LMFDB - Embedded newform 2704.2.f.o.337.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2704,2,Mod(337,2704)] 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("2704.337"); 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, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,4,0,0,0,0,0,-6,0,0,0,0,0,0,0,4,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$21.5915487066$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 169)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.6
Root			$-0.445042i$ of defining polynomial
Character	$\chi$	$=$	2704.337
Dual form			2704.2.f.o.337.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.24698 q^{3} +0.246980i q^{5} -2.35690i q^{7} +2.04892 q^{9} -4.24698i q^{11} +0.554958i q^{15} -2.15883 q^{17} +0.0881460i q^{19} -5.29590i q^{21} +1.49396 q^{23} +4.93900 q^{25} -2.13706 q^{27} +4.63102 q^{29} +6.63102i q^{31} -9.54288i q^{33} +0.582105 q^{35} -5.69202i q^{37} -11.5918i q^{41} -0.295897 q^{43} +0.506041i q^{45} -7.35690i q^{47} +1.44504 q^{49} -4.85086 q^{51} -10.3937 q^{53} +1.04892 q^{55} +0.198062i q^{57} -6.78017i q^{59} +3.47219 q^{61} -4.82908i q^{63} -7.67994i q^{67} +3.35690 q^{69} +8.66487i q^{71} -6.73556i q^{73} +11.0978 q^{75} -10.0097 q^{77} -9.97046 q^{79} -10.9487 q^{81} -1.60925i q^{83} -0.533188i q^{85} +10.4058 q^{87} +2.88471i q^{89} +14.8998i q^{93} -0.0217703 q^{95} -8.05861i q^{97} -8.70171i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 6 q^{9} + 4 q^{17} - 10 q^{23} + 10 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{35} + 26 q^{43} + 8 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 8 q^{61} + 12 q^{69} + 30 q^{75} - 16 q^{77} + 10 q^{79}+ \cdots + 6 q^{95}+O(q^{100}) $ 6 * q + 4 * q^3 - 6 * q^9 + 4 * q^17 - 10 * q^23 + 10 * q^25 - 2 * q^27 - 2 * q^29 - 8 * q^35 + 26 * q^43 + 8 * q^49 - 2 * q^51 + 2 * q^53 - 12 * q^55 + 8 * q^61 + 12 * q^69 + 30 * q^75 - 16 * q^77 + 10 * q^79 - 2 * q^81 + 36 * q^87 + 6 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$.

$n$	$677$	$1185$	$2367$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	0.246980i	0.110453i	0.998474	+	0.0552263i	$0.0175880\pi$
$5$	0.246980i	0.110453i	−0.998474	+	0.0552263i	$0.982412\pi$
$6$	0	0
$7$	− 2.35690i	− 0.890823i	−0.895326	−	0.445411i	$-0.853057\pi$
$7$	− 2.35690i	− 0.890823i	0.895326	−	0.445411i	$-0.146943\pi$
$8$	0	0
$9$	2.04892	0.682972
$10$	0	0
$11$	− 4.24698i	− 1.28051i	−0.768161	−	0.640256i	$-0.778828\pi$
$11$	− 4.24698i	− 1.28051i	0.768161	−	0.640256i	$-0.221172\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.554958i	0.143290i
$16$	0	0
$17$	−2.15883	−0.523594	−0.261797	−	0.965123i	$-0.584315\pi$
$17$	−2.15883	−0.523594	−0.261797	+	0.965123i	$0.584315\pi$
$18$	0	0
$19$	0.0881460i	0.0202221i	0.999949	+	0.0101110i	$0.00321850\pi$
$19$	0.0881460i	0.0202221i	−0.999949	+	0.0101110i	$0.996782\pi$
$20$	0	0
$21$	− 5.29590i	− 1.15566i
$22$	0	0
$23$	1.49396	0.311512	0.155756	−	0.987796i	$-0.450219\pi$
$23$	1.49396	0.311512	0.155756	+	0.987796i	$0.450219\pi$
$24$	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
$31$	6.63102i	1.19097i	0.803368	+	0.595483i	$0.203039\pi$
$31$	6.63102i	1.19097i	−0.803368	+	0.595483i	$0.796961\pi$
$32$	0	0
$33$	− 9.54288i	− 1.66120i
$34$	0	0
$35$	0.582105	0.0983937
$36$	0	0
$37$	− 5.69202i	− 0.935763i	−0.883791	−	0.467881i	$-0.845017\pi$
$37$	− 5.69202i	− 0.935763i	0.883791	−	0.467881i	$-0.154983\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 11.5918i	− 1.81033i	−0.425056	−	0.905167i	$-0.639746\pi$
$41$	− 11.5918i	− 1.81033i	0.425056	−	0.905167i	$-0.360254\pi$
$42$	0	0
$43$	−0.295897	−0.0451239	−0.0225619	−	0.999745i	$-0.507182\pi$
$43$	−0.295897	−0.0451239	−0.0225619	+	0.999745i	$0.507182\pi$
$44$	0	0
$45$	0.506041i	0.0754361i
$46$	0	0
$47$	− 7.35690i	− 1.07311i	−0.843864	−	0.536557i	$-0.819725\pi$
$47$	− 7.35690i	− 1.07311i	0.843864	−	0.536557i	$-0.180275\pi$
$48$	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
$57$	0.198062i	0.0262340i
$58$	0	0
$59$	− 6.78017i	− 0.882703i	−0.897334	−	0.441351i	$-0.854499\pi$
$59$	− 6.78017i	− 0.882703i	0.897334	−	0.441351i	$-0.145501\pi$
