LMFDB - Embedded newform 765.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [765,2,Mod(1,765)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(765, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("765.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 765 = 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	765.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:	$6.10855575463$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	765.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14510 q^{2} +2.60147 q^{4} -1.00000 q^{5} +2.74657 q^{7} -1.29021 q^{8} +2.14510 q^{10} +0.746568 q^{11} +2.00000 q^{13} -5.89167 q^{14} -2.43531 q^{16} +1.00000 q^{17} +1.25343 q^{19} -2.60147 q^{20} -1.60147 q^{22} +1.00000 q^{25} -4.29021 q^{26} +7.14510 q^{28} -2.45636 q^{29} -5.49314 q^{31} +7.80440 q^{32} -2.14510 q^{34} -2.74657 q^{35} -0.456363 q^{37} -2.68874 q^{38} +1.29021 q^{40} +3.94950 q^{41} +10.5804 q^{43} +1.94217 q^{44} +0.340706 q^{47} +0.543637 q^{49} -2.14510 q^{50} +5.20293 q^{52} +11.0368 q^{53} -0.746568 q^{55} -3.54364 q^{56} +5.26915 q^{58} -1.49314 q^{59} -5.08727 q^{61} +11.7833 q^{62} -11.8706 q^{64} -2.00000 q^{65} -8.07355 q^{67} +2.60147 q^{68} +5.89167 q^{70} +1.08727 q^{71} +13.0368 q^{73} +0.978945 q^{74} +3.26076 q^{76} +2.05050 q^{77} +8.00000 q^{79} +2.43531 q^{80} -8.47208 q^{82} +14.9863 q^{83} -1.00000 q^{85} -22.6961 q^{86} -0.963226 q^{88} +10.0735 q^{89} +5.49314 q^{91} -0.730849 q^{94} -1.25343 q^{95} +0.912726 q^{97} -1.16616 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} - 3 q^{5} + 9 q^{8} - 6 q^{11} + 6 q^{13} - 3 q^{14} + 12 q^{16} + 3 q^{17} + 12 q^{19} - 6 q^{20} - 3 q^{22} + 3 q^{25} + 15 q^{28} - 12 q^{29} + 18 q^{32} - 6 q^{37} + 3 q^{38} - 9 q^{40}+ \cdots - 21 q^{98}+O(q^{100}) $ 3 * q + 6 * q^4 - 3 * q^5 + 9 * q^8 - 6 * q^11 + 6 * q^13 - 3 * q^14 + 12 * q^16 + 3 * q^17 + 12 * q^19 - 6 * q^20 - 3 * q^22 + 3 * q^25 + 15 * q^28 - 12 * q^29 + 18 * q^32 - 6 * q^37 + 3 * q^38 - 9 * q^40 + 6 * q^43 + 3 * q^44 - 3 * q^49 + 12 * q^52 + 12 * q^53 + 6 * q^55 - 6 * q^56 - 21 * q^58 + 12 * q^59 - 6 * q^61 + 6 * q^62 + 3 * q^64 - 6 * q^65 + 18 * q^67 + 6 * q^68 + 3 * q^70 - 6 * q^71 + 18 * q^73 - 21 * q^74 + 9 * q^76 + 18 * q^77 + 24 * q^79 - 12 * q^80 + 15 * q^82 + 12 * q^83 - 3 * q^85 - 48 * q^86 - 24 * q^88 - 12 * q^89 - 39 * q^94 - 12 * q^95 + 12 * q^97 - 21 * q^98

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 $

$2$	−2.14510	−1.51682	−0.758408	−	0.651780i	$-0.774023\pi$
$2$	−2.14510	−1.51682	−0.758408	+	0.651780i	$0.774023\pi$
$3$	0	0
$3$	0	0
$4$	2.60147	1.30073
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.74657	1.03811	0.519053	−	0.854742i	$-0.326285\pi$
$7$	2.74657	1.03811	0.519053	+	0.854742i	$0.326285\pi$
$8$	−1.29021	−0.456156
$9$	0	0
$10$	2.14510	0.678341
$11$	0.746568	0.225099	0.112549	−	0.993646i	$-0.464098\pi$
$11$	0.746568	0.225099	0.112549	+	0.993646i	$0.464098\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−5.89167	−1.57462
$15$	0	0
$16$	−2.43531	−0.608827
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	1.25343	0.287557	0.143778	−	0.989610i	$-0.454075\pi$
$19$	1.25343	0.287557	0.143778	+	0.989610i	$0.454075\pi$
$20$	−2.60147	−0.581705
$21$	0	0
$22$	−1.60147	−0.341434
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.29021	−0.841378
$27$	0	0
$28$	7.14510	1.35030
$29$	−2.45636	−0.456135	−0.228068	−	0.973645i	$-0.573241\pi$
$29$	−2.45636	−0.456135	−0.228068	+	0.973645i	$0.573241\pi$
$30$	0	0
$31$	−5.49314	−0.986596	−0.493298	−	0.869860i	$-0.664209\pi$
$31$	−5.49314	−0.986596	−0.493298	+	0.869860i	$0.664209\pi$
$32$	7.80440	1.37964
$33$	0	0
$34$	−2.14510	−0.367882
$35$	−2.74657	−0.464255
$36$	0	0
$37$	−0.456363	−0.0750256	−0.0375128	−	0.999296i	$-0.511943\pi$
$37$	−0.456363	−0.0750256	−0.0375128	+	0.999296i	$0.511943\pi$
$38$	−2.68874	−0.436171
$39$	0	0
$40$	1.29021	0.203999
$41$	3.94950	0.616808	0.308404	−	0.951255i	$-0.400205\pi$
$41$	3.94950	0.616808	0.308404	+	0.951255i	$0.400205\pi$
$42$	0	0
$43$	10.5804	1.61350	0.806749	−	0.590895i	$-0.201225\pi$
$43$	10.5804	1.61350	0.806749	+	0.590895i	$0.201225\pi$
$44$	1.94217	0.292793
$45$	0	0
$46$	0	0
$47$	0.340706	0.0496971	0.0248485	−	0.999691i	$-0.492090\pi$
$47$	0.340706	0.0496971	0.0248485	+	0.999691i	$0.492090\pi$
$48$	0	0
$49$	0.543637	0.0776624
$50$	−2.14510	−0.303363
$51$	0	0
$52$	5.20293	0.721517
$53$	11.0368	1.51602	0.758009	−	0.652244i	$-0.226172\pi$
$53$	11.0368	1.51602	0.758009	+	0.652244i	$0.226172\pi$
$54$	0	0
$55$	−0.746568	−0.100667
$56$	−3.54364	−0.473538
$57$	0	0
$58$	5.26915	0.691873
$59$	−1.49314	−0.194390	−0.0971949	−	0.995265i	$-0.530987\pi$
$59$	−1.49314	−0.194390	−0.0971949	+	0.995265i	$0.530987\pi$
$60$	0	0
$61$	−5.08727	−0.651359	−0.325679	−	0.945480i	$-0.605593\pi$
$61$	−5.08727	−0.651359	−0.325679	+	0.945480i	$0.605593\pi$
$62$	11.7833	1.49649
$63$	0	0
$64$	−11.8706	−1.48383
