LMFDB - Embedded newform 8512.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, 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("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
Analytic rank:	$0$
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 532)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.79129 q^{3} -3.00000 q^{5} +1.00000 q^{7} +4.79129 q^{9} +O(q^{10})$	$q+2.79129 q^{3} -3.00000 q^{5} +1.00000 q^{7} +4.79129 q^{9} +3.79129 q^{11} +1.00000 q^{13} -8.37386 q^{15} +3.79129 q^{17} -1.00000 q^{19} +2.79129 q^{21} +4.58258 q^{23} +4.00000 q^{25} +5.00000 q^{27} -3.79129 q^{29} +7.37386 q^{31} +10.5826 q^{33} -3.00000 q^{35} -5.00000 q^{37} +2.79129 q^{39} -3.79129 q^{41} -2.00000 q^{43} -14.3739 q^{45} +10.5826 q^{47} +1.00000 q^{49} +10.5826 q^{51} +8.37386 q^{53} -11.3739 q^{55} -2.79129 q^{57} -12.1652 q^{59} +1.00000 q^{61} +4.79129 q^{63} -3.00000 q^{65} +9.37386 q^{67} +12.7913 q^{69} -12.1652 q^{71} +16.3739 q^{73} +11.1652 q^{75} +3.79129 q^{77} -10.0000 q^{79} -0.417424 q^{81} -14.3739 q^{83} -11.3739 q^{85} -10.5826 q^{87} +7.58258 q^{89} +1.00000 q^{91} +20.5826 q^{93} +3.00000 q^{95} -7.00000 q^{97} +18.1652 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 6 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q + q^3 - 6 * q^5 + 2 * q^7 + 5 * q^9	$ 2 q + q^{3} - 6 q^{5} + 2 q^{7} + 5 q^{9} + 3 q^{11} + 2 q^{13} - 3 q^{15} + 3 q^{17} - 2 q^{19} + q^{21} + 8 q^{25} + 10 q^{27} - 3 q^{29} + q^{31} + 12 q^{33} - 6 q^{35} - 10 q^{37} + q^{39} - 3 q^{41} - 4 q^{43} - 15 q^{45} + 12 q^{47} + 2 q^{49} + 12 q^{51} + 3 q^{53} - 9 q^{55} - q^{57} - 6 q^{59} + 2 q^{61} + 5 q^{63} - 6 q^{65} + 5 q^{67} + 21 q^{69} - 6 q^{71} + 19 q^{73} + 4 q^{75} + 3 q^{77} - 20 q^{79} - 10 q^{81} - 15 q^{83} - 9 q^{85} - 12 q^{87} + 6 q^{89} + 2 q^{91} + 32 q^{93} + 6 q^{95} - 14 q^{97} + 18 q^{99}+O(q^{100}) $ 2 * q + q^3 - 6 * q^5 + 2 * q^7 + 5 * q^9 + 3 * q^11 + 2 * q^13 - 3 * q^15 + 3 * q^17 - 2 * q^19 + q^21 + 8 * q^25 + 10 * q^27 - 3 * q^29 + q^31 + 12 * q^33 - 6 * q^35 - 10 * q^37 + q^39 - 3 * q^41 - 4 * q^43 - 15 * q^45 + 12 * q^47 + 2 * q^49 + 12 * q^51 + 3 * q^53 - 9 * q^55 - q^57 - 6 * q^59 + 2 * q^61 + 5 * q^63 - 6 * q^65 + 5 * q^67 + 21 * q^69 - 6 * q^71 + 19 * q^73 + 4 * q^75 + 3 * q^77 - 20 * q^79 - 10 * q^81 - 15 * q^83 - 9 * q^85 - 12 * q^87 + 6 * q^89 + 2 * q^91 + 32 * q^93 + 6 * q^95 - 14 * 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	$-0.201749\pi$
$3$	2.79129	1.61155	0.805775	+	0.592221i	$0.201749\pi$
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.79129	1.59710
$10$	0	0
$11$	3.79129	1.14312	0.571558	−	0.820562i	$-0.306339\pi$
$11$	3.79129	1.14312	0.571558	+	0.820562i	$0.306339\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	−8.37386	−2.16212
$16$	0	0
$17$	3.79129	0.919522	0.459761	−	0.888043i	$-0.347935\pi$
$17$	3.79129	0.919522	0.459761	+	0.888043i	$0.347935\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.79129	0.609109
$22$	0	0
$23$	4.58258	0.955533	0.477767	−	0.878487i	$-0.341446\pi$
$23$	4.58258	0.955533	0.477767	+	0.878487i	$0.341446\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−3.79129	−0.704024	−0.352012	−	0.935995i	$-0.614502\pi$
$29$	−3.79129	−0.704024	−0.352012	+	0.935995i	$0.614502\pi$
$30$	0	0
$31$	7.37386	1.32438	0.662192	−	0.749334i	$-0.269626\pi$
$31$	7.37386	1.32438	0.662192	+	0.749334i	$0.269626\pi$
$32$	0	0
$33$	10.5826	1.84219
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	2.79129	0.446964
$40$	0	0
$41$	−3.79129	−0.592100	−0.296050	−	0.955172i	$-0.595669\pi$
$41$	−3.79129	−0.592100	−0.296050	+	0.955172i	$0.595669\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−14.3739	−2.14273
$46$	0	0
$47$	10.5826	1.54363	0.771814	−	0.635849i	$-0.219350\pi$
$47$	10.5826	1.54363	0.771814	+	0.635849i	$0.219350\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	10.5826	1.48186
$52$	0	0
$53$	8.37386	1.15024	0.575119	−	0.818070i	$-0.304956\pi$
$53$	8.37386	1.15024	0.575119	+	0.818070i	$0.304956\pi$
$54$	0	0
$55$	−11.3739	−1.53365
$56$	0	0
$57$	−2.79129	−0.369715
$58$	0	0
$59$	−12.1652	−1.58377	−0.791884	−	0.610672i	$-0.790900\pi$
$59$	−12.1652	−1.58377	−0.791884	+	0.610672i	$0.790900\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	4.79129	0.603646
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	9.37386	1.14520	0.572600	−	0.819835i	$-0.305935\pi$
$67$	9.37386	1.14520	0.572600	+	0.819835i	$0.305935\pi$
$68$	0	0
$69$	12.7913	1.53989
$70$	0	0
$71$	−12.1652	−1.44374	−0.721869	−	0.692030i	$-0.756717\pi$
