LMFDB - Embedded newform 6016.2.a.o.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6016,2,Mod(1,6016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6016 = 2^{7} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6016.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:	$48.0380018560$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} + \cdots - 832 $ x^13 - 4x^12 - 22x^11 + 96x^10 + 163x^9 - 840x^8 - 438x^7 + 3358x^6 + 5x^5 - 6230x^4 + 1488x^3 + 4720x^2 - 1440x - 832
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$3.18041$ of defining polynomial
Character	$\chi$	$=$	6016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.18041 q^{3} -1.45922 q^{5} -4.05725 q^{7} +7.11498 q^{9} +O(q^{10})$	$q+3.18041 q^{3} -1.45922 q^{5} -4.05725 q^{7} +7.11498 q^{9} -2.02978 q^{11} +5.40999 q^{13} -4.64090 q^{15} +0.106886 q^{17} +7.30798 q^{19} -12.9037 q^{21} -1.01316 q^{23} -2.87068 q^{25} +13.0873 q^{27} +1.97186 q^{29} +1.08878 q^{31} -6.45552 q^{33} +5.92041 q^{35} -5.58522 q^{37} +17.2060 q^{39} -2.25192 q^{41} +2.62867 q^{43} -10.3823 q^{45} -1.00000 q^{47} +9.46127 q^{49} +0.339940 q^{51} -3.83968 q^{53} +2.96189 q^{55} +23.2423 q^{57} +6.49409 q^{59} +7.99033 q^{61} -28.8673 q^{63} -7.89435 q^{65} +3.82891 q^{67} -3.22226 q^{69} +11.3390 q^{71} -16.6647 q^{73} -9.12994 q^{75} +8.23532 q^{77} +12.7709 q^{79} +20.2781 q^{81} +17.6711 q^{83} -0.155969 q^{85} +6.27130 q^{87} +12.5472 q^{89} -21.9497 q^{91} +3.46277 q^{93} -10.6639 q^{95} +12.5084 q^{97} -14.4418 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9}+O(q^{10}) $ 13 * q + 4 * q^3 - 6 * q^5 - 2 * q^7 + 21 * q^9	$ 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9} + 10 q^{11} - 4 q^{13} + 14 q^{15} + 10 q^{17} + 8 q^{19} - 10 q^{21} + 18 q^{23} + 23 q^{25} + 16 q^{27} - 14 q^{29} + 4 q^{31} + 14 q^{33} + 14 q^{35} - 16 q^{37} + 12 q^{39} + 10 q^{41} + 12 q^{43} - 10 q^{45} - 13 q^{47} + 9 q^{49} + 22 q^{51} - 26 q^{53} - 2 q^{55} + 20 q^{57} + 30 q^{59} - 18 q^{61} + 12 q^{63} - 4 q^{65} + 4 q^{67} - 2 q^{69} + 36 q^{71} + 10 q^{73} + 38 q^{75} - 42 q^{77} + 21 q^{81} + 12 q^{83} - 4 q^{85} + 6 q^{87} + 50 q^{89} - 4 q^{91} - 52 q^{93} + 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) $ 13 * q + 4 * q^3 - 6 * q^5 - 2 * q^7 + 21 * q^9 + 10 * q^11 - 4 * q^13 + 14 * q^15 + 10 * q^17 + 8 * q^19 - 10 * q^21 + 18 * q^23 + 23 * q^25 + 16 * q^27 - 14 * q^29 + 4 * q^31 + 14 * q^33 + 14 * q^35 - 16 * q^37 + 12 * q^39 + 10 * q^41 + 12 * q^43 - 10 * q^45 - 13 * q^47 + 9 * q^49 + 22 * q^51 - 26 * q^53 - 2 * q^55 + 20 * q^57 + 30 * q^59 - 18 * q^61 + 12 * q^63 - 4 * q^65 + 4 * q^67 - 2 * q^69 + 36 * q^71 + 10 * q^73 + 38 * q^75 - 42 * q^77 + 21 * q^81 + 12 * q^83 - 4 * q^85 + 6 * q^87 + 50 * q^89 - 4 * q^91 - 52 * q^93 + 8 * q^95 - 10 * q^97 + 34 * 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
$3$	3.18041	1.83621	0.918104	−	0.396339i	$-0.129719\pi$
$3$	3.18041	1.83621	0.918104	+	0.396339i	$0.129719\pi$
$4$	0	0
$5$	−1.45922	−0.652582	−0.326291	−	0.945269i	$-0.605799\pi$
$5$	−1.45922	−0.652582	−0.326291	+	0.945269i	$0.605799\pi$
$6$	0	0
$7$	−4.05725	−1.53350	−0.766748	−	0.641948i	$-0.778126\pi$
$7$	−4.05725	−1.53350	−0.766748	+	0.641948i	$0.778126\pi$
$8$	0	0
$9$	7.11498	2.37166
$10$	0	0
$11$	−2.02978	−0.612001	−0.306001	−	0.952031i	$-0.598991\pi$
$11$	−2.02978	−0.612001	−0.306001	+	0.952031i	$0.598991\pi$
$12$	0	0
$13$	5.40999	1.50046	0.750230	−	0.661177i	$-0.229943\pi$
$13$	5.40999	1.50046	0.750230	+	0.661177i	$0.229943\pi$
$14$	0	0
$15$	−4.64090	−1.19828
$16$	0	0
$17$	0.106886	0.0259236	0.0129618	−	0.999916i	$-0.495874\pi$
$17$	0.106886	0.0259236	0.0129618	+	0.999916i	$0.495874\pi$
$18$	0	0
$19$	7.30798	1.67657	0.838283	−	0.545236i	$-0.183560\pi$
$19$	7.30798	1.67657	0.838283	+	0.545236i	$0.183560\pi$
$20$	0	0
$21$	−12.9037	−2.81582
$22$	0	0
$23$	−1.01316	−0.211259	−0.105629	−	0.994406i	$-0.533686\pi$
$23$	−1.01316	−0.211259	−0.105629	+	0.994406i	$0.533686\pi$
$24$	0	0
$25$	−2.87068	−0.574137
$26$	0	0
$27$	13.0873	2.51866
$28$	0	0
$29$	1.97186	0.366165	0.183082	−	0.983098i	$-0.441393\pi$
$29$	1.97186	0.366165	0.183082	+	0.983098i	$0.441393\pi$
$30$	0	0
$31$	1.08878	0.195551	0.0977757	−	0.995208i	$-0.468827\pi$
$31$	1.08878	0.195551	0.0977757	+	0.995208i	$0.468827\pi$
$32$	0	0
$33$	−6.45552	−1.12376
$34$	0	0
$35$	5.92041	1.00073
$36$	0	0
$37$	−5.58522	−0.918205	−0.459102	−	0.888383i	$-0.651829\pi$
$37$	−5.58522	−0.918205	−0.459102	+	0.888383i	$0.651829\pi$
$38$	0	0
$39$	17.2060	2.75516
$40$	0	0
$41$	−2.25192	−0.351691	−0.175846	−	0.984418i	$-0.556266\pi$
$41$	−2.25192	−0.351691	−0.175846	+	0.984418i	$0.556266\pi$
$42$	0	0
$43$	2.62867	0.400868	0.200434	−	0.979707i	$-0.435765\pi$
$43$	2.62867	0.400868	0.200434	+	0.979707i	$0.435765\pi$
$44$	0	0
$45$	−10.3823	−1.54770
$46$	0	0
