LMFDB - Embedded newform 6028.2.a.f.1.10

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:	$29$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-0.886923 q^{3} -0.640762 q^{5} +1.27780 q^{7} -2.21337 q^{9} +O(q^{10})$	$q-0.886923 q^{3} -0.640762 q^{5} +1.27780 q^{7} -2.21337 q^{9} +1.00000 q^{11} +0.262808 q^{13} +0.568307 q^{15} -4.04537 q^{17} +3.16103 q^{19} -1.13331 q^{21} -8.22348 q^{23} -4.58942 q^{25} +4.62386 q^{27} +7.92257 q^{29} -2.15513 q^{31} -0.886923 q^{33} -0.818767 q^{35} +10.1602 q^{37} -0.233090 q^{39} -1.74757 q^{41} -5.26318 q^{43} +1.41824 q^{45} +3.30774 q^{47} -5.36722 q^{49} +3.58793 q^{51} +6.13347 q^{53} -0.640762 q^{55} -2.80359 q^{57} -3.45420 q^{59} +0.846162 q^{61} -2.82824 q^{63} -0.168397 q^{65} -10.0457 q^{67} +7.29359 q^{69} +0.399049 q^{71} -3.76750 q^{73} +4.07046 q^{75} +1.27780 q^{77} +12.8736 q^{79} +2.53910 q^{81} +17.6354 q^{83} +2.59212 q^{85} -7.02671 q^{87} -5.52428 q^{89} +0.335816 q^{91} +1.91143 q^{93} -2.02547 q^{95} +2.44191 q^{97} -2.21337 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * 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
$3$	−0.886923	−0.512065	−0.256033	−	0.966668i	$-0.582415\pi$
$3$	−0.886923	−0.512065	−0.256033	+	0.966668i	$0.582415\pi$
$4$	0	0
$5$	−0.640762	−0.286558	−0.143279	−	0.989682i	$-0.545765\pi$
$5$	−0.640762	−0.286558	−0.143279	+	0.989682i	$0.545765\pi$
$6$	0	0
$7$	1.27780	0.482964	0.241482	−	0.970405i	$-0.422367\pi$
$7$	1.27780	0.482964	0.241482	+	0.970405i	$0.422367\pi$
$8$	0	0
$9$	−2.21337	−0.737789
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.262808	0.0728897	0.0364449	−	0.999336i	$-0.488397\pi$
$13$	0.262808	0.0728897	0.0364449	+	0.999336i	$0.488397\pi$
$14$	0	0
$15$	0.568307	0.146736
$16$	0	0
$17$	−4.04537	−0.981147	−0.490574	−	0.871400i	$-0.663213\pi$
$17$	−4.04537	−0.981147	−0.490574	+	0.871400i	$0.663213\pi$
$18$	0	0
$19$	3.16103	0.725190	0.362595	−	0.931947i	$-0.381891\pi$
$19$	3.16103	0.725190	0.362595	+	0.931947i	$0.381891\pi$
$20$	0	0
$21$	−1.13331	−0.247309
$22$	0	0
$23$	−8.22348	−1.71471	−0.857357	−	0.514723i	$-0.827895\pi$
$23$	−8.22348	−1.71471	−0.857357	+	0.514723i	$0.827895\pi$
$24$	0	0
$25$	−4.58942	−0.917885
$26$	0	0
$27$	4.62386	0.889861
$28$	0	0
$29$	7.92257	1.47119	0.735593	−	0.677424i	$-0.236904\pi$
$29$	7.92257	1.47119	0.735593	+	0.677424i	$0.236904\pi$
$30$	0	0
$31$	−2.15513	−0.387072	−0.193536	−	0.981093i	$-0.561996\pi$
$31$	−2.15513	−0.387072	−0.193536	+	0.981093i	$0.561996\pi$
$32$	0	0
$33$	−0.886923	−0.154393
$34$	0	0
$35$	−0.818767	−0.138397
$36$	0	0
$37$	10.1602	1.67032	0.835160	−	0.550007i	$-0.185375\pi$
$37$	10.1602	1.67032	0.835160	+	0.550007i	$0.185375\pi$
$38$	0	0
$39$	−0.233090	−0.0373243
$40$	0	0
$41$	−1.74757	−0.272925	−0.136462	−	0.990645i	$-0.543573\pi$
$41$	−1.74757	−0.272925	−0.136462	+	0.990645i	$0.543573\pi$
$42$	0	0
$43$	−5.26318	−0.802628	−0.401314	−	0.915941i	$-0.631446\pi$
$43$	−5.26318	−0.802628	−0.401314	+	0.915941i	$0.631446\pi$
$44$	0	0
$45$	1.41824	0.211419
$46$	0	0
$47$	3.30774	0.482484	0.241242	−	0.970465i	$-0.422445\pi$
$47$	3.30774	0.482484	0.241242	+	0.970465i	$0.422445\pi$
$48$	0	0
$49$	−5.36722	−0.766746
$50$	0	0
$51$	3.58793	0.502411
$52$	0	0
$53$	6.13347	0.842496	0.421248	−	0.906945i	$-0.361592\pi$
$53$	6.13347	0.842496	0.421248	+	0.906945i	$0.361592\pi$
$54$	0	0
$55$	−0.640762	−0.0864004
$56$	0	0
$57$	−2.80359	−0.371345
$58$	0	0
$59$	−3.45420	−0.449699	−0.224849	−	0.974394i	$-0.572189\pi$
$59$	−3.45420	−0.449699	−0.224849	+	0.974394i	$0.572189\pi$
$60$	0	0
$61$	0.846162	0.108340	0.0541699	−	0.998532i	$-0.482749\pi$
$61$	0.846162	0.108340	0.0541699	+	0.998532i	$0.482749\pi$
$62$	0	0
$63$	−2.82824	−0.356325
$64$	0	0
$65$	−0.168397	−0.0208871
$66$	0	0
$67$	−10.0457	−1.22727	−0.613636	−	0.789589i	$-0.710294\pi$
$67$	−10.0457	−1.22727	−0.613636	+	0.789589i	$0.710294\pi$
$68$	0	0
$69$	7.29359	0.878045
$70$	0	0
$71$	0.399049	0.0473584	0.0236792	−	0.999720i	$-0.492462\pi$
$71$	0.399049	0.0473584	0.0236792	+	0.999720i	$0.492462\pi$
$72$	0	0
$73$	−3.76750	−0.440953	−0.220476	−	0.975392i	$-0.570761\pi$
$73$	−3.76750	−0.440953	−0.220476	+	0.975392i	$0.570761\pi$
$74$	0	0
$75$	4.07046	0.470017
$76$	0	0
$77$	1.27780	0.145619
$78$	0	0
$79$	12.8736	1.44839	0.724194	−	0.689596i	$-0.242212\pi$
$79$	12.8736	1.44839	0.724194	+	0.689596i	$0.242212\pi$
$80$	0	0
$81$	2.53910	0.282122
$82$	0	0
$83$	17.6354	1.93573	0.967866	−	0.251466i	$-0.0809128\pi$
