LMFDB - Embedded newform 8024.2.a.x.1.9

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:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-0.687639 q^{3} -1.07935 q^{5} +2.48918 q^{7} -2.52715 q^{9} +O(q^{10})$	$q-0.687639 q^{3} -1.07935 q^{5} +2.48918 q^{7} -2.52715 q^{9} -0.826184 q^{11} +6.98425 q^{13} +0.742202 q^{15} -1.00000 q^{17} -3.44363 q^{19} -1.71166 q^{21} +2.11466 q^{23} -3.83501 q^{25} +3.80069 q^{27} +1.30842 q^{29} -5.43578 q^{31} +0.568116 q^{33} -2.68670 q^{35} -5.01356 q^{37} -4.80265 q^{39} +0.972877 q^{41} +2.69997 q^{43} +2.72768 q^{45} +1.68722 q^{47} -0.803959 q^{49} +0.687639 q^{51} -6.31800 q^{53} +0.891740 q^{55} +2.36798 q^{57} +1.00000 q^{59} -6.15750 q^{61} -6.29055 q^{63} -7.53844 q^{65} -11.3645 q^{67} -1.45412 q^{69} +11.5532 q^{71} +10.6855 q^{73} +2.63710 q^{75} -2.05652 q^{77} -16.4505 q^{79} +4.96796 q^{81} +10.2189 q^{83} +1.07935 q^{85} -0.899718 q^{87} -3.50779 q^{89} +17.3851 q^{91} +3.73785 q^{93} +3.71688 q^{95} +14.8839 q^{97} +2.08789 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * 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
$3$	−0.687639	−0.397009	−0.198504	−	0.980100i	$-0.563608\pi$
$3$	−0.687639	−0.397009	−0.198504	+	0.980100i	$0.563608\pi$
$4$	0	0
$5$	−1.07935	−0.482699	−0.241350	−	0.970438i	$-0.577590\pi$
$5$	−1.07935	−0.482699	−0.241350	+	0.970438i	$0.577590\pi$
$6$	0	0
$7$	2.48918	0.940823	0.470412	−	0.882447i	$-0.344105\pi$
$7$	2.48918	0.940823	0.470412	+	0.882447i	$0.344105\pi$
$8$	0	0
$9$	−2.52715	−0.842384
$10$	0	0
$11$	−0.826184	−0.249104	−0.124552	−	0.992213i	$-0.539749\pi$
$11$	−0.826184	−0.249104	−0.124552	+	0.992213i	$0.539749\pi$
$12$	0	0
$13$	6.98425	1.93708	0.968542	−	0.248851i	$-0.0800529\pi$
$13$	6.98425	1.93708	0.968542	+	0.248851i	$0.0800529\pi$
$14$	0	0
$15$	0.742202	0.191636
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−3.44363	−0.790024	−0.395012	−	0.918676i	$-0.629260\pi$
$19$	−3.44363	−0.790024	−0.395012	+	0.918676i	$0.629260\pi$
$20$	0	0
$21$	−1.71166	−0.373515
$22$	0	0
$23$	2.11466	0.440938	0.220469	−	0.975394i	$-0.429241\pi$
$23$	2.11466	0.440938	0.220469	+	0.975394i	$0.429241\pi$
$24$	0	0
$25$	−3.83501	−0.767001
$26$	0	0
$27$	3.80069	0.731442
$28$	0	0
$29$	1.30842	0.242967	0.121483	−	0.992593i	$-0.461235\pi$
$29$	1.30842	0.242967	0.121483	+	0.992593i	$0.461235\pi$
$30$	0	0
$31$	−5.43578	−0.976295	−0.488147	−	0.872761i	$-0.662327\pi$
$31$	−5.43578	−0.976295	−0.488147	+	0.872761i	$0.662327\pi$
$32$	0	0
$33$	0.568116	0.0988964
$34$	0	0
$35$	−2.68670	−0.454135
$36$	0	0
$37$	−5.01356	−0.824224	−0.412112	−	0.911133i	$-0.635209\pi$
$37$	−5.01356	−0.824224	−0.412112	+	0.911133i	$0.635209\pi$
$38$	0	0
$39$	−4.80265	−0.769039
$40$	0	0
$41$	0.972877	0.151938	0.0759690	−	0.997110i	$-0.475795\pi$
$41$	0.972877	0.151938	0.0759690	+	0.997110i	$0.475795\pi$
$42$	0	0
$43$	2.69997	0.411742	0.205871	−	0.978579i	$-0.433997\pi$
$43$	2.69997	0.411742	0.205871	+	0.978579i	$0.433997\pi$
$44$	0	0
$45$	2.72768	0.406618
$46$	0	0
$47$	1.68722	0.246107	0.123053	−	0.992400i	$-0.460731\pi$
$47$	1.68722	0.246107	0.123053	+	0.992400i	$0.460731\pi$
$48$	0	0
$49$	−0.803959	−0.114851
$50$	0	0
$51$	0.687639	0.0962887
$52$	0	0
$53$	−6.31800	−0.867844	−0.433922	−	0.900950i	$-0.642871\pi$
$53$	−6.31800	−0.867844	−0.433922	+	0.900950i	$0.642871\pi$
$54$	0	0
$55$	0.891740	0.120242
$56$	0	0
$57$	2.36798	0.313646
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−6.15750	−0.788387	−0.394194	−	0.919027i	$-0.628976\pi$
$61$	−6.15750	−0.788387	−0.394194	+	0.919027i	$0.628976\pi$
$62$	0	0
$63$	−6.29055	−0.792535
$64$	0	0
$65$	−7.53844	−0.935029
$66$	0	0
$67$	−11.3645	−1.38839	−0.694196	−	0.719786i	$-0.744240\pi$
$67$	−11.3645	−1.38839	−0.694196	+	0.719786i	$0.744240\pi$
$68$	0	0
$69$	−1.45412	−0.175056
$70$	0	0
$71$	11.5532	1.37112	0.685559	−	0.728018i	$-0.259558\pi$
$71$	11.5532	1.37112	0.685559	+	0.728018i	$0.259558\pi$
$72$	0	0
$73$	10.6855	1.25065	0.625324	−	0.780365i	$-0.284967\pi$
$73$	10.6855	1.25065	0.625324	+	0.780365i	$0.284967\pi$
$74$	0	0
$75$	2.63710	0.304506
$76$	0	0
$77$	−2.05652	−0.234363
$78$	0	0
$79$	−16.4505	−1.85082	−0.925412	−	0.378962i	$-0.876281\pi$
$79$	−16.4505	−1.85082	−0.925412	+	0.378962i	$0.876281\pi$
$80$	0	0
$81$	4.96796	0.551995
$82$	0	0
$83$	10.2189	1.12167	0.560834	−	0.827928i	$-0.310481\pi$
$83$	10.2189	1.12167	0.560834	+	0.827928i	$0.310481\pi$
$84$	0	0
$85$	1.07935	0.117072
$86$	0	0
