LMFDB - Embedded newform 2352.3.f.j.97.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,3,Mod(97,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2352.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.f (of order $2$, degree $1$, not 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:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.3
Root			$-1.60021 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	2352.97
Dual form			2352.3.f.j.97.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +0.480662i q^{5} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +0.480662i q^{5} -3.00000 q^{9} -5.76316 q^{11} -1.41991i q^{13} +0.832530 q^{15} +23.0671i q^{17} -10.6739i q^{19} +6.59980 q^{23} +24.7690 q^{25} +5.19615i q^{27} -6.20258 q^{29} -41.9320i q^{31} +9.98209i q^{33} -60.0929 q^{37} -2.45936 q^{39} +48.8250i q^{41} +51.5603 q^{43} -1.44198i q^{45} +19.2421i q^{47} +39.9534 q^{51} +82.1345 q^{53} -2.77013i q^{55} -18.4877 q^{57} -92.5800i q^{59} +4.99187i q^{61} +0.682497 q^{65} +2.20699 q^{67} -11.4312i q^{69} +80.5899 q^{71} -13.9088i q^{73} -42.9011i q^{75} +64.8885 q^{79} +9.00000 q^{81} -118.005i q^{83} -11.0875 q^{85} +10.7432i q^{87} -104.265i q^{89} -72.6284 q^{93} +5.13052 q^{95} -31.7875i q^{97} +17.2895 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} + 16 q^{23} + 72 q^{25} + 80 q^{29} + 128 q^{37} + 112 q^{43} + 96 q^{51} - 144 q^{53} - 192 q^{57} + 240 q^{65} + 64 q^{67} - 224 q^{71} + 432 q^{79} + 72 q^{81} - 96 q^{85} - 96 q^{93} + 272 q^{95}+O(q^{100}) $ 8 * q - 24 * q^9 + 16 * q^23 + 72 * q^25 + 80 * q^29 + 128 * q^37 + 112 * q^43 + 96 * q^51 - 144 * q^53 - 192 * q^57 + 240 * q^65 + 64 * q^67 - 224 * q^71 + 432 * q^79 + 72 * q^81 - 96 * q^85 - 96 * q^93 + 272 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	0.480662i	0.0961323i	0.998844	+	0.0480662i	$0.0153058\pi$
$5$	0.480662i	0.0961323i	−0.998844	+	0.0480662i	$0.984694\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−5.76316	−0.523924	−0.261962	−	0.965078i	$-0.584369\pi$
$11$	−5.76316	−0.523924	−0.261962	+	0.965078i	$0.584369\pi$
$12$	0	0
$13$	− 1.41991i	− 0.109224i	−0.998508	−	0.0546120i	$-0.982608\pi$
$13$	− 1.41991i	− 0.109224i	0.998508	−	0.0546120i	$-0.0173922\pi$
$14$	0	0
$15$	0.832530	0.0555020
$16$	0	0
$17$	23.0671i	1.35689i	0.734651	+	0.678445i	$0.237346\pi$
$17$	23.0671i	1.35689i	−0.734651	+	0.678445i	$0.762654\pi$
$18$	0	0
$19$	− 10.6739i	− 0.561783i	−0.959740	−	0.280891i	$-0.909370\pi$
$19$	− 10.6739i	− 0.561783i	0.959740	−	0.280891i	$-0.0906300\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.59980	0.286948	0.143474	−	0.989654i	$-0.454173\pi$
$23$	6.59980	0.286948	0.143474	+	0.989654i	$0.454173\pi$
$24$	0	0
$25$	24.7690	0.990759
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−6.20258	−0.213882	−0.106941	−	0.994265i	$-0.534106\pi$
$29$	−6.20258	−0.213882	−0.106941	+	0.994265i	$0.534106\pi$
$30$	0	0
$31$	− 41.9320i	− 1.35265i	−0.736605	−	0.676323i	$-0.763572\pi$
$31$	− 41.9320i	− 1.35265i	0.736605	−	0.676323i	$-0.236428\pi$
$32$	0	0
$33$	9.98209i	0.302488i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−60.0929	−1.62413	−0.812066	−	0.583566i	$-0.801657\pi$
$37$	−60.0929	−1.62413	−0.812066	+	0.583566i	$0.801657\pi$
$38$	0	0
$39$	−2.45936	−0.0630605
$40$	0	0
$41$	48.8250i	1.19085i	0.803409	+	0.595427i	$0.203017\pi$
$41$	48.8250i	1.19085i	−0.803409	+	0.595427i	$0.796983\pi$
$42$	0	0
$43$	51.5603	1.19908	0.599539	−	0.800346i	$-0.295351\pi$
$43$	51.5603	1.19908	0.599539	+	0.800346i	$0.295351\pi$
$44$	0	0
$45$	− 1.44198i	− 0.0320441i
$46$	0	0
$47$	19.2421i	0.409406i	0.978824	+	0.204703i	$0.0656229\pi$
$47$	19.2421i	0.409406i	−0.978824	+	0.204703i	$0.934377\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	39.9534	0.783401
$52$	0	0
$53$	82.1345	1.54971	0.774854	−	0.632141i	$-0.217824\pi$
$53$	82.1345	1.54971	0.774854	+	0.632141i	$0.217824\pi$
$54$	0	0
$55$	− 2.77013i	− 0.0503660i
$56$	0	0
$57$	−18.4877	−0.324345
$58$	0	0
$59$	− 92.5800i	− 1.56915i	−0.620032	−	0.784576i	$-0.712881\pi$
$59$	− 92.5800i	− 1.56915i	0.620032	−	0.784576i	$-0.287119\pi$
$60$	0	0
$61$	4.99187i	0.0818340i	0.999163	+	0.0409170i	$0.0130279\pi$
$61$	4.99187i	0.0818340i	−0.999163	+	0.0409170i	$0.986972\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.682497	0.0105000
$66$	0	0
$67$	2.20699	0.0329402	0.0164701	−	0.999864i	$-0.494757\pi$
$67$	2.20699	0.0329402	0.0164701	+	0.999864i	$0.494757\pi$
$68$	0	0
$69$	− 11.4312i	− 0.165669i
$70$	0	0
$71$	80.5899	1.13507	0.567535	−	0.823349i	$-0.307897\pi$
$71$	80.5899	1.13507	0.567535	+	0.823349i	$0.307897\pi$
$72$	0	0
$73$	− 13.9088i	− 0.190531i	−0.995452	−	0.0952657i	$-0.969630\pi$
$73$	− 13.9088i	− 0.190531i	0.995452	−	0.0952657i	$-0.0303701\pi$
$74$	0	0
