LMFDB - Embedded newform 3536.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3536,2,Mod(1,3536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3536, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3536.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3536 = 2^{4} \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3536.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:	$28.2351021547$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1768)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	3536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} -0.732051 q^{5} -4.00000 q^{7} +4.46410 q^{9} +O(q^{10})$	$q-2.73205 q^{3} -0.732051 q^{5} -4.00000 q^{7} +4.46410 q^{9} -3.46410 q^{11} +1.00000 q^{13} +2.00000 q^{15} +1.00000 q^{17} +3.46410 q^{19} +10.9282 q^{21} +2.19615 q^{23} -4.46410 q^{25} -4.00000 q^{27} +5.46410 q^{29} +1.46410 q^{31} +9.46410 q^{33} +2.92820 q^{35} +2.19615 q^{37} -2.73205 q^{39} -8.19615 q^{41} +5.46410 q^{43} -3.26795 q^{45} +6.00000 q^{47} +9.00000 q^{49} -2.73205 q^{51} +8.00000 q^{53} +2.53590 q^{55} -9.46410 q^{57} -0.535898 q^{59} -4.00000 q^{61} -17.8564 q^{63} -0.732051 q^{65} +8.00000 q^{67} -6.00000 q^{69} -1.07180 q^{71} +0.196152 q^{73} +12.1962 q^{75} +13.8564 q^{77} -7.66025 q^{79} -2.46410 q^{81} -8.00000 q^{83} -0.732051 q^{85} -14.9282 q^{87} +4.53590 q^{89} -4.00000 q^{91} -4.00000 q^{93} -2.53590 q^{95} +9.26795 q^{97} -15.4641 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 8 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 - 8 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} - 8 q^{7} + 2 q^{9} + 2 q^{13} + 4 q^{15} + 2 q^{17} + 8 q^{21} - 6 q^{23} - 2 q^{25} - 8 q^{27} + 4 q^{29} - 4 q^{31} + 12 q^{33} - 8 q^{35} - 6 q^{37} - 2 q^{39} - 6 q^{41} + 4 q^{43} - 10 q^{45} + 12 q^{47} + 18 q^{49} - 2 q^{51} + 16 q^{53} + 12 q^{55} - 12 q^{57} - 8 q^{59} - 8 q^{61} - 8 q^{63} + 2 q^{65} + 16 q^{67} - 12 q^{69} - 16 q^{71} - 10 q^{73} + 14 q^{75} + 2 q^{79} + 2 q^{81} - 16 q^{83} + 2 q^{85} - 16 q^{87} + 16 q^{89} - 8 q^{91} - 8 q^{93} - 12 q^{95} + 22 q^{97} - 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 8 * q^7 + 2 * q^9 + 2 * q^13 + 4 * q^15 + 2 * q^17 + 8 * q^21 - 6 * q^23 - 2 * q^25 - 8 * q^27 + 4 * q^29 - 4 * q^31 + 12 * q^33 - 8 * q^35 - 6 * q^37 - 2 * q^39 - 6 * q^41 + 4 * q^43 - 10 * q^45 + 12 * q^47 + 18 * q^49 - 2 * q^51 + 16 * q^53 + 12 * q^55 - 12 * q^57 - 8 * q^59 - 8 * q^61 - 8 * q^63 + 2 * q^65 + 16 * q^67 - 12 * q^69 - 16 * q^71 - 10 * q^73 + 14 * q^75 + 2 * q^79 + 2 * q^81 - 16 * q^83 + 2 * q^85 - 16 * q^87 + 16 * q^89 - 8 * q^91 - 8 * q^93 - 12 * q^95 + 22 * q^97 - 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	0	0
$5$	−0.732051	−0.327383	−0.163692	−	0.986512i	$-0.552340\pi$
$5$	−0.732051	−0.327383	−0.163692	+	0.986512i	$0.552340\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	0	0
$21$	10.9282	2.38473
$22$	0	0
$23$	2.19615	0.457929	0.228965	−	0.973435i	$-0.426466\pi$
$23$	2.19615	0.457929	0.228965	+	0.973435i	$0.426466\pi$
$24$	0	0
$25$	−4.46410	−0.892820
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	5.46410	1.01466	0.507329	−	0.861752i	$-0.330633\pi$
$29$	5.46410	1.01466	0.507329	+	0.861752i	$0.330633\pi$
$30$	0	0
$31$	1.46410	0.262960	0.131480	−	0.991319i	$-0.458027\pi$
$31$	1.46410	0.262960	0.131480	+	0.991319i	$0.458027\pi$
$32$	0	0
$33$	9.46410	1.64749
$34$	0	0
$35$	2.92820	0.494957
$36$	0	0
$37$	2.19615	0.361045	0.180523	−	0.983571i	$-0.442221\pi$
$37$	2.19615	0.361045	0.180523	+	0.983571i	$0.442221\pi$
$38$	0	0
$39$	−2.73205	−0.437478
$40$	0	0
$41$	−8.19615	−1.28002	−0.640012	−	0.768365i	$-0.721071\pi$
$41$	−8.19615	−1.28002	−0.640012	+	0.768365i	$0.721071\pi$
$42$	0	0
$43$	5.46410	0.833268	0.416634	−	0.909074i	$-0.363210\pi$
$43$	5.46410	0.833268	0.416634	+	0.909074i	$0.363210\pi$
$44$	0	0
$45$	−3.26795	−0.487157
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	−2.73205	−0.382564
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	2.53590	0.341940
$56$	0	0
$57$	−9.46410	−1.25355
$58$	0	0
$59$	−0.535898	−0.0697680	−0.0348840	−	0.999391i	$-0.511106\pi$
$59$	−0.535898	−0.0697680	−0.0348840	+	0.999391i	$0.511106\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	−17.8564	−2.24970
$64$	0	0
$65$	−0.732051	−0.0907997
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−1.07180	−0.127199	−0.0635994	−	0.997976i	$-0.520258\pi$
$71$	−1.07180	−0.127199	−0.0635994	+	0.997976i	$0.520258\pi$
$72$	0	0
$73$	0.196152	0.0229579	0.0114790	−	0.999934i	$-0.496346\pi$
