LMFDB - Embedded newform 1968.2.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1968,2,Mod(1,1968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1968, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1968.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1968 = 2^{4} \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1968.a (trivial)

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:	yes
Analytic conductor:	$15.7145591178$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 123)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	1968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.24914 q^{5} -3.30777 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.24914 q^{5} -3.30777 q^{7} +1.00000 q^{9} +1.47068 q^{11} -0.249141 q^{13} +4.24914 q^{15} -5.02760 q^{17} +2.24914 q^{19} -3.30777 q^{21} +6.24914 q^{23} +13.0552 q^{25} +1.00000 q^{27} -2.41205 q^{29} +2.89572 q^{31} +1.47068 q^{33} -14.0552 q^{35} +9.71982 q^{37} -0.249141 q^{39} +1.00000 q^{41} +10.8337 q^{43} +4.24914 q^{45} +4.08623 q^{47} +3.94137 q^{49} -5.02760 q^{51} -1.43965 q^{53} +6.24914 q^{55} +2.24914 q^{57} -2.61555 q^{59} -5.71982 q^{61} -3.30777 q^{63} -1.05863 q^{65} -15.9379 q^{67} +6.24914 q^{69} +10.5293 q^{71} +9.39400 q^{73} +13.0552 q^{75} -4.86469 q^{77} +0.560352 q^{79} +1.00000 q^{81} -3.80605 q^{83} -21.3630 q^{85} -2.41205 q^{87} +4.11727 q^{89} +0.824101 q^{91} +2.89572 q^{93} +9.55691 q^{95} -7.19051 q^{97} +1.47068 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 4 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 4 * q^5 - 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 4 q^{5} - 2 q^{7} + 3 q^{9} + 4 q^{11} + 8 q^{13} + 4 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{21} + 10 q^{23} + 5 q^{25} + 3 q^{27} - 6 q^{29} + 2 q^{31} + 4 q^{33} - 8 q^{35} + 20 q^{37} + 8 q^{39} + 3 q^{41} - 10 q^{43} + 4 q^{45} - 4 q^{47} + 11 q^{49} + 2 q^{51} + 14 q^{53} + 10 q^{55} - 2 q^{57} + 8 q^{59} - 8 q^{61} - 2 q^{63} - 4 q^{65} - 12 q^{67} + 10 q^{69} + 32 q^{71} + 4 q^{73} + 5 q^{75} + 10 q^{77} + 20 q^{79} + 3 q^{81} + 14 q^{83} - 22 q^{85} - 6 q^{87} + 14 q^{89} + 2 q^{93} + 12 q^{95} - 12 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 4 * q^5 - 2 * q^7 + 3 * q^9 + 4 * q^11 + 8 * q^13 + 4 * q^15 + 2 * q^17 - 2 * q^19 - 2 * q^21 + 10 * q^23 + 5 * q^25 + 3 * q^27 - 6 * q^29 + 2 * q^31 + 4 * q^33 - 8 * q^35 + 20 * q^37 + 8 * q^39 + 3 * q^41 - 10 * q^43 + 4 * q^45 - 4 * q^47 + 11 * q^49 + 2 * q^51 + 14 * q^53 + 10 * q^55 - 2 * q^57 + 8 * q^59 - 8 * q^61 - 2 * q^63 - 4 * q^65 - 12 * q^67 + 10 * q^69 + 32 * q^71 + 4 * q^73 + 5 * q^75 + 10 * q^77 + 20 * q^79 + 3 * q^81 + 14 * q^83 - 22 * q^85 - 6 * q^87 + 14 * q^89 + 2 * q^93 + 12 * q^95 - 12 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	4.24914	1.90027	0.950137	−	0.311834i	$-0.100943\pi$
$5$	4.24914	1.90027	0.950137	+	0.311834i	$0.100943\pi$
$6$	0	0
$7$	−3.30777	−1.25022	−0.625110	−	0.780536i	$-0.714946\pi$
$7$	−3.30777	−1.25022	−0.625110	+	0.780536i	$0.714946\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.47068	0.443428	0.221714	−	0.975112i	$-0.428835\pi$
$11$	1.47068	0.443428	0.221714	+	0.975112i	$0.428835\pi$
$12$	0	0
$13$	−0.249141	−0.0690992	−0.0345496	−	0.999403i	$-0.511000\pi$
$13$	−0.249141	−0.0690992	−0.0345496	+	0.999403i	$0.511000\pi$
$14$	0	0
$15$	4.24914	1.09712
$16$	0	0
$17$	−5.02760	−1.21937	−0.609686	−	0.792643i	$-0.708704\pi$
$17$	−5.02760	−1.21937	−0.609686	+	0.792643i	$0.708704\pi$
$18$	0	0
$19$	2.24914	0.515988	0.257994	−	0.966146i	$-0.416938\pi$
$19$	2.24914	0.515988	0.257994	+	0.966146i	$0.416938\pi$
$20$	0	0
$21$	−3.30777	−0.721815
$22$	0	0
$23$	6.24914	1.30304	0.651518	−	0.758633i	$-0.274133\pi$
$23$	6.24914	1.30304	0.651518	+	0.758633i	$0.274133\pi$
$24$	0	0
$25$	13.0552	2.61104
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.41205	−0.447906	−0.223953	−	0.974600i	$-0.571896\pi$
$29$	−2.41205	−0.447906	−0.223953	+	0.974600i	$0.571896\pi$
$30$	0	0
$31$	2.89572	0.520087	0.260044	−	0.965597i	$-0.416263\pi$
$31$	2.89572	0.520087	0.260044	+	0.965597i	$0.416263\pi$
$32$	0	0
$33$	1.47068	0.256013
$34$	0	0
$35$	−14.0552	−2.37576
$36$	0	0
$37$	9.71982	1.59793	0.798965	−	0.601378i	$-0.205381\pi$
$37$	9.71982	1.59793	0.798965	+	0.601378i	$0.205381\pi$
$38$	0	0
$39$	−0.249141	−0.0398944
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	10.8337	1.65212	0.826058	−	0.563585i	$-0.190578\pi$
$43$	10.8337	1.65212	0.826058	+	0.563585i	$0.190578\pi$
$44$	0	0
$45$	4.24914	0.633424
$46$	0	0
$47$	4.08623	0.596038	0.298019	−	0.954560i	$-0.403674\pi$
$47$	4.08623	0.596038	0.298019	+	0.954560i	$0.403674\pi$
$48$	0	0
$49$	3.94137	0.563052
$50$	0	0
$51$	−5.02760	−0.704004
$52$	0	0
$53$	−1.43965	−0.197751	−0.0988754	−	0.995100i	$-0.531525\pi$
$53$	−1.43965	−0.197751	−0.0988754	+	0.995100i	$0.531525\pi$
$54$	0	0
$55$	6.24914	0.842634
