LMFDB - Embedded newform 1932.2.a.g.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +0.618034 q^{13} -1.61803 q^{15} +6.70820 q^{17} -4.23607 q^{19} -1.00000 q^{21} -1.00000 q^{23} -2.38197 q^{25} +1.00000 q^{27} -1.76393 q^{29} -3.00000 q^{31} -3.00000 q^{33} +1.61803 q^{35} +9.47214 q^{37} +0.618034 q^{39} -11.9443 q^{41} -9.09017 q^{43} -1.61803 q^{45} -9.23607 q^{47} +1.00000 q^{49} +6.70820 q^{51} -9.32624 q^{53} +4.85410 q^{55} -4.23607 q^{57} -9.61803 q^{59} +5.85410 q^{61} -1.00000 q^{63} -1.00000 q^{65} +6.56231 q^{67} -1.00000 q^{69} -4.61803 q^{71} +1.76393 q^{73} -2.38197 q^{75} +3.00000 q^{77} -6.70820 q^{79} +1.00000 q^{81} -7.47214 q^{83} -10.8541 q^{85} -1.76393 q^{87} -16.7984 q^{89} -0.618034 q^{91} -3.00000 q^{93} +6.85410 q^{95} -12.4164 q^{97} -3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{11} - q^{13} - q^{15} - 4 q^{19} - 2 q^{21} - 2 q^{23} - 7 q^{25} + 2 q^{27} - 8 q^{29} - 6 q^{31} - 6 q^{33} + q^{35} + 10 q^{37} - q^{39} - 6 q^{41} - 7 q^{43} - q^{45} - 14 q^{47} + 2 q^{49} - 3 q^{53} + 3 q^{55} - 4 q^{57} - 17 q^{59} + 5 q^{61} - 2 q^{63} - 2 q^{65} - 7 q^{67} - 2 q^{69} - 7 q^{71} + 8 q^{73} - 7 q^{75} + 6 q^{77} + 2 q^{81} - 6 q^{83} - 15 q^{85} - 8 q^{87} - 9 q^{89} + q^{91} - 6 q^{93} + 7 q^{95} + 2 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9 - 6 * q^11 - q^13 - q^15 - 4 * q^19 - 2 * q^21 - 2 * q^23 - 7 * q^25 + 2 * q^27 - 8 * q^29 - 6 * q^31 - 6 * q^33 + q^35 + 10 * q^37 - q^39 - 6 * q^41 - 7 * q^43 - q^45 - 14 * q^47 + 2 * q^49 - 3 * q^53 + 3 * q^55 - 4 * q^57 - 17 * q^59 + 5 * q^61 - 2 * q^63 - 2 * q^65 - 7 * q^67 - 2 * q^69 - 7 * q^71 + 8 * q^73 - 7 * q^75 + 6 * q^77 + 2 * q^81 - 6 * q^83 - 15 * q^85 - 8 * q^87 - 9 * q^89 + q^91 - 6 * q^93 + 7 * q^95 + 2 * q^97 - 6 * 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	$-0.617839\pi$
$5$	−1.61803	−0.723607	−0.361803	+	0.932254i	$0.617839\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	0.618034	0.171412	0.0857059	−	0.996320i	$-0.472685\pi$
$13$	0.618034	0.171412	0.0857059	+	0.996320i	$0.472685\pi$
$14$	0	0
$15$	−1.61803	−0.417775
$16$	0	0
$17$	6.70820	1.62698	0.813489	−	0.581580i	$-0.197565\pi$
$17$	6.70820	1.62698	0.813489	+	0.581580i	$0.197565\pi$
$18$	0	0
$19$	−4.23607	−0.971821	−0.485910	−	0.874009i	$-0.661512\pi$
$19$	−4.23607	−0.971821	−0.485910	+	0.874009i	$0.661512\pi$
$20$	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
$29$	−1.76393	−0.327554	−0.163777	−	0.986497i	$-0.552368\pi$
$29$	−1.76393	−0.327554	−0.163777	+	0.986497i	$0.552368\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	1.61803	0.273498
$36$	0	0
$37$	9.47214	1.55721	0.778605	−	0.627515i	$-0.215928\pi$
$37$	9.47214	1.55721	0.778605	+	0.627515i	$0.215928\pi$
$38$	0	0
$39$	0.618034	0.0989646
$40$	0	0
$41$	−11.9443	−1.86538	−0.932691	−	0.360677i	$-0.882546\pi$
$41$	−11.9443	−1.86538	−0.932691	+	0.360677i	$0.882546\pi$
$42$	0	0
$43$	−9.09017	−1.38624	−0.693119	−	0.720823i	$-0.743764\pi$
$43$	−9.09017	−1.38624	−0.693119	+	0.720823i	$0.743764\pi$
$44$	0	0
$45$	−1.61803	−0.241202
$46$	0	0
$47$	−9.23607	−1.34722	−0.673609	−	0.739087i	$-0.735257\pi$
$47$	−9.23607	−1.34722	−0.673609	+	0.739087i	$0.735257\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.70820	0.939336
$52$	0	0
$53$	−9.32624	−1.28106	−0.640529	−	0.767934i	$-0.721285\pi$
$53$	−9.32624	−1.28106	−0.640529	+	0.767934i	$0.721285\pi$
$54$	0	0
$55$	4.85410	0.654527
$56$	0	0
$57$	−4.23607	−0.561081
$58$	0	0
$59$	−9.61803	−1.25216	−0.626081	−	0.779758i	$-0.715342\pi$
$59$	−9.61803	−1.25216	−0.626081	+	0.779758i	$0.715342\pi$
$60$	0	0
$61$	5.85410	0.749541	0.374770	−	0.927118i	$-0.377722\pi$
$61$	5.85410	0.749541	0.374770	+	0.927118i	$0.377722\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	6.56231	0.801713	0.400857	−	0.916141i	$-0.368713\pi$
$67$	6.56231	0.801713	0.400857	+	0.916141i	$0.368713\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−4.61803	−0.548060	−0.274030	−	0.961721i	$-0.588357\pi$
$71$	−4.61803	−0.548060	−0.274030	+	0.961721i	$0.588357\pi$
$72$	0	0
$73$	1.76393	0.206453	0.103226	−	0.994658i	$-0.467083\pi$
$73$	1.76393	0.206453	0.103226	+	0.994658i	$0.467083\pi$
$74$	0	0
$75$	−2.38197	−0.275046
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	−6.70820	−0.754732	−0.377366	−	0.926064i	$-0.623170\pi$
$79$	−6.70820	−0.754732	−0.377366	+	0.926064i	$0.623170\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.47214	−0.820173	−0.410087	−	0.912047i	$-0.634502\pi$
$83$	−7.47214	−0.820173	−0.410087	+	0.912047i	$0.634502\pi$
