LMFDB - Embedded newform 2028.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,2,Mod(1,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, 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("2028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.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:	$16.1936615299$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	2028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.04892 q^{5} -5.04892 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.04892 q^{5} -5.04892 q^{7} +1.00000 q^{9} -3.80194 q^{11} +3.04892 q^{15} -0.356896 q^{17} -6.13706 q^{19} -5.04892 q^{21} -7.93900 q^{23} +4.29590 q^{25} +1.00000 q^{27} +1.41789 q^{29} +5.78986 q^{31} -3.80194 q^{33} -15.3937 q^{35} +0.0271471 q^{37} -9.18598 q^{41} -1.21983 q^{43} +3.04892 q^{45} -9.56465 q^{47} +18.4916 q^{49} -0.356896 q^{51} -4.73556 q^{53} -11.5918 q^{55} -6.13706 q^{57} +13.5254 q^{59} +0.576728 q^{61} -5.04892 q^{63} -7.95108 q^{67} -7.93900 q^{69} -3.07606 q^{71} -14.0368 q^{73} +4.29590 q^{75} +19.1957 q^{77} -0.841166 q^{79} +1.00000 q^{81} -0.192685 q^{83} -1.08815 q^{85} +1.41789 q^{87} +9.21313 q^{89} +5.78986 q^{93} -18.7114 q^{95} -0.682333 q^{97} -3.80194 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 6 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 6 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 6 q^{7} + 3 q^{9} - 7 q^{11} + 3 q^{17} - 13 q^{19} - 6 q^{21} - 14 q^{23} - q^{25} + 3 q^{27} + 10 q^{29} - 6 q^{31} - 7 q^{33} - 14 q^{35} - 6 q^{37} - 13 q^{41} - 5 q^{43} - 7 q^{47} + 5 q^{49} + 3 q^{51} - 3 q^{53} - 7 q^{55} - 13 q^{57} + 6 q^{59} - q^{61} - 6 q^{63} - 33 q^{67} - 14 q^{69} + 6 q^{71} - 14 q^{73} - q^{75} + 21 q^{77} - 11 q^{79} + 3 q^{81} - 8 q^{83} - 7 q^{85} + 10 q^{87} + 7 q^{89} - 6 q^{93} - 7 q^{95} - 19 q^{97} - 7 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 6 * q^7 + 3 * q^9 - 7 * q^11 + 3 * q^17 - 13 * q^19 - 6 * q^21 - 14 * q^23 - q^25 + 3 * q^27 + 10 * q^29 - 6 * q^31 - 7 * q^33 - 14 * q^35 - 6 * q^37 - 13 * q^41 - 5 * q^43 - 7 * q^47 + 5 * q^49 + 3 * q^51 - 3 * q^53 - 7 * q^55 - 13 * q^57 + 6 * q^59 - q^61 - 6 * q^63 - 33 * q^67 - 14 * q^69 + 6 * q^71 - 14 * q^73 - q^75 + 21 * q^77 - 11 * q^79 + 3 * q^81 - 8 * q^83 - 7 * q^85 + 10 * q^87 + 7 * q^89 - 6 * q^93 - 7 * q^95 - 19 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.04892	1.36352	0.681759	−	0.731577i	$-0.261215\pi$
$5$	3.04892	1.36352	0.681759	+	0.731577i	$0.261215\pi$
$6$	0	0
$7$	−5.04892	−1.90831	−0.954156	−	0.299311i	$-0.903243\pi$
$7$	−5.04892	−1.90831	−0.954156	+	0.299311i	$0.903243\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.80194	−1.14633	−0.573164	−	0.819441i	$-0.694284\pi$
$11$	−3.80194	−1.14633	−0.573164	+	0.819441i	$0.694284\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.04892	0.787227
$16$	0	0
$17$	−0.356896	−0.0865600	−0.0432800	−	0.999063i	$-0.513781\pi$
$17$	−0.356896	−0.0865600	−0.0432800	+	0.999063i	$0.513781\pi$
$18$	0	0
$19$	−6.13706	−1.40794	−0.703969	−	0.710230i	$-0.748591\pi$
$19$	−6.13706	−1.40794	−0.703969	+	0.710230i	$0.748591\pi$
$20$	0	0
$21$	−5.04892	−1.10176
$22$	0	0
$23$	−7.93900	−1.65540	−0.827698	−	0.561174i	$-0.810350\pi$
$23$	−7.93900	−1.65540	−0.827698	+	0.561174i	$0.810350\pi$
$24$	0	0
$25$	4.29590	0.859179
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.41789	0.263296	0.131648	−	0.991296i	$-0.457973\pi$
$29$	1.41789	0.263296	0.131648	+	0.991296i	$0.457973\pi$
$30$	0	0
$31$	5.78986	1.03989	0.519944	−	0.854200i	$-0.325953\pi$
$31$	5.78986	1.03989	0.519944	+	0.854200i	$0.325953\pi$
$32$	0	0
$33$	−3.80194	−0.661832
$34$	0	0
$35$	−15.3937	−2.60202
$36$	0	0
$37$	0.0271471	0.00446295	0.00223148	−	0.999998i	$-0.499290\pi$
$37$	0.0271471	0.00446295	0.00223148	+	0.999998i	$0.499290\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.18598	−1.43461	−0.717305	−	0.696760i	$-0.754624\pi$
$41$	−9.18598	−1.43461	−0.717305	+	0.696760i	$0.754624\pi$
$42$	0	0
$43$	−1.21983	−0.186023	−0.0930114	−	0.995665i	$-0.529649\pi$
$43$	−1.21983	−0.186023	−0.0930114	+	0.995665i	$0.529649\pi$
$44$	0	0
$45$	3.04892	0.454506
$46$	0	0
$47$	−9.56465	−1.39515	−0.697574	−	0.716513i	$-0.745737\pi$
$47$	−9.56465	−1.39515	−0.697574	+	0.716513i	$0.745737\pi$
$48$	0	0
$49$	18.4916	2.64165
$50$	0	0
$51$	−0.356896	−0.0499754
$52$	0	0
$53$	−4.73556	−0.650479	−0.325240	−	0.945632i	$-0.605445\pi$
$53$	−4.73556	−0.650479	−0.325240	+	0.945632i	$0.605445\pi$
$54$	0	0
$55$	−11.5918	−1.56304
$56$	0	0
$57$	−6.13706	−0.812874
$58$	0	0
$59$	13.5254	1.76086	0.880430	−	0.474177i	$-0.157254\pi$
$59$	13.5254	1.76086	0.880430	+	0.474177i	$0.157254\pi$
$60$	0	0
$61$	0.576728	0.0738425	0.0369213	−	0.999318i	$-0.488245\pi$
$61$	0.576728	0.0738425	0.0369213	+	0.999318i	$0.488245\pi$
$62$	0	0
$63$	−5.04892	−0.636104