$60$	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
$63$	− 4.82908i	− 0.608407i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.67994i	− 0.938254i	−0.883131	−	0.469127i	$-0.844569\pi$
$67$	− 7.67994i	− 0.938254i	0.883131	−	0.469127i	$-0.155431\pi$
$68$	0	0
$69$	3.35690	0.404123
$70$	0	0
$71$	8.66487i	1.02833i	0.857691	+	0.514166i	$0.171898\pi$
$71$	8.66487i	1.02833i	−0.857691	+	0.514166i	$0.828102\pi$
$72$	0	0
$73$	− 6.73556i	− 0.788338i	−0.919038	−	0.394169i	$-0.871032\pi$
$73$	− 6.73556i	− 0.788338i	0.919038	−	0.394169i	$-0.128968\pi$
$74$	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
$83$	− 1.60925i	− 0.176638i	−0.996092	−	0.0883192i	$-0.971850\pi$
$83$	− 1.60925i	− 0.176638i	0.996092	−	0.0883192i	$-0.0281495\pi$
$84$	0	0
$85$	− 0.533188i	− 0.0578323i
$86$	0	0
$87$	10.4058	1.11562
$88$	0	0
$89$	2.88471i	0.305778i	0.988243	+	0.152889i	$0.0488577\pi$
$89$	2.88471i	0.305778i	−0.988243	+	0.152889i	$0.951142\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	14.8998i	1.54503i
$94$	0	0
$95$	−0.0217703	−0.00223358
$96$	0	0
$97$	− 8.05861i	− 0.818227i	−0.912483	−	0.409114i	$-0.865838\pi$
$97$	− 8.05861i	− 0.818227i	0.912483	−	0.409114i	$-0.134162\pi$
$98$	0	0
$99$	− 8.70171i	− 0.874555i
$100$	0	0
$101$	13.3545	1.32882	0.664411	−	0.747367i	$-0.268682\pi$
$101$	13.3545	1.32882	0.664411	+	0.747367i	$0.268682\pi$
$102$	0	0
$103$	1.36227	0.134229	0.0671144	−	0.997745i	$-0.478621\pi$
$103$	1.36227	0.134229	0.0671144	+	0.997745i	$0.478621\pi$
$104$	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
$109$	15.7017i	1.50395i	0.659191	+	0.751976i	$0.270899\pi$
$109$	15.7017i	1.50395i	−0.659191	+	0.751976i	$0.729101\pi$
$110$	0	0
$111$	− 12.7899i	− 1.21396i
$112$	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
$115$	0.368977i	0.0344073i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.08815i	0.466430i
$120$	0	0
$121$	−7.03684	−0.639712
$122$	0	0
$123$	− 26.0465i	− 2.34854i
$124$	0	0
$125$	2.45473i	0.219558i
$126$	0	0
$127$	−9.80731	−0.870258	−0.435129	−	0.900368i	$-0.643297\pi$
$127$	−9.80731	−0.870258	−0.435129	+	0.900368i	$0.643297\pi$
$128$	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
$135$	− 0.527811i	− 0.0454267i
$136$	0	0
$137$	− 6.21983i	− 0.531396i	−0.964056	−	0.265698i	$-0.914398\pi$
$137$	− 6.21983i	− 0.531396i	0.964056	−	0.265698i	$-0.0856024\pi$
$138$	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
$141$	− 16.5308i	− 1.39214i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.14377i	0.0949848i
$146$	0	0
$147$	3.24698	0.267806
$148$	0	0
$149$	4.33513i	0.355147i	0.984108	+	0.177574i	$0.0568248\pi$
$149$	4.33513i	0.355147i	−0.984108	+	0.177574i	$0.943175\pi$
$150$	0	0
$151$	3.94438i	0.320989i	0.987037	+	0.160494i	$0.0513089\pi$
$151$	3.94438i	0.320989i	−0.987037	+	0.160494i	$0.948691\pi$
$152$	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
$161$	− 3.52111i	− 0.277502i
$162$	0	0
$163$	16.1588i	1.26566i	0.774292	+	0.632829i	$0.218106\pi$
$163$	16.1588i	1.26566i	−0.774292	+	0.632829i	$0.781894\pi$
$164$	0	0
$165$	2.35690	0.183484
$166$	0	0
$167$	16.1172i	1.24719i	0.781749	+	0.623594i	$0.214328\pi$
$167$	16.1172i	1.24719i	−0.781749	+	0.623594i	$0.785672\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.180604i	0.0138111i
$172$	0	0
$173$	21.5362	1.63736	0.818682	−	0.574247i	$-0.194705\pi$
$173$	21.5362	1.63736	0.818682	+	0.574247i	$0.194705\pi$
$174$	0	0
$175$	− 11.6407i	− 0.879955i
$176$	0	0
$177$	− 15.2349i	− 1.14513i
$178$	0	0
$179$	11.4330	0.854540	0.427270	−	0.904124i	$-0.359475\pi$
$179$	11.4330	0.854540	0.427270	+	0.904124i	$0.359475\pi$
$180$	0	0
$181$	−20.9705	−1.55872	−0.779361	−	0.626575i	$-0.784456\pi$
$181$	−20.9705	−1.55872	−0.779361	+	0.626575i	$0.784456\pi$
$182$	0	0
$183$	7.80194	0.576736
$184$	0	0
$185$	1.40581	0.103357
$186$	0	0
$187$	9.16852i	0.670469i
$188$	0	0
$189$	5.03684i	0.366376i
$190$	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
$193$	13.5797i	0.977489i	0.872427	+	0.488745i	$0.162545\pi$
$193$	13.5797i	0.977489i	−0.872427	+	0.488745i	$0.837455\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 0.560335i	− 0.0399222i	−0.999801	−	0.0199611i	$-0.993646\pi$
