LMFDB - Embedded newform 8020.2.a.c.1.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.01605 q^{3} -1.00000 q^{5} +1.12256 q^{7} +1.06447 q^{9} +O(q^{10})$	$q+2.01605 q^{3} -1.00000 q^{5} +1.12256 q^{7} +1.06447 q^{9} +1.62514 q^{11} -5.08798 q^{13} -2.01605 q^{15} +0.834419 q^{17} -2.00041 q^{19} +2.26313 q^{21} -2.97509 q^{23} +1.00000 q^{25} -3.90213 q^{27} +7.67214 q^{29} +9.19552 q^{31} +3.27636 q^{33} -1.12256 q^{35} -6.56344 q^{37} -10.2576 q^{39} -11.5305 q^{41} -2.96250 q^{43} -1.06447 q^{45} -11.2435 q^{47} -5.73987 q^{49} +1.68223 q^{51} -1.53785 q^{53} -1.62514 q^{55} -4.03294 q^{57} -2.89273 q^{59} -6.38343 q^{61} +1.19493 q^{63} +5.08798 q^{65} +13.2551 q^{67} -5.99793 q^{69} -7.85264 q^{71} +14.9487 q^{73} +2.01605 q^{75} +1.82431 q^{77} -7.45354 q^{79} -11.0603 q^{81} +12.0054 q^{83} -0.834419 q^{85} +15.4674 q^{87} +11.9730 q^{89} -5.71154 q^{91} +18.5387 q^{93} +2.00041 q^{95} -12.9406 q^{97} +1.72991 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.01605	1.16397	0.581984	−	0.813200i	$-0.302276\pi$
$3$	2.01605	1.16397	0.581984	+	0.813200i	$0.302276\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.12256	0.424286	0.212143	−	0.977239i	$-0.431956\pi$
$7$	1.12256	0.424286	0.212143	+	0.977239i	$0.431956\pi$
$8$	0	0
$9$	1.06447	0.354824
$10$	0	0
$11$	1.62514	0.489997	0.244998	−	0.969523i	$-0.421213\pi$
$11$	1.62514	0.489997	0.244998	+	0.969523i	$0.421213\pi$
$12$	0	0
$13$	−5.08798	−1.41115	−0.705576	−	0.708634i	$-0.749312\pi$
$13$	−5.08798	−1.41115	−0.705576	+	0.708634i	$0.749312\pi$
$14$	0	0
$15$	−2.01605	−0.520543
$16$	0	0
$17$	0.834419	0.202376	0.101188	−	0.994867i	$-0.467736\pi$
$17$	0.834419	0.202376	0.101188	+	0.994867i	$0.467736\pi$
$18$	0	0
$19$	−2.00041	−0.458926	−0.229463	−	0.973317i	$-0.573697\pi$
$19$	−2.00041	−0.458926	−0.229463	+	0.973317i	$0.573697\pi$
$20$	0	0
$21$	2.26313	0.493856
$22$	0	0
$23$	−2.97509	−0.620348	−0.310174	−	0.950680i	$-0.600387\pi$
$23$	−2.97509	−0.620348	−0.310174	+	0.950680i	$0.600387\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.90213	−0.750965
$28$	0	0
$29$	7.67214	1.42468	0.712340	−	0.701834i	$-0.247635\pi$
$29$	7.67214	1.42468	0.712340	+	0.701834i	$0.247635\pi$
$30$	0	0
$31$	9.19552	1.65156	0.825782	−	0.563989i	$-0.190734\pi$
$31$	9.19552	1.65156	0.825782	+	0.563989i	$0.190734\pi$
$32$	0	0
$33$	3.27636	0.570341
$34$	0	0
$35$	−1.12256	−0.189747
$36$	0	0
$37$	−6.56344	−1.07902	−0.539511	−	0.841978i	$-0.681391\pi$
$37$	−6.56344	−1.07902	−0.539511	+	0.841978i	$0.681391\pi$
$38$	0	0
$39$	−10.2576	−1.64254
$40$	0	0
$41$	−11.5305	−1.80077	−0.900385	−	0.435095i	$-0.856715\pi$
$41$	−11.5305	−1.80077	−0.900385	+	0.435095i	$0.856715\pi$
$42$	0	0
$43$	−2.96250	−0.451777	−0.225888	−	0.974153i	$-0.572528\pi$
$43$	−2.96250	−0.451777	−0.225888	+	0.974153i	$0.572528\pi$
$44$	0	0
$45$	−1.06447	−0.158682
$46$	0	0
$47$	−11.2435	−1.64004	−0.820019	−	0.572336i	$-0.806037\pi$
$47$	−11.2435	−1.64004	−0.820019	+	0.572336i	$0.806037\pi$
$48$	0	0
$49$	−5.73987	−0.819981
$50$	0	0
$51$	1.68223	0.235560
$52$	0	0
$53$	−1.53785	−0.211239	−0.105620	−	0.994407i	$-0.533683\pi$
$53$	−1.53785	−0.211239	−0.105620	+	0.994407i	$0.533683\pi$
$54$	0	0
$55$	−1.62514	−0.219133
$56$	0	0
$57$	−4.03294	−0.534176
$58$	0	0
$59$	−2.89273	−0.376602	−0.188301	−	0.982111i	$-0.560298\pi$
$59$	−2.89273	−0.376602	−0.188301	+	0.982111i	$0.560298\pi$
$60$	0	0
$61$	−6.38343	−0.817315	−0.408657	−	0.912688i	$-0.634003\pi$
$61$	−6.38343	−0.817315	−0.408657	+	0.912688i	$0.634003\pi$
$62$	0	0
$63$	1.19493	0.150547
$64$	0	0
$65$	5.08798	0.631086
$66$	0	0
$67$	13.2551	1.61937	0.809686	−	0.586863i	$-0.199637\pi$
$67$	13.2551	1.61937	0.809686	+	0.586863i	$0.199637\pi$
$68$	0	0
$69$	−5.99793	−0.722066
$70$	0	0
$71$	−7.85264	−0.931937	−0.465969	−	0.884801i	$-0.654294\pi$
$71$	−7.85264	−0.931937	−0.465969	+	0.884801i	$0.654294\pi$
$72$	0	0
$73$	14.9487	1.74961	0.874807	−	0.484472i	$-0.160988\pi$
$73$	14.9487	1.74961	0.874807	+	0.484472i	$0.160988\pi$
$74$	0	0
$75$	2.01605	0.232794
$76$	0	0
$77$	1.82431	0.207899
$78$	0	0
$79$	−7.45354	−0.838589	−0.419294	−	0.907850i	$-0.637722\pi$
$79$	−7.45354	−0.838589	−0.419294	+	0.907850i	$0.637722\pi$
$80$	0	0
$81$	−11.0603	−1.22892
$82$	0	0
$83$	12.0054	1.31776	0.658879	−	0.752249i	$-0.271031\pi$
$83$	12.0054	1.31776	0.658879	+	0.752249i	$0.271031\pi$
$84$	0	0
$85$	−0.834419	−0.0905055
$86$	0	0
$87$	15.4674	1.65828
