LMFDB - Embedded newform 588.4.f.d.293.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,4,Mod(293,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("588.293");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	588.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:	$34.6931230834$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			293.9
Character	$\chi$	$=$	588.293
Dual form			588.4.f.d.293.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.02618 - 4.78483i) q^{3} -13.9873 q^{5} +(-18.7892 + 19.3899i) q^{9} +O(q^{10})$	$q+(-2.02618 - 4.78483i) q^{3} -13.9873 q^{5} +(-18.7892 + 19.3899i) q^{9} +31.7927i q^{11} +31.1041i q^{13} +(28.3408 + 66.9269i) q^{15} +123.028 q^{17} -72.4751i q^{19} +76.3535i q^{23} +70.6447 q^{25} +(130.847 + 50.6156i) q^{27} -14.1082i q^{29} -309.392i q^{31} +(152.123 - 64.4178i) q^{33} -116.720 q^{37} +(148.828 - 63.0225i) q^{39} -491.980 q^{41} +13.6538 q^{43} +(262.810 - 271.212i) q^{45} +86.8762 q^{47} +(-249.276 - 588.666i) q^{51} +449.166i q^{53} -444.695i q^{55} +(-346.781 + 146.848i) q^{57} -256.783 q^{59} +274.831i q^{61} -435.062i q^{65} +1031.97 q^{67} +(365.338 - 154.706i) q^{69} -931.763i q^{71} -1025.05i q^{73} +(-143.139 - 338.023i) q^{75} -924.621 q^{79} +(-22.9336 - 728.639i) q^{81} +991.698 q^{83} -1720.82 q^{85} +(-67.5054 + 28.5858i) q^{87} -250.787 q^{89} +(-1480.39 + 626.884i) q^{93} +1013.73i q^{95} -1059.27i q^{97} +(-616.457 - 597.359i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 64 q^{9}+O(q^{10}) $ 24 * q - 64 * q^9	$ 24 q - 64 q^{9} - 96 q^{15} + 456 q^{25} - 432 q^{37} + 688 q^{39} + 624 q^{43} - 1536 q^{51} - 1360 q^{57} - 528 q^{67} + 3744 q^{79} + 3408 q^{81} + 6912 q^{85} - 5088 q^{93} - 7736 q^{99}+O(q^{100}) $ 24 * q - 64 * q^9 - 96 * q^15 + 456 * q^25 - 432 * q^37 + 688 * q^39 + 624 * q^43 - 1536 * q^51 - 1360 * q^57 - 528 * q^67 + 3744 * q^79 + 3408 * q^81 + 6912 * q^85 - 5088 * q^93 - 7736 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−2.02618	−	4.78483i	−0.389939	−	0.920841i
$3$	−2.02618	−	4.78483i	−0.389939	−	0.920841i
$4$		0			0
$5$	−13.9873			−1.25106			−0.625531	−	0.780199i	$-0.715118\pi$
$5$	−13.9873			−1.25106			−0.625531	+	0.780199i	$0.715118\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−18.7892	+	19.3899i	−0.695895	+	0.718143i
$10$		0			0
$11$			31.7927i			0.871443i	0.900082	+	0.435721i	$0.143507\pi$
$11$			31.7927i			0.871443i	−0.900082	+	0.435721i	$0.856493\pi$
$12$		0			0
$13$			31.1041i			0.663593i	0.943351	+	0.331797i	$0.107655\pi$
$13$			31.1041i			0.663593i	−0.943351	+	0.331797i	$0.892345\pi$
$14$		0			0
$15$	28.3408	+	66.9269i	0.487838	+	1.15203i
$16$		0			0
$17$	123.028			1.75521			0.877605	−	0.479385i	$-0.159140\pi$
$17$	123.028			1.75521			0.877605	+	0.479385i	$0.159140\pi$
$18$		0			0
$19$		−	72.4751i		−	0.875102i	−0.899194	−	0.437551i	$-0.855846\pi$
$19$		−	72.4751i		−	0.875102i	0.899194	−	0.437551i	$-0.144154\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			76.3535i			0.692209i	0.938196	+	0.346104i	$0.112496\pi$
$23$			76.3535i			0.692209i	−0.938196	+	0.346104i	$0.887504\pi$
$24$		0			0
$25$	70.6447			0.565157
$26$		0			0
$27$	130.847	+	50.6156i	0.932652	+	0.360777i
$28$		0			0
$29$		−	14.1082i		−	0.0903390i	−0.998979	−	0.0451695i	$-0.985617\pi$
$29$		−	14.1082i		−	0.0903390i	0.998979	−	0.0451695i	$-0.0143828\pi$
$30$		0			0
$31$		−	309.392i		−	1.79253i	−0.443519	−	0.896265i	$-0.646270\pi$
$31$		−	309.392i		−	1.79253i	0.443519	−	0.896265i	$-0.353730\pi$
$32$		0			0
$33$	152.123	−	64.4178i	0.802460	−	0.339809i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−116.720			−0.518614			−0.259307	−	0.965795i	$-0.583494\pi$
$37$	−116.720			−0.518614			−0.259307	+	0.965795i	$0.583494\pi$
$38$		0			0
$39$	148.828	−	63.0225i	0.611064	−	0.258761i
$40$		0			0
$41$	−491.980			−1.87401			−0.937004	−	0.349319i	$-0.886413\pi$
$41$	−491.980			−1.87401			−0.937004	+	0.349319i	$0.886413\pi$
$42$		0			0
$43$	13.6538			0.0484230			0.0242115	−	0.999707i	$-0.492292\pi$
$43$	13.6538			0.0484230			0.0242115	+	0.999707i	$0.492292\pi$
$44$		0			0
$45$	262.810	−	271.212i	0.870609	−	0.898442i
$46$		0			0
$47$	86.8762			0.269621			0.134811	−	0.990871i	$-0.456957\pi$
$47$	86.8762			0.269621			0.134811	+	0.990871i	$0.456957\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−249.276	−	588.666i	−0.684424	−	1.61627i
$52$		0			0
$53$			449.166i			1.16411i	0.813151	+	0.582053i	$0.197751\pi$
$53$			449.166i			1.16411i	−0.813151	+	0.582053i	$0.802249\pi$
$54$		0			0
$55$		−	444.695i		−	1.09023i
$56$		0			0
$57$	−346.781	+	146.848i	−0.805829	+	0.341236i
$58$		0			0
$59$	−256.783			−0.566616			−0.283308	−	0.959029i	$-0.591432\pi$
$59$	−256.783			−0.566616			−0.283308	+	0.959029i	$0.591432\pi$
$60$		0			0
$61$			274.831i			0.576860i	0.957501	+	0.288430i	$0.0931332\pi$
$61$			274.831i			0.576860i	−0.957501	+	0.288430i	$0.906867\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	435.062i		−	0.830197i
$66$		0			0
$67$	1031.97			1.88171			0.940857	−	0.338803i	$-0.110022\pi$
$67$	1031.97			1.88171			0.940857	+	0.338803i	$0.110022\pi$
$68$		0			0
$69$	365.338	−	154.706i	0.637414	−	0.269919i
$70$		0			0
$71$		−	931.763i		−	1.55746i	−0.627356	−	0.778732i	$-0.715863\pi$
$71$		−	931.763i		−	1.55746i	0.627356	−	0.778732i	$-0.284137\pi$
$72$		0			0
$73$		−	1025.05i		−	1.64346i	−0.569877	−	0.821730i	$-0.693009\pi$
$73$		−	1025.05i		−	1.64346i	0.569877	−	0.821730i	$-0.306991\pi$
$74$		0			0
$75$	−143.139	−	338.023i	−0.220377	−	0.520420i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−924.621			−1.31681			−0.658405	−	0.752664i	$-0.728768\pi$