$197$	− 0.560335i	− 0.0399222i	0.999801	−	0.0199611i	$-0.00635424\pi$
$198$	0	0
$199$	11.4916	0.814616	0.407308	−	0.913291i	$-0.366468\pi$
$199$	11.4916	0.814616	0.407308	+	0.913291i	$0.366468\pi$
$200$	0	0
$201$	− 17.2567i	− 1.21719i
$202$	0	0
$203$	− 10.9148i	− 0.766071i
$204$	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
$213$	19.4698i	1.33405i
$214$	0	0
$215$	− 0.0730805i	− 0.00498405i
$216$	0	0
$217$	15.6286	1.06094
$218$	0	0
$219$	− 15.1347i	− 1.02271i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2.25906i	0.151278i	0.997135	+	0.0756390i	$0.0240996\pi$
$223$	2.25906i	0.151278i	−0.997135	+	0.0756390i	$0.975900\pi$
$224$	0	0
$225$	10.1196	0.674640
$226$	0	0
$227$	6.96615i	0.462359i	0.972911	+	0.231180i	$0.0742585\pi$
$227$	6.96615i	0.462359i	−0.972911	+	0.231180i	$0.925741\pi$
$228$	0	0
$229$	24.1739i	1.59746i	0.601692	+	0.798728i	$0.294493\pi$
$229$	24.1739i	1.59746i	−0.601692	+	0.798728i	$0.705507\pi$
$230$	0	0
$231$	−22.4916	−1.47984
$232$	0	0
$233$	3.06100	0.200533	0.100266	−	0.994961i	$-0.468031\pi$
$233$	3.06100	0.200533	0.100266	+	0.994961i	$0.468031\pi$
$234$	0	0
$235$	1.81700	0.118528
$236$	0	0
$237$	−22.4034	−1.45526
$238$	0	0
$239$	25.1468i	1.62661i	0.581839	+	0.813304i	$0.302333\pi$
$239$	25.1468i	1.62661i	−0.581839	+	0.813304i	$0.697667\pi$
$240$	0	0
$241$	20.2664i	1.30547i	0.757586	+	0.652735i	$0.226379\pi$
$241$	20.2664i	1.30547i	−0.757586	+	0.652735i	$0.773621\pi$
$242$	0	0
$243$	−18.1903	−1.16691
$244$	0	0
$245$	0.356896i	0.0228012i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 3.61596i	− 0.229152i
$250$	0	0
$251$	−23.7211	−1.49726	−0.748631	−	0.662987i	$-0.769288\pi$
$251$	−23.7211	−1.49726	−0.748631	+	0.662987i	$0.769288\pi$
$252$	0	0
$253$	− 6.34481i	− 0.398895i
$254$	0	0
$255$	− 1.19806i	− 0.0750256i
$256$	0	0
$257$	−14.2241	−0.887278	−0.443639	−	0.896206i	$-0.646313\pi$
$257$	−14.2241	−0.887278	−0.443639	+	0.896206i	$0.646313\pi$
$258$	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
$265$	− 2.56704i	− 0.157692i
$266$	0	0
$267$	6.48188i	0.396684i
$268$	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
$271$	6.44803i	0.391690i	0.980635	+	0.195845i	$0.0627449\pi$
$271$	6.44803i	0.391690i	−0.980635	+	0.195845i	$0.937255\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 20.9758i	− 1.26489i
$276$	0	0
$277$	−13.4601	−0.808739	−0.404370	−	0.914596i	$-0.632509\pi$
$277$	−13.4601	−0.808739	−0.404370	+	0.914596i	$0.632509\pi$
$278$	0	0
$279$	13.5864i	0.813398i
$280$	0	0
$281$	− 5.03684i	− 0.300472i	−0.988650	−	0.150236i	$-0.951997\pi$
$281$	− 5.03684i	− 0.300472i	0.988650	−	0.150236i	$-0.0480034\pi$
$282$	0	0
$283$	22.1280	1.31537	0.657686	−	0.753293i	$-0.271536\pi$
$283$	22.1280	1.31537	0.657686	+	0.753293i	$0.271536\pi$
$284$	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
$291$	− 18.1075i	− 1.06148i
$292$	0	0
$293$	− 14.9463i	− 0.873172i	−0.899663	−	0.436586i	$-0.856187\pi$
$293$	− 14.9463i	− 0.873172i	0.899663	−	0.436586i	$-0.143813\pi$
$294$	0	0
$295$	1.67456	0.0974968
$296$	0	0
$297$	9.07606i	0.526647i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0.697398i	0.0401974i
$302$	0	0
$303$	30.0073	1.72387
$304$	0	0
$305$	0.857560i	0.0491037i
$306$	0	0
$307$	19.1293i	1.09177i	0.837861	+	0.545883i	$0.183806\pi$
$307$	19.1293i	1.09177i	−0.837861	+	0.545883i	$0.816194\pi$
$308$	0	0
$309$	3.06100	0.174134
$310$	0	0
$311$	−0.269815	−0.0152998	−0.00764990	−	0.999971i	$-0.502435\pi$
$311$	−0.269815	−0.0152998	−0.00764990	+	0.999971i	$0.502435\pi$
$312$	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
$317$	− 13.9952i	− 0.786050i	−0.919528	−	0.393025i	$-0.871429\pi$
$317$	− 13.9952i	− 0.786050i	0.919528	−	0.393025i	$-0.128571\pi$
$318$	0	0
$319$	− 19.6679i	− 1.10119i
$320$	0	0
$321$	−7.34481	−0.409948
$322$	0	0
$323$	− 0.190293i	− 0.0105882i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	35.2814i	1.95107i
$328$	0	0
$329$	−17.3394	−0.955954
$330$	0	0
$331$	− 17.8213i	− 0.979548i	−0.871849	−	0.489774i	$-0.837079\pi$
$331$	− 17.8213i	− 0.979548i	0.871849	−	0.489774i	$-0.162921\pi$
