LMFDB - Embedded newform 8004.2.a.k.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 $ x^18 - 5x^17 - 55x^16 + 288x^15 + 1222x^14 - 6888x^13 - 13745x^12 + 88434x^11 + 76571x^10 - 655793x^9 - 114554x^8 + 2789438x^7 - 855636x^6 - 6184176x^5 + 4260960x^4 + 5001480x^3 - 5659296x^2 + 1509552*x - 115488
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$3.38212$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.38212 q^{5} +5.02591 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.38212 q^{5} +5.02591 q^{7} +1.00000 q^{9} +1.05075 q^{11} +2.71300 q^{13} +3.38212 q^{15} +1.45407 q^{17} -7.90673 q^{19} +5.02591 q^{21} -1.00000 q^{23} +6.43875 q^{25} +1.00000 q^{27} +1.00000 q^{29} -7.96415 q^{31} +1.05075 q^{33} +16.9982 q^{35} -2.92438 q^{37} +2.71300 q^{39} +12.0445 q^{41} -9.53961 q^{43} +3.38212 q^{45} -9.69287 q^{47} +18.2598 q^{49} +1.45407 q^{51} +5.88785 q^{53} +3.55378 q^{55} -7.90673 q^{57} +6.76386 q^{59} -4.05471 q^{61} +5.02591 q^{63} +9.17569 q^{65} +6.64598 q^{67} -1.00000 q^{69} -9.94477 q^{71} +9.33118 q^{73} +6.43875 q^{75} +5.28100 q^{77} -5.74980 q^{79} +1.00000 q^{81} +15.4273 q^{83} +4.91785 q^{85} +1.00000 q^{87} +0.992971 q^{89} +13.6353 q^{91} -7.96415 q^{93} -26.7415 q^{95} +11.5694 q^{97} +1.05075 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * 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
$5$	3.38212	1.51253	0.756265	−	0.654265i	$-0.227022\pi$
$5$	3.38212	1.51253	0.756265	+	0.654265i	$0.227022\pi$
$6$	0	0
$7$	5.02591	1.89962	0.949808	−	0.312834i	$-0.101278\pi$
$7$	5.02591	1.89962	0.949808	+	0.312834i	$0.101278\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.05075	0.316814	0.158407	−	0.987374i	$-0.449364\pi$
$11$	1.05075	0.316814	0.158407	+	0.987374i	$0.449364\pi$
$12$	0	0
$13$	2.71300	0.752451	0.376225	−	0.926528i	$-0.377222\pi$
$13$	2.71300	0.752451	0.376225	+	0.926528i	$0.377222\pi$
$14$	0	0
$15$	3.38212	0.873260
$16$	0	0
$17$	1.45407	0.352665	0.176332	−	0.984331i	$-0.443577\pi$
$17$	1.45407	0.352665	0.176332	+	0.984331i	$0.443577\pi$
$18$	0	0
$19$	−7.90673	−1.81393	−0.906965	−	0.421207i	$-0.861607\pi$
$19$	−7.90673	−1.81393	−0.906965	+	0.421207i	$0.861607\pi$
$20$	0	0
$21$	5.02591	1.09674
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	6.43875	1.28775
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−7.96415	−1.43040	−0.715202	−	0.698918i	$-0.753665\pi$
$31$	−7.96415	−1.43040	−0.715202	+	0.698918i	$0.753665\pi$
$32$	0	0
$33$	1.05075	0.182913
$34$	0	0
$35$	16.9982	2.87323
$36$	0	0
$37$	−2.92438	−0.480765	−0.240383	−	0.970678i	$-0.577273\pi$
$37$	−2.92438	−0.480765	−0.240383	+	0.970678i	$0.577273\pi$
$38$	0	0
$39$	2.71300	0.434428
$40$	0	0
$41$	12.0445	1.88103	0.940514	−	0.339755i	$-0.110344\pi$
$41$	12.0445	1.88103	0.940514	+	0.339755i	$0.110344\pi$
$42$	0	0
$43$	−9.53961	−1.45478	−0.727388	−	0.686226i	$-0.759266\pi$
$43$	−9.53961	−1.45478	−0.727388	+	0.686226i	$0.759266\pi$
$44$	0	0
$45$	3.38212	0.504177
$46$	0	0
$47$	−9.69287	−1.41385	−0.706925	−	0.707288i	$-0.749918\pi$
$47$	−9.69287	−1.41385	−0.706925	+	0.707288i	$0.749918\pi$
$48$	0	0
$49$	18.2598	2.60854
$50$	0	0
$51$	1.45407	0.203611
$52$	0	0
$53$	5.88785	0.808759	0.404379	−	0.914591i	$-0.367488\pi$
$53$	5.88785	0.808759	0.404379	+	0.914591i	$0.367488\pi$
$54$	0	0
$55$	3.55378	0.479191
$56$	0	0
$57$	−7.90673	−1.04727
$58$	0	0
$59$	6.76386	0.880580	0.440290	−	0.897856i	$-0.354876\pi$
$59$	6.76386	0.880580	0.440290	+	0.897856i	$0.354876\pi$
$60$	0	0
$61$	−4.05471	−0.519152	−0.259576	−	0.965723i	$-0.583583\pi$
$61$	−4.05471	−0.519152	−0.259576	+	0.965723i	$0.583583\pi$
$62$	0	0
$63$	5.02591	0.633205
$64$	0	0
$65$	9.17569	1.13810
$66$	0	0
$67$	6.64598	0.811936	0.405968	−	0.913887i	$-0.366934\pi$
$67$	6.64598	0.811936	0.405968	+	0.913887i	$0.366934\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−9.94477	−1.18023	−0.590113	−	0.807320i	$-0.700917\pi$
$71$	−9.94477	−1.18023	−0.590113	+	0.807320i	$0.700917\pi$
$72$	0	0
$73$	9.33118	1.09213	0.546066	−	0.837742i	$-0.316125\pi$
$73$	9.33118	1.09213	0.546066	+	0.837742i	$0.316125\pi$
$74$	0	0
$75$	6.43875	0.743482
$76$	0	0
$77$	5.28100	0.601825
$78$	0	0
$79$	−5.74980	−0.646903	−0.323451	−	0.946245i	$-0.604843\pi$
$79$	−5.74980	−0.646903	−0.323451	+	0.946245i	$0.604843\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.4273	1.69337	0.846683	−	0.532098i	$-0.178596\pi$
$83$	15.4273	1.69337	0.846683	+	0.532098i	$0.178596\pi$
