LMFDB - Embedded newform 4024.2.a.g.1.2

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:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.94941 q^{3} +3.18881 q^{5} +1.60634 q^{7} +5.69904 q^{9} +O(q^{10})$	$q-2.94941 q^{3} +3.18881 q^{5} +1.60634 q^{7} +5.69904 q^{9} +4.47301 q^{11} -5.83128 q^{13} -9.40510 q^{15} -6.89867 q^{17} +0.522770 q^{19} -4.73777 q^{21} +8.32229 q^{23} +5.16848 q^{25} -7.96057 q^{27} +6.94459 q^{29} +5.79520 q^{31} -13.1927 q^{33} +5.12232 q^{35} -1.08626 q^{37} +17.1989 q^{39} -1.51389 q^{41} -0.793091 q^{43} +18.1731 q^{45} +11.8669 q^{47} -4.41966 q^{49} +20.3470 q^{51} -4.60227 q^{53} +14.2635 q^{55} -1.54186 q^{57} -9.50143 q^{59} +8.00488 q^{61} +9.15461 q^{63} -18.5948 q^{65} +6.48885 q^{67} -24.5459 q^{69} -11.8039 q^{71} +14.4725 q^{73} -15.2440 q^{75} +7.18518 q^{77} +2.28374 q^{79} +6.38190 q^{81} -11.6017 q^{83} -21.9985 q^{85} -20.4825 q^{87} -1.44240 q^{89} -9.36704 q^{91} -17.0924 q^{93} +1.66701 q^{95} -3.31275 q^{97} +25.4918 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * 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
$3$	−2.94941	−1.70284	−0.851422	−	0.524481i	$-0.824259\pi$
$3$	−2.94941	−1.70284	−0.851422	+	0.524481i	$0.824259\pi$
$4$	0	0
$5$	3.18881	1.42608	0.713039	−	0.701125i	$-0.247318\pi$
$5$	3.18881	1.42608	0.713039	+	0.701125i	$0.247318\pi$
$6$	0	0
$7$	1.60634	0.607141	0.303570	−	0.952809i	$-0.401821\pi$
$7$	1.60634	0.607141	0.303570	+	0.952809i	$0.401821\pi$
$8$	0	0
$9$	5.69904	1.89968
$10$	0	0
$11$	4.47301	1.34866	0.674331	−	0.738429i	$-0.264432\pi$
$11$	4.47301	1.34866	0.674331	+	0.738429i	$0.264432\pi$
$12$	0	0
$13$	−5.83128	−1.61731	−0.808654	−	0.588285i	$-0.799803\pi$
$13$	−5.83128	−1.61731	−0.808654	+	0.588285i	$0.799803\pi$
$14$	0	0
$15$	−9.40510	−2.42839
$16$	0	0
$17$	−6.89867	−1.67317	−0.836586	−	0.547835i	$-0.815452\pi$
$17$	−6.89867	−1.67317	−0.836586	+	0.547835i	$0.815452\pi$
$18$	0	0
$19$	0.522770	0.119932	0.0599658	−	0.998200i	$-0.480901\pi$
$19$	0.522770	0.119932	0.0599658	+	0.998200i	$0.480901\pi$
$20$	0	0
$21$	−4.73777	−1.03387
$22$	0	0
$23$	8.32229	1.73532	0.867658	−	0.497161i	$-0.165624\pi$
$23$	8.32229	1.73532	0.867658	+	0.497161i	$0.165624\pi$
$24$	0	0
$25$	5.16848	1.03370
$26$	0	0
$27$	−7.96057	−1.53201
$28$	0	0
$29$	6.94459	1.28958	0.644789	−	0.764360i	$-0.276945\pi$
$29$	6.94459	1.28958	0.644789	+	0.764360i	$0.276945\pi$
$30$	0	0
$31$	5.79520	1.04085	0.520424	−	0.853908i	$-0.325774\pi$
$31$	5.79520	1.04085	0.520424	+	0.853908i	$0.325774\pi$
$32$	0	0
$33$	−13.1927	−2.29656
$34$	0	0
$35$	5.12232	0.865830
$36$	0	0
$37$	−1.08626	−0.178580	−0.0892900	−	0.996006i	$-0.528460\pi$
$37$	−1.08626	−0.178580	−0.0892900	+	0.996006i	$0.528460\pi$
$38$	0	0
$39$	17.1989	2.75402
$40$	0	0
$41$	−1.51389	−0.236430	−0.118215	−	0.992988i	$-0.537717\pi$
$41$	−1.51389	−0.236430	−0.118215	+	0.992988i	$0.537717\pi$
$42$	0	0
$43$	−0.793091	−0.120945	−0.0604726	−	0.998170i	$-0.519261\pi$
$43$	−0.793091	−0.120945	−0.0604726	+	0.998170i	$0.519261\pi$
$44$	0	0
$45$	18.1731	2.70909
$46$	0	0
$47$	11.8669	1.73097	0.865483	−	0.500938i	$-0.167012\pi$
$47$	11.8669	1.73097	0.865483	+	0.500938i	$0.167012\pi$
$48$	0	0
$49$	−4.41966	−0.631380
$50$	0	0
$51$	20.3470	2.84915
$52$	0	0
$53$	−4.60227	−0.632171	−0.316085	−	0.948731i	$-0.602369\pi$
$53$	−4.60227	−0.632171	−0.316085	+	0.948731i	$0.602369\pi$
$54$	0	0
$55$	14.2635	1.92330
$56$	0	0
$57$	−1.54186	−0.204225
$58$	0	0
$59$	−9.50143	−1.23698	−0.618490	−	0.785792i	$-0.712255\pi$
$59$	−9.50143	−1.23698	−0.618490	+	0.785792i	$0.712255\pi$
$60$	0	0
$61$	8.00488	1.02492	0.512460	−	0.858711i	$-0.328734\pi$
$61$	8.00488	1.02492	0.512460	+	0.858711i	$0.328734\pi$
$62$	0	0
$63$	9.15461	1.15337
$64$	0	0
$65$	−18.5948	−2.30641
$66$	0	0
$67$	6.48885	0.792740	0.396370	−	0.918091i	$-0.370270\pi$
$67$	6.48885	0.792740	0.396370	+	0.918091i	$0.370270\pi$
$68$	0	0
$69$	−24.5459	−2.95497
$70$	0	0
$71$	−11.8039	−1.40087	−0.700434	−	0.713717i	$-0.747010\pi$
$71$	−11.8039	−1.40087	−0.700434	+	0.713717i	$0.747010\pi$
$72$	0	0
$73$	14.4725	1.69388	0.846939	−	0.531690i	$-0.178443\pi$
$73$	14.4725	1.69388	0.846939	+	0.531690i	$0.178443\pi$
$74$	0	0
$75$	−15.2440	−1.76022
$76$	0	0
$77$	7.18518	0.818827
$78$	0	0
$79$	2.28374	0.256941	0.128471	−	0.991713i	$-0.458993\pi$
$79$	2.28374	0.256941	0.128471	+	0.991713i	$0.458993\pi$
$80$	0	0
$81$	6.38190	0.709100
$82$	0	0
$83$	−11.6017	−1.27345	−0.636723	−	0.771092i	$-0.719711\pi$
