i	0.861868	+	0.507132i	$0.169294\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3884.20	−	78.7640i	−0.543552	−	0.0110222i
$372$		0			0
$373$	2259.68	−	3913.89i	0.313678	−	0.543307i	−0.665477	−	0.746418i	$-0.731772\pi$
$373$	2259.68	−	3913.89i	0.313678	−	0.543307i	0.979156	+	0.203111i	$0.0651053\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3612.63			−0.493528
$378$		0			0
$379$	2781.32			0.376958			0.188479	−	0.982077i	$-0.439644\pi$
$379$	2781.32			0.376958			0.188479	+	0.982077i	$0.439644\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5449.44	−	9438.71i	0.727033	−	1.25926i	−0.231099	−	0.972930i	$-0.574232\pi$
$383$	5449.44	−	9438.71i	0.727033	−	1.25926i	0.958132	−	0.286327i	$-0.0924345\pi$
$384$		0			0
$385$	−5051.97	−	102.444i	−0.668759	−	0.0135611i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1922.09	+	1109.72i	−0.250524	+	0.144640i	−0.620004	−	0.784598i	$-0.712869\pi$
$389$	−1922.09	+	1109.72i	−0.250524	+	0.144640i	0.369480	+	0.929239i	$0.379536\pi$
$390$		0			0
$391$	1190.33	+	687.237i	0.153958	+	0.0888877i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2907.58			−0.370370
$396$		0			0
$397$			10873.7i			1.37464i	0.726354	+	0.687321i	$0.241214\pi$
$397$			10873.7i			1.37464i	−0.726354	+	0.687321i	$0.758786\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5301.00	+	3060.53i	0.660147	+	0.381136i	0.792333	−	0.610089i	$-0.208866\pi$
$401$	5301.00	+	3060.53i	0.660147	+	0.381136i	−0.132186	+	0.991225i	$0.542200\pi$
$402$		0			0
$403$	−2097.23	−	3632.51i	−0.259232	−	0.449003i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−11335.0	+	6544.26i	−1.38048	+	0.797020i
$408$		0			0
$409$	12155.0	+	7017.69i	1.46950	+	0.848416i	0.999415	−	0.0342045i	$-0.0108898\pi$
$409$	12155.0	+	7017.69i	1.46950	+	0.848416i	0.470085	+	0.882621i	$0.344223\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−13118.3	−	7932.75i	−1.56298	−	0.945145i
$414$		0			0
$415$	−4251.83			−0.502926
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4144.18	−	7177.94i	0.483190	−	0.836910i	−0.516624	−	0.856213i	$-0.672811\pi$
$419$	4144.18	−	7177.94i	0.483190	−	0.836910i	0.999814	+	0.0193028i	$0.00614466\pi$
$420$		0			0
$421$	7767.30	+	13453.4i	0.899181	+	1.55743i	0.828544	+	0.559924i	$0.189170\pi$
$421$	7767.30	+	13453.4i	0.899181	+	1.55743i	0.0706368	+	0.997502i	$0.477497\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	877.765	+	1520.33i	0.100183	+	0.173522i
$426$		0			0
$427$	−6129.26	−	11131.4i	−0.694651	−	1.26157i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	3167.89i		−	0.354041i	−0.984207	−	0.177021i	$-0.943354\pi$
$431$		−	3167.89i		−	0.354041i	0.984207	−	0.177021i	$-0.0566459\pi$
$432$		0			0
$433$			2187.70i			0.242804i	0.992603	+	0.121402i	$0.0387391\pi$
$433$			2187.70i			0.242804i	−0.992603	+	0.121402i	$0.961261\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−972.630	+	1684.65i	−0.106470	+	0.184411i
$438$		0			0
$439$	1703.79	−	983.684i	0.185234	−	0.106945i	−0.404516	−	0.914531i	$-0.632560\pi$
$439$	1703.79	−	983.684i	0.185234	−	0.106945i	0.589749	+	0.807586i	$0.299227\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2229.21	+	1287.04i	−0.239081	+	0.138034i	−0.614754	−	0.788719i	$-0.710745\pi$
$443$	−2229.21	+	1287.04i	−0.239081	+	0.138034i	0.375673	+	0.926752i	$0.377412\pi$
$444$		0			0
$445$	−422.783	+	732.282i	−0.0450379	+	0.0780079i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	9168.11i		−	0.963630i	−0.876273	−	0.481815i	$-0.839978\pi$
