LMFDB - Embedded newform 6036.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 4 x^{14} - 27 x^{13} + 106 x^{12} + 266 x^{11} - 1004 x^{10} - 1105 x^{9} + 4076 x^{8} + \cdots - 44 $ x^15 - 4x^14 - 27x^13 + 106x^12 + 266x^11 - 1004x^10 - 1105x^9 + 4076x^8 + 1501x^7 - 7100x^6 - 134x^5 + 5356x^4 - 1041x^3 - 1381x^2 + 543x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.01525$ of defining polynomial
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.01525 q^{5} -3.41124 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.01525 q^{5} -3.41124 q^{7} +1.00000 q^{9} -0.942703 q^{11} -0.0524594 q^{13} -4.01525 q^{15} +2.46892 q^{17} +6.45629 q^{19} -3.41124 q^{21} -1.91648 q^{23} +11.1223 q^{25} +1.00000 q^{27} -1.42443 q^{29} +3.96306 q^{31} -0.942703 q^{33} +13.6970 q^{35} +6.45415 q^{37} -0.0524594 q^{39} +4.79510 q^{41} +3.95926 q^{43} -4.01525 q^{45} -7.81224 q^{47} +4.63656 q^{49} +2.46892 q^{51} -5.37123 q^{53} +3.78519 q^{55} +6.45629 q^{57} +1.63643 q^{59} -3.15845 q^{61} -3.41124 q^{63} +0.210638 q^{65} -2.49181 q^{67} -1.91648 q^{69} +3.47255 q^{71} -11.3160 q^{73} +11.1223 q^{75} +3.21578 q^{77} -5.58817 q^{79} +1.00000 q^{81} -6.10715 q^{83} -9.91333 q^{85} -1.42443 q^{87} -11.5017 q^{89} +0.178952 q^{91} +3.96306 q^{93} -25.9236 q^{95} -4.98650 q^{97} -0.942703 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * 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
$5$	−4.01525	−1.79568	−0.897838	−	0.440326i	$-0.854863\pi$
$5$	−4.01525	−1.79568	−0.897838	+	0.440326i	$0.854863\pi$
$6$	0	0
$7$	−3.41124	−1.28933	−0.644664	−	0.764466i	$-0.723003\pi$
$7$	−3.41124	−1.28933	−0.644664	+	0.764466i	$0.723003\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.942703	−0.284236	−0.142118	−	0.989850i	$-0.545391\pi$
$11$	−0.942703	−0.284236	−0.142118	+	0.989850i	$0.545391\pi$
$12$	0	0
$13$	−0.0524594	−0.0145496	−0.00727481	−	0.999974i	$-0.502316\pi$
$13$	−0.0524594	−0.0145496	−0.00727481	+	0.999974i	$0.502316\pi$
$14$	0	0
$15$	−4.01525	−1.03673
$16$	0	0
$17$	2.46892	0.598800	0.299400	−	0.954128i	$-0.403213\pi$
$17$	2.46892	0.598800	0.299400	+	0.954128i	$0.403213\pi$
$18$	0	0
$19$	6.45629	1.48117	0.740587	−	0.671960i	$-0.234547\pi$
$19$	6.45629	1.48117	0.740587	+	0.671960i	$0.234547\pi$
$20$	0	0
$21$	−3.41124	−0.744394
$22$	0	0
$23$	−1.91648	−0.399613	−0.199807	−	0.979835i	$-0.564031\pi$
$23$	−1.91648	−0.399613	−0.199807	+	0.979835i	$0.564031\pi$
$24$	0	0
$25$	11.1223	2.22445
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.42443	−0.264510	−0.132255	−	0.991216i	$-0.542222\pi$
$29$	−1.42443	−0.264510	−0.132255	+	0.991216i	$0.542222\pi$
$30$	0	0
$31$	3.96306	0.711787	0.355894	−	0.934526i	$-0.384177\pi$
$31$	3.96306	0.711787	0.355894	+	0.934526i	$0.384177\pi$
$32$	0	0
$33$	−0.942703	−0.164103
$34$	0	0
$35$	13.6970	2.31521
$36$	0	0
$37$	6.45415	1.06106	0.530528	−	0.847667i	$-0.321994\pi$
$37$	6.45415	1.06106	0.530528	+	0.847667i	$0.321994\pi$
$38$	0	0
$39$	−0.0524594	−0.00840023
$40$	0	0
$41$	4.79510	0.748869	0.374434	−	0.927253i	$-0.377837\pi$
$41$	4.79510	0.748869	0.374434	+	0.927253i	$0.377837\pi$
$42$	0	0
$43$	3.95926	0.603782	0.301891	−	0.953342i	$-0.402382\pi$
$43$	3.95926	0.603782	0.301891	+	0.953342i	$0.402382\pi$
$44$	0	0
$45$	−4.01525	−0.598559
$46$	0	0
$47$	−7.81224	−1.13953	−0.569766	−	0.821807i	$-0.692966\pi$
$47$	−7.81224	−1.13953	−0.569766	+	0.821807i	$0.692966\pi$
$48$	0	0
$49$	4.63656	0.662365
$50$	0	0
$51$	2.46892	0.345717
$52$	0	0
$53$	−5.37123	−0.737796	−0.368898	−	0.929470i	$-0.620265\pi$
$53$	−5.37123	−0.737796	−0.368898	+	0.929470i	$0.620265\pi$
$54$	0	0
$55$	3.78519	0.510395
$56$	0	0
$57$	6.45629	0.855156
$58$	0	0
$59$	1.63643	0.213046	0.106523	−	0.994310i	$-0.466028\pi$
$59$	1.63643	0.213046	0.106523	+	0.994310i	$0.466028\pi$
$60$	0	0
$61$	−3.15845	−0.404398	−0.202199	−	0.979344i	$-0.564809\pi$
$61$	−3.15845	−0.404398	−0.202199	+	0.979344i	$0.564809\pi$
$62$	0	0
$63$	−3.41124	−0.429776
$64$	0	0
$65$	0.210638	0.0261264
$66$	0	0
$67$	−2.49181	−0.304423	−0.152211	−	0.988348i	$-0.548639\pi$
$67$	−2.49181	−0.304423	−0.152211	+	0.988348i	$0.548639\pi$
$68$	0	0
$69$	−1.91648	−0.230717
$70$	0	0
$71$	3.47255	0.412116	0.206058	−	0.978540i	$-0.433936\pi$
$71$	3.47255	0.412116	0.206058	+	0.978540i	$0.433936\pi$
$72$	0	0
$73$	−11.3160	−1.32443	−0.662217	−	0.749312i	$-0.730385\pi$
$73$	−11.3160	−1.32443	−0.662217	+	0.749312i	$0.730385\pi$
$74$	0	0
$75$	11.1223	1.28429
$76$	0	0
$77$	3.21578	0.366473
$78$	0	0
