LMFDB - Embedded newform 120.4.f.d.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [120,4,Mod(49,120)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("120.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 120 = 2^{3} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	120.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$7.08022920069$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{129})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 65x^{2} + 1024 $ x^4 + 65*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$-5.17891i$ of defining polynomial
Character	$\chi$	$=$	120.49
Dual form			120.4.f.d.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +(-0.178908 - 11.1789i) q^{5} +35.0735i q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +(-0.178908 - 11.1789i) q^{5} +35.0735i q^{7} -9.00000 q^{9} +25.6422 q^{11} +37.6422i q^{13} +(33.5367 - 0.536725i) q^{15} +95.7891i q^{17} -50.8625 q^{19} -105.220 q^{21} +110.863i q^{23} +(-124.936 + 4.00000i) q^{25} -27.0000i q^{27} +54.5047 q^{29} +198.441 q^{31} +76.9265i q^{33} +(392.083 - 6.27493i) q^{35} -266.945i q^{37} -112.927 q^{39} +103.853 q^{41} +108.000i q^{43} +(1.61018 + 100.610i) q^{45} -597.009i q^{47} -887.147 q^{49} -287.367 q^{51} -305.642i q^{53} +(-4.58760 - 286.652i) q^{55} -152.588i q^{57} +223.533 q^{59} +485.450 q^{61} -315.661i q^{63} +(420.799 - 6.73450i) q^{65} +876.166i q^{67} -332.588 q^{69} +585.597 q^{71} -1137.60i q^{73} +(-12.0000 - 374.808i) q^{75} +899.360i q^{77} -685.009 q^{79} +81.0000 q^{81} +305.725i q^{83} +(1070.82 - 17.1375i) q^{85} +163.514i q^{87} -887.175 q^{89} -1320.24 q^{91} +595.322i q^{93} +(9.09973 + 568.588i) q^{95} +556.550i q^{97} -230.780 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 22 q^{5} - 36 q^{9}+O(q^{10}) $ 4 * q + 22 * q^5 - 36 * q^9	$ 4 q + 22 q^{5} - 36 q^{9} + 148 q^{11} + 66 q^{15} + 160 q^{19} - 12 q^{21} - 100 q^{29} - 24 q^{31} + 796 q^{35} - 588 q^{39} + 688 q^{41} - 198 q^{45} - 3276 q^{49} - 468 q^{51} + 1072 q^{55} - 1332 q^{59} + 488 q^{61} + 820 q^{65} - 240 q^{69} + 616 q^{71} - 48 q^{75} - 2104 q^{79} + 324 q^{81} + 2148 q^{85} - 1368 q^{89} + 1352 q^{91} + 2944 q^{95} - 1332 q^{99}+O(q^{100}) $ 4 * q + 22 * q^5 - 36 * q^9 + 148 * q^11 + 66 * q^15 + 160 * q^19 - 12 * q^21 - 100 * q^29 - 24 * q^31 + 796 * q^35 - 588 * q^39 + 688 * q^41 - 198 * q^45 - 3276 * q^49 - 468 * q^51 + 1072 * q^55 - 1332 * q^59 + 488 * q^61 + 820 * q^65 - 240 * q^69 + 616 * q^71 - 48 * q^75 - 2104 * q^79 + 324 * q^81 + 2148 * q^85 - 1368 * q^89 + 1352 * q^91 + 2944 * q^95 - 1332 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$.

$n$	$31$	$41$	$61$	$97$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$		0			0
$5$	−0.178908	−	11.1789i	−0.0160020	−	0.999872i
$5$	−0.178908	−	11.1789i	−0.0160020	−	0.999872i
$6$		0			0
$7$			35.0735i			1.89379i	0.321545	+	0.946894i	$0.395798\pi$
$7$			35.0735i			1.89379i	−0.321545	+	0.946894i	$0.604202\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	25.6422			0.702855			0.351428	−	0.936215i	$-0.385696\pi$
$11$	25.6422			0.702855			0.351428	+	0.936215i	$0.385696\pi$
$12$		0			0
$13$			37.6422i			0.803082i	0.915841	+	0.401541i	$0.131525\pi$
$13$			37.6422i			0.803082i	−0.915841	+	0.401541i	$0.868475\pi$
$14$		0			0
$15$	33.5367	−	0.536725i	0.577276	−	0.00923879i
$16$		0			0
$17$			95.7891i			1.36660i	0.730136	+	0.683302i	$0.239457\pi$
$17$			95.7891i			1.36660i	−0.730136	+	0.683302i	$0.760543\pi$
$18$		0			0
$19$	−50.8625			−0.614140			−0.307070	−	0.951687i	$-0.599349\pi$
$19$	−50.8625			−0.614140			−0.307070	+	0.951687i	$0.599349\pi$
$20$		0			0
$21$	−105.220			−1.09338
$22$		0			0
$23$			110.863i			1.00506i	0.864559	+	0.502531i	$0.167598\pi$
$23$			110.863i			1.00506i	−0.864559	+	0.502531i	$0.832402\pi$
$24$		0			0
$25$	−124.936	+	4.00000i	−0.999488	+	0.0320000i
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$	54.5047			0.349009			0.174505	−	0.984656i	$-0.444168\pi$
$29$	54.5047			0.349009			0.174505	+	0.984656i	$0.444168\pi$
$30$		0			0
$31$	198.441			1.14971			0.574855	−	0.818255i	$-0.305059\pi$
$31$	198.441			1.14971			0.574855	+	0.818255i	$0.305059\pi$
$32$		0			0
$33$			76.9265i			0.405794i
$34$		0			0
$35$	392.083	−	6.27493i	1.89355	−	0.0303045i
$36$		0			0
$37$		−	266.945i		−	1.18610i	−0.805167	−	0.593048i	$-0.797924\pi$
$37$		−	266.945i		−	1.18610i	0.805167	−	0.593048i	$-0.202076\pi$
$38$		0			0
$39$	−112.927			−0.463659
$40$		0			0
$41$	103.853			0.395589			0.197794	−	0.980244i	$-0.436622\pi$
$41$	103.853			0.395589			0.197794	+	0.980244i	$0.436622\pi$
