LMFDB - Embedded newform 513.2.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [513,2,Mod(1,513)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("513.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 513 = 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	513.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.09632562369$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29952.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 13 $ x^4 - 8*x^2 + 13
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			1.3
Root			$1.50597$ of defining polynomial
Character	$\chi$	$=$	513.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.50597 q^{2} +0.267949 q^{4} +4.11439 q^{5} +1.26795 q^{7} -2.60842 q^{8} +6.19615 q^{10} +2.60842 q^{11} -3.46410 q^{13} +1.90949 q^{14} -4.46410 q^{16} -1.00000 q^{19} +1.10245 q^{20} +3.92820 q^{22} -1.10245 q^{23} +11.9282 q^{25} -5.21684 q^{26} +0.339746 q^{28} -8.63230 q^{29} +10.6603 q^{31} -1.50597 q^{32} +5.21684 q^{35} +4.19615 q^{37} -1.50597 q^{38} -10.7321 q^{40} -3.71087 q^{41} -2.73205 q^{43} +0.698924 q^{44} -1.66025 q^{46} -4.51791 q^{47} -5.39230 q^{49} +17.9635 q^{50} -0.928203 q^{52} -4.51791 q^{53} +10.7321 q^{55} -3.30734 q^{56} -13.0000 q^{58} -7.12633 q^{59} -7.19615 q^{61} +16.0540 q^{62} +6.66025 q^{64} -14.2527 q^{65} -5.73205 q^{67} +7.85641 q^{70} +14.2527 q^{71} +16.1244 q^{73} +6.31928 q^{74} -0.267949 q^{76} +3.30734 q^{77} -9.00000 q^{79} -18.3671 q^{80} -5.58846 q^{82} -7.52986 q^{83} -4.11439 q^{86} -6.80385 q^{88} -16.0540 q^{89} -4.39230 q^{91} -0.295400 q^{92} -6.80385 q^{94} -4.11439 q^{95} -7.66025 q^{97} -8.12066 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} + 12 q^{7} + 4 q^{10} - 4 q^{16} - 4 q^{19} - 12 q^{22} + 20 q^{25} + 36 q^{28} + 8 q^{31} - 4 q^{37} - 36 q^{40} - 4 q^{43} + 28 q^{46} + 20 q^{49} + 24 q^{52} + 36 q^{55} - 52 q^{58} - 8 q^{61}+ \cdots + 4 q^{97}+O(q^{100}) $ 4 * q + 8 * q^4 + 12 * q^7 + 4 * q^10 - 4 * q^16 - 4 * q^19 - 12 * q^22 + 20 * q^25 + 36 * q^28 + 8 * q^31 - 4 * q^37 - 36 * q^40 - 4 * q^43 + 28 * q^46 + 20 * q^49 + 24 * q^52 + 36 * q^55 - 52 * q^58 - 8 * q^61 - 8 * q^64 - 16 * q^67 - 24 * q^70 + 16 * q^73 - 8 * q^76 - 36 * q^79 + 40 * q^82 - 48 * q^88 + 24 * q^91 - 48 * q^94 + 4 * q^97

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 $

$2$	1.50597	1.06488	0.532441	−	0.846467i	$-0.321275\pi$
$2$	1.50597	1.06488	0.532441	+	0.846467i	$0.321275\pi$
$3$	0	0
$3$	0	0
$4$	0.267949	0.133975
$5$	4.11439	1.84001	0.920006	−	0.391905i	$-0.128184\pi$
$5$	4.11439	1.84001	0.920006	+	0.391905i	$0.128184\pi$
$6$	0	0
$7$	1.26795	0.479240	0.239620	−	0.970867i	$-0.422977\pi$
$7$	1.26795	0.479240	0.239620	+	0.970867i	$0.422977\pi$
$8$	−2.60842	−0.922215
$9$	0	0
$10$	6.19615	1.95940
$11$	2.60842	0.786468	0.393234	−	0.919438i	$-0.371356\pi$
$11$	2.60842	0.786468	0.393234	+	0.919438i	$0.371356\pi$
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	1.90949	0.510334
$15$	0	0
$16$	−4.46410	−1.11603
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.10245	0.246515
$21$	0	0
$22$	3.92820	0.837496
$23$	−1.10245	−0.229876	−0.114938	−	0.993373i	$-0.536667\pi$
$23$	−1.10245	−0.229876	−0.114938	+	0.993373i	$0.536667\pi$
$24$	0	0
$25$	11.9282	2.38564
$26$	−5.21684	−1.02311
$27$	0	0
$28$	0.339746	0.0642060
$29$	−8.63230	−1.60298	−0.801489	−	0.598009i	$-0.795959\pi$
$29$	−8.63230	−1.60298	−0.801489	+	0.598009i	$0.795959\pi$
$30$	0	0
$31$	10.6603	1.91464	0.957319	−	0.289033i	$-0.0933338\pi$
$31$	10.6603	1.91464	0.957319	+	0.289033i	$0.0933338\pi$
$32$	−1.50597	−0.266221
$33$	0	0
$34$	0	0
$35$	5.21684	0.881806
$36$	0	0
$37$	4.19615	0.689843	0.344922	−	0.938631i	$-0.387905\pi$
$37$	4.19615	0.689843	0.344922	+	0.938631i	$0.387905\pi$
$38$	−1.50597	−0.244301
$39$	0	0
$40$	−10.7321	−1.69689
$41$	−3.71087	−0.579540	−0.289770	−	0.957096i	$-0.593579\pi$
$41$	−3.71087	−0.579540	−0.289770	+	0.957096i	$0.593579\pi$
$42$	0	0
$43$	−2.73205	−0.416634	−0.208317	−	0.978061i	$-0.566799\pi$
$43$	−2.73205	−0.416634	−0.208317	+	0.978061i	$0.566799\pi$
$44$	0.698924	0.105367
$45$	0	0
$46$	−1.66025	−0.244791
$47$	−4.51791	−0.659005	−0.329503	−	0.944155i	$-0.606881\pi$
$47$	−4.51791	−0.659005	−0.329503	+	0.944155i	$0.606881\pi$
$48$	0	0
$49$	−5.39230	−0.770329
$50$	17.9635	2.54043
$51$	0	0
$52$	−0.928203	−0.128719
$53$	−4.51791	−0.620583	−0.310292	−	0.950641i	$-0.600427\pi$
$53$	−4.51791	−0.620583	−0.310292	+	0.950641i	$0.600427\pi$
$54$	0	0
$55$	10.7321	1.44711
$56$	−3.30734	−0.441962
$57$	0	0
$58$	−13.0000	−1.70698
$59$	−7.12633	−0.927769	−0.463885	−	0.885896i	$-0.653545\pi$
$59$	−7.12633	−0.927769	−0.463885	+	0.885896i	$0.653545\pi$
$60$	0	0
$61$	−7.19615	−0.921373	−0.460686	−	0.887563i	$-0.652397\pi$
$61$	−7.19615	−0.921373	−0.460686	+	0.887563i	$0.652397\pi$
$62$	16.0540	2.03886
$63$	0	0
$64$	6.66025	0.832532
$65$	−14.2527	−1.76783
$66$	0	0
