LMFDB - Embedded newform 6028.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, 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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.99791 q^{3} +1.37488 q^{5} +4.31566 q^{7} +5.98748 q^{9} +O(q^{10})$	$q-2.99791 q^{3} +1.37488 q^{5} +4.31566 q^{7} +5.98748 q^{9} -1.00000 q^{11} -0.410558 q^{13} -4.12178 q^{15} -0.429387 q^{17} -5.23814 q^{19} -12.9380 q^{21} +3.32340 q^{23} -3.10970 q^{25} -8.95621 q^{27} +1.27712 q^{29} +8.12439 q^{31} +2.99791 q^{33} +5.93353 q^{35} -2.84796 q^{37} +1.23082 q^{39} +5.49807 q^{41} +6.46186 q^{43} +8.23208 q^{45} +4.51777 q^{47} +11.6249 q^{49} +1.28726 q^{51} -4.76630 q^{53} -1.37488 q^{55} +15.7035 q^{57} -0.828801 q^{59} -11.2791 q^{61} +25.8399 q^{63} -0.564469 q^{65} +5.64458 q^{67} -9.96326 q^{69} +4.35032 q^{71} -0.643138 q^{73} +9.32260 q^{75} -4.31566 q^{77} +9.44903 q^{79} +8.88748 q^{81} -0.140639 q^{83} -0.590357 q^{85} -3.82869 q^{87} +4.73504 q^{89} -1.77183 q^{91} -24.3562 q^{93} -7.20184 q^{95} -0.436679 q^{97} -5.98748 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * 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
$3$	−2.99791	−1.73085	−0.865423	−	0.501042i	$-0.832950\pi$
$3$	−2.99791	−1.73085	−0.865423	+	0.501042i	$0.832950\pi$
$4$	0	0
$5$	1.37488	0.614866	0.307433	−	0.951570i	$-0.400530\pi$
$5$	1.37488	0.614866	0.307433	+	0.951570i	$0.400530\pi$
$6$	0	0
$7$	4.31566	1.63117	0.815583	−	0.578640i	$-0.196416\pi$
$7$	4.31566	1.63117	0.815583	+	0.578640i	$0.196416\pi$
$8$	0	0
$9$	5.98748	1.99583
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.410558	−0.113868	−0.0569342	−	0.998378i	$-0.518133\pi$
$13$	−0.410558	−0.113868	−0.0569342	+	0.998378i	$0.518133\pi$
$14$	0	0
$15$	−4.12178	−1.06424
$16$	0	0
$17$	−0.429387	−0.104142	−0.0520708	−	0.998643i	$-0.516582\pi$
$17$	−0.429387	−0.104142	−0.0520708	+	0.998643i	$0.516582\pi$
$18$	0	0
$19$	−5.23814	−1.20171	−0.600856	−	0.799357i	$-0.705174\pi$
$19$	−5.23814	−1.20171	−0.600856	+	0.799357i	$0.705174\pi$
$20$	0	0
$21$	−12.9380	−2.82330
$22$	0	0
$23$	3.32340	0.692976	0.346488	−	0.938054i	$-0.387374\pi$
$23$	3.32340	0.692976	0.346488	+	0.938054i	$0.387374\pi$
$24$	0	0
$25$	−3.10970	−0.621939
$26$	0	0
$27$	−8.95621	−1.72362
$28$	0	0
$29$	1.27712	0.237155	0.118578	−	0.992945i	$-0.462167\pi$
$29$	1.27712	0.237155	0.118578	+	0.992945i	$0.462167\pi$
$30$	0	0
$31$	8.12439	1.45918	0.729592	−	0.683883i	$-0.239710\pi$
$31$	8.12439	1.45918	0.729592	+	0.683883i	$0.239710\pi$
$32$	0	0
$33$	2.99791	0.521870
$34$	0	0
$35$	5.93353	1.00295
$36$	0	0
$37$	−2.84796	−0.468201	−0.234101	−	0.972212i	$-0.575214\pi$
$37$	−2.84796	−0.468201	−0.234101	+	0.972212i	$0.575214\pi$
$38$	0	0
$39$	1.23082	0.197089
$40$	0	0
$41$	5.49807	0.858655	0.429327	−	0.903149i	$-0.358751\pi$
$41$	5.49807	0.858655	0.429327	+	0.903149i	$0.358751\pi$
$42$	0	0
$43$	6.46186	0.985424	0.492712	−	0.870192i	$-0.336006\pi$
$43$	6.46186	0.985424	0.492712	+	0.870192i	$0.336006\pi$
$44$	0	0
$45$	8.23208	1.22717
$46$	0	0
$47$	4.51777	0.658984	0.329492	−	0.944158i	$-0.393122\pi$
$47$	4.51777	0.658984	0.329492	+	0.944158i	$0.393122\pi$
$48$	0	0
$49$	11.6249	1.66070
$50$	0	0
$51$	1.28726	0.180253
$52$	0	0
$53$	−4.76630	−0.654701	−0.327351	−	0.944903i	$-0.606156\pi$
$53$	−4.76630	−0.654701	−0.327351	+	0.944903i	$0.606156\pi$
$54$	0	0
$55$	−1.37488	−0.185389
$56$	0	0
$57$	15.7035	2.07998
$58$	0	0
$59$	−0.828801	−0.107901	−0.0539503	−	0.998544i	$-0.517181\pi$
$59$	−0.828801	−0.107901	−0.0539503	+	0.998544i	$0.517181\pi$
$60$	0	0
$61$	−11.2791	−1.44414	−0.722072	−	0.691818i	$-0.756810\pi$
$61$	−11.2791	−1.44414	−0.722072	+	0.691818i	$0.756810\pi$
$62$	0	0
$63$	25.8399	3.25553
$64$	0	0
$65$	−0.564469	−0.0700138
$66$	0	0
$67$	5.64458	0.689595	0.344797	−	0.938677i	$-0.387948\pi$
$67$	5.64458	0.689595	0.344797	+	0.938677i	$0.387948\pi$
$68$	0	0
$69$	−9.96326	−1.19944
$70$	0	0
$71$	4.35032	0.516287	0.258144	−	0.966107i	$-0.416889\pi$
$71$	4.35032	0.516287	0.258144	+	0.966107i	$0.416889\pi$
$72$	0	0
$73$	−0.643138	−0.0752736	−0.0376368	−	0.999291i	$-0.511983\pi$
$73$	−0.643138	−0.0752736	−0.0376368	+	0.999291i	$0.511983\pi$
$74$	0	0
$75$	9.32260	1.07648
$76$	0	0
$77$	−4.31566	−0.491815
$78$	0	0
$79$	9.44903	1.06310	0.531550	−	0.847027i	$-0.321610\pi$
$79$	9.44903	1.06310	0.531550	+	0.847027i	$0.321610\pi$
$80$	0	0
$81$	8.88748	0.987498
$82$	0	0
$83$	−0.140639	−0.0154371	−0.00771857	−	0.999970i	$-0.502457\pi$