$42$		0			0
$43$			108.000i			0.383020i	0.981491	+	0.191510i	$0.0613384\pi$
$43$			108.000i			0.383020i	−0.981491	+	0.191510i	$0.938662\pi$
$44$		0			0
$45$	1.61018	+	100.610i	0.00533402	+	0.333291i
$46$		0			0
$47$		−	597.009i		−	1.85283i	−0.376510	−	0.926413i	$-0.622876\pi$
$47$		−	597.009i		−	1.85283i	0.376510	−	0.926413i	$-0.377124\pi$
$48$		0			0
$49$	−887.147			−2.58643
$50$		0			0
$51$	−287.367			−0.789009
$52$		0			0
$53$		−	305.642i		−	0.792136i	−0.918221	−	0.396068i	$-0.870375\pi$
$53$		−	305.642i		−	0.792136i	0.918221	−	0.396068i	$-0.129625\pi$
$54$		0			0
$55$	−4.58760	−	286.652i	−0.0112471	−	0.702765i
$56$		0			0
$57$		−	152.588i		−	0.354574i
$58$		0			0
$59$	223.533			0.493246			0.246623	−	0.969111i	$-0.420679\pi$
$59$	223.533			0.493246			0.246623	+	0.969111i	$0.420679\pi$
$60$		0			0
$61$	485.450			1.01894			0.509471	−	0.860488i	$-0.329841\pi$
$61$	485.450			1.01894			0.509471	+	0.860488i	$0.329841\pi$
$62$		0			0
$63$		−	315.661i		−	0.631263i
$64$		0			0
$65$	420.799	−	6.73450i	0.802979	−	0.0128510i
$66$		0			0
$67$			876.166i			1.59762i	0.601582	+	0.798811i	$0.294537\pi$
$67$			876.166i			1.59762i	−0.601582	+	0.798811i	$0.705463\pi$
$68$		0			0
$69$	−332.588			−0.580273
$70$		0			0
$71$	585.597			0.978839			0.489420	−	0.872048i	$-0.337209\pi$
$71$	585.597			0.978839			0.489420	+	0.872048i	$0.337209\pi$
$72$		0			0
$73$		−	1137.60i		−	1.82391i	−0.410287	−	0.911957i	$-0.634571\pi$
$73$		−	1137.60i		−	1.82391i	0.410287	−	0.911957i	$-0.365429\pi$
$74$		0			0
$75$	−12.0000	−	374.808i	−0.0184752	−	0.577055i
$76$		0			0
$77$			899.360i			1.33106i
$78$		0			0
$79$	−685.009			−0.975564			−0.487782	−	0.872965i	$-0.662194\pi$
$79$	−685.009			−0.975564			−0.487782	+	0.872965i	$0.662194\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			305.725i			0.404309i	0.979354	+	0.202155i	$0.0647944\pi$
$83$			305.725i			0.404309i	−0.979354	+	0.202155i	$0.935206\pi$
$84$		0			0
$85$	1070.82	−	17.1375i	1.36643	−	0.0218685i
$86$		0			0
$87$			163.514i			0.201501i
$88$		0			0
$89$	−887.175			−1.05663			−0.528317	−	0.849047i	$-0.677177\pi$
$89$	−887.175			−1.05663			−0.528317	+	0.849047i	$0.677177\pi$
$90$		0			0
$91$	−1320.24			−1.52087
$92$		0			0
$93$			595.322i			0.663785i
$94$		0			0
$95$	9.09973	+	568.588i	0.00982750	+	0.614062i
$96$		0			0
$97$			556.550i			0.582568i	0.956637	+	0.291284i	$0.0940824\pi$
$97$			556.550i			0.582568i	−0.956637	+	0.291284i	$0.905918\pi$
$98$		0			0
$99$	−230.780			−0.234285
$100$		0			0
$101$	1591.09			1.56752			0.783760	−	0.621063i	$-0.213299\pi$
$101$	1591.09			1.56752			0.783760	+	0.621063i	$0.213299\pi$
$102$		0			0
$103$			1350.95i			1.29236i	0.763187	+	0.646178i	$0.223634\pi$
$103$			1350.95i			1.29236i	−0.763187	+	0.646178i	$0.776366\pi$
$104$		0			0
$105$	18.8248	+	1176.25i	0.0174963	+	1.09324i
$106$		0			0
$107$		−	1333.51i		−	1.20481i	−0.798190	−	0.602406i	$-0.794209\pi$
$107$		−	1333.51i		−	1.20481i	0.798190	−	0.602406i	$-0.205791\pi$
$108$		0			0
$109$	609.910			0.535952			0.267976	−	0.963426i	$-0.413645\pi$
$109$	609.910			0.535952			0.267976	+	0.963426i	$0.413645\pi$
$110$		0			0
$111$	800.836			0.684793
$112$		0			0
$113$			241.808i			0.201304i	0.994922	+	0.100652i	$0.0320929\pi$
$113$			241.808i			0.201304i	−0.994922	+	0.100652i	$0.967907\pi$
$114$		0			0
$115$	1239.32	−	19.8342i	1.00493	−	0.0160831i
$116$		0			0
$117$		−	338.780i		−	0.267694i
$118$		0			0
$119$	−3359.65			−2.58806
$120$		0			0
$121$	−673.478			−0.505994
$122$		0			0
$123$			311.559i			0.228393i
$124$		0			0
$125$	67.0677	+	1395.93i	0.0479898	+	0.998848i
$126$		0			0
$127$			1045.94i			0.730802i	0.930850	+	0.365401i	$0.119068\pi$
$127$			1045.94i			0.730802i	−0.930850	+	0.365401i	$0.880932\pi$
$128$		0			0
$129$	−324.000			−0.221137
$130$		0			0
$131$	−886.524			−0.591267			−0.295633	−	0.955301i	$-0.595531\pi$
$131$	−886.524			−0.591267			−0.295633	+	0.955301i	$0.595531\pi$
$132$		0			0
$133$		−	1783.92i		−	1.16305i
$134$		0			0
$135$	−301.831	+	4.83053i	−0.192425	+	0.00307960i
$136$		0			0
$137$			160.723i			0.100230i	0.998743	+	0.0501150i	$0.0159588\pi$
$137$			160.723i			0.100230i	−0.998743	+	0.0501150i	$0.984041\pi$
$138$		0			0
$139$	57.2655			0.0349439			0.0174719	−	0.999847i	$-0.494438\pi$
$139$	57.2655			0.0349439			0.0174719	+	0.999847i	$0.494438\pi$
$140$		0			0
$141$	1791.03			1.06973
$142$		0			0
$143$			965.228i			0.564450i
$144$		0			0
$145$	−9.75135	−	609.303i	−0.00558487	−	0.348965i
$146$		0			0
$147$		−	2661.44i		−	1.49328i
$148$		0			0
$149$	1105.15			0.607633			0.303817	−	0.952731i	$-0.401739\pi$
