LMFDB - Embedded newform 8024.2.a.bc.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.52586 q^{3} +4.15650 q^{5} +0.772198 q^{7} -0.671760 q^{9} +O(q^{10})$	$q-1.52586 q^{3} +4.15650 q^{5} +0.772198 q^{7} -0.671760 q^{9} -6.48227 q^{11} +5.01426 q^{13} -6.34222 q^{15} -1.00000 q^{17} +2.58388 q^{19} -1.17826 q^{21} -7.17372 q^{23} +12.2765 q^{25} +5.60258 q^{27} +7.88571 q^{29} +3.23815 q^{31} +9.89101 q^{33} +3.20964 q^{35} -11.5812 q^{37} -7.65105 q^{39} +10.6942 q^{41} -7.59086 q^{43} -2.79217 q^{45} +1.13700 q^{47} -6.40371 q^{49} +1.52586 q^{51} -6.30273 q^{53} -26.9435 q^{55} -3.94264 q^{57} -1.00000 q^{59} +4.62608 q^{61} -0.518732 q^{63} +20.8418 q^{65} +14.5907 q^{67} +10.9461 q^{69} +2.23313 q^{71} +4.70760 q^{73} -18.7321 q^{75} -5.00559 q^{77} -1.20354 q^{79} -6.53346 q^{81} +4.65632 q^{83} -4.15650 q^{85} -12.0325 q^{87} +14.7750 q^{89} +3.87200 q^{91} -4.94095 q^{93} +10.7399 q^{95} -3.28393 q^{97} +4.35453 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * 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
$3$	−1.52586	−0.880954	−0.440477	−	0.897764i	$-0.645191\pi$
$3$	−1.52586	−0.880954	−0.440477	+	0.897764i	$0.645191\pi$
$4$	0	0
$5$	4.15650	1.85884	0.929421	−	0.369021i	$-0.120307\pi$
$5$	4.15650	1.85884	0.929421	+	0.369021i	$0.120307\pi$
$6$	0	0
$7$	0.772198	0.291863	0.145932	−	0.989295i	$-0.453382\pi$
$7$	0.772198	0.291863	0.145932	+	0.989295i	$0.453382\pi$
$8$	0	0
$9$	−0.671760	−0.223920
$10$	0	0
$11$	−6.48227	−1.95448	−0.977239	−	0.212143i	$-0.931956\pi$
$11$	−6.48227	−1.95448	−0.977239	+	0.212143i	$0.931956\pi$
$12$	0	0
$13$	5.01426	1.39071	0.695353	−	0.718668i	$-0.255248\pi$
$13$	5.01426	1.39071	0.695353	+	0.718668i	$0.255248\pi$
$14$	0	0
$15$	−6.34222	−1.63755
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.58388	0.592783	0.296392	−	0.955066i	$-0.404217\pi$
$19$	2.58388	0.592783	0.296392	+	0.955066i	$0.404217\pi$
$20$	0	0
$21$	−1.17826	−0.257118
$22$	0	0
$23$	−7.17372	−1.49582	−0.747912	−	0.663798i	$-0.768944\pi$
$23$	−7.17372	−1.49582	−0.747912	+	0.663798i	$0.768944\pi$
$24$	0	0
$25$	12.2765	2.45529
$26$	0	0
$27$	5.60258	1.07822
$28$	0	0
$29$	7.88571	1.46434	0.732170	−	0.681122i	$-0.238508\pi$
$29$	7.88571	1.46434	0.732170	+	0.681122i	$0.238508\pi$
$30$	0	0
$31$	3.23815	0.581589	0.290794	−	0.956786i	$-0.406080\pi$
$31$	3.23815	0.581589	0.290794	+	0.956786i	$0.406080\pi$
$32$	0	0
$33$	9.89101	1.72180
$34$	0	0
$35$	3.20964	0.542528
$36$	0	0
$37$	−11.5812	−1.90395	−0.951973	−	0.306183i	$-0.900948\pi$
$37$	−11.5812	−1.90395	−0.951973	+	0.306183i	$0.900948\pi$
$38$	0	0
$39$	−7.65105	−1.22515
$40$	0	0
$41$	10.6942	1.67015	0.835076	−	0.550134i	$-0.185424\pi$
$41$	10.6942	1.67015	0.835076	+	0.550134i	$0.185424\pi$
$42$	0	0
$43$	−7.59086	−1.15759	−0.578797	−	0.815471i	$-0.696478\pi$
$43$	−7.59086	−1.15759	−0.578797	+	0.815471i	$0.696478\pi$
$44$	0	0
$45$	−2.79217	−0.416232
$46$	0	0
$47$	1.13700	0.165848	0.0829242	−	0.996556i	$-0.473574\pi$
$47$	1.13700	0.165848	0.0829242	+	0.996556i	$0.473574\pi$
$48$	0	0
$49$	−6.40371	−0.914816
$50$	0	0
$51$	1.52586	0.213663
$52$	0	0
$53$	−6.30273	−0.865747	−0.432874	−	0.901455i	$-0.642500\pi$
$53$	−6.30273	−0.865747	−0.432874	+	0.901455i	$0.642500\pi$
$54$	0	0
$55$	−26.9435	−3.63306
$56$	0	0
$57$	−3.94264	−0.522215
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	4.62608	0.592309	0.296154	−	0.955140i	$-0.404296\pi$
$61$	4.62608	0.592309	0.296154	+	0.955140i	$0.404296\pi$
$62$	0	0
$63$	−0.518732	−0.0653541
$64$	0	0
$65$	20.8418	2.58510
$66$	0	0
$67$	14.5907	1.78254	0.891270	−	0.453472i	$-0.149815\pi$
$67$	14.5907	1.78254	0.891270	+	0.453472i	$0.149815\pi$
$68$	0	0
$69$	10.9461	1.31775
$70$	0	0
$71$	2.23313	0.265024	0.132512	−	0.991181i	$-0.457696\pi$
$71$	2.23313	0.265024	0.132512	+	0.991181i	$0.457696\pi$
$72$	0	0
$73$	4.70760	0.550983	0.275491	−	0.961304i	$-0.411159\pi$
$73$	4.70760	0.550983	0.275491	+	0.961304i	$0.411159\pi$
$74$	0	0
$75$	−18.7321	−2.16300
$76$	0	0
$77$	−5.00559	−0.570440
$78$	0	0
$79$	−1.20354	−0.135409	−0.0677046	−	0.997705i	$-0.521568\pi$
$79$	−1.20354	−0.135409	−0.0677046	+	0.997705i	$0.521568\pi$
$80$	0	0
$81$	−6.53346	−0.725940
$82$	0	0
$83$	4.65632	0.511097	0.255549	−	0.966796i	$-0.417744\pi$
$83$	4.65632	0.511097	0.255549	+	0.966796i	$0.417744\pi$
$84$	0	0
$85$	−4.15650	−0.450835
$86$	0	0
$87$	−12.0325	−1.29002
$88$	0	0
$89$	14.7750	1.56615	0.783075	−	0.621928i	$-0.213650\pi$
