LMFDB - Embedded newform 1344.3.f.d.769.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,3,Mod(769,1344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1344.769"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1344.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,14,0,-6,0,36,0,0,0,24,0,0,0,0,0,0,0,-36,0,-46] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$36.6213475300$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1344.769
Dual form			1344.3.f.d.769.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -6.92820i q^{5} +7.00000 q^{7} -3.00000 q^{9} +18.0000 q^{11} -20.7846i q^{13} +12.0000 q^{15} -13.8564i q^{17} -20.7846i q^{19} +12.1244i q^{21} -18.0000 q^{23} -23.0000 q^{25} -5.19615i q^{27} -18.0000 q^{29} +41.5692i q^{31} +31.1769i q^{33} -48.4974i q^{35} -10.0000 q^{37} +36.0000 q^{39} +55.4256i q^{41} -38.0000 q^{43} +20.7846i q^{45} -27.7128i q^{47} +49.0000 q^{49} +24.0000 q^{51} -18.0000 q^{53} -124.708i q^{55} +36.0000 q^{57} -6.92820i q^{59} -20.7846i q^{61} -21.0000 q^{63} -144.000 q^{65} +26.0000 q^{67} -31.1769i q^{69} -18.0000 q^{71} -41.5692i q^{73} -39.8372i q^{75} +126.000 q^{77} -2.00000 q^{79} +9.00000 q^{81} +34.6410i q^{83} -96.0000 q^{85} -31.1769i q^{87} -145.492i q^{91} -72.0000 q^{93} -144.000 q^{95} -166.277i q^{97} -54.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{7} - 6 q^{9} + 36 q^{11} + 24 q^{15} - 36 q^{23} - 46 q^{25} - 36 q^{29} - 20 q^{37} + 72 q^{39} - 76 q^{43} + 98 q^{49} + 48 q^{51} - 36 q^{53} + 72 q^{57} - 42 q^{63} - 288 q^{65} + 52 q^{67}+ \cdots - 108 q^{99}+O(q^{100}) $ 2 * q + 14 * q^7 - 6 * q^9 + 36 * q^11 + 24 * q^15 - 36 * q^23 - 46 * q^25 - 36 * q^29 - 20 * q^37 + 72 * q^39 - 76 * q^43 + 98 * q^49 + 48 * q^51 - 36 * q^53 + 72 * q^57 - 42 * q^63 - 288 * q^65 + 52 * q^67 - 36 * q^71 + 252 * q^77 - 4 * q^79 + 18 * q^81 - 192 * q^85 - 144 * q^93 - 288 * q^95 - 108 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	− 6.92820i	− 1.38564i	−0.721110	−	0.692820i	$-0.756368\pi$
$5$	− 6.92820i	− 1.38564i	0.721110	−	0.692820i	$-0.243632\pi$
$6$	0	0
$7$	7.00000	1.00000
$7$	7.00000	1.00000
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	18.0000	1.63636	0.818182	−	0.574960i	$-0.194982\pi$
$11$	18.0000	1.63636	0.818182	+	0.574960i	$0.194982\pi$
$12$	0	0
$13$	− 20.7846i	− 1.59882i	−0.600788	−	0.799408i	$-0.705147\pi$
$13$	− 20.7846i	− 1.59882i	0.600788	−	0.799408i	$-0.294853\pi$
$14$	0	0
$15$	12.0000	0.800000
$16$	0	0
$17$	− 13.8564i	− 0.815083i	−0.913187	−	0.407541i	$-0.866386\pi$
$17$	− 13.8564i	− 0.815083i	0.913187	−	0.407541i	$-0.133614\pi$
$18$	0	0
$19$	− 20.7846i	− 1.09393i	−0.837157	−	0.546963i	$-0.815784\pi$
$19$	− 20.7846i	− 1.09393i	0.837157	−	0.546963i	$-0.184216\pi$
$20$	0	0
$21$	12.1244i	0.577350i
$22$	0	0
$23$	−18.0000	−0.782609	−0.391304	−	0.920261i	$-0.627976\pi$
$23$	−18.0000	−0.782609	−0.391304	+	0.920261i	$0.627976\pi$
$24$	0	0
$25$	−23.0000	−0.920000
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−18.0000	−0.620690	−0.310345	−	0.950624i	$-0.600445\pi$
$29$	−18.0000	−0.620690	−0.310345	+	0.950624i	$0.600445\pi$
$30$	0	0
$31$	41.5692i	1.34094i	0.741935	+	0.670471i	$0.233908\pi$
$31$	41.5692i	1.34094i	−0.741935	+	0.670471i	$0.766092\pi$
$32$	0	0
$33$	31.1769i	0.944755i
$34$	0	0
$35$	− 48.4974i	− 1.38564i
$36$	0	0
$37$	−10.0000	−0.270270	−0.135135	−	0.990827i	$-0.543147\pi$
$37$	−10.0000	−0.270270	−0.135135	+	0.990827i	$0.543147\pi$
$38$	0	0
$39$	36.0000	0.923077
$40$	0	0
$41$	55.4256i	1.35184i	0.736973	+	0.675922i	$0.236255\pi$
$41$	55.4256i	1.35184i	−0.736973	+	0.675922i	$0.763745\pi$
$42$	0	0
$43$	−38.0000	−0.883721	−0.441860	−	0.897084i	$-0.645681\pi$
$43$	−38.0000	−0.883721	−0.441860	+	0.897084i	$0.645681\pi$
$44$	0	0
$45$	20.7846i	0.461880i
$46$	0	0
$47$	− 27.7128i	− 0.589634i	−0.955554	−	0.294817i	$-0.904741\pi$
$47$	− 27.7128i	− 0.589634i	0.955554	−	0.294817i	$-0.0952587\pi$
$48$	0	0
$49$	49.0000	1.00000
$50$	0	0
$51$	24.0000	0.470588
$52$	0	0
$53$	−18.0000	−0.339623	−0.169811	−	0.985477i	$-0.554316\pi$
$53$	−18.0000	−0.339623	−0.169811	+	0.985477i	$0.554316\pi$
$54$	0	0
$55$	− 124.708i	− 2.26741i
$56$	0	0
$57$	36.0000	0.631579
$58$	0	0
$59$	− 6.92820i	− 0.117427i	−0.998275	−	0.0587136i	$-0.981300\pi$
$59$	− 6.92820i	− 0.117427i	0.998275	−	0.0587136i	$-0.0186999\pi$
$60$	0	0
$61$	− 20.7846i	− 0.340731i	−0.985381	−	0.170366i	$-0.945505\pi$
$61$	− 20.7846i	− 0.340731i	0.985381	−	0.170366i	$-0.0544949\pi$
$62$	0	0
$63$	−21.0000	−0.333333
$64$	0	0
$65$	−144.000	−2.21538
$66$	0	0
$67$	26.0000	0.388060	0.194030	−	0.980996i	$-0.437844\pi$
$67$	26.0000	0.388060	0.194030	+	0.980996i	$0.437844\pi$
$68$	0	0
$69$	− 31.1769i	− 0.451839i
$70$	0	0
$71$	−18.0000	−0.253521	−0.126761	−	0.991933i	$-0.540458\pi$
$71$	−18.0000	−0.253521	−0.126761	+	0.991933i	$0.540458\pi$
$72$	0	0
$73$	− 41.5692i	− 0.569441i	−0.958611	−	0.284721i	$-0.908099\pi$
