LMFDB - Embedded newform 6160.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1540)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} +O(q^{10})$	$q+0.732051 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} -1.00000 q^{11} -2.73205 q^{13} -0.732051 q^{15} +1.26795 q^{17} -5.46410 q^{19} -0.732051 q^{21} +1.00000 q^{25} -4.00000 q^{27} -3.46410 q^{29} +6.19615 q^{31} -0.732051 q^{33} +1.00000 q^{35} +5.46410 q^{37} -2.00000 q^{39} +8.19615 q^{41} -2.00000 q^{43} +2.46410 q^{45} -11.6603 q^{47} +1.00000 q^{49} +0.928203 q^{51} +9.46410 q^{53} +1.00000 q^{55} -4.00000 q^{57} -7.26795 q^{59} -5.26795 q^{61} +2.46410 q^{63} +2.73205 q^{65} +4.00000 q^{67} +9.46410 q^{71} -0.196152 q^{73} +0.732051 q^{75} +1.00000 q^{77} -2.00000 q^{79} +4.46410 q^{81} +4.39230 q^{83} -1.26795 q^{85} -2.53590 q^{87} +12.9282 q^{89} +2.73205 q^{91} +4.53590 q^{93} +5.46410 q^{95} +18.3923 q^{97} +2.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} + 2 q^{25} - 8 q^{27} + 2 q^{31} + 2 q^{33} + 2 q^{35} + 4 q^{37} - 4 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} - 6 q^{47} + 2 q^{49} - 12 q^{51} + 12 q^{53} + 2 q^{55} - 8 q^{57} - 18 q^{59} - 14 q^{61} - 2 q^{63} + 2 q^{65} + 8 q^{67} + 12 q^{71} + 10 q^{73} - 2 q^{75} + 2 q^{77} - 4 q^{79} + 2 q^{81} - 12 q^{83} - 6 q^{85} - 12 q^{87} + 12 q^{89} + 2 q^{91} + 16 q^{93} + 4 q^{95} + 16 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 + 2 * q^15 + 6 * q^17 - 4 * q^19 + 2 * q^21 + 2 * q^25 - 8 * q^27 + 2 * q^31 + 2 * q^33 + 2 * q^35 + 4 * q^37 - 4 * q^39 + 6 * q^41 - 4 * q^43 - 2 * q^45 - 6 * q^47 + 2 * q^49 - 12 * q^51 + 12 * q^53 + 2 * q^55 - 8 * q^57 - 18 * q^59 - 14 * q^61 - 2 * q^63 + 2 * q^65 + 8 * q^67 + 12 * q^71 + 10 * q^73 - 2 * q^75 + 2 * q^77 - 4 * q^79 + 2 * q^81 - 12 * q^83 - 6 * q^85 - 12 * q^87 + 12 * q^89 + 2 * q^91 + 16 * q^93 + 4 * q^95 + 16 * q^97 - 2 * 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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.73205	−0.757735	−0.378867	−	0.925451i	$-0.623686\pi$
$13$	−2.73205	−0.757735	−0.378867	+	0.925451i	$0.623686\pi$
$14$	0	0
$15$	−0.732051	−0.189015
$16$	0	0
$17$	1.26795	0.307523	0.153761	−	0.988108i	$-0.450861\pi$
$17$	1.26795	0.307523	0.153761	+	0.988108i	$0.450861\pi$
$18$	0	0
$19$	−5.46410	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$19$	−5.46410	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$20$	0	0
$21$	−0.732051	−0.159747
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	6.19615	1.11286	0.556431	−	0.830894i	$-0.312170\pi$
$31$	6.19615	1.11286	0.556431	+	0.830894i	$0.312170\pi$
$32$	0	0
$33$	−0.732051	−0.127434
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	5.46410	0.898293	0.449146	−	0.893458i	$-0.351728\pi$
$37$	5.46410	0.898293	0.449146	+	0.893458i	$0.351728\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	8.19615	1.28002	0.640012	−	0.768365i	$-0.278929\pi$
$41$	8.19615	1.28002	0.640012	+	0.768365i	$0.278929\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	2.46410	0.367327
$46$	0	0
$47$	−11.6603	−1.70082	−0.850411	−	0.526118i	$-0.823647\pi$
$47$	−11.6603	−1.70082	−0.850411	+	0.526118i	$0.823647\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.928203	0.129974
$52$	0	0
$53$	9.46410	1.29999	0.649997	−	0.759937i	$-0.274770\pi$
$53$	9.46410	1.29999	0.649997	+	0.759937i	$0.274770\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−7.26795	−0.946206	−0.473103	−	0.881007i	$-0.656866\pi$
$59$	−7.26795	−0.946206	−0.473103	+	0.881007i	$0.656866\pi$
$60$	0	0
$61$	−5.26795	−0.674492	−0.337246	−	0.941417i	$-0.609495\pi$
$61$	−5.26795	−0.674492	−0.337246	+	0.941417i	$0.609495\pi$
$62$	0	0
$63$	2.46410	0.310448
$64$	0	0
$65$	2.73205	0.338869
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.46410	1.12318	0.561591	−	0.827415i	$-0.310189\pi$
$71$	9.46410	1.12318	0.561591	+	0.827415i	$0.310189\pi$
$72$	0	0
$73$	−0.196152	−0.0229579	−0.0114790	−	0.999934i	$-0.503654\pi$
$73$	−0.196152	−0.0229579	−0.0114790	+	0.999934i	$0.503654\pi$
$74$	0	0
$75$	0.732051	0.0845299
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	4.39230	0.482118	0.241059	−	0.970510i	$-0.422505\pi$
$83$	4.39230	0.482118	0.241059	+	0.970510i	$0.422505\pi$
$84$	0	0
$85$	−1.26795	−0.137528
$86$	0	0
$87$	−2.53590	−0.271877
$88$	0	0
$89$	12.9282	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$89$	12.9282	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$90$	0	0
