LMFDB - Embedded newform 4640.2.a.q.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4640,2,Mod(1,4640)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4640.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4640, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4640 = 2^{5} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4640.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,4,0,0,0,-4,0,0,0,-8,0,0,0,-4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	yes
Analytic conductor:	$37.0505865379$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.517638$ of defining polynomial
Character	$\chi$	$=$	4640.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} +1.00000 q^{5} -2.44949 q^{7} -1.00000 q^{9} -1.41421 q^{11} +1.46410 q^{13} +1.41421 q^{15} -2.73205 q^{17} +2.44949 q^{19} -3.46410 q^{21} +3.48477 q^{23} +1.00000 q^{25} -5.65685 q^{27} -1.00000 q^{29} -0.378937 q^{31} -2.00000 q^{33} -2.44949 q^{35} -9.66025 q^{37} +2.07055 q^{39} -2.53590 q^{41} +6.31319 q^{43} -1.00000 q^{45} +0.656339 q^{47} -1.00000 q^{49} -3.86370 q^{51} -1.46410 q^{53} -1.41421 q^{55} +3.46410 q^{57} -5.93426 q^{59} -8.00000 q^{61} +2.44949 q^{63} +1.46410 q^{65} -6.31319 q^{67} +4.92820 q^{69} +10.2784 q^{71} +3.66025 q^{73} +1.41421 q^{75} +3.46410 q^{77} -9.14162 q^{79} -5.00000 q^{81} -6.03579 q^{83} -2.73205 q^{85} -1.41421 q^{87} -12.9282 q^{89} -3.58630 q^{91} -0.535898 q^{93} +2.44949 q^{95} -9.12436 q^{97} +1.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{9} - 8 q^{13} - 4 q^{17} + 4 q^{25} - 4 q^{29} - 8 q^{33} - 4 q^{37} - 24 q^{41} - 4 q^{45} - 4 q^{49} + 8 q^{53} - 32 q^{61} - 8 q^{65} - 8 q^{69} - 20 q^{73} - 20 q^{81} - 4 q^{85}+ \cdots + 12 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^9 - 8 * q^13 - 4 * q^17 + 4 * q^25 - 4 * q^29 - 8 * q^33 - 4 * q^37 - 24 * q^41 - 4 * q^45 - 4 * q^49 + 8 * q^53 - 32 * q^61 - 8 * q^65 - 8 * q^69 - 20 * q^73 - 20 * q^81 - 4 * q^85 - 24 * q^89 - 16 * q^93 + 12 * 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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	0	0
$17$	−2.73205	−0.662620	−0.331310	−	0.943522i	$-0.607491\pi$
$17$	−2.73205	−0.662620	−0.331310	+	0.943522i	$0.607491\pi$
$18$	0	0
$19$	2.44949	0.561951	0.280976	−	0.959715i	$-0.409342\pi$
$19$	2.44949	0.561951	0.280976	+	0.959715i	$0.409342\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	3.48477	0.726624	0.363312	−	0.931668i	$-0.381646\pi$
$23$	3.48477	0.726624	0.363312	+	0.931668i	$0.381646\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.378937	−0.0680592	−0.0340296	−	0.999421i	$-0.510834\pi$
$31$	−0.378937	−0.0680592	−0.0340296	+	0.999421i	$0.510834\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−2.44949	−0.414039
$36$	0	0
$37$	−9.66025	−1.58814	−0.794068	−	0.607829i	$-0.792041\pi$
$37$	−9.66025	−1.58814	−0.794068	+	0.607829i	$0.792041\pi$
$38$	0	0
$39$	2.07055	0.331554
$40$	0	0
$41$	−2.53590	−0.396041	−0.198020	−	0.980198i	$-0.563451\pi$
$41$	−2.53590	−0.396041	−0.198020	+	0.980198i	$0.563451\pi$
$42$	0	0
$43$	6.31319	0.962753	0.481376	−	0.876514i	$-0.340137\pi$
$43$	6.31319	0.962753	0.481376	+	0.876514i	$0.340137\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	0.656339	0.0957369	0.0478684	−	0.998854i	$-0.484757\pi$
$47$	0.656339	0.0957369	0.0478684	+	0.998854i	$0.484757\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−3.86370	−0.541027
$52$	0	0
$53$	−1.46410	−0.201110	−0.100555	−	0.994932i	$-0.532062\pi$
$53$	−1.46410	−0.201110	−0.100555	+	0.994932i	$0.532062\pi$
$54$	0	0
$55$	−1.41421	−0.190693
$56$	0	0
$57$	3.46410	0.458831
$58$	0	0
$59$	−5.93426	−0.772574	−0.386287	−	0.922379i	$-0.626243\pi$
$59$	−5.93426	−0.772574	−0.386287	+	0.922379i	$0.626243\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	2.44949	0.308607
$64$	0	0
$65$	1.46410	0.181599
$66$	0	0
$67$	−6.31319	−0.771279	−0.385640	−	0.922649i	$-0.626019\pi$
$67$	−6.31319	−0.771279	−0.385640	+	0.922649i	$0.626019\pi$
$68$	0	0
$69$	4.92820	0.593286
$70$	0	0
$71$	10.2784	1.21983	0.609913	−	0.792469i	$-0.291204\pi$
$71$	10.2784	1.21983	0.609913	+	0.792469i	$0.291204\pi$
$72$	0	0
$73$	3.66025	0.428400	0.214200	−	0.976790i	$-0.431286\pi$
$73$	3.66025	0.428400	0.214200	+	0.976790i	$0.431286\pi$
$74$	0	0
$75$	1.41421	0.163299
$76$	0	0
$77$	3.46410	0.394771
$78$	0	0
$79$	−9.14162	−1.02851	−0.514256	−	0.857637i	$-0.671932\pi$
$79$	−9.14162	−1.02851	−0.514256	+	0.857637i	$0.671932\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−6.03579	−0.662514	−0.331257	−	0.943541i	$-0.607473\pi$
