LMFDB - Embedded newform 3536.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3536,2,Mod(1,3536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3536, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3536.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3536 = 2^{4} \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3536.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:	$28.2351021547$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 221)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	3536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79129 q^{3} -1.00000 q^{5} +4.79129 q^{7} +4.79129 q^{9} +O(q^{10})$	$q-2.79129 q^{3} -1.00000 q^{5} +4.79129 q^{7} +4.79129 q^{9} -3.79129 q^{11} -1.00000 q^{13} +2.79129 q^{15} +1.00000 q^{17} -0.208712 q^{19} -13.3739 q^{21} +1.58258 q^{23} -4.00000 q^{25} -5.00000 q^{27} +9.00000 q^{29} -8.58258 q^{31} +10.5826 q^{33} -4.79129 q^{35} -8.58258 q^{37} +2.79129 q^{39} -9.00000 q^{43} -4.79129 q^{45} +5.58258 q^{47} +15.9564 q^{49} -2.79129 q^{51} -3.20871 q^{53} +3.79129 q^{55} +0.582576 q^{57} +0.582576 q^{59} +7.20871 q^{61} +22.9564 q^{63} +1.00000 q^{65} +13.5826 q^{67} -4.41742 q^{69} -2.00000 q^{71} -0.582576 q^{73} +11.1652 q^{75} -18.1652 q^{77} -0.582576 q^{79} -0.417424 q^{81} +4.58258 q^{83} -1.00000 q^{85} -25.1216 q^{87} +6.95644 q^{89} -4.79129 q^{91} +23.9564 q^{93} +0.208712 q^{95} -7.95644 q^{97} -18.1652 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 5 * q^9	$ 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 5 q^{9} - 3 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} - 5 q^{19} - 13 q^{21} - 6 q^{23} - 8 q^{25} - 10 q^{27} + 18 q^{29} - 8 q^{31} + 12 q^{33} - 5 q^{35} - 8 q^{37} + q^{39} - 18 q^{43} - 5 q^{45} + 2 q^{47} + 9 q^{49} - q^{51} - 11 q^{53} + 3 q^{55} - 8 q^{57} - 8 q^{59} + 19 q^{61} + 23 q^{63} + 2 q^{65} + 18 q^{67} - 18 q^{69} - 4 q^{71} + 8 q^{73} + 4 q^{75} - 18 q^{77} + 8 q^{79} - 10 q^{81} - 2 q^{85} - 9 q^{87} - 9 q^{89} - 5 q^{91} + 25 q^{93} + 5 q^{95} + 7 q^{97} - 18 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 5 * q^9 - 3 * q^11 - 2 * q^13 + q^15 + 2 * q^17 - 5 * q^19 - 13 * q^21 - 6 * q^23 - 8 * q^25 - 10 * q^27 + 18 * q^29 - 8 * q^31 + 12 * q^33 - 5 * q^35 - 8 * q^37 + q^39 - 18 * q^43 - 5 * q^45 + 2 * q^47 + 9 * q^49 - q^51 - 11 * q^53 + 3 * q^55 - 8 * q^57 - 8 * q^59 + 19 * q^61 + 23 * q^63 + 2 * q^65 + 18 * q^67 - 18 * q^69 - 4 * q^71 + 8 * q^73 + 4 * q^75 - 18 * q^77 + 8 * q^79 - 10 * q^81 - 2 * q^85 - 9 * q^87 - 9 * q^89 - 5 * q^91 + 25 * q^93 + 5 * q^95 + 7 * q^97 - 18 * 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
$3$	−2.79129	−1.61155	−0.805775	−	0.592221i	$-0.798251\pi$
$3$	−2.79129	−1.61155	−0.805775	+	0.592221i	$0.798251\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	4.79129	1.81094	0.905468	−	0.424414i	$-0.139520\pi$
$7$	4.79129	1.81094	0.905468	+	0.424414i	$0.139520\pi$
$8$	0	0
$9$	4.79129	1.59710
$10$	0	0
$11$	−3.79129	−1.14312	−0.571558	−	0.820562i	$-0.693661\pi$
$11$	−3.79129	−1.14312	−0.571558	+	0.820562i	$0.693661\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	2.79129	0.720707
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−0.208712	−0.0478819	−0.0239409	−	0.999713i	$-0.507621\pi$
$19$	−0.208712	−0.0478819	−0.0239409	+	0.999713i	$0.507621\pi$
$20$	0	0
$21$	−13.3739	−2.91842
$22$	0	0
$23$	1.58258	0.329990	0.164995	−	0.986294i	$-0.447239\pi$
$23$	1.58258	0.329990	0.164995	+	0.986294i	$0.447239\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−8.58258	−1.54148	−0.770738	−	0.637152i	$-0.780112\pi$
$31$	−8.58258	−1.54148	−0.770738	+	0.637152i	$0.780112\pi$
$32$	0	0
$33$	10.5826	1.84219
$34$	0	0
$35$	−4.79129	−0.809875
$36$	0	0
$37$	−8.58258	−1.41097	−0.705483	−	0.708726i	$-0.749270\pi$
$37$	−8.58258	−1.41097	−0.705483	+	0.708726i	$0.749270\pi$
$38$	0	0
$39$	2.79129	0.446964
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	0	0
$45$	−4.79129	−0.714243
$46$	0	0
$47$	5.58258	0.814302	0.407151	−	0.913361i	$-0.366522\pi$
$47$	5.58258	0.814302	0.407151	+	0.913361i	$0.366522\pi$
$48$	0	0
$49$	15.9564	2.27949
$50$	0	0
$51$	−2.79129	−0.390858
$52$	0	0
$53$	−3.20871	−0.440751	−0.220375	−	0.975415i	$-0.570728\pi$
$53$	−3.20871	−0.440751	−0.220375	+	0.975415i	$0.570728\pi$
$54$	0	0
$55$	3.79129	0.511217
$56$	0	0
$57$	0.582576	0.0771640
$58$	0	0
$59$	0.582576	0.0758449	0.0379224	−	0.999281i	$-0.487926\pi$
$59$	0.582576	0.0758449	0.0379224	+	0.999281i	$0.487926\pi$
$60$	0	0
$61$	7.20871	0.922981	0.461491	−	0.887145i	$-0.347315\pi$
$61$	7.20871	0.922981	0.461491	+	0.887145i	$0.347315\pi$
$62$	0	0
$63$	22.9564	2.89224
