LMFDB - Embedded newform 8020.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75932 q^{3} +1.00000 q^{5} -2.48714 q^{7} +4.61387 q^{9} +O(q^{10})$	$q-2.75932 q^{3} +1.00000 q^{5} -2.48714 q^{7} +4.61387 q^{9} +4.48889 q^{11} -3.07033 q^{13} -2.75932 q^{15} +3.10097 q^{17} +6.02272 q^{19} +6.86283 q^{21} -5.87763 q^{23} +1.00000 q^{25} -4.45320 q^{27} -9.61937 q^{29} -0.263331 q^{31} -12.3863 q^{33} -2.48714 q^{35} -8.76289 q^{37} +8.47205 q^{39} +6.47501 q^{41} +7.94084 q^{43} +4.61387 q^{45} -10.1529 q^{47} -0.814126 q^{49} -8.55659 q^{51} +0.387219 q^{53} +4.48889 q^{55} -16.6186 q^{57} -7.01210 q^{59} +0.690678 q^{61} -11.4754 q^{63} -3.07033 q^{65} +5.87975 q^{67} +16.2183 q^{69} +11.0447 q^{71} +0.108386 q^{73} -2.75932 q^{75} -11.1645 q^{77} +12.8460 q^{79} -1.55380 q^{81} -0.0846490 q^{83} +3.10097 q^{85} +26.5430 q^{87} +4.79442 q^{89} +7.63635 q^{91} +0.726615 q^{93} +6.02272 q^{95} -16.4068 q^{97} +20.7112 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * 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
$3$	−2.75932	−1.59310	−0.796548	−	0.604575i	$-0.793343\pi$
$3$	−2.75932	−1.59310	−0.796548	+	0.604575i	$0.793343\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.48714	−0.940051	−0.470026	−	0.882653i	$-0.655755\pi$
$7$	−2.48714	−0.940051	−0.470026	+	0.882653i	$0.655755\pi$
$8$	0	0
$9$	4.61387	1.53796
$10$	0	0
$11$	4.48889	1.35345	0.676726	−	0.736235i	$-0.263398\pi$
$11$	4.48889	1.35345	0.676726	+	0.736235i	$0.263398\pi$
$12$	0	0
$13$	−3.07033	−0.851557	−0.425779	−	0.904827i	$-0.640000\pi$
$13$	−3.07033	−0.851557	−0.425779	+	0.904827i	$0.640000\pi$
$14$	0	0
$15$	−2.75932	−0.712455
$16$	0	0
$17$	3.10097	0.752096	0.376048	−	0.926600i	$-0.377283\pi$
$17$	3.10097	0.752096	0.376048	+	0.926600i	$0.377283\pi$
$18$	0	0
$19$	6.02272	1.38171	0.690853	−	0.722995i	$-0.257235\pi$
$19$	6.02272	1.38171	0.690853	+	0.722995i	$0.257235\pi$
$20$	0	0
$21$	6.86283	1.49759
$22$	0	0
$23$	−5.87763	−1.22557	−0.612786	−	0.790249i	$-0.709951\pi$
$23$	−5.87763	−1.22557	−0.612786	+	0.790249i	$0.709951\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.45320	−0.857018
$28$	0	0
$29$	−9.61937	−1.78627	−0.893136	−	0.449786i	$-0.851500\pi$
$29$	−9.61937	−1.78627	−0.893136	+	0.449786i	$0.851500\pi$
$30$	0	0
$31$	−0.263331	−0.0472956	−0.0236478	−	0.999720i	$-0.507528\pi$
$31$	−0.263331	−0.0472956	−0.0236478	+	0.999720i	$0.507528\pi$
$32$	0	0
$33$	−12.3863	−2.15618
$34$	0	0
$35$	−2.48714	−0.420404
$36$	0	0
$37$	−8.76289	−1.44061	−0.720305	−	0.693657i	$-0.755998\pi$
$37$	−8.76289	−1.44061	−0.720305	+	0.693657i	$0.755998\pi$
$38$	0	0
$39$	8.47205	1.35661
$40$	0	0
$41$	6.47501	1.01123	0.505613	−	0.862760i	$-0.331266\pi$
$41$	6.47501	1.01123	0.505613	+	0.862760i	$0.331266\pi$
$42$	0	0
$43$	7.94084	1.21097	0.605483	−	0.795858i	$-0.292980\pi$
$43$	7.94084	1.21097	0.605483	+	0.795858i	$0.292980\pi$
$44$	0	0
$45$	4.61387	0.687795
