LMFDB - Embedded newform 338.8.b.d.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,8,Mod(337,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	338.b (of order $2$, degree $1$, not 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:	$105.586138614$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	338.337
Dual form			338.8.b.d.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.00000i q^{2} +12.0000 q^{3} -64.0000 q^{4} +210.000i q^{5} +96.0000i q^{6} +1016.00i q^{7} -512.000i q^{8} -2043.00 q^{9} -1680.00 q^{10} +1092.00i q^{11} -768.000 q^{12} -8128.00 q^{14} +2520.00i q^{15} +4096.00 q^{16} -14706.0 q^{17} -16344.0i q^{18} +39940.0i q^{19} -13440.0i q^{20} +12192.0i q^{21} -8736.00 q^{22} -68712.0 q^{23} -6144.00i q^{24} +34025.0 q^{25} -50760.0 q^{27} -65024.0i q^{28} -102570. q^{29} -20160.0 q^{30} -227552. i q^{31} +32768.0i q^{32} +13104.0i q^{33} -117648. i q^{34} -213360. q^{35} +130752. q^{36} +160526. i q^{37} -319520. q^{38} +107520. q^{40} -10842.0i q^{41} -97536.0 q^{42} +630748. q^{43} -69888.0i q^{44} -429030. i q^{45} -549696. i q^{46} +472656. i q^{47} +49152.0 q^{48} -208713. q^{49} +272200. i q^{50} -176472. q^{51} -1.49402e6 q^{53} -406080. i q^{54} -229320. q^{55} +520192. q^{56} +479280. i q^{57} -820560. i q^{58} +2.64066e6i q^{59} -161280. i q^{60} +827702. q^{61} +1.82042e6 q^{62} -2.07569e6i q^{63} -262144. q^{64} -104832. q^{66} +126004. i q^{67} +941184. q^{68} -824544. q^{69} -1.70688e6i q^{70} +1.41473e6i q^{71} +1.04602e6i q^{72} +980282. i q^{73} -1.28421e6 q^{74} +408300. q^{75} -2.55616e6i q^{76} -1.10947e6 q^{77} -3.56680e6 q^{79} +860160. i q^{80} +3.85892e6 q^{81} +86736.0 q^{82} -5.67289e6i q^{83} -780288. i q^{84} -3.08826e6i q^{85} +5.04598e6i q^{86} -1.23084e6 q^{87} +559104. q^{88} -1.19512e7i q^{89} +3.43224e6 q^{90} +4.39757e6 q^{92} -2.73062e6i q^{93} -3.78125e6 q^{94} -8.38740e6 q^{95} +393216. i q^{96} -8.68215e6i q^{97} -1.66970e6i q^{98} -2.23096e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 24 q^{3} - 128 q^{4} - 4086 q^{9} - 3360 q^{10} - 1536 q^{12} - 16256 q^{14} + 8192 q^{16} - 29412 q^{17} - 17472 q^{22} - 137424 q^{23} + 68050 q^{25} - 101520 q^{27} - 205140 q^{29} - 40320 q^{30}+ \cdots - 16774800 q^{95}+O(q^{100}) $ 2 * q + 24 * q^3 - 128 * q^4 - 4086 * q^9 - 3360 * q^10 - 1536 * q^12 - 16256 * q^14 + 8192 * q^16 - 29412 * q^17 - 17472 * q^22 - 137424 * q^23 + 68050 * q^25 - 101520 * q^27 - 205140 * q^29 - 40320 * q^30 - 426720 * q^35 + 261504 * q^36 - 639040 * q^38 + 215040 * q^40 - 195072 * q^42 + 1261496 * q^43 + 98304 * q^48 - 417426 * q^49 - 352944 * q^51 - 2988036 * q^53 - 458640 * q^55 + 1040384 * q^56 + 1655404 * q^61 + 3640832 * q^62 - 524288 * q^64 - 209664 * q^66 + 1882368 * q^68 - 1649088 * q^69 - 2568416 * q^74 + 816600 * q^75 - 2218944 * q^77 - 7133600 * q^79 + 7717842 * q^81 + 173472 * q^82 - 2461680 * q^87 + 1118208 * q^88 + 6864480 * q^90 + 8795136 * q^92 - 7562496 * q^94 - 16774800 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$.

$n$	$171$
$\chi(n)$	$-1$

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 $

$2$	8.00000i	0.707107i
$2$	8.00000i	0.707107i
$3$	12.0000	0.256600	0.128300	−	0.991735i	$-0.459048\pi$
$3$	12.0000	0.256600	0.128300	+	0.991735i	$0.459048\pi$
$4$	−64.0000	−0.500000
$5$	210.000i	0.751319i	0.926758	+	0.375659i	$0.122584\pi$
$5$	210.000i	0.751319i	−0.926758	+	0.375659i	$0.877416\pi$
$6$	96.0000i	0.181444i
$7$	1016.00i	1.11957i	0.828638	+	0.559784i	$0.189116\pi$
$7$	1016.00i	1.11957i	−0.828638	+	0.559784i	$0.810884\pi$
$8$	− 512.000i	− 0.353553i
$9$	−2043.00	−0.934156
$10$	−1680.00	−0.531263
$11$	1092.00i	0.247371i	0.992321	+	0.123685i	$0.0394713\pi$
$11$	1092.00i	0.247371i	−0.992321	+	0.123685i	$0.960529\pi$
$12$	−768.000	−0.128300
$13$	0	0
$13$	0	0
$14$	−8128.00	−0.791654
$15$	2520.00i	0.192789i
$16$	4096.00	0.250000
$17$	−14706.0	−0.725978	−0.362989	−	0.931793i	$-0.618244\pi$
$17$	−14706.0	−0.725978	−0.362989	+	0.931793i	$0.618244\pi$
$18$	− 16344.0i	− 0.660548i
$19$	39940.0i	1.33589i	0.744211	+	0.667945i	$0.232826\pi$
$19$	39940.0i	1.33589i	−0.744211	+	0.667945i	$0.767174\pi$
$20$	− 13440.0i	− 0.375659i
$21$	12192.0i	0.287281i
$22$	−8736.00	−0.174917
$23$	−68712.0	−1.17757	−0.588783	−	0.808291i	$-0.700393\pi$
$23$	−68712.0	−1.17757	−0.588783	+	0.808291i	$0.700393\pi$
$24$	− 6144.00i	− 0.0907218i
$25$	34025.0	0.435520
$26$	0	0
$27$	−50760.0	−0.496305
$28$	− 65024.0i	− 0.559784i
$29$	−102570.	−0.780957	−0.390479	−	0.920612i	$-0.627690\pi$
$29$	−102570.	−0.780957	−0.390479	+	0.920612i	$0.627690\pi$
$30$	−20160.0	−0.136322
$31$	− 227552.i	− 1.37188i	−0.727660	−	0.685938i	$-0.759392\pi$
$31$	− 227552.i	− 1.37188i	0.727660	−	0.685938i	$-0.240608\pi$
$32$	32768.0i	0.176777i
$33$	13104.0i	0.0634753i
$34$	− 117648.i	− 0.513344i
$35$	−213360.	−0.841153
$36$	130752.	0.467078
$37$	160526.i	0.521002i	0.965474	+	0.260501i	$0.0838877\pi$
$37$	160526.i	0.521002i	−0.965474	+	0.260501i	$0.916112\pi$
$38$	−319520.	−0.944616
$39$	0	0
$40$	107520.	0.265631
$41$	− 10842.0i	− 0.0245678i	−0.999925	−	0.0122839i	$-0.996090\pi$
$41$	− 10842.0i	− 0.0245678i	0.999925	−	0.0122839i	$-0.00391018\pi$
$42$	−97536.0	−0.203139
$43$	630748.	1.20981	0.604904	−	0.796299i	$-0.293212\pi$
$43$	630748.	1.20981	0.604904	+	0.796299i	$0.293212\pi$
$44$	− 69888.0i	− 0.123685i
$45$	− 429030.i	− 0.701849i
$46$	− 549696.i	− 0.832665i
$47$	472656.i	0.664053i	0.943270	+	0.332026i	$0.107732\pi$
$47$	472656.i	0.664053i	−0.943270	+	0.332026i	$0.892268\pi$
$48$	49152.0	0.0641500
$49$	−208713.	−0.253433
$50$	272200.i	0.307959i
$51$	−176472.	−0.186286
$52$	0	0
$53$	−1.49402e6	−1.37845	−0.689224	−	0.724548i	$-0.742048\pi$
$53$	−1.49402e6	−1.37845	−0.689224	+	0.724548i	$0.742048\pi$
$54$	− 406080.i	− 0.350940i
$55$	−229320.	−0.185854
$56$	520192.	0.395827
$57$	479280.i	0.342789i
$58$	− 820560.i	− 0.552220i
$59$	2.64066e6i	1.67390i	0.547277	+	0.836952i	$0.315665\pi$
$59$	2.64066e6i	1.67390i	−0.547277	+	0.836952i	$0.684335\pi$
$60$	− 161280.i	− 0.0963943i
$61$	827702.	0.466895	0.233448	−	0.972369i	$-0.424999\pi$
$61$	827702.	0.466895	0.233448	+	0.972369i	$0.424999\pi$
$62$	1.82042e6	0.970063
$63$	− 2.07569e6i	− 1.04585i
$64$	−262144.	−0.125000
$65$	0	0
$66$	−104832.	−0.0448838
$67$	126004.i	0.0511826i	0.999672	+	0.0255913i	$0.00814686\pi$
$67$	126004.i	0.0511826i	−0.999672	+	0.0255913i	$0.991853\pi$
$68$	941184.	0.362989
$69$	−824544.	−0.302164
