LMFDB - Embedded newform 484.3.d.c.241.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [484,3,Mod(241,484)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(484, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("484.241");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 484 = 2^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	484.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$13.1880447950$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 19x^{6} - 37x^{5} + 229x^{4} + 196x^{3} + 1496x^{2} + 2952x + 26896 $ x^8 - x^7 + 19x^6 - 37x^5 + 229x^4 + 196x^3 + 1496x^2 + 2952x + 26896
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 11^{3} $
Twist minimal:	no (minimal twist has level 44)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.1
Root			$-2.28040 - 1.65680i$ of defining polynomial
Character	$\chi$	$=$	484.241
Dual form			484.3.d.c.241.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.43676 q^{3} +6.56079 q^{5} -11.5169i q^{7} +10.6848 q^{9} +5.05831i q^{13} -29.1086 q^{15} +8.56914i q^{17} -10.8348i q^{19} +51.0977i q^{21} +11.7519 q^{23} +18.0440 q^{25} -7.47516 q^{27} -20.2412i q^{29} -36.4007 q^{31} -75.5600i q^{35} -36.8986 q^{37} -22.4425i q^{39} -42.3954i q^{41} -72.4430i q^{43} +70.1009 q^{45} -10.9196 q^{47} -83.6390 q^{49} -38.0192i q^{51} +72.3029 q^{53} +48.0715i q^{57} +4.47350 q^{59} -108.646i q^{61} -123.056i q^{63} +33.1866i q^{65} -37.6104 q^{67} -52.1405 q^{69} -53.6299 q^{71} +24.9469i q^{73} -80.0568 q^{75} -71.2745i q^{79} -62.9979 q^{81} -11.5389i q^{83} +56.2203i q^{85} +89.8052i q^{87} -75.6980 q^{89} +58.2561 q^{91} +161.501 q^{93} -71.0850i q^{95} +2.98620 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{5} + 44 q^{9} - 28 q^{15} + 100 q^{23} - 34 q^{25} + 60 q^{27} + 98 q^{31} - 210 q^{37} + 332 q^{45} - 110 q^{47} + 22 q^{49} + 250 q^{53} + 112 q^{59} + 10 q^{67} - 176 q^{69} - 198 q^{71}+ \cdots - 220 q^{97}+O(q^{100}) $ 8 * q + 14 * q^5 + 44 * q^9 - 28 * q^15 + 100 * q^23 - 34 * q^25 + 60 * q^27 + 98 * q^31 - 210 * q^37 + 332 * q^45 - 110 * q^47 + 22 * q^49 + 250 * q^53 + 112 * q^59 + 10 * q^67 - 176 * q^69 - 198 * q^71 - 78 * q^75 - 232 * q^81 + 18 * q^89 + 440 * q^91 + 360 * q^93 - 220 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/484\mathbb{Z}\right)^\times$.

$n$	$243$	$365$
$\chi(n)$	$1$	$-1$

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
$3$	−4.43676	−1.47892	−0.739460	−	0.673201i	$-0.764919\pi$
$3$	−4.43676	−1.47892	−0.739460	+	0.673201i	$0.764919\pi$
$4$	0	0
$5$	6.56079	1.31216	0.656079	−	0.754692i	$-0.272214\pi$
$5$	6.56079	1.31216	0.656079	+	0.754692i	$0.272214\pi$
$6$	0	0
$7$	− 11.5169i	− 1.64527i	−0.568569	−	0.822636i	$-0.692503\pi$
$7$	− 11.5169i	− 1.64527i	0.568569	−	0.822636i	$-0.307497\pi$
$8$	0	0
$9$	10.6848	1.18720
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.05831i	0.389101i	0.980892	+	0.194551i	$0.0623248\pi$
$13$	5.05831i	0.389101i	−0.980892	+	0.194551i	$0.937675\pi$
$14$	0	0
$15$	−29.1086	−1.94058
$16$	0	0
$17$	8.56914i	0.504067i	0.967718	+	0.252034i	$0.0810993\pi$
$17$	8.56914i	0.504067i	−0.967718	+	0.252034i	$0.918901\pi$
$18$	0	0
$19$	− 10.8348i	− 0.570254i	−0.958490	−	0.285127i	$-0.907964\pi$
$19$	− 10.8348i	− 0.570254i	0.958490	−	0.285127i	$-0.0920358\pi$
$20$	0	0
$21$	51.0977i	2.43322i
$22$	0	0
$23$	11.7519	0.510954	0.255477	−	0.966815i	$-0.417768\pi$
$23$	11.7519	0.510954	0.255477	+	0.966815i	$0.417768\pi$
$24$	0	0
$25$	18.0440	0.721760
$26$	0	0
$27$	−7.47516	−0.276858
$28$	0	0
$29$	− 20.2412i	− 0.697971i	−0.937128	−	0.348986i	$-0.886526\pi$
$29$	− 20.2412i	− 0.697971i	0.937128	−	0.348986i	$-0.113474\pi$
$30$	0	0
$31$	−36.4007	−1.17422	−0.587108	−	0.809508i	$-0.699734\pi$
$31$	−36.4007	−1.17422	−0.587108	+	0.809508i	$0.699734\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 75.5600i	− 2.15886i
$36$	0	0
$37$	−36.8986	−0.997260	−0.498630	−	0.866815i	$-0.666163\pi$
$37$	−36.8986	−0.997260	−0.498630	+	0.866815i	$0.666163\pi$
$38$	0	0
$39$	− 22.4425i	− 0.575449i
$40$	0	0
$41$	− 42.3954i	− 1.03404i	−0.855975	−	0.517018i	$-0.827042\pi$
$41$	− 42.3954i	− 1.03404i	0.855975	−	0.517018i	$-0.172958\pi$
$42$	0	0
$43$	− 72.4430i	− 1.68472i	−0.538915	−	0.842360i	$-0.681165\pi$
$43$	− 72.4430i	− 1.68472i	0.538915	−	0.842360i	$-0.318835\pi$
$44$	0	0
$45$	70.1009	1.55780
$46$	0	0
$47$	−10.9196	−0.232331	−0.116166	−	0.993230i	$-0.537060\pi$
$47$	−10.9196	−0.232331	−0.116166	+	0.993230i	$0.537060\pi$
$48$	0	0
$49$	−83.6390	−1.70692
$50$	0	0
$51$	− 38.0192i	− 0.745475i
$52$	0	0
$53$	72.3029	1.36421	0.682103	−	0.731256i	$-0.261066\pi$
$53$	72.3029	1.36421	0.682103	+	0.731256i	$0.261066\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	48.0715i	0.843360i
$58$	0	0
$59$	4.47350	0.0758220	0.0379110	−	0.999281i	$-0.487930\pi$
$59$	4.47350	0.0758220	0.0379110	+	0.999281i	$0.487930\pi$
$60$	0	0