$84$	0	0
$85$	4.91785	0.533416
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	0.992971	0.105255	0.0526274	−	0.998614i	$-0.483240\pi$
$89$	0.992971	0.105255	0.0526274	+	0.998614i	$0.483240\pi$
$90$	0	0
$91$	13.6353	1.42937
$92$	0	0
$93$	−7.96415	−0.825844
$94$	0	0
$95$	−26.7415	−2.74362
$96$	0	0
$97$	11.5694	1.17469	0.587346	−	0.809336i	$-0.300173\pi$
$97$	11.5694	1.17469	0.587346	+	0.809336i	$0.300173\pi$
$98$	0	0
$99$	1.05075	0.105605
$100$	0	0
$101$	−1.81461	−0.180560	−0.0902802	−	0.995916i	$-0.528776\pi$
$101$	−1.81461	−0.180560	−0.0902802	+	0.995916i	$0.528776\pi$
$102$	0	0
$103$	20.0792	1.97846	0.989230	−	0.146372i	$-0.0467598\pi$
$103$	20.0792	1.97846	0.989230	+	0.146372i	$0.0467598\pi$
$104$	0	0
$105$	16.9982	1.65886
$106$	0	0
$107$	−13.9784	−1.35134	−0.675671	−	0.737203i	$-0.736146\pi$
$107$	−13.9784	−1.35134	−0.675671	+	0.737203i	$0.736146\pi$
$108$	0	0
$109$	1.30547	0.125041	0.0625204	−	0.998044i	$-0.480086\pi$
$109$	1.30547	0.125041	0.0625204	+	0.998044i	$0.480086\pi$
$110$	0	0
$111$	−2.92438	−0.277570
$112$	0	0
$113$	−2.61994	−0.246463	−0.123232	−	0.992378i	$-0.539326\pi$
$113$	−2.61994	−0.246463	−0.123232	+	0.992378i	$0.539326\pi$
$114$	0	0
$115$	−3.38212	−0.315384
$116$	0	0
$117$	2.71300	0.250817
$118$	0	0
$119$	7.30804	0.669927
$120$	0	0
$121$	−9.89592	−0.899629
$122$	0	0
$123$	12.0445	1.08601
$124$	0	0
$125$	4.86602	0.435230
$126$	0	0
$127$	2.05300	0.182174	0.0910872	−	0.995843i	$-0.470966\pi$
$127$	2.05300	0.182174	0.0910872	+	0.995843i	$0.470966\pi$
$128$	0	0
$129$	−9.53961	−0.839916
$130$	0	0
$131$	9.76578	0.853240	0.426620	−	0.904431i	$-0.359704\pi$
$131$	9.76578	0.853240	0.426620	+	0.904431i	$0.359704\pi$
$132$	0	0
$133$	−39.7385	−3.44577
$134$	0	0
$135$	3.38212	0.291087
$136$	0	0
$137$	7.34825	0.627804	0.313902	−	0.949455i	$-0.398364\pi$
$137$	7.34825	0.627804	0.313902	+	0.949455i	$0.398364\pi$
$138$	0	0
$139$	13.5213	1.14686	0.573431	−	0.819254i	$-0.305612\pi$
$139$	13.5213	1.14686	0.573431	+	0.819254i	$0.305612\pi$
$140$	0	0
$141$	−9.69287	−0.816287
$142$	0	0
$143$	2.85069	0.238387
$144$	0	0
$145$	3.38212	0.280870
$146$	0	0
$147$	18.2598	1.50604
$148$	0	0
$149$	7.90733	0.647794	0.323897	−	0.946092i	$-0.395007\pi$
$149$	7.90733	0.647794	0.323897	+	0.946092i	$0.395007\pi$
$150$	0	0
$151$	11.2241	0.913404	0.456702	−	0.889620i	$-0.349031\pi$
$151$	11.2241	0.913404	0.456702	+	0.889620i	$0.349031\pi$
$152$	0	0
$153$	1.45407	0.117555
$154$	0	0
$155$	−26.9357	−2.16353
$156$	0	0
$157$	6.37096	0.508458	0.254229	−	0.967144i	$-0.418178\pi$
$157$	6.37096	0.508458	0.254229	+	0.967144i	$0.418178\pi$
$158$	0	0
$159$	5.88785	0.466937
$160$	0	0
$161$	−5.02591	−0.396097
$162$	0	0
$163$	6.94106	0.543666	0.271833	−	0.962344i	$-0.412370\pi$
$163$	6.94106	0.543666	0.271833	+	0.962344i	$0.412370\pi$
$164$	0	0
$165$	3.55378	0.276661
$166$	0	0
$167$	−11.5112	−0.890761	−0.445381	−	0.895341i	$-0.646932\pi$
$167$	−11.5112	−0.890761	−0.445381	+	0.895341i	$0.646932\pi$
$168$	0	0
$169$	−5.63963	−0.433818
$170$	0	0
$171$	−7.90673	−0.604643
$172$	0	0
$173$	−20.5832	−1.56491	−0.782457	−	0.622705i	$-0.786034\pi$
$173$	−20.5832	−1.56491	−0.782457	+	0.622705i	$0.786034\pi$
$174$	0	0
$175$	32.3606	2.44623
$176$	0	0
$177$	6.76386	0.508403
$178$	0	0
$179$	−16.9152	−1.26430	−0.632149	−	0.774847i	$-0.717827\pi$
$179$	−16.9152	−1.26430	−0.632149	+	0.774847i	$0.717827\pi$
$180$	0	0
$181$	14.2849	1.06179	0.530894	−	0.847438i	$-0.321856\pi$
$181$	14.2849	1.06179	0.530894	+	0.847438i	$0.321856\pi$
$182$	0	0
$183$	−4.05471	−0.299733
$184$	0	0
$185$	−9.89062	−0.727173
$186$	0	0
$187$	1.52787	0.111729
$188$	0	0
$189$	5.02591	0.365581
$190$	0	0
$191$	−17.8422	−1.29102	−0.645508	−	0.763753i	$-0.723354\pi$
$191$	−17.8422	−1.29102	−0.645508	+	0.763753i	$0.723354\pi$
$192$	0	0
$193$	−13.6783	−0.984588	−0.492294	−	0.870429i	$-0.663842\pi$
$193$	−13.6783	−0.984588	−0.492294	+	0.870429i	$0.663842\pi$
$194$	0	0
$195$	9.17569	0.657085
$196$	0	0
$197$	−9.71207	−0.691956	−0.345978	−	0.938243i	$-0.612453\pi$
$197$	−9.71207	−0.691956	−0.345978	+	0.938243i	$0.612453\pi$
$198$	0	0
$199$	−14.5081	−1.02845	−0.514226	−	0.857654i	$-0.671921\pi$
$199$	−14.5081	−1.02845	−0.514226	+	0.857654i	$0.671921\pi$
$200$	0	0
$201$	6.64598	0.468771
$202$	0	0
$203$	5.02591	0.352750
$204$	0	0
$205$	40.7358	2.84511
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−8.30803	−0.574679
$210$	0	0