$79$	−924.621			−1.31681			−0.658405	+	0.752664i	$0.728768\pi$
$80$		0			0
$81$	−22.9336	−	728.639i	−0.0314590	−	0.999505i
$82$		0			0
$83$	991.698			1.31148			0.655741	−	0.754986i	$-0.272356\pi$
$83$	991.698			1.31148			0.655741	+	0.754986i	$0.272356\pi$
$84$		0			0
$85$	−1720.82			−2.19588
$86$		0			0
$87$	−67.5054	+	28.5858i	−0.0831878	+	0.0352267i
$88$		0			0
$89$	−250.787			−0.298690			−0.149345	−	0.988785i	$-0.547716\pi$
$89$	−250.787			−0.298690			−0.149345	+	0.988785i	$0.547716\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−1480.39	+	626.884i	−1.65064	+	0.698977i
$94$		0			0
$95$			1013.73i			1.09481i
$96$		0			0
$97$		−	1059.27i		−	1.10879i	−0.832252	−	0.554397i	$-0.812949\pi$
$97$		−	1059.27i		−	1.10879i	0.832252	−	0.554397i	$-0.187051\pi$
$98$		0			0
$99$	−616.457	−	597.359i	−0.625821	−	0.606433i
$100$		0			0
$101$	1528.23			1.50559			0.752796	−	0.658254i	$-0.228705\pi$
$101$	1528.23			1.50559			0.752796	+	0.658254i	$0.228705\pi$
$102$		0			0
$103$			190.224i			0.181974i	0.995852	+	0.0909870i	$0.0290022\pi$
$103$			190.224i			0.181974i	−0.995852	+	0.0909870i	$0.970998\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			177.882i			0.160715i	0.996766	+	0.0803573i	$0.0256061\pi$
$107$			177.882i			0.160715i	−0.996766	+	0.0803573i	$0.974394\pi$
$108$		0			0
$109$	193.039			0.169631			0.0848156	−	0.996397i	$-0.472970\pi$
$109$	193.039			0.169631			0.0848156	+	0.996397i	$0.472970\pi$
$110$		0			0
$111$	236.497	+	558.487i	0.202228	+	0.477561i
$112$		0			0
$113$		−	452.079i		−	0.376354i	−0.982135	−	0.188177i	$-0.939742\pi$
$113$		−	452.079i		−	0.376354i	0.982135	−	0.188177i	$-0.0602579\pi$
$114$		0			0
$115$		−	1067.98i		−	0.865996i
$116$		0			0
$117$	−603.104	−	584.420i	−0.476555	−	0.461792i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	320.222			0.240588
$122$		0			0
$123$	996.840	+	2354.04i	0.730748	+	1.72566i
$124$		0			0
$125$	760.285			0.544015
$126$		0			0
$127$	729.629			0.509796			0.254898	−	0.966968i	$-0.417958\pi$
$127$	729.629			0.509796			0.254898	+	0.966968i	$0.417958\pi$
$128$		0			0
$129$	−27.6651	−	65.3312i	−0.0188820	−	0.0445898i
$130$		0			0
$131$	−574.432			−0.383117			−0.191558	−	0.981481i	$-0.561354\pi$
$131$	−574.432			−0.383117			−0.191558	+	0.981481i	$0.561354\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1830.20	−	707.976i	−1.16681	−	0.451355i
$136$		0			0
$137$		−	2427.76i		−	1.51400i	−0.653417	−	0.756998i	$-0.726665\pi$
$137$		−	2427.76i		−	1.51400i	0.653417	−	0.756998i	$-0.273335\pi$
$138$		0			0
$139$		−	1453.63i		−	0.887014i	−0.896271	−	0.443507i	$-0.853734\pi$
$139$		−	1453.63i		−	0.887014i	0.896271	−	0.443507i	$-0.146266\pi$
$140$		0			0
$141$	−176.027	−	415.688i	−0.105136	−	0.248278i
$142$		0			0
$143$	−988.883			−0.578284
$144$		0			0
$145$			197.336i			0.113020i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	1967.84i		−	1.08196i	−0.841036	−	0.540979i	$-0.818054\pi$
$149$		−	1967.84i		−	1.08196i	0.841036	−	0.540979i	$-0.181946\pi$
$150$		0			0
$151$	2172.81			1.17100			0.585499	−	0.810673i	$-0.300899\pi$
$151$	2172.81			1.17100			0.585499	+	0.810673i	$0.300899\pi$
$152$		0			0
$153$	−2311.59	+	2385.49i	−1.22144	+	1.26049i
$154$		0			0
$155$			4327.56i			2.24257i
$156$		0			0
$157$		−	3299.15i		−	1.67708i	−0.544844	−	0.838538i	$-0.683411\pi$
$157$		−	3299.15i		−	1.67708i	0.544844	−	0.838538i	$-0.316589\pi$
$158$		0			0
$159$	2149.18	−	910.091i	1.07196	−	0.453930i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2799.27			1.34512			0.672562	−	0.740040i	$-0.265194\pi$
$163$	2799.27			1.34512			0.672562	+	0.740040i	$0.265194\pi$
$164$		0			0
$165$	−2127.79	+	901.032i	−1.00393	+	0.425123i
$166$		0			0
$167$	−1933.50			−0.895921			−0.447960	−	0.894053i	$-0.647849\pi$
$167$	−1933.50			−0.895921			−0.447960	+	0.894053i	$0.647849\pi$
$168$		0			0
$169$	1229.54			0.559644
$170$		0			0
$171$	1405.28	+	1361.75i	0.628448	+	0.608979i
$172$		0			0
$173$	3505.01			1.54035			0.770176	−	0.637832i	$-0.220168\pi$
$173$	3505.01			1.54035			0.770176	+	0.637832i	$0.220168\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	520.290	+	1228.66i	0.220946	+	0.521763i
$178$		0			0
$179$		−	1846.25i		−	0.770925i	−0.922724	−	0.385462i	$-0.874042\pi$
$179$		−	1846.25i		−	0.770925i	0.922724	−	0.385462i	$-0.125958\pi$
$180$		0			0
$181$		−	369.402i		−	0.151699i	−0.997119	−	0.0758493i	$-0.975833\pi$
$181$		−	369.402i		−	0.151699i	0.997119	−	0.0758493i	$-0.0241668\pi$
$182$		0			0
$183$	1315.02	−	556.857i	0.531196	−	0.224940i
$184$		0			0
$185$	1632.60			0.648818
$186$		0			0
$187$			3911.38i			1.52956i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			238.688i			0.0904232i	0.998977	+	0.0452116i	$0.0143962\pi$
$191$			238.688i			0.0904232i	−0.998977	+	0.0452116i	$0.985604\pi$
$192$		0			0
$193$	−1682.10			−0.627358			−0.313679	−	0.949529i	$-0.601562\pi$
$193$	−1682.10			−0.627358			−0.313679	+	0.949529i	$0.601562\pi$
$194$		0			0
$195$	−2081.70	+	881.514i	−0.764479	+	0.323726i
$196$		0			0
$197$		−	2988.01i		−	1.08065i	−0.841458	−	0.540323i	$-0.818302\pi$
$197$		−	2988.01i		−	1.08065i	0.841458	−	0.540323i	$-0.181698\pi$
$198$		0			0
$199$			2017.24i			0.718584i	0.933225	+	0.359292i	$0.116982\pi$
$199$			2017.24i			0.718584i	−0.933225	+	0.359292i	$0.883018\pi$
$200$		0			0
$201$	−2090.95	−	4937.79i	−0.733753	−	1.73276i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	6881.47			2.34450
$206$		0			0
$207$	−1480.48	−	1434.62i	−0.497105	−	0.481705i