$83$	−11.6017	−1.27345	−0.636723	+	0.771092i	$0.719711\pi$
$84$	0	0
$85$	−21.9985	−2.38607
$86$	0	0
$87$	−20.4825	−2.19595
$88$	0	0
$89$	−1.44240	−0.152894	−0.0764470	−	0.997074i	$-0.524358\pi$
$89$	−1.44240	−0.152894	−0.0764470	+	0.997074i	$0.524358\pi$
$90$	0	0
$91$	−9.36704	−0.981933
$92$	0	0
$93$	−17.0924	−1.77240
$94$	0	0
$95$	1.66701	0.171032
$96$	0	0
$97$	−3.31275	−0.336359	−0.168179	−	0.985756i	$-0.553789\pi$
$97$	−3.31275	−0.336359	−0.168179	+	0.985756i	$0.553789\pi$
$98$	0	0
$99$	25.4918	2.56202
$100$	0	0
$101$	9.54638	0.949901	0.474950	−	0.880013i	$-0.342466\pi$
$101$	9.54638	0.949901	0.474950	+	0.880013i	$0.342466\pi$
$102$	0	0
$103$	4.73334	0.466390	0.233195	−	0.972430i	$-0.425082\pi$
$103$	4.73334	0.466390	0.233195	+	0.972430i	$0.425082\pi$
$104$	0	0
$105$	−15.1078	−1.47437
$106$	0	0
$107$	−3.95052	−0.381911	−0.190956	−	0.981599i	$-0.561159\pi$
$107$	−3.95052	−0.381911	−0.190956	+	0.981599i	$0.561159\pi$
$108$	0	0
$109$	11.3952	1.09146	0.545731	−	0.837960i	$-0.316252\pi$
$109$	11.3952	1.09146	0.545731	+	0.837960i	$0.316252\pi$
$110$	0	0
$111$	3.20383	0.304094
$112$	0	0
$113$	2.36324	0.222315	0.111157	−	0.993803i	$-0.464544\pi$
$113$	2.36324	0.222315	0.111157	+	0.993803i	$0.464544\pi$
$114$	0	0
$115$	26.5382	2.47470
$116$	0	0
$117$	−33.2327	−3.07236
$118$	0	0
$119$	−11.0816	−1.01585
$120$	0	0
$121$	9.00777	0.818889
$122$	0	0
$123$	4.46509	0.402603
$124$	0	0
$125$	0.537255	0.0480535
$126$	0	0
$127$	−3.55081	−0.315083	−0.157542	−	0.987512i	$-0.550357\pi$
$127$	−3.55081	−0.315083	−0.157542	+	0.987512i	$0.550357\pi$
$128$	0	0
$129$	2.33915	0.205951
$130$	0	0
$131$	15.3841	1.34411	0.672056	−	0.740500i	$-0.265411\pi$
$131$	15.3841	1.34411	0.672056	+	0.740500i	$0.265411\pi$
$132$	0	0
$133$	0.839748	0.0728154
$134$	0	0
$135$	−25.3847	−2.18477
$136$	0	0
$137$	−17.2253	−1.47165	−0.735827	−	0.677169i	$-0.763206\pi$
$137$	−17.2253	−1.47165	−0.735827	+	0.677169i	$0.763206\pi$
$138$	0	0
$139$	−8.65508	−0.734114	−0.367057	−	0.930198i	$-0.619635\pi$
$139$	−8.65508	−0.734114	−0.367057	+	0.930198i	$0.619635\pi$
$140$	0	0
$141$	−35.0004	−2.94757
$142$	0	0
$143$	−26.0834	−2.18120
$144$	0	0
$145$	22.1450	1.83904
$146$	0	0
$147$	13.0354	1.07514
$148$	0	0
$149$	7.76668	0.636271	0.318136	−	0.948045i	$-0.396943\pi$
$149$	7.76668	0.636271	0.318136	+	0.948045i	$0.396943\pi$
$150$	0	0
$151$	18.0246	1.46682	0.733411	−	0.679786i	$-0.237927\pi$
$151$	18.0246	1.46682	0.733411	+	0.679786i	$0.237927\pi$
$152$	0	0
$153$	−39.3158	−3.17849
$154$	0	0
$155$	18.4798	1.48433
$156$	0	0
$157$	−17.3237	−1.38258	−0.691290	−	0.722577i	$-0.742957\pi$
$157$	−17.3237	−1.38258	−0.691290	+	0.722577i	$0.742957\pi$
$158$	0	0
$159$	13.5740	1.07649
$160$	0	0
$161$	13.3684	1.05358
$162$	0	0
$163$	0.430999	0.0337584	0.0168792	−	0.999858i	$-0.494627\pi$
$163$	0.430999	0.0337584	0.0168792	+	0.999858i	$0.494627\pi$
$164$	0	0
$165$	−42.0691	−3.27507
$166$	0	0
$167$	10.6063	0.820743	0.410371	−	0.911919i	$-0.365399\pi$
$167$	10.6063	0.820743	0.410371	+	0.911919i	$0.365399\pi$
$168$	0	0
$169$	21.0039	1.61568
$170$	0	0
$171$	2.97928	0.227832
$172$	0	0
$173$	2.04186	0.155240	0.0776198	−	0.996983i	$-0.475268\pi$
$173$	2.04186	0.155240	0.0776198	+	0.996983i	$0.475268\pi$
$174$	0	0
$175$	8.30236	0.627599
$176$	0	0
$177$	28.0236	2.10639
$178$	0	0
$179$	−1.56397	−0.116896	−0.0584482	−	0.998290i	$-0.518615\pi$
$179$	−1.56397	−0.116896	−0.0584482	+	0.998290i	$0.518615\pi$
$180$	0	0
$181$	−3.66888	−0.272706	−0.136353	−	0.990660i	$-0.543538\pi$
$181$	−3.66888	−0.272706	−0.136353	+	0.990660i	$0.543538\pi$
$182$	0	0
$183$	−23.6097	−1.74528
$184$	0	0
$185$	−3.46387	−0.254669
$186$	0	0
$187$	−30.8578	−2.25654
$188$	0	0
$189$	−12.7874	−0.930147
$190$	0	0
$191$	1.90325	0.137714	0.0688571	−	0.997627i	$-0.478065\pi$
$191$	1.90325	0.137714	0.0688571	+	0.997627i	$0.478065\pi$
$192$	0	0
$193$	10.5075	0.756345	0.378172	−	0.925735i	$-0.376553\pi$
$193$	10.5075	0.756345	0.378172	+	0.925735i	$0.376553\pi$
$194$	0	0
$195$	54.8438	3.92745
$196$	0	0
$197$	13.8438	0.986332	0.493166	−	0.869935i	$-0.335839\pi$
$197$	13.8438	0.986332	0.493166	+	0.869935i	$0.335839\pi$
$198$	0	0
$199$	21.2555	1.50676	0.753381	−	0.657584i	$-0.228421\pi$
$199$	21.2555	1.50676	0.753381	+	0.657584i	$0.228421\pi$
$200$	0	0
$201$	−19.1383	−1.34991
$202$	0	0
$203$	11.1554	0.782955
$204$	0	0
$205$	−4.82750	−0.337167
$206$	0	0
$207$	47.4290	3.29654
$208$	0	0