$56$	0	0
$57$	2.24914	0.297906
$58$	0	0
$59$	−2.61555	−0.340515	−0.170258	−	0.985400i	$-0.554460\pi$
$59$	−2.61555	−0.340515	−0.170258	+	0.985400i	$0.554460\pi$
$60$	0	0
$61$	−5.71982	−0.732348	−0.366174	−	0.930546i	$-0.619333\pi$
$61$	−5.71982	−0.732348	−0.366174	+	0.930546i	$0.619333\pi$
$62$	0	0
$63$	−3.30777	−0.416740
$64$	0	0
$65$	−1.05863	−0.131307
$66$	0	0
$67$	−15.9379	−1.94713	−0.973564	−	0.228415i	$-0.926646\pi$
$67$	−15.9379	−1.94713	−0.973564	+	0.228415i	$0.926646\pi$
$68$	0	0
$69$	6.24914	0.752308
$70$	0	0
$71$	10.5293	1.24960	0.624800	−	0.780785i	$-0.285181\pi$
$71$	10.5293	1.24960	0.624800	+	0.780785i	$0.285181\pi$
$72$	0	0
$73$	9.39400	1.09949	0.549743	−	0.835334i	$-0.314726\pi$
$73$	9.39400	1.09949	0.549743	+	0.835334i	$0.314726\pi$
$74$	0	0
$75$	13.0552	1.50748
$76$	0	0
$77$	−4.86469	−0.554383
$78$	0	0
$79$	0.560352	0.0630445	0.0315223	−	0.999503i	$-0.489964\pi$
$79$	0.560352	0.0630445	0.0315223	+	0.999503i	$0.489964\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.80605	−0.417769	−0.208884	−	0.977940i	$-0.566983\pi$
$83$	−3.80605	−0.417769	−0.208884	+	0.977940i	$0.566983\pi$
$84$	0	0
$85$	−21.3630	−2.31714
$86$	0	0
$87$	−2.41205	−0.258599
$88$	0	0
$89$	4.11727	0.436429	0.218215	−	0.975901i	$-0.429977\pi$
$89$	4.11727	0.436429	0.218215	+	0.975901i	$0.429977\pi$
$90$	0	0
$91$	0.824101	0.0863892
$92$	0	0
$93$	2.89572	0.300273
$94$	0	0
$95$	9.55691	0.980519
$96$	0	0
$97$	−7.19051	−0.730085	−0.365043	−	0.930991i	$-0.618946\pi$
$97$	−7.19051	−0.730085	−0.365043	+	0.930991i	$0.618946\pi$
$98$	0	0
$99$	1.47068	0.147809
$100$	0	0
$101$	−11.1449	−1.10896	−0.554478	−	0.832199i	$-0.687082\pi$
$101$	−11.1449	−1.10896	−0.554478	+	0.832199i	$0.687082\pi$
$102$	0	0
$103$	−13.2767	−1.30820	−0.654098	−	0.756410i	$-0.726952\pi$
$103$	−13.2767	−1.30820	−0.654098	+	0.756410i	$0.726952\pi$
$104$	0	0
$105$	−14.0552	−1.37165
$106$	0	0
$107$	−9.32238	−0.901229	−0.450614	−	0.892719i	$-0.648795\pi$
$107$	−9.32238	−0.901229	−0.450614	+	0.892719i	$0.648795\pi$
$108$	0	0
$109$	−0.249141	−0.0238633	−0.0119317	−	0.999929i	$-0.503798\pi$
$109$	−0.249141	−0.0238633	−0.0119317	+	0.999929i	$0.503798\pi$
$110$	0	0
$111$	9.71982	0.922565
$112$	0	0
$113$	9.24570	0.869763	0.434881	−	0.900488i	$-0.356790\pi$
$113$	9.24570	0.869763	0.434881	+	0.900488i	$0.356790\pi$
$114$	0	0
$115$	26.5535	2.47612
$116$	0	0
$117$	−0.249141	−0.0230331
$118$	0	0
$119$	16.6302	1.52448
$120$	0	0
$121$	−8.83709	−0.803372
$122$	0	0
$123$	1.00000	0.0901670
$124$	0	0
$125$	34.2277	3.06141
$126$	0	0
$127$	0.824101	0.0731271	0.0365635	−	0.999331i	$-0.488359\pi$
$127$	0.824101	0.0731271	0.0365635	+	0.999331i	$0.488359\pi$
$128$	0	0
$129$	10.8337	0.953850
$130$	0	0
$131$	−8.13187	−0.710485	−0.355243	−	0.934774i	$-0.615602\pi$
$131$	−8.13187	−0.710485	−0.355243	+	0.934774i	$0.615602\pi$
$132$	0	0
$133$	−7.43965	−0.645099
$134$	0	0
$135$	4.24914	0.365708
$136$	0	0
$137$	−12.2035	−1.04262	−0.521308	−	0.853369i	$-0.674556\pi$
$137$	−12.2035	−1.04262	−0.521308	+	0.853369i	$0.674556\pi$
$138$	0	0
$139$	−14.2897	−1.21204	−0.606019	−	0.795450i	$-0.707235\pi$
$139$	−14.2897	−1.21204	−0.606019	+	0.795450i	$0.707235\pi$
$140$	0	0
$141$	4.08623	0.344123
$142$	0	0
$143$	−0.366407	−0.0306405
$144$	0	0
$145$	−10.2491	−0.851145
$146$	0	0
$147$	3.94137	0.325078
$148$	0	0
$149$	−22.1104	−1.81135	−0.905677	−	0.423969i	$-0.860637\pi$
$149$	−22.1104	−1.81135	−0.905677	+	0.423969i	$0.860637\pi$
$150$	0	0
$151$	−10.1173	−0.823331	−0.411666	−	0.911335i	$-0.635053\pi$
$151$	−10.1173	−0.823331	−0.411666	+	0.911335i	$0.635053\pi$
$152$	0	0
$153$	−5.02760	−0.406457
$154$	0	0
$155$	12.3043	0.988308
$156$	0	0
$157$	18.7620	1.49737	0.748686	−	0.662924i	$-0.230685\pi$
$157$	18.7620	1.49737	0.748686	+	0.662924i	$0.230685\pi$
$158$	0	0
$159$	−1.43965	−0.114172
$160$	0	0
$161$	−20.6707	−1.62908
$162$	0	0
$163$	−8.54392	−0.669212	−0.334606	−	0.942358i	$-0.608603\pi$
$163$	−8.54392	−0.669212	−0.334606	+	0.942358i	$0.608603\pi$
$164$	0	0
$165$	6.24914	0.486495
$166$	0	0
$167$	−5.05863	−0.391449	−0.195724	−	0.980659i	$-0.562706\pi$
$167$	−5.05863	−0.391449	−0.195724	+	0.980659i	$0.562706\pi$
$168$	0	0
$169$	−12.9379	−0.995225
$170$	0	0
$171$	2.24914	0.171996
$172$	0	0
$173$	18.8647	1.43426	0.717128	−	0.696942i	$-0.245456\pi$
$173$	18.8647	1.43426	0.717128	+	0.696942i	$0.245456\pi$
$174$	0	0
$175$	−43.1836	−3.26438
$176$	0	0
$177$	−2.61555	−0.196597
$178$	0	0
$179$	2.85514	0.213403	0.106701	−	0.994291i	$-0.465971\pi$
$179$	2.85514	0.213403	0.106701	+	0.994291i	$0.465971\pi$
$180$	0	0
$181$	13.4396	0.998961	0.499481	−	0.866325i	$-0.333524\pi$
$181$	13.4396	0.998961	0.499481	+	0.866325i	$0.333524\pi$
$182$	0	0
$183$	−5.71982	−0.422822
$184$	0	0
$185$	41.3009	3.03650
$186$	0	0
$187$	−7.39400	−0.540703
$188$	0	0
$189$	−3.30777	−0.240605
$190$	0	0
$191$	−13.4948	−0.976453	−0.488226	−	0.872717i	$-0.662356\pi$