$75$	− 42.9011i	− 0.572015i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	64.8885	0.821373	0.410687	−	0.911777i	$-0.365289\pi$
$79$	64.8885	0.821373	0.410687	+	0.911777i	$0.365289\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 118.005i	− 1.42174i	−0.703322	−	0.710872i	$-0.748301\pi$
$83$	− 118.005i	− 1.42174i	0.703322	−	0.710872i	$-0.251699\pi$
$84$	0	0
$85$	−11.0875	−0.130441
$86$	0	0
$87$	10.7432i	0.123485i
$88$	0	0
$89$	− 104.265i	− 1.17151i	−0.810487	−	0.585757i	$-0.800797\pi$
$89$	− 104.265i	− 1.17151i	0.810487	−	0.585757i	$-0.199203\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−72.6284	−0.780951
$94$	0	0
$95$	5.13052	0.0540055
$96$	0	0
$97$	− 31.7875i	− 0.327706i	−0.986485	−	0.163853i	$-0.947608\pi$
$97$	− 31.7875i	− 0.327706i	0.986485	−	0.163853i	$-0.0523923\pi$
$98$	0	0
$99$	17.2895	0.174641
$100$	0	0
$101$	72.5037i	0.717859i	0.933365	+	0.358929i	$0.116858\pi$
$101$	72.5037i	0.717859i	−0.933365	+	0.358929i	$0.883142\pi$
$102$	0	0
$103$	− 106.963i	− 1.03847i	−0.854631	−	0.519236i	$-0.826217\pi$
$103$	− 106.963i	− 1.03847i	0.854631	−	0.519236i	$-0.173783\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−196.619	−1.83756	−0.918782	−	0.394765i	$-0.870826\pi$
$107$	−196.619	−1.83756	−0.918782	+	0.394765i	$0.870826\pi$
$108$	0	0
$109$	42.6922	0.391672	0.195836	−	0.980637i	$-0.437258\pi$
$109$	42.6922	0.391672	0.195836	+	0.980637i	$0.437258\pi$
$110$	0	0
$111$	104.084i	0.937693i
$112$	0	0
$113$	175.501	1.55310	0.776552	−	0.630053i	$-0.216967\pi$
$113$	175.501	1.55310	0.776552	+	0.630053i	$0.216967\pi$
$114$	0	0
$115$	3.17227i	0.0275849i
$116$	0	0
$117$	4.25973i	0.0364080i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−87.7860	−0.725504
$122$	0	0
$123$	84.5674	0.687540
$124$	0	0
$125$	23.9220i	0.191376i
$126$	0	0
$127$	−31.0434	−0.244436	−0.122218	−	0.992503i	$-0.539001\pi$
$127$	−31.0434	−0.244436	−0.122218	+	0.992503i	$0.539001\pi$
$128$	0	0
$129$	− 89.3051i	− 0.692288i
$130$	0	0
$131$	46.5095i	0.355034i	0.984118	+	0.177517i	$0.0568065\pi$
$131$	46.5095i	0.355034i	−0.984118	+	0.177517i	$0.943194\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.49759	−0.0185007
$136$	0	0
$137$	45.4654	0.331864	0.165932	−	0.986137i	$-0.446937\pi$
$137$	45.4654	0.331864	0.165932	+	0.986137i	$0.446937\pi$
$138$	0	0
$139$	138.075i	0.993343i	0.867939	+	0.496672i	$0.165445\pi$
$139$	138.075i	0.993343i	−0.867939	+	0.496672i	$0.834555\pi$
$140$	0	0
$141$	33.3283	0.236371
$142$	0	0
$143$	8.18318i	0.0572250i
$144$	0	0
$145$	− 2.98134i	− 0.0205610i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	239.228	1.60556	0.802780	−	0.596276i	$-0.203354\pi$
$149$	239.228	1.60556	0.802780	+	0.596276i	$0.203354\pi$
$150$	0	0
$151$	188.146	1.24600	0.623001	−	0.782221i	$-0.285913\pi$
$151$	188.146	1.24600	0.623001	+	0.782221i	$0.285913\pi$
$152$	0	0
$153$	− 69.2014i	− 0.452297i
$154$	0	0
$155$	20.1551	0.130033
$156$	0	0
$157$	− 215.239i	− 1.37095i	−0.728097	−	0.685474i	$-0.759595\pi$
$157$	− 215.239i	− 1.37095i	0.728097	−	0.685474i	$-0.240405\pi$
$158$	0	0
$159$	− 142.261i	− 0.894724i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−149.811	−0.919083	−0.459542	−	0.888156i	$-0.651986\pi$
$163$	−149.811	−0.919083	−0.459542	+	0.888156i	$0.651986\pi$
$164$	0	0
$165$	−4.79801	−0.0290788
$166$	0	0
$167$	− 137.195i	− 0.821528i	−0.911742	−	0.410764i	$-0.865262\pi$
$167$	− 137.195i	− 0.821528i	0.911742	−	0.410764i	$-0.134738\pi$
$168$	0	0
$169$	166.984	0.988070
$170$	0	0
$171$	32.0216i	0.187261i
$172$	0	0
$173$	− 251.181i	− 1.45191i	−0.687740	−	0.725957i	$-0.741397\pi$
$173$	− 251.181i	− 1.45191i	0.687740	−	0.725957i	$-0.258603\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−160.353	−0.905951
$178$	0	0
$179$	116.793	0.652477	0.326239	−	0.945287i	$-0.394219\pi$
$179$	116.793	0.652477	0.326239	+	0.945287i	$0.394219\pi$
$180$	0	0
$181$	− 117.148i	− 0.647228i	−0.946189	−	0.323614i	$-0.895102\pi$
$181$	− 117.148i	− 0.647228i	0.946189	−	0.323614i	$-0.104898\pi$
$182$	0	0
$183$	8.64618	0.0472469
$184$	0	0
$185$	− 28.8843i	− 0.156131i
$186$	0	0
$187$	− 132.940i	− 0.710907i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−341.818	−1.78962	−0.894812	−	0.446443i	$-0.852691\pi$
$191$	−341.818	−1.78962	−0.894812	+	0.446443i	$0.852691\pi$
$192$	0	0
$193$	−224.551	−1.16348	−0.581738	−	0.813376i	$-0.697627\pi$
$193$	−224.551	−1.16348	−0.581738	+	0.813376i	$0.697627\pi$
$194$	0	0
$195$	− 1.18212i	− 0.00606215i
$196$	0	0
$197$	257.109	1.30512	0.652560	−	0.757737i	$-0.273695\pi$
$197$	257.109	1.30512	0.652560	+	0.757737i	$0.273695\pi$
$198$	0	0
$199$	245.479i	1.23356i	0.787134	+	0.616782i	$0.211564\pi$
$199$	245.479i	1.23356i	−0.787134	+	0.616782i	$0.788436\pi$
$200$	0	0
$201$	− 3.82262i	− 0.0190180i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−23.4683	−0.114480
