LMFDB - Embedded newform 336.3.m.a.127.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,3,Mod(127,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("336.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	336.m (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:	$9.15533688251$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - x^{2} - 2x + 4 $ x^4 - x^3 - x^2 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.3
Root			$-0.895644 + 1.09445i$ of defining polynomial
Character	$\chi$	$=$	336.127
Dual form			336.3.m.a.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -9.58258 q^{5} +2.64575i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -9.58258 q^{5} +2.64575i q^{7} -3.00000 q^{9} -8.03260i q^{11} +17.1652 q^{13} -16.5975i q^{15} +9.58258 q^{17} -21.1660i q^{19} -4.58258 q^{21} -41.2276i q^{23} +66.8258 q^{25} -5.19615i q^{27} -32.3303 q^{29} +20.7846i q^{31} +13.9129 q^{33} -25.3531i q^{35} -42.8348 q^{37} +29.7309i q^{39} +55.0780 q^{41} -35.0224i q^{43} +28.7477 q^{45} -12.4104i q^{47} -7.00000 q^{49} +16.5975i q^{51} -36.1561 q^{53} +76.9730i q^{55} +36.6606 q^{57} +46.2886i q^{59} +5.33939 q^{61} -7.93725i q^{63} -164.486 q^{65} -109.405i q^{67} +71.4083 q^{69} -70.3864i q^{71} +37.8258 q^{73} +115.746i q^{75} +21.2523 q^{77} -43.0152i q^{79} +9.00000 q^{81} -115.650i q^{83} -91.8258 q^{85} -55.9977i q^{87} +36.0871 q^{89} +45.4147i q^{91} -36.0000 q^{93} +202.825i q^{95} -121.826 q^{97} +24.0978i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q - 20 * q^5 - 12 * q^9	$ 4 q - 20 q^{5} - 12 q^{9} + 32 q^{13} + 20 q^{17} + 84 q^{25} - 56 q^{29} - 36 q^{33} - 208 q^{37} + 92 q^{41} + 60 q^{45} - 28 q^{49} + 112 q^{53} + 168 q^{61} - 328 q^{65} + 84 q^{69} - 32 q^{73} + 140 q^{77} + 36 q^{81} - 184 q^{85} + 236 q^{89} - 144 q^{93} - 304 q^{97}+O(q^{100}) $ 4 * q - 20 * q^5 - 12 * q^9 + 32 * q^13 + 20 * q^17 + 84 * q^25 - 56 * q^29 - 36 * q^33 - 208 * q^37 + 92 * q^41 + 60 * q^45 - 28 * q^49 + 112 * q^53 + 168 * q^61 - 328 * q^65 + 84 * q^69 - 32 * q^73 + 140 * q^77 + 36 * q^81 - 184 * q^85 + 236 * q^89 - 144 * q^93 - 304 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	−9.58258	−1.91652	−0.958258	−	0.285906i	$-0.907705\pi$
$5$	−9.58258	−1.91652	−0.958258	+	0.285906i	$0.907705\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 8.03260i	− 0.730237i	−0.930961	−	0.365118i	$-0.881029\pi$
$11$	− 8.03260i	− 0.730237i	0.930961	−	0.365118i	$-0.118971\pi$
$12$	0	0
$13$	17.1652	1.32040	0.660198	−	0.751091i	$-0.270472\pi$
$13$	17.1652	1.32040	0.660198	+	0.751091i	$0.270472\pi$
$14$	0	0
$15$	− 16.5975i	− 1.10650i
$16$	0	0
$17$	9.58258	0.563681	0.281840	−	0.959461i	$-0.409055\pi$
$17$	9.58258	0.563681	0.281840	+	0.959461i	$0.409055\pi$
$18$	0	0
$19$	− 21.1660i	− 1.11400i	−0.830512	−	0.557000i	$-0.811952\pi$
$19$	− 21.1660i	− 1.11400i	0.830512	−	0.557000i	$-0.188048\pi$
$20$	0	0
$21$	−4.58258	−0.218218
$22$	0	0
$23$	− 41.2276i	− 1.79251i	−0.443544	−	0.896253i	$-0.646279\pi$
$23$	− 41.2276i	− 1.79251i	0.443544	−	0.896253i	$-0.353721\pi$
$24$	0	0
$25$	66.8258	2.67303
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−32.3303	−1.11484	−0.557419	−	0.830231i	$-0.688208\pi$
$29$	−32.3303	−1.11484	−0.557419	+	0.830231i	$0.688208\pi$
$30$	0	0
$31$	20.7846i	0.670471i	0.942134	+	0.335236i	$0.108816\pi$
$31$	20.7846i	0.670471i	−0.942134	+	0.335236i	$0.891184\pi$
$32$	0	0
$33$	13.9129	0.421602
$34$	0	0
$35$	− 25.3531i	− 0.724375i
$36$	0	0
$37$	−42.8348	−1.15770	−0.578849	−	0.815435i	$-0.696498\pi$
$37$	−42.8348	−1.15770	−0.578849	+	0.815435i	$0.696498\pi$
$38$	0	0
$39$	29.7309i	0.762331i
$40$	0	0
$41$	55.0780	1.34337	0.671683	−	0.740838i	$-0.265572\pi$
$41$	55.0780	1.34337	0.671683	+	0.740838i	$0.265572\pi$
$42$	0	0
$43$	− 35.0224i	− 0.814475i	−0.913322	−	0.407237i	$-0.866492\pi$
$43$	− 35.0224i	− 0.814475i	0.913322	−	0.407237i	$-0.133508\pi$
$44$	0	0
$45$	28.7477	0.638838
$46$	0	0
$47$	− 12.4104i	− 0.264051i	−0.991246	−	0.132026i	$-0.957852\pi$
$47$	− 12.4104i	− 0.264051i	0.991246	−	0.132026i	$-0.0421481\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	16.5975i	0.325441i
$52$	0	0
$53$	−36.1561	−0.682190	−0.341095	−	0.940029i	$-0.610798\pi$
$53$	−36.1561	−0.682190	−0.341095	+	0.940029i	$0.610798\pi$
$54$	0	0
$55$	76.9730i	1.39951i
$56$	0	0
$57$	36.6606	0.643169
$58$	0	0
$59$	46.2886i	0.784553i	0.919847	+	0.392276i	$0.128312\pi$
$59$	46.2886i	0.784553i	−0.919847	+	0.392276i	$0.871688\pi$
$60$	0	0
$61$	5.33939	0.0875311	0.0437655	−	0.999042i	$-0.486065\pi$
$61$	5.33939	0.0875311	0.0437655	+	0.999042i	$0.486065\pi$
$62$	0	0
$63$	− 7.93725i	− 0.125988i
$64$	0	0
$65$	−164.486	−2.53056
$66$	0	0
$67$	− 109.405i	− 1.63291i	−0.577406	−	0.816457i	$-0.695935\pi$
$67$	− 109.405i	− 1.63291i	0.577406	−	0.816457i	$-0.304065\pi$
$68$	0	0
$69$	71.4083	1.03490