$61$	− 108.646i	− 1.78108i	−0.454902	−	0.890542i	$-0.650326\pi$
$61$	− 108.646i	− 1.78108i	0.454902	−	0.890542i	$-0.349674\pi$
$62$	0	0
$63$	− 123.056i	− 1.95327i
$64$	0	0
$65$	33.1866i	0.510562i
$66$	0	0
$67$	−37.6104	−0.561349	−0.280674	−	0.959803i	$-0.590558\pi$
$67$	−37.6104	−0.561349	−0.280674	+	0.959803i	$0.590558\pi$
$68$	0	0
$69$	−52.1405	−0.755659
$70$	0	0
$71$	−53.6299	−0.755350	−0.377675	−	0.925938i	$-0.623276\pi$
$71$	−53.6299	−0.755350	−0.377675	+	0.925938i	$0.623276\pi$
$72$	0	0
$73$	24.9469i	0.341738i	0.985294	+	0.170869i	$0.0546576\pi$
$73$	24.9469i	0.341738i	−0.985294	+	0.170869i	$0.945342\pi$
$74$	0	0
$75$	−80.0568	−1.06742
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 71.2745i	− 0.902209i	−0.892471	−	0.451105i	$-0.851030\pi$
$79$	− 71.2745i	− 0.902209i	0.892471	−	0.451105i	$-0.148970\pi$
$80$	0	0
$81$	−62.9979	−0.777752
$82$	0	0
$83$	− 11.5389i	− 0.139023i	−0.997581	−	0.0695115i	$-0.977856\pi$
$83$	− 11.5389i	− 0.139023i	0.997581	−	0.0695115i	$-0.0221441\pi$
$84$	0	0
$85$	56.2203i	0.661416i
$86$	0	0
$87$	89.8052i	1.03224i
$88$	0	0
$89$	−75.6980	−0.850540	−0.425270	−	0.905067i	$-0.639821\pi$
$89$	−75.6980	−0.850540	−0.425270	+	0.905067i	$0.639821\pi$
$90$	0	0
$91$	58.2561	0.640177
$92$	0	0
$93$	161.501	1.73657
$94$	0	0
$95$	− 71.0850i	− 0.748264i
$96$	0	0
$97$	2.98620	0.0307856	0.0153928	−	0.999882i	$-0.495100\pi$
$97$	2.98620	0.0307856	0.0153928	+	0.999882i	$0.495100\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	44.4427i	0.440027i	0.975497	+	0.220014i	$0.0706101\pi$
$101$	44.4427i	0.440027i	−0.975497	+	0.220014i	$0.929390\pi$
$102$	0	0
$103$	54.0041	0.524312	0.262156	−	0.965026i	$-0.415567\pi$
$103$	54.0041	0.524312	0.262156	+	0.965026i	$0.415567\pi$
$104$	0	0
$105$	335.241i	3.19278i
$106$	0	0
$107$	131.392i	1.22796i	0.789322	+	0.613980i	$0.210432\pi$
$107$	131.392i	1.22796i	−0.789322	+	0.613980i	$0.789568\pi$
$108$	0	0
$109$	6.96523i	0.0639012i	0.999489	+	0.0319506i	$0.0101719\pi$
$109$	6.96523i	0.0639012i	−0.999489	+	0.0319506i	$0.989828\pi$
$110$	0	0
$111$	163.710	1.47487
$112$	0	0
$113$	54.6655	0.483766	0.241883	−	0.970305i	$-0.422235\pi$
$113$	54.6655	0.483766	0.241883	+	0.970305i	$0.422235\pi$
$114$	0	0
$115$	77.1020	0.670452
$116$	0	0
$117$	54.0472i	0.461942i
$118$	0	0
$119$	98.6899	0.829327
$120$	0	0
$121$	0	0
$122$	0	0
$123$	188.098i	1.52925i
$124$	0	0
$125$	−45.6369	−0.365095
$126$	0	0
$127$	8.15329i	0.0641992i	0.999485	+	0.0320996i	$0.0102194\pi$
$127$	8.15329i	0.0641992i	−0.999485	+	0.0320996i	$0.989781\pi$
$128$	0	0
$129$	321.412i	2.49157i
$130$	0	0
$131$	− 162.483i	− 1.24033i	−0.784473	−	0.620163i	$-0.787066\pi$
$131$	− 162.483i	− 1.24033i	0.784473	−	0.620163i	$-0.212934\pi$
$132$	0	0
$133$	−124.784	−0.938223
$134$	0	0
$135$	−49.0430	−0.363281
$136$	0	0
$137$	234.863	1.71433	0.857165	−	0.515042i	$-0.172224\pi$
$137$	234.863	1.71433	0.857165	+	0.515042i	$0.172224\pi$
$138$	0	0
$139$	237.521i	1.70878i	0.519629	+	0.854392i	$0.326070\pi$
$139$	237.521i	1.70878i	−0.519629	+	0.854392i	$0.673930\pi$
$140$	0	0
$141$	48.4475	0.343599
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 132.798i	− 0.915849i
$146$	0	0
$147$	371.086	2.52439
$148$	0	0
$149$	− 163.988i	− 1.10059i	−0.834971	−	0.550294i	$-0.814516\pi$
$149$	− 163.988i	− 1.10059i	0.834971	−	0.550294i	$-0.185484\pi$
$150$	0	0
$151$	− 43.9622i	− 0.291140i	−0.989348	−	0.145570i	$-0.953498\pi$
$151$	− 43.9622i	− 0.291140i	0.989348	−	0.145570i	$-0.0465016\pi$
$152$	0	0
$153$	91.5598i	0.598430i
$154$	0	0
$155$	−238.817	−1.54076
$156$	0	0
$157$	8.04622	0.0512498	0.0256249	−	0.999672i	$-0.491842\pi$
$157$	8.04622	0.0512498	0.0256249	+	0.999672i	$0.491842\pi$
$158$	0	0
$159$	−320.791	−2.01755
$160$	0	0
$161$	− 135.346i	− 0.840657i
$162$	0	0
$163$	−126.237	−0.774459	−0.387229	−	0.921983i	$-0.626568\pi$
$163$	−126.237	−0.774459	−0.387229	+	0.921983i	$0.626568\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	211.983i	1.26936i	0.772774	+	0.634681i	$0.218868\pi$
$167$	211.983i	1.26936i	−0.772774	+	0.634681i	$0.781132\pi$
$168$	0	0
$169$	143.413	0.848600
$170$	0	0
$171$	− 115.768i	− 0.677007i
$172$	0	0
$173$	− 251.815i	− 1.45558i	−0.685802	−	0.727788i	$-0.740548\pi$
$173$	− 251.815i	− 1.45558i	0.685802	−	0.727788i	$-0.259452\pi$
$174$	0	0
$175$	− 207.811i	− 1.18749i
$176$	0	0
$177$	−19.8478	−0.112135
$178$	0	0
$179$	−221.779	−1.23899	−0.619495	−	0.785001i	$-0.712662\pi$
$179$	−221.779	−1.23899	−0.619495	+	0.785001i	$0.712662\pi$
$180$	0	0
$181$	−1.97299	−0.0109005	−0.00545025	−	0.999985i	$-0.501735\pi$
$181$	−1.97299	−0.0109005	−0.00545025	+	0.999985i	$0.501735\pi$
$182$	0	0
$183$	482.036i	2.63408i
$184$	0	0
$185$	−242.084	−1.30856
$186$	0	0
$187$	0	0
$188$	0	0
$189$	86.0907i	0.455506i
$190$	0	0
$191$	−20.7963	−0.108881	−0.0544407	−	0.998517i	$-0.517338\pi$
$191$	−20.7963	−0.108881	−0.0544407	+	0.998517i	$0.517338\pi$
$192$	0	0