$88$	0	0
$89$	11.9730	1.26913	0.634566	−	0.772869i	$-0.281179\pi$
$89$	11.9730	1.26913	0.634566	+	0.772869i	$0.281179\pi$
$90$	0	0
$91$	−5.71154	−0.598732
$92$	0	0
$93$	18.5387	1.92237
$94$	0	0
$95$	2.00041	0.205238
$96$	0	0
$97$	−12.9406	−1.31391	−0.656957	−	0.753928i	$-0.728157\pi$
$97$	−12.9406	−1.31391	−0.656957	+	0.753928i	$0.728157\pi$
$98$	0	0
$99$	1.72991	0.173862
$100$	0	0
$101$	−15.9320	−1.58530	−0.792649	−	0.609679i	$-0.791299\pi$
$101$	−15.9320	−1.58530	−0.792649	+	0.609679i	$0.791299\pi$
$102$	0	0
$103$	0.831194	0.0819000	0.0409500	−	0.999161i	$-0.486962\pi$
$103$	0.831194	0.0819000	0.0409500	+	0.999161i	$0.486962\pi$
$104$	0	0
$105$	−2.26313	−0.220859
$106$	0	0
$107$	−2.31881	−0.224168	−0.112084	−	0.993699i	$-0.535753\pi$
$107$	−2.31881	−0.224168	−0.112084	+	0.993699i	$0.535753\pi$
$108$	0	0
$109$	−4.39213	−0.420690	−0.210345	−	0.977627i	$-0.567459\pi$
$109$	−4.39213	−0.420690	−0.210345	+	0.977627i	$0.567459\pi$
$110$	0	0
$111$	−13.2322	−1.25595
$112$	0	0
$113$	5.04133	0.474248	0.237124	−	0.971479i	$-0.423795\pi$
$113$	5.04133	0.474248	0.237124	+	0.971479i	$0.423795\pi$
$114$	0	0
$115$	2.97509	0.277428
$116$	0	0
$117$	−5.41601	−0.500710
$118$	0	0
$119$	0.936682	0.0858655
$120$	0	0
$121$	−8.35893	−0.759903
$122$	0	0
$123$	−23.2462	−2.09604
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.1708	1.34619	0.673096	−	0.739555i	$-0.264964\pi$
$127$	15.1708	1.34619	0.673096	+	0.739555i	$0.264964\pi$
$128$	0	0
$129$	−5.97256	−0.525854
$130$	0	0
$131$	−2.93589	−0.256510	−0.128255	−	0.991741i	$-0.540938\pi$
$131$	−2.93589	−0.256510	−0.128255	+	0.991741i	$0.540938\pi$
$132$	0	0
$133$	−2.24558	−0.194716
$134$	0	0
$135$	3.90213	0.335842
$136$	0	0
$137$	2.64554	0.226024	0.113012	−	0.993594i	$-0.463950\pi$
$137$	2.64554	0.226024	0.113012	+	0.993594i	$0.463950\pi$
$138$	0	0
$139$	−20.7882	−1.76324	−0.881618	−	0.471964i	$-0.843545\pi$
$139$	−20.7882	−1.76324	−0.881618	+	0.471964i	$0.843545\pi$
$140$	0	0
$141$	−22.6676	−1.90895
$142$	0	0
$143$	−8.26866	−0.691460
$144$	0	0
$145$	−7.67214	−0.637137
$146$	0	0
$147$	−11.5719	−0.954433
$148$	0	0
$149$	7.83229	0.641646	0.320823	−	0.947139i	$-0.396041\pi$
$149$	7.83229	0.641646	0.320823	+	0.947139i	$0.396041\pi$
$150$	0	0
$151$	−3.92025	−0.319026	−0.159513	−	0.987196i	$-0.550992\pi$
$151$	−3.92025	−0.319026	−0.159513	+	0.987196i	$0.550992\pi$
$152$	0	0
$153$	0.888215	0.0718079
$154$	0	0
$155$	−9.19552	−0.738602
$156$	0	0
$157$	1.87270	0.149458	0.0747289	−	0.997204i	$-0.476191\pi$
$157$	1.87270	0.149458	0.0747289	+	0.997204i	$0.476191\pi$
$158$	0	0
$159$	−3.10038	−0.245876
$160$	0	0
$161$	−3.33970	−0.263205
$162$	0	0
$163$	−15.5607	−1.21881	−0.609404	−	0.792860i	$-0.708591\pi$
$163$	−15.5607	−1.21881	−0.609404	+	0.792860i	$0.708591\pi$
$164$	0	0
$165$	−3.27636	−0.255064
$166$	0	0
$167$	−17.0933	−1.32272	−0.661360	−	0.750069i	$-0.730020\pi$
$167$	−17.0933	−1.32272	−0.661360	+	0.750069i	$0.730020\pi$
$168$	0	0
$169$	12.8875	0.991349
$170$	0	0
$171$	−2.12938	−0.162838
$172$	0	0
$173$	1.52419	0.115882	0.0579410	−	0.998320i	$-0.481546\pi$
$173$	1.52419	0.115882	0.0579410	+	0.998320i	$0.481546\pi$
$174$	0	0
$175$	1.12256	0.0848572
$176$	0	0
$177$	−5.83191	−0.438353
$178$	0	0
$179$	−23.2680	−1.73913	−0.869565	−	0.493819i	$-0.835600\pi$
$179$	−23.2680	−1.73913	−0.869565	+	0.493819i	$0.835600\pi$
$180$	0	0
$181$	11.4090	0.848025	0.424013	−	0.905656i	$-0.360621\pi$
$181$	11.4090	0.848025	0.424013	+	0.905656i	$0.360621\pi$
$182$	0	0
$183$	−12.8693	−0.951329
$184$	0	0
$185$	6.56344	0.482554
$186$	0	0
$187$	1.35604	0.0991638
$188$	0	0
$189$	−4.38036	−0.318624
$190$	0	0
$191$	−12.4209	−0.898747	−0.449374	−	0.893344i	$-0.648353\pi$
$191$	−12.4209	−0.898747	−0.449374	+	0.893344i	$0.648353\pi$
$192$	0	0
$193$	−16.6310	−1.19713	−0.598563	−	0.801076i	$-0.704261\pi$
$193$	−16.6310	−1.19713	−0.598563	+	0.801076i	$0.704261\pi$
$194$	0	0
$195$	10.2576	0.734565
$196$	0	0
$197$	−0.212544	−0.0151432	−0.00757158	−	0.999971i	$-0.502410\pi$
$197$	−0.212544	−0.0151432	−0.00757158	+	0.999971i	$0.502410\pi$
$198$	0	0
$199$	9.29807	0.659123	0.329561	−	0.944134i	$-0.393099\pi$
$199$	9.29807	0.659123	0.329561	+	0.944134i	$0.393099\pi$
$200$	0	0
$201$	26.7231	1.88490
$202$	0	0
$203$	8.61240	0.604472
$204$	0	0
$205$	11.5305	0.805328
$206$	0	0
$207$	−3.16689	−0.220114
$208$	0	0
$209$	−3.25094	−0.224872
$210$	0	0
$211$	8.98413	0.618493	0.309247	−	0.950982i	$-0.399923\pi$
