LMFDB - Embedded newform 160.6.d.a.81.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(81,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("160.81");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	160.d (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:	$25.6614111701$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 $ x^20 - 2x^19 - 17x^18 + 78x^17 + 253x^16 - 884x^15 + 2396x^14 + 19376x^13 - 109104x^12 - 96128x^11 + 3580672x^10 - 1538048x^9 - 27930624x^8 + 79364096x^7 + 157024256x^6 - 926941184x^5 + 4244635648x^4 + 20937965568x^3 - 73014444032x^2 - 137438953472*x + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{90}\cdot 3^{4}\cdot 5^{12} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			81.13
Root			$-2.80358 - 2.85306i$ of defining polynomial
Character	$\chi$	$=$	160.81
Dual form			160.6.d.a.81.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.7455i q^{3} -25.0000i q^{5} -198.733 q^{7} +127.535 q^{9} +O(q^{10})$	$q+10.7455i q^{3} -25.0000i q^{5} -198.733 q^{7} +127.535 q^{9} +85.9303i q^{11} -407.120i q^{13} +268.637 q^{15} +1206.02 q^{17} -206.036i q^{19} -2135.48i q^{21} +2595.25 q^{23} -625.000 q^{25} +3981.57i q^{27} -6195.27i q^{29} +1862.42 q^{31} -923.363 q^{33} +4968.33i q^{35} +14708.1i q^{37} +4374.70 q^{39} +18098.0 q^{41} -9260.46i q^{43} -3188.36i q^{45} +24363.7 q^{47} +22687.9 q^{49} +12959.3i q^{51} +12764.8i q^{53} +2148.26 q^{55} +2213.96 q^{57} -20719.7i q^{59} -11368.5i q^{61} -25345.3 q^{63} -10178.0 q^{65} -62614.9i q^{67} +27887.3i q^{69} +61208.1 q^{71} +23236.4 q^{73} -6715.93i q^{75} -17077.2i q^{77} -29172.5 q^{79} -11793.1 q^{81} +48099.7i q^{83} -30150.6i q^{85} +66571.2 q^{87} +30118.4 q^{89} +80908.2i q^{91} +20012.6i q^{93} -5150.90 q^{95} -113676. q^{97} +10959.1i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 196 q^{7} - 1620 q^{9}+O(q^{10}) $ 20 * q + 196 * q^7 - 1620 * q^9	$ 20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63} + 200312 q^{71} - 105136 q^{73} - 282080 q^{79} + 65172 q^{81} + 332592 q^{87} - 3160 q^{89} - 144400 q^{95} + 147376 q^{97}+O(q^{100}) $ 20 * q + 196 * q^7 - 1620 * q^9 - 900 * q^15 + 4676 * q^23 - 12500 * q^25 - 7160 * q^31 + 5672 * q^33 + 44904 * q^39 + 11608 * q^41 - 44180 * q^47 + 18756 * q^49 + 24200 * q^55 + 5032 * q^57 - 240620 * q^63 + 200312 * q^71 - 105136 * q^73 - 282080 * q^79 + 65172 * q^81 + 332592 * q^87 - 3160 * q^89 - 144400 * q^95 + 147376 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	10.7455i	0.689323i	0.938727	+	0.344662i	$0.112006\pi$
$3$	10.7455i	0.689323i	−0.938727	+	0.344662i	$0.887994\pi$
$4$	0	0
$5$	− 25.0000i	− 0.447214i
$5$	− 25.0000i	− 0.447214i
$6$	0	0
$7$	−198.733	−1.53294	−0.766470	−	0.642280i	$-0.777989\pi$
$7$	−198.733	−1.53294	−0.766470	+	0.642280i	$0.777989\pi$
$8$	0	0
$9$	127.535	0.524833
$10$	0	0
$11$	85.9303i	0.214124i	0.994252	+	0.107062i	$0.0341443\pi$
$11$	85.9303i	0.214124i	−0.994252	+	0.107062i	$0.965856\pi$
$12$	0	0
$13$	− 407.120i	− 0.668134i	−0.942549	−	0.334067i	$-0.891579\pi$
$13$	− 407.120i	− 0.668134i	0.942549	−	0.334067i	$-0.108421\pi$
$14$	0	0
$15$	268.637	0.308275
$16$	0	0
$17$	1206.02	1.01212	0.506062	−	0.862497i	$-0.331101\pi$
$17$	1206.02	1.01212	0.506062	+	0.862497i	$0.331101\pi$
$18$	0	0
$19$	− 206.036i	− 0.130936i	−0.997855	−	0.0654680i	$-0.979146\pi$
$19$	− 206.036i	− 0.130936i	0.997855	−	0.0654680i	$-0.0208540\pi$
$20$	0	0
$21$	− 2135.48i	− 1.05669i
$22$	0	0
$23$	2595.25	1.02296	0.511482	−	0.859294i	$-0.329097\pi$
$23$	2595.25	1.02296	0.511482	+	0.859294i	$0.329097\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	3981.57i	1.05110i
$28$	0	0
$29$	− 6195.27i	− 1.36794i	−0.729512	−	0.683968i	$-0.760253\pi$
$29$	− 6195.27i	− 1.36794i	0.729512	−	0.683968i	$-0.239747\pi$
$30$	0	0
$31$	1862.42	0.348075	0.174038	−	0.984739i	$-0.444319\pi$
$31$	1862.42	0.348075	0.174038	+	0.984739i	$0.444319\pi$
$32$	0	0
$33$	−923.363	−0.147600
$34$	0	0
$35$	4968.33i	0.685552i
$36$	0	0
$37$	14708.1i	1.76625i	0.469142	+	0.883123i	$0.344563\pi$
$37$	14708.1i	1.76625i	−0.469142	+	0.883123i	$0.655437\pi$
$38$	0	0
$39$	4374.70	0.460560
$40$	0	0
$41$	18098.0	1.68140	0.840699	−	0.541503i	$-0.182144\pi$
$41$	18098.0	1.68140	0.840699	+	0.541503i	$0.182144\pi$
$42$	0	0
$43$	− 9260.46i	− 0.763768i	−0.924210	−	0.381884i	$-0.875275\pi$
$43$	− 9260.46i	− 0.763768i	0.924210	−	0.381884i	$-0.124725\pi$
$44$	0	0
$45$	− 3188.36i	− 0.234713i
$46$	0	0
$47$	24363.7	1.60879	0.804395	−	0.594095i	$-0.202490\pi$
$47$	24363.7	1.60879	0.804395	+	0.594095i	$0.202490\pi$
$48$	0	0
$49$	22687.9	1.34991
$50$	0	0
$51$	12959.3i	0.697680i
$52$	0	0
$53$	12764.8i	0.624202i	0.950049	+	0.312101i	$0.101033\pi$
$53$	12764.8i	0.624202i	−0.950049	+	0.312101i	$0.898967\pi$
$54$	0	0
$55$	2148.26	0.0957590
$56$	0	0
$57$	2213.96	0.0902572
$58$	0	0
$59$	− 20719.7i	− 0.774913i	−0.921888	−	0.387457i	$-0.873354\pi$
$59$	− 20719.7i	− 0.774913i	0.921888	−	0.387457i	$-0.126646\pi$
$60$	0	0
$61$	− 11368.5i	− 0.391183i	−0.980686	−	0.195591i	$-0.937337\pi$
$61$	− 11368.5i	− 0.391183i	0.980686	−	0.195591i	$-0.0626626\pi$
