LMFDB - Embedded newform 8036.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.879385 q^{3} +0.694593 q^{5} -2.22668 q^{9} +O(q^{10})$	$q-0.879385 q^{3} +0.694593 q^{5} -2.22668 q^{9} -4.36959 q^{11} +3.71688 q^{13} -0.610815 q^{15} -2.53209 q^{17} -2.29086 q^{19} -3.94356 q^{23} -4.51754 q^{25} +4.59627 q^{27} -0.241230 q^{29} +5.38919 q^{31} +3.84255 q^{33} -11.1061 q^{37} -3.26857 q^{39} +1.00000 q^{41} +4.98545 q^{43} -1.54664 q^{45} -9.92902 q^{47} +2.22668 q^{51} +12.8229 q^{53} -3.03508 q^{55} +2.01455 q^{57} +3.38919 q^{59} +6.24123 q^{61} +2.58172 q^{65} -7.38919 q^{67} +3.46791 q^{69} -1.43376 q^{71} +2.61081 q^{73} +3.97266 q^{75} +1.91622 q^{79} +2.63816 q^{81} -1.51754 q^{83} -1.75877 q^{85} +0.212134 q^{87} +2.32770 q^{89} -4.73917 q^{93} -1.59121 q^{95} -4.26857 q^{97} +9.72967 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3}+O(q^{10}) $ 3 * q + 3 * q^3	$ 3 q + 3 q^{3} - 6 q^{11} + 3 q^{13} - 6 q^{15} - 3 q^{17} + 9 q^{19} + 3 q^{23} + 9 q^{25} - 12 q^{29} + 12 q^{31} - 6 q^{33} - 21 q^{37} + 3 q^{41} - 3 q^{43} - 18 q^{45} + 3 q^{47} + 18 q^{53} + 36 q^{55} + 24 q^{57} + 6 q^{59} + 30 q^{61} - 24 q^{65} - 18 q^{67} + 15 q^{69} + 12 q^{71} + 12 q^{73} + 33 q^{75} + 12 q^{79} - 9 q^{81} + 18 q^{83} + 6 q^{85} - 24 q^{87} + 3 q^{89} - 6 q^{95} - 3 q^{97} - 18 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 6 * q^11 + 3 * q^13 - 6 * q^15 - 3 * q^17 + 9 * q^19 + 3 * q^23 + 9 * q^25 - 12 * q^29 + 12 * q^31 - 6 * q^33 - 21 * q^37 + 3 * q^41 - 3 * q^43 - 18 * q^45 + 3 * q^47 + 18 * q^53 + 36 * q^55 + 24 * q^57 + 6 * q^59 + 30 * q^61 - 24 * q^65 - 18 * q^67 + 15 * q^69 + 12 * q^71 + 12 * q^73 + 33 * q^75 + 12 * q^79 - 9 * q^81 + 18 * q^83 + 6 * q^85 - 24 * q^87 + 3 * q^89 - 6 * q^95 - 3 * q^97 - 18 * 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
$3$	−0.879385	−0.507713	−0.253857	−	0.967242i	$-0.581699\pi$
$3$	−0.879385	−0.507713	−0.253857	+	0.967242i	$0.581699\pi$
$4$	0	0
$5$	0.694593	0.310631	0.155316	−	0.987865i	$-0.450361\pi$
$5$	0.694593	0.310631	0.155316	+	0.987865i	$0.450361\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.22668	−0.742227
$10$	0	0
$11$	−4.36959	−1.31748	−0.658740	−	0.752371i	$-0.728910\pi$
$11$	−4.36959	−1.31748	−0.658740	+	0.752371i	$0.728910\pi$
$12$	0	0
$13$	3.71688	1.03088	0.515439	−	0.856926i	$-0.327629\pi$
$13$	3.71688	1.03088	0.515439	+	0.856926i	$0.327629\pi$
$14$	0	0
$15$	−0.610815	−0.157712
$16$	0	0
$17$	−2.53209	−0.614122	−0.307061	−	0.951690i	$-0.599346\pi$
$17$	−2.53209	−0.614122	−0.307061	+	0.951690i	$0.599346\pi$
$18$	0	0
$19$	−2.29086	−0.525559	−0.262780	−	0.964856i	$-0.584639\pi$
$19$	−2.29086	−0.525559	−0.262780	+	0.964856i	$0.584639\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.94356	−0.822290	−0.411145	−	0.911570i	$-0.634871\pi$
$23$	−3.94356	−0.822290	−0.411145	+	0.911570i	$0.634871\pi$
$24$	0	0
$25$	−4.51754	−0.903508
$26$	0	0
$27$	4.59627	0.884552
$28$	0	0
$29$	−0.241230	−0.0447952	−0.0223976	−	0.999749i	$-0.507130\pi$
$29$	−0.241230	−0.0447952	−0.0223976	+	0.999749i	$0.507130\pi$
$30$	0	0
$31$	5.38919	0.967926	0.483963	−	0.875088i	$-0.339197\pi$
$31$	5.38919	0.967926	0.483963	+	0.875088i	$0.339197\pi$
$32$	0	0
$33$	3.84255	0.668902
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.1061	−1.82583	−0.912913	−	0.408154i	$-0.866173\pi$
$37$	−11.1061	−1.82583	−0.912913	+	0.408154i	$0.866173\pi$
$38$	0	0
$39$	−3.26857	−0.523390
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	4.98545	0.760274	0.380137	−	0.924930i	$-0.375877\pi$
$43$	4.98545	0.760274	0.380137	+	0.924930i	$0.375877\pi$
$44$	0	0
$45$	−1.54664	−0.230559
$46$	0	0
$47$	−9.92902	−1.44830	−0.724148	−	0.689645i	$-0.757767\pi$
$47$	−9.92902	−1.44830	−0.724148	+	0.689645i	$0.757767\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.22668	0.311798
$52$	0	0
$53$	12.8229	1.76137	0.880684	−	0.473705i	$-0.157084\pi$
$53$	12.8229	1.76137	0.880684	+	0.473705i	$0.157084\pi$
$54$	0	0
$55$	−3.03508	−0.409250
$56$	0	0
$57$	2.01455	0.266833
$58$	0	0
$59$	3.38919	0.441234	0.220617	−	0.975360i	$-0.429193\pi$
$59$	3.38919	0.441234	0.220617	+	0.975360i	$0.429193\pi$
$60$	0	0
$61$	6.24123	0.799108	0.399554	−	0.916710i	$-0.369165\pi$
$61$	6.24123	0.799108	0.399554	+	0.916710i	$0.369165\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.58172	0.320223
$66$	0	0
$67$	−7.38919	−0.902733	−0.451366	−	0.892339i	$-0.649063\pi$
$67$	−7.38919	−0.902733	−0.451366	+	0.892339i	$0.649063\pi$
$68$	0	0
$69$	3.46791	0.417487
$70$	0	0
$71$	−1.43376	−0.170156	−0.0850782	−	0.996374i	$-0.527114\pi$
$71$	−1.43376	−0.170156	−0.0850782	+	0.996374i	$0.527114\pi$