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	13.5826	1.65938	0.829688	−	0.558228i	$-0.188518\pi$
$67$	13.5826	1.65938	0.829688	+	0.558228i	$0.188518\pi$
$68$	0	0
$69$	−4.41742	−0.531795
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	−0.582576	−0.0681853	−0.0340927	−	0.999419i	$-0.510854\pi$
$73$	−0.582576	−0.0681853	−0.0340927	+	0.999419i	$0.510854\pi$
$74$	0	0
$75$	11.1652	1.28924
$76$	0	0
$77$	−18.1652	−2.07011
$78$	0	0
$79$	−0.582576	−0.0655449	−0.0327724	−	0.999463i	$-0.510434\pi$
$79$	−0.582576	−0.0655449	−0.0327724	+	0.999463i	$0.510434\pi$
$80$	0	0
$81$	−0.417424	−0.0463805
$82$	0	0
$83$	4.58258	0.503003	0.251502	−	0.967857i	$-0.419076\pi$
$83$	4.58258	0.503003	0.251502	+	0.967857i	$0.419076\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	−25.1216	−2.69332
$88$	0	0
$89$	6.95644	0.737381	0.368691	−	0.929552i	$-0.379806\pi$
$89$	6.95644	0.737381	0.368691	+	0.929552i	$0.379806\pi$
$90$	0	0
$91$	−4.79129	−0.502263
$92$	0	0
$93$	23.9564	2.48417
$94$	0	0
$95$	0.208712	0.0214134
$96$	0	0
$97$	−7.95644	−0.807854	−0.403927	−	0.914791i	$-0.632355\pi$
$97$	−7.95644	−0.807854	−0.403927	+	0.914791i	$0.632355\pi$
$98$	0	0
$99$	−18.1652	−1.82567
$100$	0	0
$101$	−12.1652	−1.21048	−0.605239	−	0.796044i	$-0.706922\pi$
$101$	−12.1652	−1.21048	−0.605239	+	0.796044i	$0.706922\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	13.3739	1.30516
$106$	0	0
$107$	7.37386	0.712858	0.356429	−	0.934322i	$-0.383994\pi$
$107$	7.37386	0.712858	0.356429	+	0.934322i	$0.383994\pi$
$108$	0	0
$109$	5.95644	0.570523	0.285262	−	0.958450i	$-0.407919\pi$
$109$	5.95644	0.570523	0.285262	+	0.958450i	$0.407919\pi$
$110$	0	0
$111$	23.9564	2.27384
$112$	0	0
$113$	0.791288	0.0744381	0.0372190	−	0.999307i	$-0.488150\pi$
$113$	0.791288	0.0744381	0.0372190	+	0.999307i	$0.488150\pi$
$114$	0	0
$115$	−1.58258	−0.147576
$116$	0	0
$117$	−4.79129	−0.442955
$118$	0	0
$119$	4.79129	0.439217
$120$	0	0
$121$	3.37386	0.306715
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−16.9564	−1.50464	−0.752320	−	0.658797i	$-0.771065\pi$
$127$	−16.9564	−1.50464	−0.752320	+	0.658797i	$0.771065\pi$
$128$	0	0
$129$	25.1216	2.21183
$130$	0	0
$131$	−1.41742	−0.123841	−0.0619205	−	0.998081i	$-0.519723\pi$
$131$	−1.41742	−0.123841	−0.0619205	+	0.998081i	$0.519723\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	3.79129	0.323912	0.161956	−	0.986798i	$-0.448220\pi$
$137$	3.79129	0.323912	0.161956	+	0.986798i	$0.448220\pi$
$138$	0	0
$139$	−20.7477	−1.75980	−0.879900	−	0.475160i	$-0.842390\pi$
$139$	−20.7477	−1.75980	−0.879900	+	0.475160i	$0.842390\pi$
$140$	0	0
$141$	−15.5826	−1.31229
$142$	0	0
$143$	3.79129	0.317043
$144$	0	0
$145$	−9.00000	−0.747409
$146$	0	0
$147$	−44.5390	−3.67352
$148$	0	0
$149$	−18.9564	−1.55297	−0.776486	−	0.630134i	$-0.783000\pi$
$149$	−18.9564	−1.55297	−0.776486	+	0.630134i	$0.783000\pi$
$150$	0	0
$151$	8.74773	0.711880	0.355940	−	0.934509i	$-0.384161\pi$
$151$	8.74773	0.711880	0.355940	+	0.934509i	$0.384161\pi$
$152$	0	0
$153$	4.79129	0.387353
$154$	0	0
$155$	8.58258	0.689369
$156$	0	0
$157$	−14.5826	−1.16382	−0.581908	−	0.813255i	$-0.697694\pi$
$157$	−14.5826	−1.16382	−0.581908	+	0.813255i	$0.697694\pi$
$158$	0	0
$159$	8.95644	0.710292
$160$	0	0
$161$	7.58258	0.597591
$162$	0	0
$163$	−11.2087	−0.877934	−0.438967	−	0.898503i	$-0.644656\pi$
$163$	−11.2087	−0.877934	−0.438967	+	0.898503i	$0.644656\pi$
$164$	0	0
$165$	−10.5826	−0.823852
$166$	0	0
$167$	−25.3303	−1.96012	−0.980059	−	0.198708i	$-0.936326\pi$
$167$	−25.3303	−1.96012	−0.980059	+	0.198708i	$0.936326\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−10.1652	−0.772842	−0.386421	−	0.922322i	$-0.626289\pi$
$173$	−10.1652	−0.772842	−0.386421	+	0.922322i	$0.626289\pi$
$174$	0	0
$175$	−19.1652	−1.44875
$176$	0	0
$177$	−1.62614	−0.122228
$178$	0	0
$179$	18.1652	1.35773	0.678864	−	0.734264i	$-0.262473\pi$
$179$	18.1652	1.35773	0.678864	+	0.734264i	$0.262473\pi$
$180$	0	0
$181$	−15.2087	−1.13045	−0.565227	−	0.824935i	$-0.691212\pi$
$181$	−15.2087	−1.13045	−0.565227	+	0.824935i	$0.691212\pi$
$182$	0	0
$183$	−20.1216	−1.48743
$184$	0	0
$185$	8.58258	0.631004
$186$	0	0
$187$	−3.79129	−0.277246
$188$	0	0
$189$	−23.9564	−1.74257
$190$	0	0
$191$	17.7477	1.28418	0.642090	−	0.766629i	$-0.278067\pi$
$191$	17.7477	1.28418	0.642090	+	0.766629i	$0.278067\pi$
$192$	0	0
$193$	9.58258	0.689769	0.344884	−	0.938645i	$-0.387918\pi$
$193$	9.58258	0.689769	0.344884	+	0.938645i	$0.387918\pi$
$194$	0	0
$195$	−2.79129	−0.199888
$196$	0	0
$197$	−4.74773	−0.338262	−0.169131	−	0.985594i	$-0.554096\pi$
$197$	−4.74773	−0.338262	−0.169131	+	0.985594i	$0.554096\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	−37.9129	−2.67417
