LMFDB - Embedded newform 1568.3.c.d.97.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1568,3,Mod(97,1568)] 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("1568.97"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-40,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,96, 0,0,0,0,0,0,0,192,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,144] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(53)] == traces)

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

Self dual:	no
Analytic conductor:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.132892327936.16
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 16x^{6} + 84x^{4} + 152x^{2} + 49 $ x^8 + 16x^6 + 84x^4 + 152*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.7
Root			$-2.51825i$ of defining polynomial
Character	$\chi$	$=$	1568.97
Dual form			1568.3.c.d.97.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.27114i q^{3} -4.77791i q^{5} -9.24264 q^{9} +9.80697 q^{11} -4.14386i q^{13} +20.4071 q^{15} +31.8602i q^{17} -30.9343i q^{19} -13.8691 q^{23} +2.17157 q^{25} -1.03635i q^{27} -16.2843 q^{29} -51.2537i q^{31} +41.8869i q^{33} +9.85786 q^{37} +17.6990 q^{39} -13.5684i q^{41} +29.4209 q^{43} +44.1605i q^{45} -76.8805i q^{47} -136.080 q^{51} +94.3675 q^{53} -46.8568i q^{55} +132.125 q^{57} +52.1643i q^{59} +13.3283i q^{61} -19.7990 q^{65} +106.194 q^{67} -59.2371i q^{69} +122.443 q^{71} -26.1857i q^{73} +9.27509i q^{75} +114.318 q^{79} -78.7574 q^{81} +26.5375i q^{83} +152.225 q^{85} -69.5524i q^{87} +93.8006i q^{89} +218.912 q^{93} -147.801 q^{95} -130.326i q^{97} -90.6423 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 40 q^{9} + 40 q^{25} + 96 q^{29} + 192 q^{37} + 144 q^{53} + 480 q^{57} - 664 q^{81} + 720 q^{85} + 1344 q^{93}+O(q^{100}) $ 8 * q - 40 * q^9 + 40 * q^25 + 96 * q^29 + 192 * q^37 + 144 * q^53 + 480 * q^57 - 664 * q^81 + 720 * q^85 + 1344 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	4.27114i	1.42371i	0.702325	+	0.711857i	$0.252145\pi$
$3$	4.27114i	1.42371i	−0.702325	+	0.711857i	$0.747855\pi$
$4$	0	0
$5$	− 4.77791i	− 0.955582i	−0.878474	−	0.477791i	$-0.841438\pi$
$5$	− 4.77791i	− 0.955582i	0.878474	−	0.477791i	$-0.158562\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−9.24264	−1.02696
$10$	0	0
$11$	9.80697	0.891543	0.445771	−	0.895147i	$-0.352929\pi$
$11$	9.80697	0.891543	0.445771	+	0.895147i	$0.352929\pi$
$12$	0	0
$13$	− 4.14386i	− 0.318758i	−0.987217	−	0.159379i	$-0.949051\pi$
$13$	− 4.14386i	− 0.318758i	0.987217	−	0.159379i	$-0.0509493\pi$
$14$	0	0
$15$	20.4071	1.36048
$16$	0	0
$17$	31.8602i	1.87413i	0.349152	+	0.937066i	$0.386470\pi$
$17$	31.8602i	1.87413i	−0.349152	+	0.937066i	$0.613530\pi$
$18$	0	0
$19$	− 30.9343i	− 1.62812i	−0.580779	−	0.814061i	$-0.697252\pi$
$19$	− 30.9343i	− 1.62812i	0.580779	−	0.814061i	$-0.302748\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−13.8691	−0.603006	−0.301503	−	0.953465i	$-0.597488\pi$
$23$	−13.8691	−0.603006	−0.301503	+	0.953465i	$0.597488\pi$
$24$	0	0
$25$	2.17157	0.0868629
$26$	0	0
$27$	− 1.03635i	− 0.0383834i
$28$	0	0
$29$	−16.2843	−0.561527	−0.280763	−	0.959777i	$-0.590588\pi$
$29$	−16.2843	−0.561527	−0.280763	+	0.959777i	$0.590588\pi$
$30$	0	0
$31$	− 51.2537i	− 1.65334i	−0.562684	−	0.826672i	$-0.690231\pi$
$31$	− 51.2537i	− 1.65334i	0.562684	−	0.826672i	$-0.309769\pi$
$32$	0	0
$33$	41.8869i	1.26930i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.85786	0.266429	0.133214	−	0.991087i	$-0.457470\pi$
$37$	9.85786	0.266429	0.133214	+	0.991087i	$0.457470\pi$
$38$	0	0
$39$	17.6990	0.453821
$40$	0	0
$41$	− 13.5684i	− 0.330936i	−0.986215	−	0.165468i	$-0.947087\pi$
$41$	− 13.5684i	− 0.330936i	0.986215	−	0.165468i	$-0.0529134\pi$
$42$	0	0
$43$	29.4209	0.684207	0.342104	−	0.939662i	$-0.388861\pi$
$43$	29.4209	0.684207	0.342104	+	0.939662i	$0.388861\pi$
$44$	0	0
$45$	44.1605i	0.981345i
$46$	0	0
$47$	− 76.8805i	− 1.63576i	−0.575392	−	0.817878i	$-0.695150\pi$
$47$	− 76.8805i	− 1.63576i	0.575392	−	0.817878i	$-0.304850\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−136.080	−2.66823
$52$	0	0
$53$	94.3675	1.78052	0.890260	−	0.455453i	$-0.150523\pi$
$53$	94.3675	1.78052	0.890260	+	0.455453i	$0.150523\pi$
$54$	0	0
$55$	− 46.8568i	− 0.851942i
$56$	0	0
$57$	132.125	2.31798
$58$	0	0
$59$	52.1643i	0.884141i	0.896980	+	0.442070i	$0.145756\pi$
$59$	52.1643i	0.884141i	−0.896980	+	0.442070i	$0.854244\pi$
$60$	0	0
$61$	13.3283i	0.218496i	0.994015	+	0.109248i	$0.0348443\pi$
$61$	13.3283i	0.218496i	−0.994015	+	0.109248i	$0.965156\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−19.7990	−0.304600
$66$	0	0
$67$	106.194	1.58499	0.792493	−	0.609881i	$-0.208783\pi$
