LMFDB - Embedded newform 112.3.d.b.15.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,3,Mod(15,112)] 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("112.15"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	112.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$3.05177896084$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - x^{2} - 2x + 4 $ x^4 - x^3 - x^2 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			15.1
Root			$-0.895644 - 1.09445i$ of defining polynomial
Character	$\chi$	$=$	112.15
Dual form			112.3.d.b.15.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.37780i q^{3} -5.58258 q^{5} -2.64575i q^{7} -10.1652 q^{9} -1.82740i q^{11} -20.7477 q^{13} +24.4394i q^{15} +25.1652 q^{17} -30.6446i q^{19} -11.5826 q^{21} -3.27340i q^{23} +6.16515 q^{25} +5.10080i q^{27} +53.8258 q^{29} -29.1588i q^{31} -8.00000 q^{33} +14.7701i q^{35} +17.1652 q^{37} +90.8294i q^{39} +7.49545 q^{41} +57.2530i q^{43} +56.7477 q^{45} +37.2312i q^{47} -7.00000 q^{49} -110.168i q^{51} -46.0000 q^{53} +10.2016i q^{55} -134.156 q^{57} -75.1058i q^{59} +28.0871 q^{61} +26.8945i q^{63} +115.826 q^{65} -81.3906i q^{67} -14.3303 q^{69} -32.1304i q^{71} -45.6515 q^{73} -26.9898i q^{75} -4.83485 q^{77} +10.9644i q^{79} -69.1561 q^{81} +143.705i q^{83} -140.486 q^{85} -235.639i q^{87} +59.3212 q^{89} +54.8933i q^{91} -127.652 q^{93} +171.076i q^{95} +17.1652 q^{97} +18.5758i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{9} - 28 q^{13} + 64 q^{17} - 28 q^{21} - 12 q^{25} + 32 q^{29} - 32 q^{33} + 32 q^{37} - 80 q^{41} + 172 q^{45} - 28 q^{49} - 184 q^{53} - 280 q^{57} + 204 q^{61} + 280 q^{65} + 16 q^{69}+ \cdots + 32 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^9 - 28 * q^13 + 64 * q^17 - 28 * q^21 - 12 * q^25 + 32 * q^29 - 32 * q^33 + 32 * q^37 - 80 * q^41 + 172 * q^45 - 28 * q^49 - 184 * q^53 - 280 * q^57 + 204 * q^61 + 280 * q^65 + 16 * q^69 + 184 * q^73 - 56 * q^77 - 20 * q^81 - 232 * q^85 - 56 * q^89 - 144 * q^93 + 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 4.37780i	− 1.45927i	−0.683838	−	0.729634i	$-0.739691\pi$
$3$	− 4.37780i	− 1.45927i	0.683838	−	0.729634i	$-0.260309\pi$
$4$	0	0
$5$	−5.58258	−1.11652	−0.558258	−	0.829668i	$-0.688530\pi$
$5$	−5.58258	−1.11652	−0.558258	+	0.829668i	$0.688530\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	−10.1652	−1.12946
$10$	0	0
$11$	− 1.82740i	− 0.166127i	−0.996544	−	0.0830637i	$-0.973530\pi$
$11$	− 1.82740i	− 0.166127i	0.996544	−	0.0830637i	$-0.0264705\pi$
$12$	0	0
$13$	−20.7477	−1.59598	−0.797990	−	0.602671i	$-0.794103\pi$
$13$	−20.7477	−1.59598	−0.797990	+	0.602671i	$0.794103\pi$
$14$	0	0
$15$	24.4394i	1.62929i
$16$	0	0
$17$	25.1652	1.48030	0.740152	−	0.672440i	$-0.234754\pi$
$17$	25.1652	1.48030	0.740152	+	0.672440i	$0.234754\pi$
$18$	0	0
$19$	− 30.6446i	− 1.61287i	−0.591320	−	0.806437i	$-0.701393\pi$
$19$	− 30.6446i	− 1.61287i	0.591320	−	0.806437i	$-0.298607\pi$
$20$	0	0
$21$	−11.5826	−0.551551
$22$	0	0
$23$	− 3.27340i	− 0.142322i	−0.997465	−	0.0711609i	$-0.977330\pi$
$23$	− 3.27340i	− 0.142322i	0.997465	−	0.0711609i	$-0.0226704\pi$
$24$	0	0
$25$	6.16515	0.246606
$26$	0	0
$27$	5.10080i	0.188919i
$28$	0	0
$29$	53.8258	1.85606	0.928030	−	0.372505i	$-0.121501\pi$
$29$	53.8258	1.85606	0.928030	+	0.372505i	$0.121501\pi$
$30$	0	0
$31$	− 29.1588i	− 0.940607i	−0.882505	−	0.470303i	$-0.844144\pi$
$31$	− 29.1588i	− 0.940607i	0.882505	−	0.470303i	$-0.155856\pi$
$32$	0	0
$33$	−8.00000	−0.242424
$34$	0	0
$35$	14.7701i	0.422003i
$36$	0	0
$37$	17.1652	0.463923	0.231962	−	0.972725i	$-0.425486\pi$
$37$	17.1652	0.463923	0.231962	+	0.972725i	$0.425486\pi$
$38$	0	0
$39$	90.8294i	2.32896i
$40$	0	0
$41$	7.49545	0.182816	0.0914080	−	0.995814i	$-0.470863\pi$
$41$	7.49545	0.182816	0.0914080	+	0.995814i	$0.470863\pi$
$42$	0	0
$43$	57.2530i	1.33147i	0.746190	+	0.665733i	$0.231881\pi$
$43$	57.2530i	1.33147i	−0.746190	+	0.665733i	$0.768119\pi$
$44$	0	0
$45$	56.7477	1.26106
$46$	0	0
$47$	37.2312i	0.792154i	0.918217	+	0.396077i	$0.129629\pi$
$47$	37.2312i	0.792154i	−0.918217	+	0.396077i	$0.870371\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 110.168i	− 2.16016i
$52$	0	0
$53$	−46.0000	−0.867925	−0.433962	−	0.900931i	$-0.642885\pi$
$53$	−46.0000	−0.867925	−0.433962	+	0.900931i	$0.642885\pi$
$54$	0	0
$55$	10.2016i	0.185484i
$56$	0	0
$57$	−134.156	−2.35362
$58$	0	0
$59$	− 75.1058i	− 1.27298i	−0.771285	−	0.636490i	$-0.780386\pi$
$59$	− 75.1058i	− 1.27298i	0.771285	−	0.636490i	$-0.219614\pi$
$60$	0	0
$61$	28.0871	0.460445	0.230222	−	0.973138i	$-0.426055\pi$
$61$	28.0871	0.460445	0.230222	+	0.973138i	$0.426055\pi$
$62$	0	0
$63$	26.8945i	0.426896i
$64$	0	0
$65$	115.826	1.78193
$66$	0	0
$67$	− 81.3906i	− 1.21479i	−0.794402	−	0.607393i	$-0.792215\pi$
$67$	− 81.3906i	− 1.21479i	0.794402	−	0.607393i	$-0.207785\pi$
$68$	0	0
$69$	−14.3303	−0.207686
$70$	0	0
$71$	− 32.1304i	− 0.452541i	−0.974064	−	0.226271i	$-0.927347\pi$
