LMFDB - Embedded newform 4620.2.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.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:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.8220.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 20x + 12 $ x^3 - x^2 - 20*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.592845$ of defining polynomial
Character	$\chi$	$=$	4620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +6.52784 q^{13} -1.00000 q^{15} +1.59284 q^{17} +0.407155 q^{19} -1.00000 q^{21} -1.59284 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.12069 q^{29} -6.52784 q^{31} +1.00000 q^{33} +1.00000 q^{35} +6.52784 q^{37} +6.52784 q^{39} -4.52784 q^{41} +11.3064 q^{43} -1.00000 q^{45} +7.71353 q^{47} +1.00000 q^{49} +1.59284 q^{51} -2.40716 q^{53} -1.00000 q^{55} +0.407155 q^{57} +14.1207 q^{59} +0.407155 q^{61} -1.00000 q^{63} -6.52784 q^{65} +6.00000 q^{67} -1.59284 q^{69} -4.52784 q^{71} -10.2414 q^{73} +1.00000 q^{75} -1.00000 q^{77} -4.52784 q^{79} +1.00000 q^{81} -3.59284 q^{83} -1.59284 q^{85} -6.12069 q^{87} -5.30638 q^{89} -6.52784 q^{91} -6.52784 q^{93} -0.407155 q^{95} -2.12069 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{15} + 4 q^{17} + 2 q^{19} - 3 q^{21} - 4 q^{23} + 3 q^{25} + 3 q^{27} + 2 q^{29} + 3 q^{33} + 3 q^{35} + 6 q^{41} + 12 q^{43} - 3 q^{45} + 2 q^{47} + 3 q^{49} + 4 q^{51} - 8 q^{53} - 3 q^{55} + 2 q^{57} + 22 q^{59} + 2 q^{61} - 3 q^{63} + 18 q^{67} - 4 q^{69} + 6 q^{71} + 10 q^{73} + 3 q^{75} - 3 q^{77} + 6 q^{79} + 3 q^{81} - 10 q^{83} - 4 q^{85} + 2 q^{87} + 6 q^{89} - 2 q^{95} + 14 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^15 + 4 * q^17 + 2 * q^19 - 3 * q^21 - 4 * q^23 + 3 * q^25 + 3 * q^27 + 2 * q^29 + 3 * q^33 + 3 * q^35 + 6 * q^41 + 12 * q^43 - 3 * q^45 + 2 * q^47 + 3 * q^49 + 4 * q^51 - 8 * q^53 - 3 * q^55 + 2 * q^57 + 22 * q^59 + 2 * q^61 - 3 * q^63 + 18 * q^67 - 4 * q^69 + 6 * q^71 + 10 * q^73 + 3 * q^75 - 3 * q^77 + 6 * q^79 + 3 * q^81 - 10 * q^83 - 4 * q^85 + 2 * q^87 + 6 * q^89 - 2 * q^95 + 14 * q^97 + 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.52784	1.81050	0.905249	−	0.424881i	$-0.139684\pi$
$13$	6.52784	1.81050	0.905249	+	0.424881i	$0.139684\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.59284	0.386322	0.193161	−	0.981167i	$-0.438126\pi$
$17$	1.59284	0.386322	0.193161	+	0.981167i	$0.438126\pi$
$18$	0	0
$19$	0.407155	0.0934078	0.0467039	−	0.998909i	$-0.485128\pi$
$19$	0.407155	0.0934078	0.0467039	+	0.998909i	$0.485128\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.59284	−0.332131	−0.166066	−	0.986115i	$-0.553106\pi$
$23$	−1.59284	−0.332131	−0.166066	+	0.986115i	$0.553106\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.12069	−1.13658	−0.568292	−	0.822827i	$-0.692396\pi$
$29$	−6.12069	−1.13658	−0.568292	+	0.822827i	$0.692396\pi$
$30$	0	0
$31$	−6.52784	−1.17244	−0.586218	−	0.810154i	$-0.699384\pi$
$31$	−6.52784	−1.17244	−0.586218	+	0.810154i	$0.699384\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	6.52784	1.07317	0.536586	−	0.843846i	$-0.319714\pi$
$37$	6.52784	1.07317	0.536586	+	0.843846i	$0.319714\pi$
$38$	0	0
$39$	6.52784	1.04529
$40$	0	0
$41$	−4.52784	−0.707131	−0.353565	−	0.935410i	$-0.615031\pi$
$41$	−4.52784	−0.707131	−0.353565	+	0.935410i	$0.615031\pi$
$42$	0	0
$43$	11.3064	1.72421	0.862103	−	0.506732i	$-0.169147\pi$
$43$	11.3064	1.72421	0.862103	+	0.506732i	$0.169147\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	7.71353	1.12513	0.562567	−	0.826751i	$-0.309814\pi$
$47$	7.71353	1.12513	0.562567	+	0.826751i	$0.309814\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.59284	0.223043
$52$	0	0
$53$	−2.40716	−0.330648	−0.165324	−	0.986239i	$-0.552867\pi$
$53$	−2.40716	−0.330648	−0.165324	+	0.986239i	$0.552867\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0.407155	0.0539290
$58$	0	0
$59$	14.1207	1.83836	0.919179	−	0.393841i	$-0.128854\pi$
$59$	14.1207	1.83836	0.919179	+	0.393841i	$0.128854\pi$
$60$	0	0
$61$	0.407155	0.0521309	0.0260654	−	0.999660i	$-0.491702\pi$
$61$	0.407155	0.0521309	0.0260654	+	0.999660i	$0.491702\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−6.52784	−0.809680
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	0	0
$69$	−1.59284	−0.191756
$70$	0	0
$71$	−4.52784	−0.537356	−0.268678	−	0.963230i	$-0.586587\pi$
$71$	−4.52784	−0.537356	−0.268678	+	0.963230i	$0.586587\pi$
$72$	0	0
$73$	−10.2414	−1.19866	−0.599331	−	0.800501i	$-0.704567\pi$
$73$	−10.2414	−1.19866	−0.599331	+	0.800501i	$0.704567\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−4.52784	−0.509422	−0.254711	−	0.967017i	$-0.581980\pi$