$73$	0.196152	0.0229579	0.0114790	+	0.999934i	$0.496346\pi$
$74$	0	0
$75$	12.1962	1.40829
$76$	0	0
$77$	13.8564	1.57908
$78$	0	0
$79$	−7.66025	−0.861846	−0.430923	−	0.902389i	$-0.641812\pi$
$79$	−7.66025	−0.861846	−0.430923	+	0.902389i	$0.641812\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	−0.732051	−0.0794021
$86$	0	0
$87$	−14.9282	−1.60047
$88$	0	0
$89$	4.53590	0.480804	0.240402	−	0.970673i	$-0.422721\pi$
$89$	4.53590	0.480804	0.240402	+	0.970673i	$0.422721\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−2.53590	−0.260178
$96$	0	0
$97$	9.26795	0.941018	0.470509	−	0.882395i	$-0.344070\pi$
$97$	9.26795	0.941018	0.470509	+	0.882395i	$0.344070\pi$
$98$	0	0
$99$	−15.4641	−1.55420
$100$	0	0
$101$	−1.07180	−0.106648	−0.0533239	−	0.998577i	$-0.516982\pi$
$101$	−1.07180	−0.106648	−0.0533239	+	0.998577i	$0.516982\pi$
$102$	0	0
$103$	−5.46410	−0.538394	−0.269197	−	0.963085i	$-0.586758\pi$
$103$	−5.46410	−0.538394	−0.269197	+	0.963085i	$0.586758\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	0	0
$107$	5.66025	0.547197	0.273599	−	0.961844i	$-0.411786\pi$
$107$	5.66025	0.547197	0.273599	+	0.961844i	$0.411786\pi$
$108$	0	0
$109$	−11.6603	−1.11685	−0.558425	−	0.829555i	$-0.688594\pi$
$109$	−11.6603	−1.11685	−0.558425	+	0.829555i	$0.688594\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	2.39230	0.225049	0.112525	−	0.993649i	$-0.464106\pi$
$113$	2.39230	0.225049	0.112525	+	0.993649i	$0.464106\pi$
$114$	0	0
$115$	−1.60770	−0.149918
$116$	0	0
$117$	4.46410	0.412706
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	22.3923	2.01905
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−10.9282	−0.969721	−0.484861	−	0.874591i	$-0.661130\pi$
$127$	−10.9282	−0.969721	−0.484861	+	0.874591i	$0.661130\pi$
$128$	0	0
$129$	−14.9282	−1.31436
$130$	0	0
$131$	−16.1962	−1.41506	−0.707532	−	0.706681i	$-0.750192\pi$
$131$	−16.1962	−1.41506	−0.707532	+	0.706681i	$0.750192\pi$
$132$	0	0
$133$	−13.8564	−1.20150
$134$	0	0
$135$	2.92820	0.252020
$136$	0	0
$137$	−7.85641	−0.671218	−0.335609	−	0.942001i	$-0.608942\pi$
$137$	−7.85641	−0.671218	−0.335609	+	0.942001i	$0.608942\pi$
$138$	0	0
$139$	21.6603	1.83720	0.918599	−	0.395190i	$-0.129321\pi$
$139$	21.6603	1.83720	0.918599	+	0.395190i	$0.129321\pi$
$140$	0	0
$141$	−16.3923	−1.38048
$142$	0	0
$143$	−3.46410	−0.289683
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	−24.5885	−2.02802
$148$	0	0
$149$	11.4641	0.939176	0.469588	−	0.882886i	$-0.344403\pi$
$149$	11.4641	0.939176	0.469588	+	0.882886i	$0.344403\pi$
$150$	0	0
$151$	−5.07180	−0.412737	−0.206368	−	0.978474i	$-0.566165\pi$
$151$	−5.07180	−0.412737	−0.206368	+	0.978474i	$0.566165\pi$
$152$	0	0
$153$	4.46410	0.360901
$154$	0	0
$155$	−1.07180	−0.0860888
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	0	0
$159$	−21.8564	−1.73333
$160$	0	0
$161$	−8.78461	−0.692324
$162$	0	0
$163$	23.8564	1.86858	0.934289	−	0.356517i	$-0.116036\pi$
$163$	23.8564	1.86858	0.934289	+	0.356517i	$0.116036\pi$
$164$	0	0
$165$	−6.92820	−0.539360
$166$	0	0
$167$	−13.8564	−1.07224	−0.536120	−	0.844141i	$-0.680111\pi$
$167$	−13.8564	−1.07224	−0.536120	+	0.844141i	$0.680111\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	15.4641	1.18257
$172$	0	0
$173$	−2.53590	−0.192801	−0.0964004	−	0.995343i	$-0.530733\pi$
$173$	−2.53590	−0.192801	−0.0964004	+	0.995343i	$0.530733\pi$
$174$	0	0
$175$	17.8564	1.34982
$176$	0	0
$177$	1.46410	0.110049
$178$	0	0
$179$	17.8564	1.33465	0.667325	−	0.744766i	$-0.267439\pi$
$179$	17.8564	1.33465	0.667325	+	0.744766i	$0.267439\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	10.9282	0.807836
$184$	0	0
$185$	−1.60770	−0.118200
$186$	0	0
$187$	−3.46410	−0.253320
$188$	0	0
$189$	16.0000	1.16383
$190$	0	0
$191$	4.39230	0.317816	0.158908	−	0.987293i	$-0.449203\pi$
$191$	4.39230	0.317816	0.158908	+	0.987293i	$0.449203\pi$
$192$	0	0
$193$	−22.7321	−1.63629	−0.818144	−	0.575013i	$-0.804997\pi$
$193$	−22.7321	−1.63629	−0.818144	+	0.575013i	$0.804997\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	−13.1244	−0.935072	−0.467536	−	0.883974i	$-0.654858\pi$
$197$	−13.1244	−0.935072	−0.467536	+	0.883974i	$0.654858\pi$
$198$	0	0
$199$	−13.8038	−0.978529	−0.489264	−	0.872136i	$-0.662735\pi$
$199$	−13.8038	−0.978529	−0.489264	+	0.872136i	$0.662735\pi$
$200$	0	0
$201$	−21.8564	−1.54163
$202$	0	0
$203$	−21.8564	−1.53402
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	9.80385	0.681415
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−10.0526	−0.692047	−0.346023	−	0.938226i	$-0.612468\pi$
