LMFDB - Embedded newform 1984.2.c.f.993.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1984,2,Mod(993,1984)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1984, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1984.993");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1984 = 2^{6} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1984.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$15.8423197610$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 36 x^{18} + 536 x^{16} + 4280 x^{14} + 19892 x^{12} + 54784 x^{10} + 87680 x^{8} + 77472 x^{6} + \cdots + 64 $ x^20 + 36x^18 + 536x^16 + 4280x^14 + 19892x^12 + 54784x^10 + 87680x^8 + 77472x^6 + 34368x^4 + 6016*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{23} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			993.17
Root			$-2.36943i$ of defining polynomial
Character	$\chi$	$=$	1984.993
Dual form			1984.2.c.f.993.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.36943i q^{3} +0.566787i q^{5} +1.27083 q^{7} -2.61418 q^{9} +O(q^{10})$	$q+2.36943i q^{3} +0.566787i q^{5} +1.27083 q^{7} -2.61418 q^{9} +0.761006i q^{11} -5.38655i q^{13} -1.34296 q^{15} +6.36009 q^{17} +3.73742i q^{19} +3.01113i q^{21} +2.83555 q^{23} +4.67875 q^{25} +0.914181i q^{27} +3.50371i q^{29} +1.00000 q^{31} -1.80315 q^{33} +0.720287i q^{35} +7.17102i q^{37} +12.7630 q^{39} -6.21030 q^{41} +8.34920i q^{43} -1.48168i q^{45} +6.37705 q^{47} -5.38500 q^{49} +15.0698i q^{51} +0.712579i q^{53} -0.431328 q^{55} -8.85553 q^{57} +2.08124i q^{59} -12.2550i q^{61} -3.32216 q^{63} +3.05303 q^{65} -0.504875i q^{67} +6.71863i q^{69} +9.12236 q^{71} -1.19656 q^{73} +11.0860i q^{75} +0.967106i q^{77} -7.11799 q^{79} -10.0086 q^{81} -4.93075i q^{83} +3.60481i q^{85} -8.30178 q^{87} -8.82678 q^{89} -6.84537i q^{91} +2.36943i q^{93} -2.11832 q^{95} -10.6465 q^{97} -1.98940i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{9}+O(q^{10}) $ 20 * q - 12 * q^9	$ 20 q - 12 q^{9} + 28 q^{15} + 4 q^{17} + 20 q^{23} - 16 q^{25} + 20 q^{31} + 48 q^{39} + 4 q^{41} + 16 q^{47} + 8 q^{49} + 64 q^{55} - 4 q^{57} + 8 q^{63} + 12 q^{65} + 24 q^{71} - 16 q^{73} + 52 q^{79} - 20 q^{81} + 40 q^{87} + 12 q^{89} - 12 q^{97}+O(q^{100}) $ 20 * q - 12 * q^9 + 28 * q^15 + 4 * q^17 + 20 * q^23 - 16 * q^25 + 20 * q^31 + 48 * q^39 + 4 * q^41 + 16 * q^47 + 8 * q^49 + 64 * q^55 - 4 * q^57 + 8 * q^63 + 12 * q^65 + 24 * q^71 - 16 * q^73 + 52 * q^79 - 20 * q^81 + 40 * q^87 + 12 * q^89 - 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1984\mathbb{Z}\right)^\times$.

$n$	$63$	$65$	$1861$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.36943i	1.36799i	0.729487	+	0.683994i	$0.239759\pi$
$3$	2.36943i	1.36799i	−0.729487	+	0.683994i	$0.760241\pi$
$4$	0	0
$5$	0.566787i	0.253475i	0.991936	+	0.126737i	$0.0404505\pi$
$5$	0.566787i	0.253475i	−0.991936	+	0.126737i	$0.959549\pi$
$6$	0	0
$7$	1.27083	0.480327	0.240163	−	0.970732i	$-0.422799\pi$
$7$	1.27083	0.480327	0.240163	+	0.970732i	$0.422799\pi$
$8$	0	0
$9$	−2.61418	−0.871392
$10$	0	0
$11$	0.761006i	0.229452i	0.993397	+	0.114726i	$0.0365990\pi$
$11$	0.761006i	0.229452i	−0.993397	+	0.114726i	$0.963401\pi$
$12$	0	0
$13$	− 5.38655i	− 1.49396i	−0.664846	−	0.746981i	$-0.731503\pi$
$13$	− 5.38655i	− 1.49396i	0.664846	−	0.746981i	$-0.268497\pi$
$14$	0	0
$15$	−1.34296	−0.346751
$16$	0	0
$17$	6.36009	1.54255	0.771274	−	0.636503i	$-0.219620\pi$
$17$	6.36009	1.54255	0.771274	+	0.636503i	$0.219620\pi$
$18$	0	0
$19$	3.73742i	0.857422i	0.903442	+	0.428711i	$0.141032\pi$
$19$	3.73742i	0.857422i	−0.903442	+	0.428711i	$0.858968\pi$
$20$	0	0
$21$	3.01113i	0.657082i
$22$	0	0
$23$	2.83555	0.591254	0.295627	−	0.955304i	$-0.404472\pi$
$23$	2.83555	0.591254	0.295627	+	0.955304i	$0.404472\pi$
$24$	0	0
$25$	4.67875	0.935751
$26$	0	0
$27$	0.914181i	0.175934i
$28$	0	0
$29$	3.50371i	0.650623i	0.945607	+	0.325311i	$0.105469\pi$
$29$	3.50371i	0.650623i	−0.945607	+	0.325311i	$0.894531\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−1.80315	−0.313888
$34$	0	0
$35$	0.720287i	0.121751i
$36$	0	0
$37$	7.17102i	1.17891i	0.807802	+	0.589454i	$0.200657\pi$
$37$	7.17102i	1.17891i	−0.807802	+	0.589454i	$0.799343\pi$
$38$	0	0
$39$	12.7630	2.04372
$40$	0	0
$41$	−6.21030	−0.969886	−0.484943	−	0.874546i	$-0.661160\pi$
$41$	−6.21030	−0.969886	−0.484943	+	0.874546i	$0.661160\pi$
$42$	0	0
$43$	8.34920i	1.27324i	0.771177	+	0.636621i	$0.219668\pi$
$43$	8.34920i	1.27324i	−0.771177	+	0.636621i	$0.780332\pi$
$44$	0	0
$45$	− 1.48168i	− 0.220876i
$46$	0	0
$47$	6.37705	0.930189	0.465094	−	0.885261i	$-0.346020\pi$
$47$	6.37705	0.930189	0.465094	+	0.885261i	$0.346020\pi$
$48$	0	0
$49$	−5.38500	−0.769286
$50$	0	0
$51$	15.0698i	2.11019i
$52$	0	0
$53$	0.712579i	0.0978803i	0.998802	+	0.0489401i	$0.0155844\pi$
$53$	0.712579i	0.0978803i	−0.998802	+	0.0489401i	$0.984416\pi$
$54$	0	0
$55$	−0.431328	−0.0581603
$56$	0	0
$57$	−8.85553	−1.17294
$58$	0	0