$46$	0	0
$47$	−10.1529	−1.48096	−0.740478	−	0.672081i	$-0.765401\pi$
$47$	−10.1529	−1.48096	−0.740478	+	0.672081i	$0.765401\pi$
$48$	0	0
$49$	−0.814126	−0.116304
$50$	0	0
$51$	−8.55659	−1.19816
$52$	0	0
$53$	0.387219	0.0531886	0.0265943	−	0.999646i	$-0.491534\pi$
$53$	0.387219	0.0531886	0.0265943	+	0.999646i	$0.491534\pi$
$54$	0	0
$55$	4.48889	0.605282
$56$	0	0
$57$	−16.6186	−2.20119
$58$	0	0
$59$	−7.01210	−0.912898	−0.456449	−	0.889750i	$-0.650879\pi$
$59$	−7.01210	−0.912898	−0.456449	+	0.889750i	$0.650879\pi$
$60$	0	0
$61$	0.690678	0.0884323	0.0442161	−	0.999022i	$-0.485921\pi$
$61$	0.690678	0.0884323	0.0442161	+	0.999022i	$0.485921\pi$
$62$	0	0
$63$	−11.4754	−1.44576
$64$	0	0
$65$	−3.07033	−0.380828
$66$	0	0
$67$	5.87975	0.718326	0.359163	−	0.933275i	$-0.383062\pi$
$67$	5.87975	0.718326	0.359163	+	0.933275i	$0.383062\pi$
$68$	0	0
$69$	16.2183	1.95245
$70$	0	0
$71$	11.0447	1.31076	0.655382	−	0.755297i	$-0.272508\pi$
$71$	11.0447	1.31076	0.655382	+	0.755297i	$0.272508\pi$
$72$	0	0
$73$	0.108386	0.0126856	0.00634281	−	0.999980i	$-0.497981\pi$
$73$	0.108386	0.0126856	0.00634281	+	0.999980i	$0.497981\pi$
$74$	0	0
$75$	−2.75932	−0.318619
$76$	0	0
$77$	−11.1645	−1.27231
$78$	0	0
$79$	12.8460	1.44529	0.722644	−	0.691221i	$-0.242927\pi$
$79$	12.8460	1.44529	0.722644	+	0.691221i	$0.242927\pi$
$80$	0	0
$81$	−1.55380	−0.172644
$82$	0	0
$83$	−0.0846490	−0.00929144	−0.00464572	−	0.999989i	$-0.501479\pi$
$83$	−0.0846490	−0.00929144	−0.00464572	+	0.999989i	$0.501479\pi$
$84$	0	0
$85$	3.10097	0.336348
$86$	0	0
$87$	26.5430	2.84571
$88$	0	0
$89$	4.79442	0.508207	0.254104	−	0.967177i	$-0.418220\pi$
$89$	4.79442	0.508207	0.254104	+	0.967177i	$0.418220\pi$
$90$	0	0
$91$	7.63635	0.800507
$92$	0	0
$93$	0.726615	0.0753465
$94$	0	0
$95$	6.02272	0.617918
$96$	0	0
$97$	−16.4068	−1.66586	−0.832930	−	0.553378i	$-0.813339\pi$
$97$	−16.4068	−1.66586	−0.832930	+	0.553378i	$0.813339\pi$
$98$	0	0
$99$	20.7112	2.08155
$100$	0	0
$101$	−3.23908	−0.322301	−0.161150	−	0.986930i	$-0.551520\pi$
$101$	−3.23908	−0.322301	−0.161150	+	0.986930i	$0.551520\pi$
$102$	0	0
$103$	8.68051	0.855316	0.427658	−	0.903941i	$-0.359339\pi$
$103$	8.68051	0.855316	0.427658	+	0.903941i	$0.359339\pi$
$104$	0	0
$105$	6.86283	0.669744
$106$	0	0
$107$	11.0053	1.06393	0.531963	−	0.846767i	$-0.321454\pi$
$107$	11.0053	1.06393	0.531963	+	0.846767i	$0.321454\pi$
$108$	0	0
$109$	−12.6491	−1.21156	−0.605782	−	0.795630i	$-0.707140\pi$
$109$	−12.6491	−1.21156	−0.605782	+	0.795630i	$0.707140\pi$
$110$	0	0
$111$	24.1797	2.29503
$112$	0	0
$113$	5.33271	0.501659	0.250830	−	0.968031i	$-0.419297\pi$
$113$	5.33271	0.501659	0.250830	+	0.968031i	$0.419297\pi$
$114$	0	0
$115$	−5.87763	−0.548092
$116$	0	0
$117$	−14.1661	−1.30966
$118$	0	0
$119$	−7.71256	−0.707009
$120$	0	0
$121$	9.15018	0.831834
$122$	0	0
$123$	−17.8666	−1.61098
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.06878	0.361046	0.180523	−	0.983571i	$-0.442221\pi$