$62$	0	0
$63$	−25345.3	−0.804538
$64$	0	0
$65$	−10178.0	−0.298799
$66$	0	0
$67$	− 62614.9i	− 1.70408i	−0.523475	−	0.852041i	$-0.675365\pi$
$67$	− 62614.9i	− 1.70408i	0.523475	−	0.852041i	$-0.324635\pi$
$68$	0	0
$69$	27887.3i	0.705153i
$70$	0	0
$71$	61208.1	1.44100	0.720498	−	0.693457i	$-0.243913\pi$
$71$	61208.1	1.44100	0.720498	+	0.693457i	$0.243913\pi$
$72$	0	0
$73$	23236.4	0.510342	0.255171	−	0.966896i	$-0.417868\pi$
$73$	23236.4	0.510342	0.255171	+	0.966896i	$0.417868\pi$
$74$	0	0
$75$	− 6715.93i	− 0.137865i
$76$	0	0
$77$	− 17077.2i	− 0.328239i
$78$	0	0
$79$	−29172.5	−0.525904	−0.262952	−	0.964809i	$-0.584696\pi$
$79$	−29172.5	−0.525904	−0.262952	+	0.964809i	$0.584696\pi$
$80$	0	0
$81$	−11793.1	−0.199717
$82$	0	0
$83$	48099.7i	0.766386i	0.923668	+	0.383193i	$0.125176\pi$
$83$	48099.7i	0.766386i	−0.923668	+	0.383193i	$0.874824\pi$
$84$	0	0
$85$	− 30150.6i	− 0.452635i
$86$	0	0
$87$	66571.2	0.942950
$88$	0	0
$89$	30118.4	0.403048	0.201524	−	0.979484i	$-0.435411\pi$
$89$	30118.4	0.403048	0.201524	+	0.979484i	$0.435411\pi$
$90$	0	0
$91$	80908.2i	1.02421i
$92$	0	0
$93$	20012.6i	0.239937i
$94$	0	0
$95$	−5150.90	−0.0585563
$96$	0	0
$97$	−113676.	−1.22670	−0.613351	−	0.789811i	$-0.710179\pi$
$97$	−113676.	−1.22670	−0.613351	+	0.789811i	$0.710179\pi$
$98$	0	0
$99$	10959.1i	0.112379i
$100$	0	0
$101$	− 21867.4i	− 0.213302i	−0.994297	−	0.106651i	$-0.965987\pi$
$101$	− 21867.4i	− 0.213302i	0.994297	−	0.106651i	$-0.0340127\pi$
$102$	0	0
$103$	−156608.	−1.45452	−0.727262	−	0.686360i	$-0.759208\pi$
$103$	−156608.	−1.45452	−0.727262	+	0.686360i	$0.759208\pi$
$104$	0	0
$105$	−53387.1	−0.472567
$106$	0	0
$107$	91401.1i	0.771777i	0.922545	+	0.385889i	$0.126105\pi$
$107$	91401.1i	0.771777i	−0.922545	+	0.385889i	$0.873895\pi$
$108$	0	0
$109$	− 48973.9i	− 0.394819i	−0.980321	−	0.197410i	$-0.936747\pi$
$109$	− 48973.9i	− 0.394819i	0.980321	−	0.197410i	$-0.0632529\pi$
$110$	0	0
$111$	−158045.	−1.21751
$112$	0	0
$113$	−153638.	−1.13188	−0.565942	−	0.824445i	$-0.691487\pi$
$113$	−153638.	−1.13188	−0.565942	+	0.824445i	$0.691487\pi$
$114$	0	0
$115$	− 64881.4i	− 0.457483i
$116$	0	0
$117$	− 51921.8i	− 0.350659i
$118$	0	0
$119$	−239677.	−1.55152
$120$	0	0
$121$	153667.	0.954151
$122$	0	0
$123$	194472.i	1.15903i
$124$	0	0
$125$	15625.0i	0.0894427i
$126$	0	0
$127$	122092.	0.671705	0.335852	−	0.941915i	$-0.390976\pi$
$127$	122092.	0.671705	0.335852	+	0.941915i	$0.390976\pi$
$128$	0	0
$129$	99508.2	0.526483
$130$	0	0
$131$	− 179438.i	− 0.913560i	−0.889580	−	0.456780i	$-0.849003\pi$
$131$	− 179438.i	− 0.913560i	0.889580	−	0.456780i	$-0.150997\pi$
$132$	0	0
$133$	40946.2i	0.200717i
$134$	0	0
$135$	99539.3	0.470068
$136$	0	0
$137$	−327113.	−1.48900	−0.744502	−	0.667620i	$-0.767313\pi$
$137$	−327113.	−1.48900	−0.744502	+	0.667620i	$0.767313\pi$
$138$	0	0
$139$	− 359354.i	− 1.57756i	−0.614678	−	0.788778i	$-0.710714\pi$
$139$	− 359354.i	− 1.57756i	0.614678	−	0.788778i	$-0.289286\pi$
$140$	0	0
$141$	261800.i	1.10898i
$142$	0	0
$143$	34983.9	0.143063
$144$	0	0
$145$	−154882.	−0.611759
$146$	0	0
$147$	243792.i	0.930522i
$148$	0	0
$149$	− 494795.i	− 1.82583i	−0.408152	−	0.912914i	$-0.633827\pi$
$149$	− 494795.i	− 1.82583i	0.408152	−	0.912914i	$-0.366173\pi$
$150$	0	0
$151$	9960.81	0.0355511	0.0177755	−	0.999842i	$-0.494342\pi$
$151$	9960.81	0.0355511	0.0177755	+	0.999842i	$0.494342\pi$
$152$	0	0
$153$	153810.	0.531196
$154$	0	0
$155$	− 46560.5i	− 0.155664i
$156$	0	0
$157$	243819.i	0.789439i	0.918802	+	0.394719i	$0.129158\pi$
$157$	243819.i	0.789439i	−0.918802	+	0.394719i	$0.870842\pi$
$158$	0	0
$159$	−137164.	−0.430277
$160$	0	0
$161$	−515763.	−1.56814
$162$	0	0
$163$	− 268182.i	− 0.790608i	−0.918550	−	0.395304i	$-0.870639\pi$
$163$	− 268182.i	− 0.790608i	0.918550	−	0.395304i	$-0.129361\pi$
$164$	0	0
$165$	23084.1i	0.0660089i
$166$	0	0
$167$	17404.1	0.0482905	0.0241452	−	0.999708i	$-0.492314\pi$
$167$	17404.1	0.0482905	0.0241452	+	0.999708i	$0.492314\pi$
$168$	0	0
$169$	205547.	0.553597
$170$	0	0
$171$	− 26276.7i	− 0.0687195i
$172$	0	0
$173$	534365.i	1.35745i	0.734394	+	0.678723i	$0.237466\pi$
$173$	534365.i	1.35745i	−0.734394	+	0.678723i	$0.762534\pi$
$174$	0	0
$175$	124208.	0.306588
$176$	0	0
$177$	222643.	0.534166
$178$	0	0
$179$	− 342831.i	− 0.799738i	−0.916572	−	0.399869i	$-0.869056\pi$
$179$	− 342831.i	− 0.799738i	0.916572	−	0.399869i	$-0.130944\pi$
$180$	0	0
$181$	593221.i	1.34592i	0.739678	+	0.672961i	$0.234978\pi$
$181$	593221.i	1.34592i	−0.739678	+	0.672961i	$0.765022\pi$
$182$	0	0
$183$	122160.	0.269651
$184$	0	0
$185$	367701.	0.789889
$186$	0	0
$187$	103634.i	0.216720i
$188$	0	0
$189$	− 791271.i	− 1.61128i
$190$	0	0
$191$	1448.97	0.00287392	0.00143696	−	0.999999i	$-0.499543\pi$
$191$	1448.97	0.00287392	0.00143696	+	0.999999i	$0.499543\pi$
$192$	0	0
$193$	888526.	1.71703	0.858513	−	0.512791i	$-0.171388\pi$
$193$	888526.	1.71703	0.858513	+	0.512791i	$0.171388\pi$
$194$	0	0
$195$	− 109367.i	− 0.205969i
$196$	0	0
$197$	− 356565.i	− 0.654595i	−0.944921	−	0.327298i	$-0.893862\pi$
$197$	− 356565.i	− 0.654595i	0.944921	−	0.327298i	$-0.106138\pi$
$198$	0	0