$211$	−18.5664	−1.27817	−0.639083	−	0.769138i	$-0.720686\pi$
$211$	−18.5664	−1.27817	−0.639083	+	0.769138i	$0.720686\pi$
$212$	0	0
$213$	−9.94477	−0.681404
$214$	0	0
$215$	−32.2641	−2.20039
$216$	0	0
$217$	−40.0271	−2.71722
$218$	0	0
$219$	9.33118	0.630543
$220$	0	0
$221$	3.94490	0.265363
$222$	0	0
$223$	17.1142	1.14605	0.573026	−	0.819537i	$-0.305769\pi$
$223$	17.1142	1.14605	0.573026	+	0.819537i	$0.305769\pi$
$224$	0	0
$225$	6.43875	0.429250
$226$	0	0
$227$	17.1231	1.13650	0.568249	−	0.822857i	$-0.307621\pi$
$227$	17.1231	1.13650	0.568249	+	0.822857i	$0.307621\pi$
$228$	0	0
$229$	17.6674	1.16750	0.583748	−	0.811935i	$-0.301585\pi$
$229$	17.6674	1.16750	0.583748	+	0.811935i	$0.301585\pi$
$230$	0	0
$231$	5.28100	0.347464
$232$	0	0
$233$	−12.6978	−0.831859	−0.415930	−	0.909397i	$-0.636544\pi$
$233$	−12.6978	−0.831859	−0.415930	+	0.909397i	$0.636544\pi$
$234$	0	0
$235$	−32.7825	−2.13849
$236$	0	0
$237$	−5.74980	−0.373489
$238$	0	0
$239$	10.4522	0.676094	0.338047	−	0.941129i	$-0.390234\pi$
$239$	10.4522	0.676094	0.338047	+	0.941129i	$0.390234\pi$
$240$	0	0
$241$	−4.62672	−0.298033	−0.149017	−	0.988835i	$-0.547611\pi$
$241$	−4.62672	−0.298033	−0.149017	+	0.988835i	$0.547611\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	61.7568	3.94550
$246$	0	0
$247$	−21.4510	−1.36489
$248$	0	0
$249$	15.4273	0.977665
$250$	0	0
$251$	−12.3463	−0.779289	−0.389644	−	0.920965i	$-0.627402\pi$
$251$	−12.3463	−0.779289	−0.389644	+	0.920965i	$0.627402\pi$
$252$	0	0
$253$	−1.05075	−0.0660603
$254$	0	0
$255$	4.91785	0.307968
$256$	0	0
$257$	−1.77096	−0.110469	−0.0552346	−	0.998473i	$-0.517591\pi$
$257$	−1.77096	−0.110469	−0.0552346	+	0.998473i	$0.517591\pi$
$258$	0	0
$259$	−14.6977	−0.913270
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−15.9546	−0.983801	−0.491901	−	0.870651i	$-0.663698\pi$
$263$	−15.9546	−0.983801	−0.491901	+	0.870651i	$0.663698\pi$
$264$	0	0
$265$	19.9134	1.22327
$266$	0	0
$267$	0.992971	0.0607688
$268$	0	0
$269$	−13.6124	−0.829965	−0.414982	−	0.909830i	$-0.636212\pi$
$269$	−13.6124	−0.829965	−0.414982	+	0.909830i	$0.636212\pi$
$270$	0	0
$271$	0.724819	0.0440296	0.0220148	−	0.999758i	$-0.492992\pi$
$271$	0.724819	0.0440296	0.0220148	+	0.999758i	$0.492992\pi$
$272$	0	0
$273$	13.6353	0.825245
$274$	0	0
$275$	6.76554	0.407977
$276$	0	0
$277$	−24.2548	−1.45733	−0.728664	−	0.684871i	$-0.759858\pi$
$277$	−24.2548	−1.45733	−0.728664	+	0.684871i	$0.759858\pi$
$278$	0	0
$279$	−7.96415	−0.476801
$280$	0	0
$281$	−1.52351	−0.0908852	−0.0454426	−	0.998967i	$-0.514470\pi$
$281$	−1.52351	−0.0908852	−0.0454426	+	0.998967i	$0.514470\pi$
$282$	0	0
$283$	−15.5100	−0.921974	−0.460987	−	0.887407i	$-0.652505\pi$
$283$	−15.5100	−0.921974	−0.460987	+	0.887407i	$0.652505\pi$
$284$	0	0
$285$	−26.7415	−1.58403
$286$	0	0
$287$	60.5344	3.57323
$288$	0	0
$289$	−14.8857	−0.875628
$290$	0	0
$291$	11.5694	0.678209
$292$	0	0
$293$	−3.32352	−0.194162	−0.0970809	−	0.995276i	$-0.530951\pi$
$293$	−3.32352	−0.194162	−0.0970809	+	0.995276i	$0.530951\pi$
$294$	0	0
$295$	22.8762	1.33190
$296$	0	0
$297$	1.05075	0.0609709
$298$	0	0
$299$	−2.71300	−0.156897
$300$	0	0
$301$	−47.9452	−2.76352
$302$	0	0
$303$	−1.81461	−0.104247
$304$	0	0
$305$	−13.7135	−0.785234
$306$	0	0
$307$	18.0928	1.03261	0.516304	−	0.856405i	$-0.327307\pi$
$307$	18.0928	1.03261	0.516304	+	0.856405i	$0.327307\pi$
$308$	0	0
$309$	20.0792	1.14226
$310$	0	0
$311$	21.3838	1.21256	0.606281	−	0.795250i	$-0.292661\pi$
$311$	21.3838	1.21256	0.606281	+	0.795250i	$0.292661\pi$
$312$	0	0
$313$	−23.9736	−1.35507	−0.677535	−	0.735491i	$-0.736952\pi$
$313$	−23.9736	−1.35507	−0.677535	+	0.735491i	$0.736952\pi$
$314$	0	0
$315$	16.9982	0.957742
$316$	0	0
$317$	23.7744	1.33530	0.667651	−	0.744475i	$-0.267300\pi$
$317$	23.7744	1.33530	0.667651	+	0.744475i	$0.267300\pi$
$318$	0	0
$319$	1.05075	0.0588309
$320$	0	0
$321$	−13.9784	−0.780198
$322$	0	0
$323$	−11.4970	−0.639708
$324$	0	0
$325$	17.4683	0.968968
$326$	0	0
$327$	1.30547	0.0721924
$328$	0	0
$329$	−48.7155	−2.68577
$330$	0	0
$331$	−4.77219	−0.262304	−0.131152	−	0.991362i	$-0.541868\pi$
$331$	−4.77219	−0.262304	−0.131152	+	0.991362i	$0.541868\pi$
$332$	0	0
$333$	−2.92438	−0.160255
$334$	0	0
$335$	22.4775	1.22808
$336$	0	0
$337$	−22.1261	−1.20529	−0.602644	−	0.798010i	$-0.705886\pi$
$337$	−22.1261	−1.20529	−0.602644	+	0.798010i	$0.705886\pi$
$338$	0	0
$339$	−2.61994	−0.142296
$340$	0	0
$341$	−8.36836	−0.453172
$342$	0	0
