LMFDB - Embedded newform 2704.2.a.z.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2704,2,Mod(1,2704)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2704.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,2,0,-4,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$21.5915487066$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 169)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	2704.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.24698 q^{3} +0.246980 q^{5} +2.35690 q^{7} +2.04892 q^{9} +4.24698 q^{11} +0.554958 q^{15} +2.15883 q^{17} +0.0881460 q^{19} +5.29590 q^{21} -1.49396 q^{23} -4.93900 q^{25} -2.13706 q^{27} +4.63102 q^{29} +6.63102 q^{31} +9.54288 q^{33} +0.582105 q^{35} +5.69202 q^{37} -11.5918 q^{41} +0.295897 q^{43} +0.506041 q^{45} +7.35690 q^{47} -1.44504 q^{49} +4.85086 q^{51} -10.3937 q^{53} +1.04892 q^{55} +0.198062 q^{57} +6.78017 q^{59} +3.47219 q^{61} +4.82908 q^{63} -7.67994 q^{67} -3.35690 q^{69} +8.66487 q^{71} +6.73556 q^{73} -11.0978 q^{75} +10.0097 q^{77} -9.97046 q^{79} -10.9487 q^{81} -1.60925 q^{83} +0.533188 q^{85} +10.4058 q^{87} -2.88471 q^{89} +14.8998 q^{93} +0.0217703 q^{95} -8.05861 q^{97} +8.70171 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 4 q^{5} + 3 q^{7} - 3 q^{9} + 8 q^{11} + 2 q^{15} - 2 q^{17} + 4 q^{19} + 2 q^{21} + 5 q^{23} - 5 q^{25} - q^{27} - q^{29} + 5 q^{31} + 10 q^{33} - 4 q^{35} + 12 q^{37} - 7 q^{41} - 13 q^{43}+ \cdots - q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 4 * q^5 + 3 * q^7 - 3 * q^9 + 8 * q^11 + 2 * q^15 - 2 * q^17 + 4 * q^19 + 2 * q^21 + 5 * q^23 - 5 * q^25 - q^27 - q^29 + 5 * q^31 + 10 * q^33 - 4 * q^35 + 12 * q^37 - 7 * q^41 - 13 * q^43 + 11 * q^45 + 18 * q^47 - 4 * q^49 + q^51 + q^53 - 6 * q^55 + 5 * q^57 + 19 * q^59 + 4 * q^61 + 4 * q^63 + q^67 - 6 * q^69 + 27 * q^71 + 9 * q^73 - 15 * q^75 + 8 * q^77 + 5 * q^79 - q^81 + 7 * q^83 + 5 * q^85 + 18 * q^87 - 11 * q^89 + 22 * q^93 - 3 * q^95 + 7 * q^97 - 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
$3$	2.24698	1.29729	0.648647	−	0.761089i	$-0.275335\pi$
$3$	2.24698	1.29729	0.648647	+	0.761089i	$0.275335\pi$
$4$	0	0
$5$	0.246980	0.110453	0.0552263	−	0.998474i	$-0.482412\pi$
$5$	0.246980	0.110453	0.0552263	+	0.998474i	$0.482412\pi$
$6$	0	0
$7$	2.35690	0.890823	0.445411	−	0.895326i	$-0.353057\pi$
$7$	2.35690	0.890823	0.445411	+	0.895326i	$0.353057\pi$
$8$	0	0
$9$	2.04892	0.682972
$10$	0	0
$11$	4.24698	1.28051	0.640256	−	0.768161i	$-0.278828\pi$
$11$	4.24698	1.28051	0.640256	+	0.768161i	$0.278828\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.554958	0.143290
$16$	0	0
$17$	2.15883	0.523594	0.261797	−	0.965123i	$-0.415685\pi$
$17$	2.15883	0.523594	0.261797	+	0.965123i	$0.415685\pi$
$18$	0	0
$19$	0.0881460	0.0202221	0.0101110	−	0.999949i	$-0.496782\pi$
$19$	0.0881460	0.0202221	0.0101110	+	0.999949i	$0.496782\pi$
$20$	0	0
$21$	5.29590	1.15566
$22$	0	0
$23$	−1.49396	−0.311512	−0.155756	−	0.987796i	$-0.549781\pi$
$23$	−1.49396	−0.311512	−0.155756	+	0.987796i	$0.549781\pi$
$24$	0	0
$25$	−4.93900	−0.987800
$26$	0	0
$27$	−2.13706	−0.411278
$28$	0	0
$29$	4.63102	0.859959	0.429980	−	0.902839i	$-0.358521\pi$
$29$	4.63102	0.859959	0.429980	+	0.902839i	$0.358521\pi$
$30$	0	0
$31$	6.63102	1.19097	0.595483	−	0.803368i	$-0.296961\pi$
$31$	6.63102	1.19097	0.595483	+	0.803368i	$0.296961\pi$
$32$	0	0
$33$	9.54288	1.66120
$34$	0	0
$35$	0.582105	0.0983937
$36$	0	0
$37$	5.69202	0.935763	0.467881	−	0.883791i	$-0.345017\pi$
$37$	5.69202	0.935763	0.467881	+	0.883791i	$0.345017\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.5918	−1.81033	−0.905167	−	0.425056i	$-0.860254\pi$
$41$	−11.5918	−1.81033	−0.905167	+	0.425056i	$0.860254\pi$
$42$	0	0
$43$	0.295897	0.0451239	0.0225619	−	0.999745i	$-0.492818\pi$
$43$	0.295897	0.0451239	0.0225619	+	0.999745i	$0.492818\pi$
$44$	0	0
$45$	0.506041	0.0754361
$46$	0	0
$47$	7.35690	1.07311	0.536557	−	0.843864i	$-0.319725\pi$
$47$	7.35690	1.07311	0.536557	+	0.843864i	$0.319725\pi$
$48$	0	0
$49$	−1.44504	−0.206435
$50$	0	0
$51$	4.85086	0.679256
$52$	0	0
$53$	−10.3937	−1.42769	−0.713844	−	0.700304i	$-0.753048\pi$
$53$	−10.3937	−1.42769	−0.713844	+	0.700304i	$0.753048\pi$
$54$	0	0
$55$	1.04892	0.141436
$56$	0	0
$57$	0.198062	0.0262340
$58$	0	0
$59$	6.78017	0.882703	0.441351	−	0.897334i	$-0.354499\pi$
$59$	6.78017	0.882703	0.441351	+	0.897334i	$0.354499\pi$
$60$	0	0
$61$	3.47219	0.444568	0.222284	−	0.974982i	$-0.428649\pi$
$61$	3.47219	0.444568	0.222284	+	0.974982i	$0.428649\pi$
$62$	0	0
$63$	4.82908	0.608407
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.67994	−0.938254	−0.469127	−	0.883131i	$-0.655431\pi$
$67$	−7.67994	−0.938254	−0.469127	+	0.883131i	$0.655431\pi$
$68$	0	0
$69$	−3.35690	−0.404123
$70$	0	0
$71$	8.66487	1.02833	0.514166	−	0.857691i	$-0.328102\pi$
