LMFDB - Embedded newform 1932.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,2,Mod(1,1932)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, 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("1932.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1932.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:	$15.4270976705$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1932.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.23607 q^{11} +2.61803 q^{13} -1.61803 q^{15} +2.23607 q^{17} -5.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -2.38197 q^{25} -1.00000 q^{27} -3.47214 q^{29} -6.23607 q^{31} +4.23607 q^{33} -1.61803 q^{35} +3.47214 q^{37} -2.61803 q^{39} -1.47214 q^{41} +7.56231 q^{43} +1.61803 q^{45} -5.23607 q^{47} +1.00000 q^{49} -2.23607 q^{51} +4.09017 q^{53} -6.85410 q^{55} +5.47214 q^{57} -5.61803 q^{59} -10.3262 q^{61} -1.00000 q^{63} +4.23607 q^{65} -5.61803 q^{67} -1.00000 q^{69} -4.14590 q^{71} -15.9443 q^{73} +2.38197 q^{75} +4.23607 q^{77} -3.76393 q^{79} +1.00000 q^{81} +9.94427 q^{83} +3.61803 q^{85} +3.47214 q^{87} +3.85410 q^{89} -2.61803 q^{91} +6.23607 q^{93} -8.85410 q^{95} +4.23607 q^{97} -4.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} + 3 q^{13} - q^{15} - 2 q^{19} + 2 q^{21} + 2 q^{23} - 7 q^{25} - 2 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} - q^{35} - 2 q^{37} - 3 q^{39} + 6 q^{41} - 5 q^{43} + q^{45} - 6 q^{47} + 2 q^{49} - 3 q^{53} - 7 q^{55} + 2 q^{57} - 9 q^{59} - 5 q^{61} - 2 q^{63} + 4 q^{65} - 9 q^{67} - 2 q^{69} - 15 q^{71} - 14 q^{73} + 7 q^{75} + 4 q^{77} - 12 q^{79} + 2 q^{81} + 2 q^{83} + 5 q^{85} - 2 q^{87} + q^{89} - 3 q^{91} + 8 q^{93} - 11 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 + 3 * q^13 - q^15 - 2 * q^19 + 2 * q^21 + 2 * q^23 - 7 * q^25 - 2 * q^27 + 2 * q^29 - 8 * q^31 + 4 * q^33 - q^35 - 2 * q^37 - 3 * q^39 + 6 * q^41 - 5 * q^43 + q^45 - 6 * q^47 + 2 * q^49 - 3 * q^53 - 7 * q^55 + 2 * q^57 - 9 * q^59 - 5 * q^61 - 2 * q^63 + 4 * q^65 - 9 * q^67 - 2 * q^69 - 15 * q^71 - 14 * q^73 + 7 * q^75 + 4 * q^77 - 12 * q^79 + 2 * q^81 + 2 * q^83 + 5 * q^85 - 2 * q^87 + q^89 - 3 * q^91 + 8 * q^93 - 11 * q^95 + 4 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.61803	0.723607	0.361803	−	0.932254i	$-0.382161\pi$
$5$	1.61803	0.723607	0.361803	+	0.932254i	$0.382161\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.23607	−1.27722	−0.638611	−	0.769529i	$-0.720491\pi$
$11$	−4.23607	−1.27722	−0.638611	+	0.769529i	$0.720491\pi$
$12$	0	0
$13$	2.61803	0.726112	0.363056	−	0.931767i	$-0.381733\pi$
$13$	2.61803	0.726112	0.363056	+	0.931767i	$0.381733\pi$
$14$	0	0
$15$	−1.61803	−0.417775
$16$	0	0
$17$	2.23607	0.542326	0.271163	−	0.962533i	$-0.412592\pi$
$17$	2.23607	0.542326	0.271163	+	0.962533i	$0.412592\pi$
$18$	0	0
$19$	−5.47214	−1.25539	−0.627697	−	0.778458i	$-0.716002\pi$
$19$	−5.47214	−1.25539	−0.627697	+	0.778458i	$0.716002\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.38197	−0.476393
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.47214	−0.644759	−0.322380	−	0.946610i	$-0.604483\pi$
$29$	−3.47214	−0.644759	−0.322380	+	0.946610i	$0.604483\pi$
$30$	0	0
$31$	−6.23607	−1.12003	−0.560015	−	0.828482i	$-0.689205\pi$
$31$	−6.23607	−1.12003	−0.560015	+	0.828482i	$0.689205\pi$
$32$	0	0
$33$	4.23607	0.737405
$34$	0	0
$35$	−1.61803	−0.273498
$36$	0	0
$37$	3.47214	0.570816	0.285408	−	0.958406i	$-0.407871\pi$
$37$	3.47214	0.570816	0.285408	+	0.958406i	$0.407871\pi$
$38$	0	0
$39$	−2.61803	−0.419221
$40$	0	0
$41$	−1.47214	−0.229909	−0.114955	−	0.993371i	$-0.536672\pi$
$41$	−1.47214	−0.229909	−0.114955	+	0.993371i	$0.536672\pi$
$42$	0	0
$43$	7.56231	1.15324	0.576620	−	0.817012i	$-0.304371\pi$
$43$	7.56231	1.15324	0.576620	+	0.817012i	$0.304371\pi$
$44$	0	0
$45$	1.61803	0.241202
$46$	0	0
$47$	−5.23607	−0.763759	−0.381880	−	0.924212i	$-0.624723\pi$
$47$	−5.23607	−0.763759	−0.381880	+	0.924212i	$0.624723\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.23607	−0.313112
$52$	0	0
$53$	4.09017	0.561828	0.280914	−	0.959733i	$-0.409362\pi$
$53$	4.09017	0.561828	0.280914	+	0.959733i	$0.409362\pi$
$54$	0	0
$55$	−6.85410	−0.924207
$56$	0	0
$57$	5.47214	0.724802
$58$	0	0
$59$	−5.61803	−0.731406	−0.365703	−	0.930732i	$-0.619171\pi$
$59$	−5.61803	−0.731406	−0.365703	+	0.930732i	$0.619171\pi$
$60$	0	0
$61$	−10.3262	−1.32214	−0.661070	−	0.750325i	$-0.729897\pi$
$61$	−10.3262	−1.32214	−0.661070	+	0.750325i	$0.729897\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	4.23607	0.525420
$66$	0	0
$67$	−5.61803	−0.686352	−0.343176	−	0.939271i	$-0.611503\pi$
$67$	−5.61803	−0.686352	−0.343176	+	0.939271i	$0.611503\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−4.14590	−0.492028	−0.246014	−	0.969266i	$-0.579121\pi$