$83$	−6.03579	−0.662514	−0.331257	+	0.943541i	$0.607473\pi$
$84$	0	0
$85$	−2.73205	−0.296333
$86$	0	0
$87$	−1.41421	−0.151620
$88$	0	0
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	0	0
$91$	−3.58630	−0.375947
$92$	0	0
$93$	−0.535898	−0.0555701
$94$	0	0
$95$	2.44949	0.251312
$96$	0	0
$97$	−9.12436	−0.926438	−0.463219	−	0.886244i	$-0.653306\pi$
$97$	−9.12436	−0.926438	−0.463219	+	0.886244i	$0.653306\pi$
$98$	0	0
$99$	1.41421	0.142134
$100$	0	0
$101$	−0.535898	−0.0533239	−0.0266619	−	0.999645i	$-0.508488\pi$
$101$	−0.535898	−0.0533239	−0.0266619	+	0.999645i	$0.508488\pi$
$102$	0	0
$103$	12.7279	1.25412	0.627060	−	0.778971i	$-0.284258\pi$
$103$	12.7279	1.25412	0.627060	+	0.778971i	$0.284258\pi$
$104$	0	0
$105$	−3.46410	−0.338062
$106$	0	0
$107$	−17.9043	−1.73087	−0.865437	−	0.501018i	$-0.832959\pi$
$107$	−17.9043	−1.73087	−0.865437	+	0.501018i	$0.832959\pi$
$108$	0	0
$109$	2.92820	0.280471	0.140236	−	0.990118i	$-0.455214\pi$
$109$	2.92820	0.280471	0.140236	+	0.990118i	$0.455214\pi$
$110$	0	0
$111$	−13.6617	−1.29671
$112$	0	0
$113$	−14.1962	−1.33546	−0.667731	−	0.744403i	$-0.732734\pi$
$113$	−14.1962	−1.33546	−0.667731	+	0.744403i	$0.732734\pi$
$114$	0	0
$115$	3.48477	0.324956
$116$	0	0
$117$	−1.46410	−0.135356
$118$	0	0
$119$	6.69213	0.613467
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	−3.58630	−0.323366
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.00052	−0.443724	−0.221862	−	0.975078i	$-0.571213\pi$
$127$	−5.00052	−0.443724	−0.221862	+	0.975078i	$0.571213\pi$
$128$	0	0
$129$	8.92820	0.786084
$130$	0	0
$131$	−7.34847	−0.642039	−0.321019	−	0.947073i	$-0.604025\pi$
$131$	−7.34847	−0.642039	−0.321019	+	0.947073i	$0.604025\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	−5.65685	−0.486864
$136$	0	0
$137$	11.1244	0.950418	0.475209	−	0.879873i	$-0.342372\pi$
$137$	11.1244	0.950418	0.475209	+	0.879873i	$0.342372\pi$
$138$	0	0
$139$	−7.72741	−0.655430	−0.327715	−	0.944777i	$-0.606279\pi$
$139$	−7.72741	−0.655430	−0.327715	+	0.944777i	$0.606279\pi$
$140$	0	0
$141$	0.928203	0.0781688
$142$	0	0
$143$	−2.07055	−0.173148
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−1.41421	−0.116642
$148$	0	0
$149$	7.46410	0.611483	0.305742	−	0.952115i	$-0.401096\pi$
$149$	7.46410	0.611483	0.305742	+	0.952115i	$0.401096\pi$
$150$	0	0
$151$	−2.34795	−0.191074	−0.0955369	−	0.995426i	$-0.530457\pi$
$151$	−2.34795	−0.191074	−0.0955369	+	0.995426i	$0.530457\pi$
$152$	0	0
$153$	2.73205	0.220873
$154$	0	0
$155$	−0.378937	−0.0304370
$156$	0	0
$157$	−7.66025	−0.611355	−0.305677	−	0.952135i	$-0.598883\pi$
$157$	−7.66025	−0.611355	−0.305677	+	0.952135i	$0.598883\pi$
$158$	0	0
$159$	−2.07055	−0.164205
$160$	0	0
$161$	−8.53590	−0.672723
$162$	0	0
$163$	−4.24264	−0.332309	−0.166155	−	0.986100i	$-0.553135\pi$
$163$	−4.24264	−0.332309	−0.166155	+	0.986100i	$0.553135\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	7.07107	0.547176	0.273588	−	0.961847i	$-0.411790\pi$
$167$	7.07107	0.547176	0.273588	+	0.961847i	$0.411790\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	−2.44949	−0.187317
$172$	0	0
$173$	−5.85641	−0.445254	−0.222627	−	0.974904i	$-0.571463\pi$
$173$	−5.85641	−0.445254	−0.222627	+	0.974904i	$0.571463\pi$
$174$	0	0
$175$	−2.44949	−0.185164
$176$	0	0
$177$	−8.39230	−0.630804
$178$	0	0
$179$	4.34418	0.324699	0.162350	−	0.986733i	$-0.448093\pi$
$179$	4.34418	0.324699	0.162350	+	0.986733i	$0.448093\pi$
$180$	0	0
$181$	6.53590	0.485810	0.242905	−	0.970050i	$-0.421900\pi$
$181$	6.53590	0.485810	0.242905	+	0.970050i	$0.421900\pi$
$182$	0	0
$183$	−11.3137	−0.836333
$184$	0	0
$185$	−9.66025	−0.710236
$186$	0	0
$187$	3.86370	0.282542
$188$	0	0
$189$	13.8564	1.00791
$190$	0	0
$191$	−7.07107	−0.511645	−0.255822	−	0.966724i	$-0.582346\pi$
$191$	−7.07107	−0.511645	−0.255822	+	0.966724i	$0.582346\pi$
$192$	0	0
$193$	−14.7321	−1.06044	−0.530218	−	0.847861i	$-0.677890\pi$
$193$	−14.7321	−1.06044	−0.530218	+	0.847861i	$0.677890\pi$
$194$	0	0
$195$	2.07055	0.148275
$196$	0	0
$197$	15.8564	1.12972	0.564861	−	0.825186i	$-0.308930\pi$
$197$	15.8564	1.12972	0.564861	+	0.825186i	$0.308930\pi$
$198$	0	0
$199$	20.6312	1.46251	0.731253	−	0.682106i	$-0.238936\pi$
$199$	20.6312	1.46251	0.731253	+	0.682106i	$0.238936\pi$
$200$	0	0
$201$	−8.92820	−0.629747
$202$	0	0
$203$	2.44949	0.171920
$204$	0	0
$205$	−2.53590	−0.177115
$206$	0	0
$207$	−3.48477	−0.242208
$208$	0	0
$209$	−3.46410	−0.239617
$210$	0	0
$211$	11.2122	0.771878	0.385939	−	0.922524i	$-0.373878\pi$