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	9.46127	1.35161
$50$	0	0
$51$	0.339940	0.0476011
$52$	0	0
$53$	−3.83968	−0.527421	−0.263710	−	0.964602i	$-0.584946\pi$
$53$	−3.83968	−0.527421	−0.263710	+	0.964602i	$0.584946\pi$
$54$	0	0
$55$	2.96189	0.399381
$56$	0	0
$57$	23.2423	3.07852
$58$	0	0
$59$	6.49409	0.845459	0.422730	−	0.906256i	$-0.361072\pi$
$59$	6.49409	0.845459	0.422730	+	0.906256i	$0.361072\pi$
$60$	0	0
$61$	7.99033	1.02306	0.511528	−	0.859266i	$-0.329080\pi$
$61$	7.99033	1.02306	0.511528	+	0.859266i	$0.329080\pi$
$62$	0	0
$63$	−28.8673	−3.63693
$64$	0	0
$65$	−7.89435	−0.979173
$66$	0	0
$67$	3.82891	0.467776	0.233888	−	0.972264i	$-0.424855\pi$
$67$	3.82891	0.467776	0.233888	+	0.972264i	$0.424855\pi$
$68$	0	0
$69$	−3.22226	−0.387915
$70$	0	0
$71$	11.3390	1.34569	0.672846	−	0.739783i	$-0.265072\pi$
$71$	11.3390	1.34569	0.672846	+	0.739783i	$0.265072\pi$
$72$	0	0
$73$	−16.6647	−1.95046	−0.975228	−	0.221203i	$-0.929002\pi$
$73$	−16.6647	−1.95046	−0.975228	+	0.221203i	$0.929002\pi$
$74$	0	0
$75$	−9.12994	−1.05424
$76$	0	0
$77$	8.23532	0.938501
$78$	0	0
$79$	12.7709	1.43683	0.718416	−	0.695613i	$-0.244867\pi$
$79$	12.7709	1.43683	0.718416	+	0.695613i	$0.244867\pi$
$80$	0	0
$81$	20.2781	2.25312
$82$	0	0
$83$	17.6711	1.93965	0.969826	−	0.243796i	$-0.0783928\pi$
$83$	17.6711	1.93965	0.969826	+	0.243796i	$0.0783928\pi$
$84$	0	0
$85$	−0.155969	−0.0169173
$86$	0	0
$87$	6.27130	0.672354
$88$	0	0
$89$	12.5472	1.33000	0.664999	−	0.746844i	$-0.268432\pi$
$89$	12.5472	1.33000	0.664999	+	0.746844i	$0.268432\pi$
$90$	0	0
$91$	−21.9497	−2.30095
$92$	0	0
$93$	3.46277	0.359073
$94$	0	0
$95$	−10.6639	−1.09410
$96$	0	0
$97$	12.5084	1.27003	0.635017	−	0.772498i	$-0.280993\pi$
$97$	12.5084	1.27003	0.635017	+	0.772498i	$0.280993\pi$
$98$	0	0
$99$	−14.4418	−1.45146
$100$	0	0
$101$	15.2880	1.52121	0.760607	−	0.649212i	$-0.224901\pi$
$101$	15.2880	1.52121	0.760607	+	0.649212i	$0.224901\pi$
$102$	0	0
$103$	−16.9274	−1.66790	−0.833952	−	0.551838i	$-0.813927\pi$
$103$	−16.9274	−1.66790	−0.833952	+	0.551838i	$0.813927\pi$
$104$	0	0
$105$	18.8293	1.83755
$106$	0	0
$107$	19.0345	1.84013	0.920065	−	0.391766i	$-0.128136\pi$
$107$	19.0345	1.84013	0.920065	+	0.391766i	$0.128136\pi$
$108$	0	0
$109$	0.517961	0.0496117	0.0248059	−	0.999692i	$-0.492103\pi$
$109$	0.517961	0.0496117	0.0248059	+	0.999692i	$0.492103\pi$
$110$	0	0
$111$	−17.7633	−1.68602
$112$	0	0
$113$	2.99837	0.282063	0.141032	−	0.990005i	$-0.454958\pi$
$113$	2.99837	0.282063	0.141032	+	0.990005i	$0.454958\pi$
$114$	0	0
$115$	1.47842	0.137863
$116$	0	0
$117$	38.4920	3.55858
$118$	0	0
$119$	−0.433662	−0.0397537
$120$	0	0
$121$	−6.88000	−0.625454
$122$	0	0
$123$	−7.16203	−0.645778
$124$	0	0
$125$	11.4850	1.02725
$126$	0	0
$127$	−4.95573	−0.439750	−0.219875	−	0.975528i	$-0.570565\pi$
$127$	−4.95573	−0.439750	−0.219875	+	0.975528i	$0.570565\pi$
$128$	0	0
$129$	8.36023	0.736077
$130$	0	0
$131$	−3.45367	−0.301748	−0.150874	−	0.988553i	$-0.548209\pi$
$131$	−3.45367	−0.301748	−0.150874	+	0.988553i	$0.548209\pi$
$132$	0	0
$133$	−29.6503	−2.57101
$134$	0	0
$135$	−19.0973	−1.64363
$136$	0	0
$137$	−12.0416	−1.02878	−0.514390	−	0.857556i	$-0.671982\pi$
$137$	−12.0416	−1.02878	−0.514390	+	0.857556i	$0.671982\pi$
$138$	0	0
$139$	−9.30408	−0.789161	−0.394581	−	0.918861i	$-0.629110\pi$
$139$	−9.30408	−0.789161	−0.394581	+	0.918861i	$0.629110\pi$
$140$	0	0
$141$	−3.18041	−0.267839
$142$	0	0
$143$	−10.9811	−0.918284
$144$	0	0
$145$	−2.87737	−0.238952
$146$	0	0
$147$	30.0907	2.48184
$148$	0	0
$149$	6.55951	0.537376	0.268688	−	0.963227i	$-0.413410\pi$
$149$	6.55951	0.537376	0.268688	+	0.963227i	$0.413410\pi$
$150$	0	0
$151$	6.40247	0.521026	0.260513	−	0.965470i	$-0.416108\pi$
$151$	6.40247	0.521026	0.260513	+	0.965470i	$0.416108\pi$
$152$	0	0
$153$	0.760490	0.0614820
$154$	0	0
$155$	−1.58877	−0.127613
$156$	0	0
$157$	−12.3387	−0.984735	−0.492368	−	0.870387i	$-0.663868\pi$
$157$	−12.3387	−0.984735	−0.492368	+	0.870387i	$0.663868\pi$
$158$	0	0
$159$	−12.2117	−0.968455
$160$	0	0
$161$	4.11064	0.323964
$162$	0	0
$163$	−14.1632	−1.10934	−0.554672	−	0.832069i	$-0.687156\pi$
$163$	−14.1632	−1.10934	−0.554672	+	0.832069i	$0.687156\pi$
$164$	0	0
$165$	9.42001	0.733347
$166$	0	0
$167$	−3.36038	−0.260034	−0.130017	−	0.991512i	$-0.541503\pi$
$167$	−3.36038	−0.260034	−0.130017	+	0.991512i	$0.541503\pi$
$168$	0	0
$169$	16.2680	1.25138
$170$	0	0
$171$	51.9962	3.97625
$172$	0	0
$173$	−8.27769	−0.629341	−0.314670	−	0.949201i	$-0.601894\pi$
$173$	−8.27769	−0.629341	−0.314670	+	0.949201i	$0.601894\pi$
$174$	0	0
$175$	11.6471	0.880437
$176$	0	0
$177$	20.6539	1.55244
$178$	0	0
$179$	19.1053	1.42800	0.714000	−	0.700146i	$-0.246882\pi$
$179$	19.1053	1.42800	0.714000	+	0.700146i	$0.246882\pi$
$180$	0	0
$181$	24.2994	1.80616	0.903080	−	0.429473i	$-0.141301\pi$
$181$	24.2994	1.80616	0.903080	+	0.429473i	$0.141301\pi$
