LMFDB - Embedded newform 4356.3.f.m.1693.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4356,3,Mod(1693,4356)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4356.1693"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4356, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4356.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,92,0,184,0,0,0, 0,0,-104,0,0,0,0,0,144,0,0,0,0,0,0,0,0,0,80,0,-172,0,0,0,32,0,0,0,0,0, -4,0,0,0,0,0,0,0,-316,0,0,0,-132,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12, 0,-656] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	no
Analytic conductor:	$118.692403155$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 45 x^{14} + 278 x^{13} + 569 x^{12} - 3118 x^{11} + 4245 x^{10} - 22868 x^{9} + \cdots + 188375625 $ x^16 - 2x^15 - 45x^14 + 278x^13 + 569x^12 - 3118x^11 + 4245x^10 - 22868x^9 + 188656x^8 - 554550x^7 + 955785x^6 - 3424950x^5 + 11074725x^4 - 11495250x^3 + 10290375x^2 - 111172500*x + 188375625
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1693.1
Root			$-6.17368 - 0.00685009i$ of defining polynomial
Character	$\chi$	$=$	4356.1693
Dual form			4356.3.f.m.1693.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.43361 q^{5} -3.18836i q^{7} +10.7138i q^{13} +32.8304i q^{17} +25.0072i q^{19} -20.4320 q^{23} +63.9930 q^{25} -7.55416i q^{29} +5.27753 q^{31} +30.0778i q^{35} +48.5998 q^{37} -30.6668i q^{41} -51.3462i q^{43} +24.6883 q^{47} +38.8344 q^{49} +42.1056 q^{53} +3.75534 q^{59} -36.3652i q^{61} -101.070i q^{65} -72.7217 q^{67} +2.18450 q^{71} +17.8049i q^{73} +96.1361i q^{79} -29.8529i q^{83} -309.709i q^{85} -99.5260 q^{89} +34.1596 q^{91} -235.908i q^{95} +73.9702 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{5} + 92 q^{23} + 184 q^{25} - 104 q^{31} + 144 q^{37} + 80 q^{47} - 172 q^{49} + 32 q^{53} - 4 q^{59} - 316 q^{67} - 132 q^{71} - 12 q^{89} - 656 q^{91} - 296 q^{97}+O(q^{100}) $ 16 * q + 4 * q^5 + 92 * q^23 + 184 * q^25 - 104 * q^31 + 144 * q^37 + 80 * q^47 - 172 * q^49 + 32 * q^53 - 4 * q^59 - 316 * q^67 - 132 * q^71 - 12 * q^89 - 656 * q^91 - 296 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4356\mathbb{Z}\right)^\times$.

$n$	$1333$	$1937$	$2179$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−9.43361	−1.88672	−0.943361	−	0.331768i	$-0.892355\pi$
$5$	−9.43361	−1.88672	−0.943361	+	0.331768i	$0.892355\pi$
$6$	0	0
$7$	− 3.18836i	− 0.455480i	−0.973722	−	0.227740i	$-0.926866\pi$
$7$	− 3.18836i	− 0.455480i	0.973722	−	0.227740i	$-0.0731336\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	10.7138i	0.824141i	0.911152	+	0.412071i	$0.135194\pi$
$13$	10.7138i	0.824141i	−0.911152	+	0.412071i	$0.864806\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	32.8304i	1.93120i	0.260029	+	0.965601i	$0.416268\pi$
$17$	32.8304i	1.93120i	−0.260029	+	0.965601i	$0.583732\pi$
$18$	0	0
$19$	25.0072i	1.31617i	0.752945	+	0.658083i	$0.228632\pi$
$19$	25.0072i	1.31617i	−0.752945	+	0.658083i	$0.771368\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−20.4320	−0.888348	−0.444174	−	0.895941i	$-0.646503\pi$
$23$	−20.4320	−0.888348	−0.444174	+	0.895941i	$0.646503\pi$
$24$	0	0
$25$	63.9930	2.55972
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 7.55416i	− 0.260488i	−0.991482	−	0.130244i	$-0.958424\pi$
$29$	− 7.55416i	− 0.260488i	0.991482	−	0.130244i	$-0.0415761\pi$
$30$	0	0
$31$	5.27753	0.170243	0.0851215	−	0.996371i	$-0.472872\pi$
$31$	5.27753	0.170243	0.0851215	+	0.996371i	$0.472872\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	30.0778i	0.859364i
$36$	0	0
$37$	48.5998	1.31351	0.656755	−	0.754104i	$-0.271929\pi$
$37$	48.5998	1.31351	0.656755	+	0.754104i	$0.271929\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 30.6668i	− 0.747970i	−0.927435	−	0.373985i	$-0.877991\pi$
$41$	− 30.6668i	− 0.747970i	0.927435	−	0.373985i	$-0.122009\pi$
$42$	0	0
$43$	− 51.3462i	− 1.19410i	−0.802205	−	0.597049i	$-0.796340\pi$
$43$	− 51.3462i	− 1.19410i	0.802205	−	0.597049i	$-0.203660\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	24.6883	0.525282	0.262641	−	0.964894i	$-0.415407\pi$
$47$	24.6883	0.525282	0.262641	+	0.964894i	$0.415407\pi$
$48$	0	0
$49$	38.8344	0.792538
$50$	0	0
$51$	0	0
$52$	0	0
$53$	42.1056	0.794446	0.397223	−	0.917722i	$-0.369974\pi$
$53$	42.1056	0.794446	0.397223	+	0.917722i	$0.369974\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.75534	0.0636498	0.0318249	−	0.999493i	$-0.489868\pi$
$59$	3.75534	0.0636498	0.0318249	+	0.999493i	$0.489868\pi$
$60$	0	0
$61$	− 36.3652i	− 0.596151i	−0.954542	−	0.298075i	$-0.903655\pi$
