LMFDB - Embedded newform 4024.2.a.e.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.278789 q^{3} -0.533193 q^{5} +1.93823 q^{7} -2.92228 q^{9} +O(q^{10})$	$q-0.278789 q^{3} -0.533193 q^{5} +1.93823 q^{7} -2.92228 q^{9} -3.65384 q^{11} +5.69726 q^{13} +0.148648 q^{15} +6.46842 q^{17} -4.37548 q^{19} -0.540358 q^{21} -8.89803 q^{23} -4.71571 q^{25} +1.65107 q^{27} -0.939891 q^{29} +6.67842 q^{31} +1.01865 q^{33} -1.03345 q^{35} +0.254878 q^{37} -1.58833 q^{39} +1.94692 q^{41} +1.83373 q^{43} +1.55814 q^{45} -5.30907 q^{47} -3.24325 q^{49} -1.80333 q^{51} +1.23824 q^{53} +1.94820 q^{55} +1.21984 q^{57} -1.09850 q^{59} -12.6922 q^{61} -5.66405 q^{63} -3.03773 q^{65} +11.3763 q^{67} +2.48068 q^{69} -9.49053 q^{71} -0.549818 q^{73} +1.31469 q^{75} -7.08199 q^{77} -1.46883 q^{79} +8.30653 q^{81} +2.74985 q^{83} -3.44891 q^{85} +0.262032 q^{87} +8.78644 q^{89} +11.0426 q^{91} -1.86187 q^{93} +2.33298 q^{95} -8.40964 q^{97} +10.6775 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * 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
$3$	−0.278789	−0.160959	−0.0804795	−	0.996756i	$-0.525645\pi$
$3$	−0.278789	−0.160959	−0.0804795	+	0.996756i	$0.525645\pi$
$4$	0	0
$5$	−0.533193	−0.238451	−0.119226	−	0.992867i	$-0.538041\pi$
$5$	−0.533193	−0.238451	−0.119226	+	0.992867i	$0.538041\pi$
$6$	0	0
$7$	1.93823	0.732583	0.366292	−	0.930500i	$-0.380627\pi$
$7$	1.93823	0.732583	0.366292	+	0.930500i	$0.380627\pi$
$8$	0	0
$9$	−2.92228	−0.974092
$10$	0	0
$11$	−3.65384	−1.10167	−0.550837	−	0.834613i	$-0.685691\pi$
$11$	−3.65384	−1.10167	−0.550837	+	0.834613i	$0.685691\pi$
$12$	0	0
$13$	5.69726	1.58013	0.790067	−	0.613020i	$-0.210046\pi$
$13$	5.69726	1.58013	0.790067	+	0.613020i	$0.210046\pi$
$14$	0	0
$15$	0.148648	0.0383808
$16$	0	0
$17$	6.46842	1.56882	0.784411	−	0.620241i	$-0.212965\pi$
$17$	6.46842	1.56882	0.784411	+	0.620241i	$0.212965\pi$
$18$	0	0
$19$	−4.37548	−1.00380	−0.501902	−	0.864924i	$-0.667366\pi$
$19$	−4.37548	−1.00380	−0.501902	+	0.864924i	$0.667366\pi$
$20$	0	0
$21$	−0.540358	−0.117916
$22$	0	0
$23$	−8.89803	−1.85537	−0.927684	−	0.373366i	$-0.878203\pi$
$23$	−8.89803	−1.85537	−0.927684	+	0.373366i	$0.878203\pi$
$24$	0	0
$25$	−4.71571	−0.943141
$26$	0	0
$27$	1.65107	0.317748
$28$	0	0
$29$	−0.939891	−0.174533	−0.0872667	−	0.996185i	$-0.527813\pi$
$29$	−0.939891	−0.174533	−0.0872667	+	0.996185i	$0.527813\pi$
$30$	0	0
$31$	6.67842	1.19948	0.599740	−	0.800195i	$-0.295271\pi$
$31$	6.67842	1.19948	0.599740	+	0.800195i	$0.295271\pi$
$32$	0	0
$33$	1.01865	0.177324
$34$	0	0
$35$	−1.03345	−0.174685
$36$	0	0
$37$	0.254878	0.0419017	0.0209509	−	0.999781i	$-0.493331\pi$
$37$	0.254878	0.0419017	0.0209509	+	0.999781i	$0.493331\pi$
$38$	0	0
$39$	−1.58833	−0.254337
$40$	0	0
$41$	1.94692	0.304058	0.152029	−	0.988376i	$-0.451419\pi$
$41$	1.94692	0.304058	0.152029	+	0.988376i	$0.451419\pi$
$42$	0	0
$43$	1.83373	0.279641	0.139821	−	0.990177i	$-0.455347\pi$
$43$	1.83373	0.279641	0.139821	+	0.990177i	$0.455347\pi$
$44$	0	0
$45$	1.55814	0.232273
$46$	0	0
$47$	−5.30907	−0.774407	−0.387203	−	0.921994i	$-0.626559\pi$
$47$	−5.30907	−0.774407	−0.387203	+	0.921994i	$0.626559\pi$
$48$	0	0
$49$	−3.24325	−0.463322
$50$	0	0
$51$	−1.80333	−0.252516
$52$	0	0
$53$	1.23824	0.170085	0.0850424	−	0.996377i	$-0.472897\pi$
$53$	1.23824	0.170085	0.0850424	+	0.996377i	$0.472897\pi$
$54$	0	0
$55$	1.94820	0.262695
$56$	0	0
$57$	1.21984	0.161571
$58$	0	0
$59$	−1.09850	−0.143012	−0.0715062	−	0.997440i	$-0.522781\pi$
$59$	−1.09850	−0.143012	−0.0715062	+	0.997440i	$0.522781\pi$
$60$	0	0
$61$	−12.6922	−1.62507	−0.812536	−	0.582911i	$-0.801914\pi$
$61$	−12.6922	−1.62507	−0.812536	+	0.582911i	$0.801914\pi$
$62$	0	0
$63$	−5.66405	−0.713604
$64$	0	0
$65$	−3.03773	−0.376785
$66$	0	0
$67$	11.3763	1.38984	0.694920	−	0.719087i	$-0.255440\pi$
$67$	11.3763	1.38984	0.694920	+	0.719087i	$0.255440\pi$
$68$	0	0
$69$	2.48068	0.298638
$70$	0	0
$71$	−9.49053	−1.12632	−0.563159	−	0.826349i	$-0.690414\pi$
$71$	−9.49053	−1.12632	−0.563159	+	0.826349i	$0.690414\pi$
$72$	0	0
$73$	−0.549818	−0.0643514	−0.0321757	−	0.999482i	$-0.510244\pi$
$73$	−0.549818	−0.0643514	−0.0321757	+	0.999482i	$0.510244\pi$
$74$	0	0
$75$	1.31469	0.151807
$76$	0	0
$77$	−7.08199	−0.807067
$78$	0	0
$79$	−1.46883	−0.165256	−0.0826282	−	0.996580i	$-0.526331\pi$
$79$	−1.46883	−0.165256	−0.0826282	+	0.996580i	$0.526331\pi$
$80$	0	0
$81$	8.30653	0.922948
$82$	0	0
$83$	2.74985	0.301835	0.150917	−	0.988546i	$-0.451777\pi$