$70$	0	0
$71$	− 70.3864i	− 0.991358i	−0.868506	−	0.495679i	$-0.834919\pi$
$71$	− 70.3864i	− 0.991358i	0.868506	−	0.495679i	$-0.165081\pi$
$72$	0	0
$73$	37.8258	0.518161	0.259081	−	0.965856i	$-0.416580\pi$
$73$	37.8258	0.518161	0.259081	+	0.965856i	$0.416580\pi$
$74$	0	0
$75$	115.746i	1.54327i
$76$	0	0
$77$	21.2523	0.276004
$78$	0	0
$79$	− 43.0152i	− 0.544496i	−0.962227	−	0.272248i	$-0.912233\pi$
$79$	− 43.0152i	− 0.544496i	0.962227	−	0.272248i	$-0.0877672\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 115.650i	− 1.39338i	−0.717374	−	0.696688i	$-0.754656\pi$
$83$	− 115.650i	− 1.39338i	0.717374	−	0.696688i	$-0.245344\pi$
$84$	0	0
$85$	−91.8258	−1.08030
$86$	0	0
$87$	− 55.9977i	− 0.643652i
$88$	0	0
$89$	36.0871	0.405473	0.202737	−	0.979233i	$-0.435016\pi$
$89$	36.0871	0.405473	0.202737	+	0.979233i	$0.435016\pi$
$90$	0	0
$91$	45.4147i	0.499063i
$92$	0	0
$93$	−36.0000	−0.387097
$94$	0	0
$95$	202.825i	2.13500i
$96$	0	0
$97$	−121.826	−1.25594	−0.627968	−	0.778239i	$-0.716113\pi$
$97$	−121.826	−1.25594	−0.627968	+	0.778239i	$0.716113\pi$
$98$	0	0
$99$	24.0978i	0.243412i
$100$	0	0
$101$	−101.060	−1.00059	−0.500296	−	0.865854i	$-0.666776\pi$
$101$	−101.060	−1.00059	−0.500296	+	0.865854i	$0.666776\pi$
$102$	0	0
$103$	52.1522i	0.506332i	0.967423	+	0.253166i	$0.0814719\pi$
$103$	52.1522i	0.506332i	−0.967423	+	0.253166i	$0.918528\pi$
$104$	0	0
$105$	43.9129	0.418218
$106$	0	0
$107$	3.69460i	0.0345290i	0.999851	+	0.0172645i	$0.00549573\pi$
$107$	3.69460i	0.0345290i	−0.999851	+	0.0172645i	$0.994504\pi$
$108$	0	0
$109$	158.661	1.45560	0.727801	−	0.685788i	$-0.240542\pi$
$109$	158.661	1.45560	0.727801	+	0.685788i	$0.240542\pi$
$110$	0	0
$111$	− 74.1921i	− 0.668398i
$112$	0	0
$113$	−2.66061	−0.0235452	−0.0117726	−	0.999931i	$-0.503747\pi$
$113$	−2.66061	−0.0235452	−0.0117726	+	0.999931i	$0.503747\pi$
$114$	0	0
$115$	395.067i	3.43536i
$116$	0	0
$117$	−51.4955	−0.440132
$118$	0	0
$119$	25.3531i	0.213051i
$120$	0	0
$121$	56.4773	0.466754
$122$	0	0
$123$	95.3979i	0.775593i
$124$	0	0
$125$	−400.798	−3.20639
$126$	0	0
$127$	237.164i	1.86743i	0.358012	+	0.933717i	$0.383455\pi$
$127$	237.164i	1.86743i	−0.358012	+	0.933717i	$0.616545\pi$
$128$	0	0
$129$	60.6606	0.470237
$130$	0	0
$131$	83.5198i	0.637556i	0.947829	+	0.318778i	$0.103272\pi$
$131$	83.5198i	0.637556i	−0.947829	+	0.318778i	$0.896728\pi$
$132$	0	0
$133$	56.0000	0.421053
$134$	0	0
$135$	49.7925i	0.368834i
$136$	0	0
$137$	−107.495	−0.784638	−0.392319	−	0.919829i	$-0.628327\pi$
$137$	−107.495	−0.784638	−0.392319	+	0.919829i	$0.628327\pi$
$138$	0	0
$139$	− 53.5186i	− 0.385026i	−0.981294	−	0.192513i	$-0.938336\pi$
$139$	− 53.5186i	− 0.385026i	0.981294	−	0.192513i	$-0.0616638\pi$
$140$	0	0
$141$	21.4955	0.152450
$142$	0	0
$143$	− 137.881i	− 0.964202i
$144$	0	0
$145$	309.808	2.13660
$146$	0	0
$147$	− 12.1244i	− 0.0824786i
$148$	0	0
$149$	−30.1742	−0.202512	−0.101256	−	0.994860i	$-0.532286\pi$
$149$	−30.1742	−0.202512	−0.101256	+	0.994860i	$0.532286\pi$
$150$	0	0
$151$	− 280.942i	− 1.86054i	−0.366871	−	0.930272i	$-0.619571\pi$
$151$	− 280.942i	− 1.86054i	0.366871	−	0.930272i	$-0.380429\pi$
$152$	0	0
$153$	−28.7477	−0.187894
$154$	0	0
$155$	− 199.170i	− 1.28497i
$156$	0	0
$157$	58.3485	0.371646	0.185823	−	0.982583i	$-0.440505\pi$
$157$	58.3485	0.371646	0.185823	+	0.982583i	$0.440505\pi$
$158$	0	0
$159$	− 62.6241i	− 0.393862i
$160$	0	0
$161$	109.078	0.677503
$162$	0	0
$163$	− 85.3472i	− 0.523603i	−0.965122	−	0.261801i	$-0.915683\pi$
$163$	− 85.3472i	− 0.523603i	0.965122	−	0.261801i	$-0.0843165\pi$
$164$	0	0
$165$	−133.321	−0.808007
$166$	0	0
$167$	102.858i	0.615919i	0.951399	+	0.307959i	$0.0996461\pi$
$167$	102.858i	0.615919i	−0.951399	+	0.307959i	$0.900354\pi$
$168$	0	0
$169$	125.642	0.743446
$170$	0	0
$171$	63.4980i	0.371334i
$172$	0	0
$173$	−0.399241	−0.00230775	−0.00115388	−	0.999999i	$-0.500367\pi$
$173$	−0.399241	−0.00230775	−0.00115388	+	0.999999i	$0.500367\pi$
$174$	0	0
$175$	176.804i	1.01031i
$176$	0	0
$177$	−80.1742	−0.452962
$178$	0	0
$179$	− 89.1214i	− 0.497885i	−0.968518	−	0.248943i	$-0.919917\pi$
$179$	− 89.1214i	− 0.497885i	0.968518	−	0.248943i	$-0.0800830\pi$
$180$	0	0
$181$	−273.826	−1.51285	−0.756425	−	0.654081i	$-0.773056\pi$
$181$	−273.826	−1.51285	−0.756425	+	0.654081i	$0.773056\pi$
$182$	0	0
$183$	9.24810i	0.0505361i
$184$	0	0
$185$	410.468	2.21875
$186$	0	0
$187$	− 76.9730i	− 0.411621i
$188$	0	0
$189$	13.7477	0.0727393
$190$	0	0
$191$	− 240.318i	− 1.25821i	−0.777320	−	0.629105i	$-0.783422\pi$
$191$	− 240.318i	− 1.25821i	0.777320	−	0.629105i	$-0.216578\pi$
$192$	0	0
$193$	50.1742	0.259970	0.129985	−	0.991516i	$-0.458507\pi$
$193$	50.1742	0.259970	0.129985	+	0.991516i	$0.458507\pi$
$194$	0	0
$195$	− 284.899i	− 1.46102i
$196$	0	0
$197$	160.156	0.812975	0.406487	−	0.913656i	$-0.366753\pi$