$211$	8.98413	0.618493	0.309247	+	0.950982i	$0.399923\pi$
$212$	0	0
$213$	−15.8313	−1.08475
$214$	0	0
$215$	2.96250	0.202041
$216$	0	0
$217$	10.3225	0.700736
$218$	0	0
$219$	30.1374	2.03650
$220$	0	0
$221$	−4.24551	−0.285584
$222$	0	0
$223$	−24.6816	−1.65280	−0.826401	−	0.563083i	$-0.809615\pi$
$223$	−24.6816	−1.65280	−0.826401	+	0.563083i	$0.809615\pi$
$224$	0	0
$225$	1.06447	0.0709647
$226$	0	0
$227$	−22.7322	−1.50879	−0.754393	−	0.656423i	$-0.772069\pi$
$227$	−22.7322	−1.50879	−0.754393	+	0.656423i	$0.772069\pi$
$228$	0	0
$229$	−9.15852	−0.605212	−0.302606	−	0.953116i	$-0.597857\pi$
$229$	−9.15852	−0.605212	−0.302606	+	0.953116i	$0.597857\pi$
$230$	0	0
$231$	3.67790	0.241988
$232$	0	0
$233$	6.65219	0.435800	0.217900	−	0.975971i	$-0.430079\pi$
$233$	6.65219	0.435800	0.217900	+	0.975971i	$0.430079\pi$
$234$	0	0
$235$	11.2435	0.733448
$236$	0	0
$237$	−15.0267	−0.976091
$238$	0	0
$239$	−17.6825	−1.14378	−0.571892	−	0.820329i	$-0.693790\pi$
$239$	−17.6825	−1.14378	−0.571892	+	0.820329i	$0.693790\pi$
$240$	0	0
$241$	27.2381	1.75456	0.877279	−	0.479982i	$-0.159357\pi$
$241$	27.2381	1.75456	0.877279	+	0.479982i	$0.159357\pi$
$242$	0	0
$243$	−10.5918	−0.679464
$244$	0	0
$245$	5.73987	0.366707
$246$	0	0
$247$	10.1781	0.647615
$248$	0	0
$249$	24.2034	1.53383
$250$	0	0
$251$	18.4689	1.16575	0.582874	−	0.812562i	$-0.301928\pi$
$251$	18.4689	1.16575	0.582874	+	0.812562i	$0.301928\pi$
$252$	0	0
$253$	−4.83492	−0.303969
$254$	0	0
$255$	−1.68223	−0.105346
$256$	0	0
$257$	19.2871	1.20310	0.601548	−	0.798837i	$-0.294551\pi$
$257$	19.2871	1.20310	0.601548	+	0.798837i	$0.294551\pi$
$258$	0	0
$259$	−7.36782	−0.457814
$260$	0	0
$261$	8.16677	0.505510
$262$	0	0
$263$	22.0218	1.35792	0.678960	−	0.734175i	$-0.262431\pi$
$263$	22.0218	1.35792	0.678960	+	0.734175i	$0.262431\pi$
$264$	0	0
$265$	1.53785	0.0944691
$266$	0	0
$267$	24.1381	1.47723
$268$	0	0
$269$	15.5174	0.946115	0.473057	−	0.881032i	$-0.343150\pi$
$269$	15.5174	0.946115	0.473057	+	0.881032i	$0.343150\pi$
$270$	0	0
$271$	−14.3378	−0.870961	−0.435480	−	0.900198i	$-0.643421\pi$
$271$	−14.3378	−0.870961	−0.435480	+	0.900198i	$0.643421\pi$
$272$	0	0
$273$	−11.5148	−0.696906
$274$	0	0
$275$	1.62514	0.0979994
$276$	0	0
$277$	−30.5427	−1.83514	−0.917568	−	0.397579i	$-0.869850\pi$
$277$	−30.5427	−1.83514	−0.917568	+	0.397579i	$0.869850\pi$
$278$	0	0
$279$	9.78837	0.586014
$280$	0	0
$281$	−6.07422	−0.362358	−0.181179	−	0.983450i	$-0.557991\pi$
$281$	−6.07422	−0.362358	−0.181179	+	0.983450i	$0.557991\pi$
$282$	0	0
$283$	15.6389	0.929634	0.464817	−	0.885407i	$-0.346120\pi$
$283$	15.6389	0.929634	0.464817	+	0.885407i	$0.346120\pi$
$284$	0	0
$285$	4.03294	0.238891
$286$	0	0
$287$	−12.9437	−0.764042
$288$	0	0
$289$	−16.3037	−0.959044
$290$	0	0
$291$	−26.0888	−1.52936
$292$	0	0
$293$	25.8144	1.50809	0.754047	−	0.656821i	$-0.228099\pi$
$293$	25.8144	1.50809	0.754047	+	0.656821i	$0.228099\pi$
$294$	0	0
$295$	2.89273	0.168421
$296$	0	0
$297$	−6.34149	−0.367971
$298$	0	0
$299$	15.1372	0.875406
$300$	0	0
$301$	−3.32557	−0.191683
$302$	0	0
$303$	−32.1198	−1.84524
$304$	0	0
$305$	6.38343	0.365514
$306$	0	0
$307$	−15.8129	−0.902491	−0.451246	−	0.892400i	$-0.649020\pi$
$307$	−15.8129	−0.902491	−0.451246	+	0.892400i	$0.649020\pi$
$308$	0	0
$309$	1.67573	0.0953291
$310$	0	0
$311$	−1.01413	−0.0575060	−0.0287530	−	0.999587i	$-0.509154\pi$
$311$	−1.01413	−0.0575060	−0.0287530	+	0.999587i	$0.509154\pi$
$312$	0	0
$313$	29.7839	1.68349	0.841744	−	0.539877i	$-0.181529\pi$
$313$	29.7839	1.68349	0.841744	+	0.539877i	$0.181529\pi$
$314$	0	0
$315$	−1.19493	−0.0673266
$316$	0	0
$317$	7.18153	0.403355	0.201678	−	0.979452i	$-0.435361\pi$
$317$	7.18153	0.403355	0.201678	+	0.979452i	$0.435361\pi$
$318$	0	0
$319$	12.4683	0.698089
$320$	0	0
$321$	−4.67485	−0.260925
$322$	0	0
$323$	−1.66918	−0.0928759
$324$	0	0
$325$	−5.08798	−0.282230
$326$	0	0
$327$	−8.85476	−0.489670
$328$	0	0
$329$	−12.6215	−0.695846
$330$	0	0
$331$	2.69680	0.148229	0.0741147	−	0.997250i	$-0.476387\pi$
$331$	2.69680	0.148229	0.0741147	+	0.997250i	$0.476387\pi$
$332$	0	0
$333$	−6.98659	−0.382863
$334$	0	0
$335$	−13.2551	−0.724205
$336$	0	0
$337$	−9.27719	−0.505361	−0.252680	−	0.967550i	$-0.581312\pi$
$337$	−9.27719	−0.505361	−0.252680	+	0.967550i	$0.581312\pi$
$338$	0	0
$339$	10.1636	0.552011
$340$	0	0
$341$	14.9440	0.809261
$342$	0	0
$343$	−14.3012	−0.772193
$344$	0	0
