LMFDB - Embedded newform 6036.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.74974 q^{5} +1.46678 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.74974 q^{5} +1.46678 q^{7} +1.00000 q^{9} +4.30996 q^{11} +3.43074 q^{13} +2.74974 q^{15} +7.91411 q^{17} +1.80273 q^{19} -1.46678 q^{21} +1.83865 q^{23} +2.56107 q^{25} -1.00000 q^{27} +8.64332 q^{29} +3.70268 q^{31} -4.30996 q^{33} -4.03326 q^{35} +11.8125 q^{37} -3.43074 q^{39} +0.510657 q^{41} -7.96228 q^{43} -2.74974 q^{45} -4.97314 q^{47} -4.84856 q^{49} -7.91411 q^{51} -13.7075 q^{53} -11.8513 q^{55} -1.80273 q^{57} -1.90341 q^{59} -0.965055 q^{61} +1.46678 q^{63} -9.43365 q^{65} +9.78064 q^{67} -1.83865 q^{69} -2.19283 q^{71} -8.61320 q^{73} -2.56107 q^{75} +6.32176 q^{77} +4.80388 q^{79} +1.00000 q^{81} +9.61818 q^{83} -21.7617 q^{85} -8.64332 q^{87} +6.41609 q^{89} +5.03214 q^{91} -3.70268 q^{93} -4.95703 q^{95} +11.8923 q^{97} +4.30996 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.74974	−1.22972	−0.614861	−	0.788636i	$-0.710788\pi$
$5$	−2.74974	−1.22972	−0.614861	+	0.788636i	$0.710788\pi$
$6$	0	0
$7$	1.46678	0.554390	0.277195	−	0.960814i	$-0.410595\pi$
$7$	1.46678	0.554390	0.277195	+	0.960814i	$0.410595\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.30996	1.29950	0.649751	−	0.760147i	$-0.274873\pi$
$11$	4.30996	1.29950	0.649751	+	0.760147i	$0.274873\pi$
$12$	0	0
$13$	3.43074	0.951517	0.475758	−	0.879576i	$-0.342174\pi$
$13$	3.43074	0.951517	0.475758	+	0.879576i	$0.342174\pi$
$14$	0	0
$15$	2.74974	0.709980
$16$	0	0
$17$	7.91411	1.91945	0.959727	−	0.280935i	$-0.0906445\pi$
$17$	7.91411	1.91945	0.959727	+	0.280935i	$0.0906445\pi$
$18$	0	0
$19$	1.80273	0.413574	0.206787	−	0.978386i	$-0.433699\pi$
$19$	1.80273	0.413574	0.206787	+	0.978386i	$0.433699\pi$
$20$	0	0
$21$	−1.46678	−0.320077
$22$	0	0
$23$	1.83865	0.383385	0.191692	−	0.981455i	$-0.438602\pi$
$23$	1.83865	0.383385	0.191692	+	0.981455i	$0.438602\pi$
$24$	0	0
$25$	2.56107	0.512214
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.64332	1.60502	0.802512	−	0.596636i	$-0.203496\pi$
$29$	8.64332	1.60502	0.802512	+	0.596636i	$0.203496\pi$
$30$	0	0
$31$	3.70268	0.665021	0.332510	−	0.943100i	$-0.392104\pi$
$31$	3.70268	0.665021	0.332510	+	0.943100i	$0.392104\pi$
$32$	0	0
$33$	−4.30996	−0.750268
$34$	0	0
$35$	−4.03326	−0.681746
$36$	0	0
$37$	11.8125	1.94196	0.970981	−	0.239156i	$-0.0768706\pi$
$37$	11.8125	1.94196	0.970981	+	0.239156i	$0.0768706\pi$
$38$	0	0
$39$	−3.43074	−0.549358
$40$	0	0
$41$	0.510657	0.0797513	0.0398756	−	0.999205i	$-0.487304\pi$
$41$	0.510657	0.0797513	0.0398756	+	0.999205i	$0.487304\pi$
$42$	0	0
$43$	−7.96228	−1.21424	−0.607118	−	0.794612i	$-0.707675\pi$
$43$	−7.96228	−1.21424	−0.607118	+	0.794612i	$0.707675\pi$
$44$	0	0
$45$	−2.74974	−0.409907
$46$	0	0
$47$	−4.97314	−0.725407	−0.362704	−	0.931904i	$-0.618146\pi$
$47$	−4.97314	−0.725407	−0.362704	+	0.931904i	$0.618146\pi$
$48$	0	0
$49$	−4.84856	−0.692651
$50$	0	0
$51$	−7.91411	−1.10820
$52$	0	0
$53$	−13.7075	−1.88287	−0.941435	−	0.337195i	$-0.890522\pi$
$53$	−13.7075	−1.88287	−0.941435	+	0.337195i	$0.890522\pi$
$54$	0	0
$55$	−11.8513	−1.59803
$56$	0	0
$57$	−1.80273	−0.238777
$58$	0	0
$59$	−1.90341	−0.247802	−0.123901	−	0.992295i	$-0.539541\pi$
$59$	−1.90341	−0.247802	−0.123901	+	0.992295i	$0.539541\pi$
$60$	0	0
$61$	−0.965055	−0.123563	−0.0617813	−	0.998090i	$-0.519678\pi$
$61$	−0.965055	−0.123563	−0.0617813	+	0.998090i	$0.519678\pi$
$62$	0	0
$63$	1.46678	0.184797
$64$	0	0
$65$	−9.43365	−1.17010
$66$	0	0
$67$	9.78064	1.19490	0.597448	−	0.801908i	$-0.296182\pi$
$67$	9.78064	1.19490	0.597448	+	0.801908i	$0.296182\pi$
$68$	0	0
$69$	−1.83865	−0.221347
$70$	0	0
$71$	−2.19283	−0.260241	−0.130121	−	0.991498i	$-0.541536\pi$
$71$	−2.19283	−0.260241	−0.130121	+	0.991498i	$0.541536\pi$
$72$	0	0
$73$	−8.61320	−1.00810	−0.504049	−	0.863675i	$-0.668157\pi$
$73$	−8.61320	−1.00810	−0.504049	+	0.863675i	$0.668157\pi$
$74$	0	0
$75$	−2.56107	−0.295727
$76$	0	0
$77$	6.32176	0.720432
$78$	0	0
$79$	4.80388	0.540479	0.270239	−	0.962793i	$-0.412897\pi$
$79$	4.80388	0.540479	0.270239	+	0.962793i	$0.412897\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.61818	1.05573	0.527866	−	0.849328i	$-0.322992\pi$
$83$	9.61818	1.05573	0.527866	+	0.849328i	$0.322992\pi$
$84$	0	0
$85$	−21.7617	−2.36039
$86$	0	0
$87$	−8.64332	−0.926661
$88$	0	0
$89$	6.41609	0.680104	0.340052	−	0.940407i	$-0.389555\pi$
