LMFDB - Embedded newform 5096.2.a.bf.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5096,2,Mod(1,5096)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5096, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5096.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5096 = 2^{3} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5096.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,0,-2,0,-6,0,0,0,12,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$40.6917648700$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 19x^{8} + 36x^{7} + 110x^{6} - 188x^{5} - 218x^{4} + 276x^{3} + 154x^{2} - 48x - 4 $ x^10 - 2x^9 - 19x^8 + 36x^7 + 110x^6 - 188x^5 - 218x^4 + 276x^3 + 154x^2 - 48*x - 4
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			1.4
Root			$1.67486$ of defining polynomial
Character	$\chi$	$=$	5096.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67486 q^{3} +1.96101 q^{5} -0.194860 q^{9} -1.66183 q^{11} -1.00000 q^{13} -3.28441 q^{15} -5.11115 q^{17} +6.48419 q^{19} +1.12806 q^{23} -1.15444 q^{25} +5.35093 q^{27} +6.85814 q^{29} -1.16816 q^{31} +2.78333 q^{33} -2.16513 q^{37} +1.67486 q^{39} -6.12029 q^{41} +4.07691 q^{43} -0.382123 q^{45} -2.90037 q^{47} +8.56044 q^{51} +7.26079 q^{53} -3.25888 q^{55} -10.8601 q^{57} -2.21673 q^{59} -7.33909 q^{61} -1.96101 q^{65} +6.79925 q^{67} -1.88933 q^{69} -6.84306 q^{71} -15.1542 q^{73} +1.93352 q^{75} +0.576410 q^{79} -8.37745 q^{81} -9.76879 q^{83} -10.0230 q^{85} -11.4864 q^{87} +13.8886 q^{89} +1.95650 q^{93} +12.7156 q^{95} -10.7055 q^{97} +0.323826 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - 6 q^{5} + 12 q^{9} - 2 q^{11} - 10 q^{13} + 4 q^{15} - 16 q^{17} - 14 q^{19} + 8 q^{23} + 12 q^{25} - 2 q^{27} + 2 q^{29} - 8 q^{31} - 10 q^{33} - 6 q^{37} + 2 q^{39} - 22 q^{41} + 2 q^{43}+ \cdots - 24 q^{99}+O(q^{100}) $ 10 * q - 2 * q^3 - 6 * q^5 + 12 * q^9 - 2 * q^11 - 10 * q^13 + 4 * q^15 - 16 * q^17 - 14 * q^19 + 8 * q^23 + 12 * q^25 - 2 * q^27 + 2 * q^29 - 8 * q^31 - 10 * q^33 - 6 * q^37 + 2 * q^39 - 22 * q^41 + 2 * q^43 - 42 * q^45 + 36 * q^47 + 8 * q^51 - 6 * q^53 + 8 * q^55 - 16 * q^57 - 24 * q^59 - 18 * q^61 + 6 * q^65 - 14 * q^67 - 46 * q^69 - 8 * q^71 - 28 * q^73 - 14 * q^75 - 16 * q^79 + 26 * q^81 - 2 * q^83 + 16 * q^85 - 8 * q^87 - 10 * q^89 - 2 * q^93 + 10 * q^95 - 60 * q^97 - 24 * q^99

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
$3$	−1.67486	−0.966978	−0.483489	−	0.875350i	$-0.660631\pi$
$3$	−1.67486	−0.966978	−0.483489	+	0.875350i	$0.660631\pi$
$4$	0	0
$5$	1.96101	0.876991	0.438495	−	0.898733i	$-0.355512\pi$
$5$	1.96101	0.876991	0.438495	+	0.898733i	$0.355512\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.194860	−0.0649535
$10$	0	0
$11$	−1.66183	−0.501062	−0.250531	−	0.968109i	$-0.580605\pi$
$11$	−1.66183	−0.501062	−0.250531	+	0.968109i	$0.580605\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.28441	−0.848031
$16$	0	0
$17$	−5.11115	−1.23964	−0.619818	−	0.784745i	$-0.712794\pi$
$17$	−5.11115	−1.23964	−0.619818	+	0.784745i	$0.712794\pi$
$18$	0	0
$19$	6.48419	1.48758	0.743788	−	0.668416i	$-0.233027\pi$
$19$	6.48419	1.48758	0.743788	+	0.668416i	$0.233027\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.12806	0.235216	0.117608	−	0.993060i	$-0.462477\pi$
$23$	1.12806	0.235216	0.117608	+	0.993060i	$0.462477\pi$
$24$	0	0
$25$	−1.15444	−0.230888
$26$	0	0
$27$	5.35093	1.02979
$28$	0	0
$29$	6.85814	1.27352	0.636762	−	0.771060i	$-0.280273\pi$
$29$	6.85814	1.27352	0.636762	+	0.771060i	$0.280273\pi$
$30$	0	0
$31$	−1.16816	−0.209808	−0.104904	−	0.994482i	$-0.533454\pi$
$31$	−1.16816	−0.209808	−0.104904	+	0.994482i	$0.533454\pi$
$32$	0	0
$33$	2.78333	0.484516
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.16513	−0.355945	−0.177973	−	0.984035i	$-0.556954\pi$
$37$	−2.16513	−0.355945	−0.177973	+	0.984035i	$0.556954\pi$
$38$	0	0
$39$	1.67486	0.268191
$40$	0	0
$41$	−6.12029	−0.955828	−0.477914	−	0.878407i	$-0.658607\pi$
$41$	−6.12029	−0.955828	−0.477914	+	0.878407i	$0.658607\pi$
$42$	0	0
$43$	4.07691	0.621724	0.310862	−	0.950455i	$-0.399382\pi$
$43$	4.07691	0.621724	0.310862	+	0.950455i	$0.399382\pi$
$44$	0	0
$45$	−0.382123	−0.0569636
$46$	0	0
$47$	−2.90037	−0.423063	−0.211531	−	0.977371i	$-0.567845\pi$
$47$	−2.90037	−0.423063	−0.211531	+	0.977371i	$0.567845\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	8.56044	1.19870
$52$	0	0
$53$	7.26079	0.997347	0.498673	−	0.866790i	$-0.333821\pi$
$53$	7.26079	0.997347	0.498673	+	0.866790i	$0.333821\pi$
$54$	0	0
$55$	−3.25888	−0.439427
$56$	0	0
$57$	−10.8601	−1.43845
$58$	0	0
$59$	−2.21673	−0.288593	−0.144297	−	0.989534i	$-0.546092\pi$
