LMFDB - Embedded newform 960.3.n.a.161.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(161,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("960.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.n (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:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.18
Character	$\chi$	$=$	960.161
Dual form			960.3.n.a.161.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.61015 - 2.53129i) q^{3} +2.23607 q^{5} +8.16215 q^{7} +(-3.81483 - 8.15151i) q^{9} +O(q^{10})$	\(q+(1.61015 - 2.53129i) q^{3} +2.23607 q^{5} +8.16215 q^{7} +(-3.81483 - 8.15151i) q^{9} +5.16840 q^{11} -4.76739i q^{13} +(3.60041 - 5.66013i) q^{15} -1.74659i q^{17} -9.75481i q^{19} +(13.1423 - 20.6607i) q^{21} +5.74019i q^{23} +5.00000 q^{25} +(-26.7763 - 3.46871i) q^{27} +39.7010 q^{29} -8.67289 q^{31} +(8.32190 - 13.0827i) q^{33} +18.2511 q^{35} -71.0861i q^{37} +(-12.0676 - 7.67622i) q^{39} +62.7390i q^{41} +47.8315i q^{43} +(-8.53022 - 18.2273i) q^{45} -46.8289i q^{47} +17.6206 q^{49} +(-4.42113 - 2.81228i) q^{51} +81.3380 q^{53} +11.5569 q^{55} +(-24.6922 - 15.7067i) q^{57} -34.8270 q^{59} +14.4409i q^{61} +(-31.1372 - 66.5338i) q^{63} -10.6602i q^{65} +7.67810i q^{67} +(14.5301 + 9.24256i) q^{69} -41.0888i q^{71} -6.10709 q^{73} +(8.05075 - 12.6564i) q^{75} +42.1853 q^{77} -136.245 q^{79} +(-51.8941 + 62.1933i) q^{81} +116.464 q^{83} -3.90550i q^{85} +(63.9245 - 100.495i) q^{87} -73.1472i q^{89} -38.9122i q^{91} +(-13.9647 + 21.9536i) q^{93} -21.8124i q^{95} +4.86582 q^{97} +(-19.7166 - 42.1303i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 8 q^{9}+O(q^{10}) $ 24 * q - 8 * q^9	$ 24 q - 8 q^{9} + 120 q^{25} - 256 q^{33} + 104 q^{49} - 304 q^{57} + 400 q^{73} + 152 q^{81} + 208 q^{97}+O(q^{100}) $ 24 * q - 8 * q^9 + 120 * q^25 - 256 * q^33 + 104 * q^49 - 304 * q^57 + 400 * q^73 + 152 * q^81 + 208 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	1.61015	−	2.53129i	0.536717	−	0.843762i
$3$	1.61015	−	2.53129i	0.536717	−	0.843762i
$4$		0			0
$5$	2.23607			0.447214
$5$	2.23607			0.447214
$6$		0			0
$7$	8.16215			1.16602			0.583010	−	0.812465i	$-0.301875\pi$
$7$	8.16215			1.16602			0.583010	+	0.812465i	$0.301875\pi$
$8$		0			0
$9$	−3.81483	−	8.15151i	−0.423870	−	0.905723i
$10$		0			0
$11$	5.16840			0.469855			0.234927	−	0.972013i	$-0.424515\pi$
$11$	5.16840			0.469855			0.234927	+	0.972013i	$0.424515\pi$
$12$		0			0
$13$		−	4.76739i		−	0.366723i	−0.983046	−	0.183361i	$-0.941302\pi$
$13$		−	4.76739i		−	0.366723i	0.983046	−	0.183361i	$-0.0586978\pi$
$14$		0			0
$15$	3.60041	−	5.66013i	0.240027	−	0.377342i
$16$		0			0
$17$		−	1.74659i		−	0.102741i	−0.998680	−	0.0513703i	$-0.983641\pi$
$17$		−	1.74659i		−	0.102741i	0.998680	−	0.0513703i	$-0.0163589\pi$
$18$		0			0
$19$		−	9.75481i		−	0.513411i	−0.966490	−	0.256706i	$-0.917363\pi$
$19$		−	9.75481i		−	0.513411i	0.966490	−	0.256706i	$-0.0826371\pi$
$20$		0			0
$21$	13.1423	−	20.6607i	0.625823	−	0.983845i
$22$		0			0
$23$			5.74019i			0.249573i	0.992184	+	0.124787i	$0.0398246\pi$
$23$			5.74019i			0.249573i	−0.992184	+	0.124787i	$0.960175\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$		0			0
$27$	−26.7763	−	3.46871i	−0.991713	−	0.128471i
$28$		0			0
$29$	39.7010			1.36900			0.684500	−	0.729013i	$-0.260021\pi$
$29$	39.7010			1.36900			0.684500	+	0.729013i	$0.260021\pi$
$30$		0			0
$31$	−8.67289			−0.279771			−0.139885	−	0.990168i	$-0.544673\pi$
$31$	−8.67289			−0.279771			−0.139885	+	0.990168i	$0.544673\pi$
$32$		0			0
$33$	8.32190	−	13.0827i	0.252179	−	0.396446i
$34$		0			0
$35$	18.2511			0.521460
$36$		0			0
$37$		−	71.0861i		−	1.92124i	−0.277857	−	0.960622i	$-0.589624\pi$
$37$		−	71.0861i		−	1.92124i	0.277857	−	0.960622i	$-0.410376\pi$
$38$		0			0
$39$	−12.0676	−	7.67622i	−0.309427	−	0.196826i
$40$		0			0
$41$			62.7390i			1.53022i	0.643900	+	0.765110i	$0.277315\pi$
$41$			62.7390i			1.53022i	−0.643900	+	0.765110i	$0.722685\pi$
$42$		0			0
$43$			47.8315i			1.11236i	0.831061	+	0.556181i	$0.187734\pi$
$43$			47.8315i			1.11236i	−0.831061	+	0.556181i	$0.812266\pi$
$44$		0			0
$45$	−8.53022	−	18.2273i	−0.189561	−	0.405052i
$46$		0			0
$47$		−	46.8289i		−	0.996360i	−0.867074	−	0.498180i	$-0.834002\pi$
$47$		−	46.8289i		−	0.996360i	0.867074	−	0.498180i	$-0.165998\pi$
$48$		0			0
$49$	17.6206			0.359605
$50$		0			0
$51$	−4.42113	−	2.81228i	−0.0866887	−	0.0551427i
$52$		0			0
$53$	81.3380			1.53468			0.767339	−	0.641241i	$-0.221580\pi$
$53$	81.3380			1.53468			0.767339	+	0.641241i	$0.221580\pi$
$54$		0			0
$55$	11.5569			0.210125
$56$		0			0
$57$	−24.6922	−	15.7067i	−0.433197	−	0.275556i
$58$		0			0
$59$	−34.8270			−0.590288			−0.295144	−	0.955453i	$-0.595368\pi$
$59$	−34.8270			−0.590288			−0.295144	+	0.955453i	$0.595368\pi$
$60$		0			0
$61$			14.4409i			0.236736i	0.992970	+	0.118368i	$0.0377663\pi$
$61$			14.4409i			0.236736i	−0.992970	+	0.118368i	$0.962234\pi$
$62$		0			0
$63$	−31.1372	−	66.5338i	−0.494242	−	1.05609i
$64$		0			0
$65$		−	10.6602i		−	0.164003i
$66$		0			0
$67$			7.67810i			0.114598i	0.998357	+	0.0572992i	$0.0182489\pi$
$67$			7.67810i			0.114598i	−0.998357	+	0.0572992i	$0.981751\pi$
$68$		0			0
$69$	14.5301	+	9.24256i	0.210581	+	0.133950i
$70$		0			0
$71$		−	41.0888i		−	0.578715i	−0.957221	−	0.289357i	$-0.906558\pi$
$71$		−	41.0888i		−	0.578715i	0.957221	−	0.289357i	$-0.0934416\pi$
$72$		0			0
$73$	−6.10709			−0.0836588			−0.0418294	−	0.999125i	$-0.513319\pi$
$73$	−6.10709			−0.0836588			−0.0418294	+	0.999125i	$0.513319\pi$
$74$		0			0
$75$	8.05075	−	12.6564i	0.107343	−	0.168752i
$76$		0			0
$77$	42.1853			0.547860
$78$		0			0
$79$	−136.245			−1.72461			−0.862307	−	0.506386i	$-0.830981\pi$
