LMFDB - Embedded newform 4016.2.a.m.1.20

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			1.20
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)

$f(q)$	$=$	$q+2.74485 q^{3} +2.14962 q^{5} +3.54966 q^{7} +4.53421 q^{9} +O(q^{10})$	$q+2.74485 q^{3} +2.14962 q^{5} +3.54966 q^{7} +4.53421 q^{9} -6.54587 q^{11} +5.92187 q^{13} +5.90040 q^{15} +3.02967 q^{17} +5.44528 q^{19} +9.74328 q^{21} +2.97810 q^{23} -0.379115 q^{25} +4.21117 q^{27} -7.08516 q^{29} +2.90565 q^{31} -17.9674 q^{33} +7.63043 q^{35} +5.80826 q^{37} +16.2547 q^{39} -8.06047 q^{41} -8.02892 q^{43} +9.74684 q^{45} -12.1568 q^{47} +5.60008 q^{49} +8.31599 q^{51} -4.41139 q^{53} -14.0712 q^{55} +14.9465 q^{57} -10.8456 q^{59} +3.39774 q^{61} +16.0949 q^{63} +12.7298 q^{65} -12.2601 q^{67} +8.17443 q^{69} +3.79662 q^{71} -5.43917 q^{73} -1.04061 q^{75} -23.2356 q^{77} +5.75747 q^{79} -2.04360 q^{81} +2.88983 q^{83} +6.51265 q^{85} -19.4477 q^{87} +17.2847 q^{89} +21.0206 q^{91} +7.97559 q^{93} +11.7053 q^{95} +10.2466 q^{97} -29.6803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.74485	1.58474	0.792370	−	0.610041i	$-0.208847\pi$
$3$	2.74485	1.58474	0.792370	+	0.610041i	$0.208847\pi$
$4$	0	0
$5$	2.14962	0.961341	0.480671	−	0.876901i	$-0.340393\pi$
$5$	2.14962	0.961341	0.480671	+	0.876901i	$0.340393\pi$
$6$	0	0
$7$	3.54966	1.34164	0.670822	−	0.741618i	$-0.265941\pi$
$7$	3.54966	1.34164	0.670822	+	0.741618i	$0.265941\pi$
$8$	0	0
$9$	4.53421	1.51140
$10$	0	0
$11$	−6.54587	−1.97365	−0.986827	−	0.161780i	$-0.948276\pi$
$11$	−6.54587	−1.97365	−0.986827	+	0.161780i	$0.948276\pi$
$12$	0	0
$13$	5.92187	1.64243	0.821216	−	0.570617i	$-0.193296\pi$
$13$	5.92187	1.64243	0.821216	+	0.570617i	$0.193296\pi$
$14$	0	0
$15$	5.90040	1.52348
$16$	0	0
$17$	3.02967	0.734803	0.367402	−	0.930062i	$-0.380247\pi$
$17$	3.02967	0.734803	0.367402	+	0.930062i	$0.380247\pi$
$18$	0	0
$19$	5.44528	1.24923	0.624616	−	0.780932i	$-0.285255\pi$
$19$	5.44528	1.24923	0.624616	+	0.780932i	$0.285255\pi$
$20$	0	0
$21$	9.74328	2.12616
$22$	0	0
$23$	2.97810	0.620976	0.310488	−	0.950577i	$-0.399508\pi$
$23$	2.97810	0.620976	0.310488	+	0.950577i	$0.399508\pi$
$24$	0	0
$25$	−0.379115	−0.0758230
$26$	0	0
$27$	4.21117	0.810439
$28$	0	0
$29$	−7.08516	−1.31568	−0.657841	−	0.753157i	$-0.728530\pi$
$29$	−7.08516	−1.31568	−0.657841	+	0.753157i	$0.728530\pi$
$30$	0	0
$31$	2.90565	0.521871	0.260935	−	0.965356i	$-0.415969\pi$
$31$	2.90565	0.521871	0.260935	+	0.965356i	$0.415969\pi$
$32$	0	0
$33$	−17.9674	−3.12773
$34$	0	0
$35$	7.63043	1.28978
$36$	0	0
$37$	5.80826	0.954872	0.477436	−	0.878666i	$-0.341566\pi$
$37$	5.80826	0.954872	0.477436	+	0.878666i	$0.341566\pi$
$38$	0	0
$39$	16.2547	2.60283
$40$	0	0
$41$	−8.06047	−1.25883	−0.629417	−	0.777068i	$-0.716706\pi$
$41$	−8.06047	−1.25883	−0.629417	+	0.777068i	$0.716706\pi$
$42$	0	0
$43$	−8.02892	−1.22440	−0.612200	−	0.790703i	$-0.709715\pi$
$43$	−8.02892	−1.22440	−0.612200	+	0.790703i	$0.709715\pi$
$44$	0	0
$45$	9.74684	1.45297
$46$	0	0
$47$	−12.1568	−1.77325	−0.886623	−	0.462494i	$-0.846955\pi$
$47$	−12.1568	−1.77325	−0.886623	+	0.462494i	$0.846955\pi$
$48$	0	0
$49$	5.60008	0.800011
$50$	0	0
$51$	8.31599	1.16447
$52$	0	0
$53$	−4.41139	−0.605952	−0.302976	−	0.952998i	$-0.597980\pi$
$53$	−4.41139	−0.605952	−0.302976	+	0.952998i	$0.597980\pi$
$54$	0	0
$55$	−14.0712	−1.89735
$56$	0	0
$57$	14.9465	1.97971
$58$	0	0
$59$	−10.8456	−1.41198	−0.705992	−	0.708220i	$-0.749498\pi$
$59$	−10.8456	−1.41198	−0.705992	+	0.708220i	$0.749498\pi$
$60$	0	0
$61$	3.39774	0.435036	0.217518	−	0.976056i	$-0.430204\pi$
$61$	3.39774	0.435036	0.217518	+	0.976056i	$0.430204\pi$
$62$	0	0
$63$	16.0949	2.02776
$64$	0	0
$65$	12.7298	1.57894
$66$	0	0
$67$	−12.2601	−1.49780	−0.748902	−	0.662681i	$-0.769419\pi$
$67$	−12.2601	−1.49780	−0.748902	+	0.662681i	$0.769419\pi$
$68$	0	0
$69$	8.17443	0.984086
$70$	0	0
$71$	3.79662	0.450576	0.225288	−	0.974292i	$-0.427668\pi$
$71$	3.79662	0.450576	0.225288	+	0.974292i	$0.427668\pi$
$72$	0	0
$73$	−5.43917	−0.636606	−0.318303	−	0.947989i	$-0.603113\pi$
$73$	−5.43917	−0.636606	−0.318303	+	0.947989i	$0.603113\pi$
$74$	0	0
$75$	−1.04061	−0.120160
$76$	0	0
$77$	−23.2356	−2.64794
$78$	0	0
$79$	5.75747	0.647766	0.323883	−	0.946097i	$-0.395012\pi$
$79$	5.75747	0.647766	0.323883	+	0.946097i	$0.395012\pi$
$80$	0	0
$81$	−2.04360	−0.227066
$82$	0	0
$83$	2.88983	0.317201	0.158600	−	0.987343i	$-0.449302\pi$
