LMFDB - Embedded newform 8024.2.a.bc.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-2.22636 q^{3} -2.51166 q^{5} -2.56246 q^{7} +1.95667 q^{9} +O(q^{10})$	$q-2.22636 q^{3} -2.51166 q^{5} -2.56246 q^{7} +1.95667 q^{9} +5.69258 q^{11} -1.70881 q^{13} +5.59186 q^{15} -1.00000 q^{17} +6.20302 q^{19} +5.70496 q^{21} +3.26108 q^{23} +1.30845 q^{25} +2.32283 q^{27} +9.25046 q^{29} -2.12613 q^{31} -12.6737 q^{33} +6.43605 q^{35} -10.4948 q^{37} +3.80441 q^{39} +4.78841 q^{41} -1.10580 q^{43} -4.91450 q^{45} +2.99999 q^{47} -0.433779 q^{49} +2.22636 q^{51} -4.50352 q^{53} -14.2978 q^{55} -13.8102 q^{57} -1.00000 q^{59} +12.3522 q^{61} -5.01390 q^{63} +4.29194 q^{65} -6.10625 q^{67} -7.26034 q^{69} -13.6047 q^{71} -14.8093 q^{73} -2.91308 q^{75} -14.5870 q^{77} -12.8132 q^{79} -11.0415 q^{81} -6.77379 q^{83} +2.51166 q^{85} -20.5948 q^{87} +16.5139 q^{89} +4.37875 q^{91} +4.73352 q^{93} -15.5799 q^{95} +15.8715 q^{97} +11.1385 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.22636	−1.28539	−0.642694	−	0.766123i	$-0.722183\pi$
$3$	−2.22636	−1.28539	−0.642694	+	0.766123i	$0.722183\pi$
$4$	0	0
$5$	−2.51166	−1.12325	−0.561625	−	0.827392i	$-0.689824\pi$
$5$	−2.51166	−1.12325	−0.561625	+	0.827392i	$0.689824\pi$
$6$	0	0
$7$	−2.56246	−0.968520	−0.484260	−	0.874924i	$-0.660911\pi$
$7$	−2.56246	−0.968520	−0.484260	+	0.874924i	$0.660911\pi$
$8$	0	0
$9$	1.95667	0.652223
$10$	0	0
$11$	5.69258	1.71638	0.858188	−	0.513335i	$-0.171590\pi$
$11$	5.69258	1.71638	0.858188	+	0.513335i	$0.171590\pi$
$12$	0	0
$13$	−1.70881	−0.473938	−0.236969	−	0.971517i	$-0.576154\pi$
$13$	−1.70881	−0.473938	−0.236969	+	0.971517i	$0.576154\pi$
$14$	0	0
$15$	5.59186	1.44381
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	6.20302	1.42307	0.711536	−	0.702650i	$-0.248000\pi$
$19$	6.20302	1.42307	0.711536	+	0.702650i	$0.248000\pi$
$20$	0	0
$21$	5.70496	1.24492
$22$	0	0
$23$	3.26108	0.679983	0.339991	−	0.940429i	$-0.389576\pi$
$23$	3.26108	0.679983	0.339991	+	0.940429i	$0.389576\pi$
$24$	0	0
$25$	1.30845	0.261690
$26$	0	0
$27$	2.32283	0.447028
$28$	0	0
$29$	9.25046	1.71777	0.858884	−	0.512171i	$-0.171158\pi$
$29$	9.25046	1.71777	0.858884	+	0.512171i	$0.171158\pi$
$30$	0	0
$31$	−2.12613	−0.381864	−0.190932	−	0.981603i	$-0.561151\pi$
$31$	−2.12613	−0.381864	−0.190932	+	0.981603i	$0.561151\pi$
$32$	0	0
$33$	−12.6737	−2.20621
$34$	0	0
$35$	6.43605	1.08789
$36$	0	0
$37$	−10.4948	−1.72534	−0.862668	−	0.505771i	$-0.831208\pi$
$37$	−10.4948	−1.72534	−0.862668	+	0.505771i	$0.831208\pi$
$38$	0	0
$39$	3.80441	0.609194
$40$	0	0
$41$	4.78841	0.747823	0.373912	−	0.927464i	$-0.378016\pi$
$41$	4.78841	0.747823	0.373912	+	0.927464i	$0.378016\pi$
$42$	0	0
$43$	−1.10580	−0.168633	−0.0843166	−	0.996439i	$-0.526871\pi$
$43$	−1.10580	−0.168633	−0.0843166	+	0.996439i	$0.526871\pi$
$44$	0	0
$45$	−4.91450	−0.732610
$46$	0	0
$47$	2.99999	0.437594	0.218797	−	0.975770i	$-0.429787\pi$
$47$	2.99999	0.437594	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	−0.433779	−0.0619684
$50$	0	0
$51$	2.22636	0.311752
$52$	0	0
$53$	−4.50352	−0.618606	−0.309303	−	0.950964i	$-0.600096\pi$
$53$	−4.50352	−0.618606	−0.309303	+	0.950964i	$0.600096\pi$
$54$	0	0
$55$	−14.2978	−1.92792
$56$	0	0
$57$	−13.8102	−1.82920
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	12.3522	1.58154	0.790768	−	0.612116i	$-0.209682\pi$
$61$	12.3522	1.58154	0.790768	+	0.612116i	$0.209682\pi$
$62$	0	0
$63$	−5.01390	−0.631692
$64$	0	0
$65$	4.29194	0.532350
$66$	0	0
$67$	−6.10625	−0.745997	−0.372998	−	0.927832i	$-0.621670\pi$
$67$	−6.10625	−0.745997	−0.372998	+	0.927832i	$0.621670\pi$
$68$	0	0
$69$	−7.26034	−0.874042
$70$	0	0
$71$	−13.6047	−1.61458	−0.807288	−	0.590157i	$-0.799066\pi$
$71$	−13.6047	−1.61458	−0.807288	+	0.590157i	$0.799066\pi$
$72$	0	0
$73$	−14.8093	−1.73330	−0.866651	−	0.498915i	$-0.833732\pi$
$73$	−14.8093	−1.73330	−0.866651	+	0.498915i	$0.833732\pi$
$74$	0	0
$75$	−2.91308	−0.336373
$76$	0	0
$77$	−14.5870	−1.66235
$78$	0	0
$79$	−12.8132	−1.44160	−0.720798	−	0.693146i	$-0.756224\pi$
$79$	−12.8132	−1.44160	−0.720798	+	0.693146i	$0.756224\pi$
$80$	0	0
$81$	−11.0415	−1.22683
$82$	0	0
$83$	−6.77379	−0.743520	−0.371760	−	0.928329i	$-0.621245\pi$
$83$	−6.77379	−0.743520	−0.371760	+	0.928329i	$0.621245\pi$
$84$	0	0
$85$	2.51166	0.272428
$86$	0	0
$87$	−20.5948	−2.20800
$88$	0	0
$89$	16.5139	1.75047	0.875234	−	0.483700i	$-0.160707\pi$
