LMFDB - Embedded newform 8048.2.a.u.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.32813 q^{3} +1.62138 q^{5} -3.43484 q^{7} +2.42020 q^{9} +O(q^{10})$	$q-2.32813 q^{3} +1.62138 q^{5} -3.43484 q^{7} +2.42020 q^{9} +0.695688 q^{11} +2.78704 q^{13} -3.77480 q^{15} -6.87472 q^{17} +1.23169 q^{19} +7.99676 q^{21} -8.03473 q^{23} -2.37111 q^{25} +1.34986 q^{27} -8.02844 q^{29} -9.59158 q^{31} -1.61965 q^{33} -5.56920 q^{35} -0.0454896 q^{37} -6.48860 q^{39} +10.0330 q^{41} +10.6292 q^{43} +3.92407 q^{45} -1.72424 q^{47} +4.79813 q^{49} +16.0053 q^{51} -6.66671 q^{53} +1.12798 q^{55} -2.86753 q^{57} +7.83234 q^{59} +8.48076 q^{61} -8.31299 q^{63} +4.51887 q^{65} -0.128627 q^{67} +18.7059 q^{69} -9.36773 q^{71} +11.6801 q^{73} +5.52026 q^{75} -2.38958 q^{77} -9.03774 q^{79} -10.4032 q^{81} +0.939830 q^{83} -11.1466 q^{85} +18.6913 q^{87} -16.8653 q^{89} -9.57305 q^{91} +22.3305 q^{93} +1.99704 q^{95} +10.0789 q^{97} +1.68370 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.32813	−1.34415	−0.672074	−	0.740484i	$-0.734596\pi$
$3$	−2.32813	−1.34415	−0.672074	+	0.740484i	$0.734596\pi$
$4$	0	0
$5$	1.62138	0.725105	0.362553	−	0.931963i	$-0.381905\pi$
$5$	1.62138	0.725105	0.362553	+	0.931963i	$0.381905\pi$
$6$	0	0
$7$	−3.43484	−1.29825	−0.649124	−	0.760683i	$-0.724864\pi$
$7$	−3.43484	−1.29825	−0.649124	+	0.760683i	$0.724864\pi$
$8$	0	0
$9$	2.42020	0.806732
$10$	0	0
$11$	0.695688	0.209758	0.104879	−	0.994485i	$-0.466554\pi$
$11$	0.695688	0.209758	0.104879	+	0.994485i	$0.466554\pi$
$12$	0	0
$13$	2.78704	0.772987	0.386493	−	0.922292i	$-0.373686\pi$
$13$	2.78704	0.772987	0.386493	+	0.922292i	$0.373686\pi$
$14$	0	0
$15$	−3.77480	−0.974649
$16$	0	0
$17$	−6.87472	−1.66736	−0.833682	−	0.552244i	$-0.813772\pi$
$17$	−6.87472	−1.66736	−0.833682	+	0.552244i	$0.813772\pi$
$18$	0	0
$19$	1.23169	0.282569	0.141284	−	0.989969i	$-0.454877\pi$
$19$	1.23169	0.282569	0.141284	+	0.989969i	$0.454877\pi$
$20$	0	0
$21$	7.99676	1.74504
$22$	0	0
$23$	−8.03473	−1.67536	−0.837678	−	0.546164i	$-0.816087\pi$
$23$	−8.03473	−1.67536	−0.837678	+	0.546164i	$0.816087\pi$
$24$	0	0
$25$	−2.37111	−0.474222
$26$	0	0
$27$	1.34986	0.259780
$28$	0	0
$29$	−8.02844	−1.49084	−0.745422	−	0.666593i	$-0.767752\pi$
$29$	−8.02844	−1.49084	−0.745422	+	0.666593i	$0.767752\pi$
$30$	0	0
$31$	−9.59158	−1.72270	−0.861349	−	0.508013i	$-0.830380\pi$
$31$	−9.59158	−1.72270	−0.861349	+	0.508013i	$0.830380\pi$
$32$	0	0
$33$	−1.61965	−0.281946
$34$	0	0
$35$	−5.56920	−0.941366
$36$	0	0
$37$	−0.0454896	−0.00747845	−0.00373922	−	0.999993i	$-0.501190\pi$
$37$	−0.0454896	−0.00747845	−0.00373922	+	0.999993i	$0.501190\pi$
$38$	0	0
$39$	−6.48860	−1.03901
$40$	0	0
$41$	10.0330	1.56690	0.783448	−	0.621457i	$-0.213459\pi$
$41$	10.0330	1.56690	0.783448	+	0.621457i	$0.213459\pi$
$42$	0	0
$43$	10.6292	1.62094	0.810468	−	0.585782i	$-0.199213\pi$
$43$	10.6292	1.62094	0.810468	+	0.585782i	$0.199213\pi$
$44$	0	0
$45$	3.92407	0.584966
$46$	0	0
$47$	−1.72424	−0.251506	−0.125753	−	0.992062i	$-0.540135\pi$
$47$	−1.72424	−0.251506	−0.125753	+	0.992062i	$0.540135\pi$
$48$	0	0
$49$	4.79813	0.685447
$50$	0	0
$51$	16.0053	2.24118
$52$	0	0
$53$	−6.66671	−0.915743	−0.457871	−	0.889018i	$-0.651388\pi$
$53$	−6.66671	−0.915743	−0.457871	+	0.889018i	$0.651388\pi$
$54$	0	0
$55$	1.12798	0.152097
$56$	0	0
$57$	−2.86753	−0.379814
$58$	0	0
$59$	7.83234	1.01968	0.509842	−	0.860268i	$-0.329704\pi$
$59$	7.83234	1.01968	0.509842	+	0.860268i	$0.329704\pi$
$60$	0	0
$61$	8.48076	1.08585	0.542925	−	0.839781i	$-0.317317\pi$
$61$	8.48076	1.08585	0.542925	+	0.839781i	$0.317317\pi$
$62$	0	0
$63$	−8.31299	−1.04734
$64$	0	0
$65$	4.51887	0.560497
$66$	0	0
$67$	−0.128627	−0.0157143	−0.00785715	−	0.999969i	$-0.502501\pi$
$67$	−0.128627	−0.0157143	−0.00785715	+	0.999969i	$0.502501\pi$
$68$	0	0
$69$	18.7059	2.25193
$70$	0	0
$71$	−9.36773	−1.11174	−0.555872	−	0.831268i	$-0.687616\pi$
$71$	−9.36773	−1.11174	−0.555872	+	0.831268i	$0.687616\pi$
$72$	0	0
$73$	11.6801	1.36705	0.683526	−	0.729926i	$-0.260446\pi$
$73$	11.6801	1.36705	0.683526	+	0.729926i	$0.260446\pi$
$74$	0	0
$75$	5.52026	0.637425
$76$	0	0
$77$	−2.38958	−0.272318
$78$	0	0
$79$	−9.03774	−1.01682	−0.508412	−	0.861114i	$-0.669767\pi$
$79$	−9.03774	−1.01682	−0.508412	+	0.861114i	$0.669767\pi$
$80$	0	0
$81$	−10.4032	−1.15592
$82$	0	0
$83$	0.939830	0.103160	0.0515799	−	0.998669i	$-0.483574\pi$
$83$	0.939830	0.103160	0.0515799	+	0.998669i	$0.483574\pi$
$84$	0	0