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−8.07355	−0.986341	−0.493170	−	0.869933i	$-0.664162\pi$
$67$	−8.07355	−0.986341	−0.493170	+	0.869933i	$0.664162\pi$
$68$	2.60147	0.315474
$69$	0	0
$70$	5.89167	0.704189
$71$	1.08727	0.129036	0.0645179	−	0.997917i	$-0.479449\pi$
$71$	1.08727	0.129036	0.0645179	+	0.997917i	$0.479449\pi$
$72$	0	0
$73$	13.0368	1.52584	0.762919	−	0.646494i	$-0.223765\pi$
$73$	13.0368	1.52584	0.762919	+	0.646494i	$0.223765\pi$
$74$	0.978945	0.113800
$75$	0	0
$76$	3.26076	0.374035
$77$	2.05050	0.233676
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	2.43531	0.272276
$81$	0	0
$82$	−8.47208	−0.935585
$83$	14.9863	1.64496	0.822479	−	0.568796i	$-0.192591\pi$
$83$	14.9863	1.64496	0.822479	+	0.568796i	$0.192591\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	−22.6961	−2.44738
$87$	0	0
$88$	−0.963226	−0.102680
$89$	10.0735	1.06779	0.533897	−	0.845550i	$-0.320727\pi$
$89$	10.0735	1.06779	0.533897	+	0.845550i	$0.320727\pi$
$90$	0	0
$91$	5.49314	0.575837
$92$	0	0
$93$	0	0
$94$	−0.730849	−0.0753814
$95$	−1.25343	−0.128599
$96$	0	0
$97$	0.912726	0.0926733	0.0463366	−	0.998926i	$-0.485245\pi$
$97$	0.912726	0.0926733	0.0463366	+	0.998926i	$0.485245\pi$
$98$	−1.16616	−0.117800
$99$	0	0
$100$	2.60147	0.260147
$101$	14.9863	1.49119	0.745595	−	0.666399i	$-0.232165\pi$
$101$	14.9863	1.49119	0.745595	+	0.666399i	$0.232165\pi$
$102$	0	0
$103$	9.08727	0.895396	0.447698	−	0.894185i	$-0.352244\pi$
$103$	9.08727	0.895396	0.447698	+	0.894185i	$0.352244\pi$
$104$	−2.58041	−0.253030
$105$	0	0
$106$	−23.6750	−2.29952
$107$	13.4931	1.30443	0.652215	−	0.758034i	$-0.273840\pi$
$107$	13.4931	1.30443	0.652215	+	0.758034i	$0.273840\pi$
$108$	0	0
$109$	12.0735	1.15644	0.578218	−	0.815882i	$-0.303748\pi$
$109$	12.0735	1.15644	0.578218	+	0.815882i	$0.303748\pi$
$110$	1.60147	0.152694
$111$	0	0
$112$	−6.68874	−0.632027
$113$	1.08727	0.102282	0.0511411	−	0.998691i	$-0.483714\pi$
$113$	1.08727	0.102282	0.0511411	+	0.998691i	$0.483714\pi$
$114$	0	0
$115$	0	0
$116$	−6.39014	−0.593310
$117$	0	0
$118$	3.20293	0.294854
$119$	2.74657	0.251777
$120$	0	0
$121$	−10.4426	−0.949331
$122$	10.9127	0.987992
$123$	0	0
$124$	−14.2902	−1.28330
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.07355	−0.716411	−0.358206	−	0.933643i	$-0.616611\pi$
$127$	−8.07355	−0.716411	−0.358206	+	0.933643i	$0.616611\pi$
$128$	9.85490	0.871058
$129$	0	0
$130$	4.29021	0.376276
$131$	−16.0735	−1.40435	−0.702176	−	0.712003i	$-0.747788\pi$
$131$	−16.0735	−1.40435	−0.702176	+	0.712003i	$0.747788\pi$
$132$	0	0
$133$	3.44264	0.298514
$134$	17.3186	1.49610
$135$	0	0
$136$	−1.29021	−0.110634
$137$	−5.44264	−0.464996	−0.232498	−	0.972597i	$-0.574690\pi$
$137$	−5.44264	−0.464996	−0.232498	+	0.972597i	$0.574690\pi$
$138$	0	0
$139$	9.49314	0.805197	0.402599	−	0.915377i	$-0.368107\pi$
$139$	9.49314	0.805197	0.402599	+	0.915377i	$0.368107\pi$
$140$	−7.14510	−0.603871
$141$	0	0
$142$	−2.33231	−0.195724
$143$	1.49314	0.124862
$144$	0	0
$145$	2.45636	0.203990
$146$	−27.9652	−2.31442
$147$	0	0
$148$	−1.18721	−0.0975882
$149$	−10.0735	−0.825257	−0.412629	−	0.910899i	$-0.635389\pi$
$149$	−10.0735	−0.825257	−0.412629	+	0.910899i	$0.635389\pi$
$150$	0	0
$151$	−4.34071	−0.353242	−0.176621	−	0.984279i	$-0.556517\pi$
$151$	−4.34071	−0.353242	−0.176621	+	0.984279i	$0.556517\pi$
$152$	−1.61718	−0.131171
$153$	0	0
$154$	−4.39853	−0.354444
$155$	5.49314	0.441219
$156$	0	0
$157$	−12.1745	−0.971635	−0.485817	−	0.874060i	$-0.661478\pi$
$157$	−12.1745	−0.971635	−0.485817	+	0.874060i	$0.661478\pi$
$158$	−17.1608	−1.36524
$159$	0	0
$160$	−7.80440	−0.616992
$161$	0	0
$162$	0	0
$163$	−22.3133	−1.74771	−0.873854	−	0.486188i	$-0.838387\pi$
$163$	−22.3133	−1.74771	−0.873854	+	0.486188i	$0.838387\pi$
$164$	10.2745	0.802303
$165$	0	0
$166$	−32.1471	−2.49510
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	2.14510	0.164522
$171$	0	0
$172$	27.5246	2.09873
$173$	−19.0598	−1.44909	−0.724546	−	0.689227i	$-0.757950\pi$
$173$	−19.0598	−1.44909	−0.724546	+	0.689227i	$0.757950\pi$
$174$	0	0
$175$	2.74657	0.207621
$176$	−1.81812	−0.137046
$177$	0	0
$178$	−21.6088	−1.61965
$179$	17.1608	1.28266	0.641330	−	0.767265i	$-0.278383\pi$
$179$	17.1608	1.28266	0.641330	+	0.767265i	$0.278383\pi$
$180$	0	0
$181$	12.0735	0.897420	0.448710	−	0.893677i	$-0.351884\pi$
$181$	12.0735	0.897420	0.448710	+	0.893677i	$0.351884\pi$
$182$	−11.7833	−0.873439
$183$	0	0
$184$	0	0
$185$	0.456363	0.0335525
$186$	0	0
$187$	0.746568	0.0545945
$188$	0.886335	0.0646426
$189$	0	0
$190$	2.68874	0.195062
$191$	−14.1745	−1.02563	−0.512817	−	0.858498i	$-0.671398\pi$
$191$	−14.1745	−1.02563	−0.512817	+	0.858498i	$0.671398\pi$
$192$	0	0
$193$	3.08727	0.222227	0.111113	−	0.993808i	$-0.464558\pi$
$193$	3.08727	0.222227	0.111113	+	0.993808i	$0.464558\pi$
$194$	−1.95789	−0.140568
$195$	0	0
$196$	1.41425	0.101018
$197$	8.98627	0.640245	0.320123	−	0.947376i	$-0.396276\pi$
$197$	8.98627	0.640245	0.320123	+	0.947376i	$0.396276\pi$
$198$	0	0