$91$	2.73205	0.286397
$92$	0	0
$93$	4.53590	0.470351
$94$	0	0
$95$	5.46410	0.560605
$96$	0	0
$97$	18.3923	1.86746	0.933728	−	0.357984i	$-0.116536\pi$
$97$	18.3923	1.86746	0.933728	+	0.357984i	$0.116536\pi$
$98$	0	0
$99$	2.46410	0.247652
$100$	0	0
$101$	5.66025	0.563216	0.281608	−	0.959529i	$-0.409132\pi$
$101$	5.66025	0.563216	0.281608	+	0.959529i	$0.409132\pi$
$102$	0	0
$103$	1.80385	0.177738	0.0888692	−	0.996043i	$-0.471675\pi$
$103$	1.80385	0.177738	0.0888692	+	0.996043i	$0.471675\pi$
$104$	0	0
$105$	0.732051	0.0714408
$106$	0	0
$107$	−6.92820	−0.669775	−0.334887	−	0.942258i	$-0.608698\pi$
$107$	−6.92820	−0.669775	−0.334887	+	0.942258i	$0.608698\pi$
$108$	0	0
$109$	11.4641	1.09806	0.549031	−	0.835802i	$-0.314997\pi$
$109$	11.4641	1.09806	0.549031	+	0.835802i	$0.314997\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	0.928203	0.0873180	0.0436590	−	0.999046i	$-0.486098\pi$
$113$	0.928203	0.0873180	0.0436590	+	0.999046i	$0.486098\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.73205	0.622378
$118$	0	0
$119$	−1.26795	−0.116233
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.92820	−0.259836	−0.129918	−	0.991525i	$-0.541471\pi$
$127$	−2.92820	−0.259836	−0.129918	+	0.991525i	$0.541471\pi$
$128$	0	0
$129$	−1.46410	−0.128907
$130$	0	0
$131$	−18.9282	−1.65376	−0.826882	−	0.562375i	$-0.809888\pi$
$131$	−18.9282	−1.65376	−0.826882	+	0.562375i	$0.809888\pi$
$132$	0	0
$133$	5.46410	0.473798
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	−9.46410	−0.808573	−0.404286	−	0.914632i	$-0.632480\pi$
$137$	−9.46410	−0.808573	−0.404286	+	0.914632i	$0.632480\pi$
$138$	0	0
$139$	1.46410	0.124183	0.0620917	−	0.998070i	$-0.480223\pi$
$139$	1.46410	0.124183	0.0620917	+	0.998070i	$0.480223\pi$
$140$	0	0
$141$	−8.53590	−0.718852
$142$	0	0
$143$	2.73205	0.228466
$144$	0	0
$145$	3.46410	0.287678
$146$	0	0
$147$	0.732051	0.0603785
$148$	0	0
$149$	−0.928203	−0.0760414	−0.0380207	−	0.999277i	$-0.512105\pi$
$149$	−0.928203	−0.0760414	−0.0380207	+	0.999277i	$0.512105\pi$
$150$	0	0
$151$	16.9282	1.37760	0.688799	−	0.724953i	$-0.258138\pi$
$151$	16.9282	1.37760	0.688799	+	0.724953i	$0.258138\pi$
$152$	0	0
$153$	−3.12436	−0.252589
$154$	0	0
$155$	−6.19615	−0.497687
$156$	0	0
$157$	3.85641	0.307775	0.153887	−	0.988088i	$-0.450821\pi$
$157$	3.85641	0.307775	0.153887	+	0.988088i	$0.450821\pi$
$158$	0	0
$159$	6.92820	0.549442
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−16.7846	−1.31467	−0.657336	−	0.753598i	$-0.728317\pi$
$163$	−16.7846	−1.31467	−0.657336	+	0.753598i	$0.728317\pi$
$164$	0	0
$165$	0.732051	0.0569901
$166$	0	0
$167$	5.07180	0.392467	0.196234	−	0.980557i	$-0.437129\pi$
$167$	5.07180	0.392467	0.196234	+	0.980557i	$0.437129\pi$
$168$	0	0
$169$	−5.53590	−0.425838
$170$	0	0
$171$	13.4641	1.02963
$172$	0	0
$173$	17.6603	1.34268	0.671342	−	0.741148i	$-0.265718\pi$
$173$	17.6603	1.34268	0.671342	+	0.741148i	$0.265718\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−5.32051	−0.399914
$178$	0	0
$179$	20.7846	1.55351	0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$180$	0	0
$181$	6.39230	0.475136	0.237568	−	0.971371i	$-0.423650\pi$
$181$	6.39230	0.475136	0.237568	+	0.971371i	$0.423650\pi$
$182$	0	0
$183$	−3.85641	−0.285074
$184$	0	0
$185$	−5.46410	−0.401729
$186$	0	0
$187$	−1.26795	−0.0927216
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−14.3923	−1.03598	−0.517990	−	0.855386i	$-0.673320\pi$
$193$	−14.3923	−1.03598	−0.517990	+	0.855386i	$0.673320\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	15.4641	1.10177	0.550886	−	0.834581i	$-0.314290\pi$
$197$	15.4641	1.10177	0.550886	+	0.834581i	$0.314290\pi$
$198$	0	0
$199$	4.33975	0.307636	0.153818	−	0.988099i	$-0.450843\pi$
$199$	4.33975	0.307636	0.153818	+	0.988099i	$0.450843\pi$
$200$	0	0
$201$	2.92820	0.206540
$202$	0	0
$203$	3.46410	0.243132
$204$	0	0
$205$	−8.19615	−0.572444
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.46410	0.377960
$210$	0	0
$211$	22.9282	1.57844	0.789221	−	0.614109i	$-0.210484\pi$
$211$	22.9282	1.57844	0.789221	+	0.614109i	$0.210484\pi$
$212$	0	0
$213$	6.92820	0.474713
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	−6.19615	−0.420622
$218$	0	0
$219$	−0.143594	−0.00970315
$220$	0	0
$221$	−3.46410	−0.233021
$222$	0	0
$223$	−3.26795	−0.218838	−0.109419	−	0.993996i	$-0.534899\pi$