$89$	6.41609	0.680104	0.340052	+	0.940407i	$0.389555\pi$
$90$	0	0
$91$	5.03214	0.527512
$92$	0	0
$93$	−3.70268	−0.383950
$94$	0	0
$95$	−4.95703	−0.508581
$96$	0	0
$97$	11.8923	1.20748	0.603741	−	0.797181i	$-0.293676\pi$
$97$	11.8923	1.20748	0.603741	+	0.797181i	$0.293676\pi$
$98$	0	0
$99$	4.30996	0.433167
$100$	0	0
$101$	−9.25137	−0.920546	−0.460273	−	0.887777i	$-0.652248\pi$
$101$	−9.25137	−0.920546	−0.460273	+	0.887777i	$0.652248\pi$
$102$	0	0
$103$	7.66988	0.755735	0.377868	−	0.925860i	$-0.376657\pi$
$103$	7.66988	0.755735	0.377868	+	0.925860i	$0.376657\pi$
$104$	0	0
$105$	4.03326	0.393606
$106$	0	0
$107$	−14.0197	−1.35533	−0.677667	−	0.735369i	$-0.737009\pi$
$107$	−14.0197	−1.35533	−0.677667	+	0.735369i	$0.737009\pi$
$108$	0	0
$109$	5.36889	0.514246	0.257123	−	0.966379i	$-0.417225\pi$
$109$	5.36889	0.514246	0.257123	+	0.966379i	$0.417225\pi$
$110$	0	0
$111$	−11.8125	−1.12119
$112$	0	0
$113$	3.52883	0.331965	0.165982	−	0.986129i	$-0.446921\pi$
$113$	3.52883	0.331965	0.165982	+	0.986129i	$0.446921\pi$
$114$	0	0
$115$	−5.05581	−0.471457
$116$	0	0
$117$	3.43074	0.317172
$118$	0	0
$119$	11.6083	1.06413
$120$	0	0
$121$	7.57577	0.688706
$122$	0	0
$123$	−0.510657	−0.0460444
$124$	0	0
$125$	6.70642	0.599840
$126$	0	0
$127$	15.8045	1.40243	0.701213	−	0.712952i	$-0.252642\pi$
$127$	15.8045	1.40243	0.701213	+	0.712952i	$0.252642\pi$
$128$	0	0
$129$	7.96228	0.701040
$130$	0	0
$131$	−6.40460	−0.559572	−0.279786	−	0.960062i	$-0.590264\pi$
$131$	−6.40460	−0.559572	−0.279786	+	0.960062i	$0.590264\pi$
$132$	0	0
$133$	2.64420	0.229282
$134$	0	0
$135$	2.74974	0.236660
$136$	0	0
$137$	−17.1833	−1.46807	−0.734036	−	0.679111i	$-0.762366\pi$
$137$	−17.1833	−1.46807	−0.734036	+	0.679111i	$0.762366\pi$
$138$	0	0
$139$	11.1249	0.943598	0.471799	−	0.881706i	$-0.343605\pi$
$139$	11.1249	0.943598	0.471799	+	0.881706i	$0.343605\pi$
$140$	0	0
$141$	4.97314	0.418814
$142$	0	0
$143$	14.7864	1.23650
$144$	0	0
$145$	−23.7669	−1.97373
$146$	0	0
$147$	4.84856	0.399902
$148$	0	0
$149$	−2.91714	−0.238981	−0.119491	−	0.992835i	$-0.538126\pi$
$149$	−2.91714	−0.238981	−0.119491	+	0.992835i	$0.538126\pi$
$150$	0	0
$151$	3.96133	0.322368	0.161184	−	0.986924i	$-0.448469\pi$
$151$	3.96133	0.322368	0.161184	+	0.986924i	$0.448469\pi$
$152$	0	0
$153$	7.91411	0.639818
$154$	0	0
$155$	−10.1814	−0.817790
$156$	0	0
$157$	−3.65103	−0.291384	−0.145692	−	0.989330i	$-0.546541\pi$
$157$	−3.65103	−0.291384	−0.145692	+	0.989330i	$0.546541\pi$
$158$	0	0
$159$	13.7075	1.08708
$160$	0	0
$161$	2.69689	0.212545
$162$	0	0
$163$	−18.8328	−1.47510	−0.737551	−	0.675292i	$-0.764018\pi$
$163$	−18.8328	−1.47510	−0.737551	+	0.675292i	$0.764018\pi$
$164$	0	0
$165$	11.8513	0.922621
$166$	0	0
$167$	−2.46173	−0.190495	−0.0952474	−	0.995454i	$-0.530364\pi$
$167$	−2.46173	−0.190495	−0.0952474	+	0.995454i	$0.530364\pi$
$168$	0	0
$169$	−1.23001	−0.0946161
$170$	0	0
$171$	1.80273	0.137858
$172$	0	0
$173$	−9.43332	−0.717202	−0.358601	−	0.933491i	$-0.616746\pi$
$173$	−9.43332	−0.717202	−0.358601	+	0.933491i	$0.616746\pi$
$174$	0	0
$175$	3.75653	0.283967
$176$	0	0
$177$	1.90341	0.143069
$178$	0	0
$179$	−3.86292	−0.288728	−0.144364	−	0.989525i	$-0.546114\pi$
$179$	−3.86292	−0.288728	−0.144364	+	0.989525i	$0.546114\pi$
$180$	0	0
$181$	−15.5937	−1.15907	−0.579534	−	0.814948i	$-0.696765\pi$
$181$	−15.5937	−1.15907	−0.579534	+	0.814948i	$0.696765\pi$
$182$	0	0
$183$	0.965055	0.0713389
$184$	0	0
$185$	−32.4813	−2.38807
$186$	0	0
$187$	34.1095	2.49433
$188$	0	0
$189$	−1.46678	−0.106692
$190$	0	0
$191$	−22.0305	−1.59407	−0.797037	−	0.603931i	$-0.793600\pi$
$191$	−22.0305	−1.59407	−0.797037	+	0.603931i	$0.793600\pi$
$192$	0	0
$193$	21.5335	1.55001	0.775006	−	0.631953i	$-0.217747\pi$
$193$	21.5335	1.55001	0.775006	+	0.631953i	$0.217747\pi$
$194$	0	0
$195$	9.43365	0.675558
$196$	0	0
$197$	−18.0230	−1.28408	−0.642042	−	0.766669i	$-0.721913\pi$
$197$	−18.0230	−1.28408	−0.642042	+	0.766669i	$0.721913\pi$
$198$	0	0
$199$	20.6122	1.46116	0.730579	−	0.682828i	$-0.239250\pi$
$199$	20.6122	1.46116	0.730579	+	0.682828i	$0.239250\pi$
$200$	0	0
$201$	−9.78064	−0.689873
$202$	0	0
$203$	12.6778	0.889810
$204$	0	0
$205$	−1.40418	−0.0980718
$206$	0	0
$207$	1.83865	0.127795
$208$	0	0
$209$	7.76969	0.537441
$210$	0	0
$211$	28.0898	1.93378	0.966890	−	0.255194i	$-0.0821393\pi$
$211$	28.0898	1.93378	0.966890	+	0.255194i	$0.0821393\pi$