$84$	0	0
$85$	−10.8541	−1.17729
$86$	0	0
$87$	−1.76393	−0.189113
$88$	0	0
$89$	−16.7984	−1.78062	−0.890312	−	0.455351i	$-0.849514\pi$
$89$	−16.7984	−1.78062	−0.890312	+	0.455351i	$0.849514\pi$
$90$	0	0
$91$	−0.618034	−0.0647876
$92$	0	0
$93$	−3.00000	−0.311086
$94$	0	0
$95$	6.85410	0.703216
$96$	0	0
$97$	−12.4164	−1.26070	−0.630348	−	0.776313i	$-0.717088\pi$
$97$	−12.4164	−1.26070	−0.630348	+	0.776313i	$0.717088\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	5.32624	0.529980	0.264990	−	0.964251i	$-0.414631\pi$
$101$	5.32624	0.529980	0.264990	+	0.964251i	$0.414631\pi$
$102$	0	0
$103$	19.4164	1.91316	0.956578	−	0.291477i	$-0.0941468\pi$
$103$	19.4164	1.91316	0.956578	+	0.291477i	$0.0941468\pi$
$104$	0	0
$105$	1.61803	0.157904
$106$	0	0
$107$	10.6180	1.02648	0.513242	−	0.858244i	$-0.328444\pi$
$107$	10.6180	1.02648	0.513242	+	0.858244i	$0.328444\pi$
$108$	0	0
$109$	−2.09017	−0.200202	−0.100101	−	0.994977i	$-0.531917\pi$
$109$	−2.09017	−0.200202	−0.100101	+	0.994977i	$0.531917\pi$
$110$	0	0
$111$	9.47214	0.899055
$112$	0	0
$113$	3.32624	0.312906	0.156453	−	0.987685i	$-0.449994\pi$
$113$	3.32624	0.312906	0.156453	+	0.987685i	$0.449994\pi$
$114$	0	0
$115$	1.61803	0.150882
$116$	0	0
$117$	0.618034	0.0571373
$118$	0	0
$119$	−6.70820	−0.614940
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−11.9443	−1.07698
$124$	0	0
$125$	11.9443	1.06833
$126$	0	0
$127$	5.56231	0.493575	0.246787	−	0.969070i	$-0.420625\pi$
$127$	5.56231	0.493575	0.246787	+	0.969070i	$0.420625\pi$
$128$	0	0
$129$	−9.09017	−0.800345
$130$	0	0
$131$	5.94427	0.519353	0.259677	−	0.965696i	$-0.416384\pi$
$131$	5.94427	0.519353	0.259677	+	0.965696i	$0.416384\pi$
$132$	0	0
$133$	4.23607	0.367314
$134$	0	0
$135$	−1.61803	−0.139258
$136$	0	0
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	6.85410	0.581357	0.290679	−	0.956821i	$-0.406119\pi$
$139$	6.85410	0.581357	0.290679	+	0.956821i	$0.406119\pi$
$140$	0	0
$141$	−9.23607	−0.777817
$142$	0	0
$143$	−1.85410	−0.155048
$144$	0	0
$145$	2.85410	0.237020
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−1.23607	−0.101263	−0.0506313	−	0.998717i	$-0.516123\pi$
$149$	−1.23607	−0.101263	−0.0506313	+	0.998717i	$0.516123\pi$
$150$	0	0
$151$	−19.1246	−1.55634	−0.778169	−	0.628054i	$-0.783852\pi$
$151$	−19.1246	−1.55634	−0.778169	+	0.628054i	$0.783852\pi$
$152$	0	0
$153$	6.70820	0.542326
$154$	0	0
$155$	4.85410	0.389891
$156$	0	0
$157$	8.76393	0.699438	0.349719	−	0.936855i	$-0.386277\pi$
$157$	8.76393	0.699438	0.349719	+	0.936855i	$0.386277\pi$
$158$	0	0
$159$	−9.32624	−0.739619
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	10.8541	0.850159	0.425079	−	0.905156i	$-0.360246\pi$
$163$	10.8541	0.850159	0.425079	+	0.905156i	$0.360246\pi$
$164$	0	0
$165$	4.85410	0.377891
$166$	0	0
$167$	−10.5279	−0.814671	−0.407335	−	0.913279i	$-0.633542\pi$
$167$	−10.5279	−0.814671	−0.407335	+	0.913279i	$0.633542\pi$
$168$	0	0
$169$	−12.6180	−0.970618
$170$	0	0
$171$	−4.23607	−0.323940
$172$	0	0
$173$	13.4721	1.02427	0.512134	−	0.858906i	$-0.328855\pi$
$173$	13.4721	1.02427	0.512134	+	0.858906i	$0.328855\pi$
$174$	0	0
$175$	2.38197	0.180060
$176$	0	0
$177$	−9.61803	−0.722936
$178$	0	0
$179$	11.0344	0.824753	0.412376	−	0.911014i	$-0.364699\pi$
$179$	11.0344	0.824753	0.412376	+	0.911014i	$0.364699\pi$
$180$	0	0
$181$	17.4721	1.29869	0.649347	−	0.760492i	$-0.275042\pi$
$181$	17.4721	1.29869	0.649347	+	0.760492i	$0.275042\pi$
$182$	0	0
$183$	5.85410	0.432748
$184$	0	0
$185$	−15.3262	−1.12681
$186$	0	0
$187$	−20.1246	−1.47166
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−20.6525	−1.49436	−0.747180	−	0.664621i	$-0.768593\pi$
$191$	−20.6525	−1.49436	−0.747180	+	0.664621i	$0.768593\pi$
$192$	0	0
$193$	−5.70820	−0.410886	−0.205443	−	0.978669i	$-0.565863\pi$
$193$	−5.70820	−0.410886	−0.205443	+	0.978669i	$0.565863\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	1.38197	0.0984610	0.0492305	−	0.998787i	$-0.484323\pi$
$197$	1.38197	0.0984610	0.0492305	+	0.998787i	$0.484323\pi$
$198$	0	0
$199$	22.5623	1.59940	0.799700	−	0.600400i	$-0.204992\pi$
$199$	22.5623	1.59940	0.799700	+	0.600400i	$0.204992\pi$
$200$	0	0
$201$	6.56231	0.462869
$202$	0	0
$203$	1.76393	0.123804
$204$	0	0
$205$	19.3262	1.34980
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	12.7082	0.879045
$210$	0	0
$211$	−26.1246	−1.79849	−0.899246	−	0.437443i	$-0.855884\pi$
$211$	−26.1246	−1.79849	−0.899246	+	0.437443i	$0.855884\pi$
$212$	0	0
$213$	−4.61803	−0.316422
$214$	0	0
$215$	14.7082	1.00309