$193$	186.854i	0.968157i	0.875025	+	0.484078i	$0.160845\pi$
$193$	186.854i	0.968157i	−0.875025	+	0.484078i	$0.839155\pi$
$194$	0	0
$195$	− 147.241i	− 0.755081i
$196$	0	0
$197$	− 325.621i	− 1.65290i	−0.563011	−	0.826449i	$-0.690357\pi$
$197$	− 325.621i	− 1.65290i	0.563011	−	0.826449i	$-0.309643\pi$
$198$	0	0
$199$	−76.6068	−0.384959	−0.192479	−	0.981301i	$-0.561653\pi$
$199$	−76.6068	−0.384959	−0.192479	+	0.981301i	$0.561653\pi$
$200$	0	0
$201$	166.868	0.830190
$202$	0	0
$203$	−233.116	−1.14835
$204$	0	0
$205$	− 278.148i	− 1.35682i
$206$	0	0
$207$	125.567	0.606606
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 229.452i	− 1.08745i	−0.839264	−	0.543724i	$-0.817014\pi$
$211$	− 229.452i	− 1.08745i	0.839264	−	0.543724i	$-0.182986\pi$
$212$	0	0
$213$	237.943	1.11710
$214$	0	0
$215$	− 475.283i	− 2.21062i
$216$	0	0
$217$	419.223i	1.93190i
$218$	0	0
$219$	− 110.683i	− 0.505404i
$220$	0	0
$221$	−43.3454	−0.196133
$222$	0	0
$223$	340.105	1.52514	0.762568	−	0.646908i	$-0.223938\pi$
$223$	340.105	1.52514	0.762568	+	0.646908i	$0.223938\pi$
$224$	0	0
$225$	192.797	0.856875
$226$	0	0
$227$	185.842i	0.818685i	0.912381	+	0.409343i	$0.134242\pi$
$227$	185.842i	0.818685i	−0.912381	+	0.409343i	$0.865758\pi$
$228$	0	0
$229$	148.459	0.648291	0.324146	−	0.946007i	$-0.394923\pi$
$229$	148.459	0.648291	0.324146	+	0.946007i	$0.394923\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	81.4706i	0.349659i	0.984599	+	0.174830i	$0.0559375\pi$
$233$	81.4706i	0.349659i	−0.984599	+	0.174830i	$0.944063\pi$
$234$	0	0
$235$	−71.6411	−0.304856
$236$	0	0
$237$	316.228i	1.33429i
$238$	0	0
$239$	201.381i	0.842600i	0.906921	+	0.421300i	$0.138426\pi$
$239$	201.381i	0.842600i	−0.906921	+	0.421300i	$0.861574\pi$
$240$	0	0
$241$	381.337i	1.58231i	0.611616	+	0.791155i	$0.290520\pi$
$241$	381.337i	1.58231i	−0.611616	+	0.791155i	$0.709480\pi$
$242$	0	0
$243$	346.783	1.42709
$244$	0	0
$245$	−548.738	−2.23975
$246$	0	0
$247$	54.8060	0.221886
$248$	0	0
$249$	51.1954i	0.205604i
$250$	0	0
$251$	103.927	0.414052	0.207026	−	0.978335i	$-0.433621\pi$
$251$	103.927	0.414052	0.207026	+	0.978335i	$0.433621\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	− 249.436i	− 0.978181i
$256$	0	0
$257$	−160.509	−0.624547	−0.312274	−	0.949992i	$-0.601091\pi$
$257$	−160.509	−0.624547	−0.312274	+	0.949992i	$0.601091\pi$
$258$	0	0
$259$	424.958i	1.64076i
$260$	0	0
$261$	− 216.273i	− 0.828634i
$262$	0	0
$263$	203.520i	0.773840i	0.922113	+	0.386920i	$0.126461\pi$
$263$	203.520i	0.773840i	−0.922113	+	0.386920i	$0.873539\pi$
$264$	0	0
$265$	474.364	1.79005
$266$	0	0
$267$	335.854	1.25788
$268$	0	0
$269$	225.953	0.839972	0.419986	−	0.907530i	$-0.362035\pi$
$269$	225.953	0.839972	0.419986	+	0.907530i	$0.362035\pi$
$270$	0	0
$271$	− 290.788i	− 1.07302i	−0.843894	−	0.536510i	$-0.819742\pi$
$271$	− 290.788i	− 1.07302i	0.843894	−	0.536510i	$-0.180258\pi$
$272$	0	0
$273$	−258.468	−0.946770
$274$	0	0
$275$	0	0
$276$	0	0
$277$	190.661i	0.688308i	0.938913	+	0.344154i	$0.111834\pi$
$277$	190.661i	0.688308i	−0.938913	+	0.344154i	$0.888166\pi$
$278$	0	0
$279$	−388.935	−1.39403
$280$	0	0
$281$	− 21.7299i	− 0.0773306i	−0.999252	−	0.0386653i	$-0.987689\pi$
$281$	− 21.7299i	− 0.0773306i	0.999252	−	0.0386653i	$-0.0123106\pi$
$282$	0	0
$283$	54.9793i	0.194273i	0.995271	+	0.0971366i	$0.0309684\pi$
$283$	54.9793i	0.194273i	−0.995271	+	0.0971366i	$0.969032\pi$
$284$	0	0
$285$	315.387i	1.10662i
$286$	0	0
$287$	−488.264	−1.70127
$288$	0	0
$289$	215.570	0.745916
$290$	0	0
$291$	−13.2491	−0.0455294
$292$	0	0
$293$	− 438.667i	− 1.49716i	−0.663046	−	0.748578i	$-0.730737\pi$
$293$	− 438.667i	− 1.49716i	0.663046	−	0.748578i	$-0.269263\pi$
$294$	0	0
$295$	29.3497	0.0994904
$296$	0	0
$297$	0	0
$298$	0	0
$299$	59.4450i	0.198813i
$300$	0	0
$301$	−834.319	−2.77182
$302$	0	0
$303$	− 197.182i	− 0.650765i
$304$	0	0
$305$	− 712.804i	− 2.33706i
$306$	0	0
$307$	232.888i	0.758594i	0.925275	+	0.379297i	$0.123834\pi$
$307$	232.888i	0.758594i	−0.925275	+	0.379297i	$0.876166\pi$
$308$	0	0
$309$	−239.603	−0.775415
$310$	0	0
$311$	−152.125	−0.489148	−0.244574	−	0.969631i	$-0.578648\pi$
$311$	−152.125	−0.489148	−0.244574	+	0.969631i	$0.578648\pi$
$312$	0	0
$313$	100.695	0.321710	0.160855	−	0.986978i	$-0.448575\pi$
$313$	100.695	0.321710	0.160855	+	0.986978i	$0.448575\pi$
$314$	0	0
$315$	− 807.345i	− 2.56300i
$316$	0	0
$317$	334.093	1.05392	0.526960	−	0.849890i	$-0.323332\pi$
$317$	334.093	1.05392	0.526960	+	0.849890i	$0.323332\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	− 582.953i	− 1.81605i
$322$	0	0
$323$	92.8451	0.287446
$324$	0	0
$325$	91.2722i	0.280837i
$326$	0	0
$327$	− 30.9030i	− 0.0945047i
$328$	0	0
$329$	125.760i	0.382248i
$330$	0	0
$331$	168.614	0.509409	0.254705	−	0.967019i	$-0.418022\pi$
$331$	168.614	0.509409	0.254705	+	0.967019i	$0.418022\pi$
$332$	0	0
$333$	−394.255	−1.18395
$334$	0	0
$335$	−246.754	−0.736579
