LMFDB - Embedded newform 6096.2.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6096,2,Mod(1,6096)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6096, 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("6096.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6096 = 2^{4} \cdot 3 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6096.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:	$48.6768050722$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 381)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.71250$ of defining polynomial
Character	$\chi$	$=$	6096.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.98063 q^{5} +2.49235 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.98063 q^{5} +2.49235 q^{7} +1.00000 q^{9} -1.94367 q^{11} -3.49235 q^{13} -2.98063 q^{15} +4.53628 q^{17} -2.24737 q^{19} +2.49235 q^{21} -1.85555 q^{23} +3.88418 q^{25} +1.00000 q^{27} +7.87314 q^{29} +3.36839 q^{31} -1.94367 q^{33} -7.42878 q^{35} -6.58157 q^{37} -3.49235 q^{39} +1.44706 q^{41} -11.7328 q^{43} -2.98063 q^{45} +5.12509 q^{47} -0.788187 q^{49} +4.53628 q^{51} +3.51312 q^{53} +5.79338 q^{55} -2.24737 q^{57} -10.9281 q^{59} +7.59943 q^{61} +2.49235 q^{63} +10.4094 q^{65} -0.605780 q^{67} -1.85555 q^{69} -1.80895 q^{71} -4.42816 q^{73} +3.88418 q^{75} -4.84432 q^{77} +0.536277 q^{79} +1.00000 q^{81} -2.30397 q^{83} -13.5210 q^{85} +7.87314 q^{87} -8.38896 q^{89} -8.70416 q^{91} +3.36839 q^{93} +6.69860 q^{95} -13.1852 q^{97} -1.94367 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + q^{5} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + q^5 + 5 * q^9	$ 5 q + 5 q^{3} + q^{5} + 5 q^{9} - 14 q^{11} - 5 q^{13} + q^{15} + 4 q^{17} - 4 q^{19} - 15 q^{23} - 6 q^{25} + 5 q^{27} + 9 q^{29} - 3 q^{31} - 14 q^{33} - 4 q^{35} - 5 q^{37} - 5 q^{39} + 4 q^{41} - 10 q^{43} + q^{45} + 4 q^{47} - 9 q^{49} + 4 q^{51} + 3 q^{53} - 4 q^{55} - 4 q^{57} - 23 q^{59} - 15 q^{61} + 3 q^{65} - 18 q^{67} - 15 q^{69} - 12 q^{71} - 43 q^{73} - 6 q^{75} + 4 q^{77} - 16 q^{79} + 5 q^{81} - 11 q^{83} - 24 q^{85} + 9 q^{87} + 9 q^{89} - 26 q^{91} - 3 q^{93} - 16 q^{95} - 20 q^{97} - 14 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + q^5 + 5 * q^9 - 14 * q^11 - 5 * q^13 + q^15 + 4 * q^17 - 4 * q^19 - 15 * q^23 - 6 * q^25 + 5 * q^27 + 9 * q^29 - 3 * q^31 - 14 * q^33 - 4 * q^35 - 5 * q^37 - 5 * q^39 + 4 * q^41 - 10 * q^43 + q^45 + 4 * q^47 - 9 * q^49 + 4 * q^51 + 3 * q^53 - 4 * q^55 - 4 * q^57 - 23 * q^59 - 15 * q^61 + 3 * q^65 - 18 * q^67 - 15 * q^69 - 12 * q^71 - 43 * q^73 - 6 * q^75 + 4 * q^77 - 16 * q^79 + 5 * q^81 - 11 * q^83 - 24 * q^85 + 9 * q^87 + 9 * q^89 - 26 * q^91 - 3 * q^93 - 16 * q^95 - 20 * q^97 - 14 * 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
$5$	−2.98063	−1.33298	−0.666490	−	0.745514i	$-0.732204\pi$
$5$	−2.98063	−1.33298	−0.666490	+	0.745514i	$0.732204\pi$
$6$	0	0
$7$	2.49235	0.942020	0.471010	−	0.882128i	$-0.343890\pi$
$7$	2.49235	0.942020	0.471010	+	0.882128i	$0.343890\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.94367	−0.586040	−0.293020	−	0.956106i	$-0.594660\pi$
$11$	−1.94367	−0.586040	−0.293020	+	0.956106i	$0.594660\pi$
$12$	0	0
$13$	−3.49235	−0.968604	−0.484302	−	0.874901i	$-0.660926\pi$
$13$	−3.49235	−0.968604	−0.484302	+	0.874901i	$0.660926\pi$
$14$	0	0
$15$	−2.98063	−0.769596
$16$	0	0
$17$	4.53628	1.10021	0.550104	−	0.835096i	$-0.314588\pi$
$17$	4.53628	1.10021	0.550104	+	0.835096i	$0.314588\pi$
$18$	0	0
$19$	−2.24737	−0.515583	−0.257791	−	0.966201i	$-0.582995\pi$
$19$	−2.24737	−0.515583	−0.257791	+	0.966201i	$0.582995\pi$
$20$	0	0
$21$	2.49235	0.543876
$22$	0	0
$23$	−1.85555	−0.386909	−0.193454	−	0.981109i	$-0.561969\pi$
$23$	−1.85555	−0.386909	−0.193454	+	0.981109i	$0.561969\pi$
$24$	0	0
$25$	3.88418	0.776835
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.87314	1.46201	0.731003	−	0.682374i	$-0.239053\pi$
$29$	7.87314	1.46201	0.731003	+	0.682374i	$0.239053\pi$
$30$	0	0
$31$	3.36839	0.604981	0.302490	−	0.953152i	$-0.402182\pi$
$31$	3.36839	0.604981	0.302490	+	0.953152i	$0.402182\pi$
$32$	0	0
$33$	−1.94367	−0.338350
$34$	0	0
$35$	−7.42878	−1.25569
$36$	0	0
$37$	−6.58157	−1.08200	−0.541002	−	0.841022i	$-0.681955\pi$
$37$	−6.58157	−1.08200	−0.541002	+	0.841022i	$0.681955\pi$
$38$	0	0
$39$	−3.49235	−0.559224
$40$	0	0
$41$	1.44706	0.225993	0.112996	−	0.993595i	$-0.463955\pi$
$41$	1.44706	0.225993	0.112996	+	0.993595i	$0.463955\pi$
$42$	0	0
$43$	−11.7328	−1.78923	−0.894617	−	0.446834i	$-0.852551\pi$
$43$	−11.7328	−1.78923	−0.894617	+	0.446834i	$0.852551\pi$
$44$	0	0
$45$	−2.98063	−0.444327
$46$	0	0
$47$	5.12509	0.747571	0.373785	−	0.927515i	$-0.378060\pi$
$47$	5.12509	0.747571	0.373785	+	0.927515i	$0.378060\pi$
$48$	0	0
$49$	−0.788187	−0.112598
$50$	0	0
$51$	4.53628	0.635206
$52$	0	0