$71$	−12.1652	−1.44374	−0.721869	+	0.692030i	$0.756717\pi$
$72$	0	0
$73$	16.3739	1.91642	0.958208	−	0.286073i	$-0.0923499\pi$
$73$	16.3739	1.91642	0.958208	+	0.286073i	$0.0923499\pi$
$74$	0	0
$75$	11.1652	1.28924
$76$	0	0
$77$	3.79129	0.432057
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	−0.417424	−0.0463805
$82$	0	0
$83$	−14.3739	−1.57774	−0.788868	−	0.614562i	$-0.789333\pi$
$83$	−14.3739	−1.57774	−0.788868	+	0.614562i	$0.789333\pi$
$84$	0	0
$85$	−11.3739	−1.23367
$86$	0	0
$87$	−10.5826	−1.13457
$88$	0	0
$89$	7.58258	0.803751	0.401876	−	0.915694i	$-0.368358\pi$
$89$	7.58258	0.803751	0.401876	+	0.915694i	$0.368358\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	20.5826	2.13431
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	0	0
$99$	18.1652	1.82567
$100$	0	0
$101$	7.74773	0.770928	0.385464	−	0.922723i	$-0.374041\pi$
$101$	7.74773	0.770928	0.385464	+	0.922723i	$0.374041\pi$
$102$	0	0
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	0	0
$105$	−8.37386	−0.817205
$106$	0	0
$107$	−0.165151	−0.0159658	−0.00798289	−	0.999968i	$-0.502541\pi$
$107$	−0.165151	−0.0159658	−0.00798289	+	0.999968i	$0.502541\pi$
$108$	0	0
$109$	11.7477	1.12523	0.562614	−	0.826720i	$-0.309796\pi$
$109$	11.7477	1.12523	0.562614	+	0.826720i	$0.309796\pi$
$110$	0	0
$111$	−13.9564	−1.32469
$112$	0	0
$113$	8.37386	0.787747	0.393873	−	0.919165i	$-0.371135\pi$
$113$	8.37386	0.787747	0.393873	+	0.919165i	$0.371135\pi$
$114$	0	0
$115$	−13.7477	−1.28198
$116$	0	0
$117$	4.79129	0.442955
$118$	0	0
$119$	3.79129	0.347547
$120$	0	0
$121$	3.37386	0.306715
$122$	0	0
$123$	−10.5826	−0.954199
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	6.74773	0.598764	0.299382	−	0.954133i	$-0.403220\pi$
$127$	6.74773	0.598764	0.299382	+	0.954133i	$0.403220\pi$
$128$	0	0
$129$	−5.58258	−0.491518
$130$	0	0
$131$	22.1216	1.93277	0.966386	−	0.257095i	$-0.0827653\pi$
$131$	22.1216	1.93277	0.966386	+	0.257095i	$0.0827653\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−15.0000	−1.29099
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−9.74773	−0.826791	−0.413396	−	0.910551i	$-0.635657\pi$
$139$	−9.74773	−0.826791	−0.413396	+	0.910551i	$0.635657\pi$
$140$	0	0
$141$	29.5390	2.48763
$142$	0	0
$143$	3.79129	0.317043
$144$	0	0
$145$	11.3739	0.944548
$146$	0	0
$147$	2.79129	0.230222
$148$	0	0
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	13.3739	1.08835	0.544175	−	0.838972i	$-0.316843\pi$
$151$	13.3739	1.08835	0.544175	+	0.838972i	$0.316843\pi$
$152$	0	0
$153$	18.1652	1.46857
$154$	0	0
$155$	−22.1216	−1.77685
$156$	0	0
$157$	20.1216	1.60588	0.802939	−	0.596061i	$-0.203269\pi$
$157$	20.1216	1.60588	0.802939	+	0.596061i	$0.203269\pi$
$158$	0	0
$159$	23.3739	1.85367
$160$	0	0
$161$	4.58258	0.361158
$162$	0	0
$163$	−4.37386	−0.342587	−0.171294	−	0.985220i	$-0.554795\pi$
$163$	−4.37386	−0.342587	−0.171294	+	0.985220i	$0.554795\pi$
$164$	0	0
$165$	−31.7477	−2.47156
$166$	0	0
$167$	13.4174	1.03827	0.519136	−	0.854692i	$-0.326254\pi$
$167$	13.4174	1.03827	0.519136	+	0.854692i	$0.326254\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−4.79129	−0.366399
$172$	0	0
$173$	−22.7477	−1.72948	−0.864739	−	0.502222i	$-0.832516\pi$
$173$	−22.7477	−1.72948	−0.864739	+	0.502222i	$0.832516\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	−33.9564	−2.55232
$178$	0	0
$179$	15.7913	1.18030	0.590148	−	0.807295i	$-0.299069\pi$
$179$	15.7913	1.18030	0.590148	+	0.807295i	$0.299069\pi$
$180$	0	0
$181$	9.37386	0.696754	0.348377	−	0.937355i	$-0.386733\pi$
$181$	9.37386	0.696754	0.348377	+	0.937355i	$0.386733\pi$
$182$	0	0
$183$	2.79129	0.206338
$184$	0	0
$185$	15.0000	1.10282
$186$	0	0
$187$	14.3739	1.05112
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	23.5390	1.70322	0.851612	−	0.524173i	$-0.175626\pi$
$191$	23.5390	1.70322	0.851612	+	0.524173i	$0.175626\pi$
$192$	0	0
$193$	7.37386	0.530782	0.265391	−	0.964141i	$-0.414499\pi$
$193$	7.37386	0.530782	0.265391	+	0.964141i	$0.414499\pi$
$194$	0	0
$195$	−8.37386	−0.599665
$196$	0	0
$197$	2.37386	0.169131	0.0845654	−	0.996418i	$-0.473050\pi$
$197$	2.37386	0.169131	0.0845654	+	0.996418i	$0.473050\pi$
$198$	0	0
$199$	−23.7477	−1.68343	−0.841716	−	0.539921i	$-0.818454\pi$
$199$	−23.7477	−1.68343	−0.841716	+	0.539921i	$0.818454\pi$
$200$	0	0
$201$	26.1652	1.84555
$202$	0	0
$203$	−3.79129	−0.266096
$204$	0	0
$205$	11.3739	0.794385
$206$	0	0
$207$	21.9564	1.52608
$208$	0	0
$209$	−3.79129	−0.262249
$210$	0	0
$211$	24.3739	1.67797	0.838983	−	0.544158i	$-0.183151\pi$