$191$	−13.4948	−0.976453	−0.488226	+	0.872717i	$0.662356\pi$
$192$	0	0
$193$	−8.01461	−0.576904	−0.288452	−	0.957494i	$-0.593141\pi$
$193$	−8.01461	−0.576904	−0.288452	+	0.957494i	$0.593141\pi$
$194$	0	0
$195$	−1.05863	−0.0758103
$196$	0	0
$197$	13.9785	0.995928	0.497964	−	0.867198i	$-0.334081\pi$
$197$	13.9785	0.995928	0.497964	+	0.867198i	$0.334081\pi$
$198$	0	0
$199$	4.86469	0.344849	0.172424	−	0.985023i	$-0.444840\pi$
$199$	4.86469	0.344849	0.172424	+	0.985023i	$0.444840\pi$
$200$	0	0
$201$	−15.9379	−1.12417
$202$	0	0
$203$	7.97852	0.559982
$204$	0	0
$205$	4.24914	0.296773
$206$	0	0
$207$	6.24914	0.434345
$208$	0	0
$209$	3.30777	0.228803
$210$	0	0
$211$	10.6155	0.730804	0.365402	−	0.930850i	$-0.380931\pi$
$211$	10.6155	0.730804	0.365402	+	0.930850i	$0.380931\pi$
$212$	0	0
$213$	10.5293	0.721457
$214$	0	0
$215$	46.0337	3.13947
$216$	0	0
$217$	−9.57840	−0.650224
$218$	0	0
$219$	9.39400	0.634788
$220$	0	0
$221$	1.25258	0.0842575
$222$	0	0
$223$	17.9931	1.20491	0.602454	−	0.798153i	$-0.294190\pi$
$223$	17.9931	1.20491	0.602454	+	0.798153i	$0.294190\pi$
$224$	0	0
$225$	13.0552	0.870346
$226$	0	0
$227$	0.910331	0.0604208	0.0302104	−	0.999544i	$-0.490382\pi$
$227$	0.910331	0.0604208	0.0302104	+	0.999544i	$0.490382\pi$
$228$	0	0
$229$	−7.42504	−0.490660	−0.245330	−	0.969440i	$-0.578896\pi$
$229$	−7.42504	−0.490660	−0.245330	+	0.969440i	$0.578896\pi$
$230$	0	0
$231$	−4.86469	−0.320073
$232$	0	0
$233$	14.7620	0.967093	0.483546	−	0.875319i	$-0.339348\pi$
$233$	14.7620	0.967093	0.483546	+	0.875319i	$0.339348\pi$
$234$	0	0
$235$	17.3630	1.13264
$236$	0	0
$237$	0.560352	0.0363988
$238$	0	0
$239$	1.38445	0.0895528	0.0447764	−	0.998997i	$-0.485742\pi$
$239$	1.38445	0.0895528	0.0447764	+	0.998997i	$0.485742\pi$
$240$	0	0
$241$	−5.95436	−0.383554	−0.191777	−	0.981439i	$-0.561425\pi$
$241$	−5.95436	−0.383554	−0.191777	+	0.981439i	$0.561425\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	16.7474	1.06995
$246$	0	0
$247$	−0.560352	−0.0356543
$248$	0	0
$249$	−3.80605	−0.241199
$250$	0	0
$251$	−2.70683	−0.170854	−0.0854269	−	0.996344i	$-0.527225\pi$
$251$	−2.70683	−0.170854	−0.0854269	+	0.996344i	$0.527225\pi$
$252$	0	0
$253$	9.19051	0.577802
$254$	0	0
$255$	−21.3630	−1.33780
$256$	0	0
$257$	−0.793065	−0.0494700	−0.0247350	−	0.999694i	$-0.507874\pi$
$257$	−0.793065	−0.0494700	−0.0247350	+	0.999694i	$0.507874\pi$
$258$	0	0
$259$	−32.1510	−1.99776
$260$	0	0
$261$	−2.41205	−0.149302
$262$	0	0
$263$	15.5879	0.961194	0.480597	−	0.876942i	$-0.340420\pi$
$263$	15.5879	0.961194	0.480597	+	0.876942i	$0.340420\pi$
$264$	0	0
$265$	−6.11727	−0.375781
$266$	0	0
$267$	4.11727	0.251973
$268$	0	0
$269$	3.05863	0.186488	0.0932441	−	0.995643i	$-0.470276\pi$
$269$	3.05863	0.186488	0.0932441	+	0.995643i	$0.470276\pi$
$270$	0	0
$271$	−12.2802	−0.745968	−0.372984	−	0.927838i	$-0.621665\pi$
$271$	−12.2802	−0.745968	−0.372984	+	0.927838i	$0.621665\pi$
$272$	0	0
$273$	0.824101	0.0498768
$274$	0	0
$275$	19.2001	1.15781
$276$	0	0
$277$	−8.56990	−0.514916	−0.257458	−	0.966290i	$-0.582885\pi$
$277$	−8.56990	−0.514916	−0.257458	+	0.966290i	$0.582885\pi$
$278$	0	0
$279$	2.89572	0.173362
$280$	0	0
$281$	−14.5845	−0.870039	−0.435020	−	0.900421i	$-0.643259\pi$
$281$	−14.5845	−0.870039	−0.435020	+	0.900421i	$0.643259\pi$
$282$	0	0
$283$	20.9509	1.24540	0.622701	−	0.782460i	$-0.286035\pi$
$283$	20.9509	1.24540	0.622701	+	0.782460i	$0.286035\pi$
$284$	0	0
$285$	9.55691	0.566103
$286$	0	0
$287$	−3.30777	−0.195252
$288$	0	0
$289$	8.27674	0.486867
$290$	0	0
$291$	−7.19051	−0.421515
$292$	0	0
$293$	31.4086	1.83491	0.917455	−	0.397839i	$-0.130240\pi$
$293$	31.4086	1.83491	0.917455	+	0.397839i	$0.130240\pi$
$294$	0	0
$295$	−11.1138	−0.647072
$296$	0	0
$297$	1.47068	0.0853377
$298$	0	0
$299$	−1.55691	−0.0900387
$300$	0	0
$301$	−35.8353	−2.06551
$302$	0	0
$303$	−11.1449	−0.640256
$304$	0	0
$305$	−24.3043	−1.39166
$306$	0	0
$307$	−8.54392	−0.487628	−0.243814	−	0.969822i	$-0.578399\pi$
$307$	−8.54392	−0.487628	−0.243814	+	0.969822i	$0.578399\pi$
$308$	0	0
$309$	−13.2767	−0.755287
$310$	0	0
$311$	5.49484	0.311584	0.155792	−	0.987790i	$-0.450207\pi$
$311$	5.49484	0.311584	0.155792	+	0.987790i	$0.450207\pi$
$312$	0	0
$313$	5.48024	0.309761	0.154881	−	0.987933i	$-0.450501\pi$
$313$	5.48024	0.309761	0.154881	+	0.987933i	$0.450501\pi$
$314$	0	0
$315$	−14.0552	−0.791921
$316$	0	0
$317$	−11.3173	−0.635644	−0.317822	−	0.948150i	$-0.602951\pi$
$317$	−11.3173	−0.635644	−0.317822	+	0.948150i	$0.602951\pi$
$318$	0	0
$319$	−3.54736	−0.198614
$320$	0	0
$321$	−9.32238	−0.520325
$322$	0	0
$323$	−11.3078	−0.629181
$324$	0	0
$325$	−3.25258	−0.180421
$326$	0	0
$327$	−0.249141	−0.0137775
$328$	0	0
$329$	−13.5163	−0.745179
$330$	0	0
$331$	−33.8613	−1.86118	−0.930591	−	0.366060i	$-0.880707\pi$
$331$	−33.8613	−1.86118	−0.930591	+	0.366060i	$0.880707\pi$
$332$	0	0
$333$	9.71982	0.532643
$334$	0	0
