LMFDB - Embedded newform 3200.2.a.bt.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(1,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, 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("3200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.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:	$25.5521286468$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 640)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	3200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.90321 q^{3} -3.52543 q^{7} +5.42864 q^{9} +O(q^{10})$	$q+2.90321 q^{3} -3.52543 q^{7} +5.42864 q^{9} -3.80642 q^{11} -2.62222 q^{13} -5.80642 q^{17} +5.05086 q^{19} -10.2351 q^{21} -0.474572 q^{23} +7.05086 q^{27} -2.00000 q^{29} -2.75557 q^{31} -11.0509 q^{33} -7.18421 q^{37} -7.61285 q^{39} +5.18421 q^{41} -1.95407 q^{43} +5.33185 q^{47} +5.42864 q^{49} -16.8573 q^{51} -5.37778 q^{53} +14.6637 q^{57} -5.05086 q^{59} -12.2351 q^{61} -19.1383 q^{63} -7.76049 q^{67} -1.37778 q^{69} -4.85728 q^{71} +6.66370 q^{73} +13.4193 q^{77} -5.24443 q^{79} +4.18421 q^{81} +12.1476 q^{83} -5.80642 q^{87} -12.1017 q^{89} +9.24443 q^{91} -8.00000 q^{93} -13.8064 q^{97} -20.6637 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 - 4 * q^7 + 3 * q^9	$ 3 q + 2 q^{3} - 4 q^{7} + 3 q^{9} + 2 q^{11} - 8 q^{13} - 4 q^{17} + 2 q^{19} - 4 q^{21} - 8 q^{23} + 8 q^{27} - 6 q^{29} - 8 q^{31} - 20 q^{33} - 8 q^{37} + 4 q^{39} + 2 q^{41} + 14 q^{43} - 4 q^{47} + 3 q^{49} - 24 q^{51} - 16 q^{53} + 4 q^{57} - 2 q^{59} - 10 q^{61} - 24 q^{63} + 10 q^{67} - 4 q^{69} + 12 q^{71} - 20 q^{73} - 16 q^{79} - q^{81} + 30 q^{83} - 4 q^{87} - 10 q^{89} + 28 q^{91} - 24 q^{93} - 28 q^{97} - 22 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 4 * q^7 + 3 * q^9 + 2 * q^11 - 8 * q^13 - 4 * q^17 + 2 * q^19 - 4 * q^21 - 8 * q^23 + 8 * q^27 - 6 * q^29 - 8 * q^31 - 20 * q^33 - 8 * q^37 + 4 * q^39 + 2 * q^41 + 14 * q^43 - 4 * q^47 + 3 * q^49 - 24 * q^51 - 16 * q^53 + 4 * q^57 - 2 * q^59 - 10 * q^61 - 24 * q^63 + 10 * q^67 - 4 * q^69 + 12 * q^71 - 20 * q^73 - 16 * q^79 - q^81 + 30 * q^83 - 4 * q^87 - 10 * q^89 + 28 * q^91 - 24 * q^93 - 28 * q^97 - 22 * 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
$3$	2.90321	1.67617	0.838085	−	0.545540i	$-0.183675\pi$
$3$	2.90321	1.67617	0.838085	+	0.545540i	$0.183675\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.52543	−1.33249	−0.666243	−	0.745735i	$-0.732099\pi$
$7$	−3.52543	−1.33249	−0.666243	+	0.745735i	$0.732099\pi$
$8$	0	0
$9$	5.42864	1.80955
$10$	0	0
$11$	−3.80642	−1.14768	−0.573840	−	0.818967i	$-0.694547\pi$
$11$	−3.80642	−1.14768	−0.573840	+	0.818967i	$0.694547\pi$
$12$	0	0
$13$	−2.62222	−0.727272	−0.363636	−	0.931541i	$-0.618465\pi$
$13$	−2.62222	−0.727272	−0.363636	+	0.931541i	$0.618465\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.80642	−1.40826	−0.704132	−	0.710069i	$-0.748664\pi$
$17$	−5.80642	−1.40826	−0.704132	+	0.710069i	$0.748664\pi$
$18$	0	0
$19$	5.05086	1.15875	0.579373	−	0.815063i	$-0.303298\pi$
$19$	5.05086	1.15875	0.579373	+	0.815063i	$0.303298\pi$
$20$	0	0
$21$	−10.2351	−2.23347
$22$	0	0
$23$	−0.474572	−0.0989552	−0.0494776	−	0.998775i	$-0.515756\pi$
$23$	−0.474572	−0.0989552	−0.0494776	+	0.998775i	$0.515756\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	7.05086	1.35694
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−2.75557	−0.494915	−0.247457	−	0.968899i	$-0.579595\pi$
$31$	−2.75557	−0.494915	−0.247457	+	0.968899i	$0.579595\pi$
$32$	0	0
$33$	−11.0509	−1.92371
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.18421	−1.18108	−0.590538	−	0.807010i	$-0.701085\pi$
$37$	−7.18421	−1.18108	−0.590538	+	0.807010i	$0.701085\pi$
$38$	0	0
$39$	−7.61285	−1.21903
$40$	0	0
$41$	5.18421	0.809637	0.404819	−	0.914397i	$-0.367335\pi$
$41$	5.18421	0.809637	0.404819	+	0.914397i	$0.367335\pi$
$42$	0	0
$43$	−1.95407	−0.297992	−0.148996	−	0.988838i	$-0.547604\pi$
$43$	−1.95407	−0.297992	−0.148996	+	0.988838i	$0.547604\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.33185	0.777730	0.388865	−	0.921295i	$-0.372867\pi$
$47$	5.33185	0.777730	0.388865	+	0.921295i	$0.372867\pi$
$48$	0	0
$49$	5.42864	0.775520
$50$	0	0
$51$	−16.8573	−2.36049
$52$	0	0
$53$	−5.37778	−0.738695	−0.369348	−	0.929291i	$-0.620419\pi$
$53$	−5.37778	−0.738695	−0.369348	+	0.929291i	$0.620419\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	14.6637	1.94225
$58$	0	0
$59$	−5.05086	−0.657565	−0.328783	−	0.944406i	$-0.606638\pi$
$59$	−5.05086	−0.657565	−0.328783	+	0.944406i	$0.606638\pi$
$60$	0	0
$61$	−12.2351	−1.56654	−0.783270	−	0.621682i	$-0.786450\pi$
$61$	−12.2351	−1.56654	−0.783270	+	0.621682i	$0.786450\pi$
$62$	0	0
$63$	−19.1383	−2.41120
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.76049	−0.948095	−0.474047	−	0.880499i	$-0.657207\pi$