$87$	−0.899718	−0.0964599
$88$	0	0
$89$	−3.50779	−0.371825	−0.185913	−	0.982566i	$-0.559524\pi$
$89$	−3.50779	−0.371825	−0.185913	+	0.982566i	$0.559524\pi$
$90$	0	0
$91$	17.3851	1.82245
$92$	0	0
$93$	3.73785	0.387597
$94$	0	0
$95$	3.71688	0.381344
$96$	0	0
$97$	14.8839	1.51123	0.755614	−	0.655017i	$-0.227339\pi$
$97$	14.8839	1.51123	0.755614	+	0.655017i	$0.227339\pi$
$98$	0	0
$99$	2.08789	0.209841
$100$	0	0
$101$	3.99749	0.397765	0.198882	−	0.980023i	$-0.436269\pi$
$101$	3.99749	0.397765	0.198882	+	0.980023i	$0.436269\pi$
$102$	0	0
$103$	14.2350	1.40262	0.701308	−	0.712858i	$-0.252600\pi$
$103$	14.2350	1.40262	0.701308	+	0.712858i	$0.252600\pi$
$104$	0	0
$105$	1.84748	0.180295
$106$	0	0
$107$	−20.0067	−1.93412	−0.967061	−	0.254543i	$-0.918075\pi$
$107$	−20.0067	−1.93412	−0.967061	+	0.254543i	$0.918075\pi$
$108$	0	0
$109$	2.45997	0.235622	0.117811	−	0.993036i	$-0.462412\pi$
$109$	2.45997	0.235622	0.117811	+	0.993036i	$0.462412\pi$
$110$	0	0
$111$	3.44752	0.327224
$112$	0	0
$113$	−11.4793	−1.07988	−0.539940	−	0.841704i	$-0.681553\pi$
$113$	−11.4793	−1.07988	−0.539940	+	0.841704i	$0.681553\pi$
$114$	0	0
$115$	−2.28246	−0.212840
$116$	0	0
$117$	−17.6503	−1.63177
$118$	0	0
$119$	−2.48918	−0.228183
$120$	0	0
$121$	−10.3174	−0.937947
$122$	0	0
$123$	−0.668988	−0.0603206
$124$	0	0
$125$	9.53605	0.852930
$126$	0	0
$127$	−6.42766	−0.570362	−0.285181	−	0.958474i	$-0.592054\pi$
$127$	−6.42766	−0.570362	−0.285181	+	0.958474i	$0.592054\pi$
$128$	0	0
$129$	−1.85661	−0.163465
$130$	0	0
$131$	−4.59914	−0.401829	−0.200915	−	0.979609i	$-0.564391\pi$
$131$	−4.59914	−0.401829	−0.200915	+	0.979609i	$0.564391\pi$
$132$	0	0
$133$	−8.57184	−0.743273
$134$	0	0
$135$	−4.10226	−0.353067
$136$	0	0
$137$	8.59315	0.734163	0.367081	−	0.930189i	$-0.380357\pi$
$137$	8.59315	0.734163	0.367081	+	0.930189i	$0.380357\pi$
$138$	0	0
$139$	−2.31911	−0.196704	−0.0983522	−	0.995152i	$-0.531357\pi$
$139$	−2.31911	−0.196704	−0.0983522	+	0.995152i	$0.531357\pi$
$140$	0	0
$141$	−1.16020	−0.0977066
$142$	0	0
$143$	−5.77028	−0.482535
$144$	0	0
$145$	−1.41224	−0.117280
$146$	0	0
$147$	0.552834	0.0455970
$148$	0	0
$149$	6.15120	0.503926	0.251963	−	0.967737i	$-0.418924\pi$
$149$	6.15120	0.503926	0.251963	+	0.967737i	$0.418924\pi$
$150$	0	0
$151$	−22.3147	−1.81595	−0.907973	−	0.419028i	$-0.862371\pi$
$151$	−22.3147	−1.81595	−0.907973	+	0.419028i	$0.862371\pi$
$152$	0	0
$153$	2.52715	0.204308
$154$	0	0
$155$	5.86710	0.471257
$156$	0	0
$157$	−20.9931	−1.67543	−0.837716	−	0.546107i	$-0.816109\pi$
$157$	−20.9931	−1.67543	−0.837716	+	0.546107i	$0.816109\pi$
$158$	0	0
$159$	4.34450	0.344541
$160$	0	0
$161$	5.26379	0.414845
$162$	0	0
$163$	10.9269	0.855861	0.427930	−	0.903812i	$-0.359243\pi$
$163$	10.9269	0.855861	0.427930	+	0.903812i	$0.359243\pi$
$164$	0	0
$165$	−0.613195	−0.0477372
$166$	0	0
$167$	10.7429	0.831307	0.415654	−	0.909523i	$-0.363553\pi$
$167$	10.7429	0.831307	0.415654	+	0.909523i	$0.363553\pi$
$168$	0	0
$169$	35.7798	2.75229
$170$	0	0
$171$	8.70259	0.665504
$172$	0	0
$173$	5.57912	0.424173	0.212086	−	0.977251i	$-0.431974\pi$
$173$	5.57912	0.424173	0.212086	+	0.977251i	$0.431974\pi$
$174$	0	0
$175$	−9.54604	−0.721613
$176$	0	0
$177$	−0.687639	−0.0516861
$178$	0	0
$179$	4.89291	0.365713	0.182857	−	0.983140i	$-0.441466\pi$
$179$	4.89291	0.365713	0.182857	+	0.983140i	$0.441466\pi$
$180$	0	0
$181$	−6.17768	−0.459183	−0.229592	−	0.973287i	$-0.573739\pi$
$181$	−6.17768	−0.459183	−0.229592	+	0.973287i	$0.573739\pi$
$182$	0	0
$183$	4.23414	0.312996
$184$	0	0
$185$	5.41137	0.397852
$186$	0	0
$187$	0.826184	0.0604166
$188$	0	0
$189$	9.46061	0.688158
$190$	0	0
$191$	24.4092	1.76618	0.883092	−	0.469199i	$-0.155457\pi$
$191$	24.4092	1.76618	0.883092	+	0.469199i	$0.155457\pi$
$192$	0	0
$193$	−19.4071	−1.39695	−0.698477	−	0.715633i	$-0.746138\pi$
$193$	−19.4071	−1.39695	−0.698477	+	0.715633i	$0.746138\pi$
$194$	0	0
$195$	5.18373	0.371214
$196$	0	0
$197$	−25.8389	−1.84094	−0.920471	−	0.390810i	$-0.872195\pi$
$197$	−25.8389	−1.84094	−0.920471	+	0.390810i	$0.872195\pi$
$198$	0	0
$199$	4.15852	0.294790	0.147395	−	0.989078i	$-0.452911\pi$
$199$	4.15852	0.294790	0.147395	+	0.989078i	$0.452911\pi$
$200$	0	0
$201$	7.81466	0.551204
$202$	0	0
$203$	3.25689	0.228589
$204$	0	0
$205$	−1.05007	−0.0733403
$206$	0	0
$207$	−5.34408	−0.371439
$208$	0	0
$209$	2.84508	0.196798
$210$	0	0
$211$	19.3503	1.33213	0.666064	−	0.745894i	$-0.267978\pi$
$211$	19.3503	1.33213	0.666064	+	0.745894i	$0.267978\pi$