$335$	−67.7225	−3.70008
$336$	0	0
$337$	−6.77846	−0.369246	−0.184623	−	0.982809i	$-0.559106\pi$
$337$	−6.77846	−0.369246	−0.184623	+	0.982809i	$0.559106\pi$
$338$	0	0
$339$	9.24570	0.502158
$340$	0	0
$341$	4.25869	0.230621
$342$	0	0
$343$	10.1173	0.546281
$344$	0	0
$345$	26.5535	1.42959
$346$	0	0
$347$	−19.2001	−1.03071	−0.515357	−	0.856976i	$-0.672341\pi$
$347$	−19.2001	−1.03071	−0.515357	+	0.856976i	$0.672341\pi$
$348$	0	0
$349$	−6.54392	−0.350288	−0.175144	−	0.984543i	$-0.556039\pi$
$349$	−6.54392	−0.350288	−0.175144	+	0.984543i	$0.556039\pi$
$350$	0	0
$351$	−0.249141	−0.0132981
$352$	0	0
$353$	−15.9931	−0.851228	−0.425614	−	0.904905i	$-0.639942\pi$
$353$	−15.9931	−0.851228	−0.425614	+	0.904905i	$0.639942\pi$
$354$	0	0
$355$	44.7405	2.37458
$356$	0	0
$357$	16.6302	0.880161
$358$	0	0
$359$	12.1319	0.640296	0.320148	−	0.947368i	$-0.396267\pi$
$359$	12.1319	0.640296	0.320148	+	0.947368i	$0.396267\pi$
$360$	0	0
$361$	−13.9414	−0.733756
$362$	0	0
$363$	−8.83709	−0.463827
$364$	0	0
$365$	39.9164	2.08932
$366$	0	0
$367$	−6.39744	−0.333944	−0.166972	−	0.985962i	$-0.553399\pi$
$367$	−6.39744	−0.333944	−0.166972	+	0.985962i	$0.553399\pi$
$368$	0	0
$369$	1.00000	0.0520579
$370$	0	0
$371$	4.76203	0.247232
$372$	0	0
$373$	−7.54049	−0.390432	−0.195216	−	0.980760i	$-0.562541\pi$
$373$	−7.54049	−0.390432	−0.195216	+	0.980760i	$0.562541\pi$
$374$	0	0
$375$	34.2277	1.76751
$376$	0	0
$377$	0.600939	0.0309500
$378$	0	0
$379$	0.0620710	0.00318837	0.00159419	−	0.999999i	$-0.499493\pi$
$379$	0.0620710	0.00318837	0.00159419	+	0.999999i	$0.499493\pi$
$380$	0	0
$381$	0.824101	0.0422199
$382$	0	0
$383$	−23.9329	−1.22291	−0.611456	−	0.791278i	$-0.709416\pi$
$383$	−23.9329	−1.22291	−0.611456	+	0.791278i	$0.709416\pi$
$384$	0	0
$385$	−20.6707	−1.05348
$386$	0	0
$387$	10.8337	0.550706
$388$	0	0
$389$	−23.8613	−1.20981	−0.604907	−	0.796296i	$-0.706790\pi$
$389$	−23.8613	−1.20981	−0.604907	+	0.796296i	$0.706790\pi$
$390$	0	0
$391$	−31.4182	−1.58888
$392$	0	0
$393$	−8.13187	−0.410199
$394$	0	0
$395$	2.38101	0.119802
$396$	0	0
$397$	14.6302	0.734266	0.367133	−	0.930168i	$-0.380339\pi$
$397$	14.6302	0.734266	0.367133	+	0.930168i	$0.380339\pi$
$398$	0	0
$399$	−7.43965	−0.372448
$400$	0	0
$401$	−11.3224	−0.565413	−0.282706	−	0.959206i	$-0.591232\pi$
$401$	−11.3224	−0.565413	−0.282706	+	0.959206i	$0.591232\pi$
$402$	0	0
$403$	−0.721442	−0.0359376
$404$	0	0
$405$	4.24914	0.211141
$406$	0	0
$407$	14.2948	0.708566
$408$	0	0
$409$	−33.8923	−1.67587	−0.837933	−	0.545773i	$-0.816236\pi$
$409$	−33.8923	−1.67587	−0.837933	+	0.545773i	$0.816236\pi$
$410$	0	0
$411$	−12.2035	−0.601954
$412$	0	0
$413$	8.65164	0.425719
$414$	0	0
$415$	−16.1725	−0.793875
$416$	0	0
$417$	−14.2897	−0.699771
$418$	0	0
$419$	1.98539	0.0969928	0.0484964	−	0.998823i	$-0.484557\pi$
$419$	1.98539	0.0969928	0.0484964	+	0.998823i	$0.484557\pi$
$420$	0	0
$421$	−5.67418	−0.276543	−0.138271	−	0.990394i	$-0.544155\pi$
$421$	−5.67418	−0.276543	−0.138271	+	0.990394i	$0.544155\pi$
$422$	0	0
$423$	4.08623	0.198679
$424$	0	0
$425$	−65.6363	−3.18383
$426$	0	0
$427$	18.9199	0.915597
$428$	0	0
$429$	−0.366407	−0.0176903
$430$	0	0
$431$	−1.75086	−0.0843359	−0.0421680	−	0.999111i	$-0.513426\pi$
$431$	−1.75086	−0.0843359	−0.0421680	+	0.999111i	$0.513426\pi$
$432$	0	0
$433$	−34.0647	−1.63705	−0.818524	−	0.574473i	$-0.805207\pi$
$433$	−34.0647	−1.63705	−0.818524	+	0.574473i	$0.805207\pi$
$434$	0	0
$435$	−10.2491	−0.491409
$436$	0	0
$437$	14.0552	0.672351
$438$	0	0
$439$	4.32582	0.206460	0.103230	−	0.994658i	$-0.467082\pi$
$439$	4.32582	0.206460	0.103230	+	0.994658i	$0.467082\pi$
$440$	0	0
$441$	3.94137	0.187684
$442$	0	0
$443$	−36.9966	−1.75776	−0.878880	−	0.477043i	$-0.841709\pi$
$443$	−36.9966	−1.75776	−0.878880	+	0.477043i	$0.841709\pi$
$444$	0	0
$445$	17.4948	0.829335
$446$	0	0
$447$	−22.1104	−1.04579
$448$	0	0
$449$	−15.9448	−0.752482	−0.376241	−	0.926522i	$-0.622784\pi$
$449$	−15.9448	−0.752482	−0.376241	+	0.926522i	$0.622784\pi$
$450$	0	0
$451$	1.47068	0.0692518
$452$	0	0
$453$	−10.1173	−0.475351
$454$	0	0
$455$	3.50172	0.164163
$456$	0	0
$457$	−29.4396	−1.37713	−0.688564	−	0.725175i	$-0.741759\pi$
$457$	−29.4396	−1.37713	−0.688564	+	0.725175i	$0.741759\pi$
$458$	0	0
$459$	−5.02760	−0.234668
$460$	0	0
$461$	21.8759	1.01886	0.509430	−	0.860512i	$-0.329856\pi$
$461$	21.8759	1.01886	0.509430	+	0.860512i	$0.329856\pi$
$462$	0	0
$463$	18.6561	0.867024	0.433512	−	0.901148i	$-0.357274\pi$
$463$	18.6561	0.867024	0.433512	+	0.901148i	$0.357274\pi$
$464$	0	0
$465$	12.3043	0.570600
$466$	0	0
$467$	−8.67074	−0.401234	−0.200617	−	0.979670i	$-0.564295\pi$
$467$	−8.67074	−0.401234	−0.200617	+	0.979670i	$0.564295\pi$
$468$	0	0
$469$	52.7191	2.43434
$470$	0	0
$471$	18.7620	0.864509
$472$	0	0
$473$	15.9329	0.732594
$474$	0	0
$475$	29.3630	1.34727
$476$	0	0
$477$	−1.43965	−0.0659169
$478$	0	0
$479$	20.4312	0.933523	0.466762	−	0.884383i	$-0.345421\pi$