$202$	0	0
$203$	43.1216	3.02654
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.58258	0.527025
$208$	0	0
$209$	0.791288	0.0547345
$210$	0	0
$211$	−20.5826	−1.41696	−0.708481	−	0.705729i	$-0.750619\pi$
$211$	−20.5826	−1.41696	−0.708481	+	0.705729i	$0.750619\pi$
$212$	0	0
$213$	5.58258	0.382512
$214$	0	0
$215$	9.00000	0.613795
$216$	0	0
$217$	−41.1216	−2.79152
$218$	0	0
$219$	1.62614	0.109884
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	−12.7913	−0.856568	−0.428284	−	0.903644i	$-0.640882\pi$
$223$	−12.7913	−0.856568	−0.428284	+	0.903644i	$0.640882\pi$
$224$	0	0
$225$	−19.1652	−1.27768
$226$	0	0
$227$	−1.58258	−0.105039	−0.0525196	−	0.998620i	$-0.516725\pi$
$227$	−1.58258	−0.105039	−0.0525196	+	0.998620i	$0.516725\pi$
$228$	0	0
$229$	1.41742	0.0936660	0.0468330	−	0.998903i	$-0.485087\pi$
$229$	1.41742	0.0936660	0.0468330	+	0.998903i	$0.485087\pi$
$230$	0	0
$231$	50.7042	3.33609
$232$	0	0
$233$	−26.9129	−1.76312	−0.881561	−	0.472071i	$-0.843507\pi$
$233$	−26.9129	−1.76312	−0.881561	+	0.472071i	$0.843507\pi$
$234$	0	0
$235$	−5.58258	−0.364167
$236$	0	0
$237$	1.62614	0.105629
$238$	0	0
$239$	−1.74773	−0.113051	−0.0565255	−	0.998401i	$-0.518002\pi$
$239$	−1.74773	−0.113051	−0.0565255	+	0.998401i	$0.518002\pi$
$240$	0	0
$241$	−9.74773	−0.627906	−0.313953	−	0.949438i	$-0.601653\pi$
$241$	−9.74773	−0.627906	−0.313953	+	0.949438i	$0.601653\pi$
$242$	0	0
$243$	16.1652	1.03699
$244$	0	0
$245$	−15.9564	−1.01942
$246$	0	0
$247$	0.208712	0.0132800
$248$	0	0
$249$	−12.7913	−0.810615
$250$	0	0
$251$	10.3739	0.654792	0.327396	−	0.944887i	$-0.393829\pi$
$251$	10.3739	0.654792	0.327396	+	0.944887i	$0.393829\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	2.79129	0.174797
$256$	0	0
$257$	8.16515	0.509328	0.254664	−	0.967030i	$-0.418035\pi$
$257$	8.16515	0.509328	0.254664	+	0.967030i	$0.418035\pi$
$258$	0	0
$259$	−41.1216	−2.55517
$260$	0	0
$261$	43.1216	2.66916
$262$	0	0
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	3.20871	0.197110
$266$	0	0
$267$	−19.4174	−1.18833
$268$	0	0
$269$	−6.62614	−0.404003	−0.202001	−	0.979385i	$-0.564745\pi$
$269$	−6.62614	−0.404003	−0.202001	+	0.979385i	$0.564745\pi$
$270$	0	0
$271$	−30.3739	−1.84508	−0.922540	−	0.385901i	$-0.873891\pi$
$271$	−30.3739	−1.84508	−0.922540	+	0.385901i	$0.873891\pi$
$272$	0	0
$273$	13.3739	0.809423
$274$	0	0
$275$	15.1652	0.914493
$276$	0	0
$277$	−6.83485	−0.410666	−0.205333	−	0.978692i	$-0.565828\pi$
$277$	−6.83485	−0.410666	−0.205333	+	0.978692i	$0.565828\pi$
$278$	0	0
$279$	−41.1216	−2.46189
$280$	0	0
$281$	−29.1216	−1.73725	−0.868624	−	0.495471i	$-0.834995\pi$
$281$	−29.1216	−1.73725	−0.868624	+	0.495471i	$0.834995\pi$
$282$	0	0
$283$	−5.20871	−0.309626	−0.154813	−	0.987944i	$-0.549477\pi$
$283$	−5.20871	−0.309626	−0.154813	+	0.987944i	$0.549477\pi$
$284$	0	0
$285$	−0.582576	−0.0345088
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	22.2087	1.30190
$292$	0	0
$293$	17.6261	1.02973	0.514865	−	0.857271i	$-0.327842\pi$
$293$	17.6261	1.02973	0.514865	+	0.857271i	$0.327842\pi$
$294$	0	0
$295$	−0.582576	−0.0339189
$296$	0	0
$297$	18.9564	1.09996
$298$	0	0
$299$	−1.58258	−0.0915227
$300$	0	0
$301$	−43.1216	−2.48549
$302$	0	0
$303$	33.9564	1.95075
$304$	0	0
$305$	−7.20871	−0.412770
$306$	0	0
$307$	−19.7913	−1.12955	−0.564774	−	0.825245i	$-0.691037\pi$
$307$	−19.7913	−1.12955	−0.564774	+	0.825245i	$0.691037\pi$
$308$	0	0
$309$	−2.79129	−0.158791
$310$	0	0
$311$	−22.5826	−1.28054	−0.640270	−	0.768150i	$-0.721178\pi$
$311$	−22.5826	−1.28054	−0.640270	+	0.768150i	$0.721178\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	−22.9564	−1.29345
$316$	0	0
$317$	17.0000	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$317$	17.0000	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$318$	0	0
$319$	−34.1216	−1.91044
$320$	0	0
$321$	−20.5826	−1.14881
$322$	0	0
$323$	−0.208712	−0.0116131
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−16.6261	−0.919427
$328$	0	0
$329$	26.7477	1.47465
$330$	0	0
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	0	0
$333$	−41.1216	−2.25345
$334$	0	0
$335$	−13.5826	−0.742095
$336$	0	0
$337$	−6.20871	−0.338210	−0.169105	−	0.985598i	$-0.554088\pi$
$337$	−6.20871	−0.338210	−0.169105	+	0.985598i	$0.554088\pi$
$338$	0	0
$339$	−2.20871	−0.119961
$340$	0	0
$341$	32.5390	1.76209
$342$	0	0
$343$	42.9129	2.31708
$344$	0	0
$345$	4.41742	0.237826
$346$	0	0
$347$	−11.7477	−0.630651	−0.315326	−	0.948984i	$-0.602114\pi$
$347$	−11.7477	−0.630651	−0.315326	+	0.948984i	$0.602114\pi$
$348$	0	0
$349$	10.6261	0.568804	0.284402	−	0.958705i	$-0.408205\pi$