$336$	0	0
$337$	523.585i	1.55366i	0.629708	+	0.776832i	$0.283174\pi$
$337$	523.585i	1.55366i	−0.629708	+	0.776832i	$0.716826\pi$
$338$	0	0
$339$	−242.538	−0.715451
$340$	0	0
$341$	0	0
$342$	0	0
$343$	398.934i	1.16307i
$344$	0	0
$345$	−342.083	−0.991545
$346$	0	0
$347$	− 478.209i	− 1.37812i	−0.724703	−	0.689062i	$-0.758023\pi$
$347$	− 478.209i	− 1.37812i	0.724703	−	0.689062i	$-0.241977\pi$
$348$	0	0
$349$	− 9.60448i	− 0.0275200i	−0.999905	−	0.0137600i	$-0.995620\pi$
$349$	− 9.60448i	− 0.0275200i	0.999905	−	0.0137600i	$-0.00438008\pi$
$350$	0	0
$351$	− 37.8117i	− 0.107726i
$352$	0	0
$353$	691.089	1.95776	0.978880	−	0.204436i	$-0.0655361\pi$
$353$	691.089	1.95776	0.978880	+	0.204436i	$0.0655361\pi$
$354$	0	0
$355$	−351.854	−0.991139
$356$	0	0
$357$	−437.863	−1.22651
$358$	0	0
$359$	− 46.1883i	− 0.128658i	−0.997929	−	0.0643291i	$-0.979509\pi$
$359$	− 46.1883i	− 0.128658i	0.997929	−	0.0643291i	$-0.0204907\pi$
$360$	0	0
$361$	243.607	0.674810
$362$	0	0
$363$	0	0
$364$	0	0
$365$	163.671i	0.448415i
$366$	0	0
$367$	456.172	1.24298	0.621488	−	0.783424i	$-0.286528\pi$
$367$	456.172	1.24298	0.621488	+	0.783424i	$0.286528\pi$
$368$	0	0
$369$	− 452.988i	− 1.22761i
$370$	0	0
$371$	− 832.706i	− 2.24449i
$372$	0	0
$373$	99.8021i	0.267566i	0.991011	+	0.133783i	$0.0427125\pi$
$373$	99.8021i	0.267566i	−0.991011	+	0.133783i	$0.957288\pi$
$374$	0	0
$375$	202.480	0.539947
$376$	0	0
$377$	102.386	0.271581
$378$	0	0
$379$	387.237	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$379$	387.237	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$380$	0	0
$381$	− 36.1742i	− 0.0949454i
$382$	0	0
$383$	606.285	1.58299	0.791495	−	0.611175i	$-0.209303\pi$
$383$	606.285	1.58299	0.791495	+	0.611175i	$0.209303\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 774.041i	− 2.00011i
$388$	0	0
$389$	579.347	1.48932	0.744662	−	0.667442i	$-0.232611\pi$
$389$	579.347	1.48932	0.744662	+	0.667442i	$0.232611\pi$
$390$	0	0
$391$	100.704i	0.257555i
$392$	0	0
$393$	720.897i	1.83434i
$394$	0	0
$395$	− 467.617i	− 1.18384i
$396$	0	0
$397$	−163.072	−0.410761	−0.205381	−	0.978682i	$-0.565843\pi$
$397$	−163.072	−0.410761	−0.205381	+	0.978682i	$0.565843\pi$
$398$	0	0
$399$	553.635	1.38756
$400$	0	0
$401$	−747.040	−1.86294	−0.931471	−	0.363816i	$-0.881474\pi$
$401$	−747.040	−1.86294	−0.931471	+	0.363816i	$0.881474\pi$
$402$	0	0
$403$	− 184.126i	− 0.456889i
$404$	0	0
$405$	−413.316	−1.02053
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 121.932i	− 0.298122i	−0.988828	−	0.149061i	$-0.952375\pi$
$409$	− 121.932i	− 0.298122i	0.988828	−	0.149061i	$-0.0476251\pi$
$410$	0	0
$411$	−1042.03	−2.53536
$412$	0	0
$413$	− 51.5208i	− 0.124748i
$414$	0	0
$415$	− 75.7044i	− 0.182420i
$416$	0	0
$417$	− 1053.82i	− 2.52715i
$418$	0	0
$419$	330.708	0.789279	0.394640	−	0.918836i	$-0.370869\pi$
$419$	330.708	0.789279	0.394640	+	0.918836i	$0.370869\pi$
$420$	0	0
$421$	640.074	1.52037	0.760183	−	0.649709i	$-0.225109\pi$
$421$	640.074	1.52037	0.760183	+	0.649709i	$0.225109\pi$
$422$	0	0
$423$	−116.674	−0.275824
$424$	0	0
$425$	154.622i	0.363815i
$426$	0	0
$427$	−1251.27	−2.93037
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 146.091i	− 0.338959i	−0.985534	−	0.169479i	$-0.945791\pi$
$431$	− 146.091i	− 0.338959i	0.985534	−	0.169479i	$-0.0542086\pi$
$432$	0	0
$433$	91.8796	0.212193	0.106097	−	0.994356i	$-0.466165\pi$
$433$	91.8796	0.212193	0.106097	+	0.994356i	$0.466165\pi$
$434$	0	0
$435$	589.193i	1.35447i
$436$	0	0
$437$	− 127.330i	− 0.291373i
$438$	0	0
$439$	358.371i	0.816335i	0.912907	+	0.408168i	$0.133832\pi$
$439$	358.371i	0.816335i	−0.912907	+	0.408168i	$0.866168\pi$
$440$	0	0
$441$	−893.668	−2.02646
$442$	0	0
$443$	−821.312	−1.85398	−0.926989	−	0.375089i	$-0.877612\pi$
$443$	−821.312	−1.85398	−0.926989	+	0.375089i	$0.877612\pi$
$444$	0	0
$445$	−496.639	−1.11604
$446$	0	0
$447$	727.573i	1.62768i
$448$	0	0
$449$	162.139	0.361111	0.180556	−	0.983565i	$-0.442210\pi$
$449$	162.139	0.361111	0.180556	+	0.983565i	$0.442210\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	195.050i	0.430573i
$454$	0	0
$455$	382.206	0.840014
$456$	0	0
$457$	− 363.463i	− 0.795324i	−0.917532	−	0.397662i	$-0.869822\pi$
$457$	− 363.463i	− 0.795324i	0.917532	−	0.397662i	$-0.130178\pi$
$458$	0	0
$459$	− 64.0557i	− 0.139555i
$460$	0	0
$461$	487.885i	1.05832i	0.848523	+	0.529159i	$0.177493\pi$
$461$	487.885i	1.05832i	−0.848523	+	0.529159i	$0.822507\pi$
$462$	0	0
$463$	367.635	0.794028	0.397014	−	0.917812i	$-0.370046\pi$
$463$	367.635	0.794028	0.397014	+	0.917812i	$0.370046\pi$
$464$	0	0
$465$	1059.58	2.27866
$466$	0	0
$467$	412.459	0.883209	0.441605	−	0.897210i	$-0.354409\pi$
$467$	412.459	0.883209	0.441605	+	0.897210i	$0.354409\pi$
$468$	0	0
$469$	433.155i	0.923571i
$470$	0	0
$471$	−35.6991	−0.0757943
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 195.503i	− 0.411586i
$476$	0	0
$477$	772.544	1.61959
