LMFDB - Embedded newform 2320.2.a.q.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2320,2,Mod(1,2320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2320.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,3,0,1,0,6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 290)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.16425$ of defining polynomial
Character	$\chi$	$=$	2320.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.683969 q^{3} +1.00000 q^{5} -2.16425 q^{7} -2.53219 q^{9} +4.32850 q^{11} +5.64453 q^{13} -0.683969 q^{15} -4.16425 q^{17} +1.36794 q^{19} +1.48028 q^{21} +0.683969 q^{23} +1.00000 q^{25} +3.78384 q^{27} +1.00000 q^{29} -7.86068 q^{31} -2.96056 q^{33} -2.16425 q^{35} -6.32850 q^{37} -3.86068 q^{39} -2.32850 q^{41} +10.8212 q^{43} -2.53219 q^{45} +8.00000 q^{47} -2.31603 q^{49} +2.84822 q^{51} +8.49274 q^{53} +4.32850 q^{55} -0.935628 q^{57} -2.27659 q^{59} +2.68397 q^{61} +5.48028 q^{63} +5.64453 q^{65} +2.73588 q^{67} -0.467814 q^{69} -2.73588 q^{71} +15.2286 q^{73} -0.683969 q^{75} -9.36794 q^{77} +15.0374 q^{79} +5.00853 q^{81} +6.63206 q^{83} -4.16425 q^{85} -0.683969 q^{87} -5.28905 q^{89} -12.2162 q^{91} +5.37646 q^{93} +1.36794 q^{95} +0.987535 q^{97} -10.9606 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 3 q^{5} + q^{7} + 6 q^{9} - 2 q^{11} + 5 q^{13} + q^{15} - 5 q^{17} - 2 q^{19} - q^{23} + 3 q^{25} + 28 q^{27} + 3 q^{29} + 5 q^{31} + q^{35} - 4 q^{37} + 17 q^{39} + 8 q^{41} - 5 q^{43}+ \cdots - 24 q^{99}+O(q^{100}) $ 3 * q + q^3 + 3 * q^5 + q^7 + 6 * q^9 - 2 * q^11 + 5 * q^13 + q^15 - 5 * q^17 - 2 * q^19 - q^23 + 3 * q^25 + 28 * q^27 + 3 * q^29 + 5 * q^31 + q^35 - 4 * q^37 + 17 * q^39 + 8 * q^41 - 5 * q^43 + 6 * q^45 + 24 * q^47 - 10 * q^49 - 2 * q^51 + 3 * q^53 - 2 * q^55 - 30 * q^57 - q^59 + 5 * q^61 + 12 * q^63 + 5 * q^65 - 4 * q^67 - 15 * q^69 + 4 * q^71 + 11 * q^73 + q^75 - 22 * q^77 - 9 * q^79 + 35 * q^81 + 26 * q^83 - 5 * q^85 + q^87 + 8 * q^89 - 20 * q^91 + 30 * q^93 - 2 * q^95 + 21 * q^97 - 24 * q^99

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
$3$	−0.683969	−0.394890	−0.197445	−	0.980314i	$-0.563264\pi$
$3$	−0.683969	−0.394890	−0.197445	+	0.980314i	$0.563264\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.16425	−0.818009	−0.409004	−	0.912532i	$-0.634124\pi$
$7$	−2.16425	−0.818009	−0.409004	+	0.912532i	$0.634124\pi$
$8$	0	0
$9$	−2.53219	−0.844062
$10$	0	0
$11$	4.32850	1.30509	0.652545	−	0.757750i	$-0.273701\pi$
$11$	4.32850	1.30509	0.652545	+	0.757750i	$0.273701\pi$
$12$	0	0
$13$	5.64453	1.56551	0.782755	−	0.622330i	$-0.213814\pi$
$13$	5.64453	1.56551	0.782755	+	0.622330i	$0.213814\pi$
$14$	0	0
$15$	−0.683969	−0.176600
$16$	0	0
$17$	−4.16425	−1.00998	−0.504989	−	0.863126i	$-0.668504\pi$
$17$	−4.16425	−1.00998	−0.504989	+	0.863126i	$0.668504\pi$
$18$	0	0
$19$	1.36794	0.313827	0.156913	−	0.987612i	$-0.449846\pi$
$19$	1.36794	0.313827	0.156913	+	0.987612i	$0.449846\pi$
$20$	0	0
$21$	1.48028	0.323023
$22$	0	0
$23$	0.683969	0.142617	0.0713087	−	0.997454i	$-0.477282\pi$
$23$	0.683969	0.142617	0.0713087	+	0.997454i	$0.477282\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.78384	0.728201
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−7.86068	−1.41182	−0.705910	−	0.708301i	$-0.749462\pi$
$31$	−7.86068	−1.41182	−0.705910	+	0.708301i	$0.749462\pi$
$32$	0	0
$33$	−2.96056	−0.515367
$34$	0	0
$35$	−2.16425	−0.365825
$36$	0	0
$37$	−6.32850	−1.04040	−0.520199	−	0.854045i	$-0.674142\pi$
$37$	−6.32850	−1.04040	−0.520199	+	0.854045i	$0.674142\pi$
$38$	0	0
$39$	−3.86068	−0.618204
$40$	0	0
$41$	−2.32850	−0.363650	−0.181825	−	0.983331i	$-0.558200\pi$
$41$	−2.32850	−0.363650	−0.181825	+	0.983331i	$0.558200\pi$
$42$	0	0
$43$	10.8212	1.65022	0.825112	−	0.564969i	$-0.191112\pi$
$43$	10.8212	1.65022	0.825112	+	0.564969i	$0.191112\pi$
$44$	0	0
$45$	−2.53219	−0.377476
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−2.31603	−0.330862
$50$	0	0
$51$	2.84822	0.398830
$52$	0	0
$53$	8.49274	1.16657	0.583284	−	0.812268i	$-0.301767\pi$
$53$	8.49274	1.16657	0.583284	+	0.812268i	$0.301767\pi$
$54$	0	0
$55$	4.32850	0.583654
$56$	0	0
$57$	−0.935628	−0.123927
$58$	0	0
$59$	−2.27659	−0.296387	−0.148193	−	0.988958i	$-0.547346\pi$
$59$	−2.27659	−0.296387	−0.148193	+	0.988958i	$0.547346\pi$
$60$	0	0
$61$	2.68397	0.343647	0.171824	−	0.985128i	$-0.445034\pi$
$61$	2.68397	0.343647	0.171824	+	0.985128i	$0.445034\pi$
$62$	0	0
$63$	5.48028	0.690450
$64$	0	0
$65$	5.64453	0.700117
$66$	0	0
$67$	2.73588	0.334241	0.167120	−	0.985937i	$-0.446553\pi$
$67$	2.73588	0.334241	0.167120	+	0.985937i	$0.446553\pi$
$68$	0	0
$69$	−0.467814	−0.0563182
$70$	0	0
