LMFDB - Embedded newform 6028.2.a.e.1.16

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			1.16
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)

$f(q)$	$=$	$q+0.470040 q^{3} -0.875088 q^{5} +3.59259 q^{7} -2.77906 q^{9} +O(q^{10})$	$q+0.470040 q^{3} -0.875088 q^{5} +3.59259 q^{7} -2.77906 q^{9} -1.00000 q^{11} -3.99130 q^{13} -0.411326 q^{15} -2.81445 q^{17} +0.580466 q^{19} +1.68866 q^{21} -4.13918 q^{23} -4.23422 q^{25} -2.71639 q^{27} +6.99628 q^{29} -0.341316 q^{31} -0.470040 q^{33} -3.14383 q^{35} -3.23280 q^{37} -1.87607 q^{39} +3.62614 q^{41} +4.53992 q^{43} +2.43193 q^{45} +9.19931 q^{47} +5.90668 q^{49} -1.32290 q^{51} +13.3113 q^{53} +0.875088 q^{55} +0.272842 q^{57} +1.10083 q^{59} +11.6452 q^{61} -9.98403 q^{63} +3.49274 q^{65} +6.21115 q^{67} -1.94558 q^{69} +15.5081 q^{71} +8.41127 q^{73} -1.99025 q^{75} -3.59259 q^{77} -1.14183 q^{79} +7.06038 q^{81} -11.0818 q^{83} +2.46289 q^{85} +3.28853 q^{87} +5.64012 q^{89} -14.3391 q^{91} -0.160432 q^{93} -0.507959 q^{95} +14.2748 q^{97} +2.77906 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.470040	0.271378	0.135689	−	0.990752i	$-0.456675\pi$
$3$	0.470040	0.271378	0.135689	+	0.990752i	$0.456675\pi$
$4$	0	0
$5$	−0.875088	−0.391351	−0.195676	−	0.980669i	$-0.562690\pi$
$5$	−0.875088	−0.391351	−0.195676	+	0.980669i	$0.562690\pi$
$6$	0	0
$7$	3.59259	1.35787	0.678935	−	0.734198i	$-0.262442\pi$
$7$	3.59259	1.35787	0.678935	+	0.734198i	$0.262442\pi$
$8$	0	0
$9$	−2.77906	−0.926354
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.99130	−1.10699	−0.553494	−	0.832853i	$-0.686706\pi$
$13$	−3.99130	−1.10699	−0.553494	+	0.832853i	$0.686706\pi$
$14$	0	0
$15$	−0.411326	−0.106204
$16$	0	0
$17$	−2.81445	−0.682605	−0.341303	−	0.939953i	$-0.610868\pi$
$17$	−2.81445	−0.682605	−0.341303	+	0.939953i	$0.610868\pi$
$18$	0	0
$19$	0.580466	0.133168	0.0665840	−	0.997781i	$-0.478790\pi$
$19$	0.580466	0.133168	0.0665840	+	0.997781i	$0.478790\pi$
$20$	0	0
$21$	1.68866	0.368495
$22$	0	0
$23$	−4.13918	−0.863079	−0.431539	−	0.902094i	$-0.642029\pi$
$23$	−4.13918	−0.863079	−0.431539	+	0.902094i	$0.642029\pi$
$24$	0	0
$25$	−4.23422	−0.846844
$26$	0	0
$27$	−2.71639	−0.522769
$28$	0	0
$29$	6.99628	1.29918	0.649588	−	0.760286i	$-0.274941\pi$
$29$	6.99628	1.29918	0.649588	+	0.760286i	$0.274941\pi$
$30$	0	0
$31$	−0.341316	−0.0613021	−0.0306510	−	0.999530i	$-0.509758\pi$
$31$	−0.341316	−0.0613021	−0.0306510	+	0.999530i	$0.509758\pi$
$32$	0	0
$33$	−0.470040	−0.0818234
$34$	0	0
$35$	−3.14383	−0.531404
$36$	0	0
$37$	−3.23280	−0.531469	−0.265734	−	0.964046i	$-0.585614\pi$
$37$	−3.23280	−0.531469	−0.265734	+	0.964046i	$0.585614\pi$
$38$	0	0
$39$	−1.87607	−0.300412
$40$	0	0
$41$	3.62614	0.566309	0.283154	−	0.959074i	$-0.408619\pi$
$41$	3.62614	0.566309	0.283154	+	0.959074i	$0.408619\pi$
$42$	0	0
$43$	4.53992	0.692332	0.346166	−	0.938173i	$-0.387483\pi$
$43$	4.53992	0.692332	0.346166	+	0.938173i	$0.387483\pi$
$44$	0	0
$45$	2.43193	0.362530
$46$	0	0
$47$	9.19931	1.34186	0.670929	−	0.741522i	$-0.265896\pi$
$47$	9.19931	1.34186	0.670929	+	0.741522i	$0.265896\pi$
$48$	0	0
$49$	5.90668	0.843812
$50$	0	0
$51$	−1.32290	−0.185244
$52$	0	0
$53$	13.3113	1.82845	0.914224	−	0.405209i	$-0.132801\pi$
$53$	13.3113	1.82845	0.914224	+	0.405209i	$0.132801\pi$
$54$	0	0
$55$	0.875088	0.117997
$56$	0	0
$57$	0.272842	0.0361388
$58$	0	0
$59$	1.10083	0.143316	0.0716578	−	0.997429i	$-0.477171\pi$
$59$	1.10083	0.143316	0.0716578	+	0.997429i	$0.477171\pi$
$60$	0	0
$61$	11.6452	1.49102	0.745509	−	0.666495i	$-0.232206\pi$
$61$	11.6452	1.49102	0.745509	+	0.666495i	$0.232206\pi$
$62$	0	0
$63$	−9.98403	−1.25787
$64$	0	0
$65$	3.49274	0.433221
$66$	0	0
$67$	6.21115	0.758813	0.379406	−	0.925230i	$-0.376128\pi$
$67$	6.21115	0.758813	0.379406	+	0.925230i	$0.376128\pi$
$68$	0	0
$69$	−1.94558	−0.234220
$70$	0	0
$71$	15.5081	1.84047	0.920234	−	0.391368i	$-0.127998\pi$
$71$	15.5081	1.84047	0.920234	+	0.391368i	$0.127998\pi$
$72$	0	0
$73$	8.41127	0.984465	0.492233	−	0.870464i	$-0.336181\pi$
$73$	8.41127	0.984465	0.492233	+	0.870464i	$0.336181\pi$
$74$	0	0
$75$	−1.99025	−0.229814
$76$	0	0
$77$	−3.59259	−0.409413
$78$	0	0
$79$	−1.14183	−0.128466	−0.0642330	−	0.997935i	$-0.520460\pi$
$79$	−1.14183	−0.128466	−0.0642330	+	0.997935i	$0.520460\pi$
$80$	0	0
$81$	7.06038	0.784486
$82$	0	0
$83$	−11.0818	−1.21639	−0.608195	−	0.793787i	$-0.708106\pi$
$83$	−11.0818	−1.21639	−0.608195	+	0.793787i	$0.708106\pi$