$79$	−136.245			−1.72461			−0.862307	+	0.506386i	$0.830981\pi$
$80$		0			0
$81$	−51.8941	+	62.1933i	−0.640668	+	0.767818i
$82$		0			0
$83$	116.464			1.40318			0.701590	−	0.712581i	$-0.252474\pi$
$83$	116.464			1.40318			0.701590	+	0.712581i	$0.252474\pi$
$84$		0			0
$85$		−	3.90550i		−	0.0459470i
$86$		0			0
$87$	63.9245	−	100.495i	0.734765	−	1.15511i
$88$		0			0
$89$		−	73.1472i		−	0.821879i	−0.911663	−	0.410939i	$-0.865201\pi$
$89$		−	73.1472i		−	0.821879i	0.911663	−	0.410939i	$-0.134799\pi$
$90$		0			0
$91$		−	38.9122i		−	0.427606i
$92$		0			0
$93$	−13.9647	+	21.9536i	−0.150158	+	0.236060i
$94$		0			0
$95$		−	21.8124i		−	0.229605i
$96$		0			0
$97$	4.86582			0.0501631			0.0250815	−	0.999685i	$-0.492015\pi$
$97$	4.86582			0.0501631			0.0250815	+	0.999685i	$0.492015\pi$
$98$		0			0
$99$	−19.7166	−	42.1303i	−0.199157	−	0.425558i
$100$		0			0
$101$	149.808			1.48325			0.741623	−	0.670817i	$-0.234056\pi$
$101$	149.808			1.48325			0.741623	+	0.670817i	$0.234056\pi$
$102$		0			0
$103$	−98.6161			−0.957438			−0.478719	−	0.877968i	$-0.658899\pi$
$103$	−98.6161			−0.957438			−0.478719	+	0.877968i	$0.658899\pi$
$104$		0			0
$105$	29.3870	−	46.1988i	0.279877	−	0.439989i
$106$		0			0
$107$	−53.6536			−0.501435			−0.250718	−	0.968060i	$-0.580667\pi$
$107$	−53.6536			−0.501435			−0.250718	+	0.968060i	$0.580667\pi$
$108$		0			0
$109$			10.3967i			0.0953824i	0.998862	+	0.0476912i	$0.0151863\pi$
$109$			10.3967i			0.0953824i	−0.998862	+	0.0476912i	$0.984814\pi$
$110$		0			0
$111$	−179.939	−	114.459i	−1.62107	−	1.03116i
$112$		0			0
$113$		−	160.091i		−	1.41673i	−0.705844	−	0.708367i	$-0.749432\pi$
$113$		−	160.091i		−	1.41673i	0.705844	−	0.708367i	$-0.250568\pi$
$114$		0			0
$115$			12.8354i			0.111613i
$116$		0			0
$117$	−38.8614	+	18.1868i	−0.332149	+	0.155443i
$118$		0			0
$119$		−	14.2559i		−	0.119798i
$120$		0			0
$121$	−94.2876			−0.779237
$122$		0			0
$123$	158.811	+	101.019i	1.29114	+	0.821295i
$124$		0			0
$125$	11.1803			0.0894427
$126$		0			0
$127$	−175.172			−1.37931			−0.689653	−	0.724140i	$-0.742237\pi$
$127$	−175.172			−1.37931			−0.689653	+	0.724140i	$0.742237\pi$
$128$		0			0
$129$	121.075	+	77.0160i	0.938569	+	0.597023i
$130$		0			0
$131$	−189.821			−1.44901			−0.724507	−	0.689267i	$-0.757933\pi$
$131$	−189.821			−1.44901			−0.724507	+	0.689267i	$0.757933\pi$
$132$		0			0
$133$		−	79.6202i		−	0.598648i
$134$		0			0
$135$	−59.8735	−	7.75628i	−0.443508	−	0.0574539i
$136$		0			0
$137$			180.907i			1.32049i	0.751050	+	0.660246i	$0.229548\pi$
$137$			180.907i			1.32049i	−0.751050	+	0.660246i	$0.770452\pi$
$138$		0			0
$139$		−	169.479i		−	1.21927i	−0.792682	−	0.609635i	$-0.791316\pi$
$139$		−	169.479i		−	1.21927i	0.792682	−	0.609635i	$-0.208684\pi$
$140$		0			0
$141$	−118.538	−	75.4016i	−0.840692	−	0.534763i
$142$		0			0
$143$		−	24.6398i		−	0.172306i
$144$		0			0
$145$	88.7741			0.612235
$146$		0			0
$147$	28.3719	−	44.6029i	0.193006	−	0.303421i
$148$		0			0
$149$	177.189			1.18919			0.594594	−	0.804026i	$-0.297313\pi$
$149$	177.189			1.18919			0.594594	+	0.804026i	$0.297313\pi$
$150$		0			0
$151$	195.030			1.29159			0.645793	−	0.763512i	$-0.276527\pi$
$151$	195.030			1.29159			0.645793	+	0.763512i	$0.276527\pi$
$152$		0			0
$153$	−14.2374	+	6.66296i	−0.0930546	+	0.0435487i
$154$		0			0
$155$	−19.3932			−0.125117
$156$		0			0
$157$		−	19.0487i		−	0.121329i	−0.998158	−	0.0606647i	$-0.980678\pi$
$157$		−	19.0487i		−	0.121329i	0.998158	−	0.0606647i	$-0.0193220\pi$
$158$		0			0
$159$	130.966	−	205.890i	0.823688	−	1.29490i
$160$		0			0
$161$			46.8522i			0.291008i
$162$		0			0
$163$			178.468i			1.09489i	0.836841	+	0.547446i	$0.184400\pi$
$163$			178.468i			1.09489i	−0.836841	+	0.547446i	$0.815600\pi$
$164$		0			0
$165$	18.6083	−	29.2538i	0.112778	−	0.177296i
$166$		0			0
$167$			165.641i			0.991860i	0.868363	+	0.495930i	$0.165173\pi$
$167$			165.641i			0.991860i	−0.868363	+	0.495930i	$0.834827\pi$
$168$		0			0
$169$	146.272			0.865515
$170$		0			0
$171$	−79.5164	+	37.2130i	−0.465008	+	0.217620i
$172$		0			0
$173$	−15.0955			−0.0872570			−0.0436285	−	0.999048i	$-0.513892\pi$
$173$	−15.0955			−0.0872570			−0.0436285	+	0.999048i	$0.513892\pi$
$174$		0			0
$175$	40.8107			0.233204
$176$		0			0
$177$	−56.0767	+	88.1572i	−0.316818	+	0.498063i
$178$		0			0
$179$	−241.590			−1.34966			−0.674831	−	0.737972i	$-0.735784\pi$
$179$	−241.590			−1.34966			−0.674831	+	0.737972i	$0.735784\pi$
$180$		0			0
$181$			94.1790i			0.520326i	0.965565	+	0.260163i	$0.0837763\pi$
$181$			94.1790i			0.520326i	−0.965565	+	0.260163i	$0.916224\pi$
$182$		0			0
$183$	36.5541	+	23.2520i	0.199749	+	0.127060i
$184$		0			0
$185$		−	158.953i		−	0.859207i
$186$		0			0
$187$		−	9.02709i		−	0.0482732i
$188$		0			0
$189$	−218.552	−	28.3121i	−1.15636	−	0.149800i
$190$		0			0
$191$			281.488i			1.47376i	0.676025	+	0.736879i	$0.263701\pi$
$191$			281.488i			1.47376i	−0.676025	+	0.736879i	$0.736299\pi$
$192$		0			0
$193$	−165.831			−0.859227			−0.429614	−	0.903013i	$-0.641350\pi$
$193$	−165.831			−0.859227			−0.429614	+	0.903013i	$0.641350\pi$
$194$		0			0
$195$	−26.9841	−	17.1646i	−0.138380	−	0.0880233i
$196$		0			0
$197$	316.768			1.60796			0.803980	−	0.594656i	$-0.202712\pi$
$197$	316.768			1.60796			0.803980	+	0.594656i	$0.202712\pi$
$198$		0			0
$199$	176.704			0.887960			0.443980	−	0.896037i	$-0.353566\pi$
$199$	176.704			0.887960			0.443980	+	0.896037i	$0.353566\pi$
$200$		0			0
$201$	19.4355	+	12.3629i	0.0966939	+	0.0615069i
$202$		0			0
$203$	324.045			1.59628
$204$		0			0
$205$			140.289i			0.684335i
$206$		0			0
$207$	46.7912	−	21.8979i	0.226044	−	0.105787i
$208$		0			0