$449$		−	9168.11i		−	0.963630i	0.876273	−	0.481815i	$-0.160022\pi$
$450$		0			0
$451$		−	1857.93i		−	0.193983i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1408.77	+	2558.49i	0.145152	+	0.263613i
$456$		0			0
$457$	449.418	+	778.415i	0.0460019	+	0.0796777i	0.888110	−	0.459632i	$-0.152019\pi$
$457$	449.418	+	778.415i	0.0460019	+	0.0796777i	−0.842108	+	0.539310i	$0.818685\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	750.955	+	1300.69i	0.0758686	+	0.131408i	0.901464	−	0.432855i	$-0.142494\pi$
$461$	750.955	+	1300.69i	0.0758686	+	0.131408i	−0.825595	+	0.564263i	$0.809160\pi$
$462$		0			0
$463$	515.882	−	893.534i	0.0517820	−	0.0896891i	−0.838973	−	0.544174i	$-0.816843\pi$
$463$	515.882	−	893.534i	0.0517820	−	0.0896891i	0.890755	+	0.454485i	$0.150177\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8619.68			0.854114			0.427057	−	0.904225i	$-0.359550\pi$
$467$	8619.68			0.854114			0.427057	+	0.904225i	$0.359550\pi$
$468$		0			0
$469$	−5434.23	−	3286.12i	−0.535031	−	0.323537i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4364.66	+	2519.94i	0.424286	+	0.244962i
$474$		0			0
$475$	−2151.69	+	1242.28i	−0.207845	+	0.119999i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2037.51	+	3529.07i	0.194355	+	0.336634i	0.946689	−	0.322149i	$-0.104405\pi$
$479$	2037.51	+	3529.07i	0.194355	+	0.336634i	−0.752334	+	0.658782i	$0.771072\pi$
$480$		0			0
$481$	6551.77	+	3782.66i	0.621070	+	0.358575i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	9036.40i		−	0.846025i
$486$		0			0
$487$	−14817.8			−1.37876			−0.689381	−	0.724399i	$-0.742117\pi$
$487$	−14817.8			−1.37876			−0.689381	+	0.724399i	$0.742117\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6217.36	+	3589.59i	0.571457	+	0.329931i	0.757731	−	0.652567i	$-0.226308\pi$
$491$	6217.36	+	3589.59i	0.571457	+	0.329931i	−0.186274	+	0.982498i	$0.559641\pi$
$492$		0			0
$493$	2347.88	−	1355.55i	0.214489	−	0.123835i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7174.68	+	145.488i	0.647542	+	0.0131309i
$498$		0			0
$499$	−5380.59	+	9319.46i	−0.482702	+	0.836065i	−0.999803	−	0.0198598i	$-0.993678\pi$
$499$	−5380.59	+	9319.46i	−0.482702	+	0.836065i	0.517100	+	0.855925i	$0.327011\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−11285.3			−1.00037			−0.500186	−	0.865918i	$-0.666735\pi$
$503$	−11285.3			−1.00037			−0.500186	+	0.865918i	$0.666735\pi$
$504$		0			0
$505$	−11082.1			−0.976527
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5404.52	−	9360.91i	0.470631	−	0.815157i	−0.528805	−	0.848744i	$-0.677360\pi$
$509$	5404.52	−	9360.91i	0.470631	−	0.815157i	0.999436	+	0.0335864i	$0.0106929\pi$
$510$		0			0
$511$	4086.96	+	82.8755i	0.353809	+	0.00717456i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4178.84	−	2412.66i	0.357557	−	0.206436i
$516$		0			0
$517$	−8637.54	−	4986.89i	−0.734775	−	0.424223i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3085.93			0.259495			0.129747	−	0.991547i	$-0.458583\pi$
$521$	3085.93			0.259495			0.129747	+	0.991547i	$0.458583\pi$
$522$		0			0
$523$			2155.94i			0.180254i	0.995930	+	0.0901268i	$0.0287272\pi$
$523$			2155.94i			0.180254i	−0.995930	+	0.0901268i	$0.971273\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2726.01	+	1573.87i	0.225327	+	0.130092i
$528$		0			0
$529$	−3655.69	−	6331.85i	−0.300460	−	0.520412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−930.030	+	536.953i	−0.0755799	+	0.0436361i
$534$		0			0
$535$	−1470.48	−	848.982i	−0.118831	−	0.0686069i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−13812.7	+	7244.67i	−1.10382	+	0.578942i