$209$	2.33835	0.161747
$210$	0	0
$211$	−5.11727	−0.352288	−0.176144	−	0.984364i	$-0.556362\pi$
$211$	−5.11727	−0.352288	−0.176144	+	0.984364i	$0.556362\pi$
$212$	0	0
$213$	34.8146	2.38546
$214$	0	0
$215$	−2.52901	−0.172477
$216$	0	0
$217$	9.30908	0.631941
$218$	0	0
$219$	−42.6854	−2.88441
$220$	0	0
$221$	40.2281	2.70603
$222$	0	0
$223$	−10.9046	−0.730227	−0.365114	−	0.930963i	$-0.618970\pi$
$223$	−10.9046	−0.730227	−0.365114	+	0.930963i	$0.618970\pi$
$224$	0	0
$225$	29.4554	1.96369
$226$	0	0
$227$	−9.99259	−0.663231	−0.331616	−	0.943415i	$-0.607594\pi$
$227$	−9.99259	−0.663231	−0.331616	+	0.943415i	$0.607594\pi$
$228$	0	0
$229$	12.8939	0.852051	0.426025	−	0.904711i	$-0.359913\pi$
$229$	12.8939	0.852051	0.426025	+	0.904711i	$0.359913\pi$
$230$	0	0
$231$	−21.1921	−1.39434
$232$	0	0
$233$	24.6343	1.61385	0.806924	−	0.590655i	$-0.201131\pi$
$233$	24.6343	1.61385	0.806924	+	0.590655i	$0.201131\pi$
$234$	0	0
$235$	37.8413	2.46849
$236$	0	0
$237$	−6.73571	−0.437531
$238$	0	0
$239$	0.359859	0.0232773	0.0116387	−	0.999932i	$-0.496295\pi$
$239$	0.359859	0.0232773	0.0116387	+	0.999932i	$0.496295\pi$
$240$	0	0
$241$	−18.6909	−1.20399	−0.601995	−	0.798500i	$-0.705627\pi$
$241$	−18.6909	−1.20399	−0.601995	+	0.798500i	$0.705627\pi$
$242$	0	0
$243$	5.05886	0.324526
$244$	0	0
$245$	−14.0934	−0.900397
$246$	0	0
$247$	−3.04842	−0.193966
$248$	0	0
$249$	34.2181	2.16848
$250$	0	0
$251$	−1.90561	−0.120281	−0.0601405	−	0.998190i	$-0.519155\pi$
$251$	−1.90561	−0.120281	−0.0601405	+	0.998190i	$0.519155\pi$
$252$	0	0
$253$	37.2256	2.34036
$254$	0	0
$255$	64.8827	4.06311
$256$	0	0
$257$	−15.0200	−0.936923	−0.468461	−	0.883484i	$-0.655192\pi$
$257$	−15.0200	−0.936923	−0.468461	+	0.883484i	$0.655192\pi$
$258$	0	0
$259$	−1.74491	−0.108423
$260$	0	0
$261$	39.5775	2.44978
$262$	0	0
$263$	30.8327	1.90123	0.950613	−	0.310377i	$-0.100455\pi$
$263$	30.8327	1.90123	0.950613	+	0.310377i	$0.100455\pi$
$264$	0	0
$265$	−14.6758	−0.901525
$266$	0	0
$267$	4.25423	0.260355
$268$	0	0
$269$	30.1529	1.83846	0.919228	−	0.393725i	$-0.128814\pi$
$269$	30.1529	1.83846	0.919228	+	0.393725i	$0.128814\pi$
$270$	0	0
$271$	28.2837	1.71811	0.859055	−	0.511883i	$-0.171052\pi$
$271$	28.2837	1.71811	0.859055	+	0.511883i	$0.171052\pi$
$272$	0	0
$273$	27.6273	1.67208
$274$	0	0
$275$	23.1186	1.39411
$276$	0	0
$277$	−28.9547	−1.73972	−0.869860	−	0.493299i	$-0.835791\pi$
$277$	−28.9547	−1.73972	−0.869860	+	0.493299i	$0.835791\pi$
$278$	0	0
$279$	33.0270	1.97728
$280$	0	0
$281$	11.0656	0.660119	0.330060	−	0.943960i	$-0.392931\pi$
$281$	11.0656	0.660119	0.330060	+	0.943960i	$0.392931\pi$
$282$	0	0
$283$	18.7215	1.11288	0.556440	−	0.830888i	$-0.312167\pi$
$283$	18.7215	1.11288	0.556440	+	0.830888i	$0.312167\pi$
$284$	0	0
$285$	−4.91671	−0.291240
$286$	0	0
$287$	−2.43183	−0.143546
$288$	0	0
$289$	30.5916	1.79951
$290$	0	0
$291$	9.77067	0.572767
$292$	0	0
$293$	−24.6262	−1.43868	−0.719339	−	0.694659i	$-0.755555\pi$
$293$	−24.6262	−1.43868	−0.719339	+	0.694659i	$0.755555\pi$
$294$	0	0
$295$	−30.2982	−1.76403
$296$	0	0
$297$	−35.6077	−2.06617
$298$	0	0
$299$	−48.5296	−2.80654
$300$	0	0
$301$	−1.27398	−0.0734308
$302$	0	0
$303$	−28.1562	−1.61753
$304$	0	0
$305$	25.5260	1.46162
$306$	0	0
$307$	−29.7976	−1.70064	−0.850321	−	0.526265i	$-0.823592\pi$
$307$	−29.7976	−1.70064	−0.850321	+	0.526265i	$0.823592\pi$
$308$	0	0
$309$	−13.9606	−0.794189
$310$	0	0
$311$	4.35182	0.246769	0.123384	−	0.992359i	$-0.460625\pi$
$311$	4.35182	0.246769	0.123384	+	0.992359i	$0.460625\pi$
$312$	0	0
$313$	19.6804	1.11240	0.556201	−	0.831048i	$-0.312259\pi$
$313$	19.6804	1.11240	0.556201	+	0.831048i	$0.312259\pi$
$314$	0	0
$315$	29.1923	1.64480
$316$	0	0
$317$	−27.0724	−1.52054	−0.760270	−	0.649607i	$-0.774933\pi$
$317$	−27.0724	−1.52054	−0.760270	+	0.649607i	$0.774933\pi$
$318$	0	0
$319$	31.0632	1.73921
$320$	0	0
$321$	11.6517	0.650335
$322$	0	0
$323$	−3.60642	−0.200666
$324$	0	0
$325$	−30.1389	−1.67180
$326$	0	0
$327$	−33.6092	−1.85859
$328$	0	0
$329$	19.0623	1.05094
$330$	0	0
$331$	21.4058	1.17657	0.588284	−	0.808654i	$-0.299804\pi$
$331$	21.4058	1.17657	0.588284	+	0.808654i	$0.299804\pi$
$332$	0	0
$333$	−6.19063	−0.339245
$334$	0	0
$335$	20.6917	1.13051
$336$	0	0
$337$	24.2808	1.32266	0.661330	−	0.750095i	$-0.269992\pi$
$337$	24.2808	1.32266	0.661330	+	0.750095i	$0.269992\pi$
$338$	0	0
$339$	−6.97017	−0.378568
$340$	0	0
$341$	25.9219	1.40375
$342$	0	0
$343$	−18.3439	−0.990477