$479$	20.4312	0.933523	0.466762	+	0.884383i	$0.345421\pi$
$480$	0	0
$481$	−2.42160	−0.110416
$482$	0	0
$483$	−20.6707	−0.940551
$484$	0	0
$485$	−30.5535	−1.38736
$486$	0	0
$487$	7.00611	0.317477	0.158739	−	0.987321i	$-0.449257\pi$
$487$	7.00611	0.317477	0.158739	+	0.987321i	$0.449257\pi$
$488$	0	0
$489$	−8.54392	−0.386370
$490$	0	0
$491$	15.5715	0.702733	0.351366	−	0.936238i	$-0.385717\pi$
$491$	15.5715	0.702733	0.351366	+	0.936238i	$0.385717\pi$
$492$	0	0
$493$	12.1268	0.546164
$494$	0	0
$495$	6.24914	0.280878
$496$	0	0
$497$	−34.8286	−1.56228
$498$	0	0
$499$	1.61899	0.0724757	0.0362379	−	0.999343i	$-0.488463\pi$
$499$	1.61899	0.0724757	0.0362379	+	0.999343i	$0.488463\pi$
$500$	0	0
$501$	−5.05863	−0.226003
$502$	0	0
$503$	39.3725	1.75553	0.877767	−	0.479088i	$-0.159032\pi$
$503$	39.3725	1.75553	0.877767	+	0.479088i	$0.159032\pi$
$504$	0	0
$505$	−47.3561	−2.10732
$506$	0	0
$507$	−12.9379	−0.574594
$508$	0	0
$509$	−8.63971	−0.382948	−0.191474	−	0.981498i	$-0.561327\pi$
$509$	−8.63971	−0.382948	−0.191474	+	0.981498i	$0.561327\pi$
$510$	0	0
$511$	−31.0732	−1.37460
$512$	0	0
$513$	2.24914	0.0993020
$514$	0	0
$515$	−56.4147	−2.48593
$516$	0	0
$517$	6.00955	0.264300
$518$	0	0
$519$	18.8647	0.828068
$520$	0	0
$521$	30.7862	1.34877	0.674384	−	0.738381i	$-0.264409\pi$
$521$	30.7862	1.34877	0.674384	+	0.738381i	$0.264409\pi$
$522$	0	0
$523$	−17.4036	−0.761004	−0.380502	−	0.924780i	$-0.624249\pi$
$523$	−17.4036	−0.761004	−0.380502	+	0.924780i	$0.624249\pi$
$524$	0	0
$525$	−43.1836	−1.88469
$526$	0	0
$527$	−14.5585	−0.634180
$528$	0	0
$529$	16.0518	0.697902
$530$	0	0
$531$	−2.61555	−0.113505
$532$	0	0
$533$	−0.249141	−0.0107915
$534$	0	0
$535$	−39.6121	−1.71258
$536$	0	0
$537$	2.85514	0.123208
$538$	0	0
$539$	5.79650	0.249673
$540$	0	0
$541$	−9.87586	−0.424596	−0.212298	−	0.977205i	$-0.568095\pi$
$541$	−9.87586	−0.424596	−0.212298	+	0.977205i	$0.568095\pi$
$542$	0	0
$543$	13.4396	0.576750
$544$	0	0
$545$	−1.05863	−0.0453469
$546$	0	0
$547$	18.1465	0.775888	0.387944	−	0.921683i	$-0.373185\pi$
$547$	18.1465	0.775888	0.387944	+	0.921683i	$0.373185\pi$
$548$	0	0
$549$	−5.71982	−0.244116
$550$	0	0
$551$	−5.42504	−0.231114
$552$	0	0
$553$	−1.85352	−0.0788196
$554$	0	0
$555$	41.3009	1.75313
$556$	0	0
$557$	37.3725	1.58352	0.791762	−	0.610829i	$-0.209164\pi$
$557$	37.3725	1.58352	0.791762	+	0.610829i	$0.209164\pi$
$558$	0	0
$559$	−2.69910	−0.114160
$560$	0	0
$561$	−7.39400	−0.312175
$562$	0	0
$563$	−22.8742	−0.964034	−0.482017	−	0.876162i	$-0.660096\pi$
$563$	−22.8742	−0.964034	−0.482017	+	0.876162i	$0.660096\pi$
$564$	0	0
$565$	39.2863	1.65279
$566$	0	0
$567$	−3.30777	−0.138913
$568$	0	0
$569$	34.4147	1.44274	0.721370	−	0.692550i	$-0.243513\pi$
$569$	34.4147	1.44274	0.721370	+	0.692550i	$0.243513\pi$
$570$	0	0
$571$	−36.3449	−1.52099	−0.760494	−	0.649345i	$-0.775043\pi$
$571$	−36.3449	−1.52099	−0.760494	+	0.649345i	$0.775043\pi$
$572$	0	0
$573$	−13.4948	−0.563755
$574$	0	0
$575$	81.5838	3.40228
$576$	0	0
$577$	26.1725	1.08957	0.544787	−	0.838575i	$-0.316611\pi$
$577$	26.1725	1.08957	0.544787	+	0.838575i	$0.316611\pi$
$578$	0	0
$579$	−8.01461	−0.333076
$580$	0	0
$581$	12.5896	0.522303
$582$	0	0
$583$	−2.11727	−0.0876882
$584$	0	0
$585$	−1.05863	−0.0437691
$586$	0	0
$587$	17.8156	0.735329	0.367664	−	0.929959i	$-0.380158\pi$
$587$	17.8156	0.735329	0.367664	+	0.929959i	$0.380158\pi$
$588$	0	0
$589$	6.51289	0.268359
$590$	0	0
$591$	13.9785	0.574999
$592$	0	0
$593$	−32.7018	−1.34290	−0.671451	−	0.741049i	$-0.734328\pi$
$593$	−32.7018	−1.34290	−0.671451	+	0.741049i	$0.734328\pi$
$594$	0	0
$595$	70.6639	2.89694
$596$	0	0
$597$	4.86469	0.199098
$598$	0	0
$599$	−10.2784	−0.419962	−0.209981	−	0.977705i	$-0.567340\pi$
$599$	−10.2784	−0.419962	−0.209981	+	0.977705i	$0.567340\pi$
$600$	0	0
$601$	31.8398	1.29877	0.649386	−	0.760459i	$-0.275026\pi$
$601$	31.8398	1.29877	0.649386	+	0.760459i	$0.275026\pi$
$602$	0	0
$603$	−15.9379	−0.649043
$604$	0	0
$605$	−37.5500	−1.52663
$606$	0	0
$607$	−10.2086	−0.414352	−0.207176	−	0.978304i	$-0.566427\pi$
$607$	−10.2086	−0.414352	−0.207176	+	0.978304i	$0.566427\pi$
$608$	0	0
$609$	7.97852	0.323306
$610$	0	0
$611$	−1.01805	−0.0411857
$612$	0	0
$613$	−4.94137	−0.199580	−0.0997900	−	0.995009i	$-0.531817\pi$
$613$	−4.94137	−0.199580	−0.0997900	+	0.995009i	$0.531817\pi$
$614$	0	0
$615$	4.24914	0.171342
$616$	0	0
$617$	15.2526	0.614046	0.307023	−	0.951702i	$-0.400667\pi$
$617$	15.2526	0.614046	0.307023	+	0.951702i	$0.400667\pi$
$618$	0	0
$619$	20.6543	0.830167	0.415084	−	0.909783i	$-0.363752\pi$
$619$	20.6543	0.830167	0.415084	+	0.909783i	$0.363752\pi$
$620$	0	0
$621$	6.24914	0.250769
$622$	0	0
$623$	−13.6190	−0.545633
$624$	0	0
$625$	80.1621	3.20649
$626$	0	0
$627$	3.30777	0.132100
$628$	0	0
$629$	−48.8674	−1.94847
$630$	0	0
$631$	−5.83709	−0.232371	−0.116185	−	0.993228i	$-0.537067\pi$
