LMFDB - Embedded newform 2940.2.a.o.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2940,2,Mod(1,2940)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2940.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2940.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,2,0,0,0,2,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.4760181943$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2940.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} -2.00000 q^{11} -2.82843 q^{13} -1.00000 q^{15} +6.82843 q^{17} -2.58579 q^{19} -4.58579 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.65685 q^{29} +4.24264 q^{31} +2.00000 q^{33} +6.48528 q^{37} +2.82843 q^{39} +2.00000 q^{41} -9.65685 q^{43} +1.00000 q^{45} -6.48528 q^{47} -6.82843 q^{51} -7.41421 q^{53} -2.00000 q^{55} +2.58579 q^{57} +2.82843 q^{59} -9.89949 q^{61} -2.82843 q^{65} -1.17157 q^{67} +4.58579 q^{69} +6.48528 q^{71} +12.4853 q^{73} -1.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} -0.828427 q^{83} +6.82843 q^{85} +7.65685 q^{87} -0.828427 q^{89} -4.24264 q^{93} -2.58579 q^{95} -4.00000 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} - 4 q^{11} - 2 q^{15} + 8 q^{17} - 8 q^{19} - 12 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 4 q^{33} - 4 q^{37} + 4 q^{41} - 8 q^{43} + 2 q^{45} + 4 q^{47} - 8 q^{51} - 12 q^{53}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9 - 4 * q^11 - 2 * q^15 + 8 * q^17 - 8 * q^19 - 12 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 + 4 * q^33 - 4 * q^37 + 4 * q^41 - 8 * q^43 + 2 * q^45 + 4 * q^47 - 8 * q^51 - 12 * q^53 - 4 * q^55 + 8 * q^57 - 8 * q^67 + 12 * q^69 - 4 * q^71 + 8 * q^73 - 2 * q^75 - 20 * q^79 + 2 * q^81 + 4 * q^83 + 8 * q^85 + 4 * q^87 + 4 * q^89 - 8 * q^95 - 8 * q^97 - 4 * q^99

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	6.82843	1.65614	0.828068	−	0.560627i	$-0.189440\pi$
$17$	6.82843	1.65614	0.828068	+	0.560627i	$0.189440\pi$
$18$	0	0
$19$	−2.58579	−0.593220	−0.296610	−	0.954999i	$-0.595856\pi$
$19$	−2.58579	−0.593220	−0.296610	+	0.954999i	$0.595856\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.58579	−0.956203	−0.478101	−	0.878305i	$-0.658675\pi$
$23$	−4.58579	−0.956203	−0.478101	+	0.878305i	$0.658675\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	0	0
$31$	4.24264	0.762001	0.381000	−	0.924575i	$-0.375580\pi$
$31$	4.24264	0.762001	0.381000	+	0.924575i	$0.375580\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.48528	1.06617	0.533087	−	0.846061i	$-0.321032\pi$
$37$	6.48528	1.06617	0.533087	+	0.846061i	$0.321032\pi$
$38$	0	0
$39$	2.82843	0.452911
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−6.48528	−0.945976	−0.472988	−	0.881069i	$-0.656825\pi$
$47$	−6.48528	−0.945976	−0.472988	+	0.881069i	$0.656825\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.82843	−0.956171
$52$	0	0
$53$	−7.41421	−1.01842	−0.509210	−	0.860642i	$-0.670062\pi$
$53$	−7.41421	−1.01842	−0.509210	+	0.860642i	$0.670062\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	2.58579	0.342496
$58$	0	0
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	−9.89949	−1.26750	−0.633750	−	0.773538i	$-0.718485\pi$
$61$	−9.89949	−1.26750	−0.633750	+	0.773538i	$0.718485\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.82843	−0.350823
$66$	0	0
$67$	−1.17157	−0.143130	−0.0715652	−	0.997436i	$-0.522799\pi$
$67$	−1.17157	−0.143130	−0.0715652	+	0.997436i	$0.522799\pi$
$68$	0	0
$69$	4.58579	0.552064
$70$	0	0
$71$	6.48528	0.769661	0.384831	−	0.922987i	$-0.374260\pi$
$71$	6.48528	0.769661	0.384831	+	0.922987i	$0.374260\pi$
$72$	0	0
$73$	12.4853	1.46129	0.730646	−	0.682757i	$-0.239219\pi$
$73$	12.4853	1.46129	0.730646	+	0.682757i	$0.239219\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.828427	−0.0909317	−0.0454658	−	0.998966i	$-0.514477\pi$
$83$	−0.828427	−0.0909317	−0.0454658	+	0.998966i	$0.514477\pi$
$84$	0	0
$85$	6.82843	0.740647
$86$	0	0
$87$	7.65685	0.820901
$88$	0	0
$89$	−0.828427	−0.0878131	−0.0439065	−	0.999036i	$-0.513980\pi$
$89$	−0.828427	−0.0878131	−0.0439065	+	0.999036i	$0.513980\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.24264	−0.439941
$94$	0	0
$95$	−2.58579	−0.265296
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−0.343146	−0.0341443	−0.0170721	−	0.999854i	$-0.505434\pi$