$71$	−2.73588	−0.324689	−0.162344	−	0.986734i	$-0.551906\pi$
$71$	−2.73588	−0.324689	−0.162344	+	0.986734i	$0.551906\pi$
$72$	0	0
$73$	15.2286	1.78238	0.891188	−	0.453635i	$-0.149873\pi$
$73$	15.2286	1.78238	0.891188	+	0.453635i	$0.149873\pi$
$74$	0	0
$75$	−0.683969	−0.0789780
$76$	0	0
$77$	−9.36794	−1.06758
$78$	0	0
$79$	15.0374	1.69184	0.845920	−	0.533311i	$-0.179052\pi$
$79$	15.0374	1.69184	0.845920	+	0.533311i	$0.179052\pi$
$80$	0	0
$81$	5.00853	0.556503
$82$	0	0
$83$	6.63206	0.727963	0.363982	−	0.931406i	$-0.381417\pi$
$83$	6.63206	0.727963	0.363982	+	0.931406i	$0.381417\pi$
$84$	0	0
$85$	−4.16425	−0.451676
$86$	0	0
$87$	−0.683969	−0.0733292
$88$	0	0
$89$	−5.28905	−0.560639	−0.280319	−	0.959907i	$-0.590440\pi$
$89$	−5.28905	−0.560639	−0.280319	+	0.959907i	$0.590440\pi$
$90$	0	0
$91$	−12.2162	−1.28060
$92$	0	0
$93$	5.37646	0.557513
$94$	0	0
$95$	1.36794	0.140348
$96$	0	0
$97$	0.987535	0.100269	0.0501345	−	0.998742i	$-0.484035\pi$
$97$	0.987535	0.100269	0.0501345	+	0.998742i	$0.484035\pi$
$98$	0	0
$99$	−10.9606	−1.10158
$100$	0	0
$101$	12.7089	1.26458	0.632291	−	0.774731i	$-0.282115\pi$
$101$	12.7089	1.26458	0.632291	+	0.774731i	$0.282115\pi$
$102$	0	0
$103$	−9.92112	−0.977557	−0.488778	−	0.872408i	$-0.662557\pi$
$103$	−9.92112	−0.977557	−0.488778	+	0.872408i	$0.662557\pi$
$104$	0	0
$105$	1.48028	0.144460
$106$	0	0
$107$	12.1038	1.17012	0.585060	−	0.810990i	$-0.301071\pi$
$107$	12.1038	1.17012	0.585060	+	0.810990i	$0.301071\pi$
$108$	0	0
$109$	18.9855	1.81848	0.909240	−	0.416272i	$-0.136664\pi$
$109$	18.9855	1.81848	0.909240	+	0.416272i	$0.136664\pi$
$110$	0	0
$111$	4.32850	0.410843
$112$	0	0
$113$	3.72341	0.350269	0.175135	−	0.984545i	$-0.443964\pi$
$113$	3.72341	0.350269	0.175135	+	0.984545i	$0.443964\pi$
$114$	0	0
$115$	0.683969	0.0637805
$116$	0	0
$117$	−14.2930	−1.32139
$118$	0	0
$119$	9.01247	0.826171
$120$	0	0
$121$	7.73588	0.703262
$122$	0	0
$123$	1.59262	0.143602
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.73588	−0.242770	−0.121385	−	0.992606i	$-0.538734\pi$
$127$	−2.73588	−0.242770	−0.121385	+	0.992606i	$0.538734\pi$
$128$	0	0
$129$	−7.40139	−0.651656
$130$	0	0
$131$	−16.4323	−1.43570	−0.717849	−	0.696199i	$-0.754873\pi$
$131$	−16.4323	−1.43570	−0.717849	+	0.696199i	$0.754873\pi$
$132$	0	0
$133$	−2.96056	−0.256713
$134$	0	0
$135$	3.78384	0.325661
$136$	0	0
$137$	4.38040	0.374243	0.187122	−	0.982337i	$-0.440084\pi$
$137$	4.38040	0.374243	0.187122	+	0.982337i	$0.440084\pi$
$138$	0	0
$139$	2.38893	0.202626	0.101313	−	0.994855i	$-0.467696\pi$
$139$	2.38893	0.202626	0.101313	+	0.994855i	$0.467696\pi$
$140$	0	0
$141$	−5.47175	−0.460805
$142$	0	0
$143$	24.4323	2.04313
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	1.58409	0.130654
$148$	0	0
$149$	18.1038	1.48312	0.741561	−	0.670885i	$-0.234086\pi$
$149$	18.1038	1.48312	0.741561	+	0.670885i	$0.234086\pi$
$150$	0	0
$151$	−0.710947	−0.0578560	−0.0289280	−	0.999581i	$-0.509209\pi$
$151$	−0.710947	−0.0578560	−0.0289280	+	0.999581i	$0.509209\pi$
$152$	0	0
$153$	10.5447	0.852485
$154$	0	0
$155$	−7.86068	−0.631385
$156$	0	0
$157$	22.6570	1.80822	0.904112	−	0.427295i	$-0.140533\pi$
$157$	22.6570	1.80822	0.904112	+	0.427295i	$0.140533\pi$
$158$	0	0
$159$	−5.80877	−0.460666
$160$	0	0
$161$	−1.48028	−0.116662
$162$	0	0
$163$	−18.5781	−1.45515	−0.727575	−	0.686028i	$-0.759353\pi$
$163$	−18.5781	−1.45515	−0.727575	+	0.686028i	$0.759353\pi$
$164$	0	0
$165$	−2.96056	−0.230479
$166$	0	0
$167$	5.83575	0.451584	0.225792	−	0.974176i	$-0.427503\pi$
$167$	5.83575	0.451584	0.225792	+	0.974176i	$0.427503\pi$
$168$	0	0
$169$	18.8607	1.45082
$170$	0	0
$171$	−3.46387	−0.264889
$172$	0	0
$173$	−18.3015	−1.39144	−0.695719	−	0.718314i	$-0.744914\pi$
$173$	−18.3015	−1.39144	−0.695719	+	0.718314i	$0.744914\pi$
$174$	0	0
$175$	−2.16425	−0.163602
$176$	0	0
$177$	1.55712	0.117040
$178$	0	0
$179$	−23.0374	−1.72190	−0.860948	−	0.508693i	$-0.830129\pi$
$179$	−23.0374	−1.72190	−0.860948	+	0.508693i	$0.830129\pi$
$180$	0	0
$181$	−26.3784	−1.96069	−0.980344	−	0.197296i	$-0.936784\pi$
$181$	−26.3784	−1.96069	−0.980344	+	0.197296i	$0.936784\pi$
$182$	0	0
$183$	−1.83575	−0.135703
$184$	0	0
$185$	−6.32850	−0.465280
$186$	0	0
$187$	−18.0249	−1.31811
$188$	0	0
$189$	−8.18918	−0.595675
$190$	0	0
$191$	−1.45330	−0.105157	−0.0525786	−	0.998617i	$-0.516744\pi$
$191$	−1.45330	−0.105157	−0.0525786	+	0.998617i	$0.516744\pi$
$192$	0	0
$193$	−11.5656	−0.832513	−0.416257	−	0.909247i	$-0.636658\pi$
$193$	−11.5656	−0.832513	−0.416257	+	0.909247i	$0.636658\pi$
$194$	0	0
$195$	−3.86068	−0.276469
$196$	0	0
$197$	13.9645	0.994929	0.497465	−	0.867484i	$-0.334264\pi$
$197$	13.9645	0.994929	0.497465	+	0.867484i	$0.334264\pi$
