LMFDB - Embedded newform 6044.2.a.a.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$1$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.75230 q^{3} -3.12629 q^{5} -3.74178 q^{7} +0.0705422 q^{9} +O(q^{10})$	$q-1.75230 q^{3} -3.12629 q^{5} -3.74178 q^{7} +0.0705422 q^{9} -0.746097 q^{11} -6.22341 q^{13} +5.47818 q^{15} +5.22856 q^{17} +0.539019 q^{19} +6.55671 q^{21} -5.97332 q^{23} +4.77367 q^{25} +5.13328 q^{27} +4.15638 q^{29} -0.191607 q^{31} +1.30738 q^{33} +11.6979 q^{35} +10.3496 q^{37} +10.9053 q^{39} +7.45937 q^{41} +8.22128 q^{43} -0.220535 q^{45} -1.34237 q^{47} +7.00091 q^{49} -9.16198 q^{51} -5.11309 q^{53} +2.33251 q^{55} -0.944521 q^{57} -8.13011 q^{59} -0.461856 q^{61} -0.263953 q^{63} +19.4562 q^{65} -4.97365 q^{67} +10.4670 q^{69} -11.2539 q^{71} +4.17439 q^{73} -8.36488 q^{75} +2.79173 q^{77} -1.22033 q^{79} -9.20665 q^{81} +12.4705 q^{83} -16.3460 q^{85} -7.28321 q^{87} +3.30319 q^{89} +23.2866 q^{91} +0.335752 q^{93} -1.68513 q^{95} -2.81547 q^{97} -0.0526314 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * 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
$3$	−1.75230	−1.01169	−0.505844	−	0.862625i	$-0.668819\pi$
$3$	−1.75230	−1.01169	−0.505844	+	0.862625i	$0.668819\pi$
$4$	0	0
$5$	−3.12629	−1.39812	−0.699059	−	0.715064i	$-0.746398\pi$
$5$	−3.12629	−1.39812	−0.699059	+	0.715064i	$0.746398\pi$
$6$	0	0
$7$	−3.74178	−1.41426	−0.707130	−	0.707084i	$-0.750010\pi$
$7$	−3.74178	−1.41426	−0.707130	+	0.707084i	$0.750010\pi$
$8$	0	0
$9$	0.0705422	0.0235141
$10$	0	0
$11$	−0.746097	−0.224957	−0.112478	−	0.993654i	$-0.535879\pi$
$11$	−0.746097	−0.224957	−0.112478	+	0.993654i	$0.535879\pi$
$12$	0	0
$13$	−6.22341	−1.72606	−0.863032	−	0.505150i	$-0.831437\pi$
$13$	−6.22341	−1.72606	−0.863032	+	0.505150i	$0.831437\pi$
$14$	0	0
$15$	5.47818	1.41446
$16$	0	0
$17$	5.22856	1.26811	0.634056	−	0.773287i	$-0.281389\pi$
$17$	5.22856	1.26811	0.634056	+	0.773287i	$0.281389\pi$
$18$	0	0
$19$	0.539019	0.123659	0.0618297	−	0.998087i	$-0.480306\pi$
$19$	0.539019	0.123659	0.0618297	+	0.998087i	$0.480306\pi$
$20$	0	0
$21$	6.55671	1.43079
$22$	0	0
$23$	−5.97332	−1.24552	−0.622762	−	0.782412i	$-0.713989\pi$
$23$	−5.97332	−1.24552	−0.622762	+	0.782412i	$0.713989\pi$
$24$	0	0
$25$	4.77367	0.954734
$26$	0	0
$27$	5.13328	0.987900
$28$	0	0
$29$	4.15638	0.771820	0.385910	−	0.922536i	$-0.373887\pi$
$29$	4.15638	0.771820	0.385910	+	0.922536i	$0.373887\pi$
$30$	0	0
$31$	−0.191607	−0.0344136	−0.0172068	−	0.999852i	$-0.505477\pi$
$31$	−0.191607	−0.0344136	−0.0172068	+	0.999852i	$0.505477\pi$
$32$	0	0
$33$	1.30738	0.227586
$34$	0	0
$35$	11.6979	1.97730
$36$	0	0
$37$	10.3496	1.70146	0.850731	−	0.525601i	$-0.176159\pi$
$37$	10.3496	1.70146	0.850731	+	0.525601i	$0.176159\pi$
$38$	0	0
$39$	10.9053	1.74624
$40$	0	0
$41$	7.45937	1.16496	0.582479	−	0.812846i	$-0.302083\pi$
$41$	7.45937	1.16496	0.582479	+	0.812846i	$0.302083\pi$
$42$	0	0
$43$	8.22128	1.25373	0.626867	−	0.779126i	$-0.284337\pi$
$43$	8.22128	1.25373	0.626867	+	0.779126i	$0.284337\pi$
$44$	0	0
$45$	−0.220535	−0.0328755
$46$	0	0
$47$	−1.34237	−0.195805	−0.0979025	−	0.995196i	$-0.531213\pi$
$47$	−1.34237	−0.195805	−0.0979025	+	0.995196i	$0.531213\pi$
$48$	0	0
$49$	7.00091	1.00013
$50$	0	0
$51$	−9.16198	−1.28293
$52$	0	0
$53$	−5.11309	−0.702336	−0.351168	−	0.936312i	$-0.614215\pi$
$53$	−5.11309	−0.702336	−0.351168	+	0.936312i	$0.614215\pi$
$54$	0	0
$55$	2.33251	0.314516
$56$	0	0
$57$	−0.944521	−0.125105
$58$	0	0
$59$	−8.13011	−1.05845	−0.529225	−	0.848481i	$-0.677517\pi$
$59$	−8.13011	−1.05845	−0.529225	+	0.848481i	$0.677517\pi$
$60$	0	0
$61$	−0.461856	−0.0591346	−0.0295673	−	0.999563i	$-0.509413\pi$
$61$	−0.461856	−0.0591346	−0.0295673	+	0.999563i	$0.509413\pi$
$62$	0	0
$63$	−0.263953	−0.0332550
$64$	0	0
$65$	19.4562	2.41324
$66$	0	0
$67$	−4.97365	−0.607629	−0.303814	−	0.952731i	$-0.598260\pi$
$67$	−4.97365	−0.607629	−0.303814	+	0.952731i	$0.598260\pi$
$68$	0	0
$69$	10.4670	1.26008
$70$	0	0
$71$	−11.2539	−1.33560	−0.667799	−	0.744342i	$-0.732763\pi$
$71$	−11.2539	−1.33560	−0.667799	+	0.744342i	$0.732763\pi$
$72$	0	0
$73$	4.17439	0.488575	0.244288	−	0.969703i	$-0.421446\pi$
$73$	4.17439	0.488575	0.244288	+	0.969703i	$0.421446\pi$
$74$	0	0
$75$	−8.36488	−0.965894
$76$	0	0
$77$	2.79173	0.318147
$78$	0	0
$79$	−1.22033	−0.137298	−0.0686489	−	0.997641i	$-0.521869\pi$
$79$	−1.22033	−0.137298	−0.0686489	+	0.997641i	$0.521869\pi$
$80$	0	0
$81$	−9.20665	−1.02296
$82$	0	0