$478$	0	0
$479$	531.662i	1.10994i	0.831870	+	0.554970i	$0.187270\pi$
$479$	531.662i	1.10994i	−0.831870	+	0.554970i	$0.812730\pi$
$480$	0	0
$481$	− 186.645i	− 0.388035i
$482$	0	0
$483$	600.497i	1.24326i
$484$	0	0
$485$	19.5919	0.0403956
$486$	0	0
$487$	−643.438	−1.32123	−0.660614	−	0.750726i	$-0.729704\pi$
$487$	−643.438	−1.32123	−0.660614	+	0.750726i	$0.729704\pi$
$488$	0	0
$489$	560.082	1.14536
$490$	0	0
$491$	− 750.349i	− 1.52821i	−0.645095	−	0.764103i	$-0.723182\pi$
$491$	− 750.349i	− 1.52821i	0.645095	−	0.764103i	$-0.276818\pi$
$492$	0	0
$493$	173.449	0.351824
$494$	0	0
$495$	0	0
$496$	0	0
$497$	617.650i	1.24276i
$498$	0	0
$499$	184.075	0.368888	0.184444	−	0.982843i	$-0.440952\pi$
$499$	184.075	0.368888	0.184444	+	0.982843i	$0.440952\pi$
$500$	0	0
$501$	− 940.519i	− 1.87728i
$502$	0	0
$503$	443.922i	0.882550i	0.897372	+	0.441275i	$0.145474\pi$
$503$	443.922i	0.882550i	−0.897372	+	0.441275i	$0.854526\pi$
$504$	0	0
$505$	291.580i	0.577385i
$506$	0	0
$507$	−636.291	−1.25501
$508$	0	0
$509$	−498.940	−0.980235	−0.490118	−	0.871656i	$-0.663046\pi$
$509$	−498.940	−0.980235	−0.490118	+	0.871656i	$0.663046\pi$
$510$	0	0
$511$	287.311	0.562253
$512$	0	0
$513$	80.9921i	0.157879i
$514$	0	0
$515$	354.310	0.687980
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1117.24i	2.15268i
$520$	0	0
$521$	−256.146	−0.491643	−0.245822	−	0.969315i	$-0.579058\pi$
$521$	−256.146	−0.491643	−0.245822	+	0.969315i	$0.579058\pi$
$522$	0	0
$523$	730.547i	1.39684i	0.715689	+	0.698419i	$0.246113\pi$
$523$	730.547i	1.39684i	−0.715689	+	0.698419i	$0.753887\pi$
$524$	0	0
$525$	922.006i	1.75620i
$526$	0	0
$527$	− 311.923i	− 0.591884i
$528$	0	0
$529$	−390.892	−0.738926
$530$	0	0
$531$	47.7985	0.0900161
$532$	0	0
$533$	214.450	0.402344
$534$	0	0
$535$	862.033i	1.61128i
$536$	0	0
$537$	983.980	1.83237
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 352.255i	− 0.651119i	−0.945522	−	0.325560i	$-0.894447\pi$
$541$	− 352.255i	− 0.651119i	0.945522	−	0.325560i	$-0.105553\pi$
$542$	0	0
$543$	8.75369	0.0161210
$544$	0	0
$545$	45.6974i	0.0838485i
$546$	0	0
$547$	835.372i	1.52719i	0.645697	+	0.763594i	$0.276567\pi$
$547$	835.372i	1.52719i	−0.645697	+	0.763594i	$0.723433\pi$
$548$	0	0
$549$	− 1160.86i	− 2.11451i
$550$	0	0
$551$	−219.310	−0.398021
$552$	0	0
$553$	−820.862	−1.48438
$554$	0	0
$555$	1074.07	1.93526
$556$	0	0
$557$	− 392.747i	− 0.705112i	−0.935791	−	0.352556i	$-0.885313\pi$
$557$	− 392.747i	− 0.705112i	0.935791	−	0.352556i	$-0.114687\pi$
$558$	0	0
$559$	366.439	0.655527
$560$	0	0
$561$	0	0
$562$	0	0
$563$	388.716i	0.690437i	0.938522	+	0.345218i	$0.112195\pi$
$563$	388.716i	0.690437i	−0.938522	+	0.345218i	$0.887805\pi$
$564$	0	0
$565$	358.649	0.634777
$566$	0	0
$567$	725.541i	1.27961i
$568$	0	0
$569$	− 659.098i	− 1.15834i	−0.815205	−	0.579172i	$-0.803376\pi$
$569$	− 659.098i	− 1.15834i	0.815205	−	0.579172i	$-0.196624\pi$
$570$	0	0
$571$	− 893.667i	− 1.56509i	−0.622593	−	0.782546i	$-0.713921\pi$
$571$	− 893.667i	− 1.56509i	0.622593	−	0.782546i	$-0.286079\pi$
$572$	0	0
$573$	92.2683	0.161027
$574$	0	0
$575$	212.052	0.368786
$576$	0	0
$577$	−17.7492	−0.0307612	−0.0153806	−	0.999882i	$-0.504896\pi$
$577$	−17.7492	−0.0307612	−0.0153806	+	0.999882i	$0.504896\pi$
$578$	0	0
$579$	− 829.027i	− 1.43183i
$580$	0	0
$581$	−132.893	−0.228731
$582$	0	0
$583$	0	0
$584$	0	0
$585$	354.593i	0.606141i
$586$	0	0
$587$	−163.056	−0.277778	−0.138889	−	0.990308i	$-0.544353\pi$
$587$	−163.056	−0.277778	−0.138889	+	0.990308i	$0.544353\pi$
$588$	0	0
$589$	394.395i	0.669602i
$590$	0	0
$591$	1444.70i	2.44450i
$592$	0	0
$593$	513.367i	0.865711i	0.901463	+	0.432855i	$0.142494\pi$
$593$	513.367i	0.865711i	−0.901463	+	0.432855i	$0.857506\pi$
$594$	0	0
$595$	647.484	1.08821
$596$	0	0
$597$	339.886	0.569323
$598$	0	0
$599$	926.376	1.54654	0.773269	−	0.634078i	$-0.218620\pi$
$599$	926.376	1.54654	0.773269	+	0.634078i	$0.218620\pi$
$600$	0	0
$601$	− 160.568i	− 0.267168i	−0.991038	−	0.133584i	$-0.957351\pi$
$601$	− 160.568i	− 0.267168i	0.991038	−	0.133584i	$-0.0426486\pi$
$602$	0	0
$603$	−401.860	−0.666435
$604$	0	0
$605$	0	0
$606$	0	0
$607$	527.231i	0.868584i	0.900772	+	0.434292i	$0.143001\pi$
$607$	527.231i	0.868584i	−0.900772	+	0.434292i	$0.856999\pi$
$608$	0	0
$609$	1034.28	1.69832
$610$	0	0
$611$	− 55.2346i	− 0.0904004i
$612$	0	0
$613$	− 663.170i	− 1.08184i	−0.841073	−	0.540922i	$-0.818076\pi$
$613$	− 663.170i	− 1.08184i	0.841073	−	0.540922i	$-0.181924\pi$
$614$	0	0
$615$	1234.07i	2.00662i
$616$	0	0
$617$	−260.782	−0.422661	−0.211331	−	0.977415i	$-0.567780\pi$
$617$	−260.782	−0.422661	−0.211331	+	0.977415i	$0.567780\pi$
$618$	0	0
$619$	1062.72	1.71683	0.858415	−	0.512956i	$-0.171450\pi$
$619$	1062.72	1.71683	0.858415	+	0.512956i	$0.171450\pi$
$620$	0	0
$621$	−87.8476	−0.141462
$622$	0	0
$623$	871.807i	1.39937i
$624$	0	0
$625$	−750.514	−1.20082