$127$	4.06878	0.361046	0.180523	+	0.983571i	$0.442221\pi$
$128$	0	0
$129$	−21.9114	−1.92919
$130$	0	0
$131$	8.68296	0.758634	0.379317	−	0.925267i	$-0.376159\pi$
$131$	8.68296	0.758634	0.379317	+	0.925267i	$0.376159\pi$
$132$	0	0
$133$	−14.9794	−1.29887
$134$	0	0
$135$	−4.45320	−0.383270
$136$	0	0
$137$	−2.99710	−0.256060	−0.128030	−	0.991770i	$-0.540865\pi$
$137$	−2.99710	−0.256060	−0.128030	+	0.991770i	$0.540865\pi$
$138$	0	0
$139$	−13.0348	−1.10559	−0.552797	−	0.833316i	$-0.686440\pi$
$139$	−13.0348	−1.10559	−0.552797	+	0.833316i	$0.686440\pi$
$140$	0	0
$141$	28.0152	2.35930
$142$	0	0
$143$	−13.7824	−1.15254
$144$	0	0
$145$	−9.61937	−0.798845
$146$	0	0
$147$	2.24644	0.185283
$148$	0	0
$149$	0.956659	0.0783726	0.0391863	−	0.999232i	$-0.487523\pi$
$149$	0.956659	0.0783726	0.0391863	+	0.999232i	$0.487523\pi$
$150$	0	0
$151$	2.04813	0.166675	0.0833374	−	0.996521i	$-0.473442\pi$
$151$	2.04813	0.166675	0.0833374	+	0.996521i	$0.473442\pi$
$152$	0	0
$153$	14.3075	1.15669
$154$	0	0
$155$	−0.263331	−0.0211512
$156$	0	0
$157$	9.76661	0.779460	0.389730	−	0.920929i	$-0.372568\pi$
$157$	9.76661	0.779460	0.389730	+	0.920929i	$0.372568\pi$
$158$	0	0
$159$	−1.06846	−0.0847347
$160$	0	0
$161$	14.6185	1.15210
$162$	0	0
$163$	0.0864182	0.00676880	0.00338440	−	0.999994i	$-0.498923\pi$
$163$	0.0864182	0.00676880	0.00338440	+	0.999994i	$0.498923\pi$
$164$	0	0
$165$	−12.3863	−0.964274
$166$	0	0
$167$	3.56704	0.276026	0.138013	−	0.990430i	$-0.455928\pi$
$167$	3.56704	0.276026	0.138013	+	0.990430i	$0.455928\pi$
$168$	0	0
$169$	−3.57306	−0.274850
$170$	0	0
$171$	27.7881	2.12501
$172$	0	0
$173$	8.81074	0.669868	0.334934	−	0.942242i	$-0.391286\pi$
$173$	8.81074	0.669868	0.334934	+	0.942242i	$0.391286\pi$
$174$	0	0
$175$	−2.48714	−0.188010
$176$	0	0
$177$	19.3487	1.45433
$178$	0	0
$179$	−3.50295	−0.261823	−0.130911	−	0.991394i	$-0.541790\pi$
$179$	−3.50295	−0.261823	−0.130911	+	0.991394i	$0.541790\pi$
$180$	0	0
$181$	17.5077	1.30134	0.650670	−	0.759361i	$-0.274488\pi$
$181$	17.5077	1.30134	0.650670	+	0.759361i	$0.274488\pi$
$182$	0	0
$183$	−1.90580	−0.140881
$184$	0	0
$185$	−8.76289	−0.644260
$186$	0	0
$187$	13.9199	1.01793
$188$	0	0
$189$	11.0757	0.805641
$190$	0	0
$191$	−23.6930	−1.71437	−0.857183	−	0.515012i	$-0.827787\pi$
$191$	−23.6930	−1.71437	−0.857183	+	0.515012i	$0.827787\pi$
$192$	0	0
$193$	14.3133	1.03029	0.515146	−	0.857102i	$-0.327738\pi$
$193$	14.3133	1.03029	0.515146	+	0.857102i	$0.327738\pi$
$194$	0	0
$195$	8.47205	0.606696
$196$	0	0
$197$	−10.3882	−0.740131	−0.370065	−	0.929006i	$-0.620665\pi$
$197$	−10.3882	−0.740131	−0.370065	+	0.929006i	$0.620665\pi$
$198$	0	0
$199$	−10.6004	−0.751444	−0.375722	−	0.926732i	$-0.622605\pi$
$199$	−10.6004	−0.751444	−0.375722	+	0.926732i	$0.622605\pi$
$200$	0	0
$201$	−16.2241	−1.14436
$202$	0	0
$203$	23.9247	1.67919
$204$	0	0
$205$	6.47501	0.452234