$216$	0	0
$217$	3.00000	0.203653
$218$	0	0
$219$	1.76393	0.119195
$220$	0	0
$221$	4.14590	0.278883
$222$	0	0
$223$	11.7984	0.790078	0.395039	−	0.918664i	$-0.370731\pi$
$223$	11.7984	0.790078	0.395039	+	0.918664i	$0.370731\pi$
$224$	0	0
$225$	−2.38197	−0.158798
$226$	0	0
$227$	26.4508	1.75560	0.877802	−	0.479023i	$-0.159009\pi$
$227$	26.4508	1.75560	0.877802	+	0.479023i	$0.159009\pi$
$228$	0	0
$229$	24.2705	1.60384	0.801920	−	0.597431i	$-0.203812\pi$
$229$	24.2705	1.60384	0.801920	+	0.597431i	$0.203812\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	−12.0902	−0.792053	−0.396027	−	0.918239i	$-0.629611\pi$
$233$	−12.0902	−0.792053	−0.396027	+	0.918239i	$0.629611\pi$
$234$	0	0
$235$	14.9443	0.974857
$236$	0	0
$237$	−6.70820	−0.435745
$238$	0	0
$239$	−17.6180	−1.13962	−0.569808	−	0.821778i	$-0.692982\pi$
$239$	−17.6180	−1.13962	−0.569808	+	0.821778i	$0.692982\pi$
$240$	0	0
$241$	21.0000	1.35273	0.676364	−	0.736567i	$-0.263554\pi$
$241$	21.0000	1.35273	0.676364	+	0.736567i	$0.263554\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.61803	−0.103372
$246$	0	0
$247$	−2.61803	−0.166582
$248$	0	0
$249$	−7.47214	−0.473527
$250$	0	0
$251$	−22.6525	−1.42981	−0.714906	−	0.699221i	$-0.753530\pi$
$251$	−22.6525	−1.42981	−0.714906	+	0.699221i	$0.753530\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	0	0
$255$	−10.8541	−0.679710
$256$	0	0
$257$	16.6525	1.03875	0.519376	−	0.854546i	$-0.326164\pi$
$257$	16.6525	1.03875	0.519376	+	0.854546i	$0.326164\pi$
$258$	0	0
$259$	−9.47214	−0.588570
$260$	0	0
$261$	−1.76393	−0.109185
$262$	0	0
$263$	15.1803	0.936060	0.468030	−	0.883713i	$-0.344964\pi$
$263$	15.1803	0.936060	0.468030	+	0.883713i	$0.344964\pi$
$264$	0	0
$265$	15.0902	0.926982
$266$	0	0
$267$	−16.7984	−1.02804
$268$	0	0
$269$	−22.7984	−1.39004	−0.695021	−	0.718990i	$-0.744605\pi$
$269$	−22.7984	−1.39004	−0.695021	+	0.718990i	$0.744605\pi$
$270$	0	0
$271$	−21.1803	−1.28661	−0.643307	−	0.765608i	$-0.722438\pi$
$271$	−21.1803	−1.28661	−0.643307	+	0.765608i	$0.722438\pi$
$272$	0	0
$273$	−0.618034	−0.0374051
$274$	0	0
$275$	7.14590	0.430914
$276$	0	0
$277$	24.0344	1.44409	0.722045	−	0.691846i	$-0.243202\pi$
$277$	24.0344	1.44409	0.722045	+	0.691846i	$0.243202\pi$
$278$	0	0
$279$	−3.00000	−0.179605
$280$	0	0
$281$	−12.7639	−0.761432	−0.380716	−	0.924692i	$-0.624323\pi$
$281$	−12.7639	−0.761432	−0.380716	+	0.924692i	$0.624323\pi$
$282$	0	0
$283$	−14.8541	−0.882985	−0.441492	−	0.897265i	$-0.645551\pi$
$283$	−14.8541	−0.882985	−0.441492	+	0.897265i	$0.645551\pi$
$284$	0	0
$285$	6.85410	0.406002
$286$	0	0
$287$	11.9443	0.705048
$288$	0	0
$289$	28.0000	1.64706
$290$	0	0
$291$	−12.4164	−0.727863
$292$	0	0
$293$	−0.944272	−0.0551650	−0.0275825	−	0.999620i	$-0.508781\pi$
$293$	−0.944272	−0.0551650	−0.0275825	+	0.999620i	$0.508781\pi$
$294$	0	0
$295$	15.5623	0.906072
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	−0.618034	−0.0357418
$300$	0	0
$301$	9.09017	0.523949
$302$	0	0
$303$	5.32624	0.305984
$304$	0	0
$305$	−9.47214	−0.542373
$306$	0	0
$307$	−28.4164	−1.62181	−0.810905	−	0.585178i	$-0.801025\pi$
$307$	−28.4164	−1.62181	−0.810905	+	0.585178i	$0.801025\pi$
$308$	0	0
$309$	19.4164	1.10456
$310$	0	0
$311$	−15.2705	−0.865911	−0.432956	−	0.901415i	$-0.642529\pi$
$311$	−15.2705	−0.865911	−0.432956	+	0.901415i	$0.642529\pi$
$312$	0	0
$313$	17.5279	0.990733	0.495367	−	0.868684i	$-0.335034\pi$
$313$	17.5279	0.990733	0.495367	+	0.868684i	$0.335034\pi$
$314$	0	0
$315$	1.61803	0.0911659
$316$	0	0
$317$	5.61803	0.315540	0.157770	−	0.987476i	$-0.449570\pi$
$317$	5.61803	0.315540	0.157770	+	0.987476i	$0.449570\pi$
$318$	0	0
$319$	5.29180	0.296284
$320$	0	0
$321$	10.6180	0.592641
$322$	0	0
$323$	−28.4164	−1.58113
$324$	0	0
$325$	−1.47214	−0.0816594
$326$	0	0
$327$	−2.09017	−0.115587
$328$	0	0
$329$	9.23607	0.509201
$330$	0	0
$331$	7.88854	0.433594	0.216797	−	0.976217i	$-0.430439\pi$
$331$	7.88854	0.433594	0.216797	+	0.976217i	$0.430439\pi$
$332$	0	0
$333$	9.47214	0.519070
$334$	0	0
$335$	−10.6180	−0.580125
$336$	0	0
$337$	−12.6180	−0.687348	−0.343674	−	0.939089i	$-0.611672\pi$
$337$	−12.6180	−0.687348	−0.343674	+	0.939089i	$0.611672\pi$
$338$	0	0
$339$	3.32624	0.180656
$340$	0	0
$341$	9.00000	0.487377
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.61803	0.0871120
$346$	0	0
$347$	−13.4721	−0.723222	−0.361611	−	0.932329i	$-0.617773\pi$
$347$	−13.4721	−0.723222	−0.361611	+	0.932329i	$0.617773\pi$
$348$	0	0
$349$	13.1459	0.703684	0.351842	−	0.936059i	$-0.385555\pi$