$59$	−2.21673	−0.288593	−0.144297	+	0.989534i	$0.546092\pi$
$60$	0	0
$61$	−7.33909	−0.939675	−0.469837	−	0.882753i	$-0.655687\pi$
$61$	−7.33909	−0.939675	−0.469837	+	0.882753i	$0.655687\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.96101	−0.243233
$66$	0	0
$67$	6.79925	0.830661	0.415330	−	0.909671i	$-0.363666\pi$
$67$	6.79925	0.830661	0.415330	+	0.909671i	$0.363666\pi$
$68$	0	0
$69$	−1.88933	−0.227449
$70$	0	0
$71$	−6.84306	−0.812122	−0.406061	−	0.913846i	$-0.633098\pi$
$71$	−6.84306	−0.812122	−0.406061	+	0.913846i	$0.633098\pi$
$72$	0	0
$73$	−15.1542	−1.77366	−0.886832	−	0.462093i	$-0.847099\pi$
$73$	−15.1542	−1.77366	−0.886832	+	0.462093i	$0.847099\pi$
$74$	0	0
$75$	1.93352	0.223263
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.576410	0.0648512	0.0324256	−	0.999474i	$-0.489677\pi$
$79$	0.576410	0.0648512	0.0324256	+	0.999474i	$0.489677\pi$
$80$	0	0
$81$	−8.37745	−0.930828
$82$	0	0
$83$	−9.76879	−1.07226	−0.536132	−	0.844134i	$-0.680115\pi$
$83$	−9.76879	−1.07226	−0.536132	+	0.844134i	$0.680115\pi$
$84$	0	0
$85$	−10.0230	−1.08715
$86$	0	0
$87$	−11.4864	−1.23147
$88$	0	0
$89$	13.8886	1.47219	0.736095	−	0.676879i	$-0.236668\pi$
$89$	13.8886	1.47219	0.736095	+	0.676879i	$0.236668\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.95650	0.202880
$94$	0	0
$95$	12.7156	1.30459
$96$	0	0
$97$	−10.7055	−1.08698	−0.543488	−	0.839417i	$-0.682897\pi$
$97$	−10.7055	−1.08698	−0.543488	+	0.839417i	$0.682897\pi$
$98$	0	0
$99$	0.323826	0.0325457
$100$	0	0
$101$	−1.64214	−0.163399	−0.0816994	−	0.996657i	$-0.526035\pi$
$101$	−1.64214	−0.163399	−0.0816994	+	0.996657i	$0.526035\pi$
$102$	0	0
$103$	8.20389	0.808353	0.404177	−	0.914681i	$-0.367558\pi$
$103$	8.20389	0.808353	0.404177	+	0.914681i	$0.367558\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.3099	1.09337	0.546684	−	0.837339i	$-0.315890\pi$
$107$	11.3099	1.09337	0.546684	+	0.837339i	$0.315890\pi$
$108$	0	0
$109$	−10.9024	−1.04426	−0.522129	−	0.852867i	$-0.674862\pi$
$109$	−10.9024	−1.04426	−0.522129	+	0.852867i	$0.674862\pi$
$110$	0	0
$111$	3.62628	0.344191
$112$	0	0
$113$	−0.810483	−0.0762439	−0.0381219	−	0.999273i	$-0.512138\pi$
$113$	−0.810483	−0.0762439	−0.0381219	+	0.999273i	$0.512138\pi$
$114$	0	0
$115$	2.21213	0.206283
$116$	0	0
$117$	0.194860	0.0180149
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.23830	−0.748937
$122$	0	0
$123$	10.2506	0.924265
$124$	0	0
$125$	−12.0689	−1.07948
$126$	0	0
$127$	3.20808	0.284671	0.142336	−	0.989818i	$-0.454539\pi$
$127$	3.20808	0.284671	0.142336	+	0.989818i	$0.454539\pi$
$128$	0	0
$129$	−6.82824	−0.601193
$130$	0	0
$131$	−15.6228	−1.36497	−0.682484	−	0.730901i	$-0.739100\pi$
$131$	−15.6228	−1.36497	−0.682484	+	0.730901i	$0.739100\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	10.4932	0.903113
$136$	0	0
$137$	−1.65545	−0.141435	−0.0707175	−	0.997496i	$-0.522529\pi$
$137$	−1.65545	−0.141435	−0.0707175	+	0.997496i	$0.522529\pi$
$138$	0	0
$139$	−11.1733	−0.947705	−0.473852	−	0.880604i	$-0.657137\pi$
$139$	−11.1733	−0.947705	−0.473852	+	0.880604i	$0.657137\pi$
$140$	0	0
$141$	4.85770	0.409092
$142$	0	0
$143$	1.66183	0.138970
$144$	0	0
$145$	13.4489	1.11687
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.5646	−0.865486	−0.432743	−	0.901517i	$-0.642454\pi$
$149$	−10.5646	−0.865486	−0.432743	+	0.901517i	$0.642454\pi$
$150$	0	0
$151$	−8.15279	−0.663465	−0.331732	−	0.943374i	$-0.607633\pi$
$151$	−8.15279	−0.663465	−0.331732	+	0.943374i	$0.607633\pi$
$152$	0	0
$153$	0.995961	0.0805187
$154$	0	0
$155$	−2.29078	−0.184000
$156$	0	0
$157$	−16.1733	−1.29077	−0.645386	−	0.763856i	$-0.723304\pi$
$157$	−16.1733	−1.29077	−0.645386	+	0.763856i	$0.723304\pi$
$158$	0	0
$159$	−12.1608	−0.964412
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.1542	1.18697	0.593486	−	0.804844i	$-0.297751\pi$
$163$	15.1542	1.18697	0.593486	+	0.804844i	$0.297751\pi$
$164$	0	0
$165$	5.45814	0.424916
$166$	0	0
$167$	22.0967	1.70990	0.854949	−	0.518713i	$-0.173589\pi$
$167$	22.0967	1.70990	0.854949	+	0.518713i	$0.173589\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.26351	−0.0966232
$172$	0	0
$173$	2.59337	0.197170	0.0985852	−	0.995129i	$-0.468568\pi$
$173$	2.59337	0.197170	0.0985852	+	0.995129i	$0.468568\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.71270	0.279064
$178$	0	0
$179$	13.5240	1.01083	0.505414	−	0.862877i	$-0.331340\pi$
$179$	13.5240	1.01083	0.505414	+	0.862877i	$0.331340\pi$
$180$	0	0
$181$	−6.69331	−0.497509	−0.248755	−	0.968567i	$-0.580021\pi$
$181$	−6.69331	−0.497509	−0.248755	+	0.968567i	$0.580021\pi$
$182$	0	0
$183$	12.2919	0.908645
$184$	0	0
$185$	−4.24584	−0.312161
$186$	0	0
$187$	8.49389	0.621135