$197$	160.156	0.812975	0.406487	+	0.913656i	$0.366753\pi$
$198$	0	0
$199$	− 382.355i	− 1.92138i	−0.277624	−	0.960690i	$-0.589547\pi$
$199$	− 382.355i	− 1.92138i	0.277624	−	0.960690i	$-0.410453\pi$
$200$	0	0
$201$	189.495	0.942763
$202$	0	0
$203$	− 85.5379i	− 0.421369i
$204$	0	0
$205$	−527.789	−2.57458
$206$	0	0
$207$	123.683i	0.597502i
$208$	0	0
$209$	−170.018	−0.813484
$210$	0	0
$211$	126.837i	0.601123i	0.953763	+	0.300561i	$0.0971740\pi$
$211$	126.837i	0.601123i	−0.953763	+	0.300561i	$0.902826\pi$
$212$	0	0
$213$	121.913	0.572361
$214$	0	0
$215$	335.605i	1.56095i
$216$	0	0
$217$	−54.9909	−0.253414
$218$	0	0
$219$	65.5161i	0.299160i
$220$	0	0
$221$	164.486	0.744282
$222$	0	0
$223$	− 271.503i	− 1.21750i	−0.793361	−	0.608752i	$-0.791671\pi$
$223$	− 271.503i	− 1.21750i	0.793361	−	0.608752i	$-0.208329\pi$
$224$	0	0
$225$	−200.477	−0.891010
$226$	0	0
$227$	89.9870i	0.396419i	0.980160	+	0.198209i	$0.0635126\pi$
$227$	89.9870i	0.396419i	−0.980160	+	0.198209i	$0.936487\pi$
$228$	0	0
$229$	267.459	1.16794	0.583972	−	0.811774i	$-0.301498\pi$
$229$	267.459	1.16794	0.583972	+	0.811774i	$0.301498\pi$
$230$	0	0
$231$	36.8100i	0.159351i
$232$	0	0
$233$	−93.8621	−0.402842	−0.201421	−	0.979505i	$-0.564556\pi$
$233$	−93.8621	−0.402842	−0.201421	+	0.979505i	$0.564556\pi$
$234$	0	0
$235$	118.924i	0.506058i
$236$	0	0
$237$	74.5045	0.314365
$238$	0	0
$239$	− 49.9036i	− 0.208802i	−0.994535	−	0.104401i	$-0.966708\pi$
$239$	− 49.9036i	− 0.208802i	0.994535	−	0.104401i	$-0.0332925\pi$
$240$	0	0
$241$	−250.450	−1.03921	−0.519606	−	0.854406i	$-0.673921\pi$
$241$	−250.450	−1.03921	−0.519606	+	0.854406i	$0.673921\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	67.0780	0.273788
$246$	0	0
$247$	− 363.318i	− 1.47092i
$248$	0	0
$249$	200.312	0.804466
$250$	0	0
$251$	103.081i	0.410680i	0.978691	+	0.205340i	$0.0658300\pi$
$251$	103.081i	0.410680i	−0.978691	+	0.205340i	$0.934170\pi$
$252$	0	0
$253$	−331.165	−1.30895
$254$	0	0
$255$	− 159.047i	− 0.623713i
$256$	0	0
$257$	367.078	1.42832	0.714160	−	0.699983i	$-0.246809\pi$
$257$	367.078	1.42832	0.714160	+	0.699983i	$0.246809\pi$
$258$	0	0
$259$	− 113.330i	− 0.437569i
$260$	0	0
$261$	96.9909	0.371613
$262$	0	0
$263$	424.265i	1.61318i	0.591114	+	0.806588i	$0.298688\pi$
$263$	424.265i	1.61318i	−0.591114	+	0.806588i	$0.701312\pi$
$264$	0	0
$265$	346.468	1.30743
$266$	0	0
$267$	62.5047i	0.234100i
$268$	0	0
$269$	354.033	1.31611	0.658053	−	0.752972i	$-0.271380\pi$
$269$	354.033	1.31611	0.658053	+	0.752972i	$0.271380\pi$
$270$	0	0
$271$	− 32.5118i	− 0.119970i	−0.998199	−	0.0599849i	$-0.980895\pi$
$271$	− 32.5118i	− 0.119970i	0.998199	−	0.0599849i	$-0.0191053\pi$
$272$	0	0
$273$	−78.6606	−0.288134
$274$	0	0
$275$	− 536.785i	− 1.95194i
$276$	0	0
$277$	130.174	0.469943	0.234972	−	0.972002i	$-0.424500\pi$
$277$	130.174	0.469943	0.234972	+	0.972002i	$0.424500\pi$
$278$	0	0
$279$	− 62.3538i	− 0.223490i
$280$	0	0
$281$	8.46818	0.0301359	0.0150679	−	0.999886i	$-0.495204\pi$
$281$	8.46818	0.0301359	0.0150679	+	0.999886i	$0.495204\pi$
$282$	0	0
$283$	27.7128i	0.0979251i	0.998801	+	0.0489626i	$0.0155915\pi$
$283$	27.7128i	0.0979251i	−0.998801	+	0.0489626i	$0.984408\pi$
$284$	0	0
$285$	−351.303	−1.23264
$286$	0	0
$287$	145.723i	0.507745i
$288$	0	0
$289$	−197.174	−0.682264
$290$	0	0
$291$	− 211.008i	− 0.725115i
$292$	0	0
$293$	525.161	1.79236	0.896180	−	0.443691i	$-0.146331\pi$
$293$	525.161	1.79236	0.896180	+	0.443691i	$0.146331\pi$
$294$	0	0
$295$	− 443.564i	− 1.50361i
$296$	0	0
$297$	−41.7386	−0.140534
$298$	0	0
$299$	− 707.678i	− 2.36682i
$300$	0	0
$301$	92.6606	0.307843
$302$	0	0
$303$	− 175.041i	− 0.577692i
$304$	0	0
$305$	−51.1652	−0.167755
$306$	0	0
$307$	134.465i	0.437996i	0.975725	+	0.218998i	$0.0702789\pi$
$307$	134.465i	0.437996i	−0.975725	+	0.218998i	$0.929721\pi$
$308$	0	0
$309$	−90.3303	−0.292331
$310$	0	0
$311$	459.789i	1.47842i	0.673475	+	0.739210i	$0.264801\pi$
$311$	459.789i	1.47842i	−0.673475	+	0.739210i	$0.735199\pi$
$312$	0	0
$313$	−579.909	−1.85274	−0.926372	−	0.376609i	$-0.877090\pi$
$313$	−579.909	−1.85274	−0.926372	+	0.376609i	$0.877090\pi$
$314$	0	0
$315$	76.0593i	0.241458i
$316$	0	0
$317$	−28.1561	−0.0888204	−0.0444102	−	0.999013i	$-0.514141\pi$
$317$	−28.1561	−0.0888204	−0.0444102	+	0.999013i	$0.514141\pi$
$318$	0	0
$319$	259.697i	0.814096i
$320$	0	0
$321$	−6.39924	−0.0199353
$322$	0	0
$323$	− 202.825i	− 0.627941i
$324$	0	0
$325$	1147.07	3.52946
$326$	0	0
$327$	274.808i	0.840392i
$328$	0	0
$329$	32.8348	0.0998020
$330$	0	0
$331$	51.7708i	0.156407i	0.996937	+	0.0782037i	$0.0249184\pi$
$331$	51.7708i	0.156407i	−0.996937	+	0.0782037i	$0.975082\pi$
$332$	0	0
$333$	128.505	0.385900
$334$	0	0
$335$	1048.38i	3.12950i
$336$	0	0
$337$	−410.936	−1.21940	−0.609698	−	0.792634i	$-0.708709\pi$
$337$	−410.936	−1.21940	−0.609698	+	0.792634i	$0.708709\pi$
$338$	0	0