$198$	0	0
$199$	12.3285	0.873944	0.436972	−	0.899475i	$-0.356051\pi$
$199$	12.3285	0.873944	0.436972	+	0.899475i	$0.356051\pi$
$200$	0	0
$201$	−1.87126	−0.131988
$202$	0	0
$203$	−2.16425	−0.151900
$204$	0	0
$205$	−2.32850	−0.162629
$206$	0	0
$207$	−1.73194	−0.120378
$208$	0	0
$209$	5.92112	0.409572
$210$	0	0
$211$	−4.32850	−0.297986	−0.148993	−	0.988838i	$-0.547603\pi$
$211$	−4.32850	−0.297986	−0.148993	+	0.988838i	$0.547603\pi$
$212$	0	0
$213$	1.87126	0.128216
$214$	0	0
$215$	10.8212	0.738002
$216$	0	0
$217$	17.0125	1.15488
$218$	0	0
$219$	−10.4159	−0.703842
$220$	0	0
$221$	−23.5052	−1.58113
$222$	0	0
$223$	11.4198	0.764729	0.382365	−	0.924011i	$-0.375110\pi$
$223$	11.4198	0.764729	0.382365	+	0.924011i	$0.375110\pi$
$224$	0	0
$225$	−2.53219	−0.168812
$226$	0	0
$227$	28.2745	1.87665	0.938324	−	0.345758i	$-0.112378\pi$
$227$	28.2745	1.87665	0.938324	+	0.345758i	$0.112378\pi$
$228$	0	0
$229$	11.4533	0.756855	0.378428	−	0.925631i	$-0.376465\pi$
$229$	11.4533	0.756855	0.378428	+	0.925631i	$0.376465\pi$
$230$	0	0
$231$	6.40738	0.421575
$232$	0	0
$233$	−0.0788848	−0.00516792	−0.00258396	−	0.999997i	$-0.500823\pi$
$233$	−0.0788848	−0.00516792	−0.00258396	+	0.999997i	$0.500823\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	−10.2851	−0.668090
$238$	0	0
$239$	−10.9606	−0.708980	−0.354490	−	0.935060i	$-0.615345\pi$
$239$	−10.9606	−0.708980	−0.354490	+	0.935060i	$0.615345\pi$
$240$	0	0
$241$	17.3659	1.11864	0.559318	−	0.828953i	$-0.311063\pi$
$241$	17.3659	1.11864	0.559318	+	0.828953i	$0.311063\pi$
$242$	0	0
$243$	−14.7772	−0.947959
$244$	0	0
$245$	−2.31603	−0.147966
$246$	0	0
$247$	7.72136	0.491299
$248$	0	0
$249$	−4.53613	−0.287465
$250$	0	0
$251$	26.0249	1.64268	0.821340	−	0.570440i	$-0.193227\pi$
$251$	26.0249	1.64268	0.821340	+	0.570440i	$0.193227\pi$
$252$	0	0
$253$	2.96056	0.186129
$254$	0	0
$255$	2.84822	0.178362
$256$	0	0
$257$	−10.3285	−0.644274	−0.322137	−	0.946693i	$-0.604401\pi$
$257$	−10.3285	−0.644274	−0.322137	+	0.946693i	$0.604401\pi$
$258$	0	0
$259$	13.6964	0.851055
$260$	0	0
$261$	−2.53219	−0.156738
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	8.49274	0.521705
$266$	0	0
$267$	3.61755	0.221390
$268$	0	0
$269$	−13.8607	−0.845101	−0.422550	−	0.906339i	$-0.638865\pi$
$269$	−13.8607	−0.845101	−0.422550	+	0.906339i	$0.638865\pi$
$270$	0	0
$271$	−6.73588	−0.409175	−0.204588	−	0.978848i	$-0.565585\pi$
$271$	−6.73588	−0.409175	−0.204588	+	0.978848i	$0.565585\pi$
$272$	0	0
$273$	8.35547	0.505696
$274$	0	0
$275$	4.32850	0.261018
$276$	0	0
$277$	17.3929	1.04504	0.522518	−	0.852628i	$-0.324993\pi$
$277$	17.3929	1.04504	0.522518	+	0.852628i	$0.324993\pi$
$278$	0	0
$279$	19.9047	1.19166
$280$	0	0
$281$	−11.9396	−0.712255	−0.356127	−	0.934437i	$-0.615903\pi$
$281$	−11.9396	−0.712255	−0.356127	+	0.934437i	$0.615903\pi$
$282$	0	0
$283$	23.4967	1.39673	0.698366	−	0.715740i	$-0.253911\pi$
$283$	23.4967	1.39673	0.698366	+	0.715740i	$0.253911\pi$
$284$	0	0
$285$	−0.935628	−0.0554218
$286$	0	0
$287$	5.03944	0.297469
$288$	0	0
$289$	0.340961	0.0200565
$290$	0	0
$291$	−0.675443	−0.0395952
$292$	0	0
$293$	5.51373	0.322116	0.161058	−	0.986945i	$-0.448509\pi$
$293$	5.51373	0.322116	0.161058	+	0.986945i	$0.448509\pi$
$294$	0	0
$295$	−2.27659	−0.132548
$296$	0	0
$297$	16.3784	0.950369
$298$	0	0
$299$	3.86068	0.223269
$300$	0	0
$301$	−23.4198	−1.34990
$302$	0	0
$303$	−8.69249	−0.499371
$304$	0	0
$305$	2.68397	0.153684
$306$	0	0
$307$	4.65699	0.265789	0.132894	−	0.991130i	$-0.457573\pi$
$307$	4.65699	0.265789	0.132894	+	0.991130i	$0.457573\pi$
$308$	0	0
$309$	6.78574	0.386027
$310$	0	0
$311$	4.35547	0.246976	0.123488	−	0.992346i	$-0.460592\pi$
$311$	4.35547	0.246976	0.123488	+	0.992346i	$0.460592\pi$
$312$	0	0
$313$	1.34301	0.0759113	0.0379557	−	0.999279i	$-0.487915\pi$
$313$	1.34301	0.0759113	0.0379557	+	0.999279i	$0.487915\pi$
$314$	0	0
$315$	5.48028	0.308779
$316$	0	0
$317$	−11.3679	−0.638487	−0.319244	−	0.947673i	$-0.603429\pi$
$317$	−11.3679	−0.638487	−0.319244	+	0.947673i	$0.603429\pi$
$318$	0	0
$319$	4.32850	0.242349
$320$	0	0
$321$	−8.27864	−0.462068
$322$	0	0
$323$	−5.69643	−0.316958
$324$	0	0
$325$	5.64453	0.313102
$326$	0	0
$327$	−12.9855	−0.718099
$328$	0	0
$329$	−17.3140	−0.954551
$330$	0	0
$331$	−12.3285	−0.677635	−0.338818	−	0.940852i	$-0.610027\pi$
$331$	−12.3285	−0.677635	−0.338818	+	0.940852i	$0.610027\pi$
$332$	0	0
$333$	16.0249	0.878161
$334$	0	0
$335$	2.73588	0.149477
$336$	0	0
$337$	−17.8067	−0.969994	−0.484997	−	0.874516i	$-0.661179\pi$
$337$	−17.8067	−0.969994	−0.484997	+	0.874516i	$0.661179\pi$
$338$	0	0
$339$	−2.54670	−0.138318
$340$	0	0
$341$	−34.0249	−1.84255
$342$	0	0