$89$	16.5139	1.75047	0.875234	+	0.483700i	$0.160707\pi$
$90$	0	0
$91$	4.37875	0.459018
$92$	0	0
$93$	4.73352	0.490843
$94$	0	0
$95$	−15.5799	−1.59846
$96$	0	0
$97$	15.8715	1.61151	0.805753	−	0.592251i	$-0.201761\pi$
$97$	15.8715	1.61151	0.805753	+	0.592251i	$0.201761\pi$
$98$	0	0
$99$	11.1385	1.11946
$100$	0	0
$101$	10.5085	1.04564	0.522818	−	0.852444i	$-0.324881\pi$
$101$	10.5085	1.04564	0.522818	+	0.852444i	$0.324881\pi$
$102$	0	0
$103$	10.9010	1.07410	0.537051	−	0.843549i	$-0.319538\pi$
$103$	10.9010	1.07410	0.537051	+	0.843549i	$0.319538\pi$
$104$	0	0
$105$	−14.3289	−1.39836
$106$	0	0
$107$	−6.34030	−0.612940	−0.306470	−	0.951880i	$-0.599148\pi$
$107$	−6.34030	−0.612940	−0.306470	+	0.951880i	$0.599148\pi$
$108$	0	0
$109$	7.84829	0.751730	0.375865	−	0.926674i	$-0.377346\pi$
$109$	7.84829	0.751730	0.375865	+	0.926674i	$0.377346\pi$
$110$	0	0
$111$	23.3652	2.21773
$112$	0	0
$113$	2.13418	0.200767	0.100383	−	0.994949i	$-0.467993\pi$
$113$	2.13418	0.200767	0.100383	+	0.994949i	$0.467993\pi$
$114$	0	0
$115$	−8.19074	−0.763790
$116$	0	0
$117$	−3.34357	−0.309113
$118$	0	0
$119$	2.56246	0.234901
$120$	0	0
$121$	21.4054	1.94595
$122$	0	0
$123$	−10.6607	−0.961243
$124$	0	0
$125$	9.27193	0.829307
$126$	0	0
$127$	−17.2821	−1.53354	−0.766770	−	0.641922i	$-0.778137\pi$
$127$	−17.2821	−1.53354	−0.766770	+	0.641922i	$0.778137\pi$
$128$	0	0
$129$	2.46191	0.216759
$130$	0	0
$131$	−14.5037	−1.26720	−0.633598	−	0.773662i	$-0.718423\pi$
$131$	−14.5037	−1.26720	−0.633598	+	0.773662i	$0.718423\pi$
$132$	0	0
$133$	−15.8950	−1.37827
$134$	0	0
$135$	−5.83416	−0.502124
$136$	0	0
$137$	3.89309	0.332609	0.166305	−	0.986074i	$-0.446817\pi$
$137$	3.89309	0.332609	0.166305	+	0.986074i	$0.446817\pi$
$138$	0	0
$139$	−2.32813	−0.197470	−0.0987348	−	0.995114i	$-0.531480\pi$
$139$	−2.32813	−0.197470	−0.0987348	+	0.995114i	$0.531480\pi$
$140$	0	0
$141$	−6.67906	−0.562478
$142$	0	0
$143$	−9.72751	−0.813455
$144$	0	0
$145$	−23.2340	−1.92948
$146$	0	0
$147$	0.965746	0.0796534
$148$	0	0
$149$	9.45863	0.774881	0.387441	−	0.921895i	$-0.373359\pi$
$149$	9.45863	0.774881	0.387441	+	0.921895i	$0.373359\pi$
$150$	0	0
$151$	12.5420	1.02065	0.510326	−	0.859981i	$-0.329525\pi$
$151$	12.5420	1.02065	0.510326	+	0.859981i	$0.329525\pi$
$152$	0	0
$153$	−1.95667	−0.158187
$154$	0	0
$155$	5.34012	0.428929
$156$	0	0
$157$	−13.0596	−1.04227	−0.521136	−	0.853474i	$-0.674491\pi$
$157$	−13.0596	−1.04227	−0.521136	+	0.853474i	$0.674491\pi$
$158$	0	0
$159$	10.0264	0.795149
$160$	0	0
$161$	−8.35640	−0.658577
$162$	0	0
$163$	−15.7355	−1.23250	−0.616249	−	0.787551i	$-0.711349\pi$
$163$	−15.7355	−1.23250	−0.616249	+	0.787551i	$0.711349\pi$
$164$	0	0
$165$	31.8321	2.47813
$166$	0	0
$167$	14.4736	1.12000	0.560001	−	0.828492i	$-0.310801\pi$
$167$	14.4736	1.12000	0.560001	+	0.828492i	$0.310801\pi$
$168$	0	0
$169$	−10.0800	−0.775383
$170$	0	0
$171$	12.1373	0.928160
$172$	0	0
$173$	−17.0416	−1.29565	−0.647823	−	0.761791i	$-0.724320\pi$
$173$	−17.0416	−1.29565	−0.647823	+	0.761791i	$0.724320\pi$
$174$	0	0
$175$	−3.35286	−0.253452
$176$	0	0
$177$	2.22636	0.167343
$178$	0	0
$179$	10.7202	0.801268	0.400634	−	0.916238i	$-0.368790\pi$
$179$	10.7202	0.801268	0.400634	+	0.916238i	$0.368790\pi$
$180$	0	0
$181$	−19.1945	−1.42672	−0.713358	−	0.700800i	$-0.752826\pi$
$181$	−19.1945	−1.42672	−0.713358	+	0.700800i	$0.752826\pi$
$182$	0	0
$183$	−27.5004	−2.03289
$184$	0	0
$185$	26.3594	1.93798
$186$	0	0
$187$	−5.69258	−0.416282
$188$	0	0
$189$	−5.95216	−0.432956
$190$	0	0
$191$	19.0197	1.37622	0.688110	−	0.725607i	$-0.258441\pi$
$191$	19.0197	1.37622	0.688110	+	0.725607i	$0.258441\pi$
$192$	0	0
$193$	21.7525	1.56578	0.782889	−	0.622162i	$-0.213745\pi$
$193$	21.7525	1.56578	0.782889	+	0.622162i	$0.213745\pi$
$194$	0	0
$195$	−9.55541	−0.684277
$196$	0	0
$197$	4.03149	0.287232	0.143616	−	0.989634i	$-0.454127\pi$
$197$	4.03149	0.287232	0.143616	+	0.989634i	$0.454127\pi$
$198$	0	0
$199$	−14.5490	−1.03135	−0.515677	−	0.856783i	$-0.672459\pi$
$199$	−14.5490	−1.03135	−0.515677	+	0.856783i	$0.672459\pi$
$200$	0	0
$201$	13.5947	0.958895
$202$	0	0
$203$	−23.7040	−1.66369
$204$	0	0
$205$	−12.0269	−0.839992
$206$	0	0
$207$	6.38086	0.443500
$208$	0	0
$209$	35.3112	2.44253
$210$	0	0
$211$	−0.853843	−0.0587810	−0.0293905	−	0.999568i	$-0.509357\pi$
$211$	−0.853843	−0.0587810	−0.0293905	+	0.999568i	$0.509357\pi$
$212$	0	0
$213$	30.2889	2.07536
$214$	0	0