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.2145	0.883812	0.441906	−	0.897061i	$-0.354303\pi$
$191$	12.2145	0.883812	0.441906	+	0.897061i	$0.354303\pi$
$192$	0	0
$193$	25.6943	1.84951	0.924757	−	0.380558i	$-0.124268\pi$
$193$	25.6943	1.84951	0.924757	+	0.380558i	$0.124268\pi$
$194$	0	0
$195$	3.28441	0.235201
$196$	0	0
$197$	−12.7826	−0.910725	−0.455362	−	0.890306i	$-0.650490\pi$
$197$	−12.7826	−0.910725	−0.455362	+	0.890306i	$0.650490\pi$
$198$	0	0
$199$	0.332281	0.0235548	0.0117774	−	0.999931i	$-0.496251\pi$
$199$	0.332281	0.0235548	0.0117774	+	0.999931i	$0.496251\pi$
$200$	0	0
$201$	−11.3878	−0.803230
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−12.0019	−0.838252
$206$	0	0
$207$	−0.219814	−0.0152781
$208$	0	0
$209$	−10.7757	−0.745368
$210$	0	0
$211$	−21.3728	−1.47136	−0.735682	−	0.677328i	$-0.763138\pi$
$211$	−21.3728	−1.47136	−0.735682	+	0.677328i	$0.763138\pi$
$212$	0	0
$213$	11.4611	0.785304
$214$	0	0
$215$	7.99487	0.545246
$216$	0	0
$217$	0	0
$218$	0	0
$219$	25.3811	1.71509
$220$	0	0
$221$	5.11115	0.343813
$222$	0	0
$223$	−1.92719	−0.129054	−0.0645271	−	0.997916i	$-0.520554\pi$
$223$	−1.92719	−0.129054	−0.0645271	+	0.997916i	$0.520554\pi$
$224$	0	0
$225$	0.224954	0.0149970
$226$	0	0
$227$	7.46734	0.495624	0.247812	−	0.968808i	$-0.420288\pi$
$227$	7.46734	0.495624	0.247812	+	0.968808i	$0.420288\pi$
$228$	0	0
$229$	−19.7763	−1.30686	−0.653428	−	0.756989i	$-0.726670\pi$
$229$	−19.7763	−1.30686	−0.653428	+	0.756989i	$0.726670\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.97921	0.522735	0.261368	−	0.965239i	$-0.415827\pi$
$233$	7.97921	0.522735	0.261368	+	0.965239i	$0.415827\pi$
$234$	0	0
$235$	−5.68766	−0.371022
$236$	0	0
$237$	−0.965403	−0.0627097
$238$	0	0
$239$	−27.5872	−1.78447	−0.892233	−	0.451575i	$-0.850862\pi$
$239$	−27.5872	−1.78447	−0.892233	+	0.451575i	$0.850862\pi$
$240$	0	0
$241$	−1.97176	−0.127012	−0.0635062	−	0.997981i	$-0.520228\pi$
$241$	−1.97176	−0.127012	−0.0635062	+	0.997981i	$0.520228\pi$
$242$	0	0
$243$	−2.02177	−0.129697
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.48419	−0.412579
$248$	0	0
$249$	16.3613	1.03686
$250$	0	0
$251$	−24.1574	−1.52480	−0.762399	−	0.647107i	$-0.775979\pi$
$251$	−24.1574	−1.52480	−0.762399	+	0.647107i	$0.775979\pi$
$252$	0	0
$253$	−1.87465	−0.117858
$254$	0	0
$255$	16.7871	1.05125
$256$	0	0
$257$	−7.88181	−0.491654	−0.245827	−	0.969314i	$-0.579059\pi$
$257$	−7.88181	−0.491654	−0.245827	+	0.969314i	$0.579059\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.33638	−0.0827199
$262$	0	0
$263$	−20.5830	−1.26920	−0.634602	−	0.772839i	$-0.718836\pi$
$263$	−20.5830	−1.26920	−0.634602	+	0.772839i	$0.718836\pi$
$264$	0	0
$265$	14.2385	0.874664
$266$	0	0
$267$	−23.2614	−1.42357
$268$	0	0
$269$	−3.04982	−0.185951	−0.0929754	−	0.995668i	$-0.529638\pi$
$269$	−3.04982	−0.185951	−0.0929754	+	0.995668i	$0.529638\pi$
$270$	0	0
$271$	−19.2830	−1.17136	−0.585681	−	0.810542i	$-0.699173\pi$
$271$	−19.2830	−1.17136	−0.585681	+	0.810542i	$0.699173\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.91848	0.115689
$276$	0	0
$277$	13.7602	0.826772	0.413386	−	0.910556i	$-0.364346\pi$
$277$	13.7602	0.826772	0.413386	+	0.910556i	$0.364346\pi$
$278$	0	0
$279$	0.227628	0.0136278
$280$	0	0
$281$	−7.73565	−0.461470	−0.230735	−	0.973017i	$-0.574113\pi$
$281$	−7.73565	−0.461470	−0.230735	+	0.973017i	$0.574113\pi$
$282$	0	0
$283$	−10.9354	−0.650039	−0.325020	−	0.945707i	$-0.605371\pi$
$283$	−10.9354	−0.650039	−0.325020	+	0.945707i	$0.605371\pi$
$284$	0	0
$285$	−21.2967	−1.26151
$286$	0	0
$287$	0	0
$288$	0	0
$289$	9.12387	0.536698
$290$	0	0
$291$	17.9301	1.05108
$292$	0	0
$293$	−4.51209	−0.263599	−0.131799	−	0.991276i	$-0.542075\pi$
$293$	−4.51209	−0.263599	−0.131799	+	0.991276i	$0.542075\pi$
$294$	0	0
$295$	−4.34703	−0.253094
$296$	0	0
$297$	−8.89236	−0.515987
$298$	0	0
$299$	−1.12806	−0.0652373
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.75034	0.158003
$304$	0	0
$305$	−14.3920	−0.824086
$306$	0	0
$307$	−8.14085	−0.464623	−0.232311	−	0.972641i	$-0.574629\pi$
$307$	−8.14085	−0.464623	−0.232311	+	0.972641i	$0.574629\pi$
$308$	0	0
$309$	−13.7403	−0.781660
$310$	0	0
$311$	7.76461	0.440291	0.220145	−	0.975467i	$-0.429347\pi$
$311$	7.76461	0.440291	0.220145	+	0.975467i	$0.429347\pi$
$312$	0	0
$313$	−20.7421	−1.17241	−0.586205	−	0.810162i	$-0.699379\pi$
$313$	−20.7421	−1.17241	−0.586205	+	0.810162i	$0.699379\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.282737	0.0158801	0.00794005	−	0.999968i	$-0.497473\pi$
$317$	0.282737	0.0158801	0.00794005	+	0.999968i	$0.497473\pi$
$318$	0	0
$319$	−11.3971	−0.638115
$320$	0	0
$321$	−18.9424	−1.05726
$322$	0	0
$323$	−33.1417	−1.84405