$211$	11.2122	0.771878	0.385939	+	0.922524i	$0.373878\pi$
$212$	0	0
$213$	14.5359	0.995983
$214$	0	0
$215$	6.31319	0.430556
$216$	0	0
$217$	0.928203	0.0630105
$218$	0	0
$219$	5.17638	0.349787
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	0.101536	0.00679935	0.00339968	−	0.999994i	$-0.498918\pi$
$223$	0.101536	0.00679935	0.00339968	+	0.999994i	$0.498918\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	5.55532	0.368719	0.184360	−	0.982859i	$-0.440979\pi$
$227$	5.55532	0.368719	0.184360	+	0.982859i	$0.440979\pi$
$228$	0	0
$229$	−14.7846	−0.976995	−0.488497	−	0.872565i	$-0.662455\pi$
$229$	−14.7846	−0.976995	−0.488497	+	0.872565i	$0.662455\pi$
$230$	0	0
$231$	4.89898	0.322329
$232$	0	0
$233$	−8.53590	−0.559205	−0.279603	−	0.960116i	$-0.590203\pi$
$233$	−8.53590	−0.559205	−0.279603	+	0.960116i	$0.590203\pi$
$234$	0	0
$235$	0.656339	0.0428148
$236$	0	0
$237$	−12.9282	−0.839777
$238$	0	0
$239$	3.30890	0.214035	0.107017	−	0.994257i	$-0.465870\pi$
$239$	3.30890	0.214035	0.107017	+	0.994257i	$0.465870\pi$
$240$	0	0
$241$	−12.9282	−0.832779	−0.416389	−	0.909186i	$-0.636705\pi$
$241$	−12.9282	−0.832779	−0.416389	+	0.909186i	$0.636705\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	3.58630	0.228191
$248$	0	0
$249$	−8.53590	−0.540941
$250$	0	0
$251$	14.7985	0.934071	0.467036	−	0.884238i	$-0.345322\pi$
$251$	14.7985	0.934071	0.467036	+	0.884238i	$0.345322\pi$
$252$	0	0
$253$	−4.92820	−0.309833
$254$	0	0
$255$	−3.86370	−0.241954
$256$	0	0
$257$	16.7846	1.04700	0.523498	−	0.852027i	$-0.324627\pi$
$257$	16.7846	1.04700	0.523498	+	0.852027i	$0.324627\pi$
$258$	0	0
$259$	23.6627	1.47033
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	13.4858	0.831570	0.415785	−	0.909463i	$-0.363507\pi$
$263$	13.4858	0.831570	0.415785	+	0.909463i	$0.363507\pi$
$264$	0	0
$265$	−1.46410	−0.0899390
$266$	0	0
$267$	−18.2832	−1.11892
$268$	0	0
$269$	−1.46410	−0.0892679	−0.0446339	−	0.999003i	$-0.514212\pi$
$269$	−1.46410	−0.0892679	−0.0446339	+	0.999003i	$0.514212\pi$
$270$	0	0
$271$	19.6975	1.19654	0.598268	−	0.801296i	$-0.295856\pi$
$271$	19.6975	1.19654	0.598268	+	0.801296i	$0.295856\pi$
$272$	0	0
$273$	−5.07180	−0.306959
$274$	0	0
$275$	−1.41421	−0.0852803
$276$	0	0
$277$	15.3205	0.920520	0.460260	−	0.887784i	$-0.347756\pi$
$277$	15.3205	0.920520	0.460260	+	0.887784i	$0.347756\pi$
$278$	0	0
$279$	0.378937	0.0226864
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	25.3543	1.50716	0.753579	−	0.657358i	$-0.228326\pi$
$283$	25.3543	1.50716	0.753579	+	0.657358i	$0.228326\pi$
$284$	0	0
$285$	3.46410	0.205196
$286$	0	0
$287$	6.21166	0.366663
$288$	0	0
$289$	−9.53590	−0.560935
$290$	0	0
$291$	−12.9038	−0.756433
$292$	0	0
$293$	4.73205	0.276449	0.138225	−	0.990401i	$-0.455860\pi$
$293$	4.73205	0.276449	0.138225	+	0.990401i	$0.455860\pi$
$294$	0	0
$295$	−5.93426	−0.345506
$296$	0	0
$297$	8.00000	0.464207
$298$	0	0
$299$	5.10205	0.295059
$300$	0	0
$301$	−15.4641	−0.891336
$302$	0	0
$303$	−0.757875	−0.0435388
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−26.1122	−1.49030	−0.745150	−	0.666896i	$-0.767622\pi$
$307$	−26.1122	−1.49030	−0.745150	+	0.666896i	$0.767622\pi$
$308$	0	0
$309$	18.0000	1.02398
$310$	0	0
$311$	−10.1769	−0.577079	−0.288539	−	0.957468i	$-0.593170\pi$
$311$	−10.1769	−0.577079	−0.288539	+	0.957468i	$0.593170\pi$
$312$	0	0
$313$	−8.39230	−0.474361	−0.237181	−	0.971466i	$-0.576223\pi$
$313$	−8.39230	−0.474361	−0.237181	+	0.971466i	$0.576223\pi$
$314$	0	0
$315$	2.44949	0.138013
$316$	0	0
$317$	−20.5885	−1.15636	−0.578181	−	0.815908i	$-0.696237\pi$
$317$	−20.5885	−1.15636	−0.578181	+	0.815908i	$0.696237\pi$
$318$	0	0
$319$	1.41421	0.0791808
$320$	0	0
$321$	−25.3205	−1.41325
$322$	0	0
$323$	−6.69213	−0.372360
$324$	0	0
$325$	1.46410	0.0812137
$326$	0	0
$327$	4.14110	0.229004
$328$	0	0
$329$	−1.60770	−0.0886351
$330$	0	0
$331$	21.2875	1.17007	0.585034	−	0.811009i	$-0.301081\pi$
$331$	21.2875	1.17007	0.585034	+	0.811009i	$0.301081\pi$
$332$	0	0
$333$	9.66025	0.529379
$334$	0	0
$335$	−6.31319	−0.344927
$336$	0	0
$337$	26.5885	1.44837	0.724183	−	0.689608i	$-0.242217\pi$
$337$	26.5885	1.44837	0.724183	+	0.689608i	$0.242217\pi$
$338$	0	0
$339$	−20.0764	−1.09040
$340$	0	0
$341$	0.535898	0.0290205
$342$	0	0
$343$	19.5959	1.05808
$344$	0	0
$345$	4.92820	0.265326
$346$	0	0
$347$	−3.96524	−0.212865	−0.106433	−	0.994320i	$-0.533943\pi$
$347$	−3.96524	−0.212865	−0.106433	+	0.994320i	$0.533943\pi$
$348$	0	0
$349$	−7.85641	−0.420544	−0.210272	−	0.977643i	$-0.567435\pi$
$349$	−7.85641	−0.420544	−0.210272	+	0.977643i	$0.567435\pi$