$61$	− 36.3652i	− 0.596151i	0.954542	−	0.298075i	$-0.0963447\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 101.070i	− 1.55493i
$66$	0	0
$67$	−72.7217	−1.08540	−0.542699	−	0.839927i	$-0.682598\pi$
$67$	−72.7217	−1.08540	−0.542699	+	0.839927i	$0.682598\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.18450	0.0307675	0.0153838	−	0.999882i	$-0.495103\pi$
$71$	2.18450	0.0307675	0.0153838	+	0.999882i	$0.495103\pi$
$72$	0	0
$73$	17.8049i	0.243903i	0.992536	+	0.121952i	$0.0389152\pi$
$73$	17.8049i	0.243903i	−0.992536	+	0.121952i	$0.961085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	96.1361i	1.21691i	0.793587	+	0.608456i	$0.208211\pi$
$79$	96.1361i	1.21691i	−0.793587	+	0.608456i	$0.791789\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 29.8529i	− 0.359674i	−0.983696	−	0.179837i	$-0.942443\pi$
$83$	− 29.8529i	− 0.359674i	0.983696	−	0.179837i	$-0.0575570\pi$
$84$	0	0
$85$	− 309.709i	− 3.64364i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−99.5260	−1.11827	−0.559135	−	0.829077i	$-0.688867\pi$
$89$	−99.5260	−1.11827	−0.559135	+	0.829077i	$0.688867\pi$
$90$	0	0
$91$	34.1596	0.375380
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 235.908i	− 2.48324i
$96$	0	0
$97$	73.9702	0.762580	0.381290	−	0.924456i	$-0.375480\pi$
$97$	73.9702	0.762580	0.381290	+	0.924456i	$0.375480\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	153.802i	1.52279i	0.648287	+	0.761396i	$0.275486\pi$
$101$	153.802i	1.52279i	−0.648287	+	0.761396i	$0.724514\pi$
$102$	0	0
$103$	48.4212	0.470109	0.235054	−	0.971982i	$-0.424473\pi$
$103$	48.4212	0.470109	0.235054	+	0.971982i	$0.424473\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	195.686i	1.82884i	0.404765	+	0.914421i	$0.367353\pi$
$107$	195.686i	1.82884i	−0.404765	+	0.914421i	$0.632647\pi$
$108$	0	0
$109$	− 25.7099i	− 0.235871i	−0.993021	−	0.117935i	$-0.962372\pi$
$109$	− 25.7099i	− 0.235871i	0.993021	−	0.117935i	$-0.0376276\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−76.3369	−0.675547	−0.337774	−	0.941227i	$-0.609674\pi$
$113$	−76.3369	−0.675547	−0.337774	+	0.941227i	$0.609674\pi$
$114$	0	0
$115$	192.748	1.67607
$116$	0	0
$117$	0	0
$118$	0	0
$119$	104.675	0.879624
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−367.845	−2.94276
$126$	0	0
$127$	− 56.0694i	− 0.441492i	−0.975331	−	0.220746i	$-0.929151\pi$
$127$	− 56.0694i	− 0.441492i	0.975331	−	0.220746i	$-0.0708491\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 84.2041i	− 0.642780i	−0.946947	−	0.321390i	$-0.895850\pi$
$131$	− 84.2041i	− 0.642780i	0.946947	−	0.321390i	$-0.104150\pi$
$132$	0	0
$133$	79.7318	0.599487
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−51.9521	−0.379213	−0.189606	−	0.981860i	$-0.560721\pi$
$137$	−51.9521	−0.379213	−0.189606	+	0.981860i	$0.560721\pi$
$138$	0	0
$139$	186.964i	1.34506i	0.740068	+	0.672532i	$0.234793\pi$
$139$	186.964i	1.34506i	−0.740068	+	0.672532i	$0.765207\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	71.2630i	0.491469i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	72.5256i	0.486749i	0.969932	+	0.243375i	$0.0782545\pi$
$149$	72.5256i	0.486749i	−0.969932	+	0.243375i	$0.921746\pi$
$150$	0	0
$151$	213.572i	1.41438i	0.707022	+	0.707192i	$0.250038\pi$
$151$	213.572i	1.41438i	−0.707022	+	0.707192i	$0.749962\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−49.7862	−0.321201
$156$	0	0
$157$	−11.3908	−0.0725529	−0.0362765	−	0.999342i	$-0.511550\pi$
$157$	−11.3908	−0.0725529	−0.0362765	+	0.999342i	$0.511550\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	65.1446i	0.404625i
$162$	0	0
$163$	−120.865	−0.741505	−0.370753	−	0.928732i	$-0.620900\pi$
$163$	−120.865	−0.741505	−0.370753	+	0.928732i	$0.620900\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	205.052i	1.22786i	0.789362	+	0.613929i	$0.210412\pi$
$167$	205.052i	1.22786i	−0.789362	+	0.613929i	$0.789588\pi$
$168$	0	0
$169$	54.2137	0.320791
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 82.1056i	− 0.474599i	−0.971437	−	0.237299i	$-0.923738\pi$
$173$	− 82.1056i	− 0.474599i	0.971437	−	0.237299i	$-0.0762623\pi$
$174$	0	0
$175$	− 204.033i	− 1.16590i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−170.274	−0.951252	−0.475626	−	0.879648i	$-0.657778\pi$
$179$	−170.274	−0.951252	−0.475626	+	0.879648i	$0.657778\pi$
$180$	0	0
$181$	212.035	1.17147	0.585733	−	0.810504i	$-0.300807\pi$
$181$	212.035	1.17147	0.585733	+	0.810504i	$0.300807\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−458.472	−2.47823
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	55.5015	0.290584	0.145292	−	0.989389i	$-0.453588\pi$
