LMFDB - Embedded newform 4016.2.a.m.1.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.22331 q^{3} +3.75978 q^{5} -2.77250 q^{7} +7.38975 q^{9} +O(q^{10})$	$q+3.22331 q^{3} +3.75978 q^{5} -2.77250 q^{7} +7.38975 q^{9} +3.10483 q^{11} +4.47755 q^{13} +12.1189 q^{15} +4.42932 q^{17} -7.35385 q^{19} -8.93662 q^{21} -6.95315 q^{23} +9.13592 q^{25} +14.1495 q^{27} -3.27157 q^{29} -1.68860 q^{31} +10.0078 q^{33} -10.4240 q^{35} -9.33250 q^{37} +14.4326 q^{39} +1.88359 q^{41} +12.6941 q^{43} +27.7838 q^{45} +6.26180 q^{47} +0.686736 q^{49} +14.2771 q^{51} -3.80745 q^{53} +11.6735 q^{55} -23.7038 q^{57} -12.4150 q^{59} -2.70326 q^{61} -20.4880 q^{63} +16.8346 q^{65} -10.3907 q^{67} -22.4122 q^{69} +1.87846 q^{71} -9.50644 q^{73} +29.4479 q^{75} -8.60812 q^{77} +10.6601 q^{79} +23.4391 q^{81} -6.37969 q^{83} +16.6533 q^{85} -10.5453 q^{87} +0.850231 q^{89} -12.4140 q^{91} -5.44289 q^{93} -27.6488 q^{95} -5.86178 q^{97} +22.9439 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9	$ 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) $ 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9 - 8 * q^11 + 8 * q^13 - 7 * q^15 + 19 * q^17 + 9 * q^19 + 9 * q^21 - 21 * q^23 + 65 * q^25 - 5 * q^27 + 10 * q^29 + 9 * q^31 + 34 * q^33 - 12 * q^35 + 11 * q^37 + 9 * q^39 + 35 * q^41 + 9 * q^43 + 29 * q^45 - 37 * q^47 + 77 * q^49 + 17 * q^51 + 38 * q^53 + 20 * q^55 + 51 * q^57 - 17 * q^59 - 22 * q^63 + 41 * q^65 - 9 * q^67 + 8 * q^69 - 13 * q^71 + 41 * q^73 - 25 * q^75 + 36 * q^77 + 36 * q^79 + 127 * q^81 - 29 * q^83 + 34 * q^85 - 10 * q^87 + 36 * q^89 + 6 * q^91 + 36 * q^93 - 25 * q^95 + 40 * q^97 - 19 * 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
$3$	3.22331	1.86098	0.930490	−	0.366317i	$-0.119381\pi$
$3$	3.22331	1.86098	0.930490	+	0.366317i	$0.119381\pi$
$4$	0	0
$5$	3.75978	1.68142	0.840712	−	0.541483i	$-0.182137\pi$
$5$	3.75978	1.68142	0.840712	+	0.541483i	$0.182137\pi$
$6$	0	0
$7$	−2.77250	−1.04791	−0.523953	−	0.851747i	$-0.675543\pi$
$7$	−2.77250	−1.04791	−0.523953	+	0.851747i	$0.675543\pi$
$8$	0	0
$9$	7.38975	2.46325
$10$	0	0
$11$	3.10483	0.936141	0.468070	−	0.883691i	$-0.344949\pi$
$11$	3.10483	0.936141	0.468070	+	0.883691i	$0.344949\pi$
$12$	0	0
$13$	4.47755	1.24185	0.620925	−	0.783870i	$-0.286757\pi$
$13$	4.47755	1.24185	0.620925	+	0.783870i	$0.286757\pi$
$14$	0	0
$15$	12.1189	3.12910
$16$	0	0
$17$	4.42932	1.07427	0.537134	−	0.843497i	$-0.319507\pi$
$17$	4.42932	1.07427	0.537134	+	0.843497i	$0.319507\pi$
$18$	0	0
$19$	−7.35385	−1.68709	−0.843545	−	0.537059i	$-0.819535\pi$
$19$	−7.35385	−1.68709	−0.843545	+	0.537059i	$0.819535\pi$
$20$	0	0
$21$	−8.93662	−1.95013
$22$	0	0
$23$	−6.95315	−1.44983	−0.724916	−	0.688837i	$-0.758122\pi$
$23$	−6.95315	−1.44983	−0.724916	+	0.688837i	$0.758122\pi$
$24$	0	0
$25$	9.13592	1.82718
$26$	0	0
$27$	14.1495	2.72308
$28$	0	0
$29$	−3.27157	−0.607515	−0.303758	−	0.952749i	$-0.598241\pi$
$29$	−3.27157	−0.607515	−0.303758	+	0.952749i	$0.598241\pi$
$30$	0	0
$31$	−1.68860	−0.303282	−0.151641	−	0.988436i	$-0.548456\pi$
$31$	−1.68860	−0.303282	−0.151641	+	0.988436i	$0.548456\pi$
$32$	0	0
$33$	10.0078	1.74214
$34$	0	0
$35$	−10.4240	−1.76197
$36$	0	0
$37$	−9.33250	−1.53425	−0.767127	−	0.641495i	$-0.778314\pi$
$37$	−9.33250	−1.53425	−0.767127	+	0.641495i	$0.778314\pi$
$38$	0	0
$39$	14.4326	2.31106
$40$	0	0
$41$	1.88359	0.294168	0.147084	−	0.989124i	$-0.453011\pi$
$41$	1.88359	0.294168	0.147084	+	0.989124i	$0.453011\pi$
$42$	0	0
$43$	12.6941	1.93583	0.967915	−	0.251278i	$-0.0808509\pi$
$43$	12.6941	1.93583	0.967915	+	0.251278i	$0.0808509\pi$
$44$	0	0
$45$	27.7838	4.14176
$46$	0	0
$47$	6.26180	0.913378	0.456689	−	0.889627i	$-0.349035\pi$
$47$	6.26180	0.913378	0.456689	+	0.889627i	$0.349035\pi$
$48$	0	0
$49$	0.686736	0.0981052
$50$	0	0
$51$	14.2771	1.99919
$52$	0	0
$53$	−3.80745	−0.522993	−0.261496	−	0.965204i	$-0.584216\pi$
$53$	−3.80745	−0.522993	−0.261496	+	0.965204i	$0.584216\pi$
$54$	0	0
$55$	11.6735	1.57405
$56$	0	0
$57$	−23.7038	−3.13964
$58$	0	0
$59$	−12.4150	−1.61630	−0.808149	−	0.588978i	$-0.799531\pi$
$59$	−12.4150	−1.61630	−0.808149	+	0.588978i	$0.799531\pi$
$60$	0	0
$61$	−2.70326	−0.346117	−0.173059	−	0.984912i	$-0.555365\pi$
$61$	−2.70326	−0.346117	−0.173059	+	0.984912i	$0.555365\pi$
$62$	0	0
$63$	−20.4880	−2.58125
$64$	0	0
$65$	16.8346	2.08808
$66$	0	0
$67$	−10.3907	−1.26943	−0.634715	−	0.772747i	$-0.718882\pi$
$67$	−10.3907	−1.26943	−0.634715	+	0.772747i	$0.718882\pi$
$68$	0	0
$69$	−22.4122	−2.69811
$70$	0	0
$71$	1.87846	0.222932	0.111466	−	0.993768i	$-0.464445\pi$
