LMFDB - Embedded newform 162.8.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,8,Mod(1,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

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:	yes
Analytic conductor:	$50.6063741284$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.43103376.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 383x^{2} + 384x + 18612 $ x^4 - 2x^3 - 383x^2 + 384*x + 18612
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-7.05976$ of defining polynomial
Character	$\chi$	$=$	162.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.00000 q^{2} +64.0000 q^{4} -11.2463 q^{5} +1439.47 q^{7} -512.000 q^{8} +89.9705 q^{10} -4432.63 q^{11} -7230.97 q^{13} -11515.8 q^{14} +4096.00 q^{16} -13221.1 q^{17} +13506.6 q^{19} -719.764 q^{20} +35461.0 q^{22} +65738.6 q^{23} -77998.5 q^{25} +57847.8 q^{26} +92126.0 q^{28} -148498. q^{29} +138862. q^{31} -32768.0 q^{32} +105769. q^{34} -16188.7 q^{35} -121532. q^{37} -108053. q^{38} +5758.11 q^{40} -52786.9 q^{41} +751724. q^{43} -283688. q^{44} -525909. q^{46} -983593. q^{47} +1.24853e6 q^{49} +623988. q^{50} -462782. q^{52} +621759. q^{53} +49850.7 q^{55} -737008. q^{56} +1.18798e6 q^{58} -814216. q^{59} -2.46953e6 q^{61} -1.11090e6 q^{62} +262144. q^{64} +81321.8 q^{65} +3.45482e6 q^{67} -846150. q^{68} +129510. q^{70} -3.64076e6 q^{71} -5.55060e6 q^{73} +972253. q^{74} +864424. q^{76} -6.38063e6 q^{77} -2.04636e6 q^{79} -46064.9 q^{80} +422295. q^{82} -1.02652e7 q^{83} +148689. q^{85} -6.01379e6 q^{86} +2.26951e6 q^{88} -8.12374e6 q^{89} -1.04088e7 q^{91} +4.20727e6 q^{92} +7.86874e6 q^{94} -151900. q^{95} -1.32932e7 q^{97} -9.98823e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} + 256 q^{4} - 528 q^{5} + 560 q^{7} - 2048 q^{8} + 4224 q^{10} - 2160 q^{11} + 13460 q^{13} - 4480 q^{14} + 16384 q^{16} - 22560 q^{17} + 36704 q^{19} - 33792 q^{20} + 17280 q^{22} - 62640 q^{23}+ \cdots - 11595168 q^{98}+O(q^{100}) $ 4 * q - 32 * q^2 + 256 * q^4 - 528 * q^5 + 560 * q^7 - 2048 * q^8 + 4224 * q^10 - 2160 * q^11 + 13460 * q^13 - 4480 * q^14 + 16384 * q^16 - 22560 * q^17 + 36704 * q^19 - 33792 * q^20 + 17280 * q^22 - 62640 * q^23 + 94696 * q^25 - 107680 * q^26 + 35840 * q^28 - 68400 * q^29 + 227504 * q^31 - 131072 * q^32 + 180480 * q^34 + 63024 * q^35 + 523580 * q^37 - 293632 * q^38 + 270336 * q^40 - 67200 * q^41 + 562640 * q^43 - 138240 * q^44 + 501120 * q^46 - 515328 * q^47 + 1449396 * q^49 - 757568 * q^50 + 861440 * q^52 - 2498016 * q^53 + 1322784 * q^55 - 286720 * q^56 + 547200 * q^58 - 4155840 * q^59 + 2130764 * q^61 - 1820032 * q^62 + 1048576 * q^64 - 8966496 * q^65 - 1205440 * q^67 - 1443840 * q^68 - 504192 * q^70 - 12486480 * q^71 - 4820860 * q^73 - 4188640 * q^74 + 2349056 * q^76 - 18686592 * q^77 - 11471680 * q^79 - 2162688 * q^80 + 537600 * q^82 - 16811232 * q^83 - 21006396 * q^85 - 4501120 * q^86 + 1105920 * q^88 - 16857600 * q^89 - 23200544 * q^91 - 4008960 * q^92 + 4122624 * q^94 - 14090160 * q^95 - 27078040 * q^97 - 11595168 * q^98

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	−11.2463	−0.0402360	−0.0201180	−	0.999798i	$-0.506404\pi$
$5$	−11.2463	−0.0402360	−0.0201180	+	0.999798i	$0.506404\pi$
$6$	0	0
$7$	1439.47	1.58620	0.793102	−	0.609088i	$-0.208465\pi$
$7$	1439.47	1.58620	0.793102	+	0.609088i	$0.208465\pi$
$8$	−512.000	−0.353553
$9$	0	0
$10$	89.9705	0.0284512
$11$	−4432.63	−1.00412	−0.502061	−	0.864832i	$-0.667425\pi$
$11$	−4432.63	−1.00412	−0.502061	+	0.864832i	$0.667425\pi$
$12$	0	0
$13$	−7230.97	−0.912841	−0.456420	−	0.889764i	$-0.650869\pi$
$13$	−7230.97	−0.912841	−0.456420	+	0.889764i	$0.650869\pi$
$14$	−11515.8	−1.12162
$15$	0	0
$16$	4096.00	0.250000
$17$	−13221.1	−0.652674	−0.326337	−	0.945254i	$-0.605814\pi$
$17$	−13221.1	−0.652674	−0.326337	+	0.945254i	$0.605814\pi$
$18$	0	0
$19$	13506.6	0.451761	0.225881	−	0.974155i	$-0.427474\pi$
$19$	13506.6	0.451761	0.225881	+	0.974155i	$0.427474\pi$
$20$	−719.764	−0.0201180
$21$	0	0
$22$	35461.0	0.710022
$23$	65738.6	1.12661	0.563304	−	0.826250i	$-0.309530\pi$
$23$	65738.6	1.12661	0.563304	+	0.826250i	$0.309530\pi$
$24$	0	0
$25$	−77998.5	−0.998381
$26$	57847.8	0.645476
$27$	0	0
$28$	92126.0	0.793102
$29$	−148498.	−1.13065	−0.565325	−	0.824868i	$-0.691249\pi$
$29$	−148498.	−1.13065	−0.565325	+	0.824868i	$0.691249\pi$
$30$	0	0
$31$	138862.	0.837180	0.418590	−	0.908175i	$-0.362525\pi$
$31$	138862.	0.837180	0.418590	+	0.908175i	$0.362525\pi$
$32$	−32768.0	−0.176777
$33$	0	0
$34$	105769.	0.461510
$35$	−16188.7	−0.0638226
$36$	0	0
$37$	−121532.	−0.394442	−0.197221	−	0.980359i	$-0.563192\pi$
$37$	−121532.	−0.394442	−0.197221	+	0.980359i	$0.563192\pi$
$38$	−108053.	−0.319444
$39$	0	0
$40$	5758.11	0.0142256
$41$	−52786.9	−0.119614	−0.0598070	−	0.998210i	$-0.519049\pi$
$41$	−52786.9	−0.119614	−0.0598070	+	0.998210i	$0.519049\pi$
$42$	0	0
$43$	751724.	1.44185	0.720923	−	0.693015i	$-0.243718\pi$
$43$	751724.	1.44185	0.720923	+	0.693015i	$0.243718\pi$
$44$	−283688.	−0.502061
$45$	0	0
$46$	−525909.	−0.796633
$47$	−983593.	−1.38189	−0.690944	−	0.722908i	$-0.742805\pi$
$47$	−983593.	−1.38189	−0.690944	+	0.722908i	$0.742805\pi$
$48$	0	0
$49$	1.24853e6	1.51605
$50$	623988.	0.705962
$51$	0	0
$52$	−462782.	−0.456420
$53$	621759.	0.573663	0.286831	−	0.957981i	$-0.407398\pi$
$53$	621759.	0.573663	0.286831	+	0.957981i	$0.407398\pi$
$54$	0	0
$55$	49850.7	0.0404019
$56$	−737008.	−0.560808
$57$	0	0
$58$	1.18798e6	0.799490