$67$	106.194	1.58499	0.792493	+	0.609881i	$0.208783\pi$
$68$	0	0
$69$	− 59.2371i	− 0.858508i
$70$	0	0
$71$	122.443	1.72455	0.862273	−	0.506444i	$-0.169040\pi$
$71$	122.443	1.72455	0.862273	+	0.506444i	$0.169040\pi$
$72$	0	0
$73$	− 26.1857i	− 0.358708i	−0.983785	−	0.179354i	$-0.942599\pi$
$73$	− 26.1857i	− 0.358708i	0.983785	−	0.179354i	$-0.0574007\pi$
$74$	0	0
$75$	9.27509i	0.123668i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	114.318	1.44707	0.723534	−	0.690289i	$-0.242516\pi$
$79$	114.318	1.44707	0.723534	+	0.690289i	$0.242516\pi$
$80$	0	0
$81$	−78.7574	−0.972313
$82$	0	0
$83$	26.5375i	0.319728i	0.987139	+	0.159864i	$0.0511056\pi$
$83$	26.5375i	0.319728i	−0.987139	+	0.159864i	$0.948894\pi$
$84$	0	0
$85$	152.225	1.79089
$86$	0	0
$87$	− 69.5524i	− 0.799453i
$88$	0	0
$89$	93.8006i	1.05394i	0.849884	+	0.526969i	$0.176672\pi$
$89$	93.8006i	1.05394i	−0.849884	+	0.526969i	$0.823328\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	218.912	2.35389
$94$	0	0
$95$	−147.801	−1.55581
$96$	0	0
$97$	− 130.326i	− 1.34357i	−0.740747	−	0.671784i	$-0.765528\pi$
$97$	− 130.326i	− 1.34357i	0.740747	−	0.671784i	$-0.234472\pi$
$98$	0	0
$99$	−90.6423	−0.915579
$100$	0	0
$101$	146.476i	1.45025i	0.688615	+	0.725127i	$0.258219\pi$
$101$	146.476i	1.45025i	−0.688615	+	0.725127i	$0.741781\pi$
$102$	0	0
$103$	− 36.2418i	− 0.351862i	−0.984402	−	0.175931i	$-0.943706\pi$
$103$	− 36.2418i	− 0.351862i	0.984402	−	0.175931i	$-0.0562936\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	53.0970	0.496234	0.248117	−	0.968730i	$-0.420188\pi$
$107$	53.0970	0.496234	0.248117	+	0.968730i	$0.420188\pi$
$108$	0	0
$109$	−131.230	−1.20395	−0.601975	−	0.798515i	$-0.705619\pi$
$109$	−131.230	−1.20395	−0.601975	+	0.798515i	$0.705619\pi$
$110$	0	0
$111$	42.1043i	0.379318i
$112$	0	0
$113$	0.124892	0.00110524	0.000552618	−	1.00000i	$-0.499824\pi$
$113$	0.124892	0.00110524	0.000552618		1.00000i	$0.499824\pi$
$114$	0	0
$115$	66.2655i	0.576222i
$116$	0	0
$117$	38.3002i	0.327352i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−24.8234	−0.205152
$122$	0	0
$123$	57.9524	0.471158
$124$	0	0
$125$	− 129.823i	− 1.03859i
$126$	0	0
$127$	−128.188	−1.00935	−0.504675	−	0.863309i	$-0.668388\pi$
$127$	−128.188	−1.00935	−0.504675	+	0.863309i	$0.668388\pi$
$128$	0	0
$129$	125.661i	0.974115i
$130$	0	0
$131$	68.4640i	0.522626i	0.965254	+	0.261313i	$0.0841554\pi$
$131$	68.4640i	0.522626i	−0.965254	+	0.261313i	$0.915845\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.95160	−0.0366785
$136$	0	0
$137$	27.5563	0.201141	0.100571	−	0.994930i	$-0.467933\pi$
$137$	27.5563	0.201141	0.100571	+	0.994930i	$0.467933\pi$
$138$	0	0
$139$	95.9121i	0.690015i	0.938600	+	0.345007i	$0.112124\pi$
$139$	95.9121i	0.690015i	−0.938600	+	0.345007i	$0.887876\pi$
$140$	0	0
$141$	328.368	2.32885
$142$	0	0
$143$	− 40.6387i	− 0.284187i
$144$	0	0
$145$	77.8048i	0.536585i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	147.338	0.988846	0.494423	−	0.869221i	$-0.335379\pi$
$149$	147.338	0.988846	0.494423	+	0.869221i	$0.335379\pi$
$150$	0	0
$151$	−123.428	−0.817407	−0.408703	−	0.912667i	$-0.634019\pi$
$151$	−123.428	−0.817407	−0.408703	+	0.912667i	$0.634019\pi$
$152$	0	0
$153$	− 294.473i	− 1.92466i
$154$	0	0
$155$	−244.886	−1.57991
$156$	0	0
$157$	149.229i	0.950506i	0.879849	+	0.475253i	$0.157644\pi$
$157$	149.229i	0.950506i	−0.879849	+	0.475253i	$0.842356\pi$
$158$	0	0
$159$	403.057i	2.53495i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	34.1800	0.209694	0.104847	−	0.994488i	$-0.466565\pi$
$163$	34.1800	0.209694	0.104847	+	0.994488i	$0.466565\pi$
$164$	0	0
$165$	200.132	1.21292
$166$	0	0
$167$	− 136.928i	− 0.819928i	−0.912102	−	0.409964i	$-0.865541\pi$
$167$	− 136.928i	− 0.819928i	0.912102	−	0.409964i	$-0.134459\pi$
$168$	0	0
$169$	151.828	0.898393
$170$	0	0
$171$	285.915i	1.67202i
$172$	0	0
$173$	− 140.788i	− 0.813803i	−0.913472	−	0.406901i	$-0.866609\pi$
$173$	− 140.788i	− 0.813803i	0.913472	−	0.406901i	$-0.133391\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−222.801	−1.25876
$178$	0	0
$179$	−198.519	−1.10904	−0.554522	−	0.832169i	$-0.687099\pi$
$179$	−198.519	−1.10904	−0.554522	+	0.832169i	$0.687099\pi$
$180$	0	0
$181$	285.836i	1.57921i	0.613618	+	0.789603i	$0.289713\pi$
$181$	285.836i	1.57921i	−0.613618	+	0.789603i	$0.710287\pi$
$182$	0	0
$183$	−56.9269	−0.311076
$184$	0	0
$185$	− 47.1000i	− 0.254595i
$186$	0	0
$187$	312.452i	1.67087i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	295.603	1.54766	0.773830	−	0.633394i	$-0.218339\pi$
$191$	295.603	1.54766	0.773830	+	0.633394i	$0.218339\pi$
$192$	0	0
$193$	188.610	0.977255	0.488627	−	0.872493i	$-0.337498\pi$
$193$	188.610	0.977255	0.488627	+	0.872493i	$0.337498\pi$
$194$	0	0