$67$	−7.76049	−0.948095	−0.474047	+	0.880499i	$0.657207\pi$
$68$	0	0
$69$	−1.37778	−0.165866
$70$	0	0
$71$	−4.85728	−0.576453	−0.288226	−	0.957562i	$-0.593066\pi$
$71$	−4.85728	−0.576453	−0.288226	+	0.957562i	$0.593066\pi$
$72$	0	0
$73$	6.66370	0.779927	0.389964	−	0.920830i	$-0.372488\pi$
$73$	6.66370	0.779927	0.389964	+	0.920830i	$0.372488\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.4193	1.52927
$78$	0	0
$79$	−5.24443	−0.590045	−0.295022	−	0.955490i	$-0.595327\pi$
$79$	−5.24443	−0.590045	−0.295022	+	0.955490i	$0.595327\pi$
$80$	0	0
$81$	4.18421	0.464912
$82$	0	0
$83$	12.1476	1.33338	0.666689	−	0.745336i	$-0.267711\pi$
$83$	12.1476	1.33338	0.666689	+	0.745336i	$0.267711\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.80642	−0.622514
$88$	0	0
$89$	−12.1017	−1.28278	−0.641389	−	0.767216i	$-0.721642\pi$
$89$	−12.1017	−1.28278	−0.641389	+	0.767216i	$0.721642\pi$
$90$	0	0
$91$	9.24443	0.969080
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.8064	−1.40183	−0.700915	−	0.713245i	$-0.747225\pi$
$97$	−13.8064	−1.40183	−0.700915	+	0.713245i	$0.747225\pi$
$98$	0	0
$99$	−20.6637	−2.07678
$100$	0	0
$101$	19.7146	1.96167	0.980836	−	0.194836i	$-0.0624173\pi$
$101$	19.7146	1.96167	0.980836	+	0.194836i	$0.0624173\pi$
$102$	0	0
$103$	3.13828	0.309223	0.154612	−	0.987975i	$-0.450587\pi$
$103$	3.13828	0.309223	0.154612	+	0.987975i	$0.450587\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.39207	−0.521272	−0.260636	−	0.965437i	$-0.583932\pi$
$107$	−5.39207	−0.521272	−0.260636	+	0.965437i	$0.583932\pi$
$108$	0	0
$109$	8.62222	0.825858	0.412929	−	0.910763i	$-0.364506\pi$
$109$	8.62222	0.825858	0.412929	+	0.910763i	$0.364506\pi$
$110$	0	0
$111$	−20.8573	−1.97969
$112$	0	0
$113$	−5.51114	−0.518444	−0.259222	−	0.965818i	$-0.583466\pi$
$113$	−5.51114	−0.518444	−0.259222	+	0.965818i	$0.583466\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−14.2351	−1.31603
$118$	0	0
$119$	20.4701	1.87649
$120$	0	0
$121$	3.48886	0.317169
$122$	0	0
$123$	15.0509	1.35709
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.2810	−0.912291	−0.456145	−	0.889905i	$-0.650770\pi$
$127$	−10.2810	−0.912291	−0.456145	+	0.889905i	$0.650770\pi$
$128$	0	0
$129$	−5.67307	−0.499486
$130$	0	0
$131$	4.66370	0.407470	0.203735	−	0.979026i	$-0.434692\pi$
$131$	4.66370	0.407470	0.203735	+	0.979026i	$0.434692\pi$
$132$	0	0
$133$	−17.8064	−1.54401
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.3461	−0.969366	−0.484683	−	0.874690i	$-0.661065\pi$
$137$	−11.3461	−0.969366	−0.484683	+	0.874690i	$0.661065\pi$
$138$	0	0
$139$	11.8064	1.00141	0.500704	−	0.865619i	$-0.333075\pi$
$139$	11.8064	1.00141	0.500704	+	0.865619i	$0.333075\pi$
$140$	0	0
$141$	15.4795	1.30361
$142$	0	0
$143$	9.98126	0.834675
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.7605	1.29990
$148$	0	0
$149$	−5.47949	−0.448898	−0.224449	−	0.974486i	$-0.572058\pi$
$149$	−5.47949	−0.448898	−0.224449	+	0.974486i	$0.572058\pi$
$150$	0	0
$151$	23.6128	1.92159	0.960793	−	0.277266i	$-0.0894284\pi$
$151$	23.6128	1.92159	0.960793	+	0.277266i	$0.0894284\pi$
$152$	0	0
$153$	−31.5210	−2.54832
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.815792	−0.0651073	−0.0325536	−	0.999470i	$-0.510364\pi$
$157$	−0.815792	−0.0651073	−0.0325536	+	0.999470i	$0.510364\pi$
$158$	0	0
$159$	−15.6128	−1.23818
$160$	0	0
$161$	1.67307	0.131856
$162$	0	0
$163$	10.9032	0.854005	0.427003	−	0.904250i	$-0.359569\pi$
$163$	10.9032	0.854005	0.427003	+	0.904250i	$0.359569\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.57628	0.508888	0.254444	−	0.967088i	$-0.418108\pi$
$167$	6.57628	0.508888	0.254444	+	0.967088i	$0.418108\pi$
$168$	0	0
$169$	−6.12399	−0.471076
$170$	0	0
$171$	27.4193	2.09680
$172$	0	0
$173$	10.5303	0.800608	0.400304	−	0.916382i	$-0.368905\pi$
$173$	10.5303	0.800608	0.400304	+	0.916382i	$0.368905\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.6637	−1.10219
$178$	0	0
$179$	−6.29529	−0.470532	−0.235266	−	0.971931i	$-0.575596\pi$
$179$	−6.29529	−0.470532	−0.235266	+	0.971931i	$0.575596\pi$
$180$	0	0
$181$	−0.488863	−0.0363369	−0.0181684	−	0.999835i	$-0.505784\pi$
$181$	−0.488863	−0.0363369	−0.0181684	+	0.999835i	$0.505784\pi$
$182$	0	0
$183$	−35.5210	−2.62579
$184$	0	0
$185$	0	0
$186$	0	0
$187$	22.1017	1.61624
$188$	0	0
$189$	−24.8573	−1.80810
$190$	0	0
$191$	−10.4889	−0.758947	−0.379474	−	0.925203i	$-0.623895\pi$
$191$	−10.4889	−0.758947	−0.379474	+	0.925203i	$0.623895\pi$
$192$	0	0
$193$	−13.8064	−0.993808	−0.496904	−	0.867805i	$-0.665530\pi$
$193$	−13.8064	−0.993808	−0.496904	+	0.867805i	$0.665530\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.7239	−1.19153	−0.595765	−	0.803159i	$-0.703151\pi$
$197$	−16.7239	−1.19153	−0.595765	+	0.803159i	$0.703151\pi$
$198$	0	0
$199$	20.8573	1.47853	0.739267	−	0.673413i	$-0.235172\pi$