$59$	−814216.	−0.516128	−0.258064	−	0.966128i	$-0.583085\pi$
$59$	−814216.	−0.516128	−0.258064	+	0.966128i	$0.583085\pi$
$60$	0	0
$61$	−2.46953e6	−1.39303	−0.696514	−	0.717543i	$-0.745267\pi$
$61$	−2.46953e6	−1.39303	−0.696514	+	0.717543i	$0.745267\pi$
$62$	−1.11090e6	−0.591975
$63$	0	0
$64$	262144.	0.125000
$65$	81321.8	0.0367291
$66$	0	0
$67$	3.45482e6	1.40334	0.701671	−	0.712502i	$-0.252438\pi$
$67$	3.45482e6	1.40334	0.701671	+	0.712502i	$0.252438\pi$
$68$	−846150.	−0.326337
$69$	0	0
$70$	129510.	0.0451294
$71$	−3.64076e6	−1.20722	−0.603612	−	0.797278i	$-0.706272\pi$
$71$	−3.64076e6	−1.20722	−0.603612	+	0.797278i	$0.706272\pi$
$72$	0	0
$73$	−5.55060e6	−1.66997	−0.834987	−	0.550269i	$-0.814525\pi$
$73$	−5.55060e6	−1.66997	−0.834987	+	0.550269i	$0.814525\pi$
$74$	972253.	0.278913
$75$	0	0
$76$	864424.	0.225881
$77$	−6.38063e6	−1.59274
$78$	0	0
$79$	−2.04636e6	−0.466969	−0.233484	−	0.972361i	$-0.575013\pi$
$79$	−2.04636e6	−0.466969	−0.233484	+	0.972361i	$0.575013\pi$
$80$	−46064.9	−0.0100590
$81$	0	0
$82$	422295.	0.0845799
$83$	−1.02652e7	−1.97057	−0.985286	−	0.170911i	$-0.945329\pi$
$83$	−1.02652e7	−1.97057	−0.985286	+	0.170911i	$0.945329\pi$
$84$	0	0
$85$	148689.	0.0262610
$86$	−6.01379e6	−1.01954
$87$	0	0
$88$	2.26951e6	0.355011
$89$	−8.12374e6	−1.22149	−0.610747	−	0.791826i	$-0.709131\pi$
$89$	−8.12374e6	−1.22149	−0.610747	+	0.791826i	$0.709131\pi$
$90$	0	0
$91$	−1.04088e7	−1.44795
$92$	4.20727e6	0.563304
$93$	0	0
$94$	7.86874e6	0.977142
$95$	−151900.	−0.0181771
$96$	0	0
$97$	−1.32932e7	−1.47886	−0.739432	−	0.673231i	$-0.764906\pi$
$97$	−1.32932e7	−1.47886	−0.739432	+	0.673231i	$0.764906\pi$
$98$	−9.98823e6	−1.07201
$99$	0	0
$100$	−4.99191e6	−0.499191
$101$	1.33665e7	1.29090	0.645452	−	0.763801i	$-0.276669\pi$
$101$	1.33665e7	1.29090	0.645452	+	0.763801i	$0.276669\pi$
$102$	0	0
$103$	1.35323e7	1.22023	0.610114	−	0.792314i	$-0.291124\pi$
$103$	1.35323e7	1.22023	0.610114	+	0.792314i	$0.291124\pi$
$104$	3.70226e6	0.322738
$105$	0	0
$106$	−4.97407e6	−0.405641
$107$	1.89550e7	1.49583	0.747914	−	0.663796i	$-0.231056\pi$
$107$	1.89550e7	1.49583	0.747914	+	0.663796i	$0.231056\pi$
$108$	0	0
$109$	−1.33257e7	−0.985589	−0.492794	−	0.870146i	$-0.664025\pi$
$109$	−1.33257e7	−0.985589	−0.492794	+	0.870146i	$0.664025\pi$
$110$	−398806.	−0.0285685
$111$	0	0
$112$	5.89607e6	0.396551
$113$	−2.04267e7	−1.33176	−0.665878	−	0.746061i	$-0.731943\pi$
$113$	−2.04267e7	−1.33176	−0.665878	+	0.746061i	$0.731943\pi$
$114$	0	0
$115$	−739317.	−0.0453303
$116$	−9.50388e6	−0.565325
$117$	0	0
$118$	6.51373e6	0.364958
$119$	−1.90314e7	−1.03527
$120$	0	0
$121$	161021.	0.00826294
$122$	1.97562e7	0.985020
$123$	0	0
$124$	8.88719e6	0.418590
$125$	1.75581e6	0.0804069
$126$	0	0
$127$	1.13846e7	0.493178	0.246589	−	0.969120i	$-0.420690\pi$
$127$	1.13846e7	0.493178	0.246589	+	0.969120i	$0.420690\pi$
$128$	−2.09715e6	−0.0883883
$129$	0	0
$130$	−650574.	−0.0259714
$131$	−5.28179e6	−0.205273	−0.102636	−	0.994719i	$-0.532728\pi$
$131$	−5.28179e6	−0.205273	−0.102636	+	0.994719i	$0.532728\pi$
$132$	0	0
$133$	1.94424e7	0.716586
$134$	−2.76385e7	−0.992312
$135$	0	0
$136$	6.76920e6	0.230755
$137$	−4.90114e7	−1.62845	−0.814226	−	0.580548i	$-0.802838\pi$
$137$	−4.90114e7	−1.62845	−0.814226	+	0.580548i	$0.802838\pi$
$138$	0	0
$139$	−1.88794e7	−0.596262	−0.298131	−	0.954525i	$-0.596363\pi$
$139$	−1.88794e7	−0.596262	−0.298131	+	0.954525i	$0.596363\pi$
$140$	−1.03608e6	−0.0319113
$141$	0	0
$142$	2.91261e7	0.853636
$143$	3.20522e7	0.916604
$144$	0	0
$145$	1.67006e6	0.0454929
$146$	4.44048e7	1.18085
$147$	0	0
$148$	−7.77802e6	−0.197221
$149$	2.88497e7	0.714480	0.357240	−	0.934013i	$-0.383718\pi$
$149$	2.88497e7	0.714480	0.357240	+	0.934013i	$0.383718\pi$
$150$	0	0
$151$	−3.59533e7	−0.849804	−0.424902	−	0.905239i	$-0.639692\pi$
$151$	−3.59533e7	−0.849804	−0.424902	+	0.905239i	$0.639692\pi$
$152$	−6.91539e6	−0.159722
$153$	0	0
$154$	5.10451e7	1.12624
$155$	−1.56169e6	−0.0336848
$156$	0	0
$157$	−5.78090e7	−1.19219	−0.596097	−	0.802913i	$-0.703283\pi$
$157$	−5.78090e7	−1.19219	−0.596097	+	0.802913i	$0.703283\pi$
$158$	1.63709e7	0.330197
$159$	0	0
$160$	368519.	0.00711279
$161$	9.46287e7	1.78703
$162$	0	0
$163$	3.23536e7	0.585148	0.292574	−	0.956243i	$-0.405488\pi$
$163$	3.23536e7	0.585148	0.292574	+	0.956243i	$0.405488\pi$
$164$	−3.37836e6	−0.0598070
$165$	0	0
$166$	8.21213e7	1.39341
$167$	8.67745e7	1.44173	0.720866	−	0.693075i	$-0.243744\pi$
$167$	8.67745e7	1.44173	0.720866	+	0.693075i	$0.243744\pi$
$168$	0	0
$169$	−1.04615e7	−0.166722
$170$	−1.18951e6	−0.0185693
$171$	0	0
$172$	4.81103e7	0.720923
$173$	1.07724e7	0.158179	0.0790897	−	0.996868i	$-0.474799\pi$
$173$	1.07724e7	0.158179	0.0790897	+	0.996868i	$0.474799\pi$
$174$	0	0
$175$	−1.12276e8	−1.58364
$176$	−1.81560e7	−0.251031
$177$	0	0
$178$	6.49900e7	0.863726
$179$	3.61175e7	0.470687	0.235343	−	0.971912i	$-0.424379\pi$
$179$	3.61175e7	0.470687	0.235343	+	0.971912i	$0.424379\pi$
$180$	0	0
$181$	2.78118e7	0.348621	0.174310	−	0.984691i	$-0.444230\pi$
$181$	2.78118e7	0.348621	0.174310	+	0.984691i	$0.444230\pi$
$182$	8.32701e7	1.02386
$183$	0	0
$184$	−3.36582e7	−0.398316
$185$	1.36678e6	0.0158708
$186$	0	0
$187$	5.86042e7	0.655365
$188$	−6.29499e7	−0.690944
$189$	0	0
$190$	1.21520e6	0.0128531
$191$	9.46254e7	0.982632	0.491316	−	0.870981i	$-0.336516\pi$
$191$	9.46254e7	0.982632	0.491316	+	0.870981i	$0.336516\pi$
$192$	0	0