$345$	5.99793	0.322918
$346$	0	0
$347$	13.1516	0.706015	0.353008	−	0.935620i	$-0.385159\pi$
$347$	13.1516	0.706015	0.353008	+	0.935620i	$0.385159\pi$
$348$	0	0
$349$	−30.0762	−1.60994	−0.804971	−	0.593314i	$-0.797819\pi$
$349$	−30.0762	−1.60994	−0.804971	+	0.593314i	$0.797819\pi$
$350$	0	0
$351$	19.8540	1.05973
$352$	0	0
$353$	10.8572	0.577871	0.288936	−	0.957349i	$-0.406699\pi$
$353$	10.8572	0.577871	0.288936	+	0.957349i	$0.406699\pi$
$354$	0	0
$355$	7.85264	0.416775
$356$	0	0
$357$	1.88840	0.0999448
$358$	0	0
$359$	−8.65343	−0.456710	−0.228355	−	0.973578i	$-0.573335\pi$
$359$	−8.65343	−0.456710	−0.228355	+	0.973578i	$0.573335\pi$
$360$	0	0
$361$	−14.9983	−0.789387
$362$	0	0
$363$	−16.8521	−0.884504
$364$	0	0
$365$	−14.9487	−0.782451
$366$	0	0
$367$	−29.6972	−1.55018	−0.775092	−	0.631849i	$-0.782296\pi$
$367$	−29.6972	−1.55018	−0.775092	+	0.631849i	$0.782296\pi$
$368$	0	0
$369$	−12.2739	−0.638955
$370$	0	0
$371$	−1.72632	−0.0896259
$372$	0	0
$373$	13.7206	0.710426	0.355213	−	0.934785i	$-0.384408\pi$
$373$	13.7206	0.710426	0.355213	+	0.934785i	$0.384408\pi$
$374$	0	0
$375$	−2.01605	−0.104109
$376$	0	0
$377$	−39.0357	−2.01044
$378$	0	0
$379$	16.7428	0.860022	0.430011	−	0.902824i	$-0.358510\pi$
$379$	16.7428	0.860022	0.430011	+	0.902824i	$0.358510\pi$
$380$	0	0
$381$	30.5852	1.56693
$382$	0	0
$383$	16.3149	0.833651	0.416826	−	0.908986i	$-0.363143\pi$
$383$	16.3149	0.833651	0.416826	+	0.908986i	$0.363143\pi$
$384$	0	0
$385$	−1.82431	−0.0929752
$386$	0	0
$387$	−3.15349	−0.160301
$388$	0	0
$389$	22.7348	1.15270	0.576350	−	0.817203i	$-0.304476\pi$
$389$	22.7348	1.15270	0.576350	+	0.817203i	$0.304476\pi$
$390$	0	0
$391$	−2.48247	−0.125544
$392$	0	0
$393$	−5.91890	−0.298569
$394$	0	0
$395$	7.45354	0.375028
$396$	0	0
$397$	10.3034	0.517111	0.258555	−	0.965996i	$-0.416754\pi$
$397$	10.3034	0.517111	0.258555	+	0.965996i	$0.416754\pi$
$398$	0	0
$399$	−4.52720	−0.226644
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−46.7866	−2.33061
$404$	0	0
$405$	11.0603	0.549591
$406$	0	0
$407$	−10.6665	−0.528718
$408$	0	0
$409$	−35.3063	−1.74578	−0.872892	−	0.487914i	$-0.837758\pi$
$409$	−35.3063	−1.74578	−0.872892	+	0.487914i	$0.837758\pi$
$410$	0	0
$411$	5.33355	0.263085
$412$	0	0
$413$	−3.24726	−0.159787
$414$	0	0
$415$	−12.0054	−0.589320
$416$	0	0
$417$	−41.9102	−2.05235
$418$	0	0
$419$	−1.31377	−0.0641820	−0.0320910	−	0.999485i	$-0.510217\pi$
$419$	−1.31377	−0.0641820	−0.0320910	+	0.999485i	$0.510217\pi$
$420$	0	0
$421$	2.02583	0.0987331	0.0493666	−	0.998781i	$-0.484280\pi$
$421$	2.02583	0.0987331	0.0493666	+	0.998781i	$0.484280\pi$
$422$	0	0
$423$	−11.9684	−0.581925
$424$	0	0
$425$	0.834419	0.0404753
$426$	0	0
$427$	−7.16576	−0.346775
$428$	0	0
$429$	−16.6701	−0.804838
$430$	0	0
$431$	−37.9243	−1.82675	−0.913375	−	0.407120i	$-0.866533\pi$
$431$	−37.9243	−1.82675	−0.913375	+	0.407120i	$0.866533\pi$
$432$	0	0
$433$	25.7789	1.23885	0.619427	−	0.785054i	$-0.287365\pi$
$433$	25.7789	1.23885	0.619427	+	0.785054i	$0.287365\pi$
$434$	0	0
$435$	−15.4674	−0.741607
$436$	0	0
$437$	5.95140	0.284694
$438$	0	0
$439$	−1.07589	−0.0513496	−0.0256748	−	0.999670i	$-0.508173\pi$
$439$	−1.07589	−0.0513496	−0.0256748	+	0.999670i	$0.508173\pi$
$440$	0	0
$441$	−6.10992	−0.290949
$442$	0	0
$443$	−7.79083	−0.370153	−0.185077	−	0.982724i	$-0.559253\pi$
$443$	−7.79083	−0.370153	−0.185077	+	0.982724i	$0.559253\pi$
$444$	0	0
$445$	−11.9730	−0.567573
$446$	0	0
$447$	15.7903	0.746856
$448$	0	0
$449$	−8.49466	−0.400888	−0.200444	−	0.979705i	$-0.564238\pi$
$449$	−8.49466	−0.400888	−0.200444	+	0.979705i	$0.564238\pi$
$450$	0	0
$451$	−18.7387	−0.882371
$452$	0	0
$453$	−7.90344	−0.371336
$454$	0	0
$455$	5.71154	0.267761
$456$	0	0
$457$	40.2721	1.88385	0.941924	−	0.335827i	$-0.109016\pi$
$457$	40.2721	1.88385	0.941924	+	0.335827i	$0.109016\pi$
$458$	0	0
$459$	−3.25601	−0.151978
$460$	0	0
$461$	13.8138	0.643372	0.321686	−	0.946846i	$-0.395751\pi$
$461$	13.8138	0.643372	0.321686	+	0.946846i	$0.395751\pi$
$462$	0	0
$463$	−5.44468	−0.253036	−0.126518	−	0.991964i	$-0.540380\pi$
$463$	−5.44468	−0.253036	−0.126518	+	0.991964i	$0.540380\pi$
$464$	0	0
$465$	−18.5387	−0.859710
$466$	0	0
$467$	−28.6237	−1.32455	−0.662273	−	0.749263i	$-0.730408\pi$
$467$	−28.6237	−1.32455	−0.662273	+	0.749263i	$0.730408\pi$
$468$	0	0
$469$	14.8796	0.687077
$470$	0	0
$471$	3.77546	0.173964
$472$	0	0
$473$	−4.81446	−0.221369
$474$	0	0