$540$		0			0
$541$	19410.4			1.54255			0.771273	−	0.636504i	$-0.219620\pi$
$541$	19410.4			1.54255			0.771273	+	0.636504i	$0.219620\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5479.23	+	9490.31i	−0.430651	+	0.745909i
$546$		0			0
$547$	−2279.16	−	3947.62i	−0.178153	−	0.308570i	0.763095	−	0.646286i	$-0.223679\pi$
$547$	−2279.16	−	3947.62i	−0.178153	−	0.308570i	−0.941248	+	0.337716i	$0.890346\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1918.48	+	3322.90i	0.148330	+	0.256915i
$552$		0			0
$553$	−7861.91	+	4328.97i	−0.604561	+	0.332887i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	4710.38i		−	0.358322i	−0.983820	−	0.179161i	$-0.942662\pi$
$557$		−	4710.38i		−	0.358322i	0.983820	−	0.179161i	$-0.0573383\pi$
$558$		0			0
$559$		−	2913.11i		−	0.220414i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2316.26	+	4011.87i	−0.173390	+	0.300320i	−0.939603	−	0.342267i	$-0.888805\pi$
$563$	−2316.26	+	4011.87i	−0.173390	+	0.300320i	0.766213	+	0.642587i	$0.222139\pi$
$564$		0			0
$565$	−4894.81	+	2826.02i	−0.364471	+	0.210428i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6667.01	−	3849.20i	0.491205	−	0.283597i	−0.233869	−	0.972268i	$-0.575139\pi$
$569$	6667.01	−	3849.20i	0.491205	−	0.283597i	0.725074	+	0.688671i	$0.241805\pi$
$570$		0			0
$571$	−8556.34	+	14820.0i	−0.627096	+	1.08616i	0.361036	+	0.932552i	$0.382423\pi$
$571$	−8556.34	+	14820.0i	−0.627096	+	1.08616i	−0.988132	+	0.153610i	$0.950910\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			6201.77i			0.449794i
$576$		0			0
$577$			2389.01i			0.172367i	0.996279	+	0.0861834i	$0.0274671\pi$
$577$			2389.01i			0.172367i	−0.996279	+	0.0861834i	$0.972533\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−11496.7	+	6330.38i	−0.820935	+	0.452028i
$582$		0			0
$583$	4769.48	+	8260.97i	0.338819	+	0.586852i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7877.22	−	13643.7i	−0.553880	−	0.959348i	−0.997990	−	0.0633757i	$-0.979813\pi$
$587$	−7877.22	−	13643.7i	−0.553880	−	0.959348i	0.444110	−	0.895972i	$-0.353520\pi$
$588$		0			0
$589$	−2227.45	+	3858.06i	−0.155824	+	0.269896i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14439.8			0.999953			0.499976	−	0.866039i	$-0.333342\pi$
$593$	14439.8			0.999953			0.499976	+	0.866039i	$0.333342\pi$
$594$		0			0
$595$	−1875.58	−	1134.18i	−0.129229	−	0.0781458i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13069.6	+	7545.75i	0.891504	+	0.514710i	0.874434	−	0.485145i	$-0.161233\pi$
$599$	13069.6	+	7545.75i	0.891504	+	0.514710i	0.0170695	+	0.999854i	$0.494566\pi$
$600$		0			0
$601$	−21885.4	+	12635.5i	−1.48540	+	0.857595i	−0.999862	−	0.0166219i	$-0.994709\pi$
$601$	−21885.4	+	12635.5i	−1.48540	+	0.857595i	−0.485536	+	0.874217i	$0.661376\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2210.44	+	3828.59i	0.148541	+	0.257280i
$606$		0			0
$607$	16922.0	+	9769.93i	1.13154	+	0.653293i	0.944321	−	0.329025i	$-0.106720\pi$
$607$	16922.0	+	9769.93i	1.13154	+	0.653293i	0.187216	+	0.982319i	$0.440053\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			5764.97i			0.381711i
$612$		0			0
$613$	22968.2			1.51334			0.756671	−	0.653796i	$-0.226825\pi$
$613$	22968.2			1.51334			0.756671	+	0.653796i	$0.226825\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24539.5	+	14167.9i	1.60117	+	0.924437i	0.991253	+	0.131974i	$0.0421316\pi$
$617$	24539.5	+	14167.9i	1.60117	+	0.924437i	0.609920	+	0.792463i	$0.291202\pi$
$618$		0			0