$193$	−1.22223e8	−1.22378	−0.611889	−	0.790944i	$-0.709590\pi$
$193$	−1.22223e8	−1.22378	−0.611889	+	0.790944i	$0.709590\pi$
$194$	1.06346e8	1.04572
$195$	0	0
$196$	7.99058e7	0.758023
$197$	1.37768e8	1.28385	0.641926	−	0.766766i	$-0.278136\pi$
$197$	1.37768e8	1.28385	0.641926	+	0.766766i	$0.278136\pi$
$198$	0	0
$199$	−1.62646e8	−1.46305	−0.731524	−	0.681815i	$-0.761191\pi$
$199$	−1.62646e8	−1.46305	−0.731524	+	0.681815i	$0.761191\pi$
$200$	3.99352e7	0.352981
$201$	0	0
$202$	−1.06932e8	−0.912807
$203$	−2.13758e8	−1.79344
$204$	0	0
$205$	593658.	0.00481280
$206$	−1.08258e8	−0.862831
$207$	0	0
$208$	−2.96181e7	−0.228210
$209$	−5.98698e7	−0.453624
$210$	0	0
$211$	−1.55578e8	−1.14014	−0.570071	−	0.821595i	$-0.693084\pi$
$211$	−1.55578e8	−1.14014	−0.570071	+	0.821595i	$0.693084\pi$
$212$	3.97926e7	0.286831
$213$	0	0
$214$	−1.51640e8	−1.05771
$215$	−8.45413e6	−0.0580142
$216$	0	0
$217$	1.99888e8	1.32794
$218$	1.06605e8	0.696917
$219$	0	0
$220$	3.19045e6	0.0202010
$221$	9.56014e7	0.595787
$222$	0	0
$223$	−1.23943e8	−0.748435	−0.374218	−	0.927341i	$-0.622089\pi$
$223$	−1.23943e8	−0.748435	−0.374218	+	0.927341i	$0.622089\pi$
$224$	−4.71685e7	−0.280404
$225$	0	0
$226$	1.63414e8	0.941694
$227$	1.37704e8	0.781369	0.390685	−	0.920525i	$-0.372238\pi$
$227$	1.37704e8	0.781369	0.390685	+	0.920525i	$0.372238\pi$
$228$	0	0
$229$	2.35121e8	1.29380	0.646899	−	0.762575i	$-0.276065\pi$
$229$	2.35121e8	1.29380	0.646899	+	0.762575i	$0.276065\pi$
$230$	5.91454e6	0.0320533
$231$	0	0
$232$	7.60310e7	0.399745
$233$	1.44039e8	0.745992	0.372996	−	0.927833i	$-0.378330\pi$
$233$	1.44039e8	0.745992	0.372996	+	0.927833i	$0.378330\pi$
$234$	0	0
$235$	1.10618e7	0.0556017
$236$	−5.21098e7	−0.258064
$237$	0	0
$238$	1.52251e8	0.732050
$239$	−7.31386e7	−0.346541	−0.173270	−	0.984874i	$-0.555433\pi$
$239$	−7.31386e7	−0.346541	−0.173270	+	0.984874i	$0.555433\pi$
$240$	0	0
$241$	−1.22840e8	−0.565300	−0.282650	−	0.959223i	$-0.591214\pi$
$241$	−1.22840e8	−0.565300	−0.282650	+	0.959223i	$0.591214\pi$
$242$	−1.28817e6	−0.00584278
$243$	0	0
$244$	−1.58050e8	−0.696514
$245$	−1.40413e7	−0.0609997
$246$	0	0
$247$	−9.76660e7	−0.412386
$248$	−7.10975e7	−0.295988
$249$	0	0
$250$	−1.40465e7	−0.0568563
$251$	−7.30348e7	−0.291522	−0.145761	−	0.989320i	$-0.546563\pi$
$251$	−7.30348e7	−0.291522	−0.145761	+	0.989320i	$0.546563\pi$
$252$	0	0
$253$	−2.91395e8	−1.13125
$254$	−9.10766e7	−0.348729
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−1.33996e8	−0.492408	−0.246204	−	0.969218i	$-0.579183\pi$
$257$	−1.33996e8	−0.492408	−0.246204	+	0.969218i	$0.579183\pi$
$258$	0	0
$259$	−1.74941e8	−0.625666
$260$	5.20460e6	0.0183646
$261$	0	0
$262$	4.22543e7	0.145150
$263$	−3.80130e8	−1.28851	−0.644253	−	0.764812i	$-0.722832\pi$
$263$	−3.80130e8	−1.28851	−0.644253	+	0.764812i	$0.722832\pi$
$264$	0	0
$265$	−6.99250e6	−0.0230819
$266$	−1.55539e8	−0.506703
$267$	0	0
$268$	2.21108e8	0.701671
$269$	−5.57685e8	−1.74685	−0.873426	−	0.486958i	$-0.838107\pi$
$269$	−5.57685e8	−1.74685	−0.873426	+	0.486958i	$0.838107\pi$
$270$	0	0
$271$	−9.65793e7	−0.294776	−0.147388	−	0.989079i	$-0.547087\pi$
$271$	−9.65793e7	−0.294776	−0.147388	+	0.989079i	$0.547087\pi$
$272$	−5.41536e7	−0.163168
$273$	0	0
$274$	3.92091e8	1.15149
$275$	3.45738e8	1.00250
$276$	0	0
$277$	5.65800e8	1.59950	0.799749	−	0.600334i	$-0.204966\pi$
$277$	5.65800e8	1.59950	0.799749	+	0.600334i	$0.204966\pi$
$278$	1.51035e8	0.421621
$279$	0	0
$280$	8.28863e6	0.0225647
$281$	−3.99550e8	−1.07423	−0.537117	−	0.843508i	$-0.680487\pi$
$281$	−3.99550e8	−1.07423	−0.537117	+	0.843508i	$0.680487\pi$
$282$	0	0
$283$	6.78723e8	1.78008	0.890040	−	0.455882i	$-0.150676\pi$
$283$	6.78723e8	1.78008	0.890040	+	0.455882i	$0.150676\pi$
$284$	−2.33009e8	−0.603612
$285$	0	0
$286$	−2.56418e8	−0.648137
$287$	−7.59851e7	−0.189732
$288$	0	0
$289$	−2.35541e8	−0.574017
$290$	−1.33605e7	−0.0321683
$291$	0	0
$292$	−3.55238e8	−0.834987
$293$	−1.11739e8	−0.259518	−0.129759	−	0.991546i	$-0.541420\pi$
$293$	−1.11739e8	−0.259518	−0.129759	+	0.991546i	$0.541420\pi$
$294$	0	0
$295$	9.15693e6	0.0207670
$296$	6.22242e7	0.139456
$297$	0	0
$298$	−2.30798e8	−0.505214
$299$	−4.75354e8	−1.02841
$300$	0	0
$301$	1.08208e9	2.28706
$302$	2.87626e8	0.600903
$303$	0	0
$304$	5.53231e7	0.112940
$305$	2.77731e7	0.0560500
$306$	0	0
$307$	−1.53022e8	−0.301835	−0.150917	−	0.988546i	$-0.548223\pi$
$307$	−1.53022e8	−0.301835	−0.150917	+	0.988546i	$0.548223\pi$
$308$	−4.08360e8	−0.796372
$309$	0	0
$310$	1.24935e7	0.0238187
$311$	2.28368e8	0.430501	0.215250	−	0.976559i	$-0.430943\pi$
$311$	2.28368e8	0.430501	0.215250	+	0.976559i	$0.430943\pi$
$312$	0	0
$313$	−7.02744e8	−1.29536	−0.647682	−	0.761910i	$-0.724262\pi$
$313$	−7.02744e8	−1.29536	−0.647682	+	0.761910i	$0.724262\pi$
$314$	4.62472e8	0.843008
$315$	0	0
$316$	−1.30967e8	−0.233484
$317$	1.04118e9	1.83577	0.917883	−	0.396852i	$-0.129897\pi$
$317$	1.04118e9	1.83577	0.917883	+	0.396852i	$0.129897\pi$
$318$	0	0
$319$	6.58237e8	1.13531
$320$	−2.94815e6	−0.00502951
$321$	0	0
$322$	−7.57030e8	−1.26362
$323$	−1.78572e8	−0.294853
$324$	0	0
$325$	5.64005e8	0.911363
$326$	−2.58829e8	−0.413762
$327$	0	0
$328$	2.70269e7	0.0422900
$329$	−1.41585e9	−2.19196
$330$	0	0
$331$	3.21275e8	0.486944	0.243472	−	0.969908i	$-0.421714\pi$
$331$	3.21275e8	0.486944	0.243472	+	0.969908i	$0.421714\pi$
$332$	−6.56970e8	−0.985286
$333$	0	0
$334$	−6.94196e8	−1.01946
$335$	−3.88540e7	−0.0564649
$336$	0	0
$337$	−9.13646e8	−1.30039	−0.650194	−	0.759768i	$-0.725313\pi$