$79$	−4.52784	−0.509422	−0.254711	+	0.967017i	$0.581980\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.59284	−0.394366	−0.197183	−	0.980367i	$-0.563179\pi$
$83$	−3.59284	−0.394366	−0.197183	+	0.980367i	$0.563179\pi$
$84$	0	0
$85$	−1.59284	−0.172768
$86$	0	0
$87$	−6.12069	−0.656207
$88$	0	0
$89$	−5.30638	−0.562475	−0.281238	−	0.959638i	$-0.590745\pi$
$89$	−5.30638	−0.562475	−0.281238	+	0.959638i	$0.590745\pi$
$90$	0	0
$91$	−6.52784	−0.684304
$92$	0	0
$93$	−6.52784	−0.676906
$94$	0	0
$95$	−0.407155	−0.0417732
$96$	0	0
$97$	−2.12069	−0.215323	−0.107662	−	0.994188i	$-0.534336\pi$
$97$	−2.12069	−0.215323	−0.107662	+	0.994188i	$0.534336\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	3.18569	0.316988	0.158494	−	0.987360i	$-0.449336\pi$
$101$	3.18569	0.316988	0.158494	+	0.987360i	$0.449336\pi$
$102$	0	0
$103$	8.93500	0.880392	0.440196	−	0.897902i	$-0.354909\pi$
$103$	8.93500	0.880392	0.440196	+	0.897902i	$0.354909\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	9.34216	0.903140	0.451570	−	0.892236i	$-0.350864\pi$
$107$	9.34216	0.903140	0.451570	+	0.892236i	$0.350864\pi$
$108$	0	0
$109$	14.5278	1.39152	0.695758	−	0.718277i	$-0.255069\pi$
$109$	14.5278	1.39152	0.695758	+	0.718277i	$0.255069\pi$
$110$	0	0
$111$	6.52784	0.619596
$112$	0	0
$113$	−14.6485	−1.37802	−0.689009	−	0.724753i	$-0.741954\pi$
$113$	−14.6485	−1.37802	−0.689009	+	0.724753i	$0.741954\pi$
$114$	0	0
$115$	1.59284	0.148534
$116$	0	0
$117$	6.52784	0.603499
$118$	0	0
$119$	−1.59284	−0.146016
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.52784	−0.408262
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.12069	0.720595	0.360297	−	0.932838i	$-0.382675\pi$
$127$	8.12069	0.720595	0.360297	+	0.932838i	$0.382675\pi$
$128$	0	0
$129$	11.3064	0.995471
$130$	0	0
$131$	15.7135	1.37290	0.686449	−	0.727178i	$-0.259169\pi$
$131$	15.7135	1.37290	0.686449	+	0.727178i	$0.259169\pi$
$132$	0	0
$133$	−0.407155	−0.0353048
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	18.2414	1.55847	0.779233	−	0.626734i	$-0.215609\pi$
$137$	18.2414	1.55847	0.779233	+	0.626734i	$0.215609\pi$
$138$	0	0
$139$	−4.24138	−0.359749	−0.179875	−	0.983690i	$-0.557569\pi$
$139$	−4.24138	−0.359749	−0.179875	+	0.983690i	$0.557569\pi$
$140$	0	0
$141$	7.71353	0.649597
$142$	0	0
$143$	6.52784	0.545886
$144$	0	0
$145$	6.12069	0.508296
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	7.05569	0.578025	0.289012	−	0.957325i	$-0.406673\pi$
$149$	7.05569	0.578025	0.289012	+	0.957325i	$0.406673\pi$
$150$	0	0
$151$	0.814310	0.0662676	0.0331338	−	0.999451i	$-0.489451\pi$
$151$	0.814310	0.0662676	0.0331338	+	0.999451i	$0.489451\pi$
$152$	0	0
$153$	1.59284	0.128774
$154$	0	0
$155$	6.52784	0.524329
$156$	0	0
$157$	−5.30638	−0.423495	−0.211748	−	0.977324i	$-0.567915\pi$
$157$	−5.30638	−0.423495	−0.211748	+	0.977324i	$0.567915\pi$
$158$	0	0
$159$	−2.40716	−0.190900
$160$	0	0
$161$	1.59284	0.125534
$162$	0	0
$163$	2.52784	0.197996	0.0989980	−	0.995088i	$-0.468436\pi$
$163$	2.52784	0.197996	0.0989980	+	0.995088i	$0.468436\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	−9.87000	−0.763764	−0.381882	−	0.924211i	$-0.624724\pi$
$167$	−9.87000	−0.763764	−0.381882	+	0.924211i	$0.624724\pi$
$168$	0	0
$169$	29.6128	2.27790
$170$	0	0
$171$	0.407155	0.0311359
$172$	0	0
$173$	−10.2414	−0.778638	−0.389319	−	0.921103i	$-0.627290\pi$
$173$	−10.2414	−0.778638	−0.389319	+	0.921103i	$0.627290\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	14.1207	1.06138
$178$	0	0
$179$	21.5835	1.61323	0.806615	−	0.591078i	$-0.201297\pi$
$179$	21.5835	1.61323	0.806615	+	0.591078i	$0.201297\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	0.407155	0.0300978
$184$	0	0
$185$	−6.52784	−0.479937
$186$	0	0
$187$	1.59284	0.116480
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−6.37138	−0.461017	−0.230508	−	0.973070i	$-0.574039\pi$
$191$	−6.37138	−0.461017	−0.230508	+	0.973070i	$0.574039\pi$
$192$	0	0
$193$	18.6128	1.33978	0.669888	−	0.742462i	$-0.266342\pi$
$193$	18.6128	1.33978	0.669888	+	0.742462i	$0.266342\pi$
$194$	0	0
$195$	−6.52784	−0.467469
$196$	0	0
$197$	−7.42707	−0.529157	−0.264578	−	0.964364i	$-0.585233\pi$
$197$	−7.42707	−0.529157	−0.264578	+	0.964364i	$0.585233\pi$
$198$	0	0
$199$	17.1857	1.21826	0.609131	−	0.793070i	$-0.291519\pi$
$199$	17.1857	1.21826	0.609131	+	0.793070i	$0.291519\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	0	0
$203$	6.12069	0.429588
$204$	0	0
$205$	4.52784	0.316238
$206$	0	0
$207$	−1.59284	−0.110710