$71$	8.66487	1.02833	0.514166	+	0.857691i	$0.328102\pi$
$72$	0	0
$73$	6.73556	0.788338	0.394169	−	0.919038i	$-0.371032\pi$
$73$	6.73556	0.788338	0.394169	+	0.919038i	$0.371032\pi$
$74$	0	0
$75$	−11.0978	−1.28147
$76$	0	0
$77$	10.0097	1.14071
$78$	0	0
$79$	−9.97046	−1.12176	−0.560882	−	0.827896i	$-0.689538\pi$
$79$	−9.97046	−1.12176	−0.560882	+	0.827896i	$0.689538\pi$
$80$	0	0
$81$	−10.9487	−1.21652
$82$	0	0
$83$	−1.60925	−0.176638	−0.0883192	−	0.996092i	$-0.528150\pi$
$83$	−1.60925	−0.176638	−0.0883192	+	0.996092i	$0.528150\pi$
$84$	0	0
$85$	0.533188	0.0578323
$86$	0	0
$87$	10.4058	1.11562
$88$	0	0
$89$	−2.88471	−0.305778	−0.152889	−	0.988243i	$-0.548858\pi$
$89$	−2.88471	−0.305778	−0.152889	+	0.988243i	$0.548858\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	14.8998	1.54503
$94$	0	0
$95$	0.0217703	0.00223358
$96$	0	0
$97$	−8.05861	−0.818227	−0.409114	−	0.912483i	$-0.634162\pi$
$97$	−8.05861	−0.818227	−0.409114	+	0.912483i	$0.634162\pi$
$98$	0	0
$99$	8.70171	0.874555
$100$	0	0
$101$	−13.3545	−1.32882	−0.664411	−	0.747367i	$-0.731318\pi$
$101$	−13.3545	−1.32882	−0.664411	+	0.747367i	$0.731318\pi$
$102$	0	0
$103$	−1.36227	−0.134229	−0.0671144	−	0.997745i	$-0.521379\pi$
$103$	−1.36227	−0.134229	−0.0671144	+	0.997745i	$0.521379\pi$
$104$	0	0
$105$	1.30798	0.127646
$106$	0	0
$107$	−3.26875	−0.316002	−0.158001	−	0.987439i	$-0.550505\pi$
$107$	−3.26875	−0.316002	−0.158001	+	0.987439i	$0.550505\pi$
$108$	0	0
$109$	15.7017	1.50395	0.751976	−	0.659191i	$-0.229101\pi$
$109$	15.7017	1.50395	0.751976	+	0.659191i	$0.229101\pi$
$110$	0	0
$111$	12.7899	1.21396
$112$	0	0
$113$	12.0489	1.13347	0.566733	−	0.823901i	$-0.308207\pi$
$113$	12.0489	1.13347	0.566733	+	0.823901i	$0.308207\pi$
$114$	0	0
$115$	−0.368977	−0.0344073
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.08815	0.466430
$120$	0	0
$121$	7.03684	0.639712
$122$	0	0
$123$	−26.0465	−2.34854
$124$	0	0
$125$	−2.45473	−0.219558
$126$	0	0
$127$	9.80731	0.870258	0.435129	−	0.900368i	$-0.356703\pi$
$127$	9.80731	0.870258	0.435129	+	0.900368i	$0.356703\pi$
$128$	0	0
$129$	0.664874	0.0585389
$130$	0	0
$131$	6.57673	0.574611	0.287306	−	0.957839i	$-0.407240\pi$
$131$	6.57673	0.574611	0.287306	+	0.957839i	$0.407240\pi$
$132$	0	0
$133$	0.207751	0.0180143
$134$	0	0
$135$	−0.527811	−0.0454267
$136$	0	0
$137$	6.21983	0.531396	0.265698	−	0.964056i	$-0.414398\pi$
$137$	6.21983	0.531396	0.265698	+	0.964056i	$0.414398\pi$
$138$	0	0
$139$	14.7071	1.24744	0.623719	−	0.781648i	$-0.285621\pi$
$139$	14.7071	1.24744	0.623719	+	0.781648i	$0.285621\pi$
$140$	0	0
$141$	16.5308	1.39214
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.14377	0.0949848
$146$	0	0
$147$	−3.24698	−0.267806
$148$	0	0
$149$	4.33513	0.355147	0.177574	−	0.984108i	$-0.443175\pi$
$149$	4.33513	0.355147	0.177574	+	0.984108i	$0.443175\pi$
$150$	0	0
$151$	−3.94438	−0.320989	−0.160494	−	0.987037i	$-0.551309\pi$
$151$	−3.94438	−0.320989	−0.160494	+	0.987037i	$0.551309\pi$
$152$	0	0
$153$	4.42327	0.357600
$154$	0	0
$155$	1.63773	0.131545
$156$	0	0
$157$	4.45473	0.355526	0.177763	−	0.984073i	$-0.443114\pi$
$157$	4.45473	0.355526	0.177763	+	0.984073i	$0.443114\pi$
$158$	0	0
$159$	−23.3545	−1.85213
$160$	0	0
$161$	−3.52111	−0.277502
$162$	0	0
$163$	−16.1588	−1.26566	−0.632829	−	0.774292i	$-0.718106\pi$
$163$	−16.1588	−1.26566	−0.632829	+	0.774292i	$0.718106\pi$
$164$	0	0
$165$	2.35690	0.183484
$166$	0	0
$167$	−16.1172	−1.24719	−0.623594	−	0.781749i	$-0.714328\pi$
$167$	−16.1172	−1.24719	−0.623594	+	0.781749i	$0.714328\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.180604	0.0138111
$172$	0	0
$173$	−21.5362	−1.63736	−0.818682	−	0.574247i	$-0.805295\pi$
$173$	−21.5362	−1.63736	−0.818682	+	0.574247i	$0.805295\pi$
$174$	0	0
$175$	−11.6407	−0.879955
$176$	0	0
$177$	15.2349	1.14513
$178$	0	0
$179$	−11.4330	−0.854540	−0.427270	−	0.904124i	$-0.640525\pi$
$179$	−11.4330	−0.854540	−0.427270	+	0.904124i	$0.640525\pi$
$180$	0	0
$181$	20.9705	1.55872	0.779361	−	0.626575i	$-0.215544\pi$
$181$	20.9705	1.55872	0.779361	+	0.626575i	$0.215544\pi$
$182$	0	0
$183$	7.80194	0.576736
$184$	0	0
$185$	1.40581	0.103357
$186$	0	0
$187$	9.16852	0.670469
$188$	0	0
$189$	−5.03684	−0.366376
$190$	0	0
$191$	14.4373	1.04464	0.522322	−	0.852748i	$-0.325066\pi$
$191$	14.4373	1.04464	0.522322	+	0.852748i	$0.325066\pi$
$192$	0	0
$193$	−13.5797	−0.977489	−0.488745	−	0.872427i	$-0.662545\pi$
$193$	−13.5797	−0.977489	−0.488745	+	0.872427i	$0.662545\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.560335	−0.0399222	−0.0199611	−	0.999801i	$-0.506354\pi$
$197$	−0.560335	−0.0399222	−0.0199611	+	0.999801i	$0.506354\pi$
$198$	0	0
$199$	−11.4916	−0.814616	−0.407308	−	0.913291i	$-0.633532\pi$
$199$	−11.4916	−0.814616	−0.407308	+	0.913291i	$0.633532\pi$
$200$	0	0
$201$	−17.2567	−1.21719
$202$	0	0
$203$	10.9148	0.766071
$204$	0	0