$339$	− 4.60830i	− 0.0135938i
$340$	0	0
$341$	166.955	0.489603
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	−684.276	−1.98341
$346$	0	0
$347$	532.447i	1.53443i	0.641391	+	0.767214i	$0.278358\pi$
$347$	532.447i	1.53443i	−0.641391	+	0.767214i	$0.721642\pi$
$348$	0	0
$349$	−30.6242	−0.0877485	−0.0438743	−	0.999037i	$-0.513970\pi$
$349$	−30.6242	−0.0877485	−0.0438743	+	0.999037i	$0.513970\pi$
$350$	0	0
$351$	− 89.1927i	− 0.254110i
$352$	0	0
$353$	−189.023	−0.535477	−0.267739	−	0.963492i	$-0.586276\pi$
$353$	−189.023	−0.535477	−0.267739	+	0.963492i	$0.586276\pi$
$354$	0	0
$355$	674.483i	1.89995i
$356$	0	0
$357$	−43.9129	−0.123005
$358$	0	0
$359$	56.7688i	0.158130i	0.996869	+	0.0790652i	$0.0251935\pi$
$359$	56.7688i	0.158130i	−0.996869	+	0.0790652i	$0.974806\pi$
$360$	0	0
$361$	−87.0000	−0.240997
$362$	0	0
$363$	97.8215i	0.269481i
$364$	0	0
$365$	−362.468	−0.993064
$366$	0	0
$367$	− 60.4468i	− 0.164705i	−0.996603	−	0.0823526i	$-0.973757\pi$
$367$	− 60.4468i	− 0.164705i	0.996603	−	0.0823526i	$-0.0262434\pi$
$368$	0	0
$369$	−165.234	−0.447789
$370$	0	0
$371$	− 95.6599i	− 0.257844i
$372$	0	0
$373$	−370.661	−0.993728	−0.496864	−	0.867828i	$-0.665515\pi$
$373$	−370.661	−0.993728	−0.496864	+	0.867828i	$0.665515\pi$
$374$	0	0
$375$	− 694.203i	− 1.85121i
$376$	0	0
$377$	−554.955	−1.47203
$378$	0	0
$379$	337.734i	0.891119i	0.895252	+	0.445560i	$0.146995\pi$
$379$	337.734i	0.891119i	−0.895252	+	0.445560i	$0.853005\pi$
$380$	0	0
$381$	−410.780	−1.07816
$382$	0	0
$383$	− 381.433i	− 0.995908i	−0.867204	−	0.497954i	$-0.834085\pi$
$383$	− 381.433i	− 0.995908i	0.867204	−	0.497954i	$-0.165915\pi$
$384$	0	0
$385$	−203.652	−0.528965
$386$	0	0
$387$	105.067i	0.271492i
$388$	0	0
$389$	605.579	1.55676	0.778379	−	0.627795i	$-0.216042\pi$
$389$	605.579	1.55676	0.778379	+	0.627795i	$0.216042\pi$
$390$	0	0
$391$	− 395.067i	− 1.01040i
$392$	0	0
$393$	−144.661	−0.368093
$394$	0	0
$395$	412.197i	1.04354i
$396$	0	0
$397$	−82.3121	−0.207335	−0.103668	−	0.994612i	$-0.533058\pi$
$397$	−82.3121	−0.207335	−0.103668	+	0.994612i	$0.533058\pi$
$398$	0	0
$399$	96.9948i	0.243095i
$400$	0	0
$401$	−124.468	−0.310394	−0.155197	−	0.987884i	$-0.549601\pi$
$401$	−124.468	−0.310394	−0.155197	+	0.987884i	$0.549601\pi$
$402$	0	0
$403$	356.771i	0.885288i
$404$	0	0
$405$	−86.2432	−0.212946
$406$	0	0
$407$	344.075i	0.845394i
$408$	0	0
$409$	−225.826	−0.552141	−0.276071	−	0.961137i	$-0.589032\pi$
$409$	−225.826	−0.552141	−0.276071	+	0.961137i	$0.589032\pi$
$410$	0	0
$411$	− 186.188i	− 0.453011i
$412$	0	0
$413$	−122.468	−0.296533
$414$	0	0
$415$	1108.23i	2.67043i
$416$	0	0
$417$	92.6970	0.222295
$418$	0	0
$419$	162.924i	0.388840i	0.980918	+	0.194420i	$0.0622824\pi$
$419$	162.924i	0.388840i	−0.980918	+	0.194420i	$0.937718\pi$
$420$	0	0
$421$	21.5136	0.0511013	0.0255506	−	0.999674i	$-0.491866\pi$
$421$	21.5136	0.0511013	0.0255506	+	0.999674i	$0.491866\pi$
$422$	0	0
$423$	37.2312i	0.0880171i
$424$	0	0
$425$	640.363	1.50674
$426$	0	0
$427$	14.1267i	0.0330836i
$428$	0	0
$429$	238.817	0.556682
$430$	0	0
$431$	− 373.861i	− 0.867427i	−0.901051	−	0.433713i	$-0.857203\pi$
$431$	− 373.861i	− 0.867427i	0.901051	−	0.433713i	$-0.142797\pi$
$432$	0	0
$433$	618.973	1.42950	0.714749	−	0.699381i	$-0.246541\pi$
$433$	618.973	1.42950	0.714749	+	0.699381i	$0.246541\pi$
$434$	0	0
$435$	536.602i	1.23357i
$436$	0	0
$437$	−872.624	−1.99685
$438$	0	0
$439$	349.906i	0.797052i	0.917157	+	0.398526i	$0.130478\pi$
$439$	349.906i	0.797052i	−0.917157	+	0.398526i	$0.869522\pi$
$440$	0	0
$441$	21.0000	0.0476190
$442$	0	0
$443$	− 654.501i	− 1.47743i	−0.674018	−	0.738715i	$-0.735433\pi$
$443$	− 654.501i	− 1.47743i	0.674018	−	0.738715i	$-0.264567\pi$
$444$	0	0
$445$	−345.808	−0.777096
$446$	0	0
$447$	− 52.2633i	− 0.116920i
$448$	0	0
$449$	490.348	1.09209	0.546045	−	0.837756i	$-0.316133\pi$
$449$	490.348	1.09209	0.546045	+	0.837756i	$0.316133\pi$
$450$	0	0
$451$	− 442.420i	− 0.980976i
$452$	0	0
$453$	486.606	1.07419
$454$	0	0
$455$	− 435.190i	− 0.956462i
$456$	0	0
$457$	−19.9818	−0.0437239	−0.0218619	−	0.999761i	$-0.506959\pi$
$457$	−19.9818	−0.0437239	−0.0218619	+	0.999761i	$0.506959\pi$
$458$	0	0
$459$	− 49.7925i	− 0.108480i
$460$	0	0
$461$	−193.096	−0.418864	−0.209432	−	0.977823i	$-0.567161\pi$
$461$	−193.096	−0.418864	−0.209432	+	0.977823i	$0.567161\pi$
$462$	0	0
$463$	− 830.257i	− 1.79321i	−0.442830	−	0.896606i	$-0.646025\pi$
$463$	− 830.257i	− 1.79321i	0.442830	−	0.896606i	$-0.353975\pi$
$464$	0	0
$465$	344.973	0.741877
$466$	0	0
$467$	375.665i	0.804422i	0.915547	+	0.402211i	$0.131758\pi$
$467$	375.665i	0.804422i	−0.915547	+	0.402211i	$0.868242\pi$
$468$	0	0
$469$	289.459	0.617184
$470$	0	0
$471$	101.063i	0.214570i
$472$	0	0
$473$	−281.321	−0.594759
$474$	0	0
$475$	− 1414.43i	− 2.97776i
$476$	0	0
$477$	108.468	0.227397
$478$	0	0