$195$	− 84.5643i	− 0.433663i
$196$	0	0
$197$	34.0000	0.172589	0.0862944	−	0.996270i	$-0.472497\pi$
$197$	34.0000	0.172589	0.0862944	+	0.996270i	$0.472497\pi$
$198$	0	0
$199$	− 108.725i	− 0.546359i	−0.961963	−	0.273180i	$-0.911925\pi$
$199$	− 108.725i	− 0.546359i	0.961963	−	0.273180i	$-0.0880753\pi$
$200$	0	0
$201$	453.570i	2.25657i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−64.8284	−0.316236
$206$	0	0
$207$	128.188	0.619264
$208$	0	0
$209$	− 303.372i	− 1.45154i
$210$	0	0
$211$	267.865	1.26950	0.634750	−	0.772717i	$-0.281103\pi$
$211$	267.865	1.26950	0.634750	+	0.772717i	$0.281103\pi$
$212$	0	0
$213$	522.970i	2.45526i
$214$	0	0
$215$	− 140.570i	− 0.653816i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	111.843	0.510697
$220$	0	0
$221$	132.024	0.597395
$222$	0	0
$223$	134.352i	0.602477i	0.953549	+	0.301238i	$0.0974000\pi$
$223$	134.352i	0.602477i	−0.953549	+	0.301238i	$0.902600\pi$
$224$	0	0
$225$	−20.0711	−0.0892047
$226$	0	0
$227$	− 313.363i	− 1.38045i	−0.723593	−	0.690227i	$-0.757511\pi$
$227$	− 313.363i	− 1.38045i	0.723593	−	0.690227i	$-0.242489\pi$
$228$	0	0
$229$	290.338i	1.26785i	0.773393	+	0.633926i	$0.218558\pi$
$229$	290.338i	1.26785i	−0.773393	+	0.633926i	$0.781442\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	221.865	0.952210	0.476105	−	0.879388i	$-0.342048\pi$
$233$	221.865	0.952210	0.476105	+	0.879388i	$0.342048\pi$
$234$	0	0
$235$	−367.328	−1.56310
$236$	0	0
$237$	488.270i	2.06021i
$238$	0	0
$239$	−195.154	−0.816543	−0.408271	−	0.912861i	$-0.633868\pi$
$239$	−195.154	−0.816543	−0.408271	+	0.912861i	$0.633868\pi$
$240$	0	0
$241$	42.9600i	0.178257i	0.996020	+	0.0891286i	$0.0284082\pi$
$241$	42.9600i	0.178257i	−0.996020	+	0.0891286i	$0.971592\pi$
$242$	0	0
$243$	− 345.711i	− 1.42268i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−128.188	−0.518978
$248$	0	0
$249$	−113.345	−0.455202
$250$	0	0
$251$	21.6072i	0.0860843i	0.999073	+	0.0430422i	$0.0137050\pi$
$251$	21.6072i	0.0860843i	−0.999073	+	0.0430422i	$0.986295\pi$
$252$	0	0
$253$	−136.014	−0.537606
$254$	0	0
$255$	650.176i	2.54971i
$256$	0	0
$257$	16.1628i	0.0628904i	0.999505	+	0.0314452i	$0.0100110\pi$
$257$	16.1628i	0.0628904i	−0.999505	+	0.0314452i	$0.989989\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	150.510	0.576665
$262$	0	0
$263$	298.968	1.13676	0.568381	−	0.822766i	$-0.307570\pi$
$263$	298.968	1.13676	0.568381	+	0.822766i	$0.307570\pi$
$264$	0	0
$265$	− 450.880i	− 1.70143i
$266$	0	0
$267$	−400.635	−1.50051
$268$	0	0
$269$	− 176.854i	− 0.657450i	−0.944426	−	0.328725i	$-0.893381\pi$
$269$	− 176.854i	− 0.657450i	0.944426	−	0.328725i	$-0.106619\pi$
$270$	0	0
$271$	6.97250i	0.0257288i	0.999917	+	0.0128644i	$0.00409497\pi$
$271$	6.97250i	0.0257288i	−0.999917	+	0.0128644i	$0.995905\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	21.2965	0.0774420
$276$	0	0
$277$	−153.632	−0.554630	−0.277315	−	0.960779i	$-0.589445\pi$
$277$	−153.632	−0.554630	−0.277315	+	0.960779i	$0.589445\pi$
$278$	0	0
$279$	473.719i	1.69792i
$280$	0	0
$281$	36.2914	0.129151	0.0645755	−	0.997913i	$-0.479431\pi$
$281$	36.2914	0.129151	0.0645755	+	0.997913i	$0.479431\pi$
$282$	0	0
$283$	− 447.182i	− 1.58015i	−0.613011	−	0.790074i	$-0.710042\pi$
$283$	− 447.182i	− 1.58015i	0.613011	−	0.790074i	$-0.289958\pi$
$284$	0	0
$285$	− 631.281i	− 2.21502i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−726.075	−2.51237
$290$	0	0
$291$	556.641	1.91286
$292$	0	0
$293$	− 115.118i	− 0.392895i	−0.980514	−	0.196447i	$-0.937059\pi$
$293$	− 115.118i	− 0.392895i	0.980514	−	0.196447i	$-0.0629405\pi$
$294$	0	0
$295$	249.236	0.844869
$296$	0	0
$297$	− 10.1635i	− 0.0342205i
$298$	0	0
$299$	57.4718i	0.192213i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−625.618	−2.06475
$304$	0	0
$305$	63.6812	0.208791
$306$	0	0
$307$	− 294.709i	− 0.959963i	−0.877279	−	0.479982i	$-0.840643\pi$
$307$	− 294.709i	− 0.959963i	0.877279	−	0.479982i	$-0.159357\pi$
$308$	0	0
$309$	154.794	0.500951
$310$	0	0
$311$	− 81.2774i	− 0.261342i	−0.991426	−	0.130671i	$-0.958287\pi$
$311$	− 81.2774i	− 0.261342i	0.991426	−	0.130671i	$-0.0417132\pi$
$312$	0	0
$313$	− 77.5328i	− 0.247709i	−0.992300	−	0.123854i	$-0.960474\pi$
$313$	− 77.5328i	− 0.247709i	0.992300	−	0.123854i	$-0.0395256\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	560.926	1.76948	0.884741	−	0.466082i	$-0.154335\pi$
$317$	560.926	1.76948	0.884741	+	0.466082i	$0.154335\pi$
$318$	0	0
$319$	−159.699	−0.500625
$320$	0	0
$321$	226.785i	0.706495i
$322$	0	0
$323$	985.576	3.05132
$324$	0	0
$325$	− 8.99869i	− 0.0276883i
$326$	0	0
$327$	− 560.504i	− 1.71408i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	264.788	0.799964	0.399982	−	0.916523i	$-0.369016\pi$