$215$	2.77740	0.189417
$216$	0	0
$217$	5.44813	0.369843
$218$	0	0
$219$	32.9709	2.22797
$220$	0	0
$221$	1.70881	0.114947
$222$	0	0
$223$	5.56010	0.372332	0.186166	−	0.982518i	$-0.440394\pi$
$223$	5.56010	0.372332	0.186166	+	0.982518i	$0.440394\pi$
$224$	0	0
$225$	2.56020	0.170680
$226$	0	0
$227$	5.14356	0.341390	0.170695	−	0.985324i	$-0.445399\pi$
$227$	5.14356	0.341390	0.170695	+	0.985324i	$0.445399\pi$
$228$	0	0
$229$	4.73093	0.312629	0.156314	−	0.987707i	$-0.450039\pi$
$229$	4.73093	0.312629	0.156314	+	0.987707i	$0.450039\pi$
$230$	0	0
$231$	32.4759	2.13676
$232$	0	0
$233$	−2.68306	−0.175773	−0.0878867	−	0.996130i	$-0.528011\pi$
$233$	−2.68306	−0.175773	−0.0878867	+	0.996130i	$0.528011\pi$
$234$	0	0
$235$	−7.53497	−0.491527
$236$	0	0
$237$	28.5267	1.85301
$238$	0	0
$239$	−2.91110	−0.188304	−0.0941518	−	0.995558i	$-0.530014\pi$
$239$	−2.91110	−0.188304	−0.0941518	+	0.995558i	$0.530014\pi$
$240$	0	0
$241$	−5.01486	−0.323035	−0.161518	−	0.986870i	$-0.551639\pi$
$241$	−5.01486	−0.323035	−0.161518	+	0.986870i	$0.551639\pi$
$242$	0	0
$243$	17.6137	1.12992
$244$	0	0
$245$	1.08951	0.0696060
$246$	0	0
$247$	−10.5998	−0.674447
$248$	0	0
$249$	15.0809	0.955711
$250$	0	0
$251$	2.69258	0.169954	0.0849771	−	0.996383i	$-0.472918\pi$
$251$	2.69258	0.169954	0.0849771	+	0.996383i	$0.472918\pi$
$252$	0	0
$253$	18.5640	1.16711
$254$	0	0
$255$	−5.59186	−0.350176
$256$	0	0
$257$	22.7725	1.42051	0.710254	−	0.703946i	$-0.248580\pi$
$257$	22.7725	1.42051	0.710254	+	0.703946i	$0.248580\pi$
$258$	0	0
$259$	26.8926	1.67102
$260$	0	0
$261$	18.1001	1.12037
$262$	0	0
$263$	8.35110	0.514951	0.257475	−	0.966285i	$-0.417109\pi$
$263$	8.35110	0.514951	0.257475	+	0.966285i	$0.417109\pi$
$264$	0	0
$265$	11.3113	0.694849
$266$	0	0
$267$	−36.7658	−2.25003
$268$	0	0
$269$	−21.7329	−1.32508	−0.662538	−	0.749028i	$-0.730521\pi$
$269$	−21.7329	−1.32508	−0.662538	+	0.749028i	$0.730521\pi$
$270$	0	0
$271$	17.9247	1.08885	0.544425	−	0.838809i	$-0.316748\pi$
$271$	17.9247	1.08885	0.544425	+	0.838809i	$0.316748\pi$
$272$	0	0
$273$	−9.74867	−0.590017
$274$	0	0
$275$	7.44845	0.449158
$276$	0	0
$277$	−29.3718	−1.76478	−0.882390	−	0.470518i	$-0.844067\pi$
$277$	−29.3718	−1.76478	−0.882390	+	0.470518i	$0.844067\pi$
$278$	0	0
$279$	−4.16013	−0.249061
$280$	0	0
$281$	0.283778	0.0169288	0.00846438	−	0.999964i	$-0.497306\pi$
$281$	0.283778	0.0169288	0.00846438	+	0.999964i	$0.497306\pi$
$282$	0	0
$283$	−5.04183	−0.299706	−0.149853	−	0.988708i	$-0.547880\pi$
$283$	−5.04183	−0.299706	−0.149853	+	0.988708i	$0.547880\pi$
$284$	0	0
$285$	34.6864	2.05465
$286$	0	0
$287$	−12.2701	−0.724282
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−35.3356	−2.07141
$292$	0	0
$293$	−0.272994	−0.0159485	−0.00797423	−	0.999968i	$-0.502538\pi$
$293$	−0.272994	−0.0159485	−0.00797423	+	0.999968i	$0.502538\pi$
$294$	0	0
$295$	2.51166	0.146235
$296$	0	0
$297$	13.2229	0.767268
$298$	0	0
$299$	−5.57256	−0.322269
$300$	0	0
$301$	2.83358	0.163325
$302$	0	0
$303$	−23.3957	−1.34405
$304$	0	0
$305$	−31.0245	−1.77646
$306$	0	0
$307$	−2.11726	−0.120838	−0.0604191	−	0.998173i	$-0.519244\pi$
$307$	−2.11726	−0.120838	−0.0604191	+	0.998173i	$0.519244\pi$
$308$	0	0
$309$	−24.2694	−1.38064
$310$	0	0
$311$	−32.2729	−1.83003	−0.915015	−	0.403421i	$-0.867821\pi$
$311$	−32.2729	−1.83003	−0.915015	+	0.403421i	$0.867821\pi$
$312$	0	0
$313$	29.1089	1.64533	0.822667	−	0.568523i	$-0.192485\pi$
$313$	29.1089	1.64533	0.822667	+	0.568523i	$0.192485\pi$
$314$	0	0
$315$	12.5932	0.709547
$316$	0	0
$317$	17.9020	1.00547	0.502737	−	0.864439i	$-0.332326\pi$
$317$	17.9020	1.00547	0.502737	+	0.864439i	$0.332326\pi$
$318$	0	0
$319$	52.6590	2.94834
$320$	0	0
$321$	14.1158	0.787866
$322$	0	0
$323$	−6.20302	−0.345146
$324$	0	0
$325$	−2.23589	−0.124025
$326$	0	0
$327$	−17.4731	−0.966265
$328$	0	0
$329$	−7.68737	−0.423819
$330$	0	0
$331$	−3.87396	−0.212932	−0.106466	−	0.994316i	$-0.533954\pi$
$331$	−3.87396	−0.212932	−0.106466	+	0.994316i	$0.533954\pi$
$332$	0	0
$333$	−20.5349	−1.12530
$334$	0	0
$335$	15.3368	0.837940
$336$	0	0
$337$	12.7582	0.694983	0.347492	−	0.937683i	$-0.387034\pi$
$337$	12.7582	0.694983	0.347492	+	0.937683i	$0.387034\pi$
$338$	0	0
$339$	−4.75145	−0.258063
$340$	0	0
$341$	−12.1031	−0.655422
$342$	0	0
$343$	19.0488	1.02854
$344$	0	0
$345$	18.2355	0.981767
$346$	0	0
$347$	6.21194	0.333475	0.166737	−	0.986001i	$-0.446677\pi$