$71$	1.87846	0.222932	0.111466	+	0.993768i	$0.464445\pi$
$72$	0	0
$73$	−9.50644	−1.11265	−0.556323	−	0.830966i	$-0.687788\pi$
$73$	−9.50644	−1.11265	−0.556323	+	0.830966i	$0.687788\pi$
$74$	0	0
$75$	29.4479	3.40036
$76$	0	0
$77$	−8.60812	−0.980987
$78$	0	0
$79$	10.6601	1.19936	0.599679	−	0.800241i	$-0.295295\pi$
$79$	10.6601	1.19936	0.599679	+	0.800241i	$0.295295\pi$
$80$	0	0
$81$	23.4391	2.60435
$82$	0	0
$83$	−6.37969	−0.700262	−0.350131	−	0.936701i	$-0.613863\pi$
$83$	−6.37969	−0.700262	−0.350131	+	0.936701i	$0.613863\pi$
$84$	0	0
$85$	16.6533	1.80630
$86$	0	0
$87$	−10.5453	−1.13057
$88$	0	0
$89$	0.850231	0.0901244	0.0450622	−	0.998984i	$-0.485651\pi$
$89$	0.850231	0.0901244	0.0450622	+	0.998984i	$0.485651\pi$
$90$	0	0
$91$	−12.4140	−1.30134
$92$	0	0
$93$	−5.44289	−0.564401
$94$	0	0
$95$	−27.6488	−2.83671
$96$	0	0
$97$	−5.86178	−0.595173	−0.297587	−	0.954695i	$-0.596182\pi$
$97$	−5.86178	−0.595173	−0.297587	+	0.954695i	$0.596182\pi$
$98$	0	0
$99$	22.9439	2.30595
$100$	0	0
$101$	13.6852	1.36173	0.680864	−	0.732410i	$-0.261604\pi$
$101$	13.6852	1.36173	0.680864	+	0.732410i	$0.261604\pi$
$102$	0	0
$103$	−17.5993	−1.73411	−0.867055	−	0.498212i	$-0.833990\pi$
$103$	−17.5993	−1.73411	−0.867055	+	0.498212i	$0.833990\pi$
$104$	0	0
$105$	−33.5997	−3.27900
$106$	0	0
$107$	16.5494	1.59989	0.799945	−	0.600074i	$-0.204862\pi$
$107$	16.5494	1.59989	0.799945	+	0.600074i	$0.204862\pi$
$108$	0	0
$109$	−16.6137	−1.59130	−0.795650	−	0.605757i	$-0.792870\pi$
$109$	−16.6137	−1.59130	−0.795650	+	0.605757i	$0.792870\pi$
$110$	0	0
$111$	−30.0816	−2.85522
$112$	0	0
$113$	2.07454	0.195156	0.0975780	−	0.995228i	$-0.468890\pi$
$113$	2.07454	0.195156	0.0975780	+	0.995228i	$0.468890\pi$
$114$	0	0
$115$	−26.1423	−2.43778
$116$	0	0
$117$	33.0880	3.05899
$118$	0	0
$119$	−12.2803	−1.12573
$120$	0	0
$121$	−1.36005	−0.123641
$122$	0	0
$123$	6.07141	0.547441
$124$	0	0
$125$	15.5502	1.39085
$126$	0	0
$127$	−6.33345	−0.562003	−0.281001	−	0.959707i	$-0.590667\pi$
$127$	−6.33345	−0.562003	−0.281001	+	0.959707i	$0.590667\pi$
$128$	0	0
$129$	40.9170	3.60254
$130$	0	0
$131$	4.18929	0.366020	0.183010	−	0.983111i	$-0.441416\pi$
$131$	4.18929	0.366020	0.183010	+	0.983111i	$0.441416\pi$
$132$	0	0
$133$	20.3885	1.76791
$134$	0	0
$135$	53.1991	4.57865
$136$	0	0
$137$	16.3373	1.39579	0.697896	−	0.716199i	$-0.254120\pi$
$137$	16.3373	1.39579	0.697896	+	0.716199i	$0.254120\pi$
$138$	0	0
$139$	7.06378	0.599142	0.299571	−	0.954074i	$-0.403156\pi$
$139$	7.06378	0.599142	0.299571	+	0.954074i	$0.403156\pi$
$140$	0	0
$141$	20.1837	1.69978
$142$	0	0
$143$	13.9020	1.16255
$144$	0	0
$145$	−12.3004	−1.02149
$146$	0	0
$147$	2.21357	0.182572
$148$	0	0
$149$	−2.81803	−0.230862	−0.115431	−	0.993316i	$-0.536825\pi$
$149$	−2.81803	−0.230862	−0.115431	+	0.993316i	$0.536825\pi$
$150$	0	0
$151$	12.6036	1.02566	0.512832	−	0.858489i	$-0.328596\pi$
$151$	12.6036	1.02566	0.512832	+	0.858489i	$0.328596\pi$
$152$	0	0
$153$	32.7316	2.64619
$154$	0	0
$155$	−6.34876	−0.509945
$156$	0	0
$157$	3.91864	0.312741	0.156371	−	0.987698i	$-0.450021\pi$
$157$	3.91864	0.312741	0.156371	+	0.987698i	$0.450021\pi$
$158$	0	0
$159$	−12.2726	−0.973280
$160$	0	0
$161$	19.2776	1.51929
$162$	0	0
$163$	−21.3297	−1.67067	−0.835337	−	0.549738i	$-0.814727\pi$
$163$	−21.3297	−1.67067	−0.835337	+	0.549738i	$0.814727\pi$
$164$	0	0
$165$	37.6272	2.92927
$166$	0	0
$167$	−3.51502	−0.272000	−0.136000	−	0.990709i	$-0.543425\pi$
$167$	−3.51502	−0.272000	−0.136000	+	0.990709i	$0.543425\pi$
$168$	0	0
$169$	7.04849	0.542192
$170$	0	0
$171$	−54.3431	−4.15572
$172$	0	0
$173$	9.93184	0.755104	0.377552	−	0.925988i	$-0.376766\pi$
$173$	9.93184	0.755104	0.377552	+	0.925988i	$0.376766\pi$
$174$	0	0
$175$	−25.3293	−1.91472
$176$	0	0
$177$	−40.0175	−3.00790
$178$	0	0
$179$	23.1291	1.72875	0.864377	−	0.502844i	$-0.167713\pi$
$179$	23.1291	1.72875	0.864377	+	0.502844i	$0.167713\pi$
$180$	0	0
$181$	0.808312	0.0600814	0.0300407	−	0.999549i	$-0.490436\pi$
$181$	0.808312	0.0600814	0.0300407	+	0.999549i	$0.490436\pi$
$182$	0	0
$183$	−8.71346	−0.644117
$184$	0	0
$185$	−35.0881	−2.57973
$186$	0	0
$187$	13.7523	1.00567
$188$	0	0
$189$	−39.2295	−2.85353
$190$	0	0
$191$	2.59975	0.188111	0.0940557	−	0.995567i	$-0.470017\pi$
$191$	2.59975	0.188111	0.0940557	+	0.995567i	$0.470017\pi$
$192$	0	0
$193$	−1.15478	−0.0831231	−0.0415616	−	0.999136i	$-0.513233\pi$
$193$	−1.15478	−0.0831231	−0.0415616	+	0.999136i	$0.513233\pi$
$194$	0	0
$195$	54.2632	3.88587
$196$	0	0
$197$	2.82602	0.201346	0.100673	−	0.994920i	$-0.467900\pi$
$197$	2.82602	0.201346	0.100673	+	0.994920i	$0.467900\pi$