$199$	20.8573	1.47853	0.739267	+	0.673413i	$0.235172\pi$
$200$	0	0
$201$	−22.5303	−1.58917
$202$	0	0
$203$	7.05086	0.494873
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.57628	−0.179064
$208$	0	0
$209$	−19.2257	−1.32987
$210$	0	0
$211$	4.66370	0.321063	0.160531	−	0.987031i	$-0.448679\pi$
$211$	4.66370	0.321063	0.160531	+	0.987031i	$0.448679\pi$
$212$	0	0
$213$	−14.1017	−0.966233
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.71456	0.659467
$218$	0	0
$219$	19.3461	1.30729
$220$	0	0
$221$	15.2257	1.02419
$222$	0	0
$223$	−26.4558	−1.77161	−0.885807	−	0.464054i	$-0.846394\pi$
$223$	−26.4558	−1.77161	−0.885807	+	0.464054i	$0.846394\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.3225	0.817872	0.408936	−	0.912563i	$-0.365900\pi$
$227$	12.3225	0.817872	0.408936	+	0.912563i	$0.365900\pi$
$228$	0	0
$229$	13.2257	0.873979	0.436989	−	0.899467i	$-0.356045\pi$
$229$	13.2257	0.873979	0.436989	+	0.899467i	$0.356045\pi$
$230$	0	0
$231$	38.9590	2.56331
$232$	0	0
$233$	6.66370	0.436554	0.218277	−	0.975887i	$-0.429956\pi$
$233$	6.66370	0.436554	0.218277	+	0.975887i	$0.429956\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−15.2257	−0.989015
$238$	0	0
$239$	22.9590	1.48509	0.742547	−	0.669794i	$-0.233618\pi$
$239$	22.9590	1.48509	0.742547	+	0.669794i	$0.233618\pi$
$240$	0	0
$241$	−14.0415	−0.904492	−0.452246	−	0.891893i	$-0.649377\pi$
$241$	−14.0415	−0.904492	−0.452246	+	0.891893i	$0.649377\pi$
$242$	0	0
$243$	−9.00492	−0.577666
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−13.2444	−0.842723
$248$	0	0
$249$	35.2672	2.23497
$250$	0	0
$251$	24.9304	1.57359	0.786797	−	0.617212i	$-0.211738\pi$
$251$	24.9304	1.57359	0.786797	+	0.617212i	$0.211738\pi$
$252$	0	0
$253$	1.80642	0.113569
$254$	0	0
$255$	0	0
$256$	0	0
$257$	25.7146	1.60403	0.802015	−	0.597304i	$-0.203761\pi$
$257$	25.7146	1.60403	0.802015	+	0.597304i	$0.203761\pi$
$258$	0	0
$259$	25.3274	1.57377
$260$	0	0
$261$	−10.8573	−0.672049
$262$	0	0
$263$	−2.57628	−0.158860	−0.0794302	−	0.996840i	$-0.525310\pi$
$263$	−2.57628	−0.158860	−0.0794302	+	0.996840i	$0.525310\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−35.1338	−2.15016
$268$	0	0
$269$	25.7462	1.56977	0.784887	−	0.619639i	$-0.212721\pi$
$269$	25.7462	1.56977	0.784887	+	0.619639i	$0.212721\pi$
$270$	0	0
$271$	30.1847	1.83359	0.916795	−	0.399359i	$-0.130767\pi$
$271$	30.1847	1.83359	0.916795	+	0.399359i	$0.130767\pi$
$272$	0	0
$273$	26.8385	1.62434
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.5303	0.632707	0.316354	−	0.948641i	$-0.397541\pi$
$277$	10.5303	0.632707	0.316354	+	0.948641i	$0.397541\pi$
$278$	0	0
$279$	−14.9590	−0.895571
$280$	0	0
$281$	−7.93978	−0.473647	−0.236824	−	0.971553i	$-0.576106\pi$
$281$	−7.93978	−0.473647	−0.236824	+	0.971553i	$0.576106\pi$
$282$	0	0
$283$	10.9032	0.648129	0.324064	−	0.946035i	$-0.394951\pi$
$283$	10.9032	0.648129	0.324064	+	0.946035i	$0.394951\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−18.2766	−1.07883
$288$	0	0
$289$	16.7146	0.983209
$290$	0	0
$291$	−40.0830	−2.34971
$292$	0	0
$293$	−4.42864	−0.258724	−0.129362	−	0.991597i	$-0.541293\pi$
$293$	−4.42864	−0.258724	−0.129362	+	0.991597i	$0.541293\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−26.8385	−1.55733
$298$	0	0
$299$	1.24443	0.0719673
$300$	0	0
$301$	6.88892	0.397071
$302$	0	0
$303$	57.2355	3.28810
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5.27163	−0.300868	−0.150434	−	0.988620i	$-0.548067\pi$
$307$	−5.27163	−0.300868	−0.150434	+	0.988620i	$0.548067\pi$
$308$	0	0
$309$	9.11108	0.518311
$310$	0	0
$311$	0.387152	0.0219534	0.0109767	−	0.999940i	$-0.496506\pi$
$311$	0.387152	0.0219534	0.0109767	+	0.999940i	$0.496506\pi$
$312$	0	0
$313$	−11.3461	−0.641322	−0.320661	−	0.947194i	$-0.603905\pi$
$313$	−11.3461	−0.641322	−0.320661	+	0.947194i	$0.603905\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.7239	0.939309	0.469655	−	0.882850i	$-0.344378\pi$
$317$	16.7239	0.939309	0.469655	+	0.882850i	$0.344378\pi$
$318$	0	0
$319$	7.61285	0.426238
$320$	0	0
$321$	−15.6543	−0.873740
$322$	0	0
$323$	−29.3274	−1.63182
$324$	0	0
$325$	0	0
$326$	0	0
$327$	25.0321	1.38428
$328$	0	0
$329$	−18.7971	−1.03632
$330$	0	0
$331$	3.33630	0.183379	0.0916897	−	0.995788i	$-0.470773\pi$
$331$	3.33630	0.183379	0.0916897	+	0.995788i	$0.470773\pi$
$332$	0	0
$333$	−39.0005	−2.13721
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.61285	0.196804	0.0984022	−	0.995147i	$-0.468627\pi$
$337$	3.61285	0.196804	0.0984022	+	0.995147i	$0.468627\pi$
$338$	0	0
$339$	−16.0000	−0.869001
$340$	0	0
$341$	10.4889	0.568004
$342$	0	0
$343$	5.53972	0.299117
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.7605	−0.846067	−0.423034	−	0.906114i	$-0.639035\pi$
$347$	−15.7605	−0.846067	−0.423034	+	0.906114i	$0.639035\pi$
$348$	0	0
$349$	−0.285442	−0.0152794	−0.00763968	−	0.999971i	$-0.502432\pi$