$344$	0	0
$345$	−78.2720	−4.21402
$346$	0	0
$347$	−10.5150	−0.564477	−0.282238	−	0.959344i	$-0.591077\pi$
$347$	−10.5150	−0.564477	−0.282238	+	0.959344i	$0.591077\pi$
$348$	0	0
$349$	−5.80506	−0.310738	−0.155369	−	0.987857i	$-0.549657\pi$
$349$	−5.80506	−0.310738	−0.155369	+	0.987857i	$0.549657\pi$
$350$	0	0
$351$	46.4203	2.47773
$352$	0	0
$353$	−17.5867	−0.936046	−0.468023	−	0.883716i	$-0.655034\pi$
$353$	−17.5867	−0.936046	−0.468023	+	0.883716i	$0.655034\pi$
$354$	0	0
$355$	−37.6404	−1.99775
$356$	0	0
$357$	32.6843	1.72984
$358$	0	0
$359$	−1.14626	−0.0604975	−0.0302487	−	0.999542i	$-0.509630\pi$
$359$	−1.14626	−0.0604975	−0.0302487	+	0.999542i	$0.509630\pi$
$360$	0	0
$361$	−18.7267	−0.985616
$362$	0	0
$363$	−26.5676	−1.39444
$364$	0	0
$365$	46.1500	2.41560
$366$	0	0
$367$	−23.0689	−1.20419	−0.602094	−	0.798426i	$-0.705667\pi$
$367$	−23.0689	−1.20419	−0.602094	+	0.798426i	$0.705667\pi$
$368$	0	0
$369$	−8.62771	−0.449141
$370$	0	0
$371$	−7.39283	−0.383817
$372$	0	0
$373$	15.9152	0.824060	0.412030	−	0.911170i	$-0.364820\pi$
$373$	15.9152	0.824060	0.412030	+	0.911170i	$0.364820\pi$
$374$	0	0
$375$	−1.58459	−0.0818277
$376$	0	0
$377$	−40.4959	−2.08564
$378$	0	0
$379$	−18.7642	−0.963854	−0.481927	−	0.876211i	$-0.660063\pi$
$379$	−18.7642	−0.963854	−0.481927	+	0.876211i	$0.660063\pi$
$380$	0	0
$381$	10.4728	0.536538
$382$	0	0
$383$	35.9075	1.83479	0.917393	−	0.397983i	$-0.130290\pi$
$383$	35.9075	1.83479	0.917393	+	0.397983i	$0.130290\pi$
$384$	0	0
$385$	22.9121	1.16771
$386$	0	0
$387$	−4.51986	−0.229757
$388$	0	0
$389$	8.09393	0.410378	0.205189	−	0.978722i	$-0.434219\pi$
$389$	8.09393	0.410378	0.205189	+	0.978722i	$0.434219\pi$
$390$	0	0
$391$	−57.4127	−2.90348
$392$	0	0
$393$	−45.3740	−2.28881
$394$	0	0
$395$	7.28242	0.366418
$396$	0	0
$397$	7.47358	0.375088	0.187544	−	0.982256i	$-0.439947\pi$
$397$	7.47358	0.375088	0.187544	+	0.982256i	$0.439947\pi$
$398$	0	0
$399$	−2.47676	−0.123993
$400$	0	0
$401$	−0.477857	−0.0238631	−0.0119315	−	0.999929i	$-0.503798\pi$
$401$	−0.477857	−0.0238631	−0.0119315	+	0.999929i	$0.503798\pi$
$402$	0	0
$403$	−33.7934	−1.68337
$404$	0	0
$405$	20.3506	1.01123
$406$	0	0
$407$	−4.85885	−0.240844
$408$	0	0
$409$	−13.0396	−0.644768	−0.322384	−	0.946609i	$-0.604484\pi$
$409$	−13.0396	−0.644768	−0.322384	+	0.946609i	$0.604484\pi$
$410$	0	0
$411$	50.8044	2.50600
$412$	0	0
$413$	−15.2626	−0.751021
$414$	0	0
$415$	−36.9954	−1.81603
$416$	0	0
$417$	25.5274	1.25008
$418$	0	0
$419$	−16.7326	−0.817439	−0.408719	−	0.912660i	$-0.634025\pi$
$419$	−16.7326	−0.817439	−0.408719	+	0.912660i	$0.634025\pi$
$420$	0	0
$421$	32.2085	1.56975	0.784873	−	0.619657i	$-0.212728\pi$
$421$	32.2085	1.56975	0.784873	+	0.619657i	$0.212728\pi$
$422$	0	0
$423$	67.6299	3.28828
$424$	0	0
$425$	−35.6556	−1.72955
$426$	0	0
$427$	12.8586	0.622271
$428$	0	0
$429$	76.9306	3.71424
$430$	0	0
$431$	18.8705	0.908961	0.454480	−	0.890757i	$-0.349825\pi$
$431$	18.8705	0.908961	0.454480	+	0.890757i	$0.349825\pi$
$432$	0	0
$433$	−4.25219	−0.204347	−0.102174	−	0.994767i	$-0.532580\pi$
$433$	−4.25219	−0.204347	−0.102174	+	0.994767i	$0.532580\pi$
$434$	0	0
$435$	−65.3146	−3.13160
$436$	0	0
$437$	4.35064	0.208119
$438$	0	0
$439$	−10.8115	−0.516005	−0.258003	−	0.966144i	$-0.583064\pi$
$439$	−10.8115	−0.516005	−0.258003	+	0.966144i	$0.583064\pi$
$440$	0	0
$441$	−25.1878	−1.19942
$442$	0	0
$443$	−8.74121	−0.415308	−0.207654	−	0.978202i	$-0.566583\pi$
$443$	−8.74121	−0.415308	−0.207654	+	0.978202i	$0.566583\pi$
$444$	0	0
$445$	−4.59953	−0.218039
$446$	0	0
$447$	−22.9071	−1.08347
$448$	0	0
$449$	−3.09292	−0.145964	−0.0729819	−	0.997333i	$-0.523252\pi$
$449$	−3.09292	−0.145964	−0.0729819	+	0.997333i	$0.523252\pi$
$450$	0	0
$451$	−6.77164	−0.318864
$452$	0	0
$453$	−53.1620	−2.49777
$454$	0	0
$455$	−29.8697	−1.40031
$456$	0	0
$457$	1.97614	0.0924401	0.0462201	−	0.998931i	$-0.485282\pi$
$457$	1.97614	0.0924401	0.0462201	+	0.998931i	$0.485282\pi$
$458$	0	0
$459$	54.9173	2.56332
$460$	0	0
$461$	−8.79090	−0.409433	−0.204716	−	0.978821i	$-0.565627\pi$
$461$	−8.79090	−0.409433	−0.204716	+	0.978821i	$0.565627\pi$
$462$	0	0
$463$	36.4871	1.69570	0.847849	−	0.530238i	$-0.177897\pi$
$463$	36.4871	1.69570	0.847849	+	0.530238i	$0.177897\pi$
$464$	0	0
$465$	−54.5044	−2.52758
$466$	0	0
$467$	−24.3339	−1.12604	−0.563020	−	0.826443i	$-0.690361\pi$
$467$	−24.3339	−1.12604	−0.563020	+	0.826443i	$0.690361\pi$
$468$	0	0
$469$	10.4233	0.481304
