LMFDB - Embedded newform 820.2.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [820,2,Mod(1,820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(820, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("820.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 820 = 2^{2} \cdot 5 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	820.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.54773296574$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13068.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} - x + 1 $ x^4 - x^3 - 6*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.548230$ of defining polynomial
Character	$\chi$	$=$	820.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27582 q^{3} +1.00000 q^{5} -3.97526 q^{7} -1.37228 q^{9} -1.27582 q^{11} +5.97526 q^{13} -1.27582 q^{15} +3.27582 q^{17} +0.769298 q^{19} +5.07172 q^{21} +4.37228 q^{23} +1.00000 q^{25} +5.57825 q^{27} +1.67284 q^{29} -2.04512 q^{31} +1.62772 q^{33} -3.97526 q^{35} +4.87880 q^{37} -7.62337 q^{39} +1.00000 q^{41} +9.62337 q^{43} -1.37228 q^{45} -3.79590 q^{47} +8.80273 q^{49} -4.17936 q^{51} +4.90354 q^{53} -1.27582 q^{55} -0.981487 q^{57} -0.327162 q^{59} -10.1299 q^{61} +5.45518 q^{63} +5.97526 q^{65} +9.73287 q^{67} -5.57825 q^{69} -6.67471 q^{71} -11.8559 q^{73} -1.27582 q^{75} +5.07172 q^{77} +12.7595 q^{79} -3.00000 q^{81} +13.8027 q^{83} +3.27582 q^{85} -2.13424 q^{87} +13.3359 q^{89} -23.7533 q^{91} +2.60921 q^{93} +0.769298 q^{95} +10.7514 q^{97} +1.75079 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + q^{7} + 6 q^{9} + 7 q^{13} + 8 q^{17} - 3 q^{19} - 3 q^{21} + 6 q^{23} + 4 q^{25} + 7 q^{29} + 3 q^{31} + 18 q^{33} + q^{35} + 9 q^{37} + 3 q^{39} + 4 q^{41} + 5 q^{43} + 6 q^{45} + 3 q^{47}+ \cdots + 6 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + q^7 + 6 * q^9 + 7 * q^13 + 8 * q^17 - 3 * q^19 - 3 * q^21 + 6 * q^23 + 4 * q^25 + 7 * q^29 + 3 * q^31 + 18 * q^33 + q^35 + 9 * q^37 + 3 * q^39 + 4 * q^41 + 5 * q^43 + 6 * q^45 + 3 * q^47 + 3 * q^49 - 18 * q^51 + 26 * q^53 - 3 * q^57 - q^59 - 8 * q^61 + 18 * q^63 + 7 * q^65 + q^67 + 2 * q^71 + 9 * q^73 - 3 * q^77 + q^79 - 12 * q^81 + 23 * q^83 + 8 * q^85 - 21 * q^87 + 10 * q^89 - 29 * q^91 + 21 * q^93 - 3 * q^95 + 6 * q^97

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
$3$	−1.27582	−0.736595	−0.368298	−	0.929708i	$-0.620059\pi$
$3$	−1.27582	−0.736595	−0.368298	+	0.929708i	$0.620059\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.97526	−1.50251	−0.751254	−	0.660013i	$-0.770551\pi$
$7$	−3.97526	−1.50251	−0.751254	+	0.660013i	$0.770551\pi$
$8$	0	0
$9$	−1.37228	−0.457427
$10$	0	0
$11$	−1.27582	−0.384674	−0.192337	−	0.981329i	$-0.561607\pi$
$11$	−1.27582	−0.384674	−0.192337	+	0.981329i	$0.561607\pi$
$12$	0	0
$13$	5.97526	1.65724	0.828620	−	0.559811i	$-0.189126\pi$
$13$	5.97526	1.65724	0.828620	+	0.559811i	$0.189126\pi$
$14$	0	0
$15$	−1.27582	−0.329416
$16$	0	0
$17$	3.27582	0.794503	0.397252	−	0.917710i	$-0.369964\pi$
$17$	3.27582	0.794503	0.397252	+	0.917710i	$0.369964\pi$
$18$	0	0
$19$	0.769298	0.176489	0.0882446	−	0.996099i	$-0.471874\pi$
$19$	0.769298	0.176489	0.0882446	+	0.996099i	$0.471874\pi$
$20$	0	0
$21$	5.07172	1.10674
$22$	0	0
$23$	4.37228	0.911684	0.455842	−	0.890061i	$-0.349338\pi$
$23$	4.37228	0.911684	0.455842	+	0.890061i	$0.349338\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.57825	1.07353
$28$	0	0
$29$	1.67284	0.310638	0.155319	−	0.987864i	$-0.450359\pi$
$29$	1.67284	0.310638	0.155319	+	0.987864i	$0.450359\pi$
$30$	0	0
$31$	−2.04512	−0.367314	−0.183657	−	0.982990i	$-0.558794\pi$
$31$	−2.04512	−0.367314	−0.183657	+	0.982990i	$0.558794\pi$
$32$	0	0
$33$	1.62772	0.283349
$34$	0	0
$35$	−3.97526	−0.671942
$36$	0	0
$37$	4.87880	0.802070	0.401035	−	0.916063i	$-0.368650\pi$
$37$	4.87880	0.802070	0.401035	+	0.916063i	$0.368650\pi$
$38$	0	0
$39$	−7.62337	−1.22072
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	9.62337	1.46755	0.733775	−	0.679393i	$-0.237757\pi$
$43$	9.62337	1.46755	0.733775	+	0.679393i	$0.237757\pi$
$44$	0	0
$45$	−1.37228	−0.204568
$46$	0	0
$47$	−3.79590	−0.553690	−0.276845	−	0.960915i	$-0.589289\pi$
$47$	−3.79590	−0.553690	−0.276845	+	0.960915i	$0.589289\pi$
$48$	0	0
$49$	8.80273	1.25753
$50$	0	0
$51$	−4.17936	−0.585227
$52$	0	0
$53$	4.90354	0.673553	0.336776	−	0.941585i	$-0.390663\pi$
$53$	4.90354	0.673553	0.336776	+	0.941585i	$0.390663\pi$
$54$	0	0
$55$	−1.27582	−0.172032
$56$	0	0
$57$	−0.981487	−0.130001
$58$	0	0
$59$	−0.327162	−0.0425929	−0.0212964	−	0.999773i	$-0.506779\pi$
$59$	−0.327162	−0.0425929	−0.0212964	+	0.999773i	$0.506779\pi$
$60$	0	0
$61$	−10.1299	−1.29700	−0.648500	−	0.761215i	$-0.724603\pi$