$83$	2.74985	0.301835	0.150917	+	0.988546i	$0.451777\pi$
$84$	0	0
$85$	−3.44891	−0.374087
$86$	0	0
$87$	0.262032	0.0280927
$88$	0	0
$89$	8.78644	0.931360	0.465680	−	0.884953i	$-0.345810\pi$
$89$	8.78644	0.931360	0.465680	+	0.884953i	$0.345810\pi$
$90$	0	0
$91$	11.0426	1.15758
$92$	0	0
$93$	−1.86187	−0.193067
$94$	0	0
$95$	2.33298	0.239358
$96$	0	0
$97$	−8.40964	−0.853870	−0.426935	−	0.904282i	$-0.640407\pi$
$97$	−8.40964	−0.853870	−0.426935	+	0.904282i	$0.640407\pi$
$98$	0	0
$99$	10.6775	1.07313
$100$	0	0
$101$	−7.56448	−0.752694	−0.376347	−	0.926479i	$-0.622820\pi$
$101$	−7.56448	−0.752694	−0.376347	+	0.926479i	$0.622820\pi$
$102$	0	0
$103$	−7.15698	−0.705198	−0.352599	−	0.935775i	$-0.614702\pi$
$103$	−7.15698	−0.705198	−0.352599	+	0.935775i	$0.614702\pi$
$104$	0	0
$105$	0.288115	0.0281172
$106$	0	0
$107$	−15.6918	−1.51698	−0.758492	−	0.651682i	$-0.774064\pi$
$107$	−15.6918	−1.51698	−0.758492	+	0.651682i	$0.774064\pi$
$108$	0	0
$109$	5.48174	0.525055	0.262528	−	0.964924i	$-0.415444\pi$
$109$	5.48174	0.525055	0.262528	+	0.964924i	$0.415444\pi$
$110$	0	0
$111$	−0.0710573	−0.00674446
$112$	0	0
$113$	−4.51716	−0.424939	−0.212469	−	0.977168i	$-0.568151\pi$
$113$	−4.51716	−0.424939	−0.212469	+	0.977168i	$0.568151\pi$
$114$	0	0
$115$	4.74437	0.442414
$116$	0	0
$117$	−16.6490	−1.53920
$118$	0	0
$119$	12.5373	1.14929
$120$	0	0
$121$	2.35053	0.213684
$122$	0	0
$123$	−0.542781	−0.0489409
$124$	0	0
$125$	5.18034	0.463344
$126$	0	0
$127$	−0.356947	−0.0316739	−0.0158370	−	0.999875i	$-0.505041\pi$
$127$	−0.356947	−0.0316739	−0.0158370	+	0.999875i	$0.505041\pi$
$128$	0	0
$129$	−0.511224	−0.0450108
$130$	0	0
$131$	−15.5810	−1.36132	−0.680660	−	0.732600i	$-0.738307\pi$
$131$	−15.5810	−1.36132	−0.680660	+	0.732600i	$0.738307\pi$
$132$	0	0
$133$	−8.48070	−0.735370
$134$	0	0
$135$	−0.880337	−0.0757673
$136$	0	0
$137$	−14.3404	−1.22518	−0.612592	−	0.790399i	$-0.709873\pi$
$137$	−14.3404	−1.22518	−0.612592	+	0.790399i	$0.709873\pi$
$138$	0	0
$139$	0.617329	0.0523612	0.0261806	−	0.999657i	$-0.491666\pi$
$139$	0.617329	0.0523612	0.0261806	+	0.999657i	$0.491666\pi$
$140$	0	0
$141$	1.48011	0.124648
$142$	0	0
$143$	−20.8168	−1.74079
$144$	0	0
$145$	0.501143	0.0416177
$146$	0	0
$147$	0.904184	0.0745758
$148$	0	0
$149$	14.7701	1.21001	0.605005	−	0.796222i	$-0.293171\pi$
$149$	14.7701	1.21001	0.605005	+	0.796222i	$0.293171\pi$
$150$	0	0
$151$	−10.9354	−0.889909	−0.444954	−	0.895553i	$-0.646780\pi$
$151$	−10.9354	−0.889909	−0.444954	+	0.895553i	$0.646780\pi$
$152$	0	0
$153$	−18.9025	−1.52818
$154$	0	0
$155$	−3.56089	−0.286017
$156$	0	0
$157$	−10.8967	−0.869655	−0.434827	−	0.900514i	$-0.643191\pi$
$157$	−10.8967	−0.869655	−0.434827	+	0.900514i	$0.643191\pi$
$158$	0	0
$159$	−0.345207	−0.0273767
$160$	0	0
$161$	−17.2465	−1.35921
$162$	0	0
$163$	11.3000	0.885087	0.442544	−	0.896747i	$-0.354076\pi$
$163$	11.3000	0.885087	0.442544	+	0.896747i	$0.354076\pi$
$164$	0	0
$165$	−0.543137	−0.0422832
$166$	0	0
$167$	−24.2107	−1.87348	−0.936739	−	0.350028i	$-0.886172\pi$
$167$	−24.2107	−1.87348	−0.936739	+	0.350028i	$0.886172\pi$
$168$	0	0
$169$	19.4587	1.49682
$170$	0	0
$171$	12.7864	0.977798
$172$	0	0
$173$	−3.68096	−0.279858	−0.139929	−	0.990162i	$-0.544687\pi$
$173$	−3.68096	−0.279858	−0.139929	+	0.990162i	$0.544687\pi$
$174$	0	0
$175$	−9.14014	−0.690929
$176$	0	0
$177$	0.306250	0.0230191
$178$	0	0
$179$	10.3487	0.773496	0.386748	−	0.922185i	$-0.373598\pi$
$179$	10.3487	0.773496	0.386748	+	0.922185i	$0.373598\pi$
$180$	0	0
$181$	6.27377	0.466325	0.233163	−	0.972438i	$-0.425093\pi$
$181$	6.27377	0.466325	0.233163	+	0.972438i	$0.425093\pi$
$182$	0	0
$183$	3.53845	0.261570
$184$	0	0
$185$	−0.135899	−0.00999151
$186$	0	0
$187$	−23.6346	−1.72833
$188$	0	0
$189$	3.20015	0.232777
$190$	0	0
$191$	−12.8440	−0.929359	−0.464680	−	0.885479i	$-0.653830\pi$
$191$	−12.8440	−0.929359	−0.464680	+	0.885479i	$0.653830\pi$
$192$	0	0
$193$	24.6873	1.77703	0.888517	−	0.458844i	$-0.151736\pi$
$193$	24.6873	1.77703	0.888517	+	0.458844i	$0.151736\pi$
$194$	0	0
$195$	0.846888	0.0606469
$196$	0	0
$197$	−19.9025	−1.41799	−0.708996	−	0.705213i	$-0.750851\pi$
$197$	−19.9025	−1.41799	−0.708996	+	0.705213i	$0.750851\pi$
$198$	0	0
$199$	−19.4320	−1.37750	−0.688748	−	0.725001i	$-0.741839\pi$
$199$	−19.4320	−1.37750	−0.688748	+	0.725001i	$0.741839\pi$
$200$	0	0
$201$	−3.17160	−0.223707
$202$	0	0
$203$	−1.82173	−0.127860
$204$	0	0
$205$	−1.03808	−0.0725029
$206$	0	0
$207$	26.0025	1.80730