$79$	−5.58817	−0.628718	−0.314359	−	0.949304i	$-0.601790\pi$
$79$	−5.58817	−0.628718	−0.314359	+	0.949304i	$0.601790\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.10715	−0.670347	−0.335173	−	0.942157i	$-0.608795\pi$
$83$	−6.10715	−0.670347	−0.335173	+	0.942157i	$0.608795\pi$
$84$	0	0
$85$	−9.91333	−1.07525
$86$	0	0
$87$	−1.42443	−0.152715
$88$	0	0
$89$	−11.5017	−1.21918	−0.609588	−	0.792719i	$-0.708665\pi$
$89$	−11.5017	−1.21918	−0.609588	+	0.792719i	$0.708665\pi$
$90$	0	0
$91$	0.178952	0.0187592
$92$	0	0
$93$	3.96306	0.410951
$94$	0	0
$95$	−25.9236	−2.65971
$96$	0	0
$97$	−4.98650	−0.506302	−0.253151	−	0.967427i	$-0.581467\pi$
$97$	−4.98650	−0.506302	−0.253151	+	0.967427i	$0.581467\pi$
$98$	0	0
$99$	−0.942703	−0.0947452
$100$	0	0
$101$	−4.36685	−0.434518	−0.217259	−	0.976114i	$-0.569712\pi$
$101$	−4.36685	−0.434518	−0.217259	+	0.976114i	$0.569712\pi$
$102$	0	0
$103$	15.8204	1.55883	0.779417	−	0.626506i	$-0.215516\pi$
$103$	15.8204	1.55883	0.779417	+	0.626506i	$0.215516\pi$
$104$	0	0
$105$	13.6970	1.33669
$106$	0	0
$107$	−7.31076	−0.706758	−0.353379	−	0.935480i	$-0.614967\pi$
$107$	−7.31076	−0.706758	−0.353379	+	0.935480i	$0.614967\pi$
$108$	0	0
$109$	−8.19561	−0.784997	−0.392498	−	0.919753i	$-0.628389\pi$
$109$	−8.19561	−0.784997	−0.392498	+	0.919753i	$0.628389\pi$
$110$	0	0
$111$	6.45415	0.612601
$112$	0	0
$113$	5.76691	0.542505	0.271253	−	0.962508i	$-0.412562\pi$
$113$	5.76691	0.542505	0.271253	+	0.962508i	$0.412562\pi$
$114$	0	0
$115$	7.69515	0.717576
$116$	0	0
$117$	−0.0524594	−0.00484987
$118$	0	0
$119$	−8.42207	−0.772050
$120$	0	0
$121$	−10.1113	−0.919210
$122$	0	0
$123$	4.79510	0.432360
$124$	0	0
$125$	−24.5824	−2.19872
$126$	0	0
$127$	−11.5005	−1.02050	−0.510251	−	0.860026i	$-0.670447\pi$
$127$	−11.5005	−1.02050	−0.510251	+	0.860026i	$0.670447\pi$
$128$	0	0
$129$	3.95926	0.348594
$130$	0	0
$131$	−6.13043	−0.535618	−0.267809	−	0.963472i	$-0.586300\pi$
$131$	−6.13043	−0.535618	−0.267809	+	0.963472i	$0.586300\pi$
$132$	0	0
$133$	−22.0240	−1.90972
$134$	0	0
$135$	−4.01525	−0.345578
$136$	0	0
$137$	16.6493	1.42244	0.711222	−	0.702968i	$-0.248142\pi$
$137$	16.6493	1.42244	0.711222	+	0.702968i	$0.248142\pi$
$138$	0	0
$139$	−10.5493	−0.894778	−0.447389	−	0.894340i	$-0.647646\pi$
$139$	−10.5493	−0.894778	−0.447389	+	0.894340i	$0.647646\pi$
$140$	0	0
$141$	−7.81224	−0.657910
$142$	0	0
$143$	0.0494536	0.00413552
$144$	0	0
$145$	5.71945	0.474974
$146$	0	0
$147$	4.63656	0.382417
$148$	0	0
$149$	12.3008	1.00772	0.503862	−	0.863784i	$-0.331912\pi$
$149$	12.3008	1.00772	0.503862	+	0.863784i	$0.331912\pi$
$150$	0	0
$151$	−10.3433	−0.841723	−0.420862	−	0.907125i	$-0.638272\pi$
$151$	−10.3433	−0.841723	−0.420862	+	0.907125i	$0.638272\pi$
$152$	0	0
$153$	2.46892	0.199600
$154$	0	0
$155$	−15.9127	−1.27814
$156$	0	0
$157$	−8.50057	−0.678419	−0.339210	−	0.940711i	$-0.610160\pi$
$157$	−8.50057	−0.678419	−0.339210	+	0.940711i	$0.610160\pi$
$158$	0	0
$159$	−5.37123	−0.425967
$160$	0	0
$161$	6.53757	0.515232
$162$	0	0
$163$	19.4328	1.52209	0.761045	−	0.648699i	$-0.224686\pi$
$163$	19.4328	1.52209	0.761045	+	0.648699i	$0.224686\pi$
$164$	0	0
$165$	3.78519	0.294677
$166$	0	0
$167$	17.0642	1.32047	0.660233	−	0.751061i	$-0.270458\pi$
$167$	17.0642	1.32047	0.660233	+	0.751061i	$0.270458\pi$
$168$	0	0
$169$	−12.9972	−0.999788
$170$	0	0
$171$	6.45629	0.493725
$172$	0	0
$173$	−2.00293	−0.152280	−0.0761399	−	0.997097i	$-0.524260\pi$
$173$	−2.00293	−0.152280	−0.0761399	+	0.997097i	$0.524260\pi$
$174$	0	0
$175$	−37.9407	−2.86805
$176$	0	0
$177$	1.63643	0.123002
$178$	0	0
$179$	−14.2827	−1.06754	−0.533769	−	0.845630i	$-0.679225\pi$
$179$	−14.2827	−1.06754	−0.533769	+	0.845630i	$0.679225\pi$
$180$	0	0
$181$	−1.09829	−0.0816352	−0.0408176	−	0.999167i	$-0.512996\pi$
$181$	−1.09829	−0.0816352	−0.0408176	+	0.999167i	$0.512996\pi$
$182$	0	0
$183$	−3.15845	−0.233479
$184$	0	0
$185$	−25.9151	−1.90531
$186$	0	0
$187$	−2.32745	−0.170200
$188$	0	0
$189$	−3.41124	−0.248131
$190$	0	0
$191$	−6.22913	−0.450724	−0.225362	−	0.974275i	$-0.572356\pi$
$191$	−6.22913	−0.450724	−0.225362	+	0.974275i	$0.572356\pi$
$192$	0	0
$193$	25.3301	1.82330	0.911652	−	0.410963i	$-0.134807\pi$
$193$	25.3301	1.82330	0.911652	+	0.410963i	$0.134807\pi$
$194$	0	0
$195$	0.210638	0.0150841
$196$	0	0
$197$	−4.06948	−0.289938	−0.144969	−	0.989436i	$-0.546308\pi$
$197$	−4.06948	−0.289938	−0.144969	+	0.989436i	$0.546308\pi$
$198$	0	0
$199$	12.3837	0.877856	0.438928	−	0.898522i	$-0.355358\pi$