$199$	11.6677	0.827100	0.413550	−	0.910481i	$-0.364289\pi$
$199$	11.6677	0.827100	0.413550	+	0.910481i	$0.364289\pi$
$200$	−1.29021	−0.0912313
$201$	0	0
$202$	−32.1471	−2.26186
$203$	−6.74657	−0.473516
$204$	0	0
$205$	−3.94950	−0.275845
$206$	−19.4931	−1.35815
$207$	0	0
$208$	−4.87062	−0.337716
$209$	0.935772	0.0647287
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	28.7118	1.97193
$213$	0	0
$214$	−28.9442	−1.97858
$215$	−10.5804	−0.721578
$216$	0	0
$217$	−15.0873	−1.02419
$218$	−25.8990	−1.75410
$219$	0	0
$220$	−1.94217	−0.130941
$221$	2.00000	0.134535
$222$	0	0
$223$	24.8853	1.66644	0.833221	−	0.552941i	$-0.186494\pi$
$223$	24.8853	1.66644	0.833221	+	0.552941i	$0.186494\pi$
$224$	21.4353	1.43221
$225$	0	0
$226$	−2.33231	−0.155143
$227$	−17.9725	−1.19288	−0.596440	−	0.802658i	$-0.703419\pi$
$227$	−17.9725	−1.19288	−0.596440	+	0.802658i	$0.703419\pi$
$228$	0	0
$229$	−3.03677	−0.200676	−0.100338	−	0.994953i	$-0.531992\pi$
$229$	−3.03677	−0.200676	−0.100338	+	0.994953i	$0.531992\pi$
$230$	0	0
$231$	0	0
$232$	3.16921	0.208069
$233$	−4.07355	−0.266867	−0.133433	−	0.991058i	$-0.542600\pi$
$233$	−4.07355	−0.266867	−0.133433	+	0.991058i	$0.542600\pi$
$234$	0	0
$235$	−0.340706	−0.0222252
$236$	−3.88434	−0.252849
$237$	0	0
$238$	−5.89167	−0.381900
$239$	−15.6677	−1.01346	−0.506729	−	0.862105i	$-0.669146\pi$
$239$	−15.6677	−1.01346	−0.506729	+	0.862105i	$0.669146\pi$
$240$	0	0
$241$	−8.07355	−0.520063	−0.260031	−	0.965600i	$-0.583733\pi$
$241$	−8.07355	−0.520063	−0.260031	+	0.965600i	$0.583733\pi$
$242$	22.4005	1.43996
$243$	0	0
$244$	−13.2344	−0.847244
$245$	−0.543637	−0.0347317
$246$	0	0
$247$	2.50686	0.159508
$248$	7.08727	0.450042
$249$	0	0
$250$	2.14510	0.135668
$251$	−13.4931	−0.851679	−0.425840	−	0.904799i	$-0.640021\pi$
$251$	−13.4931	−0.851679	−0.425840	+	0.904799i	$0.640021\pi$
$252$	0	0
$253$	0	0
$254$	17.3186	1.08666
$255$	0	0
$256$	2.60147	0.162592
$257$	−8.17455	−0.509914	−0.254957	−	0.966952i	$-0.582061\pi$
$257$	−8.17455	−0.509914	−0.254957	+	0.966952i	$0.582061\pi$
$258$	0	0
$259$	−1.25343	−0.0778845
$260$	−5.20293	−0.322672
$261$	0	0
$262$	34.4794	2.13015
$263$	13.8338	0.853031	0.426516	−	0.904480i	$-0.359741\pi$
$263$	13.8338	0.853031	0.426516	+	0.904480i	$0.359741\pi$
$264$	0	0
$265$	−11.0368	−0.677984
$266$	−7.38481	−0.452792
$267$	0	0
$268$	−21.0031	−1.28297
$269$	−12.5299	−0.763962	−0.381981	−	0.924170i	$-0.624758\pi$
$269$	−12.5299	−0.763962	−0.381981	+	0.924170i	$0.624758\pi$
$270$	0	0
$271$	13.1608	0.799463	0.399731	−	0.916632i	$-0.369103\pi$
$271$	13.1608	0.799463	0.399731	+	0.916632i	$0.369103\pi$
$272$	−2.43531	−0.147662
$273$	0	0
$274$	11.6750	0.705313
$275$	0.746568	0.0450198
$276$	0	0
$277$	3.08727	0.185496	0.0927482	−	0.995690i	$-0.470435\pi$
$277$	3.08727	0.185496	0.0927482	+	0.995690i	$0.470435\pi$
$278$	−20.3638	−1.22134
$279$	0	0
$280$	3.54364	0.211773
$281$	−2.98627	−0.178146	−0.0890731	−	0.996025i	$-0.528390\pi$
$281$	−2.98627	−0.178146	−0.0890731	+	0.996025i	$0.528390\pi$
$282$	0	0
$283$	23.3270	1.38664	0.693322	−	0.720627i	$-0.256146\pi$
$283$	23.3270	1.38664	0.693322	+	0.720627i	$0.256146\pi$
$284$	2.82851	0.167841
$285$	0	0
$286$	−3.20293	−0.189393
$287$	10.8476	0.640312
$288$	0	0
$289$	1.00000	0.0588235
$290$	−5.26915	−0.309415
$291$	0	0
$292$	33.9147	1.98471
$293$	−24.0966	−1.40774	−0.703869	−	0.710330i	$-0.748546\pi$
$293$	−24.0966	−1.40774	−0.703869	+	0.710330i	$0.748546\pi$
$294$	0	0
$295$	1.49314	0.0869338
$296$	0.588802	0.0342234
$297$	0	0
$298$	21.6088	1.25176
$299$	0	0
$300$	0	0
$301$	29.0598	1.67498
$302$	9.31126	0.535803
$303$	0	0
$304$	−3.05249	−0.175072
$305$	5.08727	0.291296
$306$	0	0
$307$	24.0735	1.37395	0.686975	−	0.726681i	$-0.258938\pi$
$307$	24.0735	1.37395	0.686975	+	0.726681i	$0.258938\pi$
$308$	5.33431	0.303950
$309$	0	0
$310$	−11.7833	−0.669249
$311$	−4.84757	−0.274880	−0.137440	−	0.990510i	$-0.543888\pi$
$311$	−4.84757	−0.274880	−0.137440	+	0.990510i	$0.543888\pi$
$312$	0	0
$313$	−31.9220	−1.80434	−0.902170	−	0.431380i	$-0.858027\pi$
$313$	−31.9220	−1.80434	−0.902170	+	0.431380i	$0.858027\pi$
$314$	26.1157	1.47379
$315$	0	0
$316$	20.8117	1.17075
$317$	23.1608	1.30084	0.650421	−	0.759574i	$-0.274593\pi$
$317$	23.1608	1.30084	0.650421	+	0.759574i	$0.274593\pi$
$318$	0	0
$319$	−1.83384	−0.102675
$320$	11.8706	0.663588
$321$	0	0
$322$	0	0
$323$	1.25343	0.0697428
$324$	0	0
$325$	2.00000	0.110940
$326$	47.8642	2.65095
$327$	0	0
$328$	−5.09567	−0.281361
$329$	0.935772	0.0515908
$330$	0	0
$331$	19.2260	1.05676	0.528378	−	0.849009i	$-0.322801\pi$
$331$	19.2260	1.05676	0.528378	+	0.849009i	$0.322801\pi$
$332$	38.9863	2.13965
$333$	0	0
$334$	25.7412	1.40850
$335$	8.07355	0.441105
$336$	0	0
$337$	−31.1103	−1.69469	−0.847344	−	0.531045i	$-0.821799\pi$
$337$	−31.1103	−1.69469	−0.847344	+	0.531045i	$0.821799\pi$
$338$	19.3059	1.05010
$339$	0	0
$340$	−2.60147	−0.141084
$341$	−4.10100	−0.222082
$342$	0	0
$343$	−17.7328	−0.957483
$344$	−13.6509	−0.736007