$101$	−0.343146	−0.0341443	−0.0170721	+	0.999854i	$0.505434\pi$
$102$	0	0
$103$	2.82843	0.278693	0.139347	−	0.990244i	$-0.455500\pi$
$103$	2.82843	0.278693	0.139347	+	0.990244i	$0.455500\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.07107	−0.876933	−0.438467	−	0.898747i	$-0.644478\pi$
$107$	−9.07107	−0.876933	−0.438467	+	0.898747i	$0.644478\pi$
$108$	0	0
$109$	−15.6569	−1.49965	−0.749827	−	0.661634i	$-0.769863\pi$
$109$	−15.6569	−1.49965	−0.749827	+	0.661634i	$0.769863\pi$
$110$	0	0
$111$	−6.48528	−0.615556
$112$	0	0
$113$	2.24264	0.210970	0.105485	−	0.994421i	$-0.466361\pi$
$113$	2.24264	0.210970	0.105485	+	0.994421i	$0.466361\pi$
$114$	0	0
$115$	−4.58579	−0.427627
$116$	0	0
$117$	−2.82843	−0.261488
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.1421	−1.25491	−0.627456	−	0.778652i	$-0.715904\pi$
$127$	−14.1421	−1.25491	−0.627456	+	0.778652i	$0.715904\pi$
$128$	0	0
$129$	9.65685	0.850239
$130$	0	0
$131$	11.3137	0.988483	0.494242	−	0.869325i	$-0.335446\pi$
$131$	11.3137	0.988483	0.494242	+	0.869325i	$0.335446\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	11.8995	1.01664	0.508321	−	0.861168i	$-0.330266\pi$
$137$	11.8995	1.01664	0.508321	+	0.861168i	$0.330266\pi$
$138$	0	0
$139$	−23.5563	−1.99802	−0.999012	−	0.0444473i	$-0.985847\pi$
$139$	−23.5563	−1.99802	−0.999012	+	0.0444473i	$0.985847\pi$
$140$	0	0
$141$	6.48528	0.546159
$142$	0	0
$143$	5.65685	0.473050
$144$	0	0
$145$	−7.65685	−0.635867
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.34315	−0.683497	−0.341749	−	0.939791i	$-0.611019\pi$
$149$	−8.34315	−0.683497	−0.341749	+	0.939791i	$0.611019\pi$
$150$	0	0
$151$	−17.3137	−1.40897	−0.704485	−	0.709719i	$-0.748822\pi$
$151$	−17.3137	−1.40897	−0.704485	+	0.709719i	$0.748822\pi$
$152$	0	0
$153$	6.82843	0.552046
$154$	0	0
$155$	4.24264	0.340777
$156$	0	0
$157$	18.1421	1.44790	0.723950	−	0.689852i	$-0.242325\pi$
$157$	18.1421	1.44790	0.723950	+	0.689852i	$0.242325\pi$
$158$	0	0
$159$	7.41421	0.587985
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.1421	0.794393	0.397197	−	0.917734i	$-0.369983\pi$
$163$	10.1421	0.794393	0.397197	+	0.917734i	$0.369983\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−2.58579	−0.197740
$172$	0	0
$173$	−14.1421	−1.07521	−0.537603	−	0.843198i	$-0.680670\pi$
$173$	−14.1421	−1.07521	−0.537603	+	0.843198i	$0.680670\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.82843	−0.212598
$178$	0	0
$179$	6.48528	0.484733	0.242366	−	0.970185i	$-0.422076\pi$
$179$	6.48528	0.484733	0.242366	+	0.970185i	$0.422076\pi$
$180$	0	0
$181$	−15.7574	−1.17124	−0.585618	−	0.810587i	$-0.699148\pi$
$181$	−15.7574	−1.17124	−0.585618	+	0.810587i	$0.699148\pi$
$182$	0	0
$183$	9.89949	0.731792
$184$	0	0
$185$	6.48528	0.476807
$186$	0	0
$187$	−13.6569	−0.998688
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.9706	−1.37266	−0.686331	−	0.727289i	$-0.740780\pi$
$191$	−18.9706	−1.37266	−0.686331	+	0.727289i	$0.740780\pi$
$192$	0	0
$193$	7.17157	0.516221	0.258111	−	0.966115i	$-0.416900\pi$
$193$	7.17157	0.516221	0.258111	+	0.966115i	$0.416900\pi$
$194$	0	0
$195$	2.82843	0.202548
$196$	0	0
$197$	−1.07107	−0.0763104	−0.0381552	−	0.999272i	$-0.512148\pi$
$197$	−1.07107	−0.0763104	−0.0381552	+	0.999272i	$0.512148\pi$
$198$	0	0
$199$	0.242641	0.0172003	0.00860017	−	0.999963i	$-0.497262\pi$
$199$	0.242641	0.0172003	0.00860017	+	0.999963i	$0.497262\pi$
$200$	0	0
$201$	1.17157	0.0826364
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	−4.58579	−0.318734
$208$	0	0
$209$	5.17157	0.357725
$210$	0	0
$211$	−6.34315	−0.436680	−0.218340	−	0.975873i	$-0.570064\pi$
$211$	−6.34315	−0.436680	−0.218340	+	0.975873i	$0.570064\pi$
$212$	0	0
$213$	−6.48528	−0.444364
$214$	0	0
$215$	−9.65685	−0.658592
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−12.4853	−0.843677
$220$	0	0
$221$	−19.3137	−1.29918
$222$	0	0
$223$	−10.8284	−0.725125	−0.362563	−	0.931959i	$-0.618098\pi$
$223$	−10.8284	−0.725125	−0.362563	+	0.931959i	$0.618098\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	2.58579	0.170874	0.0854368	−	0.996344i	$-0.472771\pi$
$229$	2.58579	0.170874	0.0854368	+	0.996344i	$0.472771\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.7279	−1.48896	−0.744478	−	0.667647i	$-0.767301\pi$
$233$	−22.7279	−1.48896	−0.744478	+	0.667647i	$0.767301\pi$
$234$	0	0
$235$	−6.48528	−0.423053