$70$	− 1.70688e6i	− 0.594785i
$71$	1.41473e6i	0.469104i	0.972104	+	0.234552i	$0.0753622\pi$
$71$	1.41473e6i	0.469104i	−0.972104	+	0.234552i	$0.924638\pi$
$72$	1.04602e6i	0.330274i
$73$	980282.i	0.294931i	0.989067	+	0.147466i	$0.0471116\pi$
$73$	980282.i	0.294931i	−0.989067	+	0.147466i	$0.952888\pi$
$74$	−1.28421e6	−0.368404
$75$	408300.	0.111754
$76$	− 2.55616e6i	− 0.667945i
$77$	−1.10947e6	−0.276948
$78$	0	0
$79$	−3.56680e6	−0.813924	−0.406962	−	0.913445i	$-0.633412\pi$
$79$	−3.56680e6	−0.813924	−0.406962	+	0.913445i	$0.633412\pi$
$80$	860160.i	0.187830i
$81$	3.85892e6	0.806805
$82$	86736.0	0.0173720
$83$	− 5.67289e6i	− 1.08901i	−0.838758	−	0.544504i	$-0.816718\pi$
$83$	− 5.67289e6i	− 1.08901i	0.838758	−	0.544504i	$-0.183282\pi$
$84$	− 780288.i	− 0.143641i
$85$	− 3.08826e6i	− 0.545441i
$86$	5.04598e6i	0.855463i
$87$	−1.23084e6	−0.200394
$88$	559104.	0.0874587
$89$	− 1.19512e7i	− 1.79699i	−0.438982	−	0.898496i	$-0.644661\pi$
$89$	− 1.19512e7i	− 1.79699i	0.438982	−	0.898496i	$-0.355339\pi$
$90$	3.43224e6	0.496282
$91$	0	0
$92$	4.39757e6	0.588783
$93$	− 2.73062e6i	− 0.352023i
$94$	−3.78125e6	−0.469556
$95$	−8.38740e6	−1.00368
$96$	393216.i	0.0453609i
$97$	− 8.68215e6i	− 0.965886i	−0.875652	−	0.482943i	$-0.839568\pi$
$97$	− 8.68215e6i	− 0.965886i	0.875652	−	0.482943i	$-0.160432\pi$
$98$	− 1.66970e6i	− 0.179204i
$99$	− 2.23096e6i	− 0.231083i
$100$	−2.17760e6	−0.217760
$101$	1.00795e7	0.973455	0.486727	−	0.873554i	$-0.338190\pi$
$101$	1.00795e7	0.973455	0.486727	+	0.873554i	$0.338190\pi$
$102$	− 1.41178e6i	− 0.131724i
$103$	−3.74799e6	−0.337962	−0.168981	−	0.985619i	$-0.554048\pi$
$103$	−3.74799e6	−0.337962	−0.168981	+	0.985619i	$0.554048\pi$
$104$	0	0
$105$	−2.56032e6	−0.215840
$106$	− 1.19521e7i	− 0.974710i
$107$	−1.79856e7	−1.41932	−0.709661	−	0.704543i	$-0.751152\pi$
$107$	−1.79856e7	−1.41932	−0.709661	+	0.704543i	$0.751152\pi$
$108$	3.24864e6	0.248152
$109$	− 1.22570e7i	− 0.906552i	−0.891370	−	0.453276i	$-0.850255\pi$
$109$	− 1.22570e7i	− 0.906552i	0.891370	−	0.453276i	$-0.149745\pi$
$110$	− 1.83456e6i	− 0.131419i
$111$	1.92631e6i	0.133689i
$112$	4.16154e6i	0.279892i
$113$	1.65950e7	1.08194	0.540968	−	0.841043i	$-0.318058\pi$
$113$	1.65950e7	1.08194	0.540968	+	0.841043i	$0.318058\pi$
$114$	−3.83424e6	−0.242389
$115$	− 1.44295e7i	− 0.884727i
$116$	6.56448e6	0.390479
$117$	0	0
$118$	−2.11253e7	−1.18363
$119$	− 1.49413e7i	− 0.812782i
$120$	1.29024e6	0.0681610
$121$	1.82947e7	0.938808
$122$	6.62162e6i	0.330145i
$123$	− 130104.i	− 0.00630410i
$124$	1.45633e7i	0.685938i
$125$	2.35515e7i	1.07853i
$126$	1.66055e7	0.739529
$127$	−1.16826e6	−0.0506087	−0.0253043	−	0.999680i	$-0.508055\pi$
$127$	−1.16826e6	−0.0506087	−0.0253043	+	0.999680i	$0.508055\pi$
$128$	− 2.09715e6i	− 0.0883883i
$129$	7.56898e6	0.310437
$130$	0	0
$131$	−7.92383e6	−0.307954	−0.153977	−	0.988074i	$-0.549208\pi$
$131$	−7.92383e6	−0.307954	−0.153977	+	0.988074i	$0.549208\pi$
$132$	− 838656.i	− 0.0317377i
$133$	−4.05790e7	−1.49562
$134$	−1.00803e6	−0.0361916
$135$	− 1.06596e7i	− 0.372883i
$136$	7.52947e6i	0.256672i
$137$	− 315654.i	− 0.0104879i	−0.999986	−	0.00524396i	$-0.998331\pi$
$137$	− 315654.i	− 0.0104879i	0.999986	−	0.00524396i	$-0.00166921\pi$
$138$	− 6.59635e6i	− 0.213662i
$139$	3.92038e7	1.23816	0.619079	−	0.785329i	$-0.287506\pi$
$139$	3.92038e7	1.23816	0.619079	+	0.785329i	$0.287506\pi$
$140$	1.36550e7	0.420576
$141$	5.67187e6i	0.170396i
$142$	−1.13178e7	−0.331706
$143$	0	0
$144$	−8.36813e6	−0.233539
$145$	− 2.15397e7i	− 0.586748i
$146$	−7.84226e6	−0.208548
$147$	−2.50456e6	−0.0650309
$148$	− 1.02737e7i	− 0.260501i
$149$	2.18860e7i	0.542020i	0.962577	+	0.271010i	$0.0873577\pi$
$149$	2.18860e7i	0.542020i	−0.962577	+	0.271010i	$0.912642\pi$
$150$	3.26640e6i	0.0790224i
$151$	− 2.94154e7i	− 0.695274i	−0.937629	−	0.347637i	$-0.886984\pi$
$151$	− 2.94154e7i	− 0.695274i	0.937629	−	0.347637i	$-0.113016\pi$
$152$	2.04493e7	0.472308
$153$	3.00444e7	0.678177
$154$	− 8.87578e6i	− 0.195832i
$155$	4.77859e7	1.03072
$156$	0	0
$157$	6.05550e7	1.24882	0.624412	−	0.781095i	$-0.285339\pi$
$157$	6.05550e7	1.24882	0.624412	+	0.781095i	$0.285339\pi$
$158$	− 2.85344e7i	− 0.575531i
$159$	−1.79282e7	−0.353710
$160$	−6.88128e6	−0.132816
$161$	− 6.98114e7i	− 1.31837i
$162$	3.08714e7i	0.570497i
$163$	5.70853e7i	1.03245i	0.856454	+	0.516223i	$0.172663\pi$
$163$	5.70853e7i	1.03245i	−0.856454	+	0.516223i	$0.827337\pi$
$164$	693888.i	0.0122839i
$165$	−2.75184e6	−0.0476902
$166$	4.53831e7	0.770045
$167$	− 8.77265e7i	− 1.45755i	−0.684754	−	0.728775i	$-0.740090\pi$
$167$	− 8.77265e7i	− 1.45755i	0.684754	−	0.728775i	$-0.259910\pi$
$168$	6.24230e6	0.101569
$169$	0	0
$170$	2.47061e7	0.385685
$171$	− 8.15974e7i	− 1.24793i
$172$	−4.03679e7	−0.604904
$173$	−8.56954e6	−0.125833	−0.0629167	−	0.998019i	$-0.520040\pi$
$173$	−8.56954e6	−0.125833	−0.0629167	+	0.998019i	$0.520040\pi$
$174$	− 9.84672e6i	− 0.141700i
$175$	3.45694e7i	0.487594i
$176$	4.47283e6i	0.0618427i
$177$	3.16879e7i	0.429524i
$178$	9.56095e7	1.27067
$179$	−1.88041e7	−0.245056	−0.122528	−	0.992465i	$-0.539100\pi$
$179$	−1.88041e7	−0.245056	−0.122528	+	0.992465i	$0.539100\pi$
$180$	2.74579e7i	0.350925i
$181$	5.99625e7	0.751631	0.375816	−	0.926694i	$-0.377363\pi$
$181$	5.99625e7	0.751631	0.375816	+	0.926694i	$0.377363\pi$
$182$	0	0
$183$	9.93242e6	0.119805
$184$	3.51805e7i	0.416332i
$185$	−3.37105e7	−0.391439
$186$	2.18450e7	0.248918
$187$	− 1.60590e7i	− 0.179586i
$188$	− 3.02500e7i	− 0.332026i
$189$	− 5.15722e7i	− 0.555647i
$190$	− 6.70992e7i	− 0.709708i
$191$	9.39861e7	0.975993	0.487997	−	0.872845i	$-0.337728\pi$
$191$	9.39861e7	0.975993	0.487997	+	0.872845i	$0.337728\pi$
$192$	−3.14573e6	−0.0320750
$193$	− 3.51946e7i	− 0.352391i	−0.984355	−	0.176196i	$-0.943621\pi$
$193$	− 3.51946e7i	− 0.352391i	0.984355	−	0.176196i	$-0.0563791\pi$
$194$	6.94572e7	0.682985
$195$	0	0
$196$	1.33576e7	0.126717
$197$	− 1.02985e8i	− 0.959718i	−0.877346	−	0.479859i	$-0.840688\pi$
$197$	− 1.02985e8i	− 0.959718i	0.877346	−	0.479859i	$-0.159312\pi$
$198$	1.78476e7	0.163400
$199$	−8.36376e7	−0.752342	−0.376171	−	0.926550i	$-0.622760\pi$
$199$	−8.36376e7	−0.752342	−0.376171	+	0.926550i	$0.622760\pi$
$200$	− 1.74208e7i	− 0.153980i
$201$	1.51205e6i	0.0131335i
$202$	8.06363e7i	0.688337i
$203$	− 1.04211e8i	− 0.874335i
$204$	1.12942e7	0.0931430
$205$	2.27682e6	0.0184582
$206$	− 2.99839e7i	− 0.238975i
$207$	1.40379e8	1.10003
$208$	0	0
$209$	−4.36145e7	−0.330460
$210$	− 2.04826e7i	− 0.152622i