$345$	0	0
$346$	40.8853	2.19801
$347$	0.811724	0.0435757	0.0217878	−	0.999763i	$-0.493064\pi$
$347$	0.811724	0.0435757	0.0217878	+	0.999763i	$0.493064\pi$
$348$	0	0
$349$	17.1103	0.915894	0.457947	−	0.888979i	$-0.348585\pi$
$349$	17.1103	0.915894	0.457947	+	0.888979i	$0.348585\pi$
$350$	−5.89167	−0.314923
$351$	0	0
$352$	5.82651	0.310554
$353$	26.7045	1.42133	0.710667	−	0.703528i	$-0.248393\pi$
$353$	26.7045	1.42133	0.710667	+	0.703528i	$0.248393\pi$
$354$	0	0
$355$	−1.08727	−0.0577065
$356$	26.2060	1.38891
$357$	0	0
$358$	−36.8117	−1.94556
$359$	−1.49314	−0.0788047	−0.0394024	−	0.999223i	$-0.512545\pi$
$359$	−1.49314	−0.0788047	−0.0394024	+	0.999223i	$0.512545\pi$
$360$	0	0
$361$	−17.4289	−0.917311
$362$	−25.8990	−1.36122
$363$	0	0
$364$	14.2902	0.749010
$365$	−13.0368	−0.682376
$366$	0	0
$367$	−12.1471	−0.634073	−0.317037	−	0.948413i	$-0.602688\pi$
$367$	−12.1471	−0.634073	−0.317037	+	0.948413i	$0.602688\pi$
$368$	0	0
$369$	0	0
$370$	−0.978945	−0.0508929
$371$	30.3133	1.57379
$372$	0	0
$373$	0.0735473	0.00380813	0.00190407	−	0.999998i	$-0.499394\pi$
$373$	0.0735473	0.00380813	0.00190407	+	0.999998i	$0.499394\pi$
$374$	−1.60147	−0.0828098
$375$	0	0
$376$	−0.439581	−0.0226696
$377$	−4.91273	−0.253018
$378$	0	0
$379$	23.6677	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$379$	23.6677	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$380$	−3.26076	−0.167273
$381$	0	0
$382$	30.4059	1.55570
$383$	23.6593	1.20893	0.604467	−	0.796630i	$-0.293386\pi$
$383$	23.6593	1.20893	0.604467	+	0.796630i	$0.293386\pi$
$384$	0	0
$385$	−2.05050	−0.104503
$386$	−6.62252	−0.337077
$387$	0	0
$388$	2.37442	0.120543
$389$	−22.8853	−1.16033	−0.580165	−	0.814499i	$-0.697012\pi$
$389$	−22.8853	−1.16033	−0.580165	+	0.814499i	$0.697012\pi$
$390$	0	0
$391$	0	0
$392$	−0.701404	−0.0354262
$393$	0	0
$394$	−19.2765	−0.971135
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−3.44264	−0.172781	−0.0863905	−	0.996261i	$-0.527533\pi$
$397$	−3.44264	−0.172781	−0.0863905	+	0.996261i	$0.527533\pi$
$398$	−25.0284	−1.25456
$399$	0	0
$400$	−2.43531	−0.121765
$401$	−6.12405	−0.305820	−0.152910	−	0.988240i	$-0.548865\pi$
$401$	−6.12405	−0.305820	−0.152910	+	0.988240i	$0.548865\pi$
$402$	0	0
$403$	−10.9863	−0.547265
$404$	38.9863	1.93964
$405$	0	0
$406$	14.4721	0.718237
$407$	−0.340706	−0.0168882
$408$	0	0
$409$	32.7780	1.62077	0.810384	−	0.585899i	$-0.199258\pi$
$409$	32.7780	1.62077	0.810384	+	0.585899i	$0.199258\pi$
$410$	8.47208	0.418406
$411$	0	0
$412$	23.6402	1.16467
$413$	−4.10100	−0.201797
$414$	0	0
$415$	−14.9863	−0.735647
$416$	15.6088	0.765284
$417$	0	0
$418$	−2.00733	−0.0981816
$419$	−18.7191	−0.914489	−0.457244	−	0.889341i	$-0.651163\pi$
$419$	−18.7191	−0.914489	−0.457244	+	0.889341i	$0.651163\pi$
$420$	0	0
$421$	−22.9358	−1.11782	−0.558911	−	0.829228i	$-0.688781\pi$
$421$	−22.9358	−1.11782	−0.558911	+	0.829228i	$0.688781\pi$
$422$	−42.9021	−2.08844
$423$	0	0
$424$	−14.2397	−0.691541
$425$	1.00000	0.0485071
$426$	0	0
$427$	−13.9725	−0.676179
$428$	35.1019	1.69672
$429$	0	0
$430$	22.6961	1.09450
$431$	−12.3133	−0.593108	−0.296554	−	0.955016i	$-0.595838\pi$
$431$	−12.3133	−0.593108	−0.296554	+	0.955016i	$0.595838\pi$
$432$	0	0
$433$	−35.0598	−1.68487	−0.842434	−	0.538800i	$-0.818878\pi$
$433$	−35.0598	−1.68487	−0.842434	+	0.538800i	$0.818878\pi$
$434$	32.3638	1.55351
$435$	0	0
$436$	31.4089	1.50421
$437$	0	0
$438$	0	0
$439$	−18.9863	−0.906165	−0.453083	−	0.891468i	$-0.649676\pi$
$439$	−18.9863	−0.906165	−0.453083	+	0.891468i	$0.649676\pi$
$440$	0.963226	0.0459200
$441$	0	0
$442$	−4.29021	−0.204064
$443$	17.9725	0.853901	0.426951	−	0.904275i	$-0.359588\pi$
$443$	17.9725	0.853901	0.426951	+	0.904275i	$0.359588\pi$
$444$	0	0
$445$	−10.0735	−0.477532
$446$	−53.3815	−2.52769
$447$	0	0
$448$	−32.6035	−1.54037
$449$	−38.1471	−1.80027	−0.900136	−	0.435608i	$-0.856533\pi$
$449$	−38.1471	−1.80027	−0.900136	+	0.435608i	$0.856533\pi$
$450$	0	0
$451$	2.94857	0.138843
$452$	2.82851	0.133042
$453$	0	0
$454$	38.5530	1.80938
$455$	−5.49314	−0.257522
$456$	0	0
$457$	−20.0735	−0.939001	−0.469500	−	0.882932i	$-0.655566\pi$
$457$	−20.0735	−0.939001	−0.469500	+	0.882932i	$0.655566\pi$
$458$	6.51419	0.304388
$459$	0	0
$460$	0	0
$461$	29.1608	1.35815	0.679077	−	0.734067i	$-0.262380\pi$
$461$	29.1608	1.35815	0.679077	+	0.734067i	$0.262380\pi$
$462$	0	0
$463$	9.89900	0.460045	0.230023	−	0.973185i	$-0.426120\pi$
$463$	9.89900	0.460045	0.230023	+	0.973185i	$0.426120\pi$
$464$	5.98200	0.277707
$465$	0	0
$466$	8.73818	0.404788
$467$	−11.9074	−0.551008	−0.275504	−	0.961300i	$-0.588845\pi$
$467$	−11.9074	−0.551008	−0.275504	+	0.961300i	$0.588845\pi$
$468$	0	0
$469$	−22.1745	−1.02393
$470$	0.730849	0.0337116
$471$	0	0
$472$	1.92645	0.0886722
$473$	7.89900	0.363196
$474$	0	0
$475$	1.25343	0.0575114
$476$	7.14510	0.327495
$477$	0	0
$478$	33.6088	1.53723
$479$	−25.8990	−1.18336	−0.591678	−	0.806175i	$-0.701534\pi$
$479$	−25.8990	−1.18336	−0.591678	+	0.806175i	$0.701534\pi$