$67$	−5.73205	−0.700281	−0.350141	−	0.936697i	$-0.613866\pi$
$67$	−5.73205	−0.700281	−0.350141	+	0.936697i	$0.613866\pi$
$68$	0	0
$69$	0	0
$70$	7.85641	0.939020
$71$	14.2527	1.69148	0.845740	−	0.533595i	$-0.179159\pi$
$71$	14.2527	1.69148	0.845740	+	0.533595i	$0.179159\pi$
$72$	0	0
$73$	16.1244	1.88721	0.943607	−	0.331069i	$-0.107409\pi$
$73$	16.1244	1.88721	0.943607	+	0.331069i	$0.107409\pi$
$74$	6.31928	0.734602
$75$	0	0
$76$	−0.267949	−0.0307359
$77$	3.30734	0.376907
$78$	0	0
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	−18.3671	−2.05350
$81$	0	0
$82$	−5.58846	−0.617142
$83$	−7.52986	−0.826509	−0.413255	−	0.910616i	$-0.635608\pi$
$83$	−7.52986	−0.826509	−0.413255	+	0.910616i	$0.635608\pi$
$84$	0	0
$85$	0	0
$86$	−4.11439	−0.443666
$87$	0	0
$88$	−6.80385	−0.725293
$89$	−16.0540	−1.70172	−0.850862	−	0.525389i	$-0.823920\pi$
$89$	−16.0540	−1.70172	−0.850862	+	0.525389i	$0.823920\pi$
$90$	0	0
$91$	−4.39230	−0.460439
$92$	−0.295400	−0.0307976
$93$	0	0
$94$	−6.80385	−0.701763
$95$	−4.11439	−0.422127
$96$	0	0
$97$	−7.66025	−0.777781	−0.388890	−	0.921284i	$-0.627142\pi$
$97$	−7.66025	−0.777781	−0.388890	+	0.921284i	$0.627142\pi$
$98$	−8.12066	−0.820310
$99$	0	0
$100$	3.19615	0.319615
$101$	−9.33123	−0.928492	−0.464246	−	0.885706i	$-0.653675\pi$
$101$	−9.33123	−0.928492	−0.464246	+	0.885706i	$0.653675\pi$
$102$	0	0
$103$	4.53590	0.446935	0.223468	−	0.974711i	$-0.428262\pi$
$103$	4.53590	0.446935	0.223468	+	0.974711i	$0.428262\pi$
$104$	9.03583	0.886036
$105$	0	0
$106$	−6.80385	−0.660848
$107$	13.1502	1.27128	0.635640	−	0.771986i	$-0.280736\pi$
$107$	13.1502	1.27128	0.635640	+	0.771986i	$0.280736\pi$
$108$	0	0
$109$	−8.73205	−0.836379	−0.418189	−	0.908360i	$-0.637335\pi$
$109$	−8.73205	−0.836379	−0.418189	+	0.908360i	$0.637335\pi$
$110$	16.1622	1.54100
$111$	0	0
$112$	−5.66025	−0.534844
$113$	−6.42741	−0.604640	−0.302320	−	0.953207i	$-0.597761\pi$
$113$	−6.42741	−0.604640	−0.302320	+	0.953207i	$0.597761\pi$
$114$	0	0
$115$	−4.53590	−0.422975
$116$	−2.31302	−0.214758
$117$	0	0
$118$	−10.7321	−0.987965
$119$	0	0
$120$	0	0
$121$	−4.19615	−0.381468
$122$	−10.8372	−0.981154
$123$	0	0
$124$	2.85641	0.256513
$125$	28.5053	2.54959
$126$	0	0
$127$	10.9282	0.969721	0.484861	−	0.874591i	$-0.338870\pi$
$127$	10.9282	0.969721	0.484861	+	0.874591i	$0.338870\pi$
$128$	13.0421	1.15277
$129$	0	0
$130$	−21.4641	−1.88253
$131$	4.92144	0.429988	0.214994	−	0.976615i	$-0.431027\pi$
$131$	4.92144	0.429988	0.214994	+	0.976615i	$0.431027\pi$
$132$	0	0
$133$	−1.26795	−0.109945
$134$	−8.63230	−0.745717
$135$	0	0
$136$	0	0
$137$	14.2527	1.21769	0.608844	−	0.793290i	$-0.291634\pi$
$137$	14.2527	1.21769	0.608844	+	0.793290i	$0.291634\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	1.39785	0.118140
$141$	0	0
$142$	21.4641	1.80123
$143$	−9.03583	−0.755614
$144$	0	0
$145$	−35.5167	−2.94950
$146$	24.2828	2.00966
$147$	0	0
$148$	1.12436	0.0924215
$149$	9.03583	0.740244	0.370122	−	0.928983i	$-0.379316\pi$
$149$	9.03583	0.740244	0.370122	+	0.928983i	$0.379316\pi$
$150$	0	0
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	2.60842	0.211571
$153$	0	0
$154$	4.98076	0.401361
$155$	43.8604	3.52296
$156$	0	0
$157$	19.5885	1.56333	0.781665	−	0.623699i	$-0.214371\pi$
$157$	19.5885	1.56333	0.781665	+	0.623699i	$0.214371\pi$
$158$	−13.5537	−1.07828
$159$	0	0
$160$	−6.19615	−0.489849
$161$	−1.39785	−0.110166
$162$	0	0
$163$	24.7321	1.93716	0.968582	−	0.248695i	$-0.0800017\pi$
$163$	24.7321	1.93716	0.968582	+	0.248695i	$0.0800017\pi$
$164$	−0.994324	−0.0776436
$165$	0	0
$166$	−11.3397	−0.880135
$167$	−19.4695	−1.50660	−0.753298	−	0.657680i	$-0.771538\pi$
$167$	−19.4695	−1.50660	−0.753298	+	0.657680i	$0.771538\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	−0.732051	−0.0558184
$173$	−6.72281	−0.511126	−0.255563	−	0.966792i	$-0.582261\pi$
$173$	−6.72281	−0.511126	−0.255563	+	0.966792i	$0.582261\pi$
$174$	0	0
$175$	15.1244	1.14329
$176$	−11.6442	−0.877718
$177$	0	0
$178$	−24.1769	−1.81214
$179$	6.31928	0.472326	0.236163	−	0.971714i	$-0.424110\pi$
$179$	6.31928	0.472326	0.236163	+	0.971714i	$0.424110\pi$
$180$	0	0
$181$	−21.6603	−1.60999	−0.804997	−	0.593279i	$-0.797833\pi$
$181$	−21.6603	−1.60999	−0.804997	+	0.593279i	$0.797833\pi$
$182$	−6.61468	−0.490313
$183$	0	0
$184$	2.87564	0.211995
$185$	17.2646	1.26932
$186$	0	0
$187$	0	0
$188$	−1.21057	−0.0882900
$189$	0	0
$190$	−6.19615	−0.449516
$191$	7.82526	0.566216	0.283108	−	0.959088i	$-0.408635\pi$
$191$	7.82526	0.566216	0.283108	+	0.959088i	$0.408635\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−11.5361	−0.828245
$195$	0	0
$196$	−1.44486	−0.103205
$197$	−22.4814	−1.60174	−0.800868	−	0.598841i	$-0.795628\pi$
$197$	−22.4814	−1.60174	−0.800868	+	0.598841i	$0.795628\pi$
$198$	0	0
$199$	19.4641	1.37977	0.689887	−	0.723917i	$-0.257660\pi$
$199$	19.4641	1.37977	0.689887	+	0.723917i	$0.257660\pi$
$200$	−31.1137	−2.20007
$201$	0	0