$212$	0	0
$213$	2.19283	0.150250
$214$	0	0
$215$	21.8942	1.49317
$216$	0	0
$217$	5.43101	0.368681
$218$	0	0
$219$	8.61320	0.582026
$220$	0	0
$221$	27.1513	1.82639
$222$	0	0
$223$	−17.7735	−1.19020	−0.595102	−	0.803650i	$-0.702888\pi$
$223$	−17.7735	−1.19020	−0.595102	+	0.803650i	$0.702888\pi$
$224$	0	0
$225$	2.56107	0.170738
$226$	0	0
$227$	23.8817	1.58508	0.792541	−	0.609819i	$-0.208758\pi$
$227$	23.8817	1.58508	0.792541	+	0.609819i	$0.208758\pi$
$228$	0	0
$229$	5.85550	0.386942	0.193471	−	0.981106i	$-0.438025\pi$
$229$	5.85550	0.386942	0.193471	+	0.981106i	$0.438025\pi$
$230$	0	0
$231$	−6.32176	−0.415941
$232$	0	0
$233$	17.3891	1.13920	0.569599	−	0.821923i	$-0.307098\pi$
$233$	17.3891	1.13920	0.569599	+	0.821923i	$0.307098\pi$
$234$	0	0
$235$	13.6749	0.892049
$236$	0	0
$237$	−4.80388	−0.312045
$238$	0	0
$239$	−29.4336	−1.90390	−0.951951	−	0.306250i	$-0.900926\pi$
$239$	−29.4336	−1.90390	−0.951951	+	0.306250i	$0.900926\pi$
$240$	0	0
$241$	18.9487	1.22059	0.610296	−	0.792173i	$-0.291050\pi$
$241$	18.9487	1.22059	0.610296	+	0.792173i	$0.291050\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	13.3323	0.851768
$246$	0	0
$247$	6.18470	0.393523
$248$	0	0
$249$	−9.61818	−0.609527
$250$	0	0
$251$	−29.0508	−1.83367	−0.916834	−	0.399268i	$-0.869264\pi$
$251$	−29.0508	−1.83367	−0.916834	+	0.399268i	$0.869264\pi$
$252$	0	0
$253$	7.92451	0.498210
$254$	0	0
$255$	21.7617	1.36277
$256$	0	0
$257$	5.09395	0.317752	0.158876	−	0.987299i	$-0.449213\pi$
$257$	5.09395	0.317752	0.158876	+	0.987299i	$0.449213\pi$
$258$	0	0
$259$	17.3263	1.07661
$260$	0	0
$261$	8.64332	0.535008
$262$	0	0
$263$	−0.308249	−0.0190075	−0.00950374	−	0.999955i	$-0.503025\pi$
$263$	−0.308249	−0.0190075	−0.00950374	+	0.999955i	$0.503025\pi$
$264$	0	0
$265$	37.6921	2.31541
$266$	0	0
$267$	−6.41609	−0.392658
$268$	0	0
$269$	22.9962	1.40210	0.701052	−	0.713110i	$-0.252714\pi$
$269$	22.9962	1.40210	0.701052	+	0.713110i	$0.252714\pi$
$270$	0	0
$271$	13.0506	0.792766	0.396383	−	0.918085i	$-0.370265\pi$
$271$	13.0506	0.792766	0.396383	+	0.918085i	$0.370265\pi$
$272$	0	0
$273$	−5.03214	−0.304559
$274$	0	0
$275$	11.0381	0.665624
$276$	0	0
$277$	26.2985	1.58012	0.790062	−	0.613026i	$-0.210048\pi$
$277$	26.2985	1.58012	0.790062	+	0.613026i	$0.210048\pi$
$278$	0	0
$279$	3.70268	0.221674
$280$	0	0
$281$	−25.9918	−1.55054	−0.775270	−	0.631630i	$-0.782386\pi$
$281$	−25.9918	−1.55054	−0.775270	+	0.631630i	$0.782386\pi$
$282$	0	0
$283$	−27.6377	−1.64289	−0.821447	−	0.570285i	$-0.806833\pi$
$283$	−27.6377	−1.64289	−0.821447	+	0.570285i	$0.806833\pi$
$284$	0	0
$285$	4.95703	0.293629
$286$	0	0
$287$	0.749022	0.0442133
$288$	0	0
$289$	45.6331	2.68430
$290$	0	0
$291$	−11.8923	−0.697140
$292$	0	0
$293$	0.689548	0.0402838	0.0201419	−	0.999797i	$-0.493588\pi$
$293$	0.689548	0.0402838	0.0201419	+	0.999797i	$0.493588\pi$
$294$	0	0
$295$	5.23387	0.304728
$296$	0	0
$297$	−4.30996	−0.250089
$298$	0	0
$299$	6.30793	0.364797
$300$	0	0
$301$	−11.6789	−0.673161
$302$	0	0
$303$	9.25137	0.531478
$304$	0	0
$305$	2.65365	0.151948
$306$	0	0
$307$	−28.1808	−1.60836	−0.804180	−	0.594385i	$-0.797395\pi$
$307$	−28.1808	−1.60836	−0.804180	+	0.594385i	$0.797395\pi$
$308$	0	0
$309$	−7.66988	−0.436324
$310$	0	0
$311$	11.2398	0.637352	0.318676	−	0.947864i	$-0.396762\pi$
$311$	11.2398	0.637352	0.318676	+	0.947864i	$0.396762\pi$
$312$	0	0
$313$	2.55110	0.144196	0.0720982	−	0.997398i	$-0.477030\pi$
$313$	2.55110	0.144196	0.0720982	+	0.997398i	$0.477030\pi$
$314$	0	0
$315$	−4.03326	−0.227249
$316$	0	0
$317$	3.47272	0.195047	0.0975237	−	0.995233i	$-0.468908\pi$
$317$	3.47272	0.195047	0.0975237	+	0.995233i	$0.468908\pi$
$318$	0	0
$319$	37.2524	2.08573
$320$	0	0
$321$	14.0197	0.782503
$322$	0	0
$323$	14.2670	0.793837
$324$	0	0
$325$	8.78638	0.487381
$326$	0	0
$327$	−5.36889	−0.296900
$328$	0	0
$329$	−7.29450	−0.402159
$330$	0	0
$331$	−27.7879	−1.52736	−0.763680	−	0.645595i	$-0.776609\pi$
$331$	−27.7879	−1.52736	−0.763680	+	0.645595i	$0.776609\pi$
$332$	0	0
$333$	11.8125	0.647321
$334$	0	0
$335$	−26.8942	−1.46939
$336$	0	0
$337$	18.5986	1.01313	0.506565	−	0.862202i	$-0.330915\pi$
$337$	18.5986	1.01313	0.506565	+	0.862202i	$0.330915\pi$
$338$	0	0
$339$	−3.52883	−0.191660
$340$	0	0
$341$	15.9584	0.864196
$342$	0	0
$343$	−17.3792	−0.938390
$344$	0	0
$345$	5.05581	0.272196
$346$	0	0
$347$	−28.8362	−1.54801	−0.774004	−	0.633181i	$-0.781749\pi$