$349$	−0.285442	−0.0152794	−0.00763968	+	0.999971i	$0.502432\pi$
$350$	0	0
$351$	−18.4889	−0.986862
$352$	0	0
$353$	4.38715	0.233505	0.116752	−	0.993161i	$-0.462752\pi$
$353$	4.38715	0.233505	0.116752	+	0.993161i	$0.462752\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	59.4291	3.14532
$358$	0	0
$359$	−20.5906	−1.08673	−0.543364	−	0.839497i	$-0.682850\pi$
$359$	−20.5906	−1.08673	−0.543364	+	0.839497i	$0.682850\pi$
$360$	0	0
$361$	6.51114	0.342691
$362$	0	0
$363$	10.1289	0.531630
$364$	0	0
$365$	0	0
$366$	0	0
$367$	16.5575	0.864297	0.432148	−	0.901802i	$-0.357756\pi$
$367$	16.5575	0.864297	0.432148	+	0.901802i	$0.357756\pi$
$368$	0	0
$369$	28.1432	1.46508
$370$	0	0
$371$	18.9590	0.984302
$372$	0	0
$373$	24.8988	1.28921	0.644605	−	0.764516i	$-0.277022\pi$
$373$	24.8988	1.28921	0.644605	+	0.764516i	$0.277022\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.24443	0.270102
$378$	0	0
$379$	9.31756	0.478611	0.239305	−	0.970944i	$-0.423080\pi$
$379$	9.31756	0.478611	0.239305	+	0.970944i	$0.423080\pi$
$380$	0	0
$381$	−29.8479	−1.52915
$382$	0	0
$383$	−32.5575	−1.66361	−0.831806	−	0.555066i	$-0.812693\pi$
$383$	−32.5575	−1.66361	−0.831806	+	0.555066i	$0.812693\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.6079	−0.539231
$388$	0	0
$389$	−18.9906	−0.962863	−0.481432	−	0.876484i	$-0.659883\pi$
$389$	−18.9906	−0.962863	−0.481432	+	0.876484i	$0.659883\pi$
$390$	0	0
$391$	2.75557	0.139355
$392$	0	0
$393$	13.5397	0.682988
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−32.9906	−1.65575	−0.827876	−	0.560911i	$-0.810451\pi$
$397$	−32.9906	−1.65575	−0.827876	+	0.560911i	$0.810451\pi$
$398$	0	0
$399$	−51.6958	−2.58803
$400$	0	0
$401$	−16.1017	−0.804081	−0.402041	−	0.915622i	$-0.631699\pi$
$401$	−16.1017	−0.804081	−0.402041	+	0.915622i	$0.631699\pi$
$402$	0	0
$403$	7.22570	0.359938
$404$	0	0
$405$	0	0
$406$	0	0
$407$	27.3461	1.35550
$408$	0	0
$409$	−33.3876	−1.65091	−0.825456	−	0.564466i	$-0.809082\pi$
$409$	−33.3876	−1.65091	−0.825456	+	0.564466i	$0.809082\pi$
$410$	0	0
$411$	−32.9403	−1.62482
$412$	0	0
$413$	17.8064	0.876197
$414$	0	0
$415$	0	0
$416$	0	0
$417$	34.2766	1.67853
$418$	0	0
$419$	27.4193	1.33952	0.669760	−	0.742578i	$-0.266397\pi$
$419$	27.4193	1.33952	0.669760	+	0.742578i	$0.266397\pi$
$420$	0	0
$421$	−12.2351	−0.596301	−0.298150	−	0.954519i	$-0.596370\pi$
$421$	−12.2351	−0.596301	−0.298150	+	0.954519i	$0.596370\pi$
$422$	0	0
$423$	28.9447	1.40734
$424$	0	0
$425$	0	0
$426$	0	0
$427$	43.1338	2.08739
$428$	0	0
$429$	28.9777	1.39906
$430$	0	0
$431$	−28.4701	−1.37136	−0.685679	−	0.727904i	$-0.740495\pi$
$431$	−28.4701	−1.37136	−0.685679	+	0.727904i	$0.740495\pi$
$432$	0	0
$433$	−34.0098	−1.63441	−0.817204	−	0.576348i	$-0.804477\pi$
$433$	−34.0098	−1.63441	−0.817204	+	0.576348i	$0.804477\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.39700	−0.114664
$438$	0	0
$439$	−14.8385	−0.708205	−0.354103	−	0.935207i	$-0.615214\pi$
$439$	−14.8385	−0.708205	−0.354103	+	0.935207i	$0.615214\pi$
$440$	0	0
$441$	29.4701	1.40334
$442$	0	0
$443$	−13.0968	−0.622247	−0.311124	−	0.950369i	$-0.600705\pi$
$443$	−13.0968	−0.622247	−0.311124	+	0.950369i	$0.600705\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−15.9081	−0.752429
$448$	0	0
$449$	−21.3876	−1.00934	−0.504672	−	0.863311i	$-0.668387\pi$
$449$	−21.3876	−1.00934	−0.504672	+	0.863311i	$0.668387\pi$
$450$	0	0
$451$	−19.7333	−0.929205
$452$	0	0
$453$	68.5531	3.22091
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.9813	−0.841128	−0.420564	−	0.907263i	$-0.638168\pi$
$457$	−17.9813	−0.841128	−0.420564	+	0.907263i	$0.638168\pi$
$458$	0	0
$459$	−40.9403	−1.91093
$460$	0	0
$461$	−10.7368	−0.500064	−0.250032	−	0.968238i	$-0.580441\pi$
$461$	−10.7368	−0.500064	−0.250032	+	0.968238i	$0.580441\pi$
$462$	0	0
$463$	9.30327	0.432360	0.216180	−	0.976354i	$-0.430640\pi$
$463$	9.30327	0.432360	0.216180	+	0.976354i	$0.430640\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.70964	0.403034	0.201517	−	0.979485i	$-0.435413\pi$
$467$	8.70964	0.403034	0.201517	+	0.979485i	$0.435413\pi$
$468$	0	0
$469$	27.3590	1.26332
$470$	0	0
$471$	−2.36842	−0.109131
$472$	0	0
$473$	7.43801	0.342000
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−29.1941	−1.33670
$478$	0	0
$479$	23.2257	1.06121	0.530605	−	0.847619i	$-0.321965\pi$
$479$	23.2257	1.06121	0.530605	+	0.847619i	$0.321965\pi$
$480$	0	0
$481$	18.8385	0.858964
$482$	0	0
$483$	4.85728	0.221014
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−32.8528	−1.48870	−0.744352	−	0.667787i	$-0.767241\pi$
$487$	−32.8528	−1.48870	−0.744352	+	0.667787i	$0.767241\pi$
$488$	0	0
$489$	31.6543	1.43146
$490$	0	0
$491$	−2.94914	−0.133093	−0.0665465	−	0.997783i	$-0.521198\pi$
$491$	−2.94914	−0.133093	−0.0665465	+	0.997783i	$0.521198\pi$
$492$	0	0