$59$	2.08124i	0.270954i	0.990780	+	0.135477i	$0.0432568\pi$
$59$	2.08124i	0.270954i	−0.990780	+	0.135477i	$0.956743\pi$
$60$	0	0
$61$	− 12.2550i	− 1.56909i	−0.620071	−	0.784545i	$-0.712896\pi$
$61$	− 12.2550i	− 1.56909i	0.620071	−	0.784545i	$-0.287104\pi$
$62$	0	0
$63$	−3.32216	−0.418553
$64$	0	0
$65$	3.05303	0.378682
$66$	0	0
$67$	− 0.504875i	− 0.0616803i	−0.999524	−	0.0308402i	$-0.990182\pi$
$67$	− 0.504875i	− 0.0616803i	0.999524	−	0.0308402i	$-0.00981828\pi$
$68$	0	0
$69$	6.71863i	0.808828i
$70$	0	0
$71$	9.12236	1.08262	0.541312	−	0.840822i	$-0.317928\pi$
$71$	9.12236	1.08262	0.541312	+	0.840822i	$0.317928\pi$
$72$	0	0
$73$	−1.19656	−0.140047	−0.0700236	−	0.997545i	$-0.522307\pi$
$73$	−1.19656	−0.140047	−0.0700236	+	0.997545i	$0.522307\pi$
$74$	0	0
$75$	11.0860i	1.28010i
$76$	0	0
$77$	0.967106i	0.110212i
$78$	0	0
$79$	−7.11799	−0.800836	−0.400418	−	0.916333i	$-0.631135\pi$
$79$	−7.11799	−0.800836	−0.400418	+	0.916333i	$0.631135\pi$
$80$	0	0
$81$	−10.0086	−1.11207
$82$	0	0
$83$	− 4.93075i	− 0.541220i	−0.962689	−	0.270610i	$-0.912775\pi$
$83$	− 4.93075i	− 0.541220i	0.962689	−	0.270610i	$-0.0872254\pi$
$84$	0	0
$85$	3.60481i	0.390997i
$86$	0	0
$87$	−8.30178	−0.890044
$88$	0	0
$89$	−8.82678	−0.935637	−0.467818	−	0.883825i	$-0.654960\pi$
$89$	−8.82678	−0.935637	−0.467818	+	0.883825i	$0.654960\pi$
$90$	0	0
$91$	− 6.84537i	− 0.717590i
$92$	0	0
$93$	2.36943i	0.245698i
$94$	0	0
$95$	−2.11832	−0.217335
$96$	0	0
$97$	−10.6465	−1.08098	−0.540492	−	0.841349i	$-0.681762\pi$
$97$	−10.6465	−1.08098	−0.540492	+	0.841349i	$0.681762\pi$
$98$	0	0
$99$	− 1.98940i	− 0.199943i
$100$	0	0
$101$	− 15.2076i	− 1.51321i	−0.653872	−	0.756605i	$-0.726857\pi$
$101$	− 15.2076i	− 1.51321i	0.653872	−	0.756605i	$-0.273143\pi$
$102$	0	0
$103$	7.13029	0.702568	0.351284	−	0.936269i	$-0.385745\pi$
$103$	7.13029	0.702568	0.351284	+	0.936269i	$0.385745\pi$
$104$	0	0
$105$	−1.70667	−0.166554
$106$	0	0
$107$	− 4.82328i	− 0.466284i	−0.972443	−	0.233142i	$-0.925099\pi$
$107$	− 4.82328i	− 0.466284i	0.972443	−	0.233142i	$-0.0749006\pi$
$108$	0	0
$109$	3.60023i	0.344840i	0.985024	+	0.172420i	$0.0551586\pi$
$109$	3.60023i	0.344840i	−0.985024	+	0.172420i	$0.944841\pi$
$110$	0	0
$111$	−16.9912	−1.61273
$112$	0	0
$113$	2.94246	0.276804	0.138402	−	0.990376i	$-0.455803\pi$
$113$	2.94246	0.276804	0.138402	+	0.990376i	$0.455803\pi$
$114$	0	0
$115$	1.60715i	0.149868i
$116$	0	0
$117$	14.0814i	1.30183i
$118$	0	0
$119$	8.08256	0.740927
$120$	0	0
$121$	10.4209	0.947352
$122$	0	0
$123$	− 14.7148i	− 1.32679i
$124$	0	0
$125$	5.48579i	0.490664i
$126$	0	0
$127$	3.52200	0.312527	0.156264	−	0.987715i	$-0.450055\pi$
$127$	3.52200	0.312527	0.156264	+	0.987715i	$0.450055\pi$
$128$	0	0
$129$	−19.7828	−1.74178
$130$	0	0
$131$	7.53511i	0.658346i	0.944270	+	0.329173i	$0.106770\pi$
$131$	7.53511i	0.658346i	−0.944270	+	0.329173i	$0.893230\pi$
$132$	0	0
$133$	4.74960i	0.411843i
$134$	0	0
$135$	−0.518146	−0.0445949
$136$	0	0
$137$	16.2730	1.39029	0.695147	−	0.718868i	$-0.255339\pi$
$137$	16.2730	1.39029	0.695147	+	0.718868i	$0.255339\pi$
$138$	0	0
$139$	16.3174i	1.38402i	0.721888	+	0.692010i	$0.243275\pi$
$139$	16.3174i	1.38402i	−0.721888	+	0.692010i	$0.756725\pi$
$140$	0	0
$141$	15.1100i	1.27249i
$142$	0	0
$143$	4.09920	0.342792
$144$	0	0
$145$	−1.98586	−0.164916
$146$	0	0
$147$	− 12.7594i	− 1.05237i
$148$	0	0
$149$	20.1040i	1.64698i	0.567330	+	0.823490i	$0.307976\pi$
$149$	20.1040i	1.64698i	−0.567330	+	0.823490i	$0.692024\pi$
$150$	0	0
$151$	−14.5986	−1.18802	−0.594008	−	0.804459i	$-0.702455\pi$
$151$	−14.5986	−1.18802	−0.594008	+	0.804459i	$0.702455\pi$
$152$	0	0
$153$	−16.6264	−1.34416
$154$	0	0
$155$	0.566787i	0.0455254i
$156$	0	0
$157$	− 1.03578i	− 0.0826646i	−0.999145	−	0.0413323i	$-0.986840\pi$
$157$	− 1.03578i	− 0.0826646i	0.999145	−	0.0413323i	$-0.0131602\pi$
$158$	0	0
$159$	−1.68840	−0.133899
$160$	0	0
$161$	3.60349	0.283995
$162$	0	0
$163$	− 6.78054i	− 0.531093i	−0.964098	−	0.265546i	$-0.914448\pi$
$163$	− 6.78054i	− 0.531093i	0.964098	−	0.265546i	$-0.0855523\pi$
$164$	0	0
$165$	− 1.02200i	− 0.0795626i
$166$	0	0
$167$	5.48466	0.424416	0.212208	−	0.977225i	$-0.431935\pi$
$167$	5.48466	0.424416	0.212208	+	0.977225i	$0.431935\pi$
$168$	0	0
$169$	−16.0150	−1.23192
$170$	0	0
$171$	− 9.77026i	− 0.747151i
$172$	0	0
$173$	2.11364i	0.160697i	0.996767	+	0.0803486i	$0.0256034\pi$
$173$	2.11364i	0.160697i	−0.996767	+	0.0803486i	$0.974397\pi$
$174$	0	0
$175$	5.94588	0.449466
$176$	0	0
$177$	−4.93134	−0.370662
$178$	0	0
$179$	− 2.13605i	− 0.159656i	−0.996809	−	0.0798279i	$-0.974563\pi$
$179$	− 2.13605i	− 0.159656i	0.996809	−	0.0798279i	$-0.0254371\pi$
$180$	0	0
$181$	− 1.10457i	− 0.0821022i	−0.999157	−	0.0410511i	$-0.986929\pi$
$181$	− 1.10457i	− 0.0821022i	0.999157	−	0.0410511i	$-0.0130706\pi$
$182$	0	0
$183$	29.0373	2.14650
$184$	0	0
$185$	−4.06444	−0.298823
$186$	0	0
$187$	4.84006i	0.353941i
$188$	0	0
$189$	1.16176i	0.0845060i