$211$	−9.74010e7	−0.713797	−0.356899	−	0.934143i	$-0.616166\pi$
$211$	−9.74010e7	−0.713797	−0.356899	+	0.934143i	$0.616166\pi$
$212$	9.56172e7	0.689224
$213$	1.69767e7i	0.120372i
$214$	− 1.43885e8i	− 1.00361i
$215$	1.32457e8i	0.908951i
$216$	2.59891e7i	0.175470i
$217$	2.31193e8	1.53591
$218$	9.80562e7	0.641029
$219$	1.17634e7i	0.0756794i
$220$	1.46765e7	0.0929271
$221$	0	0
$222$	−1.54105e7	−0.0945325
$223$	1.46457e7i	0.0884390i	0.999022	+	0.0442195i	$0.0140801\pi$
$223$	1.46457e7i	0.0884390i	−0.999022	+	0.0442195i	$0.985920\pi$
$224$	−3.32923e7	−0.197914
$225$	−6.95131e7	−0.406844
$226$	1.32760e8i	0.765045i
$227$	1.84541e8i	1.04713i	0.851985	+	0.523567i	$0.175399\pi$
$227$	1.84541e8i	1.04713i	−0.851985	+	0.523567i	$0.824601\pi$
$228$	− 3.06739e7i	− 0.171395i
$229$	− 8.75461e6i	− 0.0481740i	−0.999710	−	0.0240870i	$-0.992332\pi$
$229$	− 8.75461e6i	− 0.0481740i	0.999710	−	0.0240870i	$-0.00766787\pi$
$230$	1.15436e8	0.625597
$231$	−1.33137e7	−0.0710650
$232$	5.25158e7i	0.276110i
$233$	1.19556e8	0.619193	0.309597	−	0.950868i	$-0.399806\pi$
$233$	1.19556e8	0.619193	0.309597	+	0.950868i	$0.399806\pi$
$234$	0	0
$235$	−9.92578e7	−0.498915
$236$	− 1.69002e8i	− 0.836952i
$237$	−4.28016e7	−0.208853
$238$	1.19530e8	0.574723
$239$	− 3.96209e8i	− 1.87729i	−0.344883	−	0.938646i	$-0.612081\pi$
$239$	− 3.96209e8i	− 1.87729i	0.344883	−	0.938646i	$-0.387919\pi$
$240$	1.03219e7i	0.0481971i
$241$	− 2.56606e8i	− 1.18089i	−0.807080	−	0.590443i	$-0.798953\pi$
$241$	− 2.56606e8i	− 1.18089i	0.807080	−	0.590443i	$-0.201047\pi$
$242$	1.46358e8i	0.663837i
$243$	1.57319e8	0.703331
$244$	−5.29729e7	−0.233448
$245$	− 4.38297e7i	− 0.190409i
$246$	1.04083e6	0.00445767
$247$	0	0
$248$	−1.16507e8	−0.485031
$249$	− 6.80747e7i	− 0.279440i
$250$	−1.88412e8	−0.762638
$251$	7.34775e7	0.293290	0.146645	−	0.989189i	$-0.453153\pi$
$251$	7.34775e7	0.293290	0.146645	+	0.989189i	$0.453153\pi$
$252$	1.32844e8i	0.522926i
$253$	− 7.50335e7i	− 0.291295i
$254$	− 9.34605e6i	− 0.0357857i
$255$	− 3.70591e7i	− 0.139960i
$256$	1.67772e7	0.0625000
$257$	2.02701e8	0.744886	0.372443	−	0.928055i	$-0.378520\pi$
$257$	2.02701e8	0.744886	0.372443	+	0.928055i	$0.378520\pi$
$258$	6.05518e7i	0.219512i
$259$	−1.63094e8	−0.583297
$260$	0	0
$261$	2.09551e8	0.729536
$262$	− 6.33906e7i	− 0.217756i
$263$	1.54254e8	0.522867	0.261434	−	0.965221i	$-0.415805\pi$
$263$	1.54254e8	0.522867	0.261434	+	0.965221i	$0.415805\pi$
$264$	6.70925e6	0.0224419
$265$	− 3.13744e8i	− 1.03565i
$266$	− 3.24632e8i	− 1.05756i
$267$	− 1.43414e8i	− 0.461108i
$268$	− 8.06426e6i	− 0.0255913i
$269$	−6.24018e8	−1.95463	−0.977315	−	0.211793i	$-0.932070\pi$
$269$	−6.24018e8	−1.95463	−0.977315	+	0.211793i	$0.932070\pi$
$270$	8.52768e7	0.263668
$271$	− 3.87983e8i	− 1.18419i	−0.805869	−	0.592094i	$-0.798302\pi$
$271$	− 3.87983e8i	− 1.18419i	0.805869	−	0.592094i	$-0.201698\pi$
$272$	−6.02358e7	−0.181494
$273$	0	0
$274$	2.52523e6	0.00741608
$275$	3.71553e7i	0.107735i
$276$	5.27708e7	0.151082
$277$	−4.53952e8	−1.28331	−0.641654	−	0.766994i	$-0.721752\pi$
$277$	−4.53952e8	−1.28331	−0.641654	+	0.766994i	$0.721752\pi$
$278$	3.13630e8i	0.875510i
$279$	4.64889e8i	1.28155i
$280$	1.09240e8i	0.297392i
$281$	3.33770e8i	0.897377i	0.893688	+	0.448689i	$0.148109\pi$
$281$	3.33770e8i	0.897377i	−0.893688	+	0.448689i	$0.851891\pi$
$282$	−4.53750e7	−0.120488
$283$	−5.37695e8	−1.41021	−0.705104	−	0.709104i	$-0.749100\pi$
$283$	−5.37695e8	−1.41021	−0.705104	+	0.709104i	$0.749100\pi$
$284$	− 9.05426e7i	− 0.234552i
$285$	−1.00649e8	−0.257544
$286$	0	0
$287$	1.10155e7	0.0275053
$288$	− 6.69450e7i	− 0.165137i
$289$	−1.94072e8	−0.472956
$290$	1.72318e8	0.414894
$291$	− 1.04186e8i	− 0.247847i
$292$	− 6.27380e7i	− 0.147466i
$293$	3.35600e8i	0.779445i	0.920932	+	0.389722i	$0.127429\pi$
$293$	3.35600e8i	0.779445i	−0.920932	+	0.389722i	$0.872571\pi$
$294$	− 2.00364e7i	− 0.0459838i
$295$	−5.54539e8	−1.25764
$296$	8.21893e7	0.184202
$297$	− 5.54299e7i	− 0.122771i
$298$	−1.75088e8	−0.383266
$299$	0	0
$300$	−2.61312e7	−0.0558772
$301$	6.40840e8i	1.35446i
$302$	2.35324e8	0.491633
$303$	1.20954e8	0.249789
$304$	1.63594e8i	0.333972i
$305$	1.73817e8i	0.350787i
$306$	2.40355e8i	0.479543i
$307$	2.15029e8i	0.424143i	0.977254	+	0.212072i	$0.0680210\pi$
$307$	2.15029e8i	0.424143i	−0.977254	+	0.212072i	$0.931979\pi$
$308$	7.10062e7	0.138474
$309$	−4.49759e7	−0.0867212
$310$	3.82287e8i	0.728826i
$311$	−7.92062e8	−1.49313	−0.746565	−	0.665313i	$-0.768298\pi$
$311$	−7.92062e8	−1.49313	−0.746565	+	0.665313i	$0.768298\pi$
$312$	0	0
$313$	−1.18457e8	−0.218352	−0.109176	−	0.994022i	$-0.534821\pi$
$313$	−1.18457e8	−0.218352	−0.109176	+	0.994022i	$0.534821\pi$
$314$	4.84440e8i	0.883051i
$315$	4.35894e8	0.785768
$316$	2.28275e8	0.406962
$317$	5.07310e7i	0.0894470i	0.998999	+	0.0447235i	$0.0142407\pi$
$317$	5.07310e7i	0.0894470i	−0.998999	+	0.0447235i	$0.985759\pi$
$318$	− 1.43426e8i	− 0.250111i
$319$	− 1.12006e8i	− 0.193186i
$320$	− 5.50502e7i	− 0.0939149i
$321$	−2.15827e8	−0.364198
$322$	5.58491e8	0.932225
$323$	− 5.87358e8i	− 0.969826i
$324$	−2.46971e8	−0.403402
$325$	0	0
$326$	−4.56682e8	−0.730050
$327$	− 1.47084e8i	− 0.232621i
$328$	−5.55110e6	−0.00868602
$329$	−4.80218e8	−0.743453
$330$	− 2.20147e7i	− 0.0337221i
$331$	− 2.73757e8i	− 0.414923i	−0.978243	−	0.207461i	$-0.933480\pi$
$331$	− 2.73757e8i	− 0.414923i	0.978243	−	0.207461i	$-0.0665201\pi$
$332$	3.63065e8i	0.544504i
$333$	− 3.27955e8i	− 0.486697i
$334$	7.01812e8	1.03064
$335$	−2.64608e7	−0.0384545
$336$	4.99384e7i	0.0718203i
$337$	9.18512e7	0.130732	0.0653658	−	0.997861i	$-0.479179\pi$
$337$	9.18512e7	0.130732	0.0653658	+	0.997861i	$0.479179\pi$
$338$	0	0
$339$	1.99140e8	0.277625
$340$	1.97649e8i	0.272720i
$341$	2.48487e8	0.339362
$342$	6.52779e8	0.882419
$343$	6.24667e8i	0.835833i
$344$	− 3.22943e8i	− 0.427732i
$345$	− 1.73154e8i	− 0.227021i
$346$	− 6.85563e7i	− 0.0889777i
$347$	−1.36700e9	−1.75637	−0.878187	−	0.478318i	$-0.841247\pi$
$347$	−1.36700e9	−1.75637	−0.878187	+	0.478318i	$0.841247\pi$
$348$	7.87738e7	0.100197
$349$	1.13143e9i	1.42475i	0.701797	+	0.712377i	$0.252381\pi$
$349$	1.13143e9i	1.42475i	−0.701797	+	0.712377i	$0.747619\pi$
$350$	−2.76555e8	−0.344781
$351$	0	0
$352$	−3.57827e7	−0.0437294
$353$	4.48395e7i	0.0542562i	0.999632	+	0.0271281i	$0.00863620\pi$
$353$	4.48395e7i	0.0542562i	−0.999632	+	0.0271281i	$0.991364\pi$
$354$	−2.53503e8	−0.303719
$355$	−2.97093e8	−0.352446
$356$	7.64876e8i	0.898496i