$209$		−	50.4168i		−	0.241229i
$210$		0			0
$211$			102.122i			0.483993i	0.970277	+	0.241996i	$0.0778022\pi$
$211$			102.122i			0.483993i	−0.970277	+	0.241996i	$0.922198\pi$
$212$		0			0
$213$	−104.007	−	66.1591i	−0.488298	−	0.310606i
$214$		0			0
$215$			106.955i			0.497463i
$216$		0			0
$217$	−70.7894			−0.326219
$218$		0			0
$219$	−9.83333	+	15.4588i	−0.0449011	+	0.0705881i
$220$		0			0
$221$	−8.32669			−0.0376773
$222$		0			0
$223$	307.107			1.37716			0.688580	−	0.725160i	$-0.258234\pi$
$223$	307.107			1.37716			0.688580	+	0.725160i	$0.258234\pi$
$224$		0			0
$225$	−19.0742	−	40.7575i	−0.0847741	−	0.181145i
$226$		0			0
$227$	243.151			1.07115			0.535575	−	0.844488i	$-0.320095\pi$
$227$	243.151			1.07115			0.535575	+	0.844488i	$0.320095\pi$
$228$		0			0
$229$		−	357.133i		−	1.55953i	−0.626070	−	0.779767i	$-0.715338\pi$
$229$		−	357.133i		−	1.55953i	0.626070	−	0.779767i	$-0.284662\pi$
$230$		0			0
$231$	67.9246	−	106.783i	0.294046	−	0.462264i
$232$		0			0
$233$			183.981i			0.789618i	0.918763	+	0.394809i	$0.129189\pi$
$233$			183.981i			0.789618i	−0.918763	+	0.394809i	$0.870811\pi$
$234$		0			0
$235$		−	104.713i		−	0.445586i
$236$		0			0
$237$	−219.374	+	344.874i	−0.925629	+	1.45516i
$238$		0			0
$239$			347.730i			1.45494i	0.686141	+	0.727469i	$0.259303\pi$
$239$			347.730i			1.45494i	−0.686141	+	0.727469i	$0.740697\pi$
$240$		0			0
$241$	−339.671			−1.40942			−0.704711	−	0.709495i	$-0.748923\pi$
$241$	−339.671			−1.40942			−0.704711	+	0.709495i	$0.748923\pi$
$242$		0			0
$243$	73.8717	+	231.499i	0.303999	+	0.952672i
$244$		0			0
$245$	39.4009			0.160820
$246$		0			0
$247$	−46.5050			−0.188280
$248$		0			0
$249$	187.524	−	294.804i	0.753110	−	1.18395i
$250$		0			0
$251$	75.7547			0.301812			0.150906	−	0.988548i	$-0.451781\pi$
$251$	75.7547			0.301812			0.150906	+	0.988548i	$0.451781\pi$
$252$		0			0
$253$			29.6676i			0.117263i
$254$		0			0
$255$	−9.88594	−	6.28844i	−0.0387684	−	0.0246605i
$256$		0			0
$257$		−	51.9361i		−	0.202086i	−0.994882	−	0.101043i	$-0.967782\pi$
$257$		−	51.9361i		−	0.202086i	0.994882	−	0.101043i	$-0.0322179\pi$
$258$		0			0
$259$		−	580.215i		−	2.24021i
$260$		0			0
$261$	−151.453	−	323.623i	−0.580278	−	1.23993i
$262$		0			0
$263$			315.740i			1.20053i	0.799801	+	0.600266i	$0.204938\pi$
$263$			315.740i			1.20053i	−0.799801	+	0.600266i	$0.795062\pi$
$264$		0			0
$265$	181.877			0.686329
$266$		0			0
$267$	−185.157	−	117.778i	−0.693470	−	0.441116i
$268$		0			0
$269$	−335.479			−1.24713			−0.623567	−	0.781770i	$-0.714317\pi$
$269$	−335.479			−1.24713			−0.623567	+	0.781770i	$0.714317\pi$
$270$		0			0
$271$	58.0706			0.214283			0.107141	−	0.994244i	$-0.465830\pi$
$271$	58.0706			0.214283			0.107141	+	0.994244i	$0.465830\pi$
$272$		0			0
$273$	−98.4979	−	62.6544i	−0.360798	−	0.229503i
$274$		0			0
$275$	25.8420			0.0939710
$276$		0			0
$277$		−	2.02211i		−	0.00730004i	−0.999993	−	0.00365002i	$-0.998838\pi$
$277$		−	2.02211i		−	0.00730004i	0.999993	−	0.00365002i	$-0.00116184\pi$
$278$		0			0
$279$	33.0856	+	70.6971i	0.118586	+	0.253395i
$280$		0			0
$281$			260.289i			0.926296i	0.886281	+	0.463148i	$0.153280\pi$
$281$			260.289i			0.926296i	−0.886281	+	0.463148i	$0.846720\pi$
$282$		0			0
$283$			270.345i			0.955282i	0.878555	+	0.477641i	$0.158508\pi$
$283$			270.345i			0.955282i	−0.878555	+	0.477641i	$0.841492\pi$
$284$		0			0
$285$	−55.2135	−	35.1213i	−0.193732	−	0.123233i
$286$		0			0
$287$			512.085i			1.78427i
$288$		0			0
$289$	285.949			0.989444
$290$		0			0
$291$	7.83470	−	12.3168i	0.0269234	−	0.0423257i
$292$		0			0
$293$	228.079			0.778426			0.389213	−	0.921148i	$-0.372747\pi$
$293$	228.079			0.778426			0.389213	+	0.921148i	$0.372747\pi$
$294$		0			0
$295$	−77.8756			−0.263985
$296$		0			0
$297$	−138.390	−	17.9277i	−0.465961	−	0.0603626i
$298$		0			0
$299$	27.3657			0.0915242
$300$		0			0
$301$			390.408i			1.29704i
$302$		0			0
$303$	241.213	−	379.207i	0.796083	−	1.25151i
$304$		0			0
$305$			32.2908i			0.105872i
$306$		0			0
$307$			450.897i			1.46872i	0.678760	+	0.734360i	$0.262518\pi$
$307$			450.897i			1.46872i	−0.678760	+	0.734360i	$0.737482\pi$
$308$		0			0
$309$	−158.787	+	249.626i	−0.513873	+	0.807850i
$310$		0			0
$311$			103.595i			0.333103i	0.986033	+	0.166551i	$0.0532632\pi$
$311$			103.595i			0.333103i	−0.986033	+	0.166551i	$0.946737\pi$
$312$		0			0
$313$	−356.467			−1.13887			−0.569436	−	0.822036i	$-0.692838\pi$
$313$	−356.467			−1.13887			−0.569436	+	0.822036i	$0.692838\pi$
$314$		0			0
$315$	−69.6249	−	148.774i	−0.221032	−	0.472299i
$316$		0			0
$317$	−377.664			−1.19137			−0.595685	−	0.803218i	$-0.703119\pi$
$317$	−377.664			−1.19137			−0.595685	+	0.803218i	$0.703119\pi$
$318$		0			0
$319$	205.191			0.643231
$320$		0			0
$321$	−86.3903	+	135.813i	−0.269129	+	0.423092i
$322$		0			0
$323$	−17.0377			−0.0527482
$324$		0			0
$325$		−	23.8370i		−	0.0733445i
$326$		0			0
$327$	26.3170	+	16.7402i	0.0804801	+	0.0511933i
$328$		0			0
$329$		−	382.225i		−	1.16178i
$330$		0			0
$331$			185.939i			0.561749i	0.959744	+	0.280875i	$0.0906245\pi$
$331$			185.939i			0.561749i	−0.959744	+	0.280875i	$0.909376\pi$
$332$		0			0
$333$	−579.458	+	271.181i	−1.74012	+	0.814359i
$334$		0			0
$335$			17.1687i			0.0512500i
$336$		0			0
$337$	403.318			1.19679			0.598395	−	0.801201i	$-0.295805\pi$
$337$	403.318			1.19679			0.598395	+	0.801201i	$0.295805\pi$
$338$		0			0
$339$	−405.236	−	257.770i	−1.19539	−	0.760385i
$340$		0			0
$341$	−44.8250			−0.131452
$342$		0			0
$343$	−256.123			−0.746714
$344$		0			0
$345$	32.4902	+	20.6670i	0.0941745	+	0.0599044i
$346$		0			0
$347$	−505.149			−1.45576			−0.727880	−	0.685704i	$-0.759494\pi$