$331$	264.788	0.799964	0.399982	+	0.916523i	$0.369016\pi$
$332$	0	0
$333$	−91.1127	−0.273612
$334$	0	0
$335$	− 507.386i	− 1.51458i
$336$	0	0
$337$	−256.887	−0.762277	−0.381138	−	0.924518i	$-0.624468\pi$
$337$	−256.887	−0.762277	−0.381138	+	0.924518i	$0.624468\pi$
$338$	0	0
$339$	0.533430i	0.00157354i
$340$	0	0
$341$	− 502.643i	− 1.47403i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−283.029	−0.820375
$346$	0	0
$347$	−200.610	−0.578126	−0.289063	−	0.957310i	$-0.593344\pi$
$347$	−200.610	−0.578126	−0.289063	+	0.957310i	$0.593344\pi$
$348$	0	0
$349$	− 30.0575i	− 0.0861248i	−0.999072	−	0.0430624i	$-0.986289\pi$
$349$	− 30.0575i	− 0.0861248i	0.999072	−	0.0430624i	$-0.0137114\pi$
$350$	0	0
$351$	−4.29450	−0.0122350
$352$	0	0
$353$	− 544.609i	− 1.54280i	−0.636350	−	0.771400i	$-0.719557\pi$
$353$	− 544.609i	− 1.54280i	0.636350	−	0.771400i	$-0.280443\pi$
$354$	0	0
$355$	− 585.021i	− 1.64795i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−567.819	−1.58167	−0.790834	−	0.612031i	$-0.790353\pi$
$359$	−567.819	−1.58167	−0.790834	+	0.612031i	$0.790353\pi$
$360$	0	0
$361$	−595.933	−1.65078
$362$	0	0
$363$	− 106.024i	− 0.292077i
$364$	0	0
$365$	−125.113	−0.342775
$366$	0	0
$367$	− 181.964i	− 0.495813i	−0.968784	−	0.247907i	$-0.920257\pi$
$367$	− 181.964i	− 0.495813i	0.968784	−	0.247907i	$-0.0797426\pi$
$368$	0	0
$369$	125.408i	0.339858i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	384.926	1.03197	0.515987	−	0.856597i	$-0.327425\pi$
$373$	384.926	1.03197	0.515987	+	0.856597i	$0.327425\pi$
$374$	0	0
$375$	554.494	1.47865
$376$	0	0
$377$	67.4797i	0.178991i
$378$	0	0
$379$	471.840	1.24496	0.622480	−	0.782636i	$-0.286125\pi$
$379$	471.840	1.24496	0.622480	+	0.782636i	$0.286125\pi$
$380$	0	0
$381$	− 547.507i	− 1.43703i
$382$	0	0
$383$	360.597i	0.941507i	0.882265	+	0.470753i	$0.156018\pi$
$383$	360.597i	0.941507i	−0.882265	+	0.470753i	$0.843982\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−271.927	−0.702653
$388$	0	0
$389$	85.8924	0.220803	0.110401	−	0.993887i	$-0.464786\pi$
$389$	85.8924	0.220803	0.110401	+	0.993887i	$0.464786\pi$
$390$	0	0
$391$	− 441.874i	− 1.13011i
$392$	0	0
$393$	−292.419	−0.744069
$394$	0	0
$395$	− 546.203i	− 1.38279i
$396$	0	0
$397$	426.194i	1.07354i	0.843729	+	0.536769i	$0.180355\pi$
$397$	426.194i	1.07354i	−0.843729	+	0.536769i	$0.819645\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	192.759	0.480697	0.240348	−	0.970687i	$-0.422738\pi$
$401$	192.759	0.480697	0.240348	+	0.970687i	$0.422738\pi$
$402$	0	0
$403$	−212.388	−0.527018
$404$	0	0
$405$	376.296i	0.929125i
$406$	0	0
$407$	96.6758	0.237533
$408$	0	0
$409$	− 613.813i	− 1.50076i	−0.661004	−	0.750382i	$-0.729869\pi$
$409$	− 613.813i	− 1.50076i	0.661004	−	0.750382i	$-0.270131\pi$
$410$	0	0
$411$	117.697i	0.286367i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	126.794	0.305527
$416$	0	0
$417$	−409.654	−0.982383
$418$	0	0
$419$	457.797i	1.09259i	0.837592	+	0.546297i	$0.183963\pi$
$419$	457.797i	1.09259i	−0.837592	+	0.546297i	$0.816037\pi$
$420$	0	0
$421$	−215.103	−0.510933	−0.255466	−	0.966818i	$-0.582229\pi$
$421$	−215.103	−0.510933	−0.255466	+	0.966818i	$0.582229\pi$
$422$	0	0
$423$	710.579i	1.67986i
$424$	0	0
$425$	69.1868i	0.162793i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	173.574	0.404600
$430$	0	0
$431$	157.897	0.366351	0.183175	−	0.983080i	$-0.441362\pi$
$431$	157.897	0.366351	0.183175	+	0.983080i	$0.441362\pi$
$432$	0	0
$433$	679.390i	1.56903i	0.620110	+	0.784515i	$0.287088\pi$
$433$	679.390i	1.56903i	−0.620110	+	0.784515i	$0.712912\pi$
$434$	0	0
$435$	−332.315	−0.763943
$436$	0	0
$437$	429.033i	0.981769i
$438$	0	0
$439$	− 617.932i	− 1.40759i	−0.710403	−	0.703795i	$-0.751487\pi$
$439$	− 617.932i	− 1.40759i	0.710403	−	0.703795i	$-0.248513\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	240.126	0.542046	0.271023	−	0.962573i	$-0.412638\pi$
$443$	240.126	0.542046	0.271023	+	0.962573i	$0.412638\pi$
$444$	0	0
$445$	448.171	1.00713
$446$	0	0
$447$	629.302i	1.40783i
$448$	0	0
$449$	−641.823	−1.42945	−0.714725	−	0.699405i	$-0.753448\pi$
$449$	−641.823	−1.42945	−0.714725	+	0.699405i	$0.753448\pi$
$450$	0	0
$451$	− 133.065i	− 0.295043i
$452$	0	0
$453$	− 527.180i	− 1.16375i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−641.698	−1.40415	−0.702077	−	0.712101i	$-0.747744\pi$
$457$	−641.698	−1.40415	−0.702077	+	0.712101i	$0.747744\pi$
$458$	0	0
$459$	33.0184	0.0719356
$460$	0	0
$461$	597.836i	1.29682i	0.761290	+	0.648412i	$0.224566\pi$
$461$	597.836i	1.29682i	−0.761290	+	0.648412i	$0.775434\pi$
$462$	0	0
$463$	−617.550	−1.33380	−0.666901	−	0.745146i	$-0.732380\pi$
$463$	−617.550	−1.33380	−0.666901	+	0.745146i	$0.732380\pi$
$464$	0	0
$465$	− 1045.94i	− 2.24933i
$466$	0	0