$84$	0	0
$85$	2.46289	0.267138
$86$	0	0
$87$	3.28853	0.352567
$88$	0	0
$89$	5.64012	0.597852	0.298926	−	0.954276i	$-0.403372\pi$
$89$	5.64012	0.597852	0.298926	+	0.954276i	$0.403372\pi$
$90$	0	0
$91$	−14.3391	−1.50315
$92$	0	0
$93$	−0.160432	−0.0166360
$94$	0	0
$95$	−0.507959	−0.0521155
$96$	0	0
$97$	14.2748	1.44939	0.724695	−	0.689069i	$-0.241981\pi$
$97$	14.2748	1.44939	0.724695	+	0.689069i	$0.241981\pi$
$98$	0	0
$99$	2.77906	0.279306
$100$	0	0
$101$	−11.2016	−1.11461	−0.557303	−	0.830310i	$-0.688164\pi$
$101$	−11.2016	−1.11461	−0.557303	+	0.830310i	$0.688164\pi$
$102$	0	0
$103$	−5.61309	−0.553074	−0.276537	−	0.961003i	$-0.589187\pi$
$103$	−5.61309	−0.553074	−0.276537	+	0.961003i	$0.589187\pi$
$104$	0	0
$105$	−1.47773	−0.144211
$106$	0	0
$107$	−1.80814	−0.174799	−0.0873995	−	0.996173i	$-0.527856\pi$
$107$	−1.80814	−0.174799	−0.0873995	+	0.996173i	$0.527856\pi$
$108$	0	0
$109$	−6.43735	−0.616587	−0.308293	−	0.951291i	$-0.599758\pi$
$109$	−6.43735	−0.616587	−0.308293	+	0.951291i	$0.599758\pi$
$110$	0	0
$111$	−1.51954	−0.144229
$112$	0	0
$113$	−0.517854	−0.0487156	−0.0243578	−	0.999703i	$-0.507754\pi$
$113$	−0.517854	−0.0487156	−0.0243578	+	0.999703i	$0.507754\pi$
$114$	0	0
$115$	3.62215	0.337767
$116$	0	0
$117$	11.0921	1.02546
$118$	0	0
$119$	−10.1112	−0.926889
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.70443	0.153683
$124$	0	0
$125$	8.08076	0.722765
$126$	0	0
$127$	13.5799	1.20502	0.602510	−	0.798111i	$-0.294167\pi$
$127$	13.5799	1.20502	0.602510	+	0.798111i	$0.294167\pi$
$128$	0	0
$129$	2.13394	0.187883
$130$	0	0
$131$	8.23907	0.719851	0.359925	−	0.932981i	$-0.382802\pi$
$131$	8.23907	0.719851	0.359925	+	0.932981i	$0.382802\pi$
$132$	0	0
$133$	2.08537	0.180825
$134$	0	0
$135$	2.37708	0.204586
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	−4.70849	−0.399369	−0.199684	−	0.979860i	$-0.563992\pi$
$139$	−4.70849	−0.399369	−0.199684	+	0.979860i	$0.563992\pi$
$140$	0	0
$141$	4.32404	0.364150
$142$	0	0
$143$	3.99130	0.333770
$144$	0	0
$145$	−6.12236	−0.508434
$146$	0	0
$147$	2.77637	0.228992
$148$	0	0
$149$	24.0247	1.96818	0.984090	−	0.177669i	$-0.0568557\pi$
$149$	24.0247	1.96818	0.984090	+	0.177669i	$0.0568557\pi$
$150$	0	0
$151$	−8.02103	−0.652742	−0.326371	−	0.945242i	$-0.605826\pi$
$151$	−8.02103	−0.652742	−0.326371	+	0.945242i	$0.605826\pi$
$152$	0	0
$153$	7.82154	0.632334
$154$	0	0
$155$	0.298681	0.0239906
$156$	0	0
$157$	2.18937	0.174731	0.0873653	−	0.996176i	$-0.472155\pi$
$157$	2.18937	0.174731	0.0873653	+	0.996176i	$0.472155\pi$
$158$	0	0
$159$	6.25684	0.496200
$160$	0	0
$161$	−14.8704	−1.17195
$162$	0	0
$163$	0.507884	0.0397805	0.0198903	−	0.999802i	$-0.493668\pi$
$163$	0.507884	0.0397805	0.0198903	+	0.999802i	$0.493668\pi$
$164$	0	0
$165$	0.411326	0.0320217
$166$	0	0
$167$	1.31821	0.102006	0.0510031	−	0.998698i	$-0.483758\pi$
$167$	1.31821	0.102006	0.0510031	+	0.998698i	$0.483758\pi$
$168$	0	0
$169$	2.93051	0.225424
$170$	0	0
$171$	−1.61315	−0.123361
$172$	0	0
$173$	3.37260	0.256414	0.128207	−	0.991747i	$-0.459078\pi$
$173$	3.37260	0.256414	0.128207	+	0.991747i	$0.459078\pi$
$174$	0	0
$175$	−15.2118	−1.14990
$176$	0	0
$177$	0.517433	0.0388926
$178$	0	0
$179$	−15.8101	−1.18170	−0.590851	−	0.806781i	$-0.701208\pi$
$179$	−15.8101	−1.18170	−0.590851	+	0.806781i	$0.701208\pi$
$180$	0	0
$181$	5.21518	0.387641	0.193821	−	0.981037i	$-0.437912\pi$
$181$	5.21518	0.387641	0.193821	+	0.981037i	$0.437912\pi$
$182$	0	0
$183$	5.47372	0.404629
$184$	0	0
$185$	2.82898	0.207991
$186$	0	0
$187$	2.81445	0.205813
$188$	0	0
$189$	−9.75886	−0.709853
$190$	0	0
$191$	−9.14127	−0.661439	−0.330720	−	0.943729i	$-0.607291\pi$
$191$	−9.14127	−0.661439	−0.330720	+	0.943729i	$0.607291\pi$
$192$	0	0
$193$	−14.8973	−1.07233	−0.536167	−	0.844112i	$-0.680128\pi$
$193$	−14.8973	−1.07233	−0.536167	+	0.844112i	$0.680128\pi$
$194$	0	0
$195$	1.64173	0.117567
$196$	0	0
$197$	14.0640	1.00202	0.501009	−	0.865442i	$-0.332962\pi$
$197$	14.0640	1.00202	0.501009	+	0.865442i	$0.332962\pi$
$198$	0	0
$199$	4.77328	0.338369	0.169184	−	0.985584i	$-0.445887\pi$
$199$	4.77328	0.338369	0.169184	+	0.985584i	$0.445887\pi$
$200$	0	0
$201$	2.91949	0.205925
$202$	0	0
$203$	25.1347	1.76411
$204$	0	0
$205$	−3.17320	−0.221626
$206$	0	0
$207$	11.5030	0.799517
$208$	0	0
$209$	−0.580466	−0.0401517
$210$	0	0
$211$	−7.89817	−0.543732	−0.271866	−	0.962335i	$-0.587641\pi$
$211$	−7.89817	−0.543732	−0.271866	+	0.962335i	$0.587641\pi$