$206$	0	0
$207$	−19.7994	−0.0956492
$208$	0	0
$209$	61.5152i	0.294331i
$210$	0	0
$211$	95.8210	0.454128	0.227064	−	0.973880i	$-0.427087\pi$
$211$	95.8210	0.454128	0.227064	+	0.973880i	$0.427087\pi$
$212$	0	0
$213$	− 139.586i	− 0.655333i
$214$	0	0
$215$	24.7831i	0.115270i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−24.0907	−0.110003
$220$	0	0
$221$	32.7533	0.148205
$222$	0	0
$223$	94.2091i	0.422462i	0.977436	+	0.211231i	$0.0677473\pi$
$223$	94.2091i	0.422462i	−0.977436	+	0.211231i	$0.932253\pi$
$224$	0	0
$225$	−74.3069	−0.330253
$226$	0	0
$227$	224.253i	0.987899i	0.869491	+	0.493949i	$0.164447\pi$
$227$	224.253i	0.987899i	−0.869491	+	0.493949i	$0.835553\pi$
$228$	0	0
$229$	− 366.724i	− 1.60142i	−0.599055	−	0.800708i	$-0.704457\pi$
$229$	− 366.724i	− 1.60142i	0.599055	−	0.800708i	$-0.295543\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	69.6741	0.299030	0.149515	−	0.988759i	$-0.452229\pi$
$233$	69.6741	0.299030	0.149515	+	0.988759i	$0.452229\pi$
$234$	0	0
$235$	−9.24893	−0.0393572
$236$	0	0
$237$	− 112.390i	− 0.474220i
$238$	0	0
$239$	214.544	0.897674	0.448837	−	0.893614i	$-0.351838\pi$
$239$	214.544	0.897674	0.448837	+	0.893614i	$0.351838\pi$
$240$	0	0
$241$	− 163.395i	− 0.677989i	−0.940788	−	0.338994i	$-0.889913\pi$
$241$	− 163.395i	− 0.677989i	0.940788	−	0.338994i	$-0.110087\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−15.1560	−0.0613601
$248$	0	0
$249$	−204.390	−0.820844
$250$	0	0
$251$	− 330.546i	− 1.31692i	−0.752617	−	0.658458i	$-0.771209\pi$
$251$	− 330.546i	− 1.31692i	0.752617	−	0.658458i	$-0.228791\pi$
$252$	0	0
$253$	−38.0357	−0.150339
$254$	0	0
$255$	19.2041i	0.0753101i
$256$	0	0
$257$	− 160.529i	− 0.624626i	−0.949979	−	0.312313i	$-0.898896\pi$
$257$	− 160.529i	− 0.624626i	0.949979	−	0.312313i	$-0.101104\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	18.6077	0.0712940
$262$	0	0
$263$	106.515	0.405002	0.202501	−	0.979282i	$-0.435093\pi$
$263$	106.515	0.405002	0.202501	+	0.979282i	$0.435093\pi$
$264$	0	0
$265$	39.4789i	0.148977i
$266$	0	0
$267$	−180.592	−0.676374
$268$	0	0
$269$	155.454i	0.577895i	0.957345	+	0.288947i	$0.0933052\pi$
$269$	155.454i	0.577895i	−0.957345	+	0.288947i	$0.906695\pi$
$270$	0	0
$271$	− 516.080i	− 1.90435i	−0.305549	−	0.952176i	$-0.598840\pi$
$271$	− 516.080i	− 1.90435i	0.305549	−	0.952176i	$-0.401160\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−142.748	−0.519082
$276$	0	0
$277$	200.202	0.722750	0.361375	−	0.932421i	$-0.382308\pi$
$277$	200.202	0.722750	0.361375	+	0.932421i	$0.382308\pi$
$278$	0	0
$279$	125.796i	0.450882i
$280$	0	0
$281$	228.093	0.811720	0.405860	−	0.913935i	$-0.366972\pi$
$281$	228.093	0.811720	0.405860	+	0.913935i	$0.366972\pi$
$282$	0	0
$283$	− 260.710i	− 0.921236i	−0.887598	−	0.460618i	$-0.847628\pi$
$283$	− 260.710i	− 0.921236i	0.887598	−	0.460618i	$-0.152372\pi$
$284$	0	0
$285$	− 8.88632i	− 0.0311801i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−243.092	−0.841150
$290$	0	0
$291$	−55.0576	−0.189201
$292$	0	0
$293$	− 349.885i	− 1.19415i	−0.802186	−	0.597074i	$-0.796330\pi$
$293$	− 349.885i	− 1.19415i	0.802186	−	0.597074i	$-0.203670\pi$
$294$	0	0
$295$	44.4996	0.150846
$296$	0	0
$297$	− 29.9463i	− 0.100829i
$298$	0	0
$299$	− 9.37113i	− 0.0313416i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	125.580	0.414456
$304$	0	0
$305$	−2.39940	−0.00786689
$306$	0	0
$307$	− 146.898i	− 0.478495i	−0.970959	−	0.239247i	$-0.923099\pi$
$307$	− 146.898i	− 0.478495i	0.970959	−	0.239247i	$-0.0769007\pi$
$308$	0	0
$309$	−185.265	−0.599562
$310$	0	0
$311$	− 80.7340i	− 0.259595i	−0.991541	−	0.129797i	$-0.958567\pi$
$311$	− 80.7340i	− 0.259595i	0.991541	−	0.129797i	$-0.0414327\pi$
$312$	0	0
$313$	− 154.339i	− 0.493096i	−0.969131	−	0.246548i	$-0.920704\pi$
$313$	− 154.339i	− 0.493096i	0.969131	−	0.246548i	$-0.0792963\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−188.545	−0.594780	−0.297390	−	0.954756i	$-0.596116\pi$
$317$	−188.545	−0.594780	−0.297390	+	0.954756i	$0.596116\pi$
$318$	0	0
$319$	35.7465	0.112058
$320$	0	0
$321$	340.555i	1.06092i
$322$	0	0
$323$	246.216	0.762277
$324$	0	0
$325$	− 35.1697i	− 0.108215i
$326$	0	0
$327$	− 73.9451i	− 0.226132i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−146.787	−0.443466	−0.221733	−	0.975107i	$-0.571171\pi$
$331$	−146.787	−0.443466	−0.221733	+	0.975107i	$0.571171\pi$
$332$	0	0
$333$	180.279	0.541377
$334$	0	0
$335$	1.06082i	0.00316662i
$336$	0	0
$337$	101.231	0.300388	0.150194	−	0.988657i	$-0.452010\pi$
$337$	101.231	0.300388	0.150194	+	0.988657i	$0.452010\pi$
$338$	0	0
$339$	− 303.976i	− 0.896685i
$340$	0	0
$341$	241.661i	0.708684i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	5.49453	0.0159262
$346$	0	0
$347$	−351.689	−1.01351	−0.506756	−	0.862089i	$-0.669156\pi$
