LMFDB - Embedded newform 2760.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(1,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, 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("2760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2760.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:	$22.0387109579$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.54764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.36007$ of defining polynomial
Character	$\chi$	$=$	2760.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +0.512641 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +0.512641 q^{7} +1.00000 q^{9} -1.76485 q^{11} -4.95530 q^{13} +1.00000 q^{15} +7.97235 q^{17} +1.93002 q^{19} -0.512641 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -9.16280 q^{29} +4.20750 q^{31} +1.76485 q^{33} -0.512641 q^{35} +6.37267 q^{37} +4.95530 q^{39} +4.27749 q^{41} -12.8853 q^{43} -1.00000 q^{45} -1.93002 q^{47} -6.73720 q^{49} -7.97235 q^{51} +4.60782 q^{53} +1.76485 q^{55} -1.93002 q^{57} -11.1628 q^{59} +8.65016 q^{61} +0.512641 q^{63} +4.95530 q^{65} -11.1822 q^{67} -1.00000 q^{69} +2.27749 q^{71} -8.95530 q^{73} -1.00000 q^{75} -0.904733 q^{77} +8.58018 q^{79} +1.00000 q^{81} -6.91296 q^{83} -7.97235 q^{85} +9.16280 q^{87} -1.69486 q^{89} -2.54029 q^{91} -4.20750 q^{93} -1.93002 q^{95} -14.4150 q^{97} -1.76485 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^9	$ 4 q - 4 q^{3} - 4 q^{5} + 4 q^{9} - 2 q^{13} + 4 q^{15} + 2 q^{17} - 6 q^{19} + 4 q^{23} + 4 q^{25} - 4 q^{27} + 4 q^{29} - 6 q^{31} - 4 q^{37} + 2 q^{39} + 8 q^{41} - 20 q^{43} - 4 q^{45} + 6 q^{47} + 10 q^{49} - 2 q^{51} - 4 q^{53} + 6 q^{57} - 4 q^{59} - 4 q^{61} + 2 q^{65} - 26 q^{67} - 4 q^{69} - 18 q^{73} - 4 q^{75} + 6 q^{77} - 18 q^{79} + 4 q^{81} - 26 q^{83} - 2 q^{85} - 4 q^{87} + 14 q^{89} - 38 q^{91} + 6 q^{93} + 6 q^{95} - 12 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^9 - 2 * q^13 + 4 * q^15 + 2 * q^17 - 6 * q^19 + 4 * q^23 + 4 * q^25 - 4 * q^27 + 4 * q^29 - 6 * q^31 - 4 * q^37 + 2 * q^39 + 8 * q^41 - 20 * q^43 - 4 * q^45 + 6 * q^47 + 10 * q^49 - 2 * q^51 - 4 * q^53 + 6 * q^57 - 4 * q^59 - 4 * q^61 + 2 * q^65 - 26 * q^67 - 4 * q^69 - 18 * q^73 - 4 * q^75 + 6 * q^77 - 18 * q^79 + 4 * q^81 - 26 * q^83 - 2 * q^85 - 4 * q^87 + 14 * q^89 - 38 * q^91 + 6 * q^93 + 6 * q^95 - 12 * q^97

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
$7$	0.512641	0.193760	0.0968801	−	0.995296i	$-0.469114\pi$
$7$	0.512641	0.193760	0.0968801	+	0.995296i	$0.469114\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.76485	−0.532122	−0.266061	−	0.963956i	$-0.585722\pi$
$11$	−1.76485	−0.532122	−0.266061	+	0.963956i	$0.585722\pi$
$12$	0	0
$13$	−4.95530	−1.37435	−0.687176	−	0.726491i	$-0.741150\pi$
$13$	−4.95530	−1.37435	−0.687176	+	0.726491i	$0.741150\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	7.97235	1.93358	0.966790	−	0.255573i	$-0.0822643\pi$
$17$	7.97235	1.93358	0.966790	+	0.255573i	$0.0822643\pi$
$18$	0	0
$19$	1.93002	0.442776	0.221388	−	0.975186i	$-0.428941\pi$
$19$	1.93002	0.442776	0.221388	+	0.975186i	$0.428941\pi$
$20$	0	0
$21$	−0.512641	−0.111867
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−9.16280	−1.70149	−0.850745	−	0.525579i	$-0.823849\pi$
$29$	−9.16280	−1.70149	−0.850745	+	0.525579i	$0.823849\pi$
$30$	0	0
$31$	4.20750	0.755690	0.377845	−	0.925869i	$-0.376665\pi$
$31$	4.20750	0.755690	0.377845	+	0.925869i	$0.376665\pi$
$32$	0	0
$33$	1.76485	0.307221
$34$	0	0
$35$	−0.512641	−0.0866522
$36$	0	0
$37$	6.37267	1.04766	0.523830	−	0.851823i	$-0.324503\pi$
$37$	6.37267	1.04766	0.523830	+	0.851823i	$0.324503\pi$
$38$	0	0
$39$	4.95530	0.793483
$40$	0	0
$41$	4.27749	0.668031	0.334016	−	0.942567i	$-0.391596\pi$
$41$	4.27749	0.668031	0.334016	+	0.942567i	$0.391596\pi$
$42$	0	0
$43$	−12.8853	−1.96499	−0.982496	−	0.186284i	$-0.940356\pi$
$43$	−12.8853	−1.96499	−0.982496	+	0.186284i	$0.940356\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−1.93002	−0.281522	−0.140761	−	0.990044i	$-0.544955\pi$
$47$	−1.93002	−0.281522	−0.140761	+	0.990044i	$0.544955\pi$
$48$	0	0
$49$	−6.73720	−0.962457
$50$	0	0
$51$	−7.97235	−1.11635
$52$	0	0
$53$	4.60782	0.632933	0.316467	−	0.948604i	$-0.397503\pi$
$53$	4.60782	0.632933	0.316467	+	0.948604i	$0.397503\pi$
$54$	0	0
$55$	1.76485	0.237972
$56$	0	0
$57$	−1.93002	−0.255637
$58$	0	0
$59$	−11.1628	−1.45327	−0.726637	−	0.687022i	$-0.758918\pi$
$59$	−11.1628	−1.45327	−0.726637	+	0.687022i	$0.758918\pi$
$60$	0	0
$61$	8.65016	1.10754	0.553770	−	0.832670i	$-0.313189\pi$
$61$	8.65016	1.10754	0.553770	+	0.832670i	$0.313189\pi$
$62$	0	0
$63$	0.512641	0.0645867
$64$	0	0
$65$	4.95530	0.614629
$66$	0	0
$67$	−11.1822	−1.36613	−0.683063	−	0.730360i	$-0.739353\pi$