$347$	−28.8362	−1.54801	−0.774004	+	0.633181i	$0.781749\pi$
$348$	0	0
$349$	−29.5016	−1.57918	−0.789592	−	0.613632i	$-0.789708\pi$
$349$	−29.5016	−1.57918	−0.789592	+	0.613632i	$0.789708\pi$
$350$	0	0
$351$	−3.43074	−0.183119
$352$	0	0
$353$	12.0505	0.641383	0.320691	−	0.947184i	$-0.396085\pi$
$353$	12.0505	0.641383	0.320691	+	0.947184i	$0.396085\pi$
$354$	0	0
$355$	6.02971	0.320024
$356$	0	0
$357$	−11.6083	−0.614374
$358$	0	0
$359$	18.6952	0.986694	0.493347	−	0.869833i	$-0.335773\pi$
$359$	18.6952	0.986694	0.493347	+	0.869833i	$0.335773\pi$
$360$	0	0
$361$	−15.7502	−0.828956
$362$	0	0
$363$	−7.57577	−0.397625
$364$	0	0
$365$	23.6841	1.23968
$366$	0	0
$367$	1.79781	0.0938450	0.0469225	−	0.998899i	$-0.485059\pi$
$367$	1.79781	0.0938450	0.0469225	+	0.998899i	$0.485059\pi$
$368$	0	0
$369$	0.510657	0.0265838
$370$	0	0
$371$	−20.1059	−1.04385
$372$	0	0
$373$	−3.23876	−0.167697	−0.0838484	−	0.996479i	$-0.526721\pi$
$373$	−3.23876	−0.167697	−0.0838484	+	0.996479i	$0.526721\pi$
$374$	0	0
$375$	−6.70642	−0.346318
$376$	0	0
$377$	29.6530	1.52721
$378$	0	0
$379$	24.2653	1.24643	0.623213	−	0.782052i	$-0.285827\pi$
$379$	24.2653	1.24643	0.623213	+	0.782052i	$0.285827\pi$
$380$	0	0
$381$	−15.8045	−0.809691
$382$	0	0
$383$	3.78210	0.193256	0.0966282	−	0.995321i	$-0.469194\pi$
$383$	3.78210	0.193256	0.0966282	+	0.995321i	$0.469194\pi$
$384$	0	0
$385$	−17.3832	−0.885930
$386$	0	0
$387$	−7.96228	−0.404745
$388$	0	0
$389$	−22.4387	−1.13769	−0.568843	−	0.822446i	$-0.692609\pi$
$389$	−22.4387	−1.13769	−0.568843	+	0.822446i	$0.692609\pi$
$390$	0	0
$391$	14.5513	0.735890
$392$	0	0
$393$	6.40460	0.323069
$394$	0	0
$395$	−13.2094	−0.664638
$396$	0	0
$397$	−3.46414	−0.173860	−0.0869300	−	0.996214i	$-0.527706\pi$
$397$	−3.46414	−0.173860	−0.0869300	+	0.996214i	$0.527706\pi$
$398$	0	0
$399$	−2.64420	−0.132376
$400$	0	0
$401$	31.5980	1.57793	0.788963	−	0.614440i	$-0.210618\pi$
$401$	31.5980	1.57793	0.788963	+	0.614440i	$0.210618\pi$
$402$	0	0
$403$	12.7029	0.632778
$404$	0	0
$405$	−2.74974	−0.136636
$406$	0	0
$407$	50.9114	2.52358
$408$	0	0
$409$	19.6908	0.973649	0.486824	−	0.873500i	$-0.338155\pi$
$409$	19.6908	0.973649	0.486824	+	0.873500i	$0.338155\pi$
$410$	0	0
$411$	17.1833	0.847592
$412$	0	0
$413$	−2.79188	−0.137379
$414$	0	0
$415$	−26.4475	−1.29826
$416$	0	0
$417$	−11.1249	−0.544787
$418$	0	0
$419$	−31.8208	−1.55455	−0.777274	−	0.629162i	$-0.783398\pi$
$419$	−31.8208	−1.55455	−0.777274	+	0.629162i	$0.783398\pi$
$420$	0	0
$421$	2.42733	0.118301	0.0591503	−	0.998249i	$-0.481161\pi$
$421$	2.42733	0.118301	0.0591503	+	0.998249i	$0.481161\pi$
$422$	0	0
$423$	−4.97314	−0.241802
$424$	0	0
$425$	20.2686	0.983172
$426$	0	0
$427$	−1.41552	−0.0685019
$428$	0	0
$429$	−14.7864	−0.713892
$430$	0	0
$431$	−24.4858	−1.17944	−0.589719	−	0.807608i	$-0.700762\pi$
$431$	−24.4858	−1.17944	−0.589719	+	0.807608i	$0.700762\pi$
$432$	0	0
$433$	23.9179	1.14942	0.574711	−	0.818356i	$-0.305114\pi$
$433$	23.9179	1.14942	0.574711	+	0.818356i	$0.305114\pi$
$434$	0	0
$435$	23.7669	1.13953
$436$	0	0
$437$	3.31459	0.158558
$438$	0	0
$439$	−5.17762	−0.247114	−0.123557	−	0.992337i	$-0.539430\pi$
$439$	−5.17762	−0.247114	−0.123557	+	0.992337i	$0.539430\pi$
$440$	0	0
$441$	−4.84856	−0.230884
$442$	0	0
$443$	−7.28618	−0.346177	−0.173088	−	0.984906i	$-0.555375\pi$
$443$	−7.28618	−0.346177	−0.173088	+	0.984906i	$0.555375\pi$
$444$	0	0
$445$	−17.6426	−0.836339
$446$	0	0
$447$	2.91714	0.137976
$448$	0	0
$449$	−17.8547	−0.842617	−0.421309	−	0.906917i	$-0.638429\pi$
$449$	−17.8547	−0.842617	−0.421309	+	0.906917i	$0.638429\pi$
$450$	0	0
$451$	2.20091	0.103637
$452$	0	0
$453$	−3.96133	−0.186119
$454$	0	0
$455$	−13.8371	−0.648692
$456$	0	0
$457$	−41.2178	−1.92809	−0.964044	−	0.265744i	$-0.914382\pi$
$457$	−41.2178	−1.92809	−0.964044	+	0.265744i	$0.914382\pi$
$458$	0	0
$459$	−7.91411	−0.369399
$460$	0	0
$461$	30.9818	1.44297	0.721484	−	0.692431i	$-0.243460\pi$
$461$	30.9818	1.44297	0.721484	+	0.692431i	$0.243460\pi$
$462$	0	0
$463$	28.4680	1.32302	0.661510	−	0.749937i	$-0.269916\pi$
$463$	28.4680	1.32302	0.661510	+	0.749937i	$0.269916\pi$
$464$	0	0
$465$	10.1814	0.472151
$466$	0	0
$467$	−6.18707	−0.286303	−0.143152	−	0.989701i	$-0.545724\pi$
$467$	−6.18707	−0.286303	−0.143152	+	0.989701i	$0.545724\pi$
$468$	0	0
$469$	14.3460	0.662439
$470$	0	0
$471$	3.65103	0.168231
$472$	0	0
$473$	−34.3171	−1.57790
$474$	0	0