$73$	− 41.5692i	− 0.569441i	0.958611	−	0.284721i	$-0.0919009\pi$
$74$	0	0
$75$	− 39.8372i	− 0.531162i
$76$	0	0
$77$	126.000	1.63636
$78$	0	0
$79$	−2.00000	−0.0253165	−0.0126582	−	0.999920i	$-0.504029\pi$
$79$	−2.00000	−0.0253165	−0.0126582	+	0.999920i	$0.504029\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	34.6410i	0.417362i	0.977984	+	0.208681i	$0.0669170\pi$
$83$	34.6410i	0.417362i	−0.977984	+	0.208681i	$0.933083\pi$
$84$	0	0
$85$	−96.0000	−1.12941
$86$	0	0
$87$	− 31.1769i	− 0.358355i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	− 145.492i	− 1.59882i
$92$	0	0
$93$	−72.0000	−0.774194
$94$	0	0
$95$	−144.000	−1.51579
$96$	0	0
$97$	− 166.277i	− 1.71419i	−0.515155	−	0.857097i	$-0.672265\pi$
$97$	− 166.277i	− 1.71419i	0.515155	−	0.857097i	$-0.327735\pi$
$98$	0	0
$99$	−54.0000	−0.545455
$100$	0	0
$101$	187.061i	1.85209i	0.377408	+	0.926047i	$0.376815\pi$
$101$	187.061i	1.85209i	−0.377408	+	0.926047i	$0.623185\pi$
$102$	0	0
$103$	− 83.1384i	− 0.807169i	−0.914942	−	0.403585i	$-0.867764\pi$
$103$	− 83.1384i	− 0.807169i	0.914942	−	0.403585i	$-0.132236\pi$
$104$	0	0
$105$	84.0000	0.800000
$106$	0	0
$107$	−126.000	−1.17757	−0.588785	−	0.808290i	$-0.700394\pi$
$107$	−126.000	−1.17757	−0.588785	+	0.808290i	$0.700394\pi$
$108$	0	0
$109$	134.000	1.22936	0.614679	−	0.788777i	$-0.289286\pi$
$109$	134.000	1.22936	0.614679	+	0.788777i	$0.289286\pi$
$110$	0	0
$111$	− 17.3205i	− 0.156041i
$112$	0	0
$113$	−126.000	−1.11504	−0.557522	−	0.830162i	$-0.688248\pi$
$113$	−126.000	−1.11504	−0.557522	+	0.830162i	$0.688248\pi$
$114$	0	0
$115$	124.708i	1.08441i
$116$	0	0
$117$	62.3538i	0.532939i
$118$	0	0
$119$	− 96.9948i	− 0.815083i
$120$	0	0
$121$	203.000	1.67769
$122$	0	0
$123$	−96.0000	−0.780488
$124$	0	0
$125$	− 13.8564i	− 0.110851i
$126$	0	0
$127$	46.0000	0.362205	0.181102	−	0.983464i	$-0.442033\pi$
$127$	46.0000	0.362205	0.181102	+	0.983464i	$0.442033\pi$
$128$	0	0
$129$	− 65.8179i	− 0.510217i
$130$	0	0
$131$	− 187.061i	− 1.42795i	−0.700171	−	0.713975i	$-0.746893\pi$
$131$	− 187.061i	− 1.42795i	0.700171	−	0.713975i	$-0.253107\pi$
$132$	0	0
$133$	− 145.492i	− 1.09393i
$134$	0	0
$135$	−36.0000	−0.266667
$136$	0	0
$137$	162.000	1.18248	0.591241	−	0.806495i	$-0.298638\pi$
$137$	162.000	1.18248	0.591241	+	0.806495i	$0.298638\pi$
$138$	0	0
$139$	− 62.3538i	− 0.448589i	−0.974521	−	0.224294i	$-0.927992\pi$
$139$	− 62.3538i	− 0.448589i	0.974521	−	0.224294i	$-0.0720077\pi$
$140$	0	0
$141$	48.0000	0.340426
$142$	0	0
$143$	− 374.123i	− 2.61624i
$144$	0	0
$145$	124.708i	0.860053i
$146$	0	0
$147$	84.8705i	0.577350i
$148$	0	0
$149$	−162.000	−1.08725	−0.543624	−	0.839329i	$-0.682948\pi$
$149$	−162.000	−1.08725	−0.543624	+	0.839329i	$0.682948\pi$
$150$	0	0
$151$	254.000	1.68212	0.841060	−	0.540942i	$-0.181932\pi$
$151$	254.000	1.68212	0.841060	+	0.540942i	$0.181932\pi$
$152$	0	0
$153$	41.5692i	0.271694i
$154$	0	0
$155$	288.000	1.85806
$156$	0	0
$157$	− 20.7846i	− 0.132386i	−0.997807	−	0.0661930i	$-0.978915\pi$
$157$	− 20.7846i	− 0.132386i	0.997807	−	0.0661930i	$-0.0210853\pi$
$158$	0	0
$159$	− 31.1769i	− 0.196081i
$160$	0	0
$161$	−126.000	−0.782609
$162$	0	0
$163$	26.0000	0.159509	0.0797546	−	0.996815i	$-0.474586\pi$
$163$	26.0000	0.159509	0.0797546	+	0.996815i	$0.474586\pi$
$164$	0	0
$165$	216.000	1.30909
$166$	0	0
$167$	180.133i	1.07864i	0.842100	+	0.539321i	$0.181319\pi$
$167$	180.133i	1.07864i	−0.842100	+	0.539321i	$0.818681\pi$
$168$	0	0
$169$	−263.000	−1.55621
$170$	0	0
$171$	62.3538i	0.364642i
$172$	0	0
$173$	− 187.061i	− 1.08128i	−0.841254	−	0.540640i	$-0.818182\pi$
$173$	− 187.061i	− 1.08128i	0.841254	−	0.540640i	$-0.181818\pi$
$174$	0	0
$175$	−161.000	−0.920000
$176$	0	0
$177$	12.0000	0.0677966
$178$	0	0
$179$	306.000	1.70950	0.854749	−	0.519042i	$-0.173711\pi$
$179$	306.000	1.70950	0.854749	+	0.519042i	$0.173711\pi$
$180$	0	0
$181$	187.061i	1.03349i	0.856140	+	0.516744i	$0.172856\pi$
$181$	187.061i	1.03349i	−0.856140	+	0.516744i	$0.827144\pi$
$182$	0	0
$183$	36.0000	0.196721
$184$	0	0
$185$	69.2820i	0.374497i
$186$	0	0
$187$	− 249.415i	− 1.33377i
$188$	0	0
$189$	− 36.3731i	− 0.192450i
$190$	0	0
$191$	270.000	1.41361	0.706806	−	0.707407i	$-0.250135\pi$
$191$	270.000	1.41361	0.706806	+	0.707407i	$0.250135\pi$
$192$	0	0
$193$	274.000	1.41969	0.709845	−	0.704358i	$-0.248765\pi$
$193$	274.000	1.41969	0.709845	+	0.704358i	$0.248765\pi$
$194$	0	0
$195$	− 249.415i	− 1.27905i
$196$	0	0
$197$	−162.000	−0.822335	−0.411168	−	0.911560i	$-0.634879\pi$
$197$	−162.000	−0.822335	−0.411168	+	0.911560i	$0.634879\pi$
$198$	0	0
$199$	− 83.1384i	− 0.417781i	−0.977939	−	0.208891i	$-0.933015\pi$
$199$	− 83.1384i	− 0.417781i	0.977939	−	0.208891i	$-0.0669852\pi$
$200$	0	0
$201$	45.0333i	0.224046i
$202$	0	0
$203$	−126.000	−0.620690
$204$	0	0
$205$	384.000	1.87317
$206$	0	0
$207$	54.0000	0.260870
$208$	0	0
$209$	− 374.123i	− 1.79006i
$210$	0	0
$211$	10.0000	0.0473934	0.0236967	−	0.999719i	$-0.492456\pi$