$347$	−351.689	−1.01351	−0.506756	+	0.862089i	$0.669156\pi$
$348$	0	0
$349$	− 88.3780i	− 0.253232i	−0.991952	−	0.126616i	$-0.959588\pi$
$349$	− 88.3780i	− 0.253232i	0.991952	−	0.126616i	$-0.0404116\pi$
$350$	0	0
$351$	7.37808	0.0210202
$352$	0	0
$353$	520.542i	1.47462i	0.675552	+	0.737312i	$0.263905\pi$
$353$	520.542i	1.47462i	−0.675552	+	0.737312i	$0.736095\pi$
$354$	0	0
$355$	38.7365i	0.109117i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	514.265	1.43249	0.716246	−	0.697848i	$-0.245859\pi$
$359$	514.265	1.43249	0.716246	+	0.697848i	$0.245859\pi$
$360$	0	0
$361$	247.069	0.684400
$362$	0	0
$363$	152.050i	0.418870i
$364$	0	0
$365$	6.68542	0.0183162
$366$	0	0
$367$	− 417.686i	− 1.13811i	−0.822300	−	0.569054i	$-0.807310\pi$
$367$	− 417.686i	− 1.13811i	0.822300	−	0.569054i	$-0.192690\pi$
$368$	0	0
$369$	− 146.475i	− 0.396951i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−140.314	−0.376176	−0.188088	−	0.982152i	$-0.560229\pi$
$373$	−140.314	−0.376176	−0.188088	+	0.982152i	$0.560229\pi$
$374$	0	0
$375$	41.4342	0.110491
$376$	0	0
$377$	8.80712i	0.0233611i
$378$	0	0
$379$	−153.298	−0.404480	−0.202240	−	0.979336i	$-0.564822\pi$
$379$	−153.298	−0.404480	−0.202240	+	0.979336i	$0.564822\pi$
$380$	0	0
$381$	53.7687i	0.141125i
$382$	0	0
$383$	− 121.717i	− 0.317799i	−0.987295	−	0.158900i	$-0.949205\pi$
$383$	− 121.717i	− 0.317799i	0.987295	−	0.158900i	$-0.0507946\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−154.681	−0.399692
$388$	0	0
$389$	401.271	1.03155	0.515773	−	0.856725i	$-0.327505\pi$
$389$	401.271	1.03155	0.515773	+	0.856725i	$0.327505\pi$
$390$	0	0
$391$	152.238i	0.389356i
$392$	0	0
$393$	80.5568	0.204979
$394$	0	0
$395$	31.1894i	0.0789605i
$396$	0	0
$397$	60.2251i	0.151700i	0.997119	+	0.0758502i	$0.0241671\pi$
$397$	60.2251i	0.151700i	−0.997119	+	0.0758502i	$0.975833\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−374.308	−0.933436	−0.466718	−	0.884406i	$-0.654564\pi$
$401$	−374.308	−0.933436	−0.466718	+	0.884406i	$0.654564\pi$
$402$	0	0
$403$	−59.5398	−0.147741
$404$	0	0
$405$	4.32595i	0.0106814i
$406$	0	0
$407$	346.325	0.850921
$408$	0	0
$409$	614.609i	1.50271i	0.659897	+	0.751356i	$0.270600\pi$
$409$	614.609i	1.50271i	−0.659897	+	0.751356i	$0.729400\pi$
$410$	0	0
$411$	− 78.7484i	− 0.191602i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	56.7203	0.136675
$416$	0	0
$417$	239.152	0.573507
$418$	0	0
$419$	− 129.067i	− 0.308035i	−0.988068	−	0.154017i	$-0.950779\pi$
$419$	− 129.067i	− 0.308035i	0.988068	−	0.154017i	$-0.0492212\pi$
$420$	0	0
$421$	−697.880	−1.65767	−0.828836	−	0.559492i	$-0.810996\pi$
$421$	−697.880	−1.65767	−0.828836	+	0.559492i	$0.810996\pi$
$422$	0	0
$423$	− 57.7263i	− 0.136469i
$424$	0	0
$425$	571.349i	1.34435i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	14.1737	0.0330389
$430$	0	0
$431$	277.092	0.642906	0.321453	−	0.946926i	$-0.395829\pi$
$431$	277.092	0.642906	0.321453	+	0.946926i	$0.395829\pi$
$432$	0	0
$433$	822.794i	1.90022i	0.311919	+	0.950109i	$0.399028\pi$
$433$	822.794i	1.90022i	−0.311919	+	0.950109i	$0.600972\pi$
$434$	0	0
$435$	−5.16384	−0.0118709
$436$	0	0
$437$	− 70.4454i	− 0.161202i
$438$	0	0
$439$	365.827i	0.833320i	0.909062	+	0.416660i	$0.136799\pi$
$439$	365.827i	0.833320i	−0.909062	+	0.416660i	$0.863201\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	573.392	1.29434	0.647169	−	0.762346i	$-0.275953\pi$
$443$	573.392	1.29434	0.647169	+	0.762346i	$0.275953\pi$
$444$	0	0
$445$	50.1161	0.112620
$446$	0	0
$447$	− 414.356i	− 0.926970i
$448$	0	0
$449$	227.961	0.507708	0.253854	−	0.967243i	$-0.418302\pi$
$449$	227.961	0.507708	0.253854	+	0.967243i	$0.418302\pi$
$450$	0	0
$451$	− 281.386i	− 0.623917i
$452$	0	0
$453$	− 325.879i	− 0.719380i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−343.124	−0.750818	−0.375409	−	0.926859i	$-0.622498\pi$
$457$	−343.124	−0.750818	−0.375409	+	0.926859i	$0.622498\pi$
$458$	0	0
$459$	−119.860	−0.261134
$460$	0	0
$461$	− 611.314i	− 1.32606i	−0.748593	−	0.663030i	$-0.769270\pi$
$461$	− 611.314i	− 1.32606i	0.748593	−	0.663030i	$-0.230730\pi$
$462$	0	0
$463$	67.2682	0.145288	0.0726439	−	0.997358i	$-0.476856\pi$
$463$	67.2682	0.145288	0.0726439	+	0.997358i	$0.476856\pi$
$464$	0	0
$465$	− 34.9097i	− 0.0750746i
$466$	0	0
$467$	93.7498i	0.200749i	0.994950	+	0.100375i	$0.0320041\pi$
$467$	93.7498i	0.200749i	−0.994950	+	0.100375i	$0.967996\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−372.804	−0.791517
$472$	0	0
$473$	−297.150	−0.628225
$474$	0	0
$475$	− 264.381i	− 0.556591i
$476$	0	0
$477$	−246.403	−0.516569
$478$	0	0
$479$	353.796i	0.738613i	0.929308	+	0.369306i	$0.120405\pi$
$479$	353.796i	0.738613i	−0.929308	+	0.369306i	$0.879595\pi$
$480$	0	0
$481$	85.3265i	0.177394i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.2790	0.0315032