$182$	0	0
$183$	25.4125	1.87854
$184$	0	0
$185$	8.15005	0.599204
$186$	0	0
$187$	−0.216954	−0.0158653
$188$	0	0
$189$	−53.0985	−3.86235
$190$	0	0
$191$	24.2239	1.75278	0.876388	−	0.481605i	$-0.159946\pi$
$191$	24.2239	1.75278	0.876388	+	0.481605i	$0.159946\pi$
$192$	0	0
$193$	−5.07978	−0.365651	−0.182825	−	0.983145i	$-0.558524\pi$
$193$	−5.07978	−0.365651	−0.182825	+	0.983145i	$0.558524\pi$
$194$	0	0
$195$	−25.1072	−1.79797
$196$	0	0
$197$	−21.1181	−1.50460	−0.752300	−	0.658821i	$-0.771055\pi$
$197$	−21.1181	−1.50460	−0.752300	+	0.658821i	$0.771055\pi$
$198$	0	0
$199$	18.6005	1.31855	0.659277	−	0.751900i	$-0.270863\pi$
$199$	18.6005	1.31855	0.659277	+	0.751900i	$0.270863\pi$
$200$	0	0
$201$	12.1775	0.858935
$202$	0	0
$203$	−8.00031	−0.561512
$204$	0	0
$205$	3.28605	0.229507
$206$	0	0
$207$	−7.20862	−0.501034
$208$	0	0
$209$	−14.8336	−1.02606
$210$	0	0
$211$	−11.9473	−0.822485	−0.411242	−	0.911526i	$-0.634905\pi$
$211$	−11.9473	−0.822485	−0.411242	+	0.911526i	$0.634905\pi$
$212$	0	0
$213$	36.0626	2.47097
$214$	0	0
$215$	−3.83580	−0.261599
$216$	0	0
$217$	−4.41747	−0.299877
$218$	0	0
$219$	−53.0005	−3.58144
$220$	0	0
$221$	0.578250	0.0388973
$222$	0	0
$223$	3.45381	0.231284	0.115642	−	0.993291i	$-0.463107\pi$
$223$	3.45381	0.231284	0.115642	+	0.993291i	$0.463107\pi$
$224$	0	0
$225$	−20.4249	−1.36166
$226$	0	0
$227$	−2.61154	−0.173334	−0.0866669	−	0.996237i	$-0.527622\pi$
$227$	−2.61154	−0.173334	−0.0866669	+	0.996237i	$0.527622\pi$
$228$	0	0
$229$	0.305737	0.0202037	0.0101018	−	0.999949i	$-0.496784\pi$
$229$	0.305737	0.0202037	0.0101018	+	0.999949i	$0.496784\pi$
$230$	0	0
$231$	26.1917	1.72328
$232$	0	0
$233$	0.487328	0.0319259	0.0159630	−	0.999873i	$-0.494919\pi$
$233$	0.487328	0.0319259	0.0159630	+	0.999873i	$0.494919\pi$
$234$	0	0
$235$	1.45922	0.0951888
$236$	0	0
$237$	40.6165	2.63832
$238$	0	0
$239$	18.2723	1.18194	0.590969	−	0.806694i	$-0.298746\pi$
$239$	18.2723	1.18194	0.590969	+	0.806694i	$0.298746\pi$
$240$	0	0
$241$	−14.1469	−0.911280	−0.455640	−	0.890164i	$-0.650590\pi$
$241$	−14.1469	−0.911280	−0.455640	+	0.890164i	$0.650590\pi$
$242$	0	0
$243$	25.2305	1.61854
$244$	0	0
$245$	−13.8060	−0.882036
$246$	0	0
$247$	39.5361	2.51562
$248$	0	0
$249$	56.2012	3.56161
$250$	0	0
$251$	−2.61574	−0.165104	−0.0825521	−	0.996587i	$-0.526307\pi$
$251$	−2.61574	−0.165104	−0.0825521	+	0.996587i	$0.526307\pi$
$252$	0	0
$253$	2.05649	0.129290
$254$	0	0
$255$	−0.496046	−0.0310636
$256$	0	0
$257$	−13.1456	−0.820000	−0.410000	−	0.912085i	$-0.634471\pi$
$257$	−13.1456	−0.820000	−0.410000	+	0.912085i	$0.634471\pi$
$258$	0	0
$259$	22.6606	1.40806
$260$	0	0
$261$	14.0297	0.868418
$262$	0	0
$263$	−5.71157	−0.352190	−0.176095	−	0.984373i	$-0.556347\pi$
$263$	−5.71157	−0.352190	−0.176095	+	0.984373i	$0.556347\pi$
$264$	0	0
$265$	5.60293	0.344185
$266$	0	0
$267$	39.9051	2.44215
$268$	0	0
$269$	2.51021	0.153050	0.0765252	−	0.997068i	$-0.475617\pi$
$269$	2.51021	0.153050	0.0765252	+	0.997068i	$0.475617\pi$
$270$	0	0
$271$	9.56391	0.580966	0.290483	−	0.956880i	$-0.406184\pi$
$271$	9.56391	0.580966	0.290483	+	0.956880i	$0.406184\pi$
$272$	0	0
$273$	−69.8089	−4.22502
$274$	0	0
$275$	5.82685	0.351372
$276$	0	0
$277$	5.50843	0.330970	0.165485	−	0.986212i	$-0.447081\pi$
$277$	5.50843	0.330970	0.165485	+	0.986212i	$0.447081\pi$
$278$	0	0
$279$	7.74668	0.463782
$280$	0	0
$281$	−25.2477	−1.50615	−0.753077	−	0.657932i	$-0.771431\pi$
$281$	−25.2477	−1.50615	−0.753077	+	0.657932i	$0.771431\pi$
$282$	0	0
$283$	−10.2704	−0.610514	−0.305257	−	0.952270i	$-0.598742\pi$
$283$	−10.2704	−0.610514	−0.305257	+	0.952270i	$0.598742\pi$
$284$	0	0
$285$	−33.9156	−2.00899
$286$	0	0
$287$	9.13661	0.539317
$288$	0	0
$289$	−16.9886	−0.999328
$290$	0	0
$291$	39.7818	2.33205
$292$	0	0
$293$	16.1167	0.941549	0.470774	−	0.882254i	$-0.343975\pi$
$293$	16.1167	0.941549	0.470774	+	0.882254i	$0.343975\pi$
$294$	0	0
$295$	−9.47629	−0.551731
$296$	0	0
$297$	−26.5644	−1.54142
$298$	0	0
$299$	−5.48118	−0.316985
$300$	0	0
$301$	−10.6652	−0.614730
$302$	0	0
$303$	48.6221	2.79327
$304$	0	0
$305$	−11.6596	−0.667628
$306$	0	0
$307$	−4.93801	−0.281827	−0.140914	−	0.990022i	$-0.545004\pi$
$307$	−4.93801	−0.281827	−0.140914	+	0.990022i	$0.545004\pi$
$308$	0	0
$309$	−53.8359	−3.06262
$310$	0	0
$311$	17.8321	1.01117	0.505584	−	0.862777i	$-0.331277\pi$
$311$	17.8321	1.01117	0.505584	+	0.862777i	$0.331277\pi$
$312$	0	0
$313$	17.4098	0.984058	0.492029	−	0.870579i	$-0.336255\pi$
$313$	17.4098	0.984058	0.492029	+	0.870579i	$0.336255\pi$
$314$	0	0
$315$	42.1236	2.37340
$316$	0	0
$317$	−31.1986	−1.75229	−0.876144	−	0.482049i	$-0.839893\pi$
$317$	−31.1986	−1.75229	−0.876144	+	0.482049i	$0.839893\pi$
$318$	0	0
$319$	−4.00243	−0.224093
$320$	0	0
$321$	60.5373	3.37886
$322$	0	0
$323$	0.781118	0.0434626
$324$	0	0
$325$	−15.5304	−0.861470
$326$	0	0