$89$	14.7750	1.56615	0.783075	+	0.621928i	$0.213650\pi$
$90$	0	0
$91$	3.87200	0.405896
$92$	0	0
$93$	−4.94095	−0.512353
$94$	0	0
$95$	10.7399	1.10189
$96$	0	0
$97$	−3.28393	−0.333433	−0.166716	−	0.986005i	$-0.553316\pi$
$97$	−3.28393	−0.333433	−0.166716	+	0.986005i	$0.553316\pi$
$98$	0	0
$99$	4.35453	0.437647
$100$	0	0
$101$	−4.45381	−0.443171	−0.221586	−	0.975141i	$-0.571123\pi$
$101$	−4.45381	−0.443171	−0.221586	+	0.975141i	$0.571123\pi$
$102$	0	0
$103$	13.3492	1.31533	0.657667	−	0.753309i	$-0.271543\pi$
$103$	13.3492	1.31533	0.657667	+	0.753309i	$0.271543\pi$
$104$	0	0
$105$	−4.89745	−0.477942
$106$	0	0
$107$	5.51567	0.533220	0.266610	−	0.963804i	$-0.414096\pi$
$107$	5.51567	0.533220	0.266610	+	0.963804i	$0.414096\pi$
$108$	0	0
$109$	2.67973	0.256671	0.128336	−	0.991731i	$-0.459037\pi$
$109$	2.67973	0.256671	0.128336	+	0.991731i	$0.459037\pi$
$110$	0	0
$111$	17.6713	1.67729
$112$	0	0
$113$	−12.3054	−1.15760	−0.578799	−	0.815470i	$-0.696478\pi$
$113$	−12.3054	−1.15760	−0.578799	+	0.815470i	$0.696478\pi$
$114$	0	0
$115$	−29.8175	−2.78050
$116$	0	0
$117$	−3.36838	−0.311407
$118$	0	0
$119$	−0.772198	−0.0707873
$120$	0	0
$121$	31.0198	2.81998
$122$	0	0
$123$	−16.3178	−1.47133
$124$	0	0
$125$	30.2446	2.70516
$126$	0	0
$127$	−10.1891	−0.904139	−0.452070	−	0.891983i	$-0.649314\pi$
$127$	−10.1891	−0.904139	−0.452070	+	0.891983i	$0.649314\pi$
$128$	0	0
$129$	11.5826	1.01979
$130$	0	0
$131$	10.5360	0.920534	0.460267	−	0.887781i	$-0.347754\pi$
$131$	10.5360	0.920534	0.460267	+	0.887781i	$0.347754\pi$
$132$	0	0
$133$	1.99527	0.173012
$134$	0	0
$135$	23.2871	2.00424
$136$	0	0
$137$	−14.5444	−1.24261	−0.621306	−	0.783568i	$-0.713397\pi$
$137$	−14.5444	−1.24261	−0.621306	+	0.783568i	$0.713397\pi$
$138$	0	0
$139$	21.7175	1.84205	0.921026	−	0.389500i	$-0.127352\pi$
$139$	21.7175	1.84205	0.921026	+	0.389500i	$0.127352\pi$
$140$	0	0
$141$	−1.73490	−0.146105
$142$	0	0
$143$	−32.5038	−2.71810
$144$	0	0
$145$	32.7769	2.72198
$146$	0	0
$147$	9.77115	0.805911
$148$	0	0
$149$	−0.427615	−0.0350316	−0.0175158	−	0.999847i	$-0.505576\pi$
$149$	−0.427615	−0.0350316	−0.0175158	+	0.999847i	$0.505576\pi$
$150$	0	0
$151$	−3.93727	−0.320411	−0.160205	−	0.987084i	$-0.551216\pi$
$151$	−3.93727	−0.320411	−0.160205	+	0.987084i	$0.551216\pi$
$152$	0	0
$153$	0.671760	0.0543086
$154$	0	0
$155$	13.4594	1.08108
$156$	0	0
$157$	18.7285	1.49470	0.747349	−	0.664431i	$-0.231326\pi$
$157$	18.7285	1.49470	0.747349	+	0.664431i	$0.231326\pi$
$158$	0	0
$159$	9.61707	0.762683
$160$	0	0
$161$	−5.53953	−0.436576
$162$	0	0
$163$	12.8649	1.00766	0.503829	−	0.863804i	$-0.331924\pi$
$163$	12.8649	1.00766	0.503829	+	0.863804i	$0.331924\pi$
$164$	0	0
$165$	41.1120	3.20056
$166$	0	0
$167$	−9.55003	−0.739003	−0.369502	−	0.929230i	$-0.620472\pi$
$167$	−9.55003	−0.739003	−0.369502	+	0.929230i	$0.620472\pi$
$168$	0	0
$169$	12.1428	0.934062
$170$	0	0
$171$	−1.73575	−0.132736
$172$	0	0
$173$	19.2201	1.46128	0.730640	−	0.682763i	$-0.239222\pi$
$173$	19.2201	1.46128	0.730640	+	0.682763i	$0.239222\pi$
$174$	0	0
$175$	9.47986	0.716610
$176$	0	0
$177$	1.52586	0.114690
$178$	0	0
$179$	18.9553	1.41679	0.708395	−	0.705817i	$-0.249420\pi$
$179$	18.9553	1.41679	0.708395	+	0.705817i	$0.249420\pi$
$180$	0	0
$181$	4.36959	0.324789	0.162394	−	0.986726i	$-0.448078\pi$
$181$	4.36959	0.324789	0.162394	+	0.986726i	$0.448078\pi$
$182$	0	0
$183$	−7.05874	−0.521797
$184$	0	0
$185$	−48.1374	−3.53913
$186$	0	0
$187$	6.48227	0.474030
$188$	0	0
$189$	4.32630	0.314692
$190$	0	0
$191$	−2.65015	−0.191758	−0.0958792	−	0.995393i	$-0.530566\pi$
$191$	−2.65015	−0.191758	−0.0958792	+	0.995393i	$0.530566\pi$
$192$	0	0
$193$	−1.64143	−0.118153	−0.0590763	−	0.998253i	$-0.518816\pi$
$193$	−1.64143	−0.118153	−0.0590763	+	0.998253i	$0.518816\pi$
$194$	0	0
$195$	−31.8015	−2.27736
$196$	0	0
$197$	−18.7807	−1.33807	−0.669036	−	0.743230i	$-0.733293\pi$
$197$	−18.7807	−1.33807	−0.669036	+	0.743230i	$0.733293\pi$
$198$	0	0
$199$	1.87835	0.133153	0.0665765	−	0.997781i	$-0.478792\pi$
$199$	1.87835	0.133153	0.0665765	+	0.997781i	$0.478792\pi$
$200$	0	0
$201$	−22.2634	−1.57034
$202$	0	0
$203$	6.08933	0.427387
$204$	0	0
$205$	44.4504	3.10455
$206$	0	0
$207$	4.81902	0.334945
$208$	0	0
$209$	−16.7494	−1.15858
$210$	0	0
$211$	−10.9969	−0.757057	−0.378528	−	0.925590i	$-0.623570\pi$
$211$	−10.9969	−0.757057	−0.378528	+	0.925590i	$0.623570\pi$
$212$	0	0