$83$	−0.140639	−0.0154371	−0.00771857	+	0.999970i	$0.502457\pi$
$84$	0	0
$85$	−0.590357	−0.0640332
$86$	0	0
$87$	−3.82869	−0.410479
$88$	0	0
$89$	4.73504	0.501913	0.250957	−	0.967998i	$-0.419255\pi$
$89$	4.73504	0.501913	0.250957	+	0.967998i	$0.419255\pi$
$90$	0	0
$91$	−1.77183	−0.185738
$92$	0	0
$93$	−24.3562	−2.52562
$94$	0	0
$95$	−7.20184	−0.738893
$96$	0	0
$97$	−0.436679	−0.0443380	−0.0221690	−	0.999754i	$-0.507057\pi$
$97$	−0.436679	−0.0443380	−0.0221690	+	0.999754i	$0.507057\pi$
$98$	0	0
$99$	−5.98748	−0.601764
$100$	0	0
$101$	11.1419	1.10866	0.554332	−	0.832296i	$-0.312974\pi$
$101$	11.1419	1.10866	0.554332	+	0.832296i	$0.312974\pi$
$102$	0	0
$103$	5.96618	0.587866	0.293933	−	0.955826i	$-0.405036\pi$
$103$	5.96618	0.587866	0.293933	+	0.955826i	$0.405036\pi$
$104$	0	0
$105$	−17.7882	−1.73595
$106$	0	0
$107$	−7.10256	−0.686631	−0.343315	−	0.939220i	$-0.611550\pi$
$107$	−7.10256	−0.686631	−0.343315	+	0.939220i	$0.611550\pi$
$108$	0	0
$109$	5.32153	0.509710	0.254855	−	0.966979i	$-0.417972\pi$
$109$	5.32153	0.509710	0.254855	+	0.966979i	$0.417972\pi$
$110$	0	0
$111$	8.53792	0.810384
$112$	0	0
$113$	0.286464	0.0269483	0.0134742	−	0.999909i	$-0.495711\pi$
$113$	0.286464	0.0269483	0.0134742	+	0.999909i	$0.495711\pi$
$114$	0	0
$115$	4.56928	0.426088
$116$	0	0
$117$	−2.45821	−0.227261
$118$	0	0
$119$	−1.85309	−0.169872
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−16.4827	−1.48620
$124$	0	0
$125$	−11.1499	−0.997276
$126$	0	0
$127$	−3.05723	−0.271285	−0.135643	−	0.990758i	$-0.543310\pi$
$127$	−3.05723	−0.271285	−0.135643	+	0.990758i	$0.543310\pi$
$128$	0	0
$129$	−19.3721	−1.70562
$130$	0	0
$131$	3.56643	0.311600	0.155800	−	0.987789i	$-0.450204\pi$
$131$	3.56643	0.311600	0.155800	+	0.987789i	$0.450204\pi$
$132$	0	0
$133$	−22.6061	−1.96019
$134$	0	0
$135$	−12.3137	−1.05980
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	1.16097	0.0984718	0.0492359	−	0.998787i	$-0.484321\pi$
$139$	1.16097	0.0984718	0.0492359	+	0.998787i	$0.484321\pi$
$140$	0	0
$141$	−13.5439	−1.14060
$142$	0	0
$143$	0.410558	0.0343326
$144$	0	0
$145$	1.75589	0.145819
$146$	0	0
$147$	−34.8505	−2.87442
$148$	0	0
$149$	−15.1600	−1.24195	−0.620976	−	0.783830i	$-0.713263\pi$
$149$	−15.1600	−1.24195	−0.620976	+	0.783830i	$0.713263\pi$
$150$	0	0
$151$	4.96328	0.403906	0.201953	−	0.979395i	$-0.435271\pi$
$151$	4.96328	0.403906	0.201953	+	0.979395i	$0.435271\pi$
$152$	0	0
$153$	−2.57095	−0.207849
$154$	0	0
$155$	11.1701	0.897203
$156$	0	0
$157$	−11.7172	−0.935137	−0.467568	−	0.883957i	$-0.654870\pi$
$157$	−11.7172	−0.935137	−0.467568	+	0.883957i	$0.654870\pi$
$158$	0	0
$159$	14.2889	1.13319
$160$	0	0
$161$	14.3427	1.13036
$162$	0	0
$163$	15.2112	1.19143	0.595716	−	0.803195i	$-0.296868\pi$
$163$	15.2112	1.19143	0.595716	+	0.803195i	$0.296868\pi$
$164$	0	0
$165$	4.12178	0.320880
$166$	0	0
$167$	−12.2547	−0.948296	−0.474148	−	0.880445i	$-0.657244\pi$
$167$	−12.2547	−0.948296	−0.474148	+	0.880445i	$0.657244\pi$
$168$	0	0
$169$	−12.8314	−0.987034
$170$	0	0
$171$	−31.3633	−2.39841
$172$	0	0
$173$	18.6051	1.41452	0.707261	−	0.706953i	$-0.249931\pi$
$173$	18.6051	1.41452	0.707261	+	0.706953i	$0.249931\pi$
$174$	0	0
$175$	−13.4204	−1.01449
$176$	0	0
$177$	2.48467	0.186759
$178$	0	0
$179$	−5.01743	−0.375020	−0.187510	−	0.982263i	$-0.560042\pi$
$179$	−5.01743	−0.375020	−0.187510	+	0.982263i	$0.560042\pi$
$180$	0	0
$181$	9.92030	0.737370	0.368685	−	0.929554i	$-0.379808\pi$
$181$	9.92030	0.737370	0.368685	+	0.929554i	$0.379808\pi$
$182$	0	0
$183$	33.8138	2.49959
$184$	0	0
$185$	−3.91561	−0.287881
$186$	0	0
$187$	0.429387	0.0313999
$188$	0	0
$189$	−38.6519	−2.81152
$190$	0	0
$191$	−26.1284	−1.89059	−0.945293	−	0.326223i	$-0.894224\pi$
$191$	−26.1284	−1.89059	−0.945293	+	0.326223i	$0.894224\pi$
$192$	0	0
$193$	−18.3237	−1.31897	−0.659485	−	0.751717i	$-0.729226\pi$
$193$	−18.3237	−1.31897	−0.659485	+	0.751717i	$0.729226\pi$
$194$	0	0
$195$	1.69223	0.121183
$196$	0	0
$197$	10.3444	0.737007	0.368503	−	0.929626i	$-0.379870\pi$
$197$	10.3444	0.737007	0.368503	+	0.929626i	$0.379870\pi$
$198$	0	0
$199$	5.16745	0.366311	0.183155	−	0.983084i	$-0.441369\pi$
$199$	5.16745	0.366311	0.183155	+	0.983084i	$0.441369\pi$
$200$	0	0
$201$	−16.9219	−1.19358
$202$	0	0
$203$	5.51162	0.386840
$204$	0	0
$205$	7.55920	0.527958
$206$	0	0
$207$	19.8988	1.38306
$208$	0	0
$209$	5.23814	0.362330
$210$	0	0
$211$	1.52577	0.105038	0.0525190	−	0.998620i	$-0.483275\pi$