$619$	−17347.3	+	10015.5i	−1.12641	+	0.650332i	−0.943029	−	0.332710i	$-0.892037\pi$
$619$	−17347.3	+	10015.5i	−1.12641	+	0.650332i	−0.183379	+	0.983042i	$0.558703\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−52.9157	+	2609.51i	−0.00340292	+	0.167813i
$624$		0			0
$625$	−1710.61	+	2962.87i	−0.109479	+	0.189624i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5677.39			−0.359893
$630$		0			0
$631$	7219.42			0.455469			0.227734	−	0.973723i	$-0.426868\pi$
$631$	7219.42			0.455469			0.227734	+	0.973723i	$0.426868\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−961.643	+	1665.61i	−0.0600970	+	0.104091i
$636$		0			0
$637$	7618.45	+	4820.54i	0.473868	+	0.299838i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	14946.4	−	8629.29i	0.920976	−	0.531726i	0.0370299	−	0.999314i	$-0.488210\pi$
$641$	14946.4	−	8629.29i	0.920976	−	0.531726i	0.883946	+	0.467588i	$0.154877\pi$
$642$		0			0
$643$	−21951.3	−	12673.6i	−1.34631	−	0.777291i	−0.358584	−	0.933498i	$-0.616740\pi$
$643$	−21951.3	−	12673.6i	−1.34631	−	0.777291i	−0.987724	+	0.156206i	$0.950074\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6799.46			−0.413160			−0.206580	−	0.978430i	$-0.566233\pi$
$647$	−6799.46			−0.413160			−0.206580	+	0.978430i	$0.566233\pi$
$648$		0			0
$649$			37640.9i			2.27663i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19452.8	−	11231.1i	−1.16577	−	0.673058i	−0.213090	−	0.977033i	$-0.568353\pi$
$653$	−19452.8	−	11231.1i	−1.16577	−	0.673058i	−0.952680	+	0.303975i	$0.901686\pi$
$654$		0			0
$655$	−1841.42	−	3189.43i	−0.109848	−	0.190262i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3236.46	+	1868.57i	−0.191312	+	0.110454i	−0.592597	−	0.805499i	$-0.701897\pi$
$659$	−3236.46	+	1868.57i	−0.191312	+	0.110454i	0.401285	+	0.915953i	$0.368564\pi$
$660$		0			0
$661$	−10205.8	−	5892.33i	−0.600545	−	0.346725i	0.168711	−	0.985666i	$-0.446040\pi$
$661$	−10205.8	−	5892.33i	−0.600545	−	0.346725i	−0.769256	+	0.638941i	$0.779373\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1605.17	−	2654.46i	0.0936030	−	0.154790i
$666$		0			0
$667$	9577.50			0.555986
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−15600.4	+	27020.6i	−0.897534	+	1.55457i
$672$		0			0
$673$	5712.35	+	9894.08i	0.327184	+	0.566699i	0.981952	−	0.189131i	$-0.0605670\pi$
$673$	5712.35	+	9894.08i	0.327184	+	0.566699i	−0.654768	+	0.755830i	$0.727234\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15013.5	+	26004.1i	0.852312	+	1.47625i	0.879116	+	0.476607i	$0.158134\pi$
$677$	15013.5	+	26004.1i	0.852312	+	1.47625i	−0.0268044	+	0.999641i	$0.508533\pi$
$678$		0			0
$679$	−13453.9	−	24433.9i	−0.760404	−	1.38098i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			18482.4i			1.03545i	0.855548	+	0.517723i	$0.173220\pi$
$683$			18482.4i			1.03545i	−0.855548	+	0.517723i	$0.826780\pi$
$684$		0			0
$685$		−	10502.1i		−	0.585786i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2756.82	−	4774.94i	0.152433	−	0.264022i
$690$		0			0
$691$	16408.4	−	9473.38i	0.903334	−	0.521540i	0.0250537	−	0.999686i	$-0.492024\pi$
$691$	16408.4	−	9473.38i	0.903334	−	0.521540i	0.878280	+	0.478146i	$0.158691\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6724.25	−	3882.25i	0.367001	−	0.211888i
$696$		0			0
$697$	402.956	−	697.941i	0.0218982	−	0.0379288i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	9045.09i		−	0.487344i	−0.969858	−	0.243672i	$-0.921648\pi$
$701$		−	9045.09i		−	0.487344i	0.969858	−	0.243672i	$-0.0783521\pi$
$702$		0			0