$208$	0	0
$209$	0.407155	0.0281635
$210$	0	0
$211$	−4.24138	−0.291989	−0.145994	−	0.989285i	$-0.546638\pi$
$211$	−4.24138	−0.291989	−0.145994	+	0.989285i	$0.546638\pi$
$212$	0	0
$213$	−4.52784	−0.310243
$214$	0	0
$215$	−11.3064	−0.771089
$216$	0	0
$217$	6.52784	0.443139
$218$	0	0
$219$	−10.2414	−0.692048
$220$	0	0
$221$	10.3978	0.699435
$222$	0	0
$223$	−3.87931	−0.259778	−0.129889	−	0.991529i	$-0.541462\pi$
$223$	−3.87931	−0.259778	−0.129889	+	0.991529i	$0.541462\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	13.2215	0.877539	0.438770	−	0.898600i	$-0.355414\pi$
$227$	13.2215	0.877539	0.438770	+	0.898600i	$0.355414\pi$
$228$	0	0
$229$	−2.52784	−0.167045	−0.0835223	−	0.996506i	$-0.526617\pi$
$229$	−2.52784	−0.167045	−0.0835223	+	0.996506i	$0.526617\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	−3.47216	−0.227468	−0.113734	−	0.993511i	$-0.536281\pi$
$233$	−3.47216	−0.227468	−0.113734	+	0.993511i	$0.536281\pi$
$234$	0	0
$235$	−7.71353	−0.503176
$236$	0	0
$237$	−4.52784	−0.294115
$238$	0	0
$239$	2.25069	0.145585	0.0727925	−	0.997347i	$-0.476809\pi$
$239$	2.25069	0.145585	0.0727925	+	0.997347i	$0.476809\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	2.65784	0.169115
$248$	0	0
$249$	−3.59284	−0.227687
$250$	0	0
$251$	−14.8143	−0.935071	−0.467535	−	0.883974i	$-0.654858\pi$
$251$	−14.8143	−0.935071	−0.467535	+	0.883974i	$0.654858\pi$
$252$	0	0
$253$	−1.59284	−0.100141
$254$	0	0
$255$	−1.59284	−0.0997478
$256$	0	0
$257$	12.0849	0.753836	0.376918	−	0.926247i	$-0.376984\pi$
$257$	12.0849	0.753836	0.376918	+	0.926247i	$0.376984\pi$
$258$	0	0
$259$	−6.52784	−0.405621
$260$	0	0
$261$	−6.12069	−0.378861
$262$	0	0
$263$	−15.4271	−0.951274	−0.475637	−	0.879642i	$-0.657782\pi$
$263$	−15.4271	−0.951274	−0.475637	+	0.879642i	$0.657782\pi$
$264$	0	0
$265$	2.40716	0.147870
$266$	0	0
$267$	−5.30638	−0.324745
$268$	0	0
$269$	18.3621	1.11956	0.559778	−	0.828643i	$-0.310887\pi$
$269$	18.3621	1.11956	0.559778	+	0.828643i	$0.310887\pi$
$270$	0	0
$271$	15.8342	0.961861	0.480930	−	0.876759i	$-0.340299\pi$
$271$	15.8342	0.961861	0.480930	+	0.876759i	$0.340299\pi$
$272$	0	0
$273$	−6.52784	−0.395083
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−26.1114	−1.56888	−0.784440	−	0.620204i	$-0.787050\pi$
$277$	−26.1114	−1.56888	−0.784440	+	0.620204i	$0.787050\pi$
$278$	0	0
$279$	−6.52784	−0.390812
$280$	0	0
$281$	−13.1857	−0.786592	−0.393296	−	0.919412i	$-0.628665\pi$
$281$	−13.1857	−0.786592	−0.393296	+	0.919412i	$0.628665\pi$
$282$	0	0
$283$	26.3978	1.56919	0.784595	−	0.620009i	$-0.212871\pi$
$283$	26.3978	1.56919	0.784595	+	0.620009i	$0.212871\pi$
$284$	0	0
$285$	−0.407155	−0.0241178
$286$	0	0
$287$	4.52784	0.267270
$288$	0	0
$289$	−14.4628	−0.850756
$290$	0	0
$291$	−2.12069	−0.124317
$292$	0	0
$293$	−2.16578	−0.126526	−0.0632630	−	0.997997i	$-0.520151\pi$
$293$	−2.16578	−0.126526	−0.0632630	+	0.997997i	$0.520151\pi$
$294$	0	0
$295$	−14.1207	−0.822138
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−10.3978	−0.601323
$300$	0	0
$301$	−11.3064	−0.651689
$302$	0	0
$303$	3.18569	0.183013
$304$	0	0
$305$	−0.407155	−0.0233136
$306$	0	0
$307$	−5.87000	−0.335019	−0.167509	−	0.985870i	$-0.553572\pi$
$307$	−5.87000	−0.335019	−0.167509	+	0.985870i	$0.553572\pi$
$308$	0	0
$309$	8.93500	0.508294
$310$	0	0
$311$	19.8700	1.12672	0.563362	−	0.826210i	$-0.309508\pi$
$311$	19.8700	1.12672	0.563362	+	0.826210i	$0.309508\pi$
$312$	0	0
$313$	−12.4921	−0.706094	−0.353047	−	0.935606i	$-0.614854\pi$
$313$	−12.4921	−0.706094	−0.353047	+	0.935606i	$0.614854\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	19.2971	1.08383	0.541916	−	0.840433i	$-0.317699\pi$
$317$	19.2971	1.08383	0.541916	+	0.840433i	$0.317699\pi$
$318$	0	0
$319$	−6.12069	−0.342693
$320$	0	0
$321$	9.34216	0.521428
$322$	0	0
$323$	0.648535	0.0360854
$324$	0	0
$325$	6.52784	0.362100
$326$	0	0
$327$	14.5278	0.803392
$328$	0	0
$329$	−7.71353	−0.425261
$330$	0	0
$331$	2.77853	0.152722	0.0763610	−	0.997080i	$-0.475670\pi$
$331$	2.77853	0.152722	0.0763610	+	0.997080i	$0.475670\pi$
$332$	0	0
$333$	6.52784	0.357724
$334$	0	0
$335$	−6.00000	−0.327815
$336$	0	0
$337$	12.1207	0.660256	0.330128	−	0.943936i	$-0.392908\pi$
$337$	12.1207	0.660256	0.330128	+	0.943936i	$0.392908\pi$
$338$	0	0
$339$	−14.6485	−0.795599
$340$	0	0
$341$	−6.52784	−0.353503
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.59284	0.0857559
$346$	0	0
$347$	−1.05569	−0.0566724	−0.0283362	−	0.999598i	$-0.509021\pi$
$347$	−1.05569	−0.0566724	−0.0283362	+	0.999598i	$0.509021\pi$
$348$	0	0