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	−8.34315	−0.539673	−0.269837	−	0.962906i	$-0.586970\pi$
$239$	−8.34315	−0.539673	−0.269837	+	0.962906i	$0.586970\pi$
$240$	0	0
$241$	−26.5858	−1.71254	−0.856271	−	0.516528i	$-0.827224\pi$
$241$	−26.5858	−1.71254	−0.856271	+	0.516528i	$0.827224\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.31371	0.465360
$248$	0	0
$249$	0.828427	0.0524994
$250$	0	0
$251$	−28.4853	−1.79798	−0.898988	−	0.437974i	$-0.855696\pi$
$251$	−28.4853	−1.79798	−0.898988	+	0.437974i	$0.855696\pi$
$252$	0	0
$253$	9.17157	0.576612
$254$	0	0
$255$	−6.82843	−0.427613
$256$	0	0
$257$	−1.31371	−0.0819469	−0.0409734	−	0.999160i	$-0.513046\pi$
$257$	−1.31371	−0.0819469	−0.0409734	+	0.999160i	$0.513046\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−7.65685	−0.473947
$262$	0	0
$263$	−2.92893	−0.180606	−0.0903028	−	0.995914i	$-0.528783\pi$
$263$	−2.92893	−0.180606	−0.0903028	+	0.995914i	$0.528783\pi$
$264$	0	0
$265$	−7.41421	−0.455452
$266$	0	0
$267$	0.828427	0.0506989
$268$	0	0
$269$	20.8284	1.26993	0.634966	−	0.772540i	$-0.281014\pi$
$269$	20.8284	1.26993	0.634966	+	0.772540i	$0.281014\pi$
$270$	0	0
$271$	−11.7574	−0.714209	−0.357104	−	0.934064i	$-0.616236\pi$
$271$	−11.7574	−0.714209	−0.357104	+	0.934064i	$0.616236\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−17.3137	−1.04028	−0.520140	−	0.854081i	$-0.674120\pi$
$277$	−17.3137	−1.04028	−0.520140	+	0.854081i	$0.674120\pi$
$278$	0	0
$279$	4.24264	0.254000
$280$	0	0
$281$	30.9706	1.84755	0.923774	−	0.382937i	$-0.125087\pi$
$281$	30.9706	1.84755	0.923774	+	0.382937i	$0.125087\pi$
$282$	0	0
$283$	−8.48528	−0.504398	−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528	−0.504398	−0.252199	+	0.967675i	$0.581154\pi$
$284$	0	0
$285$	2.58579	0.153169
$286$	0	0
$287$	0	0
$288$	0	0
$289$	29.6274	1.74279
$290$	0	0
$291$	4.00000	0.234484
$292$	0	0
$293$	−7.65685	−0.447318	−0.223659	−	0.974667i	$-0.571800\pi$
$293$	−7.65685	−0.447318	−0.223659	+	0.974667i	$0.571800\pi$
$294$	0	0
$295$	2.82843	0.164677
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	12.9706	0.750107
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0.343146	0.0197132
$304$	0	0
$305$	−9.89949	−0.566843
$306$	0	0
$307$	9.65685	0.551146	0.275573	−	0.961280i	$-0.411132\pi$
$307$	9.65685	0.551146	0.275573	+	0.961280i	$0.411132\pi$
$308$	0	0
$309$	−2.82843	−0.160904
$310$	0	0
$311$	−31.3137	−1.77564	−0.887819	−	0.460193i	$-0.847780\pi$
$311$	−31.3137	−1.77564	−0.887819	+	0.460193i	$0.847780\pi$
$312$	0	0
$313$	−11.5147	−0.650850	−0.325425	−	0.945568i	$-0.605507\pi$
$313$	−11.5147	−0.650850	−0.325425	+	0.945568i	$0.605507\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.8995	1.11767	0.558833	−	0.829280i	$-0.311249\pi$
$317$	19.8995	1.11767	0.558833	+	0.829280i	$0.311249\pi$
$318$	0	0
$319$	15.3137	0.857403
$320$	0	0
$321$	9.07107	0.506298
$322$	0	0
$323$	−17.6569	−0.982454
$324$	0	0
$325$	−2.82843	−0.156893
$326$	0	0
$327$	15.6569	0.865826
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.6274	0.584136	0.292068	−	0.956398i	$-0.405657\pi$
$331$	10.6274	0.584136	0.292068	+	0.956398i	$0.405657\pi$
$332$	0	0
$333$	6.48528	0.355391
$334$	0	0
$335$	−1.17157	−0.0640099
$336$	0	0
$337$	−5.79899	−0.315891	−0.157946	−	0.987448i	$-0.550487\pi$
$337$	−5.79899	−0.315891	−0.157946	+	0.987448i	$0.550487\pi$
$338$	0	0
$339$	−2.24264	−0.121804
$340$	0	0
$341$	−8.48528	−0.459504
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.58579	0.246890
$346$	0	0
$347$	15.4142	0.827478	0.413739	−	0.910395i	$-0.364223\pi$
$347$	15.4142	0.827478	0.413739	+	0.910395i	$0.364223\pi$
$348$	0	0
$349$	13.2132	0.707287	0.353643	−	0.935380i	$-0.384943\pi$
$349$	13.2132	0.707287	0.353643	+	0.935380i	$0.384943\pi$
$350$	0	0
$351$	2.82843	0.150970
$352$	0	0
$353$	26.2843	1.39897	0.699485	−	0.714647i	$-0.253413\pi$
$353$	26.2843	1.39897	0.699485	+	0.714647i	$0.253413\pi$
$354$	0	0
$355$	6.48528	0.344203
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.97056	−0.367892	−0.183946	−	0.982936i	$-0.558887\pi$
$359$	−6.97056	−0.367892	−0.183946	+	0.982936i	$0.558887\pi$
$360$	0	0
$361$	−12.3137	−0.648090
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	12.4853	0.653509
$366$	0	0
$367$	25.1716	1.31395	0.656973	−	0.753914i	$-0.271837\pi$