$61$	−10.1299	−1.29700	−0.648500	+	0.761215i	$0.724603\pi$
$62$	0	0
$63$	5.45518	0.687288
$64$	0	0
$65$	5.97526	0.741140
$66$	0	0
$67$	9.73287	1.18906	0.594530	−	0.804074i	$-0.297338\pi$
$67$	9.73287	1.18906	0.594530	+	0.804074i	$0.297338\pi$
$68$	0	0
$69$	−5.57825	−0.671542
$70$	0	0
$71$	−6.67471	−0.792142	−0.396071	−	0.918220i	$-0.629627\pi$
$71$	−6.67471	−0.792142	−0.396071	+	0.918220i	$0.629627\pi$
$72$	0	0
$73$	−11.8559	−1.38763	−0.693816	−	0.720152i	$-0.744072\pi$
$73$	−11.8559	−1.38763	−0.693816	+	0.720152i	$0.744072\pi$
$74$	0	0
$75$	−1.27582	−0.147319
$76$	0	0
$77$	5.07172	0.577977
$78$	0	0
$79$	12.7595	1.43555	0.717777	−	0.696273i	$-0.245160\pi$
$79$	12.7595	1.43555	0.717777	+	0.696273i	$0.245160\pi$
$80$	0	0
$81$	−3.00000	−0.333333
$82$	0	0
$83$	13.8027	1.51505	0.757523	−	0.652808i	$-0.226409\pi$
$83$	13.8027	1.51505	0.757523	+	0.652808i	$0.226409\pi$
$84$	0	0
$85$	3.27582	0.355313
$86$	0	0
$87$	−2.13424	−0.228815
$88$	0	0
$89$	13.3359	1.41360	0.706799	−	0.707415i	$-0.250139\pi$
$89$	13.3359	1.41360	0.706799	+	0.707415i	$0.250139\pi$
$90$	0	0
$91$	−23.7533	−2.49002
$92$	0	0
$93$	2.60921	0.270562
$94$	0	0
$95$	0.769298	0.0789283
$96$	0	0
$97$	10.7514	1.09164	0.545819	−	0.837903i	$-0.316219\pi$
$97$	10.7514	1.09164	0.545819	+	0.837903i	$0.316219\pi$
$98$	0	0
$99$	1.75079	0.175961
$100$	0	0
$101$	12.6500	1.25872	0.629360	−	0.777114i	$-0.283317\pi$
$101$	12.6500	1.25872	0.629360	+	0.777114i	$0.283317\pi$
$102$	0	0
$103$	2.56520	0.252757	0.126378	−	0.991982i	$-0.459665\pi$
$103$	2.56520	0.252757	0.126378	+	0.991982i	$0.459665\pi$
$104$	0	0
$105$	5.07172	0.494950
$106$	0	0
$107$	−15.7576	−1.52335	−0.761673	−	0.647962i	$-0.775622\pi$
$107$	−15.7576	−1.52335	−0.761673	+	0.647962i	$0.775622\pi$
$108$	0	0
$109$	0.372281	0.0356581	0.0178290	−	0.999841i	$-0.494325\pi$
$109$	0.372281	0.0356581	0.0178290	+	0.999841i	$0.494325\pi$
$110$	0	0
$111$	−6.22448	−0.590801
$112$	0	0
$113$	1.07608	0.101229	0.0506144	−	0.998718i	$-0.483882\pi$
$113$	1.07608	0.101229	0.0506144	+	0.998718i	$0.483882\pi$
$114$	0	0
$115$	4.37228	0.407717
$116$	0	0
$117$	−8.19974	−0.758067
$118$	0	0
$119$	−13.0223	−1.19375
$120$	0	0
$121$	−9.37228	−0.852026
$122$	0	0
$123$	−1.27582	−0.115037
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.57390	0.849546	0.424773	−	0.905300i	$-0.360354\pi$
$127$	9.57390	0.849546	0.424773	+	0.905300i	$0.360354\pi$
$128$	0	0
$129$	−12.2777	−1.08099
$130$	0	0
$131$	−8.23804	−0.719761	−0.359880	−	0.932998i	$-0.617183\pi$
$131$	−8.23804	−0.719761	−0.359880	+	0.932998i	$0.617183\pi$
$132$	0	0
$133$	−3.05816	−0.265176
$134$	0	0
$135$	5.57825	0.480099
$136$	0	0
$137$	−14.3432	−1.22542	−0.612711	−	0.790307i	$-0.709921\pi$
$137$	−14.3432	−1.22542	−0.612711	+	0.790307i	$0.709921\pi$
$138$	0	0
$139$	−8.74456	−0.741704	−0.370852	−	0.928692i	$-0.620934\pi$
$139$	−8.74456	−0.741704	−0.370852	+	0.928692i	$0.620934\pi$
$140$	0	0
$141$	4.84289	0.407845
$142$	0	0
$143$	−7.62337	−0.637498
$144$	0	0
$145$	1.67284	0.138922
$146$	0	0
$147$	−11.2307	−0.926293
$148$	0	0
$149$	6.41740	0.525734	0.262867	−	0.964832i	$-0.415332\pi$
$149$	6.41740	0.525734	0.262867	+	0.964832i	$0.415332\pi$
$150$	0	0
$151$	−5.75079	−0.467992	−0.233996	−	0.972238i	$-0.575180\pi$
$151$	−5.75079	−0.467992	−0.233996	+	0.972238i	$0.575180\pi$
$152$	0	0
$153$	−4.49535	−0.363427
$154$	0	0
$155$	−2.04512	−0.164268
$156$	0	0
$157$	−23.2671	−1.85692	−0.928459	−	0.371435i	$-0.878866\pi$
$157$	−23.2671	−1.85692	−0.928459	+	0.371435i	$0.878866\pi$
$158$	0	0
$159$	−6.25604	−0.496136
$160$	0	0
$161$	−17.3810	−1.36981
$162$	0	0
$163$	−14.8429	−1.16259	−0.581293	−	0.813695i	$-0.697453\pi$
$163$	−14.8429	−1.16259	−0.581293	+	0.813695i	$0.697453\pi$
$164$	0	0
$165$	1.62772	0.126718
$166$	0	0
$167$	−10.9938	−0.850724	−0.425362	−	0.905023i	$-0.639853\pi$
$167$	−10.9938	−0.850724	−0.425362	+	0.905023i	$0.639853\pi$
$168$	0	0
$169$	22.7038	1.74644
$170$	0	0
$171$	−1.05569	−0.0807309
$172$	0	0
$173$	8.52008	0.647770	0.323885	−	0.946096i	$-0.395011\pi$
$173$	8.52008	0.647770	0.323885	+	0.946096i	$0.395011\pi$
$174$	0	0
$175$	−3.97526	−0.300502
$176$	0	0
$177$	0.417400	0.0313737
$178$	0	0
$179$	5.22635	0.390636	0.195318	−	0.980740i	$-0.437426\pi$
$179$	5.22635	0.390636	0.195318	+	0.980740i	$0.437426\pi$
$180$	0	0
$181$	−6.56520	−0.487988	−0.243994	−	0.969777i	$-0.578458\pi$
$181$	−6.56520	−0.487988	−0.243994	+	0.969777i	$0.578458\pi$
$182$	0	0
$183$	12.9239	0.955364
$184$	0	0
$185$	4.87880	0.358697
$186$	0	0
$187$	−4.17936	−0.305625
$188$	0	0
$189$	−22.1750	−1.61299
$190$	0	0
$191$	23.6394	1.71049	0.855243	−	0.518227i	$-0.173408\pi$
$191$	23.6394	1.71049	0.855243	+	0.518227i	$0.173408\pi$
$192$	0	0