$211$	−10.0526	−0.692047	−0.346023	+	0.938226i	$0.612468\pi$
$212$	0	0
$213$	2.92820	0.200637
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	−5.85641	−0.397559
$218$	0	0
$219$	−0.535898	−0.0362127
$220$	0	0
$221$	1.00000	0.0672673
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	−19.9282	−1.32855
$226$	0	0
$227$	7.46410	0.495410	0.247705	−	0.968836i	$-0.420324\pi$
$227$	7.46410	0.495410	0.247705	+	0.968836i	$0.420324\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	−37.8564	−2.49077
$232$	0	0
$233$	14.3923	0.942871	0.471436	−	0.881900i	$-0.343736\pi$
$233$	14.3923	0.942871	0.471436	+	0.881900i	$0.343736\pi$
$234$	0	0
$235$	−4.39230	−0.286522
$236$	0	0
$237$	20.9282	1.35943
$238$	0	0
$239$	−24.7846	−1.60318	−0.801592	−	0.597872i	$-0.796013\pi$
$239$	−24.7846	−1.60318	−0.801592	+	0.597872i	$0.796013\pi$
$240$	0	0
$241$	−23.5167	−1.51484	−0.757421	−	0.652927i	$-0.773541\pi$
$241$	−23.5167	−1.51484	−0.757421	+	0.652927i	$0.773541\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	−6.58846	−0.420921
$246$	0	0
$247$	3.46410	0.220416
$248$	0	0
$249$	21.8564	1.38509
$250$	0	0
$251$	−12.3923	−0.782195	−0.391098	−	0.920349i	$-0.627905\pi$
$251$	−12.3923	−0.782195	−0.391098	+	0.920349i	$0.627905\pi$
$252$	0	0
$253$	−7.60770	−0.478292
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	24.9282	1.55498	0.777489	−	0.628896i	$-0.216493\pi$
$257$	24.9282	1.55498	0.777489	+	0.628896i	$0.216493\pi$
$258$	0	0
$259$	−8.78461	−0.545849
$260$	0	0
$261$	24.3923	1.50985
$262$	0	0
$263$	−25.8564	−1.59437	−0.797187	−	0.603732i	$-0.793680\pi$
$263$	−25.8564	−1.59437	−0.797187	+	0.603732i	$0.793680\pi$
$264$	0	0
$265$	−5.85641	−0.359756
$266$	0	0
$267$	−12.3923	−0.758397
$268$	0	0
$269$	12.3923	0.755572	0.377786	−	0.925893i	$-0.376685\pi$
$269$	12.3923	0.755572	0.377786	+	0.925893i	$0.376685\pi$
$270$	0	0
$271$	2.92820	0.177876	0.0889378	−	0.996037i	$-0.471653\pi$
$271$	2.92820	0.177876	0.0889378	+	0.996037i	$0.471653\pi$
$272$	0	0
$273$	10.9282	0.661405
$274$	0	0
$275$	15.4641	0.932520
$276$	0	0
$277$	−12.3923	−0.744581	−0.372291	−	0.928116i	$-0.621428\pi$
$277$	−12.3923	−0.744581	−0.372291	+	0.928116i	$0.621428\pi$
$278$	0	0
$279$	6.53590	0.391294
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	−27.5167	−1.63570	−0.817848	−	0.575435i	$-0.804833\pi$
$283$	−27.5167	−1.63570	−0.817848	+	0.575435i	$0.804833\pi$
$284$	0	0
$285$	6.92820	0.410391
$286$	0	0
$287$	32.7846	1.93521
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−25.3205	−1.48431
$292$	0	0
$293$	−11.0718	−0.646821	−0.323411	−	0.946259i	$-0.604830\pi$
$293$	−11.0718	−0.646821	−0.323411	+	0.946259i	$0.604830\pi$
$294$	0	0
$295$	0.392305	0.0228409
$296$	0	0
$297$	13.8564	0.804030
$298$	0	0
$299$	2.19615	0.127007
$300$	0	0
$301$	−21.8564	−1.25978
$302$	0	0
$303$	2.92820	0.168221
$304$	0	0
$305$	2.92820	0.167668
$306$	0	0
$307$	−22.3923	−1.27800	−0.638998	−	0.769208i	$-0.720651\pi$
$307$	−22.3923	−1.27800	−0.638998	+	0.769208i	$0.720651\pi$
$308$	0	0
$309$	14.9282	0.849236
$310$	0	0
$311$	2.87564	0.163063	0.0815314	−	0.996671i	$-0.474019\pi$
$311$	2.87564	0.163063	0.0815314	+	0.996671i	$0.474019\pi$
$312$	0	0
$313$	17.3205	0.979013	0.489506	−	0.872000i	$-0.337177\pi$
$313$	17.3205	0.979013	0.489506	+	0.872000i	$0.337177\pi$
$314$	0	0
$315$	13.0718	0.736512
$316$	0	0
$317$	−15.2679	−0.857533	−0.428767	−	0.903415i	$-0.641052\pi$
$317$	−15.2679	−0.857533	−0.428767	+	0.903415i	$0.641052\pi$
$318$	0	0
$319$	−18.9282	−1.05978
$320$	0	0
$321$	−15.4641	−0.863122
$322$	0	0
$323$	3.46410	0.192748
$324$	0	0
$325$	−4.46410	−0.247624
$326$	0	0
$327$	31.8564	1.76166
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−16.7846	−0.922566	−0.461283	−	0.887253i	$-0.652611\pi$
$331$	−16.7846	−0.922566	−0.461283	+	0.887253i	$0.652611\pi$
$332$	0	0
$333$	9.80385	0.537248
$334$	0	0
$335$	−5.85641	−0.319970
$336$	0	0
$337$	−6.78461	−0.369581	−0.184791	−	0.982778i	$-0.559161\pi$
$337$	−6.78461	−0.369581	−0.184791	+	0.982778i	$0.559161\pi$
$338$	0	0
$339$	−6.53590	−0.354981
$340$	0	0
$341$	−5.07180	−0.274653
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	4.39230	0.236474
$346$	0	0
$347$	19.5167	1.04771	0.523855	−	0.851808i	$-0.324494\pi$
$347$	19.5167	1.04771	0.523855	+	0.851808i	$0.324494\pi$
$348$	0	0
$349$	4.53590	0.242801	0.121401	−	0.992604i	$-0.461261\pi$
$349$	4.53590	0.242801	0.121401	+	0.992604i	$0.461261\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−24.2487	−1.29063	−0.645314	−	0.763917i	$-0.723274\pi$