$337$	−9.13646e8	−1.30039	−0.650194	+	0.759768i	$0.725313\pi$
$338$	8.36923e7	0.117890
$339$	0	0
$340$	9.51607e6	0.0131305
$341$	−6.15525e8	−0.840631
$342$	0	0
$343$	6.11754e8	0.818554
$344$	−3.84883e8	−0.509770
$345$	0	0
$346$	−8.61789e7	−0.111850
$347$	5.97984e8	0.768310	0.384155	−	0.923269i	$-0.374493\pi$
$347$	5.97984e8	0.768310	0.384155	+	0.923269i	$0.374493\pi$
$348$	0	0
$349$	6.50546e8	0.819199	0.409599	−	0.912265i	$-0.365668\pi$
$349$	6.50546e8	0.819199	0.409599	+	0.912265i	$0.365668\pi$
$350$	8.98212e8	1.11980
$351$	0	0
$352$	1.45248e8	0.177506
$353$	6.17246e7	0.0746873	0.0373437	−	0.999302i	$-0.488110\pi$
$353$	6.17246e7	0.0746873	0.0373437	+	0.999302i	$0.488110\pi$
$354$	0	0
$355$	4.09451e7	0.0485739
$356$	−5.19920e8	−0.610747
$357$	0	0
$358$	−2.88940e8	−0.332826
$359$	−1.70542e9	−1.94537	−0.972683	−	0.232137i	$-0.925428\pi$
$359$	−1.70542e9	−1.94537	−0.972683	+	0.232137i	$0.925428\pi$
$360$	0	0
$361$	−7.11443e8	−0.795912
$362$	−2.22494e8	−0.246512
$363$	0	0
$364$	−6.66161e8	−0.723976
$365$	6.24238e7	0.0671932
$366$	0	0
$367$	2.43018e8	0.256630	0.128315	−	0.991733i	$-0.459043\pi$
$367$	2.43018e8	0.256630	0.128315	+	0.991733i	$0.459043\pi$
$368$	2.69265e8	0.281652
$369$	0	0
$370$	−1.09343e7	−0.0112223
$371$	8.95003e8	0.909946
$372$	0	0
$373$	−6.58541e8	−0.657055	−0.328528	−	0.944494i	$-0.606552\pi$
$373$	−6.58541e8	−0.657055	−0.328528	+	0.944494i	$0.606552\pi$
$374$	−4.68834e8	−0.463413
$375$	0	0
$376$	5.03599e8	0.488571
$377$	1.07379e9	1.03210
$378$	0	0
$379$	1.55665e7	0.0146877	0.00734387	−	0.999973i	$-0.497662\pi$
$379$	1.55665e7	0.0146877	0.00734387	+	0.999973i	$0.497662\pi$
$380$	−9.72158e6	−0.00908854
$381$	0	0
$382$	−7.57003e8	−0.694826
$383$	8.67880e8	0.789340	0.394670	−	0.918823i	$-0.370859\pi$
$383$	8.67880e8	0.789340	0.394670	+	0.918823i	$0.370859\pi$
$384$	0	0
$385$	7.17586e7	0.0640857
$386$	9.77784e8	0.865342
$387$	0	0
$388$	−8.50765e8	−0.739432
$389$	1.78512e9	1.53760	0.768799	−	0.639490i	$-0.220855\pi$
$389$	1.78512e9	1.53760	0.768799	+	0.639490i	$0.220855\pi$
$390$	0	0
$391$	−8.69136e8	−0.735308
$392$	−6.39247e8	−0.536003
$393$	0	0
$394$	−1.10214e9	−0.907821
$395$	2.30140e7	0.0187890
$396$	0	0
$397$	1.49554e9	1.19958	0.599792	−	0.800156i	$-0.295250\pi$
$397$	1.49554e9	1.19958	0.599792	+	0.800156i	$0.295250\pi$
$398$	1.30117e9	1.03453
$399$	0	0
$400$	−3.19482e8	−0.249595
$401$	1.04217e9	0.807109	0.403554	−	0.914956i	$-0.367775\pi$
$401$	1.04217e9	0.807109	0.403554	+	0.914956i	$0.367775\pi$
$402$	0	0
$403$	−1.00411e9	−0.764212
$404$	8.55458e8	0.645452
$405$	0	0
$406$	1.71007e9	1.26815
$407$	5.38704e8	0.396068
$408$	0	0
$409$	1.38845e9	1.00345	0.501727	−	0.865026i	$-0.332698\pi$
$409$	1.38845e9	1.00345	0.501727	+	0.865026i	$0.332698\pi$
$410$	−4.74926e6	−0.00340316
$411$	0	0
$412$	8.66066e8	0.610114
$413$	−1.17204e9	−0.818685
$414$	0	0
$415$	1.15445e8	0.0792880
$416$	2.36945e8	0.161369
$417$	0	0
$418$	4.78959e8	0.320761
$419$	1.07093e9	0.711235	0.355617	−	0.934632i	$-0.384271\pi$
$419$	1.07093e9	0.711235	0.355617	+	0.934632i	$0.384271\pi$
$420$	0	0
$421$	6.67885e8	0.436229	0.218114	−	0.975923i	$-0.430009\pi$
$421$	6.67885e8	0.436229	0.218114	+	0.975923i	$0.430009\pi$
$422$	1.24462e9	0.806203
$423$	0	0
$424$	−3.18341e8	−0.202820
$425$	1.03123e9	0.651617
$426$	0	0
$427$	−3.55481e9	−2.20963
$428$	1.21312e9	0.747914
$429$	0	0
$430$	6.76330e7	0.0410222
$431$	8.88266e8	0.534408	0.267204	−	0.963640i	$-0.413900\pi$
$431$	8.88266e8	0.534408	0.267204	+	0.963640i	$0.413900\pi$
$432$	0	0
$433$	1.04197e9	0.616806	0.308403	−	0.951256i	$-0.400205\pi$
$433$	1.04197e9	0.616806	0.308403	+	0.951256i	$0.400205\pi$
$434$	−1.59910e9	−0.938994
$435$	0	0
$436$	−8.52842e8	−0.492794
$437$	8.87906e8	0.508958
$438$	0	0
$439$	2.72360e9	1.53645	0.768223	−	0.640183i	$-0.221141\pi$
$439$	2.72360e9	1.53645	0.768223	+	0.640183i	$0.221141\pi$
$440$	−2.55236e7	−0.0142842
$441$	0	0
$442$	−7.64811e8	−0.421285
$443$	−1.41866e9	−0.775292	−0.387646	−	0.921808i	$-0.626712\pi$
$443$	−1.41866e9	−0.775292	−0.387646	+	0.921808i	$0.626712\pi$
$444$	0	0
$445$	9.13622e7	0.0491481
$446$	9.91543e8	0.529224
$447$	0	0
$448$	3.77348e8	0.198276
$449$	3.79133e9	1.97665	0.988324	−	0.152368i	$-0.0486899\pi$
$449$	3.79133e9	1.97665	0.988324	+	0.152368i	$0.0486899\pi$
$450$	0	0
$451$	2.33984e8	0.120107
$452$	−1.30731e9	−0.665878
$453$	0	0
$454$	−1.10163e9	−0.552511
$455$	1.17060e8	0.0582599
$456$	0	0
$457$	1.87296e9	0.917958	0.458979	−	0.888447i	$-0.348215\pi$
$457$	1.87296e9	0.917958	0.458979	+	0.888447i	$0.348215\pi$
$458$	−1.88097e9	−0.914854
$459$	0	0
$460$	−4.73163e7	−0.0226651
$461$	−5.47394e8	−0.260224	−0.130112	−	0.991499i	$-0.541534\pi$
$461$	−5.47394e8	−0.260224	−0.130112	+	0.991499i	$0.541534\pi$
$462$	0	0
$463$	−1.12278e9	−0.525726	−0.262863	−	0.964833i	$-0.584667\pi$
$463$	−1.12278e9	−0.525726	−0.262863	+	0.964833i	$0.584667\pi$
$464$	−6.08248e8	−0.282662
$465$	0	0
$466$	−1.15231e9	−0.527496
$467$	−2.32612e9	−1.05687	−0.528436	−	0.848973i	$-0.677221\pi$
$467$	−2.32612e9	−1.05687	−0.528436	+	0.848973i	$0.677221\pi$
$468$	0	0
$469$	4.97310e9	2.22599
$470$	−8.84943e7	−0.0393163
$471$	0	0
$472$	4.16879e8	0.182479
$473$	−3.33211e9	−1.44779
$474$	0	0
$475$	−1.05350e9	−0.451030
$476$	−1.21801e9	−0.517637
$477$	0	0
$478$	5.85109e8	0.245041
$479$	1.76813e9	0.735091	0.367545	−	0.930006i	$-0.380198\pi$
$479$	1.76813e9	0.735091	0.367545	+	0.930006i	$0.380198\pi$