$199$	12.3837	0.877856	0.438928	+	0.898522i	$0.355358\pi$
$200$	0	0
$201$	−2.49181	−0.175759
$202$	0	0
$203$	4.85907	0.341040
$204$	0	0
$205$	−19.2535	−1.34473
$206$	0	0
$207$	−1.91648	−0.133204
$208$	0	0
$209$	−6.08636	−0.421002
$210$	0	0
$211$	5.37012	0.369694	0.184847	−	0.982767i	$-0.440821\pi$
$211$	5.37012	0.369694	0.184847	+	0.982767i	$0.440821\pi$
$212$	0	0
$213$	3.47255	0.237935
$214$	0	0
$215$	−15.8974	−1.08420
$216$	0	0
$217$	−13.5190	−0.917727
$218$	0	0
$219$	−11.3160	−0.764663
$220$	0	0
$221$	−0.129518	−0.00871232
$222$	0	0
$223$	−2.15780	−0.144497	−0.0722486	−	0.997387i	$-0.523018\pi$
$223$	−2.15780	−0.144497	−0.0722486	+	0.997387i	$0.523018\pi$
$224$	0	0
$225$	11.1223	0.741484
$226$	0	0
$227$	12.5187	0.830893	0.415446	−	0.909618i	$-0.363625\pi$
$227$	12.5187	0.830893	0.415446	+	0.909618i	$0.363625\pi$
$228$	0	0
$229$	0.848259	0.0560545	0.0280273	−	0.999607i	$-0.491077\pi$
$229$	0.848259	0.0560545	0.0280273	+	0.999607i	$0.491077\pi$
$230$	0	0
$231$	3.21578	0.211583
$232$	0	0
$233$	14.9314	0.978190	0.489095	−	0.872230i	$-0.337327\pi$
$233$	14.9314	0.978190	0.489095	+	0.872230i	$0.337327\pi$
$234$	0	0
$235$	31.3681	2.04623
$236$	0	0
$237$	−5.58817	−0.362990
$238$	0	0
$239$	−14.7836	−0.956268	−0.478134	−	0.878287i	$-0.658687\pi$
$239$	−14.7836	−0.956268	−0.478134	+	0.878287i	$0.658687\pi$
$240$	0	0
$241$	−5.09764	−0.328368	−0.164184	−	0.986430i	$-0.552499\pi$
$241$	−5.09764	−0.328368	−0.164184	+	0.986430i	$0.552499\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−18.6170	−1.18939
$246$	0	0
$247$	−0.338693	−0.0215505
$248$	0	0
$249$	−6.10715	−0.387025
$250$	0	0
$251$	−26.6622	−1.68290	−0.841452	−	0.540331i	$-0.818299\pi$
$251$	−26.6622	−1.68290	−0.841452	+	0.540331i	$0.818299\pi$
$252$	0	0
$253$	1.80667	0.113584
$254$	0	0
$255$	−9.91333	−0.620797
$256$	0	0
$257$	−13.5964	−0.848120	−0.424060	−	0.905634i	$-0.639395\pi$
$257$	−13.5964	−0.848120	−0.424060	+	0.905634i	$0.639395\pi$
$258$	0	0
$259$	−22.0167	−1.36805
$260$	0	0
$261$	−1.42443	−0.0881700
$262$	0	0
$263$	−29.7649	−1.83539	−0.917693	−	0.397291i	$-0.869950\pi$
$263$	−29.7649	−1.83539	−0.917693	+	0.397291i	$0.869950\pi$
$264$	0	0
$265$	21.5669	1.32484
$266$	0	0
$267$	−11.5017	−0.703891
$268$	0	0
$269$	14.1573	0.863183	0.431591	−	0.902069i	$-0.357952\pi$
$269$	14.1573	0.863183	0.431591	+	0.902069i	$0.357952\pi$
$270$	0	0
$271$	−2.79431	−0.169742	−0.0848710	−	0.996392i	$-0.527048\pi$
$271$	−2.79431	−0.169742	−0.0848710	+	0.996392i	$0.527048\pi$
$272$	0	0
$273$	0.178952	0.0108306
$274$	0	0
$275$	−10.4850	−0.632269
$276$	0	0
$277$	−14.4653	−0.869134	−0.434567	−	0.900640i	$-0.643099\pi$
$277$	−14.4653	−0.869134	−0.434567	+	0.900640i	$0.643099\pi$
$278$	0	0
$279$	3.96306	0.237262
$280$	0	0
$281$	−13.6796	−0.816056	−0.408028	−	0.912969i	$-0.633783\pi$
$281$	−13.6796	−0.816056	−0.408028	+	0.912969i	$0.633783\pi$
$282$	0	0
$283$	29.1116	1.73050	0.865252	−	0.501337i	$-0.167158\pi$
$283$	29.1116	1.73050	0.865252	+	0.501337i	$0.167158\pi$
$284$	0	0
$285$	−25.9236	−1.53558
$286$	0	0
$287$	−16.3572	−0.965537
$288$	0	0
$289$	−10.9045	−0.641438
$290$	0	0
$291$	−4.98650	−0.292314
$292$	0	0
$293$	−13.3512	−0.779986	−0.389993	−	0.920818i	$-0.627523\pi$
$293$	−13.3512	−0.779986	−0.389993	+	0.920818i	$0.627523\pi$
$294$	0	0
$295$	−6.57070	−0.382561
$296$	0	0
$297$	−0.942703	−0.0547011
$298$	0	0
$299$	0.100537	0.00581422
$300$	0	0
$301$	−13.5060	−0.778472
$302$	0	0
$303$	−4.36685	−0.250869
$304$	0	0
$305$	12.6820	0.726168
$306$	0	0
$307$	20.6026	1.17585	0.587926	−	0.808914i	$-0.299944\pi$
$307$	20.6026	1.17585	0.587926	+	0.808914i	$0.299944\pi$
$308$	0	0
$309$	15.8204	0.899993
$310$	0	0
$311$	−16.5899	−0.940727	−0.470363	−	0.882473i	$-0.655877\pi$
$311$	−16.5899	−0.940727	−0.470363	+	0.882473i	$0.655877\pi$
$312$	0	0
$313$	4.08435	0.230861	0.115431	−	0.993316i	$-0.463175\pi$
$313$	4.08435	0.230861	0.115431	+	0.993316i	$0.463175\pi$
$314$	0	0
$315$	13.6970	0.771738
$316$	0	0
$317$	−11.3455	−0.637228	−0.318614	−	0.947885i	$-0.603217\pi$
$317$	−11.3455	−0.637228	−0.318614	+	0.947885i	$0.603217\pi$
$318$	0	0
$319$	1.34281	0.0751831
$320$	0	0
$321$	−7.31076	−0.408047
$322$	0	0
$323$	15.9400	0.886928
$324$	0	0
$325$	−0.583468	−0.0323650
$326$	0	0
$327$	−8.19561	−0.453218
$328$	0	0
$329$	26.6494	1.46923
$330$	0	0
$331$	−16.1841	−0.889558	−0.444779	−	0.895640i	$-0.646718\pi$
$331$	−16.1841	−0.889558	−0.444779	+	0.895640i	$0.646718\pi$
$332$	0	0
$333$	6.45415	0.353685
$334$	0	0
$335$	10.0052	0.546645
$336$	0	0