$191$	55.5015	0.290584	0.145292	+	0.989389i	$0.453588\pi$
$192$	0	0
$193$	167.281i	0.866739i	0.901216	+	0.433370i	$0.142676\pi$
$193$	167.281i	0.866739i	−0.901216	+	0.433370i	$0.857324\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	72.3636i	0.367328i	0.982989	+	0.183664i	$0.0587958\pi$
$197$	72.3636i	0.367328i	−0.982989	+	0.183664i	$0.941204\pi$
$198$	0	0
$199$	317.651	1.59624	0.798118	−	0.602501i	$-0.205829\pi$
$199$	317.651	1.59624	0.798118	+	0.602501i	$0.205829\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−24.0854	−0.118647
$204$	0	0
$205$	289.298i	1.41121i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 248.577i	− 1.17809i	−0.808100	−	0.589046i	$-0.799504\pi$
$211$	− 248.577i	− 1.17809i	0.808100	−	0.589046i	$-0.200496\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	484.380i	2.25293i
$216$	0	0
$217$	− 16.8267i	− 0.0775423i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−351.740	−1.59158
$222$	0	0
$223$	98.7957	0.443030	0.221515	−	0.975157i	$-0.428900\pi$
$223$	98.7957	0.443030	0.221515	+	0.975157i	$0.428900\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	411.271i	1.81177i	0.423527	+	0.905884i	$0.360792\pi$
$227$	411.271i	1.81177i	−0.423527	+	0.905884i	$0.639208\pi$
$228$	0	0
$229$	−300.824	−1.31364	−0.656821	−	0.754046i	$-0.728099\pi$
$229$	−300.824	−1.31364	−0.656821	+	0.754046i	$0.728099\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.4018i	0.0832697i	0.999133	+	0.0416348i	$0.0132566\pi$
$233$	19.4018i	0.0832697i	−0.999133	+	0.0416348i	$0.986743\pi$
$234$	0	0
$235$	−232.899	−0.991061
$236$	0	0
$237$	0	0
$238$	0	0
$239$	18.0215i	0.0754039i	0.999289	+	0.0377020i	$0.0120038\pi$
$239$	18.0215i	0.0754039i	−0.999289	+	0.0377020i	$0.987996\pi$
$240$	0	0
$241$	26.6523i	0.110591i	0.998470	+	0.0552953i	$0.0176100\pi$
$241$	26.6523i	0.110591i	−0.998470	+	0.0552953i	$0.982390\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−366.348	−1.49530
$246$	0	0
$247$	−267.923	−1.08471
$248$	0	0
$249$	0	0
$250$	0	0
$251$	183.977	0.732975	0.366487	−	0.930423i	$-0.380560\pi$
$251$	183.977	0.732975	0.366487	+	0.930423i	$0.380560\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−40.8892	−0.159102	−0.0795510	−	0.996831i	$-0.525349\pi$
$257$	−40.8892	−0.159102	−0.0795510	+	0.996831i	$0.525349\pi$
$258$	0	0
$259$	− 154.954i	− 0.598277i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 518.488i	− 1.97144i	−0.168402	−	0.985718i	$-0.553861\pi$
$263$	− 518.488i	− 1.97144i	0.168402	−	0.985718i	$-0.446139\pi$
$264$	0	0
$265$	−397.208	−1.49890
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−281.603	−1.04685	−0.523425	−	0.852072i	$-0.675346\pi$
$269$	−281.603	−1.04685	−0.523425	+	0.852072i	$0.675346\pi$
$270$	0	0
$271$	− 128.615i	− 0.474595i	−0.971437	−	0.237298i	$-0.923738\pi$
$271$	− 128.615i	− 0.474595i	0.971437	−	0.237298i	$-0.0762617\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 317.542i	− 1.14636i	−0.819429	−	0.573180i	$-0.805709\pi$
$277$	− 317.542i	− 1.14636i	0.819429	−	0.573180i	$-0.194291\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 270.149i	− 0.961385i	−0.876889	−	0.480692i	$-0.840385\pi$
$281$	− 270.149i	− 0.961385i	0.876889	−	0.480692i	$-0.159615\pi$
$282$	0	0
$283$	411.351i	1.45354i	0.686882	+	0.726769i	$0.258979\pi$
$283$	411.351i	1.45354i	−0.686882	+	0.726769i	$0.741021\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−97.7767	−0.340685
$288$	0	0
$289$	−788.837	−2.72954
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 235.474i	− 0.803667i	−0.915713	−	0.401834i	$-0.868373\pi$
$293$	− 235.474i	− 0.803667i	0.915713	−	0.401834i	$-0.131627\pi$
$294$	0	0
$295$	−35.4264	−0.120089
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 218.905i	− 0.732125i
$300$	0	0
$301$	−163.710	−0.543888
$302$	0	0
$303$	0	0
$304$	0	0
$305$	343.055i	1.12477i
$306$	0	0
$307$	202.665i	0.660146i	0.943955	+	0.330073i	$0.107073\pi$
$307$	202.665i	0.660146i	−0.943955	+	0.330073i	$0.892927\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.49354	−0.00801780	−0.00400890	−	0.999992i	$-0.501276\pi$
$311$	−2.49354	−0.00801780	−0.00400890	+	0.999992i	$0.501276\pi$
$312$	0	0
$313$	−433.480	−1.38492	−0.692460	−	0.721456i	$-0.743473\pi$
$313$	−433.480	−1.38492	−0.692460	+	0.721456i	$0.743473\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−247.195	−0.779795	−0.389897	−	0.920858i	$-0.627490\pi$
$317$	−247.195	−0.779795	−0.389897	+	0.920858i	$0.627490\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−820.996	−2.54178
$324$	0	0
$325$	685.611i	2.10957i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 78.7151i	− 0.239255i