$367$	25.1716	1.31395	0.656973	+	0.753914i	$0.271837\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	25.3137	1.31069	0.655347	−	0.755328i	$-0.272522\pi$
$373$	25.3137	1.31069	0.655347	+	0.755328i	$0.272522\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	21.6569	1.11538
$378$	0	0
$379$	−35.9411	−1.84617	−0.923086	−	0.384594i	$-0.874341\pi$
$379$	−35.9411	−1.84617	−0.923086	+	0.384594i	$0.874341\pi$
$380$	0	0
$381$	14.1421	0.724524
$382$	0	0
$383$	32.1421	1.64239	0.821193	−	0.570650i	$-0.193309\pi$
$383$	32.1421	1.64239	0.821193	+	0.570650i	$0.193309\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.65685	−0.490885
$388$	0	0
$389$	12.8284	0.650427	0.325214	−	0.945641i	$-0.394564\pi$
$389$	12.8284	0.650427	0.325214	+	0.945641i	$0.394564\pi$
$390$	0	0
$391$	−31.3137	−1.58360
$392$	0	0
$393$	−11.3137	−0.570701
$394$	0	0
$395$	−10.0000	−0.503155
$396$	0	0
$397$	−28.2843	−1.41955	−0.709773	−	0.704430i	$-0.751203\pi$
$397$	−28.2843	−1.41955	−0.709773	+	0.704430i	$0.751203\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.5147	1.07439	0.537197	−	0.843457i	$-0.319483\pi$
$401$	21.5147	1.07439	0.537197	+	0.843457i	$0.319483\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−12.9706	−0.642927
$408$	0	0
$409$	−0.727922	−0.0359934	−0.0179967	−	0.999838i	$-0.505729\pi$
$409$	−0.727922	−0.0359934	−0.0179967	+	0.999838i	$0.505729\pi$
$410$	0	0
$411$	−11.8995	−0.586959
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−0.828427	−0.0406659
$416$	0	0
$417$	23.5563	1.15356
$418$	0	0
$419$	−0.970563	−0.0474151	−0.0237075	−	0.999719i	$-0.507547\pi$
$419$	−0.970563	−0.0474151	−0.0237075	+	0.999719i	$0.507547\pi$
$420$	0	0
$421$	24.2843	1.18354	0.591771	−	0.806106i	$-0.298429\pi$
$421$	24.2843	1.18354	0.591771	+	0.806106i	$0.298429\pi$
$422$	0	0
$423$	−6.48528	−0.315325
$424$	0	0
$425$	6.82843	0.331227
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.65685	−0.273115
$430$	0	0
$431$	21.5147	1.03633	0.518164	−	0.855281i	$-0.326616\pi$
$431$	21.5147	1.03633	0.518164	+	0.855281i	$0.326616\pi$
$432$	0	0
$433$	−18.8284	−0.904836	−0.452418	−	0.891806i	$-0.649439\pi$
$433$	−18.8284	−0.904836	−0.452418	+	0.891806i	$0.649439\pi$
$434$	0	0
$435$	7.65685	0.367118
$436$	0	0
$437$	11.8579	0.567239
$438$	0	0
$439$	0.727922	0.0347418	0.0173709	−	0.999849i	$-0.494470\pi$
$439$	0.727922	0.0347418	0.0173709	+	0.999849i	$0.494470\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−23.4142	−1.11244	−0.556221	−	0.831034i	$-0.687749\pi$
$443$	−23.4142	−1.11244	−0.556221	+	0.831034i	$0.687749\pi$
$444$	0	0
$445$	−0.828427	−0.0392712
$446$	0	0
$447$	8.34315	0.394617
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	17.3137	0.813469
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.4853	1.23893	0.619465	−	0.785025i	$-0.287350\pi$
$457$	26.4853	1.23893	0.619465	+	0.785025i	$0.287350\pi$
$458$	0	0
$459$	−6.82843	−0.318724
$460$	0	0
$461$	39.4558	1.83764	0.918821	−	0.394675i	$-0.129143\pi$
$461$	39.4558	1.83764	0.918821	+	0.394675i	$0.129143\pi$
$462$	0	0
$463$	−34.1421	−1.58672	−0.793360	−	0.608753i	$-0.791670\pi$
$463$	−34.1421	−1.58672	−0.793360	+	0.608753i	$0.791670\pi$
$464$	0	0
$465$	−4.24264	−0.196748
$466$	0	0
$467$	37.9411	1.75571	0.877853	−	0.478930i	$-0.158975\pi$
$467$	37.9411	1.75571	0.877853	+	0.478930i	$0.158975\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−18.1421	−0.835945
$472$	0	0
$473$	19.3137	0.888045
$474$	0	0
$475$	−2.58579	−0.118644
$476$	0	0
$477$	−7.41421	−0.339474
$478$	0	0
$479$	−33.9411	−1.55081	−0.775405	−	0.631464i	$-0.782454\pi$
$479$	−33.9411	−1.55081	−0.775405	+	0.631464i	$0.782454\pi$
$480$	0	0
$481$	−18.3431	−0.836375
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.00000	−0.181631
$486$	0	0
$487$	36.2843	1.64420	0.822099	−	0.569345i	$-0.192803\pi$
$487$	36.2843	1.64420	0.822099	+	0.569345i	$0.192803\pi$
$488$	0	0
$489$	−10.1421	−0.458643
$490$	0	0
$491$	33.1127	1.49436	0.747178	−	0.664624i	$-0.231408\pi$
$491$	33.1127	1.49436	0.747178	+	0.664624i	$0.231408\pi$
$492$	0	0
$493$	−52.2843	−2.35477
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.2843	1.17665	0.588323	−	0.808626i	$-0.299788\pi$
$499$	26.2843	1.17665	0.588323	+	0.808626i	$0.299788\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	27.1716	1.21152	0.605760	−	0.795647i	$-0.292869\pi$