$347$	6.21194	0.333475	0.166737	+	0.986001i	$0.446677\pi$
$348$	0	0
$349$	−4.81635	−0.257813	−0.128907	−	0.991657i	$-0.541147\pi$
$349$	−4.81635	−0.257813	−0.128907	+	0.991657i	$0.541147\pi$
$350$	0	0
$351$	−3.96926	−0.211863
$352$	0	0
$353$	22.9509	1.22156	0.610778	−	0.791802i	$-0.290857\pi$
$353$	22.9509	1.22156	0.610778	+	0.791802i	$0.290857\pi$
$354$	0	0
$355$	34.1703	1.81357
$356$	0	0
$357$	−5.70496	−0.301939
$358$	0	0
$359$	−7.51927	−0.396852	−0.198426	−	0.980116i	$-0.563583\pi$
$359$	−7.51927	−0.396852	−0.198426	+	0.980116i	$0.563583\pi$
$360$	0	0
$361$	19.4775	1.02513
$362$	0	0
$363$	−47.6561	−2.50130
$364$	0	0
$365$	37.1961	1.94693
$366$	0	0
$367$	33.9304	1.77115	0.885577	−	0.464493i	$-0.153763\pi$
$367$	33.9304	1.77115	0.885577	+	0.464493i	$0.153763\pi$
$368$	0	0
$369$	9.36933	0.487748
$370$	0	0
$371$	11.5401	0.599133
$372$	0	0
$373$	9.90999	0.513120	0.256560	−	0.966528i	$-0.417411\pi$
$373$	9.90999	0.513120	0.256560	+	0.966528i	$0.417411\pi$
$374$	0	0
$375$	−20.6426	−1.06598
$376$	0	0
$377$	−15.8072	−0.814114
$378$	0	0
$379$	6.63192	0.340659	0.170329	−	0.985387i	$-0.445517\pi$
$379$	6.63192	0.340659	0.170329	+	0.985387i	$0.445517\pi$
$380$	0	0
$381$	38.4762	1.97119
$382$	0	0
$383$	−24.2159	−1.23737	−0.618687	−	0.785638i	$-0.712335\pi$
$383$	−24.2159	−1.23737	−0.618687	+	0.785638i	$0.712335\pi$
$384$	0	0
$385$	36.6377	1.86723
$386$	0	0
$387$	−2.16369	−0.109987
$388$	0	0
$389$	−23.2030	−1.17644	−0.588218	−	0.808702i	$-0.700170\pi$
$389$	−23.2030	−1.17644	−0.588218	+	0.808702i	$0.700170\pi$
$390$	0	0
$391$	−3.26108	−0.164920
$392$	0	0
$393$	32.2905	1.62884
$394$	0	0
$395$	32.1824	1.61927
$396$	0	0
$397$	18.5562	0.931310	0.465655	−	0.884966i	$-0.345819\pi$
$397$	18.5562	0.931310	0.465655	+	0.884966i	$0.345819\pi$
$398$	0	0
$399$	35.3880	1.77162
$400$	0	0
$401$	−2.84382	−0.142014	−0.0710068	−	0.997476i	$-0.522621\pi$
$401$	−2.84382	−0.142014	−0.0710068	+	0.997476i	$0.522621\pi$
$402$	0	0
$403$	3.63314	0.180980
$404$	0	0
$405$	27.7324	1.37803
$406$	0	0
$407$	−59.7425	−2.96133
$408$	0	0
$409$	18.9405	0.936546	0.468273	−	0.883584i	$-0.344876\pi$
$409$	18.9405	0.936546	0.468273	+	0.883584i	$0.344876\pi$
$410$	0	0
$411$	−8.66741	−0.427532
$412$	0	0
$413$	2.56246	0.126091
$414$	0	0
$415$	17.0135	0.835158
$416$	0	0
$417$	5.18326	0.253825
$418$	0	0
$419$	−15.4635	−0.755439	−0.377720	−	0.925920i	$-0.623292\pi$
$419$	−15.4635	−0.755439	−0.377720	+	0.925920i	$0.623292\pi$
$420$	0	0
$421$	−23.5349	−1.14702	−0.573510	−	0.819199i	$-0.694418\pi$
$421$	−23.5349	−1.14702	−0.573510	+	0.819199i	$0.694418\pi$
$422$	0	0
$423$	5.87000	0.285409
$424$	0	0
$425$	−1.30845	−0.0634691
$426$	0	0
$427$	−31.6520	−1.53175
$428$	0	0
$429$	21.6569	1.04561
$430$	0	0
$431$	−1.95955	−0.0943881	−0.0471940	−	0.998886i	$-0.515028\pi$
$431$	−1.95955	−0.0943881	−0.0471940	+	0.998886i	$0.515028\pi$
$432$	0	0
$433$	0.618925	0.0297436	0.0148718	−	0.999889i	$-0.495266\pi$
$433$	0.618925	0.0297436	0.0148718	+	0.999889i	$0.495266\pi$
$434$	0	0
$435$	51.7273	2.48013
$436$	0	0
$437$	20.2286	0.967664
$438$	0	0
$439$	−28.0298	−1.33779	−0.668896	−	0.743356i	$-0.733233\pi$
$439$	−28.0298	−1.33779	−0.668896	+	0.743356i	$0.733233\pi$
$440$	0	0
$441$	−0.848762	−0.0404172
$442$	0	0
$443$	35.9170	1.70647	0.853233	−	0.521530i	$-0.174639\pi$
$443$	35.9170	1.70647	0.853233	+	0.521530i	$0.174639\pi$
$444$	0	0
$445$	−41.4773	−1.96621
$446$	0	0
$447$	−21.0583	−0.996023
$448$	0	0
$449$	−5.53758	−0.261335	−0.130667	−	0.991426i	$-0.541712\pi$
$449$	−5.53758	−0.261335	−0.130667	+	0.991426i	$0.541712\pi$
$450$	0	0
$451$	27.2584	1.28355
$452$	0	0
$453$	−27.9230	−1.31194
$454$	0	0
$455$	−10.9980	−0.515592
$456$	0	0
$457$	17.7332	0.829526	0.414763	−	0.909930i	$-0.363865\pi$
$457$	17.7332	0.829526	0.414763	+	0.909930i	$0.363865\pi$
$458$	0	0
$459$	−2.32283	−0.108420
$460$	0	0
$461$	20.7622	0.966993	0.483497	−	0.875346i	$-0.339367\pi$
$461$	20.7622	0.966993	0.483497	+	0.875346i	$0.339367\pi$
$462$	0	0
$463$	−5.95054	−0.276545	−0.138272	−	0.990394i	$-0.544155\pi$
$463$	−5.95054	−0.276545	−0.138272	+	0.990394i	$0.544155\pi$
$464$	0	0
$465$	−11.8890	−0.551340
$466$	0	0
$467$	14.4978	0.670878	0.335439	−	0.942062i	$-0.391115\pi$
$467$	14.4978	0.670878	0.335439	+	0.942062i	$0.391115\pi$
$468$	0	0
$469$	15.6470	0.722513
$470$	0	0
$471$	29.0754	1.33972
$472$	0	0
$473$	−6.29486	−0.289438
$474$	0	0
$475$	8.11635	0.372403
$476$	0	0