$67$	−11.1822	−1.36613	−0.683063	+	0.730360i	$0.739353\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	2.27749	0.270288	0.135144	−	0.990826i	$-0.456850\pi$
$71$	2.27749	0.270288	0.135144	+	0.990826i	$0.456850\pi$
$72$	0	0
$73$	−8.95530	−1.04814	−0.524069	−	0.851676i	$-0.675587\pi$
$73$	−8.95530	−1.04814	−0.524069	+	0.851676i	$0.675587\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−0.904733	−0.103104
$78$	0	0
$79$	8.58018	0.965345	0.482673	−	0.875801i	$-0.339666\pi$
$79$	8.58018	0.965345	0.482673	+	0.875801i	$0.339666\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.91296	−0.758796	−0.379398	−	0.925234i	$-0.623869\pi$
$83$	−6.91296	−0.758796	−0.379398	+	0.925234i	$0.623869\pi$
$84$	0	0
$85$	−7.97235	−0.864723
$86$	0	0
$87$	9.16280	0.982355
$88$	0	0
$89$	−1.69486	−0.179655	−0.0898276	−	0.995957i	$-0.528632\pi$
$89$	−1.69486	−0.179655	−0.0898276	+	0.995957i	$0.528632\pi$
$90$	0	0
$91$	−2.54029	−0.266295
$92$	0	0
$93$	−4.20750	−0.436298
$94$	0	0
$95$	−1.93002	−0.198015
$96$	0	0
$97$	−14.4150	−1.46362	−0.731811	−	0.681507i	$-0.761325\pi$
$97$	−14.4150	−1.46362	−0.731811	+	0.681507i	$0.761325\pi$
$98$	0	0
$99$	−1.76485	−0.177374
$100$	0	0
$101$	9.63311	0.958530	0.479265	−	0.877670i	$-0.340903\pi$
$101$	9.63311	0.958530	0.479265	+	0.877670i	$0.340903\pi$
$102$	0	0
$103$	−7.83483	−0.771989	−0.385994	−	0.922501i	$-0.626142\pi$
$103$	−7.83483	−0.771989	−0.385994	+	0.922501i	$0.626142\pi$
$104$	0	0
$105$	0.512641	0.0500287
$106$	0	0
$107$	2.44266	0.236141	0.118070	−	0.993005i	$-0.462329\pi$
$107$	2.44266	0.236141	0.118070	+	0.993005i	$0.462329\pi$
$108$	0	0
$109$	3.76485	0.360607	0.180303	−	0.983611i	$-0.442292\pi$
$109$	3.76485	0.360607	0.180303	+	0.983611i	$0.442292\pi$
$110$	0	0
$111$	−6.37267	−0.604867
$112$	0	0
$113$	−2.53206	−0.238196	−0.119098	−	0.992882i	$-0.538000\pi$
$113$	−2.53206	−0.238196	−0.119098	+	0.992882i	$0.538000\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−4.95530	−0.458117
$118$	0	0
$119$	4.08696	0.374651
$120$	0	0
$121$	−7.88531	−0.716847
$122$	0	0
$123$	−4.27749	−0.385688
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.4344	−1.10338	−0.551689	−	0.834050i	$-0.686016\pi$
$127$	−12.4344	−1.10338	−0.551689	+	0.834050i	$0.686016\pi$
$128$	0	0
$129$	12.8853	1.13449
$130$	0	0
$131$	−1.94944	−0.170323	−0.0851615	−	0.996367i	$-0.527141\pi$
$131$	−1.94944	−0.170323	−0.0851615	+	0.996367i	$0.527141\pi$
$132$	0	0
$133$	0.989405	0.0857923
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	9.22456	0.788107	0.394054	−	0.919087i	$-0.371072\pi$
$137$	9.22456	0.788107	0.394054	+	0.919087i	$0.371072\pi$
$138$	0	0
$139$	−17.5631	−1.48968	−0.744842	−	0.667241i	$-0.767475\pi$
$139$	−17.5631	−1.48968	−0.744842	+	0.667241i	$0.767475\pi$
$140$	0	0
$141$	1.93002	0.162537
$142$	0	0
$143$	8.74534	0.731322
$144$	0	0
$145$	9.16280	0.760929
$146$	0	0
$147$	6.73720	0.555675
$148$	0	0
$149$	14.3956	1.17933	0.589666	−	0.807647i	$-0.299259\pi$
$149$	14.3956	1.17933	0.589666	+	0.807647i	$0.299259\pi$
$150$	0	0
$151$	−14.9359	−1.21546	−0.607732	−	0.794142i	$-0.707921\pi$
$151$	−14.9359	−1.21546	−0.607732	+	0.794142i	$0.707921\pi$
$152$	0	0
$153$	7.97235	0.644526
$154$	0	0
$155$	−4.20750	−0.337955
$156$	0	0
$157$	−18.4573	−1.47306	−0.736528	−	0.676407i	$-0.763536\pi$
$157$	−18.4573	−1.47306	−0.736528	+	0.676407i	$0.763536\pi$
$158$	0	0
$159$	−4.60782	−0.365424
$160$	0	0
$161$	0.512641	0.0404018
$162$	0	0
$163$	−3.58499	−0.280798	−0.140399	−	0.990095i	$-0.544839\pi$
$163$	−3.58499	−0.280798	−0.140399	+	0.990095i	$0.544839\pi$
$164$	0	0
$165$	−1.76485	−0.137393
$166$	0	0
$167$	−0.120549	−0.00932835	−0.00466418	−	0.999989i	$-0.501485\pi$
$167$	−0.120549	−0.00932835	−0.00466418	+	0.999989i	$0.501485\pi$
$168$	0	0
$169$	11.5550	0.888844
$170$	0	0
$171$	1.93002	0.147592
$172$	0	0
$173$	−11.0253	−0.838237	−0.419118	−	0.907932i	$-0.637661\pi$
$173$	−11.0253	−0.838237	−0.419118	+	0.907932i	$0.637661\pi$
$174$	0	0
$175$	0.512641	0.0387520
$176$	0	0
$177$	11.1628	0.839048
$178$	0	0
$179$	16.2750	1.21645	0.608227	−	0.793763i	$-0.291881\pi$
$179$	16.2750	1.21645	0.608227	+	0.793763i	$0.291881\pi$
$180$	0	0
$181$	12.7543	0.948016	0.474008	−	0.880520i	$-0.342807\pi$
$181$	12.7543	0.948016	0.474008	+	0.880520i	$0.342807\pi$
$182$	0	0
$183$	−8.65016	−0.639438
$184$	0	0
$185$	−6.37267	−0.468528
$186$	0	0
$187$	−14.0700	−1.02890
$188$	0	0
$189$	−0.512641	−0.0372892
$190$	0	0
$191$	−14.9000	−1.07813	−0.539063	−	0.842265i	$-0.681222\pi$
$191$	−14.9000	−1.07813	−0.539063	+	0.842265i	$0.681222\pi$
$192$	0	0
$193$	−17.9447	−1.29169	−0.645844	−	0.763469i	$-0.723494\pi$
$193$	−17.9447	−1.29169	−0.645844	+	0.763469i	$0.723494\pi$
$194$	0	0
$195$	−4.95530	−0.354856