$83$	12.4705	1.36882	0.684410	−	0.729098i	$-0.260060\pi$
$83$	12.4705	1.36882	0.684410	+	0.729098i	$0.260060\pi$
$84$	0	0
$85$	−16.3460	−1.77297
$86$	0	0
$87$	−7.28321	−0.780842
$88$	0	0
$89$	3.30319	0.350137	0.175069	−	0.984556i	$-0.443985\pi$
$89$	3.30319	0.350137	0.175069	+	0.984556i	$0.443985\pi$
$90$	0	0
$91$	23.2866	2.44110
$92$	0	0
$93$	0.335752	0.0348158
$94$	0	0
$95$	−1.68513	−0.172890
$96$	0	0
$97$	−2.81547	−0.285867	−0.142934	−	0.989732i	$-0.545654\pi$
$97$	−2.81547	−0.285867	−0.142934	+	0.989732i	$0.545654\pi$
$98$	0	0
$99$	−0.0526314	−0.00528965
$100$	0	0
$101$	−18.0705	−1.79808	−0.899041	−	0.437865i	$-0.855735\pi$
$101$	−18.0705	−1.79808	−0.899041	+	0.437865i	$0.855735\pi$
$102$	0	0
$103$	12.3106	1.21300	0.606500	−	0.795084i	$-0.292573\pi$
$103$	12.3106	1.21300	0.606500	+	0.795084i	$0.292573\pi$
$104$	0	0
$105$	−20.4981	−2.00041
$106$	0	0
$107$	11.8873	1.14919	0.574593	−	0.818439i	$-0.305160\pi$
$107$	11.8873	1.14919	0.574593	+	0.818439i	$0.305160\pi$
$108$	0	0
$109$	−15.5999	−1.49420	−0.747098	−	0.664714i	$-0.768553\pi$
$109$	−15.5999	−1.49420	−0.747098	+	0.664714i	$0.768553\pi$
$110$	0	0
$111$	−18.1356	−1.72135
$112$	0	0
$113$	−10.6096	−0.998069	−0.499034	−	0.866582i	$-0.666312\pi$
$113$	−10.6096	−0.998069	−0.499034	+	0.866582i	$0.666312\pi$
$114$	0	0
$115$	18.6743	1.74139
$116$	0	0
$117$	−0.439013	−0.0405868
$118$	0	0
$119$	−19.5641	−1.79344
$120$	0	0
$121$	−10.4433	−0.949394
$122$	0	0
$123$	−13.0710	−1.17857
$124$	0	0
$125$	0.707573	0.0632872
$126$	0	0
$127$	12.0587	1.07004	0.535018	−	0.844840i	$-0.320305\pi$
$127$	12.0587	1.07004	0.535018	+	0.844840i	$0.320305\pi$
$128$	0	0
$129$	−14.4061	−1.26839
$130$	0	0
$131$	10.2844	0.898554	0.449277	−	0.893392i	$-0.351682\pi$
$131$	10.2844	0.898554	0.449277	+	0.893392i	$0.351682\pi$
$132$	0	0
$133$	−2.01689	−0.174886
$134$	0	0
$135$	−16.0481	−1.38120
$136$	0	0
$137$	19.9693	1.70609	0.853047	−	0.521833i	$-0.174752\pi$
$137$	19.9693	1.70609	0.853047	+	0.521833i	$0.174752\pi$
$138$	0	0
$139$	0.418892	0.0355300	0.0177650	−	0.999842i	$-0.494345\pi$
$139$	0.418892	0.0355300	0.0177650	+	0.999842i	$0.494345\pi$
$140$	0	0
$141$	2.35223	0.198094
$142$	0	0
$143$	4.64327	0.388290
$144$	0	0
$145$	−12.9940	−1.07910
$146$	0	0
$147$	−12.2677	−1.01182
$148$	0	0
$149$	23.6840	1.94027	0.970134	−	0.242568i	$-0.0779898\pi$
$149$	23.6840	1.94027	0.970134	+	0.242568i	$0.0779898\pi$
$150$	0	0
$151$	10.7123	0.871752	0.435876	−	0.900007i	$-0.356439\pi$
$151$	10.7123	0.871752	0.435876	+	0.900007i	$0.356439\pi$
$152$	0	0
$153$	0.368834	0.0298185
$154$	0	0
$155$	0.599017	0.0481142
$156$	0	0
$157$	8.87355	0.708187	0.354093	−	0.935210i	$-0.384790\pi$
$157$	8.87355	0.708187	0.354093	+	0.935210i	$0.384790\pi$
$158$	0	0
$159$	8.95964	0.710546
$160$	0	0
$161$	22.3508	1.76149
$162$	0	0
$163$	−23.8161	−1.86542	−0.932709	−	0.360631i	$-0.882561\pi$
$163$	−23.8161	−1.86542	−0.932709	+	0.360631i	$0.882561\pi$
$164$	0	0
$165$	−4.08725	−0.318192
$166$	0	0
$167$	7.36935	0.570257	0.285129	−	0.958489i	$-0.407964\pi$
$167$	7.36935	0.570257	0.285129	+	0.958489i	$0.407964\pi$
$168$	0	0
$169$	25.7308	1.97930
$170$	0	0
$171$	0.0380236	0.00290774
$172$	0	0
$173$	−10.7329	−0.816005	−0.408002	−	0.912981i	$-0.633775\pi$
$173$	−10.7329	−0.816005	−0.408002	+	0.912981i	$0.633775\pi$
$174$	0	0
$175$	−17.8620	−1.35024
$176$	0	0
$177$	14.2464	1.07082
$178$	0	0
$179$	−19.4773	−1.45580	−0.727901	−	0.685682i	$-0.759504\pi$
$179$	−19.4773	−1.45580	−0.727901	+	0.685682i	$0.759504\pi$
$180$	0	0
$181$	−12.9956	−0.965958	−0.482979	−	0.875632i	$-0.660445\pi$
$181$	−12.9956	−0.965958	−0.482979	+	0.875632i	$0.660445\pi$
$182$	0	0
$183$	0.809308	0.0598258
$184$	0	0
$185$	−32.3558	−2.37885
$186$	0	0
$187$	−3.90101	−0.285270
$188$	0	0
$189$	−19.2076	−1.39715
$190$	0	0
$191$	−7.68890	−0.556349	−0.278174	−	0.960531i	$-0.589729\pi$
$191$	−7.68890	−0.556349	−0.278174	+	0.960531i	$0.589729\pi$
$192$	0	0
$193$	8.30236	0.597617	0.298808	−	0.954313i	$-0.403411\pi$
$193$	8.30236	0.597617	0.298808	+	0.954313i	$0.403411\pi$
$194$	0	0
$195$	−34.0930	−2.44145
$196$	0	0
$197$	−13.2871	−0.946664	−0.473332	−	0.880884i	$-0.656949\pi$
$197$	−13.2871	−0.946664	−0.473332	+	0.880884i	$0.656949\pi$
$198$	0	0
$199$	16.9326	1.20032	0.600159	−	0.799881i	$-0.295104\pi$
$199$	16.9326	1.20032	0.600159	+	0.799881i	$0.295104\pi$
$200$	0	0
$201$	8.71532	0.614731
$202$	0	0
$203$	−15.5523	−1.09155
$204$	0	0
$205$	−23.3201	−1.62875
$206$	0	0
$207$	−0.421371	−0.0292873