$211$	1.52577	0.105038	0.0525190	+	0.998620i	$0.483275\pi$
$212$	0	0
$213$	−13.0419	−0.893614
$214$	0	0
$215$	8.88430	0.605904
$216$	0	0
$217$	35.0621	2.38017
$218$	0	0
$219$	1.92807	0.130287
$220$	0	0
$221$	0.176288	0.0118584
$222$	0	0
$223$	4.93872	0.330721	0.165361	−	0.986233i	$-0.447121\pi$
$223$	4.93872	0.330721	0.165361	+	0.986233i	$0.447121\pi$
$224$	0	0
$225$	−18.6193	−1.24128
$226$	0	0
$227$	6.01395	0.399160	0.199580	−	0.979882i	$-0.436042\pi$
$227$	6.01395	0.399160	0.199580	+	0.979882i	$0.436042\pi$
$228$	0	0
$229$	−6.71914	−0.444013	−0.222006	−	0.975045i	$-0.571261\pi$
$229$	−6.71914	−0.444013	−0.222006	+	0.975045i	$0.571261\pi$
$230$	0	0
$231$	12.9380	0.851256
$232$	0	0
$233$	−8.61386	−0.564313	−0.282157	−	0.959368i	$-0.591050\pi$
$233$	−8.61386	−0.564313	−0.282157	+	0.959368i	$0.591050\pi$
$234$	0	0
$235$	6.21140	0.405187
$236$	0	0
$237$	−28.3274	−1.84006
$238$	0	0
$239$	8.18516	0.529454	0.264727	−	0.964323i	$-0.414718\pi$
$239$	8.18516	0.529454	0.264727	+	0.964323i	$0.414718\pi$
$240$	0	0
$241$	25.3822	1.63501	0.817506	−	0.575920i	$-0.195356\pi$
$241$	25.3822	1.63501	0.817506	+	0.575920i	$0.195356\pi$
$242$	0	0
$243$	0.224720	0.0144158
$244$	0	0
$245$	15.9829	1.02111
$246$	0	0
$247$	2.15056	0.136837
$248$	0	0
$249$	0.421623	0.0267193
$250$	0	0
$251$	23.2594	1.46812	0.734061	−	0.679084i	$-0.237623\pi$
$251$	23.2594	1.46812	0.734061	+	0.679084i	$0.237623\pi$
$252$	0	0
$253$	−3.32340	−0.208940
$254$	0	0
$255$	1.76984	0.110832
$256$	0	0
$257$	9.55054	0.595746	0.297873	−	0.954605i	$-0.403723\pi$
$257$	9.55054	0.595746	0.297873	+	0.954605i	$0.403723\pi$
$258$	0	0
$259$	−12.2908	−0.763714
$260$	0	0
$261$	7.64673	0.473321
$262$	0	0
$263$	27.0886	1.67036	0.835178	−	0.549980i	$-0.185365\pi$
$263$	27.0886	1.67036	0.835178	+	0.549980i	$0.185365\pi$
$264$	0	0
$265$	−6.55310	−0.402554
$266$	0	0
$267$	−14.1952	−0.868735
$268$	0	0
$269$	−1.27681	−0.0778485	−0.0389242	−	0.999242i	$-0.512393\pi$
$269$	−1.27681	−0.0778485	−0.0389242	+	0.999242i	$0.512393\pi$
$270$	0	0
$271$	3.95462	0.240226	0.120113	−	0.992760i	$-0.461674\pi$
$271$	3.95462	0.240226	0.120113	+	0.992760i	$0.461674\pi$
$272$	0	0
$273$	5.31179	0.321484
$274$	0	0
$275$	3.10970	0.187522
$276$	0	0
$277$	24.2670	1.45806	0.729031	−	0.684480i	$-0.239971\pi$
$277$	24.2670	1.45806	0.729031	+	0.684480i	$0.239971\pi$
$278$	0	0
$279$	48.6446	2.91228
$280$	0	0
$281$	7.64138	0.455846	0.227923	−	0.973679i	$-0.426806\pi$
$281$	7.64138	0.455846	0.227923	+	0.973679i	$0.426806\pi$
$282$	0	0
$283$	22.3187	1.32671	0.663354	−	0.748306i	$-0.269133\pi$
$283$	22.3187	1.32671	0.663354	+	0.748306i	$0.269133\pi$
$284$	0	0
$285$	21.5905	1.27891
$286$	0	0
$287$	23.7278	1.40061
$288$	0	0
$289$	−16.8156	−0.989155
$290$	0	0
$291$	1.30913	0.0767423
$292$	0	0
$293$	13.5946	0.794203	0.397101	−	0.917775i	$-0.370016\pi$
$293$	13.5946	0.794203	0.397101	+	0.917775i	$0.370016\pi$
$294$	0	0
$295$	−1.13950	−0.0663445
$296$	0	0
$297$	8.95621	0.519692
$298$	0	0
$299$	−1.36445	−0.0789080
$300$	0	0
$301$	27.8872	1.60739
$302$	0	0
$303$	−33.4025	−1.91893
$304$	0	0
$305$	−15.5075	−0.887956
$306$	0	0
$307$	−10.2882	−0.587179	−0.293590	−	0.955932i	$-0.594850\pi$
$307$	−10.2882	−0.587179	−0.293590	+	0.955932i	$0.594850\pi$
$308$	0	0
$309$	−17.8861	−1.01750
$310$	0	0
$311$	29.1114	1.65076	0.825379	−	0.564578i	$-0.190961\pi$
$311$	29.1114	1.65076	0.825379	+	0.564578i	$0.190961\pi$
$312$	0	0
$313$	−32.5579	−1.84028	−0.920141	−	0.391588i	$-0.871926\pi$
$313$	−32.5579	−1.84028	−0.920141	+	0.391588i	$0.871926\pi$
$314$	0	0
$315$	35.5269	2.00171
$316$	0	0
$317$	−25.6078	−1.43828	−0.719139	−	0.694866i	$-0.755464\pi$
$317$	−25.6078	−1.43828	−0.719139	+	0.694866i	$0.755464\pi$
$318$	0	0
$319$	−1.27712	−0.0715050
$320$	0	0
$321$	21.2929	1.18845
$322$	0	0
$323$	2.24919	0.125148
$324$	0	0
$325$	1.27671	0.0708192
$326$	0	0
$327$	−15.9535	−0.882229
$328$	0	0
$329$	19.4972	1.07491
$330$	0	0
$331$	11.4351	0.628530	0.314265	−	0.949335i	$-0.398242\pi$
$331$	11.4351	0.628530	0.314265	+	0.949335i	$0.398242\pi$
$332$	0	0
$333$	−17.0521	−0.934448
$334$	0	0
$335$	7.76063	0.424009
$336$	0	0
$337$	−4.88623	−0.266170	−0.133085	−	0.991105i	$-0.542488\pi$
$337$	−4.88623	−0.266170	−0.133085	+	0.991105i	$0.542488\pi$
$338$	0	0
$339$	−0.858796	−0.0466434
$340$	0	0
$341$	−8.12439	−0.439960
$342$	0	0
$343$	19.9596	1.07772
$344$	0	0
$345$	−13.6983	−0.737492
$346$	0	0