$479$	291.509i	0.608577i	0.952580	+	0.304289i	$0.0984187\pi$
$479$	291.509i	0.608577i	−0.952580	+	0.304289i	$0.901581\pi$
$480$	0	0
$481$	−735.267	−1.52862
$482$	0	0
$483$	188.929i	0.391157i
$484$	0	0
$485$	1167.40	2.40702
$486$	0	0
$487$	254.914i	0.523438i	0.965144	+	0.261719i	$0.0842893\pi$
$487$	254.914i	0.523438i	−0.965144	+	0.261719i	$0.915711\pi$
$488$	0	0
$489$	147.826	0.302302
$490$	0	0
$491$	655.025i	1.33406i	0.745029	+	0.667032i	$0.232436\pi$
$491$	655.025i	1.33406i	−0.745029	+	0.667032i	$0.767564\pi$
$492$	0	0
$493$	−309.808	−0.628413
$494$	0	0
$495$	− 230.919i	− 0.466503i
$496$	0	0
$497$	186.225	0.374698
$498$	0	0
$499$	− 134.909i	− 0.270359i	−0.990821	−	0.135180i	$-0.956839\pi$
$499$	− 134.909i	− 0.270359i	0.990821	−	0.135180i	$-0.0431611\pi$
$500$	0	0
$501$	−178.156	−0.355601
$502$	0	0
$503$	1002.84i	1.99372i	0.0791778	+	0.996861i	$0.474771\pi$
$503$	1002.84i	1.99372i	−0.0791778	+	0.996861i	$0.525229\pi$
$504$	0	0
$505$	968.414	1.91765
$506$	0	0
$507$	217.619i	0.429229i
$508$	0	0
$509$	−196.922	−0.386880	−0.193440	−	0.981112i	$-0.561965\pi$
$509$	−196.922	−0.386880	−0.193440	+	0.981112i	$0.561965\pi$
$510$	0	0
$511$	100.078i	0.195846i
$512$	0	0
$513$	−109.982	−0.214390
$514$	0	0
$515$	− 499.753i	− 0.970393i
$516$	0	0
$517$	−99.6879	−0.192820
$518$	0	0
$519$	− 0.691506i	− 0.00133238i
$520$	0	0
$521$	282.831	0.542862	0.271431	−	0.962458i	$-0.412503\pi$
$521$	282.831	0.542862	0.271431	+	0.962458i	$0.412503\pi$
$522$	0	0
$523$	− 451.859i	− 0.863975i	−0.901880	−	0.431987i	$-0.857812\pi$
$523$	− 451.859i	− 0.863975i	0.901880	−	0.431987i	$-0.142188\pi$
$524$	0	0
$525$	−306.234	−0.583303
$526$	0	0
$527$	199.170i	0.377932i
$528$	0	0
$529$	−1170.72	−2.21307
$530$	0	0
$531$	− 138.866i	− 0.261518i
$532$	0	0
$533$	945.423	1.77378
$534$	0	0
$535$	− 35.4038i	− 0.0661754i
$536$	0	0
$537$	154.363	0.287454
$538$	0	0
$539$	56.2282i	0.104320i
$540$	0	0
$541$	94.1379	0.174007	0.0870036	−	0.996208i	$-0.472271\pi$
$541$	94.1379	0.174007	0.0870036	+	0.996208i	$0.472271\pi$
$542$	0	0
$543$	− 474.280i	− 0.873444i
$544$	0	0
$545$	−1520.38	−2.78968
$546$	0	0
$547$	834.356i	1.52533i	0.646793	+	0.762665i	$0.276110\pi$
$547$	834.356i	1.52533i	−0.646793	+	0.762665i	$0.723890\pi$
$548$	0	0
$549$	−16.0182	−0.0291770
$550$	0	0
$551$	684.304i	1.24193i
$552$	0	0
$553$	113.808	0.205800
$554$	0	0
$555$	710.952i	1.28099i
$556$	0	0
$557$	−624.395	−1.12100	−0.560499	−	0.828155i	$-0.689390\pi$
$557$	−624.395	−1.12100	−0.560499	+	0.828155i	$0.689390\pi$
$558$	0	0
$559$	− 601.165i	− 1.07543i
$560$	0	0
$561$	133.321	0.237649
$562$	0	0
$563$	− 186.697i	− 0.331610i	−0.986159	−	0.165805i	$-0.946978\pi$
$563$	− 186.697i	− 0.331610i	0.986159	−	0.165805i	$-0.0530223\pi$
$564$	0	0
$565$	25.4955	0.0451247
$566$	0	0
$567$	23.8118i	0.0419961i
$568$	0	0
$569$	670.871	1.17904	0.589518	−	0.807755i	$-0.299318\pi$
$569$	670.871	1.17904	0.589518	+	0.807755i	$0.299318\pi$
$570$	0	0
$571$	532.725i	0.932969i	0.884529	+	0.466485i	$0.154480\pi$
$571$	532.725i	0.932969i	−0.884529	+	0.466485i	$0.845520\pi$
$572$	0	0
$573$	416.243	0.726428
$574$	0	0
$575$	− 2755.07i	− 4.79142i
$576$	0	0
$577$	−783.670	−1.35818	−0.679090	−	0.734055i	$-0.737625\pi$
$577$	−783.670	−1.35818	−0.679090	+	0.734055i	$0.737625\pi$
$578$	0	0
$579$	86.9043i	0.150094i
$580$	0	0
$581$	305.982	0.526647
$582$	0	0
$583$	290.427i	0.498160i
$584$	0	0
$585$	493.459	0.843520
$586$	0	0
$587$	− 36.4054i	− 0.0620195i	−0.999519	−	0.0310097i	$-0.990128\pi$
$587$	− 36.4054i	− 0.0620195i	0.999519	−	0.0310097i	$-0.00987229\pi$
$588$	0	0
$589$	439.927	0.746905
$590$	0	0
$591$	277.398i	0.469371i
$592$	0	0
$593$	−379.354	−0.639720	−0.319860	−	0.947465i	$-0.603636\pi$
$593$	−379.354	−0.639720	−0.319860	+	0.947465i	$0.603636\pi$
$594$	0	0
$595$	− 242.948i	− 0.408316i
$596$	0	0
$597$	662.258	1.10931
$598$	0	0
$599$	− 778.304i	− 1.29934i	−0.760217	−	0.649669i	$-0.774907\pi$
$599$	− 778.304i	− 1.29934i	0.760217	−	0.649669i	$-0.225093\pi$
$600$	0	0
$601$	241.724	0.402203	0.201102	−	0.979570i	$-0.435548\pi$
$601$	241.724	0.402203	0.201102	+	0.979570i	$0.435548\pi$
$602$	0	0
$603$	328.216i	0.544305i
$604$	0	0
$605$	−541.198	−0.894542
$606$	0	0
$607$	612.733i	1.00944i	0.863282	+	0.504722i	$0.168405\pi$
$607$	612.733i	1.00944i	−0.863282	+	0.504722i	$0.831595\pi$
$608$	0	0
$609$	148.156	0.243278
$610$	0	0
$611$	− 213.027i	− 0.348652i
$612$	0	0
$613$	1071.84	1.74851	0.874255	−	0.485467i	$-0.161351\pi$
$613$	1071.84	1.74851	0.874255	+	0.485467i	$0.161351\pi$
$614$	0	0
$615$	− 914.158i	− 1.48644i
$616$	0	0
$617$	−681.753	−1.10495	−0.552474	−	0.833530i	$-0.686316\pi$
$617$	−681.753	−1.10495	−0.552474	+	0.833530i	$0.686316\pi$
$618$	0	0
$619$	− 1102.38i	− 1.78091i	−0.455076	−	0.890453i	$-0.650388\pi$
$619$	− 1102.38i	− 1.78091i	0.455076	−	0.890453i	$-0.349612\pi$
$620$	0	0
$621$	−214.225	−0.344968