$213$	−3.40744	−0.233474
$214$	0	0
$215$	−31.5514	−2.15179
$216$	0	0
$217$	2.50049	0.169745
$218$	0	0
$219$	−7.18313	−0.485391
$220$	0	0
$221$	−5.01426	−0.337296
$222$	0	0
$223$	11.1407	0.746034	0.373017	−	0.927824i	$-0.378323\pi$
$223$	11.1407	0.746034	0.373017	+	0.927824i	$0.378323\pi$
$224$	0	0
$225$	−8.24684	−0.549789
$226$	0	0
$227$	−11.1542	−0.740333	−0.370167	−	0.928965i	$-0.620699\pi$
$227$	−11.1542	−0.740333	−0.370167	+	0.928965i	$0.620699\pi$
$228$	0	0
$229$	−25.5978	−1.69155	−0.845774	−	0.533541i	$-0.820861\pi$
$229$	−25.5978	−1.69155	−0.845774	+	0.533541i	$0.820861\pi$
$230$	0	0
$231$	7.63782	0.502532
$232$	0	0
$233$	1.91425	0.125407	0.0627033	−	0.998032i	$-0.480028\pi$
$233$	1.91425	0.125407	0.0627033	+	0.998032i	$0.480028\pi$
$234$	0	0
$235$	4.72593	0.308286
$236$	0	0
$237$	1.83643	0.119289
$238$	0	0
$239$	12.2185	0.790349	0.395175	−	0.918606i	$-0.370684\pi$
$239$	12.2185	0.790349	0.395175	+	0.918606i	$0.370684\pi$
$240$	0	0
$241$	−29.7336	−1.91531	−0.957655	−	0.287918i	$-0.907037\pi$
$241$	−29.7336	−1.91531	−0.957655	+	0.287918i	$0.907037\pi$
$242$	0	0
$243$	−6.83862	−0.438698
$244$	0	0
$245$	−26.6170	−1.70050
$246$	0	0
$247$	12.9563	0.824387
$248$	0	0
$249$	−7.10488	−0.450253
$250$	0	0
$251$	23.1134	1.45890	0.729451	−	0.684033i	$-0.239776\pi$
$251$	23.1134	1.45890	0.729451	+	0.684033i	$0.239776\pi$
$252$	0	0
$253$	46.5020	2.92355
$254$	0	0
$255$	6.34222	0.397165
$256$	0	0
$257$	14.3285	0.893786	0.446893	−	0.894588i	$-0.352531\pi$
$257$	14.3285	0.893786	0.446893	+	0.894588i	$0.352531\pi$
$258$	0	0
$259$	−8.94301	−0.555692
$260$	0	0
$261$	−5.29731	−0.327895
$262$	0	0
$263$	−5.12173	−0.315820	−0.157910	−	0.987454i	$-0.550476\pi$
$263$	−5.12173	−0.315820	−0.157910	+	0.987454i	$0.550476\pi$
$264$	0	0
$265$	−26.1973	−1.60929
$266$	0	0
$267$	−22.5446	−1.37971
$268$	0	0
$269$	−21.9370	−1.33752	−0.668762	−	0.743476i	$-0.733176\pi$
$269$	−21.9370	−1.33752	−0.668762	+	0.743476i	$0.733176\pi$
$270$	0	0
$271$	5.09748	0.309650	0.154825	−	0.987942i	$-0.450519\pi$
$271$	5.09748	0.309650	0.154825	+	0.987942i	$0.450519\pi$
$272$	0	0
$273$	−5.90812	−0.357576
$274$	0	0
$275$	−79.5793	−4.79881
$276$	0	0
$277$	13.0673	0.785136	0.392568	−	0.919723i	$-0.371587\pi$
$277$	13.0673	0.785136	0.392568	+	0.919723i	$0.371587\pi$
$278$	0	0
$279$	−2.17526	−0.130229
$280$	0	0
$281$	8.93261	0.532875	0.266437	−	0.963852i	$-0.414153\pi$
$281$	8.93261	0.532875	0.266437	+	0.963852i	$0.414153\pi$
$282$	0	0
$283$	−2.35741	−0.140133	−0.0700667	−	0.997542i	$-0.522321\pi$
$283$	−2.35741	−0.140133	−0.0700667	+	0.997542i	$0.522321\pi$
$284$	0	0
$285$	−16.3876	−0.970715
$286$	0	0
$287$	8.25803	0.487456
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	5.01081	0.293739
$292$	0	0
$293$	−15.6375	−0.913552	−0.456776	−	0.889582i	$-0.650996\pi$
$293$	−15.6375	−0.913552	−0.456776	+	0.889582i	$0.650996\pi$
$294$	0	0
$295$	−4.15650	−0.242001
$296$	0	0
$297$	−36.3174	−2.10735
$298$	0	0
$299$	−35.9709	−2.08025
$300$	0	0
$301$	−5.86165	−0.337860
$302$	0	0
$303$	6.79588	0.390413
$304$	0	0
$305$	19.2283	1.10101
$306$	0	0
$307$	23.2646	1.32778	0.663892	−	0.747829i	$-0.268904\pi$
$307$	23.2646	1.32778	0.663892	+	0.747829i	$0.268904\pi$
$308$	0	0
$309$	−20.3689	−1.15875
$310$	0	0
$311$	7.68498	0.435775	0.217888	−	0.975974i	$-0.430083\pi$
$311$	7.68498	0.435775	0.217888	+	0.975974i	$0.430083\pi$
$312$	0	0
$313$	13.3520	0.754700	0.377350	−	0.926071i	$-0.376835\pi$
$313$	13.3520	0.754700	0.377350	+	0.926071i	$0.376835\pi$
$314$	0	0
$315$	−2.15611	−0.121483
$316$	0	0
$317$	−7.16217	−0.402267	−0.201134	−	0.979564i	$-0.564463\pi$
$317$	−7.16217	−0.402267	−0.201134	+	0.979564i	$0.564463\pi$
$318$	0	0
$319$	−51.1173	−2.86202
$320$	0	0
$321$	−8.41613	−0.469742
$322$	0	0
$323$	−2.58388	−0.143771
$324$	0	0
$325$	61.5574	3.41459
$326$	0	0
$327$	−4.08888	−0.226116
$328$	0	0
$329$	0.877988	0.0484051
$330$	0	0
$331$	1.22871	0.0675362	0.0337681	−	0.999430i	$-0.489249\pi$
$331$	1.22871	0.0675362	0.0337681	+	0.999430i	$0.489249\pi$
$332$	0	0
$333$	7.77982	0.426331
$334$	0	0
$335$	60.6463	3.31346
$336$	0	0
$337$	33.5870	1.82960	0.914801	−	0.403906i	$-0.132348\pi$
$337$	33.5870	1.82960	0.914801	+	0.403906i	$0.132348\pi$
$338$	0	0
$339$	18.7763	1.01979
$340$	0	0
$341$	−20.9906	−1.13670
$342$	0	0
$343$	−10.3503	−0.558865
$344$	0	0
$345$	45.4973	2.44949
$346$	0	0
$347$	−2.30868	−0.123937	−0.0619683	−	0.998078i	$-0.519738\pi$