$212$	0	0
$213$	7.28940	0.499462
$214$	0	0
$215$	−3.97283	−0.270945
$216$	0	0
$217$	−1.22621	−0.0832403
$218$	0	0
$219$	3.95363	0.267162
$220$	0	0
$221$	11.2333	0.755636
$222$	0	0
$223$	4.93244	0.330301	0.165150	−	0.986268i	$-0.447189\pi$
$223$	4.93244	0.330301	0.165150	+	0.986268i	$0.447189\pi$
$224$	0	0
$225$	11.7672	0.784478
$226$	0	0
$227$	4.61563	0.306350	0.153175	−	0.988199i	$-0.451050\pi$
$227$	4.61563	0.306350	0.153175	+	0.988199i	$0.451050\pi$
$228$	0	0
$229$	2.63156	0.173899	0.0869493	−	0.996213i	$-0.472288\pi$
$229$	2.63156	0.173899	0.0869493	+	0.996213i	$0.472288\pi$
$230$	0	0
$231$	−1.68866	−0.111106
$232$	0	0
$233$	−1.95241	−0.127906	−0.0639532	−	0.997953i	$-0.520371\pi$
$233$	−1.95241	−0.127906	−0.0639532	+	0.997953i	$0.520371\pi$
$234$	0	0
$235$	−8.05021	−0.525138
$236$	0	0
$237$	−0.536706	−0.0348628
$238$	0	0
$239$	−9.81016	−0.634567	−0.317283	−	0.948331i	$-0.602771\pi$
$239$	−9.81016	−0.634567	−0.317283	+	0.948331i	$0.602771\pi$
$240$	0	0
$241$	−22.5172	−1.45046	−0.725230	−	0.688507i	$-0.758266\pi$
$241$	−22.5172	−1.45046	−0.725230	+	0.688507i	$0.758266\pi$
$242$	0	0
$243$	11.4678	0.735661
$244$	0	0
$245$	−5.16887	−0.330227
$246$	0	0
$247$	−2.31682	−0.147415
$248$	0	0
$249$	−5.20891	−0.330101
$250$	0	0
$251$	−22.5205	−1.42148	−0.710741	−	0.703454i	$-0.751640\pi$
$251$	−22.5205	−1.42148	−0.710741	+	0.703454i	$0.751640\pi$
$252$	0	0
$253$	4.13918	0.260228
$254$	0	0
$255$	1.15766	0.0724954
$256$	0	0
$257$	15.5945	0.972759	0.486380	−	0.873748i	$-0.338317\pi$
$257$	15.5945	0.972759	0.486380	+	0.873748i	$0.338317\pi$
$258$	0	0
$259$	−11.6141	−0.721666
$260$	0	0
$261$	−19.4431	−1.20350
$262$	0	0
$263$	1.02323	0.0630953	0.0315477	−	0.999502i	$-0.489956\pi$
$263$	1.02323	0.0630953	0.0315477	+	0.999502i	$0.489956\pi$
$264$	0	0
$265$	−11.6486	−0.715566
$266$	0	0
$267$	2.65108	0.162243
$268$	0	0
$269$	7.07772	0.431536	0.215768	−	0.976445i	$-0.430774\pi$
$269$	7.07772	0.431536	0.215768	+	0.976445i	$0.430774\pi$
$270$	0	0
$271$	9.91473	0.602277	0.301138	−	0.953580i	$-0.402633\pi$
$271$	9.91473	0.602277	0.301138	+	0.953580i	$0.402633\pi$
$272$	0	0
$273$	−6.73995	−0.407920
$274$	0	0
$275$	4.23422	0.255333
$276$	0	0
$277$	−12.1765	−0.731614	−0.365807	−	0.930691i	$-0.619207\pi$
$277$	−12.1765	−0.731614	−0.365807	+	0.930691i	$0.619207\pi$
$278$	0	0
$279$	0.948537	0.0567874
$280$	0	0
$281$	22.5439	1.34485	0.672427	−	0.740163i	$-0.265252\pi$
$281$	22.5439	1.34485	0.672427	+	0.740163i	$0.265252\pi$
$282$	0	0
$283$	0.283137	0.0168308	0.00841538	−	0.999965i	$-0.497321\pi$
$283$	0.283137	0.0168308	0.00841538	+	0.999965i	$0.497321\pi$
$284$	0	0
$285$	−0.238761	−0.0141430
$286$	0	0
$287$	13.0272	0.768974
$288$	0	0
$289$	−9.07886	−0.534050
$290$	0	0
$291$	6.70974	0.393332
$292$	0	0
$293$	3.10973	0.181672	0.0908362	−	0.995866i	$-0.471046\pi$
$293$	3.10973	0.181672	0.0908362	+	0.995866i	$0.471046\pi$
$294$	0	0
$295$	−0.963322	−0.0560868
$296$	0	0
$297$	2.71639	0.157621
$298$	0	0
$299$	16.5207	0.955418
$300$	0	0
$301$	16.3101	0.940097
$302$	0	0
$303$	−5.26522	−0.302479
$304$	0	0
$305$	−10.1906	−0.583512
$306$	0	0
$307$	−12.0352	−0.686885	−0.343442	−	0.939174i	$-0.611593\pi$
$307$	−12.0352	−0.686885	−0.343442	+	0.939174i	$0.611593\pi$
$308$	0	0
$309$	−2.63837	−0.150092
$310$	0	0
$311$	−14.8145	−0.840054	−0.420027	−	0.907512i	$-0.637979\pi$
$311$	−14.8145	−0.840054	−0.420027	+	0.907512i	$0.637979\pi$
$312$	0	0
$313$	5.15370	0.291304	0.145652	−	0.989336i	$-0.453472\pi$
$313$	5.15370	0.291304	0.145652	+	0.989336i	$0.453472\pi$
$314$	0	0
$315$	8.73690	0.492269
$316$	0	0
$317$	−7.84008	−0.440343	−0.220171	−	0.975461i	$-0.570662\pi$
$317$	−7.84008	−0.440343	−0.220171	+	0.975461i	$0.570662\pi$
$318$	0	0
$319$	−6.99628	−0.391716
$320$	0	0
$321$	−0.849895	−0.0474365
$322$	0	0
$323$	−1.63369	−0.0909012
$324$	0	0
$325$	16.9001	0.937447
$326$	0	0
$327$	−3.02581	−0.167328
$328$	0	0
$329$	33.0493	1.82207
$330$	0	0
$331$	12.8336	0.705400	0.352700	−	0.935736i	$-0.385264\pi$
$331$	12.8336	0.705400	0.352700	+	0.935736i	$0.385264\pi$
$332$	0	0
$333$	8.98415	0.492328
$334$	0	0
$335$	−5.43530	−0.296962
$336$	0	0
$337$	−33.1404	−1.80527	−0.902636	−	0.430405i	$-0.858371\pi$
$337$	−33.1404	−1.80527	−0.902636	+	0.430405i	$0.858371\pi$
$338$	0	0
$339$	−0.243412	−0.0132203
$340$	0	0
$341$	0.341316	0.0184833
$342$	0	0
$343$	−3.92784	−0.212083
$344$	0	0
$345$	1.70255	0.0916624
$346$	0	0
$347$	−3.44342	−0.184853	−0.0924264	−	0.995720i	$-0.529462\pi$