$85$	−11.1466	−1.20902
$86$	0	0
$87$	18.6913	2.00391
$88$	0	0
$89$	−16.8653	−1.78772	−0.893861	−	0.448344i	$-0.852014\pi$
$89$	−16.8653	−1.78772	−0.893861	+	0.448344i	$0.852014\pi$
$90$	0	0
$91$	−9.57305	−1.00353
$92$	0	0
$93$	22.3305	2.31556
$94$	0	0
$95$	1.99704	0.204892
$96$	0	0
$97$	10.0789	1.02336	0.511678	−	0.859177i	$-0.329024\pi$
$97$	10.0789	1.02336	0.511678	+	0.859177i	$0.329024\pi$
$98$	0	0
$99$	1.68370	0.169219
$100$	0	0
$101$	−16.9764	−1.68921	−0.844607	−	0.535386i	$-0.820166\pi$
$101$	−16.9764	−1.68921	−0.844607	+	0.535386i	$0.820166\pi$
$102$	0	0
$103$	3.81703	0.376103	0.188052	−	0.982159i	$-0.439783\pi$
$103$	3.81703	0.376103	0.188052	+	0.982159i	$0.439783\pi$
$104$	0	0
$105$	12.9658	1.26534
$106$	0	0
$107$	12.3085	1.18991	0.594953	−	0.803761i	$-0.297171\pi$
$107$	12.3085	1.18991	0.594953	+	0.803761i	$0.297171\pi$
$108$	0	0
$109$	0.884124	0.0846837	0.0423419	−	0.999103i	$-0.486518\pi$
$109$	0.884124	0.0846837	0.0423419	+	0.999103i	$0.486518\pi$
$110$	0	0
$111$	0.105906	0.0100521
$112$	0	0
$113$	−14.4313	−1.35758	−0.678789	−	0.734333i	$-0.737495\pi$
$113$	−14.4313	−1.35758	−0.678789	+	0.734333i	$0.737495\pi$
$114$	0	0
$115$	−13.0274	−1.21481
$116$	0	0
$117$	6.74519	0.623593
$118$	0	0
$119$	23.6136	2.16465
$120$	0	0
$121$	−10.5160	−0.956002
$122$	0	0
$123$	−23.3582	−2.10614
$124$	0	0
$125$	−11.9514	−1.06897
$126$	0	0
$127$	12.0092	1.06564	0.532821	−	0.846228i	$-0.321132\pi$
$127$	12.0092	1.06564	0.532821	+	0.846228i	$0.321132\pi$
$128$	0	0
$129$	−24.7462	−2.17878
$130$	0	0
$131$	−8.12927	−0.710258	−0.355129	−	0.934817i	$-0.615563\pi$
$131$	−8.12927	−0.710258	−0.355129	+	0.934817i	$0.615563\pi$
$132$	0	0
$133$	−4.23065	−0.366844
$134$	0	0
$135$	2.18864	0.188368
$136$	0	0
$137$	−1.79780	−0.153596	−0.0767981	−	0.997047i	$-0.524470\pi$
$137$	−1.79780	−0.153596	−0.0767981	+	0.997047i	$0.524470\pi$
$138$	0	0
$139$	2.97405	0.252256	0.126128	−	0.992014i	$-0.459745\pi$
$139$	2.97405	0.252256	0.126128	+	0.992014i	$0.459745\pi$
$140$	0	0
$141$	4.01425	0.338061
$142$	0	0
$143$	1.93891	0.162140
$144$	0	0
$145$	−13.0172	−1.08102
$146$	0	0
$147$	−11.1707	−0.921342
$148$	0	0
$149$	−0.000281231 0	−2.30393e−5 0	−1.15197e−5	−	1.00000i	$-0.500004\pi$
$149$	−0.000281231 0	−2.30393e−5 0	−1.15197e−5		1.00000i	$0.500004\pi$
$150$	0	0
$151$	−5.88133	−0.478615	−0.239308	−	0.970944i	$-0.576920\pi$
$151$	−5.88133	−0.478615	−0.239308	+	0.970944i	$0.576920\pi$
$152$	0	0
$153$	−16.6382	−1.34512
$154$	0	0
$155$	−15.5516	−1.24914
$156$	0	0
$157$	9.54018	0.761390	0.380695	−	0.924701i	$-0.375685\pi$
$157$	9.54018	0.761390	0.380695	+	0.924701i	$0.375685\pi$
$158$	0	0
$159$	15.5210	1.23089
$160$	0	0
$161$	27.5980	2.17503
$162$	0	0
$163$	1.44103	0.112870	0.0564351	−	0.998406i	$-0.482027\pi$
$163$	1.44103	0.112870	0.0564351	+	0.998406i	$0.482027\pi$
$164$	0	0
$165$	−2.62608	−0.204440
$166$	0	0
$167$	11.4521	0.886186	0.443093	−	0.896476i	$-0.353881\pi$
$167$	11.4521	0.886186	0.443093	+	0.896476i	$0.353881\pi$
$168$	0	0
$169$	−5.23239	−0.402492
$170$	0	0
$171$	2.98093	0.227957
$172$	0	0
$173$	−9.03758	−0.687114	−0.343557	−	0.939132i	$-0.611632\pi$
$173$	−9.03758	−0.687114	−0.343557	+	0.939132i	$0.611632\pi$
$174$	0	0
$175$	8.14439	0.615658
$176$	0	0
$177$	−18.2347	−1.37061
$178$	0	0
$179$	−8.49163	−0.634694	−0.317347	−	0.948309i	$-0.602792\pi$
$179$	−8.49163	−0.634694	−0.317347	+	0.948309i	$0.602792\pi$
$180$	0	0
$181$	1.18999	0.0884509	0.0442255	−	0.999022i	$-0.485918\pi$
$181$	1.18999	0.0884509	0.0442255	+	0.999022i	$0.485918\pi$
$182$	0	0
$183$	−19.7443	−1.45954
$184$	0	0
$185$	−0.0737562	−0.00542266
$186$	0	0
$187$	−4.78266	−0.349743
$188$	0	0
$189$	−4.63654	−0.337259
$190$	0	0
$191$	−17.6433	−1.27662	−0.638311	−	0.769778i	$-0.720367\pi$
$191$	−17.6433	−1.27662	−0.638311	+	0.769778i	$0.720367\pi$
$192$	0	0
$193$	17.2303	1.24026	0.620132	−	0.784497i	$-0.287079\pi$
$193$	17.2303	1.24026	0.620132	+	0.784497i	$0.287079\pi$
$194$	0	0
$195$	−10.5205	−0.753390
$196$	0	0
$197$	−7.14985	−0.509406	−0.254703	−	0.967019i	$-0.581978\pi$
$197$	−7.14985	−0.509406	−0.254703	+	0.967019i	$0.581978\pi$
$198$	0	0
$199$	7.14453	0.506462	0.253231	−	0.967406i	$-0.418507\pi$
$199$	7.14453	0.506462	0.253231	+	0.967406i	$0.418507\pi$
$200$	0	0
$201$	0.299461	0.0211223
$202$	0	0
$203$	27.5764	1.93549
$204$	0	0
$205$	16.2674	1.13616
$206$	0	0
$207$	−19.4456	−1.35156
$208$	0	0
$209$	0.856871	0.0592710
$210$	0	0
$211$	28.6407	1.97171	0.985855	−	0.167602i	$-0.0536022\pi$