$622$	0	0
$623$	95.4775i	0.153254i
$624$	0	0
$625$	2170.04	3.47206
$626$	0	0
$627$	− 294.480i	− 0.469665i
$628$	0	0
$629$	−410.468	−0.652573
$630$	0	0
$631$	− 151.578i	− 0.240219i	−0.992761	−	0.120109i	$-0.961675\pi$
$631$	− 151.578i	− 0.240219i	0.992761	−	0.120109i	$-0.0383245\pi$
$632$	0	0
$633$	−219.688	−0.347058
$634$	0	0
$635$	− 2272.64i	− 3.57897i
$636$	0	0
$637$	−120.156	−0.188628
$638$	0	0
$639$	211.159i	0.330453i
$640$	0	0
$641$	−161.550	−0.252028	−0.126014	−	0.992028i	$-0.540218\pi$
$641$	−161.550	−0.252028	−0.126014	+	0.992028i	$0.540218\pi$
$642$	0	0
$643$	− 274.395i	− 0.426742i	−0.976971	−	0.213371i	$-0.931556\pi$
$643$	− 274.395i	− 0.426742i	0.976971	−	0.213371i	$-0.0684444\pi$
$644$	0	0
$645$	−581.285	−0.901217
$646$	0	0
$647$	− 499.355i	− 0.771800i	−0.922541	−	0.385900i	$-0.873891\pi$
$647$	− 499.355i	− 0.771800i	0.922541	−	0.385900i	$-0.126109\pi$
$648$	0	0
$649$	371.818	0.572909
$650$	0	0
$651$	− 95.2470i	− 0.146309i
$652$	0	0
$653$	511.670	0.783568	0.391784	−	0.920057i	$-0.371858\pi$
$653$	511.670	0.783568	0.391784	+	0.920057i	$0.371858\pi$
$654$	0	0
$655$	− 800.335i	− 1.22189i
$656$	0	0
$657$	−113.477	−0.172720
$658$	0	0
$659$	479.834i	0.728124i	0.931375	+	0.364062i	$0.118610\pi$
$659$	479.834i	0.728124i	−0.931375	+	0.364062i	$0.881390\pi$
$660$	0	0
$661$	231.633	0.350429	0.175214	−	0.984530i	$-0.443938\pi$
$661$	231.633	0.350429	0.175214	+	0.984530i	$0.443938\pi$
$662$	0	0
$663$	284.899i	0.429712i
$664$	0	0
$665$	−536.624	−0.806954
$666$	0	0
$667$	1332.90i	1.99835i
$668$	0	0
$669$	470.258	0.702926
$670$	0	0
$671$	− 42.8892i	− 0.0639184i
$672$	0	0
$673$	−655.423	−0.973882	−0.486941	−	0.873435i	$-0.661887\pi$
$673$	−655.423	−0.973882	−0.486941	+	0.873435i	$0.661887\pi$
$674$	0	0
$675$	− 347.237i	− 0.514425i
$676$	0	0
$677$	−773.960	−1.14322	−0.571610	−	0.820525i	$-0.693681\pi$
$677$	−773.960	−1.14322	−0.571610	+	0.820525i	$0.693681\pi$
$678$	0	0
$679$	− 322.321i	− 0.474699i
$680$	0	0
$681$	−155.862	−0.228872
$682$	0	0
$683$	585.425i	0.857137i	0.903509	+	0.428569i	$0.140982\pi$
$683$	585.425i	0.857137i	−0.903509	+	0.428569i	$0.859018\pi$
$684$	0	0
$685$	1030.08	1.50377
$686$	0	0
$687$	463.253i	0.674313i
$688$	0	0
$689$	−620.624	−0.900761
$690$	0	0
$691$	− 779.995i	− 1.12879i	−0.825504	−	0.564396i	$-0.809109\pi$
$691$	− 779.995i	− 1.12879i	0.825504	−	0.564396i	$-0.190891\pi$
$692$	0	0
$693$	−63.7568	−0.0920012
$694$	0	0
$695$	512.846i	0.737908i
$696$	0	0
$697$	527.789	0.757230
$698$	0	0
$699$	− 162.574i	− 0.232581i
$700$	0	0
$701$	1299.98	1.85447	0.927234	−	0.374483i	$-0.122180\pi$
$701$	1299.98	1.85447	0.927234	+	0.374483i	$0.122180\pi$
$702$	0	0
$703$	906.643i	1.28968i
$704$	0	0
$705$	−205.982	−0.292173
$706$	0	0
$707$	− 267.379i	− 0.378188i
$708$	0	0
$709$	−545.891	−0.769945	−0.384972	−	0.922928i	$-0.625789\pi$
$709$	−545.891	−0.769945	−0.384972	+	0.922928i	$0.625789\pi$
$710$	0	0
$711$	129.046i	0.181499i
$712$	0	0
$713$	856.900	1.20182
$714$	0	0
$715$	1321.25i	1.84791i
$716$	0	0
$717$	86.4356	0.120552
$718$	0	0
$719$	− 610.922i	− 0.849683i	−0.905268	−	0.424842i	$-0.860330\pi$
$719$	− 610.922i	− 0.849683i	0.905268	−	0.424842i	$-0.139670\pi$
$720$	0	0
$721$	−137.982	−0.191376
$722$	0	0
$723$	− 433.792i	− 0.599989i
$724$	0	0
$725$	−2160.50	−2.98000
$726$	0	0
$727$	1068.12i	1.46922i	0.678491	+	0.734608i	$0.262634\pi$
$727$	1068.12i	1.46922i	−0.678491	+	0.734608i	$0.737366\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 335.605i	− 0.459104i
$732$	0	0
$733$	445.056	0.607171	0.303585	−	0.952804i	$-0.401816\pi$
$733$	445.056	0.607171	0.303585	+	0.952804i	$0.401816\pi$
$734$	0	0
$735$	116.183i	0.158072i
$736$	0	0
$737$	−878.809	−1.19241
$738$	0	0
$739$	− 1194.50i	− 1.61637i	−0.588929	−	0.808184i	$-0.700450\pi$
$739$	− 1194.50i	− 1.61637i	0.588929	−	0.808184i	$-0.299550\pi$
$740$	0	0
$741$	629.285	0.849237
$742$	0	0
$743$	− 86.0536i	− 0.115819i	−0.998322	−	0.0579096i	$-0.981556\pi$
$743$	− 86.0536i	− 0.115819i	0.998322	−	0.0579096i	$-0.0184435\pi$
$744$	0	0
$745$	289.147	0.388117
$746$	0	0
$747$	346.951i	0.464459i
$748$	0	0
$749$	−9.77500	−0.0130507
$750$	0	0
$751$	408.383i	0.543785i	0.962328	+	0.271893i	$0.0876495\pi$
$751$	408.383i	0.543785i	−0.962328	+	0.271893i	$0.912350\pi$
$752$	0	0
$753$	−178.541	−0.237106
$754$	0	0
$755$	2692.15i	3.56576i
$756$	0	0
$757$	−62.6970	−0.0828229	−0.0414115	−	0.999142i	$-0.513185\pi$
$757$	−62.6970	−0.0828229	−0.0414115	+	0.999142i	$0.513185\pi$
$758$	0	0
$759$	− 573.595i	− 0.755724i
$760$	0	0
$761$	−20.6099	−0.0270826	−0.0135413	−	0.999908i	$-0.504310\pi$
$761$	−20.6099	−0.0270826	−0.0135413	+	0.999908i	$0.504310\pi$
$762$	0	0
$763$	419.777i	0.550166i
$764$	0	0
$765$	275.477	0.360101
$766$	0	0
$767$	794.551i	1.03592i
$768$	0	0
$769$	950.936	1.23659	0.618294	−	0.785947i	$-0.287824\pi$
$769$	950.936	1.23659	0.618294	+	0.785947i	$0.287824\pi$