$324$	0	0
$325$	1.15444	0.0640367
$326$	0	0
$327$	18.2599	1.00977
$328$	0	0
$329$	0	0
$330$	0	0
$331$	19.8450	1.09078	0.545390	−	0.838182i	$-0.316381\pi$
$331$	19.8450	1.09078	0.545390	+	0.838182i	$0.316381\pi$
$332$	0	0
$333$	0.421898	0.0231199
$334$	0	0
$335$	13.3334	0.728481
$336$	0	0
$337$	3.91104	0.213048	0.106524	−	0.994310i	$-0.466028\pi$
$337$	3.91104	0.213048	0.106524	+	0.994310i	$0.466028\pi$
$338$	0	0
$339$	1.35744	0.0737261
$340$	0	0
$341$	1.94129	0.105127
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.70500	−0.199471
$346$	0	0
$347$	−14.5491	−0.781034	−0.390517	−	0.920596i	$-0.627704\pi$
$347$	−14.5491	−0.781034	−0.390517	+	0.920596i	$0.627704\pi$
$348$	0	0
$349$	15.6952	0.840144	0.420072	−	0.907491i	$-0.362005\pi$
$349$	15.6952	0.840144	0.420072	+	0.907491i	$0.362005\pi$
$350$	0	0
$351$	−5.35093	−0.285611
$352$	0	0
$353$	−4.47548	−0.238206	−0.119103	−	0.992882i	$-0.538002\pi$
$353$	−4.47548	−0.238206	−0.119103	+	0.992882i	$0.538002\pi$
$354$	0	0
$355$	−13.4193	−0.712223
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.6654	−0.826788	−0.413394	−	0.910552i	$-0.635657\pi$
$359$	−15.6654	−0.826788	−0.413394	+	0.910552i	$0.635657\pi$
$360$	0	0
$361$	23.0448	1.21288
$362$	0	0
$363$	13.7980	0.724205
$364$	0	0
$365$	−29.7175	−1.55549
$366$	0	0
$367$	−32.2933	−1.68569	−0.842847	−	0.538153i	$-0.819122\pi$
$367$	−32.2933	−1.68569	−0.842847	+	0.538153i	$0.819122\pi$
$368$	0	0
$369$	1.19260	0.0620844
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−28.2758	−1.46407	−0.732033	−	0.681269i	$-0.761428\pi$
$373$	−28.2758	−1.46407	−0.732033	+	0.681269i	$0.761428\pi$
$374$	0	0
$375$	20.2137	1.04383
$376$	0	0
$377$	−6.85814	−0.353212
$378$	0	0
$379$	5.86299	0.301161	0.150581	−	0.988598i	$-0.451886\pi$
$379$	5.86299	0.301161	0.150581	+	0.988598i	$0.451886\pi$
$380$	0	0
$381$	−5.37308	−0.275271
$382$	0	0
$383$	14.9135	0.762047	0.381023	−	0.924565i	$-0.375572\pi$
$383$	14.9135	0.762047	0.381023	+	0.924565i	$0.375572\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.794429	−0.0403831
$388$	0	0
$389$	−10.7317	−0.544119	−0.272060	−	0.962280i	$-0.587705\pi$
$389$	−10.7317	−0.544119	−0.272060	+	0.962280i	$0.587705\pi$
$390$	0	0
$391$	−5.76568	−0.291583
$392$	0	0
$393$	26.1659	1.31989
$394$	0	0
$395$	1.13035	0.0568739
$396$	0	0
$397$	−29.5225	−1.48169	−0.740847	−	0.671674i	$-0.765576\pi$
$397$	−29.5225	−1.48169	−0.740847	+	0.671674i	$0.765576\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.3814	−0.568360	−0.284180	−	0.958771i	$-0.591721\pi$
$401$	−11.3814	−0.568360	−0.284180	+	0.958771i	$0.591721\pi$
$402$	0	0
$403$	1.16816	0.0581902
$404$	0	0
$405$	−16.4283	−0.816327
$406$	0	0
$407$	3.59809	0.178351
$408$	0	0
$409$	−15.7081	−0.776714	−0.388357	−	0.921509i	$-0.626957\pi$
$409$	−15.7081	−0.776714	−0.388357	+	0.921509i	$0.626957\pi$
$410$	0	0
$411$	2.77265	0.136765
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−19.1567	−0.940365
$416$	0	0
$417$	18.7136	0.916410
$418$	0	0
$419$	34.8495	1.70251	0.851256	−	0.524751i	$-0.175842\pi$
$419$	34.8495	1.70251	0.851256	+	0.524751i	$0.175842\pi$
$420$	0	0
$421$	1.28023	0.0623948	0.0311974	−	0.999513i	$-0.490068\pi$
$421$	1.28023	0.0623948	0.0311974	+	0.999513i	$0.490068\pi$
$422$	0	0
$423$	0.565168	0.0274794
$424$	0	0
$425$	5.90051	0.286217
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.78333	−0.134381
$430$	0	0
$431$	−36.4027	−1.75346	−0.876729	−	0.480985i	$-0.840279\pi$
$431$	−36.4027	−1.75346	−0.876729	+	0.480985i	$0.840279\pi$
$432$	0	0
$433$	−12.0832	−0.580680	−0.290340	−	0.956924i	$-0.593768\pi$
$433$	−12.0832	−0.580680	−0.290340	+	0.956924i	$0.593768\pi$
$434$	0	0
$435$	−22.5249	−1.07999
$436$	0	0
$437$	7.31455	0.349902
$438$	0	0
$439$	16.5632	0.790516	0.395258	−	0.918570i	$-0.370655\pi$
$439$	16.5632	0.790516	0.395258	+	0.918570i	$0.370655\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.9088	−0.660825	−0.330413	−	0.943837i	$-0.607188\pi$
$443$	−13.9088	−0.660825	−0.330413	+	0.943837i	$0.607188\pi$
$444$	0	0
$445$	27.2357	1.29110
$446$	0	0
$447$	17.6942	0.836906
$448$	0	0
$449$	37.0558	1.74877	0.874385	−	0.485233i	$-0.161265\pi$
$449$	37.0558	1.74877	0.874385	+	0.485233i	$0.161265\pi$
$450$	0	0
$451$	10.1709	0.478929
$452$	0	0
$453$	13.6547	0.641556
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.9900	−1.02865	−0.514323	−	0.857596i	$-0.671957\pi$
$457$	−21.9900	−1.02865	−0.514323	+	0.857596i	$0.671957\pi$
$458$	0	0
$459$	−27.3494	−1.27656
$460$	0	0
$461$	20.0228	0.932556	0.466278	−	0.884638i	$-0.345595\pi$
$461$	20.0228	0.932556	0.466278	+	0.884638i	$0.345595\pi$
$462$	0	0
$463$	9.26388	0.430529	0.215264	−	0.976556i	$-0.430939\pi$