$349$	23.0199	1.23223	0.616114	−	0.787657i	$-0.288706\pi$
$349$	23.0199	1.23223	0.616114	+	0.787657i	$0.288706\pi$
$350$	0	0
$351$	6.52784	0.348431
$352$	0	0
$353$	−11.0557	−0.588435	−0.294217	−	0.955738i	$-0.595059\pi$
$353$	−11.0557	−0.588435	−0.294217	+	0.955738i	$0.595059\pi$
$354$	0	0
$355$	4.52784	0.240313
$356$	0	0
$357$	−1.59284	−0.0843023
$358$	0	0
$359$	−6.56362	−0.346415	−0.173207	−	0.984885i	$-0.555413\pi$
$359$	−6.56362	−0.346415	−0.173207	+	0.984885i	$0.555413\pi$
$360$	0	0
$361$	−18.8342	−0.991275
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	10.2414	0.536058
$366$	0	0
$367$	16.1207	0.841493	0.420747	−	0.907178i	$-0.361768\pi$
$367$	16.1207	0.841493	0.420747	+	0.907178i	$0.361768\pi$
$368$	0	0
$369$	−4.52784	−0.235710
$370$	0	0
$371$	2.40716	0.124973
$372$	0	0
$373$	8.93500	0.462637	0.231318	−	0.972878i	$-0.425696\pi$
$373$	8.93500	0.462637	0.231318	+	0.972878i	$0.425696\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−39.9549	−2.05778
$378$	0	0
$379$	18.2772	0.938834	0.469417	−	0.882976i	$-0.344464\pi$
$379$	18.2772	0.938834	0.469417	+	0.882976i	$0.344464\pi$
$380$	0	0
$381$	8.12069	0.416036
$382$	0	0
$383$	−20.5278	−1.04892	−0.524462	−	0.851434i	$-0.675733\pi$
$383$	−20.5278	−1.04892	−0.524462	+	0.851434i	$0.675733\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	11.3064	0.574736
$388$	0	0
$389$	−32.8992	−1.66806	−0.834028	−	0.551721i	$-0.813971\pi$
$389$	−32.8992	−1.66806	−0.834028	+	0.551721i	$0.813971\pi$
$390$	0	0
$391$	−2.53716	−0.128309
$392$	0	0
$393$	15.7135	0.792643
$394$	0	0
$395$	4.52784	0.227821
$396$	0	0
$397$	−19.2971	−0.968492	−0.484246	−	0.874932i	$-0.660906\pi$
$397$	−19.2971	−0.968492	−0.484246	+	0.874932i	$0.660906\pi$
$398$	0	0
$399$	−0.407155	−0.0203832
$400$	0	0
$401$	−24.3978	−1.21837	−0.609185	−	0.793028i	$-0.708503\pi$
$401$	−24.3978	−1.21837	−0.609185	+	0.793028i	$0.708503\pi$
$402$	0	0
$403$	−42.6128	−2.12269
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	6.52784	0.323573
$408$	0	0
$409$	13.6286	0.673892	0.336946	−	0.941524i	$-0.390606\pi$
$409$	13.6286	0.673892	0.336946	+	0.941524i	$0.390606\pi$
$410$	0	0
$411$	18.2414	0.899781
$412$	0	0
$413$	−14.1207	−0.694834
$414$	0	0
$415$	3.59284	0.176366
$416$	0	0
$417$	−4.24138	−0.207701
$418$	0	0
$419$	−5.87931	−0.287223	−0.143612	−	0.989634i	$-0.545872\pi$
$419$	−5.87931	−0.287223	−0.143612	+	0.989634i	$0.545872\pi$
$420$	0	0
$421$	−24.2772	−1.18320	−0.591598	−	0.806233i	$-0.701503\pi$
$421$	−24.2772	−1.18320	−0.591598	+	0.806233i	$0.701503\pi$
$422$	0	0
$423$	7.71353	0.375045
$424$	0	0
$425$	1.59284	0.0772643
$426$	0	0
$427$	−0.407155	−0.0197036
$428$	0	0
$429$	6.52784	0.315167
$430$	0	0
$431$	−10.6843	−0.514645	−0.257323	−	0.966326i	$-0.582840\pi$
$431$	−10.6843	−0.514645	−0.257323	+	0.966326i	$0.582840\pi$
$432$	0	0
$433$	8.68431	0.417341	0.208671	−	0.977986i	$-0.433086\pi$
$433$	8.68431	0.417341	0.208671	+	0.977986i	$0.433086\pi$
$434$	0	0
$435$	6.12069	0.293465
$436$	0	0
$437$	−0.648535	−0.0310236
$438$	0	0
$439$	9.46284	0.451637	0.225818	−	0.974169i	$-0.427494\pi$
$439$	9.46284	0.451637	0.225818	+	0.974169i	$0.427494\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−4.68431	−0.222558	−0.111279	−	0.993789i	$-0.535495\pi$
$443$	−4.68431	−0.222558	−0.111279	+	0.993789i	$0.535495\pi$
$444$	0	0
$445$	5.30638	0.251547
$446$	0	0
$447$	7.05569	0.333723
$448$	0	0
$449$	−29.7135	−1.40227	−0.701134	−	0.713029i	$-0.747323\pi$
$449$	−29.7135	−1.40227	−0.701134	+	0.713029i	$0.747323\pi$
$450$	0	0
$451$	−4.52784	−0.213208
$452$	0	0
$453$	0.814310	0.0382596
$454$	0	0
$455$	6.52784	0.306030
$456$	0	0
$457$	−0.622070	−0.0290992	−0.0145496	−	0.999894i	$-0.504631\pi$
$457$	−0.622070	−0.0290992	−0.0145496	+	0.999894i	$0.504631\pi$
$458$	0	0
$459$	1.59284	0.0743476
$460$	0	0
$461$	40.7241	1.89671	0.948356	−	0.317208i	$-0.102745\pi$
$461$	40.7241	1.89671	0.948356	+	0.317208i	$0.102745\pi$
$462$	0	0
$463$	−41.1671	−1.91320	−0.956598	−	0.291411i	$-0.905875\pi$
$463$	−41.1671	−1.91320	−0.956598	+	0.291411i	$0.905875\pi$
$464$	0	0
$465$	6.52784	0.302722
$466$	0	0
$467$	−23.4271	−1.08408	−0.542038	−	0.840354i	$-0.682347\pi$
$467$	−23.4271	−1.08408	−0.542038	+	0.840354i	$0.682347\pi$
$468$	0	0
$469$	−6.00000	−0.277054
$470$	0	0
$471$	−5.30638	−0.244505
$472$	0	0
$473$	11.3064	0.519868
$474$	0	0
$475$	0.407155	0.0186816
$476$	0	0
$477$	−2.40716	−0.110216
$478$	0	0
$479$	−14.3978	−0.657854	−0.328927	−	0.944355i	$-0.606687\pi$
$479$	−14.3978	−0.657854	−0.328927	+	0.944355i	$0.606687\pi$
$480$	0	0
$481$	42.6128	1.94297
$482$	0	0