$211$	10.0000	0.0473934	0.0236967	+	0.999719i	$0.492456\pi$
$212$	0	0
$213$	− 31.1769i	− 0.146370i
$214$	0	0
$215$	263.272i	1.22452i
$216$	0	0
$217$	290.985i	1.34094i
$218$	0	0
$219$	72.0000	0.328767
$220$	0	0
$221$	−288.000	−1.30317
$222$	0	0
$223$	− 41.5692i	− 0.186409i	−0.995647	−	0.0932045i	$-0.970289\pi$
$223$	− 41.5692i	− 0.186409i	0.995647	−	0.0932045i	$-0.0297110\pi$
$224$	0	0
$225$	69.0000	0.306667
$226$	0	0
$227$	− 187.061i	− 0.824059i	−0.911170	−	0.412030i	$-0.864820\pi$
$227$	− 187.061i	− 0.824059i	0.911170	−	0.412030i	$-0.135180\pi$
$228$	0	0
$229$	270.200i	1.17991i	0.807435	+	0.589956i	$0.200855\pi$
$229$	270.200i	1.17991i	−0.807435	+	0.589956i	$0.799145\pi$
$230$	0	0
$231$	218.238i	0.944755i
$232$	0	0
$233$	−126.000	−0.540773	−0.270386	−	0.962752i	$-0.587151\pi$
$233$	−126.000	−0.540773	−0.270386	+	0.962752i	$0.587151\pi$
$234$	0	0
$235$	−192.000	−0.817021
$236$	0	0
$237$	− 3.46410i	− 0.0146165i
$238$	0	0
$239$	−18.0000	−0.0753138	−0.0376569	−	0.999291i	$-0.511989\pi$
$239$	−18.0000	−0.0753138	−0.0376569	+	0.999291i	$0.511989\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	− 339.482i	− 1.38564i
$246$	0	0
$247$	−432.000	−1.74899
$248$	0	0
$249$	−60.0000	−0.240964
$250$	0	0
$251$	48.4974i	0.193217i	0.995322	+	0.0966084i	$0.0307994\pi$
$251$	48.4974i	0.193217i	−0.995322	+	0.0966084i	$0.969201\pi$
$252$	0	0
$253$	−324.000	−1.28063
$254$	0	0
$255$	− 166.277i	− 0.652066i
$256$	0	0
$257$	374.123i	1.45573i	0.685720	+	0.727866i	$0.259488\pi$
$257$	374.123i	1.45573i	−0.685720	+	0.727866i	$0.740512\pi$
$258$	0	0
$259$	−70.0000	−0.270270
$260$	0	0
$261$	54.0000	0.206897
$262$	0	0
$263$	270.000	1.02662	0.513308	−	0.858204i	$-0.328420\pi$
$263$	270.000	1.02662	0.513308	+	0.858204i	$0.328420\pi$
$264$	0	0
$265$	124.708i	0.470595i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 76.2102i	− 0.283309i	−0.989916	−	0.141655i	$-0.954758\pi$
$269$	− 76.2102i	− 0.283309i	0.989916	−	0.141655i	$-0.0452423\pi$
$270$	0	0
$271$	207.846i	0.766960i	0.923549	+	0.383480i	$0.125274\pi$
$271$	207.846i	0.766960i	−0.923549	+	0.383480i	$0.874726\pi$
$272$	0	0
$273$	252.000	0.923077
$274$	0	0
$275$	−414.000	−1.50545
$276$	0	0
$277$	358.000	1.29242	0.646209	−	0.763160i	$-0.276353\pi$
$277$	358.000	1.29242	0.646209	+	0.763160i	$0.276353\pi$
$278$	0	0
$279$	− 124.708i	− 0.446981i
$280$	0	0
$281$	−414.000	−1.47331	−0.736655	−	0.676269i	$-0.763596\pi$
$281$	−414.000	−1.47331	−0.736655	+	0.676269i	$0.763596\pi$
$282$	0	0
$283$	270.200i	0.954770i	0.878694	+	0.477385i	$0.158415\pi$
$283$	270.200i	0.954770i	−0.878694	+	0.477385i	$0.841585\pi$
$284$	0	0
$285$	− 249.415i	− 0.875141i
$286$	0	0
$287$	387.979i	1.35184i
$288$	0	0
$289$	97.0000	0.335640
$290$	0	0
$291$	288.000	0.989691
$292$	0	0
$293$	− 117.779i	− 0.401978i	−0.979594	−	0.200989i	$-0.935585\pi$
$293$	− 117.779i	− 0.401978i	0.979594	−	0.200989i	$-0.0644155\pi$
$294$	0	0
$295$	−48.0000	−0.162712
$296$	0	0
$297$	− 93.5307i	− 0.314918i
$298$	0	0
$299$	374.123i	1.25125i
$300$	0	0
$301$	−266.000	−0.883721
$302$	0	0
$303$	−324.000	−1.06931
$304$	0	0
$305$	−144.000	−0.472131
$306$	0	0
$307$	62.3538i	0.203107i	0.994830	+	0.101553i	$0.0323813\pi$
$307$	62.3538i	0.203107i	−0.994830	+	0.101553i	$0.967619\pi$
$308$	0	0
$309$	144.000	0.466019
$310$	0	0
$311$	− 318.697i	− 1.02475i	−0.858762	−	0.512375i	$-0.828766\pi$
$311$	− 318.697i	− 1.02475i	0.858762	−	0.512375i	$-0.171234\pi$
$312$	0	0
$313$	374.123i	1.19528i	0.801764	+	0.597641i	$0.203895\pi$
$313$	374.123i	1.19528i	−0.801764	+	0.597641i	$0.796105\pi$
$314$	0	0
$315$	145.492i	0.461880i
$316$	0	0
$317$	−450.000	−1.41956	−0.709779	−	0.704424i	$-0.751205\pi$
$317$	−450.000	−1.41956	−0.709779	+	0.704424i	$0.751205\pi$
$318$	0	0
$319$	−324.000	−1.01567
$320$	0	0
$321$	− 218.238i	− 0.679870i
$322$	0	0
$323$	−288.000	−0.891641
$324$	0	0
$325$	478.046i	1.47091i
$326$	0	0
$327$	232.095i	0.709770i
$328$	0	0
$329$	− 193.990i	− 0.589634i
$330$	0	0
$331$	250.000	0.755287	0.377644	−	0.925951i	$-0.376734\pi$
$331$	250.000	0.755287	0.377644	+	0.925951i	$0.376734\pi$
$332$	0	0
$333$	30.0000	0.0900901
$334$	0	0
$335$	− 180.133i	− 0.537711i
$336$	0	0
$337$	−14.0000	−0.0415430	−0.0207715	−	0.999784i	$-0.506612\pi$
$337$	−14.0000	−0.0415430	−0.0207715	+	0.999784i	$0.506612\pi$
$338$	0	0
$339$	− 218.238i	− 0.643771i
$340$	0	0
$341$	748.246i	2.19427i
$342$	0	0
$343$	343.000	1.00000
$344$	0	0
$345$	−216.000	−0.626087
$346$	0	0
$347$	−414.000	−1.19308	−0.596542	−	0.802582i	$-0.703459\pi$
$347$	−414.000	−1.19308	−0.596542	+	0.802582i	$0.703459\pi$
$348$	0	0
$349$	− 436.477i	− 1.25065i	−0.780364	−	0.625325i	$-0.784966\pi$
$349$	− 436.477i	− 1.25065i	0.780364	−	0.625325i	$-0.215034\pi$
$350$	0	0
$351$	−108.000	−0.307692
$352$	0	0
$353$	− 595.825i	− 1.68789i	−0.536429	−	0.843945i	$-0.680227\pi$
$353$	− 595.825i	− 1.68789i	0.536429	−	0.843945i	$-0.319773\pi$
$354$	0	0