$350$	0	0
$351$	−8.28221	−0.442072
$352$	0	0
$353$	7.32051	0.389631	0.194816	−	0.980840i	$-0.437589\pi$
$353$	7.32051	0.389631	0.194816	+	0.980840i	$0.437589\pi$
$354$	0	0
$355$	10.2784	0.545523
$356$	0	0
$357$	9.46410	0.500893
$358$	0	0
$359$	−3.96524	−0.209277	−0.104639	−	0.994510i	$-0.533369\pi$
$359$	−3.96524	−0.209277	−0.104639	+	0.994510i	$0.533369\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	−12.7279	−0.668043
$364$	0	0
$365$	3.66025	0.191586
$366$	0	0
$367$	−7.62587	−0.398067	−0.199034	−	0.979993i	$-0.563780\pi$
$367$	−7.62587	−0.398067	−0.199034	+	0.979993i	$0.563780\pi$
$368$	0	0
$369$	2.53590	0.132014
$370$	0	0
$371$	3.58630	0.186192
$372$	0	0
$373$	16.2487	0.841326	0.420663	−	0.907217i	$-0.361797\pi$
$373$	16.2487	0.841326	0.420663	+	0.907217i	$0.361797\pi$
$374$	0	0
$375$	1.41421	0.0730297
$376$	0	0
$377$	−1.46410	−0.0754051
$378$	0	0
$379$	28.1827	1.44765	0.723825	−	0.689984i	$-0.242382\pi$
$379$	28.1827	1.44765	0.723825	+	0.689984i	$0.242382\pi$
$380$	0	0
$381$	−7.07180	−0.362299
$382$	0	0
$383$	18.6622	0.953593	0.476796	−	0.879014i	$-0.341798\pi$
$383$	18.6622	0.953593	0.476796	+	0.879014i	$0.341798\pi$
$384$	0	0
$385$	3.46410	0.176547
$386$	0	0
$387$	−6.31319	−0.320918
$388$	0	0
$389$	13.4641	0.682657	0.341329	−	0.939944i	$-0.389123\pi$
$389$	13.4641	0.682657	0.341329	+	0.939944i	$0.389123\pi$
$390$	0	0
$391$	−9.52056	−0.481475
$392$	0	0
$393$	−10.3923	−0.524222
$394$	0	0
$395$	−9.14162	−0.459965
$396$	0	0
$397$	19.8564	0.996564	0.498282	−	0.867015i	$-0.333964\pi$
$397$	19.8564	0.996564	0.498282	+	0.867015i	$0.333964\pi$
$398$	0	0
$399$	−8.48528	−0.424795
$400$	0	0
$401$	−0.679492	−0.0339322	−0.0169661	−	0.999856i	$-0.505401\pi$
$401$	−0.679492	−0.0339322	−0.0169661	+	0.999856i	$0.505401\pi$
$402$	0	0
$403$	−0.554803	−0.0276367
$404$	0	0
$405$	−5.00000	−0.248452
$406$	0	0
$407$	13.6617	0.677183
$408$	0	0
$409$	20.7846	1.02773	0.513866	−	0.857870i	$-0.328213\pi$
$409$	20.7846	1.02773	0.513866	+	0.857870i	$0.328213\pi$
$410$	0	0
$411$	15.7322	0.776013
$412$	0	0
$413$	14.5359	0.715265
$414$	0	0
$415$	−6.03579	−0.296285
$416$	0	0
$417$	−10.9282	−0.535156
$418$	0	0
$419$	35.5312	1.73581	0.867906	−	0.496728i	$-0.165465\pi$
$419$	35.5312	1.73581	0.867906	+	0.496728i	$0.165465\pi$
$420$	0	0
$421$	−8.39230	−0.409016	−0.204508	−	0.978865i	$-0.565559\pi$
$421$	−8.39230	−0.409016	−0.204508	+	0.978865i	$0.565559\pi$
$422$	0	0
$423$	−0.656339	−0.0319123
$424$	0	0
$425$	−2.73205	−0.132524
$426$	0	0
$427$	19.5959	0.948313
$428$	0	0
$429$	−2.92820	−0.141375
$430$	0	0
$431$	−14.9000	−0.717708	−0.358854	−	0.933394i	$-0.616832\pi$
$431$	−14.9000	−0.717708	−0.358854	+	0.933394i	$0.616832\pi$
$432$	0	0
$433$	−24.1962	−1.16279	−0.581396	−	0.813620i	$-0.697493\pi$
$433$	−24.1962	−1.16279	−0.581396	+	0.813620i	$0.697493\pi$
$434$	0	0
$435$	−1.41421	−0.0678064
$436$	0	0
$437$	8.53590	0.408327
$438$	0	0
$439$	3.10583	0.148233	0.0741166	−	0.997250i	$-0.476386\pi$
$439$	3.10583	0.148233	0.0741166	+	0.997250i	$0.476386\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	9.89949	0.470339	0.235170	−	0.971954i	$-0.424435\pi$
$443$	9.89949	0.470339	0.235170	+	0.971954i	$0.424435\pi$
$444$	0	0
$445$	−12.9282	−0.612856
$446$	0	0
$447$	10.5558	0.499274
$448$	0	0
$449$	−30.9282	−1.45959	−0.729796	−	0.683665i	$-0.760385\pi$
$449$	−30.9282	−1.45959	−0.729796	+	0.683665i	$0.760385\pi$
$450$	0	0
$451$	3.58630	0.168872
$452$	0	0
$453$	−3.32051	−0.156011
$454$	0	0
$455$	−3.58630	−0.168128
$456$	0	0
$457$	−5.46410	−0.255600	−0.127800	−	0.991800i	$-0.540792\pi$
$457$	−5.46410	−0.255600	−0.127800	+	0.991800i	$0.540792\pi$
$458$	0	0
$459$	15.4548	0.721369
$460$	0	0
$461$	−29.3205	−1.36559	−0.682796	−	0.730609i	$-0.739236\pi$
$461$	−29.3205	−1.36559	−0.682796	+	0.730609i	$0.739236\pi$
$462$	0	0
$463$	−6.51626	−0.302837	−0.151418	−	0.988470i	$-0.548384\pi$
$463$	−6.51626	−0.302837	−0.151418	+	0.988470i	$0.548384\pi$
$464$	0	0
$465$	−0.535898	−0.0248517
$466$	0	0
$467$	14.7985	0.684792	0.342396	−	0.939556i	$-0.388762\pi$
$467$	14.7985	0.684792	0.342396	+	0.939556i	$0.388762\pi$
$468$	0	0
$469$	15.4641	0.714066
$470$	0	0
$471$	−10.8332	−0.499169
$472$	0	0
$473$	−8.92820	−0.410519
$474$	0	0
$475$	2.44949	0.112390
$476$	0	0
$477$	1.46410	0.0670366
$478$	0	0
$479$	−24.5964	−1.12384	−0.561920	−	0.827192i	$-0.689937\pi$
$479$	−24.5964	−1.12384	−0.561920	+	0.827192i	$0.689937\pi$
$480$	0	0
$481$	−14.1436	−0.644892
$482$	0	0
$483$	−12.0716	−0.549276
$484$	0	0
$485$	−9.12436	−0.414316
$486$	0	0