$332$	0	0
$333$	− 11.6625i	− 0.639100i
$334$	0	0
$335$	1.89679	0.103633
$336$	0	0
$337$	27.8485	1.51700	0.758501	−	0.651672i	$-0.225932\pi$
$337$	27.8485	1.51700	0.758501	+	0.651672i	$0.225932\pi$
$338$	0	0
$339$	27.0737	1.47044
$340$	0	0
$341$	28.1618	1.52505
$342$	0	0
$343$	− 19.9041i	− 1.07472i
$344$	0	0
$345$	0.829085i	0.0446364i
$346$	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
$349$	14.1860i	0.759358i	0.925118	+	0.379679i	$0.123966\pi$
$349$	14.1860i	0.759358i	−0.925118	+	0.379679i	$0.876034\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 7.16852i	− 0.381542i	−0.981635	−	0.190771i	$-0.938901\pi$
$353$	− 7.16852i	− 0.381542i	0.981635	−	0.190771i	$-0.0610988\pi$
$354$	0	0
$355$	−2.14005	−0.113582
$356$	0	0
$357$	11.4330i	0.605096i
$358$	0	0
$359$	19.8853i	1.04951i	0.851255	+	0.524753i	$0.175842\pi$
$359$	19.8853i	1.04951i	−0.851255	+	0.524753i	$0.824158\pi$
$360$	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
$369$	− 23.7506i	− 1.23641i
$370$	0	0
$371$	24.4969i	1.27182i
$372$	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
$375$	5.51573i	0.284831i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2.40880i	0.123732i	0.998084	+	0.0618658i	$0.0197051\pi$
$379$	2.40880i	0.123732i	−0.998084	+	0.0618658i	$0.980295\pi$
$380$	0	0
$381$	−22.0368	−1.12898
$382$	0	0
$383$	30.3913i	1.55292i	0.630164	+	0.776462i	$0.282988\pi$
$383$	30.3913i	1.55292i	−0.630164	+	0.776462i	$0.717012\pi$
$384$	0	0
$385$	− 2.47219i	− 0.125994i
$386$	0	0
$387$	−0.606268	−0.0308184
$388$	0	0
$389$	15.9409	0.808237	0.404118	−	0.914707i	$-0.367578\pi$
$389$	15.9409	0.808237	0.404118	+	0.914707i	$0.367578\pi$
$390$	0	0
$391$	−3.22521	−0.163106
$392$	0	0
$393$	14.7778	0.745440
$394$	0	0
$395$	− 2.46250i	− 0.123902i
$396$	0	0
$397$	− 16.9148i	− 0.848931i	−0.905444	−	0.424466i	$-0.860462\pi$
$397$	− 16.9148i	− 0.848931i	0.905444	−	0.424466i	$-0.139538\pi$
$398$	0	0
$399$	0.466812	0.0233698
$400$	0	0
$401$	− 26.6625i	− 1.33146i	−0.746192	−	0.665730i	$-0.768120\pi$
$401$	− 26.6625i	− 1.33146i	0.746192	−	0.665730i	$-0.231880\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 2.70410i	− 0.134368i
$406$	0	0
$407$	−24.1739	−1.19826
$408$	0	0
$409$	28.5163i	1.41004i	0.709187	+	0.705021i	$0.249062\pi$
$409$	28.5163i	1.41004i	−0.709187	+	0.705021i	$0.750938\pi$
$410$	0	0
$411$	− 13.9758i	− 0.689377i
$412$	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
$421$	− 11.6606i	− 0.568301i	−0.958780	−	0.284151i	$-0.908288\pi$
$421$	− 11.6606i	− 0.568301i	0.958780	−	0.284151i	$-0.0917115\pi$
$422$	0	0
$423$	− 15.0737i	− 0.732907i
$424$	0	0
$425$	−10.6625	−0.517206
$426$	0	0
$427$	− 8.18359i	− 0.396032i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 4.34913i	− 0.209490i	−0.994499	−	0.104745i	$-0.966597\pi$
$431$	− 4.34913i	− 0.209490i	0.994499	−	0.104745i	$-0.0334026\pi$
$432$	0	0
$433$	14.3884	0.691460	0.345730	−	0.938334i	$-0.387631\pi$
$433$	14.3884	0.691460	0.345730	+	0.938334i	$0.387631\pi$
$434$	0	0
$435$	2.57002i	0.123223i
$436$	0	0
$437$	0.131687i	0.00629942i
$438$	0	0
$439$	−20.2325	−0.965645	−0.482822	−	0.875718i	$-0.660388\pi$
$439$	−20.2325	−0.965645	−0.482822	+	0.875718i	$0.660388\pi$
$440$	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
$447$	9.74094i	0.460731i
$448$	0	0
$449$	− 12.4916i	− 0.589513i	−0.955572	−	0.294757i	$-0.904761\pi$
$449$	− 12.4916i	− 0.589513i	0.955572	−	0.294757i	$-0.0952386\pi$
$450$	0	0
$451$	−49.2301	−2.31816
$452$	0	0
$453$	8.86294i	0.416417i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.98121i	0.279789i	0.990166	+	0.139895i	$0.0446764\pi$
$457$	5.98121i	0.279789i	−0.990166	+	0.139895i	$0.955324\pi$
$458$	0	0
$459$	4.61356	0.215343
$460$	0	0
$461$	− 2.05669i	− 0.0957895i	−0.998852	−	0.0478947i	$-0.984749\pi$
$461$	− 2.05669i	− 0.0957895i	0.998852	−	0.0478947i	$-0.0152512\pi$
$462$	0	0
$463$	− 8.44935i	− 0.392675i	−0.980536	−	0.196337i	$-0.937095\pi$
$463$	− 8.44935i	− 0.392675i	0.980536	−	0.196337i	$-0.0629048\pi$
$464$	0	0
$465$	−3.67994	−0.170653
$466$	0	0
$467$	33.5139	1.55084	0.775420	−	0.631446i	$-0.217538\pi$