$355$	124.708i	0.351289i
$356$	0	0
$357$	168.000	0.470588
$358$	0	0
$359$	−18.0000	−0.0501393	−0.0250696	−	0.999686i	$-0.507981\pi$
$359$	−18.0000	−0.0501393	−0.0250696	+	0.999686i	$0.507981\pi$
$360$	0	0
$361$	−71.0000	−0.196676
$362$	0	0
$363$	351.606i	0.968612i
$364$	0	0
$365$	−288.000	−0.789041
$366$	0	0
$367$	457.261i	1.24594i	0.782244	+	0.622972i	$0.214075\pi$
$367$	457.261i	1.24594i	−0.782244	+	0.622972i	$0.785925\pi$
$368$	0	0
$369$	− 166.277i	− 0.450615i
$370$	0	0
$371$	−126.000	−0.339623
$372$	0	0
$373$	454.000	1.21716	0.608579	−	0.793493i	$-0.291740\pi$
$373$	454.000	1.21716	0.608579	+	0.793493i	$0.291740\pi$
$374$	0	0
$375$	24.0000	0.0640000
$376$	0	0
$377$	374.123i	0.992369i
$378$	0	0
$379$	−134.000	−0.353562	−0.176781	−	0.984250i	$-0.556568\pi$
$379$	−134.000	−0.353562	−0.176781	+	0.984250i	$0.556568\pi$
$380$	0	0
$381$	79.6743i	0.209119i
$382$	0	0
$383$	193.990i	0.506500i	0.967401	+	0.253250i	$0.0814996\pi$
$383$	193.990i	0.506500i	−0.967401	+	0.253250i	$0.918500\pi$
$384$	0	0
$385$	− 872.954i	− 2.26741i
$386$	0	0
$387$	114.000	0.294574
$388$	0	0
$389$	−594.000	−1.52699	−0.763496	−	0.645812i	$-0.776519\pi$
$389$	−594.000	−1.52699	−0.763496	+	0.645812i	$0.776519\pi$
$390$	0	0
$391$	249.415i	0.637891i
$392$	0	0
$393$	324.000	0.824427
$394$	0	0
$395$	13.8564i	0.0350795i
$396$	0	0
$397$	− 353.338i	− 0.890021i	−0.895525	−	0.445011i	$-0.853200\pi$
$397$	− 353.338i	− 0.890021i	0.895525	−	0.445011i	$-0.146800\pi$
$398$	0	0
$399$	252.000	0.631579
$400$	0	0
$401$	162.000	0.403990	0.201995	−	0.979387i	$-0.435258\pi$
$401$	162.000	0.403990	0.201995	+	0.979387i	$0.435258\pi$
$402$	0	0
$403$	864.000	2.14392
$404$	0	0
$405$	− 62.3538i	− 0.153960i
$406$	0	0
$407$	−180.000	−0.442260
$408$	0	0
$409$	− 124.708i	− 0.304909i	−0.988311	−	0.152454i	$-0.951282\pi$
$409$	− 124.708i	− 0.304909i	0.988311	−	0.152454i	$-0.0487177\pi$
$410$	0	0
$411$	280.592i	0.682706i
$412$	0	0
$413$	− 48.4974i	− 0.117427i
$414$	0	0
$415$	240.000	0.578313
$416$	0	0
$417$	108.000	0.258993
$418$	0	0
$419$	505.759i	1.20706i	0.797340	+	0.603531i	$0.206240\pi$
$419$	505.759i	1.20706i	−0.797340	+	0.603531i	$0.793760\pi$
$420$	0	0
$421$	454.000	1.07838	0.539192	−	0.842183i	$-0.318730\pi$
$421$	454.000	1.07838	0.539192	+	0.842183i	$0.318730\pi$
$422$	0	0
$423$	83.1384i	0.196545i
$424$	0	0
$425$	318.697i	0.749876i
$426$	0	0
$427$	− 145.492i	− 0.340731i
$428$	0	0
$429$	648.000	1.51049
$430$	0	0
$431$	270.000	0.626450	0.313225	−	0.949679i	$-0.398591\pi$
$431$	270.000	0.626450	0.313225	+	0.949679i	$0.398591\pi$
$432$	0	0
$433$	− 166.277i	− 0.384011i	−0.981394	−	0.192006i	$-0.938501\pi$
$433$	− 166.277i	− 0.384011i	0.981394	−	0.192006i	$-0.0614992\pi$
$434$	0	0
$435$	−216.000	−0.496552
$436$	0	0
$437$	374.123i	0.856117i
$438$	0	0
$439$	− 498.831i	− 1.13629i	−0.822929	−	0.568144i	$-0.807662\pi$
$439$	− 498.831i	− 1.13629i	0.822929	−	0.568144i	$-0.192338\pi$
$440$	0	0
$441$	−147.000	−0.333333
$442$	0	0
$443$	306.000	0.690745	0.345372	−	0.938466i	$-0.387753\pi$
$443$	306.000	0.690745	0.345372	+	0.938466i	$0.387753\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 280.592i	− 0.627723i
$448$	0	0
$449$	162.000	0.360802	0.180401	−	0.983593i	$-0.442260\pi$
$449$	162.000	0.360802	0.180401	+	0.983593i	$0.442260\pi$
$450$	0	0
$451$	997.661i	2.21211i
$452$	0	0
$453$	439.941i	0.971172i
$454$	0	0
$455$	−1008.00	−2.21538
$456$	0	0
$457$	−158.000	−0.345733	−0.172867	−	0.984945i	$-0.555303\pi$
$457$	−158.000	−0.345733	−0.172867	+	0.984945i	$0.555303\pi$
$458$	0	0
$459$	−72.0000	−0.156863
$460$	0	0
$461$	− 408.764i	− 0.886690i	−0.896351	−	0.443345i	$-0.853792\pi$
$461$	− 408.764i	− 0.886690i	0.896351	−	0.443345i	$-0.146208\pi$
$462$	0	0
$463$	382.000	0.825054	0.412527	−	0.910945i	$-0.364646\pi$
$463$	382.000	0.825054	0.412527	+	0.910945i	$0.364646\pi$
$464$	0	0
$465$	498.831i	1.07275i
$466$	0	0
$467$	561.184i	1.20168i	0.799369	+	0.600840i	$0.205167\pi$
$467$	561.184i	1.20168i	−0.799369	+	0.600840i	$0.794833\pi$
$468$	0	0
$469$	182.000	0.388060
$470$	0	0
$471$	36.0000	0.0764331
$472$	0	0
$473$	−684.000	−1.44609
$474$	0	0
$475$	478.046i	1.00641i
$476$	0	0
$477$	54.0000	0.113208
$478$	0	0
$479$	775.959i	1.61996i	0.586460	+	0.809978i	$0.300521\pi$
$479$	775.959i	1.61996i	−0.586460	+	0.809978i	$0.699479\pi$
$480$	0	0
$481$	207.846i	0.432112i
$482$	0	0
$483$	− 218.238i	− 0.451839i
$484$	0	0
$485$	−1152.00	−2.37526
$486$	0	0
$487$	542.000	1.11294	0.556468	−	0.830869i	$-0.312156\pi$
$487$	542.000	1.11294	0.556468	+	0.830869i	$0.312156\pi$
$488$	0	0
$489$	45.0333i	0.0920927i
$490$	0	0
$491$	594.000	1.20978	0.604888	−	0.796311i	$-0.293218\pi$
$491$	594.000	1.20978	0.604888	+	0.796311i	$0.293218\pi$
$492$	0	0
$493$	249.415i	0.505913i
$494$	0	0
$495$	374.123i	0.755804i
$496$	0	0
$497$	−126.000	−0.253521
$498$	0	0
$499$	−662.000	−1.32665	−0.663327	−	0.748330i	$-0.730856\pi$
$499$	−662.000	−1.32665	−0.663327	+	0.748330i	$0.730856\pi$