$327$	1.64733	0.0910974
$328$	0	0
$329$	4.05725	0.223683
$330$	0	0
$331$	−19.7595	−1.08608	−0.543040	−	0.839707i	$-0.682727\pi$
$331$	−19.7595	−1.08608	−0.543040	+	0.839707i	$0.682727\pi$
$332$	0	0
$333$	−39.7388	−2.17767
$334$	0	0
$335$	−5.58722	−0.305262
$336$	0	0
$337$	24.1786	1.31709	0.658545	−	0.752541i	$-0.271172\pi$
$337$	24.1786	1.31709	0.658545	+	0.752541i	$0.271172\pi$
$338$	0	0
$339$	9.53604	0.517927
$340$	0	0
$341$	−2.20999	−0.119678
$342$	0	0
$343$	−9.98598	−0.539192
$344$	0	0
$345$	4.70198	0.253146
$346$	0	0
$347$	26.3533	1.41472	0.707360	−	0.706853i	$-0.249886\pi$
$347$	26.3533	1.41472	0.707360	+	0.706853i	$0.249886\pi$
$348$	0	0
$349$	−20.9932	−1.12374	−0.561871	−	0.827225i	$-0.689918\pi$
$349$	−20.9932	−1.12374	−0.561871	+	0.827225i	$0.689918\pi$
$350$	0	0
$351$	70.8023	3.77915
$352$	0	0
$353$	4.36468	0.232308	0.116154	−	0.993231i	$-0.462943\pi$
$353$	4.36468	0.232308	0.116154	+	0.993231i	$0.462943\pi$
$354$	0	0
$355$	−16.5461	−0.878174
$356$	0	0
$357$	−1.37922	−0.0729961
$358$	0	0
$359$	−3.25940	−0.172024	−0.0860122	−	0.996294i	$-0.527412\pi$
$359$	−3.25940	−0.172024	−0.0860122	+	0.996294i	$0.527412\pi$
$360$	0	0
$361$	34.4066	1.81087
$362$	0	0
$363$	−21.8812	−1.14846
$364$	0	0
$365$	24.3174	1.27283
$366$	0	0
$367$	18.7606	0.979297	0.489648	−	0.871920i	$-0.337125\pi$
$367$	18.7606	0.979297	0.489648	+	0.871920i	$0.337125\pi$
$368$	0	0
$369$	−16.0224	−0.834093
$370$	0	0
$371$	15.5785	0.808798
$372$	0	0
$373$	23.9257	1.23882	0.619412	−	0.785066i	$-0.287371\pi$
$373$	23.9257	1.23882	0.619412	+	0.785066i	$0.287371\pi$
$374$	0	0
$375$	36.5271	1.88625
$376$	0	0
$377$	10.6677	0.549415
$378$	0	0
$379$	−26.2323	−1.34746	−0.673732	−	0.738976i	$-0.735310\pi$
$379$	−26.2323	−1.34746	−0.673732	+	0.738976i	$0.735310\pi$
$380$	0	0
$381$	−15.7612	−0.807473
$382$	0	0
$383$	1.76933	0.0904084	0.0452042	−	0.998978i	$-0.485606\pi$
$383$	1.76933	0.0904084	0.0452042	+	0.998978i	$0.485606\pi$
$384$	0	0
$385$	−12.0171	−0.612449
$386$	0	0
$387$	18.7029	0.950723
$388$	0	0
$389$	−2.51512	−0.127522	−0.0637608	−	0.997965i	$-0.520309\pi$
$389$	−2.51512	−0.127522	−0.0637608	+	0.997965i	$0.520309\pi$
$390$	0	0
$391$	−0.108292	−0.00547658
$392$	0	0
$393$	−10.9841	−0.554073
$394$	0	0
$395$	−18.6354	−0.937651
$396$	0	0
$397$	−28.1310	−1.41186	−0.705928	−	0.708284i	$-0.749470\pi$
$397$	−28.1310	−1.41186	−0.705928	+	0.708284i	$0.749470\pi$
$398$	0	0
$399$	−94.3000	−4.72090
$400$	0	0
$401$	−17.4170	−0.869764	−0.434882	−	0.900487i	$-0.643210\pi$
$401$	−17.4170	−0.869764	−0.434882	+	0.900487i	$0.643210\pi$
$402$	0	0
$403$	5.89031	0.293417
$404$	0	0
$405$	−29.5901	−1.47034
$406$	0	0
$407$	11.3368	0.561943
$408$	0	0
$409$	33.1284	1.63809	0.819047	−	0.573726i	$-0.194503\pi$
$409$	33.1284	1.63809	0.819047	+	0.573726i	$0.194503\pi$
$410$	0	0
$411$	−38.2971	−1.88905
$412$	0	0
$413$	−26.3482	−1.29651
$414$	0	0
$415$	−25.7859	−1.26578
$416$	0	0
$417$	−29.5907	−1.44906
$418$	0	0
$419$	18.5995	0.908647	0.454323	−	0.890837i	$-0.349881\pi$
$419$	18.5995	0.908647	0.454323	+	0.890837i	$0.349881\pi$
$420$	0	0
$421$	9.94782	0.484827	0.242414	−	0.970173i	$-0.422061\pi$
$421$	9.94782	0.484827	0.242414	+	0.970173i	$0.422061\pi$
$422$	0	0
$423$	−7.11498	−0.345942
$424$	0	0
$425$	−0.306835	−0.0148837
$426$	0	0
$427$	−32.4187	−1.56885
$428$	0	0
$429$	−34.9243	−1.68616
$430$	0	0
$431$	17.7134	0.853223	0.426611	−	0.904435i	$-0.359707\pi$
$431$	17.7134	0.853223	0.426611	+	0.904435i	$0.359707\pi$
$432$	0	0
$433$	25.7462	1.23728	0.618642	−	0.785673i	$-0.287683\pi$
$433$	25.7462	1.23728	0.618642	+	0.785673i	$0.287683\pi$
$434$	0	0
$435$	−9.15120	−0.438766
$436$	0	0
$437$	−7.40416	−0.354189
$438$	0	0
$439$	−23.6449	−1.12851	−0.564255	−	0.825601i	$-0.690836\pi$
$439$	−23.6449	−1.12851	−0.564255	+	0.825601i	$0.690836\pi$
$440$	0	0
$441$	67.3168	3.20556
$442$	0	0
$443$	−38.5442	−1.83129	−0.915645	−	0.401987i	$-0.868320\pi$
$443$	−38.5442	−1.83129	−0.915645	+	0.401987i	$0.868320\pi$
$444$	0	0
$445$	−18.3091	−0.867932
$446$	0	0
$447$	20.8619	0.986734
$448$	0	0
$449$	−0.766661	−0.0361810	−0.0180905	−	0.999836i	$-0.505759\pi$
$449$	−0.766661	−0.0361810	−0.0180905	+	0.999836i	$0.505759\pi$
$450$	0	0
$451$	4.57090	0.215235
$452$	0	0
$453$	20.3625	0.956712
$454$	0	0
$455$	32.0293	1.50156
$456$	0	0
$457$	−22.7540	−1.06439	−0.532194	−	0.846622i	$-0.678632\pi$
$457$	−22.7540	−1.06439	−0.532194	+	0.846622i	$0.678632\pi$
$458$	0	0
$459$	1.39885	0.0652926
$460$	0	0
$461$	−8.26367	−0.384878	−0.192439	−	0.981309i	$-0.561640\pi$
$461$	−8.26367	−0.384878	−0.192439	+	0.981309i	$0.561640\pi$
$462$	0	0
$463$	−29.4153	−1.36705	−0.683523	−	0.729929i	$-0.739553\pi$
$463$	−29.4153	−1.36705	−0.683523	+	0.729929i	$0.739553\pi$
$464$	0	0
$465$	−5.05294	−0.234325
$466$	0	0
$467$	−20.8895	−0.966651	−0.483325	−	0.875441i	$-0.660571\pi$
$467$	−20.8895	−0.966651	−0.483325	+	0.875441i	$0.660571\pi$