$703$		−	8035.08i		−	0.431079i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−29965.2	+	16499.6i	−1.59400	+	0.877698i
$708$		0			0
$709$	−1866.89	−	3233.55i	−0.0988893	−	0.171281i	0.812336	−	0.583190i	$-0.198196\pi$
$709$	−1866.89	−	3233.55i	−0.0988893	−	0.171281i	−0.911225	+	0.411909i	$0.864862\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5560.00	+	9630.20i	0.292039	+	0.505826i
$714$		0			0
$715$	3585.64	−	6210.51i	0.187546	−	0.324839i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	35410.2			1.83669			0.918343	−	0.395785i	$-0.129527\pi$
$719$	35410.2			1.83669			0.918343	+	0.395785i	$0.129527\pi$
$720$		0			0
$721$	7707.23	−	12745.4i	0.398103	−	0.658339i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	10593.9	+	6116.37i	0.542685	+	0.313319i
$726$		0			0
$727$	2328.92	−	1344.60i	0.118810	−	0.0685951i	−0.439417	−	0.898283i	$-0.644815\pi$
$727$	2328.92	−	1344.60i	0.118810	−	0.0685951i	0.558227	+	0.829688i	$0.311482\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1093.07	+	1893.25i	0.0553060	+	0.0957928i
$732$		0			0
$733$	10520.4	+	6073.96i	0.530123	+	0.306066i	0.741066	−	0.671432i	$-0.234320\pi$
$733$	10520.4	+	6073.96i	0.530123	+	0.306066i	−0.210944	+	0.977498i	$0.567654\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			15592.7i			0.779326i
$738$		0			0
$739$	−33517.3			−1.66841			−0.834204	−	0.551456i	$-0.814072\pi$
$739$	−33517.3			−1.66841			−0.834204	+	0.551456i	$0.814072\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−21381.5	−	12344.6i	−1.05573	−	0.609529i	−0.131485	−	0.991318i	$-0.541975\pi$
$743$	−21381.5	−	12344.6i	−1.05573	−	0.609529i	−0.924249	+	0.381790i	$0.875308\pi$
$744$		0			0
$745$	−14995.2	+	8657.49i	−0.737426	+	0.425753i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5240.10	−	106.259i	−0.255633	−	0.00518373i
$750$		0			0
$751$	−4794.74	+	8304.74i	−0.232973	+	0.403521i	−0.958682	−	0.284481i	$-0.908179\pi$
$751$	−4794.74	+	8304.74i	−0.232973	+	0.403521i	0.725709	+	0.688002i	$0.241512\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−18487.4			−0.891159
$756$		0			0
$757$	−13621.8			−0.654022			−0.327011	−	0.945021i	$-0.606041\pi$
$757$	−13621.8			−0.654022			−0.327011	+	0.945021i	$0.606041\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−839.906	+	1454.76i	−0.0400086	+	0.0692970i	−0.885336	−	0.464951i	$-0.846072\pi$
$761$	−839.906	+	1454.76i	−0.0400086	+	0.0692970i	0.845328	+	0.534248i	$0.179405\pi$
$762$		0			0
$763$	−685.783	+	33819.0i	−0.0325387	+	1.60463i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	18842.0	−	10878.5i	0.887023	−	0.512123i
$768$		0			0
$769$	−30695.1	−	17721.8i	−1.43939	−	0.831034i	−0.441586	−	0.897219i	$-0.645584\pi$
$769$	−30695.1	−	17721.8i	−1.43939	−	0.831034i	−0.997807	+	0.0661849i	$0.978917\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13901.4			0.646829			0.323415	−	0.946257i	$-0.395169\pi$
$773$	13901.4			0.646829			0.323415	+	0.946257i	$0.395169\pi$
$774$		0			0
$775$			14202.9i			0.658300i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	987.778	+	570.294i	0.0454311	+	0.0262297i
$780$		0			0
$781$	−8809.91	−	15259.2i	−0.403641	−	0.699126i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6540.73	−	3776.29i	0.297387	−	0.171696i
$786$		0			0
$787$	3906.63	+	2255.49i	0.176946	+	0.102160i	0.585857	−	0.810415i	$-0.300758\pi$
$787$	3906.63	+	2255.49i	0.176946	+	0.102160i	−0.408911	+	0.912574i	$0.634091\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−9027.72	+	14929.1i	−0.405801	+	0.671070i
$792$		0			0
$793$	18034.4			0.807592