$347$	−3.44342	−0.184853	−0.0924264	+	0.995720i	$0.529462\pi$
$348$	0	0
$349$	−2.34473	−0.125511	−0.0627553	−	0.998029i	$-0.519989\pi$
$349$	−2.34473	−0.125511	−0.0627553	+	0.998029i	$0.519989\pi$
$350$	0	0
$351$	10.8419	0.578700
$352$	0	0
$353$	28.5918	1.52179	0.760895	−	0.648875i	$-0.224760\pi$
$353$	28.5918	1.52179	0.760895	+	0.648875i	$0.224760\pi$
$354$	0	0
$355$	−13.5709	−0.720270
$356$	0	0
$357$	−4.75265	−0.251537
$358$	0	0
$359$	22.0216	1.16226	0.581129	−	0.813811i	$-0.302611\pi$
$359$	22.0216	1.16226	0.581129	+	0.813811i	$0.302611\pi$
$360$	0	0
$361$	−18.6631	−0.982266
$362$	0	0
$363$	0.470040	0.0246707
$364$	0	0
$365$	−7.36061	−0.385272
$366$	0	0
$367$	32.7005	1.70695	0.853475	−	0.521134i	$-0.174491\pi$
$367$	32.7005	1.70695	0.853475	+	0.521134i	$0.174491\pi$
$368$	0	0
$369$	−10.0773	−0.524602
$370$	0	0
$371$	47.8220	2.48280
$372$	0	0
$373$	−13.0093	−0.673596	−0.336798	−	0.941577i	$-0.609344\pi$
$373$	−13.0093	−0.673596	−0.336798	+	0.941577i	$0.609344\pi$
$374$	0	0
$375$	3.79828	0.196142
$376$	0	0
$377$	−27.9243	−1.43817
$378$	0	0
$379$	12.1713	0.625197	0.312599	−	0.949885i	$-0.398801\pi$
$379$	12.1713	0.625197	0.312599	+	0.949885i	$0.398801\pi$
$380$	0	0
$381$	6.38309	0.327015
$382$	0	0
$383$	14.7847	0.755464	0.377732	−	0.925915i	$-0.376704\pi$
$383$	14.7847	0.755464	0.377732	+	0.925915i	$0.376704\pi$
$384$	0	0
$385$	3.14383	0.160224
$386$	0	0
$387$	−12.6167	−0.641345
$388$	0	0
$389$	14.7864	0.749700	0.374850	−	0.927086i	$-0.377694\pi$
$389$	14.7864	0.749700	0.374850	+	0.927086i	$0.377694\pi$
$390$	0	0
$391$	11.6495	0.589142
$392$	0	0
$393$	3.87269	0.195351
$394$	0	0
$395$	0.999203	0.0502753
$396$	0	0
$397$	23.3105	1.16992	0.584961	−	0.811062i	$-0.301110\pi$
$397$	23.3105	1.16992	0.584961	+	0.811062i	$0.301110\pi$
$398$	0	0
$399$	0.980209	0.0490718
$400$	0	0
$401$	23.1759	1.15735	0.578675	−	0.815558i	$-0.303570\pi$
$401$	23.1759	1.15735	0.578675	+	0.815558i	$0.303570\pi$
$402$	0	0
$403$	1.36229	0.0678607
$404$	0	0
$405$	−6.17845	−0.307010
$406$	0	0
$407$	3.23280	0.160244
$408$	0	0
$409$	−14.3775	−0.710921	−0.355460	−	0.934691i	$-0.615676\pi$
$409$	−14.3775	−0.710921	−0.355460	+	0.934691i	$0.615676\pi$
$410$	0	0
$411$	−0.470040	−0.0231853
$412$	0	0
$413$	3.95482	0.194604
$414$	0	0
$415$	9.69759	0.476036
$416$	0	0
$417$	−2.21318	−0.108380
$418$	0	0
$419$	−12.6172	−0.616391	−0.308196	−	0.951323i	$-0.599725\pi$
$419$	−12.6172	−0.616391	−0.308196	+	0.951323i	$0.599725\pi$
$420$	0	0
$421$	−22.2424	−1.08403	−0.542015	−	0.840369i	$-0.682338\pi$
$421$	−22.2424	−1.08403	−0.542015	+	0.840369i	$0.682338\pi$
$422$	0	0
$423$	−25.5655	−1.24304
$424$	0	0
$425$	11.9170	0.578060
$426$	0	0
$427$	41.8365	2.02461
$428$	0	0
$429$	1.87607	0.0905776
$430$	0	0
$431$	9.75761	0.470007	0.235004	−	0.971994i	$-0.424490\pi$
$431$	9.75761	0.470007	0.235004	+	0.971994i	$0.424490\pi$
$432$	0	0
$433$	−24.2385	−1.16483	−0.582413	−	0.812893i	$-0.697891\pi$
$433$	−24.2385	−1.16483	−0.582413	+	0.812893i	$0.697891\pi$
$434$	0	0
$435$	−2.87775	−0.137978
$436$	0	0
$437$	−2.40265	−0.114934
$438$	0	0
$439$	12.9068	0.616007	0.308004	−	0.951385i	$-0.400339\pi$
$439$	12.9068	0.616007	0.308004	+	0.951385i	$0.400339\pi$
$440$	0	0
$441$	−16.4150	−0.781669
$442$	0	0
$443$	−6.33398	−0.300936	−0.150468	−	0.988615i	$-0.548078\pi$
$443$	−6.33398	−0.300936	−0.150468	+	0.988615i	$0.548078\pi$
$444$	0	0
$445$	−4.93560	−0.233970
$446$	0	0
$447$	11.2926	0.534120
$448$	0	0
$449$	10.9620	0.517330	0.258665	−	0.965967i	$-0.416717\pi$
$449$	10.9620	0.517330	0.258665	+	0.965967i	$0.416717\pi$
$450$	0	0
$451$	−3.62614	−0.170748
$452$	0	0
$453$	−3.77020	−0.177140
$454$	0	0
$455$	12.5480	0.588259
$456$	0	0
$457$	1.71507	0.0802277	0.0401138	−	0.999195i	$-0.487228\pi$
$457$	1.71507	0.0802277	0.0401138	+	0.999195i	$0.487228\pi$
$458$	0	0
$459$	7.64515	0.356845
$460$	0	0
$461$	2.50488	0.116664	0.0583320	−	0.998297i	$-0.481422\pi$
$461$	2.50488	0.116664	0.0583320	+	0.998297i	$0.481422\pi$
$462$	0	0
$463$	3.62463	0.168451	0.0842255	−	0.996447i	$-0.473158\pi$
$463$	3.62463	0.168451	0.0842255	+	0.996447i	$0.473158\pi$
$464$	0	0
$465$	0.140392	0.00651052
$466$	0	0
$467$	−24.1233	−1.11629	−0.558147	−	0.829742i	$-0.688488\pi$
$467$	−24.1233	−1.11629	−0.558147	+	0.829742i	$0.688488\pi$
$468$	0	0
$469$	22.3141	1.03037
$470$	0	0
$471$	1.02909	0.0474179
$472$	0	0
$473$	−4.53992	−0.208746
$474$	0	0
$475$	−2.45782	−0.112773
$476$	0	0