$480$	0	0
$481$	−0.912726	−0.0416167
$482$	17.3186	0.788840
$483$	0	0
$484$	−27.1662	−1.23483
$485$	−0.912726	−0.0414447
$486$	0	0
$487$	−3.18828	−0.144475	−0.0722373	−	0.997387i	$-0.523014\pi$
$487$	−3.18828	−0.144475	−0.0722373	+	0.997387i	$0.523014\pi$
$488$	6.56363	0.297122
$489$	0	0
$490$	1.16616	0.0526816
$491$	−4.47941	−0.202153	−0.101076	−	0.994879i	$-0.532229\pi$
$491$	−4.47941	−0.202153	−0.101076	+	0.994879i	$0.532229\pi$
$492$	0	0
$493$	−2.45636	−0.110629
$494$	−5.37748	−0.241944
$495$	0	0
$496$	13.3775	0.600667
$497$	2.98627	0.133953
$498$	0	0
$499$	40.8285	1.82773	0.913867	−	0.406013i	$-0.133081\pi$
$499$	40.8285	1.82773	0.913867	+	0.406013i	$0.133081\pi$
$500$	−2.60147	−0.116341
$501$	0	0
$502$	28.9442	1.29184
$503$	11.3186	0.504671	0.252335	−	0.967640i	$-0.418801\pi$
$503$	11.3186	0.504671	0.252335	+	0.967640i	$0.418801\pi$
$504$	0	0
$505$	−14.9863	−0.666880
$506$	0	0
$507$	0	0
$508$	−21.0031	−0.931860
$509$	−38.9863	−1.72804	−0.864018	−	0.503461i	$-0.832060\pi$
$509$	−38.9863	−1.72804	−0.864018	+	0.503461i	$0.832060\pi$
$510$	0	0
$511$	35.8064	1.58398
$512$	−25.2902	−1.11768
$513$	0	0
$514$	17.5352	0.773447
$515$	−9.08727	−0.400433
$516$	0	0
$517$	0.254360	0.0111868
$518$	2.68874	0.118136
$519$	0	0
$520$	2.58041	0.113158
$521$	−6.12405	−0.268299	−0.134150	−	0.990961i	$-0.542830\pi$
$521$	−6.12405	−0.268299	−0.134150	+	0.990961i	$0.542830\pi$
$522$	0	0
$523$	−3.59414	−0.157161	−0.0785803	−	0.996908i	$-0.525039\pi$
$523$	−3.59414	−0.157161	−0.0785803	+	0.996908i	$0.525039\pi$
$524$	−41.8148	−1.82669
$525$	0	0
$526$	−29.6750	−1.29389
$527$	−5.49314	−0.239285
$528$	0	0
$529$	−23.0000	−1.00000
$530$	23.6750	1.02838
$531$	0	0
$532$	8.95590	0.388287
$533$	7.89900	0.342144
$534$	0	0
$535$	−13.4931	−0.583359
$536$	10.4165	0.449926
$537$	0	0
$538$	26.8779	1.15879
$539$	0.405862	0.0174817
$540$	0	0
$541$	−42.1471	−1.81205	−0.906023	−	0.423229i	$-0.860896\pi$
$541$	−42.1471	−1.81205	−0.906023	+	0.423229i	$0.860896\pi$
$542$	−28.2313	−1.21264
$543$	0	0
$544$	7.80440	0.334611
$545$	−12.0735	−0.517174
$546$	0	0
$547$	21.1524	0.904413	0.452206	−	0.891913i	$-0.350637\pi$
$547$	21.1524	0.904413	0.452206	+	0.891913i	$0.350637\pi$
$548$	−14.1588	−0.604835
$549$	0	0
$550$	−1.60147	−0.0682867
$551$	−3.07888	−0.131165
$552$	0	0
$553$	21.9725	0.934368
$554$	−6.62252	−0.281364
$555$	0	0
$556$	24.6961	1.04735
$557$	35.9725	1.52421	0.762103	−	0.647456i	$-0.224167\pi$
$557$	35.9725	1.52421	0.762103	+	0.647456i	$0.224167\pi$
$558$	0	0
$559$	21.1608	0.895007
$560$	6.68874	0.282651
$561$	0	0
$562$	6.40586	0.270215
$563$	−11.2260	−0.473119	−0.236559	−	0.971617i	$-0.576020\pi$
$563$	−11.2260	−0.473119	−0.236559	+	0.971617i	$0.576020\pi$
$564$	0	0
$565$	−1.08727	−0.0457420
$566$	−50.0388	−2.10329
$567$	0	0
$568$	−1.40281	−0.0588605
$569$	−4.91273	−0.205952	−0.102976	−	0.994684i	$-0.532837\pi$
$569$	−4.91273	−0.205952	−0.102976	+	0.994684i	$0.532837\pi$
$570$	0	0
$571$	22.3049	0.933429	0.466715	−	0.884408i	$-0.345437\pi$
$571$	22.3049	0.933429	0.466715	+	0.884408i	$0.345437\pi$
$572$	3.88434	0.162413
$573$	0	0
$574$	−23.2692	−0.971236
$575$	0	0
$576$	0	0
$577$	7.16082	0.298109	0.149054	−	0.988829i	$-0.452377\pi$
$577$	7.16082	0.298109	0.149054	+	0.988829i	$0.452377\pi$
$578$	−2.14510	−0.0892245
$579$	0	0
$580$	6.39014	0.265336
$581$	41.1608	1.70764
$582$	0	0
$583$	8.23970	0.341254
$584$	−16.8201	−0.696021
$585$	0	0
$586$	51.6897	2.13528
$587$	19.4280	0.801879	0.400939	−	0.916105i	$-0.368684\pi$
$587$	19.4280	0.801879	0.400939	+	0.916105i	$0.368684\pi$
$588$	0	0
$589$	−6.88527	−0.283703
$590$	−3.20293	−0.131863
$591$	0	0
$592$	1.11138	0.0456776
$593$	−32.4289	−1.33170	−0.665848	−	0.746088i	$-0.731930\pi$
$593$	−32.4289	−1.33170	−0.665848	+	0.746088i	$0.731930\pi$
$594$	0	0
$595$	−2.74657	−0.112598
$596$	−26.2060	−1.07344
$597$	0	0
$598$	0	0
$599$	8.14709	0.332881	0.166441	−	0.986051i	$-0.446773\pi$
$599$	8.14709	0.332881	0.166441	+	0.986051i	$0.446773\pi$
$600$	0	0
$601$	13.1334	0.535721	0.267861	−	0.963458i	$-0.413683\pi$
$601$	13.1334	0.535721	0.267861	+	0.963458i	$0.413683\pi$
$602$	−62.3363	−2.54064
$603$	0	0
$604$	−11.2922	−0.459473
$605$	10.4426	0.424554
$606$	0	0
$607$	−6.26716	−0.254376	−0.127188	−	0.991879i	$-0.540595\pi$
$607$	−6.26716	−0.254376	−0.127188	+	0.991879i	$0.540595\pi$
$608$	9.78228	0.396724
$609$	0	0
$610$	−10.9127	−0.441843
$611$	0.681412	0.0275670
$612$	0	0
$613$	−19.0137	−0.767957	−0.383979	−	0.923342i	$-0.625446\pi$
$613$	−19.0137	−0.767957	−0.383979	+	0.923342i	$0.625446\pi$
$614$	−51.6402	−2.08403
$615$	0	0
$616$	−2.64557	−0.106593
$617$	−25.0873	−1.00998	−0.504988	−	0.863126i	$-0.668503\pi$
$617$	−25.0873	−1.00998	−0.504988	+	0.863126i	$0.668503\pi$
$618$	0	0
$619$	8.81172	0.354173	0.177087	−	0.984195i	$-0.443333\pi$
$619$	8.81172	0.354173	0.177087	+	0.984195i	$0.443333\pi$
$620$	14.2902	0.573908
$621$	0	0
$622$	10.3985	0.416943
$623$	27.6677	1.10848