$337$	26.6985	1.45436	0.727181	−	0.686446i	$-0.240830\pi$
$337$	26.6985	1.45436	0.727181	+	0.686446i	$0.240830\pi$
$338$	0	0
$339$	5.76691	0.313216
$340$	0	0
$341$	−3.73599	−0.202315
$342$	0	0
$343$	8.06227	0.435321
$344$	0	0
$345$	7.69515	0.414293
$346$	0	0
$347$	−11.9383	−0.640879	−0.320440	−	0.947269i	$-0.603831\pi$
$347$	−11.9383	−0.640879	−0.320440	+	0.947269i	$0.603831\pi$
$348$	0	0
$349$	30.9836	1.65851	0.829256	−	0.558868i	$-0.188764\pi$
$349$	30.9836	1.65851	0.829256	+	0.558868i	$0.188764\pi$
$350$	0	0
$351$	−0.0524594	−0.00280008
$352$	0	0
$353$	−27.1829	−1.44680	−0.723399	−	0.690430i	$-0.757421\pi$
$353$	−27.1829	−1.44680	−0.723399	+	0.690430i	$0.757421\pi$
$354$	0	0
$355$	−13.9432	−0.740027
$356$	0	0
$357$	−8.42207	−0.445743
$358$	0	0
$359$	25.8032	1.36184	0.680920	−	0.732358i	$-0.261580\pi$
$359$	25.8032	1.36184	0.680920	+	0.732358i	$0.261580\pi$
$360$	0	0
$361$	22.6837	1.19388
$362$	0	0
$363$	−10.1113	−0.530706
$364$	0	0
$365$	45.4365	2.37826
$366$	0	0
$367$	−16.8585	−0.880005	−0.440002	−	0.897997i	$-0.645022\pi$
$367$	−16.8585	−0.880005	−0.440002	+	0.897997i	$0.645022\pi$
$368$	0	0
$369$	4.79510	0.249623
$370$	0	0
$371$	18.3226	0.951260
$372$	0	0
$373$	−8.49097	−0.439646	−0.219823	−	0.975540i	$-0.570548\pi$
$373$	−8.49097	−0.439646	−0.219823	+	0.975540i	$0.570548\pi$
$374$	0	0
$375$	−24.5824	−1.26943
$376$	0	0
$377$	0.0747248	0.00384852
$378$	0	0
$379$	−9.73575	−0.500092	−0.250046	−	0.968234i	$-0.580446\pi$
$379$	−9.73575	−0.500092	−0.250046	+	0.968234i	$0.580446\pi$
$380$	0	0
$381$	−11.5005	−0.589187
$382$	0	0
$383$	−0.901542	−0.0460667	−0.0230333	−	0.999735i	$-0.507332\pi$
$383$	−0.901542	−0.0460667	−0.0230333	+	0.999735i	$0.507332\pi$
$384$	0	0
$385$	−12.9122	−0.658066
$386$	0	0
$387$	3.95926	0.201261
$388$	0	0
$389$	−15.3424	−0.777891	−0.388945	−	0.921261i	$-0.627161\pi$
$389$	−15.3424	−0.777891	−0.388945	+	0.921261i	$0.627161\pi$
$390$	0	0
$391$	−4.73162	−0.239289
$392$	0	0
$393$	−6.13043	−0.309239
$394$	0	0
$395$	22.4379	1.12897
$396$	0	0
$397$	11.6130	0.582842	0.291421	−	0.956595i	$-0.405872\pi$
$397$	11.6130	0.582842	0.291421	+	0.956595i	$0.405872\pi$
$398$	0	0
$399$	−22.0240	−1.10258
$400$	0	0
$401$	−33.6967	−1.68273	−0.841365	−	0.540467i	$-0.818248\pi$
$401$	−33.6967	−1.68273	−0.841365	+	0.540467i	$0.818248\pi$
$402$	0	0
$403$	−0.207900	−0.0103562
$404$	0	0
$405$	−4.01525	−0.199520
$406$	0	0
$407$	−6.08435	−0.301590
$408$	0	0
$409$	9.24538	0.457155	0.228577	−	0.973526i	$-0.426593\pi$
$409$	9.24538	0.457155	0.228577	+	0.973526i	$0.426593\pi$
$410$	0	0
$411$	16.6493	0.821248
$412$	0	0
$413$	−5.58227	−0.274686
$414$	0	0
$415$	24.5217	1.20373
$416$	0	0
$417$	−10.5493	−0.516600
$418$	0	0
$419$	−32.0393	−1.56522	−0.782611	−	0.622510i	$-0.786113\pi$
$419$	−32.0393	−1.56522	−0.782611	+	0.622510i	$0.786113\pi$
$420$	0	0
$421$	−2.80651	−0.136781	−0.0683905	−	0.997659i	$-0.521786\pi$
$421$	−2.80651	−0.136781	−0.0683905	+	0.997659i	$0.521786\pi$
$422$	0	0
$423$	−7.81224	−0.379844
$424$	0	0
$425$	27.4599	1.33200
$426$	0	0
$427$	10.7742	0.521402
$428$	0	0
$429$	0.0494536	0.00238764
$430$	0	0
$431$	−0.919689	−0.0442998	−0.0221499	−	0.999755i	$-0.507051\pi$
$431$	−0.919689	−0.0442998	−0.0221499	+	0.999755i	$0.507051\pi$
$432$	0	0
$433$	−18.2734	−0.878165	−0.439082	−	0.898447i	$-0.644696\pi$
$433$	−18.2734	−0.878165	−0.439082	+	0.898447i	$0.644696\pi$
$434$	0	0
$435$	5.71945	0.274227
$436$	0	0
$437$	−12.3733	−0.591897
$438$	0	0
$439$	−23.9702	−1.14403	−0.572017	−	0.820242i	$-0.693839\pi$
$439$	−23.9702	−1.14403	−0.572017	+	0.820242i	$0.693839\pi$
$440$	0	0
$441$	4.63656	0.220788
$442$	0	0
$443$	7.76147	0.368759	0.184379	−	0.982855i	$-0.440973\pi$
$443$	7.76147	0.368759	0.184379	+	0.982855i	$0.440973\pi$
$444$	0	0
$445$	46.1822	2.18924
$446$	0	0
$447$	12.3008	0.581810
$448$	0	0
$449$	5.53865	0.261385	0.130692	−	0.991423i	$-0.458280\pi$
$449$	5.53865	0.261385	0.130692	+	0.991423i	$0.458280\pi$
$450$	0	0
$451$	−4.52035	−0.212855
$452$	0	0
$453$	−10.3433	−0.485969
$454$	0	0
$455$	−0.718536	−0.0336855
$456$	0	0
$457$	34.0935	1.59483	0.797413	−	0.603433i	$-0.206201\pi$
$457$	34.0935	1.59483	0.797413	+	0.603433i	$0.206201\pi$
$458$	0	0
$459$	2.46892	0.115239
$460$	0	0
$461$	−30.5906	−1.42475	−0.712373	−	0.701801i	$-0.752380\pi$
$461$	−30.5906	−1.42475	−0.712373	+	0.701801i	$0.752380\pi$
$462$	0	0
$463$	−22.4639	−1.04398	−0.521992	−	0.852950i	$-0.674811\pi$
$463$	−22.4639	−1.04398	−0.521992	+	0.852950i	$0.674811\pi$
$464$	0	0