$199$	406304.	0.727307	0.363654	−	0.931534i	$-0.381529\pi$
$199$	406304.	0.727307	0.363654	+	0.931534i	$0.381529\pi$
$200$	0	0
$201$	672827.	1.17466
$202$	0	0
$203$	1.23121e6i	2.09696i
$204$	0	0
$205$	− 452450.i	− 0.751944i
$206$	0	0
$207$	330984.	0.536886
$208$	0	0
$209$	17704.7	0.0280365
$210$	0	0
$211$	582646.i	0.900945i	0.892790	+	0.450473i	$0.148744\pi$
$211$	582646.i	0.900945i	−0.892790	+	0.450473i	$0.851256\pi$
$212$	0	0
$213$	657711.i	0.993312i
$214$	0	0
$215$	−231512.	−0.341568
$216$	0	0
$217$	−370125.	−0.533579
$218$	0	0
$219$	249686.i	0.351791i
$220$	0	0
$221$	− 490996.i	− 0.676234i
$222$	0	0
$223$	789020.	1.06249	0.531246	−	0.847218i	$-0.321724\pi$
$223$	789020.	1.06249	0.531246	+	0.847218i	$0.321724\pi$
$224$	0	0
$225$	−79709.1	−0.104967
$226$	0	0
$227$	− 872412.i	− 1.12372i	−0.827233	−	0.561858i	$-0.810087\pi$
$227$	− 872412.i	− 1.12372i	0.827233	−	0.561858i	$-0.189913\pi$
$228$	0	0
$229$	404072.i	0.509178i	0.967049	+	0.254589i	$0.0819402\pi$
$229$	404072.i	0.509178i	−0.967049	+	0.254589i	$0.918060\pi$
$230$	0	0
$231$	183503.	0.226263
$232$	0	0
$233$	−198854.	−0.239963	−0.119982	−	0.992776i	$-0.538284\pi$
$233$	−198854.	−0.239963	−0.119982	+	0.992776i	$0.538284\pi$
$234$	0	0
$235$	− 609093.i	− 0.719472i
$236$	0	0
$237$	− 313473.i	− 0.362518i
$238$	0	0
$239$	388433.	0.439868	0.219934	−	0.975515i	$-0.429416\pi$
$239$	388433.	0.439868	0.219934	+	0.975515i	$0.429416\pi$
$240$	0	0
$241$	−714818.	−0.792780	−0.396390	−	0.918082i	$-0.629737\pi$
$241$	−714818.	−0.792780	−0.396390	+	0.918082i	$0.629737\pi$
$242$	0	0
$243$	840800.i	0.913434i
$244$	0	0
$245$	− 567197.i	− 0.603696i
$246$	0	0
$247$	−83881.3	−0.0874828
$248$	0	0
$249$	−516855.	−0.528287
$250$	0	0
$251$	1.57948e6i	1.58244i	0.611529	+	0.791222i	$0.290555\pi$
$251$	1.57948e6i	1.58244i	−0.611529	+	0.791222i	$0.709445\pi$
$252$	0	0
$253$	223011.i	0.219041i
$254$	0	0
$255$	323983.	0.312012
$256$	0	0
$257$	1.80546e6	1.70512	0.852558	−	0.522632i	$-0.175050\pi$
$257$	1.80546e6	1.70512	0.852558	+	0.522632i	$0.175050\pi$
$258$	0	0
$259$	− 2.92298e6i	− 2.70755i
$260$	0	0
$261$	− 790111.i	− 0.717938i
$262$	0	0
$263$	−185503.	−0.165372	−0.0826860	−	0.996576i	$-0.526350\pi$
$263$	−185503.	−0.165372	−0.0826860	+	0.996576i	$0.526350\pi$
$264$	0	0
$265$	319121.	0.279152
$266$	0	0
$267$	323637.i	0.277830i
$268$	0	0
$269$	− 1.12615e6i	− 0.948887i	−0.880286	−	0.474444i	$-0.842649\pi$
$269$	− 1.12615e6i	− 0.948887i	0.880286	−	0.474444i	$-0.157351\pi$
$270$	0	0
$271$	−157610.	−0.130365	−0.0651825	−	0.997873i	$-0.520763\pi$
$271$	−157610.	−0.130365	−0.0651825	+	0.997873i	$0.520763\pi$
$272$	0	0
$273$	−869398.	−0.706012
$274$	0	0
$275$	− 53706.4i	− 0.0428247i
$276$	0	0
$277$	1.93510e6i	1.51532i	0.652650	+	0.757660i	$0.273657\pi$
$277$	1.93510e6i	1.51532i	−0.652650	+	0.757660i	$0.726343\pi$
$278$	0	0
$279$	237523.	0.182682
$280$	0	0
$281$	394102.	0.297744	0.148872	−	0.988856i	$-0.452436\pi$
$281$	394102.	0.297744	0.148872	+	0.988856i	$0.452436\pi$
$282$	0	0
$283$	538592.i	0.399755i	0.979821	+	0.199878i	$0.0640545\pi$
$283$	538592.i	0.399755i	−0.979821	+	0.199878i	$0.935946\pi$
$284$	0	0
$285$	− 55348.9i	− 0.0403642i
$286$	0	0
$287$	−3.59667e6	−2.57748
$288$	0	0
$289$	34635.4	0.0243936
$290$	0	0
$291$	− 1.22150e6i	− 0.845594i
$292$	0	0
$293$	1.25662e6i	0.855134i	0.903984	+	0.427567i	$0.140629\pi$
$293$	1.25662e6i	0.855134i	−0.903984	+	0.427567i	$0.859371\pi$
$294$	0	0
$295$	−517992.	−0.346552
$296$	0	0
$297$	−342138.	−0.225066
$298$	0	0
$299$	− 1.05658e6i	− 0.683477i
$300$	0	0
$301$	1.84036e6i	1.17081i
$302$	0	0
$303$	234976.	0.147034
$304$	0	0
$305$	−284213.	−0.174942
$306$	0	0
$307$	− 227401.i	− 0.137704i	−0.997627	−	0.0688519i	$-0.978066\pi$
$307$	− 227401.i	− 0.137704i	0.997627	−	0.0688519i	$-0.0219336\pi$
$308$	0	0
$309$	− 1.68283e6i	− 1.00264i
$310$	0	0
$311$	−1.13530e6	−0.665593	−0.332796	−	0.942999i	$-0.607992\pi$
$311$	−1.13530e6	−0.665593	−0.332796	+	0.942999i	$0.607992\pi$
$312$	0	0
$313$	−2.32737e6	−1.34278	−0.671390	−	0.741105i	$-0.734302\pi$
$313$	−2.32737e6	−1.34278	−0.671390	+	0.741105i	$0.734302\pi$
$314$	0	0
$315$	633633.i	0.359800i
$316$	0	0
$317$	1.50212e6i	0.839570i	0.907624	+	0.419785i	$0.137894\pi$
$317$	1.50212e6i	0.839570i	−0.907624	+	0.419785i	$0.862106\pi$
$318$	0	0
$319$	532362.	0.292907
$320$	0	0
$321$	−982149.	−0.532004
$322$	0	0
$323$	− 248484.i	− 0.132523i
$324$	0	0
$325$	254450.i	0.133627i
$326$	0	0
$327$	526248.	0.272158
$328$	0	0
$329$	−4.84188e6	−2.46618
$330$	0	0
$331$	2.72477e6i	1.36697i	0.729964	+	0.683486i	$0.239537\pi$
$331$	2.72477e6i	1.36697i	−0.729964	+	0.683486i	$0.760463\pi$
$332$	0	0
$333$	1.87578e6i	0.926984i
$334$	0	0
$335$	−1.56537e6	−0.762089
$336$	0	0
$337$	2.04270e6	0.979784	0.489892	−	0.871783i	$-0.337036\pi$
$337$	2.04270e6	0.979784	0.489892	+	0.871783i	$0.337036\pi$
$338$	0	0
$339$	− 1.65091e6i	− 0.780234i
$340$	0	0
$341$	160038.i	0.0745312i
$342$	0	0
$343$	−1.16872e6	−0.536385
$344$	0	0
$345$	697182.	0.315354
$346$	0	0
$347$	2.57456e6i	1.14783i	0.818914	+	0.573916i	$0.194577\pi$
$347$	2.57456e6i	1.14783i	−0.818914	+	0.573916i	$0.805423\pi$