$83$	17.6354	1.93573	0.967866	+	0.251466i	$0.0809128\pi$
$84$	0	0
$85$	2.59212	0.281155
$86$	0	0
$87$	−7.02671	−0.753343
$88$	0	0
$89$	−5.52428	−0.585572	−0.292786	−	0.956178i	$-0.594582\pi$
$89$	−5.52428	−0.585572	−0.292786	+	0.956178i	$0.594582\pi$
$90$	0	0
$91$	0.335816	0.0352031
$92$	0	0
$93$	1.91143	0.198206
$94$	0	0
$95$	−2.02547	−0.207809
$96$	0	0
$97$	2.44191	0.247938	0.123969	−	0.992286i	$-0.460438\pi$
$97$	2.44191	0.247938	0.123969	+	0.992286i	$0.460438\pi$
$98$	0	0
$99$	−2.21337	−0.222452
$100$	0	0
$101$	12.0571	1.19972	0.599862	−	0.800104i	$-0.295222\pi$
$101$	12.0571	1.19972	0.599862	+	0.800104i	$0.295222\pi$
$102$	0	0
$103$	−6.58992	−0.649324	−0.324662	−	0.945830i	$-0.605251\pi$
$103$	−6.58992	−0.649324	−0.324662	+	0.945830i	$0.605251\pi$
$104$	0	0
$105$	0.726183	0.0708682
$106$	0	0
$107$	10.3466	1.00024	0.500122	−	0.865955i	$-0.333288\pi$
$107$	10.3466	1.00024	0.500122	+	0.865955i	$0.333288\pi$
$108$	0	0
$109$	−3.43655	−0.329162	−0.164581	−	0.986364i	$-0.552627\pi$
$109$	−3.43655	−0.329162	−0.164581	+	0.986364i	$0.552627\pi$
$110$	0	0
$111$	−9.01128	−0.855313
$112$	0	0
$113$	5.42512	0.510352	0.255176	−	0.966895i	$-0.417867\pi$
$113$	5.42512	0.510352	0.255176	+	0.966895i	$0.417867\pi$
$114$	0	0
$115$	5.26929	0.491364
$116$	0	0
$117$	−0.581690	−0.0537773
$118$	0	0
$119$	−5.16918	−0.473858
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.54996	0.139755
$124$	0	0
$125$	6.14454	0.549585
$126$	0	0
$127$	−7.07298	−0.627626	−0.313813	−	0.949485i	$-0.601606\pi$
$127$	−7.07298	−0.627626	−0.313813	+	0.949485i	$0.601606\pi$
$128$	0	0
$129$	4.66804	0.410998
$130$	0	0
$131$	−5.69672	−0.497724	−0.248862	−	0.968539i	$-0.580057\pi$
$131$	−5.69672	−0.497724	−0.248862	+	0.968539i	$0.580057\pi$
$132$	0	0
$133$	4.03917	0.350240
$134$	0	0
$135$	−2.96279	−0.254997
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	−8.63036	−0.732017	−0.366009	−	0.930611i	$-0.619276\pi$
$139$	−8.63036	−0.732017	−0.366009	+	0.930611i	$0.619276\pi$
$140$	0	0
$141$	−2.93371	−0.247063
$142$	0	0
$143$	0.262808	0.0219771
$144$	0	0
$145$	−5.07649	−0.421579
$146$	0	0
$147$	4.76031	0.392624
$148$	0	0
$149$	−12.5302	−1.02651	−0.513256	−	0.858235i	$-0.671561\pi$
$149$	−12.5302	−1.02651	−0.513256	+	0.858235i	$0.671561\pi$
$150$	0	0
$151$	14.0887	1.14653	0.573263	−	0.819372i	$-0.305677\pi$
$151$	14.0887	1.14653	0.573263	+	0.819372i	$0.305677\pi$
$152$	0	0
$153$	8.95390	0.723880
$154$	0	0
$155$	1.38092	0.110918
$156$	0	0
$157$	9.63394	0.768872	0.384436	−	0.923152i	$-0.374396\pi$
$157$	9.63394	0.768872	0.384436	+	0.923152i	$0.374396\pi$
$158$	0	0
$159$	−5.43991	−0.431413
$160$	0	0
$161$	−10.5080	−0.828144
$162$	0	0
$163$	−8.62863	−0.675847	−0.337923	−	0.941174i	$-0.609724\pi$
$163$	−8.62863	−0.675847	−0.337923	+	0.941174i	$0.609724\pi$
$164$	0	0
$165$	0.568307	0.0442426
$166$	0	0
$167$	5.48018	0.424069	0.212034	−	0.977262i	$-0.431991\pi$
$167$	5.48018	0.424069	0.212034	+	0.977262i	$0.431991\pi$
$168$	0	0
$169$	−12.9309	−0.994687
$170$	0	0
$171$	−6.99652	−0.535038
$172$	0	0
$173$	12.6432	0.961241	0.480621	−	0.876929i	$-0.340411\pi$
$173$	12.6432	0.961241	0.480621	+	0.876929i	$0.340411\pi$
$174$	0	0
$175$	−5.86437	−0.443305
$176$	0	0
$177$	3.06361	0.230275
$178$	0	0
$179$	−7.60170	−0.568178	−0.284089	−	0.958798i	$-0.591691\pi$
$179$	−7.60170	−0.568178	−0.284089	+	0.958798i	$0.591691\pi$
$180$	0	0
$181$	3.41223	0.253629	0.126814	−	0.991926i	$-0.459525\pi$
$181$	3.41223	0.253629	0.126814	+	0.991926i	$0.459525\pi$
$182$	0	0
$183$	−0.750480	−0.0554771
$184$	0	0
$185$	−6.51025	−0.478643
$186$	0	0
$187$	−4.04537	−0.295827
$188$	0	0
$189$	5.90837	0.429771
$190$	0	0
$191$	14.3032	1.03495	0.517473	−	0.855700i	$-0.326873\pi$
$191$	14.3032	1.03495	0.517473	+	0.855700i	$0.326873\pi$
$192$	0	0
$193$	−15.3532	−1.10515	−0.552573	−	0.833465i	$-0.686354\pi$
$193$	−15.3532	−1.10515	−0.552573	+	0.833465i	$0.686354\pi$
$194$	0	0
$195$	0.149355	0.0106956
$196$	0	0
$197$	26.1625	1.86400	0.932001	−	0.362455i	$-0.118061\pi$
$197$	26.1625	1.86400	0.932001	+	0.362455i	$0.118061\pi$
$198$	0	0
$199$	−20.6542	−1.46414	−0.732069	−	0.681230i	$-0.761445\pi$
$199$	−20.6542	−1.46414	−0.732069	+	0.681230i	$0.761445\pi$
$200$	0	0
$201$	8.90972	0.628444
$202$	0	0
$203$	10.1235	0.710529
$204$	0	0
$205$	1.11978	0.0782087
$206$	0	0
$207$	18.2016	1.26510
$208$	0	0
$209$	3.16103	0.218653
$210$	0	0
$211$	16.6648	1.14725	0.573625	−	0.819118i	$-0.305537\pi$