$149$	1105.15			0.607633			0.303817	+	0.952731i	$0.401739\pi$
$150$		0			0
$151$	2289.63			1.23396			0.616980	−	0.786979i	$-0.288356\pi$
$151$	2289.63			1.23396			0.616980	+	0.786979i	$0.288356\pi$
$152$		0			0
$153$		−	862.102i		−	0.455535i
$154$		0			0
$155$	−35.5027	−	2218.35i	−0.0183977	−	1.14956i
$156$		0			0
$157$			161.514i			0.0821034i	0.999157	+	0.0410517i	$0.0130708\pi$
$157$			161.514i			0.0821034i	−0.999157	+	0.0410517i	$0.986929\pi$
$158$		0			0
$159$	916.927			0.457340
$160$		0			0
$161$	−3888.33			−1.90338
$162$		0			0
$163$			1594.15i			0.766032i	0.923742	+	0.383016i	$0.125115\pi$
$163$			1594.15i			0.766032i	−0.923742	+	0.383016i	$0.874885\pi$
$164$		0			0
$165$	859.955	−	13.7628i	0.405742	−	0.00649353i
$166$		0			0
$167$		−	1017.39i		−	0.471427i	−0.971823	−	0.235713i	$-0.924257\pi$
$167$		−	1017.39i		−	0.471427i	0.971823	−	0.235713i	$-0.0757427\pi$
$168$		0			0
$169$	780.066			0.355060
$170$		0			0
$171$	457.763			0.204713
$172$		0			0
$173$		−	2444.49i		−	1.07428i	−0.843492	−	0.537142i	$-0.819504\pi$
$173$		−	2444.49i		−	1.07428i	0.843492	−	0.537142i	$-0.180496\pi$
$174$		0			0
$175$	−140.294	−	4381.94i	−0.0606012	−	1.89282i
$176$		0			0
$177$			670.599i			0.284776i
$178$		0			0
$179$	−222.780			−0.0930242			−0.0465121	−	0.998918i	$-0.514811\pi$
$179$	−222.780			−0.0930242			−0.0465121	+	0.998918i	$0.514811\pi$
$180$		0			0
$181$	−100.034			−0.0410798			−0.0205399	−	0.999789i	$-0.506539\pi$
$181$	−100.034			−0.0410798			−0.0205399	+	0.999789i	$0.506539\pi$
$182$		0			0
$183$			1456.35i			0.588287i
$184$		0			0
$185$	−2984.16	+	47.7588i	−1.18594	+	0.0189800i
$186$		0			0
$187$			2456.24i			0.960525i
$188$		0			0
$189$	946.983			0.364460
$190$		0			0
$191$	702.403			0.266095			0.133047	−	0.991110i	$-0.457524\pi$
$191$	702.403			0.266095			0.133047	+	0.991110i	$0.457524\pi$
$192$		0			0
$193$		−	4126.63i		−	1.53907i	−0.638602	−	0.769537i	$-0.720487\pi$
$193$		−	4126.63i		−	1.53907i	0.638602	−	0.769537i	$-0.279513\pi$
$194$		0			0
$195$	20.2035	+	1262.40i	0.00741950	+	0.463600i
$196$		0			0
$197$		−	3104.45i		−	1.12276i	−0.827559	−	0.561378i	$-0.810271\pi$
$197$		−	3104.45i		−	1.12276i	0.827559	−	0.561378i	$-0.189729\pi$
$198$		0			0
$199$	367.616			0.130953			0.0654764	−	0.997854i	$-0.479143\pi$
$199$	367.616			0.130953			0.0654764	+	0.997854i	$0.479143\pi$
$200$		0			0
$201$	−2628.50			−0.922388
$202$		0			0
$203$			1911.67i			0.660950i
$204$		0			0
$205$	−18.5802	−	1160.96i	−0.00633023	−	0.395538i
$206$		0			0
$207$		−	997.763i		−	0.335021i
$208$		0			0
$209$	−1304.23			−0.431652
$210$		0			0
$211$	2594.85			0.846619			0.423310	−	0.905985i	$-0.360868\pi$
$211$	2594.85			0.846619			0.423310	+	0.905985i	$0.360868\pi$
$212$		0			0
$213$			1756.79i			0.565133i
$214$		0			0
$215$	1207.32	−	19.3221i	0.382971	−	0.00612910i
$216$		0			0
$217$			6960.00i			2.17731i
$218$		0			0
$219$	3412.79			1.05304
$220$		0			0
$221$	−3605.71			−1.09749
$222$		0			0
$223$		−	1834.41i		−	0.550857i	−0.961321	−	0.275429i	$-0.911180\pi$
$223$		−	1834.41i		−	0.550857i	0.961321	−	0.275429i	$-0.0888198\pi$
$224$		0			0
$225$	1124.42	−	36.0000i	0.333163	−	0.0106667i
$226$		0			0
$227$		−	175.811i		−	0.0514053i	−0.999670	−	0.0257027i	$-0.991818\pi$
$227$		−	175.811i		−	0.0514053i	0.999670	−	0.0257027i	$-0.00818231\pi$
$228$		0			0
$229$	−1622.35			−0.468158			−0.234079	−	0.972218i	$-0.575207\pi$
$229$	−1622.35			−0.468158			−0.234079	+	0.972218i	$0.575207\pi$
$230$		0			0
$231$	−2698.08			−0.768487
$232$		0			0
$233$			3965.65i			1.11501i	0.830172	+	0.557507i	$0.188242\pi$
$233$			3965.65i			1.11501i	−0.830172	+	0.557507i	$0.811758\pi$
$234$		0			0
$235$	−6673.91	+	106.810i	−1.85259	+	0.0296490i
$236$		0			0
$237$		−	2055.03i		−	0.563242i
$238$		0			0
$239$	6323.25			1.71137			0.855685	−	0.517497i	$-0.173136\pi$
$239$	6323.25			1.71137			0.855685	+	0.517497i	$0.173136\pi$
$240$		0			0
$241$	3407.51			0.910775			0.455388	−	0.890293i	$-0.349501\pi$
$241$	3407.51			0.910775			0.455388	+	0.890293i	$0.349501\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	158.718	+	9917.33i	0.0413882	+	2.58610i
$246$		0			0
$247$		−	1914.58i		−	0.493205i
$248$		0			0
$249$	−917.175			−0.233428
$250$		0			0
$251$	1345.81			0.338433			0.169216	−	0.985579i	$-0.445876\pi$
$251$	1345.81			0.338433			0.169216	+	0.985579i	$0.445876\pi$
$252$		0			0
$253$			2842.76i			0.706414i
$254$		0			0
$255$	51.4124	+	3212.45i	0.0126258	+	0.788908i
$256$		0			0
$257$		−	4697.19i		−	1.14009i	−0.821614	−	0.570044i	$-0.806926\pi$