$208$	0	0
$209$	−0.402160	−0.0278180
$210$	0	0
$211$	−11.8866	−0.818304	−0.409152	−	0.912466i	$-0.634175\pi$
$211$	−11.8866	−0.818304	−0.409152	+	0.912466i	$0.634175\pi$
$212$	0	0
$213$	19.7203	1.35121
$214$	0	0
$215$	−25.7021	−1.75287
$216$	0	0
$217$	0.716950	0.0486697
$218$	0	0
$219$	−7.31477	−0.494286
$220$	0	0
$221$	−32.5395	−2.18884
$222$	0	0
$223$	−3.79409	−0.254071	−0.127036	−	0.991898i	$-0.540546\pi$
$223$	−3.79409	−0.254071	−0.127036	+	0.991898i	$0.540546\pi$
$224$	0	0
$225$	0.336745	0.0224497
$226$	0	0
$227$	−0.523216	−0.0347270	−0.0173635	−	0.999849i	$-0.505527\pi$
$227$	−0.523216	−0.0347270	−0.0173635	+	0.999849i	$0.505527\pi$
$228$	0	0
$229$	21.8815	1.44597	0.722984	−	0.690865i	$-0.242770\pi$
$229$	21.8815	1.44597	0.722984	+	0.690865i	$0.242770\pi$
$230$	0	0
$231$	−4.89194	−0.321866
$232$	0	0
$233$	24.3897	1.59782	0.798910	−	0.601450i	$-0.205410\pi$
$233$	24.3897	1.59782	0.798910	+	0.601450i	$0.205410\pi$
$234$	0	0
$235$	4.19664	0.273758
$236$	0	0
$237$	2.13838	0.138903
$238$	0	0
$239$	−18.3647	−1.18791	−0.593957	−	0.804497i	$-0.702435\pi$
$239$	−18.3647	−1.18791	−0.593957	+	0.804497i	$0.702435\pi$
$240$	0	0
$241$	12.6318	0.813683	0.406842	−	0.913499i	$-0.366630\pi$
$241$	12.6318	0.813683	0.406842	+	0.913499i	$0.366630\pi$
$242$	0	0
$243$	0.732946	0.0470185
$244$	0	0
$245$	−21.8869	−1.39830
$246$	0	0
$247$	−3.35454	−0.213444
$248$	0	0
$249$	−21.8521	−1.38482
$250$	0	0
$251$	−12.6113	−0.796017	−0.398009	−	0.917382i	$-0.630299\pi$
$251$	−12.6113	−0.796017	−0.398009	+	0.917382i	$0.630299\pi$
$252$	0	0
$253$	4.45668	0.280189
$254$	0	0
$255$	28.6430	1.79369
$256$	0	0
$257$	4.28099	0.267041	0.133520	−	0.991046i	$-0.457372\pi$
$257$	4.28099	0.267041	0.133520	+	0.991046i	$0.457372\pi$
$258$	0	0
$259$	−38.7259	−2.40631
$260$	0	0
$261$	0.293200	0.0181486
$262$	0	0
$263$	−2.79250	−0.172193	−0.0860965	−	0.996287i	$-0.527439\pi$
$263$	−2.79250	−0.172193	−0.0860965	+	0.996287i	$0.527439\pi$
$264$	0	0
$265$	15.9850	0.981949
$266$	0	0
$267$	−5.78816	−0.354230
$268$	0	0
$269$	11.6824	0.712288	0.356144	−	0.934431i	$-0.384091\pi$
$269$	11.6824	0.712288	0.356144	+	0.934431i	$0.384091\pi$
$270$	0	0
$271$	−20.1953	−1.22678	−0.613389	−	0.789781i	$-0.710194\pi$
$271$	−20.1953	−1.22678	−0.613389	+	0.789781i	$0.710194\pi$
$272$	0	0
$273$	−40.8051	−2.46964
$274$	0	0
$275$	−3.56162	−0.214774
$276$	0	0
$277$	−5.30385	−0.318678	−0.159339	−	0.987224i	$-0.550936\pi$
$277$	−5.30385	−0.318678	−0.159339	+	0.987224i	$0.550936\pi$
$278$	0	0
$279$	−0.0135164	−0.000809203 0
$280$	0	0
$281$	19.1290	1.14114	0.570570	−	0.821249i	$-0.306722\pi$
$281$	19.1290	1.14114	0.570570	+	0.821249i	$0.306722\pi$
$282$	0	0
$283$	−31.7393	−1.88671	−0.943353	−	0.331790i	$-0.892348\pi$
$283$	−31.7393	−1.88671	−0.943353	+	0.331790i	$0.892348\pi$
$284$	0	0
$285$	2.95284	0.174911
$286$	0	0
$287$	−27.9113	−1.64755
$288$	0	0
$289$	10.3378	0.608107
$290$	0	0
$291$	4.93353	0.289209
$292$	0	0
$293$	−20.8246	−1.21658	−0.608292	−	0.793714i	$-0.708145\pi$
$293$	−20.8246	−1.21658	−0.608292	+	0.793714i	$0.708145\pi$
$294$	0	0
$295$	25.4171	1.47984
$296$	0	0
$297$	−3.82992	−0.222235
$298$	0	0
$299$	37.1744	2.14985
$300$	0	0
$301$	−30.7622	−1.77311
$302$	0	0
$303$	31.6649	1.81910
$304$	0	0
$305$	1.44389	0.0826771
$306$	0	0
$307$	6.81772	0.389108	0.194554	−	0.980892i	$-0.437674\pi$
$307$	6.81772	0.389108	0.194554	+	0.980892i	$0.437674\pi$
$308$	0	0
$309$	−21.5718	−1.22718
$310$	0	0
$311$	−1.20327	−0.0682313	−0.0341157	−	0.999418i	$-0.510861\pi$
$311$	−1.20327	−0.0682313	−0.0341157	+	0.999418i	$0.510861\pi$
$312$	0	0
$313$	26.9323	1.52230	0.761151	−	0.648575i	$-0.224635\pi$
$313$	26.9323	1.52230	0.761151	+	0.648575i	$0.224635\pi$
$314$	0	0
$315$	0.825194	0.0464944
$316$	0	0
$317$	24.3667	1.36857	0.684285	−	0.729214i	$-0.260114\pi$
$317$	24.3667	1.36857	0.684285	+	0.729214i	$0.260114\pi$
$318$	0	0
$319$	−3.10106	−0.173626
$320$	0	0
$321$	−20.8300	−1.16262
$322$	0	0
$323$	2.81829	0.156814
$324$	0	0
$325$	−29.7085	−1.64793
$326$	0	0
$327$	27.3356	1.51166
$328$	0	0
$329$	5.02286	0.276919
$330$	0	0
$331$	−22.6930	−1.24732	−0.623661	−	0.781695i	$-0.714356\pi$
$331$	−22.6930	−1.24732	−0.623661	+	0.781695i	$0.714356\pi$
$332$	0	0
$333$	0.730084	0.0400083
$334$	0	0
$335$	15.5491	0.849536
$336$	0	0
$337$	16.7697	0.913503	0.456752	−	0.889594i	$-0.349013\pi$
$337$	16.7697	0.913503	0.456752	+	0.889594i	$0.349013\pi$
$338$	0	0
$339$	18.5912	1.00974
$340$	0	0
$341$	0.142957	0.00774156
$342$	0	0