$211$	16.6648	1.14725	0.573625	+	0.819118i	$0.305537\pi$
$212$	0	0
$213$	−0.353925	−0.0242506
$214$	0	0
$215$	3.37245	0.229999
$216$	0	0
$217$	−2.75382	−0.186942
$218$	0	0
$219$	3.34148	0.225796
$220$	0	0
$221$	−1.06316	−0.0715156
$222$	0	0
$223$	15.8025	1.05822	0.529108	−	0.848554i	$-0.322527\pi$
$223$	15.8025	1.05822	0.529108	+	0.848554i	$0.322527\pi$
$224$	0	0
$225$	10.1581	0.677206
$226$	0	0
$227$	15.8390	1.05127	0.525634	−	0.850711i	$-0.323828\pi$
$227$	15.8390	1.05127	0.525634	+	0.850711i	$0.323828\pi$
$228$	0	0
$229$	3.24155	0.214208	0.107104	−	0.994248i	$-0.465842\pi$
$229$	3.24155	0.214208	0.107104	+	0.994248i	$0.465842\pi$
$230$	0	0
$231$	−1.13331	−0.0745664
$232$	0	0
$233$	1.06421	0.0697190	0.0348595	−	0.999392i	$-0.488902\pi$
$233$	1.06421	0.0697190	0.0348595	+	0.999392i	$0.488902\pi$
$234$	0	0
$235$	−2.11948	−0.138259
$236$	0	0
$237$	−11.4179	−0.741669
$238$	0	0
$239$	11.7459	0.759782	0.379891	−	0.925031i	$-0.375961\pi$
$239$	11.7459	0.759782	0.379891	+	0.925031i	$0.375961\pi$
$240$	0	0
$241$	12.5467	0.808207	0.404104	−	0.914713i	$-0.367584\pi$
$241$	12.5467	0.808207	0.404104	+	0.914713i	$0.367584\pi$
$242$	0	0
$243$	−16.1236	−1.03433
$244$	0	0
$245$	3.43911	0.219717
$246$	0	0
$247$	0.830743	0.0528589
$248$	0	0
$249$	−15.6412	−0.991221
$250$	0	0
$251$	12.9481	0.817279	0.408639	−	0.912696i	$-0.366003\pi$
$251$	12.9481	0.817279	0.408639	+	0.912696i	$0.366003\pi$
$252$	0	0
$253$	−8.22348	−0.517006
$254$	0	0
$255$	−2.29901	−0.143970
$256$	0	0
$257$	1.89974	0.118502	0.0592512	−	0.998243i	$-0.481129\pi$
$257$	1.89974	0.118502	0.0592512	+	0.998243i	$0.481129\pi$
$258$	0	0
$259$	12.9827	0.806704
$260$	0	0
$261$	−17.5356	−1.08542
$262$	0	0
$263$	24.4075	1.50503	0.752514	−	0.658576i	$-0.228841\pi$
$263$	24.4075	1.50503	0.752514	+	0.658576i	$0.228841\pi$
$264$	0	0
$265$	−3.93009	−0.241424
$266$	0	0
$267$	4.89961	0.299851
$268$	0	0
$269$	−3.22993	−0.196932	−0.0984661	−	0.995140i	$-0.531394\pi$
$269$	−3.22993	−0.196932	−0.0984661	+	0.995140i	$0.531394\pi$
$270$	0	0
$271$	−1.31953	−0.0801559	−0.0400779	−	0.999197i	$-0.512761\pi$
$271$	−1.31953	−0.0801559	−0.0400779	+	0.999197i	$0.512761\pi$
$272$	0	0
$273$	−0.297843	−0.0180263
$274$	0	0
$275$	−4.58942	−0.276753
$276$	0	0
$277$	13.5302	0.812949	0.406475	−	0.913662i	$-0.366758\pi$
$277$	13.5302	0.812949	0.406475	+	0.913662i	$0.366758\pi$
$278$	0	0
$279$	4.77009	0.285578
$280$	0	0
$281$	15.3596	0.916276	0.458138	−	0.888881i	$-0.348516\pi$
$281$	15.3596	0.916276	0.458138	+	0.888881i	$0.348516\pi$
$282$	0	0
$283$	17.6262	1.04777	0.523885	−	0.851789i	$-0.324482\pi$
$283$	17.6262	1.04777	0.523885	+	0.851789i	$0.324482\pi$
$284$	0	0
$285$	1.79643	0.106412
$286$	0	0
$287$	−2.23305	−0.131813
$288$	0	0
$289$	−0.634953	−0.0373502
$290$	0	0
$291$	−2.16578	−0.126960
$292$	0	0
$293$	10.9496	0.639685	0.319842	−	0.947471i	$-0.396370\pi$
$293$	10.9496	0.639685	0.319842	+	0.947471i	$0.396370\pi$
$294$	0	0
$295$	2.21332	0.128865
$296$	0	0
$297$	4.62386	0.268303
$298$	0	0
$299$	−2.16119	−0.124985
$300$	0	0
$301$	−6.72530	−0.387640
$302$	0	0
$303$	−10.6937	−0.614336
$304$	0	0
$305$	−0.542188	−0.0310456
$306$	0	0
$307$	29.1131	1.66157	0.830787	−	0.556590i	$-0.187890\pi$
$307$	29.1131	1.66157	0.830787	+	0.556590i	$0.187890\pi$
$308$	0	0
$309$	5.84475	0.332496
$310$	0	0
$311$	10.7035	0.606941	0.303470	−	0.952841i	$-0.401855\pi$
$311$	10.7035	0.606941	0.303470	+	0.952841i	$0.401855\pi$
$312$	0	0
$313$	−16.8299	−0.951284	−0.475642	−	0.879639i	$-0.657784\pi$
$313$	−16.8299	−0.951284	−0.475642	+	0.879639i	$0.657784\pi$
$314$	0	0
$315$	1.81223	0.102108
$316$	0	0
$317$	4.23930	0.238103	0.119052	−	0.992888i	$-0.462015\pi$
$317$	4.23930	0.238103	0.119052	+	0.992888i	$0.462015\pi$
$318$	0	0
$319$	7.92257	0.443579
$320$	0	0
$321$	−9.17665	−0.512191
$322$	0	0
$323$	−12.7875	−0.711518
$324$	0	0
$325$	−1.20614	−0.0669044
$326$	0	0
$327$	3.04796	0.168552
$328$	0	0
$329$	4.22664	0.233022
$330$	0	0
$331$	22.6381	1.24430	0.622150	−	0.782898i	$-0.286259\pi$
$331$	22.6381	1.24430	0.622150	+	0.782898i	$0.286259\pi$
$332$	0	0
$333$	−22.4882	−1.23234
$334$	0	0
$335$	6.43688	0.351684
$336$	0	0
$337$	−1.27266	−0.0693259	−0.0346630	−	0.999399i	$-0.511036\pi$
$337$	−1.27266	−0.0693259	−0.0346630	+	0.999399i	$0.511036\pi$
$338$	0	0
$339$	−4.81166	−0.261333
$340$	0	0
$341$	−2.15513	−0.116707
$342$	0	0
$343$	−15.8029	−0.853274
$344$	0	0
$345$	−4.67346	−0.251611
$346$	0	0
$347$	6.83587	0.366969	0.183484	−	0.983023i	$-0.441262\pi$