$71$	−4.14590	−0.492028	−0.246014	+	0.969266i	$0.579121\pi$
$72$	0	0
$73$	−15.9443	−1.86614	−0.933068	−	0.359700i	$-0.882879\pi$
$73$	−15.9443	−1.86614	−0.933068	+	0.359700i	$0.882879\pi$
$74$	0	0
$75$	2.38197	0.275046
$76$	0	0
$77$	4.23607	0.482745
$78$	0	0
$79$	−3.76393	−0.423475	−0.211738	−	0.977327i	$-0.567912\pi$
$79$	−3.76393	−0.423475	−0.211738	+	0.977327i	$0.567912\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.94427	1.09153	0.545763	−	0.837940i	$-0.316240\pi$
$83$	9.94427	1.09153	0.545763	+	0.837940i	$0.316240\pi$
$84$	0	0
$85$	3.61803	0.392431
$86$	0	0
$87$	3.47214	0.372252
$88$	0	0
$89$	3.85410	0.408534	0.204267	−	0.978915i	$-0.434519\pi$
$89$	3.85410	0.408534	0.204267	+	0.978915i	$0.434519\pi$
$90$	0	0
$91$	−2.61803	−0.274445
$92$	0	0
$93$	6.23607	0.646650
$94$	0	0
$95$	−8.85410	−0.908412
$96$	0	0
$97$	4.23607	0.430108	0.215054	−	0.976602i	$-0.431007\pi$
$97$	4.23607	0.430108	0.215054	+	0.976602i	$0.431007\pi$
$98$	0	0
$99$	−4.23607	−0.425741
$100$	0	0
$101$	−3.61803	−0.360008	−0.180004	−	0.983666i	$-0.557611\pi$
$101$	−3.61803	−0.360008	−0.180004	+	0.983666i	$0.557611\pi$
$102$	0	0
$103$	8.94427	0.881305	0.440653	−	0.897678i	$-0.354747\pi$
$103$	8.94427	0.881305	0.440653	+	0.897678i	$0.354747\pi$
$104$	0	0
$105$	1.61803	0.157904
$106$	0	0
$107$	−12.6180	−1.21983	−0.609916	−	0.792466i	$-0.708797\pi$
$107$	−12.6180	−1.21983	−0.609916	+	0.792466i	$0.708797\pi$
$108$	0	0
$109$	−4.38197	−0.419716	−0.209858	−	0.977732i	$-0.567300\pi$
$109$	−4.38197	−0.419716	−0.209858	+	0.977732i	$0.567300\pi$
$110$	0	0
$111$	−3.47214	−0.329561
$112$	0	0
$113$	−12.5623	−1.18176	−0.590881	−	0.806759i	$-0.701220\pi$
$113$	−12.5623	−1.18176	−0.590881	+	0.806759i	$0.701220\pi$
$114$	0	0
$115$	1.61803	0.150882
$116$	0	0
$117$	2.61803	0.242037
$118$	0	0
$119$	−2.23607	−0.204980
$120$	0	0
$121$	6.94427	0.631297
$122$	0	0
$123$	1.47214	0.132738
$124$	0	0
$125$	−11.9443	−1.06833
$126$	0	0
$127$	−13.8541	−1.22935	−0.614676	−	0.788779i	$-0.710713\pi$
$127$	−13.8541	−1.22935	−0.614676	+	0.788779i	$0.710713\pi$
$128$	0	0
$129$	−7.56231	−0.665824
$130$	0	0
$131$	17.9443	1.56780	0.783899	−	0.620888i	$-0.213228\pi$
$131$	17.9443	1.56780	0.783899	+	0.620888i	$0.213228\pi$
$132$	0	0
$133$	5.47214	0.474494
$134$	0	0
$135$	−1.61803	−0.139258
$136$	0	0
$137$	4.23607	0.361912	0.180956	−	0.983491i	$-0.442081\pi$
$137$	4.23607	0.361912	0.180956	+	0.983491i	$0.442081\pi$
$138$	0	0
$139$	17.0344	1.44484	0.722421	−	0.691453i	$-0.243029\pi$
$139$	17.0344	1.44484	0.722421	+	0.691453i	$0.243029\pi$
$140$	0	0
$141$	5.23607	0.440956
$142$	0	0
$143$	−11.0902	−0.927407
$144$	0	0
$145$	−5.61803	−0.466552
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−16.6525	−1.36422	−0.682112	−	0.731248i	$-0.738938\pi$
$149$	−16.6525	−1.36422	−0.682112	+	0.731248i	$0.738938\pi$
$150$	0	0
$151$	−15.7082	−1.27832	−0.639158	−	0.769076i	$-0.720717\pi$
$151$	−15.7082	−1.27832	−0.639158	+	0.769076i	$0.720717\pi$
$152$	0	0
$153$	2.23607	0.180775
$154$	0	0
$155$	−10.0902	−0.810462
$156$	0	0
$157$	−12.7639	−1.01867	−0.509336	−	0.860568i	$-0.670109\pi$
$157$	−12.7639	−1.01867	−0.509336	+	0.860568i	$0.670109\pi$
$158$	0	0
$159$	−4.09017	−0.324372
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−16.5623	−1.29726	−0.648630	−	0.761104i	$-0.724658\pi$
$163$	−16.5623	−1.29726	−0.648630	+	0.761104i	$0.724658\pi$
$164$	0	0
$165$	6.85410	0.533591
$166$	0	0
$167$	20.8885	1.61640	0.808202	−	0.588905i	$-0.200441\pi$
$167$	20.8885	1.61640	0.808202	+	0.588905i	$0.200441\pi$
$168$	0	0
$169$	−6.14590	−0.472761
$170$	0	0
$171$	−5.47214	−0.418465
$172$	0	0
$173$	2.05573	0.156294	0.0781471	−	0.996942i	$-0.475100\pi$
$173$	2.05573	0.156294	0.0781471	+	0.996942i	$0.475100\pi$
$174$	0	0
$175$	2.38197	0.180060
$176$	0	0
$177$	5.61803	0.422277
$178$	0	0
$179$	21.7984	1.62929	0.814643	−	0.579962i	$-0.196933\pi$
$179$	21.7984	1.62929	0.814643	+	0.579962i	$0.196933\pi$
$180$	0	0
$181$	0.236068	0.0175468	0.00877340	−	0.999962i	$-0.497207\pi$
$181$	0.236068	0.0175468	0.00877340	+	0.999962i	$0.497207\pi$
$182$	0	0
$183$	10.3262	0.763337
$184$	0	0
$185$	5.61803	0.413046
$186$	0	0
$187$	−9.47214	−0.692671
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−19.1246	−1.38381	−0.691904	−	0.721989i	$-0.743228\pi$
$191$	−19.1246	−1.38381	−0.691904	+	0.721989i	$0.743228\pi$
$192$	0	0
$193$	−17.7082	−1.27466	−0.637332	−	0.770589i	$-0.719962\pi$
$193$	−17.7082	−1.27466	−0.637332	+	0.770589i	$0.719962\pi$
$194$	0	0
$195$	−4.23607	−0.303351
$196$	0	0
$197$	−8.61803	−0.614009	−0.307005	−	0.951708i	$-0.599327\pi$