$487$	−11.4896	−0.520642	−0.260321	−	0.965522i	$-0.583828\pi$
$487$	−11.4896	−0.520642	−0.260321	+	0.965522i	$0.583828\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	0	0
$491$	−29.9759	−1.35279	−0.676396	−	0.736538i	$-0.736459\pi$
$491$	−29.9759	−1.35279	−0.676396	+	0.736538i	$0.736459\pi$
$492$	0	0
$493$	2.73205	0.123045
$494$	0	0
$495$	1.41421	0.0635642
$496$	0	0
$497$	−25.1769	−1.12934
$498$	0	0
$499$	30.9096	1.38370	0.691852	−	0.722039i	$-0.256795\pi$
$499$	30.9096	1.38370	0.691852	+	0.722039i	$0.256795\pi$
$500$	0	0
$501$	10.0000	0.446767
$502$	0	0
$503$	17.0721	0.761207	0.380604	−	0.924738i	$-0.375716\pi$
$503$	17.0721	0.761207	0.380604	+	0.924738i	$0.375716\pi$
$504$	0	0
$505$	−0.535898	−0.0238472
$506$	0	0
$507$	−15.3533	−0.681863
$508$	0	0
$509$	−5.85641	−0.259581	−0.129790	−	0.991541i	$-0.541430\pi$
$509$	−5.85641	−0.259581	−0.129790	+	0.991541i	$0.541430\pi$
$510$	0	0
$511$	−8.96575	−0.396622
$512$	0	0
$513$	−13.8564	−0.611775
$514$	0	0
$515$	12.7279	0.560859
$516$	0	0
$517$	−0.928203	−0.0408223
$518$	0	0
$519$	−8.28221	−0.363549
$520$	0	0
$521$	20.7846	0.910590	0.455295	−	0.890341i	$-0.349534\pi$
$521$	20.7846	0.910590	0.455295	+	0.890341i	$0.349534\pi$
$522$	0	0
$523$	−43.4345	−1.89926	−0.949629	−	0.313378i	$-0.898539\pi$
$523$	−43.4345	−1.89926	−0.949629	+	0.313378i	$0.898539\pi$
$524$	0	0
$525$	−3.46410	−0.151186
$526$	0	0
$527$	1.03528	0.0450973
$528$	0	0
$529$	−10.8564	−0.472018
$530$	0	0
$531$	5.93426	0.257525
$532$	0	0
$533$	−3.71281	−0.160820
$534$	0	0
$535$	−17.9043	−0.774071
$536$	0	0
$537$	6.14359	0.265116
$538$	0	0
$539$	1.41421	0.0609145
$540$	0	0
$541$	13.7128	0.589560	0.294780	−	0.955565i	$-0.404754\pi$
$541$	13.7128	0.589560	0.294780	+	0.955565i	$0.404754\pi$
$542$	0	0
$543$	9.24316	0.396662
$544$	0	0
$545$	2.92820	0.125430
$546$	0	0
$547$	−13.5601	−0.579789	−0.289895	−	0.957059i	$-0.593620\pi$
$547$	−13.5601	−0.579789	−0.289895	+	0.957059i	$0.593620\pi$
$548$	0	0
$549$	8.00000	0.341432
$550$	0	0
$551$	−2.44949	−0.104352
$552$	0	0
$553$	22.3923	0.952218
$554$	0	0
$555$	−13.6617	−0.579905
$556$	0	0
$557$	29.3205	1.24235	0.621175	−	0.783672i	$-0.286656\pi$
$557$	29.3205	1.24235	0.621175	+	0.783672i	$0.286656\pi$
$558$	0	0
$559$	9.24316	0.390944
$560$	0	0
$561$	5.46410	0.230695
$562$	0	0
$563$	9.89949	0.417214	0.208607	−	0.978000i	$-0.433107\pi$
$563$	9.89949	0.417214	0.208607	+	0.978000i	$0.433107\pi$
$564$	0	0
$565$	−14.1962	−0.597237
$566$	0	0
$567$	12.2474	0.514344
$568$	0	0
$569$	−36.1051	−1.51361	−0.756803	−	0.653643i	$-0.773240\pi$
$569$	−36.1051	−1.51361	−0.756803	+	0.653643i	$0.773240\pi$
$570$	0	0
$571$	17.5254	0.733414	0.366707	−	0.930337i	$-0.380485\pi$
$571$	17.5254	0.733414	0.366707	+	0.930337i	$0.380485\pi$
$572$	0	0
$573$	−10.0000	−0.417756
$574$	0	0
$575$	3.48477	0.145325
$576$	0	0
$577$	7.12436	0.296591	0.148295	−	0.988943i	$-0.452621\pi$
$577$	7.12436	0.296591	0.148295	+	0.988943i	$0.452621\pi$
$578$	0	0
$579$	−20.8343	−0.865843
$580$	0	0
$581$	14.7846	0.613369
$582$	0	0
$583$	2.07055	0.0857535
$584$	0	0
$585$	−1.46410	−0.0605332
$586$	0	0
$587$	11.9700	0.494057	0.247028	−	0.969008i	$-0.420546\pi$
$587$	11.9700	0.494057	0.247028	+	0.969008i	$0.420546\pi$
$588$	0	0
$589$	−0.928203	−0.0382459
$590$	0	0
$591$	22.4243	0.922414
$592$	0	0
$593$	9.07180	0.372534	0.186267	−	0.982499i	$-0.440361\pi$
$593$	9.07180	0.372534	0.186267	+	0.982499i	$0.440361\pi$
$594$	0	0
$595$	6.69213	0.274351
$596$	0	0
$597$	29.1769	1.19413
$598$	0	0
$599$	−3.68784	−0.150681	−0.0753405	−	0.997158i	$-0.524004\pi$
$599$	−3.68784	−0.150681	−0.0753405	+	0.997158i	$0.524004\pi$
$600$	0	0
$601$	−17.4641	−0.712376	−0.356188	−	0.934414i	$-0.615924\pi$
$601$	−17.4641	−0.712376	−0.356188	+	0.934414i	$0.615924\pi$
$602$	0	0
$603$	6.31319	0.257093
$604$	0	0
$605$	−9.00000	−0.365902
$606$	0	0
$607$	−43.0827	−1.74867	−0.874337	−	0.485319i	$-0.838704\pi$
$607$	−43.0827	−1.74867	−0.874337	+	0.485319i	$0.838704\pi$
$608$	0	0
$609$	3.46410	0.140372
$610$	0	0
$611$	0.960947	0.0388757
$612$	0	0
$613$	−15.0718	−0.608744	−0.304372	−	0.952553i	$-0.598447\pi$
$613$	−15.0718	−0.608744	−0.304372	+	0.952553i	$0.598447\pi$
$614$	0	0
$615$	−3.58630	−0.144614
$616$	0	0
$617$	18.8756	0.759905	0.379952	−	0.925006i	$-0.375940\pi$
$617$	18.8756	0.759905	0.379952	+	0.925006i	$0.375940\pi$
$618$	0	0
$619$	−23.0807	−0.927691	−0.463846	−	0.885916i	$-0.653531\pi$
$619$	−23.0807	−0.927691	−0.463846	+	0.885916i	$0.653531\pi$
$620$	0	0
$621$	−19.7128	−0.791048
$622$	0	0
$623$	31.6675	1.26873