$349$	10.6261	0.568804	0.284402	+	0.958705i	$0.408205\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	22.3303	1.18852	0.594261	−	0.804272i	$-0.297445\pi$
$353$	22.3303	1.18852	0.594261	+	0.804272i	$0.297445\pi$
$354$	0	0
$355$	2.00000	0.106149
$356$	0	0
$357$	−13.3739	−0.707820
$358$	0	0
$359$	16.7913	0.886210	0.443105	−	0.896470i	$-0.353877\pi$
$359$	16.7913	0.886210	0.443105	+	0.896470i	$0.353877\pi$
$360$	0	0
$361$	−18.9564	−0.997707
$362$	0	0
$363$	−9.41742	−0.494287
$364$	0	0
$365$	0.582576	0.0304934
$366$	0	0
$367$	2.04356	0.106673	0.0533365	−	0.998577i	$-0.483014\pi$
$367$	2.04356	0.106673	0.0533365	+	0.998577i	$0.483014\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−15.3739	−0.798171
$372$	0	0
$373$	27.1216	1.40430	0.702151	−	0.712028i	$-0.252223\pi$
$373$	27.1216	1.40430	0.702151	+	0.712028i	$0.252223\pi$
$374$	0	0
$375$	−25.1216	−1.29727
$376$	0	0
$377$	−9.00000	−0.463524
$378$	0	0
$379$	18.7477	0.963006	0.481503	−	0.876444i	$-0.340091\pi$
$379$	18.7477	0.963006	0.481503	+	0.876444i	$0.340091\pi$
$380$	0	0
$381$	47.3303	2.42480
$382$	0	0
$383$	−28.9129	−1.47738	−0.738690	−	0.674046i	$-0.764555\pi$
$383$	−28.9129	−1.47738	−0.738690	+	0.674046i	$0.764555\pi$
$384$	0	0
$385$	18.1652	0.925782
$386$	0	0
$387$	−43.1216	−2.19199
$388$	0	0
$389$	−1.04356	−0.0529106	−0.0264553	−	0.999650i	$-0.508422\pi$
$389$	−1.04356	−0.0529106	−0.0264553	+	0.999650i	$0.508422\pi$
$390$	0	0
$391$	1.58258	0.0800343
$392$	0	0
$393$	3.95644	0.199576
$394$	0	0
$395$	0.582576	0.0293126
$396$	0	0
$397$	−17.1652	−0.861494	−0.430747	−	0.902473i	$-0.641750\pi$
$397$	−17.1652	−0.861494	−0.430747	+	0.902473i	$0.641750\pi$
$398$	0	0
$399$	2.79129	0.139739
$400$	0	0
$401$	−18.9564	−0.946639	−0.473320	−	0.880891i	$-0.656944\pi$
$401$	−18.9564	−0.946639	−0.473320	+	0.880891i	$0.656944\pi$
$402$	0	0
$403$	8.58258	0.427529
$404$	0	0
$405$	0.417424	0.0207420
$406$	0	0
$407$	32.5390	1.61290
$408$	0	0
$409$	−5.16515	−0.255400	−0.127700	−	0.991813i	$-0.540760\pi$
$409$	−5.16515	−0.255400	−0.127700	+	0.991813i	$0.540760\pi$
$410$	0	0
$411$	−10.5826	−0.522000
$412$	0	0
$413$	2.79129	0.137350
$414$	0	0
$415$	−4.58258	−0.224950
$416$	0	0
$417$	57.9129	2.83601
$418$	0	0
$419$	10.6261	0.519121	0.259560	−	0.965727i	$-0.416422\pi$
$419$	10.6261	0.519121	0.259560	+	0.965727i	$0.416422\pi$
$420$	0	0
$421$	−6.20871	−0.302594	−0.151297	−	0.988488i	$-0.548345\pi$
$421$	−6.20871	−0.302594	−0.151297	+	0.988488i	$0.548345\pi$
$422$	0	0
$423$	26.7477	1.30052
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	34.5390	1.67146
$428$	0	0
$429$	−10.5826	−0.510932
$430$	0	0
$431$	20.8348	1.00358	0.501790	−	0.864990i	$-0.332675\pi$
$431$	20.8348	1.00358	0.501790	+	0.864990i	$0.332675\pi$
$432$	0	0
$433$	−9.16515	−0.440449	−0.220225	−	0.975449i	$-0.570679\pi$
$433$	−9.16515	−0.440449	−0.220225	+	0.975449i	$0.570679\pi$
$434$	0	0
$435$	25.1216	1.20449
$436$	0	0
$437$	−0.330303	−0.0158005
$438$	0	0
$439$	18.3739	0.876937	0.438468	−	0.898747i	$-0.355521\pi$
$439$	18.3739	0.876937	0.438468	+	0.898747i	$0.355521\pi$
$440$	0	0
$441$	76.4519	3.64057
$442$	0	0
$443$	−0.582576	−0.0276790	−0.0138395	−	0.999904i	$-0.504405\pi$
$443$	−0.582576	−0.0276790	−0.0138395	+	0.999904i	$0.504405\pi$
$444$	0	0
$445$	−6.95644	−0.329767
$446$	0	0
$447$	52.9129	2.50269
$448$	0	0
$449$	−15.1216	−0.713632	−0.356816	−	0.934175i	$-0.616138\pi$
$449$	−15.1216	−0.713632	−0.356816	+	0.934175i	$0.616138\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−24.4174	−1.14723
$454$	0	0
$455$	4.79129	0.224619
$456$	0	0
$457$	24.7042	1.15561	0.577806	−	0.816174i	$-0.303909\pi$
$457$	24.7042	1.15561	0.577806	+	0.816174i	$0.303909\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	0	0
$461$	27.7477	1.29234	0.646170	−	0.763193i	$-0.276370\pi$
$461$	27.7477	1.29234	0.646170	+	0.763193i	$0.276370\pi$
$462$	0	0
$463$	−15.7477	−0.731859	−0.365929	−	0.930643i	$-0.619249\pi$
$463$	−15.7477	−0.731859	−0.365929	+	0.930643i	$0.619249\pi$
$464$	0	0
$465$	−23.9564	−1.11095
$466$	0	0
$467$	−4.79129	−0.221714	−0.110857	−	0.993836i	$-0.535360\pi$
$467$	−4.79129	−0.221714	−0.110857	+	0.993836i	$0.535360\pi$
$468$	0	0
$469$	65.0780	3.00502
$470$	0	0
$471$	40.7042	1.87555
$472$	0	0
$473$	34.1216	1.56891
$474$	0	0
$475$	0.834849	0.0383055
$476$	0	0
$477$	−15.3739	−0.703921
$478$	0	0
$479$	−35.1216	−1.60475	−0.802373	−	0.596823i	$-0.796430\pi$
$479$	−35.1216	−1.60475	−0.802373	+	0.596823i	$0.796430\pi$
$480$	0	0
$481$	8.58258	0.391332
$482$	0	0
$483$	−21.1652	−0.963048
$484$	0	0
$485$	7.95644	0.361283
$486$	0	0
$487$	23.0000	1.04223	0.521115	−	0.853487i	$-0.325516\pi$
$487$	23.0000	1.04223	0.521115	+	0.853487i	$0.325516\pi$