$190$	0	0
$191$	−20.2961	−1.46857	−0.734287	−	0.678839i	$-0.762483\pi$
$191$	−20.2961	−1.46857	−0.734287	+	0.678839i	$0.762483\pi$
$192$	0	0
$193$	−5.22912	−0.376400	−0.188200	−	0.982131i	$-0.560265\pi$
$193$	−5.22912	−0.376400	−0.188200	+	0.982131i	$0.560265\pi$
$194$	0	0
$195$	7.23392i	0.518032i
$196$	0	0
$197$	− 13.9558i	− 0.994308i	−0.867662	−	0.497154i	$-0.834378\pi$
$197$	− 13.9558i	− 0.994308i	0.867662	−	0.497154i	$-0.165622\pi$
$198$	0	0
$199$	8.48641	0.601586	0.300793	−	0.953689i	$-0.402749\pi$
$199$	8.48641	0.601586	0.300793	+	0.953689i	$0.402749\pi$
$200$	0	0
$201$	1.19626	0.0843779
$202$	0	0
$203$	4.45261i	0.312512i
$204$	0	0
$205$	− 3.51992i	− 0.245842i
$206$	0	0
$207$	−7.41263	−0.515214
$208$	0	0
$209$	−2.84420	−0.196737
$210$	0	0
$211$	20.5903i	1.41749i	0.705463	+	0.708747i	$0.250739\pi$
$211$	20.5903i	1.41749i	−0.705463	+	0.708747i	$0.749261\pi$
$212$	0	0
$213$	21.6147i	1.48102i
$214$	0	0
$215$	−4.73222	−0.322735
$216$	0	0
$217$	1.27083	0.0862693
$218$	0	0
$219$	− 2.83517i	− 0.191583i
$220$	0	0
$221$	− 34.2590i	− 2.30451i
$222$	0	0
$223$	−13.9083	−0.931372	−0.465686	−	0.884950i	$-0.654192\pi$
$223$	−13.9083	−0.931372	−0.465686	+	0.884950i	$0.654192\pi$
$224$	0	0
$225$	−12.2311	−0.815406
$226$	0	0
$227$	− 19.7365i	− 1.30995i	−0.755648	−	0.654977i	$-0.772678\pi$
$227$	− 19.7365i	− 1.30995i	0.755648	−	0.654977i	$-0.227322\pi$
$228$	0	0
$229$	22.2461i	1.47006i	0.678034	+	0.735031i	$0.262832\pi$
$229$	22.2461i	1.47006i	−0.678034	+	0.735031i	$0.737168\pi$
$230$	0	0
$231$	−2.29148	−0.150769
$232$	0	0
$233$	−10.5240	−0.689453	−0.344726	−	0.938703i	$-0.612028\pi$
$233$	−10.5240	−0.689453	−0.344726	+	0.938703i	$0.612028\pi$
$234$	0	0
$235$	3.61443i	0.235779i
$236$	0	0
$237$	− 16.8655i	− 1.09553i
$238$	0	0
$239$	−3.50956	−0.227015	−0.113507	−	0.993537i	$-0.536209\pi$
$239$	−3.50956	−0.227015	−0.113507	+	0.993537i	$0.536209\pi$
$240$	0	0
$241$	4.42577	0.285089	0.142545	−	0.989788i	$-0.454472\pi$
$241$	4.42577	0.285089	0.142545	+	0.989788i	$0.454472\pi$
$242$	0	0
$243$	− 20.9721i	− 1.34536i
$244$	0	0
$245$	− 3.05215i	− 0.194995i
$246$	0	0
$247$	20.1318	1.28096
$248$	0	0
$249$	11.6830	0.740383
$250$	0	0
$251$	18.0575i	1.13978i	0.821721	+	0.569890i	$0.193014\pi$
$251$	18.0575i	1.13978i	−0.821721	+	0.569890i	$0.806986\pi$
$252$	0	0
$253$	2.15787i	0.135664i
$254$	0	0
$255$	−8.54134	−0.534879
$256$	0	0
$257$	25.8589	1.61303	0.806515	−	0.591213i	$-0.201351\pi$
$257$	25.8589	1.61303	0.806515	+	0.591213i	$0.201351\pi$
$258$	0	0
$259$	9.11311i	0.566261i
$260$	0	0
$261$	− 9.15932i	− 0.566948i
$262$	0	0
$263$	−31.4441	−1.93893	−0.969465	−	0.245232i	$-0.921136\pi$
$263$	−31.4441	−1.93893	−0.969465	+	0.245232i	$0.921136\pi$
$264$	0	0
$265$	−0.403880	−0.0248102
$266$	0	0
$267$	− 20.9144i	− 1.27994i
$268$	0	0
$269$	− 23.0282i	− 1.40405i	−0.712150	−	0.702027i	$-0.752278\pi$
$269$	− 23.0282i	− 1.40405i	0.712150	−	0.702027i	$-0.247722\pi$
$270$	0	0
$271$	26.7341	1.62398	0.811989	−	0.583672i	$-0.198385\pi$
$271$	26.7341	1.62398	0.811989	+	0.583672i	$0.198385\pi$
$272$	0	0
$273$	16.2196	0.981655
$274$	0	0
$275$	3.56056i	0.214710i
$276$	0	0
$277$	12.7824i	0.768017i	0.923329	+	0.384009i	$0.125457\pi$
$277$	12.7824i	0.768017i	−0.923329	+	0.384009i	$0.874543\pi$
$278$	0	0
$279$	−2.61418	−0.156507
$280$	0	0
$281$	28.5113	1.70084	0.850422	−	0.526101i	$-0.176347\pi$
$281$	28.5113	1.70084	0.850422	+	0.526101i	$0.176347\pi$
$282$	0	0
$283$	20.2637i	1.20455i	0.798288	+	0.602276i	$0.205739\pi$
$283$	20.2637i	1.20455i	−0.798288	+	0.602276i	$0.794261\pi$
$284$	0	0
$285$	− 5.01920i	− 0.297312i
$286$	0	0
$287$	−7.89221	−0.465862
$288$	0	0
$289$	23.4507	1.37945
$290$	0	0
$291$	− 25.2260i	− 1.47877i
$292$	0	0
$293$	− 7.49787i	− 0.438030i	−0.975721	−	0.219015i	$-0.929716\pi$
$293$	− 7.49787i	− 0.438030i	0.975721	−	0.219015i	$-0.0702844\pi$
$294$	0	0
$295$	−1.17962	−0.0686801
$296$	0	0
$297$	−0.695697	−0.0403684
$298$	0	0
$299$	− 15.2739i	− 0.883310i
$300$	0	0
$301$	10.6104i	0.611572i
$302$	0	0
$303$	36.0332	2.07005
$304$	0	0
$305$	6.94597	0.397725
$306$	0	0
$307$	11.1887i	0.638573i	0.947658	+	0.319287i	$0.103443\pi$
$307$	11.1887i	0.638573i	−0.947658	+	0.319287i	$0.896557\pi$
$308$	0	0
$309$	16.8947i	0.961105i
$310$	0	0
$311$	19.4910	1.10524	0.552618	−	0.833435i	$-0.313629\pi$
$311$	19.4910	1.10524	0.552618	+	0.833435i	$0.313629\pi$
$312$	0	0
$313$	4.10150	0.231830	0.115915	−	0.993259i	$-0.463020\pi$
$313$	4.10150	0.231830	0.115915	+	0.993259i	$0.463020\pi$
$314$	0	0
$315$	− 1.88296i	− 0.106093i
$316$	0	0
$317$	− 21.6366i	− 1.21523i	−0.794231	−	0.607616i	$-0.792126\pi$
$317$	− 21.6366i	− 1.21523i	0.794231	−	0.607616i	$-0.207874\pi$
$318$	0	0
$319$	−2.66634	−0.149287
$320$	0	0
$321$	11.4284	0.637871
$322$	0	0
$323$	23.7703i	1.32261i
$324$	0	0
$325$	− 25.2024i	− 1.39798i
$326$	0	0
$327$	−8.53048	−0.471737
$328$	0	0
$329$	8.10412	0.446795
$330$	0	0