$468$	0	0
$469$	−15.5349	−0.717333
$470$	0	0
$471$	−39.2421	−1.80818
$472$	0	0
$473$	−5.33561	−0.245332
$474$	0	0
$475$	−20.9789	−0.962578
$476$	0	0
$477$	−27.3193	−1.25086
$478$	0	0
$479$	−37.6909	−1.72214	−0.861071	−	0.508486i	$-0.830206\pi$
$479$	−37.6909	−1.72214	−0.861071	+	0.508486i	$0.830206\pi$
$480$	0	0
$481$	−30.2160	−1.37773
$482$	0	0
$483$	13.0735	0.594866
$484$	0	0
$485$	−18.2525	−0.828801
$486$	0	0
$487$	−26.0373	−1.17986	−0.589931	−	0.807453i	$-0.700845\pi$
$487$	−26.0373	−1.17986	−0.589931	+	0.807453i	$0.700845\pi$
$488$	0	0
$489$	−45.0446	−2.03699
$490$	0	0
$491$	28.8519	1.30207	0.651034	−	0.759048i	$-0.274335\pi$
$491$	28.8519	1.30207	0.651034	+	0.759048i	$0.274335\pi$
$492$	0	0
$493$	0.210763	0.00949229
$494$	0	0
$495$	21.0738	0.947196
$496$	0	0
$497$	−46.0051	−2.06361
$498$	0	0
$499$	−29.0133	−1.29881	−0.649406	−	0.760442i	$-0.724982\pi$
$499$	−29.0133	−1.29881	−0.649406	+	0.760442i	$0.724982\pi$
$500$	0	0
$501$	−10.6874	−0.477476
$502$	0	0
$503$	24.0037	1.07027	0.535137	−	0.844765i	$-0.320260\pi$
$503$	24.0037	1.07027	0.535137	+	0.844765i	$0.320260\pi$
$504$	0	0
$505$	−22.3085	−0.992717
$506$	0	0
$507$	51.7388	2.29780
$508$	0	0
$509$	−34.5360	−1.53078	−0.765391	−	0.643566i	$-0.777454\pi$
$509$	−34.5360	−1.53078	−0.765391	+	0.643566i	$0.777454\pi$
$510$	0	0
$511$	67.6128	2.99102
$512$	0	0
$513$	95.6419	4.22269
$514$	0	0
$515$	24.7007	1.08844
$516$	0	0
$517$	2.02978	0.0892696
$518$	0	0
$519$	−26.3264	−1.15560
$520$	0	0
$521$	−5.76191	−0.252434	−0.126217	−	0.992003i	$-0.540284\pi$
$521$	−5.76191	−0.252434	−0.126217	+	0.992003i	$0.540284\pi$
$522$	0	0
$523$	−20.8912	−0.913507	−0.456754	−	0.889593i	$-0.650988\pi$
$523$	−20.8912	−0.913507	−0.456754	+	0.889593i	$0.650988\pi$
$524$	0	0
$525$	37.0425	1.61667
$526$	0	0
$527$	0.116375	0.00506939
$528$	0	0
$529$	−21.9735	−0.955370
$530$	0	0
$531$	46.2054	2.00514
$532$	0	0
$533$	−12.1829	−0.527699
$534$	0	0
$535$	−27.7754	−1.20084
$536$	0	0
$537$	60.7627	2.62210
$538$	0	0
$539$	−19.2043	−0.827187
$540$	0	0
$541$	−2.37755	−0.102219	−0.0511094	−	0.998693i	$-0.516276\pi$
$541$	−2.37755	−0.102219	−0.0511094	+	0.998693i	$0.516276\pi$
$542$	0	0
$543$	77.2819	3.31648
$544$	0	0
$545$	−0.755818	−0.0323757
$546$	0	0
$547$	−12.7628	−0.545697	−0.272848	−	0.962057i	$-0.587966\pi$
$547$	−12.7628	−0.545697	−0.272848	+	0.962057i	$0.587966\pi$
$548$	0	0
$549$	56.8510	2.42634
$550$	0	0
$551$	14.4103	0.613899
$552$	0	0
$553$	−51.8145	−2.20338
$554$	0	0
$555$	25.9205	1.10026
$556$	0	0
$557$	−4.05977	−0.172018	−0.0860090	−	0.996294i	$-0.527411\pi$
$557$	−4.05977	−0.172018	−0.0860090	+	0.996294i	$0.527411\pi$
$558$	0	0
$559$	14.2211	0.601487
$560$	0	0
$561$	−0.690002	−0.0291319
$562$	0	0
$563$	−43.4354	−1.83059	−0.915293	−	0.402789i	$-0.868041\pi$
$563$	−43.4354	−1.83059	−0.915293	+	0.402789i	$0.868041\pi$
$564$	0	0
$565$	−4.37528	−0.184069
$566$	0	0
$567$	−82.2731	−3.45515
$568$	0	0
$569$	0.518581	0.0217400	0.0108700	−	0.999941i	$-0.496540\pi$
$569$	0.518581	0.0217400	0.0108700	+	0.999941i	$0.496540\pi$
$570$	0	0
$571$	−17.1849	−0.719165	−0.359582	−	0.933113i	$-0.617081\pi$
$571$	−17.1849	−0.719165	−0.359582	+	0.933113i	$0.617081\pi$
$572$	0	0
$573$	77.0417	3.21846
$574$	0	0
$575$	2.90846	0.121291
$576$	0	0
$577$	32.6860	1.36074	0.680368	−	0.732870i	$-0.261820\pi$
$577$	32.6860	1.36074	0.680368	+	0.732870i	$0.261820\pi$
$578$	0	0
$579$	−16.1558	−0.671411
$580$	0	0
$581$	−71.6960	−2.97445
$582$	0	0
$583$	7.79370	0.322782
$584$	0	0
$585$	−56.1682	−2.32227
$586$	0	0
$587$	36.2411	1.49583	0.747916	−	0.663794i	$-0.231055\pi$
$587$	36.2411	1.49583	0.747916	+	0.663794i	$0.231055\pi$
$588$	0	0
$589$	7.95681	0.327855
$590$	0	0
$591$	−67.1640	−2.76276
$592$	0	0
$593$	25.5340	1.04856	0.524278	−	0.851547i	$-0.324335\pi$
$593$	25.5340	1.04856	0.524278	+	0.851547i	$0.324335\pi$
$594$	0	0
$595$	0.632807	0.0259425
$596$	0	0
$597$	59.1571	2.42114
$598$	0	0
$599$	21.9720	0.897752	0.448876	−	0.893594i	$-0.351825\pi$
$599$	21.9720	0.897752	0.448876	+	0.893594i	$0.351825\pi$
$600$	0	0
$601$	−43.3573	−1.76858	−0.884290	−	0.466939i	$-0.845357\pi$
$601$	−43.3573	−1.76858	−0.884290	+	0.466939i	$0.845357\pi$
$602$	0	0
$603$	27.2427	1.10941
$604$	0	0
$605$	10.0394	0.408160
$606$	0	0
$607$	3.21713	0.130579	0.0652896	−	0.997866i	$-0.479203\pi$
$607$	3.21713	0.130579	0.0652896	+	0.997866i	$0.479203\pi$
$608$	0	0
$609$	−25.4442	−1.03105
$610$	0	0
$611$	−5.40999	−0.218865
$612$	0	0
$613$	−28.3121	−1.14351	−0.571757	−	0.820423i	$-0.693738\pi$
$613$	−28.3121	−1.14351	−0.571757	+	0.820423i	$0.693738\pi$
$614$	0	0
$615$	10.4510	0.421423
$616$	0	0
$617$	−34.2998	−1.38086	−0.690429	−	0.723400i	$-0.742578\pi$
$617$	−34.2998	−1.38086	−0.690429	+	0.723400i	$0.742578\pi$
$618$	0	0
$619$	30.6276	1.23103	0.615514	−	0.788126i	$-0.288948\pi$