$71$	− 32.1304i	− 0.452541i	0.974064	−	0.226271i	$-0.0726533\pi$
$72$	0	0
$73$	−45.6515	−0.625363	−0.312682	−	0.949858i	$-0.601227\pi$
$73$	−45.6515	−0.625363	−0.312682	+	0.949858i	$0.601227\pi$
$74$	0	0
$75$	− 26.9898i	− 0.359864i
$76$	0	0
$77$	−4.83485	−0.0627902
$78$	0	0
$79$	10.9644i	0.138790i	0.997589	+	0.0693950i	$0.0221069\pi$
$79$	10.9644i	0.138790i	−0.997589	+	0.0693950i	$0.977893\pi$
$80$	0	0
$81$	−69.1561	−0.853779
$82$	0	0
$83$	143.705i	1.73138i	0.500579	+	0.865691i	$0.333120\pi$
$83$	143.705i	1.73138i	−0.500579	+	0.865691i	$0.666880\pi$
$84$	0	0
$85$	−140.486	−1.65278
$86$	0	0
$87$	− 235.639i	− 2.70849i
$88$	0	0
$89$	59.3212	0.666530	0.333265	−	0.942833i	$-0.391850\pi$
$89$	59.3212	0.666530	0.333265	+	0.942833i	$0.391850\pi$
$90$	0	0
$91$	54.8933i	0.603223i
$92$	0	0
$93$	−127.652	−1.37260
$94$	0	0
$95$	171.076i	1.80080i
$96$	0	0
$97$	17.1652	0.176960	0.0884802	−	0.996078i	$-0.471799\pi$
$97$	17.1652	0.176960	0.0884802	+	0.996078i	$0.471799\pi$
$98$	0	0
$99$	18.5758i	0.187634i
$100$	0	0
$101$	8.74773	0.0866112	0.0433056	−	0.999062i	$-0.486211\pi$
$101$	8.74773	0.0866112	0.0433056	+	0.999062i	$0.486211\pi$
$102$	0	0
$103$	− 54.7424i	− 0.531480i	−0.964045	−	0.265740i	$-0.914384\pi$
$103$	− 54.7424i	− 0.531480i	0.964045	−	0.265740i	$-0.0856162\pi$
$104$	0	0
$105$	64.6606	0.615815
$106$	0	0
$107$	127.377i	1.19044i	0.803561	+	0.595222i	$0.202936\pi$
$107$	127.377i	1.19044i	−0.803561	+	0.595222i	$0.797064\pi$
$108$	0	0
$109$	−48.5045	−0.444996	−0.222498	−	0.974933i	$-0.571421\pi$
$109$	−48.5045	−0.444996	−0.222498	+	0.974933i	$0.571421\pi$
$110$	0	0
$111$	− 75.1456i	− 0.676988i
$112$	0	0
$113$	−36.5045	−0.323049	−0.161525	−	0.986869i	$-0.551641\pi$
$113$	−36.5045	−0.323049	−0.161525	+	0.986869i	$0.551641\pi$
$114$	0	0
$115$	18.2740i	0.158904i
$116$	0	0
$117$	210.904	1.80260
$118$	0	0
$119$	− 66.5807i	− 0.559502i
$120$	0	0
$121$	117.661	0.972402
$122$	0	0
$123$	− 32.8136i	− 0.266777i
$124$	0	0
$125$	105.147	0.841176
$126$	0	0
$127$	41.9506i	0.330320i	0.986267	+	0.165160i	$0.0528140\pi$
$127$	41.9506i	0.330320i	−0.986267	+	0.165160i	$0.947186\pi$
$128$	0	0
$129$	250.642	1.94296
$130$	0	0
$131$	− 47.3930i	− 0.361779i	−0.983503	−	0.180889i	$-0.942102\pi$
$131$	− 47.3930i	− 0.361779i	0.983503	−	0.180889i	$-0.0578976\pi$
$132$	0	0
$133$	−81.0780	−0.609609
$134$	0	0
$135$	− 28.4756i	− 0.210930i
$136$	0	0
$137$	153.652	1.12154	0.560772	−	0.827970i	$-0.310504\pi$
$137$	153.652	1.12154	0.560772	+	0.827970i	$0.310504\pi$
$138$	0	0
$139$	147.439i	1.06071i	0.847775	+	0.530356i	$0.177942\pi$
$139$	147.439i	1.06071i	−0.847775	+	0.530356i	$0.822058\pi$
$140$	0	0
$141$	162.991	1.15596
$142$	0	0
$143$	37.9144i	0.265136i
$144$	0	0
$145$	−300.486	−2.07232
$146$	0	0
$147$	30.6446i	0.208467i
$148$	0	0
$149$	78.3121	0.525585	0.262792	−	0.964852i	$-0.415357\pi$
$149$	78.3121	0.525585	0.262792	+	0.964852i	$0.415357\pi$
$150$	0	0
$151$	− 154.327i	− 1.02204i	−0.859570	−	0.511018i	$-0.829268\pi$
$151$	− 154.327i	− 1.02204i	0.859570	−	0.511018i	$-0.170732\pi$
$152$	0	0
$153$	−255.808	−1.67194
$154$	0	0
$155$	162.781i	1.05020i
$156$	0	0
$157$	−161.408	−1.02808	−0.514039	−	0.857767i	$-0.671851\pi$
$157$	−161.408	−1.02808	−0.514039	+	0.857767i	$0.671851\pi$
$158$	0	0
$159$	201.379i	1.26653i
$160$	0	0
$161$	−8.66061	−0.0537926
$162$	0	0
$163$	− 139.788i	− 0.857594i	−0.903401	−	0.428797i	$-0.858938\pi$
$163$	− 139.788i	− 0.857594i	0.903401	−	0.428797i	$-0.141062\pi$
$164$	0	0
$165$	44.6606	0.270670
$166$	0	0
$167$	− 115.348i	− 0.690709i	−0.938472	−	0.345355i	$-0.887759\pi$
$167$	− 115.348i	− 0.690709i	0.938472	−	0.345355i	$-0.112241\pi$
$168$	0	0
$169$	261.468	1.54715
$170$	0	0
$171$	311.507i	1.82168i
$172$	0	0
$173$	−85.7205	−0.495494	−0.247747	−	0.968825i	$-0.579690\pi$
$173$	−85.7205	−0.495494	−0.247747	+	0.968825i	$0.579690\pi$
$174$	0	0
$175$	− 16.3115i	− 0.0932083i
$176$	0	0
$177$	−328.798	−1.85762
$178$	0	0
$179$	112.758i	0.629934i	0.949103	+	0.314967i	$0.101994\pi$
$179$	112.758i	0.629934i	−0.949103	+	0.314967i	$0.898006\pi$
$180$	0	0
$181$	65.9311	0.364260	0.182130	−	0.983274i	$-0.441701\pi$
$181$	65.9311	0.364260	0.182130	+	0.983274i	$0.441701\pi$
$182$	0	0
$183$	− 122.960i	− 0.671912i
$184$	0	0
$185$	−95.8258	−0.517977
$186$	0	0
$187$	− 45.9868i	− 0.245919i
$188$	0	0
$189$	13.4955	0.0714045
$190$	0	0
$191$	− 207.243i	− 1.08504i	−0.840043	−	0.542520i	$-0.817470\pi$
$191$	− 207.243i	− 1.08504i	0.840043	−	0.542520i	$-0.182530\pi$
$192$	0	0
$193$	127.183	0.658981	0.329491	−	0.944159i	$-0.393123\pi$
$193$	127.183	0.658981	0.329491	+	0.944159i	$0.393123\pi$
$194$	0	0
$195$	− 507.062i	− 2.60032i
$196$	0	0
$197$	227.670	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$197$	227.670	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$198$	0	0
$199$	171.378i	0.861194i	0.902544	+	0.430597i	$0.141697\pi$
$199$	171.378i	0.861194i	−0.902544	+	0.430597i	$0.858303\pi$