$357$	− 1.79296e8i	− 0.208560i
$358$	− 1.50432e8i	− 0.173281i
$359$	3.98281e8i	0.454317i	0.973858	+	0.227158i	$0.0729435\pi$
$359$	3.98281e8i	0.454317i	−0.973858	+	0.227158i	$0.927057\pi$
$360$	−2.19663e8	−0.248141
$361$	−7.01332e8	−0.784600
$362$	4.79700e8i	0.531483i
$363$	2.19536e8	0.240898
$364$	0	0
$365$	−2.05859e8	−0.221588
$366$	7.94594e7i	0.0847152i
$367$	1.63472e9	1.72628	0.863140	−	0.504964i	$-0.168494\pi$
$367$	1.63472e9	1.72628	0.863140	+	0.504964i	$0.168494\pi$
$368$	−2.81444e8	−0.294391
$369$	2.21502e7i	0.0229501i
$370$	− 2.69684e8i	− 0.276789i
$371$	− 1.51792e9i	− 1.54327i
$372$	1.74760e8i	0.176012i
$373$	−1.54633e9	−1.54284	−0.771421	−	0.636325i	$-0.780454\pi$
$373$	−1.54633e9	−1.54284	−0.771421	+	0.636325i	$0.780454\pi$
$374$	1.28472e8	0.126986
$375$	2.82618e8i	0.276752i
$376$	2.42000e8	0.234778
$377$	0	0
$378$	4.12577e8	0.392902
$379$	1.05688e9i	0.997216i	0.866828	+	0.498608i	$0.166155\pi$
$379$	1.05688e9i	0.997216i	−0.866828	+	0.498608i	$0.833845\pi$
$380$	5.36794e8	0.501839
$381$	−1.40191e7	−0.0129862
$382$	7.51889e8i	0.690132i
$383$	− 2.24910e8i	− 0.204556i	−0.994756	−	0.102278i	$-0.967387\pi$
$383$	− 2.24910e8i	− 0.204556i	0.994756	−	0.102278i	$-0.0326132\pi$
$384$	− 2.51658e7i	− 0.0226805i
$385$	− 2.32989e8i	− 0.208077i
$386$	2.81556e8	0.249178
$387$	−1.28862e9	−1.13015
$388$	5.55657e8i	0.482943i
$389$	−1.01788e9	−0.876746	−0.438373	−	0.898793i	$-0.644445\pi$
$389$	−1.01788e9	−0.876746	−0.438373	+	0.898793i	$0.644445\pi$
$390$	0	0
$391$	1.01048e9	0.854887
$392$	1.06861e8i	0.0896021i
$393$	−9.50859e7	−0.0790210
$394$	8.23883e8	0.678623
$395$	− 7.49028e8i	− 0.611517i
$396$	1.42781e8i	0.115541i
$397$	− 1.47565e9i	− 1.18363i	−0.806072	−	0.591817i	$-0.798411\pi$
$397$	− 1.47565e9i	− 1.18363i	0.806072	−	0.591817i	$-0.201589\pi$
$398$	− 6.69100e8i	− 0.531986i
$399$	−4.86948e8	−0.383776
$400$	1.39366e8	0.108880
$401$	2.74912e8i	0.212906i	0.994318	+	0.106453i	$0.0339494\pi$
$401$	2.74912e8i	0.212906i	−0.994318	+	0.106453i	$0.966051\pi$
$402$	−1.20964e7	−0.00928676
$403$	0	0
$404$	−6.45090e8	−0.486727
$405$	8.10373e8i	0.606167i
$406$	8.33689e8	0.618248
$407$	−1.75294e8	−0.128881
$408$	9.03537e7i	0.0658620i
$409$	1.63427e9i	1.18112i	0.806995	+	0.590558i	$0.201092\pi$
$409$	1.63427e9i	1.18112i	−0.806995	+	0.590558i	$0.798908\pi$
$410$	1.82146e7i	0.0130519i
$411$	− 3.78785e6i	− 0.00269120i
$412$	2.39871e8	0.168981
$413$	−2.68291e9	−1.87405
$414$	1.12303e9i	0.777839i
$415$	1.19131e9	0.818192
$416$	0	0
$417$	4.70445e8	0.317712
$418$	− 3.48916e8i	− 0.233670i
$419$	−1.11280e9	−0.739039	−0.369519	−	0.929223i	$-0.620478\pi$
$419$	−1.11280e9	−0.739039	−0.369519	+	0.929223i	$0.620478\pi$
$420$	1.63860e8	0.107920
$421$	− 9.22528e8i	− 0.602549i	−0.953537	−	0.301274i	$-0.902588\pi$
$421$	− 9.22528e8i	− 0.602549i	0.953537	−	0.301274i	$-0.0974120\pi$
$422$	− 7.79208e8i	− 0.504731i
$423$	− 9.65636e8i	− 0.620329i
$424$	7.64937e8i	0.487355i
$425$	−5.00372e8	−0.316178
$426$	−1.35814e8	−0.0851159
$427$	8.40945e8i	0.522721i
$428$	1.15108e9	0.709661
$429$	0	0
$430$	−1.05966e9	−0.642726
$431$	9.81508e8i	0.590505i	0.955419	+	0.295252i	$0.0954037\pi$
$431$	9.81508e8i	0.590505i	−0.955419	+	0.295252i	$0.904596\pi$
$432$	−2.07913e8	−0.124076
$433$	−2.84998e9	−1.68707	−0.843537	−	0.537071i	$-0.819531\pi$
$433$	−2.84998e9	−1.68707	−0.843537	+	0.537071i	$0.819531\pi$
$434$	1.84954e9i	1.08605i
$435$	− 2.58476e8i	− 0.150560i
$436$	7.84450e8i	0.453276i
$437$	− 2.74436e9i	− 1.57310i
$438$	−9.41071e7	−0.0535134
$439$	1.05622e9	0.595838	0.297919	−	0.954591i	$-0.403708\pi$
$439$	1.05622e9	0.595838	0.297919	+	0.954591i	$0.403708\pi$
$440$	1.17412e8i	0.0657094i
$441$	4.26401e8	0.236746
$442$	0	0
$443$	1.82325e9	0.996401	0.498201	−	0.867062i	$-0.333994\pi$
$443$	1.82325e9	0.996401	0.498201	+	0.867062i	$0.333994\pi$
$444$	− 1.23284e8i	− 0.0668446i
$445$	2.50975e9	1.35011
$446$	−1.17166e8	−0.0625358
$447$	2.62633e8i	0.139082i
$448$	− 2.66338e8i	− 0.139946i
$449$	1.84846e9i	0.963713i	0.876250	+	0.481856i	$0.160037\pi$
$449$	1.84846e9i	0.963713i	−0.876250	+	0.481856i	$0.839963\pi$
$450$	− 5.56105e8i	− 0.287682i
$451$	1.18395e7	0.00607735
$452$	−1.06208e9	−0.540968
$453$	− 3.52985e8i	− 0.178407i
$454$	−1.47633e9	−0.740435
$455$	0	0
$456$	2.45391e8	0.121194
$457$	2.98066e9i	1.46085i	0.682993	+	0.730425i	$0.260678\pi$
$457$	2.98066e9i	1.46085i	−0.682993	+	0.730425i	$0.739322\pi$
$458$	7.00369e7	0.0340642
$459$	7.46477e8	0.360306
$460$	9.23489e8i	0.442364i
$461$	2.52781e9i	1.20169i	0.799367	+	0.600843i	$0.205168\pi$
$461$	2.52781e9i	1.20169i	−0.799367	+	0.600843i	$0.794832\pi$
$462$	− 1.06509e8i	− 0.0502505i
$463$	− 8.90291e8i	− 0.416868i	−0.978036	−	0.208434i	$-0.933163\pi$
$463$	− 8.90291e8i	− 0.416868i	0.978036	−	0.208434i	$-0.0668366\pi$
$464$	−4.20127e8	−0.195239
$465$	5.73431e8	0.264482
$466$	9.56450e8i	0.437836i
$467$	−2.65667e9	−1.20706	−0.603529	−	0.797341i	$-0.706239\pi$
$467$	−2.65667e9	−1.20706	−0.603529	+	0.797341i	$0.706239\pi$
$468$	0	0
$469$	−1.28020e8	−0.0573024
$470$	− 7.94062e8i	− 0.352786i
$471$	7.26660e8	0.320448
$472$	1.35202e9	0.591814
$473$	6.88777e8i	0.299271i
$474$	− 3.42413e8i	− 0.147681i
$475$	1.35896e9i	0.581806i
$476$	9.56243e8i	0.406391i
$477$	3.05228e9	1.28769
$478$	3.16967e9	1.32745
$479$	1.30093e9i	0.540855i	0.962740	+	0.270428i	$0.0871651\pi$
$479$	1.30093e9i	0.540855i	−0.962740	+	0.270428i	$0.912835\pi$
$480$	−8.25754e7	−0.0340805
$481$	0	0
$482$	2.05285e9	0.835012
$483$	− 8.37737e8i	− 0.338293i
$484$	−1.17086e9	−0.469404
$485$	1.82325e9	0.725689
$486$	1.25855e9i	0.497330i
$487$	1.07447e9i	0.421542i	0.977535	+	0.210771i	$0.0675975\pi$
$487$	1.07447e9i	0.421542i	−0.977535	+	0.210771i	$0.932402\pi$
$488$	− 4.23783e8i	− 0.165072i
$489$	6.85024e8i	0.264926i
$490$	3.50638e8	0.134640
$491$	7.83344e8	0.298653	0.149327	−	0.988788i	$-0.452289\pi$
$491$	7.83344e8	0.298653	0.149327	+	0.988788i	$0.452289\pi$
$492$	8.32666e6i	0.00315205i
$493$	1.50839e9	0.566958
$494$	0	0
$495$	4.68501e8	0.173617
$496$	− 9.32053e8i	− 0.342969i
$497$	−1.43736e9	−0.525193
$498$	5.44598e8	0.197594
$499$	6.23188e8i	0.224526i	0.993679	+	0.112263i	$0.0358100\pi$
$499$	6.23188e8i	0.224526i	−0.993679	+	0.112263i	$0.964190\pi$
$500$	− 1.50730e9i	− 0.539267i
$501$	− 1.05272e9i	− 0.374007i
$502$	5.87820e8i	0.207387i
$503$	−2.70927e9	−0.949215	−0.474607	−	0.880198i	$-0.657410\pi$
$503$	−2.70927e9	−0.949215	−0.474607	+	0.880198i	$0.657410\pi$
$504$	−1.06275e9	−0.369764