$477$	−8.81190	−0.403469
$478$	0	0
$479$	−2.65486	−0.121304	−0.0606519	−	0.998159i	$-0.519318\pi$
$479$	−2.65486	−0.121304	−0.0606519	+	0.998159i	$0.519318\pi$
$480$	0	0
$481$	17.9336	0.817701
$482$	0	0
$483$	18.6043	0.846527
$484$	0	0
$485$	−39.8639	−1.81012
$486$	0	0
$487$	−18.3872	−0.833204	−0.416602	−	0.909089i	$-0.636779\pi$
$487$	−18.3872	−0.833204	−0.416602	+	0.909089i	$0.636779\pi$
$488$	0	0
$489$	35.0328	1.58424
$490$	0	0
$491$	22.9139	1.03409	0.517045	−	0.855958i	$-0.327032\pi$
$491$	22.9139	1.03409	0.517045	+	0.855958i	$0.327032\pi$
$492$	0	0
$493$	−9.25046	−0.416620
$494$	0	0
$495$	−27.9761	−1.25743
$496$	0	0
$497$	34.8615	1.56375
$498$	0	0
$499$	12.7969	0.572866	0.286433	−	0.958100i	$-0.407530\pi$
$499$	12.7969	0.572866	0.286433	+	0.958100i	$0.407530\pi$
$500$	0	0
$501$	−32.2235	−1.43964
$502$	0	0
$503$	−26.0504	−1.16153	−0.580765	−	0.814071i	$-0.697247\pi$
$503$	−26.0504	−1.16153	−0.580765	+	0.814071i	$0.697247\pi$
$504$	0	0
$505$	−26.3939	−1.17451
$506$	0	0
$507$	22.4416	0.996669
$508$	0	0
$509$	−39.6149	−1.75590	−0.877949	−	0.478753i	$-0.841089\pi$
$509$	−39.6149	−1.75590	−0.877949	+	0.478753i	$0.841089\pi$
$510$	0	0
$511$	37.9484	1.67874
$512$	0	0
$513$	14.4085	0.636153
$514$	0	0
$515$	−27.3795	−1.20649
$516$	0	0
$517$	17.0777	0.751076
$518$	0	0
$519$	37.9406	1.66541
$520$	0	0
$521$	18.2999	0.801732	0.400866	−	0.916137i	$-0.368709\pi$
$521$	18.2999	0.801732	0.400866	+	0.916137i	$0.368709\pi$
$522$	0	0
$523$	29.7991	1.30302	0.651511	−	0.758639i	$-0.274136\pi$
$523$	29.7991	1.30302	0.651511	+	0.758639i	$0.274136\pi$
$524$	0	0
$525$	7.46466	0.325784
$526$	0	0
$527$	2.12613	0.0926156
$528$	0	0
$529$	−12.3653	−0.537624
$530$	0	0
$531$	−1.95667	−0.0849122
$532$	0	0
$533$	−8.18246	−0.354422
$534$	0	0
$535$	15.9247	0.688485
$536$	0	0
$537$	−23.8671	−1.02994
$538$	0	0
$539$	−2.46932	−0.106361
$540$	0	0
$541$	−2.51986	−0.108337	−0.0541686	−	0.998532i	$-0.517251\pi$
$541$	−2.51986	−0.108337	−0.0541686	+	0.998532i	$0.517251\pi$
$542$	0	0
$543$	42.7338	1.83388
$544$	0	0
$545$	−19.7123	−0.844380
$546$	0	0
$547$	−32.8877	−1.40618	−0.703088	−	0.711103i	$-0.748196\pi$
$547$	−32.8877	−1.40618	−0.703088	+	0.711103i	$0.748196\pi$
$548$	0	0
$549$	24.1692	1.03151
$550$	0	0
$551$	57.3808	2.44451
$552$	0	0
$553$	32.8333	1.39621
$554$	0	0
$555$	−58.6855	−2.49106
$556$	0	0
$557$	34.5097	1.46222	0.731112	−	0.682257i	$-0.239002\pi$
$557$	34.5097	1.46222	0.731112	+	0.682257i	$0.239002\pi$
$558$	0	0
$559$	1.88960	0.0799216
$560$	0	0
$561$	12.6737	0.535085
$562$	0	0
$563$	7.84005	0.330419	0.165209	−	0.986259i	$-0.447170\pi$
$563$	7.84005	0.330419	0.165209	+	0.986259i	$0.447170\pi$
$564$	0	0
$565$	−5.36034	−0.225511
$566$	0	0
$567$	28.2933	1.18821
$568$	0	0
$569$	35.2102	1.47609	0.738045	−	0.674752i	$-0.235749\pi$
$569$	35.2102	1.47609	0.738045	+	0.674752i	$0.235749\pi$
$570$	0	0
$571$	35.7665	1.49678	0.748391	−	0.663258i	$-0.230827\pi$
$571$	35.7665	1.49678	0.748391	+	0.663258i	$0.230827\pi$
$572$	0	0
$573$	−42.3447	−1.76898
$574$	0	0
$575$	4.26696	0.177945
$576$	0	0
$577$	−23.0765	−0.960687	−0.480343	−	0.877080i	$-0.659488\pi$
$577$	−23.0765	−0.960687	−0.480343	+	0.877080i	$0.659488\pi$
$578$	0	0
$579$	−48.4288	−2.01263
$580$	0	0
$581$	17.3576	0.720114
$582$	0	0
$583$	−25.6366	−1.06176
$584$	0	0
$585$	8.39792	0.347211
$586$	0	0
$587$	−16.4063	−0.677161	−0.338580	−	0.940938i	$-0.609947\pi$
$587$	−16.4063	−0.677161	−0.338580	+	0.940938i	$0.609947\pi$
$588$	0	0
$589$	−13.1884	−0.543420
$590$	0	0
$591$	−8.97554	−0.369204
$592$	0	0
$593$	25.8443	1.06130	0.530650	−	0.847591i	$-0.321948\pi$
$593$	25.8443	1.06130	0.530650	+	0.847591i	$0.321948\pi$
$594$	0	0
$595$	−6.43605	−0.263852
$596$	0	0
$597$	32.3913	1.32569
$598$	0	0
$599$	14.3591	0.586698	0.293349	−	0.956005i	$-0.405230\pi$
$599$	14.3591	0.586698	0.293349	+	0.956005i	$0.405230\pi$
$600$	0	0
$601$	6.84671	0.279283	0.139642	−	0.990202i	$-0.455405\pi$
$601$	6.84671	0.279283	0.139642	+	0.990202i	$0.455405\pi$
$602$	0	0
$603$	−11.9479	−0.486556
$604$	0	0
$605$	−53.7632	−2.18579
$606$	0	0
$607$	−6.10026	−0.247602	−0.123801	−	0.992307i	$-0.539508\pi$
$607$	−6.10026	−0.247602	−0.123801	+	0.992307i	$0.539508\pi$
$608$	0	0
$609$	52.7735	2.13849
$610$	0	0
$611$	−5.12641	−0.207392
$612$	0	0
$613$	0.524598	0.0211883	0.0105942	−	0.999944i	$-0.496628\pi$
$613$	0.524598	0.0211883	0.0105942	+	0.999944i	$0.496628\pi$
$614$	0	0