$208$		0			0
$209$	2304.18			0.762601
$210$		0			0
$211$	4685.01			1.52858			0.764288	−	0.644875i	$-0.223091\pi$
$211$	4685.01			1.52858			0.764288	+	0.644875i	$0.223091\pi$
$212$		0			0
$213$	−4458.33	+	1887.92i	−1.43418	+	0.607316i
$214$		0			0
$215$	−190.980			−0.0605802
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−4904.67	+	2076.93i	−1.51336	+	0.640849i
$220$		0			0
$221$			3826.66i			1.16475i
$222$		0			0
$223$		−	17.2808i		−	0.00518927i	−0.999997	−	0.00259463i	$-0.999174\pi$
$223$		−	17.2808i		−	0.00518927i	0.999997	−	0.00259463i	$-0.000825899\pi$
$224$		0			0
$225$	−1327.36	+	1369.79i	−0.393290	+	0.405864i
$226$		0			0
$227$	−1463.87			−0.428021			−0.214011	−	0.976831i	$-0.568653\pi$
$227$	−1463.87			−0.428021			−0.214011	+	0.976831i	$0.568653\pi$
$228$		0			0
$229$			116.073i			0.0334948i	0.999860	+	0.0167474i	$0.00533111\pi$
$229$			116.073i			0.0334948i	−0.999860	+	0.0167474i	$0.994669\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3750.59i			1.05455i	0.849696	+	0.527274i	$0.176786\pi$
$233$			3750.59i			1.05455i	−0.849696	+	0.527274i	$0.823214\pi$
$234$		0			0
$235$	−1215.16			−0.337313
$236$		0			0
$237$	1873.45	+	4424.15i	0.513475	+	1.21257i
$238$		0			0
$239$			5423.48i			1.46785i	0.679232	+	0.733924i	$0.262313\pi$
$239$			5423.48i			1.46785i	−0.679232	+	0.733924i	$0.737687\pi$
$240$		0			0
$241$			4542.33i			1.21410i	0.794665	+	0.607048i	$0.207646\pi$
$241$			4542.33i			1.21410i	−0.794665	+	0.607048i	$0.792354\pi$
$242$		0			0
$243$	−3439.95	+	1586.09i	−0.908118	+	0.418714i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2254.27			0.580712
$248$		0			0
$249$	−2009.36	−	4745.11i	−0.511398	−	1.20767i
$250$		0			0
$251$	1115.63			0.280549			0.140275	−	0.990113i	$-0.455201\pi$
$251$	1115.63			0.280549			0.140275	+	0.990113i	$0.455201\pi$
$252$		0			0
$253$	−2427.49			−0.603220
$254$		0			0
$255$	3486.70	+	8233.84i	0.856257	+	2.02205i
$256$		0			0
$257$	−6379.29			−1.54836			−0.774182	−	0.632963i	$-0.781838\pi$
$257$	−6379.29			−1.54836			−0.774182	+	0.632963i	$0.781838\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	273.556	+	265.082i	0.0648763	+	0.0628665i
$262$		0			0
$263$		−	721.223i		−	0.169097i	−0.996419	−	0.0845485i	$-0.973055\pi$
$263$		−	721.223i		−	0.169097i	0.996419	−	0.0845485i	$-0.0269448\pi$
$264$		0			0
$265$		−	6282.62i		−	1.45637i
$266$		0			0
$267$	508.140	+	1199.97i	0.116471	+	0.275046i
$268$		0			0
$269$	−4746.65			−1.07587			−0.537933	−	0.842987i	$-0.680795\pi$
$269$	−4746.65			−1.07587			−0.537933	+	0.842987i	$0.680795\pi$
$270$		0			0
$271$			8226.80i			1.84407i	0.387108	+	0.922034i	$0.373474\pi$
$271$			8226.80i			1.84407i	−0.387108	+	0.922034i	$0.626526\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			2245.99i			0.492502i
$276$		0			0
$277$	−5103.76			−1.10706			−0.553529	−	0.832830i	$-0.686719\pi$
$277$	−5103.76			−1.10706			−0.553529	+	0.832830i	$0.686719\pi$
$278$		0			0
$279$	5999.07	+	5813.22i	1.28729	+	1.24741i
$280$		0			0
$281$		−	6818.51i		−	1.44754i	−0.690042	−	0.723769i	$-0.742408\pi$
$281$		−	6818.51i		−	1.44754i	0.690042	−	0.723769i	$-0.257592\pi$
$282$		0			0
$283$			2676.20i			0.562134i	0.959688	+	0.281067i	$0.0906883\pi$
$283$			2676.20i			0.562134i	−0.959688	+	0.281067i	$0.909312\pi$
$284$		0			0
$285$	4850.53	−	2054.00i	1.00814	−	0.426908i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	10222.8			2.08076
$290$		0			0
$291$	−5068.45	+	2146.28i	−1.02102	+	0.432362i
$292$		0			0
$293$	−1743.71			−0.347674			−0.173837	−	0.984774i	$-0.555617\pi$
$293$	−1743.71			−0.347674			−0.173837	+	0.984774i	$0.555617\pi$
$294$		0			0
$295$	3591.71			0.708872
$296$		0			0
$297$	−1609.21	+	4160.00i	−0.314397	+	0.812753i
$298$		0			0
$299$	−2374.90			−0.459345
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−3096.48	−	7312.33i	−0.587089	−	1.38641i
$304$		0			0
$305$		−	3844.14i		−	0.721688i
$306$		0			0
$307$		−	1503.17i		−	0.279448i	−0.990191	−	0.139724i	$-0.955378\pi$
$307$		−	1503.17i		−	0.279448i	0.990191	−	0.139724i	$-0.0446215\pi$
$308$		0			0
$309$	910.189	−	385.428i	0.167569	−	0.0709587i
$310$		0			0
$311$	−8741.45			−1.59383			−0.796916	−	0.604090i	$-0.793537\pi$
$311$	−8741.45			−1.59383			−0.796916	+	0.604090i	$0.793537\pi$
$312$		0			0
$313$		−	3730.75i		−	0.673720i	−0.941555	−	0.336860i	$-0.890635\pi$
$313$		−	3730.75i		−	0.673720i	0.941555	−	0.336860i	$-0.109365\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			4980.17i			0.882379i	0.897414	+	0.441190i	$0.145443\pi$
$317$			4980.17i			0.882379i	−0.897414	+	0.441190i	$0.854557\pi$
$318$		0			0
$319$	448.539			0.0787252
$320$		0			0
$321$	851.133	−	360.421i	0.147993	−	0.0626689i
$322$		0			0
$323$		−	8916.43i		−	1.53599i
$324$		0			0
$325$			2197.34i			0.375035i
$326$		0			0
$327$	−391.132	−	923.659i	−0.0661457	−	0.156203i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6194.54			1.02865			0.514324	−	0.857596i	$-0.328043\pi$
$331$	6194.54			1.02865			0.514324	+	0.857596i	$0.328043\pi$
$332$		0			0
$333$	2193.08	−	2263.19i	0.360901	−	0.372439i
$334$		0			0
$335$	−14434.4			−2.35414
$336$		0			0
$337$	−2866.79			−0.463395			−0.231697	−	0.972788i	$-0.574428\pi$
$337$	−2866.79			−0.463395			−0.231697	+	0.972788i	$0.574428\pi$
$338$		0			0
$339$	−2163.12	+	915.994i	−0.346562	+	0.146755i
$340$		0			0
$341$	9836.42			1.56209
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−5110.10	+	2163.92i	−0.797445	+	0.337685i
$346$		0			0
$347$		−	1447.61i		−	0.223953i	−0.993711	−	0.111976i	$-0.964282\pi$