$343$	−0.00340172	−0.000183675 0
$344$	0	0
$345$	−32.7229	−1.76174
$346$	0	0
$347$	28.6610	1.53861	0.769303	−	0.638884i	$-0.220604\pi$
$347$	28.6610	1.53861	0.769303	+	0.638884i	$0.220604\pi$
$348$	0	0
$349$	15.5896	0.834490	0.417245	−	0.908794i	$-0.362996\pi$
$349$	15.5896	0.834490	0.417245	+	0.908794i	$0.362996\pi$
$350$	0	0
$351$	−31.9465	−1.70518
$352$	0	0
$353$	−3.78989	−0.201716	−0.100858	−	0.994901i	$-0.532159\pi$
$353$	−3.78989	−0.201716	−0.100858	+	0.994901i	$0.532159\pi$
$354$	0	0
$355$	35.1831	1.86732
$356$	0	0
$357$	34.2821	1.81440
$358$	0	0
$359$	−23.2102	−1.22499	−0.612493	−	0.790476i	$-0.709833\pi$
$359$	−23.2102	−1.22499	−0.612493	+	0.790476i	$0.709833\pi$
$360$	0	0
$361$	−18.7095	−0.984708
$362$	0	0
$363$	18.2998	0.960492
$364$	0	0
$365$	−13.0503	−0.683086
$366$	0	0
$367$	19.2782	1.00631	0.503156	−	0.864195i	$-0.332172\pi$
$367$	19.2782	1.00631	0.503156	+	0.864195i	$0.332172\pi$
$368$	0	0
$369$	0.526201	0.0273929
$370$	0	0
$371$	19.1320	0.993286
$372$	0	0
$373$	24.1080	1.24827	0.624133	−	0.781318i	$-0.285452\pi$
$373$	24.1080	1.24827	0.624133	+	0.781318i	$0.285452\pi$
$374$	0	0
$375$	−1.23988	−0.0640270
$376$	0	0
$377$	−25.8669	−1.33221
$378$	0	0
$379$	−8.25260	−0.423908	−0.211954	−	0.977280i	$-0.567983\pi$
$379$	−8.25260	−0.423908	−0.211954	+	0.977280i	$0.567983\pi$
$380$	0	0
$381$	−21.1304	−1.08254
$382$	0	0
$383$	38.0758	1.94558	0.972791	−	0.231685i	$-0.0744239\pi$
$383$	38.0758	1.94558	0.972791	+	0.231685i	$0.0744239\pi$
$384$	0	0
$385$	−8.72775	−0.444807
$386$	0	0
$387$	0.579948	0.0294804
$388$	0	0
$389$	−16.8509	−0.854374	−0.427187	−	0.904163i	$-0.640495\pi$
$389$	−16.8509	−0.854374	−0.427187	+	0.904163i	$0.640495\pi$
$390$	0	0
$391$	−31.2318	−1.57946
$392$	0	0
$393$	−18.0214	−0.909057
$394$	0	0
$395$	3.81510	0.191958
$396$	0	0
$397$	−20.6045	−1.03411	−0.517055	−	0.855952i	$-0.672972\pi$
$397$	−20.6045	−1.03411	−0.517055	+	0.855952i	$0.672972\pi$
$398$	0	0
$399$	3.53419	0.176931
$400$	0	0
$401$	14.5490	0.726540	0.363270	−	0.931684i	$-0.381660\pi$
$401$	14.5490	0.726540	0.363270	+	0.931684i	$0.381660\pi$
$402$	0	0
$403$	1.19245	0.0594000
$404$	0	0
$405$	28.7826	1.43022
$406$	0	0
$407$	−7.72180	−0.382755
$408$	0	0
$409$	13.9696	0.690751	0.345375	−	0.938465i	$-0.387752\pi$
$409$	13.9696	0.690751	0.345375	+	0.938465i	$0.387752\pi$
$410$	0	0
$411$	−34.9922	−1.72604
$412$	0	0
$413$	30.4211	1.49692
$414$	0	0
$415$	−38.9865	−1.91377
$416$	0	0
$417$	−0.734023	−0.0359453
$418$	0	0
$419$	−16.1060	−0.786831	−0.393415	−	0.919361i	$-0.628707\pi$
$419$	−16.1060	−0.786831	−0.393415	+	0.919361i	$0.628707\pi$
$420$	0	0
$421$	11.9173	0.580814	0.290407	−	0.956903i	$-0.406209\pi$
$421$	11.9173	0.580814	0.290407	+	0.956903i	$0.406209\pi$
$422$	0	0
$423$	−0.0946939	−0.00460417
$424$	0	0
$425$	24.9594	1.21071
$426$	0	0
$427$	1.72816	0.0836316
$428$	0	0
$429$	−8.13638	−0.392828
$430$	0	0
$431$	−5.41704	−0.260930	−0.130465	−	0.991453i	$-0.541647\pi$
$431$	−5.41704	−0.260930	−0.130465	+	0.991453i	$0.541647\pi$
$432$	0	0
$433$	−1.10067	−0.0528950	−0.0264475	−	0.999650i	$-0.508419\pi$
$433$	−1.10067	−0.0528950	−0.0264475	+	0.999650i	$0.508419\pi$
$434$	0	0
$435$	22.7694	1.09171
$436$	0	0
$437$	−3.21973	−0.154021
$438$	0	0
$439$	−28.2352	−1.34759	−0.673797	−	0.738916i	$-0.735338\pi$
$439$	−28.2352	−1.34759	−0.673797	+	0.738916i	$0.735338\pi$
$440$	0	0
$441$	0.493860	0.0235171
$442$	0	0
$443$	10.2942	0.489090	0.244545	−	0.969638i	$-0.421361\pi$
$443$	10.2942	0.489090	0.244545	+	0.969638i	$0.421361\pi$
$444$	0	0
$445$	−10.3267	−0.489533
$446$	0	0
$447$	−41.5014	−1.96295
$448$	0	0
$449$	−18.5568	−0.875751	−0.437875	−	0.899036i	$-0.644269\pi$
$449$	−18.5568	−0.875751	−0.437875	+	0.899036i	$0.644269\pi$
$450$	0	0
$451$	−5.56541	−0.262065
$452$	0	0
$453$	−18.7711	−0.881942
$454$	0	0
$455$	−72.8007	−3.41295
$456$	0	0
$457$	24.1408	1.12926	0.564630	−	0.825344i	$-0.309019\pi$
$457$	24.1408	1.12926	0.564630	+	0.825344i	$0.309019\pi$
$458$	0	0
$459$	26.8396	1.25277
$460$	0	0
$461$	−1.27031	−0.0591643	−0.0295821	−	0.999562i	$-0.509418\pi$
$461$	−1.27031	−0.0591643	−0.0295821	+	0.999562i	$0.509418\pi$
$462$	0	0
$463$	−34.9007	−1.62197	−0.810986	−	0.585065i	$-0.801069\pi$
$463$	−34.9007	−1.62197	−0.810986	+	0.585065i	$0.801069\pi$
$464$	0	0
$465$	−1.04966	−0.0486766
$466$	0	0
$467$	−22.4888	−1.04066	−0.520329	−	0.853966i	$-0.674191\pi$
$467$	−22.4888	−1.04066	−0.520329	+	0.853966i	$0.674191\pi$
$468$	0	0
$469$	18.6103	0.859344
$470$	0	0