$211$	28.6407	1.97171	0.985855	+	0.167602i	$0.0536022\pi$
$212$	0	0
$213$	21.8093	1.49435
$214$	0	0
$215$	17.2340	1.17535
$216$	0	0
$217$	32.9455	2.23649
$218$	0	0
$219$	−27.1928	−1.83752
$220$	0	0
$221$	−19.1601	−1.28885
$222$	0	0
$223$	−19.3571	−1.29624	−0.648122	−	0.761536i	$-0.724445\pi$
$223$	−19.3571	−1.29624	−0.648122	+	0.761536i	$0.724445\pi$
$224$	0	0
$225$	−5.73856	−0.382570
$226$	0	0
$227$	7.85883	0.521609	0.260804	−	0.965392i	$-0.416012\pi$
$227$	7.85883	0.521609	0.260804	+	0.965392i	$0.416012\pi$
$228$	0	0
$229$	6.47613	0.427955	0.213977	−	0.976839i	$-0.431358\pi$
$229$	6.47613	0.427955	0.213977	+	0.976839i	$0.431358\pi$
$230$	0	0
$231$	5.56325	0.366035
$232$	0	0
$233$	27.3747	1.79337	0.896687	−	0.442664i	$-0.145967\pi$
$233$	27.3747	1.79337	0.896687	+	0.442664i	$0.145967\pi$
$234$	0	0
$235$	−2.79565	−0.182368
$236$	0	0
$237$	21.0410	1.36676
$238$	0	0
$239$	−5.27155	−0.340988	−0.170494	−	0.985359i	$-0.554536\pi$
$239$	−5.27155	−0.340988	−0.170494	+	0.985359i	$0.554536\pi$
$240$	0	0
$241$	−1.48633	−0.0957427	−0.0478714	−	0.998854i	$-0.515244\pi$
$241$	−1.48633	−0.0957427	−0.0478714	+	0.998854i	$0.515244\pi$
$242$	0	0
$243$	20.1705	1.29394
$244$	0	0
$245$	7.77961	0.497021
$246$	0	0
$247$	3.43277	0.218422
$248$	0	0
$249$	−2.18805	−0.138662
$250$	0	0
$251$	−0.130529	−0.00823890	−0.00411945	−	0.999992i	$-0.501311\pi$
$251$	−0.130529	−0.00823890	−0.00411945	+	0.999992i	$0.501311\pi$
$252$	0	0
$253$	−5.58967	−0.351419
$254$	0	0
$255$	25.9507	1.62509
$256$	0	0
$257$	16.6727	1.04001	0.520007	−	0.854162i	$-0.325929\pi$
$257$	16.6727	1.04001	0.520007	+	0.854162i	$0.325929\pi$
$258$	0	0
$259$	0.156250	0.00970888
$260$	0	0
$261$	−19.4304	−1.20271
$262$	0	0
$263$	25.3669	1.56419	0.782094	−	0.623160i	$-0.214151\pi$
$263$	25.3669	1.56419	0.782094	+	0.623160i	$0.214151\pi$
$264$	0	0
$265$	−10.8093	−0.664010
$266$	0	0
$267$	39.2647	2.40296
$268$	0	0
$269$	27.5757	1.68132	0.840661	−	0.541561i	$-0.182167\pi$
$269$	27.5757	1.68132	0.840661	+	0.541561i	$0.182167\pi$
$270$	0	0
$271$	−8.30935	−0.504757	−0.252379	−	0.967629i	$-0.581213\pi$
$271$	−8.30935	−0.504757	−0.252379	+	0.967629i	$0.581213\pi$
$272$	0	0
$273$	22.2873	1.34889
$274$	0	0
$275$	−1.64955	−0.0994719
$276$	0	0
$277$	−14.4793	−0.869974	−0.434987	−	0.900437i	$-0.643247\pi$
$277$	−14.4793	−0.869974	−0.434987	+	0.900437i	$0.643247\pi$
$278$	0	0
$279$	−23.2135	−1.38976
$280$	0	0
$281$	−25.7545	−1.53639	−0.768193	−	0.640219i	$-0.778844\pi$
$281$	−25.7545	−1.53639	−0.768193	+	0.640219i	$0.778844\pi$
$282$	0	0
$283$	0.0245165	0.00145735	0.000728677	−	1.00000i	$-0.499768\pi$
$283$	0.0245165	0.00145735	0.000728677		1.00000i	$0.499768\pi$
$284$	0	0
$285$	−4.64937	−0.275405
$286$	0	0
$287$	−34.4619	−2.03422
$288$	0	0
$289$	30.2618	1.78011
$290$	0	0
$291$	−23.4650	−1.37554
$292$	0	0
$293$	2.58192	0.150837	0.0754186	−	0.997152i	$-0.475971\pi$
$293$	2.58192	0.150837	0.0754186	+	0.997152i	$0.475971\pi$
$294$	0	0
$295$	12.6992	0.739378
$296$	0	0
$297$	0.939080	0.0544909
$298$	0	0
$299$	−22.3931	−1.29503
$300$	0	0
$301$	−36.5096	−2.10438
$302$	0	0
$303$	39.5233	2.27055
$304$	0	0
$305$	13.7506	0.787355
$306$	0	0
$307$	12.9548	0.739367	0.369683	−	0.929158i	$-0.379466\pi$
$307$	12.9548	0.739367	0.369683	+	0.929158i	$0.379466\pi$
$308$	0	0
$309$	−8.88655	−0.505538
$310$	0	0
$311$	8.10729	0.459722	0.229861	−	0.973223i	$-0.426173\pi$
$311$	8.10729	0.459722	0.229861	+	0.973223i	$0.426173\pi$
$312$	0	0
$313$	10.0345	0.567183	0.283592	−	0.958945i	$-0.408474\pi$
$313$	10.0345	0.567183	0.283592	+	0.958945i	$0.408474\pi$
$314$	0	0
$315$	−13.4786	−0.759431
$316$	0	0
$317$	34.0789	1.91406	0.957030	−	0.289990i	$-0.0936518\pi$
$317$	34.0789	1.91406	0.957030	+	0.289990i	$0.0936518\pi$
$318$	0	0
$319$	−5.58529	−0.312716
$320$	0	0
$321$	−28.6557	−1.59941
$322$	0	0
$323$	−8.46751	−0.471145
$324$	0	0
$325$	−6.60839	−0.366567
$326$	0	0
$327$	−2.05836	−0.113827
$328$	0	0
$329$	5.92248	0.326517
$330$	0	0
$331$	−8.68978	−0.477634	−0.238817	−	0.971065i	$-0.576760\pi$
$331$	−8.68978	−0.477634	−0.238817	+	0.971065i	$0.576760\pi$
$332$	0	0
$333$	−0.110094	−0.00603311
$334$	0	0
$335$	−0.208554	−0.0113945
$336$	0	0
$337$	−24.8121	−1.35160	−0.675799	−	0.737086i	$-0.736201\pi$
$337$	−24.8121	−1.35160	−0.675799	+	0.737086i	$0.736201\pi$
$338$	0	0
$339$	33.5979	1.82479
$340$	0	0
$341$	−6.67275	−0.361350
$342$	0	0
$343$	7.56308	0.408368
$344$	0	0
$345$	30.3295	1.63288
$346$	0	0