$330$	0	0
$331$	−581.456	−1.75666	−0.878332	−	0.478050i	$-0.841344\pi$
$331$	−581.456	−1.75666	−0.878332	+	0.478050i	$0.841344\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	686.028	2.04784
$336$	0	0
$337$	18.3203i	0.0543629i	0.999631	+	0.0271814i	$0.00865318\pi$
$337$	18.3203i	0.0543629i	−0.999631	+	0.0271814i	$0.991347\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 280.048i	− 0.816465i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	283.283i	0.816379i	0.912897	+	0.408189i	$0.133840\pi$
$347$	283.283i	0.816379i	−0.912897	+	0.408189i	$0.866160\pi$
$348$	0	0
$349$	− 398.818i	− 1.14274i	−0.820691	−	0.571372i	$-0.806411\pi$
$349$	− 398.818i	− 1.14274i	0.820691	−	0.571372i	$-0.193589\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	395.028	1.11906	0.559530	−	0.828810i	$-0.310982\pi$
$353$	395.028	1.11906	0.559530	+	0.828810i	$0.310982\pi$
$354$	0	0
$355$	−20.6077	−0.0580498
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 148.680i	− 0.414151i	−0.978325	−	0.207075i	$-0.933605\pi$
$359$	− 148.680i	− 0.414151i	0.978325	−	0.207075i	$-0.0663946\pi$
$360$	0	0
$361$	−264.358	−0.732293
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 167.965i	− 0.460177i
$366$	0	0
$367$	−483.684	−1.31794	−0.658970	−	0.752169i	$-0.729008\pi$
$367$	−483.684	−1.31794	−0.658970	+	0.752169i	$0.729008\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 134.248i	− 0.361854i
$372$	0	0
$373$	− 170.877i	− 0.458115i	−0.973413	−	0.229057i	$-0.926436\pi$
$373$	− 170.877i	− 0.458115i	0.973413	−	0.229057i	$-0.0735644\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	80.9341	0.214679
$378$	0	0
$379$	−435.307	−1.14857	−0.574283	−	0.818657i	$-0.694719\pi$
$379$	−435.307	−1.14857	−0.574283	+	0.818657i	$0.694719\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−158.942	−0.414991	−0.207496	−	0.978236i	$-0.566531\pi$
$383$	−158.942	−0.414991	−0.207496	+	0.978236i	$0.566531\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	443.319	1.13964	0.569819	−	0.821770i	$-0.307013\pi$
$389$	443.319	1.13964	0.569819	+	0.821770i	$0.307013\pi$
$390$	0	0
$391$	− 670.791i	− 1.71558i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 906.910i	− 2.29598i
$396$	0	0
$397$	−145.331	−0.366074	−0.183037	−	0.983106i	$-0.558593\pi$
$397$	−145.331	−0.366074	−0.183037	+	0.983106i	$0.558593\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	620.853	1.54826	0.774131	−	0.633026i	$-0.218187\pi$
$401$	620.853	1.54826	0.774131	+	0.633026i	$0.218187\pi$
$402$	0	0
$403$	56.5426i	0.140304i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	571.379i	1.39701i	0.715603	+	0.698507i	$0.246152\pi$
$409$	571.379i	1.39701i	−0.715603	+	0.698507i	$0.753848\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 11.9734i	− 0.0289912i
$414$	0	0
$415$	281.621i	0.678604i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−271.765	−0.648604	−0.324302	−	0.945954i	$-0.605129\pi$
$419$	−271.765	−0.648604	−0.324302	+	0.945954i	$0.605129\pi$
$420$	0	0
$421$	−434.059	−1.03102	−0.515510	−	0.856884i	$-0.672397\pi$
$421$	−434.059	−1.03102	−0.515510	+	0.856884i	$0.672397\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2100.92i	4.94334i
$426$	0	0
$427$	−115.945	−0.271535
$428$	0	0
$429$	0	0
$430$	0	0
$431$	425.188i	0.986516i	0.869883	+	0.493258i	$0.164194\pi$
$431$	425.188i	0.986516i	−0.869883	+	0.493258i	$0.835806\pi$
$432$	0	0
$433$	−230.771	−0.532958	−0.266479	−	0.963841i	$-0.585860\pi$
$433$	−230.771	−0.532958	−0.266479	+	0.963841i	$0.585860\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 510.946i	− 1.16921i
$438$	0	0
$439$	115.222i	0.262465i	0.991352	+	0.131233i	$0.0418934\pi$
$439$	115.222i	0.262465i	−0.991352	+	0.131233i	$0.958107\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	158.779	0.358416	0.179208	−	0.983811i	$-0.442646\pi$
$443$	158.779	0.358416	0.179208	+	0.983811i	$0.442646\pi$
$444$	0	0
$445$	938.890	2.10986
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−407.707	−0.908033	−0.454016	−	0.890993i	$-0.650009\pi$
$449$	−407.707	−0.908033	−0.454016	+	0.890993i	$0.650009\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−322.248	−0.708238
$456$	0	0
$457$	797.114i	1.74423i	0.489300	+	0.872116i	$0.337252\pi$
$457$	797.114i	1.74423i	−0.489300	+	0.872116i	$0.662748\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 380.724i	− 0.825866i	−0.910761	−	0.412933i	$-0.864504\pi$
$461$	− 380.724i	− 0.825866i	0.910761	−	0.412933i	$-0.135496\pi$
$462$	0	0
$463$	702.102	1.51642	0.758209	−	0.652011i	$-0.226075\pi$