$223$	−3.26795	−0.218838	−0.109419	+	0.993996i	$0.534899\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	−5.07180	−0.336627	−0.168313	−	0.985734i	$-0.553832\pi$
$227$	−5.07180	−0.336627	−0.168313	+	0.985734i	$0.553832\pi$
$228$	0	0
$229$	−14.3923	−0.951070	−0.475535	−	0.879697i	$-0.657746\pi$
$229$	−14.3923	−0.951070	−0.475535	+	0.879697i	$0.657746\pi$
$230$	0	0
$231$	0.732051	0.0481654
$232$	0	0
$233$	11.0718	0.725338	0.362669	−	0.931918i	$-0.381866\pi$
$233$	11.0718	0.725338	0.362669	+	0.931918i	$0.381866\pi$
$234$	0	0
$235$	11.6603	0.760631
$236$	0	0
$237$	−1.46410	−0.0951036
$238$	0	0
$239$	5.07180	0.328067	0.164034	−	0.986455i	$-0.447549\pi$
$239$	5.07180	0.328067	0.164034	+	0.986455i	$0.447549\pi$
$240$	0	0
$241$	−21.6603	−1.39526	−0.697630	−	0.716458i	$-0.745762\pi$
$241$	−21.6603	−1.39526	−0.697630	+	0.716458i	$0.745762\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	14.9282	0.949859
$248$	0	0
$249$	3.21539	0.203767
$250$	0	0
$251$	−6.58846	−0.415860	−0.207930	−	0.978144i	$-0.566673\pi$
$251$	−6.58846	−0.415860	−0.207930	+	0.978144i	$0.566673\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−0.928203	−0.0581263
$256$	0	0
$257$	31.8564	1.98715	0.993574	−	0.113184i	$-0.0361050\pi$
$257$	31.8564	1.98715	0.993574	+	0.113184i	$0.0361050\pi$
$258$	0	0
$259$	−5.46410	−0.339523
$260$	0	0
$261$	8.53590	0.528359
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−9.46410	−0.581375
$266$	0	0
$267$	9.46410	0.579194
$268$	0	0
$269$	−10.3923	−0.633630	−0.316815	−	0.948487i	$-0.602613\pi$
$269$	−10.3923	−0.633630	−0.316815	+	0.948487i	$0.602613\pi$
$270$	0	0
$271$	20.3923	1.23874	0.619372	−	0.785098i	$-0.287387\pi$
$271$	20.3923	1.23874	0.619372	+	0.785098i	$0.287387\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	25.3205	1.52136	0.760681	−	0.649126i	$-0.224865\pi$
$277$	25.3205	1.52136	0.760681	+	0.649126i	$0.224865\pi$
$278$	0	0
$279$	−15.2679	−0.914068
$280$	0	0
$281$	−22.3923	−1.33581	−0.667906	−	0.744245i	$-0.732809\pi$
$281$	−22.3923	−1.33581	−0.667906	+	0.744245i	$0.732809\pi$
$282$	0	0
$283$	5.85641	0.348127	0.174064	−	0.984734i	$-0.444310\pi$
$283$	5.85641	0.348127	0.174064	+	0.984734i	$0.444310\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	−8.19615	−0.483804
$288$	0	0
$289$	−15.3923	−0.905430
$290$	0	0
$291$	13.4641	0.789280
$292$	0	0
$293$	−5.66025	−0.330676	−0.165338	−	0.986237i	$-0.552871\pi$
$293$	−5.66025	−0.330676	−0.165338	+	0.986237i	$0.552871\pi$
$294$	0	0
$295$	7.26795	0.423156
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	0	0
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	4.14359	0.238043
$304$	0	0
$305$	5.26795	0.301642
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	1.32051	0.0751211
$310$	0	0
$311$	21.1244	1.19785	0.598926	−	0.800804i	$-0.295594\pi$
$311$	21.1244	1.19785	0.598926	+	0.800804i	$0.295594\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	−2.46410	−0.138836
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	3.46410	0.193952
$320$	0	0
$321$	−5.07180	−0.283080
$322$	0	0
$323$	−6.92820	−0.385496
$324$	0	0
$325$	−2.73205	−0.151547
$326$	0	0
$327$	8.39230	0.464096
$328$	0	0
$329$	11.6603	0.642851
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	0	0
$333$	−13.4641	−0.737828
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−30.7846	−1.67694	−0.838472	−	0.544944i	$-0.816551\pi$
$337$	−30.7846	−1.67694	−0.838472	+	0.544944i	$0.816551\pi$
$338$	0	0
$339$	0.679492	0.0369049
$340$	0	0
$341$	−6.19615	−0.335540
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.14359	0.222440	0.111220	−	0.993796i	$-0.464524\pi$
$347$	4.14359	0.222440	0.111220	+	0.993796i	$0.464524\pi$
$348$	0	0
$349$	8.58846	0.459730	0.229865	−	0.973223i	$-0.426172\pi$
$349$	8.58846	0.459730	0.229865	+	0.973223i	$0.426172\pi$
$350$	0	0
$351$	10.9282	0.583304
$352$	0	0
$353$	−26.7846	−1.42560	−0.712800	−	0.701367i	$-0.752573\pi$
$353$	−26.7846	−1.42560	−0.712800	+	0.701367i	$0.752573\pi$
$354$	0	0
$355$	−9.46410	−0.502302
$356$	0	0
$357$	−0.928203	−0.0491257
$358$	0	0
$359$	−19.8564	−1.04798	−0.523991	−	0.851724i	$-0.675557\pi$
$359$	−19.8564	−1.04798	−0.523991	+	0.851724i	$0.675557\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	0	0
$363$	0.732051	0.0384227
$364$	0	0
$365$	0.196152	0.0102671
$366$	0	0
$367$	18.1962	0.949831	0.474916	−	0.880031i	$-0.342479\pi$
$367$	18.1962	0.949831	0.474916	+	0.880031i	$0.342479\pi$
$368$	0	0