$770$	0	0
$771$	635.798i	0.824640i
$772$	0	0
$773$	−1018.77	−1.31794	−0.658969	−	0.752170i	$-0.729007\pi$
$773$	−1018.77	−1.31794	−0.658969	+	0.752170i	$0.729007\pi$
$774$	0	0
$775$	1388.95i	1.79219i
$776$	0	0
$777$	196.294	0.252631
$778$	0	0
$779$	− 1165.78i	− 1.49651i
$780$	0	0
$781$	−565.386	−0.723926
$782$	0	0
$783$	167.993i	0.214551i
$784$	0	0
$785$	−559.129	−0.712266
$786$	0	0
$787$	516.597i	0.656413i	0.944606	+	0.328207i	$0.106444\pi$
$787$	516.597i	0.656413i	−0.944606	+	0.328207i	$0.893556\pi$
$788$	0	0
$789$	−734.849	−0.931368
$790$	0	0
$791$	− 7.03930i	− 0.00889924i
$792$	0	0
$793$	91.6515	0.115576
$794$	0	0
$795$	600.100i	0.754843i
$796$	0	0
$797$	407.673	0.511510	0.255755	−	0.966742i	$-0.417676\pi$
$797$	407.673	0.511510	0.255755	+	0.966742i	$0.417676\pi$
$798$	0	0
$799$	− 118.924i	− 0.148841i
$800$	0	0
$801$	−108.261	−0.135158
$802$	0	0
$803$	− 303.839i	− 0.378380i
$804$	0	0
$805$	−1045.25	−1.29845
$806$	0	0
$807$	613.202i	0.759854i
$808$	0	0
$809$	940.221	1.16220	0.581101	−	0.813832i	$-0.302622\pi$
$809$	940.221	1.16220	0.581101	+	0.813832i	$0.302622\pi$
$810$	0	0
$811$	880.359i	1.08552i	0.839887	+	0.542762i	$0.182621\pi$
$811$	880.359i	1.08552i	−0.839887	+	0.542762i	$0.817379\pi$
$812$	0	0
$813$	56.3121	0.0692646
$814$	0	0
$815$	817.846i	1.00349i
$816$	0	0
$817$	−741.285	−0.907325
$818$	0	0
$819$	− 136.244i	− 0.166354i
$820$	0	0
$821$	567.532	0.691269	0.345634	−	0.938369i	$-0.387664\pi$
$821$	567.532	0.691269	0.345634	+	0.938369i	$0.387664\pi$
$822$	0	0
$823$	− 1280.21i	− 1.55554i	−0.628550	−	0.777770i	$-0.716351\pi$
$823$	− 1280.21i	− 1.55554i	0.628550	−	0.777770i	$-0.283649\pi$
$824$	0	0
$825$	929.739	1.12696
$826$	0	0
$827$	− 1473.95i	− 1.78229i	−0.453720	−	0.891144i	$-0.649903\pi$
$827$	− 1473.95i	− 1.78229i	0.453720	−	0.891144i	$-0.350097\pi$
$828$	0	0
$829$	70.1742	0.0846493	0.0423246	−	0.999104i	$-0.486524\pi$
$829$	70.1742	0.0846493	0.0423246	+	0.999104i	$0.486524\pi$
$830$	0	0
$831$	225.468i	0.271322i
$832$	0	0
$833$	−67.0780	−0.0805258
$834$	0	0
$835$	− 985.649i	− 1.18042i
$836$	0	0
$837$	108.000	0.129032
$838$	0	0
$839$	648.883i	0.773401i	0.922205	+	0.386700i	$0.126385\pi$
$839$	648.883i	0.773401i	−0.922205	+	0.386700i	$0.873615\pi$
$840$	0	0
$841$	204.248	0.242864
$842$	0	0
$843$	14.6673i	0.0173990i
$844$	0	0
$845$	−1203.98	−1.42483
$846$	0	0
$847$	149.425i	0.176417i
$848$	0	0
$849$	−48.0000	−0.0565371
$850$	0	0
$851$	1765.98i	2.07518i
$852$	0	0
$853$	−687.873	−0.806416	−0.403208	−	0.915108i	$-0.632105\pi$
$853$	−687.873	−0.806416	−0.403208	+	0.915108i	$0.632105\pi$
$854$	0	0
$855$	− 608.475i	− 0.711666i
$856$	0	0
$857$	−253.786	−0.296133	−0.148066	−	0.988977i	$-0.547305\pi$
$857$	−253.786	−0.296133	−0.148066	+	0.988977i	$0.547305\pi$
$858$	0	0
$859$	− 802.305i	− 0.933999i	−0.884258	−	0.467000i	$-0.845335\pi$
$859$	− 802.305i	− 0.933999i	0.884258	−	0.467000i	$-0.154665\pi$
$860$	0	0
$861$	−252.399	−0.293147
$862$	0	0
$863$	548.210i	0.635238i	0.948218	+	0.317619i	$0.102883\pi$
$863$	548.210i	0.635238i	−0.948218	+	0.317619i	$0.897117\pi$
$864$	0	0
$865$	3.82576	0.00442284
$866$	0	0
$867$	− 341.516i	− 0.393905i
$868$	0	0
$869$	−345.524	−0.397611
$870$	0	0
$871$	− 1877.96i	− 2.15609i
$872$	0	0
$873$	365.477	0.418645
$874$	0	0
$875$	− 1060.41i	− 1.21190i
$876$	0	0
$877$	−607.176	−0.692333	−0.346166	−	0.938173i	$-0.612517\pi$
$877$	−607.176	−0.692333	−0.346166	+	0.938173i	$0.612517\pi$
$878$	0	0
$879$	909.606i	1.03482i
$880$	0	0
$881$	383.492	0.435291	0.217646	−	0.976028i	$-0.430162\pi$
$881$	383.492	0.435291	0.217646	+	0.976028i	$0.430162\pi$
$882$	0	0
$883$	826.204i	0.935678i	0.883814	+	0.467839i	$0.154967\pi$
$883$	826.204i	0.935678i	−0.883814	+	0.467839i	$0.845033\pi$
$884$	0	0
$885$	768.276	0.868108
$886$	0	0
$887$	628.513i	0.708583i	0.935135	+	0.354291i	$0.115278\pi$
$887$	628.513i	0.708583i	−0.935135	+	0.354291i	$0.884722\pi$
$888$	0	0
$889$	−627.477	−0.705824
$890$	0	0
$891$	− 72.2934i	− 0.0811374i
$892$	0	0
$893$	−262.679	−0.294153
$894$	0	0
$895$	854.013i	0.954204i
$896$	0	0
$897$	1225.73	1.36648
$898$	0	0
$899$	− 671.973i	− 0.747467i
$900$	0	0
$901$	−346.468	−0.384537
$902$	0	0
$903$	160.493i	0.177733i
$904$	0	0
$905$	2623.96	2.89940
$906$	0	0
$907$	895.201i	0.986991i	0.869748	+	0.493495i	$0.164281\pi$
$907$	895.201i	0.986991i	−0.869748	+	0.493495i	$0.835719\pi$
$908$	0	0
$909$	303.180	0.333531
$910$	0	0
$911$	174.930i	0.192019i	0.995380	+	0.0960097i	$0.0306080\pi$
$911$	174.930i	0.192019i	−0.995380	+	0.0960097i	$0.969392\pi$
$912$	0	0
$913$	−928.973	−1.01749
$914$	0	0
$915$	− 88.6206i	− 0.0968532i
$916$	0	0
$917$	−220.973	−0.240974
$918$	0	0
$919$	336.971i	0.366672i	0.983050	+	0.183336i	$0.0586896\pi$
$919$	336.971i	0.366672i	−0.983050	+	0.183336i	$0.941310\pi$
$920$	0	0
$921$	−232.900	−0.252877
$922$	0	0