$348$	0	0
$349$	− 1.97991e6i	− 0.870125i	−0.900400	−	0.435062i	$-0.856726\pi$
$349$	− 1.97991e6i	− 0.870125i	0.900400	−	0.435062i	$-0.143274\pi$
$350$	0	0
$351$	1.62098e6	0.702278
$352$	0	0
$353$	−1.02870e6	−0.439394	−0.219697	−	0.975568i	$-0.570507\pi$
$353$	−1.02870e6	−0.439394	−0.219697	+	0.975568i	$0.570507\pi$
$354$	0	0
$355$	− 1.53020e6i	− 0.644433i
$356$	0	0
$357$	− 2.57544e6i	− 1.06950i
$358$	0	0
$359$	979196.	0.400990	0.200495	−	0.979695i	$-0.435745\pi$
$359$	979196.	0.400990	0.200495	+	0.979695i	$0.435745\pi$
$360$	0	0
$361$	2.43365e6	0.982856
$362$	0	0
$363$	1.65123e6i	0.657719i
$364$	0	0
$365$	− 580909.i	− 0.228232i
$366$	0	0
$367$	1.07920e6	0.418250	0.209125	−	0.977889i	$-0.432938\pi$
$367$	1.07920e6	0.418250	0.209125	+	0.977889i	$0.432938\pi$
$368$	0	0
$369$	2.30812e6	0.882454
$370$	0	0
$371$	− 2.53679e6i	− 0.956864i
$372$	0	0
$373$	− 3.65291e6i	− 1.35946i	−0.733461	−	0.679731i	$-0.762096\pi$
$373$	− 3.65291e6i	− 1.35946i	0.733461	−	0.679731i	$-0.237904\pi$
$374$	0	0
$375$	−167898.	−0.0616550
$376$	0	0
$377$	−2.52222e6	−0.913964
$378$	0	0
$379$	2.99413e6i	1.07071i	0.844627	+	0.535355i	$0.179822\pi$
$379$	2.99413e6i	1.07071i	−0.844627	+	0.535355i	$0.820178\pi$
$380$	0	0
$381$	1.31194e6i	0.463022i
$382$	0	0
$383$	−4.15754e6	−1.44824	−0.724119	−	0.689675i	$-0.757753\pi$
$383$	−4.15754e6	−1.44824	−0.724119	+	0.689675i	$0.757753\pi$
$384$	0	0
$385$	−426930.	−0.146793
$386$	0	0
$387$	− 1.18103e6i	− 0.400851i
$388$	0	0
$389$	− 1.08421e6i	− 0.363278i	−0.983365	−	0.181639i	$-0.941860\pi$
$389$	− 1.08421e6i	− 0.363278i	0.983365	−	0.181639i	$-0.0581402\pi$
$390$	0	0
$391$	3.12994e6	1.03537
$392$	0	0
$393$	1.92815e6	0.629739
$394$	0	0
$395$	729313.i	0.235191i
$396$	0	0
$397$	− 3.11966e6i	− 0.993415i	−0.867918	−	0.496708i	$-0.834542\pi$
$397$	− 3.11966e6i	− 0.993415i	0.867918	−	0.496708i	$-0.165458\pi$
$398$	0	0
$399$	−439987.	−0.138359
$400$	0	0
$401$	−3.18432e6	−0.988908	−0.494454	−	0.869204i	$-0.664632\pi$
$401$	−3.18432e6	−0.988908	−0.494454	+	0.869204i	$0.664632\pi$
$402$	0	0
$403$	− 758228.i	− 0.232561i
$404$	0	0
$405$	294827.i	0.0893160i
$406$	0	0
$407$	−1.26387e6	−0.378195
$408$	0	0
$409$	3.76338e6	1.11242	0.556211	−	0.831041i	$-0.312255\pi$
$409$	3.76338e6	1.11242	0.556211	+	0.831041i	$0.312255\pi$
$410$	0	0
$411$	− 3.51499e6i	− 1.02641i
$412$	0	0
$413$	4.11769e6i	1.18790i
$414$	0	0
$415$	1.20249e6	0.342738
$416$	0	0
$417$	3.86143e6	1.08745
$418$	0	0
$419$	2.73449e6i	0.760924i	0.924797	+	0.380462i	$0.124235\pi$
$419$	2.73449e6i	0.760924i	−0.924797	+	0.380462i	$0.875765\pi$
$420$	0	0
$421$	450371.i	0.123841i	0.998081	+	0.0619206i	$0.0197225\pi$
$421$	450371.i	0.123841i	−0.998081	+	0.0619206i	$0.980277\pi$
$422$	0	0
$423$	3.10722e6	0.844346
$424$	0	0
$425$	−753765.	−0.202425
$426$	0	0
$427$	2.25930e6i	0.599659i
$428$	0	0
$429$	375919.i	0.0986169i
$430$	0	0
$431$	−517327.	−0.134144	−0.0670721	−	0.997748i	$-0.521366\pi$
$431$	−517327.	−0.134144	−0.0670721	+	0.997748i	$0.521366\pi$
$432$	0	0
$433$	2.24867e6	0.576375	0.288188	−	0.957574i	$-0.406947\pi$
$433$	2.24867e6	0.576375	0.288188	+	0.957574i	$0.406947\pi$
$434$	0	0
$435$	− 1.66428e6i	− 0.421700i
$436$	0	0
$437$	− 534716.i	− 0.133943i
$438$	0	0
$439$	−862979.	−0.213717	−0.106858	−	0.994274i	$-0.534079\pi$
$439$	−862979.	−0.213717	−0.106858	+	0.994274i	$0.534079\pi$
$440$	0	0
$441$	2.89349e6	0.708476
$442$	0	0
$443$	972329.i	0.235399i	0.993049	+	0.117699i	$0.0375519\pi$
$443$	972329.i	0.235399i	−0.993049	+	0.117699i	$0.962448\pi$
$444$	0	0
$445$	− 752959.i	− 0.180248i
$446$	0	0
$447$	5.31682e6	1.25859
$448$	0	0
$449$	−28297.9	−0.00662428	−0.00331214	−	0.999995i	$-0.501054\pi$
$449$	−28297.9	−0.00662428	−0.00331214	+	0.999995i	$0.501054\pi$
$450$	0	0
$451$	1.55516e6i	0.360027i
$452$	0	0
$453$	107034.i	0.0245062i
$454$	0	0
$455$	2.02270e6	0.458041
$456$	0	0
$457$	456269.	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$457$	456269.	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$458$	0	0
$459$	4.80187e6i	1.06385i
$460$	0	0
$461$	− 11198.5i	− 0.00245418i	−0.999999	−	0.00122709i	$-0.999609\pi$
$461$	− 11198.5i	− 0.00245418i	0.999999	−	0.00122709i	$-0.000390595\pi$
$462$	0	0
$463$	7.93015e6	1.71921	0.859605	−	0.510959i	$-0.170710\pi$
$463$	7.93015e6	1.71921	0.859605	+	0.510959i	$0.170710\pi$
$464$	0	0
$465$	500315.	0.107303
$466$	0	0
$467$	− 6.59151e6i	− 1.39860i	−0.714830	−	0.699298i	$-0.753496\pi$
$467$	− 6.59151e6i	− 1.39860i	0.714830	−	0.699298i	$-0.246504\pi$
$468$	0	0
$469$	1.24437e7i	2.61226i
$470$	0	0
$471$	−2.61995e6	−0.544179
$472$	0	0
$473$	795754.	0.163541
$474$	0	0
$475$	128772.i	0.0261872i
$476$	0	0
$477$	1.62796e6i	0.327602i
$478$	0	0
$479$	1.60702e6	0.320023	0.160012	−	0.987115i	$-0.448847\pi$
$479$	1.60702e6	0.320023	0.160012	+	0.987115i	$0.448847\pi$
$480$	0	0
$481$	5.98794e6	1.18009
$482$	0	0
$483$	− 5.54213e6i	− 1.08096i
$484$	0	0
$485$	2.84190e6i	0.548597i
$486$	0	0
$487$	1.81230e6	0.346264	0.173132	−	0.984899i	$-0.444611\pi$
$487$	1.81230e6	0.346264	0.173132	+	0.984899i	$0.444611\pi$
$488$	0	0
$489$	2.88175e6	0.544985
$490$	0	0