$343$	20.1622	1.08866
$344$	0	0
$345$	−0.467814	−0.0251862
$346$	0	0
$347$	−23.2891	−1.25022	−0.625111	−	0.780536i	$-0.714946\pi$
$347$	−23.2891	−1.25022	−0.625111	+	0.780536i	$0.714946\pi$
$348$	0	0
$349$	−23.6964	−1.26844	−0.634221	−	0.773152i	$-0.718679\pi$
$349$	−23.6964	−1.26844	−0.634221	+	0.773152i	$0.718679\pi$
$350$	0	0
$351$	21.3580	1.14001
$352$	0	0
$353$	−6.43231	−0.342357	−0.171179	−	0.985240i	$-0.554758\pi$
$353$	−6.43231	−0.342357	−0.171179	+	0.985240i	$0.554758\pi$
$354$	0	0
$355$	−2.73588	−0.145205
$356$	0	0
$357$	−6.16425	−0.326247
$358$	0	0
$359$	12.3555	0.652097	0.326048	−	0.945353i	$-0.394283\pi$
$359$	12.3555	0.652097	0.326048	+	0.945353i	$0.394283\pi$
$360$	0	0
$361$	−17.1287	−0.901513
$362$	0	0
$363$	−5.29110	−0.277711
$364$	0	0
$365$	15.2286	0.797102
$366$	0	0
$367$	−18.4743	−0.964350	−0.482175	−	0.876075i	$-0.660153\pi$
$367$	−18.4743	−0.964350	−0.482175	+	0.876075i	$0.660153\pi$
$368$	0	0
$369$	5.89619	0.306943
$370$	0	0
$371$	−18.3804	−0.954263
$372$	0	0
$373$	2.57163	0.133154	0.0665769	−	0.997781i	$-0.478792\pi$
$373$	2.57163	0.133154	0.0665769	+	0.997781i	$0.478792\pi$
$374$	0	0
$375$	−0.683969	−0.0353200
$376$	0	0
$377$	5.64453	0.290708
$378$	0	0
$379$	−23.7214	−1.21848	−0.609242	−	0.792984i	$-0.708526\pi$
$379$	−23.7214	−1.21848	−0.609242	+	0.792984i	$0.708526\pi$
$380$	0	0
$381$	1.87126	0.0958673
$382$	0	0
$383$	−5.55712	−0.283955	−0.141978	−	0.989870i	$-0.545346\pi$
$383$	−5.55712	−0.283955	−0.141978	+	0.989870i	$0.545346\pi$
$384$	0	0
$385$	−9.36794	−0.477434
$386$	0	0
$387$	−27.4014	−1.39289
$388$	0	0
$389$	27.4718	1.39287	0.696437	−	0.717618i	$-0.254768\pi$
$389$	27.4718	1.39287	0.696437	+	0.717618i	$0.254768\pi$
$390$	0	0
$391$	−2.84822	−0.144041
$392$	0	0
$393$	11.2392	0.566942
$394$	0	0
$395$	15.0374	0.756613
$396$	0	0
$397$	−14.5716	−0.731329	−0.365665	−	0.930747i	$-0.619158\pi$
$397$	−14.5716	−0.731329	−0.365665	+	0.930747i	$0.619158\pi$
$398$	0	0
$399$	2.02493	0.101373
$400$	0	0
$401$	−10.0270	−0.500723	−0.250362	−	0.968152i	$-0.580550\pi$
$401$	−10.0270	−0.500723	−0.250362	+	0.968152i	$0.580550\pi$
$402$	0	0
$403$	−44.3698	−2.21022
$404$	0	0
$405$	5.00853	0.248876
$406$	0	0
$407$	−27.3929	−1.35781
$408$	0	0
$409$	−39.0893	−1.93284	−0.966421	−	0.256965i	$-0.917278\pi$
$409$	−39.0893	−1.93284	−0.966421	+	0.256965i	$0.917278\pi$
$410$	0	0
$411$	−2.99606	−0.147785
$412$	0	0
$413$	4.92710	0.242447
$414$	0	0
$415$	6.63206	0.325555
$416$	0	0
$417$	−1.63395	−0.0800151
$418$	0	0
$419$	−31.8607	−1.55650	−0.778248	−	0.627957i	$-0.783891\pi$
$419$	−31.8607	−1.55650	−0.778248	+	0.627957i	$0.783891\pi$
$420$	0	0
$421$	36.1287	1.76081	0.880404	−	0.474225i	$-0.157272\pi$
$421$	36.1287	1.76081	0.880404	+	0.474225i	$0.157272\pi$
$422$	0	0
$423$	−20.2575	−0.984953
$424$	0	0
$425$	−4.16425	−0.201996
$426$	0	0
$427$	−5.80877	−0.281106
$428$	0	0
$429$	−16.7109	−0.806812
$430$	0	0
$431$	9.36794	0.451238	0.225619	−	0.974216i	$-0.427560\pi$
$431$	9.36794	0.451238	0.225619	+	0.974216i	$0.427560\pi$
$432$	0	0
$433$	−10.8148	−0.519724	−0.259862	−	0.965646i	$-0.583677\pi$
$433$	−10.8148	−0.519724	−0.259862	+	0.965646i	$0.583677\pi$
$434$	0	0
$435$	−0.683969	−0.0327938
$436$	0	0
$437$	0.935628	0.0447571
$438$	0	0
$439$	−29.6964	−1.41733	−0.708667	−	0.705543i	$-0.750703\pi$
$439$	−29.6964	−1.41733	−0.708667	+	0.705543i	$0.750703\pi$
$440$	0	0
$441$	5.86462	0.279268
$442$	0	0
$443$	23.2621	1.10521	0.552607	−	0.833442i	$-0.313633\pi$
$443$	23.2621	1.10521	0.552607	+	0.833442i	$0.313633\pi$
$444$	0	0
$445$	−5.28905	−0.250725
$446$	0	0
$447$	−12.3825	−0.585670
$448$	0	0
$449$	−5.72136	−0.270008	−0.135004	−	0.990845i	$-0.543105\pi$
$449$	−5.72136	−0.270008	−0.135004	+	0.990845i	$0.543105\pi$
$450$	0	0
$451$	−10.0789	−0.474596
$452$	0	0
$453$	0.486265	0.0228467
$454$	0	0
$455$	−12.2162	−0.572702
$456$	0	0
$457$	−31.9710	−1.49554	−0.747770	−	0.663958i	$-0.768875\pi$
$457$	−31.9710	−1.49554	−0.747770	+	0.663958i	$0.768875\pi$
$458$	0	0
$459$	−15.7569	−0.735468
$460$	0	0
$461$	6.18918	0.288259	0.144129	−	0.989559i	$-0.453962\pi$
$461$	6.18918	0.288259	0.144129	+	0.989559i	$0.453962\pi$
$462$	0	0
$463$	12.0000	0.557687	0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000	0.557687	0.278844	+	0.960337i	$0.410049\pi$
$464$	0	0
$465$	5.37646	0.249328
$466$	0	0
$467$	15.4284	0.713940	0.356970	−	0.934116i	$-0.383810\pi$
$467$	15.4284	0.713940	0.356970	+	0.934116i	$0.383810\pi$
$468$	0	0
$469$	−5.92112	−0.273412
$470$	0	0
$471$	−15.4967	−0.714049
$472$	0	0
$473$	46.8397	2.15369
$474$	0	0
$475$	1.36794	0.0627653
$476$	0	0
$477$	−21.5052	−0.984656
$478$	0	0
$479$	−32.0314	−1.46355	−0.731776	−	0.681545i	$-0.761308\pi$