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.95108	−0.971379	−0.485690	−	0.874131i	$-0.661431\pi$
$67$	−7.95108	−0.971379	−0.485690	+	0.874131i	$0.661431\pi$
$68$	0	0
$69$	−7.93900	−0.955743
$70$	0	0
$71$	−3.07606	−0.365062	−0.182531	−	0.983200i	$-0.558429\pi$
$71$	−3.07606	−0.365062	−0.182531	+	0.983200i	$0.558429\pi$
$72$	0	0
$73$	−14.0368	−1.64289	−0.821444	−	0.570290i	$-0.806831\pi$
$73$	−14.0368	−1.64289	−0.821444	+	0.570290i	$0.806831\pi$
$74$	0	0
$75$	4.29590	0.496047
$76$	0	0
$77$	19.1957	2.18755
$78$	0	0
$79$	−0.841166	−0.0946386	−0.0473193	−	0.998880i	$-0.515068\pi$
$79$	−0.841166	−0.0946386	−0.0473193	+	0.998880i	$0.515068\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.192685	−0.0211500	−0.0105750	−	0.999944i	$-0.503366\pi$
$83$	−0.192685	−0.0211500	−0.0105750	+	0.999944i	$0.503366\pi$
$84$	0	0
$85$	−1.08815	−0.118026
$86$	0	0
$87$	1.41789	0.152014
$88$	0	0
$89$	9.21313	0.976590	0.488295	−	0.872679i	$-0.337619\pi$
$89$	9.21313	0.976590	0.488295	+	0.872679i	$0.337619\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.78986	0.600380
$94$	0	0
$95$	−18.7114	−1.91975
$96$	0	0
$97$	−0.682333	−0.0692804	−0.0346402	−	0.999400i	$-0.511029\pi$
$97$	−0.682333	−0.0692804	−0.0346402	+	0.999400i	$0.511029\pi$
$98$	0	0
$99$	−3.80194	−0.382109
$100$	0	0
$101$	6.75063	0.671713	0.335856	−	0.941913i	$-0.390974\pi$
$101$	6.75063	0.671713	0.335856	+	0.941913i	$0.390974\pi$
$102$	0	0
$103$	7.78017	0.766603	0.383301	−	0.923623i	$-0.374787\pi$
$103$	7.78017	0.766603	0.383301	+	0.923623i	$0.374787\pi$
$104$	0	0
$105$	−15.3937	−1.50227
$106$	0	0
$107$	8.51573	0.823247	0.411623	−	0.911354i	$-0.364962\pi$
$107$	8.51573	0.823247	0.411623	+	0.911354i	$0.364962\pi$
$108$	0	0
$109$	−1.08815	−0.104225	−0.0521127	−	0.998641i	$-0.516596\pi$
$109$	−1.08815	−0.104225	−0.0521127	+	0.998641i	$0.516596\pi$
$110$	0	0
$111$	0.0271471	0.00257669
$112$	0	0
$113$	3.67456	0.345674	0.172837	−	0.984950i	$-0.444707\pi$
$113$	3.67456	0.345674	0.172837	+	0.984950i	$0.444707\pi$
$114$	0	0
$115$	−24.2054	−2.25716
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.80194	0.165183
$120$	0	0
$121$	3.45473	0.314066
$122$	0	0
$123$	−9.18598	−0.828272
$124$	0	0
$125$	−2.14675	−0.192011
$126$	0	0
$127$	11.8116	1.04811	0.524056	−	0.851684i	$-0.324418\pi$
$127$	11.8116	1.04811	0.524056	+	0.851684i	$0.324418\pi$
$128$	0	0
$129$	−1.21983	−0.107400
$130$	0	0
$131$	12.5579	1.09719	0.548596	−	0.836087i	$-0.315162\pi$
$131$	12.5579	1.09719	0.548596	+	0.836087i	$0.315162\pi$
$132$	0	0
$133$	30.9855	2.68679
$134$	0	0
$135$	3.04892	0.262409
$136$	0	0
$137$	−6.65817	−0.568846	−0.284423	−	0.958699i	$-0.591802\pi$
$137$	−6.65817	−0.568846	−0.284423	+	0.958699i	$0.591802\pi$
$138$	0	0
$139$	−6.17390	−0.523663	−0.261832	−	0.965114i	$-0.584327\pi$
$139$	−6.17390	−0.523663	−0.261832	+	0.965114i	$0.584327\pi$
$140$	0	0
$141$	−9.56465	−0.805489
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.32304	0.359009
$146$	0	0
$147$	18.4916	1.52516
$148$	0	0
$149$	−0.146752	−0.0120224	−0.00601120	−	0.999982i	$-0.501913\pi$
$149$	−0.146752	−0.0120224	−0.00601120	+	0.999982i	$0.501913\pi$
$150$	0	0
$151$	9.36658	0.762242	0.381121	−	0.924525i	$-0.375538\pi$
$151$	9.36658	0.762242	0.381121	+	0.924525i	$0.375538\pi$
$152$	0	0
$153$	−0.356896	−0.0288533
$154$	0	0
$155$	17.6528	1.41791
$156$	0	0
$157$	−4.03146	−0.321745	−0.160873	−	0.986975i	$-0.551431\pi$
$157$	−4.03146	−0.321745	−0.160873	+	0.986975i	$0.551431\pi$
$158$	0	0
$159$	−4.73556	−0.375554
$160$	0	0
$161$	40.0834	3.15901
$162$	0	0
$163$	10.4209	0.816226	0.408113	−	0.912931i	$-0.366187\pi$
$163$	10.4209	0.816226	0.408113	+	0.912931i	$0.366187\pi$
$164$	0	0
$165$	−11.5918	−0.902420
$166$	0	0
$167$	10.3056	0.797470	0.398735	−	0.917066i	$-0.369449\pi$
$167$	10.3056	0.797470	0.398735	+	0.917066i	$0.369449\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−6.13706	−0.469313
$172$	0	0
$173$	−22.3032	−1.69568	−0.847840	−	0.530252i	$-0.822097\pi$
$173$	−22.3032	−1.69568	−0.847840	+	0.530252i	$0.822097\pi$
$174$	0	0
$175$	−21.6896	−1.63958
$176$	0	0
$177$	13.5254	1.01663
$178$	0	0
$179$	−17.3773	−1.29884	−0.649422	−	0.760429i	$-0.724989\pi$
$179$	−17.3773	−1.29884	−0.649422	+	0.760429i	$0.724989\pi$
$180$	0	0
$181$	−24.4644	−1.81843	−0.909213	−	0.416331i	$-0.863316\pi$
$181$	−24.4644	−1.81843	−0.909213	+	0.416331i	$0.863316\pi$
$182$	0	0
$183$	0.576728	0.0426330
$184$	0	0
$185$	0.0827692	0.00608531
$186$	0	0
$187$	1.35690	0.0992261
$188$	0	0
$189$	−5.04892	−0.367255
$190$	0	0
$191$	21.4088	1.54909	0.774543	−	0.632521i	$-0.217980\pi$
$191$	21.4088	1.54909	0.774543	+	0.632521i	$0.217980\pi$
$192$	0	0
$193$	−14.4276	−1.03852	−0.519260	−	0.854616i	$-0.673792\pi$
$193$	−14.4276	−1.03852	−0.519260	+	0.854616i	$0.673792\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.04652	−0.145809	−0.0729044	−	0.997339i	$-0.523227\pi$