$211$	24.3739	1.67797	0.838983	+	0.544158i	$0.183151\pi$
$212$	0	0
$213$	−33.9564	−2.32666
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	7.37386	0.500570
$218$	0	0
$219$	45.7042	3.08840
$220$	0	0
$221$	3.79129	0.255030
$222$	0	0
$223$	−23.7477	−1.59027	−0.795133	−	0.606435i	$-0.792599\pi$
$223$	−23.7477	−1.59027	−0.795133	+	0.606435i	$0.792599\pi$
$224$	0	0
$225$	19.1652	1.27768
$226$	0	0
$227$	3.79129	0.251637	0.125818	−	0.992053i	$-0.459844\pi$
$227$	3.79129	0.251637	0.125818	+	0.992053i	$0.459844\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	10.5826	0.696282
$232$	0	0
$233$	−28.1216	−1.84231	−0.921153	−	0.389200i	$-0.872752\pi$
$233$	−28.1216	−1.84231	−0.921153	+	0.389200i	$0.872752\pi$
$234$	0	0
$235$	−31.7477	−2.07099
$236$	0	0
$237$	−27.9129	−1.81314
$238$	0	0
$239$	−7.74773	−0.501159	−0.250579	−	0.968096i	$-0.580621\pi$
$239$	−7.74773	−0.501159	−0.250579	+	0.968096i	$0.580621\pi$
$240$	0	0
$241$	−8.74773	−0.563491	−0.281745	−	0.959489i	$-0.590913\pi$
$241$	−8.74773	−0.563491	−0.281745	+	0.959489i	$0.590913\pi$
$242$	0	0
$243$	−16.1652	−1.03699
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	−40.1216	−2.54260
$250$	0	0
$251$	−9.79129	−0.618021	−0.309010	−	0.951059i	$-0.599998\pi$
$251$	−9.79129	−0.618021	−0.309010	+	0.951059i	$0.599998\pi$
$252$	0	0
$253$	17.3739	1.09229
$254$	0	0
$255$	−31.7477	−1.98812
$256$	0	0
$257$	11.2087	0.699180	0.349590	−	0.936903i	$-0.386321\pi$
$257$	11.2087	0.699180	0.349590	+	0.936903i	$0.386321\pi$
$258$	0	0
$259$	−5.00000	−0.310685
$260$	0	0
$261$	−18.1652	−1.12439
$262$	0	0
$263$	6.79129	0.418769	0.209384	−	0.977833i	$-0.432854\pi$
$263$	6.79129	0.418769	0.209384	+	0.977833i	$0.432854\pi$
$264$	0	0
$265$	−25.1216	−1.54321
$266$	0	0
$267$	21.1652	1.29529
$268$	0	0
$269$	9.95644	0.607055	0.303527	−	0.952823i	$-0.401836\pi$
$269$	9.95644	0.607055	0.303527	+	0.952823i	$0.401836\pi$
$270$	0	0
$271$	−14.1216	−0.857826	−0.428913	−	0.903346i	$-0.641103\pi$
$271$	−14.1216	−0.857826	−0.428913	+	0.903346i	$0.641103\pi$
$272$	0	0
$273$	2.79129	0.168936
$274$	0	0
$275$	15.1652	0.914493
$276$	0	0
$277$	−3.25227	−0.195410	−0.0977051	−	0.995215i	$-0.531150\pi$
$277$	−3.25227	−0.195410	−0.0977051	+	0.995215i	$0.531150\pi$
$278$	0	0
$279$	35.3303	2.11517
$280$	0	0
$281$	19.4174	1.15835	0.579173	−	0.815205i	$-0.303375\pi$
$281$	19.4174	1.15835	0.579173	+	0.815205i	$0.303375\pi$
$282$	0	0
$283$	6.37386	0.378887	0.189443	−	0.981892i	$-0.439332\pi$
$283$	6.37386	0.378887	0.189443	+	0.981892i	$0.439332\pi$
$284$	0	0
$285$	8.37386	0.496025
$286$	0	0
$287$	−3.79129	−0.223793
$288$	0	0
$289$	−2.62614	−0.154479
$290$	0	0
$291$	−19.5390	−1.14540
$292$	0	0
$293$	−19.7477	−1.15367	−0.576837	−	0.816859i	$-0.695713\pi$
$293$	−19.7477	−1.15367	−0.576837	+	0.816859i	$0.695713\pi$
$294$	0	0
$295$	36.4955	2.12485
$296$	0	0
$297$	18.9564	1.09996
$298$	0	0
$299$	4.58258	0.265017
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	21.6261	1.24239
$304$	0	0
$305$	−3.00000	−0.171780
$306$	0	0
$307$	27.3739	1.56231	0.781154	−	0.624338i	$-0.214631\pi$
$307$	27.3739	1.56231	0.781154	+	0.624338i	$0.214631\pi$
$308$	0	0
$309$	−36.2867	−2.06428
$310$	0	0
$311$	26.2087	1.48616	0.743080	−	0.669203i	$-0.233364\pi$
$311$	26.2087	1.48616	0.743080	+	0.669203i	$0.233364\pi$
$312$	0	0
$313$	12.7477	0.720544	0.360272	−	0.932847i	$-0.382684\pi$
$313$	12.7477	0.720544	0.360272	+	0.932847i	$0.382684\pi$
$314$	0	0
$315$	−14.3739	−0.809875
$316$	0	0
$317$	−14.8348	−0.833208	−0.416604	−	0.909088i	$-0.636780\pi$
$317$	−14.8348	−0.833208	−0.416604	+	0.909088i	$0.636780\pi$
$318$	0	0
$319$	−14.3739	−0.804782
$320$	0	0
$321$	−0.460985	−0.0257297
$322$	0	0
$323$	−3.79129	−0.210953
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	32.7913	1.81336
$328$	0	0
$329$	10.5826	0.583436
$330$	0	0
$331$	−28.3739	−1.55957	−0.779784	−	0.626048i	$-0.784671\pi$
$331$	−28.3739	−1.55957	−0.779784	+	0.626048i	$0.784671\pi$
$332$	0	0
$333$	−23.9564	−1.31280
$334$	0	0
$335$	−28.1216	−1.53645
$336$	0	0
$337$	1.37386	0.0748391	0.0374196	−	0.999300i	$-0.488086\pi$
$337$	1.37386	0.0748391	0.0374196	+	0.999300i	$0.488086\pi$
$338$	0	0
$339$	23.3739	1.26949
$340$	0	0
$341$	27.9564	1.51393
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−38.3739	−2.06598
$346$	0	0
$347$	−0.791288	−0.0424786	−0.0212393	−	0.999774i	$-0.506761\pi$
$347$	−0.791288	−0.0424786	−0.0212393	+	0.999774i	$0.506761\pi$
$348$	0	0
$349$	1.62614	0.0870451	0.0435225	−	0.999052i	$-0.486142\pi$