$205$	−2.86294	−0.199956
$206$	0	0
$207$	−3.06100	−0.212754
$208$	0	0
$209$	0.374354	0.0258946
$210$	0	0
$211$	−8.78448	−0.604748	−0.302374	−	0.953189i	$-0.597779\pi$
$211$	−8.78448	−0.604748	−0.302374	+	0.953189i	$0.597779\pi$
$212$	0	0
$213$	19.4698	1.33405
$214$	0	0
$215$	0.0730805	0.00498405
$216$	0	0
$217$	15.6286	1.06094
$218$	0	0
$219$	15.1347	1.02271
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2.25906	0.151278	0.0756390	−	0.997135i	$-0.475900\pi$
$223$	2.25906	0.151278	0.0756390	+	0.997135i	$0.475900\pi$
$224$	0	0
$225$	−10.1196	−0.674640
$226$	0	0
$227$	6.96615	0.462359	0.231180	−	0.972911i	$-0.425741\pi$
$227$	6.96615	0.462359	0.231180	+	0.972911i	$0.425741\pi$
$228$	0	0
$229$	−24.1739	−1.59746	−0.798728	−	0.601692i	$-0.794493\pi$
$229$	−24.1739	−1.59746	−0.798728	+	0.601692i	$0.794493\pi$
$230$	0	0
$231$	22.4916	1.47984
$232$	0	0
$233$	−3.06100	−0.200533	−0.100266	−	0.994961i	$-0.531969\pi$
$233$	−3.06100	−0.200533	−0.100266	+	0.994961i	$0.531969\pi$
$234$	0	0
$235$	1.81700	0.118528
$236$	0	0
$237$	−22.4034	−1.45526
$238$	0	0
$239$	25.1468	1.62661	0.813304	−	0.581839i	$-0.197667\pi$
$239$	25.1468	1.62661	0.813304	+	0.581839i	$0.197667\pi$
$240$	0	0
$241$	−20.2664	−1.30547	−0.652735	−	0.757586i	$-0.726379\pi$
$241$	−20.2664	−1.30547	−0.652735	+	0.757586i	$0.726379\pi$
$242$	0	0
$243$	−18.1903	−1.16691
$244$	0	0
$245$	−0.356896	−0.0228012
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−3.61596	−0.229152
$250$	0	0
$251$	23.7211	1.49726	0.748631	−	0.662987i	$-0.230712\pi$
$251$	23.7211	1.49726	0.748631	+	0.662987i	$0.230712\pi$
$252$	0	0
$253$	−6.34481	−0.398895
$254$	0	0
$255$	1.19806	0.0750256
$256$	0	0
$257$	14.2241	0.887278	0.443639	−	0.896206i	$-0.353687\pi$
$257$	14.2241	0.887278	0.443639	+	0.896206i	$0.353687\pi$
$258$	0	0
$259$	13.4155	0.833599
$260$	0	0
$261$	9.48858	0.587329
$262$	0	0
$263$	17.0954	1.05415	0.527075	−	0.849819i	$-0.323289\pi$
$263$	17.0954	1.05415	0.527075	+	0.849819i	$0.323289\pi$
$264$	0	0
$265$	−2.56704	−0.157692
$266$	0	0
$267$	−6.48188	−0.396684
$268$	0	0
$269$	−6.46681	−0.394288	−0.197144	−	0.980374i	$-0.563167\pi$
$269$	−6.46681	−0.394288	−0.197144	+	0.980374i	$0.563167\pi$
$270$	0	0
$271$	−6.44803	−0.391690	−0.195845	−	0.980635i	$-0.562745\pi$
$271$	−6.44803	−0.391690	−0.195845	+	0.980635i	$0.562745\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−20.9758	−1.26489
$276$	0	0
$277$	13.4601	0.808739	0.404370	−	0.914596i	$-0.367491\pi$
$277$	13.4601	0.808739	0.404370	+	0.914596i	$0.367491\pi$
$278$	0	0
$279$	13.5864	0.813398
$280$	0	0
$281$	5.03684	0.300472	0.150236	−	0.988650i	$-0.451997\pi$
$281$	5.03684	0.300472	0.150236	+	0.988650i	$0.451997\pi$
$282$	0	0
$283$	−22.1280	−1.31537	−0.657686	−	0.753293i	$-0.728464\pi$
$283$	−22.1280	−1.31537	−0.657686	+	0.753293i	$0.728464\pi$
$284$	0	0
$285$	0.0489173	0.00289761
$286$	0	0
$287$	−27.3207	−1.61269
$288$	0	0
$289$	−12.3394	−0.725849
$290$	0	0
$291$	−18.1075	−1.06148
$292$	0	0
$293$	14.9463	0.873172	0.436586	−	0.899663i	$-0.356187\pi$
$293$	14.9463	0.873172	0.436586	+	0.899663i	$0.356187\pi$
$294$	0	0
$295$	1.67456	0.0974968
$296$	0	0
$297$	−9.07606	−0.526647
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0.697398	0.0401974
$302$	0	0
$303$	−30.0073	−1.72387
$304$	0	0
$305$	0.857560	0.0491037
$306$	0	0
$307$	−19.1293	−1.09177	−0.545883	−	0.837861i	$-0.683806\pi$
$307$	−19.1293	−1.09177	−0.545883	+	0.837861i	$0.683806\pi$
$308$	0	0
$309$	−3.06100	−0.174134
$310$	0	0
$311$	0.269815	0.0152998	0.00764990	−	0.999971i	$-0.497565\pi$
$311$	0.269815	0.0152998	0.00764990	+	0.999971i	$0.497565\pi$
$312$	0	0
$313$	−23.3937	−1.32229	−0.661146	−	0.750257i	$-0.729930\pi$
$313$	−23.3937	−1.32229	−0.661146	+	0.750257i	$0.729930\pi$
$314$	0	0
$315$	1.19269	0.0672002
$316$	0	0
$317$	−13.9952	−0.786050	−0.393025	−	0.919528i	$-0.628571\pi$
$317$	−13.9952	−0.786050	−0.393025	+	0.919528i	$0.628571\pi$
$318$	0	0
$319$	19.6679	1.10119
$320$	0	0
$321$	−7.34481	−0.409948
$322$	0	0
$323$	0.190293	0.0105882
$324$	0	0
$325$	0	0
$326$	0	0
$327$	35.2814	1.95107
$328$	0	0
$329$	17.3394	0.955954
$330$	0	0
$331$	−17.8213	−0.979548	−0.489774	−	0.871849i	$-0.662921\pi$
$331$	−17.8213	−0.979548	−0.489774	+	0.871849i	$0.662921\pi$
$332$	0	0
$333$	11.6625	0.639100
$334$	0	0
$335$	−1.89679	−0.103633
$336$	0	0
$337$	−27.8485	−1.51700	−0.758501	−	0.651672i	$-0.774068\pi$
$337$	−27.8485	−1.51700	−0.758501	+	0.651672i	$0.774068\pi$
$338$	0	0
$339$	27.0737	1.47044
$340$	0	0
$341$	28.1618	1.52505
$342$	0	0
$343$	−19.9041	−1.07472
$344$	0	0
$345$	−0.829085	−0.0446364
$346$	0	0
$347$	−1.50365	−0.0807200	−0.0403600	−	0.999185i	$-0.512850\pi$
$347$	−1.50365	−0.0807200	−0.0403600	+	0.999185i	$0.512850\pi$
$348$	0	0
$349$	−14.1860	−0.759358	−0.379679	−	0.925118i	$-0.623966\pi$