$463$	9.26388	0.430529	0.215264	+	0.976556i	$0.430939\pi$
$464$	0	0
$465$	3.83672	0.177924
$466$	0	0
$467$	10.4544	0.483773	0.241886	−	0.970305i	$-0.422234\pi$
$467$	10.4544	0.483773	0.241886	+	0.970305i	$0.422234\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	27.0880	1.24815
$472$	0	0
$473$	−6.77516	−0.311522
$474$	0	0
$475$	−7.48560	−0.343463
$476$	0	0
$477$	−1.41484	−0.0647812
$478$	0	0
$479$	−15.8626	−0.724779	−0.362389	−	0.932027i	$-0.618039\pi$
$479$	−15.8626	−0.724779	−0.362389	+	0.932027i	$0.618039\pi$
$480$	0	0
$481$	2.16513	0.0987215
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−20.9935	−0.953268
$486$	0	0
$487$	−3.18402	−0.144282	−0.0721408	−	0.997394i	$-0.522983\pi$
$487$	−3.18402	−0.144282	−0.0721408	+	0.997394i	$0.522983\pi$
$488$	0	0
$489$	−25.3812	−1.14778
$490$	0	0
$491$	−27.8310	−1.25599	−0.627997	−	0.778216i	$-0.716125\pi$
$491$	−27.8310	−1.25599	−0.627997	+	0.778216i	$0.716125\pi$
$492$	0	0
$493$	−35.0530	−1.57871
$494$	0	0
$495$	0.635026	0.0285423
$496$	0	0
$497$	0	0
$498$	0	0
$499$	5.71618	0.255891	0.127946	−	0.991781i	$-0.459162\pi$
$499$	5.71618	0.255891	0.127946	+	0.991781i	$0.459162\pi$
$500$	0	0
$501$	−37.0088	−1.65343
$502$	0	0
$503$	2.09686	0.0934945	0.0467473	−	0.998907i	$-0.485114\pi$
$503$	2.09686	0.0934945	0.0467473	+	0.998907i	$0.485114\pi$
$504$	0	0
$505$	−3.22025	−0.143299
$506$	0	0
$507$	−1.67486	−0.0743829
$508$	0	0
$509$	−12.4440	−0.551573	−0.275786	−	0.961219i	$-0.588938\pi$
$509$	−12.4440	−0.551573	−0.275786	+	0.961219i	$0.588938\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	34.6964	1.53189
$514$	0	0
$515$	16.0879	0.708918
$516$	0	0
$517$	4.81994	0.211981
$518$	0	0
$519$	−4.34352	−0.190659
$520$	0	0
$521$	−3.25659	−0.142674	−0.0713368	−	0.997452i	$-0.522727\pi$
$521$	−3.25659	−0.142674	−0.0713368	+	0.997452i	$0.522727\pi$
$522$	0	0
$523$	−17.4891	−0.764744	−0.382372	−	0.924009i	$-0.624893\pi$
$523$	−17.4891	−0.764744	−0.382372	+	0.924009i	$0.624893\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.97065	0.260086
$528$	0	0
$529$	−21.7275	−0.944673
$530$	0	0
$531$	0.431953	0.0187452
$532$	0	0
$533$	6.12029	0.265099
$534$	0	0
$535$	22.1788	0.958873
$536$	0	0
$537$	−22.6507	−0.977449
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−43.6895	−1.87836	−0.939179	−	0.343429i	$-0.888411\pi$
$541$	−43.6895	−1.87836	−0.939179	+	0.343429i	$0.888411\pi$
$542$	0	0
$543$	11.2103	0.481081
$544$	0	0
$545$	−21.3797	−0.915804
$546$	0	0
$547$	22.1623	0.947589	0.473795	−	0.880635i	$-0.342884\pi$
$547$	22.1623	0.947589	0.473795	+	0.880635i	$0.342884\pi$
$548$	0	0
$549$	1.43010	0.0610352
$550$	0	0
$551$	44.4695	1.89446
$552$	0	0
$553$	0	0
$554$	0	0
$555$	7.11117	0.301853
$556$	0	0
$557$	22.5647	0.956099	0.478049	−	0.878333i	$-0.341344\pi$
$557$	22.5647	0.956099	0.478049	+	0.878333i	$0.341344\pi$
$558$	0	0
$559$	−4.07691	−0.172435
$560$	0	0
$561$	−14.2260	−0.600624
$562$	0	0
$563$	−9.71333	−0.409368	−0.204684	−	0.978828i	$-0.565617\pi$
$563$	−9.71333	−0.409368	−0.204684	+	0.978828i	$0.565617\pi$
$564$	0	0
$565$	−1.58937	−0.0668651
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21.6896	0.909275	0.454637	−	0.890677i	$-0.349769\pi$
$569$	21.6896	0.909275	0.454637	+	0.890677i	$0.349769\pi$
$570$	0	0
$571$	37.4493	1.56720	0.783601	−	0.621264i	$-0.213381\pi$
$571$	37.4493	1.56720	0.783601	+	0.621264i	$0.213381\pi$
$572$	0	0
$573$	−20.4576	−0.854626
$574$	0	0
$575$	−1.30227	−0.0543085
$576$	0	0
$577$	30.3682	1.26424	0.632122	−	0.774869i	$-0.282184\pi$
$577$	30.3682	1.26424	0.632122	+	0.774869i	$0.282184\pi$
$578$	0	0
$579$	−43.0342	−1.78844
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−12.0662	−0.499733
$584$	0	0
$585$	0.382123	0.0157989
$586$	0	0
$587$	−12.3621	−0.510240	−0.255120	−	0.966909i	$-0.582115\pi$
$587$	−12.3621	−0.510240	−0.255120	+	0.966909i	$0.582115\pi$
$588$	0	0
$589$	−7.57458	−0.312105
$590$	0	0
$591$	21.4091	0.880651
$592$	0	0
$593$	17.0465	0.700015	0.350007	−	0.936747i	$-0.386179\pi$
$593$	17.0465	0.700015	0.350007	+	0.936747i	$0.386179\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.556523	−0.0227770
$598$	0	0
$599$	−37.1138	−1.51643	−0.758214	−	0.652006i	$-0.773928\pi$
$599$	−37.1138	−1.51643	−0.758214	+	0.652006i	$0.773928\pi$
$600$	0	0
$601$	42.0688	1.71602	0.858011	−	0.513631i	$-0.171700\pi$
$601$	42.0688	1.71602	0.858011	+	0.513631i	$0.171700\pi$
$602$	0	0
$603$	−1.32491	−0.0539543
$604$	0	0
$605$	−16.1554	−0.656811
$606$	0	0
$607$	13.0426	0.529385	0.264692	−	0.964333i	$-0.414730\pi$
$607$	13.0426	0.529385	0.264692	+	0.964333i	$0.414730\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.90037	0.117337
$612$	0	0