$475$	4.61692	0.211839
$476$	0	0
$477$	−13.7075	−0.627623
$478$	0	0
$479$	30.4611	1.39180	0.695902	−	0.718137i	$-0.255005\pi$
$479$	30.4611	1.39180	0.695902	+	0.718137i	$0.255005\pi$
$480$	0	0
$481$	40.5256	1.84781
$482$	0	0
$483$	−2.69689	−0.122713
$484$	0	0
$485$	−32.7008	−1.48487
$486$	0	0
$487$	19.1358	0.867127	0.433564	−	0.901123i	$-0.357256\pi$
$487$	19.1358	0.867127	0.433564	+	0.901123i	$0.357256\pi$
$488$	0	0
$489$	18.8328	0.851650
$490$	0	0
$491$	−18.4030	−0.830516	−0.415258	−	0.909704i	$-0.636309\pi$
$491$	−18.4030	−0.830516	−0.415258	+	0.909704i	$0.636309\pi$
$492$	0	0
$493$	68.4042	3.08077
$494$	0	0
$495$	−11.8513	−0.532675
$496$	0	0
$497$	−3.21640	−0.144275
$498$	0	0
$499$	1.98963	0.0890681	0.0445340	−	0.999008i	$-0.485820\pi$
$499$	1.98963	0.0890681	0.0445340	+	0.999008i	$0.485820\pi$
$500$	0	0
$501$	2.46173	0.109982
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	25.4389	1.13202
$506$	0	0
$507$	1.23001	0.0546266
$508$	0	0
$509$	22.3251	0.989544	0.494772	−	0.869023i	$-0.335251\pi$
$509$	22.3251	0.989544	0.494772	+	0.869023i	$0.335251\pi$
$510$	0	0
$511$	−12.6337	−0.558880
$512$	0	0
$513$	−1.80273	−0.0795924
$514$	0	0
$515$	−21.0902	−0.929344
$516$	0	0
$517$	−21.4341	−0.942669
$518$	0	0
$519$	9.43332	0.414077
$520$	0	0
$521$	26.1222	1.14444	0.572218	−	0.820102i	$-0.306083\pi$
$521$	26.1222	1.14444	0.572218	+	0.820102i	$0.306083\pi$
$522$	0	0
$523$	−31.9411	−1.39669	−0.698343	−	0.715763i	$-0.746079\pi$
$523$	−31.9411	−1.39669	−0.698343	+	0.715763i	$0.746079\pi$
$524$	0	0
$525$	−3.75653	−0.163948
$526$	0	0
$527$	29.3034	1.27648
$528$	0	0
$529$	−19.6194	−0.853016
$530$	0	0
$531$	−1.90341	−0.0826008
$532$	0	0
$533$	1.75193	0.0758847
$534$	0	0
$535$	38.5505	1.66668
$536$	0	0
$537$	3.86292	0.166697
$538$	0	0
$539$	−20.8971	−0.900102
$540$	0	0
$541$	−36.5743	−1.57245	−0.786225	−	0.617940i	$-0.787967\pi$
$541$	−36.5743	−1.57245	−0.786225	+	0.617940i	$0.787967\pi$
$542$	0	0
$543$	15.5937	0.669188
$544$	0	0
$545$	−14.7630	−0.632379
$546$	0	0
$547$	−23.0369	−0.984988	−0.492494	−	0.870316i	$-0.663915\pi$
$547$	−23.0369	−0.984988	−0.492494	+	0.870316i	$0.663915\pi$
$548$	0	0
$549$	−0.965055	−0.0411875
$550$	0	0
$551$	15.5816	0.663796
$552$	0	0
$553$	7.04623	0.299636
$554$	0	0
$555$	32.4813	1.37875
$556$	0	0
$557$	−13.3429	−0.565356	−0.282678	−	0.959215i	$-0.591223\pi$
$557$	−13.3429	−0.565356	−0.282678	+	0.959215i	$0.591223\pi$
$558$	0	0
$559$	−27.3165	−1.15537
$560$	0	0
$561$	−34.1095	−1.44010
$562$	0	0
$563$	−16.2420	−0.684519	−0.342260	−	0.939605i	$-0.611192\pi$
$563$	−16.2420	−0.684519	−0.342260	+	0.939605i	$0.611192\pi$
$564$	0	0
$565$	−9.70338	−0.408224
$566$	0	0
$567$	1.46678	0.0615989
$568$	0	0
$569$	−44.4645	−1.86405	−0.932024	−	0.362397i	$-0.881958\pi$
$569$	−44.4645	−1.86405	−0.932024	+	0.362397i	$0.881958\pi$
$570$	0	0
$571$	−13.2134	−0.552963	−0.276481	−	0.961019i	$-0.589168\pi$
$571$	−13.2134	−0.552963	−0.276481	+	0.961019i	$0.589168\pi$
$572$	0	0
$573$	22.0305	0.920339
$574$	0	0
$575$	4.70891	0.196375
$576$	0	0
$577$	−12.4856	−0.519781	−0.259891	−	0.965638i	$-0.583687\pi$
$577$	−12.4856	−0.519781	−0.259891	+	0.965638i	$0.583687\pi$
$578$	0	0
$579$	−21.5335	−0.894900
$580$	0	0
$581$	14.1077	0.585288
$582$	0	0
$583$	−59.0788	−2.44679
$584$	0	0
$585$	−9.43365	−0.390033
$586$	0	0
$587$	31.3818	1.29527	0.647633	−	0.761952i	$-0.275759\pi$
$587$	31.3818	1.29527	0.647633	+	0.761952i	$0.275759\pi$
$588$	0	0
$589$	6.67492	0.275035
$590$	0	0
$591$	18.0230	0.741366
$592$	0	0
$593$	41.2451	1.69373	0.846867	−	0.531804i	$-0.178486\pi$
$593$	41.2451	1.69373	0.846867	+	0.531804i	$0.178486\pi$
$594$	0	0
$595$	−31.9197	−1.30858
$596$	0	0
$597$	−20.6122	−0.843600
$598$	0	0
$599$	37.1847	1.51933	0.759664	−	0.650316i	$-0.225364\pi$
$599$	37.1847	1.51933	0.759664	+	0.650316i	$0.225364\pi$
$600$	0	0
$601$	3.40436	0.138867	0.0694335	−	0.997587i	$-0.477881\pi$
$601$	3.40436	0.138867	0.0694335	+	0.997587i	$0.477881\pi$
$602$	0	0
$603$	9.78064	0.398298
$604$	0	0
$605$	−20.8314	−0.846917
$606$	0	0
$607$	15.9840	0.648769	0.324384	−	0.945925i	$-0.394843\pi$
$607$	15.9840	0.648769	0.324384	+	0.945925i	$0.394843\pi$
$608$	0	0
$609$	−12.6778	−0.513732
$610$	0	0
$611$	−17.0616	−0.690237
$612$	0	0
$613$	14.8589	0.600145	0.300072	−	0.953916i	$-0.402989\pi$
$613$	14.8589	0.600145	0.300072	+	0.953916i	$0.402989\pi$
$614$	0	0
$615$	1.40418	0.0566218
$616$	0	0