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 316.189i	− 0.502686i
$630$	0	0
$631$	522.733	0.828420	0.414210	−	0.910181i	$-0.364058\pi$
$631$	522.733	0.828420	0.414210	+	0.910181i	$0.364058\pi$
$632$	0	0
$633$	1018.02i	1.60825i
$634$	0	0
$635$	53.4921i	0.0842395i
$636$	0	0
$637$	− 423.072i	− 0.664164i
$638$	0	0
$639$	−573.026	−0.896754
$640$	0	0
$641$	−1110.91	−1.73309	−0.866547	−	0.499095i	$-0.833666\pi$
$641$	−1110.91	−1.73309	−0.866547	+	0.499095i	$0.833666\pi$
$642$	0	0
$643$	−885.748	−1.37752	−0.688762	−	0.724987i	$-0.741846\pi$
$643$	−885.748	−1.37752	−0.688762	+	0.724987i	$0.741846\pi$
$644$	0	0
$645$	2108.72i	3.26933i
$646$	0	0
$647$	−114.389	−0.176799	−0.0883993	−	0.996085i	$-0.528175\pi$
$647$	−114.389	−0.176799	−0.0883993	+	0.996085i	$0.528175\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 1859.99i	− 2.85713i
$652$	0	0
$653$	−163.928	−0.251038	−0.125519	−	0.992091i	$-0.540060\pi$
$653$	−163.928	−0.251038	−0.125519	+	0.992091i	$0.540060\pi$
$654$	0	0
$655$	− 1066.02i	− 1.62750i
$656$	0	0
$657$	266.553i	0.405713i
$658$	0	0
$659$	116.651i	0.177012i	0.996076	+	0.0885062i	$0.0282093\pi$
$659$	116.651i	0.177012i	−0.996076	+	0.0885062i	$0.971791\pi$
$660$	0	0
$661$	205.962	0.311591	0.155796	−	0.987789i	$-0.450206\pi$
$661$	205.962	0.311591	0.155796	+	0.987789i	$0.450206\pi$
$662$	0	0
$663$	192.313	0.290065
$664$	0	0
$665$	−818.679	−1.23110
$666$	0	0
$667$	− 237.873i	− 0.356631i
$668$	0	0
$669$	−1508.96	−2.25555
$670$	0	0
$671$	0	0
$672$	0	0
$673$	27.9419i	0.0415184i	0.999785	+	0.0207592i	$0.00660834\pi$
$673$	27.9419i	0.0415184i	−0.999785	+	0.0207592i	$0.993392\pi$
$674$	0	0
$675$	−134.882	−0.199825
$676$	0	0
$677$	− 540.241i	− 0.797993i	−0.916952	−	0.398997i	$-0.869359\pi$
$677$	− 540.241i	− 0.797993i	0.916952	−	0.398997i	$-0.130641\pi$
$678$	0	0
$679$	− 34.3918i	− 0.0506507i
$680$	0	0
$681$	− 824.534i	− 1.21077i
$682$	0	0
$683$	−1117.89	−1.63674	−0.818369	−	0.574694i	$-0.805121\pi$
$683$	−1117.89	−1.63674	−0.818369	+	0.574694i	$0.805121\pi$
$684$	0	0
$685$	1540.89	2.24947
$686$	0	0
$687$	−658.675	−0.958771
$688$	0	0
$689$	365.731i	0.530814i
$690$	0	0
$691$	500.242	0.723939	0.361969	−	0.932190i	$-0.382104\pi$
$691$	500.242	0.723939	0.361969	+	0.932190i	$0.382104\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1558.33i	2.24219i
$696$	0	0
$697$	363.293	0.521223
$698$	0	0
$699$	− 361.466i	− 0.517118i
$700$	0	0
$701$	646.819i	0.922709i	0.887216	+	0.461355i	$0.152636\pi$
$701$	646.819i	0.922709i	−0.887216	+	0.461355i	$0.847364\pi$
$702$	0	0
$703$	399.790i	0.568691i
$704$	0	0
$705$	317.854	0.450857
$706$	0	0
$707$	511.842	0.723964
$708$	0	0
$709$	−789.315	−1.11328	−0.556640	−	0.830754i	$-0.687910\pi$
$709$	−789.315	−1.11328	−0.556640	+	0.830754i	$0.687910\pi$
$710$	0	0
$711$	− 761.556i	− 1.07111i
$712$	0	0
$713$	−427.779	−0.599970
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 893.481i	− 1.24614i
$718$	0	0
$719$	−247.723	−0.344539	−0.172269	−	0.985050i	$-0.555110\pi$
$719$	−247.723	−0.344539	−0.172269	+	0.985050i	$0.555110\pi$
$720$	0	0
$721$	− 621.960i	− 0.862635i
$722$	0	0
$723$	− 1691.90i	− 2.34011i
$724$	0	0
$725$	− 365.231i	− 0.503768i
$726$	0	0
$727$	−133.030	−0.182986	−0.0914928	−	0.995806i	$-0.529164\pi$
$727$	−133.030	−0.182986	−0.0914928	+	0.995806i	$0.529164\pi$
$728$	0	0
$729$	−971.611	−1.33280
$730$	0	0
$731$	620.774	0.849212
$732$	0	0
$733$	412.846i	0.563227i	0.959528	+	0.281614i	$0.0908696\pi$
$733$	412.846i	0.563227i	−0.959528	+	0.281614i	$0.909130\pi$
$734$	0	0
$735$	2434.62	3.31240
$736$	0	0
$737$	0	0
$738$	0	0
$739$	79.1777i	0.107142i	0.998564	+	0.0535708i	$0.0170603\pi$
$739$	79.1777i	0.107142i	−0.998564	+	0.0535708i	$0.982940\pi$
$740$	0	0
$741$	−243.161	−0.328152
$742$	0	0
$743$	− 513.850i	− 0.691589i	−0.938310	−	0.345794i	$-0.887609\pi$
$743$	− 513.850i	− 0.691589i	0.938310	−	0.345794i	$-0.112391\pi$
$744$	0	0
$745$	− 1075.89i	− 1.44415i
$746$	0	0
$747$	− 123.291i	− 0.165049i
$748$	0	0
$749$	1513.22	2.02033
$750$	0	0
$751$	−48.6547	−0.0647865	−0.0323932	−	0.999475i	$-0.510313\pi$
$751$	−48.6547	−0.0647865	−0.0323932	+	0.999475i	$0.510313\pi$
$752$	0	0
$753$	−461.099	−0.612350
$754$	0	0
$755$	− 288.427i	− 0.382022i
$756$	0	0
$757$	−204.410	−0.270027	−0.135014	−	0.990844i	$-0.543108\pi$
$757$	−204.410	−0.270027	−0.135014	+	0.990844i	$0.543108\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	372.239i	0.489144i	0.969631	+	0.244572i	$0.0786475\pi$
$761$	372.239i	0.489144i	−0.969631	+	0.244572i	$0.921353\pi$
$762$	0	0
$763$	80.2178	0.105135
$764$	0	0
$765$	600.705i	0.785235i
$766$	0	0
$767$	22.6284i	0.0295024i
$768$	0	0
$769$	− 1491.70i	− 1.93979i	−0.243531	−	0.969893i	$-0.578306\pi$
$769$	− 1491.70i	− 1.93979i	0.243531	−	0.969893i	$-0.421694\pi$
$770$	0	0
$771$	712.138	0.923655
$772$	0	0
$773$	−679.260	−0.878732	−0.439366	−	0.898308i	$-0.644797\pi$
$773$	−679.260	−0.878732	−0.439366	+	0.898308i	$0.644797\pi$
