LMFDB - Embedded newform 8024.2.a.y.1.14

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:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q+0.453475 q^{3} +2.43721 q^{5} +1.57711 q^{7} -2.79436 q^{9} +O(q^{10})$	$q+0.453475 q^{3} +2.43721 q^{5} +1.57711 q^{7} -2.79436 q^{9} -1.43482 q^{11} -3.06317 q^{13} +1.10521 q^{15} -1.00000 q^{17} +3.99994 q^{19} +0.715178 q^{21} -6.82452 q^{23} +0.939999 q^{25} -2.62760 q^{27} -3.52783 q^{29} +7.83929 q^{31} -0.650656 q^{33} +3.84374 q^{35} -6.03921 q^{37} -1.38907 q^{39} +1.39294 q^{41} +7.98117 q^{43} -6.81045 q^{45} -8.67044 q^{47} -4.51274 q^{49} -0.453475 q^{51} +1.22706 q^{53} -3.49696 q^{55} +1.81387 q^{57} -1.00000 q^{59} +5.68615 q^{61} -4.40700 q^{63} -7.46558 q^{65} -5.05183 q^{67} -3.09475 q^{69} +3.31175 q^{71} +3.07942 q^{73} +0.426266 q^{75} -2.26286 q^{77} -14.6823 q^{79} +7.19153 q^{81} -2.49480 q^{83} -2.43721 q^{85} -1.59978 q^{87} +14.8134 q^{89} -4.83094 q^{91} +3.55492 q^{93} +9.74869 q^{95} -15.8517 q^{97} +4.00941 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * 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
$3$	0.453475	0.261814	0.130907	−	0.991395i	$-0.458211\pi$
$3$	0.453475	0.261814	0.130907	+	0.991395i	$0.458211\pi$
$4$	0	0
$5$	2.43721	1.08995	0.544977	−	0.838451i	$-0.316538\pi$
$5$	2.43721	1.08995	0.544977	+	0.838451i	$0.316538\pi$
$6$	0	0
$7$	1.57711	0.596090	0.298045	−	0.954552i	$-0.403665\pi$
$7$	1.57711	0.596090	0.298045	+	0.954552i	$0.403665\pi$
$8$	0	0
$9$	−2.79436	−0.931453
$10$	0	0
$11$	−1.43482	−0.432615	−0.216307	−	0.976325i	$-0.569401\pi$
$11$	−1.43482	−0.432615	−0.216307	+	0.976325i	$0.569401\pi$
$12$	0	0
$13$	−3.06317	−0.849569	−0.424785	−	0.905294i	$-0.639650\pi$
$13$	−3.06317	−0.849569	−0.424785	+	0.905294i	$0.639650\pi$
$14$	0	0
$15$	1.10521	0.285365
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	3.99994	0.917648	0.458824	−	0.888527i	$-0.348271\pi$
$19$	3.99994	0.917648	0.458824	+	0.888527i	$0.348271\pi$
$20$	0	0
$21$	0.715178	0.156065
$22$	0	0
$23$	−6.82452	−1.42301	−0.711505	−	0.702681i	$-0.751986\pi$
$23$	−6.82452	−1.42301	−0.711505	+	0.702681i	$0.751986\pi$
$24$	0	0
$25$	0.939999	0.188000
$26$	0	0
$27$	−2.62760	−0.505682
$28$	0	0
$29$	−3.52783	−0.655101	−0.327550	−	0.944834i	$-0.606223\pi$
$29$	−3.52783	−0.655101	−0.327550	+	0.944834i	$0.606223\pi$
$30$	0	0
$31$	7.83929	1.40798	0.703989	−	0.710211i	$-0.251400\pi$
$31$	7.83929	1.40798	0.703989	+	0.710211i	$0.251400\pi$
$32$	0	0
$33$	−0.650656	−0.113265
$34$	0	0
$35$	3.84374	0.649711
$36$	0	0
$37$	−6.03921	−0.992839	−0.496420	−	0.868083i	$-0.665352\pi$
$37$	−6.03921	−0.992839	−0.496420	+	0.868083i	$0.665352\pi$
$38$	0	0
$39$	−1.38907	−0.222429
$40$	0	0
$41$	1.39294	0.217540	0.108770	−	0.994067i	$-0.465309\pi$
$41$	1.39294	0.217540	0.108770	+	0.994067i	$0.465309\pi$
$42$	0	0
$43$	7.98117	1.21712	0.608559	−	0.793509i	$-0.291748\pi$
$43$	7.98117	1.21712	0.608559	+	0.793509i	$0.291748\pi$
$44$	0	0
$45$	−6.81045	−1.01524
$46$	0	0
$47$	−8.67044	−1.26471	−0.632357	−	0.774677i	$-0.717912\pi$
$47$	−8.67044	−1.26471	−0.632357	+	0.774677i	$0.717912\pi$
$48$	0	0
$49$	−4.51274	−0.644677
$50$	0	0
$51$	−0.453475	−0.0634992
$52$	0	0
$53$	1.22706	0.168550	0.0842750	−	0.996443i	$-0.473143\pi$
$53$	1.22706	0.168550	0.0842750	+	0.996443i	$0.473143\pi$
$54$	0	0
$55$	−3.49696	−0.471530
$56$	0	0
$57$	1.81387	0.240253
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	5.68615	0.728036	0.364018	−	0.931392i	$-0.381405\pi$
$61$	5.68615	0.728036	0.364018	+	0.931392i	$0.381405\pi$
$62$	0	0
$63$	−4.40700	−0.555230
$64$	0	0
$65$	−7.46558	−0.925992
$66$	0	0
$67$	−5.05183	−0.617179	−0.308589	−	0.951195i	$-0.599857\pi$
$67$	−5.05183	−0.617179	−0.308589	+	0.951195i	$0.599857\pi$
$68$	0	0
$69$	−3.09475	−0.372564
$70$	0	0
$71$	3.31175	0.393033	0.196516	−	0.980501i	$-0.437037\pi$
$71$	3.31175	0.393033	0.196516	+	0.980501i	$0.437037\pi$
$72$	0	0
$73$	3.07942	0.360419	0.180209	−	0.983628i	$-0.442322\pi$
$73$	3.07942	0.360419	0.180209	+	0.983628i	$0.442322\pi$
$74$	0	0
$75$	0.426266	0.0492210
$76$	0	0
$77$	−2.26286	−0.257877
$78$	0	0
$79$	−14.6823	−1.65189	−0.825946	−	0.563749i	$-0.809359\pi$
$79$	−14.6823	−1.65189	−0.825946	+	0.563749i	$0.809359\pi$
$80$	0	0
$81$	7.19153	0.799059
$82$	0	0
$83$	−2.49480	−0.273840	−0.136920	−	0.990582i	$-0.543720\pi$
$83$	−2.49480	−0.273840	−0.136920	+	0.990582i	$0.543720\pi$
$84$	0	0
$85$	−2.43721	−0.264353
$86$	0	0