$471$	−15.5491	−0.716464
$472$	0	0
$473$	−6.13388	−0.282036
$474$	0	0
$475$	2.57310	0.118062
$476$	0	0
$477$	−0.360689	−0.0165148
$478$	0	0
$479$	15.5097	0.708654	0.354327	−	0.935122i	$-0.384710\pi$
$479$	15.5097	0.708654	0.354327	+	0.935122i	$0.384710\pi$
$480$	0	0
$481$	−64.4098	−2.93683
$482$	0	0
$483$	−39.1653	−1.78208
$484$	0	0
$485$	8.80196	0.399676
$486$	0	0
$487$	12.6809	0.574626	0.287313	−	0.957837i	$-0.407238\pi$
$487$	12.6809	0.574626	0.287313	+	0.957837i	$0.407238\pi$
$488$	0	0
$489$	41.7328	1.88722
$490$	0	0
$491$	−14.7505	−0.665682	−0.332841	−	0.942983i	$-0.608007\pi$
$491$	−14.7505	−0.665682	−0.332841	+	0.942983i	$0.608007\pi$
$492$	0	0
$493$	21.7319	0.978754
$494$	0	0
$495$	0.164541	0.00739556
$496$	0	0
$497$	42.1098	1.88888
$498$	0	0
$499$	−37.7690	−1.69077	−0.845387	−	0.534155i	$-0.820630\pi$
$499$	−37.7690	−1.69077	−0.845387	+	0.534155i	$0.820630\pi$
$500$	0	0
$501$	−12.9133	−0.576923
$502$	0	0
$503$	16.9020	0.753625	0.376812	−	0.926290i	$-0.377020\pi$
$503$	16.9020	0.753625	0.376812	+	0.926290i	$0.377020\pi$
$504$	0	0
$505$	56.4936	2.51393
$506$	0	0
$507$	−45.0881	−2.00243
$508$	0	0
$509$	−4.67842	−0.207367	−0.103684	−	0.994610i	$-0.533063\pi$
$509$	−4.67842	−0.207367	−0.103684	+	0.994610i	$0.533063\pi$
$510$	0	0
$511$	−15.6196	−0.690972
$512$	0	0
$513$	2.76693	0.122163
$514$	0	0
$515$	−38.4865	−1.69592
$516$	0	0
$517$	1.00154	0.0440476
$518$	0	0
$519$	18.8072	0.825543
$520$	0	0
$521$	−22.2318	−0.973993	−0.486996	−	0.873404i	$-0.661908\pi$
$521$	−22.2318	−0.973993	−0.486996	+	0.873404i	$0.661908\pi$
$522$	0	0
$523$	13.8089	0.603822	0.301911	−	0.953336i	$-0.402375\pi$
$523$	13.8089	0.603822	0.301911	+	0.953336i	$0.402375\pi$
$524$	0	0
$525$	31.2995	1.36602
$526$	0	0
$527$	−1.00183	−0.0436402
$528$	0	0
$529$	12.6806	0.551328
$530$	0	0
$531$	−0.573516	−0.0248885
$532$	0	0
$533$	−46.4227	−2.01079
$534$	0	0
$535$	−37.1630	−1.60670
$536$	0	0
$537$	34.1300	1.47282
$538$	0	0
$539$	−5.22336	−0.224986
$540$	0	0
$541$	2.86707	0.123265	0.0616324	−	0.998099i	$-0.480369\pi$
$541$	2.86707	0.123265	0.0616324	+	0.998099i	$0.480369\pi$
$542$	0	0
$543$	22.7722	0.977249
$544$	0	0
$545$	48.7696	2.08906
$546$	0	0
$547$	−40.7227	−1.74118	−0.870589	−	0.492010i	$-0.836262\pi$
$547$	−40.7227	−1.74118	−0.870589	+	0.492010i	$0.836262\pi$
$548$	0	0
$549$	−0.0325804	−0.00139050
$550$	0	0
$551$	2.24037	0.0954428
$552$	0	0
$553$	4.56620	0.194175
$554$	0	0
$555$	56.6969	2.40665
$556$	0	0
$557$	−17.4620	−0.739889	−0.369945	−	0.929054i	$-0.620623\pi$
$557$	−17.4620	−0.739889	−0.369945	+	0.929054i	$0.620623\pi$
$558$	0	0
$559$	−51.1644	−2.16402
$560$	0	0
$561$	6.83573	0.288605
$562$	0	0
$563$	−0.0750359	−0.00316238	−0.00158119	−	0.999999i	$-0.500503\pi$
$563$	−0.0750359	−0.00316238	−0.00158119	+	0.999999i	$0.500503\pi$
$564$	0	0
$565$	33.1687	1.39542
$566$	0	0
$567$	34.4493	1.44673
$568$	0	0
$569$	35.5337	1.48965	0.744826	−	0.667259i	$-0.232533\pi$
$569$	35.5337	1.48965	0.744826	+	0.667259i	$0.232533\pi$
$570$	0	0
$571$	−3.50567	−0.146708	−0.0733539	−	0.997306i	$-0.523370\pi$
$571$	−3.50567	−0.146708	−0.0733539	+	0.997306i	$0.523370\pi$
$572$	0	0
$573$	13.4732	0.562852
$574$	0	0
$575$	−28.5147	−1.18914
$576$	0	0
$577$	36.6311	1.52497	0.762486	−	0.647004i	$-0.223978\pi$
$577$	36.6311	1.52497	0.762486	+	0.647004i	$0.223978\pi$
$578$	0	0
$579$	−14.5482	−0.604602
$580$	0	0
$581$	−46.6620	−1.93587
$582$	0	0
$583$	3.81486	0.157995
$584$	0	0
$585$	1.37248	0.0567451
$586$	0	0
$587$	17.1757	0.708919	0.354459	−	0.935071i	$-0.384665\pi$
$587$	17.1757	0.708919	0.354459	+	0.935071i	$0.384665\pi$
$588$	0	0
$589$	−0.103280	−0.00425556
$590$	0	0
$591$	23.2829	0.957730
$592$	0	0
$593$	5.46499	0.224420	0.112210	−	0.993685i	$-0.464207\pi$
$593$	5.46499	0.224420	0.112210	+	0.993685i	$0.464207\pi$
$594$	0	0
$595$	61.1630	2.50744
$596$	0	0
$597$	−29.6709	−1.21435
$598$	0	0
$599$	19.6906	0.804535	0.402268	−	0.915522i	$-0.368222\pi$
$599$	19.6906	0.804535	0.402268	+	0.915522i	$0.368222\pi$
$600$	0	0
$601$	29.2782	1.19428	0.597141	−	0.802136i	$-0.296303\pi$
$601$	29.2782	1.19428	0.597141	+	0.802136i	$0.296303\pi$
$602$	0	0
$603$	−0.350853	−0.0142878
$604$	0	0
$605$	32.6489	1.32737
$606$	0	0
$607$	20.8264	0.845317	0.422658	−	0.906289i	$-0.361097\pi$
$607$	20.8264	0.845317	0.422658	+	0.906289i	$0.361097\pi$
$608$	0	0
$609$	27.2522	1.10431
$610$	0	0
$611$	8.35413	0.337972
$612$	0	0
$613$	44.3603	1.79170	0.895849	−	0.444359i	$-0.146569\pi$