$465$	−15.9127	−0.737934
$466$	0	0
$467$	−3.61860	−0.167449	−0.0837244	−	0.996489i	$-0.526682\pi$
$467$	−3.61860	−0.167449	−0.0837244	+	0.996489i	$0.526682\pi$
$468$	0	0
$469$	8.50016	0.392501
$470$	0	0
$471$	−8.50057	−0.391686
$472$	0	0
$473$	−3.73241	−0.171616
$474$	0	0
$475$	71.8086	3.29480
$476$	0	0
$477$	−5.37123	−0.245932
$478$	0	0
$479$	−24.3435	−1.11228	−0.556142	−	0.831088i	$-0.687719\pi$
$479$	−24.3435	−1.11228	−0.556142	+	0.831088i	$0.687719\pi$
$480$	0	0
$481$	−0.338581	−0.0154380
$482$	0	0
$483$	6.53757	0.297470
$484$	0	0
$485$	20.0220	0.909154
$486$	0	0
$487$	3.68270	0.166879	0.0834395	−	0.996513i	$-0.473409\pi$
$487$	3.68270	0.166879	0.0834395	+	0.996513i	$0.473409\pi$
$488$	0	0
$489$	19.4328	0.878780
$490$	0	0
$491$	19.1346	0.863531	0.431765	−	0.901986i	$-0.357891\pi$
$491$	19.1346	0.863531	0.431765	+	0.901986i	$0.357891\pi$
$492$	0	0
$493$	−3.51680	−0.158389
$494$	0	0
$495$	3.78519	0.170132
$496$	0	0
$497$	−11.8457	−0.531352
$498$	0	0
$499$	17.3020	0.774543	0.387272	−	0.921966i	$-0.373418\pi$
$499$	17.3020	0.774543	0.387272	+	0.921966i	$0.373418\pi$
$500$	0	0
$501$	17.0642	0.762371
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	17.5340	0.780254
$506$	0	0
$507$	−12.9972	−0.577228
$508$	0	0
$509$	22.8033	1.01074	0.505368	−	0.862904i	$-0.331357\pi$
$509$	22.8033	1.01074	0.505368	+	0.862904i	$0.331357\pi$
$510$	0	0
$511$	38.6015	1.70763
$512$	0	0
$513$	6.45629	0.285052
$514$	0	0
$515$	−63.5230	−2.79916
$516$	0	0
$517$	7.36462	0.323896
$518$	0	0
$519$	−2.00293	−0.0879188
$520$	0	0
$521$	−15.7216	−0.688778	−0.344389	−	0.938827i	$-0.611914\pi$
$521$	−15.7216	−0.688778	−0.344389	+	0.938827i	$0.611914\pi$
$522$	0	0
$523$	32.3305	1.41372	0.706858	−	0.707356i	$-0.250112\pi$
$523$	32.3305	1.41372	0.706858	+	0.707356i	$0.250112\pi$
$524$	0	0
$525$	−37.9407	−1.65587
$526$	0	0
$527$	9.78448	0.426218
$528$	0	0
$529$	−19.3271	−0.840309
$530$	0	0
$531$	1.63643	0.0710152
$532$	0	0
$533$	−0.251548	−0.0108958
$534$	0	0
$535$	29.3546	1.26911
$536$	0	0
$537$	−14.2827	−0.616344
$538$	0	0
$539$	−4.37090	−0.188268
$540$	0	0
$541$	−2.43532	−0.104703	−0.0523513	−	0.998629i	$-0.516672\pi$
$541$	−2.43532	−0.104703	−0.0523513	+	0.998629i	$0.516672\pi$
$542$	0	0
$543$	−1.09829	−0.0471321
$544$	0	0
$545$	32.9074	1.40960
$546$	0	0
$547$	5.13952	0.219750	0.109875	−	0.993945i	$-0.464955\pi$
$547$	5.13952	0.219750	0.109875	+	0.993945i	$0.464955\pi$
$548$	0	0
$549$	−3.15845	−0.134799
$550$	0	0
$551$	−9.19653	−0.391785
$552$	0	0
$553$	19.0626	0.810623
$554$	0	0
$555$	−25.9151	−1.10003
$556$	0	0
$557$	1.82366	0.0772710	0.0386355	−	0.999253i	$-0.487699\pi$
$557$	1.82366	0.0772710	0.0386355	+	0.999253i	$0.487699\pi$
$558$	0	0
$559$	−0.207701	−0.00878480
$560$	0	0
$561$	−2.32745	−0.0982652
$562$	0	0
$563$	−14.0557	−0.592379	−0.296189	−	0.955129i	$-0.595716\pi$
$563$	−14.0557	−0.592379	−0.296189	+	0.955129i	$0.595716\pi$
$564$	0	0
$565$	−23.1556	−0.974164
$566$	0	0
$567$	−3.41124	−0.143259
$568$	0	0
$569$	19.8848	0.833616	0.416808	−	0.908994i	$-0.363149\pi$
$569$	19.8848	0.833616	0.416808	+	0.908994i	$0.363149\pi$
$570$	0	0
$571$	−5.24281	−0.219405	−0.109702	−	0.993964i	$-0.534990\pi$
$571$	−5.24281	−0.219405	−0.109702	+	0.993964i	$0.534990\pi$
$572$	0	0
$573$	−6.22913	−0.260226
$574$	0	0
$575$	−21.3156	−0.888921
$576$	0	0
$577$	−35.1315	−1.46254	−0.731272	−	0.682086i	$-0.761073\pi$
$577$	−35.1315	−1.46254	−0.731272	+	0.682086i	$0.761073\pi$
$578$	0	0
$579$	25.3301	1.05269
$580$	0	0
$581$	20.8329	0.864296
$582$	0	0
$583$	5.06348	0.209708
$584$	0	0
$585$	0.210638	0.00870880
$586$	0	0
$587$	−28.1708	−1.16273	−0.581366	−	0.813642i	$-0.697482\pi$
$587$	−28.1708	−1.16273	−0.581366	+	0.813642i	$0.697482\pi$
$588$	0	0
$589$	25.5867	1.05428
$590$	0	0
$591$	−4.06948	−0.167396
$592$	0	0
$593$	10.3812	0.426304	0.213152	−	0.977019i	$-0.431627\pi$
$593$	10.3812	0.426304	0.213152	+	0.977019i	$0.431627\pi$
$594$	0	0
$595$	33.8167	1.38635
$596$	0	0
$597$	12.3837	0.506830
$598$	0	0
$599$	−13.0820	−0.534516	−0.267258	−	0.963625i	$-0.586118\pi$
$599$	−13.0820	−0.534516	−0.267258	+	0.963625i	$0.586118\pi$
$600$	0	0
$601$	−3.47942	−0.141928	−0.0709642	−	0.997479i	$-0.522608\pi$
$601$	−3.47942	−0.141928	−0.0709642	+	0.997479i	$0.522608\pi$
$602$	0	0
$603$	−2.49181	−0.101474
$604$	0	0
$605$	40.5995	1.65060
$606$	0	0
$607$	−6.26320	−0.254216	−0.127108	−	0.991889i	$-0.540569\pi$
$607$	−6.26320	−0.254216	−0.127108	+	0.991889i	$0.540569\pi$
$608$	0	0
$609$	4.85907	0.196900
$610$	0	0