$193$	−11.2579	−0.810362	−0.405181	−	0.914237i	$-0.632791\pi$
$193$	−11.2579	−0.810362	−0.405181	+	0.914237i	$0.632791\pi$
$194$	0	0
$195$	−7.62337	−0.545921
$196$	0	0
$197$	−0.264130	−0.0188185	−0.00940925	−	0.999956i	$-0.502995\pi$
$197$	−0.264130	−0.0188185	−0.00940925	+	0.999956i	$0.502995\pi$
$198$	0	0
$199$	−3.55786	−0.252210	−0.126105	−	0.992017i	$-0.540248\pi$
$199$	−3.55786	−0.252210	−0.126105	+	0.992017i	$0.540248\pi$
$200$	0	0
$201$	−12.4174	−0.875856
$202$	0	0
$203$	−6.64997	−0.466737
$204$	0	0
$205$	1.00000	0.0698430
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−0.981487	−0.0678909
$210$	0	0
$211$	−4.14098	−0.285077	−0.142538	−	0.989789i	$-0.545526\pi$
$211$	−4.14098	−0.285077	−0.142538	+	0.989789i	$0.545526\pi$
$212$	0	0
$213$	8.51573	0.583488
$214$	0	0
$215$	9.62337	0.656308
$216$	0	0
$217$	8.12989	0.551893
$218$	0	0
$219$	15.1261	1.02212
$220$	0	0
$221$	19.5739	1.31668
$222$	0	0
$223$	14.1695	0.948863	0.474431	−	0.880293i	$-0.342654\pi$
$223$	14.1695	0.948863	0.474431	+	0.880293i	$0.342654\pi$
$224$	0	0
$225$	−1.37228	−0.0914854
$226$	0	0
$227$	5.76443	0.382599	0.191299	−	0.981532i	$-0.438730\pi$
$227$	5.76443	0.382599	0.191299	+	0.981532i	$0.438730\pi$
$228$	0	0
$229$	−23.2919	−1.53917	−0.769584	−	0.638545i	$-0.779537\pi$
$229$	−23.2919	−1.53917	−0.769584	+	0.638545i	$0.779537\pi$
$230$	0	0
$231$	−6.47061	−0.425735
$232$	0	0
$233$	15.3742	1.00719	0.503597	−	0.863939i	$-0.332010\pi$
$233$	15.3742	1.00719	0.503597	+	0.863939i	$0.332010\pi$
$234$	0	0
$235$	−3.79590	−0.247617
$236$	0	0
$237$	−16.2788	−1.05742
$238$	0	0
$239$	−4.07360	−0.263499	−0.131749	−	0.991283i	$-0.542059\pi$
$239$	−4.07360	−0.263499	−0.131749	+	0.991283i	$0.542059\pi$
$240$	0	0
$241$	7.20161	0.463897	0.231948	−	0.972728i	$-0.425490\pi$
$241$	7.20161	0.463897	0.231948	+	0.972728i	$0.425490\pi$
$242$	0	0
$243$	−12.9073	−0.828002
$244$	0	0
$245$	8.80273	0.562386
$246$	0	0
$247$	4.59676	0.292485
$248$	0	0
$249$	−17.6098	−1.11598
$250$	0	0
$251$	10.0272	0.632912	0.316456	−	0.948607i	$-0.397507\pi$
$251$	10.0272	0.632912	0.316456	+	0.948607i	$0.397507\pi$
$252$	0	0
$253$	−5.57825	−0.350701
$254$	0	0
$255$	−4.17936	−0.261722
$256$	0	0
$257$	20.8225	1.29887	0.649436	−	0.760416i	$-0.275005\pi$
$257$	20.8225	1.29887	0.649436	+	0.760416i	$0.275005\pi$
$258$	0	0
$259$	−19.3945	−1.20512
$260$	0	0
$261$	−2.29560	−0.142094
$262$	0	0
$263$	28.5226	1.75878	0.879388	−	0.476106i	$-0.157952\pi$
$263$	28.5226	1.75878	0.879388	+	0.476106i	$0.157952\pi$
$264$	0	0
$265$	4.90354	0.301222
$266$	0	0
$267$	−17.0142	−1.04125
$268$	0	0
$269$	12.7391	0.776716	0.388358	−	0.921509i	$-0.373042\pi$
$269$	12.7391	0.776716	0.388358	+	0.921509i	$0.373042\pi$
$270$	0	0
$271$	−12.4255	−0.754795	−0.377397	−	0.926051i	$-0.623181\pi$
$271$	−12.4255	−0.754795	−0.377397	+	0.926051i	$0.623181\pi$
$272$	0	0
$273$	30.3049	1.83414
$274$	0	0
$275$	−1.27582	−0.0769349
$276$	0	0
$277$	11.8739	0.713431	0.356715	−	0.934213i	$-0.383897\pi$
$277$	11.8739	0.713431	0.356715	+	0.934213i	$0.383897\pi$
$278$	0	0
$279$	2.80648	0.168019
$280$	0	0
$281$	10.7842	0.643332	0.321666	−	0.946853i	$-0.395757\pi$
$281$	10.7842	0.643332	0.321666	+	0.946853i	$0.395757\pi$
$282$	0	0
$283$	24.7304	1.47007	0.735035	−	0.678030i	$-0.237166\pi$
$283$	24.7304	1.47007	0.735035	+	0.678030i	$0.237166\pi$
$284$	0	0
$285$	−0.981487	−0.0581383
$286$	0	0
$287$	−3.97526	−0.234652
$288$	0	0
$289$	−6.26900	−0.368765
$290$	0	0
$291$	−13.7168	−0.804095
$292$	0	0
$293$	30.1144	1.75930	0.879650	−	0.475622i	$-0.157777\pi$
$293$	30.1144	1.75930	0.879650	+	0.475622i	$0.157777\pi$
$294$	0	0
$295$	−0.327162	−0.0190481
$296$	0	0
$297$	−7.11684	−0.412961
$298$	0	0
$299$	26.1255	1.51088
$300$	0	0
$301$	−38.2554	−2.20501
$302$	0	0
$303$	−16.1391	−0.927167
$304$	0	0
$305$	−10.1299	−0.580036
$306$	0	0
$307$	−33.8831	−1.93381	−0.966907	−	0.255131i	$-0.917882\pi$
$307$	−33.8831	−1.93381	−0.966907	+	0.255131i	$0.917882\pi$
$308$	0	0
$309$	−3.27274	−0.186180
$310$	0	0
$311$	20.9388	1.18733	0.593666	−	0.804711i	$-0.297680\pi$
$311$	20.9388	1.18733	0.593666	+	0.804711i	$0.297680\pi$
$312$	0	0
$313$	14.9579	0.845469	0.422734	−	0.906254i	$-0.361070\pi$
$313$	14.9579	0.845469	0.422734	+	0.906254i	$0.361070\pi$
$314$	0	0
$315$	5.45518	0.307365
$316$	0	0
$317$	8.45570	0.474919	0.237460	−	0.971397i	$-0.423685\pi$
$317$	8.45570	0.474919	0.237460	+	0.971397i	$0.423685\pi$
$318$	0	0
$319$	−2.13424	−0.119495
$320$	0	0
$321$	20.1039	1.12209
$322$	0	0
$323$	2.52008	0.140221
$324$	0	0
$325$	5.97526	0.331448
$326$	0	0
$327$	−0.474964	−0.0262656
$328$	0	0
$329$	15.0897	0.831923
$330$	0	0
$331$	−9.92205	−0.545365	−0.272683	−	0.962104i	$-0.587911\pi$
$331$	−9.92205	−0.545365	−0.272683	+	0.962104i	$0.587911\pi$