$198$	0	0
$199$	−14.7864	−1.04818	−0.524090	−	0.851663i	$-0.675594\pi$
$199$	−14.7864	−1.04818	−0.524090	+	0.851663i	$0.675594\pi$
$200$	0	0
$201$	−33.4926	−2.36238
$202$	0	0
$203$	9.07042	0.636618
$204$	0	0
$205$	7.08189	0.494621
$206$	0	0
$207$	−51.3820	−3.57130
$208$	0	0
$209$	−22.8324	−1.57935
$210$	0	0
$211$	12.1453	0.836120	0.418060	−	0.908419i	$-0.362710\pi$
$211$	12.1453	0.836120	0.418060	+	0.908419i	$0.362710\pi$
$212$	0	0
$213$	6.05486	0.414872
$214$	0	0
$215$	47.7269	3.25495
$216$	0	0
$217$	4.68164	0.317810
$218$	0	0
$219$	−30.6422	−2.07061
$220$	0	0
$221$	19.8325	1.33408
$222$	0	0
$223$	2.85543	0.191214	0.0956070	−	0.995419i	$-0.469521\pi$
$223$	2.85543	0.191214	0.0956070	+	0.995419i	$0.469521\pi$
$224$	0	0
$225$	67.5122	4.50081
$226$	0	0
$227$	20.9269	1.38897	0.694484	−	0.719508i	$-0.255633\pi$
$227$	20.9269	1.38897	0.694484	+	0.719508i	$0.255633\pi$
$228$	0	0
$229$	−20.0644	−1.32589	−0.662946	−	0.748668i	$-0.730694\pi$
$229$	−20.0644	−1.32589	−0.662946	+	0.748668i	$0.730694\pi$
$230$	0	0
$231$	−27.7467	−1.82560
$232$	0	0
$233$	−10.2478	−0.671358	−0.335679	−	0.941976i	$-0.608966\pi$
$233$	−10.2478	−0.671358	−0.335679	+	0.941976i	$0.608966\pi$
$234$	0	0
$235$	23.5430	1.53577
$236$	0	0
$237$	34.3609	2.23198
$238$	0	0
$239$	−18.8365	−1.21843	−0.609215	−	0.793005i	$-0.708516\pi$
$239$	−18.8365	−1.21843	−0.609215	+	0.793005i	$0.708516\pi$
$240$	0	0
$241$	−2.45668	−0.158248	−0.0791242	−	0.996865i	$-0.525212\pi$
$241$	−2.45668	−0.158248	−0.0791242	+	0.996865i	$0.525212\pi$
$242$	0	0
$243$	33.1030	2.12356
$244$	0	0
$245$	2.58198	0.164956
$246$	0	0
$247$	−32.9273	−2.09511
$248$	0	0
$249$	−20.5637	−1.30317
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−21.5883	−1.35725
$254$	0	0
$255$	53.6787	3.36149
$256$	0	0
$257$	15.7369	0.981643	0.490821	−	0.871260i	$-0.336697\pi$
$257$	15.7369	0.981643	0.490821	+	0.871260i	$0.336697\pi$
$258$	0	0
$259$	25.8743	1.60775
$260$	0	0
$261$	−24.1761	−1.49646
$262$	0	0
$263$	−11.8527	−0.730867	−0.365433	−	0.930838i	$-0.619079\pi$
$263$	−11.8527	−0.730867	−0.365433	+	0.930838i	$0.619079\pi$
$264$	0	0
$265$	−14.3152	−0.879373
$266$	0	0
$267$	2.74056	0.167720
$268$	0	0
$269$	−11.6604	−0.710950	−0.355475	−	0.934686i	$-0.615681\pi$
$269$	−11.6604	−0.710950	−0.355475	+	0.934686i	$0.615681\pi$
$270$	0	0
$271$	−17.2990	−1.05084	−0.525419	−	0.850844i	$-0.676091\pi$
$271$	−17.2990	−1.05084	−0.525419	+	0.850844i	$0.676091\pi$
$272$	0	0
$273$	−40.0142	−2.42177
$274$	0	0
$275$	28.3655	1.71050
$276$	0	0
$277$	17.3802	1.04427	0.522137	−	0.852862i	$-0.325135\pi$
$277$	17.3802	1.04427	0.522137	+	0.852862i	$0.325135\pi$
$278$	0	0
$279$	−12.4783	−0.747058
$280$	0	0
$281$	9.52419	0.568165	0.284083	−	0.958800i	$-0.408311\pi$
$281$	9.52419	0.568165	0.284083	+	0.958800i	$0.408311\pi$
$282$	0	0
$283$	8.33173	0.495270	0.247635	−	0.968853i	$-0.420347\pi$
$283$	8.33173	0.495270	0.247635	+	0.968853i	$0.420347\pi$
$284$	0	0
$285$	−89.1209	−5.27907
$286$	0	0
$287$	−5.22226	−0.308260
$288$	0	0
$289$	2.61889	0.154052
$290$	0	0
$291$	−18.8943	−1.10761
$292$	0	0
$293$	0.887628	0.0518558	0.0259279	−	0.999664i	$-0.491746\pi$
$293$	0.887628	0.0518558	0.0259279	+	0.999664i	$0.491746\pi$
$294$	0	0
$295$	−46.6777	−2.71768
$296$	0	0
$297$	43.9318	2.54918
$298$	0	0
$299$	−31.1331	−1.80047
$300$	0	0
$301$	−35.1943	−2.02857
$302$	0	0
$303$	44.1117	2.53415
$304$	0	0
$305$	−10.1637	−0.581970
$306$	0	0
$307$	11.9827	0.683890	0.341945	−	0.939720i	$-0.388914\pi$
$307$	11.9827	0.683890	0.341945	+	0.939720i	$0.388914\pi$
$308$	0	0
$309$	−56.7280	−3.22715
$310$	0	0
$311$	1.59452	0.0904167	0.0452084	−	0.998978i	$-0.485605\pi$
$311$	1.59452	0.0904167	0.0452084	+	0.998978i	$0.485605\pi$
$312$	0	0
$313$	13.3501	0.754590	0.377295	−	0.926093i	$-0.376854\pi$
$313$	13.3501	0.754590	0.377295	+	0.926093i	$0.376854\pi$
$314$	0	0
$315$	−77.0305	−4.34018
$316$	0	0
$317$	18.3151	1.02868	0.514338	−	0.857588i	$-0.328038\pi$
$317$	18.3151	1.02868	0.514338	+	0.857588i	$0.328038\pi$
$318$	0	0
$319$	−10.1577	−0.568720
$320$	0	0
$321$	53.3438	2.97736
$322$	0	0
$323$	−32.5726	−1.81239
$324$	0	0
$325$	40.9066	2.26909
$326$	0	0
$327$	−53.5510	−2.96138
$328$	0	0
$329$	−17.3608	−0.957133
$330$	0	0
$331$	9.22503	0.507053	0.253527	−	0.967328i	$-0.418409\pi$
$331$	9.22503	0.507053	0.253527	+	0.967328i	$0.418409\pi$
$332$	0	0
$333$	−68.9648	−3.77925
$334$	0	0
$335$	−39.0668	−2.13445
$336$	0	0
$337$	13.6737	0.744853	0.372427	−	0.928062i	$-0.378526\pi$
$337$	13.6737	0.744853	0.372427	+	0.928062i	$0.378526\pi$
$338$	0	0
$339$	6.68688	0.363182