$369$	−20.1962	−1.05137
$370$	0	0
$371$	−9.46410	−0.491352
$372$	0	0
$373$	−2.39230	−0.123869	−0.0619344	−	0.998080i	$-0.519727\pi$
$373$	−2.39230	−0.123869	−0.0619344	+	0.998080i	$0.519727\pi$
$374$	0	0
$375$	−0.732051	−0.0378029
$376$	0	0
$377$	9.46410	0.487426
$378$	0	0
$379$	6.53590	0.335727	0.167863	−	0.985810i	$-0.446313\pi$
$379$	6.53590	0.335727	0.167863	+	0.985810i	$0.446313\pi$
$380$	0	0
$381$	−2.14359	−0.109820
$382$	0	0
$383$	0.339746	0.0173602	0.00868010	−	0.999962i	$-0.497237\pi$
$383$	0.339746	0.0173602	0.00868010	+	0.999962i	$0.497237\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	4.92820	0.250515
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−13.8564	−0.698963
$394$	0	0
$395$	2.00000	0.100631
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	−12.9282	−0.645604	−0.322802	−	0.946467i	$-0.604625\pi$
$401$	−12.9282	−0.645604	−0.322802	+	0.946467i	$0.604625\pi$
$402$	0	0
$403$	−16.9282	−0.843254
$404$	0	0
$405$	−4.46410	−0.221823
$406$	0	0
$407$	−5.46410	−0.270845
$408$	0	0
$409$	11.1244	0.550064	0.275032	−	0.961435i	$-0.411312\pi$
$409$	11.1244	0.550064	0.275032	+	0.961435i	$0.411312\pi$
$410$	0	0
$411$	−6.92820	−0.341743
$412$	0	0
$413$	7.26795	0.357632
$414$	0	0
$415$	−4.39230	−0.215610
$416$	0	0
$417$	1.07180	0.0524861
$418$	0	0
$419$	−32.4449	−1.58504	−0.792518	−	0.609849i	$-0.791230\pi$
$419$	−32.4449	−1.58504	−0.792518	+	0.609849i	$0.791230\pi$
$420$	0	0
$421$	−6.53590	−0.318540	−0.159270	−	0.987235i	$-0.550914\pi$
$421$	−6.53590	−0.318540	−0.159270	+	0.987235i	$0.550914\pi$
$422$	0	0
$423$	28.7321	1.39700
$424$	0	0
$425$	1.26795	0.0615046
$426$	0	0
$427$	5.26795	0.254934
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	24.9282	1.20075	0.600375	−	0.799719i	$-0.295018\pi$
$431$	24.9282	1.20075	0.600375	+	0.799719i	$0.295018\pi$
$432$	0	0
$433$	30.3923	1.46056	0.730280	−	0.683147i	$-0.239389\pi$
$433$	30.3923	1.46056	0.730280	+	0.683147i	$0.239389\pi$
$434$	0	0
$435$	2.53590	0.121587
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−19.3205	−0.922118	−0.461059	−	0.887370i	$-0.652530\pi$
$439$	−19.3205	−0.922118	−0.461059	+	0.887370i	$0.652530\pi$
$440$	0	0
$441$	−2.46410	−0.117338
$442$	0	0
$443$	7.60770	0.361453	0.180726	−	0.983533i	$-0.442155\pi$
$443$	7.60770	0.361453	0.180726	+	0.983533i	$0.442155\pi$
$444$	0	0
$445$	−12.9282	−0.612856
$446$	0	0
$447$	−0.679492	−0.0321389
$448$	0	0
$449$	16.3923	0.773601	0.386800	−	0.922163i	$-0.373580\pi$
$449$	16.3923	0.773601	0.386800	+	0.922163i	$0.373580\pi$
$450$	0	0
$451$	−8.19615	−0.385942
$452$	0	0
$453$	12.3923	0.582241
$454$	0	0
$455$	−2.73205	−0.128081
$456$	0	0
$457$	30.3923	1.42169	0.710846	−	0.703348i	$-0.248312\pi$
$457$	30.3923	1.42169	0.710846	+	0.703348i	$0.248312\pi$
$458$	0	0
$459$	−5.07180	−0.236731
$460$	0	0
$461$	−12.5885	−0.586303	−0.293151	−	0.956066i	$-0.594704\pi$
$461$	−12.5885	−0.586303	−0.293151	+	0.956066i	$0.594704\pi$
$462$	0	0
$463$	34.9282	1.62325	0.811626	−	0.584178i	$-0.198583\pi$
$463$	34.9282	1.62325	0.811626	+	0.584178i	$0.198583\pi$
$464$	0	0
$465$	−4.53590	−0.210347
$466$	0	0
$467$	−2.87564	−0.133069	−0.0665345	−	0.997784i	$-0.521194\pi$
$467$	−2.87564	−0.133069	−0.0665345	+	0.997784i	$0.521194\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	2.82309	0.130081
$472$	0	0
$473$	2.00000	0.0919601
$474$	0	0
$475$	−5.46410	−0.250710
$476$	0	0
$477$	−23.3205	−1.06777
$478$	0	0
$479$	−37.1769	−1.69866	−0.849328	−	0.527865i	$-0.822993\pi$
$479$	−37.1769	−1.69866	−0.849328	+	0.527865i	$0.822993\pi$
$480$	0	0
$481$	−14.9282	−0.680667
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−18.3923	−0.835152
$486$	0	0
$487$	36.7846	1.66687	0.833435	−	0.552618i	$-0.186371\pi$
$487$	36.7846	1.66687	0.833435	+	0.552618i	$0.186371\pi$
$488$	0	0
$489$	−12.2872	−0.555646
$490$	0	0
$491$	11.0718	0.499663	0.249832	−	0.968289i	$-0.419625\pi$
$491$	11.0718	0.499663	0.249832	+	0.968289i	$0.419625\pi$
$492$	0	0
$493$	−4.39230	−0.197819
$494$	0	0
$495$	−2.46410	−0.110753
$496$	0	0
$497$	−9.46410	−0.424523
$498$	0	0
$499$	3.32051	0.148646	0.0743232	−	0.997234i	$-0.476320\pi$
$499$	3.32051	0.148646	0.0743232	+	0.997234i	$0.476320\pi$
$500$	0	0
$501$	3.71281	0.165876
$502$	0	0
$503$	25.1769	1.12258	0.561292	−	0.827618i	$-0.310305\pi$
$503$	25.1769	1.12258	0.561292	+	0.827618i	$0.310305\pi$
$504$	0	0
$505$	−5.66025	−0.251878
$506$	0	0
$507$	−4.05256	−0.179980
$508$	0	0