$200$	0	0
$201$	−356.312	−1.77270
$202$	0	0
$203$	− 142.410i	− 0.701525i
$204$	0	0
$205$	−41.8439	−0.204117
$206$	0	0
$207$	33.2746i	0.160747i
$208$	0	0
$209$	−56.0000	−0.267943
$210$	0	0
$211$	− 149.910i	− 0.710473i	−0.934776	−	0.355237i	$-0.884400\pi$
$211$	− 149.910i	− 0.710473i	0.934776	−	0.355237i	$-0.115600\pi$
$212$	0	0
$213$	−140.661	−0.660378
$214$	0	0
$215$	− 319.619i	− 1.48660i
$216$	0	0
$217$	−77.1470	−0.355516
$218$	0	0
$219$	199.853i	0.912572i
$220$	0	0
$221$	−522.120	−2.36253
$222$	0	0
$223$	− 85.2676i	− 0.382366i	−0.981554	−	0.191183i	$-0.938768\pi$
$223$	− 85.2676i	− 0.382366i	0.981554	−	0.191183i	$-0.0612324\pi$
$224$	0	0
$225$	−62.6697	−0.278532
$226$	0	0
$227$	154.510i	0.680660i	0.940306	+	0.340330i	$0.110539\pi$
$227$	154.510i	0.680660i	−0.940306	+	0.340330i	$0.889461\pi$
$228$	0	0
$229$	330.555	1.44347	0.721736	−	0.692168i	$-0.243344\pi$
$229$	330.555	1.44347	0.721736	+	0.692168i	$0.243344\pi$
$230$	0	0
$231$	21.1660i	0.0916278i
$232$	0	0
$233$	213.339	0.915620	0.457810	−	0.889050i	$-0.348634\pi$
$233$	213.339	0.915620	0.457810	+	0.889050i	$0.348634\pi$
$234$	0	0
$235$	− 207.846i	− 0.884451i
$236$	0	0
$237$	48.0000	0.202532
$238$	0	0
$239$	− 266.545i	− 1.11525i	−0.830092	−	0.557626i	$-0.811712\pi$
$239$	− 266.545i	− 1.11525i	0.830092	−	0.557626i	$-0.188288\pi$
$240$	0	0
$241$	26.4864	0.109902	0.0549510	−	0.998489i	$-0.482500\pi$
$241$	26.4864	0.109902	0.0549510	+	0.998489i	$0.482500\pi$
$242$	0	0
$243$	348.659i	1.43481i
$244$	0	0
$245$	39.0780	0.159502
$246$	0	0
$247$	635.806i	2.57411i
$248$	0	0
$249$	629.111	2.52655
$250$	0	0
$251$	372.796i	1.48524i	0.669711	+	0.742622i	$0.266418\pi$
$251$	372.796i	1.48524i	−0.669711	+	0.742622i	$0.733582\pi$
$252$	0	0
$253$	−5.98182	−0.0236435
$254$	0	0
$255$	615.021i	2.41185i
$256$	0	0
$257$	−206.000	−0.801556	−0.400778	−	0.916175i	$-0.631260\pi$
$257$	−206.000	−0.801556	−0.400778	+	0.916175i	$0.631260\pi$
$258$	0	0
$259$	− 45.4147i	− 0.175346i
$260$	0	0
$261$	−547.147	−2.09635
$262$	0	0
$263$	− 465.493i	− 1.76994i	−0.465652	−	0.884968i	$-0.654180\pi$
$263$	− 465.493i	− 1.76994i	0.465652	−	0.884968i	$-0.345820\pi$
$264$	0	0
$265$	256.798	0.969051
$266$	0	0
$267$	− 259.697i	− 0.972646i
$268$	0	0
$269$	−390.417	−1.45137	−0.725683	−	0.688029i	$-0.758476\pi$
$269$	−390.417	−1.45137	−0.725683	+	0.688029i	$0.758476\pi$
$270$	0	0
$271$	133.543i	0.492778i	0.969171	+	0.246389i	$0.0792441\pi$
$271$	133.543i	0.492778i	−0.969171	+	0.246389i	$0.920756\pi$
$272$	0	0
$273$	240.312	0.880264
$274$	0	0
$275$	− 11.2662i	− 0.0409680i
$276$	0	0
$277$	306.624	1.10695	0.553473	−	0.832867i	$-0.313302\pi$
$277$	306.624	1.10695	0.553473	+	0.832867i	$0.313302\pi$
$278$	0	0
$279$	296.404i	1.06238i
$280$	0	0
$281$	−112.642	−0.400863	−0.200431	−	0.979708i	$-0.564234\pi$
$281$	−112.642	−0.400863	−0.200431	+	0.979708i	$0.564234\pi$
$282$	0	0
$283$	− 474.368i	− 1.67621i	−0.545507	−	0.838106i	$-0.683663\pi$
$283$	− 474.368i	− 1.67621i	0.545507	−	0.838106i	$-0.316337\pi$
$284$	0	0
$285$	748.936	2.62785
$286$	0	0
$287$	− 19.8311i	− 0.0690979i
$288$	0	0
$289$	344.285	1.19130
$290$	0	0
$291$	− 75.1456i	− 0.258232i
$292$	0	0
$293$	17.0962	0.0583489	0.0291744	−	0.999574i	$-0.490712\pi$
$293$	17.0962	0.0583489	0.0291744	+	0.999574i	$0.490712\pi$
$294$	0	0
$295$	419.284i	1.42130i
$296$	0	0
$297$	9.32121	0.0313845
$298$	0	0
$299$	67.9156i	0.227143i
$300$	0	0
$301$	151.477	0.503247
$302$	0	0
$303$	− 38.2958i	− 0.126389i
$304$	0	0
$305$	−156.798	−0.514093
$306$	0	0
$307$	− 508.468i	− 1.65625i	−0.560544	−	0.828124i	$-0.689408\pi$
$307$	− 508.468i	− 1.65625i	0.560544	−	0.828124i	$-0.310592\pi$
$308$	0	0
$309$	−239.652	−0.775571
$310$	0	0
$311$	31.2084i	0.100349i	0.998740	+	0.0501743i	$0.0159777\pi$
$311$	31.2084i	0.100349i	−0.998740	+	0.0501743i	$0.984022\pi$
$312$	0	0
$313$	−175.183	−0.559691	−0.279846	−	0.960045i	$-0.590283\pi$
$313$	−175.183	−0.559691	−0.279846	+	0.960045i	$0.590283\pi$
$314$	0	0
$315$	− 150.140i	− 0.476636i
$316$	0	0
$317$	217.652	0.686598	0.343299	−	0.939226i	$-0.388456\pi$
$317$	217.652	0.686598	0.343299	+	0.939226i	$0.388456\pi$
$318$	0	0
$319$	− 98.3612i	− 0.308342i
$320$	0	0
$321$	557.633	1.73718
$322$	0	0
$323$	− 771.176i	− 2.38754i
$324$	0	0
$325$	−127.913	−0.393578
$326$	0	0
$327$	212.343i	0.649368i
$328$	0	0
$329$	98.5045	0.299406
$330$	0	0
$331$	601.467i	1.81712i	0.417754	+	0.908560i	$0.362817\pi$
$331$	601.467i	1.81712i	−0.417754	+	0.908560i	$0.637183\pi$
$332$	0	0
$333$	−174.486	−0.523983
$334$	0	0
$335$	454.369i	1.35633i
$336$	0	0
$337$	−445.477	−1.32189	−0.660946	−	0.750434i	$-0.729845\pi$
$337$	−445.477	−1.32189	−0.660946	+	0.750434i	$0.729845\pi$
$338$	0	0
$339$	159.810i	0.471415i
$340$	0	0
$341$	−53.2848	−0.156261
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	80.0000	0.231884
$346$	0	0
$347$	404.426i	1.16549i	0.812654	+	0.582746i	$0.198022\pi$