$491$	292083.i	0.0546768i	0.999626	+	0.0273384i	$0.00870317\pi$
$491$	292083.i	0.0546768i	−0.999626	+	0.0273384i	$0.991297\pi$
$492$	0	0
$493$	− 7.47164e6i	− 1.38452i
$494$	0	0
$495$	273977.	0.0502575
$496$	0	0
$497$	−1.21641e7	−2.20896
$498$	0	0
$499$	− 8.47619e6i	− 1.52388i	−0.647650	−	0.761938i	$-0.724248\pi$
$499$	− 8.47619e6i	− 1.52388i	0.647650	−	0.761938i	$-0.275752\pi$
$500$	0	0
$501$	187016.i	0.0332877i
$502$	0	0
$503$	−2.07725e6	−0.366074	−0.183037	−	0.983106i	$-0.558593\pi$
$503$	−2.07725e6	−0.366074	−0.183037	+	0.983106i	$0.558593\pi$
$504$	0	0
$505$	−546686.	−0.0953914
$506$	0	0
$507$	2.20870e6i	0.381607i
$508$	0	0
$509$	739081.i	0.126444i	0.997999	+	0.0632219i	$0.0201376\pi$
$509$	739081.i	0.126444i	−0.997999	+	0.0632219i	$0.979862\pi$
$510$	0	0
$511$	−4.61784e6	−0.782324
$512$	0	0
$513$	820347.	0.137627
$514$	0	0
$515$	3.91520e6i	0.650483i
$516$	0	0
$517$	2.09358e6i	0.344480i
$518$	0	0
$519$	−5.74201e6	−0.935719
$520$	0	0
$521$	−7.18449e6	−1.15958	−0.579791	−	0.814765i	$-0.696866\pi$
$521$	−7.18449e6	−1.15958	−0.579791	+	0.814765i	$0.696866\pi$
$522$	0	0
$523$	1.94687e6i	0.311231i	0.987818	+	0.155616i	$0.0497361\pi$
$523$	1.94687e6i	0.311231i	−0.987818	+	0.155616i	$0.950264\pi$
$524$	0	0
$525$	1.33468e6i	0.211338i
$526$	0	0
$527$	2.24612e6	0.352295
$528$	0	0
$529$	299003.	0.0464554
$530$	0	0
$531$	− 2.64247e6i	− 0.406700i
$532$	0	0
$533$	− 7.36804e6i	− 1.12340i
$534$	0	0
$535$	2.28503e6	0.345149
$536$	0	0
$537$	3.68389e6	0.551278
$538$	0	0
$539$	1.94958e6i	0.289047i
$540$	0	0
$541$	− 1.18585e7i	− 1.74195i	−0.491330	−	0.870973i	$-0.663489\pi$
$541$	− 1.18585e7i	− 1.74195i	0.491330	−	0.870973i	$-0.336511\pi$
$542$	0	0
$543$	−6.37445e6	−0.927776
$544$	0	0
$545$	−1.22435e6	−0.176569
$546$	0	0
$547$	2.10343e6i	0.300580i	0.988642	+	0.150290i	$0.0480207\pi$
$547$	2.10343e6i	0.300580i	−0.988642	+	0.150290i	$0.951979\pi$
$548$	0	0
$549$	− 1.44988e6i	− 0.205306i
$550$	0	0
$551$	−1.27645e6	−0.179112
$552$	0	0
$553$	5.79755e6	0.806179
$554$	0	0
$555$	3.95113e6i	0.544489i
$556$	0	0
$557$	4.61967e6i	0.630918i	0.948939	+	0.315459i	$0.102158\pi$
$557$	4.61967e6i	0.630918i	−0.948939	+	0.315459i	$0.897842\pi$
$558$	0	0
$559$	−3.77012e6	−0.510300
$560$	0	0
$561$	−1.11360e6	−0.149390
$562$	0	0
$563$	− 1.23357e7i	− 1.64019i	−0.572227	−	0.820096i	$-0.693920\pi$
$563$	− 1.23357e7i	− 1.64019i	0.572227	−	0.820096i	$-0.306080\pi$
$564$	0	0
$565$	3.84095e6i	0.506194i
$566$	0	0
$567$	2.34367e6	0.306154
$568$	0	0
$569$	7.85222e6	1.01674	0.508372	−	0.861138i	$-0.330247\pi$
$569$	7.85222e6	1.01674	0.508372	+	0.861138i	$0.330247\pi$
$570$	0	0
$571$	1.14825e7i	1.47383i	0.675984	+	0.736916i	$0.263719\pi$
$571$	1.14825e7i	1.47383i	−0.675984	+	0.736916i	$0.736281\pi$
$572$	0	0
$573$	15569.8i	0.00198106i
$574$	0	0
$575$	−1.62203e6	−0.204593
$576$	0	0
$577$	−1.10508e7	−1.38183	−0.690913	−	0.722938i	$-0.742791\pi$
$577$	−1.10508e7	−1.38183	−0.690913	+	0.722938i	$0.742791\pi$
$578$	0	0
$579$	9.54765e6i	1.18359i
$580$	0	0
$581$	− 9.55901e6i	− 1.17482i
$582$	0	0
$583$	−1.09689e6	−0.133656
$584$	0	0
$585$	−1.29805e6	−0.156820
$586$	0	0
$587$	− 1.22974e6i	− 0.147305i	−0.997284	−	0.0736524i	$-0.976534\pi$
$587$	− 1.22974e6i	− 0.147305i	0.997284	−	0.0736524i	$-0.0234655\pi$
$588$	0	0
$589$	− 383725.i	− 0.0455756i
$590$	0	0
$591$	3.83146e6	0.451228
$592$	0	0
$593$	4.14520e6	0.484071	0.242035	−	0.970267i	$-0.422185\pi$
$593$	4.14520e6	0.484071	0.242035	+	0.970267i	$0.422185\pi$
$594$	0	0
$595$	5.99192e6i	0.693863i
$596$	0	0
$597$	4.36593e6i	0.501350i
$598$	0	0
$599$	−1.54313e7	−1.75726	−0.878629	−	0.477505i	$-0.841541\pi$
$599$	−1.54313e7	−1.75726	−0.878629	+	0.477505i	$0.841541\pi$
$600$	0	0
$601$	−1.20526e7	−1.36111	−0.680556	−	0.732696i	$-0.738262\pi$
$601$	−1.20526e7	−1.36111	−0.680556	+	0.732696i	$0.738262\pi$
$602$	0	0
$603$	− 7.98556e6i	− 0.894359i
$604$	0	0
$605$	− 3.84167e6i	− 0.426709i
$606$	0	0
$607$	−7.95296e6	−0.876107	−0.438053	−	0.898949i	$-0.644332\pi$
$607$	−7.95296e6	−0.876107	−0.438053	+	0.898949i	$0.644332\pi$
$608$	0	0
$609$	−1.32299e7	−1.44549
$610$	0	0
$611$	− 9.91895e6i	− 1.07489i
$612$	0	0
$613$	− 1.04301e7i	− 1.12108i	−0.828128	−	0.560538i	$-0.810594\pi$
$613$	− 1.04301e7i	− 1.12108i	0.828128	−	0.560538i	$-0.189406\pi$
$614$	0	0
$615$	4.86179e6	0.518333
$616$	0	0
$617$	−6.91606e6	−0.731384	−0.365692	−	0.930736i	$-0.619168\pi$
$617$	−6.91606e6	−0.731384	−0.365692	+	0.930736i	$0.619168\pi$
$618$	0	0
$619$	1.15145e7i	1.20786i	0.797037	+	0.603931i	$0.206400\pi$
$619$	1.15145e7i	1.20786i	−0.797037	+	0.603931i	$0.793600\pi$
$620$	0	0
$621$	1.03332e7i	1.07524i
$622$	0	0
$623$	−5.98552e6	−0.617848
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	190246.i	0.0193262i
$628$	0	0
$629$	1.77383e7i	1.78766i
$630$	0	0
$631$	−5.76766e6	−0.576669	−0.288334	−	0.957530i	$-0.593101\pi$
$631$	−5.76766e6	−0.576669	−0.288334	+	0.957530i	$0.593101\pi$
$632$	0	0
$633$	−6.26081e6	−0.621042
$634$	0	0
$635$	− 3.05230e6i	− 0.300396i
$636$	0	0
$637$	− 9.23668e6i	− 0.901919i
$638$	0	0
$639$	7.80614e6	0.756283
$640$	0	0
$641$	−518990.	−0.0498901	−0.0249450	−	0.999689i	$-0.507941\pi$