$480$	0	0
$481$	8.78792e8	0.360063
$482$	9.82718e8	0.399728
$483$	0	0
$484$	1.03054e7	0.00413147
$485$	1.49500e8	0.0595037
$486$	0	0
$487$	−1.97414e9	−0.774509	−0.387254	−	0.921973i	$-0.626577\pi$
$487$	−1.97414e9	−0.774509	−0.387254	+	0.921973i	$0.626577\pi$
$488$	1.26440e9	0.492510
$489$	0	0
$490$	1.12331e8	0.0431333
$491$	3.50381e9	1.33584	0.667920	−	0.744233i	$-0.267185\pi$
$491$	3.50381e9	1.33584	0.667920	+	0.744233i	$0.267185\pi$
$492$	0	0
$493$	1.96331e9	0.737945
$494$	7.81328e8	0.291601
$495$	0	0
$496$	5.68780e8	0.209295
$497$	−5.24076e9	−1.91490
$498$	0	0
$499$	8.74654e8	0.315126	0.157563	−	0.987509i	$-0.449636\pi$
$499$	8.74654e8	0.315126	0.157563	+	0.987509i	$0.449636\pi$
$500$	1.12372e8	0.0402035
$501$	0	0
$502$	5.84279e8	0.206137
$503$	2.61478e9	0.916111	0.458055	−	0.888924i	$-0.348546\pi$
$503$	2.61478e9	0.916111	0.458055	+	0.888924i	$0.348546\pi$
$504$	0	0
$505$	−1.50324e8	−0.0519409
$506$	2.33116e9	0.799917
$507$	0	0
$508$	7.28612e8	0.246589
$509$	−8.52572e8	−0.286562	−0.143281	−	0.989682i	$-0.545765\pi$
$509$	−8.52572e8	−0.286562	−0.143281	+	0.989682i	$0.545765\pi$
$510$	0	0
$511$	−7.98992e9	−2.64892
$512$	−1.34218e8	−0.0441942
$513$	0	0
$514$	1.07197e9	0.348185
$515$	−1.52188e8	−0.0490971
$516$	0	0
$517$	4.35990e9	1.38759
$518$	1.39953e9	0.442412
$519$	0	0
$520$	−4.16368e7	−0.0129857
$521$	2.66940e9	0.826954	0.413477	−	0.910515i	$-0.364314\pi$
$521$	2.66940e9	0.826954	0.413477	+	0.910515i	$0.364314\pi$
$522$	0	0
$523$	−4.41770e9	−1.35033	−0.675165	−	0.737667i	$-0.735928\pi$
$523$	−4.41770e9	−1.35033	−0.675165	+	0.737667i	$0.735928\pi$
$524$	−3.38035e8	−0.102636
$525$	0	0
$526$	3.04104e9	0.911112
$527$	−1.83591e9	−0.546405
$528$	0	0
$529$	9.16739e8	0.269247
$530$	5.59400e7	0.0163214
$531$	0	0
$532$	1.24431e9	0.358293
$533$	3.81700e8	0.109189
$534$	0	0
$535$	−2.13174e8	−0.0601862
$536$	−1.76887e9	−0.496156
$537$	0	0
$538$	4.46148e9	1.23521
$539$	−5.53426e9	−1.52230
$540$	0	0
$541$	−3.86240e9	−1.04874	−0.524368	−	0.851492i	$-0.675699\pi$
$541$	−3.86240e9	−1.04874	−0.524368	+	0.851492i	$0.675699\pi$
$542$	7.72635e8	0.208438
$543$	0	0
$544$	4.33229e8	0.115378
$545$	1.49865e8	0.0396562
$546$	0	0
$547$	−2.16921e9	−0.566692	−0.283346	−	0.959018i	$-0.591444\pi$
$547$	−2.16921e9	−0.566692	−0.283346	+	0.959018i	$0.591444\pi$
$548$	−3.13673e9	−0.814226
$549$	0	0
$550$	−2.76591e9	−0.708873
$551$	−2.00571e9	−0.510784
$552$	0	0
$553$	−2.94568e9	−0.740708
$554$	−4.52640e9	−1.13102
$555$	0	0
$556$	−1.20828e9	−0.298131
$557$	−2.25645e8	−0.0553265	−0.0276632	−	0.999617i	$-0.508807\pi$
$557$	−2.25645e8	−0.0553265	−0.0276632	+	0.999617i	$0.508807\pi$
$558$	0	0
$559$	−5.43570e9	−1.31618
$560$	−6.63090e7	−0.0159557
$561$	0	0
$562$	3.19640e9	0.759598
$563$	−7.80802e8	−0.184400	−0.0922001	−	0.995740i	$-0.529390\pi$
$563$	−7.80802e8	−0.184400	−0.0922001	+	0.995740i	$0.529390\pi$
$564$	0	0
$565$	2.29726e8	0.0535846
$566$	−5.42978e9	−1.25871
$567$	0	0
$568$	1.86407e9	0.426818
$569$	6.69214e8	0.152290	0.0761452	−	0.997097i	$-0.475739\pi$
$569$	6.69214e8	0.152290	0.0761452	+	0.997097i	$0.475739\pi$
$570$	0	0
$571$	−1.33652e9	−0.300435	−0.150217	−	0.988653i	$-0.547997\pi$
$571$	−1.33652e9	−0.300435	−0.150217	+	0.988653i	$0.547997\pi$
$572$	2.05134e9	0.458302
$573$	0	0
$574$	6.07880e8	0.134161
$575$	−5.12751e9	−1.12478
$576$	0	0
$577$	−7.13624e9	−1.54651	−0.773257	−	0.634092i	$-0.781374\pi$
$577$	−7.13624e9	−1.54651	−0.773257	+	0.634092i	$0.781374\pi$
$578$	1.88433e9	0.405891
$579$	0	0
$580$	1.06884e8	0.0227464
$581$	−1.47764e10	−3.12573
$582$	0	0
$583$	−2.75603e9	−0.576028
$584$	2.84191e9	0.590425
$585$	0	0
$586$	8.93912e8	0.183507
$587$	−2.85470e9	−0.582542	−0.291271	−	0.956641i	$-0.594078\pi$
$587$	−2.85470e9	−0.582542	−0.291271	+	0.956641i	$0.594078\pi$
$588$	0	0
$589$	1.87556e9	0.378205
$590$	−7.32555e7	−0.0146845
$591$	0	0
$592$	−4.97793e8	−0.0986105
$593$	6.55017e9	1.28991	0.644957	−	0.764219i	$-0.276875\pi$
$593$	6.55017e9	1.28991	0.644957	+	0.764219i	$0.276875\pi$
$594$	0	0
$595$	2.14033e8	0.0416553
$596$	1.84638e9	0.357240
$597$	0	0
$598$	3.80283e9	0.727199
$599$	7.51415e9	1.42852	0.714259	−	0.699881i	$-0.246764\pi$
$599$	7.51415e9	1.42852	0.714259	+	0.699881i	$0.246764\pi$
$600$	0	0
$601$	−3.40742e9	−0.640274	−0.320137	−	0.947371i	$-0.603729\pi$
$601$	−3.40742e9	−0.640274	−0.320137	+	0.947371i	$0.603729\pi$
$602$	−8.65667e9	−1.61720
$603$	0	0
$604$	−2.30101e9	−0.424902
$605$	−1.81090e6	−0.000332468 0
$606$	0	0
$607$	4.94302e9	0.897081	0.448540	−	0.893763i	$-0.351944\pi$
$607$	4.94302e9	0.897081	0.448540	+	0.893763i	$0.351944\pi$
$608$	−4.42585e8	−0.0798609
$609$	0	0
$610$	−2.22185e8	−0.0396333
$611$	7.11233e9	1.26144
$612$	0	0
$613$	4.05250e9	0.710577	0.355288	−	0.934757i	$-0.384383\pi$
$613$	4.05250e9	0.710577	0.355288	+	0.934757i	$0.384383\pi$
$614$	1.22417e9	0.213429
$615$	0	0
$616$	3.26688e9	0.563120
$617$	−6.15589e9	−1.05510	−0.527549	−	0.849525i	$-0.676889\pi$
$617$	−6.15589e9	−1.05510	−0.527549	+	0.849525i	$0.676889\pi$
$618$	0	0
$619$	3.64125e8	0.0617068	0.0308534	−	0.999524i	$-0.490177\pi$
$619$	3.64125e8	0.0617068	0.0308534	+	0.999524i	$0.490177\pi$
$620$	−9.99482e7	−0.0168424
$621$	0	0
$622$	−1.82694e9	−0.304410
$623$	−1.16939e10	−1.93754
$624$	0	0
$625$	6.07389e9	0.995146
$626$	5.62195e9	0.915961
$627$	0	0
$628$	−3.69978e9	−0.596097
$629$	1.60678e9	0.257442
$630$	0	0
$631$	3.22087e9	0.510353	0.255177	−	0.966894i	$-0.417866\pi$