$483$	1.59284	0.0724770
$484$	0	0
$485$	2.12069	0.0962956
$486$	0	0
$487$	19.7984	0.897153	0.448577	−	0.893744i	$-0.351931\pi$
$487$	19.7984	0.897153	0.448577	+	0.893744i	$0.351931\pi$
$488$	0	0
$489$	2.52784	0.114313
$490$	0	0
$491$	25.1764	1.13619	0.568097	−	0.822962i	$-0.307680\pi$
$491$	25.1764	1.13619	0.568097	+	0.822962i	$0.307680\pi$
$492$	0	0
$493$	−9.74931	−0.439087
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	4.52784	0.203102
$498$	0	0
$499$	−42.1870	−1.88855	−0.944274	−	0.329159i	$-0.893235\pi$
$499$	−42.1870	−1.88855	−0.944274	+	0.329159i	$0.893235\pi$
$500$	0	0
$501$	−9.87000	−0.440959
$502$	0	0
$503$	−18.2772	−0.814938	−0.407469	−	0.913219i	$-0.633589\pi$
$503$	−18.2772	−0.814938	−0.407469	+	0.913219i	$0.633589\pi$
$504$	0	0
$505$	−3.18569	−0.141761
$506$	0	0
$507$	29.6128	1.31515
$508$	0	0
$509$	−14.1207	−0.625889	−0.312944	−	0.949771i	$-0.601315\pi$
$509$	−14.1207	−0.625889	−0.312944	+	0.949771i	$0.601315\pi$
$510$	0	0
$511$	10.2414	0.453052
$512$	0	0
$513$	0.407155	0.0179763
$514$	0	0
$515$	−8.93500	−0.393723
$516$	0	0
$517$	7.71353	0.339241
$518$	0	0
$519$	−10.2414	−0.449547
$520$	0	0
$521$	−15.6778	−0.686855	−0.343428	−	0.939179i	$-0.611588\pi$
$521$	−15.6778	−0.686855	−0.343428	+	0.939179i	$0.611588\pi$
$522$	0	0
$523$	−2.94431	−0.128746	−0.0643728	−	0.997926i	$-0.520505\pi$
$523$	−2.94431	−0.128746	−0.0643728	+	0.997926i	$0.520505\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	−10.3978	−0.452937
$528$	0	0
$529$	−20.4628	−0.889689
$530$	0	0
$531$	14.1207	0.612786
$532$	0	0
$533$	−29.5571	−1.28026
$534$	0	0
$535$	−9.34216	−0.403897
$536$	0	0
$537$	21.5835	0.931398
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	28.6392	1.23130	0.615648	−	0.788021i	$-0.288894\pi$
$541$	28.6392	1.23130	0.615648	+	0.788021i	$0.288894\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	−14.5278	−0.622305
$546$	0	0
$547$	2.56362	0.109612	0.0548062	−	0.998497i	$-0.482546\pi$
$547$	2.56362	0.109612	0.0548062	+	0.998497i	$0.482546\pi$
$548$	0	0
$549$	0.407155	0.0173770
$550$	0	0
$551$	−2.49207	−0.106166
$552$	0	0
$553$	4.52784	0.192544
$554$	0	0
$555$	−6.52784	−0.277092
$556$	0	0
$557$	−38.6128	−1.63608	−0.818038	−	0.575165i	$-0.804938\pi$
$557$	−38.6128	−1.63608	−0.818038	+	0.575165i	$0.804938\pi$
$558$	0	0
$559$	73.8063	3.12167
$560$	0	0
$561$	1.59284	0.0672500
$562$	0	0
$563$	38.9257	1.64052	0.820261	−	0.571989i	$-0.193828\pi$
$563$	38.9257	1.64052	0.820261	+	0.571989i	$0.193828\pi$
$564$	0	0
$565$	14.6485	0.616268
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	2.12069	0.0889039	0.0444520	−	0.999012i	$-0.485846\pi$
$569$	2.12069	0.0889039	0.0444520	+	0.999012i	$0.485846\pi$
$570$	0	0
$571$	42.6392	1.78440	0.892198	−	0.451644i	$-0.149162\pi$
$571$	42.6392	1.78440	0.892198	+	0.451644i	$0.149162\pi$
$572$	0	0
$573$	−6.37138	−0.266168
$574$	0	0
$575$	−1.59284	−0.0664262
$576$	0	0
$577$	29.7400	1.23809	0.619046	−	0.785355i	$-0.287519\pi$
$577$	29.7400	1.23809	0.619046	+	0.785355i	$0.287519\pi$
$578$	0	0
$579$	18.6128	0.773520
$580$	0	0
$581$	3.59284	0.149056
$582$	0	0
$583$	−2.40716	−0.0996942
$584$	0	0
$585$	−6.52784	−0.269893
$586$	0	0
$587$	−35.9549	−1.48402	−0.742009	−	0.670390i	$-0.766127\pi$
$587$	−35.9549	−1.48402	−0.742009	+	0.670390i	$0.766127\pi$
$588$	0	0
$589$	−2.65784	−0.109515
$590$	0	0
$591$	−7.42707	−0.305509
$592$	0	0
$593$	3.55707	0.146071	0.0730357	−	0.997329i	$-0.476731\pi$
$593$	3.55707	0.146071	0.0730357	+	0.997329i	$0.476731\pi$
$594$	0	0
$595$	1.59284	0.0653003
$596$	0	0
$597$	17.1857	0.703363
$598$	0	0
$599$	−20.9841	−0.857389	−0.428694	−	0.903450i	$-0.641026\pi$
$599$	−20.9841	−0.857389	−0.428694	+	0.903450i	$0.641026\pi$
$600$	0	0
$601$	−39.5742	−1.61427	−0.807133	−	0.590369i	$-0.798982\pi$
$601$	−39.5742	−1.61427	−0.807133	+	0.590369i	$0.798982\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−24.2414	−0.983927	−0.491964	−	0.870616i	$-0.663721\pi$
$607$	−24.2414	−0.983927	−0.491964	+	0.870616i	$0.663721\pi$
$608$	0	0
$609$	6.12069	0.248023
$610$	0	0
$611$	50.3528	2.03705
$612$	0	0
$613$	−0.814310	−0.0328897	−0.0164448	−	0.999865i	$-0.505235\pi$
$613$	−0.814310	−0.0328897	−0.0164448	+	0.999865i	$0.505235\pi$
$614$	0	0
$615$	4.52784	0.182580
$616$	0	0
$617$	40.0398	1.61194	0.805971	−	0.591954i	$-0.201643\pi$
$617$	40.0398	1.61194	0.805971	+	0.591954i	$0.201643\pi$
$618$	0	0
$619$	40.1114	1.61221	0.806106	−	0.591771i	$-0.201571\pi$
$619$	40.1114	1.61221	0.806106	+	0.591771i	$0.201571\pi$
$620$	0	0
$621$	−1.59284	−0.0639187
$622$	0	0
$623$	5.30638	0.212596