$53$	3.51312	0.482564	0.241282	−	0.970455i	$-0.422432\pi$
$53$	3.51312	0.482564	0.241282	+	0.970455i	$0.422432\pi$
$54$	0	0
$55$	5.79338	0.781179
$56$	0	0
$57$	−2.24737	−0.297672
$58$	0	0
$59$	−10.9281	−1.42272	−0.711359	−	0.702829i	$-0.751920\pi$
$59$	−10.9281	−1.42272	−0.711359	+	0.702829i	$0.751920\pi$
$60$	0	0
$61$	7.59943	0.973008	0.486504	−	0.873678i	$-0.338272\pi$
$61$	7.59943	0.973008	0.486504	+	0.873678i	$0.338272\pi$
$62$	0	0
$63$	2.49235	0.314007
$64$	0	0
$65$	10.4094	1.29113
$66$	0	0
$67$	−0.605780	−0.0740078	−0.0370039	−	0.999315i	$-0.511781\pi$
$67$	−0.605780	−0.0740078	−0.0370039	+	0.999315i	$0.511781\pi$
$68$	0	0
$69$	−1.85555	−0.223382
$70$	0	0
$71$	−1.80895	−0.214683	−0.107342	−	0.994222i	$-0.534234\pi$
$71$	−1.80895	−0.214683	−0.107342	+	0.994222i	$0.534234\pi$
$72$	0	0
$73$	−4.42816	−0.518277	−0.259139	−	0.965840i	$-0.583439\pi$
$73$	−4.42816	−0.518277	−0.259139	+	0.965840i	$0.583439\pi$
$74$	0	0
$75$	3.88418	0.448506
$76$	0	0
$77$	−4.84432	−0.552061
$78$	0	0
$79$	0.536277	0.0603359	0.0301679	−	0.999545i	$-0.490396\pi$
$79$	0.536277	0.0603359	0.0301679	+	0.999545i	$0.490396\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.30397	−0.252894	−0.126447	−	0.991973i	$-0.540357\pi$
$83$	−2.30397	−0.252894	−0.126447	+	0.991973i	$0.540357\pi$
$84$	0	0
$85$	−13.5210	−1.46656
$86$	0	0
$87$	7.87314	0.844089
$88$	0	0
$89$	−8.38896	−0.889228	−0.444614	−	0.895722i	$-0.646659\pi$
$89$	−8.38896	−0.889228	−0.444614	+	0.895722i	$0.646659\pi$
$90$	0	0
$91$	−8.70416	−0.912444
$92$	0	0
$93$	3.36839	0.349286
$94$	0	0
$95$	6.69860	0.687261
$96$	0	0
$97$	−13.1852	−1.33875	−0.669377	−	0.742923i	$-0.733439\pi$
$97$	−13.1852	−1.33875	−0.669377	+	0.742923i	$0.733439\pi$
$98$	0	0
$99$	−1.94367	−0.195347
$100$	0	0
$101$	2.48023	0.246792	0.123396	−	0.992358i	$-0.460622\pi$
$101$	2.48023	0.246792	0.123396	+	0.992358i	$0.460622\pi$
$102$	0	0
$103$	−8.99229	−0.886037	−0.443018	−	0.896513i	$-0.646092\pi$
$103$	−8.99229	−0.886037	−0.443018	+	0.896513i	$0.646092\pi$
$104$	0	0
$105$	−7.42878	−0.724975
$106$	0	0
$107$	−0.866778	−0.0837946	−0.0418973	−	0.999122i	$-0.513340\pi$
$107$	−0.866778	−0.0837946	−0.0418973	+	0.999122i	$0.513340\pi$
$108$	0	0
$109$	9.87287	0.945649	0.472825	−	0.881157i	$-0.343234\pi$
$109$	9.87287	0.945649	0.472825	+	0.881157i	$0.343234\pi$
$110$	0	0
$111$	−6.58157	−0.624695
$112$	0	0
$113$	16.0821	1.51287	0.756436	−	0.654067i	$-0.226939\pi$
$113$	16.0821	1.51287	0.756436	+	0.654067i	$0.226939\pi$
$114$	0	0
$115$	5.53071	0.515741
$116$	0	0
$117$	−3.49235	−0.322868
$118$	0	0
$119$	11.3060	1.03642
$120$	0	0
$121$	−7.22213	−0.656557
$122$	0	0
$123$	1.44706	0.130477
$124$	0	0
$125$	3.32586	0.297474
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	0	0
$129$	−11.7328	−1.03301
$130$	0	0
$131$	−17.6006	−1.53777	−0.768884	−	0.639388i	$-0.779188\pi$
$131$	−17.6006	−1.53777	−0.768884	+	0.639388i	$0.779188\pi$
$132$	0	0
$133$	−5.60124	−0.485689
$134$	0	0
$135$	−2.98063	−0.256532
$136$	0	0
$137$	1.92188	0.164197	0.0820986	−	0.996624i	$-0.473838\pi$
$137$	1.92188	0.164197	0.0820986	+	0.996624i	$0.473838\pi$
$138$	0	0
$139$	−10.8500	−0.920283	−0.460142	−	0.887846i	$-0.652201\pi$
$139$	−10.8500	−0.920283	−0.460142	+	0.887846i	$0.652201\pi$
$140$	0	0
$141$	5.12509	0.431610
$142$	0	0
$143$	6.78799	0.567640
$144$	0	0
$145$	−23.4669	−1.94882
$146$	0	0
$147$	−0.788187	−0.0650086
$148$	0	0
$149$	−22.5796	−1.84979	−0.924896	−	0.380219i	$-0.875849\pi$
$149$	−22.5796	−1.84979	−0.924896	+	0.380219i	$0.875849\pi$
$150$	0	0
$151$	−22.5109	−1.83191	−0.915954	−	0.401282i	$-0.868565\pi$
$151$	−22.5109	−1.83191	−0.915954	+	0.401282i	$0.868565\pi$
$152$	0	0
$153$	4.53628	0.366736
$154$	0	0
$155$	−10.0399	−0.806427
$156$	0	0
$157$	−15.7880	−1.26002	−0.630011	−	0.776586i	$-0.716950\pi$
$157$	−15.7880	−1.26002	−0.630011	+	0.776586i	$0.716950\pi$
$158$	0	0
$159$	3.51312	0.278608
$160$	0	0
$161$	−4.62468	−0.364476
$162$	0	0
$163$	−17.0517	−1.33559	−0.667796	−	0.744345i	$-0.732762\pi$
$163$	−17.0517	−1.33559	−0.667796	+	0.744345i	$0.732762\pi$
$164$	0	0
$165$	5.79338	0.451014
$166$	0	0
$167$	8.95076	0.692631	0.346315	−	0.938118i	$-0.387433\pi$
$167$	8.95076	0.692631	0.346315	+	0.938118i	$0.387433\pi$
$168$	0	0
$169$	−0.803486	−0.0618066
$170$	0	0
$171$	−2.24737	−0.171861
$172$	0	0
$173$	−5.32614	−0.404939	−0.202469	−	0.979289i	$-0.564897\pi$
$173$	−5.32614	−0.404939	−0.202469	+	0.979289i	$0.564897\pi$
$174$	0	0
$175$	9.68073	0.731794
$176$	0	0
$177$	−10.9281	−0.821406
$178$	0	0
$179$	−19.5811	−1.46356	−0.731780	−	0.681541i	$-0.761310\pi$
$179$	−19.5811	−1.46356	−0.731780	+	0.681541i	$0.761310\pi$
$180$	0	0
$181$	19.1351	1.42230	0.711152	−	0.703038i	$-0.248174\pi$
$181$	19.1351	1.42230	0.711152	+	0.703038i	$0.248174\pi$
$182$	0	0
$183$	7.59943	0.561766
$184$	0	0
$185$	19.6172	1.44229
$186$	0	0