$477$	−36.9929	−1.69379
$478$	0	0
$479$	3.16192	0.144472	0.0722360	−	0.997388i	$-0.476987\pi$
$479$	3.16192	0.144472	0.0722360	+	0.997388i	$0.476987\pi$
$480$	0	0
$481$	12.9031	0.588330
$482$	0	0
$483$	−6.98966	−0.318041
$484$	0	0
$485$	−12.4917	−0.567221
$486$	0	0
$487$	21.6970	0.983183	0.491592	−	0.870826i	$-0.336415\pi$
$487$	21.6970	0.983183	0.491592	+	0.870826i	$0.336415\pi$
$488$	0	0
$489$	0.238725	0.0107955
$490$	0	0
$491$	31.9297	1.44097	0.720485	−	0.693471i	$-0.243919\pi$
$491$	31.9297	1.44097	0.720485	+	0.693471i	$0.243919\pi$
$492$	0	0
$493$	−19.6907	−0.886824
$494$	0	0
$495$	−2.43193	−0.109307
$496$	0	0
$497$	55.7141	2.49912
$498$	0	0
$499$	−18.7255	−0.838268	−0.419134	−	0.907924i	$-0.637666\pi$
$499$	−18.7255	−0.838268	−0.419134	+	0.907924i	$0.637666\pi$
$500$	0	0
$501$	0.619611	0.0276822
$502$	0	0
$503$	−21.6056	−0.963347	−0.481673	−	0.876351i	$-0.659971\pi$
$503$	−21.6056	−0.963347	−0.481673	+	0.876351i	$0.659971\pi$
$504$	0	0
$505$	9.80243	0.436202
$506$	0	0
$507$	1.37746	0.0611749
$508$	0	0
$509$	−34.4780	−1.52821	−0.764106	−	0.645091i	$-0.776820\pi$
$509$	−34.4780	−1.52821	−0.764106	+	0.645091i	$0.776820\pi$
$510$	0	0
$511$	30.2182	1.33678
$512$	0	0
$513$	−1.57677	−0.0696161
$514$	0	0
$515$	4.91195	0.216446
$516$	0	0
$517$	−9.19931	−0.404585
$518$	0	0
$519$	1.58525	0.0695850
$520$	0	0
$521$	33.0516	1.44802	0.724008	−	0.689792i	$-0.242298\pi$
$521$	33.0516	1.44802	0.724008	+	0.689792i	$0.242298\pi$
$522$	0	0
$523$	16.8900	0.738550	0.369275	−	0.929320i	$-0.379606\pi$
$523$	16.8900	0.738550	0.369275	+	0.929320i	$0.379606\pi$
$524$	0	0
$525$	−7.15015	−0.312058
$526$	0	0
$527$	0.960616	0.0418451
$528$	0	0
$529$	−5.86719	−0.255095
$530$	0	0
$531$	−3.05927	−0.132761
$532$	0	0
$533$	−14.4730	−0.626897
$534$	0	0
$535$	1.58228	0.0684078
$536$	0	0
$537$	−7.43137	−0.320687
$538$	0	0
$539$	−5.90668	−0.254419
$540$	0	0
$541$	−23.9062	−1.02781	−0.513904	−	0.857848i	$-0.671801\pi$
$541$	−23.9062	−1.02781	−0.513904	+	0.857848i	$0.671801\pi$
$542$	0	0
$543$	2.45134	0.105197
$544$	0	0
$545$	5.63325	0.241302
$546$	0	0
$547$	33.8302	1.44648	0.723238	−	0.690598i	$-0.242653\pi$
$547$	33.8302	1.44648	0.723238	+	0.690598i	$0.242653\pi$
$548$	0	0
$549$	−32.3628	−1.38121
$550$	0	0
$551$	4.06110	0.173009
$552$	0	0
$553$	−4.10213	−0.174440
$554$	0	0
$555$	1.32973	0.0564441
$556$	0	0
$557$	−8.83167	−0.374210	−0.187105	−	0.982340i	$-0.559910\pi$
$557$	−8.83167	−0.374210	−0.187105	+	0.982340i	$0.559910\pi$
$558$	0	0
$559$	−18.1202	−0.766404
$560$	0	0
$561$	1.32290	0.0558531
$562$	0	0
$563$	5.17816	0.218233	0.109117	−	0.994029i	$-0.465198\pi$
$563$	5.17816	0.218233	0.109117	+	0.994029i	$0.465198\pi$
$564$	0	0
$565$	0.453168	0.0190649
$566$	0	0
$567$	25.3650	1.06523
$568$	0	0
$569$	31.7231	1.32990	0.664950	−	0.746888i	$-0.268453\pi$
$569$	31.7231	1.32990	0.664950	+	0.746888i	$0.268453\pi$
$570$	0	0
$571$	−14.0793	−0.589201	−0.294601	−	0.955620i	$-0.595187\pi$
$571$	−14.0793	−0.589201	−0.294601	+	0.955620i	$0.595187\pi$
$572$	0	0
$573$	−4.29676	−0.179500
$574$	0	0
$575$	17.5262	0.730893
$576$	0	0
$577$	1.32874	0.0553162	0.0276581	−	0.999617i	$-0.491195\pi$
$577$	1.32874	0.0553162	0.0276581	+	0.999617i	$0.491195\pi$
$578$	0	0
$579$	−7.00234	−0.291007
$580$	0	0
$581$	−39.8125	−1.65170
$582$	0	0
$583$	−13.3113	−0.551298
$584$	0	0
$585$	−9.70655	−0.401317
$586$	0	0
$587$	35.4750	1.46421	0.732104	−	0.681193i	$-0.238538\pi$
$587$	35.4750	1.46421	0.732104	+	0.681193i	$0.238538\pi$
$588$	0	0
$589$	−0.198122	−0.00816347
$590$	0	0
$591$	6.61063	0.271925
$592$	0	0
$593$	29.3933	1.20704	0.603520	−	0.797348i	$-0.293764\pi$
$593$	29.3933	1.20704	0.603520	+	0.797348i	$0.293764\pi$
$594$	0	0
$595$	8.84816	0.362739
$596$	0	0
$597$	2.24363	0.0918257
$598$	0	0
$599$	1.80962	0.0739393	0.0369696	−	0.999316i	$-0.488230\pi$
$599$	1.80962	0.0739393	0.0369696	+	0.999316i	$0.488230\pi$
$600$	0	0
$601$	−14.5679	−0.594238	−0.297119	−	0.954840i	$-0.596026\pi$
$601$	−14.5679	−0.594238	−0.297119	+	0.954840i	$0.596026\pi$
$602$	0	0
$603$	−17.2612	−0.702929
$604$	0	0
$605$	−0.875088	−0.0355774
$606$	0	0
$607$	9.18732	0.372902	0.186451	−	0.982464i	$-0.440301\pi$
$607$	9.18732	0.372902	0.186451	+	0.982464i	$0.440301\pi$
$608$	0	0
$609$	11.8143	0.478740
$610$	0	0
$611$	−36.7172	−1.48542
$612$	0	0
$613$	25.2891	1.02142	0.510709	−	0.859754i	$-0.329383\pi$
$613$	25.2891	1.02142	0.510709	+	0.859754i	$0.329383\pi$
$614$	0	0
$615$	−1.49153	−0.0601442
$616$	0	0