$197$	−8.61803	−0.614009	−0.307005	+	0.951708i	$0.599327\pi$
$198$	0	0
$199$	−21.3262	−1.51178	−0.755888	−	0.654700i	$-0.772795\pi$
$199$	−21.3262	−1.51178	−0.755888	+	0.654700i	$0.772795\pi$
$200$	0	0
$201$	5.61803	0.396266
$202$	0	0
$203$	3.47214	0.243696
$204$	0	0
$205$	−2.38197	−0.166364
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	23.1803	1.60342
$210$	0	0
$211$	2.23607	0.153937	0.0769686	−	0.997034i	$-0.475476\pi$
$211$	2.23607	0.153937	0.0769686	+	0.997034i	$0.475476\pi$
$212$	0	0
$213$	4.14590	0.284072
$214$	0	0
$215$	12.2361	0.834493
$216$	0	0
$217$	6.23607	0.423332
$218$	0	0
$219$	15.9443	1.07741
$220$	0	0
$221$	5.85410	0.393790
$222$	0	0
$223$	−25.2148	−1.68851	−0.844253	−	0.535944i	$-0.819956\pi$
$223$	−25.2148	−1.68851	−0.844253	+	0.535944i	$0.819956\pi$
$224$	0	0
$225$	−2.38197	−0.158798
$226$	0	0
$227$	−10.0902	−0.669708	−0.334854	−	0.942270i	$-0.608687\pi$
$227$	−10.0902	−0.669708	−0.334854	+	0.942270i	$0.608687\pi$
$228$	0	0
$229$	12.3820	0.818223	0.409112	−	0.912484i	$-0.365839\pi$
$229$	12.3820	0.818223	0.409112	+	0.912484i	$0.365839\pi$
$230$	0	0
$231$	−4.23607	−0.278713
$232$	0	0
$233$	2.09017	0.136932	0.0684658	−	0.997653i	$-0.478190\pi$
$233$	2.09017	0.136932	0.0684658	+	0.997653i	$0.478190\pi$
$234$	0	0
$235$	−8.47214	−0.552661
$236$	0	0
$237$	3.76393	0.244494
$238$	0	0
$239$	7.03444	0.455020	0.227510	−	0.973776i	$-0.426942\pi$
$239$	7.03444	0.455020	0.227510	+	0.973776i	$0.426942\pi$
$240$	0	0
$241$	−11.2918	−0.727369	−0.363684	−	0.931522i	$-0.618481\pi$
$241$	−11.2918	−0.727369	−0.363684	+	0.931522i	$0.618481\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.61803	0.103372
$246$	0	0
$247$	−14.3262	−0.911557
$248$	0	0
$249$	−9.94427	−0.630193
$250$	0	0
$251$	−0.180340	−0.0113830	−0.00569148	−	0.999984i	$-0.501812\pi$
$251$	−0.180340	−0.0113830	−0.00569148	+	0.999984i	$0.501812\pi$
$252$	0	0
$253$	−4.23607	−0.266319
$254$	0	0
$255$	−3.61803	−0.226570
$256$	0	0
$257$	12.2918	0.766741	0.383371	−	0.923595i	$-0.374763\pi$
$257$	12.2918	0.766741	0.383371	+	0.923595i	$0.374763\pi$
$258$	0	0
$259$	−3.47214	−0.215748
$260$	0	0
$261$	−3.47214	−0.214920
$262$	0	0
$263$	9.94427	0.613190	0.306595	−	0.951840i	$-0.400810\pi$
$263$	9.94427	0.613190	0.306595	+	0.951840i	$0.400810\pi$
$264$	0	0
$265$	6.61803	0.406543
$266$	0	0
$267$	−3.85410	−0.235867
$268$	0	0
$269$	23.0902	1.40783	0.703916	−	0.710283i	$-0.251433\pi$
$269$	23.0902	1.40783	0.703916	+	0.710283i	$0.251433\pi$
$270$	0	0
$271$	31.3607	1.90503	0.952513	−	0.304499i	$-0.0984889\pi$
$271$	31.3607	1.90503	0.952513	+	0.304499i	$0.0984889\pi$
$272$	0	0
$273$	2.61803	0.158451
$274$	0	0
$275$	10.0902	0.608460
$276$	0	0
$277$	8.61803	0.517807	0.258904	−	0.965903i	$-0.416639\pi$
$277$	8.61803	0.517807	0.258904	+	0.965903i	$0.416639\pi$
$278$	0	0
$279$	−6.23607	−0.373344
$280$	0	0
$281$	22.6525	1.35133	0.675667	−	0.737207i	$-0.263856\pi$
$281$	22.6525	1.35133	0.675667	+	0.737207i	$0.263856\pi$
$282$	0	0
$283$	−7.90983	−0.470191	−0.235095	−	0.971972i	$-0.575540\pi$
$283$	−7.90983	−0.470191	−0.235095	+	0.971972i	$0.575540\pi$
$284$	0	0
$285$	8.85410	0.524472
$286$	0	0
$287$	1.47214	0.0868974
$288$	0	0
$289$	−12.0000	−0.705882
$290$	0	0
$291$	−4.23607	−0.248323
$292$	0	0
$293$	24.9443	1.45726	0.728630	−	0.684908i	$-0.240157\pi$
$293$	24.9443	1.45726	0.728630	+	0.684908i	$0.240157\pi$
$294$	0	0
$295$	−9.09017	−0.529250
$296$	0	0
$297$	4.23607	0.245802
$298$	0	0
$299$	2.61803	0.151405
$300$	0	0
$301$	−7.56231	−0.435884
$302$	0	0
$303$	3.61803	0.207851
$304$	0	0
$305$	−16.7082	−0.956709
$306$	0	0
$307$	10.2361	0.584203	0.292102	−	0.956387i	$-0.405645\pi$
$307$	10.2361	0.584203	0.292102	+	0.956387i	$0.405645\pi$
$308$	0	0
$309$	−8.94427	−0.508822
$310$	0	0
$311$	30.9787	1.75664	0.878321	−	0.478072i	$-0.158664\pi$
$311$	30.9787	1.75664	0.878321	+	0.478072i	$0.158664\pi$
$312$	0	0
$313$	−2.47214	−0.139733	−0.0698667	−	0.997556i	$-0.522257\pi$
$313$	−2.47214	−0.139733	−0.0698667	+	0.997556i	$0.522257\pi$
$314$	0	0
$315$	−1.61803	−0.0911659
$316$	0	0
$317$	4.96556	0.278894	0.139447	−	0.990230i	$-0.455468\pi$
$317$	4.96556	0.278894	0.139447	+	0.990230i	$0.455468\pi$
$318$	0	0
$319$	14.7082	0.823501
$320$	0	0
$321$	12.6180	0.704270
$322$	0	0
$323$	−12.2361	−0.680833
$324$	0	0
$325$	−6.23607	−0.345915
$326$	0	0
$327$	4.38197	0.242323
$328$	0	0
$329$	5.23607	0.288674
$330$	0	0
$331$	18.9443	1.04127	0.520636	−	0.853779i	$-0.325695\pi$
$331$	18.9443	1.04127	0.520636	+	0.853779i	$0.325695\pi$
$332$	0	0
$333$	3.47214	0.190272
$334$	0	0
$335$	−9.09017	−0.496649
$336$	0	0
$337$	5.56231	0.302998	0.151499	−	0.988457i	$-0.451590\pi$
$337$	5.56231	0.302998	0.151499	+	0.988457i	$0.451590\pi$