$196$	0	0
$197$	−6.96999	−0.496591	−0.248295	−	0.968684i	$-0.579870\pi$
$197$	−6.96999	−0.496591	−0.248295	+	0.968684i	$0.579870\pi$
$198$	0	0
$199$	−21.4655	−1.52165	−0.760824	−	0.648958i	$-0.775205\pi$
$199$	−21.4655	−1.52165	−0.760824	+	0.648958i	$0.775205\pi$
$200$	0	0
$201$	11.1822	0.788733
$202$	0	0
$203$	−4.69723	−0.329681
$204$	0	0
$205$	−4.27749	−0.298753
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−3.40618	−0.235611
$210$	0	0
$211$	−25.3175	−1.74293	−0.871463	−	0.490462i	$-0.836828\pi$
$211$	−25.3175	−1.74293	−0.871463	+	0.490462i	$0.836828\pi$
$212$	0	0
$213$	−2.27749	−0.156051
$214$	0	0
$215$	12.8853	0.878771
$216$	0	0
$217$	2.15694	0.146423
$218$	0	0
$219$	8.95530	0.605143
$220$	0	0
$221$	−39.5054	−2.65742
$222$	0	0
$223$	−14.3644	−0.961914	−0.480957	−	0.876744i	$-0.659711\pi$
$223$	−14.3644	−0.961914	−0.480957	+	0.876744i	$0.659711\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−16.2498	−1.07854	−0.539270	−	0.842133i	$-0.681300\pi$
$227$	−16.2498	−1.07854	−0.539270	+	0.842133i	$0.681300\pi$
$228$	0	0
$229$	14.8048	0.978330	0.489165	−	0.872191i	$-0.337302\pi$
$229$	14.8048	0.978330	0.489165	+	0.872191i	$0.337302\pi$
$230$	0	0
$231$	0.904733	0.0595271
$232$	0	0
$233$	−29.8047	−1.95257	−0.976287	−	0.216482i	$-0.930542\pi$
$233$	−29.8047	−1.95257	−0.976287	+	0.216482i	$0.930542\pi$
$234$	0	0
$235$	1.93002	0.125900
$236$	0	0
$237$	−8.58018	−0.557342
$238$	0	0
$239$	−11.1075	−0.718485	−0.359242	−	0.933244i	$-0.616965\pi$
$239$	−11.1075	−0.718485	−0.359242	+	0.933244i	$0.616965\pi$
$240$	0	0
$241$	20.3062	1.30804	0.654018	−	0.756479i	$-0.273082\pi$
$241$	20.3062	1.30804	0.654018	+	0.756479i	$0.273082\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.73720	0.430424
$246$	0	0
$247$	−9.56380	−0.608530
$248$	0	0
$249$	6.91296	0.438091
$250$	0	0
$251$	2.21573	0.139856	0.0699279	−	0.997552i	$-0.477723\pi$
$251$	2.21573	0.139856	0.0699279	+	0.997552i	$0.477723\pi$
$252$	0	0
$253$	−1.76485	−0.110955
$254$	0	0
$255$	7.97235	0.499248
$256$	0	0
$257$	2.90473	0.181192	0.0905961	−	0.995888i	$-0.471123\pi$
$257$	2.90473	0.181192	0.0905961	+	0.995888i	$0.471123\pi$
$258$	0	0
$259$	3.26689	0.202995
$260$	0	0
$261$	−9.16280	−0.567163
$262$	0	0
$263$	3.75662	0.231643	0.115822	−	0.993270i	$-0.463050\pi$
$263$	3.75662	0.231643	0.115822	+	0.993270i	$0.463050\pi$
$264$	0	0
$265$	−4.60782	−0.283056
$266$	0	0
$267$	1.69486	0.103724
$268$	0	0
$269$	7.58254	0.462316	0.231158	−	0.972916i	$-0.425749\pi$
$269$	7.58254	0.462316	0.231158	+	0.972916i	$0.425749\pi$
$270$	0	0
$271$	−15.0423	−0.913752	−0.456876	−	0.889530i	$-0.651032\pi$
$271$	−15.0423	−0.913752	−0.456876	+	0.889530i	$0.651032\pi$
$272$	0	0
$273$	2.54029	0.153745
$274$	0	0
$275$	−1.76485	−0.106424
$276$	0	0
$277$	8.05056	0.483712	0.241856	−	0.970312i	$-0.422244\pi$
$277$	8.05056	0.483712	0.241856	+	0.970312i	$0.422244\pi$
$278$	0	0
$279$	4.20750	0.251897
$280$	0	0
$281$	−14.3704	−0.857266	−0.428633	−	0.903479i	$-0.641005\pi$
$281$	−14.3704	−0.857266	−0.428633	+	0.903479i	$0.641005\pi$
$282$	0	0
$283$	24.0287	1.42836	0.714179	−	0.699963i	$-0.246800\pi$
$283$	24.0287	1.42836	0.714179	+	0.699963i	$0.246800\pi$
$284$	0	0
$285$	1.93002	0.114324
$286$	0	0
$287$	2.19282	0.129438
$288$	0	0
$289$	46.5584	2.73873
$290$	0	0
$291$	14.4150	0.845023
$292$	0	0
$293$	8.46786	0.494697	0.247349	−	0.968927i	$-0.420441\pi$
$293$	8.46786	0.494697	0.247349	+	0.968927i	$0.420441\pi$
$294$	0	0
$295$	11.1628	0.649923
$296$	0	0
$297$	1.76485	0.102407
$298$	0	0
$299$	−4.95530	−0.286572
$300$	0	0
$301$	−6.60554	−0.380737
$302$	0	0
$303$	−9.63311	−0.553408
$304$	0	0
$305$	−8.65016	−0.495307
$306$	0	0
$307$	17.9594	1.02500	0.512498	−	0.858688i	$-0.328720\pi$
$307$	17.9594	1.02500	0.512498	+	0.858688i	$0.328720\pi$
$308$	0	0
$309$	7.83483	0.445708
$310$	0	0
$311$	23.8894	1.35464	0.677322	−	0.735687i	$-0.263140\pi$
$311$	23.8894	1.35464	0.677322	+	0.735687i	$0.263140\pi$
$312$	0	0
$313$	25.4184	1.43673	0.718367	−	0.695664i	$-0.244890\pi$
$313$	25.4184	1.43673	0.718367	+	0.695664i	$0.244890\pi$
$314$	0	0
$315$	−0.512641	−0.0288841
$316$	0	0
$317$	24.0147	1.34880	0.674400	−	0.738367i	$-0.264403\pi$
$317$	24.0147	1.34880	0.674400	+	0.738367i	$0.264403\pi$
$318$	0	0
$319$	16.1709	0.905399
$320$	0	0
$321$	−2.44266	−0.136336
$322$	0	0
$323$	15.3868	0.856142
$324$	0	0
$325$	−4.95530	−0.274870
$326$	0	0
$327$	−3.76485	−0.208197
$328$	0	0
$329$	−0.989405	−0.0545477
$330$	0	0
$331$	−12.7625	−0.701489	−0.350745	−	0.936471i	$-0.614072\pi$
$331$	−12.7625	−0.701489	−0.350745	+	0.936471i	$0.614072\pi$
$332$	0	0
$333$	6.37267	0.349220
$334$	0	0
$335$	11.1822	0.610950
$336$	0	0
$337$	−5.38973	−0.293597	−0.146799	−	0.989166i	$-0.546897\pi$