$347$	6.83587	0.366969	0.183484	+	0.983023i	$0.441262\pi$
$348$	0	0
$349$	−19.3361	−1.03504	−0.517520	−	0.855671i	$-0.673145\pi$
$349$	−19.3361	−1.03504	−0.517520	+	0.855671i	$0.673145\pi$
$350$	0	0
$351$	1.21518	0.0648618
$352$	0	0
$353$	22.8049	1.21378	0.606892	−	0.794784i	$-0.292416\pi$
$353$	22.8049	1.21378	0.606892	+	0.794784i	$0.292416\pi$
$354$	0	0
$355$	−0.255695	−0.0135709
$356$	0	0
$357$	4.58467	0.242646
$358$	0	0
$359$	8.13012	0.429091	0.214546	−	0.976714i	$-0.431173\pi$
$359$	8.13012	0.429091	0.214546	+	0.976714i	$0.431173\pi$
$360$	0	0
$361$	−9.00789	−0.474099
$362$	0	0
$363$	−0.886923	−0.0465514
$364$	0	0
$365$	2.41407	0.126358
$366$	0	0
$367$	19.2254	1.00356	0.501779	−	0.864996i	$-0.332679\pi$
$367$	19.2254	1.00356	0.501779	+	0.864996i	$0.332679\pi$
$368$	0	0
$369$	3.86802	0.201361
$370$	0	0
$371$	7.83735	0.406895
$372$	0	0
$373$	2.75392	0.142592	0.0712962	−	0.997455i	$-0.477286\pi$
$373$	2.75392	0.142592	0.0712962	+	0.997455i	$0.477286\pi$
$374$	0	0
$375$	−5.44973	−0.281423
$376$	0	0
$377$	2.08211	0.107234
$378$	0	0
$379$	32.3525	1.66184	0.830919	−	0.556394i	$-0.187815\pi$
$379$	32.3525	1.66184	0.830919	+	0.556394i	$0.187815\pi$
$380$	0	0
$381$	6.27319	0.321385
$382$	0	0
$383$	−2.37969	−0.121597	−0.0607983	−	0.998150i	$-0.519365\pi$
$383$	−2.37969	−0.121597	−0.0607983	+	0.998150i	$0.519365\pi$
$384$	0	0
$385$	−0.818767	−0.0417282
$386$	0	0
$387$	11.6494	0.592170
$388$	0	0
$389$	−21.7986	−1.10523	−0.552617	−	0.833435i	$-0.686371\pi$
$389$	−21.7986	−1.10523	−0.552617	+	0.833435i	$0.686371\pi$
$390$	0	0
$391$	33.2670	1.68239
$392$	0	0
$393$	5.05255	0.254867
$394$	0	0
$395$	−8.24889	−0.415047
$396$	0	0
$397$	−8.24413	−0.413761	−0.206881	−	0.978366i	$-0.566331\pi$
$397$	−8.24413	−0.413761	−0.206881	+	0.978366i	$0.566331\pi$
$398$	0	0
$399$	−3.58243	−0.179346
$400$	0	0
$401$	−37.3336	−1.86435	−0.932176	−	0.362005i	$-0.882092\pi$
$401$	−37.3336	−1.86435	−0.932176	+	0.362005i	$0.882092\pi$
$402$	0	0
$403$	−0.566384	−0.0282136
$404$	0	0
$405$	−1.62696	−0.0808443
$406$	0	0
$407$	10.1602	0.503621
$408$	0	0
$409$	6.10525	0.301885	0.150943	−	0.988543i	$-0.451769\pi$
$409$	6.10525	0.301885	0.150943	+	0.988543i	$0.451769\pi$
$410$	0	0
$411$	−0.886923	−0.0437487
$412$	0	0
$413$	−4.41378	−0.217188
$414$	0	0
$415$	−11.3001	−0.554699
$416$	0	0
$417$	7.65446	0.374841
$418$	0	0
$419$	−10.0093	−0.488985	−0.244492	−	0.969651i	$-0.578621\pi$
$419$	−10.0093	−0.488985	−0.244492	+	0.969651i	$0.578621\pi$
$420$	0	0
$421$	0.229935	0.0112063	0.00560317	−	0.999984i	$-0.498216\pi$
$421$	0.229935	0.0112063	0.00560317	+	0.999984i	$0.498216\pi$
$422$	0	0
$423$	−7.32125	−0.355971
$424$	0	0
$425$	18.5659	0.900580
$426$	0	0
$427$	1.08123	0.0523242
$428$	0	0
$429$	−0.233090	−0.0112537
$430$	0	0
$431$	−0.619946	−0.0298617	−0.0149309	−	0.999889i	$-0.504753\pi$
$431$	−0.619946	−0.0298617	−0.0149309	+	0.999889i	$0.504753\pi$
$432$	0	0
$433$	−13.4297	−0.645390	−0.322695	−	0.946503i	$-0.604589\pi$
$433$	−13.4297	−0.645390	−0.322695	+	0.946503i	$0.604589\pi$
$434$	0	0
$435$	4.50245	0.215876
$436$	0	0
$437$	−25.9947	−1.24349
$438$	0	0
$439$	−17.1770	−0.819815	−0.409907	−	0.912127i	$-0.634439\pi$
$439$	−17.1770	−0.819815	−0.409907	+	0.912127i	$0.634439\pi$
$440$	0	0
$441$	11.8796	0.565697
$442$	0	0
$443$	5.22647	0.248317	0.124159	−	0.992262i	$-0.460377\pi$
$443$	5.22647	0.248317	0.124159	+	0.992262i	$0.460377\pi$
$444$	0	0
$445$	3.53975	0.167800
$446$	0	0
$447$	11.1133	0.525641
$448$	0	0
$449$	15.4970	0.731347	0.365674	−	0.930743i	$-0.380839\pi$
$449$	15.4970	0.731347	0.365674	+	0.930743i	$0.380839\pi$
$450$	0	0
$451$	−1.74757	−0.0822899
$452$	0	0
$453$	−12.4956	−0.587096
$454$	0	0
$455$	−0.215178	−0.0100877
$456$	0	0
$457$	16.3038	0.762661	0.381330	−	0.924439i	$-0.375466\pi$
$457$	16.3038	0.762661	0.381330	+	0.924439i	$0.375466\pi$
$458$	0	0
$459$	−18.7052	−0.873085
$460$	0	0
$461$	−2.56626	−0.119523	−0.0597613	−	0.998213i	$-0.519034\pi$
$461$	−2.56626	−0.119523	−0.0597613	+	0.998213i	$0.519034\pi$
$462$	0	0
$463$	−0.747195	−0.0347251	−0.0173625	−	0.999849i	$-0.505527\pi$
$463$	−0.747195	−0.0347251	−0.0173625	+	0.999849i	$0.505527\pi$
$464$	0	0
$465$	−1.22477	−0.0567975
$466$	0	0
$467$	8.45523	0.391261	0.195631	−	0.980678i	$-0.437325\pi$
$467$	8.45523	0.391261	0.195631	+	0.980678i	$0.437325\pi$
$468$	0	0
$469$	−12.8364	−0.592728
$470$	0	0
$471$	−8.54456	−0.393712
$472$	0	0
$473$	−5.26318	−0.242001
$474$	0	0
$475$	−14.5073	−0.665641