$202$	−14.0526	−0.988735
$203$	−10.9453	−0.768211
$204$	0	0
$205$	−15.2679	−1.06636
$206$	6.83093	0.475934
$207$	0	0
$208$	15.4641	1.07224
$209$	−2.60842	−0.180428
$210$	0	0
$211$	−10.1244	−0.696989	−0.348495	−	0.937311i	$-0.613307\pi$
$211$	−10.1244	−0.696989	−0.348495	+	0.937311i	$0.613307\pi$
$212$	−1.21057	−0.0831424
$213$	0	0
$214$	19.8038	1.35376
$215$	−11.2407	−0.766611
$216$	0	0
$217$	13.5167	0.917571
$218$	−13.1502	−0.890645
$219$	0	0
$220$	2.87564	0.193876
$221$	0	0
$222$	0	0
$223$	15.5359	1.04036	0.520180	−	0.854056i	$-0.325865\pi$
$223$	15.5359	1.04036	0.520180	+	0.854056i	$0.325865\pi$
$224$	−1.90949	−0.127583
$225$	0	0
$226$	−9.67949	−0.643870
$227$	5.21684	0.346254	0.173127	−	0.984900i	$-0.444613\pi$
$227$	5.21684	0.346254	0.173127	+	0.984900i	$0.444613\pi$
$228$	0	0
$229$	20.3205	1.34282	0.671408	−	0.741087i	$-0.265690\pi$
$229$	20.3205	1.34282	0.671408	+	0.741087i	$0.265690\pi$
$230$	−6.83093	−0.450418
$231$	0	0
$232$	22.5167	1.47829
$233$	9.33123	0.611309	0.305655	−	0.952142i	$-0.401125\pi$
$233$	9.33123	0.611309	0.305655	+	0.952142i	$0.401125\pi$
$234$	0	0
$235$	−18.5885	−1.21258
$236$	−1.90949	−0.124298
$237$	0	0
$238$	0	0
$239$	−0.698924	−0.0452096	−0.0226048	−	0.999744i	$-0.507196\pi$
$239$	−0.698924	−0.0452096	−0.0226048	+	0.999744i	$0.507196\pi$
$240$	0	0
$241$	−2.19615	−0.141467	−0.0707333	−	0.997495i	$-0.522534\pi$
$241$	−2.19615	−0.141467	−0.0707333	+	0.997495i	$0.522534\pi$
$242$	−6.31928	−0.406219
$243$	0	0
$244$	−1.92820	−0.123441
$245$	−22.1860	−1.41741
$246$	0	0
$247$	3.46410	0.220416
$248$	−27.8064	−1.76571
$249$	0	0
$250$	42.9282	2.71502
$251$	7.12633	0.449810	0.224905	−	0.974381i	$-0.427793\pi$
$251$	7.12633	0.449810	0.224905	+	0.974381i	$0.427793\pi$
$252$	0	0
$253$	−2.87564	−0.180790
$254$	16.4576	1.03264
$255$	0	0
$256$	6.32051	0.395032
$257$	15.3551	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$257$	15.3551	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$258$	0	0
$259$	5.32051	0.330600
$260$	−3.81899	−0.236844
$261$	0	0
$262$	7.41154	0.457887
$263$	−27.2948	−1.68307	−0.841533	−	0.540205i	$-0.818347\pi$
$263$	−27.2948	−1.68307	−0.841533	+	0.540205i	$0.818347\pi$
$264$	0	0
$265$	−18.5885	−1.14188
$266$	−1.90949	−0.117079
$267$	0	0
$268$	−1.53590	−0.0938199
$269$	17.5600	1.07065	0.535326	−	0.844645i	$-0.320189\pi$
$269$	17.5600	1.07065	0.535326	+	0.844645i	$0.320189\pi$
$270$	0	0
$271$	17.6603	1.07278	0.536392	−	0.843969i	$-0.319787\pi$
$271$	17.6603	1.07278	0.536392	+	0.843969i	$0.319787\pi$
$272$	0	0
$273$	0	0
$274$	21.4641	1.29669
$275$	31.1137	1.87623
$276$	0	0
$277$	−0.607695	−0.0365129	−0.0182564	−	0.999833i	$-0.505812\pi$
$277$	−0.607695	−0.0365129	−0.0182564	+	0.999833i	$0.505812\pi$
$278$	15.0597	0.903221
$279$	0	0
$280$	−13.6077	−0.813215
$281$	31.0056	1.84964	0.924820	−	0.380405i	$-0.124215\pi$
$281$	31.0056	1.84964	0.924820	+	0.380405i	$0.124215\pi$
$282$	0	0
$283$	15.0718	0.895925	0.447963	−	0.894052i	$-0.352150\pi$
$283$	15.0718	0.895925	0.447963	+	0.894052i	$0.352150\pi$
$284$	3.81899	0.226615
$285$	0	0
$286$	−13.6077	−0.804640
$287$	−4.70519	−0.277739
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−53.4871	−3.14087
$291$	0	0
$292$	4.32051	0.252839
$293$	3.41547	0.199534	0.0997668	−	0.995011i	$-0.468190\pi$
$293$	3.41547	0.199534	0.0997668	+	0.995011i	$0.468190\pi$
$294$	0	0
$295$	−29.3205	−1.70711
$296$	−10.9453	−0.636184
$297$	0	0
$298$	13.6077	0.788273
$299$	3.81899	0.220858
$300$	0	0
$301$	−3.46410	−0.199667
$302$	10.5418	0.606612
$303$	0	0
$304$	4.46410	0.256034
$305$	−29.6078	−1.69534
$306$	0	0
$307$	−3.00000	−0.171219	−0.0856095	−	0.996329i	$-0.527284\pi$
$307$	−3.00000	−0.171219	−0.0856095	+	0.996329i	$0.527284\pi$
$308$	0.886200	0.0504959
$309$	0	0
$310$	66.0526	3.75153
$311$	−0.403524	−0.0228817	−0.0114409	−	0.999935i	$-0.503642\pi$
$311$	−0.403524	−0.0228817	−0.0114409	+	0.999935i	$0.503642\pi$
$312$	0	0
$313$	−10.8038	−0.610670	−0.305335	−	0.952245i	$-0.598768\pi$
$313$	−10.8038	−0.610670	−0.305335	+	0.952245i	$0.598768\pi$
$314$	29.4997	1.66476
$315$	0	0
$316$	−2.41154	−0.135660
$317$	1.90949	0.107248	0.0536240	−	0.998561i	$-0.482923\pi$
$317$	1.90949	0.107248	0.0536240	+	0.998561i	$0.482923\pi$
$318$	0	0
$319$	−22.5167	−1.26069
$320$	27.4029	1.53187
$321$	0	0
$322$	−2.10512	−0.117314
$323$	0	0
$324$	0	0
$325$	−41.3205	−2.29205
$326$	37.2458	2.06285
$327$	0	0
$328$	9.67949	0.534461
$329$	−5.72848	−0.315822
$330$	0	0
$331$	−1.60770	−0.0883669	−0.0441835	−	0.999023i	$-0.514069\pi$
$331$	−1.60770	−0.0883669	−0.0441835	+	0.999023i	$0.514069\pi$
$332$	−2.01762	−0.110731
$333$	0	0
$334$	−29.3205	−1.60435
$335$	−23.5839	−1.28853
$336$	0	0
$337$	−1.80385	−0.0982618	−0.0491309	−	0.998792i	$-0.515645\pi$
$337$	−1.80385	−0.0982618	−0.0491309	+	0.998792i	$0.515645\pi$
$338$	−1.50597	−0.0819140
$339$	0	0
$340$	0	0
$341$	27.8064	1.50580
$342$	0	0
$343$	−15.7128	−0.848412