$87$	−1.59978	−0.171515
$88$	0	0
$89$	14.8134	1.57022	0.785109	−	0.619357i	$-0.212607\pi$
$89$	14.8134	1.57022	0.785109	+	0.619357i	$0.212607\pi$
$90$	0	0
$91$	−4.83094	−0.506420
$92$	0	0
$93$	3.55492	0.368628
$94$	0	0
$95$	9.74869	1.00019
$96$	0	0
$97$	−15.8517	−1.60949	−0.804746	−	0.593620i	$-0.797698\pi$
$97$	−15.8517	−1.60949	−0.804746	+	0.593620i	$0.797698\pi$
$98$	0	0
$99$	4.00941	0.402961
$100$	0	0
$101$	−11.4735	−1.14166	−0.570828	−	0.821070i	$-0.693378\pi$
$101$	−11.4735	−1.14166	−0.570828	+	0.821070i	$0.693378\pi$
$102$	0	0
$103$	−15.2243	−1.50010	−0.750049	−	0.661383i	$-0.769970\pi$
$103$	−15.2243	−1.50010	−0.750049	+	0.661383i	$0.769970\pi$
$104$	0	0
$105$	1.74304	0.170103
$106$	0	0
$107$	16.0961	1.55607	0.778035	−	0.628221i	$-0.216217\pi$
$107$	16.0961	1.55607	0.778035	+	0.628221i	$0.216217\pi$
$108$	0	0
$109$	−6.86840	−0.657873	−0.328937	−	0.944352i	$-0.606690\pi$
$109$	−6.86840	−0.657873	−0.328937	+	0.944352i	$0.606690\pi$
$110$	0	0
$111$	−2.73863	−0.259939
$112$	0	0
$113$	9.61190	0.904211	0.452106	−	0.891964i	$-0.350673\pi$
$113$	9.61190	0.904211	0.452106	+	0.891964i	$0.350673\pi$
$114$	0	0
$115$	−16.6328	−1.55102
$116$	0	0
$117$	8.55959	0.791334
$118$	0	0
$119$	−1.57711	−0.144573
$120$	0	0
$121$	−8.94129	−0.812844
$122$	0	0
$123$	0.631662	0.0569551
$124$	0	0
$125$	−9.89508	−0.885043
$126$	0	0
$127$	−18.6578	−1.65561	−0.827807	−	0.561013i	$-0.810412\pi$
$127$	−18.6578	−1.65561	−0.827807	+	0.561013i	$0.810412\pi$
$128$	0	0
$129$	3.61926	0.318658
$130$	0	0
$131$	−10.6672	−0.931994	−0.465997	−	0.884786i	$-0.654304\pi$
$131$	−10.6672	−0.931994	−0.465997	+	0.884786i	$0.654304\pi$
$132$	0	0
$133$	6.30832	0.547001
$134$	0	0
$135$	−6.40401	−0.551170
$136$	0	0
$137$	−14.8766	−1.27099	−0.635497	−	0.772103i	$-0.719205\pi$
$137$	−14.8766	−1.27099	−0.635497	+	0.772103i	$0.719205\pi$
$138$	0	0
$139$	−19.9484	−1.69200	−0.845999	−	0.533184i	$-0.820995\pi$
$139$	−19.9484	−1.69200	−0.845999	+	0.533184i	$0.820995\pi$
$140$	0	0
$141$	−3.93183	−0.331120
$142$	0	0
$143$	4.39509	0.367536
$144$	0	0
$145$	−8.59806	−0.714030
$146$	0	0
$147$	−2.04641	−0.168785
$148$	0	0
$149$	11.4861	0.940976	0.470488	−	0.882406i	$-0.344078\pi$
$149$	11.4861	0.940976	0.470488	+	0.882406i	$0.344078\pi$
$150$	0	0
$151$	−0.0206329	−0.00167908	−0.000839542	−	1.00000i	$-0.500267\pi$
$151$	−0.0206329	−0.00167908	−0.000839542		1.00000i	$0.500267\pi$
$152$	0	0
$153$	2.79436	0.225911
$154$	0	0
$155$	19.1060	1.53463
$156$	0	0
$157$	2.61670	0.208835	0.104418	−	0.994534i	$-0.466702\pi$
$157$	2.61670	0.208835	0.104418	+	0.994534i	$0.466702\pi$
$158$	0	0
$159$	0.556442	0.0441288
$160$	0	0
$161$	−10.7630	−0.848242
$162$	0	0
$163$	−17.5424	−1.37403	−0.687013	−	0.726645i	$-0.741079\pi$
$163$	−17.5424	−1.37403	−0.687013	+	0.726645i	$0.741079\pi$
$164$	0	0
$165$	−1.58579	−0.123453
$166$	0	0
$167$	18.6026	1.43951	0.719756	−	0.694227i	$-0.244254\pi$
$167$	18.6026	1.43951	0.719756	+	0.694227i	$0.244254\pi$
$168$	0	0
$169$	−3.61701	−0.278232
$170$	0	0
$171$	−11.1773	−0.854747
$172$	0	0
$173$	23.9698	1.82239	0.911193	−	0.411979i	$-0.135162\pi$
$173$	23.9698	1.82239	0.911193	+	0.411979i	$0.135162\pi$
$174$	0	0
$175$	1.48248	0.112065
$176$	0	0
$177$	−0.453475	−0.0340853
$178$	0	0
$179$	−6.49846	−0.485718	−0.242859	−	0.970062i	$-0.578085\pi$
$179$	−6.49846	−0.485718	−0.242859	+	0.970062i	$0.578085\pi$
$180$	0	0
$181$	−10.6417	−0.790989	−0.395494	−	0.918468i	$-0.629427\pi$
$181$	−10.6417	−0.790989	−0.395494	+	0.918468i	$0.629427\pi$
$182$	0	0
$183$	2.57853	0.190610
$184$	0	0
$185$	−14.7188	−1.08215
$186$	0	0
$187$	1.43482	0.104924
$188$	0	0
$189$	−4.14400	−0.301432
$190$	0	0
$191$	−6.36528	−0.460576	−0.230288	−	0.973123i	$-0.573967\pi$
$191$	−6.36528	−0.460576	−0.230288	+	0.973123i	$0.573967\pi$
$192$	0	0
$193$	17.9200	1.28991	0.644956	−	0.764220i	$-0.276876\pi$
$193$	17.9200	1.28991	0.644956	+	0.764220i	$0.276876\pi$
$194$	0	0
$195$	−3.38546	−0.242438
$196$	0	0
$197$	−11.9517	−0.851522	−0.425761	−	0.904836i	$-0.639993\pi$
$197$	−11.9517	−0.851522	−0.425761	+	0.904836i	$0.639993\pi$
$198$	0	0
$199$	−9.46122	−0.670688	−0.335344	−	0.942096i	$-0.608853\pi$
$199$	−9.46122	−0.670688	−0.335344	+	0.942096i	$0.608853\pi$
$200$	0	0
$201$	−2.29088	−0.161586
$202$	0	0
$203$	−5.56375	−0.390499
$204$	0	0
$205$	3.39488	0.237109
$206$	0	0
$207$	19.0702	1.32547
$208$	0	0
$209$	−5.73919	−0.396988
$210$	0	0
$211$	−13.2497	−0.912149	−0.456074	−	0.889942i	$-0.650745\pi$