$493$	11.6128	0.523016
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17.1240	0.768116
$498$	0	0
$499$	−35.0321	−1.56825	−0.784127	−	0.620601i	$-0.786889\pi$
$499$	−35.0321	−1.56825	−0.784127	+	0.620601i	$0.786889\pi$
$500$	0	0
$501$	19.0923	0.852983
$502$	0	0
$503$	−16.2908	−0.726373	−0.363186	−	0.931717i	$-0.618311\pi$
$503$	−16.2908	−0.726373	−0.363186	+	0.931717i	$0.618311\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−17.7792	−0.789603
$508$	0	0
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	−23.4924	−1.03924
$512$	0	0
$513$	35.6128	1.57235
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−20.2953	−0.892586
$518$	0	0
$519$	30.5718	1.34195
$520$	0	0
$521$	11.7146	0.513224	0.256612	−	0.966514i	$-0.417394\pi$
$521$	11.7146	0.513224	0.256612	+	0.966514i	$0.417394\pi$
$522$	0	0
$523$	−38.0370	−1.66324	−0.831622	−	0.555342i	$-0.812587\pi$
$523$	−38.0370	−1.66324	−0.831622	+	0.555342i	$0.812587\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	−22.7748	−0.990208
$530$	0	0
$531$	−27.4193	−1.18990
$532$	0	0
$533$	−13.5941	−0.588826
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−18.2766	−0.788691
$538$	0	0
$539$	−20.6637	−0.890049
$540$	0	0
$541$	−11.5111	−0.494902	−0.247451	−	0.968900i	$-0.579593\pi$
$541$	−11.5111	−0.494902	−0.247451	+	0.968900i	$0.579593\pi$
$542$	0	0
$543$	−1.41927	−0.0609068
$544$	0	0
$545$	0	0
$546$	0	0
$547$	30.2208	1.29215	0.646073	−	0.763275i	$-0.276410\pi$
$547$	30.2208	1.29215	0.646073	+	0.763275i	$0.276410\pi$
$548$	0	0
$549$	−66.4197	−2.83473
$550$	0	0
$551$	−10.1017	−0.430347
$552$	0	0
$553$	18.4889	0.786226
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.75605	−0.413377	−0.206688	−	0.978407i	$-0.566269\pi$
$557$	−9.75605	−0.413377	−0.206688	+	0.978407i	$0.566269\pi$
$558$	0	0
$559$	5.12399	0.216721
$560$	0	0
$561$	64.1659	2.70909
$562$	0	0
$563$	−4.50622	−0.189914	−0.0949572	−	0.995481i	$-0.530271\pi$
$563$	−4.50622	−0.189914	−0.0949572	+	0.995481i	$0.530271\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−14.7511	−0.619489
$568$	0	0
$569$	5.30465	0.222383	0.111191	−	0.993799i	$-0.464533\pi$
$569$	5.30465	0.222383	0.111191	+	0.993799i	$0.464533\pi$
$570$	0	0
$571$	16.3970	0.686193	0.343096	−	0.939300i	$-0.388524\pi$
$571$	16.3970	0.686193	0.343096	+	0.939300i	$0.388524\pi$
$572$	0	0
$573$	−30.4514	−1.27213
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−39.8163	−1.65757	−0.828786	−	0.559565i	$-0.810968\pi$
$577$	−39.8163	−1.65757	−0.828786	+	0.559565i	$0.810968\pi$
$578$	0	0
$579$	−40.0830	−1.66579
$580$	0	0
$581$	−42.8256	−1.77671
$582$	0	0
$583$	20.4701	0.847786
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.1699	−0.626130	−0.313065	−	0.949732i	$-0.601356\pi$
$587$	−15.1699	−0.626130	−0.313065	+	0.949732i	$0.601356\pi$
$588$	0	0
$589$	−13.9180	−0.573480
$590$	0	0
$591$	−48.5531	−1.99721
$592$	0	0
$593$	8.00000	0.328521	0.164260	−	0.986417i	$-0.447476\pi$
$593$	8.00000	0.328521	0.164260	+	0.986417i	$0.447476\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	60.5531	2.47827
$598$	0	0
$599$	1.83500	0.0749762	0.0374881	−	0.999297i	$-0.488064\pi$
$599$	1.83500	0.0749762	0.0374881	+	0.999297i	$0.488064\pi$
$600$	0	0
$601$	−36.1432	−1.47431	−0.737156	−	0.675723i	$-0.763832\pi$
$601$	−36.1432	−1.47431	−0.737156	+	0.675723i	$0.763832\pi$
$602$	0	0
$603$	−42.1289	−1.71562
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.5353	0.711735	0.355867	−	0.934536i	$-0.384185\pi$
$607$	17.5353	0.711735	0.355867	+	0.934536i	$0.384185\pi$
$608$	0	0
$609$	20.4701	0.829491
$610$	0	0
$611$	−13.9813	−0.565621
$612$	0	0
$613$	−33.9309	−1.37046	−0.685228	−	0.728329i	$-0.740297\pi$
$613$	−33.9309	−1.37046	−0.685228	+	0.728329i	$0.740297\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.68598	0.389943	0.194971	−	0.980809i	$-0.437539\pi$
$617$	9.68598	0.389943	0.194971	+	0.980809i	$0.437539\pi$
$618$	0	0
$619$	41.4005	1.66403	0.832014	−	0.554755i	$-0.187188\pi$
$619$	41.4005	1.66403	0.832014	+	0.554755i	$0.187188\pi$
$620$	0	0
$621$	−3.34614	−0.134276
$622$	0	0
$623$	42.6637	1.70929
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−55.8163	−2.22909
$628$	0	0
$629$	41.7146	1.66327
$630$	0	0
$631$	−27.8163	−1.10735	−0.553674	−	0.832733i	$-0.686775\pi$
$631$	−27.8163	−1.10735	−0.553674	+	0.832733i	$0.686775\pi$
$632$	0	0
$633$	13.5397	0.538155
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−14.2351	−0.564014
$638$	0	0
$639$	−26.3684	−1.04312
$640$	0	0
$641$	42.8988	1.69440	0.847200	−	0.531275i	$-0.178287\pi$
$641$	42.8988	1.69440	0.847200	+	0.531275i	$0.178287\pi$
$642$	0	0
$643$	27.9639	1.10279	0.551395	−	0.834245i	$-0.314096\pi$
$643$	27.9639	1.10279	0.551395	+	0.834245i	$0.314096\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.13828	−0.280635	−0.140317	−	0.990107i	$-0.544812\pi$