$347$		−	1447.61i		−	0.223953i	0.993711	−	0.111976i	$-0.0357181\pi$
$348$		0			0
$349$		−	5343.79i		−	0.819618i	−0.912171	−	0.409809i	$-0.865595\pi$
$349$		−	5343.79i		−	0.819618i	0.912171	−	0.409809i	$-0.134405\pi$
$350$		0			0
$351$	−1574.35	+	4069.89i	−0.239409	+	0.618902i
$352$		0			0
$353$	4402.80			0.663845			0.331922	−	0.943307i	$-0.392303\pi$
$353$	4402.80			0.663845			0.331922	+	0.943307i	$0.392303\pi$
$354$		0			0
$355$			13032.9i			1.94849i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	5358.74i		−	0.787810i	−0.919151	−	0.393905i	$-0.871124\pi$
$359$		−	5358.74i		−	0.787810i	0.919151	−	0.393905i	$-0.128876\pi$
$360$		0			0
$361$	1606.36			0.234197
$362$		0			0
$363$	−648.828	−	1532.21i	−0.0938144	−	0.221543i
$364$		0			0
$365$			14337.6i			2.05607i
$366$		0			0
$367$		−	8333.84i		−	1.18535i	−0.805442	−	0.592674i	$-0.798072\pi$
$367$		−	8333.84i		−	1.18535i	0.805442	−	0.592674i	$-0.201928\pi$
$368$		0			0
$369$	9243.89	−	9539.42i	1.30411	−	1.34581i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	7195.96			0.998908			0.499454	−	0.866340i	$-0.333534\pi$
$373$	7195.96			0.998908			0.499454	+	0.866340i	$0.333534\pi$
$374$		0			0
$375$	−1540.47	−	3637.83i	−0.212133	−	0.500951i
$376$		0			0
$377$	438.823			0.0599483
$378$		0			0
$379$	12312.1			1.66869			0.834343	−	0.551245i	$-0.185847\pi$
$379$	12312.1			1.66869			0.834343	+	0.551245i	$0.185847\pi$
$380$		0			0
$381$	−1478.36	−	3491.15i	−0.198789	−	0.469441i
$382$		0			0
$383$	−10730.4			−1.43158			−0.715791	−	0.698315i	$-0.753934\pi$
$383$	−10730.4			−1.43158			−0.715791	+	0.698315i	$0.753934\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−256.544	+	264.746i	−0.0336973	+	0.0347746i
$388$		0			0
$389$			10016.2i			1.30551i	0.757570	+	0.652754i	$0.226386\pi$
$389$			10016.2i			1.30551i	−0.757570	+	0.652754i	$0.773614\pi$
$390$		0			0
$391$			9393.58i			1.21497i
$392$		0			0
$393$	1163.90	+	2748.56i	0.149392	+	0.352790i
$394$		0			0
$395$	12932.9			1.64741
$396$		0			0
$397$		−	3742.22i		−	0.473090i	−0.971621	−	0.236545i	$-0.923985\pi$
$397$		−	3742.22i		−	0.473090i	0.971621	−	0.236545i	$-0.0760150\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	9634.62i		−	1.19983i	−0.800065	−	0.599913i	$-0.795202\pi$
$401$		−	9634.62i		−	1.19983i	0.800065	−	0.599913i	$-0.204798\pi$
$402$		0			0
$403$	9623.35			1.18951
$404$		0			0
$405$	320.779	+	10191.7i	0.0393571	+	1.25044i
$406$		0			0
$407$		−	3710.86i		−	0.451942i
$408$		0			0
$409$		−	5251.24i		−	0.634858i	−0.948282	−	0.317429i	$-0.897180\pi$
$409$		−	5251.24i		−	0.634858i	0.948282	−	0.317429i	$-0.102820\pi$
$410$		0			0
$411$	−11616.4	+	4919.08i	−1.39415	+	0.590366i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−13871.2			−1.64075
$416$		0			0
$417$	−6955.36	+	2945.31i	−0.816799	+	0.345881i
$418$		0			0
$419$	11867.1			1.38364			0.691819	−	0.722071i	$-0.256809\pi$
$419$	11867.1			1.38364			0.691819	+	0.722071i	$0.256809\pi$
$420$		0			0
$421$	−8108.04			−0.938626			−0.469313	−	0.883032i	$-0.655498\pi$
$421$	−8108.04			−0.938626			−0.469313	+	0.883032i	$0.655498\pi$
$422$		0			0
$423$	−1632.33	+	1684.52i	−0.187628	+	0.193627i
$424$		0			0
$425$	8691.24			0.991969
$426$		0			0
$427$		0			0
$428$		0			0
$429$	2003.66	+	4731.64i	0.225495	+	0.532507i
$430$		0			0
$431$		−	450.719i		−	0.0503721i	−0.999683	−	0.0251860i	$-0.991982\pi$
$431$		−	450.719i		−	0.0503721i	0.999683	−	0.0251860i	$-0.00801782\pi$
$432$		0			0
$433$		−	8133.13i		−	0.902664i	−0.892356	−	0.451332i	$-0.850949\pi$
$433$		−	8133.13i		−	0.902664i	0.892356	−	0.451332i	$-0.149051\pi$
$434$		0			0
$435$	944.219	−	399.838i	0.104073	−	0.0440708i
$436$		0			0
$437$	5533.73			0.605753
$438$		0			0
$439$		−	9256.43i		−	1.00634i	−0.864186	−	0.503172i	$-0.832166\pi$
$439$		−	9256.43i		−	1.00634i	0.864186	−	0.503172i	$-0.167834\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			10224.1i			1.09652i	0.836306	+	0.548262i	$0.184710\pi$
$443$			10224.1i			1.09652i	−0.836306	+	0.548262i	$0.815290\pi$
$444$		0			0
$445$	3507.84			0.373679
$446$		0			0
$447$	−9415.78	+	3987.20i	−0.996311	+	0.421897i
$448$		0			0
$449$			2896.50i			0.304442i	0.988346	+	0.152221i	$0.0486425\pi$
$449$			2896.50i			0.304442i	−0.988346	+	0.152221i	$0.951357\pi$
$450$		0			0
$451$		−	15641.4i		−	1.63309i
$452$		0			0
$453$	−4402.51	−	10396.5i	−0.456618	−	1.07830i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7990.29			−0.817878			−0.408939	−	0.912562i	$-0.634101\pi$
$457$	−7990.29			−0.817878			−0.408939	+	0.912562i	$0.634101\pi$
$458$		0			0
$459$	16097.8	+	6227.11i	1.63700	+	0.633239i
$460$		0			0
$461$	4639.50			0.468727			0.234363	−	0.972149i	$-0.424699\pi$
$461$	4639.50			0.468727			0.234363	+	0.972149i	$0.424699\pi$
$462$		0			0
$463$	4926.63			0.494514			0.247257	−	0.968950i	$-0.420471\pi$
$463$	4926.63			0.494514			0.247257	+	0.968950i	$0.420471\pi$
$464$		0			0
$465$	20706.6	−	8768.42i	2.06505	−	0.874464i
$466$		0			0
$467$	6809.28			0.674724			0.337362	−	0.941375i	$-0.390465\pi$
$467$	6809.28			0.674724			0.337362	+	0.941375i	$0.390465\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−15785.9	+	6684.68i	−1.54432	+	0.653957i
$472$		0			0
$473$			434.092i			0.0421979i
$474$		0			0
$475$		−	5119.98i		−	0.494570i
$476$		0			0
$477$	−8709.26	−	8439.45i	−0.835995	−	0.810097i
$478$		0			0
$479$	−2232.91			−0.212994			−0.106497	−	0.994313i	$-0.533963\pi$
$479$	−2232.91			−0.212994			−0.106497	+	0.994313i	$0.533963\pi$
$480$		0			0
$481$		−	3630.48i		−	0.344149i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			14816.4i			1.38717i