$923$	− 1208.19i	− 1.30899i
$924$	0	0
$925$	−2862.47	−3.09456
$926$	0	0
$927$	− 156.457i	− 0.168777i
$928$	0	0
$929$	−552.225	−0.594429	−0.297215	−	0.954811i	$-0.596058\pi$
$929$	−552.225	−0.594429	−0.297215	+	0.954811i	$0.596058\pi$
$930$	0	0
$931$	148.162i	0.159143i
$932$	0	0
$933$	−796.377	−0.853566
$934$	0	0
$935$	737.600i	0.788877i
$936$	0	0
$937$	1178.83	1.25809	0.629043	−	0.777370i	$-0.283447\pi$
$937$	1178.83	1.25809	0.629043	+	0.777370i	$0.283447\pi$
$938$	0	0
$939$	− 1004.43i	− 1.06968i
$940$	0	0
$941$	−272.958	−0.290073	−0.145036	−	0.989426i	$-0.546330\pi$
$941$	−272.958	−0.290073	−0.145036	+	0.989426i	$0.546330\pi$
$942$	0	0
$943$	− 2270.74i	− 2.40799i
$944$	0	0
$945$	−131.739	−0.139406
$946$	0	0
$947$	− 531.366i	− 0.561104i	−0.959839	−	0.280552i	$-0.909482\pi$
$947$	− 531.366i	− 0.561104i	0.959839	−	0.280552i	$-0.0905175\pi$
$948$	0	0
$949$	649.285	0.684178
$950$	0	0
$951$	− 48.7677i	− 0.0512805i
$952$	0	0
$953$	−510.000	−0.535152	−0.267576	−	0.963537i	$-0.586223\pi$
$953$	−510.000	−0.535152	−0.267576	+	0.963537i	$0.586223\pi$
$954$	0	0
$955$	2302.87i	2.41138i
$956$	0	0
$957$	−449.808	−0.470018
$958$	0	0
$959$	− 284.406i	− 0.296565i
$960$	0	0
$961$	529.000	0.550468
$962$	0	0
$963$	− 11.0838i	− 0.0115097i
$964$	0	0
$965$	−480.798	−0.498237
$966$	0	0
$967$	580.079i	0.599875i	0.953959	+	0.299937i	$0.0969658\pi$
$967$	580.079i	0.599875i	−0.953959	+	0.299937i	$0.903034\pi$
$968$	0	0
$969$	351.303	0.362542
$970$	0	0
$971$	1351.19i	1.39155i	0.718261	+	0.695773i	$0.244938\pi$
$971$	1351.19i	1.39155i	−0.718261	+	0.695773i	$0.755062\pi$
$972$	0	0
$973$	141.597	0.145526
$974$	0	0
$975$	1986.79i	2.03773i
$976$	0	0
$977$	−63.5318	−0.0650275	−0.0325137	−	0.999471i	$-0.510351\pi$
$977$	−63.5318	−0.0650275	−0.0325137	+	0.999471i	$0.510351\pi$
$978$	0	0
$979$	− 289.874i	− 0.296091i
$980$	0	0
$981$	−475.982	−0.485201
$982$	0	0
$983$	− 1473.52i	− 1.49900i	−0.662005	−	0.749499i	$-0.730294\pi$
$983$	− 1473.52i	− 1.49900i	0.662005	−	0.749499i	$-0.269706\pi$
$984$	0	0
$985$	−1534.71	−1.55808
$986$	0	0
$987$	56.8716i	0.0576207i
$988$	0	0
$989$	−1443.89	−1.45995
$990$	0	0
$991$	− 999.028i	− 1.00810i	−0.863674	−	0.504050i	$-0.831843\pi$
$991$	− 999.028i	− 1.00810i	0.863674	−	0.504050i	$-0.168157\pi$
$992$	0	0
$993$	−89.6697	−0.0903018
$994$	0	0
$995$	3663.94i	3.68235i
$996$	0	0
$997$	−1076.88	−1.08012	−0.540061	−	0.841626i	$-0.681599\pi$
$997$	−1076.88	−1.08012	−0.540061	+	0.841626i	$0.681599\pi$
$998$	0	0
$999$	222.576i	0.222799i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.m.a.127.3	yes	4
3.2	odd	2		1008.3.m.f.127.4		4
4.3	odd	2	inner	336.3.m.a.127.1	✓	4
7.6	odd	2		2352.3.m.k.1471.2		4
8.3	odd	2		1344.3.m.c.127.4		4
8.5	even	2		1344.3.m.c.127.2		4
12.11	even	2		1008.3.m.f.127.3		4
28.27	even	2		2352.3.m.k.1471.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.m.a.127.1	✓	4	4.3	odd	2	inner
336.3.m.a.127.3	yes	4	1.1	even	1	trivial
1008.3.m.f.127.3		4	12.11	even	2
1008.3.m.f.127.4		4	3.2	odd	2
1344.3.m.c.127.2		4	8.5	even	2
1344.3.m.c.127.4		4	8.3	odd	2
2352.3.m.k.1471.2		4	7.6	odd	2
2352.3.m.k.1471.4		4	28.27	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	336.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -9.58258 q^{5} +2.64575i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -9.58258 q^{5} +2.64575i q^{7} -3.00000 q^{9} -8.03260i q^{11} +17.1652 q^{13} -16.5975i q^{15} +9.58258 q^{17} -21.1660i q^{19} -4.58258 q^{21} -41.2276i q^{23} +66.8258 q^{25} -5.19615i q^{27} -32.3303 q^{29} +20.7846i q^{31} +13.9129 q^{33} -25.3531i q^{35} -42.8348 q^{37} +29.7309i q^{39} +55.0780 q^{41} -35.0224i q^{43} +28.7477 q^{45} -12.4104i q^{47} -7.00000 q^{49} +16.5975i q^{51} -36.1561 q^{53} +76.9730i q^{55} +36.6606 q^{57} +46.2886i q^{59} +5.33939 q^{61} -7.93725i q^{63} -164.486 q^{65} -109.405i q^{67} +71.4083 q^{69} -70.3864i q^{71} +37.8258 q^{73} +115.746i q^{75} +21.2523 q^{77} -43.0152i q^{79} +9.00000 q^{81} -115.650i q^{83} -91.8258 q^{85} -55.9977i q^{87} +36.0871 q^{89} +45.4147i q^{91} -36.0000 q^{93} +202.825i q^{95} -121.826 q^{97} +24.0978i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q - 20 * q^5 - 12 * q^9	\( 4 q - 20 q^{5} - 12 q^{9} + 32 q^{13} + 20 q^{17} + 84 q^{25} - 56 q^{29} - 36 q^{33} - 208 q^{37} + 92 q^{41} + 60 q^{45} - 28 q^{49} + 112 q^{53} + 168 q^{61} - 328 q^{65} + 84 q^{69} - 32 q^{73} + 140 q^{77} + 36 q^{81} - 184 q^{85} + 236 q^{89} - 144 q^{93} - 304 q^{97}+O(q^{100}) \) 4 * q - 20 * q^5 - 12 * q^9 + 32 * q^13 + 20 * q^17 + 84 * q^25 - 56 * q^29 - 36 * q^33 - 208 * q^37 + 92 * q^41 + 60 * q^45 - 28 * q^49 + 112 * q^53 + 168 * q^61 - 328 * q^65 + 84 * q^69 - 32 * q^73 + 140 * q^77 + 36 * q^81 - 184 * q^85 + 236 * q^89 - 144 * q^93 - 304 * q^97

Introduction

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