$615$	26.7761	1.07972
$616$	0	0
$617$	1.53590	0.0618330	0.0309165	−	0.999522i	$-0.490157\pi$
$617$	1.53590	0.0618330	0.0309165	+	0.999522i	$0.490157\pi$
$618$	0	0
$619$	15.9210	0.639918	0.319959	−	0.947431i	$-0.396331\pi$
$619$	15.9210	0.639918	0.319959	+	0.947431i	$0.396331\pi$
$620$	0	0
$621$	7.57493	0.303971
$622$	0	0
$623$	−42.3162	−1.69536
$624$	0	0
$625$	−29.8302	−1.19321
$626$	0	0
$627$	−78.6154	−3.13959
$628$	0	0
$629$	10.4948	0.418455
$630$	0	0
$631$	12.8400	0.511153	0.255577	−	0.966789i	$-0.417735\pi$
$631$	12.8400	0.511153	0.255577	+	0.966789i	$0.417735\pi$
$632$	0	0
$633$	1.90096	0.0755564
$634$	0	0
$635$	43.4068	1.72255
$636$	0	0
$637$	0.741244	0.0293691
$638$	0	0
$639$	−26.6198	−1.05306
$640$	0	0
$641$	−43.5420	−1.71980	−0.859902	−	0.510459i	$-0.829476\pi$
$641$	−43.5420	−1.71980	−0.859902	+	0.510459i	$0.829476\pi$
$642$	0	0
$643$	43.0560	1.69796	0.848982	−	0.528422i	$-0.177216\pi$
$643$	43.0560	1.69796	0.848982	+	0.528422i	$0.177216\pi$
$644$	0	0
$645$	−6.18349	−0.243475
$646$	0	0
$647$	23.6756	0.930784	0.465392	−	0.885105i	$-0.345913\pi$
$647$	23.6756	0.930784	0.465392	+	0.885105i	$0.345913\pi$
$648$	0	0
$649$	−5.69258	−0.223453
$650$	0	0
$651$	−12.1295	−0.475392
$652$	0	0
$653$	−33.6041	−1.31503	−0.657514	−	0.753442i	$-0.728392\pi$
$653$	−33.6041	−1.31503	−0.657514	+	0.753442i	$0.728392\pi$
$654$	0	0
$655$	36.4285	1.42338
$656$	0	0
$657$	−28.9770	−1.13050
$658$	0	0
$659$	19.1014	0.744085	0.372042	−	0.928216i	$-0.378658\pi$
$659$	19.1014	0.744085	0.372042	+	0.928216i	$0.378658\pi$
$660$	0	0
$661$	11.9285	0.463966	0.231983	−	0.972720i	$-0.425479\pi$
$661$	11.9285	0.463966	0.231983	+	0.972720i	$0.425479\pi$
$662$	0	0
$663$	−3.80441	−0.147751
$664$	0	0
$665$	39.9229	1.54815
$666$	0	0
$667$	30.1665	1.16805
$668$	0	0
$669$	−12.3788	−0.478591
$670$	0	0
$671$	70.3158	2.71451
$672$	0	0
$673$	14.5229	0.559816	0.279908	−	0.960027i	$-0.409696\pi$
$673$	14.5229	0.559816	0.279908	+	0.960027i	$0.409696\pi$
$674$	0	0
$675$	3.03930	0.116983
$676$	0	0
$677$	−11.8758	−0.456426	−0.228213	−	0.973611i	$-0.573288\pi$
$677$	−11.8758	−0.456426	−0.228213	+	0.973611i	$0.573288\pi$
$678$	0	0
$679$	−40.6701	−1.56078
$680$	0	0
$681$	−11.4514	−0.438819
$682$	0	0
$683$	12.4622	0.476852	0.238426	−	0.971161i	$-0.423369\pi$
$683$	12.4622	0.476852	0.238426	+	0.971161i	$0.423369\pi$
$684$	0	0
$685$	−9.77813	−0.373603
$686$	0	0
$687$	−10.5327	−0.401849
$688$	0	0
$689$	7.69564	0.293181
$690$	0	0
$691$	16.6774	0.634438	0.317219	−	0.948352i	$-0.397251\pi$
$691$	16.6774	0.634438	0.317219	+	0.948352i	$0.397251\pi$
$692$	0	0
$693$	−28.5420	−1.08422
$694$	0	0
$695$	5.84749	0.221808
$696$	0	0
$697$	−4.78841	−0.181374
$698$	0	0
$699$	5.97346	0.225937
$700$	0	0
$701$	−41.8740	−1.58156	−0.790780	−	0.612100i	$-0.790325\pi$
$701$	−41.8740	−1.58156	−0.790780	+	0.612100i	$0.790325\pi$
$702$	0	0
$703$	−65.0996	−2.45528
$704$	0	0
$705$	16.7755	0.631803
$706$	0	0
$707$	−26.9277	−1.01272
$708$	0	0
$709$	−39.0679	−1.46723	−0.733613	−	0.679568i	$-0.762167\pi$
$709$	−39.0679	−1.46723	−0.733613	+	0.679568i	$0.762167\pi$
$710$	0	0
$711$	−25.0712	−0.940242
$712$	0	0
$713$	−6.93348	−0.259661
$714$	0	0
$715$	24.4322	0.913713
$716$	0	0
$717$	6.48116	0.242043
$718$	0	0
$719$	−32.4942	−1.21183	−0.605914	−	0.795530i	$-0.707192\pi$
$719$	−32.4942	−1.21183	−0.605914	+	0.795530i	$0.707192\pi$
$720$	0	0
$721$	−27.9333	−1.04029
$722$	0	0
$723$	11.1649	0.415226
$724$	0	0
$725$	12.1038	0.449522
$726$	0	0
$727$	40.0701	1.48612	0.743058	−	0.669227i	$-0.233375\pi$
$727$	40.0701	1.48612	0.743058	+	0.669227i	$0.233375\pi$
$728$	0	0
$729$	−6.09015	−0.225561
$730$	0	0
$731$	1.10580	0.0408996
$732$	0	0
$733$	31.3745	1.15884	0.579422	−	0.815027i	$-0.303278\pi$
$733$	31.3745	1.15884	0.579422	+	0.815027i	$0.303278\pi$
$734$	0	0
$735$	−2.42563	−0.0894707
$736$	0	0
$737$	−34.7603	−1.28041
$738$	0	0
$739$	−31.5689	−1.16128	−0.580640	−	0.814160i	$-0.697198\pi$
$739$	−31.5689	−1.16128	−0.580640	+	0.814160i	$0.697198\pi$
$740$	0	0
$741$	23.5989	0.866926
$742$	0	0
$743$	−14.0072	−0.513875	−0.256938	−	0.966428i	$-0.582713\pi$
$743$	−14.0072	−0.513875	−0.256938	+	0.966428i	$0.582713\pi$
$744$	0	0
$745$	−23.7569	−0.870385
$746$	0	0
$747$	−13.2541	−0.484941
$748$	0	0
$749$	16.2468	0.593645
$750$	0	0
$751$	30.1945	1.10181	0.550907	−	0.834566i	$-0.314282\pi$
$751$	30.1945	1.10181	0.550907	+	0.834566i	$0.314282\pi$
$752$	0	0