$631$	−5.83709	−0.232371	−0.116185	+	0.993228i	$0.537067\pi$
$632$	0	0
$633$	10.6155	0.421930
$634$	0	0
$635$	3.50172	0.138961
$636$	0	0
$637$	−0.981954	−0.0389064
$638$	0	0
$639$	10.5293	0.416533
$640$	0	0
$641$	42.6328	1.68390	0.841948	−	0.539559i	$-0.181409\pi$
$641$	42.6328	1.68390	0.841948	+	0.539559i	$0.181409\pi$
$642$	0	0
$643$	−30.7259	−1.21171	−0.605856	−	0.795574i	$-0.707169\pi$
$643$	−30.7259	−1.21171	−0.605856	+	0.795574i	$0.707169\pi$
$644$	0	0
$645$	46.0337	1.81258
$646$	0	0
$647$	−18.3595	−0.721788	−0.360894	−	0.932607i	$-0.617528\pi$
$647$	−18.3595	−0.721788	−0.360894	+	0.932607i	$0.617528\pi$
$648$	0	0
$649$	−3.84664	−0.150994
$650$	0	0
$651$	−9.57840	−0.375407
$652$	0	0
$653$	−35.5811	−1.39240	−0.696198	−	0.717850i	$-0.745126\pi$
$653$	−35.5811	−1.39240	−0.696198	+	0.717850i	$0.745126\pi$
$654$	0	0
$655$	−34.5535	−1.35012
$656$	0	0
$657$	9.39400	0.366495
$658$	0	0
$659$	6.38101	0.248569	0.124285	−	0.992247i	$-0.460336\pi$
$659$	6.38101	0.248569	0.124285	+	0.992247i	$0.460336\pi$
$660$	0	0
$661$	30.9053	1.20208	0.601038	−	0.799220i	$-0.294754\pi$
$661$	30.9053	1.20208	0.601038	+	0.799220i	$0.294754\pi$
$662$	0	0
$663$	1.25258	0.0486461
$664$	0	0
$665$	−31.6121	−1.22587
$666$	0	0
$667$	−15.0732	−0.583638
$668$	0	0
$669$	17.9931	0.695654
$670$	0	0
$671$	−8.41205	−0.324744
$672$	0	0
$673$	36.7259	1.41568	0.707840	−	0.706372i	$-0.249670\pi$
$673$	36.7259	1.41568	0.707840	+	0.706372i	$0.249670\pi$
$674$	0	0
$675$	13.0552	0.502495
$676$	0	0
$677$	−5.66281	−0.217639	−0.108820	−	0.994062i	$-0.534707\pi$
$677$	−5.66281	−0.217639	−0.108820	+	0.994062i	$0.534707\pi$
$678$	0	0
$679$	23.7846	0.912768
$680$	0	0
$681$	0.910331	0.0348840
$682$	0	0
$683$	29.1690	1.11612	0.558061	−	0.829800i	$-0.311546\pi$
$683$	29.1690	1.11612	0.558061	+	0.829800i	$0.311546\pi$
$684$	0	0
$685$	−51.8544	−1.98125
$686$	0	0
$687$	−7.42504	−0.283283
$688$	0	0
$689$	0.358675	0.0136644
$690$	0	0
$691$	−1.68879	−0.0642445	−0.0321223	−	0.999484i	$-0.510227\pi$
$691$	−1.68879	−0.0642445	−0.0321223	+	0.999484i	$0.510227\pi$
$692$	0	0
$693$	−4.86469	−0.184794
$694$	0	0
$695$	−60.7191	−2.30321
$696$	0	0
$697$	−5.02760	−0.190434
$698$	0	0
$699$	14.7620	0.558351
$700$	0	0
$701$	−20.3810	−0.769780	−0.384890	−	0.922962i	$-0.625761\pi$
$701$	−20.3810	−0.769780	−0.384890	+	0.922962i	$0.625761\pi$
$702$	0	0
$703$	21.8613	0.824513
$704$	0	0
$705$	17.3630	0.653927
$706$	0	0
$707$	36.8647	1.38644
$708$	0	0
$709$	−40.8363	−1.53364	−0.766820	−	0.641862i	$-0.778162\pi$
$709$	−40.8363	−1.53364	−0.766820	+	0.641862i	$0.778162\pi$
$710$	0	0
$711$	0.560352	0.0210148
$712$	0	0
$713$	18.0958	0.677692
$714$	0	0
$715$	−1.55691	−0.0582253
$716$	0	0
$717$	1.38445	0.0517033
$718$	0	0
$719$	−9.14486	−0.341046	−0.170523	−	0.985354i	$-0.554546\pi$
$719$	−9.14486	−0.341046	−0.170523	+	0.985354i	$0.554546\pi$
$720$	0	0
$721$	43.9164	1.63553
$722$	0	0
$723$	−5.95436	−0.221445
$724$	0	0
$725$	−31.4898	−1.16950
$726$	0	0
$727$	2.13875	0.0793218	0.0396609	−	0.999213i	$-0.487372\pi$
$727$	2.13875	0.0793218	0.0396609	+	0.999213i	$0.487372\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−54.4672	−2.01454
$732$	0	0
$733$	15.3388	0.566552	0.283276	−	0.959038i	$-0.408579\pi$
$733$	15.3388	0.566552	0.283276	+	0.959038i	$0.408579\pi$
$734$	0	0
$735$	16.7474	0.617738
$736$	0	0
$737$	−23.4396	−0.863411
$738$	0	0
$739$	15.8111	0.581621	0.290811	−	0.956781i	$-0.406075\pi$
$739$	15.8111	0.581621	0.290811	+	0.956781i	$0.406075\pi$
$740$	0	0
$741$	−0.560352	−0.0205850
$742$	0	0
$743$	20.2017	0.741128	0.370564	−	0.928807i	$-0.379164\pi$
$743$	20.2017	0.741128	0.370564	+	0.928807i	$0.379164\pi$
$744$	0	0
$745$	−93.9502	−3.44207
$746$	0	0
$747$	−3.80605	−0.139256
$748$	0	0
$749$	30.8363	1.12673
$750$	0	0
$751$	−32.9053	−1.20073	−0.600365	−	0.799726i	$-0.704978\pi$
$751$	−32.9053	−1.20073	−0.600365	+	0.799726i	$0.704978\pi$
$752$	0	0
$753$	−2.70683	−0.0986425
$754$	0	0
$755$	−42.9897	−1.56455
$756$	0	0
$757$	11.3009	0.410738	0.205369	−	0.978685i	$-0.434161\pi$
$757$	11.3009	0.410738	0.205369	+	0.978685i	$0.434161\pi$
$758$	0	0
$759$	9.19051	0.333594
$760$	0	0
$761$	−2.93449	−0.106375	−0.0531876	−	0.998585i	$-0.516938\pi$
$761$	−2.93449	−0.106375	−0.0531876	+	0.998585i	$0.516938\pi$
$762$	0	0
$763$	0.824101	0.0298344
$764$	0	0
$765$	−21.3630	−0.772380
$766$	0	0
$767$	0.651639	0.0235293
$768$	0	0
$769$	−22.7785	−0.821412	−0.410706	−	0.911768i	$-0.634718\pi$
$769$	−22.7785	−0.821412	−0.410706	+	0.911768i	$0.634718\pi$
$770$	0	0
$771$	−0.793065	−0.0285615
$772$	0	0
$773$	10.5553	0.379648	0.189824	−	0.981818i	$-0.439208\pi$
$773$	10.5553	0.379648	0.189824	+	0.981818i	$0.439208\pi$
$774$	0	0
$775$	37.8042	1.35797
$776$	0	0
$777$	−32.1510	−1.15341
$778$	0	0
$779$	2.24914	0.0805838
$780$	0	0
$781$	15.4853	0.554107
$782$	0	0
$783$	−2.41205	−0.0861996
$784$	0	0
$785$	79.7225	2.84542
$786$	0	0
$787$	23.0682	0.822292	0.411146	−	0.911570i	$-0.365129\pi$