$347$	404.426i	1.16549i	−0.812654	+	0.582746i	$0.801978\pi$
$348$	0	0
$349$	157.895	0.452420	0.226210	−	0.974079i	$-0.427366\pi$
$349$	157.895	0.452420	0.226210	+	0.974079i	$0.427366\pi$
$350$	0	0
$351$	− 105.830i	− 0.301510i
$352$	0	0
$353$	−321.267	−0.910104	−0.455052	−	0.890465i	$-0.650379\pi$
$353$	−321.267	−0.910104	−0.455052	+	0.890465i	$0.650379\pi$
$354$	0	0
$355$	179.370i	0.505269i
$356$	0	0
$357$	−291.477	−0.816463
$358$	0	0
$359$	− 173.968i	− 0.484590i	−0.970203	−	0.242295i	$-0.922100\pi$
$359$	− 173.968i	− 0.484590i	0.970203	−	0.242295i	$-0.0779002\pi$
$360$	0	0
$361$	−578.092	−1.60136
$362$	0	0
$363$	− 515.095i	− 1.41899i
$364$	0	0
$365$	254.853	0.698227
$366$	0	0
$367$	278.050i	0.757630i	0.925472	+	0.378815i	$0.123668\pi$
$367$	278.050i	0.757630i	−0.925472	+	0.378815i	$0.876332\pi$
$368$	0	0
$369$	−76.1924	−0.206484
$370$	0	0
$371$	121.705i	0.328045i
$372$	0	0
$373$	−342.973	−0.919498	−0.459749	−	0.888049i	$-0.652061\pi$
$373$	−342.973	−0.919498	−0.459749	+	0.888049i	$0.652061\pi$
$374$	0	0
$375$	− 460.313i	− 1.22750i
$376$	0	0
$377$	−1116.76	−2.96223
$378$	0	0
$379$	− 145.731i	− 0.384515i	−0.981345	−	0.192257i	$-0.938419\pi$
$379$	− 145.731i	− 0.384515i	0.981345	−	0.192257i	$-0.0615808\pi$
$380$	0	0
$381$	183.652	0.482025
$382$	0	0
$383$	− 224.071i	− 0.585040i	−0.956259	−	0.292520i	$-0.905506\pi$
$383$	− 224.071i	− 0.585040i	0.956259	−	0.292520i	$-0.0944939\pi$
$384$	0	0
$385$	26.9909	0.0701063
$386$	0	0
$387$	− 581.986i	− 1.50384i
$388$	0	0
$389$	99.8076	0.256575	0.128287	−	0.991737i	$-0.459052\pi$
$389$	99.8076	0.256575	0.128287	+	0.991737i	$0.459052\pi$
$390$	0	0
$391$	− 82.3756i	− 0.210679i
$392$	0	0
$393$	−207.477	−0.527932
$394$	0	0
$395$	− 61.2096i	− 0.154961i
$396$	0	0
$397$	−163.354	−0.411470	−0.205735	−	0.978608i	$-0.565959\pi$
$397$	−163.354	−0.411470	−0.205735	+	0.978608i	$0.565959\pi$
$398$	0	0
$399$	354.944i	0.889583i
$400$	0	0
$401$	−123.459	−0.307878	−0.153939	−	0.988080i	$-0.549196\pi$
$401$	−123.459	−0.307878	−0.153939	+	0.988080i	$0.549196\pi$
$402$	0	0
$403$	604.979i	1.50119i
$404$	0	0
$405$	386.069	0.953257
$406$	0	0
$407$	− 31.3676i	− 0.0770703i
$408$	0	0
$409$	7.84394	0.0191783	0.00958917	−	0.999954i	$-0.496948\pi$
$409$	7.84394	0.0191783	0.00958917	+	0.999954i	$0.496948\pi$
$410$	0	0
$411$	− 672.656i	− 1.63663i
$412$	0	0
$413$	−198.711	−0.481141
$414$	0	0
$415$	− 802.242i	− 1.93311i
$416$	0	0
$417$	645.459	1.54786
$418$	0	0
$419$	364.565i	0.870083i	0.900410	+	0.435041i	$0.143266\pi$
$419$	364.565i	0.870083i	−0.900410	+	0.435041i	$0.856734\pi$
$420$	0	0
$421$	708.570	1.68306	0.841532	−	0.540208i	$-0.181654\pi$
$421$	708.570	1.68306	0.841532	+	0.540208i	$0.181654\pi$
$422$	0	0
$423$	− 378.461i	− 0.894707i
$424$	0	0
$425$	155.147	0.365052
$426$	0	0
$427$	− 74.3115i	− 0.174032i
$428$	0	0
$429$	165.982	0.386904
$430$	0	0
$431$	− 80.9463i	− 0.187810i	−0.995581	−	0.0939052i	$-0.970065\pi$
$431$	− 80.9463i	− 0.187810i	0.995581	−	0.0939052i	$-0.0299350\pi$
$432$	0	0
$433$	295.495	0.682438	0.341219	−	0.939984i	$-0.389160\pi$
$433$	295.495	0.682438	0.341219	+	0.939984i	$0.389160\pi$
$434$	0	0
$435$	1315.47i	3.02407i
$436$	0	0
$437$	−100.312	−0.229547
$438$	0	0
$439$	− 11.5680i	− 0.0263508i	−0.999913	−	0.0131754i	$-0.995806\pi$
$439$	− 11.5680i	− 0.0263508i	0.999913	−	0.0131754i	$-0.00419398\pi$
$440$	0	0
$441$	71.1561	0.161352
$442$	0	0
$443$	− 284.660i	− 0.642573i	−0.946982	−	0.321287i	$-0.895885\pi$
$443$	− 284.660i	− 0.642573i	0.946982	−	0.321287i	$-0.104115\pi$
$444$	0	0
$445$	−331.165	−0.744191
$446$	0	0
$447$	− 342.835i	− 0.766969i
$448$	0	0
$449$	154.624	0.344375	0.172187	−	0.985064i	$-0.444917\pi$
$449$	154.624	0.344375	0.172187	+	0.985064i	$0.444917\pi$
$450$	0	0
$451$	− 13.6972i	− 0.0303707i
$452$	0	0
$453$	−675.615	−1.49142
$454$	0	0
$455$	− 306.446i	− 0.673508i
$456$	0	0
$457$	410.450	0.898140	0.449070	−	0.893497i	$-0.351755\pi$
$457$	410.450	0.898140	0.449070	+	0.893497i	$0.351755\pi$
$458$	0	0
$459$	128.362i	0.279657i
$460$	0	0
$461$	−595.078	−1.29084	−0.645421	−	0.763827i	$-0.723318\pi$
$461$	−595.078	−1.29084	−0.645421	+	0.763827i	$0.723318\pi$
$462$	0	0
$463$	511.924i	1.10567i	0.833291	+	0.552834i	$0.186454\pi$
$463$	511.924i	1.10567i	−0.833291	+	0.552834i	$0.813546\pi$
$464$	0	0
$465$	712.624	1.53253
$466$	0	0
$467$	365.122i	0.781846i	0.920423	+	0.390923i	$0.127844\pi$
$467$	365.122i	0.781846i	−0.920423	+	0.390923i	$0.872156\pi$
$468$	0	0
$469$	−215.339	−0.459146
$470$	0	0
$471$	706.614i	1.50024i
$472$	0	0
$473$	104.624	0.221193
$474$	0	0
$475$	− 188.929i	− 0.397745i
$476$	0	0
$477$	467.597	0.980287
$478$	0	0
$479$	− 202.745i	− 0.423268i	−0.977349	−	0.211634i	$-0.932122\pi$
$479$	− 202.745i	− 0.423268i	0.977349	−	0.211634i	$-0.0678785\pi$
$480$	0	0
$481$	−356.138	−0.740411
$482$	0	0
$483$	37.9144i	0.0784978i
$484$	0	0
$485$	−95.8258	−0.197579
$486$	0	0
$487$	821.120i	1.68608i	0.537853	+	0.843039i	$0.319236\pi$