$197$	−2.04652	−0.145809	−0.0729044	+	0.997339i	$0.523227\pi$
$198$	0	0
$199$	−5.55496	−0.393781	−0.196890	−	0.980426i	$-0.563084\pi$
$199$	−5.55496	−0.393781	−0.196890	+	0.980426i	$0.563084\pi$
$200$	0	0
$201$	−7.95108	−0.560826
$202$	0	0
$203$	−7.15883	−0.502452
$204$	0	0
$205$	−28.0073	−1.95611
$206$	0	0
$207$	−7.93900	−0.551799
$208$	0	0
$209$	23.3327	1.61396
$210$	0	0
$211$	−21.8092	−1.50141	−0.750705	−	0.660638i	$-0.770286\pi$
$211$	−21.8092	−1.50141	−0.750705	+	0.660638i	$0.770286\pi$
$212$	0	0
$213$	−3.07606	−0.210768
$214$	0	0
$215$	−3.71917	−0.253645
$216$	0	0
$217$	−29.2325	−1.98443
$218$	0	0
$219$	−14.0368	−0.948521
$220$	0	0
$221$	0	0
$222$	0	0
$223$	25.6015	1.71440	0.857201	−	0.514982i	$-0.172201\pi$
$223$	25.6015	1.71440	0.857201	+	0.514982i	$0.172201\pi$
$224$	0	0
$225$	4.29590	0.286393
$226$	0	0
$227$	2.72587	0.180923	0.0904613	−	0.995900i	$-0.471166\pi$
$227$	2.72587	0.180923	0.0904613	+	0.995900i	$0.471166\pi$
$228$	0	0
$229$	−18.1075	−1.19658	−0.598289	−	0.801280i	$-0.704153\pi$
$229$	−18.1075	−1.19658	−0.598289	+	0.801280i	$0.704153\pi$
$230$	0	0
$231$	19.1957	1.26298
$232$	0	0
$233$	−18.9051	−1.23852	−0.619259	−	0.785187i	$-0.712567\pi$
$233$	−18.9051	−1.23852	−0.619259	+	0.785187i	$0.712567\pi$
$234$	0	0
$235$	−29.1618	−1.90231
$236$	0	0
$237$	−0.841166	−0.0546396
$238$	0	0
$239$	−20.3099	−1.31374	−0.656869	−	0.754005i	$-0.728120\pi$
$239$	−20.3099	−1.31374	−0.656869	+	0.754005i	$0.728120\pi$
$240$	0	0
$241$	−15.7942	−1.01739	−0.508696	−	0.860946i	$-0.669872\pi$
$241$	−15.7942	−1.01739	−0.508696	+	0.860946i	$0.669872\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	56.3793	3.60194
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.192685	−0.0122109
$250$	0	0
$251$	−7.88231	−0.497527	−0.248764	−	0.968564i	$-0.580024\pi$
$251$	−7.88231	−0.497527	−0.248764	+	0.968564i	$0.580024\pi$
$252$	0	0
$253$	30.1836	1.89763
$254$	0	0
$255$	−1.08815	−0.0681423
$256$	0	0
$257$	18.5972	1.16006	0.580030	−	0.814595i	$-0.303041\pi$
$257$	18.5972	1.16006	0.580030	+	0.814595i	$0.303041\pi$
$258$	0	0
$259$	−0.137063	−0.00851670
$260$	0	0
$261$	1.41789	0.0877655
$262$	0	0
$263$	1.25667	0.0774895	0.0387447	−	0.999249i	$-0.487664\pi$
$263$	1.25667	0.0774895	0.0387447	+	0.999249i	$0.487664\pi$
$264$	0	0
$265$	−14.4383	−0.886940
$266$	0	0
$267$	9.21313	0.563834
$268$	0	0
$269$	−18.8944	−1.15201	−0.576006	−	0.817446i	$-0.695389\pi$
$269$	−18.8944	−1.15201	−0.576006	+	0.817446i	$0.695389\pi$
$270$	0	0
$271$	20.6082	1.25186	0.625929	−	0.779880i	$-0.284720\pi$
$271$	20.6082	1.25186	0.625929	+	0.779880i	$0.284720\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−16.3327	−0.984901
$276$	0	0
$277$	11.8890	0.714342	0.357171	−	0.934039i	$-0.383741\pi$
$277$	11.8890	0.714342	0.357171	+	0.934039i	$0.383741\pi$
$278$	0	0
$279$	5.78986	0.346630
$280$	0	0
$281$	7.74333	0.461928	0.230964	−	0.972962i	$-0.425812\pi$
$281$	7.74333	0.461928	0.230964	+	0.972962i	$0.425812\pi$
$282$	0	0
$283$	−12.2349	−0.727289	−0.363645	−	0.931538i	$-0.618468\pi$
$283$	−12.2349	−0.727289	−0.363645	+	0.931538i	$0.618468\pi$
$284$	0	0
$285$	−18.7114	−1.10837
$286$	0	0
$287$	46.3793	2.73768
$288$	0	0
$289$	−16.8726	−0.992507
$290$	0	0
$291$	−0.682333	−0.0399991
$292$	0	0
$293$	18.5985	1.08654	0.543268	−	0.839559i	$-0.317187\pi$
$293$	18.5985	1.08654	0.543268	+	0.839559i	$0.317187\pi$
$294$	0	0
$295$	41.2379	2.40096
$296$	0	0
$297$	−3.80194	−0.220611
$298$	0	0
$299$	0	0
$300$	0	0
$301$	6.15883	0.354989
$302$	0	0
$303$	6.75063	0.387813
$304$	0	0
$305$	1.75840	0.100686
$306$	0	0
$307$	−3.32006	−0.189486	−0.0947429	−	0.995502i	$-0.530203\pi$
$307$	−3.32006	−0.189486	−0.0947429	+	0.995502i	$0.530203\pi$
$308$	0	0
$309$	7.78017	0.442598
$310$	0	0
$311$	20.2881	1.15043	0.575217	−	0.818001i	$-0.304918\pi$
$311$	20.2881	1.15043	0.575217	+	0.818001i	$0.304918\pi$
$312$	0	0
$313$	−14.3870	−0.813203	−0.406601	−	0.913606i	$-0.633286\pi$
$313$	−14.3870	−0.813203	−0.406601	+	0.913606i	$0.633286\pi$
$314$	0	0
$315$	−15.3937	−0.867339
$316$	0	0
$317$	12.4450	0.698983	0.349492	−	0.936939i	$-0.386354\pi$
$317$	12.4450	0.698983	0.349492	+	0.936939i	$0.386354\pi$
$318$	0	0
$319$	−5.39075	−0.301824
$320$	0	0
$321$	8.51573	0.475302
$322$	0	0
$323$	2.19029	0.121871
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.08815	−0.0601746
$328$	0	0
$329$	48.2911	2.66238
$330$	0	0
$331$	3.31873	0.182414	0.0912070	−	0.995832i	$-0.470928\pi$
$331$	3.31873	0.182414	0.0912070	+	0.995832i	$0.470928\pi$
$332$	0	0
$333$	0.0271471	0.00148765
$334$	0	0
$335$	−24.2422	−1.32449
$336$	0	0
$337$	−12.3515	−0.672830	−0.336415	−	0.941714i	$-0.609214\pi$
$337$	−12.3515	−0.672830	−0.336415	+	0.941714i	$0.609214\pi$
$338$	0	0
$339$	3.67456	0.199575
$340$	0	0
$341$	−22.0127	−1.19205
$342$	0	0
$343$	−58.0200	−3.13278
$344$	0	0
$345$	−24.2054	−1.30317
$346$	0	0
$347$	10.4969	0.563505	0.281753	−	0.959487i	$-0.409084\pi$