$611$	0.409826	0.0165798
$612$	0	0
$613$	−20.4156	−0.824580	−0.412290	−	0.911053i	$-0.635271\pi$
$613$	−20.4156	−0.824580	−0.412290	+	0.911053i	$0.635271\pi$
$614$	0	0
$615$	−19.2535	−0.776378
$616$	0	0
$617$	22.1908	0.893369	0.446684	−	0.894692i	$-0.352605\pi$
$617$	22.1908	0.893369	0.446684	+	0.894692i	$0.352605\pi$
$618$	0	0
$619$	1.70540	0.0685457	0.0342729	−	0.999413i	$-0.489088\pi$
$619$	1.70540	0.0685457	0.0342729	+	0.999413i	$0.489088\pi$
$620$	0	0
$621$	−1.91648	−0.0769056
$622$	0	0
$623$	39.2350	1.57192
$624$	0	0
$625$	43.0935	1.72374
$626$	0	0
$627$	−6.08636	−0.243066
$628$	0	0
$629$	15.9348	0.635361
$630$	0	0
$631$	13.3902	0.533056	0.266528	−	0.963827i	$-0.414124\pi$
$631$	13.3902	0.533056	0.266528	+	0.963827i	$0.414124\pi$
$632$	0	0
$633$	5.37012	0.213443
$634$	0	0
$635$	46.1773	1.83249
$636$	0	0
$637$	−0.243231	−0.00963717
$638$	0	0
$639$	3.47255	0.137372
$640$	0	0
$641$	−35.7869	−1.41350	−0.706748	−	0.707465i	$-0.749839\pi$
$641$	−35.7869	−1.41350	−0.706748	+	0.707465i	$0.749839\pi$
$642$	0	0
$643$	18.2097	0.718120	0.359060	−	0.933315i	$-0.383097\pi$
$643$	18.2097	0.718120	0.359060	+	0.933315i	$0.383097\pi$
$644$	0	0
$645$	−15.8974	−0.625961
$646$	0	0
$647$	31.5405	1.23998	0.619992	−	0.784608i	$-0.287136\pi$
$647$	31.5405	1.23998	0.619992	+	0.784608i	$0.287136\pi$
$648$	0	0
$649$	−1.54267	−0.0605551
$650$	0	0
$651$	−13.5190	−0.529850
$652$	0	0
$653$	−4.10196	−0.160522	−0.0802611	−	0.996774i	$-0.525575\pi$
$653$	−4.10196	−0.160522	−0.0802611	+	0.996774i	$0.525575\pi$
$654$	0	0
$655$	24.6152	0.961796
$656$	0	0
$657$	−11.3160	−0.441478
$658$	0	0
$659$	−0.307285	−0.0119701	−0.00598505	−	0.999982i	$-0.501905\pi$
$659$	−0.307285	−0.0119701	−0.00598505	+	0.999982i	$0.501905\pi$
$660$	0	0
$661$	37.7726	1.46918	0.734592	−	0.678509i	$-0.237374\pi$
$661$	37.7726	1.46918	0.734592	+	0.678509i	$0.237374\pi$
$662$	0	0
$663$	−0.129518	−0.00503006
$664$	0	0
$665$	88.4318	3.42924
$666$	0	0
$667$	2.72989	0.105702
$668$	0	0
$669$	−2.15780	−0.0834255
$670$	0	0
$671$	2.97748	0.114944
$672$	0	0
$673$	−36.4713	−1.40586	−0.702932	−	0.711257i	$-0.748126\pi$
$673$	−36.4713	−1.40586	−0.702932	+	0.711257i	$0.748126\pi$
$674$	0	0
$675$	11.1223	0.428096
$676$	0	0
$677$	−9.31919	−0.358165	−0.179083	−	0.983834i	$-0.557313\pi$
$677$	−9.31919	−0.358165	−0.179083	+	0.983834i	$0.557313\pi$
$678$	0	0
$679$	17.0101	0.652789
$680$	0	0
$681$	12.5187	0.479716
$682$	0	0
$683$	22.6181	0.865459	0.432730	−	0.901524i	$-0.357550\pi$
$683$	22.6181	0.865459	0.432730	+	0.901524i	$0.357550\pi$
$684$	0	0
$685$	−66.8511	−2.55425
$686$	0	0
$687$	0.848259	0.0323631
$688$	0	0
$689$	0.281772	0.0107347
$690$	0	0
$691$	−39.0277	−1.48468	−0.742341	−	0.670022i	$-0.766285\pi$
$691$	−39.0277	−1.48468	−0.742341	+	0.670022i	$0.766285\pi$
$692$	0	0
$693$	3.21578	0.122158
$694$	0	0
$695$	42.3580	1.60673
$696$	0	0
$697$	11.8387	0.448423
$698$	0	0
$699$	14.9314	0.564759
$700$	0	0
$701$	−21.5411	−0.813598	−0.406799	−	0.913518i	$-0.633355\pi$
$701$	−21.5411	−0.813598	−0.406799	+	0.913518i	$0.633355\pi$
$702$	0	0
$703$	41.6699	1.57161
$704$	0	0
$705$	31.3681	1.18139
$706$	0	0
$707$	14.8964	0.560236
$708$	0	0
$709$	29.6541	1.11368	0.556841	−	0.830619i	$-0.312013\pi$
$709$	29.6541	1.11368	0.556841	+	0.830619i	$0.312013\pi$
$710$	0	0
$711$	−5.58817	−0.209573
$712$	0	0
$713$	−7.59512	−0.284440
$714$	0	0
$715$	−0.198569	−0.00742605
$716$	0	0
$717$	−14.7836	−0.552102
$718$	0	0
$719$	−1.61658	−0.0602882	−0.0301441	−	0.999546i	$-0.509597\pi$
$719$	−1.61658	−0.0602882	−0.0301441	+	0.999546i	$0.509597\pi$
$720$	0	0
$721$	−53.9673	−2.00985
$722$	0	0
$723$	−5.09764	−0.189583
$724$	0	0
$725$	−15.8429	−0.588390
$726$	0	0
$727$	15.6379	0.579978	0.289989	−	0.957030i	$-0.406348\pi$
$727$	15.6379	0.579978	0.289989	+	0.957030i	$0.406348\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.77509	0.361545
$732$	0	0
$733$	−26.0045	−0.960499	−0.480249	−	0.877132i	$-0.659454\pi$
$733$	−26.0045	−0.960499	−0.480249	+	0.877132i	$0.659454\pi$
$734$	0	0
$735$	−18.6170	−0.686697
$736$	0	0
$737$	2.34904	0.0865278
$738$	0	0
$739$	−30.6613	−1.12790	−0.563948	−	0.825811i	$-0.690718\pi$
$739$	−30.6613	−1.12790	−0.563948	+	0.825811i	$0.690718\pi$
$740$	0	0
$741$	−0.338693	−0.0124422
$742$	0	0
$743$	−26.0081	−0.954145	−0.477072	−	0.878864i	$-0.658302\pi$
$743$	−26.0081	−0.954145	−0.477072	+	0.878864i	$0.658302\pi$
$744$	0	0
$745$	−49.3910	−1.80955
$746$	0	0
$747$	−6.10715	−0.223449
$748$	0	0
$749$	24.9388	0.911242
$750$	0	0