$187$	−8.81704	−0.644766
$188$	0	0
$189$	2.49235	0.181292
$190$	0	0
$191$	17.2812	1.25042	0.625212	−	0.780455i	$-0.285013\pi$
$191$	17.2812	1.25042	0.625212	+	0.780455i	$0.285013\pi$
$192$	0	0
$193$	18.8248	1.35504	0.677519	−	0.735505i	$-0.263055\pi$
$193$	18.8248	1.35504	0.677519	+	0.735505i	$0.263055\pi$
$194$	0	0
$195$	10.4094	0.745434
$196$	0	0
$197$	21.3799	1.52326	0.761628	−	0.648015i	$-0.224400\pi$
$197$	21.3799	1.52326	0.761628	+	0.648015i	$0.224400\pi$
$198$	0	0
$199$	−26.9170	−1.90809	−0.954047	−	0.299658i	$-0.903127\pi$
$199$	−26.9170	−1.90809	−0.954047	+	0.299658i	$0.903127\pi$
$200$	0	0
$201$	−0.605780	−0.0427284
$202$	0	0
$203$	19.6226	1.37724
$204$	0	0
$205$	−4.31316	−0.301244
$206$	0	0
$207$	−1.85555	−0.128970
$208$	0	0
$209$	4.36816	0.302152
$210$	0	0
$211$	−3.54184	−0.243831	−0.121915	−	0.992541i	$-0.538904\pi$
$211$	−3.54184	−0.243831	−0.121915	+	0.992541i	$0.538904\pi$
$212$	0	0
$213$	−1.80895	−0.123947
$214$	0	0
$215$	34.9711	2.38501
$216$	0	0
$217$	8.39521	0.569904
$218$	0	0
$219$	−4.42816	−0.299228
$220$	0	0
$221$	−15.8423	−1.06567
$222$	0	0
$223$	17.4252	1.16688	0.583438	−	0.812158i	$-0.301707\pi$
$223$	17.4252	1.16688	0.583438	+	0.812158i	$0.301707\pi$
$224$	0	0
$225$	3.88418	0.258945
$226$	0	0
$227$	10.8821	0.722267	0.361134	−	0.932514i	$-0.382390\pi$
$227$	10.8821	0.722267	0.361134	+	0.932514i	$0.382390\pi$
$228$	0	0
$229$	−12.0378	−0.795480	−0.397740	−	0.917498i	$-0.630205\pi$
$229$	−12.0378	−0.795480	−0.397740	+	0.917498i	$0.630205\pi$
$230$	0	0
$231$	−4.84432	−0.318733
$232$	0	0
$233$	−13.3785	−0.876452	−0.438226	−	0.898865i	$-0.644393\pi$
$233$	−13.3785	−0.876452	−0.438226	+	0.898865i	$0.644393\pi$
$234$	0	0
$235$	−15.2760	−0.996496
$236$	0	0
$237$	0.536277	0.0348349
$238$	0	0
$239$	−30.1290	−1.94889	−0.974443	−	0.224637i	$-0.927880\pi$
$239$	−30.1290	−1.94889	−0.974443	+	0.224637i	$0.927880\pi$
$240$	0	0
$241$	26.4367	1.70294	0.851470	−	0.524403i	$-0.175712\pi$
$241$	26.4367	1.70294	0.851470	+	0.524403i	$0.175712\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.34930	0.150091
$246$	0	0
$247$	7.84862	0.499395
$248$	0	0
$249$	−2.30397	−0.146008
$250$	0	0
$251$	21.4096	1.35136	0.675680	−	0.737195i	$-0.263850\pi$
$251$	21.4096	1.35136	0.675680	+	0.737195i	$0.263850\pi$
$252$	0	0
$253$	3.60658	0.226744
$254$	0	0
$255$	−13.5210	−0.846717
$256$	0	0
$257$	−1.84014	−0.114785	−0.0573923	−	0.998352i	$-0.518279\pi$
$257$	−1.84014	−0.114785	−0.0573923	+	0.998352i	$0.518279\pi$
$258$	0	0
$259$	−16.4036	−1.01927
$260$	0	0
$261$	7.87314	0.487335
$262$	0	0
$263$	−2.13565	−0.131690	−0.0658449	−	0.997830i	$-0.520974\pi$
$263$	−2.13565	−0.131690	−0.0658449	+	0.997830i	$0.520974\pi$
$264$	0	0
$265$	−10.4713	−0.643248
$266$	0	0
$267$	−8.38896	−0.513396
$268$	0	0
$269$	−22.0930	−1.34703	−0.673517	−	0.739172i	$-0.735217\pi$
$269$	−22.0930	−1.34703	−0.673517	+	0.739172i	$0.735217\pi$
$270$	0	0
$271$	16.7849	1.01961	0.509804	−	0.860291i	$-0.329718\pi$
$271$	16.7849	1.01961	0.509804	+	0.860291i	$0.329718\pi$
$272$	0	0
$273$	−8.70416	−0.526800
$274$	0	0
$275$	−7.54957	−0.455256
$276$	0	0
$277$	−20.6239	−1.23917	−0.619584	−	0.784931i	$-0.712699\pi$
$277$	−20.6239	−1.23917	−0.619584	+	0.784931i	$0.712699\pi$
$278$	0	0
$279$	3.36839	0.201660
$280$	0	0
$281$	12.9127	0.770308	0.385154	−	0.922852i	$-0.374148\pi$
$281$	12.9127	0.770308	0.385154	+	0.922852i	$0.374148\pi$
$282$	0	0
$283$	−5.12835	−0.304849	−0.152424	−	0.988315i	$-0.548708\pi$
$283$	−5.12835	−0.304849	−0.152424	+	0.988315i	$0.548708\pi$
$284$	0	0
$285$	6.69860	0.396791
$286$	0	0
$287$	3.60658	0.212890
$288$	0	0
$289$	3.57781	0.210459
$290$	0	0
$291$	−13.1852	−0.772930
$292$	0	0
$293$	13.8495	0.809099	0.404550	−	0.914516i	$-0.367428\pi$
$293$	13.8495	0.809099	0.404550	+	0.914516i	$0.367428\pi$
$294$	0	0
$295$	32.5727	1.89645
$296$	0	0
$297$	−1.94367	−0.112783
$298$	0	0
$299$	6.48023	0.374761
$300$	0	0
$301$	−29.2422	−1.68549
$302$	0	0
$303$	2.48023	0.142485
$304$	0	0
$305$	−22.6511	−1.29700
$306$	0	0
$307$	2.97082	0.169554	0.0847769	−	0.996400i	$-0.472982\pi$
$307$	2.97082	0.169554	0.0847769	+	0.996400i	$0.472982\pi$
$308$	0	0
$309$	−8.99229	−0.511554
$310$	0	0
$311$	−20.8146	−1.18029	−0.590145	−	0.807297i	$-0.700929\pi$
$311$	−20.8146	−1.18029	−0.590145	+	0.807297i	$0.700929\pi$
$312$	0	0
$313$	−28.0654	−1.58635	−0.793175	−	0.608994i	$-0.791573\pi$
$313$	−28.0654	−1.58635	−0.793175	+	0.608994i	$0.791573\pi$
$314$	0	0
$315$	−7.42878	−0.418565
$316$	0	0
$317$	17.9162	1.00627	0.503137	−	0.864207i	$-0.332179\pi$
$317$	17.9162	1.00627	0.503137	+	0.864207i	$0.332179\pi$
$318$	0	0
$319$	−15.3028	−0.856793
$320$	0	0
$321$	−0.866778	−0.0483788
$322$	0	0
$323$	−10.1947	−0.567249
$324$	0	0
$325$	−13.5649	−0.752446
$326$	0	0
$327$	9.87287	0.545971
$328$	0	0
$329$	12.7735	0.704226
$330$	0	0