$617$	23.5784	0.949231	0.474616	−	0.880193i	$-0.342587\pi$
$617$	23.5784	0.949231	0.474616	+	0.880193i	$0.342587\pi$
$618$	0	0
$619$	−5.89002	−0.236740	−0.118370	−	0.992970i	$-0.537767\pi$
$619$	−5.89002	−0.236740	−0.118370	+	0.992970i	$0.537767\pi$
$620$	0	0
$621$	−1.83865	−0.0737825
$622$	0	0
$623$	9.41099	0.377043
$624$	0	0
$625$	−31.2463	−1.24985
$626$	0	0
$627$	−7.76969	−0.310291
$628$	0	0
$629$	93.4854	3.72751
$630$	0	0
$631$	26.9322	1.07215	0.536076	−	0.844169i	$-0.319906\pi$
$631$	26.9322	1.07215	0.536076	+	0.844169i	$0.319906\pi$
$632$	0	0
$633$	−28.0898	−1.11647
$634$	0	0
$635$	−43.4584	−1.72459
$636$	0	0
$637$	−16.6342	−0.659069
$638$	0	0
$639$	−2.19283	−0.0867470
$640$	0	0
$641$	41.2328	1.62860	0.814299	−	0.580445i	$-0.197122\pi$
$641$	41.2328	1.62860	0.814299	+	0.580445i	$0.197122\pi$
$642$	0	0
$643$	16.7823	0.661828	0.330914	−	0.943661i	$-0.392643\pi$
$643$	16.7823	0.661828	0.330914	+	0.943661i	$0.392643\pi$
$644$	0	0
$645$	−21.8942	−0.862083
$646$	0	0
$647$	−38.2542	−1.50393	−0.751964	−	0.659204i	$-0.770893\pi$
$647$	−38.2542	−1.50393	−0.751964	+	0.659204i	$0.770893\pi$
$648$	0	0
$649$	−8.20360	−0.322020
$650$	0	0
$651$	−5.43101	−0.212858
$652$	0	0
$653$	43.2135	1.69108	0.845538	−	0.533916i	$-0.179280\pi$
$653$	43.2135	1.69108	0.845538	+	0.533916i	$0.179280\pi$
$654$	0	0
$655$	17.6110	0.688118
$656$	0	0
$657$	−8.61320	−0.336033
$658$	0	0
$659$	−5.52402	−0.215185	−0.107593	−	0.994195i	$-0.534314\pi$
$659$	−5.52402	−0.215185	−0.107593	+	0.994195i	$0.534314\pi$
$660$	0	0
$661$	37.8576	1.47249	0.736245	−	0.676716i	$-0.236597\pi$
$661$	37.8576	1.47249	0.736245	+	0.676716i	$0.236597\pi$
$662$	0	0
$663$	−27.1513	−1.05447
$664$	0	0
$665$	−7.27088	−0.281952
$666$	0	0
$667$	15.8920	0.615342
$668$	0	0
$669$	17.7735	0.687165
$670$	0	0
$671$	−4.15935	−0.160570
$672$	0	0
$673$	−42.3798	−1.63362	−0.816810	−	0.576907i	$-0.804259\pi$
$673$	−42.3798	−1.63362	−0.816810	+	0.576907i	$0.804259\pi$
$674$	0	0
$675$	−2.56107	−0.0985757
$676$	0	0
$677$	31.6977	1.21824	0.609122	−	0.793077i	$-0.291522\pi$
$677$	31.6977	1.21824	0.609122	+	0.793077i	$0.291522\pi$
$678$	0	0
$679$	17.4434	0.669416
$680$	0	0
$681$	−23.8817	−0.915147
$682$	0	0
$683$	−41.2166	−1.57711	−0.788554	−	0.614966i	$-0.789170\pi$
$683$	−41.2166	−1.57711	−0.788554	+	0.614966i	$0.789170\pi$
$684$	0	0
$685$	47.2497	1.80532
$686$	0	0
$687$	−5.85550	−0.223401
$688$	0	0
$689$	−47.0269	−1.79158
$690$	0	0
$691$	24.7586	0.941861	0.470931	−	0.882170i	$-0.343918\pi$
$691$	24.7586	0.941861	0.470931	+	0.882170i	$0.343918\pi$
$692$	0	0
$693$	6.32176	0.240144
$694$	0	0
$695$	−30.5905	−1.16036
$696$	0	0
$697$	4.04140	0.153079
$698$	0	0
$699$	−17.3891	−0.657717
$700$	0	0
$701$	47.4789	1.79325	0.896627	−	0.442787i	$-0.146010\pi$
$701$	47.4789	1.79325	0.896627	+	0.442787i	$0.146010\pi$
$702$	0	0
$703$	21.2947	0.803146
$704$	0	0
$705$	−13.6749	−0.515025
$706$	0	0
$707$	−13.5697	−0.510342
$708$	0	0
$709$	−29.4616	−1.10645	−0.553226	−	0.833031i	$-0.686603\pi$
$709$	−29.4616	−1.10645	−0.553226	+	0.833031i	$0.686603\pi$
$710$	0	0
$711$	4.80388	0.180160
$712$	0	0
$713$	6.80793	0.254959
$714$	0	0
$715$	−40.6587	−1.52055
$716$	0	0
$717$	29.4336	1.09922
$718$	0	0
$719$	−38.3768	−1.43121	−0.715607	−	0.698503i	$-0.753850\pi$
$719$	−38.3768	−1.43121	−0.715607	+	0.698503i	$0.753850\pi$
$720$	0	0
$721$	11.2500	0.418972
$722$	0	0
$723$	−18.9487	−0.704710
$724$	0	0
$725$	22.1362	0.822116
$726$	0	0
$727$	−30.6862	−1.13809	−0.569045	−	0.822306i	$-0.692687\pi$
$727$	−30.6862	−1.13809	−0.569045	+	0.822306i	$0.692687\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−63.0144	−2.33067
$732$	0	0
$733$	19.8661	0.733772	0.366886	−	0.930266i	$-0.380424\pi$
$733$	19.8661	0.733772	0.366886	+	0.930266i	$0.380424\pi$
$734$	0	0
$735$	−13.3323	−0.491768
$736$	0	0
$737$	42.1542	1.55277
$738$	0	0
$739$	35.5555	1.30793	0.653966	−	0.756524i	$-0.273104\pi$
$739$	35.5555	1.30793	0.653966	+	0.756524i	$0.273104\pi$
$740$	0	0
$741$	−6.18470	−0.227200
$742$	0	0
$743$	−4.47104	−0.164027	−0.0820133	−	0.996631i	$-0.526135\pi$
$743$	−4.47104	−0.164027	−0.0820133	+	0.996631i	$0.526135\pi$
$744$	0	0
$745$	8.02137	0.293880
$746$	0	0
$747$	9.61818	0.351911
$748$	0	0
$749$	−20.5638	−0.751384
$750$	0	0
$751$	29.0420	1.05976	0.529878	−	0.848074i	$-0.322238\pi$
$751$	29.0420	1.05976	0.529878	+	0.848074i	$0.322238\pi$
$752$	0	0
$753$	29.0508	1.05867
$754$	0	0