$774$	0	0
$775$	−656.814	−0.847502
$776$	0	0
$777$	− 1885.43i	− 2.42656i
$778$	0	0
$779$	−459.347	−0.589663
$780$	0	0
$781$	0	0
$782$	0	0
$783$	151.306i	0.193239i
$784$	0	0
$785$	52.7896	0.0672478
$786$	0	0
$787$	530.519i	0.674103i	0.941486	+	0.337052i	$0.109430\pi$
$787$	530.519i	0.674103i	−0.941486	+	0.337052i	$0.890570\pi$
$788$	0	0
$789$	− 902.969i	− 1.14445i
$790$	0	0
$791$	− 629.577i	− 0.795926i
$792$	0	0
$793$	549.566	0.693022
$794$	0	0
$795$	−2104.64	−2.64735
$796$	0	0
$797$	894.517	1.12236	0.561178	−	0.827695i	$-0.310348\pi$
$797$	894.517	1.12236	0.561178	+	0.827695i	$0.310348\pi$
$798$	0	0
$799$	− 93.5714i	− 0.117111i
$800$	0	0
$801$	−808.820	−1.00976
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 887.976i	− 1.10308i
$806$	0	0
$807$	−1002.50	−1.24225
$808$	0	0
$809$	− 1079.13i	− 1.33391i	−0.745099	−	0.666954i	$-0.767598\pi$
$809$	− 1079.13i	− 1.33391i	0.745099	−	0.666954i	$-0.232402\pi$
$810$	0	0
$811$	1189.16i	1.46628i	0.680076	+	0.733142i	$0.261947\pi$
$811$	1189.16i	1.46628i	−0.680076	+	0.733142i	$0.738053\pi$
$812$	0	0
$813$	1290.16i	1.58691i
$814$	0	0
$815$	−828.213	−1.01621
$816$	0	0
$817$	−784.907	−0.960719
$818$	0	0
$819$	622.456	0.760020
$820$	0	0
$821$	− 1063.96i	− 1.29593i	−0.761670	−	0.647965i	$-0.775620\pi$
$821$	− 1063.96i	− 1.29593i	0.761670	−	0.647965i	$-0.224380\pi$
$822$	0	0
$823$	1029.49	1.25090	0.625449	−	0.780265i	$-0.284916\pi$
$823$	1029.49	1.25090	0.625449	+	0.780265i	$0.284916\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1113.21i	− 1.34609i	−0.739603	−	0.673044i	$-0.764986\pi$
$827$	− 1113.21i	− 1.34609i	0.739603	−	0.673044i	$-0.235014\pi$
$828$	0	0
$829$	−1293.00	−1.55971	−0.779853	−	0.625963i	$-0.784706\pi$
$829$	−1293.00	−1.55971	−0.779853	+	0.625963i	$0.784706\pi$
$830$	0	0
$831$	− 845.918i	− 1.01795i
$832$	0	0
$833$	− 716.714i	− 0.860401i
$834$	0	0
$835$	1390.78i	1.66560i
$836$	0	0
$837$	272.101	0.325091
$838$	0	0
$839$	1230.48	1.46660	0.733302	−	0.679904i	$-0.237978\pi$
$839$	1230.48	1.46660	0.733302	+	0.679904i	$0.237978\pi$
$840$	0	0
$841$	431.295	0.512836
$842$	0	0
$843$	96.4104i	0.114366i
$844$	0	0
$845$	940.906	1.11350
$846$	0	0
$847$	0	0
$848$	0	0
$849$	− 243.930i	− 0.287314i
$850$	0	0
$851$	−433.630	−0.509553
$852$	0	0
$853$	866.309i	1.01560i	0.861474	+	0.507801i	$0.169542\pi$
$853$	866.309i	1.01560i	−0.861474	+	0.507801i	$0.830458\pi$
$854$	0	0
$855$	− 759.531i	− 0.888341i
$856$	0	0
$857$	1219.98i	1.42354i	0.702411	+	0.711772i	$0.252107\pi$
$857$	1219.98i	1.42354i	−0.702411	+	0.711772i	$0.747893\pi$
$858$	0	0
$859$	−469.893	−0.547023	−0.273512	−	0.961869i	$-0.588185\pi$
$859$	−469.893	−0.547023	−0.273512	+	0.961869i	$0.588185\pi$
$860$	0	0
$861$	2166.31	2.51604
$862$	0	0
$863$	4.23962	0.00491265	0.00245633	−	0.999997i	$-0.499218\pi$
$863$	4.23962	0.00491265	0.00245633	+	0.999997i	$0.499218\pi$
$864$	0	0
$865$	− 1652.10i	− 1.90995i
$866$	0	0
$867$	−956.431	−1.10315
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 190.245i	− 0.218421i
$872$	0	0
$873$	31.9071	0.0365488
$874$	0	0
$875$	525.596i	0.600681i
$876$	0	0
$877$	425.554i	0.485239i	0.970122	+	0.242619i	$0.0780066\pi$
$877$	425.554i	0.485239i	−0.970122	+	0.242619i	$0.921993\pi$
$878$	0	0
$879$	1946.26i	2.21417i
$880$	0	0
$881$	−1669.31	−1.89479	−0.947396	−	0.320064i	$-0.896296\pi$
$881$	−1669.31	−1.89479	−0.947396	+	0.320064i	$0.896296\pi$
$882$	0	0
$883$	−213.204	−0.241455	−0.120727	−	0.992686i	$-0.538523\pi$
$883$	−213.204	−0.241455	−0.120727	+	0.992686i	$0.538523\pi$
$884$	0	0
$885$	−130.217	−0.147138
$886$	0	0
$887$	523.852i	0.590589i	0.955406	+	0.295294i	$0.0954177\pi$
$887$	523.852i	0.590589i	−0.955406	+	0.295294i	$0.904582\pi$
$888$	0	0
$889$	93.9006	0.105625
$890$	0	0
$891$	0	0
$892$	0	0
$893$	118.312i	0.132488i
$894$	0	0
$895$	−1455.05	−1.62575
$896$	0	0
$897$	− 263.743i	− 0.294028i
$898$	0	0
$899$	736.793i	0.819569i
$900$	0	0
$901$	619.574i	0.687651i
$902$	0	0
$903$	3701.67	4.09930
$904$	0	0
$905$	−12.9444	−0.0143032
$906$	0	0
$907$	872.013	0.961426	0.480713	−	0.876878i	$-0.340378\pi$
$907$	872.013	0.961426	0.480713	+	0.876878i	$0.340378\pi$
$908$	0	0
$909$	474.863i	0.522401i
$910$	0	0
$911$	−907.086	−0.995704	−0.497852	−	0.867262i	$-0.665878\pi$
$911$	−907.086	−0.995704	−0.497852	+	0.867262i	$0.665878\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3162.54i	3.45633i
$916$	0	0
$917$	−1871.30	−2.04067
$918$	0	0
$919$	1204.69i	1.31087i	0.755253	+	0.655434i	$0.227514\pi$
$919$	1204.69i	1.31087i	−0.755253	+	0.655434i	$0.772486\pi$
$920$	0	0
$921$	− 1033.27i	− 1.12190i
$922$	0	0
$923$	− 271.277i	− 0.293908i
$924$	0	0
$925$	−665.798	−0.719782
$926$	0	0
$927$	577.025	0.622464
$928$	0	0
$929$	−1714.81	−1.84587	−0.922934	−	0.384958i	$-0.874216\pi$
$929$	−1714.81	−1.84587	−0.922934	+	0.384958i	$0.874216\pi$
$930$	0	0
$931$	906.214i	0.973377i
$932$	0	0
$933$	674.943	0.723411