$212$	0	0
$213$	−7.94446	−0.544345
$214$	0	0
$215$	−2.91421	−0.198748
$216$	0	0
$217$	−13.5307	−0.918521
$218$	0	0
$219$	−7.34780	−0.496518
$220$	0	0
$221$	−6.98425	−0.469812
$222$	0	0
$223$	−27.5779	−1.84675	−0.923375	−	0.383898i	$-0.874581\pi$
$223$	−27.5779	−1.84675	−0.923375	+	0.383898i	$0.874581\pi$
$224$	0	0
$225$	9.69165	0.646110
$226$	0	0
$227$	−25.4942	−1.69211	−0.846056	−	0.533094i	$-0.821029\pi$
$227$	−25.4942	−1.69211	−0.846056	+	0.533094i	$0.821029\pi$
$228$	0	0
$229$	13.5796	0.897364	0.448682	−	0.893692i	$-0.351894\pi$
$229$	13.5796	0.897364	0.448682	+	0.893692i	$0.351894\pi$
$230$	0	0
$231$	1.41415	0.0930440
$232$	0	0
$233$	−20.4105	−1.33713	−0.668567	−	0.743652i	$-0.733092\pi$
$233$	−20.4105	−1.33713	−0.668567	+	0.743652i	$0.733092\pi$
$234$	0	0
$235$	−1.82110	−0.118796
$236$	0	0
$237$	11.3120	0.734793
$238$	0	0
$239$	−18.0128	−1.16515	−0.582576	−	0.812776i	$-0.697955\pi$
$239$	−18.0128	−1.16515	−0.582576	+	0.812776i	$0.697955\pi$
$240$	0	0
$241$	3.18805	0.205360	0.102680	−	0.994714i	$-0.467258\pi$
$241$	3.18805	0.205360	0.102680	+	0.994714i	$0.467258\pi$
$242$	0	0
$243$	−14.8182	−0.950589
$244$	0	0
$245$	0.867752	0.0554386
$246$	0	0
$247$	−24.0512	−1.53034
$248$	0	0
$249$	−7.02690	−0.445312
$250$	0	0
$251$	14.2490	0.899390	0.449695	−	0.893182i	$-0.351533\pi$
$251$	14.2490	0.899390	0.449695	+	0.893182i	$0.351533\pi$
$252$	0	0
$253$	−1.74710	−0.109839
$254$	0	0
$255$	−0.742202	−0.0464785
$256$	0	0
$257$	−5.34794	−0.333595	−0.166798	−	0.985991i	$-0.553343\pi$
$257$	−5.34794	−0.333595	−0.166798	+	0.985991i	$0.553343\pi$
$258$	0	0
$259$	−12.4797	−0.775449
$260$	0	0
$261$	−3.30657	−0.204671
$262$	0	0
$263$	6.68563	0.412254	0.206127	−	0.978525i	$-0.433914\pi$
$263$	6.68563	0.412254	0.206127	+	0.978525i	$0.433914\pi$
$264$	0	0
$265$	6.81932	0.418907
$266$	0	0
$267$	2.41210	0.147618
$268$	0	0
$269$	4.35478	0.265516	0.132758	−	0.991149i	$-0.457617\pi$
$269$	4.35478	0.265516	0.132758	+	0.991149i	$0.457617\pi$
$270$	0	0
$271$	−16.3785	−0.994922	−0.497461	−	0.867486i	$-0.665734\pi$
$271$	−16.3785	−0.994922	−0.497461	+	0.867486i	$0.665734\pi$
$272$	0	0
$273$	−11.9547	−0.723530
$274$	0	0
$275$	3.16842	0.191063
$276$	0	0
$277$	−25.7325	−1.54612	−0.773059	−	0.634334i	$-0.781275\pi$
$277$	−25.7325	−1.54612	−0.773059	+	0.634334i	$0.781275\pi$
$278$	0	0
$279$	13.7370	0.822415
$280$	0	0
$281$	−26.5351	−1.58295	−0.791474	−	0.611203i	$-0.790686\pi$
$281$	−26.5351	−1.58295	−0.791474	+	0.611203i	$0.790686\pi$
$282$	0	0
$283$	11.1686	0.663907	0.331953	−	0.943296i	$-0.392292\pi$
$283$	11.1686	0.663907	0.331953	+	0.943296i	$0.392292\pi$
$284$	0	0
$285$	−2.55587	−0.151397
$286$	0	0
$287$	2.42167	0.142947
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−10.2347	−0.599971
$292$	0	0
$293$	−0.302734	−0.0176859	−0.00884294	−	0.999961i	$-0.502815\pi$
$293$	−0.302734	−0.0176859	−0.00884294	+	0.999961i	$0.502815\pi$
$294$	0	0
$295$	−1.07935	−0.0628421
$296$	0	0
$297$	−3.14007	−0.182205
$298$	0	0
$299$	14.7693	0.854133
$300$	0	0
$301$	6.72074	0.387377
$302$	0	0
$303$	−2.74883	−0.157916
$304$	0	0
$305$	6.64609	0.380554
$306$	0	0
$307$	6.61464	0.377517	0.188759	−	0.982024i	$-0.439554\pi$
$307$	6.61464	0.377517	0.188759	+	0.982024i	$0.439554\pi$
$308$	0	0
$309$	−9.78854	−0.556851
$310$	0	0
$311$	−13.7338	−0.778774	−0.389387	−	0.921074i	$-0.627313\pi$
$311$	−13.7338	−0.778774	−0.389387	+	0.921074i	$0.627313\pi$
$312$	0	0
$313$	−2.53278	−0.143161	−0.0715805	−	0.997435i	$-0.522804\pi$
$313$	−2.53278	−0.143161	−0.0715805	+	0.997435i	$0.522804\pi$
$314$	0	0
$315$	6.78969	0.382556
$316$	0	0
$317$	−21.7642	−1.22240	−0.611200	−	0.791476i	$-0.709313\pi$
$317$	−21.7642	−1.22240	−0.611200	+	0.791476i	$0.709313\pi$
$318$	0	0
$319$	−1.08099	−0.0605240
$320$	0	0
$321$	13.7574	0.767863
$322$	0	0
$323$	3.44363	0.191609
$324$	0	0
$325$	−26.7847	−1.48575
$326$	0	0
$327$	−1.69157	−0.0935441
$328$	0	0
$329$	4.19981	0.231543
$330$	0	0
$331$	0.686915	0.0377563	0.0188781	−	0.999822i	$-0.493991\pi$
$331$	0.686915	0.0377563	0.0188781	+	0.999822i	$0.493991\pi$
$332$	0	0
$333$	12.6700	0.694313
$334$	0	0
$335$	12.2662	0.670176
$336$	0	0
$337$	−3.32940	−0.181364	−0.0906819	−	0.995880i	$-0.528905\pi$
$337$	−3.32940	−0.181364	−0.0906819	+	0.995880i	$0.528905\pi$
$338$	0	0
$339$	7.89360	0.428722
$340$	0	0
$341$	4.49095	0.243199
$342$	0	0
$343$	−19.4255	−1.04888
$344$	0	0
$345$	1.56951	0.0844994
$346$	0	0