$353$	−24.2487	−1.29063	−0.645314	+	0.763917i	$0.723274\pi$
$354$	0	0
$355$	0.784610	0.0416428
$356$	0	0
$357$	10.9282	0.578382
$358$	0	0
$359$	0.928203	0.0489887	0.0244943	−	0.999700i	$-0.492202\pi$
$359$	0.928203	0.0489887	0.0244943	+	0.999700i	$0.492202\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	−2.73205	−0.143395
$364$	0	0
$365$	−0.143594	−0.00751603
$366$	0	0
$367$	26.1962	1.36743	0.683714	−	0.729750i	$-0.260364\pi$
$367$	26.1962	1.36743	0.683714	+	0.729750i	$0.260364\pi$
$368$	0	0
$369$	−36.5885	−1.90472
$370$	0	0
$371$	−32.0000	−1.66136
$372$	0	0
$373$	28.7846	1.49041	0.745205	−	0.666835i	$-0.232351\pi$
$373$	28.7846	1.49041	0.745205	+	0.666835i	$0.232351\pi$
$374$	0	0
$375$	−18.9282	−0.977448
$376$	0	0
$377$	5.46410	0.281416
$378$	0	0
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	0	0
$381$	29.8564	1.52959
$382$	0	0
$383$	−0.143594	−0.00733729	−0.00366864	−	0.999993i	$-0.501168\pi$
$383$	−0.143594	−0.00733729	−0.00366864	+	0.999993i	$0.501168\pi$
$384$	0	0
$385$	−10.1436	−0.516965
$386$	0	0
$387$	24.3923	1.23993
$388$	0	0
$389$	28.9282	1.46672	0.733359	−	0.679842i	$-0.237951\pi$
$389$	28.9282	1.46672	0.733359	+	0.679842i	$0.237951\pi$
$390$	0	0
$391$	2.19615	0.111064
$392$	0	0
$393$	44.2487	2.23205
$394$	0	0
$395$	5.60770	0.282154
$396$	0	0
$397$	−14.1962	−0.712484	−0.356242	−	0.934394i	$-0.615942\pi$
$397$	−14.1962	−0.712484	−0.356242	+	0.934394i	$0.615942\pi$
$398$	0	0
$399$	37.8564	1.89519
$400$	0	0
$401$	30.4449	1.52034	0.760172	−	0.649722i	$-0.225115\pi$
$401$	30.4449	1.52034	0.760172	+	0.649722i	$0.225115\pi$
$402$	0	0
$403$	1.46410	0.0729321
$404$	0	0
$405$	1.80385	0.0896339
$406$	0	0
$407$	−7.60770	−0.377099
$408$	0	0
$409$	31.1769	1.54160	0.770800	−	0.637078i	$-0.219857\pi$
$409$	31.1769	1.54160	0.770800	+	0.637078i	$0.219857\pi$
$410$	0	0
$411$	21.4641	1.05875
$412$	0	0
$413$	2.14359	0.105479
$414$	0	0
$415$	5.85641	0.287480
$416$	0	0
$417$	−59.1769	−2.89791
$418$	0	0
$419$	−3.12436	−0.152635	−0.0763174	−	0.997084i	$-0.524316\pi$
$419$	−3.12436	−0.152635	−0.0763174	+	0.997084i	$0.524316\pi$
$420$	0	0
$421$	−22.3923	−1.09133	−0.545667	−	0.838002i	$-0.683724\pi$
$421$	−22.3923	−1.09133	−0.545667	+	0.838002i	$0.683724\pi$
$422$	0	0
$423$	26.7846	1.30231
$424$	0	0
$425$	−4.46410	−0.216541
$426$	0	0
$427$	16.0000	0.774294
$428$	0	0
$429$	9.46410	0.456931
$430$	0	0
$431$	41.1769	1.98342	0.991711	−	0.128488i	$-0.0410123\pi$
$431$	41.1769	1.98342	0.991711	+	0.128488i	$0.0410123\pi$
$432$	0	0
$433$	12.3923	0.595536	0.297768	−	0.954638i	$-0.403758\pi$
$433$	12.3923	0.595536	0.297768	+	0.954638i	$0.403758\pi$
$434$	0	0
$435$	10.9282	0.523967
$436$	0	0
$437$	7.60770	0.363925
$438$	0	0
$439$	−29.1244	−1.39003	−0.695015	−	0.718995i	$-0.744602\pi$
$439$	−29.1244	−1.39003	−0.695015	+	0.718995i	$0.744602\pi$
$440$	0	0
$441$	40.1769	1.91319
$442$	0	0
$443$	33.4641	1.58993	0.794964	−	0.606657i	$-0.207490\pi$
$443$	33.4641	1.58993	0.794964	+	0.606657i	$0.207490\pi$
$444$	0	0
$445$	−3.32051	−0.157407
$446$	0	0
$447$	−31.3205	−1.48141
$448$	0	0
$449$	−22.0526	−1.04072	−0.520362	−	0.853946i	$-0.674203\pi$
$449$	−22.0526	−1.04072	−0.520362	+	0.853946i	$0.674203\pi$
$450$	0	0
$451$	28.3923	1.33694
$452$	0	0
$453$	13.8564	0.651031
$454$	0	0
$455$	2.92820	0.137276
$456$	0	0
$457$	−12.9282	−0.604756	−0.302378	−	0.953188i	$-0.597780\pi$
$457$	−12.9282	−0.604756	−0.302378	+	0.953188i	$0.597780\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	3.85641	0.179611	0.0898054	−	0.995959i	$-0.471375\pi$
$461$	3.85641	0.179611	0.0898054	+	0.995959i	$0.471375\pi$
$462$	0	0
$463$	8.92820	0.414929	0.207464	−	0.978243i	$-0.433479\pi$
$463$	8.92820	0.414929	0.207464	+	0.978243i	$0.433479\pi$
$464$	0	0
$465$	2.92820	0.135792
$466$	0	0
$467$	16.0000	0.740392	0.370196	−	0.928954i	$-0.379291\pi$
$467$	16.0000	0.740392	0.370196	+	0.928954i	$0.379291\pi$
$468$	0	0
$469$	−32.0000	−1.47762
$470$	0	0
$471$	−10.9282	−0.503545
$472$	0	0
$473$	−18.9282	−0.870320
$474$	0	0
$475$	−15.4641	−0.709542
$476$	0	0
$477$	35.7128	1.63518
$478$	0	0
$479$	−34.9282	−1.59591	−0.797955	−	0.602717i	$-0.794085\pi$
$479$	−34.9282	−1.59591	−0.797955	+	0.602717i	$0.794085\pi$
$480$	0	0
$481$	2.19615	0.100136
$482$	0	0
$483$	24.0000	1.09204
$484$	0	0
$485$	−6.78461	−0.308073
$486$	0	0
$487$	36.3923	1.64909	0.824546	−	0.565794i	$-0.191430\pi$
$487$	36.3923	1.64909	0.824546	+	0.565794i	$0.191430\pi$
$488$	0	0
$489$	−65.1769	−2.94740
$490$	0	0
$491$	−2.92820	−0.132148	−0.0660740	−	0.997815i	$-0.521047\pi$