$487$	821.120i	1.68608i	−0.537853	+	0.843039i	$0.680764\pi$
$488$	0	0
$489$	−611.964	−1.25146
$490$	0	0
$491$	316.505i	0.644613i	0.946635	+	0.322307i	$0.104458\pi$
$491$	316.505i	0.644613i	−0.946635	+	0.322307i	$0.895542\pi$
$492$	0	0
$493$	1354.53	2.74753
$494$	0	0
$495$	− 103.701i	− 0.209497i
$496$	0	0
$497$	−85.0091	−0.171044
$498$	0	0
$499$	176.415i	0.353538i	0.984252	+	0.176769i	$0.0565646\pi$
$499$	176.415i	0.353538i	−0.984252	+	0.176769i	$0.943435\pi$
$500$	0	0
$501$	−504.973	−1.00793
$502$	0	0
$503$	883.918i	1.75729i	0.477474	+	0.878646i	$0.341553\pi$
$503$	883.918i	1.75729i	−0.477474	+	0.878646i	$0.658447\pi$
$504$	0	0
$505$	−48.8348	−0.0967027
$506$	0	0
$507$	− 1144.66i	− 2.25770i
$508$	0	0
$509$	519.216	1.02007	0.510035	−	0.860153i	$-0.329632\pi$
$509$	519.216	1.02007	0.510035	+	0.860153i	$0.329632\pi$
$510$	0	0
$511$	120.783i	0.236365i
$512$	0	0
$513$	156.312	0.304702
$514$	0	0
$515$	305.604i	0.593405i
$516$	0	0
$517$	68.0364	0.131598
$518$	0	0
$519$	375.267i	0.723058i
$520$	0	0
$521$	1037.40	1.99118	0.995590	−	0.0938137i	$-0.0299058\pi$
$521$	1037.40	1.99118	0.995590	+	0.0938137i	$0.0299058\pi$
$522$	0	0
$523$	525.900i	1.00555i	0.864419	+	0.502773i	$0.167687\pi$
$523$	525.900i	1.00555i	−0.864419	+	0.502773i	$0.832313\pi$
$524$	0	0
$525$	−71.4083	−0.136016
$526$	0	0
$527$	− 733.786i	− 1.39238i
$528$	0	0
$529$	518.285	0.979745
$530$	0	0
$531$	763.462i	1.43778i
$532$	0	0
$533$	−155.514	−0.291770
$534$	0	0
$535$	− 711.094i	− 1.32915i
$536$	0	0
$537$	493.633	0.919243
$538$	0	0
$539$	12.7918i	0.0237325i
$540$	0	0
$541$	−776.221	−1.43479	−0.717395	−	0.696667i	$-0.754666\pi$
$541$	−776.221	−1.43479	−0.717395	+	0.696667i	$0.754666\pi$
$542$	0	0
$543$	− 288.633i	− 0.531553i
$544$	0	0
$545$	270.780	0.496845
$546$	0	0
$547$	− 153.041i	− 0.279782i	−0.990167	−	0.139891i	$-0.955325\pi$
$547$	− 153.041i	− 0.279782i	0.990167	−	0.139891i	$-0.0446752\pi$
$548$	0	0
$549$	−285.510	−0.520054
$550$	0	0
$551$	− 1649.47i	− 2.99359i
$552$	0	0
$553$	29.0091	0.0524577
$554$	0	0
$555$	419.506i	0.755867i
$556$	0	0
$557$	481.579	0.864594	0.432297	−	0.901731i	$-0.357703\pi$
$557$	481.579	0.864594	0.432297	+	0.901731i	$0.357703\pi$
$558$	0	0
$559$	− 1187.87i	− 2.12499i
$560$	0	0
$561$	−201.321	−0.358861
$562$	0	0
$563$	489.909i	0.870176i	0.900388	+	0.435088i	$0.143283\pi$
$563$	489.909i	0.870176i	−0.900388	+	0.435088i	$0.856717\pi$
$564$	0	0
$565$	203.789	0.360689
$566$	0	0
$567$	182.970i	0.322698i
$568$	0	0
$569$	97.8985	0.172054	0.0860268	−	0.996293i	$-0.472583\pi$
$569$	97.8985	0.172054	0.0860268	+	0.996293i	$0.472583\pi$
$570$	0	0
$571$	− 752.680i	− 1.31818i	−0.752065	−	0.659089i	$-0.770942\pi$
$571$	− 752.680i	− 1.31818i	0.752065	−	0.659089i	$-0.229058\pi$
$572$	0	0
$573$	−907.267	−1.58336
$574$	0	0
$575$	− 20.1810i	− 0.0350974i
$576$	0	0
$577$	−924.533	−1.60231	−0.801155	−	0.598456i	$-0.795781\pi$
$577$	−924.533	−1.60231	−0.801155	+	0.598456i	$0.795781\pi$
$578$	0	0
$579$	− 556.783i	− 0.961629i
$580$	0	0
$581$	380.207	0.654401
$582$	0	0
$583$	84.0604i	0.144186i
$584$	0	0
$585$	−1177.39	−2.01263
$586$	0	0
$587$	1110.24i	1.89138i	0.325077	+	0.945688i	$0.394610\pi$
$587$	1110.24i	1.89138i	−0.325077	+	0.945688i	$0.605390\pi$
$588$	0	0
$589$	−893.561	−1.51708
$590$	0	0
$591$	− 996.693i	− 1.68645i
$592$	0	0
$593$	44.9909	0.0758700	0.0379350	−	0.999280i	$-0.487922\pi$
$593$	44.9909	0.0758700	0.0379350	+	0.999280i	$0.487922\pi$
$594$	0	0
$595$	371.692i	0.624692i
$596$	0	0
$597$	750.258	1.25671
$598$	0	0
$599$	988.382i	1.65005i	0.565094	+	0.825026i	$0.308840\pi$
$599$	988.382i	1.65005i	−0.565094	+	0.825026i	$0.691160\pi$
$600$	0	0
$601$	732.570	1.21892	0.609459	−	0.792818i	$-0.291387\pi$
$601$	732.570	1.21892	0.609459	+	0.792818i	$0.291387\pi$
$602$	0	0
$603$	827.348i	1.37205i
$604$	0	0
$605$	−656.849	−1.08570
$606$	0	0
$607$	145.874i	0.240319i	0.992755	+	0.120160i	$0.0383406\pi$
$607$	145.874i	0.240319i	−0.992755	+	0.120160i	$0.961659\pi$
$608$	0	0
$609$	−623.441	−1.02371
$610$	0	0
$611$	− 772.463i	− 1.26426i
$612$	0	0
$613$	100.577	0.164074	0.0820369	−	0.996629i	$-0.473857\pi$
$613$	100.577	0.164074	0.0820369	+	0.996629i	$0.473857\pi$
$614$	0	0
$615$	183.184i	0.297861i
$616$	0	0
$617$	−342.102	−0.554459	−0.277230	−	0.960804i	$-0.589416\pi$
$617$	−342.102	−0.554459	−0.277230	+	0.960804i	$0.589416\pi$
$618$	0	0
$619$	− 512.123i	− 0.827340i	−0.910427	−	0.413670i	$-0.864247\pi$
$619$	− 512.123i	− 0.827340i	0.910427	−	0.413670i	$-0.135753\pi$
$620$	0	0
$621$	16.6970	0.0268872
$622$	0	0
$623$	− 156.949i	− 0.251925i
$624$	0	0
$625$	−741.120	−1.18579
$626$	0	0
$627$	245.157i	0.391000i
$628$	0	0
$629$	431.964	0.686747
$630$	0	0
$631$	− 530.517i	− 0.840755i	−0.907349	−	0.420378i	$-0.861898\pi$
$631$	− 530.517i	− 0.840755i	0.907349	−	0.420378i	$-0.138102\pi$
$632$	0	0
$633$	−656.276	−1.03677
$634$	0	0
$635$	− 234.193i	− 0.368807i
$636$	0	0
$637$	145.234	0.227997
$638$	0	0
$639$	326.611i	0.511128i