$347$	11.7072	0.628474	0.314237	−	0.949345i	$-0.398251\pi$
$347$	11.7072	0.628474	0.314237	+	0.949345i	$0.398251\pi$
$348$	0	0
$349$	−6.20955	−0.332390	−0.166195	−	0.986093i	$-0.553148\pi$
$349$	−6.20955	−0.332390	−0.166195	+	0.986093i	$0.553148\pi$
$350$	0	0
$351$	3.67704	0.196266
$352$	0	0
$353$	29.6901	1.58024	0.790122	−	0.612950i	$-0.210017\pi$
$353$	29.6901	1.58024	0.790122	+	0.612950i	$0.210017\pi$
$354$	0	0
$355$	5.98117	0.317448
$356$	0	0
$357$	5.55540	0.294023
$358$	0	0
$359$	0.971238	0.0512600	0.0256300	−	0.999671i	$-0.491841\pi$
$359$	0.971238	0.0512600	0.0256300	+	0.999671i	$0.491841\pi$
$360$	0	0
$361$	8.43816	0.444114
$362$	0	0
$363$	−2.99791	−0.157350
$364$	0	0
$365$	−0.884239	−0.0462832
$366$	0	0
$367$	−33.3504	−1.74088	−0.870438	−	0.492279i	$-0.836164\pi$
$367$	−33.3504	−1.74088	−0.870438	+	0.492279i	$0.836164\pi$
$368$	0	0
$369$	32.9196	1.71373
$370$	0	0
$371$	−20.5697	−1.06793
$372$	0	0
$373$	22.5561	1.16791	0.583954	−	0.811786i	$-0.301505\pi$
$373$	22.5561	1.16791	0.583954	+	0.811786i	$0.301505\pi$
$374$	0	0
$375$	33.4264	1.72613
$376$	0	0
$377$	−0.524332	−0.0270045
$378$	0	0
$379$	11.6842	0.600175	0.300088	−	0.953912i	$-0.402984\pi$
$379$	11.6842	0.600175	0.300088	+	0.953912i	$0.402984\pi$
$380$	0	0
$381$	9.16531	0.469553
$382$	0	0
$383$	36.5931	1.86982	0.934909	−	0.354887i	$-0.115481\pi$
$383$	36.5931	1.86982	0.934909	+	0.354887i	$0.115481\pi$
$384$	0	0
$385$	−5.93353	−0.302401
$386$	0	0
$387$	38.6903	1.96674
$388$	0	0
$389$	35.1517	1.78226	0.891132	−	0.453745i	$-0.149912\pi$
$389$	35.1517	1.78226	0.891132	+	0.453745i	$0.149912\pi$
$390$	0	0
$391$	−1.42702	−0.0721677
$392$	0	0
$393$	−10.6918	−0.539332
$394$	0	0
$395$	12.9913	0.653664
$396$	0	0
$397$	11.1561	0.559909	0.279954	−	0.960013i	$-0.409681\pi$
$397$	11.1561	0.559909	0.279954	+	0.960013i	$0.409681\pi$
$398$	0	0
$399$	67.7710	3.39279
$400$	0	0
$401$	−8.24256	−0.411614	−0.205807	−	0.978593i	$-0.565982\pi$
$401$	−8.24256	−0.411614	−0.205807	+	0.978593i	$0.565982\pi$
$402$	0	0
$403$	−3.33553	−0.166155
$404$	0	0
$405$	12.2192	0.607179
$406$	0	0
$407$	2.84796	0.141168
$408$	0	0
$409$	−5.28798	−0.261474	−0.130737	−	0.991417i	$-0.541734\pi$
$409$	−5.28798	−0.261474	−0.130737	+	0.991417i	$0.541734\pi$
$410$	0	0
$411$	2.99791	0.147876
$412$	0	0
$413$	−3.57682	−0.176004
$414$	0	0
$415$	−0.193362	−0.00949177
$416$	0	0
$417$	−3.48047	−0.170440
$418$	0	0
$419$	−10.2779	−0.502107	−0.251054	−	0.967973i	$-0.580777\pi$
$419$	−10.2779	−0.502107	−0.251054	+	0.967973i	$0.580777\pi$
$420$	0	0
$421$	−8.39994	−0.409388	−0.204694	−	0.978826i	$-0.565620\pi$
$421$	−8.39994	−0.409388	−0.204694	+	0.978826i	$0.565620\pi$
$422$	0	0
$423$	27.0500	1.31522
$424$	0	0
$425$	1.33526	0.0647698
$426$	0	0
$427$	−48.6769	−2.35564
$428$	0	0
$429$	−1.23082	−0.0594244
$430$	0	0
$431$	−0.374868	−0.0180568	−0.00902839	−	0.999959i	$-0.502874\pi$
$431$	−0.374868	−0.0180568	−0.00902839	+	0.999959i	$0.502874\pi$
$432$	0	0
$433$	−5.17250	−0.248575	−0.124287	−	0.992246i	$-0.539664\pi$
$433$	−5.17250	−0.248575	−0.124287	+	0.992246i	$0.539664\pi$
$434$	0	0
$435$	−5.26401	−0.252390
$436$	0	0
$437$	−17.4084	−0.832758
$438$	0	0
$439$	−31.7650	−1.51606	−0.758029	−	0.652220i	$-0.773838\pi$
$439$	−31.7650	−1.51606	−0.758029	+	0.652220i	$0.773838\pi$
$440$	0	0
$441$	69.6040	3.31448
$442$	0	0
$443$	−20.5543	−0.976563	−0.488282	−	0.872686i	$-0.662376\pi$
$443$	−20.5543	−0.976563	−0.488282	+	0.872686i	$0.662376\pi$
$444$	0	0
$445$	6.51013	0.308610
$446$	0	0
$447$	45.4482	2.14963
$448$	0	0
$449$	9.02562	0.425945	0.212973	−	0.977058i	$-0.431685\pi$
$449$	9.02562	0.425945	0.212973	+	0.977058i	$0.431685\pi$
$450$	0	0
$451$	−5.49807	−0.258894
$452$	0	0
$453$	−14.8795	−0.699099
$454$	0	0
$455$	−2.43606	−0.114204
$456$	0	0
$457$	−6.00775	−0.281031	−0.140515	−	0.990079i	$-0.544876\pi$
$457$	−6.00775	−0.281031	−0.140515	+	0.990079i	$0.544876\pi$
$458$	0	0
$459$	3.84568	0.179501
$460$	0	0
$461$	7.21025	0.335815	0.167907	−	0.985803i	$-0.446299\pi$
$461$	7.21025	0.335815	0.167907	+	0.985803i	$0.446299\pi$
$462$	0	0
$463$	−35.5955	−1.65427	−0.827133	−	0.562007i	$-0.810030\pi$
$463$	−35.5955	−1.65427	−0.827133	+	0.562007i	$0.810030\pi$
$464$	0	0
$465$	−33.4869	−1.55292
$466$	0	0
$467$	11.2126	0.518857	0.259429	−	0.965762i	$-0.416466\pi$
$467$	11.2126	0.518857	0.259429	+	0.965762i	$0.416466\pi$
$468$	0	0
$469$	24.3601	1.12484
$470$	0	0
$471$	35.1272	1.61858
$472$	0	0
$473$	−6.46186	−0.297117
$474$	0	0