$475$	−2.00041	−0.0917853
$476$	0	0
$477$	−1.63699	−0.0749527
$478$	0	0
$479$	14.4753	0.661394	0.330697	−	0.943737i	$-0.392716\pi$
$479$	14.4753	0.661394	0.330697	+	0.943737i	$0.392716\pi$
$480$	0	0
$481$	33.3946	1.52266
$482$	0	0
$483$	−6.73301	−0.306363
$484$	0	0
$485$	12.9406	0.587600
$486$	0	0
$487$	−3.41190	−0.154608	−0.0773040	−	0.997008i	$-0.524631\pi$
$487$	−3.41190	−0.154608	−0.0773040	+	0.997008i	$0.524631\pi$
$488$	0	0
$489$	−31.3712	−1.41865
$490$	0	0
$491$	6.51421	0.293982	0.146991	−	0.989138i	$-0.453041\pi$
$491$	6.51421	0.293982	0.146991	+	0.989138i	$0.453041\pi$
$492$	0	0
$493$	6.40178	0.288322
$494$	0	0
$495$	−1.72991	−0.0777536
$496$	0	0
$497$	−8.81503	−0.395408
$498$	0	0
$499$	12.9276	0.578719	0.289360	−	0.957220i	$-0.406558\pi$
$499$	12.9276	0.578719	0.289360	+	0.957220i	$0.406558\pi$
$500$	0	0
$501$	−34.4610	−1.53960
$502$	0	0
$503$	21.5742	0.961946	0.480973	−	0.876735i	$-0.340284\pi$
$503$	21.5742	0.961946	0.480973	+	0.876735i	$0.340284\pi$
$504$	0	0
$505$	15.9320	0.708967
$506$	0	0
$507$	25.9820	1.15390
$508$	0	0
$509$	−13.7778	−0.610690	−0.305345	−	0.952242i	$-0.598772\pi$
$509$	−13.7778	−0.610690	−0.305345	+	0.952242i	$0.598772\pi$
$510$	0	0
$511$	16.7808	0.742337
$512$	0	0
$513$	7.80587	0.344638
$514$	0	0
$515$	−0.831194	−0.0366268
$516$	0	0
$517$	−18.2723	−0.803614
$518$	0	0
$519$	3.07285	0.134883
$520$	0	0
$521$	−16.8562	−0.738483	−0.369241	−	0.929333i	$-0.620382\pi$
$521$	−16.8562	−0.738483	−0.369241	+	0.929333i	$0.620382\pi$
$522$	0	0
$523$	42.8866	1.87530	0.937650	−	0.347582i	$-0.112997\pi$
$523$	42.8866	1.87530	0.937650	+	0.347582i	$0.112997\pi$
$524$	0	0
$525$	2.26313	0.0987712
$526$	0	0
$527$	7.67292	0.334238
$528$	0	0
$529$	−14.1489	−0.615168
$530$	0	0
$531$	−3.07923	−0.133627
$532$	0	0
$533$	58.6672	2.54116
$534$	0	0
$535$	2.31881	0.100251
$536$	0	0
$537$	−46.9095	−2.02429
$538$	0	0
$539$	−9.32806	−0.401788
$540$	0	0
$541$	6.32762	0.272045	0.136023	−	0.990706i	$-0.456568\pi$
$541$	6.32762	0.272045	0.136023	+	0.990706i	$0.456568\pi$
$542$	0	0
$543$	23.0012	0.987075
$544$	0	0
$545$	4.39213	0.188138
$546$	0	0
$547$	22.2309	0.950526	0.475263	−	0.879844i	$-0.342353\pi$
$547$	22.2309	0.950526	0.475263	+	0.879844i	$0.342353\pi$
$548$	0	0
$549$	−6.79498	−0.290003
$550$	0	0
$551$	−15.3475	−0.653823
$552$	0	0
$553$	−8.36701	−0.355802
$554$	0	0
$555$	13.2322	0.561677
$556$	0	0
$557$	−13.0749	−0.554002	−0.277001	−	0.960870i	$-0.589341\pi$
$557$	−13.0749	−0.554002	−0.277001	+	0.960870i	$0.589341\pi$
$558$	0	0
$559$	15.0731	0.637526
$560$	0	0
$561$	2.73386	0.115424
$562$	0	0
$563$	21.3746	0.900831	0.450416	−	0.892819i	$-0.351276\pi$
$563$	21.3746	0.900831	0.450416	+	0.892819i	$0.351276\pi$
$564$	0	0
$565$	−5.04133	−0.212090
$566$	0	0
$567$	−12.4158	−0.521415
$568$	0	0
$569$	−13.6404	−0.571837	−0.285918	−	0.958254i	$-0.592299\pi$
$569$	−13.6404	−0.571837	−0.285918	+	0.958254i	$0.592299\pi$
$570$	0	0
$571$	42.7662	1.78971	0.894855	−	0.446357i	$-0.147279\pi$
$571$	42.7662	1.78971	0.894855	+	0.446357i	$0.147279\pi$
$572$	0	0
$573$	−25.0413	−1.04611
$574$	0	0
$575$	−2.97509	−0.124070
$576$	0	0
$577$	18.9385	0.788420	0.394210	−	0.919020i	$-0.371018\pi$
$577$	18.9385	0.788420	0.394210	+	0.919020i	$0.371018\pi$
$578$	0	0
$579$	−33.5290	−1.39342
$580$	0	0
$581$	13.4767	0.559107
$582$	0	0
$583$	−2.49921	−0.103507
$584$	0	0
$585$	5.41601	0.223924
$586$	0	0
$587$	−33.0230	−1.36300	−0.681502	−	0.731817i	$-0.738673\pi$
$587$	−33.0230	−1.36300	−0.681502	+	0.731817i	$0.738673\pi$
$588$	0	0
$589$	−18.3948	−0.757947
$590$	0	0
$591$	−0.428501	−0.0176262
$592$	0	0
$593$	−20.6734	−0.848953	−0.424476	−	0.905439i	$-0.639542\pi$
$593$	−20.6734	−0.848953	−0.424476	+	0.905439i	$0.639542\pi$
$594$	0	0
$595$	−0.936682	−0.0384002
$596$	0	0
$597$	18.7454	0.767198
$598$	0	0
$599$	−20.2383	−0.826916	−0.413458	−	0.910523i	$-0.635679\pi$
$599$	−20.2383	−0.826916	−0.413458	+	0.910523i	$0.635679\pi$
$600$	0	0
$601$	11.3044	0.461115	0.230557	−	0.973059i	$-0.425945\pi$
$601$	11.3044	0.461115	0.230557	+	0.973059i	$0.425945\pi$
$602$	0	0
$603$	14.1097	0.574592
$604$	0	0
$605$	8.35893	0.339839
$606$	0	0
$607$	33.1772	1.34662	0.673310	−	0.739360i	$-0.264872\pi$
$607$	33.1772	1.34662	0.673310	+	0.739360i	$0.264872\pi$
$608$	0	0
$609$	17.3631	0.703587
$610$	0	0
$611$	57.2069	2.31434
$612$	0	0
$613$	13.5991	0.549263	0.274631	−	0.961550i	$-0.411444\pi$
$613$	13.5991	0.549263	0.274631	+	0.961550i	$0.411444\pi$