$331$	− 19.9126i	− 1.09450i	−0.836970	−	0.547249i	$-0.815675\pi$
$331$	− 19.9126i	− 1.09450i	0.836970	−	0.547249i	$-0.184325\pi$
$332$	0	0
$333$	− 18.7463i	− 1.02729i
$334$	0	0
$335$	0.286157	0.0156344
$336$	0	0
$337$	−4.69438	−0.255719	−0.127859	−	0.991792i	$-0.540811\pi$
$337$	−4.69438	−0.255719	−0.127859	+	0.991792i	$0.540811\pi$
$338$	0	0
$339$	6.97195i	0.378664i
$340$	0	0
$341$	0.761006i	0.0412108i
$342$	0	0
$343$	−15.7392	−0.849836
$344$	0	0
$345$	−3.80803	−0.205017
$346$	0	0
$347$	− 26.0768i	− 1.39988i	−0.714204	−	0.699938i	$-0.753211\pi$
$347$	− 26.0768i	− 1.39988i	0.714204	−	0.699938i	$-0.246789\pi$
$348$	0	0
$349$	− 32.7751i	− 1.75441i	−0.480116	−	0.877205i	$-0.659405\pi$
$349$	− 32.7751i	− 1.75441i	0.480116	−	0.877205i	$-0.340595\pi$
$350$	0	0
$351$	4.92429	0.262839
$352$	0	0
$353$	25.0717	1.33443	0.667217	−	0.744863i	$-0.267485\pi$
$353$	25.0717	1.33443	0.667217	+	0.744863i	$0.267485\pi$
$354$	0	0
$355$	5.17043i	0.274418i
$356$	0	0
$357$	19.1510i	1.01358i
$358$	0	0
$359$	−22.0494	−1.16372	−0.581860	−	0.813289i	$-0.697675\pi$
$359$	−22.0494	−1.16372	−0.581860	+	0.813289i	$0.697675\pi$
$360$	0	0
$361$	5.03172	0.264827
$362$	0	0
$363$	24.6915i	1.29597i
$364$	0	0
$365$	− 0.678197i	− 0.0354985i
$366$	0	0
$367$	20.0626	1.04726	0.523630	−	0.851946i	$-0.324578\pi$
$367$	20.0626	1.04726	0.523630	+	0.851946i	$0.324578\pi$
$368$	0	0
$369$	16.2348	0.845151
$370$	0	0
$371$	0.905564i	0.0470145i
$372$	0	0
$373$	22.3269i	1.15604i	0.816022	+	0.578021i	$0.196175\pi$
$373$	22.3269i	1.15604i	−0.816022	+	0.578021i	$0.803825\pi$
$374$	0	0
$375$	−12.9982	−0.671223
$376$	0	0
$377$	18.8729	0.972005
$378$	0	0
$379$	7.53634i	0.387116i	0.981089	+	0.193558i	$0.0620028\pi$
$379$	7.53634i	0.387116i	−0.981089	+	0.193558i	$0.937997\pi$
$380$	0	0
$381$	8.34512i	0.427534i
$382$	0	0
$383$	1.40043	0.0715585	0.0357793	−	0.999360i	$-0.488609\pi$
$383$	1.40043	0.0715585	0.0357793	+	0.999360i	$0.488609\pi$
$384$	0	0
$385$	−0.548143	−0.0279359
$386$	0	0
$387$	− 21.8263i	− 1.10949i
$388$	0	0
$389$	7.64795i	0.387766i	0.981025	+	0.193883i	$0.0621083\pi$
$389$	7.64795i	0.387766i	−0.981025	+	0.193883i	$0.937892\pi$
$390$	0	0
$391$	18.0344	0.912037
$392$	0	0
$393$	−17.8539	−0.900609
$394$	0	0
$395$	− 4.03438i	− 0.202992i
$396$	0	0
$397$	− 14.2616i	− 0.715768i	−0.933766	−	0.357884i	$-0.883498\pi$
$397$	− 14.2616i	− 0.715768i	0.933766	−	0.357884i	$-0.116502\pi$
$398$	0	0
$399$	−11.2538	−0.563396
$400$	0	0
$401$	16.7122	0.834566	0.417283	−	0.908777i	$-0.362982\pi$
$401$	16.7122	0.834566	0.417283	+	0.908777i	$0.362982\pi$
$402$	0	0
$403$	− 5.38655i	− 0.268323i
$404$	0	0
$405$	− 5.67275i	− 0.281881i
$406$	0	0
$407$	−5.45719	−0.270503
$408$	0	0
$409$	15.9962	0.790962	0.395481	−	0.918474i	$-0.370578\pi$
$409$	15.9962	0.790962	0.395481	+	0.918474i	$0.370578\pi$
$410$	0	0
$411$	38.5576i	1.90191i
$412$	0	0
$413$	2.64489i	0.130147i
$414$	0	0
$415$	2.79468	0.137186
$416$	0	0
$417$	−38.6628	−1.89332
$418$	0	0
$419$	− 21.1557i	− 1.03353i	−0.856129	−	0.516763i	$-0.827137\pi$
$419$	− 21.1557i	− 1.03353i	0.856129	−	0.516763i	$-0.172863\pi$
$420$	0	0
$421$	− 33.4459i	− 1.63005i	−0.579424	−	0.815026i	$-0.696722\pi$
$421$	− 33.4459i	− 1.63005i	0.579424	−	0.815026i	$-0.303278\pi$
$422$	0	0
$423$	−16.6707	−0.810559
$424$	0	0
$425$	29.7573	1.44344
$426$	0	0
$427$	− 15.5740i	− 0.753676i
$428$	0	0
$429$	9.71275i	0.468936i
$430$	0	0
$431$	−7.18273	−0.345980	−0.172990	−	0.984924i	$-0.555343\pi$
$431$	−7.18273	−0.345980	−0.172990	+	0.984924i	$0.555343\pi$
$432$	0	0
$433$	−10.2213	−0.491205	−0.245602	−	0.969371i	$-0.578986\pi$
$433$	−10.2213	−0.491205	−0.245602	+	0.969371i	$0.578986\pi$
$434$	0	0
$435$	− 4.70534i	− 0.225604i
$436$	0	0
$437$	10.5976i	0.506954i
$438$	0	0
$439$	−33.2015	−1.58462	−0.792312	−	0.610117i	$-0.791123\pi$
$439$	−33.2015	−1.58462	−0.792312	+	0.610117i	$0.791123\pi$
$440$	0	0
$441$	14.0773	0.670350
$442$	0	0
$443$	− 18.9952i	− 0.902489i	−0.892400	−	0.451244i	$-0.850980\pi$
$443$	− 18.9952i	− 0.902489i	0.892400	−	0.451244i	$-0.149020\pi$
$444$	0	0
$445$	− 5.00290i	− 0.237160i
$446$	0	0
$447$	−47.6348	−2.25305
$448$	0	0
$449$	−15.9438	−0.752435	−0.376218	−	0.926531i	$-0.622776\pi$
$449$	−15.9438	−0.752435	−0.376218	+	0.926531i	$0.622776\pi$
$450$	0	0
$451$	− 4.72608i	− 0.222542i
$452$	0	0
$453$	− 34.5902i	− 1.62519i
$454$	0	0
$455$	3.87987	0.181891
$456$	0	0
$457$	0.0761677	0.00356298	0.00178149	−	0.999998i	$-0.499433\pi$
$457$	0.0761677	0.00356298	0.00178149	+	0.999998i	$0.499433\pi$
$458$	0	0
$459$	5.81427i	0.271387i
$460$	0	0
$461$	− 25.4496i	− 1.18531i	−0.805457	−	0.592654i	$-0.798080\pi$
$461$	− 25.4496i	− 1.18531i	0.805457	−	0.592654i	$-0.201920\pi$
$462$	0	0
$463$	35.2737	1.63931	0.819654	−	0.572859i	$-0.194166\pi$
$463$	35.2737	1.63931	0.819654	+	0.572859i	$0.194166\pi$
$464$	0	0
$465$	−1.34296	−0.0622782
$466$	0	0
$467$	− 3.98356i	− 0.184337i	−0.995743	−	0.0921686i	$-0.970620\pi$