$332$	0	0
$333$	−6.69509	−0.366889
$334$	0	0
$335$	9.73287	0.531764
$336$	0	0
$337$	19.8750	1.08266	0.541329	−	0.840811i	$-0.317921\pi$
$337$	19.8750	1.08266	0.541329	+	0.840811i	$0.317921\pi$
$338$	0	0
$339$	−1.37288	−0.0745647
$340$	0	0
$341$	2.60921	0.141296
$342$	0	0
$343$	−7.16632	−0.386945
$344$	0	0
$345$	−5.57825	−0.300323
$346$	0	0
$347$	−25.8639	−1.38845	−0.694223	−	0.719760i	$-0.744252\pi$
$347$	−25.8639	−1.38845	−0.694223	+	0.719760i	$0.744252\pi$
$348$	0	0
$349$	−17.4250	−0.932738	−0.466369	−	0.884590i	$-0.654438\pi$
$349$	−17.4250	−0.932738	−0.466369	+	0.884590i	$0.654438\pi$
$350$	0	0
$351$	33.3315	1.77910
$352$	0	0
$353$	−23.2424	−1.23707	−0.618534	−	0.785758i	$-0.712273\pi$
$353$	−23.2424	−1.23707	−0.618534	+	0.785758i	$0.712273\pi$
$354$	0	0
$355$	−6.67471	−0.354257
$356$	0	0
$357$	16.6141	0.879309
$358$	0	0
$359$	−10.9908	−0.580071	−0.290036	−	0.957016i	$-0.593667\pi$
$359$	−10.9908	−0.580071	−0.290036	+	0.957016i	$0.593667\pi$
$360$	0	0
$361$	−18.4082	−0.968852
$362$	0	0
$363$	11.9574	0.627598
$364$	0	0
$365$	−11.8559	−0.620568
$366$	0	0
$367$	14.3283	0.747930	0.373965	−	0.927443i	$-0.377998\pi$
$367$	14.3283	0.747930	0.373965	+	0.927443i	$0.377998\pi$
$368$	0	0
$369$	−1.37228	−0.0714381
$370$	0	0
$371$	−19.4929	−1.01202
$372$	0	0
$373$	−22.9647	−1.18907	−0.594533	−	0.804071i	$-0.702663\pi$
$373$	−22.9647	−1.18907	−0.594533	+	0.804071i	$0.702663\pi$
$374$	0	0
$375$	−1.27582	−0.0658831
$376$	0	0
$377$	9.99565	0.514802
$378$	0	0
$379$	−13.6190	−0.699562	−0.349781	−	0.936832i	$-0.613744\pi$
$379$	−13.6190	−0.699562	−0.349781	+	0.936832i	$0.613744\pi$
$380$	0	0
$381$	−12.2146	−0.625772
$382$	0	0
$383$	28.7785	1.47051	0.735257	−	0.677789i	$-0.237062\pi$
$383$	28.7785	1.47051	0.735257	+	0.677789i	$0.237062\pi$
$384$	0	0
$385$	5.07172	0.258479
$386$	0	0
$387$	−13.2060	−0.671297
$388$	0	0
$389$	−29.4082	−1.49105	−0.745527	−	0.666475i	$-0.767802\pi$
$389$	−29.4082	−1.49105	−0.745527	+	0.666475i	$0.767802\pi$
$390$	0	0
$391$	14.3228	0.724336
$392$	0	0
$393$	10.5103	0.530173
$394$	0	0
$395$	12.7595	0.641999
$396$	0	0
$397$	4.30678	0.216151	0.108076	−	0.994143i	$-0.465531\pi$
$397$	4.30678	0.216151	0.108076	+	0.994143i	$0.465531\pi$
$398$	0	0
$399$	3.90167	0.195328
$400$	0	0
$401$	−11.5919	−0.578871	−0.289436	−	0.957197i	$-0.593468\pi$
$401$	−11.5919	−0.578871	−0.289436	+	0.957197i	$0.593468\pi$
$402$	0	0
$403$	−12.2201	−0.608728
$404$	0	0
$405$	−3.00000	−0.149071
$406$	0	0
$407$	−6.22448	−0.308536
$408$	0	0
$409$	19.4614	0.962304	0.481152	−	0.876637i	$-0.340219\pi$
$409$	19.4614	0.962304	0.481152	+	0.876637i	$0.340219\pi$
$410$	0	0
$411$	18.2993	0.902640
$412$	0	0
$413$	1.30056	0.0639962
$414$	0	0
$415$	13.8027	0.677549
$416$	0	0
$417$	11.1565	0.546336
$418$	0	0
$419$	−10.7718	−0.526235	−0.263118	−	0.964764i	$-0.584751\pi$
$419$	−10.7718	−0.526235	−0.263118	+	0.964764i	$0.584751\pi$
$420$	0	0
$421$	14.2967	0.696780	0.348390	−	0.937350i	$-0.386729\pi$
$421$	14.2967	0.696780	0.348390	+	0.937350i	$0.386729\pi$
$422$	0	0
$423$	5.20905	0.253273
$424$	0	0
$425$	3.27582	0.158901
$426$	0	0
$427$	40.2690	1.94875
$428$	0	0
$429$	9.72605	0.469578
$430$	0	0
$431$	4.47496	0.215551	0.107776	−	0.994175i	$-0.465627\pi$
$431$	4.47496	0.215551	0.107776	+	0.994175i	$0.465627\pi$
$432$	0	0
$433$	−22.6772	−1.08980	−0.544898	−	0.838502i	$-0.683432\pi$
$433$	−22.6772	−1.08980	−0.544898	+	0.838502i	$0.683432\pi$
$434$	0	0
$435$	−2.13424	−0.102329
$436$	0	0
$437$	3.36359	0.160902
$438$	0	0
$439$	17.1046	0.816356	0.408178	−	0.912902i	$-0.366164\pi$
$439$	17.1046	0.816356	0.408178	+	0.912902i	$0.366164\pi$
$440$	0	0
$441$	−12.0798	−0.575229
$442$	0	0
$443$	−25.0315	−1.18928	−0.594642	−	0.803990i	$-0.702706\pi$
$443$	−25.0315	−1.18928	−0.594642	+	0.803990i	$0.702706\pi$
$444$	0	0
$445$	13.3359	0.632180
$446$	0	0
$447$	−8.18745	−0.387253
$448$	0	0
$449$	36.7033	1.73213	0.866067	−	0.499928i	$-0.166640\pi$
$449$	36.7033	1.73213	0.866067	+	0.499928i	$0.166640\pi$
$450$	0	0
$451$	−1.27582	−0.0600761
$452$	0	0
$453$	7.33697	0.344721
$454$	0	0
$455$	−23.7533	−1.11357
$456$	0	0
$457$	−19.0878	−0.892888	−0.446444	−	0.894812i	$-0.647310\pi$
$457$	−19.0878	−0.892888	−0.446444	+	0.894812i	$0.647310\pi$
$458$	0	0
$459$	18.2733	0.852926
$460$	0	0
$461$	32.3365	1.50606	0.753029	−	0.657987i	$-0.228592\pi$
$461$	32.3365	1.50606	0.753029	+	0.657987i	$0.228592\pi$
$462$	0	0
$463$	−6.44588	−0.299565	−0.149783	−	0.988719i	$-0.547857\pi$
$463$	−6.44588	−0.299565	−0.149783	+	0.988719i	$0.547857\pi$
$464$	0	0
$465$	2.60921	0.120999
$466$	0	0
$467$	20.3049	0.939599	0.469799	−	0.882773i	$-0.344326\pi$
$467$	20.3049	0.939599	0.469799	+	0.882773i	$0.344326\pi$
$468$	0	0