$614$	0	0
$615$	23.2462	0.937377
$616$	0	0
$617$	0.240361	0.00967656	0.00483828	−	0.999988i	$-0.498460\pi$
$617$	0.240361	0.00967656	0.00483828	+	0.999988i	$0.498460\pi$
$618$	0	0
$619$	35.9909	1.44660	0.723298	−	0.690536i	$-0.242625\pi$
$619$	35.9909	1.44660	0.723298	+	0.690536i	$0.242625\pi$
$620$	0	0
$621$	11.6092	0.465860
$622$	0	0
$623$	13.4403	0.538475
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.55407	−0.261745
$628$	0	0
$629$	−5.47666	−0.218369
$630$	0	0
$631$	1.88128	0.0748927	0.0374463	−	0.999299i	$-0.488078\pi$
$631$	1.88128	0.0748927	0.0374463	+	0.999299i	$0.488078\pi$
$632$	0	0
$633$	18.1125	0.719907
$634$	0	0
$635$	−15.1708	−0.602036
$636$	0	0
$637$	29.2043	1.15712
$638$	0	0
$639$	−8.35891	−0.330673
$640$	0	0
$641$	21.4815	0.848467	0.424234	−	0.905553i	$-0.360544\pi$
$641$	21.4815	0.848467	0.424234	+	0.905553i	$0.360544\pi$
$642$	0	0
$643$	−1.12255	−0.0442690	−0.0221345	−	0.999755i	$-0.507046\pi$
$643$	−1.12255	−0.0442690	−0.0221345	+	0.999755i	$0.507046\pi$
$644$	0	0
$645$	5.97256	0.235169
$646$	0	0
$647$	−30.6366	−1.20445	−0.602224	−	0.798327i	$-0.705719\pi$
$647$	−30.6366	−1.20445	−0.602224	+	0.798327i	$0.705719\pi$
$648$	0	0
$649$	−4.70108	−0.184534
$650$	0	0
$651$	20.8107	0.815635
$652$	0	0
$653$	−19.6429	−0.768687	−0.384344	−	0.923190i	$-0.625572\pi$
$653$	−19.6429	−0.768687	−0.384344	+	0.923190i	$0.625572\pi$
$654$	0	0
$655$	2.93589	0.114715
$656$	0	0
$657$	15.9125	0.620804
$658$	0	0
$659$	−28.9859	−1.12913	−0.564566	−	0.825388i	$-0.690956\pi$
$659$	−28.9859	−1.12913	−0.564566	+	0.825388i	$0.690956\pi$
$660$	0	0
$661$	−27.0740	−1.05306	−0.526528	−	0.850158i	$-0.676506\pi$
$661$	−27.0740	−1.05306	−0.526528	+	0.850158i	$0.676506\pi$
$662$	0	0
$663$	−8.55917	−0.332411
$664$	0	0
$665$	2.24558	0.0870797
$666$	0	0
$667$	−22.8253	−0.883798
$668$	0	0
$669$	−49.7594	−1.92381
$670$	0	0
$671$	−10.3739	−0.400482
$672$	0	0
$673$	−41.6821	−1.60673	−0.803363	−	0.595489i	$-0.796958\pi$
$673$	−41.6821	−1.60673	−0.803363	+	0.595489i	$0.796958\pi$
$674$	0	0
$675$	−3.90213	−0.150193
$676$	0	0
$677$	−26.7799	−1.02924	−0.514618	−	0.857420i	$-0.672066\pi$
$677$	−26.7799	−1.02924	−0.514618	+	0.857420i	$0.672066\pi$
$678$	0	0
$679$	−14.5265	−0.557476
$680$	0	0
$681$	−45.8292	−1.75618
$682$	0	0
$683$	−20.2144	−0.773483	−0.386741	−	0.922188i	$-0.626399\pi$
$683$	−20.2144	−0.773483	−0.386741	+	0.922188i	$0.626399\pi$
$684$	0	0
$685$	−2.64554	−0.101081
$686$	0	0
$687$	−18.4641	−0.704448
$688$	0	0
$689$	7.82453	0.298091
$690$	0	0
$691$	47.6706	1.81348	0.906738	−	0.421695i	$-0.138565\pi$
$691$	47.6706	1.81348	0.906738	+	0.421695i	$0.138565\pi$
$692$	0	0
$693$	1.94192	0.0737674
$694$	0	0
$695$	20.7882	0.788543
$696$	0	0
$697$	−9.62131	−0.364433
$698$	0	0
$699$	13.4112	0.507257
$700$	0	0
$701$	−6.73030	−0.254200	−0.127100	−	0.991890i	$-0.540567\pi$
$701$	−6.73030	−0.254200	−0.127100	+	0.991890i	$0.540567\pi$
$702$	0	0
$703$	13.1296	0.495192
$704$	0	0
$705$	22.6676	0.853710
$706$	0	0
$707$	−17.8846	−0.672620
$708$	0	0
$709$	−10.7229	−0.402706	−0.201353	−	0.979519i	$-0.564534\pi$
$709$	−10.7229	−0.402706	−0.201353	+	0.979519i	$0.564534\pi$
$710$	0	0
$711$	−7.93408	−0.297551
$712$	0	0
$713$	−27.3575	−1.02455
$714$	0	0
$715$	8.26866	0.309230
$716$	0	0
$717$	−35.6488	−1.33133
$718$	0	0
$719$	−19.1859	−0.715515	−0.357758	−	0.933814i	$-0.616459\pi$
$719$	−19.1859	−0.715515	−0.357758	+	0.933814i	$0.616459\pi$
$720$	0	0
$721$	0.933062	0.0347490
$722$	0	0
$723$	54.9134	2.04225
$724$	0	0
$725$	7.67214	0.284936
$726$	0	0
$727$	−13.4492	−0.498802	−0.249401	−	0.968400i	$-0.580234\pi$
$727$	−13.4492	−0.498802	−0.249401	+	0.968400i	$0.580234\pi$
$728$	0	0
$729$	11.8273	0.438049
$730$	0	0
$731$	−2.47197	−0.0914290
$732$	0	0
$733$	49.2324	1.81844	0.909221	−	0.416314i	$-0.136678\pi$
$733$	49.2324	1.81844	0.909221	+	0.416314i	$0.136678\pi$
$734$	0	0
$735$	11.5719	0.426835
$736$	0	0
$737$	21.5414	0.793487
$738$	0	0
$739$	12.6449	0.465149	0.232574	−	0.972579i	$-0.425285\pi$
$739$	12.6449	0.465149	0.232574	+	0.972579i	$0.425285\pi$
$740$	0	0
$741$	20.5195	0.753803
$742$	0	0
$743$	−35.8819	−1.31638	−0.658189	−	0.752852i	$-0.728677\pi$
$743$	−35.8819	−1.31638	−0.658189	+	0.752852i	$0.728677\pi$
$744$	0	0
$745$	−7.83229	−0.286953
$746$	0	0
$747$	12.7794	0.467572
$748$	0	0
$749$	−2.60299	−0.0951114
$750$	0	0
$751$	−20.7938	−0.758776	−0.379388	−	0.925238i	$-0.623865\pi$