$83$	2.88983	0.317201	0.158600	+	0.987343i	$0.449302\pi$
$84$	0	0
$85$	6.51265	0.706396
$86$	0	0
$87$	−19.4477	−2.08501
$88$	0	0
$89$	17.2847	1.83218	0.916090	−	0.400973i	$-0.131328\pi$
$89$	17.2847	1.83218	0.916090	+	0.400973i	$0.131328\pi$
$90$	0	0
$91$	21.0206	2.20356
$92$	0	0
$93$	7.97559	0.827030
$94$	0	0
$95$	11.7053	1.20094
$96$	0	0
$97$	10.2466	1.04039	0.520194	−	0.854048i	$-0.325859\pi$
$97$	10.2466	1.04039	0.520194	+	0.854048i	$0.325859\pi$
$98$	0	0
$99$	−29.6803	−2.98298
$100$	0	0
$101$	−15.2411	−1.51654	−0.758271	−	0.651939i	$-0.773956\pi$
$101$	−15.2411	−1.51654	−0.758271	+	0.651939i	$0.773956\pi$
$102$	0	0
$103$	−4.10404	−0.404383	−0.202191	−	0.979346i	$-0.564806\pi$
$103$	−4.10404	−0.404383	−0.202191	+	0.979346i	$0.564806\pi$
$104$	0	0
$105$	20.9444	2.04396
$106$	0	0
$107$	4.91806	0.475447	0.237723	−	0.971333i	$-0.423599\pi$
$107$	4.91806	0.475447	0.237723	+	0.971333i	$0.423599\pi$
$108$	0	0
$109$	9.54452	0.914200	0.457100	−	0.889415i	$-0.348888\pi$
$109$	9.54452	0.914200	0.457100	+	0.889415i	$0.348888\pi$
$110$	0	0
$111$	15.9428	1.51322
$112$	0	0
$113$	7.77713	0.731611	0.365805	−	0.930691i	$-0.380794\pi$
$113$	7.77713	0.731611	0.365805	+	0.930691i	$0.380794\pi$
$114$	0	0
$115$	6.40179	0.596970
$116$	0	0
$117$	26.8510	2.48238
$118$	0	0
$119$	10.7543	0.985845
$120$	0	0
$121$	31.8484	2.89531
$122$	0	0
$123$	−22.1248	−1.99492
$124$	0	0
$125$	−11.5631	−1.03423
$126$	0	0
$127$	−14.2833	−1.26744	−0.633720	−	0.773562i	$-0.718473\pi$
$127$	−14.2833	−1.26744	−0.633720	+	0.773562i	$0.718473\pi$
$128$	0	0
$129$	−22.0382	−1.94036
$130$	0	0
$131$	−7.04115	−0.615188	−0.307594	−	0.951518i	$-0.599524\pi$
$131$	−7.04115	−0.615188	−0.307594	+	0.951518i	$0.599524\pi$
$132$	0	0
$133$	19.3289	1.67603
$134$	0	0
$135$	9.05242	0.779109
$136$	0	0
$137$	−0.449893	−0.0384370	−0.0192185	−	0.999815i	$-0.506118\pi$
$137$	−0.449893	−0.0384370	−0.0192185	+	0.999815i	$0.506118\pi$
$138$	0	0
$139$	3.85143	0.326674	0.163337	−	0.986570i	$-0.447774\pi$
$139$	3.85143	0.326674	0.163337	+	0.986570i	$0.447774\pi$
$140$	0	0
$141$	−33.3685	−2.81013
$142$	0	0
$143$	−38.7638	−3.24159
$144$	0	0
$145$	−15.2304	−1.26482
$146$	0	0
$147$	15.3714	1.26781
$148$	0	0
$149$	8.40916	0.688905	0.344453	−	0.938804i	$-0.388065\pi$
$149$	8.40916	0.688905	0.344453	+	0.938804i	$0.388065\pi$
$150$	0	0
$151$	−3.60603	−0.293454	−0.146727	−	0.989177i	$-0.546874\pi$
$151$	−3.60603	−0.293454	−0.146727	+	0.989177i	$0.546874\pi$
$152$	0	0
$153$	13.7371	1.11058
$154$	0	0
$155$	6.24606	0.501696
$156$	0	0
$157$	22.2108	1.77261	0.886306	−	0.463100i	$-0.153263\pi$
$157$	22.2108	1.77261	0.886306	+	0.463100i	$0.153263\pi$
$158$	0	0
$159$	−12.1086	−0.960276
$160$	0	0
$161$	10.5712	0.833129
$162$	0	0
$163$	12.1388	0.950781	0.475390	−	0.879775i	$-0.342307\pi$
$163$	12.1388	0.950781	0.475390	+	0.879775i	$0.342307\pi$
$164$	0	0
$165$	−38.6232	−3.00681
$166$	0	0
$167$	1.16030	0.0897865	0.0448932	−	0.998992i	$-0.485705\pi$
$167$	1.16030	0.0897865	0.0448932	+	0.998992i	$0.485705\pi$
$168$	0	0
$169$	22.0686	1.69758
$170$	0	0
$171$	24.6900	1.88809
$172$	0	0
$173$	−24.8853	−1.89199	−0.945997	−	0.324175i	$-0.894913\pi$
$173$	−24.8853	−1.89199	−0.945997	+	0.324175i	$0.894913\pi$
$174$	0	0
$175$	−1.34573	−0.101728
$176$	0	0
$177$	−29.7697	−2.23763
$178$	0	0
$179$	−18.2481	−1.36393	−0.681964	−	0.731386i	$-0.738874\pi$
$179$	−18.2481	−1.36393	−0.681964	+	0.731386i	$0.738874\pi$
$180$	0	0
$181$	−13.0683	−0.971361	−0.485681	−	0.874136i	$-0.661428\pi$
$181$	−13.0683	−0.971361	−0.485681	+	0.874136i	$0.661428\pi$
$182$	0	0
$183$	9.32629	0.689419
$184$	0	0
$185$	12.4856	0.917958
$186$	0	0
$187$	−19.8318	−1.45025
$188$	0	0
$189$	14.9482	1.08732
$190$	0	0
$191$	5.53780	0.400701	0.200351	−	0.979724i	$-0.435792\pi$
$191$	5.53780	0.400701	0.200351	+	0.979724i	$0.435792\pi$
$192$	0	0
$193$	−10.2677	−0.739086	−0.369543	−	0.929214i	$-0.620486\pi$
$193$	−10.2677	−0.739086	−0.369543	+	0.929214i	$0.620486\pi$
$194$	0	0
$195$	34.9414	2.50221
$196$	0	0
$197$	25.7190	1.83240	0.916202	−	0.400716i	$-0.131239\pi$
$197$	25.7190	1.83240	0.916202	+	0.400716i	$0.131239\pi$
$198$	0	0
$199$	−3.78676	−0.268436	−0.134218	−	0.990952i	$-0.542852\pi$
$199$	−3.78676	−0.268436	−0.134218	+	0.990952i	$0.542852\pi$
$200$	0	0
$201$	−33.6520	−2.37363
$202$	0	0
$203$	−25.1499	−1.76518
$204$	0	0
$205$	−17.3270	−1.21017
$206$	0	0
$207$	13.5033	0.938544
$208$	0	0
$209$	−35.6441	−2.46555
$210$	0	0