$640$	0	0
$641$	−1054.80	−1.64555	−0.822776	−	0.568366i	$-0.807576\pi$
$641$	−1054.80	−1.64555	−0.822776	+	0.568366i	$0.807576\pi$
$642$	0	0
$643$	− 107.316i	− 0.166899i	−0.996512	−	0.0834493i	$-0.973406\pi$
$643$	− 107.316i	− 0.166899i	0.996512	−	0.0834493i	$-0.0265937\pi$
$644$	0	0
$645$	−1399.23	−2.16935
$646$	0	0
$647$	202.142i	0.312429i	0.987723	+	0.156215i	$0.0499291\pi$
$647$	202.142i	0.312429i	−0.987723	+	0.156215i	$0.950071\pi$
$648$	0	0
$649$	−137.248	−0.211477
$650$	0	0
$651$	337.734i	0.518793i
$652$	0	0
$653$	1278.73	1.95823	0.979116	−	0.203301i	$-0.0651671\pi$
$653$	1278.73	1.95823	0.979116	+	0.203301i	$0.0651671\pi$
$654$	0	0
$655$	264.575i	0.403931i
$656$	0	0
$657$	464.055	0.706324
$658$	0	0
$659$	− 835.500i	− 1.26783i	−0.773403	−	0.633915i	$-0.781447\pi$
$659$	− 835.500i	− 1.26783i	0.773403	−	0.633915i	$-0.218553\pi$
$660$	0	0
$661$	998.381	1.51041	0.755205	−	0.655489i	$-0.227537\pi$
$661$	998.381	1.51041	0.755205	+	0.655489i	$0.227537\pi$
$662$	0	0
$663$	2285.74i	3.44757i
$664$	0	0
$665$	452.624	0.680638
$666$	0	0
$667$	− 176.193i	− 0.264158i
$668$	0	0
$669$	−373.285	−0.557974
$670$	0	0
$671$	− 51.3264i	− 0.0764925i
$672$	0	0
$673$	−572.330	−0.850416	−0.425208	−	0.905096i	$-0.639799\pi$
$673$	−572.330	−0.850416	−0.425208	+	0.905096i	$0.639799\pi$
$674$	0	0
$675$	31.4472i	0.0465885i
$676$	0	0
$677$	1079.77	1.59493	0.797465	−	0.603365i	$-0.206174\pi$
$677$	1079.77	1.59493	0.797465	+	0.603365i	$0.206174\pi$
$678$	0	0
$679$	− 45.4147i	− 0.0668847i
$680$	0	0
$681$	676.414	0.993265
$682$	0	0
$683$	− 444.708i	− 0.651110i	−0.945523	−	0.325555i	$-0.894449\pi$
$683$	− 444.708i	− 0.651110i	0.945523	−	0.325555i	$-0.105551\pi$
$684$	0	0
$685$	−857.771	−1.25222
$686$	0	0
$687$	− 1447.11i	− 2.10641i
$688$	0	0
$689$	954.395	1.38519
$690$	0	0
$691$	214.227i	0.310025i	0.987913	+	0.155012i	$0.0495417\pi$
$691$	214.227i	0.310025i	−0.987913	+	0.155012i	$0.950458\pi$
$692$	0	0
$693$	49.1470	0.0709191
$694$	0	0
$695$	− 823.090i	− 1.18430i
$696$	0	0
$697$	188.624	0.270623
$698$	0	0
$699$	− 933.958i	− 1.33613i
$700$	0	0
$701$	573.477	0.818085	0.409042	−	0.912515i	$-0.365863\pi$
$701$	573.477	0.818085	0.409042	+	0.912515i	$0.365863\pi$
$702$	0	0
$703$	− 526.019i	− 0.748250i
$704$	0	0
$705$	−909.909	−1.29065
$706$	0	0
$707$	− 23.1443i	− 0.0327359i
$708$	0	0
$709$	−436.817	−0.616102	−0.308051	−	0.951370i	$-0.599677\pi$
$709$	−436.817	−0.616102	−0.308051	+	0.951370i	$0.599677\pi$
$710$	0	0
$711$	− 111.455i	− 0.156758i
$712$	0	0
$713$	−95.4485	−0.133869
$714$	0	0
$715$	− 211.660i	− 0.296028i
$716$	0	0
$717$	−1166.88	−1.62745
$718$	0	0
$719$	1279.32i	1.77930i	0.456639	+	0.889652i	$0.349053\pi$
$719$	1279.32i	1.77930i	−0.456639	+	0.889652i	$0.650947\pi$
$720$	0	0
$721$	−144.835	−0.200881
$722$	0	0
$723$	− 115.952i	− 0.160376i
$724$	0	0
$725$	331.844	0.457716
$726$	0	0
$727$	166.834i	0.229483i	0.993395	+	0.114741i	$0.0366040\pi$
$727$	166.834i	0.229483i	−0.993395	+	0.114741i	$0.963396\pi$
$728$	0	0
$729$	903.955	1.23999
$730$	0	0
$731$	1440.78i	1.97097i
$732$	0	0
$733$	105.299	0.143655	0.0718276	−	0.997417i	$-0.477117\pi$
$733$	105.299	0.143655	0.0718276	+	0.997417i	$0.477117\pi$
$734$	0	0
$735$	− 171.076i	− 0.232756i
$736$	0	0
$737$	−148.733	−0.201809
$738$	0	0
$739$	− 898.713i	− 1.21612i	−0.793891	−	0.608060i	$-0.791948\pi$
$739$	− 898.713i	− 1.21612i	0.793891	−	0.608060i	$-0.208052\pi$
$740$	0	0
$741$	2783.43	3.75632
$742$	0	0
$743$	1154.15i	1.55337i	0.629891	+	0.776683i	$0.283099\pi$
$743$	1154.15i	1.55337i	−0.629891	+	0.776683i	$0.716901\pi$
$744$	0	0
$745$	−437.183	−0.586823
$746$	0	0
$747$	− 1460.78i	− 1.95553i
$748$	0	0
$749$	337.009	0.449945
$750$	0	0
$751$	561.025i	0.747038i	0.927623	+	0.373519i	$0.121849\pi$
$751$	561.025i	0.747038i	−0.927623	+	0.373519i	$0.878151\pi$
$752$	0	0
$753$	1632.03	2.16737
$754$	0	0
$755$	861.545i	1.14112i
$756$	0	0
$757$	−860.120	−1.13622	−0.568111	−	0.822952i	$-0.692325\pi$
$757$	−860.120	−1.13622	−0.568111	+	0.822952i	$0.692325\pi$
$758$	0	0
$759$	26.1872i	0.0345023i
$760$	0	0
$761$	1336.47	1.75620	0.878100	−	0.478477i	$-0.158811\pi$
$761$	1336.47	1.75620	0.878100	+	0.478477i	$0.158811\pi$
$762$	0	0
$763$	128.331i	0.168193i
$764$	0	0
$765$	1428.07	1.86675
$766$	0	0
$767$	1558.28i	2.03165i
$768$	0	0
$769$	−716.323	−0.931499	−0.465749	−	0.884917i	$-0.654215\pi$
$769$	−716.323	−0.931499	−0.465749	+	0.884917i	$0.654215\pi$
$770$	0	0
$771$	901.827i	1.16969i
$772$	0	0
$773$	−98.6644	−0.127638	−0.0638191	−	0.997961i	$-0.520328\pi$
$773$	−98.6644	−0.127638	−0.0638191	+	0.997961i	$0.520328\pi$
$774$	0	0
$775$	− 179.768i	− 0.231959i
$776$	0	0
$777$	−198.817	−0.255877
$778$	0	0
$779$	− 229.695i	− 0.294859i
$780$	0	0
$781$	−58.7152	−0.0751795
$782$	0	0
$783$	274.555i	0.350644i
$784$	0	0
$785$	901.074	1.14787
$786$	0	0
$787$	673.903i	0.856293i	0.903709	+	0.428147i	$0.140833\pi$
$787$	673.903i	0.856293i	−0.903709	+	0.428147i	$0.859167\pi$