$208$	0	0
$209$	15.9873	1.10586
$210$	0	0
$211$	−26.2368	−1.80622	−0.903109	−	0.429412i	$-0.858721\pi$
$211$	−26.2368	−1.80622	−0.903109	+	0.429412i	$0.858721\pi$
$212$	0	0
$213$	2.64586	0.181291
$214$	0	0
$215$	−0.977732	−0.0666808
$216$	0	0
$217$	12.9443	0.878719
$218$	0	0
$219$	0.153283	0.0103579
$220$	0	0
$221$	36.8522	2.47895
$222$	0	0
$223$	2.20020	0.147336	0.0736681	−	0.997283i	$-0.476529\pi$
$223$	2.20020	0.147336	0.0736681	+	0.997283i	$0.476529\pi$
$224$	0	0
$225$	13.7806	0.918706
$226$	0	0
$227$	−8.32572	−0.552598	−0.276299	−	0.961072i	$-0.589108\pi$
$227$	−8.32572	−0.552598	−0.276299	+	0.961072i	$0.589108\pi$
$228$	0	0
$229$	9.16853	0.605873	0.302937	−	0.953011i	$-0.402033\pi$
$229$	9.16853	0.605873	0.302937	+	0.953011i	$0.402033\pi$
$230$	0	0
$231$	1.97438	0.129905
$232$	0	0
$233$	−27.9523	−1.83121	−0.915607	−	0.402074i	$-0.868289\pi$
$233$	−27.9523	−1.83121	−0.915607	+	0.402074i	$0.868289\pi$
$234$	0	0
$235$	2.83076	0.184658
$236$	0	0
$237$	0.409494	0.0265995
$238$	0	0
$239$	7.06396	0.456930	0.228465	−	0.973552i	$-0.426629\pi$
$239$	7.06396	0.456930	0.228465	+	0.973552i	$0.426629\pi$
$240$	0	0
$241$	−13.7543	−0.885995	−0.442998	−	0.896523i	$-0.646085\pi$
$241$	−13.7543	−0.885995	−0.442998	+	0.896523i	$0.646085\pi$
$242$	0	0
$243$	−7.26897	−0.466305
$244$	0	0
$245$	1.72928	0.110480
$246$	0	0
$247$	−24.9282	−1.58615
$248$	0	0
$249$	−0.766627	−0.0485830
$250$	0	0
$251$	−6.33227	−0.399690	−0.199845	−	0.979828i	$-0.564044\pi$
$251$	−6.33227	−0.399690	−0.199845	+	0.979828i	$0.564044\pi$
$252$	0	0
$253$	32.5120	2.04401
$254$	0	0
$255$	0.961520	0.0602127
$256$	0	0
$257$	−1.48751	−0.0927882	−0.0463941	−	0.998923i	$-0.514773\pi$
$257$	−1.48751	−0.0927882	−0.0463941	+	0.998923i	$0.514773\pi$
$258$	0	0
$259$	0.494013	0.0306965
$260$	0	0
$261$	2.74662	0.170012
$262$	0	0
$263$	−16.3815	−1.01012	−0.505062	−	0.863083i	$-0.668531\pi$
$263$	−16.3815	−1.01012	−0.505062	+	0.863083i	$0.668531\pi$
$264$	0	0
$265$	−0.660218	−0.0405569
$266$	0	0
$267$	−2.44956	−0.149911
$268$	0	0
$269$	29.9069	1.82345	0.911726	−	0.410798i	$-0.134750\pi$
$269$	29.9069	1.82345	0.911726	+	0.410798i	$0.134750\pi$
$270$	0	0
$271$	20.5497	1.24830	0.624152	−	0.781303i	$-0.285444\pi$
$271$	20.5497	1.24830	0.624152	+	0.781303i	$0.285444\pi$
$272$	0	0
$273$	−3.07856	−0.186323
$274$	0	0
$275$	17.2304	1.03903
$276$	0	0
$277$	16.7625	1.00716	0.503579	−	0.863949i	$-0.332016\pi$
$277$	16.7625	1.00716	0.503579	+	0.863949i	$0.332016\pi$
$278$	0	0
$279$	−19.5162	−1.16840
$280$	0	0
$281$	25.6396	1.52953	0.764766	−	0.644308i	$-0.222855\pi$
$281$	25.6396	1.52953	0.764766	+	0.644308i	$0.222855\pi$
$282$	0	0
$283$	−12.4338	−0.739112	−0.369556	−	0.929209i	$-0.620490\pi$
$283$	−12.4338	−0.739112	−0.369556	+	0.929209i	$0.620490\pi$
$284$	0	0
$285$	−0.650408	−0.0385269
$286$	0	0
$287$	3.77359	0.222748
$288$	0	0
$289$	24.8405	1.46120
$290$	0	0
$291$	2.34452	0.137438
$292$	0	0
$293$	12.9618	0.757236	0.378618	−	0.925553i	$-0.376399\pi$
$293$	12.9618	0.757236	0.378618	+	0.925553i	$0.376399\pi$
$294$	0	0
$295$	0.585711	0.0341014
$296$	0	0
$297$	−6.03273	−0.350054
$298$	0	0
$299$	−50.6944	−2.93173
$300$	0	0
$301$	3.55420	0.204861
$302$	0	0
$303$	2.10890	0.121153
$304$	0	0
$305$	6.76740	0.387500
$306$	0	0
$307$	14.0655	0.802759	0.401379	−	0.915912i	$-0.368531\pi$
$307$	14.0655	0.802759	0.401379	+	0.915912i	$0.368531\pi$
$308$	0	0
$309$	1.99529	0.113508
$310$	0	0
$311$	−10.2484	−0.581133	−0.290566	−	0.956855i	$-0.593844\pi$
$311$	−10.2484	−0.581133	−0.290566	+	0.956855i	$0.593844\pi$
$312$	0	0
$313$	7.08174	0.400284	0.200142	−	0.979767i	$-0.435860\pi$
$313$	7.08174	0.400284	0.200142	+	0.979767i	$0.435860\pi$
$314$	0	0
$315$	3.02003	0.170159
$316$	0	0
$317$	−22.9770	−1.29052	−0.645260	−	0.763963i	$-0.723251\pi$
$317$	−22.9770	−1.29052	−0.645260	+	0.763963i	$0.723251\pi$
$318$	0	0
$319$	3.43421	0.192279
$320$	0	0
$321$	4.37471	0.244172
$322$	0	0
$323$	−28.3025	−1.57479
$324$	0	0
$325$	−26.8666	−1.49029
$326$	0	0
$327$	−1.52825	−0.0845124
$328$	0	0
$329$	−10.2902	−0.567318
$330$	0	0
$331$	−22.6362	−1.24420	−0.622098	−	0.782939i	$-0.713720\pi$
$331$	−22.6362	−1.24420	−0.622098	+	0.782939i	$0.713720\pi$
$332$	0	0
$333$	−0.744825	−0.0408161
$334$	0	0
$335$	−6.06577	−0.331409
$336$	0	0
$337$	−6.23572	−0.339681	−0.169841	−	0.985472i	$-0.554325\pi$
$337$	−6.23572	−0.339681	−0.169841	+	0.985472i	$0.554325\pi$
$338$	0	0
$339$	1.25934	0.0683977
$340$	0	0
$341$	−24.4019	−1.32144