$257$		−	4697.19i		−	1.14009i	0.821614	−	0.570044i	$-0.193074\pi$
$258$		0			0
$259$	9362.70			2.24621
$260$		0			0
$261$	−490.542			−0.116336
$262$		0			0
$263$			4700.66i			1.10211i	0.834468	+	0.551056i	$0.185775\pi$
$263$			4700.66i			1.10211i	−0.834468	+	0.551056i	$0.814225\pi$
$264$		0			0
$265$	−3416.75	+	54.6819i	−0.792034	+	0.0126758i
$266$		0			0
$267$		−	2661.53i		−	0.610048i
$268$		0			0
$269$	7962.67			1.80481			0.902403	−	0.430894i	$-0.141802\pi$
$269$	7962.67			1.80481			0.902403	+	0.430894i	$0.141802\pi$
$270$		0			0
$271$	−6122.73			−1.37243			−0.686217	−	0.727397i	$-0.740730\pi$
$271$	−6122.73			−1.37243			−0.686217	+	0.727397i	$0.740730\pi$
$272$		0			0
$273$		−	3960.72i		−	0.878073i
$274$		0			0
$275$	−3203.63	+	102.569i	−0.702495	+	0.0224914i
$276$		0			0
$277$			8417.57i			1.82586i	0.408117	+	0.912930i	$0.366186\pi$
$277$			8417.57i			1.82586i	−0.408117	+	0.912930i	$0.633814\pi$
$278$		0			0
$279$	−1785.97			−0.383237
$280$		0			0
$281$	3030.99			0.643466			0.321733	−	0.946830i	$-0.395735\pi$
$281$	3030.99			0.643466			0.321733	+	0.946830i	$0.395735\pi$
$282$		0			0
$283$			2890.81i			0.607211i	0.952798	+	0.303606i	$0.0981905\pi$
$283$			2890.81i			0.607211i	−0.952798	+	0.303606i	$0.901809\pi$
$284$		0			0
$285$	−1705.76	+	27.2992i	−0.354529	+	0.00567391i
$286$		0			0
$287$			3642.49i			0.749161i
$288$		0			0
$289$	−4262.55			−0.867606
$290$		0			0
$291$	−1669.65			−0.336346
$292$		0			0
$293$		−	8966.75i		−	1.78786i	−0.448206	−	0.893930i	$-0.647937\pi$
$293$		−	8966.75i		−	1.78786i	0.448206	−	0.893930i	$-0.352063\pi$
$294$		0			0
$295$	−39.9919	−	2498.86i	−0.00789295	−	0.493183i
$296$		0			0
$297$		−	692.339i		−	0.135265i
$298$		0			0
$299$	−4173.11			−0.807147
$300$		0			0
$301$	−3787.93			−0.725358
$302$		0			0
$303$			4773.28i			0.905009i
$304$		0			0
$305$	−86.8511	−	5426.80i	−0.0163052	−	1.01881i
$306$		0			0
$307$			3298.42i			0.613194i	0.951839	+	0.306597i	$0.0991904\pi$
$307$			3298.42i			0.613194i	−0.951839	+	0.306597i	$0.900810\pi$
$308$		0			0
$309$	−4052.84			−0.746142
$310$		0			0
$311$	−3394.92			−0.618998			−0.309499	−	0.950900i	$-0.600161\pi$
$311$	−3394.92			−0.618998			−0.309499	+	0.950900i	$0.600161\pi$
$312$		0			0
$313$		−	5946.95i		−	1.07393i	−0.843603	−	0.536967i	$-0.819570\pi$
$313$		−	5946.95i		−	1.07393i	0.843603	−	0.536967i	$-0.180430\pi$
$314$		0			0
$315$	−3528.75	+	56.4744i	−0.631182	+	0.0101015i
$316$		0			0
$317$		−	2520.83i		−	0.446637i	−0.974746	−	0.223318i	$-0.928311\pi$
$317$		−	2520.83i		−	0.446637i	0.974746	−	0.223318i	$-0.0716889\pi$
$318$		0			0
$319$	1397.62			0.245303
$320$		0			0
$321$	4000.52			0.695599
$322$		0			0
$323$		−	4872.08i		−	0.839286i
$324$		0			0
$325$	−150.569	−	4702.86i	−0.0256986	−	0.802671i
$326$		0			0
$327$			1829.73i			0.309432i
$328$		0			0
$329$	20939.2			3.50886
$330$		0			0
$331$	−4586.86			−0.761682			−0.380841	−	0.924641i	$-0.624365\pi$
$331$	−4586.86			−0.761682			−0.380841	+	0.924641i	$0.624365\pi$
$332$		0			0
$333$			2402.51i			0.395365i
$334$		0			0
$335$	9794.58	−	156.753i	1.59742	−	0.0255652i
$336$		0			0
$337$			8582.12i			1.38723i	0.720344	+	0.693617i	$0.243984\pi$
$337$			8582.12i			1.38723i	−0.720344	+	0.693617i	$0.756016\pi$
$338$		0			0
$339$	−725.424			−0.116223
$340$		0			0
$341$	5088.45			0.808080
$342$		0			0
$343$		−	19085.1i		−	3.00437i
$344$		0			0
$345$	59.5027	+	3717.97i	0.00928556	+	0.580199i
$346$		0			0
$347$		−	2539.38i		−	0.392857i	−0.980518	−	0.196428i	$-0.937066\pi$
$347$		−	2539.38i		−	0.392857i	0.980518	−	0.196428i	$-0.0629343\pi$
$348$		0			0
$349$	9002.82			1.38083			0.690415	−	0.723413i	$-0.257428\pi$
$349$	9002.82			1.38083			0.690415	+	0.723413i	$0.257428\pi$
$350$		0			0
$351$	1016.34			0.154553
$352$		0			0
$353$			3928.08i			0.592267i	0.955146	+	0.296134i	$0.0956974\pi$
$353$			3928.08i			0.592267i	−0.955146	+	0.296134i	$0.904303\pi$
$354$		0			0
$355$	−104.768	−	6546.34i	−0.0156634	−	0.978714i
$356$		0			0
$357$		−	10079.0i		−	1.49422i
$358$		0			0
$359$	−10001.7			−1.47039			−0.735197	−	0.677854i	$-0.762910\pi$
$359$	−10001.7			−1.47039			−0.735197	+	0.677854i	$0.762910\pi$
$360$		0			0
$361$	−4272.00			−0.622832
$362$		0			0
$363$		−	2020.44i		−	0.292136i
$364$		0			0
$365$	−12717.1	+	203.526i	−1.82368	+	0.0291863i
$366$		0			0
$367$		−	5967.79i		−	0.848818i	−0.905471	−	0.424409i	$-0.860482\pi$
$367$		−	5967.79i		−	0.848818i	0.905471	−	0.424409i	$-0.139518\pi$
$368$		0			0
$369$	−934.678			−0.131863
$370$		0			0
$371$	10719.9			1.50014
$372$		0			0
$373$			6931.55i			0.962204i	0.876665	+	0.481102i	$0.159763\pi$