$347$	−2.30868	−0.123937	−0.0619683	+	0.998078i	$0.519738\pi$
$348$	0	0
$349$	0.181700	0.00972620	0.00486310	−	0.999988i	$-0.498452\pi$
$349$	0.181700	0.00972620	0.00486310	+	0.999988i	$0.498452\pi$
$350$	0	0
$351$	28.0928	1.49948
$352$	0	0
$353$	31.4124	1.67191	0.835956	−	0.548796i	$-0.184914\pi$
$353$	31.4124	1.67191	0.835956	+	0.548796i	$0.184914\pi$
$354$	0	0
$355$	9.28200	0.492637
$356$	0	0
$357$	1.17826	0.0623603
$358$	0	0
$359$	12.6943	0.669980	0.334990	−	0.942222i	$-0.391267\pi$
$359$	12.6943	0.669980	0.334990	+	0.942222i	$0.391267\pi$
$360$	0	0
$361$	−12.3236	−0.648608
$362$	0	0
$363$	−47.3318	−2.48427
$364$	0	0
$365$	19.5671	1.02419
$366$	0	0
$367$	20.8306	1.08735	0.543673	−	0.839297i	$-0.317033\pi$
$367$	20.8306	1.08735	0.543673	+	0.839297i	$0.317033\pi$
$368$	0	0
$369$	−7.18393	−0.373981
$370$	0	0
$371$	−4.86696	−0.252680
$372$	0	0
$373$	10.8617	0.562397	0.281199	−	0.959650i	$-0.409268\pi$
$373$	10.8617	0.562397	0.281199	+	0.959650i	$0.409268\pi$
$374$	0	0
$375$	−46.1490	−2.38312
$376$	0	0
$377$	39.5410	2.03647
$378$	0	0
$379$	−33.8062	−1.73651	−0.868255	−	0.496118i	$-0.834758\pi$
$379$	−33.8062	−1.73651	−0.868255	+	0.496118i	$0.834758\pi$
$380$	0	0
$381$	15.5472	0.796505
$382$	0	0
$383$	−24.2060	−1.23687	−0.618433	−	0.785837i	$-0.712232\pi$
$383$	−24.2060	−1.23687	−0.618433	+	0.785837i	$0.712232\pi$
$384$	0	0
$385$	−20.8057	−1.06036
$386$	0	0
$387$	5.09924	0.259209
$388$	0	0
$389$	35.2842	1.78898	0.894491	−	0.447086i	$-0.147538\pi$
$389$	35.2842	1.78898	0.894491	+	0.447086i	$0.147538\pi$
$390$	0	0
$391$	7.17372	0.362791
$392$	0	0
$393$	−16.0764	−0.810948
$394$	0	0
$395$	−5.00252	−0.251704
$396$	0	0
$397$	35.8978	1.80166	0.900830	−	0.434172i	$-0.142959\pi$
$397$	35.8978	1.80166	0.900830	+	0.434172i	$0.142959\pi$
$398$	0	0
$399$	−3.04450	−0.152415
$400$	0	0
$401$	−27.4112	−1.36885	−0.684426	−	0.729083i	$-0.739947\pi$
$401$	−27.4112	−1.36885	−0.684426	+	0.729083i	$0.739947\pi$
$402$	0	0
$403$	16.2369	0.808819
$404$	0	0
$405$	−27.1563	−1.34941
$406$	0	0
$407$	75.0727	3.72122
$408$	0	0
$409$	−14.9564	−0.739547	−0.369774	−	0.929122i	$-0.620565\pi$
$409$	−14.9564	−0.739547	−0.369774	+	0.929122i	$0.620565\pi$
$410$	0	0
$411$	22.1927	1.09468
$412$	0	0
$413$	−0.772198	−0.0379974
$414$	0	0
$415$	19.3540	0.950049
$416$	0	0
$417$	−33.1378	−1.62276
$418$	0	0
$419$	6.37839	0.311605	0.155802	−	0.987788i	$-0.450204\pi$
$419$	6.37839	0.311605	0.155802	+	0.987788i	$0.450204\pi$
$420$	0	0
$421$	15.3914	0.750133	0.375067	−	0.926998i	$-0.377620\pi$
$421$	15.3914	0.750133	0.375067	+	0.926998i	$0.377620\pi$
$422$	0	0
$423$	−0.763790	−0.0371368
$424$	0	0
$425$	−12.2765	−0.595496
$426$	0	0
$427$	3.57225	0.172873
$428$	0	0
$429$	49.5961	2.39452
$430$	0	0
$431$	−1.17008	−0.0563610	−0.0281805	−	0.999603i	$-0.508971\pi$
$431$	−1.17008	−0.0563610	−0.0281805	+	0.999603i	$0.508971\pi$
$432$	0	0
$433$	14.2465	0.684642	0.342321	−	0.939583i	$-0.388787\pi$
$433$	14.2465	0.684642	0.342321	+	0.939583i	$0.388787\pi$
$434$	0	0
$435$	−50.0129	−2.39794
$436$	0	0
$437$	−18.5361	−0.886700
$438$	0	0
$439$	27.2057	1.29846	0.649230	−	0.760593i	$-0.275091\pi$
$439$	27.2057	1.29846	0.649230	+	0.760593i	$0.275091\pi$
$440$	0	0
$441$	4.30176	0.204846
$442$	0	0
$443$	−15.8924	−0.755069	−0.377535	−	0.925996i	$-0.623228\pi$
$443$	−15.8924	−0.755069	−0.377535	+	0.925996i	$0.623228\pi$
$444$	0	0
$445$	61.4123	2.91122
$446$	0	0
$447$	0.652480	0.0308612
$448$	0	0
$449$	−18.3870	−0.867738	−0.433869	−	0.900976i	$-0.642852\pi$
$449$	−18.3870	−0.867738	−0.433869	+	0.900976i	$0.642852\pi$
$450$	0	0
$451$	−69.3226	−3.26427
$452$	0	0
$453$	6.00772	0.282267
$454$	0	0
$455$	16.0940	0.754497
$456$	0	0
$457$	−25.1206	−1.17509	−0.587546	−	0.809191i	$-0.699906\pi$
$457$	−25.1206	−1.17509	−0.587546	+	0.809191i	$0.699906\pi$
$458$	0	0
$459$	−5.60258	−0.261506
$460$	0	0
$461$	−18.2202	−0.848599	−0.424300	−	0.905522i	$-0.639480\pi$
$461$	−18.2202	−0.848599	−0.424300	+	0.905522i	$0.639480\pi$
$462$	0	0
$463$	38.8317	1.80466	0.902332	−	0.431042i	$-0.141854\pi$
$463$	38.8317	1.80466	0.902332	+	0.431042i	$0.141854\pi$
$464$	0	0
$465$	−20.5371	−0.952383
$466$	0	0
$467$	−17.0536	−0.789144	−0.394572	−	0.918865i	$-0.629107\pi$
$467$	−17.0536	−0.789144	−0.394572	+	0.918865i	$0.629107\pi$
$468$	0	0
$469$	11.2669	0.520258
$470$	0	0
$471$	−28.5770	−1.31676
$472$	0	0
$473$	49.2060	2.26249
$474$	0	0
$475$	31.7209	1.45546
$476$	0	0
$477$	4.23392	0.193858