$342$	0	0
$343$	−19.8538	−1.07201
$344$	0	0
$345$	−1.32268	−0.0712106
$346$	0	0
$347$	15.9815	0.857932	0.428966	−	0.903321i	$-0.358878\pi$
$347$	15.9815	0.857932	0.428966	+	0.903321i	$0.358878\pi$
$348$	0	0
$349$	−7.07839	−0.378897	−0.189449	−	0.981891i	$-0.560670\pi$
$349$	−7.07839	−0.378897	−0.189449	+	0.981891i	$0.560670\pi$
$350$	0	0
$351$	9.40655	0.502084
$352$	0	0
$353$	−13.6725	−0.727714	−0.363857	−	0.931455i	$-0.618540\pi$
$353$	−13.6725	−0.727714	−0.363857	+	0.931455i	$0.618540\pi$
$354$	0	0
$355$	5.06028	0.268572
$356$	0	0
$357$	−3.49527	−0.184989
$358$	0	0
$359$	−15.0638	−0.795038	−0.397519	−	0.917594i	$-0.630129\pi$
$359$	−15.0638	−0.795038	−0.397519	+	0.917594i	$0.630129\pi$
$360$	0	0
$361$	0.144848	0.00762360
$362$	0	0
$363$	−0.655301	−0.0343944
$364$	0	0
$365$	0.293159	0.0153446
$366$	0	0
$367$	3.64083	0.190050	0.0950248	−	0.995475i	$-0.469707\pi$
$367$	3.64083	0.190050	0.0950248	+	0.995475i	$0.469707\pi$
$368$	0	0
$369$	−5.68944	−0.296181
$370$	0	0
$371$	2.39999	0.124601
$372$	0	0
$373$	8.53608	0.441981	0.220991	−	0.975276i	$-0.429071\pi$
$373$	8.53608	0.441981	0.220991	+	0.975276i	$0.429071\pi$
$374$	0	0
$375$	−1.44422	−0.0745794
$376$	0	0
$377$	−5.35480	−0.275786
$378$	0	0
$379$	29.9016	1.53594	0.767971	−	0.640485i	$-0.221267\pi$
$379$	29.9016	1.53594	0.767971	+	0.640485i	$0.221267\pi$
$380$	0	0
$381$	0.0995129	0.00509820
$382$	0	0
$383$	−23.6657	−1.20926	−0.604629	−	0.796507i	$-0.706679\pi$
$383$	−23.6657	−1.20926	−0.604629	+	0.796507i	$0.706679\pi$
$384$	0	0
$385$	3.77606	0.192446
$386$	0	0
$387$	−5.35867	−0.272396
$388$	0	0
$389$	−22.0050	−1.11570	−0.557849	−	0.829942i	$-0.688373\pi$
$389$	−22.0050	−1.11570	−0.557849	+	0.829942i	$0.688373\pi$
$390$	0	0
$391$	−57.5562	−2.91074
$392$	0	0
$393$	4.34382	0.219117
$394$	0	0
$395$	0.783170	0.0394056
$396$	0	0
$397$	−29.9740	−1.50435	−0.752175	−	0.658963i	$-0.770995\pi$
$397$	−29.9740	−1.50435	−0.752175	+	0.658963i	$0.770995\pi$
$398$	0	0
$399$	2.36433	0.118365
$400$	0	0
$401$	−8.98771	−0.448825	−0.224412	−	0.974494i	$-0.572046\pi$
$401$	−8.98771	−0.448825	−0.224412	+	0.974494i	$0.572046\pi$
$402$	0	0
$403$	38.0487	1.89534
$404$	0	0
$405$	−4.42898	−0.220078
$406$	0	0
$407$	−0.931284	−0.0461620
$408$	0	0
$409$	19.2282	0.950772	0.475386	−	0.879777i	$-0.342308\pi$
$409$	19.2282	0.950772	0.475386	+	0.879777i	$0.342308\pi$
$410$	0	0
$411$	3.99795	0.197204
$412$	0	0
$413$	−2.12915	−0.104768
$414$	0	0
$415$	−1.46620	−0.0719728
$416$	0	0
$417$	−0.172105	−0.00842801
$418$	0	0
$419$	−19.6843	−0.961643	−0.480822	−	0.876818i	$-0.659662\pi$
$419$	−19.6843	−0.961643	−0.480822	+	0.876818i	$0.659662\pi$
$420$	0	0
$421$	−13.3581	−0.651033	−0.325517	−	0.945536i	$-0.605538\pi$
$421$	−13.3581	−0.651033	−0.325517	+	0.945536i	$0.605538\pi$
$422$	0	0
$423$	15.5146	0.754344
$424$	0	0
$425$	−30.5032	−1.47962
$426$	0	0
$427$	−24.6005	−1.19050
$428$	0	0
$429$	5.80351	0.280196
$430$	0	0
$431$	40.3852	1.94529	0.972644	−	0.232301i	$-0.0746253\pi$
$431$	40.3852	1.94529	0.972644	+	0.232301i	$0.0746253\pi$
$432$	0	0
$433$	−28.9912	−1.39323	−0.696614	−	0.717446i	$-0.745311\pi$
$433$	−28.9912	−1.39323	−0.696614	+	0.717446i	$0.745311\pi$
$434$	0	0
$435$	−0.139713	−0.00669874
$436$	0	0
$437$	38.9332	1.86243
$438$	0	0
$439$	−1.09475	−0.0522497	−0.0261248	−	0.999659i	$-0.508317\pi$
$439$	−1.09475	−0.0522497	−0.0261248	+	0.999659i	$0.508317\pi$
$440$	0	0
$441$	9.47768	0.451318
$442$	0	0
$443$	−5.38223	−0.255718	−0.127859	−	0.991792i	$-0.540810\pi$
$443$	−5.38223	−0.255718	−0.127859	+	0.991792i	$0.540810\pi$
$444$	0	0
$445$	−4.68486	−0.222084
$446$	0	0
$447$	−4.11773	−0.194762
$448$	0	0
$449$	−40.5743	−1.91482	−0.957410	−	0.288733i	$-0.906766\pi$
$449$	−40.5743	−1.91482	−0.957410	+	0.288733i	$0.906766\pi$
$450$	0	0
$451$	−7.11373	−0.334973
$452$	0	0
$453$	3.04867	0.143239
$454$	0	0
$455$	−5.88784	−0.276026
$456$	0	0
$457$	12.8573	0.601438	0.300719	−	0.953713i	$-0.402773\pi$
$457$	12.8573	0.601438	0.300719	+	0.953713i	$0.402773\pi$
$458$	0	0
$459$	10.6798	0.498490
$460$	0	0
$461$	28.2492	1.31569	0.657847	−	0.753152i	$-0.271467\pi$
$461$	28.2492	1.31569	0.657847	+	0.753152i	$0.271467\pi$
$462$	0	0
$463$	5.58957	0.259769	0.129885	−	0.991529i	$-0.458539\pi$
$463$	5.58957	0.259769	0.129885	+	0.991529i	$0.458539\pi$
$464$	0	0
$465$	0.992737	0.0460371
$466$	0	0
$467$	23.4778	1.08642	0.543211	−	0.839596i	$-0.317209\pi$
$467$	23.4778	1.08642	0.543211	+	0.839596i	$0.317209\pi$
$468$	0	0