$349$	−14.1860	−0.759358	−0.379679	+	0.925118i	$0.623966\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.16852	−0.381542	−0.190771	−	0.981635i	$-0.561099\pi$
$353$	−7.16852	−0.381542	−0.190771	+	0.981635i	$0.561099\pi$
$354$	0	0
$355$	2.14005	0.113582
$356$	0	0
$357$	11.4330	0.605096
$358$	0	0
$359$	−19.8853	−1.04951	−0.524753	−	0.851255i	$-0.675842\pi$
$359$	−19.8853	−1.04951	−0.524753	+	0.851255i	$0.675842\pi$
$360$	0	0
$361$	−18.9922	−0.999591
$362$	0	0
$363$	15.8116	0.829895
$364$	0	0
$365$	1.66355	0.0870740
$366$	0	0
$367$	−1.08383	−0.0565757	−0.0282878	−	0.999600i	$-0.509006\pi$
$367$	−1.08383	−0.0565757	−0.0282878	+	0.999600i	$0.509006\pi$
$368$	0	0
$369$	−23.7506	−1.23641
$370$	0	0
$371$	−24.4969	−1.27182
$372$	0	0
$373$	−6.13036	−0.317418	−0.158709	−	0.987325i	$-0.550733\pi$
$373$	−6.13036	−0.317418	−0.158709	+	0.987325i	$0.550733\pi$
$374$	0	0
$375$	−5.51573	−0.284831
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2.40880	0.123732	0.0618658	−	0.998084i	$-0.480295\pi$
$379$	2.40880	0.123732	0.0618658	+	0.998084i	$0.480295\pi$
$380$	0	0
$381$	22.0368	1.12898
$382$	0	0
$383$	30.3913	1.55292	0.776462	−	0.630164i	$-0.217012\pi$
$383$	30.3913	1.55292	0.776462	+	0.630164i	$0.217012\pi$
$384$	0	0
$385$	2.47219	0.125994
$386$	0	0
$387$	0.606268	0.0308184
$388$	0	0
$389$	−15.9409	−0.808237	−0.404118	−	0.914707i	$-0.632422\pi$
$389$	−15.9409	−0.808237	−0.404118	+	0.914707i	$0.632422\pi$
$390$	0	0
$391$	−3.22521	−0.163106
$392$	0	0
$393$	14.7778	0.745440
$394$	0	0
$395$	−2.46250	−0.123902
$396$	0	0
$397$	16.9148	0.848931	0.424466	−	0.905444i	$-0.360462\pi$
$397$	16.9148	0.848931	0.424466	+	0.905444i	$0.360462\pi$
$398$	0	0
$399$	0.466812	0.0233698
$400$	0	0
$401$	26.6625	1.33146	0.665730	−	0.746192i	$-0.268120\pi$
$401$	26.6625	1.33146	0.665730	+	0.746192i	$0.268120\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2.70410	−0.134368
$406$	0	0
$407$	24.1739	1.19826
$408$	0	0
$409$	28.5163	1.41004	0.705021	−	0.709187i	$-0.250938\pi$
$409$	28.5163	1.41004	0.705021	+	0.709187i	$0.250938\pi$
$410$	0	0
$411$	13.9758	0.689377
$412$	0	0
$413$	15.9801	0.786332
$414$	0	0
$415$	−0.397452	−0.0195102
$416$	0	0
$417$	33.0465	1.61830
$418$	0	0
$419$	29.6093	1.44651	0.723253	−	0.690583i	$-0.242646\pi$
$419$	29.6093	1.44651	0.723253	+	0.690583i	$0.242646\pi$
$420$	0	0
$421$	−11.6606	−0.568301	−0.284151	−	0.958780i	$-0.591712\pi$
$421$	−11.6606	−0.568301	−0.284151	+	0.958780i	$0.591712\pi$
$422$	0	0
$423$	15.0737	0.732907
$424$	0	0
$425$	−10.6625	−0.517206
$426$	0	0
$427$	8.18359	0.396032
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.34913	−0.209490	−0.104745	−	0.994499i	$-0.533403\pi$
$431$	−4.34913	−0.209490	−0.104745	+	0.994499i	$0.533403\pi$
$432$	0	0
$433$	−14.3884	−0.691460	−0.345730	−	0.938334i	$-0.612369\pi$
$433$	−14.3884	−0.691460	−0.345730	+	0.938334i	$0.612369\pi$
$434$	0	0
$435$	2.57002	0.123223
$436$	0	0
$437$	−0.131687	−0.00629942
$438$	0	0
$439$	20.2325	0.965645	0.482822	−	0.875718i	$-0.339612\pi$
$439$	20.2325	0.965645	0.482822	+	0.875718i	$0.339612\pi$
$440$	0	0
$441$	−2.96077	−0.140989
$442$	0	0
$443$	−8.12200	−0.385888	−0.192944	−	0.981210i	$-0.561804\pi$
$443$	−8.12200	−0.385888	−0.192944	+	0.981210i	$0.561804\pi$
$444$	0	0
$445$	−0.712464	−0.0337740
$446$	0	0
$447$	9.74094	0.460731
$448$	0	0
$449$	12.4916	0.589513	0.294757	−	0.955572i	$-0.404761\pi$
$449$	12.4916	0.589513	0.294757	+	0.955572i	$0.404761\pi$
$450$	0	0
$451$	−49.2301	−2.31816
$452$	0	0
$453$	−8.86294	−0.416417
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.98121	0.279789	0.139895	−	0.990166i	$-0.455324\pi$
$457$	5.98121	0.279789	0.139895	+	0.990166i	$0.455324\pi$
$458$	0	0
$459$	−4.61356	−0.215343
$460$	0	0
$461$	−2.05669	−0.0957895	−0.0478947	−	0.998852i	$-0.515251\pi$
$461$	−2.05669	−0.0957895	−0.0478947	+	0.998852i	$0.515251\pi$
$462$	0	0
$463$	8.44935	0.392675	0.196337	−	0.980536i	$-0.437095\pi$
$463$	8.44935	0.392675	0.196337	+	0.980536i	$0.437095\pi$
$464$	0	0
$465$	3.67994	0.170653
$466$	0	0
$467$	−33.5139	−1.55084	−0.775420	−	0.631446i	$-0.782462\pi$
$467$	−33.5139	−1.55084	−0.775420	+	0.631446i	$0.782462\pi$
$468$	0	0
$469$	−18.1008	−0.835818
$470$	0	0
$471$	10.0097	0.461222
$472$	0	0
$473$	1.25667	0.0577817
$474$	0	0
$475$	−0.435353	−0.0199754
$476$	0	0
$477$	−21.2959	−0.975072
$478$	0	0
$479$	24.7313	1.13000	0.565000	−	0.825091i	$-0.308876\pi$
$479$	24.7313	1.13000	0.565000	+	0.825091i	$0.308876\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−7.91185	−0.360002
$484$	0	0
$485$	−1.99031	−0.0903754
$486$	0	0
$487$	37.7555	1.71087	0.855433	−	0.517913i	$-0.173291\pi$
$487$	37.7555	1.71087	0.855433	+	0.517913i	$0.173291\pi$
$488$	0	0
$489$	−36.3086	−1.64193
$490$	0	0
$491$	−31.3110	−1.41304	−0.706522	−	0.707691i	$-0.749737\pi$
$491$	−31.3110	−1.41304	−0.706522	+	0.707691i	$0.749737\pi$
$492$	0	0