$475$	16.2890	0.747393
$476$	0	0
$477$	−28.5381	−1.30667
$478$	0	0
$479$	−21.0867	−0.963474	−0.481737	−	0.876316i	$-0.659994\pi$
$479$	−21.0867	−0.963474	−0.481737	+	0.876316i	$0.659994\pi$
$480$	0	0
$481$	1.16925	0.0533133
$482$	0	0
$483$	−42.9980	−1.95648
$484$	0	0
$485$	−0.600383	−0.0272620
$486$	0	0
$487$	37.3391	1.69200	0.845998	−	0.533186i	$-0.179005\pi$
$487$	37.3391	1.69200	0.845998	+	0.533186i	$0.179005\pi$
$488$	0	0
$489$	−45.6018	−2.06219
$490$	0	0
$491$	11.3046	0.510171	0.255086	−	0.966918i	$-0.417896\pi$
$491$	11.3046	0.510171	0.255086	+	0.966918i	$0.417896\pi$
$492$	0	0
$493$	−0.548379	−0.0246977
$494$	0	0
$495$	−8.23208	−0.370005
$496$	0	0
$497$	18.7745	0.842151
$498$	0	0
$499$	−0.359420	−0.0160899	−0.00804493	−	0.999968i	$-0.502561\pi$
$499$	−0.359420	−0.0160899	−0.00804493	+	0.999968i	$0.502561\pi$
$500$	0	0
$501$	36.7385	1.64135
$502$	0	0
$503$	8.98912	0.400805	0.200403	−	0.979714i	$-0.435775\pi$
$503$	8.98912	0.400805	0.200403	+	0.979714i	$0.435775\pi$
$504$	0	0
$505$	15.3189	0.681680
$506$	0	0
$507$	38.4675	1.70840
$508$	0	0
$509$	26.5210	1.17552	0.587762	−	0.809034i	$-0.300009\pi$
$509$	26.5210	1.17552	0.587762	+	0.809034i	$0.300009\pi$
$510$	0	0
$511$	−2.77556	−0.122784
$512$	0	0
$513$	46.9139	2.07130
$514$	0	0
$515$	8.20280	0.361459
$516$	0	0
$517$	−4.51777	−0.198691
$518$	0	0
$519$	−55.7766	−2.44832
$520$	0	0
$521$	13.0029	0.569666	0.284833	−	0.958577i	$-0.408062\pi$
$521$	13.0029	0.569666	0.284833	+	0.958577i	$0.408062\pi$
$522$	0	0
$523$	13.1971	0.577068	0.288534	−	0.957470i	$-0.406832\pi$
$523$	13.1971	0.577068	0.288534	+	0.957470i	$0.406832\pi$
$524$	0	0
$525$	40.2332	1.75592
$526$	0	0
$527$	−3.48851	−0.151962
$528$	0	0
$529$	−11.9550	−0.519784
$530$	0	0
$531$	−4.96243	−0.215351
$532$	0	0
$533$	−2.25728	−0.0977736
$534$	0	0
$535$	−9.76519	−0.422186
$536$	0	0
$537$	15.0418	0.649102
$538$	0	0
$539$	−11.6249	−0.500721
$540$	0	0
$541$	−9.58120	−0.411928	−0.205964	−	0.978560i	$-0.566033\pi$
$541$	−9.58120	−0.411928	−0.205964	+	0.978560i	$0.566033\pi$
$542$	0	0
$543$	−29.7402	−1.27627
$544$	0	0
$545$	7.31648	0.313403
$546$	0	0
$547$	20.9677	0.896513	0.448257	−	0.893905i	$-0.352045\pi$
$547$	20.9677	0.896513	0.448257	+	0.893905i	$0.352045\pi$
$548$	0	0
$549$	−67.5335	−2.88226
$550$	0	0
$551$	−6.68974	−0.284992
$552$	0	0
$553$	40.7788	1.73409
$554$	0	0
$555$	11.7386	0.498278
$556$	0	0
$557$	35.4932	1.50389	0.751947	−	0.659224i	$-0.229115\pi$
$557$	35.4932	1.50389	0.751947	+	0.659224i	$0.229115\pi$
$558$	0	0
$559$	−2.65297	−0.112209
$560$	0	0
$561$	−1.28726	−0.0543484
$562$	0	0
$563$	−38.0466	−1.60347	−0.801737	−	0.597677i	$-0.796090\pi$
$563$	−38.0466	−1.60347	−0.801737	+	0.597677i	$0.796090\pi$
$564$	0	0
$565$	0.393855	0.0165696
$566$	0	0
$567$	38.3554	1.61077
$568$	0	0
$569$	−3.83176	−0.160636	−0.0803179	−	0.996769i	$-0.525594\pi$
$569$	−3.83176	−0.160636	−0.0803179	+	0.996769i	$0.525594\pi$
$570$	0	0
$571$	−31.5629	−1.32087	−0.660433	−	0.750885i	$-0.729627\pi$
$571$	−31.5629	−1.32087	−0.660433	+	0.750885i	$0.729627\pi$
$572$	0	0
$573$	78.3307	3.27231
$574$	0	0
$575$	−10.3348	−0.430989
$576$	0	0
$577$	−11.2706	−0.469202	−0.234601	−	0.972092i	$-0.575378\pi$
$577$	−11.2706	−0.469202	−0.234601	+	0.972092i	$0.575378\pi$
$578$	0	0
$579$	54.9329	2.28294
$580$	0	0
$581$	−0.606950	−0.0251805
$582$	0	0
$583$	4.76630	0.197400
$584$	0	0
$585$	−3.37975	−0.139735
$586$	0	0
$587$	−4.81115	−0.198577	−0.0992887	−	0.995059i	$-0.531657\pi$
$587$	−4.81115	−0.198577	−0.0992887	+	0.995059i	$0.531657\pi$
$588$	0	0
$589$	−42.5567	−1.75352
$590$	0	0
$591$	−31.0116	−1.27565
$592$	0	0
$593$	36.4511	1.49687	0.748434	−	0.663209i	$-0.230806\pi$
$593$	36.4511	1.49687	0.748434	+	0.663209i	$0.230806\pi$
$594$	0	0
$595$	−2.54778	−0.104449
$596$	0	0
$597$	−15.4916	−0.634028
$598$	0	0
$599$	30.3414	1.23972	0.619859	−	0.784713i	$-0.287190\pi$
$599$	30.3414	1.23972	0.619859	+	0.784713i	$0.287190\pi$
$600$	0	0
$601$	−31.5996	−1.28897	−0.644487	−	0.764616i	$-0.722929\pi$
$601$	−31.5996	−1.28897	−0.644487	+	0.764616i	$0.722929\pi$
$602$	0	0
$603$	33.7968	1.37631
$604$	0	0
$605$	1.37488	0.0558969
$606$	0	0
$607$	3.86066	0.156699	0.0783497	−	0.996926i	$-0.475035\pi$
$607$	3.86066	0.156699	0.0783497	+	0.996926i	$0.475035\pi$
$608$	0	0
$609$	−16.5233	−0.669560
$610$	0	0
$611$	−1.85481	−0.0750374
$612$	0	0
$613$	2.24143	0.0905306	0.0452653	−	0.998975i	$-0.485587\pi$
$613$	2.24143	0.0905306	0.0452653	+	0.998975i	$0.485587\pi$