$470$	0	0
$471$	51.0947	2.35432
$472$	0	0
$473$	−3.54750	−0.163114
$474$	0	0
$475$	2.70193	0.123973
$476$	0	0
$477$	−26.2285	−1.20092
$478$	0	0
$479$	17.6823	0.807925	0.403963	−	0.914775i	$-0.367633\pi$
$479$	17.6823	0.807925	0.403963	+	0.914775i	$0.367633\pi$
$480$	0	0
$481$	6.33429	0.288819
$482$	0	0
$483$	−39.4291	−1.79409
$484$	0	0
$485$	−10.5637	−0.479674
$486$	0	0
$487$	36.4080	1.64980	0.824901	−	0.565277i	$-0.191231\pi$
$487$	36.4080	1.64980	0.824901	+	0.565277i	$0.191231\pi$
$488$	0	0
$489$	−1.27119	−0.0574853
$490$	0	0
$491$	14.0076	0.632155	0.316077	−	0.948733i	$-0.397634\pi$
$491$	14.0076	0.632155	0.316077	+	0.948733i	$0.397634\pi$
$492$	0	0
$493$	−47.9084	−2.15769
$494$	0	0
$495$	81.2884	3.65364
$496$	0	0
$497$	−18.9612	−0.850524
$498$	0	0
$499$	6.51181	0.291509	0.145754	−	0.989321i	$-0.453439\pi$
$499$	6.51181	0.291509	0.145754	+	0.989321i	$0.453439\pi$
$500$	0	0
$501$	−31.2824	−1.39760
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	30.4416	1.35463
$506$	0	0
$507$	−61.9491	−2.75126
$508$	0	0
$509$	−0.885805	−0.0392626	−0.0196313	−	0.999807i	$-0.506249\pi$
$509$	−0.885805	−0.0392626	−0.0196313	+	0.999807i	$0.506249\pi$
$510$	0	0
$511$	23.2478	1.02842
$512$	0	0
$513$	−4.16155	−0.183737
$514$	0	0
$515$	15.0937	0.665108
$516$	0	0
$517$	53.0807	2.33449
$518$	0	0
$519$	−6.02228	−0.264349
$520$	0	0
$521$	3.15011	0.138009	0.0690044	−	0.997616i	$-0.478018\pi$
$521$	3.15011	0.138009	0.0690044	+	0.997616i	$0.478018\pi$
$522$	0	0
$523$	17.4507	0.763066	0.381533	−	0.924355i	$-0.375396\pi$
$523$	17.4507	0.763066	0.381533	+	0.924355i	$0.375396\pi$
$524$	0	0
$525$	−24.4871	−1.06870
$526$	0	0
$527$	−39.9791	−1.74152
$528$	0	0
$529$	46.2605	2.01132
$530$	0	0
$531$	−54.1490	−2.34987
$532$	0	0
$533$	8.82792	0.382380
$534$	0	0
$535$	−12.5974	−0.544635
$536$	0	0
$537$	4.61278	0.199056
$538$	0	0
$539$	−19.7692	−0.851518
$540$	0	0
$541$	−33.3142	−1.43229	−0.716144	−	0.697953i	$-0.754095\pi$
$541$	−33.3142	−1.43229	−0.716144	+	0.697953i	$0.754095\pi$
$542$	0	0
$543$	10.8210	0.464375
$544$	0	0
$545$	36.3371	1.55651
$546$	0	0
$547$	12.0720	0.516161	0.258081	−	0.966123i	$-0.416910\pi$
$547$	12.0720	0.516161	0.258081	+	0.966123i	$0.416910\pi$
$548$	0	0
$549$	45.6201	1.94702
$550$	0	0
$551$	3.63042	0.154661
$552$	0	0
$553$	3.66848	0.156000
$554$	0	0
$555$	10.2164	0.433662
$556$	0	0
$557$	30.8158	1.30571	0.652854	−	0.757484i	$-0.273572\pi$
$557$	30.8158	1.30571	0.652854	+	0.757484i	$0.273572\pi$
$558$	0	0
$559$	4.62474	0.195606
$560$	0	0
$561$	91.0123	3.84254
$562$	0	0
$563$	15.5450	0.655144	0.327572	−	0.944826i	$-0.393770\pi$
$563$	15.5450	0.655144	0.327572	+	0.944826i	$0.393770\pi$
$564$	0	0
$565$	7.53591	0.317038
$566$	0	0
$567$	10.2515	0.430523
$568$	0	0
$569$	34.2673	1.43656	0.718281	−	0.695753i	$-0.244929\pi$
$569$	34.2673	1.43656	0.718281	+	0.695753i	$0.244929\pi$
$570$	0	0
$571$	−20.1754	−0.844312	−0.422156	−	0.906523i	$-0.638727\pi$
$571$	−20.1754	−0.844312	−0.422156	+	0.906523i	$0.638727\pi$
$572$	0	0
$573$	−5.61346	−0.234506
$574$	0	0
$575$	43.0136	1.79379
$576$	0	0
$577$	10.5452	0.439001	0.219500	−	0.975612i	$-0.429557\pi$
$577$	10.5452	0.439001	0.219500	+	0.975612i	$0.429557\pi$
$578$	0	0
$579$	−30.9909	−1.28794
$580$	0	0
$581$	−18.6362	−0.773161
$582$	0	0
$583$	−20.5860	−0.852585
$584$	0	0
$585$	−105.973	−4.38143
$586$	0	0
$587$	14.9470	0.616930	0.308465	−	0.951236i	$-0.400185\pi$
$587$	14.9470	0.616930	0.308465	+	0.951236i	$0.400185\pi$
$588$	0	0
$589$	3.02956	0.124831
$590$	0	0
$591$	−40.8312	−1.67957
$592$	0	0
$593$	−6.62158	−0.271916	−0.135958	−	0.990715i	$-0.543411\pi$
$593$	−6.62158	−0.271916	−0.135958	+	0.990715i	$0.543411\pi$
$594$	0	0
$595$	−35.3372	−1.44868
$596$	0	0
$597$	−62.6913	−2.56578
$598$	0	0
$599$	14.7155	0.601258	0.300629	−	0.953741i	$-0.402803\pi$
$599$	14.7155	0.601258	0.300629	+	0.953741i	$0.402803\pi$
$600$	0	0
$601$	26.8205	1.09403	0.547016	−	0.837122i	$-0.315764\pi$
$601$	26.8205	1.09403	0.547016	+	0.837122i	$0.315764\pi$
$602$	0	0
$603$	36.9802	1.50595
$604$	0	0
$605$	28.7240	1.16780
$606$	0	0
$607$	23.4418	0.951472	0.475736	−	0.879588i	$-0.342182\pi$
$607$	23.4418	0.951472	0.475736	+	0.879588i	$0.342182\pi$
$608$	0	0
$609$	−32.9019	−1.33325
$610$	0	0
$611$	−69.1993	−2.79950
$612$	0	0
$613$	19.9169	0.804436	0.402218	−	0.915544i	$-0.368239\pi$
$613$	19.9169	0.804436	0.402218	+	0.915544i	$0.368239\pi$
$614$	0	0
$615$	14.2383	0.574143
$616$	0	0