$347$	−505.149			−1.45576			−0.727880	+	0.685704i	$0.759494\pi$
$348$		0			0
$349$			524.781i			1.50367i	0.659351	+	0.751836i	$0.270831\pi$
$349$			524.781i			1.50367i	−0.659351	+	0.751836i	$0.729169\pi$
$350$		0			0
$351$	−16.5367	+	127.653i	−0.0471132	+	0.363684i
$352$		0			0
$353$			552.296i			1.56458i	0.622917	+	0.782288i	$0.285948\pi$
$353$			552.296i			1.56458i	−0.622917	+	0.782288i	$0.714052\pi$
$354$		0			0
$355$		−	91.8772i		−	0.258809i
$356$		0			0
$357$	−36.0859	−	22.9542i	−0.101081	−	0.0642975i
$358$		0			0
$359$			9.42319i			0.0262485i	0.999914	+	0.0131242i	$0.00417769\pi$
$359$			9.42319i			0.0262485i	−0.999914	+	0.0131242i	$0.995822\pi$
$360$		0			0
$361$	265.844			0.736409
$362$		0			0
$363$	−151.817	+	238.669i	−0.418229	+	0.657491i
$364$		0			0
$365$	−13.6559			−0.0374133
$366$		0			0
$367$	83.5907			0.227768			0.113884	−	0.993494i	$-0.463671\pi$
$367$	83.5907			0.227768			0.113884	+	0.993494i	$0.463671\pi$
$368$		0			0
$369$	511.418	−	239.339i	1.38596	−	0.648615i
$370$		0			0
$371$	663.892			1.78947
$372$		0			0
$373$			354.792i			0.951185i	0.879666	+	0.475592i	$0.157766\pi$
$373$			354.792i			0.951185i	−0.879666	+	0.475592i	$0.842234\pi$
$374$		0			0
$375$	18.0020	−	28.3007i	0.0480054	−	0.0754684i
$376$		0			0
$377$		−	189.270i		−	0.502043i
$378$		0			0
$379$		−	375.899i		−	0.991818i	−0.868374	−	0.495909i	$-0.834835\pi$
$379$		−	375.899i		−	0.991818i	0.868374	−	0.495909i	$-0.165165\pi$
$380$		0			0
$381$	−282.053	+	443.410i	−0.740297	+	1.16381i
$382$		0			0
$383$			596.645i			1.55782i	0.627136	+	0.778910i	$0.284227\pi$
$383$			596.645i			1.55782i	−0.627136	+	0.778910i	$0.715773\pi$
$384$		0			0
$385$	94.3291			0.245011
$386$		0			0
$387$	389.899	−	182.469i	1.00749	−	0.471497i
$388$		0			0
$389$	353.985			0.909987			0.454993	−	0.890495i	$-0.349642\pi$
$389$	353.985			0.909987			0.454993	+	0.890495i	$0.349642\pi$
$390$		0			0
$391$	10.0258			0.0256413
$392$		0			0
$393$	−305.640	+	480.491i	−0.777710	+	1.22262i
$394$		0			0
$395$	−304.652			−0.771271
$396$		0			0
$397$			184.801i			0.465494i	0.972537	+	0.232747i	$0.0747714\pi$
$397$			184.801i			0.465494i	−0.972537	+	0.232747i	$0.925229\pi$
$398$		0			0
$399$	−201.542	−	128.201i	−0.505117	−	0.321305i
$400$		0			0
$401$			106.165i			0.264751i	0.991200	+	0.132375i	$0.0422604\pi$
$401$			106.165i			0.264751i	−0.991200	+	0.132375i	$0.957740\pi$
$402$		0			0
$403$			41.3471i			0.102598i
$404$		0			0
$405$	−116.039	+	139.068i	−0.286515	+	0.343379i
$406$		0			0
$407$		−	367.401i		−	0.902706i
$408$		0			0
$409$	192.636			0.470992			0.235496	−	0.971875i	$-0.424329\pi$
$409$	192.636			0.470992			0.235496	+	0.971875i	$0.424329\pi$
$410$		0			0
$411$	457.928	+	291.288i	1.11418	+	0.708730i
$412$		0			0
$413$	−284.263			−0.688289
$414$		0			0
$415$	260.421			0.627521
$416$		0			0
$417$	−428.999	−	272.886i	−1.02877	−	0.654403i
$418$		0			0
$419$	−616.975			−1.47249			−0.736247	−	0.676713i	$-0.763404\pi$
$419$	−616.975			−1.47249			−0.736247	+	0.676713i	$0.763404\pi$
$420$		0			0
$421$		−	173.681i		−	0.412545i	−0.978495	−	0.206272i	$-0.933867\pi$
$421$		−	173.681i		−	0.412545i	0.978495	−	0.206272i	$-0.0661333\pi$
$422$		0			0
$423$	−381.726	+	178.645i	−0.902426	+	0.422328i
$424$		0			0
$425$		−	8.73296i		−	0.0205481i
$426$		0			0
$427$			117.869i			0.276039i
$428$		0			0
$429$	−62.3704	−	39.6738i	−0.145386	−	0.0924797i
$430$		0			0
$431$			217.302i			0.504182i	0.967704	+	0.252091i	$0.0811182\pi$
$431$			217.302i			0.504182i	−0.967704	+	0.252091i	$0.918882\pi$
$432$		0			0
$433$	−683.809			−1.57924			−0.789618	−	0.613599i	$-0.789721\pi$
$433$	−683.809			−1.57924			−0.789618	+	0.613599i	$0.789721\pi$
$434$		0			0
$435$	142.940	−	224.713i	0.328597	−	0.516581i
$436$		0			0
$437$	55.9945			0.128134
$438$		0			0
$439$	−515.364			−1.17395			−0.586975	−	0.809605i	$-0.699681\pi$
$439$	−515.364			−1.17395			−0.586975	+	0.809605i	$0.699681\pi$
$440$		0			0
$441$	−67.2198	−	143.635i	−0.152426	−	0.325702i
$442$		0			0
$443$	56.6514			0.127881			0.0639407	−	0.997954i	$-0.479633\pi$
$443$	56.6514			0.127881			0.0639407	+	0.997954i	$0.479633\pi$
$444$		0			0
$445$		−	163.562i		−	0.367555i
$446$		0			0
$447$	285.301	−	448.516i	0.638257	−	1.00339i
$448$		0			0
$449$		−	526.462i		−	1.17252i	−0.810122	−	0.586261i	$-0.800599\pi$
$449$		−	526.462i		−	1.17252i	0.810122	−	0.586261i	$-0.199401\pi$
$450$		0			0
$451$			324.261i			0.718981i
$452$		0			0
$453$	314.027	−	493.676i	0.693216	−	1.08979i
$454$		0			0
$455$		−	87.0103i		−	0.191231i
$456$		0			0
$457$	−62.7659			−0.137343			−0.0686717	−	0.997639i	$-0.521876\pi$
$457$	−62.7659			−0.137343			−0.0686717	+	0.997639i	$0.521876\pi$
$458$		0			0
$459$	−6.05843	+	46.7672i	−0.0131992	+	0.101889i
$460$		0			0
$461$	−171.672			−0.372391			−0.186195	−	0.982513i	$-0.559616\pi$
$461$	−171.672			−0.372391			−0.186195	+	0.982513i	$0.559616\pi$
$462$		0			0
$463$	−174.511			−0.376914			−0.188457	−	0.982081i	$-0.560349\pi$
$463$	−174.511			−0.376914			−0.188457	+	0.982081i	$0.560349\pi$
$464$		0			0
$465$	−31.2259	+	49.0897i	−0.0671525	+	0.105569i
$466$		0			0
$467$	339.644			0.727289			0.363644	−	0.931538i	$-0.381532\pi$
$467$	339.644			0.727289			0.363644	+	0.931538i	$0.381532\pi$
$468$		0			0
$469$			62.6697i			0.133624i
$470$		0			0
$471$	−48.2177	−	30.6713i	−0.102373	−	0.0651195i
$472$		0			0
$473$			247.213i			0.522648i
$474$		0			0
$475$		−	48.7741i		−	0.102682i
$476$		0			0
$477$	−310.291	−	663.027i	−0.650505	−	1.38999i
$478$		0			0
$479$			375.145i			0.783185i	0.920139	+	0.391592i	$0.128076\pi$
$479$			375.145i			0.783185i	−0.920139	+	0.391592i	$0.871924\pi$
$480$		0			0
$481$	−338.895			−0.704564
$482$		0			0