$337$	−5.38973	−0.293597	−0.146799	+	0.989166i	$0.546897\pi$
$338$	0	0
$339$	2.53206	0.137523
$340$	0	0
$341$	−7.42560	−0.402119
$342$	0	0
$343$	−7.04225	−0.380246
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	7.52969	0.404215	0.202108	−	0.979363i	$-0.435221\pi$
$347$	7.52969	0.404215	0.202108	+	0.979363i	$0.435221\pi$
$348$	0	0
$349$	−5.59723	−0.299613	−0.149806	−	0.988715i	$-0.547865\pi$
$349$	−5.59723	−0.299613	−0.149806	+	0.988715i	$0.547865\pi$
$350$	0	0
$351$	4.95530	0.264494
$352$	0	0
$353$	−37.3150	−1.98608	−0.993039	−	0.117788i	$-0.962420\pi$
$353$	−37.3150	−1.98608	−0.993039	+	0.117788i	$0.962420\pi$
$354$	0	0
$355$	−2.27749	−0.120877
$356$	0	0
$357$	−4.08696	−0.216305
$358$	0	0
$359$	15.1799	0.801167	0.400583	−	0.916260i	$-0.368808\pi$
$359$	15.1799	0.801167	0.400583	+	0.916260i	$0.368808\pi$
$360$	0	0
$361$	−15.2750	−0.803949
$362$	0	0
$363$	7.88531	0.413872
$364$	0	0
$365$	8.95530	0.468742
$366$	0	0
$367$	−35.6389	−1.86033	−0.930167	−	0.367136i	$-0.880338\pi$
$367$	−35.6389	−1.86033	−0.930167	+	0.367136i	$0.880338\pi$
$368$	0	0
$369$	4.27749	0.222677
$370$	0	0
$371$	2.36216	0.122637
$372$	0	0
$373$	−0.864846	−0.0447800	−0.0223900	−	0.999749i	$-0.507128\pi$
$373$	−0.864846	−0.0447800	−0.0223900	+	0.999749i	$0.507128\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	45.4044	2.33845
$378$	0	0
$379$	−16.8300	−0.864500	−0.432250	−	0.901754i	$-0.642280\pi$
$379$	−16.8300	−0.864500	−0.432250	+	0.901754i	$0.642280\pi$
$380$	0	0
$381$	12.4344	0.637035
$382$	0	0
$383$	18.3622	0.938263	0.469131	−	0.883128i	$-0.344567\pi$
$383$	18.3622	0.938263	0.469131	+	0.883128i	$0.344567\pi$
$384$	0	0
$385$	0.904733	0.0461095
$386$	0	0
$387$	−12.8853	−0.654997
$388$	0	0
$389$	22.0294	1.11693	0.558467	−	0.829527i	$-0.311390\pi$
$389$	22.0294	1.11693	0.558467	+	0.829527i	$0.311390\pi$
$390$	0	0
$391$	7.97235	0.403179
$392$	0	0
$393$	1.94944	0.0983360
$394$	0	0
$395$	−8.58018	−0.431716
$396$	0	0
$397$	23.8553	1.19726	0.598632	−	0.801024i	$-0.295711\pi$
$397$	23.8553	1.19726	0.598632	+	0.801024i	$0.295711\pi$
$398$	0	0
$399$	−0.989405	−0.0495322
$400$	0	0
$401$	18.5413	0.925910	0.462955	−	0.886382i	$-0.346789\pi$
$401$	18.5413	0.925910	0.462955	+	0.886382i	$0.346789\pi$
$402$	0	0
$403$	−20.8494	−1.03858
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−11.2468	−0.557483
$408$	0	0
$409$	24.5884	1.21582	0.607909	−	0.794007i	$-0.292008\pi$
$409$	24.5884	1.21582	0.607909	+	0.794007i	$0.292008\pi$
$410$	0	0
$411$	−9.22456	−0.455014
$412$	0	0
$413$	−5.72251	−0.281586
$414$	0	0
$415$	6.91296	0.339344
$416$	0	0
$417$	17.5631	0.860070
$418$	0	0
$419$	5.86598	0.286572	0.143286	−	0.989681i	$-0.454233\pi$
$419$	5.86598	0.286572	0.143286	+	0.989681i	$0.454233\pi$
$420$	0	0
$421$	−13.3955	−0.652857	−0.326429	−	0.945222i	$-0.605845\pi$
$421$	−13.3955	−0.652857	−0.326429	+	0.945222i	$0.605845\pi$
$422$	0	0
$423$	−1.93002	−0.0938406
$424$	0	0
$425$	7.97235	0.386716
$426$	0	0
$427$	4.43443	0.214597
$428$	0	0
$429$	−8.74534	−0.422229
$430$	0	0
$431$	39.3509	1.89547	0.947733	−	0.319065i	$-0.103369\pi$
$431$	39.3509	1.89547	0.947733	+	0.319065i	$0.103369\pi$
$432$	0	0
$433$	−37.0034	−1.77827	−0.889135	−	0.457644i	$-0.848693\pi$
$433$	−37.0034	−1.77827	−0.889135	+	0.457644i	$0.848693\pi$
$434$	0	0
$435$	−9.16280	−0.439323
$436$	0	0
$437$	1.93002	0.0923252
$438$	0	0
$439$	−7.75434	−0.370094	−0.185047	−	0.982730i	$-0.559244\pi$
$439$	−7.75434	−0.370094	−0.185047	+	0.982730i	$0.559244\pi$
$440$	0	0
$441$	−6.73720	−0.320819
$442$	0	0
$443$	−28.6706	−1.36218	−0.681091	−	0.732198i	$-0.738494\pi$
$443$	−28.6706	−1.36218	−0.681091	+	0.732198i	$0.738494\pi$
$444$	0	0
$445$	1.69486	0.0803442
$446$	0	0
$447$	−14.3956	−0.680888
$448$	0	0
$449$	−17.5943	−0.830325	−0.415162	−	0.909747i	$-0.636275\pi$
$449$	−17.5943	−0.830325	−0.415162	+	0.909747i	$0.636275\pi$
$450$	0	0
$451$	−7.54911	−0.355474
$452$	0	0
$453$	14.9359	0.701749
$454$	0	0
$455$	2.54029	0.119091
$456$	0	0
$457$	20.1569	0.942902	0.471451	−	0.881892i	$-0.343730\pi$
$457$	20.1569	0.942902	0.471451	+	0.881892i	$0.343730\pi$
$458$	0	0
$459$	−7.97235	−0.372118
$460$	0	0
$461$	27.6307	1.28689	0.643444	−	0.765493i	$-0.277505\pi$
$461$	27.6307	1.28689	0.643444	+	0.765493i	$0.277505\pi$
$462$	0	0
$463$	25.0398	1.16370	0.581849	−	0.813297i	$-0.302329\pi$
$463$	25.0398	1.16370	0.581849	+	0.813297i	$0.302329\pi$
$464$	0	0
$465$	4.20750	0.195118
$466$	0	0
$467$	8.06176	0.373054	0.186527	−	0.982450i	$-0.440277\pi$
$467$	8.06176	0.373054	0.186527	+	0.982450i	$0.440277\pi$
$468$	0	0
$469$	−5.73247	−0.264701
$470$	0	0
$471$	18.4573	0.850470
$472$	0	0
$473$	22.7406	1.04561
$474$	0	0
$475$	1.93002	0.0885552
$476$	0	0
$477$	4.60782	0.210978
$478$	0	0