$338$	0	0
$339$	12.5623	0.682291
$340$	0	0
$341$	26.4164	1.43053
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.61803	−0.0871120
$346$	0	0
$347$	−3.18034	−0.170730	−0.0853648	−	0.996350i	$-0.527206\pi$
$347$	−3.18034	−0.170730	−0.0853648	+	0.996350i	$0.527206\pi$
$348$	0	0
$349$	11.9098	0.637519	0.318759	−	0.947836i	$-0.396734\pi$
$349$	11.9098	0.637519	0.318759	+	0.947836i	$0.396734\pi$
$350$	0	0
$351$	−2.61803	−0.139740
$352$	0	0
$353$	25.9443	1.38087	0.690437	−	0.723392i	$-0.257418\pi$
$353$	25.9443	1.38087	0.690437	+	0.723392i	$0.257418\pi$
$354$	0	0
$355$	−6.70820	−0.356034
$356$	0	0
$357$	2.23607	0.118345
$358$	0	0
$359$	−7.32624	−0.386664	−0.193332	−	0.981133i	$-0.561929\pi$
$359$	−7.32624	−0.386664	−0.193332	+	0.981133i	$0.561929\pi$
$360$	0	0
$361$	10.9443	0.576014
$362$	0	0
$363$	−6.94427	−0.364480
$364$	0	0
$365$	−25.7984	−1.35035
$366$	0	0
$367$	12.7984	0.668070	0.334035	−	0.942561i	$-0.391590\pi$
$367$	12.7984	0.668070	0.334035	+	0.942561i	$0.391590\pi$
$368$	0	0
$369$	−1.47214	−0.0766363
$370$	0	0
$371$	−4.09017	−0.212351
$372$	0	0
$373$	8.05573	0.417110	0.208555	−	0.978011i	$-0.433124\pi$
$373$	8.05573	0.417110	0.208555	+	0.978011i	$0.433124\pi$
$374$	0	0
$375$	11.9443	0.616800
$376$	0	0
$377$	−9.09017	−0.468168
$378$	0	0
$379$	−26.8328	−1.37831	−0.689155	−	0.724614i	$-0.742018\pi$
$379$	−26.8328	−1.37831	−0.689155	+	0.724614i	$0.742018\pi$
$380$	0	0
$381$	13.8541	0.709767
$382$	0	0
$383$	9.47214	0.484004	0.242002	−	0.970276i	$-0.422196\pi$
$383$	9.47214	0.484004	0.242002	+	0.970276i	$0.422196\pi$
$384$	0	0
$385$	6.85410	0.349317
$386$	0	0
$387$	7.56231	0.384414
$388$	0	0
$389$	26.4164	1.33937	0.669683	−	0.742648i	$-0.266430\pi$
$389$	26.4164	1.33937	0.669683	+	0.742648i	$0.266430\pi$
$390$	0	0
$391$	2.23607	0.113083
$392$	0	0
$393$	−17.9443	−0.905169
$394$	0	0
$395$	−6.09017	−0.306430
$396$	0	0
$397$	5.81966	0.292080	0.146040	−	0.989279i	$-0.453347\pi$
$397$	5.81966	0.292080	0.146040	+	0.989279i	$0.453347\pi$
$398$	0	0
$399$	−5.47214	−0.273949
$400$	0	0
$401$	33.3607	1.66595	0.832976	−	0.553308i	$-0.186635\pi$
$401$	33.3607	1.66595	0.832976	+	0.553308i	$0.186635\pi$
$402$	0	0
$403$	−16.3262	−0.813268
$404$	0	0
$405$	1.61803	0.0804008
$406$	0	0
$407$	−14.7082	−0.729059
$408$	0	0
$409$	0.236068	0.0116728	0.00583641	−	0.999983i	$-0.498142\pi$
$409$	0.236068	0.0116728	0.00583641	+	0.999983i	$0.498142\pi$
$410$	0	0
$411$	−4.23607	−0.208950
$412$	0	0
$413$	5.61803	0.276445
$414$	0	0
$415$	16.0902	0.789835
$416$	0	0
$417$	−17.0344	−0.834180
$418$	0	0
$419$	−11.2148	−0.547878	−0.273939	−	0.961747i	$-0.588327\pi$
$419$	−11.2148	−0.547878	−0.273939	+	0.961747i	$0.588327\pi$
$420$	0	0
$421$	−20.5066	−0.999429	−0.499715	−	0.866190i	$-0.666562\pi$
$421$	−20.5066	−0.999429	−0.499715	+	0.866190i	$0.666562\pi$
$422$	0	0
$423$	−5.23607	−0.254586
$424$	0	0
$425$	−5.32624	−0.258360
$426$	0	0
$427$	10.3262	0.499722
$428$	0	0
$429$	11.0902	0.535438
$430$	0	0
$431$	−2.43769	−0.117420	−0.0587098	−	0.998275i	$-0.518699\pi$
$431$	−2.43769	−0.117420	−0.0587098	+	0.998275i	$0.518699\pi$
$432$	0	0
$433$	−10.2918	−0.494592	−0.247296	−	0.968940i	$-0.579542\pi$
$433$	−10.2918	−0.494592	−0.247296	+	0.968940i	$0.579542\pi$
$434$	0	0
$435$	5.61803	0.269364
$436$	0	0
$437$	−5.47214	−0.261768
$438$	0	0
$439$	29.1803	1.39270	0.696351	−	0.717702i	$-0.254806\pi$
$439$	29.1803	1.39270	0.696351	+	0.717702i	$0.254806\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	21.8885	1.03996	0.519978	−	0.854180i	$-0.325940\pi$
$443$	21.8885	1.03996	0.519978	+	0.854180i	$0.325940\pi$
$444$	0	0
$445$	6.23607	0.295618
$446$	0	0
$447$	16.6525	0.787635
$448$	0	0
$449$	−2.09017	−0.0986412	−0.0493206	−	0.998783i	$-0.515706\pi$
$449$	−2.09017	−0.0986412	−0.0493206	+	0.998783i	$0.515706\pi$
$450$	0	0
$451$	6.23607	0.293645
$452$	0	0
$453$	15.7082	0.738036
$454$	0	0
$455$	−4.23607	−0.198590
$456$	0	0
$457$	2.85410	0.133509	0.0667546	−	0.997769i	$-0.478736\pi$
$457$	2.85410	0.133509	0.0667546	+	0.997769i	$0.478736\pi$
$458$	0	0
$459$	−2.23607	−0.104371
$460$	0	0
$461$	−4.56231	−0.212488	−0.106244	−	0.994340i	$-0.533882\pi$
$461$	−4.56231	−0.212488	−0.106244	+	0.994340i	$0.533882\pi$
$462$	0	0
$463$	−1.11146	−0.0516537	−0.0258269	−	0.999666i	$-0.508222\pi$
$463$	−1.11146	−0.0516537	−0.0258269	+	0.999666i	$0.508222\pi$
$464$	0	0
$465$	10.0902	0.467920
$466$	0	0
$467$	−12.4164	−0.574563	−0.287281	−	0.957846i	$-0.592751\pi$
$467$	−12.4164	−0.574563	−0.287281	+	0.957846i	$0.592751\pi$
$468$	0	0
$469$	5.61803	0.259417
$470$	0	0
$471$	12.7639	0.588131
$472$	0	0
$473$	−32.0344	−1.47295
$474$	0	0
$475$	13.0344	0.598061
$476$	0	0
$477$	4.09017	0.187276
$478$	0	0