$486$		0			0
$487$	−121.678			−0.0113219			−0.00566095	−	0.999984i	$-0.501802\pi$
$487$	−121.678			−0.0113219			−0.00566095	+	0.999984i	$0.501802\pi$
$488$		0			0
$489$	−5671.82	−	13394.0i	−0.524516	−	1.23865i
$490$		0			0
$491$			9638.92i			0.885944i	0.896535	+	0.442972i	$0.146076\pi$
$491$			9638.92i			0.885944i	−0.896535	+	0.442972i	$0.853924\pi$
$492$		0			0
$493$		−	1735.70i		−	0.158564i
$494$		0			0
$495$	8622.57	+	8355.45i	0.782941	+	0.758686i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−3751.14			−0.336521			−0.168261	−	0.985743i	$-0.553815\pi$
$499$	−3751.14			−0.336521			−0.168261	+	0.985743i	$0.553815\pi$
$500$		0			0
$501$	3917.62	+	9251.47i	0.349354	+	0.825000i
$502$		0			0
$503$	13881.9			1.23054			0.615272	−	0.788315i	$-0.289046\pi$
$503$	13881.9			1.23054			0.615272	+	0.788315i	$0.289046\pi$
$504$		0			0
$505$	−21375.9			−1.88359
$506$		0			0
$507$	−2491.27	−	5883.13i	−0.218227	−	0.515343i
$508$		0			0
$509$	−6193.26			−0.539316			−0.269658	−	0.962956i	$-0.586911\pi$
$509$	−6193.26			−0.539316			−0.269658	+	0.962956i	$0.586911\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	3668.37	−	9483.19i	0.315717	−	0.816166i
$514$		0			0
$515$		−	2660.72i		−	0.227661i
$516$		0			0
$517$			2762.03i			0.234959i
$518$		0			0
$519$	−7101.78	−	16770.9i	−0.600643	−	1.41842i
$520$		0			0
$521$	−4513.23			−0.379517			−0.189758	−	0.981831i	$-0.560770\pi$
$521$	−4513.23			−0.379517			−0.189758	+	0.981831i	$0.560770\pi$
$522$		0			0
$523$			621.022i			0.0519224i	0.999663	+	0.0259612i	$0.00826463\pi$
$523$			621.022i			0.0519224i	−0.999663	+	0.0259612i	$0.991735\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	38063.7i		−	3.14627i
$528$		0			0
$529$	6337.15			0.520847
$530$		0			0
$531$	4824.75	−	4979.00i	0.394306	−	0.406912i
$532$		0			0
$533$		−	15302.6i		−	1.24358i
$534$		0			0
$535$		−	2488.09i		−	0.201064i
$536$		0			0
$537$	−8834.01	+	3740.85i	−0.709899	+	0.300613i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−10203.6			−0.810882			−0.405441	−	0.914121i	$-0.632882\pi$
$541$	−10203.6			−0.810882			−0.405441	+	0.914121i	$0.632882\pi$
$542$		0			0
$543$	−1767.53	+	748.476i	−0.139690	+	0.0591532i
$544$		0			0
$545$	−2700.10			−0.212219
$546$		0			0
$547$	−6359.52			−0.497100			−0.248550	−	0.968619i	$-0.579954\pi$
$547$	−6359.52			−0.497100			−0.248550	+	0.968619i	$0.579954\pi$
$548$		0			0
$549$	−5328.93	−	5163.84i	−0.414268	−	0.401434i
$550$		0			0
$551$	−1022.49			−0.0790558
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−3307.95	−	7811.73i	−0.252999	−	0.597458i
$556$		0			0
$557$		−	21829.6i		−	1.66060i	−0.557320	−	0.830298i	$-0.688170\pi$
$557$		−	21829.6i		−	1.66060i	0.557320	−	0.830298i	$-0.311830\pi$
$558$		0			0
$559$			424.689i			0.0321332i
$560$		0			0
$561$	18715.3	−	7925.17i	1.40849	−	0.596436i
$562$		0			0
$563$	624.001			0.0467114			0.0233557	−	0.999727i	$-0.492565\pi$
$563$	624.001			0.0467114			0.0233557	+	0.999727i	$0.492565\pi$
$564$		0			0
$565$			6323.36i			0.470842i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	14085.5i		−	1.03777i	−0.854843	−	0.518887i	$-0.826346\pi$
$569$		−	14085.5i		−	1.03777i	0.854843	−	0.518887i	$-0.173654\pi$
$570$		0			0
$571$	8211.01			0.601786			0.300893	−	0.953658i	$-0.402715\pi$
$571$	8211.01			0.601786			0.300893	+	0.953658i	$0.402715\pi$
$572$		0			0
$573$	1142.08	−	483.625i	0.0832654	−	0.0352595i
$574$		0			0
$575$			5393.96i			0.391207i
$576$		0			0
$577$		−	11141.8i		−	0.803880i	−0.915666	−	0.401940i	$-0.868336\pi$
$577$		−	11141.8i		−	0.803880i	0.915666	−	0.401940i	$-0.131664\pi$
$578$		0			0
$579$	3408.24	+	8048.55i	0.244631	+	0.577697i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−14280.2			−1.01445
$584$		0			0
$585$	8435.79	+	8174.46i	0.596200	+	0.577730i
$586$		0			0
$587$	5198.19			0.365507			0.182753	−	0.983159i	$-0.441499\pi$
$587$	5198.19			0.365507			0.182753	+	0.983159i	$0.441499\pi$
$588$		0			0
$589$	−22423.2			−1.56865
$590$		0			0
$591$	−14297.1	+	6054.26i	−0.995102	+	0.421385i
$592$		0			0
$593$	10970.3			0.759690			0.379845	−	0.925050i	$-0.375977\pi$
$593$	10970.3			0.759690			0.379845	+	0.925050i	$0.375977\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	9652.14	−	4087.29i	0.661701	−	0.280204i
$598$		0			0
$599$			25818.0i			1.76109i	0.473958	+	0.880547i	$0.342825\pi$
$599$			25818.0i			1.76109i	−0.473958	+	0.880547i	$0.657175\pi$
$600$		0			0
$601$		−	11968.5i		−	0.812323i	−0.913801	−	0.406162i	$-0.866867\pi$
$601$		−	11968.5i		−	0.812323i	0.913801	−	0.406162i	$-0.133133\pi$
$602$		0			0
$603$	−19389.8	+	20009.7i	−1.30948	+	1.35134i
$604$		0			0
$605$	−4479.04			−0.300990
$606$		0			0
$607$		−	6250.07i		−	0.417929i	−0.977923	−	0.208964i	$-0.932991\pi$
$607$		−	6250.07i		−	0.417929i	0.977923	−	0.208964i	$-0.0670092\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			2702.20i			0.178919i
$612$		0			0
$613$	4939.66			0.325467			0.162733	−	0.986670i	$-0.447969\pi$
$613$	4939.66			0.325467			0.162733	+	0.986670i	$0.447969\pi$
$614$		0			0
$615$	−13943.1	−	32926.7i	−0.914212	−	2.15891i
$616$		0			0
$617$		−	14836.2i		−	0.968042i	−0.875056	−	0.484021i	$-0.839176\pi$
$617$		−	14836.2i		−	0.968042i	0.875056	−	0.484021i	$-0.160824\pi$
$618$		0			0
$619$			12164.1i			0.789848i	0.918714	+	0.394924i	$0.129229\pi$
$619$			12164.1i			0.789848i	−0.918714	+	0.394924i	$0.870771\pi$
$620$		0			0
$621$	−3864.68	+	9990.66i	−0.249733	+	0.645590i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−19464.9			−1.24575
$626$		0			0
$627$	−4668.69	−	11025.1i	−0.297368	−	0.702234i