$72$	0	0
$73$	2.61081	0.305573	0.152786	−	0.988259i	$-0.451175\pi$
$73$	2.61081	0.305573	0.152786	+	0.988259i	$0.451175\pi$
$74$	0	0
$75$	3.97266	0.458723
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.91622	0.215592	0.107796	−	0.994173i	$-0.465621\pi$
$79$	1.91622	0.215592	0.107796	+	0.994173i	$0.465621\pi$
$80$	0	0
$81$	2.63816	0.293128
$82$	0	0
$83$	−1.51754	−0.166572	−0.0832859	−	0.996526i	$-0.526541\pi$
$83$	−1.51754	−0.166572	−0.0832859	+	0.996526i	$0.526541\pi$
$84$	0	0
$85$	−1.75877	−0.190765
$86$	0	0
$87$	0.212134	0.0227431
$88$	0	0
$89$	2.32770	0.246735	0.123368	−	0.992361i	$-0.460631\pi$
$89$	2.32770	0.246735	0.123368	+	0.992361i	$0.460631\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.73917	−0.491429
$94$	0	0
$95$	−1.59121	−0.163255
$96$	0	0
$97$	−4.26857	−0.433408	−0.216704	−	0.976237i	$-0.569531\pi$
$97$	−4.26857	−0.433408	−0.216704	+	0.976237i	$0.569531\pi$
$98$	0	0
$99$	9.72967	0.977869
$100$	0	0
$101$	0.638156	0.0634989	0.0317494	−	0.999496i	$-0.489892\pi$
$101$	0.638156	0.0634989	0.0317494	+	0.999496i	$0.489892\pi$
$102$	0	0
$103$	17.7297	1.74696	0.873478	−	0.486863i	$-0.161859\pi$
$103$	17.7297	1.74696	0.873478	+	0.486863i	$0.161859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.0496	−1.74492	−0.872462	−	0.488682i	$-0.837478\pi$
$107$	−18.0496	−1.74492	−0.872462	+	0.488682i	$0.837478\pi$
$108$	0	0
$109$	−5.51754	−0.528485	−0.264242	−	0.964456i	$-0.585122\pi$
$109$	−5.51754	−0.528485	−0.264242	+	0.964456i	$0.585122\pi$
$110$	0	0
$111$	9.76651	0.926996
$112$	0	0
$113$	17.8307	1.67737	0.838685	−	0.544617i	$-0.183325\pi$
$113$	17.8307	1.67737	0.838685	+	0.544617i	$0.183325\pi$
$114$	0	0
$115$	−2.73917	−0.255429
$116$	0	0
$117$	−8.27631	−0.765145
$118$	0	0
$119$	0	0
$120$	0	0
$121$	8.09327	0.735752
$122$	0	0
$123$	−0.879385	−0.0792915
$124$	0	0
$125$	−6.61081	−0.591289
$126$	0	0
$127$	−2.76558	−0.245405	−0.122703	−	0.992443i	$-0.539156\pi$
$127$	−2.76558	−0.245405	−0.122703	+	0.992443i	$0.539156\pi$
$128$	0	0
$129$	−4.38413	−0.386001
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.19253	0.274770
$136$	0	0
$137$	7.14796	0.610691	0.305346	−	0.952242i	$-0.401228\pi$
$137$	7.14796	0.610691	0.305346	+	0.952242i	$0.401228\pi$
$138$	0	0
$139$	−12.0155	−1.01914	−0.509570	−	0.860429i	$-0.670195\pi$
$139$	−12.0155	−1.01914	−0.509570	+	0.860429i	$0.670195\pi$
$140$	0	0
$141$	8.73143	0.735319
$142$	0	0
$143$	−16.2412	−1.35816
$144$	0	0
$145$	−0.167556	−0.0139148
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.85204	0.397495	0.198747	−	0.980051i	$-0.436313\pi$
$149$	4.85204	0.397495	0.198747	+	0.980051i	$0.436313\pi$
$150$	0	0
$151$	20.5526	1.67255	0.836274	−	0.548311i	$-0.184729\pi$
$151$	20.5526	1.67255	0.836274	+	0.548311i	$0.184729\pi$
$152$	0	0
$153$	5.63816	0.455818
$154$	0	0
$155$	3.74329	0.300668
$156$	0	0
$157$	15.5963	1.24472	0.622359	−	0.782732i	$-0.286174\pi$
$157$	15.5963	1.24472	0.622359	+	0.782732i	$0.286174\pi$
$158$	0	0
$159$	−11.2763	−0.894270
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−0.963163	−0.0754408	−0.0377204	−	0.999288i	$-0.512010\pi$
$163$	−0.963163	−0.0754408	−0.0377204	+	0.999288i	$0.512010\pi$
$164$	0	0
$165$	2.66901	0.207782
$166$	0	0
$167$	2.03415	0.157407	0.0787036	−	0.996898i	$-0.474922\pi$
$167$	2.03415	0.157407	0.0787036	+	0.996898i	$0.474922\pi$
$168$	0	0
$169$	0.815207	0.0627083
$170$	0	0
$171$	5.10101	0.390084
$172$	0	0
$173$	−12.9513	−0.984669	−0.492335	−	0.870406i	$-0.663856\pi$
$173$	−12.9513	−0.984669	−0.492335	+	0.870406i	$0.663856\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.98040	−0.224021
$178$	0	0
$179$	−21.9864	−1.64334	−0.821670	−	0.569964i	$-0.806957\pi$
$179$	−21.9864	−1.64334	−0.821670	+	0.569964i	$0.806957\pi$
$180$	0	0
$181$	5.85978	0.435554	0.217777	−	0.975999i	$-0.430119\pi$
$181$	5.85978	0.435554	0.217777	+	0.975999i	$0.430119\pi$
$182$	0	0
$183$	−5.48845	−0.405718
$184$	0	0
$185$	−7.71419	−0.567159
$186$	0	0
$187$	11.0642	0.809093
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.45336	0.177519	0.0887596	−	0.996053i	$-0.471710\pi$
$191$	2.45336	0.177519	0.0887596	+	0.996053i	$0.471710\pi$
$192$	0	0
$193$	22.0547	1.58753	0.793765	−	0.608224i	$-0.208118\pi$
$193$	22.0547	1.58753	0.793765	+	0.608224i	$0.208118\pi$
$194$	0	0
$195$	−2.27033	−0.162581
$196$	0	0
$197$	20.0942	1.43165	0.715826	−	0.698278i	$-0.246050\pi$
$197$	20.0942	1.43165	0.715826	+	0.698278i	$0.246050\pi$
$198$	0	0
$199$	18.8357	1.33523	0.667615	−	0.744507i	$-0.267315\pi$
$199$	18.8357	1.33523	0.667615	+	0.744507i	$0.267315\pi$
$200$	0	0
$201$	6.49794	0.458329