$619$	30.6276	1.23103	0.615514	+	0.788126i	$0.288948\pi$
$620$	0	0
$621$	−13.2596	−0.532088
$622$	0	0
$623$	−50.9070	−2.03955
$624$	0	0
$625$	−2.40575	−0.0962299
$626$	0	0
$627$	−47.1768	−1.88406
$628$	0	0
$629$	−0.596980	−0.0238032
$630$	0	0
$631$	1.48865	0.0592623	0.0296311	−	0.999561i	$-0.490567\pi$
$631$	1.48865	0.0592623	0.0296311	+	0.999561i	$0.490567\pi$
$632$	0	0
$633$	−37.9972	−1.51025
$634$	0	0
$635$	7.23149	0.286973
$636$	0	0
$637$	51.1853	2.02804
$638$	0	0
$639$	80.6768	3.19153
$640$	0	0
$641$	38.2965	1.51262	0.756310	−	0.654214i	$-0.227000\pi$
$641$	38.2965	1.51262	0.756310	+	0.654214i	$0.227000\pi$
$642$	0	0
$643$	−37.9178	−1.49533	−0.747666	−	0.664075i	$-0.768826\pi$
$643$	−37.9178	−1.49533	−0.747666	+	0.664075i	$0.768826\pi$
$644$	0	0
$645$	−12.1994	−0.480351
$646$	0	0
$647$	−27.8115	−1.09338	−0.546691	−	0.837334i	$-0.684113\pi$
$647$	−27.8115	−1.09338	−0.546691	+	0.837334i	$0.684113\pi$
$648$	0	0
$649$	−13.1816	−0.517422
$650$	0	0
$651$	−14.0493	−0.550637
$652$	0	0
$653$	−33.1847	−1.29862	−0.649308	−	0.760525i	$-0.724941\pi$
$653$	−33.1847	−1.29862	−0.649308	+	0.760525i	$0.724941\pi$
$654$	0	0
$655$	5.03965	0.196916
$656$	0	0
$657$	−118.569	−4.62582
$658$	0	0
$659$	6.47719	0.252315	0.126158	−	0.992010i	$-0.459735\pi$
$659$	6.47719	0.252315	0.126158	+	0.992010i	$0.459735\pi$
$660$	0	0
$661$	−10.6435	−0.413983	−0.206991	−	0.978343i	$-0.566367\pi$
$661$	−10.6435	−0.413983	−0.206991	+	0.978343i	$0.566367\pi$
$662$	0	0
$663$	1.83907	0.0714236
$664$	0	0
$665$	43.2662	1.67779
$666$	0	0
$667$	−1.99781	−0.0773554
$668$	0	0
$669$	10.9845	0.424686
$670$	0	0
$671$	−16.2186	−0.626112
$672$	0	0
$673$	−34.7167	−1.33823	−0.669115	−	0.743159i	$-0.733327\pi$
$673$	−34.7167	−1.33823	−0.669115	+	0.743159i	$0.733327\pi$
$674$	0	0
$675$	−37.5696	−1.44605
$676$	0	0
$677$	19.2706	0.740631	0.370315	−	0.928906i	$-0.379250\pi$
$677$	19.2706	0.740631	0.370315	+	0.928906i	$0.379250\pi$
$678$	0	0
$679$	−50.7496	−1.94759
$680$	0	0
$681$	−8.30575	−0.318277
$682$	0	0
$683$	23.4901	0.898822	0.449411	−	0.893325i	$-0.351634\pi$
$683$	23.4901	0.898822	0.449411	+	0.893325i	$0.351634\pi$
$684$	0	0
$685$	17.5713	0.671363
$686$	0	0
$687$	0.972368	0.0370982
$688$	0	0
$689$	−20.7726	−0.791374
$690$	0	0
$691$	6.23229	0.237087	0.118544	−	0.992949i	$-0.462177\pi$
$691$	6.23229	0.237087	0.118544	+	0.992949i	$0.462177\pi$
$692$	0	0
$693$	58.5942	2.22581
$694$	0	0
$695$	13.5767	0.514992
$696$	0	0
$697$	−0.240698	−0.00911710
$698$	0	0
$699$	1.54990	0.0586227
$700$	0	0
$701$	−20.5685	−0.776860	−0.388430	−	0.921478i	$-0.626982\pi$
$701$	−20.5685	−0.776860	−0.388430	+	0.921478i	$0.626982\pi$
$702$	0	0
$703$	−40.8167	−1.53943
$704$	0	0
$705$	4.64090	0.174787
$706$	0	0
$707$	−62.0273	−2.33278
$708$	0	0
$709$	−38.8033	−1.45729	−0.728643	−	0.684893i	$-0.759849\pi$
$709$	−38.8033	−1.45729	−0.728643	+	0.684893i	$0.759849\pi$
$710$	0	0
$711$	90.8644	3.40768
$712$	0	0
$713$	−1.10311	−0.0413119
$714$	0	0
$715$	16.0238	0.599255
$716$	0	0
$717$	58.1134	2.17028
$718$	0	0
$719$	14.5394	0.542227	0.271114	−	0.962547i	$-0.412608\pi$
$719$	14.5394	0.542227	0.271114	+	0.962547i	$0.412608\pi$
$720$	0	0
$721$	68.6785	2.55772
$722$	0	0
$723$	−44.9928	−1.67330
$724$	0	0
$725$	−5.66058	−0.210229
$726$	0	0
$727$	24.9338	0.924742	0.462371	−	0.886686i	$-0.346999\pi$
$727$	24.9338	0.924742	0.462371	+	0.886686i	$0.346999\pi$
$728$	0	0
$729$	19.4090	0.718853
$730$	0	0
$731$	0.280967	0.0103919
$732$	0	0
$733$	18.3941	0.679401	0.339700	−	0.940534i	$-0.389674\pi$
$733$	18.3941	0.679401	0.339700	+	0.940534i	$0.389674\pi$
$734$	0	0
$735$	−43.9088	−1.61960
$736$	0	0
$737$	−7.77185	−0.286280
$738$	0	0
$739$	8.39288	0.308737	0.154369	−	0.988013i	$-0.450666\pi$
$739$	8.39288	0.308737	0.154369	+	0.988013i	$0.450666\pi$
$740$	0	0
$741$	125.741	4.61920
$742$	0	0
$743$	−33.4135	−1.22582	−0.612912	−	0.790151i	$-0.710002\pi$
$743$	−33.4135	−1.22582	−0.612912	+	0.790151i	$0.710002\pi$
$744$	0	0
$745$	−9.57175	−0.350682
$746$	0	0
$747$	125.729	4.60020
$748$	0	0
$749$	−77.2275	−2.82183
$750$	0	0
$751$	−34.3613	−1.25386	−0.626930	−	0.779075i	$-0.715689\pi$
$751$	−34.3613	−1.25386	−0.626930	+	0.779075i	$0.715689\pi$
$752$	0	0
$753$	−8.31912	−0.303166
$754$	0	0
$755$	−9.34260	−0.340012
$756$	0	0
$757$	34.1908	1.24269	0.621343	−	0.783539i	$-0.286588\pi$
$757$	34.1908	1.24269	0.621343	+	0.783539i	$0.286588\pi$
$758$	0	0
$759$	6.54048	0.237404
$760$	0	0
$761$	31.3635	1.13693	0.568463	−	0.822709i	$-0.307538\pi$
$761$	31.3635	1.13693	0.568463	+	0.822709i	$0.307538\pi$
$762$	0	0
$763$	−2.10150	−0.0760794
$764$	0	0
$765$	−1.10972	−0.0401220
$766$	0	0
$767$	35.1330	1.26858
$768$	0	0
$769$	−11.4622	−0.413337	−0.206668	−	0.978411i	$-0.566262\pi$
$769$	−11.4622	−0.413337	−0.206668	+	0.978411i	$0.566262\pi$
$770$	0	0
$771$	−41.8084	−1.50569
$772$	0	0
$773$	31.4988	1.13293	0.566466	−	0.824085i	$-0.308310\pi$