$493$	9.99761	0.450270
$494$	0	0
$495$	2.14914	0.0965969
$496$	0	0
$497$	20.4222	0.916061
$498$	0	0
$499$	−21.4873	−0.961902	−0.480951	−	0.876748i	$-0.659708\pi$
$499$	−21.4873	−0.961902	−0.480951	+	0.876748i	$0.659708\pi$
$500$	0	0
$501$	−36.2150	−1.61797
$502$	0	0
$503$	−37.5924	−1.67616	−0.838081	−	0.545546i	$-0.816322\pi$
$503$	−37.5924	−1.67616	−0.838081	+	0.545546i	$0.816322\pi$
$504$	0	0
$505$	−3.29829	−0.146772
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17.1075	−0.758278	−0.379139	−	0.925340i	$-0.623780\pi$
$509$	−17.1075	−0.758278	−0.379139	+	0.925340i	$0.623780\pi$
$510$	0	0
$511$	15.8750	0.702269
$512$	0	0
$513$	−0.188374	−0.00831690
$514$	0	0
$515$	−0.336454	−0.0148259
$516$	0	0
$517$	31.2446	1.37414
$518$	0	0
$519$	−48.3913	−2.12414
$520$	0	0
$521$	−19.8465	−0.869493	−0.434746	−	0.900553i	$-0.643162\pi$
$521$	−19.8465	−0.869493	−0.434746	+	0.900553i	$0.643162\pi$
$522$	0	0
$523$	11.4300	0.499798	0.249899	−	0.968272i	$-0.419603\pi$
$523$	11.4300	0.499798	0.249899	+	0.968272i	$0.419603\pi$
$524$	0	0
$525$	−26.1564	−1.14156
$526$	0	0
$527$	14.3153	0.623583
$528$	0	0
$529$	−20.7681	−0.902960
$530$	0	0
$531$	13.8920	0.602862
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−0.807315	−0.0349033
$536$	0	0
$537$	−25.6896	−1.10859
$538$	0	0
$539$	−6.13706	−0.264342
$540$	0	0
$541$	16.1884	0.695993	0.347996	−	0.937496i	$-0.386862\pi$
$541$	16.1884	0.695993	0.347996	+	0.937496i	$0.386862\pi$
$542$	0	0
$543$	47.1202	2.02212
$544$	0	0
$545$	3.87800	0.166115
$546$	0	0
$547$	−5.33081	−0.227929	−0.113965	−	0.993485i	$-0.536355\pi$
$547$	−5.33081	−0.227929	−0.113965	+	0.993485i	$0.536355\pi$
$548$	0	0
$549$	7.11423	0.303628
$550$	0	0
$551$	0.408206	0.0173902
$552$	0	0
$553$	−23.4993	−0.999293
$554$	0	0
$555$	3.15883	0.134085
$556$	0	0
$557$	7.39075	0.313156	0.156578	−	0.987666i	$-0.449954\pi$
$557$	7.39075	0.313156	0.156578	+	0.987666i	$0.449954\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	20.6015	0.869795
$562$	0	0
$563$	9.47889	0.399488	0.199744	−	0.979848i	$-0.435989\pi$
$563$	9.47889	0.399488	0.199744	+	0.979848i	$0.435989\pi$
$564$	0	0
$565$	2.97584	0.125194
$566$	0	0
$567$	−25.8049	−1.08370
$568$	0	0
$569$	−10.1438	−0.425249	−0.212624	−	0.977134i	$-0.568201\pi$
$569$	−10.1438	−0.425249	−0.212624	+	0.977134i	$0.568201\pi$
$570$	0	0
$571$	14.0925	0.589751	0.294876	−	0.955536i	$-0.404722\pi$
$571$	14.0925	0.589751	0.294876	+	0.955536i	$0.404722\pi$
$572$	0	0
$573$	32.4403	1.35521
$574$	0	0
$575$	7.37867	0.307712
$576$	0	0
$577$	25.1545	1.04720	0.523598	−	0.851965i	$-0.324589\pi$
$577$	25.1545	1.04720	0.523598	+	0.851965i	$0.324589\pi$
$578$	0	0
$579$	−30.5133	−1.26809
$580$	0	0
$581$	−3.79284	−0.157354
$582$	0	0
$583$	−44.1420	−1.82817
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−43.8353	−1.80928	−0.904639	−	0.426180i	$-0.859859\pi$
$587$	−43.8353	−1.80928	−0.904639	+	0.426180i	$0.859859\pi$
$588$	0	0
$589$	0.584498	0.0240838
$590$	0	0
$591$	−1.25906	−0.0517909
$592$	0	0
$593$	−24.9965	−1.02648	−0.513242	−	0.858244i	$-0.671556\pi$
$593$	−24.9965	−1.02648	−0.513242	+	0.858244i	$0.671556\pi$
$594$	0	0
$595$	1.25667	0.0515184
$596$	0	0
$597$	−25.8213	−1.05680
$598$	0	0
$599$	6.24027	0.254971	0.127485	−	0.991840i	$-0.459309\pi$
$599$	6.24027	0.254971	0.127485	+	0.991840i	$0.459309\pi$
$600$	0	0
$601$	6.32975	0.258196	0.129098	−	0.991632i	$-0.458792\pi$
$601$	6.32975	0.258196	0.129098	+	0.991632i	$0.458792\pi$
$602$	0	0
$603$	−15.7356	−0.640802
$604$	0	0
$605$	1.73795	0.0706579
$606$	0	0
$607$	43.6480	1.77162	0.885809	−	0.464050i	$-0.153604\pi$
$607$	43.6480	1.77162	0.885809	+	0.464050i	$0.153604\pi$
$608$	0	0
$609$	24.5254	0.993820
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−25.9541	−1.04827	−0.524137	−	0.851634i	$-0.675612\pi$
$613$	−25.9541	−1.04827	−0.524137	+	0.851634i	$0.675612\pi$
$614$	0	0
$615$	−6.43296	−0.259402
$616$	0	0
$617$	45.9396	1.84946	0.924729	−	0.380626i	$-0.124291\pi$
$617$	45.9396	1.84946	0.924729	+	0.380626i	$0.124291\pi$
$618$	0	0
$619$	−6.73556	−0.270725	−0.135363	−	0.990796i	$-0.543220\pi$
$619$	−6.73556	−0.270725	−0.135363	+	0.990796i	$0.543220\pi$
$620$	0	0
$621$	3.19269	0.128118
$622$	0	0
$623$	−6.79895	−0.272394
$624$	0	0
$625$	24.0887	0.963549
$626$	0	0
$627$	0.841166	0.0335930
$628$	0	0
$629$	12.2881	0.489960
$630$	0	0
$631$	−45.0998	−1.79539	−0.897696	−	0.440614i	$-0.854761\pi$
$631$	−45.0998	−1.79539	−0.897696	+	0.440614i	$0.854761\pi$
$632$	0	0
$633$	−19.7385	−0.784537
$634$	0	0
$635$	2.42221	0.0961223
$636$	0	0
$637$	0	0
$638$	0	0
$639$	17.7536	0.702322
$640$	0	0
$641$	32.5821	1.28692	0.643458	−	0.765482i	$-0.277499\pi$
$641$	32.5821	1.28692	0.643458	+	0.765482i	$0.277499\pi$
$642$	0	0
$643$	−25.5754	−1.00860	−0.504298	−	0.863530i	$-0.668249\pi$
$643$	−25.5754	−1.00860	−0.504298	+	0.863530i	$0.668249\pi$