$755$	−10.8926	−0.396423
$756$	0	0
$757$	23.0360	0.837259	0.418629	−	0.908157i	$-0.362511\pi$
$757$	23.0360	0.837259	0.418629	+	0.908157i	$0.362511\pi$
$758$	0	0
$759$	−7.92451	−0.287641
$760$	0	0
$761$	−6.51065	−0.236011	−0.118005	−	0.993013i	$-0.537650\pi$
$761$	−6.51065	−0.236011	−0.118005	+	0.993013i	$0.537650\pi$
$762$	0	0
$763$	7.87497	0.285093
$764$	0	0
$765$	−21.7617	−0.786798
$766$	0	0
$767$	−6.53009	−0.235788
$768$	0	0
$769$	−35.7317	−1.28852	−0.644259	−	0.764807i	$-0.722834\pi$
$769$	−35.7317	−1.28852	−0.644259	+	0.764807i	$0.722834\pi$
$770$	0	0
$771$	−5.09395	−0.183454
$772$	0	0
$773$	−10.0259	−0.360606	−0.180303	−	0.983611i	$-0.557708\pi$
$773$	−10.0259	−0.360606	−0.180303	+	0.983611i	$0.557708\pi$
$774$	0	0
$775$	9.48283	0.340633
$776$	0	0
$777$	−17.3263	−0.621578
$778$	0	0
$779$	0.920576	0.0329831
$780$	0	0
$781$	−9.45102	−0.338184
$782$	0	0
$783$	−8.64332	−0.308887
$784$	0	0
$785$	10.0394	0.358321
$786$	0	0
$787$	−22.3144	−0.795423	−0.397712	−	0.917511i	$-0.630196\pi$
$787$	−22.3144	−0.795423	−0.397712	+	0.917511i	$0.630196\pi$
$788$	0	0
$789$	0.308249	0.0109740
$790$	0	0
$791$	5.17602	0.184038
$792$	0	0
$793$	−3.31085	−0.117572
$794$	0	0
$795$	−37.6921	−1.33680
$796$	0	0
$797$	−7.31156	−0.258989	−0.129494	−	0.991580i	$-0.541335\pi$
$797$	−7.31156	−0.258989	−0.129494	+	0.991580i	$0.541335\pi$
$798$	0	0
$799$	−39.3580	−1.39239
$800$	0	0
$801$	6.41609	0.226701
$802$	0	0
$803$	−37.1225	−1.31003
$804$	0	0
$805$	−7.41576	−0.261371
$806$	0	0
$807$	−22.9962	−0.809505
$808$	0	0
$809$	−11.1191	−0.390925	−0.195463	−	0.980711i	$-0.562621\pi$
$809$	−11.1191	−0.390925	−0.195463	+	0.980711i	$0.562621\pi$
$810$	0	0
$811$	−22.4291	−0.787594	−0.393797	−	0.919197i	$-0.628839\pi$
$811$	−22.4291	−0.787594	−0.393797	+	0.919197i	$0.628839\pi$
$812$	0	0
$813$	−13.0506	−0.457704
$814$	0	0
$815$	51.7854	1.81396
$816$	0	0
$817$	−14.3538	−0.502177
$818$	0	0
$819$	5.03214	0.175837
$820$	0	0
$821$	6.28870	0.219477	0.109739	−	0.993960i	$-0.464999\pi$
$821$	6.28870	0.219477	0.109739	+	0.993960i	$0.464999\pi$
$822$	0	0
$823$	28.1969	0.982883	0.491441	−	0.870911i	$-0.336470\pi$
$823$	28.1969	0.982883	0.491441	+	0.870911i	$0.336470\pi$
$824$	0	0
$825$	−11.0381	−0.384298
$826$	0	0
$827$	−18.4390	−0.641187	−0.320594	−	0.947217i	$-0.603882\pi$
$827$	−18.4390	−0.641187	−0.320594	+	0.947217i	$0.603882\pi$
$828$	0	0
$829$	−6.48726	−0.225312	−0.112656	−	0.993634i	$-0.535936\pi$
$829$	−6.48726	−0.225312	−0.112656	+	0.993634i	$0.535936\pi$
$830$	0	0
$831$	−26.2985	−0.912286
$832$	0	0
$833$	−38.3720	−1.32951
$834$	0	0
$835$	6.76913	0.234255
$836$	0	0
$837$	−3.70268	−0.127983
$838$	0	0
$839$	27.8022	0.959838	0.479919	−	0.877313i	$-0.340666\pi$
$839$	27.8022	0.959838	0.479919	+	0.877313i	$0.340666\pi$
$840$	0	0
$841$	45.7069	1.57610
$842$	0	0
$843$	25.9918	0.895205
$844$	0	0
$845$	3.38221	0.116351
$846$	0	0
$847$	11.1120	0.381812
$848$	0	0
$849$	27.6377	0.948525
$850$	0	0
$851$	21.7190	0.744519
$852$	0	0
$853$	−43.0165	−1.47286	−0.736428	−	0.676516i	$-0.763489\pi$
$853$	−43.0165	−1.47286	−0.736428	+	0.676516i	$0.763489\pi$
$854$	0	0
$855$	−4.95703	−0.169527
$856$	0	0
$857$	5.19274	0.177381	0.0886904	−	0.996059i	$-0.471732\pi$
$857$	5.19274	0.177381	0.0886904	+	0.996059i	$0.471732\pi$
$858$	0	0
$859$	9.82796	0.335326	0.167663	−	0.985844i	$-0.446378\pi$
$859$	9.82796	0.335326	0.167663	+	0.985844i	$0.446378\pi$
$860$	0	0
$861$	−0.749022	−0.0255266
$862$	0	0
$863$	−38.4544	−1.30900	−0.654502	−	0.756060i	$-0.727122\pi$
$863$	−38.4544	−1.30900	−0.654502	+	0.756060i	$0.727122\pi$
$864$	0	0
$865$	25.9392	0.881959
$866$	0	0
$867$	−45.6331	−1.54978
$868$	0	0
$869$	20.7045	0.702353
$870$	0	0
$871$	33.5549	1.13696
$872$	0	0
$873$	11.8923	0.402494
$874$	0	0
$875$	9.83683	0.332546
$876$	0	0
$877$	−11.9864	−0.404752	−0.202376	−	0.979308i	$-0.564866\pi$
$877$	−11.9864	−0.404752	−0.202376	+	0.979308i	$0.564866\pi$
$878$	0	0
$879$	−0.689548	−0.0232579
$880$	0	0
$881$	−43.4989	−1.46551	−0.732757	−	0.680490i	$-0.761767\pi$
$881$	−43.4989	−1.46551	−0.732757	+	0.680490i	$0.761767\pi$
$882$	0	0
$883$	−37.7763	−1.27127	−0.635637	−	0.771988i	$-0.719262\pi$
$883$	−37.7763	−1.27127	−0.635637	+	0.771988i	$0.719262\pi$
$884$	0	0
$885$	−5.23387	−0.175935
$886$	0	0
$887$	−5.59671	−0.187919	−0.0939596	−	0.995576i	$-0.529952\pi$
$887$	−5.59671	−0.187919	−0.0939596	+	0.995576i	$0.529952\pi$
$888$	0	0
$889$	23.1818	0.777492