$505$	2.11670e9i	0.731375i
$506$	6.00268e8	0.205977
$507$	0	0
$508$	7.47684e7	0.0253043
$509$	− 3.49943e9i	− 1.17621i	−0.808784	−	0.588106i	$-0.799874\pi$
$509$	− 3.49943e9i	− 1.17621i	0.808784	−	0.588106i	$-0.200126\pi$
$510$	2.96473e8	0.0989668
$511$	−9.95967e8	−0.330196
$512$	1.34218e8i	0.0441942i
$513$	− 2.02735e9i	− 0.663008i
$514$	1.62161e9i	0.526714i
$515$	− 7.87078e8i	− 0.253918i
$516$	−4.84414e8	−0.155218
$517$	−5.16140e8	−0.164267
$518$	− 1.30476e9i	− 0.412453i
$519$	−1.02835e8	−0.0322889
$520$	0	0
$521$	−1.37683e9	−0.426530	−0.213265	−	0.976994i	$-0.568410\pi$
$521$	−1.37683e9	−0.426530	−0.213265	+	0.976994i	$0.568410\pi$
$522$	1.67640e9i	0.515860i
$523$	−2.86154e9	−0.874669	−0.437334	−	0.899299i	$-0.644077\pi$
$523$	−2.86154e9	−0.874669	−0.437334	+	0.899299i	$0.644077\pi$
$524$	5.07125e8	0.153977
$525$	4.14833e8i	0.125117i
$526$	1.23403e9i	0.369723i
$527$	3.34638e9i	0.995951i
$528$	5.36740e7i	0.0158688i
$529$	1.31651e9	0.386661
$530$	2.50995e9	0.732318
$531$	− 5.39487e9i	− 1.56369i
$532$	2.59706e9	0.747810
$533$	0	0
$534$	1.14731e9	0.326053
$535$	− 3.77697e9i	− 1.06636i
$536$	6.45140e7	0.0180958
$537$	−2.25649e8	−0.0628815
$538$	− 4.99215e9i	− 1.38213i
$539$	− 2.27915e8i	− 0.0626919i
$540$	6.82214e8i	0.186442i
$541$	5.34467e9i	1.45121i	0.688111	+	0.725605i	$0.258440\pi$
$541$	5.34467e9i	1.45121i	−0.688111	+	0.725605i	$0.741560\pi$
$542$	3.10387e9	0.837347
$543$	7.19550e8	0.192869
$544$	− 4.81886e8i	− 0.128336i
$545$	2.57398e9	0.681109
$546$	0	0
$547$	−3.37135e9	−0.880740	−0.440370	−	0.897816i	$-0.645153\pi$
$547$	−3.37135e9	−0.880740	−0.440370	+	0.897816i	$0.645153\pi$
$548$	2.02019e7i	0.00524396i
$549$	−1.69100e9	−0.436153
$550$	−2.97242e8	−0.0761801
$551$	− 4.09665e9i	− 1.04327i
$552$	4.22167e8i	0.106831i
$553$	− 3.62387e9i	− 0.911244i
$554$	− 3.63162e9i	− 0.907436i
$555$	−4.04526e8	−0.100443
$556$	−2.50904e9	−0.619079
$557$	− 5.61106e9i	− 1.37579i	−0.725811	−	0.687894i	$-0.758535\pi$
$557$	− 5.61106e9i	− 1.37579i	0.725811	−	0.687894i	$-0.241465\pi$
$558$	−3.71911e9	−0.906190
$559$	0	0
$560$	−8.73923e8	−0.210288
$561$	− 1.92707e8i	− 0.0460817i
$562$	−2.67016e9	−0.634542
$563$	−6.69690e9	−1.58159	−0.790795	−	0.612081i	$-0.790333\pi$
$563$	−6.69690e9	−1.58159	−0.790795	+	0.612081i	$0.790333\pi$
$564$	− 3.63000e8i	− 0.0851980i
$565$	3.48494e9i	0.812879i
$566$	− 4.30156e9i	− 0.997168i
$567$	3.92066e9i	0.903273i
$568$	7.24341e8	0.165853
$569$	−1.96850e9	−0.447964	−0.223982	−	0.974593i	$-0.571906\pi$
$569$	−1.96850e9	−0.447964	−0.223982	+	0.974593i	$0.571906\pi$
$570$	− 8.05190e8i	− 0.182111i
$571$	−1.02926e9	−0.231365	−0.115682	−	0.993286i	$-0.536906\pi$
$571$	−1.02926e9	−0.231365	−0.115682	+	0.993286i	$0.536906\pi$
$572$	0	0
$573$	1.12783e9	0.250440
$574$	8.81238e7i	0.0194492i
$575$	−2.33793e9	−0.512853
$576$	5.35560e8	0.116770
$577$	− 3.31179e9i	− 0.717708i	−0.933394	−	0.358854i	$-0.883168\pi$
$577$	− 3.31179e9i	− 0.717708i	0.933394	−	0.358854i	$-0.116832\pi$
$578$	− 1.55258e9i	− 0.334431i
$579$	− 4.22335e8i	− 0.0904236i
$580$	1.37854e9i	0.293374i
$581$	5.76366e9	1.21922
$582$	8.33486e8	0.175254
$583$	− 1.63147e9i	− 0.340988i
$584$	5.01904e8	0.104274
$585$	0	0
$586$	−2.68480e9	−0.551151
$587$	5.59411e8i	0.114156i	0.998370	+	0.0570778i	$0.0181783\pi$
$587$	5.59411e8i	0.114156i	−0.998370	+	0.0570778i	$0.981822\pi$
$588$	1.60292e8	0.0325155
$589$	9.08843e9	1.83267
$590$	− 4.43631e9i	− 0.889282i
$591$	− 1.23582e9i	− 0.246264i
$592$	6.57514e8i	0.130250i
$593$	− 3.02459e9i	− 0.595628i	−0.954624	−	0.297814i	$-0.903742\pi$
$593$	− 3.02459e9i	− 0.595628i	0.954624	−	0.297814i	$-0.0962575\pi$
$594$	4.43439e8	0.0868124
$595$	3.13767e9	0.610658
$596$	− 1.40071e9i	− 0.271010i
$597$	−1.00365e9	−0.193051
$598$	0	0
$599$	−5.63246e9	−1.07079	−0.535395	−	0.844602i	$-0.679837\pi$
$599$	−5.63246e9	−1.07079	−0.535395	+	0.844602i	$0.679837\pi$
$600$	− 2.09050e8i	− 0.0395112i
$601$	3.40792e8	0.0640366	0.0320183	−	0.999487i	$-0.489807\pi$
$601$	3.40792e8	0.0640366	0.0320183	+	0.999487i	$0.489807\pi$
$602$	−5.12672e9	−0.957749
$603$	− 2.57426e8i	− 0.0478126i
$604$	1.88259e9i	0.347637i
$605$	3.84189e9i	0.705344i
$606$	9.67636e8i	0.176627i
$607$	3.85420e9	0.699477	0.349739	−	0.936847i	$-0.386270\pi$
$607$	3.85420e9	0.699477	0.349739	+	0.936847i	$0.386270\pi$
$608$	−1.30875e9	−0.236154
$609$	− 1.25053e9i	− 0.224355i
$610$	−1.39054e9	−0.248044
$611$	0	0
$612$	−1.92284e9	−0.339088
$613$	− 9.22245e9i	− 1.61709i	−0.588434	−	0.808545i	$-0.700255\pi$
$613$	− 9.22245e9i	− 1.61709i	0.588434	−	0.808545i	$-0.299745\pi$
$614$	−1.72023e9	−0.299915
$615$	2.73218e7	0.00473639
$616$	5.68050e8i	0.0979160i
$617$	− 6.53611e9i	− 1.12027i	−0.828402	−	0.560133i	$-0.810750\pi$
$617$	− 6.53611e9i	− 1.12027i	0.828402	−	0.560133i	$-0.189250\pi$
$618$	− 3.59807e8i	− 0.0613211i
$619$	1.36559e9i	0.231420i	0.993283	+	0.115710i	$0.0369144\pi$
$619$	1.36559e9i	0.231420i	−0.993283	+	0.115710i	$0.963086\pi$
$620$	−3.05830e9	−0.515358
$621$	3.48782e9	0.584431
$622$	− 6.33649e9i	− 1.05580i
$623$	1.21424e10	2.01186
$624$	0	0
$625$	−2.28761e9	−0.374802
$626$	− 9.47659e8i	− 0.154398i
$627$	−5.23374e8	−0.0847960
$628$	−3.87552e9	−0.624412
$629$	− 2.36070e9i	− 0.378236i
$630$	3.48716e9i	0.555622i
$631$	1.54079e9i	0.244141i	0.992521	+	0.122070i	$0.0389533\pi$
$631$	1.54079e9i	0.244141i	−0.992521	+	0.122070i	$0.961047\pi$
$632$	1.82620e9i	0.287766i
$633$	−1.16881e9	−0.183160
$634$	−4.05848e8	−0.0632486
$635$	− 2.45334e8i	− 0.0380233i
$636$	1.14741e9	0.176855
$637$	0	0
$638$	8.96052e8	0.136603
$639$	− 2.89029e9i	− 0.438216i
$640$	4.40402e8	0.0664078
$641$	4.54018e9	0.680879	0.340440	−	0.940266i	$-0.389424\pi$
$641$	4.54018e9	0.680879	0.340440	+	0.940266i	$0.389424\pi$
$642$	− 1.72661e9i	− 0.257527i
$643$	− 1.14054e10i	− 1.69189i	−0.533272	−	0.845944i	$-0.679038\pi$
$643$	− 1.14054e10i	− 1.69189i	0.533272	−	0.845944i	$-0.320962\pi$
$644$	4.46793e9i	0.659183i
$645$	1.58948e9i	0.233237i
$646$	4.69886e9	0.685770
$647$	1.26393e10	1.83468	0.917338	−	0.398109i	$-0.130334\pi$
$647$	1.26393e10	1.83468	0.917338	+	0.398109i	$0.130334\pi$
$648$	− 1.97577e9i	− 0.285248i
$649$	−2.88360e9	−0.414075
$650$	0	0
$651$	2.77431e9	0.394114
$652$	− 3.65346e9i	− 0.516223i
$653$	−1.05004e10	−1.47575	−0.737873	−	0.674940i	$-0.764170\pi$
$653$	−1.05004e10	−1.47575	−0.737873	+	0.674940i	$0.764170\pi$
$654$	1.17667e9	0.164488
$655$	− 1.66400e9i	− 0.231371i
$656$	− 4.44088e7i	− 0.00614194i
$657$	− 2.00272e9i	− 0.275512i