$647$	−7.13828	−0.280635	−0.140317	+	0.990107i	$0.544812\pi$
$648$	0	0
$649$	19.2257	0.754675
$650$	0	0
$651$	28.2034	1.10538
$652$	0	0
$653$	17.7649	0.695196	0.347598	−	0.937644i	$-0.386997\pi$
$653$	17.7649	0.695196	0.347598	+	0.937644i	$0.386997\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	36.1748	1.41131
$658$	0	0
$659$	20.1936	0.786630	0.393315	−	0.919404i	$-0.371328\pi$
$659$	20.1936	0.786630	0.393315	+	0.919404i	$0.371328\pi$
$660$	0	0
$661$	22.0701	0.858426	0.429213	−	0.903203i	$-0.358791\pi$
$661$	22.0701	0.858426	0.429213	+	0.903203i	$0.358791\pi$
$662$	0	0
$663$	44.2034	1.71672
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.949145	0.0367510
$668$	0	0
$669$	−76.8069	−2.96953
$670$	0	0
$671$	46.5718	1.79789
$672$	0	0
$673$	26.0098	1.00261	0.501303	−	0.865272i	$-0.332854\pi$
$673$	26.0098	1.00261	0.501303	+	0.865272i	$0.332854\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.86665	0.302340	0.151170	−	0.988508i	$-0.451696\pi$
$677$	7.86665	0.302340	0.151170	+	0.988508i	$0.451696\pi$
$678$	0	0
$679$	48.6735	1.86792
$680$	0	0
$681$	35.7748	1.37089
$682$	0	0
$683$	18.2494	0.698292	0.349146	−	0.937068i	$-0.386472\pi$
$683$	18.2494	0.698292	0.349146	+	0.937068i	$0.386472\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	38.3970	1.46494
$688$	0	0
$689$	14.1017	0.537232
$690$	0	0
$691$	−25.2543	−0.960718	−0.480359	−	0.877072i	$-0.659494\pi$
$691$	−25.2543	−0.960718	−0.480359	+	0.877072i	$0.659494\pi$
$692$	0	0
$693$	72.8484	2.76728
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−30.1017	−1.14018
$698$	0	0
$699$	19.3461	0.731738
$700$	0	0
$701$	34.8069	1.31464	0.657319	−	0.753612i	$-0.271690\pi$
$701$	34.8069	1.31464	0.657319	+	0.753612i	$0.271690\pi$
$702$	0	0
$703$	−36.2864	−1.36857
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−69.5022	−2.61390
$708$	0	0
$709$	−7.51114	−0.282087	−0.141043	−	0.990003i	$-0.545046\pi$
$709$	−7.51114	−0.282087	−0.141043	+	0.990003i	$0.545046\pi$
$710$	0	0
$711$	−28.4701	−1.06771
$712$	0	0
$713$	1.30772	0.0489744
$714$	0	0
$715$	0	0
$716$	0	0
$717$	66.6548	2.48927
$718$	0	0
$719$	26.7556	0.997814	0.498907	−	0.866655i	$-0.333735\pi$
$719$	26.7556	0.997814	0.498907	+	0.866655i	$0.333735\pi$
$720$	0	0
$721$	−11.0638	−0.412036
$722$	0	0
$723$	−40.7654	−1.51608
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−25.6271	−0.950458	−0.475229	−	0.879862i	$-0.657635\pi$
$727$	−25.6271	−0.950458	−0.475229	+	0.879862i	$0.657635\pi$
$728$	0	0
$729$	−38.6958	−1.43318
$730$	0	0
$731$	11.3461	0.419652
$732$	0	0
$733$	19.6543	0.725949	0.362975	−	0.931799i	$-0.381761\pi$
$733$	19.6543	0.725949	0.362975	+	0.931799i	$0.381761\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	29.5397	1.08811
$738$	0	0
$739$	−32.5433	−1.19712	−0.598562	−	0.801077i	$-0.704261\pi$
$739$	−32.5433	−1.19712	−0.598562	+	0.801077i	$0.704261\pi$
$740$	0	0
$741$	−38.4514	−1.41255
$742$	0	0
$743$	23.1383	0.848861	0.424430	−	0.905461i	$-0.360474\pi$
$743$	23.1383	0.848861	0.424430	+	0.905461i	$0.360474\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	65.9452	2.41281
$748$	0	0
$749$	19.0094	0.694587
$750$	0	0
$751$	−2.48886	−0.0908199	−0.0454099	−	0.998968i	$-0.514459\pi$
$751$	−2.48886	−0.0908199	−0.0454099	+	0.998968i	$0.514459\pi$
$752$	0	0
$753$	72.3783	2.63761
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−21.2859	−0.773650	−0.386825	−	0.922153i	$-0.626428\pi$
$757$	−21.2859	−0.773650	−0.386825	+	0.922153i	$0.626428\pi$
$758$	0	0
$759$	5.24443	0.190361
$760$	0	0
$761$	19.1240	0.693244	0.346622	−	0.938005i	$-0.387329\pi$
$761$	19.1240	0.693244	0.346622	+	0.938005i	$0.387329\pi$
$762$	0	0
$763$	−30.3970	−1.10045
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.2444	0.478229
$768$	0	0
$769$	−33.9625	−1.22472	−0.612360	−	0.790579i	$-0.709780\pi$
$769$	−33.9625	−1.22472	−0.612360	+	0.790579i	$0.709780\pi$
$770$	0	0
$771$	74.6548	2.68863
$772$	0	0
$773$	0.133353	0.00479638	0.00239819	−	0.999997i	$-0.499237\pi$
$773$	0.133353	0.00479638	0.00239819	+	0.999997i	$0.499237\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	73.5308	2.63790
$778$	0	0
$779$	26.1847	0.938164
$780$	0	0
$781$	18.4889	0.661584
$782$	0	0
$783$	−14.1017	−0.503954
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.12537	−0.0401150	−0.0200575	−	0.999799i	$-0.506385\pi$
$787$	−1.12537	−0.0401150	−0.0200575	+	0.999799i	$0.506385\pi$
$788$	0	0
$789$	−7.47949	−0.266277
$790$	0	0
$791$	19.4291	0.690820
$792$	0	0
$793$	32.0830	1.13930
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.11108	0.181044	0.0905218	−	0.995894i	$-0.471147\pi$
$797$	5.11108	0.181044	0.0905218	+	0.995894i	$0.471147\pi$
$798$	0	0
$799$	−30.9590	−1.09525
$800$	0	0
$801$	−65.6958	−2.32125
$802$	0	0
$803$	−25.3649	−0.895107
$804$	0	0
$805$	0	0
$806$	0	0
$807$	74.7467	2.63121