$751$	25.0453	0.913918	0.456959	−	0.889488i	$-0.348939\pi$
$751$	25.0453	0.913918	0.456959	+	0.889488i	$0.348939\pi$
$752$	0	0
$753$	−26.6622	−0.971625
$754$	0	0
$755$	41.5309	1.51146
$756$	0	0
$757$	5.14742	0.187086	0.0935430	−	0.995615i	$-0.470181\pi$
$757$	5.14742	0.187086	0.0935430	+	0.995615i	$0.470181\pi$
$758$	0	0
$759$	1.80667	0.0655779
$760$	0	0
$761$	−36.4361	−1.32081	−0.660404	−	0.750910i	$-0.729615\pi$
$761$	−36.4361	−1.32081	−0.660404	+	0.750910i	$0.729615\pi$
$762$	0	0
$763$	27.9572	1.01212
$764$	0	0
$765$	−9.91333	−0.358417
$766$	0	0
$767$	−0.0858464	−0.00309973
$768$	0	0
$769$	−7.30809	−0.263537	−0.131768	−	0.991281i	$-0.542065\pi$
$769$	−7.30809	−0.263537	−0.131768	+	0.991281i	$0.542065\pi$
$770$	0	0
$771$	−13.5964	−0.489663
$772$	0	0
$773$	24.0791	0.866067	0.433033	−	0.901378i	$-0.357443\pi$
$773$	24.0791	0.866067	0.433033	+	0.901378i	$0.357443\pi$
$774$	0	0
$775$	44.0782	1.58334
$776$	0	0
$777$	−22.0167	−0.789844
$778$	0	0
$779$	30.9586	1.10921
$780$	0	0
$781$	−3.27358	−0.117138
$782$	0	0
$783$	−1.42443	−0.0509050
$784$	0	0
$785$	34.1319	1.21822
$786$	0	0
$787$	−30.8437	−1.09946	−0.549729	−	0.835343i	$-0.685269\pi$
$787$	−30.8437	−1.09946	−0.549729	+	0.835343i	$0.685269\pi$
$788$	0	0
$789$	−29.7649	−1.05966
$790$	0	0
$791$	−19.6723	−0.699467
$792$	0	0
$793$	0.165690	0.00588384
$794$	0	0
$795$	21.5669	0.764898
$796$	0	0
$797$	21.1040	0.747541	0.373771	−	0.927521i	$-0.378065\pi$
$797$	21.1040	0.747541	0.373771	+	0.927521i	$0.378065\pi$
$798$	0	0
$799$	−19.2878	−0.682353
$800$	0	0
$801$	−11.5017	−0.406392
$802$	0	0
$803$	10.6676	0.376451
$804$	0	0
$805$	−26.2500	−0.925190
$806$	0	0
$807$	14.1573	0.498359
$808$	0	0
$809$	−42.6345	−1.49895	−0.749474	−	0.662034i	$-0.769694\pi$
$809$	−42.6345	−1.49895	−0.749474	+	0.662034i	$0.769694\pi$
$810$	0	0
$811$	−30.1661	−1.05928	−0.529638	−	0.848224i	$-0.677672\pi$
$811$	−30.1661	−1.05928	−0.529638	+	0.848224i	$0.677672\pi$
$812$	0	0
$813$	−2.79431	−0.0980006
$814$	0	0
$815$	−78.0274	−2.73318
$816$	0	0
$817$	25.5621	0.894306
$818$	0	0
$819$	0.178952	0.00625308
$820$	0	0
$821$	9.00173	0.314163	0.157081	−	0.987586i	$-0.449792\pi$
$821$	9.00173	0.314163	0.157081	+	0.987586i	$0.449792\pi$
$822$	0	0
$823$	8.24095	0.287262	0.143631	−	0.989631i	$-0.454122\pi$
$823$	8.24095	0.287262	0.143631	+	0.989631i	$0.454122\pi$
$824$	0	0
$825$	−10.4850	−0.365040
$826$	0	0
$827$	−15.8965	−0.552775	−0.276387	−	0.961046i	$-0.589137\pi$
$827$	−15.8965	−0.552775	−0.276387	+	0.961046i	$0.589137\pi$
$828$	0	0
$829$	−22.8729	−0.794410	−0.397205	−	0.917730i	$-0.630020\pi$
$829$	−22.8729	−0.794410	−0.397205	+	0.917730i	$0.630020\pi$
$830$	0	0
$831$	−14.4653	−0.501795
$832$	0	0
$833$	11.4473	0.396625
$834$	0	0
$835$	−68.5170	−2.37113
$836$	0	0
$837$	3.96306	0.136984
$838$	0	0
$839$	−3.82681	−0.132116	−0.0660581	−	0.997816i	$-0.521042\pi$
$839$	−3.82681	−0.132116	−0.0660581	+	0.997816i	$0.521042\pi$
$840$	0	0
$841$	−26.9710	−0.930034
$842$	0	0
$843$	−13.6796	−0.471150
$844$	0	0
$845$	52.1873	1.79530
$846$	0	0
$847$	34.4921	1.18516
$848$	0	0
$849$	29.1116	0.999107
$850$	0	0
$851$	−12.3692	−0.424012
$852$	0	0
$853$	−36.3471	−1.24450	−0.622250	−	0.782819i	$-0.713781\pi$
$853$	−36.3471	−1.24450	−0.622250	+	0.782819i	$0.713781\pi$
$854$	0	0
$855$	−25.9236	−0.886570
$856$	0	0
$857$	−15.6160	−0.533432	−0.266716	−	0.963775i	$-0.585938\pi$
$857$	−15.6160	−0.533432	−0.266716	+	0.963775i	$0.585938\pi$
$858$	0	0
$859$	18.0884	0.617169	0.308585	−	0.951197i	$-0.400145\pi$
$859$	18.0884	0.617169	0.308585	+	0.951197i	$0.400145\pi$
$860$	0	0
$861$	−16.3572	−0.557453
$862$	0	0
$863$	7.47419	0.254424	0.127212	−	0.991876i	$-0.459397\pi$
$863$	7.47419	0.254424	0.127212	+	0.991876i	$0.459397\pi$
$864$	0	0
$865$	8.04226	0.273445
$866$	0	0
$867$	−10.9045	−0.370335
$868$	0	0
$869$	5.26798	0.178704
$870$	0	0
$871$	0.130719	0.00442924
$872$	0	0
$873$	−4.98650	−0.168767
$874$	0	0
$875$	83.8566	2.83487
$876$	0	0
$877$	−22.9554	−0.775150	−0.387575	−	0.921838i	$-0.626687\pi$
$877$	−22.9554	−0.775150	−0.387575	+	0.921838i	$0.626687\pi$
$878$	0	0
$879$	−13.3512	−0.450325
$880$	0	0
$881$	−3.84769	−0.129632	−0.0648160	−	0.997897i	$-0.520646\pi$
$881$	−3.84769	−0.129632	−0.0648160	+	0.997897i	$0.520646\pi$
$882$	0	0
$883$	−21.1281	−0.711016	−0.355508	−	0.934673i	$-0.615692\pi$
$883$	−21.1281	−0.711016	−0.355508	+	0.934673i	$0.615692\pi$
$884$	0	0
$885$	−6.57070	−0.220872
$886$	0	0
$887$	−15.7929	−0.530275	−0.265137	−	0.964211i	$-0.585417\pi$