$658$	− 3.84175e9i	− 0.525700i
$659$	9.64818e9	1.31325	0.656624	−	0.754219i	$-0.271984\pi$
$659$	9.64818e9	1.31325	0.656624	+	0.754219i	$0.271984\pi$
$660$	1.76118e8	0.0238451
$661$	− 6.58299e9i	− 0.886580i	−0.896378	−	0.443290i	$-0.853811\pi$
$661$	− 6.58299e9i	− 0.886580i	0.896378	−	0.443290i	$-0.146189\pi$
$662$	2.19006e9	0.293395
$663$	0	0
$664$	−2.90452e9	−0.385023
$665$	− 8.52160e9i	− 1.12369i
$666$	2.62364e9	0.344147
$667$	7.04779e9	0.919629
$668$	5.61450e9i	0.728775i
$669$	1.75749e8i	0.0226935i
$670$	− 2.11687e8i	− 0.0271914i
$671$	9.03851e8i	0.115496i
$672$	−3.99507e8	−0.0507846
$673$	8.54649e9	1.08077	0.540387	−	0.841416i	$-0.318278\pi$
$673$	8.54649e9	1.08077	0.540387	+	0.841416i	$0.318278\pi$
$674$	7.34810e8i	0.0924411i
$675$	−1.72711e9	−0.216151
$676$	0	0
$677$	8.71305e9	1.07922	0.539610	−	0.841915i	$-0.318572\pi$
$677$	8.71305e9	1.07922	0.539610	+	0.841915i	$0.318572\pi$
$678$	1.59312e9i	0.196311i
$679$	8.82106e9	1.08138
$680$	−1.58119e9	−0.192842
$681$	2.21449e9i	0.268695i
$682$	1.98789e9i	0.239965i
$683$	1.46109e10i	1.75470i	0.479849	+	0.877351i	$0.340692\pi$
$683$	1.46109e10i	1.75470i	−0.479849	+	0.877351i	$0.659308\pi$
$684$	5.22223e9i	0.623965i
$685$	6.62873e7	0.00787977
$686$	−4.99734e9	−0.591023
$687$	− 1.05055e8i	− 0.0123615i
$688$	2.58354e9	0.302452
$689$	0	0
$690$	1.38523e9	0.160528
$691$	1.47348e10i	1.69891i	0.527662	+	0.849454i	$0.323069\pi$
$691$	1.47348e10i	1.69891i	−0.527662	+	0.849454i	$0.676931\pi$
$692$	5.48451e8	0.0629167
$693$	2.26665e9	0.258713
$694$	− 1.09360e10i	− 1.24194i
$695$	8.23279e9i	0.930252i
$696$	6.30190e8i	0.0708499i
$697$	1.59442e8i	0.0178357i
$698$	−9.05146e9	−1.00745
$699$	1.43467e9	0.158885
$700$	− 2.21244e9i	− 0.243797i
$701$	−1.31502e9	−0.144185	−0.0720923	−	0.997398i	$-0.522968\pi$
$701$	−1.31502e9	−0.144185	−0.0720923	+	0.997398i	$0.522968\pi$
$702$	0	0
$703$	−6.41141e9	−0.696001
$704$	− 2.86261e8i	− 0.0309213i
$705$	−1.19109e9	−0.128022
$706$	−3.58716e8	−0.0383649
$707$	1.02408e10i	1.08985i
$708$	− 2.02803e9i	− 0.214762i
$709$	6.64028e8i	0.0699721i	0.999388	+	0.0349860i	$0.0111387\pi$
$709$	6.64028e8i	0.0699721i	−0.999388	+	0.0349860i	$0.988861\pi$
$710$	− 2.37674e9i	− 0.249217i
$711$	7.28697e9	0.760332
$712$	−6.11901e9	−0.635333
$713$	1.56356e10i	1.61547i
$714$	1.43436e9	0.147474
$715$	0	0
$716$	1.20346e9	0.122528
$717$	− 4.75451e9i	− 0.481713i
$718$	−3.18624e9	−0.321250
$719$	−4.95034e9	−0.496689	−0.248344	−	0.968672i	$-0.579886\pi$
$719$	−4.95034e9	−0.496689	−0.248344	+	0.968672i	$0.579886\pi$
$720$	− 1.75731e9i	− 0.175462i
$721$	− 3.80796e9i	− 0.378372i
$722$	− 5.61065e9i	− 0.554796i
$723$	− 3.07928e9i	− 0.303015i
$724$	−3.83760e9	−0.375816
$725$	−3.48994e9	−0.340123
$726$	1.75629e9i	0.170341i
$727$	−8.81101e9	−0.850463	−0.425231	−	0.905085i	$-0.639807\pi$
$727$	−8.81101e9	−0.850463	−0.425231	+	0.905085i	$0.639807\pi$
$728$	0	0
$729$	−6.55163e9	−0.626330
$730$	− 1.64687e9i	− 0.156686i
$731$	−9.27578e9	−0.878293
$732$	−6.35675e8	−0.0599027
$733$	1.49414e8i	0.0140129i	0.999975	+	0.00700643i	$0.00223023\pi$
$733$	1.49414e8i	0.0140129i	−0.999975	+	0.00700643i	$0.997770\pi$
$734$	1.30777e10i	1.22066i
$735$	− 5.25957e8i	− 0.0488590i
$736$	− 2.25155e9i	− 0.208166i
$737$	−1.37596e8	−0.0126611
$738$	−1.77202e8	−0.0162282
$739$	− 4.70806e9i	− 0.429127i	−0.976710	−	0.214564i	$-0.931167\pi$
$739$	− 4.70806e9i	− 0.429127i	0.976710	−	0.214564i	$-0.0688329\pi$
$740$	2.15747e9	0.195719
$741$	0	0
$742$	1.21434e10	1.09125
$743$	− 1.69676e9i	− 0.151761i	−0.997117	−	0.0758805i	$-0.975823\pi$
$743$	− 1.69676e9i	− 0.151761i	0.997117	−	0.0758805i	$-0.0241768\pi$
$744$	−1.39808e9	−0.124459
$745$	−4.59607e9	−0.407230
$746$	− 1.23707e10i	− 1.09095i
$747$	1.15897e10i	1.01730i
$748$	1.02777e9i	0.0897928i
$749$	− 1.82733e10i	− 1.58903i
$750$	−2.26094e9	−0.195693
$751$	−1.06650e10	−0.918800	−0.459400	−	0.888229i	$-0.651936\pi$
$751$	−1.06650e10	−0.918800	−0.459400	+	0.888229i	$0.651936\pi$
$752$	1.93600e9i	0.166013i
$753$	8.81731e8	0.0752581
$754$	0	0
$755$	6.17724e9	0.522373
$756$	3.30062e9i	0.277824i
$757$	6.22876e9	0.521874	0.260937	−	0.965356i	$-0.415968\pi$
$757$	6.22876e9	0.521874	0.260937	+	0.965356i	$0.415968\pi$
$758$	−8.45506e9	−0.705138
$759$	− 9.00402e8i	− 0.0747464i
$760$	4.29435e9i	0.354854i
$761$	− 8.38334e9i	− 0.689558i	−0.938684	−	0.344779i	$-0.887954\pi$
$761$	− 8.38334e9i	− 0.689558i	0.938684	−	0.344779i	$-0.112046\pi$
$762$	− 1.12153e8i	− 0.00918263i
$763$	1.24531e10	1.01495
$764$	−6.01511e9	−0.487997
$765$	6.30932e9i	0.509527i
$766$	1.79928e9	0.144643
$767$	0	0
$768$	2.01327e8	0.0160375
$769$	1.18649e10i	0.940852i	0.882439	+	0.470426i	$0.155900\pi$
$769$	1.18649e10i	0.940852i	−0.882439	+	0.470426i	$0.844100\pi$
$770$	1.86391e9	0.147132
$771$	2.43241e9	0.191138
$772$	2.25245e9i	0.176196i
$773$	− 5.56680e9i	− 0.433488i	−0.976228	−	0.216744i	$-0.930456\pi$
$773$	− 5.56680e9i	− 0.433488i	0.976228	−	0.216744i	$-0.0695438\pi$
$774$	− 1.03089e10i	− 0.799136i
$775$	− 7.74246e9i	− 0.597479i
$776$	−4.44526e9	−0.341492
$777$	−1.95713e9	−0.149674
$778$	− 8.14306e9i	− 0.619953i
$779$	4.33029e8	0.0328198
$780$	0	0
$781$	−1.54488e9	−0.116042
$782$	8.08383e9i	0.604496i
$783$	5.20645e9	0.387593
$784$	−8.54888e8	−0.0633583
$785$	1.27165e10i	0.938264i
$786$	− 7.60687e8i	− 0.0558763i
$787$	1.34611e8i	0.00984395i	0.999988	+	0.00492198i	$0.00156672\pi$
$787$	1.34611e8i	0.00984395i	−0.999988	+	0.00492198i	$0.998433\pi$
$788$	6.59106e9i	0.479859i
$789$	1.85105e9	0.134168
$790$	5.99222e9	0.432408
$791$	1.68605e10i	1.21130i
$792$	−1.14225e9	−0.0817001
$793$	0	0
$794$	1.18052e10	0.836955
$795$	− 3.76493e9i	− 0.265749i
$796$	5.35280e9	0.376171
$797$	7.41548e9	0.518842	0.259421	−	0.965764i	$-0.416468\pi$
$797$	7.41548e9	0.518842	0.259421	+	0.965764i	$0.416468\pi$
$798$	− 3.89559e9i	− 0.271371i
$799$	− 6.95088e9i	− 0.482088i
$800$	1.11493e9i	0.0769898i
$801$	2.44163e10i	1.67867i
$802$	−2.19930e9	−0.150548
$803$	−1.07047e9	−0.0729574
$804$	− 9.67711e7i	− 0.00656673i
$805$	1.46604e10	0.990513
$806$	0	0
$807$	−7.48822e9	−0.501558
$808$	− 5.16072e9i	− 0.344168i
$809$	−1.41542e10	−0.939863	−0.469932	−	0.882703i	$-0.655721\pi$
$809$	−1.41542e10	−0.939863	−0.469932	+	0.882703i	$0.655721\pi$
$810$	−6.48299e9	−0.428625
$811$	2.63708e10i	1.73600i	0.496563	+	0.868001i	$0.334595\pi$
$811$	2.63708e10i	1.73600i	−0.496563	+	0.868001i	$0.665405\pi$
$812$	6.66951e9i	0.437168i
$813$	− 4.65580e9i	− 0.303863i