$463$	702.102	1.51642	0.758209	+	0.652011i	$0.226075\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.8377	0.0638923	0.0319461	−	0.999490i	$-0.489829\pi$
$467$	29.8377	0.0638923	0.0319461	+	0.999490i	$0.489829\pi$
$468$	0	0
$469$	231.863i	0.494377i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1600.28i	3.36902i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 346.379i	− 0.723128i	−0.932347	−	0.361564i	$-0.882243\pi$
$479$	− 346.379i	− 0.723128i	0.932347	−	0.361564i	$-0.117757\pi$
$480$	0	0
$481$	520.691i	1.08252i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−697.806	−1.43878
$486$	0	0
$487$	676.614	1.38935	0.694675	−	0.719324i	$-0.255548\pi$
$487$	676.614	1.38935	0.694675	+	0.719324i	$0.255548\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	147.029i	0.299448i	0.988728	+	0.149724i	$0.0478385\pi$
$491$	147.029i	0.299448i	−0.988728	+	0.149724i	$0.952162\pi$
$492$	0	0
$493$	248.006	0.503056
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.96496i	− 0.0140140i
$498$	0	0
$499$	−839.154	−1.68167	−0.840836	−	0.541290i	$-0.817936\pi$
$499$	−839.154	−1.68167	−0.840836	+	0.541290i	$0.817936\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 296.831i	− 0.590121i	−0.955479	−	0.295061i	$-0.904660\pi$
$503$	− 296.831i	− 0.590121i	0.955479	−	0.295061i	$-0.0953398\pi$
$504$	0	0
$505$	− 1450.91i	− 2.87309i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	421.228	0.827560	0.413780	−	0.910377i	$-0.364208\pi$
$509$	421.228	0.827560	0.413780	+	0.910377i	$0.364208\pi$
$510$	0	0
$511$	56.7685	0.111093
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−456.787	−0.886965
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	169.070	0.324510	0.162255	−	0.986749i	$-0.448123\pi$
$521$	169.070	0.324510	0.162255	+	0.986749i	$0.448123\pi$
$522$	0	0
$523$	− 399.217i	− 0.763322i	−0.924302	−	0.381661i	$-0.875352\pi$
$523$	− 399.217i	− 0.763322i	0.924302	−	0.381661i	$-0.124648\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	173.264i	0.328774i
$528$	0	0
$529$	−111.533	−0.210838
$530$	0	0
$531$	0	0
$532$	0	0
$533$	328.559	0.616433
$534$	0	0
$535$	− 1846.03i	− 3.45051i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 57.8425i	− 0.106918i	−0.998570	−	0.0534589i	$-0.982975\pi$
$541$	− 57.8425i	− 0.106918i	0.998570	−	0.0534589i	$-0.0170246\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	242.537i	0.445023i
$546$	0	0
$547$	64.5030i	0.117921i	0.998260	+	0.0589607i	$0.0187786\pi$
$547$	64.5030i	0.117921i	−0.998260	+	0.0589607i	$0.981221\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	188.908	0.342846
$552$	0	0
$553$	306.517	0.554279
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 956.474i	− 1.71719i	−0.512655	−	0.858595i	$-0.671338\pi$
$557$	− 956.474i	− 1.71719i	0.512655	−	0.858595i	$-0.328662\pi$
$558$	0	0
$559$	550.115	0.984105
$560$	0	0
$561$	0	0
$562$	0	0
$563$	819.742i	1.45603i	0.685564	+	0.728013i	$0.259556\pi$
$563$	819.742i	1.45603i	−0.685564	+	0.728013i	$0.740444\pi$
$564$	0	0
$565$	720.132	1.27457
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 980.298i	− 1.72284i	−0.507890	−	0.861422i	$-0.669574\pi$
$569$	− 980.298i	− 1.72284i	0.507890	−	0.861422i	$-0.330426\pi$
$570$	0	0
$571$	− 579.244i	− 1.01444i	−0.861817	−	0.507219i	$-0.830674\pi$
$571$	− 579.244i	− 1.01444i	0.861817	−	0.507219i	$-0.169326\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1307.51	−2.27392
$576$	0	0
$577$	−645.740	−1.11913	−0.559567	−	0.828785i	$-0.689033\pi$
$577$	−645.740	−1.11913	−0.559567	+	0.828785i	$0.689033\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−95.1819	−0.163824
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	372.950	0.635349	0.317674	−	0.948200i	$-0.397098\pi$
$587$	372.950	0.635349	0.317674	+	0.948200i	$0.397098\pi$
$588$	0	0
$589$	131.976i	0.224068i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 199.689i	− 0.336743i	−0.985724	−	0.168372i	$-0.946149\pi$
$593$	− 199.689i	− 0.336743i	0.985724	−	0.168372i	$-0.0538509\pi$
$594$	0	0
$595$	−987.465	−1.65961
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−356.856	−0.595753	−0.297877	−	0.954604i	$-0.596278\pi$
$599$	−356.856	−0.595753	−0.297877	+	0.954604i	$0.596278\pi$
$600$	0	0
$601$	− 549.397i	− 0.914139i	−0.889431	−	0.457069i	$-0.848899\pi$
$601$	− 549.397i	− 0.914139i	0.889431	−	0.457069i	$-0.151101\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	625.736i	1.03087i	0.856930	+	0.515433i	$0.172369\pi$
$607$	625.736i	1.03087i	−0.856930	+	0.515433i	$0.827631\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	264.506i	0.432907i
$612$	0	0
$613$	− 707.024i	− 1.15338i	−0.816962	−	0.576691i	$-0.804344\pi$