$478$	0	0
$479$	−0.590199	−0.0269669	−0.0134834	−	0.999909i	$-0.504292\pi$
$479$	−0.590199	−0.0269669	−0.0134834	+	0.999909i	$0.504292\pi$
$480$	0	0
$481$	−58.0714	−2.64783
$482$	0	0
$483$	8.45253	0.384604
$484$	0	0
$485$	−13.6497	−0.619799
$486$	0	0
$487$	−9.40897	−0.426361	−0.213180	−	0.977013i	$-0.568382\pi$
$487$	−9.40897	−0.426361	−0.213180	+	0.977013i	$0.568382\pi$
$488$	0	0
$489$	−19.6300	−0.887700
$490$	0	0
$491$	14.2768	0.644303	0.322151	−	0.946688i	$-0.395594\pi$
$491$	14.2768	0.644303	0.322151	+	0.946688i	$0.395594\pi$
$492$	0	0
$493$	−7.88571	−0.355155
$494$	0	0
$495$	18.0996	0.813516
$496$	0	0
$497$	1.72442	0.0773508
$498$	0	0
$499$	31.3542	1.40361	0.701803	−	0.712371i	$-0.252379\pi$
$499$	31.3542	1.40361	0.701803	+	0.712371i	$0.252379\pi$
$500$	0	0
$501$	14.5720	0.651028
$502$	0	0
$503$	−29.0834	−1.29677	−0.648383	−	0.761315i	$-0.724554\pi$
$503$	−29.0834	−1.29677	−0.648383	+	0.761315i	$0.724554\pi$
$504$	0	0
$505$	−18.5123	−0.823785
$506$	0	0
$507$	−18.5282	−0.822866
$508$	0	0
$509$	−20.5583	−0.911229	−0.455614	−	0.890177i	$-0.650580\pi$
$509$	−20.5583	−0.911229	−0.455614	+	0.890177i	$0.650580\pi$
$510$	0	0
$511$	3.63520	0.160812
$512$	0	0
$513$	14.4764	0.639149
$514$	0	0
$515$	55.4858	2.44500
$516$	0	0
$517$	−7.37033	−0.324147
$518$	0	0
$519$	−29.3272	−1.28732
$520$	0	0
$521$	−7.37002	−0.322886	−0.161443	−	0.986882i	$-0.551615\pi$
$521$	−7.37002	−0.322886	−0.161443	+	0.986882i	$0.551615\pi$
$522$	0	0
$523$	−11.9220	−0.521311	−0.260655	−	0.965432i	$-0.583939\pi$
$523$	−11.9220	−0.521311	−0.260655	+	0.965432i	$0.583939\pi$
$524$	0	0
$525$	−14.4649	−0.631301
$526$	0	0
$527$	−3.23815	−0.141056
$528$	0	0
$529$	28.4623	1.23749
$530$	0	0
$531$	0.671760	0.0291519
$532$	0	0
$533$	53.6235	2.32269
$534$	0	0
$535$	22.9259	0.991172
$536$	0	0
$537$	−28.9232	−1.24813
$538$	0	0
$539$	41.5106	1.78799
$540$	0	0
$541$	−0.316013	−0.0135865	−0.00679323	−	0.999977i	$-0.502162\pi$
$541$	−0.316013	−0.0135865	−0.00679323	+	0.999977i	$0.502162\pi$
$542$	0	0
$543$	−6.66736	−0.286124
$544$	0	0
$545$	11.1383	0.477111
$546$	0	0
$547$	−8.41834	−0.359942	−0.179971	−	0.983672i	$-0.557600\pi$
$547$	−8.41834	−0.359942	−0.179971	+	0.983672i	$0.557600\pi$
$548$	0	0
$549$	−3.10762	−0.132630
$550$	0	0
$551$	20.3757	0.868036
$552$	0	0
$553$	−0.929373	−0.0395210
$554$	0	0
$555$	73.4508	3.11781
$556$	0	0
$557$	11.8687	0.502893	0.251446	−	0.967871i	$-0.419094\pi$
$557$	11.8687	0.502893	0.251446	+	0.967871i	$0.419094\pi$
$558$	0	0
$559$	−38.0625	−1.60987
$560$	0	0
$561$	−9.89101	−0.417599
$562$	0	0
$563$	8.24461	0.347469	0.173735	−	0.984793i	$-0.444417\pi$
$563$	8.24461	0.347469	0.173735	+	0.984793i	$0.444417\pi$
$564$	0	0
$565$	−51.1475	−2.15179
$566$	0	0
$567$	−5.04512	−0.211875
$568$	0	0
$569$	27.7157	1.16190	0.580950	−	0.813939i	$-0.302681\pi$
$569$	27.7157	1.16190	0.580950	+	0.813939i	$0.302681\pi$
$570$	0	0
$571$	−9.77612	−0.409118	−0.204559	−	0.978854i	$-0.565576\pi$
$571$	−9.77612	−0.409118	−0.204559	+	0.978854i	$0.565576\pi$
$572$	0	0
$573$	4.04376	0.168930
$574$	0	0
$575$	−88.0679	−3.67269
$576$	0	0
$577$	11.3746	0.473530	0.236765	−	0.971567i	$-0.423913\pi$
$577$	11.3746	0.473530	0.236765	+	0.971567i	$0.423913\pi$
$578$	0	0
$579$	2.50459	0.104087
$580$	0	0
$581$	3.59560	0.149171
$582$	0	0
$583$	40.8560	1.69208
$584$	0	0
$585$	−14.0007	−0.578856
$586$	0	0
$587$	−17.3141	−0.714628	−0.357314	−	0.933984i	$-0.616307\pi$
$587$	−17.3141	−0.714628	−0.357314	+	0.933984i	$0.616307\pi$
$588$	0	0
$589$	8.36700	0.344756
$590$	0	0
$591$	28.6567	1.17878
$592$	0	0
$593$	24.3598	1.00034	0.500168	−	0.865929i	$-0.333272\pi$
$593$	24.3598	1.00034	0.500168	+	0.865929i	$0.333272\pi$
$594$	0	0
$595$	−3.20964	−0.131582
$596$	0	0
$597$	−2.86610	−0.117302
$598$	0	0
$599$	3.37002	0.137695	0.0688477	−	0.997627i	$-0.478068\pi$
$599$	3.37002	0.137695	0.0688477	+	0.997627i	$0.478068\pi$
$600$	0	0
$601$	17.2833	0.705001	0.352501	−	0.935812i	$-0.385331\pi$
$601$	17.2833	0.705001	0.352501	+	0.935812i	$0.385331\pi$
$602$	0	0
$603$	−9.80147	−0.399147
$604$	0	0
$605$	128.934	5.24190
$606$	0	0
$607$	9.78059	0.396982	0.198491	−	0.980103i	$-0.436396\pi$
$607$	9.78059	0.396982	0.198491	+	0.980103i	$0.436396\pi$
$608$	0	0
$609$	−9.29145	−0.376508
$610$	0	0
$611$	5.70121	0.230646
$612$	0	0
$613$	13.3130	0.537705	0.268853	−	0.963181i	$-0.413356\pi$
$613$	13.3130	0.537705	0.268853	+	0.963181i	$0.413356\pi$