$488$	0	0
$489$	31.2867	1.41484
$490$	0	0
$491$	12.9564	0.584716	0.292358	−	0.956309i	$-0.405560\pi$
$491$	12.9564	0.584716	0.292358	+	0.956309i	$0.405560\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	0	0
$495$	18.1652	0.816463
$496$	0	0
$497$	−9.58258	−0.429837
$498$	0	0
$499$	13.1652	0.589353	0.294677	−	0.955597i	$-0.404788\pi$
$499$	13.1652	0.589353	0.294677	+	0.955597i	$0.404788\pi$
$500$	0	0
$501$	70.7042	3.15883
$502$	0	0
$503$	−31.3739	−1.39889	−0.699446	−	0.714686i	$-0.746570\pi$
$503$	−31.3739	−1.39889	−0.699446	+	0.714686i	$0.746570\pi$
$504$	0	0
$505$	12.1652	0.541342
$506$	0	0
$507$	−2.79129	−0.123965
$508$	0	0
$509$	−17.4174	−0.772014	−0.386007	−	0.922496i	$-0.626146\pi$
$509$	−17.4174	−0.772014	−0.386007	+	0.922496i	$0.626146\pi$
$510$	0	0
$511$	−2.79129	−0.123479
$512$	0	0
$513$	1.04356	0.0460743
$514$	0	0
$515$	−1.00000	−0.0440653
$516$	0	0
$517$	−21.1652	−0.930842
$518$	0	0
$519$	28.3739	1.24547
$520$	0	0
$521$	13.4174	0.587828	0.293914	−	0.955832i	$-0.405042\pi$
$521$	13.4174	0.587828	0.293914	+	0.955832i	$0.405042\pi$
$522$	0	0
$523$	24.8693	1.08746	0.543730	−	0.839260i	$-0.317012\pi$
$523$	24.8693	1.08746	0.543730	+	0.839260i	$0.317012\pi$
$524$	0	0
$525$	53.4955	2.33473
$526$	0	0
$527$	−8.58258	−0.373863
$528$	0	0
$529$	−20.4955	−0.891107
$530$	0	0
$531$	2.79129	0.121132
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−7.37386	−0.318800
$536$	0	0
$537$	−50.7042	−2.18805
$538$	0	0
$539$	−60.4955	−2.60572
$540$	0	0
$541$	1.79129	0.0770135	0.0385067	−	0.999258i	$-0.487740\pi$
$541$	1.79129	0.0770135	0.0385067	+	0.999258i	$0.487740\pi$
$542$	0	0
$543$	42.4519	1.82179
$544$	0	0
$545$	−5.95644	−0.255146
$546$	0	0
$547$	35.8693	1.53366	0.766831	−	0.641849i	$-0.221833\pi$
$547$	35.8693	1.53366	0.766831	+	0.641849i	$0.221833\pi$
$548$	0	0
$549$	34.5390	1.47409
$550$	0	0
$551$	−1.87841	−0.0800229
$552$	0	0
$553$	−2.79129	−0.118698
$554$	0	0
$555$	−23.9564	−1.01689
$556$	0	0
$557$	14.4174	0.610886	0.305443	−	0.952210i	$-0.401195\pi$
$557$	14.4174	0.610886	0.305443	+	0.952210i	$0.401195\pi$
$558$	0	0
$559$	9.00000	0.380659
$560$	0	0
$561$	10.5826	0.446797
$562$	0	0
$563$	−39.9564	−1.68396	−0.841982	−	0.539506i	$-0.818611\pi$
$563$	−39.9564	−1.68396	−0.841982	+	0.539506i	$0.818611\pi$
$564$	0	0
$565$	−0.791288	−0.0332897
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−21.3739	−0.896039	−0.448019	−	0.894024i	$-0.647870\pi$
$569$	−21.3739	−0.896039	−0.448019	+	0.894024i	$0.647870\pi$
$570$	0	0
$571$	−35.1652	−1.47162	−0.735808	−	0.677190i	$-0.763197\pi$
$571$	−35.1652	−1.47162	−0.735808	+	0.677190i	$0.763197\pi$
$572$	0	0
$573$	−49.5390	−2.06952
$574$	0	0
$575$	−6.33030	−0.263992
$576$	0	0
$577$	30.2867	1.26085	0.630427	−	0.776249i	$-0.282880\pi$
$577$	30.2867	1.26085	0.630427	+	0.776249i	$0.282880\pi$
$578$	0	0
$579$	−26.7477	−1.11160
$580$	0	0
$581$	21.9564	0.910907
$582$	0	0
$583$	12.1652	0.503829
$584$	0	0
$585$	4.79129	0.198095
$586$	0	0
$587$	21.9564	0.906239	0.453120	−	0.891450i	$-0.350311\pi$
$587$	21.9564	0.906239	0.453120	+	0.891450i	$0.350311\pi$
$588$	0	0
$589$	1.79129	0.0738087
$590$	0	0
$591$	13.2523	0.545126
$592$	0	0
$593$	−11.6261	−0.477428	−0.238714	−	0.971090i	$-0.576726\pi$
$593$	−11.6261	−0.477428	−0.238714	+	0.971090i	$0.576726\pi$
$594$	0	0
$595$	−4.79129	−0.196424
$596$	0	0
$597$	30.7042	1.25664
$598$	0	0
$599$	−5.00000	−0.204294	−0.102147	−	0.994769i	$-0.532571\pi$
$599$	−5.00000	−0.204294	−0.102147	+	0.994769i	$0.532571\pi$
$600$	0	0
$601$	−35.4519	−1.44611	−0.723056	−	0.690789i	$-0.757263\pi$
$601$	−35.4519	−1.44611	−0.723056	+	0.690789i	$0.757263\pi$
$602$	0	0
$603$	65.0780	2.65018
$604$	0	0
$605$	−3.37386	−0.137167
$606$	0	0
$607$	−1.41742	−0.0575315	−0.0287657	−	0.999586i	$-0.509158\pi$
$607$	−1.41742	−0.0575315	−0.0287657	+	0.999586i	$0.509158\pi$
$608$	0	0
$609$	−120.365	−4.87743
$610$	0	0
$611$	−5.58258	−0.225847
$612$	0	0
$613$	−43.7477	−1.76695	−0.883477	−	0.468474i	$-0.844804\pi$
$613$	−43.7477	−1.76695	−0.883477	+	0.468474i	$0.844804\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.5826	0.828623	0.414312	−	0.910135i	$-0.364022\pi$
$617$	20.5826	0.828623	0.414312	+	0.910135i	$0.364022\pi$
$618$	0	0
$619$	−29.8348	−1.19916	−0.599582	−	0.800313i	$-0.704666\pi$
$619$	−29.8348	−1.19916	−0.599582	+	0.800313i	$0.704666\pi$
$620$	0	0
$621$	−7.91288	−0.317533
$622$	0	0
$623$	33.3303	1.33535
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−2.20871	−0.0882075
$628$	0	0
$629$	−8.58258	−0.342210
$630$	0	0
$631$	−13.8348	−0.550757	−0.275378	−	0.961336i	$-0.588803\pi$
$631$	−13.8348	−0.550757	−0.275378	+	0.961336i	$0.588803\pi$