$211$	−13.2497	−0.912149	−0.456074	+	0.889942i	$0.650745\pi$
$212$	0	0
$213$	1.50180	0.102901
$214$	0	0
$215$	19.4518	1.32660
$216$	0	0
$217$	12.3634	0.839281
$218$	0	0
$219$	1.39644	0.0943627
$220$	0	0
$221$	3.06317	0.206051
$222$	0	0
$223$	22.8774	1.53199	0.765993	−	0.642849i	$-0.222248\pi$
$223$	22.8774	1.53199	0.765993	+	0.642849i	$0.222248\pi$
$224$	0	0
$225$	−2.62670	−0.175113
$226$	0	0
$227$	−7.94528	−0.527347	−0.263673	−	0.964612i	$-0.584934\pi$
$227$	−7.94528	−0.527347	−0.263673	+	0.964612i	$0.584934\pi$
$228$	0	0
$229$	14.4559	0.955271	0.477635	−	0.878558i	$-0.341494\pi$
$229$	14.4559	0.955271	0.477635	+	0.878558i	$0.341494\pi$
$230$	0	0
$231$	−1.02615	−0.0675159
$232$	0	0
$233$	−3.71087	−0.243107	−0.121554	−	0.992585i	$-0.538788\pi$
$233$	−3.71087	−0.243107	−0.121554	+	0.992585i	$0.538788\pi$
$234$	0	0
$235$	−21.1317	−1.37848
$236$	0	0
$237$	−6.65808	−0.432489
$238$	0	0
$239$	−15.5117	−1.00337	−0.501685	−	0.865050i	$-0.667286\pi$
$239$	−15.5117	−1.00337	−0.501685	+	0.865050i	$0.667286\pi$
$240$	0	0
$241$	−5.60096	−0.360790	−0.180395	−	0.983594i	$-0.557738\pi$
$241$	−5.60096	−0.360790	−0.180395	+	0.983594i	$0.557738\pi$
$242$	0	0
$243$	11.1440	0.714886
$244$	0	0
$245$	−10.9985	−0.702668
$246$	0	0
$247$	−12.2525	−0.779606
$248$	0	0
$249$	−1.13133	−0.0716951
$250$	0	0
$251$	−2.03446	−0.128414	−0.0642069	−	0.997937i	$-0.520452\pi$
$251$	−2.03446	−0.128414	−0.0642069	+	0.997937i	$0.520452\pi$
$252$	0	0
$253$	9.79196	0.615615
$254$	0	0
$255$	−1.10521	−0.0692112
$256$	0	0
$257$	−13.7256	−0.856177	−0.428088	−	0.903737i	$-0.640813\pi$
$257$	−13.7256	−0.856177	−0.428088	+	0.903737i	$0.640813\pi$
$258$	0	0
$259$	−9.52447	−0.591822
$260$	0	0
$261$	9.85802	0.610196
$262$	0	0
$263$	19.8680	1.22511	0.612557	−	0.790427i	$-0.290141\pi$
$263$	19.8680	1.22511	0.612557	+	0.790427i	$0.290141\pi$
$264$	0	0
$265$	2.99061	0.183712
$266$	0	0
$267$	6.71751	0.411105
$268$	0	0
$269$	−11.6357	−0.709441	−0.354721	−	0.934972i	$-0.615424\pi$
$269$	−11.6357	−0.709441	−0.354721	+	0.934972i	$0.615424\pi$
$270$	0	0
$271$	−19.5163	−1.18553	−0.592765	−	0.805376i	$-0.701964\pi$
$271$	−19.5163	−1.18553	−0.592765	+	0.805376i	$0.701964\pi$
$272$	0	0
$273$	−2.19071	−0.132588
$274$	0	0
$275$	−1.34873	−0.0813315
$276$	0	0
$277$	11.3188	0.680079	0.340040	−	0.940411i	$-0.389560\pi$
$277$	11.3188	0.680079	0.340040	+	0.940411i	$0.389560\pi$
$278$	0	0
$279$	−21.9058	−1.31147
$280$	0	0
$281$	−16.4310	−0.980193	−0.490096	−	0.871668i	$-0.663038\pi$
$281$	−16.4310	−0.980193	−0.490096	+	0.871668i	$0.663038\pi$
$282$	0	0
$283$	17.8720	1.06238	0.531190	−	0.847253i	$-0.321745\pi$
$283$	17.8720	1.06238	0.531190	+	0.847253i	$0.321745\pi$
$284$	0	0
$285$	4.42079	0.261865
$286$	0	0
$287$	2.19681	0.129674
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−7.18833	−0.421387
$292$	0	0
$293$	−0.884128	−0.0516513	−0.0258257	−	0.999666i	$-0.508221\pi$
$293$	−0.884128	−0.0516513	−0.0258257	+	0.999666i	$0.508221\pi$
$294$	0	0
$295$	−2.43721	−0.141900
$296$	0	0
$297$	3.77013	0.218765
$298$	0	0
$299$	20.9046	1.20895
$300$	0	0
$301$	12.5872	0.725511
$302$	0	0
$303$	−5.20295	−0.298902
$304$	0	0
$305$	13.8583	0.793526
$306$	0	0
$307$	24.0731	1.37392	0.686961	−	0.726694i	$-0.258944\pi$
$307$	24.0731	1.37392	0.686961	+	0.726694i	$0.258944\pi$
$308$	0	0
$309$	−6.90385	−0.392746
$310$	0	0
$311$	−21.7987	−1.23609	−0.618046	−	0.786142i	$-0.712075\pi$
$311$	−21.7987	−1.23609	−0.618046	+	0.786142i	$0.712075\pi$
$312$	0	0
$313$	32.5090	1.83752	0.918758	−	0.394820i	$-0.129193\pi$
$313$	32.5090	1.83752	0.918758	+	0.394820i	$0.129193\pi$
$314$	0	0
$315$	−10.7408	−0.605175
$316$	0	0
$317$	13.6868	0.768728	0.384364	−	0.923182i	$-0.374421\pi$
$317$	13.6868	0.768728	0.384364	+	0.923182i	$0.374421\pi$
$318$	0	0
$319$	5.06180	0.283406
$320$	0	0
$321$	7.29919	0.407401
$322$	0	0
$323$	−3.99994	−0.222562
$324$	0	0
$325$	−2.87937	−0.159719
$326$	0	0
$327$	−3.11465	−0.172240
$328$	0	0
$329$	−13.6742	−0.753883
$330$	0	0
$331$	−3.58772	−0.197199	−0.0985994	−	0.995127i	$-0.531436\pi$
$331$	−3.58772	−0.197199	−0.0985994	+	0.995127i	$0.531436\pi$
$332$	0	0
$333$	16.8757	0.924784
$334$	0	0
$335$	−12.3124	−0.672697
$336$	0	0
$337$	30.1021	1.63977	0.819884	−	0.572529i	$-0.194038\pi$
$337$	30.1021	1.63977	0.819884	+	0.572529i	$0.194038\pi$
$338$	0	0
$339$	4.35876	0.236735
$340$	0	0
$341$	−11.2480	−0.609112
$342$	0	0
$343$	−18.1568	−0.980375
$344$	0	0
$345$	−7.54256	−0.406078