$340$	0	0
$341$	−5.24282	−0.283914
$342$	0	0
$343$	17.5035	0.945100
$344$	0	0
$345$	−84.2648	−4.53667
$346$	0	0
$347$	14.9536	0.802749	0.401374	−	0.915914i	$-0.368533\pi$
$347$	14.9536	0.802749	0.401374	+	0.915914i	$0.368533\pi$
$348$	0	0
$349$	−20.0471	−1.07310	−0.536549	−	0.843869i	$-0.680272\pi$
$349$	−20.0471	−1.07310	−0.536549	+	0.843869i	$0.680272\pi$
$350$	0	0
$351$	63.3553	3.38165
$352$	0	0
$353$	8.51713	0.453321	0.226661	−	0.973974i	$-0.427219\pi$
$353$	8.51713	0.453321	0.226661	+	0.973974i	$0.427219\pi$
$354$	0	0
$355$	7.06259	0.374843
$356$	0	0
$357$	−39.5832	−2.09496
$358$	0	0
$359$	21.8634	1.15391	0.576954	−	0.816777i	$-0.304241\pi$
$359$	21.8634	1.15391	0.576954	+	0.816777i	$0.304241\pi$
$360$	0	0
$361$	35.0792	1.84627
$362$	0	0
$363$	−4.38386	−0.230093
$364$	0	0
$365$	−35.7421	−1.87083
$366$	0	0
$367$	24.0931	1.25765	0.628825	−	0.777547i	$-0.283536\pi$
$367$	24.0931	1.25765	0.628825	+	0.777547i	$0.283536\pi$
$368$	0	0
$369$	13.9193	0.724609
$370$	0	0
$371$	10.5561	0.548047
$372$	0	0
$373$	−22.2178	−1.15040	−0.575198	−	0.818015i	$-0.695075\pi$
$373$	−22.2178	−1.15040	−0.575198	+	0.818015i	$0.695075\pi$
$374$	0	0
$375$	50.1230	2.58834
$376$	0	0
$377$	−14.6486	−0.754443
$378$	0	0
$379$	−11.6686	−0.599375	−0.299688	−	0.954037i	$-0.596882\pi$
$379$	−11.6686	−0.599375	−0.299688	+	0.954037i	$0.596882\pi$
$380$	0	0
$381$	−20.4147	−1.04588
$382$	0	0
$383$	−8.85833	−0.452639	−0.226320	−	0.974053i	$-0.572669\pi$
$383$	−8.85833	−0.452639	−0.226320	+	0.974053i	$0.572669\pi$
$384$	0	0
$385$	−32.3646	−1.64945
$386$	0	0
$387$	93.8061	4.76843
$388$	0	0
$389$	8.94080	0.453317	0.226658	−	0.973974i	$-0.427220\pi$
$389$	8.94080	0.453317	0.226658	+	0.973974i	$0.427220\pi$
$390$	0	0
$391$	−30.7977	−1.55751
$392$	0	0
$393$	13.5034	0.681156
$394$	0	0
$395$	40.0797	2.01663
$396$	0	0
$397$	−13.7562	−0.690406	−0.345203	−	0.938528i	$-0.612190\pi$
$397$	−13.7562	−0.690406	−0.345203	+	0.938528i	$0.612190\pi$
$398$	0	0
$399$	65.7186	3.29005
$400$	0	0
$401$	−11.6690	−0.582721	−0.291361	−	0.956613i	$-0.594108\pi$
$401$	−11.6690	−0.582721	−0.291361	+	0.956613i	$0.594108\pi$
$402$	0	0
$403$	−7.56080	−0.376630
$404$	0	0
$405$	88.1258	4.37901
$406$	0	0
$407$	−28.9758	−1.43628
$408$	0	0
$409$	−4.31748	−0.213486	−0.106743	−	0.994287i	$-0.534042\pi$
$409$	−4.31748	−0.213486	−0.106743	+	0.994287i	$0.534042\pi$
$410$	0	0
$411$	52.6603	2.59754
$412$	0	0
$413$	34.4206	1.69373
$414$	0	0
$415$	−23.9862	−1.17744
$416$	0	0
$417$	22.7688	1.11499
$418$	0	0
$419$	−3.36574	−0.164427	−0.0822136	−	0.996615i	$-0.526199\pi$
$419$	−3.36574	−0.164427	−0.0822136	+	0.996615i	$0.526199\pi$
$420$	0	0
$421$	−24.9856	−1.21773	−0.608863	−	0.793276i	$-0.708374\pi$
$421$	−24.9856	−1.21773	−0.608863	+	0.793276i	$0.708374\pi$
$422$	0	0
$423$	46.2731	2.24988
$424$	0	0
$425$	40.4659	1.96289
$426$	0	0
$427$	7.49478	0.362698
$428$	0	0
$429$	44.8106	2.16348
$430$	0	0
$431$	34.0224	1.63880	0.819402	−	0.573220i	$-0.194306\pi$
$431$	34.0224	1.63880	0.819402	+	0.573220i	$0.194306\pi$
$432$	0	0
$433$	−12.6317	−0.607040	−0.303520	−	0.952825i	$-0.598162\pi$
$433$	−12.6317	−0.607040	−0.303520	+	0.952825i	$0.598162\pi$
$434$	0	0
$435$	−39.6480	−1.90097
$436$	0	0
$437$	51.1325	2.44600
$438$	0	0
$439$	9.35109	0.446303	0.223152	−	0.974784i	$-0.428365\pi$
$439$	9.35109	0.446303	0.223152	+	0.974784i	$0.428365\pi$
$440$	0	0
$441$	5.07481	0.241657
$442$	0	0
$443$	0.130665	0.00620809	0.00310404	−	0.999995i	$-0.499012\pi$
$443$	0.130665	0.00620809	0.00310404	+	0.999995i	$0.499012\pi$
$444$	0	0
$445$	3.19668	0.151537
$446$	0	0
$447$	−9.08338	−0.429629
$448$	0	0
$449$	−30.4939	−1.43910	−0.719548	−	0.694443i	$-0.755651\pi$
$449$	−30.4939	−1.43910	−0.719548	+	0.694443i	$0.755651\pi$
$450$	0	0
$451$	5.84823	0.275382
$452$	0	0
$453$	40.6252	1.90874
$454$	0	0
$455$	−46.6739	−2.18811
$456$	0	0
$457$	21.8644	1.02277	0.511386	−	0.859351i	$-0.329132\pi$
$457$	21.8644	1.02277	0.511386	+	0.859351i	$0.329132\pi$
$458$	0	0
$459$	62.6728	2.92532
$460$	0	0
$461$	18.0982	0.842916	0.421458	−	0.906848i	$-0.361518\pi$
$461$	18.0982	0.842916	0.421458	+	0.906848i	$0.361518\pi$
$462$	0	0
$463$	−5.13962	−0.238859	−0.119429	−	0.992843i	$-0.538106\pi$
$463$	−5.13962	−0.238859	−0.119429	+	0.992843i	$0.538106\pi$
$464$	0	0
$465$	−20.4641	−0.948998
$466$	0	0
$467$	−37.2312	−1.72285	−0.861427	−	0.507881i	$-0.830429\pi$
$467$	−37.2312	−1.72285	−0.861427	+	0.507881i	$0.830429\pi$
$468$	0	0
$469$	28.8082	1.33024
$470$	0	0
$471$	12.6310	0.582005
$472$	0	0
$473$	39.4129	1.81221
$474$	0	0
$475$	−67.1842	−3.08262
$476$	0	0