$347$	−14.3599	−0.770877	−0.385439	−	0.922733i	$-0.625950\pi$
$347$	−14.3599	−0.770877	−0.385439	+	0.922733i	$0.625950\pi$
$348$	0	0
$349$	1.08809	0.0582443	0.0291222	−	0.999576i	$-0.490729\pi$
$349$	1.08809	0.0582443	0.0291222	+	0.999576i	$0.490729\pi$
$350$	0	0
$351$	3.76211	0.200807
$352$	0	0
$353$	20.6043	1.09666	0.548329	−	0.836262i	$-0.315264\pi$
$353$	20.6043	1.09666	0.548329	+	0.836262i	$0.315264\pi$
$354$	0	0
$355$	−15.1887	−0.806132
$356$	0	0
$357$	−54.9755	−2.90961
$358$	0	0
$359$	20.5135	1.08266	0.541330	−	0.840810i	$-0.317921\pi$
$359$	20.5135	1.08266	0.541330	+	0.840810i	$0.317921\pi$
$360$	0	0
$361$	−17.4829	−0.920155
$362$	0	0
$363$	24.4827	1.28501
$364$	0	0
$365$	18.9379	0.991257
$366$	0	0
$367$	−25.5549	−1.33395	−0.666976	−	0.745079i	$-0.732412\pi$
$367$	−25.5549	−1.33395	−0.666976	+	0.745079i	$0.732412\pi$
$368$	0	0
$369$	24.2819	1.26407
$370$	0	0
$371$	22.8991	1.18886
$372$	0	0
$373$	0.136423	0.00706371	0.00353186	−	0.999994i	$-0.498876\pi$
$373$	0.136423	0.00706371	0.00353186	+	0.999994i	$0.498876\pi$
$374$	0	0
$375$	27.8245	1.43685
$376$	0	0
$377$	−22.3756	−1.15240
$378$	0	0
$379$	−4.01675	−0.206327	−0.103163	−	0.994664i	$-0.532896\pi$
$379$	−4.01675	−0.206327	−0.103163	+	0.994664i	$0.532896\pi$
$380$	0	0
$381$	−27.9589	−1.43238
$382$	0	0
$383$	21.1948	1.08301	0.541503	−	0.840699i	$-0.317856\pi$
$383$	21.1948	1.08301	0.541503	+	0.840699i	$0.317856\pi$
$384$	0	0
$385$	−3.87443	−0.197459
$386$	0	0
$387$	25.7247	1.30766
$388$	0	0
$389$	−11.4478	−0.580427	−0.290213	−	0.956962i	$-0.593726\pi$
$389$	−11.4478	−0.580427	−0.290213	+	0.956962i	$0.593726\pi$
$390$	0	0
$391$	55.2365	2.79343
$392$	0	0
$393$	18.9260	0.954691
$394$	0	0
$395$	−14.6536	−0.737305
$396$	0	0
$397$	35.4863	1.78101	0.890503	−	0.454977i	$-0.150352\pi$
$397$	35.4863	1.78101	0.890503	+	0.454977i	$0.150352\pi$
$398$	0	0
$399$	9.84951	0.493092
$400$	0	0
$401$	7.10338	0.354726	0.177363	−	0.984146i	$-0.443243\pi$
$401$	7.10338	0.354726	0.177363	+	0.984146i	$0.443243\pi$
$402$	0	0
$403$	−26.7321	−1.33162
$404$	0	0
$405$	−16.8677	−0.838160
$406$	0	0
$407$	−0.0316466	−0.00156866
$408$	0	0
$409$	16.6490	0.823241	0.411621	−	0.911355i	$-0.364963\pi$
$409$	16.6490	0.823241	0.411621	+	0.911355i	$0.364963\pi$
$410$	0	0
$411$	4.18551	0.206456
$412$	0	0
$413$	−26.9028	−1.32380
$414$	0	0
$415$	1.52383	0.0748017
$416$	0	0
$417$	−6.92398	−0.339069
$418$	0	0
$419$	−18.7726	−0.917100	−0.458550	−	0.888668i	$-0.651631\pi$
$419$	−18.7726	−0.917100	−0.458550	+	0.888668i	$0.651631\pi$
$420$	0	0
$421$	−2.14317	−0.104452	−0.0522259	−	0.998635i	$-0.516632\pi$
$421$	−2.14317	−0.104452	−0.0522259	+	0.998635i	$0.516632\pi$
$422$	0	0
$423$	−4.17299	−0.202898
$424$	0	0
$425$	16.3007	0.790701
$426$	0	0
$427$	−29.1300	−1.40970
$428$	0	0
$429$	−4.51405	−0.217940
$430$	0	0
$431$	−9.41826	−0.453662	−0.226831	−	0.973934i	$-0.572836\pi$
$431$	−9.41826	−0.453662	−0.226831	+	0.973934i	$0.572836\pi$
$432$	0	0
$433$	−20.1683	−0.969228	−0.484614	−	0.874728i	$-0.661040\pi$
$433$	−20.1683	−0.969228	−0.484614	+	0.874728i	$0.661040\pi$
$434$	0	0
$435$	30.3057	1.45305
$436$	0	0
$437$	−9.89628	−0.473403
$438$	0	0
$439$	16.7247	0.798227	0.399114	−	0.916901i	$-0.369318\pi$
$439$	16.7247	0.798227	0.399114	+	0.916901i	$0.369318\pi$
$440$	0	0
$441$	11.6124	0.552972
$442$	0	0
$443$	3.17947	0.151061	0.0755305	−	0.997143i	$-0.475935\pi$
$443$	3.17947	0.151061	0.0755305	+	0.997143i	$0.475935\pi$
$444$	0	0
$445$	−27.3452	−1.29629
$446$	0	0
$447$	0.000654743 0	3.09683e−5 0
$448$	0	0
$449$	19.2597	0.908921	0.454460	−	0.890767i	$-0.349832\pi$
$449$	19.2597	0.908921	0.454460	+	0.890767i	$0.349832\pi$
$450$	0	0
$451$	6.97986	0.328669
$452$	0	0
$453$	13.6925	0.643330
$454$	0	0
$455$	−15.5216	−0.727663
$456$	0	0
$457$	31.4676	1.47199	0.735997	−	0.676985i	$-0.236714\pi$
$457$	31.4676	1.47199	0.735997	+	0.676985i	$0.236714\pi$
$458$	0	0
$459$	−9.27989	−0.433148
$460$	0	0
$461$	4.06341	0.189252	0.0946260	−	0.995513i	$-0.469834\pi$
$461$	4.06341	0.189252	0.0946260	+	0.995513i	$0.469834\pi$
$462$	0	0
$463$	−13.6213	−0.633035	−0.316517	−	0.948587i	$-0.602514\pi$
$463$	−13.6213	−0.633035	−0.316517	+	0.948587i	$0.602514\pi$
$464$	0	0
$465$	36.2063	1.67903
$466$	0	0
$467$	−3.22261	−0.149124	−0.0745622	−	0.997216i	$-0.523756\pi$
$467$	−3.22261	−0.149124	−0.0745622	+	0.997216i	$0.523756\pi$
$468$	0	0
$469$	0.441814	0.0204011
$470$	0	0
$471$	−22.2108	−1.02342
$472$	0	0
$473$	7.39461	0.340004