$347$	10.4969	0.563505	0.281753	+	0.959487i	$0.409084\pi$
$348$	0	0
$349$	−14.1086	−0.755215	−0.377608	−	0.925966i	$-0.623253\pi$
$349$	−14.1086	−0.755215	−0.377608	+	0.925966i	$0.623253\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.5187	−0.772753	−0.386377	−	0.922341i	$-0.626273\pi$
$353$	−14.5187	−0.772753	−0.386377	+	0.922341i	$0.626273\pi$
$354$	0	0
$355$	−9.37867	−0.497768
$356$	0	0
$357$	1.80194	0.0953687
$358$	0	0
$359$	24.7144	1.30438	0.652188	−	0.758058i	$-0.273851\pi$
$359$	24.7144	1.30438	0.652188	+	0.758058i	$0.273851\pi$
$360$	0	0
$361$	18.6635	0.982292
$362$	0	0
$363$	3.45473	0.181326
$364$	0	0
$365$	−42.7972	−2.24011
$366$	0	0
$367$	19.5066	1.01824	0.509119	−	0.860696i	$-0.329971\pi$
$367$	19.5066	1.01824	0.509119	+	0.860696i	$0.329971\pi$
$368$	0	0
$369$	−9.18598	−0.478203
$370$	0	0
$371$	23.9095	1.24132
$372$	0	0
$373$	−3.80194	−0.196857	−0.0984284	−	0.995144i	$-0.531382\pi$
$373$	−3.80194	−0.196857	−0.0984284	+	0.995144i	$0.531382\pi$
$374$	0	0
$375$	−2.14675	−0.110858
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−2.20344	−0.113183	−0.0565915	−	0.998397i	$-0.518023\pi$
$379$	−2.20344	−0.113183	−0.0565915	+	0.998397i	$0.518023\pi$
$380$	0	0
$381$	11.8116	0.605128
$382$	0	0
$383$	30.9691	1.58245	0.791224	−	0.611526i	$-0.209444\pi$
$383$	30.9691	1.58245	0.791224	+	0.611526i	$0.209444\pi$
$384$	0	0
$385$	58.5260	2.98276
$386$	0	0
$387$	−1.21983	−0.0620076
$388$	0	0
$389$	24.4470	1.23951	0.619755	−	0.784795i	$-0.287232\pi$
$389$	24.4470	1.23951	0.619755	+	0.784795i	$0.287232\pi$
$390$	0	0
$391$	2.83340	0.143291
$392$	0	0
$393$	12.5579	0.633464
$394$	0	0
$395$	−2.56465	−0.129041
$396$	0	0
$397$	−0.728857	−0.0365803	−0.0182901	−	0.999833i	$-0.505822\pi$
$397$	−0.728857	−0.0365803	−0.0182901	+	0.999833i	$0.505822\pi$
$398$	0	0
$399$	30.9855	1.55122
$400$	0	0
$401$	−10.2597	−0.512343	−0.256171	−	0.966631i	$-0.582461\pi$
$401$	−10.2597	−0.512343	−0.256171	+	0.966631i	$0.582461\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.04892	0.151502
$406$	0	0
$407$	−0.103211	−0.00511600
$408$	0	0
$409$	22.7700	1.12590	0.562952	−	0.826489i	$-0.309666\pi$
$409$	22.7700	1.12590	0.562952	+	0.826489i	$0.309666\pi$
$410$	0	0
$411$	−6.65817	−0.328423
$412$	0	0
$413$	−68.2887	−3.36027
$414$	0	0
$415$	−0.587482	−0.0288384
$416$	0	0
$417$	−6.17390	−0.302337
$418$	0	0
$419$	19.0930	0.932756	0.466378	−	0.884585i	$-0.345559\pi$
$419$	19.0930	0.932756	0.466378	+	0.884585i	$0.345559\pi$
$420$	0	0
$421$	15.8629	0.773112	0.386556	−	0.922266i	$-0.373665\pi$
$421$	15.8629	0.773112	0.386556	+	0.922266i	$0.373665\pi$
$422$	0	0
$423$	−9.56465	−0.465049
$424$	0	0
$425$	−1.53319	−0.0743705
$426$	0	0
$427$	−2.91185	−0.140914
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.7144	−1.19045	−0.595225	−	0.803559i	$-0.702937\pi$
$431$	−24.7144	−1.19045	−0.595225	+	0.803559i	$0.702937\pi$
$432$	0	0
$433$	16.2000	0.778521	0.389261	−	0.921128i	$-0.372731\pi$
$433$	16.2000	0.778521	0.389261	+	0.921128i	$0.372731\pi$
$434$	0	0
$435$	4.32304	0.207274
$436$	0	0
$437$	48.7222	2.33070
$438$	0	0
$439$	−22.0049	−1.05024	−0.525118	−	0.851029i	$-0.675979\pi$
$439$	−22.0049	−1.05024	−0.525118	+	0.851029i	$0.675979\pi$
$440$	0	0
$441$	18.4916	0.880551
$442$	0	0
$443$	4.33273	0.205854	0.102927	−	0.994689i	$-0.467179\pi$
$443$	4.33273	0.205854	0.102927	+	0.994689i	$0.467179\pi$
$444$	0	0
$445$	28.0901	1.33160
$446$	0	0
$447$	−0.146752	−0.00694113
$448$	0	0
$449$	16.0935	0.759500	0.379750	−	0.925089i	$-0.376010\pi$
$449$	16.0935	0.759500	0.379750	+	0.925089i	$0.376010\pi$
$450$	0	0
$451$	34.9245	1.64453
$452$	0	0
$453$	9.36658	0.440081
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.08383	0.0506996	0.0253498	−	0.999679i	$-0.491930\pi$
$457$	1.08383	0.0506996	0.0253498	+	0.999679i	$0.491930\pi$
$458$	0	0
$459$	−0.356896	−0.0166585
$460$	0	0
$461$	34.1642	1.59119	0.795593	−	0.605832i	$-0.207159\pi$
$461$	34.1642	1.59119	0.795593	+	0.605832i	$0.207159\pi$
$462$	0	0
$463$	−23.7560	−1.10404	−0.552018	−	0.833832i	$-0.686142\pi$
$463$	−23.7560	−1.10404	−0.552018	+	0.833832i	$0.686142\pi$
$464$	0	0
$465$	17.6528	0.818629
$466$	0	0
$467$	−42.8189	−1.98142	−0.990712	−	0.135979i	$-0.956582\pi$
$467$	−42.8189	−1.98142	−0.990712	+	0.135979i	$0.956582\pi$
$468$	0	0
$469$	40.1444	1.85369
$470$	0	0
$471$	−4.03146	−0.185760
$472$	0	0
$473$	4.63773	0.213243
$474$	0	0
$475$	−26.3642	−1.20967
$476$	0	0
$477$	−4.73556	−0.216826
$478$	0	0
$479$	−5.66594	−0.258883	−0.129442	−	0.991587i	$-0.541319\pi$
$479$	−5.66594	−0.258883	−0.129442	+	0.991587i	$0.541319\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	40.0834	1.82386
$484$	0	0
$485$	−2.08038	−0.0944650
$486$	0	0
$487$	−12.2513	−0.555159	−0.277580	−	0.960703i	$-0.589532\pi$
$487$	−12.2513	−0.555159	−0.277580	+	0.960703i	$0.589532\pi$
$488$	0	0
$489$	10.4209	0.471248