$479$	−32.0314	−1.46355	−0.731776	+	0.681545i	$0.761308\pi$
$480$	0	0
$481$	−35.7214	−1.62875
$482$	0	0
$483$	1.01247	0.0460688
$484$	0	0
$485$	0.987535	0.0448417
$486$	0	0
$487$	−24.8916	−1.12795	−0.563973	−	0.825793i	$-0.690728\pi$
$487$	−24.8916	−1.12795	−0.563973	+	0.825793i	$0.690728\pi$
$488$	0	0
$489$	12.7069	0.574624
$490$	0	0
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	0	0
$493$	−4.16425	−0.187548
$494$	0	0
$495$	−10.9606	−0.492640
$496$	0	0
$497$	5.92112	0.265598
$498$	0	0
$499$	24.8547	1.11265	0.556324	−	0.830965i	$-0.312211\pi$
$499$	24.8547	1.11265	0.556324	+	0.830965i	$0.312211\pi$
$500$	0	0
$501$	−3.99147	−0.178326
$502$	0	0
$503$	19.3929	0.864685	0.432343	−	0.901709i	$-0.357687\pi$
$503$	19.3929	0.864685	0.432343	+	0.901709i	$0.357687\pi$
$504$	0	0
$505$	12.7089	0.565539
$506$	0	0
$507$	−12.9001	−0.572915
$508$	0	0
$509$	17.6715	0.783276	0.391638	−	0.920119i	$-0.371909\pi$
$509$	17.6715	0.783276	0.391638	+	0.920119i	$0.371909\pi$
$510$	0	0
$511$	−32.9585	−1.45800
$512$	0	0
$513$	5.17607	0.228529
$514$	0	0
$515$	−9.92112	−0.437177
$516$	0	0
$517$	34.6280	1.52294
$518$	0	0
$519$	12.5177	0.549465
$520$	0	0
$521$	−2.31004	−0.101205	−0.0506024	−	0.998719i	$-0.516114\pi$
$521$	−2.31004	−0.101205	−0.0506024	+	0.998719i	$0.516114\pi$
$522$	0	0
$523$	1.36794	0.0598158	0.0299079	−	0.999553i	$-0.490479\pi$
$523$	1.36794	0.0598158	0.0299079	+	0.999553i	$0.490479\pi$
$524$	0	0
$525$	1.48028	0.0646047
$526$	0	0
$527$	32.7338	1.42591
$528$	0	0
$529$	−22.5322	−0.979660
$530$	0	0
$531$	5.76475	0.250169
$532$	0	0
$533$	−13.1433	−0.569298
$534$	0	0
$535$	12.1038	0.523294
$536$	0	0
$537$	15.7569	0.679959
$538$	0	0
$539$	−10.0249	−0.431804
$540$	0	0
$541$	0.717425	0.0308445	0.0154223	−	0.999881i	$-0.495091\pi$
$541$	0.717425	0.0308445	0.0154223	+	0.999881i	$0.495091\pi$
$542$	0	0
$543$	18.0420	0.774256
$544$	0	0
$545$	18.9855	0.813249
$546$	0	0
$547$	−31.9460	−1.36591	−0.682957	−	0.730458i	$-0.739306\pi$
$547$	−31.9460	−1.36591	−0.682957	+	0.730458i	$0.739306\pi$
$548$	0	0
$549$	−6.79631	−0.290059
$550$	0	0
$551$	1.36794	0.0582761
$552$	0	0
$553$	−32.5447	−1.38394
$554$	0	0
$555$	4.32850	0.183734
$556$	0	0
$557$	−3.72341	−0.157766	−0.0788830	−	0.996884i	$-0.525135\pi$
$557$	−3.72341	−0.157766	−0.0788830	+	0.996884i	$0.525135\pi$
$558$	0	0
$559$	61.0808	2.58344
$560$	0	0
$561$	12.3285	0.520510
$562$	0	0
$563$	−23.2621	−0.980380	−0.490190	−	0.871616i	$-0.663073\pi$
$563$	−23.2621	−0.980380	−0.490190	+	0.871616i	$0.663073\pi$
$564$	0	0
$565$	3.72341	0.156645
$566$	0	0
$567$	−10.8397	−0.455224
$568$	0	0
$569$	14.2745	0.598420	0.299210	−	0.954187i	$-0.403277\pi$
$569$	14.2745	0.598420	0.299210	+	0.954187i	$0.403277\pi$
$570$	0	0
$571$	−6.82977	−0.285817	−0.142908	−	0.989736i	$-0.545645\pi$
$571$	−6.82977	−0.285817	−0.142908	+	0.989736i	$0.545645\pi$
$572$	0	0
$573$	0.994013	0.0415255
$574$	0	0
$575$	0.683969	0.0285235
$576$	0	0
$577$	−42.9500	−1.78803	−0.894016	−	0.448036i	$-0.852124\pi$
$577$	−42.9500	−1.78803	−0.894016	+	0.448036i	$0.852124\pi$
$578$	0	0
$579$	7.91054	0.328751
$580$	0	0
$581$	−14.3534	−0.595480
$582$	0	0
$583$	36.7608	1.52248
$584$	0	0
$585$	−14.2930	−0.590943
$586$	0	0
$587$	1.08930	0.0449603	0.0224802	−	0.999747i	$-0.492844\pi$
$587$	1.08930	0.0449603	0.0224802	+	0.999747i	$0.492844\pi$
$588$	0	0
$589$	−10.7529	−0.443067
$590$	0	0
$591$	−9.55128	−0.392887
$592$	0	0
$593$	7.21017	0.296086	0.148043	−	0.988981i	$-0.452703\pi$
$593$	7.21017	0.296086	0.148043	+	0.988981i	$0.452703\pi$
$594$	0	0
$595$	9.01247	0.369475
$596$	0	0
$597$	−8.43231	−0.345112
$598$	0	0
$599$	42.3179	1.72906	0.864532	−	0.502578i	$-0.167615\pi$
$599$	42.3179	1.72906	0.864532	+	0.502578i	$0.167615\pi$
$600$	0	0
$601$	−16.6819	−0.680470	−0.340235	−	0.940340i	$-0.610507\pi$
$601$	−16.6819	−0.680470	−0.340235	+	0.940340i	$0.610507\pi$
$602$	0	0
$603$	−6.92775	−0.282120
$604$	0	0
$605$	7.73588	0.314508
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	1.48028	0.0599839
$610$	0	0
$611$	45.1562	1.82682
$612$	0	0
$613$	37.1996	1.50248	0.751239	−	0.660031i	$-0.229457\pi$
$613$	37.1996	1.50248	0.751239	+	0.660031i	$0.229457\pi$
$614$	0	0
$615$	1.59262	0.0642206
$616$	0	0
$617$	4.77138	0.192089	0.0960443	−	0.995377i	$-0.469381\pi$
$617$	4.77138	0.192089	0.0960443	+	0.995377i	$0.469381\pi$
$618$	0	0
$619$	−6.80279	−0.273427	−0.136714	−	0.990611i	$-0.543654\pi$
$619$	−6.80279	−0.273427	−0.136714	+	0.990611i	$0.543654\pi$
$620$	0	0
$621$	2.58803	0.103854
$622$	0	0
$623$	11.4468	0.458607
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.04986	−0.161736
$628$	0	0
$629$	26.3534	1.05078
$630$	0	0