$331$	16.8466	0.925975	0.462988	−	0.886365i	$-0.346777\pi$
$331$	16.8466	0.925975	0.462988	+	0.886365i	$0.346777\pi$
$332$	0	0
$333$	−6.58157	−0.360668
$334$	0	0
$335$	1.80561	0.0986509
$336$	0	0
$337$	−34.3624	−1.87184	−0.935920	−	0.352211i	$-0.885430\pi$
$337$	−34.3624	−1.87184	−0.935920	+	0.352211i	$0.885430\pi$
$338$	0	0
$339$	16.0821	0.873457
$340$	0	0
$341$	−6.54705	−0.354543
$342$	0	0
$343$	−19.4109	−1.04809
$344$	0	0
$345$	5.53071	0.297763
$346$	0	0
$347$	−11.7399	−0.630231	−0.315115	−	0.949053i	$-0.602043\pi$
$347$	−11.7399	−0.630231	−0.315115	+	0.949053i	$0.602043\pi$
$348$	0	0
$349$	−2.81977	−0.150939	−0.0754695	−	0.997148i	$-0.524046\pi$
$349$	−2.81977	−0.150939	−0.0754695	+	0.997148i	$0.524046\pi$
$350$	0	0
$351$	−3.49235	−0.186408
$352$	0	0
$353$	−15.4602	−0.822862	−0.411431	−	0.911441i	$-0.634971\pi$
$353$	−15.4602	−0.822862	−0.411431	+	0.911441i	$0.634971\pi$
$354$	0	0
$355$	5.39182	0.286168
$356$	0	0
$357$	11.3060	0.598377
$358$	0	0
$359$	−22.0346	−1.16294	−0.581472	−	0.813567i	$-0.697523\pi$
$359$	−22.0346	−1.16294	−0.581472	+	0.813567i	$0.697523\pi$
$360$	0	0
$361$	−13.9493	−0.734174
$362$	0	0
$363$	−7.22213	−0.379064
$364$	0	0
$365$	13.1987	0.690853
$366$	0	0
$367$	−16.4982	−0.861199	−0.430599	−	0.902543i	$-0.641698\pi$
$367$	−16.4982	−0.861199	−0.430599	+	0.902543i	$0.641698\pi$
$368$	0	0
$369$	1.44706	0.0753309
$370$	0	0
$371$	8.75592	0.454585
$372$	0	0
$373$	−35.5556	−1.84100	−0.920500	−	0.390743i	$-0.872218\pi$
$373$	−35.5556	−1.84100	−0.920500	+	0.390743i	$0.872218\pi$
$374$	0	0
$375$	3.32586	0.171747
$376$	0	0
$377$	−27.4958	−1.41610
$378$	0	0
$379$	11.0970	0.570013	0.285007	−	0.958526i	$-0.408004\pi$
$379$	11.0970	0.570013	0.285007	+	0.958526i	$0.408004\pi$
$380$	0	0
$381$	−1.00000	−0.0512316
$382$	0	0
$383$	33.8793	1.73115	0.865576	−	0.500777i	$-0.166952\pi$
$383$	33.8793	1.73115	0.865576	+	0.500777i	$0.166952\pi$
$384$	0	0
$385$	14.4391	0.735887
$386$	0	0
$387$	−11.7328	−0.596411
$388$	0	0
$389$	2.72379	0.138101	0.0690507	−	0.997613i	$-0.478003\pi$
$389$	2.72379	0.138101	0.0690507	+	0.997613i	$0.478003\pi$
$390$	0	0
$391$	−8.41728	−0.425680
$392$	0	0
$393$	−17.6006	−0.887831
$394$	0	0
$395$	−1.59844	−0.0804265
$396$	0	0
$397$	−16.6770	−0.836994	−0.418497	−	0.908218i	$-0.637443\pi$
$397$	−16.6770	−0.836994	−0.418497	+	0.908218i	$0.637443\pi$
$398$	0	0
$399$	−5.60124	−0.280413
$400$	0	0
$401$	5.90801	0.295032	0.147516	−	0.989060i	$-0.452872\pi$
$401$	5.90801	0.295032	0.147516	+	0.989060i	$0.452872\pi$
$402$	0	0
$403$	−11.7636	−0.585987
$404$	0	0
$405$	−2.98063	−0.148109
$406$	0	0
$407$	12.7924	0.634097
$408$	0	0
$409$	−10.2716	−0.507899	−0.253950	−	0.967217i	$-0.581730\pi$
$409$	−10.2716	−0.507899	−0.253950	+	0.967217i	$0.581730\pi$
$410$	0	0
$411$	1.92188	0.0947993
$412$	0	0
$413$	−27.2367	−1.34023
$414$	0	0
$415$	6.86730	0.337102
$416$	0	0
$417$	−10.8500	−0.531326
$418$	0	0
$419$	−0.982385	−0.0479926	−0.0239963	−	0.999712i	$-0.507639\pi$
$419$	−0.982385	−0.0479926	−0.0239963	+	0.999712i	$0.507639\pi$
$420$	0	0
$421$	16.2694	0.792924	0.396462	−	0.918051i	$-0.370238\pi$
$421$	16.2694	0.792924	0.396462	+	0.918051i	$0.370238\pi$
$422$	0	0
$423$	5.12509	0.249190
$424$	0	0
$425$	17.6197	0.854681
$426$	0	0
$427$	18.9405	0.916593
$428$	0	0
$429$	6.78799	0.327727
$430$	0	0
$431$	29.4579	1.41894	0.709468	−	0.704738i	$-0.248935\pi$
$431$	29.4579	1.41894	0.709468	+	0.704738i	$0.248935\pi$
$432$	0	0
$433$	36.8755	1.77212	0.886061	−	0.463568i	$-0.153431\pi$
$433$	36.8755	1.77212	0.886061	+	0.463568i	$0.153431\pi$
$434$	0	0
$435$	−23.4669	−1.12515
$436$	0	0
$437$	4.17011	0.199483
$438$	0	0
$439$	−28.1619	−1.34410	−0.672048	−	0.740507i	$-0.734585\pi$
$439$	−28.1619	−1.34410	−0.672048	+	0.740507i	$0.734585\pi$
$440$	0	0
$441$	−0.788187	−0.0375327
$442$	0	0
$443$	−8.90892	−0.423275	−0.211638	−	0.977348i	$-0.567880\pi$
$443$	−8.90892	−0.423275	−0.211638	+	0.977348i	$0.567880\pi$
$444$	0	0
$445$	25.0044	1.18532
$446$	0	0
$447$	−22.5796	−1.06798
$448$	0	0
$449$	5.64422	0.266367	0.133184	−	0.991091i	$-0.457480\pi$
$449$	5.64422	0.266367	0.133184	+	0.991091i	$0.457480\pi$
$450$	0	0
$451$	−2.81261	−0.132441
$452$	0	0
$453$	−22.5109	−1.05765
$454$	0	0
$455$	25.9439	1.21627
$456$	0	0
$457$	1.69002	0.0790556	0.0395278	−	0.999218i	$-0.487415\pi$
$457$	1.69002	0.0790556	0.0395278	+	0.999218i	$0.487415\pi$
$458$	0	0
$459$	4.53628	0.211735
$460$	0	0
$461$	−2.03146	−0.0946145	−0.0473072	−	0.998880i	$-0.515064\pi$
$461$	−2.03146	−0.0946145	−0.0473072	+	0.998880i	$0.515064\pi$
$462$	0	0
$463$	21.6428	1.00583	0.502913	−	0.864337i	$-0.332261\pi$
$463$	21.6428	1.00583	0.502913	+	0.864337i	$0.332261\pi$
$464$	0	0
$465$	−10.0399	−0.465591
$466$	0	0
$467$	−11.3847	−0.526819	−0.263410	−	0.964684i	$-0.584847\pi$
$467$	−11.3847	−0.526819	−0.263410	+	0.964684i	$0.584847\pi$
$468$	0	0