$617$	−23.5382	−0.947611	−0.473806	−	0.880629i	$-0.657120\pi$
$617$	−23.5382	−0.947611	−0.473806	+	0.880629i	$0.657120\pi$
$618$	0	0
$619$	−33.1458	−1.33224	−0.666121	−	0.745844i	$-0.732047\pi$
$619$	−33.1458	−1.33224	−0.666121	+	0.745844i	$0.732047\pi$
$620$	0	0
$621$	−66.2501	−2.65853
$622$	0	0
$623$	−2.31699	−0.0928282
$624$	0	0
$625$	−24.1292	−0.965168
$626$	0	0
$627$	−6.89677	−0.275430
$628$	0	0
$629$	7.49375	0.298795
$630$	0	0
$631$	49.8817	1.98576	0.992879	−	0.119128i	$-0.0380098\pi$
$631$	49.8817	1.98576	0.992879	+	0.119128i	$0.0380098\pi$
$632$	0	0
$633$	15.0929	0.599891
$634$	0	0
$635$	−11.3228	−0.449333
$636$	0	0
$637$	25.7723	1.02114
$638$	0	0
$639$	−67.2710	−2.66120
$640$	0	0
$641$	−25.2648	−0.997898	−0.498949	−	0.866631i	$-0.666281\pi$
$641$	−25.2648	−0.997898	−0.498949	+	0.866631i	$0.666281\pi$
$642$	0	0
$643$	−25.9432	−1.02310	−0.511550	−	0.859253i	$-0.670929\pi$
$643$	−25.9432	−1.02310	−0.511550	+	0.859253i	$0.670929\pi$
$644$	0	0
$645$	7.45911	0.293702
$646$	0	0
$647$	33.7480	1.32677	0.663385	−	0.748278i	$-0.269119\pi$
$647$	33.7480	1.32677	0.663385	+	0.748278i	$0.269119\pi$
$648$	0	0
$649$	−42.4999	−1.66827
$650$	0	0
$651$	−27.4563	−1.07610
$652$	0	0
$653$	−27.8775	−1.09093	−0.545466	−	0.838133i	$-0.683647\pi$
$653$	−27.8775	−1.09093	−0.545466	+	0.838133i	$0.683647\pi$
$654$	0	0
$655$	49.0568	1.91681
$656$	0	0
$657$	82.4793	3.21782
$658$	0	0
$659$	−27.1598	−1.05799	−0.528997	−	0.848624i	$-0.677432\pi$
$659$	−27.1598	−1.05799	−0.528997	+	0.848624i	$0.677432\pi$
$660$	0	0
$661$	−5.64026	−0.219381	−0.109690	−	0.993966i	$-0.534986\pi$
$661$	−5.64026	−0.219381	−0.109690	+	0.993966i	$0.534986\pi$
$662$	0	0
$663$	−118.649	−4.60796
$664$	0	0
$665$	2.67779	0.103840
$666$	0	0
$667$	57.7949	2.23783
$668$	0	0
$669$	32.1622	1.24346
$670$	0	0
$671$	35.8059	1.38227
$672$	0	0
$673$	9.72662	0.374934	0.187467	−	0.982271i	$-0.439972\pi$
$673$	9.72662	0.374934	0.187467	+	0.982271i	$0.439972\pi$
$674$	0	0
$675$	−41.1441	−1.58364
$676$	0	0
$677$	−36.4417	−1.40057	−0.700284	−	0.713864i	$-0.746943\pi$
$677$	−36.4417	−1.40057	−0.700284	+	0.713864i	$0.746943\pi$
$678$	0	0
$679$	−5.32142	−0.204217
$680$	0	0
$681$	29.4723	1.12938
$682$	0	0
$683$	6.06955	0.232245	0.116122	−	0.993235i	$-0.462954\pi$
$683$	6.06955	0.232245	0.116122	+	0.993235i	$0.462954\pi$
$684$	0	0
$685$	−54.9280	−2.09869
$686$	0	0
$687$	−38.0293	−1.45091
$688$	0	0
$689$	26.8372	1.02241
$690$	0	0
$691$	27.1711	1.03364	0.516818	−	0.856095i	$-0.327116\pi$
$691$	27.1711	1.03364	0.516818	+	0.856095i	$0.327116\pi$
$692$	0	0
$693$	40.9486	1.55551
$694$	0	0
$695$	−27.5994	−1.04690
$696$	0	0
$697$	10.4438	0.395588
$698$	0	0
$699$	−72.6568	−2.74813
$700$	0	0
$701$	0.379084	0.0143178	0.00715891	−	0.999974i	$-0.497721\pi$
$701$	0.379084	0.0143178	0.00715891	+	0.999974i	$0.497721\pi$
$702$	0	0
$703$	−0.567864	−0.0214174
$704$	0	0
$705$	−111.609	−4.20346
$706$	0	0
$707$	15.3348	0.576723
$708$	0	0
$709$	33.9045	1.27331	0.636656	−	0.771148i	$-0.280317\pi$
$709$	33.9045	1.27331	0.636656	+	0.771148i	$0.280317\pi$
$710$	0	0
$711$	13.0151	0.488106
$712$	0	0
$713$	48.2293	1.80620
$714$	0	0
$715$	−83.1748	−3.11056
$716$	0	0
$717$	−1.06137	−0.0396377
$718$	0	0
$719$	1.40800	0.0525096	0.0262548	−	0.999655i	$-0.491642\pi$
$719$	1.40800	0.0525096	0.0262548	+	0.999655i	$0.491642\pi$
$720$	0	0
$721$	7.60336	0.283164
$722$	0	0
$723$	55.1273	2.05021
$724$	0	0
$725$	35.8930	1.33303
$726$	0	0
$727$	−0.794811	−0.0294779	−0.0147390	−	0.999891i	$-0.504692\pi$
$727$	−0.794811	−0.0294779	−0.0147390	+	0.999891i	$0.504692\pi$
$728$	0	0
$729$	−34.0664	−1.26172
$730$	0	0
$731$	5.47127	0.202362
$732$	0	0
$733$	13.2811	0.490548	0.245274	−	0.969454i	$-0.421122\pi$
$733$	13.2811	0.490548	0.245274	+	0.969454i	$0.421122\pi$
$734$	0	0
$735$	41.5674	1.53324
$736$	0	0
$737$	29.0247	1.06914
$738$	0	0
$739$	−37.6483	−1.38492	−0.692458	−	0.721458i	$-0.743472\pi$
$739$	−37.6483	−1.38492	−0.692458	+	0.721458i	$0.743472\pi$
$740$	0	0
$741$	8.99105	0.330294
$742$	0	0
$743$	13.0217	0.477721	0.238860	−	0.971054i	$-0.423226\pi$
$743$	13.0217	0.477721	0.238860	+	0.971054i	$0.423226\pi$
$744$	0	0
$745$	24.7664	0.907372
$746$	0	0
$747$	−66.1182	−2.41914
$748$	0	0
$749$	−6.34589	−0.231874
$750$	0	0
$751$	−1.91787	−0.0699842	−0.0349921	−	0.999388i	$-0.511141\pi$
$751$	−1.91787	−0.0699842	−0.0349921	+	0.999388i	$0.511141\pi$
$752$	0	0
$753$	5.62043	0.204820
$754$	0	0
$755$	57.4770	2.09180
$756$	0	0