$617$	5.63401	0.226817	0.113408	−	0.993548i	$-0.463823\pi$
$617$	5.63401	0.226817	0.113408	+	0.993548i	$0.463823\pi$
$618$	0	0
$619$	7.32351	0.294357	0.147178	−	0.989110i	$-0.452981\pi$
$619$	7.32351	0.294357	0.147178	+	0.989110i	$0.452981\pi$
$620$	0	0
$621$	11.2436	0.451191
$622$	0	0
$623$	20.2626	0.811805
$624$	0	0
$625$	14.0997	0.563989
$626$	0	0
$627$	−0.272842	−0.0108963
$628$	0	0
$629$	9.09856	0.362783
$630$	0	0
$631$	29.2071	1.16272	0.581358	−	0.813648i	$-0.302522\pi$
$631$	29.2071	1.16272	0.581358	+	0.813648i	$0.302522\pi$
$632$	0	0
$633$	−3.71245	−0.147557
$634$	0	0
$635$	−11.8836	−0.471586
$636$	0	0
$637$	−23.5754	−0.934090
$638$	0	0
$639$	−43.0979	−1.70493
$640$	0	0
$641$	−5.09374	−0.201191	−0.100595	−	0.994927i	$-0.532075\pi$
$641$	−5.09374	−0.201191	−0.100595	+	0.994927i	$0.532075\pi$
$642$	0	0
$643$	0.295855	0.0116674	0.00583370	−	0.999983i	$-0.498143\pi$
$643$	0.295855	0.0116674	0.00583370	+	0.999983i	$0.498143\pi$
$644$	0	0
$645$	−1.86739	−0.0735284
$646$	0	0
$647$	27.4556	1.07939	0.539695	−	0.841861i	$-0.318540\pi$
$647$	27.4556	1.07939	0.539695	+	0.841861i	$0.318540\pi$
$648$	0	0
$649$	−1.10083	−0.0432113
$650$	0	0
$651$	−0.576365	−0.0225895
$652$	0	0
$653$	13.9238	0.544879	0.272439	−	0.962173i	$-0.412170\pi$
$653$	13.9238	0.544879	0.272439	+	0.962173i	$0.412170\pi$
$654$	0	0
$655$	−7.20991	−0.281715
$656$	0	0
$657$	−23.3755	−0.911963
$658$	0	0
$659$	39.3883	1.53435	0.767175	−	0.641438i	$-0.221662\pi$
$659$	39.3883	1.53435	0.767175	+	0.641438i	$0.221662\pi$
$660$	0	0
$661$	−40.1285	−1.56082	−0.780408	−	0.625270i	$-0.784989\pi$
$661$	−40.1285	−1.56082	−0.780408	+	0.625270i	$0.784989\pi$
$662$	0	0
$663$	5.28011	0.205063
$664$	0	0
$665$	−1.82489	−0.0707661
$666$	0	0
$667$	−28.9588	−1.12129
$668$	0	0
$669$	2.31844	0.0896362
$670$	0	0
$671$	−11.6452	−0.449559
$672$	0	0
$673$	20.1002	0.774807	0.387403	−	0.921910i	$-0.373372\pi$
$673$	20.1002	0.774807	0.387403	+	0.921910i	$0.373372\pi$
$674$	0	0
$675$	11.5018	0.442704
$676$	0	0
$677$	−7.07634	−0.271966	−0.135983	−	0.990711i	$-0.543419\pi$
$677$	−7.07634	−0.271966	−0.135983	+	0.990711i	$0.543419\pi$
$678$	0	0
$679$	51.2836	1.96808
$680$	0	0
$681$	2.16953	0.0831366
$682$	0	0
$683$	−18.2778	−0.699381	−0.349690	−	0.936865i	$-0.613713\pi$
$683$	−18.2778	−0.699381	−0.349690	+	0.936865i	$0.613713\pi$
$684$	0	0
$685$	0.875088	0.0334354
$686$	0	0
$687$	1.23694	0.0471922
$688$	0	0
$689$	−53.1295	−2.02407
$690$	0	0
$691$	7.84419	0.298407	0.149203	−	0.988807i	$-0.452329\pi$
$691$	7.84419	0.298407	0.149203	+	0.988807i	$0.452329\pi$
$692$	0	0
$693$	9.98403	0.379262
$694$	0	0
$695$	4.12034	0.156293
$696$	0	0
$697$	−10.2056	−0.386565
$698$	0	0
$699$	−0.917708	−0.0347109
$700$	0	0
$701$	21.3767	0.807388	0.403694	−	0.914894i	$-0.367726\pi$
$701$	21.3767	0.807388	0.403694	+	0.914894i	$0.367726\pi$
$702$	0	0
$703$	−1.87653	−0.0707746
$704$	0	0
$705$	−3.78392	−0.142511
$706$	0	0
$707$	−40.2429	−1.51349
$708$	0	0
$709$	29.4344	1.10543	0.552716	−	0.833370i	$-0.313591\pi$
$709$	29.4344	1.10543	0.552716	+	0.833370i	$0.313591\pi$
$710$	0	0
$711$	3.17322	0.119005
$712$	0	0
$713$	1.41277	0.0529085
$714$	0	0
$715$	−3.49274	−0.130621
$716$	0	0
$717$	−4.61117	−0.172207
$718$	0	0
$719$	−1.68037	−0.0626672	−0.0313336	−	0.999509i	$-0.509975\pi$
$719$	−1.68037	−0.0626672	−0.0313336	+	0.999509i	$0.509975\pi$
$720$	0	0
$721$	−20.1655	−0.751003
$722$	0	0
$723$	−10.5840	−0.393622
$724$	0	0
$725$	−29.6238	−1.10020
$726$	0	0
$727$	16.6317	0.616835	0.308418	−	0.951251i	$-0.400201\pi$
$727$	16.6317	0.616835	0.308418	+	0.951251i	$0.400201\pi$
$728$	0	0
$729$	−15.7908	−0.584845
$730$	0	0
$731$	−12.7774	−0.472589
$732$	0	0
$733$	−44.4259	−1.64091	−0.820455	−	0.571711i	$-0.806280\pi$
$733$	−44.4259	−1.64091	−0.820455	+	0.571711i	$0.806280\pi$
$734$	0	0
$735$	−2.42957	−0.0896161
$736$	0	0
$737$	−6.21115	−0.228791
$738$	0	0
$739$	−42.1574	−1.55078	−0.775392	−	0.631480i	$-0.782448\pi$
$739$	−42.1574	−1.55078	−0.775392	+	0.631480i	$0.782448\pi$
$740$	0	0
$741$	−1.08900	−0.0400052
$742$	0	0
$743$	−34.5801	−1.26862	−0.634310	−	0.773079i	$-0.718716\pi$
$743$	−34.5801	−1.26862	−0.634310	+	0.773079i	$0.718716\pi$
$744$	0	0
$745$	−21.0237	−0.770250
$746$	0	0
$747$	30.7971	1.12681
$748$	0	0
$749$	−6.49588	−0.237354
$750$	0	0
$751$	34.5213	1.25970	0.629850	−	0.776717i	$-0.283116\pi$
$751$	34.5213	1.25970	0.629850	+	0.776717i	$0.283116\pi$
$752$	0	0
$753$	−10.5855	−0.385758
$754$	0	0