$347$	29.2046	1.56779	0.783894	−	0.620895i	$-0.213231\pi$
$347$	29.2046	1.56779	0.783894	+	0.620895i	$0.213231\pi$
$348$	0	0
$349$	−29.6794	−1.58870	−0.794351	−	0.607459i	$-0.792189\pi$
$349$	−29.6794	−1.58870	−0.794351	+	0.607459i	$0.792189\pi$
$350$	0	0
$351$	26.5450	1.41686
$352$	0	0
$353$	6.15873	0.327796	0.163898	−	0.986477i	$-0.447593\pi$
$353$	6.15873	0.327796	0.163898	+	0.986477i	$0.447593\pi$
$354$	0	0
$355$	−12.4700	−0.661837
$356$	0	0
$357$	1.71166	0.0905907
$358$	0	0
$359$	2.77947	0.146695	0.0733475	−	0.997306i	$-0.476632\pi$
$359$	2.77947	0.146695	0.0733475	+	0.997306i	$0.476632\pi$
$360$	0	0
$361$	−7.14139	−0.375862
$362$	0	0
$363$	7.09466	0.372373
$364$	0	0
$365$	−11.5334	−0.603687
$366$	0	0
$367$	−14.2055	−0.741521	−0.370760	−	0.928729i	$-0.620903\pi$
$367$	−14.2055	−0.741521	−0.370760	+	0.928729i	$0.620903\pi$
$368$	0	0
$369$	−2.45861	−0.127990
$370$	0	0
$371$	−15.7267	−0.816488
$372$	0	0
$373$	−26.4118	−1.36755	−0.683774	−	0.729693i	$-0.739663\pi$
$373$	−26.4118	−1.36755	−0.683774	+	0.729693i	$0.739663\pi$
$374$	0	0
$375$	−6.55736	−0.338621
$376$	0	0
$377$	9.13831	0.470647
$378$	0	0
$379$	12.8702	0.661097	0.330549	−	0.943789i	$-0.392766\pi$
$379$	12.8702	0.661097	0.330549	+	0.943789i	$0.392766\pi$
$380$	0	0
$381$	4.41991	0.226439
$382$	0	0
$383$	−9.88568	−0.505135	−0.252567	−	0.967579i	$-0.581275\pi$
$383$	−9.88568	−0.505135	−0.252567	+	0.967579i	$0.581275\pi$
$384$	0	0
$385$	2.21971	0.113127
$386$	0	0
$387$	−6.82325	−0.346845
$388$	0	0
$389$	11.9971	0.608278	0.304139	−	0.952628i	$-0.401631\pi$
$389$	11.9971	0.608278	0.304139	+	0.952628i	$0.401631\pi$
$390$	0	0
$391$	−2.11466	−0.106943
$392$	0	0
$393$	3.16255	0.159530
$394$	0	0
$395$	17.7558	0.893392
$396$	0	0
$397$	15.5674	0.781304	0.390652	−	0.920538i	$-0.372250\pi$
$397$	15.5674	0.781304	0.390652	+	0.920538i	$0.372250\pi$
$398$	0	0
$399$	5.89433	0.295086
$400$	0	0
$401$	23.0335	1.15024	0.575119	−	0.818069i	$-0.304956\pi$
$401$	23.0335	1.15024	0.575119	+	0.818069i	$0.304956\pi$
$402$	0	0
$403$	−37.9649	−1.89116
$404$	0	0
$405$	−5.36216	−0.266448
$406$	0	0
$407$	4.14212	0.205317
$408$	0	0
$409$	−38.2641	−1.89204	−0.946018	−	0.324113i	$-0.894934\pi$
$409$	−38.2641	−1.89204	−0.946018	+	0.324113i	$0.894934\pi$
$410$	0	0
$411$	−5.90899	−0.291469
$412$	0	0
$413$	2.48918	0.122485
$414$	0	0
$415$	−11.0297	−0.541428
$416$	0	0
$417$	1.59471	0.0780933
$418$	0	0
$419$	36.0298	1.76017	0.880087	−	0.474813i	$-0.157484\pi$
$419$	36.0298	1.76017	0.880087	+	0.474813i	$0.157484\pi$
$420$	0	0
$421$	−18.6873	−0.910764	−0.455382	−	0.890296i	$-0.650497\pi$
$421$	−18.6873	−0.910764	−0.455382	+	0.890296i	$0.650497\pi$
$422$	0	0
$423$	−4.26387	−0.207317
$424$	0	0
$425$	3.83501	0.186025
$426$	0	0
$427$	−15.3272	−0.741733
$428$	0	0
$429$	3.96787	0.191571
$430$	0	0
$431$	−34.8822	−1.68022	−0.840108	−	0.542419i	$-0.817509\pi$
$431$	−34.8822	−1.68022	−0.840108	+	0.542419i	$0.817509\pi$
$432$	0	0
$433$	−15.0144	−0.721547	−0.360773	−	0.932654i	$-0.617487\pi$
$433$	−15.0144	−0.721547	−0.360773	+	0.932654i	$0.617487\pi$
$434$	0	0
$435$	0.971109	0.0465611
$436$	0	0
$437$	−7.28213	−0.348351
$438$	0	0
$439$	−18.4792	−0.881963	−0.440982	−	0.897516i	$-0.645370\pi$
$439$	−18.4792	−0.881963	−0.440982	+	0.897516i	$0.645370\pi$
$440$	0	0
$441$	2.03173	0.0967489
$442$	0	0
$443$	11.9068	0.565709	0.282854	−	0.959163i	$-0.408719\pi$
$443$	11.9068	0.565709	0.282854	+	0.959163i	$0.408719\pi$
$444$	0	0
$445$	3.78613	0.179480
$446$	0	0
$447$	−4.22981	−0.200063
$448$	0	0
$449$	−18.6379	−0.879577	−0.439789	−	0.898101i	$-0.644947\pi$
$449$	−18.6379	−0.879577	−0.439789	+	0.898101i	$0.644947\pi$
$450$	0	0
$451$	−0.803776	−0.0378483
$452$	0	0
$453$	15.3445	0.720946
$454$	0	0
$455$	−18.7646	−0.879697
$456$	0	0
$457$	−24.4320	−1.14288	−0.571441	−	0.820643i	$-0.693616\pi$
$457$	−24.4320	−1.14288	−0.571441	+	0.820643i	$0.693616\pi$
$458$	0	0
$459$	−3.80069	−0.177401
$460$	0	0
$461$	12.8836	0.600047	0.300024	−	0.953932i	$-0.403005\pi$
$461$	12.8836	0.600047	0.300024	+	0.953932i	$0.403005\pi$
$462$	0	0
$463$	−1.75485	−0.0815548	−0.0407774	−	0.999168i	$-0.512983\pi$
$463$	−1.75485	−0.0815548	−0.0407774	+	0.999168i	$0.512983\pi$
$464$	0	0
$465$	−4.03445	−0.187093
$466$	0	0
$467$	−14.5898	−0.675134	−0.337567	−	0.941302i	$-0.609604\pi$
$467$	−14.5898	−0.675134	−0.337567	+	0.941302i	$0.609604\pi$
$468$	0	0
$469$	−28.2883	−1.30623
$470$	0	0
$471$	14.4357	0.665160
$472$	0	0
$473$	−2.23068	−0.102567
$474$	0	0