$486$	0	0
$487$	−952.677	−1.95621	−0.978107	−	0.208101i	$-0.933272\pi$
$487$	−952.677	−1.95621	−0.978107	+	0.208101i	$0.933272\pi$
$488$	0	0
$489$	259.479i	0.530633i
$490$	0	0
$491$	818.649	1.66731	0.833655	−	0.552286i	$-0.186244\pi$
$491$	818.649	1.66731	0.833655	+	0.552286i	$0.186244\pi$
$492$	0	0
$493$	− 143.076i	− 0.290214i
$494$	0	0
$495$	8.31039i	0.0167887i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	24.0821	0.0482607	0.0241304	−	0.999709i	$-0.492318\pi$
$499$	24.0821	0.0482607	0.0241304	+	0.999709i	$0.492318\pi$
$500$	0	0
$501$	−237.629	−0.474310
$502$	0	0
$503$	420.445i	0.835874i	0.908476	+	0.417937i	$0.137247\pi$
$503$	420.445i	0.835874i	−0.908476	+	0.417937i	$0.862753\pi$
$504$	0	0
$505$	−34.8498	−0.0690094
$506$	0	0
$507$	− 289.225i	− 0.570463i
$508$	0	0
$509$	239.869i	0.471255i	0.971843	+	0.235627i	$0.0757145\pi$
$509$	239.869i	0.471255i	−0.971843	+	0.235627i	$0.924285\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	55.4631	0.108115
$514$	0	0
$515$	51.4128	0.0998307
$516$	0	0
$517$	− 110.895i	− 0.214498i
$518$	0	0
$519$	−435.059	−0.838263
$520$	0	0
$521$	− 26.8565i	− 0.0515479i	−0.999668	−	0.0257739i	$-0.991795\pi$
$521$	− 26.8565i	− 0.0515479i	0.999668	−	0.0257739i	$-0.00820501\pi$
$522$	0	0
$523$	223.737i	0.427796i	0.976856	+	0.213898i	$0.0686160\pi$
$523$	223.737i	0.427796i	−0.976856	+	0.213898i	$0.931384\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	967.252	1.83539
$528$	0	0
$529$	−485.443	−0.917661
$530$	0	0
$531$	277.740i	0.523051i
$532$	0	0
$533$	69.3272	0.130070
$534$	0	0
$535$	− 94.5074i	− 0.176649i
$536$	0	0
$537$	− 202.292i	− 0.376708i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	986.190	1.82290	0.911451	−	0.411410i	$-0.134963\pi$
$541$	986.190	1.82290	0.911451	+	0.411410i	$0.134963\pi$
$542$	0	0
$543$	−202.907	−0.373678
$544$	0	0
$545$	20.5205i	0.0376523i
$546$	0	0
$547$	−735.369	−1.34437	−0.672183	−	0.740385i	$-0.734643\pi$
$547$	−735.369	−1.34437	−0.672183	+	0.740385i	$0.734643\pi$
$548$	0	0
$549$	− 14.9756i	− 0.0272780i
$550$	0	0
$551$	66.2056i	0.120155i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−50.0291	−0.0901426
$556$	0	0
$557$	418.568	0.751469	0.375735	−	0.926727i	$-0.377390\pi$
$557$	418.568	0.751469	0.375735	+	0.926727i	$0.377390\pi$
$558$	0	0
$559$	− 73.2111i	− 0.130968i
$560$	0	0
$561$	−230.258	−0.410442
$562$	0	0
$563$	746.738i	1.32635i	0.748462	+	0.663177i	$0.230792\pi$
$563$	746.738i	1.32635i	−0.748462	+	0.663177i	$0.769208\pi$
$564$	0	0
$565$	84.3565i	0.149304i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−288.710	−0.507398	−0.253699	−	0.967283i	$-0.581647\pi$
$569$	−288.710	−0.507398	−0.253699	+	0.967283i	$0.581647\pi$
$570$	0	0
$571$	916.207	1.60457	0.802283	−	0.596944i	$-0.203618\pi$
$571$	916.207	1.60457	0.802283	+	0.596944i	$0.203618\pi$
$572$	0	0
$573$	592.047i	1.03324i
$574$	0	0
$575$	163.470	0.284296
$576$	0	0
$577$	− 992.068i	− 1.71936i	−0.510836	−	0.859678i	$-0.670664\pi$
$577$	− 992.068i	− 1.71936i	0.510836	−	0.859678i	$-0.329336\pi$
$578$	0	0
$579$	388.934i	0.671733i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−473.354	−0.811929
$584$	0	0
$585$	−2.04749	−0.00349998
$586$	0	0
$587$	154.965i	0.263996i	0.991250	+	0.131998i	$0.0421392\pi$
$587$	154.965i	0.263996i	−0.991250	+	0.131998i	$0.957861\pi$
$588$	0	0
$589$	−447.577	−0.759893
$590$	0	0
$591$	− 445.325i	− 0.753512i
$592$	0	0
$593$	1004.17i	1.69337i	0.532094	+	0.846685i	$0.321405\pi$
$593$	1004.17i	1.69337i	−0.532094	+	0.846685i	$0.678595\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	425.182	0.712198
$598$	0	0
$599$	796.917	1.33041	0.665206	−	0.746660i	$-0.268344\pi$
$599$	796.917	1.33041	0.665206	+	0.746660i	$0.268344\pi$
$600$	0	0
$601$	467.002i	0.777041i	0.921440	+	0.388521i	$0.127014\pi$
$601$	467.002i	0.777041i	−0.921440	+	0.388521i	$0.872986\pi$
$602$	0	0
$603$	−6.62098	−0.0109801
$604$	0	0
$605$	− 42.1953i	− 0.0697444i
$606$	0	0
$607$	910.216i	1.49953i	0.661703	+	0.749766i	$0.269834\pi$
$607$	910.216i	1.49953i	−0.661703	+	0.749766i	$0.730166\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.3221	0.0447170
$612$	0	0
$613$	775.800	1.26558	0.632790	−	0.774324i	$-0.281910\pi$
$613$	775.800	1.26558	0.632790	+	0.774324i	$0.281910\pi$
$614$	0	0
$615$	40.6483i	0.0660948i
$616$	0	0
$617$	974.803	1.57991	0.789954	−	0.613166i	$-0.210104\pi$
$617$	974.803	1.57991	0.789954	+	0.613166i	$0.210104\pi$
$618$	0	0
$619$	− 199.260i	− 0.321906i	−0.986962	−	0.160953i	$-0.948543\pi$
$619$	− 199.260i	− 0.321906i	0.986962	−	0.160953i	$-0.0514568\pi$
$620$	0	0
$621$	34.2935i	0.0552231i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	607.726	0.972361
$626$	0	0
$627$	106.548	0.169932
$628$	0	0
$629$	− 1386.17i	− 2.20377i
$630$	0	0