$751$	−20.7938	−0.758776	−0.379388	+	0.925238i	$0.623865\pi$
$752$	0	0
$753$	37.2343	1.35689
$754$	0	0
$755$	3.92025	0.142673
$756$	0	0
$757$	−42.6011	−1.54837	−0.774183	−	0.632962i	$-0.781839\pi$
$757$	−42.6011	−1.54837	−0.774183	+	0.632962i	$0.781839\pi$
$758$	0	0
$759$	−9.74745	−0.353810
$760$	0	0
$761$	50.1211	1.81689	0.908444	−	0.418006i	$-0.137271\pi$
$761$	50.1211	1.81689	0.908444	+	0.418006i	$0.137271\pi$
$762$	0	0
$763$	−4.93041	−0.178493
$764$	0	0
$765$	−0.888215	−0.0321135
$766$	0	0
$767$	14.7182	0.531442
$768$	0	0
$769$	−32.5924	−1.17531	−0.587656	−	0.809111i	$-0.699949\pi$
$769$	−32.5924	−1.17531	−0.587656	+	0.809111i	$0.699949\pi$
$770$	0	0
$771$	38.8838	1.40037
$772$	0	0
$773$	−35.5590	−1.27897	−0.639485	−	0.768804i	$-0.720852\pi$
$773$	−35.5590	−1.27897	−0.639485	+	0.768804i	$0.720852\pi$
$774$	0	0
$775$	9.19552	0.330313
$776$	0	0
$777$	−14.8539	−0.532882
$778$	0	0
$779$	23.0659	0.826420
$780$	0	0
$781$	−12.7616	−0.456646
$782$	0	0
$783$	−29.9377	−1.06989
$784$	0	0
$785$	−1.87270	−0.0668396
$786$	0	0
$787$	18.3610	0.654499	0.327249	−	0.944938i	$-0.393878\pi$
$787$	18.3610	0.654499	0.327249	+	0.944938i	$0.393878\pi$
$788$	0	0
$789$	44.3970	1.58058
$790$	0	0
$791$	5.65917	0.201217
$792$	0	0
$793$	32.4788	1.15336
$794$	0	0
$795$	3.10038	0.109959
$796$	0	0
$797$	−5.45646	−0.193278	−0.0966389	−	0.995320i	$-0.530809\pi$
$797$	−5.45646	−0.193278	−0.0966389	+	0.995320i	$0.530809\pi$
$798$	0	0
$799$	−9.38183	−0.331905
$800$	0	0
$801$	12.7449	0.450318
$802$	0	0
$803$	24.2937	0.857305
$804$	0	0
$805$	3.33970	0.117709
$806$	0	0
$807$	31.2840	1.10125
$808$	0	0
$809$	−18.0052	−0.633029	−0.316515	−	0.948588i	$-0.602513\pi$
$809$	−18.0052	−0.633029	−0.316515	+	0.948588i	$0.602513\pi$
$810$	0	0
$811$	42.3050	1.48553	0.742764	−	0.669553i	$-0.233514\pi$
$811$	42.3050	1.48553	0.742764	+	0.669553i	$0.233514\pi$
$812$	0	0
$813$	−28.9058	−1.01377
$814$	0	0
$815$	15.5607	0.545067
$816$	0	0
$817$	5.92622	0.207332
$818$	0	0
$819$	−6.07977	−0.212444
$820$	0	0
$821$	−24.2877	−0.847647	−0.423824	−	0.905745i	$-0.639312\pi$
$821$	−24.2877	−0.847647	−0.423824	+	0.905745i	$0.639312\pi$
$822$	0	0
$823$	29.4480	1.02649	0.513246	−	0.858242i	$-0.328443\pi$
$823$	29.4480	1.02649	0.513246	+	0.858242i	$0.328443\pi$
$824$	0	0
$825$	3.27636	0.114068
$826$	0	0
$827$	−23.7750	−0.826739	−0.413370	−	0.910563i	$-0.635648\pi$
$827$	−23.7750	−0.826739	−0.413370	+	0.910563i	$0.635648\pi$
$828$	0	0
$829$	−17.0659	−0.592723	−0.296362	−	0.955076i	$-0.595773\pi$
$829$	−17.0659	−0.592723	−0.296362	+	0.955076i	$0.595773\pi$
$830$	0	0
$831$	−61.5758	−2.13604
$832$	0	0
$833$	−4.78946	−0.165945
$834$	0	0
$835$	17.0933	0.591538
$836$	0	0
$837$	−35.8821	−1.24027
$838$	0	0
$839$	−3.92574	−0.135532	−0.0677658	−	0.997701i	$-0.521587\pi$
$839$	−3.92574	−0.135532	−0.0677658	+	0.997701i	$0.521587\pi$
$840$	0	0
$841$	29.8617	1.02971
$842$	0	0
$843$	−12.2460	−0.421773
$844$	0	0
$845$	−12.8875	−0.443345
$846$	0	0
$847$	−9.38337	−0.322416
$848$	0	0
$849$	31.5288	1.08207
$850$	0	0
$851$	19.5268	0.669370
$852$	0	0
$853$	55.6184	1.90434	0.952169	−	0.305573i	$-0.0988479\pi$
$853$	55.6184	1.90434	0.952169	+	0.305573i	$0.0988479\pi$
$854$	0	0
$855$	2.12938	0.0728233
$856$	0	0
$857$	39.6671	1.35500	0.677502	−	0.735521i	$-0.263063\pi$
$857$	39.6671	1.35500	0.677502	+	0.735521i	$0.263063\pi$
$858$	0	0
$859$	−10.1955	−0.347866	−0.173933	−	0.984757i	$-0.555648\pi$
$859$	−10.1955	−0.347866	−0.173933	+	0.984757i	$0.555648\pi$
$860$	0	0
$861$	−26.0952	−0.889321
$862$	0	0
$863$	−8.41158	−0.286334	−0.143167	−	0.989699i	$-0.545729\pi$
$863$	−8.41158	−0.286334	−0.143167	+	0.989699i	$0.545729\pi$
$864$	0	0
$865$	−1.52419	−0.0518240
$866$	0	0
$867$	−32.8692	−1.11630
$868$	0	0
$869$	−12.1130	−0.410906
$870$	0	0
$871$	−67.4419	−2.28518
$872$	0	0
$873$	−13.7748	−0.466208
$874$	0	0
$875$	−1.12256	−0.0379493
$876$	0	0
$877$	10.4532	0.352980	0.176490	−	0.984302i	$-0.443526\pi$
$877$	10.4532	0.352980	0.176490	+	0.984302i	$0.443526\pi$
$878$	0	0
$879$	52.0432	1.75537
$880$	0	0
$881$	−52.9959	−1.78548	−0.892738	−	0.450576i	$-0.851219\pi$
$881$	−52.9959	−1.78548	−0.892738	+	0.450576i	$0.851219\pi$
$882$	0	0
$883$	−7.49312	−0.252164	−0.126082	−	0.992020i	$-0.540240\pi$
$883$	−7.49312	−0.252164	−0.126082	+	0.992020i	$0.540240\pi$
$884$	0	0
$885$	5.83191	0.196037
$886$	0	0
$887$	−50.4637	−1.69440	−0.847202	−	0.531271i	$-0.821715\pi$