$503$	27.1716	1.21152	0.605760	+	0.795647i	$0.292869\pi$
$504$	0	0
$505$	−0.343146	−0.0152698
$506$	0	0
$507$	5.00000	0.222058
$508$	0	0
$509$	21.5147	0.953623	0.476812	−	0.879006i	$-0.341792\pi$
$509$	21.5147	0.953623	0.476812	+	0.879006i	$0.341792\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.58579	0.114165
$514$	0	0
$515$	2.82843	0.124635
$516$	0	0
$517$	12.9706	0.570445
$518$	0	0
$519$	14.1421	0.620771
$520$	0	0
$521$	−41.1127	−1.80118	−0.900590	−	0.434670i	$-0.856865\pi$
$521$	−41.1127	−1.80118	−0.900590	+	0.434670i	$0.856865\pi$
$522$	0	0
$523$	18.8284	0.823310	0.411655	−	0.911340i	$-0.364951\pi$
$523$	18.8284	0.823310	0.411655	+	0.911340i	$0.364951\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.9706	1.26198
$528$	0	0
$529$	−1.97056	−0.0856766
$530$	0	0
$531$	2.82843	0.122743
$532$	0	0
$533$	−5.65685	−0.245026
$534$	0	0
$535$	−9.07107	−0.392176
$536$	0	0
$537$	−6.48528	−0.279861
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−20.2843	−0.872089	−0.436044	−	0.899925i	$-0.643621\pi$
$541$	−20.2843	−0.872089	−0.436044	+	0.899925i	$0.643621\pi$
$542$	0	0
$543$	15.7574	0.676213
$544$	0	0
$545$	−15.6569	−0.670666
$546$	0	0
$547$	−35.1127	−1.50131	−0.750655	−	0.660694i	$-0.770262\pi$
$547$	−35.1127	−1.50131	−0.750655	+	0.660694i	$0.770262\pi$
$548$	0	0
$549$	−9.89949	−0.422500
$550$	0	0
$551$	19.7990	0.843465
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−6.48528	−0.275285
$556$	0	0
$557$	27.6985	1.17362	0.586811	−	0.809724i	$-0.300383\pi$
$557$	27.6985	1.17362	0.586811	+	0.809724i	$0.300383\pi$
$558$	0	0
$559$	27.3137	1.15525
$560$	0	0
$561$	13.6569	0.576593
$562$	0	0
$563$	21.5147	0.906737	0.453369	−	0.891323i	$-0.350222\pi$
$563$	21.5147	0.906737	0.453369	+	0.891323i	$0.350222\pi$
$564$	0	0
$565$	2.24264	0.0943486
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.5147	0.734255	0.367128	−	0.930171i	$-0.380341\pi$
$569$	17.5147	0.734255	0.367128	+	0.930171i	$0.380341\pi$
$570$	0	0
$571$	−36.2843	−1.51845	−0.759225	−	0.650829i	$-0.774422\pi$
$571$	−36.2843	−1.51845	−0.759225	+	0.650829i	$0.774422\pi$
$572$	0	0
$573$	18.9706	0.792507
$574$	0	0
$575$	−4.58579	−0.191241
$576$	0	0
$577$	−2.82843	−0.117749	−0.0588745	−	0.998265i	$-0.518751\pi$
$577$	−2.82843	−0.117749	−0.0588745	+	0.998265i	$0.518751\pi$
$578$	0	0
$579$	−7.17157	−0.298040
$580$	0	0
$581$	0	0
$582$	0	0
$583$	14.8284	0.614131
$584$	0	0
$585$	−2.82843	−0.116941
$586$	0	0
$587$	−11.1716	−0.461100	−0.230550	−	0.973060i	$-0.574053\pi$
$587$	−11.1716	−0.461100	−0.230550	+	0.973060i	$0.574053\pi$
$588$	0	0
$589$	−10.9706	−0.452034
$590$	0	0
$591$	1.07107	0.0440578
$592$	0	0
$593$	−30.9706	−1.27181	−0.635904	−	0.771768i	$-0.719373\pi$
$593$	−30.9706	−1.27181	−0.635904	+	0.771768i	$0.719373\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.242641	−0.00993062
$598$	0	0
$599$	−14.0000	−0.572024	−0.286012	−	0.958226i	$-0.592330\pi$
$599$	−14.0000	−0.572024	−0.286012	+	0.958226i	$0.592330\pi$
$600$	0	0
$601$	−28.2426	−1.15204	−0.576021	−	0.817435i	$-0.695395\pi$
$601$	−28.2426	−1.15204	−0.576021	+	0.817435i	$0.695395\pi$
$602$	0	0
$603$	−1.17157	−0.0477101
$604$	0	0
$605$	−7.00000	−0.284590
$606$	0	0
$607$	29.4558	1.19558	0.597788	−	0.801654i	$-0.296046\pi$
$607$	29.4558	1.19558	0.597788	+	0.801654i	$0.296046\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18.3431	0.742084
$612$	0	0
$613$	14.6863	0.593174	0.296587	−	0.955006i	$-0.404152\pi$
$613$	14.6863	0.593174	0.296587	+	0.955006i	$0.404152\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	−23.6985	−0.954065	−0.477033	−	0.878886i	$-0.658288\pi$
$617$	−23.6985	−0.954065	−0.477033	+	0.878886i	$0.658288\pi$
$618$	0	0
$619$	33.8995	1.36254	0.681268	−	0.732034i	$-0.261429\pi$
$619$	33.8995	1.36254	0.681268	+	0.732034i	$0.261429\pi$
$620$	0	0
$621$	4.58579	0.184021
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.17157	−0.206533
$628$	0	0
$629$	44.2843	1.76573
$630$	0	0
$631$	29.3137	1.16696	0.583480	−	0.812127i	$-0.301691\pi$
$631$	29.3137	1.16696	0.583480	+	0.812127i	$0.301691\pi$
$632$	0	0
$633$	6.34315	0.252117
$634$	0	0
$635$	−14.1421	−0.561214
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.48528	0.256554
$640$	0	0
$641$	42.9706	1.69724	0.848618	−	0.529007i	$-0.177435\pi$
$641$	42.9706	1.69724	0.848618	+	0.529007i	$0.177435\pi$
$642$	0	0