$479$	−15.1803	−0.693607	−0.346804	−	0.937938i	$-0.612733\pi$
$479$	−15.1803	−0.693607	−0.346804	+	0.937938i	$0.612733\pi$
$480$	0	0
$481$	9.09017	0.414476
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	6.85410	0.311229
$486$	0	0
$487$	29.4721	1.33551	0.667755	−	0.744381i	$-0.267255\pi$
$487$	29.4721	1.33551	0.667755	+	0.744381i	$0.267255\pi$
$488$	0	0
$489$	16.5623	0.748973
$490$	0	0
$491$	5.85410	0.264192	0.132096	−	0.991237i	$-0.457829\pi$
$491$	5.85410	0.264192	0.132096	+	0.991237i	$0.457829\pi$
$492$	0	0
$493$	−7.76393	−0.349670
$494$	0	0
$495$	−6.85410	−0.308069
$496$	0	0
$497$	4.14590	0.185969
$498$	0	0
$499$	32.2705	1.44463	0.722313	−	0.691566i	$-0.243079\pi$
$499$	32.2705	1.44463	0.722313	+	0.691566i	$0.243079\pi$
$500$	0	0
$501$	−20.8885	−0.933231
$502$	0	0
$503$	−24.5623	−1.09518	−0.547590	−	0.836747i	$-0.684454\pi$
$503$	−24.5623	−1.09518	−0.547590	+	0.836747i	$0.684454\pi$
$504$	0	0
$505$	−5.85410	−0.260504
$506$	0	0
$507$	6.14590	0.272949
$508$	0	0
$509$	15.7082	0.696254	0.348127	−	0.937447i	$-0.386818\pi$
$509$	15.7082	0.696254	0.348127	+	0.937447i	$0.386818\pi$
$510$	0	0
$511$	15.9443	0.705333
$512$	0	0
$513$	5.47214	0.241601
$514$	0	0
$515$	14.4721	0.637719
$516$	0	0
$517$	22.1803	0.975490
$518$	0	0
$519$	−2.05573	−0.0902364
$520$	0	0
$521$	−0.472136	−0.0206847	−0.0103423	−	0.999947i	$-0.503292\pi$
$521$	−0.472136	−0.0206847	−0.0103423	+	0.999947i	$0.503292\pi$
$522$	0	0
$523$	9.41641	0.411751	0.205875	−	0.978578i	$-0.433996\pi$
$523$	9.41641	0.411751	0.205875	+	0.978578i	$0.433996\pi$
$524$	0	0
$525$	−2.38197	−0.103958
$526$	0	0
$527$	−13.9443	−0.607422
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.61803	−0.243802
$532$	0	0
$533$	−3.85410	−0.166940
$534$	0	0
$535$	−20.4164	−0.882678
$536$	0	0
$537$	−21.7984	−0.940669
$538$	0	0
$539$	−4.23607	−0.182460
$540$	0	0
$541$	−15.7082	−0.675348	−0.337674	−	0.941263i	$-0.609640\pi$
$541$	−15.7082	−0.675348	−0.337674	+	0.941263i	$0.609640\pi$
$542$	0	0
$543$	−0.236068	−0.0101306
$544$	0	0
$545$	−7.09017	−0.303710
$546$	0	0
$547$	−8.09017	−0.345911	−0.172955	−	0.984930i	$-0.555332\pi$
$547$	−8.09017	−0.345911	−0.172955	+	0.984930i	$0.555332\pi$
$548$	0	0
$549$	−10.3262	−0.440713
$550$	0	0
$551$	19.0000	0.809427
$552$	0	0
$553$	3.76393	0.160059
$554$	0	0
$555$	−5.61803	−0.238472
$556$	0	0
$557$	−26.6525	−1.12930	−0.564651	−	0.825330i	$-0.690989\pi$
$557$	−26.6525	−1.12930	−0.564651	+	0.825330i	$0.690989\pi$
$558$	0	0
$559$	19.7984	0.837382
$560$	0	0
$561$	9.47214	0.399914
$562$	0	0
$563$	0.506578	0.0213497	0.0106749	−	0.999943i	$-0.496602\pi$
$563$	0.506578	0.0213497	0.0106749	+	0.999943i	$0.496602\pi$
$564$	0	0
$565$	−20.3262	−0.855131
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	17.5279	0.734806	0.367403	−	0.930062i	$-0.380247\pi$
$569$	17.5279	0.734806	0.367403	+	0.930062i	$0.380247\pi$
$570$	0	0
$571$	16.7082	0.699217	0.349608	−	0.936896i	$-0.386315\pi$
$571$	16.7082	0.699217	0.349608	+	0.936896i	$0.386315\pi$
$572$	0	0
$573$	19.1246	0.798942
$574$	0	0
$575$	−2.38197	−0.0993348
$576$	0	0
$577$	−13.0557	−0.543517	−0.271759	−	0.962365i	$-0.587605\pi$
$577$	−13.0557	−0.543517	−0.271759	+	0.962365i	$0.587605\pi$
$578$	0	0
$579$	17.7082	0.735928
$580$	0	0
$581$	−9.94427	−0.412558
$582$	0	0
$583$	−17.3262	−0.717579
$584$	0	0
$585$	4.23607	0.175140
$586$	0	0
$587$	12.0344	0.496715	0.248357	−	0.968668i	$-0.420109\pi$
$587$	12.0344	0.496715	0.248357	+	0.968668i	$0.420109\pi$
$588$	0	0
$589$	34.1246	1.40608
$590$	0	0
$591$	8.61803	0.354499
$592$	0	0
$593$	7.23607	0.297150	0.148575	−	0.988901i	$-0.452531\pi$
$593$	7.23607	0.297150	0.148575	+	0.988901i	$0.452531\pi$
$594$	0	0
$595$	−3.61803	−0.148325
$596$	0	0
$597$	21.3262	0.872825
$598$	0	0
$599$	−36.3951	−1.48706	−0.743532	−	0.668700i	$-0.766851\pi$
$599$	−36.3951	−1.48706	−0.743532	+	0.668700i	$0.766851\pi$
$600$	0	0
$601$	−25.6869	−1.04779	−0.523896	−	0.851782i	$-0.675522\pi$
$601$	−25.6869	−1.04779	−0.523896	+	0.851782i	$0.675522\pi$
$602$	0	0
$603$	−5.61803	−0.228784
$604$	0	0
$605$	11.2361	0.456811
$606$	0	0
$607$	−37.6312	−1.52740	−0.763701	−	0.645570i	$-0.776620\pi$
$607$	−37.6312	−1.52740	−0.763701	+	0.645570i	$0.776620\pi$
$608$	0	0
$609$	−3.47214	−0.140698
$610$	0	0
$611$	−13.7082	−0.554575
$612$	0	0
$613$	−22.2361	−0.898106	−0.449053	−	0.893505i	$-0.648239\pi$
$613$	−22.2361	−0.898106	−0.449053	+	0.893505i	$0.648239\pi$
$614$	0	0
$615$	2.38197	0.0960501
$616$	0	0
$617$	−27.0902	−1.09061	−0.545305	−	0.838238i	$-0.683586\pi$
$617$	−27.0902	−1.09061	−0.545305	+	0.838238i	$0.683586\pi$
$618$	0	0
$619$	−22.0344	−0.885639	−0.442819	−	0.896611i	$-0.646022\pi$