$500$	0	0
$501$	−312.000	−0.622754
$502$	0	0
$503$	− 13.8564i	− 0.0275475i	−0.999905	−	0.0137738i	$-0.995616\pi$
$503$	− 13.8564i	− 0.0275475i	0.999905	−	0.0137738i	$-0.00438446\pi$
$504$	0	0
$505$	1296.00	2.56634
$506$	0	0
$507$	− 455.529i	− 0.898480i
$508$	0	0
$509$	256.344i	0.503622i	0.967776	+	0.251811i	$0.0810261\pi$
$509$	256.344i	0.503622i	−0.967776	+	0.251811i	$0.918974\pi$
$510$	0	0
$511$	− 290.985i	− 0.569441i
$512$	0	0
$513$	−108.000	−0.210526
$514$	0	0
$515$	−576.000	−1.11845
$516$	0	0
$517$	− 498.831i	− 0.964856i
$518$	0	0
$519$	324.000	0.624277
$520$	0	0
$521$	886.810i	1.70213i	0.525060	+	0.851065i	$0.324043\pi$
$521$	886.810i	1.70213i	−0.525060	+	0.851065i	$0.675957\pi$
$522$	0	0
$523$	− 644.323i	− 1.23197i	−0.787756	−	0.615987i	$-0.788757\pi$
$523$	− 644.323i	− 1.23197i	0.787756	−	0.615987i	$-0.211243\pi$
$524$	0	0
$525$	− 278.860i	− 0.531162i
$526$	0	0
$527$	576.000	1.09298
$528$	0	0
$529$	−205.000	−0.387524
$530$	0	0
$531$	20.7846i	0.0391424i
$532$	0	0
$533$	1152.00	2.16135
$534$	0	0
$535$	872.954i	1.63169i
$536$	0	0
$537$	530.008i	0.986979i
$538$	0	0
$539$	882.000	1.63636
$540$	0	0
$541$	−458.000	−0.846580	−0.423290	−	0.905994i	$-0.639125\pi$
$541$	−458.000	−0.846580	−0.423290	+	0.905994i	$0.639125\pi$
$542$	0	0
$543$	−324.000	−0.596685
$544$	0	0
$545$	− 928.379i	− 1.70345i
$546$	0	0
$547$	506.000	0.925046	0.462523	−	0.886607i	$-0.346944\pi$
$547$	506.000	0.925046	0.462523	+	0.886607i	$0.346944\pi$
$548$	0	0
$549$	62.3538i	0.113577i
$550$	0	0
$551$	374.123i	0.678989i
$552$	0	0
$553$	−14.0000	−0.0253165
$554$	0	0
$555$	−120.000	−0.216216
$556$	0	0
$557$	558.000	1.00180	0.500898	−	0.865507i	$-0.333003\pi$
$557$	558.000	1.00180	0.500898	+	0.865507i	$0.333003\pi$
$558$	0	0
$559$	789.815i	1.41291i
$560$	0	0
$561$	432.000	0.770053
$562$	0	0
$563$	− 187.061i	− 0.332258i	−0.986104	−	0.166129i	$-0.946873\pi$
$563$	− 187.061i	− 0.332258i	0.986104	−	0.166129i	$-0.0531269\pi$
$564$	0	0
$565$	872.954i	1.54505i
$566$	0	0
$567$	63.0000	0.111111
$568$	0	0
$569$	738.000	1.29701	0.648506	−	0.761209i	$-0.275394\pi$
$569$	738.000	1.29701	0.648506	+	0.761209i	$0.275394\pi$
$570$	0	0
$571$	266.000	0.465849	0.232925	−	0.972495i	$-0.425170\pi$
$571$	266.000	0.465849	0.232925	+	0.972495i	$0.425170\pi$
$572$	0	0
$573$	467.654i	0.816150i
$574$	0	0
$575$	414.000	0.720000
$576$	0	0
$577$	83.1384i	0.144087i	0.997401	+	0.0720437i	$0.0229521\pi$
$577$	83.1384i	0.144087i	−0.997401	+	0.0720437i	$0.977048\pi$
$578$	0	0
$579$	474.582i	0.819658i
$580$	0	0
$581$	242.487i	0.417362i
$582$	0	0
$583$	−324.000	−0.555746
$584$	0	0
$585$	432.000	0.738462
$586$	0	0
$587$	− 561.184i	− 0.956021i	−0.878354	−	0.478011i	$-0.841358\pi$
$587$	− 561.184i	− 0.956021i	0.878354	−	0.478011i	$-0.158642\pi$
$588$	0	0
$589$	864.000	1.46689
$590$	0	0
$591$	− 280.592i	− 0.474775i
$592$	0	0
$593$	263.272i	0.443966i	0.975051	+	0.221983i	$0.0712529\pi$
$593$	263.272i	0.443966i	−0.975051	+	0.221983i	$0.928747\pi$
$594$	0	0
$595$	−672.000	−1.12941
$596$	0	0
$597$	144.000	0.241206
$598$	0	0
$599$	−18.0000	−0.0300501	−0.0150250	−	0.999887i	$-0.504783\pi$
$599$	−18.0000	−0.0300501	−0.0150250	+	0.999887i	$0.504783\pi$
$600$	0	0
$601$	− 41.5692i	− 0.0691668i	−0.999402	−	0.0345834i	$-0.988990\pi$
$601$	− 41.5692i	− 0.0691668i	0.999402	−	0.0345834i	$-0.0110104\pi$
$602$	0	0
$603$	−78.0000	−0.129353
$604$	0	0
$605$	− 1406.43i	− 2.32467i
$606$	0	0
$607$	− 374.123i	− 0.616348i	−0.951330	−	0.308174i	$-0.900282\pi$
$607$	− 374.123i	− 0.616348i	0.951330	−	0.308174i	$-0.0997178\pi$
$608$	0	0
$609$	− 218.238i	− 0.358355i
$610$	0	0
$611$	−576.000	−0.942717
$612$	0	0
$613$	−794.000	−1.29527	−0.647635	−	0.761951i	$-0.724242\pi$
$613$	−794.000	−1.29527	−0.647635	+	0.761951i	$0.724242\pi$
$614$	0	0
$615$	665.108i	1.08148i
$616$	0	0
$617$	−702.000	−1.13776	−0.568882	−	0.822419i	$-0.692624\pi$
$617$	−702.000	−1.13776	−0.568882	+	0.822419i	$0.692624\pi$
$618$	0	0
$619$	103.923i	0.167889i	0.996470	+	0.0839443i	$0.0267518\pi$
$619$	103.923i	0.167889i	−0.996470	+	0.0839443i	$0.973248\pi$
$620$	0	0
$621$	93.5307i	0.150613i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−671.000	−1.07360
$626$	0	0
$627$	648.000	1.03349
$628$	0	0
$629$	138.564i	0.220293i
$630$	0	0
$631$	−530.000	−0.839937	−0.419968	−	0.907539i	$-0.637959\pi$
$631$	−530.000	−0.839937	−0.419968	+	0.907539i	$0.637959\pi$
$632$	0	0
$633$	17.3205i	0.0273626i
$634$	0	0
$635$	− 318.697i	− 0.501886i
$636$	0	0
$637$	− 1018.45i	− 1.59882i
$638$	0	0
$639$	54.0000	0.0845070
$640$	0	0
$641$	−990.000	−1.54446	−0.772231	−	0.635342i	$-0.780859\pi$
$641$	−990.000	−1.54446	−0.772231	+	0.635342i	$0.780859\pi$
$642$	0	0
$643$	− 436.477i	− 0.678813i	−0.940640	−	0.339407i	$-0.889774\pi$
$643$	− 436.477i	− 0.678813i	0.940640	−	0.339407i	$-0.110226\pi$
$644$	0	0
$645$	−456.000	−0.706977
$646$	0	0
$647$	374.123i	0.578243i	0.957292	+	0.289121i	$0.0933631\pi$