$343$	56.5906	3.05561
$344$	0	0
$345$	−3.38212	−0.182087
$346$	0	0
$347$	−0.889149	−0.0477320	−0.0238660	−	0.999715i	$-0.507598\pi$
$347$	−0.889149	−0.0477320	−0.0238660	+	0.999715i	$0.507598\pi$
$348$	0	0
$349$	33.2622	1.78048	0.890242	−	0.455488i	$-0.150535\pi$
$349$	33.2622	1.78048	0.890242	+	0.455488i	$0.150535\pi$
$350$	0	0
$351$	2.71300	0.144809
$352$	0	0
$353$	−30.3127	−1.61338	−0.806691	−	0.590974i	$-0.798744\pi$
$353$	−30.3127	−1.61338	−0.806691	+	0.590974i	$0.798744\pi$
$354$	0	0
$355$	−33.6344	−1.78513
$356$	0	0
$357$	7.30804	0.386783
$358$	0	0
$359$	−26.9507	−1.42240	−0.711202	−	0.702988i	$-0.751849\pi$
$359$	−26.9507	−1.42240	−0.711202	+	0.702988i	$0.751849\pi$
$360$	0	0
$361$	43.5164	2.29034
$362$	0	0
$363$	−9.89592	−0.519401
$364$	0	0
$365$	31.5592	1.65188
$366$	0	0
$367$	−7.44023	−0.388377	−0.194188	−	0.980964i	$-0.562207\pi$
$367$	−7.44023	−0.388377	−0.194188	+	0.980964i	$0.562207\pi$
$368$	0	0
$369$	12.0445	0.627009
$370$	0	0
$371$	29.5918	1.53633
$372$	0	0
$373$	12.5143	0.647968	0.323984	−	0.946062i	$-0.394978\pi$
$373$	12.5143	0.647968	0.323984	+	0.946062i	$0.394978\pi$
$374$	0	0
$375$	4.86602	0.251280
$376$	0	0
$377$	2.71300	0.139727
$378$	0	0
$379$	−22.1663	−1.13861	−0.569304	−	0.822127i	$-0.692787\pi$
$379$	−22.1663	−1.13861	−0.569304	+	0.822127i	$0.692787\pi$
$380$	0	0
$381$	2.05300	0.105178
$382$	0	0
$383$	−5.40983	−0.276429	−0.138215	−	0.990402i	$-0.544136\pi$
$383$	−5.40983	−0.276429	−0.138215	+	0.990402i	$0.544136\pi$
$384$	0	0
$385$	17.8610	0.910279
$386$	0	0
$387$	−9.53961	−0.484926
$388$	0	0
$389$	−37.8091	−1.91700	−0.958499	−	0.285096i	$-0.907975\pi$
$389$	−37.8091	−1.91700	−0.958499	+	0.285096i	$0.907975\pi$
$390$	0	0
$391$	−1.45407	−0.0735356
$392$	0	0
$393$	9.76578	0.492618
$394$	0	0
$395$	−19.4465	−0.978460
$396$	0	0
$397$	6.00182	0.301223	0.150611	−	0.988593i	$-0.451876\pi$
$397$	6.00182	0.301223	0.150611	+	0.988593i	$0.451876\pi$
$398$	0	0
$399$	−39.7385	−1.98942
$400$	0	0
$401$	−16.6339	−0.830656	−0.415328	−	0.909672i	$-0.636333\pi$
$401$	−16.6339	−0.830656	−0.415328	+	0.909672i	$0.636333\pi$
$402$	0	0
$403$	−21.6067	−1.07631
$404$	0	0
$405$	3.38212	0.168059
$406$	0	0
$407$	−3.07281	−0.152313
$408$	0	0
$409$	22.4151	1.10836	0.554178	−	0.832398i	$-0.313033\pi$
$409$	22.4151	1.10836	0.554178	+	0.832398i	$0.313033\pi$
$410$	0	0
$411$	7.34825	0.362463
$412$	0	0
$413$	33.9946	1.67276
$414$	0	0
$415$	52.1770	2.56127
$416$	0	0
$417$	13.5213	0.662141
$418$	0	0
$419$	29.2002	1.42652	0.713261	−	0.700899i	$-0.247218\pi$
$419$	29.2002	1.42652	0.713261	+	0.700899i	$0.247218\pi$
$420$	0	0
$421$	−24.8564	−1.21143	−0.605713	−	0.795683i	$-0.707112\pi$
$421$	−24.8564	−1.21143	−0.605713	+	0.795683i	$0.707112\pi$
$422$	0	0
$423$	−9.69287	−0.471283
$424$	0	0
$425$	9.36241	0.454143
$426$	0	0
$427$	−20.3786	−0.986190
$428$	0	0
$429$	2.85069	0.137633
$430$	0	0
$431$	−23.2580	−1.12030	−0.560148	−	0.828392i	$-0.689256\pi$
$431$	−23.2580	−1.12030	−0.560148	+	0.828392i	$0.689256\pi$
$432$	0	0
$433$	31.2676	1.50263	0.751313	−	0.659946i	$-0.229421\pi$
$433$	31.2676	1.50263	0.751313	+	0.659946i	$0.229421\pi$
$434$	0	0
$435$	3.38212	0.162160
$436$	0	0
$437$	7.90673	0.378230
$438$	0	0
$439$	−36.7963	−1.75619	−0.878095	−	0.478486i	$-0.841186\pi$
$439$	−36.7963	−1.75619	−0.878095	+	0.478486i	$0.841186\pi$
$440$	0	0
$441$	18.2598	0.869513
$442$	0	0
$443$	−40.6240	−1.93010	−0.965052	−	0.262060i	$-0.915598\pi$
$443$	−40.6240	−1.93010	−0.965052	+	0.262060i	$0.915598\pi$
$444$	0	0
$445$	3.35835	0.159201
$446$	0	0
$447$	7.90733	0.374004
$448$	0	0
$449$	−2.19308	−0.103498	−0.0517489	−	0.998660i	$-0.516480\pi$
$449$	−2.19308	−0.103498	−0.0517489	+	0.998660i	$0.516480\pi$
$450$	0	0
$451$	12.6558	0.595936
$452$	0	0
$453$	11.2241	0.527354
$454$	0	0
$455$	46.1162	2.16196
$456$	0	0
$457$	−28.9154	−1.35261	−0.676303	−	0.736623i	$-0.736419\pi$
$457$	−28.9154	−1.35261	−0.676303	+	0.736623i	$0.736419\pi$
$458$	0	0
$459$	1.45407	0.0678703
$460$	0	0
$461$	−21.3089	−0.992456	−0.496228	−	0.868192i	$-0.665282\pi$
$461$	−21.3089	−0.992456	−0.496228	+	0.868192i	$0.665282\pi$
$462$	0	0
$463$	−15.1075	−0.702107	−0.351053	−	0.936355i	$-0.614176\pi$
$463$	−15.1075	−0.702107	−0.351053	+	0.936355i	$0.614176\pi$
$464$	0	0
$465$	−26.9357	−1.24911
$466$	0	0
$467$	29.0453	1.34406	0.672029	−	0.740525i	$-0.265423\pi$
$467$	29.0453	1.34406	0.672029	+	0.740525i	$0.265423\pi$
$468$	0	0
$469$	33.4021	1.54237
$470$	0	0