$890$	0	0
$891$	4.30996	0.144389
$892$	0	0
$893$	−8.96523	−0.300010
$894$	0	0
$895$	10.6220	0.355055
$896$	0	0
$897$	−6.30793	−0.210616
$898$	0	0
$899$	32.0034	1.06737
$900$	0	0
$901$	−108.483	−3.61408
$902$	0	0
$903$	11.6789	0.388650
$904$	0	0
$905$	42.8785	1.42533
$906$	0	0
$907$	5.32909	0.176950	0.0884748	−	0.996078i	$-0.471801\pi$
$907$	5.32909	0.176950	0.0884748	+	0.996078i	$0.471801\pi$
$908$	0	0
$909$	−9.25137	−0.306849
$910$	0	0
$911$	−30.1263	−0.998129	−0.499064	−	0.866565i	$-0.666323\pi$
$911$	−30.1263	−0.998129	−0.499064	+	0.866565i	$0.666323\pi$
$912$	0	0
$913$	41.4540	1.37193
$914$	0	0
$915$	−2.65365	−0.0877270
$916$	0	0
$917$	−9.39413	−0.310221
$918$	0	0
$919$	−16.1951	−0.534226	−0.267113	−	0.963665i	$-0.586070\pi$
$919$	−16.1951	−0.534226	−0.267113	+	0.963665i	$0.586070\pi$
$920$	0	0
$921$	28.1808	0.928588
$922$	0	0
$923$	−7.52304	−0.247624
$924$	0	0
$925$	30.2527	0.994701
$926$	0	0
$927$	7.66988	0.251912
$928$	0	0
$929$	−21.0043	−0.689130	−0.344565	−	0.938762i	$-0.611974\pi$
$929$	−21.0043	−0.689130	−0.344565	+	0.938762i	$0.611974\pi$
$930$	0	0
$931$	−8.74063	−0.286463
$932$	0	0
$933$	−11.2398	−0.367976
$934$	0	0
$935$	−93.7923	−3.06734
$936$	0	0
$937$	−18.1860	−0.594111	−0.297056	−	0.954860i	$-0.596005\pi$
$937$	−18.1860	−0.594111	−0.297056	+	0.954860i	$0.596005\pi$
$938$	0	0
$939$	−2.55110	−0.0832519
$940$	0	0
$941$	28.7973	0.938764	0.469382	−	0.882995i	$-0.344477\pi$
$941$	28.7973	0.938764	0.469382	+	0.882995i	$0.344477\pi$
$942$	0	0
$943$	0.938920	0.0305754
$944$	0	0
$945$	4.03326	0.131202
$946$	0	0
$947$	−16.7495	−0.544287	−0.272143	−	0.962257i	$-0.587732\pi$
$947$	−16.7495	−0.544287	−0.272143	+	0.962257i	$0.587732\pi$
$948$	0	0
$949$	−29.5497	−0.959222
$950$	0	0
$951$	−3.47272	−0.112611
$952$	0	0
$953$	29.7130	0.962500	0.481250	−	0.876583i	$-0.340183\pi$
$953$	29.7130	0.962500	0.481250	+	0.876583i	$0.340183\pi$
$954$	0	0
$955$	60.5783	1.96027
$956$	0	0
$957$	−37.2524	−1.20420
$958$	0	0
$959$	−25.2042	−0.813885
$960$	0	0
$961$	−17.2902	−0.557747
$962$	0	0
$963$	−14.0197	−0.451778
$964$	0	0
$965$	−59.2114	−1.90608
$966$	0	0
$967$	13.7163	0.441087	0.220544	−	0.975377i	$-0.429217\pi$
$967$	13.7163	0.441087	0.220544	+	0.975377i	$0.429217\pi$
$968$	0	0
$969$	−14.2670	−0.458322
$970$	0	0
$971$	32.1725	1.03246	0.516232	−	0.856449i	$-0.327334\pi$
$971$	32.1725	1.03246	0.516232	+	0.856449i	$0.327334\pi$
$972$	0	0
$973$	16.3177	0.523122
$974$	0	0
$975$	−8.78638	−0.281389
$976$	0	0
$977$	−14.8657	−0.475596	−0.237798	−	0.971315i	$-0.576426\pi$
$977$	−14.8657	−0.475596	−0.237798	+	0.971315i	$0.576426\pi$
$978$	0	0
$979$	27.6531	0.883797
$980$	0	0
$981$	5.36889	0.171415
$982$	0	0
$983$	32.4742	1.03577	0.517883	−	0.855452i	$-0.326720\pi$
$983$	32.4742	1.03577	0.517883	+	0.855452i	$0.326720\pi$
$984$	0	0
$985$	49.5585	1.57907
$986$	0	0
$987$	7.29450	0.232187
$988$	0	0
$989$	−14.6398	−0.465520
$990$	0	0
$991$	−31.9577	−1.01517	−0.507585	−	0.861602i	$-0.669462\pi$
$991$	−31.9577	−1.01517	−0.507585	+	0.861602i	$0.669462\pi$
$992$	0	0
$993$	27.7879	0.881821
$994$	0	0
$995$	−56.6781	−1.79682
$996$	0	0
$997$	40.1961	1.27302	0.636511	−	0.771267i	$-0.280377\pi$
$997$	40.1961	1.27302	0.636511	+	0.771267i	$0.280377\pi$
$998$	0	0
$999$	−11.8125	−0.373731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.4	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.4	✓	26	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.74974 q^{5} +1.46678 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.74974 q^{5} +1.46678 q^{7} +1.00000 q^{9} +4.30996 q^{11} +3.43074 q^{13} +2.74974 q^{15} +7.91411 q^{17} +1.80273 q^{19} -1.46678 q^{21} +1.83865 q^{23} +2.56107 q^{25} -1.00000 q^{27} +8.64332 q^{29} +3.70268 q^{31} -4.30996 q^{33} -4.03326 q^{35} +11.8125 q^{37} -3.43074 q^{39} +0.510657 q^{41} -7.96228 q^{43} -2.74974 q^{45} -4.97314 q^{47} -4.84856 q^{49} -7.91411 q^{51} -13.7075 q^{53} -11.8513 q^{55} -1.80273 q^{57} -1.90341 q^{59} -0.965055 q^{61} +1.46678 q^{63} -9.43365 q^{65} +9.78064 q^{67} -1.83865 q^{69} -2.19283 q^{71} -8.61320 q^{73} -2.56107 q^{75} +6.32176 q^{77} +4.80388 q^{79} +1.00000 q^{81} +9.61818 q^{83} -21.7617 q^{85} -8.64332 q^{87} +6.41609 q^{89} +5.03214 q^{91} -3.70268 q^{93} -4.95703 q^{95} +11.8923 q^{97} +4.30996 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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