$773$	31.4988	1.13293	0.566466	+	0.824085i	$0.308310\pi$
$774$	0	0
$775$	−3.12555	−0.112273
$776$	0	0
$777$	72.0700	2.58550
$778$	0	0
$779$	−16.4570	−0.589634
$780$	0	0
$781$	−23.0157	−0.823565
$782$	0	0
$783$	25.8063	0.922243
$784$	0	0
$785$	18.0048	0.642620
$786$	0	0
$787$	46.4451	1.65559	0.827794	−	0.561033i	$-0.189596\pi$
$787$	46.4451	1.65559	0.827794	+	0.561033i	$0.189596\pi$
$788$	0	0
$789$	−18.1651	−0.646695
$790$	0	0
$791$	−12.1651	−0.432543
$792$	0	0
$793$	43.2276	1.53506
$794$	0	0
$795$	17.8196	0.631996
$796$	0	0
$797$	−40.4667	−1.43341	−0.716703	−	0.697379i	$-0.754349\pi$
$797$	−40.4667	−1.43341	−0.716703	+	0.697379i	$0.754349\pi$
$798$	0	0
$799$	−0.106886	−0.00378134
$800$	0	0
$801$	89.2730	3.15430
$802$	0	0
$803$	33.8256	1.19368
$804$	0	0
$805$	−5.99832	−0.211413
$806$	0	0
$807$	7.98350	0.281033
$808$	0	0
$809$	−44.6559	−1.57002	−0.785009	−	0.619484i	$-0.787342\pi$
$809$	−44.6559	−1.57002	−0.785009	+	0.619484i	$0.787342\pi$
$810$	0	0
$811$	−19.2845	−0.677170	−0.338585	−	0.940936i	$-0.609948\pi$
$811$	−19.2845	−0.677170	−0.338585	+	0.940936i	$0.609948\pi$
$812$	0	0
$813$	30.4171	1.06677
$814$	0	0
$815$	20.6671	0.723938
$816$	0	0
$817$	19.2103	0.672082
$818$	0	0
$819$	−156.172	−5.45708
$820$	0	0
$821$	−23.8982	−0.834053	−0.417027	−	0.908894i	$-0.636928\pi$
$821$	−23.8982	−0.834053	−0.417027	+	0.908894i	$0.636928\pi$
$822$	0	0
$823$	−18.9797	−0.661592	−0.330796	−	0.943702i	$-0.607317\pi$
$823$	−18.9797	−0.661592	−0.330796	+	0.943702i	$0.607317\pi$
$824$	0	0
$825$	18.5318	0.645193
$826$	0	0
$827$	−16.2913	−0.566504	−0.283252	−	0.959045i	$-0.591413\pi$
$827$	−16.2913	−0.566504	−0.283252	+	0.959045i	$0.591413\pi$
$828$	0	0
$829$	8.20673	0.285031	0.142516	−	0.989793i	$-0.454481\pi$
$829$	8.20673	0.285031	0.142516	+	0.989793i	$0.454481\pi$
$830$	0	0
$831$	17.5191	0.607730
$832$	0	0
$833$	1.01127	0.0350386
$834$	0	0
$835$	4.90352	0.169693
$836$	0	0
$837$	14.2493	0.492527
$838$	0	0
$839$	31.9591	1.10335	0.551676	−	0.834058i	$-0.313988\pi$
$839$	31.9591	1.10335	0.551676	+	0.834058i	$0.313988\pi$
$840$	0	0
$841$	−25.1118	−0.865924
$842$	0	0
$843$	−80.2981	−2.76561
$844$	0	0
$845$	−23.7385	−0.816629
$846$	0	0
$847$	27.9139	0.959132
$848$	0	0
$849$	−32.6641	−1.12103
$850$	0	0
$851$	5.65872	0.193979
$852$	0	0
$853$	30.1210	1.03132	0.515661	−	0.856792i	$-0.327546\pi$
$853$	30.1210	1.03132	0.515661	+	0.856792i	$0.327546\pi$
$854$	0	0
$855$	−75.8737	−2.59483
$856$	0	0
$857$	4.56463	0.155925	0.0779625	−	0.996956i	$-0.475159\pi$
$857$	4.56463	0.155925	0.0779625	+	0.996956i	$0.475159\pi$
$858$	0	0
$859$	14.8525	0.506761	0.253381	−	0.967367i	$-0.418457\pi$
$859$	14.8525	0.506761	0.253381	+	0.967367i	$0.418457\pi$
$860$	0	0
$861$	29.0581	0.990299
$862$	0	0
$863$	−27.8901	−0.949388	−0.474694	−	0.880151i	$-0.657441\pi$
$863$	−27.8901	−0.949388	−0.474694	+	0.880151i	$0.657441\pi$
$864$	0	0
$865$	12.0789	0.410696
$866$	0	0
$867$	−54.0306	−1.83497
$868$	0	0
$869$	−25.9220	−0.879344
$870$	0	0
$871$	20.7144	0.701880
$872$	0	0
$873$	88.9970	3.01209
$874$	0	0
$875$	−46.5977	−1.57529
$876$	0	0
$877$	−13.0636	−0.441125	−0.220563	−	0.975373i	$-0.570789\pi$
$877$	−13.0636	−0.441125	−0.220563	+	0.975373i	$0.570789\pi$
$878$	0	0
$879$	51.2577	1.72888
$880$	0	0
$881$	−12.5316	−0.422202	−0.211101	−	0.977464i	$-0.567705\pi$
$881$	−12.5316	−0.422202	−0.211101	+	0.977464i	$0.567705\pi$
$882$	0	0
$883$	41.0319	1.38083	0.690416	−	0.723413i	$-0.257428\pi$
$883$	41.0319	1.38083	0.690416	+	0.723413i	$0.257428\pi$
$884$	0	0
$885$	−30.1385	−1.01309
$886$	0	0
$887$	34.8695	1.17080	0.585402	−	0.810743i	$-0.300937\pi$
$887$	34.8695	1.17080	0.585402	+	0.810743i	$0.300937\pi$
$888$	0	0
$889$	20.1066	0.674355
$890$	0	0
$891$	−41.1600	−1.37891
$892$	0	0
$893$	−7.30798	−0.244552
$894$	0	0
$895$	−27.8788	−0.931886
$896$	0	0
$897$	−17.4324	−0.582051
$898$	0	0
$899$	2.14692	0.0716040
$900$	0	0
$901$	−0.410407	−0.0136726
$902$	0	0
$903$	−33.9195	−1.12877
$904$	0	0
$905$	−35.4581	−1.17867
$906$	0	0
$907$	1.00403	0.0333382	0.0166691	−	0.999861i	$-0.494694\pi$
$907$	1.00403	0.0333382	0.0166691	+	0.999861i	$0.494694\pi$
$908$	0	0
$909$	108.774	3.60781
$910$	0	0
$911$	11.5623	0.383075	0.191538	−	0.981485i	$-0.438653\pi$
$911$	11.5623	0.383075	0.191538	+	0.981485i	$0.438653\pi$
$912$	0	0
$913$	−35.8684	−1.18707
$914$	0	0
$915$	−37.0823	−1.22590
$916$	0	0
$917$	14.0124	0.462730
$918$	0	0
$919$	41.9705	1.38448	0.692239	−	0.721668i	$-0.256624\pi$
$919$	41.9705	1.38448	0.692239	+	0.721668i	$0.256624\pi$
$920$	0	0
$921$	−15.7049	−0.517494
$922$	0	0
$923$	61.3439	2.01916
$924$	0	0
$925$	16.0334	0.527175
$926$	0	0
$927$	−120.438	−3.95570
$928$	0	0
$929$	16.4059	0.538261	0.269131	−	0.963104i	$-0.413264\pi$
$929$	16.4059	0.538261	0.269131	+	0.963104i	$0.413264\pi$
$930$	0	0
$931$	69.1428	2.26606
$932$	0	0
$933$	56.7135	1.85672
$934$	0	0