$788$	0	0
$789$	−2037.84	−2.58281
$790$	0	0
$791$	96.5819i	0.122101i
$792$	0	0
$793$	−582.744	−0.734860
$794$	0	0
$795$	− 1124.21i	− 1.41410i
$796$	0	0
$797$	−1097.76	−1.37736	−0.688681	−	0.725065i	$-0.741810\pi$
$797$	−1097.76	−1.37736	−0.688681	+	0.725065i	$0.741810\pi$
$798$	0	0
$799$	936.929i	1.17263i
$800$	0	0
$801$	−603.009	−0.752820
$802$	0	0
$803$	83.4236i	0.103890i
$804$	0	0
$805$	48.3485	0.0600602
$806$	0	0
$807$	1709.17i	2.11793i
$808$	0	0
$809$	−340.853	−0.421326	−0.210663	−	0.977559i	$-0.567562\pi$
$809$	−340.853	−0.421326	−0.210663	+	0.977559i	$0.567562\pi$
$810$	0	0
$811$	112.655i	0.138909i	0.997585	+	0.0694547i	$0.0221259\pi$
$811$	112.655i	0.138909i	−0.997585	+	0.0694547i	$0.977874\pi$
$812$	0	0
$813$	584.624	0.719095
$814$	0	0
$815$	780.376i	0.957517i
$816$	0	0
$817$	1754.50	2.14749
$818$	0	0
$819$	− 557.999i	− 0.681317i
$820$	0	0
$821$	−1267.91	−1.54435	−0.772174	−	0.635412i	$-0.780830\pi$
$821$	−1267.91	−1.54435	−0.772174	+	0.635412i	$0.780830\pi$
$822$	0	0
$823$	− 743.669i	− 0.903608i	−0.892117	−	0.451804i	$-0.850781\pi$
$823$	− 743.669i	− 0.903608i	0.892117	−	0.451804i	$-0.149219\pi$
$824$	0	0
$825$	−49.3212	−0.0597833
$826$	0	0
$827$	− 1473.29i	− 1.78149i	−0.454503	−	0.890745i	$-0.650183\pi$
$827$	− 1473.29i	− 1.78149i	0.454503	−	0.890745i	$-0.349817\pi$
$828$	0	0
$829$	−379.840	−0.458191	−0.229095	−	0.973404i	$-0.573577\pi$
$829$	−379.840	−0.458191	−0.229095	+	0.973404i	$0.573577\pi$
$830$	0	0
$831$	− 1342.34i	− 1.61533i
$832$	0	0
$833$	−176.156	−0.211472
$834$	0	0
$835$	643.941i	0.771187i
$836$	0	0
$837$	148.733	0.177698
$838$	0	0
$839$	− 1080.75i	− 1.28814i	−0.764965	−	0.644072i	$-0.777244\pi$
$839$	− 1080.75i	− 1.28814i	0.764965	−	0.644072i	$-0.222756\pi$
$840$	0	0
$841$	2056.21	2.44496
$842$	0	0
$843$	493.126i	0.584966i
$844$	0	0
$845$	−1459.67	−1.72742
$846$	0	0
$847$	− 311.301i	− 0.367533i
$848$	0	0
$849$	−2076.69	−2.44604
$850$	0	0
$851$	− 56.1884i	− 0.0660264i
$852$	0	0
$853$	−125.372	−0.146978	−0.0734888	−	0.997296i	$-0.523413\pi$
$853$	−125.372	−0.146978	−0.0734888	+	0.997296i	$0.523413\pi$
$854$	0	0
$855$	− 1739.01i	− 2.03393i
$856$	0	0
$857$	721.717	0.842143	0.421072	−	0.907027i	$-0.361654\pi$
$857$	721.717	0.842143	0.421072	+	0.907027i	$0.361654\pi$
$858$	0	0
$859$	− 104.742i	− 0.121935i	−0.998140	−	0.0609676i	$-0.980581\pi$
$859$	− 104.742i	− 0.121935i	0.998140	−	0.0609676i	$-0.0194186\pi$
$860$	0	0
$861$	−86.8167	−0.100832
$862$	0	0
$863$	425.005i	0.492474i	0.969210	+	0.246237i	$0.0791941\pi$
$863$	425.005i	0.492474i	−0.969210	+	0.246237i	$0.920806\pi$
$864$	0	0
$865$	478.541	0.553226
$866$	0	0
$867$	− 1507.21i	− 1.73842i
$868$	0	0
$869$	20.0364	0.0230568
$870$	0	0
$871$	1688.67i	1.93877i
$872$	0	0
$873$	−174.486	−0.199870
$874$	0	0
$875$	− 278.193i	− 0.317935i
$876$	0	0
$877$	−469.368	−0.535197	−0.267599	−	0.963530i	$-0.586230\pi$
$877$	−469.368	−0.535197	−0.267599	+	0.963530i	$0.586230\pi$
$878$	0	0
$879$	− 74.8438i	− 0.0851466i
$880$	0	0
$881$	−1116.11	−1.26687	−0.633435	−	0.773796i	$-0.718356\pi$
$881$	−1116.11	−1.26687	−0.633435	+	0.773796i	$0.718356\pi$
$882$	0	0
$883$	271.281i	0.307227i	0.988131	+	0.153613i	$0.0490910\pi$
$883$	271.281i	0.307227i	−0.988131	+	0.153613i	$0.950909\pi$
$884$	0	0
$885$	1835.54	2.07406
$886$	0	0
$887$	− 211.421i	− 0.238355i	−0.992873	−	0.119178i	$-0.961974\pi$
$887$	− 211.421i	− 0.238355i	0.992873	−	0.119178i	$-0.0380258\pi$
$888$	0	0
$889$	110.991	0.124849
$890$	0	0
$891$	126.376i	0.141836i
$892$	0	0
$893$	1140.94	1.27764
$894$	0	0
$895$	− 629.481i	− 0.703331i
$896$	0	0
$897$	297.321	0.331462
$898$	0	0
$899$	− 1569.50i	− 1.74582i
$900$	0	0
$901$	−1157.60	−1.28479
$902$	0	0
$903$	− 663.138i	− 0.734372i
$904$	0	0
$905$	−368.065	−0.406702
$906$	0	0
$907$	− 410.290i	− 0.452359i	−0.974086	−	0.226179i	$-0.927376\pi$
$907$	− 410.290i	− 0.452359i	0.974086	−	0.226179i	$-0.0726236\pi$
$908$	0	0
$909$	−88.9220	−0.0978239
$910$	0	0
$911$	− 608.982i	− 0.668476i	−0.942489	−	0.334238i	$-0.891521\pi$
$911$	− 608.982i	− 0.668476i	0.942489	−	0.334238i	$-0.108479\pi$
$912$	0	0
$913$	262.606	0.287630
$914$	0	0
$915$	686.433i	0.750200i
$916$	0	0
$917$	−125.390	−0.136740
$918$	0	0
$919$	− 497.942i	− 0.541830i	−0.962603	−	0.270915i	$-0.912674\pi$
$919$	− 497.942i	− 0.541830i	0.962603	−	0.270915i	$-0.0873262\pi$
$920$	0	0
$921$	−2225.97	−2.41691
$922$	0	0
$923$	666.633i	0.722246i
$924$	0	0
$925$	105.826	0.114406
$926$	0	0
$927$	556.465i	0.600286i
$928$	0	0
$929$	−65.4045	−0.0704032	−0.0352016	−	0.999380i	$-0.511207\pi$
$929$	−65.4045	−0.0704032	−0.0352016	+	0.999380i	$0.511207\pi$
$930$	0	0
$931$	214.512i	0.230411i
$932$	0	0
$933$	136.624	0.146435
$934$	0	0
$935$	256.725i	0.274572i
$936$	0	0
$937$	−600.918	−0.641321	−0.320661	−	0.947194i	$-0.603905\pi$
$937$	−600.918	−0.641321	−0.320661	+	0.947194i	$0.603905\pi$
$938$	0	0
$939$	766.918i	0.816739i
$940$	0	0