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 893.655i	− 0.953741i	−0.878974	−	0.476870i	$-0.841771\pi$
$937$	− 893.655i	− 0.953741i	0.878974	−	0.476870i	$-0.158229\pi$
$938$	0	0
$939$	−446.761	−0.475784
$940$	0	0
$941$	1456.50i	1.54782i	0.633295	+	0.773910i	$0.281702\pi$
$941$	1456.50i	1.54782i	−0.633295	+	0.773910i	$0.718298\pi$
$942$	0	0
$943$	− 498.228i	− 0.528344i
$944$	0	0
$945$	564.823i	0.597697i
$946$	0	0
$947$	1026.37	1.08381	0.541906	−	0.840439i	$-0.317703\pi$
$947$	1026.37	1.08381	0.541906	+	0.840439i	$0.317703\pi$
$948$	0	0
$949$	−126.189	−0.132971
$950$	0	0
$951$	−1482.29	−1.55866
$952$	0	0
$953$	− 77.0539i	− 0.0808541i	−0.999182	−	0.0404270i	$-0.987128\pi$
$953$	− 77.0539i	− 0.0808541i	0.999182	−	0.0404270i	$-0.0128718\pi$
$954$	0	0
$955$	−136.440	−0.142870
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 2704.90i	− 2.82054i
$960$	0	0
$961$	364.012	0.378784
$962$	0	0
$963$	1403.90i	1.45784i
$964$	0	0
$965$	1225.91i	1.27037i
$966$	0	0
$967$	637.935i	0.659705i	0.944032	+	0.329853i	$0.106999\pi$
$967$	637.935i	0.659705i	−0.944032	+	0.329853i	$0.893001\pi$
$968$	0	0
$969$	−411.931	−0.425110
$970$	0	0
$971$	1344.13	1.38428	0.692139	−	0.721764i	$-0.256668\pi$
$971$	1344.13	1.38428	0.692139	+	0.721764i	$0.256668\pi$
$972$	0	0
$973$	2735.50	2.81141
$974$	0	0
$975$	− 404.953i	− 0.415336i
$976$	0	0
$977$	766.956	0.785012	0.392506	−	0.919750i	$-0.371608\pi$
$977$	766.956	0.785012	0.392506	+	0.919750i	$0.371608\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	74.4223i	0.0758637i
$982$	0	0
$983$	641.293	0.652383	0.326192	−	0.945304i	$-0.394235\pi$
$983$	641.293	0.652383	0.326192	+	0.945304i	$0.394235\pi$
$984$	0	0
$985$	− 2136.33i	− 2.16887i
$986$	0	0
$987$	− 557.965i	− 0.565314i
$988$	0	0
$989$	− 851.345i	− 0.860814i
$990$	0	0
$991$	−1547.49	−1.56154	−0.780770	−	0.624818i	$-0.785173\pi$
$991$	−1547.49	−1.56154	−0.780770	+	0.624818i	$0.785173\pi$
$992$	0	0
$993$	−748.102	−0.753375
$994$	0	0
$995$	−502.601	−0.505127
$996$	0	0
$997$	− 1943.59i	− 1.94943i	−0.223443	−	0.974717i	$-0.571730\pi$
$997$	− 1943.59i	− 1.94943i	0.223443	−	0.974717i	$-0.428270\pi$
$998$	0	0
$999$	275.823	0.276099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	484.3.d.c.241.1		8
3.2	odd	2		4356.3.f.g.1693.1		8
11.2	odd	10		484.3.f.a.161.1		8
11.3	even	5		484.3.f.d.233.2		8
11.4	even	5		484.3.f.e.457.2		8
11.5	even	5		484.3.f.a.481.1		8
11.6	odd	10		44.3.f.a.41.1	yes	8
11.7	odd	10		484.3.f.d.457.2		8
11.8	odd	10		484.3.f.e.233.2		8
11.9	even	5		44.3.f.a.29.1	✓	8
11.10	odd	2	inner	484.3.d.c.241.2		8
33.17	even	10		396.3.t.a.217.2		8
33.20	odd	10		396.3.t.a.73.2		8
33.32	even	2		4356.3.f.g.1693.2		8
44.31	odd	10		176.3.n.c.161.2		8
44.39	even	10		176.3.n.c.129.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.3.f.a.29.1	✓	8	11.9	even	5
44.3.f.a.41.1	yes	8	11.6	odd	10
176.3.n.c.129.2		8	44.39	even	10
176.3.n.c.161.2		8	44.31	odd	10
396.3.t.a.73.2		8	33.20	odd	10
396.3.t.a.217.2		8	33.17	even	10
484.3.d.c.241.1		8	1.1	even	1	trivial
484.3.d.c.241.2		8	11.10	odd	2	inner
484.3.f.a.161.1		8	11.2	odd	10
484.3.f.a.481.1		8	11.5	even	5
484.3.f.d.233.2		8	11.3	even	5
484.3.f.d.457.2		8	11.7	odd	10
484.3.f.e.233.2		8	11.8	odd	10
484.3.f.e.457.2		8	11.4	even	5
4356.3.f.g.1693.1		8	3.2	odd	2
4356.3.f.g.1693.2		8	33.32	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 484 = 2^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	484.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.43676 q^{3} +6.56079 q^{5} -11.5169i q^{7} +10.6848 q^{9} +5.05831i q^{13} -29.1086 q^{15} +8.56914i q^{17} -10.8348i q^{19} +51.0977i q^{21} +11.7519 q^{23} +18.0440 q^{25} -7.47516 q^{27} -20.2412i q^{29} -36.4007 q^{31} -75.5600i q^{35} -36.8986 q^{37} -22.4425i q^{39} -42.3954i q^{41} -72.4430i q^{43} +70.1009 q^{45} -10.9196 q^{47} -83.6390 q^{49} -38.0192i q^{51} +72.3029 q^{53} +48.0715i q^{57} +4.47350 q^{59} -108.646i q^{61} -123.056i q^{63} +33.1866i q^{65} -37.6104 q^{67} -52.1405 q^{69} -53.6299 q^{71} +24.9469i q^{73} -80.0568 q^{75} -71.2745i q^{79} -62.9979 q^{81} -11.5389i q^{83} +56.2203i q^{85} +89.8052i q^{87} -75.6980 q^{89} +58.2561 q^{91} +161.501 q^{93} -71.0850i q^{95} +2.98620 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{5} + 44 q^{9} - 28 q^{15} + 100 q^{23} - 34 q^{25} + 60 q^{27} + 98 q^{31} - 210 q^{37} + 332 q^{45} - 110 q^{47} + 22 q^{49} + 250 q^{53} + 112 q^{59} + 10 q^{67} - 176 q^{69} - 198 q^{71}+ \cdots - 220 q^{97}+O(q^{100}) \) 8 * q + 14 * q^5 + 44 * q^9 - 28 * q^15 + 100 * q^23 - 34 * q^25 + 60 * q^27 + 98 * q^31 - 210 * q^37 + 332 * q^45 - 110 * q^47 + 22 * q^49 + 250 * q^53 + 112 * q^59 + 10 * q^67 - 176 * q^69 - 198 * q^71 - 78 * q^75 - 232 * q^81 + 18 * q^89 + 440 * q^91 + 360 * q^93 - 220 * q^97

Introduction

L-functions

Modular forms

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