$477$	−28.1361	−1.28826
$478$	0	0
$479$	−14.8097	−0.676673	−0.338336	−	0.941025i	$-0.609864\pi$
$479$	−14.8097	−0.676673	−0.338336	+	0.941025i	$0.609864\pi$
$480$	0	0
$481$	−41.7868	−1.90531
$482$	0	0
$483$	62.1377	2.82736
$484$	0	0
$485$	−22.0390	−1.00074
$486$	0	0
$487$	3.89280	0.176400	0.0881999	−	0.996103i	$-0.471889\pi$
$487$	3.89280	0.176400	0.0881999	+	0.996103i	$0.471889\pi$
$488$	0	0
$489$	−68.7524	−3.10909
$490$	0	0
$491$	8.69062	0.392202	0.196101	−	0.980584i	$-0.437172\pi$
$491$	8.69062	0.392202	0.196101	+	0.980584i	$0.437172\pi$
$492$	0	0
$493$	−14.4908	−0.652634
$494$	0	0
$495$	86.2639	3.87727
$496$	0	0
$497$	−5.20802	−0.233612
$498$	0	0
$499$	12.4014	0.555162	0.277581	−	0.960702i	$-0.410467\pi$
$499$	12.4014	0.555162	0.277581	+	0.960702i	$0.410467\pi$
$500$	0	0
$501$	−11.3300	−0.506187
$502$	0	0
$503$	−17.5673	−0.783288	−0.391644	−	0.920117i	$-0.628093\pi$
$503$	−17.5673	−0.783288	−0.391644	+	0.920117i	$0.628093\pi$
$504$	0	0
$505$	51.4533	2.28964
$506$	0	0
$507$	22.7195	1.00901
$508$	0	0
$509$	9.89580	0.438624	0.219312	−	0.975655i	$-0.429619\pi$
$509$	9.89580	0.438624	0.219312	+	0.975655i	$0.429619\pi$
$510$	0	0
$511$	26.3566	1.16595
$512$	0	0
$513$	−104.054	−4.59408
$514$	0	0
$515$	−66.1694	−2.91577
$516$	0	0
$517$	19.4418	0.855050
$518$	0	0
$519$	32.0134	1.40523
$520$	0	0
$521$	10.0674	0.441061	0.220530	−	0.975380i	$-0.429221\pi$
$521$	10.0674	0.441061	0.220530	+	0.975380i	$0.429221\pi$
$522$	0	0
$523$	−15.9541	−0.697625	−0.348813	−	0.937192i	$-0.613415\pi$
$523$	−15.9541	−0.697625	−0.348813	+	0.937192i	$0.613415\pi$
$524$	0	0
$525$	−81.6443	−3.56325
$526$	0	0
$527$	−7.47936	−0.325806
$528$	0	0
$529$	25.3463	1.10201
$530$	0	0
$531$	−91.7439	−3.98134
$532$	0	0
$533$	8.43389	0.365312
$534$	0	0
$535$	62.2220	2.69009
$536$	0	0
$537$	74.5525	3.21718
$538$	0	0
$539$	2.13220	0.0918403
$540$	0	0
$541$	30.1012	1.29415	0.647077	−	0.762425i	$-0.275991\pi$
$541$	30.1012	1.29415	0.647077	+	0.762425i	$0.275991\pi$
$542$	0	0
$543$	2.60544	0.111810
$544$	0	0
$545$	−62.4636	−2.67565
$546$	0	0
$547$	10.3752	0.443612	0.221806	−	0.975091i	$-0.428805\pi$
$547$	10.3752	0.443612	0.221806	+	0.975091i	$0.428805\pi$
$548$	0	0
$549$	−19.9764	−0.852573
$550$	0	0
$551$	24.0586	1.02493
$552$	0	0
$553$	−29.5552	−1.25681
$554$	0	0
$555$	−113.100	−4.80083
$556$	0	0
$557$	−2.72370	−0.115407	−0.0577035	−	0.998334i	$-0.518378\pi$
$557$	−2.72370	−0.115407	−0.0577035	+	0.998334i	$0.518378\pi$
$558$	0	0
$559$	56.8385	2.40401
$560$	0	0
$561$	44.3279	1.87153
$562$	0	0
$563$	4.58352	0.193172	0.0965861	−	0.995325i	$-0.469208\pi$
$563$	4.58352	0.193172	0.0965861	+	0.995325i	$0.469208\pi$
$564$	0	0
$565$	7.79980	0.328140
$566$	0	0
$567$	−64.9849	−2.72911
$568$	0	0
$569$	−17.9937	−0.754337	−0.377169	−	0.926145i	$-0.623102\pi$
$569$	−17.9937	−0.754337	−0.377169	+	0.926145i	$0.623102\pi$
$570$	0	0
$571$	−13.4526	−0.562974	−0.281487	−	0.959565i	$-0.590828\pi$
$571$	−13.4526	−0.562974	−0.281487	+	0.959565i	$0.590828\pi$
$572$	0	0
$573$	8.37981	0.350072
$574$	0	0
$575$	−63.5235	−2.64911
$576$	0	0
$577$	8.74774	0.364173	0.182086	−	0.983283i	$-0.441715\pi$
$577$	8.74774	0.364173	0.182086	+	0.983283i	$0.441715\pi$
$578$	0	0
$579$	−3.72223	−0.154691
$580$	0	0
$581$	17.6877	0.733808
$582$	0	0
$583$	−11.8215	−0.489595
$584$	0	0
$585$	124.403	5.14345
$586$	0	0
$587$	39.0164	1.61038	0.805189	−	0.593019i	$-0.202064\pi$
$587$	39.0164	1.61038	0.805189	+	0.593019i	$0.202064\pi$
$588$	0	0
$589$	12.4177	0.511663
$590$	0	0
$591$	9.10916	0.374701
$592$	0	0
$593$	−11.4472	−0.470081	−0.235041	−	0.971986i	$-0.575522\pi$
$593$	−11.4472	−0.470081	−0.235041	+	0.971986i	$0.575522\pi$
$594$	0	0
$595$	−46.1711	−1.89283
$596$	0	0
$597$	−47.6611	−1.95064
$598$	0	0
$599$	0.583741	0.0238510	0.0119255	−	0.999929i	$-0.496204\pi$
$599$	0.583741	0.0238510	0.0119255	+	0.999929i	$0.496204\pi$
$600$	0	0
$601$	−35.9613	−1.46689	−0.733446	−	0.679747i	$-0.762089\pi$
$601$	−35.9613	−1.46689	−0.733446	+	0.679747i	$0.762089\pi$
$602$	0	0
$603$	−76.7848	−3.12692
$604$	0	0
$605$	−5.11347	−0.207892
$606$	0	0
$607$	−32.6887	−1.32679	−0.663397	−	0.748268i	$-0.730886\pi$
$607$	−32.6887	−1.32679	−0.663397	+	0.748268i	$0.730886\pi$
$608$	0	0
$609$	29.2368	1.18473
$610$	0	0
$611$	28.0376	1.13428
$612$	0	0
$613$	−30.3872	−1.22733	−0.613664	−	0.789567i	$-0.710305\pi$
$613$	−30.3872	−1.22733	−0.613664	+	0.789567i	$0.710305\pi$
$614$	0	0
$615$	22.8272	0.920480
$616$	0	0
$617$	21.3003	0.857519	0.428759	−	0.903419i	$-0.358951\pi$
$617$	21.3003	0.857519	0.428759	+	0.903419i	$0.358951\pi$
$618$	0	0