$644$	0	0
$645$	0.164210	0.00646578
$646$	0	0
$647$	30.1715	1.18616	0.593082	−	0.805142i	$-0.297911\pi$
$647$	30.1715	1.18616	0.593082	+	0.805142i	$0.297911\pi$
$648$	0	0
$649$	28.7952	1.13031
$650$	0	0
$651$	35.1172	1.37635
$652$	0	0
$653$	36.9028	1.44412	0.722058	−	0.691832i	$-0.243196\pi$
$653$	36.9028	1.44412	0.722058	+	0.691832i	$0.243196\pi$
$654$	0	0
$655$	1.62432	0.0634673
$656$	0	0
$657$	13.8006	0.538413
$658$	0	0
$659$	−23.6866	−0.922701	−0.461350	−	0.887218i	$-0.652635\pi$
$659$	−23.6866	−0.922701	−0.461350	+	0.887218i	$0.652635\pi$
$660$	0	0
$661$	−31.7590	−1.23528	−0.617641	−	0.786460i	$-0.711911\pi$
$661$	−31.7590	−1.23528	−0.617641	+	0.786460i	$0.711911\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.0513102	0.00198973
$666$	0	0
$667$	−6.91856	−0.267888
$668$	0	0
$669$	5.07606	0.196252
$670$	0	0
$671$	14.7463	0.569275
$672$	0	0
$673$	−7.50232	−0.289193	−0.144597	−	0.989491i	$-0.546188\pi$
$673$	−7.50232	−0.289193	−0.144597	+	0.989491i	$0.546188\pi$
$674$	0	0
$675$	10.5550	0.406261
$676$	0	0
$677$	−35.0315	−1.34637	−0.673184	−	0.739475i	$-0.735074\pi$
$677$	−35.0315	−1.34637	−0.673184	+	0.739475i	$0.735074\pi$
$678$	0	0
$679$	−18.9933	−0.728896
$680$	0	0
$681$	15.6528	0.599816
$682$	0	0
$683$	24.0834	0.921524	0.460762	−	0.887524i	$-0.347576\pi$
$683$	24.0834	0.921524	0.460762	+	0.887524i	$0.347576\pi$
$684$	0	0
$685$	1.53617	0.0586941
$686$	0	0
$687$	−54.3183	−2.07237
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−2.01447	−0.0766342	−0.0383171	−	0.999266i	$-0.512200\pi$
$691$	−2.01447	−0.0766342	−0.0383171	+	0.999266i	$0.512200\pi$
$692$	0	0
$693$	20.5090	0.779073
$694$	0	0
$695$	3.63235	0.137783
$696$	0	0
$697$	−25.0248	−0.947880
$698$	0	0
$699$	−6.87800	−0.260150
$700$	0	0
$701$	−48.8189	−1.84387	−0.921933	−	0.387350i	$-0.873390\pi$
$701$	−48.8189	−1.84387	−0.921933	+	0.387350i	$0.873390\pi$
$702$	0	0
$703$	0.501729	0.0189231
$704$	0	0
$705$	4.08277	0.153766
$706$	0	0
$707$	−31.4752	−1.18375
$708$	0	0
$709$	−20.8060	−0.781385	−0.390693	−	0.920521i	$-0.627764\pi$
$709$	−20.8060	−0.781385	−0.390693	+	0.920521i	$0.627764\pi$
$710$	0	0
$711$	−20.4286	−0.766134
$712$	0	0
$713$	−9.90648	−0.371000
$714$	0	0
$715$	0	0
$716$	0	0
$717$	56.5042	2.11019
$718$	0	0
$719$	−21.4306	−0.799225	−0.399613	−	0.916684i	$-0.630855\pi$
$719$	−21.4306	−0.799225	−0.399613	+	0.916684i	$0.630855\pi$
$720$	0	0
$721$	−3.21073	−0.119574
$722$	0	0
$723$	−45.5381	−1.69358
$724$	0	0
$725$	−22.8726	−0.849468
$726$	0	0
$727$	−13.4862	−0.500175	−0.250088	−	0.968223i	$-0.580459\pi$
$727$	−13.4862	−0.500175	−0.250088	+	0.968223i	$0.580459\pi$
$728$	0	0
$729$	−8.02715	−0.297302
$730$	0	0
$731$	0.638792	0.0236266
$732$	0	0
$733$	−43.5424	−1.60828	−0.804138	−	0.594443i	$-0.797373\pi$
$733$	−43.5424	−1.60828	−0.804138	+	0.594443i	$0.797373\pi$
$734$	0	0
$735$	−0.801938	−0.0295799
$736$	0	0
$737$	−32.6165	−1.20145
$738$	0	0
$739$	−20.0543	−0.737709	−0.368855	−	0.929487i	$-0.620250\pi$
$739$	−20.0543	−0.737709	−0.368855	+	0.929487i	$0.620250\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.1685	1.21684	0.608418	−	0.793617i	$-0.291805\pi$
$743$	33.1685	1.21684	0.608418	+	0.793617i	$0.291805\pi$
$744$	0	0
$745$	1.07069	0.0392270
$746$	0	0
$747$	−3.29722	−0.120639
$748$	0	0
$749$	−7.70410	−0.281502
$750$	0	0
$751$	−39.2814	−1.43340	−0.716700	−	0.697382i	$-0.754348\pi$
$751$	−39.2814	−1.43340	−0.716700	+	0.697382i	$0.754348\pi$
$752$	0	0
$753$	53.3008	1.94239
$754$	0	0
$755$	−0.974181	−0.0354541
$756$	0	0
$757$	−46.6426	−1.69526	−0.847628	−	0.530592i	$-0.821970\pi$
$757$	−46.6426	−1.69526	−0.847628	+	0.530592i	$0.821970\pi$
$758$	0	0
$759$	−14.2567	−0.517484
$760$	0	0
$761$	−21.8984	−0.793818	−0.396909	−	0.917858i	$-0.629917\pi$
$761$	−21.8984	−0.793818	−0.396909	+	0.917858i	$0.629917\pi$
$762$	0	0
$763$	37.0073	1.33975
$764$	0	0
$765$	1.09246	0.0394979
$766$	0	0
$767$	0	0
$768$	0	0
$769$	46.7096	1.68439	0.842196	−	0.539172i	$-0.181263\pi$
$769$	46.7096	1.68439	0.842196	+	0.539172i	$0.181263\pi$
$770$	0	0
$771$	31.9614	1.15106
$772$	0	0
$773$	−30.2416	−1.08771	−0.543857	−	0.839178i	$-0.683037\pi$
$773$	−30.2416	−1.08771	−0.543857	+	0.839178i	$0.683037\pi$
$774$	0	0
$775$	−32.7506	−1.17644
$776$	0	0
$777$	30.1444	1.08142
$778$	0	0
$779$	−1.02177	−0.0366087
$780$	0	0
$781$	36.7995	1.31679
$782$	0	0
$783$	−9.89679	−0.353682
$784$	0	0
$785$	1.10023	0.0392688
$786$	0	0
$787$	−28.7023	−1.02313	−0.511563	−	0.859246i	$-0.670933\pi$
$787$	−28.7023	−1.02313	−0.511563	+	0.859246i	$0.670933\pi$
$788$	0	0
$789$	38.4131	1.36754
$790$	0	0
$791$	28.3980	1.00972
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−5.76809	−0.204573
$796$	0	0
$797$	−18.5418	−0.656785	−0.328392	−	0.944541i	$-0.606507\pi$
$797$	−18.5418	−0.656785	−0.328392	+	0.944541i	$0.606507\pi$
$798$	0	0
$799$	15.8823	0.561876