$808$	0	0
$809$	18.7368	0.658752	0.329376	−	0.944199i	$-0.393162\pi$
$809$	18.7368	0.658752	0.329376	+	0.944199i	$0.393162\pi$
$810$	0	0
$811$	−37.5210	−1.31754	−0.658770	−	0.752344i	$-0.728923\pi$
$811$	−37.5210	−1.31754	−0.658770	+	0.752344i	$0.728923\pi$
$812$	0	0
$813$	87.6325	3.07341
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−9.86971	−0.345297
$818$	0	0
$819$	50.1847	1.75359
$820$	0	0
$821$	18.4001	0.642166	0.321083	−	0.947051i	$-0.395953\pi$
$821$	18.4001	0.642166	0.321083	+	0.947051i	$0.395953\pi$
$822$	0	0
$823$	0.649413	0.0226371	0.0113186	−	0.999936i	$-0.496397\pi$
$823$	0.649413	0.0226371	0.0113186	+	0.999936i	$0.496397\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29.8622	−1.03841	−0.519205	−	0.854650i	$-0.673772\pi$
$827$	−29.8622	−1.03841	−0.519205	+	0.854650i	$0.673772\pi$
$828$	0	0
$829$	21.1753	0.735449	0.367725	−	0.929935i	$-0.380137\pi$
$829$	21.1753	0.735449	0.367725	+	0.929935i	$0.380137\pi$
$830$	0	0
$831$	30.5718	1.06053
$832$	0	0
$833$	−31.5210	−1.09214
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−19.4291	−0.671568
$838$	0	0
$839$	−43.8163	−1.51271	−0.756353	−	0.654164i	$-0.773021\pi$
$839$	−43.8163	−1.51271	−0.756353	+	0.654164i	$0.773021\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−23.0509	−0.793914
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−12.2997	−0.422624
$848$	0	0
$849$	31.6543	1.08637
$850$	0	0
$851$	3.40943	0.116874
$852$	0	0
$853$	−37.7846	−1.29372	−0.646860	−	0.762608i	$-0.723918\pi$
$853$	−37.7846	−1.29372	−0.646860	+	0.762608i	$0.723918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.5299	0.667128	0.333564	−	0.942727i	$-0.391749\pi$
$857$	19.5299	0.667128	0.333564	+	0.942727i	$0.391749\pi$
$858$	0	0
$859$	−33.2543	−1.13462	−0.567311	−	0.823504i	$-0.692016\pi$
$859$	−33.2543	−1.13462	−0.567311	+	0.823504i	$0.692016\pi$
$860$	0	0
$861$	−53.0607	−1.80830
$862$	0	0
$863$	44.9733	1.53091	0.765454	−	0.643490i	$-0.222514\pi$
$863$	44.9733	1.53091	0.765454	+	0.643490i	$0.222514\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	48.5259	1.64803
$868$	0	0
$869$	19.9625	0.677182
$870$	0	0
$871$	20.3497	0.689523
$872$	0	0
$873$	−74.9501	−2.53668
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.30819	0.280548	0.140274	−	0.990113i	$-0.455202\pi$
$877$	8.30819	0.280548	0.140274	+	0.990113i	$0.455202\pi$
$878$	0	0
$879$	−12.8573	−0.433665
$880$	0	0
$881$	−39.3689	−1.32637	−0.663186	−	0.748455i	$-0.730796\pi$
$881$	−39.3689	−1.32637	−0.663186	+	0.748455i	$0.730796\pi$
$882$	0	0
$883$	7.29036	0.245340	0.122670	−	0.992447i	$-0.460854\pi$
$883$	7.29036	0.245340	0.122670	+	0.992447i	$0.460854\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.1097	0.373027	0.186514	−	0.982452i	$-0.440281\pi$
$887$	11.1097	0.373027	0.186514	+	0.982452i	$0.440281\pi$
$888$	0	0
$889$	36.2449	1.21562
$890$	0	0
$891$	−15.9269	−0.533570
$892$	0	0
$893$	26.9304	0.901192
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.61285	0.120629
$898$	0	0
$899$	5.51114	0.183807
$900$	0	0
$901$	31.2257	1.04028
$902$	0	0
$903$	20.0000	0.665558
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−17.5383	−0.582351	−0.291175	−	0.956670i	$-0.594046\pi$
$907$	−17.5383	−0.582351	−0.291175	+	0.956670i	$0.594046\pi$
$908$	0	0
$909$	107.023	3.54974
$910$	0	0
$911$	44.4701	1.47336	0.736681	−	0.676241i	$-0.236392\pi$
$911$	44.4701	1.47336	0.736681	+	0.676241i	$0.236392\pi$
$912$	0	0
$913$	−46.2391	−1.53029
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−16.4415	−0.542948
$918$	0	0
$919$	7.87955	0.259922	0.129961	−	0.991519i	$-0.458515\pi$
$919$	7.87955	0.259922	0.129961	+	0.991519i	$0.458515\pi$
$920$	0	0
$921$	−15.3047	−0.504306
$922$	0	0
$923$	12.7368	0.419238
$924$	0	0
$925$	0	0
$926$	0	0
$927$	17.0366	0.559554
$928$	0	0
$929$	15.7560	0.516939	0.258470	−	0.966019i	$-0.416782\pi$
$929$	15.7560	0.516939	0.258470	+	0.966019i	$0.416782\pi$
$930$	0	0
$931$	27.4193	0.898630
$932$	0	0
$933$	1.12399	0.0367976
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.2766	0.335720	0.167860	−	0.985811i	$-0.446314\pi$
$937$	10.2766	0.335720	0.167860	+	0.985811i	$0.446314\pi$
$938$	0	0
$939$	−32.9403	−1.07496
$940$	0	0
$941$	2.53341	0.0825869	0.0412934	−	0.999147i	$-0.486852\pi$
$941$	2.53341	0.0825869	0.0412934	+	0.999147i	$0.486852\pi$
$942$	0	0
$943$	−2.46028	−0.0801178
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.06821	−0.0347121	−0.0173560	−	0.999849i	$-0.505525\pi$
$947$	−1.06821	−0.0347121	−0.0173560	+	0.999849i	$0.505525\pi$
$948$	0	0
$949$	−17.4737	−0.567219
$950$	0	0
$951$	48.5531	1.57444
$952$	0	0
$953$	51.6958	1.67459	0.837296	−	0.546750i	$-0.184135\pi$
$953$	51.6958	1.67459	0.837296	+	0.546750i	$0.184135\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	22.1017	0.714447
$958$	0	0
$959$	40.0000	1.29167