$373$			6931.55i			0.962204i	−0.876665	+	0.481102i	$0.840237\pi$
$374$		0			0
$375$	−4187.80	+	201.203i	−0.576685	+	0.0277069i
$376$		0			0
$377$			2051.68i			0.280283i
$378$		0			0
$379$	−9711.70			−1.31624			−0.658122	−	0.752911i	$-0.728649\pi$
$379$	−9711.70			−1.31624			−0.658122	+	0.752911i	$0.728649\pi$
$380$		0			0
$381$	−3137.81			−0.421929
$382$		0			0
$383$			5664.84i			0.755769i	0.925853	+	0.377885i	$0.123348\pi$
$383$			5664.84i			0.755769i	−0.925853	+	0.377885i	$0.876652\pi$
$384$		0			0
$385$	10053.9	−	160.903i	1.33089	−	0.0212997i
$386$		0			0
$387$		−	972.000i		−	0.127673i
$388$		0			0
$389$	−8918.17			−1.16239			−0.581195	−	0.813765i	$-0.697414\pi$
$389$	−8918.17			−1.16239			−0.581195	+	0.813765i	$0.697414\pi$
$390$		0			0
$391$	−10619.4			−1.37352
$392$		0			0
$393$		−	2659.57i		−	0.341368i
$394$		0			0
$395$	122.554	+	7657.66i	0.0156110	+	0.975439i
$396$		0			0
$397$		−	12849.9i		−	1.62448i	−0.583321	−	0.812242i	$-0.698247\pi$
$397$		−	12849.9i		−	1.62448i	0.583321	−	0.812242i	$-0.301753\pi$
$398$		0			0
$399$	5351.77			0.671488
$400$		0			0
$401$	3563.08			0.443721			0.221860	−	0.975078i	$-0.428787\pi$
$401$	3563.08			0.443721			0.221860	+	0.975078i	$0.428787\pi$
$402$		0			0
$403$			7469.74i			0.923311i
$404$		0			0
$405$	−14.4916	−	905.492i	−0.00177801	−	0.111097i
$406$		0			0
$407$		−	6845.06i		−	0.833654i
$408$		0			0
$409$	2026.69			0.245020			0.122510	−	0.992467i	$-0.460906\pi$
$409$	2026.69			0.245020			0.122510	+	0.992467i	$0.460906\pi$
$410$		0			0
$411$	−482.169			−0.0578678
$412$		0			0
$413$			7840.07i			0.934104i
$414$		0			0
$415$	3417.67	−	54.6968i	0.404258	−	0.00646978i
$416$		0			0
$417$			171.796i			0.0201748i
$418$		0			0
$419$	1670.39			0.194759			0.0973793	−	0.995247i	$-0.468954\pi$
$419$	1670.39			0.194759			0.0973793	+	0.995247i	$0.468954\pi$
$420$		0			0
$421$	6079.66			0.703812			0.351906	−	0.936035i	$-0.385534\pi$
$421$	6079.66			0.703812			0.351906	+	0.936035i	$0.385534\pi$
$422$		0			0
$423$			5373.08i			0.617608i
$424$		0			0
$425$	−383.156	−	11967.5i	−0.0437313	−	1.36590i
$426$		0			0
$427$			17026.4i			1.92966i
$428$		0			0
$429$	−2895.68			−0.325886
$430$		0			0
$431$	1719.37			0.192155			0.0960777	−	0.995374i	$-0.469370\pi$
$431$	1719.37			0.192155			0.0960777	+	0.995374i	$0.469370\pi$
$432$		0			0
$433$			12024.9i			1.33459i	0.744792	+	0.667296i	$0.232549\pi$
$433$			12024.9i			1.33459i	−0.744792	+	0.667296i	$0.767451\pi$
$434$		0			0
$435$	1827.91	−	29.2540i	0.201475	−	0.00322442i
$436$		0			0
$437$		−	5638.75i		−	0.617249i
$438$		0			0
$439$	7542.97			0.820060			0.410030	−	0.912072i	$-0.365518\pi$
$439$	7542.97			0.820060			0.410030	+	0.912072i	$0.365518\pi$
$440$		0			0
$441$	7984.32			0.862145
$442$		0			0
$443$		−	4578.13i		−	0.491001i	−0.969396	−	0.245501i	$-0.921048\pi$
$443$		−	4578.13i		−	0.491001i	0.969396	−	0.245501i	$-0.0789524\pi$
$444$		0			0
$445$	158.723	+	9917.65i	0.0169083	+	1.05650i
$446$		0			0
$447$			3315.45i			0.350817i
$448$		0			0
$449$	−6875.21			−0.722630			−0.361315	−	0.932444i	$-0.617672\pi$
$449$	−6875.21			−0.722630			−0.361315	+	0.932444i	$0.617672\pi$
$450$		0			0
$451$	2663.02			0.278042
$452$		0			0
$453$			6868.90i			0.712427i
$454$		0			0
$455$	236.202	+	14758.9i	0.0243370	+	1.52067i
$456$		0			0
$457$			7351.97i			0.752540i	0.926510	+	0.376270i	$0.122793\pi$
$457$			7351.97i			0.752540i	−0.926510	+	0.376270i	$0.877207\pi$
$458$		0			0
$459$	2586.31			0.263003
$460$		0			0
$461$	−15614.5			−1.57752			−0.788762	−	0.614698i	$-0.789278\pi$
$461$	−15614.5			−1.57752			−0.788762	+	0.614698i	$0.789278\pi$
$462$		0			0
$463$		−	1684.73i		−	0.169106i	−0.996419	−	0.0845530i	$-0.973054\pi$
$463$		−	1684.73i		−	0.169106i	0.996419	−	0.0845530i	$-0.0269462\pi$
$464$		0			0
$465$	6655.05	−	106.508i	0.663700	−	0.0106219i
$466$		0			0
$467$		−	10235.2i		−	1.01420i	−0.861888	−	0.507099i	$-0.830718\pi$
$467$		−	10235.2i		−	1.01420i	0.861888	−	0.507099i	$-0.169282\pi$
$468$		0			0
$469$	−30730.2			−3.02556
$470$		0			0
$471$	−484.542			−0.0474024
$472$		0			0
$473$			2769.36i			0.269207i
$474$		0			0
$475$	6354.56	−	203.450i	0.613826	−	0.0196525i
$476$		0			0
$477$			2750.78i			0.264045i
$478$		0			0
$479$	−12247.2			−1.16825			−0.584123	−	0.811665i	$-0.698561\pi$
$479$	−12247.2			−1.16825			−0.584123	+	0.811665i	$0.698561\pi$
$480$		0			0
$481$	10048.4			0.952532
$482$		0			0
$483$		−	11665.0i		−	1.09891i
$484$		0			0
$485$	6221.62	−	99.5714i	0.582493	−	0.00932228i
$486$		0			0
$487$			7895.91i			0.734698i	0.930083	+	0.367349i	$0.119735\pi$