$479$	4.92119	0.224855	0.112427	−	0.993660i	$-0.464137\pi$
$479$	4.92119	0.224855	0.112427	+	0.993660i	$0.464137\pi$
$480$	0	0
$481$	−31.5785	−1.43986
$482$	0	0
$483$	−0.512641	−0.0233260
$484$	0	0
$485$	14.4150	0.654552
$486$	0	0
$487$	7.96412	0.360889	0.180444	−	0.983585i	$-0.442246\pi$
$487$	7.96412	0.360889	0.180444	+	0.983585i	$0.442246\pi$
$488$	0	0
$489$	3.58499	0.162119
$490$	0	0
$491$	37.9587	1.71305	0.856526	−	0.516103i	$-0.172618\pi$
$491$	37.9587	1.71305	0.856526	+	0.516103i	$0.172618\pi$
$492$	0	0
$493$	−73.0491	−3.28997
$494$	0	0
$495$	1.76485	0.0793240
$496$	0	0
$497$	1.16753	0.0523711
$498$	0	0
$499$	−31.8887	−1.42754	−0.713768	−	0.700382i	$-0.753013\pi$
$499$	−31.8887	−1.42754	−0.713768	+	0.700382i	$0.753013\pi$
$500$	0	0
$501$	0.120549	0.00538573
$502$	0	0
$503$	−35.3990	−1.57836	−0.789182	−	0.614160i	$-0.789495\pi$
$503$	−35.3990	−1.57836	−0.789182	+	0.614160i	$0.789495\pi$
$504$	0	0
$505$	−9.63311	−0.428668
$506$	0	0
$507$	−11.5550	−0.513175
$508$	0	0
$509$	−16.0341	−0.710699	−0.355350	−	0.934733i	$-0.615638\pi$
$509$	−16.0341	−0.710699	−0.355350	+	0.934733i	$0.615638\pi$
$510$	0	0
$511$	−4.59085	−0.203087
$512$	0	0
$513$	−1.93002	−0.0852123
$514$	0	0
$515$	7.83483	0.345244
$516$	0	0
$517$	3.40618	0.149804
$518$	0	0
$519$	11.0253	0.483956
$520$	0	0
$521$	16.0399	0.702720	0.351360	−	0.936240i	$-0.385719\pi$
$521$	16.0399	0.702720	0.351360	+	0.936240i	$0.385719\pi$
$522$	0	0
$523$	31.1098	1.36034	0.680168	−	0.733056i	$-0.261907\pi$
$523$	31.1098	1.36034	0.680168	+	0.733056i	$0.261907\pi$
$524$	0	0
$525$	−0.512641	−0.0223735
$526$	0	0
$527$	33.5437	1.46119
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−11.1628	−0.484424
$532$	0	0
$533$	−21.1962	−0.918111
$534$	0	0
$535$	−2.44266	−0.105605
$536$	0	0
$537$	−16.2750	−0.702320
$538$	0	0
$539$	11.8901	0.512144
$540$	0	0
$541$	24.5160	1.05402	0.527012	−	0.849858i	$-0.323312\pi$
$541$	24.5160	1.05402	0.527012	+	0.849858i	$0.323312\pi$
$542$	0	0
$543$	−12.7543	−0.547337
$544$	0	0
$545$	−3.76485	−0.161268
$546$	0	0
$547$	−2.22937	−0.0953211	−0.0476606	−	0.998864i	$-0.515177\pi$
$547$	−2.22937	−0.0953211	−0.0476606	+	0.998864i	$0.515177\pi$
$548$	0	0
$549$	8.65016	0.369180
$550$	0	0
$551$	−17.6844	−0.753379
$552$	0	0
$553$	4.39855	0.187045
$554$	0	0
$555$	6.37267	0.270505
$556$	0	0
$557$	−12.5525	−0.531868	−0.265934	−	0.963991i	$-0.585680\pi$
$557$	−12.5525	−0.531868	−0.265934	+	0.963991i	$0.585680\pi$
$558$	0	0
$559$	63.8506	2.70059
$560$	0	0
$561$	14.0700	0.594035
$562$	0	0
$563$	−27.8829	−1.17513	−0.587563	−	0.809178i	$-0.699913\pi$
$563$	−27.8829	−1.17513	−0.587563	+	0.809178i	$0.699913\pi$
$564$	0	0
$565$	2.53206	0.106525
$566$	0	0
$567$	0.512641	0.0215289
$568$	0	0
$569$	−22.8717	−0.958830	−0.479415	−	0.877588i	$-0.659151\pi$
$569$	−22.8717	−0.958830	−0.479415	+	0.877588i	$0.659151\pi$
$570$	0	0
$571$	30.9847	1.29667	0.648334	−	0.761356i	$-0.275466\pi$
$571$	30.9847	1.29667	0.648334	+	0.761356i	$0.275466\pi$
$572$	0	0
$573$	14.9000	0.622456
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−29.2274	−1.21675	−0.608376	−	0.793649i	$-0.708179\pi$
$577$	−29.2274	−1.21675	−0.608376	+	0.793649i	$0.708179\pi$
$578$	0	0
$579$	17.9447	0.745756
$580$	0	0
$581$	−3.54387	−0.147024
$582$	0	0
$583$	−8.13211	−0.336798
$584$	0	0
$585$	4.95530	0.204876
$586$	0	0
$587$	−4.36461	−0.180147	−0.0900734	−	0.995935i	$-0.528710\pi$
$587$	−4.36461	−0.180147	−0.0900734	+	0.995935i	$0.528710\pi$
$588$	0	0
$589$	8.12055	0.334601
$590$	0	0
$591$	6.96999	0.286707
$592$	0	0
$593$	−28.7765	−1.18171	−0.590854	−	0.806778i	$-0.701209\pi$
$593$	−28.7765	−1.18171	−0.590854	+	0.806778i	$0.701209\pi$
$594$	0	0
$595$	−4.08696	−0.167549
$596$	0	0
$597$	21.4655	0.878524
$598$	0	0
$599$	15.9612	0.652155	0.326078	−	0.945343i	$-0.394273\pi$
$599$	15.9612	0.652155	0.326078	+	0.945343i	$0.394273\pi$
$600$	0	0
$601$	−0.571948	−0.0233302	−0.0116651	−	0.999932i	$-0.503713\pi$
$601$	−0.571948	−0.0233302	−0.0116651	+	0.999932i	$0.503713\pi$
$602$	0	0
$603$	−11.1822	−0.455375
$604$	0	0
$605$	7.88531	0.320584
$606$	0	0
$607$	24.5356	0.995868	0.497934	−	0.867215i	$-0.334092\pi$
$607$	24.5356	0.995868	0.497934	+	0.867215i	$0.334092\pi$
$608$	0	0
$609$	4.69723	0.190341
$610$	0	0
$611$	9.56380	0.386910
$612$	0	0
$613$	−11.8689	−0.479382	−0.239691	−	0.970849i	$-0.577046\pi$
$613$	−11.8689	−0.479382	−0.239691	+	0.970849i	$0.577046\pi$
$614$	0	0
$615$	4.27749	0.172485
$616$	0	0
$617$	−30.5614	−1.23036	−0.615179	−	0.788388i	$-0.710916\pi$
$617$	−30.5614	−1.23036	−0.615179	+	0.788388i	$0.710916\pi$
$618$	0	0
$619$	10.9583	0.440450	0.220225	−	0.975449i	$-0.429321\pi$
$619$	10.9583	0.440450	0.220225	+	0.975449i	$0.429321\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−0.868857	−0.0348100