$475$	13.2064	0.605949
$476$	0	0
$477$	15.9665	0.731058
$478$	0	0
$479$	25.7661	1.17728	0.588641	−	0.808394i	$-0.299663\pi$
$479$	25.7661	1.17728	0.588641	+	0.808394i	$0.299663\pi$
$480$	0	0
$481$	−35.0160	−1.59659
$482$	0	0
$483$	−3.61959	−0.164697
$484$	0	0
$485$	−16.0649	−0.729469
$486$	0	0
$487$	−0.781180	−0.0353987	−0.0176993	−	0.999843i	$-0.505634\pi$
$487$	−0.781180	−0.0353987	−0.0176993	+	0.999843i	$0.505634\pi$
$488$	0	0
$489$	−7.51376	−0.339784
$490$	0	0
$491$	21.4547	0.968236	0.484118	−	0.875003i	$-0.339141\pi$
$491$	21.4547	0.968236	0.484118	+	0.875003i	$0.339141\pi$
$492$	0	0
$493$	−1.30842	−0.0589281
$494$	0	0
$495$	−2.25356	−0.101290
$496$	0	0
$497$	28.7581	1.28998
$498$	0	0
$499$	24.8409	1.11203	0.556015	−	0.831172i	$-0.312330\pi$
$499$	24.8409	1.11203	0.556015	+	0.831172i	$0.312330\pi$
$500$	0	0
$501$	−7.38721	−0.330036
$502$	0	0
$503$	35.7997	1.59623	0.798115	−	0.602505i	$-0.205831\pi$
$503$	35.7997	1.59623	0.798115	+	0.602505i	$0.205831\pi$
$504$	0	0
$505$	−4.31468	−0.192001
$506$	0	0
$507$	−24.6036	−1.09268
$508$	0	0
$509$	−30.2638	−1.34142	−0.670710	−	0.741720i	$-0.734011\pi$
$509$	−30.2638	−1.34142	−0.670710	+	0.741720i	$0.734011\pi$
$510$	0	0
$511$	26.5983	1.17664
$512$	0	0
$513$	−13.0882	−0.577857
$514$	0	0
$515$	−15.3645	−0.677042
$516$	0	0
$517$	−1.39396	−0.0613062
$518$	0	0
$519$	−3.83642	−0.168400
$520$	0	0
$521$	−34.1966	−1.49818	−0.749091	−	0.662468i	$-0.769509\pi$
$521$	−34.1966	−1.49818	−0.749091	+	0.662468i	$0.769509\pi$
$522$	0	0
$523$	20.2602	0.885918	0.442959	−	0.896542i	$-0.353929\pi$
$523$	20.2602	0.885918	0.442959	+	0.896542i	$0.353929\pi$
$524$	0	0
$525$	6.56423	0.286487
$526$	0	0
$527$	5.43578	0.236786
$528$	0	0
$529$	−18.5282	−0.805574
$530$	0	0
$531$	−2.52715	−0.109669
$532$	0	0
$533$	6.79482	0.294316
$534$	0	0
$535$	21.5942	0.933600
$536$	0	0
$537$	−3.36456	−0.145191
$538$	0	0
$539$	0.664218	0.0286099
$540$	0	0
$541$	−7.84623	−0.337336	−0.168668	−	0.985673i	$-0.553947\pi$
$541$	−7.84623	−0.337336	−0.168668	+	0.985673i	$0.553947\pi$
$542$	0	0
$543$	4.24801	0.182300
$544$	0	0
$545$	−2.65516	−0.113735
$546$	0	0
$547$	18.1123	0.774427	0.387214	−	0.921990i	$-0.373438\pi$
$547$	18.1123	0.774427	0.387214	+	0.921990i	$0.373438\pi$
$548$	0	0
$549$	15.5609	0.664125
$550$	0	0
$551$	−4.50571	−0.191950
$552$	0	0
$553$	−40.9483	−1.74130
$554$	0	0
$555$	−3.72107	−0.157951
$556$	0	0
$557$	35.1908	1.49108	0.745542	−	0.666459i	$-0.232191\pi$
$557$	35.1908	1.49108	0.745542	+	0.666459i	$0.232191\pi$
$558$	0	0
$559$	18.8573	0.797579
$560$	0	0
$561$	−0.568116	−0.0239859
$562$	0	0
$563$	−16.8124	−0.708557	−0.354279	−	0.935140i	$-0.615274\pi$
$563$	−16.8124	−0.708557	−0.354279	+	0.935140i	$0.615274\pi$
$564$	0	0
$565$	12.3901	0.521257
$566$	0	0
$567$	12.3662	0.519330
$568$	0	0
$569$	−34.4755	−1.44529	−0.722644	−	0.691221i	$-0.757073\pi$
$569$	−34.4755	−1.44529	−0.722644	+	0.691221i	$0.757073\pi$
$570$	0	0
$571$	−18.1446	−0.759329	−0.379665	−	0.925124i	$-0.623961\pi$
$571$	−18.1446	−0.759329	−0.379665	+	0.925124i	$0.623961\pi$
$572$	0	0
$573$	−16.7847	−0.701191
$574$	0	0
$575$	−8.10975	−0.338200
$576$	0	0
$577$	−14.6716	−0.610785	−0.305392	−	0.952227i	$-0.598788\pi$
$577$	−14.6716	−0.610785	−0.305392	+	0.952227i	$0.598788\pi$
$578$	0	0
$579$	13.3451	0.554602
$580$	0	0
$581$	25.4367	1.05529
$582$	0	0
$583$	5.21983	0.216183
$584$	0	0
$585$	19.0508	0.787653
$586$	0	0
$587$	26.6153	1.09853	0.549266	−	0.835648i	$-0.314907\pi$
$587$	26.6153	1.09853	0.549266	+	0.835648i	$0.314907\pi$
$588$	0	0
$589$	18.7188	0.771296
$590$	0	0
$591$	17.7678	0.730870
$592$	0	0
$593$	−17.2672	−0.709080	−0.354540	−	0.935041i	$-0.615363\pi$
$593$	−17.2672	−0.709080	−0.354540	+	0.935041i	$0.615363\pi$
$594$	0	0
$595$	2.68670	0.110144
$596$	0	0
$597$	−2.85956	−0.117034
$598$	0	0
$599$	−31.7350	−1.29666	−0.648328	−	0.761361i	$-0.724531\pi$
$599$	−31.7350	−1.29666	−0.648328	+	0.761361i	$0.724531\pi$
$600$	0	0
$601$	6.54287	0.266889	0.133445	−	0.991056i	$-0.457396\pi$
$601$	6.54287	0.266889	0.133445	+	0.991056i	$0.457396\pi$
$602$	0	0
$603$	28.7198	1.16956
$604$	0	0
$605$	11.1361	0.452746
$606$	0	0
$607$	−17.8251	−0.723497	−0.361749	−	0.932276i	$-0.617820\pi$
$607$	−17.8251	−0.723497	−0.361749	+	0.932276i	$0.617820\pi$
$608$	0	0
$609$	−2.23956	−0.0907517
$610$	0	0
$611$	11.7840	0.476730
$612$	0	0
$613$	−36.0902	−1.45767	−0.728835	−	0.684689i	$-0.759938\pi$
$613$	−36.0902	−1.45767	−0.728835	+	0.684689i	$0.759938\pi$