$469$	22.0500	1.01817
$470$	0	0
$471$	3.03789	0.139979
$472$	0	0
$473$	−6.70015	−0.308073
$474$	0	0
$475$	20.6335	0.946729
$476$	0	0
$477$	−3.61847	−0.165678
$478$	0	0
$479$	9.58953	0.438157	0.219078	−	0.975707i	$-0.429695\pi$
$479$	9.58953	0.438157	0.219078	+	0.975707i	$0.429695\pi$
$480$	0	0
$481$	1.45211	0.0662104
$482$	0	0
$483$	4.80813	0.218777
$484$	0	0
$485$	4.48396	0.203606
$486$	0	0
$487$	−9.27219	−0.420163	−0.210081	−	0.977684i	$-0.567373\pi$
$487$	−9.27219	−0.420163	−0.210081	+	0.977684i	$0.567373\pi$
$488$	0	0
$489$	−3.15033	−0.142463
$490$	0	0
$491$	12.1352	0.547655	0.273828	−	0.961779i	$-0.411710\pi$
$491$	12.1352	0.547655	0.273828	+	0.961779i	$0.411710\pi$
$492$	0	0
$493$	−6.07961	−0.273812
$494$	0	0
$495$	−5.69318	−0.255889
$496$	0	0
$497$	−18.3948	−0.825122
$498$	0	0
$499$	−24.6687	−1.10432	−0.552162	−	0.833737i	$-0.686197\pi$
$499$	−24.6687	−1.10432	−0.552162	+	0.833737i	$0.686197\pi$
$500$	0	0
$501$	6.74968	0.301553
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	4.03332	0.179481
$506$	0	0
$507$	−5.42488	−0.240927
$508$	0	0
$509$	30.0468	1.33180	0.665902	−	0.746040i	$-0.268047\pi$
$509$	30.0468	1.33180	0.665902	+	0.746040i	$0.268047\pi$
$510$	0	0
$511$	−1.06568	−0.0471427
$512$	0	0
$513$	−7.22421	−0.318957
$514$	0	0
$515$	3.81605	0.168155
$516$	0	0
$517$	19.3985	0.853143
$518$	0	0
$519$	1.02621	0.0450457
$520$	0	0
$521$	−25.7233	−1.12696	−0.563479	−	0.826130i	$-0.690537\pi$
$521$	−25.7233	−1.12696	−0.563479	+	0.826130i	$0.690537\pi$
$522$	0	0
$523$	40.5812	1.77449	0.887247	−	0.461296i	$-0.152615\pi$
$523$	40.5812	1.77449	0.887247	+	0.461296i	$0.152615\pi$
$524$	0	0
$525$	2.54817	0.111211
$526$	0	0
$527$	43.1989	1.88177
$528$	0	0
$529$	56.1750	2.44239
$530$	0	0
$531$	3.21012	0.139307
$532$	0	0
$533$	11.0921	0.480453
$534$	0	0
$535$	8.36676	0.361726
$536$	0	0
$537$	−2.88510	−0.124501
$538$	0	0
$539$	11.8503	0.510429
$540$	0	0
$541$	43.0528	1.85098	0.925491	−	0.378768i	$-0.123652\pi$
$541$	43.0528	1.85098	0.925491	+	0.378768i	$0.123652\pi$
$542$	0	0
$543$	−1.74906	−0.0750593
$544$	0	0
$545$	−2.92282	−0.125200
$546$	0	0
$547$	14.8613	0.635425	0.317713	−	0.948187i	$-0.397085\pi$
$547$	14.8613	0.635425	0.317713	+	0.948187i	$0.397085\pi$
$548$	0	0
$549$	37.0902	1.58297
$550$	0	0
$551$	4.11248	0.175197
$552$	0	0
$553$	−2.84694	−0.121064
$554$	0	0
$555$	0.0378872	0.00160822
$556$	0	0
$557$	7.73500	0.327742	0.163871	−	0.986482i	$-0.447602\pi$
$557$	7.73500	0.327742	0.163871	+	0.986482i	$0.447602\pi$
$558$	0	0
$559$	10.4472	0.441871
$560$	0	0
$561$	6.58906	0.278190
$562$	0	0
$563$	−14.4246	−0.607925	−0.303963	−	0.952684i	$-0.598310\pi$
$563$	−14.4246	−0.607925	−0.303963	+	0.952684i	$0.598310\pi$
$564$	0	0
$565$	2.40852	0.101327
$566$	0	0
$567$	16.1000	0.676136
$568$	0	0
$569$	−11.0590	−0.463618	−0.231809	−	0.972761i	$-0.574464\pi$
$569$	−11.0590	−0.463618	−0.231809	+	0.972761i	$0.574464\pi$
$570$	0	0
$571$	−9.00340	−0.376780	−0.188390	−	0.982094i	$-0.560327\pi$
$571$	−9.00340	−0.376780	−0.188390	+	0.982094i	$0.560327\pi$
$572$	0	0
$573$	3.58077	0.149589
$574$	0	0
$575$	41.9605	1.74987
$576$	0	0
$577$	−11.3685	−0.473277	−0.236638	−	0.971598i	$-0.576046\pi$
$577$	−11.3685	−0.473277	−0.236638	+	0.971598i	$0.576046\pi$
$578$	0	0
$579$	−6.88257	−0.286030
$580$	0	0
$581$	5.32984	0.221119
$582$	0	0
$583$	−4.52431	−0.187378
$584$	0	0
$585$	8.87710	0.367023
$586$	0	0
$587$	−27.8389	−1.14904	−0.574518	−	0.818492i	$-0.694810\pi$
$587$	−27.8389	−1.14904	−0.574518	+	0.818492i	$0.694810\pi$
$588$	0	0
$589$	−29.2213	−1.20404
$590$	0	0
$591$	5.54859	0.228239
$592$	0	0
$593$	21.0177	0.863092	0.431546	−	0.902091i	$-0.357968\pi$
$593$	21.0177	0.863092	0.431546	+	0.902091i	$0.357968\pi$
$594$	0	0
$595$	−6.68480	−0.274050
$596$	0	0
$597$	5.41742	0.221720
$598$	0	0
$599$	32.6730	1.33498	0.667491	−	0.744618i	$-0.267368\pi$
$599$	32.6730	1.33498	0.667491	+	0.744618i	$0.267368\pi$
$600$	0	0
$601$	−20.8085	−0.848798	−0.424399	−	0.905475i	$-0.639515\pi$
$601$	−20.8085	−0.848798	−0.424399	+	0.905475i	$0.639515\pi$
$602$	0	0
$603$	−33.2448	−1.35383
$604$	0	0
$605$	−1.25328	−0.0509532
$606$	0	0
$607$	−8.77008	−0.355966	−0.177983	−	0.984034i	$-0.556957\pi$
$607$	−8.77008	−0.355966	−0.177983	+	0.984034i	$0.556957\pi$
$608$	0	0
$609$	0.507878	0.0205803
$610$	0	0
$611$	−30.2471	−1.22367
$612$	0	0
$613$	34.8737	1.40854	0.704268	−	0.709934i	$-0.251275\pi$
$613$	34.8737	1.40854	0.704268	+	0.709934i	$0.251275\pi$
$614$	0	0
$615$	0.289407	0.0116700