$753$	−5.99465	−0.218457
$754$	0	0
$755$	−31.5013	−1.14645
$756$	0	0
$757$	−6.19975	−0.225334	−0.112667	−	0.993633i	$-0.535939\pi$
$757$	−6.19975	−0.225334	−0.112667	+	0.993633i	$0.535939\pi$
$758$	0	0
$759$	−41.3300	−1.50018
$760$	0	0
$761$	−5.53598	−0.200679	−0.100339	−	0.994953i	$-0.531993\pi$
$761$	−5.53598	−0.200679	−0.100339	+	0.994953i	$0.531993\pi$
$762$	0	0
$763$	−20.1110	−0.728065
$764$	0	0
$765$	4.91450	0.177684
$766$	0	0
$767$	1.70881	0.0617014
$768$	0	0
$769$	−3.99981	−0.144237	−0.0721185	−	0.997396i	$-0.522976\pi$
$769$	−3.99981	−0.144237	−0.0721185	+	0.997396i	$0.522976\pi$
$770$	0	0
$771$	−50.6997	−1.82590
$772$	0	0
$773$	11.3754	0.409144	0.204572	−	0.978852i	$-0.434420\pi$
$773$	11.3754	0.409144	0.204572	+	0.978852i	$0.434420\pi$
$774$	0	0
$775$	−2.78193	−0.0999299
$776$	0	0
$777$	−59.8725	−2.14791
$778$	0	0
$779$	29.7026	1.06421
$780$	0	0
$781$	−77.4456	−2.77122
$782$	0	0
$783$	21.4872	0.767890
$784$	0	0
$785$	32.8014	1.17073
$786$	0	0
$787$	−20.8816	−0.744350	−0.372175	−	0.928163i	$-0.621388\pi$
$787$	−20.8816	−0.744350	−0.372175	+	0.928163i	$0.621388\pi$
$788$	0	0
$789$	−18.5925	−0.661912
$790$	0	0
$791$	−5.46876	−0.194447
$792$	0	0
$793$	−21.1075	−0.749549
$794$	0	0
$795$	−25.1831	−0.893151
$796$	0	0
$797$	−14.5690	−0.516060	−0.258030	−	0.966137i	$-0.583073\pi$
$797$	−14.5690	−0.516060	−0.258030	+	0.966137i	$0.583073\pi$
$798$	0	0
$799$	−2.99999	−0.106132
$800$	0	0
$801$	32.3122	1.14170
$802$	0	0
$803$	−84.3033	−2.97500
$804$	0	0
$805$	20.9885	0.739746
$806$	0	0
$807$	48.3851	1.70324
$808$	0	0
$809$	40.0564	1.40831	0.704153	−	0.710048i	$-0.251327\pi$
$809$	40.0564	1.40831	0.704153	+	0.710048i	$0.251327\pi$
$810$	0	0
$811$	15.4711	0.543265	0.271632	−	0.962401i	$-0.412437\pi$
$811$	15.4711	0.543265	0.271632	+	0.962401i	$0.412437\pi$
$812$	0	0
$813$	−39.9069	−1.39960
$814$	0	0
$815$	39.5223	1.38440
$816$	0	0
$817$	−6.85932	−0.239977
$818$	0	0
$819$	8.56778	0.299382
$820$	0	0
$821$	37.3930	1.30502	0.652512	−	0.757778i	$-0.273715\pi$
$821$	37.3930	1.30502	0.652512	+	0.757778i	$0.273715\pi$
$822$	0	0
$823$	2.84403	0.0991368	0.0495684	−	0.998771i	$-0.484215\pi$
$823$	2.84403	0.0991368	0.0495684	+	0.998771i	$0.484215\pi$
$824$	0	0
$825$	−16.5829	−0.577343
$826$	0	0
$827$	−40.1060	−1.39462	−0.697310	−	0.716769i	$-0.745620\pi$
$827$	−40.1060	−1.39462	−0.697310	+	0.716769i	$0.745620\pi$
$828$	0	0
$829$	38.7233	1.34492	0.672458	−	0.740135i	$-0.265239\pi$
$829$	38.7233	1.34492	0.672458	+	0.740135i	$0.265239\pi$
$830$	0	0
$831$	65.3921	2.26843
$832$	0	0
$833$	0.433779	0.0150295
$834$	0	0
$835$	−36.3528	−1.25804
$836$	0	0
$837$	−4.93863	−0.170704
$838$	0	0
$839$	3.91250	0.135075	0.0675373	−	0.997717i	$-0.478486\pi$
$839$	3.91250	0.135075	0.0675373	+	0.997717i	$0.478486\pi$
$840$	0	0
$841$	56.5710	1.95072
$842$	0	0
$843$	−0.631790	−0.0217600
$844$	0	0
$845$	25.3175	0.870949
$846$	0	0
$847$	−54.8506	−1.88469
$848$	0	0
$849$	11.2249	0.385238
$850$	0	0
$851$	−34.2244	−1.17320
$852$	0	0
$853$	−31.2812	−1.07105	−0.535523	−	0.844520i	$-0.679886\pi$
$853$	−31.2812	−1.07105	−0.535523	+	0.844520i	$0.679886\pi$
$854$	0	0
$855$	−30.4847	−1.04256
$856$	0	0
$857$	22.7345	0.776597	0.388298	−	0.921534i	$-0.373063\pi$
$857$	22.7345	0.776597	0.388298	+	0.921534i	$0.373063\pi$
$858$	0	0
$859$	−5.65356	−0.192897	−0.0964486	−	0.995338i	$-0.530748\pi$
$859$	−5.65356	−0.192897	−0.0964486	+	0.995338i	$0.530748\pi$
$860$	0	0
$861$	27.3177	0.930984
$862$	0	0
$863$	−21.8913	−0.745190	−0.372595	−	0.927994i	$-0.621532\pi$
$863$	−21.8913	−0.745190	−0.372595	+	0.927994i	$0.621532\pi$
$864$	0	0
$865$	42.8027	1.45533
$866$	0	0
$867$	−2.22636	−0.0756111
$868$	0	0
$869$	−72.9400	−2.47432
$870$	0	0
$871$	10.4344	0.353556
$872$	0	0
$873$	31.0553	1.05106
$874$	0	0
$875$	−23.7590	−0.803200
$876$	0	0
$877$	38.5189	1.30069	0.650346	−	0.759638i	$-0.274624\pi$
$877$	38.5189	1.30069	0.650346	+	0.759638i	$0.274624\pi$
$878$	0	0
$879$	0.607782	0.0205000
$880$	0	0
$881$	30.2491	1.01912	0.509560	−	0.860435i	$-0.329808\pi$
$881$	30.2491	1.01912	0.509560	+	0.860435i	$0.329808\pi$
$882$	0	0
$883$	3.40506	0.114589	0.0572947	−	0.998357i	$-0.481753\pi$
$883$	3.40506	0.114589	0.0572947	+	0.998357i	$0.481753\pi$
$884$	0	0
$885$	−5.59186	−0.187968
$886$	0	0
$887$	32.9040	1.10481	0.552405	−	0.833576i	$-0.313710\pi$
$887$	32.9040	1.10481	0.552405	+	0.833576i	$0.313710\pi$
$888$	0	0
$889$	44.2848	1.48526