$483$	118.597	+	75.4392i	0.245541	+	0.156189i
$484$		0			0
$485$	10.8803			0.0224336
$486$		0			0
$487$	449.237			0.922458			0.461229	−	0.887281i	$-0.347409\pi$
$487$	449.237			0.922458			0.461229	+	0.887281i	$0.347409\pi$
$488$		0			0
$489$	451.753	+	287.360i	0.923829	+	0.587647i
$490$		0			0
$491$	296.423			0.603713			0.301857	−	0.953353i	$-0.402394\pi$
$491$	296.423			0.603713			0.301857	+	0.953353i	$0.402394\pi$
$492$		0			0
$493$		−	69.3414i		−	0.140652i
$494$		0			0
$495$	−44.0876	−	94.2061i	−0.0890659	−	0.190315i
$496$		0			0
$497$		−	335.372i		−	0.674794i
$498$		0			0
$499$		−	847.011i		−	1.69742i	−0.528861	−	0.848709i	$-0.677381\pi$
$499$		−	847.011i		−	1.69742i	0.528861	−	0.848709i	$-0.322619\pi$
$500$		0			0
$501$	419.284	+	266.706i	0.836894	+	0.532348i
$502$		0			0
$503$		−	600.517i		−	1.19387i	−0.802289	−	0.596935i	$-0.796385\pi$
$503$		−	600.517i		−	1.19387i	0.802289	−	0.596935i	$-0.203615\pi$
$504$		0			0
$505$	334.980			0.663328
$506$		0			0
$507$	235.520	−	370.256i	0.464536	−	0.730289i
$508$		0			0
$509$	384.142			0.754699			0.377349	−	0.926071i	$-0.376836\pi$
$509$	384.142			0.754699			0.377349	+	0.926071i	$0.376836\pi$
$510$		0			0
$511$	−49.8470			−0.0975479
$512$		0			0
$513$	−33.8366	+	261.197i	−0.0659584	+	0.509157i
$514$		0			0
$515$	−220.512			−0.428179
$516$		0			0
$517$		−	242.031i		−	0.468145i
$518$		0			0
$519$	−24.3060	+	38.2110i	−0.0468323	+	0.0736242i
$520$		0			0
$521$		−	969.901i		−	1.86161i	−0.365511	−	0.930807i	$-0.619106\pi$
$521$		−	969.901i		−	1.86161i	0.365511	−	0.930807i	$-0.380894\pi$
$522$		0			0
$523$			403.371i			0.771263i	0.922653	+	0.385632i	$0.126016\pi$
$523$			403.371i			0.771263i	−0.922653	+	0.385632i	$0.873984\pi$
$524$		0			0
$525$	65.7114	−	103.304i	0.125165	−	0.196769i
$526$		0			0
$527$			15.1480i			0.0287438i
$528$		0			0
$529$	496.050			0.937713
$530$		0			0
$531$	132.859	+	283.893i	0.250206	+	0.534638i
$532$		0			0
$533$	299.102			0.561166
$534$		0			0
$535$	−119.973			−0.224249
$536$		0			0
$537$	−388.995	+	611.533i	−0.724386	+	1.13879i
$538$		0			0
$539$	91.0705			0.168962
$540$		0			0
$541$		−	834.729i		−	1.54294i	−0.636267	−	0.771469i	$-0.719522\pi$
$541$		−	834.729i		−	1.54294i	0.636267	−	0.771469i	$-0.280478\pi$
$542$		0			0
$543$	238.394	+	151.642i	0.439031	+	0.279268i
$544$		0			0
$545$			23.2477i			0.0426563i
$546$		0			0
$547$			795.220i			1.45378i	0.686751	+	0.726892i	$0.259036\pi$
$547$			795.220i			1.45378i	−0.686751	+	0.726892i	$0.740964\pi$
$548$		0			0
$549$	117.715	−	55.0896i	0.214417	−	0.100345i
$550$		0			0
$551$		−	387.276i		−	0.702860i
$552$		0			0
$553$	−1112.05			−2.01094
$554$		0			0
$555$	−402.356	−	255.939i	−0.724966	−	0.461151i
$556$		0			0
$557$	214.852			0.385731			0.192866	−	0.981225i	$-0.438222\pi$
$557$	214.852			0.385731			0.192866	+	0.981225i	$0.438222\pi$
$558$		0			0
$559$	228.032			0.407928
$560$		0			0
$561$	−22.8502	−	14.5350i	−0.0407311	−	0.0259090i
$562$		0			0
$563$	−526.184			−0.934607			−0.467304	−	0.884097i	$-0.654775\pi$
$563$	−526.184			−0.934607			−0.467304	+	0.884097i	$0.654775\pi$
$564$		0			0
$565$		−	357.974i		−	0.633583i
$566$		0			0
$567$	−423.567	+	507.630i	−0.747032	+	0.895292i
$568$		0			0
$569$			255.284i			0.448653i	0.974514	+	0.224327i	$0.0720183\pi$
$569$			255.284i			0.448653i	−0.974514	+	0.224327i	$0.927982\pi$
$570$		0			0
$571$			190.752i			0.334066i	0.985951	+	0.167033i	$0.0534187\pi$
$571$			190.752i			0.334066i	−0.985951	+	0.167033i	$0.946581\pi$
$572$		0			0
$573$	712.526	+	453.237i	1.24350	+	0.790990i
$574$		0			0
$575$			28.7009i			0.0499147i
$576$		0			0
$577$	1027.35			1.78050			0.890250	−	0.455471i	$-0.150529\pi$
$577$	1027.35			1.78050			0.890250	+	0.455471i	$0.150529\pi$
$578$		0			0
$579$	−267.013	+	419.766i	−0.461162	+	0.724984i
$580$		0			0
$581$	950.596			1.63614
$582$		0			0
$583$	420.387			0.721076
$584$		0			0
$585$	−86.8968	+	40.6669i	−0.148542	+	0.0695161i
$586$		0			0
$587$	236.995			0.403739			0.201869	−	0.979412i	$-0.435298\pi$
$587$	236.995			0.403739			0.201869	+	0.979412i	$0.435298\pi$
$588$		0			0
$589$			84.6025i			0.143637i
$590$		0			0
$591$	510.044	−	801.831i	0.863019	−	1.35674i
$592$		0			0
$593$			1102.66i			1.85946i	0.368246	+	0.929728i	$0.379958\pi$
$593$			1102.66i			1.85946i	−0.368246	+	0.929728i	$0.620042\pi$
$594$		0			0
$595$		−	31.8772i		−	0.0535752i
$596$		0			0
$597$	284.520	−	447.289i	0.476583	−	0.749228i
$598$		0			0
$599$		−	315.104i		−	0.526050i	−0.964789	−	0.263025i	$-0.915280\pi$
$599$		−	315.104i		−	0.526050i	0.964789	−	0.263025i	$-0.0847201\pi$
$600$		0			0
$601$	163.211			0.271566			0.135783	−	0.990739i	$-0.456645\pi$
$601$	163.211			0.271566			0.135783	+	0.990739i	$0.456645\pi$
$602$		0			0
$603$	62.5880	−	29.2906i	0.103794	−	0.0485749i
$604$		0			0
$605$	−210.834			−0.348485
$606$		0			0
$607$	416.694			0.686482			0.343241	−	0.939247i	$-0.388475\pi$
$607$	416.694			0.686482			0.343241	+	0.939247i	$0.388475\pi$
$608$		0			0
$609$	521.761	−	820.251i	0.856751	−	1.34688i
$610$		0			0
$611$	−223.252			−0.365388
$612$		0			0
$613$		−	376.371i		−	0.613982i	−0.951712	−	0.306991i	$-0.900678\pi$
$613$		−	376.371i		−	0.613982i	0.951712	−	0.306991i	$-0.0993222\pi$
$614$		0			0
$615$	355.111	+	225.886i	0.577416	+	0.367294i
$616$		0			0
$617$			256.481i			0.415691i	0.978162	+	0.207846i	$0.0666451\pi$
$617$			256.481i			0.415691i	−0.978162	+	0.207846i	$0.933355\pi$
$618$		0			0
$619$		−	943.976i		−	1.52500i	−0.646987	−	0.762501i	$-0.723971\pi$
$619$		−	943.976i		−	1.52500i	0.646987	−	0.762501i	$-0.276029\pi$
$620$		0			0
$621$	19.9111	−	153.701i	0.0320629	−	0.247505i
$622$		0			0