$469$	−38.6907	−1.78657
$470$	0	0
$471$	29.6847	1.36780
$472$	0	0
$473$	−12.2777	−0.564529
$474$	0	0
$475$	0.769298	0.0352978
$476$	0	0
$477$	−6.72904	−0.308101
$478$	0	0
$479$	31.9177	1.45836	0.729178	−	0.684324i	$-0.239902\pi$
$479$	31.9177	1.45836	0.729178	+	0.684324i	$0.239902\pi$
$480$	0	0
$481$	29.1521	1.32922
$482$	0	0
$483$	22.1750	1.00900
$484$	0	0
$485$	10.7514	0.488195
$486$	0	0
$487$	−34.0049	−1.54091	−0.770455	−	0.637494i	$-0.779971\pi$
$487$	−34.0049	−1.54091	−0.770455	+	0.637494i	$0.779971\pi$
$488$	0	0
$489$	18.9369	0.856355
$490$	0	0
$491$	10.9196	0.492793	0.246397	−	0.969169i	$-0.420753\pi$
$491$	10.9196	0.492793	0.246397	+	0.969169i	$0.420753\pi$
$492$	0	0
$493$	5.47992	0.246803
$494$	0	0
$495$	1.75079	0.0786919
$496$	0	0
$497$	26.5337	1.19020
$498$	0	0
$499$	10.0612	0.450399	0.225199	−	0.974313i	$-0.427697\pi$
$499$	10.0612	0.450399	0.225199	+	0.974313i	$0.427697\pi$
$500$	0	0
$501$	14.0261	0.626639
$502$	0	0
$503$	−20.3334	−0.906620	−0.453310	−	0.891353i	$-0.649757\pi$
$503$	−20.3334	−0.906620	−0.453310	+	0.891353i	$0.649757\pi$
$504$	0	0
$505$	12.6500	0.562916
$506$	0	0
$507$	−28.9660	−1.28642
$508$	0	0
$509$	−11.6271	−0.515362	−0.257681	−	0.966230i	$-0.582958\pi$
$509$	−11.6271	−0.515362	−0.257681	+	0.966230i	$0.582958\pi$
$510$	0	0
$511$	47.1305	2.08493
$512$	0	0
$513$	4.29134	0.189467
$514$	0	0
$515$	2.56520	0.113036
$516$	0	0
$517$	4.84289	0.212990
$518$	0	0
$519$	−10.8701	−0.477144
$520$	0	0
$521$	−43.6228	−1.91115	−0.955576	−	0.294746i	$-0.904765\pi$
$521$	−43.6228	−1.91115	−0.955576	+	0.294746i	$0.904765\pi$
$522$	0	0
$523$	18.9016	0.826508	0.413254	−	0.910616i	$-0.364392\pi$
$523$	18.9016	0.826508	0.413254	+	0.910616i	$0.364392\pi$
$524$	0	0
$525$	5.07172	0.221348
$526$	0	0
$527$	−6.69944	−0.291832
$528$	0	0
$529$	−3.88316	−0.168833
$530$	0	0
$531$	0.448959	0.0194831
$532$	0	0
$533$	5.97526	0.258817
$534$	0	0
$535$	−15.7576	−0.681261
$536$	0	0
$537$	−6.66789	−0.287740
$538$	0	0
$539$	−11.2307	−0.483741
$540$	0	0
$541$	−22.1299	−0.951438	−0.475719	−	0.879597i	$-0.657812\pi$
$541$	−22.1299	−0.951438	−0.475719	+	0.879597i	$0.657812\pi$
$542$	0	0
$543$	8.37602	0.359449
$544$	0	0
$545$	0.372281	0.0159468
$546$	0	0
$547$	33.8437	1.44705	0.723527	−	0.690296i	$-0.242520\pi$
$547$	33.8437	1.44705	0.723527	+	0.690296i	$0.242520\pi$
$548$	0	0
$549$	13.9011	0.593283
$550$	0	0
$551$	1.28691	0.0548243
$552$	0	0
$553$	−50.7223	−2.15693
$554$	0	0
$555$	−6.22448	−0.264214
$556$	0	0
$557$	−3.39650	−0.143914	−0.0719572	−	0.997408i	$-0.522924\pi$
$557$	−3.39650	−0.143914	−0.0719572	+	0.997408i	$0.522924\pi$
$558$	0	0
$559$	57.5022	2.43208
$560$	0	0
$561$	5.33211	0.225122
$562$	0	0
$563$	32.9617	1.38917	0.694585	−	0.719411i	$-0.255588\pi$
$563$	32.9617	1.38917	0.694585	+	0.719411i	$0.255588\pi$
$564$	0	0
$565$	1.07608	0.0452709
$566$	0	0
$567$	11.9258	0.500836
$568$	0	0
$569$	−38.9809	−1.63416	−0.817082	−	0.576522i	$-0.804410\pi$
$569$	−38.9809	−1.63416	−0.817082	+	0.576522i	$0.804410\pi$
$570$	0	0
$571$	−39.2041	−1.64064	−0.820320	−	0.571905i	$-0.806205\pi$
$571$	−39.2041	−1.64064	−0.820320	+	0.571905i	$0.806205\pi$
$572$	0	0
$573$	−30.1596	−1.25994
$574$	0	0
$575$	4.37228	0.182337
$576$	0	0
$577$	−28.6344	−1.19207	−0.596034	−	0.802959i	$-0.703258\pi$
$577$	−28.6344	−1.19207	−0.596034	+	0.802959i	$0.703258\pi$
$578$	0	0
$579$	14.3631	0.596909
$580$	0	0
$581$	−54.8695	−2.27637
$582$	0	0
$583$	−6.25604	−0.259099
$584$	0	0
$585$	−8.19974	−0.339018
$586$	0	0
$587$	−28.4323	−1.17353	−0.586764	−	0.809758i	$-0.699598\pi$
$587$	−28.4323	−1.17353	−0.586764	+	0.809758i	$0.699598\pi$
$588$	0	0
$589$	−1.57331	−0.0648270
$590$	0	0
$591$	0.336983	0.0138616
$592$	0	0
$593$	−31.1411	−1.27881	−0.639405	−	0.768870i	$-0.720819\pi$
$593$	−31.1411	−1.27881	−0.639405	+	0.768870i	$0.720819\pi$
$594$	0	0
$595$	−13.0223	−0.533860
$596$	0	0
$597$	4.53920	0.185777
$598$	0	0
$599$	15.4917	0.632976	0.316488	−	0.948597i	$-0.397496\pi$
$599$	15.4917	0.632976	0.316488	+	0.948597i	$0.397496\pi$
$600$	0	0
$601$	5.62225	0.229336	0.114668	−	0.993404i	$-0.463420\pi$
$601$	5.62225	0.229336	0.114668	+	0.993404i	$0.463420\pi$
$602$	0	0
$603$	−13.3562	−0.543908
$604$	0	0
$605$	−9.37228	−0.381037
$606$	0	0
$607$	21.9109	0.889335	0.444668	−	0.895696i	$-0.353322\pi$
$607$	21.9109	0.889335	0.444668	+	0.895696i	$0.353322\pi$
$608$	0	0
$609$	8.48417	0.343796
$610$	0	0
$611$	−22.6815	−0.917597
$612$	0	0
$613$	24.8918	1.00537	0.502686	−	0.864469i	$-0.332345\pi$
$613$	24.8918	1.00537	0.502686	+	0.864469i	$0.332345\pi$
$614$	0	0
$615$	−1.27582	−0.0514461
$616$	0	0
$617$	38.3637	1.54446	0.772231	−	0.635341i	$-0.219141\pi$
$617$	38.3637	1.54446	0.772231	+	0.635341i	$0.219141\pi$
$618$	0	0