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.694593	0.0485125
$206$	0	0
$207$	8.78106	0.610326
$208$	0	0
$209$	10.0101	0.692413
$210$	0	0
$211$	2.48246	0.170900	0.0854498	−	0.996342i	$-0.472767\pi$
$211$	2.48246	0.170900	0.0854498	+	0.996342i	$0.472767\pi$
$212$	0	0
$213$	1.26083	0.0863906
$214$	0	0
$215$	3.46286	0.236165
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2.29591	−0.155143
$220$	0	0
$221$	−9.41147	−0.633084
$222$	0	0
$223$	10.4534	0.700009	0.350004	−	0.936748i	$-0.386180\pi$
$223$	10.4534	0.700009	0.350004	+	0.936748i	$0.386180\pi$
$224$	0	0
$225$	10.0591	0.670608
$226$	0	0
$227$	−26.5526	−1.76236	−0.881180	−	0.472781i	$-0.843250\pi$
$227$	−26.5526	−1.76236	−0.881180	+	0.472781i	$0.843250\pi$
$228$	0	0
$229$	−2.22937	−0.147321	−0.0736605	−	0.997283i	$-0.523468\pi$
$229$	−2.22937	−0.147321	−0.0736605	+	0.997283i	$0.523468\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.4534	0.946871	0.473436	−	0.880828i	$-0.343014\pi$
$233$	14.4534	0.946871	0.473436	+	0.880828i	$0.343014\pi$
$234$	0	0
$235$	−6.89662	−0.449886
$236$	0	0
$237$	−1.68510	−0.109459
$238$	0	0
$239$	−5.56212	−0.359784	−0.179892	−	0.983686i	$-0.557575\pi$
$239$	−5.56212	−0.359784	−0.179892	+	0.983686i	$0.557575\pi$
$240$	0	0
$241$	−28.1830	−1.81543	−0.907715	−	0.419588i	$-0.862174\pi$
$241$	−28.1830	−1.81543	−0.907715	+	0.419588i	$0.862174\pi$
$242$	0	0
$243$	−16.1088	−1.03338
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.51485	−0.541787
$248$	0	0
$249$	1.33450	0.0845707
$250$	0	0
$251$	−4.95130	−0.312524	−0.156262	−	0.987716i	$-0.549944\pi$
$251$	−4.95130	−0.312524	−0.156262	+	0.987716i	$0.549944\pi$
$252$	0	0
$253$	17.2317	1.08335
$254$	0	0
$255$	1.54664	0.0968542
$256$	0	0
$257$	−2.14022	−0.133503	−0.0667515	−	0.997770i	$-0.521263\pi$
$257$	−2.14022	−0.133503	−0.0667515	+	0.997770i	$0.521263\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0.537141	0.0332482
$262$	0	0
$263$	−27.8135	−1.71505	−0.857525	−	0.514441i	$-0.827999\pi$
$263$	−27.8135	−1.71505	−0.857525	+	0.514441i	$0.827999\pi$
$264$	0	0
$265$	8.90673	0.547136
$266$	0	0
$267$	−2.04694	−0.125271
$268$	0	0
$269$	16.5972	1.01195	0.505975	−	0.862548i	$-0.331133\pi$
$269$	16.5972	1.01195	0.505975	+	0.862548i	$0.331133\pi$
$270$	0	0
$271$	7.40467	0.449801	0.224901	−	0.974382i	$-0.427794\pi$
$271$	7.40467	0.449801	0.224901	+	0.974382i	$0.427794\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	19.7398	1.19035
$276$	0	0
$277$	25.1807	1.51296	0.756480	−	0.654017i	$-0.226917\pi$
$277$	25.1807	1.51296	0.756480	+	0.654017i	$0.226917\pi$
$278$	0	0
$279$	−12.0000	−0.718421
$280$	0	0
$281$	25.8871	1.54430	0.772148	−	0.635442i	$-0.219182\pi$
$281$	25.8871	1.54430	0.772148	+	0.635442i	$0.219182\pi$
$282$	0	0
$283$	3.59121	0.213476	0.106738	−	0.994287i	$-0.465959\pi$
$283$	3.59121	0.213476	0.106738	+	0.994287i	$0.465959\pi$
$284$	0	0
$285$	1.39929	0.0828868
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−10.5885	−0.622854
$290$	0	0
$291$	3.75372	0.220047
$292$	0	0
$293$	7.55674	0.441470	0.220735	−	0.975334i	$-0.429154\pi$
$293$	7.55674	0.441470	0.220735	+	0.975334i	$0.429154\pi$
$294$	0	0
$295$	2.35410	0.137061
$296$	0	0
$297$	−20.0838	−1.16538
$298$	0	0
$299$	−14.6578	−0.847680
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−0.561185	−0.0322392
$304$	0	0
$305$	4.33511	0.248228
$306$	0	0
$307$	−0.167556	−0.00956294	−0.00478147	−	0.999989i	$-0.501522\pi$
$307$	−0.167556	−0.00956294	−0.00478147	+	0.999989i	$0.501522\pi$
$308$	0	0
$309$	−15.5912	−0.886953
$310$	0	0
$311$	−7.99495	−0.453352	−0.226676	−	0.973970i	$-0.572786\pi$
$311$	−7.99495	−0.453352	−0.226676	+	0.973970i	$0.572786\pi$
$312$	0	0
$313$	11.2713	0.637089	0.318545	−	0.947908i	$-0.396806\pi$
$313$	11.2713	0.637089	0.318545	+	0.947908i	$0.396806\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.9905	−0.954282	−0.477141	−	0.878827i	$-0.658327\pi$
$317$	−16.9905	−0.954282	−0.477141	+	0.878827i	$0.658327\pi$
$318$	0	0
$319$	1.05407	0.0590168
$320$	0	0
$321$	15.8726	0.885921
$322$	0	0
$323$	5.80066	0.322757
$324$	0	0
$325$	−16.7912	−0.931406
$326$	0	0
$327$	4.85204	0.268319
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.53714	−0.139454	−0.0697269	−	0.997566i	$-0.522213\pi$
$331$	−2.53714	−0.139454	−0.0697269	+	0.997566i	$0.522213\pi$
$332$	0	0
$333$	24.7297	1.35518
$334$	0	0
$335$	−5.13247	−0.280417
$336$	0	0
$337$	30.9941	1.68836	0.844179	−	0.536062i	$-0.180089\pi$
$337$	30.9941	1.68836	0.844179	+	0.536062i	$0.180089\pi$
$338$	0	0
$339$	−15.6800	−0.851623
$340$	0	0
$341$	−23.5485	−1.27522
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.40879	0.129685