$467$	33.5139	1.55084	0.775420	+	0.631446i	$0.217538\pi$
$468$	0	0
$469$	−18.1008	−0.835818
$470$	0	0
$471$	10.0097	0.461222
$472$	0	0
$473$	1.25667i	0.0577817i
$474$	0	0
$475$	0.435353i	0.0199754i
$476$	0	0
$477$	−21.2959	−0.975072
$478$	0	0
$479$	− 24.7313i	− 1.13000i	−0.825091	−	0.565000i	$-0.808876\pi$
$479$	− 24.7313i	− 1.13000i	0.825091	−	0.565000i	$-0.191124\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 7.91185i	− 0.360002i
$484$	0	0
$485$	1.99031	0.0903754
$486$	0	0
$487$	37.7555i	1.71087i	0.517913	+	0.855433i	$0.326709\pi$
$487$	37.7555i	1.71087i	−0.517913	+	0.855433i	$0.673291\pi$
$488$	0	0
$489$	36.3086i	1.64193i
$490$	0	0
$491$	31.3110	1.41304	0.706522	−	0.707691i	$-0.250263\pi$
$491$	31.3110	1.41304	0.706522	+	0.707691i	$0.250263\pi$
$492$	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
$499$	− 21.4873i	− 0.961902i	−0.876748	−	0.480951i	$-0.840292\pi$
$499$	− 21.4873i	− 0.961902i	0.876748	−	0.480951i	$-0.159708\pi$
$500$	0	0
$501$	36.2150i	1.61797i
$502$	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
$505$	3.29829i	0.146772i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 17.1075i	− 0.758278i	−0.925340	−	0.379139i	$-0.876220\pi$
$509$	− 17.1075i	− 0.758278i	0.925340	−	0.379139i	$-0.123780\pi$
$510$	0	0
$511$	−15.8750	−0.702269
$512$	0	0
$513$	− 0.188374i	− 0.00831690i
$514$	0	0
$515$	0.336454i	0.0148259i
$516$	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
$525$	− 26.1564i	− 1.14156i
$526$	0	0
$527$	− 14.3153i	− 0.623583i
$528$	0	0
$529$	−20.7681	−0.902960
$530$	0	0
$531$	− 13.8920i	− 0.602862i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 0.807315i	− 0.0349033i
$536$	0	0
$537$	25.6896	1.10859
$538$	0	0
$539$	− 6.13706i	− 0.264342i
$540$	0	0
$541$	− 16.1884i	− 0.695993i	−0.937496	−	0.347996i	$-0.886862\pi$
$541$	− 16.1884i	− 0.695993i	0.937496	−	0.347996i	$-0.113138\pi$
$542$	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
$551$	0.408206i	0.0173902i
$552$	0	0
$553$	23.4993i	0.999293i
$554$	0	0
$555$	3.15883	0.134085
$556$	0	0
$557$	− 7.39075i	− 0.313156i	−0.987666	−	0.156578i	$-0.949954\pi$
$557$	− 7.39075i	− 0.313156i	0.987666	−	0.156578i	$-0.0500463\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	20.6015i	0.869795i
$562$	0	0
$563$	−9.47889	−0.399488	−0.199744	−	0.979848i	$-0.564011\pi$
$563$	−9.47889	−0.399488	−0.199744	+	0.979848i	$0.564011\pi$
$564$	0	0
$565$	2.97584i	0.125194i
$566$	0	0
$567$	25.8049i	1.08370i
$568$	0	0
$569$	10.1438	0.425249	0.212624	−	0.977134i	$-0.431799\pi$
$569$	10.1438	0.425249	0.212624	+	0.977134i	$0.431799\pi$
$570$	0	0
$571$	−14.0925	−0.589751	−0.294876	−	0.955536i	$-0.595278\pi$
$571$	−14.0925	−0.589751	−0.294876	+	0.955536i	$0.595278\pi$
$572$	0	0
$573$	32.4403	1.35521
$574$	0	0
$575$	7.37867	0.307712
$576$	0	0
$577$	25.1545i	1.04720i	0.851965	+	0.523598i	$0.175411\pi$
$577$	25.1545i	1.04720i	−0.851965	+	0.523598i	$0.824589\pi$
$578$	0	0
$579$	30.5133i	1.26809i
$580$	0	0
$581$	−3.79284	−0.157354
$582$	0	0
$583$	44.1420i	1.82817i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 43.8353i	− 1.80928i	−0.426180	−	0.904639i	$-0.640141\pi$
$587$	− 43.8353i	− 1.80928i	0.426180	−	0.904639i	$-0.359859\pi$
$588$	0	0
$589$	−0.584498	−0.0240838
$590$	0	0
$591$	− 1.25906i	− 0.0517909i
$592$	0	0
$593$	24.9965i	1.02648i	0.858244	+	0.513242i	$0.171556\pi$
$593$	24.9965i	1.02648i	−0.858244	+	0.513242i	$0.828444\pi$
$594$	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
$603$	− 15.7356i	− 0.640802i
$604$	0	0
$605$	− 1.73795i	− 0.0706579i
$606$	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
$609$	− 24.5254i	− 0.993820i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 25.9541i	− 1.04827i	−0.851634	−	0.524137i	$-0.824388\pi$
$613$	− 25.9541i	− 1.04827i	0.851634	−	0.524137i	$-0.175612\pi$
$614$	0	0
$615$	6.43296	0.259402
$616$	0	0
$617$	45.9396i	1.84946i	0.380626	+	0.924729i	$0.375709\pi$
$617$	45.9396i	1.84946i	−0.380626	+	0.924729i	$0.624291\pi$
$618$	0	0
$619$	6.73556i	0.270725i	0.990796	+	0.135363i	$0.0432199\pi$
$619$	6.73556i	0.270725i	−0.990796	+	0.135363i	$0.956780\pi$
$620$	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