$346$	0	0
$347$	−33.0184	−1.77252	−0.886260	−	0.463187i	$-0.846706\pi$
$347$	−33.0184	−1.77252	−0.886260	+	0.463187i	$0.846706\pi$
$348$	0	0
$349$	24.4164	1.30698	0.653490	−	0.756936i	$-0.273304\pi$
$349$	24.4164	1.30698	0.653490	+	0.756936i	$0.273304\pi$
$350$	0	0
$351$	8.04877	0.429612
$352$	0	0
$353$	−1.87403	−0.0997445	−0.0498722	−	0.998756i	$-0.515881\pi$
$353$	−1.87403	−0.0997445	−0.0498722	+	0.998756i	$0.515881\pi$
$354$	0	0
$355$	8.07144	0.428388
$356$	0	0
$357$	−0.715178	−0.0378512
$358$	0	0
$359$	2.04466	0.107913	0.0539565	−	0.998543i	$-0.482817\pi$
$359$	2.04466	0.107913	0.0539565	+	0.998543i	$0.482817\pi$
$360$	0	0
$361$	−3.00051	−0.157922
$362$	0	0
$363$	−4.05465	−0.212814
$364$	0	0
$365$	7.50519	0.392840
$366$	0	0
$367$	−24.1148	−1.25878	−0.629390	−	0.777089i	$-0.716695\pi$
$367$	−24.1148	−1.25878	−0.629390	+	0.777089i	$0.716695\pi$
$368$	0	0
$369$	−3.89237	−0.202629
$370$	0	0
$371$	1.93521	0.100471
$372$	0	0
$373$	−21.3010	−1.10292	−0.551462	−	0.834200i	$-0.685930\pi$
$373$	−21.3010	−1.10292	−0.551462	+	0.834200i	$0.685930\pi$
$374$	0	0
$375$	−4.48717	−0.231717
$376$	0	0
$377$	10.8063	0.556554
$378$	0	0
$379$	13.9252	0.715292	0.357646	−	0.933857i	$-0.383580\pi$
$379$	13.9252	0.715292	0.357646	+	0.933857i	$0.383580\pi$
$380$	0	0
$381$	−8.46086	−0.433463
$382$	0	0
$383$	−2.48053	−0.126749	−0.0633745	−	0.997990i	$-0.520186\pi$
$383$	−2.48053	−0.126749	−0.0633745	+	0.997990i	$0.520186\pi$
$384$	0	0
$385$	−5.51508	−0.281074
$386$	0	0
$387$	−22.3023	−1.13369
$388$	0	0
$389$	27.4212	1.39031	0.695154	−	0.718861i	$-0.255336\pi$
$389$	27.4212	1.39031	0.695154	+	0.718861i	$0.255336\pi$
$390$	0	0
$391$	6.82452	0.345131
$392$	0	0
$393$	−4.83729	−0.244009
$394$	0	0
$395$	−35.7840	−1.80049
$396$	0	0
$397$	5.07589	0.254751	0.127376	−	0.991855i	$-0.459345\pi$
$397$	5.07589	0.254751	0.127376	+	0.991855i	$0.459345\pi$
$398$	0	0
$399$	2.86067	0.143212
$400$	0	0
$401$	17.6730	0.882546	0.441273	−	0.897373i	$-0.354527\pi$
$401$	17.6730	0.882546	0.441273	+	0.897373i	$0.354527\pi$
$402$	0	0
$403$	−24.0130	−1.19617
$404$	0	0
$405$	17.5273	0.870938
$406$	0	0
$407$	8.66518	0.429517
$408$	0	0
$409$	22.5746	1.11624	0.558121	−	0.829760i	$-0.311523\pi$
$409$	22.5746	1.11624	0.558121	+	0.829760i	$0.311523\pi$
$410$	0	0
$411$	−6.74617	−0.332764
$412$	0	0
$413$	−1.57711	−0.0776043
$414$	0	0
$415$	−6.08035	−0.298473
$416$	0	0
$417$	−9.04609	−0.442989
$418$	0	0
$419$	10.3638	0.506304	0.253152	−	0.967427i	$-0.418533\pi$
$419$	10.3638	0.506304	0.253152	+	0.967427i	$0.418533\pi$
$420$	0	0
$421$	−3.64460	−0.177627	−0.0888136	−	0.996048i	$-0.528308\pi$
$421$	−3.64460	−0.177627	−0.0888136	+	0.996048i	$0.528308\pi$
$422$	0	0
$423$	24.2283	1.17802
$424$	0	0
$425$	−0.939999	−0.0455967
$426$	0	0
$427$	8.96765	0.433975
$428$	0	0
$429$	1.99307	0.0962261
$430$	0	0
$431$	19.3289	0.931043	0.465521	−	0.885037i	$-0.345867\pi$
$431$	19.3289	0.931043	0.465521	+	0.885037i	$0.345867\pi$
$432$	0	0
$433$	−29.8067	−1.43242	−0.716209	−	0.697885i	$-0.754124\pi$
$433$	−29.8067	−1.43242	−0.716209	+	0.697885i	$0.754124\pi$
$434$	0	0
$435$	−3.89901	−0.186943
$436$	0	0
$437$	−27.2976	−1.30582
$438$	0	0
$439$	28.0908	1.34070	0.670350	−	0.742045i	$-0.266144\pi$
$439$	28.0908	1.34070	0.670350	+	0.742045i	$0.266144\pi$
$440$	0	0
$441$	12.6102	0.600487
$442$	0	0
$443$	2.28491	0.108559	0.0542797	−	0.998526i	$-0.482714\pi$
$443$	2.28491	0.108559	0.0542797	+	0.998526i	$0.482714\pi$
$444$	0	0
$445$	36.1034	1.71147
$446$	0	0
$447$	5.20865	0.246361
$448$	0	0
$449$	17.1490	0.809312	0.404656	−	0.914469i	$-0.367391\pi$
$449$	17.1490	0.809312	0.404656	+	0.914469i	$0.367391\pi$
$450$	0	0
$451$	−1.99862	−0.0941111
$452$	0	0
$453$	−0.00935652	−0.000439608 0
$454$	0	0
$455$	−11.7740	−0.551974
$456$	0	0
$457$	8.30477	0.388481	0.194240	−	0.980954i	$-0.437776\pi$
$457$	8.30477	0.388481	0.194240	+	0.980954i	$0.437776\pi$
$458$	0	0
$459$	2.62760	0.122646
$460$	0	0
$461$	30.1785	1.40555	0.702776	−	0.711411i	$-0.251944\pi$
$461$	30.1785	1.40555	0.702776	+	0.711411i	$0.251944\pi$
$462$	0	0
$463$	8.36001	0.388523	0.194261	−	0.980950i	$-0.437769\pi$
$463$	8.36001	0.388523	0.194261	+	0.980950i	$0.437769\pi$
$464$	0	0
$465$	8.66410	0.401788
$466$	0	0
$467$	1.57523	0.0728929	0.0364464	−	0.999336i	$-0.488396\pi$
$467$	1.57523	0.0728929	0.0364464	+	0.999336i	$0.488396\pi$
$468$	0	0
$469$	−7.96726	−0.367894
$470$	0	0
$471$	1.18661	0.0546760
$472$	0	0
$473$	−11.4516	−0.526543