$476$	0	0
$477$	−13.5756	−0.621585
$478$	0	0
$479$	11.1737	0.510538	0.255269	−	0.966870i	$-0.417836\pi$
$479$	11.1737	0.510538	0.255269	+	0.966870i	$0.417836\pi$
$480$	0	0
$481$	2.67017	0.121749
$482$	0	0
$483$	9.31976	0.424064
$484$	0	0
$485$	−1.56468	−0.0710486
$486$	0	0
$487$	−34.1601	−1.54794	−0.773971	−	0.633221i	$-0.781732\pi$
$487$	−34.1601	−1.54794	−0.773971	+	0.633221i	$0.781732\pi$
$488$	0	0
$489$	7.65293	0.346078
$490$	0	0
$491$	2.05437	0.0927125	0.0463562	−	0.998925i	$-0.485239\pi$
$491$	2.05437	0.0927125	0.0463562	+	0.998925i	$0.485239\pi$
$492$	0	0
$493$	−32.0498	−1.44345
$494$	0	0
$495$	1.41824	0.0637453
$496$	0	0
$497$	0.509905	0.0228724
$498$	0	0
$499$	29.6066	1.32537	0.662686	−	0.748897i	$-0.269416\pi$
$499$	29.6066	1.32537	0.662686	+	0.748897i	$0.269416\pi$
$500$	0	0
$501$	−4.86049	−0.217151
$502$	0	0
$503$	28.9210	1.28952	0.644761	−	0.764384i	$-0.276957\pi$
$503$	28.9210	1.28952	0.644761	+	0.764384i	$0.276957\pi$
$504$	0	0
$505$	−7.72571	−0.343790
$506$	0	0
$507$	11.4687	0.509345
$508$	0	0
$509$	35.4852	1.57285	0.786427	−	0.617684i	$-0.211929\pi$
$509$	35.4852	1.57285	0.786427	+	0.617684i	$0.211929\pi$
$510$	0	0
$511$	−4.81412	−0.212964
$512$	0	0
$513$	14.6161	0.645319
$514$	0	0
$515$	4.22257	0.186069
$516$	0	0
$517$	3.30774	0.145474
$518$	0	0
$519$	−11.2135	−0.492218
$520$	0	0
$521$	−23.4820	−1.02877	−0.514383	−	0.857561i	$-0.671979\pi$
$521$	−23.4820	−1.02877	−0.514383	+	0.857561i	$0.671979\pi$
$522$	0	0
$523$	20.4467	0.894070	0.447035	−	0.894516i	$-0.352480\pi$
$523$	20.4467	0.894070	0.447035	+	0.894516i	$0.352480\pi$
$524$	0	0
$525$	5.20125	0.227001
$526$	0	0
$527$	8.71829	0.379775
$528$	0	0
$529$	44.6256	1.94024
$530$	0	0
$531$	7.64542	0.331783
$532$	0	0
$533$	−0.459275	−0.0198934
$534$	0	0
$535$	−6.62972	−0.286628
$536$	0	0
$537$	6.74212	0.290944
$538$	0	0
$539$	−5.36722	−0.231183
$540$	0	0
$541$	15.5876	0.670164	0.335082	−	0.942189i	$-0.391236\pi$
$541$	15.5876	0.670164	0.335082	+	0.942189i	$0.391236\pi$
$542$	0	0
$543$	−3.02638	−0.129875
$544$	0	0
$545$	2.20201	0.0943239
$546$	0	0
$547$	17.2112	0.735899	0.367950	−	0.929846i	$-0.380060\pi$
$547$	17.2112	0.735899	0.367950	+	0.929846i	$0.380060\pi$
$548$	0	0
$549$	−1.87287	−0.0799320
$550$	0	0
$551$	25.0435	1.06689
$552$	0	0
$553$	16.4499	0.699519
$554$	0	0
$555$	5.77409	0.245096
$556$	0	0
$557$	−22.9091	−0.970689	−0.485345	−	0.874323i	$-0.661306\pi$
$557$	−22.9091	−0.970689	−0.485345	+	0.874323i	$0.661306\pi$
$558$	0	0
$559$	−1.38320	−0.0585033
$560$	0	0
$561$	3.58793	0.151483
$562$	0	0
$563$	−30.0261	−1.26545	−0.632724	−	0.774377i	$-0.718063\pi$
$563$	−30.0261	−1.26545	−0.632724	+	0.774377i	$0.718063\pi$
$564$	0	0
$565$	−3.47621	−0.146245
$566$	0	0
$567$	3.24447	0.136255
$568$	0	0
$569$	−12.8104	−0.537040	−0.268520	−	0.963274i	$-0.586535\pi$
$569$	−12.8104	−0.537040	−0.268520	+	0.963274i	$0.586535\pi$
$570$	0	0
$571$	29.1232	1.21877	0.609385	−	0.792875i	$-0.291416\pi$
$571$	29.1232	1.21877	0.609385	+	0.792875i	$0.291416\pi$
$572$	0	0
$573$	−12.6859	−0.529960
$574$	0	0
$575$	37.7410	1.57391
$576$	0	0
$577$	−12.3965	−0.516073	−0.258036	−	0.966135i	$-0.583075\pi$
$577$	−12.3965	−0.516073	−0.258036	+	0.966135i	$0.583075\pi$
$578$	0	0
$579$	13.6171	0.565907
$580$	0	0
$581$	22.5345	0.934888
$582$	0	0
$583$	6.13347	0.254022
$584$	0	0
$585$	0.372725	0.0154103
$586$	0	0
$587$	−22.8851	−0.944570	−0.472285	−	0.881446i	$-0.656571\pi$
$587$	−22.8851	−0.944570	−0.472285	+	0.881446i	$0.656571\pi$
$588$	0	0
$589$	−6.81242	−0.280701
$590$	0	0
$591$	−23.2041	−0.954491
$592$	0	0
$593$	34.7497	1.42700	0.713499	−	0.700656i	$-0.247109\pi$
$593$	34.7497	1.42700	0.713499	+	0.700656i	$0.247109\pi$
$594$	0	0
$595$	3.31222	0.135788
$596$	0	0
$597$	18.3187	0.749734
$598$	0	0
$599$	19.6930	0.804632	0.402316	−	0.915501i	$-0.368205\pi$
$599$	19.6930	0.804632	0.402316	+	0.915501i	$0.368205\pi$
$600$	0	0
$601$	−37.9216	−1.54686	−0.773428	−	0.633884i	$-0.781459\pi$
$601$	−37.9216	−1.54686	−0.773428	+	0.633884i	$0.781459\pi$
$602$	0	0
$603$	22.2347	0.905469
$604$	0	0
$605$	−0.640762	−0.0260507
$606$	0	0
$607$	33.0588	1.34181	0.670907	−	0.741541i	$-0.265905\pi$
$607$	33.0588	1.34181	0.670907	+	0.741541i	$0.265905\pi$
$608$	0	0
$609$	−8.97874	−0.363837
$610$	0	0
$611$	0.869300	0.0351681
$612$	0	0
$613$	−12.9572	−0.523337	−0.261668	−	0.965158i	$-0.584273\pi$
$613$	−12.9572	−0.523337	−0.261668	+	0.965158i	$0.584273\pi$
$614$	0	0
$615$	−0.993156	−0.0400479