$469$	−1.50982	−0.0697168
$470$	0	0
$471$	−15.7880	−0.727474
$472$	0	0
$473$	22.8047	1.04856
$474$	0	0
$475$	−8.72919	−0.400523
$476$	0	0
$477$	3.51312	0.160855
$478$	0	0
$479$	−30.9588	−1.41454	−0.707272	−	0.706941i	$-0.750075\pi$
$479$	−30.9588	−1.41454	−0.707272	+	0.706941i	$0.750075\pi$
$480$	0	0
$481$	22.9851	1.04803
$482$	0	0
$483$	−4.62468	−0.210430
$484$	0	0
$485$	39.3003	1.78453
$486$	0	0
$487$	13.8723	0.628614	0.314307	−	0.949321i	$-0.398228\pi$
$487$	13.8723	0.628614	0.314307	+	0.949321i	$0.398228\pi$
$488$	0	0
$489$	−17.0517	−0.771104
$490$	0	0
$491$	−13.9144	−0.627948	−0.313974	−	0.949432i	$-0.601660\pi$
$491$	−13.9144	−0.627948	−0.313974	+	0.949432i	$0.601660\pi$
$492$	0	0
$493$	35.7147	1.60851
$494$	0	0
$495$	5.79338	0.260393
$496$	0	0
$497$	−4.50854	−0.202236
$498$	0	0
$499$	5.79082	0.259233	0.129616	−	0.991564i	$-0.458625\pi$
$499$	5.79082	0.259233	0.129616	+	0.991564i	$0.458625\pi$
$500$	0	0
$501$	8.95076	0.399890
$502$	0	0
$503$	−21.4775	−0.957632	−0.478816	−	0.877915i	$-0.658934\pi$
$503$	−21.4775	−0.957632	−0.478816	+	0.877915i	$0.658934\pi$
$504$	0	0
$505$	−7.39264	−0.328968
$506$	0	0
$507$	−0.803486	−0.0356841
$508$	0	0
$509$	44.5553	1.97488	0.987441	−	0.157991i	$-0.0505018\pi$
$509$	44.5553	1.97488	0.987441	+	0.157991i	$0.0505018\pi$
$510$	0	0
$511$	−11.0365	−0.488228
$512$	0	0
$513$	−2.24737	−0.0992239
$514$	0	0
$515$	26.8027	1.18107
$516$	0	0
$517$	−9.96150	−0.438106
$518$	0	0
$519$	−5.32614	−0.233791
$520$	0	0
$521$	−17.8423	−0.781687	−0.390844	−	0.920457i	$-0.627817\pi$
$521$	−17.8423	−0.781687	−0.390844	+	0.920457i	$0.627817\pi$
$522$	0	0
$523$	5.54698	0.242553	0.121276	−	0.992619i	$-0.461301\pi$
$523$	5.54698	0.242553	0.121276	+	0.992619i	$0.461301\pi$
$524$	0	0
$525$	9.68073	0.422502
$526$	0	0
$527$	15.2800	0.665605
$528$	0	0
$529$	−19.5569	−0.850302
$530$	0	0
$531$	−10.9281	−0.474239
$532$	0	0
$533$	−5.05364	−0.218897
$534$	0	0
$535$	2.58355	0.111697
$536$	0	0
$537$	−19.5811	−0.844987
$538$	0	0
$539$	1.53198	0.0659870
$540$	0	0
$541$	23.4622	1.00872	0.504360	−	0.863493i	$-0.331728\pi$
$541$	23.4622	1.00872	0.504360	+	0.863493i	$0.331728\pi$
$542$	0	0
$543$	19.1351	0.821168
$544$	0	0
$545$	−29.4274	−1.26053
$546$	0	0
$547$	−14.2196	−0.607985	−0.303992	−	0.952674i	$-0.598320\pi$
$547$	−14.2196	−0.607985	−0.303992	+	0.952674i	$0.598320\pi$
$548$	0	0
$549$	7.59943	0.324336
$550$	0	0
$551$	−17.6939	−0.753785
$552$	0	0
$553$	1.33659	0.0568376
$554$	0	0
$555$	19.6172	0.832706
$556$	0	0
$557$	2.04726	0.0867452	0.0433726	−	0.999059i	$-0.486190\pi$
$557$	2.04726	0.0867452	0.0433726	+	0.999059i	$0.486190\pi$
$558$	0	0
$559$	40.9750	1.73306
$560$	0	0
$561$	−8.81704	−0.372256
$562$	0	0
$563$	−34.9445	−1.47273	−0.736367	−	0.676582i	$-0.763460\pi$
$563$	−34.9445	−1.47273	−0.736367	+	0.676582i	$0.763460\pi$
$564$	0	0
$565$	−47.9347	−2.01663
$566$	0	0
$567$	2.49235	0.104669
$568$	0	0
$569$	7.52522	0.315474	0.157737	−	0.987481i	$-0.449580\pi$
$569$	7.52522	0.315474	0.157737	+	0.987481i	$0.449580\pi$
$570$	0	0
$571$	19.0334	0.796522	0.398261	−	0.917272i	$-0.369614\pi$
$571$	19.0334	0.796522	0.398261	+	0.917272i	$0.369614\pi$
$572$	0	0
$573$	17.2812	0.721932
$574$	0	0
$575$	−7.20728	−0.300564
$576$	0	0
$577$	−21.0187	−0.875019	−0.437510	−	0.899214i	$-0.644139\pi$
$577$	−21.0187	−0.875019	−0.437510	+	0.899214i	$0.644139\pi$
$578$	0	0
$579$	18.8248	0.782332
$580$	0	0
$581$	−5.74231	−0.238231
$582$	0	0
$583$	−6.82835	−0.282801
$584$	0	0
$585$	10.4094	0.430376
$586$	0	0
$587$	−27.1603	−1.12103	−0.560513	−	0.828145i	$-0.689396\pi$
$587$	−27.1603	−1.12103	−0.560513	+	0.828145i	$0.689396\pi$
$588$	0	0
$589$	−7.57003	−0.311918
$590$	0	0
$591$	21.3799	0.879452
$592$	0	0
$593$	−3.72771	−0.153079	−0.0765394	−	0.997067i	$-0.524387\pi$
$593$	−3.72771	−0.153079	−0.0765394	+	0.997067i	$0.524387\pi$
$594$	0	0
$595$	−33.6990	−1.38153
$596$	0	0
$597$	−26.9170	−1.10164
$598$	0	0
$599$	−6.87209	−0.280786	−0.140393	−	0.990096i	$-0.544837\pi$
$599$	−6.87209	−0.280786	−0.140393	+	0.990096i	$0.544837\pi$
$600$	0	0
$601$	46.5836	1.90019	0.950093	−	0.311966i	$-0.100988\pi$
$601$	46.5836	1.90019	0.950093	+	0.311966i	$0.100988\pi$
$602$	0	0
$603$	−0.605780	−0.0246693
$604$	0	0
$605$	21.5265	0.875178
$606$	0	0
$607$	25.9091	1.05162	0.525808	−	0.850603i	$-0.323763\pi$
$607$	25.9091	1.05162	0.525808	+	0.850603i	$0.323763\pi$
$608$	0	0
$609$	19.6226	0.795149
$610$	0	0
$611$	−17.8986	−0.724100
$612$	0	0
$613$	38.0304	1.53603	0.768017	−	0.640430i	$-0.221244\pi$
$613$	38.0304	1.53603	0.768017	+	0.640430i	$0.221244\pi$
$614$	0	0
$615$	−4.31316	−0.173923
$616$	0	0
$617$	12.4331	0.500538	0.250269	−	0.968176i	$-0.419481\pi$
$617$	12.4331	0.500538	0.250269	+	0.968176i	$0.419481\pi$
$618$	0	0
$619$	44.2294	1.77773	0.888864	−	0.458171i	$-0.151495\pi$
$619$	44.2294	1.77773	0.888864	+	0.458171i	$0.151495\pi$