$467$	− 566.522i	− 1.21311i	−0.795042	−	0.606555i	$-0.792551\pi$
$467$	− 566.522i	− 1.21311i	0.795042	−	0.606555i	$-0.207449\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−637.380	−1.35325
$472$	0	0
$473$	288.530	0.610000
$474$	0	0
$475$	− 67.1762i	− 0.141424i
$476$	0	0
$477$	−872.205	−1.82852
$478$	0	0
$479$	− 326.176i	− 0.680953i	−0.940253	−	0.340476i	$-0.889412\pi$
$479$	− 326.176i	− 0.680953i	0.940253	−	0.340476i	$-0.110588\pi$
$480$	0	0
$481$	− 40.8496i	− 0.0849264i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−622.686	−1.28389
$486$	0	0
$487$	629.448	1.29250	0.646251	−	0.763125i	$-0.276336\pi$
$487$	629.448	1.29250	0.646251	+	0.763125i	$0.276336\pi$
$488$	0	0
$489$	145.988i	0.298543i
$490$	0	0
$491$	−48.7462	−0.0992793	−0.0496397	−	0.998767i	$-0.515807\pi$
$491$	−48.7462	−0.0992793	−0.0496397	+	0.998767i	$0.515807\pi$
$492$	0	0
$493$	− 518.821i	− 1.05238i
$494$	0	0
$495$	433.081i	0.874911i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−112.924	−0.226302	−0.113151	−	0.993578i	$-0.536094\pi$
$499$	−112.924	−0.226302	−0.113151	+	0.993578i	$0.536094\pi$
$500$	0	0
$501$	584.839	1.16734
$502$	0	0
$503$	424.287i	0.843513i	0.906709	+	0.421756i	$0.138586\pi$
$503$	424.287i	0.843513i	−0.906709	+	0.421756i	$0.861414\pi$
$504$	0	0
$505$	699.848	1.38584
$506$	0	0
$507$	648.481i	1.27905i
$508$	0	0
$509$	− 622.871i	− 1.22372i	−0.790968	−	0.611858i	$-0.790422\pi$
$509$	− 622.871i	− 1.22372i	0.790968	−	0.611858i	$-0.209578\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−32.0589	−0.0624929
$514$	0	0
$515$	−173.160	−0.336233
$516$	0	0
$517$	− 753.965i	− 1.45835i
$518$	0	0
$519$	601.325	1.15862
$520$	0	0
$521$	733.863i	1.40857i	0.709920	+	0.704283i	$0.248731\pi$
$521$	733.863i	1.40857i	−0.709920	+	0.704283i	$0.751269\pi$
$522$	0	0
$523$	655.306i	1.25297i	0.779432	+	0.626487i	$0.215508\pi$
$523$	655.306i	1.25297i	−0.779432	+	0.626487i	$0.784492\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1632.96	3.09859
$528$	0	0
$529$	−336.647	−0.636383
$530$	0	0
$531$	− 482.136i	− 0.907977i
$532$	0	0
$533$	−56.2254	−0.105489
$534$	0	0
$535$	− 253.693i	− 0.474192i
$536$	0	0
$537$	− 847.902i	− 1.57896i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−460.794	−0.851745	−0.425872	−	0.904783i	$-0.640033\pi$
$541$	−460.794	−0.851745	−0.425872	+	0.904783i	$0.640033\pi$
$542$	0	0
$543$	−1220.85	−2.24834
$544$	0	0
$545$	627.007i	1.15047i
$546$	0	0
$547$	−556.449	−1.01727	−0.508637	−	0.860981i	$-0.669850\pi$
$547$	−556.449	−1.01727	−0.508637	+	0.860981i	$0.669850\pi$
$548$	0	0
$549$	− 123.188i	− 0.224387i
$550$	0	0
$551$	503.743i	0.914234i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	201.171	0.362470
$556$	0	0
$557$	−301.294	−0.540922	−0.270461	−	0.962731i	$-0.587176\pi$
$557$	−301.294	−0.540922	−0.270461	+	0.962731i	$0.587176\pi$
$558$	0	0
$559$	− 121.916i	− 0.218097i
$560$	0	0
$561$	−1334.53	−2.37884
$562$	0	0
$563$	160.356i	0.284825i	0.989807	+	0.142412i	$0.0454859\pi$
$563$	160.356i	0.284825i	−0.989807	+	0.142412i	$0.954514\pi$
$564$	0	0
$565$	− 0.596721i	− 0.00105614i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−392.818	−0.690366	−0.345183	−	0.938535i	$-0.612183\pi$
$569$	−392.818	−0.690366	−0.345183	+	0.938535i	$0.612183\pi$
$570$	0	0
$571$	−596.254	−1.04423	−0.522114	−	0.852876i	$-0.674856\pi$
$571$	−596.254	−1.04423	−0.522114	+	0.852876i	$0.674856\pi$
$572$	0	0
$573$	1262.56i	2.20342i
$574$	0	0
$575$	−30.1179	−0.0523789
$576$	0	0
$577$	− 134.056i	− 0.232332i	−0.993230	−	0.116166i	$-0.962939\pi$
$577$	− 134.056i	− 0.232332i	0.993230	−	0.116166i	$-0.0370605\pi$
$578$	0	0
$579$	805.581i	1.39133i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	925.459	1.58741
$584$	0	0
$585$	182.995	0.312812
$586$	0	0
$587$	− 493.505i	− 0.840725i	−0.907356	−	0.420362i	$-0.861903\pi$
$587$	− 493.505i	− 0.840725i	0.907356	−	0.420362i	$-0.138097\pi$
$588$	0	0
$589$	−1585.50	−2.69185
$590$	0	0
$591$	145.219i	0.245717i
$592$	0	0
$593$	183.742i	0.309852i	0.987926	+	0.154926i	$0.0495140\pi$
$593$	183.742i	0.309852i	−0.987926	+	0.154926i	$0.950486\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	464.382	0.777859
$598$	0	0
$599$	−852.918	−1.42390	−0.711951	−	0.702229i	$-0.752188\pi$
$599$	−852.918	−1.42390	−0.711951	+	0.702229i	$0.752188\pi$
$600$	0	0
$601$	− 733.964i	− 1.22124i	−0.791925	−	0.610619i	$-0.790921\pi$
$601$	− 733.964i	− 1.22124i	0.791925	−	0.610619i	$-0.209079\pi$
$602$	0	0
$603$	−981.513	−1.62772
$604$	0	0
$605$	118.604i	0.196039i
$606$	0	0
$607$	450.981i	0.742966i	0.928440	+	0.371483i	$0.121151\pi$
$607$	450.981i	0.742966i	−0.928440	+	0.371483i	$0.878849\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−318.582	−0.521411
$612$	0	0