$211$	8.33696	0.573940	0.286970	−	0.957940i	$-0.407352\pi$
$211$	8.33696	0.573940	0.286970	+	0.957940i	$0.407352\pi$
$212$	0	0
$213$	10.4211	0.714045
$214$	0	0
$215$	−17.2592	−1.17707
$216$	0	0
$217$	10.3141	0.700165
$218$	0	0
$219$	−14.9297	−1.00886
$220$	0	0
$221$	17.9413	1.20686
$222$	0	0
$223$	−9.94350	−0.665866	−0.332933	−	0.942950i	$-0.608038\pi$
$223$	−9.94350	−0.665866	−0.332933	+	0.942950i	$0.608038\pi$
$224$	0	0
$225$	−1.71899	−0.114599
$226$	0	0
$227$	23.6972	1.57284	0.786418	−	0.617695i	$-0.211933\pi$
$227$	23.6972	1.57284	0.786418	+	0.617695i	$0.211933\pi$
$228$	0	0
$229$	4.54781	0.300528	0.150264	−	0.988646i	$-0.451988\pi$
$229$	4.54781	0.300528	0.150264	+	0.988646i	$0.451988\pi$
$230$	0	0
$231$	−63.7783	−4.19630
$232$	0	0
$233$	10.0120	0.655909	0.327955	−	0.944694i	$-0.393641\pi$
$233$	10.0120	0.655909	0.327955	+	0.944694i	$0.393641\pi$
$234$	0	0
$235$	−26.1325	−1.70469
$236$	0	0
$237$	15.8034	1.02654
$238$	0	0
$239$	−25.3052	−1.63685	−0.818427	−	0.574610i	$-0.805154\pi$
$239$	−25.3052	−1.63685	−0.818427	+	0.574610i	$0.805154\pi$
$240$	0	0
$241$	−16.7184	−1.07693	−0.538464	−	0.842649i	$-0.680995\pi$
$241$	−16.7184	−1.07693	−0.538464	+	0.842649i	$0.680995\pi$
$242$	0	0
$243$	−18.2429	−1.17028
$244$	0	0
$245$	12.0381	0.769084
$246$	0	0
$247$	32.2463	2.05178
$248$	0	0
$249$	7.93216	0.502680
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−19.4942	−1.22559
$254$	0	0
$255$	17.8763	1.11945
$256$	0	0
$257$	21.7416	1.35620	0.678101	−	0.734969i	$-0.262803\pi$
$257$	21.7416	1.35620	0.678101	+	0.734969i	$0.262803\pi$
$258$	0	0
$259$	20.6173	1.28110
$260$	0	0
$261$	−32.1256	−1.98852
$262$	0	0
$263$	1.27198	0.0784339	0.0392169	−	0.999231i	$-0.487514\pi$
$263$	1.27198	0.0784339	0.0392169	+	0.999231i	$0.487514\pi$
$264$	0	0
$265$	−9.48284	−0.582526
$266$	0	0
$267$	47.4441	2.90353
$268$	0	0
$269$	6.77658	0.413175	0.206588	−	0.978428i	$-0.433764\pi$
$269$	6.77658	0.413175	0.206588	+	0.978428i	$0.433764\pi$
$270$	0	0
$271$	−2.07518	−0.126058	−0.0630292	−	0.998012i	$-0.520076\pi$
$271$	−2.07518	−0.126058	−0.0630292	+	0.998012i	$0.520076\pi$
$272$	0	0
$273$	57.6985	3.49207
$274$	0	0
$275$	2.48164	0.149648
$276$	0	0
$277$	10.9588	0.658451	0.329226	−	0.944251i	$-0.393212\pi$
$277$	10.9588	0.658451	0.329226	+	0.944251i	$0.393212\pi$
$278$	0	0
$279$	13.1748	0.788757
$280$	0	0
$281$	2.68447	0.160142	0.0800711	−	0.996789i	$-0.474485\pi$
$281$	2.68447	0.160142	0.0800711	+	0.996789i	$0.474485\pi$
$282$	0	0
$283$	−9.10824	−0.541428	−0.270714	−	0.962660i	$-0.587260\pi$
$283$	−9.10824	−0.541428	−0.270714	+	0.962660i	$0.587260\pi$
$284$	0	0
$285$	32.1293	1.90318
$286$	0	0
$287$	−28.6119	−1.68891
$288$	0	0
$289$	−7.82110	−0.460065
$290$	0	0
$291$	28.1255	1.64875
$292$	0	0
$293$	−6.16669	−0.360262	−0.180131	−	0.983643i	$-0.557652\pi$
$293$	−6.16669	−0.360262	−0.180131	+	0.983643i	$0.557652\pi$
$294$	0	0
$295$	−23.3141	−1.35740
$296$	0	0
$297$	−27.5657	−1.59953
$298$	0	0
$299$	17.6359	1.01991
$300$	0	0
$301$	−28.4999	−1.64271
$302$	0	0
$303$	−41.8344	−2.40333
$304$	0	0
$305$	7.30387	0.418218
$306$	0	0
$307$	1.52823	0.0872205	0.0436102	−	0.999049i	$-0.486114\pi$
$307$	1.52823	0.0872205	0.0436102	+	0.999049i	$0.486114\pi$
$308$	0	0
$309$	−11.2650	−0.640842
$310$	0	0
$311$	−1.12444	−0.0637610	−0.0318805	−	0.999492i	$-0.510150\pi$
$311$	−1.12444	−0.0637610	−0.0318805	+	0.999492i	$0.510150\pi$
$312$	0	0
$313$	33.9524	1.91910	0.959550	−	0.281537i	$-0.0908441\pi$
$313$	33.9524	1.91910	0.959550	+	0.281537i	$0.0908441\pi$
$314$	0	0
$315$	34.5980	1.94937
$316$	0	0
$317$	11.9388	0.670551	0.335275	−	0.942120i	$-0.391171\pi$
$317$	11.9388	0.670551	0.335275	+	0.942120i	$0.391171\pi$
$318$	0	0
$319$	46.3785	2.59670
$320$	0	0
$321$	13.4993	0.753459
$322$	0	0
$323$	16.4974	0.917940
$324$	0	0
$325$	−2.24507	−0.124534
$326$	0	0
$327$	26.1983	1.44877
$328$	0	0
$329$	−43.1523	−2.37907
$330$	0	0
$331$	9.86021	0.541966	0.270983	−	0.962584i	$-0.412651\pi$
$331$	9.86021	0.541966	0.270983	+	0.962584i	$0.412651\pi$
$332$	0	0
$333$	26.3358	1.44320
$334$	0	0
$335$	−26.3545	−1.43990
$336$	0	0
$337$	11.8956	0.647996	0.323998	−	0.946058i	$-0.394973\pi$
$337$	11.8956	0.647996	0.323998	+	0.946058i	$0.394973\pi$
$338$	0	0
$339$	21.3471	1.15941
$340$	0	0
$341$	−19.0200	−1.02999
$342$	0	0
$343$	−4.96924	−0.268314
$344$	0	0
$345$	17.5720	0.946042
$346$	0	0
$347$	−11.6786	−0.626942	−0.313471	−	0.949598i	$-0.601492\pi$