$613$	5.37525	0.217104	0.108552	−	0.994091i	$-0.465379\pi$
$613$	5.37525	0.217104	0.108552	+	0.994091i	$0.465379\pi$
$614$	0	0
$615$	20.1015	0.810572
$616$	0	0
$617$	10.9605	0.441255	0.220627	−	0.975358i	$-0.429190\pi$
$617$	10.9605	0.441255	0.220627	+	0.975358i	$0.429190\pi$
$618$	0	0
$619$	−7.48189	−0.300723	−0.150361	−	0.988631i	$-0.548044\pi$
$619$	−7.48189	−0.300723	−0.150361	+	0.988631i	$0.548044\pi$
$620$	0	0
$621$	6.03616	0.242223
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−17.8951	−0.715803
$626$	0	0
$627$	18.0477	0.720754
$628$	0	0
$629$	11.0663	0.441243
$630$	0	0
$631$	4.71079	0.187534	0.0937668	−	0.995594i	$-0.470109\pi$
$631$	4.71079	0.187534	0.0937668	+	0.995594i	$0.470109\pi$
$632$	0	0
$633$	35.7963	1.42278
$634$	0	0
$635$	6.29109	0.249654
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.33344	0.0527501
$640$	0	0
$641$	−19.4710	−0.769057	−0.384528	−	0.923113i	$-0.625636\pi$
$641$	−19.4710	−0.769057	−0.384528	+	0.923113i	$0.625636\pi$
$642$	0	0
$643$	48.8114	1.92493	0.962467	−	0.271400i	$-0.0874865\pi$
$643$	48.8114	1.92493	0.962467	+	0.271400i	$0.0874865\pi$
$644$	0	0
$645$	−13.3903	−0.527241
$646$	0	0
$647$	3.52355	0.138525	0.0692625	−	0.997598i	$-0.477935\pi$
$647$	3.52355	0.138525	0.0692625	+	0.997598i	$0.477935\pi$
$648$	0	0
$649$	3.68384	0.144603
$650$	0	0
$651$	0	0
$652$	0	0
$653$	40.5848	1.58820	0.794102	−	0.607784i	$-0.207941\pi$
$653$	40.5848	1.58820	0.794102	+	0.607784i	$0.207941\pi$
$654$	0	0
$655$	−30.6364	−1.19706
$656$	0	0
$657$	2.95295	0.115206
$658$	0	0
$659$	−40.8875	−1.59275	−0.796375	−	0.604804i	$-0.793252\pi$
$659$	−40.8875	−1.59275	−0.796375	+	0.604804i	$0.793252\pi$
$660$	0	0
$661$	11.7704	0.457816	0.228908	−	0.973448i	$-0.426485\pi$
$661$	11.7704	0.457816	0.228908	+	0.973448i	$0.426485\pi$
$662$	0	0
$663$	−8.56044	−0.332460
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.73638	0.299554
$668$	0	0
$669$	3.22777	0.124793
$670$	0	0
$671$	12.1964	0.470835
$672$	0	0
$673$	42.0668	1.62156	0.810778	−	0.585353i	$-0.199044\pi$
$673$	42.0668	1.62156	0.810778	+	0.585353i	$0.199044\pi$
$674$	0	0
$675$	−6.17731	−0.237765
$676$	0	0
$677$	−39.2209	−1.50738	−0.753692	−	0.657228i	$-0.771729\pi$
$677$	−39.2209	−1.50738	−0.753692	+	0.657228i	$0.771729\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.5067	−0.479258
$682$	0	0
$683$	−26.1992	−1.00248	−0.501242	−	0.865307i	$-0.667123\pi$
$683$	−26.1992	−1.00248	−0.501242	+	0.865307i	$0.667123\pi$
$684$	0	0
$685$	−3.24636	−0.124037
$686$	0	0
$687$	33.1225	1.26370
$688$	0	0
$689$	−7.26079	−0.276614
$690$	0	0
$691$	23.9778	0.912160	0.456080	−	0.889939i	$-0.349253\pi$
$691$	23.9778	0.912160	0.456080	+	0.889939i	$0.349253\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−21.9109	−0.831128
$696$	0	0
$697$	31.2817	1.18488
$698$	0	0
$699$	−13.3640	−0.505473
$700$	0	0
$701$	43.3312	1.63660	0.818298	−	0.574795i	$-0.194918\pi$
$701$	43.3312	1.63660	0.818298	+	0.574795i	$0.194918\pi$
$702$	0	0
$703$	−14.0391	−0.529496
$704$	0	0
$705$	9.52601	0.358770
$706$	0	0
$707$	0	0
$708$	0	0
$709$	9.45761	0.355188	0.177594	−	0.984104i	$-0.443169\pi$
$709$	9.45761	0.355188	0.177594	+	0.984104i	$0.443169\pi$
$710$	0	0
$711$	−0.112320	−0.00421231
$712$	0	0
$713$	−1.31775	−0.0493503
$714$	0	0
$715$	3.25888	0.121875
$716$	0	0
$717$	46.2045	1.72554
$718$	0	0
$719$	29.0483	1.08332	0.541660	−	0.840598i	$-0.317796\pi$
$719$	29.0483	1.08332	0.541660	+	0.840598i	$0.317796\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.30242	0.122818
$724$	0	0
$725$	−7.91729	−0.294041
$726$	0	0
$727$	−19.7593	−0.732833	−0.366417	−	0.930451i	$-0.619416\pi$
$727$	−19.7593	−0.732833	−0.366417	+	0.930451i	$0.619416\pi$
$728$	0	0
$729$	28.5185	1.05624
$730$	0	0
$731$	−20.8377	−0.770711
$732$	0	0
$733$	−11.1158	−0.410571	−0.205286	−	0.978702i	$-0.565812\pi$
$733$	−11.1158	−0.410571	−0.205286	+	0.978702i	$0.565812\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.2992	−0.416212
$738$	0	0
$739$	−34.8652	−1.28254	−0.641269	−	0.767316i	$-0.721592\pi$
$739$	−34.8652	−1.28254	−0.641269	+	0.767316i	$0.721592\pi$
$740$	0	0
$741$	10.8601	0.398955
$742$	0	0
$743$	16.7148	0.613206	0.306603	−	0.951837i	$-0.400808\pi$
$743$	16.7148	0.613206	0.306603	+	0.951837i	$0.400808\pi$
$744$	0	0
$745$	−20.7173	−0.759023
$746$	0	0
$747$	1.90355	0.0696473
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−18.8535	−0.687975	−0.343987	−	0.938974i	$-0.611778\pi$
$751$	−18.8535	−0.687975	−0.343987	+	0.938974i	$0.611778\pi$
$752$	0	0
$753$	40.4601	1.47445
$754$	0	0
$755$	−15.9877	−0.581853
$756$	0	0
$757$	−30.3541	−1.10324	−0.551619	−	0.834096i	$-0.685990\pi$
$757$	−30.3541	−1.10324	−0.551619	+	0.834096i	$0.685990\pi$