$613$	−1109.53	−1.81001	−0.905003	−	0.425406i	$-0.860131\pi$
$613$	−1109.53	−1.81001	−0.905003	+	0.425406i	$0.860131\pi$
$614$	0	0
$615$	− 276.891i	− 0.450230i
$616$	0	0
$617$	−445.799	−0.722527	−0.361263	−	0.932464i	$-0.617654\pi$
$617$	−445.799	−0.722527	−0.361263	+	0.932464i	$0.617654\pi$
$618$	0	0
$619$	775.713i	1.25317i	0.779353	+	0.626586i	$0.215548\pi$
$619$	775.713i	1.25317i	−0.779353	+	0.626586i	$0.784452\pi$
$620$	0	0
$621$	14.3733i	0.0231455i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−565.995	−0.905592
$626$	0	0
$627$	1295.74	2.06658
$628$	0	0
$629$	314.074i	0.499323i
$630$	0	0
$631$	−16.2487	−0.0257507	−0.0128754	−	0.999917i	$-0.504098\pi$
$631$	−16.2487	−0.0257507	−0.0128754	+	0.999917i	$0.504098\pi$
$632$	0	0
$633$	1144.09i	1.80741i
$634$	0	0
$635$	612.469i	0.964517i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1131.69	−1.77104
$640$	0	0
$641$	638.642	0.996321	0.498160	−	0.867085i	$-0.334009\pi$
$641$	638.642	0.996321	0.498160	+	0.867085i	$0.334009\pi$
$642$	0	0
$643$	863.209i	1.34247i	0.741244	+	0.671235i	$0.234236\pi$
$643$	863.209i	1.34247i	−0.741244	+	0.671235i	$0.765764\pi$
$644$	0	0
$645$	600.396	0.930847
$646$	0	0
$647$	− 567.433i	− 0.877022i	−0.898726	−	0.438511i	$-0.855506\pi$
$647$	− 567.433i	− 0.877022i	0.898726	−	0.438511i	$-0.144494\pi$
$648$	0	0
$649$	511.574i	0.788249i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−575.460	−0.881256	−0.440628	−	0.897690i	$-0.645244\pi$
$653$	−575.460	−0.881256	−0.440628	+	0.897690i	$0.645244\pi$
$654$	0	0
$655$	327.115	0.499412
$656$	0	0
$657$	242.025i	0.368378i
$658$	0	0
$659$	1.68261	0.00255328	0.00127664	−	0.999999i	$-0.499594\pi$
$659$	1.68261	0.00255328	0.00127664	+	0.999999i	$0.499594\pi$
$660$	0	0
$661$	− 1034.33i	− 1.56480i	−0.622778	−	0.782399i	$-0.713996\pi$
$661$	− 1034.33i	− 1.56480i	0.622778	−	0.782399i	$-0.286004\pi$
$662$	0	0
$663$	563.895i	0.850520i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	225.849	0.338604
$668$	0	0
$669$	−573.838	−0.857754
$670$	0	0
$671$	130.710i	0.194799i
$672$	0	0
$673$	−22.6488	−0.0336536	−0.0168268	−	0.999858i	$-0.505356\pi$
$673$	−22.6488	−0.0336536	−0.0168268	+	0.999858i	$0.505356\pi$
$674$	0	0
$675$	− 2.25051i	− 0.00333410i
$676$	0	0
$677$	88.5067i	0.130734i	0.997861	+	0.0653669i	$0.0208218\pi$
$677$	88.5067i	0.130734i	−0.997861	+	0.0653669i	$0.979178\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1338.42	1.96537
$682$	0	0
$683$	−129.173	−0.189126	−0.0945631	−	0.995519i	$-0.530145\pi$
$683$	−129.173	−0.189126	−0.0945631	+	0.995519i	$0.530145\pi$
$684$	0	0
$685$	− 131.662i	− 0.192207i
$686$	0	0
$687$	−1240.08	−1.80506
$688$	0	0
$689$	− 391.046i	− 0.567556i
$690$	0	0
$691$	47.9884i	0.0694477i	0.999397	+	0.0347239i	$0.0110552\pi$
$691$	47.9884i	0.0694477i	−0.999397	+	0.0347239i	$0.988945\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	458.259	0.659366
$696$	0	0
$697$	432.291	0.620217
$698$	0	0
$699$	947.617i	1.35567i
$700$	0	0
$701$	244.701	0.349074	0.174537	−	0.984651i	$-0.444157\pi$
$701$	244.701	0.349074	0.174537	+	0.984651i	$0.444157\pi$
$702$	0	0
$703$	− 304.946i	− 0.433779i
$704$	0	0
$705$	− 1568.91i	− 2.22541i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−384.270	−0.541989	−0.270994	−	0.962581i	$-0.587352\pi$
$709$	−384.270	−0.541989	−0.270994	+	0.962581i	$0.587352\pi$
$710$	0	0
$711$	−1056.60	−1.48608
$712$	0	0
$713$	710.845i	0.996977i
$714$	0	0
$715$	−194.168	−0.271564
$716$	0	0
$717$	− 833.529i	− 1.16252i
$718$	0	0
$719$	− 814.153i	− 1.13234i	−0.824288	−	0.566171i	$-0.808424\pi$
$719$	− 814.153i	− 1.13234i	0.824288	−	0.566171i	$-0.191576\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−183.488	−0.253787
$724$	0	0
$725$	−35.3625	−0.0487758
$726$	0	0
$727$	− 1287.87i	− 1.77149i	−0.464173	−	0.885745i	$-0.653648\pi$
$727$	− 1287.87i	− 1.77149i	0.464173	−	0.885745i	$-0.346352\pi$
$728$	0	0
$729$	767.764	1.05317
$730$	0	0
$731$	937.357i	1.28229i
$732$	0	0
$733$	328.280i	0.447858i	0.974605	+	0.223929i	$0.0718885\pi$
$733$	328.280i	0.447858i	−0.974605	+	0.223929i	$0.928112\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1041.44	1.41308
$738$	0	0
$739$	−729.609	−0.987292	−0.493646	−	0.869663i	$-0.664336\pi$
$739$	−729.609	−0.987292	−0.493646	+	0.869663i	$0.664336\pi$
$740$	0	0
$741$	− 547.507i	− 0.738876i
$742$	0	0
$743$	−334.423	−0.450098	−0.225049	−	0.974347i	$-0.572254\pi$
$743$	−334.423	−0.450098	−0.225049	+	0.974347i	$0.572254\pi$
$744$	0	0
$745$	− 703.968i	− 0.944924i
$746$	0	0
$747$	− 245.276i	− 0.328348i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1239.69	1.65072	0.825360	−	0.564607i	$-0.190972\pi$
$751$	1239.69	1.65072	0.825360	+	0.564607i	$0.190972\pi$
$752$	0	0
$753$	−92.2872	−0.122559