$467$	− 3.98356i	− 0.184337i	0.995743	−	0.0921686i	$-0.0293799\pi$
$468$	0	0
$469$	− 0.641608i	− 0.0296267i
$470$	0	0
$471$	2.45421	0.113084
$472$	0	0
$473$	−6.35379	−0.292148
$474$	0	0
$475$	17.4864i	0.802333i
$476$	0	0
$477$	− 1.86281i	− 0.0852921i
$478$	0	0
$479$	4.38491	0.200351	0.100176	−	0.994970i	$-0.468060\pi$
$479$	4.38491	0.200351	0.100176	+	0.994970i	$0.468060\pi$
$480$	0	0
$481$	38.6271	1.76124
$482$	0	0
$483$	8.53821i	0.388502i
$484$	0	0
$485$	− 6.03427i	− 0.274002i
$486$	0	0
$487$	−36.8293	−1.66889	−0.834447	−	0.551088i	$-0.814213\pi$
$487$	−36.8293	−1.66889	−0.834447	+	0.551088i	$0.814213\pi$
$488$	0	0
$489$	16.0660	0.726528
$490$	0	0
$491$	− 9.96732i	− 0.449819i	−0.974380	−	0.224909i	$-0.927791\pi$
$491$	− 9.96732i	− 0.449819i	0.974380	−	0.224909i	$-0.0722086\pi$
$492$	0	0
$493$	22.2839i	1.00362i
$494$	0	0
$495$	1.12757	0.0506804
$496$	0	0
$497$	11.5929	0.520014
$498$	0	0
$499$	− 31.9294i	− 1.42936i	−0.699454	−	0.714678i	$-0.746573\pi$
$499$	− 31.9294i	− 1.42936i	0.699454	−	0.714678i	$-0.253427\pi$
$500$	0	0
$501$	12.9955i	0.580595i
$502$	0	0
$503$	13.7882	0.614787	0.307394	−	0.951582i	$-0.400543\pi$
$503$	13.7882	0.614787	0.307394	+	0.951582i	$0.400543\pi$
$504$	0	0
$505$	8.61945	0.383561
$506$	0	0
$507$	− 37.9463i	− 1.68525i
$508$	0	0
$509$	− 16.0288i	− 0.710464i	−0.934778	−	0.355232i	$-0.884402\pi$
$509$	− 16.0288i	− 0.710464i	0.934778	−	0.355232i	$-0.115598\pi$
$510$	0	0
$511$	−1.52062	−0.0672685
$512$	0	0
$513$	−3.41668	−0.150850
$514$	0	0
$515$	4.04135i	0.178083i
$516$	0	0
$517$	4.85297i	0.213434i
$518$	0	0
$519$	−5.00812	−0.219832
$520$	0	0
$521$	−10.7539	−0.471138	−0.235569	−	0.971858i	$-0.575695\pi$
$521$	−10.7539	−0.471138	−0.235569	+	0.971858i	$0.575695\pi$
$522$	0	0
$523$	9.39885i	0.410983i	0.978659	+	0.205492i	$0.0658793\pi$
$523$	9.39885i	0.410983i	−0.978659	+	0.205492i	$0.934121\pi$
$524$	0	0
$525$	14.0883i	0.614864i
$526$	0	0
$527$	6.36009	0.277050
$528$	0	0
$529$	−14.9596	−0.650419
$530$	0	0
$531$	− 5.44073i	− 0.236107i
$532$	0	0
$533$	33.4521i	1.44897i
$534$	0	0
$535$	2.73377	0.118191
$536$	0	0
$537$	5.06121	0.218407
$538$	0	0
$539$	− 4.09802i	− 0.176514i
$540$	0	0
$541$	− 16.7483i	− 0.720065i	−0.932940	−	0.360032i	$-0.882766\pi$
$541$	− 16.7483i	− 0.720065i	0.932940	−	0.360032i	$-0.117234\pi$
$542$	0	0
$543$	2.61720	0.112315
$544$	0	0
$545$	−2.04056	−0.0874082
$546$	0	0
$547$	21.9821i	0.939885i	0.882697	+	0.469942i	$0.155725\pi$
$547$	21.9821i	0.939885i	−0.882697	+	0.469942i	$0.844275\pi$
$548$	0	0
$549$	32.0367i	1.36729i
$550$	0	0
$551$	−13.0948	−0.557858
$552$	0	0
$553$	−9.04572	−0.384663
$554$	0	0
$555$	− 9.63038i	− 0.408787i
$556$	0	0
$557$	22.1584i	0.938880i	0.882964	+	0.469440i	$0.155544\pi$
$557$	22.1584i	0.938880i	−0.882964	+	0.469440i	$0.844456\pi$
$558$	0	0
$559$	44.9734	1.90217
$560$	0	0
$561$	−11.4682	−0.484187
$562$	0	0
$563$	40.7030i	1.71543i	0.514128	+	0.857714i	$0.328116\pi$
$563$	40.7030i	1.71543i	−0.514128	+	0.857714i	$0.671884\pi$
$564$	0	0
$565$	1.66775i	0.0701628i
$566$	0	0
$567$	−12.7192	−0.534156
$568$	0	0
$569$	6.07071	0.254497	0.127249	−	0.991871i	$-0.459385\pi$
$569$	6.07071	0.254497	0.127249	+	0.991871i	$0.459385\pi$
$570$	0	0
$571$	− 27.7122i	− 1.15972i	−0.814716	−	0.579861i	$-0.803107\pi$
$571$	− 27.7122i	− 1.15972i	0.814716	−	0.579861i	$-0.196893\pi$
$572$	0	0
$573$	− 48.0901i	− 2.00899i
$574$	0	0
$575$	13.2668	0.553266
$576$	0	0
$577$	−25.4087	−1.05778	−0.528888	−	0.848691i	$-0.677391\pi$
$577$	−25.4087	−1.05778	−0.528888	+	0.848691i	$0.677391\pi$
$578$	0	0
$579$	− 12.3900i	− 0.514911i
$580$	0	0
$581$	− 6.26612i	− 0.259963i
$582$	0	0
$583$	−0.542277	−0.0224588
$584$	0	0
$585$	−7.98115	−0.329980
$586$	0	0
$587$	− 40.4764i	− 1.67064i	−0.549765	−	0.835320i	$-0.685283\pi$
$587$	− 40.4764i	− 1.67064i	0.549765	−	0.835320i	$-0.314717\pi$
$588$	0	0
$589$	3.73742i	0.153998i
$590$	0	0
$591$	33.0672	1.36020
$592$	0	0
$593$	−14.1387	−0.580607	−0.290304	−	0.956935i	$-0.593756\pi$
$593$	−14.1387	−0.580607	−0.290304	+	0.956935i	$0.593756\pi$
$594$	0	0
$595$	4.58109i	0.187806i
$596$	0	0
$597$	20.1079i	0.822962i
$598$	0	0
$599$	−44.2407	−1.80763	−0.903813	−	0.427928i	$-0.859244\pi$
$599$	−44.2407	−1.80763	−0.903813	+	0.427928i	$0.859244\pi$
$600$	0	0
$601$	−5.24353	−0.213888	−0.106944	−	0.994265i	$-0.534107\pi$
$601$	−5.24353	−0.213888	−0.106944	+	0.994265i	$0.534107\pi$
$602$	0	0
$603$	1.31983i	0.0537477i
$604$	0	0
$605$	5.90641i	0.240130i
$606$	0	0
$607$	−14.7149	−0.597260	−0.298630	−	0.954369i	$-0.596530\pi$
$607$	−14.7149	−0.597260	−0.298630	+	0.954369i	$0.596530\pi$
$608$	0	0
$609$	−10.5501	−0.427512
$610$	0	0
$611$	− 34.3503i	− 1.38967i
$612$	0	0
$613$	− 19.1128i	− 0.771960i	−0.922507	−	0.385980i	$-0.873863\pi$
$613$	− 19.1128i	− 0.771960i	0.922507	−	0.385980i	$-0.126137\pi$
$614$	0	0
$615$	8.34018	0.336309
$616$	0	0
$617$	−7.97361	−0.321005	−0.160503	−	0.987035i	$-0.551312\pi$
$617$	−7.97361	−0.321005	−0.160503	+	0.987035i	$0.551312\pi$