$349$	13.1459	0.703684	0.351842	+	0.936059i	$0.385555\pi$
$350$	0	0
$351$	0.618034	0.0329882
$352$	0	0
$353$	−15.0000	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$354$	0	0
$355$	7.47214	0.396580
$356$	0	0
$357$	−6.70820	−0.355036
$358$	0	0
$359$	27.0344	1.42682	0.713412	−	0.700745i	$-0.247149\pi$
$359$	27.0344	1.42682	0.713412	+	0.700745i	$0.247149\pi$
$360$	0	0
$361$	−1.05573	−0.0555646
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−2.85410	−0.149391
$366$	0	0
$367$	−30.7984	−1.60766	−0.803831	−	0.594858i	$-0.797208\pi$
$367$	−30.7984	−1.60766	−0.803831	+	0.594858i	$0.797208\pi$
$368$	0	0
$369$	−11.9443	−0.621794
$370$	0	0
$371$	9.32624	0.484194
$372$	0	0
$373$	−6.88854	−0.356675	−0.178338	−	0.983969i	$-0.557072\pi$
$373$	−6.88854	−0.356675	−0.178338	+	0.983969i	$0.557072\pi$
$374$	0	0
$375$	11.9443	0.616800
$376$	0	0
$377$	−1.09017	−0.0561466
$378$	0	0
$379$	−14.4721	−0.743384	−0.371692	−	0.928356i	$-0.621222\pi$
$379$	−14.4721	−0.743384	−0.371692	+	0.928356i	$0.621222\pi$
$380$	0	0
$381$	5.56231	0.284966
$382$	0	0
$383$	12.0557	0.616019	0.308009	−	0.951383i	$-0.400337\pi$
$383$	12.0557	0.616019	0.308009	+	0.951383i	$0.400337\pi$
$384$	0	0
$385$	−4.85410	−0.247388
$386$	0	0
$387$	−9.09017	−0.462079
$388$	0	0
$389$	−21.6525	−1.09782	−0.548912	−	0.835880i	$-0.684958\pi$
$389$	−21.6525	−1.09782	−0.548912	+	0.835880i	$0.684958\pi$
$390$	0	0
$391$	−6.70820	−0.339248
$392$	0	0
$393$	5.94427	0.299849
$394$	0	0
$395$	10.8541	0.546129
$396$	0	0
$397$	−15.1246	−0.759083	−0.379541	−	0.925175i	$-0.623918\pi$
$397$	−15.1246	−0.759083	−0.379541	+	0.925175i	$0.623918\pi$
$398$	0	0
$399$	4.23607	0.212069
$400$	0	0
$401$	−16.2361	−0.810791	−0.405395	−	0.914141i	$-0.632866\pi$
$401$	−16.2361	−0.810791	−0.405395	+	0.914141i	$0.632866\pi$
$402$	0	0
$403$	−1.85410	−0.0923594
$404$	0	0
$405$	−1.61803	−0.0804008
$406$	0	0
$407$	−28.4164	−1.40855
$408$	0	0
$409$	36.4164	1.80068	0.900338	−	0.435192i	$-0.143319\pi$
$409$	36.4164	1.80068	0.900338	+	0.435192i	$0.143319\pi$
$410$	0	0
$411$	−9.00000	−0.443937
$412$	0	0
$413$	9.61803	0.473273
$414$	0	0
$415$	12.0902	0.593483
$416$	0	0
$417$	6.85410	0.335647
$418$	0	0
$419$	27.2148	1.32953	0.664765	−	0.747053i	$-0.268532\pi$
$419$	27.2148	1.32953	0.664765	+	0.747053i	$0.268532\pi$
$420$	0	0
$421$	−13.3820	−0.652197	−0.326099	−	0.945336i	$-0.605734\pi$
$421$	−13.3820	−0.652197	−0.326099	+	0.945336i	$0.605734\pi$
$422$	0	0
$423$	−9.23607	−0.449073
$424$	0	0
$425$	−15.9787	−0.775081
$426$	0	0
$427$	−5.85410	−0.283300
$428$	0	0
$429$	−1.85410	−0.0895169
$430$	0	0
$431$	31.8541	1.53436	0.767179	−	0.641433i	$-0.221660\pi$
$431$	31.8541	1.53436	0.767179	+	0.641433i	$0.221660\pi$
$432$	0	0
$433$	14.2918	0.686820	0.343410	−	0.939186i	$-0.388418\pi$
$433$	14.2918	0.686820	0.343410	+	0.939186i	$0.388418\pi$
$434$	0	0
$435$	2.85410	0.136844
$436$	0	0
$437$	4.23607	0.202639
$438$	0	0
$439$	−4.88854	−0.233317	−0.116659	−	0.993172i	$-0.537218\pi$
$439$	−4.88854	−0.233317	−0.116659	+	0.993172i	$0.537218\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	7.41641	0.352364	0.176182	−	0.984358i	$-0.443625\pi$
$443$	7.41641	0.352364	0.176182	+	0.984358i	$0.443625\pi$
$444$	0	0
$445$	27.1803	1.28847
$446$	0	0
$447$	−1.23607	−0.0584640
$448$	0	0
$449$	12.0902	0.570570	0.285285	−	0.958443i	$-0.407912\pi$
$449$	12.0902	0.570570	0.285285	+	0.958443i	$0.407912\pi$
$450$	0	0
$451$	35.8328	1.68730
$452$	0	0
$453$	−19.1246	−0.898553
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	39.9787	1.87013	0.935063	−	0.354482i	$-0.115343\pi$
$457$	39.9787	1.87013	0.935063	+	0.354482i	$0.115343\pi$
$458$	0	0
$459$	6.70820	0.313112
$460$	0	0
$461$	31.7984	1.48100	0.740499	−	0.672058i	$-0.234589\pi$
$461$	31.7984	1.48100	0.740499	+	0.672058i	$0.234589\pi$
$462$	0	0
$463$	15.8328	0.735813	0.367907	−	0.929863i	$-0.380075\pi$
$463$	15.8328	0.735813	0.367907	+	0.929863i	$0.380075\pi$
$464$	0	0
$465$	4.85410	0.225104
$466$	0	0
$467$	10.8885	0.503862	0.251931	−	0.967745i	$-0.418934\pi$
$467$	10.8885	0.503862	0.251931	+	0.967745i	$0.418934\pi$
$468$	0	0
$469$	−6.56231	−0.303019
$470$	0	0
$471$	8.76393	0.403821
$472$	0	0
$473$	27.2705	1.25390
$474$	0	0
$475$	10.0902	0.462969
$476$	0	0
$477$	−9.32624	−0.427019
$478$	0	0
$479$	18.2361	0.833227	0.416614	−	0.909084i	$-0.363217\pi$
$479$	18.2361	0.833227	0.416614	+	0.909084i	$0.363217\pi$
$480$	0	0
$481$	5.85410	0.266924
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	20.0902	0.912248
$486$	0	0
$487$	−29.0000	−1.31412	−0.657058	−	0.753840i	$-0.728199\pi$