$347$	−11.6786	−0.626942	−0.313471	+	0.949598i	$0.601492\pi$
$348$	0	0
$349$	−24.0249	−1.28602	−0.643012	−	0.765856i	$-0.722315\pi$
$349$	−24.0249	−1.28602	−0.643012	+	0.765856i	$0.722315\pi$
$350$	0	0
$351$	24.9380	1.33109
$352$	0	0
$353$	1.37513	0.0731906	0.0365953	−	0.999330i	$-0.488349\pi$
$353$	1.37513	0.0731906	0.0365953	+	0.999330i	$0.488349\pi$
$354$	0	0
$355$	8.16130	0.433157
$356$	0	0
$357$	29.5189	1.56231
$358$	0	0
$359$	32.1819	1.69850	0.849249	−	0.527993i	$-0.177055\pi$
$359$	32.1819	1.69850	0.849249	+	0.527993i	$0.177055\pi$
$360$	0	0
$361$	10.6511	0.560582
$362$	0	0
$363$	87.4191	4.58831
$364$	0	0
$365$	−11.6922	−0.611996
$366$	0	0
$367$	−8.98147	−0.468829	−0.234414	−	0.972137i	$-0.575317\pi$
$367$	−8.98147	−0.468829	−0.234414	+	0.972137i	$0.575317\pi$
$368$	0	0
$369$	−36.5478	−1.90260
$370$	0	0
$371$	−15.6589	−0.812972
$372$	0	0
$373$	−26.9056	−1.39312	−0.696559	−	0.717499i	$-0.745287\pi$
$373$	−26.9056	−1.39312	−0.696559	+	0.717499i	$0.745287\pi$
$374$	0	0
$375$	−31.7389	−1.63899
$376$	0	0
$377$	−41.9574	−2.16092
$378$	0	0
$379$	−10.2662	−0.527340	−0.263670	−	0.964613i	$-0.584933\pi$
$379$	−10.2662	−0.527340	−0.263670	+	0.964613i	$0.584933\pi$
$380$	0	0
$381$	−39.2056	−2.00856
$382$	0	0
$383$	25.9393	1.32544	0.662718	−	0.748869i	$-0.269403\pi$
$383$	25.9393	1.32544	0.662718	+	0.748869i	$0.269403\pi$
$384$	0	0
$385$	−49.9478	−2.54558
$386$	0	0
$387$	−36.4048	−1.85056
$388$	0	0
$389$	16.6478	0.844079	0.422039	−	0.906577i	$-0.361314\pi$
$389$	16.6478	0.844079	0.422039	+	0.906577i	$0.361314\pi$
$390$	0	0
$391$	9.02265	0.456295
$392$	0	0
$393$	−19.3269	−0.974914
$394$	0	0
$395$	12.3764	0.622724
$396$	0	0
$397$	−22.9366	−1.15116	−0.575578	−	0.817747i	$-0.695223\pi$
$397$	−22.9366	−1.15116	−0.575578	+	0.817747i	$0.695223\pi$
$398$	0	0
$399$	53.0549	2.65607
$400$	0	0
$401$	14.4812	0.723154	0.361577	−	0.932342i	$-0.382238\pi$
$401$	14.4812	0.723154	0.361577	+	0.932342i	$0.382238\pi$
$402$	0	0
$403$	17.2069	0.857138
$404$	0	0
$405$	−4.39296	−0.218288
$406$	0	0
$407$	−38.0201	−1.88459
$408$	0	0
$409$	−16.3102	−0.806486	−0.403243	−	0.915093i	$-0.632117\pi$
$409$	−16.3102	−0.806486	−0.403243	+	0.915093i	$0.632117\pi$
$410$	0	0
$411$	−1.23489	−0.0609126
$412$	0	0
$413$	−38.4984	−1.89438
$414$	0	0
$415$	6.21206	0.304938
$416$	0	0
$417$	10.5716	0.517693
$418$	0	0
$419$	2.69777	0.131795	0.0658973	−	0.997826i	$-0.479009\pi$
$419$	2.69777	0.131795	0.0658973	+	0.997826i	$0.479009\pi$
$420$	0	0
$421$	13.6445	0.664991	0.332495	−	0.943105i	$-0.392109\pi$
$421$	13.6445	0.664991	0.332495	+	0.943105i	$0.392109\pi$
$422$	0	0
$423$	−55.1212	−2.68009
$424$	0	0
$425$	−1.14859	−0.0557150
$426$	0	0
$427$	12.0608	0.583664
$428$	0	0
$429$	−106.401	−5.13708
$430$	0	0
$431$	0.166970	0.00804268	0.00402134	−	0.999992i	$-0.498720\pi$
$431$	0.166970	0.00804268	0.00402134	+	0.999992i	$0.498720\pi$
$432$	0	0
$433$	12.0726	0.580171	0.290085	−	0.957001i	$-0.406316\pi$
$433$	12.0726	0.580171	0.290085	+	0.957001i	$0.406316\pi$
$434$	0	0
$435$	−41.8053	−2.00441
$436$	0	0
$437$	16.2166	0.775744
$438$	0	0
$439$	−7.96878	−0.380329	−0.190165	−	0.981752i	$-0.560902\pi$
$439$	−7.96878	−0.380329	−0.190165	+	0.981752i	$0.560902\pi$
$440$	0	0
$441$	25.3919	1.20914
$442$	0	0
$443$	−12.3635	−0.587407	−0.293704	−	0.955897i	$-0.594888\pi$
$443$	−12.3635	−0.587407	−0.293704	+	0.955897i	$0.594888\pi$
$444$	0	0
$445$	37.1557	1.76135
$446$	0	0
$447$	23.0819	1.09174
$448$	0	0
$449$	3.54161	0.167139	0.0835694	−	0.996502i	$-0.473368\pi$
$449$	3.54161	0.167139	0.0835694	+	0.996502i	$0.473368\pi$
$450$	0	0
$451$	52.7628	2.48450
$452$	0	0
$453$	−9.89800	−0.465049
$454$	0	0
$455$	45.1865	2.11837
$456$	0	0
$457$	−18.9259	−0.885314	−0.442657	−	0.896691i	$-0.645964\pi$
$457$	−18.9259	−0.885314	−0.442657	+	0.896691i	$0.645964\pi$
$458$	0	0
$459$	12.7584	0.595513
$460$	0	0
$461$	17.2051	0.801320	0.400660	−	0.916227i	$-0.368781\pi$
$461$	17.2051	0.801320	0.400660	+	0.916227i	$0.368781\pi$
$462$	0	0
$463$	−16.2631	−0.755809	−0.377905	−	0.925844i	$-0.623355\pi$
$463$	−16.2631	−0.755809	−0.377905	+	0.925844i	$0.623355\pi$
$464$	0	0
$465$	17.1445	0.795058
$466$	0	0
$467$	6.23047	0.288312	0.144156	−	0.989555i	$-0.453953\pi$
$467$	6.23047	0.288312	0.144156	+	0.989555i	$0.453953\pi$
$468$	0	0
$469$	−43.5190	−2.00952
$470$	0	0
$471$	60.9652	2.80913
$472$	0	0
$473$	52.5563	2.41654
$474$	0	0
$475$	−2.06439	−0.0947206
$476$	0	0
$477$	−20.0022	−0.915837
$478$	0	0