$800$	0	0
$801$	−5.91053	−0.208838
$802$	0	0
$803$	28.6058	1.00948
$804$	0	0
$805$	−0.869641	−0.0306508
$806$	0	0
$807$	−14.5308	−0.511508
$808$	0	0
$809$	−10.0677	−0.353962	−0.176981	−	0.984214i	$-0.556633\pi$
$809$	−10.0677	−0.353962	−0.176981	+	0.984214i	$0.556633\pi$
$810$	0	0
$811$	10.0285	0.352147	0.176074	−	0.984377i	$-0.443660\pi$
$811$	10.0285	0.352147	0.176074	+	0.984377i	$0.443660\pi$
$812$	0	0
$813$	−14.4886	−0.508137
$814$	0	0
$815$	−3.99090	−0.139795
$816$	0	0
$817$	0.0260821	0.000912498 0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.1704	−0.913355	−0.456677	−	0.889632i	$-0.650961\pi$
$821$	−26.1704	−0.913355	−0.456677	+	0.889632i	$0.650961\pi$
$822$	0	0
$823$	−1.82238	−0.0635242	−0.0317621	−	0.999495i	$-0.510112\pi$
$823$	−1.82238	−0.0635242	−0.0317621	+	0.999495i	$0.510112\pi$
$824$	0	0
$825$	−47.1323	−1.64094
$826$	0	0
$827$	32.2941	1.12298	0.561488	−	0.827485i	$-0.310229\pi$
$827$	32.2941	1.12298	0.561488	+	0.827485i	$0.310229\pi$
$828$	0	0
$829$	15.1002	0.524453	0.262226	−	0.965006i	$-0.415543\pi$
$829$	15.1002	0.524453	0.262226	+	0.965006i	$0.415543\pi$
$830$	0	0
$831$	30.2446	1.04917
$832$	0	0
$833$	−3.11960	−0.108088
$834$	0	0
$835$	−3.98062	−0.137755
$836$	0	0
$837$	−14.1709	−0.489818
$838$	0	0
$839$	32.9965	1.13917	0.569584	−	0.821933i	$-0.307104\pi$
$839$	32.9965	1.13917	0.569584	+	0.821933i	$0.307104\pi$
$840$	0	0
$841$	−7.55363	−0.260470
$842$	0	0
$843$	11.3177	0.389801
$844$	0	0
$845$	0	0
$846$	0	0
$847$	16.5851	0.569870
$848$	0	0
$849$	−49.7211	−1.70642
$850$	0	0
$851$	−8.50365	−0.291501
$852$	0	0
$853$	−37.7802	−1.29357	−0.646784	−	0.762673i	$-0.723887\pi$
$853$	−37.7802	−1.29357	−0.646784	+	0.762673i	$0.723887\pi$
$854$	0	0
$855$	0.0446055	0.00152547
$856$	0	0
$857$	27.3623	0.934677	0.467339	−	0.884078i	$-0.345213\pi$
$857$	27.3623	0.934677	0.467339	+	0.884078i	$0.345213\pi$
$858$	0	0
$859$	20.0629	0.684538	0.342269	−	0.939602i	$-0.388805\pi$
$859$	20.0629	0.684538	0.342269	+	0.939602i	$0.388805\pi$
$860$	0	0
$861$	−61.3889	−2.09213
$862$	0	0
$863$	6.14483	0.209173	0.104586	−	0.994516i	$-0.466648\pi$
$863$	6.14483	0.209173	0.104586	+	0.994516i	$0.466648\pi$
$864$	0	0
$865$	−5.31900	−0.180851
$866$	0	0
$867$	−27.7265	−0.941640
$868$	0	0
$869$	−42.3443	−1.43643
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−16.5114	−0.558827
$874$	0	0
$875$	−5.78554	−0.195587
$876$	0	0
$877$	−13.5077	−0.456123	−0.228061	−	0.973647i	$-0.573239\pi$
$877$	−13.5077	−0.456123	−0.228061	+	0.973647i	$0.573239\pi$
$878$	0	0
$879$	33.5840	1.13276
$880$	0	0
$881$	−5.23431	−0.176348	−0.0881741	−	0.996105i	$-0.528103\pi$
$881$	−5.23431	−0.176348	−0.0881741	+	0.996105i	$0.528103\pi$
$882$	0	0
$883$	4.57301	0.153894	0.0769470	−	0.997035i	$-0.475483\pi$
$883$	4.57301	0.153894	0.0769470	+	0.997035i	$0.475483\pi$
$884$	0	0
$885$	3.76271	0.126482
$886$	0	0
$887$	1.64071	0.0550897	0.0275448	−	0.999621i	$-0.491231\pi$
$887$	1.64071	0.0550897	0.0275448	+	0.999621i	$0.491231\pi$
$888$	0	0
$889$	23.1148	0.775246
$890$	0	0
$891$	−46.4989	−1.55777
$892$	0	0
$893$	0.648481	0.0217006
$894$	0	0
$895$	−2.82371	−0.0943861
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30.7084	1.02418
$900$	0	0
$901$	−22.4383	−0.747529
$902$	0	0
$903$	1.56704	0.0521478
$904$	0	0
$905$	5.17928	0.172165
$906$	0	0
$907$	−8.10215	−0.269027	−0.134514	−	0.990912i	$-0.542947\pi$
$907$	−8.10215	−0.269027	−0.134514	+	0.990912i	$0.542947\pi$
$908$	0	0
$909$	−27.3623	−0.907549
$910$	0	0
$911$	9.18119	0.304187	0.152093	−	0.988366i	$-0.451399\pi$
$911$	9.18119	0.304187	0.152093	+	0.988366i	$0.451399\pi$
$912$	0	0
$913$	−6.83446	−0.226188
$914$	0	0
$915$	1.92692	0.0637020
$916$	0	0
$917$	15.5007	0.511877
$918$	0	0
$919$	−27.5036	−0.907262	−0.453631	−	0.891190i	$-0.649872\pi$
$919$	−27.5036	−0.907262	−0.453631	+	0.891190i	$0.649872\pi$
$920$	0	0
$921$	−42.9831	−1.41634
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−28.1129	−0.924346
$926$	0	0
$927$	−2.79118	−0.0916745
$928$	0	0
$929$	24.2131	0.794407	0.397203	−	0.917731i	$-0.369981\pi$
$929$	24.2131	0.794407	0.397203	+	0.917731i	$0.369981\pi$
$930$	0	0
$931$	−0.127375	−0.00417454
$932$	0	0
$933$	0.606268	0.0198483
$934$	0	0
$935$	2.26444	0.0740550
$936$	0	0
$937$	11.1830	0.365333	0.182666	−	0.983175i	$-0.441527\pi$
$937$	11.1830	0.365333	0.182666	+	0.983175i	$0.441527\pi$
$938$	0	0
$939$	−52.5652	−1.71540
$940$	0	0
$941$	−15.9638	−0.520404	−0.260202	−	0.965554i	$-0.583789\pi$
$941$	−15.9638	−0.520404	−0.260202	+	0.965554i	$0.583789\pi$
$942$	0	0
$943$	17.3177	0.563941
$944$	0	0
$945$	−1.24400	−0.0404672
$946$	0	0
$947$	−6.51466	−0.211698	−0.105849	−	0.994382i	$-0.533756\pi$
$947$	−6.51466	−0.211698	−0.105849	+	0.994382i	$0.533756\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−31.4470	−1.01974
$952$	0	0
$953$	47.6469	1.54344	0.771718	−	0.635965i	$-0.219398\pi$