$641$	−518990.	−0.0498901	−0.0249450	+	0.999689i	$0.507941\pi$
$642$	0	0
$643$	6.52874e6i	0.622733i	0.950290	+	0.311367i	$0.100787\pi$
$643$	6.52874e6i	0.622733i	−0.950290	+	0.311367i	$0.899213\pi$
$644$	0	0
$645$	− 2.48770e6i	− 0.235450i
$646$	0	0
$647$	6.41651e6	0.602613	0.301306	−	0.953527i	$-0.402577\pi$
$647$	6.41651e6	0.602613	0.301306	+	0.953527i	$0.402577\pi$
$648$	0	0
$649$	1.78045e6	0.165927
$650$	0	0
$651$	− 3.97717e6i	− 0.367808i
$652$	0	0
$653$	9.11065e6i	0.836116i	0.908420	+	0.418058i	$0.137289\pi$
$653$	9.11065e6i	0.836116i	−0.908420	+	0.418058i	$0.862711\pi$
$654$	0	0
$655$	−4.48596e6	−0.408557
$656$	0	0
$657$	2.96344e6	0.267844
$658$	0	0
$659$	1.97306e7i	1.76981i	0.465775	+	0.884903i	$0.345776\pi$
$659$	1.97306e7i	1.76981i	−0.465775	+	0.884903i	$0.654224\pi$
$660$	0	0
$661$	− 9.04530e6i	− 0.805228i	−0.915370	−	0.402614i	$-0.868102\pi$
$661$	− 9.04530e6i	− 0.805228i	0.915370	−	0.402614i	$-0.131898\pi$
$662$	0	0
$663$	5.27599e6	0.466144
$664$	0	0
$665$	1.02365e6	0.0897634
$666$	0	0
$667$	− 1.60783e7i	− 1.39935i
$668$	0	0
$669$	8.47840e6i	0.732401i
$670$	0	0
$671$	976900.	0.0837614
$672$	0	0
$673$	−1.11387e7	−0.947974	−0.473987	−	0.880532i	$-0.657186\pi$
$673$	−1.11387e7	−0.947974	−0.473987	+	0.880532i	$0.657186\pi$
$674$	0	0
$675$	− 2.48848e6i	− 0.210221i
$676$	0	0
$677$	− 1.13338e7i	− 0.950395i	−0.879879	−	0.475198i	$-0.842377\pi$
$677$	− 1.13338e7i	− 0.950395i	0.879879	−	0.475198i	$-0.157623\pi$
$678$	0	0
$679$	2.25912e7	1.88046
$680$	0	0
$681$	9.37449e6	0.774604
$682$	0	0
$683$	− 1.06005e7i	− 0.869514i	−0.900548	−	0.434757i	$-0.856834\pi$
$683$	− 1.06005e7i	− 0.869514i	0.900548	−	0.434757i	$-0.143166\pi$
$684$	0	0
$685$	8.17782e6i	0.665903i
$686$	0	0
$687$	−4.34195e6	−0.350988
$688$	0	0
$689$	5.19681e6	0.417051
$690$	0	0
$691$	− 1.18528e7i	− 0.944331i	−0.881510	−	0.472165i	$-0.843473\pi$
$691$	− 1.18528e7i	− 0.944331i	0.881510	−	0.472165i	$-0.156527\pi$
$692$	0	0
$693$	− 2.17793e6i	− 0.172271i
$694$	0	0
$695$	−8.98384e6	−0.705505
$696$	0	0
$697$	2.18266e7	1.70178
$698$	0	0
$699$	− 2.13678e6i	− 0.165412i
$700$	0	0
$701$	1.42728e7i	1.09702i	0.836144	+	0.548511i	$0.184805\pi$
$701$	1.42728e7i	1.09702i	−0.836144	+	0.548511i	$0.815195\pi$
$702$	0	0
$703$	3.03039e6	0.231265
$704$	0	0
$705$	6.54500e6	0.495949
$706$	0	0
$707$	4.34578e6i	0.326979i
$708$	0	0
$709$	1.40052e7i	1.04635i	0.852227	+	0.523173i	$0.175252\pi$
$709$	1.40052e7i	1.04635i	−0.852227	+	0.523173i	$0.824748\pi$
$710$	0	0
$711$	−3.72050e6	−0.276012
$712$	0	0
$713$	4.83345e6	0.356069
$714$	0	0
$715$	− 874598.i	− 0.0639799i
$716$	0	0
$717$	4.17391e6i	0.303211i
$718$	0	0
$719$	−7.28309e6	−0.525404	−0.262702	−	0.964877i	$-0.584614\pi$
$719$	−7.28309e6	−0.525404	−0.262702	+	0.964877i	$0.584614\pi$
$720$	0	0
$721$	3.11232e7	2.22970
$722$	0	0
$723$	− 7.68107e6i	− 0.546482i
$724$	0	0
$725$	3.87205e6i	0.273587i
$726$	0	0
$727$	−3.41888e6	−0.239910	−0.119955	−	0.992779i	$-0.538275\pi$
$727$	−3.41888e6	−0.239910	−0.119955	+	0.992779i	$0.538275\pi$
$728$	0	0
$729$	−1.19005e7	−0.829368
$730$	0	0
$731$	− 1.11683e7i	− 0.773028i
$732$	0	0
$733$	− 2.11502e6i	− 0.145397i	−0.997354	−	0.0726983i	$-0.976839\pi$
$733$	− 2.11502e6i	− 0.145397i	0.997354	−	0.0726983i	$-0.0231610\pi$
$734$	0	0
$735$	6.09481e6	0.416142
$736$	0	0
$737$	5.38051e6	0.364884
$738$	0	0
$739$	− 4.97420e6i	− 0.335052i	−0.985868	−	0.167526i	$-0.946422\pi$
$739$	− 4.97420e6i	− 0.335052i	0.985868	−	0.167526i	$-0.0535778\pi$
$740$	0	0
$741$	− 901345.i	− 0.0603039i
$742$	0	0
$743$	−6.04483e6	−0.401709	−0.200855	−	0.979621i	$-0.564372\pi$
$743$	−6.04483e6	−0.401709	−0.200855	+	0.979621i	$0.564372\pi$
$744$	0	0
$745$	−1.23699e7	−0.816535
$746$	0	0
$747$	6.13437e6i	0.402225i
$748$	0	0
$749$	− 1.81644e7i	− 1.18309i
$750$	0	0
$751$	2.14521e7	1.38794	0.693970	−	0.720004i	$-0.255860\pi$
$751$	2.14521e7	1.38794	0.693970	+	0.720004i	$0.255860\pi$
$752$	0	0
$753$	−1.69722e7	−1.09082
$754$	0	0
$755$	− 249020.i	− 0.0158989i
$756$	0	0
$757$	− 1.22790e7i	− 0.778794i	−0.921070	−	0.389397i	$-0.872683\pi$
$757$	− 1.22790e7i	− 0.778794i	0.921070	−	0.389397i	$-0.127317\pi$
$758$	0	0
$759$	−2.39636e6	−0.150990
$760$	0	0
$761$	−1.38648e7	−0.867867	−0.433933	−	0.900945i	$-0.642875\pi$
$761$	−1.38648e7	−0.867867	−0.433933	+	0.900945i	$0.642875\pi$
$762$	0	0
$763$	9.73273e6i	0.605234i
$764$	0	0
$765$	− 3.84524e6i	− 0.237558i
$766$	0	0
$767$	−8.43539e6	−0.517746
$768$	0	0
$769$	1.92248e6	0.117232	0.0586158	−	0.998281i	$-0.481331\pi$
$769$	1.92248e6	0.117232	0.0586158	+	0.998281i	$0.481331\pi$
$770$	0	0
$771$	1.94005e7i	1.17538i
$772$	0	0
$773$	− 2.66728e7i	− 1.60554i	−0.596291	−	0.802768i	$-0.703360\pi$
$773$	− 2.66728e7i	− 1.60554i	0.596291	−	0.802768i	$-0.296640\pi$
$774$	0	0
$775$	−1.16401e6	−0.0696151
$776$	0	0
$777$	3.14088e7	1.86638
$778$	0	0
$779$	− 3.72883e6i	− 0.220155i
$780$	0	0
$781$	5.25963e6i	0.308551i
$782$	0	0
$783$	2.46669e7	1.43784
$784$	0	0
$785$	6.09548e6	0.353048
$786$	0	0
$787$	7.22979e6i	0.416092i	0.978119	+	0.208046i	$0.0667103\pi$
$787$	7.22979e6i	0.416092i	−0.978119	+	0.208046i	$0.933290\pi$