$647$	374.123i	0.578243i	−0.957292	+	0.289121i	$0.906637\pi$
$648$	0	0
$649$	− 124.708i	− 0.192154i
$650$	0	0
$651$	−504.000	−0.774194
$652$	0	0
$653$	−18.0000	−0.0275651	−0.0137825	−	0.999905i	$-0.504387\pi$
$653$	−18.0000	−0.0275651	−0.0137825	+	0.999905i	$0.504387\pi$
$654$	0	0
$655$	−1296.00	−1.97863
$656$	0	0
$657$	124.708i	0.189814i
$658$	0	0
$659$	882.000	1.33839	0.669196	−	0.743086i	$-0.266639\pi$
$659$	882.000	1.33839	0.669196	+	0.743086i	$0.266639\pi$
$660$	0	0
$661$	− 1226.29i	− 1.85521i	−0.373566	−	0.927604i	$-0.621865\pi$
$661$	− 1226.29i	− 1.85521i	0.373566	−	0.927604i	$-0.378135\pi$
$662$	0	0
$663$	− 498.831i	− 0.752384i
$664$	0	0
$665$	−1008.00	−1.51579
$666$	0	0
$667$	324.000	0.485757
$668$	0	0
$669$	72.0000	0.107623
$670$	0	0
$671$	− 374.123i	− 0.557560i
$672$	0	0
$673$	706.000	1.04903	0.524517	−	0.851400i	$-0.324246\pi$
$673$	706.000	1.04903	0.524517	+	0.851400i	$0.324246\pi$
$674$	0	0
$675$	119.512i	0.177054i
$676$	0	0
$677$	187.061i	0.276309i	0.990411	+	0.138155i	$0.0441171\pi$
$677$	187.061i	0.276309i	−0.990411	+	0.138155i	$0.955883\pi$
$678$	0	0
$679$	− 1163.94i	− 1.71419i
$680$	0	0
$681$	324.000	0.475771
$682$	0	0
$683$	−558.000	−0.816984	−0.408492	−	0.912762i	$-0.633945\pi$
$683$	−558.000	−0.816984	−0.408492	+	0.912762i	$0.633945\pi$
$684$	0	0
$685$	− 1122.37i	− 1.63849i
$686$	0	0
$687$	−468.000	−0.681223
$688$	0	0
$689$	374.123i	0.542994i
$690$	0	0
$691$	− 602.754i	− 0.872292i	−0.899876	−	0.436146i	$-0.856343\pi$
$691$	− 602.754i	− 0.872292i	0.899876	−	0.436146i	$-0.143657\pi$
$692$	0	0
$693$	−378.000	−0.545455
$694$	0	0
$695$	−432.000	−0.621583
$696$	0	0
$697$	768.000	1.10187
$698$	0	0
$699$	− 218.238i	− 0.312215i
$700$	0	0
$701$	414.000	0.590585	0.295292	−	0.955407i	$-0.404583\pi$
$701$	414.000	0.590585	0.295292	+	0.955407i	$0.404583\pi$
$702$	0	0
$703$	207.846i	0.295656i
$704$	0	0
$705$	− 332.554i	− 0.471707i
$706$	0	0
$707$	1309.43i	1.85209i
$708$	0	0
$709$	182.000	0.256700	0.128350	−	0.991729i	$-0.459032\pi$
$709$	182.000	0.256700	0.128350	+	0.991729i	$0.459032\pi$
$710$	0	0
$711$	6.00000	0.00843882
$712$	0	0
$713$	− 748.246i	− 1.04943i
$714$	0	0
$715$	−2592.00	−3.62517
$716$	0	0
$717$	− 31.1769i	− 0.0434824i
$718$	0	0
$719$	− 1136.23i	− 1.58029i	−0.612923	−	0.790143i	$-0.710006\pi$
$719$	− 1136.23i	− 1.58029i	0.612923	−	0.790143i	$-0.289994\pi$
$720$	0	0
$721$	− 581.969i	− 0.807169i
$722$	0	0
$723$	0	0
$724$	0	0
$725$	414.000	0.571034
$726$	0	0
$727$	1247.08i	1.71537i	0.514173	+	0.857687i	$0.328099\pi$
$727$	1247.08i	1.71537i	−0.514173	+	0.857687i	$0.671901\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	526.543i	0.720306i
$732$	0	0
$733$	1392.57i	1.89982i	0.312522	+	0.949911i	$0.398826\pi$
$733$	1392.57i	1.89982i	−0.312522	+	0.949911i	$0.601174\pi$
$734$	0	0
$735$	588.000	0.800000
$736$	0	0
$737$	468.000	0.635007
$738$	0	0
$739$	−1222.00	−1.65359	−0.826793	−	0.562506i	$-0.809837\pi$
$739$	−1222.00	−1.65359	−0.826793	+	0.562506i	$0.809837\pi$
$740$	0	0
$741$	− 748.246i	− 1.00978i
$742$	0	0
$743$	270.000	0.363392	0.181696	−	0.983355i	$-0.441841\pi$
$743$	270.000	0.363392	0.181696	+	0.983355i	$0.441841\pi$
$744$	0	0
$745$	1122.37i	1.50654i
$746$	0	0
$747$	− 103.923i	− 0.139121i
$748$	0	0
$749$	−882.000	−1.17757
$750$	0	0
$751$	206.000	0.274301	0.137150	−	0.990550i	$-0.456206\pi$
$751$	206.000	0.274301	0.137150	+	0.990550i	$0.456206\pi$
$752$	0	0
$753$	−84.0000	−0.111554
$754$	0	0
$755$	− 1759.76i	− 2.33081i
$756$	0	0
$757$	−1162.00	−1.53501	−0.767503	−	0.641045i	$-0.778501\pi$
$757$	−1162.00	−1.53501	−0.767503	+	0.641045i	$0.778501\pi$
$758$	0	0
$759$	− 561.184i	− 0.739373i
$760$	0	0
$761$	554.256i	0.728326i	0.931335	+	0.364163i	$0.118645\pi$
$761$	554.256i	0.728326i	−0.931335	+	0.364163i	$0.881355\pi$
$762$	0	0
$763$	938.000	1.22936
$764$	0	0
$765$	288.000	0.376471
$766$	0	0
$767$	−144.000	−0.187744
$768$	0	0
$769$	− 1330.22i	− 1.72980i	−0.501946	−	0.864899i	$-0.667382\pi$
$769$	− 1330.22i	− 1.72980i	0.501946	−	0.864899i	$-0.332618\pi$
$770$	0	0
$771$	−648.000	−0.840467
$772$	0	0
$773$	1323.29i	1.71188i	0.517071	+	0.855942i	$0.327022\pi$
$773$	1323.29i	1.71188i	−0.517071	+	0.855942i	$0.672978\pi$
$774$	0	0
$775$	− 956.092i	− 1.23367i
$776$	0	0
$777$	− 121.244i	− 0.156041i
$778$	0	0
$779$	1152.00	1.47882
$780$	0	0
$781$	−324.000	−0.414853
$782$	0	0
$783$	93.5307i	0.119452i
$784$	0	0
$785$	−144.000	−0.183439
$786$	0	0
$787$	− 769.031i	− 0.977167i	−0.872517	−	0.488584i	$-0.837514\pi$
$787$	− 769.031i	− 0.977167i	0.872517	−	0.488584i	$-0.162486\pi$
$788$	0	0
$789$	467.654i	0.592717i
$790$	0	0
$791$	−882.000	−1.11504
$792$	0	0
$793$	−432.000	−0.544767
$794$	0	0
$795$	−216.000	−0.271698
$796$	0	0
$797$	− 187.061i	− 0.234707i	−0.993090	−	0.117354i	$-0.962559\pi$
$797$	− 187.061i	− 0.234707i	0.993090	−	0.117354i	$-0.0374411\pi$
$798$	0	0
$799$	−384.000	−0.480601
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 748.246i	− 0.931813i