$471$	6.37096	0.293559
$472$	0	0
$473$	−10.0238	−0.460894
$474$	0	0
$475$	−50.9095	−2.33589
$476$	0	0
$477$	5.88785	0.269586
$478$	0	0
$479$	6.69532	0.305917	0.152959	−	0.988233i	$-0.451120\pi$
$479$	6.69532	0.305917	0.152959	+	0.988233i	$0.451120\pi$
$480$	0	0
$481$	−7.93385	−0.361752
$482$	0	0
$483$	−5.02591	−0.228687
$484$	0	0
$485$	39.1290	1.77676
$486$	0	0
$487$	2.29189	0.103856	0.0519278	−	0.998651i	$-0.483463\pi$
$487$	2.29189	0.103856	0.0519278	+	0.998651i	$0.483463\pi$
$488$	0	0
$489$	6.94106	0.313885
$490$	0	0
$491$	30.6252	1.38210	0.691048	−	0.722809i	$-0.257150\pi$
$491$	30.6252	1.38210	0.691048	+	0.722809i	$0.257150\pi$
$492$	0	0
$493$	1.45407	0.0654882
$494$	0	0
$495$	3.55378	0.159730
$496$	0	0
$497$	−49.9815	−2.24198
$498$	0	0
$499$	17.0316	0.762440	0.381220	−	0.924484i	$-0.375504\pi$
$499$	17.0316	0.762440	0.381220	+	0.924484i	$0.375504\pi$
$500$	0	0
$501$	−11.5112	−0.514281
$502$	0	0
$503$	−18.3824	−0.819630	−0.409815	−	0.912169i	$-0.634407\pi$
$503$	−18.3824	−0.819630	−0.409815	+	0.912169i	$0.634407\pi$
$504$	0	0
$505$	−6.13723	−0.273103
$506$	0	0
$507$	−5.63963	−0.250465
$508$	0	0
$509$	−30.8150	−1.36585	−0.682925	−	0.730488i	$-0.739293\pi$
$509$	−30.8150	−1.36585	−0.682925	+	0.730488i	$0.739293\pi$
$510$	0	0
$511$	46.8977	2.07463
$512$	0	0
$513$	−7.90673	−0.349091
$514$	0	0
$515$	67.9102	2.99248
$516$	0	0
$517$	−10.1848	−0.447928
$518$	0	0
$519$	−20.5832	−0.903503
$520$	0	0
$521$	3.79672	0.166337	0.0831686	−	0.996535i	$-0.473496\pi$
$521$	3.79672	0.166337	0.0831686	+	0.996535i	$0.473496\pi$
$522$	0	0
$523$	25.2036	1.10208	0.551038	−	0.834480i	$-0.314232\pi$
$523$	25.2036	1.10208	0.551038	+	0.834480i	$0.314232\pi$
$524$	0	0
$525$	32.3606	1.41233
$526$	0	0
$527$	−11.5805	−0.504453
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.76386	0.293527
$532$	0	0
$533$	32.6766	1.41538
$534$	0	0
$535$	−47.2766	−2.04395
$536$	0	0
$537$	−16.9152	−0.729943
$538$	0	0
$539$	19.1865	0.826422
$540$	0	0
$541$	30.1192	1.29493	0.647463	−	0.762097i	$-0.275830\pi$
$541$	30.1192	1.29493	0.647463	+	0.762097i	$0.275830\pi$
$542$	0	0
$543$	14.2849	0.613023
$544$	0	0
$545$	4.41524	0.189128
$546$	0	0
$547$	−29.0510	−1.24213	−0.621066	−	0.783758i	$-0.713300\pi$
$547$	−29.0510	−1.24213	−0.621066	+	0.783758i	$0.713300\pi$
$548$	0	0
$549$	−4.05471	−0.173051
$550$	0	0
$551$	−7.90673	−0.336838
$552$	0	0
$553$	−28.8980	−1.22887
$554$	0	0
$555$	−9.89062	−0.419833
$556$	0	0
$557$	26.5897	1.12664	0.563322	−	0.826238i	$-0.309523\pi$
$557$	26.5897	1.12664	0.563322	+	0.826238i	$0.309523\pi$
$558$	0	0
$559$	−25.8810	−1.09465
$560$	0	0
$561$	1.52787	0.0645068
$562$	0	0
$563$	−23.1711	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$563$	−23.1711	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$564$	0	0
$565$	−8.86096	−0.372783
$566$	0	0
$567$	5.02591	0.211068
$568$	0	0
$569$	−11.0508	−0.463276	−0.231638	−	0.972802i	$-0.574408\pi$
$569$	−11.0508	−0.463276	−0.231638	+	0.972802i	$0.574408\pi$
$570$	0	0
$571$	−24.0413	−1.00610	−0.503049	−	0.864258i	$-0.667789\pi$
$571$	−24.0413	−1.00610	−0.503049	+	0.864258i	$0.667789\pi$
$572$	0	0
$573$	−17.8422	−0.745369
$574$	0	0
$575$	−6.43875	−0.268514
$576$	0	0
$577$	30.4232	1.26654	0.633268	−	0.773933i	$-0.281713\pi$
$577$	30.4232	1.26654	0.633268	+	0.773933i	$0.281713\pi$
$578$	0	0
$579$	−13.6783	−0.568452
$580$	0	0
$581$	77.5362	3.21674
$582$	0	0
$583$	6.18668	0.256226
$584$	0	0
$585$	9.17569	0.379368
$586$	0	0
$587$	35.0879	1.44823	0.724117	−	0.689677i	$-0.242247\pi$
$587$	35.0879	1.44823	0.724117	+	0.689677i	$0.242247\pi$
$588$	0	0
$589$	62.9704	2.59465
$590$	0	0
$591$	−9.71207	−0.399501
$592$	0	0
$593$	16.7693	0.688633	0.344317	−	0.938854i	$-0.388111\pi$
$593$	16.7693	0.688633	0.344317	+	0.938854i	$0.388111\pi$
$594$	0	0
$595$	24.7167	1.01329
$596$	0	0
$597$	−14.5081	−0.593778
$598$	0	0
$599$	−22.3202	−0.911980	−0.455990	−	0.889985i	$-0.650715\pi$
$599$	−22.3202	−0.911980	−0.455990	+	0.889985i	$0.650715\pi$
$600$	0	0
$601$	5.78084	0.235805	0.117903	−	0.993025i	$-0.462383\pi$
$601$	5.78084	0.235805	0.117903	+	0.993025i	$0.462383\pi$
$602$	0	0
$603$	6.64598	0.270645
$604$	0	0
$605$	−33.4692	−1.36072
$606$	0	0
$607$	9.75675	0.396014	0.198007	−	0.980201i	$-0.436553\pi$
$607$	9.75675	0.396014	0.198007	+	0.980201i	$0.436553\pi$
$608$	0	0
$609$	5.02591	0.203660
$610$	0	0
$611$	−26.2968	−1.06385
$612$	0	0
$613$	10.5200	0.424898	0.212449	−	0.977172i	$-0.431856\pi$