$787$	23.0682	0.822292	0.411146	+	0.911570i	$0.365129\pi$
$788$	0	0
$789$	15.5879	0.554946
$790$	0	0
$791$	−30.5827	−1.08740
$792$	0	0
$793$	1.42504	0.0506047
$794$	0	0
$795$	−6.11727	−0.216957
$796$	0	0
$797$	0.0360915	0.00127843	0.000639213	−	1.00000i	$-0.499797\pi$
$797$	0.0360915	0.00127843	0.000639213		1.00000i	$0.499797\pi$
$798$	0	0
$799$	−20.5439	−0.726792
$800$	0	0
$801$	4.11727	0.145476
$802$	0	0
$803$	13.8156	0.487542
$804$	0	0
$805$	−87.8329	−3.09570
$806$	0	0
$807$	3.05863	0.107669
$808$	0	0
$809$	33.1430	1.16525	0.582624	−	0.812742i	$-0.302026\pi$
$809$	33.1430	1.16525	0.582624	+	0.812742i	$0.302026\pi$
$810$	0	0
$811$	1.10428	0.0387764	0.0193882	−	0.999812i	$-0.493828\pi$
$811$	1.10428	0.0387764	0.0193882	+	0.999812i	$0.493828\pi$
$812$	0	0
$813$	−12.2802	−0.430685
$814$	0	0
$815$	−36.3043	−1.27169
$816$	0	0
$817$	24.3664	0.852473
$818$	0	0
$819$	0.824101	0.0287964
$820$	0	0
$821$	−7.42504	−0.259136	−0.129568	−	0.991571i	$-0.541359\pi$
$821$	−7.42504	−0.259136	−0.129568	+	0.991571i	$0.541359\pi$
$822$	0	0
$823$	−42.1656	−1.46980	−0.734900	−	0.678176i	$-0.762771\pi$
$823$	−42.1656	−1.46980	−0.734900	+	0.678176i	$0.762771\pi$
$824$	0	0
$825$	19.2001	0.668460
$826$	0	0
$827$	54.4863	1.89468	0.947338	−	0.320235i	$-0.103762\pi$
$827$	54.4863	1.89468	0.947338	+	0.320235i	$0.103762\pi$
$828$	0	0
$829$	24.0388	0.834901	0.417450	−	0.908700i	$-0.362924\pi$
$829$	24.0388	0.834901	0.417450	+	0.908700i	$0.362924\pi$
$830$	0	0
$831$	−8.56990	−0.297287
$832$	0	0
$833$	−19.8156	−0.686570
$834$	0	0
$835$	−21.4948	−0.743860
$836$	0	0
$837$	2.89572	0.100091
$838$	0	0
$839$	13.6190	0.470180	0.235090	−	0.971974i	$-0.424462\pi$
$839$	13.6190	0.470180	0.235090	+	0.971974i	$0.424462\pi$
$840$	0	0
$841$	−23.1820	−0.799380
$842$	0	0
$843$	−14.5845	−0.502317
$844$	0	0
$845$	−54.9751	−1.89120
$846$	0	0
$847$	29.2311	1.00439
$848$	0	0
$849$	20.9509	0.719034
$850$	0	0
$851$	60.7405	2.08216
$852$	0	0
$853$	−6.93449	−0.237432	−0.118716	−	0.992928i	$-0.537878\pi$
$853$	−6.93449	−0.237432	−0.118716	+	0.992928i	$0.537878\pi$
$854$	0	0
$855$	9.55691	0.326840
$856$	0	0
$857$	−34.9345	−1.19334	−0.596670	−	0.802487i	$-0.703510\pi$
$857$	−34.9345	−1.19334	−0.596670	+	0.802487i	$0.703510\pi$
$858$	0	0
$859$	−4.39057	−0.149804	−0.0749021	−	0.997191i	$-0.523864\pi$
$859$	−4.39057	−0.149804	−0.0749021	+	0.997191i	$0.523864\pi$
$860$	0	0
$861$	−3.30777	−0.112729
$862$	0	0
$863$	−15.4611	−0.526303	−0.263152	−	0.964755i	$-0.584762\pi$
$863$	−15.4611	−0.526303	−0.263152	+	0.964755i	$0.584762\pi$
$864$	0	0
$865$	80.1587	2.72548
$866$	0	0
$867$	8.27674	0.281093
$868$	0	0
$869$	0.824101	0.0279557
$870$	0	0
$871$	3.97078	0.134545
$872$	0	0
$873$	−7.19051	−0.243362
$874$	0	0
$875$	−113.217	−3.82744
$876$	0	0
$877$	−55.2338	−1.86511	−0.932556	−	0.361025i	$-0.882427\pi$
$877$	−55.2338	−1.86511	−0.932556	+	0.361025i	$0.882427\pi$
$878$	0	0
$879$	31.4086	1.05939
$880$	0	0
$881$	−28.4431	−0.958272	−0.479136	−	0.877741i	$-0.659050\pi$
$881$	−28.4431	−0.958272	−0.479136	+	0.877741i	$0.659050\pi$
$882$	0	0
$883$	24.1319	0.812102	0.406051	−	0.913850i	$-0.366905\pi$
$883$	24.1319	0.812102	0.406051	+	0.913850i	$0.366905\pi$
$884$	0	0
$885$	−11.1138	−0.373587
$886$	0	0
$887$	2.06025	0.0691765	0.0345882	−	0.999402i	$-0.488988\pi$
$887$	2.06025	0.0691765	0.0345882	+	0.999402i	$0.488988\pi$
$888$	0	0
$889$	−2.72594	−0.0914250
$890$	0	0
$891$	1.47068	0.0492697
$892$	0	0
$893$	9.19051	0.307549
$894$	0	0
$895$	12.1319	0.405524
$896$	0	0
$897$	−1.55691	−0.0519839
$898$	0	0
$899$	−6.98463	−0.232950
$900$	0	0
$901$	7.23797	0.241132
$902$	0	0
$903$	−35.8353	−1.19252
$904$	0	0
$905$	57.1070	1.89830
$906$	0	0
$907$	19.5208	0.648178	0.324089	−	0.946027i	$-0.394942\pi$
$907$	19.5208	0.648178	0.324089	+	0.946027i	$0.394942\pi$
$908$	0	0
$909$	−11.1449	−0.369652
$910$	0	0
$911$	39.9785	1.32455	0.662274	−	0.749262i	$-0.269592\pi$
$911$	39.9785	1.32455	0.662274	+	0.749262i	$0.269592\pi$
$912$	0	0
$913$	−5.59750	−0.185250
$914$	0	0
$915$	−24.3043	−0.803477
$916$	0	0
$917$	26.8984	0.888263
$918$	0	0
$919$	26.4508	0.872532	0.436266	−	0.899818i	$-0.356301\pi$
$919$	26.4508	0.872532	0.436266	+	0.899818i	$0.356301\pi$
$920$	0	0
$921$	−8.54392	−0.281532
$922$	0	0
$923$	−2.62328	−0.0863463
$924$	0	0
$925$	126.894	4.17226
$926$	0	0
$927$	−13.2767	−0.436065
$928$	0	0
$929$	33.2913	1.09225	0.546127	−	0.837703i	$-0.316102\pi$
$929$	33.2913	1.09225	0.546127	+	0.837703i	$0.316102\pi$
$930$	0	0
$931$	8.86469	0.290528
$932$	0	0
$933$	5.49484	0.179893
$934$	0	0
$935$	−31.4182	−1.02748
$936$	0	0
$937$	−38.7811	−1.26692	−0.633462	−	0.773774i	$-0.718367\pi$
$937$	−38.7811	−1.26692	−0.633462	+	0.773774i	$0.718367\pi$
$938$	0	0
$939$	5.48024	0.178841
$940$	0	0
$941$	15.6482	0.510117	0.255058	−	0.966926i	$-0.417905\pi$
$941$	15.6482	0.510117	0.255058	+	0.966926i	$0.417905\pi$
$942$	0	0