$757$	−22.1913	−0.806556	−0.403278	−	0.915078i	$-0.632129\pi$
$757$	−22.1913	−0.806556	−0.403278	+	0.915078i	$0.632129\pi$
$758$	0	0
$759$	−109.794	−3.98526
$760$	0	0
$761$	6.25553	0.226763	0.113381	−	0.993552i	$-0.463832\pi$
$761$	6.25553	0.226763	0.113381	+	0.993552i	$0.463832\pi$
$762$	0	0
$763$	18.3046	0.662672
$764$	0	0
$765$	−125.370	−4.53277
$766$	0	0
$767$	55.4055	2.00058
$768$	0	0
$769$	−30.4720	−1.09885	−0.549424	−	0.835544i	$-0.685153\pi$
$769$	−30.4720	−1.09885	−0.549424	+	0.835544i	$0.685153\pi$
$770$	0	0
$771$	44.3002	1.59543
$772$	0	0
$773$	28.7185	1.03293	0.516467	−	0.856307i	$-0.327247\pi$
$773$	28.7185	1.03293	0.516467	+	0.856307i	$0.327247\pi$
$774$	0	0
$775$	29.9524	1.07592
$776$	0	0
$777$	5.14645	0.184628
$778$	0	0
$779$	−0.791416	−0.0283554
$780$	0	0
$781$	−52.7990	−1.88930
$782$	0	0
$783$	−55.2829	−1.97565
$784$	0	0
$785$	−55.2418	−1.97167
$786$	0	0
$787$	−42.2008	−1.50430	−0.752148	−	0.658994i	$-0.770982\pi$
$787$	−42.2008	−1.50430	−0.752148	+	0.658994i	$0.770982\pi$
$788$	0	0
$789$	−90.9384	−3.23749
$790$	0	0
$791$	3.79618	0.134976
$792$	0	0
$793$	−46.6787	−1.65761
$794$	0	0
$795$	43.2849	1.53516
$796$	0	0
$797$	54.7494	1.93932	0.969662	−	0.244449i	$-0.0786070\pi$
$797$	54.7494	1.93932	0.969662	+	0.244449i	$0.0786070\pi$
$798$	0	0
$799$	−81.8658	−2.89621
$800$	0	0
$801$	−8.22029	−0.290450
$802$	0	0
$803$	64.7356	2.28447
$804$	0	0
$805$	42.6294	1.50249
$806$	0	0
$807$	−88.9335	−3.13061
$808$	0	0
$809$	−38.9670	−1.37001	−0.685004	−	0.728540i	$-0.740199\pi$
$809$	−38.9670	−1.37001	−0.685004	+	0.728540i	$0.740199\pi$
$810$	0	0
$811$	−6.49059	−0.227915	−0.113958	−	0.993486i	$-0.536353\pi$
$811$	−6.49059	−0.227915	−0.113958	+	0.993486i	$0.536353\pi$
$812$	0	0
$813$	−83.4202	−2.92567
$814$	0	0
$815$	1.37437	0.0481421
$816$	0	0
$817$	−0.414604	−0.0145052
$818$	0	0
$819$	−53.3831	−1.86536
$820$	0	0
$821$	−37.1602	−1.29690	−0.648451	−	0.761257i	$-0.724583\pi$
$821$	−37.1602	−1.29690	−0.648451	+	0.761257i	$0.724583\pi$
$822$	0	0
$823$	25.1004	0.874946	0.437473	−	0.899232i	$-0.355874\pi$
$823$	25.1004	0.874946	0.437473	+	0.899232i	$0.355874\pi$
$824$	0	0
$825$	−68.1864	−2.37395
$826$	0	0
$827$	57.0288	1.98309	0.991543	−	0.129777i	$-0.0414262\pi$
$827$	57.0288	1.98309	0.991543	+	0.129777i	$0.0414262\pi$
$828$	0	0
$829$	−53.2748	−1.85031	−0.925156	−	0.379588i	$-0.876066\pi$
$829$	−53.2748	−1.85031	−0.925156	+	0.379588i	$0.876066\pi$
$830$	0	0
$831$	85.3994	2.96247
$832$	0	0
$833$	30.4898	1.05641
$834$	0	0
$835$	33.8215	1.17044
$836$	0	0
$837$	−46.1331	−1.59459
$838$	0	0
$839$	9.16578	0.316438	0.158219	−	0.987404i	$-0.449425\pi$
$839$	9.16578	0.316438	0.158219	+	0.987404i	$0.449425\pi$
$840$	0	0
$841$	19.2274	0.663012
$842$	0	0
$843$	−32.6371	−1.12408
$844$	0	0
$845$	66.9773	2.30409
$846$	0	0
$847$	14.4696	0.497181
$848$	0	0
$849$	−55.2175	−1.89506
$850$	0	0
$851$	−9.04017	−0.309893
$852$	0	0
$853$	−44.2542	−1.51523	−0.757617	−	0.652699i	$-0.773637\pi$
$853$	−44.2542	−1.51523	−0.757617	+	0.652699i	$0.773637\pi$
$854$	0	0
$855$	9.50036	0.324905
$856$	0	0
$857$	−18.9921	−0.648757	−0.324379	−	0.945927i	$-0.605155\pi$
$857$	−18.9921	−0.648757	−0.324379	+	0.945927i	$0.605155\pi$
$858$	0	0
$859$	−50.8342	−1.73444	−0.867220	−	0.497925i	$-0.834095\pi$
$859$	−50.8342	−1.73444	−0.867220	+	0.497925i	$0.834095\pi$
$860$	0	0
$861$	7.17246	0.244437
$862$	0	0
$863$	−43.6746	−1.48670	−0.743351	−	0.668902i	$-0.766765\pi$
$863$	−43.6746	−1.48670	−0.743351	+	0.668902i	$0.766765\pi$
$864$	0	0
$865$	6.51108	0.221384
$866$	0	0
$867$	−90.2273	−3.06428
$868$	0	0
$869$	10.2152	0.346527
$870$	0	0
$871$	−37.8384	−1.28210
$872$	0	0
$873$	−18.8795	−0.638974
$874$	0	0
$875$	0.863015	0.0291752
$876$	0	0
$877$	−50.4919	−1.70499	−0.852495	−	0.522736i	$-0.824912\pi$
$877$	−50.4919	−1.70499	−0.852495	+	0.522736i	$0.824912\pi$
$878$	0	0
$879$	72.6328	2.44984
$880$	0	0
$881$	5.18968	0.174845	0.0874225	−	0.996171i	$-0.472137\pi$
$881$	5.18968	0.174845	0.0874225	+	0.996171i	$0.472137\pi$
$882$	0	0
$883$	47.7367	1.60647	0.803233	−	0.595665i	$-0.203111\pi$
$883$	47.7367	1.60647	0.803233	+	0.595665i	$0.203111\pi$
$884$	0	0
$885$	89.3619	3.00387
$886$	0	0
$887$	4.39400	0.147536	0.0737681	−	0.997275i	$-0.476498\pi$
$887$	4.39400	0.147536	0.0737681	+	0.997275i	$0.476498\pi$
$888$	0	0
$889$	−5.70382	−0.191300
$890$	0	0
$891$	28.5463	0.956336
$892$	0	0
$893$	6.20366	0.207598
$894$	0	0
$895$	−4.98719	−0.166703