$344$	7.12633	0.384226
$345$	0	0
$346$	−10.1244	−0.544289
$347$	−13.7410	−0.737656	−0.368828	−	0.929498i	$-0.620241\pi$
$347$	−13.7410	−0.737656	−0.368828	+	0.929498i	$0.620241\pi$
$348$	0	0
$349$	26.1769	1.40122	0.700609	−	0.713545i	$-0.252912\pi$
$349$	26.1769	1.40122	0.700609	+	0.713545i	$0.252912\pi$
$350$	22.7768	1.21747
$351$	0	0
$352$	−3.92820	−0.209374
$353$	−2.20489	−0.117355	−0.0586774	−	0.998277i	$-0.518688\pi$
$353$	−2.20489	−0.117355	−0.0586774	+	0.998277i	$0.518688\pi$
$354$	0	0
$355$	58.6410	3.11234
$356$	−4.30167	−0.227988
$357$	0	0
$358$	9.51666	0.502971
$359$	−34.8246	−1.83797	−0.918986	−	0.394289i	$-0.870991\pi$
$359$	−34.8246	−1.83797	−0.918986	+	0.394289i	$0.870991\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−32.6197	−1.71445
$363$	0	0
$364$	−1.17691	−0.0616871
$365$	66.3419	3.47249
$366$	0	0
$367$	17.4641	0.911619	0.455809	−	0.890077i	$-0.349350\pi$
$367$	17.4641	0.911619	0.455809	+	0.890077i	$0.349350\pi$
$368$	4.92144	0.256548
$369$	0	0
$370$	26.0000	1.35168
$371$	−5.72848	−0.297408
$372$	0	0
$373$	1.66025	0.0859647	0.0429823	−	0.999076i	$-0.486314\pi$
$373$	1.66025	0.0859647	0.0429823	+	0.999076i	$0.486314\pi$
$374$	0	0
$375$	0	0
$376$	11.7846	0.607745
$377$	29.9032	1.54009
$378$	0	0
$379$	10.8564	0.557656	0.278828	−	0.960341i	$-0.410054\pi$
$379$	10.8564	0.557656	0.278828	+	0.960341i	$0.410054\pi$
$380$	−1.10245	−0.0565544
$381$	0	0
$382$	11.7846	0.602953
$383$	−13.1502	−0.671945	−0.335972	−	0.941872i	$-0.609065\pi$
$383$	−13.1502	−0.671945	−0.335972	+	0.941872i	$0.609065\pi$
$384$	0	0
$385$	13.6077	0.693512
$386$	15.0597	0.766519
$387$	0	0
$388$	−2.05256	−0.104203
$389$	−21.3790	−1.08396	−0.541979	−	0.840392i	$-0.682325\pi$
$389$	−21.3790	−1.08396	−0.541979	+	0.840392i	$0.682325\pi$
$390$	0	0
$391$	0	0
$392$	14.0654	0.710409
$393$	0	0
$394$	−33.8564	−1.70566
$395$	−37.0295	−1.86316
$396$	0	0
$397$	−2.85641	−0.143359	−0.0716795	−	0.997428i	$-0.522836\pi$
$397$	−2.85641	−0.143359	−0.0716795	+	0.997428i	$0.522836\pi$
$398$	29.3124	1.46930
$399$	0	0
$400$	−53.2487	−2.66244
$401$	−4.11439	−0.205463	−0.102731	−	0.994709i	$-0.532758\pi$
$401$	−4.11439	−0.205463	−0.102731	+	0.994709i	$0.532758\pi$
$402$	0	0
$403$	−36.9282	−1.83952
$404$	−2.50029	−0.124394
$405$	0	0
$406$	−16.4833	−0.818054
$407$	10.9453	0.542539
$408$	0	0
$409$	31.1244	1.53900	0.769500	−	0.638647i	$-0.220505\pi$
$409$	31.1244	1.53900	0.769500	+	0.638647i	$0.220505\pi$
$410$	−22.9931	−1.13555
$411$	0	0
$412$	1.21539	0.0598780
$413$	−9.03583	−0.444624
$414$	0	0
$415$	−30.9808	−1.52079
$416$	5.21684	0.255776
$417$	0	0
$418$	−3.92820	−0.192135
$419$	−1.50597	−0.0735715	−0.0367858	−	0.999323i	$-0.511712\pi$
$419$	−1.50597	−0.0735715	−0.0367858	+	0.999323i	$0.511712\pi$
$420$	0	0
$421$	−1.46410	−0.0713559	−0.0356780	−	0.999363i	$-0.511359\pi$
$421$	−1.46410	−0.0713559	−0.0356780	+	0.999363i	$0.511359\pi$
$422$	−15.2470	−0.742212
$423$	0	0
$424$	11.7846	0.572311
$425$	0	0
$426$	0	0
$427$	−9.12436	−0.441559
$428$	3.52359	0.170319
$429$	0	0
$430$	−16.9282	−0.816350
$431$	−17.5600	−0.845836	−0.422918	−	0.906168i	$-0.638994\pi$
$431$	−17.5600	−0.845836	−0.422918	+	0.906168i	$0.638994\pi$
$432$	0	0
$433$	11.5167	0.553455	0.276728	−	0.960948i	$-0.410750\pi$
$433$	11.5167	0.553455	0.276728	+	0.960948i	$0.410750\pi$
$434$	20.3557	0.977105
$435$	0	0
$436$	−2.33975	−0.112054
$437$	1.10245	0.0527372
$438$	0	0
$439$	−13.7846	−0.657904	−0.328952	−	0.944347i	$-0.606695\pi$
$439$	−13.7846	−0.657904	−0.328952	+	0.944347i	$0.606695\pi$
$440$	−27.9937	−1.33455
$441$	0	0
$442$	0	0
$443$	−8.33690	−0.396098	−0.198049	−	0.980192i	$-0.563461\pi$
$443$	−8.33690	−0.396098	−0.198049	+	0.980192i	$0.563461\pi$
$444$	0	0
$445$	−66.0526	−3.13119
$446$	23.3966	1.10786
$447$	0	0
$448$	8.44486	0.398982
$449$	−11.3488	−0.535585	−0.267793	−	0.963477i	$-0.586294\pi$
$449$	−11.3488	−0.535585	−0.267793	+	0.963477i	$0.586294\pi$
$450$	0	0
$451$	−9.67949	−0.455789
$452$	−1.72222	−0.0810064
$453$	0	0
$454$	7.85641	0.368719
$455$	−18.0717	−0.847212
$456$	0	0
$457$	−15.5885	−0.729197	−0.364599	−	0.931165i	$-0.618794\pi$
$457$	−15.5885	−0.729197	−0.364599	+	0.931165i	$0.618794\pi$
$458$	30.6021	1.42994
$459$	0	0
$460$	−1.21539	−0.0566679
$461$	−10.4337	−0.485945	−0.242972	−	0.970033i	$-0.578122\pi$
$461$	−10.4337	−0.485945	−0.242972	+	0.970033i	$0.578122\pi$
$462$	0	0
$463$	−2.39230	−0.111180	−0.0555899	−	0.998454i	$-0.517704\pi$
$463$	−2.39230	−0.111180	−0.0555899	+	0.998454i	$0.517704\pi$
$464$	38.5355	1.78896
$465$	0	0
$466$	14.0526	0.650972
$467$	35.2281	1.63016	0.815082	−	0.579346i	$-0.196692\pi$
$467$	35.2281	1.63016	0.815082	+	0.579346i	$0.196692\pi$
$468$	0	0
$469$	−7.26795	−0.335603
$470$	−27.9937	−1.29125
$471$	0	0
$472$	18.5885	0.855603
$473$	−7.12633	−0.327669
$474$	0	0
$475$	−11.9282	−0.547303
$476$	0	0
$477$	0	0
$478$	−1.05256	−0.0481429
$479$	6.31928	0.288735	0.144368	−	0.989524i	$-0.453885\pi$