$935$	0.316583	0.0103534
$936$	0	0
$937$	21.9613	0.717446	0.358723	−	0.933444i	$-0.383212\pi$
$937$	21.9613	0.717446	0.358723	+	0.933444i	$0.383212\pi$
$938$	0	0
$939$	55.3701	1.80694
$940$	0	0
$941$	−19.8290	−0.646406	−0.323203	−	0.946330i	$-0.604760\pi$
$941$	−19.8290	−0.646406	−0.323203	+	0.946330i	$0.604760\pi$
$942$	0	0
$943$	2.28156	0.0742978
$944$	0	0
$945$	77.4823	2.52050
$946$	0	0
$947$	−26.4809	−0.860513	−0.430256	−	0.902707i	$-0.641577\pi$
$947$	−26.4809	−0.860513	−0.430256	+	0.902707i	$0.641577\pi$
$948$	0	0
$949$	−90.1558	−2.92658
$950$	0	0
$951$	−99.2243	−3.21757
$952$	0	0
$953$	1.95808	0.0634285	0.0317143	−	0.999497i	$-0.489903\pi$
$953$	1.95808	0.0634285	0.0317143	+	0.999497i	$0.489903\pi$
$954$	0	0
$955$	−35.3479	−1.14383
$956$	0	0
$957$	−12.7294	−0.411482
$958$	0	0
$959$	48.8556	1.57763
$960$	0	0
$961$	−29.8146	−0.961760
$962$	0	0
$963$	135.430	4.36417
$964$	0	0
$965$	7.41251	0.238617
$966$	0	0
$967$	−17.3590	−0.558228	−0.279114	−	0.960258i	$-0.590041\pi$
$967$	−17.3590	−0.558228	−0.279114	+	0.960258i	$0.590041\pi$
$968$	0	0
$969$	2.48427	0.0798064
$970$	0	0
$971$	60.5170	1.94208	0.971042	−	0.238908i	$-0.0767894\pi$
$971$	60.5170	1.94208	0.971042	+	0.238908i	$0.0767894\pi$
$972$	0	0
$973$	37.7490	1.21018
$974$	0	0
$975$	−49.3929	−1.58184
$976$	0	0
$977$	35.8288	1.14626	0.573132	−	0.819463i	$-0.305728\pi$
$977$	35.8288	1.14626	0.573132	+	0.819463i	$0.305728\pi$
$978$	0	0
$979$	−25.4680	−0.813960
$980$	0	0
$981$	3.68529	0.117662
$982$	0	0
$983$	9.08037	0.289619	0.144809	−	0.989460i	$-0.453743\pi$
$983$	9.08037	0.289619	0.144809	+	0.989460i	$0.453743\pi$
$984$	0	0
$985$	30.8158	0.981874
$986$	0	0
$987$	12.9037	0.410729
$988$	0	0
$989$	−2.66326	−0.0846868
$990$	0	0
$991$	−2.09371	−0.0665088	−0.0332544	−	0.999447i	$-0.510587\pi$
$991$	−2.09371	−0.0665088	−0.0332544	+	0.999447i	$0.510587\pi$
$992$	0	0
$993$	−62.8433	−1.99427
$994$	0	0
$995$	−27.1422	−0.860464
$996$	0	0
$997$	−4.11408	−0.130294	−0.0651471	−	0.997876i	$-0.520752\pi$
$997$	−4.11408	−0.130294	−0.0651471	+	0.997876i	$0.520752\pi$
$998$	0	0
$999$	−73.0956	−2.31264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6016.2.a.o.1.12	yes	13
4.3	odd	2		6016.2.a.m.1.2	✓	13
8.3	odd	2		6016.2.a.p.1.12	yes	13
8.5	even	2		6016.2.a.n.1.2	yes	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6016.2.a.m.1.2	✓	13	4.3	odd	2
6016.2.a.n.1.2	yes	13	8.5	even	2
6016.2.a.o.1.12	yes	13	1.1	even	1	trivial
6016.2.a.p.1.12	yes	13	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6016 = 2^{7} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6016.a (trivial)

\(f(q)\)	\(=\)	\(q+3.18041 q^{3} -1.45922 q^{5} -4.05725 q^{7} +7.11498 q^{9} +O(q^{10})\)	\(q+3.18041 q^{3} -1.45922 q^{5} -4.05725 q^{7} +7.11498 q^{9} -2.02978 q^{11} +5.40999 q^{13} -4.64090 q^{15} +0.106886 q^{17} +7.30798 q^{19} -12.9037 q^{21} -1.01316 q^{23} -2.87068 q^{25} +13.0873 q^{27} +1.97186 q^{29} +1.08878 q^{31} -6.45552 q^{33} +5.92041 q^{35} -5.58522 q^{37} +17.2060 q^{39} -2.25192 q^{41} +2.62867 q^{43} -10.3823 q^{45} -1.00000 q^{47} +9.46127 q^{49} +0.339940 q^{51} -3.83968 q^{53} +2.96189 q^{55} +23.2423 q^{57} +6.49409 q^{59} +7.99033 q^{61} -28.8673 q^{63} -7.89435 q^{65} +3.82891 q^{67} -3.22226 q^{69} +11.3390 q^{71} -16.6647 q^{73} -9.12994 q^{75} +8.23532 q^{77} +12.7709 q^{79} +20.2781 q^{81} +17.6711 q^{83} -0.155969 q^{85} +6.27130 q^{87} +12.5472 q^{89} -21.9497 q^{91} +3.46277 q^{93} -10.6639 q^{95} +12.5084 q^{97} -14.4418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9}+O(q^{10}) \) 13 * q + 4 * q^3 - 6 * q^5 - 2 * q^7 + 21 * q^9	\( 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9} + 10 q^{11} - 4 q^{13} + 14 q^{15} + 10 q^{17} + 8 q^{19} - 10 q^{21} + 18 q^{23} + 23 q^{25} + 16 q^{27} - 14 q^{29} + 4 q^{31} + 14 q^{33} + 14 q^{35} - 16 q^{37} + 12 q^{39} + 10 q^{41} + 12 q^{43} - 10 q^{45} - 13 q^{47} + 9 q^{49} + 22 q^{51} - 26 q^{53} - 2 q^{55} + 20 q^{57} + 30 q^{59} - 18 q^{61} + 12 q^{63} - 4 q^{65} + 4 q^{67} - 2 q^{69} + 36 q^{71} + 10 q^{73} + 38 q^{75} - 42 q^{77} + 21 q^{81} + 12 q^{83} - 4 q^{85} + 6 q^{87} + 50 q^{89} - 4 q^{91} - 52 q^{93} + 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) \) 13 * q + 4 * q^3 - 6 * q^5 - 2 * q^7 + 21 * q^9 + 10 * q^11 - 4 * q^13 + 14 * q^15 + 10 * q^17 + 8 * q^19 - 10 * q^21 + 18 * q^23 + 23 * q^25 + 16 * q^27 - 14 * q^29 + 4 * q^31 + 14 * q^33 + 14 * q^35 - 16 * q^37 + 12 * q^39 + 10 * q^41 + 12 * q^43 - 10 * q^45 - 13 * q^47 + 9 * q^49 + 22 * q^51 - 26 * q^53 - 2 * q^55 + 20 * q^57 + 30 * q^59 - 18 * q^61 + 12 * q^63 - 4 * q^65 + 4 * q^67 - 2 * q^69 + 36 * q^71 + 10 * q^73 + 38 * q^75 - 42 * q^77 + 21 * q^81 + 12 * q^83 - 4 * q^85 + 6 * q^87 + 50 * q^89 - 4 * q^91 - 52 * q^93 + 8 * q^95 - 10 * q^97 + 34 * q^99

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