$643$	24.2843	0.957678	0.478839	−	0.877903i	$-0.341058\pi$
$643$	24.2843	0.957678	0.478839	+	0.877903i	$0.341058\pi$
$644$	0	0
$645$	9.65685	0.380238
$646$	0	0
$647$	12.6863	0.498750	0.249375	−	0.968407i	$-0.419775\pi$
$647$	12.6863	0.498750	0.249375	+	0.968407i	$0.419775\pi$
$648$	0	0
$649$	−5.65685	−0.222051
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.4142	0.759737	0.379868	−	0.925041i	$-0.375969\pi$
$653$	19.4142	0.759737	0.379868	+	0.925041i	$0.375969\pi$
$654$	0	0
$655$	11.3137	0.442063
$656$	0	0
$657$	12.4853	0.487097
$658$	0	0
$659$	−29.7990	−1.16080	−0.580402	−	0.814330i	$-0.697105\pi$
$659$	−29.7990	−1.16080	−0.580402	+	0.814330i	$0.697105\pi$
$660$	0	0
$661$	−44.2426	−1.72084	−0.860420	−	0.509586i	$-0.829799\pi$
$661$	−44.2426	−1.72084	−0.860420	+	0.509586i	$0.829799\pi$
$662$	0	0
$663$	19.3137	0.750082
$664$	0	0
$665$	0	0
$666$	0	0
$667$	35.1127	1.35957
$668$	0	0
$669$	10.8284	0.418651
$670$	0	0
$671$	19.7990	0.764332
$672$	0	0
$673$	35.1716	1.35576	0.677882	−	0.735170i	$-0.262898\pi$
$673$	35.1716	1.35576	0.677882	+	0.735170i	$0.262898\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−9.02944	−0.347029	−0.173515	−	0.984831i	$-0.555512\pi$
$677$	−9.02944	−0.347029	−0.173515	+	0.984831i	$0.555512\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	−33.5563	−1.28400	−0.641999	−	0.766705i	$-0.721895\pi$
$683$	−33.5563	−1.28400	−0.641999	+	0.766705i	$0.721895\pi$
$684$	0	0
$685$	11.8995	0.454656
$686$	0	0
$687$	−2.58579	−0.0986539
$688$	0	0
$689$	20.9706	0.798915
$690$	0	0
$691$	44.7279	1.70153	0.850765	−	0.525546i	$-0.176139\pi$
$691$	44.7279	1.70153	0.850765	+	0.525546i	$0.176139\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−23.5563	−0.893543
$696$	0	0
$697$	13.6569	0.517290
$698$	0	0
$699$	22.7279	0.859649
$700$	0	0
$701$	2.97056	0.112197	0.0560983	−	0.998425i	$-0.482134\pi$
$701$	2.97056	0.112197	0.0560983	+	0.998425i	$0.482134\pi$
$702$	0	0
$703$	−16.7696	−0.632476
$704$	0	0
$705$	6.48528	0.244250
$706$	0	0
$707$	0	0
$708$	0	0
$709$	31.3137	1.17601	0.588006	−	0.808857i	$-0.299913\pi$
$709$	31.3137	1.17601	0.588006	+	0.808857i	$0.299913\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	−19.4558	−0.728627
$714$	0	0
$715$	5.65685	0.211554
$716$	0	0
$717$	8.34315	0.311580
$718$	0	0
$719$	4.48528	0.167273	0.0836364	−	0.996496i	$-0.473347\pi$
$719$	4.48528	0.167273	0.0836364	+	0.996496i	$0.473347\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	26.5858	0.988736
$724$	0	0
$725$	−7.65685	−0.284368
$726$	0	0
$727$	−35.5980	−1.32026	−0.660128	−	0.751153i	$-0.729498\pi$
$727$	−35.5980	−1.32026	−0.660128	+	0.751153i	$0.729498\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−65.9411	−2.43892
$732$	0	0
$733$	−7.51472	−0.277562	−0.138781	−	0.990323i	$-0.544318\pi$
$733$	−7.51472	−0.277562	−0.138781	+	0.990323i	$0.544318\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.34315	0.0863109
$738$	0	0
$739$	38.6274	1.42093	0.710466	−	0.703731i	$-0.248484\pi$
$739$	38.6274	1.42093	0.710466	+	0.703731i	$0.248484\pi$
$740$	0	0
$741$	−7.31371	−0.268676
$742$	0	0
$743$	30.0416	1.10212	0.551060	−	0.834465i	$-0.314223\pi$
$743$	30.0416	1.10212	0.551060	+	0.834465i	$0.314223\pi$
$744$	0	0
$745$	−8.34315	−0.305669
$746$	0	0
$747$	−0.828427	−0.0303106
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.34315	0.0855026	0.0427513	−	0.999086i	$-0.486388\pi$
$751$	2.34315	0.0855026	0.0427513	+	0.999086i	$0.486388\pi$
$752$	0	0
$753$	28.4853	1.03806
$754$	0	0
$755$	−17.3137	−0.630110
$756$	0	0
$757$	−4.82843	−0.175492	−0.0877461	−	0.996143i	$-0.527966\pi$
$757$	−4.82843	−0.175492	−0.0877461	+	0.996143i	$0.527966\pi$
$758$	0	0
$759$	−9.17157	−0.332907
$760$	0	0
$761$	10.4853	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$761$	10.4853	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	6.82843	0.246882
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−29.2132	−1.05346	−0.526728	−	0.850034i	$-0.676581\pi$
$769$	−29.2132	−1.05346	−0.526728	+	0.850034i	$0.676581\pi$
$770$	0	0
$771$	1.31371	0.0473121
$772$	0	0
$773$	13.4558	0.483973	0.241987	−	0.970280i	$-0.422201\pi$
$773$	13.4558	0.483973	0.241987	+	0.970280i	$0.422201\pi$
$774$	0	0
$775$	4.24264	0.152400
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.17157	−0.185291
$780$	0	0
$781$	−12.9706	−0.464123
$782$	0	0