$620$	0	0
$621$	−1.85555	−0.0744606
$622$	0	0
$623$	−20.9082	−0.837671
$624$	0	0
$625$	−29.3341	−1.17336
$626$	0	0
$627$	4.36816	0.174448
$628$	0	0
$629$	−29.8558	−1.19043
$630$	0	0
$631$	−18.9265	−0.753450	−0.376725	−	0.926325i	$-0.622950\pi$
$631$	−18.9265	−0.753450	−0.376725	+	0.926325i	$0.622950\pi$
$632$	0	0
$633$	−3.54184	−0.140776
$634$	0	0
$635$	2.98063	0.118283
$636$	0	0
$637$	2.75263	0.109063
$638$	0	0
$639$	−1.80895	−0.0715611
$640$	0	0
$641$	−30.5891	−1.20820	−0.604099	−	0.796910i	$-0.706467\pi$
$641$	−30.5891	−1.20820	−0.604099	+	0.796910i	$0.706467\pi$
$642$	0	0
$643$	5.97532	0.235644	0.117822	−	0.993035i	$-0.462409\pi$
$643$	5.97532	0.235644	0.117822	+	0.993035i	$0.462409\pi$
$644$	0	0
$645$	34.9711	1.37699
$646$	0	0
$647$	−47.5449	−1.86918	−0.934592	−	0.355721i	$-0.884235\pi$
$647$	−47.5449	−1.86918	−0.934592	+	0.355721i	$0.884235\pi$
$648$	0	0
$649$	21.2407	0.833769
$650$	0	0
$651$	8.39521	0.329034
$652$	0	0
$653$	34.0921	1.33413	0.667063	−	0.745002i	$-0.267551\pi$
$653$	34.0921	1.33413	0.667063	+	0.745002i	$0.267551\pi$
$654$	0	0
$655$	52.4608	2.04981
$656$	0	0
$657$	−4.42816	−0.172759
$658$	0	0
$659$	10.4016	0.405189	0.202594	−	0.979263i	$-0.435063\pi$
$659$	10.4016	0.405189	0.202594	+	0.979263i	$0.435063\pi$
$660$	0	0
$661$	13.9384	0.542141	0.271070	−	0.962560i	$-0.412622\pi$
$661$	13.9384	0.542141	0.271070	+	0.962560i	$0.412622\pi$
$662$	0	0
$663$	−15.8423	−0.615263
$664$	0	0
$665$	16.6952	0.647414
$666$	0	0
$667$	−14.6090	−0.565662
$668$	0	0
$669$	17.4252	0.673696
$670$	0	0
$671$	−14.7708	−0.570221
$672$	0	0
$673$	34.4795	1.32909	0.664543	−	0.747250i	$-0.268626\pi$
$673$	34.4795	1.32909	0.664543	+	0.747250i	$0.268626\pi$
$674$	0	0
$675$	3.88418	0.149502
$676$	0	0
$677$	−17.9850	−0.691219	−0.345609	−	0.938379i	$-0.612328\pi$
$677$	−17.9850	−0.691219	−0.345609	+	0.938379i	$0.612328\pi$
$678$	0	0
$679$	−32.8622	−1.26113
$680$	0	0
$681$	10.8821	0.417001
$682$	0	0
$683$	10.4054	0.398152	0.199076	−	0.979984i	$-0.436206\pi$
$683$	10.4054	0.398152	0.199076	+	0.979984i	$0.436206\pi$
$684$	0	0
$685$	−5.72841	−0.218871
$686$	0	0
$687$	−12.0378	−0.459271
$688$	0	0
$689$	−12.2690	−0.467413
$690$	0	0
$691$	6.48584	0.246733	0.123367	−	0.992361i	$-0.460631\pi$
$691$	6.48584	0.246733	0.123367	+	0.992361i	$0.460631\pi$
$692$	0	0
$693$	−4.84432	−0.184020
$694$	0	0
$695$	32.3398	1.22672
$696$	0	0
$697$	6.56426	0.248639
$698$	0	0
$699$	−13.3785	−0.506020
$700$	0	0
$701$	25.4982	0.963055	0.481528	−	0.876431i	$-0.340082\pi$
$701$	25.4982	0.963055	0.481528	+	0.876431i	$0.340082\pi$
$702$	0	0
$703$	14.7912	0.557862
$704$	0	0
$705$	−15.2760	−0.575327
$706$	0	0
$707$	6.18159	0.232483
$708$	0	0
$709$	41.6178	1.56299	0.781495	−	0.623911i	$-0.214457\pi$
$709$	41.6178	1.56299	0.781495	+	0.623911i	$0.214457\pi$
$710$	0	0
$711$	0.536277	0.0201120
$712$	0	0
$713$	−6.25021	−0.234072
$714$	0	0
$715$	−20.2325	−0.756653
$716$	0	0
$717$	−30.1290	−1.12519
$718$	0	0
$719$	18.2708	0.681387	0.340693	−	0.940174i	$-0.389338\pi$
$719$	18.2708	0.681387	0.340693	+	0.940174i	$0.389338\pi$
$720$	0	0
$721$	−22.4119	−0.834664
$722$	0	0
$723$	26.4367	0.983193
$724$	0	0
$725$	30.5807	1.13574
$726$	0	0
$727$	37.4813	1.39011	0.695053	−	0.718959i	$-0.255381\pi$
$727$	37.4813	1.39011	0.695053	+	0.718959i	$0.255381\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−53.2232	−1.96853
$732$	0	0
$733$	15.0688	0.556579	0.278289	−	0.960497i	$-0.410233\pi$
$733$	15.0688	0.556579	0.278289	+	0.960497i	$0.410233\pi$
$734$	0	0
$735$	2.34930	0.0866552
$736$	0	0
$737$	1.17744	0.0433715
$738$	0	0
$739$	−7.00612	−0.257724	−0.128862	−	0.991663i	$-0.541132\pi$
$739$	−7.00612	−0.257724	−0.128862	+	0.991663i	$0.541132\pi$
$740$	0	0
$741$	7.84862	0.288326
$742$	0	0
$743$	24.6885	0.905733	0.452867	−	0.891578i	$-0.350401\pi$
$743$	24.6885	0.905733	0.452867	+	0.891578i	$0.350401\pi$
$744$	0	0
$745$	67.3015	2.46574
$746$	0	0
$747$	−2.30397	−0.0842980
$748$	0	0
$749$	−2.16032	−0.0789362
$750$	0	0
$751$	−25.8423	−0.943000	−0.471500	−	0.881866i	$-0.656287\pi$
$751$	−25.8423	−0.943000	−0.471500	+	0.881866i	$0.656287\pi$
$752$	0	0
$753$	21.4096	0.780208
$754$	0	0
$755$	67.0967	2.44190
$756$	0	0
$757$	−39.9510	−1.45204	−0.726022	−	0.687672i	$-0.758633\pi$
$757$	−39.9510	−1.45204	−0.726022	+	0.687672i	$0.758633\pi$
$758$	0	0
$759$	3.60658	0.130911
$760$	0	0
$761$	−12.6209	−0.457506	−0.228753	−	0.973485i	$-0.573465\pi$
$761$	−12.6209	−0.457506	−0.228753	+	0.973485i	$0.573465\pi$
$762$	0	0
$763$	24.6066	0.890820
$764$	0	0
$765$	−13.5210	−0.488852
$766$	0	0
$767$	38.1648	1.37805
$768$	0	0
$769$	−33.8677	−1.22130	−0.610650	−	0.791901i	$-0.709092\pi$
$769$	−33.8677	−1.22130	−0.610650	+	0.791901i	$0.709092\pi$
$770$	0	0
$771$	−1.84014	−0.0662709
$772$	0	0
$773$	10.1726	0.365884	0.182942	−	0.983124i	$-0.441438\pi$