$490$	0	0
$491$	−4.96615	−0.224119	−0.112060	−	0.993701i	$-0.535745\pi$
$491$	−4.96615	−0.224119	−0.112060	+	0.993701i	$0.535745\pi$
$492$	0	0
$493$	−0.506041	−0.0227909
$494$	0	0
$495$	−11.5918	−0.521012
$496$	0	0
$497$	15.5308	0.696651
$498$	0	0
$499$	16.2500	0.727448	0.363724	−	0.931507i	$-0.381505\pi$
$499$	16.2500	0.727448	0.363724	+	0.931507i	$0.381505\pi$
$500$	0	0
$501$	10.3056	0.460420
$502$	0	0
$503$	−12.2459	−0.546018	−0.273009	−	0.962011i	$-0.588019\pi$
$503$	−12.2459	−0.546018	−0.273009	+	0.962011i	$0.588019\pi$
$504$	0	0
$505$	20.5821	0.915892
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.84787	0.170554	0.0852769	−	0.996357i	$-0.472822\pi$
$509$	3.84787	0.170554	0.0852769	+	0.996357i	$0.472822\pi$
$510$	0	0
$511$	70.8708	3.13514
$512$	0	0
$513$	−6.13706	−0.270958
$514$	0	0
$515$	23.7211	1.04528
$516$	0	0
$517$	36.3642	1.59930
$518$	0	0
$519$	−22.3032	−0.979002
$520$	0	0
$521$	−9.42626	−0.412972	−0.206486	−	0.978450i	$-0.566203\pi$
$521$	−9.42626	−0.412972	−0.206486	+	0.978450i	$0.566203\pi$
$522$	0	0
$523$	−40.5937	−1.77504	−0.887520	−	0.460770i	$-0.847573\pi$
$523$	−40.5937	−1.77504	−0.887520	+	0.460770i	$0.847573\pi$
$524$	0	0
$525$	−21.6896	−0.946613
$526$	0	0
$527$	−2.06638	−0.0900127
$528$	0	0
$529$	40.0277	1.74034
$530$	0	0
$531$	13.5254	0.586953
$532$	0	0
$533$	0	0
$534$	0	0
$535$	25.9638	1.12251
$536$	0	0
$537$	−17.3773	−0.749887
$538$	0	0
$539$	−70.3038	−3.02820
$540$	0	0
$541$	4.32006	0.185734	0.0928669	−	0.995679i	$-0.470397\pi$
$541$	4.32006	0.185734	0.0928669	+	0.995679i	$0.470397\pi$
$542$	0	0
$543$	−24.4644	−1.04987
$544$	0	0
$545$	−3.31767	−0.142113
$546$	0	0
$547$	35.8702	1.53370	0.766850	−	0.641826i	$-0.221823\pi$
$547$	35.8702	1.53370	0.766850	+	0.641826i	$0.221823\pi$
$548$	0	0
$549$	0.576728	0.0246142
$550$	0	0
$551$	−8.70171	−0.370705
$552$	0	0
$553$	4.24698	0.180600
$554$	0	0
$555$	0.0827692	0.00351336
$556$	0	0
$557$	30.4795	1.29146	0.645729	−	0.763567i	$-0.276554\pi$
$557$	30.4795	1.29146	0.645729	+	0.763567i	$0.276554\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.35690	0.0572882
$562$	0	0
$563$	−32.7711	−1.38114	−0.690568	−	0.723268i	$-0.742639\pi$
$563$	−32.7711	−1.38114	−0.690568	+	0.723268i	$0.742639\pi$
$564$	0	0
$565$	11.2034	0.471332
$566$	0	0
$567$	−5.04892	−0.212035
$568$	0	0
$569$	−2.91185	−0.122071	−0.0610356	−	0.998136i	$-0.519440\pi$
$569$	−2.91185	−0.122071	−0.0610356	+	0.998136i	$0.519440\pi$
$570$	0	0
$571$	−22.8592	−0.956628	−0.478314	−	0.878189i	$-0.658752\pi$
$571$	−22.8592	−0.956628	−0.478314	+	0.878189i	$0.658752\pi$
$572$	0	0
$573$	21.4088	0.894365
$574$	0	0
$575$	−34.1051	−1.42228
$576$	0	0
$577$	−7.78315	−0.324017	−0.162008	−	0.986789i	$-0.551797\pi$
$577$	−7.78315	−0.324017	−0.162008	+	0.986789i	$0.551797\pi$
$578$	0	0
$579$	−14.4276	−0.599590
$580$	0	0
$581$	0.972853	0.0403607
$582$	0	0
$583$	18.0043	0.745662
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.70304	−0.111566	−0.0557832	−	0.998443i	$-0.517766\pi$
$587$	−2.70304	−0.111566	−0.0557832	+	0.998443i	$0.517766\pi$
$588$	0	0
$589$	−35.5327	−1.46410
$590$	0	0
$591$	−2.04652	−0.0841828
$592$	0	0
$593$	−38.8961	−1.59727	−0.798635	−	0.601816i	$-0.794444\pi$
$593$	−38.8961	−1.59727	−0.798635	+	0.601816i	$0.794444\pi$
$594$	0	0
$595$	5.49396	0.225230
$596$	0	0
$597$	−5.55496	−0.227349
$598$	0	0
$599$	13.5308	0.552853	0.276427	−	0.961035i	$-0.410850\pi$
$599$	13.5308	0.552853	0.276427	+	0.961035i	$0.410850\pi$
$600$	0	0
$601$	−26.2513	−1.07081	−0.535406	−	0.844595i	$-0.679841\pi$
$601$	−26.2513	−1.07081	−0.535406	+	0.844595i	$0.679841\pi$
$602$	0	0
$603$	−7.95108	−0.323793
$604$	0	0
$605$	10.5332	0.428235
$606$	0	0
$607$	−38.1183	−1.54717	−0.773587	−	0.633691i	$-0.781539\pi$
$607$	−38.1183	−1.54717	−0.773587	+	0.633691i	$0.781539\pi$
$608$	0	0
$609$	−7.15883	−0.290091
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−19.7918	−0.799382	−0.399691	−	0.916650i	$-0.630883\pi$
$613$	−19.7918	−0.799382	−0.399691	+	0.916650i	$0.630883\pi$
$614$	0	0
$615$	−28.0073	−1.12936
$616$	0	0
$617$	48.8068	1.96489	0.982445	−	0.186554i	$-0.0597319\pi$
$617$	48.8068	1.96489	0.982445	+	0.186554i	$0.0597319\pi$
$618$	0	0
$619$	24.8702	0.999619	0.499810	−	0.866135i	$-0.333403\pi$
$619$	24.8702	0.999619	0.499810	+	0.866135i	$0.333403\pi$
$620$	0	0
$621$	−7.93900	−0.318581
$622$	0	0
$623$	−46.5163	−1.86364
$624$	0	0
$625$	−28.0248	−1.12099
$626$	0	0
$627$	23.3327	0.931820
$628$	0	0
$629$	−0.00968868	−0.000386313 0
$630$	0	0
$631$	46.8689	1.86582	0.932911	−	0.360108i	$-0.117260\pi$
$631$	46.8689	1.86582	0.932911	+	0.360108i	$0.117260\pi$
$632$	0	0
$633$	−21.8092	−0.866839
$634$	0	0
$635$	36.0127	1.42912
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.07606	−0.121687
$640$	0	0
$641$	18.1105	0.715322	0.357661	−	0.933851i	$-0.383574\pi$
$641$	18.1105	0.715322	0.357661	+	0.933851i	$0.383574\pi$