$632$	0	0
$633$	57.4519	2.28351
$634$	0	0
$635$	16.9564	0.672896
$636$	0	0
$637$	−15.9564	−0.632217
$638$	0	0
$639$	−9.58258	−0.379081
$640$	0	0
$641$	2.58258	0.102006	0.0510028	−	0.998699i	$-0.483758\pi$
$641$	2.58258	0.102006	0.0510028	+	0.998699i	$0.483758\pi$
$642$	0	0
$643$	−28.7042	−1.13198	−0.565991	−	0.824411i	$-0.691506\pi$
$643$	−28.7042	−1.13198	−0.565991	+	0.824411i	$0.691506\pi$
$644$	0	0
$645$	−25.1216	−0.989162
$646$	0	0
$647$	25.2087	0.991057	0.495528	−	0.868592i	$-0.334974\pi$
$647$	25.2087	0.991057	0.495528	+	0.868592i	$0.334974\pi$
$648$	0	0
$649$	−2.20871	−0.0866995
$650$	0	0
$651$	114.782	4.49867
$652$	0	0
$653$	42.0780	1.64664	0.823320	−	0.567577i	$-0.192119\pi$
$653$	42.0780	1.64664	0.823320	+	0.567577i	$0.192119\pi$
$654$	0	0
$655$	1.41742	0.0553834
$656$	0	0
$657$	−2.79129	−0.108899
$658$	0	0
$659$	−10.9129	−0.425105	−0.212553	−	0.977150i	$-0.568178\pi$
$659$	−10.9129	−0.425105	−0.212553	+	0.977150i	$0.568178\pi$
$660$	0	0
$661$	38.7477	1.50711	0.753556	−	0.657384i	$-0.228337\pi$
$661$	38.7477	1.50711	0.753556	+	0.657384i	$0.228337\pi$
$662$	0	0
$663$	2.79129	0.108405
$664$	0	0
$665$	1.00000	0.0387783
$666$	0	0
$667$	14.2432	0.551498
$668$	0	0
$669$	35.7042	1.38040
$670$	0	0
$671$	−27.3303	−1.05507
$672$	0	0
$673$	31.9129	1.23015	0.615076	−	0.788468i	$-0.289126\pi$
$673$	31.9129	1.23015	0.615076	+	0.788468i	$0.289126\pi$
$674$	0	0
$675$	20.0000	0.769800
$676$	0	0
$677$	−45.8693	−1.76290	−0.881451	−	0.472276i	$-0.843432\pi$
$677$	−45.8693	−1.76290	−0.881451	+	0.472276i	$0.843432\pi$
$678$	0	0
$679$	−38.1216	−1.46297
$680$	0	0
$681$	4.41742	0.169276
$682$	0	0
$683$	−13.9129	−0.532361	−0.266181	−	0.963923i	$-0.585762\pi$
$683$	−13.9129	−0.532361	−0.266181	+	0.963923i	$0.585762\pi$
$684$	0	0
$685$	−3.79129	−0.144858
$686$	0	0
$687$	−3.95644	−0.150948
$688$	0	0
$689$	3.20871	0.122242
$690$	0	0
$691$	−4.08712	−0.155481	−0.0777407	−	0.996974i	$-0.524771\pi$
$691$	−4.08712	−0.155481	−0.0777407	+	0.996974i	$0.524771\pi$
$692$	0	0
$693$	−87.0345	−3.30617
$694$	0	0
$695$	20.7477	0.787006
$696$	0	0
$697$	0	0
$698$	0	0
$699$	75.1216	2.84136
$700$	0	0
$701$	−26.9564	−1.01813	−0.509065	−	0.860728i	$-0.670009\pi$
$701$	−26.9564	−1.01813	−0.509065	+	0.860728i	$0.670009\pi$
$702$	0	0
$703$	1.79129	0.0675597
$704$	0	0
$705$	15.5826	0.586874
$706$	0	0
$707$	−58.2867	−2.19210
$708$	0	0
$709$	−30.5390	−1.14692	−0.573458	−	0.819235i	$-0.694399\pi$
$709$	−30.5390	−1.14692	−0.573458	+	0.819235i	$0.694399\pi$
$710$	0	0
$711$	−2.79129	−0.104681
$712$	0	0
$713$	−13.5826	−0.508671
$714$	0	0
$715$	−3.79129	−0.141786
$716$	0	0
$717$	4.87841	0.182188
$718$	0	0
$719$	17.1652	0.640152	0.320076	−	0.947392i	$-0.396292\pi$
$719$	17.1652	0.640152	0.320076	+	0.947392i	$0.396292\pi$
$720$	0	0
$721$	4.79129	0.178437
$722$	0	0
$723$	27.2087	1.01190
$724$	0	0
$725$	−36.0000	−1.33701
$726$	0	0
$727$	−32.3303	−1.19906	−0.599532	−	0.800351i	$-0.704647\pi$
$727$	−32.3303	−1.19906	−0.599532	+	0.800351i	$0.704647\pi$
$728$	0	0
$729$	−43.8693	−1.62479
$730$	0	0
$731$	−9.00000	−0.332877
$732$	0	0
$733$	22.1652	0.818689	0.409344	−	0.912380i	$-0.365758\pi$
$733$	22.1652	0.818689	0.409344	+	0.912380i	$0.365758\pi$
$734$	0	0
$735$	44.5390	1.64285
$736$	0	0
$737$	−51.4955	−1.89686
$738$	0	0
$739$	−12.7042	−0.467330	−0.233665	−	0.972317i	$-0.575072\pi$
$739$	−12.7042	−0.467330	−0.233665	+	0.972317i	$0.575072\pi$
$740$	0	0
$741$	−0.582576	−0.0214015
$742$	0	0
$743$	−22.7477	−0.834533	−0.417267	−	0.908784i	$-0.637012\pi$
$743$	−22.7477	−0.834533	−0.417267	+	0.908784i	$0.637012\pi$
$744$	0	0
$745$	18.9564	0.694510
$746$	0	0
$747$	21.9564	0.803344
$748$	0	0
$749$	35.3303	1.29094
$750$	0	0
$751$	−14.3303	−0.522920	−0.261460	−	0.965214i	$-0.584204\pi$
$751$	−14.3303	−0.522920	−0.261460	+	0.965214i	$0.584204\pi$
$752$	0	0
$753$	−28.9564	−1.05523
$754$	0	0
$755$	−8.74773	−0.318362
$756$	0	0
$757$	5.37386	0.195316	0.0976582	−	0.995220i	$-0.468865\pi$
$757$	5.37386	0.195316	0.0976582	+	0.995220i	$0.468865\pi$
$758$	0	0
$759$	16.7477	0.607904
$760$	0	0
$761$	−2.91288	−0.105592	−0.0527959	−	0.998605i	$-0.516813\pi$
$761$	−2.91288	−0.105592	−0.0527959	+	0.998605i	$0.516813\pi$
$762$	0	0
$763$	28.5390	1.03318
$764$	0	0
$765$	−4.79129	−0.173229
$766$	0	0
$767$	−0.582576	−0.0210356
$768$	0	0
$769$	−40.7913	−1.47097	−0.735486	−	0.677540i	$-0.763046\pi$
$769$	−40.7913	−1.47097	−0.735486	+	0.677540i	$0.763046\pi$
$770$	0	0
$771$	−22.7913	−0.820808
$772$	0	0
$773$	9.79129	0.352168	0.176084	−	0.984375i	$-0.443657\pi$
$773$	9.79129	0.352168	0.176084	+	0.984375i	$0.443657\pi$
$774$	0	0
$775$	34.3303	1.23318
$776$	0	0
$777$	114.782	4.11779
$778$	0	0