$613$	− 707.024i	− 1.15338i	0.816962	−	0.576691i	$-0.195656\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−444.514	−0.720445	−0.360222	−	0.932866i	$-0.617299\pi$
$617$	−444.514	−0.720445	−0.360222	+	0.932866i	$0.617299\pi$
$618$	0	0
$619$	−413.690	−0.668320	−0.334160	−	0.942516i	$-0.608453\pi$
$619$	−413.690	−0.668320	−0.334160	+	0.942516i	$0.608453\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	317.325i	0.509350i
$624$	0	0
$625$	1870.28	2.99245
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1595.55i	2.53665i
$630$	0	0
$631$	−57.8885	−0.0917409	−0.0458705	−	0.998947i	$-0.514606\pi$
$631$	−57.8885	−0.0917409	−0.0458705	+	0.998947i	$0.514606\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	528.937i	0.832972i
$636$	0	0
$637$	416.065i	0.653163i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1075.85	1.67840	0.839200	−	0.543823i	$-0.183024\pi$
$641$	1075.85	1.67840	0.839200	+	0.543823i	$0.183024\pi$
$642$	0	0
$643$	507.262	0.788899	0.394450	−	0.918918i	$-0.370935\pi$
$643$	507.262	0.788899	0.394450	+	0.918918i	$0.370935\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	917.342	1.41784	0.708920	−	0.705289i	$-0.249183\pi$
$647$	917.342	1.41784	0.708920	+	0.705289i	$0.249183\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−627.147	−0.960409	−0.480205	−	0.877157i	$-0.659438\pi$
$653$	−627.147	−0.960409	−0.480205	+	0.877157i	$0.659438\pi$
$654$	0	0
$655$	794.349i	1.21275i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 297.301i	− 0.451140i	−0.974227	−	0.225570i	$-0.927576\pi$
$659$	− 297.301i	− 0.451140i	0.974227	−	0.225570i	$-0.0724245\pi$
$660$	0	0
$661$	−470.110	−0.711210	−0.355605	−	0.934636i	$-0.615725\pi$
$661$	−470.110	−0.711210	−0.355605	+	0.934636i	$0.615725\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−752.159	−1.13107
$666$	0	0
$667$	154.347i	0.231404i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 415.011i	− 0.616659i	−0.951280	−	0.308329i	$-0.900230\pi$
$673$	− 415.011i	− 0.616659i	0.951280	−	0.308329i	$-0.0997699\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 755.920i	− 1.11657i	−0.829648	−	0.558286i	$-0.811459\pi$
$677$	− 755.920i	− 1.11657i	0.829648	−	0.558286i	$-0.188541\pi$
$678$	0	0
$679$	− 235.844i	− 0.347340i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−293.903	−0.430311	−0.215156	−	0.976580i	$-0.569026\pi$
$683$	−293.903	−0.430311	−0.215156	+	0.976580i	$0.569026\pi$
$684$	0	0
$685$	490.096	0.715469
$686$	0	0
$687$	0	0
$688$	0	0
$689$	451.113i	0.654736i
$690$	0	0
$691$	−481.482	−0.696791	−0.348395	−	0.937348i	$-0.613273\pi$
$691$	−481.482	−0.696791	−0.348395	+	0.937348i	$0.613273\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1763.74i	− 2.53776i
$696$	0	0
$697$	1006.80	1.44448
$698$	0	0
$699$	0	0
$700$	0	0
$701$	967.437i	1.38008i	0.723770	+	0.690041i	$0.242407\pi$
$701$	967.437i	1.38008i	−0.723770	+	0.690041i	$0.757593\pi$
$702$	0	0
$703$	1215.34i	1.72880i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	490.376	0.693601
$708$	0	0
$709$	−281.105	−0.396480	−0.198240	−	0.980153i	$-0.563523\pi$
$709$	−281.105	−0.396480	−0.198240	+	0.980153i	$0.563523\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−107.831	−0.151235
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	488.441	0.679333	0.339667	−	0.940546i	$-0.389686\pi$
$719$	488.441	0.679333	0.339667	+	0.940546i	$0.389686\pi$
$720$	0	0
$721$	− 154.384i	− 0.214125i
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 483.414i	− 0.666777i
$726$	0	0
$727$	−450.038	−0.619035	−0.309517	−	0.950894i	$-0.600167\pi$
$727$	−450.038	−0.619035	−0.309517	+	0.950894i	$0.600167\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1685.72	2.30604
$732$	0	0
$733$	677.704i	0.924562i	0.886733	+	0.462281i	$0.152969\pi$
$733$	677.704i	0.924562i	−0.886733	+	0.462281i	$0.847031\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	398.745i	0.539573i	0.962920	+	0.269787i	$0.0869532\pi$
$739$	398.745i	0.539573i	−0.962920	+	0.269787i	$0.913047\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1168.88i	1.57319i	0.617471	+	0.786594i	$0.288157\pi$
$743$	1168.88i	1.57319i	−0.617471	+	0.786594i	$0.711843\pi$
$744$	0	0
$745$	− 684.179i	− 0.918361i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	623.918	0.833001
$750$	0	0
$751$	−428.785	−0.570952	−0.285476	−	0.958386i	$-0.592152\pi$
$751$	−428.785	−0.570952	−0.285476	+	0.958386i	$0.592152\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 2014.75i	− 2.66855i
$756$	0	0
$757$	−1322.77	−1.74738	−0.873691	−	0.486482i	$-0.838280\pi$