$487$	−29.0000	−1.31412	−0.657058	+	0.753840i	$0.728199\pi$
$488$	0	0
$489$	10.8541	0.490839
$490$	0	0
$491$	−5.09017	−0.229716	−0.114858	−	0.993382i	$-0.536641\pi$
$491$	−5.09017	−0.229716	−0.114858	+	0.993382i	$0.536641\pi$
$492$	0	0
$493$	−11.8328	−0.532923
$494$	0	0
$495$	4.85410	0.218176
$496$	0	0
$497$	4.61803	0.207147
$498$	0	0
$499$	−3.50658	−0.156976	−0.0784880	−	0.996915i	$-0.525009\pi$
$499$	−3.50658	−0.156976	−0.0784880	+	0.996915i	$0.525009\pi$
$500$	0	0
$501$	−10.5279	−0.470350
$502$	0	0
$503$	−11.2148	−0.500042	−0.250021	−	0.968240i	$-0.580438\pi$
$503$	−11.2148	−0.500042	−0.250021	+	0.968240i	$0.580438\pi$
$504$	0	0
$505$	−8.61803	−0.383497
$506$	0	0
$507$	−12.6180	−0.560387
$508$	0	0
$509$	21.2361	0.941272	0.470636	−	0.882327i	$-0.344024\pi$
$509$	21.2361	0.941272	0.470636	+	0.882327i	$0.344024\pi$
$510$	0	0
$511$	−1.76393	−0.0780318
$512$	0	0
$513$	−4.23607	−0.187027
$514$	0	0
$515$	−31.4164	−1.38437
$516$	0	0
$517$	27.7082	1.21861
$518$	0	0
$519$	13.4721	0.591361
$520$	0	0
$521$	22.3607	0.979639	0.489820	−	0.871824i	$-0.337063\pi$
$521$	22.3607	0.979639	0.489820	+	0.871824i	$0.337063\pi$
$522$	0	0
$523$	18.0000	0.787085	0.393543	−	0.919306i	$-0.371249\pi$
$523$	18.0000	0.787085	0.393543	+	0.919306i	$0.371249\pi$
$524$	0	0
$525$	2.38197	0.103958
$526$	0	0
$527$	−20.1246	−0.876642
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−9.61803	−0.417387
$532$	0	0
$533$	−7.38197	−0.319748
$534$	0	0
$535$	−17.1803	−0.742771
$536$	0	0
$537$	11.0344	0.476171
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−3.12461	−0.134338	−0.0671688	−	0.997742i	$-0.521397\pi$
$541$	−3.12461	−0.134338	−0.0671688	+	0.997742i	$0.521397\pi$
$542$	0	0
$543$	17.4721	0.749801
$544$	0	0
$545$	3.38197	0.144868
$546$	0	0
$547$	−11.1459	−0.476564	−0.238282	−	0.971196i	$-0.576584\pi$
$547$	−11.1459	−0.476564	−0.238282	+	0.971196i	$0.576584\pi$
$548$	0	0
$549$	5.85410	0.249847
$550$	0	0
$551$	7.47214	0.318324
$552$	0	0
$553$	6.70820	0.285262
$554$	0	0
$555$	−15.3262	−0.650563
$556$	0	0
$557$	23.2361	0.984544	0.492272	−	0.870441i	$-0.336167\pi$
$557$	23.2361	0.984544	0.492272	+	0.870441i	$0.336167\pi$
$558$	0	0
$559$	−5.61803	−0.237618
$560$	0	0
$561$	−20.1246	−0.849662
$562$	0	0
$563$	−30.6180	−1.29040	−0.645198	−	0.764015i	$-0.723225\pi$
$563$	−30.6180	−1.29040	−0.645198	+	0.764015i	$0.723225\pi$
$564$	0	0
$565$	−5.38197	−0.226421
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−31.4164	−1.31704	−0.658522	−	0.752561i	$-0.728818\pi$
$569$	−31.4164	−1.31704	−0.658522	+	0.752561i	$0.728818\pi$
$570$	0	0
$571$	−12.7082	−0.531822	−0.265911	−	0.963998i	$-0.585673\pi$
$571$	−12.7082	−0.531822	−0.265911	+	0.963998i	$0.585673\pi$
$572$	0	0
$573$	−20.6525	−0.862770
$574$	0	0
$575$	2.38197	0.0993348
$576$	0	0
$577$	−33.7771	−1.40616	−0.703079	−	0.711111i	$-0.748192\pi$
$577$	−33.7771	−1.40616	−0.703079	+	0.711111i	$0.748192\pi$
$578$	0	0
$579$	−5.70820	−0.237225
$580$	0	0
$581$	7.47214	0.309996
$582$	0	0
$583$	27.9787	1.15876
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	24.6180	1.01610	0.508048	−	0.861329i	$-0.330367\pi$
$587$	24.6180	1.01610	0.508048	+	0.861329i	$0.330367\pi$
$588$	0	0
$589$	12.7082	0.523632
$590$	0	0
$591$	1.38197	0.0568465
$592$	0	0
$593$	6.65248	0.273184	0.136592	−	0.990627i	$-0.456385\pi$
$593$	6.65248	0.273184	0.136592	+	0.990627i	$0.456385\pi$
$594$	0	0
$595$	10.8541	0.444975
$596$	0	0
$597$	22.5623	0.923414
$598$	0	0
$599$	33.7426	1.37869	0.689344	−	0.724435i	$-0.257899\pi$
$599$	33.7426	1.37869	0.689344	+	0.724435i	$0.257899\pi$
$600$	0	0
$601$	29.7984	1.21550	0.607751	−	0.794128i	$-0.292072\pi$
$601$	29.7984	1.21550	0.607751	+	0.794128i	$0.292072\pi$
$602$	0	0
$603$	6.56231	0.267238
$604$	0	0
$605$	3.23607	0.131565
$606$	0	0
$607$	15.5623	0.631655	0.315827	−	0.948817i	$-0.397718\pi$
$607$	15.5623	0.631655	0.315827	+	0.948817i	$0.397718\pi$
$608$	0	0
$609$	1.76393	0.0714781
$610$	0	0
$611$	−5.70820	−0.230929
$612$	0	0
$613$	−22.7082	−0.917176	−0.458588	−	0.888649i	$-0.651645\pi$
$613$	−22.7082	−0.917176	−0.458588	+	0.888649i	$0.651645\pi$
$614$	0	0
$615$	19.3262	0.779309
$616$	0	0
$617$	16.1459	0.650009	0.325005	−	0.945712i	$-0.394634\pi$
$617$	16.1459	0.650009	0.325005	+	0.945712i	$0.394634\pi$
$618$	0	0
$619$	−37.8541	−1.52148	−0.760742	−	0.649054i	$-0.775165\pi$
$619$	−37.8541	−1.52148	−0.760742	+	0.649054i	$0.775165\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	16.7984	0.673013
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0	0