$346$	0	0
$347$	−1.65951	−0.0890872	−0.0445436	−	0.999007i	$-0.514183\pi$
$347$	−1.65951	−0.0890872	−0.0445436	+	0.999007i	$0.514183\pi$
$348$	0	0
$349$	34.9959	1.87329	0.936643	−	0.350285i	$-0.113915\pi$
$349$	34.9959	1.87329	0.936643	+	0.350285i	$0.113915\pi$
$350$	0	0
$351$	17.0838	0.911865
$352$	0	0
$353$	−2.58172	−0.137411	−0.0687055	−	0.997637i	$-0.521887\pi$
$353$	−2.58172	−0.137411	−0.0687055	+	0.997637i	$0.521887\pi$
$354$	0	0
$355$	−0.995881	−0.0528559
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.6236	−0.982916	−0.491458	−	0.870901i	$-0.663536\pi$
$359$	−18.6236	−0.982916	−0.491458	+	0.870901i	$0.663536\pi$
$360$	0	0
$361$	−13.7520	−0.723788
$362$	0	0
$363$	−7.11711	−0.373551
$364$	0	0
$365$	1.81345	0.0949205
$366$	0	0
$367$	−8.16756	−0.426343	−0.213171	−	0.977015i	$-0.568379\pi$
$367$	−8.16756	−0.426343	−0.213171	+	0.977015i	$0.568379\pi$
$368$	0	0
$369$	−2.22668	−0.115916
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3.68004	0.190545	0.0952727	−	0.995451i	$-0.469628\pi$
$373$	3.68004	0.190545	0.0952727	+	0.995451i	$0.469628\pi$
$374$	0	0
$375$	5.81345	0.300205
$376$	0	0
$377$	−0.896622	−0.0461784
$378$	0	0
$379$	−4.20027	−0.215754	−0.107877	−	0.994164i	$-0.534405\pi$
$379$	−4.20027	−0.215754	−0.107877	+	0.994164i	$0.534405\pi$
$380$	0	0
$381$	2.43201	0.124596
$382$	0	0
$383$	37.4475	1.91348	0.956739	−	0.290949i	$-0.0939709\pi$
$383$	37.4475	1.91348	0.956739	+	0.290949i	$0.0939709\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.1010	−0.564296
$388$	0	0
$389$	6.28312	0.318567	0.159283	−	0.987233i	$-0.449082\pi$
$389$	6.28312	0.318567	0.159283	+	0.987233i	$0.449082\pi$
$390$	0	0
$391$	9.98545	0.504986
$392$	0	0
$393$	−12.3114	−0.621028
$394$	0	0
$395$	1.33099	0.0669696
$396$	0	0
$397$	35.6459	1.78902	0.894508	−	0.447052i	$-0.147526\pi$
$397$	35.6459	1.78902	0.894508	+	0.447052i	$0.147526\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.8494	0.741541	0.370771	−	0.928724i	$-0.379094\pi$
$401$	14.8494	0.741541	0.370771	+	0.928724i	$0.379094\pi$
$402$	0	0
$403$	20.0310	0.997813
$404$	0	0
$405$	1.83244	0.0910549
$406$	0	0
$407$	48.5289	2.40549
$408$	0	0
$409$	−14.4979	−0.716877	−0.358439	−	0.933553i	$-0.616691\pi$
$409$	−14.4979	−0.716877	−0.358439	+	0.933553i	$0.616691\pi$
$410$	0	0
$411$	−6.28581	−0.310056
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.05407	−0.0517424
$416$	0	0
$417$	10.5662	0.517431
$418$	0	0
$419$	7.06418	0.345108	0.172554	−	0.985000i	$-0.444798\pi$
$419$	7.06418	0.345108	0.172554	+	0.985000i	$0.444798\pi$
$420$	0	0
$421$	5.47834	0.266998	0.133499	−	0.991049i	$-0.457379\pi$
$421$	5.47834	0.266998	0.133499	+	0.991049i	$0.457379\pi$
$422$	0	0
$423$	22.1088	1.07496
$424$	0	0
$425$	11.4388	0.554864
$426$	0	0
$427$	0	0
$428$	0	0
$429$	14.2823	0.689556
$430$	0	0
$431$	7.34998	0.354036	0.177018	−	0.984208i	$-0.443355\pi$
$431$	7.34998	0.354036	0.177018	+	0.984208i	$0.443355\pi$
$432$	0	0
$433$	−12.6108	−0.606037	−0.303019	−	0.952985i	$-0.597994\pi$
$433$	−12.6108	−0.606037	−0.303019	+	0.952985i	$0.597994\pi$
$434$	0	0
$435$	0.147347	0.00706472
$436$	0	0
$437$	9.03415	0.432162
$438$	0	0
$439$	37.2799	1.77927	0.889637	−	0.456668i	$-0.150957\pi$
$439$	37.2799	1.77927	0.889637	+	0.456668i	$0.150957\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.3628	−0.729908	−0.364954	−	0.931026i	$-0.618915\pi$
$443$	−15.3628	−0.729908	−0.364954	+	0.931026i	$0.618915\pi$
$444$	0	0
$445$	1.61680	0.0766437
$446$	0	0
$447$	−4.26682	−0.201813
$448$	0	0
$449$	−23.1070	−1.09049	−0.545243	−	0.838278i	$-0.683563\pi$
$449$	−23.1070	−1.09049	−0.545243	+	0.838278i	$0.683563\pi$
$450$	0	0
$451$	−4.36959	−0.205756
$452$	0	0
$453$	−18.0737	−0.849175
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15.9026	−0.743893	−0.371946	−	0.928254i	$-0.621309\pi$
$457$	−15.9026	−0.743893	−0.371946	+	0.928254i	$0.621309\pi$
$458$	0	0
$459$	−11.6382	−0.543223
$460$	0	0
$461$	10.6655	0.496742	0.248371	−	0.968665i	$-0.420105\pi$
$461$	10.6655	0.496742	0.248371	+	0.968665i	$0.420105\pi$
$462$	0	0
$463$	−22.7547	−1.05750	−0.528749	−	0.848778i	$-0.677339\pi$
$463$	−22.7547	−1.05750	−0.528749	+	0.848778i	$0.677339\pi$
$464$	0	0
$465$	−3.29179	−0.152653
$466$	0	0
$467$	5.38919	0.249382	0.124691	−	0.992196i	$-0.460206\pi$
$467$	5.38919	0.249382	0.124691	+	0.992196i	$0.460206\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−13.7151	−0.631960
$472$	0	0
$473$	−21.7844	−1.00165
$474$	0	0
$475$	10.3491	0.474847
$476$	0	0
$477$	−28.5526	−1.30733
$478$	0	0
$479$	−22.9495	−1.04859	−0.524296	−	0.851536i	$-0.675671\pi$