$614$	0	0
$615$	−22.6618	−0.913813
$616$	0	0
$617$	5.18870	0.208889	0.104445	−	0.994531i	$-0.466694\pi$
$617$	5.18870	0.208889	0.104445	+	0.994531i	$0.466694\pi$
$618$	0	0
$619$	4.49504	0.180671	0.0903354	−	0.995911i	$-0.471206\pi$
$619$	4.49504	0.180671	0.0903354	+	0.995911i	$0.471206\pi$
$620$	0	0
$621$	−29.7650	−1.19443
$622$	0	0
$623$	20.4348	0.818704
$624$	0	0
$625$	0.218702	0.00874809
$626$	0	0
$627$	−15.7035	−0.627137
$628$	0	0
$629$	1.22288	0.0487592
$630$	0	0
$631$	35.3029	1.40539	0.702693	−	0.711494i	$-0.251981\pi$
$631$	35.3029	1.40539	0.702693	+	0.711494i	$0.251981\pi$
$632$	0	0
$633$	−4.57411	−0.181805
$634$	0	0
$635$	−4.20333	−0.166804
$636$	0	0
$637$	−4.77271	−0.189102
$638$	0	0
$639$	26.0474	1.03042
$640$	0	0
$641$	1.37580	0.0543407	0.0271703	−	0.999631i	$-0.491350\pi$
$641$	1.37580	0.0543407	0.0271703	+	0.999631i	$0.491350\pi$
$642$	0	0
$643$	4.54665	0.179302	0.0896512	−	0.995973i	$-0.471425\pi$
$643$	4.54665	0.179302	0.0896512	+	0.995973i	$0.471425\pi$
$644$	0	0
$645$	−26.6344	−1.04873
$646$	0	0
$647$	22.4542	0.882766	0.441383	−	0.897319i	$-0.354488\pi$
$647$	22.4542	0.882766	0.441383	+	0.897319i	$0.354488\pi$
$648$	0	0
$649$	0.828801	0.0325333
$650$	0	0
$651$	−105.113	−4.11971
$652$	0	0
$653$	−20.6420	−0.807785	−0.403892	−	0.914807i	$-0.632343\pi$
$653$	−20.6420	−0.807785	−0.403892	+	0.914807i	$0.632343\pi$
$654$	0	0
$655$	4.90342	0.191592
$656$	0	0
$657$	−3.85078	−0.150233
$658$	0	0
$659$	−31.8850	−1.24206	−0.621032	−	0.783786i	$-0.713286\pi$
$659$	−31.8850	−1.24206	−0.621032	+	0.783786i	$0.713286\pi$
$660$	0	0
$661$	13.8872	0.540148	0.270074	−	0.962840i	$-0.412952\pi$
$661$	13.8872	0.540148	0.270074	+	0.962840i	$0.412952\pi$
$662$	0	0
$663$	−0.528497	−0.0205251
$664$	0	0
$665$	−31.0807	−1.20526
$666$	0	0
$667$	4.24438	0.164343
$668$	0	0
$669$	−14.8059	−0.572428
$670$	0	0
$671$	11.2791	0.435426
$672$	0	0
$673$	7.72133	0.297636	0.148818	−	0.988865i	$-0.452453\pi$
$673$	7.72133	0.297636	0.148818	+	0.988865i	$0.452453\pi$
$674$	0	0
$675$	27.8511	1.07199
$676$	0	0
$677$	−33.6364	−1.29275	−0.646377	−	0.763018i	$-0.723717\pi$
$677$	−33.6364	−1.29275	−0.646377	+	0.763018i	$0.723717\pi$
$678$	0	0
$679$	−1.88456	−0.0723227
$680$	0	0
$681$	−18.0293	−0.690884
$682$	0	0
$683$	0.715725	0.0273865	0.0136932	−	0.999906i	$-0.495641\pi$
$683$	0.715725	0.0273865	0.0136932	+	0.999906i	$0.495641\pi$
$684$	0	0
$685$	−1.37488	−0.0525316
$686$	0	0
$687$	20.1434	0.768518
$688$	0	0
$689$	1.95684	0.0745497
$690$	0	0
$691$	−18.0658	−0.687257	−0.343628	−	0.939106i	$-0.611656\pi$
$691$	−18.0658	−0.687257	−0.343628	+	0.939106i	$0.611656\pi$
$692$	0	0
$693$	−25.8399	−0.981578
$694$	0	0
$695$	1.59619	0.0605470
$696$	0	0
$697$	−2.36080	−0.0894217
$698$	0	0
$699$	25.8236	0.976739
$700$	0	0
$701$	20.6792	0.781041	0.390521	−	0.920594i	$-0.372295\pi$
$701$	20.6792	0.781041	0.390521	+	0.920594i	$0.372295\pi$
$702$	0	0
$703$	14.9180	0.562643
$704$	0	0
$705$	−18.6212	−0.701317
$706$	0	0
$707$	48.0848	1.80841
$708$	0	0
$709$	−48.0807	−1.80571	−0.902854	−	0.429947i	$-0.858532\pi$
$709$	−48.0807	−1.80571	−0.902854	+	0.429947i	$0.858532\pi$
$710$	0	0
$711$	56.5759	2.12176
$712$	0	0
$713$	27.0006	1.01118
$714$	0	0
$715$	0.564469	0.0211100
$716$	0	0
$717$	−24.5384	−0.916403
$718$	0	0
$719$	17.5328	0.653863	0.326931	−	0.945048i	$-0.393985\pi$
$719$	17.5328	0.653863	0.326931	+	0.945048i	$0.393985\pi$
$720$	0	0
$721$	25.7480	0.958906
$722$	0	0
$723$	−76.0937	−2.82995
$724$	0	0
$725$	−3.97146	−0.147496
$726$	0	0
$727$	31.3755	1.16365	0.581826	−	0.813313i	$-0.302338\pi$
$727$	31.3755	1.16365	0.581826	+	0.813313i	$0.302338\pi$
$728$	0	0
$729$	−27.3361	−1.01245
$730$	0	0
$731$	−2.77464	−0.102624
$732$	0	0
$733$	−30.9765	−1.14414	−0.572071	−	0.820204i	$-0.693860\pi$
$733$	−30.9765	−1.14414	−0.572071	+	0.820204i	$0.693860\pi$
$734$	0	0
$735$	−47.9154	−1.76739
$736$	0	0
$737$	−5.64458	−0.207921
$738$	0	0
$739$	−34.4523	−1.26735	−0.633673	−	0.773601i	$-0.718454\pi$
$739$	−34.4523	−1.26735	−0.633673	+	0.773601i	$0.718454\pi$
$740$	0	0
$741$	−6.44720	−0.236844
$742$	0	0
$743$	12.1801	0.446844	0.223422	−	0.974722i	$-0.428277\pi$
$743$	12.1801	0.446844	0.223422	+	0.974722i	$0.428277\pi$
$744$	0	0
$745$	−20.8432	−0.763634
$746$	0	0
$747$	−0.842073	−0.0308098
$748$	0	0
$749$	−30.6523	−1.12001
$750$	0	0
$751$	12.7717	0.466047	0.233024	−	0.972471i	$-0.425138\pi$
$751$	12.7717	0.466047	0.233024	+	0.972471i	$0.425138\pi$