$788$	0	0
$789$	− 1.99332e6i	− 0.113995i
$790$	0	0
$791$	3.05329e7	1.73511
$792$	0	0
$793$	−4.62835e6	−0.261362
$794$	0	0
$795$	3.42911e6i	0.192426i
$796$	0	0
$797$	4.41257e6i	0.246062i	0.992403	+	0.123031i	$0.0392615\pi$
$797$	4.41257e6i	0.246062i	−0.992403	+	0.123031i	$0.960738\pi$
$798$	0	0
$799$	2.93832e7	1.62829
$800$	0	0
$801$	3.84113e6	0.211533
$802$	0	0
$803$	1.99671e6i	0.109276i
$804$	0	0
$805$	1.28941e7i	0.701295i
$806$	0	0
$807$	1.21010e7	0.654090
$808$	0	0
$809$	6.04388e6	0.324672	0.162336	−	0.986736i	$-0.448097\pi$
$809$	6.04388e6	0.324672	0.162336	+	0.986736i	$0.448097\pi$
$810$	0	0
$811$	− 2.40664e7i	− 1.28487i	−0.766340	−	0.642435i	$-0.777924\pi$
$811$	− 2.40664e7i	− 1.28487i	0.766340	−	0.642435i	$-0.222076\pi$
$812$	0	0
$813$	− 1.69360e6i	− 0.0898636i
$814$	0	0
$815$	−6.70456e6	−0.353571
$816$	0	0
$817$	−1.90799e6	−0.100005
$818$	0	0
$819$	1.03186e7i	0.537540i
$820$	0	0
$821$	1.16527e7i	0.603350i	0.953411	+	0.301675i	$0.0975458\pi$
$821$	1.16527e7i	0.603350i	−0.953411	+	0.301675i	$0.902454\pi$
$822$	0	0
$823$	161156.	0.00829369	0.00414684	−	0.999991i	$-0.498680\pi$
$823$	161156.	0.00829369	0.00414684	+	0.999991i	$0.498680\pi$
$824$	0	0
$825$	577102.	0.0295201
$826$	0	0
$827$	2.85594e7i	1.45206i	0.687661	+	0.726031i	$0.258637\pi$
$827$	2.85594e7i	1.45206i	−0.687661	+	0.726031i	$0.741363\pi$
$828$	0	0
$829$	− 1.49564e7i	− 0.755859i	−0.925834	−	0.377929i	$-0.876636\pi$
$829$	− 1.49564e7i	− 0.755859i	0.925834	−	0.377929i	$-0.123364\pi$
$830$	0	0
$831$	−2.07936e7	−1.04455
$832$	0	0
$833$	2.73621e7	1.36627
$834$	0	0
$835$	− 435104.i	− 0.0215962i
$836$	0	0
$837$	7.41536e6i	0.365863i
$838$	0	0
$839$	1.76445e7	0.865374	0.432687	−	0.901544i	$-0.357565\pi$
$839$	1.76445e7	0.865374	0.432687	+	0.901544i	$0.357565\pi$
$840$	0	0
$841$	−1.78703e7	−0.871246
$842$	0	0
$843$	4.23482e6i	0.205242i
$844$	0	0
$845$	− 5.13866e6i	− 0.247576i
$846$	0	0
$847$	−3.05387e7	−1.46266
$848$	0	0
$849$	−5.78744e6	−0.275561
$850$	0	0
$851$	3.81711e7i	1.80681i
$852$	0	0
$853$	3.43356e7i	1.61574i	0.589361	+	0.807870i	$0.299380\pi$
$853$	3.43356e7i	1.61574i	−0.589361	+	0.807870i	$0.700620\pi$
$854$	0	0
$855$	−656917.	−0.0307323
$856$	0	0
$857$	−1.27760e7	−0.594216	−0.297108	−	0.954844i	$-0.596022\pi$
$857$	−1.27760e7	−0.594216	−0.297108	+	0.954844i	$0.596022\pi$
$858$	0	0
$859$	− 3.23768e7i	− 1.49710i	−0.663077	−	0.748551i	$-0.730750\pi$
$859$	− 3.23768e7i	− 1.49710i	0.663077	−	0.748551i	$-0.269250\pi$
$860$	0	0
$861$	− 3.86480e7i	− 1.77672i
$862$	0	0
$863$	−2.57138e7	−1.17527	−0.587637	−	0.809125i	$-0.699942\pi$
$863$	−2.57138e7	−1.17527	−0.587637	+	0.809125i	$0.699942\pi$
$864$	0	0
$865$	1.33591e7	0.607068
$866$	0	0
$867$	372174.i	0.0168151i
$868$	0	0
$869$	− 2.50680e6i	− 0.112608i
$870$	0	0
$871$	−2.54917e7	−1.13856
$872$	0	0
$873$	−1.44976e7	−0.643814
$874$	0	0
$875$	− 3.10521e6i	− 0.137110i
$876$	0	0
$877$	4.69271e6i	0.206027i	0.994680	+	0.103014i	$0.0328485\pi$
$877$	4.69271e6i	0.206027i	−0.994680	+	0.103014i	$0.967151\pi$
$878$	0	0
$879$	−1.35030e7	−0.589464
$880$	0	0
$881$	446984.	0.0194023	0.00970113	−	0.999953i	$-0.496912\pi$
$881$	446984.	0.0194023	0.00970113	+	0.999953i	$0.496912\pi$
$882$	0	0
$883$	3.04436e7i	1.31400i	0.753892	+	0.656999i	$0.228174\pi$
$883$	3.04436e7i	1.31400i	−0.753892	+	0.656999i	$0.771826\pi$
$884$	0	0
$885$	− 5.56608e6i	− 0.238886i
$886$	0	0
$887$	1.40477e7	0.599508	0.299754	−	0.954016i	$-0.403095\pi$
$887$	1.40477e7	0.599508	0.299754	+	0.954016i	$0.403095\pi$
$888$	0	0
$889$	−2.42638e7	−1.02968
$890$	0	0
$891$	− 1.01338e6i	− 0.0427640i
$892$	0	0
$893$	− 5.01980e6i	− 0.210648i
$894$	0	0
$895$	−8.57078e6	−0.357654
$896$	0	0
$897$	1.13535e7	0.471137
$898$	0	0
$899$	− 1.15382e7i	− 0.476145i
$900$	0	0
$901$	1.53947e7i	0.631769i
$902$	0	0
$903$	−1.97756e7	−0.807067
$904$	0	0
$905$	1.48305e7	0.601915
$906$	0	0
$907$	− 3.03623e7i	− 1.22551i	−0.790273	−	0.612755i	$-0.790061\pi$
$907$	− 3.03623e7i	− 1.22551i	0.790273	−	0.612755i	$-0.209939\pi$
$908$	0	0
$909$	− 2.78885e6i	− 0.111948i
$910$	0	0
$911$	1.87844e7	0.749896	0.374948	−	0.927046i	$-0.377661\pi$
$911$	1.87844e7	0.749896	0.374948	+	0.927046i	$0.377661\pi$
$912$	0	0
$913$	−4.13322e6	−0.164101
$914$	0	0
$915$	− 3.05401e6i	− 0.120592i
$916$	0	0
$917$	3.56604e7i	1.40043i
$918$	0	0
$919$	−4.38053e7	−1.71095	−0.855476	−	0.517842i	$-0.826735\pi$
$919$	−4.38053e7	−1.71095	−0.855476	+	0.517842i	$0.826735\pi$
$920$	0	0
$921$	2.44353e6	0.0949224
$922$	0	0
$923$	− 2.49190e7i	− 0.962779i
$924$	0	0
$925$	− 9.19253e6i	− 0.353249i
$926$	0	0
$927$	−1.99729e7	−0.763383
$928$	0	0
$929$	−1.88925e7	−0.718208	−0.359104	−	0.933298i	$-0.616918\pi$
$929$	−1.88925e7	−0.718208	−0.359104	+	0.933298i	$0.616918\pi$
$930$	0	0
$931$	− 4.67452e6i	− 0.176751i
$932$	0	0
$933$	− 1.21993e7i	− 0.458809i
$934$	0	0
$935$	2.59085e6	0.0969199
$936$	0	0
$937$	−3.00651e7	−1.11870	−0.559350	−	0.828931i	$-0.688949\pi$
$937$	−3.00651e7	−1.11870	−0.559350	+	0.828931i	$0.688949\pi$
$938$	0	0
$939$	− 2.50087e7i	− 0.925609i
$940$	0	0
$941$	368874.i	0.0135801i	0.999977	+	0.00679006i	$0.00216136\pi$