$487$			7895.91i			0.734698i	−0.930083	+	0.367349i	$0.880265\pi$
$488$		0			0
$489$	−4782.44			−0.442269
$490$		0			0
$491$	−7625.48			−0.700882			−0.350441	−	0.936585i	$-0.613968\pi$
$491$	−7625.48			−0.700882			−0.350441	+	0.936585i	$0.613968\pi$
$492$		0			0
$493$			5220.96i			0.476958i
$494$		0			0
$495$	41.2884	+	2579.86i	0.00374904	+	0.234255i
$496$		0			0
$497$			20538.9i			1.85371i
$498$		0			0
$499$	8655.23			0.776476			0.388238	−	0.921559i	$-0.373084\pi$
$499$	8655.23			0.776476			0.388238	+	0.921559i	$0.373084\pi$
$500$		0			0
$501$	3052.18			0.272178
$502$		0			0
$503$			118.441i			0.0104990i	0.999986	+	0.00524951i	$0.00167098\pi$
$503$			118.441i			0.0104990i	−0.999986	+	0.00524951i	$0.998329\pi$
$504$		0			0
$505$	−284.660	−	17786.7i	−0.0250835	−	1.56732i
$506$		0			0
$507$			2340.20i			0.204994i
$508$		0			0
$509$	−5359.43			−0.466704			−0.233352	−	0.972392i	$-0.574970\pi$
$509$	−5359.43			−0.466704			−0.233352	+	0.972392i	$0.574970\pi$
$510$		0			0
$511$	39899.5			3.45411
$512$		0			0
$513$			1373.29i			0.118191i
$514$		0			0
$515$	15102.1	−	241.695i	1.29219	−	0.0206803i
$516$		0			0
$517$		−	15308.6i		−	1.30227i
$518$		0			0
$519$	7333.47			0.620238
$520$		0			0
$521$	−10862.4			−0.913414			−0.456707	−	0.889617i	$-0.650971\pi$
$521$	−10862.4			−0.913414			−0.456707	+	0.889617i	$0.650971\pi$
$522$		0			0
$523$		−	9553.39i		−	0.798740i	−0.916790	−	0.399370i	$-0.869229\pi$
$523$		−	9553.39i		−	0.798740i	0.916790	−	0.399370i	$-0.130771\pi$
$524$		0			0
$525$	13145.8	−	420.881i	1.09282	−	0.0349881i
$526$		0			0
$527$			19008.5i			1.57120i
$528$		0			0
$529$	−123.501			−0.0101505
$530$		0			0
$531$	−2011.80			−0.164415
$532$		0			0
$533$			3909.26i			0.317690i
$534$		0			0
$535$	−14907.1	+	238.575i	−1.20466	+	0.0192795i
$536$		0			0
$537$		−	668.339i		−	0.0537076i
$538$		0			0
$539$	−22748.4			−1.81789
$540$		0			0
$541$	−6132.47			−0.487348			−0.243674	−	0.969857i	$-0.578353\pi$
$541$	−6132.47			−0.487348			−0.243674	+	0.969857i	$0.578353\pi$
$542$		0			0
$543$		−	300.101i		−	0.0237174i
$544$		0			0
$545$	−109.118	−	6818.12i	−0.00857633	−	0.535883i
$546$		0			0
$547$			2853.85i			0.223075i	0.993760	+	0.111537i	$0.0355775\pi$
$547$			2853.85i			0.223075i	−0.993760	+	0.111537i	$0.964422\pi$
$548$		0			0
$549$	−4369.05			−0.339648
$550$		0			0
$551$	−2772.25			−0.214341
$552$		0			0
$553$		−	24025.6i		−	1.84751i
$554$		0			0
$555$	−143.276	−	8952.48i	−0.0109581	−	0.684705i
$556$		0			0
$557$		−	5192.60i		−	0.395005i	−0.980302	−	0.197502i	$-0.936717\pi$
$557$		−	5192.60i		−	0.395005i	0.980302	−	0.197502i	$-0.0632830\pi$
$558$		0			0
$559$	−4065.36			−0.307596
$560$		0			0
$561$	−7368.72			−0.554559
$562$		0			0
$563$		−	10907.2i		−	0.816492i	−0.912872	−	0.408246i	$-0.866141\pi$
$563$		−	10907.2i		−	0.816492i	0.912872	−	0.408246i	$-0.133859\pi$
$564$		0			0
$565$	2703.15	−	43.2615i	0.201278	−	0.00322128i
$566$		0			0
$567$			2840.95i			0.210421i
$568$		0			0
$569$	155.257			0.0114389			0.00571945	−	0.999984i	$-0.498179\pi$
$569$	155.257			0.0114389			0.00571945	+	0.999984i	$0.498179\pi$
$570$		0			0
$571$	−4925.15			−0.360965			−0.180483	−	0.983578i	$-0.557766\pi$
$571$	−4925.15			−0.360965			−0.180483	+	0.983578i	$0.557766\pi$
$572$		0			0
$573$			2107.21i			0.153630i
$574$		0			0
$575$	−443.450	−	13850.7i	−0.0321620	−	1.00455i
$576$		0			0
$577$		−	5292.05i		−	0.381822i	−0.981607	−	0.190911i	$-0.938856\pi$
$577$		−	5292.05i		−	0.381822i	0.981607	−	0.190911i	$-0.0611441\pi$
$578$		0			0
$579$	12379.9			0.888585
$580$		0			0
$581$	−10722.8			−0.765677
$582$		0			0
$583$		−	7837.33i		−	0.556757i
$584$		0			0
$585$	−3787.19	+	60.6105i	−0.267660	+	0.00428365i
$586$		0			0
$587$		−	19658.5i		−	1.38227i	−0.722727	−	0.691134i	$-0.757112\pi$
$587$		−	19658.5i		−	1.38227i	0.722727	−	0.691134i	$-0.242888\pi$
$588$		0			0
$589$	−10093.2			−0.706083
$590$		0			0
$591$	9313.36			0.648224
$592$		0			0
$593$		−	6578.08i		−	0.455530i	−0.973716	−	0.227765i	$-0.926858\pi$
$593$		−	6578.08i		−	0.455530i	0.973716	−	0.227765i	$-0.0731418\pi$
$594$		0			0
$595$	601.070	+	37557.3i	0.0414142	+	2.58773i
$596$		0			0
$597$			1102.85i			0.0756056i
$598$		0			0
$599$	−16915.9			−1.15386			−0.576931	−	0.816793i	$-0.695750\pi$
$599$	−16915.9			−1.15386			−0.576931	+	0.816793i	$0.695750\pi$
$600$		0			0
$601$	19801.4			1.34395			0.671977	−	0.740572i	$-0.265445\pi$
$601$	19801.4			1.34395			0.671977	+	0.740572i	$0.265445\pi$
$602$		0			0
$603$		−	7885.49i		−	0.532541i
$604$		0			0
$605$	120.491	+	7528.75i	0.00809695	+	0.505930i
$606$		0			0