$631$	−751.062	−1.19027	−0.595136	−	0.803625i	$-0.702902\pi$
$631$	−751.062	−1.19027	−0.595136	+	0.803625i	$0.702902\pi$
$632$	0	0
$633$	− 165.967i	− 0.262191i
$634$	0	0
$635$	− 14.9214i	− 0.0234982i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−241.770	−0.378356
$640$	0	0
$641$	−1138.28	−1.77578	−0.887891	−	0.460053i	$-0.847830\pi$
$641$	−1138.28	−1.77578	−0.887891	+	0.460053i	$0.847830\pi$
$642$	0	0
$643$	647.823i	1.00750i	0.863849	+	0.503751i	$0.168047\pi$
$643$	647.823i	1.00750i	−0.863849	+	0.503751i	$0.831953\pi$
$644$	0	0
$645$	42.9255	0.0665512
$646$	0	0
$647$	882.943i	1.36467i	0.731039	+	0.682336i	$0.239036\pi$
$647$	882.943i	1.36467i	−0.731039	+	0.682336i	$0.760964\pi$
$648$	0	0
$649$	533.554i	0.822116i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	545.117	0.834788	0.417394	−	0.908726i	$-0.362944\pi$
$653$	545.117	0.834788	0.417394	+	0.908726i	$0.362944\pi$
$654$	0	0
$655$	−22.3553	−0.0341302
$656$	0	0
$657$	41.7264i	0.0635104i
$658$	0	0
$659$	−698.290	−1.05962	−0.529810	−	0.848116i	$-0.677737\pi$
$659$	−698.290	−1.05962	−0.529810	+	0.848116i	$0.677737\pi$
$660$	0	0
$661$	571.725i	0.864940i	0.901648	+	0.432470i	$0.142358\pi$
$661$	571.725i	0.864940i	−0.901648	+	0.432470i	$0.857642\pi$
$662$	0	0
$663$	− 56.7303i	− 0.0855661i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−40.9358	−0.0613730
$668$	0	0
$669$	163.175	0.243909
$670$	0	0
$671$	− 28.7690i	− 0.0428748i
$672$	0	0
$673$	221.015	0.328403	0.164202	−	0.986427i	$-0.447495\pi$
$673$	221.015	0.328403	0.164202	+	0.986427i	$0.447495\pi$
$674$	0	0
$675$	128.703i	0.190672i
$676$	0	0
$677$	470.666i	0.695222i	0.937639	+	0.347611i	$0.113007\pi$
$677$	470.666i	0.695222i	−0.937639	+	0.347611i	$0.886993\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	388.418	0.570364
$682$	0	0
$683$	−215.960	−0.316193	−0.158097	−	0.987424i	$-0.550536\pi$
$683$	−215.960	−0.316193	−0.158097	+	0.987424i	$0.550536\pi$
$684$	0	0
$685$	21.8535i	0.0319029i
$686$	0	0
$687$	−635.185	−0.924578
$688$	0	0
$689$	− 116.624i	− 0.169265i
$690$	0	0
$691$	− 80.1141i	− 0.115939i	−0.998318	−	0.0579697i	$-0.981537\pi$
$691$	− 80.1141i	− 0.115939i	0.998318	−	0.0579697i	$-0.0184627\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−66.3672	−0.0954924
$696$	0	0
$697$	−1126.25	−1.61586
$698$	0	0
$699$	− 120.679i	− 0.172645i
$700$	0	0
$701$	−821.973	−1.17257	−0.586286	−	0.810104i	$-0.699411\pi$
$701$	−821.973	−1.17257	−0.586286	+	0.810104i	$0.699411\pi$
$702$	0	0
$703$	641.423i	0.912409i
$704$	0	0
$705$	16.0196i	0.0227229i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	575.456	0.811644	0.405822	−	0.913952i	$-0.366985\pi$
$709$	575.456	0.811644	0.405822	+	0.913952i	$0.366985\pi$
$710$	0	0
$711$	−194.665	−0.273791
$712$	0	0
$713$	− 276.743i	− 0.388139i
$714$	0	0
$715$	−3.93334	−0.00550117
$716$	0	0
$717$	− 371.601i	− 0.518272i
$718$	0	0
$719$	1213.93i	1.68836i	0.536063	+	0.844178i	$0.319911\pi$
$719$	1213.93i	1.68836i	−0.536063	+	0.844178i	$0.680089\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−283.009	−0.391437
$724$	0	0
$725$	−153.632	−0.211906
$726$	0	0
$727$	379.498i	0.522005i	0.965338	+	0.261003i	$0.0840532\pi$
$727$	379.498i	0.522005i	−0.965338	+	0.261003i	$0.915947\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	1189.35i	1.62702i
$732$	0	0
$733$	− 1250.46i	− 1.70595i	−0.521954	−	0.852973i	$-0.674797\pi$
$733$	− 1250.46i	− 1.70595i	0.521954	−	0.852973i	$-0.325203\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.7193	−0.0172582
$738$	0	0
$739$	−501.234	−0.678260	−0.339130	−	0.940740i	$-0.610133\pi$
$739$	−501.234	−0.678260	−0.339130	+	0.940740i	$0.610133\pi$
$740$	0	0
$741$	26.2509i	0.0354263i
$742$	0	0
$743$	−444.584	−0.598363	−0.299181	−	0.954196i	$-0.596714\pi$
$743$	−444.584	−0.598363	−0.299181	+	0.954196i	$0.596714\pi$
$744$	0	0
$745$	114.988i	0.154346i
$746$	0	0
$747$	354.014i	0.473914i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−239.335	−0.318688	−0.159344	−	0.987223i	$-0.550938\pi$
$751$	−239.335	−0.318688	−0.159344	+	0.987223i	$0.550938\pi$
$752$	0	0
$753$	−572.522	−0.760322
$754$	0	0
$755$	90.4347i	0.119781i
$756$	0	0
$757$	249.486	0.329572	0.164786	−	0.986329i	$-0.447307\pi$
$757$	249.486	0.329572	0.164786	+	0.986329i	$0.447307\pi$
$758$	0	0
$759$	65.8797i	0.0867981i
$760$	0	0
$761$	1375.68i	1.80773i	0.427821	+	0.903864i	$0.359281\pi$
$761$	1375.68i	1.80773i	−0.427821	+	0.903864i	$0.640719\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	33.2624	0.0434803
$766$	0	0
$767$	−131.455	−0.171389
$768$	0	0
$769$	528.594i	0.687379i	0.939083	+	0.343689i	$0.111677\pi$
$769$	528.594i	0.687379i	−0.939083	+	0.343689i	$0.888323\pi$
$770$	0	0
$771$	−278.044	−0.360628
$772$	0	0
$773$	61.9934i	0.0801984i	0.999196	+	0.0400992i	$0.0127674\pi$