$616$	0	0
$617$	20.2002	0.813230	0.406615	−	0.913600i	$-0.366709\pi$
$617$	20.2002	0.813230	0.406615	+	0.913600i	$0.366709\pi$
$618$	0	0
$619$	36.7621	1.47759	0.738796	−	0.673929i	$-0.235395\pi$
$619$	36.7621	1.47759	0.738796	+	0.673929i	$0.235395\pi$
$620$	0	0
$621$	−38.0242	−1.52586
$622$	0	0
$623$	−7.05893	−0.282810
$624$	0	0
$625$	19.0099	0.760397
$626$	0	0
$627$	−2.80359	−0.111965
$628$	0	0
$629$	−41.1017	−1.63883
$630$	0	0
$631$	2.23391	0.0889304	0.0444652	−	0.999011i	$-0.485842\pi$
$631$	2.23391	0.0889304	0.0444652	+	0.999011i	$0.485842\pi$
$632$	0	0
$633$	−14.7804	−0.587466
$634$	0	0
$635$	4.53210	0.179851
$636$	0	0
$637$	−1.41055	−0.0558879
$638$	0	0
$639$	−0.883242	−0.0349405
$640$	0	0
$641$	−0.981954	−0.0387849	−0.0193924	−	0.999812i	$-0.506173\pi$
$641$	−0.981954	−0.0387849	−0.0193924	+	0.999812i	$0.506173\pi$
$642$	0	0
$643$	−38.7875	−1.52963	−0.764814	−	0.644251i	$-0.777169\pi$
$643$	−38.7875	−1.52963	−0.764814	+	0.644251i	$0.777169\pi$
$644$	0	0
$645$	−2.99110	−0.117775
$646$	0	0
$647$	33.9860	1.33613	0.668063	−	0.744105i	$-0.267124\pi$
$647$	33.9860	1.33613	0.668063	+	0.744105i	$0.267124\pi$
$648$	0	0
$649$	−3.45420	−0.135589
$650$	0	0
$651$	2.44243	0.0957264
$652$	0	0
$653$	−11.0598	−0.432804	−0.216402	−	0.976304i	$-0.569432\pi$
$653$	−11.0598	−0.432804	−0.216402	+	0.976304i	$0.569432\pi$
$654$	0	0
$655$	3.65024	0.142627
$656$	0	0
$657$	8.33886	0.325330
$658$	0	0
$659$	24.5140	0.954930	0.477465	−	0.878651i	$-0.341556\pi$
$659$	24.5140	0.954930	0.477465	+	0.878651i	$0.341556\pi$
$660$	0	0
$661$	−4.56158	−0.177425	−0.0887125	−	0.996057i	$-0.528275\pi$
$661$	−4.56158	−0.177425	−0.0887125	+	0.996057i	$0.528275\pi$
$662$	0	0
$663$	0.942937	0.0366206
$664$	0	0
$665$	−2.58815	−0.100364
$666$	0	0
$667$	−65.1511	−2.52266
$668$	0	0
$669$	−14.0156	−0.541876
$670$	0	0
$671$	0.846162	0.0326657
$672$	0	0
$673$	−12.2373	−0.471712	−0.235856	−	0.971788i	$-0.575789\pi$
$673$	−12.2373	−0.471712	−0.235856	+	0.971788i	$0.575789\pi$
$674$	0	0
$675$	−21.2208	−0.816790
$676$	0	0
$677$	14.9429	0.574303	0.287152	−	0.957885i	$-0.407292\pi$
$677$	14.9429	0.574303	0.287152	+	0.957885i	$0.407292\pi$
$678$	0	0
$679$	3.12027	0.119745
$680$	0	0
$681$	−14.0479	−0.538318
$682$	0	0
$683$	−48.2354	−1.84568	−0.922839	−	0.385187i	$-0.874137\pi$
$683$	−48.2354	−1.84568	−0.922839	+	0.385187i	$0.874137\pi$
$684$	0	0
$685$	−0.640762	−0.0244823
$686$	0	0
$687$	−2.87501	−0.109688
$688$	0	0
$689$	1.61192	0.0614093
$690$	0	0
$691$	−7.84303	−0.298363	−0.149181	−	0.988810i	$-0.547664\pi$
$691$	−7.84303	−0.298363	−0.149181	+	0.988810i	$0.547664\pi$
$692$	0	0
$693$	−2.82824	−0.107436
$694$	0	0
$695$	5.53001	0.209765
$696$	0	0
$697$	7.06958	0.267779
$698$	0	0
$699$	−0.943875	−0.0357006
$700$	0	0
$701$	22.5920	0.853289	0.426644	−	0.904419i	$-0.359696\pi$
$701$	22.5920	0.853289	0.426644	+	0.904419i	$0.359696\pi$
$702$	0	0
$703$	32.1166	1.21130
$704$	0	0
$705$	1.87981	0.0707978
$706$	0	0
$707$	15.4065	0.579422
$708$	0	0
$709$	−15.5252	−0.583062	−0.291531	−	0.956561i	$-0.594165\pi$
$709$	−15.5252	−0.583062	−0.291531	+	0.956561i	$0.594165\pi$
$710$	0	0
$711$	−28.4939	−1.06861
$712$	0	0
$713$	17.7226	0.663718
$714$	0	0
$715$	−0.168397	−0.00629770
$716$	0	0
$717$	−10.4177	−0.389058
$718$	0	0
$719$	−17.3855	−0.648370	−0.324185	−	0.945994i	$-0.605090\pi$
$719$	−17.3855	−0.648370	−0.324185	+	0.945994i	$0.605090\pi$
$720$	0	0
$721$	−8.42061	−0.313600
$722$	0	0
$723$	−11.1280	−0.413855
$724$	0	0
$725$	−36.3600	−1.35038
$726$	0	0
$727$	19.7325	0.731837	0.365919	−	0.930647i	$-0.380755\pi$
$727$	19.7325	0.731837	0.365919	+	0.930647i	$0.380755\pi$
$728$	0	0
$729$	6.68304	0.247520
$730$	0	0
$731$	21.2915	0.787496
$732$	0	0
$733$	45.0689	1.66466	0.832330	−	0.554281i	$-0.187007\pi$
$733$	45.0689	1.66466	0.832330	+	0.554281i	$0.187007\pi$
$734$	0	0
$735$	−3.05023	−0.112509
$736$	0	0
$737$	−10.0457	−0.370037
$738$	0	0
$739$	−39.5676	−1.45552	−0.727759	−	0.685833i	$-0.759438\pi$
$739$	−39.5676	−1.45552	−0.727759	+	0.685833i	$0.759438\pi$
$740$	0	0
$741$	−0.736805	−0.0270672
$742$	0	0
$743$	−10.4586	−0.383687	−0.191844	−	0.981426i	$-0.561447\pi$
$743$	−10.4586	−0.383687	−0.191844	+	0.981426i	$0.561447\pi$
$744$	0	0
$745$	8.02887	0.294155
$746$	0	0
$747$	−39.0335	−1.42816
$748$	0	0
$749$	13.2209	0.483082
$750$	0	0
$751$	10.6782	0.389652	0.194826	−	0.980838i	$-0.437586\pi$
$751$	10.6782	0.389652	0.194826	+	0.980838i	$0.437586\pi$
$752$	0	0
$753$	−11.4840	−0.418500