$960$	0	0
$961$	−23.4068	−0.755059
$962$	0	0
$963$	−29.2716	−0.943265
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−52.2623	−1.68064	−0.840320	−	0.542090i	$-0.817633\pi$
$967$	−52.2623	−1.68064	−0.840320	+	0.542090i	$0.817633\pi$
$968$	0	0
$969$	−85.1437	−2.73521
$970$	0	0
$971$	−59.3560	−1.90482	−0.952412	−	0.304813i	$-0.901406\pi$
$971$	−59.3560	−1.90482	−0.952412	+	0.304813i	$0.901406\pi$
$972$	0	0
$973$	−41.6227	−1.33436
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−47.8707	−1.53152	−0.765759	−	0.643128i	$-0.777637\pi$
$977$	−47.8707	−1.53152	−0.765759	+	0.643128i	$0.777637\pi$
$978$	0	0
$979$	46.0642	1.47222
$980$	0	0
$981$	46.8069	1.49443
$982$	0	0
$983$	6.01429	0.191826	0.0959130	−	0.995390i	$-0.469423\pi$
$983$	6.01429	0.191826	0.0959130	+	0.995390i	$0.469423\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−54.5718	−1.73704
$988$	0	0
$989$	0.927346	0.0294879
$990$	0	0
$991$	−19.4291	−0.617186	−0.308593	−	0.951194i	$-0.599858\pi$
$991$	−19.4291	−0.617186	−0.308593	+	0.951194i	$0.599858\pi$
$992$	0	0
$993$	9.68598	0.307375
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−32.1334	−1.01767	−0.508837	−	0.860863i	$-0.669924\pi$
$997$	−32.1334	−1.01767	−0.508837	+	0.860863i	$0.669924\pi$
$998$	0	0
$999$	−50.6548	−1.60265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.a.bt.1.3		3
4.3	odd	2		3200.2.a.bq.1.1		3
5.2	odd	4		640.2.c.b.129.1	yes	6
5.3	odd	4		640.2.c.b.129.6	yes	6
5.4	even	2		3200.2.a.br.1.1		3
8.3	odd	2		3200.2.a.bv.1.3		3
8.5	even	2		3200.2.a.bo.1.1		3
20.3	even	4		640.2.c.a.129.1	✓	6
20.7	even	4		640.2.c.a.129.6	yes	6
20.19	odd	2		3200.2.a.bs.1.3		3
40.3	even	4		640.2.c.d.129.6	yes	6
40.13	odd	4		640.2.c.c.129.1	yes	6
40.19	odd	2		3200.2.a.bp.1.1		3
40.27	even	4		640.2.c.d.129.1	yes	6
40.29	even	2		3200.2.a.bu.1.3		3
40.37	odd	4		640.2.c.c.129.6	yes	6
80.3	even	4		1280.2.f.l.129.6		6
80.13	odd	4		1280.2.f.j.129.2		6
80.27	even	4		1280.2.f.l.129.5		6
80.37	odd	4		1280.2.f.j.129.1		6
80.43	even	4		1280.2.f.i.129.1		6
80.53	odd	4		1280.2.f.k.129.5		6
80.67	even	4		1280.2.f.i.129.2		6
80.77	odd	4		1280.2.f.k.129.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.c.a.129.1	✓	6	20.3	even	4
640.2.c.a.129.6	yes	6	20.7	even	4
640.2.c.b.129.1	yes	6	5.2	odd	4
640.2.c.b.129.6	yes	6	5.3	odd	4
640.2.c.c.129.1	yes	6	40.13	odd	4
640.2.c.c.129.6	yes	6	40.37	odd	4
640.2.c.d.129.1	yes	6	40.27	even	4
640.2.c.d.129.6	yes	6	40.3	even	4
1280.2.f.i.129.1		6	80.43	even	4
1280.2.f.i.129.2		6	80.67	even	4
1280.2.f.j.129.1		6	80.37	odd	4
1280.2.f.j.129.2		6	80.13	odd	4
1280.2.f.k.129.5		6	80.53	odd	4
1280.2.f.k.129.6		6	80.77	odd	4
1280.2.f.l.129.5		6	80.27	even	4
1280.2.f.l.129.6		6	80.3	even	4
3200.2.a.bo.1.1		3	8.5	even	2
3200.2.a.bp.1.1		3	40.19	odd	2
3200.2.a.bq.1.1		3	4.3	odd	2
3200.2.a.br.1.1		3	5.4	even	2
3200.2.a.bs.1.3		3	20.19	odd	2
3200.2.a.bt.1.3		3	1.1	even	1	trivial
3200.2.a.bu.1.3		3	40.29	even	2
3200.2.a.bv.1.3		3	8.3	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:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.a (trivial)

\(f(q)\)	\(=\)	\(q+2.90321 q^{3} -3.52543 q^{7} +5.42864 q^{9} +O(q^{10})\)	\(q+2.90321 q^{3} -3.52543 q^{7} +5.42864 q^{9} -3.80642 q^{11} -2.62222 q^{13} -5.80642 q^{17} +5.05086 q^{19} -10.2351 q^{21} -0.474572 q^{23} +7.05086 q^{27} -2.00000 q^{29} -2.75557 q^{31} -11.0509 q^{33} -7.18421 q^{37} -7.61285 q^{39} +5.18421 q^{41} -1.95407 q^{43} +5.33185 q^{47} +5.42864 q^{49} -16.8573 q^{51} -5.37778 q^{53} +14.6637 q^{57} -5.05086 q^{59} -12.2351 q^{61} -19.1383 q^{63} -7.76049 q^{67} -1.37778 q^{69} -4.85728 q^{71} +6.66370 q^{73} +13.4193 q^{77} -5.24443 q^{79} +4.18421 q^{81} +12.1476 q^{83} -5.80642 q^{87} -12.1017 q^{89} +9.24443 q^{91} -8.00000 q^{93} -13.8064 q^{97} -20.6637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 - 4 * q^7 + 3 * q^9	\( 3 q + 2 q^{3} - 4 q^{7} + 3 q^{9} + 2 q^{11} - 8 q^{13} - 4 q^{17} + 2 q^{19} - 4 q^{21} - 8 q^{23} + 8 q^{27} - 6 q^{29} - 8 q^{31} - 20 q^{33} - 8 q^{37} + 4 q^{39} + 2 q^{41} + 14 q^{43} - 4 q^{47} + 3 q^{49} - 24 q^{51} - 16 q^{53} + 4 q^{57} - 2 q^{59} - 10 q^{61} - 24 q^{63} + 10 q^{67} - 4 q^{69} + 12 q^{71} - 20 q^{73} - 16 q^{79} - q^{81} + 30 q^{83} - 4 q^{87} - 10 q^{89} + 28 q^{91} - 24 q^{93} - 28 q^{97} - 22 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 4 * q^7 + 3 * q^9 + 2 * q^11 - 8 * q^13 - 4 * q^17 + 2 * q^19 - 4 * q^21 - 8 * q^23 + 8 * q^27 - 6 * q^29 - 8 * q^31 - 20 * q^33 - 8 * q^37 + 4 * q^39 + 2 * q^41 + 14 * q^43 - 4 * q^47 + 3 * q^49 - 24 * q^51 - 16 * q^53 + 4 * q^57 - 2 * q^59 - 10 * q^61 - 24 * q^63 + 10 * q^67 - 4 * q^69 + 12 * q^71 - 20 * q^73 - 16 * q^79 - q^81 + 30 * q^83 - 4 * q^87 - 10 * q^89 + 28 * q^91 - 24 * q^93 - 28 * q^97 - 22 * q^99

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