$614$	0	0
$615$	0.722071	0.0291167
$616$	0	0
$617$	−10.3874	−0.418180	−0.209090	−	0.977896i	$-0.567050\pi$
$617$	−10.3874	−0.418180	−0.209090	+	0.977896i	$0.567050\pi$
$618$	0	0
$619$	29.2200	1.17445	0.587226	−	0.809423i	$-0.300220\pi$
$619$	29.2200	1.17445	0.587226	+	0.809423i	$0.300220\pi$
$620$	0	0
$621$	8.03717	0.322521
$622$	0	0
$623$	−8.73155	−0.349822
$624$	0	0
$625$	8.88232	0.355293
$626$	0	0
$627$	−1.95638	−0.0781305
$628$	0	0
$629$	5.01356	0.199904
$630$	0	0
$631$	−15.7259	−0.626039	−0.313020	−	0.949747i	$-0.601341\pi$
$631$	−15.7259	−0.626039	−0.313020	+	0.949747i	$0.601341\pi$
$632$	0	0
$633$	−13.3060	−0.528867
$634$	0	0
$635$	6.93768	0.275313
$636$	0	0
$637$	−5.61506	−0.222477
$638$	0	0
$639$	−29.1968	−1.15501
$640$	0	0
$641$	−27.1890	−1.07390	−0.536950	−	0.843614i	$-0.680424\pi$
$641$	−27.1890	−1.07390	−0.536950	+	0.843614i	$0.680424\pi$
$642$	0	0
$643$	−33.4082	−1.31749	−0.658745	−	0.752366i	$-0.728912\pi$
$643$	−33.4082	−1.31749	−0.658745	+	0.752366i	$0.728912\pi$
$644$	0	0
$645$	2.00393	0.0789045
$646$	0	0
$647$	48.3054	1.89908	0.949540	−	0.313645i	$-0.101550\pi$
$647$	48.3054	1.89908	0.949540	+	0.313645i	$0.101550\pi$
$648$	0	0
$649$	−0.826184	−0.0324306
$650$	0	0
$651$	9.30421	0.364661
$652$	0	0
$653$	−9.71407	−0.380141	−0.190070	−	0.981770i	$-0.560872\pi$
$653$	−9.71407	−0.380141	−0.190070	+	0.981770i	$0.560872\pi$
$654$	0	0
$655$	4.96408	0.193963
$656$	0	0
$657$	−27.0040	−1.05353
$658$	0	0
$659$	35.0655	1.36596	0.682979	−	0.730438i	$-0.260684\pi$
$659$	35.0655	1.36596	0.682979	+	0.730438i	$0.260684\pi$
$660$	0	0
$661$	35.3190	1.37375	0.686875	−	0.726776i	$-0.258982\pi$
$661$	35.3190	1.37375	0.686875	+	0.726776i	$0.258982\pi$
$662$	0	0
$663$	4.80265	0.186519
$664$	0	0
$665$	9.25200	0.358777
$666$	0	0
$667$	2.76686	0.107133
$668$	0	0
$669$	18.9636	0.733176
$670$	0	0
$671$	5.08723	0.196390
$672$	0	0
$673$	−16.5694	−0.638703	−0.319352	−	0.947636i	$-0.603465\pi$
$673$	−16.5694	−0.638703	−0.319352	+	0.947636i	$0.603465\pi$
$674$	0	0
$675$	−14.5757	−0.561017
$676$	0	0
$677$	−42.1784	−1.62105	−0.810523	−	0.585706i	$-0.800817\pi$
$677$	−42.1784	−1.62105	−0.810523	+	0.585706i	$0.800817\pi$
$678$	0	0
$679$	37.0487	1.42180
$680$	0	0
$681$	17.5308	0.671783
$682$	0	0
$683$	23.9548	0.916604	0.458302	−	0.888796i	$-0.348458\pi$
$683$	23.9548	0.916604	0.458302	+	0.888796i	$0.348458\pi$
$684$	0	0
$685$	−9.27501	−0.354380
$686$	0	0
$687$	−9.33785	−0.356261
$688$	0	0
$689$	−44.1265	−1.68109
$690$	0	0
$691$	−8.61232	−0.327628	−0.163814	−	0.986491i	$-0.552380\pi$
$691$	−8.61232	−0.327628	−0.163814	+	0.986491i	$0.552380\pi$
$692$	0	0
$693$	5.19715	0.197423
$694$	0	0
$695$	2.50313	0.0949490
$696$	0	0
$697$	−0.972877	−0.0368504
$698$	0	0
$699$	14.0350	0.530854
$700$	0	0
$701$	−25.9354	−0.979568	−0.489784	−	0.871844i	$-0.662924\pi$
$701$	−25.9354	−0.979568	−0.489784	+	0.871844i	$0.662924\pi$
$702$	0	0
$703$	17.2649	0.651156
$704$	0	0
$705$	1.25226	0.0471629
$706$	0	0
$707$	9.95048	0.374226
$708$	0	0
$709$	22.0209	0.827013	0.413507	−	0.910501i	$-0.364304\pi$
$709$	22.0209	0.827013	0.413507	+	0.910501i	$0.364304\pi$
$710$	0	0
$711$	41.5729	1.55911
$712$	0	0
$713$	−11.4948	−0.430485
$714$	0	0
$715$	6.22814	0.232919
$716$	0	0
$717$	12.3863	0.462575
$718$	0	0
$719$	4.98844	0.186037	0.0930186	−	0.995664i	$-0.470348\pi$
$719$	4.98844	0.186037	0.0930186	+	0.995664i	$0.470348\pi$
$720$	0	0
$721$	35.4336	1.31961
$722$	0	0
$723$	−2.19223	−0.0815298
$724$	0	0
$725$	−5.01779	−0.186356
$726$	0	0
$727$	1.22405	0.0453973	0.0226987	−	0.999742i	$-0.492774\pi$
$727$	1.22405	0.0453973	0.0226987	+	0.999742i	$0.492774\pi$
$728$	0	0
$729$	−4.71429	−0.174603
$730$	0	0
$731$	−2.69997	−0.0998622
$732$	0	0
$733$	−15.5907	−0.575855	−0.287927	−	0.957652i	$-0.592966\pi$
$733$	−15.5907	−0.575855	−0.287927	+	0.957652i	$0.592966\pi$
$734$	0	0
$735$	−0.596700	−0.0220096
$736$	0	0
$737$	9.38916	0.345854
$738$	0	0
$739$	10.4612	0.384822	0.192411	−	0.981314i	$-0.438369\pi$
$739$	10.4612	0.384822	0.192411	+	0.981314i	$0.438369\pi$
$740$	0	0
$741$	16.5386	0.607559
$742$	0	0
$743$	11.8425	0.434460	0.217230	−	0.976120i	$-0.430298\pi$
$743$	11.8425	0.434460	0.217230	+	0.976120i	$0.430298\pi$
$744$	0	0
$745$	−6.63929	−0.243245
$746$	0	0
$747$	−25.8247	−0.944876
$748$	0	0
$749$	−49.8004	−1.81967
$750$	0	0
$751$	−18.9031	−0.689782	−0.344891	−	0.938643i	$-0.612084\pi$
$751$	−18.9031	−0.689782	−0.344891	+	0.938643i	$0.612084\pi$
$752$	0	0