$618$	0	0
$619$	− 26.6933i	− 1.07290i	−0.843933	−	0.536448i	$-0.819766\pi$
$619$	− 26.6933i	− 1.07290i	0.843933	−	0.536448i	$-0.180234\pi$
$620$	0	0
$621$	2.59221i	0.104022i
$622$	0	0
$623$	−11.2173	−0.449412
$624$	0	0
$625$	20.2845	0.811380
$626$	0	0
$627$	− 6.73911i	− 0.269134i
$628$	0	0
$629$	45.6083i	1.81852i
$630$	0	0
$631$	22.4187	0.892476	0.446238	−	0.894914i	$-0.352763\pi$
$631$	22.4187	0.892476	0.446238	+	0.894914i	$0.352763\pi$
$632$	0	0
$633$	−48.7872	−1.93912
$634$	0	0
$635$	1.99623i	0.0792178i
$636$	0	0
$637$	29.0066i	1.14928i
$638$	0	0
$639$	−23.8475	−0.943391
$640$	0	0
$641$	14.2953	0.564632	0.282316	−	0.959322i	$-0.408897\pi$
$641$	14.2953	0.564632	0.282316	+	0.959322i	$0.408897\pi$
$642$	0	0
$643$	− 2.02931i	− 0.0800283i	−0.999199	−	0.0400141i	$-0.987260\pi$
$643$	− 2.02931i	− 0.0800283i	0.999199	−	0.0400141i	$-0.0127403\pi$
$644$	0	0
$645$	− 11.2126i	− 0.441497i
$646$	0	0
$647$	−48.9436	−1.92417	−0.962086	−	0.272746i	$-0.912068\pi$
$647$	−48.9436	−1.92417	−0.962086	+	0.272746i	$0.912068\pi$
$648$	0	0
$649$	−1.58384	−0.0621710
$650$	0	0
$651$	3.01113i	0.118015i
$652$	0	0
$653$	− 41.8528i	− 1.63783i	−0.573918	−	0.818913i	$-0.694577\pi$
$653$	− 41.8528i	− 1.63783i	0.573918	−	0.818913i	$-0.305423\pi$
$654$	0	0
$655$	−4.27080	−0.166874
$656$	0	0
$657$	3.12803	0.122036
$658$	0	0
$659$	− 35.1649i	− 1.36983i	−0.728624	−	0.684914i	$-0.759840\pi$
$659$	− 35.1649i	− 1.36983i	0.728624	−	0.684914i	$-0.240160\pi$
$660$	0	0
$661$	30.2671i	1.17725i	0.808405	+	0.588626i	$0.200331\pi$
$661$	30.2671i	1.17725i	−0.808405	+	0.588626i	$0.799669\pi$
$662$	0	0
$663$	81.1740	3.15254
$664$	0	0
$665$	−2.69201	−0.104392
$666$	0	0
$667$	9.93496i	0.384683i
$668$	0	0
$669$	− 32.9548i	− 1.27411i
$670$	0	0
$671$	9.32612	0.360031
$672$	0	0
$673$	13.5001	0.520392	0.260196	−	0.965556i	$-0.416213\pi$
$673$	13.5001	0.520392	0.260196	+	0.965556i	$0.416213\pi$
$674$	0	0
$675$	4.27723i	0.164631i
$676$	0	0
$677$	9.01497i	0.346473i	0.984880	+	0.173237i	$0.0554225\pi$
$677$	9.01497i	0.346473i	−0.984880	+	0.173237i	$0.944577\pi$
$678$	0	0
$679$	−13.5298	−0.519226
$680$	0	0
$681$	46.7641	1.79200
$682$	0	0
$683$	− 35.1202i	− 1.34384i	−0.740626	−	0.671918i	$-0.765471\pi$
$683$	− 35.1202i	− 1.34384i	0.740626	−	0.671918i	$-0.234529\pi$
$684$	0	0
$685$	9.22331i	0.352404i
$686$	0	0
$687$	−52.7104	−2.01103
$688$	0	0
$689$	3.83835	0.146229
$690$	0	0
$691$	− 17.1134i	− 0.651023i	−0.945538	−	0.325512i	$-0.894463\pi$
$691$	− 17.1134i	− 0.651023i	0.945538	−	0.325512i	$-0.105537\pi$
$692$	0	0
$693$	− 2.52818i	− 0.0960378i
$694$	0	0
$695$	−9.24847	−0.350814
$696$	0	0
$697$	−39.4981	−1.49610
$698$	0	0
$699$	− 24.9359i	− 0.943163i
$700$	0	0
$701$	− 20.2368i	− 0.764333i	−0.924094	−	0.382166i	$-0.875178\pi$
$701$	− 20.2368i	− 0.764333i	0.924094	−	0.382166i	$-0.124822\pi$
$702$	0	0
$703$	−26.8011	−1.01082
$704$	0	0
$705$	−8.56412	−0.322543
$706$	0	0
$707$	− 19.3262i	− 0.726835i
$708$	0	0
$709$	− 33.6797i	− 1.26487i	−0.774614	−	0.632435i	$-0.782056\pi$
$709$	− 33.6797i	− 1.26487i	0.774614	−	0.632435i	$-0.217944\pi$
$710$	0	0
$711$	18.6077	0.697842
$712$	0	0
$713$	2.83555	0.106192
$714$	0	0
$715$	2.32337i	0.0868892i
$716$	0	0
$717$	− 8.31564i	− 0.310553i
$718$	0	0
$719$	52.0290	1.94035	0.970176	−	0.242400i	$-0.0779346\pi$
$719$	52.0290	1.94035	0.970176	+	0.242400i	$0.0779346\pi$
$720$	0	0
$721$	9.06135	0.337462
$722$	0	0
$723$	10.4865i	0.389999i
$724$	0	0
$725$	16.3930i	0.608821i
$726$	0	0
$727$	−28.3503	−1.05146	−0.525728	−	0.850653i	$-0.676207\pi$
$727$	−28.3503	−1.05146	−0.525728	+	0.850653i	$0.676207\pi$
$728$	0	0
$729$	19.6660	0.728371
$730$	0	0
$731$	53.1017i	1.96404i
$732$	0	0
$733$	41.9066i	1.54786i	0.633273	+	0.773928i	$0.281711\pi$
$733$	41.9066i	1.54786i	−0.633273	+	0.773928i	$0.718289\pi$
$734$	0	0
$735$	7.23184	0.266750
$736$	0	0
$737$	0.384213	0.0141527
$738$	0	0
$739$	22.8447i	0.840354i	0.907442	+	0.420177i	$0.138032\pi$
$739$	22.8447i	0.840354i	−0.907442	+	0.420177i	$0.861968\pi$
$740$	0	0
$741$	47.7008i	1.75233i
$742$	0	0
$743$	−52.7289	−1.93443	−0.967217	−	0.253950i	$-0.918270\pi$
$743$	−52.7289	−1.93443	−0.967217	+	0.253950i	$0.918270\pi$
$744$	0	0
$745$	−11.3947	−0.417468
$746$	0	0
$747$	12.8898i	0.471615i
$748$	0	0
$749$	− 6.12954i	− 0.223969i
$750$	0	0
$751$	−22.6662	−0.827103	−0.413551	−	0.910481i	$-0.635712\pi$
$751$	−22.6662	−0.827103	−0.413551	+	0.910481i	$0.635712\pi$
$752$	0	0
$753$	−42.7859	−1.55921
$754$	0	0
$755$	− 8.27428i	− 0.301132i
$756$	0	0
$757$	− 22.6344i	− 0.822661i	−0.911486	−	0.411330i	$-0.865064\pi$
$757$	− 22.6344i	− 0.822661i	0.911486	−	0.411330i	$-0.134936\pi$
$758$	0	0
$759$	−5.11292	−0.185587
$760$	0	0
$761$	34.8649	1.26385	0.631925	−	0.775029i	$-0.282265\pi$
$761$	34.8649	1.26385	0.631925	+	0.775029i	$0.282265\pi$
$762$	0	0
$763$	4.57527i	0.165636i
$764$	0	0
$765$	− 9.42362i	− 0.340712i
$766$	0	0
$767$	11.2107	0.404795