$479$	−22.9495	−1.04859	−0.524296	+	0.851536i	$0.675671\pi$
$480$	0	0
$481$	−41.2799	−1.88220
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.96492	−0.134630
$486$	0	0
$487$	−12.0888	−0.547797	−0.273899	−	0.961759i	$-0.588313\pi$
$487$	−12.0888	−0.547797	−0.273899	+	0.961759i	$0.588313\pi$
$488$	0	0
$489$	0.846992	0.0383023
$490$	0	0
$491$	6.92396	0.312474	0.156237	−	0.987720i	$-0.450064\pi$
$491$	6.92396	0.312474	0.156237	+	0.987720i	$0.450064\pi$
$492$	0	0
$493$	0.610815	0.0275097
$494$	0	0
$495$	6.75816	0.303757
$496$	0	0
$497$	0	0
$498$	0	0
$499$	32.0155	1.43321	0.716605	−	0.697479i	$-0.245695\pi$
$499$	32.0155	1.43321	0.716605	+	0.697479i	$0.245695\pi$
$500$	0	0
$501$	−1.78880	−0.0799177
$502$	0	0
$503$	19.3429	0.862455	0.431228	−	0.902243i	$-0.358081\pi$
$503$	19.3429	0.862455	0.431228	+	0.902243i	$0.358081\pi$
$504$	0	0
$505$	0.443258	0.0197247
$506$	0	0
$507$	−0.716881	−0.0318378
$508$	0	0
$509$	25.8708	1.14670	0.573352	−	0.819309i	$-0.305643\pi$
$509$	25.8708	1.14670	0.573352	+	0.819309i	$0.305643\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−10.5294	−0.464884
$514$	0	0
$515$	12.3149	0.542659
$516$	0	0
$517$	43.3857	1.90810
$518$	0	0
$519$	11.3892	0.499930
$520$	0	0
$521$	−35.8316	−1.56981	−0.784906	−	0.619615i	$-0.787289\pi$
$521$	−35.8316	−1.56981	−0.784906	+	0.619615i	$0.787289\pi$
$522$	0	0
$523$	42.3168	1.85038	0.925192	−	0.379500i	$-0.123904\pi$
$523$	42.3168	1.85038	0.925192	+	0.379500i	$0.123904\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13.6459	−0.594425
$528$	0	0
$529$	−7.44831	−0.323840
$530$	0	0
$531$	−7.54664	−0.327496
$532$	0	0
$533$	3.71688	0.160996
$534$	0	0
$535$	−12.5371	−0.542028
$536$	0	0
$537$	19.3345	0.834345
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.67593	0.373007	0.186504	−	0.982454i	$-0.440284\pi$
$541$	8.67593	0.373007	0.186504	+	0.982454i	$0.440284\pi$
$542$	0	0
$543$	−5.15301	−0.221137
$544$	0	0
$545$	−3.83244	−0.164164
$546$	0	0
$547$	12.7392	0.544688	0.272344	−	0.962200i	$-0.412201\pi$
$547$	12.7392	0.544688	0.272344	+	0.962200i	$0.412201\pi$
$548$	0	0
$549$	−13.8972	−0.593119
$550$	0	0
$551$	0.552623	0.0235425
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6.78375	0.287954
$556$	0	0
$557$	−16.0547	−0.680259	−0.340129	−	0.940379i	$-0.610471\pi$
$557$	−16.0547	−0.680259	−0.340129	+	0.940379i	$0.610471\pi$
$558$	0	0
$559$	18.5303	0.783750
$560$	0	0
$561$	−9.72967	−0.410787
$562$	0	0
$563$	19.5749	0.824984	0.412492	−	0.910961i	$-0.364658\pi$
$563$	19.5749	0.824984	0.412492	+	0.910961i	$0.364658\pi$
$564$	0	0
$565$	12.3851	0.521044
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.6450	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$569$	−12.6450	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$570$	0	0
$571$	−24.3797	−1.02026	−0.510129	−	0.860098i	$-0.670402\pi$
$571$	−24.3797	−1.02026	−0.510129	+	0.860098i	$0.670402\pi$
$572$	0	0
$573$	−2.15745	−0.0901288
$574$	0	0
$575$	17.8152	0.742946
$576$	0	0
$577$	25.3191	1.05405	0.527025	−	0.849850i	$-0.323308\pi$
$577$	25.3191	1.05405	0.527025	+	0.849850i	$0.323308\pi$
$578$	0	0
$579$	−19.3946	−0.806010
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−56.0310	−2.32057
$584$	0	0
$585$	−5.74867	−0.237678
$586$	0	0
$587$	−29.6509	−1.22383	−0.611913	−	0.790925i	$-0.709600\pi$
$587$	−29.6509	−1.22383	−0.611913	+	0.790925i	$0.709600\pi$
$588$	0	0
$589$	−12.3459	−0.508703
$590$	0	0
$591$	−17.6705	−0.726869
$592$	0	0
$593$	12.0574	0.495137	0.247568	−	0.968870i	$-0.420368\pi$
$593$	12.0574	0.495137	0.247568	+	0.968870i	$0.420368\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.5639	−0.677914
$598$	0	0
$599$	17.0087	0.694956	0.347478	−	0.937688i	$-0.387038\pi$
$599$	17.0087	0.694956	0.347478	+	0.937688i	$0.387038\pi$
$600$	0	0
$601$	−43.7948	−1.78643	−0.893213	−	0.449633i	$-0.851555\pi$
$601$	−43.7948	−1.78643	−0.893213	+	0.449633i	$0.851555\pi$
$602$	0	0
$603$	16.4534	0.670033
$604$	0	0
$605$	5.62153	0.228548
$606$	0	0
$607$	37.5175	1.52279	0.761395	−	0.648288i	$-0.224515\pi$
$607$	37.5175	1.52279	0.761395	+	0.648288i	$0.224515\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−36.9050	−1.49302
$612$	0	0
$613$	−14.9085	−0.602148	−0.301074	−	0.953601i	$-0.597345\pi$
$613$	−14.9085	−0.602148	−0.301074	+	0.953601i	$0.597345\pi$
$614$	0	0
$615$	−0.610815	−0.0246304
$616$	0	0
$617$	−6.22256	−0.250511	−0.125255	−	0.992125i	$-0.539975\pi$
$617$	−6.22256	−0.250511	−0.125255	+	0.992125i	$0.539975\pi$
$618$	0	0
$619$	21.9554	0.882463	0.441231	−	0.897393i	$-0.354542\pi$
$619$	21.9554	0.882463	0.441231	+	0.897393i	$0.354542\pi$
$620$	0	0