$890$	0	0
$891$	−62.8543	−2.10570
$892$	0	0
$893$	18.6090	0.622727
$894$	0	0
$895$	−26.9256	−0.900024
$896$	0	0
$897$	12.4065	0.414241
$898$	0	0
$899$	−19.6677	−0.655953
$900$	0	0
$901$	4.50352	0.150034
$902$	0	0
$903$	−6.30856	−0.209936
$904$	0	0
$905$	48.2101	1.60256
$906$	0	0
$907$	32.8928	1.09219	0.546093	−	0.837725i	$-0.316115\pi$
$907$	32.8928	1.09219	0.546093	+	0.837725i	$0.316115\pi$
$908$	0	0
$909$	20.5617	0.681989
$910$	0	0
$911$	−46.9786	−1.55647	−0.778236	−	0.627972i	$-0.783885\pi$
$911$	−46.9786	−1.55647	−0.778236	+	0.627972i	$0.783885\pi$
$912$	0	0
$913$	−38.5603	−1.27616
$914$	0	0
$915$	69.0717	2.28344
$916$	0	0
$917$	37.1653	1.22731
$918$	0	0
$919$	59.0475	1.94780	0.973899	−	0.226980i	$-0.0728852\pi$
$919$	59.0475	1.94780	0.973899	+	0.226980i	$0.0728852\pi$
$920$	0	0
$921$	4.71377	0.155324
$922$	0	0
$923$	23.2477	0.765208
$924$	0	0
$925$	−13.7319	−0.451503
$926$	0	0
$927$	21.3296	0.700555
$928$	0	0
$929$	−24.6618	−0.809128	−0.404564	−	0.914510i	$-0.632577\pi$
$929$	−24.6618	−0.809128	−0.404564	+	0.914510i	$0.632577\pi$
$930$	0	0
$931$	−2.69074	−0.0881854
$932$	0	0
$933$	71.8511	2.35230
$934$	0	0
$935$	14.2978	0.467589
$936$	0	0
$937$	−0.215975	−0.00705560	−0.00352780	−	0.999994i	$-0.501123\pi$
$937$	−0.215975	−0.00705560	−0.00352780	+	0.999994i	$0.501123\pi$
$938$	0	0
$939$	−64.8069	−2.11489
$940$	0	0
$941$	52.8247	1.72204	0.861018	−	0.508575i	$-0.169827\pi$
$941$	52.8247	1.72204	0.861018	+	0.508575i	$0.169827\pi$
$942$	0	0
$943$	15.6154	0.508507
$944$	0	0
$945$	14.9498	0.486317
$946$	0	0
$947$	−3.39648	−0.110371	−0.0551853	−	0.998476i	$-0.517575\pi$
$947$	−3.39648	−0.110371	−0.0551853	+	0.998476i	$0.517575\pi$
$948$	0	0
$949$	25.3063	0.821477
$950$	0	0
$951$	−39.8562	−1.29243
$952$	0	0
$953$	49.2804	1.59635	0.798174	−	0.602427i	$-0.205800\pi$
$953$	49.2804	1.59635	0.798174	+	0.602427i	$0.205800\pi$
$954$	0	0
$955$	−47.7711	−1.54584
$956$	0	0
$957$	−117.238	−3.78976
$958$	0	0
$959$	−9.97590	−0.322139
$960$	0	0
$961$	−26.4796	−0.854180
$962$	0	0
$963$	−12.4059	−0.399774
$964$	0	0
$965$	−54.6349	−1.75876
$966$	0	0
$967$	−56.2290	−1.80820	−0.904101	−	0.427320i	$-0.859458\pi$
$967$	−56.2290	−1.80820	−0.904101	+	0.427320i	$0.859458\pi$
$968$	0	0
$969$	13.8102	0.443646
$970$	0	0
$971$	−5.65119	−0.181355	−0.0906776	−	0.995880i	$-0.528903\pi$
$971$	−5.65119	−0.181355	−0.0906776	+	0.995880i	$0.528903\pi$
$972$	0	0
$973$	5.96576	0.191253
$974$	0	0
$975$	4.97788	0.159420
$976$	0	0
$977$	47.7889	1.52890	0.764452	−	0.644681i	$-0.223010\pi$
$977$	47.7889	1.52890	0.764452	+	0.644681i	$0.223010\pi$
$978$	0	0
$979$	94.0065	3.00446
$980$	0	0
$981$	15.3565	0.490296
$982$	0	0
$983$	−17.9428	−0.572288	−0.286144	−	0.958187i	$-0.592374\pi$
$983$	−17.9428	−0.572288	−0.286144	+	0.958187i	$0.592374\pi$
$984$	0	0
$985$	−10.1257	−0.322633
$986$	0	0
$987$	17.1148	0.544772
$988$	0	0
$989$	−3.60611	−0.114668
$990$	0	0
$991$	6.45291	0.204984	0.102492	−	0.994734i	$-0.467318\pi$
$991$	6.45291	0.204984	0.102492	+	0.994734i	$0.467318\pi$
$992$	0	0
$993$	8.62482	0.273700
$994$	0	0
$995$	36.5422	1.15847
$996$	0	0
$997$	56.6701	1.79476	0.897380	−	0.441259i	$-0.145468\pi$
$997$	56.6701	1.79476	0.897380	+	0.441259i	$0.145468\pi$
$998$	0	0
$999$	−24.3776	−0.771274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bc.1.7	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bc.1.7	✓	33	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.22636 q^{3} -2.51166 q^{5} -2.56246 q^{7} +1.95667 q^{9} +O(q^{10})\)	\(q-2.22636 q^{3} -2.51166 q^{5} -2.56246 q^{7} +1.95667 q^{9} +5.69258 q^{11} -1.70881 q^{13} +5.59186 q^{15} -1.00000 q^{17} +6.20302 q^{19} +5.70496 q^{21} +3.26108 q^{23} +1.30845 q^{25} +2.32283 q^{27} +9.25046 q^{29} -2.12613 q^{31} -12.6737 q^{33} +6.43605 q^{35} -10.4948 q^{37} +3.80441 q^{39} +4.78841 q^{41} -1.10580 q^{43} -4.91450 q^{45} +2.99999 q^{47} -0.433779 q^{49} +2.22636 q^{51} -4.50352 q^{53} -14.2978 q^{55} -13.8102 q^{57} -1.00000 q^{59} +12.3522 q^{61} -5.01390 q^{63} +4.29194 q^{65} -6.10625 q^{67} -7.26034 q^{69} -13.6047 q^{71} -14.8093 q^{73} -2.91308 q^{75} -14.5870 q^{77} -12.8132 q^{79} -11.0415 q^{81} -6.77379 q^{83} +2.51166 q^{85} -20.5948 q^{87} +16.5139 q^{89} +4.37875 q^{91} +4.73352 q^{93} -15.5799 q^{95} +15.8715 q^{97} +11.1385 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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