$941$	−1482.87	−1.57584	−0.787921	−	0.615776i	$-0.788843\pi$
$941$	−1482.87	−1.57584	−0.787921	+	0.615776i	$0.788843\pi$
$942$	0	0
$943$	− 24.5356i	− 0.0260187i
$944$	0	0
$945$	−75.3394	−0.0797242
$946$	0	0
$947$	806.865i	0.852023i	0.904718	+	0.426011i	$0.140082\pi$
$947$	806.865i	0.852023i	−0.904718	+	0.426011i	$0.859918\pi$
$948$	0	0
$949$	947.165	0.998067
$950$	0	0
$951$	− 952.835i	− 1.00193i
$952$	0	0
$953$	899.945	0.944329	0.472164	−	0.881510i	$-0.343473\pi$
$953$	899.945	0.944329	0.472164	+	0.881510i	$0.343473\pi$
$954$	0	0
$955$	1156.95i	1.21146i
$956$	0	0
$957$	−430.606	−0.449954
$958$	0	0
$959$	− 406.524i	− 0.423904i
$960$	0	0
$961$	110.764	0.115259
$962$	0	0
$963$	− 1294.81i	− 1.34456i
$964$	0	0
$965$	−710.011	−0.735762
$966$	0	0
$967$	1725.92i	1.78482i	0.451228	+	0.892409i	$0.350986\pi$
$967$	1725.92i	1.78482i	−0.451228	+	0.892409i	$0.649014\pi$
$968$	0	0
$969$	−3376.06	−3.48406
$970$	0	0
$971$	657.951i	0.677601i	0.940858	+	0.338800i	$0.110021\pi$
$971$	657.951i	0.677601i	−0.940858	+	0.338800i	$0.889979\pi$
$972$	0	0
$973$	390.087	0.400912
$974$	0	0
$975$	559.977i	0.574336i
$976$	0	0
$977$	46.9364	0.0480413	0.0240207	−	0.999711i	$-0.492353\pi$
$977$	46.9364	0.0480413	0.0240207	+	0.999711i	$0.492353\pi$
$978$	0	0
$979$	− 108.404i	− 0.110729i
$980$	0	0
$981$	493.056	0.502606
$982$	0	0
$983$	377.095i	0.383616i	0.981432	+	0.191808i	$0.0614351\pi$
$983$	377.095i	0.383616i	−0.981432	+	0.191808i	$0.938565\pi$
$984$	0	0
$985$	−1270.98	−1.29034
$986$	0	0
$987$	− 431.233i	− 0.436913i
$988$	0	0
$989$	187.412	0.189497
$990$	0	0
$991$	− 926.283i	− 0.934696i	−0.884074	−	0.467348i	$-0.845210\pi$
$991$	− 926.283i	− 0.934696i	0.884074	−	0.467348i	$-0.154790\pi$
$992$	0	0
$993$	2633.10	2.65166
$994$	0	0
$995$	− 956.729i	− 0.961537i
$996$	0	0
$997$	−56.6462	−0.0568167	−0.0284083	−	0.999596i	$-0.509044\pi$
$997$	−56.6462	−0.0568167	−0.0284083	+	0.999596i	$0.509044\pi$
$998$	0	0
$999$	87.5560i	0.0876437i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.3.d.b.15.1	✓	4
3.2	odd	2		1008.3.m.d.127.3		4
4.3	odd	2	inner	112.3.d.b.15.4	yes	4
7.2	even	3		784.3.r.o.655.2		4
7.3	odd	6		784.3.r.n.79.2		4
7.4	even	3		784.3.r.j.79.1		4
7.5	odd	6		784.3.r.i.655.1		4
7.6	odd	2		784.3.d.j.687.4		4
8.3	odd	2		448.3.d.b.127.1		4
8.5	even	2		448.3.d.b.127.4		4
12.11	even	2		1008.3.m.d.127.4		4
16.3	odd	4		1792.3.g.e.127.2		8
16.5	even	4		1792.3.g.e.127.1		8
16.11	odd	4		1792.3.g.e.127.7		8
16.13	even	4		1792.3.g.e.127.8		8
28.3	even	6		784.3.r.i.79.1		4
28.11	odd	6		784.3.r.o.79.2		4
28.19	even	6		784.3.r.n.655.2		4
28.23	odd	6		784.3.r.j.655.1		4
28.27	even	2		784.3.d.j.687.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.d.b.15.1	✓	4	1.1	even	1	trivial
112.3.d.b.15.4	yes	4	4.3	odd	2	inner
448.3.d.b.127.1		4	8.3	odd	2
448.3.d.b.127.4		4	8.5	even	2
784.3.d.j.687.1		4	28.27	even	2
784.3.d.j.687.4		4	7.6	odd	2
784.3.r.i.79.1		4	28.3	even	6
784.3.r.i.655.1		4	7.5	odd	6
784.3.r.j.79.1		4	7.4	even	3
784.3.r.j.655.1		4	28.23	odd	6
784.3.r.n.79.2		4	7.3	odd	6
784.3.r.n.655.2		4	28.19	even	6
784.3.r.o.79.2		4	28.11	odd	6
784.3.r.o.655.2		4	7.2	even	3
1008.3.m.d.127.3		4	3.2	odd	2
1008.3.m.d.127.4		4	12.11	even	2
1792.3.g.e.127.1		8	16.5	even	4
1792.3.g.e.127.2		8	16.3	odd	4
1792.3.g.e.127.7		8	16.11	odd	4
1792.3.g.e.127.8		8	16.13	even	4

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	112.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.37780i q^{3} -5.58258 q^{5} -2.64575i q^{7} -10.1652 q^{9} -1.82740i q^{11} -20.7477 q^{13} +24.4394i q^{15} +25.1652 q^{17} -30.6446i q^{19} -11.5826 q^{21} -3.27340i q^{23} +6.16515 q^{25} +5.10080i q^{27} +53.8258 q^{29} -29.1588i q^{31} -8.00000 q^{33} +14.7701i q^{35} +17.1652 q^{37} +90.8294i q^{39} +7.49545 q^{41} +57.2530i q^{43} +56.7477 q^{45} +37.2312i q^{47} -7.00000 q^{49} -110.168i q^{51} -46.0000 q^{53} +10.2016i q^{55} -134.156 q^{57} -75.1058i q^{59} +28.0871 q^{61} +26.8945i q^{63} +115.826 q^{65} -81.3906i q^{67} -14.3303 q^{69} -32.1304i q^{71} -45.6515 q^{73} -26.9898i q^{75} -4.83485 q^{77} +10.9644i q^{79} -69.1561 q^{81} +143.705i q^{83} -140.486 q^{85} -235.639i q^{87} +59.3212 q^{89} +54.8933i q^{91} -127.652 q^{93} +171.076i q^{95} +17.1652 q^{97} +18.5758i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{9} - 28 q^{13} + 64 q^{17} - 28 q^{21} - 12 q^{25} + 32 q^{29} - 32 q^{33} + 32 q^{37} - 80 q^{41} + 172 q^{45} - 28 q^{49} - 184 q^{53} - 280 q^{57} + 204 q^{61} + 280 q^{65} + 16 q^{69}+ \cdots + 32 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^9 - 28 * q^13 + 64 * q^17 - 28 * q^21 - 12 * q^25 + 32 * q^29 - 32 * q^33 + 32 * q^37 - 80 * q^41 + 172 * q^45 - 28 * q^49 - 184 * q^53 - 280 * q^57 + 204 * q^61 + 280 * q^65 + 16 * q^69 + 184 * q^73 - 56 * q^77 - 20 * q^81 - 232 * q^85 - 56 * q^89 - 144 * q^93 + 32 * q^97

Introduction

L-functions

Modular forms

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