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.40618	0.136030
$628$	0	0
$629$	50.8052	2.02574
$630$	0	0
$631$	−8.21396	−0.326993	−0.163496	−	0.986544i	$-0.552277\pi$
$631$	−8.21396	−0.326993	−0.163496	+	0.986544i	$0.552277\pi$
$632$	0	0
$633$	25.3175	1.00628
$634$	0	0
$635$	12.4344	0.493445
$636$	0	0
$637$	33.3848	1.32276
$638$	0	0
$639$	2.27749	0.0900961
$640$	0	0
$641$	18.0401	0.712539	0.356270	−	0.934383i	$-0.384048\pi$
$641$	18.0401	0.712539	0.356270	+	0.934383i	$0.384048\pi$
$642$	0	0
$643$	8.39787	0.331180	0.165590	−	0.986195i	$-0.447047\pi$
$643$	8.39787	0.331180	0.165590	+	0.986195i	$0.447047\pi$
$644$	0	0
$645$	−12.8853	−0.507359
$646$	0	0
$647$	8.57440	0.337094	0.168547	−	0.985694i	$-0.446092\pi$
$647$	8.57440	0.337094	0.168547	+	0.985694i	$0.446092\pi$
$648$	0	0
$649$	19.7006	0.773318
$650$	0	0
$651$	−2.15694	−0.0845371
$652$	0	0
$653$	43.9206	1.71874	0.859372	−	0.511351i	$-0.170855\pi$
$653$	43.9206	1.71874	0.859372	+	0.511351i	$0.170855\pi$
$654$	0	0
$655$	1.94944	0.0761707
$656$	0	0
$657$	−8.95530	−0.349379
$658$	0	0
$659$	42.7170	1.66402	0.832009	−	0.554762i	$-0.187191\pi$
$659$	42.7170	1.66402	0.832009	+	0.554762i	$0.187191\pi$
$660$	0	0
$661$	−47.7017	−1.85538	−0.927690	−	0.373351i	$-0.878209\pi$
$661$	−47.7017	−1.85538	−0.927690	+	0.373351i	$0.878209\pi$
$662$	0	0
$663$	39.5054	1.53426
$664$	0	0
$665$	−0.989405	−0.0383675
$666$	0	0
$667$	−9.16280	−0.354785
$668$	0	0
$669$	14.3644	0.555361
$670$	0	0
$671$	−15.2662	−0.589346
$672$	0	0
$673$	19.8912	0.766748	0.383374	−	0.923593i	$-0.374762\pi$
$673$	19.8912	0.766748	0.383374	+	0.923593i	$0.374762\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−12.3457	−0.474484	−0.237242	−	0.971451i	$-0.576243\pi$
$677$	−12.3457	−0.474484	−0.237242	+	0.971451i	$0.576243\pi$
$678$	0	0
$679$	−7.38973	−0.283592
$680$	0	0
$681$	16.2498	0.622695
$682$	0	0
$683$	42.9504	1.64345	0.821726	−	0.569883i	$-0.193012\pi$
$683$	42.9504	1.64345	0.821726	+	0.569883i	$0.193012\pi$
$684$	0	0
$685$	−9.22456	−0.352452
$686$	0	0
$687$	−14.8048	−0.564839
$688$	0	0
$689$	−22.8331	−0.869874
$690$	0	0
$691$	−33.2156	−1.26358	−0.631791	−	0.775138i	$-0.717680\pi$
$691$	−33.2156	−1.26358	−0.631791	+	0.775138i	$0.717680\pi$
$692$	0	0
$693$	−0.904733	−0.0343680
$694$	0	0
$695$	17.5631	0.666207
$696$	0	0
$697$	34.1016	1.29169
$698$	0	0
$699$	29.8047	1.12732
$700$	0	0
$701$	−6.62480	−0.250215	−0.125108	−	0.992143i	$-0.539928\pi$
$701$	−6.62480	−0.250215	−0.125108	+	0.992143i	$0.539928\pi$
$702$	0	0
$703$	12.2994	0.463879
$704$	0	0
$705$	−1.93002	−0.0726886
$706$	0	0
$707$	4.93833	0.185725
$708$	0	0
$709$	3.88836	0.146030	0.0730152	−	0.997331i	$-0.476738\pi$
$709$	3.88836	0.146030	0.0730152	+	0.997331i	$0.476738\pi$
$710$	0	0
$711$	8.58018	0.321782
$712$	0	0
$713$	4.20750	0.157572
$714$	0	0
$715$	−8.74534	−0.327057
$716$	0	0
$717$	11.1075	0.414817
$718$	0	0
$719$	29.1969	1.08886	0.544430	−	0.838806i	$-0.316746\pi$
$719$	29.1969	1.08886	0.544430	+	0.838806i	$0.316746\pi$
$720$	0	0
$721$	−4.01646	−0.149581
$722$	0	0
$723$	−20.3062	−0.755195
$724$	0	0
$725$	−9.16280	−0.340298
$726$	0	0
$727$	−1.16175	−0.0430871	−0.0215435	−	0.999768i	$-0.506858\pi$
$727$	−1.16175	−0.0430871	−0.0215435	+	0.999768i	$0.506858\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−102.726	−3.79947
$732$	0	0
$733$	3.29226	0.121602	0.0608012	−	0.998150i	$-0.480634\pi$
$733$	3.29226	0.121602	0.0608012	+	0.998150i	$0.480634\pi$
$734$	0	0
$735$	−6.73720	−0.248505
$736$	0	0
$737$	19.7349	0.726945
$738$	0	0
$739$	−13.4232	−0.493779	−0.246889	−	0.969044i	$-0.579408\pi$
$739$	−13.4232	−0.493779	−0.246889	+	0.969044i	$0.579408\pi$
$740$	0	0
$741$	9.56380	0.351335
$742$	0	0
$743$	10.6279	0.389901	0.194950	−	0.980813i	$-0.437545\pi$
$743$	10.6279	0.389901	0.194950	+	0.980813i	$0.437545\pi$
$744$	0	0
$745$	−14.3956	−0.527414
$746$	0	0
$747$	−6.91296	−0.252932
$748$	0	0
$749$	1.25221	0.0457546
$750$	0	0
$751$	−4.92993	−0.179896	−0.0899479	−	0.995946i	$-0.528670\pi$
$751$	−4.92993	−0.179896	−0.0899479	+	0.995946i	$0.528670\pi$
$752$	0	0
$753$	−2.21573	−0.0807458
$754$	0	0
$755$	14.9359	0.543572
$756$	0	0
$757$	−31.2580	−1.13609	−0.568045	−	0.822997i	$-0.692300\pi$
$757$	−31.2580	−1.13609	−0.568045	+	0.822997i	$0.692300\pi$
$758$	0	0
$759$	1.76485	0.0640599
$760$	0	0
$761$	−10.6972	−0.387774	−0.193887	−	0.981024i	$-0.562110\pi$
$761$	−10.6972	−0.387774	−0.193887	+	0.981024i	$0.562110\pi$
$762$	0	0
$763$	1.93002	0.0698713
$764$	0	0
$765$	−7.97235	−0.288241
$766$	0	0
$767$	55.3150	1.99731
$768$	0	0
$769$	40.2897	1.45288	0.726442	−	0.687227i	$-0.241172\pi$
$769$	40.2897	1.45288	0.726442	+	0.687227i	$0.241172\pi$
$770$	0	0
$771$	−2.90473	−0.104611
$772$	0	0