$479$	−31.2103	−1.42603	−0.713017	−	0.701147i	$-0.752672\pi$
$479$	−31.2103	−1.42603	−0.713017	+	0.701147i	$0.752672\pi$
$480$	0	0
$481$	34.3958	1.56831
$482$	0	0
$483$	29.0164	1.32029
$484$	0	0
$485$	22.0264	1.00017
$486$	0	0
$487$	−14.6600	−0.664309	−0.332155	−	0.943225i	$-0.607776\pi$
$487$	−14.6600	−0.664309	−0.332155	+	0.943225i	$0.607776\pi$
$488$	0	0
$489$	33.3191	1.50674
$490$	0	0
$491$	−36.8141	−1.66140	−0.830699	−	0.556722i	$-0.812059\pi$
$491$	−36.8141	−1.66140	−0.830699	+	0.556722i	$0.812059\pi$
$492$	0	0
$493$	−21.4657	−0.966767
$494$	0	0
$495$	−63.8015	−2.86767
$496$	0	0
$497$	13.4767	0.604512
$498$	0	0
$499$	8.75326	0.391850	0.195925	−	0.980619i	$-0.437229\pi$
$499$	8.75326	0.391850	0.195925	+	0.980619i	$0.437229\pi$
$500$	0	0
$501$	3.18484	0.142288
$502$	0	0
$503$	28.1327	1.25437	0.627187	−	0.778869i	$-0.284206\pi$
$503$	28.1327	1.25437	0.627187	+	0.778869i	$0.284206\pi$
$504$	0	0
$505$	−32.7626	−1.45791
$506$	0	0
$507$	60.5750	2.69023
$508$	0	0
$509$	−22.3215	−0.989385	−0.494692	−	0.869068i	$-0.664719\pi$
$509$	−22.3215	−0.989385	−0.494692	+	0.869068i	$0.664719\pi$
$510$	0	0
$511$	−19.3072	−0.854100
$512$	0	0
$513$	22.9310	1.01243
$514$	0	0
$515$	−8.82214	−0.388750
$516$	0	0
$517$	79.5765	3.49977
$518$	0	0
$519$	−68.3064	−2.99832
$520$	0	0
$521$	−29.1435	−1.27680	−0.638399	−	0.769705i	$-0.720403\pi$
$521$	−29.1435	−1.27680	−0.638399	+	0.769705i	$0.720403\pi$
$522$	0	0
$523$	23.0289	1.00698	0.503491	−	0.864000i	$-0.332049\pi$
$523$	23.0289	1.00698	0.503491	+	0.864000i	$0.332049\pi$
$524$	0	0
$525$	−3.69383	−0.161212
$526$	0	0
$527$	8.80317	0.383472
$528$	0	0
$529$	−14.1309	−0.614389
$530$	0	0
$531$	−49.1764	−2.13407
$532$	0	0
$533$	−47.7331	−2.06755
$534$	0	0
$535$	10.5720	0.457066
$536$	0	0
$537$	−50.0883	−2.16147
$538$	0	0
$539$	−36.6574	−1.57894
$540$	0	0
$541$	−31.0582	−1.33530	−0.667649	−	0.744476i	$-0.732699\pi$
$541$	−31.0582	−1.33530	−0.667649	+	0.744476i	$0.732699\pi$
$542$	0	0
$543$	−35.8706	−1.53936
$544$	0	0
$545$	20.5171	0.878858
$546$	0	0
$547$	0.154505	0.00660614	0.00330307	−	0.999995i	$-0.498949\pi$
$547$	0.154505	0.00660614	0.00330307	+	0.999995i	$0.498949\pi$
$548$	0	0
$549$	15.4061	0.657515
$550$	0	0
$551$	−38.5807	−1.64359
$552$	0	0
$553$	20.4371	0.869072
$554$	0	0
$555$	34.2710	1.45472
$556$	0	0
$557$	39.0419	1.65426	0.827129	−	0.562012i	$-0.189973\pi$
$557$	39.0419	1.65426	0.827129	+	0.562012i	$0.189973\pi$
$558$	0	0
$559$	−47.5463	−2.01099
$560$	0	0
$561$	−54.4354	−2.29826
$562$	0	0
$563$	31.9413	1.34616	0.673082	−	0.739567i	$-0.264970\pi$
$563$	31.9413	1.34616	0.673082	+	0.739567i	$0.264970\pi$
$564$	0	0
$565$	16.7179	0.703327
$566$	0	0
$567$	−7.25407	−0.304642
$568$	0	0
$569$	0.966749	0.0405282	0.0202641	−	0.999795i	$-0.493549\pi$
$569$	0.966749	0.0405282	0.0202641	+	0.999795i	$0.493549\pi$
$570$	0	0
$571$	−3.10421	−0.129907	−0.0649535	−	0.997888i	$-0.520690\pi$
$571$	−3.10421	−0.129907	−0.0649535	+	0.997888i	$0.520690\pi$
$572$	0	0
$573$	15.2004	0.635007
$574$	0	0
$575$	−1.12904	−0.0470843
$576$	0	0
$577$	−43.6469	−1.81705	−0.908523	−	0.417835i	$-0.862789\pi$
$577$	−43.6469	−1.81705	−0.908523	+	0.417835i	$0.862789\pi$
$578$	0	0
$579$	−28.1833	−1.17126
$580$	0	0
$581$	10.2579	0.425571
$582$	0	0
$583$	28.8764	1.19594
$584$	0	0
$585$	57.7196	2.38641
$586$	0	0
$587$	37.7925	1.55986	0.779931	−	0.625865i	$-0.215254\pi$
$587$	37.7925	1.55986	0.779931	+	0.625865i	$0.215254\pi$
$588$	0	0
$589$	15.8221	0.651938
$590$	0	0
$591$	70.5949	2.90389
$592$	0	0
$593$	−16.4832	−0.676882	−0.338441	−	0.940988i	$-0.609900\pi$
$593$	−16.4832	−0.676882	−0.338441	+	0.940988i	$0.609900\pi$
$594$	0	0
$595$	23.1177	0.947733
$596$	0	0
$597$	−10.3941	−0.425401
$598$	0	0
$599$	8.27806	0.338232	0.169116	−	0.985596i	$-0.445909\pi$
$599$	8.27806	0.338232	0.169116	+	0.985596i	$0.445909\pi$
$600$	0	0
$601$	24.0477	0.980928	0.490464	−	0.871461i	$-0.336827\pi$
$601$	24.0477	0.980928	0.490464	+	0.871461i	$0.336827\pi$
$602$	0	0
$603$	−55.5896	−2.26378
$604$	0	0
$605$	68.4621	2.78338
$606$	0	0
$607$	12.1529	0.493271	0.246635	−	0.969108i	$-0.420675\pi$
$607$	12.1529	0.493271	0.246635	+	0.969108i	$0.420675\pi$
$608$	0	0
$609$	−69.0328	−2.79735
$610$	0	0
$611$	−71.9908	−2.91244
$612$	0	0
$613$	−12.4177	−0.501547	−0.250774	−	0.968046i	$-0.580685\pi$
$613$	−12.4177	−0.501547	−0.250774	+	0.968046i	$0.580685\pi$
$614$	0	0
$615$	−47.5600	−1.91780
$616$	0	0