$628$		0			0
$629$	−14359.8			−0.910276
$630$		0			0
$631$	8372.13			0.528192			0.264096	−	0.964496i	$-0.414926\pi$
$631$	8372.13			0.528192			0.264096	+	0.964496i	$0.414926\pi$
$632$		0			0
$633$	−9492.68	−	22417.0i	−0.596051	−	1.40757i
$634$		0			0
$635$	−10205.5			−0.637787
$636$		0			0
$637$		0			0
$638$		0			0
$639$	18066.8	+	17507.1i	1.11848	+	1.08383i
$640$		0			0
$641$		−	14979.4i		−	0.923011i	−0.887137	−	0.461505i	$-0.847309\pi$
$641$		−	14979.4i		−	0.923011i	0.887137	−	0.461505i	$-0.152691\pi$
$642$		0			0
$643$			4861.62i			0.298170i	0.988824	+	0.149085i	$0.0476329\pi$
$643$			4861.62i			0.298170i	−0.988824	+	0.149085i	$0.952367\pi$
$644$		0			0
$645$	386.960	+	913.807i	0.0236226	+	0.0557847i
$646$		0			0
$647$	27184.9			1.65185			0.825925	−	0.563780i	$-0.190653\pi$
$647$	27184.9			1.65185			0.825925	+	0.563780i	$0.190653\pi$
$648$		0			0
$649$		−	8163.85i		−	0.493774i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			4336.82i			0.259897i	0.991521	+	0.129949i	$0.0414812\pi$
$653$			4336.82i			0.259897i	−0.991521	+	0.129949i	$0.958519\pi$
$654$		0			0
$655$	8034.75			0.479303
$656$		0			0
$657$	19875.5	+	19259.8i	1.18024	+	1.14368i
$658$		0			0
$659$		−	15562.8i		−	0.919939i	−0.887935	−	0.459969i	$-0.847860\pi$
$659$		−	15562.8i		−	0.919939i	0.887935	−	0.459969i	$-0.152140\pi$
$660$		0			0
$661$			14164.7i			0.833498i	0.909022	+	0.416749i	$0.136831\pi$
$661$			14164.7i			0.833498i	−0.909022	+	0.416749i	$0.863169\pi$
$662$		0			0
$663$	18309.9	−	7753.50i	1.07254	−	0.454179i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1077.21			0.0625334
$668$		0			0
$669$	−82.6856	+	35.0140i	−0.00477849	+	0.00202350i
$670$		0			0
$671$	−8737.62			−0.502700
$672$		0			0
$673$	21045.6			1.20542			0.602710	−	0.797960i	$-0.294087\pi$
$673$	21045.6			1.20542			0.602710	+	0.797960i	$0.294087\pi$
$674$		0			0
$675$	9243.68	+	3575.72i	0.527095	+	0.203896i
$676$		0			0
$677$	−18074.2			−1.02607			−0.513034	−	0.858368i	$-0.671479\pi$
$677$	−18074.2			−1.02607			−0.513034	+	0.858368i	$0.671479\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	2966.08	+	7004.39i	0.166902	+	0.394139i
$682$		0			0
$683$			24743.0i			1.38618i	0.720849	+	0.693092i	$0.243752\pi$
$683$			24743.0i			1.38618i	−0.720849	+	0.693092i	$0.756248\pi$
$684$		0			0
$685$			33957.8i			1.89410i
$686$		0			0
$687$	555.388	−	235.184i	0.0308433	−	0.0130609i
$688$		0			0
$689$	−13970.9			−0.772494
$690$		0			0
$691$		−	23314.5i		−	1.28354i	−0.766898	−	0.641769i	$-0.778201\pi$
$691$		−	23314.5i		−	1.28354i	0.766898	−	0.641769i	$-0.221799\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			20332.3i			1.10971i
$696$		0			0
$697$	−60527.0			−3.28928
$698$		0			0
$699$	17945.9	−	7599.38i	0.971070	−	0.411209i
$700$		0			0
$701$			901.167i			0.0485544i	0.999705	+	0.0242772i	$0.00772843\pi$
$701$			901.167i			0.0485544i	−0.999705	+	0.0242772i	$0.992272\pi$
$702$		0			0
$703$			8459.32i			0.453840i
$704$		0			0
$705$	2462.14	+	5814.35i	0.131531	+	0.310612i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−5835.30			−0.309096			−0.154548	−	0.987985i	$-0.549392\pi$
$709$	−5835.30			−0.309096			−0.154548	+	0.987985i	$0.549392\pi$
$710$		0			0
$711$	17372.9	−	17928.3i	0.916362	−	0.945658i
$712$		0			0
$713$	23623.1			1.24080
$714$		0			0
$715$	13831.8			0.723469
$716$		0			0
$717$	25950.4	−	10988.9i	1.35165	−	0.572371i
$718$		0			0
$719$	18468.8			0.957957			0.478978	−	0.877827i	$-0.341007\pi$
$719$	18468.8			0.957957			0.478978	+	0.877827i	$0.341007\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	21734.3	−	9203.58i	1.11799	−	0.473423i
$724$		0			0
$725$		−	996.670i		−	0.0510557i
$726$		0			0
$727$			32300.3i			1.64780i	0.566735	+	0.823900i	$0.308206\pi$
$727$			32300.3i			1.64780i	−0.566735	+	0.823900i	$0.691794\pi$
$728$		0			0
$729$	14559.1	+	13245.9i	0.739680	+	0.672959i
$730$		0			0
$731$	1679.80			0.0849924
$732$		0			0
$733$			5992.12i			0.301943i	0.988538	+	0.150971i	$0.0482401\pi$
$733$			5992.12i			0.301943i	−0.988538	+	0.150971i	$0.951760\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			32809.1i			1.63981i
$738$		0			0
$739$	−1307.80			−0.0650988			−0.0325494	−	0.999470i	$-0.510363\pi$
$739$	−1307.80			−0.0650988			−0.0325494	+	0.999470i	$0.510363\pi$
$740$		0			0
$741$	−4567.56	−	10786.3i	−0.226442	−	0.534743i
$742$		0			0
$743$			30175.4i			1.48995i	0.667095	+	0.744973i	$0.267538\pi$
$743$			30175.4i			1.48995i	−0.667095	+	0.744973i	$0.732462\pi$
$744$		0			0
$745$			27524.8i			1.35360i
$746$		0			0
$747$	−18633.2	+	19228.9i	−0.912655	+	0.941832i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1072.71			0.0521219			0.0260610	−	0.999660i	$-0.491704\pi$
$751$	1072.71			0.0521219			0.0260610	+	0.999660i	$0.491704\pi$
$752$		0			0
$753$	−2260.47	−	5338.09i	−0.109397	−	0.258341i
$754$		0			0
$755$	−30391.7			−1.46499
$756$		0			0
$757$	−7773.72			−0.373237			−0.186619	−	0.982432i	$-0.559753\pi$
$757$	−7773.72			−0.373237			−0.186619	+	0.982432i	$0.559753\pi$
$758$		0			0
$759$	4918.53	+	11615.1i	0.235219	+	0.555470i
$760$		0			0
$761$	−20863.6			−0.993832			−0.496916	−	0.867799i	$-0.665534\pi$
$761$	−20863.6			−0.993832			−0.496916	+	0.867799i	$0.665534\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	32332.9	−	33366.5i	1.52810	−	1.57695i
$766$		0			0
$767$		−	7987.01i		−	0.376003i
$768$		0			0
$769$			1423.15i			0.0667364i	0.999443	+	0.0333682i	$0.0106234\pi$
$769$			1423.15i			0.0667364i	−0.999443	+	0.0333682i	$0.989377\pi$
$770$		0			0
$771$	12925.6	+	30523.8i	0.603767	+	1.42580i