$773$	−9.24976	−0.332691	−0.166345	−	0.986068i	$-0.553197\pi$
$773$	−9.24976	−0.332691	−0.166345	+	0.986068i	$0.553197\pi$
$774$	0	0
$775$	4.20750	0.151138
$776$	0	0
$777$	−3.26689	−0.117199
$778$	0	0
$779$	8.25562	0.295788
$780$	0	0
$781$	−4.01942	−0.143826
$782$	0	0
$783$	9.16280	0.327452
$784$	0	0
$785$	18.4573	0.658771
$786$	0	0
$787$	−48.8129	−1.73999	−0.869996	−	0.493059i	$-0.835879\pi$
$787$	−48.8129	−1.73999	−0.869996	+	0.493059i	$0.835879\pi$
$788$	0	0
$789$	−3.75662	−0.133739
$790$	0	0
$791$	−1.29804	−0.0461529
$792$	0	0
$793$	−42.8641	−1.52215
$794$	0	0
$795$	4.60782	0.163423
$796$	0	0
$797$	−28.7594	−1.01871	−0.509354	−	0.860557i	$-0.670116\pi$
$797$	−28.7594	−1.01871	−0.509354	+	0.860557i	$0.670116\pi$
$798$	0	0
$799$	−15.3868	−0.544345
$800$	0	0
$801$	−1.69486	−0.0598850
$802$	0	0
$803$	15.8047	0.557737
$804$	0	0
$805$	−0.512641	−0.0180682
$806$	0	0
$807$	−7.58254	−0.266918
$808$	0	0
$809$	18.1034	0.636482	0.318241	−	0.948010i	$-0.396908\pi$
$809$	18.1034	0.636482	0.318241	+	0.948010i	$0.396908\pi$
$810$	0	0
$811$	−9.59740	−0.337010	−0.168505	−	0.985701i	$-0.553894\pi$
$811$	−9.59740	−0.337010	−0.168505	+	0.985701i	$0.553894\pi$
$812$	0	0
$813$	15.0423	0.527555
$814$	0	0
$815$	3.58499	0.125577
$816$	0	0
$817$	−24.8689	−0.870051
$818$	0	0
$819$	−2.54029	−0.0887649
$820$	0	0
$821$	−34.3044	−1.19723	−0.598616	−	0.801036i	$-0.704283\pi$
$821$	−34.3044	−1.19723	−0.598616	+	0.801036i	$0.704283\pi$
$822$	0	0
$823$	24.0504	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$823$	24.0504	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$824$	0	0
$825$	1.76485	0.0614441
$826$	0	0
$827$	−29.3621	−1.02102	−0.510510	−	0.859872i	$-0.670543\pi$
$827$	−29.3621	−1.02102	−0.510510	+	0.859872i	$0.670543\pi$
$828$	0	0
$829$	−26.8472	−0.932440	−0.466220	−	0.884669i	$-0.654385\pi$
$829$	−26.8472	−0.932440	−0.466220	+	0.884669i	$0.654385\pi$
$830$	0	0
$831$	−8.05056	−0.279271
$832$	0	0
$833$	−53.7113	−1.86099
$834$	0	0
$835$	0.120549	0.00417177
$836$	0	0
$837$	−4.20750	−0.145433
$838$	0	0
$839$	−16.7453	−0.578114	−0.289057	−	0.957312i	$-0.593342\pi$
$839$	−16.7453	−0.578114	−0.289057	+	0.957312i	$0.593342\pi$
$840$	0	0
$841$	54.9569	1.89507
$842$	0	0
$843$	14.3704	0.494942
$844$	0	0
$845$	−11.5550	−0.397503
$846$	0	0
$847$	−4.04234	−0.138896
$848$	0	0
$849$	−24.0287	−0.824663
$850$	0	0
$851$	6.37267	0.218452
$852$	0	0
$853$	24.3421	0.833456	0.416728	−	0.909031i	$-0.363177\pi$
$853$	24.3421	0.833456	0.416728	+	0.909031i	$0.363177\pi$
$854$	0	0
$855$	−1.93002	−0.0660051
$856$	0	0
$857$	23.0921	0.788812	0.394406	−	0.918936i	$-0.370950\pi$
$857$	23.0921	0.788812	0.394406	+	0.918936i	$0.370950\pi$
$858$	0	0
$859$	16.4532	0.561375	0.280687	−	0.959799i	$-0.409438\pi$
$859$	16.4532	0.561375	0.280687	+	0.959799i	$0.409438\pi$
$860$	0	0
$861$	−2.19282	−0.0747310
$862$	0	0
$863$	−19.6144	−0.667681	−0.333840	−	0.942630i	$-0.608345\pi$
$863$	−19.6144	−0.667681	−0.333840	+	0.942630i	$0.608345\pi$
$864$	0	0
$865$	11.0253	0.374871
$866$	0	0
$867$	−46.5584	−1.58121
$868$	0	0
$869$	−15.1427	−0.513681
$870$	0	0
$871$	55.4112	1.87754
$872$	0	0
$873$	−14.4150	−0.487874
$874$	0	0
$875$	−0.512641	−0.0173304
$876$	0	0
$877$	38.1305	1.28758	0.643788	−	0.765204i	$-0.277362\pi$
$877$	38.1305	1.28758	0.643788	+	0.765204i	$0.277362\pi$
$878$	0	0
$879$	−8.46786	−0.285614
$880$	0	0
$881$	13.2051	0.444892	0.222446	−	0.974945i	$-0.428596\pi$
$881$	13.2051	0.444892	0.222446	+	0.974945i	$0.428596\pi$
$882$	0	0
$883$	33.3362	1.12185	0.560926	−	0.827866i	$-0.310445\pi$
$883$	33.3362	1.12185	0.560926	+	0.827866i	$0.310445\pi$
$884$	0	0
$885$	−11.1628	−0.375233
$886$	0	0
$887$	−17.2839	−0.580335	−0.290168	−	0.956976i	$-0.593711\pi$
$887$	−17.2839	−0.580335	−0.290168	+	0.956976i	$0.593711\pi$
$888$	0	0
$889$	−6.37440	−0.213790
$890$	0	0
$891$	−1.76485	−0.0591246
$892$	0	0
$893$	−3.72496	−0.124651
$894$	0	0
$895$	−16.2750	−0.544015
$896$	0	0
$897$	4.95530	0.165453
$898$	0	0
$899$	−38.5525	−1.28580
$900$	0	0
$901$	36.7352	1.22383
$902$	0	0
$903$	6.60554	0.219819
$904$	0	0
$905$	−12.7543	−0.423966
$906$	0	0
$907$	−38.6731	−1.28412	−0.642059	−	0.766655i	$-0.721920\pi$
$907$	−38.6731	−1.28412	−0.642059	+	0.766655i	$0.721920\pi$
$908$	0	0
$909$	9.63311	0.319510
$910$	0	0
$911$	0.0503989	0.00166979	0.000834895	−	1.00000i	$-0.499734\pi$
$911$	0.0503989	0.00166979	0.000834895		1.00000i	$0.499734\pi$
$912$	0	0
$913$	12.2003	0.403772
$914$	0	0
$915$	8.65016	0.285966
$916$	0	0
$917$	−0.999361	−0.0330018
$918$	0	0
$919$	−6.47921	−0.213730	−0.106865	−	0.994274i	$-0.534081\pi$
$919$	−6.47921	−0.213730	−0.106865	+	0.994274i	$0.534081\pi$
$920$	0	0
$921$	−17.9594	−0.591782
$922$	0	0
$923$	−11.2856	−0.371471
$924$	0	0