$491$	−2.92820	−0.132148	−0.0660740	+	0.997815i	$0.521047\pi$
$492$	0	0
$493$	5.46410	0.246091
$494$	0	0
$495$	11.3205	0.508819
$496$	0	0
$497$	4.28719	0.192307
$498$	0	0
$499$	−3.07180	−0.137513	−0.0687563	−	0.997633i	$-0.521903\pi$
$499$	−3.07180	−0.137513	−0.0687563	+	0.997633i	$0.521903\pi$
$500$	0	0
$501$	37.8564	1.69130
$502$	0	0
$503$	−38.9808	−1.73807	−0.869033	−	0.494754i	$-0.835258\pi$
$503$	−38.9808	−1.73807	−0.869033	+	0.494754i	$0.835258\pi$
$504$	0	0
$505$	0.784610	0.0349147
$506$	0	0
$507$	−2.73205	−0.121335
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−0.784610	−0.0347091
$512$	0	0
$513$	−13.8564	−0.611775
$514$	0	0
$515$	4.00000	0.176261
$516$	0	0
$517$	−20.7846	−0.914106
$518$	0	0
$519$	6.92820	0.304114
$520$	0	0
$521$	−42.3923	−1.85724	−0.928620	−	0.371031i	$-0.879004\pi$
$521$	−42.3923	−1.85724	−0.928620	+	0.371031i	$0.879004\pi$
$522$	0	0
$523$	−22.2487	−0.972868	−0.486434	−	0.873717i	$-0.661703\pi$
$523$	−22.2487	−0.972868	−0.486434	+	0.873717i	$0.661703\pi$
$524$	0	0
$525$	−48.7846	−2.12913
$526$	0	0
$527$	1.46410	0.0637773
$528$	0	0
$529$	−18.1769	−0.790301
$530$	0	0
$531$	−2.39230	−0.103817
$532$	0	0
$533$	−8.19615	−0.355015
$534$	0	0
$535$	−4.14359	−0.179143
$536$	0	0
$537$	−48.7846	−2.10521
$538$	0	0
$539$	−31.1769	−1.34288
$540$	0	0
$541$	−42.9808	−1.84789	−0.923944	−	0.382529i	$-0.875053\pi$
$541$	−42.9808	−1.84789	−0.923944	+	0.382529i	$0.875053\pi$
$542$	0	0
$543$	21.8564	0.937948
$544$	0	0
$545$	8.53590	0.365638
$546$	0	0
$547$	29.6603	1.26818	0.634090	−	0.773259i	$-0.281375\pi$
$547$	29.6603	1.26818	0.634090	+	0.773259i	$0.281375\pi$
$548$	0	0
$549$	−17.8564	−0.762093
$550$	0	0
$551$	18.9282	0.806369
$552$	0	0
$553$	30.6410	1.30299
$554$	0	0
$555$	4.39230	0.186443
$556$	0	0
$557$	−23.8564	−1.01083	−0.505414	−	0.862877i	$-0.668660\pi$
$557$	−23.8564	−1.01083	−0.505414	+	0.862877i	$0.668660\pi$
$558$	0	0
$559$	5.46410	0.231107
$560$	0	0
$561$	9.46410	0.399575
$562$	0	0
$563$	35.3205	1.48858	0.744291	−	0.667855i	$-0.232788\pi$
$563$	35.3205	1.48858	0.744291	+	0.667855i	$0.232788\pi$
$564$	0	0
$565$	−1.75129	−0.0736773
$566$	0	0
$567$	9.85641	0.413930
$568$	0	0
$569$	24.3923	1.02258	0.511289	−	0.859409i	$-0.329168\pi$
$569$	24.3923	1.02258	0.511289	+	0.859409i	$0.329168\pi$
$570$	0	0
$571$	38.4449	1.60887	0.804434	−	0.594042i	$-0.202469\pi$
$571$	38.4449	1.60887	0.804434	+	0.594042i	$0.202469\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	−9.80385	−0.408849
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	0	0
$579$	62.1051	2.58100
$580$	0	0
$581$	32.0000	1.32758
$582$	0	0
$583$	−27.7128	−1.14775
$584$	0	0
$585$	−3.26795	−0.135113
$586$	0	0
$587$	−17.3205	−0.714894	−0.357447	−	0.933933i	$-0.616353\pi$
$587$	−17.3205	−0.714894	−0.357447	+	0.933933i	$0.616353\pi$
$588$	0	0
$589$	5.07180	0.208980
$590$	0	0
$591$	35.8564	1.47494
$592$	0	0
$593$	−0.535898	−0.0220067	−0.0110034	−	0.999939i	$-0.503503\pi$
$593$	−0.535898	−0.0220067	−0.0110034	+	0.999939i	$0.503503\pi$
$594$	0	0
$595$	2.92820	0.120045
$596$	0	0
$597$	37.7128	1.54348
$598$	0	0
$599$	−14.2487	−0.582187	−0.291093	−	0.956695i	$-0.594019\pi$
$599$	−14.2487	−0.582187	−0.291093	+	0.956695i	$0.594019\pi$
$600$	0	0
$601$	12.5359	0.511350	0.255675	−	0.966763i	$-0.417702\pi$
$601$	12.5359	0.511350	0.255675	+	0.966763i	$0.417702\pi$
$602$	0	0
$603$	35.7128	1.45434
$604$	0	0
$605$	−0.732051	−0.0297621
$606$	0	0
$607$	−15.2679	−0.619707	−0.309853	−	0.950784i	$-0.600280\pi$
$607$	−15.2679	−0.619707	−0.309853	+	0.950784i	$0.600280\pi$
$608$	0	0
$609$	59.7128	2.41969
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	−26.7846	−1.08182	−0.540910	−	0.841080i	$-0.681920\pi$
$613$	−26.7846	−1.08182	−0.540910	+	0.841080i	$0.681920\pi$
$614$	0	0
$615$	−16.3923	−0.661002
$616$	0	0
$617$	17.6603	0.710975	0.355488	−	0.934681i	$-0.384315\pi$
$617$	17.6603	0.710975	0.355488	+	0.934681i	$0.384315\pi$
$618$	0	0
$619$	−15.1769	−0.610012	−0.305006	−	0.952350i	$-0.598658\pi$
$619$	−15.1769	−0.610012	−0.305006	+	0.952350i	$0.598658\pi$
$620$	0	0
$621$	−8.78461	−0.352514
$622$	0	0
$623$	−18.1436	−0.726908
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	32.7846	1.30929
$628$	0	0
$629$	2.19615	0.0875663
$630$	0	0
$631$	−24.6410	−0.980943	−0.490472	−	0.871457i	$-0.663175\pi$
$631$	−24.6410	−0.980943	−0.490472	+	0.871457i	$0.663175\pi$
$632$	0	0
$633$	27.4641	1.09160
$634$	0	0
$635$	8.00000	0.317470
$636$	0	0
$637$	9.00000	0.356593
$638$	0	0
$639$	−4.78461	−0.189276