$804$	0	0
$805$	872.954i	1.08441i
$806$	0	0
$807$	132.000	0.163569
$808$	0	0
$809$	450.000	0.556242	0.278121	−	0.960546i	$-0.410288\pi$
$809$	450.000	0.556242	0.278121	+	0.960546i	$0.410288\pi$
$810$	0	0
$811$	852.169i	1.05076i	0.850867	+	0.525382i	$0.176077\pi$
$811$	852.169i	1.05076i	−0.850867	+	0.525382i	$0.823923\pi$
$812$	0	0
$813$	−360.000	−0.442804
$814$	0	0
$815$	− 180.133i	− 0.221022i
$816$	0	0
$817$	789.815i	0.966726i
$818$	0	0
$819$	436.477i	0.532939i
$820$	0	0
$821$	846.000	1.03045	0.515225	−	0.857055i	$-0.327708\pi$
$821$	846.000	1.03045	0.515225	+	0.857055i	$0.327708\pi$
$822$	0	0
$823$	1390.00	1.68894	0.844471	−	0.535601i	$-0.179915\pi$
$823$	1390.00	1.68894	0.844471	+	0.535601i	$0.179915\pi$
$824$	0	0
$825$	− 717.069i	− 0.869175i
$826$	0	0
$827$	1314.00	1.58888	0.794438	−	0.607346i	$-0.207766\pi$
$827$	1314.00	1.58888	0.794438	+	0.607346i	$0.207766\pi$
$828$	0	0
$829$	478.046i	0.576654i	0.957532	+	0.288327i	$0.0930990\pi$
$829$	478.046i	0.576654i	−0.957532	+	0.288327i	$0.906901\pi$
$830$	0	0
$831$	620.074i	0.746178i
$832$	0	0
$833$	− 678.964i	− 0.815083i
$834$	0	0
$835$	1248.00	1.49461
$836$	0	0
$837$	216.000	0.258065
$838$	0	0
$839$	− 900.666i	− 1.07350i	−0.843741	−	0.536750i	$-0.819652\pi$
$839$	− 900.666i	− 1.07350i	0.843741	−	0.536750i	$-0.180348\pi$
$840$	0	0
$841$	−517.000	−0.614744
$842$	0	0
$843$	− 717.069i	− 0.850616i
$844$	0	0
$845$	1822.12i	2.15635i
$846$	0	0
$847$	1421.00	1.67769
$848$	0	0
$849$	−468.000	−0.551237
$850$	0	0
$851$	180.000	0.211516
$852$	0	0
$853$	1517.28i	1.77875i	0.457175	+	0.889377i	$0.348861\pi$
$853$	1517.28i	1.77875i	−0.457175	+	0.889377i	$0.651139\pi$
$854$	0	0
$855$	432.000	0.505263
$856$	0	0
$857$	775.959i	0.905436i	0.891654	+	0.452718i	$0.149546\pi$
$857$	775.959i	0.905436i	−0.891654	+	0.452718i	$0.850454\pi$
$858$	0	0
$859$	1184.72i	1.37919i	0.724196	+	0.689594i	$0.242211\pi$
$859$	1184.72i	1.37919i	−0.724196	+	0.689594i	$0.757789\pi$
$860$	0	0
$861$	−672.000	−0.780488
$862$	0	0
$863$	558.000	0.646582	0.323291	−	0.946300i	$-0.395211\pi$
$863$	558.000	0.646582	0.323291	+	0.946300i	$0.395211\pi$
$864$	0	0
$865$	−1296.00	−1.49827
$866$	0	0
$867$	168.009i	0.193782i
$868$	0	0
$869$	−36.0000	−0.0414269
$870$	0	0
$871$	− 540.400i	− 0.620436i
$872$	0	0
$873$	498.831i	0.571398i
$874$	0	0
$875$	− 96.9948i	− 0.110851i
$876$	0	0
$877$	1174.00	1.33865	0.669327	−	0.742968i	$-0.266582\pi$
$877$	1174.00	1.33865	0.669327	+	0.742968i	$0.266582\pi$
$878$	0	0
$879$	204.000	0.232082
$880$	0	0
$881$	1122.37i	1.27397i	0.770876	+	0.636986i	$0.219819\pi$
$881$	1122.37i	1.27397i	−0.770876	+	0.636986i	$0.780181\pi$
$882$	0	0
$883$	1450.00	1.64213	0.821065	−	0.570835i	$-0.193381\pi$
$883$	1450.00	1.64213	0.821065	+	0.570835i	$0.193381\pi$
$884$	0	0
$885$	− 83.1384i	− 0.0939417i
$886$	0	0
$887$	− 69.2820i	− 0.0781083i	−0.999237	−	0.0390541i	$-0.987566\pi$
$887$	− 69.2820i	− 0.0781083i	0.999237	−	0.0390541i	$-0.0124345\pi$
$888$	0	0
$889$	322.000	0.362205
$890$	0	0
$891$	162.000	0.181818
$892$	0	0
$893$	−576.000	−0.645017
$894$	0	0
$895$	− 2120.03i	− 2.36875i
$896$	0	0
$897$	−648.000	−0.722408
$898$	0	0
$899$	− 748.246i	− 0.832309i
$900$	0	0
$901$	249.415i	0.276821i
$902$	0	0
$903$	− 460.726i	− 0.510217i
$904$	0	0
$905$	1296.00	1.43204
$906$	0	0
$907$	826.000	0.910695	0.455347	−	0.890314i	$-0.349515\pi$
$907$	826.000	0.910695	0.455347	+	0.890314i	$0.349515\pi$
$908$	0	0
$909$	− 561.184i	− 0.617365i
$910$	0	0
$911$	−594.000	−0.652031	−0.326015	−	0.945364i	$-0.605706\pi$
$911$	−594.000	−0.652031	−0.326015	+	0.945364i	$0.605706\pi$
$912$	0	0
$913$	623.538i	0.682955i
$914$	0	0
$915$	− 249.415i	− 0.272585i
$916$	0	0
$917$	− 1309.43i	− 1.42795i
$918$	0	0
$919$	−82.0000	−0.0892274	−0.0446137	−	0.999004i	$-0.514206\pi$
$919$	−82.0000	−0.0892274	−0.0446137	+	0.999004i	$0.514206\pi$
$920$	0	0
$921$	−108.000	−0.117264
$922$	0	0
$923$	374.123i	0.405334i
$924$	0	0
$925$	230.000	0.248649
$926$	0	0
$927$	249.415i	0.269056i
$928$	0	0
$929$	− 845.241i	− 0.909839i	−0.890532	−	0.454920i	$-0.849668\pi$
$929$	− 845.241i	− 0.909839i	0.890532	−	0.454920i	$-0.150332\pi$
$930$	0	0
$931$	− 1018.45i	− 1.09393i
$932$	0	0
$933$	552.000	0.591640
$934$	0	0
$935$	−1728.00	−1.84813
$936$	0	0
$937$	− 872.954i	− 0.931647i	−0.884878	−	0.465824i	$-0.845758\pi$
$937$	− 872.954i	− 0.931647i	0.884878	−	0.465824i	$-0.154242\pi$
$938$	0	0
$939$	−648.000	−0.690096
$940$	0	0
$941$	− 491.902i	− 0.522744i	−0.965238	−	0.261372i	$-0.915825\pi$
$941$	− 491.902i	− 0.522744i	0.965238	−	0.261372i	$-0.0841750\pi$
$942$	0	0
$943$	− 997.661i	− 1.05797i
$944$	0	0
$945$	−252.000	−0.266667
$946$	0	0
$947$	−1710.00	−1.80570	−0.902851	−	0.429953i	$-0.858530\pi$
$947$	−1710.00	−1.80570	−0.902851	+	0.429953i	$0.858530\pi$
$948$	0	0
$949$	−864.000	−0.910432
$950$	0	0
$951$	− 779.423i	− 0.819582i
$952$	0	0
$953$	−126.000	−0.132214	−0.0661070	−	0.997813i	$-0.521058\pi$