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.407155	0.0162602
$628$	0	0
$629$	10.3978	0.414589
$630$	0	0
$631$	4.64853	0.185055	0.0925276	−	0.995710i	$-0.470505\pi$
$631$	4.64853	0.185055	0.0925276	+	0.995710i	$0.470505\pi$
$632$	0	0
$633$	−4.24138	−0.168580
$634$	0	0
$635$	−8.12069	−0.322260
$636$	0	0
$637$	6.52784	0.258643
$638$	0	0
$639$	−4.52784	−0.179119
$640$	0	0
$641$	−3.62862	−0.143322	−0.0716609	−	0.997429i	$-0.522830\pi$
$641$	−3.62862	−0.143322	−0.0716609	+	0.997429i	$0.522830\pi$
$642$	0	0
$643$	−6.56362	−0.258844	−0.129422	−	0.991590i	$-0.541312\pi$
$643$	−6.56362	−0.258844	−0.129422	+	0.991590i	$0.541312\pi$
$644$	0	0
$645$	−11.3064	−0.445188
$646$	0	0
$647$	−5.05569	−0.198760	−0.0993798	−	0.995050i	$-0.531686\pi$
$647$	−5.05569	−0.198760	−0.0993798	+	0.995050i	$0.531686\pi$
$648$	0	0
$649$	14.1207	0.554286
$650$	0	0
$651$	6.52784	0.255846
$652$	0	0
$653$	−43.7042	−1.71028	−0.855139	−	0.518398i	$-0.826528\pi$
$653$	−43.7042	−1.71028	−0.855139	+	0.518398i	$0.826528\pi$
$654$	0	0
$655$	−15.7135	−0.613979
$656$	0	0
$657$	−10.2414	−0.399554
$658$	0	0
$659$	−9.99069	−0.389182	−0.194591	−	0.980884i	$-0.562338\pi$
$659$	−9.99069	−0.389182	−0.194591	+	0.980884i	$0.562338\pi$
$660$	0	0
$661$	9.47216	0.368424	0.184212	−	0.982887i	$-0.441027\pi$
$661$	9.47216	0.368424	0.184212	+	0.982887i	$0.441027\pi$
$662$	0	0
$663$	10.3978	0.403819
$664$	0	0
$665$	0.407155	0.0157888
$666$	0	0
$667$	9.74931	0.377495
$668$	0	0
$669$	−3.87931	−0.149983
$670$	0	0
$671$	0.407155	0.0157180
$672$	0	0
$673$	−34.7334	−1.33888	−0.669438	−	0.742868i	$-0.733465\pi$
$673$	−34.7334	−1.33888	−0.669438	+	0.742868i	$0.733465\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−38.8899	−1.49466	−0.747330	−	0.664453i	$-0.768665\pi$
$677$	−38.8899	−1.49466	−0.747330	+	0.664453i	$0.768665\pi$
$678$	0	0
$679$	2.12069	0.0813846
$680$	0	0
$681$	13.2215	0.506648
$682$	0	0
$683$	1.18569	0.0453692	0.0226846	−	0.999743i	$-0.492779\pi$
$683$	1.18569	0.0453692	0.0226846	+	0.999743i	$0.492779\pi$
$684$	0	0
$685$	−18.2414	−0.696967
$686$	0	0
$687$	−2.52784	−0.0964433
$688$	0	0
$689$	−15.7135	−0.598638
$690$	0	0
$691$	−40.1114	−1.52591	−0.762954	−	0.646452i	$-0.776252\pi$
$691$	−40.1114	−1.52591	−0.762954	+	0.646452i	$0.776252\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	4.24138	0.160885
$696$	0	0
$697$	−7.21216	−0.273180
$698$	0	0
$699$	−3.47216	−0.131329
$700$	0	0
$701$	−23.9907	−0.906116	−0.453058	−	0.891481i	$-0.649667\pi$
$701$	−23.9907	−0.906116	−0.453058	+	0.891481i	$0.649667\pi$
$702$	0	0
$703$	2.65784	0.100243
$704$	0	0
$705$	−7.71353	−0.290509
$706$	0	0
$707$	−3.18569	−0.119810
$708$	0	0
$709$	−12.7785	−0.479908	−0.239954	−	0.970784i	$-0.577132\pi$
$709$	−12.7785	−0.479908	−0.239954	+	0.970784i	$0.577132\pi$
$710$	0	0
$711$	−4.52784	−0.169807
$712$	0	0
$713$	10.3978	0.389402
$714$	0	0
$715$	−6.52784	−0.244128
$716$	0	0
$717$	2.25069	0.0840536
$718$	0	0
$719$	10.1207	0.377438	0.188719	−	0.982031i	$-0.439566\pi$
$719$	10.1207	0.377438	0.188719	+	0.982031i	$0.439566\pi$
$720$	0	0
$721$	−8.93500	−0.332757
$722$	0	0
$723$	20.0000	0.743808
$724$	0	0
$725$	−6.12069	−0.227317
$726$	0	0
$727$	−8.12069	−0.301180	−0.150590	−	0.988596i	$-0.548117\pi$
$727$	−8.12069	−0.301180	−0.150590	+	0.988596i	$0.548117\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.0093	0.666098
$732$	0	0
$733$	−32.6843	−1.20722	−0.603611	−	0.797279i	$-0.706272\pi$
$733$	−32.6843	−1.20722	−0.603611	+	0.797279i	$0.706272\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	6.00000	0.221013
$738$	0	0
$739$	28.2414	1.03888	0.519438	−	0.854508i	$-0.326141\pi$
$739$	28.2414	1.03888	0.519438	+	0.854508i	$0.326141\pi$
$740$	0	0
$741$	2.65784	0.0976384
$742$	0	0
$743$	−14.6128	−0.536090	−0.268045	−	0.963406i	$-0.586378\pi$
$743$	−14.6128	−0.536090	−0.268045	+	0.963406i	$0.586378\pi$
$744$	0	0
$745$	−7.05569	−0.258500
$746$	0	0
$747$	−3.59284	−0.131455
$748$	0	0
$749$	−9.34216	−0.341355
$750$	0	0
$751$	−42.1870	−1.53942	−0.769712	−	0.638391i	$-0.779600\pi$
$751$	−42.1870	−1.53942	−0.769712	+	0.638391i	$0.779600\pi$
$752$	0	0
$753$	−14.8143	−0.539863
$754$	0	0
$755$	−0.814310	−0.0296358
$756$	0	0
$757$	52.2812	1.90019	0.950096	−	0.311956i	$-0.100984\pi$
$757$	52.2812	1.90019	0.950096	+	0.311956i	$0.100984\pi$
$758$	0	0
$759$	−1.59284	−0.0578166
$760$	0	0
$761$	2.44293	0.0885562	0.0442781	−	0.999019i	$-0.485901\pi$
$761$	2.44293	0.0885562	0.0442781	+	0.999019i	$0.485901\pi$
$762$	0	0
$763$	−14.5278	−0.525943
$764$	0	0
$765$	−1.59284	−0.0575894
$766$	0	0
$767$	92.1777	3.32834