$755$	7.01911	0.255452
$756$	0	0
$757$	−42.7248	−1.55286	−0.776430	−	0.630203i	$-0.782972\pi$
$757$	−42.7248	−1.55286	−0.776430	+	0.630203i	$0.782972\pi$
$758$	0	0
$759$	1.94558	0.0706200
$760$	0	0
$761$	23.9226	0.867193	0.433597	−	0.901107i	$-0.357244\pi$
$761$	23.9226	0.867193	0.433597	+	0.901107i	$0.357244\pi$
$762$	0	0
$763$	−23.1268	−0.837245
$764$	0	0
$765$	−6.84454	−0.247465
$766$	0	0
$767$	−4.39374	−0.158649
$768$	0	0
$769$	−25.1451	−0.906756	−0.453378	−	0.891318i	$-0.649781\pi$
$769$	−25.1451	−0.906756	−0.453378	+	0.891318i	$0.649781\pi$
$770$	0	0
$771$	7.33004	0.263985
$772$	0	0
$773$	−45.6208	−1.64087	−0.820433	−	0.571742i	$-0.806268\pi$
$773$	−45.6208	−1.64087	−0.820433	+	0.571742i	$0.806268\pi$
$774$	0	0
$775$	1.44521	0.0519133
$776$	0	0
$777$	−5.45909	−0.195844
$778$	0	0
$779$	2.10485	0.0754142
$780$	0	0
$781$	−15.5081	−0.554922
$782$	0	0
$783$	−19.0046	−0.679169
$784$	0	0
$785$	−1.91589	−0.0683810
$786$	0	0
$787$	−24.6900	−0.880103	−0.440052	−	0.897972i	$-0.645040\pi$
$787$	−24.6900	−0.880103	−0.440052	+	0.897972i	$0.645040\pi$
$788$	0	0
$789$	0.480960	0.0171226
$790$	0	0
$791$	−1.86043	−0.0661494
$792$	0	0
$793$	−46.4796	−1.65054
$794$	0	0
$795$	−5.47529	−0.194188
$796$	0	0
$797$	9.16848	0.324764	0.162382	−	0.986728i	$-0.448082\pi$
$797$	9.16848	0.324764	0.162382	+	0.986728i	$0.448082\pi$
$798$	0	0
$799$	−25.8910	−0.915959
$800$	0	0
$801$	−15.6742	−0.553822
$802$	0	0
$803$	−8.41127	−0.296827
$804$	0	0
$805$	13.0129	0.458644
$806$	0	0
$807$	3.32681	0.117109
$808$	0	0
$809$	35.6478	1.25331	0.626655	−	0.779297i	$-0.284424\pi$
$809$	35.6478	1.25331	0.626655	+	0.779297i	$0.284424\pi$
$810$	0	0
$811$	18.5709	0.652114	0.326057	−	0.945350i	$-0.394280\pi$
$811$	18.5709	0.652114	0.326057	+	0.945350i	$0.394280\pi$
$812$	0	0
$813$	4.66032	0.163444
$814$	0	0
$815$	−0.444443	−0.0155682
$816$	0	0
$817$	2.63527	0.0921965
$818$	0	0
$819$	39.8493	1.39245
$820$	0	0
$821$	−0.253487	−0.00884675	−0.00442338	−	0.999990i	$-0.501408\pi$
$821$	−0.253487	−0.00884675	−0.00442338	+	0.999990i	$0.501408\pi$
$822$	0	0
$823$	−6.49632	−0.226448	−0.113224	−	0.993570i	$-0.536118\pi$
$823$	−6.49632	−0.226448	−0.113224	+	0.993570i	$0.536118\pi$
$824$	0	0
$825$	1.99025	0.0692917
$826$	0	0
$827$	14.4300	0.501781	0.250890	−	0.968015i	$-0.419277\pi$
$827$	14.4300	0.501781	0.250890	+	0.968015i	$0.419277\pi$
$828$	0	0
$829$	−16.5231	−0.573873	−0.286936	−	0.957950i	$-0.592637\pi$
$829$	−16.5231	−0.573873	−0.286936	+	0.957950i	$0.592637\pi$
$830$	0	0
$831$	−5.72343	−0.198544
$832$	0	0
$833$	−16.6241	−0.575990
$834$	0	0
$835$	−1.15355	−0.0399203
$836$	0	0
$837$	0.927145	0.0320468
$838$	0	0
$839$	−53.5765	−1.84967	−0.924833	−	0.380374i	$-0.875795\pi$
$839$	−53.5765	−1.84967	−0.924833	+	0.380374i	$0.875795\pi$
$840$	0	0
$841$	19.9479	0.687858
$842$	0	0
$843$	10.5965	0.364963
$844$	0	0
$845$	−2.56445	−0.0882199
$846$	0	0
$847$	3.59259	0.123443
$848$	0	0
$849$	0.133086	0.00456749
$850$	0	0
$851$	13.3811	0.458699
$852$	0	0
$853$	9.15647	0.313512	0.156756	−	0.987637i	$-0.449896\pi$
$853$	9.15647	0.313512	0.156756	+	0.987637i	$0.449896\pi$
$854$	0	0
$855$	1.41165	0.0482774
$856$	0	0
$857$	1.56975	0.0536215	0.0268107	−	0.999641i	$-0.491465\pi$
$857$	1.56975	0.0536215	0.0268107	+	0.999641i	$0.491465\pi$
$858$	0	0
$859$	−4.39888	−0.150088	−0.0750440	−	0.997180i	$-0.523910\pi$
$859$	−4.39888	−0.150088	−0.0750440	+	0.997180i	$0.523910\pi$
$860$	0	0
$861$	6.12332	0.208682
$862$	0	0
$863$	−5.12520	−0.174464	−0.0872320	−	0.996188i	$-0.527802\pi$
$863$	−5.12520	−0.174464	−0.0872320	+	0.996188i	$0.527802\pi$
$864$	0	0
$865$	−2.95132	−0.100348
$866$	0	0
$867$	−4.26742	−0.144929
$868$	0	0
$869$	1.14183	0.0387339
$870$	0	0
$871$	−24.7906	−0.839997
$872$	0	0
$873$	−39.6707	−1.34265
$874$	0	0
$875$	29.0308	0.981421
$876$	0	0
$877$	1.14261	0.0385832	0.0192916	−	0.999814i	$-0.493859\pi$
$877$	1.14261	0.0385832	0.0192916	+	0.999814i	$0.493859\pi$
$878$	0	0
$879$	1.46170	0.0493018
$880$	0	0
$881$	4.85308	0.163505	0.0817523	−	0.996653i	$-0.473948\pi$
$881$	4.85308	0.163505	0.0817523	+	0.996653i	$0.473948\pi$
$882$	0	0
$883$	−25.5717	−0.860556	−0.430278	−	0.902697i	$-0.641584\pi$
$883$	−25.5717	−0.860556	−0.430278	+	0.902697i	$0.641584\pi$
$884$	0	0
$885$	−0.452799	−0.0152207
$886$	0	0
$887$	56.0159	1.88083	0.940415	−	0.340028i	$-0.110436\pi$
$887$	56.0159	1.88083	0.940415	+	0.340028i	$0.110436\pi$
$888$	0	0
$889$	48.7869	1.63626
$890$	0	0