$623$		−	597.038i		−	0.958328i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−127.619	−	81.1786i	−0.203540	−	0.129472i
$628$		0			0
$629$	−124.158			−0.197390
$630$		0			0
$631$	−979.595			−1.55245			−0.776224	−	0.630457i	$-0.782868\pi$
$631$	−979.595			−1.55245			−0.776224	+	0.630457i	$0.782868\pi$
$632$		0			0
$633$	258.501	+	164.432i	0.408375	+	0.259767i
$634$		0			0
$635$	−391.696			−0.616844
$636$		0			0
$637$		−	84.0045i		−	0.131875i
$638$		0			0
$639$	−334.935	+	156.747i	−0.524155	+	0.245300i
$640$		0			0
$641$			194.583i			0.303562i	0.988414	+	0.151781i	$0.0485008\pi$
$641$			194.583i			0.303562i	−0.988414	+	0.151781i	$0.951499\pi$
$642$		0			0
$643$		−	364.353i		−	0.566646i	−0.959025	−	0.283323i	$-0.908563\pi$
$643$		−	364.353i		−	0.566646i	0.959025	−	0.283323i	$-0.0914368\pi$
$644$		0			0
$645$	270.733	+	172.213i	0.419741	+	0.266997i
$646$		0			0
$647$		−	772.411i		−	1.19384i	−0.802302	−	0.596918i	$-0.796392\pi$
$647$		−	772.411i		−	1.19384i	0.802302	−	0.596918i	$-0.203608\pi$
$648$		0			0
$649$	−180.000			−0.277350
$650$		0			0
$651$	−113.982	+	179.188i	−0.175087	+	0.275251i
$652$		0			0
$653$	356.314			0.545657			0.272828	−	0.962063i	$-0.412041\pi$
$653$	356.314			0.545657			0.272828	+	0.962063i	$0.412041\pi$
$654$		0			0
$655$	−424.452			−0.648019
$656$		0			0
$657$	23.2975	+	49.7820i	0.0354605	+	0.0757717i
$658$		0			0
$659$	−498.693			−0.756742			−0.378371	−	0.925654i	$-0.623516\pi$
$659$	−498.693			−0.756742			−0.378371	+	0.925654i	$0.623516\pi$
$660$		0			0
$661$		−	184.687i		−	0.279405i	−0.990194	−	0.139703i	$-0.955385\pi$
$661$		−	184.687i		−	0.279405i	0.990194	−	0.139703i	$-0.0446147\pi$
$662$		0			0
$663$	−13.4072	+	21.0772i	−0.0202221	+	0.0317907i
$664$		0			0
$665$		−	178.036i		−	0.267724i
$666$		0			0
$667$			227.891i			0.341666i
$668$		0			0
$669$	494.488	−	777.376i	0.739145	−	1.16200i
$670$		0			0
$671$			74.6364i			0.111232i
$672$		0			0
$673$	−371.348			−0.551781			−0.275890	−	0.961189i	$-0.588973\pi$
$673$	−371.348			−0.551781			−0.275890	+	0.961189i	$0.588973\pi$
$674$		0			0
$675$	−133.881	−	17.3436i	−0.198343	−	0.0256942i
$676$		0			0
$677$	270.915			0.400170			0.200085	−	0.979779i	$-0.435878\pi$
$677$	270.915			0.400170			0.200085	+	0.979779i	$0.435878\pi$
$678$		0			0
$679$	39.7155			0.0584912
$680$		0			0
$681$	391.510	−	615.485i	0.574904	−	0.903796i
$682$		0			0
$683$	581.504			0.851397			0.425698	−	0.904865i	$-0.360028\pi$
$683$	581.504			0.851397			0.425698	+	0.904865i	$0.360028\pi$
$684$		0			0
$685$			404.521i			0.590542i
$686$		0			0
$687$	−904.007	−	575.038i	−1.31588	−	0.837028i
$688$		0			0
$689$		−	387.770i		−	0.562801i
$690$		0			0
$691$		−	142.675i		−	0.206476i	−0.994657	−	0.103238i	$-0.967080\pi$
$691$		−	142.675i		−	0.206476i	0.994657	−	0.103238i	$-0.0329203\pi$
$692$		0			0
$693$	−160.930	−	343.873i	−0.232222	−	0.496210i
$694$		0			0
$695$		−	378.966i		−	0.545274i
$696$		0			0
$697$	109.579			0.157216
$698$		0			0
$699$	465.709	+	296.237i	0.666250	+	0.423801i
$700$		0			0
$701$	−268.963			−0.383685			−0.191843	−	0.981426i	$-0.561446\pi$
$701$	−268.963			−0.383685			−0.191843	+	0.981426i	$0.561446\pi$
$702$		0			0
$703$	−693.431			−0.986389
$704$		0			0
$705$	−265.058	−	168.603i	−0.375969	−	0.239153i
$706$		0			0
$707$	1222.75			1.72950
$708$		0			0
$709$		−	1204.06i		−	1.69825i	−0.528190	−	0.849126i	$-0.677129\pi$
$709$		−	1204.06i		−	1.69825i	0.528190	−	0.849126i	$-0.322871\pi$
$710$		0			0
$711$	519.750	+	1110.60i	0.731013	+	1.56202i
$712$		0			0
$713$		−	49.7840i		−	0.0698233i
$714$		0			0
$715$		−	55.0963i		−	0.0770577i
$716$		0			0
$717$	880.205	+	559.898i	1.22762	+	0.780889i
$718$		0			0
$719$		−	1420.67i		−	1.97589i	−0.154800	−	0.987946i	$-0.549473\pi$
$719$		−	1420.67i		−	1.97589i	0.154800	−	0.987946i	$-0.450527\pi$
$720$		0			0
$721$	−804.919			−1.11639
$722$		0			0
$723$	−546.921	+	859.804i	−0.756460	+	1.18922i
$724$		0			0
$725$	198.505			0.273800
$726$		0			0
$727$	−532.985			−0.733130			−0.366565	−	0.930393i	$-0.619466\pi$
$727$	−532.985			−0.733130			−0.366565	+	0.930393i	$0.619466\pi$
$728$		0			0
$729$	704.936	+	185.758i	0.966990	+	0.254812i
$730$		0			0
$731$	83.5422			0.114285
$732$		0			0
$733$		−	538.634i		−	0.734835i	−0.930056	−	0.367418i	$-0.880242\pi$
$733$		−	538.634i		−	0.734835i	0.930056	−	0.367418i	$-0.119758\pi$
$734$		0			0
$735$	63.4414	−	99.7351i	0.0863149	−	0.135694i
$736$		0			0
$737$			39.6835i			0.0538446i
$738$		0			0
$739$		−	792.423i		−	1.07229i	−0.844126	−	0.536145i	$-0.819880\pi$
$739$		−	792.423i		−	1.07229i	0.844126	−	0.536145i	$-0.180120\pi$
$740$		0			0
$741$	−74.8801	+	117.718i	−0.101053	+	0.158863i
$742$		0			0
$743$		−	657.405i		−	0.884798i	−0.896818	−	0.442399i	$-0.854128\pi$
$743$		−	657.405i		−	0.884798i	0.896818	−	0.442399i	$-0.145872\pi$
$744$		0			0
$745$	396.206			0.531821
$746$		0			0
$747$	−444.291	−	949.357i	−0.594766	−	1.27089i
$748$		0			0
$749$	−437.928			−0.584684
$750$		0			0
$751$	−735.089			−0.978814			−0.489407	−	0.872055i	$-0.662787\pi$
$751$	−735.089			−0.978814			−0.489407	+	0.872055i	$0.662787\pi$
$752$		0			0
$753$	121.977	−	191.757i	0.161987	−	0.254657i
$754$		0			0
$755$	436.099			0.577615
$756$		0			0
$757$			613.660i			0.810648i	0.914173	+	0.405324i	$0.132841\pi$
$757$			613.660i			0.810648i	−0.914173	+	0.405324i	$0.867159\pi$
$758$		0			0
$759$	75.0972	+	47.7693i	0.0989423	+	0.0629371i
$760$		0			0
$761$		−	76.5910i		−	0.100645i	−0.998733	−	0.0503226i	$-0.983975\pi$
$761$		−	76.5910i		−	0.100645i	0.998733	−	0.0503226i	$-0.0160249\pi$
$762$		0			0
$763$			84.8593i			0.111218i
$764$		0			0
$765$	−31.8357	+	14.8988i	−0.0416153	+	0.0194756i
$766$		0			0