$814$	− 1.40236e9i	− 0.0911324i
$815$	−1.19879e10	−0.775697
$816$	−7.22829e8	−0.0465715
$817$	2.51921e10i	1.61617i
$818$	−1.30742e10	−0.835176
$819$	0	0
$820$	−1.45716e8	−0.00922912
$821$	− 8.06264e9i	− 0.508483i	−0.967141	−	0.254241i	$-0.918174\pi$
$821$	− 8.06264e9i	− 0.508483i	0.967141	−	0.254241i	$-0.0818258\pi$
$822$	3.03028e7	0.00190297
$823$	2.34202e10	1.46451	0.732253	−	0.681033i	$-0.238469\pi$
$823$	2.34202e10	1.46451	0.732253	+	0.681033i	$0.238469\pi$
$824$	1.91897e9i	0.119488i
$825$	4.45864e8i	0.0276448i
$826$	− 2.14633e10i	− 1.32515i
$827$	5.55722e9i	0.341655i	0.985301	+	0.170828i	$0.0546442\pi$
$827$	5.55722e9i	0.341655i	−0.985301	+	0.170828i	$0.945356\pi$
$828$	−8.98423e9	−0.550015
$829$	−2.84256e10	−1.73288	−0.866440	−	0.499281i	$-0.833597\pi$
$829$	−2.84256e10	−1.73288	−0.866440	+	0.499281i	$0.833597\pi$
$830$	9.53046e9i	0.578549i
$831$	−5.44743e9	−0.329297
$832$	0	0
$833$	3.06933e9	0.183987
$834$	3.76356e9i	0.224656i
$835$	1.84226e10	1.09508
$836$	2.79133e9	0.165230
$837$	1.15505e10i	0.680868i
$838$	− 8.90238e9i	− 0.522579i
$839$	1.04036e10i	0.608156i	0.952647	+	0.304078i	$0.0983484\pi$
$839$	1.04036e10i	0.608156i	−0.952647	+	0.304078i	$0.901652\pi$
$840$	1.31088e9i	0.0763109i
$841$	−6.72927e9	−0.390105
$842$	7.38023e9	0.426066
$843$	4.00524e9i	0.230267i
$844$	6.23367e9	0.356899
$845$	0	0
$846$	7.72509e9	0.438639
$847$	1.85874e10i	1.05106i
$848$	−6.11950e9	−0.344612
$849$	−6.45234e9	−0.361860
$850$	− 4.00297e9i	− 0.223572i
$851$	− 1.10301e10i	− 0.613514i
$852$	− 1.08651e9i	− 0.0601860i
$853$	− 1.80580e10i	− 0.996205i	−0.867118	−	0.498102i	$-0.834030\pi$
$853$	− 1.80580e10i	− 0.996205i	0.867118	−	0.498102i	$-0.165970\pi$
$854$	−6.72756e9	−0.369620
$855$	1.71355e10	0.937593
$856$	9.20861e9i	0.501806i
$857$	6.34034e9	0.344096	0.172048	−	0.985089i	$-0.444962\pi$
$857$	6.34034e9	0.344096	0.172048	+	0.985089i	$0.444962\pi$
$858$	0	0
$859$	1.21489e10	0.653973	0.326987	−	0.945029i	$-0.393967\pi$
$859$	1.21489e10	0.653973	0.326987	+	0.945029i	$0.393967\pi$
$860$	− 8.47725e9i	− 0.454476i
$861$	1.32186e8	0.00705786
$862$	−7.85206e9	−0.417550
$863$	2.87111e10i	1.52059i	0.649578	+	0.760295i	$0.274946\pi$
$863$	2.87111e10i	1.52059i	−0.649578	+	0.760295i	$0.725054\pi$
$864$	− 1.66330e9i	− 0.0877351i
$865$	− 1.79960e9i	− 0.0945411i
$866$	− 2.27998e10i	− 1.19294i
$867$	−2.32887e9	−0.121361
$868$	−1.47963e10	−0.767954
$869$	− 3.89495e9i	− 0.201341i
$870$	2.06781e9	0.106462
$871$	0	0
$872$	−6.27560e9	−0.320514
$873$	1.77376e10i	0.902289i
$874$	2.19549e10	1.11235
$875$	−2.39283e10	−1.20749
$876$	− 7.52857e8i	− 0.0378397i
$877$	− 2.46021e10i	− 1.23161i	−0.787898	−	0.615806i	$-0.788831\pi$
$877$	− 2.46021e10i	− 1.23161i	0.787898	−	0.615806i	$-0.211169\pi$
$878$	8.44975e9i	0.421321i
$879$	4.02720e9i	0.200006i
$880$	−9.39295e8	−0.0464636
$881$	1.25378e10	0.617738	0.308869	−	0.951105i	$-0.400049\pi$
$881$	1.25378e10	0.617738	0.308869	+	0.951105i	$0.400049\pi$
$882$	3.41121e9i	0.167405i
$883$	−1.93097e10	−0.943873	−0.471937	−	0.881633i	$-0.656445\pi$
$883$	−1.93097e10	−0.943873	−0.471937	+	0.881633i	$0.656445\pi$
$884$	0	0
$885$	−6.65446e9	−0.322709
$886$	1.45860e10i	0.704562i
$887$	3.20268e10	1.54092	0.770462	−	0.637486i	$-0.220026\pi$
$887$	3.20268e10	1.54092	0.770462	+	0.637486i	$0.220026\pi$
$888$	9.86272e8	0.0472663
$889$	− 1.18695e9i	− 0.0566599i
$890$	2.00780e10i	0.954675i
$891$	4.21394e9i	0.199580i
$892$	− 9.37327e8i	− 0.0442195i
$893$	−1.88779e10	−0.887101
$894$	−2.10106e9	−0.0983461
$895$	− 3.94885e9i	− 0.184115i
$896$	2.13071e9	0.0989568
$897$	0	0
$898$	−1.47877e10	−0.681448
$899$	2.33400e10i	1.07138i
$900$	4.44884e9	0.203422
$901$	2.19710e10	1.00072
$902$	9.47157e7i	0.00429733i
$903$	7.69008e9i	0.347555i
$904$	− 8.49662e9i	− 0.382522i
$905$	1.25921e10i	0.564715i
$906$	2.82388e9	0.126153
$907$	−2.33703e9	−0.104002	−0.0520008	−	0.998647i	$-0.516560\pi$
$907$	−2.33703e9	−0.104002	−0.0520008	+	0.998647i	$0.516560\pi$
$908$	− 1.18106e10i	− 0.523567i
$909$	−2.05925e10	−0.909359
$910$	0	0
$911$	2.20343e10	0.965573	0.482786	−	0.875738i	$-0.339625\pi$
$911$	2.20343e10	0.965573	0.482786	+	0.875738i	$0.339625\pi$
$912$	1.96313e9i	0.0856973i
$913$	6.19480e9	0.269389
$914$	−2.38453e10	−1.03298
$915$	2.08581e9i	0.0900121i
$916$	5.60295e8i	0.0240870i
$917$	− 8.05061e9i	− 0.344775i
$918$	5.97181e9i	0.254775i
$919$	−1.43277e10	−0.608938	−0.304469	−	0.952522i	$-0.598479\pi$
$919$	−1.43277e10	−0.608938	−0.304469	+	0.952522i	$0.598479\pi$
$920$	−7.38791e9	−0.312798
$921$	2.58035e9i	0.108835i
$922$	−2.02225e10	−0.849720
$923$	0	0
$924$	8.52074e8	0.0355325
$925$	5.46190e9i	0.226907i
$926$	7.12233e9	0.294770
$927$	7.65715e9	0.315710
$928$	− 3.36101e9i	− 0.138055i
$929$	− 1.31280e10i	− 0.537208i	−0.963251	−	0.268604i	$-0.913438\pi$
$929$	− 1.31280e10i	− 0.537208i	0.963251	−	0.268604i	$-0.0865623\pi$
$930$	4.58745e9i	0.187017i
$931$	− 8.33600e9i	− 0.338558i
$932$	−7.65160e9	−0.309597
$933$	−9.50474e9	−0.383137
$934$	− 2.12533e10i	− 0.853519i
$935$	3.37238e9	0.134926
$936$	0	0
$937$	−3.87626e10	−1.53930	−0.769652	−	0.638463i	$-0.779571\pi$
$937$	−3.87626e10	−1.53930	−0.769652	+	0.638463i	$0.779571\pi$
$938$	− 1.02416e9i	− 0.0405189i
$939$	−1.42149e9	−0.0560291
$940$	6.35250e9	0.249458
$941$	− 2.06279e10i	− 0.807035i	−0.914972	−	0.403517i	$-0.867788\pi$
$941$	− 2.06279e10i	− 0.807035i	0.914972	−	0.403517i	$-0.132212\pi$
$942$	5.81328e9i	0.226591i
$943$	7.44976e8i	0.0289302i
$944$	1.08161e10i	0.418476i
$945$	1.08302e10	0.417468
$946$	−5.51021e9	−0.211617
$947$	− 2.11705e10i	− 0.810040i	−0.914308	−	0.405020i	$-0.867264\pi$
$947$	− 2.11705e10i	− 0.810040i	0.914308	−	0.405020i	$-0.132736\pi$
$948$	2.73930e9	0.104427
$949$	0	0
$950$	−1.08717e10	−0.411399
$951$	6.08771e8i	0.0229521i
$952$	−7.64994e9	−0.287362
$953$	−2.14876e10	−0.804196	−0.402098	−	0.915597i	$-0.631719\pi$
$953$	−2.14876e10	−0.804196	−0.402098	+	0.915597i	$0.631719\pi$
$954$	2.44182e10i	0.910531i
$955$	1.97371e10i	0.733282i
$956$	2.53574e10i	0.938646i
$957$	− 1.34408e9i	− 0.0495715i
$958$	−1.04075e10	−0.382442
$959$	3.20704e8	0.0117419
$960$	− 6.60603e8i	− 0.0240986i
$961$	−2.42673e10	−0.882043
$962$	0	0
$963$	3.67445e10	1.32587
$964$	1.64228e10i	0.590443i
$965$	7.39086e9	0.264758
$966$	6.70189e9	0.239209
$967$	− 3.92625e10i	− 1.39632i	−0.715941	−	0.698161i	$-0.754002\pi$
$967$	− 3.92625e10i	− 1.39632i	0.715941	−	0.698161i	$-0.245998\pi$
$968$	− 9.36689e9i	− 0.331919i
$969$	− 7.04829e9i	− 0.248857i