$474$	0	0
$475$	−2.92047	−0.134000
$476$	0	0
$477$	−16.1347	−0.738759
$478$	0	0
$479$	6.83879	0.312472	0.156236	−	0.987720i	$-0.450064\pi$
$479$	6.83879	0.312472	0.156236	+	0.987720i	$0.450064\pi$
$480$	0	0
$481$	−0.126782	−0.00578074
$482$	0	0
$483$	−64.2518	−2.92356
$484$	0	0
$485$	16.3418	0.742041
$486$	0	0
$487$	22.5419	1.02147	0.510734	−	0.859739i	$-0.329374\pi$
$487$	22.5419	1.02147	0.510734	+	0.859739i	$0.329374\pi$
$488$	0	0
$489$	−3.35491	−0.151714
$490$	0	0
$491$	24.5019	1.10575	0.552877	−	0.833263i	$-0.313530\pi$
$491$	24.5019	1.10575	0.552877	+	0.833263i	$0.313530\pi$
$492$	0	0
$493$	55.1933	2.48578
$494$	0	0
$495$	2.72993	0.122701
$496$	0	0
$497$	32.1767	1.44332
$498$	0	0
$499$	4.84063	0.216697	0.108348	−	0.994113i	$-0.465444\pi$
$499$	4.84063	0.216697	0.108348	+	0.994113i	$0.465444\pi$
$500$	0	0
$501$	−26.6619	−1.19117
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−27.5253	−1.22486
$506$	0	0
$507$	12.1817	0.541008
$508$	0	0
$509$	2.61918	0.116093	0.0580467	−	0.998314i	$-0.481513\pi$
$509$	2.61918	0.116093	0.0580467	+	0.998314i	$0.481513\pi$
$510$	0	0
$511$	−40.1193	−1.77477
$512$	0	0
$513$	1.66260	0.0734057
$514$	0	0
$515$	6.18887	0.272714
$516$	0	0
$517$	−1.19953	−0.0527553
$518$	0	0
$519$	21.0407	0.923583
$520$	0	0
$521$	−27.8461	−1.21996	−0.609980	−	0.792417i	$-0.708823\pi$
$521$	−27.8461	−1.21996	−0.609980	+	0.792417i	$0.708823\pi$
$522$	0	0
$523$	27.2966	1.19360	0.596800	−	0.802390i	$-0.296439\pi$
$523$	27.2966	1.19360	0.596800	+	0.802390i	$0.296439\pi$
$524$	0	0
$525$	−18.9612	−0.827535
$526$	0	0
$527$	65.9394	2.87237
$528$	0	0
$529$	41.5568	1.80682
$530$	0	0
$531$	18.9558	0.822612
$532$	0	0
$533$	27.9625	1.21119
$534$	0	0
$535$	19.9568	0.862807
$536$	0	0
$537$	19.7696	0.853123
$538$	0	0
$539$	3.33800	0.143778
$540$	0	0
$541$	−30.1441	−1.29600	−0.647998	−	0.761642i	$-0.724393\pi$
$541$	−30.1441	−1.29600	−0.647998	+	0.761642i	$0.724393\pi$
$542$	0	0
$543$	−2.77044	−0.118891
$544$	0	0
$545$	1.43351	0.0614046
$546$	0	0
$547$	15.0768	0.644636	0.322318	−	0.946631i	$-0.395538\pi$
$547$	15.0768	0.644636	0.322318	+	0.946631i	$0.395538\pi$
$548$	0	0
$549$	20.5251	0.875990
$550$	0	0
$551$	−9.88854	−0.421266
$552$	0	0
$553$	31.0432	1.32009
$554$	0	0
$555$	0.171714	0.00728886
$556$	0	0
$557$	38.1097	1.61476	0.807380	−	0.590032i	$-0.200885\pi$
$557$	38.1097	1.61476	0.807380	+	0.590032i	$0.200885\pi$
$558$	0	0
$559$	29.6240	1.25296
$560$	0	0
$561$	11.1347	0.470106
$562$	0	0
$563$	−44.0769	−1.85762	−0.928811	−	0.370554i	$-0.879168\pi$
$563$	−44.0769	−1.85762	−0.928811	+	0.370554i	$0.879168\pi$
$564$	0	0
$565$	−23.3986	−0.984387
$566$	0	0
$567$	35.7335	1.50066
$568$	0	0
$569$	23.1026	0.968511	0.484256	−	0.874927i	$-0.339090\pi$
$569$	23.1026	0.968511	0.484256	+	0.874927i	$0.339090\pi$
$570$	0	0
$571$	22.2356	0.930533	0.465266	−	0.885171i	$-0.345959\pi$
$571$	22.2356	0.930533	0.465266	+	0.885171i	$0.345959\pi$
$572$	0	0
$573$	41.0759	1.71597
$574$	0	0
$575$	19.0512	0.794491
$576$	0	0
$577$	11.9626	0.498010	0.249005	−	0.968502i	$-0.419896\pi$
$577$	11.9626	0.498010	0.249005	+	0.968502i	$0.419896\pi$
$578$	0	0
$579$	−40.1144	−1.66710
$580$	0	0
$581$	−3.22817	−0.133927
$582$	0	0
$583$	−4.63795	−0.192084
$584$	0	0
$585$	10.9366	0.452171
$586$	0	0
$587$	−1.64277	−0.0678046	−0.0339023	−	0.999425i	$-0.510793\pi$
$587$	−1.64277	−0.0678046	−0.0339023	+	0.999425i	$0.510793\pi$
$588$	0	0
$589$	−11.8138	−0.486780
$590$	0	0
$591$	16.6458	0.684717
$592$	0	0
$593$	13.6181	0.559227	0.279614	−	0.960113i	$-0.409794\pi$
$593$	13.6181	0.559227	0.279614	+	0.960113i	$0.409794\pi$
$594$	0	0
$595$	38.2867	1.56960
$596$	0	0
$597$	−16.6334	−0.680760
$598$	0	0
$599$	−15.2924	−0.624832	−0.312416	−	0.949945i	$-0.601138\pi$
$599$	−15.2924	−0.624832	−0.312416	+	0.949945i	$0.601138\pi$
$600$	0	0
$601$	19.4107	0.791779	0.395890	−	0.918298i	$-0.370436\pi$
$601$	19.4107	0.791779	0.395890	+	0.918298i	$0.370436\pi$
$602$	0	0
$603$	−0.311303	−0.0126772
$604$	0	0
$605$	−17.0505	−0.693202
$606$	0	0
$607$	0.827856	0.0336016	0.0168008	−	0.999859i	$-0.494652\pi$
$607$	0.827856	0.0336016	0.0168008	+	0.999859i	$0.494652\pi$
$608$	0	0
$609$	−64.2015	−2.60158
$610$	0	0
$611$	−4.80552	−0.194411
$612$	0	0
$613$	−4.99298	−0.201665	−0.100832	−	0.994903i	$-0.532151\pi$
$613$	−4.99298	−0.201665	−0.100832	+	0.994903i	$0.532151\pi$
$614$	0	0
$615$	−37.8727	−1.52717
$616$	0	0
$617$	1.11916	0.0450557	0.0225279	−	0.999746i	$-0.492829\pi$