$752$	0	0
$753$	−69.7297	−2.54109
$754$	0	0
$755$	6.82393	0.248348
$756$	0	0
$757$	−4.23276	−0.153842	−0.0769211	−	0.997037i	$-0.524509\pi$
$757$	−4.23276	−0.153842	−0.0769211	+	0.997037i	$0.524509\pi$
$758$	0	0
$759$	9.96326	0.361643
$760$	0	0
$761$	22.9620	0.832373	0.416187	−	0.909279i	$-0.363366\pi$
$761$	22.9620	0.832373	0.416187	+	0.909279i	$0.363366\pi$
$762$	0	0
$763$	22.9659	0.831421
$764$	0	0
$765$	−3.53475	−0.127799
$766$	0	0
$767$	0.340271	0.0122865
$768$	0	0
$769$	−46.9279	−1.69226	−0.846132	−	0.532974i	$-0.821074\pi$
$769$	−46.9279	−1.69226	−0.846132	+	0.532974i	$0.821074\pi$
$770$	0	0
$771$	−28.6317	−1.03114
$772$	0	0
$773$	26.1802	0.941636	0.470818	−	0.882230i	$-0.343959\pi$
$773$	26.1802	0.941636	0.470818	+	0.882230i	$0.343959\pi$
$774$	0	0
$775$	−25.2644	−0.907524
$776$	0	0
$777$	36.8468	1.32187
$778$	0	0
$779$	−28.7997	−1.03186
$780$	0	0
$781$	−4.35032	−0.155667
$782$	0	0
$783$	−11.4381	−0.408766
$784$	0	0
$785$	−16.1098	−0.574984
$786$	0	0
$787$	44.0482	1.57015	0.785075	−	0.619401i	$-0.212624\pi$
$787$	44.0482	1.57015	0.785075	+	0.619401i	$0.212624\pi$
$788$	0	0
$789$	−81.2093	−2.89113
$790$	0	0
$791$	1.23628	0.0439572
$792$	0	0
$793$	4.63074	0.164442
$794$	0	0
$795$	19.6456	0.696758
$796$	0	0
$797$	28.2720	1.00145	0.500723	−	0.865607i	$-0.333067\pi$
$797$	28.2720	1.00145	0.500723	+	0.865607i	$0.333067\pi$
$798$	0	0
$799$	−1.93987	−0.0686277
$800$	0	0
$801$	28.3510	1.00173
$802$	0	0
$803$	0.643138	0.0226958
$804$	0	0
$805$	19.7195	0.695020
$806$	0	0
$807$	3.82776	0.134744
$808$	0	0
$809$	28.2256	0.992360	0.496180	−	0.868220i	$-0.334736\pi$
$809$	28.2256	0.992360	0.496180	+	0.868220i	$0.334736\pi$
$810$	0	0
$811$	43.5015	1.52755	0.763773	−	0.645485i	$-0.223345\pi$
$811$	43.5015	1.52755	0.763773	+	0.645485i	$0.223345\pi$
$812$	0	0
$813$	−11.8556	−0.415794
$814$	0	0
$815$	20.9136	0.732572
$816$	0	0
$817$	−33.8482	−1.18420
$818$	0	0
$819$	−10.6088	−0.370701
$820$	0	0
$821$	−9.15720	−0.319589	−0.159794	−	0.987150i	$-0.551083\pi$
$821$	−9.15720	−0.319589	−0.159794	+	0.987150i	$0.551083\pi$
$822$	0	0
$823$	−21.3524	−0.744297	−0.372148	−	0.928173i	$-0.621379\pi$
$823$	−21.3524	−0.744297	−0.372148	+	0.928173i	$0.621379\pi$
$824$	0	0
$825$	−9.32260	−0.324571
$826$	0	0
$827$	43.6416	1.51757	0.758783	−	0.651344i	$-0.225794\pi$
$827$	43.6416	1.51757	0.758783	+	0.651344i	$0.225794\pi$
$828$	0	0
$829$	48.2356	1.67529	0.837646	−	0.546213i	$-0.183931\pi$
$829$	48.2356	1.67529	0.837646	+	0.546213i	$0.183931\pi$
$830$	0	0
$831$	−72.7503	−2.52368
$832$	0	0
$833$	−4.99159	−0.172948
$834$	0	0
$835$	−16.8488	−0.583075
$836$	0	0
$837$	−72.7637	−2.51508
$838$	0	0
$839$	20.7845	0.717560	0.358780	−	0.933422i	$-0.383193\pi$
$839$	20.7845	0.717560	0.358780	+	0.933422i	$0.383193\pi$
$840$	0	0
$841$	−27.3690	−0.943757
$842$	0	0
$843$	−22.9082	−0.789000
$844$	0	0
$845$	−17.6417	−0.606894
$846$	0	0
$847$	4.31566	0.148288
$848$	0	0
$849$	−66.9094	−2.29633
$850$	0	0
$851$	−9.46489	−0.324452
$852$	0	0
$853$	−53.4398	−1.82975	−0.914873	−	0.403743i	$-0.867709\pi$
$853$	−53.4398	−1.82975	−0.914873	+	0.403743i	$0.867709\pi$
$854$	0	0
$855$	−43.1209	−1.47470
$856$	0	0
$857$	28.3111	0.967089	0.483545	−	0.875320i	$-0.339349\pi$
$857$	28.3111	0.967089	0.483545	+	0.875320i	$0.339349\pi$
$858$	0	0
$859$	−35.7257	−1.21895	−0.609473	−	0.792807i	$-0.708619\pi$
$859$	−35.7257	−1.21895	−0.609473	+	0.792807i	$0.708619\pi$
$860$	0	0
$861$	−71.1339	−2.42424
$862$	0	0
$863$	−11.4950	−0.391295	−0.195648	−	0.980674i	$-0.562681\pi$
$863$	−11.4950	−0.391295	−0.195648	+	0.980674i	$0.562681\pi$
$864$	0	0
$865$	25.5799	0.869742
$866$	0	0
$867$	50.4118	1.71207
$868$	0	0
$869$	−9.44903	−0.320536
$870$	0	0
$871$	−2.31743	−0.0785230
$872$	0	0
$873$	−2.61461	−0.0884911
$874$	0	0
$875$	−48.1191	−1.62672
$876$	0	0
$877$	34.6928	1.17149	0.585746	−	0.810495i	$-0.300802\pi$
$877$	34.6928	1.17149	0.585746	+	0.810495i	$0.300802\pi$
$878$	0	0
$879$	−40.7553	−1.37464
$880$	0	0
$881$	19.7057	0.663901	0.331951	−	0.943297i	$-0.392293\pi$
$881$	19.7057	0.663901	0.331951	+	0.943297i	$0.392293\pi$
$882$	0	0
$883$	23.5150	0.791342	0.395671	−	0.918392i	$-0.370512\pi$
$883$	23.5150	0.791342	0.395671	+	0.918392i	$0.370512\pi$
$884$	0	0
$885$	3.41613	0.114832
$886$	0	0
$887$	−13.5980	−0.456577	−0.228288	−	0.973594i	$-0.573313\pi$
$887$	−13.5980	−0.456577	−0.228288	+	0.973594i	$0.573313\pi$
$888$	0	0
$889$	−13.1940	−0.442511
$890$	0	0
$891$	−8.88748	−0.297742