$619$	−26.4706	−1.06394	−0.531972	−	0.846762i	$-0.678549\pi$
$619$	−26.4706	−1.06394	−0.531972	+	0.846762i	$0.678549\pi$
$620$	0	0
$621$	24.3897	0.978724
$622$	0	0
$623$	−53.0135	−2.12394
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.25220	0.0500081
$628$	0	0
$629$	15.9821	0.637248
$630$	0	0
$631$	13.1707	0.524316	0.262158	−	0.965025i	$-0.415566\pi$
$631$	13.1707	0.524316	0.262158	+	0.965025i	$0.415566\pi$
$632$	0	0
$633$	5.28315	0.209986
$634$	0	0
$635$	9.57390	0.379928
$636$	0	0
$637$	52.5986	2.08403
$638$	0	0
$639$	9.15958	0.362347
$640$	0	0
$641$	−3.59189	−0.141871	−0.0709356	−	0.997481i	$-0.522598\pi$
$641$	−3.59189	−0.141871	−0.0709356	+	0.997481i	$0.522598\pi$
$642$	0	0
$643$	18.7155	0.738066	0.369033	−	0.929416i	$-0.379689\pi$
$643$	18.7155	0.738066	0.369033	+	0.929416i	$0.379689\pi$
$644$	0	0
$645$	−12.2777	−0.483434
$646$	0	0
$647$	8.58320	0.337440	0.168720	−	0.985664i	$-0.446037\pi$
$647$	8.58320	0.337440	0.168720	+	0.985664i	$0.446037\pi$
$648$	0	0
$649$	0.417400	0.0163844
$650$	0	0
$651$	−10.3723	−0.406522
$652$	0	0
$653$	−29.5905	−1.15797	−0.578984	−	0.815339i	$-0.696550\pi$
$653$	−29.5905	−1.15797	−0.578984	+	0.815339i	$0.696550\pi$
$654$	0	0
$655$	−8.23804	−0.321887
$656$	0	0
$657$	16.2697	0.634741
$658$	0	0
$659$	20.8589	0.812548	0.406274	−	0.913751i	$-0.366828\pi$
$659$	20.8589	0.812548	0.406274	+	0.913751i	$0.366828\pi$
$660$	0	0
$661$	25.6370	0.997164	0.498582	−	0.866842i	$-0.333854\pi$
$661$	25.6370	0.997164	0.498582	+	0.866842i	$0.333854\pi$
$662$	0	0
$663$	−24.9728	−0.969862
$664$	0	0
$665$	−3.05816	−0.118591
$666$	0	0
$667$	7.31412	0.283204
$668$	0	0
$669$	−18.0778	−0.698928
$670$	0	0
$671$	12.9239	0.498923
$672$	0	0
$673$	29.6698	1.14369	0.571844	−	0.820362i	$-0.306228\pi$
$673$	29.6698	1.14369	0.571844	+	0.820362i	$0.306228\pi$
$674$	0	0
$675$	5.57825	0.214707
$676$	0	0
$677$	−17.7228	−0.681143	−0.340572	−	0.940219i	$-0.610621\pi$
$677$	−17.7228	−0.681143	−0.340572	+	0.940219i	$0.610621\pi$
$678$	0	0
$679$	−42.7396	−1.64020
$680$	0	0
$681$	−7.35438	−0.281820
$682$	0	0
$683$	11.5975	0.443767	0.221883	−	0.975073i	$-0.428780\pi$
$683$	11.5975	0.443767	0.221883	+	0.975073i	$0.428780\pi$
$684$	0	0
$685$	−14.3432	−0.548025
$686$	0	0
$687$	29.7162	1.13374
$688$	0	0
$689$	29.2999	1.11624
$690$	0	0
$691$	−1.60298	−0.0609803	−0.0304902	−	0.999535i	$-0.509707\pi$
$691$	−1.60298	−0.0609803	−0.0304902	+	0.999535i	$0.509707\pi$
$692$	0	0
$693$	−6.95983	−0.264382
$694$	0	0
$695$	−8.74456	−0.331700
$696$	0	0
$697$	3.27582	0.124081
$698$	0	0
$699$	−19.6147	−0.741895
$700$	0	0
$701$	−28.2572	−1.06726	−0.533629	−	0.845719i	$-0.679172\pi$
$701$	−28.2572	−1.06726	−0.533629	+	0.845719i	$0.679172\pi$
$702$	0	0
$703$	3.75326	0.141557
$704$	0	0
$705$	4.84289	0.182394
$706$	0	0
$707$	−50.2870	−1.89124
$708$	0	0
$709$	−1.08590	−0.0407817	−0.0203909	−	0.999792i	$-0.506491\pi$
$709$	−1.08590	−0.0407817	−0.0203909	+	0.999792i	$0.506491\pi$
$710$	0	0
$711$	−17.5096	−0.656661
$712$	0	0
$713$	−8.94184	−0.334874
$714$	0	0
$715$	−7.62337	−0.285098
$716$	0	0
$717$	5.19718	0.194092
$718$	0	0
$719$	44.3074	1.65239	0.826193	−	0.563387i	$-0.190502\pi$
$719$	44.3074	1.65239	0.826193	+	0.563387i	$0.190502\pi$
$720$	0	0
$721$	−10.1974	−0.379769
$722$	0	0
$723$	−9.18797	−0.341704
$724$	0	0
$725$	1.67284	0.0621276
$726$	0	0
$727$	28.5214	1.05780	0.528901	−	0.848684i	$-0.322604\pi$
$727$	28.5214	1.05780	0.528901	+	0.848684i	$0.322604\pi$
$728$	0	0
$729$	25.4674	0.943236
$730$	0	0
$731$	31.5244	1.16597
$732$	0	0
$733$	−46.4897	−1.71714	−0.858569	−	0.512699i	$-0.828646\pi$
$733$	−46.4897	−1.71714	−0.858569	+	0.512699i	$0.828646\pi$
$734$	0	0
$735$	−11.2307	−0.414251
$736$	0	0
$737$	−12.4174	−0.457401
$738$	0	0
$739$	−41.4379	−1.52432	−0.762159	−	0.647389i	$-0.775861\pi$
$739$	−41.4379	−1.52432	−0.762159	+	0.647389i	$0.775861\pi$
$740$	0	0
$741$	−5.86464	−0.215443
$742$	0	0
$743$	−1.00869	−0.0370053	−0.0185027	−	0.999829i	$-0.505890\pi$
$743$	−1.00869	−0.0370053	−0.0185027	+	0.999829i	$0.505890\pi$
$744$	0	0
$745$	6.41740	0.235115
$746$	0	0
$747$	−18.9412	−0.693023
$748$	0	0
$749$	62.6407	2.28884
$750$	0	0
$751$	−23.1046	−0.843100	−0.421550	−	0.906805i	$-0.638514\pi$
$751$	−23.1046	−0.843100	−0.421550	+	0.906805i	$0.638514\pi$
$752$	0	0
$753$	−12.7929	−0.466200
$754$	0	0
$755$	−5.75079	−0.209292
$756$	0	0
$757$	−8.58941	−0.312187	−0.156094	−	0.987742i	$-0.549890\pi$
$757$	−8.58941	−0.312187	−0.156094	+	0.987742i	$0.549890\pi$
$758$	0	0
$759$	7.11684	0.258325
$760$	0	0
$761$	−33.9870	−1.23203	−0.616014	−	0.787735i	$-0.711254\pi$
$761$	−33.9870	−1.23203	−0.616014	+	0.787735i	$0.711254\pi$
$762$	0	0
$763$	−1.47992	−0.0535766
$764$	0	0
$765$	−4.49535	−0.162530