$619$	−24.4120	−0.981202	−0.490601	−	0.871384i	$-0.663223\pi$
$619$	−24.4120	−0.981202	−0.490601	+	0.871384i	$0.663223\pi$
$620$	0	0
$621$	−98.3838	−3.94801
$622$	0	0
$623$	−2.35726	−0.0944418
$624$	0	0
$625$	12.7855	0.511420
$626$	0	0
$627$	−73.5961	−2.93915
$628$	0	0
$629$	−41.3367	−1.64820
$630$	0	0
$631$	−3.00910	−0.119790	−0.0598951	−	0.998205i	$-0.519077\pi$
$631$	−3.00910	−0.119790	−0.0598951	+	0.998205i	$0.519077\pi$
$632$	0	0
$633$	39.1482	1.55600
$634$	0	0
$635$	−23.8124	−0.944965
$636$	0	0
$637$	3.07490	0.121832
$638$	0	0
$639$	13.8813	0.549137
$640$	0	0
$641$	−9.80611	−0.387318	−0.193659	−	0.981069i	$-0.562036\pi$
$641$	−9.80611	−0.387318	−0.193659	+	0.981069i	$0.562036\pi$
$642$	0	0
$643$	4.96520	0.195808	0.0979041	−	0.995196i	$-0.468786\pi$
$643$	4.96520	0.195808	0.0979041	+	0.995196i	$0.468786\pi$
$644$	0	0
$645$	153.839	6.05740
$646$	0	0
$647$	−48.5818	−1.90995	−0.954973	−	0.296692i	$-0.904117\pi$
$647$	−48.5818	−1.90995	−0.954973	+	0.296692i	$0.904117\pi$
$648$	0	0
$649$	−38.5465	−1.51308
$650$	0	0
$651$	15.0904	0.591439
$652$	0	0
$653$	25.7049	1.00591	0.502955	−	0.864313i	$-0.332246\pi$
$653$	25.7049	1.00591	0.502955	+	0.864313i	$0.332246\pi$
$654$	0	0
$655$	15.7508	0.615434
$656$	0	0
$657$	−70.2502	−2.74072
$658$	0	0
$659$	−39.9697	−1.55700	−0.778499	−	0.627646i	$-0.784018\pi$
$659$	−39.9697	−1.55700	−0.778499	+	0.627646i	$0.784018\pi$
$660$	0	0
$661$	−12.9957	−0.505475	−0.252738	−	0.967535i	$-0.581331\pi$
$661$	−12.9957	−0.505475	−0.252738	+	0.967535i	$0.581331\pi$
$662$	0	0
$663$	63.9264	2.48270
$664$	0	0
$665$	76.6563	2.97261
$666$	0	0
$667$	22.7477	0.880795
$668$	0	0
$669$	9.20395	0.355845
$670$	0	0
$671$	−8.39316	−0.324014
$672$	0	0
$673$	11.3607	0.437921	0.218960	−	0.975734i	$-0.429733\pi$
$673$	11.3607	0.437921	0.218960	+	0.975734i	$0.429733\pi$
$674$	0	0
$675$	129.269	4.97557
$676$	0	0
$677$	35.2994	1.35667	0.678333	−	0.734754i	$-0.262703\pi$
$677$	35.2994	1.35667	0.678333	+	0.734754i	$0.262703\pi$
$678$	0	0
$679$	16.2518	0.623685
$680$	0	0
$681$	67.4540	2.58484
$682$	0	0
$683$	−25.4225	−0.972765	−0.486382	−	0.873746i	$-0.661684\pi$
$683$	−25.4225	−0.972765	−0.486382	+	0.873746i	$0.661684\pi$
$684$	0	0
$685$	61.4247	2.34692
$686$	0	0
$687$	−64.6738	−2.46746
$688$	0	0
$689$	−17.0480	−0.649479
$690$	0	0
$691$	35.7061	1.35832	0.679162	−	0.733989i	$-0.262344\pi$
$691$	35.7061	1.35832	0.679162	+	0.733989i	$0.262344\pi$
$692$	0	0
$693$	−63.6118	−2.41641
$694$	0	0
$695$	26.5583	1.00741
$696$	0	0
$697$	8.34304	0.316015
$698$	0	0
$699$	−33.0320	−1.24938
$700$	0	0
$701$	6.24282	0.235788	0.117894	−	0.993026i	$-0.462386\pi$
$701$	6.24282	0.235788	0.117894	+	0.993026i	$0.462386\pi$
$702$	0	0
$703$	68.6298	2.58842
$704$	0	0
$705$	75.8864	2.85805
$706$	0	0
$707$	−37.9422	−1.42696
$708$	0	0
$709$	3.69284	0.138688	0.0693438	−	0.997593i	$-0.477909\pi$
$709$	3.69284	0.138688	0.0693438	+	0.997593i	$0.477909\pi$
$710$	0	0
$711$	78.7756	2.95432
$712$	0	0
$713$	11.7411	0.439708
$714$	0	0
$715$	52.2685	1.95473
$716$	0	0
$717$	−60.7159	−2.26748
$718$	0	0
$719$	9.68705	0.361266	0.180633	−	0.983551i	$-0.442185\pi$
$719$	9.68705	0.361266	0.180633	+	0.983551i	$0.442185\pi$
$720$	0	0
$721$	48.7940	1.81718
$722$	0	0
$723$	−7.91864	−0.294497
$724$	0	0
$725$	−29.8888	−1.11004
$726$	0	0
$727$	−20.7493	−0.769550	−0.384775	−	0.923010i	$-0.625721\pi$
$727$	−20.7493	−0.769550	−0.384775	+	0.923010i	$0.625721\pi$
$728$	0	0
$729$	36.3841	1.34756
$730$	0	0
$731$	56.2262	2.07960
$732$	0	0
$733$	−28.7562	−1.06213	−0.531066	−	0.847330i	$-0.678209\pi$
$733$	−28.7562	−1.06213	−0.531066	+	0.847330i	$0.678209\pi$
$734$	0	0
$735$	8.32251	0.306981
$736$	0	0
$737$	−32.2614	−1.18836
$738$	0	0
$739$	−34.8692	−1.28268	−0.641342	−	0.767255i	$-0.721622\pi$
$739$	−34.8692	−1.28268	−0.641342	+	0.767255i	$0.721622\pi$
$740$	0	0
$741$	−106.135	−3.89896
$742$	0	0
$743$	0.118574	0.00435004	0.00217502	−	0.999998i	$-0.499308\pi$
$743$	0.118574	0.00435004	0.00217502	+	0.999998i	$0.499308\pi$
$744$	0	0
$745$	−10.5951	−0.388176
$746$	0	0
$747$	−47.1443	−1.72492
$748$	0	0
$749$	−45.8831	−1.67653
$750$	0	0
$751$	34.0536	1.24263	0.621317	−	0.783559i	$-0.286598\pi$
$751$	34.0536	1.24263	0.621317	+	0.783559i	$0.286598\pi$
$752$	0	0
$753$	3.22331	0.117464
$754$	0	0
$755$	47.3866	1.72457
$756$	0	0
$757$	−27.1626	−0.987242	−0.493621	−	0.869677i	$-0.664327\pi$
$757$	−27.1626	−0.987242	−0.493621	+	0.869677i	$0.664327\pi$
$758$	0	0
$759$	−69.5860	−2.52581
$760$	0	0
$761$	−51.8146	−1.87828	−0.939139	−	0.343538i	$-0.888374\pi$
$761$	−51.8146	−1.87828	−0.939139	+	0.343538i	$0.888374\pi$