$607$		−	22498.4i		−	1.50442i	−0.658926	−	0.752208i	$-0.728989\pi$
$607$		−	22498.4i		−	1.50442i	0.658926	−	0.752208i	$-0.271011\pi$
$608$		0			0
$609$	−5735.01			−0.381600
$610$		0			0
$611$	22472.7			1.48797
$612$		0			0
$613$			23829.3i			1.57008i	0.619446	+	0.785039i	$0.287357\pi$
$613$			23829.3i			1.57008i	−0.619446	+	0.785039i	$0.712643\pi$
$614$		0			0
$615$	3482.89	−	55.7406i	0.228364	−	0.00365476i
$616$		0			0
$617$		−	4277.52i		−	0.279103i	−0.990215	−	0.139551i	$-0.955434\pi$
$617$		−	4277.52i		−	0.279103i	0.990215	−	0.139551i	$-0.0445660\pi$
$618$		0			0
$619$	−4995.99			−0.324404			−0.162202	−	0.986758i	$-0.551860\pi$
$619$	−4995.99			−0.324404			−0.162202	+	0.986758i	$0.551860\pi$
$620$		0			0
$621$	2993.29			0.193424
$622$		0			0
$623$		−	31116.3i		−	2.00104i
$624$		0			0
$625$	15593.0	−	999.488i	0.997952	−	0.0639672i
$626$		0			0
$627$		−	3912.68i		−	0.249214i
$628$		0			0
$629$	25570.5			1.62092
$630$		0			0
$631$	−11328.0			−0.714675			−0.357337	−	0.933975i	$-0.616315\pi$
$631$	−11328.0			−0.714675			−0.357337	+	0.933975i	$0.616315\pi$
$632$		0			0
$633$			7784.54i			0.488796i
$634$		0			0
$635$	11692.4	−	187.127i	0.730708	−	0.0116943i
$636$		0			0
$637$		−	33394.1i		−	2.07712i
$638$		0			0
$639$	−5270.37			−0.326280
$640$		0			0
$641$	5955.35			0.366961			0.183481	−	0.983023i	$-0.441264\pi$
$641$	5955.35			0.366961			0.183481	+	0.983023i	$0.441264\pi$
$642$		0			0
$643$			15727.7i			0.964605i	0.876005	+	0.482302i	$0.160199\pi$
$643$			15727.7i			0.964605i	−0.876005	+	0.482302i	$0.839801\pi$
$644$		0			0
$645$	57.9663	+	3621.97i	0.00353864	+	0.221108i
$646$		0			0
$647$			9582.24i			0.582252i	0.956685	+	0.291126i	$0.0940298\pi$
$647$			9582.24i			0.582252i	−0.956685	+	0.291126i	$0.905970\pi$
$648$		0			0
$649$	5731.87			0.346681
$650$		0			0
$651$	−20880.0			−1.25707
$652$		0			0
$653$			8313.70i			0.498224i	0.968475	+	0.249112i	$0.0801388\pi$
$653$			8313.70i			0.498224i	−0.968475	+	0.249112i	$0.919861\pi$
$654$		0			0
$655$	158.606	+	9910.37i	0.00946148	+	0.591191i
$656$		0			0
$657$			10238.4i			0.607971i
$658$		0			0
$659$	−15095.5			−0.892317			−0.446158	−	0.894954i	$-0.647208\pi$
$659$	−15095.5			−0.892317			−0.446158	+	0.894954i	$0.647208\pi$
$660$		0			0
$661$	8266.98			0.486457			0.243229	−	0.969969i	$-0.421793\pi$
$661$	8266.98			0.486457			0.243229	+	0.969969i	$0.421793\pi$
$662$		0			0
$663$		−	10817.1i		−	0.633639i
$664$		0			0
$665$	−19942.3	+	319.159i	−1.16290	+	0.0186112i
$666$		0			0
$667$			6042.53i			0.350776i
$668$		0			0
$669$	5503.23			0.318038
$670$		0			0
$671$	12448.0			0.716170
$672$		0			0
$673$			11186.7i			0.640735i	0.947293	+	0.320367i	$0.103806\pi$
$673$			11186.7i			0.640735i	−0.947293	+	0.320367i	$0.896194\pi$
$674$		0			0
$675$	108.000	+	3373.27i	0.00615840	+	0.192352i
$676$		0			0
$677$		−	7675.84i		−	0.435756i	−0.975976	−	0.217878i	$-0.930087\pi$
$677$		−	7675.84i		−	0.435756i	0.975976	−	0.217878i	$-0.0699134\pi$
$678$		0			0
$679$	−19520.1			−1.10326
$680$		0			0
$681$	527.434			0.0296789
$682$		0			0
$683$			10550.9i			0.591099i	0.955327	+	0.295549i	$0.0955027\pi$
$683$			10550.9i			0.591099i	−0.955327	+	0.295549i	$0.904497\pi$
$684$		0			0
$685$	1796.71	−	28.7547i	0.100217	−	0.00160388i
$686$		0			0
$687$		−	4867.06i		−	0.270291i
$688$		0			0
$689$	11505.0			0.636150
$690$		0			0
$691$	−26950.9			−1.48373			−0.741867	−	0.670547i	$-0.766060\pi$
$691$	−26950.9			−1.48373			−0.741867	+	0.670547i	$0.766060\pi$
$692$		0			0
$693$		−	8094.24i		−	0.443686i
$694$		0			0
$695$	−10.2453	−	640.166i	−0.000559173	−	0.0349394i
$696$		0			0
$697$			9947.99i			0.540613i
$698$		0			0
$699$	−11897.0			−0.643754
$700$		0			0
$701$	12791.6			0.689204			0.344602	−	0.938749i	$-0.388014\pi$
$701$	12791.6			0.689204			0.344602	+	0.938749i	$0.388014\pi$
$702$		0			0
$703$			13577.5i			0.728429i
$704$		0			0
$705$	−320.430	−	20021.7i	−0.0171179	−	1.06959i
$706$		0			0
$707$			55805.1i			2.96855i
$708$		0			0
$709$	16238.3			0.860142			0.430071	−	0.902795i	$-0.358489\pi$
$709$	16238.3			0.860142			0.430071	+	0.902795i	$0.358489\pi$
$710$		0			0
$711$	6165.08			0.325188
$712$		0			0
$713$			21999.6i			1.15553i
$714$		0			0
$715$	10790.2	−	172.687i	0.564378	−	0.00903236i
$716$		0			0
$717$			18969.8i			0.988060i
$718$		0			0
$719$	24285.8			1.25968			0.629839	−	0.776726i	$-0.283121\pi$
$719$	24285.8			1.25968			0.629839	+	0.776726i	$0.283121\pi$
$720$		0			0
$721$	−47382.3			−2.44745
$722$		0			0
$723$			10222.5i			0.525836i
$724$		0			0
$725$	−6809.60	+	218.019i	−0.348831	+	0.0111683i
$726$		0			0