$631$	−17.0893	−0.680314	−0.340157	−	0.940369i	$-0.610480\pi$
$631$	−17.0893	−0.680314	−0.340157	+	0.940369i	$0.610480\pi$
$632$	0	0
$633$	2.96056	0.117672
$634$	0	0
$635$	−2.73588	−0.108570
$636$	0	0
$637$	−13.0729	−0.517967
$638$	0	0
$639$	6.92775	0.274058
$640$	0	0
$641$	11.1433	0.440132	0.220066	−	0.975485i	$-0.429373\pi$
$641$	11.1433	0.440132	0.220066	+	0.975485i	$0.429373\pi$
$642$	0	0
$643$	24.9356	0.983365	0.491683	−	0.870775i	$-0.336382\pi$
$643$	24.9356	0.983365	0.491683	+	0.870775i	$0.336382\pi$
$644$	0	0
$645$	−7.40139	−0.291430
$646$	0	0
$647$	−0.814761	−0.0320316	−0.0160158	−	0.999872i	$-0.505098\pi$
$647$	−0.814761	−0.0320316	−0.0160158	+	0.999872i	$0.505098\pi$
$648$	0	0
$649$	−9.85420	−0.386811
$650$	0	0
$651$	−11.6360	−0.456051
$652$	0	0
$653$	−29.9959	−1.17383	−0.586915	−	0.809648i	$-0.699658\pi$
$653$	−29.9959	−1.17383	−0.586915	+	0.809648i	$0.699658\pi$
$654$	0	0
$655$	−16.4323	−0.642064
$656$	0	0
$657$	−38.5617	−1.50444
$658$	0	0
$659$	−27.6715	−1.07793	−0.538964	−	0.842329i	$-0.681184\pi$
$659$	−27.6715	−1.07793	−0.538964	+	0.842329i	$0.681184\pi$
$660$	0	0
$661$	−6.55318	−0.254889	−0.127445	−	0.991846i	$-0.540677\pi$
$661$	−6.55318	−0.254889	−0.127445	+	0.991846i	$0.540677\pi$
$662$	0	0
$663$	16.0768	0.624373
$664$	0	0
$665$	−2.96056	−0.114805
$666$	0	0
$667$	0.683969	0.0264834
$668$	0	0
$669$	−7.81082	−0.301984
$670$	0	0
$671$	11.6175	0.448491
$672$	0	0
$673$	−18.9855	−0.731837	−0.365918	−	0.930647i	$-0.619245\pi$
$673$	−18.9855	−0.731837	−0.365918	+	0.930647i	$0.619245\pi$
$674$	0	0
$675$	3.78384	0.145640
$676$	0	0
$677$	33.6175	1.29203	0.646014	−	0.763326i	$-0.276435\pi$
$677$	33.6175	1.29203	0.646014	+	0.763326i	$0.276435\pi$
$678$	0	0
$679$	−2.13727	−0.0820209
$680$	0	0
$681$	−19.3389	−0.741069
$682$	0	0
$683$	18.6819	0.714844	0.357422	−	0.933943i	$-0.383656\pi$
$683$	18.6819	0.714844	0.357422	+	0.933943i	$0.383656\pi$
$684$	0	0
$685$	4.38040	0.167367
$686$	0	0
$687$	−7.83370	−0.298874
$688$	0	0
$689$	47.9375	1.82627
$690$	0	0
$691$	−38.7174	−1.47288	−0.736440	−	0.676503i	$-0.763495\pi$
$691$	−38.7174	−1.47288	−0.736440	+	0.676503i	$0.763495\pi$
$692$	0	0
$693$	23.7214	0.901100
$694$	0	0
$695$	2.38893	0.0906172
$696$	0	0
$697$	9.69643	0.367279
$698$	0	0
$699$	0.0539548	0.00204076
$700$	0	0
$701$	28.7318	1.08518	0.542592	−	0.839996i	$-0.317443\pi$
$701$	28.7318	1.08518	0.542592	+	0.839996i	$0.317443\pi$
$702$	0	0
$703$	−8.65699	−0.326505
$704$	0	0
$705$	−5.47175	−0.206078
$706$	0	0
$707$	−27.5052	−1.03444
$708$	0	0
$709$	−47.2102	−1.77302	−0.886508	−	0.462714i	$-0.846876\pi$
$709$	−47.2102	−1.77302	−0.886508	+	0.462714i	$0.846876\pi$
$710$	0	0
$711$	−38.0775	−1.42802
$712$	0	0
$713$	−5.37646	−0.201350
$714$	0	0
$715$	24.4323	0.913717
$716$	0	0
$717$	7.49668	0.279969
$718$	0	0
$719$	14.6321	0.545684	0.272842	−	0.962059i	$-0.412036\pi$
$719$	14.6321	0.545684	0.272842	+	0.962059i	$0.412036\pi$
$720$	0	0
$721$	21.4718	0.799650
$722$	0	0
$723$	−11.8777	−0.441738
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	1.59262	0.0590670	0.0295335	−	0.999564i	$-0.490598\pi$
$727$	1.59262	0.0590670	0.0295335	+	0.999564i	$0.490598\pi$
$728$	0	0
$729$	−4.91842	−0.182164
$730$	0	0
$731$	−45.0623	−1.66669
$732$	0	0
$733$	−12.3036	−0.454443	−0.227221	−	0.973843i	$-0.572964\pi$
$733$	−12.3036	−0.454443	−0.227221	+	0.973843i	$0.572964\pi$
$734$	0	0
$735$	1.58409	0.0584302
$736$	0	0
$737$	11.8422	0.436214
$738$	0	0
$739$	25.3140	0.931190	0.465595	−	0.884998i	$-0.345840\pi$
$739$	25.3140	0.931190	0.465595	+	0.884998i	$0.345840\pi$
$740$	0	0
$741$	−5.28117	−0.194009
$742$	0	0
$743$	16.6570	0.611086	0.305543	−	0.952178i	$-0.401162\pi$
$743$	16.6570	0.611086	0.305543	+	0.952178i	$0.401162\pi$
$744$	0	0
$745$	18.1038	0.663272
$746$	0	0
$747$	−16.7936	−0.614446
$748$	0	0
$749$	−26.1957	−0.957168
$750$	0	0
$751$	13.3140	0.485834	0.242917	−	0.970047i	$-0.421896\pi$
$751$	13.3140	0.485834	0.242917	+	0.970047i	$0.421896\pi$
$752$	0	0
$753$	−17.8002	−0.648677
$754$	0	0
$755$	−0.710947	−0.0258740
$756$	0	0
$757$	−20.4572	−0.743531	−0.371766	−	0.928327i	$-0.621247\pi$
$757$	−20.4572	−0.743531	−0.371766	+	0.928327i	$0.621247\pi$
$758$	0	0
$759$	−2.02493	−0.0735003
$760$	0	0
$761$	30.9670	1.12255	0.561277	−	0.827628i	$-0.310310\pi$
$761$	30.9670	1.12255	0.561277	+	0.827628i	$0.310310\pi$
$762$	0	0
$763$	−41.0893	−1.48753
$764$	0	0
$765$	10.5447	0.381243
$766$	0	0
$767$	−12.8503	−0.463996
$768$	0	0
$769$	−4.18270	−0.150832	−0.0754160	−	0.997152i	$-0.524028\pi$
$769$	−4.18270	−0.150832	−0.0754160	+	0.997152i	$0.524028\pi$
$770$	0	0
$771$	7.06437	0.254417
$772$	0	0
$773$	42.0499	1.51243	0.756214	−	0.654324i	$-0.227047\pi$