$624$	0	0
$625$	1.00000	0.0400000
$626$	68.4761	2.73685
$627$	0	0
$628$	−31.6717	−1.26384
$629$	−0.456363	−0.0181964
$630$	0	0
$631$	−34.3133	−1.36599	−0.682994	−	0.730424i	$-0.739323\pi$
$631$	−34.3133	−1.36599	−0.682994	+	0.730424i	$0.739323\pi$
$632$	−10.3216	−0.410573
$633$	0	0
$634$	−49.6823	−1.97314
$635$	8.07355	0.320389
$636$	0	0
$637$	1.08727	0.0430794
$638$	3.93378	0.155740
$639$	0	0
$640$	−9.85490	−0.389549
$641$	47.0368	1.85784	0.928920	−	0.370279i	$-0.120738\pi$
$641$	47.0368	1.85784	0.928920	+	0.370279i	$0.120738\pi$
$642$	0	0
$643$	18.1662	0.716403	0.358202	−	0.933644i	$-0.383390\pi$
$643$	18.1662	0.716403	0.358202	+	0.933644i	$0.383390\pi$
$644$	0	0
$645$	0	0
$646$	−2.68874	−0.105787
$647$	32.9588	1.29574	0.647872	−	0.761749i	$-0.275659\pi$
$647$	32.9588	1.29574	0.647872	+	0.761749i	$0.275659\pi$
$648$	0	0
$649$	−1.11473	−0.0437569
$650$	−4.29021	−0.168276
$651$	0	0
$652$	−58.0472	−2.27330
$653$	28.0735	1.09860	0.549301	−	0.835624i	$-0.314894\pi$
$653$	28.0735	1.09860	0.549301	+	0.835624i	$0.314894\pi$
$654$	0	0
$655$	16.0735	0.628045
$656$	−9.61825	−0.375529
$657$	0	0
$658$	−2.00733	−0.0782538
$659$	−15.6677	−0.610326	−0.305163	−	0.952300i	$-0.598711\pi$
$659$	−15.6677	−0.610326	−0.305163	+	0.952300i	$0.598711\pi$
$660$	0	0
$661$	−12.8622	−0.500283	−0.250141	−	0.968209i	$-0.580477\pi$
$661$	−12.8622	−0.500283	−0.250141	+	0.968209i	$0.580477\pi$
$662$	−41.2417	−1.60290
$663$	0	0
$664$	−19.3354	−0.750358
$665$	−3.44264	−0.133500
$666$	0	0
$667$	0	0
$668$	−31.2176	−1.20784
$669$	0	0
$670$	−17.3186	−0.669075
$671$	−3.79800	−0.146620
$672$	0	0
$673$	−35.0873	−1.35252	−0.676258	−	0.736665i	$-0.736399\pi$
$673$	−35.0873	−1.35252	−0.676258	+	0.736665i	$0.736399\pi$
$674$	66.7348	2.57053
$675$	0	0
$676$	−23.4132	−0.900507
$677$	−49.3354	−1.89611	−0.948056	−	0.318103i	$-0.896954\pi$
$677$	−49.3354	−1.89611	−0.948056	+	0.318103i	$0.896954\pi$
$678$	0	0
$679$	2.50686	0.0962046
$680$	1.29021	0.0494771
$681$	0	0
$682$	8.79707	0.336857
$683$	−1.49314	−0.0571333	−0.0285666	−	0.999592i	$-0.509094\pi$
$683$	−1.49314	−0.0571333	−0.0285666	+	0.999592i	$0.509094\pi$
$684$	0	0
$685$	5.44264	0.207952
$686$	38.0388	1.45233
$687$	0	0
$688$	−25.7666	−0.982341
$689$	22.0735	0.840935
$690$	0	0
$691$	−40.6265	−1.54551	−0.772753	−	0.634707i	$-0.781121\pi$
$691$	−40.6265	−1.54551	−0.772753	+	0.634707i	$0.781121\pi$
$692$	−49.5835	−1.88488
$693$	0	0
$694$	−1.74123	−0.0660963
$695$	−9.49314	−0.360095
$696$	0	0
$697$	3.94950	0.149598
$698$	−36.7034	−1.38924
$699$	0	0
$700$	7.14510	0.270059
$701$	−41.1608	−1.55462	−0.777311	−	0.629116i	$-0.783417\pi$
$701$	−41.1608	−1.55462	−0.777311	+	0.629116i	$0.783417\pi$
$702$	0	0
$703$	−0.572020	−0.0215741
$704$	−8.86223	−0.334008
$705$	0	0
$706$	−57.2838	−2.15590
$707$	41.1608	1.54801
$708$	0	0
$709$	−43.2069	−1.62267	−0.811335	−	0.584582i	$-0.801259\pi$
$709$	−43.2069	−1.62267	−0.811335	+	0.584582i	$0.801259\pi$
$710$	2.33231	0.0875302
$711$	0	0
$712$	−12.9969	−0.487081
$713$	0	0
$714$	0	0
$715$	−1.49314	−0.0558401
$716$	44.6433	1.66840
$717$	0	0
$718$	3.20293	0.119532
$719$	−48.3133	−1.80178	−0.900890	−	0.434047i	$-0.857085\pi$
$719$	−48.3133	−1.80178	−0.900890	+	0.434047i	$0.857085\pi$
$720$	0	0
$721$	24.9588	0.929515
$722$	37.3868	1.39139
$723$	0	0
$724$	31.4089	1.16730
$725$	−2.45636	−0.0912270
$726$	0	0
$727$	−2.91273	−0.108027	−0.0540135	−	0.998540i	$-0.517201\pi$
$727$	−2.91273	−0.108027	−0.0540135	+	0.998540i	$0.517201\pi$
$728$	−7.08727	−0.262672
$729$	0	0
$730$	27.9652	1.03504
$731$	10.5804	0.391331
$732$	0	0
$733$	14.8117	0.547084	0.273542	−	0.961860i	$-0.411805\pi$
$733$	14.8117	0.547084	0.273542	+	0.961860i	$0.411805\pi$
$734$	26.0568	0.961773
$735$	0	0
$736$	0	0
$737$	−6.02745	−0.222024
$738$	0	0
$739$	−35.3354	−1.29983	−0.649916	−	0.760006i	$-0.725196\pi$
$739$	−35.3354	−1.29983	−0.649916	+	0.760006i	$0.725196\pi$
$740$	1.18721	0.0436428
$741$	0	0
$742$	−65.0250	−2.38714
$743$	−2.98627	−0.109556	−0.0547779	−	0.998499i	$-0.517445\pi$
$743$	−2.98627	−0.109556	−0.0547779	+	0.998499i	$0.517445\pi$
$744$	0	0
$745$	10.0735	0.369066
$746$	−0.157766	−0.00577624
$747$	0	0
$748$	1.94217	0.0710128
$749$	37.0598	1.35414
$750$	0	0
$751$	−9.34604	−0.341042	−0.170521	−	0.985354i	$-0.554545\pi$
$751$	−9.34604	−0.341042	−0.170521	+	0.985354i	$0.554545\pi$
$752$	−0.829724	−0.0302569
$753$	0	0
$754$	10.5383	0.383782
$755$	4.34071	0.157974
$756$	0	0
$757$	−19.8255	−0.720568	−0.360284	−	0.932843i	$-0.617320\pi$
$757$	−19.8255	−0.720568	−0.360284	+	0.932843i	$0.617320\pi$
$758$	−50.7696	−1.84404
$759$	0	0
$760$	1.61718	0.0586614
$761$	46.0735	1.67016	0.835082	−	0.550125i	$-0.185420\pi$
$761$	46.0735	1.67016	0.835082	+	0.550125i	$0.185420\pi$
$762$	0	0
$763$	33.1608	1.20050
$764$	−36.8746	−1.33408
$765$	0	0
$766$	−50.7516	−1.83373
$767$	−2.98627	−0.107828
$768$	0	0
$769$	−9.44264	−0.340510	−0.170255	−	0.985400i	$-0.554459\pi$
$769$	−9.44264	−0.340510	−0.170255	+	0.985400i	$0.554459\pi$