$757$	−1322.77	−1.74738	−0.873691	+	0.486482i	$0.838280\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 663.677i	− 0.872112i	−0.899919	−	0.436056i	$-0.856375\pi$
$761$	− 663.677i	− 0.872112i	0.899919	−	0.436056i	$-0.143625\pi$
$762$	0	0
$763$	−81.9725	−0.107434
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40.2341i	0.0524564i
$768$	0	0
$769$	− 215.995i	− 0.280878i	−0.990089	−	0.140439i	$-0.955149\pi$
$769$	− 215.995i	− 0.280878i	0.990089	−	0.140439i	$-0.0448514\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	206.459	0.267088	0.133544	−	0.991043i	$-0.457364\pi$
$773$	206.459	0.267088	0.133544	+	0.991043i	$0.457364\pi$
$774$	0	0
$775$	337.725	0.435774
$776$	0	0
$777$	0	0
$778$	0	0
$779$	766.888	0.984452
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	107.456	0.136887
$786$	0	0
$787$	947.296i	1.20368i	0.798617	+	0.601840i	$0.205566\pi$
$787$	947.296i	1.20368i	−0.798617	+	0.601840i	$0.794434\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	243.389i	0.307698i
$792$	0	0
$793$	389.611	0.491313
$794$	0	0
$795$	0	0
$796$	0	0
$797$	845.877	1.06133	0.530663	−	0.847583i	$-0.321943\pi$
$797$	845.877	1.06133	0.530663	+	0.847583i	$0.321943\pi$
$798$	0	0
$799$	810.526i	1.01443i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 614.549i	− 0.763415i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 754.562i	− 0.932710i	−0.884598	−	0.466355i	$-0.845567\pi$
$809$	− 754.562i	− 0.932710i	0.884598	−	0.466355i	$-0.154433\pi$
$810$	0	0
$811$	407.368i	0.502303i	0.967948	+	0.251152i	$0.0808093\pi$
$811$	407.368i	0.502303i	−0.967948	+	0.251152i	$0.919191\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1140.20	1.39901
$816$	0	0
$817$	1284.02	1.57163
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1280.50i	1.55968i	0.625980	+	0.779839i	$0.284699\pi$
$821$	1280.50i	1.55968i	−0.625980	+	0.779839i	$0.715301\pi$
$822$	0	0
$823$	−988.194	−1.20072	−0.600361	−	0.799729i	$-0.704976\pi$
$823$	−988.194	−1.20072	−0.600361	+	0.799729i	$0.704976\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	586.967i	0.709755i	0.934913	+	0.354877i	$0.115477\pi$
$827$	586.967i	0.709755i	−0.934913	+	0.354877i	$0.884523\pi$
$828$	0	0
$829$	−517.567	−0.624326	−0.312163	−	0.950028i	$-0.601054\pi$
$829$	−517.567	−0.624326	−0.312163	+	0.950028i	$0.601054\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1274.95i	1.53055i
$834$	0	0
$835$	− 1934.38i	− 2.31663i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−190.017	−0.226481	−0.113240	−	0.993568i	$-0.536123\pi$
$839$	−190.017	−0.226481	−0.113240	+	0.993568i	$0.536123\pi$
$840$	0	0
$841$	783.935	0.932146
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−511.431	−0.605243
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−992.992	−1.16685
$852$	0	0
$853$	− 1450.28i	− 1.70021i	−0.526612	−	0.850106i	$-0.676538\pi$
$853$	− 1450.28i	− 1.70021i	0.526612	−	0.850106i	$-0.323462\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	468.523i	0.546701i	0.961914	+	0.273351i	$0.0881319\pi$
$857$	468.523i	0.546701i	−0.961914	+	0.273351i	$0.911868\pi$
$858$	0	0
$859$	−836.529	−0.973841	−0.486920	−	0.873446i	$-0.661880\pi$
$859$	−836.529	−0.973841	−0.486920	+	0.873446i	$0.661880\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1614.22	−1.87048	−0.935238	−	0.354019i	$-0.884815\pi$
$863$	−1614.22	−1.87048	−0.935238	+	0.354019i	$0.884815\pi$
$864$	0	0
$865$	774.552i	0.895436i
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 779.128i	− 0.894521i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1172.82i	1.34037i
$876$	0	0
$877$	803.570i	0.916271i	0.888882	+	0.458135i	$0.151483\pi$
$877$	803.570i	0.916271i	−0.888882	+	0.458135i	$0.848517\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−991.644	−1.12559	−0.562794	−	0.826597i	$-0.690274\pi$
$881$	−991.644	−1.12559	−0.562794	+	0.826597i	$0.690274\pi$
$882$	0	0
$883$	−1075.09	−1.21754	−0.608770	−	0.793347i	$-0.708337\pi$
$883$	−1075.09	−1.21754	−0.608770	+	0.793347i	$0.708337\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 384.505i	− 0.433489i	−0.976228	−	0.216744i	$-0.930456\pi$
$887$	− 384.505i	− 0.433489i	0.976228	−	0.216744i	$-0.0695438\pi$
$888$	0	0
$889$	−178.770	−0.201091
$890$	0	0
$891$	0	0
$892$	0	0
$893$	617.383i	0.691358i
$894$	0	0
$895$	1606.30	1.79475
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 39.8673i	− 0.0443463i
$900$	0	0
$901$	1382.35i	1.53423i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2000.26	−2.21023
$906$	0	0
$907$	−758.683	−0.836475	−0.418238	−	0.908338i	$-0.637352\pi$