$479$	6.31928	0.288735	0.144368	+	0.989524i	$0.453885\pi$
$480$	0	0
$481$	−14.5359	−0.662780
$482$	−3.30734	−0.150645
$483$	0	0
$484$	−1.12436	−0.0511071
$485$	−31.5173	−1.43113
$486$	0	0
$487$	28.6603	1.29872	0.649360	−	0.760481i	$-0.275037\pi$
$487$	28.6603	1.29872	0.649360	+	0.760481i	$0.275037\pi$
$488$	18.7706	0.849704
$489$	0	0
$490$	−33.4115	−1.50938
$491$	−14.1445	−0.638334	−0.319167	−	0.947698i	$-0.603403\pi$
$491$	−14.1445	−0.638334	−0.319167	+	0.947698i	$0.603403\pi$
$492$	0	0
$493$	0	0
$494$	5.21684	0.234717
$495$	0	0
$496$	−47.5885	−2.13678
$497$	18.0717	0.810624
$498$	0	0
$499$	−6.39230	−0.286159	−0.143079	−	0.989711i	$-0.545700\pi$
$499$	−6.39230	−0.286159	−0.143079	+	0.989711i	$0.545700\pi$
$500$	7.63798	0.341581
$501$	0	0
$502$	10.7321	0.478995
$503$	22.0779	0.984406	0.492203	−	0.870481i	$-0.336192\pi$
$503$	22.0779	0.984406	0.492203	+	0.870481i	$0.336192\pi$
$504$	0	0
$505$	−38.3923	−1.70844
$506$	−4.33064	−0.192520
$507$	0	0
$508$	2.92820	0.129918
$509$	−1.50597	−0.0667510	−0.0333755	−	0.999443i	$-0.510626\pi$
$509$	−1.50597	−0.0667510	−0.0333755	+	0.999443i	$0.510626\pi$
$510$	0	0
$511$	20.4449	0.904428
$512$	−16.5657	−0.732107
$513$	0	0
$514$	23.1244	1.01997
$515$	18.6625	0.822366
$516$	0	0
$517$	−11.7846	−0.518287
$518$	8.01253	0.352050
$519$	0	0
$520$	37.1769	1.63032
$521$	−18.3671	−0.804675	−0.402338	−	0.915491i	$-0.631802\pi$
$521$	−18.3671	−0.804675	−0.402338	+	0.915491i	$0.631802\pi$
$522$	0	0
$523$	18.5167	0.809677	0.404839	−	0.914388i	$-0.367328\pi$
$523$	18.5167	0.809677	0.404839	+	0.914388i	$0.367328\pi$
$524$	1.31870	0.0576075
$525$	0	0
$526$	−41.1051	−1.79227
$527$	0	0
$528$	0	0
$529$	−21.7846	−0.947157
$530$	−27.9937	−1.21597
$531$	0	0
$532$	−0.339746	−0.0147299
$533$	12.8548	0.556804
$534$	0	0
$535$	54.1051	2.33917
$536$	14.9516	0.645810
$537$	0	0
$538$	26.4449	1.14012
$539$	−14.0654	−0.605839
$540$	0	0
$541$	−3.19615	−0.137413	−0.0687067	−	0.997637i	$-0.521887\pi$
$541$	−3.19615	−0.137413	−0.0687067	+	0.997637i	$0.521887\pi$
$542$	26.5958	1.14239
$543$	0	0
$544$	0	0
$545$	−35.9271	−1.53895
$546$	0	0
$547$	−6.14359	−0.262681	−0.131341	−	0.991337i	$-0.541928\pi$
$547$	−6.14359	−0.262681	−0.131341	+	0.991337i	$0.541928\pi$
$548$	3.81899	0.163139
$549$	0	0
$550$	46.8564	1.99796
$551$	8.63230	0.367748
$552$	0	0
$553$	−11.4115	−0.485268
$554$	−0.915171	−0.0388819
$555$	0	0
$556$	2.67949	0.113636
$557$	18.9579	0.803270	0.401635	−	0.915800i	$-0.368442\pi$
$557$	18.9579	0.803270	0.401635	+	0.915800i	$0.368442\pi$
$558$	0	0
$559$	9.46410	0.400289
$560$	−23.2885	−0.984118
$561$	0	0
$562$	46.6936	1.96965
$563$	−31.2219	−1.31584	−0.657922	−	0.753086i	$-0.728565\pi$
$563$	−31.2219	−1.31584	−0.657922	+	0.753086i	$0.728565\pi$
$564$	0	0
$565$	−26.4449	−1.11254
$566$	22.6977	0.954055
$567$	0	0
$568$	−37.1769	−1.55991
$569$	22.7768	0.954855	0.477427	−	0.878671i	$-0.341569\pi$
$569$	22.7768	0.954855	0.477427	+	0.878671i	$0.341569\pi$
$570$	0	0
$571$	−32.0526	−1.34136	−0.670679	−	0.741748i	$-0.733997\pi$
$571$	−32.0526	−1.34136	−0.670679	+	0.741748i	$0.733997\pi$
$572$	−2.42114	−0.101233
$573$	0	0
$574$	−7.08588	−0.295759
$575$	−13.1502	−0.548402
$576$	0	0
$577$	−32.5359	−1.35449	−0.677244	−	0.735759i	$-0.736826\pi$
$577$	−32.5359	−1.35449	−0.677244	+	0.735759i	$0.736826\pi$
$578$	−25.6015	−1.06488
$579$	0	0
$580$	−9.51666	−0.395158
$581$	−9.54747	−0.396096
$582$	0	0
$583$	−11.7846	−0.488069
$584$	−42.0591	−1.74042
$585$	0	0
$586$	5.14359	0.212480
$587$	−32.6197	−1.34636	−0.673180	−	0.739479i	$-0.735072\pi$
$587$	−32.6197	−1.34636	−0.673180	+	0.739479i	$0.735072\pi$
$588$	0	0
$589$	−10.6603	−0.439248
$590$	−44.1558	−1.81787
$591$	0	0
$592$	−18.7321	−0.769883
$593$	7.12633	0.292643	0.146322	−	0.989237i	$-0.453257\pi$
$593$	7.12633	0.292643	0.146322	+	0.989237i	$0.453257\pi$
$594$	0	0
$595$	0	0
$596$	2.42114	0.0991739
$597$	0	0
$598$	5.75129	0.235188
$599$	31.8127	1.29983	0.649915	−	0.760007i	$-0.274804\pi$
$599$	31.8127	1.29983	0.649915	+	0.760007i	$0.274804\pi$
$600$	0	0
$601$	−30.9808	−1.26373	−0.631866	−	0.775078i	$-0.717711\pi$
$601$	−30.9808	−1.26373	−0.631866	+	0.775078i	$0.717711\pi$
$602$	−5.21684	−0.212622
$603$	0	0
$604$	1.87564	0.0763189
$605$	−17.2646	−0.701906
$606$	0	0
$607$	29.4449	1.19513	0.597565	−	0.801820i	$-0.296135\pi$
$607$	29.4449	1.19513	0.597565	+	0.801820i	$0.296135\pi$
$608$	1.50597	0.0610752
$609$	0	0
$610$	−44.5885	−1.80533
$611$	15.6505	0.633152
$612$	0	0
$613$	17.0718	0.689523	0.344762	−	0.938690i	$-0.387960\pi$
$613$	17.0718	0.689523	0.344762	+	0.938690i	$0.387960\pi$
$614$	−4.51791	−0.182328
$615$	0	0
$616$	−8.62693	−0.347589
$617$	8.74043	0.351876	0.175938	−	0.984401i	$-0.443704\pi$
$617$	8.74043	0.351876	0.175938	+	0.984401i	$0.443704\pi$
$618$	0	0
$619$	−0.535898	−0.0215396	−0.0107698	−	0.999942i	$-0.503428\pi$
$619$	−0.535898	−0.0215396	−0.0107698	+	0.999942i	$0.503428\pi$