$474$	0	0
$475$	3.75994	0.172518
$476$	0	0
$477$	−3.42885	−0.156996
$478$	0	0
$479$	−19.8890	−0.908754	−0.454377	−	0.890810i	$-0.650138\pi$
$479$	−19.8890	−0.908754	−0.454377	+	0.890810i	$0.650138\pi$
$480$	0	0
$481$	18.4991	0.843486
$482$	0	0
$483$	−4.88075	−0.222082
$484$	0	0
$485$	−38.6338	−1.75427
$486$	0	0
$487$	−30.1816	−1.36766	−0.683830	−	0.729642i	$-0.739687\pi$
$487$	−30.1816	−1.36766	−0.683830	+	0.729642i	$0.739687\pi$
$488$	0	0
$489$	−7.95504	−0.359739
$490$	0	0
$491$	13.9373	0.628981	0.314491	−	0.949261i	$-0.398166\pi$
$491$	13.9373	0.628981	0.314491	+	0.949261i	$0.398166\pi$
$492$	0	0
$493$	3.52783	0.158885
$494$	0	0
$495$	9.77177	0.439208
$496$	0	0
$497$	5.22298	0.234283
$498$	0	0
$499$	−16.7489	−0.749784	−0.374892	−	0.927068i	$-0.622320\pi$
$499$	−16.7489	−0.749784	−0.374892	+	0.927068i	$0.622320\pi$
$500$	0	0
$501$	8.43582	0.376885
$502$	0	0
$503$	39.9450	1.78106	0.890530	−	0.454925i	$-0.150334\pi$
$503$	39.9450	1.78106	0.890530	+	0.454925i	$0.150334\pi$
$504$	0	0
$505$	−27.9634	−1.24435
$506$	0	0
$507$	−1.64023	−0.0728450
$508$	0	0
$509$	14.8624	0.658764	0.329382	−	0.944197i	$-0.393160\pi$
$509$	14.8624	0.658764	0.329382	+	0.944197i	$0.393160\pi$
$510$	0	0
$511$	4.85657	0.214842
$512$	0	0
$513$	−10.5102	−0.464038
$514$	0	0
$515$	−37.1049	−1.63504
$516$	0	0
$517$	12.4405	0.547134
$518$	0	0
$519$	10.8697	0.477126
$520$	0	0
$521$	−30.2768	−1.32645	−0.663226	−	0.748419i	$-0.730813\pi$
$521$	−30.2768	−1.32645	−0.663226	+	0.748419i	$0.730813\pi$
$522$	0	0
$523$	8.87499	0.388076	0.194038	−	0.980994i	$-0.437841\pi$
$523$	8.87499	0.388076	0.194038	+	0.980994i	$0.437841\pi$
$524$	0	0
$525$	0.672267	0.0293401
$526$	0	0
$527$	−7.83929	−0.341485
$528$	0	0
$529$	23.5740	1.02496
$530$	0	0
$531$	2.79436	0.121265
$532$	0	0
$533$	−4.26680	−0.184816
$534$	0	0
$535$	39.2296	1.69605
$536$	0	0
$537$	−2.94689	−0.127168
$538$	0	0
$539$	6.47497	0.278897
$540$	0	0
$541$	4.49305	0.193171	0.0965857	−	0.995325i	$-0.469208\pi$
$541$	4.49305	0.193171	0.0965857	+	0.995325i	$0.469208\pi$
$542$	0	0
$543$	−4.82573	−0.207092
$544$	0	0
$545$	−16.7397	−0.717052
$546$	0	0
$547$	−36.0040	−1.53942	−0.769710	−	0.638394i	$-0.779599\pi$
$547$	−36.0040	−1.53942	−0.769710	+	0.638394i	$0.779599\pi$
$548$	0	0
$549$	−15.8891	−0.678132
$550$	0	0
$551$	−14.1111	−0.601152
$552$	0	0
$553$	−23.1556	−0.984676
$554$	0	0
$555$	−6.67462	−0.283322
$556$	0	0
$557$	−12.0897	−0.512259	−0.256129	−	0.966643i	$-0.582447\pi$
$557$	−12.0897	−0.512259	−0.256129	+	0.966643i	$0.582447\pi$
$558$	0	0
$559$	−24.4477	−1.03403
$560$	0	0
$561$	0.650656	0.0274707
$562$	0	0
$563$	24.8784	1.04850	0.524250	−	0.851565i	$-0.324346\pi$
$563$	24.8784	1.04850	0.524250	+	0.851565i	$0.324346\pi$
$564$	0	0
$565$	23.4262	0.985549
$566$	0	0
$567$	11.3418	0.476311
$568$	0	0
$569$	−4.46486	−0.187177	−0.0935884	−	0.995611i	$-0.529834\pi$
$569$	−4.46486	−0.187177	−0.0935884	+	0.995611i	$0.529834\pi$
$570$	0	0
$571$	−9.29575	−0.389015	−0.194508	−	0.980901i	$-0.562311\pi$
$571$	−9.29575	−0.389015	−0.194508	+	0.980901i	$0.562311\pi$
$572$	0	0
$573$	−2.88650	−0.120585
$574$	0	0
$575$	−6.41504	−0.267526
$576$	0	0
$577$	21.7856	0.906948	0.453474	−	0.891269i	$-0.350184\pi$
$577$	21.7856	0.906948	0.453474	+	0.891269i	$0.350184\pi$
$578$	0	0
$579$	8.12629	0.337717
$580$	0	0
$581$	−3.93456	−0.163233
$582$	0	0
$583$	−1.76061	−0.0729172
$584$	0	0
$585$	20.8615	0.862518
$586$	0	0
$587$	19.8082	0.817574	0.408787	−	0.912630i	$-0.365952\pi$
$587$	19.8082	0.817574	0.408787	+	0.912630i	$0.365952\pi$
$588$	0	0
$589$	31.3567	1.29203
$590$	0	0
$591$	−5.41979	−0.222940
$592$	0	0
$593$	4.36921	0.179422	0.0897111	−	0.995968i	$-0.471406\pi$
$593$	4.36921	0.179422	0.0897111	+	0.995968i	$0.471406\pi$
$594$	0	0
$595$	−3.84374	−0.157578
$596$	0	0
$597$	−4.29043	−0.175596
$598$	0	0
$599$	−16.4978	−0.674082	−0.337041	−	0.941490i	$-0.609426\pi$
$599$	−16.4978	−0.674082	−0.337041	+	0.941490i	$0.609426\pi$
$600$	0	0
$601$	−16.6233	−0.678080	−0.339040	−	0.940772i	$-0.610102\pi$
$601$	−16.6233	−0.678080	−0.339040	+	0.940772i	$0.610102\pi$
$602$	0	0
$603$	14.1166	0.574873
$604$	0	0
$605$	−21.7918	−0.885963
$606$	0	0
$607$	−14.3661	−0.583100	−0.291550	−	0.956556i	$-0.594171\pi$
$607$	−14.3661	−0.583100	−0.291550	+	0.956556i	$0.594171\pi$
$608$	0	0
$609$	−2.52302	−0.102238
$610$	0	0
$611$	26.5590	1.07446
$612$	0	0
$613$	−42.3099	−1.70888	−0.854440	−	0.519551i	$-0.826099\pi$