$617$	1.11916	0.0450557	0.0225279	+	0.999746i	$0.492829\pi$
$618$	0	0
$619$	1.74728	0.0702291	0.0351146	−	0.999383i	$-0.488820\pi$
$619$	1.74728	0.0702291	0.0351146	+	0.999383i	$0.488820\pi$
$620$	0	0
$621$	−10.8457	−0.435224
$622$	0	0
$623$	57.9298	2.32091
$624$	0	0
$625$	−7.52228	−0.300891
$626$	0	0
$627$	−1.99491	−0.0796690
$628$	0	0
$629$	0.312729	0.0124693
$630$	0	0
$631$	25.1527	1.00131	0.500657	−	0.865646i	$-0.333092\pi$
$631$	25.1527	1.00131	0.500657	+	0.865646i	$0.333092\pi$
$632$	0	0
$633$	−66.6794	−2.65027
$634$	0	0
$635$	19.4715	0.772703
$636$	0	0
$637$	13.3726	0.529841
$638$	0	0
$639$	−22.6718	−0.896881
$640$	0	0
$641$	−33.9530	−1.34106	−0.670531	−	0.741881i	$-0.733934\pi$
$641$	−33.9530	−1.34106	−0.670531	+	0.741881i	$0.733934\pi$
$642$	0	0
$643$	10.4416	0.411778	0.205889	−	0.978575i	$-0.433991\pi$
$643$	10.4416	0.411778	0.205889	+	0.978575i	$0.433991\pi$
$644$	0	0
$645$	−40.1231	−1.57984
$646$	0	0
$647$	−1.28801	−0.0506370	−0.0253185	−	0.999679i	$-0.508060\pi$
$647$	−1.28801	−0.0506370	−0.0253185	+	0.999679i	$0.508060\pi$
$648$	0	0
$649$	5.44887	0.213887
$650$	0	0
$651$	−76.7016	−3.00617
$652$	0	0
$653$	−9.35507	−0.366092	−0.183046	−	0.983104i	$-0.558596\pi$
$653$	−9.35507	−0.366092	−0.183046	+	0.983104i	$0.558596\pi$
$654$	0	0
$655$	−13.1807	−0.515012
$656$	0	0
$657$	28.2681	1.10284
$658$	0	0
$659$	−3.13916	−0.122284	−0.0611422	−	0.998129i	$-0.519474\pi$
$659$	−3.13916	−0.122284	−0.0611422	+	0.998129i	$0.519474\pi$
$660$	0	0
$661$	25.5931	0.995454	0.497727	−	0.867334i	$-0.334168\pi$
$661$	25.5931	0.995454	0.497727	+	0.867334i	$0.334168\pi$
$662$	0	0
$663$	44.6073	1.73241
$664$	0	0
$665$	−6.85951	−0.266001
$666$	0	0
$667$	64.5063	2.49770
$668$	0	0
$669$	45.0658	1.74234
$670$	0	0
$671$	5.89996	0.227766
$672$	0	0
$673$	−13.4391	−0.518040	−0.259020	−	0.965872i	$-0.583400\pi$
$673$	−13.4391	−0.518040	−0.259020	+	0.965872i	$0.583400\pi$
$674$	0	0
$675$	−3.20066	−0.123193
$676$	0	0
$677$	37.5397	1.44277	0.721383	−	0.692536i	$-0.243507\pi$
$677$	37.5397	1.44277	0.721383	+	0.692536i	$0.243507\pi$
$678$	0	0
$679$	−34.6194	−1.32857
$680$	0	0
$681$	−18.2964	−0.701119
$682$	0	0
$683$	19.9976	0.765188	0.382594	−	0.923917i	$-0.375031\pi$
$683$	19.9976	0.765188	0.382594	+	0.923917i	$0.375031\pi$
$684$	0	0
$685$	−2.91492	−0.111373
$686$	0	0
$687$	−15.0773	−0.575234
$688$	0	0
$689$	−18.5804	−0.707857
$690$	0	0
$691$	−37.6109	−1.43078	−0.715392	−	0.698723i	$-0.753752\pi$
$691$	−37.6109	−1.43078	−0.715392	+	0.698723i	$0.753752\pi$
$692$	0	0
$693$	−5.78325	−0.219688
$694$	0	0
$695$	4.82208	0.182912
$696$	0	0
$697$	−68.9743	−2.61259
$698$	0	0
$699$	−63.7319	−2.41056
$700$	0	0
$701$	36.2517	1.36921	0.684605	−	0.728914i	$-0.259975\pi$
$701$	36.2517	1.36921	0.684605	+	0.728914i	$0.259975\pi$
$702$	0	0
$703$	−0.0560290	−0.00211317
$704$	0	0
$705$	6.50864	0.245130
$706$	0	0
$707$	58.3112	2.19302
$708$	0	0
$709$	7.64415	0.287082	0.143541	−	0.989644i	$-0.454151\pi$
$709$	7.64415	0.287082	0.143541	+	0.989644i	$0.454151\pi$
$710$	0	0
$711$	−21.8731	−0.820305
$712$	0	0
$713$	77.0657	2.88613
$714$	0	0
$715$	3.14372	0.117569
$716$	0	0
$717$	12.2729	0.458338
$718$	0	0
$719$	15.7591	0.587716	0.293858	−	0.955849i	$-0.405061\pi$
$719$	15.7591	0.587716	0.293858	+	0.955849i	$0.405061\pi$
$720$	0	0
$721$	−13.1109	−0.488275
$722$	0	0
$723$	3.46036	0.128692
$724$	0	0
$725$	19.0363	0.706992
$726$	0	0
$727$	−30.7425	−1.14018	−0.570088	−	0.821584i	$-0.693091\pi$
$727$	−30.7425	−1.14018	−0.570088	+	0.821584i	$0.693091\pi$
$728$	0	0
$729$	−15.7500	−0.583332
$730$	0	0
$731$	−73.0727	−2.70269
$732$	0	0
$733$	−31.7320	−1.17205	−0.586023	−	0.810294i	$-0.699307\pi$
$733$	−31.7320	−1.17205	−0.586023	+	0.810294i	$0.699307\pi$
$734$	0	0
$735$	−18.1120	−0.668070
$736$	0	0
$737$	−0.0894844	−0.00329620
$738$	0	0
$739$	−9.79630	−0.360363	−0.180181	−	0.983633i	$-0.557668\pi$
$739$	−9.79630	−0.360363	−0.180181	+	0.983633i	$0.557668\pi$
$740$	0	0
$741$	−7.99193	−0.293591
$742$	0	0
$743$	−49.5259	−1.81693	−0.908464	−	0.417963i	$-0.862744\pi$
$743$	−49.5259	−1.81693	−0.908464	+	0.417963i	$0.862744\pi$
$744$	0	0
$745$	−0.000455984 0	−1.67060e−5 0
$746$	0	0
$747$	2.27458	0.0832224
$748$	0	0
$749$	−42.2776	−1.54479
$750$	0	0
$751$	−23.7905	−0.868128	−0.434064	−	0.900882i	$-0.642921\pi$
$751$	−23.7905	−0.868128	−0.434064	+	0.900882i	$0.642921\pi$
$752$	0	0
$753$	0.303888	0.0110743
$754$	0	0
$755$	−9.53589	−0.347047
$756$	0	0