$758$	0	0
$759$	3.13976	0.113966
$760$	0	0
$761$	12.3954	0.449333	0.224666	−	0.974436i	$-0.427871\pi$
$761$	12.3954	0.449333	0.224666	+	0.974436i	$0.427871\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.95309	0.0706142
$766$	0	0
$767$	2.21673	0.0800414
$768$	0	0
$769$	−21.6913	−0.782209	−0.391104	−	0.920346i	$-0.627907\pi$
$769$	−21.6913	−0.782209	−0.391104	+	0.920346i	$0.627907\pi$
$770$	0	0
$771$	13.2009	0.475418
$772$	0	0
$773$	−35.0601	−1.26102	−0.630512	−	0.776180i	$-0.717155\pi$
$773$	−35.0601	−1.26102	−0.630512	+	0.776180i	$0.717155\pi$
$774$	0	0
$775$	1.34857	0.0484420
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−39.6851	−1.42187
$780$	0	0
$781$	11.3720	0.406923
$782$	0	0
$783$	36.6974	1.31146
$784$	0	0
$785$	−31.7161	−1.13200
$786$	0	0
$787$	−41.2144	−1.46914	−0.734568	−	0.678536i	$-0.762615\pi$
$787$	−41.2144	−1.46914	−0.734568	+	0.678536i	$0.762615\pi$
$788$	0	0
$789$	34.4736	1.22729
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.33909	0.260619
$794$	0	0
$795$	−23.8474	−0.845781
$796$	0	0
$797$	4.92718	0.174530	0.0872649	−	0.996185i	$-0.472187\pi$
$797$	4.92718	0.174530	0.0872649	+	0.996185i	$0.472187\pi$
$798$	0	0
$799$	14.8242	0.524444
$800$	0	0
$801$	−2.70634	−0.0956238
$802$	0	0
$803$	25.1838	0.888715
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.10800	0.179810
$808$	0	0
$809$	−21.4872	−0.755450	−0.377725	−	0.925918i	$-0.623293\pi$
$809$	−21.4872	−0.755450	−0.377725	+	0.925918i	$0.623293\pi$
$810$	0	0
$811$	−2.66185	−0.0934703	−0.0467352	−	0.998907i	$-0.514882\pi$
$811$	−2.66185	−0.0934703	−0.0467352	+	0.998907i	$0.514882\pi$
$812$	0	0
$813$	32.2963	1.13268
$814$	0	0
$815$	29.7176	1.04096
$816$	0	0
$817$	26.4355	0.924861
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.84059	−0.0642370	−0.0321185	−	0.999484i	$-0.510225\pi$
$821$	−1.84059	−0.0642370	−0.0321185	+	0.999484i	$0.510225\pi$
$822$	0	0
$823$	−6.58793	−0.229641	−0.114820	−	0.993386i	$-0.536629\pi$
$823$	−6.58793	−0.229641	−0.114820	+	0.993386i	$0.536629\pi$
$824$	0	0
$825$	−3.21318	−0.111869
$826$	0	0
$827$	3.52260	0.122493	0.0612465	−	0.998123i	$-0.480492\pi$
$827$	3.52260	0.122493	0.0612465	+	0.998123i	$0.480492\pi$
$828$	0	0
$829$	45.7282	1.58821	0.794103	−	0.607783i	$-0.207941\pi$
$829$	45.7282	1.58821	0.794103	+	0.607783i	$0.207941\pi$
$830$	0	0
$831$	−23.0464	−0.799471
$832$	0	0
$833$	0	0
$834$	0	0
$835$	43.3319	1.49956
$836$	0	0
$837$	−6.25075	−0.216057
$838$	0	0
$839$	−31.8369	−1.09913	−0.549566	−	0.835451i	$-0.685207\pi$
$839$	−31.8369	−1.09913	−0.549566	+	0.835451i	$0.685207\pi$
$840$	0	0
$841$	18.0341	0.621864
$842$	0	0
$843$	12.9561	0.446231
$844$	0	0
$845$	1.96101	0.0674608
$846$	0	0
$847$	0	0
$848$	0	0
$849$	18.3151	0.628574
$850$	0	0
$851$	−2.44239	−0.0837242
$852$	0	0
$853$	−17.1818	−0.588294	−0.294147	−	0.955760i	$-0.595036\pi$
$853$	−17.1818	−0.588294	−0.294147	+	0.955760i	$0.595036\pi$
$854$	0	0
$855$	−2.47776	−0.0847377
$856$	0	0
$857$	16.5113	0.564015	0.282008	−	0.959412i	$-0.409000\pi$
$857$	16.5113	0.564015	0.282008	+	0.959412i	$0.409000\pi$
$858$	0	0
$859$	23.5720	0.804268	0.402134	−	0.915581i	$-0.368269\pi$
$859$	23.5720	0.804268	0.402134	+	0.915581i	$0.368269\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.0887	1.39868	0.699338	−	0.714791i	$-0.253478\pi$
$863$	41.0887	1.39868	0.699338	+	0.714791i	$0.253478\pi$
$864$	0	0
$865$	5.08563	0.172917
$866$	0	0
$867$	−15.2812	−0.518975
$868$	0	0
$869$	−0.957898	−0.0324945
$870$	0	0
$871$	−6.79925	−0.230384
$872$	0	0
$873$	2.08607	0.0706029
$874$	0	0
$875$	0	0
$876$	0	0
$877$	48.1917	1.62732	0.813659	−	0.581343i	$-0.197473\pi$
$877$	48.1917	1.62732	0.813659	+	0.581343i	$0.197473\pi$
$878$	0	0
$879$	7.55709	0.254894
$880$	0	0
$881$	4.84370	0.163188	0.0815942	−	0.996666i	$-0.473999\pi$
$881$	4.84370	0.163188	0.0815942	+	0.996666i	$0.473999\pi$
$882$	0	0
$883$	13.8975	0.467688	0.233844	−	0.972274i	$-0.424869\pi$
$883$	13.8975	0.467688	0.233844	+	0.972274i	$0.424869\pi$
$884$	0	0
$885$	7.28064	0.244736
$886$	0	0
$887$	33.1608	1.11343	0.556715	−	0.830704i	$-0.312062\pi$
$887$	33.1608	1.11343	0.556715	+	0.830704i	$0.312062\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	13.9219	0.466402
$892$	0	0
$893$	−18.8066	−0.629338
$894$	0	0
$895$	26.5206	0.886487
$896$	0	0
$897$	1.88933	0.0630830
$898$	0	0
$899$	−8.01141	−0.267195
$900$	0	0
$901$	−37.1110	−1.23635
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.1256	−0.436311
$906$	0	0
$907$	48.9307	1.62472	0.812359	−	0.583157i	$-0.198183\pi$
$907$	48.9307	1.62472	0.812359	+	0.583157i	$0.198183\pi$
$908$	0	0
$909$	0.319988	0.0106133
$910$	0	0
$911$	51.1552	1.69485	0.847423	−	0.530919i	$-0.178153\pi$