$613$	−42.3099	−1.70888	−0.854440	+	0.519551i	$0.826099\pi$
$614$	0	0
$615$	1.53949	0.0620784
$616$	0	0
$617$	−1.74462	−0.0702356	−0.0351178	−	0.999383i	$-0.511181\pi$
$617$	−1.74462	−0.0702356	−0.0351178	+	0.999383i	$0.511181\pi$
$618$	0	0
$619$	−9.61399	−0.386419	−0.193209	−	0.981158i	$-0.561890\pi$
$619$	−9.61399	−0.386419	−0.193209	+	0.981158i	$0.561890\pi$
$620$	0	0
$621$	17.9321	0.719590
$622$	0	0
$623$	23.3623	0.935991
$624$	0	0
$625$	−28.8164	−1.15266
$626$	0	0
$627$	−2.60258	−0.103937
$628$	0	0
$629$	6.03921	0.240799
$630$	0	0
$631$	−25.1841	−1.00256	−0.501282	−	0.865284i	$-0.667138\pi$
$631$	−25.1841	−1.00256	−0.501282	+	0.865284i	$0.667138\pi$
$632$	0	0
$633$	−6.00842	−0.238813
$634$	0	0
$635$	−45.4731	−1.80454
$636$	0	0
$637$	13.8233	0.547698
$638$	0	0
$639$	−9.25423	−0.366092
$640$	0	0
$641$	−47.0536	−1.85851	−0.929253	−	0.369445i	$-0.879548\pi$
$641$	−47.0536	−1.85851	−0.929253	+	0.369445i	$0.879548\pi$
$642$	0	0
$643$	−1.71605	−0.0676746	−0.0338373	−	0.999427i	$-0.510773\pi$
$643$	−1.71605	−0.0676746	−0.0338373	+	0.999427i	$0.510773\pi$
$644$	0	0
$645$	8.82091	0.347323
$646$	0	0
$647$	5.87748	0.231068	0.115534	−	0.993304i	$-0.463142\pi$
$647$	5.87748	0.231068	0.115534	+	0.993304i	$0.463142\pi$
$648$	0	0
$649$	1.43482	0.0563216
$650$	0	0
$651$	5.60649	0.219736
$652$	0	0
$653$	15.6602	0.612832	0.306416	−	0.951898i	$-0.400870\pi$
$653$	15.6602	0.612832	0.306416	+	0.951898i	$0.400870\pi$
$654$	0	0
$655$	−25.9981	−1.01583
$656$	0	0
$657$	−8.60501	−0.335713
$658$	0	0
$659$	11.3607	0.442549	0.221274	−	0.975212i	$-0.428978\pi$
$659$	11.3607	0.442549	0.221274	+	0.975212i	$0.428978\pi$
$660$	0	0
$661$	−41.2976	−1.60629	−0.803146	−	0.595782i	$-0.796842\pi$
$661$	−41.2976	−1.60629	−0.803146	+	0.595782i	$0.796842\pi$
$662$	0	0
$663$	1.38907	0.0539470
$664$	0	0
$665$	15.3747	0.596206
$666$	0	0
$667$	24.0757	0.932215
$668$	0	0
$669$	10.3743	0.401095
$670$	0	0
$671$	−8.15860	−0.314959
$672$	0	0
$673$	−26.5471	−1.02331	−0.511657	−	0.859190i	$-0.670968\pi$
$673$	−26.5471	−1.02331	−0.511657	+	0.859190i	$0.670968\pi$
$674$	0	0
$675$	−2.46994	−0.0950681
$676$	0	0
$677$	38.4424	1.47746	0.738731	−	0.674000i	$-0.235425\pi$
$677$	38.4424	1.47746	0.738731	+	0.674000i	$0.235425\pi$
$678$	0	0
$679$	−24.9997	−0.959401
$680$	0	0
$681$	−3.60299	−0.138067
$682$	0	0
$683$	−30.1923	−1.15528	−0.577638	−	0.816293i	$-0.696026\pi$
$683$	−30.1923	−1.15528	−0.577638	+	0.816293i	$0.696026\pi$
$684$	0	0
$685$	−36.2574	−1.38532
$686$	0	0
$687$	6.55538	0.250103
$688$	0	0
$689$	−3.75870	−0.143195
$690$	0	0
$691$	8.49880	0.323309	0.161655	−	0.986847i	$-0.448317\pi$
$691$	8.49880	0.323309	0.161655	+	0.986847i	$0.448317\pi$
$692$	0	0
$693$	6.32326	0.240201
$694$	0	0
$695$	−48.6184	−1.84420
$696$	0	0
$697$	−1.39294	−0.0527613
$698$	0	0
$699$	−1.68279	−0.0636489
$700$	0	0
$701$	−30.6404	−1.15727	−0.578637	−	0.815585i	$-0.696415\pi$
$701$	−30.6404	−1.15727	−0.578637	+	0.815585i	$0.696415\pi$
$702$	0	0
$703$	−24.1564	−0.911077
$704$	0	0
$705$	−9.58270	−0.360905
$706$	0	0
$707$	−18.0949	−0.680530
$708$	0	0
$709$	−29.0445	−1.09079	−0.545395	−	0.838179i	$-0.683620\pi$
$709$	−29.0445	−1.09079	−0.545395	+	0.838179i	$0.683620\pi$
$710$	0	0
$711$	41.0278	1.53866
$712$	0	0
$713$	−53.4994	−2.00357
$714$	0	0
$715$	10.7118	0.400598
$716$	0	0
$717$	−7.03418	−0.262696
$718$	0	0
$719$	−51.2759	−1.91227	−0.956134	−	0.292930i	$-0.905370\pi$
$719$	−51.2759	−1.91227	−0.956134	+	0.292930i	$0.905370\pi$
$720$	0	0
$721$	−24.0104	−0.894193
$722$	0	0
$723$	−2.53990	−0.0944598
$724$	0	0
$725$	−3.31615	−0.123159
$726$	0	0
$727$	44.4988	1.65037	0.825185	−	0.564862i	$-0.191071\pi$
$727$	44.4988	1.65037	0.825185	+	0.564862i	$0.191071\pi$
$728$	0	0
$729$	−16.5211	−0.611892
$730$	0	0
$731$	−7.98117	−0.295194
$732$	0	0
$733$	−30.7666	−1.13639	−0.568195	−	0.822894i	$-0.692358\pi$
$733$	−30.7666	−1.13639	−0.568195	+	0.822894i	$0.692358\pi$
$734$	0	0
$735$	−4.98755	−0.183968
$736$	0	0
$737$	7.24847	0.267001
$738$	0	0
$739$	−3.24740	−0.119458	−0.0597288	−	0.998215i	$-0.519024\pi$
$739$	−3.24740	−0.119458	−0.0597288	+	0.998215i	$0.519024\pi$
$740$	0	0
$741$	−5.55619	−0.204112
$742$	0	0
$743$	38.0248	1.39500	0.697498	−	0.716587i	$-0.254297\pi$
$743$	38.0248	1.39500	0.697498	+	0.716587i	$0.254297\pi$
$744$	0	0
$745$	27.9940	1.02562
$746$	0	0
$747$	6.97137	0.255069
$748$	0	0
$749$	25.3853	0.927558
$750$	0	0
$751$	−20.1695	−0.735996	−0.367998	−	0.929827i	$-0.619957\pi$