$762$	0	0
$763$	46.0613	1.66753
$764$	0	0
$765$	123.063	4.44937
$766$	0	0
$767$	−55.5889	−2.00720
$768$	0	0
$769$	24.9906	0.901185	0.450593	−	0.892730i	$-0.351213\pi$
$769$	24.9906	0.901185	0.450593	+	0.892730i	$0.351213\pi$
$770$	0	0
$771$	50.7251	1.82682
$772$	0	0
$773$	10.9431	0.393597	0.196799	−	0.980444i	$-0.436945\pi$
$773$	10.9431	0.393597	0.196799	+	0.980444i	$0.436945\pi$
$774$	0	0
$775$	−15.4269	−0.554152
$776$	0	0
$777$	83.4011	2.99200
$778$	0	0
$779$	−13.8517	−0.496288
$780$	0	0
$781$	5.83229	0.208696
$782$	0	0
$783$	−46.2912	−1.65431
$784$	0	0
$785$	14.7332	0.525851
$786$	0	0
$787$	−51.5882	−1.83892	−0.919460	−	0.393184i	$-0.871374\pi$
$787$	−51.5882	−1.83892	−0.919460	+	0.393184i	$0.871374\pi$
$788$	0	0
$789$	−38.2049	−1.36013
$790$	0	0
$791$	−5.75165	−0.204505
$792$	0	0
$793$	−12.1040	−0.429826
$794$	0	0
$795$	−46.1422	−1.63650
$796$	0	0
$797$	−10.5218	−0.372702	−0.186351	−	0.982483i	$-0.559666\pi$
$797$	−10.5218	−0.372702	−0.186351	+	0.982483i	$0.559666\pi$
$798$	0	0
$799$	27.7355	0.981213
$800$	0	0
$801$	6.28300	0.221999
$802$	0	0
$803$	−29.5159	−1.04159
$804$	0	0
$805$	72.4794	2.55456
$806$	0	0
$807$	−37.5853	−1.32306
$808$	0	0
$809$	50.9378	1.79088	0.895440	−	0.445183i	$-0.146861\pi$
$809$	50.9378	1.79088	0.895440	+	0.445183i	$0.146861\pi$
$810$	0	0
$811$	34.0594	1.19599	0.597993	−	0.801501i	$-0.295965\pi$
$811$	34.0594	1.19599	0.597993	+	0.801501i	$0.295965\pi$
$812$	0	0
$813$	−55.7600	−1.95559
$814$	0	0
$815$	−80.1951	−2.80911
$816$	0	0
$817$	−93.3504	−3.26592
$818$	0	0
$819$	−91.7363	−3.20553
$820$	0	0
$821$	32.4208	1.13149	0.565747	−	0.824579i	$-0.308588\pi$
$821$	32.4208	1.13149	0.565747	+	0.824579i	$0.308588\pi$
$822$	0	0
$823$	38.5185	1.34267	0.671335	−	0.741154i	$-0.265721\pi$
$823$	38.5185	1.34267	0.671335	+	0.741154i	$0.265721\pi$
$824$	0	0
$825$	91.4308	3.18321
$826$	0	0
$827$	26.7278	0.929417	0.464708	−	0.885464i	$-0.346159\pi$
$827$	26.7278	0.929417	0.464708	+	0.885464i	$0.346159\pi$
$828$	0	0
$829$	−11.7868	−0.409373	−0.204687	−	0.978828i	$-0.565618\pi$
$829$	−11.7868	−0.409373	−0.204687	+	0.978828i	$0.565618\pi$
$830$	0	0
$831$	56.0218	1.94337
$832$	0	0
$833$	3.04178	0.105391
$834$	0	0
$835$	−13.2157	−0.457347
$836$	0	0
$837$	−23.8929	−0.825860
$838$	0	0
$839$	46.5868	1.60836	0.804178	−	0.594388i	$-0.202606\pi$
$839$	46.5868	1.60836	0.804178	+	0.594388i	$0.202606\pi$
$840$	0	0
$841$	−18.2968	−0.630925
$842$	0	0
$843$	30.6994	1.05734
$844$	0	0
$845$	26.5008	0.911654
$846$	0	0
$847$	3.77072	0.129564
$848$	0	0
$849$	26.8558	0.921687
$850$	0	0
$851$	64.8903	2.22441
$852$	0	0
$853$	23.2073	0.794602	0.397301	−	0.917688i	$-0.369947\pi$
$853$	23.2073	0.794602	0.397301	+	0.917688i	$0.369947\pi$
$854$	0	0
$855$	−204.318	−6.98753
$856$	0	0
$857$	−24.8878	−0.850150	−0.425075	−	0.905158i	$-0.639752\pi$
$857$	−24.8878	−0.850150	−0.425075	+	0.905158i	$0.639752\pi$
$858$	0	0
$859$	−1.52674	−0.0520918	−0.0260459	−	0.999661i	$-0.508292\pi$
$859$	−1.52674	−0.0520918	−0.0260459	+	0.999661i	$0.508292\pi$
$860$	0	0
$861$	−16.8330	−0.573666
$862$	0	0
$863$	6.08038	0.206979	0.103489	−	0.994631i	$-0.466999\pi$
$863$	6.08038	0.206979	0.103489	+	0.994631i	$0.466999\pi$
$864$	0	0
$865$	37.3415	1.26965
$866$	0	0
$867$	8.44150	0.286688
$868$	0	0
$869$	33.0978	1.12277
$870$	0	0
$871$	−46.5250	−1.57644
$872$	0	0
$873$	−43.3171	−1.46606
$874$	0	0
$875$	−43.1127	−1.45748
$876$	0	0
$877$	−28.5567	−0.964293	−0.482146	−	0.876091i	$-0.660143\pi$
$877$	−28.5567	−0.964293	−0.482146	+	0.876091i	$0.660143\pi$
$878$	0	0
$879$	2.86110	0.0965026
$880$	0	0
$881$	−46.0754	−1.55232	−0.776160	−	0.630536i	$-0.782835\pi$
$881$	−46.0754	−1.55232	−0.776160	+	0.630536i	$0.782835\pi$
$882$	0	0
$883$	15.4676	0.520526	0.260263	−	0.965538i	$-0.416191\pi$
$883$	15.4676	0.520526	0.260263	+	0.965538i	$0.416191\pi$
$884$	0	0
$885$	−150.457	−5.05755
$886$	0	0
$887$	−9.81698	−0.329622	−0.164811	−	0.986325i	$-0.552701\pi$
$887$	−9.81698	−0.329622	−0.164811	+	0.986325i	$0.552701\pi$
$888$	0	0
$889$	17.5595	0.588926
$890$	0	0
$891$	72.7744	2.43803
$892$	0	0
$893$	−46.0484	−1.54095
$894$	0	0
$895$	86.9604	2.90677
$896$	0	0
$897$	−100.352	−3.35065
$898$	0	0
$899$	5.52438	0.184248
$900$	0	0
$901$	−16.8644	−0.561835
$902$	0	0
$903$	−113.442	−3.77512
$904$	0	0
$905$	3.03907	0.101022
$906$	0	0
$907$	44.0144	1.46147	0.730736	−	0.682660i	$-0.239177\pi$
$907$	44.0144	1.46147	0.730736	+	0.682660i	$0.239177\pi$
$908$	0	0
$909$	101.130	3.35428
$910$	0	0
$911$	45.5179	1.50807	0.754037	−	0.656832i	$-0.228104\pi$