$616$	0	0
$617$	19.6845	0.792466	0.396233	−	0.918150i	$-0.370317\pi$
$617$	19.6845	0.792466	0.396233	+	0.918150i	$0.370317\pi$
$618$	0	0
$619$	−21.3438	−0.857879	−0.428939	−	0.903333i	$-0.641113\pi$
$619$	−21.3438	−0.857879	−0.428939	+	0.903333i	$0.641113\pi$
$620$	0	0
$621$	−14.6912	−0.589539
$622$	0	0
$623$	17.0302	0.682299
$624$	0	0
$625$	20.8164	0.832656
$626$	0	0
$627$	−4.45709	−0.177999
$628$	0	0
$629$	1.64866	0.0657364
$630$	0	0
$631$	49.9643	1.98905	0.994524	−	0.104512i	$-0.0333281\pi$
$631$	49.9643	1.98905	0.994524	+	0.104512i	$0.0333281\pi$
$632$	0	0
$633$	7.31455	0.290727
$634$	0	0
$635$	0.190321	0.00755268
$636$	0	0
$637$	−18.4776	−0.732111
$638$	0	0
$639$	27.7339	1.09714
$640$	0	0
$641$	−30.3561	−1.19899	−0.599497	−	0.800377i	$-0.704633\pi$
$641$	−30.3561	−1.19899	−0.599497	+	0.800377i	$0.704633\pi$
$642$	0	0
$643$	38.9321	1.53533	0.767665	−	0.640851i	$-0.221418\pi$
$643$	38.9321	1.53533	0.767665	+	0.640851i	$0.221418\pi$
$644$	0	0
$645$	0.272581	0.0107329
$646$	0	0
$647$	−19.0572	−0.749215	−0.374608	−	0.927183i	$-0.622223\pi$
$647$	−19.0572	−0.749215	−0.374608	+	0.927183i	$0.622223\pi$
$648$	0	0
$649$	4.01373	0.157553
$650$	0	0
$651$	−3.60874	−0.141438
$652$	0	0
$653$	−32.4775	−1.27094	−0.635472	−	0.772124i	$-0.719194\pi$
$653$	−32.4775	−1.27094	−0.635472	+	0.772124i	$0.719194\pi$
$654$	0	0
$655$	8.30768	0.324608
$656$	0	0
$657$	1.60672	0.0626842
$658$	0	0
$659$	−13.3863	−0.521454	−0.260727	−	0.965413i	$-0.583962\pi$
$659$	−13.3863	−0.521454	−0.260727	+	0.965413i	$0.583962\pi$
$660$	0	0
$661$	−14.9688	−0.582220	−0.291110	−	0.956690i	$-0.594025\pi$
$661$	−14.9688	−0.582220	−0.291110	+	0.956690i	$0.594025\pi$
$662$	0	0
$663$	−10.2740	−0.399009
$664$	0	0
$665$	4.52185	0.175350
$666$	0	0
$667$	8.36318	0.323824
$668$	0	0
$669$	−0.613392	−0.0237151
$670$	0	0
$671$	46.3753	1.79030
$672$	0	0
$673$	6.08519	0.234567	0.117283	−	0.993098i	$-0.462581\pi$
$673$	6.08519	0.234567	0.117283	+	0.993098i	$0.462581\pi$
$674$	0	0
$675$	−7.78595	−0.299681
$676$	0	0
$677$	26.3791	1.01383	0.506916	−	0.861995i	$-0.330785\pi$
$677$	26.3791	1.01383	0.506916	+	0.861995i	$0.330785\pi$
$678$	0	0
$679$	−16.2998	−0.625531
$680$	0	0
$681$	2.32112	0.0889456
$682$	0	0
$683$	8.12382	0.310850	0.155425	−	0.987848i	$-0.450325\pi$
$683$	8.12382	0.310850	0.155425	+	0.987848i	$0.450325\pi$
$684$	0	0
$685$	7.64620	0.292146
$686$	0	0
$687$	−2.55609	−0.0975208
$688$	0	0
$689$	7.05455	0.268757
$690$	0	0
$691$	5.85362	0.222682	0.111341	−	0.993782i	$-0.464485\pi$
$691$	5.85362	0.222682	0.111341	+	0.993782i	$0.464485\pi$
$692$	0	0
$693$	20.6955	0.786158
$694$	0	0
$695$	−0.329155	−0.0124856
$696$	0	0
$697$	12.5935	0.477013
$698$	0	0
$699$	7.79279	0.294751
$700$	0	0
$701$	33.6736	1.27184	0.635918	−	0.771757i	$-0.280622\pi$
$701$	33.6736	1.27184	0.635918	+	0.771757i	$0.280622\pi$
$702$	0	0
$703$	−1.11522	−0.0420611
$704$	0	0
$705$	−0.789184	−0.0297224
$706$	0	0
$707$	−14.6617	−0.551411
$708$	0	0
$709$	11.1190	0.417581	0.208791	−	0.977960i	$-0.433047\pi$
$709$	11.1190	0.417581	0.208791	+	0.977960i	$0.433047\pi$
$710$	0	0
$711$	4.29233	0.160975
$712$	0	0
$713$	−59.4248	−2.22548
$714$	0	0
$715$	11.0994	0.415094
$716$	0	0
$717$	−1.96936	−0.0735470
$718$	0	0
$719$	12.7212	0.474422	0.237211	−	0.971458i	$-0.423767\pi$
$719$	12.7212	0.474422	0.237211	+	0.971458i	$0.423767\pi$
$720$	0	0
$721$	−13.8719	−0.516616
$722$	0	0
$723$	3.83456	0.142609
$724$	0	0
$725$	4.43225	0.164610
$726$	0	0
$727$	−36.8102	−1.36521	−0.682607	−	0.730785i	$-0.739154\pi$
$727$	−36.8102	−1.36521	−0.682607	+	0.730785i	$0.739154\pi$
$728$	0	0
$729$	−22.8931	−0.847892
$730$	0	0
$731$	11.8613	0.438708
$732$	0	0
$733$	2.36736	0.0874403	0.0437202	−	0.999044i	$-0.486079\pi$
$733$	2.36736	0.0874403	0.0437202	+	0.999044i	$0.486079\pi$
$734$	0	0
$735$	−0.482104	−0.0177827
$736$	0	0
$737$	−41.5672	−1.53115
$738$	0	0
$739$	25.2838	0.930079	0.465039	−	0.885290i	$-0.346040\pi$
$739$	25.2838	0.930079	0.465039	+	0.885290i	$0.346040\pi$
$740$	0	0
$741$	6.94973	0.255305
$742$	0	0
$743$	−38.3022	−1.40517	−0.702586	−	0.711599i	$-0.747971\pi$
$743$	−38.3022	−1.40517	−0.702586	+	0.711599i	$0.747971\pi$
$744$	0	0
$745$	−7.87529	−0.288528
$746$	0	0
$747$	−8.03581	−0.294015
$748$	0	0
$749$	−30.4144	−1.11132
$750$	0	0
$751$	12.1161	0.442121	0.221061	−	0.975260i	$-0.429048\pi$
$751$	12.1161	0.442121	0.221061	+	0.975260i	$0.429048\pi$
$752$	0	0
$753$	1.76537	0.0643336
$754$	0	0
$755$	5.83066	0.212200