$642$	0	0
$643$	−23.5319	−0.928006	−0.464003	−	0.885834i	$-0.653587\pi$
$643$	−23.5319	−0.928006	−0.464003	+	0.885834i	$0.653587\pi$
$644$	0	0
$645$	−3.71917	−0.146442
$646$	0	0
$647$	4.04593	0.159062	0.0795310	−	0.996832i	$-0.474658\pi$
$647$	4.04593	0.159062	0.0795310	+	0.996832i	$0.474658\pi$
$648$	0	0
$649$	−51.4228	−2.01852
$650$	0	0
$651$	−29.2325	−1.14571
$652$	0	0
$653$	10.2252	0.400143	0.200072	−	0.979781i	$-0.435882\pi$
$653$	10.2252	0.400143	0.200072	+	0.979781i	$0.435882\pi$
$654$	0	0
$655$	38.2881	1.49604
$656$	0	0
$657$	−14.0368	−0.547629
$658$	0	0
$659$	−25.8756	−1.00797	−0.503985	−	0.863712i	$-0.668133\pi$
$659$	−25.8756	−1.00797	−0.503985	+	0.863712i	$0.668133\pi$
$660$	0	0
$661$	7.34588	0.285722	0.142861	−	0.989743i	$-0.454370\pi$
$661$	7.34588	0.285722	0.142861	+	0.989743i	$0.454370\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	94.4723	3.66348
$666$	0	0
$667$	−11.2567	−0.435860
$668$	0	0
$669$	25.6015	0.989811
$670$	0	0
$671$	−2.19269	−0.0846477
$672$	0	0
$673$	34.8334	1.34273	0.671364	−	0.741127i	$-0.265709\pi$
$673$	34.8334	1.34273	0.671364	+	0.741127i	$0.265709\pi$
$674$	0	0
$675$	4.29590	0.165349
$676$	0	0
$677$	−17.2000	−0.661049	−0.330524	−	0.943797i	$-0.607226\pi$
$677$	−17.2000	−0.661049	−0.330524	+	0.943797i	$0.607226\pi$
$678$	0	0
$679$	3.44504	0.132209
$680$	0	0
$681$	2.72587	0.104456
$682$	0	0
$683$	−33.8412	−1.29490	−0.647448	−	0.762110i	$-0.724164\pi$
$683$	−33.8412	−1.29490	−0.647448	+	0.762110i	$0.724164\pi$
$684$	0	0
$685$	−20.3002	−0.775631
$686$	0	0
$687$	−18.1075	−0.690845
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−17.7101	−0.673723	−0.336861	−	0.941554i	$-0.609365\pi$
$691$	−17.7101	−0.673723	−0.336861	+	0.941554i	$0.609365\pi$
$692$	0	0
$693$	19.1957	0.729183
$694$	0	0
$695$	−18.8237	−0.714024
$696$	0	0
$697$	3.27844	0.124180
$698$	0	0
$699$	−18.9051	−0.715058
$700$	0	0
$701$	9.22952	0.348594	0.174297	−	0.984693i	$-0.444235\pi$
$701$	9.22952	0.348594	0.174297	+	0.984693i	$0.444235\pi$
$702$	0	0
$703$	−0.166603	−0.00628356
$704$	0	0
$705$	−29.1618	−1.09830
$706$	0	0
$707$	−34.0834	−1.28184
$708$	0	0
$709$	−0.636399	−0.0239005	−0.0119502	−	0.999929i	$-0.503804\pi$
$709$	−0.636399	−0.0239005	−0.0119502	+	0.999929i	$0.503804\pi$
$710$	0	0
$711$	−0.841166	−0.0315462
$712$	0	0
$713$	−45.9657	−1.72143
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−20.3099	−0.758487
$718$	0	0
$719$	8.43967	0.314746	0.157373	−	0.987539i	$-0.449697\pi$
$719$	8.43967	0.314746	0.157373	+	0.987539i	$0.449697\pi$
$720$	0	0
$721$	−39.2814	−1.46292
$722$	0	0
$723$	−15.7942	−0.587391
$724$	0	0
$725$	6.09113	0.226219
$726$	0	0
$727$	−19.9922	−0.741471	−0.370735	−	0.928739i	$-0.620894\pi$
$727$	−19.9922	−0.741471	−0.370735	+	0.928739i	$0.620894\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.435353	0.0161021
$732$	0	0
$733$	3.93767	0.145441	0.0727206	−	0.997352i	$-0.476832\pi$
$733$	3.93767	0.145441	0.0727206	+	0.997352i	$0.476832\pi$
$734$	0	0
$735$	56.3793	2.07958
$736$	0	0
$737$	30.2295	1.11352
$738$	0	0
$739$	−23.7530	−0.873769	−0.436884	−	0.899518i	$-0.643918\pi$
$739$	−23.7530	−0.873769	−0.436884	+	0.899518i	$0.643918\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−37.0707	−1.35999	−0.679996	−	0.733216i	$-0.738018\pi$
$743$	−37.0707	−1.35999	−0.679996	+	0.733216i	$0.738018\pi$
$744$	0	0
$745$	−0.447435	−0.0163927
$746$	0	0
$747$	−0.192685	−0.00704999
$748$	0	0
$749$	−42.9952	−1.57101
$750$	0	0
$751$	−10.5597	−0.385331	−0.192665	−	0.981265i	$-0.561713\pi$
$751$	−10.5597	−0.385331	−0.192665	+	0.981265i	$0.561713\pi$
$752$	0	0
$753$	−7.88231	−0.287247
$754$	0	0
$755$	28.5579	1.03933
$756$	0	0
$757$	−2.90323	−0.105520	−0.0527598	−	0.998607i	$-0.516802\pi$
$757$	−2.90323	−0.105520	−0.0527598	+	0.998607i	$0.516802\pi$
$758$	0	0
$759$	30.1836	1.09559
$760$	0	0
$761$	23.2239	0.841865	0.420933	−	0.907092i	$-0.361703\pi$
$761$	23.2239	0.841865	0.420933	+	0.907092i	$0.361703\pi$
$762$	0	0
$763$	5.49396	0.198895
$764$	0	0
$765$	−1.08815	−0.0393420
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−39.7259	−1.43255	−0.716276	−	0.697817i	$-0.754155\pi$
$769$	−39.7259	−1.43255	−0.716276	+	0.697817i	$0.754155\pi$
$770$	0	0
$771$	18.5972	0.669761
$772$	0	0
$773$	3.17092	0.114050	0.0570249	−	0.998373i	$-0.481839\pi$
$773$	3.17092	0.114050	0.0570249	+	0.998373i	$0.481839\pi$
$774$	0	0
$775$	24.8726	0.893451
$776$	0	0
$777$	−0.137063	−0.00491712
$778$	0	0
$779$	56.3749	2.01984
$780$	0	0
$781$	11.6950	0.418480
$782$	0	0
$783$	1.41789	0.0506714
$784$	0	0
$785$	−12.2916	−0.438705
$786$	0	0
$787$	−7.49502	−0.267169	−0.133584	−	0.991037i	$-0.542649\pi$
$787$	−7.49502	−0.267169	−0.133584	+	0.991037i	$0.542649\pi$
$788$	0	0
$789$	1.25667	0.0447386
$790$	0	0
$791$	−18.5526	−0.659653
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−14.4383	−0.512075