$773$	42.0499	1.51243	0.756214	+	0.654324i	$0.227047\pi$
$774$	0	0
$775$	−7.86068	−0.282364
$776$	0	0
$777$	−9.36794	−0.336073
$778$	0	0
$779$	−3.18524	−0.114123
$780$	0	0
$781$	−11.8422	−0.423748
$782$	0	0
$783$	3.78384	0.135224
$784$	0	0
$785$	22.6570	0.808663
$786$	0	0
$787$	33.0524	1.17819	0.589095	−	0.808063i	$-0.299484\pi$
$787$	33.0524	1.17819	0.589095	+	0.808063i	$0.299484\pi$
$788$	0	0
$789$	−10.9435	−0.389599
$790$	0	0
$791$	−8.05839	−0.286523
$792$	0	0
$793$	15.1497	0.537983
$794$	0	0
$795$	−5.80877	−0.206016
$796$	0	0
$797$	11.9041	0.421664	0.210832	−	0.977522i	$-0.432383\pi$
$797$	11.9041	0.421664	0.210832	+	0.977522i	$0.432383\pi$
$798$	0	0
$799$	−33.3140	−1.17856
$800$	0	0
$801$	13.3929	0.473214
$802$	0	0
$803$	65.9170	2.32616
$804$	0	0
$805$	−1.48028	−0.0521730
$806$	0	0
$807$	9.48028	0.333722
$808$	0	0
$809$	−53.6674	−1.88685	−0.943423	−	0.331592i	$-0.892414\pi$
$809$	−53.6674	−1.88685	−0.943423	+	0.331592i	$0.892414\pi$
$810$	0	0
$811$	−47.3034	−1.66105	−0.830524	−	0.556983i	$-0.811959\pi$
$811$	−47.3034	−1.66105	−0.830524	+	0.556983i	$0.811959\pi$
$812$	0	0
$813$	4.60713	0.161579
$814$	0	0
$815$	−18.5781	−0.650763
$816$	0	0
$817$	14.8028	0.517884
$818$	0	0
$819$	30.9336	1.08091
$820$	0	0
$821$	31.3679	1.09475	0.547374	−	0.836888i	$-0.315627\pi$
$821$	31.3679	1.09475	0.547374	+	0.836888i	$0.315627\pi$
$822$	0	0
$823$	23.4598	0.817757	0.408878	−	0.912589i	$-0.365920\pi$
$823$	23.4598	0.817757	0.408878	+	0.912589i	$0.365920\pi$
$824$	0	0
$825$	−2.96056	−0.103073
$826$	0	0
$827$	13.1788	0.458270	0.229135	−	0.973395i	$-0.426410\pi$
$827$	13.1788	0.458270	0.229135	+	0.973395i	$0.426410\pi$
$828$	0	0
$829$	−20.2312	−0.702657	−0.351329	−	0.936252i	$-0.614270\pi$
$829$	−20.2312	−0.702657	−0.351329	+	0.936252i	$0.614270\pi$
$830$	0	0
$831$	−11.8962	−0.412674
$832$	0	0
$833$	9.64453	0.334163
$834$	0	0
$835$	5.83575	0.201955
$836$	0	0
$837$	−29.7436	−1.02809
$838$	0	0
$839$	33.3638	1.15185	0.575924	−	0.817503i	$-0.304642\pi$
$839$	33.3638	1.15185	0.575924	+	0.817503i	$0.304642\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	8.16630	0.281262
$844$	0	0
$845$	18.8607	0.648827
$846$	0	0
$847$	−16.7424	−0.575274
$848$	0	0
$849$	−16.0710	−0.551556
$850$	0	0
$851$	−4.32850	−0.148379
$852$	0	0
$853$	−55.0852	−1.88608	−0.943041	−	0.332677i	$-0.892048\pi$
$853$	−55.0852	−1.88608	−0.943041	+	0.332677i	$0.892048\pi$
$854$	0	0
$855$	−3.46387	−0.118462
$856$	0	0
$857$	−8.02493	−0.274126	−0.137063	−	0.990562i	$-0.543766\pi$
$857$	−8.02493	−0.274126	−0.137063	+	0.990562i	$0.543766\pi$
$858$	0	0
$859$	30.0748	1.02614	0.513069	−	0.858347i	$-0.328508\pi$
$859$	30.0748	1.02614	0.513069	+	0.858347i	$0.328508\pi$
$860$	0	0
$861$	−3.44682	−0.117467
$862$	0	0
$863$	20.5802	0.700557	0.350278	−	0.936646i	$-0.386087\pi$
$863$	20.5802	0.700557	0.350278	+	0.936646i	$0.386087\pi$
$864$	0	0
$865$	−18.3015	−0.622270
$866$	0	0
$867$	−0.233207	−0.00792012
$868$	0	0
$869$	65.0893	2.20800
$870$	0	0
$871$	15.4427	0.523257
$872$	0	0
$873$	−2.50062	−0.0846332
$874$	0	0
$875$	−2.16425	−0.0731649
$876$	0	0
$877$	−19.7234	−0.666012	−0.333006	−	0.942925i	$-0.608063\pi$
$877$	−19.7234	−0.666012	−0.333006	+	0.942925i	$0.608063\pi$
$878$	0	0
$879$	−3.77122	−0.127200
$880$	0	0
$881$	10.1707	0.342660	0.171330	−	0.985214i	$-0.445193\pi$
$881$	10.1707	0.342660	0.171330	+	0.985214i	$0.445193\pi$
$882$	0	0
$883$	13.2102	0.444558	0.222279	−	0.974983i	$-0.428650\pi$
$883$	13.2102	0.444558	0.222279	+	0.974983i	$0.428650\pi$
$884$	0	0
$885$	1.55712	0.0523419
$886$	0	0
$887$	−17.5216	−0.588318	−0.294159	−	0.955756i	$-0.595040\pi$
$887$	−17.5216	−0.588318	−0.294159	+	0.955756i	$0.595040\pi$
$888$	0	0
$889$	5.92112	0.198588
$890$	0	0
$891$	21.6794	0.726287
$892$	0	0
$893$	10.9435	0.366210
$894$	0	0
$895$	−23.0374	−0.770055
$896$	0	0
$897$	−2.64059	−0.0881666
$898$	0	0
$899$	−7.86068	−0.262168
$900$	0	0
$901$	−35.3659	−1.17821
$902$	0	0
$903$	16.0185	0.533061
$904$	0	0
$905$	−26.3784	−0.876846
$906$	0	0
$907$	25.0210	0.830808	0.415404	−	0.909637i	$-0.363640\pi$
$907$	25.0210	0.830808	0.415404	+	0.909637i	$0.363640\pi$
$908$	0	0
$909$	−32.1813	−1.06739
$910$	0	0
$911$	49.0623	1.62551	0.812754	−	0.582607i	$-0.197967\pi$
$911$	49.0623	1.62551	0.812754	+	0.582607i	$0.197967\pi$
$912$	0	0
$913$	28.7069	0.950058
$914$	0	0
$915$	−1.83575	−0.0606881
$916$	0	0
$917$	35.5636	1.17441
$918$	0	0
$919$	−33.3140	−1.09893	−0.549463	−	0.835518i	$-0.685168\pi$
$919$	−33.3140	−1.09893	−0.549463	+	0.835518i	$0.685168\pi$
$920$	0	0
$921$	−3.18524	−0.104957
$922$	0	0
$923$	−15.4427	−0.508304
$924$	0	0
$925$	−6.32850	−0.208080
$926$	0	0
$927$	25.1221	0.825118
$928$	0	0