$751$	−20.1695	−0.735996	−0.367998	+	0.929827i	$0.619957\pi$
$752$	0	0
$753$	−0.922576	−0.0336206
$754$	0	0
$755$	−0.0502868	−0.00183012
$756$	0	0
$757$	18.1157	0.658425	0.329213	−	0.944256i	$-0.393217\pi$
$757$	18.1157	0.658425	0.329213	+	0.944256i	$0.393217\pi$
$758$	0	0
$759$	4.44041	0.161177
$760$	0	0
$761$	44.8732	1.62665	0.813327	−	0.581807i	$-0.197654\pi$
$761$	44.8732	1.62665	0.813327	+	0.581807i	$0.197654\pi$
$762$	0	0
$763$	−10.8322	−0.392152
$764$	0	0
$765$	6.81045	0.246232
$766$	0	0
$767$	3.06317	0.110605
$768$	0	0
$769$	−21.9293	−0.790790	−0.395395	−	0.918511i	$-0.629392\pi$
$769$	−21.9293	−0.790790	−0.395395	+	0.918511i	$0.629392\pi$
$770$	0	0
$771$	−6.22420	−0.224159
$772$	0	0
$773$	16.1495	0.580858	0.290429	−	0.956897i	$-0.406202\pi$
$773$	16.1495	0.580858	0.290429	+	0.956897i	$0.406202\pi$
$774$	0	0
$775$	7.36893	0.264700
$776$	0	0
$777$	−4.31911	−0.154947
$778$	0	0
$779$	5.57166	0.199625
$780$	0	0
$781$	−4.75177	−0.170032
$782$	0	0
$783$	9.26971	0.331272
$784$	0	0
$785$	6.37745	0.227621
$786$	0	0
$787$	45.3748	1.61744	0.808718	−	0.588196i	$-0.200162\pi$
$787$	45.3748	1.61744	0.808718	+	0.588196i	$0.200162\pi$
$788$	0	0
$789$	9.00965	0.320752
$790$	0	0
$791$	15.1590	0.538991
$792$	0	0
$793$	−17.4176	−0.618517
$794$	0	0
$795$	1.35617	0.0480983
$796$	0	0
$797$	−44.6951	−1.58318	−0.791592	−	0.611050i	$-0.790747\pi$
$797$	−44.6951	−1.58318	−0.791592	+	0.611050i	$0.790747\pi$
$798$	0	0
$799$	8.67044	0.306738
$800$	0	0
$801$	−41.3940	−1.46259
$802$	0	0
$803$	−4.41841	−0.155922
$804$	0	0
$805$	−26.2317	−0.924545
$806$	0	0
$807$	−5.27650	−0.185742
$808$	0	0
$809$	8.56691	0.301197	0.150598	−	0.988595i	$-0.451880\pi$
$809$	8.56691	0.301197	0.150598	+	0.988595i	$0.451880\pi$
$810$	0	0
$811$	−42.1858	−1.48134	−0.740671	−	0.671868i	$-0.765492\pi$
$811$	−42.1858	−1.48134	−0.740671	+	0.671868i	$0.765492\pi$
$812$	0	0
$813$	−8.85015	−0.310388
$814$	0	0
$815$	−42.7545	−1.49763
$816$	0	0
$817$	31.9242	1.11689
$818$	0	0
$819$	13.4994	0.471706
$820$	0	0
$821$	−22.6151	−0.789273	−0.394636	−	0.918837i	$-0.629129\pi$
$821$	−22.6151	−0.789273	−0.394636	+	0.918837i	$0.629129\pi$
$822$	0	0
$823$	9.32448	0.325031	0.162515	−	0.986706i	$-0.448039\pi$
$823$	9.32448	0.325031	0.162515	+	0.986706i	$0.448039\pi$
$824$	0	0
$825$	−0.611616	−0.0212937
$826$	0	0
$827$	−29.4618	−1.02449	−0.512244	−	0.858840i	$-0.671186\pi$
$827$	−29.4618	−1.02449	−0.512244	+	0.858840i	$0.671186\pi$
$828$	0	0
$829$	20.7991	0.722381	0.361191	−	0.932492i	$-0.382370\pi$
$829$	20.7991	0.722381	0.361191	+	0.932492i	$0.382370\pi$
$830$	0	0
$831$	5.13278	0.178054
$832$	0	0
$833$	4.51274	0.156357
$834$	0	0
$835$	45.3385	1.56900
$836$	0	0
$837$	−20.5985	−0.711988
$838$	0	0
$839$	35.3721	1.22118	0.610590	−	0.791947i	$-0.290932\pi$
$839$	35.3721	1.22118	0.610590	+	0.791947i	$0.290932\pi$
$840$	0	0
$841$	−16.5544	−0.570843
$842$	0	0
$843$	−7.45106	−0.256628
$844$	0	0
$845$	−8.81543	−0.303260
$846$	0	0
$847$	−14.1014	−0.484528
$848$	0	0
$849$	8.10451	0.278146
$850$	0	0
$851$	41.2147	1.41282
$852$	0	0
$853$	14.3840	0.492498	0.246249	−	0.969207i	$-0.420802\pi$
$853$	14.3840	0.492498	0.246249	+	0.969207i	$0.420802\pi$
$854$	0	0
$855$	−27.2414	−0.931635
$856$	0	0
$857$	41.6732	1.42353	0.711765	−	0.702418i	$-0.247896\pi$
$857$	41.6732	1.42353	0.711765	+	0.702418i	$0.247896\pi$
$858$	0	0
$859$	44.4605	1.51697	0.758487	−	0.651688i	$-0.225939\pi$
$859$	44.4605	1.51697	0.758487	+	0.651688i	$0.225939\pi$
$860$	0	0
$861$	0.996198	0.0339503
$862$	0	0
$863$	−13.1056	−0.446121	−0.223061	−	0.974805i	$-0.571605\pi$
$863$	−13.1056	−0.446121	−0.223061	+	0.974805i	$0.571605\pi$
$864$	0	0
$865$	58.4194	1.98632
$866$	0	0
$867$	0.453475	0.0154008
$868$	0	0
$869$	21.0665	0.714633
$870$	0	0
$871$	15.4746	0.524336
$872$	0	0
$873$	44.2952	1.49917
$874$	0	0
$875$	−15.6056	−0.527565
$876$	0	0
$877$	40.6575	1.37291	0.686454	−	0.727174i	$-0.259166\pi$
$877$	40.6575	1.37291	0.686454	+	0.727174i	$0.259166\pi$
$878$	0	0
$879$	−0.400930	−0.0135230
$880$	0	0
$881$	18.7742	0.632518	0.316259	−	0.948673i	$-0.397573\pi$
$881$	18.7742	0.632518	0.316259	+	0.948673i	$0.397573\pi$
$882$	0	0
$883$	−53.0066	−1.78381	−0.891907	−	0.452219i	$-0.850632\pi$
$883$	−53.0066	−1.78381	−0.891907	+	0.452219i	$0.850632\pi$
$884$	0	0
$885$	−1.10521	−0.0371514
$886$	0	0
$887$	−29.8268	−1.00148	−0.500742	−	0.865596i	$-0.666940\pi$
$887$	−29.8268	−1.00148	−0.500742	+	0.865596i	$0.666940\pi$
$888$	0	0