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	0	0
$643$	−3.85641	−0.152082	−0.0760409	−	0.997105i	$-0.524228\pi$
$643$	−3.85641	−0.152082	−0.0760409	+	0.997105i	$0.524228\pi$
$644$	0	0
$645$	10.9282	0.430298
$646$	0	0
$647$	−12.7846	−0.502615	−0.251307	−	0.967907i	$-0.580861\pi$
$647$	−12.7846	−0.502615	−0.251307	+	0.967907i	$0.580861\pi$
$648$	0	0
$649$	1.85641	0.0728703
$650$	0	0
$651$	16.0000	0.627089
$652$	0	0
$653$	47.3205	1.85179	0.925897	−	0.377775i	$-0.123311\pi$
$653$	47.3205	1.85179	0.925897	+	0.377775i	$0.123311\pi$
$654$	0	0
$655$	11.8564	0.463268
$656$	0	0
$657$	0.875644	0.0341621
$658$	0	0
$659$	−0.784610	−0.0305641	−0.0152820	−	0.999883i	$-0.504865\pi$
$659$	−0.784610	−0.0305641	−0.0152820	+	0.999883i	$0.504865\pi$
$660$	0	0
$661$	18.7846	0.730637	0.365318	−	0.930883i	$-0.380960\pi$
$661$	18.7846	0.730637	0.365318	+	0.930883i	$0.380960\pi$
$662$	0	0
$663$	−2.73205	−0.106104
$664$	0	0
$665$	10.1436	0.393352
$666$	0	0
$667$	12.0000	0.464642
$668$	0	0
$669$	−65.5692	−2.53505
$670$	0	0
$671$	13.8564	0.534921
$672$	0	0
$673$	−11.0718	−0.426786	−0.213393	−	0.976966i	$-0.568452\pi$
$673$	−11.0718	−0.426786	−0.213393	+	0.976966i	$0.568452\pi$
$674$	0	0
$675$	17.8564	0.687293
$676$	0	0
$677$	−31.7128	−1.21882	−0.609411	−	0.792854i	$-0.708594\pi$
$677$	−31.7128	−1.21882	−0.609411	+	0.792854i	$0.708594\pi$
$678$	0	0
$679$	−37.0718	−1.42268
$680$	0	0
$681$	−20.3923	−0.781435
$682$	0	0
$683$	35.5692	1.36102	0.680509	−	0.732740i	$-0.261759\pi$
$683$	35.5692	1.36102	0.680509	+	0.732740i	$0.261759\pi$
$684$	0	0
$685$	5.75129	0.219745
$686$	0	0
$687$	71.0333	2.71009
$688$	0	0
$689$	8.00000	0.304776
$690$	0	0
$691$	−12.5359	−0.476888	−0.238444	−	0.971156i	$-0.576637\pi$
$691$	−12.5359	−0.476888	−0.238444	+	0.971156i	$0.576637\pi$
$692$	0	0
$693$	61.8564	2.34973
$694$	0	0
$695$	−15.8564	−0.601468
$696$	0	0
$697$	−8.19615	−0.310451
$698$	0	0
$699$	−39.3205	−1.48724
$700$	0	0
$701$	46.6410	1.76161	0.880803	−	0.473482i	$-0.157003\pi$
$701$	46.6410	1.76161	0.880803	+	0.473482i	$0.157003\pi$
$702$	0	0
$703$	7.60770	0.286930
$704$	0	0
$705$	12.0000	0.451946
$706$	0	0
$707$	4.28719	0.161236
$708$	0	0
$709$	39.3731	1.47869	0.739343	−	0.673329i	$-0.235136\pi$
$709$	39.3731	1.47869	0.739343	+	0.673329i	$0.235136\pi$
$710$	0	0
$711$	−34.1962	−1.28246
$712$	0	0
$713$	3.21539	0.120417
$714$	0	0
$715$	2.53590	0.0948372
$716$	0	0
$717$	67.7128	2.52878
$718$	0	0
$719$	11.2679	0.420224	0.210112	−	0.977677i	$-0.432617\pi$
$719$	11.2679	0.420224	0.210112	+	0.977677i	$0.432617\pi$
$720$	0	0
$721$	21.8564	0.813975
$722$	0	0
$723$	64.2487	2.38944
$724$	0	0
$725$	−24.3923	−0.905907
$726$	0	0
$727$	3.32051	0.123151	0.0615754	−	0.998102i	$-0.480388\pi$
$727$	3.32051	0.123151	0.0615754	+	0.998102i	$0.480388\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	5.46410	0.202097
$732$	0	0
$733$	8.24871	0.304673	0.152337	−	0.988329i	$-0.451320\pi$
$733$	8.24871	0.304673	0.152337	+	0.988329i	$0.451320\pi$
$734$	0	0
$735$	18.0000	0.663940
$736$	0	0
$737$	−27.7128	−1.02081
$738$	0	0
$739$	42.6410	1.56858	0.784288	−	0.620397i	$-0.213029\pi$
$739$	42.6410	1.56858	0.784288	+	0.620397i	$0.213029\pi$
$740$	0	0
$741$	−9.46410	−0.347672
$742$	0	0
$743$	−22.2487	−0.816226	−0.408113	−	0.912931i	$-0.633813\pi$
$743$	−22.2487	−0.816226	−0.408113	+	0.912931i	$0.633813\pi$
$744$	0	0
$745$	−8.39230	−0.307470
$746$	0	0
$747$	−35.7128	−1.30666
$748$	0	0
$749$	−22.6410	−0.827285
$750$	0	0
$751$	−33.1244	−1.20872	−0.604362	−	0.796709i	$-0.706572\pi$
$751$	−33.1244	−1.20872	−0.604362	+	0.796709i	$0.706572\pi$
$752$	0	0
$753$	33.8564	1.23380
$754$	0	0
$755$	3.71281	0.135123
$756$	0	0
$757$	2.92820	0.106427	0.0532137	−	0.998583i	$-0.483054\pi$
$757$	2.92820	0.106427	0.0532137	+	0.998583i	$0.483054\pi$
$758$	0	0
$759$	20.7846	0.754434
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	46.6410	1.68852
$764$	0	0
$765$	−3.26795	−0.118153
$766$	0	0
$767$	−0.535898	−0.0193502
$768$	0	0
$769$	−10.7846	−0.388903	−0.194451	−	0.980912i	$-0.562293\pi$
$769$	−10.7846	−0.388903	−0.194451	+	0.980912i	$0.562293\pi$
$770$	0	0
$771$	−68.1051	−2.45275
$772$	0	0
$773$	19.4641	0.700075	0.350038	−	0.936736i	$-0.386169\pi$
$773$	19.4641	0.700075	0.350038	+	0.936736i	$0.386169\pi$
$774$	0	0
$775$	−6.53590	−0.234776
$776$	0	0
$777$	24.0000	0.860995
$778$	0	0
$779$	−28.3923	−1.01726
$780$	0	0
$781$	3.71281	0.132855
$782$	0	0
$783$	−21.8564	−0.781084
$784$	0	0
$785$	−2.92820	−0.104512
$786$	0	0
$787$	−23.1769	−0.826168	−0.413084	−	0.910693i	$-0.635548\pi$