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.89898	−0.195646
$628$	0	0
$629$	26.3923	1.05233
$630$	0	0
$631$	−5.65685	−0.225196	−0.112598	−	0.993641i	$-0.535917\pi$
$631$	−5.65685	−0.225196	−0.112598	+	0.993641i	$0.535917\pi$
$632$	0	0
$633$	15.8564	0.630236
$634$	0	0
$635$	−5.00052	−0.198439
$636$	0	0
$637$	−1.46410	−0.0580098
$638$	0	0
$639$	−10.2784	−0.406608
$640$	0	0
$641$	1.46410	0.0578285	0.0289143	−	0.999582i	$-0.490795\pi$
$641$	1.46410	0.0578285	0.0289143	+	0.999582i	$0.490795\pi$
$642$	0	0
$643$	−25.1512	−0.991868	−0.495934	−	0.868360i	$-0.665174\pi$
$643$	−25.1512	−0.991868	−0.495934	+	0.868360i	$0.665174\pi$
$644$	0	0
$645$	8.92820	0.351548
$646$	0	0
$647$	−34.1170	−1.34128	−0.670639	−	0.741784i	$-0.733980\pi$
$647$	−34.1170	−1.34128	−0.670639	+	0.741784i	$0.733980\pi$
$648$	0	0
$649$	8.39230	0.329427
$650$	0	0
$651$	1.31268	0.0514479
$652$	0	0
$653$	35.6603	1.39549	0.697747	−	0.716344i	$-0.254186\pi$
$653$	35.6603	1.39549	0.697747	+	0.716344i	$0.254186\pi$
$654$	0	0
$655$	−7.34847	−0.287128
$656$	0	0
$657$	−3.66025	−0.142800
$658$	0	0
$659$	4.52004	0.176076	0.0880379	−	0.996117i	$-0.471940\pi$
$659$	4.52004	0.176076	0.0880379	+	0.996117i	$0.471940\pi$
$660$	0	0
$661$	−13.0718	−0.508434	−0.254217	−	0.967147i	$-0.581818\pi$
$661$	−13.0718	−0.508434	−0.254217	+	0.967147i	$0.581818\pi$
$662$	0	0
$663$	−5.65685	−0.219694
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	−3.48477	−0.134931
$668$	0	0
$669$	0.143594	0.00555165
$670$	0	0
$671$	11.3137	0.436761
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	0	0
$675$	−5.65685	−0.217732
$676$	0	0
$677$	16.7321	0.643065	0.321532	−	0.946899i	$-0.395802\pi$
$677$	16.7321	0.643065	0.321532	+	0.946899i	$0.395802\pi$
$678$	0	0
$679$	22.3500	0.857715
$680$	0	0
$681$	7.85641	0.301058
$682$	0	0
$683$	−25.3543	−0.970156	−0.485078	−	0.874471i	$-0.661209\pi$
$683$	−25.3543	−0.970156	−0.485078	+	0.874471i	$0.661209\pi$
$684$	0	0
$685$	11.1244	0.425040
$686$	0	0
$687$	−20.9086	−0.797713
$688$	0	0
$689$	−2.14359	−0.0816644
$690$	0	0
$691$	10.0754	0.383285	0.191642	−	0.981465i	$-0.438619\pi$
$691$	10.0754	0.383285	0.191642	+	0.981465i	$0.438619\pi$
$692$	0	0
$693$	−3.46410	−0.131590
$694$	0	0
$695$	−7.72741	−0.293117
$696$	0	0
$697$	6.92820	0.262424
$698$	0	0
$699$	−12.0716	−0.456589
$700$	0	0
$701$	−5.07180	−0.191559	−0.0957796	−	0.995403i	$-0.530534\pi$
$701$	−5.07180	−0.191559	−0.0957796	+	0.995403i	$0.530534\pi$
$702$	0	0
$703$	−23.6627	−0.892455
$704$	0	0
$705$	0.928203	0.0349582
$706$	0	0
$707$	1.31268	0.0493683
$708$	0	0
$709$	38.6410	1.45119	0.725597	−	0.688120i	$-0.241564\pi$
$709$	38.6410	1.45119	0.725597	+	0.688120i	$0.241564\pi$
$710$	0	0
$711$	9.14162	0.342838
$712$	0	0
$713$	−1.32051	−0.0494534
$714$	0	0
$715$	−2.07055	−0.0774343
$716$	0	0
$717$	4.67949	0.174759
$718$	0	0
$719$	−9.72363	−0.362630	−0.181315	−	0.983425i	$-0.558035\pi$
$719$	−9.72363	−0.362630	−0.181315	+	0.983425i	$0.558035\pi$
$720$	0	0
$721$	−31.1769	−1.16109
$722$	0	0
$723$	−18.2832	−0.679961
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	34.9492	1.29619	0.648097	−	0.761558i	$-0.275565\pi$
$727$	34.9492	1.29619	0.648097	+	0.761558i	$0.275565\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−17.2480	−0.637939
$732$	0	0
$733$	16.9808	0.627199	0.313599	−	0.949555i	$-0.398465\pi$
$733$	16.9808	0.627199	0.313599	+	0.949555i	$0.398465\pi$
$734$	0	0
$735$	−1.41421	−0.0521641
$736$	0	0
$737$	8.92820	0.328875
$738$	0	0
$739$	18.8652	0.693969	0.346985	−	0.937871i	$-0.387206\pi$
$739$	18.8652	0.693969	0.346985	+	0.937871i	$0.387206\pi$
$740$	0	0
$741$	5.07180	0.186317
$742$	0	0
$743$	−32.1208	−1.17840	−0.589198	−	0.807988i	$-0.700556\pi$
$743$	−32.1208	−1.17840	−0.589198	+	0.807988i	$0.700556\pi$
$744$	0	0
$745$	7.46410	0.273464
$746$	0	0
$747$	6.03579	0.220838
$748$	0	0
$749$	43.8564	1.60248
$750$	0	0
$751$	21.6937	0.791614	0.395807	−	0.918334i	$-0.370465\pi$
$751$	21.6937	0.791614	0.395807	+	0.918334i	$0.370465\pi$
$752$	0	0
$753$	20.9282	0.762666
$754$	0	0
$755$	−2.34795	−0.0854508
$756$	0	0
$757$	3.26795	0.118776	0.0593878	−	0.998235i	$-0.481085\pi$
$757$	3.26795	0.118776	0.0593878	+	0.998235i	$0.481085\pi$
$758$	0	0
$759$	−6.96953	−0.252978
$760$	0	0
$761$	−24.6410	−0.893236	−0.446618	−	0.894725i	$-0.647372\pi$
$761$	−24.6410	−0.893236	−0.446618	+	0.894725i	$0.647372\pi$
$762$	0	0
$763$	−7.17260	−0.259666
$764$	0	0
$765$	2.73205	0.0987775
$766$	0	0
$767$	−8.68835	−0.313718
$768$	0	0
$769$	6.39230	0.230512	0.115256	−	0.993336i	$-0.463231\pi$
$769$	6.39230	0.230512	0.115256	+	0.993336i	$0.463231\pi$