$627$	12.7082	0.507517
$628$	0	0
$629$	63.5410	2.53355
$630$	0	0
$631$	30.7771	1.22522	0.612608	−	0.790387i	$-0.290120\pi$
$631$	30.7771	1.22522	0.612608	+	0.790387i	$0.290120\pi$
$632$	0	0
$633$	−26.1246	−1.03836
$634$	0	0
$635$	−9.00000	−0.357154
$636$	0	0
$637$	0.618034	0.0244874
$638$	0	0
$639$	−4.61803	−0.182687
$640$	0	0
$641$	−40.3262	−1.59279	−0.796395	−	0.604776i	$-0.793262\pi$
$641$	−40.3262	−1.59279	−0.796395	+	0.604776i	$0.793262\pi$
$642$	0	0
$643$	−8.32624	−0.328355	−0.164177	−	0.986431i	$-0.552497\pi$
$643$	−8.32624	−0.328355	−0.164177	+	0.986431i	$0.552497\pi$
$644$	0	0
$645$	14.7082	0.579135
$646$	0	0
$647$	−47.3951	−1.86329	−0.931647	−	0.363364i	$-0.881628\pi$
$647$	−47.3951	−1.86329	−0.931647	+	0.363364i	$0.881628\pi$
$648$	0	0
$649$	28.8541	1.13262
$650$	0	0
$651$	3.00000	0.117579
$652$	0	0
$653$	−19.1459	−0.749237	−0.374618	−	0.927179i	$-0.622226\pi$
$653$	−19.1459	−0.749237	−0.374618	+	0.927179i	$0.622226\pi$
$654$	0	0
$655$	−9.61803	−0.375808
$656$	0	0
$657$	1.76393	0.0688175
$658$	0	0
$659$	−40.3050	−1.57006	−0.785029	−	0.619459i	$-0.787352\pi$
$659$	−40.3050	−1.57006	−0.785029	+	0.619459i	$0.787352\pi$
$660$	0	0
$661$	42.7771	1.66384	0.831918	−	0.554899i	$-0.187243\pi$
$661$	42.7771	1.66384	0.831918	+	0.554899i	$0.187243\pi$
$662$	0	0
$663$	4.14590	0.161013
$664$	0	0
$665$	−6.85410	−0.265791
$666$	0	0
$667$	1.76393	0.0682997
$668$	0	0
$669$	11.7984	0.456151
$670$	0	0
$671$	−17.5623	−0.677985
$672$	0	0
$673$	18.3050	0.705604	0.352802	−	0.935698i	$-0.385229\pi$
$673$	18.3050	0.705604	0.352802	+	0.935698i	$0.385229\pi$
$674$	0	0
$675$	−2.38197	−0.0916819
$676$	0	0
$677$	−11.0902	−0.426230	−0.213115	−	0.977027i	$-0.568361\pi$
$677$	−11.0902	−0.426230	−0.213115	+	0.977027i	$0.568361\pi$
$678$	0	0
$679$	12.4164	0.476498
$680$	0	0
$681$	26.4508	1.01360
$682$	0	0
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	0	0
$685$	14.5623	0.556397
$686$	0	0
$687$	24.2705	0.925978
$688$	0	0
$689$	−5.76393	−0.219588
$690$	0	0
$691$	17.3820	0.661241	0.330621	−	0.943764i	$-0.392742\pi$
$691$	17.3820	0.661241	0.330621	+	0.943764i	$0.392742\pi$
$692$	0	0
$693$	3.00000	0.113961
$694$	0	0
$695$	−11.0902	−0.420674
$696$	0	0
$697$	−80.1246	−3.03494
$698$	0	0
$699$	−12.0902	−0.457292
$700$	0	0
$701$	1.90983	0.0721333	0.0360666	−	0.999349i	$-0.488517\pi$
$701$	1.90983	0.0721333	0.0360666	+	0.999349i	$0.488517\pi$
$702$	0	0
$703$	−40.1246	−1.51333
$704$	0	0
$705$	14.9443	0.562834
$706$	0	0
$707$	−5.32624	−0.200314
$708$	0	0
$709$	−11.0902	−0.416500	−0.208250	−	0.978076i	$-0.566777\pi$
$709$	−11.0902	−0.416500	−0.208250	+	0.978076i	$0.566777\pi$
$710$	0	0
$711$	−6.70820	−0.251577
$712$	0	0
$713$	3.00000	0.112351
$714$	0	0
$715$	3.00000	0.112194
$716$	0	0
$717$	−17.6180	−0.657958
$718$	0	0
$719$	27.6525	1.03126	0.515632	−	0.856810i	$-0.327557\pi$
$719$	27.6525	1.03126	0.515632	+	0.856810i	$0.327557\pi$
$720$	0	0
$721$	−19.4164	−0.723105
$722$	0	0
$723$	21.0000	0.780998
$724$	0	0
$725$	4.20163	0.156044
$726$	0	0
$727$	−32.7082	−1.21308	−0.606540	−	0.795053i	$-0.707443\pi$
$727$	−32.7082	−1.21308	−0.606540	+	0.795053i	$0.707443\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−60.9787	−2.25538
$732$	0	0
$733$	47.5967	1.75803	0.879013	−	0.476798i	$-0.158203\pi$
$733$	47.5967	1.75803	0.879013	+	0.476798i	$0.158203\pi$
$734$	0	0
$735$	−1.61803	−0.0596821
$736$	0	0
$737$	−19.6869	−0.725177
$738$	0	0
$739$	28.7082	1.05605	0.528024	−	0.849229i	$-0.322933\pi$
$739$	28.7082	1.05605	0.528024	+	0.849229i	$0.322933\pi$
$740$	0	0
$741$	−2.61803	−0.0961759
$742$	0	0
$743$	−20.3820	−0.747742	−0.373871	−	0.927481i	$-0.621970\pi$
$743$	−20.3820	−0.747742	−0.373871	+	0.927481i	$0.621970\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	−7.47214	−0.273391
$748$	0	0
$749$	−10.6180	−0.387975
$750$	0	0
$751$	38.2705	1.39651	0.698255	−	0.715849i	$-0.253960\pi$
$751$	38.2705	1.39651	0.698255	+	0.715849i	$0.253960\pi$
$752$	0	0
$753$	−22.6525	−0.825502
$754$	0	0
$755$	30.9443	1.12618
$756$	0	0
$757$	−17.4721	−0.635036	−0.317518	−	0.948252i	$-0.602849\pi$
$757$	−17.4721	−0.635036	−0.317518	+	0.948252i	$0.602849\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	−26.4721	−0.959614	−0.479807	−	0.877374i	$-0.659293\pi$
$761$	−26.4721	−0.959614	−0.479807	+	0.877374i	$0.659293\pi$
$762$	0	0
$763$	2.09017	0.0756692
$764$	0	0
$765$	−10.8541	−0.392431
$766$	0	0
$767$	−5.94427	−0.214635
$768$	0	0
$769$	5.12461	0.184798	0.0923991	−	0.995722i	$-0.470546\pi$
$769$	5.12461	0.184798	0.0923991	+	0.995722i	$0.470546\pi$