$773$	10.1726	0.365884	0.182942	+	0.983124i	$0.441438\pi$
$774$	0	0
$775$	13.0834	0.469970
$776$	0	0
$777$	−16.4036	−0.588475
$778$	0	0
$779$	−3.25208	−0.116518
$780$	0	0
$781$	3.51601	0.125813
$782$	0	0
$783$	7.87314	0.281363
$784$	0	0
$785$	47.0583	1.67958
$786$	0	0
$787$	6.49794	0.231627	0.115813	−	0.993271i	$-0.463053\pi$
$787$	6.49794	0.231627	0.115813	+	0.993271i	$0.463053\pi$
$788$	0	0
$789$	−2.13565	−0.0760312
$790$	0	0
$791$	40.0821	1.42516
$792$	0	0
$793$	−26.5399	−0.942459
$794$	0	0
$795$	−10.4713	−0.371379
$796$	0	0
$797$	13.4596	0.476762	0.238381	−	0.971172i	$-0.423383\pi$
$797$	13.4596	0.476762	0.238381	+	0.971172i	$0.423383\pi$
$798$	0	0
$799$	23.2488	0.822484
$800$	0	0
$801$	−8.38896	−0.296409
$802$	0	0
$803$	8.60691	0.303731
$804$	0	0
$805$	13.7845	0.485839
$806$	0	0
$807$	−22.0930	−0.777710
$808$	0	0
$809$	−25.6210	−0.900787	−0.450393	−	0.892830i	$-0.648716\pi$
$809$	−25.6210	−0.900787	−0.450393	+	0.892830i	$0.648716\pi$
$810$	0	0
$811$	23.9897	0.842391	0.421196	−	0.906970i	$-0.361611\pi$
$811$	23.9897	0.842391	0.421196	+	0.906970i	$0.361611\pi$
$812$	0	0
$813$	16.7849	0.588671
$814$	0	0
$815$	50.8248	1.78032
$816$	0	0
$817$	26.3680	0.922498
$818$	0	0
$819$	−8.70416	−0.304148
$820$	0	0
$821$	0.726367	0.0253504	0.0126752	−	0.999920i	$-0.495965\pi$
$821$	0.726367	0.0253504	0.0126752	+	0.999920i	$0.495965\pi$
$822$	0	0
$823$	3.66517	0.127760	0.0638798	−	0.997958i	$-0.479653\pi$
$823$	3.66517	0.127760	0.0638798	+	0.997958i	$0.479653\pi$
$824$	0	0
$825$	−7.54957	−0.262842
$826$	0	0
$827$	−32.7090	−1.13740	−0.568702	−	0.822544i	$-0.692554\pi$
$827$	−32.7090	−1.13740	−0.568702	+	0.822544i	$0.692554\pi$
$828$	0	0
$829$	−19.6772	−0.683418	−0.341709	−	0.939806i	$-0.611006\pi$
$829$	−19.6772	−0.683418	−0.341709	+	0.939806i	$0.611006\pi$
$830$	0	0
$831$	−20.6239	−0.715434
$832$	0	0
$833$	−3.57544	−0.123882
$834$	0	0
$835$	−26.6789	−0.923263
$836$	0	0
$837$	3.36839	0.116429
$838$	0	0
$839$	47.7471	1.64841	0.824206	−	0.566290i	$-0.191622\pi$
$839$	47.7471	1.64841	0.824206	+	0.566290i	$0.191622\pi$
$840$	0	0
$841$	32.9863	1.13746
$842$	0	0
$843$	12.9127	0.444737
$844$	0	0
$845$	2.39490	0.0823869
$846$	0	0
$847$	−18.0001	−0.618490
$848$	0	0
$849$	−5.12835	−0.176005
$850$	0	0
$851$	12.2124	0.418636
$852$	0	0
$853$	−21.7797	−0.745722	−0.372861	−	0.927887i	$-0.621623\pi$
$853$	−21.7797	−0.745722	−0.372861	+	0.927887i	$0.621623\pi$
$854$	0	0
$855$	6.69860	0.229087
$856$	0	0
$857$	43.0744	1.47139	0.735696	−	0.677311i	$-0.236855\pi$
$857$	43.0744	1.47139	0.735696	+	0.677311i	$0.236855\pi$
$858$	0	0
$859$	−4.24922	−0.144982	−0.0724908	−	0.997369i	$-0.523095\pi$
$859$	−4.24922	−0.144982	−0.0724908	+	0.997369i	$0.523095\pi$
$860$	0	0
$861$	3.60658	0.122912
$862$	0	0
$863$	−38.2608	−1.30241	−0.651207	−	0.758900i	$-0.725737\pi$
$863$	−38.2608	−1.30241	−0.651207	+	0.758900i	$0.725737\pi$
$864$	0	0
$865$	15.8753	0.539775
$866$	0	0
$867$	3.57781	0.121509
$868$	0	0
$869$	−1.04235	−0.0353592
$870$	0	0
$871$	2.11560	0.0716842
$872$	0	0
$873$	−13.1852	−0.446252
$874$	0	0
$875$	8.28922	0.280227
$876$	0	0
$877$	27.5995	0.931968	0.465984	−	0.884793i	$-0.345700\pi$
$877$	27.5995	0.931968	0.465984	+	0.884793i	$0.345700\pi$
$878$	0	0
$879$	13.8495	0.467134
$880$	0	0
$881$	16.9324	0.570466	0.285233	−	0.958458i	$-0.407929\pi$
$881$	16.9324	0.570466	0.285233	+	0.958458i	$0.407929\pi$
$882$	0	0
$883$	−37.5631	−1.26410	−0.632049	−	0.774928i	$-0.717786\pi$
$883$	−37.5631	−1.26410	−0.632049	+	0.774928i	$0.717786\pi$
$884$	0	0
$885$	32.5727	1.09492
$886$	0	0
$887$	10.4133	0.349644	0.174822	−	0.984600i	$-0.444065\pi$
$887$	10.4133	0.349644	0.174822	+	0.984600i	$0.444065\pi$
$888$	0	0
$889$	−2.49235	−0.0835908
$890$	0	0
$891$	−1.94367	−0.0651155
$892$	0	0
$893$	−11.5180	−0.385434
$894$	0	0
$895$	58.3641	1.95090
$896$	0	0
$897$	6.48023	0.216368
$898$	0	0
$899$	26.5198	0.884485
$900$	0	0
$901$	15.9365	0.530921
$902$	0	0
$903$	−29.2422	−0.973121
$904$	0	0
$905$	−57.0349	−1.89590
$906$	0	0
$907$	−16.9448	−0.562642	−0.281321	−	0.959614i	$-0.590773\pi$
$907$	−16.9448	−0.562642	−0.281321	+	0.959614i	$0.590773\pi$
$908$	0	0
$909$	2.48023	0.0822639
$910$	0	0
$911$	41.9676	1.39045	0.695224	−	0.718793i	$-0.255305\pi$
$911$	41.9676	1.39045	0.695224	+	0.718793i	$0.255305\pi$
$912$	0	0
$913$	4.47817	0.148206
$914$	0	0
$915$	−22.6511	−0.748823
$916$	0	0
$917$	−43.8668	−1.44861
$918$	0	0
$919$	−31.6581	−1.04431	−0.522153	−	0.852852i	$-0.674871\pi$
$919$	−31.6581	−1.04431	−0.522153	+	0.852852i	$0.674871\pi$
$920$	0	0
$921$	2.97082	0.0978920
$922$	0	0
$923$	6.31750	0.207943
$924$	0	0
$925$	−25.5640	−0.840538
$926$	0	0
$927$	−8.99229	−0.295346
$928$	0	0
$929$	15.9324	0.522724	0.261362	−	0.965241i	$-0.415828\pi$
$929$	15.9324	0.522724	0.261362	+	0.965241i	$0.415828\pi$
$930$	0	0
$931$	1.77135	0.0580537
$932$	0	0