$614$	0	0
$615$	−67.8249	−2.73496
$616$	0	0
$617$	−4.13715	−0.166555	−0.0832777	−	0.996526i	$-0.526539\pi$
$617$	−4.13715	−0.166555	−0.0832777	+	0.996526i	$0.526539\pi$
$618$	0	0
$619$	26.4700	1.06392	0.531959	−	0.846770i	$-0.321456\pi$
$619$	26.4700	1.06392	0.531959	+	0.846770i	$0.321456\pi$
$620$	0	0
$621$	−40.1913	−1.61282
$622$	0	0
$623$	11.4092	0.457102
$624$	0	0
$625$	64.3293	2.57317
$626$	0	0
$627$	25.5572	1.02066
$628$	0	0
$629$	11.5812	0.461775
$630$	0	0
$631$	−26.3490	−1.04894	−0.524468	−	0.851430i	$-0.675736\pi$
$631$	−26.3490	−1.04894	−0.524468	+	0.851430i	$0.675736\pi$
$632$	0	0
$633$	16.7797	0.666932
$634$	0	0
$635$	−42.3511	−1.68065
$636$	0	0
$637$	−32.1099	−1.27224
$638$	0	0
$639$	−1.50013	−0.0593442
$640$	0	0
$641$	24.6306	0.972852	0.486426	−	0.873722i	$-0.338300\pi$
$641$	24.6306	0.972852	0.486426	+	0.873722i	$0.338300\pi$
$642$	0	0
$643$	−9.50775	−0.374949	−0.187475	−	0.982269i	$-0.560030\pi$
$643$	−9.50775	−0.374949	−0.187475	+	0.982269i	$0.560030\pi$
$644$	0	0
$645$	48.1429	1.89562
$646$	0	0
$647$	31.0453	1.22052	0.610258	−	0.792203i	$-0.291066\pi$
$647$	31.0453	1.22052	0.610258	+	0.792203i	$0.291066\pi$
$648$	0	0
$649$	6.48227	0.254451
$650$	0	0
$651$	−3.81539	−0.149537
$652$	0	0
$653$	−25.1971	−0.986038	−0.493019	−	0.870019i	$-0.664107\pi$
$653$	−25.1971	−0.986038	−0.493019	+	0.870019i	$0.664107\pi$
$654$	0	0
$655$	43.7928	1.71113
$656$	0	0
$657$	−3.16238	−0.123376
$658$	0	0
$659$	41.1061	1.60127	0.800633	−	0.599156i	$-0.204497\pi$
$659$	41.1061	1.60127	0.800633	+	0.599156i	$0.204497\pi$
$660$	0	0
$661$	28.6310	1.11362	0.556809	−	0.830641i	$-0.312026\pi$
$661$	28.6310	1.11362	0.556809	+	0.830641i	$0.312026\pi$
$662$	0	0
$663$	7.65105	0.297142
$664$	0	0
$665$	8.29333	0.321602
$666$	0	0
$667$	−56.5699	−2.19039
$668$	0	0
$669$	−16.9991	−0.657222
$670$	0	0
$671$	−29.9875	−1.15765
$672$	0	0
$673$	44.8909	1.73042	0.865209	−	0.501412i	$-0.167186\pi$
$673$	44.8909	1.73042	0.865209	+	0.501412i	$0.167186\pi$
$674$	0	0
$675$	68.7799	2.64734
$676$	0	0
$677$	−21.1105	−0.811342	−0.405671	−	0.914019i	$-0.632962\pi$
$677$	−21.1105	−0.811342	−0.405671	+	0.914019i	$0.632962\pi$
$678$	0	0
$679$	−2.53585	−0.0973169
$680$	0	0
$681$	17.0198	0.652200
$682$	0	0
$683$	−45.8286	−1.75358	−0.876791	−	0.480872i	$-0.840320\pi$
$683$	−45.8286	−1.75358	−0.876791	+	0.480872i	$0.840320\pi$
$684$	0	0
$685$	−60.4537	−2.30982
$686$	0	0
$687$	39.0586	1.49018
$688$	0	0
$689$	−31.6036	−1.20400
$690$	0	0
$691$	0.884985	0.0336664	0.0168332	−	0.999858i	$-0.494642\pi$
$691$	0.884985	0.0336664	0.0168332	+	0.999858i	$0.494642\pi$
$692$	0	0
$693$	3.36256	0.127733
$694$	0	0
$695$	90.2687	3.42409
$696$	0	0
$697$	−10.6942	−0.405071
$698$	0	0
$699$	−2.92087	−0.110478
$700$	0	0
$701$	−15.6256	−0.590171	−0.295086	−	0.955471i	$-0.595348\pi$
$701$	−15.6256	−0.590171	−0.295086	+	0.955471i	$0.595348\pi$
$702$	0	0
$703$	−29.9246	−1.12863
$704$	0	0
$705$	−7.21110	−0.271586
$706$	0	0
$707$	−3.43923	−0.129345
$708$	0	0
$709$	41.0064	1.54003	0.770014	−	0.638027i	$-0.220249\pi$
$709$	41.0064	1.54003	0.770014	+	0.638027i	$0.220249\pi$
$710$	0	0
$711$	0.808492	0.0303208
$712$	0	0
$713$	−23.2296	−0.869955
$714$	0	0
$715$	−135.102	−5.05252
$716$	0	0
$717$	−18.6437	−0.696262
$718$	0	0
$719$	−14.8663	−0.554418	−0.277209	−	0.960810i	$-0.589409\pi$
$719$	−14.8663	−0.554418	−0.277209	+	0.960810i	$0.589409\pi$
$720$	0	0
$721$	10.3082	0.383898
$722$	0	0
$723$	45.3692	1.68730
$724$	0	0
$725$	96.8087	3.59538
$726$	0	0
$727$	−21.1658	−0.784997	−0.392499	−	0.919753i	$-0.628389\pi$
$727$	−21.1658	−0.784997	−0.392499	+	0.919753i	$0.628389\pi$
$728$	0	0
$729$	30.0351	1.11241
$730$	0	0
$731$	7.59086	0.280758
$732$	0	0
$733$	−19.3199	−0.713597	−0.356798	−	0.934181i	$-0.616132\pi$
$733$	−19.3199	−0.713597	−0.356798	+	0.934181i	$0.616132\pi$
$734$	0	0
$735$	40.6137	1.49806
$736$	0	0
$737$	−94.5810	−3.48394
$738$	0	0
$739$	−47.4178	−1.74429	−0.872146	−	0.489245i	$-0.837272\pi$
$739$	−47.4178	−1.74429	−0.872146	+	0.489245i	$0.837272\pi$
$740$	0	0
$741$	−19.7694	−0.726247
$742$	0	0
$743$	23.9880	0.880036	0.440018	−	0.897989i	$-0.354972\pi$
$743$	23.9880	0.880036	0.440018	+	0.897989i	$0.354972\pi$
$744$	0	0
$745$	−1.77738	−0.0651182
$746$	0	0
$747$	−3.12793	−0.114445
$748$	0	0
$749$	4.25919	0.155627
$750$	0	0
$751$	11.2507	0.410544	0.205272	−	0.978705i	$-0.434192\pi$
$751$	11.2507	0.410544	0.205272	+	0.978705i	$0.434192\pi$