$619$	−22.0344	−0.885639	−0.442819	+	0.896611i	$0.646022\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−3.85410	−0.154411
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0	0
$627$	−23.1803	−0.925734
$628$	0	0
$629$	7.76393	0.309568
$630$	0	0
$631$	−0.527864	−0.0210139	−0.0105070	−	0.999945i	$-0.503345\pi$
$631$	−0.527864	−0.0210139	−0.0105070	+	0.999945i	$0.503345\pi$
$632$	0	0
$633$	−2.23607	−0.0888757
$634$	0	0
$635$	−22.4164	−0.889568
$636$	0	0
$637$	2.61803	0.103730
$638$	0	0
$639$	−4.14590	−0.164009
$640$	0	0
$641$	−28.1459	−1.11170	−0.555848	−	0.831284i	$-0.687606\pi$
$641$	−28.1459	−1.11170	−0.555848	+	0.831284i	$0.687606\pi$
$642$	0	0
$643$	−26.9787	−1.06394	−0.531968	−	0.846764i	$-0.678547\pi$
$643$	−26.9787	−1.06394	−0.531968	+	0.846764i	$0.678547\pi$
$644$	0	0
$645$	−12.2361	−0.481795
$646$	0	0
$647$	−42.0902	−1.65474	−0.827368	−	0.561661i	$-0.810163\pi$
$647$	−42.0902	−1.65474	−0.827368	+	0.561661i	$0.810163\pi$
$648$	0	0
$649$	23.7984	0.934168
$650$	0	0
$651$	−6.23607	−0.244411
$652$	0	0
$653$	19.0344	0.744875	0.372438	−	0.928057i	$-0.378522\pi$
$653$	19.0344	0.744875	0.372438	+	0.928057i	$0.378522\pi$
$654$	0	0
$655$	29.0344	1.13447
$656$	0	0
$657$	−15.9443	−0.622045
$658$	0	0
$659$	17.0689	0.664909	0.332455	−	0.943119i	$-0.392123\pi$
$659$	17.0689	0.664909	0.332455	+	0.943119i	$0.392123\pi$
$660$	0	0
$661$	−31.1803	−1.21277	−0.606387	−	0.795169i	$-0.707382\pi$
$661$	−31.1803	−1.21277	−0.606387	+	0.795169i	$0.707382\pi$
$662$	0	0
$663$	−5.85410	−0.227354
$664$	0	0
$665$	8.85410	0.343347
$666$	0	0
$667$	−3.47214	−0.134442
$668$	0	0
$669$	25.2148	0.974860
$670$	0	0
$671$	43.7426	1.68867
$672$	0	0
$673$	−7.58359	−0.292326	−0.146163	−	0.989261i	$-0.546692\pi$
$673$	−7.58359	−0.292326	−0.146163	+	0.989261i	$0.546692\pi$
$674$	0	0
$675$	2.38197	0.0916819
$676$	0	0
$677$	24.9787	0.960010	0.480005	−	0.877266i	$-0.340635\pi$
$677$	24.9787	0.960010	0.480005	+	0.877266i	$0.340635\pi$
$678$	0	0
$679$	−4.23607	−0.162565
$680$	0	0
$681$	10.0902	0.386656
$682$	0	0
$683$	−20.2361	−0.774312	−0.387156	−	0.922014i	$-0.626542\pi$
$683$	−20.2361	−0.774312	−0.387156	+	0.922014i	$0.626542\pi$
$684$	0	0
$685$	6.85410	0.261882
$686$	0	0
$687$	−12.3820	−0.472401
$688$	0	0
$689$	10.7082	0.407950
$690$	0	0
$691$	−17.5623	−0.668102	−0.334051	−	0.942555i	$-0.608416\pi$
$691$	−17.5623	−0.668102	−0.334051	+	0.942555i	$0.608416\pi$
$692$	0	0
$693$	4.23607	0.160915
$694$	0	0
$695$	27.5623	1.04550
$696$	0	0
$697$	−3.29180	−0.124686
$698$	0	0
$699$	−2.09017	−0.0790575
$700$	0	0
$701$	5.21478	0.196960	0.0984798	−	0.995139i	$-0.468602\pi$
$701$	5.21478	0.196960	0.0984798	+	0.995139i	$0.468602\pi$
$702$	0	0
$703$	−19.0000	−0.716599
$704$	0	0
$705$	8.47214	0.319079
$706$	0	0
$707$	3.61803	0.136070
$708$	0	0
$709$	−25.2705	−0.949054	−0.474527	−	0.880241i	$-0.657381\pi$
$709$	−25.2705	−0.949054	−0.474527	+	0.880241i	$0.657381\pi$
$710$	0	0
$711$	−3.76393	−0.141158
$712$	0	0
$713$	−6.23607	−0.233543
$714$	0	0
$715$	−17.9443	−0.671078
$716$	0	0
$717$	−7.03444	−0.262706
$718$	0	0
$719$	−6.23607	−0.232566	−0.116283	−	0.993216i	$-0.537098\pi$
$719$	−6.23607	−0.232566	−0.116283	+	0.993216i	$0.537098\pi$
$720$	0	0
$721$	−8.94427	−0.333102
$722$	0	0
$723$	11.2918	0.419946
$724$	0	0
$725$	8.27051	0.307159
$726$	0	0
$727$	36.5279	1.35474	0.677372	−	0.735641i	$-0.263119\pi$
$727$	36.5279	1.35474	0.677372	+	0.735641i	$0.263119\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.9098	0.625433
$732$	0	0
$733$	7.59675	0.280592	0.140296	−	0.990110i	$-0.455195\pi$
$733$	7.59675	0.280592	0.140296	+	0.990110i	$0.455195\pi$
$734$	0	0
$735$	−1.61803	−0.0596821
$736$	0	0
$737$	23.7984	0.876624
$738$	0	0
$739$	−24.5967	−0.904806	−0.452403	−	0.891814i	$-0.649433\pi$
$739$	−24.5967	−0.904806	−0.452403	+	0.891814i	$0.649433\pi$
$740$	0	0
$741$	14.3262	0.526288
$742$	0	0
$743$	−53.7984	−1.97367	−0.986835	−	0.161727i	$-0.948293\pi$
$743$	−53.7984	−1.97367	−0.986835	+	0.161727i	$0.948293\pi$
$744$	0	0
$745$	−26.9443	−0.987162
$746$	0	0
$747$	9.94427	0.363842
$748$	0	0
$749$	12.6180	0.461053
$750$	0	0
$751$	−8.96556	−0.327158	−0.163579	−	0.986530i	$-0.552304\pi$
$751$	−8.96556	−0.327158	−0.163579	+	0.986530i	$0.552304\pi$
$752$	0	0
$753$	0.180340	0.00657195
$754$	0	0
$755$	−25.4164	−0.924998
$756$	0	0
$757$	35.0000	1.27210	0.636048	−	0.771649i	$-0.280568\pi$
$757$	35.0000	1.27210	0.636048	+	0.771649i	$0.280568\pi$
$758$	0	0
$759$	4.23607	0.153760
$760$	0	0
$761$	−42.8328	−1.55269	−0.776344	−	0.630309i	$-0.782928\pi$
$761$	−42.8328	−1.55269	−0.776344	+	0.630309i	$0.782928\pi$
$762$	0	0
$763$	4.38197	0.158638
$764$	0	0
$765$	3.61803	0.130810
$766$	0	0
$767$	−14.7082	−0.531082
$768$	0	0