$891$	−7.06038	−0.236532
$892$	0	0
$893$	5.33989	0.178692
$894$	0	0
$895$	13.8352	0.462461
$896$	0	0
$897$	7.76540	0.259279
$898$	0	0
$899$	−2.38794	−0.0796422
$900$	0	0
$901$	−37.4640	−1.24811
$902$	0	0
$903$	7.66638	0.255121
$904$	0	0
$905$	−4.56374	−0.151704
$906$	0	0
$907$	−18.0224	−0.598423	−0.299212	−	0.954187i	$-0.596724\pi$
$907$	−18.0224	−0.598423	−0.299212	+	0.954187i	$0.596724\pi$
$908$	0	0
$909$	31.1301	1.03252
$910$	0	0
$911$	−19.3011	−0.639475	−0.319737	−	0.947506i	$-0.603595\pi$
$911$	−19.3011	−0.639475	−0.319737	+	0.947506i	$0.603595\pi$
$912$	0	0
$913$	11.0818	0.366756
$914$	0	0
$915$	−4.78999	−0.158352
$916$	0	0
$917$	29.5996	0.977464
$918$	0	0
$919$	5.77739	0.190578	0.0952892	−	0.995450i	$-0.469622\pi$
$919$	5.77739	0.190578	0.0952892	+	0.995450i	$0.469622\pi$
$920$	0	0
$921$	−5.65702	−0.186405
$922$	0	0
$923$	−61.8974	−2.03738
$924$	0	0
$925$	13.6884	0.450071
$926$	0	0
$927$	15.5991	0.512342
$928$	0	0
$929$	46.8256	1.53630	0.768148	−	0.640272i	$-0.221178\pi$
$929$	46.8256	1.53630	0.768148	+	0.640272i	$0.221178\pi$
$930$	0	0
$931$	3.42863	0.112369
$932$	0	0
$933$	−6.96341	−0.227972
$934$	0	0
$935$	−2.46289	−0.0805453
$936$	0	0
$937$	46.8842	1.53164	0.765820	−	0.643055i	$-0.222333\pi$
$937$	46.8842	1.53164	0.765820	+	0.643055i	$0.222333\pi$
$938$	0	0
$939$	2.42244	0.0790534
$940$	0	0
$941$	0.996705	0.0324916	0.0162458	−	0.999868i	$-0.494829\pi$
$941$	0.996705	0.0324916	0.0162458	+	0.999868i	$0.494829\pi$
$942$	0	0
$943$	−15.0093	−0.488769
$944$	0	0
$945$	8.53987	0.277802
$946$	0	0
$947$	25.0386	0.813645	0.406823	−	0.913507i	$-0.366637\pi$
$947$	25.0386	0.813645	0.406823	+	0.913507i	$0.366637\pi$
$948$	0	0
$949$	−33.5720	−1.08979
$950$	0	0
$951$	−3.68515	−0.119499
$952$	0	0
$953$	−44.9863	−1.45725	−0.728625	−	0.684912i	$-0.759840\pi$
$953$	−44.9863	−1.45725	−0.728625	+	0.684912i	$0.759840\pi$
$954$	0	0
$955$	7.99942	0.258855
$956$	0	0
$957$	−3.28853	−0.106303
$958$	0	0
$959$	−3.59259	−0.116011
$960$	0	0
$961$	−30.8835	−0.996242
$962$	0	0
$963$	5.02492	0.161926
$964$	0	0
$965$	13.0365	0.419659
$966$	0	0
$967$	−22.9108	−0.736763	−0.368382	−	0.929675i	$-0.620088\pi$
$967$	−22.9108	−0.736763	−0.368382	+	0.929675i	$0.620088\pi$
$968$	0	0
$969$	−0.767901	−0.0246685
$970$	0	0
$971$	2.50235	0.0803043	0.0401522	−	0.999194i	$-0.487216\pi$
$971$	2.50235	0.0803043	0.0401522	+	0.999194i	$0.487216\pi$
$972$	0	0
$973$	−16.9156	−0.542291
$974$	0	0
$975$	7.94370	0.254402
$976$	0	0
$977$	−7.56616	−0.242063	−0.121031	−	0.992649i	$-0.538620\pi$
$977$	−7.56616	−0.242063	−0.121031	+	0.992649i	$0.538620\pi$
$978$	0	0
$979$	−5.64012	−0.180259
$980$	0	0
$981$	17.8898	0.571178
$982$	0	0
$983$	−12.8764	−0.410693	−0.205346	−	0.978689i	$-0.565832\pi$
$983$	−12.8764	−0.410693	−0.205346	+	0.978689i	$0.565832\pi$
$984$	0	0
$985$	−12.3072	−0.392141
$986$	0	0
$987$	15.5345	0.494468
$988$	0	0
$989$	−18.7916	−0.597537
$990$	0	0
$991$	18.2519	0.579790	0.289895	−	0.957058i	$-0.406380\pi$
$991$	18.2519	0.579790	0.289895	+	0.957058i	$0.406380\pi$
$992$	0	0
$993$	6.03232	0.191430
$994$	0	0
$995$	−4.17704	−0.132421
$996$	0	0
$997$	29.9896	0.949780	0.474890	−	0.880045i	$-0.342488\pi$
$997$	29.9896	0.949780	0.474890	+	0.880045i	$0.342488\pi$
$998$	0	0
$999$	8.78154	0.277836

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.e.1.16	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.e.1.16	✓	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)

\(f(q)\)	\(=\)	\(q+0.470040 q^{3} -0.875088 q^{5} +3.59259 q^{7} -2.77906 q^{9} +O(q^{10})\)	\(q+0.470040 q^{3} -0.875088 q^{5} +3.59259 q^{7} -2.77906 q^{9} -1.00000 q^{11} -3.99130 q^{13} -0.411326 q^{15} -2.81445 q^{17} +0.580466 q^{19} +1.68866 q^{21} -4.13918 q^{23} -4.23422 q^{25} -2.71639 q^{27} +6.99628 q^{29} -0.341316 q^{31} -0.470040 q^{33} -3.14383 q^{35} -3.23280 q^{37} -1.87607 q^{39} +3.62614 q^{41} +4.53992 q^{43} +2.43193 q^{45} +9.19931 q^{47} +5.90668 q^{49} -1.32290 q^{51} +13.3113 q^{53} +0.875088 q^{55} +0.272842 q^{57} +1.10083 q^{59} +11.6452 q^{61} -9.98403 q^{63} +3.49274 q^{65} +6.21115 q^{67} -1.94558 q^{69} +15.5081 q^{71} +8.41127 q^{73} -1.99025 q^{75} -3.59259 q^{77} -1.14183 q^{79} +7.06038 q^{81} -11.0818 q^{83} +2.46289 q^{85} +3.28853 q^{87} +5.64012 q^{89} -14.3391 q^{91} -0.160432 q^{93} -0.507959 q^{95} +14.2748 q^{97} +2.77906 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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