$787$	−23.1769	−0.826168	−0.413084	+	0.910693i	$0.635548\pi$
$788$	0	0
$789$	70.6410	2.51489
$790$	0	0
$791$	−9.56922	−0.340242
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	16.0000	0.567462
$796$	0	0
$797$	4.14359	0.146774	0.0733868	−	0.997304i	$-0.476619\pi$
$797$	4.14359	0.146774	0.0733868	+	0.997304i	$0.476619\pi$
$798$	0	0
$799$	6.00000	0.212265
$800$	0	0
$801$	20.2487	0.715453
$802$	0	0
$803$	−0.679492	−0.0239787
$804$	0	0
$805$	6.43078	0.226655
$806$	0	0
$807$	−33.8564	−1.19180
$808$	0	0
$809$	53.3205	1.87465	0.937325	−	0.348457i	$-0.113294\pi$
$809$	53.3205	1.87465	0.937325	+	0.348457i	$0.113294\pi$
$810$	0	0
$811$	−39.1769	−1.37569	−0.687844	−	0.725859i	$-0.741443\pi$
$811$	−39.1769	−1.37569	−0.687844	+	0.725859i	$0.741443\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	0	0
$815$	−17.4641	−0.611741
$816$	0	0
$817$	18.9282	0.662214
$818$	0	0
$819$	−17.8564	−0.623953
$820$	0	0
$821$	8.44486	0.294728	0.147364	−	0.989082i	$-0.452921\pi$
$821$	8.44486	0.294728	0.147364	+	0.989082i	$0.452921\pi$
$822$	0	0
$823$	−15.2679	−0.532207	−0.266104	−	0.963944i	$-0.585736\pi$
$823$	−15.2679	−0.532207	−0.266104	+	0.963944i	$0.585736\pi$
$824$	0	0
$825$	−42.2487	−1.47091
$826$	0	0
$827$	13.3205	0.463199	0.231600	−	0.972811i	$-0.425604\pi$
$827$	13.3205	0.463199	0.231600	+	0.972811i	$0.425604\pi$
$828$	0	0
$829$	−38.9282	−1.35203	−0.676016	−	0.736887i	$-0.736295\pi$
$829$	−38.9282	−1.35203	−0.676016	+	0.736887i	$0.736295\pi$
$830$	0	0
$831$	33.8564	1.17447
$832$	0	0
$833$	9.00000	0.311832
$834$	0	0
$835$	10.1436	0.351034
$836$	0	0
$837$	−5.85641	−0.202427
$838$	0	0
$839$	−54.2487	−1.87287	−0.936437	−	0.350836i	$-0.885897\pi$
$839$	−54.2487	−1.87287	−0.936437	+	0.350836i	$0.885897\pi$
$840$	0	0
$841$	0.856406	0.0295313
$842$	0	0
$843$	71.0333	2.44652
$844$	0	0
$845$	−0.732051	−0.0251833
$846$	0	0
$847$	−4.00000	−0.137442
$848$	0	0
$849$	75.1769	2.58007
$850$	0	0
$851$	4.82309	0.165333
$852$	0	0
$853$	17.5167	0.599759	0.299880	−	0.953977i	$-0.403054\pi$
$853$	17.5167	0.599759	0.299880	+	0.953977i	$0.403054\pi$
$854$	0	0
$855$	−11.3205	−0.387153
$856$	0	0
$857$	−47.9615	−1.63833	−0.819167	−	0.573555i	$-0.805564\pi$
$857$	−47.9615	−1.63833	−0.819167	+	0.573555i	$0.805564\pi$
$858$	0	0
$859$	−46.2487	−1.57799	−0.788993	−	0.614402i	$-0.789397\pi$
$859$	−46.2487	−1.57799	−0.788993	+	0.614402i	$0.789397\pi$
$860$	0	0
$861$	−89.5692	−3.05251
$862$	0	0
$863$	−32.7846	−1.11600	−0.558001	−	0.829841i	$-0.688431\pi$
$863$	−32.7846	−1.11600	−0.558001	+	0.829841i	$0.688431\pi$
$864$	0	0
$865$	1.85641	0.0631197
$866$	0	0
$867$	−2.73205	−0.0927853
$868$	0	0
$869$	26.5359	0.900169
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	41.3731	1.40027
$874$	0	0
$875$	−27.7128	−0.936864
$876$	0	0
$877$	14.9808	0.505864	0.252932	−	0.967484i	$-0.418605\pi$
$877$	14.9808	0.505864	0.252932	+	0.967484i	$0.418605\pi$
$878$	0	0
$879$	30.2487	1.02026
$880$	0	0
$881$	36.2487	1.22125	0.610625	−	0.791920i	$-0.290918\pi$
$881$	36.2487	1.22125	0.610625	+	0.791920i	$0.290918\pi$
$882$	0	0
$883$	−29.1769	−0.981881	−0.490941	−	0.871193i	$-0.663347\pi$
$883$	−29.1769	−0.981881	−0.490941	+	0.871193i	$0.663347\pi$
$884$	0	0
$885$	−1.07180	−0.0360281
$886$	0	0
$887$	30.5885	1.02706	0.513530	−	0.858072i	$-0.328338\pi$
$887$	30.5885	1.02706	0.513530	+	0.858072i	$0.328338\pi$
$888$	0	0
$889$	43.7128	1.46608
$890$	0	0
$891$	8.53590	0.285963
$892$	0	0
$893$	20.7846	0.695530
$894$	0	0
$895$	−13.0718	−0.436942
$896$	0	0
$897$	−6.00000	−0.200334
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	59.7128	1.98712
$904$	0	0
$905$	5.85641	0.194674
$906$	0	0
$907$	−50.0526	−1.66197	−0.830984	−	0.556296i	$-0.812222\pi$
$907$	−50.0526	−1.66197	−0.830984	+	0.556296i	$0.812222\pi$
$908$	0	0
$909$	−4.78461	−0.158695
$910$	0	0
$911$	19.6603	0.651373	0.325687	−	0.945478i	$-0.394405\pi$
$911$	19.6603	0.651373	0.325687	+	0.945478i	$0.394405\pi$
$912$	0	0
$913$	27.7128	0.917160
$914$	0	0
$915$	−8.00000	−0.264472
$916$	0	0
$917$	64.7846	2.13938
$918$	0	0
$919$	−42.2487	−1.39366	−0.696828	−	0.717238i	$-0.745406\pi$
$919$	−42.2487	−1.39366	−0.696828	+	0.717238i	$0.745406\pi$
$920$	0	0
$921$	61.1769	2.01585
$922$	0	0
$923$	−1.07180	−0.0352786
$924$	0	0
$925$	−9.80385	−0.322349
$926$	0	0
$927$	−24.3923	−0.801148
$928$	0	0
$929$	49.2679	1.61643	0.808214	−	0.588888i	$-0.200434\pi$
$929$	49.2679	1.61643	0.808214	+	0.588888i	$0.200434\pi$
$930$	0	0
$931$	31.1769	1.02178
$932$	0	0
$933$	−7.85641	−0.257207
$934$	0	0
$935$	2.53590	0.0829327
$936$	0	0