$753$	−9.79818	−0.357066
$754$	0	0
$755$	24.0854	0.876556
$756$	0	0
$757$	−16.5113	−0.600112	−0.300056	−	0.953922i	$-0.597005\pi$
$757$	−16.5113	−0.600112	−0.300056	+	0.953922i	$0.597005\pi$
$758$	0	0
$759$	1.20137	0.0436071
$760$	0	0
$761$	28.4898	1.03276	0.516378	−	0.856361i	$-0.327280\pi$
$761$	28.4898	1.03276	0.516378	+	0.856361i	$0.327280\pi$
$762$	0	0
$763$	6.12332	0.221679
$764$	0	0
$765$	−2.72768	−0.0986194
$766$	0	0
$767$	6.98425	0.252187
$768$	0	0
$769$	−27.0939	−0.977030	−0.488515	−	0.872556i	$-0.662461\pi$
$769$	−27.0939	−0.977030	−0.488515	+	0.872556i	$0.662461\pi$
$770$	0	0
$771$	3.67745	0.132440
$772$	0	0
$773$	15.4021	0.553974	0.276987	−	0.960874i	$-0.410664\pi$
$773$	15.4021	0.553974	0.276987	+	0.960874i	$0.410664\pi$
$774$	0	0
$775$	20.8463	0.748820
$776$	0	0
$777$	8.58151	0.307860
$778$	0	0
$779$	−3.35023	−0.120035
$780$	0	0
$781$	−9.54510	−0.341551
$782$	0	0
$783$	4.97288	0.177716
$784$	0	0
$785$	22.6589	0.808729
$786$	0	0
$787$	−35.1855	−1.25423	−0.627114	−	0.778928i	$-0.715764\pi$
$787$	−35.1855	−1.25423	−0.627114	+	0.778928i	$0.715764\pi$
$788$	0	0
$789$	−4.59730	−0.163668
$790$	0	0
$791$	−28.5740	−1.01598
$792$	0	0
$793$	−43.0055	−1.52717
$794$	0	0
$795$	−4.68923	−0.166310
$796$	0	0
$797$	−21.3892	−0.757645	−0.378823	−	0.925469i	$-0.623671\pi$
$797$	−21.3892	−0.757645	−0.378823	+	0.925469i	$0.623671\pi$
$798$	0	0
$799$	−1.68722	−0.0596897
$800$	0	0
$801$	8.86473	0.313220
$802$	0	0
$803$	−8.82823	−0.311541
$804$	0	0
$805$	−5.68146	−0.200245
$806$	0	0
$807$	−2.99452	−0.105412
$808$	0	0
$809$	13.2974	0.467512	0.233756	−	0.972295i	$-0.424898\pi$
$809$	13.2974	0.467512	0.233756	+	0.972295i	$0.424898\pi$
$810$	0	0
$811$	−35.2915	−1.23925	−0.619627	−	0.784896i	$-0.712716\pi$
$811$	−35.2915	−1.23925	−0.619627	+	0.784896i	$0.712716\pi$
$812$	0	0
$813$	11.2625	0.394993
$814$	0	0
$815$	−11.7939	−0.413123
$816$	0	0
$817$	−9.29772	−0.325286
$818$	0	0
$819$	−43.9348	−1.53521
$820$	0	0
$821$	27.7471	0.968381	0.484190	−	0.874963i	$-0.339114\pi$
$821$	27.7471	0.968381	0.484190	+	0.874963i	$0.339114\pi$
$822$	0	0
$823$	−2.35926	−0.0822385	−0.0411193	−	0.999154i	$-0.513092\pi$
$823$	−2.35926	−0.0822385	−0.0411193	+	0.999154i	$0.513092\pi$
$824$	0	0
$825$	−2.17873	−0.0758537
$826$	0	0
$827$	−25.9414	−0.902071	−0.451035	−	0.892506i	$-0.648945\pi$
$827$	−25.9414	−0.902071	−0.451035	+	0.892506i	$0.648945\pi$
$828$	0	0
$829$	41.2188	1.43159	0.715794	−	0.698311i	$-0.246065\pi$
$829$	41.2188	1.43159	0.715794	+	0.698311i	$0.246065\pi$
$830$	0	0
$831$	17.6947	0.613822
$832$	0	0
$833$	0.803959	0.0278555
$834$	0	0
$835$	−11.5953	−0.401271
$836$	0	0
$837$	−20.6597	−0.714103
$838$	0	0
$839$	−52.9300	−1.82734	−0.913672	−	0.406451i	$-0.866766\pi$
$839$	−52.9300	−1.82734	−0.913672	+	0.406451i	$0.866766\pi$
$840$	0	0
$841$	−27.2880	−0.940967
$842$	0	0
$843$	18.2465	0.628444
$844$	0	0
$845$	−38.6189	−1.32853
$846$	0	0
$847$	−25.6820	−0.882443
$848$	0	0
$849$	−7.67999	−0.263577
$850$	0	0
$851$	−10.6020	−0.363431
$852$	0	0
$853$	−3.62977	−0.124281	−0.0621405	−	0.998067i	$-0.519793\pi$
$853$	−3.62977	−0.124281	−0.0621405	+	0.998067i	$0.519793\pi$
$854$	0	0
$855$	−9.39312	−0.321238
$856$	0	0
$857$	9.60900	0.328237	0.164119	−	0.986441i	$-0.447522\pi$
$857$	9.60900	0.328237	0.164119	+	0.986441i	$0.447522\pi$
$858$	0	0
$859$	46.1865	1.57586	0.787931	−	0.615764i	$-0.211152\pi$
$859$	46.1865	1.57586	0.787931	+	0.615764i	$0.211152\pi$
$860$	0	0
$861$	−1.66524	−0.0567511
$862$	0	0
$863$	−27.3184	−0.929930	−0.464965	−	0.885329i	$-0.653933\pi$
$863$	−27.3184	−0.929930	−0.464965	+	0.885329i	$0.653933\pi$
$864$	0	0
$865$	−6.02181	−0.204748
$866$	0	0
$867$	−0.687639	−0.0233534
$868$	0	0
$869$	13.5911	0.461048
$870$	0	0
$871$	−79.3725	−2.68943
$872$	0	0
$873$	−37.6138	−1.27303
$874$	0	0
$875$	23.7370	0.802457
$876$	0	0
$877$	8.00346	0.270258	0.135129	−	0.990828i	$-0.456855\pi$
$877$	8.00346	0.270258	0.135129	+	0.990828i	$0.456855\pi$
$878$	0	0
$879$	0.208171	0.00702145
$880$	0	0
$881$	13.8374	0.466195	0.233097	−	0.972453i	$-0.425114\pi$
$881$	13.8374	0.466195	0.233097	+	0.972453i	$0.425114\pi$
$882$	0	0
$883$	−10.4121	−0.350396	−0.175198	−	0.984533i	$-0.556057\pi$
$883$	−10.4121	−0.350396	−0.175198	+	0.984533i	$0.556057\pi$
$884$	0	0
$885$	0.742202	0.0249488
$886$	0	0
$887$	16.7726	0.563170	0.281585	−	0.959536i	$-0.409140\pi$
$887$	16.7726	0.563170	0.281585	+	0.959536i	$0.409140\pi$
$888$	0	0
$889$	−15.9996	−0.536610
$890$	0	0