$754$	0	0
$755$	589.730i	0.781099i
$756$	0	0
$757$	827.710	1.09341	0.546704	−	0.837326i	$-0.315882\pi$
$757$	827.710	1.09341	0.546704	+	0.837326i	$0.315882\pi$
$758$	0	0
$759$	− 580.936i	− 0.765397i
$760$	0	0
$761$	425.191i	0.558726i	0.960186	+	0.279363i	$0.0901233\pi$
$761$	425.191i	0.558726i	−0.960186	+	0.279363i	$0.909877\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1406.96	−1.83917
$766$	0	0
$767$	216.162	0.281827
$768$	0	0
$769$	− 869.454i	− 1.13063i	−0.824875	−	0.565315i	$-0.808755\pi$
$769$	− 869.454i	− 1.13063i	0.824875	−	0.565315i	$-0.191245\pi$
$770$	0	0
$771$	−69.0337	−0.0895379
$772$	0	0
$773$	423.918i	0.548406i	0.961672	+	0.274203i	$0.0884140\pi$
$773$	423.918i	0.548406i	−0.961672	+	0.274203i	$0.911586\pi$
$774$	0	0
$775$	− 111.301i	− 0.143614i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−419.728	−0.538804
$780$	0	0
$781$	1200.79	1.53751
$782$	0	0
$783$	16.8762i	0.0215533i
$784$	0	0
$785$	713.005	0.908287
$786$	0	0
$787$	− 225.647i	− 0.286717i	−0.989671	−	0.143359i	$-0.954210\pi$
$787$	− 225.647i	− 0.286717i	0.989671	−	0.143359i	$-0.0457903\pi$
$788$	0	0
$789$	1276.94i	1.61842i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	55.2304	0.0696475
$794$	0	0
$795$	1925.77	2.42235
$796$	0	0
$797$	323.178i	0.405493i	0.979231	+	0.202747i	$0.0649868\pi$
$797$	323.178i	0.405493i	−0.979231	+	0.202747i	$0.935013\pi$
$798$	0	0
$799$	2449.43	3.06562
$800$	0	0
$801$	− 866.965i	− 1.08235i
$802$	0	0
$803$	− 256.802i	− 0.319803i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	755.369	0.936021
$808$	0	0
$809$	−17.0669	−0.0210963	−0.0105481	−	0.999944i	$-0.503358\pi$
$809$	−17.0669	−0.0210963	−0.0105481	+	0.999944i	$0.503358\pi$
$810$	0	0
$811$	890.878i	1.09849i	0.835660	+	0.549246i	$0.185085\pi$
$811$	890.878i	1.09849i	−0.835660	+	0.549246i	$0.814915\pi$
$812$	0	0
$813$	−29.7805	−0.0366304
$814$	0	0
$815$	− 163.309i	− 0.200379i
$816$	0	0
$817$	− 910.116i	− 1.11397i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1194.03	1.45436	0.727179	−	0.686447i	$-0.240831\pi$
$821$	1194.03	1.45436	0.727179	+	0.686447i	$0.240831\pi$
$822$	0	0
$823$	−572.409	−0.695515	−0.347757	−	0.937585i	$-0.613057\pi$
$823$	−572.409	−0.695515	−0.347757	+	0.937585i	$0.613057\pi$
$824$	0	0
$825$	90.9605i	0.110255i
$826$	0	0
$827$	126.794	0.153318	0.0766588	−	0.997057i	$-0.475575\pi$
$827$	126.794	0.153318	0.0766588	+	0.997057i	$0.475575\pi$
$828$	0	0
$829$	254.978i	0.307572i	0.988104	+	0.153786i	$0.0491467\pi$
$829$	254.978i	0.307572i	−0.988104	+	0.153786i	$0.950853\pi$
$830$	0	0
$831$	− 656.186i	− 0.789634i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−654.229	−0.783508
$836$	0	0
$837$	−53.1169	−0.0634610
$838$	0	0
$839$	− 247.475i	− 0.294964i	−0.989065	−	0.147482i	$-0.952883\pi$
$839$	− 247.475i	− 0.294964i	0.989065	−	0.147482i	$-0.0471168\pi$
$840$	0	0
$841$	−575.823	−0.684688
$842$	0	0
$843$	155.006i	0.183874i
$844$	0	0
$845$	− 725.423i	− 0.858488i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1909.98	2.24968
$850$	0	0
$851$	−136.720	−0.160658
$852$	0	0
$853$	− 1493.89i	− 1.75133i	−0.482915	−	0.875667i	$-0.660422\pi$
$853$	− 1493.89i	− 1.75133i	0.482915	−	0.875667i	$-0.339578\pi$
$854$	0	0
$855$	1366.08	1.59775
$856$	0	0
$857$	691.307i	0.806660i	0.915055	+	0.403330i	$0.132147\pi$
$857$	691.307i	0.806660i	−0.915055	+	0.403330i	$0.867853\pi$
$858$	0	0
$859$	− 777.755i	− 0.905419i	−0.891658	−	0.452710i	$-0.850457\pi$
$859$	− 777.755i	− 0.905419i	0.891658	−	0.452710i	$-0.149543\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−925.459	−1.07237	−0.536187	−	0.844099i	$-0.680136\pi$
$863$	−925.459	−1.07237	−0.536187	+	0.844099i	$0.680136\pi$
$864$	0	0
$865$	−672.672	−0.777655
$866$	0	0
$867$	− 3101.17i	− 3.57690i
$868$	0	0
$869$	1121.12	1.29012
$870$	0	0
$871$	− 440.053i	− 0.505228i
$872$	0	0
$873$	1204.56i	1.37979i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1143.54	−1.30392	−0.651960	−	0.758254i	$-0.726053\pi$
$877$	−1143.54	−1.30392	−0.651960	+	0.758254i	$0.726053\pi$
$878$	0	0
$879$	491.686	0.559370
$880$	0	0
$881$	1165.65i	1.32310i	0.749902	+	0.661549i	$0.230101\pi$
$881$	1165.65i	1.32310i	−0.749902	+	0.661549i	$0.769899\pi$
$882$	0	0
$883$	874.911	0.990839	0.495420	−	0.868654i	$-0.335014\pi$
$883$	874.911	0.990839	0.495420	+	0.868654i	$0.335014\pi$
$884$	0	0
$885$	1064.52i	1.20285i
$886$	0	0
$887$	− 924.388i	− 1.04215i	−0.853511	−	0.521075i	$-0.825531\pi$
$887$	− 924.388i	− 1.04215i	0.853511	−	0.521075i	$-0.174469\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−772.371	−0.866858
$892$	0	0
$893$	−2378.25	−2.66321
$894$	0	0
$895$	948.506i	1.05978i
$896$	0	0
$897$	−245.470	−0.273657
$898$	0	0
$899$	834.629i	0.928397i
$900$	0	0