$887$	−50.4637	−1.69440	−0.847202	+	0.531271i	$0.821715\pi$
$888$	0	0
$889$	17.0301	0.571171
$890$	0	0
$891$	−17.9745	−0.602169
$892$	0	0
$893$	22.4917	0.752657
$894$	0	0
$895$	23.2680	0.777762
$896$	0	0
$897$	30.5174	1.01895
$898$	0	0
$899$	70.5493	2.35295
$900$	0	0
$901$	−1.28321	−0.0427499
$902$	0	0
$903$	−6.70453	−0.223113
$904$	0	0
$905$	−11.4090	−0.379248
$906$	0	0
$907$	−45.5488	−1.51242	−0.756212	−	0.654327i	$-0.772952\pi$
$907$	−45.5488	−1.51242	−0.756212	+	0.654327i	$0.772952\pi$
$908$	0	0
$909$	−16.9592	−0.562501
$910$	0	0
$911$	1.49889	0.0496603	0.0248301	−	0.999692i	$-0.492096\pi$
$911$	1.49889	0.0496603	0.0248301	+	0.999692i	$0.492096\pi$
$912$	0	0
$913$	19.5103	0.645698
$914$	0	0
$915$	12.8693	0.425447
$916$	0	0
$917$	−3.29570	−0.108833
$918$	0	0
$919$	22.4844	0.741691	0.370845	−	0.928695i	$-0.379068\pi$
$919$	22.4844	0.741691	0.370845	+	0.928695i	$0.379068\pi$
$920$	0	0
$921$	−31.8797	−1.05047
$922$	0	0
$923$	39.9541	1.31511
$924$	0	0
$925$	−6.56344	−0.215804
$926$	0	0
$927$	0.884782	0.0290601
$928$	0	0
$929$	28.0047	0.918805	0.459403	−	0.888228i	$-0.348064\pi$
$929$	28.0047	0.918805	0.459403	+	0.888228i	$0.348064\pi$
$930$	0	0
$931$	11.4821	0.376311
$932$	0	0
$933$	−2.04454	−0.0669353
$934$	0	0
$935$	−1.35604	−0.0443474
$936$	0	0
$937$	54.9115	1.79388	0.896940	−	0.442153i	$-0.145785\pi$
$937$	54.9115	1.79388	0.896940	+	0.442153i	$0.145785\pi$
$938$	0	0
$939$	60.0460	1.95953
$940$	0	0
$941$	−42.1112	−1.37279	−0.686393	−	0.727231i	$-0.740807\pi$
$941$	−42.1112	−1.37279	−0.686393	+	0.727231i	$0.740807\pi$
$942$	0	0
$943$	34.3044	1.11710
$944$	0	0
$945$	4.38036	0.142493
$946$	0	0
$947$	9.26480	0.301066	0.150533	−	0.988605i	$-0.451901\pi$
$947$	9.26480	0.301066	0.150533	+	0.988605i	$0.451901\pi$
$948$	0	0
$949$	−76.0587	−2.46897
$950$	0	0
$951$	14.4784	0.469493
$952$	0	0
$953$	−51.6330	−1.67256	−0.836279	−	0.548304i	$-0.815274\pi$
$953$	−51.6330	−1.67256	−0.836279	+	0.548304i	$0.815274\pi$
$954$	0	0
$955$	12.4209	0.401932
$956$	0	0
$957$	25.1367	0.812554
$958$	0	0
$959$	2.96977	0.0958988
$960$	0	0
$961$	53.5576	1.72767
$962$	0	0
$963$	−2.46831	−0.0795401
$964$	0	0
$965$	16.6310	0.535371
$966$	0	0
$967$	21.2718	0.684055	0.342027	−	0.939690i	$-0.388886\pi$
$967$	21.2718	0.684055	0.342027	+	0.939690i	$0.388886\pi$
$968$	0	0
$969$	−3.36516	−0.108105
$970$	0	0
$971$	6.09720	0.195668	0.0978342	−	0.995203i	$-0.468809\pi$
$971$	6.09720	0.195668	0.0978342	+	0.995203i	$0.468809\pi$
$972$	0	0
$973$	−23.3360	−0.748117
$974$	0	0
$975$	−10.2576	−0.328507
$976$	0	0
$977$	45.8621	1.46726	0.733629	−	0.679550i	$-0.237825\pi$
$977$	45.8621	1.46726	0.733629	+	0.679550i	$0.237825\pi$
$978$	0	0
$979$	19.4577	0.621871
$980$	0	0
$981$	−4.67529	−0.149271
$982$	0	0
$983$	43.9077	1.40044	0.700220	−	0.713928i	$-0.253085\pi$
$983$	43.9077	1.40044	0.700220	+	0.713928i	$0.253085\pi$
$984$	0	0
$985$	0.212544	0.00677222
$986$	0	0
$987$	−25.4456	−0.809943
$988$	0	0
$989$	8.81369	0.280259
$990$	0	0
$991$	36.3765	1.15554	0.577769	−	0.816201i	$-0.303924\pi$
$991$	36.3765	1.15554	0.577769	+	0.816201i	$0.303924\pi$
$992$	0	0
$993$	5.43689	0.172534
$994$	0	0
$995$	−9.29807	−0.294769
$996$	0	0
$997$	33.9245	1.07440	0.537200	−	0.843455i	$-0.319482\pi$
$997$	33.9245	1.07440	0.537200	+	0.843455i	$0.319482\pi$
$998$	0	0
$999$	25.6114	0.810308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.23	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.23	✓	28	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q+2.01605 q^{3} -1.00000 q^{5} +1.12256 q^{7} +1.06447 q^{9} +O(q^{10})\)	\(q+2.01605 q^{3} -1.00000 q^{5} +1.12256 q^{7} +1.06447 q^{9} +1.62514 q^{11} -5.08798 q^{13} -2.01605 q^{15} +0.834419 q^{17} -2.00041 q^{19} +2.26313 q^{21} -2.97509 q^{23} +1.00000 q^{25} -3.90213 q^{27} +7.67214 q^{29} +9.19552 q^{31} +3.27636 q^{33} -1.12256 q^{35} -6.56344 q^{37} -10.2576 q^{39} -11.5305 q^{41} -2.96250 q^{43} -1.06447 q^{45} -11.2435 q^{47} -5.73987 q^{49} +1.68223 q^{51} -1.53785 q^{53} -1.62514 q^{55} -4.03294 q^{57} -2.89273 q^{59} -6.38343 q^{61} +1.19493 q^{63} +5.08798 q^{65} +13.2551 q^{67} -5.99793 q^{69} -7.85264 q^{71} +14.9487 q^{73} +2.01605 q^{75} +1.82431 q^{77} -7.45354 q^{79} -11.0603 q^{81} +12.0054 q^{83} -0.834419 q^{85} +15.4674 q^{87} +11.9730 q^{89} -5.71154 q^{91} +18.5387 q^{93} +2.00041 q^{95} -12.9406 q^{97} +1.72991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more