$768$	0	0
$769$	−1.46284	−0.0527515	−0.0263758	−	0.999652i	$-0.508397\pi$
$769$	−1.46284	−0.0527515	−0.0263758	+	0.999652i	$0.508397\pi$
$770$	0	0
$771$	12.0849	0.435228
$772$	0	0
$773$	−2.57293	−0.0925419	−0.0462709	−	0.998929i	$-0.514734\pi$
$773$	−2.57293	−0.0925419	−0.0462709	+	0.998929i	$0.514734\pi$
$774$	0	0
$775$	−6.52784	−0.234487
$776$	0	0
$777$	−6.52784	−0.234185
$778$	0	0
$779$	−1.84353	−0.0660515
$780$	0	0
$781$	−4.52784	−0.162019
$782$	0	0
$783$	−6.12069	−0.218736
$784$	0	0
$785$	5.30638	0.189393
$786$	0	0
$787$	−6.11138	−0.217847	−0.108924	−	0.994050i	$-0.534740\pi$
$787$	−6.11138	−0.217847	−0.108924	+	0.994050i	$0.534740\pi$
$788$	0	0
$789$	−15.4271	−0.549218
$790$	0	0
$791$	14.6485	0.520842
$792$	0	0
$793$	2.65784	0.0943828
$794$	0	0
$795$	2.40716	0.0853730
$796$	0	0
$797$	−28.6843	−1.01605	−0.508025	−	0.861342i	$-0.669624\pi$
$797$	−28.6843	−1.01605	−0.508025	+	0.861342i	$0.669624\pi$
$798$	0	0
$799$	12.2865	0.434664
$800$	0	0
$801$	−5.30638	−0.187492
$802$	0	0
$803$	−10.2414	−0.361410
$804$	0	0
$805$	−1.59284	−0.0561404
$806$	0	0
$807$	18.3621	0.646375
$808$	0	0
$809$	−36.6843	−1.28975	−0.644876	−	0.764287i	$-0.723091\pi$
$809$	−36.6843	−1.28975	−0.644876	+	0.764287i	$0.723091\pi$
$810$	0	0
$811$	−6.11138	−0.214600	−0.107300	−	0.994227i	$-0.534220\pi$
$811$	−6.11138	−0.214600	−0.107300	+	0.994227i	$0.534220\pi$
$812$	0	0
$813$	15.8342	0.555331
$814$	0	0
$815$	−2.52784	−0.0885465
$816$	0	0
$817$	4.60345	0.161054
$818$	0	0
$819$	−6.52784	−0.228101
$820$	0	0
$821$	−16.4921	−0.575577	−0.287789	−	0.957694i	$-0.592920\pi$
$821$	−16.4921	−0.575577	−0.287789	+	0.957694i	$0.592920\pi$
$822$	0	0
$823$	5.95491	0.207575	0.103788	−	0.994599i	$-0.466904\pi$
$823$	5.95491	0.207575	0.103788	+	0.994599i	$0.466904\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	41.5384	1.44443	0.722217	−	0.691667i	$-0.243123\pi$
$827$	41.5384	1.44443	0.722217	+	0.691667i	$0.243123\pi$
$828$	0	0
$829$	−20.6843	−0.718396	−0.359198	−	0.933261i	$-0.616950\pi$
$829$	−20.6843	−0.718396	−0.359198	+	0.933261i	$0.616950\pi$
$830$	0	0
$831$	−26.1114	−0.905794
$832$	0	0
$833$	1.59284	0.0551888
$834$	0	0
$835$	9.87000	0.341565
$836$	0	0
$837$	−6.52784	−0.225635
$838$	0	0
$839$	−16.4921	−0.569369	−0.284685	−	0.958621i	$-0.591889\pi$
$839$	−16.4921	−0.569369	−0.284685	+	0.958621i	$0.591889\pi$
$840$	0	0
$841$	8.46284	0.291822
$842$	0	0
$843$	−13.1857	−0.454139
$844$	0	0
$845$	−29.6128	−1.01871
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	26.3978	0.905972
$850$	0	0
$851$	−10.3978	−0.356434
$852$	0	0
$853$	−15.9151	−0.544922	−0.272461	−	0.962167i	$-0.587838\pi$
$853$	−15.9151	−0.544922	−0.272461	+	0.962167i	$0.587838\pi$
$854$	0	0
$855$	−0.407155	−0.0139244
$856$	0	0
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	10.0451	0.342734	0.171367	−	0.985207i	$-0.445182\pi$
$859$	10.0451	0.342734	0.171367	+	0.985207i	$0.445182\pi$
$860$	0	0
$861$	4.52784	0.154309
$862$	0	0
$863$	−8.77853	−0.298825	−0.149412	−	0.988775i	$-0.547738\pi$
$863$	−8.77853	−0.298825	−0.149412	+	0.988775i	$0.547738\pi$
$864$	0	0
$865$	10.2414	0.348217
$866$	0	0
$867$	−14.4628	−0.491184
$868$	0	0
$869$	−4.52784	−0.153597
$870$	0	0
$871$	39.1671	1.32713
$872$	0	0
$873$	−2.12069	−0.0717745
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−6.23207	−0.210442	−0.105221	−	0.994449i	$-0.533555\pi$
$877$	−6.23207	−0.210442	−0.105221	+	0.994449i	$0.533555\pi$
$878$	0	0
$879$	−2.16578	−0.0730498
$880$	0	0
$881$	−32.7334	−1.10282	−0.551409	−	0.834235i	$-0.685910\pi$
$881$	−32.7334	−1.10282	−0.551409	+	0.834235i	$0.685910\pi$
$882$	0	0
$883$	−7.38724	−0.248600	−0.124300	−	0.992245i	$-0.539669\pi$
$883$	−7.38724	−0.248600	−0.124300	+	0.992245i	$0.539669\pi$
$884$	0	0
$885$	−14.1207	−0.474662
$886$	0	0
$887$	9.22147	0.309626	0.154813	−	0.987944i	$-0.450522\pi$
$887$	9.22147	0.309626	0.154813	+	0.987944i	$0.450522\pi$
$888$	0	0
$889$	−8.12069	−0.272359
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	3.14060	0.105096
$894$	0	0
$895$	−21.5835	−0.721458
$896$	0	0
$897$	−10.3978	−0.347174
$898$	0	0
$899$	39.9549	1.33257
$900$	0	0
$901$	−3.83422	−0.127737
$902$	0	0
$903$	−11.3064	−0.376253
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−15.5835	−0.517443	−0.258721	−	0.965952i	$-0.583301\pi$
$907$	−15.5835	−0.517443	−0.258721	+	0.965952i	$0.583301\pi$
$908$	0	0
$909$	3.18569	0.105663
$910$	0	0
$911$	−53.0557	−1.75781	−0.878907	−	0.476994i	$-0.841726\pi$
$911$	−53.0557	−1.75781	−0.878907	+	0.476994i	$0.841726\pi$
$912$	0	0
$913$	−3.59284	−0.118906
$914$	0	0