$621$	−18.1257	−0.727358
$622$	0	0
$623$	0	0
$624$	0	0
$625$	17.9959	0.719835
$626$	0	0
$627$	−8.80274	−0.351548
$628$	0	0
$629$	28.1215	1.12128
$630$	0	0
$631$	6.22493	0.247810	0.123905	−	0.992294i	$-0.460458\pi$
$631$	6.22493	0.247810	0.123905	+	0.992294i	$0.460458\pi$
$632$	0	0
$633$	−2.18304	−0.0867680
$634$	0	0
$635$	−1.92095	−0.0762306
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.19253	0.126295
$640$	0	0
$641$	36.2121	1.43029	0.715147	−	0.698974i	$-0.246360\pi$
$641$	36.2121	1.43029	0.715147	+	0.698974i	$0.246360\pi$
$642$	0	0
$643$	−9.92364	−0.391350	−0.195675	−	0.980669i	$-0.562690\pi$
$643$	−9.92364	−0.391350	−0.195675	+	0.980669i	$0.562690\pi$
$644$	0	0
$645$	−3.04519	−0.119904
$646$	0	0
$647$	−45.6769	−1.79574	−0.897871	−	0.440258i	$-0.854887\pi$
$647$	−45.6769	−1.79574	−0.897871	+	0.440258i	$0.854887\pi$
$648$	0	0
$649$	−14.8093	−0.581317
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.6209	0.572161	0.286080	−	0.958206i	$-0.407648\pi$
$653$	14.6209	0.572161	0.286080	+	0.958206i	$0.407648\pi$
$654$	0	0
$655$	9.72430	0.379960
$656$	0	0
$657$	−5.81345	−0.226804
$658$	0	0
$659$	0.990505	0.0385846	0.0192923	−	0.999814i	$-0.493859\pi$
$659$	0.990505	0.0385846	0.0192923	+	0.999814i	$0.493859\pi$
$660$	0	0
$661$	47.1397	1.83352	0.916761	−	0.399436i	$-0.130794\pi$
$661$	47.1397	1.83352	0.916761	+	0.399436i	$0.130794\pi$
$662$	0	0
$663$	8.27631	0.321425
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.951304	0.0368346
$668$	0	0
$669$	−9.19253	−0.355404
$670$	0	0
$671$	−27.2716	−1.05281
$672$	0	0
$673$	−13.4338	−0.517834	−0.258917	−	0.965900i	$-0.583366\pi$
$673$	−13.4338	−0.517834	−0.258917	+	0.965900i	$0.583366\pi$
$674$	0	0
$675$	−20.7638	−0.799200
$676$	0	0
$677$	33.5039	1.28766	0.643830	−	0.765168i	$-0.277344\pi$
$677$	33.5039	1.28766	0.643830	+	0.765168i	$0.277344\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	23.3500	0.894773
$682$	0	0
$683$	19.4047	0.742499	0.371249	−	0.928533i	$-0.378929\pi$
$683$	19.4047	0.742499	0.371249	+	0.928533i	$0.378929\pi$
$684$	0	0
$685$	4.96492	0.189700
$686$	0	0
$687$	1.96048	0.0747968
$688$	0	0
$689$	47.6614	1.81575
$690$	0	0
$691$	40.0574	1.52385	0.761927	−	0.647663i	$-0.224253\pi$
$691$	40.0574	1.52385	0.761927	+	0.647663i	$0.224253\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.34587	−0.316577
$696$	0	0
$697$	−2.53209	−0.0959097
$698$	0	0
$699$	−12.7101	−0.480739
$700$	0	0
$701$	−18.2885	−0.690747	−0.345373	−	0.938465i	$-0.612248\pi$
$701$	−18.2885	−0.690747	−0.345373	+	0.938465i	$0.612248\pi$
$702$	0	0
$703$	25.4424	0.959580
$704$	0	0
$705$	6.06479	0.228413
$706$	0	0
$707$	0	0
$708$	0	0
$709$	46.1147	1.73188	0.865938	−	0.500152i	$-0.166722\pi$
$709$	46.1147	1.73188	0.865938	+	0.500152i	$0.166722\pi$
$710$	0	0
$711$	−4.26682	−0.160018
$712$	0	0
$713$	−21.2526	−0.795916
$714$	0	0
$715$	−11.2810	−0.421887
$716$	0	0
$717$	4.89124	0.182667
$718$	0	0
$719$	14.0392	0.523574	0.261787	−	0.965126i	$-0.415688\pi$
$719$	14.0392	0.523574	0.261787	+	0.965126i	$0.415688\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	24.7837	0.921717
$724$	0	0
$725$	1.08976	0.0404728
$726$	0	0
$727$	17.4834	0.648423	0.324212	−	0.945985i	$-0.394901\pi$
$727$	17.4834	0.648423	0.324212	+	0.945985i	$0.394901\pi$
$728$	0	0
$729$	6.25133	0.231531
$730$	0	0
$731$	−12.6236	−0.466901
$732$	0	0
$733$	2.97502	0.109885	0.0549425	−	0.998490i	$-0.482502\pi$
$733$	2.97502	0.109885	0.0549425	+	0.998490i	$0.482502\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.2877	1.18933
$738$	0	0
$739$	−16.2439	−0.597542	−0.298771	−	0.954325i	$-0.596577\pi$
$739$	−16.2439	−0.597542	−0.298771	+	0.954325i	$0.596577\pi$
$740$	0	0
$741$	7.48784	0.275073
$742$	0	0
$743$	35.1138	1.28820	0.644100	−	0.764941i	$-0.277232\pi$
$743$	35.1138	1.28820	0.644100	+	0.764941i	$0.277232\pi$
$744$	0	0
$745$	3.37019	0.123474
$746$	0	0
$747$	3.37908	0.123634
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4.95130	0.180676	0.0903378	−	0.995911i	$-0.471205\pi$
$751$	4.95130	0.180676	0.0903378	+	0.995911i	$0.471205\pi$
$752$	0	0
$753$	4.35410	0.158672
$754$	0	0
$755$	14.2757	0.519546
$756$	0	0
$757$	−12.2020	−0.443490	−0.221745	−	0.975105i	$-0.571175\pi$
$757$	−12.2020	−0.443490	−0.221745	+	0.975105i	$0.571175\pi$
$758$	0	0
$759$	−15.1533	−0.550031
$760$	0	0
$761$	−20.3696	−0.738397	−0.369198	−	0.929351i	$-0.620368\pi$
$761$	−20.3696	−0.738397	−0.369198	+	0.929351i	$0.620368\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.91622	0.141591
$766$	0	0
$767$	12.5972	0.454859
$768$	0	0
$769$	6.34049	0.228644	0.114322	−	0.993444i	$-0.463530\pi$