$629$	12.2881i	0.489960i
$630$	0	0
$631$	45.0998i	1.79539i	0.440614	+	0.897696i	$0.354761\pi$
$631$	45.0998i	1.79539i	−0.440614	+	0.897696i	$0.645239\pi$
$632$	0	0
$633$	−19.7385	−0.784537
$634$	0	0
$635$	− 2.42221i	− 0.0961223i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	17.7536i	0.702322i
$640$	0	0
$641$	−32.5821	−1.28692	−0.643458	−	0.765482i	$-0.722501\pi$
$641$	−32.5821	−1.28692	−0.643458	+	0.765482i	$0.722501\pi$
$642$	0	0
$643$	− 25.5754i	− 1.00860i	−0.863530	−	0.504298i	$-0.831751\pi$
$643$	− 25.5754i	− 1.00860i	0.863530	−	0.504298i	$-0.168249\pi$
$644$	0	0
$645$	− 0.164210i	− 0.00646578i
$646$	0	0
$647$	−30.1715	−1.18616	−0.593082	−	0.805142i	$-0.702089\pi$
$647$	−30.1715	−1.18616	−0.593082	+	0.805142i	$0.702089\pi$
$648$	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
$655$	1.62432i	0.0634673i
$656$	0	0
$657$	− 13.8006i	− 0.538413i
$658$	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
$661$	31.7590i	1.23528i	0.786460	+	0.617641i	$0.211911\pi$
$661$	31.7590i	1.23528i	−0.786460	+	0.617641i	$0.788089\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.0513102i	0.00198973i
$666$	0	0
$667$	6.91856	0.267888
$668$	0	0
$669$	5.07606i	0.196252i
$670$	0	0
$671$	− 14.7463i	− 0.569275i
$672$	0	0
$673$	7.50232	0.289193	0.144597	−	0.989491i	$-0.453812\pi$
$673$	7.50232	0.289193	0.144597	+	0.989491i	$0.453812\pi$
$674$	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
$681$	15.6528i	0.599816i
$682$	0	0
$683$	− 24.0834i	− 0.921524i	−0.887524	−	0.460762i	$-0.847576\pi$
$683$	− 24.0834i	− 0.921524i	0.887524	−	0.460762i	$-0.152424\pi$
$684$	0	0
$685$	1.53617	0.0586941
$686$	0	0
$687$	54.3183i	2.07237i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 2.01447i	− 0.0766342i	−0.999266	−	0.0383171i	$-0.987800\pi$
$691$	− 2.01447i	− 0.0766342i	0.999266	−	0.0383171i	$-0.0121997\pi$
$692$	0	0
$693$	−20.5090	−0.779073
$694$	0	0
$695$	3.63235i	0.137783i
$696$	0	0
$697$	25.0248i	0.947880i
$698$	0	0
$699$	6.87800	0.260150
$700$	0	0
$701$	48.8189	1.84387	0.921933	−	0.387350i	$-0.126610\pi$
$701$	48.8189	1.84387	0.921933	+	0.387350i	$0.126610\pi$
$702$	0	0
$703$	0.501729	0.0189231
$704$	0	0
$705$	4.08277	0.153766
$706$	0	0
$707$	− 31.4752i	− 1.18375i
$708$	0	0
$709$	20.8060i	0.781385i	0.920521	+	0.390693i	$0.127764\pi$
$709$	20.8060i	0.781385i	−0.920521	+	0.390693i	$0.872236\pi$
$710$	0	0
$711$	−20.4286	−0.766134
$712$	0	0
$713$	9.90648i	0.371000i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	56.5042i	2.11019i
$718$	0	0
$719$	21.4306	0.799225	0.399613	−	0.916684i	$-0.369145\pi$
$719$	21.4306	0.799225	0.399613	+	0.916684i	$0.369145\pi$
$720$	0	0
$721$	− 3.21073i	− 0.119574i
$722$	0	0
$723$	45.5381i	1.69358i
$724$	0	0
$725$	22.8726	0.849468
$726$	0	0
$727$	13.4862	0.500175	0.250088	−	0.968223i	$-0.419541\pi$
$727$	13.4862	0.500175	0.250088	+	0.968223i	$0.419541\pi$
$728$	0	0
$729$	−8.02715	−0.297302
$730$	0	0
$731$	0.638792	0.0236266
$732$	0	0
$733$	− 43.5424i	− 1.60828i	−0.594443	−	0.804138i	$-0.702627\pi$
$733$	− 43.5424i	− 1.60828i	0.594443	−	0.804138i	$-0.297373\pi$
$734$	0	0
$735$	0.801938i	0.0295799i
$736$	0	0
$737$	−32.6165	−1.20145
$738$	0	0
$739$	20.0543i	0.737709i	0.929487	+	0.368855i	$0.120250\pi$
$739$	20.0543i	0.737709i	−0.929487	+	0.368855i	$0.879750\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.1685i	1.21684i	0.793617	+	0.608418i	$0.208195\pi$
$743$	33.1685i	1.21684i	−0.793617	+	0.608418i	$0.791805\pi$
$744$	0	0
$745$	−1.07069	−0.0392270
$746$	0	0
$747$	− 3.29722i	− 0.120639i
$748$	0	0
$749$	7.70410i	0.281502i
$750$	0	0
$751$	39.2814	1.43340	0.716700	−	0.697382i	$-0.245652\pi$
$751$	39.2814	1.43340	0.716700	+	0.697382i	$0.245652\pi$
$752$	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
$759$	− 14.2567i	− 0.517484i
$760$	0	0
$761$	21.8984i	0.793818i	0.917858	+	0.396909i	$0.129917\pi$
$761$	21.8984i	0.793818i	−0.917858	+	0.396909i	$0.870083\pi$
$762$	0	0
$763$	37.0073	1.33975
$764$	0	0
$765$	− 1.09246i	− 0.0394979i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	46.7096i	1.68439i	0.539172	+	0.842196i	$0.318737\pi$