$769$	7.81966	0.281984	0.140992	−	0.990011i	$-0.454971\pi$
$769$	7.81966	0.281984	0.140992	+	0.990011i	$0.454971\pi$
$770$	0	0
$771$	−12.2918	−0.442678
$772$	0	0
$773$	15.4721	0.556494	0.278247	−	0.960510i	$-0.410247\pi$
$773$	15.4721	0.556494	0.278247	+	0.960510i	$0.410247\pi$
$774$	0	0
$775$	14.8541	0.533575
$776$	0	0
$777$	3.47214	0.124562
$778$	0	0
$779$	8.05573	0.288626
$780$	0	0
$781$	17.5623	0.628429
$782$	0	0
$783$	3.47214	0.124084
$784$	0	0
$785$	−20.6525	−0.737118
$786$	0	0
$787$	38.4508	1.37062	0.685312	−	0.728249i	$-0.259666\pi$
$787$	38.4508	1.37062	0.685312	+	0.728249i	$0.259666\pi$
$788$	0	0
$789$	−9.94427	−0.354025
$790$	0	0
$791$	12.5623	0.446664
$792$	0	0
$793$	−27.0344	−0.960021
$794$	0	0
$795$	−6.61803	−0.234717
$796$	0	0
$797$	−38.5410	−1.36519	−0.682596	−	0.730795i	$-0.739149\pi$
$797$	−38.5410	−1.36519	−0.682596	+	0.730795i	$0.739149\pi$
$798$	0	0
$799$	−11.7082	−0.414206
$800$	0	0
$801$	3.85410	0.136178
$802$	0	0
$803$	67.5410	2.38347
$804$	0	0
$805$	−1.61803	−0.0570282
$806$	0	0
$807$	−23.0902	−0.812812
$808$	0	0
$809$	18.6180	0.654575	0.327288	−	0.944925i	$-0.393865\pi$
$809$	18.6180	0.654575	0.327288	+	0.944925i	$0.393865\pi$
$810$	0	0
$811$	32.4164	1.13829	0.569147	−	0.822236i	$-0.307274\pi$
$811$	32.4164	1.13829	0.569147	+	0.822236i	$0.307274\pi$
$812$	0	0
$813$	−31.3607	−1.09987
$814$	0	0
$815$	−26.7984	−0.938706
$816$	0	0
$817$	−41.3820	−1.44777
$818$	0	0
$819$	−2.61803	−0.0914815
$820$	0	0
$821$	0.875388	0.0305513	0.0152756	−	0.999883i	$-0.495137\pi$
$821$	0.875388	0.0305513	0.0152756	+	0.999883i	$0.495137\pi$
$822$	0	0
$823$	15.7295	0.548296	0.274148	−	0.961688i	$-0.411604\pi$
$823$	15.7295	0.548296	0.274148	+	0.961688i	$0.411604\pi$
$824$	0	0
$825$	−10.0902	−0.351295
$826$	0	0
$827$	−3.43769	−0.119540	−0.0597702	−	0.998212i	$-0.519037\pi$
$827$	−3.43769	−0.119540	−0.0597702	+	0.998212i	$0.519037\pi$
$828$	0	0
$829$	20.1246	0.698957	0.349478	−	0.936944i	$-0.386359\pi$
$829$	20.1246	0.698957	0.349478	+	0.936944i	$0.386359\pi$
$830$	0	0
$831$	−8.61803	−0.298956
$832$	0	0
$833$	2.23607	0.0774752
$834$	0	0
$835$	33.7984	1.16964
$836$	0	0
$837$	6.23607	0.215550
$838$	0	0
$839$	29.9230	1.03306	0.516528	−	0.856270i	$-0.327224\pi$
$839$	29.9230	1.03306	0.516528	+	0.856270i	$0.327224\pi$
$840$	0	0
$841$	−16.9443	−0.584285
$842$	0	0
$843$	−22.6525	−0.780193
$844$	0	0
$845$	−9.94427	−0.342093
$846$	0	0
$847$	−6.94427	−0.238608
$848$	0	0
$849$	7.90983	0.271465
$850$	0	0
$851$	3.47214	0.119023
$852$	0	0
$853$	2.65248	0.0908190	0.0454095	−	0.998968i	$-0.485541\pi$
$853$	2.65248	0.0908190	0.0454095	+	0.998968i	$0.485541\pi$
$854$	0	0
$855$	−8.85410	−0.302804
$856$	0	0
$857$	41.1246	1.40479	0.702395	−	0.711787i	$-0.252114\pi$
$857$	41.1246	1.40479	0.702395	+	0.711787i	$0.252114\pi$
$858$	0	0
$859$	2.94427	0.100457	0.0502286	−	0.998738i	$-0.484005\pi$
$859$	2.94427	0.100457	0.0502286	+	0.998738i	$0.484005\pi$
$860$	0	0
$861$	−1.47214	−0.0501703
$862$	0	0
$863$	−37.1246	−1.26374	−0.631868	−	0.775076i	$-0.717712\pi$
$863$	−37.1246	−1.26374	−0.631868	+	0.775076i	$0.717712\pi$
$864$	0	0
$865$	3.32624	0.113095
$866$	0	0
$867$	12.0000	0.407541
$868$	0	0
$869$	15.9443	0.540872
$870$	0	0
$871$	−14.7082	−0.498368
$872$	0	0
$873$	4.23607	0.143369
$874$	0	0
$875$	11.9443	0.403790
$876$	0	0
$877$	51.1935	1.72868	0.864341	−	0.502907i	$-0.167736\pi$
$877$	51.1935	1.72868	0.864341	+	0.502907i	$0.167736\pi$
$878$	0	0
$879$	−24.9443	−0.841349
$880$	0	0
$881$	−22.2918	−0.751030	−0.375515	−	0.926816i	$-0.622534\pi$
$881$	−22.2918	−0.751030	−0.375515	+	0.926816i	$0.622534\pi$
$882$	0	0
$883$	9.74265	0.327866	0.163933	−	0.986471i	$-0.447582\pi$
$883$	9.74265	0.327866	0.163933	+	0.986471i	$0.447582\pi$
$884$	0	0
$885$	9.09017	0.305563
$886$	0	0
$887$	−0.437694	−0.0146963	−0.00734816	−	0.999973i	$-0.502339\pi$
$887$	−0.437694	−0.0146963	−0.00734816	+	0.999973i	$0.502339\pi$
$888$	0	0
$889$	13.8541	0.464652
$890$	0	0
$891$	−4.23607	−0.141914
$892$	0	0
$893$	28.6525	0.958819
$894$	0	0
$895$	35.2705	1.17896
$896$	0	0
$897$	−2.61803	−0.0874136
$898$	0	0
$899$	21.6525	0.722151
$900$	0	0
$901$	9.14590	0.304694
$902$	0	0
$903$	7.56231	0.251658
$904$	0	0
$905$	0.381966	0.0126970
$906$	0	0
$907$	47.0344	1.56175	0.780877	−	0.624685i	$-0.214773\pi$
$907$	47.0344	1.56175	0.780877	+	0.624685i	$0.214773\pi$
$908$	0	0
$909$	−3.61803	−0.120003
$910$	0	0
$911$	−34.5410	−1.14440	−0.572198	−	0.820116i	$-0.693909\pi$
$911$	−34.5410	−1.14440	−0.572198	+	0.820116i	$0.693909\pi$
$912$	0	0
$913$	−42.1246	−1.39412
$914$	0	0
$915$	16.7082	0.552356
$916$	0	0
$917$	−17.9443	−0.592572
$918$	0	0
$919$	20.4164	0.673475	0.336738	−	0.941599i	$-0.390676\pi$