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15620.6	+	27055.6i	−0.694240	+	1.20246i	0.276196	+	0.961101i	$0.410926\pi$
$797$	−15620.6	+	27055.6i	−0.694240	+	1.20246i	−0.970436	+	0.241358i	$0.922407\pi$
$798$		0			0
$799$	−2163.16	−	3746.70i	−0.0957785	−	0.165893i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5018.45	−	8692.21i	−0.220544	−	0.381994i
$804$		0			0
$805$	−3734.81	−	6782.85i	−0.163521	−	0.296974i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			42276.5i			1.83728i	0.395091	+	0.918642i	$0.370713\pi$
$809$			42276.5i			1.83728i	−0.395091	+	0.918642i	$0.629287\pi$
$810$		0			0
$811$		−	29402.1i		−	1.27305i	−0.771254	−	0.636527i	$-0.780370\pi$
$811$		−	29402.1i		−	1.27305i	0.771254	−	0.636527i	$-0.219630\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8720.74	−	15104.8i	0.374815	−	0.649198i
$816$		0			0
$817$	−2679.48	+	1547.00i	−0.114741	+	0.0662455i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−36104.3	+	20844.8i	−1.53477	+	0.886103i	−0.535643	+	0.844445i	$0.679931\pi$
$821$	−36104.3	+	20844.8i	−1.53477	+	0.886103i	−0.999132	+	0.0416579i	$0.986736\pi$
$822$		0			0
$823$	4162.09	−	7208.95i	0.176284	−	0.305332i	−0.764321	−	0.644836i	$-0.776926\pi$
$823$	4162.09	−	7208.95i	0.176284	−	0.305332i	0.940605	+	0.339504i	$0.110259\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			7260.76i			0.305298i	0.988280	+	0.152649i	$0.0487804\pi$
$827$			7260.76i			0.305298i	−0.988280	+	0.152649i	$0.951220\pi$
$828$		0			0
$829$			26509.9i			1.11065i	0.831634	+	0.555324i	$0.187406\pi$
$829$			26509.9i			1.11065i	−0.831634	+	0.555324i	$0.812594\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6760.08	−	274.275i	−0.281180	−	0.0114082i
$834$		0			0
$835$	1720.63	+	2980.22i	0.0713113	+	0.123515i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−4438.40	−	7687.53i	−0.182635	−	0.316332i	0.760142	−	0.649757i	$-0.225129\pi$
$839$	−4438.40	−	7687.53i	−0.182635	−	0.316332i	−0.942777	+	0.333424i	$0.891796\pi$
$840$		0			0
$841$	−2748.88	+	4761.19i	−0.112710	+	0.195219i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9036.78			0.367899
$846$		0			0
$847$	11677.1	+	7061.25i	0.473708	+	0.286455i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−17369.5	−	10028.3i	−0.699669	−	0.403954i
$852$		0			0
$853$	13168.8	−	7603.02i	0.528596	−	0.305185i	−0.211849	−	0.977302i	$-0.567948\pi$
$853$	13168.8	−	7603.02i	0.528596	−	0.305185i	0.740444	+	0.672118i	$0.234615\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−11826.9	−	20484.8i	−0.471410	−	0.816506i	0.528055	−	0.849210i	$-0.322922\pi$
$857$	−11826.9	−	20484.8i	−0.471410	−	0.816506i	−0.999465	+	0.0327038i	$0.989588\pi$
$858$		0			0
$859$	25545.4	+	14748.7i	1.01467	+	0.585818i	0.912555	−	0.408954i	$-0.134107\pi$
$859$	25545.4	+	14748.7i	1.01467	+	0.585818i	0.102112	+	0.994773i	$0.467440\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	36862.0i		−	1.45399i	−0.686640	−	0.726997i	$-0.740915\pi$
$863$		−	36862.0i		−	1.45399i	0.686640	−	0.726997i	$-0.259085\pi$
$864$		0			0
$865$	−23173.4			−0.910888
$866$		0			0
$867$		0			0
$868$		0			0
$869$	19084.1	+	11018.2i	0.744976	+	0.430112i
$870$		0			0
$871$	7805.27	−	4506.38i	0.303641	−	0.175307i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	482.109	−	23775.0i	0.0186266	−	0.918560i
$876$		0			0
$877$	4353.93	−	7541.22i	0.167642	−	0.290364i	−0.769949	−	0.638106i	$-0.779718\pi$
$877$	4353.93	−	7541.22i	0.167642	−	0.290364i	0.937590	+	0.347742i	$0.113052\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18950.2			−0.724685			−0.362342	−	0.932045i	$-0.618023\pi$