$953$	47.6469	1.54344	0.771718	+	0.635965i	$0.219398\pi$
$954$	0	0
$955$	3.56571	0.115384
$956$	0	0
$957$	44.1933	1.42857
$958$	0	0
$959$	14.6595	0.473380
$960$	0	0
$961$	12.9705	0.418402
$962$	0	0
$963$	−6.69740	−0.215821
$964$	0	0
$965$	−3.35391	−0.107966
$966$	0	0
$967$	43.8122	1.40891	0.704453	−	0.709751i	$-0.251192\pi$
$967$	43.8122	1.40891	0.704453	+	0.709751i	$0.251192\pi$
$968$	0	0
$969$	0.427583	0.0137360
$970$	0	0
$971$	−4.29483	−0.137828	−0.0689139	−	0.997623i	$-0.521953\pi$
$971$	−4.29483	−0.137828	−0.0689139	+	0.997623i	$0.521953\pi$
$972$	0	0
$973$	34.6631	1.11125
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.8019	0.857470	0.428735	−	0.903430i	$-0.358959\pi$
$977$	26.8019	0.857470	0.428735	+	0.903430i	$0.358959\pi$
$978$	0	0
$979$	−12.2513	−0.391553
$980$	0	0
$981$	32.1715	1.02716
$982$	0	0
$983$	−27.2495	−0.869124	−0.434562	−	0.900642i	$-0.643097\pi$
$983$	−27.2495	−0.869124	−0.434562	+	0.900642i	$0.643097\pi$
$984$	0	0
$985$	−0.138391	−0.00440951
$986$	0	0
$987$	38.9614	1.24015
$988$	0	0
$989$	−0.442058	−0.0140566
$990$	0	0
$991$	−24.3889	−0.774740	−0.387370	−	0.921924i	$-0.626616\pi$
$991$	−24.3889	−0.774740	−0.387370	+	0.921924i	$0.626616\pi$
$992$	0	0
$993$	−40.0441	−1.27076
$994$	0	0
$995$	−2.83818	−0.0899764
$996$	0	0
$997$	31.3207	0.991935	0.495967	−	0.868341i	$-0.334814\pi$
$997$	31.3207	0.991935	0.495967	+	0.868341i	$0.334814\pi$
$998$	0	0
$999$	−12.1642	−0.384859

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.a.z.1.3		3
4.3	odd	2		169.2.a.b.1.3	✓	3
12.11	even	2		1521.2.a.r.1.1		3
13.5	odd	4		2704.2.f.o.337.5		6
13.8	odd	4		2704.2.f.o.337.6		6
13.12	even	2		2704.2.a.ba.1.3		3
20.19	odd	2		4225.2.a.bg.1.1		3
28.27	even	2		8281.2.a.bf.1.3		3
52.3	odd	6		169.2.c.c.22.1		6
52.7	even	12		169.2.e.b.23.5		12
52.11	even	12		169.2.e.b.147.2		12
52.15	even	12		169.2.e.b.147.5		12
52.19	even	12		169.2.e.b.23.2		12
52.23	odd	6		169.2.c.b.22.3		6
52.31	even	4		169.2.b.b.168.2		6
52.35	odd	6		169.2.c.c.146.1		6
52.43	odd	6		169.2.c.b.146.3		6
52.47	even	4		169.2.b.b.168.5		6
52.51	odd	2		169.2.a.c.1.1	yes	3
156.47	odd	4		1521.2.b.l.1351.2		6
156.83	odd	4		1521.2.b.l.1351.5		6
156.155	even	2		1521.2.a.o.1.3		3
260.259	odd	2		4225.2.a.bb.1.3		3
364.363	even	2		8281.2.a.bj.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.3	✓	3	4.3	odd	2
169.2.a.c.1.1	yes	3	52.51	odd	2
169.2.b.b.168.2		6	52.31	even	4
169.2.b.b.168.5		6	52.47	even	4
169.2.c.b.22.3		6	52.23	odd	6
169.2.c.b.146.3		6	52.43	odd	6
169.2.c.c.22.1		6	52.3	odd	6
169.2.c.c.146.1		6	52.35	odd	6
169.2.e.b.23.2		12	52.19	even	12
169.2.e.b.23.5		12	52.7	even	12
169.2.e.b.147.2		12	52.11	even	12
169.2.e.b.147.5		12	52.15	even	12
1521.2.a.o.1.3		3	156.155	even	2
1521.2.a.r.1.1		3	12.11	even	2
1521.2.b.l.1351.2		6	156.47	odd	4
1521.2.b.l.1351.5		6	156.83	odd	4
2704.2.a.z.1.3		3	1.1	even	1	trivial
2704.2.a.ba.1.3		3	13.12	even	2
2704.2.f.o.337.5		6	13.5	odd	4
2704.2.f.o.337.6		6	13.8	odd	4
4225.2.a.bb.1.3		3	260.259	odd	2
4225.2.a.bg.1.1		3	20.19	odd	2
8281.2.a.bf.1.3		3	28.27	even	2
8281.2.a.bj.1.1		3	364.363	even	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:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.a (trivial)

\(f(q)\)	\(=\)	\(q+2.24698 q^{3} +0.246980 q^{5} +2.35690 q^{7} +2.04892 q^{9} +4.24698 q^{11} +0.554958 q^{15} +2.15883 q^{17} +0.0881460 q^{19} +5.29590 q^{21} -1.49396 q^{23} -4.93900 q^{25} -2.13706 q^{27} +4.63102 q^{29} +6.63102 q^{31} +9.54288 q^{33} +0.582105 q^{35} +5.69202 q^{37} -11.5918 q^{41} +0.295897 q^{43} +0.506041 q^{45} +7.35690 q^{47} -1.44504 q^{49} +4.85086 q^{51} -10.3937 q^{53} +1.04892 q^{55} +0.198062 q^{57} +6.78017 q^{59} +3.47219 q^{61} +4.82908 q^{63} -7.67994 q^{67} -3.35690 q^{69} +8.66487 q^{71} +6.73556 q^{73} -11.0978 q^{75} +10.0097 q^{77} -9.97046 q^{79} -10.9487 q^{81} -1.60925 q^{83} +0.533188 q^{85} +10.4058 q^{87} -2.88471 q^{89} +14.8998 q^{93} +0.0217703 q^{95} -8.05861 q^{97} +8.70171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 4 q^{5} + 3 q^{7} - 3 q^{9} + 8 q^{11} + 2 q^{15} - 2 q^{17} + 4 q^{19} + 2 q^{21} + 5 q^{23} - 5 q^{25} - q^{27} - q^{29} + 5 q^{31} + 10 q^{33} - 4 q^{35} + 12 q^{37} - 7 q^{41} - 13 q^{43}+ \cdots - q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 4 * q^5 + 3 * q^7 - 3 * q^9 + 8 * q^11 + 2 * q^15 - 2 * q^17 + 4 * q^19 + 2 * q^21 + 5 * q^23 - 5 * q^25 - q^27 - q^29 + 5 * q^31 + 10 * q^33 - 4 * q^35 + 12 * q^37 - 7 * q^41 - 13 * q^43 + 11 * q^45 + 18 * q^47 - 4 * q^49 + q^51 + q^53 - 6 * q^55 + 5 * q^57 + 19 * q^59 + 4 * q^61 + 4 * q^63 + q^67 - 6 * q^69 + 27 * q^71 + 9 * q^73 - 15 * q^75 + 8 * q^77 + 5 * q^79 - q^81 + 7 * q^83 + 5 * q^85 + 18 * q^87 - 11 * q^89 + 22 * q^93 - 3 * q^95 + 7 * q^97 - q^99

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