$613$	10.5200	0.424898	0.212449	+	0.977172i	$0.431856\pi$
$614$	0	0
$615$	40.7358	1.64263
$616$	0	0
$617$	35.9684	1.44803	0.724017	−	0.689782i	$-0.242294\pi$
$617$	35.9684	1.44803	0.724017	+	0.689782i	$0.242294\pi$
$618$	0	0
$619$	−36.3287	−1.46017	−0.730087	−	0.683355i	$-0.760520\pi$
$619$	−36.3287	−1.46017	−0.730087	+	0.683355i	$0.760520\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	4.99058	0.199944
$624$	0	0
$625$	−15.7363	−0.629451
$626$	0	0
$627$	−8.30803	−0.331791
$628$	0	0
$629$	−4.25227	−0.169549
$630$	0	0
$631$	10.4902	0.417609	0.208805	−	0.977957i	$-0.433043\pi$
$631$	10.4902	0.417609	0.208805	+	0.977957i	$0.433043\pi$
$632$	0	0
$633$	−18.5664	−0.737949
$634$	0	0
$635$	6.94350	0.275545
$636$	0	0
$637$	49.5388	1.96280
$638$	0	0
$639$	−9.94477	−0.393409
$640$	0	0
$641$	−19.3910	−0.765897	−0.382949	−	0.923770i	$-0.625091\pi$
$641$	−19.3910	−0.765897	−0.382949	+	0.923770i	$0.625091\pi$
$642$	0	0
$643$	12.6644	0.499436	0.249718	−	0.968319i	$-0.419662\pi$
$643$	12.6644	0.499436	0.249718	+	0.968319i	$0.419662\pi$
$644$	0	0
$645$	−32.2641	−1.27040
$646$	0	0
$647$	−50.7194	−1.99399	−0.996993	−	0.0774914i	$-0.975309\pi$
$647$	−50.7194	−1.99399	−0.996993	+	0.0774914i	$0.975309\pi$
$648$	0	0
$649$	7.10716	0.278980
$650$	0	0
$651$	−40.0271	−1.56879
$652$	0	0
$653$	−10.5332	−0.412196	−0.206098	−	0.978531i	$-0.566077\pi$
$653$	−10.5332	−0.412196	−0.206098	+	0.978531i	$0.566077\pi$
$654$	0	0
$655$	33.0290	1.29055
$656$	0	0
$657$	9.33118	0.364044
$658$	0	0
$659$	−11.7617	−0.458171	−0.229085	−	0.973406i	$-0.573574\pi$
$659$	−11.7617	−0.458171	−0.229085	+	0.973406i	$0.573574\pi$
$660$	0	0
$661$	7.27828	0.283092	0.141546	−	0.989932i	$-0.454793\pi$
$661$	7.27828	0.283092	0.141546	+	0.989932i	$0.454793\pi$
$662$	0	0
$663$	3.94490	0.153207
$664$	0	0
$665$	−134.401	−5.21183
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	17.1142	0.661673
$670$	0	0
$671$	−4.26050	−0.164475
$672$	0	0
$673$	7.53032	0.290273	0.145136	−	0.989412i	$-0.453638\pi$
$673$	7.53032	0.290273	0.145136	+	0.989412i	$0.453638\pi$
$674$	0	0
$675$	6.43875	0.247827
$676$	0	0
$677$	4.71281	0.181128	0.0905640	−	0.995891i	$-0.471133\pi$
$677$	4.71281	0.181128	0.0905640	+	0.995891i	$0.471133\pi$
$678$	0	0
$679$	58.1466	2.23146
$680$	0	0
$681$	17.1231	0.656157
$682$	0	0
$683$	−19.8186	−0.758339	−0.379169	−	0.925327i	$-0.623790\pi$
$683$	−19.8186	−0.758339	−0.379169	+	0.925327i	$0.623790\pi$
$684$	0	0
$685$	24.8527	0.949572
$686$	0	0
$687$	17.6674	0.674055
$688$	0	0
$689$	15.9737	0.608551
$690$	0	0
$691$	0.429088	0.0163233	0.00816163	−	0.999967i	$-0.497402\pi$
$691$	0.429088	0.0163233	0.00816163	+	0.999967i	$0.497402\pi$
$692$	0	0
$693$	5.28100	0.200608
$694$	0	0
$695$	45.7307	1.73466
$696$	0	0
$697$	17.5135	0.663372
$698$	0	0
$699$	−12.6978	−0.480274
$700$	0	0
$701$	5.47129	0.206648	0.103324	−	0.994648i	$-0.467052\pi$
$701$	5.47129	0.206648	0.103324	+	0.994648i	$0.467052\pi$
$702$	0	0
$703$	23.1223	0.872074
$704$	0	0
$705$	−32.7825	−1.23466
$706$	0	0
$707$	−9.12007	−0.342995
$708$	0	0
$709$	−43.6496	−1.63929	−0.819647	−	0.572869i	$-0.805830\pi$
$709$	−43.6496	−1.63929	−0.819647	+	0.572869i	$0.805830\pi$
$710$	0	0
$711$	−5.74980	−0.215634
$712$	0	0
$713$	7.96415	0.298260
$714$	0	0
$715$	9.64140	0.360568
$716$	0	0
$717$	10.4522	0.390343
$718$	0	0
$719$	−33.0264	−1.23168	−0.615839	−	0.787872i	$-0.711183\pi$
$719$	−33.0264	−1.23168	−0.615839	+	0.787872i	$0.711183\pi$
$720$	0	0
$721$	100.916	3.75831
$722$	0	0
$723$	−4.62672	−0.172070
$724$	0	0
$725$	6.43875	0.239129
$726$	0	0
$727$	−7.25956	−0.269242	−0.134621	−	0.990897i	$-0.542982\pi$
$727$	−7.25956	−0.269242	−0.134621	+	0.990897i	$0.542982\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−13.8713	−0.513048
$732$	0	0
$733$	−32.7402	−1.20929	−0.604644	−	0.796496i	$-0.706684\pi$
$733$	−32.7402	−1.20929	−0.604644	+	0.796496i	$0.706684\pi$
$734$	0	0
$735$	61.7568	2.27793
$736$	0	0
$737$	6.98329	0.257233
$738$	0	0
$739$	13.2172	0.486202	0.243101	−	0.970001i	$-0.421835\pi$
$739$	13.2172	0.486202	0.243101	+	0.970001i	$0.421835\pi$
$740$	0	0
$741$	−21.4510	−0.788021
$742$	0	0
$743$	41.0280	1.50517	0.752585	−	0.658495i	$-0.228806\pi$
$743$	41.0280	1.50517	0.752585	+	0.658495i	$0.228806\pi$
$744$	0	0
$745$	26.7435	0.979808
$746$	0	0
$747$	15.4273	0.564455
$748$	0	0
$749$	−70.2541	−2.56703
$750$	0	0
$751$	7.58003	0.276599	0.138300	−	0.990390i	$-0.455836\pi$