$349$	1.62614	0.0870451	0.0435225	+	0.999052i	$0.486142\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−8.53901	−0.454486	−0.227243	−	0.973838i	$-0.572971\pi$
$353$	−8.53901	−0.454486	−0.227243	+	0.973838i	$0.572971\pi$
$354$	0	0
$355$	36.4955	1.93698
$356$	0	0
$357$	10.5826	0.560089
$358$	0	0
$359$	−0.626136	−0.0330462	−0.0165231	−	0.999863i	$-0.505260\pi$
$359$	−0.626136	−0.0330462	−0.0165231	+	0.999863i	$0.505260\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	9.41742	0.494287
$364$	0	0
$365$	−49.1216	−2.57114
$366$	0	0
$367$	−22.4955	−1.17425	−0.587127	−	0.809495i	$-0.699741\pi$
$367$	−22.4955	−1.17425	−0.587127	+	0.809495i	$0.699741\pi$
$368$	0	0
$369$	−18.1652	−0.945640
$370$	0	0
$371$	8.37386	0.434749
$372$	0	0
$373$	−16.3739	−0.847807	−0.423903	−	0.905707i	$-0.639340\pi$
$373$	−16.3739	−0.847807	−0.423903	+	0.905707i	$0.639340\pi$
$374$	0	0
$375$	8.37386	0.432424
$376$	0	0
$377$	−3.79129	−0.195261
$378$	0	0
$379$	−21.7477	−1.11711	−0.558553	−	0.829469i	$-0.688643\pi$
$379$	−21.7477	−1.11711	−0.558553	+	0.829469i	$0.688643\pi$
$380$	0	0
$381$	18.8348	0.964939
$382$	0	0
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	0	0
$385$	−11.3739	−0.579666
$386$	0	0
$387$	−9.58258	−0.487110
$388$	0	0
$389$	18.6261	0.944383	0.472191	−	0.881496i	$-0.343463\pi$
$389$	18.6261	0.944383	0.472191	+	0.881496i	$0.343463\pi$
$390$	0	0
$391$	17.3739	0.878634
$392$	0	0
$393$	61.7477	3.11476
$394$	0	0
$395$	30.0000	1.50946
$396$	0	0
$397$	5.25227	0.263604	0.131802	−	0.991276i	$-0.457924\pi$
$397$	5.25227	0.263604	0.131802	+	0.991276i	$0.457924\pi$
$398$	0	0
$399$	−2.79129	−0.139739
$400$	0	0
$401$	−12.6261	−0.630519	−0.315260	−	0.949005i	$-0.602092\pi$
$401$	−12.6261	−0.630519	−0.315260	+	0.949005i	$0.602092\pi$
$402$	0	0
$403$	7.37386	0.367318
$404$	0	0
$405$	1.25227	0.0622259
$406$	0	0
$407$	−18.9564	−0.939636
$408$	0	0
$409$	27.1216	1.34108	0.670538	−	0.741875i	$-0.266063\pi$
$409$	27.1216	1.34108	0.670538	+	0.741875i	$0.266063\pi$
$410$	0	0
$411$	16.7477	0.826104
$412$	0	0
$413$	−12.1652	−0.598608
$414$	0	0
$415$	43.1216	2.11676
$416$	0	0
$417$	−27.2087	−1.33242
$418$	0	0
$419$	25.7477	1.25786	0.628929	−	0.777462i	$-0.283493\pi$
$419$	25.7477	1.25786	0.628929	+	0.777462i	$0.283493\pi$
$420$	0	0
$421$	19.0000	0.926003	0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000	0.926003	0.463002	+	0.886357i	$0.346772\pi$
$422$	0	0
$423$	50.7042	2.46532
$424$	0	0
$425$	15.1652	0.735618
$426$	0	0
$427$	1.00000	0.0483934
$428$	0	0
$429$	10.5826	0.510932
$430$	0	0
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	0	0
$433$	9.74773	0.468446	0.234223	−	0.972183i	$-0.424745\pi$
$433$	9.74773	0.468446	0.234223	+	0.972183i	$0.424745\pi$
$434$	0	0
$435$	31.7477	1.52219
$436$	0	0
$437$	−4.58258	−0.219214
$438$	0	0
$439$	41.4955	1.98047	0.990235	−	0.139408i	$-0.0445200\pi$
$439$	41.4955	1.98047	0.990235	+	0.139408i	$0.0445200\pi$
$440$	0	0
$441$	4.79129	0.228157
$442$	0	0
$443$	14.0436	0.667230	0.333615	−	0.942709i	$-0.391732\pi$
$443$	14.0436	0.667230	0.333615	+	0.942709i	$0.391732\pi$
$444$	0	0
$445$	−22.7477	−1.07835
$446$	0	0
$447$	−8.37386	−0.396070
$448$	0	0
$449$	25.1216	1.18556	0.592781	−	0.805364i	$-0.298030\pi$
$449$	25.1216	1.18556	0.592781	+	0.805364i	$0.298030\pi$
$450$	0	0
$451$	−14.3739	−0.676839
$452$	0	0
$453$	37.3303	1.75393
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	−39.8693	−1.86501	−0.932504	−	0.361160i	$-0.882381\pi$
$457$	−39.8693	−1.86501	−0.932504	+	0.361160i	$0.882381\pi$
$458$	0	0
$459$	18.9564	0.884811
$460$	0	0
$461$	11.5390	0.537426	0.268713	−	0.963220i	$-0.413402\pi$
$461$	11.5390	0.537426	0.268713	+	0.963220i	$0.413402\pi$
$462$	0	0
$463$	−34.4955	−1.60314	−0.801570	−	0.597901i	$-0.796002\pi$
$463$	−34.4955	−1.60314	−0.801570	+	0.597901i	$0.796002\pi$
$464$	0	0
$465$	−61.7477	−2.86348
$466$	0	0
$467$	−34.2867	−1.58660	−0.793301	−	0.608830i	$-0.791639\pi$
$467$	−34.2867	−1.58660	−0.793301	+	0.608830i	$0.791639\pi$
$468$	0	0
$469$	9.37386	0.432845
$470$	0	0
$471$	56.1652	2.58795
$472$	0	0
$473$	−7.58258	−0.348647
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	40.1216	1.83704
$478$	0	0
$479$	12.6261	0.576903	0.288451	−	0.957495i	$-0.406860\pi$
$479$	12.6261	0.576903	0.288451	+	0.957495i	$0.406860\pi$
$480$	0	0
$481$	−5.00000	−0.227980
$482$	0	0
$483$	12.7913	0.582024
$484$	0	0
$485$	21.0000	0.953561
$486$	0	0
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	0	0
$489$	−12.2087	−0.552097
$490$	0	0
$491$	15.1652	0.684394	0.342197	−	0.939628i	$-0.388829\pi$