$770$	0	0
$771$	23.7370	0.854868
$772$	0	0
$773$	48.5885	1.74761	0.873803	−	0.486281i	$-0.161647\pi$
$773$	48.5885	1.74761	0.873803	+	0.486281i	$0.161647\pi$
$774$	0	0
$775$	−0.378937	−0.0136118
$776$	0	0
$777$	33.4641	1.20052
$778$	0	0
$779$	−6.21166	−0.222556
$780$	0	0
$781$	−14.5359	−0.520135
$782$	0	0
$783$	5.65685	0.202159
$784$	0	0
$785$	−7.66025	−0.273406
$786$	0	0
$787$	22.6002	0.805611	0.402805	−	0.915286i	$-0.368035\pi$
$787$	22.6002	0.805611	0.402805	+	0.915286i	$0.368035\pi$
$788$	0	0
$789$	19.0718	0.678974
$790$	0	0
$791$	34.7733	1.23640
$792$	0	0
$793$	−11.7128	−0.415934
$794$	0	0
$795$	−2.07055	−0.0734349
$796$	0	0
$797$	−37.5167	−1.32891	−0.664454	−	0.747329i	$-0.731336\pi$
$797$	−37.5167	−1.32891	−0.664454	+	0.747329i	$0.731336\pi$
$798$	0	0
$799$	−1.79315	−0.0634371
$800$	0	0
$801$	12.9282	0.456796
$802$	0	0
$803$	−5.17638	−0.182671
$804$	0	0
$805$	−8.53590	−0.300851
$806$	0	0
$807$	−2.07055	−0.0728869
$808$	0	0
$809$	−0.143594	−0.00504848	−0.00252424	−	0.999997i	$-0.500803\pi$
$809$	−0.143594	−0.00504848	−0.00252424	+	0.999997i	$0.500803\pi$
$810$	0	0
$811$	−28.0812	−0.986064	−0.493032	−	0.870011i	$-0.664112\pi$
$811$	−28.0812	−0.986064	−0.493032	+	0.870011i	$0.664112\pi$
$812$	0	0
$813$	27.8564	0.976967
$814$	0	0
$815$	−4.24264	−0.148613
$816$	0	0
$817$	15.4641	0.541020
$818$	0	0
$819$	3.58630	0.125316
$820$	0	0
$821$	−12.2487	−0.427483	−0.213741	−	0.976890i	$-0.568565\pi$
$821$	−12.2487	−0.427483	−0.213741	+	0.976890i	$0.568565\pi$
$822$	0	0
$823$	1.96902	0.0686356	0.0343178	−	0.999411i	$-0.489074\pi$
$823$	1.96902	0.0686356	0.0343178	+	0.999411i	$0.489074\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	−16.6660	−0.579532	−0.289766	−	0.957098i	$-0.593577\pi$
$827$	−16.6660	−0.579532	−0.289766	+	0.957098i	$0.593577\pi$
$828$	0	0
$829$	−36.3923	−1.26396	−0.631978	−	0.774986i	$-0.717757\pi$
$829$	−36.3923	−1.26396	−0.631978	+	0.774986i	$0.717757\pi$
$830$	0	0
$831$	21.6665	0.751602
$832$	0	0
$833$	2.73205	0.0946600
$834$	0	0
$835$	7.07107	0.244704
$836$	0	0
$837$	2.14359	0.0740934
$838$	0	0
$839$	−33.0817	−1.14211	−0.571054	−	0.820913i	$-0.693465\pi$
$839$	−33.0817	−1.14211	−0.571054	+	0.820913i	$0.693465\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−16.9706	−0.584497
$844$	0	0
$845$	−10.8564	−0.373472
$846$	0	0
$847$	22.0454	0.757489
$848$	0	0
$849$	35.8564	1.23059
$850$	0	0
$851$	−33.6637	−1.15398
$852$	0	0
$853$	2.05256	0.0702783	0.0351391	−	0.999382i	$-0.488813\pi$
$853$	2.05256	0.0702783	0.0351391	+	0.999382i	$0.488813\pi$
$854$	0	0
$855$	−2.44949	−0.0837708
$856$	0	0
$857$	28.0000	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$857$	28.0000	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$858$	0	0
$859$	−7.07107	−0.241262	−0.120631	−	0.992697i	$-0.538492\pi$
$859$	−7.07107	−0.241262	−0.120631	+	0.992697i	$0.538492\pi$
$860$	0	0
$861$	8.78461	0.299379
$862$	0	0
$863$	−42.0475	−1.43131	−0.715656	−	0.698453i	$-0.753872\pi$
$863$	−42.0475	−1.43131	−0.715656	+	0.698453i	$0.753872\pi$
$864$	0	0
$865$	−5.85641	−0.199124
$866$	0	0
$867$	−13.4858	−0.458002
$868$	0	0
$869$	12.9282	0.438559
$870$	0	0
$871$	−9.24316	−0.313192
$872$	0	0
$873$	9.12436	0.308813
$874$	0	0
$875$	−2.44949	−0.0828079
$876$	0	0
$877$	21.4641	0.724791	0.362396	−	0.932024i	$-0.381959\pi$
$877$	21.4641	0.724791	0.362396	+	0.932024i	$0.381959\pi$
$878$	0	0
$879$	6.69213	0.225720
$880$	0	0
$881$	22.6410	0.762795	0.381398	−	0.924411i	$-0.375443\pi$
$881$	22.6410	0.762795	0.381398	+	0.924411i	$0.375443\pi$
$882$	0	0
$883$	20.7327	0.697712	0.348856	−	0.937176i	$-0.386570\pi$
$883$	20.7327	0.697712	0.348856	+	0.937176i	$0.386570\pi$
$884$	0	0
$885$	−8.39230	−0.282104
$886$	0	0
$887$	16.3142	0.547778	0.273889	−	0.961761i	$-0.411690\pi$
$887$	16.3142	0.547778	0.273889	+	0.961761i	$0.411690\pi$
$888$	0	0
$889$	12.2487	0.410809
$890$	0	0
$891$	7.07107	0.236890
$892$	0	0
$893$	1.60770	0.0537995
$894$	0	0
$895$	4.34418	0.145210
$896$	0	0
$897$	7.21539	0.240915
$898$	0	0
$899$	0.378937	0.0126383
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	−21.8695	−0.727773
$904$	0	0
$905$	6.53590	0.217261
$906$	0	0
$907$	23.4868	0.779867	0.389934	−	0.920843i	$-0.372498\pi$
$907$	23.4868	0.779867	0.389934	+	0.920843i	$0.372498\pi$
$908$	0	0
$909$	0.535898	0.0177746
$910$	0	0
$911$	44.4698	1.47335	0.736674	−	0.676248i	$-0.236395\pi$
$911$	44.4698	1.47335	0.736674	+	0.676248i	$0.236395\pi$
$912$	0	0
$913$	8.53590	0.282497
$914$	0	0
$915$	−11.3137	−0.374020
$916$	0	0