$889$	−29.4254	−0.986895
$890$	0	0
$891$	−10.3186	−0.345685
$892$	0	0
$893$	−34.6812	−1.16056
$894$	0	0
$895$	−15.8381	−0.529410
$896$	0	0
$897$	9.47973	0.316519
$898$	0	0
$899$	−27.6556	−0.922367
$900$	0	0
$901$	−1.22706	−0.0408794
$902$	0	0
$903$	5.70796	0.189949
$904$	0	0
$905$	−25.9360	−0.862142
$906$	0	0
$907$	−13.9145	−0.462025	−0.231012	−	0.972951i	$-0.574204\pi$
$907$	−13.9145	−0.462025	−0.231012	+	0.972951i	$0.574204\pi$
$908$	0	0
$909$	32.0611	1.06340
$910$	0	0
$911$	−34.0797	−1.12911	−0.564556	−	0.825395i	$-0.690953\pi$
$911$	−34.0797	−1.12911	−0.564556	+	0.825395i	$0.690953\pi$
$912$	0	0
$913$	3.57959	0.118467
$914$	0	0
$915$	6.28441	0.207756
$916$	0	0
$917$	−16.8232	−0.555552
$918$	0	0
$919$	30.3312	1.00054	0.500268	−	0.865871i	$-0.333235\pi$
$919$	30.3312	1.00054	0.500268	+	0.865871i	$0.333235\pi$
$920$	0	0
$921$	10.9165	0.359712
$922$	0	0
$923$	−10.1445	−0.333909
$924$	0	0
$925$	−5.67685	−0.186654
$926$	0	0
$927$	42.5422	1.39727
$928$	0	0
$929$	51.9229	1.70354	0.851768	−	0.523919i	$-0.175531\pi$
$929$	51.9229	1.70354	0.851768	+	0.523919i	$0.175531\pi$
$930$	0	0
$931$	−18.0507	−0.591587
$932$	0	0
$933$	−9.88517	−0.323626
$934$	0	0
$935$	3.49696	0.114363
$936$	0	0
$937$	44.2471	1.44549	0.722745	−	0.691115i	$-0.242880\pi$
$937$	44.2471	1.44549	0.722745	+	0.691115i	$0.242880\pi$
$938$	0	0
$939$	14.7420	0.481088
$940$	0	0
$941$	44.5218	1.45137	0.725684	−	0.688028i	$-0.241524\pi$
$941$	44.5218	1.45137	0.725684	+	0.688028i	$0.241524\pi$
$942$	0	0
$943$	−9.50612	−0.309562
$944$	0	0
$945$	−10.0998	−0.328547
$946$	0	0
$947$	−4.36074	−0.141705	−0.0708524	−	0.997487i	$-0.522572\pi$
$947$	−4.36074	−0.141705	−0.0708524	+	0.997487i	$0.522572\pi$
$948$	0	0
$949$	−9.43277	−0.306201
$950$	0	0
$951$	6.20663	0.201264
$952$	0	0
$953$	41.1795	1.33393	0.666967	−	0.745087i	$-0.267592\pi$
$953$	41.1795	1.33393	0.666967	+	0.745087i	$0.267592\pi$
$954$	0	0
$955$	−15.5135	−0.502006
$956$	0	0
$957$	2.29540	0.0741997
$958$	0	0
$959$	−23.4620	−0.757626
$960$	0	0
$961$	30.4545	0.982402
$962$	0	0
$963$	−44.9783	−1.44941
$964$	0	0
$965$	43.6749	1.40594
$966$	0	0
$967$	−28.1059	−0.903826	−0.451913	−	0.892062i	$-0.649258\pi$
$967$	−28.1059	−0.903826	−0.451913	+	0.892062i	$0.649258\pi$
$968$	0	0
$969$	−1.81387	−0.0582700
$970$	0	0
$971$	1.54825	0.0496857	0.0248428	−	0.999691i	$-0.492091\pi$
$971$	1.54825	0.0496857	0.0248428	+	0.999691i	$0.492091\pi$
$972$	0	0
$973$	−31.4607	−1.00858
$974$	0	0
$975$	−1.30572	−0.0418167
$976$	0	0
$977$	37.7224	1.20685	0.603423	−	0.797421i	$-0.293803\pi$
$977$	37.7224	1.20685	0.603423	+	0.797421i	$0.293803\pi$
$978$	0	0
$979$	−21.2546	−0.679300
$980$	0	0
$981$	19.1928	0.612778
$982$	0	0
$983$	4.83845	0.154323	0.0771614	−	0.997019i	$-0.475414\pi$
$983$	4.83845	0.154323	0.0771614	+	0.997019i	$0.475414\pi$
$984$	0	0
$985$	−29.1288	−0.928119
$986$	0	0
$987$	−6.20091	−0.197377
$988$	0	0
$989$	−54.4676	−1.73197
$990$	0	0
$991$	6.16037	0.195691	0.0978454	−	0.995202i	$-0.468805\pi$
$991$	6.16037	0.195691	0.0978454	+	0.995202i	$0.468805\pi$
$992$	0	0
$993$	−1.62694	−0.0516294
$994$	0	0
$995$	−23.0590	−0.731020
$996$	0	0
$997$	−31.3698	−0.993491	−0.496745	−	0.867896i	$-0.665472\pi$
$997$	−31.3698	−0.993491	−0.496745	+	0.867896i	$0.665472\pi$
$998$	0	0
$999$	15.8686	0.502061

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.14	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.14	✓	23	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+0.453475 q^{3} +2.43721 q^{5} +1.57711 q^{7} -2.79436 q^{9} +O(q^{10})\)	\(q+0.453475 q^{3} +2.43721 q^{5} +1.57711 q^{7} -2.79436 q^{9} -1.43482 q^{11} -3.06317 q^{13} +1.10521 q^{15} -1.00000 q^{17} +3.99994 q^{19} +0.715178 q^{21} -6.82452 q^{23} +0.939999 q^{25} -2.62760 q^{27} -3.52783 q^{29} +7.83929 q^{31} -0.650656 q^{33} +3.84374 q^{35} -6.03921 q^{37} -1.38907 q^{39} +1.39294 q^{41} +7.98117 q^{43} -6.81045 q^{45} -8.67044 q^{47} -4.51274 q^{49} -0.453475 q^{51} +1.22706 q^{53} -3.49696 q^{55} +1.81387 q^{57} -1.00000 q^{59} +5.68615 q^{61} -4.40700 q^{63} -7.46558 q^{65} -5.05183 q^{67} -3.09475 q^{69} +3.31175 q^{71} +3.07942 q^{73} +0.426266 q^{75} -2.26286 q^{77} -14.6823 q^{79} +7.19153 q^{81} -2.49480 q^{83} -2.43721 q^{85} -1.59978 q^{87} +14.8134 q^{89} -4.83094 q^{91} +3.55492 q^{93} +9.74869 q^{95} -15.8517 q^{97} +4.00941 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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