$613$	44.3603	1.79170	0.895849	+	0.444359i	$0.146569\pi$
$614$	0	0
$615$	40.8638	1.64779
$616$	0	0
$617$	−48.3693	−1.94727	−0.973637	−	0.228103i	$-0.926748\pi$
$617$	−48.3693	−1.94727	−0.973637	+	0.228103i	$0.926748\pi$
$618$	0	0
$619$	−27.4269	−1.10238	−0.551190	−	0.834380i	$-0.685826\pi$
$619$	−27.4269	−1.10238	−0.551190	+	0.834380i	$0.685826\pi$
$620$	0	0
$621$	−30.6627	−1.23045
$622$	0	0
$623$	−12.3598	−0.495185
$624$	0	0
$625$	−26.0804	−1.04322
$626$	0	0
$627$	0.704704	0.0281432
$628$	0	0
$629$	54.1134	2.15764
$630$	0	0
$631$	−35.1087	−1.39766	−0.698828	−	0.715290i	$-0.746295\pi$
$631$	−35.1087	−1.39766	−0.698828	+	0.715290i	$0.746295\pi$
$632$	0	0
$633$	20.8288	0.827869
$634$	0	0
$635$	−37.6990	−1.49604
$636$	0	0
$637$	−43.5695	−1.72629
$638$	0	0
$639$	−0.793879	−0.0314054
$640$	0	0
$641$	35.6571	1.40837	0.704185	−	0.710016i	$-0.251313\pi$
$641$	35.6571	1.40837	0.704185	+	0.710016i	$0.251313\pi$
$642$	0	0
$643$	−29.9510	−1.18115	−0.590576	−	0.806982i	$-0.701099\pi$
$643$	−29.9510	−1.18115	−0.590576	+	0.806982i	$0.701099\pi$
$644$	0	0
$645$	45.0377	1.77336
$646$	0	0
$647$	−21.7154	−0.853720	−0.426860	−	0.904318i	$-0.640380\pi$
$647$	−21.7154	−0.853720	−0.426860	+	0.904318i	$0.640380\pi$
$648$	0	0
$649$	6.06585	0.238106
$650$	0	0
$651$	−1.25631	−0.0492386
$652$	0	0
$653$	−32.3495	−1.26593	−0.632967	−	0.774179i	$-0.718163\pi$
$653$	−32.3495	−1.26593	−0.632967	+	0.774179i	$0.718163\pi$
$654$	0	0
$655$	−32.1521	−1.25628
$656$	0	0
$657$	0.294471	0.0114884
$658$	0	0
$659$	12.2878	0.478664	0.239332	−	0.970938i	$-0.423072\pi$
$659$	12.2878	0.478664	0.239332	+	0.970938i	$0.423072\pi$
$660$	0	0
$661$	−41.6979	−1.62186	−0.810929	−	0.585144i	$-0.801038\pi$
$661$	−41.6979	−1.62186	−0.810929	+	0.585144i	$0.801038\pi$
$662$	0	0
$663$	57.0188	2.21443
$664$	0	0
$665$	6.30537	0.244512
$666$	0	0
$667$	−24.8274	−0.961320
$668$	0	0
$669$	6.64838	0.257041
$670$	0	0
$671$	0.344589	0.0133027
$672$	0	0
$673$	−41.9990	−1.61894	−0.809471	−	0.587160i	$-0.800246\pi$
$673$	−41.9990	−1.61894	−0.809471	+	0.587160i	$0.800246\pi$
$674$	0	0
$675$	24.5046	0.943182
$676$	0	0
$677$	3.11452	0.119701	0.0598505	−	0.998207i	$-0.480938\pi$
$677$	3.11452	0.119701	0.0598505	+	0.998207i	$0.480938\pi$
$678$	0	0
$679$	10.5349	0.404291
$680$	0	0
$681$	0.916829	0.0351330
$682$	0	0
$683$	29.7648	1.13892	0.569458	−	0.822020i	$-0.307153\pi$
$683$	29.7648	1.13892	0.569458	+	0.822020i	$0.307153\pi$
$684$	0	0
$685$	−62.4299	−2.38532
$686$	0	0
$687$	−38.3428	−1.46287
$688$	0	0
$689$	31.8208	1.21228
$690$	0	0
$691$	13.7777	0.524127	0.262063	−	0.965051i	$-0.415597\pi$
$691$	13.7777	0.524127	0.262063	+	0.965051i	$0.415597\pi$
$692$	0	0
$693$	0.196935	0.00748094
$694$	0	0
$695$	−1.30958	−0.0496751
$696$	0	0
$697$	39.0017	1.47730
$698$	0	0
$699$	−42.7379	−1.61650
$700$	0	0
$701$	−18.2276	−0.688448	−0.344224	−	0.938888i	$-0.611858\pi$
$701$	−18.2276	−0.688448	−0.344224	+	0.938888i	$0.611858\pi$
$702$	0	0
$703$	5.57863	0.210402
$704$	0	0
$705$	−7.35375	−0.276958
$706$	0	0
$707$	67.6158	2.54295
$708$	0	0
$709$	16.2898	0.611778	0.305889	−	0.952067i	$-0.401046\pi$
$709$	16.2898	0.611778	0.305889	+	0.952067i	$0.401046\pi$
$710$	0	0
$711$	−0.0860847	−0.00322843
$712$	0	0
$713$	1.14453	0.0428629
$714$	0	0
$715$	−14.5162	−0.542875
$716$	0	0
$717$	32.1804	1.20180
$718$	0	0
$719$	46.8213	1.74614	0.873070	−	0.487594i	$-0.162126\pi$
$719$	46.8213	1.74614	0.873070	+	0.487594i	$0.162126\pi$
$720$	0	0
$721$	−46.0635	−1.71550
$722$	0	0
$723$	−22.1346	−0.823194
$724$	0	0
$725$	19.8412	0.736883
$726$	0	0
$727$	5.56992	0.206577	0.103288	−	0.994651i	$-0.467064\pi$
$727$	5.56992	0.206577	0.103288	+	0.994651i	$0.467064\pi$
$728$	0	0
$729$	26.3356	0.975393
$730$	0	0
$731$	42.9855	1.58987
$732$	0	0
$733$	8.24361	0.304485	0.152242	−	0.988343i	$-0.451351\pi$
$733$	8.24361	0.304485	0.152242	+	0.988343i	$0.451351\pi$
$734$	0	0
$735$	38.3522	1.41464
$736$	0	0
$737$	3.71083	0.136690
$738$	0	0
$739$	−38.4219	−1.41337	−0.706686	−	0.707527i	$-0.749811\pi$
$739$	−38.4219	−1.41337	−0.706686	+	0.707527i	$0.749811\pi$
$740$	0	0
$741$	5.87814	0.215939
$742$	0	0
$743$	−39.0600	−1.43297	−0.716486	−	0.697601i	$-0.754251\pi$
$743$	−39.0600	−1.43297	−0.716486	+	0.697601i	$0.754251\pi$
$744$	0	0
$745$	−74.0430	−2.71272
$746$	0	0
$747$	0.879700	0.0321865
$748$	0	0
$749$	−44.4796	−1.62525
$750$	0	0
$751$	14.1361	0.515834	0.257917	−	0.966167i	$-0.416964\pi$
$751$	14.1361	0.515834	0.257917	+	0.966167i	$0.416964\pi$
$752$	0	0
$753$	22.0987	0.805322