$917$	18.0000	0.594412
$918$	0	0
$919$	0.832204	0.0274519	0.0137259	−	0.999906i	$-0.495631\pi$
$919$	0.832204	0.0274519	0.0137259	+	0.999906i	$0.495631\pi$
$920$	0	0
$921$	−36.9282	−1.21683
$922$	0	0
$923$	15.0487	0.495333
$924$	0	0
$925$	−9.66025	−0.317627
$926$	0	0
$927$	−12.7279	−0.418040
$928$	0	0
$929$	10.3923	0.340960	0.170480	−	0.985361i	$-0.445468\pi$
$929$	10.3923	0.340960	0.170480	+	0.985361i	$0.445468\pi$
$930$	0	0
$931$	−2.44949	−0.0802788
$932$	0	0
$933$	−14.3923	−0.471183
$934$	0	0
$935$	3.86370	0.126357
$936$	0	0
$937$	46.9282	1.53308	0.766539	−	0.642198i	$-0.221977\pi$
$937$	46.9282	1.53308	0.766539	+	0.642198i	$0.221977\pi$
$938$	0	0
$939$	−11.8685	−0.387314
$940$	0	0
$941$	−7.85641	−0.256112	−0.128056	−	0.991767i	$-0.540874\pi$
$941$	−7.85641	−0.256112	−0.128056	+	0.991767i	$0.540874\pi$
$942$	0	0
$943$	−8.83701	−0.287773
$944$	0	0
$945$	13.8564	0.450749
$946$	0	0
$947$	51.3650	1.66914	0.834568	−	0.550904i	$-0.185717\pi$
$947$	51.3650	1.66914	0.834568	+	0.550904i	$0.185717\pi$
$948$	0	0
$949$	5.35898	0.173960
$950$	0	0
$951$	−29.1165	−0.944166
$952$	0	0
$953$	3.85641	0.124921	0.0624606	−	0.998047i	$-0.480105\pi$
$953$	3.85641	0.124921	0.0624606	+	0.998047i	$0.480105\pi$
$954$	0	0
$955$	−7.07107	−0.228814
$956$	0	0
$957$	2.00000	0.0646508
$958$	0	0
$959$	−27.2490	−0.879916
$960$	0	0
$961$	−30.8564	−0.995368
$962$	0	0
$963$	17.9043	0.576958
$964$	0	0
$965$	−14.7321	−0.474241
$966$	0	0
$967$	30.0502	0.966350	0.483175	−	0.875524i	$-0.339484\pi$
$967$	30.0502	0.966350	0.483175	+	0.875524i	$0.339484\pi$
$968$	0	0
$969$	−9.46410	−0.304031
$970$	0	0
$971$	−3.76217	−0.120734	−0.0603668	−	0.998176i	$-0.519227\pi$
$971$	−3.76217	−0.120734	−0.0603668	+	0.998176i	$0.519227\pi$
$972$	0	0
$973$	18.9282	0.606810
$974$	0	0
$975$	2.07055	0.0663107
$976$	0	0
$977$	20.0000	0.639857	0.319928	−	0.947442i	$-0.396341\pi$
$977$	20.0000	0.639857	0.319928	+	0.947442i	$0.396341\pi$
$978$	0	0
$979$	18.2832	0.584335
$980$	0	0
$981$	−2.92820	−0.0934903
$982$	0	0
$983$	−8.38375	−0.267400	−0.133700	−	0.991022i	$-0.542686\pi$
$983$	−8.38375	−0.267400	−0.133700	+	0.991022i	$0.542686\pi$
$984$	0	0
$985$	15.8564	0.505227
$986$	0	0
$987$	−2.27362	−0.0723703
$988$	0	0
$989$	22.0000	0.699559
$990$	0	0
$991$	−19.8733	−0.631297	−0.315648	−	0.948876i	$-0.602222\pi$
$991$	−19.8733	−0.631297	−0.315648	+	0.948876i	$0.602222\pi$
$992$	0	0
$993$	30.1051	0.955357
$994$	0	0
$995$	20.6312	0.654053
$996$	0	0
$997$	−30.9808	−0.981171	−0.490585	−	0.871393i	$-0.663217\pi$
$997$	−30.9808	−0.981171	−0.490585	+	0.871393i	$0.663217\pi$
$998$	0	0
$999$	54.6466	1.72894

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4640.2.a.q.1.3	yes	4
4.3	odd	2	inner	4640.2.a.q.1.2	✓	4
8.3	odd	2		9280.2.a.ca.1.4		4
8.5	even	2		9280.2.a.ca.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.q.1.2	✓	4	4.3	odd	2	inner
4640.2.a.q.1.3	yes	4	1.1	even	1	trivial
9280.2.a.ca.1.1		4	8.5	even	2
9280.2.a.ca.1.4		4	8.3	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:	\( N \)	\(=\)	\( 4640 = 2^{5} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4640.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} +1.00000 q^{5} -2.44949 q^{7} -1.00000 q^{9} -1.41421 q^{11} +1.46410 q^{13} +1.41421 q^{15} -2.73205 q^{17} +2.44949 q^{19} -3.46410 q^{21} +3.48477 q^{23} +1.00000 q^{25} -5.65685 q^{27} -1.00000 q^{29} -0.378937 q^{31} -2.00000 q^{33} -2.44949 q^{35} -9.66025 q^{37} +2.07055 q^{39} -2.53590 q^{41} +6.31319 q^{43} -1.00000 q^{45} +0.656339 q^{47} -1.00000 q^{49} -3.86370 q^{51} -1.46410 q^{53} -1.41421 q^{55} +3.46410 q^{57} -5.93426 q^{59} -8.00000 q^{61} +2.44949 q^{63} +1.46410 q^{65} -6.31319 q^{67} +4.92820 q^{69} +10.2784 q^{71} +3.66025 q^{73} +1.41421 q^{75} +3.46410 q^{77} -9.14162 q^{79} -5.00000 q^{81} -6.03579 q^{83} -2.73205 q^{85} -1.41421 q^{87} -12.9282 q^{89} -3.58630 q^{91} -0.535898 q^{93} +2.44949 q^{95} -9.12436 q^{97} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{9} - 8 q^{13} - 4 q^{17} + 4 q^{25} - 4 q^{29} - 8 q^{33} - 4 q^{37} - 24 q^{41} - 4 q^{45} - 4 q^{49} + 8 q^{53} - 32 q^{61} - 8 q^{65} - 8 q^{69} - 20 q^{73} - 20 q^{81} - 4 q^{85}+ \cdots + 12 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^9 - 8 * q^13 - 4 * q^17 + 4 * q^25 - 4 * q^29 - 8 * q^33 - 4 * q^37 - 24 * q^41 - 4 * q^45 - 4 * q^49 + 8 * q^53 - 32 * q^61 - 8 * q^65 - 8 * q^69 - 20 * q^73 - 20 * q^81 - 4 * q^85 - 24 * q^89 - 16 * q^93 + 12 * q^97

Introduction

L-functions

Modular forms

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