$779$	0	0
$780$	0	0
$781$	7.58258	0.271326
$782$	0	0
$783$	−45.0000	−1.60817
$784$	0	0
$785$	14.5826	0.520474
$786$	0	0
$787$	−31.2432	−1.11370	−0.556850	−	0.830613i	$-0.687990\pi$
$787$	−31.2432	−1.11370	−0.556850	+	0.830613i	$0.687990\pi$
$788$	0	0
$789$	−5.58258	−0.198745
$790$	0	0
$791$	3.79129	0.134803
$792$	0	0
$793$	−7.20871	−0.255989
$794$	0	0
$795$	−8.95644	−0.317652
$796$	0	0
$797$	44.8693	1.58935	0.794676	−	0.607033i	$-0.207641\pi$
$797$	44.8693	1.58935	0.794676	+	0.607033i	$0.207641\pi$
$798$	0	0
$799$	5.58258	0.197497
$800$	0	0
$801$	33.3303	1.17767
$802$	0	0
$803$	2.20871	0.0779438
$804$	0	0
$805$	−7.58258	−0.267251
$806$	0	0
$807$	18.4955	0.651071
$808$	0	0
$809$	−34.3303	−1.20699	−0.603495	−	0.797367i	$-0.706226\pi$
$809$	−34.3303	−1.20699	−0.603495	+	0.797367i	$0.706226\pi$
$810$	0	0
$811$	−30.1652	−1.05924	−0.529621	−	0.848234i	$-0.677666\pi$
$811$	−30.1652	−1.05924	−0.529621	+	0.848234i	$0.677666\pi$
$812$	0	0
$813$	84.7822	2.97344
$814$	0	0
$815$	11.2087	0.392624
$816$	0	0
$817$	1.87841	0.0657172
$818$	0	0
$819$	−22.9564	−0.802163
$820$	0	0
$821$	46.4083	1.61966	0.809831	−	0.586663i	$-0.199559\pi$
$821$	46.4083	1.61966	0.809831	+	0.586663i	$0.199559\pi$
$822$	0	0
$823$	−7.41742	−0.258555	−0.129278	−	0.991608i	$-0.541266\pi$
$823$	−7.41742	−0.258555	−0.129278	+	0.991608i	$0.541266\pi$
$824$	0	0
$825$	−42.3303	−1.47375
$826$	0	0
$827$	−37.4519	−1.30233	−0.651165	−	0.758936i	$-0.725719\pi$
$827$	−37.4519	−1.30233	−0.651165	+	0.758936i	$0.725719\pi$
$828$	0	0
$829$	6.41742	0.222886	0.111443	−	0.993771i	$-0.464453\pi$
$829$	6.41742	0.222886	0.111443	+	0.993771i	$0.464453\pi$
$830$	0	0
$831$	19.0780	0.661810
$832$	0	0
$833$	15.9564	0.552858
$834$	0	0
$835$	25.3303	0.876591
$836$	0	0
$837$	42.9129	1.48329
$838$	0	0
$839$	8.20871	0.283396	0.141698	−	0.989910i	$-0.454744\pi$
$839$	8.20871	0.283396	0.141698	+	0.989910i	$0.454744\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	81.2867	2.79966
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	16.1652	0.555441
$848$	0	0
$849$	14.5390	0.498978
$850$	0	0
$851$	−13.5826	−0.465605
$852$	0	0
$853$	−44.1652	−1.51219	−0.756093	−	0.654464i	$-0.772894\pi$
$853$	−44.1652	−1.51219	−0.756093	+	0.654464i	$0.772894\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	−28.0780	−0.959127	−0.479564	−	0.877507i	$-0.659205\pi$
$857$	−28.0780	−0.959127	−0.479564	+	0.877507i	$0.659205\pi$
$858$	0	0
$859$	−18.5390	−0.632543	−0.316272	−	0.948669i	$-0.602431\pi$
$859$	−18.5390	−0.632543	−0.316272	+	0.948669i	$0.602431\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	0	0
$865$	10.1652	0.345626
$866$	0	0
$867$	−2.79129	−0.0947971
$868$	0	0
$869$	2.20871	0.0749254
$870$	0	0
$871$	−13.5826	−0.460228
$872$	0	0
$873$	−38.1216	−1.29022
$874$	0	0
$875$	43.1216	1.45778
$876$	0	0
$877$	27.3739	0.924350	0.462175	−	0.886789i	$-0.347069\pi$
$877$	27.3739	0.924350	0.462175	+	0.886789i	$0.347069\pi$
$878$	0	0
$879$	−49.1996	−1.65946
$880$	0	0
$881$	−1.46099	−0.0492218	−0.0246109	−	0.999697i	$-0.507835\pi$
$881$	−1.46099	−0.0492218	−0.0246109	+	0.999697i	$0.507835\pi$
$882$	0	0
$883$	19.0000	0.639401	0.319700	−	0.947519i	$-0.396418\pi$
$883$	19.0000	0.639401	0.319700	+	0.947519i	$0.396418\pi$
$884$	0	0
$885$	1.62614	0.0546620
$886$	0	0
$887$	50.9564	1.71095	0.855475	−	0.517844i	$-0.173265\pi$
$887$	50.9564	1.71095	0.855475	+	0.517844i	$0.173265\pi$
$888$	0	0
$889$	−81.2432	−2.72481
$890$	0	0
$891$	1.58258	0.0530183
$892$	0	0
$893$	−1.16515	−0.0389903
$894$	0	0
$895$	−18.1652	−0.607194
$896$	0	0
$897$	4.41742	0.147494
$898$	0	0
$899$	−77.2432	−2.57620
$900$	0	0
$901$	−3.20871	−0.106898
$902$	0	0
$903$	120.365	4.00549
$904$	0	0
$905$	15.2087	0.505555
$906$	0	0
$907$	−22.6261	−0.751289	−0.375644	−	0.926764i	$-0.622579\pi$
$907$	−22.6261	−0.751289	−0.375644	+	0.926764i	$0.622579\pi$
$908$	0	0
$909$	−58.2867	−1.93325
$910$	0	0
$911$	23.8348	0.789684	0.394842	−	0.918749i	$-0.370799\pi$
$911$	23.8348	0.789684	0.394842	+	0.918749i	$0.370799\pi$
$912$	0	0
$913$	−17.3739	−0.574991
$914$	0	0
$915$	20.1216	0.665199
$916$	0	0
$917$	−6.79129	−0.224268
$918$	0	0
$919$	34.4174	1.13533	0.567663	−	0.823261i	$-0.307848\pi$
$919$	34.4174	1.13533	0.567663	+	0.823261i	$0.307848\pi$
$920$	0	0
$921$	55.2432	1.82032
$922$	0	0
$923$	2.00000	0.0658308
$924$	0	0
$925$	34.3303	1.12877
$926$	0	0
$927$	4.79129	0.157367
$928$	0	0
$929$	−4.62614	−0.151779	−0.0758893	−	0.997116i	$-0.524180\pi$
$929$	−4.62614	−0.151779	−0.0758893	+	0.997116i	$0.524180\pi$
$930$	0	0
$931$	−3.33030	−0.109146
$932$	0	0
$933$	63.0345	2.06366
$934$	0	0
$935$	3.79129	0.123988
$936$	0	0