$915$	−0.407155	−0.0134601
$916$	0	0
$917$	−15.7135	−0.518907
$918$	0	0
$919$	54.0398	1.78261	0.891305	−	0.453405i	$-0.149791\pi$
$919$	54.0398	1.78261	0.891305	+	0.453405i	$0.149791\pi$
$920$	0	0
$921$	−5.87000	−0.193423
$922$	0	0
$923$	−29.5571	−0.972883
$924$	0	0
$925$	6.52784	0.214634
$926$	0	0
$927$	8.93500	0.293464
$928$	0	0
$929$	1.66845	0.0547401	0.0273700	−	0.999625i	$-0.491287\pi$
$929$	1.66845	0.0547401	0.0273700	+	0.999625i	$0.491287\pi$
$930$	0	0
$931$	0.407155	0.0133440
$932$	0	0
$933$	19.8700	0.650515
$934$	0	0
$935$	−1.59284	−0.0520916
$936$	0	0
$937$	18.0265	0.588899	0.294449	−	0.955667i	$-0.404864\pi$
$937$	18.0265	0.588899	0.294449	+	0.955667i	$0.404864\pi$
$938$	0	0
$939$	−12.4921	−0.407663
$940$	0	0
$941$	19.1406	0.623966	0.311983	−	0.950088i	$-0.399007\pi$
$941$	19.1406	0.623966	0.311983	+	0.950088i	$0.399007\pi$
$942$	0	0
$943$	7.21216	0.234860
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−6.90853	−0.224497	−0.112249	−	0.993680i	$-0.535805\pi$
$947$	−6.90853	−0.224497	−0.112249	+	0.993680i	$0.535805\pi$
$948$	0	0
$949$	−66.8541	−2.17018
$950$	0	0
$951$	19.2971	0.625750
$952$	0	0
$953$	5.29707	0.171589	0.0857945	−	0.996313i	$-0.472657\pi$
$953$	5.29707	0.171589	0.0857945	+	0.996313i	$0.472657\pi$
$954$	0	0
$955$	6.37138	0.206173
$956$	0	0
$957$	−6.12069	−0.197854
$958$	0	0
$959$	−18.2414	−0.589045
$960$	0	0
$961$	11.6128	0.374605
$962$	0	0
$963$	9.34216	0.301047
$964$	0	0
$965$	−18.6128	−0.599166
$966$	0	0
$967$	14.5636	0.468334	0.234167	−	0.972196i	$-0.424764\pi$
$967$	14.5636	0.468334	0.234167	+	0.972196i	$0.424764\pi$
$968$	0	0
$969$	0.648535	0.0208339
$970$	0	0
$971$	31.6778	1.01659	0.508294	−	0.861184i	$-0.330276\pi$
$971$	31.6778	1.01659	0.508294	+	0.861184i	$0.330276\pi$
$972$	0	0
$973$	4.24138	0.135972
$974$	0	0
$975$	6.52784	0.209058
$976$	0	0
$977$	−43.6327	−1.39593	−0.697966	−	0.716130i	$-0.745912\pi$
$977$	−43.6327	−1.39593	−0.697966	+	0.716130i	$0.745912\pi$
$978$	0	0
$979$	−5.30638	−0.169593
$980$	0	0
$981$	14.5278	0.463838
$982$	0	0
$983$	12.2414	0.390439	0.195220	−	0.980760i	$-0.437458\pi$
$983$	12.2414	0.390439	0.195220	+	0.980760i	$0.437458\pi$
$984$	0	0
$985$	7.42707	0.236646
$986$	0	0
$987$	−7.71353	−0.245525
$988$	0	0
$989$	−18.0093	−0.572663
$990$	0	0
$991$	40.3885	1.28298	0.641492	−	0.767130i	$-0.278316\pi$
$991$	40.3885	1.28298	0.641492	+	0.767130i	$0.278316\pi$
$992$	0	0
$993$	2.77853	0.0881741
$994$	0	0
$995$	−17.1857	−0.544823
$996$	0	0
$997$	12.8541	0.407095	0.203547	−	0.979065i	$-0.434753\pi$
$997$	12.8541	0.407095	0.203547	+	0.979065i	$0.434753\pi$
$998$	0	0
$999$	6.52784	0.206532

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.a.u.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.u.1.3	✓	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:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +6.52784 q^{13} -1.00000 q^{15} +1.59284 q^{17} +0.407155 q^{19} -1.00000 q^{21} -1.59284 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.12069 q^{29} -6.52784 q^{31} +1.00000 q^{33} +1.00000 q^{35} +6.52784 q^{37} +6.52784 q^{39} -4.52784 q^{41} +11.3064 q^{43} -1.00000 q^{45} +7.71353 q^{47} +1.00000 q^{49} +1.59284 q^{51} -2.40716 q^{53} -1.00000 q^{55} +0.407155 q^{57} +14.1207 q^{59} +0.407155 q^{61} -1.00000 q^{63} -6.52784 q^{65} +6.00000 q^{67} -1.59284 q^{69} -4.52784 q^{71} -10.2414 q^{73} +1.00000 q^{75} -1.00000 q^{77} -4.52784 q^{79} +1.00000 q^{81} -3.59284 q^{83} -1.59284 q^{85} -6.12069 q^{87} -5.30638 q^{89} -6.52784 q^{91} -6.52784 q^{93} -0.407155 q^{95} -2.12069 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{15} + 4 q^{17} + 2 q^{19} - 3 q^{21} - 4 q^{23} + 3 q^{25} + 3 q^{27} + 2 q^{29} + 3 q^{33} + 3 q^{35} + 6 q^{41} + 12 q^{43} - 3 q^{45} + 2 q^{47} + 3 q^{49} + 4 q^{51} - 8 q^{53} - 3 q^{55} + 2 q^{57} + 22 q^{59} + 2 q^{61} - 3 q^{63} + 18 q^{67} - 4 q^{69} + 6 q^{71} + 10 q^{73} + 3 q^{75} - 3 q^{77} + 6 q^{79} + 3 q^{81} - 10 q^{83} - 4 q^{85} + 2 q^{87} + 6 q^{89} - 2 q^{95} + 14 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^15 + 4 * q^17 + 2 * q^19 - 3 * q^21 - 4 * q^23 + 3 * q^25 + 3 * q^27 + 2 * q^29 + 3 * q^33 + 3 * q^35 + 6 * q^41 + 12 * q^43 - 3 * q^45 + 2 * q^47 + 3 * q^49 + 4 * q^51 - 8 * q^53 - 3 * q^55 + 2 * q^57 + 22 * q^59 + 2 * q^61 - 3 * q^63 + 18 * q^67 - 4 * q^69 + 6 * q^71 + 10 * q^73 + 3 * q^75 - 3 * q^77 + 6 * q^79 + 3 * q^81 - 10 * q^83 - 4 * q^85 + 2 * q^87 + 6 * q^89 - 2 * q^95 + 14 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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