$796$	0	0
$797$	−39.6775	−1.40545	−0.702725	−	0.711461i	$-0.748034\pi$
$797$	−39.6775	−1.40545	−0.702725	+	0.711461i	$0.748034\pi$
$798$	0	0
$799$	3.41358	0.120764
$800$	0	0
$801$	9.21313	0.325530
$802$	0	0
$803$	53.3672	1.88329
$804$	0	0
$805$	122.211	4.30737
$806$	0	0
$807$	−18.8944	−0.665114
$808$	0	0
$809$	50.0180	1.75854	0.879270	−	0.476323i	$-0.158031\pi$
$809$	50.0180	1.75854	0.879270	+	0.476323i	$0.158031\pi$
$810$	0	0
$811$	28.7590	1.00986	0.504932	−	0.863159i	$-0.331517\pi$
$811$	28.7590	1.00986	0.504932	+	0.863159i	$0.331517\pi$
$812$	0	0
$813$	20.6082	0.722761
$814$	0	0
$815$	31.7724	1.11294
$816$	0	0
$817$	7.48619	0.261909
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.7200	1.35134	0.675669	−	0.737205i	$-0.263855\pi$
$821$	38.7200	1.35134	0.675669	+	0.737205i	$0.263855\pi$
$822$	0	0
$823$	−3.09246	−0.107796	−0.0538982	−	0.998546i	$-0.517165\pi$
$823$	−3.09246	−0.107796	−0.0538982	+	0.998546i	$0.517165\pi$
$824$	0	0
$825$	−16.3327	−0.568633
$826$	0	0
$827$	4.80194	0.166980	0.0834899	−	0.996509i	$-0.473393\pi$
$827$	4.80194	0.166980	0.0834899	+	0.996509i	$0.473393\pi$
$828$	0	0
$829$	−20.9812	−0.728708	−0.364354	−	0.931261i	$-0.618710\pi$
$829$	−20.9812	−0.728708	−0.364354	+	0.931261i	$0.618710\pi$
$830$	0	0
$831$	11.8890	0.412425
$832$	0	0
$833$	−6.59956	−0.228661
$834$	0	0
$835$	31.4209	1.08736
$836$	0	0
$837$	5.78986	0.200127
$838$	0	0
$839$	−33.4263	−1.15400	−0.577001	−	0.816743i	$-0.695777\pi$
$839$	−33.4263	−1.15400	−0.577001	+	0.816743i	$0.695777\pi$
$840$	0	0
$841$	−26.9896	−0.930675
$842$	0	0
$843$	7.74333	0.266695
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−17.4426	−0.599337
$848$	0	0
$849$	−12.2349	−0.419901
$850$	0	0
$851$	−0.215521	−0.00738795
$852$	0	0
$853$	51.1527	1.75144	0.875718	−	0.482823i	$-0.160389\pi$
$853$	51.1527	1.75144	0.875718	+	0.482823i	$0.160389\pi$
$854$	0	0
$855$	−18.7114	−0.639916
$856$	0	0
$857$	12.7004	0.433837	0.216918	−	0.976190i	$-0.430399\pi$
$857$	12.7004	0.433837	0.216918	+	0.976190i	$0.430399\pi$
$858$	0	0
$859$	−34.0455	−1.16162	−0.580808	−	0.814041i	$-0.697263\pi$
$859$	−34.0455	−1.16162	−0.580808	+	0.814041i	$0.697263\pi$
$860$	0	0
$861$	46.3793	1.58060
$862$	0	0
$863$	23.8775	0.812801	0.406400	−	0.913695i	$-0.366784\pi$
$863$	23.8775	0.812801	0.406400	+	0.913695i	$0.366784\pi$
$864$	0	0
$865$	−68.0006	−2.31209
$866$	0	0
$867$	−16.8726	−0.573024
$868$	0	0
$869$	3.19806	0.108487
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−0.682333	−0.0230935
$874$	0	0
$875$	10.8388	0.366417
$876$	0	0
$877$	−15.5047	−0.523557	−0.261778	−	0.965128i	$-0.584309\pi$
$877$	−15.5047	−0.523557	−0.261778	+	0.965128i	$0.584309\pi$
$878$	0	0
$879$	18.5985	0.627312
$880$	0	0
$881$	41.9318	1.41272	0.706359	−	0.707853i	$-0.250336\pi$
$881$	41.9318	1.41272	0.706359	+	0.707853i	$0.250336\pi$
$882$	0	0
$883$	58.8219	1.97951	0.989757	−	0.142760i	$-0.0455976\pi$
$883$	58.8219	1.97951	0.989757	+	0.142760i	$0.0455976\pi$
$884$	0	0
$885$	41.2379	1.38620
$886$	0	0
$887$	10.2924	0.345586	0.172793	−	0.984958i	$-0.444721\pi$
$887$	10.2924	0.345586	0.172793	+	0.984958i	$0.444721\pi$
$888$	0	0
$889$	−59.6359	−2.00012
$890$	0	0
$891$	−3.80194	−0.127370
$892$	0	0
$893$	58.6988	1.96428
$894$	0	0
$895$	−52.9821	−1.77100
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.20941	0.273799
$900$	0	0
$901$	1.69010	0.0563055
$902$	0	0
$903$	6.15883	0.204953
$904$	0	0
$905$	−74.5900	−2.47946
$906$	0	0
$907$	−22.7006	−0.753763	−0.376881	−	0.926262i	$-0.623004\pi$
$907$	−22.7006	−0.753763	−0.376881	+	0.926262i	$0.623004\pi$
$908$	0	0
$909$	6.75063	0.223904
$910$	0	0
$911$	−37.3019	−1.23587	−0.617933	−	0.786231i	$-0.712030\pi$
$911$	−37.3019	−1.23587	−0.617933	+	0.786231i	$0.712030\pi$
$912$	0	0
$913$	0.732578	0.0242448
$914$	0	0
$915$	1.75840	0.0581308
$916$	0	0
$917$	−63.4040	−2.09378
$918$	0	0
$919$	−9.48427	−0.312857	−0.156429	−	0.987689i	$-0.549998\pi$
$919$	−9.48427	−0.312857	−0.156429	+	0.987689i	$0.549998\pi$
$920$	0	0
$921$	−3.32006	−0.109400
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0.116621	0.00383448
$926$	0	0
$927$	7.78017	0.255534
$928$	0	0
$929$	−5.32245	−0.174624	−0.0873120	−	0.996181i	$-0.527828\pi$
$929$	−5.32245	−0.174624	−0.0873120	+	0.996181i	$0.527828\pi$
$930$	0	0
$931$	−113.484	−3.71929
$932$	0	0
$933$	20.2881	0.664203
$934$	0	0
$935$	4.13706	0.135296
$936$	0	0
$937$	−20.2097	−0.660221	−0.330111	−	0.943942i	$-0.607086\pi$
$937$	−20.2097	−0.660221	−0.330111	+	0.943942i	$0.607086\pi$
$938$	0	0
$939$	−14.3870	−0.469503
$940$	0	0
$941$	43.1135	1.40546	0.702730	−	0.711457i	$-0.251964\pi$
$941$	43.1135	1.40546	0.702730	+	0.711457i	$0.251964\pi$
$942$	0	0
$943$	72.9275	2.37485
$944$	0	0
$945$	−15.3937	−0.500758
$946$	0	0
$947$	−55.9627	−1.81854	−0.909272	−	0.416203i	$-0.863360\pi$
$947$	−55.9627	−1.81854	−0.909272	+	0.416203i	$0.863360\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	12.4450	0.403558