$881$	−18950.2			−0.724685			−0.362342	+	0.932045i	$0.618023\pi$
$882$		0			0
$883$	5510.89			0.210030			0.105015	−	0.994471i	$-0.466511\pi$
$883$	5510.89			0.210030			0.105015	+	0.994471i	$0.466511\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12652.8	−	21915.3i	0.478962	−	0.829586i	−0.520747	−	0.853711i	$-0.674347\pi$
$887$	12652.8	−	21915.3i	0.478962	−	0.829586i	0.999709	+	0.0241247i	$0.00767987\pi$
$888$		0			0
$889$	−120.359	+	5935.47i	−0.00454075	+	0.223925i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5302.61	−	3061.46i	0.198707	−	0.114723i
$894$		0			0
$895$	11682.2	+	6744.71i	0.436304	+	0.251900i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	21933.8			0.813717
$900$		0			0
$901$			4137.70i			0.152993i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−864.669	−	499.217i	−0.0317598	−	0.0183365i
$906$		0			0
$907$	9256.41	+	16032.6i	0.338869	+	0.586938i	0.984220	−	0.176948i	$-0.0566224\pi$
$907$	9256.41	+	16032.6i	0.338869	+	0.586938i	−0.645351	+	0.763886i	$0.723289\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	27338.9	−	15784.1i	0.994266	−	0.574040i	0.0877196	−	0.996145i	$-0.472042\pi$
$911$	27338.9	−	15784.1i	0.994266	−	0.574040i	0.906547	+	0.422105i	$0.138709\pi$
$912$		0			0
$913$	27907.2	+	16112.2i	1.01160	+	0.584050i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9727.70	−	5882.42i	−0.350313	−	0.211837i
$918$		0			0
$919$	2352.27			0.0844332			0.0422166	−	0.999108i	$-0.486558\pi$
$919$	2352.27			0.0844332			0.0422166	+	0.999108i	$0.486558\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5092.23	+	8820.01i	−0.181596	+	0.314533i
$924$		0			0
$925$	−12808.5	−	22185.0i	−0.455287	−	0.788580i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4728.40	−	8189.83i	−0.166990	−	0.289235i	0.770370	−	0.637597i	$-0.220071\pi$
$929$	−4728.40	−	8189.83i	−0.166990	−	0.289235i	−0.937360	+	0.348362i	$0.886738\pi$
$930$		0			0
$931$	388.174	−	9567.38i	0.0136648	−	0.336797i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			5381.68i			0.188235i
$936$		0			0
$937$		−	34791.9i		−	1.21302i	−0.795075	−	0.606511i	$-0.792569\pi$
$937$		−	34791.9i		−	1.21302i	0.795075	−	0.606511i	$-0.207431\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5515.38	−	9552.92i	0.191069	−	0.330942i	−0.754535	−	0.656259i	$-0.772138\pi$
$941$	5515.38	−	9552.92i	0.191069	−	0.330942i	0.945605	+	0.325317i	$0.105471\pi$
$942$		0			0
$943$	2465.62	−	1423.52i	0.0851448	−	0.0491584i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	26402.6	−	15243.5i	0.905985	−	0.523071i	0.0268478	−	0.999640i	$-0.491453\pi$
$947$	26402.6	−	15243.5i	0.905985	−	0.523071i	0.879137	+	0.476569i	$0.158120\pi$
$948$		0			0
$949$	−2900.72	+	5024.20i	−0.0992218	+	0.171857i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	21378.9i		−	0.726685i	−0.931656	−	0.363343i	$-0.881635\pi$
$953$		−	21378.9i		−	0.726685i	0.931656	−	0.363343i	$-0.118365\pi$
$954$		0			0
$955$			12185.7i			0.412902i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15636.1	−	28396.9i	−0.526502	−	0.956189i
$960$		0			0
$961$	−2162.36	−	3745.31i	−0.0725842	−	0.125720i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1506.69	+	2609.67i	0.0502613	+	0.0870550i
$966$		0			0
$967$	1999.31	−	3462.91i	0.0664877	−	0.115160i	−0.830865	−	0.556474i	$-0.812154\pi$
$967$	1999.31	−	3462.91i	0.0664877	−	0.115160i	0.897353	+	0.441314i	$0.145487\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	48223.6			1.59379			0.796895	−	0.604118i	$-0.206475\pi$