$491$	15.1652	0.684394	0.342197	+	0.939628i	$0.388829\pi$
$492$	0	0
$493$	−14.3739	−0.647366
$494$	0	0
$495$	−54.4955	−2.44939
$496$	0	0
$497$	−12.1652	−0.545682
$498$	0	0
$499$	−17.6261	−0.789054	−0.394527	−	0.918884i	$-0.629092\pi$
$499$	−17.6261	−0.789054	−0.394527	+	0.918884i	$0.629092\pi$
$500$	0	0
$501$	37.4519	1.67323
$502$	0	0
$503$	−6.33030	−0.282254	−0.141127	−	0.989991i	$-0.545073\pi$
$503$	−6.33030	−0.282254	−0.141127	+	0.989991i	$0.545073\pi$
$504$	0	0
$505$	−23.2432	−1.03431
$506$	0	0
$507$	−33.4955	−1.48759
$508$	0	0
$509$	−0.330303	−0.0146404	−0.00732021	−	0.999973i	$-0.502330\pi$
$509$	−0.330303	−0.0146404	−0.00732021	+	0.999973i	$0.502330\pi$
$510$	0	0
$511$	16.3739	0.724337
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	39.0000	1.71855
$516$	0	0
$517$	40.1216	1.76455
$518$	0	0
$519$	−63.4955	−2.78714
$520$	0	0
$521$	−13.4174	−0.587828	−0.293914	−	0.955832i	$-0.594958\pi$
$521$	−13.4174	−0.587828	−0.293914	+	0.955832i	$0.594958\pi$
$522$	0	0
$523$	−0.252273	−0.0110311	−0.00551556	−	0.999985i	$-0.501756\pi$
$523$	−0.252273	−0.0110311	−0.00551556	+	0.999985i	$0.501756\pi$
$524$	0	0
$525$	11.1652	0.487287
$526$	0	0
$527$	27.9564	1.21780
$528$	0	0
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	−58.2867	−2.52943
$532$	0	0
$533$	−3.79129	−0.164219
$534$	0	0
$535$	0.495454	0.0214204
$536$	0	0
$537$	44.0780	1.90211
$538$	0	0
$539$	3.79129	0.163302
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	26.1652	1.12285
$544$	0	0
$545$	−35.2432	−1.50965
$546$	0	0
$547$	12.3739	0.529068	0.264534	−	0.964376i	$-0.414782\pi$
$547$	12.3739	0.529068	0.264534	+	0.964376i	$0.414782\pi$
$548$	0	0
$549$	4.79129	0.204487
$550$	0	0
$551$	3.79129	0.161514
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	41.8693	1.77725
$556$	0	0
$557$	−0.626136	−0.0265303	−0.0132651	−	0.999912i	$-0.504223\pi$
$557$	−0.626136	−0.0265303	−0.0132651	+	0.999912i	$0.504223\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	40.1216	1.69393
$562$	0	0
$563$	12.1652	0.512700	0.256350	−	0.966584i	$-0.417480\pi$
$563$	12.1652	0.512700	0.256350	+	0.966584i	$0.417480\pi$
$564$	0	0
$565$	−25.1216	−1.05687
$566$	0	0
$567$	−0.417424	−0.0175302
$568$	0	0
$569$	30.1652	1.26459	0.632294	−	0.774728i	$-0.282113\pi$
$569$	30.1652	1.26459	0.632294	+	0.774728i	$0.282113\pi$
$570$	0	0
$571$	−32.4955	−1.35989	−0.679946	−	0.733262i	$-0.737997\pi$
$571$	−32.4955	−1.35989	−0.679946	+	0.733262i	$0.737997\pi$
$572$	0	0
$573$	65.7042	2.74483
$574$	0	0
$575$	18.3303	0.764426
$576$	0	0
$577$	−23.1216	−0.962564	−0.481282	−	0.876566i	$-0.659829\pi$
$577$	−23.1216	−0.962564	−0.481282	+	0.876566i	$0.659829\pi$
$578$	0	0
$579$	20.5826	0.855383
$580$	0	0
$581$	−14.3739	−0.596328
$582$	0	0
$583$	31.7477	1.31486
$584$	0	0
$585$	−14.3739	−0.594286
$586$	0	0
$587$	9.16515	0.378286	0.189143	−	0.981950i	$-0.439429\pi$
$587$	9.16515	0.378286	0.189143	+	0.981950i	$0.439429\pi$
$588$	0	0
$589$	−7.37386	−0.303835
$590$	0	0
$591$	6.62614	0.272563
$592$	0	0
$593$	−28.9129	−1.18731	−0.593655	−	0.804720i	$-0.702316\pi$
$593$	−28.9129	−1.18731	−0.593655	+	0.804720i	$0.702316\pi$
$594$	0	0
$595$	−11.3739	−0.466283
$596$	0	0
$597$	−66.2867	−2.71294
$598$	0	0
$599$	−20.3739	−0.832453	−0.416227	−	0.909261i	$-0.636648\pi$
$599$	−20.3739	−0.832453	−0.416227	+	0.909261i	$0.636648\pi$
$600$	0	0
$601$	11.6261	0.474240	0.237120	−	0.971480i	$-0.423797\pi$
$601$	11.6261	0.474240	0.237120	+	0.971480i	$0.423797\pi$
$602$	0	0
$603$	44.9129	1.82899
$604$	0	0
$605$	−10.1216	−0.411501
$606$	0	0
$607$	26.4955	1.07542	0.537709	−	0.843131i	$-0.319290\pi$
$607$	26.4955	1.07542	0.537709	+	0.843131i	$0.319290\pi$
$608$	0	0
$609$	−10.5826	−0.428828
$610$	0	0
$611$	10.5826	0.428125
$612$	0	0
$613$	−34.3739	−1.38835	−0.694174	−	0.719808i	$-0.744230\pi$
$613$	−34.3739	−1.38835	−0.694174	+	0.719808i	$0.744230\pi$
$614$	0	0
$615$	31.7477	1.28019
$616$	0	0
$617$	−6.95644	−0.280056	−0.140028	−	0.990148i	$-0.544719\pi$
$617$	−6.95644	−0.280056	−0.140028	+	0.990148i	$0.544719\pi$
$618$	0	0
$619$	−4.37386	−0.175800	−0.0879002	−	0.996129i	$-0.528016\pi$
$619$	−4.37386	−0.175800	−0.0879002	+	0.996129i	$0.528016\pi$
$620$	0	0
$621$	22.9129	0.919462
$622$	0	0
$623$	7.58258	0.303789
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−10.5826	−0.422627
$628$	0	0
$629$	−18.9564	−0.755843
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	0	0
$633$	68.0345	2.70413
$634$	0	0
$635$	−20.2432	−0.803326
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−58.2867	−2.30579
$640$	0	0