$509$	−12.2487	−0.542915	−0.271457	−	0.962450i	$-0.587506\pi$
$509$	−12.2487	−0.542915	−0.271457	+	0.962450i	$0.587506\pi$
$510$	0	0
$511$	0.196152	0.00867727
$512$	0	0
$513$	21.8564	0.964984
$514$	0	0
$515$	−1.80385	−0.0794870
$516$	0	0
$517$	11.6603	0.512817
$518$	0	0
$519$	12.9282	0.567485
$520$	0	0
$521$	34.3923	1.50675	0.753377	−	0.657589i	$-0.228424\pi$
$521$	34.3923	1.50675	0.753377	+	0.657589i	$0.228424\pi$
$522$	0	0
$523$	−16.7846	−0.733940	−0.366970	−	0.930233i	$-0.619605\pi$
$523$	−16.7846	−0.733940	−0.366970	+	0.930233i	$0.619605\pi$
$524$	0	0
$525$	−0.732051	−0.0319493
$526$	0	0
$527$	7.85641	0.342230
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	17.9090	0.777183
$532$	0	0
$533$	−22.3923	−0.969918
$534$	0	0
$535$	6.92820	0.299532
$536$	0	0
$537$	15.2154	0.656593
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−11.8564	−0.509747	−0.254873	−	0.966974i	$-0.582034\pi$
$541$	−11.8564	−0.509747	−0.254873	+	0.966974i	$0.582034\pi$
$542$	0	0
$543$	4.67949	0.200816
$544$	0	0
$545$	−11.4641	−0.491068
$546$	0	0
$547$	−28.7846	−1.23074	−0.615371	−	0.788238i	$-0.710994\pi$
$547$	−28.7846	−1.23074	−0.615371	+	0.788238i	$0.710994\pi$
$548$	0	0
$549$	12.9808	0.554005
$550$	0	0
$551$	18.9282	0.806369
$552$	0	0
$553$	2.00000	0.0850487
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	22.3923	0.948792	0.474396	−	0.880311i	$-0.342666\pi$
$557$	22.3923	0.948792	0.474396	+	0.880311i	$0.342666\pi$
$558$	0	0
$559$	5.46410	0.231107
$560$	0	0
$561$	−0.928203	−0.0391888
$562$	0	0
$563$	9.46410	0.398864	0.199432	−	0.979912i	$-0.436090\pi$
$563$	9.46410	0.398864	0.199432	+	0.979912i	$0.436090\pi$
$564$	0	0
$565$	−0.928203	−0.0390498
$566$	0	0
$567$	−4.46410	−0.187475
$568$	0	0
$569$	43.1769	1.81007	0.905035	−	0.425337i	$-0.139844\pi$
$569$	43.1769	1.81007	0.905035	+	0.425337i	$0.139844\pi$
$570$	0	0
$571$	−37.5692	−1.57222	−0.786111	−	0.618085i	$-0.787909\pi$
$571$	−37.5692	−1.57222	−0.786111	+	0.618085i	$0.787909\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−47.1769	−1.96400	−0.982000	−	0.188879i	$-0.939515\pi$
$577$	−47.1769	−1.96400	−0.982000	+	0.188879i	$0.939515\pi$
$578$	0	0
$579$	−10.5359	−0.437857
$580$	0	0
$581$	−4.39230	−0.182224
$582$	0	0
$583$	−9.46410	−0.391963
$584$	0	0
$585$	−6.73205	−0.278336
$586$	0	0
$587$	−21.8038	−0.899941	−0.449971	−	0.893043i	$-0.648566\pi$
$587$	−21.8038	−0.899941	−0.449971	+	0.893043i	$0.648566\pi$
$588$	0	0
$589$	−33.8564	−1.39503
$590$	0	0
$591$	11.3205	0.465663
$592$	0	0
$593$	39.1244	1.60664	0.803322	−	0.595544i	$-0.203064\pi$
$593$	39.1244	1.60664	0.803322	+	0.595544i	$0.203064\pi$
$594$	0	0
$595$	1.26795	0.0519808
$596$	0	0
$597$	3.17691	0.130022
$598$	0	0
$599$	−16.3923	−0.669771	−0.334886	−	0.942259i	$-0.608698\pi$
$599$	−16.3923	−0.669771	−0.334886	+	0.942259i	$0.608698\pi$
$600$	0	0
$601$	2.33975	0.0954402	0.0477201	−	0.998861i	$-0.484804\pi$
$601$	2.33975	0.0954402	0.0477201	+	0.998861i	$0.484804\pi$
$602$	0	0
$603$	−9.85641	−0.401384
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−21.8564	−0.887124	−0.443562	−	0.896244i	$-0.646285\pi$
$607$	−21.8564	−0.887124	−0.443562	+	0.896244i	$0.646285\pi$
$608$	0	0
$609$	2.53590	0.102760
$610$	0	0
$611$	31.8564	1.28877
$612$	0	0
$613$	34.7846	1.40494	0.702469	−	0.711715i	$-0.252081\pi$
$613$	34.7846	1.40494	0.702469	+	0.711715i	$0.252081\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	0.679492	0.0273553	0.0136777	−	0.999906i	$-0.495646\pi$
$617$	0.679492	0.0273553	0.0136777	+	0.999906i	$0.495646\pi$
$618$	0	0
$619$	10.5885	0.425586	0.212793	−	0.977097i	$-0.431744\pi$
$619$	10.5885	0.425586	0.212793	+	0.977097i	$0.431744\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.9282	−0.517958
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.00000	0.159745
$628$	0	0
$629$	6.92820	0.276246
$630$	0	0
$631$	−38.9282	−1.54971	−0.774854	−	0.632141i	$-0.782176\pi$
$631$	−38.9282	−1.54971	−0.774854	+	0.632141i	$0.782176\pi$
$632$	0	0
$633$	16.7846	0.667128
$634$	0	0
$635$	2.92820	0.116202
$636$	0	0
$637$	−2.73205	−0.108248
$638$	0	0
$639$	−23.3205	−0.922545
$640$	0	0
$641$	37.1769	1.46840	0.734200	−	0.678933i	$-0.237557\pi$
$641$	37.1769	1.46840	0.734200	+	0.678933i	$0.237557\pi$
$642$	0	0
$643$	−17.1244	−0.675319	−0.337659	−	0.941268i	$-0.609635\pi$
$643$	−17.1244	−0.675319	−0.337659	+	0.941268i	$0.609635\pi$
$644$	0	0
$645$	1.46410	0.0576489
$646$	0	0