$952$	0	0
$953$	−15.1830	−0.491826	−0.245913	−	0.969292i	$-0.579088\pi$
$953$	−15.1830	−0.491826	−0.245913	+	0.969292i	$0.579088\pi$
$954$	0	0
$955$	65.2737	2.11221
$956$	0	0
$957$	−5.39075	−0.174258
$958$	0	0
$959$	33.6165	1.08553
$960$	0	0
$961$	2.52243	0.0813688
$962$	0	0
$963$	8.51573	0.274416
$964$	0	0
$965$	−43.9885	−1.41604
$966$	0	0
$967$	15.4064	0.495437	0.247718	−	0.968832i	$-0.420319\pi$
$967$	15.4064	0.495437	0.247718	+	0.968832i	$0.420319\pi$
$968$	0	0
$969$	2.19029	0.0703623
$970$	0	0
$971$	6.00133	0.192592	0.0962959	−	0.995353i	$-0.469300\pi$
$971$	6.00133	0.192592	0.0962959	+	0.995353i	$0.469300\pi$
$972$	0	0
$973$	31.1715	0.999313
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.3931	−0.652434	−0.326217	−	0.945295i	$-0.605774\pi$
$977$	−20.3931	−0.652434	−0.326217	+	0.945295i	$0.605774\pi$
$978$	0	0
$979$	−35.0277	−1.11949
$980$	0	0
$981$	−1.08815	−0.0347418
$982$	0	0
$983$	29.9764	0.956100	0.478050	−	0.878333i	$-0.341344\pi$
$983$	29.9764	0.956100	0.478050	+	0.878333i	$0.341344\pi$
$984$	0	0
$985$	−6.23968	−0.198813
$986$	0	0
$987$	48.2911	1.53712
$988$	0	0
$989$	9.68425	0.307941
$990$	0	0
$991$	36.4722	1.15858	0.579289	−	0.815122i	$-0.303330\pi$
$991$	36.4722	1.15858	0.579289	+	0.815122i	$0.303330\pi$
$992$	0	0
$993$	3.31873	0.105317
$994$	0	0
$995$	−16.9366	−0.536927
$996$	0	0
$997$	−21.9597	−0.695471	−0.347735	−	0.937593i	$-0.613049\pi$
$997$	−21.9597	−0.695471	−0.347735	+	0.937593i	$0.613049\pi$
$998$	0	0
$999$	0.0271471	0.000858895 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.a.k.1.3	✓	3
3.2	odd	2		6084.2.a.z.1.1		3
4.3	odd	2		8112.2.a.cd.1.3		3
13.2	odd	12		2028.2.q.i.1837.1		12
13.3	even	3		2028.2.i.k.529.3		6
13.4	even	6		2028.2.i.j.2005.1		6
13.5	odd	4		2028.2.b.g.337.1		6
13.6	odd	12		2028.2.q.i.361.1		12
13.7	odd	12		2028.2.q.i.361.6		12
13.8	odd	4		2028.2.b.g.337.6		6
13.9	even	3		2028.2.i.k.2005.3		6
13.10	even	6		2028.2.i.j.529.1		6
13.11	odd	12		2028.2.q.i.1837.6		12
13.12	even	2		2028.2.a.l.1.1	yes	3
39.5	even	4		6084.2.b.q.4393.6		6
39.8	even	4		6084.2.b.q.4393.1		6
39.38	odd	2		6084.2.a.ba.1.3		3
52.51	odd	2		8112.2.a.ca.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2028.2.a.k.1.3	✓	3	1.1	even	1	trivial
2028.2.a.l.1.1	yes	3	13.12	even	2
2028.2.b.g.337.1		6	13.5	odd	4
2028.2.b.g.337.6		6	13.8	odd	4
2028.2.i.j.529.1		6	13.10	even	6
2028.2.i.j.2005.1		6	13.4	even	6
2028.2.i.k.529.3		6	13.3	even	3
2028.2.i.k.2005.3		6	13.9	even	3
2028.2.q.i.361.1		12	13.6	odd	12
2028.2.q.i.361.6		12	13.7	odd	12
2028.2.q.i.1837.1		12	13.2	odd	12
2028.2.q.i.1837.6		12	13.11	odd	12
6084.2.a.z.1.1		3	3.2	odd	2
6084.2.a.ba.1.3		3	39.38	odd	2
6084.2.b.q.4393.1		6	39.8	even	4
6084.2.b.q.4393.6		6	39.5	even	4
8112.2.a.ca.1.1		3	52.51	odd	2
8112.2.a.cd.1.3		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.04892 q^{5} -5.04892 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.04892 q^{5} -5.04892 q^{7} +1.00000 q^{9} -3.80194 q^{11} +3.04892 q^{15} -0.356896 q^{17} -6.13706 q^{19} -5.04892 q^{21} -7.93900 q^{23} +4.29590 q^{25} +1.00000 q^{27} +1.41789 q^{29} +5.78986 q^{31} -3.80194 q^{33} -15.3937 q^{35} +0.0271471 q^{37} -9.18598 q^{41} -1.21983 q^{43} +3.04892 q^{45} -9.56465 q^{47} +18.4916 q^{49} -0.356896 q^{51} -4.73556 q^{53} -11.5918 q^{55} -6.13706 q^{57} +13.5254 q^{59} +0.576728 q^{61} -5.04892 q^{63} -7.95108 q^{67} -7.93900 q^{69} -3.07606 q^{71} -14.0368 q^{73} +4.29590 q^{75} +19.1957 q^{77} -0.841166 q^{79} +1.00000 q^{81} -0.192685 q^{83} -1.08815 q^{85} +1.41789 q^{87} +9.21313 q^{89} +5.78986 q^{93} -18.7114 q^{95} -0.682333 q^{97} -3.80194 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 6 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 6 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 6 q^{7} + 3 q^{9} - 7 q^{11} + 3 q^{17} - 13 q^{19} - 6 q^{21} - 14 q^{23} - q^{25} + 3 q^{27} + 10 q^{29} - 6 q^{31} - 7 q^{33} - 14 q^{35} - 6 q^{37} - 13 q^{41} - 5 q^{43} - 7 q^{47} + 5 q^{49} + 3 q^{51} - 3 q^{53} - 7 q^{55} - 13 q^{57} + 6 q^{59} - q^{61} - 6 q^{63} - 33 q^{67} - 14 q^{69} + 6 q^{71} - 14 q^{73} - q^{75} + 21 q^{77} - 11 q^{79} + 3 q^{81} - 8 q^{83} - 7 q^{85} + 10 q^{87} + 7 q^{89} - 6 q^{93} - 7 q^{95} - 19 q^{97} - 7 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 6 * q^7 + 3 * q^9 - 7 * q^11 + 3 * q^17 - 13 * q^19 - 6 * q^21 - 14 * q^23 - q^25 + 3 * q^27 + 10 * q^29 - 6 * q^31 - 7 * q^33 - 14 * q^35 - 6 * q^37 - 13 * q^41 - 5 * q^43 - 7 * q^47 + 5 * q^49 + 3 * q^51 - 3 * q^53 - 7 * q^55 - 13 * q^57 + 6 * q^59 - q^61 - 6 * q^63 - 33 * q^67 - 14 * q^69 + 6 * q^71 - 14 * q^73 - q^75 + 21 * q^77 - 11 * q^79 + 3 * q^81 - 8 * q^83 - 7 * q^85 + 10 * q^87 + 7 * q^89 - 6 * q^93 - 7 * q^95 - 19 * q^97 - 7 * q^99

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