$617$	44.3680	1.78619	0.893093	−	0.449871i	$-0.148530\pi$
$617$	44.3680	1.78619	0.893093	+	0.449871i	$0.148530\pi$
$618$	0	0
$619$	−39.4107	−1.58405	−0.792025	−	0.610489i	$-0.790973\pi$
$619$	−39.4107	−1.58405	−0.792025	+	0.610489i	$0.790973\pi$
$620$	0	0
$621$	12.5413	0.503263
$622$	0	0
$623$	61.3550	2.45813
$624$	0	0
$625$	−22.9607	−0.918428
$626$	0	0
$627$	−97.8377	−3.90726
$628$	0	0
$629$	17.5971	0.701643
$630$	0	0
$631$	2.28040	0.0907815	0.0453907	−	0.998969i	$-0.485547\pi$
$631$	2.28040	0.0907815	0.0453907	+	0.998969i	$0.485547\pi$
$632$	0	0
$633$	22.8837	0.909546
$634$	0	0
$635$	−30.7038	−1.21844
$636$	0	0
$637$	33.1630	1.31396
$638$	0	0
$639$	17.2146	0.681001
$640$	0	0
$641$	−17.2719	−0.682201	−0.341100	−	0.940027i	$-0.610800\pi$
$641$	−17.2719	−0.682201	−0.341100	+	0.940027i	$0.610800\pi$
$642$	0	0
$643$	−26.1244	−1.03025	−0.515123	−	0.857116i	$-0.672254\pi$
$643$	−26.1244	−1.03025	−0.515123	+	0.857116i	$0.672254\pi$
$644$	0	0
$645$	−47.3738	−1.86534
$646$	0	0
$647$	−12.5710	−0.494216	−0.247108	−	0.968988i	$-0.579480\pi$
$647$	−12.5710	−0.494216	−0.247108	+	0.968988i	$0.579480\pi$
$648$	0	0
$649$	70.9942	2.78677
$650$	0	0
$651$	28.3106	1.10958
$652$	0	0
$653$	28.5660	1.11787	0.558936	−	0.829211i	$-0.311210\pi$
$653$	28.5660	1.11787	0.558936	+	0.829211i	$0.311210\pi$
$654$	0	0
$655$	−15.1358	−0.591406
$656$	0	0
$657$	−24.6623	−0.962168
$658$	0	0
$659$	0.868932	0.0338488	0.0169244	−	0.999857i	$-0.494613\pi$
$659$	0.868932	0.0338488	0.0169244	+	0.999857i	$0.494613\pi$
$660$	0	0
$661$	−24.5283	−0.954041	−0.477021	−	0.878892i	$-0.658283\pi$
$661$	−24.5283	−0.954041	−0.477021	+	0.878892i	$0.658283\pi$
$662$	0	0
$663$	49.2463	1.91257
$664$	0	0
$665$	41.5498	1.61123
$666$	0	0
$667$	−21.1003	−0.817007
$668$	0	0
$669$	−27.2934	−1.05522
$670$	0	0
$671$	−22.2412	−0.858611
$672$	0	0
$673$	30.0673	1.15901	0.579504	−	0.814969i	$-0.303246\pi$
$673$	30.0673	1.15901	0.579504	+	0.814969i	$0.303246\pi$
$674$	0	0
$675$	−1.59652	−0.0614500
$676$	0	0
$677$	34.7007	1.33366	0.666828	−	0.745212i	$-0.267652\pi$
$677$	34.7007	1.33366	0.666828	+	0.745212i	$0.267652\pi$
$678$	0	0
$679$	36.3721	1.39583
$680$	0	0
$681$	65.0452	2.49254
$682$	0	0
$683$	4.97088	0.190205	0.0951026	−	0.995467i	$-0.469682\pi$
$683$	4.97088	0.190205	0.0951026	+	0.995467i	$0.469682\pi$
$684$	0	0
$685$	−0.967101	−0.0369510
$686$	0	0
$687$	12.4831	0.476258
$688$	0	0
$689$	−26.1237	−0.995235
$690$	0	0
$691$	15.9682	0.607460	0.303730	−	0.952758i	$-0.401768\pi$
$691$	15.9682	0.607460	0.303730	+	0.952758i	$0.401768\pi$
$692$	0	0
$693$	−105.355	−4.00211
$694$	0	0
$695$	8.27912	0.314045
$696$	0	0
$697$	−24.4206	−0.924995
$698$	0	0
$699$	27.4815	1.03945
$700$	0	0
$701$	−1.29023	−0.0487311	−0.0243656	−	0.999703i	$-0.507757\pi$
$701$	−1.29023	−0.0487311	−0.0243656	+	0.999703i	$0.507757\pi$
$702$	0	0
$703$	31.6276	1.19286
$704$	0	0
$705$	−71.7297	−2.70150
$706$	0	0
$707$	−54.1006	−2.03466
$708$	0	0
$709$	−35.7870	−1.34401	−0.672004	−	0.740547i	$-0.734566\pi$
$709$	−35.7870	−1.34401	−0.672004	+	0.740547i	$0.734566\pi$
$710$	0	0
$711$	26.1056	0.979035
$712$	0	0
$713$	8.65332	0.324069
$714$	0	0
$715$	−83.3276	−3.11628
$716$	0	0
$717$	−69.4589	−2.59399
$718$	0	0
$719$	15.0809	0.562424	0.281212	−	0.959646i	$-0.409264\pi$
$719$	15.0809	0.562424	0.281212	+	0.959646i	$0.409264\pi$
$720$	0	0
$721$	−14.5679	−0.542538
$722$	0	0
$723$	−45.8895	−1.70665
$724$	0	0
$725$	2.68609	0.0997590
$726$	0	0
$727$	4.95659	0.183830	0.0919150	−	0.995767i	$-0.470701\pi$
$727$	4.95659	0.183830	0.0919150	+	0.995767i	$0.470701\pi$
$728$	0	0
$729$	−43.9431	−1.62752
$730$	0	0
$731$	−24.3250	−0.899692
$732$	0	0
$733$	9.73805	0.359683	0.179842	−	0.983696i	$-0.442441\pi$
$733$	9.73805	0.359683	0.179842	+	0.983696i	$0.442441\pi$
$734$	0	0
$735$	33.0427	1.21880
$736$	0	0
$737$	80.2527	2.95615
$738$	0	0
$739$	−4.94296	−0.181830	−0.0909149	−	0.995859i	$-0.528979\pi$
$739$	−4.94296	−0.181830	−0.0909149	+	0.995859i	$0.528979\pi$
$740$	0	0
$741$	88.5112	3.25154
$742$	0	0
$743$	29.6999	1.08958	0.544792	−	0.838571i	$-0.316609\pi$
$743$	29.6999	1.08958	0.544792	+	0.838571i	$0.316609\pi$
$744$	0	0
$745$	18.0765	0.662273
$746$	0	0
$747$	13.1031	0.479418
$748$	0	0
$749$	17.4574	0.637880
$750$	0	0
$751$	23.2689	0.849095	0.424548	−	0.905406i	$-0.360433\pi$
$751$	23.2689	0.849095	0.424548	+	0.905406i	$0.360433\pi$
$752$	0	0
$753$	2.74485	0.100028
$754$	0	0
$755$	−7.75160	−0.282110
$756$	0	0