$911$	51.1552	1.69485	0.847423	+	0.530919i	$0.178153\pi$
$912$	0	0
$913$	16.2341	0.537271
$914$	0	0
$915$	24.1046	0.796873
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−55.4938	−1.83057	−0.915286	−	0.402805i	$-0.868035\pi$
$919$	−55.4938	−1.83057	−0.915286	+	0.402805i	$0.868035\pi$
$920$	0	0
$921$	13.6347	0.449280
$922$	0	0
$923$	6.84306	0.225242
$924$	0	0
$925$	2.49951	0.0821833
$926$	0	0
$927$	−1.59861	−0.0525054
$928$	0	0
$929$	8.84546	0.290210	0.145105	−	0.989416i	$-0.453648\pi$
$929$	8.84546	0.290210	0.145105	+	0.989416i	$0.453648\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−13.0046	−0.425751
$934$	0	0
$935$	16.6566	0.544729
$936$	0	0
$937$	−53.3264	−1.74210	−0.871048	−	0.491197i	$-0.836559\pi$
$937$	−53.3264	−1.74210	−0.871048	+	0.491197i	$0.836559\pi$
$938$	0	0
$939$	34.7400	1.13370
$940$	0	0
$941$	5.56337	0.181361	0.0906804	−	0.995880i	$-0.471096\pi$
$941$	5.56337	0.181361	0.0906804	+	0.995880i	$0.471096\pi$
$942$	0	0
$943$	−6.90404	−0.224826
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.92116	0.0624293	0.0312146	−	0.999513i	$-0.490062\pi$
$947$	1.92116	0.0624293	0.0312146	+	0.999513i	$0.490062\pi$
$948$	0	0
$949$	15.1542	0.491926
$950$	0	0
$951$	−0.473544	−0.0153557
$952$	0	0
$953$	54.6606	1.77063	0.885315	−	0.464991i	$-0.153942\pi$
$953$	54.6606	1.77063	0.885315	+	0.464991i	$0.153942\pi$
$954$	0	0
$955$	23.9528	0.775094
$956$	0	0
$957$	19.0885	0.617043
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.6354	−0.955981
$962$	0	0
$963$	−2.20385	−0.0710181
$964$	0	0
$965$	50.3867	1.62201
$966$	0	0
$967$	17.4272	0.560422	0.280211	−	0.959938i	$-0.409596\pi$
$967$	17.4272	0.560422	0.280211	+	0.959938i	$0.409596\pi$
$968$	0	0
$969$	55.5075	1.78316
$970$	0	0
$971$	57.0217	1.82991	0.914957	−	0.403552i	$-0.132225\pi$
$971$	57.0217	1.82991	0.914957	+	0.403552i	$0.132225\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.93352	−0.0619221
$976$	0	0
$977$	−4.51785	−0.144539	−0.0722694	−	0.997385i	$-0.523024\pi$
$977$	−4.51785	−0.144539	−0.0722694	+	0.997385i	$0.523024\pi$
$978$	0	0
$979$	−23.0806	−0.737658
$980$	0	0
$981$	2.12444	0.0678282
$982$	0	0
$983$	22.3081	0.711517	0.355758	−	0.934578i	$-0.384223\pi$
$983$	22.3081	0.711517	0.355758	+	0.934578i	$0.384223\pi$
$984$	0	0
$985$	−25.0669	−0.798697
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.59900	0.146240
$990$	0	0
$991$	−34.8858	−1.10819	−0.554093	−	0.832455i	$-0.686935\pi$
$991$	−34.8858	−1.10819	−0.554093	+	0.832455i	$0.686935\pi$
$992$	0	0
$993$	−33.2375	−1.05476
$994$	0	0
$995$	0.651607	0.0206573
$996$	0	0
$997$	43.7563	1.38577	0.692887	−	0.721046i	$-0.256338\pi$
$997$	43.7563	1.38577	0.692887	+	0.721046i	$0.256338\pi$
$998$	0	0
$999$	−11.5855	−0.366548

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5096.2.a.bf.1.4	✓	10
7.6	odd	2		5096.2.a.bg.1.7	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5096.2.a.bf.1.4	✓	10	1.1	even	1	trivial
5096.2.a.bg.1.7	yes	10	7.6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5096 = 2^{3} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5096.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67486 q^{3} +1.96101 q^{5} -0.194860 q^{9} -1.66183 q^{11} -1.00000 q^{13} -3.28441 q^{15} -5.11115 q^{17} +6.48419 q^{19} +1.12806 q^{23} -1.15444 q^{25} +5.35093 q^{27} +6.85814 q^{29} -1.16816 q^{31} +2.78333 q^{33} -2.16513 q^{37} +1.67486 q^{39} -6.12029 q^{41} +4.07691 q^{43} -0.382123 q^{45} -2.90037 q^{47} +8.56044 q^{51} +7.26079 q^{53} -3.25888 q^{55} -10.8601 q^{57} -2.21673 q^{59} -7.33909 q^{61} -1.96101 q^{65} +6.79925 q^{67} -1.88933 q^{69} -6.84306 q^{71} -15.1542 q^{73} +1.93352 q^{75} +0.576410 q^{79} -8.37745 q^{81} -9.76879 q^{83} -10.0230 q^{85} -11.4864 q^{87} +13.8886 q^{89} +1.95650 q^{93} +12.7156 q^{95} -10.7055 q^{97} +0.323826 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 6 q^{5} + 12 q^{9} - 2 q^{11} - 10 q^{13} + 4 q^{15} - 16 q^{17} - 14 q^{19} + 8 q^{23} + 12 q^{25} - 2 q^{27} + 2 q^{29} - 8 q^{31} - 10 q^{33} - 6 q^{37} + 2 q^{39} - 22 q^{41} + 2 q^{43}+ \cdots - 24 q^{99}+O(q^{100}) \) 10 * q - 2 * q^3 - 6 * q^5 + 12 * q^9 - 2 * q^11 - 10 * q^13 + 4 * q^15 - 16 * q^17 - 14 * q^19 + 8 * q^23 + 12 * q^25 - 2 * q^27 + 2 * q^29 - 8 * q^31 - 10 * q^33 - 6 * q^37 + 2 * q^39 - 22 * q^41 + 2 * q^43 - 42 * q^45 + 36 * q^47 + 8 * q^51 - 6 * q^53 + 8 * q^55 - 16 * q^57 - 24 * q^59 - 18 * q^61 + 6 * q^65 - 14 * q^67 - 46 * q^69 - 8 * q^71 - 28 * q^73 - 14 * q^75 - 16 * q^79 + 26 * q^81 - 2 * q^83 + 16 * q^85 - 8 * q^87 - 10 * q^89 - 2 * q^93 + 10 * q^95 - 60 * q^97 - 24 * q^99

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