$754$	0	0
$755$	−9.02753	−0.328546
$756$	0	0
$757$	−2.28553	−0.0830690	−0.0415345	−	0.999137i	$-0.513225\pi$
$757$	−2.28553	−0.0830690	−0.0415345	+	0.999137i	$0.513225\pi$
$758$	0	0
$759$	7.29359	0.264741
$760$	0	0
$761$	−28.0583	−1.01711	−0.508557	−	0.861028i	$-0.669821\pi$
$761$	−28.0583	−1.01711	−0.508557	+	0.861028i	$0.669821\pi$
$762$	0	0
$763$	−4.39123	−0.158973
$764$	0	0
$765$	−5.73732	−0.207433
$766$	0	0
$767$	−0.907791	−0.0327784
$768$	0	0
$769$	−9.98572	−0.360094	−0.180047	−	0.983658i	$-0.557625\pi$
$769$	−9.98572	−0.360094	−0.180047	+	0.983658i	$0.557625\pi$
$770$	0	0
$771$	−1.68492	−0.0606809
$772$	0	0
$773$	11.6886	0.420409	0.210205	−	0.977657i	$-0.432587\pi$
$773$	11.6886	0.420409	0.210205	+	0.977657i	$0.432587\pi$
$774$	0	0
$775$	9.89079	0.355288
$776$	0	0
$777$	−11.5146	−0.413085
$778$	0	0
$779$	−5.52412	−0.197922
$780$	0	0
$781$	0.399049	0.0142791
$782$	0	0
$783$	36.6328	1.30915
$784$	0	0
$785$	−6.17306	−0.220326
$786$	0	0
$787$	26.3872	0.940602	0.470301	−	0.882506i	$-0.344145\pi$
$787$	26.3872	0.940602	0.470301	+	0.882506i	$0.344145\pi$
$788$	0	0
$789$	−21.6475	−0.770673
$790$	0	0
$791$	6.93222	0.246481
$792$	0	0
$793$	0.222378	0.00789687
$794$	0	0
$795$	3.48569	0.123625
$796$	0	0
$797$	−44.9758	−1.59312	−0.796562	−	0.604557i	$-0.793350\pi$
$797$	−44.9758	−1.59312	−0.796562	+	0.604557i	$0.793350\pi$
$798$	0	0
$799$	−13.3811	−0.473388
$800$	0	0
$801$	12.2273	0.432029
$802$	0	0
$803$	−3.76750	−0.132952
$804$	0	0
$805$	6.73311	0.237311
$806$	0	0
$807$	2.86470	0.100842
$808$	0	0
$809$	2.88356	0.101380	0.0506902	−	0.998714i	$-0.483858\pi$
$809$	2.88356	0.101380	0.0506902	+	0.998714i	$0.483858\pi$
$810$	0	0
$811$	−41.1161	−1.44378	−0.721890	−	0.692007i	$-0.756727\pi$
$811$	−41.1161	−1.44378	−0.721890	+	0.692007i	$0.756727\pi$
$812$	0	0
$813$	1.17032	0.0410450
$814$	0	0
$815$	5.52890	0.193669
$816$	0	0
$817$	−16.6371	−0.582058
$818$	0	0
$819$	−0.743284	−0.0259725
$820$	0	0
$821$	−3.34436	−0.116719	−0.0583594	−	0.998296i	$-0.518587\pi$
$821$	−3.34436	−0.116719	−0.0583594	+	0.998296i	$0.518587\pi$
$822$	0	0
$823$	−4.47178	−0.155877	−0.0779383	−	0.996958i	$-0.524834\pi$
$823$	−4.47178	−0.155877	−0.0779383	+	0.996958i	$0.524834\pi$
$824$	0	0
$825$	4.07046	0.141715
$826$	0	0
$827$	−8.47385	−0.294665	−0.147332	−	0.989087i	$-0.547069\pi$
$827$	−8.47385	−0.294665	−0.147332	+	0.989087i	$0.547069\pi$
$828$	0	0
$829$	45.3314	1.57442	0.787212	−	0.616682i	$-0.211524\pi$
$829$	45.3314	1.57442	0.787212	+	0.616682i	$0.211524\pi$
$830$	0	0
$831$	−12.0002	−0.416283
$832$	0	0
$833$	21.7124	0.752291
$834$	0	0
$835$	−3.51149	−0.121520
$836$	0	0
$837$	−9.96499	−0.344441
$838$	0	0
$839$	15.9943	0.552185	0.276093	−	0.961131i	$-0.410960\pi$
$839$	15.9943	0.552185	0.276093	+	0.961131i	$0.410960\pi$
$840$	0	0
$841$	33.7672	1.16439
$842$	0	0
$843$	−13.6228	−0.469193
$844$	0	0
$845$	8.28565	0.285035
$846$	0	0
$847$	1.27780	0.0439058
$848$	0	0
$849$	−15.6331	−0.536527
$850$	0	0
$851$	−83.5519	−2.86412
$852$	0	0
$853$	43.9412	1.50452	0.752259	−	0.658867i	$-0.228964\pi$
$853$	43.9412	1.50452	0.752259	+	0.658867i	$0.228964\pi$
$854$	0	0
$855$	4.48311	0.153319
$856$	0	0
$857$	−4.27875	−0.146159	−0.0730796	−	0.997326i	$-0.523283\pi$
$857$	−4.27875	−0.146159	−0.0730796	+	0.997326i	$0.523283\pi$
$858$	0	0
$859$	−7.51001	−0.256238	−0.128119	−	0.991759i	$-0.540894\pi$
$859$	−7.51001	−0.256238	−0.128119	+	0.991759i	$0.540894\pi$
$860$	0	0
$861$	1.98054	0.0674967
$862$	0	0
$863$	14.4032	0.490290	0.245145	−	0.969486i	$-0.421164\pi$
$863$	14.4032	0.490290	0.245145	+	0.969486i	$0.421164\pi$
$864$	0	0
$865$	−8.10126	−0.275451
$866$	0	0
$867$	0.563154	0.0191257
$868$	0	0
$869$	12.8736	0.436706
$870$	0	0
$871$	−2.64008	−0.0894556
$872$	0	0
$873$	−5.40484	−0.182926
$874$	0	0
$875$	7.85150	0.265429
$876$	0	0
$877$	−22.1682	−0.748566	−0.374283	−	0.927315i	$-0.622111\pi$
$877$	−22.1682	−0.748566	−0.374283	+	0.927315i	$0.622111\pi$
$878$	0	0
$879$	−9.71148	−0.327560
$880$	0	0
$881$	−33.4600	−1.12730	−0.563648	−	0.826015i	$-0.690602\pi$
$881$	−33.4600	−1.12730	−0.563648	+	0.826015i	$0.690602\pi$
$882$	0	0
$883$	11.4268	0.384543	0.192271	−	0.981342i	$-0.438415\pi$
$883$	11.4268	0.384543	0.192271	+	0.981342i	$0.438415\pi$
$884$	0	0
$885$	−1.96305	−0.0659871
$886$	0	0
$887$	3.62074	0.121573	0.0607863	−	0.998151i	$-0.480639\pi$
$887$	3.62074	0.121573	0.0607863	+	0.998151i	$0.480639\pi$
$888$	0	0
$889$	−9.03786	−0.303120
$890$	0	0
$891$	2.53910	0.0850631