$769$	6.34049	0.228644	0.114322	+	0.993444i	$0.463530\pi$
$770$	0	0
$771$	1.88207	0.0677812
$772$	0	0
$773$	52.0205	1.87105	0.935524	−	0.353262i	$-0.114928\pi$
$773$	52.0205	1.87105	0.935524	+	0.353262i	$0.114928\pi$
$774$	0	0
$775$	−24.3459	−0.874529
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.29086	−0.0820786
$780$	0	0
$781$	6.26495	0.224177
$782$	0	0
$783$	−1.10876	−0.0396237
$784$	0	0
$785$	10.8331	0.386648
$786$	0	0
$787$	44.1539	1.57392	0.786959	−	0.617005i	$-0.211654\pi$
$787$	44.1539	1.57392	0.786959	+	0.617005i	$0.211654\pi$
$788$	0	0
$789$	24.4587	0.870754
$790$	0	0
$791$	0	0
$792$	0	0
$793$	23.1979	0.823782
$794$	0	0
$795$	−7.83244	−0.277788
$796$	0	0
$797$	15.9162	0.563782	0.281891	−	0.959447i	$-0.409038\pi$
$797$	15.9162	0.563782	0.281891	+	0.959447i	$0.409038\pi$
$798$	0	0
$799$	25.1411	0.889430
$800$	0	0
$801$	−5.18304	−0.183134
$802$	0	0
$803$	−11.4082	−0.402586
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.5953	−0.513780
$808$	0	0
$809$	26.8331	0.943400	0.471700	−	0.881759i	$-0.343641\pi$
$809$	26.8331	0.943400	0.471700	+	0.881759i	$0.343641\pi$
$810$	0	0
$811$	−33.0196	−1.15947	−0.579737	−	0.814803i	$-0.696845\pi$
$811$	−33.0196	−1.15947	−0.579737	+	0.814803i	$0.696845\pi$
$812$	0	0
$813$	−6.51155	−0.228370
$814$	0	0
$815$	−0.669006	−0.0234343
$816$	0	0
$817$	−11.4210	−0.399569
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.2327	−0.636324	−0.318162	−	0.948036i	$-0.603066\pi$
$821$	−18.2327	−0.636324	−0.318162	+	0.948036i	$0.603066\pi$
$822$	0	0
$823$	−7.77238	−0.270928	−0.135464	−	0.990782i	$-0.543253\pi$
$823$	−7.77238	−0.270928	−0.135464	+	0.990782i	$0.543253\pi$
$824$	0	0
$825$	−17.3589	−0.604358
$826$	0	0
$827$	−9.00599	−0.313169	−0.156584	−	0.987665i	$-0.550048\pi$
$827$	−9.00599	−0.313169	−0.156584	+	0.987665i	$0.550048\pi$
$828$	0	0
$829$	24.0702	0.835991	0.417996	−	0.908449i	$-0.362733\pi$
$829$	24.0702	0.835991	0.417996	+	0.908449i	$0.362733\pi$
$830$	0	0
$831$	−22.1435	−0.768150
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.41290	0.0488956
$836$	0	0
$837$	24.7701	0.856181
$838$	0	0
$839$	51.3191	1.77173	0.885867	−	0.463940i	$-0.153565\pi$
$839$	51.3191	1.77173	0.885867	+	0.463940i	$0.153565\pi$
$840$	0	0
$841$	−28.9418	−0.997993
$842$	0	0
$843$	−22.7648	−0.784060
$844$	0	0
$845$	0.566237	0.0194792
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.15806	−0.108384
$850$	0	0
$851$	43.7975	1.50136
$852$	0	0
$853$	29.0114	0.993330	0.496665	−	0.867942i	$-0.334558\pi$
$853$	29.0114	0.993330	0.496665	+	0.867942i	$0.334558\pi$
$854$	0	0
$855$	3.54313	0.121172
$856$	0	0
$857$	−16.1385	−0.551279	−0.275640	−	0.961261i	$-0.588890\pi$
$857$	−16.1385	−0.551279	−0.275640	+	0.961261i	$0.588890\pi$
$858$	0	0
$859$	−44.1492	−1.50635	−0.753176	−	0.657819i	$-0.771479\pi$
$859$	−44.1492	−1.50635	−0.753176	+	0.657819i	$0.771479\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36.2567	1.23419	0.617096	−	0.786888i	$-0.288309\pi$
$863$	36.2567	1.23419	0.617096	+	0.786888i	$0.288309\pi$
$864$	0	0
$865$	−8.99588	−0.305869
$866$	0	0
$867$	9.31139	0.316231
$868$	0	0
$869$	−8.37309	−0.284038
$870$	0	0
$871$	−27.4647	−0.930607
$872$	0	0
$873$	9.50475	0.321687
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.578097	0.0195209	0.00976047	−	0.999952i	$-0.496893\pi$
$877$	0.578097	0.0195209	0.00976047	+	0.999952i	$0.496893\pi$
$878$	0	0
$879$	−6.64529	−0.224140
$880$	0	0
$881$	44.1046	1.48592	0.742961	−	0.669334i	$-0.233421\pi$
$881$	44.1046	1.48592	0.742961	+	0.669334i	$0.233421\pi$
$882$	0	0
$883$	−38.9959	−1.31232	−0.656158	−	0.754624i	$-0.727819\pi$
$883$	−38.9959	−1.31232	−0.656158	+	0.754624i	$0.727819\pi$
$884$	0	0
$885$	−2.07016	−0.0695878
$886$	0	0
$887$	9.36278	0.314371	0.157186	−	0.987569i	$-0.449758\pi$
$887$	9.36278	0.314371	0.157186	+	0.987569i	$0.449758\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−11.5276	−0.386191
$892$	0	0
$893$	22.7460	0.761165
$894$	0	0
$895$	−15.2716	−0.510473
$896$	0	0
$897$	12.8898	0.430378
$898$	0	0
$899$	−1.30003	−0.0433584
$900$	0	0
$901$	−32.4688	−1.08169
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.07016	0.135297
$906$	0	0
$907$	−31.2844	−1.03878	−0.519390	−	0.854537i	$-0.673841\pi$
$907$	−31.2844	−1.03878	−0.519390	+	0.854537i	$0.673841\pi$
$908$	0	0
$909$	−1.42097	−0.0471306
$910$	0	0
$911$	21.6355	0.716815	0.358408	−	0.933565i	$-0.383320\pi$
$911$	21.6355	0.716815	0.358408	+	0.933565i	$0.383320\pi$
$912$	0	0
$913$	6.63102	0.219455
$914$	0	0
$915$	−3.81223	−0.126029
$916$	0	0
$917$	0	0
$918$	0	0
$919$	14.4296	0.475990	0.237995	−	0.971266i	$-0.423510\pi$