$933$	−20.8146	−0.681440
$934$	0	0
$935$	26.2804	0.859460
$936$	0	0
$937$	−26.2574	−0.857791	−0.428895	−	0.903354i	$-0.641097\pi$
$937$	−26.2574	−0.857791	−0.428895	+	0.903354i	$0.641097\pi$
$938$	0	0
$939$	−28.0654	−0.915879
$940$	0	0
$941$	−34.8878	−1.13731	−0.568654	−	0.822576i	$-0.692536\pi$
$941$	−34.8878	−1.13731	−0.568654	+	0.822576i	$0.692536\pi$
$942$	0	0
$943$	−2.68509	−0.0874385
$944$	0	0
$945$	−7.42878	−0.241658
$946$	0	0
$947$	−15.3419	−0.498546	−0.249273	−	0.968433i	$-0.580192\pi$
$947$	−15.3419	−0.498546	−0.249273	+	0.968433i	$0.580192\pi$
$948$	0	0
$949$	15.4647	0.502005
$950$	0	0
$951$	17.9162	0.580972
$952$	0	0
$953$	−32.2332	−1.04414	−0.522068	−	0.852904i	$-0.674839\pi$
$953$	−32.2332	−1.04414	−0.522068	+	0.852904i	$0.674839\pi$
$954$	0	0
$955$	−51.5089	−1.66679
$956$	0	0
$957$	−15.3028	−0.494670
$958$	0	0
$959$	4.79000	0.154677
$960$	0	0
$961$	−19.6539	−0.633998
$962$	0	0
$963$	−0.866778	−0.0279315
$964$	0	0
$965$	−56.1098	−1.80624
$966$	0	0
$967$	−21.9359	−0.705411	−0.352706	−	0.935734i	$-0.614738\pi$
$967$	−21.9359	−0.705411	−0.352706	+	0.935734i	$0.614738\pi$
$968$	0	0
$969$	−10.1947	−0.327501
$970$	0	0
$971$	−55.9118	−1.79429	−0.897147	−	0.441732i	$-0.854364\pi$
$971$	−55.9118	−1.79429	−0.897147	+	0.441732i	$0.854364\pi$
$972$	0	0
$973$	−27.0420	−0.866925
$974$	0	0
$975$	−13.5649	−0.434425
$976$	0	0
$977$	24.7888	0.793064	0.396532	−	0.918021i	$-0.370214\pi$
$977$	24.7888	0.793064	0.396532	+	0.918021i	$0.370214\pi$
$978$	0	0
$979$	16.3054	0.521123
$980$	0	0
$981$	9.87287	0.315216
$982$	0	0
$983$	−20.3512	−0.649103	−0.324551	−	0.945868i	$-0.605213\pi$
$983$	−20.3512	−0.649103	−0.324551	+	0.945868i	$0.605213\pi$
$984$	0	0
$985$	−63.7257	−2.03047
$986$	0	0
$987$	12.7735	0.406585
$988$	0	0
$989$	21.7708	0.692270
$990$	0	0
$991$	40.5877	1.28931	0.644656	−	0.764473i	$-0.277001\pi$
$991$	40.5877	1.28931	0.644656	+	0.764473i	$0.277001\pi$
$992$	0	0
$993$	16.8466	0.534612
$994$	0	0
$995$	80.2296	2.54345
$996$	0	0
$997$	55.4482	1.75606	0.878031	−	0.478605i	$-0.158857\pi$
$997$	55.4482	1.75606	0.878031	+	0.478605i	$0.158857\pi$
$998$	0	0
$999$	−6.58157	−0.208232

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6096.2.a.bf.1.1		5
4.3	odd	2		381.2.a.d.1.2	✓	5
12.11	even	2		1143.2.a.g.1.4		5
20.19	odd	2		9525.2.a.j.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
381.2.a.d.1.2	✓	5	4.3	odd	2
1143.2.a.g.1.4		5	12.11	even	2
6096.2.a.bf.1.1		5	1.1	even	1	trivial
9525.2.a.j.1.4		5	20.19	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:	\( N \)	\(=\)	\( 6096 = 2^{4} \cdot 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6096.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.98063 q^{5} +2.49235 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.98063 q^{5} +2.49235 q^{7} +1.00000 q^{9} -1.94367 q^{11} -3.49235 q^{13} -2.98063 q^{15} +4.53628 q^{17} -2.24737 q^{19} +2.49235 q^{21} -1.85555 q^{23} +3.88418 q^{25} +1.00000 q^{27} +7.87314 q^{29} +3.36839 q^{31} -1.94367 q^{33} -7.42878 q^{35} -6.58157 q^{37} -3.49235 q^{39} +1.44706 q^{41} -11.7328 q^{43} -2.98063 q^{45} +5.12509 q^{47} -0.788187 q^{49} +4.53628 q^{51} +3.51312 q^{53} +5.79338 q^{55} -2.24737 q^{57} -10.9281 q^{59} +7.59943 q^{61} +2.49235 q^{63} +10.4094 q^{65} -0.605780 q^{67} -1.85555 q^{69} -1.80895 q^{71} -4.42816 q^{73} +3.88418 q^{75} -4.84432 q^{77} +0.536277 q^{79} +1.00000 q^{81} -2.30397 q^{83} -13.5210 q^{85} +7.87314 q^{87} -8.38896 q^{89} -8.70416 q^{91} +3.36839 q^{93} +6.69860 q^{95} -13.1852 q^{97} -1.94367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + q^{5} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + q^5 + 5 * q^9	\( 5 q + 5 q^{3} + q^{5} + 5 q^{9} - 14 q^{11} - 5 q^{13} + q^{15} + 4 q^{17} - 4 q^{19} - 15 q^{23} - 6 q^{25} + 5 q^{27} + 9 q^{29} - 3 q^{31} - 14 q^{33} - 4 q^{35} - 5 q^{37} - 5 q^{39} + 4 q^{41} - 10 q^{43} + q^{45} + 4 q^{47} - 9 q^{49} + 4 q^{51} + 3 q^{53} - 4 q^{55} - 4 q^{57} - 23 q^{59} - 15 q^{61} + 3 q^{65} - 18 q^{67} - 15 q^{69} - 12 q^{71} - 43 q^{73} - 6 q^{75} + 4 q^{77} - 16 q^{79} + 5 q^{81} - 11 q^{83} - 24 q^{85} + 9 q^{87} + 9 q^{89} - 26 q^{91} - 3 q^{93} - 16 q^{95} - 20 q^{97} - 14 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + q^5 + 5 * q^9 - 14 * q^11 - 5 * q^13 + q^15 + 4 * q^17 - 4 * q^19 - 15 * q^23 - 6 * q^25 + 5 * q^27 + 9 * q^29 - 3 * q^31 - 14 * q^33 - 4 * q^35 - 5 * q^37 - 5 * q^39 + 4 * q^41 - 10 * q^43 + q^45 + 4 * q^47 - 9 * q^49 + 4 * q^51 + 3 * q^53 - 4 * q^55 - 4 * q^57 - 23 * q^59 - 15 * q^61 + 3 * q^65 - 18 * q^67 - 15 * q^69 - 12 * q^71 - 43 * q^73 - 6 * q^75 + 4 * q^77 - 16 * q^79 + 5 * q^81 - 11 * q^83 - 24 * q^85 + 9 * q^87 + 9 * q^89 - 26 * q^91 - 3 * q^93 - 16 * q^95 - 20 * q^97 - 14 * q^99

Introduction

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