$772$		0			0
$773$	−5328.29			−0.247924			−0.123962	−	0.992287i	$-0.539560\pi$
$773$	−5328.29			−0.247924			−0.123962	+	0.992287i	$0.539560\pi$
$774$		0			0
$775$		−	21856.9i		−	1.01306i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			35656.3i			1.63995i
$780$		0			0
$781$	29623.3			1.35724
$782$		0			0
$783$	714.096	−	1846.02i	0.0325922	−	0.0842548i
$784$		0			0
$785$			46146.2i			2.09813i
$786$		0			0
$787$		−	11526.7i		−	0.522088i	−0.965327	−	0.261044i	$-0.915933\pi$
$787$		−	11526.7i		−	0.522088i	0.965327	−	0.261044i	$-0.0840668\pi$
$788$		0			0
$789$	−3450.93	+	1461.33i	−0.155711	+	0.0659374i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−8548.35			−0.382800
$794$		0			0
$795$	−30061.2	+	12729.7i	−1.34109	+	0.567895i
$796$		0			0
$797$	−42557.4			−1.89142			−0.945709	−	0.325013i	$-0.894631\pi$
$797$	−42557.4			−1.89142			−0.945709	+	0.325013i	$0.894631\pi$
$798$		0			0
$799$	10688.2			0.473242
$800$		0			0
$801$	4712.08	−	4862.73i	0.207857	−	0.214502i
$802$		0			0
$803$	32589.0			1.43218
$804$		0			0
$805$		0			0
$806$		0			0
$807$	9617.57	+	22711.9i	0.419522	+	0.990702i
$808$		0			0
$809$			19587.8i			0.851261i	0.904897	+	0.425630i	$0.139948\pi$
$809$			19587.8i			0.851261i	−0.904897	+	0.425630i	$0.860052\pi$
$810$		0			0
$811$			26034.2i			1.12723i	0.826038	+	0.563615i	$0.190590\pi$
$811$			26034.2i			1.12723i	−0.826038	+	0.563615i	$0.809410\pi$
$812$		0			0
$813$	39363.8	−	16669.0i	1.69809	−	0.719074i
$814$		0			0
$815$	−39154.2			−1.68284
$816$		0			0
$817$		−	989.562i		−	0.0423750i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	29114.8i		−	1.23765i	−0.785528	−	0.618827i	$-0.787608\pi$
$821$		−	29114.8i		−	1.23765i	0.785528	−	0.618827i	$-0.212392\pi$
$822$		0			0
$823$	−29582.4			−1.25295			−0.626475	−	0.779442i	$-0.715503\pi$
$823$	−29582.4			−1.25295			−0.626475	+	0.779442i	$0.715503\pi$
$824$		0			0
$825$	10746.7	−	4550.78i	0.453516	−	0.192046i
$826$		0			0
$827$		−	33550.7i		−	1.41073i	−0.708845	−	0.705364i	$-0.750784\pi$
$827$		−	33550.7i		−	1.41073i	0.708845	−	0.705364i	$-0.249216\pi$
$828$		0			0
$829$			47159.6i			1.97578i	0.155156	+	0.987890i	$0.450412\pi$
$829$			47159.6i			1.97578i	−0.155156	+	0.987890i	$0.549588\pi$
$830$		0			0
$831$	10341.1	+	24420.6i	0.431685	+	1.01942i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	27044.5			1.12085
$836$		0			0
$837$	15660.1	−	40483.2i	0.646704	−	1.67181i
$838$		0			0
$839$	4790.07			0.197106			0.0985528	−	0.995132i	$-0.468579\pi$
$839$	4790.07			0.197106			0.0985528	+	0.995132i	$0.468579\pi$
$840$		0			0
$841$	24190.0			0.991839
$842$		0			0
$843$	−32625.4	+	13815.5i	−1.33295	+	0.564451i
$844$		0			0
$845$	−17197.9			−0.700149
$846$		0			0
$847$		0			0
$848$		0			0
$849$	12805.2	−	5422.48i	0.517636	−	0.219198i
$850$		0			0
$851$		−	8912.00i		−	0.358989i
$852$		0			0
$853$			4925.15i			0.197695i	0.995103	+	0.0988475i	$0.0315156\pi$
$853$			4925.15i			0.197695i	−0.995103	+	0.0988475i	$0.968484\pi$
$854$		0			0
$855$	−19656.1	−	19047.2i	−0.786228	−	0.761871i
$856$		0			0
$857$	−4931.78			−0.196577			−0.0982885	−	0.995158i	$-0.531337\pi$
$857$	−4931.78			−0.196577			−0.0982885	+	0.995158i	$0.531337\pi$
$858$		0			0
$859$		−	35166.4i		−	1.39681i	−0.715701	−	0.698407i	$-0.753893\pi$
$859$		−	35166.4i		−	1.39681i	0.715701	−	0.698407i	$-0.246107\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	31267.6i		−	1.23333i	−0.787227	−	0.616664i	$-0.788484\pi$
$863$		−	31267.6i		−	1.23333i	0.787227	−	0.616664i	$-0.211516\pi$
$864$		0			0
$865$	−49025.6			−1.92708
$866$		0			0
$867$	−20713.2	−	48914.2i	−0.811369	−	1.91605i
$868$		0			0
$869$		−	29396.2i		−	1.14752i
$870$		0			0
$871$			32098.4i			1.24869i
$872$		0			0
$873$	20539.2	+	19902.9i	0.796273	+	0.771605i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	12797.0			0.492731			0.246366	−	0.969177i	$-0.420764\pi$
$877$	12797.0			0.492731			0.246366	+	0.969177i	$0.420764\pi$
$878$		0			0
$879$	3533.06	+	8343.33i	0.135571	+	0.320152i
$880$		0			0
$881$	2497.90			0.0955235			0.0477618	−	0.998859i	$-0.484791\pi$
$881$	2497.90			0.0955235			0.0477618	+	0.998859i	$0.484791\pi$
$882$		0			0
$883$	−8139.20			−0.310199			−0.155100	−	0.987899i	$-0.549570\pi$
$883$	−8139.20			−0.310199			−0.155100	+	0.987899i	$0.549570\pi$
$884$		0			0
$885$	−7277.45	−	17185.7i	−0.276417	−	0.652759i
$886$		0			0
$887$	5856.99			0.221712			0.110856	−	0.993836i	$-0.464641\pi$
$887$	5856.99			0.221712			0.110856	+	0.993836i	$0.464641\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	23165.4	−	729.122i	0.871011	−	0.0274147i
$892$		0			0
$893$		−	6296.36i		−	0.235946i
$894$		0			0
$895$			25824.1i			0.964475i
$896$		0			0
$897$	4811.98	+	11363.5i	0.179116	+	0.422984i
$898$		0			0
$899$	−4364.97			−0.161935
$900$		0			0
$901$			55259.7i			2.04325i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			5166.94i			0.189784i
$906$		0			0
$907$	−22542.4			−0.825258			−0.412629	−	0.910899i	$-0.635389\pi$
$907$	−22542.4			−0.825258			−0.412629	+	0.910899i	$0.635389\pi$
$908$		0			0
$909$	−28714.2	+	29632.2i	−1.04773	+	1.08123i
$910$		0			0
$911$		−	14306.2i		−	0.520291i	−0.965569	−	0.260146i	$-0.916229\pi$
$911$		−	14306.2i		−	0.520291i	0.965569	−	0.260146i	$-0.0837706\pi$
$912$		0			0
$913$			31528.8i			1.14288i
$914$		0			0
$915$	−18393.5	+	7788.92i	−0.664560	+	0.281414i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	20971.7			0.752767			0.376384	−	0.926464i	$-0.377167\pi$
$919$	20971.7			0.752767			0.376384	+	0.926464i	$0.377167\pi$
$920$		0			0