$929$	−1.68996	−0.0554457	−0.0277228	−	0.999616i	$-0.508826\pi$
$929$	−1.68996	−0.0554457	−0.0277228	+	0.999616i	$0.508826\pi$
$930$	0	0
$931$	−3.16819	−0.103833
$932$	0	0
$933$	−2.97901	−0.0975284
$934$	0	0
$935$	−18.0249	−0.589478
$936$	0	0
$937$	23.1931	0.757686	0.378843	−	0.925461i	$-0.376322\pi$
$937$	23.1931	0.757686	0.378843	+	0.925461i	$0.376322\pi$
$938$	0	0
$939$	−0.918576	−0.0299766
$940$	0	0
$941$	53.4598	1.74274	0.871370	−	0.490627i	$-0.163232\pi$
$941$	53.4598	1.74274	0.871370	+	0.490627i	$0.163232\pi$
$942$	0	0
$943$	−1.59262	−0.0518628
$944$	0	0
$945$	−8.18918	−0.266394
$946$	0	0
$947$	17.3949	0.565259	0.282629	−	0.959229i	$-0.408793\pi$
$947$	17.3949	0.565259	0.282629	+	0.959229i	$0.408793\pi$
$948$	0	0
$949$	85.9584	2.79033
$950$	0	0
$951$	7.77532	0.252132
$952$	0	0
$953$	60.1786	1.94938	0.974688	−	0.223569i	$-0.0717708\pi$
$953$	60.1786	1.94938	0.974688	+	0.223569i	$0.0717708\pi$
$954$	0	0
$955$	−1.45330	−0.0470277
$956$	0	0
$957$	−2.96056	−0.0957012
$958$	0	0
$959$	−9.48028	−0.306134
$960$	0	0
$961$	30.7903	0.993236
$962$	0	0
$963$	−30.6491	−0.987654
$964$	0	0
$965$	−11.5656	−0.372311
$966$	0	0
$967$	−5.09340	−0.163793	−0.0818963	−	0.996641i	$-0.526098\pi$
$967$	−5.09340	−0.163793	−0.0818963	+	0.996641i	$0.526098\pi$
$968$	0	0
$969$	3.89619	0.125164
$970$	0	0
$971$	11.4468	0.367346	0.183673	−	0.982987i	$-0.441201\pi$
$971$	11.4468	0.367346	0.183673	+	0.982987i	$0.441201\pi$
$972$	0	0
$973$	−5.17023	−0.165750
$974$	0	0
$975$	−3.86068	−0.123641
$976$	0	0
$977$	−41.9959	−1.34357	−0.671784	−	0.740747i	$-0.734472\pi$
$977$	−41.9959	−1.34357	−0.671784	+	0.740747i	$0.734472\pi$
$978$	0	0
$979$	−22.8936	−0.731684
$980$	0	0
$981$	−48.0748	−1.53491
$982$	0	0
$983$	2.75293	0.0878048	0.0439024	−	0.999036i	$-0.486021\pi$
$983$	2.75293	0.0878048	0.0439024	+	0.999036i	$0.486021\pi$
$984$	0	0
$985$	13.9645	0.444946
$986$	0	0
$987$	11.8422	0.376942
$988$	0	0
$989$	7.40139	0.235351
$990$	0	0
$991$	61.7422	1.96131	0.980653	−	0.195755i	$-0.0627158\pi$
$991$	61.7422	1.96131	0.980653	+	0.195755i	$0.0627158\pi$
$992$	0	0
$993$	8.43231	0.267591
$994$	0	0
$995$	12.3285	0.390840
$996$	0	0
$997$	45.4967	1.44089	0.720447	−	0.693510i	$-0.243937\pi$
$997$	45.4967	1.44089	0.720447	+	0.693510i	$0.243937\pi$
$998$	0	0
$999$	−23.9460	−0.757619

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.a.q.1.2		3
4.3	odd	2		290.2.a.d.1.2	✓	3
8.3	odd	2		9280.2.a.bp.1.2		3
8.5	even	2		9280.2.a.bn.1.2		3
12.11	even	2		2610.2.a.w.1.3		3
20.3	even	4		1450.2.b.j.349.2		6
20.7	even	4		1450.2.b.j.349.5		6
20.19	odd	2		1450.2.a.r.1.2		3
116.115	odd	2		8410.2.a.w.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
290.2.a.d.1.2	✓	3	4.3	odd	2
1450.2.a.r.1.2		3	20.19	odd	2
1450.2.b.j.349.2		6	20.3	even	4
1450.2.b.j.349.5		6	20.7	even	4
2320.2.a.q.1.2		3	1.1	even	1	trivial
2610.2.a.w.1.3		3	12.11	even	2
8410.2.a.w.1.2		3	116.115	odd	2
9280.2.a.bn.1.2		3	8.5	even	2
9280.2.a.bp.1.2		3	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.a (trivial)

\(f(q)\)	\(=\)	\(q-0.683969 q^{3} +1.00000 q^{5} -2.16425 q^{7} -2.53219 q^{9} +4.32850 q^{11} +5.64453 q^{13} -0.683969 q^{15} -4.16425 q^{17} +1.36794 q^{19} +1.48028 q^{21} +0.683969 q^{23} +1.00000 q^{25} +3.78384 q^{27} +1.00000 q^{29} -7.86068 q^{31} -2.96056 q^{33} -2.16425 q^{35} -6.32850 q^{37} -3.86068 q^{39} -2.32850 q^{41} +10.8212 q^{43} -2.53219 q^{45} +8.00000 q^{47} -2.31603 q^{49} +2.84822 q^{51} +8.49274 q^{53} +4.32850 q^{55} -0.935628 q^{57} -2.27659 q^{59} +2.68397 q^{61} +5.48028 q^{63} +5.64453 q^{65} +2.73588 q^{67} -0.467814 q^{69} -2.73588 q^{71} +15.2286 q^{73} -0.683969 q^{75} -9.36794 q^{77} +15.0374 q^{79} +5.00853 q^{81} +6.63206 q^{83} -4.16425 q^{85} -0.683969 q^{87} -5.28905 q^{89} -12.2162 q^{91} +5.37646 q^{93} +1.36794 q^{95} +0.987535 q^{97} -10.9606 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 3 q^{5} + q^{7} + 6 q^{9} - 2 q^{11} + 5 q^{13} + q^{15} - 5 q^{17} - 2 q^{19} - q^{23} + 3 q^{25} + 28 q^{27} + 3 q^{29} + 5 q^{31} + q^{35} - 4 q^{37} + 17 q^{39} + 8 q^{41} - 5 q^{43}+ \cdots - 24 q^{99}+O(q^{100}) \) 3 * q + q^3 + 3 * q^5 + q^7 + 6 * q^9 - 2 * q^11 + 5 * q^13 + q^15 - 5 * q^17 - 2 * q^19 - q^23 + 3 * q^25 + 28 * q^27 + 3 * q^29 + 5 * q^31 + q^35 - 4 * q^37 + 17 * q^39 + 8 * q^41 - 5 * q^43 + 6 * q^45 + 24 * q^47 - 10 * q^49 - 2 * q^51 + 3 * q^53 - 2 * q^55 - 30 * q^57 - q^59 + 5 * q^61 + 12 * q^63 + 5 * q^65 - 4 * q^67 - 15 * q^69 + 4 * q^71 + 11 * q^73 + q^75 - 22 * q^77 - 9 * q^79 + 35 * q^81 + 26 * q^83 - 5 * q^85 + q^87 + 8 * q^89 - 20 * q^91 + 30 * q^93 - 2 * q^95 + 21 * q^97 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more