$766$	0	0
$767$	−1.95488	−0.0705867
$768$	0	0
$769$	−26.2598	−0.946952	−0.473476	−	0.880807i	$-0.657001\pi$
$769$	−26.2598	−0.946952	−0.473476	+	0.880807i	$0.657001\pi$
$770$	0	0
$771$	−26.5658	−0.956744
$772$	0	0
$773$	−31.3822	−1.12874	−0.564370	−	0.825522i	$-0.690881\pi$
$773$	−31.3822	−1.12874	−0.564370	+	0.825522i	$0.690881\pi$
$774$	0	0
$775$	−2.04512	−0.0734628
$776$	0	0
$777$	24.7440	0.887684
$778$	0	0
$779$	0.769298	0.0275630
$780$	0	0
$781$	8.51573	0.304717
$782$	0	0
$783$	9.33150	0.333481
$784$	0	0
$785$	−23.2671	−0.830439
$786$	0	0
$787$	−15.9722	−0.569347	−0.284673	−	0.958625i	$-0.591885\pi$
$787$	−15.9722	−0.569347	−0.284673	+	0.958625i	$0.591885\pi$
$788$	0	0
$789$	−36.3897	−1.29551
$790$	0	0
$791$	−4.27769	−0.152097
$792$	0	0
$793$	−60.5288	−2.14944
$794$	0	0
$795$	−6.25604	−0.221879
$796$	0	0
$797$	27.9419	0.989753	0.494877	−	0.868963i	$-0.335213\pi$
$797$	27.9419	0.989753	0.494877	+	0.868963i	$0.335213\pi$
$798$	0	0
$799$	−12.4347	−0.439908
$800$	0	0
$801$	−18.3005	−0.646618
$802$	0	0
$803$	15.1261	0.533787
$804$	0	0
$805$	−17.3810	−0.612599
$806$	0	0
$807$	−16.2528	−0.572126
$808$	0	0
$809$	−40.7044	−1.43109	−0.715545	−	0.698567i	$-0.753822\pi$
$809$	−40.7044	−1.43109	−0.715545	+	0.698567i	$0.753822\pi$
$810$	0	0
$811$	50.2906	1.76594	0.882972	−	0.469426i	$-0.155539\pi$
$811$	50.2906	1.76594	0.882972	+	0.469426i	$0.155539\pi$
$812$	0	0
$813$	15.8527	0.555979
$814$	0	0
$815$	−14.8429	−0.519924
$816$	0	0
$817$	7.40324	0.259007
$818$	0	0
$819$	32.5961	1.13900
$820$	0	0
$821$	45.8924	1.60165	0.800827	−	0.598896i	$-0.204394\pi$
$821$	45.8924	1.60165	0.800827	+	0.598896i	$0.204394\pi$
$822$	0	0
$823$	−36.0861	−1.25788	−0.628942	−	0.777453i	$-0.716512\pi$
$823$	−36.0861	−1.25788	−0.628942	+	0.777453i	$0.716512\pi$
$824$	0	0
$825$	1.62772	0.0566699
$826$	0	0
$827$	−43.6878	−1.51917	−0.759586	−	0.650407i	$-0.774598\pi$
$827$	−43.6878	−1.51917	−0.759586	+	0.650407i	$0.774598\pi$
$828$	0	0
$829$	35.3315	1.22711	0.613557	−	0.789651i	$-0.289738\pi$
$829$	35.3315	1.22711	0.613557	+	0.789651i	$0.289738\pi$
$830$	0	0
$831$	−15.1489	−0.525510
$832$	0	0
$833$	28.8362	0.999114
$834$	0	0
$835$	−10.9938	−0.380455
$836$	0	0
$837$	−11.4082	−0.394324
$838$	0	0
$839$	8.05186	0.277981	0.138990	−	0.990294i	$-0.455614\pi$
$839$	8.05186	0.277981	0.138990	+	0.990294i	$0.455614\pi$
$840$	0	0
$841$	−26.2016	−0.903504
$842$	0	0
$843$	−13.7587	−0.473876
$844$	0	0
$845$	22.7038	0.781034
$846$	0	0
$847$	37.2573	1.28018
$848$	0	0
$849$	−31.5515	−1.08285
$850$	0	0
$851$	21.3315	0.731234
$852$	0	0
$853$	31.4793	1.07783	0.538915	−	0.842360i	$-0.318834\pi$
$853$	31.4793	1.07783	0.538915	+	0.842360i	$0.318834\pi$
$854$	0	0
$855$	−1.05569	−0.0361040
$856$	0	0
$857$	6.32543	0.216073	0.108036	−	0.994147i	$-0.465544\pi$
$857$	6.32543	0.216073	0.108036	+	0.994147i	$0.465544\pi$
$858$	0	0
$859$	−27.8516	−0.950284	−0.475142	−	0.879909i	$-0.657603\pi$
$859$	−27.8516	−0.950284	−0.475142	+	0.879909i	$0.657603\pi$
$860$	0	0
$861$	5.07172	0.172844
$862$	0	0
$863$	−10.1001	−0.343810	−0.171905	−	0.985114i	$-0.554992\pi$
$863$	−10.1001	−0.343810	−0.171905	+	0.985114i	$0.554992\pi$
$864$	0	0
$865$	8.52008	0.289692
$866$	0	0
$867$	7.99812	0.271630
$868$	0	0
$869$	−16.2788	−0.552221
$870$	0	0
$871$	58.1565	1.97056
$872$	0	0
$873$	−14.7539	−0.499345
$874$	0	0
$875$	−3.97526	−0.134388
$876$	0	0
$877$	−17.4602	−0.589589	−0.294794	−	0.955561i	$-0.595251\pi$
$877$	−17.4602	−0.589589	−0.294794	+	0.955561i	$0.595251\pi$
$878$	0	0
$879$	−38.4205	−1.29589
$880$	0	0
$881$	19.0908	0.643184	0.321592	−	0.946878i	$-0.395782\pi$
$881$	19.0908	0.643184	0.321592	+	0.946878i	$0.395782\pi$
$882$	0	0
$883$	22.2802	0.749787	0.374894	−	0.927068i	$-0.377679\pi$
$883$	22.2802	0.749787	0.374894	+	0.927068i	$0.377679\pi$
$884$	0	0
$885$	0.417400	0.0140308
$886$	0	0
$887$	9.05569	0.304060	0.152030	−	0.988376i	$-0.451419\pi$
$887$	9.05569	0.304060	0.152030	+	0.988376i	$0.451419\pi$
$888$	0	0
$889$	−38.0588	−1.27645
$890$	0	0
$891$	3.82746	0.128225
$892$	0	0
$893$	−2.92018	−0.0977202
$894$	0	0
$895$	5.22635	0.174698
$896$	0	0
$897$	−33.3315	−1.11291
$898$	0	0
$899$	−3.42115	−0.114102
$900$	0	0
$901$	16.0631	0.535140
$902$	0	0
$903$	48.8071	1.62420
$904$	0	0
$905$	−6.56520	−0.218235
$906$	0	0
$907$	−28.9326	−0.960691	−0.480346	−	0.877079i	$-0.659489\pi$
$907$	−28.9326	−0.960691	−0.480346	+	0.877079i	$0.659489\pi$
$908$	0	0
$909$	−17.3593	−0.575772
$910$	0	0
$911$	−11.6859	−0.387170	−0.193585	−	0.981083i	$-0.562012\pi$
$911$	−11.6859	−0.387170	−0.193585	+	0.981083i	$0.562012\pi$
$912$	0	0
$913$	−17.6098	−0.582800
$914$	0	0
$915$	12.9239	0.427252
$916$	0	0
$917$	32.7484	1.08145
$918$	0	0