$767$			166.034i			0.216472i
$768$		0			0
$769$	−577.971			−0.751588			−0.375794	−	0.926703i	$-0.622630\pi$
$769$	−577.971			−0.751588			−0.375794	+	0.926703i	$0.622630\pi$
$770$		0			0
$771$	−131.465	−	83.6249i	−0.170512	−	0.108463i
$772$		0			0
$773$	−696.300			−0.900776			−0.450388	−	0.892833i	$-0.648714\pi$
$773$	−696.300			−0.900776			−0.450388	+	0.892833i	$0.648714\pi$
$774$		0			0
$775$	−43.3645			−0.0559541
$776$		0			0
$777$	−1468.69	−	934.233i	−1.89021	−	1.20236i
$778$		0			0
$779$	612.008			0.785632
$780$		0			0
$781$		−	212.363i		−	0.271912i
$782$		0			0
$783$	−1063.04	−	137.711i	−1.35765	−	0.175876i
$784$		0			0
$785$		−	42.5942i		−	0.0542601i
$786$		0			0
$787$			712.691i			0.905579i	0.891617	+	0.452790i	$0.149571\pi$
$787$			712.691i			0.905579i	−0.891617	+	0.452790i	$0.850429\pi$
$788$		0			0
$789$	799.228	+	508.388i	1.01296	+	0.644345i
$790$		0			0
$791$		−	1306.69i		−	1.65194i
$792$		0			0
$793$	68.8455			0.0868165
$794$		0			0
$795$	292.850	−	460.384i	0.368364	−	0.579099i
$796$		0			0
$797$	140.555			0.176355			0.0881774	−	0.996105i	$-0.471896\pi$
$797$	140.555			0.176355			0.0881774	+	0.996105i	$0.471896\pi$
$798$		0			0
$799$	−81.7910			−0.102367
$800$		0			0
$801$	−596.260	+	279.044i	−0.744394	+	0.348370i
$802$		0			0
$803$	−31.5639			−0.0393075
$804$		0			0
$805$			104.765i			0.130143i
$806$		0			0
$807$	−540.171	+	849.193i	−0.669357	+	1.05228i
$808$		0			0
$809$			169.051i			0.208963i	0.994527	+	0.104482i	$0.0333183\pi$
$809$			169.051i			0.208963i	−0.994527	+	0.104482i	$0.966682\pi$
$810$		0			0
$811$			237.265i			0.292559i	0.989243	+	0.146279i	$0.0467298\pi$
$811$			237.265i			0.292559i	−0.989243	+	0.146279i	$0.953270\pi$
$812$		0			0
$813$	93.5024	−	146.993i	0.115009	−	0.180804i
$814$		0			0
$815$			399.065i			0.489651i
$816$		0			0
$817$	466.588			0.571099
$818$		0			0
$819$	−317.193	+	148.443i	−0.387293	+	0.181250i
$820$		0			0
$821$	−1086.80			−1.32375			−0.661874	−	0.749615i	$-0.730239\pi$
$821$	−1086.80			−1.32375			−0.661874	+	0.749615i	$0.730239\pi$
$822$		0			0
$823$	959.801			1.16622			0.583111	−	0.812392i	$-0.301835\pi$
$823$	959.801			1.16622			0.583111	+	0.812392i	$0.301835\pi$
$824$		0			0
$825$	41.6095	−	65.4136i	0.0504358	−	0.0792892i
$826$		0			0
$827$	−1107.81			−1.33955			−0.669776	−	0.742563i	$-0.733610\pi$
$827$	−1107.81			−1.33955			−0.669776	+	0.742563i	$0.733610\pi$
$828$		0			0
$829$			1451.55i			1.75097i	0.483249	+	0.875483i	$0.339457\pi$
$829$			1451.55i			1.75097i	−0.483249	+	0.875483i	$0.660543\pi$
$830$		0			0
$831$	−5.11854	−	3.25590i	−0.00615950	−	0.00391805i
$832$		0			0
$833$		−	30.7761i		−	0.0369460i
$834$		0			0
$835$			370.384i			0.443573i
$836$		0			0
$837$	232.228	+	30.0838i	0.277452	+	0.0359424i
$838$		0			0
$839$		−	636.757i		−	0.758948i	−0.925202	−	0.379474i	$-0.876105\pi$
$839$		−	636.757i		−	0.758948i	0.925202	−	0.379474i	$-0.123895\pi$
$840$		0			0
$841$	735.167			0.874158
$842$		0			0
$843$	658.867	+	419.105i	0.781574	+	0.497159i
$844$		0			0
$845$	327.074			0.387070
$846$		0			0
$847$	−769.589			−0.908606
$848$		0			0
$849$	684.320	+	435.296i	0.806031	+	0.512716i
$850$		0			0
$851$	408.047			0.479492
$852$		0			0
$853$			1114.31i			1.30634i	0.757212	+	0.653169i	$0.226561\pi$
$853$			1114.31i			1.30634i	−0.757212	+	0.653169i	$0.773439\pi$
$854$		0			0
$855$	−177.804	+	83.2108i	−0.207958	+	0.0973225i
$856$		0			0
$857$			946.338i			1.10425i	0.833763	+	0.552123i	$0.186182\pi$
$857$			946.338i			1.10425i	−0.833763	+	0.552123i	$0.813818\pi$
$858$		0			0
$859$			162.123i			0.188735i	0.995537	+	0.0943673i	$0.0300828\pi$
$859$			162.123i			0.188735i	−0.995537	+	0.0943673i	$0.969917\pi$
$860$		0			0
$861$	1296.23	+	824.534i	1.50550	+	0.957647i
$862$		0			0
$863$			463.268i			0.536811i	0.963306	+	0.268405i	$0.0864967\pi$
$863$			463.268i			0.536811i	−0.963306	+	0.268405i	$0.913503\pi$
$864$		0			0
$865$	−33.7545			−0.0390225
$866$		0			0
$867$	460.422	−	723.820i	0.531051	−	0.834856i
$868$		0			0
$869$	−704.166			−0.810318
$870$		0			0
$871$	36.6045			0.0420258
$872$		0			0
$873$	−18.5623	−	39.6638i	−0.0212626	−	0.0454339i
$874$		0			0
$875$	91.2556			0.104292
$876$		0			0
$877$			832.201i			0.948918i	0.880278	+	0.474459i	$0.157356\pi$
$877$			832.201i			0.948918i	−0.880278	+	0.474459i	$0.842644\pi$
$878$		0			0
$879$	367.241	−	577.333i	0.417794	−	0.656807i
$880$		0			0
$881$			1415.53i			1.60673i	0.595484	+	0.803367i	$0.296960\pi$
$881$			1415.53i			1.60673i	−0.595484	+	0.803367i	$0.703040\pi$
$882$		0			0
$883$		−	762.559i		−	0.863600i	−0.901969	−	0.431800i	$-0.857879\pi$
$883$		−	762.559i		−	0.863600i	0.901969	−	0.431800i	$-0.142121\pi$
$884$		0			0
$885$	−125.391	+	197.125i	−0.141685	+	0.222741i
$886$		0			0
$887$		−	63.9277i		−	0.0720718i	−0.999350	−	0.0360359i	$-0.988527\pi$
$887$		−	63.9277i		−	0.0720718i	0.999350	−	0.0360359i	$-0.0114731\pi$
$888$		0			0
$889$	−1429.78			−1.60830
$890$		0			0
$891$	−268.210	+	321.440i	−0.301021	+	0.360763i
$892$		0			0
$893$	−456.808			−0.511543
$894$		0			0
$895$	−540.211			−0.603587
$896$		0			0
$897$	44.0629	−	69.2705i	0.0491226	−	0.0772247i
$898$		0			0
$899$	−344.322			−0.383006
$900$		0			0
$901$		−	142.064i		−	0.157674i
$902$		0			0
$903$	988.235	+	628.616i	1.09439	+	0.696141i
$904$		0			0
$905$			210.591i			0.232697i
$906$		0			0
$907$			839.954i			0.926079i	0.886338	+	0.463039i	$0.153241\pi$
$907$			839.954i			0.926079i	−0.886338	+	0.463039i	$0.846759\pi$
$908$		0			0
$909$	−571.492	−	1221.16i	−0.628704	−	1.34341i
$910$		0			0
$911$		−	1032.36i		−	1.13322i	−0.823987	−	0.566609i	$-0.808255\pi$
$911$		−	1032.36i		−	1.13322i	0.823987	−	0.566609i	$-0.191745\pi$