$891$	−4.10445	−0.137504
$892$	0	0
$893$	−5.81018	−0.194430
$894$	0	0
$895$	−5.28115	−0.176529
$896$	0	0
$897$	−10.1560	−0.339098
$898$	0	0
$899$	−7.11226	−0.237207
$900$	0	0
$901$	6.31800	0.210483
$902$	0	0
$903$	−4.62144	−0.153792
$904$	0	0
$905$	6.66787	0.221647
$906$	0	0
$907$	−26.3337	−0.874398	−0.437199	−	0.899365i	$-0.644029\pi$
$907$	−26.3337	−0.874398	−0.437199	+	0.899365i	$0.644029\pi$
$908$	0	0
$909$	−10.1023	−0.335071
$910$	0	0
$911$	1.43716	0.0476153	0.0238076	−	0.999717i	$-0.492421\pi$
$911$	1.43716	0.0476153	0.0238076	+	0.999717i	$0.492421\pi$
$912$	0	0
$913$	−8.44268	−0.279412
$914$	0	0
$915$	−4.57011	−0.151083
$916$	0	0
$917$	−11.4481	−0.378050
$918$	0	0
$919$	3.47598	0.114662	0.0573310	−	0.998355i	$-0.481741\pi$
$919$	3.47598	0.114662	0.0573310	+	0.998355i	$0.481741\pi$
$920$	0	0
$921$	−4.54848	−0.149878
$922$	0	0
$923$	80.6908	2.65597
$924$	0	0
$925$	19.2270	0.632181
$926$	0	0
$927$	−35.9740	−1.18154
$928$	0	0
$929$	−23.4325	−0.768796	−0.384398	−	0.923167i	$-0.625591\pi$
$929$	−23.4325	−0.768796	−0.384398	+	0.923167i	$0.625591\pi$
$930$	0	0
$931$	2.76854	0.0907353
$932$	0	0
$933$	9.44391	0.309180
$934$	0	0
$935$	−0.891740	−0.0291630
$936$	0	0
$937$	53.0537	1.73319	0.866594	−	0.499014i	$-0.166304\pi$
$937$	53.0537	1.73319	0.866594	+	0.499014i	$0.166304\pi$
$938$	0	0
$939$	1.74164	0.0568362
$940$	0	0
$941$	31.1854	1.01661	0.508307	−	0.861176i	$-0.330271\pi$
$941$	31.1854	1.01661	0.508307	+	0.861176i	$0.330271\pi$
$942$	0	0
$943$	2.05731	0.0669952
$944$	0	0
$945$	−10.2113	−0.332173
$946$	0	0
$947$	−12.2933	−0.399479	−0.199739	−	0.979849i	$-0.564010\pi$
$947$	−12.2933	−0.399479	−0.199739	+	0.979849i	$0.564010\pi$
$948$	0	0
$949$	74.6306	2.42261
$950$	0	0
$951$	14.9659	0.485303
$952$	0	0
$953$	−53.0892	−1.71973	−0.859864	−	0.510524i	$-0.829452\pi$
$953$	−53.0892	−1.71973	−0.859864	+	0.510524i	$0.829452\pi$
$954$	0	0
$955$	−26.3460	−0.852536
$956$	0	0
$957$	0.743333	0.0240285
$958$	0	0
$959$	21.3899	0.690717
$960$	0	0
$961$	−1.45230	−0.0468483
$962$	0	0
$963$	50.5600	1.62927
$964$	0	0
$965$	20.9470	0.674308
$966$	0	0
$967$	26.3149	0.846229	0.423115	−	0.906076i	$-0.360937\pi$
$967$	26.3149	0.846229	0.423115	+	0.906076i	$0.360937\pi$
$968$	0	0
$969$	−2.36798	−0.0760704
$970$	0	0
$971$	−11.1469	−0.357721	−0.178860	−	0.983874i	$-0.557241\pi$
$971$	−11.1469	−0.357721	−0.178860	+	0.983874i	$0.557241\pi$
$972$	0	0
$973$	−5.77269	−0.185064
$974$	0	0
$975$	18.4182	0.589854
$976$	0	0
$977$	27.0789	0.866330	0.433165	−	0.901315i	$-0.357397\pi$
$977$	27.0789	0.866330	0.433165	+	0.901315i	$0.357397\pi$
$978$	0	0
$979$	2.89808	0.0926231
$980$	0	0
$981$	−6.21672	−0.198485
$982$	0	0
$983$	29.6235	0.944844	0.472422	−	0.881373i	$-0.343380\pi$
$983$	29.6235	0.944844	0.472422	+	0.881373i	$0.343380\pi$
$984$	0	0
$985$	27.8891	0.888622
$986$	0	0
$987$	−2.88796	−0.0919246
$988$	0	0
$989$	5.70954	0.181553
$990$	0	0
$991$	9.11783	0.289637	0.144819	−	0.989458i	$-0.453740\pi$
$991$	9.11783	0.289637	0.144819	+	0.989458i	$0.453740\pi$
$992$	0	0
$993$	−0.472350	−0.0149896
$994$	0	0
$995$	−4.48849	−0.142295
$996$	0	0
$997$	23.1554	0.733339	0.366669	−	0.930351i	$-0.380498\pi$
$997$	23.1554	0.733339	0.366669	+	0.930351i	$0.380498\pi$
$998$	0	0
$999$	−19.0550	−0.602872

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.x.1.9	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.x.1.9	✓	22	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)

\(f(q)\)	\(=\)	\(q-0.687639 q^{3} -1.07935 q^{5} +2.48918 q^{7} -2.52715 q^{9} +O(q^{10})\)	\(q-0.687639 q^{3} -1.07935 q^{5} +2.48918 q^{7} -2.52715 q^{9} -0.826184 q^{11} +6.98425 q^{13} +0.742202 q^{15} -1.00000 q^{17} -3.44363 q^{19} -1.71166 q^{21} +2.11466 q^{23} -3.83501 q^{25} +3.80069 q^{27} +1.30842 q^{29} -5.43578 q^{31} +0.568116 q^{33} -2.68670 q^{35} -5.01356 q^{37} -4.80265 q^{39} +0.972877 q^{41} +2.69997 q^{43} +2.72768 q^{45} +1.68722 q^{47} -0.803959 q^{49} +0.687639 q^{51} -6.31800 q^{53} +0.891740 q^{55} +2.36798 q^{57} +1.00000 q^{59} -6.15750 q^{61} -6.29055 q^{63} -7.53844 q^{65} -11.3645 q^{67} -1.45412 q^{69} +11.5532 q^{71} +10.6855 q^{73} +2.63710 q^{75} -2.05652 q^{77} -16.4505 q^{79} +4.96796 q^{81} +10.2189 q^{83} +1.07935 q^{85} -0.899718 q^{87} -3.50779 q^{89} +17.3851 q^{91} +3.73785 q^{93} +3.71688 q^{95} +14.8839 q^{97} +2.08789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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