$631$	3.22087e9	0.510353	0.255177	+	0.966894i	$0.417866\pi$
$632$	1.04774e9	0.165098
$633$	0	0
$634$	−8.32942e9	−1.29808
$635$	−1.28034e8	−0.0198435
$636$	0	0
$637$	−9.02808e9	−1.38391
$638$	−5.26590e9	−0.802786
$639$	0	0
$640$	2.35852e7	0.00355640
$641$	1.24441e10	1.86621	0.933106	−	0.359603i	$-0.117088\pi$
$641$	1.24441e10	1.86621	0.933106	+	0.359603i	$0.117088\pi$
$642$	0	0
$643$	−4.83009e8	−0.0716502	−0.0358251	−	0.999358i	$-0.511406\pi$
$643$	−4.83009e8	−0.0716502	−0.0358251	+	0.999358i	$0.511406\pi$
$644$	6.05624e9	0.893516
$645$	0	0
$646$	1.42858e9	0.208492
$647$	−4.60241e9	−0.668068	−0.334034	−	0.942561i	$-0.608410\pi$
$647$	−4.60241e9	−0.668068	−0.334034	+	0.942561i	$0.608410\pi$
$648$	0	0
$649$	3.60912e9	0.518256
$650$	−4.51204e9	−0.644431
$651$	0	0
$652$	2.07063e9	0.292574
$653$	−5.68364e9	−0.798787	−0.399393	−	0.916780i	$-0.630779\pi$
$653$	−5.68364e9	−0.798787	−0.399393	+	0.916780i	$0.630779\pi$
$654$	0	0
$655$	5.94007e7	0.00825937
$656$	−2.16215e8	−0.0299035
$657$	0	0
$658$	1.13268e10	1.54995
$659$	−4.51142e9	−0.614065	−0.307032	−	0.951699i	$-0.599336\pi$
$659$	−4.51142e9	−0.614065	−0.307032	+	0.951699i	$0.599336\pi$
$660$	0	0
$661$	9.76185e8	0.131470	0.0657351	−	0.997837i	$-0.479061\pi$
$661$	9.76185e8	0.131470	0.0657351	+	0.997837i	$0.479061\pi$
$662$	−2.57020e9	−0.344322
$663$	0	0
$664$	5.25576e9	0.696703
$665$	−2.18655e8	−0.0288326
$666$	0	0
$667$	−9.76206e9	−1.27380
$668$	5.55357e9	0.720866
$669$	0	0
$670$	3.10832e8	0.0399267
$671$	1.09465e10	1.39877
$672$	0	0
$673$	1.25246e10	1.58384	0.791921	−	0.610623i	$-0.209081\pi$
$673$	1.25246e10	1.58384	0.791921	+	0.610623i	$0.209081\pi$
$674$	7.30917e9	0.919514
$675$	0	0
$676$	−6.69539e8	−0.0833608
$677$	−1.04491e10	−1.29425	−0.647126	−	0.762383i	$-0.724029\pi$
$677$	−1.04491e10	−1.29425	−0.647126	+	0.762383i	$0.724029\pi$
$678$	0	0
$679$	−1.91352e10	−2.34578
$680$	−7.61286e7	−0.00928467
$681$	0	0
$682$	4.92420e9	0.594416
$683$	−1.12280e9	−0.134844	−0.0674218	−	0.997725i	$-0.521477\pi$
$683$	−1.12280e9	−0.134844	−0.0674218	+	0.997725i	$0.521477\pi$
$684$	0	0
$685$	5.51197e8	0.0655225
$686$	−4.89403e9	−0.578805
$687$	0	0
$688$	3.07906e9	0.360462
$689$	−4.49592e9	−0.523663
$690$	0	0
$691$	7.46043e9	0.860183	0.430091	−	0.902785i	$-0.358481\pi$
$691$	7.46043e9	0.860183	0.430091	+	0.902785i	$0.358481\pi$
$692$	6.89431e8	0.0790897
$693$	0	0
$694$	−4.78387e9	−0.543277
$695$	2.12324e8	0.0239912
$696$	0	0
$697$	6.97900e8	0.0780690
$698$	−5.20437e9	−0.579261
$699$	0	0
$700$	−7.18569e9	−0.791818
$701$	1.79013e10	1.96278	0.981392	−	0.192014i	$-0.0615020\pi$
$701$	1.79013e10	1.96278	0.981392	+	0.192014i	$0.0615020\pi$
$702$	0	0
$703$	−1.64148e9	−0.178194
$704$	−1.16199e9	−0.125515
$705$	0	0
$706$	−4.93797e8	−0.0528119
$707$	1.92407e10	2.04764
$708$	0	0
$709$	7.55041e9	0.795626	0.397813	−	0.917467i	$-0.369769\pi$
$709$	7.55041e9	0.795626	0.397813	+	0.917467i	$0.369769\pi$
$710$	−3.27561e8	−0.0343469
$711$	0	0
$712$	4.15936e9	0.431863
$713$	9.12862e9	0.943174
$714$	0	0
$715$	−3.60469e8	−0.0368805
$716$	2.31152e9	0.235343
$717$	0	0
$718$	1.36434e10	1.37558
$719$	3.79868e9	0.381137	0.190569	−	0.981674i	$-0.438967\pi$
$719$	3.79868e9	0.381137	0.190569	+	0.981674i	$0.438967\pi$
$720$	0	0
$721$	1.94793e10	1.93553
$722$	5.69154e9	0.562795
$723$	0	0
$724$	1.77995e9	0.174310
$725$	1.15826e10	1.12882
$726$	0	0
$727$	−3.93525e9	−0.379841	−0.189920	−	0.981799i	$-0.560823\pi$
$727$	−3.93525e9	−0.379841	−0.189920	+	0.981799i	$0.560823\pi$
$728$	5.32929e9	0.511928
$729$	0	0
$730$	−4.99390e8	−0.0475127
$731$	−9.93862e9	−0.941055
$732$	0	0
$733$	−1.13491e10	−1.06439	−0.532193	−	0.846623i	$-0.678632\pi$
$733$	−1.13491e10	−1.06439	−0.532193	+	0.846623i	$0.678632\pi$
$734$	−1.94415e9	−0.181465
$735$	0	0
$736$	−2.15412e9	−0.199158
$737$	−1.53139e10	−1.40913
$738$	0	0
$739$	1.61420e10	1.47130	0.735652	−	0.677359i	$-0.236876\pi$
$739$	1.61420e10	1.47130	0.735652	+	0.677359i	$0.236876\pi$
$740$	8.74741e7	0.00793539
$741$	0	0
$742$	−7.16002e9	−0.643429
$743$	5.93148e9	0.530521	0.265260	−	0.964177i	$-0.414542\pi$
$743$	5.93148e9	0.530521	0.265260	+	0.964177i	$0.414542\pi$
$744$	0	0
$745$	−3.24453e8	−0.0287478
$746$	5.26833e9	0.464608
$747$	0	0
$748$	3.75067e9	0.327682
$749$	2.72852e10	2.37269
$750$	0	0
$751$	−3.34926e9	−0.288542	−0.144271	−	0.989538i	$-0.546084\pi$
$751$	−3.34926e9	−0.288542	−0.144271	+	0.989538i	$0.546084\pi$
$752$	−4.02880e9	−0.345472
$753$	0	0
$754$	−8.59029e9	−0.729807
$755$	4.04342e8	0.0341928
$756$	0	0
$757$	−3.71611e9	−0.311353	−0.155677	−	0.987808i	$-0.549756\pi$
$757$	−3.71611e9	−0.311353	−0.155677	+	0.987808i	$0.549756\pi$
$758$	−1.24532e8	−0.0103858
$759$	0	0
$760$	7.77727e7	0.00642657
$761$	−1.44344e9	−0.118728	−0.0593641	−	0.998236i	$-0.518907\pi$
$761$	−1.44344e9	−0.118728	−0.0593641	+	0.998236i	$0.518907\pi$
$762$	0	0
$763$	−1.91819e10	−1.56335
$764$	6.05602e9	0.491316
$765$	0	0
$766$	−6.94304e9	−0.558148
$767$	5.88758e9	0.471143
$768$	0	0
$769$	1.02499e10	0.812788	0.406394	−	0.913698i	$-0.366786\pi$
$769$	1.02499e10	0.812788	0.406394	+	0.913698i	$0.366786\pi$
$770$	−5.74069e8	−0.0453155
$771$	0	0
$772$	−7.82228e9	−0.611889
$773$	2.47581e10	1.92792	0.963959	−	0.266049i	$-0.0857185\pi$
$773$	2.47581e10	1.92792	0.963959	+	0.266049i	$0.0857185\pi$
$774$	0	0
$775$	−1.08311e10	−0.835824
$776$	6.80612e9	0.522858
$777$	0	0
$778$	−1.42809e10	−1.08725
$779$	−7.12972e8	−0.0540370
$780$	0	0
$781$	1.61381e10	1.21220