$783$	7.65685	0.273634
$784$	0	0
$785$	18.1421	0.647521
$786$	0	0
$787$	−23.1127	−0.823879	−0.411939	−	0.911211i	$-0.635148\pi$
$787$	−23.1127	−0.823879	−0.411939	+	0.911211i	$0.635148\pi$
$788$	0	0
$789$	2.92893	0.104273
$790$	0	0
$791$	0	0
$792$	0	0
$793$	28.0000	0.994309
$794$	0	0
$795$	7.41421	0.262955
$796$	0	0
$797$	47.1127	1.66882	0.834409	−	0.551146i	$-0.185809\pi$
$797$	47.1127	1.66882	0.834409	+	0.551146i	$0.185809\pi$
$798$	0	0
$799$	−44.2843	−1.56666
$800$	0	0
$801$	−0.828427	−0.0292710
$802$	0	0
$803$	−24.9706	−0.881192
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−20.8284	−0.733195
$808$	0	0
$809$	22.4853	0.790540	0.395270	−	0.918565i	$-0.370651\pi$
$809$	22.4853	0.790540	0.395270	+	0.918565i	$0.370651\pi$
$810$	0	0
$811$	−17.8995	−0.628536	−0.314268	−	0.949334i	$-0.601759\pi$
$811$	−17.8995	−0.628536	−0.314268	+	0.949334i	$0.601759\pi$
$812$	0	0
$813$	11.7574	0.412349
$814$	0	0
$815$	10.1421	0.355264
$816$	0	0
$817$	24.9706	0.873609
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−44.4264	−1.55049	−0.775246	−	0.631659i	$-0.782374\pi$
$821$	−44.4264	−1.55049	−0.775246	+	0.631659i	$0.782374\pi$
$822$	0	0
$823$	−25.4558	−0.887335	−0.443667	−	0.896191i	$-0.646323\pi$
$823$	−25.4558	−0.887335	−0.443667	+	0.896191i	$0.646323\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	−5.75736	−0.200203	−0.100101	−	0.994977i	$-0.531917\pi$
$827$	−5.75736	−0.200203	−0.100101	+	0.994977i	$0.531917\pi$
$828$	0	0
$829$	11.0711	0.384514	0.192257	−	0.981345i	$-0.438419\pi$
$829$	11.0711	0.384514	0.192257	+	0.981345i	$0.438419\pi$
$830$	0	0
$831$	17.3137	0.600606
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	−4.24264	−0.146647
$838$	0	0
$839$	−15.5147	−0.535628	−0.267814	−	0.963471i	$-0.586301\pi$
$839$	−15.5147	−0.535628	−0.267814	+	0.963471i	$0.586301\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	0	0
$843$	−30.9706	−1.06668
$844$	0	0
$845$	−5.00000	−0.172005
$846$	0	0
$847$	0	0
$848$	0	0
$849$	8.48528	0.291214
$850$	0	0
$851$	−29.7401	−1.01948
$852$	0	0
$853$	16.2843	0.557563	0.278781	−	0.960355i	$-0.410070\pi$
$853$	16.2843	0.557563	0.278781	+	0.960355i	$0.410070\pi$
$854$	0	0
$855$	−2.58579	−0.0884320
$856$	0	0
$857$	−25.4558	−0.869555	−0.434778	−	0.900538i	$-0.643173\pi$
$857$	−25.4558	−0.869555	−0.434778	+	0.900538i	$0.643173\pi$
$858$	0	0
$859$	−29.8995	−1.02016	−0.510079	−	0.860128i	$-0.670384\pi$
$859$	−29.8995	−1.02016	−0.510079	+	0.860128i	$0.670384\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.1005	−0.548068	−0.274034	−	0.961720i	$-0.588358\pi$
$863$	−16.1005	−0.548068	−0.274034	+	0.961720i	$0.588358\pi$
$864$	0	0
$865$	−14.1421	−0.480847
$866$	0	0
$867$	−29.6274	−1.00620
$868$	0	0
$869$	20.0000	0.678454
$870$	0	0
$871$	3.31371	0.112281
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.0294	−0.439973	−0.219986	−	0.975503i	$-0.570601\pi$
$877$	−13.0294	−0.439973	−0.219986	+	0.975503i	$0.570601\pi$
$878$	0	0
$879$	7.65685	0.258259
$880$	0	0
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	0	0
$883$	16.9706	0.571105	0.285552	−	0.958363i	$-0.407823\pi$
$883$	16.9706	0.571105	0.285552	+	0.958363i	$0.407823\pi$
$884$	0	0
$885$	−2.82843	−0.0950765
$886$	0	0
$887$	32.8284	1.10227	0.551135	−	0.834416i	$-0.314195\pi$
$887$	32.8284	1.10227	0.551135	+	0.834416i	$0.314195\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	16.7696	0.561172
$894$	0	0
$895$	6.48528	0.216779
$896$	0	0
$897$	−12.9706	−0.433074
$898$	0	0
$899$	−32.4853	−1.08344
$900$	0	0
$901$	−50.6274	−1.68664
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−15.7574	−0.523792
$906$	0	0
$907$	22.3431	0.741892	0.370946	−	0.928654i	$-0.379033\pi$
$907$	22.3431	0.741892	0.370946	+	0.928654i	$0.379033\pi$
$908$	0	0
$909$	−0.343146	−0.0113814
$910$	0	0
$911$	19.9411	0.660679	0.330339	−	0.943862i	$-0.392837\pi$
$911$	19.9411	0.660679	0.330339	+	0.943862i	$0.392837\pi$
$912$	0	0
$913$	1.65685	0.0548339
$914$	0	0
$915$	9.89949	0.327267
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−28.2843	−0.933012	−0.466506	−	0.884518i	$-0.654487\pi$
$919$	−28.2843	−0.933012	−0.466506	+	0.884518i	$0.654487\pi$
$920$	0	0
$921$	−9.65685	−0.318204
$922$	0	0
$923$	−18.3431	−0.603772
$924$	0	0
$925$	6.48528	0.213235
$926$	0	0
$927$	2.82843	0.0928977
$928$	0	0
$929$	−24.8284	−0.814594	−0.407297	−	0.913296i	$-0.633529\pi$