$641$	−20.2087	−0.798196	−0.399098	−	0.916908i	$-0.630677\pi$
$641$	−20.2087	−0.798196	−0.399098	+	0.916908i	$0.630677\pi$
$642$	0	0
$643$	−23.0000	−0.907031	−0.453516	−	0.891248i	$-0.649830\pi$
$643$	−23.0000	−0.907031	−0.453516	+	0.891248i	$0.649830\pi$
$644$	0	0
$645$	16.7477	0.659441
$646$	0	0
$647$	−13.4174	−0.527493	−0.263747	−	0.964592i	$-0.584958\pi$
$647$	−13.4174	−0.527493	−0.263747	+	0.964592i	$0.584958\pi$
$648$	0	0
$649$	−46.1216	−1.81043
$650$	0	0
$651$	20.5826	0.806695
$652$	0	0
$653$	−19.4174	−0.759863	−0.379931	−	0.925015i	$-0.624052\pi$
$653$	−19.4174	−0.759863	−0.379931	+	0.925015i	$0.624052\pi$
$654$	0	0
$655$	−66.3648	−2.59309
$656$	0	0
$657$	78.4519	3.06070
$658$	0	0
$659$	36.9564	1.43962	0.719809	−	0.694172i	$-0.244229\pi$
$659$	36.9564	1.43962	0.719809	+	0.694172i	$0.244229\pi$
$660$	0	0
$661$	34.0000	1.32245	0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000	1.32245	0.661223	+	0.750189i	$0.270038\pi$
$662$	0	0
$663$	10.5826	0.410993
$664$	0	0
$665$	3.00000	0.116335
$666$	0	0
$667$	−17.3739	−0.672719
$668$	0	0
$669$	−66.2867	−2.56279
$670$	0	0
$671$	3.79129	0.146361
$672$	0	0
$673$	−17.1216	−0.659989	−0.329994	−	0.943983i	$-0.607047\pi$
$673$	−17.1216	−0.659989	−0.329994	+	0.943983i	$0.607047\pi$
$674$	0	0
$675$	20.0000	0.769800
$676$	0	0
$677$	21.9564	0.843855	0.421927	−	0.906630i	$-0.361354\pi$
$677$	21.9564	0.843855	0.421927	+	0.906630i	$0.361354\pi$
$678$	0	0
$679$	−7.00000	−0.268635
$680$	0	0
$681$	10.5826	0.405525
$682$	0	0
$683$	36.1652	1.38382	0.691911	−	0.721983i	$-0.256769\pi$
$683$	36.1652	1.38382	0.691911	+	0.721983i	$0.256769\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	61.4083	2.34287
$688$	0	0
$689$	8.37386	0.319019
$690$	0	0
$691$	−48.7477	−1.85445	−0.927225	−	0.374504i	$-0.877813\pi$
$691$	−48.7477	−1.85445	−0.927225	+	0.374504i	$0.877813\pi$
$692$	0	0
$693$	18.1652	0.690037
$694$	0	0
$695$	29.2432	1.10926
$696$	0	0
$697$	−14.3739	−0.544449
$698$	0	0
$699$	−78.4955	−2.96897
$700$	0	0
$701$	−42.3303	−1.59879	−0.799397	−	0.600804i	$-0.794847\pi$
$701$	−42.3303	−1.59879	−0.799397	+	0.600804i	$0.794847\pi$
$702$	0	0
$703$	5.00000	0.188579
$704$	0	0
$705$	−88.6170	−3.33751
$706$	0	0
$707$	7.74773	0.291383
$708$	0	0
$709$	−31.2432	−1.17336	−0.586681	−	0.809818i	$-0.699566\pi$
$709$	−31.2432	−1.17336	−0.586681	+	0.809818i	$0.699566\pi$
$710$	0	0
$711$	−47.9129	−1.79687
$712$	0	0
$713$	33.7913	1.26549
$714$	0	0
$715$	−11.3739	−0.425358
$716$	0	0
$717$	−21.6261	−0.807643
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	−13.0000	−0.484145
$722$	0	0
$723$	−24.4174	−0.908094
$724$	0	0
$725$	−15.1652	−0.563220
$726$	0	0
$727$	11.0000	0.407967	0.203984	−	0.978974i	$-0.434611\pi$
$727$	11.0000	0.407967	0.203984	+	0.978974i	$0.434611\pi$
$728$	0	0
$729$	−43.8693	−1.62479
$730$	0	0
$731$	−7.58258	−0.280452
$732$	0	0
$733$	−41.4955	−1.53267	−0.766335	−	0.642441i	$-0.777922\pi$
$733$	−41.4955	−1.53267	−0.766335	+	0.642441i	$0.777922\pi$
$734$	0	0
$735$	−8.37386	−0.308875
$736$	0	0
$737$	35.5390	1.30910
$738$	0	0
$739$	35.7477	1.31500	0.657501	−	0.753454i	$-0.271614\pi$
$739$	35.7477	1.31500	0.657501	+	0.753454i	$0.271614\pi$
$740$	0	0
$741$	−2.79129	−0.102541
$742$	0	0
$743$	21.3303	0.782533	0.391266	−	0.920277i	$-0.372037\pi$
$743$	21.3303	0.782533	0.391266	+	0.920277i	$0.372037\pi$
$744$	0	0
$745$	9.00000	0.329734
$746$	0	0
$747$	−68.8693	−2.51980
$748$	0	0
$749$	−0.165151	−0.00603450
$750$	0	0
$751$	11.6261	0.424244	0.212122	−	0.977243i	$-0.431963\pi$
$751$	11.6261	0.424244	0.212122	+	0.977243i	$0.431963\pi$
$752$	0	0
$753$	−27.3303	−0.995972
$754$	0	0
$755$	−40.1216	−1.46017
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	48.4955	1.76027
$760$	0	0
$761$	−39.3303	−1.42572	−0.712861	−	0.701305i	$-0.752601\pi$
$761$	−39.3303	−1.42572	−0.712861	+	0.701305i	$0.752601\pi$
$762$	0	0
$763$	11.7477	0.425296
$764$	0	0
$765$	−54.4955	−1.97029
$766$	0	0
$767$	−12.1652	−0.439258
$768$	0	0
$769$	−13.4955	−0.486659	−0.243329	−	0.969944i	$-0.578240\pi$
$769$	−13.4955	−0.486659	−0.243329	+	0.969944i	$0.578240\pi$
$770$	0	0
$771$	31.2867	1.12676
$772$	0	0
$773$	−16.7477	−0.602374	−0.301187	−	0.953565i	$-0.597383\pi$
$773$	−16.7477	−0.602374	−0.301187	+	0.953565i	$0.597383\pi$
$774$	0	0
$775$	29.4955	1.05951
$776$	0	0
$777$	−13.9564	−0.500684
$778$	0	0
$779$	3.79129	0.135837
$780$	0	0
$781$	−46.1216	−1.65036
$782$	0	0
$783$	−18.9564	−0.677448
$784$	0	0
$785$	−60.3648	−2.15451
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	0	0
$789$	18.9564	0.674867