$782$	6.95309e9	0.519941
$783$	0	0
$784$	5.11397e9	0.379011
$785$	6.50139e8	0.0479692
$786$	0	0
$787$	−2.09697e10	−1.53349	−0.766744	−	0.641953i	$-0.778125\pi$
$787$	−2.09697e10	−1.53349	−0.766744	+	0.641953i	$0.778125\pi$
$788$	8.81712e9	0.641926
$789$	0	0
$790$	−1.84112e8	−0.0132858
$791$	−2.94037e10	−2.11244
$792$	0	0
$793$	1.78571e10	1.27161
$794$	−1.19643e10	−0.848234
$795$	0	0
$796$	−1.04094e10	−0.731524
$797$	−1.63579e10	−1.14452	−0.572259	−	0.820073i	$-0.693933\pi$
$797$	−1.63579e10	−1.14452	−0.572259	+	0.820073i	$0.693933\pi$
$798$	0	0
$799$	1.30042e10	0.901922
$800$	2.55586e9	0.176491
$801$	0	0
$802$	−8.33734e9	−0.570712
$803$	2.46037e10	1.67686
$804$	0	0
$805$	−1.06422e9	−0.0719031
$806$	8.03288e9	0.540379
$807$	0	0
$808$	−6.84367e9	−0.456404
$809$	−8.79414e9	−0.583947	−0.291974	−	0.956426i	$-0.594312\pi$
$809$	−8.79414e9	−0.583947	−0.291974	+	0.956426i	$0.594312\pi$
$810$	0	0
$811$	8.84977e8	0.0582585	0.0291292	−	0.999576i	$-0.490727\pi$
$811$	8.84977e8	0.0582585	0.0291292	+	0.999576i	$0.490727\pi$
$812$	−1.36805e10	−0.896721
$813$	0	0
$814$	−4.30963e9	−0.280063
$815$	−3.63859e8	−0.0235440
$816$	0	0
$817$	1.01533e10	0.651370
$818$	−1.11076e10	−0.709549
$819$	0	0
$820$	3.79941e7	0.00240640
$821$	−2.05048e10	−1.29317	−0.646584	−	0.762843i	$-0.723803\pi$
$821$	−2.05048e10	−1.29317	−0.646584	+	0.762843i	$0.723803\pi$
$822$	0	0
$823$	4.85079e9	0.303328	0.151664	−	0.988432i	$-0.451537\pi$
$823$	4.85079e9	0.303328	0.151664	+	0.988432i	$0.451537\pi$
$824$	−6.92853e9	−0.431416
$825$	0	0
$826$	9.37631e9	0.578898
$827$	−1.02887e10	−0.632545	−0.316272	−	0.948668i	$-0.602431\pi$
$827$	−1.02887e10	−0.632545	−0.316272	+	0.948668i	$0.602431\pi$
$828$	0	0
$829$	3.17520e9	0.193566	0.0967832	−	0.995305i	$-0.469145\pi$
$829$	3.17520e9	0.193566	0.0967832	+	0.995305i	$0.469145\pi$
$830$	−9.23562e8	−0.0560651
$831$	0	0
$832$	−1.89556e9	−0.114105
$833$	−1.65069e10	−0.989483
$834$	0	0
$835$	−9.75893e8	−0.0580096
$836$	−3.83167e9	−0.226812
$837$	0	0
$838$	−8.56746e9	−0.502919
$839$	2.44147e10	1.42720	0.713601	−	0.700553i	$-0.247063\pi$
$839$	2.44147e10	1.42720	0.713601	+	0.700553i	$0.247063\pi$
$840$	0	0
$841$	4.80181e9	0.278368
$842$	−5.34308e9	−0.308460
$843$	0	0
$844$	−9.95698e9	−0.570071
$845$	1.17654e8	0.00670822
$846$	0	0
$847$	2.31785e8	0.0131067
$848$	2.54672e9	0.143416
$849$	0	0
$850$	−8.24981e9	−0.460763
$851$	−7.98932e9	−0.444382
$852$	0	0
$853$	−2.10145e9	−0.115931	−0.0579653	−	0.998319i	$-0.518461\pi$
$853$	−2.10145e9	−0.115931	−0.0579653	+	0.998319i	$0.518461\pi$
$854$	2.84385e10	1.56244
$855$	0	0
$856$	−9.70498e9	−0.528855
$857$	4.33590e9	0.235313	0.117657	−	0.993054i	$-0.462462\pi$
$857$	4.33590e9	0.235313	0.117657	+	0.993054i	$0.462462\pi$
$858$	0	0
$859$	2.04143e10	1.09890	0.549451	−	0.835526i	$-0.314837\pi$
$859$	2.04143e10	1.09890	0.549451	+	0.835526i	$0.314837\pi$
$860$	−5.41064e8	−0.0290071
$861$	0	0
$862$	−7.10613e9	−0.377883
$863$	−7.33916e9	−0.388695	−0.194347	−	0.980933i	$-0.562259\pi$
$863$	−7.33916e9	−0.388695	−0.194347	+	0.980933i	$0.562259\pi$
$864$	0	0
$865$	−1.21149e8	−0.00636451
$866$	−8.33578e9	−0.436148
$867$	0	0
$868$	1.27928e10	0.663969
$869$	9.07077e9	0.468894
$870$	0	0
$871$	−2.49817e10	−1.28103
$872$	6.82273e9	0.348458
$873$	0	0
$874$	−7.10325e9	−0.359888
$875$	2.52744e9	0.127542
$876$	0	0
$877$	7.64808e9	0.382872	0.191436	−	0.981505i	$-0.438686\pi$
$877$	7.64808e9	0.382872	0.191436	+	0.981505i	$0.438686\pi$
$878$	−2.17888e10	−1.08643
$879$	0	0
$880$	2.04189e8	0.0101005
$881$	−2.38502e9	−0.117511	−0.0587553	−	0.998272i	$-0.518713\pi$
$881$	−2.38502e9	−0.117511	−0.0587553	+	0.998272i	$0.518713\pi$
$882$	0	0
$883$	−1.51552e10	−0.740797	−0.370399	−	0.928873i	$-0.620779\pi$
$883$	−1.51552e10	−0.740797	−0.370399	+	0.928873i	$0.620779\pi$
$884$	6.11849e9	0.297894
$885$	0	0
$886$	1.13493e10	0.548214
$887$	3.90717e10	1.87988	0.939938	−	0.341344i	$-0.110882\pi$
$887$	3.90717e10	1.87988	0.939938	+	0.341344i	$0.110882\pi$
$888$	0	0
$889$	1.63877e10	0.782281
$890$	−7.30898e8	−0.0347529
$891$	0	0
$892$	−7.93234e9	−0.374218
$893$	−1.32850e10	−0.624284
$894$	0	0
$895$	−4.06189e8	−0.0189386
$896$	−3.01879e9	−0.140202
$897$	0	0
$898$	−3.03306e10	−1.39770
$899$	−2.06208e10	−0.946557
$900$	0	0
$901$	−8.22033e9	−0.374415
$902$	−1.87188e9	−0.0849286
$903$	0	0
$904$	1.04585e10	0.470847
$905$	−3.12780e8	−0.0140271
$906$	0	0
$907$	−9.68126e9	−0.430830	−0.215415	−	0.976523i	$-0.569110\pi$
$907$	−9.68126e9	−0.430830	−0.215415	+	0.976523i	$0.569110\pi$
$908$	8.81306e9	0.390685
$909$	0	0
$910$	−9.36482e8	−0.0411960
$911$	−3.22816e9	−0.141462	−0.0707312	−	0.997495i	$-0.522533\pi$
$911$	−3.22816e9	−0.141462	−0.0707312	+	0.997495i	$0.522533\pi$
$912$	0	0
$913$	4.55016e10	1.97870
$914$	−1.49837e10	−0.649095
$915$	0	0
$916$	1.50477e10	0.646899
$917$	−7.60298e9	−0.325605
$918$	0	0
$919$	−2.75829e10	−1.17229	−0.586145	−	0.810206i	$-0.699355\pi$
$919$	−2.75829e10	−1.17229	−0.586145	+	0.810206i	$0.699355\pi$
$920$	3.78530e8	0.0160267
$921$	0	0
$922$	4.37915e9	0.184006
$923$	2.63262e10	1.10200
$924$	0	0
$925$	9.47928e9	0.393803
$926$	8.98221e9	0.371745
$927$	0	0
$928$	4.86599e9	0.199872
$929$	−2.61114e10	−1.06850	−0.534251	−	0.845326i	$-0.679406\pi$
$929$	−2.61114e10	−1.06850	−0.534251	+	0.845326i	$0.679406\pi$
$930$	0	0
$931$	1.68634e10	0.684891
$932$	9.21849e9	0.372996
$933$	0	0
$934$	1.86089e10	0.747321
$935$	−6.59081e8	−0.0263693
$936$	0	0