$620$	11.7524	0.471987
$621$	0	0
$622$	−0.607695	−0.0243664
$623$	−20.3557	−0.815534
$624$	0	0
$625$	57.6410	2.30564
$626$	−16.2703	−0.650291
$627$	0	0
$628$	5.24871	0.209446
$629$	0	0
$630$	0	0
$631$	−20.9282	−0.833139	−0.416569	−	0.909104i	$-0.636768\pi$
$631$	−20.9282	−0.833139	−0.416569	+	0.909104i	$0.636768\pi$
$632$	23.4758	0.933816
$633$	0	0
$634$	2.87564	0.114206
$635$	44.9629	1.78430
$636$	0	0
$637$	18.6795	0.740108
$638$	−33.9094	−1.34249
$639$	0	0
$640$	53.6603	2.12111
$641$	−28.1018	−1.10995	−0.554977	−	0.831866i	$-0.687273\pi$
$641$	−28.1018	−1.10995	−0.554977	+	0.831866i	$0.687273\pi$
$642$	0	0
$643$	21.5167	0.848534	0.424267	−	0.905537i	$-0.360532\pi$
$643$	21.5167	0.848534	0.424267	+	0.905537i	$0.360532\pi$
$644$	−0.374552	−0.0147594
$645$	0	0
$646$	0	0
$647$	29.2043	1.14814	0.574069	−	0.818807i	$-0.305364\pi$
$647$	29.2043	1.14814	0.574069	+	0.818807i	$0.305364\pi$
$648$	0	0
$649$	−18.5885	−0.729661
$650$	−62.2275	−2.44076
$651$	0	0
$652$	6.62693	0.259531
$653$	−46.3607	−1.81424	−0.907118	−	0.420877i	$-0.861723\pi$
$653$	−46.3607	−1.81424	−0.907118	+	0.420877i	$0.861723\pi$
$654$	0	0
$655$	20.2487	0.791183
$656$	16.5657	0.646781
$657$	0	0
$658$	−8.62693	−0.336313
$659$	16.9692	0.661026	0.330513	−	0.943801i	$-0.392778\pi$
$659$	16.9692	0.661026	0.330513	+	0.943801i	$0.392778\pi$
$660$	0	0
$661$	−30.2487	−1.17654	−0.588269	−	0.808665i	$-0.700190\pi$
$661$	−30.2487	−1.17654	−0.588269	+	0.808665i	$0.700190\pi$
$662$	−2.42114	−0.0941004
$663$	0	0
$664$	19.6410	0.762219
$665$	−5.21684	−0.202300
$666$	0	0
$667$	9.51666	0.368487
$668$	−5.21684	−0.201845
$669$	0	0
$670$	−35.5167	−1.37213
$671$	−18.7706	−0.724630
$672$	0	0
$673$	9.85641	0.379937	0.189968	−	0.981790i	$-0.439161\pi$
$673$	9.85641	0.379937	0.189968	+	0.981790i	$0.439161\pi$
$674$	−2.71654	−0.104637
$675$	0	0
$676$	−0.267949	−0.0103057
$677$	10.1383	0.389646	0.194823	−	0.980838i	$-0.437587\pi$
$677$	10.1383	0.389646	0.194823	+	0.980838i	$0.437587\pi$
$678$	0	0
$679$	−9.71281	−0.372744
$680$	0	0
$681$	0	0
$682$	41.8756	1.60350
$683$	17.5600	0.671915	0.335957	−	0.941877i	$-0.390940\pi$
$683$	17.5600	0.671915	0.335957	+	0.941877i	$0.390940\pi$
$684$	0	0
$685$	58.6410	2.24056
$686$	−23.6630	−0.903459
$687$	0	0
$688$	12.1962	0.464974
$689$	15.6505	0.596237
$690$	0	0
$691$	−32.0526	−1.21934	−0.609668	−	0.792657i	$-0.708697\pi$
$691$	−32.0526	−1.21934	−0.609668	+	0.792657i	$0.708697\pi$
$692$	−1.80137	−0.0684779
$693$	0	0
$694$	−20.6936	−0.785517
$695$	41.1439	1.56068
$696$	0	0
$697$	0	0
$698$	39.4217	1.49213
$699$	0	0
$700$	4.05256	0.153172
$701$	31.8127	1.20155	0.600774	−	0.799419i	$-0.294859\pi$
$701$	31.8127	1.20155	0.600774	+	0.799419i	$0.294859\pi$
$702$	0	0
$703$	−4.19615	−0.158261
$704$	17.3727	0.654759
$705$	0	0
$706$	−3.32051	−0.124969
$707$	−11.8315	−0.444970
$708$	0	0
$709$	29.7128	1.11589	0.557944	−	0.829879i	$-0.311590\pi$
$709$	29.7128	1.11589	0.557944	+	0.829879i	$0.311590\pi$
$710$	88.3117	3.31428
$711$	0	0
$712$	41.8756	1.56936
$713$	−11.7524	−0.440130
$714$	0	0
$715$	−37.1769	−1.39034
$716$	1.69325	0.0632796
$717$	0	0
$718$	−52.4449	−1.95722
$719$	−3.71087	−0.138392	−0.0691960	−	0.997603i	$-0.522043\pi$
$719$	−3.71087	−0.138392	−0.0691960	+	0.997603i	$0.522043\pi$
$720$	0	0
$721$	5.75129	0.214189
$722$	1.50597	0.0560464
$723$	0	0
$724$	−5.80385	−0.215698
$725$	−102.968	−3.82413
$726$	0	0
$727$	38.2487	1.41857	0.709283	−	0.704924i	$-0.249019\pi$
$727$	38.2487	1.41857	0.709283	+	0.704924i	$0.249019\pi$
$728$	11.4570	0.424624
$729$	0	0
$730$	99.9090	3.69780
$731$	0	0
$732$	0	0
$733$	−16.6603	−0.615361	−0.307680	−	0.951490i	$-0.599553\pi$
$733$	−16.6603	−0.615361	−0.307680	+	0.951490i	$0.599553\pi$
$734$	26.3004	0.970767
$735$	0	0
$736$	1.66025	0.0611978
$737$	−14.9516	−0.550749
$738$	0	0
$739$	−33.3731	−1.22765	−0.613824	−	0.789443i	$-0.710370\pi$
$739$	−33.3731	−1.22765	−0.613824	+	0.789443i	$0.710370\pi$
$740$	4.62604	0.170057
$741$	0	0
$742$	−8.62693	−0.316705
$743$	−23.2885	−0.854372	−0.427186	−	0.904164i	$-0.640495\pi$
$743$	−23.2885	−0.854372	−0.427186	+	0.904164i	$0.640495\pi$
$744$	0	0
$745$	37.1769	1.36206
$746$	2.50029	0.0915423
$747$	0	0
$748$	0	0
$749$	16.6738	0.609248
$750$	0	0
$751$	−28.5359	−1.04129	−0.520645	−	0.853773i	$-0.674308\pi$
$751$	−28.5359	−1.04129	−0.520645	+	0.853773i	$0.674308\pi$
$752$	20.1684	0.735467
$753$	0	0
$754$	45.0333	1.64002
$755$	28.8007	1.04817
$756$	0	0
$757$	−13.9282	−0.506229	−0.253115	−	0.967436i	$-0.581455\pi$
$757$	−13.9282	−0.506229	−0.253115	+	0.967436i	$0.581455\pi$
$758$	16.3494	0.593838
$759$	0	0
$760$	10.7321	0.389292
$761$	49.5889	1.79760	0.898799	−	0.438362i	$-0.144441\pi$
$761$	49.5889	1.79760	0.898799	+	0.438362i	$0.144441\pi$
$762$	0	0
$763$	−11.0718	−0.400826
$764$	2.09677	0.0758585
$765$	0	0
$766$	−19.8038	−0.715542
$767$	24.6863	0.891372
$768$	0	0
$769$	14.7846	0.533147	0.266573	−	0.963815i	$-0.414109\pi$