$757$	−3.18511	−0.115765	−0.0578824	−	0.998323i	$-0.518435\pi$
$757$	−3.18511	−0.115765	−0.0578824	+	0.998323i	$0.518435\pi$
$758$	0	0
$759$	13.0135	0.472359
$760$	0	0
$761$	12.5094	0.453465	0.226732	−	0.973957i	$-0.427196\pi$
$761$	12.5094	0.453465	0.226732	+	0.973957i	$0.427196\pi$
$762$	0	0
$763$	−3.03682	−0.109940
$764$	0	0
$765$	−26.9769	−0.975352
$766$	0	0
$767$	21.8291	0.788202
$768$	0	0
$769$	−43.5670	−1.57107	−0.785534	−	0.618819i	$-0.787612\pi$
$769$	−43.5670	−1.57107	−0.785534	+	0.618819i	$0.787612\pi$
$770$	0	0
$771$	−38.8162	−1.39793
$772$	0	0
$773$	−12.4604	−0.448170	−0.224085	−	0.974570i	$-0.571939\pi$
$773$	−12.4604	−0.448170	−0.224085	+	0.974570i	$0.571939\pi$
$774$	0	0
$775$	22.7427	0.816942
$776$	0	0
$777$	−0.363770	−0.0130502
$778$	0	0
$779$	12.3576	0.442756
$780$	0	0
$781$	−6.51702	−0.233197
$782$	0	0
$783$	−10.8372	−0.387292
$784$	0	0
$785$	15.4683	0.552088
$786$	0	0
$787$	−4.52401	−0.161264	−0.0806318	−	0.996744i	$-0.525694\pi$
$787$	−4.52401	−0.161264	−0.0806318	+	0.996744i	$0.525694\pi$
$788$	0	0
$789$	−59.0574	−2.10250
$790$	0	0
$791$	49.5690	1.76247
$792$	0	0
$793$	23.6362	0.839347
$794$	0	0
$795$	25.1655	0.892527
$796$	0	0
$797$	5.17026	0.183140	0.0915700	−	0.995799i	$-0.470811\pi$
$797$	5.17026	0.183140	0.0915700	+	0.995799i	$0.470811\pi$
$798$	0	0
$799$	11.8536	0.419352
$800$	0	0
$801$	−40.8175	−1.44221
$802$	0	0
$803$	8.12571	0.286750
$804$	0	0
$805$	44.7470	1.57712
$806$	0	0
$807$	−64.2000	−2.25995
$808$	0	0
$809$	−32.0872	−1.12813	−0.564063	−	0.825732i	$-0.690762\pi$
$809$	−32.0872	−1.12813	−0.564063	+	0.825732i	$0.690762\pi$
$810$	0	0
$811$	0.161609	0.00567486	0.00283743	−	0.999996i	$-0.499097\pi$
$811$	0.161609	0.00567486	0.00283743	+	0.999996i	$0.499097\pi$
$812$	0	0
$813$	19.3453	0.678468
$814$	0	0
$815$	2.33647	0.0818428
$816$	0	0
$817$	13.0918	0.458026
$818$	0	0
$819$	−23.1687	−0.809579
$820$	0	0
$821$	28.5388	0.996009	0.498005	−	0.867174i	$-0.334066\pi$
$821$	28.5388	0.996009	0.498005	+	0.867174i	$0.334066\pi$
$822$	0	0
$823$	−25.6473	−0.894010	−0.447005	−	0.894531i	$-0.647509\pi$
$823$	−25.6473	−0.894010	−0.447005	+	0.894531i	$0.647509\pi$
$824$	0	0
$825$	3.84038	0.133705
$826$	0	0
$827$	5.41254	0.188212	0.0941062	−	0.995562i	$-0.470001\pi$
$827$	5.41254	0.188212	0.0941062	+	0.995562i	$0.470001\pi$
$828$	0	0
$829$	42.5786	1.47882	0.739408	−	0.673258i	$-0.235106\pi$
$829$	42.5786	1.47882	0.739408	+	0.673258i	$0.235106\pi$
$830$	0	0
$831$	33.7096	1.16937
$832$	0	0
$833$	−32.9858	−1.14289
$834$	0	0
$835$	18.5682	0.642578
$836$	0	0
$837$	−12.9473	−0.447523
$838$	0	0
$839$	39.9785	1.38021	0.690106	−	0.723708i	$-0.257564\pi$
$839$	39.9785	1.38021	0.690106	+	0.723708i	$0.257564\pi$
$840$	0	0
$841$	35.4559	1.22262
$842$	0	0
$843$	59.9599	2.06513
$844$	0	0
$845$	−8.48372	−0.291849
$846$	0	0
$847$	36.1208	1.24113
$848$	0	0
$849$	−0.0570776	−0.00195890
$850$	0	0
$851$	0.365497	0.0125291
$852$	0	0
$853$	−13.3730	−0.457884	−0.228942	−	0.973440i	$-0.573527\pi$
$853$	−13.3730	−0.457884	−0.228942	+	0.973440i	$0.573527\pi$
$854$	0	0
$855$	4.83323	0.165293
$856$	0	0
$857$	34.8819	1.19154	0.595772	−	0.803154i	$-0.296846\pi$
$857$	34.8819	1.19154	0.595772	+	0.803154i	$0.296846\pi$
$858$	0	0
$859$	−6.79181	−0.231733	−0.115867	−	0.993265i	$-0.536965\pi$
$859$	−6.79181	−0.231733	−0.115867	+	0.993265i	$0.536965\pi$
$860$	0	0
$861$	80.2317	2.73429
$862$	0	0
$863$	−13.5354	−0.460749	−0.230375	−	0.973102i	$-0.573995\pi$
$863$	−13.5354	−0.460749	−0.230375	+	0.973102i	$0.573995\pi$
$864$	0	0
$865$	−14.6534	−0.498230
$866$	0	0
$867$	−70.4535	−2.39273
$868$	0	0
$869$	−6.28745	−0.213287
$870$	0	0
$871$	−0.358489	−0.0121469
$872$	0	0
$873$	24.3929	0.825575
$874$	0	0
$875$	41.0512	1.38778
$876$	0	0
$877$	−34.3058	−1.15842	−0.579212	−	0.815177i	$-0.696640\pi$
$877$	−34.3058	−1.15842	−0.579212	+	0.815177i	$0.696640\pi$
$878$	0	0
$879$	−6.01105	−0.202748
$880$	0	0
$881$	−5.40002	−0.181931	−0.0909657	−	0.995854i	$-0.528995\pi$
$881$	−5.40002	−0.181931	−0.0909657	+	0.995854i	$0.528995\pi$
$882$	0	0
$883$	30.0119	1.00998	0.504990	−	0.863125i	$-0.331496\pi$
$883$	30.0119	1.00998	0.504990	+	0.863125i	$0.331496\pi$
$884$	0	0
$885$	−29.5655	−0.993834
$886$	0	0
$887$	−15.3759	−0.516272	−0.258136	−	0.966109i	$-0.583108\pi$
$887$	−15.3759	−0.516272	−0.258136	+	0.966109i	$0.583108\pi$
$888$	0	0
$889$	−41.2496	−1.38347
$890$	0	0
$891$	−7.23741	−0.242462
$892$	0	0
$893$	−2.12372	−0.0710676
$894$	0	0
$895$	−13.7682	−0.460220
$896$	0	0