$887$	−15.7929	−0.530275	−0.265137	+	0.964211i	$0.585417\pi$
$888$	0	0
$889$	39.2308	1.31576
$890$	0	0
$891$	−0.942703	−0.0315817
$892$	0	0
$893$	−50.4381	−1.68785
$894$	0	0
$895$	57.3487	1.91695
$896$	0	0
$897$	0.100537	0.00335684
$898$	0	0
$899$	−5.64511	−0.188275
$900$	0	0
$901$	−13.2611	−0.441792
$902$	0	0
$903$	−13.5060	−0.449451
$904$	0	0
$905$	4.40991	0.146590
$906$	0	0
$907$	33.3706	1.10805	0.554027	−	0.832499i	$-0.313091\pi$
$907$	33.3706	1.10805	0.554027	+	0.832499i	$0.313091\pi$
$908$	0	0
$909$	−4.36685	−0.144839
$910$	0	0
$911$	−14.0757	−0.466349	−0.233174	−	0.972435i	$-0.574911\pi$
$911$	−14.0757	−0.466349	−0.233174	+	0.972435i	$0.574911\pi$
$912$	0	0
$913$	5.75722	0.190536
$914$	0	0
$915$	12.6820	0.419253
$916$	0	0
$917$	20.9124	0.690587
$918$	0	0
$919$	14.5296	0.479288	0.239644	−	0.970861i	$-0.422969\pi$
$919$	14.5296	0.479288	0.239644	+	0.970861i	$0.422969\pi$
$920$	0	0
$921$	20.6026	0.678879
$922$	0	0
$923$	−0.182168	−0.00599613
$924$	0	0
$925$	71.7848	2.36027
$926$	0	0
$927$	15.8204	0.519611
$928$	0	0
$929$	20.2952	0.665863	0.332932	−	0.942951i	$-0.391962\pi$
$929$	20.2952	0.665863	0.332932	+	0.942951i	$0.391962\pi$
$930$	0	0
$931$	29.9350	0.981079
$932$	0	0
$933$	−16.5899	−0.543129
$934$	0	0
$935$	9.34532	0.305625
$936$	0	0
$937$	−53.8327	−1.75864	−0.879319	−	0.476234i	$-0.842002\pi$
$937$	−53.8327	−1.75864	−0.879319	+	0.476234i	$0.842002\pi$
$938$	0	0
$939$	4.08435	0.133288
$940$	0	0
$941$	38.4127	1.25222	0.626109	−	0.779735i	$-0.284646\pi$
$941$	38.4127	1.25222	0.626109	+	0.779735i	$0.284646\pi$
$942$	0	0
$943$	−9.18970	−0.299258
$944$	0	0
$945$	13.6970	0.445563
$946$	0	0
$947$	48.5325	1.57710	0.788548	−	0.614973i	$-0.210833\pi$
$947$	48.5325	1.57710	0.788548	+	0.614973i	$0.210833\pi$
$948$	0	0
$949$	0.593630	0.0192700
$950$	0	0
$951$	−11.3455	−0.367904
$952$	0	0
$953$	−42.4334	−1.37455	−0.687276	−	0.726396i	$-0.741194\pi$
$953$	−42.4334	−1.37455	−0.687276	+	0.726396i	$0.741194\pi$
$954$	0	0
$955$	25.0115	0.809354
$956$	0	0
$957$	1.34281	0.0434070
$958$	0	0
$959$	−56.7947	−1.83400
$960$	0	0
$961$	−15.2941	−0.493359
$962$	0	0
$963$	−7.31076	−0.235586
$964$	0	0
$965$	−101.707	−3.27406
$966$	0	0
$967$	20.7743	0.668057	0.334029	−	0.942563i	$-0.391592\pi$
$967$	20.7743	0.668057	0.334029	+	0.942563i	$0.391592\pi$
$968$	0	0
$969$	15.9400	0.512068
$970$	0	0
$971$	−0.881250	−0.0282807	−0.0141403	−	0.999900i	$-0.504501\pi$
$971$	−0.881250	−0.0282807	−0.0141403	+	0.999900i	$0.504501\pi$
$972$	0	0
$973$	35.9861	1.15366
$974$	0	0
$975$	−0.583468	−0.0186859
$976$	0	0
$977$	−32.8981	−1.05250	−0.526252	−	0.850328i	$-0.676403\pi$
$977$	−32.8981	−1.05250	−0.526252	+	0.850328i	$0.676403\pi$
$978$	0	0
$979$	10.8427	0.346533
$980$	0	0
$981$	−8.19561	−0.261666
$982$	0	0
$983$	−17.2826	−0.551230	−0.275615	−	0.961268i	$-0.588882\pi$
$983$	−17.2826	−0.551230	−0.275615	+	0.961268i	$0.588882\pi$
$984$	0	0
$985$	16.3400	0.520635
$986$	0	0
$987$	26.6494	0.848261
$988$	0	0
$989$	−7.58784	−0.241279
$990$	0	0
$991$	−18.2559	−0.579916	−0.289958	−	0.957039i	$-0.593641\pi$
$991$	−18.2559	−0.579916	−0.289958	+	0.957039i	$0.593641\pi$
$992$	0	0
$993$	−16.1841	−0.513587
$994$	0	0
$995$	−49.7236	−1.57634
$996$	0	0
$997$	−17.2774	−0.547181	−0.273591	−	0.961846i	$-0.588211\pi$
$997$	−17.2774	−0.547181	−0.273591	+	0.961846i	$0.588211\pi$
$998$	0	0
$999$	6.45415	0.204200

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.1	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.g.1.1	✓	15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.01525 q^{5} -3.41124 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.01525 q^{5} -3.41124 q^{7} +1.00000 q^{9} -0.942703 q^{11} -0.0524594 q^{13} -4.01525 q^{15} +2.46892 q^{17} +6.45629 q^{19} -3.41124 q^{21} -1.91648 q^{23} +11.1223 q^{25} +1.00000 q^{27} -1.42443 q^{29} +3.96306 q^{31} -0.942703 q^{33} +13.6970 q^{35} +6.45415 q^{37} -0.0524594 q^{39} +4.79510 q^{41} +3.95926 q^{43} -4.01525 q^{45} -7.81224 q^{47} +4.63656 q^{49} +2.46892 q^{51} -5.37123 q^{53} +3.78519 q^{55} +6.45629 q^{57} +1.63643 q^{59} -3.15845 q^{61} -3.41124 q^{63} +0.210638 q^{65} -2.49181 q^{67} -1.91648 q^{69} +3.47255 q^{71} -11.3160 q^{73} +11.1223 q^{75} +3.21578 q^{77} -5.58817 q^{79} +1.00000 q^{81} -6.10715 q^{83} -9.91333 q^{85} -1.42443 q^{87} -11.5017 q^{89} +0.178952 q^{91} +3.96306 q^{93} -25.9236 q^{95} -4.98650 q^{97} -0.942703 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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