$647$	20.4449	0.803771	0.401885	−	0.915690i	$-0.368355\pi$
$647$	20.4449	0.803771	0.401885	+	0.915690i	$0.368355\pi$
$648$	0	0
$649$	7.26795	0.285292
$650$	0	0
$651$	−4.53590	−0.177776
$652$	0	0
$653$	4.14359	0.162151	0.0810757	−	0.996708i	$-0.474164\pi$
$653$	4.14359	0.162151	0.0810757	+	0.996708i	$0.474164\pi$
$654$	0	0
$655$	18.9282	0.739586
$656$	0	0
$657$	0.483340	0.0188569
$658$	0	0
$659$	19.8564	0.773496	0.386748	−	0.922185i	$-0.373598\pi$
$659$	19.8564	0.773496	0.386748	+	0.922185i	$0.373598\pi$
$660$	0	0
$661$	10.7846	0.419473	0.209736	−	0.977758i	$-0.432739\pi$
$661$	10.7846	0.419473	0.209736	+	0.977758i	$0.432739\pi$
$662$	0	0
$663$	−2.53590	−0.0984861
$664$	0	0
$665$	−5.46410	−0.211889
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−2.39230	−0.0924918
$670$	0	0
$671$	5.26795	0.203367
$672$	0	0
$673$	−39.5692	−1.52528	−0.762641	−	0.646822i	$-0.776097\pi$
$673$	−39.5692	−1.52528	−0.762641	+	0.646822i	$0.776097\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	18.3397	0.704854	0.352427	−	0.935839i	$-0.385357\pi$
$677$	18.3397	0.704854	0.352427	+	0.935839i	$0.385357\pi$
$678$	0	0
$679$	−18.3923	−0.705832
$680$	0	0
$681$	−3.71281	−0.142275
$682$	0	0
$683$	−32.1051	−1.22847	−0.614234	−	0.789124i	$-0.710535\pi$
$683$	−32.1051	−1.22847	−0.614234	+	0.789124i	$0.710535\pi$
$684$	0	0
$685$	9.46410	0.361605
$686$	0	0
$687$	−10.5359	−0.401970
$688$	0	0
$689$	−25.8564	−0.985051
$690$	0	0
$691$	1.12436	0.0427725	0.0213863	−	0.999771i	$-0.493192\pi$
$691$	1.12436	0.0427725	0.0213863	+	0.999771i	$0.493192\pi$
$692$	0	0
$693$	−2.46410	−0.0936035
$694$	0	0
$695$	−1.46410	−0.0555365
$696$	0	0
$697$	10.3923	0.393637
$698$	0	0
$699$	8.10512	0.306564
$700$	0	0
$701$	36.2487	1.36909	0.684547	−	0.728968i	$-0.260000\pi$
$701$	36.2487	1.36909	0.684547	+	0.728968i	$0.260000\pi$
$702$	0	0
$703$	−29.8564	−1.12606
$704$	0	0
$705$	8.53590	0.321481
$706$	0	0
$707$	−5.66025	−0.212876
$708$	0	0
$709$	−16.9282	−0.635752	−0.317876	−	0.948132i	$-0.602970\pi$
$709$	−16.9282	−0.635752	−0.317876	+	0.948132i	$0.602970\pi$
$710$	0	0
$711$	4.92820	0.184822
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−2.73205	−0.102173
$716$	0	0
$717$	3.71281	0.138658
$718$	0	0
$719$	20.4449	0.762465	0.381232	−	0.924479i	$-0.375500\pi$
$719$	20.4449	0.762465	0.381232	+	0.924479i	$0.375500\pi$
$720$	0	0
$721$	−1.80385	−0.0671788
$722$	0	0
$723$	−15.8564	−0.589706
$724$	0	0
$725$	−3.46410	−0.128654
$726$	0	0
$727$	31.3731	1.16356	0.581781	−	0.813345i	$-0.302356\pi$
$727$	31.3731	1.16356	0.581781	+	0.813345i	$0.302356\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−2.53590	−0.0937936
$732$	0	0
$733$	−16.5885	−0.612709	−0.306354	−	0.951918i	$-0.599109\pi$
$733$	−16.5885	−0.612709	−0.306354	+	0.951918i	$0.599109\pi$
$734$	0	0
$735$	−0.732051	−0.0270021
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−25.0718	−0.922281	−0.461140	−	0.887327i	$-0.652560\pi$
$739$	−25.0718	−0.922281	−0.461140	+	0.887327i	$0.652560\pi$
$740$	0	0
$741$	10.9282	0.401458
$742$	0	0
$743$	32.7846	1.20275	0.601375	−	0.798967i	$-0.294620\pi$
$743$	32.7846	1.20275	0.601375	+	0.798967i	$0.294620\pi$
$744$	0	0
$745$	0.928203	0.0340067
$746$	0	0
$747$	−10.8231	−0.395996
$748$	0	0
$749$	6.92820	0.253151
$750$	0	0
$751$	−45.1769	−1.64853	−0.824265	−	0.566205i	$-0.808411\pi$
$751$	−45.1769	−1.64853	−0.824265	+	0.566205i	$0.808411\pi$
$752$	0	0
$753$	−4.82309	−0.175763
$754$	0	0
$755$	−16.9282	−0.616080
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.3397	0.664815	0.332408	−	0.943136i	$-0.392139\pi$
$761$	18.3397	0.664815	0.332408	+	0.943136i	$0.392139\pi$
$762$	0	0
$763$	−11.4641	−0.415028
$764$	0	0
$765$	3.12436	0.112961
$766$	0	0
$767$	19.8564	0.716973
$768$	0	0
$769$	20.5885	0.742439	0.371219	−	0.928545i	$-0.378940\pi$
$769$	20.5885	0.742439	0.371219	+	0.928545i	$0.378940\pi$
$770$	0	0
$771$	23.3205	0.839868
$772$	0	0
$773$	−12.2487	−0.440556	−0.220278	−	0.975437i	$-0.570696\pi$
$773$	−12.2487	−0.440556	−0.220278	+	0.975437i	$0.570696\pi$
$774$	0	0
$775$	6.19615	0.222572
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	−44.7846	−1.60458
$780$	0	0
$781$	−9.46410	−0.338652
$782$	0	0
$783$	13.8564	0.495188
$784$	0	0
$785$	−3.85641	−0.137641
$786$	0	0
$787$	3.32051	0.118363	0.0591817	−	0.998247i	$-0.481151\pi$
$787$	3.32051	0.118363	0.0591817	+	0.998247i	$0.481151\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.928203	−0.0330031