$897$	52.1341	1.74071
$898$	0	0
$899$	77.0054	2.56828
$900$	0	0
$901$	45.8318	1.52688
$902$	0	0
$903$	84.9991	2.82859
$904$	0	0
$905$	1.92942	0.0641362
$906$	0	0
$907$	−1.35116	−0.0448646	−0.0224323	−	0.999748i	$-0.507141\pi$
$907$	−1.35116	−0.0448646	−0.0224323	+	0.999748i	$0.507141\pi$
$908$	0	0
$909$	−41.0862	−1.36274
$910$	0	0
$911$	−12.2879	−0.407116	−0.203558	−	0.979063i	$-0.565251\pi$
$911$	−12.2879	−0.407116	−0.203558	+	0.979063i	$0.565251\pi$
$912$	0	0
$913$	0.653829	0.0216386
$914$	0	0
$915$	−32.0131	−1.05832
$916$	0	0
$917$	27.9227	0.922090
$918$	0	0
$919$	26.0910	0.860664	0.430332	−	0.902671i	$-0.358396\pi$
$919$	26.0910	0.860664	0.430332	+	0.902671i	$0.358396\pi$
$920$	0	0
$921$	−30.1604	−0.993818
$922$	0	0
$923$	−26.1083	−0.859364
$924$	0	0
$925$	0.107861	0.00354645
$926$	0	0
$927$	9.23796	0.303415
$928$	0	0
$929$	32.0841	1.05264	0.526322	−	0.850285i	$-0.323571\pi$
$929$	32.0841	1.05264	0.526322	+	0.850285i	$0.323571\pi$
$930$	0	0
$931$	5.90980	0.193686
$932$	0	0
$933$	−18.8748	−0.617935
$934$	0	0
$935$	−7.75454	−0.253601
$936$	0	0
$937$	23.3834	0.763903	0.381952	−	0.924182i	$-0.375252\pi$
$937$	23.3834	0.763903	0.381952	+	0.924182i	$0.375252\pi$
$938$	0	0
$939$	−23.3616	−0.762378
$940$	0	0
$941$	12.8727	0.419639	0.209820	−	0.977740i	$-0.432712\pi$
$941$	12.8727	0.419639	0.209820	+	0.977740i	$0.432712\pi$
$942$	0	0
$943$	−80.6126	−2.62511
$944$	0	0
$945$	−7.51762	−0.244548
$946$	0	0
$947$	−22.3917	−0.727634	−0.363817	−	0.931470i	$-0.618527\pi$
$947$	−22.3917	−0.727634	−0.363817	+	0.931470i	$0.618527\pi$
$948$	0	0
$949$	32.5529	1.05671
$950$	0	0
$951$	−79.3401	−2.57278
$952$	0	0
$953$	28.6568	0.928285	0.464143	−	0.885760i	$-0.346362\pi$
$953$	28.6568	0.928285	0.464143	+	0.885760i	$0.346362\pi$
$954$	0	0
$955$	−28.6065	−0.925686
$956$	0	0
$957$	13.0033	0.420337
$958$	0	0
$959$	6.17515	0.199406
$960$	0	0
$961$	60.9984	1.96769
$962$	0	0
$963$	29.7889	0.959935
$964$	0	0
$965$	27.9370	0.899322
$966$	0	0
$967$	−35.0100	−1.12585	−0.562924	−	0.826509i	$-0.690323\pi$
$967$	−35.0100	−1.12585	−0.562924	+	0.826509i	$0.690323\pi$
$968$	0	0
$969$	19.7135	0.633288
$970$	0	0
$971$	−29.8123	−0.956724	−0.478362	−	0.878163i	$-0.658769\pi$
$971$	−29.8123	−0.956724	−0.478362	+	0.878163i	$0.658769\pi$
$972$	0	0
$973$	−10.2154	−0.327490
$974$	0	0
$975$	15.3852	0.492721
$976$	0	0
$977$	−39.4944	−1.26354	−0.631769	−	0.775157i	$-0.717671\pi$
$977$	−39.4944	−1.26354	−0.631769	+	0.775157i	$0.717671\pi$
$978$	0	0
$979$	−11.7330	−0.374989
$980$	0	0
$981$	2.13975	0.0683171
$982$	0	0
$983$	−10.7749	−0.343667	−0.171834	−	0.985126i	$-0.554969\pi$
$983$	−10.7749	−0.343667	−0.171834	+	0.985126i	$0.554969\pi$
$984$	0	0
$985$	−11.5927	−0.369373
$986$	0	0
$987$	−13.7883	−0.438887
$988$	0	0
$989$	−85.4027	−2.71565
$990$	0	0
$991$	25.6135	0.813640	0.406820	−	0.913508i	$-0.366638\pi$
$991$	25.6135	0.813640	0.406820	+	0.913508i	$0.366638\pi$
$992$	0	0
$993$	20.2310	0.642010
$994$	0	0
$995$	11.5840	0.367239
$996$	0	0
$997$	−8.06918	−0.255553	−0.127777	−	0.991803i	$-0.540784\pi$
$997$	−8.06918	−0.255553	−0.127777	+	0.991803i	$0.540784\pi$
$998$	0	0
$999$	−0.0614045	−0.00194275

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.u.1.6		26
4.3	odd	2		503.2.a.f.1.8	✓	26
12.11	even	2		4527.2.a.o.1.19		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.f.1.8	✓	26	4.3	odd	2
4527.2.a.o.1.19		26	12.11	even	2
8048.2.a.u.1.6		26	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-2.32813 q^{3} +1.62138 q^{5} -3.43484 q^{7} +2.42020 q^{9} +O(q^{10})\)	\(q-2.32813 q^{3} +1.62138 q^{5} -3.43484 q^{7} +2.42020 q^{9} +0.695688 q^{11} +2.78704 q^{13} -3.77480 q^{15} -6.87472 q^{17} +1.23169 q^{19} +7.99676 q^{21} -8.03473 q^{23} -2.37111 q^{25} +1.34986 q^{27} -8.02844 q^{29} -9.59158 q^{31} -1.61965 q^{33} -5.56920 q^{35} -0.0454896 q^{37} -6.48860 q^{39} +10.0330 q^{41} +10.6292 q^{43} +3.92407 q^{45} -1.72424 q^{47} +4.79813 q^{49} +16.0053 q^{51} -6.66671 q^{53} +1.12798 q^{55} -2.86753 q^{57} +7.83234 q^{59} +8.48076 q^{61} -8.31299 q^{63} +4.51887 q^{65} -0.128627 q^{67} +18.7059 q^{69} -9.36773 q^{71} +11.6801 q^{73} +5.52026 q^{75} -2.38958 q^{77} -9.03774 q^{79} -10.4032 q^{81} +0.939830 q^{83} -11.1466 q^{85} +18.6913 q^{87} -16.8653 q^{89} -9.57305 q^{91} +22.3305 q^{93} +1.99704 q^{95} +10.0789 q^{97} +1.68370 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * q^99

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