$971$	48223.6			1.59379			0.796895	+	0.604118i	$0.206475\pi$
$972$		0			0
$973$	12401.8	−	20508.8i	0.408618	−	0.675727i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3157.95	+	1823.24i	0.103410	+	0.0597038i	0.550813	−	0.834629i	$-0.314318\pi$
$977$	3157.95	+	1823.24i	0.103410	+	0.0597038i	−0.447403	+	0.894332i	$0.647651\pi$
$978$		0			0
$979$	5549.94	−	3204.26i	0.181182	−	0.104605i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	10473.4	+	18140.5i	0.339828	+	0.588600i	0.984400	−	0.175944i	$-0.0562978\pi$
$983$	10473.4	+	18140.5i	0.339828	+	0.588600i	−0.644572	+	0.764544i	$0.722964\pi$
$984$		0			0
$985$	20519.9	+	11847.2i	0.663776	+	0.383232i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			7722.99i			0.248308i
$990$		0			0
$991$	6732.72			0.215814			0.107907	−	0.994161i	$-0.465585\pi$
$991$	6732.72			0.215814			0.107907	+	0.994161i	$0.465585\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3272.03	−	1889.11i	−0.104252	−	0.0601897i
$996$		0			0
$997$	35961.0	−	20762.1i	1.14232	−	0.659520i	0.195318	−	0.980740i	$-0.437426\pi$
$997$	35961.0	−	20762.1i	1.14232	−	0.659520i	0.947005	+	0.321220i	$0.104093\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.x.a.125.9		48
3.2	odd	2		252.4.x.a.41.10	✓	48
7.6	odd	2	inner	756.4.x.a.125.16		48
9.2	odd	6	inner	756.4.x.a.629.16		48
9.4	even	3		2268.4.f.a.1133.32		48
9.5	odd	6		2268.4.f.a.1133.17		48
9.7	even	3		252.4.x.a.209.15	yes	48
21.20	even	2		252.4.x.a.41.15	yes	48
63.13	odd	6		2268.4.f.a.1133.18		48
63.20	even	6	inner	756.4.x.a.629.9		48
63.34	odd	6		252.4.x.a.209.10	yes	48
63.41	even	6		2268.4.f.a.1133.31		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.10	✓	48	3.2	odd	2
252.4.x.a.41.15	yes	48	21.20	even	2
252.4.x.a.209.10	yes	48	63.34	odd	6
252.4.x.a.209.15	yes	48	9.7	even	3
756.4.x.a.125.9		48	1.1	even	1	trivial
756.4.x.a.125.16		48	7.6	odd	2	inner
756.4.x.a.629.9		48	63.20	even	6	inner
756.4.x.a.629.16		48	9.2	odd	6	inner
2268.4.f.a.1133.17		48	9.5	odd	6
2268.4.f.a.1133.18		48	63.13	odd	6
2268.4.f.a.1133.31		48	63.41	even	6
2268.4.f.a.1133.32		48	9.4	even	3

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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.x (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.99997 + 5.19610i) q^{5} +(-0.375477 + 18.5165i) q^{7} +(39.3810 - 22.7366i) q^{11} +(-22.7627 - 13.1421i) q^{13} +19.7249 q^{17} +27.9162i q^{19} +(60.3466 + 34.8411i) q^{23} +(44.5004 + 77.0769i) q^{25} +(119.031 - 68.7227i) q^{29} +(138.202 + 79.7908i) q^{31} +(-95.0869 - 57.4998i) q^{35} -287.829 q^{37} +(20.4288 - 35.3837i) q^{41} +(55.4158 + 95.9830i) q^{43} +(-109.666 - 189.948i) q^{47} +(-342.718 - 13.9050i) q^{49} +209.770i q^{53} +272.837i q^{55} +(-413.880 + 716.860i) q^{59} +(-594.209 + 343.066i) q^{61} +(136.575 - 78.8515i) q^{65} +(-171.449 + 296.958i) q^{67} -387.476i q^{71} -220.721i q^{73} +(406.215 + 737.734i) q^{77} +(242.301 + 419.677i) q^{79} +(354.323 + 613.706i) q^{83} +(-59.1741 + 102.493i) q^{85} +140.929 q^{89} +(251.891 - 416.550i) q^{91} +(-145.055 - 83.7476i) q^{95} +(-1304.31 + 753.041i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) \) 48 * q + 6 * q^7 + 12 * q^11 + 408 * q^23 - 600 * q^25 + 84 * q^29 + 336 * q^37 + 84 * q^43 + 318 * q^49 - 2964 * q^65 - 588 * q^67 - 2400 * q^77 + 204 * q^79 - 360 * q^85 - 1080 * q^91 - 300 * q^95

Introduction

L-functions

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