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	0	0
$939$	−47.3205	−1.54425
$940$	0	0
$941$	−53.1244	−1.73180	−0.865902	−	0.500213i	$-0.833255\pi$
$941$	−53.1244	−1.73180	−0.865902	+	0.500213i	$0.833255\pi$
$942$	0	0
$943$	−18.0000	−0.586161
$944$	0	0
$945$	−11.7128	−0.381018
$946$	0	0
$947$	−2.67949	−0.0870718	−0.0435359	−	0.999052i	$-0.513862\pi$
$947$	−2.67949	−0.0870718	−0.0435359	+	0.999052i	$0.513862\pi$
$948$	0	0
$949$	0.196152	0.00636738
$950$	0	0
$951$	41.7128	1.35263
$952$	0	0
$953$	−25.1769	−0.815560	−0.407780	−	0.913080i	$-0.633697\pi$
$953$	−25.1769	−0.815560	−0.407780	+	0.913080i	$0.633697\pi$
$954$	0	0
$955$	−3.21539	−0.104048
$956$	0	0
$957$	51.7128	1.67164
$958$	0	0
$959$	31.4256	1.01479
$960$	0	0
$961$	−28.8564	−0.930852
$962$	0	0
$963$	25.2679	0.814248
$964$	0	0
$965$	16.6410	0.535693
$966$	0	0
$967$	−11.8564	−0.381276	−0.190638	−	0.981660i	$-0.561056\pi$
$967$	−11.8564	−0.381276	−0.190638	+	0.981660i	$0.561056\pi$
$968$	0	0
$969$	−9.46410	−0.304031
$970$	0	0
$971$	28.1051	0.901936	0.450968	−	0.892540i	$-0.351079\pi$
$971$	28.1051	0.901936	0.450968	+	0.892540i	$0.351079\pi$
$972$	0	0
$973$	−86.6410	−2.77758
$974$	0	0
$975$	12.1962	0.390589
$976$	0	0
$977$	31.1769	0.997438	0.498719	−	0.866764i	$-0.333804\pi$
$977$	31.1769	0.997438	0.498719	+	0.866764i	$0.333804\pi$
$978$	0	0
$979$	−15.7128	−0.502184
$980$	0	0
$981$	−52.0526	−1.66191
$982$	0	0
$983$	−53.4641	−1.70524	−0.852620	−	0.522531i	$-0.824988\pi$
$983$	−53.4641	−1.70524	−0.852620	+	0.522531i	$0.824988\pi$
$984$	0	0
$985$	9.60770	0.306127
$986$	0	0
$987$	65.5692	2.08709
$988$	0	0
$989$	12.0000	0.381578
$990$	0	0
$991$	13.9090	0.441833	0.220916	−	0.975293i	$-0.429095\pi$
$991$	13.9090	0.441833	0.220916	+	0.975293i	$0.429095\pi$
$992$	0	0
$993$	45.8564	1.45521
$994$	0	0
$995$	10.1051	0.320354
$996$	0	0
$997$	−58.9282	−1.86627	−0.933137	−	0.359520i	$-0.882941\pi$
$997$	−58.9282	−1.86627	−0.933137	+	0.359520i	$0.882941\pi$
$998$	0	0
$999$	−8.78461	−0.277933

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3536.2.a.o.1.1		2
4.3	odd	2		1768.2.a.l.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1768.2.a.l.1.2	✓	2	4.3	odd	2
3536.2.a.o.1.1		2	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:	\( N \)	\(=\)	\( 3536 = 2^{4} \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3536.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} -0.732051 q^{5} -4.00000 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q-2.73205 q^{3} -0.732051 q^{5} -4.00000 q^{7} +4.46410 q^{9} -3.46410 q^{11} +1.00000 q^{13} +2.00000 q^{15} +1.00000 q^{17} +3.46410 q^{19} +10.9282 q^{21} +2.19615 q^{23} -4.46410 q^{25} -4.00000 q^{27} +5.46410 q^{29} +1.46410 q^{31} +9.46410 q^{33} +2.92820 q^{35} +2.19615 q^{37} -2.73205 q^{39} -8.19615 q^{41} +5.46410 q^{43} -3.26795 q^{45} +6.00000 q^{47} +9.00000 q^{49} -2.73205 q^{51} +8.00000 q^{53} +2.53590 q^{55} -9.46410 q^{57} -0.535898 q^{59} -4.00000 q^{61} -17.8564 q^{63} -0.732051 q^{65} +8.00000 q^{67} -6.00000 q^{69} -1.07180 q^{71} +0.196152 q^{73} +12.1962 q^{75} +13.8564 q^{77} -7.66025 q^{79} -2.46410 q^{81} -8.00000 q^{83} -0.732051 q^{85} -14.9282 q^{87} +4.53590 q^{89} -4.00000 q^{91} -4.00000 q^{93} -2.53590 q^{95} +9.26795 q^{97} -15.4641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 8 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - 8 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} - 8 q^{7} + 2 q^{9} + 2 q^{13} + 4 q^{15} + 2 q^{17} + 8 q^{21} - 6 q^{23} - 2 q^{25} - 8 q^{27} + 4 q^{29} - 4 q^{31} + 12 q^{33} - 8 q^{35} - 6 q^{37} - 2 q^{39} - 6 q^{41} + 4 q^{43} - 10 q^{45} + 12 q^{47} + 18 q^{49} - 2 q^{51} + 16 q^{53} + 12 q^{55} - 12 q^{57} - 8 q^{59} - 8 q^{61} - 8 q^{63} + 2 q^{65} + 16 q^{67} - 12 q^{69} - 16 q^{71} - 10 q^{73} + 14 q^{75} + 2 q^{79} + 2 q^{81} - 16 q^{83} + 2 q^{85} - 16 q^{87} + 16 q^{89} - 8 q^{91} - 8 q^{93} - 12 q^{95} + 22 q^{97} - 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 8 * q^7 + 2 * q^9 + 2 * q^13 + 4 * q^15 + 2 * q^17 + 8 * q^21 - 6 * q^23 - 2 * q^25 - 8 * q^27 + 4 * q^29 - 4 * q^31 + 12 * q^33 - 8 * q^35 - 6 * q^37 - 2 * q^39 - 6 * q^41 + 4 * q^43 - 10 * q^45 + 12 * q^47 + 18 * q^49 - 2 * q^51 + 16 * q^53 + 12 * q^55 - 12 * q^57 - 8 * q^59 - 8 * q^61 - 8 * q^63 + 2 * q^65 + 16 * q^67 - 12 * q^69 - 16 * q^71 - 10 * q^73 + 14 * q^75 + 2 * q^79 + 2 * q^81 - 16 * q^83 + 2 * q^85 - 16 * q^87 + 16 * q^89 - 8 * q^91 - 8 * q^93 - 12 * q^95 + 22 * q^97 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more