$937$	2.33086e10	0.925608	0.462804	−	0.886461i	$-0.346843\pi$
$937$	2.33086e10	0.925608	0.462804	+	0.886461i	$0.346843\pi$
$938$	−3.97848e10	−1.57401
$939$	0	0
$940$	7.07955e8	0.0278008
$941$	−2.34937e10	−0.919155	−0.459577	−	0.888138i	$-0.651999\pi$
$941$	−2.34937e10	−0.919155	−0.459577	+	0.888138i	$0.651999\pi$
$942$	0	0
$943$	−3.47013e9	−0.134758
$944$	−3.33503e9	−0.129032
$945$	0	0
$946$	2.66569e10	1.02374
$947$	−3.34690e10	−1.28061	−0.640306	−	0.768120i	$-0.721193\pi$
$947$	−3.34690e10	−1.28061	−0.640306	+	0.768120i	$0.721193\pi$
$948$	0	0
$949$	4.01362e10	1.52442
$950$	8.42797e9	0.318926
$951$	0	0
$952$	9.74406e9	0.366025
$953$	−3.61551e10	−1.35315	−0.676573	−	0.736375i	$-0.736536\pi$
$953$	−3.61551e10	−1.35315	−0.676573	+	0.736375i	$0.736536\pi$
$954$	0	0
$955$	−1.06419e9	−0.0395372
$956$	−4.68087e9	−0.173270
$957$	0	0
$958$	−1.41451e10	−0.519788
$959$	−7.05504e10	−2.58306
$960$	0	0
$961$	−8.22986e9	−0.299130
$962$	−7.03033e9	−0.254603
$963$	0	0
$964$	−7.86174e9	−0.282650
$965$	1.37456e9	0.0492400
$966$	0	0
$967$	−2.58691e10	−0.920003	−0.460002	−	0.887918i	$-0.652151\pi$
$967$	−2.58691e10	−0.920003	−0.460002	+	0.887918i	$0.652151\pi$
$968$	−8.24429e7	−0.00292139
$969$	0	0
$970$	−1.19600e9	−0.0420754
$971$	−2.99112e10	−1.04850	−0.524248	−	0.851565i	$-0.675654\pi$
$971$	−2.99112e10	−1.04850	−0.524248	+	0.851565i	$0.675654\pi$
$972$	0	0
$973$	−2.71764e10	−0.945794
$974$	1.57931e10	0.547660
$975$	0	0
$976$	−1.01152e10	−0.348257
$977$	2.94364e10	1.00984	0.504921	−	0.863166i	$-0.331522\pi$
$977$	2.94364e10	1.00984	0.504921	+	0.863166i	$0.331522\pi$
$978$	0	0
$979$	3.60095e10	1.22653
$980$	−8.98646e8	−0.0304998
$981$	0	0
$982$	−2.80304e10	−0.944582
$983$	−4.97374e10	−1.67011	−0.835057	−	0.550163i	$-0.814565\pi$
$983$	−4.97374e10	−1.67011	−0.835057	+	0.550163i	$0.814565\pi$
$984$	0	0
$985$	−1.54938e9	−0.0516571
$986$	−1.57065e10	−0.521806
$987$	0	0
$988$	−6.25062e9	−0.206193
$989$	4.94173e10	1.62440
$990$	0	0
$991$	3.46904e9	0.113227	0.0566137	−	0.998396i	$-0.481970\pi$
$991$	3.46904e9	0.113227	0.0566137	+	0.998396i	$0.481970\pi$
$992$	−4.55024e9	−0.147994
$993$	0	0
$994$	4.19261e10	1.35404
$995$	1.82917e9	0.0588673
$996$	0	0
$997$	−3.43622e10	−1.09812	−0.549058	−	0.835784i	$-0.685013\pi$
$997$	−3.43622e10	−1.09812	−0.549058	+	0.835784i	$0.685013\pi$
$998$	−6.99723e9	−0.222828
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.8.a.g.1.3	✓	4
3.2	odd	2		162.8.a.j.1.2	yes	4
9.2	odd	6		162.8.c.q.109.3		8
9.4	even	3		162.8.c.r.55.2		8
9.5	odd	6		162.8.c.q.55.3		8
9.7	even	3		162.8.c.r.109.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.8.a.g.1.3	✓	4	1.1	even	1	trivial
162.8.a.j.1.2	yes	4	3.2	odd	2
162.8.c.q.55.3		8	9.5	odd	6
162.8.c.q.109.3		8	9.2	odd	6
162.8.c.r.55.2		8	9.4	even	3
162.8.c.r.109.2		8	9.7	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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +64.0000 q^{4} -11.2463 q^{5} +1439.47 q^{7} -512.000 q^{8} +89.9705 q^{10} -4432.63 q^{11} -7230.97 q^{13} -11515.8 q^{14} +4096.00 q^{16} -13221.1 q^{17} +13506.6 q^{19} -719.764 q^{20} +35461.0 q^{22} +65738.6 q^{23} -77998.5 q^{25} +57847.8 q^{26} +92126.0 q^{28} -148498. q^{29} +138862. q^{31} -32768.0 q^{32} +105769. q^{34} -16188.7 q^{35} -121532. q^{37} -108053. q^{38} +5758.11 q^{40} -52786.9 q^{41} +751724. q^{43} -283688. q^{44} -525909. q^{46} -983593. q^{47} +1.24853e6 q^{49} +623988. q^{50} -462782. q^{52} +621759. q^{53} +49850.7 q^{55} -737008. q^{56} +1.18798e6 q^{58} -814216. q^{59} -2.46953e6 q^{61} -1.11090e6 q^{62} +262144. q^{64} +81321.8 q^{65} +3.45482e6 q^{67} -846150. q^{68} +129510. q^{70} -3.64076e6 q^{71} -5.55060e6 q^{73} +972253. q^{74} +864424. q^{76} -6.38063e6 q^{77} -2.04636e6 q^{79} -46064.9 q^{80} +422295. q^{82} -1.02652e7 q^{83} +148689. q^{85} -6.01379e6 q^{86} +2.26951e6 q^{88} -8.12374e6 q^{89} -1.04088e7 q^{91} +4.20727e6 q^{92} +7.86874e6 q^{94} -151900. q^{95} -1.32932e7 q^{97} -9.98823e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} + 256 q^{4} - 528 q^{5} + 560 q^{7} - 2048 q^{8} + 4224 q^{10} - 2160 q^{11} + 13460 q^{13} - 4480 q^{14} + 16384 q^{16} - 22560 q^{17} + 36704 q^{19} - 33792 q^{20} + 17280 q^{22} - 62640 q^{23}+ \cdots - 11595168 q^{98}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 - 528 * q^5 + 560 * q^7 - 2048 * q^8 + 4224 * q^10 - 2160 * q^11 + 13460 * q^13 - 4480 * q^14 + 16384 * q^16 - 22560 * q^17 + 36704 * q^19 - 33792 * q^20 + 17280 * q^22 - 62640 * q^23 + 94696 * q^25 - 107680 * q^26 + 35840 * q^28 - 68400 * q^29 + 227504 * q^31 - 131072 * q^32 + 180480 * q^34 + 63024 * q^35 + 523580 * q^37 - 293632 * q^38 + 270336 * q^40 - 67200 * q^41 + 562640 * q^43 - 138240 * q^44 + 501120 * q^46 - 515328 * q^47 + 1449396 * q^49 - 757568 * q^50 + 861440 * q^52 - 2498016 * q^53 + 1322784 * q^55 - 286720 * q^56 + 547200 * q^58 - 4155840 * q^59 + 2130764 * q^61 - 1820032 * q^62 + 1048576 * q^64 - 8966496 * q^65 - 1205440 * q^67 - 1443840 * q^68 - 504192 * q^70 - 12486480 * q^71 - 4820860 * q^73 - 4188640 * q^74 + 2349056 * q^76 - 18686592 * q^77 - 11471680 * q^79 - 2162688 * q^80 + 537600 * q^82 - 16811232 * q^83 - 21006396 * q^85 - 4501120 * q^86 + 1105920 * q^88 - 16857600 * q^89 - 23200544 * q^91 - 4008960 * q^92 + 4122624 * q^94 - 14090160 * q^95 - 27078040 * q^97 - 11595168 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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