$757$	−11.0623	−0.402065	−0.201032	−	0.979585i	$-0.564430\pi$
$757$	−11.0623	−0.402065	−0.201032	+	0.979585i	$0.564430\pi$
$758$	0	0
$759$	−53.5087	−1.94224
$760$	0	0
$761$	51.7916	1.87744	0.938721	−	0.344678i	$-0.112012\pi$
$761$	51.7916	1.87744	0.938721	+	0.344678i	$0.112012\pi$
$762$	0	0
$763$	33.8798	1.22653
$764$	0	0
$765$	29.5297	1.06765
$766$	0	0
$767$	−64.2266	−2.31909
$768$	0	0
$769$	1.44430	0.0520828	0.0260414	−	0.999661i	$-0.491710\pi$
$769$	1.44430	0.0520828	0.0260414	+	0.999661i	$0.491710\pi$
$770$	0	0
$771$	59.6773	2.14923
$772$	0	0
$773$	−9.47720	−0.340871	−0.170436	−	0.985369i	$-0.554517\pi$
$773$	−9.47720	−0.340871	−0.170436	+	0.985369i	$0.554517\pi$
$774$	0	0
$775$	−1.10158	−0.0395698
$776$	0	0
$777$	56.5915	2.03021
$778$	0	0
$779$	−43.8915	−1.57258
$780$	0	0
$781$	−24.8522	−0.889280
$782$	0	0
$783$	−29.8368	−1.06628
$784$	0	0
$785$	47.7448	1.70408
$786$	0	0
$787$	33.7921	1.20456	0.602280	−	0.798285i	$-0.294259\pi$
$787$	33.7921	1.20456	0.602280	+	0.798285i	$0.294259\pi$
$788$	0	0
$789$	3.49141	0.124297
$790$	0	0
$791$	27.6062	0.981562
$792$	0	0
$793$	20.1210	0.714518
$794$	0	0
$795$	−26.0290	−0.923153
$796$	0	0
$797$	−29.8090	−1.05589	−0.527944	−	0.849279i	$-0.677037\pi$
$797$	−29.8090	−1.05589	−0.527944	+	0.849279i	$0.677037\pi$
$798$	0	0
$799$	−36.8310	−1.30299
$800$	0	0
$801$	78.3726	2.76916
$802$	0	0
$803$	35.6041	1.25644
$804$	0	0
$805$	22.7242	0.800922
$806$	0	0
$807$	18.6007	0.654776
$808$	0	0
$809$	−14.7449	−0.518404	−0.259202	−	0.965823i	$-0.583460\pi$
$809$	−14.7449	−0.518404	−0.259202	+	0.965823i	$0.583460\pi$
$810$	0	0
$811$	16.1003	0.565359	0.282679	−	0.959214i	$-0.408777\pi$
$811$	16.1003	0.565359	0.282679	+	0.959214i	$0.408777\pi$
$812$	0	0
$813$	−5.69606	−0.199770
$814$	0	0
$815$	26.0938	0.914025
$816$	0	0
$817$	−43.7197	−1.52956
$818$	0	0
$819$	95.3119	3.33047
$820$	0	0
$821$	−7.80936	−0.272549	−0.136274	−	0.990671i	$-0.543513\pi$
$821$	−7.80936	−0.272549	−0.136274	+	0.990671i	$0.543513\pi$
$822$	0	0
$823$	14.5503	0.507193	0.253597	−	0.967310i	$-0.418386\pi$
$823$	14.5503	0.507193	0.253597	+	0.967310i	$0.418386\pi$
$824$	0	0
$825$	6.81173	0.237154
$826$	0	0
$827$	−30.9267	−1.07543	−0.537714	−	0.843127i	$-0.680712\pi$
$827$	−30.9267	−1.07543	−0.537714	+	0.843127i	$0.680712\pi$
$828$	0	0
$829$	−22.6097	−0.785268	−0.392634	−	0.919695i	$-0.628436\pi$
$829$	−22.6097	−0.785268	−0.392634	+	0.919695i	$0.628436\pi$
$830$	0	0
$831$	30.0803	1.04347
$832$	0	0
$833$	16.9664	0.587851
$834$	0	0
$835$	2.49420	0.0863154
$836$	0	0
$837$	12.2362	0.422945
$838$	0	0
$839$	1.27665	0.0440747	0.0220374	−	0.999757i	$-0.492985\pi$
$839$	1.27665	0.0440747	0.0220374	+	0.999757i	$0.492985\pi$
$840$	0	0
$841$	21.1995	0.731018
$842$	0	0
$843$	7.36847	0.253784
$844$	0	0
$845$	47.4392	1.63196
$846$	0	0
$847$	113.051	3.88448
$848$	0	0
$849$	−25.0008	−0.858024
$850$	0	0
$851$	17.2976	0.592953
$852$	0	0
$853$	−1.86031	−0.0636959	−0.0318480	−	0.999493i	$-0.510139\pi$
$853$	−1.86031	−0.0636959	−0.0318480	+	0.999493i	$0.510139\pi$
$854$	0	0
$855$	53.0743	1.81510
$856$	0	0
$857$	−6.76520	−0.231095	−0.115547	−	0.993302i	$-0.536862\pi$
$857$	−6.76520	−0.231095	−0.115547	+	0.993302i	$0.536862\pi$
$858$	0	0
$859$	29.5523	1.00831	0.504155	−	0.863613i	$-0.331804\pi$
$859$	29.5523	1.00831	0.504155	+	0.863613i	$0.331804\pi$
$860$	0	0
$861$	−78.5354	−2.67648
$862$	0	0
$863$	56.5475	1.92490	0.962450	−	0.271458i	$-0.0875059\pi$
$863$	56.5475	1.92490	0.962450	+	0.271458i	$0.0875059\pi$
$864$	0	0
$865$	−53.4940	−1.81885
$866$	0	0
$867$	−21.4677	−0.729083
$868$	0	0
$869$	−37.6877	−1.27847
$870$	0	0
$871$	−72.6025	−2.46004
$872$	0	0
$873$	46.4604	1.57245
$874$	0	0
$875$	−41.0450	−1.38757
$876$	0	0
$877$	3.54396	0.119671	0.0598355	−	0.998208i	$-0.480942\pi$
$877$	3.54396	0.119671	0.0598355	+	0.998208i	$0.480942\pi$
$878$	0	0
$879$	−16.9266	−0.570922
$880$	0	0
$881$	−2.30576	−0.0776830	−0.0388415	−	0.999245i	$-0.512367\pi$
$881$	−2.30576	−0.0776830	−0.0388415	+	0.999245i	$0.512367\pi$
$882$	0	0
$883$	−37.0916	−1.24823	−0.624116	−	0.781332i	$-0.714541\pi$
$883$	−37.0916	−1.24823	−0.624116	+	0.781332i	$0.714541\pi$
$884$	0	0
$885$	−63.9937	−2.15112
$886$	0	0
$887$	28.5188	0.957567	0.478784	−	0.877933i	$-0.341078\pi$
$887$	28.5188	0.957567	0.478784	+	0.877933i	$0.341078\pi$
$888$	0	0
$889$	−50.7009	−1.70046
$890$	0	0
$891$	13.3771	0.448150
$892$	0	0
$893$	−66.1969	−2.21520
$894$	0	0
$895$	−39.2266	−1.31120