$756$	0	0
$757$	−10.5424	−0.383171	−0.191586	−	0.981476i	$-0.561363\pi$
$757$	−10.5424	−0.383171	−0.191586	+	0.981476i	$0.561363\pi$
$758$	0	0
$759$	−9.06398	−0.329002
$760$	0	0
$761$	52.0276	1.88600	0.942999	−	0.332796i	$-0.107992\pi$
$761$	52.0276	1.88600	0.942999	+	0.332796i	$0.107992\pi$
$762$	0	0
$763$	10.6249	0.384647
$764$	0	0
$765$	10.0787	0.364396
$766$	0	0
$767$	−6.25843	−0.225979
$768$	0	0
$769$	17.4385	0.628848	0.314424	−	0.949283i	$-0.398189\pi$
$769$	17.4385	0.628848	0.314424	+	0.949283i	$0.398189\pi$
$770$	0	0
$771$	0.414701	0.0149351
$772$	0	0
$773$	19.9057	0.715960	0.357980	−	0.933729i	$-0.383466\pi$
$773$	19.9057	0.715960	0.357980	+	0.933729i	$0.383466\pi$
$774$	0	0
$775$	−31.4935	−1.13128
$776$	0	0
$777$	−0.137726	−0.00494088
$778$	0	0
$779$	−8.51872	−0.305215
$780$	0	0
$781$	34.6768	1.24083
$782$	0	0
$783$	−1.55182	−0.0554576
$784$	0	0
$785$	5.81006	0.207370
$786$	0	0
$787$	−2.53287	−0.0902869	−0.0451435	−	0.998981i	$-0.514374\pi$
$787$	−2.53287	−0.0902869	−0.0451435	+	0.998981i	$0.514374\pi$
$788$	0	0
$789$	4.56698	0.162589
$790$	0	0
$791$	−8.75531	−0.311303
$792$	0	0
$793$	−72.3108	−2.56783
$794$	0	0
$795$	0.184062	0.00652800
$796$	0	0
$797$	21.3327	0.755643	0.377821	−	0.925879i	$-0.376673\pi$
$797$	21.3327	0.755643	0.377821	+	0.925879i	$0.376673\pi$
$798$	0	0
$799$	−34.3413	−1.21491
$800$	0	0
$801$	−25.6764	−0.907231
$802$	0	0
$803$	2.00895	0.0708942
$804$	0	0
$805$	9.19568	0.324105
$806$	0	0
$807$	−8.33771	−0.293501
$808$	0	0
$809$	38.4011	1.35011	0.675055	−	0.737767i	$-0.264120\pi$
$809$	38.4011	1.35011	0.675055	+	0.737767i	$0.264120\pi$
$810$	0	0
$811$	32.1146	1.12770	0.563848	−	0.825879i	$-0.309320\pi$
$811$	32.1146	1.12770	0.563848	+	0.825879i	$0.309320\pi$
$812$	0	0
$813$	−5.72903	−0.200926
$814$	0	0
$815$	−6.02510	−0.211050
$816$	0	0
$817$	−8.02346	−0.280705
$818$	0	0
$819$	−32.2696	−1.12759
$820$	0	0
$821$	39.2674	1.37044	0.685221	−	0.728335i	$-0.259706\pi$
$821$	39.2674	1.37044	0.685221	+	0.728335i	$0.259706\pi$
$822$	0	0
$823$	20.6875	0.721121	0.360560	−	0.932736i	$-0.382585\pi$
$823$	20.6875	0.721121	0.360560	+	0.932736i	$0.382585\pi$
$824$	0	0
$825$	−4.80366	−0.167242
$826$	0	0
$827$	36.8318	1.28077	0.640384	−	0.768055i	$-0.278775\pi$
$827$	36.8318	1.28077	0.640384	+	0.768055i	$0.278775\pi$
$828$	0	0
$829$	44.0684	1.53056	0.765280	−	0.643698i	$-0.222601\pi$
$829$	44.0684	1.53056	0.765280	+	0.643698i	$0.222601\pi$
$830$	0	0
$831$	−4.67319	−0.162111
$832$	0	0
$833$	−20.9787	−0.726870
$834$	0	0
$835$	12.9090	0.446733
$836$	0	0
$837$	11.0265	0.381132
$838$	0	0
$839$	17.5740	0.606722	0.303361	−	0.952876i	$-0.401891\pi$
$839$	17.5740	0.606722	0.303361	+	0.952876i	$0.401891\pi$
$840$	0	0
$841$	−28.1166	−0.969538
$842$	0	0
$843$	−7.14805	−0.246192
$844$	0	0
$845$	−10.3752	−0.356919
$846$	0	0
$847$	4.55587	0.156541
$848$	0	0
$849$	3.46640	0.118967
$850$	0	0
$851$	−2.26791	−0.0777431
$852$	0	0
$853$	−4.84226	−0.165796	−0.0828978	−	0.996558i	$-0.526418\pi$
$853$	−4.84226	−0.165796	−0.0828978	+	0.996558i	$0.526418\pi$
$854$	0	0
$855$	−6.81760	−0.233157
$856$	0	0
$857$	−4.42035	−0.150996	−0.0754981	−	0.997146i	$-0.524055\pi$
$857$	−4.42035	−0.150996	−0.0754981	+	0.997146i	$0.524055\pi$
$858$	0	0
$859$	−21.9824	−0.750030	−0.375015	−	0.927019i	$-0.622362\pi$
$859$	−21.9824	−0.750030	−0.375015	+	0.927019i	$0.622362\pi$
$860$	0	0
$861$	−1.05204	−0.0358533
$862$	0	0
$863$	20.8318	0.709124	0.354562	−	0.935033i	$-0.384630\pi$
$863$	20.8318	0.709124	0.354562	+	0.935033i	$0.384630\pi$
$864$	0	0
$865$	1.96266	0.0667324
$866$	0	0
$867$	−6.92526	−0.235194
$868$	0	0
$869$	5.36687	0.182059
$870$	0	0
$871$	64.8138	2.19613
$872$	0	0
$873$	24.5753	0.831748
$874$	0	0
$875$	10.0407	0.339438
$876$	0	0
$877$	−21.7896	−0.735784	−0.367892	−	0.929869i	$-0.619920\pi$
$877$	−21.7896	−0.735784	−0.367892	+	0.929869i	$0.619920\pi$
$878$	0	0
$879$	−3.61361	−0.121884
$880$	0	0
$881$	−45.1565	−1.52136	−0.760680	−	0.649127i	$-0.775134\pi$
$881$	−45.1565	−1.52136	−0.760680	+	0.649127i	$0.775134\pi$
$882$	0	0
$883$	−37.9428	−1.27688	−0.638438	−	0.769673i	$-0.720419\pi$
$883$	−37.9428	−1.27688	−0.638438	+	0.769673i	$0.720419\pi$
$884$	0	0
$885$	−0.163290	−0.00548893
$886$	0	0
$887$	44.2579	1.48603	0.743017	−	0.669272i	$-0.233394\pi$
$887$	44.2579	1.48603	0.743017	+	0.669272i	$0.233394\pi$
$888$	0	0
$889$	−0.691846	−0.0232038
$890$	0	0
$891$	−30.3507	−1.01679
$892$	0	0
$893$	23.2297	0.777353
$894$	0	0
$895$	−5.51783	−0.184441