$919$	20.4164	0.673475	0.336738	+	0.941599i	$0.390676\pi$
$920$	0	0
$921$	−10.2361	−0.337290
$922$	0	0
$923$	−10.8541	−0.357267
$924$	0	0
$925$	−8.27051	−0.271933
$926$	0	0
$927$	8.94427	0.293768
$928$	0	0
$929$	7.50658	0.246283	0.123141	−	0.992389i	$-0.460703\pi$
$929$	7.50658	0.246283	0.123141	+	0.992389i	$0.460703\pi$
$930$	0	0
$931$	−5.47214	−0.179342
$932$	0	0
$933$	−30.9787	−1.01420
$934$	0	0
$935$	−15.3262	−0.501222
$936$	0	0
$937$	22.5279	0.735953	0.367977	−	0.929835i	$-0.380051\pi$
$937$	22.5279	0.735953	0.367977	+	0.929835i	$0.380051\pi$
$938$	0	0
$939$	2.47214	0.0806751
$940$	0	0
$941$	28.0000	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$942$	0	0
$943$	−1.47214	−0.0479393
$944$	0	0
$945$	1.61803	0.0526346
$946$	0	0
$947$	59.8885	1.94612	0.973058	−	0.230560i	$-0.0740558\pi$
$947$	59.8885	1.94612	0.973058	+	0.230560i	$0.0740558\pi$
$948$	0	0
$949$	−41.7426	−1.35502
$950$	0	0
$951$	−4.96556	−0.161019
$952$	0	0
$953$	−33.5623	−1.08719	−0.543595	−	0.839348i	$-0.682937\pi$
$953$	−33.5623	−1.08719	−0.543595	+	0.839348i	$0.682937\pi$
$954$	0	0
$955$	−30.9443	−1.00133
$956$	0	0
$957$	−14.7082	−0.475449
$958$	0	0
$959$	−4.23607	−0.136790
$960$	0	0
$961$	7.88854	0.254469
$962$	0	0
$963$	−12.6180	−0.406610
$964$	0	0
$965$	−28.6525	−0.922356
$966$	0	0
$967$	7.52786	0.242080	0.121040	−	0.992648i	$-0.461377\pi$
$967$	7.52786	0.242080	0.121040	+	0.992648i	$0.461377\pi$
$968$	0	0
$969$	12.2361	0.393079
$970$	0	0
$971$	−13.6738	−0.438812	−0.219406	−	0.975634i	$-0.570412\pi$
$971$	−13.6738	−0.438812	−0.219406	+	0.975634i	$0.570412\pi$
$972$	0	0
$973$	−17.0344	−0.546099
$974$	0	0
$975$	6.23607	0.199714
$976$	0	0
$977$	−57.0902	−1.82648	−0.913238	−	0.407426i	$-0.866426\pi$
$977$	−57.0902	−1.82648	−0.913238	+	0.407426i	$0.866426\pi$
$978$	0	0
$979$	−16.3262	−0.521789
$980$	0	0
$981$	−4.38197	−0.139905
$982$	0	0
$983$	−48.7771	−1.55575	−0.777874	−	0.628421i	$-0.783702\pi$
$983$	−48.7771	−1.55575	−0.777874	+	0.628421i	$0.783702\pi$
$984$	0	0
$985$	−13.9443	−0.444301
$986$	0	0
$987$	−5.23607	−0.166666
$988$	0	0
$989$	7.56231	0.240467
$990$	0	0
$991$	59.1591	1.87925	0.939625	−	0.342207i	$-0.111174\pi$
$991$	59.1591	1.87925	0.939625	+	0.342207i	$0.111174\pi$
$992$	0	0
$993$	−18.9443	−0.601178
$994$	0	0
$995$	−34.5066	−1.09393
$996$	0	0
$997$	28.1246	0.890715	0.445358	−	0.895353i	$-0.353077\pi$
$997$	28.1246	0.890715	0.445358	+	0.895353i	$0.353077\pi$
$998$	0	0
$999$	−3.47214	−0.109854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.a.d.1.2	✓	2
3.2	odd	2		5796.2.a.i.1.1		2
4.3	odd	2		7728.2.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.d.1.2	✓	2	1.1	even	1	trivial
5796.2.a.i.1.1		2	3.2	odd	2
7728.2.a.bo.1.2		2	4.3	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:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1932.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.23607 q^{11} +2.61803 q^{13} -1.61803 q^{15} +2.23607 q^{17} -5.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -2.38197 q^{25} -1.00000 q^{27} -3.47214 q^{29} -6.23607 q^{31} +4.23607 q^{33} -1.61803 q^{35} +3.47214 q^{37} -2.61803 q^{39} -1.47214 q^{41} +7.56231 q^{43} +1.61803 q^{45} -5.23607 q^{47} +1.00000 q^{49} -2.23607 q^{51} +4.09017 q^{53} -6.85410 q^{55} +5.47214 q^{57} -5.61803 q^{59} -10.3262 q^{61} -1.00000 q^{63} +4.23607 q^{65} -5.61803 q^{67} -1.00000 q^{69} -4.14590 q^{71} -15.9443 q^{73} +2.38197 q^{75} +4.23607 q^{77} -3.76393 q^{79} +1.00000 q^{81} +9.94427 q^{83} +3.61803 q^{85} +3.47214 q^{87} +3.85410 q^{89} -2.61803 q^{91} +6.23607 q^{93} -8.85410 q^{95} +4.23607 q^{97} -4.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} + 3 q^{13} - q^{15} - 2 q^{19} + 2 q^{21} + 2 q^{23} - 7 q^{25} - 2 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} - q^{35} - 2 q^{37} - 3 q^{39} + 6 q^{41} - 5 q^{43} + q^{45} - 6 q^{47} + 2 q^{49} - 3 q^{53} - 7 q^{55} + 2 q^{57} - 9 q^{59} - 5 q^{61} - 2 q^{63} + 4 q^{65} - 9 q^{67} - 2 q^{69} - 15 q^{71} - 14 q^{73} + 7 q^{75} + 4 q^{77} - 12 q^{79} + 2 q^{81} + 2 q^{83} + 5 q^{85} - 2 q^{87} + q^{89} - 3 q^{91} + 8 q^{93} - 11 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 + 3 * q^13 - q^15 - 2 * q^19 + 2 * q^21 + 2 * q^23 - 7 * q^25 - 2 * q^27 + 2 * q^29 - 8 * q^31 + 4 * q^33 - q^35 - 2 * q^37 - 3 * q^39 + 6 * q^41 - 5 * q^43 + q^45 - 6 * q^47 + 2 * q^49 - 3 * q^53 - 7 * q^55 + 2 * q^57 - 9 * q^59 - 5 * q^61 - 2 * q^63 + 4 * q^65 - 9 * q^67 - 2 * q^69 - 15 * q^71 - 14 * q^73 + 7 * q^75 + 4 * q^77 - 12 * q^79 + 2 * q^81 + 2 * q^83 + 5 * q^85 - 2 * q^87 + q^89 - 3 * q^91 + 8 * q^93 - 11 * q^95 + 4 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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