$751$	7.58003	0.276599	0.138300	+	0.990390i	$0.455836\pi$
$752$	0	0
$753$	−12.3463	−0.449923
$754$	0	0
$755$	37.9613	1.38155
$756$	0	0
$757$	49.0922	1.78429	0.892144	−	0.451752i	$-0.149201\pi$
$757$	49.0922	1.78429	0.892144	+	0.451752i	$0.149201\pi$
$758$	0	0
$759$	−1.05075	−0.0381399
$760$	0	0
$761$	41.0563	1.48829	0.744144	−	0.668019i	$-0.232858\pi$
$761$	41.0563	1.48829	0.744144	+	0.668019i	$0.232858\pi$
$762$	0	0
$763$	6.56115	0.237530
$764$	0	0
$765$	4.91785	0.177805
$766$	0	0
$767$	18.3504	0.662593
$768$	0	0
$769$	5.87561	0.211880	0.105940	−	0.994373i	$-0.466215\pi$
$769$	5.87561	0.211880	0.105940	+	0.994373i	$0.466215\pi$
$770$	0	0
$771$	−1.77096	−0.0637794
$772$	0	0
$773$	12.5710	0.452147	0.226073	−	0.974110i	$-0.427411\pi$
$773$	12.5710	0.452147	0.226073	+	0.974110i	$0.427411\pi$
$774$	0	0
$775$	−51.2792	−1.84200
$776$	0	0
$777$	−14.6977	−0.527276
$778$	0	0
$779$	−95.2323	−3.41205
$780$	0	0
$781$	−10.4495	−0.373913
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	21.5474	0.769059
$786$	0	0
$787$	14.1131	0.503077	0.251539	−	0.967847i	$-0.419063\pi$
$787$	14.1131	0.503077	0.251539	+	0.967847i	$0.419063\pi$
$788$	0	0
$789$	−15.9546	−0.567998
$790$	0	0
$791$	−13.1676	−0.468185
$792$	0	0
$793$	−11.0004	−0.390637
$794$	0	0
$795$	19.9134	0.706257
$796$	0	0
$797$	−31.6487	−1.12105	−0.560527	−	0.828136i	$-0.689401\pi$
$797$	−31.6487	−1.12105	−0.560527	+	0.828136i	$0.689401\pi$
$798$	0	0
$799$	−14.0941	−0.498615
$800$	0	0
$801$	0.992971	0.0350849
$802$	0	0
$803$	9.80477	0.346003
$804$	0	0
$805$	−16.9982	−0.599109
$806$	0	0
$807$	−13.6124	−0.479180
$808$	0	0
$809$	30.3471	1.06695	0.533474	−	0.845817i	$-0.320886\pi$
$809$	30.3471	1.06695	0.533474	+	0.845817i	$0.320886\pi$
$810$	0	0
$811$	18.2442	0.640641	0.320321	−	0.947309i	$-0.396209\pi$
$811$	18.2442	0.640641	0.320321	+	0.947309i	$0.396209\pi$
$812$	0	0
$813$	0.724819	0.0254205
$814$	0	0
$815$	23.4755	0.822311
$816$	0	0
$817$	75.4271	2.63886
$818$	0	0
$819$	13.6353	0.476456
$820$	0	0
$821$	−5.56376	−0.194177	−0.0970883	−	0.995276i	$-0.530953\pi$
$821$	−5.56376	−0.194177	−0.0970883	+	0.995276i	$0.530953\pi$
$822$	0	0
$823$	−18.7615	−0.653986	−0.326993	−	0.945027i	$-0.606035\pi$
$823$	−18.7615	−0.653986	−0.326993	+	0.945027i	$0.606035\pi$
$824$	0	0
$825$	6.76554	0.235546
$826$	0	0
$827$	54.5187	1.89580	0.947900	−	0.318567i	$-0.103202\pi$
$827$	54.5187	1.89580	0.947900	+	0.318567i	$0.103202\pi$
$828$	0	0
$829$	−51.5950	−1.79197	−0.895985	−	0.444084i	$-0.853529\pi$
$829$	−51.5950	−1.79197	−0.895985	+	0.444084i	$0.853529\pi$
$830$	0	0
$831$	−24.2548	−0.841389
$832$	0	0
$833$	26.5510	0.919939
$834$	0	0
$835$	−38.9322	−1.34730
$836$	0	0
$837$	−7.96415	−0.275281
$838$	0	0
$839$	−20.3909	−0.703971	−0.351986	−	0.936005i	$-0.614493\pi$
$839$	−20.3909	−0.703971	−0.351986	+	0.936005i	$0.614493\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−1.52351	−0.0524726
$844$	0	0
$845$	−19.0739	−0.656163
$846$	0	0
$847$	−49.7360	−1.70895
$848$	0	0
$849$	−15.5100	−0.532302
$850$	0	0
$851$	2.92438	0.100247
$852$	0	0
$853$	47.7052	1.63339	0.816697	−	0.577066i	$-0.195803\pi$
$853$	47.7052	1.63339	0.816697	+	0.577066i	$0.195803\pi$
$854$	0	0
$855$	−26.7415	−0.914541
$856$	0	0
$857$	−35.9637	−1.22850	−0.614249	−	0.789112i	$-0.710541\pi$
$857$	−35.9637	−1.22850	−0.614249	+	0.789112i	$0.710541\pi$
$858$	0	0
$859$	−21.0556	−0.718409	−0.359205	−	0.933259i	$-0.616952\pi$
$859$	−21.0556	−0.718409	−0.359205	+	0.933259i	$0.616952\pi$
$860$	0	0
$861$	60.5344	2.06301
$862$	0	0
$863$	4.49816	0.153119	0.0765596	−	0.997065i	$-0.475606\pi$
$863$	4.49816	0.153119	0.0765596	+	0.997065i	$0.475606\pi$
$864$	0	0
$865$	−69.6149	−2.36698
$866$	0	0
$867$	−14.8857	−0.505544
$868$	0	0
$869$	−6.04162	−0.204948
$870$	0	0
$871$	18.0305	0.610942
$872$	0	0
$873$	11.5694	0.391564
$874$	0	0
$875$	24.4562	0.826769
$876$	0	0
$877$	37.7445	1.27454	0.637271	−	0.770639i	$-0.280063\pi$
$877$	37.7445	1.27454	0.637271	+	0.770639i	$0.280063\pi$
$878$	0	0
$879$	−3.32352	−0.112099
$880$	0	0
$881$	0.384055	0.0129391	0.00646957	−	0.999979i	$-0.497941\pi$
$881$	0.384055	0.0129391	0.00646957	+	0.999979i	$0.497941\pi$
$882$	0	0
$883$	−22.4250	−0.754662	−0.377331	−	0.926078i	$-0.623158\pi$
$883$	−22.4250	−0.754662	−0.377331	+	0.926078i	$0.623158\pi$
$884$	0	0
$885$	22.8762	0.768975
$886$	0	0
$887$	2.33787	0.0784979	0.0392490	−	0.999229i	$-0.487503\pi$
$887$	2.33787	0.0784979	0.0392490	+	0.999229i	$0.487503\pi$