$907$	−758.683	−0.836475	−0.418238	+	0.908338i	$0.637352\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−968.642	−1.06327	−0.531637	−	0.846972i	$-0.678423\pi$
$911$	−968.642	−1.06327	−0.531637	+	0.846972i	$0.678423\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−268.473	−0.292773
$918$	0	0
$919$	− 1046.68i	− 1.13893i	−0.822015	−	0.569466i	$-0.807150\pi$
$919$	− 1046.68i	− 1.13893i	0.822015	−	0.569466i	$-0.192850\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	23.4043i	0.0253568i
$924$	0	0
$925$	3110.05	3.36222
$926$	0	0
$927$	0	0
$928$	0	0
$929$	92.4683	0.0995353	0.0497677	−	0.998761i	$-0.484152\pi$
$929$	92.4683	0.0995353	0.0497677	+	0.998761i	$0.484152\pi$
$930$	0	0
$931$	971.137i	1.04311i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 941.367i	− 1.00466i	−0.864676	−	0.502331i	$-0.832476\pi$
$937$	− 941.367i	− 1.00466i	0.864676	−	0.502331i	$-0.167524\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 495.866i	− 0.526957i	−0.964665	−	0.263478i	$-0.915130\pi$
$941$	− 495.866i	− 0.526957i	0.964665	−	0.263478i	$-0.0848698\pi$
$942$	0	0
$943$	626.583i	0.664457i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−865.407	−0.913841	−0.456920	−	0.889508i	$-0.651048\pi$
$947$	−865.407	−0.913841	−0.456920	+	0.889508i	$0.651048\pi$
$948$	0	0
$949$	−190.759	−0.201011
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1357.22i	− 1.42415i	−0.702102	−	0.712076i	$-0.747755\pi$
$953$	− 1357.22i	− 1.42415i	0.702102	−	0.712076i	$-0.252245\pi$
$954$	0	0
$955$	−523.580	−0.548251
$956$	0	0
$957$	0	0
$958$	0	0
$959$	165.642i	0.172724i
$960$	0	0
$961$	−933.148	−0.971017
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1578.06i	− 1.63530i
$966$	0	0
$967$	1423.32i	1.47190i	0.677038	+	0.735948i	$0.263263\pi$
$967$	1423.32i	1.47190i	−0.677038	+	0.735948i	$0.736737\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1667.99	−1.71781	−0.858903	−	0.512139i	$-0.828853\pi$
$971$	−1667.99	−1.71781	−0.858903	+	0.512139i	$0.828853\pi$
$972$	0	0
$973$	596.108	0.612650
$974$	0	0
$975$	0	0
$976$	0	0
$977$	822.327	0.841685	0.420843	−	0.907134i	$-0.361734\pi$
$977$	822.327	0.841685	0.420843	+	0.907134i	$0.361734\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	685.082	0.696930	0.348465	−	0.937322i	$-0.386703\pi$
$983$	685.082	0.696930	0.348465	+	0.937322i	$0.386703\pi$
$984$	0	0
$985$	− 682.650i	− 0.693045i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1049.11i	1.06077i
$990$	0	0
$991$	1425.94	1.43888	0.719442	−	0.694552i	$-0.244397\pi$
$991$	1425.94	1.43888	0.719442	+	0.694552i	$0.244397\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2996.60	−3.01165
$996$	0	0
$997$	− 860.170i	− 0.862759i	−0.902171	−	0.431379i	$-0.858027\pi$
$997$	− 860.170i	− 0.862759i	0.902171	−	0.431379i	$-0.141973\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4356.3.f.m.1693.1		16
3.2	odd	2		1452.3.f.d.241.7		16
11.3	even	5		396.3.t.c.145.1		16
11.7	odd	10		396.3.t.c.325.1		16
11.10	odd	2	inner	4356.3.f.m.1693.2		16
33.14	odd	10		132.3.l.a.13.4	✓	16
33.29	even	10		132.3.l.a.61.4	yes	16
33.32	even	2		1452.3.f.d.241.8		16
132.47	even	10		528.3.bf.a.145.2		16
132.95	odd	10		528.3.bf.a.193.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.3.l.a.13.4	✓	16	33.14	odd	10
132.3.l.a.61.4	yes	16	33.29	even	10
396.3.t.c.145.1		16	11.3	even	5
396.3.t.c.325.1		16	11.7	odd	10
528.3.bf.a.145.2		16	132.47	even	10
528.3.bf.a.193.2		16	132.95	odd	10
1452.3.f.d.241.7		16	3.2	odd	2
1452.3.f.d.241.8		16	33.32	even	2
4356.3.f.m.1693.1		16	1.1	even	1	trivial
4356.3.f.m.1693.2		16	11.10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4356.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.43361 q^{5} -3.18836i q^{7} +10.7138i q^{13} +32.8304i q^{17} +25.0072i q^{19} -20.4320 q^{23} +63.9930 q^{25} -7.55416i q^{29} +5.27753 q^{31} +30.0778i q^{35} +48.5998 q^{37} -30.6668i q^{41} -51.3462i q^{43} +24.6883 q^{47} +38.8344 q^{49} +42.1056 q^{53} +3.75534 q^{59} -36.3652i q^{61} -101.070i q^{65} -72.7217 q^{67} +2.18450 q^{71} +17.8049i q^{73} +96.1361i q^{79} -29.8529i q^{83} -309.709i q^{85} -99.5260 q^{89} +34.1596 q^{91} -235.908i q^{95} +73.9702 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{5} + 92 q^{23} + 184 q^{25} - 104 q^{31} + 144 q^{37} + 80 q^{47} - 172 q^{49} + 32 q^{53} - 4 q^{59} - 316 q^{67} - 132 q^{71} - 12 q^{89} - 656 q^{91} - 296 q^{97}+O(q^{100}) \) 16 * q + 4 * q^5 + 92 * q^23 + 184 * q^25 - 104 * q^31 + 144 * q^37 + 80 * q^47 - 172 * q^49 + 32 * q^53 - 4 * q^59 - 316 * q^67 - 132 * q^71 - 12 * q^89 - 656 * q^91 - 296 * q^97

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