$970$	1.45860e10i	0.513139i
$971$	−5.62647e10	−1.97228	−0.986140	−	0.165917i	$-0.946941\pi$
$971$	−5.62647e10	−1.97228	−0.986140	+	0.165917i	$0.946941\pi$
$972$	−1.00684e10	−0.351665
$973$	3.98310e10i	1.38620i
$974$	−8.59573e9	−0.298076
$975$	0	0
$976$	3.39027e9	0.116724
$977$	8.43437e9i	0.289349i	0.989479	+	0.144674i	$0.0462135\pi$
$977$	8.43437e9i	0.289349i	−0.989479	+	0.144674i	$0.953787\pi$
$978$	−5.48019e9	−0.187331
$979$	1.30507e10	0.444523
$980$	2.80510e9i	0.0952045i
$981$	2.50411e10i	0.846861i
$982$	6.26675e9i	0.211180i
$983$	− 2.24230e10i	− 0.752932i	−0.926430	−	0.376466i	$-0.877139\pi$
$983$	− 2.24230e10i	− 0.752932i	0.926430	−	0.376466i	$-0.122861\pi$
$984$	−6.66132e7	−0.00222883
$985$	2.16269e10	0.721054
$986$	1.20672e10i	0.400900i
$987$	−5.76262e9	−0.190770
$988$	0	0
$989$	−4.33400e10	−1.42463
$990$	3.74801e9i	0.122766i
$991$	3.46728e10	1.13170	0.565849	−	0.824509i	$-0.308548\pi$
$991$	3.46728e10	1.13170	0.565849	+	0.824509i	$0.308548\pi$
$992$	7.45642e9	0.242516
$993$	− 3.28508e9i	− 0.106469i
$994$	− 1.14989e10i	− 0.371368i
$995$	− 1.75639e10i	− 0.565249i
$996$	4.35678e9i	0.139720i
$997$	−2.96474e10	−0.947444	−0.473722	−	0.880674i	$-0.657090\pi$
$997$	−2.96474e10	−0.947444	−0.473722	+	0.880674i	$0.657090\pi$
$998$	−4.98550e9	−0.158764
$999$	− 8.14830e9i	− 0.258576i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.8.b.d.337.2		2
13.5	odd	4		338.8.a.d.1.1		1
13.8	odd	4		2.8.a.a.1.1	✓	1
13.12	even	2	inner	338.8.b.d.337.1		2
39.8	even	4		18.8.a.b.1.1		1
52.47	even	4		16.8.a.b.1.1		1
65.8	even	4		50.8.b.c.49.2		2
65.34	odd	4		50.8.a.g.1.1		1
65.47	even	4		50.8.b.c.49.1		2
91.34	even	4		98.8.a.a.1.1		1
91.47	even	12		98.8.c.e.67.1		2
91.60	odd	12		98.8.c.d.79.1		2
91.73	even	12		98.8.c.e.79.1		2
91.86	odd	12		98.8.c.d.67.1		2
104.21	odd	4		64.8.a.c.1.1		1
104.99	even	4		64.8.a.e.1.1		1
117.34	odd	12		162.8.c.l.109.1		2
117.47	even	12		162.8.c.a.109.1		2
117.86	even	12		162.8.c.a.55.1		2
117.112	odd	12		162.8.c.l.55.1		2
143.21	even	4		242.8.a.e.1.1		1
156.47	odd	4		144.8.a.i.1.1		1
195.8	odd	4		450.8.c.g.199.1		2
195.47	odd	4		450.8.c.g.199.2		2
195.164	even	4		450.8.a.c.1.1		1
208.21	odd	4		256.8.b.b.129.1		2
208.99	even	4		256.8.b.f.129.1		2
208.125	odd	4		256.8.b.b.129.2		2
208.203	even	4		256.8.b.f.129.2		2
221.203	odd	4		578.8.a.b.1.1		1
260.47	odd	4		400.8.c.j.49.2		2
260.99	even	4		400.8.a.l.1.1		1
260.203	odd	4		400.8.c.j.49.1		2
312.125	even	4		576.8.a.g.1.1		1
312.203	odd	4		576.8.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.8.a.a.1.1	✓	1	13.8	odd	4
16.8.a.b.1.1		1	52.47	even	4
18.8.a.b.1.1		1	39.8	even	4
50.8.a.g.1.1		1	65.34	odd	4
50.8.b.c.49.1		2	65.47	even	4
50.8.b.c.49.2		2	65.8	even	4
64.8.a.c.1.1		1	104.21	odd	4
64.8.a.e.1.1		1	104.99	even	4
98.8.a.a.1.1		1	91.34	even	4
98.8.c.d.67.1		2	91.86	odd	12
98.8.c.d.79.1		2	91.60	odd	12
98.8.c.e.67.1		2	91.47	even	12
98.8.c.e.79.1		2	91.73	even	12
144.8.a.i.1.1		1	156.47	odd	4
162.8.c.a.55.1		2	117.86	even	12
162.8.c.a.109.1		2	117.47	even	12
162.8.c.l.55.1		2	117.112	odd	12
162.8.c.l.109.1		2	117.34	odd	12
242.8.a.e.1.1		1	143.21	even	4
256.8.b.b.129.1		2	208.21	odd	4
256.8.b.b.129.2		2	208.125	odd	4
256.8.b.f.129.1		2	208.99	even	4
256.8.b.f.129.2		2	208.203	even	4
338.8.a.d.1.1		1	13.5	odd	4
338.8.b.d.337.1		2	13.12	even	2	inner
338.8.b.d.337.2		2	1.1	even	1	trivial
400.8.a.l.1.1		1	260.99	even	4
400.8.c.j.49.1		2	260.203	odd	4
400.8.c.j.49.2		2	260.47	odd	4
450.8.a.c.1.1		1	195.164	even	4
450.8.c.g.199.1		2	195.8	odd	4
450.8.c.g.199.2		2	195.47	odd	4
576.8.a.f.1.1		1	312.203	odd	4
576.8.a.g.1.1		1	312.125	even	4
578.8.a.b.1.1		1	221.203	odd	4

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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	338.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{2} +12.0000 q^{3} -64.0000 q^{4} +210.000i q^{5} +96.0000i q^{6} +1016.00i q^{7} -512.000i q^{8} -2043.00 q^{9} -1680.00 q^{10} +1092.00i q^{11} -768.000 q^{12} -8128.00 q^{14} +2520.00i q^{15} +4096.00 q^{16} -14706.0 q^{17} -16344.0i q^{18} +39940.0i q^{19} -13440.0i q^{20} +12192.0i q^{21} -8736.00 q^{22} -68712.0 q^{23} -6144.00i q^{24} +34025.0 q^{25} -50760.0 q^{27} -65024.0i q^{28} -102570. q^{29} -20160.0 q^{30} -227552. i q^{31} +32768.0i q^{32} +13104.0i q^{33} -117648. i q^{34} -213360. q^{35} +130752. q^{36} +160526. i q^{37} -319520. q^{38} +107520. q^{40} -10842.0i q^{41} -97536.0 q^{42} +630748. q^{43} -69888.0i q^{44} -429030. i q^{45} -549696. i q^{46} +472656. i q^{47} +49152.0 q^{48} -208713. q^{49} +272200. i q^{50} -176472. q^{51} -1.49402e6 q^{53} -406080. i q^{54} -229320. q^{55} +520192. q^{56} +479280. i q^{57} -820560. i q^{58} +2.64066e6i q^{59} -161280. i q^{60} +827702. q^{61} +1.82042e6 q^{62} -2.07569e6i q^{63} -262144. q^{64} -104832. q^{66} +126004. i q^{67} +941184. q^{68} -824544. q^{69} -1.70688e6i q^{70} +1.41473e6i q^{71} +1.04602e6i q^{72} +980282. i q^{73} -1.28421e6 q^{74} +408300. q^{75} -2.55616e6i q^{76} -1.10947e6 q^{77} -3.56680e6 q^{79} +860160. i q^{80} +3.85892e6 q^{81} +86736.0 q^{82} -5.67289e6i q^{83} -780288. i q^{84} -3.08826e6i q^{85} +5.04598e6i q^{86} -1.23084e6 q^{87} +559104. q^{88} -1.19512e7i q^{89} +3.43224e6 q^{90} +4.39757e6 q^{92} -2.73062e6i q^{93} -3.78125e6 q^{94} -8.38740e6 q^{95} +393216. i q^{96} -8.68215e6i q^{97} -1.66970e6i q^{98} -2.23096e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 24 q^{3} - 128 q^{4} - 4086 q^{9} - 3360 q^{10} - 1536 q^{12} - 16256 q^{14} + 8192 q^{16} - 29412 q^{17} - 17472 q^{22} - 137424 q^{23} + 68050 q^{25} - 101520 q^{27} - 205140 q^{29} - 40320 q^{30}+ \cdots - 16774800 q^{95}+O(q^{100}) \) 2 * q + 24 * q^3 - 128 * q^4 - 4086 * q^9 - 3360 * q^10 - 1536 * q^12 - 16256 * q^14 + 8192 * q^16 - 29412 * q^17 - 17472 * q^22 - 137424 * q^23 + 68050 * q^25 - 101520 * q^27 - 205140 * q^29 - 40320 * q^30 - 426720 * q^35 + 261504 * q^36 - 639040 * q^38 + 215040 * q^40 - 195072 * q^42 + 1261496 * q^43 + 98304 * q^48 - 417426 * q^49 - 352944 * q^51 - 2988036 * q^53 - 458640 * q^55 + 1040384 * q^56 + 1655404 * q^61 + 3640832 * q^62 - 524288 * q^64 - 209664 * q^66 + 1882368 * q^68 - 1649088 * q^69 - 2568416 * q^74 + 816600 * q^75 - 2218944 * q^77 - 7133600 * q^79 + 7717842 * q^81 + 173472 * q^82 - 2461680 * q^87 + 1118208 * q^88 + 6864480 * q^90 + 8795136 * q^92 - 7562496 * q^94 - 16774800 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more