$912$		0			0
$913$	601.933			0.659291
$914$		0			0
$915$	81.7374	+	51.9931i	0.0893305	+	0.0568231i
$916$		0			0
$917$	−1549.35			−1.68958
$918$		0			0
$919$	1199.54			1.30526			0.652632	−	0.757675i	$-0.273665\pi$
$919$	1199.54			1.30526			0.652632	+	0.757675i	$0.273665\pi$
$920$		0			0
$921$	1141.35	+	726.012i	1.23925	+	0.788287i
$922$		0			0
$923$	−195.886			−0.212228
$924$		0			0
$925$		−	355.430i		−	0.384249i
$926$		0			0
$927$	376.204	+	803.870i	0.405829	+	0.867173i
$928$		0			0
$929$		−	1098.06i		−	1.18198i	−0.806678	−	0.590991i	$-0.798737\pi$
$929$		−	1098.06i		−	1.18198i	0.806678	−	0.590991i	$-0.201263\pi$
$930$		0			0
$931$		−	171.886i		−	0.184625i
$932$		0			0
$933$	262.229	+	166.804i	0.281060	+	0.178782i
$934$		0			0
$935$		−	20.1852i		−	0.0215884i
$936$		0			0
$937$	672.270			0.717470			0.358735	−	0.933439i	$-0.383208\pi$
$937$	672.270			0.717470			0.358735	+	0.933439i	$0.383208\pi$
$938$		0			0
$939$	−573.965	+	902.320i	−0.611252	+	0.960937i
$940$		0			0
$941$	81.9534			0.0870919			0.0435459	−	0.999051i	$-0.486135\pi$
$941$	81.9534			0.0870919			0.0435459	+	0.999051i	$0.486135\pi$
$942$		0			0
$943$	−360.134			−0.381902
$944$		0			0
$945$	−488.697	−	63.3079i	−0.517139	−	0.0669925i
$946$		0			0
$947$	−1168.46			−1.23386			−0.616928	−	0.787020i	$-0.711623\pi$
$947$	−1168.46			−1.23386			−0.616928	+	0.787020i	$0.711623\pi$
$948$		0			0
$949$			29.1149i			0.0306796i
$950$		0			0
$951$	−608.096	+	955.976i	−0.639428	+	1.00523i
$952$		0			0
$953$		−	1422.35i		−	1.49250i	−0.665669	−	0.746248i	$-0.731854\pi$
$953$		−	1422.35i		−	1.49250i	0.665669	−	0.746248i	$-0.268146\pi$
$954$		0			0
$955$			629.425i			0.659084i
$956$		0			0
$957$	330.388	−	519.396i	0.345233	−	0.542734i
$958$		0			0
$959$			1476.59i			1.53972i
$960$		0			0
$961$	−885.781			−0.921728
$962$		0			0
$963$	204.679	+	437.357i	0.212544	+	0.454161i
$964$		0			0
$965$	−370.809			−0.384258
$966$		0			0
$967$	403.027			0.416781			0.208390	−	0.978046i	$-0.433178\pi$
$967$	403.027			0.416781			0.208390	+	0.978046i	$0.433178\pi$
$968$		0			0
$969$	−27.4332	+	43.1273i	−0.0283109	+	0.0445070i
$970$		0			0
$971$	−327.768			−0.337558			−0.168779	−	0.985654i	$-0.553982\pi$
$971$	−327.768			−0.337558			−0.168779	+	0.985654i	$0.553982\pi$
$972$		0			0
$973$		−	1383.31i		−	1.42170i
$974$		0			0
$975$	−60.3382	−	38.3811i	−0.0618854	−	0.0393652i
$976$		0			0
$977$		−	653.254i		−	0.668633i	−0.942461	−	0.334316i	$-0.891495\pi$
$977$		−	653.254i		−	0.668633i	0.942461	−	0.334316i	$-0.108505\pi$
$978$		0			0
$979$		−	378.054i		−	0.386164i
$980$		0			0
$981$	84.7486	−	39.6616i	0.0863900	−	0.0404298i
$982$		0			0
$983$		−	870.902i		−	0.885963i	−0.896531	−	0.442982i	$-0.853921\pi$
$983$		−	870.902i		−	0.885963i	0.896531	−	0.442982i	$-0.146079\pi$
$984$		0			0
$985$	708.315			0.719102
$986$		0			0
$987$	−967.521	−	615.439i	−0.980264	−	0.623545i
$988$		0			0
$989$	−274.562			−0.277616
$990$		0			0
$991$	1795.14			1.81145			0.905724	−	0.423869i	$-0.139328\pi$
$991$	1795.14			1.81145			0.905724	+	0.423869i	$0.139328\pi$
$992$		0			0
$993$	470.665	+	299.390i	0.473983	+	0.301500i
$994$		0			0
$995$	395.122			0.397108
$996$		0			0
$997$			877.803i			0.880444i	0.897889	+	0.440222i	$0.145100\pi$
$997$			877.803i			0.880444i	−0.897889	+	0.440222i	$0.854900\pi$
$998$		0			0
$999$	−246.577	+	1903.42i	−0.246824	+	1.90532i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.n.a.161.18	yes	24
3.2	odd	2	inner	960.3.n.a.161.5	✓	24
4.3	odd	2	inner	960.3.n.a.161.8	yes	24
8.3	odd	2	inner	960.3.n.a.161.17	yes	24
8.5	even	2	inner	960.3.n.a.161.7	yes	24
12.11	even	2	inner	960.3.n.a.161.19	yes	24
24.5	odd	2	inner	960.3.n.a.161.20	yes	24
24.11	even	2	inner	960.3.n.a.161.6	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.3.n.a.161.5	✓	24	3.2	odd	2	inner
960.3.n.a.161.6	yes	24	24.11	even	2	inner
960.3.n.a.161.7	yes	24	8.5	even	2	inner
960.3.n.a.161.8	yes	24	4.3	odd	2	inner
960.3.n.a.161.17	yes	24	8.3	odd	2	inner
960.3.n.a.161.18	yes	24	1.1	even	1	trivial
960.3.n.a.161.19	yes	24	12.11	even	2	inner
960.3.n.a.161.20	yes	24	24.5	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.61015 - 2.53129i) q^{3} +2.23607 q^{5} +8.16215 q^{7} +(-3.81483 - 8.15151i) q^{9} +O(q^{10})\)	\(q+(1.61015 - 2.53129i) q^{3} +2.23607 q^{5} +8.16215 q^{7} +(-3.81483 - 8.15151i) q^{9} +5.16840 q^{11} -4.76739i q^{13} +(3.60041 - 5.66013i) q^{15} -1.74659i q^{17} -9.75481i q^{19} +(13.1423 - 20.6607i) q^{21} +5.74019i q^{23} +5.00000 q^{25} +(-26.7763 - 3.46871i) q^{27} +39.7010 q^{29} -8.67289 q^{31} +(8.32190 - 13.0827i) q^{33} +18.2511 q^{35} -71.0861i q^{37} +(-12.0676 - 7.67622i) q^{39} +62.7390i q^{41} +47.8315i q^{43} +(-8.53022 - 18.2273i) q^{45} -46.8289i q^{47} +17.6206 q^{49} +(-4.42113 - 2.81228i) q^{51} +81.3380 q^{53} +11.5569 q^{55} +(-24.6922 - 15.7067i) q^{57} -34.8270 q^{59} +14.4409i q^{61} +(-31.1372 - 66.5338i) q^{63} -10.6602i q^{65} +7.67810i q^{67} +(14.5301 + 9.24256i) q^{69} -41.0888i q^{71} -6.10709 q^{73} +(8.05075 - 12.6564i) q^{75} +42.1853 q^{77} -136.245 q^{79} +(-51.8941 + 62.1933i) q^{81} +116.464 q^{83} -3.90550i q^{85} +(63.9245 - 100.495i) q^{87} -73.1472i q^{89} -38.9122i q^{91} +(-13.9647 + 21.9536i) q^{93} -21.8124i q^{95} +4.86582 q^{97} +(-19.7166 - 42.1303i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 8 q^{9}+O(q^{10}) \) 24 * q - 8 * q^9	\( 24 q - 8 q^{9} + 120 q^{25} - 256 q^{33} + 104 q^{49} - 304 q^{57} + 400 q^{73} + 152 q^{81} + 208 q^{97}+O(q^{100}) \) 24 * q - 8 * q^9 + 120 * q^25 - 256 * q^33 + 104 * q^49 - 304 * q^57 + 400 * q^73 + 152 * q^81 + 208 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more