$937$	−0.252273	−0.00824140	−0.00412070	−	0.999992i	$-0.501312\pi$
$937$	−0.252273	−0.00824140	−0.00412070	+	0.999992i	$0.501312\pi$
$938$	0	0
$939$	2.79129	0.0910902
$940$	0	0
$941$	−29.7913	−0.971168	−0.485584	−	0.874190i	$-0.661393\pi$
$941$	−29.7913	−0.971168	−0.485584	+	0.874190i	$0.661393\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	23.9564	0.779303
$946$	0	0
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	0.582576	0.0189112
$950$	0	0
$951$	−47.4519	−1.53873
$952$	0	0
$953$	27.5826	0.893487	0.446744	−	0.894662i	$-0.352584\pi$
$953$	27.5826	0.893487	0.446744	+	0.894662i	$0.352584\pi$
$954$	0	0
$955$	−17.7477	−0.574303
$956$	0	0
$957$	95.2432	3.07877
$958$	0	0
$959$	18.1652	0.586583
$960$	0	0
$961$	42.6606	1.37615
$962$	0	0
$963$	35.3303	1.13850
$964$	0	0
$965$	−9.58258	−0.308474
$966$	0	0
$967$	44.5826	1.43368	0.716839	−	0.697238i	$-0.245588\pi$
$967$	44.5826	1.43368	0.716839	+	0.697238i	$0.245588\pi$
$968$	0	0
$969$	0.582576	0.0187150
$970$	0	0
$971$	−26.0436	−0.835778	−0.417889	−	0.908498i	$-0.637230\pi$
$971$	−26.0436	−0.835778	−0.417889	+	0.908498i	$0.637230\pi$
$972$	0	0
$973$	−99.4083	−3.18688
$974$	0	0
$975$	−11.1652	−0.357571
$976$	0	0
$977$	33.9564	1.08636	0.543181	−	0.839615i	$-0.317220\pi$
$977$	33.9564	1.08636	0.543181	+	0.839615i	$0.317220\pi$
$978$	0	0
$979$	−26.3739	−0.842912
$980$	0	0
$981$	28.5390	0.911181
$982$	0	0
$983$	28.0780	0.895550	0.447775	−	0.894146i	$-0.352217\pi$
$983$	28.0780	0.895550	0.447775	+	0.894146i	$0.352217\pi$
$984$	0	0
$985$	4.74773	0.151275
$986$	0	0
$987$	−74.6606	−2.37647
$988$	0	0
$989$	−14.2432	−0.452907
$990$	0	0
$991$	7.79129	0.247498	0.123749	−	0.992314i	$-0.460508\pi$
$991$	7.79129	0.247498	0.123749	+	0.992314i	$0.460508\pi$
$992$	0	0
$993$	−30.7042	−0.974367
$994$	0	0
$995$	11.0000	0.348723
$996$	0	0
$997$	−42.0780	−1.33262	−0.666312	−	0.745673i	$-0.732128\pi$
$997$	−42.0780	−1.33262	−0.666312	+	0.745673i	$0.732128\pi$
$998$	0	0
$999$	42.9129	1.35770

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3536.2.a.r.1.1		2
4.3	odd	2		221.2.a.d.1.2	✓	2
12.11	even	2		1989.2.a.h.1.1		2
20.19	odd	2		5525.2.a.p.1.1		2
52.51	odd	2		2873.2.a.i.1.1		2
68.67	odd	2		3757.2.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.a.d.1.2	✓	2	4.3	odd	2
1989.2.a.h.1.1		2	12.11	even	2
2873.2.a.i.1.1		2	52.51	odd	2
3536.2.a.r.1.1		2	1.1	even	1	trivial
3757.2.a.g.1.2		2	68.67	odd	2
5525.2.a.p.1.1		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3536 = 2^{4} \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3536.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79129 q^{3} -1.00000 q^{5} +4.79129 q^{7} +4.79129 q^{9} +O(q^{10})\)	\(q-2.79129 q^{3} -1.00000 q^{5} +4.79129 q^{7} +4.79129 q^{9} -3.79129 q^{11} -1.00000 q^{13} +2.79129 q^{15} +1.00000 q^{17} -0.208712 q^{19} -13.3739 q^{21} +1.58258 q^{23} -4.00000 q^{25} -5.00000 q^{27} +9.00000 q^{29} -8.58258 q^{31} +10.5826 q^{33} -4.79129 q^{35} -8.58258 q^{37} +2.79129 q^{39} -9.00000 q^{43} -4.79129 q^{45} +5.58258 q^{47} +15.9564 q^{49} -2.79129 q^{51} -3.20871 q^{53} +3.79129 q^{55} +0.582576 q^{57} +0.582576 q^{59} +7.20871 q^{61} +22.9564 q^{63} +1.00000 q^{65} +13.5826 q^{67} -4.41742 q^{69} -2.00000 q^{71} -0.582576 q^{73} +11.1652 q^{75} -18.1652 q^{77} -0.582576 q^{79} -0.417424 q^{81} +4.58258 q^{83} -1.00000 q^{85} -25.1216 q^{87} +6.95644 q^{89} -4.79129 q^{91} +23.9564 q^{93} +0.208712 q^{95} -7.95644 q^{97} -18.1652 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 5 * q^9	\( 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 5 q^{9} - 3 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} - 5 q^{19} - 13 q^{21} - 6 q^{23} - 8 q^{25} - 10 q^{27} + 18 q^{29} - 8 q^{31} + 12 q^{33} - 5 q^{35} - 8 q^{37} + q^{39} - 18 q^{43} - 5 q^{45} + 2 q^{47} + 9 q^{49} - q^{51} - 11 q^{53} + 3 q^{55} - 8 q^{57} - 8 q^{59} + 19 q^{61} + 23 q^{63} + 2 q^{65} + 18 q^{67} - 18 q^{69} - 4 q^{71} + 8 q^{73} + 4 q^{75} - 18 q^{77} + 8 q^{79} - 10 q^{81} - 2 q^{85} - 9 q^{87} - 9 q^{89} - 5 q^{91} + 25 q^{93} + 5 q^{95} + 7 q^{97} - 18 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 5 * q^9 - 3 * q^11 - 2 * q^13 + q^15 + 2 * q^17 - 5 * q^19 - 13 * q^21 - 6 * q^23 - 8 * q^25 - 10 * q^27 + 18 * q^29 - 8 * q^31 + 12 * q^33 - 5 * q^35 - 8 * q^37 + q^39 - 18 * q^43 - 5 * q^45 + 2 * q^47 + 9 * q^49 - q^51 - 11 * q^53 + 3 * q^55 - 8 * q^57 - 8 * q^59 + 19 * q^61 + 23 * q^63 + 2 * q^65 + 18 * q^67 - 18 * q^69 - 4 * q^71 + 8 * q^73 + 4 * q^75 - 18 * q^77 + 8 * q^79 - 10 * q^81 - 2 * q^85 - 9 * q^87 - 9 * q^89 - 5 * q^91 + 25 * q^93 + 5 * q^95 + 7 * q^97 - 18 * q^99

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