$943$	6.24914	0.203500
$944$	0	0
$945$	−14.0552	−0.457216
$946$	0	0
$947$	47.5500	1.54517	0.772584	−	0.634912i	$-0.218964\pi$
$947$	47.5500	1.54517	0.772584	+	0.634912i	$0.218964\pi$
$948$	0	0
$949$	−2.34043	−0.0759735
$950$	0	0
$951$	−11.3173	−0.366989
$952$	0	0
$953$	−29.6819	−0.961491	−0.480746	−	0.876860i	$-0.659634\pi$
$953$	−29.6819	−0.961491	−0.480746	+	0.876860i	$0.659634\pi$
$954$	0	0
$955$	−57.3415	−1.85553
$956$	0	0
$957$	−3.54736	−0.114670
$958$	0	0
$959$	40.3664	1.30350
$960$	0	0
$961$	−22.6148	−0.729509
$962$	0	0
$963$	−9.32238	−0.300410
$964$	0	0
$965$	−34.0552	−1.09628
$966$	0	0
$967$	1.21199	0.0389750	0.0194875	−	0.999810i	$-0.493797\pi$
$967$	1.21199	0.0389750	0.0194875	+	0.999810i	$0.493797\pi$
$968$	0	0
$969$	−11.3078	−0.363258
$970$	0	0
$971$	−12.5362	−0.402306	−0.201153	−	0.979560i	$-0.564469\pi$
$971$	−12.5362	−0.402306	−0.201153	+	0.979560i	$0.564469\pi$
$972$	0	0
$973$	47.2672	1.51532
$974$	0	0
$975$	−3.25258	−0.104166
$976$	0	0
$977$	15.2069	0.486513	0.243256	−	0.969962i	$-0.421784\pi$
$977$	15.2069	0.486513	0.243256	+	0.969962i	$0.421784\pi$
$978$	0	0
$979$	6.05520	0.193525
$980$	0	0
$981$	−0.249141	−0.00795445
$982$	0	0
$983$	−39.0303	−1.24487	−0.622436	−	0.782671i	$-0.713857\pi$
$983$	−39.0303	−1.24487	−0.622436	+	0.782671i	$0.713857\pi$
$984$	0	0
$985$	59.3967	1.89254
$986$	0	0
$987$	−13.5163	−0.430229
$988$	0	0
$989$	67.7010	2.15277
$990$	0	0
$991$	8.91539	0.283207	0.141603	−	0.989923i	$-0.454774\pi$
$991$	8.91539	0.283207	0.141603	+	0.989923i	$0.454774\pi$
$992$	0	0
$993$	−33.8613	−1.07455
$994$	0	0
$995$	20.6707	0.655307
$996$	0	0
$997$	−33.6336	−1.06519	−0.532593	−	0.846371i	$-0.678782\pi$
$997$	−33.6336	−1.06519	−0.532593	+	0.846371i	$0.678782\pi$
$998$	0	0
$999$	9.71982	0.307522

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1968.2.a.w.1.3		3
3.2	odd	2		5904.2.a.bd.1.1		3
4.3	odd	2		123.2.a.d.1.2	✓	3
8.3	odd	2		7872.2.a.bx.1.1		3
8.5	even	2		7872.2.a.bs.1.1		3
12.11	even	2		369.2.a.e.1.2		3
20.19	odd	2		3075.2.a.t.1.2		3
28.27	even	2		6027.2.a.s.1.2		3
60.59	even	2		9225.2.a.bx.1.2		3
164.163	odd	2		5043.2.a.n.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.a.d.1.2	✓	3	4.3	odd	2
369.2.a.e.1.2		3	12.11	even	2
1968.2.a.w.1.3		3	1.1	even	1	trivial
3075.2.a.t.1.2		3	20.19	odd	2
5043.2.a.n.1.2		3	164.163	odd	2
5904.2.a.bd.1.1		3	3.2	odd	2
6027.2.a.s.1.2		3	28.27	even	2
7872.2.a.bs.1.1		3	8.5	even	2
7872.2.a.bx.1.1		3	8.3	odd	2
9225.2.a.bx.1.2		3	60.59	even	2

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:	\( N \)	\(=\)	\( 1968 = 2^{4} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1968.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.24914 q^{5} -3.30777 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.24914 q^{5} -3.30777 q^{7} +1.00000 q^{9} +1.47068 q^{11} -0.249141 q^{13} +4.24914 q^{15} -5.02760 q^{17} +2.24914 q^{19} -3.30777 q^{21} +6.24914 q^{23} +13.0552 q^{25} +1.00000 q^{27} -2.41205 q^{29} +2.89572 q^{31} +1.47068 q^{33} -14.0552 q^{35} +9.71982 q^{37} -0.249141 q^{39} +1.00000 q^{41} +10.8337 q^{43} +4.24914 q^{45} +4.08623 q^{47} +3.94137 q^{49} -5.02760 q^{51} -1.43965 q^{53} +6.24914 q^{55} +2.24914 q^{57} -2.61555 q^{59} -5.71982 q^{61} -3.30777 q^{63} -1.05863 q^{65} -15.9379 q^{67} +6.24914 q^{69} +10.5293 q^{71} +9.39400 q^{73} +13.0552 q^{75} -4.86469 q^{77} +0.560352 q^{79} +1.00000 q^{81} -3.80605 q^{83} -21.3630 q^{85} -2.41205 q^{87} +4.11727 q^{89} +0.824101 q^{91} +2.89572 q^{93} +9.55691 q^{95} -7.19051 q^{97} +1.47068 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 4 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 4 * q^5 - 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 4 q^{5} - 2 q^{7} + 3 q^{9} + 4 q^{11} + 8 q^{13} + 4 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{21} + 10 q^{23} + 5 q^{25} + 3 q^{27} - 6 q^{29} + 2 q^{31} + 4 q^{33} - 8 q^{35} + 20 q^{37} + 8 q^{39} + 3 q^{41} - 10 q^{43} + 4 q^{45} - 4 q^{47} + 11 q^{49} + 2 q^{51} + 14 q^{53} + 10 q^{55} - 2 q^{57} + 8 q^{59} - 8 q^{61} - 2 q^{63} - 4 q^{65} - 12 q^{67} + 10 q^{69} + 32 q^{71} + 4 q^{73} + 5 q^{75} + 10 q^{77} + 20 q^{79} + 3 q^{81} + 14 q^{83} - 22 q^{85} - 6 q^{87} + 14 q^{89} + 2 q^{93} + 12 q^{95} - 12 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 4 * q^5 - 2 * q^7 + 3 * q^9 + 4 * q^11 + 8 * q^13 + 4 * q^15 + 2 * q^17 - 2 * q^19 - 2 * q^21 + 10 * q^23 + 5 * q^25 + 3 * q^27 - 6 * q^29 + 2 * q^31 + 4 * q^33 - 8 * q^35 + 20 * q^37 + 8 * q^39 + 3 * q^41 - 10 * q^43 + 4 * q^45 - 4 * q^47 + 11 * q^49 + 2 * q^51 + 14 * q^53 + 10 * q^55 - 2 * q^57 + 8 * q^59 - 8 * q^61 - 2 * q^63 - 4 * q^65 - 12 * q^67 + 10 * q^69 + 32 * q^71 + 4 * q^73 + 5 * q^75 + 10 * q^77 + 20 * q^79 + 3 * q^81 + 14 * q^83 - 22 * q^85 - 6 * q^87 + 14 * q^89 + 2 * q^93 + 12 * q^95 - 12 * q^97 + 4 * q^99

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