$911$	45.5179	1.50807	0.754037	+	0.656832i	$0.228104\pi$
$912$	0	0
$913$	−19.8078	−0.655543
$914$	0	0
$915$	−32.7607	−1.08303
$916$	0	0
$917$	−11.6148	−0.383554
$918$	0	0
$919$	20.8385	0.687399	0.343699	−	0.939080i	$-0.388320\pi$
$919$	20.8385	0.687399	0.343699	+	0.939080i	$0.388320\pi$
$920$	0	0
$921$	38.6241	1.27271
$922$	0	0
$923$	8.41090	0.276848
$924$	0	0
$925$	−85.2610	−2.80337
$926$	0	0
$927$	−130.054	−4.27155
$928$	0	0
$929$	17.9436	0.588709	0.294355	−	0.955696i	$-0.404895\pi$
$929$	17.9436	0.588709	0.294355	+	0.955696i	$0.404895\pi$
$930$	0	0
$931$	−5.05016	−0.165512
$932$	0	0
$933$	5.13962	0.168264
$934$	0	0
$935$	51.7055	1.69095
$936$	0	0
$937$	33.7032	1.10104	0.550518	−	0.834823i	$-0.314430\pi$
$937$	33.7032	1.10104	0.550518	+	0.834823i	$0.314430\pi$
$938$	0	0
$939$	43.0314	1.40428
$940$	0	0
$941$	32.4041	1.05634	0.528171	−	0.849138i	$-0.322878\pi$
$941$	32.4041	1.05634	0.528171	+	0.849138i	$0.322878\pi$
$942$	0	0
$943$	−13.0969	−0.426494
$944$	0	0
$945$	−147.494	−4.79799
$946$	0	0
$947$	−47.3302	−1.53803	−0.769013	−	0.639233i	$-0.779252\pi$
$947$	−47.3302	−1.53803	−0.769013	+	0.639233i	$0.779252\pi$
$948$	0	0
$949$	−42.5656	−1.38174
$950$	0	0
$951$	59.0351	1.91435
$952$	0	0
$953$	26.0818	0.844872	0.422436	−	0.906393i	$-0.361175\pi$
$953$	26.0818	0.844872	0.422436	+	0.906393i	$0.361175\pi$
$954$	0	0
$955$	9.77449	0.316295
$956$	0	0
$957$	−32.7413	−1.05838
$958$	0	0
$959$	−45.2952	−1.46266
$960$	0	0
$961$	−28.1486	−0.908020
$962$	0	0
$963$	122.296	3.94092
$964$	0	0
$965$	−4.34173	−0.139765
$966$	0	0
$967$	6.72792	0.216355	0.108178	−	0.994132i	$-0.465498\pi$
$967$	6.72792	0.216355	0.108178	+	0.994132i	$0.465498\pi$
$968$	0	0
$969$	−104.992	−3.37282
$970$	0	0
$971$	0.627768	0.0201460	0.0100730	−	0.999949i	$-0.496794\pi$
$971$	0.627768	0.0201460	0.0100730	+	0.999949i	$0.496794\pi$
$972$	0	0
$973$	−19.5843	−0.627844
$974$	0	0
$975$	131.855	4.22273
$976$	0	0
$977$	43.6678	1.39706	0.698528	−	0.715583i	$-0.253839\pi$
$977$	43.6678	1.39706	0.698528	+	0.715583i	$0.253839\pi$
$978$	0	0
$979$	2.63982	0.0843691
$980$	0	0
$981$	−122.771	−3.91977
$982$	0	0
$983$	9.64944	0.307769	0.153885	−	0.988089i	$-0.450822\pi$
$983$	9.64944	0.307769	0.153885	+	0.988089i	$0.450822\pi$
$984$	0	0
$985$	10.6252	0.338548
$986$	0	0
$987$	−55.9594	−1.78121
$988$	0	0
$989$	−88.2639	−2.80663
$990$	0	0
$991$	5.24138	0.166498	0.0832489	−	0.996529i	$-0.473470\pi$
$991$	5.24138	0.166498	0.0832489	+	0.996529i	$0.473470\pi$
$992$	0	0
$993$	29.7351	0.943616
$994$	0	0
$995$	−55.5935	−1.76243
$996$	0	0
$997$	−11.7113	−0.370901	−0.185451	−	0.982654i	$-0.559374\pi$
$997$	−11.7113	−0.370901	−0.185451	+	0.982654i	$0.559374\pi$
$998$	0	0
$999$	−132.050	−4.17789

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.m.1.22		23
4.3	odd	2		2008.2.a.d.1.2	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.2	✓	23	4.3	odd	2
4016.2.a.m.1.22		23	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+3.22331 q^{3} +3.75978 q^{5} -2.77250 q^{7} +7.38975 q^{9} +O(q^{10})\)	\(q+3.22331 q^{3} +3.75978 q^{5} -2.77250 q^{7} +7.38975 q^{9} +3.10483 q^{11} +4.47755 q^{13} +12.1189 q^{15} +4.42932 q^{17} -7.35385 q^{19} -8.93662 q^{21} -6.95315 q^{23} +9.13592 q^{25} +14.1495 q^{27} -3.27157 q^{29} -1.68860 q^{31} +10.0078 q^{33} -10.4240 q^{35} -9.33250 q^{37} +14.4326 q^{39} +1.88359 q^{41} +12.6941 q^{43} +27.7838 q^{45} +6.26180 q^{47} +0.686736 q^{49} +14.2771 q^{51} -3.80745 q^{53} +11.6735 q^{55} -23.7038 q^{57} -12.4150 q^{59} -2.70326 q^{61} -20.4880 q^{63} +16.8346 q^{65} -10.3907 q^{67} -22.4122 q^{69} +1.87846 q^{71} -9.50644 q^{73} +29.4479 q^{75} -8.60812 q^{77} +10.6601 q^{79} +23.4391 q^{81} -6.37969 q^{83} +16.6533 q^{85} -10.5453 q^{87} +0.850231 q^{89} -12.4140 q^{91} -5.44289 q^{93} -27.6488 q^{95} -5.86178 q^{97} +22.9439 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) \) 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9 - 8 * q^11 + 8 * q^13 - 7 * q^15 + 19 * q^17 + 9 * q^19 + 9 * q^21 - 21 * q^23 + 65 * q^25 - 5 * q^27 + 10 * q^29 + 9 * q^31 + 34 * q^33 - 12 * q^35 + 11 * q^37 + 9 * q^39 + 35 * q^41 + 9 * q^43 + 29 * q^45 - 37 * q^47 + 77 * q^49 + 17 * q^51 + 38 * q^53 + 20 * q^55 + 51 * q^57 - 17 * q^59 - 22 * q^63 + 41 * q^65 - 9 * q^67 + 8 * q^69 - 13 * q^71 + 41 * q^73 - 25 * q^75 + 36 * q^77 + 36 * q^79 + 127 * q^81 - 29 * q^83 + 34 * q^85 - 10 * q^87 + 36 * q^89 + 6 * q^91 + 36 * q^93 - 25 * q^95 + 40 * q^97 - 19 * q^99

Introduction

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Database

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