$925$	6.37267	0.209532
$926$	0	0
$927$	−7.83483	−0.257330
$928$	0	0
$929$	30.2563	0.992677	0.496338	−	0.868129i	$-0.334677\pi$
$929$	30.2563	0.992677	0.496338	+	0.868129i	$0.334677\pi$
$930$	0	0
$931$	−13.0029	−0.426153
$932$	0	0
$933$	−23.8894	−0.782104
$934$	0	0
$935$	14.0700	0.460138
$936$	0	0
$937$	−21.2274	−0.693468	−0.346734	−	0.937963i	$-0.612709\pi$
$937$	−21.2274	−0.693468	−0.346734	+	0.937963i	$0.612709\pi$
$938$	0	0
$939$	−25.4184	−0.829499
$940$	0	0
$941$	9.59495	0.312786	0.156393	−	0.987695i	$-0.450013\pi$
$941$	9.59495	0.312786	0.156393	+	0.987695i	$0.450013\pi$
$942$	0	0
$943$	4.27749	0.139294
$944$	0	0
$945$	0.512641	0.0166762
$946$	0	0
$947$	35.2403	1.14516	0.572578	−	0.819850i	$-0.305943\pi$
$947$	35.2403	1.14516	0.572578	+	0.819850i	$0.305943\pi$
$948$	0	0
$949$	44.3762	1.44051
$950$	0	0
$951$	−24.0147	−0.778730
$952$	0	0
$953$	48.6717	1.57663	0.788315	−	0.615272i	$-0.210954\pi$
$953$	48.6717	1.57663	0.788315	+	0.615272i	$0.210954\pi$
$954$	0	0
$955$	14.9000	0.482153
$956$	0	0
$957$	−16.1709	−0.522733
$958$	0	0
$959$	4.72889	0.152704
$960$	0	0
$961$	−13.2969	−0.428933
$962$	0	0
$963$	2.44266	0.0787135
$964$	0	0
$965$	17.9447	0.577660
$966$	0	0
$967$	22.7765	0.732443	0.366221	−	0.930528i	$-0.380651\pi$
$967$	22.7765	0.732443	0.366221	+	0.930528i	$0.380651\pi$
$968$	0	0
$969$	−15.3868	−0.494294
$970$	0	0
$971$	−50.3120	−1.61459	−0.807294	−	0.590150i	$-0.799069\pi$
$971$	−50.3120	−1.61459	−0.807294	+	0.590150i	$0.799069\pi$
$972$	0	0
$973$	−9.00358	−0.288641
$974$	0	0
$975$	4.95530	0.158697
$976$	0	0
$977$	−38.6836	−1.23760	−0.618799	−	0.785549i	$-0.712380\pi$
$977$	−38.6836	−1.23760	−0.618799	+	0.785549i	$0.712380\pi$
$978$	0	0
$979$	2.99117	0.0955984
$980$	0	0
$981$	3.76485	0.120202
$982$	0	0
$983$	12.6584	0.403740	0.201870	−	0.979412i	$-0.435298\pi$
$983$	12.6584	0.403740	0.201870	+	0.979412i	$0.435298\pi$
$984$	0	0
$985$	6.96999	0.222082
$986$	0	0
$987$	0.989405	0.0314931
$988$	0	0
$989$	−12.8853	−0.409729
$990$	0	0
$991$	11.7148	0.372133	0.186067	−	0.982537i	$-0.440426\pi$
$991$	11.7148	0.372133	0.186067	+	0.982537i	$0.440426\pi$
$992$	0	0
$993$	12.7625	0.405005
$994$	0	0
$995$	21.4655	0.680502
$996$	0	0
$997$	−1.34389	−0.0425615	−0.0212808	−	0.999774i	$-0.506774\pi$
$997$	−1.34389	−0.0425615	−0.0212808	+	0.999774i	$0.506774\pi$
$998$	0	0
$999$	−6.37267	−0.201622

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2760.2.a.v.1.3	✓	4
3.2	odd	2		8280.2.a.bq.1.3		4
4.3	odd	2		5520.2.a.cb.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.v.1.3	✓	4	1.1	even	1	trivial
5520.2.a.cb.1.2		4	4.3	odd	2
8280.2.a.bq.1.3		4	3.2	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:	\( N \)	\(=\)	\( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2760.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +0.512641 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +0.512641 q^{7} +1.00000 q^{9} -1.76485 q^{11} -4.95530 q^{13} +1.00000 q^{15} +7.97235 q^{17} +1.93002 q^{19} -0.512641 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -9.16280 q^{29} +4.20750 q^{31} +1.76485 q^{33} -0.512641 q^{35} +6.37267 q^{37} +4.95530 q^{39} +4.27749 q^{41} -12.8853 q^{43} -1.00000 q^{45} -1.93002 q^{47} -6.73720 q^{49} -7.97235 q^{51} +4.60782 q^{53} +1.76485 q^{55} -1.93002 q^{57} -11.1628 q^{59} +8.65016 q^{61} +0.512641 q^{63} +4.95530 q^{65} -11.1822 q^{67} -1.00000 q^{69} +2.27749 q^{71} -8.95530 q^{73} -1.00000 q^{75} -0.904733 q^{77} +8.58018 q^{79} +1.00000 q^{81} -6.91296 q^{83} -7.97235 q^{85} +9.16280 q^{87} -1.69486 q^{89} -2.54029 q^{91} -4.20750 q^{93} -1.93002 q^{95} -14.4150 q^{97} -1.76485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^9	\( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{9} - 2 q^{13} + 4 q^{15} + 2 q^{17} - 6 q^{19} + 4 q^{23} + 4 q^{25} - 4 q^{27} + 4 q^{29} - 6 q^{31} - 4 q^{37} + 2 q^{39} + 8 q^{41} - 20 q^{43} - 4 q^{45} + 6 q^{47} + 10 q^{49} - 2 q^{51} - 4 q^{53} + 6 q^{57} - 4 q^{59} - 4 q^{61} + 2 q^{65} - 26 q^{67} - 4 q^{69} - 18 q^{73} - 4 q^{75} + 6 q^{77} - 18 q^{79} + 4 q^{81} - 26 q^{83} - 2 q^{85} - 4 q^{87} + 14 q^{89} - 38 q^{91} + 6 q^{93} + 6 q^{95} - 12 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^9 - 2 * q^13 + 4 * q^15 + 2 * q^17 - 6 * q^19 + 4 * q^23 + 4 * q^25 - 4 * q^27 + 4 * q^29 - 6 * q^31 - 4 * q^37 + 2 * q^39 + 8 * q^41 - 20 * q^43 - 4 * q^45 + 6 * q^47 + 10 * q^49 - 2 * q^51 - 4 * q^53 + 6 * q^57 - 4 * q^59 - 4 * q^61 + 2 * q^65 - 26 * q^67 - 4 * q^69 - 18 * q^73 - 4 * q^75 + 6 * q^77 - 18 * q^79 + 4 * q^81 - 26 * q^83 - 2 * q^85 - 4 * q^87 + 14 * q^89 - 38 * q^91 + 6 * q^93 + 6 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

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