$896$	0	0
$897$	143.134	4.77910
$898$	0	0
$899$	40.2453	1.34226
$900$	0	0
$901$	31.7496	1.05773
$902$	0	0
$903$	3.75748	0.125041
$904$	0	0
$905$	−11.6993	−0.388899
$906$	0	0
$907$	−27.3076	−0.906734	−0.453367	−	0.891324i	$-0.649777\pi$
$907$	−27.3076	−0.906734	−0.453367	+	0.891324i	$0.649777\pi$
$908$	0	0
$909$	54.4052	1.80451
$910$	0	0
$911$	−31.9975	−1.06012	−0.530062	−	0.847959i	$-0.677831\pi$
$911$	−31.9975	−1.06012	−0.530062	+	0.847959i	$0.677831\pi$
$912$	0	0
$913$	−51.8942	−1.71745
$914$	0	0
$915$	−75.2867	−2.48890
$916$	0	0
$917$	24.7121	0.816065
$918$	0	0
$919$	−14.6874	−0.484491	−0.242246	−	0.970215i	$-0.577884\pi$
$919$	−14.6874	−0.484491	−0.242246	+	0.970215i	$0.577884\pi$
$920$	0	0
$921$	87.8855	2.89593
$922$	0	0
$923$	68.8320	2.26563
$924$	0	0
$925$	−5.61431	−0.184598
$926$	0	0
$927$	26.9755	0.885990
$928$	0	0
$929$	−45.6698	−1.49838	−0.749188	−	0.662357i	$-0.769556\pi$
$929$	−45.6698	−1.49838	−0.749188	+	0.662357i	$0.769556\pi$
$930$	0	0
$931$	−2.31047	−0.0757225
$932$	0	0
$933$	−12.8353	−0.420209
$934$	0	0
$935$	−98.3995	−3.21801
$936$	0	0
$937$	−35.9028	−1.17289	−0.586447	−	0.809987i	$-0.699474\pi$
$937$	−35.9028	−1.17289	−0.586447	+	0.809987i	$0.699474\pi$
$938$	0	0
$939$	−58.0456	−1.89425
$940$	0	0
$941$	38.6432	1.25973	0.629866	−	0.776704i	$-0.283110\pi$
$941$	38.6432	1.25973	0.629866	+	0.776704i	$0.283110\pi$
$942$	0	0
$943$	−12.5990	−0.410281
$944$	0	0
$945$	−40.7766	−1.32646
$946$	0	0
$947$	58.3581	1.89638	0.948191	−	0.317699i	$-0.102910\pi$
$947$	58.3581	1.89638	0.948191	+	0.317699i	$0.102910\pi$
$948$	0	0
$949$	−84.3933	−2.73952
$950$	0	0
$951$	79.8478	2.58924
$952$	0	0
$953$	−27.0813	−0.877250	−0.438625	−	0.898670i	$-0.644534\pi$
$953$	−27.0813	−0.877250	−0.438625	+	0.898670i	$0.644534\pi$
$954$	0	0
$955$	6.06908	0.196391
$956$	0	0
$957$	−91.6182	−2.96160
$958$	0	0
$959$	−27.6697	−0.893501
$960$	0	0
$961$	2.58432	0.0833651
$962$	0	0
$963$	−22.5142	−0.725509
$964$	0	0
$965$	33.5063	1.07861
$966$	0	0
$967$	5.61957	0.180713	0.0903567	−	0.995909i	$-0.471199\pi$
$967$	5.61957	0.180713	0.0903567	+	0.995909i	$0.471199\pi$
$968$	0	0
$969$	10.6368	0.341704
$970$	0	0
$971$	−0.839638	−0.0269453	−0.0134726	−	0.999909i	$-0.504289\pi$
$971$	−0.839638	−0.0269453	−0.0134726	+	0.999909i	$0.504289\pi$
$972$	0	0
$973$	−13.9030	−0.445711
$974$	0	0
$975$	88.8920	2.84682
$976$	0	0
$977$	2.18909	0.0700352	0.0350176	−	0.999387i	$-0.488851\pi$
$977$	2.18909	0.0700352	0.0350176	+	0.999387i	$0.488851\pi$
$978$	0	0
$979$	−6.45186	−0.206202
$980$	0	0
$981$	64.9417	2.07343
$982$	0	0
$983$	−28.3362	−0.903784	−0.451892	−	0.892073i	$-0.649251\pi$
$983$	−28.3362	−0.903784	−0.451892	+	0.892073i	$0.649251\pi$
$984$	0	0
$985$	44.1453	1.40659
$986$	0	0
$987$	−56.2227	−1.78959
$988$	0	0
$989$	−6.60033	−0.209878
$990$	0	0
$991$	−12.6410	−0.401554	−0.200777	−	0.979637i	$-0.564347\pi$
$991$	−12.6410	−0.401554	−0.200777	+	0.979637i	$0.564347\pi$
$992$	0	0
$993$	−63.1345	−2.00351
$994$	0	0
$995$	67.7797	2.14876
$996$	0	0
$997$	−30.6953	−0.972129	−0.486064	−	0.873923i	$-0.661568\pi$
$997$	−30.6953	−0.972129	−0.486064	+	0.873923i	$0.661568\pi$
$998$	0	0
$999$	8.64725	0.273587

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.2	✓	33
4.3	odd	2		8048.2.a.x.1.32		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.2	✓	33	1.1	even	1	trivial
8048.2.a.x.1.32		33	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)

\(f(q)\)	\(=\)	\(q-2.94941 q^{3} +3.18881 q^{5} +1.60634 q^{7} +5.69904 q^{9} +O(q^{10})\)	\(q-2.94941 q^{3} +3.18881 q^{5} +1.60634 q^{7} +5.69904 q^{9} +4.47301 q^{11} -5.83128 q^{13} -9.40510 q^{15} -6.89867 q^{17} +0.522770 q^{19} -4.73777 q^{21} +8.32229 q^{23} +5.16848 q^{25} -7.96057 q^{27} +6.94459 q^{29} +5.79520 q^{31} -13.1927 q^{33} +5.12232 q^{35} -1.08626 q^{37} +17.1989 q^{39} -1.51389 q^{41} -0.793091 q^{43} +18.1731 q^{45} +11.8669 q^{47} -4.41966 q^{49} +20.3470 q^{51} -4.60227 q^{53} +14.2635 q^{55} -1.54186 q^{57} -9.50143 q^{59} +8.00488 q^{61} +9.15461 q^{63} -18.5948 q^{65} +6.48885 q^{67} -24.5459 q^{69} -11.8039 q^{71} +14.4725 q^{73} -15.2440 q^{75} +7.18518 q^{77} +2.28374 q^{79} +6.38190 q^{81} -11.6017 q^{83} -21.9985 q^{85} -20.4825 q^{87} -1.44240 q^{89} -9.36704 q^{91} -17.0924 q^{93} +1.66701 q^{95} -3.31275 q^{97} +25.4918 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

Introduction

L-functions

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