$769$	46.7096i	1.68439i	−0.539172	+	0.842196i	$0.681263\pi$
$770$	0	0
$771$	−31.9614	−1.15106
$772$	0	0
$773$	− 30.2416i	− 1.08771i	−0.839178	−	0.543857i	$-0.816963\pi$
$773$	− 30.2416i	− 1.08771i	0.839178	−	0.543857i	$-0.183037\pi$
$774$	0	0
$775$	32.7506i	1.17644i
$776$	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
$785$	1.10023i	0.0392688i
$786$	0	0
$787$	28.7023i	1.02313i	0.859246	+	0.511563i	$0.170933\pi$
$787$	28.7023i	1.02313i	−0.859246	+	0.511563i	$0.829067\pi$
$788$	0	0
$789$	38.4131	1.36754
$790$	0	0
$791$	− 28.3980i	− 1.00972i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 5.76809i	− 0.204573i
$796$	0	0
$797$	18.5418	0.656785	0.328392	−	0.944541i	$-0.393493\pi$
$797$	18.5418	0.656785	0.328392	+	0.944541i	$0.393493\pi$
$798$	0	0
$799$	15.8823i	0.561876i
$800$	0	0
$801$	5.91053i	0.208838i
$802$	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
$811$	10.0285i	0.352147i	0.984377	+	0.176074i	$0.0563397\pi$
$811$	10.0285i	0.352147i	−0.984377	+	0.176074i	$0.943660\pi$
$812$	0	0
$813$	14.4886i	0.508137i
$814$	0	0
$815$	−3.99090	−0.139795
$816$	0	0
$817$	− 0.0260821i	0 0.000912498i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 26.1704i	− 0.913355i	−0.889632	−	0.456677i	$-0.849039\pi$
$821$	− 26.1704i	− 0.913355i	0.889632	−	0.456677i	$-0.150961\pi$
$822$	0	0
$823$	1.82238	0.0635242	0.0317621	−	0.999495i	$-0.489888\pi$
$823$	1.82238	0.0635242	0.0317621	+	0.999495i	$0.489888\pi$
$824$	0	0
$825$	− 47.1323i	− 1.64094i
$826$	0	0
$827$	− 32.2941i	− 1.12298i	−0.827485	−	0.561488i	$-0.810229\pi$
$827$	− 32.2941i	− 1.12298i	0.827485	−	0.561488i	$-0.189771\pi$
$828$	0	0
$829$	−15.1002	−0.524453	−0.262226	−	0.965006i	$-0.584457\pi$
$829$	−15.1002	−0.524453	−0.262226	+	0.965006i	$0.584457\pi$
$830$	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
$837$	− 14.1709i	− 0.489818i
$838$	0	0
$839$	− 32.9965i	− 1.13917i	−0.821933	−	0.569584i	$-0.807104\pi$
$839$	− 32.9965i	− 1.13917i	0.821933	−	0.569584i	$-0.192896\pi$
$840$	0	0
$841$	−7.55363	−0.260470
$842$	0	0
$843$	− 11.3177i	− 0.389801i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	16.5851i	0.569870i
$848$	0	0
$849$	49.7211	1.70642
$850$	0	0
$851$	− 8.50365i	− 0.291501i
$852$	0	0
$853$	37.7802i	1.29357i	0.762673	+	0.646784i	$0.223887\pi$
$853$	37.7802i	1.29357i	−0.762673	+	0.646784i	$0.776113\pi$
$854$	0	0
$855$	−0.0446055	−0.00152547
$856$	0	0
$857$	−27.3623	−0.934677	−0.467339	−	0.884078i	$-0.654787\pi$
$857$	−27.3623	−0.934677	−0.467339	+	0.884078i	$0.654787\pi$
$858$	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
$863$	6.14483i	0.209173i	0.994516	+	0.104586i	$0.0333518\pi$
$863$	6.14483i	0.209173i	−0.994516	+	0.104586i	$0.966648\pi$
$864$	0	0
$865$	5.31900i	0.180851i
$866$	0	0
$867$	−27.7265	−0.941640
$868$	0	0
$869$	42.3443i	1.43643i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 16.5114i	− 0.558827i
$874$	0	0
$875$	5.78554	0.195587
$876$	0	0
$877$	− 13.5077i	− 0.456123i	−0.973647	−	0.228061i	$-0.926761\pi$
$877$	− 13.5077i	− 0.456123i	0.973647	−	0.228061i	$-0.0732386\pi$
$878$	0	0
$879$	− 33.5840i	− 1.13276i
$880$	0	0
$881$	5.23431	0.176348	0.0881741	−	0.996105i	$-0.471897\pi$
$881$	5.23431	0.176348	0.0881741	+	0.996105i	$0.471897\pi$
$882$	0	0
$883$	−4.57301	−0.153894	−0.0769470	−	0.997035i	$-0.524517\pi$
$883$	−4.57301	−0.153894	−0.0769470	+	0.997035i	$0.524517\pi$
$884$	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
$889$	23.1148i	0.775246i
$890$	0	0
$891$	46.4989i	1.55777i
$892$	0	0
$893$	0.648481	0.0217006
$894$	0	0
$895$	2.82371i	0.0943861i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30.7084i	1.02418i
$900$	0	0
$901$	22.4383	0.747529
$902$	0	0
$903$	1.56704i	0.0521478i
$904$	0	0
$905$	− 5.17928i	− 0.172165i
$906$	0	0
$907$	8.10215	0.269027	0.134514	−	0.990912i	$-0.457053\pi$
$907$	8.10215	0.269027	0.134514	+	0.990912i	$0.457053\pi$
$908$	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
$915$	1.92692i	0.0637020i
$916$	0	0
$917$	− 15.5007i	− 0.511877i
$918$	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
$921$	42.9831i	1.41634i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 28.1129i	− 0.924346i