$752$	0	0
$753$	−35.2677	−1.28523
$754$	0	0
$755$	−16.3653	−0.595593
$756$	0	0
$757$	−20.9380	−0.761004	−0.380502	−	0.924780i	$-0.624249\pi$
$757$	−20.9380	−0.761004	−0.380502	+	0.924780i	$0.624249\pi$
$758$	0	0
$759$	−70.9554	−2.57552
$760$	0	0
$761$	27.5830	0.999884	0.499942	−	0.866059i	$-0.333355\pi$
$761$	27.5830	0.999884	0.499942	+	0.866059i	$0.333355\pi$
$762$	0	0
$763$	2.06928	0.0749130
$764$	0	0
$765$	2.79217	0.100951
$766$	0	0
$767$	−5.01426	−0.181054
$768$	0	0
$769$	25.8138	0.930868	0.465434	−	0.885083i	$-0.345898\pi$
$769$	25.8138	0.930868	0.465434	+	0.885083i	$0.345898\pi$
$770$	0	0
$771$	−21.8632	−0.787384
$772$	0	0
$773$	26.9317	0.968666	0.484333	−	0.874884i	$-0.339062\pi$
$773$	26.9317	0.968666	0.484333	+	0.874884i	$0.339062\pi$
$774$	0	0
$775$	39.7530	1.42797
$776$	0	0
$777$	13.6458	0.489539
$778$	0	0
$779$	27.6325	0.990038
$780$	0	0
$781$	−14.4758	−0.517983
$782$	0	0
$783$	44.1803	1.57888
$784$	0	0
$785$	77.8450	2.77841
$786$	0	0
$787$	−35.0677	−1.25003	−0.625013	−	0.780614i	$-0.714907\pi$
$787$	−35.0677	−1.25003	−0.625013	+	0.780614i	$0.714907\pi$
$788$	0	0
$789$	7.81504	0.278223
$790$	0	0
$791$	−9.50223	−0.337860
$792$	0	0
$793$	23.1964	0.823727
$794$	0	0
$795$	39.9733	1.41771
$796$	0	0
$797$	−35.5221	−1.25826	−0.629129	−	0.777301i	$-0.716588\pi$
$797$	−35.5221	−1.25826	−0.629129	+	0.777301i	$0.716588\pi$
$798$	0	0
$799$	−1.13700	−0.0402241
$800$	0	0
$801$	−9.92527	−0.350692
$802$	0	0
$803$	−30.5159	−1.07688
$804$	0	0
$805$	−23.0250	−0.811526
$806$	0	0
$807$	33.4728	1.17830
$808$	0	0
$809$	−44.3653	−1.55980	−0.779901	−	0.625903i	$-0.784731\pi$
$809$	−44.3653	−1.55980	−0.779901	+	0.625903i	$0.784731\pi$
$810$	0	0
$811$	29.2715	1.02786	0.513931	−	0.857831i	$-0.328189\pi$
$811$	29.2715	1.02786	0.513931	+	0.857831i	$0.328189\pi$
$812$	0	0
$813$	−7.77803	−0.272788
$814$	0	0
$815$	53.4729	1.87308
$816$	0	0
$817$	−19.6139	−0.686203
$818$	0	0
$819$	−2.60106	−0.0908883
$820$	0	0
$821$	−22.3502	−0.780028	−0.390014	−	0.920809i	$-0.627530\pi$
$821$	−22.3502	−0.780028	−0.390014	+	0.920809i	$0.627530\pi$
$822$	0	0
$823$	−26.1876	−0.912841	−0.456421	−	0.889764i	$-0.650869\pi$
$823$	−26.1876	−0.912841	−0.456421	+	0.889764i	$0.650869\pi$
$824$	0	0
$825$	121.427	4.22754
$826$	0	0
$827$	6.70992	0.233327	0.116663	−	0.993172i	$-0.462780\pi$
$827$	6.70992	0.233327	0.116663	+	0.993172i	$0.462780\pi$
$828$	0	0
$829$	−11.9705	−0.415753	−0.207877	−	0.978155i	$-0.566655\pi$
$829$	−11.9705	−0.415753	−0.207877	+	0.978155i	$0.566655\pi$
$830$	0	0
$831$	−19.9388	−0.691668
$832$	0	0
$833$	6.40371	0.221875
$834$	0	0
$835$	−39.6947	−1.37369
$836$	0	0
$837$	18.1420	0.627079
$838$	0	0
$839$	8.64531	0.298469	0.149235	−	0.988802i	$-0.452319\pi$
$839$	8.64531	0.298469	0.149235	+	0.988802i	$0.452319\pi$
$840$	0	0
$841$	33.1844	1.14429
$842$	0	0
$843$	−13.6299	−0.469438
$844$	0	0
$845$	50.4716	1.73627
$846$	0	0
$847$	23.9534	0.823049
$848$	0	0
$849$	3.59707	0.123451
$850$	0	0
$851$	83.0806	2.84797
$852$	0	0
$853$	44.9979	1.54070	0.770350	−	0.637621i	$-0.220082\pi$
$853$	44.9979	1.54070	0.770350	+	0.637621i	$0.220082\pi$
$854$	0	0
$855$	−7.21464	−0.246735
$856$	0	0
$857$	−42.2148	−1.44203	−0.721014	−	0.692920i	$-0.756324\pi$
$857$	−42.2148	−1.44203	−0.721014	+	0.692920i	$0.756324\pi$
$858$	0	0
$859$	14.5774	0.497376	0.248688	−	0.968584i	$-0.420001\pi$
$859$	14.5774	0.497376	0.248688	+	0.968584i	$0.420001\pi$
$860$	0	0
$861$	−12.6006	−0.429427
$862$	0	0
$863$	19.9224	0.678165	0.339083	−	0.940757i	$-0.389883\pi$
$863$	19.9224	0.678165	0.339083	+	0.940757i	$0.389883\pi$
$864$	0	0
$865$	79.8884	2.71629
$866$	0	0
$867$	−1.52586	−0.0518208
$868$	0	0
$869$	7.80169	0.264654
$870$	0	0
$871$	73.1617	2.47899
$872$	0	0
$873$	2.20601	0.0746623
$874$	0	0
$875$	23.3548	0.789537
$876$	0	0
$877$	−9.64040	−0.325533	−0.162767	−	0.986665i	$-0.552042\pi$
$877$	−9.64040	−0.325533	−0.162767	+	0.986665i	$0.552042\pi$
$878$	0	0
$879$	23.8606	0.804797
$880$	0	0
$881$	25.1535	0.847442	0.423721	−	0.905793i	$-0.360724\pi$
$881$	25.1535	0.847442	0.423721	+	0.905793i	$0.360724\pi$
$882$	0	0
$883$	33.7922	1.13720	0.568599	−	0.822615i	$-0.307486\pi$
$883$	33.7922	1.13720	0.568599	+	0.822615i	$0.307486\pi$
$884$	0	0
$885$	6.34222	0.213191
$886$	0	0
$887$	37.6415	1.26388	0.631940	−	0.775018i	$-0.282259\pi$
$887$	37.6415	1.26388	0.631940	+	0.775018i	$0.282259\pi$
$888$	0	0
$889$	−7.86803	−0.263885
$890$	0	0
$891$	42.3516	1.41883