$792$	0	0
$793$	14.3923	0.511086
$794$	0	0
$795$	−6.92820	−0.245718
$796$	0	0
$797$	−9.21539	−0.326426	−0.163213	−	0.986591i	$-0.552186\pi$
$797$	−9.21539	−0.326426	−0.163213	+	0.986591i	$0.552186\pi$
$798$	0	0
$799$	−14.7846	−0.523042
$800$	0	0
$801$	−31.8564	−1.12559
$802$	0	0
$803$	0.196152	0.00692207
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.60770	−0.267804
$808$	0	0
$809$	9.21539	0.323996	0.161998	−	0.986791i	$-0.448206\pi$
$809$	9.21539	0.323996	0.161998	+	0.986791i	$0.448206\pi$
$810$	0	0
$811$	−18.1436	−0.637108	−0.318554	−	0.947905i	$-0.603197\pi$
$811$	−18.1436	−0.637108	−0.318554	+	0.947905i	$0.603197\pi$
$812$	0	0
$813$	14.9282	0.523555
$814$	0	0
$815$	16.7846	0.587939
$816$	0	0
$817$	10.9282	0.382329
$818$	0	0
$819$	−6.73205	−0.235237
$820$	0	0
$821$	31.8564	1.11180	0.555898	−	0.831250i	$-0.312374\pi$
$821$	31.8564	1.11180	0.555898	+	0.831250i	$0.312374\pi$
$822$	0	0
$823$	39.3205	1.37063	0.685313	−	0.728248i	$-0.259665\pi$
$823$	39.3205	1.37063	0.685313	+	0.728248i	$0.259665\pi$
$824$	0	0
$825$	−0.732051	−0.0254867
$826$	0	0
$827$	−31.8564	−1.10776	−0.553878	−	0.832598i	$-0.686853\pi$
$827$	−31.8564	−1.10776	−0.553878	+	0.832598i	$0.686853\pi$
$828$	0	0
$829$	−11.8564	−0.411790	−0.205895	−	0.978574i	$-0.566011\pi$
$829$	−11.8564	−0.411790	−0.205895	+	0.978574i	$0.566011\pi$
$830$	0	0
$831$	18.5359	0.643003
$832$	0	0
$833$	1.26795	0.0439318
$834$	0	0
$835$	−5.07180	−0.175517
$836$	0	0
$837$	−24.7846	−0.856681
$838$	0	0
$839$	25.5167	0.880933	0.440466	−	0.897769i	$-0.354813\pi$
$839$	25.5167	0.880933	0.440466	+	0.897769i	$0.354813\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	−16.3923	−0.564581
$844$	0	0
$845$	5.53590	0.190441
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	4.28719	0.147136
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−25.3731	−0.868757	−0.434379	−	0.900730i	$-0.643032\pi$
$853$	−25.3731	−0.868757	−0.434379	+	0.900730i	$0.643032\pi$
$854$	0	0
$855$	−13.4641	−0.460463
$856$	0	0
$857$	5.66025	0.193351	0.0966753	−	0.995316i	$-0.469179\pi$
$857$	5.66025	0.193351	0.0966753	+	0.995316i	$0.469179\pi$
$858$	0	0
$859$	42.1962	1.43971	0.719857	−	0.694122i	$-0.244207\pi$
$859$	42.1962	1.43971	0.719857	+	0.694122i	$0.244207\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	21.4641	0.730647	0.365323	−	0.930881i	$-0.380958\pi$
$863$	21.4641	0.730647	0.365323	+	0.930881i	$0.380958\pi$
$864$	0	0
$865$	−17.6603	−0.600467
$866$	0	0
$867$	−11.2679	−0.382680
$868$	0	0
$869$	2.00000	0.0678454
$870$	0	0
$871$	−10.9282	−0.370288
$872$	0	0
$873$	−45.3205	−1.53387
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	5.71281	0.192908	0.0964540	−	0.995337i	$-0.469250\pi$
$877$	5.71281	0.192908	0.0964540	+	0.995337i	$0.469250\pi$
$878$	0	0
$879$	−4.14359	−0.139760
$880$	0	0
$881$	20.5359	0.691872	0.345936	−	0.938258i	$-0.387561\pi$
$881$	20.5359	0.691872	0.345936	+	0.938258i	$0.387561\pi$
$882$	0	0
$883$	20.3923	0.686256	0.343128	−	0.939289i	$-0.388514\pi$
$883$	20.3923	0.686256	0.343128	+	0.939289i	$0.388514\pi$
$884$	0	0
$885$	5.32051	0.178847
$886$	0	0
$887$	−35.3205	−1.18595	−0.592973	−	0.805222i	$-0.702046\pi$
$887$	−35.3205	−1.18595	−0.592973	+	0.805222i	$0.702046\pi$
$888$	0	0
$889$	2.92820	0.0982088
$890$	0	0
$891$	−4.46410	−0.149553
$892$	0	0
$893$	63.7128	2.13207
$894$	0	0
$895$	−20.7846	−0.694753
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−21.4641	−0.715868
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	1.46410	0.0487223
$904$	0	0
$905$	−6.39230	−0.212487
$906$	0	0
$907$	14.1436	0.469630	0.234815	−	0.972040i	$-0.424552\pi$
$907$	14.1436	0.469630	0.234815	+	0.972040i	$0.424552\pi$
$908$	0	0
$909$	−13.9474	−0.462607
$910$	0	0
$911$	54.2487	1.79734	0.898670	−	0.438625i	$-0.144534\pi$
$911$	54.2487	1.79734	0.898670	+	0.438625i	$0.144534\pi$
$912$	0	0
$913$	−4.39230	−0.145364
$914$	0	0
$915$	3.85641	0.127489
$916$	0	0
$917$	18.9282	0.625064
$918$	0	0
$919$	24.7846	0.817569	0.408784	−	0.912631i	$-0.365953\pi$
$919$	24.7846	0.817569	0.408784	+	0.912631i	$0.365953\pi$
$920$	0	0
$921$	−14.6410	−0.482438
$922$	0	0
$923$	−25.8564	−0.851074
$924$	0	0
$925$	5.46410	0.179659
$926$	0	0
$927$	−4.44486	−0.145988
$928$	0	0
$929$	−34.3923	−1.12837	−0.564187	−	0.825647i	$-0.690810\pi$
$929$	−34.3923	−1.12837	−0.564187	+	0.825647i	$0.690810\pi$
$930$	0	0
$931$	−5.46410	−0.179079