$921$	−7192.41	+	3045.70i	−0.257327	+	0.108968i
$922$		0			0
$923$	28981.6			1.03352
$924$		0			0
$925$	−8245.67			−0.293098
$926$		0			0
$927$	−3688.42	−	3574.15i	−0.130683	−	0.126635i
$928$		0			0
$929$	21866.6			0.772251			0.386125	−	0.922446i	$-0.373813\pi$
$929$	21866.6			0.772251			0.386125	+	0.922446i	$0.373813\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	17711.8	+	41826.3i	0.621497	+	1.46767i
$934$		0			0
$935$		−	54709.7i		−	1.91358i
$936$		0			0
$937$		−	32284.1i		−	1.12559i	−0.826597	−	0.562794i	$-0.809726\pi$
$937$		−	32284.1i		−	1.12559i	0.826597	−	0.562794i	$-0.190274\pi$
$938$		0			0
$939$	−17851.0	+	7559.17i	−0.620388	+	0.262709i
$940$		0			0
$941$	23799.4			0.824484			0.412242	−	0.911074i	$-0.364746\pi$
$941$	23799.4			0.824484			0.412242	+	0.911074i	$0.364746\pi$
$942$		0			0
$943$		−	37564.3i		−	1.29720i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	9756.32i		−	0.334781i	−0.985891	−	0.167391i	$-0.946466\pi$
$947$		−	9756.32i		−	0.334781i	0.985891	−	0.167391i	$-0.0535341\pi$
$948$		0			0
$949$	31883.1			1.09059
$950$		0			0
$951$	23829.3	−	10090.7i	0.812531	−	0.344074i
$952$		0			0
$953$			36031.4i			1.22473i	0.790574	+	0.612367i	$0.209782\pi$
$953$			36031.4i			1.22473i	−0.790574	+	0.612367i	$0.790218\pi$
$954$		0			0
$955$		−	3338.60i		−	0.113125i
$956$		0			0
$957$	−908.821	−	2146.18i	−0.0306980	−	0.0724934i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−65932.4			−2.21317
$962$		0			0
$963$	−3449.10	−	3342.25i	−0.115416	−	0.111841i
$964$		0			0
$965$	23528.0			0.784864
$966$		0			0
$967$	−24872.5			−0.827142			−0.413571	−	0.910472i	$-0.635719\pi$
$967$	−24872.5			−0.827142			−0.413571	+	0.910472i	$0.635719\pi$
$968$		0			0
$969$	−42663.6	+	18066.3i	−1.41440	+	0.598941i
$970$		0			0
$971$	−25012.8			−0.826674			−0.413337	−	0.910578i	$-0.635637\pi$
$971$	−25012.8			−0.826674			−0.413337	+	0.910578i	$0.635637\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	10513.9	−	4452.20i	0.345347	−	0.146241i
$976$		0			0
$977$			2629.12i			0.0860931i	0.999073	+	0.0430465i	$0.0137064\pi$
$977$			2629.12i			0.0860931i	−0.999073	+	0.0430465i	$0.986294\pi$
$978$		0			0
$979$		−	7973.21i		−	0.260291i
$980$		0			0
$981$	−3627.05	+	3743.00i	−0.118046	+	0.121819i
$982$		0			0
$983$	−1613.81			−0.0523626			−0.0261813	−	0.999657i	$-0.508335\pi$
$983$	−1613.81			−0.0523626			−0.0261813	+	0.999657i	$0.508335\pi$
$984$		0			0
$985$			41794.2i			1.35195i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1042.52i			0.0335188i
$990$		0			0
$991$	−33326.9			−1.06828			−0.534140	−	0.845396i	$-0.679364\pi$
$991$	−33326.9			−1.06828			−0.534140	+	0.845396i	$0.679364\pi$
$992$		0			0
$993$	−12551.3	−	29639.8i	−0.401110	−	0.947221i
$994$		0			0
$995$		−	28215.7i		−	0.898993i
$996$		0			0
$997$			47449.9i			1.50727i	0.657291	+	0.753637i	$0.271702\pi$
$997$			47449.9i			1.50727i	−0.657291	+	0.753637i	$0.728298\pi$
$998$		0			0
$999$	−15272.6	−	5907.87i	−0.483686	−	0.187104i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.4.f.d.293.9	✓	24
3.2	odd	2	inner	588.4.f.d.293.15	yes	24
7.2	even	3		588.4.k.e.521.23		48
7.3	odd	6		588.4.k.e.509.17		48
7.4	even	3		588.4.k.e.509.8		48
7.5	odd	6		588.4.k.e.521.2		48
7.6	odd	2	inner	588.4.f.d.293.16	yes	24
21.2	odd	6		588.4.k.e.521.17		48
21.5	even	6		588.4.k.e.521.8		48
21.11	odd	6		588.4.k.e.509.2		48
21.17	even	6		588.4.k.e.509.23		48
21.20	even	2	inner	588.4.f.d.293.10	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.4.f.d.293.9	✓	24	1.1	even	1	trivial
588.4.f.d.293.10	yes	24	21.20	even	2	inner
588.4.f.d.293.15	yes	24	3.2	odd	2	inner
588.4.f.d.293.16	yes	24	7.6	odd	2	inner
588.4.k.e.509.2		48	21.11	odd	6
588.4.k.e.509.8		48	7.4	even	3
588.4.k.e.509.17		48	7.3	odd	6
588.4.k.e.509.23		48	21.17	even	6
588.4.k.e.521.2		48	7.5	odd	6
588.4.k.e.521.8		48	21.5	even	6
588.4.k.e.521.17		48	21.2	odd	6
588.4.k.e.521.23		48	7.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	588.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.02618 - 4.78483i) q^{3} -13.9873 q^{5} +(-18.7892 + 19.3899i) q^{9} +O(q^{10})\)	\(q+(-2.02618 - 4.78483i) q^{3} -13.9873 q^{5} +(-18.7892 + 19.3899i) q^{9} +31.7927i q^{11} +31.1041i q^{13} +(28.3408 + 66.9269i) q^{15} +123.028 q^{17} -72.4751i q^{19} +76.3535i q^{23} +70.6447 q^{25} +(130.847 + 50.6156i) q^{27} -14.1082i q^{29} -309.392i q^{31} +(152.123 - 64.4178i) q^{33} -116.720 q^{37} +(148.828 - 63.0225i) q^{39} -491.980 q^{41} +13.6538 q^{43} +(262.810 - 271.212i) q^{45} +86.8762 q^{47} +(-249.276 - 588.666i) q^{51} +449.166i q^{53} -444.695i q^{55} +(-346.781 + 146.848i) q^{57} -256.783 q^{59} +274.831i q^{61} -435.062i q^{65} +1031.97 q^{67} +(365.338 - 154.706i) q^{69} -931.763i q^{71} -1025.05i q^{73} +(-143.139 - 338.023i) q^{75} -924.621 q^{79} +(-22.9336 - 728.639i) q^{81} +991.698 q^{83} -1720.82 q^{85} +(-67.5054 + 28.5858i) q^{87} -250.787 q^{89} +(-1480.39 + 626.884i) q^{93} +1013.73i q^{95} -1059.27i q^{97} +(-616.457 - 597.359i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 64 q^{9}+O(q^{10}) \) 24 * q - 64 * q^9	\( 24 q - 64 q^{9} - 96 q^{15} + 456 q^{25} - 432 q^{37} + 688 q^{39} + 624 q^{43} - 1536 q^{51} - 1360 q^{57} - 528 q^{67} + 3744 q^{79} + 3408 q^{81} + 6912 q^{85} - 5088 q^{93} - 7736 q^{99}+O(q^{100}) \) 24 * q - 64 * q^9 - 96 * q^15 + 456 * q^25 - 432 * q^37 + 688 * q^39 + 624 * q^43 - 1536 * q^51 - 1360 * q^57 - 528 * q^67 + 3744 * q^79 + 3408 * q^81 + 6912 * q^85 - 5088 * q^93 - 7736 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more