$769$	14.7846	0.533147	0.266573	+	0.963815i	$0.414109\pi$
$770$	20.4928	0.738509
$771$	0	0
$772$	2.67949	0.0964370
$773$	38.2401	1.37540	0.687700	−	0.725995i	$-0.258620\pi$
$773$	38.2401	1.37540	0.687700	+	0.725995i	$0.258620\pi$
$774$	0	0
$775$	127.158	4.56764
$776$	19.9811	0.717281
$777$	0	0
$778$	−32.1962	−1.15429
$779$	3.71087	0.132956
$780$	0	0
$781$	37.1769	1.33029
$782$	0	0
$783$	0	0
$784$	24.0718	0.859707
$785$	80.5945	2.87654
$786$	0	0
$787$	−35.9808	−1.28258	−0.641288	−	0.767300i	$-0.721600\pi$
$787$	−35.9808	−1.28258	−0.641288	+	0.767300i	$0.721600\pi$
$788$	−6.02388	−0.214592
$789$	0	0
$790$	−55.7654	−1.98404
$791$	−8.14963	−0.289767
$792$	0	0
$793$	24.9282	0.885226
$794$	−4.30167	−0.152660
$795$	0	0
$796$	5.21539	0.184855
$797$	−42.0591	−1.48981	−0.744904	−	0.667171i	$-0.767505\pi$
$797$	−42.0591	−1.48981	−0.744904	+	0.667171i	$0.767505\pi$
$798$	0	0
$799$	0	0
$800$	−17.9635	−0.635107
$801$	0	0
$802$	−6.19615	−0.218794
$803$	42.0591	1.48423
$804$	0	0
$805$	−5.75129	−0.202706
$806$	−55.6128	−1.95888
$807$	0	0
$808$	24.3397	0.856269
$809$	−28.8007	−1.01258	−0.506290	−	0.862363i	$-0.668983\pi$
$809$	−28.8007	−1.01258	−0.506290	+	0.862363i	$0.668983\pi$
$810$	0	0
$811$	−12.3205	−0.432632	−0.216316	−	0.976323i	$-0.569404\pi$
$811$	−12.3205	−0.432632	−0.216316	+	0.976323i	$0.569404\pi$
$812$	−2.93279	−0.102921
$813$	0	0
$814$	16.4833	0.577741
$815$	101.757	3.56440
$816$	0	0
$817$	2.73205	0.0955824
$818$	46.8724	1.63885
$819$	0	0
$820$	−4.09103	−0.142865
$821$	−27.9937	−0.976986	−0.488493	−	0.872568i	$-0.662453\pi$
$821$	−27.9937	−0.976986	−0.488493	+	0.872568i	$0.662453\pi$
$822$	0	0
$823$	16.3397	0.569568	0.284784	−	0.958592i	$-0.408078\pi$
$823$	16.3397	0.569568	0.284784	+	0.958592i	$0.408078\pi$
$824$	−11.8315	−0.412171
$825$	0	0
$826$	−13.6077	−0.473472
$827$	−15.9459	−0.554494	−0.277247	−	0.960799i	$-0.589422\pi$
$827$	−15.9459	−0.554494	−0.277247	+	0.960799i	$0.589422\pi$
$828$	0	0
$829$	39.8564	1.38427	0.692135	−	0.721768i	$-0.256670\pi$
$829$	39.8564	1.38427	0.692135	+	0.721768i	$0.256670\pi$
$830$	−46.6561	−1.61946
$831$	0	0
$832$	−23.0718	−0.799871
$833$	0	0
$834$	0	0
$835$	−80.1051	−2.77215
$836$	−0.698924	−0.0241728
$837$	0	0
$838$	−2.26795	−0.0783450
$839$	−19.4695	−0.672162	−0.336081	−	0.941833i	$-0.609102\pi$
$839$	−19.4695	−0.672162	−0.336081	+	0.941833i	$0.609102\pi$
$840$	0	0
$841$	45.5167	1.56954
$842$	−2.20489	−0.0759857
$843$	0	0
$844$	−2.71281	−0.0933789
$845$	−4.11439	−0.141539
$846$	0	0
$847$	−5.32051	−0.182815
$848$	20.1684	0.692587
$849$	0	0
$850$	0	0
$851$	−4.62604	−0.158579
$852$	0	0
$853$	−32.7128	−1.12007	−0.560033	−	0.828471i	$-0.689211\pi$
$853$	−32.7128	−1.12007	−0.560033	+	0.828471i	$0.689211\pi$
$854$	−13.7410	−0.470208
$855$	0	0
$856$	−34.3013	−1.17239
$857$	50.5832	1.72789	0.863945	−	0.503585i	$-0.167986\pi$
$857$	50.5832	1.72789	0.863945	+	0.503585i	$0.167986\pi$
$858$	0	0
$859$	12.3923	0.422820	0.211410	−	0.977397i	$-0.432194\pi$
$859$	12.3923	0.422820	0.211410	+	0.977397i	$0.432194\pi$
$860$	−3.01194	−0.102706
$861$	0	0
$862$	−26.4449	−0.900716
$863$	−21.6744	−0.737805	−0.368903	−	0.929468i	$-0.620266\pi$
$863$	−21.6744	−0.737805	−0.368903	+	0.929468i	$0.620266\pi$
$864$	0	0
$865$	−27.6603	−0.940477
$866$	17.3438	0.589365
$867$	0	0
$868$	3.62178	0.122931
$869$	−23.4758	−0.796361
$870$	0	0
$871$	19.8564	0.672809
$872$	22.7768	0.771321
$873$	0	0
$874$	1.66025	0.0561589
$875$	36.1433	1.22187
$876$	0	0
$877$	9.26795	0.312956	0.156478	−	0.987681i	$-0.449986\pi$
$877$	9.26795	0.312956	0.156478	+	0.987681i	$0.449986\pi$
$878$	−20.7592	−0.700590
$879$	0	0
$880$	−47.9090	−1.61501
$881$	−29.0961	−0.980273	−0.490137	−	0.871646i	$-0.663053\pi$
$881$	−29.0961	−0.980273	−0.490137	+	0.871646i	$0.663053\pi$
$882$	0	0
$883$	43.5167	1.46445	0.732226	−	0.681062i	$-0.238481\pi$
$883$	43.5167	1.46445	0.732226	+	0.681062i	$0.238481\pi$
$884$	0	0
$885$	0	0
$886$	−12.5551	−0.421798
$887$	38.6436	1.29753	0.648763	−	0.760991i	$-0.275287\pi$
$887$	38.6436	1.29753	0.648763	+	0.760991i	$0.275287\pi$
$888$	0	0
$889$	13.8564	0.464729
$890$	−99.4732	−3.33435
$891$	0	0
$892$	4.16283	0.139382
$893$	4.51791	0.151186
$894$	0	0
$895$	26.0000	0.869084
$896$	16.5367	0.552453
$897$	0	0
$898$	−17.0910	−0.570335
$899$	−92.0225	−3.06912
$900$	0	0
$901$	0	0
$902$	−14.5770	−0.485362
$903$	0	0
$904$	16.7654	0.557608
$905$	−89.1187	−2.96241
$906$	0	0
$907$	40.2487	1.33644	0.668218	−	0.743965i	$-0.267057\pi$
$907$	40.2487	1.33644	0.668218	+	0.743965i	$0.267057\pi$
$908$	1.39785	0.0463892
$909$	0	0
$910$	−27.2154	−0.902181
$911$	−39.2344	−1.29989	−0.649947	−	0.759980i	$-0.725209\pi$
$911$	−39.2344	−1.29989	−0.649947	+	0.759980i	$0.725209\pi$
$912$	0	0
$913$	−19.6410	−0.650023
$914$	−23.4758	−0.776509
$915$	0	0
$916$	5.44486	0.179903
$917$	6.24013	0.206067
$918$	0	0
$919$	28.8756	0.952520	0.476260	−	0.879305i	$-0.341992\pi$