$773$	61.9934i	0.0801984i	−0.999196	+	0.0400992i	$0.987233\pi$
$774$	0	0
$775$	− 1038.61i	− 1.34015i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	521.152	0.669001
$780$	0	0
$781$	−464.453	−0.594690
$782$	0	0
$783$	− 32.2296i	− 0.0411616i
$784$	0	0
$785$	103.457	0.131792
$786$	0	0
$787$	163.568i	0.207837i	0.994586	+	0.103919i	$0.0331381\pi$
$787$	163.568i	0.207837i	−0.994586	+	0.103919i	$0.966862\pi$
$788$	0	0
$789$	− 184.490i	− 0.233828i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.08802	0.00893823
$794$	0	0
$795$	68.3794	0.0860119
$796$	0	0
$797$	− 533.394i	− 0.669253i	−0.942351	−	0.334626i	$-0.891390\pi$
$797$	− 533.394i	− 0.669253i	0.942351	−	0.334626i	$-0.108610\pi$
$798$	0	0
$799$	−443.860	−0.555519
$800$	0	0
$801$	312.794i	0.390505i
$802$	0	0
$803$	80.1586i	0.0998239i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	269.254	0.333648
$808$	0	0
$809$	−550.274	−0.680191	−0.340095	−	0.940391i	$-0.610459\pi$
$809$	−550.274	−0.680191	−0.340095	+	0.940391i	$0.610459\pi$
$810$	0	0
$811$	− 415.532i	− 0.512370i	−0.966628	−	0.256185i	$-0.917534\pi$
$811$	− 415.532i	− 0.512370i	0.966628	−	0.256185i	$-0.0824656\pi$
$812$	0	0
$813$	−893.876	−1.09948
$814$	0	0
$815$	− 72.0082i	− 0.0883536i
$816$	0	0
$817$	− 550.348i	− 0.673621i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	869.191	1.05870	0.529349	−	0.848404i	$-0.322436\pi$
$821$	869.191	1.05870	0.529349	+	0.848404i	$0.322436\pi$
$822$	0	0
$823$	421.783	0.512494	0.256247	−	0.966611i	$-0.417514\pi$
$823$	421.783	0.512494	0.256247	+	0.966611i	$0.417514\pi$
$824$	0	0
$825$	247.246i	0.299692i
$826$	0	0
$827$	−70.7290	−0.0855248	−0.0427624	−	0.999085i	$-0.513616\pi$
$827$	−70.7290	−0.0855248	−0.0427624	+	0.999085i	$0.513616\pi$
$828$	0	0
$829$	− 1268.85i	− 1.53058i	−0.643685	−	0.765290i	$-0.722595\pi$
$829$	− 1268.85i	− 1.53058i	0.643685	−	0.765290i	$-0.277405\pi$
$830$	0	0
$831$	− 346.759i	− 0.417280i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	65.9445	0.0789754
$836$	0	0
$837$	217.885	0.260317
$838$	0	0
$839$	− 613.254i	− 0.730935i	−0.930824	−	0.365467i	$-0.880909\pi$
$839$	− 613.254i	− 0.730935i	0.930824	−	0.365467i	$-0.119091\pi$
$840$	0	0
$841$	−802.528	−0.954254
$842$	0	0
$843$	− 395.069i	− 0.468647i
$844$	0	0
$845$	80.2627i	0.0949855i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−451.563	−0.531876
$850$	0	0
$851$	−396.601	−0.466041
$852$	0	0
$853$	− 668.244i	− 0.783404i	−0.920092	−	0.391702i	$-0.871886\pi$
$853$	− 668.244i	− 0.783404i	0.920092	−	0.391702i	$-0.128114\pi$
$854$	0	0
$855$	−15.3916	−0.0180018
$856$	0	0
$857$	519.712i	0.606431i	0.952922	+	0.303216i	$0.0980602\pi$
$857$	519.712i	0.606431i	−0.952922	+	0.303216i	$0.901940\pi$
$858$	0	0
$859$	715.553i	0.833007i	0.909134	+	0.416504i	$0.136745\pi$
$859$	715.553i	0.833007i	−0.909134	+	0.416504i	$0.863255\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	342.716	0.397122	0.198561	−	0.980089i	$-0.436373\pi$
$863$	342.716	0.397122	0.198561	+	0.980089i	$0.436373\pi$
$864$	0	0
$865$	120.733	0.139576
$866$	0	0
$867$	421.048i	0.485638i
$868$	0	0
$869$	−373.963	−0.430337
$870$	0	0
$871$	− 3.13374i	− 0.00359786i
$872$	0	0
$873$	95.3625i	0.109235i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−158.550	−0.180787	−0.0903934	−	0.995906i	$-0.528812\pi$
$877$	−158.550	−0.180787	−0.0903934	+	0.995906i	$0.528812\pi$
$878$	0	0
$879$	−606.019	−0.689442
$880$	0	0
$881$	1100.63i	1.24930i	0.780906	+	0.624648i	$0.214758\pi$
$881$	1100.63i	1.24930i	−0.780906	+	0.624648i	$0.785242\pi$
$882$	0	0
$883$	−255.888	−0.289794	−0.144897	−	0.989447i	$-0.546285\pi$
$883$	−255.888	−0.289794	−0.144897	+	0.989447i	$0.546285\pi$
$884$	0	0
$885$	− 77.0756i	− 0.0870911i
$886$	0	0
$887$	1569.23i	1.76915i	0.466400	+	0.884574i	$0.345551\pi$
$887$	1569.23i	1.76915i	−0.466400	+	0.884574i	$0.654449\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−51.8685	−0.0582138
$892$	0	0
$893$	205.388	0.229997
$894$	0	0
$895$	56.1381i	0.0627241i
$896$	0	0
$897$	−16.2313	−0.0180951
$898$	0	0
$899$	260.087i	0.289307i
$900$	0	0
$901$	1894.61i	2.10278i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	56.3087	0.0622196
$906$	0	0
$907$	−876.460	−0.966328	−0.483164	−	0.875530i	$-0.660513\pi$
$907$	−876.460	−0.966328	−0.483164	+	0.875530i	$0.660513\pi$
$908$	0	0
$909$	− 217.511i	− 0.239286i
$910$	0	0
$911$	1778.49	1.95224	0.976118	−	0.217241i	$-0.0697057\pi$
$911$	1778.49	1.95224	0.976118	+	0.217241i	$0.0697057\pi$
$912$	0	0
$913$	680.080i	0.744885i
$914$	0	0
$915$	4.15589i	0.00454195i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	264.863	0.288208	0.144104	−	0.989563i	$-0.453970\pi$
$919$	264.863	0.288208	0.144104	+	0.989563i	$0.453970\pi$
$920$	0	0
$921$	−254.435	−0.276259
$922$	0	0
$923$	− 114.431i	− 0.123977i
$924$	0	0
$925$	−1488.44	−1.60912