$892$	0	0
$893$	10.4559	0.349892
$894$	0	0
$895$	4.87088	0.162816
$896$	0	0
$897$	1.91681	0.0640005
$898$	0	0
$899$	−17.0742	−0.569455
$900$	0	0
$901$	−24.8122	−0.826613
$902$	0	0
$903$	5.96482	0.198497
$904$	0	0
$905$	−2.18643	−0.0726793
$906$	0	0
$907$	−3.85747	−0.128085	−0.0640426	−	0.997947i	$-0.520399\pi$
$907$	−3.85747	−0.128085	−0.0640426	+	0.997947i	$0.520399\pi$
$908$	0	0
$909$	−26.6867	−0.885143
$910$	0	0
$911$	29.9177	0.991217	0.495608	−	0.868546i	$-0.334945\pi$
$911$	29.9177	0.991217	0.495608	+	0.868546i	$0.334945\pi$
$912$	0	0
$913$	17.6354	0.583645
$914$	0	0
$915$	0.480879	0.0158974
$916$	0	0
$917$	−7.27927	−0.240383
$918$	0	0
$919$	−6.36139	−0.209843	−0.104921	−	0.994481i	$-0.533459\pi$
$919$	−6.36139	−0.209843	−0.104921	+	0.994481i	$0.533459\pi$
$920$	0	0
$921$	−25.8211	−0.850835
$922$	0	0
$923$	0.104873	0.00345194
$924$	0	0
$925$	−46.6293	−1.53316
$926$	0	0
$927$	14.5859	0.479064
$928$	0	0
$929$	13.5451	0.444399	0.222199	−	0.975001i	$-0.428676\pi$
$929$	13.5451	0.444399	0.222199	+	0.975001i	$0.428676\pi$
$930$	0	0
$931$	−16.9660	−0.556037
$932$	0	0
$933$	−9.49319	−0.310793
$934$	0	0
$935$	2.59212	0.0847715
$936$	0	0
$937$	−15.6981	−0.512836	−0.256418	−	0.966566i	$-0.582542\pi$
$937$	−15.6981	−0.512836	−0.256418	+	0.966566i	$0.582542\pi$
$938$	0	0
$939$	14.9269	0.487120
$940$	0	0
$941$	−44.7162	−1.45771	−0.728854	−	0.684669i	$-0.759947\pi$
$941$	−44.7162	−1.45771	−0.728854	+	0.684669i	$0.759947\pi$
$942$	0	0
$943$	14.3711	0.467988
$944$	0	0
$945$	−3.78586	−0.123154
$946$	0	0
$947$	−42.1281	−1.36898	−0.684489	−	0.729023i	$-0.739975\pi$
$947$	−42.1281	−1.36898	−0.684489	+	0.729023i	$0.739975\pi$
$948$	0	0
$949$	−0.990128	−0.0321409
$950$	0	0
$951$	−3.75994	−0.121924
$952$	0	0
$953$	−0.509882	−0.0165167	−0.00825834	−	0.999966i	$-0.502629\pi$
$953$	−0.509882	−0.0165167	−0.00825834	+	0.999966i	$0.502629\pi$
$954$	0	0
$955$	−9.16498	−0.296572
$956$	0	0
$957$	−7.02671	−0.227141
$958$	0	0
$959$	1.27780	0.0412624
$960$	0	0
$961$	−26.3554	−0.850175
$962$	0	0
$963$	−22.9009	−0.737970
$964$	0	0
$965$	9.83774	0.316688
$966$	0	0
$967$	28.0845	0.903137	0.451568	−	0.892237i	$-0.350865\pi$
$967$	28.0845	0.903137	0.451568	+	0.892237i	$0.350865\pi$
$968$	0	0
$969$	11.3416	0.364344
$970$	0	0
$971$	29.6189	0.950517	0.475258	−	0.879846i	$-0.342355\pi$
$971$	29.6189	0.950517	0.475258	+	0.879846i	$0.342355\pi$
$972$	0	0
$973$	−11.0279	−0.353538
$974$	0	0
$975$	1.06975	0.0342594
$976$	0	0
$977$	17.8442	0.570886	0.285443	−	0.958396i	$-0.407859\pi$
$977$	17.8442	0.570886	0.285443	+	0.958396i	$0.407859\pi$
$978$	0	0
$979$	−5.52428	−0.176557
$980$	0	0
$981$	7.60635	0.242852
$982$	0	0
$983$	1.41945	0.0452734	0.0226367	−	0.999744i	$-0.492794\pi$
$983$	1.41945	0.0452734	0.0226367	+	0.999744i	$0.492794\pi$
$984$	0	0
$985$	−16.7640	−0.534144
$986$	0	0
$987$	−3.74870	−0.119322
$988$	0	0
$989$	43.2817	1.37628
$990$	0	0
$991$	−40.2998	−1.28017	−0.640083	−	0.768306i	$-0.721100\pi$
$991$	−40.2998	−1.28017	−0.640083	+	0.768306i	$0.721100\pi$
$992$	0	0
$993$	−20.0782	−0.637163
$994$	0	0
$995$	13.2344	0.419560
$996$	0	0
$997$	−53.9602	−1.70894	−0.854469	−	0.519503i	$-0.826117\pi$
$997$	−53.9602	−1.70894	−0.854469	+	0.519503i	$0.826117\pi$
$998$	0	0
$999$	46.9791	1.48635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.f.1.10	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.f.1.10	✓	29	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)

\(f(q)\)	\(=\)	\(q-0.886923 q^{3} -0.640762 q^{5} +1.27780 q^{7} -2.21337 q^{9} +O(q^{10})\)	\(q-0.886923 q^{3} -0.640762 q^{5} +1.27780 q^{7} -2.21337 q^{9} +1.00000 q^{11} +0.262808 q^{13} +0.568307 q^{15} -4.04537 q^{17} +3.16103 q^{19} -1.13331 q^{21} -8.22348 q^{23} -4.58942 q^{25} +4.62386 q^{27} +7.92257 q^{29} -2.15513 q^{31} -0.886923 q^{33} -0.818767 q^{35} +10.1602 q^{37} -0.233090 q^{39} -1.74757 q^{41} -5.26318 q^{43} +1.41824 q^{45} +3.30774 q^{47} -5.36722 q^{49} +3.58793 q^{51} +6.13347 q^{53} -0.640762 q^{55} -2.80359 q^{57} -3.45420 q^{59} +0.846162 q^{61} -2.82824 q^{63} -0.168397 q^{65} -10.0457 q^{67} +7.29359 q^{69} +0.399049 q^{71} -3.76750 q^{73} +4.07046 q^{75} +1.27780 q^{77} +12.8736 q^{79} +2.53910 q^{81} +17.6354 q^{83} +2.59212 q^{85} -7.02671 q^{87} -5.52428 q^{89} +0.335816 q^{91} +1.91143 q^{93} -2.02547 q^{95} +2.44191 q^{97} -2.21337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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