$770$	0	0
$771$	16.6525	0.599724
$772$	0	0
$773$	−16.0557	−0.577484	−0.288742	−	0.957407i	$-0.593237\pi$
$773$	−16.0557	−0.577484	−0.288742	+	0.957407i	$0.593237\pi$
$774$	0	0
$775$	7.14590	0.256688
$776$	0	0
$777$	−9.47214	−0.339811
$778$	0	0
$779$	50.5967	1.81282
$780$	0	0
$781$	13.8541	0.495739
$782$	0	0
$783$	−1.76393	−0.0630378
$784$	0	0
$785$	−14.1803	−0.506118
$786$	0	0
$787$	24.0902	0.858722	0.429361	−	0.903133i	$-0.358739\pi$
$787$	24.0902	0.858722	0.429361	+	0.903133i	$0.358739\pi$
$788$	0	0
$789$	15.1803	0.540434
$790$	0	0
$791$	−3.32624	−0.118267
$792$	0	0
$793$	3.61803	0.128480
$794$	0	0
$795$	15.0902	0.535193
$796$	0	0
$797$	21.8197	0.772892	0.386446	−	0.922312i	$-0.373703\pi$
$797$	21.8197	0.772892	0.386446	+	0.922312i	$0.373703\pi$
$798$	0	0
$799$	−61.9574	−2.19190
$800$	0	0
$801$	−16.7984	−0.593541
$802$	0	0
$803$	−5.29180	−0.186743
$804$	0	0
$805$	−1.61803	−0.0570282
$806$	0	0
$807$	−22.7984	−0.802541
$808$	0	0
$809$	−42.2148	−1.48419	−0.742096	−	0.670293i	$-0.766168\pi$
$809$	−42.2148	−1.48419	−0.742096	+	0.670293i	$0.766168\pi$
$810$	0	0
$811$	−40.7082	−1.42946	−0.714729	−	0.699401i	$-0.753450\pi$
$811$	−40.7082	−1.42946	−0.714729	+	0.699401i	$0.753450\pi$
$812$	0	0
$813$	−21.1803	−0.742827
$814$	0	0
$815$	−17.5623	−0.615181
$816$	0	0
$817$	38.5066	1.34717
$818$	0	0
$819$	−0.618034	−0.0215959
$820$	0	0
$821$	−51.7082	−1.80463	−0.902314	−	0.431079i	$-0.858133\pi$
$821$	−51.7082	−1.80463	−0.902314	+	0.431079i	$0.858133\pi$
$822$	0	0
$823$	−27.9098	−0.972876	−0.486438	−	0.873715i	$-0.661704\pi$
$823$	−27.9098	−0.972876	−0.486438	+	0.873715i	$0.661704\pi$
$824$	0	0
$825$	7.14590	0.248788
$826$	0	0
$827$	50.9230	1.77077	0.885383	−	0.464863i	$-0.153896\pi$
$827$	50.9230	1.77077	0.885383	+	0.464863i	$0.153896\pi$
$828$	0	0
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	0	0
$831$	24.0344	0.833746
$832$	0	0
$833$	6.70820	0.232425
$834$	0	0
$835$	17.0344	0.589501
$836$	0	0
$837$	−3.00000	−0.103695
$838$	0	0
$839$	−11.0902	−0.382875	−0.191438	−	0.981505i	$-0.561315\pi$
$839$	−11.0902	−0.382875	−0.191438	+	0.981505i	$0.561315\pi$
$840$	0	0
$841$	−25.8885	−0.892708
$842$	0	0
$843$	−12.7639	−0.439613
$844$	0	0
$845$	20.4164	0.702346
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−14.8541	−0.509791
$850$	0	0
$851$	−9.47214	−0.324701
$852$	0	0
$853$	2.29180	0.0784696	0.0392348	−	0.999230i	$-0.487508\pi$
$853$	2.29180	0.0784696	0.0392348	+	0.999230i	$0.487508\pi$
$854$	0	0
$855$	6.85410	0.234405
$856$	0	0
$857$	−12.1803	−0.416072	−0.208036	−	0.978121i	$-0.566707\pi$
$857$	−12.1803	−0.416072	−0.208036	+	0.978121i	$0.566707\pi$
$858$	0	0
$859$	−35.3050	−1.20459	−0.602295	−	0.798274i	$-0.705747\pi$
$859$	−35.3050	−1.20459	−0.602295	+	0.798274i	$0.705747\pi$
$860$	0	0
$861$	11.9443	0.407060
$862$	0	0
$863$	50.6525	1.72423	0.862115	−	0.506712i	$-0.169139\pi$
$863$	50.6525	1.72423	0.862115	+	0.506712i	$0.169139\pi$
$864$	0	0
$865$	−21.7984	−0.741167
$866$	0	0
$867$	28.0000	0.950930
$868$	0	0
$869$	20.1246	0.682681
$870$	0	0
$871$	4.05573	0.137423
$872$	0	0
$873$	−12.4164	−0.420232
$874$	0	0
$875$	−11.9443	−0.403790
$876$	0	0
$877$	−3.41641	−0.115364	−0.0576819	−	0.998335i	$-0.518371\pi$
$877$	−3.41641	−0.115364	−0.0576819	+	0.998335i	$0.518371\pi$
$878$	0	0
$879$	−0.944272	−0.0318495
$880$	0	0
$881$	−9.70820	−0.327078	−0.163539	−	0.986537i	$-0.552291\pi$
$881$	−9.70820	−0.327078	−0.163539	+	0.986537i	$0.552291\pi$
$882$	0	0
$883$	−21.6738	−0.729380	−0.364690	−	0.931129i	$-0.618825\pi$
$883$	−21.6738	−0.729380	−0.364690	+	0.931129i	$0.618825\pi$
$884$	0	0
$885$	15.5623	0.523121
$886$	0	0
$887$	−3.27051	−0.109813	−0.0549065	−	0.998492i	$-0.517486\pi$
$887$	−3.27051	−0.109813	−0.0549065	+	0.998492i	$0.517486\pi$
$888$	0	0
$889$	−5.56231	−0.186554
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	39.1246	1.30926
$894$	0	0
$895$	−17.8541	−0.596797
$896$	0	0
$897$	−0.618034	−0.0206356
$898$	0	0
$899$	5.29180	0.176491
$900$	0	0
$901$	−62.5623	−2.08425
$902$	0	0
$903$	9.09017	0.302502
$904$	0	0
$905$	−28.2705	−0.939744
$906$	0	0
$907$	−52.5623	−1.74530	−0.872651	−	0.488344i	$-0.837601\pi$
$907$	−52.5623	−1.74530	−0.872651	+	0.488344i	$0.837601\pi$
$908$	0	0
$909$	5.32624	0.176660
$910$	0	0
$911$	−33.8197	−1.12050	−0.560248	−	0.828325i	$-0.689294\pi$
$911$	−33.8197	−1.12050	−0.560248	+	0.828325i	$0.689294\pi$
$912$	0	0
$913$	22.4164	0.741875
$914$	0	0
$915$	−9.47214	−0.313139
$916$	0	0
$917$	−5.94427	−0.196297
$918$	0	0