$919$	28.8756	0.952520	0.476260	+	0.879305i	$0.341992\pi$
$920$	11.8315	0.390074
$921$	0	0
$922$	−15.7128	−0.517474
$923$	−49.3727	−1.62512
$924$	0	0
$925$	50.0526	1.64572
$926$	−3.60274	−0.118393
$927$	0	0
$928$	13.0000	0.426746
$929$	−33.4268	−1.09670	−0.548348	−	0.836250i	$-0.684743\pi$
$929$	−33.4268	−1.09670	−0.548348	+	0.836250i	$0.684743\pi$
$930$	0	0
$931$	5.39230	0.176726
$932$	2.50029	0.0818999
$933$	0	0
$934$	53.0526	1.73593
$935$	0	0
$936$	0	0
$937$	−25.9282	−0.847037	−0.423519	−	0.905887i	$-0.639205\pi$
$937$	−25.9282	−0.847037	−0.423519	+	0.905887i	$0.639205\pi$
$938$	−10.9453	−0.357377
$939$	0	0
$940$	−4.98076	−0.162455
$941$	11.5361	0.376067	0.188033	−	0.982163i	$-0.439789\pi$
$941$	11.5361	0.376067	0.188033	+	0.982163i	$0.439789\pi$
$942$	0	0
$943$	4.09103	0.133222
$944$	31.8127	1.03541
$945$	0	0
$946$	−10.7321	−0.348929
$947$	−57.4142	−1.86571	−0.932855	−	0.360252i	$-0.882691\pi$
$947$	−57.4142	−1.86571	−0.932855	+	0.360252i	$0.882691\pi$
$948$	0	0
$949$	−55.8564	−1.81318
$950$	−17.9635	−0.582814
$951$	0	0
$952$	0	0
$953$	8.92770	0.289197	0.144598	−	0.989490i	$-0.453811\pi$
$953$	8.92770	0.289197	0.144598	+	0.989490i	$0.453811\pi$
$954$	0	0
$955$	32.1962	1.04184
$956$	−0.187276	−0.00605694
$957$	0	0
$958$	9.51666	0.307469
$959$	18.0717	0.583564
$960$	0	0
$961$	82.6410	2.66584
$962$	−21.8906	−0.705783
$963$	0	0
$964$	−0.588457	−0.0189529
$965$	41.1439	1.32447
$966$	0	0
$967$	9.41154	0.302655	0.151327	−	0.988484i	$-0.451645\pi$
$967$	9.41154	0.302655	0.151327	+	0.988484i	$0.451645\pi$
$968$	10.9453	0.351796
$969$	0	0
$970$	−47.4641	−1.52398
$971$	23.2885	0.747363	0.373682	−	0.927557i	$-0.378095\pi$
$971$	23.2885	0.747363	0.373682	+	0.927557i	$0.378095\pi$
$972$	0	0
$973$	12.6795	0.406486
$974$	43.1615	1.38298
$975$	0	0
$976$	32.1244	1.02828
$977$	−20.1684	−0.645245	−0.322623	−	0.946528i	$-0.604564\pi$
$977$	−20.1684	−0.645245	−0.322623	+	0.946528i	$0.604564\pi$
$978$	0	0
$979$	−41.8756	−1.33835
$980$	−5.94473	−0.189898
$981$	0	0
$982$	−21.3013	−0.679751
$983$	35.4154	1.12958	0.564788	−	0.825236i	$-0.308958\pi$
$983$	35.4154	1.12958	0.564788	+	0.825236i	$0.308958\pi$
$984$	0	0
$985$	−92.4974	−2.94721
$986$	0	0
$987$	0	0
$988$	0.928203	0.0295301
$989$	3.01194	0.0957742
$990$	0	0
$991$	−17.4641	−0.554765	−0.277383	−	0.960760i	$-0.589467\pi$
$991$	−17.4641	−0.554765	−0.277383	+	0.960760i	$0.589467\pi$
$992$	−16.0540	−0.509716
$993$	0	0
$994$	27.2154	0.863220
$995$	80.0829	2.53880
$996$	0	0
$997$	32.9282	1.04285	0.521423	−	0.853298i	$-0.325401\pi$
$997$	32.9282	1.04285	0.521423	+	0.853298i	$0.325401\pi$
$998$	−9.62663	−0.304726
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.a.i.1.3	yes	4
3.2	odd	2	inner	513.2.a.i.1.2	✓	4
4.3	odd	2		8208.2.a.bt.1.4		4
12.11	even	2		8208.2.a.bt.1.1		4
19.18	odd	2		9747.2.a.bf.1.2		4
57.56	even	2		9747.2.a.bf.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.i.1.2	✓	4	3.2	odd	2	inner
513.2.a.i.1.3	yes	4	1.1	even	1	trivial
8208.2.a.bt.1.1		4	12.11	even	2
8208.2.a.bt.1.4		4	4.3	odd	2
9747.2.a.bf.1.2		4	19.18	odd	2
9747.2.a.bf.1.3		4	57.56	even	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:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.a (trivial)

\(f(q)\)	\(=\)	\(q+1.50597 q^{2} +0.267949 q^{4} +4.11439 q^{5} +1.26795 q^{7} -2.60842 q^{8} +6.19615 q^{10} +2.60842 q^{11} -3.46410 q^{13} +1.90949 q^{14} -4.46410 q^{16} -1.00000 q^{19} +1.10245 q^{20} +3.92820 q^{22} -1.10245 q^{23} +11.9282 q^{25} -5.21684 q^{26} +0.339746 q^{28} -8.63230 q^{29} +10.6603 q^{31} -1.50597 q^{32} +5.21684 q^{35} +4.19615 q^{37} -1.50597 q^{38} -10.7321 q^{40} -3.71087 q^{41} -2.73205 q^{43} +0.698924 q^{44} -1.66025 q^{46} -4.51791 q^{47} -5.39230 q^{49} +17.9635 q^{50} -0.928203 q^{52} -4.51791 q^{53} +10.7321 q^{55} -3.30734 q^{56} -13.0000 q^{58} -7.12633 q^{59} -7.19615 q^{61} +16.0540 q^{62} +6.66025 q^{64} -14.2527 q^{65} -5.73205 q^{67} +7.85641 q^{70} +14.2527 q^{71} +16.1244 q^{73} +6.31928 q^{74} -0.267949 q^{76} +3.30734 q^{77} -9.00000 q^{79} -18.3671 q^{80} -5.58846 q^{82} -7.52986 q^{83} -4.11439 q^{86} -6.80385 q^{88} -16.0540 q^{89} -4.39230 q^{91} -0.295400 q^{92} -6.80385 q^{94} -4.11439 q^{95} -7.66025 q^{97} -8.12066 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} + 12 q^{7} + 4 q^{10} - 4 q^{16} - 4 q^{19} - 12 q^{22} + 20 q^{25} + 36 q^{28} + 8 q^{31} - 4 q^{37} - 36 q^{40} - 4 q^{43} + 28 q^{46} + 20 q^{49} + 24 q^{52} + 36 q^{55} - 52 q^{58} - 8 q^{61}+ \cdots + 4 q^{97}+O(q^{100}) \) 4 * q + 8 * q^4 + 12 * q^7 + 4 * q^10 - 4 * q^16 - 4 * q^19 - 12 * q^22 + 20 * q^25 + 36 * q^28 + 8 * q^31 - 4 * q^37 - 36 * q^40 - 4 * q^43 + 28 * q^46 + 20 * q^49 + 24 * q^52 + 36 * q^55 - 52 * q^58 - 8 * q^61 - 8 * q^64 - 16 * q^67 - 24 * q^70 + 16 * q^73 - 8 * q^76 - 36 * q^79 + 40 * q^82 - 48 * q^88 + 24 * q^91 - 48 * q^94 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more