$892$	0	0
$893$	2.93787	0.0983121
$894$	0	0
$895$	78.7878	2.63359
$896$	0	0
$897$	54.8865	1.83261
$898$	0	0
$899$	25.5351	0.851644
$900$	0	0
$901$	6.30273	0.209975
$902$	0	0
$903$	8.94403	0.297639
$904$	0	0
$905$	18.1622	0.603731
$906$	0	0
$907$	−23.6518	−0.785346	−0.392673	−	0.919678i	$-0.628450\pi$
$907$	−23.6518	−0.785346	−0.392673	+	0.919678i	$0.628450\pi$
$908$	0	0
$909$	2.99189	0.0992349
$910$	0	0
$911$	33.0601	1.09533	0.547664	−	0.836698i	$-0.315517\pi$
$911$	33.0601	1.09533	0.547664	+	0.836698i	$0.315517\pi$
$912$	0	0
$913$	−30.1835	−0.998928
$914$	0	0
$915$	−29.3396	−0.969938
$916$	0	0
$917$	8.13587	0.268670
$918$	0	0
$919$	−25.9485	−0.855962	−0.427981	−	0.903788i	$-0.640775\pi$
$919$	−25.9485	−0.855962	−0.427981	+	0.903788i	$0.640775\pi$
$920$	0	0
$921$	−35.4985	−1.16972
$922$	0	0
$923$	11.1975	0.368570
$924$	0	0
$925$	−142.177	−4.67474
$926$	0	0
$927$	−8.96745	−0.294530
$928$	0	0
$929$	33.9905	1.11519	0.557597	−	0.830112i	$-0.311724\pi$
$929$	33.9905	1.11519	0.557597	+	0.830112i	$0.311724\pi$
$930$	0	0
$931$	−16.5464	−0.542288
$932$	0	0
$933$	−11.7262	−0.383898
$934$	0	0
$935$	26.9435	0.881148
$936$	0	0
$937$	−17.3944	−0.568250	−0.284125	−	0.958787i	$-0.591703\pi$
$937$	−17.3944	−0.568250	−0.284125	+	0.958787i	$0.591703\pi$
$938$	0	0
$939$	−20.3733	−0.664856
$940$	0	0
$941$	−17.4788	−0.569791	−0.284896	−	0.958559i	$-0.591959\pi$
$941$	−17.4788	−0.569791	−0.284896	+	0.958559i	$0.591959\pi$
$942$	0	0
$943$	−76.7171	−2.49825
$944$	0	0
$945$	17.9823	0.584963
$946$	0	0
$947$	13.1978	0.428870	0.214435	−	0.976738i	$-0.431209\pi$
$947$	13.1978	0.428870	0.214435	+	0.976738i	$0.431209\pi$
$948$	0	0
$949$	23.6051	0.766255
$950$	0	0
$951$	10.9284	0.354379
$952$	0	0
$953$	8.22376	0.266394	0.133197	−	0.991090i	$-0.457476\pi$
$953$	8.22376	0.266394	0.133197	+	0.991090i	$0.457476\pi$
$954$	0	0
$955$	−11.0154	−0.356449
$956$	0	0
$957$	77.9977	2.52131
$958$	0	0
$959$	−11.2311	−0.362673
$960$	0	0
$961$	−20.5144	−0.661754
$962$	0	0
$963$	−3.70521	−0.119399
$964$	0	0
$965$	−6.82259	−0.219627
$966$	0	0
$967$	−56.1212	−1.80474	−0.902369	−	0.430965i	$-0.858173\pi$
$967$	−56.1212	−1.80474	−0.902369	+	0.430965i	$0.858173\pi$
$968$	0	0
$969$	3.94264	0.126656
$970$	0	0
$971$	−12.0304	−0.386075	−0.193038	−	0.981191i	$-0.561834\pi$
$971$	−12.0304	−0.386075	−0.193038	+	0.981191i	$0.561834\pi$
$972$	0	0
$973$	16.7702	0.537628
$974$	0	0
$975$	−93.9278	−3.00810
$976$	0	0
$977$	−0.251287	−0.00803939	−0.00401969	−	0.999992i	$-0.501280\pi$
$977$	−0.251287	−0.00803939	−0.00401969	+	0.999992i	$0.501280\pi$
$978$	0	0
$979$	−95.7757	−3.06100
$980$	0	0
$981$	−1.80013	−0.0574738
$982$	0	0
$983$	0.922439	0.0294212	0.0147106	−	0.999892i	$-0.495317\pi$
$983$	0.922439	0.0294212	0.0147106	+	0.999892i	$0.495317\pi$
$984$	0	0
$985$	−78.0621	−2.48726
$986$	0	0
$987$	−1.33968	−0.0426426
$988$	0	0
$989$	54.4547	1.73156
$990$	0	0
$991$	34.2105	1.08673	0.543366	−	0.839496i	$-0.317150\pi$
$991$	34.2105	1.08673	0.543366	+	0.839496i	$0.317150\pi$
$992$	0	0
$993$	−1.87484	−0.0594963
$994$	0	0
$995$	7.80737	0.247510
$996$	0	0
$997$	−25.2791	−0.800598	−0.400299	−	0.916385i	$-0.631094\pi$
$997$	−25.2791	−0.800598	−0.400299	+	0.916385i	$0.631094\pi$
$998$	0	0
$999$	−64.8849	−2.05287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bc.1.10	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bc.1.10	✓	33	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.52586 q^{3} +4.15650 q^{5} +0.772198 q^{7} -0.671760 q^{9} +O(q^{10})\)	\(q-1.52586 q^{3} +4.15650 q^{5} +0.772198 q^{7} -0.671760 q^{9} -6.48227 q^{11} +5.01426 q^{13} -6.34222 q^{15} -1.00000 q^{17} +2.58388 q^{19} -1.17826 q^{21} -7.17372 q^{23} +12.2765 q^{25} +5.60258 q^{27} +7.88571 q^{29} +3.23815 q^{31} +9.89101 q^{33} +3.20964 q^{35} -11.5812 q^{37} -7.65105 q^{39} +10.6942 q^{41} -7.59086 q^{43} -2.79217 q^{45} +1.13700 q^{47} -6.40371 q^{49} +1.52586 q^{51} -6.30273 q^{53} -26.9435 q^{55} -3.94264 q^{57} -1.00000 q^{59} +4.62608 q^{61} -0.518732 q^{63} +20.8418 q^{65} +14.5907 q^{67} +10.9461 q^{69} +2.23313 q^{71} +4.70760 q^{73} -18.7321 q^{75} -5.00559 q^{77} -1.20354 q^{79} -6.53346 q^{81} +4.65632 q^{83} -4.15650 q^{85} -12.0325 q^{87} +14.7750 q^{89} +3.87200 q^{91} -4.94095 q^{93} +10.7399 q^{95} -3.28393 q^{97} +4.35453 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

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