$790$	0	0
$791$	8.37386	0.297740
$792$	0	0
$793$	1.00000	0.0355110
$794$	0	0
$795$	−70.1216	−2.48696
$796$	0	0
$797$	32.8693	1.16429	0.582145	−	0.813085i	$-0.302213\pi$
$797$	32.8693	1.16429	0.582145	+	0.813085i	$0.302213\pi$
$798$	0	0
$799$	40.1216	1.41940
$800$	0	0
$801$	36.3303	1.28367
$802$	0	0
$803$	62.0780	2.19069
$804$	0	0
$805$	−13.7477	−0.484544
$806$	0	0
$807$	27.7913	0.978300
$808$	0	0
$809$	−22.9129	−0.805574	−0.402787	−	0.915294i	$-0.631958\pi$
$809$	−22.9129	−0.805574	−0.402787	+	0.915294i	$0.631958\pi$
$810$	0	0
$811$	31.4955	1.10595	0.552977	−	0.833196i	$-0.313492\pi$
$811$	31.4955	1.10595	0.552977	+	0.833196i	$0.313492\pi$
$812$	0	0
$813$	−39.4174	−1.38243
$814$	0	0
$815$	13.1216	0.459629
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	4.79129	0.167421
$820$	0	0
$821$	−42.3303	−1.47734	−0.738669	−	0.674068i	$-0.764545\pi$
$821$	−42.3303	−1.47734	−0.738669	+	0.674068i	$0.764545\pi$
$822$	0	0
$823$	−54.2432	−1.89080	−0.945399	−	0.325915i	$-0.894328\pi$
$823$	−54.2432	−1.89080	−0.945399	+	0.325915i	$0.894328\pi$
$824$	0	0
$825$	42.3303	1.47375
$826$	0	0
$827$	−39.6606	−1.37913	−0.689567	−	0.724222i	$-0.742199\pi$
$827$	−39.6606	−1.37913	−0.689567	+	0.724222i	$0.742199\pi$
$828$	0	0
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	−9.07803	−0.314913
$832$	0	0
$833$	3.79129	0.131360
$834$	0	0
$835$	−40.2523	−1.39299
$836$	0	0
$837$	36.8693	1.27439
$838$	0	0
$839$	8.83485	0.305013	0.152506	−	0.988302i	$-0.451266\pi$
$839$	8.83485	0.305013	0.152506	+	0.988302i	$0.451266\pi$
$840$	0	0
$841$	−14.6261	−0.504350
$842$	0	0
$843$	54.1996	1.86673
$844$	0	0
$845$	36.0000	1.23844
$846$	0	0
$847$	3.37386	0.115927
$848$	0	0
$849$	17.7913	0.610595
$850$	0	0
$851$	−22.9129	−0.785443
$852$	0	0
$853$	−9.12159	−0.312317	−0.156159	−	0.987732i	$-0.549911\pi$
$853$	−9.12159	−0.312317	−0.156159	+	0.987732i	$0.549911\pi$
$854$	0	0
$855$	14.3739	0.491576
$856$	0	0
$857$	−11.7042	−0.399807	−0.199903	−	0.979816i	$-0.564063\pi$
$857$	−11.7042	−0.399807	−0.199903	+	0.979816i	$0.564063\pi$
$858$	0	0
$859$	−7.37386	−0.251593	−0.125796	−	0.992056i	$-0.540149\pi$
$859$	−7.37386	−0.251593	−0.125796	+	0.992056i	$0.540149\pi$
$860$	0	0
$861$	−10.5826	−0.360653
$862$	0	0
$863$	−38.7042	−1.31751	−0.658753	−	0.752360i	$-0.728916\pi$
$863$	−38.7042	−1.31751	−0.658753	+	0.752360i	$0.728916\pi$
$864$	0	0
$865$	68.2432	2.32034
$866$	0	0
$867$	−7.33030	−0.248950
$868$	0	0
$869$	−37.9129	−1.28611
$870$	0	0
$871$	9.37386	0.317621
$872$	0	0
$873$	−33.5390	−1.13512
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	38.7477	1.30842	0.654209	−	0.756314i	$-0.273002\pi$
$877$	38.7477	1.30842	0.654209	+	0.756314i	$0.273002\pi$
$878$	0	0
$879$	−55.1216	−1.85921
$880$	0	0
$881$	2.37386	0.0799775	0.0399887	−	0.999200i	$-0.487268\pi$
$881$	2.37386	0.0799775	0.0399887	+	0.999200i	$0.487268\pi$
$882$	0	0
$883$	−24.2523	−0.816154	−0.408077	−	0.912948i	$-0.633801\pi$
$883$	−24.2523	−0.816154	−0.408077	+	0.912948i	$0.633801\pi$
$884$	0	0
$885$	101.869	3.42430
$886$	0	0
$887$	4.58258	0.153868	0.0769339	−	0.997036i	$-0.475487\pi$
$887$	4.58258	0.153868	0.0769339	+	0.997036i	$0.475487\pi$
$888$	0	0
$889$	6.74773	0.226312
$890$	0	0
$891$	−1.58258	−0.0530183
$892$	0	0
$893$	−10.5826	−0.354132
$894$	0	0
$895$	−47.3739	−1.58353
$896$	0	0
$897$	12.7913	0.427089
$898$	0	0
$899$	−27.9564	−0.932399
$900$	0	0
$901$	31.7477	1.05767
$902$	0	0
$903$	−5.58258	−0.185776
$904$	0	0
$905$	−28.1216	−0.934793
$906$	0	0
$907$	−31.2432	−1.03741	−0.518706	−	0.854952i	$-0.673586\pi$
$907$	−31.2432	−1.03741	−0.518706	+	0.854952i	$0.673586\pi$
$908$	0	0
$909$	37.1216	1.23125
$910$	0	0
$911$	3.49545	0.115810	0.0579048	−	0.998322i	$-0.481558\pi$
$911$	3.49545	0.115810	0.0579048	+	0.998322i	$0.481558\pi$
$912$	0	0
$913$	−54.4955	−1.80354
$914$	0	0
$915$	−8.37386	−0.276831
$916$	0	0
$917$	22.1216	0.730519
$918$	0	0
$919$	−34.4955	−1.13790	−0.568950	−	0.822372i	$-0.692650\pi$
$919$	−34.4955	−1.13790	−0.568950	+	0.822372i	$0.692650\pi$
$920$	0	0
$921$	76.4083	2.51774
$922$	0	0
$923$	−12.1652	−0.400421
$924$	0	0
$925$	−20.0000	−0.657596
$926$	0	0
$927$	−62.2867	−2.04577
$928$	0	0
$929$	47.8693	1.57054	0.785271	−	0.619153i	$-0.212524\pi$
$929$	47.8693	1.57054	0.785271	+	0.619153i	$0.212524\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	73.1561	2.39502
$934$	0	0
$935$	−43.1216	−1.41023
$936$	0	0
$937$	−15.3739	−0.502242	−0.251121	−	0.967956i	$-0.580799\pi$
$937$	−15.3739	−0.502242	−0.251121	+	0.967956i	$0.580799\pi$
$938$	0	0