$932$	0	0
$933$	15.4641	0.506272
$934$	0	0
$935$	1.26795	0.0414664
$936$	0	0
$937$	12.9808	0.424063	0.212032	−	0.977263i	$-0.431992\pi$
$937$	12.9808	0.424063	0.212032	+	0.977263i	$0.431992\pi$
$938$	0	0
$939$	1.46410	0.0477792
$940$	0	0
$941$	−16.9808	−0.553557	−0.276779	−	0.960934i	$-0.589267\pi$
$941$	−16.9808	−0.553557	−0.276779	+	0.960934i	$0.589267\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	56.1051	1.82317	0.911586	−	0.411110i	$-0.134859\pi$
$947$	56.1051	1.82317	0.911586	+	0.411110i	$0.134859\pi$
$948$	0	0
$949$	0.535898	0.0173960
$950$	0	0
$951$	4.39230	0.142430
$952$	0	0
$953$	3.46410	0.112213	0.0561066	−	0.998425i	$-0.482131\pi$
$953$	3.46410	0.112213	0.0561066	+	0.998425i	$0.482131\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.53590	0.0819740
$958$	0	0
$959$	9.46410	0.305612
$960$	0	0
$961$	7.39230	0.238461
$962$	0	0
$963$	17.0718	0.550131
$964$	0	0
$965$	14.3923	0.463305
$966$	0	0
$967$	−7.07180	−0.227414	−0.113707	−	0.993514i	$-0.536272\pi$
$967$	−7.07180	−0.227414	−0.113707	+	0.993514i	$0.536272\pi$
$968$	0	0
$969$	−5.07180	−0.162930
$970$	0	0
$971$	−52.0526	−1.67045	−0.835223	−	0.549911i	$-0.814662\pi$
$971$	−52.0526	−1.67045	−0.835223	+	0.549911i	$0.814662\pi$
$972$	0	0
$973$	−1.46410	−0.0469369
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	16.1436	0.516479	0.258240	−	0.966081i	$-0.416858\pi$
$977$	16.1436	0.516479	0.258240	+	0.966081i	$0.416858\pi$
$978$	0	0
$979$	−12.9282	−0.413187
$980$	0	0
$981$	−28.2487	−0.901912
$982$	0	0
$983$	−54.5885	−1.74110	−0.870551	−	0.492079i	$-0.836237\pi$
$983$	−54.5885	−1.74110	−0.870551	+	0.492079i	$0.836237\pi$
$984$	0	0
$985$	−15.4641	−0.492727
$986$	0	0
$987$	8.53590	0.271701
$988$	0	0
$989$	0	0
$990$	0	0
$991$	44.3923	1.41017	0.705084	−	0.709124i	$-0.250909\pi$
$991$	44.3923	1.41017	0.705084	+	0.709124i	$0.250909\pi$
$992$	0	0
$993$	11.7128	0.371695
$994$	0	0
$995$	−4.33975	−0.137579
$996$	0	0
$997$	1.66025	0.0525808	0.0262904	−	0.999654i	$-0.491631\pi$
$997$	1.66025	0.0525808	0.0262904	+	0.999654i	$0.491631\pi$
$998$	0	0
$999$	−21.8564	−0.691506

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.s.1.2		2
4.3	odd	2		1540.2.a.e.1.1	✓	2
20.3	even	4		7700.2.e.l.1849.2		4
20.7	even	4		7700.2.e.l.1849.3		4
20.19	odd	2		7700.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1540.2.a.e.1.1	✓	2	4.3	odd	2
6160.2.a.s.1.2		2	1.1	even	1	trivial
7700.2.a.n.1.2		2	20.19	odd	2
7700.2.e.l.1849.2		4	20.3	even	4
7700.2.e.l.1849.3		4	20.7	even	4

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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q+0.732051 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} -1.00000 q^{11} -2.73205 q^{13} -0.732051 q^{15} +1.26795 q^{17} -5.46410 q^{19} -0.732051 q^{21} +1.00000 q^{25} -4.00000 q^{27} -3.46410 q^{29} +6.19615 q^{31} -0.732051 q^{33} +1.00000 q^{35} +5.46410 q^{37} -2.00000 q^{39} +8.19615 q^{41} -2.00000 q^{43} +2.46410 q^{45} -11.6603 q^{47} +1.00000 q^{49} +0.928203 q^{51} +9.46410 q^{53} +1.00000 q^{55} -4.00000 q^{57} -7.26795 q^{59} -5.26795 q^{61} +2.46410 q^{63} +2.73205 q^{65} +4.00000 q^{67} +9.46410 q^{71} -0.196152 q^{73} +0.732051 q^{75} +1.00000 q^{77} -2.00000 q^{79} +4.46410 q^{81} +4.39230 q^{83} -1.26795 q^{85} -2.53590 q^{87} +12.9282 q^{89} +2.73205 q^{91} +4.53590 q^{93} +5.46410 q^{95} +18.3923 q^{97} +2.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} + 2 q^{25} - 8 q^{27} + 2 q^{31} + 2 q^{33} + 2 q^{35} + 4 q^{37} - 4 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} - 6 q^{47} + 2 q^{49} - 12 q^{51} + 12 q^{53} + 2 q^{55} - 8 q^{57} - 18 q^{59} - 14 q^{61} - 2 q^{63} + 2 q^{65} + 8 q^{67} + 12 q^{71} + 10 q^{73} - 2 q^{75} + 2 q^{77} - 4 q^{79} + 2 q^{81} - 12 q^{83} - 6 q^{85} - 12 q^{87} + 12 q^{89} + 2 q^{91} + 16 q^{93} + 4 q^{95} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 + 2 * q^15 + 6 * q^17 - 4 * q^19 + 2 * q^21 + 2 * q^25 - 8 * q^27 + 2 * q^31 + 2 * q^33 + 2 * q^35 + 4 * q^37 - 4 * q^39 + 6 * q^41 - 4 * q^43 - 2 * q^45 - 6 * q^47 + 2 * q^49 - 12 * q^51 + 12 * q^53 + 2 * q^55 - 8 * q^57 - 18 * q^59 - 14 * q^61 - 2 * q^63 + 2 * q^65 + 8 * q^67 + 12 * q^71 + 10 * q^73 - 2 * q^75 + 2 * q^77 - 4 * q^79 + 2 * q^81 - 12 * q^83 - 6 * q^85 - 12 * q^87 + 12 * q^89 + 2 * q^91 + 16 * q^93 + 4 * q^95 + 16 * q^97 - 2 * q^99

Introduction

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