$727$		−	4466.82i		−	0.227875i	−0.993488	−	0.113937i	$-0.963654\pi$
$727$		−	4466.82i		−	0.227875i	0.993488	−	0.113937i	$-0.0363464\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$	−10345.2			−0.523436
$732$		0			0
$733$		−	30802.1i		−	1.55212i	−0.630661	−	0.776059i	$-0.717216\pi$
$733$		−	30802.1i		−	1.55212i	0.630661	−	0.776059i	$-0.282784\pi$
$734$		0			0
$735$	−29752.0	+	476.154i	−1.49309	+	0.0238955i
$736$		0			0
$737$			22466.8i			1.12290i
$738$		0			0
$739$	12920.5			0.643151			0.321576	−	0.946884i	$-0.395788\pi$
$739$	12920.5			0.643151			0.321576	+	0.946884i	$0.395788\pi$
$740$		0			0
$741$	5743.73			0.284752
$742$		0			0
$743$			2571.29i			0.126960i	0.997983	+	0.0634802i	$0.0202200\pi$
$743$			2571.29i			0.126960i	−0.997983	+	0.0634802i	$0.979780\pi$
$744$		0			0
$745$	−197.720	−	12354.4i	−0.00972337	−	0.607555i
$746$		0			0
$747$		−	2751.53i		−	0.134770i
$748$		0			0
$749$	46770.7			2.28166
$750$		0			0
$751$	−13427.4			−0.652426			−0.326213	−	0.945296i	$-0.605773\pi$
$751$	−13427.4			−0.652426			−0.326213	+	0.945296i	$0.605773\pi$
$752$		0			0
$753$			4037.42i			0.195394i
$754$		0			0
$755$	−409.635	−	25595.6i	−0.0197459	−	1.23380i
$756$		0			0
$757$		−	7103.66i		−	0.341066i	−0.985352	−	0.170533i	$-0.945451\pi$
$757$		−	7103.66i		−	0.341066i	0.985352	−	0.170533i	$-0.0545489\pi$
$758$		0			0
$759$	−8528.27			−0.407848
$760$		0			0
$761$	19205.2			0.914831			0.457416	−	0.889253i	$-0.348775\pi$
$761$	19205.2			0.914831			0.457416	+	0.889253i	$0.348775\pi$
$762$		0			0
$763$			21391.6i			1.01498i
$764$		0			0
$765$	−9637.36	+	154.237i	−0.455476	+	0.00728949i
$766$		0			0
$767$			8414.27i			0.396117i
$768$		0			0
$769$	−19508.1			−0.914799			−0.457399	−	0.889261i	$-0.651219\pi$
$769$	−19508.1			−0.914799			−0.457399	+	0.889261i	$0.651219\pi$
$770$		0			0
$771$	14091.6			0.658230
$772$		0			0
$773$			38852.5i			1.80780i	0.427746	+	0.903899i	$0.359308\pi$
$773$			38852.5i			1.80780i	−0.427746	+	0.903899i	$0.640692\pi$
$774$		0			0
$775$	−24792.4	+	793.763i	−1.14912	+	0.0367907i
$776$		0			0
$777$			28088.1i			1.29685i
$778$		0			0
$779$	−5282.23			−0.242947
$780$		0			0
$781$	15016.0			0.687982
$782$		0			0
$783$		−	1471.63i		−	0.0671669i
$784$		0			0
$785$	1805.55	−	28.8962i	0.0820929	−	0.00131382i
$786$		0			0
$787$		−	24851.1i		−	1.12560i	−0.826594	−	0.562799i	$-0.809725\pi$
$787$		−	24851.1i		−	1.12560i	0.826594	−	0.562799i	$-0.190275\pi$
$788$		0			0
$789$	−14102.0			−0.636304
$790$		0			0
$791$	−8481.04			−0.381228

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	120.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +(-0.178908 - 11.1789i) q^{5} +35.0735i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +(-0.178908 - 11.1789i) q^{5} +35.0735i q^{7} -9.00000 q^{9} +25.6422 q^{11} +37.6422i q^{13} +(33.5367 - 0.536725i) q^{15} +95.7891i q^{17} -50.8625 q^{19} -105.220 q^{21} +110.863i q^{23} +(-124.936 + 4.00000i) q^{25} -27.0000i q^{27} +54.5047 q^{29} +198.441 q^{31} +76.9265i q^{33} +(392.083 - 6.27493i) q^{35} -266.945i q^{37} -112.927 q^{39} +103.853 q^{41} +108.000i q^{43} +(1.61018 + 100.610i) q^{45} -597.009i q^{47} -887.147 q^{49} -287.367 q^{51} -305.642i q^{53} +(-4.58760 - 286.652i) q^{55} -152.588i q^{57} +223.533 q^{59} +485.450 q^{61} -315.661i q^{63} +(420.799 - 6.73450i) q^{65} +876.166i q^{67} -332.588 q^{69} +585.597 q^{71} -1137.60i q^{73} +(-12.0000 - 374.808i) q^{75} +899.360i q^{77} -685.009 q^{79} +81.0000 q^{81} +305.725i q^{83} +(1070.82 - 17.1375i) q^{85} +163.514i q^{87} -887.175 q^{89} -1320.24 q^{91} +595.322i q^{93} +(9.09973 + 568.588i) q^{95} +556.550i q^{97} -230.780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 22 q^{5} - 36 q^{9}+O(q^{10}) \) 4 * q + 22 * q^5 - 36 * q^9	\( 4 q + 22 q^{5} - 36 q^{9} + 148 q^{11} + 66 q^{15} + 160 q^{19} - 12 q^{21} - 100 q^{29} - 24 q^{31} + 796 q^{35} - 588 q^{39} + 688 q^{41} - 198 q^{45} - 3276 q^{49} - 468 q^{51} + 1072 q^{55} - 1332 q^{59} + 488 q^{61} + 820 q^{65} - 240 q^{69} + 616 q^{71} - 48 q^{75} - 2104 q^{79} + 324 q^{81} + 2148 q^{85} - 1368 q^{89} + 1352 q^{91} + 2944 q^{95} - 1332 q^{99}+O(q^{100}) \) 4 * q + 22 * q^5 - 36 * q^9 + 148 * q^11 + 66 * q^15 + 160 * q^19 - 12 * q^21 - 100 * q^29 - 24 * q^31 + 796 * q^35 - 588 * q^39 + 688 * q^41 - 198 * q^45 - 3276 * q^49 - 468 * q^51 + 1072 * q^55 - 1332 * q^59 + 488 * q^61 + 820 * q^65 - 240 * q^69 + 616 * q^71 - 48 * q^75 - 2104 * q^79 + 324 * q^81 + 2148 * q^85 - 1368 * q^89 + 1352 * q^91 + 2944 * q^95 - 1332 * q^99

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