$770$	4.39853	0.158512
$771$	0	0
$772$	8.03144	0.289058
$773$	11.2849	0.405889	0.202944	−	0.979190i	$-0.434949\pi$
$773$	11.2849	0.405889	0.202944	+	0.979190i	$0.434949\pi$
$774$	0	0
$775$	−5.49314	−0.197319
$776$	−1.17760	−0.0422735
$777$	0	0
$778$	49.0913	1.76001
$779$	4.95043	0.177367
$780$	0	0
$781$	0.811724	0.0290458
$782$	0	0
$783$	0	0
$784$	−1.32392	−0.0472830
$785$	12.1745	0.434528
$786$	0	0
$787$	−21.6318	−0.771092	−0.385546	−	0.922689i	$-0.625987\pi$
$787$	−21.6318	−0.771092	−0.385546	+	0.922689i	$0.625987\pi$
$788$	23.3775	0.832788
$789$	0	0
$790$	17.1608	0.610555
$791$	2.98627	0.106180
$792$	0	0
$793$	−10.1745	−0.361309
$794$	7.38481	0.262077
$795$	0	0
$796$	30.3531	1.07584
$797$	18.9358	0.670739	0.335370	−	0.942087i	$-0.391139\pi$
$797$	18.9358	0.670739	0.335370	+	0.942087i	$0.391139\pi$
$798$	0	0
$799$	0.340706	0.0120533
$800$	7.80440	0.275927
$801$	0	0
$802$	13.1367	0.463873
$803$	9.73284	0.343465
$804$	0	0
$805$	0	0
$806$	23.5667	0.830101
$807$	0	0
$808$	−19.3354	−0.680216
$809$	40.3216	1.41763	0.708817	−	0.705393i	$-0.249229\pi$
$809$	40.3216	1.41763	0.708817	+	0.705393i	$0.249229\pi$
$810$	0	0
$811$	−8.34910	−0.293176	−0.146588	−	0.989198i	$-0.546829\pi$
$811$	−8.34910	−0.293176	−0.146588	+	0.989198i	$0.546829\pi$
$812$	−17.5510	−0.615918
$813$	0	0
$814$	0.730849	0.0256163
$815$	22.3133	0.781599
$816$	0	0
$817$	13.2618	0.463972
$818$	−70.3122	−2.45841
$819$	0	0
$820$	−10.2745	−0.358801
$821$	−21.7980	−0.760755	−0.380378	−	0.924831i	$-0.624206\pi$
$821$	−21.7980	−0.760755	−0.380378	+	0.924831i	$0.624206\pi$
$822$	0	0
$823$	−53.5308	−1.86597	−0.932984	−	0.359918i	$-0.882805\pi$
$823$	−53.5308	−1.86597	−0.932984	+	0.359918i	$0.882805\pi$
$824$	−11.7245	−0.408441
$825$	0	0
$826$	8.79707	0.306089
$827$	20.1471	0.700583	0.350292	−	0.936641i	$-0.386083\pi$
$827$	20.1471	0.700583	0.350292	+	0.936641i	$0.386083\pi$
$828$	0	0
$829$	12.1976	0.423640	0.211820	−	0.977309i	$-0.432061\pi$
$829$	12.1976	0.423640	0.211820	+	0.977309i	$0.432061\pi$
$830$	32.1471	1.11584
$831$	0	0
$832$	−23.7412	−0.823079
$833$	0.543637	0.0188359
$834$	0	0
$835$	12.0000	0.415277
$836$	2.43438	0.0841948
$837$	0	0
$838$	40.1544	1.38711
$839$	24.9947	0.862912	0.431456	−	0.902134i	$-0.358000\pi$
$839$	24.9947	0.862912	0.431456	+	0.902134i	$0.358000\pi$
$840$	0	0
$841$	−22.9663	−0.791941
$842$	49.1996	1.69553
$843$	0	0
$844$	52.0293	1.79092
$845$	9.00000	0.309609
$846$	0	0
$847$	−28.6814	−0.985505
$848$	−26.8779	−0.922992
$849$	0	0
$850$	−2.14510	−0.0735764
$851$	0	0
$852$	0	0
$853$	−6.73818	−0.230711	−0.115355	−	0.993324i	$-0.536801\pi$
$853$	−6.73818	−0.230711	−0.115355	+	0.993324i	$0.536801\pi$
$854$	29.9725	1.02564
$855$	0	0
$856$	−17.4089	−0.595025
$857$	−30.8117	−1.05251	−0.526254	−	0.850327i	$-0.676404\pi$
$857$	−30.8117	−1.05251	−0.526254	+	0.850327i	$0.676404\pi$
$858$	0	0
$859$	17.7328	0.605037	0.302518	−	0.953144i	$-0.402173\pi$
$859$	17.7328	0.605037	0.302518	+	0.953144i	$0.402173\pi$
$860$	−27.5246	−0.938580
$861$	0	0
$862$	26.4132	0.899637
$863$	10.8476	0.369256	0.184628	−	0.982809i	$-0.440892\pi$
$863$	10.8476	0.369256	0.184628	+	0.982809i	$0.440892\pi$
$864$	0	0
$865$	19.0598	0.648053
$866$	75.2069	2.55563
$867$	0	0
$868$	−39.2490	−1.33220
$869$	5.97255	0.202605
$870$	0	0
$871$	−16.1471	−0.547123
$872$	−15.5774	−0.527516
$873$	0	0
$874$	0	0
$875$	−2.74657	−0.0928510
$876$	0	0
$877$	9.36909	0.316372	0.158186	−	0.987409i	$-0.449435\pi$
$877$	9.36909	0.316372	0.158186	+	0.987409i	$0.449435\pi$
$878$	40.7275	1.37449
$879$	0	0
$880$	1.81812	0.0612889
$881$	49.4594	1.66633	0.833165	−	0.553024i	$-0.186526\pi$
$881$	49.4594	1.66633	0.833165	+	0.553024i	$0.186526\pi$
$882$	0	0
$883$	40.5530	1.36472	0.682358	−	0.731018i	$-0.260955\pi$
$883$	40.5530	1.36472	0.682358	+	0.731018i	$0.260955\pi$
$884$	5.20293	0.174994
$885$	0	0
$886$	−38.5530	−1.29521
$887$	−17.1608	−0.576204	−0.288102	−	0.957600i	$-0.593024\pi$
$887$	−17.1608	−0.576204	−0.288102	+	0.957600i	$0.593024\pi$
$888$	0	0
$889$	−22.1745	−0.743710
$890$	21.6088	0.724328
$891$	0	0
$892$	64.7382	2.16759
$893$	0.427052	0.0142907
$894$	0	0
$895$	−17.1608	−0.573623
$896$	27.0671	0.904250
$897$	0	0
$898$	81.8294	2.73068
$899$	13.4931	0.450021
$900$	0	0
$901$	11.0368	0.367688
$902$	−6.32499	−0.210599
$903$	0	0
$904$	−1.40281	−0.0466567
$905$	−12.0735	−0.401338
$906$	0	0
$907$	36.1387	1.19997	0.599983	−	0.800013i	$-0.295174\pi$
$907$	36.1387	1.19997	0.599983	+	0.800013i	$0.295174\pi$
$908$	−46.7550	−1.55162
$909$	0	0
$910$	11.7833	0.390614
$911$	−37.8064	−1.25258	−0.626291	−	0.779590i	$-0.715428\pi$
$911$	−37.8064	−1.25258	−0.626291	+	0.779590i	$0.715428\pi$
$912$	0	0
$913$	11.1883	0.370278
$914$	43.0598	1.42429
$915$	0	0
$916$	−7.90006	−0.261025
$917$	−44.1471	−1.45787
$918$	0	0
$919$	−46.3133	−1.52773	−0.763867	−	0.645374i	$-0.776701\pi$
$919$	−46.3133	−1.52773	−0.763867	+	0.645374i	$0.776701\pi$
$920$	0	0
$921$	0	0
$922$	−62.5530	−2.06007