$888$	0	0
$889$	10.3182	0.346062
$890$	0	0
$891$	1.05075	0.0352016
$892$	0	0
$893$	76.6389	2.56462
$894$	0	0
$895$	−57.2091	−1.91229
$896$	0	0
$897$	−2.71300	−0.0905844
$898$	0	0
$899$	−7.96415	−0.265619
$900$	0	0
$901$	8.56137	0.285220
$902$	0	0
$903$	−47.9452	−1.59552
$904$	0	0
$905$	48.3132	1.60599
$906$	0	0
$907$	54.8940	1.82272	0.911362	−	0.411606i	$-0.135032\pi$
$907$	54.8940	1.82272	0.911362	+	0.411606i	$0.135032\pi$
$908$	0	0
$909$	−1.81461	−0.0601868
$910$	0	0
$911$	−8.97130	−0.297233	−0.148616	−	0.988895i	$-0.547482\pi$
$911$	−8.97130	−0.297233	−0.148616	+	0.988895i	$0.547482\pi$
$912$	0	0
$913$	16.2103	0.536482
$914$	0	0
$915$	−13.7135	−0.453355
$916$	0	0
$917$	49.0819	1.62083
$918$	0	0
$919$	−1.59782	−0.0527073	−0.0263537	−	0.999653i	$-0.508390\pi$
$919$	−1.59782	−0.0527073	−0.0263537	+	0.999653i	$0.508390\pi$
$920$	0	0
$921$	18.0928	0.596177
$922$	0	0
$923$	−26.9802	−0.888063
$924$	0	0
$925$	−18.8294	−0.619105
$926$	0	0
$927$	20.0792	0.659486
$928$	0	0
$929$	−32.3329	−1.06081	−0.530403	−	0.847746i	$-0.677959\pi$
$929$	−32.3329	−1.06081	−0.530403	+	0.847746i	$0.677959\pi$
$930$	0	0
$931$	−144.375	−4.73171
$932$	0	0
$933$	21.3838	0.700073
$934$	0	0
$935$	5.16745	0.168994
$936$	0	0
$937$	−13.3260	−0.435341	−0.217670	−	0.976022i	$-0.569846\pi$
$937$	−13.3260	−0.435341	−0.217670	+	0.976022i	$0.569846\pi$
$938$	0	0
$939$	−23.9736	−0.782350
$940$	0	0
$941$	−30.6053	−0.997706	−0.498853	−	0.866687i	$-0.666245\pi$
$941$	−30.6053	−0.997706	−0.498853	+	0.866687i	$0.666245\pi$
$942$	0	0
$943$	−12.0445	−0.392221
$944$	0	0
$945$	16.9982	0.552953
$946$	0	0
$947$	59.1780	1.92303	0.961514	−	0.274756i	$-0.0885970\pi$
$947$	59.1780	1.92303	0.961514	+	0.274756i	$0.0885970\pi$
$948$	0	0
$949$	25.3155	0.821775
$950$	0	0
$951$	23.7744	0.770937
$952$	0	0
$953$	32.9377	1.06696	0.533478	−	0.845814i	$-0.320885\pi$
$953$	32.9377	1.06696	0.533478	+	0.845814i	$0.320885\pi$
$954$	0	0
$955$	−60.3445	−1.95270
$956$	0	0
$957$	1.05075	0.0339660
$958$	0	0
$959$	36.9317	1.19259
$960$	0	0
$961$	32.4277	1.04606
$962$	0	0
$963$	−13.9784	−0.450447
$964$	0	0
$965$	−46.2618	−1.48922
$966$	0	0
$967$	32.2923	1.03845	0.519225	−	0.854638i	$-0.326221\pi$
$967$	32.2923	1.03845	0.519225	+	0.854638i	$0.326221\pi$
$968$	0	0
$969$	−11.4970	−0.369336
$970$	0	0
$971$	12.2316	0.392531	0.196266	−	0.980551i	$-0.437119\pi$
$971$	12.2316	0.392531	0.196266	+	0.980551i	$0.437119\pi$
$972$	0	0
$973$	67.9569	2.17860
$974$	0	0
$975$	17.4683	0.559434
$976$	0	0
$977$	15.1731	0.485431	0.242715	−	0.970098i	$-0.421962\pi$
$977$	15.1731	0.485431	0.242715	+	0.970098i	$0.421962\pi$
$978$	0	0
$979$	1.04337	0.0333462
$980$	0	0
$981$	1.30547	0.0416803
$982$	0	0
$983$	−12.4828	−0.398140	−0.199070	−	0.979985i	$-0.563792\pi$
$983$	−12.4828	−0.398140	−0.199070	+	0.979985i	$0.563792\pi$
$984$	0	0
$985$	−32.8474	−1.04661
$986$	0	0
$987$	−48.7155	−1.55063
$988$	0	0
$989$	9.53961	0.303342
$990$	0	0
$991$	15.4215	0.489881	0.244940	−	0.969538i	$-0.421232\pi$
$991$	15.4215	0.489881	0.244940	+	0.969538i	$0.421232\pi$
$992$	0	0
$993$	−4.77219	−0.151441
$994$	0	0
$995$	−49.0682	−1.55557
$996$	0	0
$997$	42.8801	1.35803	0.679013	−	0.734127i	$-0.262408\pi$
$997$	42.8801	1.35803	0.679013	+	0.734127i	$0.262408\pi$
$998$	0	0
$999$	−2.92438	−0.0925234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.15	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.15	✓	18	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.38212 q^{5} +5.02591 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.38212 q^{5} +5.02591 q^{7} +1.00000 q^{9} +1.05075 q^{11} +2.71300 q^{13} +3.38212 q^{15} +1.45407 q^{17} -7.90673 q^{19} +5.02591 q^{21} -1.00000 q^{23} +6.43875 q^{25} +1.00000 q^{27} +1.00000 q^{29} -7.96415 q^{31} +1.05075 q^{33} +16.9982 q^{35} -2.92438 q^{37} +2.71300 q^{39} +12.0445 q^{41} -9.53961 q^{43} +3.38212 q^{45} -9.69287 q^{47} +18.2598 q^{49} +1.45407 q^{51} +5.88785 q^{53} +3.55378 q^{55} -7.90673 q^{57} +6.76386 q^{59} -4.05471 q^{61} +5.02591 q^{63} +9.17569 q^{65} +6.64598 q^{67} -1.00000 q^{69} -9.94477 q^{71} +9.33118 q^{73} +6.43875 q^{75} +5.28100 q^{77} -5.74980 q^{79} +1.00000 q^{81} +15.4273 q^{83} +4.91785 q^{85} +1.00000 q^{87} +0.992971 q^{89} +13.6353 q^{91} -7.96415 q^{93} -26.7415 q^{95} +11.5694 q^{97} +1.05075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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