$754$	0	0
$755$	−33.4896	−1.21881
$756$	0	0
$757$	−47.3731	−1.72180	−0.860902	−	0.508770i	$-0.830100\pi$
$757$	−47.3731	−1.72180	−0.860902	+	0.508770i	$0.830100\pi$
$758$	0	0
$759$	−7.80942	−0.283464
$760$	0	0
$761$	48.7333	1.76658	0.883291	−	0.468825i	$-0.155322\pi$
$761$	48.7333	1.76658	0.883291	+	0.468825i	$0.155322\pi$
$762$	0	0
$763$	58.3712	2.11318
$764$	0	0
$765$	−1.15308	−0.0416898
$766$	0	0
$767$	50.5970	1.82695
$768$	0	0
$769$	15.3624	0.553981	0.276990	−	0.960873i	$-0.410663\pi$
$769$	15.3624	0.553981	0.276990	+	0.960873i	$0.410663\pi$
$770$	0	0
$771$	−7.50156	−0.270162
$772$	0	0
$773$	30.7098	1.10456	0.552278	−	0.833660i	$-0.313759\pi$
$773$	30.7098	1.10456	0.552278	+	0.833660i	$0.313759\pi$
$774$	0	0
$775$	−0.914667	−0.0328558
$776$	0	0
$777$	67.8592	2.43444
$778$	0	0
$779$	4.02074	0.144058
$780$	0	0
$781$	8.39654	0.300452
$782$	0	0
$783$	21.3358	0.762481
$784$	0	0
$785$	−27.7413	−0.990128
$786$	0	0
$787$	−0.452236	−0.0161205	−0.00806024	−	0.999968i	$-0.502566\pi$
$787$	−0.452236	−0.0161205	−0.00806024	+	0.999968i	$0.502566\pi$
$788$	0	0
$789$	4.89329	0.174206
$790$	0	0
$791$	39.6988	1.41153
$792$	0	0
$793$	2.87432	0.102070
$794$	0	0
$795$	−28.0104	−0.993427
$796$	0	0
$797$	−33.6217	−1.19094	−0.595470	−	0.803377i	$-0.703034\pi$
$797$	−33.6217	−1.19094	−0.595470	+	0.803377i	$0.703034\pi$
$798$	0	0
$799$	−7.01867	−0.248303
$800$	0	0
$801$	0.233014	0.00823315
$802$	0	0
$803$	−3.11450	−0.109908
$804$	0	0
$805$	−69.8751	−2.46278
$806$	0	0
$807$	−20.4710	−0.720614
$808$	0	0
$809$	9.69594	0.340891	0.170446	−	0.985367i	$-0.445479\pi$
$809$	9.69594	0.340891	0.170446	+	0.985367i	$0.445479\pi$
$810$	0	0
$811$	38.9501	1.36772	0.683861	−	0.729613i	$-0.260300\pi$
$811$	38.9501	1.36772	0.683861	+	0.729613i	$0.260300\pi$
$812$	0	0
$813$	35.3882	1.24112
$814$	0	0
$815$	74.4558	2.60807
$816$	0	0
$817$	4.43143	0.155036
$818$	0	0
$819$	1.64269	0.0574003
$820$	0	0
$821$	43.4484	1.51636	0.758180	−	0.652046i	$-0.226089\pi$
$821$	43.4484	1.51636	0.758180	+	0.652046i	$0.226089\pi$
$822$	0	0
$823$	−38.3112	−1.33544	−0.667722	−	0.744410i	$-0.732731\pi$
$823$	−38.3112	−1.33544	−0.667722	+	0.744410i	$0.732731\pi$
$824$	0	0
$825$	6.24102	0.217284
$826$	0	0
$827$	−3.57448	−0.124297	−0.0621484	−	0.998067i	$-0.519795\pi$
$827$	−3.57448	−0.124297	−0.0621484	+	0.998067i	$0.519795\pi$
$828$	0	0
$829$	−9.20895	−0.319840	−0.159920	−	0.987130i	$-0.551124\pi$
$829$	−9.20895	−0.319840	−0.159920	+	0.987130i	$0.551124\pi$
$830$	0	0
$831$	9.29392	0.322403
$832$	0	0
$833$	36.6047	1.26828
$834$	0	0
$835$	−23.0387	−0.797287
$836$	0	0
$837$	−0.983570	−0.0339972
$838$	0	0
$839$	16.6352	0.574311	0.287156	−	0.957884i	$-0.407290\pi$
$839$	16.6352	0.574311	0.287156	+	0.957884i	$0.407290\pi$
$840$	0	0
$841$	−11.7245	−0.404294
$842$	0	0
$843$	−33.5197	−1.15448
$844$	0	0
$845$	−80.4420	−2.76729
$846$	0	0
$847$	39.0767	1.34269
$848$	0	0
$849$	55.6167	1.90876
$850$	0	0
$851$	−61.8214	−2.11921
$852$	0	0
$853$	−5.31682	−0.182045	−0.0910223	−	0.995849i	$-0.529013\pi$
$853$	−5.31682	−0.182045	−0.0910223	+	0.995849i	$0.529013\pi$
$854$	0	0
$855$	−0.118873	−0.00406536
$856$	0	0
$857$	−18.8770	−0.644825	−0.322412	−	0.946599i	$-0.604494\pi$
$857$	−18.8770	−0.644825	−0.322412	+	0.946599i	$0.604494\pi$
$858$	0	0
$859$	−19.8491	−0.677242	−0.338621	−	0.940923i	$-0.609960\pi$
$859$	−19.8491	−0.677242	−0.338621	+	0.940923i	$0.609960\pi$
$860$	0	0
$861$	48.9089	1.66681
$862$	0	0
$863$	41.5768	1.41529	0.707646	−	0.706567i	$-0.249757\pi$
$863$	41.5768	1.41529	0.707646	+	0.706567i	$0.249757\pi$
$864$	0	0
$865$	33.5540	1.14087
$866$	0	0
$867$	−18.1149	−0.615215
$868$	0	0
$869$	0.910484	0.0308861
$870$	0	0
$871$	30.9531	1.04881
$872$	0	0
$873$	−0.198609	−0.00672191
$874$	0	0
$875$	−2.64758	−0.0895045
$876$	0	0
$877$	−11.4855	−0.387837	−0.193919	−	0.981018i	$-0.562120\pi$
$877$	−11.4855	−0.387837	−0.193919	+	0.981018i	$0.562120\pi$
$878$	0	0
$879$	36.4908	1.23080
$880$	0	0
$881$	15.2075	0.512352	0.256176	−	0.966630i	$-0.417537\pi$
$881$	15.2075	0.512352	0.256176	+	0.966630i	$0.417537\pi$
$882$	0	0
$883$	−35.2500	−1.18626	−0.593128	−	0.805108i	$-0.702107\pi$
$883$	−35.2500	−1.18626	−0.593128	+	0.805108i	$0.702107\pi$
$884$	0	0
$885$	−44.5382	−1.49714
$886$	0	0
$887$	24.4293	0.820255	0.410127	−	0.912028i	$-0.365484\pi$
$887$	24.4293	0.820255	0.410127	+	0.912028i	$0.365484\pi$
$888$	0	0
$889$	−45.1210	−1.51331
$890$	0	0
$891$	6.86905	0.230122
$892$	0	0
$893$	−0.723563	−0.0242131