$926$	0	0
$927$	320.888i	0.346157i
$928$	0	0
$929$	− 789.390i	− 0.849720i	−0.905259	−	0.424860i	$-0.860323\pi$
$929$	− 789.390i	− 0.849720i	0.905259	−	0.424860i	$-0.139677\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−139.835	−0.149877
$934$	0	0
$935$	63.8989	0.0683411
$936$	0	0
$937$	1446.06i	1.54329i	0.636053	+	0.771645i	$0.280566\pi$
$937$	1446.06i	1.54329i	−0.636053	+	0.771645i	$0.719434\pi$
$938$	0	0
$939$	−267.323	−0.284689
$940$	0	0
$941$	1201.10i	1.27640i	0.769869	+	0.638202i	$0.220322\pi$
$941$	1201.10i	1.27640i	−0.769869	+	0.638202i	$0.779678\pi$
$942$	0	0
$943$	322.235i	0.341713i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	838.720	0.885660	0.442830	−	0.896606i	$-0.353974\pi$
$947$	838.720	0.885660	0.442830	+	0.896606i	$0.353974\pi$
$948$	0	0
$949$	−19.7492	−0.0208106
$950$	0	0
$951$	326.570i	0.343396i
$952$	0	0
$953$	1377.68	1.44562	0.722812	−	0.691045i	$-0.242849\pi$
$953$	1377.68	1.44562	0.722812	+	0.691045i	$0.242849\pi$
$954$	0	0
$955$	− 164.299i	− 0.172041i
$956$	0	0
$957$	− 61.9147i	− 0.0646967i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−797.295	−0.829652
$962$	0	0
$963$	589.858	0.612521
$964$	0	0
$965$	− 107.933i	− 0.111848i
$966$	0	0
$967$	−848.834	−0.877802	−0.438901	−	0.898536i	$-0.644632\pi$
$967$	−848.834	−0.877802	−0.438901	+	0.898536i	$0.644632\pi$
$968$	0	0
$969$	− 426.458i	− 0.440101i
$970$	0	0
$971$	− 1262.08i	− 1.29977i	−0.760031	−	0.649887i	$-0.774816\pi$
$971$	− 1262.08i	− 1.29977i	0.760031	−	0.649887i	$-0.225184\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−60.9158	−0.0624777
$976$	0	0
$977$	644.168	0.659332	0.329666	−	0.944098i	$-0.393064\pi$
$977$	644.168	0.659332	0.329666	+	0.944098i	$0.393064\pi$
$978$	0	0
$979$	600.895i	0.613784i
$980$	0	0
$981$	−128.077	−0.130557
$982$	0	0
$983$	− 1213.77i	− 1.23476i	−0.786667	−	0.617378i	$-0.788195\pi$
$983$	− 1213.77i	− 1.23476i	0.786667	−	0.617378i	$-0.211805\pi$
$984$	0	0
$985$	123.582i	0.125464i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	340.288	0.344072
$990$	0	0
$991$	−1691.12	−1.70648	−0.853241	−	0.521517i	$-0.825366\pi$
$991$	−1691.12	−1.70648	−0.853241	+	0.521517i	$0.825366\pi$
$992$	0	0
$993$	254.243i	0.256035i
$994$	0	0
$995$	−117.992	−0.118585
$996$	0	0
$997$	1129.58i	1.13297i	0.824071	+	0.566487i	$0.191698\pi$
$997$	1129.58i	1.13297i	−0.824071	+	0.566487i	$0.808302\pi$
$998$	0	0
$999$	− 312.252i	− 0.312564i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.f.j.97.3		8
4.3	odd	2		588.3.d.c.97.7	yes	8
7.6	odd	2	inner	2352.3.f.j.97.6		8
12.11	even	2		1764.3.d.h.685.4		8
28.3	even	6		588.3.m.e.313.3		8
28.11	odd	6		588.3.m.f.313.2		8
28.19	even	6		588.3.m.f.325.2		8
28.23	odd	6		588.3.m.e.325.3		8
28.27	even	2		588.3.d.c.97.2	✓	8
84.11	even	6		1764.3.z.l.901.3		8
84.23	even	6		1764.3.z.m.325.2		8
84.47	odd	6		1764.3.z.l.325.3		8
84.59	odd	6		1764.3.z.m.901.2		8
84.83	odd	2		1764.3.d.h.685.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.3.d.c.97.2	✓	8	28.27	even	2
588.3.d.c.97.7	yes	8	4.3	odd	2
588.3.m.e.313.3		8	28.3	even	6
588.3.m.e.325.3		8	28.23	odd	6
588.3.m.f.313.2		8	28.11	odd	6
588.3.m.f.325.2		8	28.19	even	6
1764.3.d.h.685.4		8	12.11	even	2
1764.3.d.h.685.5		8	84.83	odd	2
1764.3.z.l.325.3		8	84.47	odd	6
1764.3.z.l.901.3		8	84.11	even	6
1764.3.z.m.325.2		8	84.23	even	6
1764.3.z.m.901.2		8	84.59	odd	6
2352.3.f.j.97.3		8	1.1	even	1	trivial
2352.3.f.j.97.6		8	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +0.480662i q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +0.480662i q^{5} -3.00000 q^{9} -5.76316 q^{11} -1.41991i q^{13} +0.832530 q^{15} +23.0671i q^{17} -10.6739i q^{19} +6.59980 q^{23} +24.7690 q^{25} +5.19615i q^{27} -6.20258 q^{29} -41.9320i q^{31} +9.98209i q^{33} -60.0929 q^{37} -2.45936 q^{39} +48.8250i q^{41} +51.5603 q^{43} -1.44198i q^{45} +19.2421i q^{47} +39.9534 q^{51} +82.1345 q^{53} -2.77013i q^{55} -18.4877 q^{57} -92.5800i q^{59} +4.99187i q^{61} +0.682497 q^{65} +2.20699 q^{67} -11.4312i q^{69} +80.5899 q^{71} -13.9088i q^{73} -42.9011i q^{75} +64.8885 q^{79} +9.00000 q^{81} -118.005i q^{83} -11.0875 q^{85} +10.7432i q^{87} -104.265i q^{89} -72.6284 q^{93} +5.13052 q^{95} -31.7875i q^{97} +17.2895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} + 16 q^{23} + 72 q^{25} + 80 q^{29} + 128 q^{37} + 112 q^{43} + 96 q^{51} - 144 q^{53} - 192 q^{57} + 240 q^{65} + 64 q^{67} - 224 q^{71} + 432 q^{79} + 72 q^{81} - 96 q^{85} - 96 q^{93} + 272 q^{95}+O(q^{100}) \) 8 * q - 24 * q^9 + 16 * q^23 + 72 * q^25 + 80 * q^29 + 128 * q^37 + 112 * q^43 + 96 * q^51 - 144 * q^53 - 192 * q^57 + 240 * q^65 + 64 * q^67 - 224 * q^71 + 432 * q^79 + 72 * q^81 - 96 * q^85 - 96 * q^93 + 272 * q^95

Introduction

L-functions

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