$941$	368874.i	0.0135801i	−0.999977	+	0.00679006i	$0.997839\pi$
$942$	0	0
$943$	4.69689e7	1.72001
$944$	0	0
$945$	−1.97818e7	−0.720586
$946$	0	0
$947$	− 3.31548e6i	− 0.120135i	−0.998194	−	0.0600677i	$-0.980868\pi$
$947$	− 3.31548e6i	− 0.120135i	0.998194	−	0.0600677i	$-0.0191317\pi$
$948$	0	0
$949$	− 9.45999e6i	− 0.340977i
$950$	0	0
$951$	−1.61410e7	−0.578735
$952$	0	0
$953$	1.89980e7	0.677604	0.338802	−	0.940858i	$-0.389978\pi$
$953$	1.89980e7	0.677604	0.338802	+	0.940858i	$0.389978\pi$
$954$	0	0
$955$	− 36224.1i	− 0.00128526i
$956$	0	0
$957$	5.72049e6i	0.201908i
$958$	0	0
$959$	6.50082e7	2.28256
$960$	0	0
$961$	−2.51605e7	−0.878844
$962$	0	0
$963$	1.16568e7i	0.405054i
$964$	0	0
$965$	− 2.22132e7i	− 0.767878i
$966$	0	0
$967$	−1.00131e7	−0.344354	−0.172177	−	0.985066i	$-0.555080\pi$
$967$	−1.00131e7	−0.344354	−0.172177	+	0.985066i	$0.555080\pi$
$968$	0	0
$969$	2.67008e6	0.0913514
$970$	0	0
$971$	2.66607e6i	0.0907453i	0.998970	+	0.0453726i	$0.0144475\pi$
$971$	2.66607e6i	0.0907453i	−0.998970	+	0.0453726i	$0.985552\pi$
$972$	0	0
$973$	7.14155e7i	2.41830i
$974$	0	0
$975$	−2.73419e6	−0.0921121
$976$	0	0
$977$	−3.15935e6	−0.105891	−0.0529457	−	0.998597i	$-0.516861\pi$
$977$	−3.15935e6	−0.105891	−0.0529457	+	0.998597i	$0.516861\pi$
$978$	0	0
$979$	2.58808e6i	0.0863021i
$980$	0	0
$981$	− 6.24586e6i	− 0.207214i
$982$	0	0
$983$	−2.50171e6	−0.0825758	−0.0412879	−	0.999147i	$-0.513146\pi$
$983$	−2.50171e6	−0.0825758	−0.0412879	+	0.999147i	$0.513146\pi$
$984$	0	0
$985$	−8.91412e6	−0.292744
$986$	0	0
$987$	− 5.20284e7i	− 1.69999i
$988$	0	0
$989$	− 2.40333e7i	− 0.781307i
$990$	0	0
$991$	−4.14173e7	−1.33967	−0.669835	−	0.742510i	$-0.733635\pi$
$991$	−4.14173e7	−1.33967	−0.669835	+	0.742510i	$0.733635\pi$
$992$	0	0
$993$	−2.92790e7	−0.942286
$994$	0	0
$995$	− 1.01576e7i	− 0.325262i
$996$	0	0
$997$	− 3.80654e7i	− 1.21281i	−0.795156	−	0.606405i	$-0.792611\pi$
$997$	− 3.80654e7i	− 1.21281i	0.795156	−	0.606405i	$-0.207389\pi$
$998$	0	0
$999$	−5.85612e7	−1.85651

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.d.a.81.13		20
4.3	odd	2		40.6.d.a.21.10	yes	20
5.2	odd	4		800.6.f.b.49.13		20
5.3	odd	4		800.6.f.c.49.8		20
5.4	even	2		800.6.d.c.401.8		20
8.3	odd	2		40.6.d.a.21.9	✓	20
8.5	even	2	inner	160.6.d.a.81.8		20
12.11	even	2		360.6.k.b.181.11		20
20.3	even	4		200.6.f.b.149.19		20
20.7	even	4		200.6.f.c.149.2		20
20.19	odd	2		200.6.d.b.101.11		20
24.11	even	2		360.6.k.b.181.12		20
40.3	even	4		200.6.f.c.149.1		20
40.13	odd	4		800.6.f.b.49.14		20
40.19	odd	2		200.6.d.b.101.12		20
40.27	even	4		200.6.f.b.149.20		20
40.29	even	2		800.6.d.c.401.13		20
40.37	odd	4		800.6.f.c.49.7		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.9	✓	20	8.3	odd	2
40.6.d.a.21.10	yes	20	4.3	odd	2
160.6.d.a.81.8		20	8.5	even	2	inner
160.6.d.a.81.13		20	1.1	even	1	trivial
200.6.d.b.101.11		20	20.19	odd	2
200.6.d.b.101.12		20	40.19	odd	2
200.6.f.b.149.19		20	20.3	even	4
200.6.f.b.149.20		20	40.27	even	4
200.6.f.c.149.1		20	40.3	even	4
200.6.f.c.149.2		20	20.7	even	4
360.6.k.b.181.11		20	12.11	even	2
360.6.k.b.181.12		20	24.11	even	2
800.6.d.c.401.8		20	5.4	even	2
800.6.d.c.401.13		20	40.29	even	2
800.6.f.b.49.13		20	5.2	odd	4
800.6.f.b.49.14		20	40.13	odd	4
800.6.f.c.49.7		20	40.37	odd	4
800.6.f.c.49.8		20	5.3	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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	160.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+10.7455i q^{3} -25.0000i q^{5} -198.733 q^{7} +127.535 q^{9} +O(q^{10})\)	\(q+10.7455i q^{3} -25.0000i q^{5} -198.733 q^{7} +127.535 q^{9} +85.9303i q^{11} -407.120i q^{13} +268.637 q^{15} +1206.02 q^{17} -206.036i q^{19} -2135.48i q^{21} +2595.25 q^{23} -625.000 q^{25} +3981.57i q^{27} -6195.27i q^{29} +1862.42 q^{31} -923.363 q^{33} +4968.33i q^{35} +14708.1i q^{37} +4374.70 q^{39} +18098.0 q^{41} -9260.46i q^{43} -3188.36i q^{45} +24363.7 q^{47} +22687.9 q^{49} +12959.3i q^{51} +12764.8i q^{53} +2148.26 q^{55} +2213.96 q^{57} -20719.7i q^{59} -11368.5i q^{61} -25345.3 q^{63} -10178.0 q^{65} -62614.9i q^{67} +27887.3i q^{69} +61208.1 q^{71} +23236.4 q^{73} -6715.93i q^{75} -17077.2i q^{77} -29172.5 q^{79} -11793.1 q^{81} +48099.7i q^{83} -30150.6i q^{85} +66571.2 q^{87} +30118.4 q^{89} +80908.2i q^{91} +20012.6i q^{93} -5150.90 q^{95} -113676. q^{97} +10959.1i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 196 q^{7} - 1620 q^{9}+O(q^{10}) \) 20 * q + 196 * q^7 - 1620 * q^9	\( 20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63} + 200312 q^{71} - 105136 q^{73} - 282080 q^{79} + 65172 q^{81} + 332592 q^{87} - 3160 q^{89} - 144400 q^{95} + 147376 q^{97}+O(q^{100}) \) 20 * q + 196 * q^7 - 1620 * q^9 - 900 * q^15 + 4676 * q^23 - 12500 * q^25 - 7160 * q^31 + 5672 * q^33 + 44904 * q^39 + 11608 * q^41 - 44180 * q^47 + 18756 * q^49 + 24200 * q^55 + 5032 * q^57 - 240620 * q^63 + 200312 * q^71 - 105136 * q^73 - 282080 * q^79 + 65172 * q^81 + 332592 * q^87 - 3160 * q^89 - 144400 * q^95 + 147376 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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