$929$	−24.8284	−0.814594	−0.407297	+	0.913296i	$0.633529\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	31.3137	1.02516
$934$	0	0
$935$	−13.6569	−0.446627
$936$	0	0
$937$	6.34315	0.207222	0.103611	−	0.994618i	$-0.466960\pi$
$937$	6.34315	0.207222	0.103611	+	0.994618i	$0.466960\pi$
$938$	0	0
$939$	11.5147	0.375769
$940$	0	0
$941$	−39.2548	−1.27967	−0.639836	−	0.768512i	$-0.720998\pi$
$941$	−39.2548	−1.27967	−0.639836	+	0.768512i	$0.720998\pi$
$942$	0	0
$943$	−9.17157	−0.298668
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.0711	1.20465	0.602324	−	0.798252i	$-0.294242\pi$
$947$	37.0711	1.20465	0.602324	+	0.798252i	$0.294242\pi$
$948$	0	0
$949$	−35.3137	−1.14633
$950$	0	0
$951$	−19.8995	−0.645285
$952$	0	0
$953$	−2.04163	−0.0661349	−0.0330675	−	0.999453i	$-0.510528\pi$
$953$	−2.04163	−0.0661349	−0.0330675	+	0.999453i	$0.510528\pi$
$954$	0	0
$955$	−18.9706	−0.613873
$956$	0	0
$957$	−15.3137	−0.495022
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−13.0000	−0.419355
$962$	0	0
$963$	−9.07107	−0.292311
$964$	0	0
$965$	7.17157	0.230861
$966$	0	0
$967$	−59.7990	−1.92301	−0.961503	−	0.274795i	$-0.911390\pi$
$967$	−59.7990	−1.92301	−0.961503	+	0.274795i	$0.911390\pi$
$968$	0	0
$969$	17.6569	0.567220
$970$	0	0
$971$	−3.51472	−0.112793	−0.0563963	−	0.998408i	$-0.517961\pi$
$971$	−3.51472	−0.112793	−0.0563963	+	0.998408i	$0.517961\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.82843	0.0905822
$976$	0	0
$977$	−18.0416	−0.577203	−0.288601	−	0.957449i	$-0.593190\pi$
$977$	−18.0416	−0.577203	−0.288601	+	0.957449i	$0.593190\pi$
$978$	0	0
$979$	1.65685	0.0529533
$980$	0	0
$981$	−15.6569	−0.499885
$982$	0	0
$983$	23.5980	0.752659	0.376329	−	0.926486i	$-0.377186\pi$
$983$	23.5980	0.752659	0.376329	+	0.926486i	$0.377186\pi$
$984$	0	0
$985$	−1.07107	−0.0341271
$986$	0	0
$987$	0	0
$988$	0	0
$989$	44.2843	1.40816
$990$	0	0
$991$	11.3137	0.359392	0.179696	−	0.983722i	$-0.442489\pi$
$991$	11.3137	0.359392	0.179696	+	0.983722i	$0.442489\pi$
$992$	0	0
$993$	−10.6274	−0.337251
$994$	0	0
$995$	0.242641	0.00769223
$996$	0	0
$997$	19.3137	0.611671	0.305836	−	0.952084i	$-0.401064\pi$
$997$	19.3137	0.611671	0.305836	+	0.952084i	$0.401064\pi$
$998$	0	0
$999$	−6.48528	−0.205185

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2940.2.a.o.1.1	✓	2
3.2	odd	2		8820.2.a.bh.1.1		2
7.2	even	3		2940.2.q.r.361.1		4
7.3	odd	6		2940.2.q.p.961.2		4
7.4	even	3		2940.2.q.r.961.1		4
7.5	odd	6		2940.2.q.p.361.2		4
7.6	odd	2		2940.2.a.q.1.2	yes	2
21.20	even	2		8820.2.a.bm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2940.2.a.o.1.1	✓	2	1.1	even	1	trivial
2940.2.a.q.1.2	yes	2	7.6	odd	2
2940.2.q.p.361.2		4	7.5	odd	6
2940.2.q.p.961.2		4	7.3	odd	6
2940.2.q.r.361.1		4	7.2	even	3
2940.2.q.r.961.1		4	7.4	even	3
8820.2.a.bh.1.1		2	3.2	odd	2
8820.2.a.bm.1.2		2	21.20	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2940.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} -2.00000 q^{11} -2.82843 q^{13} -1.00000 q^{15} +6.82843 q^{17} -2.58579 q^{19} -4.58579 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.65685 q^{29} +4.24264 q^{31} +2.00000 q^{33} +6.48528 q^{37} +2.82843 q^{39} +2.00000 q^{41} -9.65685 q^{43} +1.00000 q^{45} -6.48528 q^{47} -6.82843 q^{51} -7.41421 q^{53} -2.00000 q^{55} +2.58579 q^{57} +2.82843 q^{59} -9.89949 q^{61} -2.82843 q^{65} -1.17157 q^{67} +4.58579 q^{69} +6.48528 q^{71} +12.4853 q^{73} -1.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} -0.828427 q^{83} +6.82843 q^{85} +7.65685 q^{87} -0.828427 q^{89} -4.24264 q^{93} -2.58579 q^{95} -4.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} - 4 q^{11} - 2 q^{15} + 8 q^{17} - 8 q^{19} - 12 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 4 q^{33} - 4 q^{37} + 4 q^{41} - 8 q^{43} + 2 q^{45} + 4 q^{47} - 8 q^{51} - 12 q^{53}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9 - 4 * q^11 - 2 * q^15 + 8 * q^17 - 8 * q^19 - 12 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 + 4 * q^33 - 4 * q^37 + 4 * q^41 - 8 * q^43 + 2 * q^45 + 4 * q^47 - 8 * q^51 - 12 * q^53 - 4 * q^55 + 8 * q^57 - 8 * q^67 + 12 * q^69 - 4 * q^71 + 8 * q^73 - 2 * q^75 - 20 * q^79 + 2 * q^81 + 4 * q^83 + 8 * q^85 + 4 * q^87 + 4 * q^89 - 8 * q^95 - 8 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more