$894$	0	0
$895$	60.8916	2.03538
$896$	0	0
$897$	−65.1406	−2.17498
$898$	0	0
$899$	−0.796390	−0.0265611
$900$	0	0
$901$	−26.7341	−0.890641
$902$	0	0
$903$	53.9045	1.79383
$904$	0	0
$905$	40.6281	1.35052
$906$	0	0
$907$	27.7752	0.922261	0.461131	−	0.887332i	$-0.347444\pi$
$907$	27.7752	0.922261	0.461131	+	0.887332i	$0.347444\pi$
$908$	0	0
$909$	−1.27473	−0.0422802
$910$	0	0
$911$	−39.9486	−1.32356	−0.661778	−	0.749700i	$-0.730198\pi$
$911$	−39.9486	−1.32356	−0.661778	+	0.749700i	$0.730198\pi$
$912$	0	0
$913$	−9.30423	−0.307925
$914$	0	0
$915$	−2.53013	−0.0836435
$916$	0	0
$917$	−38.4820	−1.27079
$918$	0	0
$919$	−6.57457	−0.216875	−0.108438	−	0.994103i	$-0.534585\pi$
$919$	−6.57457	−0.216875	−0.108438	+	0.994103i	$0.534585\pi$
$920$	0	0
$921$	−11.9467	−0.393656
$922$	0	0
$923$	70.0379	2.30533
$924$	0	0
$925$	49.4055	1.62444
$926$	0	0
$927$	0.868417	0.0285226
$928$	0	0
$929$	−12.6692	−0.415664	−0.207832	−	0.978164i	$-0.566641\pi$
$929$	−12.6692	−0.415664	−0.207832	+	0.978164i	$0.566641\pi$
$930$	0	0
$931$	3.77362	0.123675
$932$	0	0
$933$	2.10849	0.0690289
$934$	0	0
$935$	12.1957	0.398841
$936$	0	0
$937$	16.7836	0.548295	0.274148	−	0.961688i	$-0.411604\pi$
$937$	16.7836	0.548295	0.274148	+	0.961688i	$0.411604\pi$
$938$	0	0
$939$	−47.1933	−1.54010
$940$	0	0
$941$	38.0207	1.23944	0.619719	−	0.784824i	$-0.287247\pi$
$941$	38.0207	1.23944	0.619719	+	0.784824i	$0.287247\pi$
$942$	0	0
$943$	−44.5572	−1.45098
$944$	0	0
$945$	60.0484	1.95338
$946$	0	0
$947$	−0.314183	−0.0102096	−0.00510479	−	0.999987i	$-0.501625\pi$
$947$	−0.314183	−0.0102096	−0.00510479	+	0.999987i	$0.501625\pi$
$948$	0	0
$949$	−25.9789	−0.843312
$950$	0	0
$951$	−42.6977	−1.38457
$952$	0	0
$953$	20.4941	0.663869	0.331935	−	0.943302i	$-0.392299\pi$
$953$	20.4941	0.663869	0.331935	+	0.943302i	$0.392299\pi$
$954$	0	0
$955$	24.0377	0.777842
$956$	0	0
$957$	5.43398	0.175656
$958$	0	0
$959$	−74.7208	−2.41286
$960$	0	0
$961$	−30.9633	−0.998816
$962$	0	0
$963$	0.838555	0.0270221
$964$	0	0
$965$	−25.9556	−0.835539
$966$	0	0
$967$	−19.5676	−0.629252	−0.314626	−	0.949216i	$-0.601879\pi$
$967$	−19.5676	−0.629252	−0.314626	+	0.949216i	$0.601879\pi$
$968$	0	0
$969$	−4.93848	−0.158647
$970$	0	0
$971$	17.7614	0.569992	0.284996	−	0.958529i	$-0.408008\pi$
$971$	17.7614	0.569992	0.284996	+	0.958529i	$0.408008\pi$
$972$	0	0
$973$	−1.56740	−0.0502486
$974$	0	0
$975$	52.0581	1.66719
$976$	0	0
$977$	−27.8556	−0.891180	−0.445590	−	0.895237i	$-0.647006\pi$
$977$	−27.8556	−0.891180	−0.445590	+	0.895237i	$0.647006\pi$
$978$	0	0
$979$	−2.46450	−0.0787657
$980$	0	0
$981$	−1.10045	−0.0351346
$982$	0	0
$983$	−17.4966	−0.558056	−0.279028	−	0.960283i	$-0.590012\pi$
$983$	−17.4966	−0.558056	−0.279028	+	0.960283i	$0.590012\pi$
$984$	0	0
$985$	41.5392	1.32355
$986$	0	0
$987$	−8.80153	−0.280156
$988$	0	0
$989$	−49.1084	−1.56155
$990$	0	0
$991$	32.0373	1.01770	0.508848	−	0.860856i	$-0.330071\pi$
$991$	32.0373	1.01770	0.508848	+	0.860856i	$0.330071\pi$
$992$	0	0
$993$	39.7649	1.26190
$994$	0	0
$995$	−52.9361	−1.67819
$996$	0	0
$997$	17.9713	0.569157	0.284579	−	0.958653i	$-0.408146\pi$
$997$	17.9713	0.569157	0.284579	+	0.958653i	$0.408146\pi$
$998$	0	0
$999$	53.1273	1.68087

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.a.1.15	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.15	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-1.75230 q^{3} -3.12629 q^{5} -3.74178 q^{7} +0.0705422 q^{9} +O(q^{10})\)	\(q-1.75230 q^{3} -3.12629 q^{5} -3.74178 q^{7} +0.0705422 q^{9} -0.746097 q^{11} -6.22341 q^{13} +5.47818 q^{15} +5.22856 q^{17} +0.539019 q^{19} +6.55671 q^{21} -5.97332 q^{23} +4.77367 q^{25} +5.13328 q^{27} +4.15638 q^{29} -0.191607 q^{31} +1.30738 q^{33} +11.6979 q^{35} +10.3496 q^{37} +10.9053 q^{39} +7.45937 q^{41} +8.22128 q^{43} -0.220535 q^{45} -1.34237 q^{47} +7.00091 q^{49} -9.16198 q^{51} -5.11309 q^{53} +2.33251 q^{55} -0.944521 q^{57} -8.13011 q^{59} -0.461856 q^{61} -0.263953 q^{63} +19.4562 q^{65} -4.97365 q^{67} +10.4670 q^{69} -11.2539 q^{71} +4.17439 q^{73} -8.36488 q^{75} +2.79173 q^{77} -1.22033 q^{79} -9.20665 q^{81} +12.4705 q^{83} -16.3460 q^{85} -7.28321 q^{87} +3.30319 q^{89} +23.2866 q^{91} +0.335752 q^{93} -1.68513 q^{95} -2.81547 q^{97} -0.0526314 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * q^99

Introduction

L-functions

Modular forms

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Database

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