$901$	3006.57i	3.33693i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1365.70	1.50906
$906$	0	0
$907$	1099.61	1.21235	0.606177	−	0.795330i	$-0.292702\pi$
$907$	1099.61	1.21235	0.606177	+	0.795330i	$0.292702\pi$
$908$	0	0
$909$	− 1353.82i	− 1.48935i
$910$	0	0
$911$	−349.277	−0.383400	−0.191700	−	0.981454i	$-0.561400\pi$
$911$	−349.277	−0.383400	−0.191700	+	0.981454i	$0.561400\pi$
$912$	0	0
$913$	260.252i	0.285052i
$914$	0	0
$915$	271.992i	0.297258i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1693.77	−1.84306	−0.921528	−	0.388312i	$-0.873058\pi$
$919$	−1693.77	−1.84306	−0.921528	+	0.388312i	$0.873058\pi$
$920$	0	0
$921$	1258.74	1.36671
$922$	0	0
$923$	− 507.386i	− 0.549714i
$924$	0	0
$925$	21.4071	0.0231428
$926$	0	0
$927$	334.970i	0.361349i
$928$	0	0
$929$	− 1203.10i	− 1.29504i	−0.762047	−	0.647522i	$-0.775805\pi$
$929$	− 1203.10i	− 1.29504i	0.762047	−	0.647522i	$-0.224195\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	347.147	0.372076
$934$	0	0
$935$	1492.87	1.59665
$936$	0	0
$937$	− 1305.21i	− 1.39297i	−0.717571	−	0.696486i	$-0.754746\pi$
$937$	− 1305.21i	− 1.39297i	0.717571	−	0.696486i	$-0.245254\pi$
$938$	0	0
$939$	331.154	0.352666
$940$	0	0
$941$	− 140.120i	− 0.148905i	−0.997225	−	0.0744526i	$-0.976279\pi$
$941$	− 140.120i	− 0.148905i	0.997225	−	0.0744526i	$-0.0237209\pi$
$942$	0	0
$943$	188.182i	0.199556i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−219.646	−0.231939	−0.115970	−	0.993253i	$-0.536998\pi$
$947$	−219.646	−0.231939	−0.115970	+	0.993253i	$0.536998\pi$
$948$	0	0
$949$	−108.510	−0.114341
$950$	0	0
$951$	2395.79i	2.51924i
$952$	0	0
$953$	1733.15	1.81862	0.909310	−	0.416119i	$-0.136610\pi$
$953$	1733.15	1.81862	0.909310	+	0.416119i	$0.136610\pi$
$954$	0	0
$955$	− 1412.36i	− 1.47892i
$956$	0	0
$957$	− 682.098i	− 0.712746i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1665.94	−1.73355
$962$	0	0
$963$	−490.757	−0.509612
$964$	0	0
$965$	− 901.162i	− 0.933847i
$966$	0	0
$967$	−1863.97	−1.92758	−0.963791	−	0.266660i	$-0.914080\pi$
$967$	−1863.97	−1.92758	−0.963791	+	0.266660i	$0.914080\pi$
$968$	0	0
$969$	4209.53i	4.34420i
$970$	0	0
$971$	− 1050.54i	− 1.08192i	−0.841048	−	0.540960i	$-0.818061\pi$
$971$	− 1050.54i	− 1.08192i	0.841048	−	0.540960i	$-0.181939\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	38.4347	0.0394202
$976$	0	0
$977$	−633.233	−0.648141	−0.324070	−	0.946033i	$-0.605051\pi$
$977$	−633.233	−0.648141	−0.324070	+	0.946033i	$0.605051\pi$
$978$	0	0
$979$	919.899i	0.939631i
$980$	0	0
$981$	1212.92	1.23641
$982$	0	0
$983$	− 125.246i	− 0.127412i	−0.997969	−	0.0637061i	$-0.979708\pi$
$983$	− 125.246i	− 0.127412i	0.997969	−	0.0637061i	$-0.0202920\pi$
$984$	0	0
$985$	− 162.449i	− 0.164923i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−408.043	−0.412581
$990$	0	0
$991$	169.795	0.171337	0.0856685	−	0.996324i	$-0.472697\pi$
$991$	169.795	0.171337	0.0856685	+	0.996324i	$0.472697\pi$
$992$	0	0
$993$	1130.95i	1.13892i
$994$	0	0
$995$	−519.481	−0.522091
$996$	0	0
$997$	21.1203i	0.0211839i	0.999944	+	0.0105919i	$0.00337158\pi$
$997$	21.1203i	0.0211839i	−0.999944	+	0.0105919i	$0.996628\pi$
$998$	0	0
$999$	− 10.2162i	− 0.0102264i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.c.d.97.7	yes	8
4.3	odd	2	inner	1568.3.c.d.97.1	✓	8
7.6	odd	2	inner	1568.3.c.d.97.2	yes	8
28.27	even	2	inner	1568.3.c.d.97.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1568.3.c.d.97.1	✓	8	4.3	odd	2	inner
1568.3.c.d.97.2	yes	8	7.6	odd	2	inner
1568.3.c.d.97.7	yes	8	1.1	even	1	trivial
1568.3.c.d.97.8	yes	8	28.27	even	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.27114i q^{3} -4.77791i q^{5} -9.24264 q^{9} +9.80697 q^{11} -4.14386i q^{13} +20.4071 q^{15} +31.8602i q^{17} -30.9343i q^{19} -13.8691 q^{23} +2.17157 q^{25} -1.03635i q^{27} -16.2843 q^{29} -51.2537i q^{31} +41.8869i q^{33} +9.85786 q^{37} +17.6990 q^{39} -13.5684i q^{41} +29.4209 q^{43} +44.1605i q^{45} -76.8805i q^{47} -136.080 q^{51} +94.3675 q^{53} -46.8568i q^{55} +132.125 q^{57} +52.1643i q^{59} +13.3283i q^{61} -19.7990 q^{65} +106.194 q^{67} -59.2371i q^{69} +122.443 q^{71} -26.1857i q^{73} +9.27509i q^{75} +114.318 q^{79} -78.7574 q^{81} +26.5375i q^{83} +152.225 q^{85} -69.5524i q^{87} +93.8006i q^{89} +218.912 q^{93} -147.801 q^{95} -130.326i q^{97} -90.6423 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{9} + 40 q^{25} + 96 q^{29} + 192 q^{37} + 144 q^{53} + 480 q^{57} - 664 q^{81} + 720 q^{85} + 1344 q^{93}+O(q^{100}) \) 8 * q - 40 * q^9 + 40 * q^25 + 96 * q^29 + 192 * q^37 + 144 * q^53 + 480 * q^57 - 664 * q^81 + 720 * q^85 + 1344 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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