$768$	0	0
$769$	−14.3810	−0.518594	−0.259297	−	0.965798i	$-0.583491\pi$
$769$	−14.3810	−0.518594	−0.259297	+	0.965798i	$0.583491\pi$
$770$	0	0
$771$	61.2706i	2.20661i
$772$	0	0
$773$	6.37250i	0.229203i	0.993412	+	0.114601i	$0.0365591\pi$
$773$	6.37250i	0.229203i	−0.993412	+	0.114601i	$0.963441\pi$
$774$	0	0
$775$	4.67875	0.168066
$776$	0	0
$777$	−21.5928	−0.774639
$778$	0	0
$779$	− 23.2105i	− 0.831602i
$780$	0	0
$781$	6.94217i	0.248410i
$782$	0	0
$783$	−3.20303	−0.114467
$784$	0	0
$785$	0.587069	0.0209534
$786$	0	0
$787$	− 38.4012i	− 1.36886i	−0.729080	−	0.684428i	$-0.760052\pi$
$787$	− 38.4012i	− 1.36886i	0.729080	−	0.684428i	$-0.239948\pi$
$788$	0	0
$789$	− 74.5046i	− 2.65243i
$790$	0	0
$791$	3.73936	0.132956
$792$	0	0
$793$	−66.0122	−2.34416
$794$	0	0
$795$	− 0.956964i	− 0.0339400i
$796$	0	0
$797$	1.78669i	0.0632878i	0.999499	+	0.0316439i	$0.0100743\pi$
$797$	1.78669i	0.0632878i	−0.999499	+	0.0316439i	$0.989926\pi$
$798$	0	0
$799$	40.5586	1.43486
$800$	0	0
$801$	23.0748	0.815307
$802$	0	0
$803$	− 0.910593i	− 0.0321341i
$804$	0	0
$805$	2.04241i	0.0719856i
$806$	0	0
$807$	54.5636	1.92073
$808$	0	0
$809$	47.6122	1.67395	0.836977	−	0.547238i	$-0.184321\pi$
$809$	47.6122	1.67395	0.836977	+	0.547238i	$0.184321\pi$
$810$	0	0
$811$	55.1334i	1.93599i	0.250961	+	0.967997i	$0.419254\pi$
$811$	55.1334i	1.93599i	−0.250961	+	0.967997i	$0.580746\pi$
$812$	0	0
$813$	63.3444i	2.22158i
$814$	0	0
$815$	3.84312	0.134619
$816$	0	0
$817$	−31.2044	−1.09171
$818$	0	0
$819$	17.8950i	0.625302i
$820$	0	0
$821$	− 29.6216i	− 1.03380i	−0.856046	−	0.516900i	$-0.827086\pi$
$821$	− 29.6216i	− 1.03380i	0.856046	−	0.516900i	$-0.172914\pi$
$822$	0	0
$823$	12.9823	0.452533	0.226267	−	0.974065i	$-0.427348\pi$
$823$	12.9823	0.452533	0.226267	+	0.974065i	$0.427348\pi$
$824$	0	0
$825$	−8.43648	−0.293720
$826$	0	0
$827$	25.8831i	0.900044i	0.893018	+	0.450022i	$0.148584\pi$
$827$	25.8831i	0.900044i	−0.893018	+	0.450022i	$0.851416\pi$
$828$	0	0
$829$	11.1454i	0.387096i	0.981091	+	0.193548i	$0.0619995\pi$
$829$	11.1454i	0.387096i	−0.981091	+	0.193548i	$0.938000\pi$
$830$	0	0
$831$	−30.2868	−1.05064
$832$	0	0
$833$	−34.2491	−1.18666
$834$	0	0
$835$	3.10863i	0.107579i
$836$	0	0
$837$	0.914181i	0.0315987i
$838$	0	0
$839$	−18.1301	−0.625922	−0.312961	−	0.949766i	$-0.601321\pi$
$839$	−18.1301	−0.625922	−0.312961	+	0.949766i	$0.601321\pi$
$840$	0	0
$841$	16.7240	0.576690
$842$	0	0
$843$	67.5555i	2.32673i
$844$	0	0
$845$	− 9.07707i	− 0.312261i
$846$	0	0
$847$	13.2431	0.455039
$848$	0	0
$849$	−48.0133	−1.64781
$850$	0	0
$851$	20.3338i	0.697033i
$852$	0	0
$853$	8.88031i	0.304056i	0.988376	+	0.152028i	$0.0485804\pi$
$853$	8.88031i	0.304056i	−0.988376	+	0.152028i	$0.951420\pi$
$854$	0	0
$855$	5.53766	0.189384
$856$	0	0
$857$	−53.7859	−1.83729	−0.918645	−	0.395084i	$-0.870715\pi$
$857$	−53.7859	−1.83729	−0.918645	+	0.395084i	$0.870715\pi$
$858$	0	0
$859$	42.1025i	1.43652i	0.695775	+	0.718260i	$0.255061\pi$
$859$	42.1025i	1.43652i	−0.695775	+	0.718260i	$0.744939\pi$
$860$	0	0
$861$	− 18.7000i	− 0.637294i
$862$	0	0
$863$	12.0655	0.410713	0.205357	−	0.978687i	$-0.434165\pi$
$863$	12.0655	0.410713	0.205357	+	0.978687i	$0.434165\pi$
$864$	0	0
$865$	−1.19798	−0.0407327
$866$	0	0
$867$	55.5647i	1.88708i
$868$	0	0
$869$	− 5.41683i	− 0.183753i
$870$	0	0
$871$	−2.71954	−0.0921480
$872$	0	0
$873$	27.8317	0.941960
$874$	0	0
$875$	6.97148i	0.235679i
$876$	0	0
$877$	16.4420i	0.555207i	0.960696	+	0.277603i	$0.0895401\pi$
$877$	16.4420i	0.555207i	−0.960696	+	0.277603i	$0.910460\pi$
$878$	0	0
$879$	17.7656	0.599220
$880$	0	0
$881$	42.1498	1.42006	0.710031	−	0.704170i	$-0.248681\pi$
$881$	42.1498	1.42006	0.710031	+	0.704170i	$0.248681\pi$
$882$	0	0
$883$	13.7831i	0.463838i	0.972735	+	0.231919i	$0.0745004\pi$
$883$	13.7831i	0.463838i	−0.972735	+	0.231919i	$0.925500\pi$
$884$	0	0
$885$	− 2.79502i	− 0.0939536i
$886$	0	0
$887$	27.6359	0.927923	0.463961	−	0.885855i	$-0.346428\pi$
$887$	27.6359	0.927923	0.463961	+	0.885855i	$0.346428\pi$
$888$	0	0
$889$	4.47585	0.150115
$890$	0	0
$891$	− 7.61661i	− 0.255166i
$892$	0	0
$893$	23.8337i	0.797564i
$894$	0	0
$895$	1.21068	0.0404687
$896$	0	0
$897$	36.1903	1.20836
$898$	0	0
$899$	3.50371i	0.116855i
$900$	0	0
$901$	4.53207i	0.150985i
$902$	0	0
$903$	−25.1405	−0.836624
$904$	0	0
$905$	0.626057	0.0208108
$906$	0	0
$907$	19.5456i	0.649000i	0.945886	+	0.324500i	$0.105196\pi$
$907$	19.5456i	0.649000i	−0.945886	+	0.324500i	$0.894804\pi$
$908$	0	0
$909$	39.7553i	1.31860i
$910$	0	0
$911$	51.9784	1.72212	0.861061	−	0.508502i	$-0.169800\pi$
$911$	51.9784	1.72212	0.861061	+	0.508502i	$0.169800\pi$
$912$	0	0
$913$	3.75233	0.124184
$914$	0	0
$915$	16.4579i	0.544083i
$916$	0	0
$917$	9.57581i	0.316221i
$918$	0	0
$919$	−54.7497	−1.80602	−0.903012	−	0.429614i	$-0.858650\pi$
$919$	−54.7497	−1.80602	−0.903012	+	0.429614i	$0.858650\pi$
$920$	0	0
$921$	−26.5108	−0.873561
$922$	0	0
$923$	− 49.1381i	− 1.61740i