$896$	0	0
$897$	48.4079	1.61629
$898$	0	0
$899$	−20.5870	−0.686616
$900$	0	0
$901$	−13.3651	−0.445255
$902$	0	0
$903$	−78.2281	−2.60327
$904$	0	0
$905$	−28.0920	−0.933810
$906$	0	0
$907$	−39.7450	−1.31971	−0.659855	−	0.751393i	$-0.729382\pi$
$907$	−39.7450	−1.31971	−0.659855	+	0.751393i	$0.729382\pi$
$908$	0	0
$909$	−69.1061	−2.29210
$910$	0	0
$911$	−36.7963	−1.21912	−0.609558	−	0.792742i	$-0.708653\pi$
$911$	−36.7963	−1.21912	−0.609558	+	0.792742i	$0.708653\pi$
$912$	0	0
$913$	−18.9165	−0.626044
$914$	0	0
$915$	20.0480	0.662767
$916$	0	0
$917$	−24.9937	−0.825365
$918$	0	0
$919$	7.53441	0.248537	0.124269	−	0.992249i	$-0.460342\pi$
$919$	7.53441	0.248537	0.124269	+	0.992249i	$0.460342\pi$
$920$	0	0
$921$	4.19475	0.138222
$922$	0	0
$923$	22.4831	0.740040
$924$	0	0
$925$	−2.20200	−0.0724013
$926$	0	0
$927$	−18.6086	−0.611185
$928$	0	0
$929$	12.5221	0.410838	0.205419	−	0.978674i	$-0.434144\pi$
$929$	12.5221	0.410838	0.205419	+	0.978674i	$0.434144\pi$
$930$	0	0
$931$	30.4940	0.999400
$932$	0	0
$933$	−3.08641	−0.101045
$934$	0	0
$935$	−42.6310	−1.39418
$936$	0	0
$937$	40.8994	1.33613	0.668063	−	0.744105i	$-0.267124\pi$
$937$	40.8994	1.33613	0.668063	+	0.744105i	$0.267124\pi$
$938$	0	0
$939$	93.1942	3.04128
$940$	0	0
$941$	9.09750	0.296570	0.148285	−	0.988945i	$-0.452625\pi$
$941$	9.09750	0.296570	0.148285	+	0.988945i	$0.452625\pi$
$942$	0	0
$943$	−24.0048	−0.781705
$944$	0	0
$945$	32.1330	1.04529
$946$	0	0
$947$	−22.4429	−0.729295	−0.364648	−	0.931146i	$-0.618811\pi$
$947$	−22.4429	−0.729295	−0.364648	+	0.931146i	$0.618811\pi$
$948$	0	0
$949$	−32.2101	−1.04558
$950$	0	0
$951$	32.7703	1.06265
$952$	0	0
$953$	28.8722	0.935263	0.467631	−	0.883924i	$-0.345107\pi$
$953$	28.8722	0.935263	0.467631	+	0.883924i	$0.345107\pi$
$954$	0	0
$955$	11.9042	0.385210
$956$	0	0
$957$	127.302	4.11510
$958$	0	0
$959$	−1.59697	−0.0515688
$960$	0	0
$961$	−22.5572	−0.727651
$962$	0	0
$963$	22.2995	0.718591
$964$	0	0
$965$	−22.0717	−0.710513
$966$	0	0
$967$	−10.8350	−0.348430	−0.174215	−	0.984708i	$-0.555739\pi$
$967$	−10.8350	−0.348430	−0.174215	+	0.984708i	$0.555739\pi$
$968$	0	0
$969$	45.2829	1.45470
$970$	0	0
$971$	30.1539	0.967686	0.483843	−	0.875155i	$-0.339241\pi$
$971$	30.1539	0.967686	0.483843	+	0.875155i	$0.339241\pi$
$972$	0	0
$973$	13.6712	0.438280
$974$	0	0
$975$	−6.16239	−0.197354
$976$	0	0
$977$	−18.4510	−0.590301	−0.295150	−	0.955451i	$-0.595370\pi$
$977$	−18.4510	−0.590301	−0.295150	+	0.955451i	$0.595370\pi$
$978$	0	0
$979$	−113.144	−3.61609
$980$	0	0
$981$	43.2768	1.38172
$982$	0	0
$983$	12.1888	0.388763	0.194382	−	0.980926i	$-0.437730\pi$
$983$	12.1888	0.388763	0.194382	+	0.980926i	$0.437730\pi$
$984$	0	0
$985$	55.2862	1.76157
$986$	0	0
$987$	−118.447	−3.77020
$988$	0	0
$989$	−23.9109	−0.760323
$990$	0	0
$991$	4.65661	0.147922	0.0739610	−	0.997261i	$-0.476436\pi$
$991$	4.65661	0.147922	0.0739610	+	0.997261i	$0.476436\pi$
$992$	0	0
$993$	27.0648	0.858876
$994$	0	0
$995$	−8.14010	−0.258059
$996$	0	0
$997$	−57.5641	−1.82307	−0.911536	−	0.411220i	$-0.865103\pi$
$997$	−57.5641	−1.82307	−0.911536	+	0.411220i	$0.865103\pi$
$998$	0	0
$999$	24.4595	0.773866

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.m.1.20		23
4.3	odd	2		2008.2.a.d.1.4	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.4	✓	23	4.3	odd	2
4016.2.a.m.1.20		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)

\(f(q)\)	\(=\)	\(q+2.74485 q^{3} +2.14962 q^{5} +3.54966 q^{7} +4.53421 q^{9} +O(q^{10})\)	\(q+2.74485 q^{3} +2.14962 q^{5} +3.54966 q^{7} +4.53421 q^{9} -6.54587 q^{11} +5.92187 q^{13} +5.90040 q^{15} +3.02967 q^{17} +5.44528 q^{19} +9.74328 q^{21} +2.97810 q^{23} -0.379115 q^{25} +4.21117 q^{27} -7.08516 q^{29} +2.90565 q^{31} -17.9674 q^{33} +7.63043 q^{35} +5.80826 q^{37} +16.2547 q^{39} -8.06047 q^{41} -8.02892 q^{43} +9.74684 q^{45} -12.1568 q^{47} +5.60008 q^{49} +8.31599 q^{51} -4.41139 q^{53} -14.0712 q^{55} +14.9465 q^{57} -10.8456 q^{59} +3.39774 q^{61} +16.0949 q^{63} +12.7298 q^{65} -12.2601 q^{67} +8.17443 q^{69} +3.79662 q^{71} -5.43917 q^{73} -1.04061 q^{75} -23.2356 q^{77} +5.75747 q^{79} -2.04360 q^{81} +2.88983 q^{83} +6.51265 q^{85} -19.4477 q^{87} +17.2847 q^{89} +21.0206 q^{91} +7.97559 q^{93} +11.7053 q^{95} +10.2466 q^{97} -29.6803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) \) 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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