$926$	0	0
$927$	2.79118	0.0916745
$928$	0	0
$929$	24.2131i	0.794407i	0.917731	+	0.397203i	$0.130019\pi$
$929$	24.2131i	0.794407i	−0.917731	+	0.397203i	$0.869981\pi$
$930$	0	0
$931$	0.127375i	0.00417454i
$932$	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
$941$	− 15.9638i	− 0.520404i	−0.965554	−	0.260202i	$-0.916211\pi$
$941$	− 15.9638i	− 0.520404i	0.965554	−	0.260202i	$-0.0837891\pi$
$942$	0	0
$943$	− 17.3177i	− 0.563941i
$944$	0	0
$945$	−1.24400	−0.0404672
$946$	0	0
$947$	6.51466i	0.211698i	0.994382	+	0.105849i	$0.0337560\pi$
$947$	6.51466i	0.211698i	−0.994382	+	0.105849i	$0.966244\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 31.4470i	− 1.01974i
$952$	0	0
$953$	−47.6469	−1.54344	−0.771718	−	0.635965i	$-0.780602\pi$
$953$	−47.6469	−1.54344	−0.771718	+	0.635965i	$0.780602\pi$
$954$	0	0
$955$	3.56571i	0.115384i
$956$	0	0
$957$	− 44.1933i	− 1.42857i
$958$	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
$967$	43.8122i	1.40891i	0.709751	+	0.704453i	$0.248808\pi$
$967$	43.8122i	1.40891i	−0.709751	+	0.704453i	$0.751192\pi$
$968$	0	0
$969$	− 0.427583i	− 0.0137360i
$970$	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
$973$	− 34.6631i	− 1.11125i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.8019i	0.857470i	0.903430	+	0.428735i	$0.141041\pi$
$977$	26.8019i	0.857470i	−0.903430	+	0.428735i	$0.858959\pi$
$978$	0	0
$979$	12.2513	0.391553
$980$	0	0
$981$	32.1715i	1.02716i
$982$	0	0
$983$	27.2495i	0.869124i	0.900642	+	0.434562i	$0.143097\pi$
$983$	27.2495i	0.869124i	−0.900642	+	0.434562i	$0.856903\pi$
$984$	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
$993$	− 40.0441i	− 1.27076i
$994$	0	0
$995$	2.83818i	0.0899764i
$996$	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
$999$	12.1642i	0.384859i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.3	✓	3	52.31	even	4
169.2.a.c.1.1	yes	3	52.47	even	4
169.2.b.b.168.2		6	52.51	odd	2
169.2.b.b.168.5		6	4.3	odd	2
169.2.c.b.22.3		6	52.11	even	12
169.2.c.b.146.3		6	52.7	even	12
169.2.c.c.22.1		6	52.15	even	12
169.2.c.c.146.1		6	52.19	even	12
169.2.e.b.23.2		12	52.43	odd	6
169.2.e.b.23.5		12	52.35	odd	6
169.2.e.b.147.2		12	52.3	odd	6
169.2.e.b.147.5		12	52.23	odd	6
1521.2.a.o.1.3		3	156.47	odd	4
1521.2.a.r.1.1		3	156.83	odd	4
1521.2.b.l.1351.2		6	12.11	even	2
1521.2.b.l.1351.5		6	156.155	even	2
2704.2.a.z.1.3		3	13.5	odd	4
2704.2.a.ba.1.3		3	13.8	odd	4
2704.2.f.o.337.5		6	13.12	even	2	inner
2704.2.f.o.337.6		6	1.1	even	1	trivial
4225.2.a.bb.1.3		3	260.99	even	4
4225.2.a.bg.1.1		3	260.239	even	4
8281.2.a.bf.1.3		3	364.83	odd	4
8281.2.a.bj.1.1		3	364.307	odd	4

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.24698 q^{3} +0.246980i q^{5} -2.35690i q^{7} +2.04892 q^{9} -4.24698i q^{11} +0.554958i q^{15} -2.15883 q^{17} +0.0881460i q^{19} -5.29590i q^{21} +1.49396 q^{23} +4.93900 q^{25} -2.13706 q^{27} +4.63102 q^{29} +6.63102i q^{31} -9.54288i q^{33} +0.582105 q^{35} -5.69202i q^{37} -11.5918i q^{41} -0.295897 q^{43} +0.506041i q^{45} -7.35690i q^{47} +1.44504 q^{49} -4.85086 q^{51} -10.3937 q^{53} +1.04892 q^{55} +0.198062i q^{57} -6.78017i q^{59} +3.47219 q^{61} -4.82908i q^{63} -7.67994i q^{67} +3.35690 q^{69} +8.66487i q^{71} -6.73556i q^{73} +11.0978 q^{75} -10.0097 q^{77} -9.97046 q^{79} -10.9487 q^{81} -1.60925i q^{83} -0.533188i q^{85} +10.4058 q^{87} +2.88471i q^{89} +14.8998i q^{93} -0.0217703 q^{95} -8.05861i q^{97} -8.70171i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 6 q^{9} + 4 q^{17} - 10 q^{23} + 10 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{35} + 26 q^{43} + 8 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 8 q^{61} + 12 q^{69} + 30 q^{75} - 16 q^{77} + 10 q^{79}+ \cdots + 6 q^{95}+O(q^{100}) \) 6 * q + 4 * q^3 - 6 * q^9 + 4 * q^17 - 10 * q^23 + 10 * q^25 - 2 * q^27 - 2 * q^29 - 8 * q^35 + 26 * q^43 + 8 * q^49 - 2 * q^51 + 2 * q^53 - 12 * q^55 + 8 * q^61 + 12 * q^69 + 30 * q^75 - 16 * q^77 + 10 * q^79 - 2 * q^81 + 36 * q^87 + 6 * q^95

Introduction

L-functions

Modular forms

Varieties

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