$939$	35.5826	1.16119
$940$	0	0
$941$	−27.4955	−0.896326	−0.448163	−	0.893952i	$-0.647922\pi$
$941$	−27.4955	−0.896326	−0.448163	+	0.893952i	$0.647922\pi$
$942$	0	0
$943$	−17.3739	−0.565771
$944$	0	0
$945$	−15.0000	−0.487950
$946$	0	0
$947$	52.6170	1.70982	0.854912	−	0.518773i	$-0.173611\pi$
$947$	52.6170	1.70982	0.854912	+	0.518773i	$0.173611\pi$
$948$	0	0
$949$	16.3739	0.531518
$950$	0	0
$951$	−41.4083	−1.34276
$952$	0	0
$953$	21.4610	0.695189	0.347595	−	0.937645i	$-0.386999\pi$
$953$	21.4610	0.695189	0.347595	+	0.937645i	$0.386999\pi$
$954$	0	0
$955$	−70.6170	−2.28511
$956$	0	0
$957$	−40.1216	−1.29695
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	23.3739	0.753996
$962$	0	0
$963$	−0.791288	−0.0254989
$964$	0	0
$965$	−22.1216	−0.712119
$966$	0	0
$967$	12.1216	0.389804	0.194902	−	0.980823i	$-0.437561\pi$
$967$	12.1216	0.389804	0.194902	+	0.980823i	$0.437561\pi$
$968$	0	0
$969$	−10.5826	−0.339961
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	0	0
$973$	−9.74773	−0.312498
$974$	0	0
$975$	11.1652	0.357571
$976$	0	0
$977$	−34.4174	−1.10111	−0.550555	−	0.834799i	$-0.685584\pi$
$977$	−34.4174	−1.10111	−0.550555	+	0.834799i	$0.685584\pi$
$978$	0	0
$979$	28.7477	0.918781
$980$	0	0
$981$	56.2867	1.79710
$982$	0	0
$983$	8.66970	0.276520	0.138260	−	0.990396i	$-0.455849\pi$
$983$	8.66970	0.276520	0.138260	+	0.990396i	$0.455849\pi$
$984$	0	0
$985$	−7.12159	−0.226913
$986$	0	0
$987$	29.5390	0.940237
$988$	0	0
$989$	−9.16515	−0.291435
$990$	0	0
$991$	−8.74773	−0.277881	−0.138940	−	0.990301i	$-0.544370\pi$
$991$	−8.74773	−0.277881	−0.138940	+	0.990301i	$0.544370\pi$
$992$	0	0
$993$	−79.1996	−2.51332
$994$	0	0
$995$	71.2432	2.25856
$996$	0	0
$997$	2.87841	0.0911601	0.0455801	−	0.998961i	$-0.485486\pi$
$997$	2.87841	0.0911601	0.0455801	+	0.998961i	$0.485486\pi$
$998$	0	0
$999$	−25.0000	−0.790965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.t.1.2		2
4.3	odd	2		8512.2.a.m.1.1		2
8.3	odd	2		2128.2.a.k.1.2		2
8.5	even	2		532.2.a.c.1.1	✓	2
24.5	odd	2		4788.2.a.g.1.2		2
56.13	odd	2		3724.2.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.a.c.1.1	✓	2	8.5	even	2
2128.2.a.k.1.2		2	8.3	odd	2
3724.2.a.e.1.2		2	56.13	odd	2
4788.2.a.g.1.2		2	24.5	odd	2
8512.2.a.m.1.1		2	4.3	odd	2
8512.2.a.t.1.2		2	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:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q+2.79129 q^{3} -3.00000 q^{5} +1.00000 q^{7} +4.79129 q^{9} +O(q^{10})\)	\(q+2.79129 q^{3} -3.00000 q^{5} +1.00000 q^{7} +4.79129 q^{9} +3.79129 q^{11} +1.00000 q^{13} -8.37386 q^{15} +3.79129 q^{17} -1.00000 q^{19} +2.79129 q^{21} +4.58258 q^{23} +4.00000 q^{25} +5.00000 q^{27} -3.79129 q^{29} +7.37386 q^{31} +10.5826 q^{33} -3.00000 q^{35} -5.00000 q^{37} +2.79129 q^{39} -3.79129 q^{41} -2.00000 q^{43} -14.3739 q^{45} +10.5826 q^{47} +1.00000 q^{49} +10.5826 q^{51} +8.37386 q^{53} -11.3739 q^{55} -2.79129 q^{57} -12.1652 q^{59} +1.00000 q^{61} +4.79129 q^{63} -3.00000 q^{65} +9.37386 q^{67} +12.7913 q^{69} -12.1652 q^{71} +16.3739 q^{73} +11.1652 q^{75} +3.79129 q^{77} -10.0000 q^{79} -0.417424 q^{81} -14.3739 q^{83} -11.3739 q^{85} -10.5826 q^{87} +7.58258 q^{89} +1.00000 q^{91} +20.5826 q^{93} +3.00000 q^{95} -7.00000 q^{97} +18.1652 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 6 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q + q^3 - 6 * q^5 + 2 * q^7 + 5 * q^9	\( 2 q + q^{3} - 6 q^{5} + 2 q^{7} + 5 q^{9} + 3 q^{11} + 2 q^{13} - 3 q^{15} + 3 q^{17} - 2 q^{19} + q^{21} + 8 q^{25} + 10 q^{27} - 3 q^{29} + q^{31} + 12 q^{33} - 6 q^{35} - 10 q^{37} + q^{39} - 3 q^{41} - 4 q^{43} - 15 q^{45} + 12 q^{47} + 2 q^{49} + 12 q^{51} + 3 q^{53} - 9 q^{55} - q^{57} - 6 q^{59} + 2 q^{61} + 5 q^{63} - 6 q^{65} + 5 q^{67} + 21 q^{69} - 6 q^{71} + 19 q^{73} + 4 q^{75} + 3 q^{77} - 20 q^{79} - 10 q^{81} - 15 q^{83} - 9 q^{85} - 12 q^{87} + 6 q^{89} + 2 q^{91} + 32 q^{93} + 6 q^{95} - 14 q^{97} + 18 q^{99}+O(q^{100}) \) 2 * q + q^3 - 6 * q^5 + 2 * q^7 + 5 * q^9 + 3 * q^11 + 2 * q^13 - 3 * q^15 + 3 * q^17 - 2 * q^19 + q^21 + 8 * q^25 + 10 * q^27 - 3 * q^29 + q^31 + 12 * q^33 - 6 * q^35 - 10 * q^37 + q^39 - 3 * q^41 - 4 * q^43 - 15 * q^45 + 12 * q^47 + 2 * q^49 + 12 * q^51 + 3 * q^53 - 9 * q^55 - q^57 - 6 * q^59 + 2 * q^61 + 5 * q^63 - 6 * q^65 + 5 * q^67 + 21 * q^69 - 6 * q^71 + 19 * q^73 + 4 * q^75 + 3 * q^77 - 20 * q^79 - 10 * q^81 - 15 * q^83 - 9 * q^85 - 12 * q^87 + 6 * q^89 + 2 * q^91 + 32 * q^93 + 6 * q^95 - 14 * q^97 + 18 * q^99

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