$924$	0	0
$925$	33.5514i	1.10316i
$926$	0	0
$927$	−18.6398	−0.612212
$928$	0	0
$929$	37.2755	1.22297	0.611484	−	0.791257i	$-0.290573\pi$
$929$	37.2755	1.22297	0.611484	+	0.791257i	$0.290573\pi$
$930$	0	0
$931$	− 20.1260i	− 0.659603i
$932$	0	0
$933$	46.1826i	1.51195i
$934$	0	0
$935$	−2.74328	−0.0897150
$936$	0	0
$937$	−35.5952	−1.16284	−0.581422	−	0.813602i	$-0.697503\pi$
$937$	−35.5952	−1.16284	−0.581422	+	0.813602i	$0.697503\pi$
$938$	0	0
$939$	9.71819i	0.317141i
$940$	0	0
$941$	− 9.88749i	− 0.322323i	−0.986928	−	0.161162i	$-0.948476\pi$
$941$	− 9.88749i	− 0.322323i	0.986928	−	0.161162i	$-0.0515240\pi$
$942$	0	0
$943$	−17.6096	−0.573449
$944$	0	0
$945$	−0.658473	−0.0214201
$946$	0	0
$947$	− 0.468259i	− 0.0152164i	−0.999971	−	0.00760818i	$-0.997578\pi$
$947$	− 0.468259i	− 0.0152164i	0.999971	−	0.00760818i	$-0.00242178\pi$
$948$	0	0
$949$	6.44536i	0.209225i
$950$	0	0
$951$	51.2663	1.66242
$952$	0	0
$953$	9.82333	0.318209	0.159104	−	0.987262i	$-0.449139\pi$
$953$	9.82333	0.318209	0.159104	+	0.987262i	$0.449139\pi$
$954$	0	0
$955$	− 11.5036i	− 0.372246i
$956$	0	0
$957$	− 6.31771i	− 0.204222i
$958$	0	0
$959$	20.6801	0.667795
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	12.6089i	0.406316i
$964$	0	0
$965$	− 2.96380i	− 0.0954080i
$966$	0	0
$967$	26.6971	0.858522	0.429261	−	0.903180i	$-0.358774\pi$
$967$	26.6971	0.858522	0.429261	+	0.903180i	$0.358774\pi$
$968$	0	0
$969$	−56.3219	−1.80932
$970$	0	0
$971$	− 12.1001i	− 0.388310i	−0.980971	−	0.194155i	$-0.937804\pi$
$971$	− 12.1001i	− 0.388310i	0.980971	−	0.194155i	$-0.0621965\pi$
$972$	0	0
$973$	20.7365i	0.664782i
$974$	0	0
$975$	59.7151	1.91241
$976$	0	0
$977$	52.2575	1.67187	0.835933	−	0.548832i	$-0.184927\pi$
$977$	52.2575	1.67187	0.835933	+	0.548832i	$0.184927\pi$
$978$	0	0
$979$	− 6.71723i	− 0.214684i
$980$	0	0
$981$	− 9.41164i	− 0.300491i
$982$	0	0
$983$	39.5545	1.26159	0.630796	−	0.775949i	$-0.282728\pi$
$983$	39.5545	1.26159	0.630796	+	0.775949i	$0.282728\pi$
$984$	0	0
$985$	7.90995	0.252032
$986$	0	0
$987$	19.2021i	0.611210i
$988$	0	0
$989$	23.6746i	0.752809i
$990$	0	0
$991$	43.0919	1.36886	0.684430	−	0.729079i	$-0.260051\pi$
$991$	43.0919	1.36886	0.684430	+	0.729079i	$0.260051\pi$
$992$	0	0
$993$	47.1815	1.49726
$994$	0	0
$995$	4.80999i	0.152487i
$996$	0	0
$997$	− 1.40691i	− 0.0445572i	−0.999752	−	0.0222786i	$-0.992908\pi$
$997$	− 1.40691i	− 0.0445572i	0.999752	−	0.0222786i	$-0.00709209\pi$
$998$	0	0
$999$	−6.55561	−0.207410

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1984.2.c.f.993.17	yes	20
4.3	odd	2		1984.2.c.e.993.4	✓	20
8.3	odd	2		1984.2.c.e.993.17	yes	20
8.5	even	2	inner	1984.2.c.f.993.4	yes	20
16.3	odd	4		7936.2.a.z.1.9		10
16.5	even	4		7936.2.a.x.1.9		10
16.11	odd	4		7936.2.a.w.1.2		10
16.13	even	4		7936.2.a.y.1.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1984.2.c.e.993.4	✓	20	4.3	odd	2
1984.2.c.e.993.17	yes	20	8.3	odd	2
1984.2.c.f.993.4	yes	20	8.5	even	2	inner
1984.2.c.f.993.17	yes	20	1.1	even	1	trivial
7936.2.a.w.1.2		10	16.11	odd	4
7936.2.a.x.1.9		10	16.5	even	4
7936.2.a.y.1.2		10	16.13	even	4
7936.2.a.z.1.9		10	16.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1984 = 2^{6} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1984.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.36943i q^{3} +0.566787i q^{5} +1.27083 q^{7} -2.61418 q^{9} +O(q^{10})\)	\(q+2.36943i q^{3} +0.566787i q^{5} +1.27083 q^{7} -2.61418 q^{9} +0.761006i q^{11} -5.38655i q^{13} -1.34296 q^{15} +6.36009 q^{17} +3.73742i q^{19} +3.01113i q^{21} +2.83555 q^{23} +4.67875 q^{25} +0.914181i q^{27} +3.50371i q^{29} +1.00000 q^{31} -1.80315 q^{33} +0.720287i q^{35} +7.17102i q^{37} +12.7630 q^{39} -6.21030 q^{41} +8.34920i q^{43} -1.48168i q^{45} +6.37705 q^{47} -5.38500 q^{49} +15.0698i q^{51} +0.712579i q^{53} -0.431328 q^{55} -8.85553 q^{57} +2.08124i q^{59} -12.2550i q^{61} -3.32216 q^{63} +3.05303 q^{65} -0.504875i q^{67} +6.71863i q^{69} +9.12236 q^{71} -1.19656 q^{73} +11.0860i q^{75} +0.967106i q^{77} -7.11799 q^{79} -10.0086 q^{81} -4.93075i q^{83} +3.60481i q^{85} -8.30178 q^{87} -8.82678 q^{89} -6.84537i q^{91} +2.36943i q^{93} -2.11832 q^{95} -10.6465 q^{97} -1.98940i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{9}+O(q^{10}) \) 20 * q - 12 * q^9	\( 20 q - 12 q^{9} + 28 q^{15} + 4 q^{17} + 20 q^{23} - 16 q^{25} + 20 q^{31} + 48 q^{39} + 4 q^{41} + 16 q^{47} + 8 q^{49} + 64 q^{55} - 4 q^{57} + 8 q^{63} + 12 q^{65} + 24 q^{71} - 16 q^{73} + 52 q^{79} - 20 q^{81} + 40 q^{87} + 12 q^{89} - 12 q^{97}+O(q^{100}) \) 20 * q - 12 * q^9 + 28 * q^15 + 4 * q^17 + 20 * q^23 - 16 * q^25 + 20 * q^31 + 48 * q^39 + 4 * q^41 + 16 * q^47 + 8 * q^49 + 64 * q^55 - 4 * q^57 + 8 * q^63 + 12 * q^65 + 24 * q^71 - 16 * q^73 + 52 * q^79 - 20 * q^81 + 40 * q^87 + 12 * q^89 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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