$953$	−126.000	−0.132214	−0.0661070	+	0.997813i	$0.521058\pi$
$954$	0	0
$955$	− 1870.61i	− 1.95876i
$956$	0	0
$957$	− 561.184i	− 0.586400i
$958$	0	0
$959$	1134.00	1.18248
$960$	0	0
$961$	−767.000	−0.798127
$962$	0	0
$963$	378.000	0.392523
$964$	0	0
$965$	− 1898.33i	− 1.96718i
$966$	0	0
$967$	766.000	0.792141	0.396070	−	0.918220i	$-0.370374\pi$
$967$	766.000	0.792141	0.396070	+	0.918220i	$0.370374\pi$
$968$	0	0
$969$	− 498.831i	− 0.514789i
$970$	0	0
$971$	1572.70i	1.61967i	0.586656	+	0.809836i	$0.300444\pi$
$971$	1572.70i	1.61967i	−0.586656	+	0.809836i	$0.699556\pi$
$972$	0	0
$973$	− 436.477i	− 0.448589i
$974$	0	0
$975$	−828.000	−0.849231
$976$	0	0
$977$	1602.00	1.63971	0.819857	−	0.572569i	$-0.194053\pi$
$977$	1602.00	1.63971	0.819857	+	0.572569i	$0.194053\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−402.000	−0.409786
$982$	0	0
$983$	− 845.241i	− 0.859858i	−0.902863	−	0.429929i	$-0.858539\pi$
$983$	− 845.241i	− 0.859858i	0.902863	−	0.429929i	$-0.141461\pi$
$984$	0	0
$985$	1122.37i	1.13946i
$986$	0	0
$987$	336.000	0.340426
$988$	0	0
$989$	684.000	0.691608
$990$	0	0
$991$	62.0000	0.0625631	0.0312815	−	0.999511i	$-0.490041\pi$
$991$	62.0000	0.0625631	0.0312815	+	0.999511i	$0.490041\pi$
$992$	0	0
$993$	433.013i	0.436065i
$994$	0	0
$995$	−576.000	−0.578894
$996$	0	0
$997$	− 62.3538i	− 0.0625415i	−0.999511	−	0.0312707i	$-0.990045\pi$
$997$	− 62.3538i	− 0.0625415i	0.999511	−	0.0312707i	$-0.00995540\pi$
$998$	0	0
$999$	51.9615i	0.0520135i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.3.f.d.769.2		2
4.3	odd	2		1344.3.f.a.769.1		2
7.6	odd	2	inner	1344.3.f.d.769.1		2
8.3	odd	2		84.3.d.a.13.2	yes	2
8.5	even	2		336.3.f.b.97.1		2
24.5	odd	2		1008.3.f.f.433.1		2
24.11	even	2		252.3.d.a.181.1		2
28.27	even	2		1344.3.f.a.769.2		2
40.3	even	4		2100.3.p.b.349.2		4
40.19	odd	2		2100.3.j.c.601.1		2
40.27	even	4		2100.3.p.b.349.3		4
56.3	even	6		588.3.m.b.313.1		2
56.11	odd	6		588.3.m.c.313.1		2
56.13	odd	2		336.3.f.b.97.2		2
56.19	even	6		588.3.m.c.325.1		2
56.27	even	2		84.3.d.a.13.1	✓	2
56.51	odd	6		588.3.m.b.325.1		2
168.11	even	6		1764.3.z.g.901.1		2
168.59	odd	6		1764.3.z.a.901.1		2
168.83	odd	2		252.3.d.a.181.2		2
168.107	even	6		1764.3.z.a.325.1		2
168.125	even	2		1008.3.f.f.433.2		2
168.131	odd	6		1764.3.z.g.325.1		2
280.27	odd	4		2100.3.p.b.349.1		4
280.83	odd	4		2100.3.p.b.349.4		4
280.139	even	2		2100.3.j.c.601.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.d.a.13.1	✓	2	56.27	even	2
84.3.d.a.13.2	yes	2	8.3	odd	2
252.3.d.a.181.1		2	24.11	even	2
252.3.d.a.181.2		2	168.83	odd	2
336.3.f.b.97.1		2	8.5	even	2
336.3.f.b.97.2		2	56.13	odd	2
588.3.m.b.313.1		2	56.3	even	6
588.3.m.b.325.1		2	56.51	odd	6
588.3.m.c.313.1		2	56.11	odd	6
588.3.m.c.325.1		2	56.19	even	6
1008.3.f.f.433.1		2	24.5	odd	2
1008.3.f.f.433.2		2	168.125	even	2
1344.3.f.a.769.1		2	4.3	odd	2
1344.3.f.a.769.2		2	28.27	even	2
1344.3.f.d.769.1		2	7.6	odd	2	inner
1344.3.f.d.769.2		2	1.1	even	1	trivial
1764.3.z.a.325.1		2	168.107	even	6
1764.3.z.a.901.1		2	168.59	odd	6
1764.3.z.g.325.1		2	168.131	odd	6
1764.3.z.g.901.1		2	168.11	even	6
2100.3.j.c.601.1		2	40.19	odd	2
2100.3.j.c.601.2		2	280.139	even	2
2100.3.p.b.349.1		4	280.27	odd	4
2100.3.p.b.349.2		4	40.3	even	4
2100.3.p.b.349.3		4	40.27	even	4
2100.3.p.b.349.4		4	280.83	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1344.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -6.92820i q^{5} +7.00000 q^{7} -3.00000 q^{9} +18.0000 q^{11} -20.7846i q^{13} +12.0000 q^{15} -13.8564i q^{17} -20.7846i q^{19} +12.1244i q^{21} -18.0000 q^{23} -23.0000 q^{25} -5.19615i q^{27} -18.0000 q^{29} +41.5692i q^{31} +31.1769i q^{33} -48.4974i q^{35} -10.0000 q^{37} +36.0000 q^{39} +55.4256i q^{41} -38.0000 q^{43} +20.7846i q^{45} -27.7128i q^{47} +49.0000 q^{49} +24.0000 q^{51} -18.0000 q^{53} -124.708i q^{55} +36.0000 q^{57} -6.92820i q^{59} -20.7846i q^{61} -21.0000 q^{63} -144.000 q^{65} +26.0000 q^{67} -31.1769i q^{69} -18.0000 q^{71} -41.5692i q^{73} -39.8372i q^{75} +126.000 q^{77} -2.00000 q^{79} +9.00000 q^{81} +34.6410i q^{83} -96.0000 q^{85} -31.1769i q^{87} -145.492i q^{91} -72.0000 q^{93} -144.000 q^{95} -166.277i q^{97} -54.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{7} - 6 q^{9} + 36 q^{11} + 24 q^{15} - 36 q^{23} - 46 q^{25} - 36 q^{29} - 20 q^{37} + 72 q^{39} - 76 q^{43} + 98 q^{49} + 48 q^{51} - 36 q^{53} + 72 q^{57} - 42 q^{63} - 288 q^{65} + 52 q^{67}+ \cdots - 108 q^{99}+O(q^{100}) \) 2 * q + 14 * q^7 - 6 * q^9 + 36 * q^11 + 24 * q^15 - 36 * q^23 - 46 * q^25 - 36 * q^29 - 20 * q^37 + 72 * q^39 - 76 * q^43 + 98 * q^49 + 48 * q^51 - 36 * q^53 + 72 * q^57 - 42 * q^63 - 288 * q^65 + 52 * q^67 - 36 * q^71 + 252 * q^77 - 4 * q^79 + 18 * q^81 - 192 * q^85 - 144 * q^93 - 288 * q^95 - 108 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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