$896$	0	0
$897$	14.1330	0.471889
$898$	0	0
$899$	−6.27699	−0.209349
$900$	0	0
$901$	8.00943	0.266833
$902$	0	0
$903$	−0.990872	−0.0329742
$904$	0	0
$905$	−3.34513	−0.111196
$906$	0	0
$907$	−2.79259	−0.0927266	−0.0463633	−	0.998925i	$-0.514763\pi$
$907$	−2.79259	−0.0927266	−0.0463633	+	0.998925i	$0.514763\pi$
$908$	0	0
$909$	22.1055	0.733193
$910$	0	0
$911$	47.6731	1.57948	0.789740	−	0.613442i	$-0.210215\pi$
$911$	47.6731	1.57948	0.789740	+	0.613442i	$0.210215\pi$
$912$	0	0
$913$	−10.0475	−0.332523
$914$	0	0
$915$	−1.88668	−0.0623716
$916$	0	0
$917$	−30.1996	−0.997280
$918$	0	0
$919$	27.7545	0.915537	0.457769	−	0.889071i	$-0.348649\pi$
$919$	27.7545	0.915537	0.457769	+	0.889071i	$0.348649\pi$
$920$	0	0
$921$	−3.92130	−0.129211
$922$	0	0
$923$	−54.0699	−1.77973
$924$	0	0
$925$	−1.20193	−0.0395192
$926$	0	0
$927$	20.9147	0.686928
$928$	0	0
$929$	7.05342	0.231415	0.115708	−	0.993283i	$-0.463086\pi$
$929$	7.05342	0.231415	0.115708	+	0.993283i	$0.463086\pi$
$930$	0	0
$931$	14.1908	0.465085
$932$	0	0
$933$	2.85714	0.0935385
$934$	0	0
$935$	12.6018	0.412122
$936$	0	0
$937$	−22.1812	−0.724627	−0.362314	−	0.932056i	$-0.618013\pi$
$937$	−22.1812	−0.724627	−0.362314	+	0.932056i	$0.618013\pi$
$938$	0	0
$939$	−1.97431	−0.0644293
$940$	0	0
$941$	−24.2794	−0.791487	−0.395744	−	0.918361i	$-0.629513\pi$
$941$	−24.2794	−0.791487	−0.395744	+	0.918361i	$0.629513\pi$
$942$	0	0
$943$	−17.3238	−0.564140
$944$	0	0
$945$	−1.70630	−0.0555059
$946$	0	0
$947$	18.0102	0.585254	0.292627	−	0.956227i	$-0.405471\pi$
$947$	18.0102	0.585254	0.292627	+	0.956227i	$0.405471\pi$
$948$	0	0
$949$	−3.13246	−0.101684
$950$	0	0
$951$	6.40575	0.207721
$952$	0	0
$953$	22.0273	0.713535	0.356767	−	0.934193i	$-0.383879\pi$
$953$	22.0273	0.713535	0.356767	+	0.934193i	$0.383879\pi$
$954$	0	0
$955$	6.84833	0.221607
$956$	0	0
$957$	−0.957421	−0.0309490
$958$	0	0
$959$	−27.7951	−0.897549
$960$	0	0
$961$	13.6013	0.438753
$962$	0	0
$963$	45.8558	1.47768
$964$	0	0
$965$	−13.1631	−0.423735
$966$	0	0
$967$	−37.6765	−1.21160	−0.605798	−	0.795618i	$-0.707146\pi$
$967$	−37.6765	−1.21160	−0.605798	+	0.795618i	$0.707146\pi$
$968$	0	0
$969$	7.89042	0.253477
$970$	0	0
$971$	−7.57685	−0.243153	−0.121576	−	0.992582i	$-0.538795\pi$
$971$	−7.57685	−0.243153	−0.121576	+	0.992582i	$0.538795\pi$
$972$	0	0
$973$	1.19653	0.0383589
$974$	0	0
$975$	7.49011	0.239876
$976$	0	0
$977$	21.1260	0.675881	0.337940	−	0.941168i	$-0.390270\pi$
$977$	21.1260	0.675881	0.337940	+	0.941168i	$0.390270\pi$
$978$	0	0
$979$	−32.1042	−1.02605
$980$	0	0
$981$	−16.0192	−0.511452
$982$	0	0
$983$	−23.5187	−0.750130	−0.375065	−	0.926999i	$-0.622380\pi$
$983$	−23.5187	−0.750130	−0.375065	+	0.926999i	$0.622380\pi$
$984$	0	0
$985$	10.6118	0.338121
$986$	0	0
$987$	2.86880	0.0913149
$988$	0	0
$989$	−16.3166	−0.518838
$990$	0	0
$991$	−38.5701	−1.22522	−0.612610	−	0.790385i	$-0.709880\pi$
$991$	−38.5701	−1.22522	−0.612610	+	0.790385i	$0.709880\pi$
$992$	0	0
$993$	6.31072	0.200265
$994$	0	0
$995$	10.3610	0.328465
$996$	0	0
$997$	57.5477	1.82255	0.911277	−	0.411795i	$-0.135098\pi$
$997$	57.5477	1.82255	0.911277	+	0.411795i	$0.135098\pi$
$998$	0	0
$999$	0.420821	0.0133142

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.16	✓	29
4.3	odd	2		8048.2.a.w.1.14		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.16	✓	29	1.1	even	1	trivial
8048.2.a.w.1.14		29	4.3	odd	2

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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.278789 q^{3} -0.533193 q^{5} +1.93823 q^{7} -2.92228 q^{9} +O(q^{10})\)	\(q-0.278789 q^{3} -0.533193 q^{5} +1.93823 q^{7} -2.92228 q^{9} -3.65384 q^{11} +5.69726 q^{13} +0.148648 q^{15} +6.46842 q^{17} -4.37548 q^{19} -0.540358 q^{21} -8.89803 q^{23} -4.71571 q^{25} +1.65107 q^{27} -0.939891 q^{29} +6.67842 q^{31} +1.01865 q^{33} -1.03345 q^{35} +0.254878 q^{37} -1.58833 q^{39} +1.94692 q^{41} +1.83373 q^{43} +1.55814 q^{45} -5.30907 q^{47} -3.24325 q^{49} -1.80333 q^{51} +1.23824 q^{53} +1.94820 q^{55} +1.21984 q^{57} -1.09850 q^{59} -12.6922 q^{61} -5.66405 q^{63} -3.03773 q^{65} +11.3763 q^{67} +2.48068 q^{69} -9.49053 q^{71} -0.549818 q^{73} +1.31469 q^{75} -7.08199 q^{77} -1.46883 q^{79} +8.30653 q^{81} +2.74985 q^{83} -3.44891 q^{85} +0.262032 q^{87} +8.78644 q^{89} +11.0426 q^{91} -1.86187 q^{93} +2.33298 q^{95} -8.40964 q^{97} +10.6775 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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