$923$	2.17455	0.0715761
$924$	0	0
$925$	−0.456363	−0.0150051
$926$	−21.2344	−0.697805
$927$	0	0
$928$	−19.1704	−0.629300
$929$	4.76122	0.156211	0.0781053	−	0.996945i	$-0.475113\pi$
$929$	4.76122	0.156211	0.0781053	+	0.996945i	$0.475113\pi$
$930$	0	0
$931$	0.681412	0.0223324
$932$	−10.5972	−0.347123
$933$	0	0
$934$	25.5426	0.835779
$935$	−0.746568	−0.0244154
$936$	0	0
$937$	−10.2481	−0.334791	−0.167395	−	0.985890i	$-0.553536\pi$
$937$	−10.2481	−0.334791	−0.167395	+	0.985890i	$0.553536\pi$
$938$	47.5667	1.55311
$939$	0	0
$940$	−0.886335	−0.0289091
$941$	35.9725	1.17267	0.586336	−	0.810068i	$-0.300570\pi$
$941$	35.9725	1.17267	0.586336	+	0.810068i	$0.300570\pi$
$942$	0	0
$943$	0	0
$944$	3.63625	0.118350
$945$	0	0
$946$	−16.9442	−0.550902
$947$	44.9588	1.46097	0.730483	−	0.682931i	$-0.239295\pi$
$947$	44.9588	1.46097	0.730483	+	0.682931i	$0.239295\pi$
$948$	0	0
$949$	26.0735	0.846383
$950$	−2.68874	−0.0872342
$951$	0	0
$952$	−3.54364	−0.114850
$953$	−12.7780	−0.413920	−0.206960	−	0.978349i	$-0.566357\pi$
$953$	−12.7780	−0.413920	−0.206960	+	0.978349i	$0.566357\pi$
$954$	0	0
$955$	14.1745	0.458678
$956$	−40.7589	−1.31824
$957$	0	0
$958$	55.5560	1.79493
$959$	−14.9486	−0.482715
$960$	0	0
$961$	−0.825451	−0.0266275
$962$	1.95789	0.0631249
$963$	0	0
$964$	−21.0031	−0.676463
$965$	−3.08727	−0.0993829
$966$	0	0
$967$	−27.5941	−0.887368	−0.443684	−	0.896183i	$-0.646329\pi$
$967$	−27.5941	−0.887368	−0.443684	+	0.896183i	$0.646329\pi$
$968$	13.4731	0.433043
$969$	0	0
$970$	1.95789	0.0628641
$971$	21.8255	0.700412	0.350206	−	0.936673i	$-0.386112\pi$
$971$	21.8255	0.700412	0.350206	+	0.936673i	$0.386112\pi$
$972$	0	0
$973$	26.0735	0.835880
$974$	6.83918	0.219141
$975$	0	0
$976$	12.3891	0.396565
$977$	−51.7705	−1.65629	−0.828143	−	0.560517i	$-0.810603\pi$
$977$	−51.7705	−1.65629	−0.828143	+	0.560517i	$0.810603\pi$
$978$	0	0
$979$	7.52059	0.240359
$980$	−1.41425	−0.0451767
$981$	0	0
$982$	9.60879	0.306629
$983$	19.4657	0.620859	0.310429	−	0.950596i	$-0.399527\pi$
$983$	19.4657	0.620859	0.310429	+	0.950596i	$0.399527\pi$
$984$	0	0
$985$	−8.98627	−0.286326
$986$	5.26915	0.167804
$987$	0	0
$988$	6.52152	0.207477
$989$	0	0
$990$	0	0
$991$	−28.8117	−0.915235	−0.457617	−	0.889149i	$-0.651297\pi$
$991$	−28.8117	−0.915235	−0.457617	+	0.889149i	$0.651297\pi$
$992$	−42.8706	−1.36114
$993$	0	0
$994$	−6.40586	−0.203182
$995$	−11.6677	−0.369890
$996$	0	0
$997$	−56.8516	−1.80051	−0.900253	−	0.435366i	$-0.856619\pi$
$997$	−56.8516	−1.80051	−0.900253	+	0.435366i	$0.856619\pi$
$998$	−87.5813	−2.77234
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	765.2.a.k.1.1	✓	3
3.2	odd	2		765.2.a.l.1.3	yes	3
5.4	even	2		3825.2.a.be.1.3		3
15.14	odd	2		3825.2.a.bf.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
765.2.a.k.1.1	✓	3	1.1	even	1	trivial
765.2.a.l.1.3	yes	3	3.2	odd	2
3825.2.a.be.1.3		3	5.4	even	2
3825.2.a.bf.1.1		3	15.14	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	765.a (trivial)

\(f(q)\)	\(=\)	\(q-2.14510 q^{2} +2.60147 q^{4} -1.00000 q^{5} +2.74657 q^{7} -1.29021 q^{8} +2.14510 q^{10} +0.746568 q^{11} +2.00000 q^{13} -5.89167 q^{14} -2.43531 q^{16} +1.00000 q^{17} +1.25343 q^{19} -2.60147 q^{20} -1.60147 q^{22} +1.00000 q^{25} -4.29021 q^{26} +7.14510 q^{28} -2.45636 q^{29} -5.49314 q^{31} +7.80440 q^{32} -2.14510 q^{34} -2.74657 q^{35} -0.456363 q^{37} -2.68874 q^{38} +1.29021 q^{40} +3.94950 q^{41} +10.5804 q^{43} +1.94217 q^{44} +0.340706 q^{47} +0.543637 q^{49} -2.14510 q^{50} +5.20293 q^{52} +11.0368 q^{53} -0.746568 q^{55} -3.54364 q^{56} +5.26915 q^{58} -1.49314 q^{59} -5.08727 q^{61} +11.7833 q^{62} -11.8706 q^{64} -2.00000 q^{65} -8.07355 q^{67} +2.60147 q^{68} +5.89167 q^{70} +1.08727 q^{71} +13.0368 q^{73} +0.978945 q^{74} +3.26076 q^{76} +2.05050 q^{77} +8.00000 q^{79} +2.43531 q^{80} -8.47208 q^{82} +14.9863 q^{83} -1.00000 q^{85} -22.6961 q^{86} -0.963226 q^{88} +10.0735 q^{89} +5.49314 q^{91} -0.730849 q^{94} -1.25343 q^{95} +0.912726 q^{97} -1.16616 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} - 3 q^{5} + 9 q^{8} - 6 q^{11} + 6 q^{13} - 3 q^{14} + 12 q^{16} + 3 q^{17} + 12 q^{19} - 6 q^{20} - 3 q^{22} + 3 q^{25} + 15 q^{28} - 12 q^{29} + 18 q^{32} - 6 q^{37} + 3 q^{38} - 9 q^{40}+ \cdots - 21 q^{98}+O(q^{100}) \) 3 * q + 6 * q^4 - 3 * q^5 + 9 * q^8 - 6 * q^11 + 6 * q^13 - 3 * q^14 + 12 * q^16 + 3 * q^17 + 12 * q^19 - 6 * q^20 - 3 * q^22 + 3 * q^25 + 15 * q^28 - 12 * q^29 + 18 * q^32 - 6 * q^37 + 3 * q^38 - 9 * q^40 + 6 * q^43 + 3 * q^44 - 3 * q^49 + 12 * q^52 + 12 * q^53 + 6 * q^55 - 6 * q^56 - 21 * q^58 + 12 * q^59 - 6 * q^61 + 6 * q^62 + 3 * q^64 - 6 * q^65 + 18 * q^67 + 6 * q^68 + 3 * q^70 - 6 * q^71 + 18 * q^73 - 21 * q^74 + 9 * q^76 + 18 * q^77 + 24 * q^79 - 12 * q^80 + 15 * q^82 + 12 * q^83 - 3 * q^85 - 48 * q^86 - 24 * q^88 - 12 * q^89 - 39 * q^94 - 12 * q^95 + 12 * q^97 - 21 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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