$919$	−26.3050	−0.867720	−0.433860	−	0.900980i	$-0.642849\pi$
$919$	−26.3050	−0.867720	−0.433860	+	0.900980i	$0.642849\pi$
$920$	0	0
$921$	−28.4164	−0.936352
$922$	0	0
$923$	−2.85410	−0.0939439
$924$	0	0
$925$	−22.5623	−0.741844
$926$	0	0
$927$	19.4164	0.637719
$928$	0	0
$929$	−6.96556	−0.228533	−0.114266	−	0.993450i	$-0.536452\pi$
$929$	−6.96556	−0.228533	−0.114266	+	0.993450i	$0.536452\pi$
$930$	0	0
$931$	−4.23607	−0.138832
$932$	0	0
$933$	−15.2705	−0.499934
$934$	0	0
$935$	32.5623	1.06490
$936$	0	0
$937$	−27.2918	−0.891584	−0.445792	−	0.895137i	$-0.647078\pi$
$937$	−27.2918	−0.891584	−0.445792	+	0.895137i	$0.647078\pi$
$938$	0	0
$939$	17.5279	0.572000
$940$	0	0
$941$	−36.0000	−1.17357	−0.586783	−	0.809744i	$-0.699606\pi$
$941$	−36.0000	−1.17357	−0.586783	+	0.809744i	$0.699606\pi$
$942$	0	0
$943$	11.9443	0.388959
$944$	0	0
$945$	1.61803	0.0526346
$946$	0	0
$947$	7.30495	0.237379	0.118690	−	0.992931i	$-0.462131\pi$
$947$	7.30495	0.237379	0.118690	+	0.992931i	$0.462131\pi$
$948$	0	0
$949$	1.09017	0.0353884
$950$	0	0
$951$	5.61803	0.182177
$952$	0	0
$953$	−18.9098	−0.612549	−0.306275	−	0.951943i	$-0.599083\pi$
$953$	−18.9098	−0.612549	−0.306275	+	0.951943i	$0.599083\pi$
$954$	0	0
$955$	33.4164	1.08133
$956$	0	0
$957$	5.29180	0.171059
$958$	0	0
$959$	9.00000	0.290625
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	10.6180	0.342161
$964$	0	0
$965$	9.23607	0.297320
$966$	0	0
$967$	−54.0000	−1.73652	−0.868261	−	0.496107i	$-0.834762\pi$
$967$	−54.0000	−1.73652	−0.868261	+	0.496107i	$0.834762\pi$
$968$	0	0
$969$	−28.4164	−0.912867
$970$	0	0
$971$	−4.79837	−0.153987	−0.0769936	−	0.997032i	$-0.524532\pi$
$971$	−4.79837	−0.153987	−0.0769936	+	0.997032i	$0.524532\pi$
$972$	0	0
$973$	−6.85410	−0.219732
$974$	0	0
$975$	−1.47214	−0.0471461
$976$	0	0
$977$	50.2837	1.60872	0.804358	−	0.594144i	$-0.202509\pi$
$977$	50.2837	1.60872	0.804358	+	0.594144i	$0.202509\pi$
$978$	0	0
$979$	50.3951	1.61064
$980$	0	0
$981$	−2.09017	−0.0667340
$982$	0	0
$983$	−36.5279	−1.16506	−0.582529	−	0.812810i	$-0.697937\pi$
$983$	−36.5279	−1.16506	−0.582529	+	0.812810i	$0.697937\pi$
$984$	0	0
$985$	−2.23607	−0.0712470
$986$	0	0
$987$	9.23607	0.293987
$988$	0	0
$989$	9.09017	0.289051
$990$	0	0
$991$	0.686918	0.0218207	0.0109103	−	0.999940i	$-0.496527\pi$
$991$	0.686918	0.0218207	0.0109103	+	0.999940i	$0.496527\pi$
$992$	0	0
$993$	7.88854	0.250335
$994$	0	0
$995$	−36.5066	−1.15734
$996$	0	0
$997$	−6.52786	−0.206740	−0.103370	−	0.994643i	$-0.532962\pi$
$997$	−6.52786	−0.206740	−0.103370	+	0.994643i	$0.532962\pi$
$998$	0	0
$999$	9.47214	0.299685

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.g.1.1	✓	2	1.1	even	1	trivial
5796.2.a.l.1.2		2	3.2	odd	2
7728.2.a.ba.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +0.618034 q^{13} -1.61803 q^{15} +6.70820 q^{17} -4.23607 q^{19} -1.00000 q^{21} -1.00000 q^{23} -2.38197 q^{25} +1.00000 q^{27} -1.76393 q^{29} -3.00000 q^{31} -3.00000 q^{33} +1.61803 q^{35} +9.47214 q^{37} +0.618034 q^{39} -11.9443 q^{41} -9.09017 q^{43} -1.61803 q^{45} -9.23607 q^{47} +1.00000 q^{49} +6.70820 q^{51} -9.32624 q^{53} +4.85410 q^{55} -4.23607 q^{57} -9.61803 q^{59} +5.85410 q^{61} -1.00000 q^{63} -1.00000 q^{65} +6.56231 q^{67} -1.00000 q^{69} -4.61803 q^{71} +1.76393 q^{73} -2.38197 q^{75} +3.00000 q^{77} -6.70820 q^{79} +1.00000 q^{81} -7.47214 q^{83} -10.8541 q^{85} -1.76393 q^{87} -16.7984 q^{89} -0.618034 q^{91} -3.00000 q^{93} +6.85410 q^{95} -12.4164 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} - 6 q^{11} - q^{13} - q^{15} - 4 q^{19} - 2 q^{21} - 2 q^{23} - 7 q^{25} + 2 q^{27} - 8 q^{29} - 6 q^{31} - 6 q^{33} + q^{35} + 10 q^{37} - q^{39} - 6 q^{41} - 7 q^{43} - q^{45} - 14 q^{47} + 2 q^{49} - 3 q^{53} + 3 q^{55} - 4 q^{57} - 17 q^{59} + 5 q^{61} - 2 q^{63} - 2 q^{65} - 7 q^{67} - 2 q^{69} - 7 q^{71} + 8 q^{73} - 7 q^{75} + 6 q^{77} + 2 q^{81} - 6 q^{83} - 15 q^{85} - 8 q^{87} - 9 q^{89} + q^{91} - 6 q^{93} + 7 q^{95} + 2 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9 - 6 * q^11 - q^13 - q^15 - 4 * q^19 - 2 * q^21 - 2 * q^23 - 7 * q^25 + 2 * q^27 - 8 * q^29 - 6 * q^31 - 6 * q^33 + q^35 + 10 * q^37 - q^39 - 6 * q^41 - 7 * q^43 - q^45 - 14 * q^47 + 2 * q^49 - 3 * q^53 + 3 * q^55 - 4 * q^57 - 17 * q^59 + 5 * q^61 - 2 * q^63 - 2 * q^65 - 7 * q^67 - 2 * q^69 - 7 * q^71 + 8 * q^73 - 7 * q^75 + 6 * q^77 + 2 * q^81 - 6 * q^83 - 15 * q^85 - 8 * q^87 - 9 * q^89 + q^91 - 6 * q^93 + 7 * q^95 + 2 * q^97 - 6 * q^99

Introduction

L-functions

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