$919$	−3.36980	−0.111159	−0.0555797	−	0.998454i	$-0.517701\pi$
$919$	−3.36980	−0.111159	−0.0555797	+	0.998454i	$0.517701\pi$
$920$	0	0
$921$	43.2288	1.42444
$922$	0	0
$923$	−39.8831	−1.31277
$924$	0	0
$925$	4.87880	0.160414
$926$	0	0
$927$	−3.52018	−0.115618
$928$	0	0
$929$	−55.2140	−1.81151	−0.905757	−	0.423798i	$-0.860697\pi$
$929$	−55.2140	−1.81151	−0.905757	+	0.423798i	$0.860697\pi$
$930$	0	0
$931$	6.77192	0.221941
$932$	0	0
$933$	−26.7142	−0.874584
$934$	0	0
$935$	−4.17936	−0.136680
$936$	0	0
$937$	−4.62524	−0.151100	−0.0755499	−	0.997142i	$-0.524071\pi$
$937$	−4.62524	−0.151100	−0.0755499	+	0.997142i	$0.524071\pi$
$938$	0	0
$939$	−19.0836	−0.622768
$940$	0	0
$941$	−18.1255	−0.590873	−0.295436	−	0.955362i	$-0.595465\pi$
$941$	−18.1255	−0.590873	−0.295436	+	0.955362i	$0.595465\pi$
$942$	0	0
$943$	4.37228	0.142381
$944$	0	0
$945$	−22.1750	−0.721353
$946$	0	0
$947$	5.86523	0.190594	0.0952972	−	0.995449i	$-0.469620\pi$
$947$	5.86523	0.190594	0.0952972	+	0.995449i	$0.469620\pi$
$948$	0	0
$949$	−70.8424	−2.29964
$950$	0	0
$951$	−10.7880	−0.349823
$952$	0	0
$953$	−32.1255	−1.04065	−0.520324	−	0.853969i	$-0.674189\pi$
$953$	−32.1255	−1.04065	−0.520324	+	0.853969i	$0.674189\pi$
$954$	0	0
$955$	23.6394	0.764953
$956$	0	0
$957$	2.72291	0.0880192
$958$	0	0
$959$	57.0180	1.84121
$960$	0	0
$961$	−26.8175	−0.865080
$962$	0	0
$963$	21.6239	0.696819
$964$	0	0
$965$	−11.2579	−0.362405
$966$	0	0
$967$	−30.2975	−0.974301	−0.487150	−	0.873318i	$-0.661964\pi$
$967$	−30.2975	−0.974301	−0.487150	+	0.873318i	$0.661964\pi$
$968$	0	0
$969$	−3.21517	−0.103286
$970$	0	0
$971$	56.7149	1.82007	0.910035	−	0.414532i	$-0.136055\pi$
$971$	56.7149	1.82007	0.910035	+	0.414532i	$0.136055\pi$
$972$	0	0
$973$	34.7619	1.11442
$974$	0	0
$975$	−7.62337	−0.244143
$976$	0	0
$977$	55.6937	1.78180	0.890900	−	0.454199i	$-0.150075\pi$
$977$	55.6937	1.78180	0.890900	+	0.454199i	$0.150075\pi$
$978$	0	0
$979$	−17.0142	−0.543775
$980$	0	0
$981$	−0.510875	−0.0163110
$982$	0	0
$983$	11.1108	0.354378	0.177189	−	0.984177i	$-0.443300\pi$
$983$	11.1108	0.354378	0.177189	+	0.984177i	$0.443300\pi$
$984$	0	0
$985$	−0.264130	−0.00841589
$986$	0	0
$987$	−19.2518	−0.612791
$988$	0	0
$989$	42.0761	1.33794
$990$	0	0
$991$	34.6291	1.10003	0.550014	−	0.835155i	$-0.314622\pi$
$991$	34.6291	1.10003	0.550014	+	0.835155i	$0.314622\pi$
$992$	0	0
$993$	12.6588	0.401714
$994$	0	0
$995$	−3.55786	−0.112792
$996$	0	0
$997$	−15.2165	−0.481912	−0.240956	−	0.970536i	$-0.577461\pi$
$997$	−15.2165	−0.481912	−0.240956	+	0.970536i	$0.577461\pi$
$998$	0	0
$999$	27.2152	0.861050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	820.2.a.d.1.2	✓	4
3.2	odd	2		7380.2.a.t.1.1		4
4.3	odd	2		3280.2.a.be.1.3		4
5.2	odd	4		4100.2.d.e.1149.5		8
5.3	odd	4		4100.2.d.e.1149.4		8
5.4	even	2		4100.2.a.d.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
820.2.a.d.1.2	✓	4	1.1	even	1	trivial
3280.2.a.be.1.3		4	4.3	odd	2
4100.2.a.d.1.3		4	5.4	even	2
4100.2.d.e.1149.4		8	5.3	odd	4
4100.2.d.e.1149.5		8	5.2	odd	4
7380.2.a.t.1.1		4	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	820.a (trivial)

\(f(q)\)	\(=\)	\(q-1.27582 q^{3} +1.00000 q^{5} -3.97526 q^{7} -1.37228 q^{9} -1.27582 q^{11} +5.97526 q^{13} -1.27582 q^{15} +3.27582 q^{17} +0.769298 q^{19} +5.07172 q^{21} +4.37228 q^{23} +1.00000 q^{25} +5.57825 q^{27} +1.67284 q^{29} -2.04512 q^{31} +1.62772 q^{33} -3.97526 q^{35} +4.87880 q^{37} -7.62337 q^{39} +1.00000 q^{41} +9.62337 q^{43} -1.37228 q^{45} -3.79590 q^{47} +8.80273 q^{49} -4.17936 q^{51} +4.90354 q^{53} -1.27582 q^{55} -0.981487 q^{57} -0.327162 q^{59} -10.1299 q^{61} +5.45518 q^{63} +5.97526 q^{65} +9.73287 q^{67} -5.57825 q^{69} -6.67471 q^{71} -11.8559 q^{73} -1.27582 q^{75} +5.07172 q^{77} +12.7595 q^{79} -3.00000 q^{81} +13.8027 q^{83} +3.27582 q^{85} -2.13424 q^{87} +13.3359 q^{89} -23.7533 q^{91} +2.60921 q^{93} +0.769298 q^{95} +10.7514 q^{97} +1.75079 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + q^{7} + 6 q^{9} + 7 q^{13} + 8 q^{17} - 3 q^{19} - 3 q^{21} + 6 q^{23} + 4 q^{25} + 7 q^{29} + 3 q^{31} + 18 q^{33} + q^{35} + 9 q^{37} + 3 q^{39} + 4 q^{41} + 5 q^{43} + 6 q^{45} + 3 q^{47}+ \cdots + 6 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + q^7 + 6 * q^9 + 7 * q^13 + 8 * q^17 - 3 * q^19 - 3 * q^21 + 6 * q^23 + 4 * q^25 + 7 * q^29 + 3 * q^31 + 18 * q^33 + q^35 + 9 * q^37 + 3 * q^39 + 4 * q^41 + 5 * q^43 + 6 * q^45 + 3 * q^47 + 3 * q^49 - 18 * q^51 + 26 * q^53 - 3 * q^57 - q^59 - 8 * q^61 + 18 * q^63 + 7 * q^65 + q^67 + 2 * q^71 + 9 * q^73 - 3 * q^77 + q^79 - 12 * q^81 + 23 * q^83 + 8 * q^85 - 21 * q^87 + 10 * q^89 - 29 * q^91 + 21 * q^93 - 3 * q^95 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more