$892$	0	0
$893$	−23.6647	−0.791910
$894$	0	0
$895$	−6.89838	−0.230587
$896$	0	0
$897$	4.09050	0.136578
$898$	0	0
$899$	10.3758	0.346053
$900$	0	0
$901$	2.04659	0.0681817
$902$	0	0
$903$	−83.6034	−2.78215
$904$	0	0
$905$	13.6392	0.453384
$906$	0	0
$907$	35.0469	1.16371	0.581857	−	0.813291i	$-0.302326\pi$
$907$	35.0469	1.16371	0.581857	+	0.813291i	$0.302326\pi$
$908$	0	0
$909$	66.7121	2.21270
$910$	0	0
$911$	−16.8173	−0.557182	−0.278591	−	0.960410i	$-0.589867\pi$
$911$	−16.8173	−0.557182	−0.278591	+	0.960410i	$0.589867\pi$
$912$	0	0
$913$	0.140639	0.00465447
$914$	0	0
$915$	46.4901	1.53691
$916$	0	0
$917$	15.3915	0.508272
$918$	0	0
$919$	−7.73987	−0.255315	−0.127657	−	0.991818i	$-0.540746\pi$
$919$	−7.73987	−0.255315	−0.127657	+	0.991818i	$0.540746\pi$
$920$	0	0
$921$	30.8432	1.01632
$922$	0	0
$923$	−1.78606	−0.0587888
$924$	0	0
$925$	8.85628	0.291193
$926$	0	0
$927$	35.7224	1.17328
$928$	0	0
$929$	−24.4395	−0.801835	−0.400918	−	0.916114i	$-0.631309\pi$
$929$	−24.4395	−0.801835	−0.400918	+	0.916114i	$0.631309\pi$
$930$	0	0
$931$	−60.8930	−1.99569
$932$	0	0
$933$	−87.2736	−2.85721
$934$	0	0
$935$	0.590357	0.0193067
$936$	0	0
$937$	10.0725	0.329054	0.164527	−	0.986373i	$-0.447390\pi$
$937$	10.0725	0.329054	0.164527	+	0.986373i	$0.447390\pi$
$938$	0	0
$939$	97.6057	3.18524
$940$	0	0
$941$	−7.59507	−0.247592	−0.123796	−	0.992308i	$-0.539507\pi$
$941$	−7.59507	−0.247592	−0.123796	+	0.992308i	$0.539507\pi$
$942$	0	0
$943$	18.2723	0.595027
$944$	0	0
$945$	−53.1419	−1.72871
$946$	0	0
$947$	17.1019	0.555738	0.277869	−	0.960619i	$-0.410372\pi$
$947$	17.1019	0.555738	0.277869	+	0.960619i	$0.410372\pi$
$948$	0	0
$949$	0.264045	0.00857128
$950$	0	0
$951$	76.7700	2.48944
$952$	0	0
$953$	−5.13924	−0.166476	−0.0832381	−	0.996530i	$-0.526526\pi$
$953$	−5.13924	−0.166476	−0.0832381	+	0.996530i	$0.526526\pi$
$954$	0	0
$955$	−35.9235	−1.16246
$956$	0	0
$957$	3.82869	0.123764
$958$	0	0
$959$	−4.31566	−0.139360
$960$	0	0
$961$	35.0057	1.12922
$962$	0	0
$963$	−42.5265	−1.37040
$964$	0	0
$965$	−25.1930	−0.810991
$966$	0	0
$967$	48.5968	1.56277	0.781384	−	0.624050i	$-0.214514\pi$
$967$	48.5968	1.56277	0.781384	+	0.624050i	$0.214514\pi$
$968$	0	0
$969$	−6.74288	−0.216612
$970$	0	0
$971$	−56.8363	−1.82396	−0.911982	−	0.410231i	$-0.865448\pi$
$971$	−56.8363	−1.82396	−0.911982	+	0.410231i	$0.865448\pi$
$972$	0	0
$973$	5.01033	0.160624
$974$	0	0
$975$	−3.82747	−0.122577
$976$	0	0
$977$	15.0803	0.482461	0.241231	−	0.970468i	$-0.422449\pi$
$977$	15.0803	0.482461	0.241231	+	0.970468i	$0.422449\pi$
$978$	0	0
$979$	−4.73504	−0.151333
$980$	0	0
$981$	31.8625	1.01729
$982$	0	0
$983$	−14.5215	−0.463165	−0.231582	−	0.972815i	$-0.574390\pi$
$983$	−14.5215	−0.463165	−0.231582	+	0.972815i	$0.574390\pi$
$984$	0	0
$985$	14.2223	0.453161
$986$	0	0
$987$	−58.4508	−1.86051
$988$	0	0
$989$	21.4753	0.682876
$990$	0	0
$991$	33.6968	1.07041	0.535207	−	0.844721i	$-0.320234\pi$
$991$	33.6968	1.07041	0.535207	+	0.844721i	$0.320234\pi$
$992$	0	0
$993$	−34.2814	−1.08789
$994$	0	0
$995$	7.10464	0.225232
$996$	0	0
$997$	20.0876	0.636180	0.318090	−	0.948061i	$-0.396959\pi$
$997$	20.0876	0.636180	0.318090	+	0.948061i	$0.396959\pi$
$998$	0	0
$999$	25.5069	0.807002

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.e.1.1	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.e.1.1	✓	27	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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-2.99791 q^{3} +1.37488 q^{5} +4.31566 q^{7} +5.98748 q^{9} +O(q^{10})\)	\(q-2.99791 q^{3} +1.37488 q^{5} +4.31566 q^{7} +5.98748 q^{9} -1.00000 q^{11} -0.410558 q^{13} -4.12178 q^{15} -0.429387 q^{17} -5.23814 q^{19} -12.9380 q^{21} +3.32340 q^{23} -3.10970 q^{25} -8.95621 q^{27} +1.27712 q^{29} +8.12439 q^{31} +2.99791 q^{33} +5.93353 q^{35} -2.84796 q^{37} +1.23082 q^{39} +5.49807 q^{41} +6.46186 q^{43} +8.23208 q^{45} +4.51777 q^{47} +11.6249 q^{49} +1.28726 q^{51} -4.76630 q^{53} -1.37488 q^{55} +15.7035 q^{57} -0.828801 q^{59} -11.2791 q^{61} +25.8399 q^{63} -0.564469 q^{65} +5.64458 q^{67} -9.96326 q^{69} +4.35032 q^{71} -0.643138 q^{73} +9.32260 q^{75} -4.31566 q^{77} +9.44903 q^{79} +8.88748 q^{81} -0.140639 q^{83} -0.590357 q^{85} -3.82869 q^{87} +4.73504 q^{89} -1.77183 q^{91} -24.3562 q^{93} -7.20184 q^{95} -0.436679 q^{97} -5.98748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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