$919$	14.4296	0.475990	0.237995	+	0.971266i	$0.423510\pi$
$920$	0	0
$921$	0.147347	0.00485523
$922$	0	0
$923$	−5.32913	−0.175410
$924$	0	0
$925$	50.1721	1.64965
$926$	0	0
$927$	−39.4783	−1.29664
$928$	0	0
$929$	−19.2145	−0.630407	−0.315204	−	0.949024i	$-0.602073\pi$
$929$	−19.2145	−0.630407	−0.315204	+	0.949024i	$0.602073\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	7.03064	0.230173
$934$	0	0
$935$	7.68510	0.251330
$936$	0	0
$937$	24.0627	0.786096	0.393048	−	0.919518i	$-0.371421\pi$
$937$	24.0627	0.786096	0.393048	+	0.919518i	$0.371421\pi$
$938$	0	0
$939$	−9.91178	−0.323459
$940$	0	0
$941$	50.3424	1.64111	0.820557	−	0.571565i	$-0.193663\pi$
$941$	50.3424	1.64111	0.820557	+	0.571565i	$0.193663\pi$
$942$	0	0
$943$	−3.94356	−0.128420
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.1807	−1.27320	−0.636600	−	0.771194i	$-0.719660\pi$
$947$	−39.1807	−1.27320	−0.636600	+	0.771194i	$0.719660\pi$
$948$	0	0
$949$	9.70409	0.315008
$950$	0	0
$951$	14.9412	0.484502
$952$	0	0
$953$	−40.3533	−1.30717	−0.653586	−	0.756853i	$-0.726736\pi$
$953$	−40.3533	−1.30717	−0.653586	+	0.756853i	$0.726736\pi$
$954$	0	0
$955$	1.70409	0.0551430
$956$	0	0
$957$	−0.926936	−0.0299636
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.95668	−0.0631187
$962$	0	0
$963$	40.1908	1.29513
$964$	0	0
$965$	15.3190	0.493137
$966$	0	0
$967$	2.62630	0.0844560	0.0422280	−	0.999108i	$-0.486554\pi$
$967$	2.62630	0.0844560	0.0422280	+	0.999108i	$0.486554\pi$
$968$	0	0
$969$	−5.10101	−0.163868
$970$	0	0
$971$	−42.5577	−1.36574	−0.682870	−	0.730540i	$-0.739269\pi$
$971$	−42.5577	−1.36574	−0.682870	+	0.730540i	$0.739269\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	14.7659	0.472887
$976$	0	0
$977$	−38.0101	−1.21605	−0.608025	−	0.793917i	$-0.708038\pi$
$977$	−38.0101	−1.21605	−0.608025	+	0.793917i	$0.708038\pi$
$978$	0	0
$979$	−10.1711	−0.325069
$980$	0	0
$981$	12.2858	0.392256
$982$	0	0
$983$	−11.5276	−0.367675	−0.183837	−	0.982957i	$-0.558852\pi$
$983$	−11.5276	−0.367675	−0.183837	+	0.982957i	$0.558852\pi$
$984$	0	0
$985$	13.9573	0.444716
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−19.6604	−0.625166
$990$	0	0
$991$	10.9750	0.348633	0.174317	−	0.984690i	$-0.444228\pi$
$991$	10.9750	0.348633	0.174317	+	0.984690i	$0.444228\pi$
$992$	0	0
$993$	2.23112	0.0708026
$994$	0	0
$995$	13.0832	0.414764
$996$	0	0
$997$	−35.7279	−1.13151	−0.565757	−	0.824572i	$-0.691416\pi$
$997$	−35.7279	−1.13151	−0.565757	+	0.824572i	$0.691416\pi$
$998$	0	0
$999$	−51.0464	−1.61504

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.h.1.1		3
7.6	odd	2		1148.2.a.b.1.3	✓	3
28.27	even	2		4592.2.a.v.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.b.1.3	✓	3	7.6	odd	2
4592.2.a.v.1.1		3	28.27	even	2
8036.2.a.h.1.1		3	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-0.879385 q^{3} +0.694593 q^{5} -2.22668 q^{9} +O(q^{10})\)	\(q-0.879385 q^{3} +0.694593 q^{5} -2.22668 q^{9} -4.36959 q^{11} +3.71688 q^{13} -0.610815 q^{15} -2.53209 q^{17} -2.29086 q^{19} -3.94356 q^{23} -4.51754 q^{25} +4.59627 q^{27} -0.241230 q^{29} +5.38919 q^{31} +3.84255 q^{33} -11.1061 q^{37} -3.26857 q^{39} +1.00000 q^{41} +4.98545 q^{43} -1.54664 q^{45} -9.92902 q^{47} +2.22668 q^{51} +12.8229 q^{53} -3.03508 q^{55} +2.01455 q^{57} +3.38919 q^{59} +6.24123 q^{61} +2.58172 q^{65} -7.38919 q^{67} +3.46791 q^{69} -1.43376 q^{71} +2.61081 q^{73} +3.97266 q^{75} +1.91622 q^{79} +2.63816 q^{81} -1.51754 q^{83} -1.75877 q^{85} +0.212134 q^{87} +2.32770 q^{89} -4.73917 q^{93} -1.59121 q^{95} -4.26857 q^{97} +9.72967 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3}+O(q^{10}) \) 3 * q + 3 * q^3	\( 3 q + 3 q^{3} - 6 q^{11} + 3 q^{13} - 6 q^{15} - 3 q^{17} + 9 q^{19} + 3 q^{23} + 9 q^{25} - 12 q^{29} + 12 q^{31} - 6 q^{33} - 21 q^{37} + 3 q^{41} - 3 q^{43} - 18 q^{45} + 3 q^{47} + 18 q^{53} + 36 q^{55} + 24 q^{57} + 6 q^{59} + 30 q^{61} - 24 q^{65} - 18 q^{67} + 15 q^{69} + 12 q^{71} + 12 q^{73} + 33 q^{75} + 12 q^{79} - 9 q^{81} + 18 q^{83} + 6 q^{85} - 24 q^{87} + 3 q^{89} - 6 q^{95} - 3 q^{97} - 18 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 6 * q^11 + 3 * q^13 - 6 * q^15 - 3 * q^17 + 9 * q^19 + 3 * q^23 + 9 * q^25 - 12 * q^29 + 12 * q^31 - 6 * q^33 - 21 * q^37 + 3 * q^41 - 3 * q^43 - 18 * q^45 + 3 * q^47 + 18 * q^53 + 36 * q^55 + 24 * q^57 + 6 * q^59 + 30 * q^61 - 24 * q^65 - 18 * q^67 + 15 * q^69 + 12 * q^71 + 12 * q^73 + 33 * q^75 + 12 * q^79 - 9 * q^81 + 18 * q^83 + 6 * q^85 - 24 * q^87 + 3 * q^89 - 6 * q^95 - 3 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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