$206$	0	0
$207$	−27.1186	−1.88488
$208$	0	0
$209$	27.0353	1.87007
$210$	0	0
$211$	8.71242	0.599788	0.299894	−	0.953973i	$-0.403049\pi$
$211$	8.71242	0.599788	0.299894	+	0.953973i	$0.403049\pi$
$212$	0	0
$213$	−30.4759	−2.08817
$214$	0	0
$215$	7.94084	0.541561
$216$	0	0
$217$	0.654941	0.0444603
$218$	0	0
$219$	−0.299072	−0.0202094
$220$	0	0
$221$	−9.52102	−0.640453
$222$	0	0
$223$	−23.1917	−1.55303	−0.776514	−	0.630100i	$-0.783014\pi$
$223$	−23.1917	−1.55303	−0.776514	+	0.630100i	$0.783014\pi$
$224$	0	0
$225$	4.61387	0.307591
$226$	0	0
$227$	−18.8219	−1.24925	−0.624627	−	0.780923i	$-0.714749\pi$
$227$	−18.8219	−1.24925	−0.624627	+	0.780923i	$0.714749\pi$
$228$	0	0
$229$	−20.6024	−1.36144	−0.680722	−	0.732542i	$-0.738334\pi$
$229$	−20.6024	−1.36144	−0.680722	+	0.732542i	$0.738334\pi$
$230$	0	0
$231$	30.8065	2.02692
$232$	0	0
$233$	−11.4114	−0.747586	−0.373793	−	0.927512i	$-0.621943\pi$
$233$	−11.4114	−0.747586	−0.373793	+	0.927512i	$0.621943\pi$
$234$	0	0
$235$	−10.1529	−0.662303
$236$	0	0
$237$	−35.4463	−2.30248
$238$	0	0
$239$	18.3786	1.18881	0.594406	−	0.804165i	$-0.297387\pi$
$239$	18.3786	1.18881	0.594406	+	0.804165i	$0.297387\pi$
$240$	0	0
$241$	−13.7705	−0.887035	−0.443517	−	0.896266i	$-0.646270\pi$
$241$	−13.7705	−0.887035	−0.443517	+	0.896266i	$0.646270\pi$
$242$	0	0
$243$	17.6470	1.13206
$244$	0	0
$245$	−0.814126	−0.0520126
$246$	0	0
$247$	−18.4917	−1.17660
$248$	0	0
$249$	0.233574	0.0148022
$250$	0	0
$251$	−22.0189	−1.38982	−0.694910	−	0.719097i	$-0.744556\pi$
$251$	−22.0189	−1.38982	−0.694910	+	0.719097i	$0.744556\pi$
$252$	0	0
$253$	−26.3841	−1.65875
$254$	0	0
$255$	−8.55659	−0.535834
$256$	0	0
$257$	−12.9423	−0.807318	−0.403659	−	0.914910i	$-0.632262\pi$
$257$	−12.9423	−0.807318	−0.403659	+	0.914910i	$0.632262\pi$
$258$	0	0
$259$	21.7945	1.35425
$260$	0	0
$261$	−44.3826	−2.74721
$262$	0	0
$263$	22.7053	1.40007	0.700035	−	0.714108i	$-0.253168\pi$
$263$	22.7053	1.40007	0.700035	+	0.714108i	$0.253168\pi$
$264$	0	0
$265$	0.387219	0.0237867
$266$	0	0
$267$	−13.2294	−0.809623
$268$	0	0
$269$	20.7637	1.26599	0.632994	−	0.774157i	$-0.281826\pi$
$269$	20.7637	1.26599	0.632994	+	0.774157i	$0.281826\pi$
$270$	0	0
$271$	28.4170	1.72621	0.863106	−	0.505022i	$-0.168516\pi$
$271$	28.4170	1.72621	0.863106	+	0.505022i	$0.168516\pi$
$272$	0	0
$273$	−21.0712	−1.27529
$274$	0	0
$275$	4.48889	0.270691
$276$	0	0
$277$	−25.5767	−1.53676	−0.768378	−	0.639996i	$-0.778936\pi$
$277$	−25.5767	−1.53676	−0.768378	+	0.639996i	$0.778936\pi$
$278$	0	0
$279$	−1.21498	−0.0727387
$280$	0	0
$281$	−17.9711	−1.07207	−0.536033	−	0.844197i	$-0.680078\pi$
$281$	−17.9711	−1.07207	−0.536033	+	0.844197i	$0.680078\pi$
$282$	0	0
$283$	−3.38332	−0.201117	−0.100559	−	0.994931i	$-0.532063\pi$
$283$	−3.38332	−0.201117	−0.100559	+	0.994931i	$0.532063\pi$
$284$	0	0
$285$	−16.6186	−0.984403
$286$	0	0
$287$	−16.1043	−0.950604
$288$	0	0
$289$	−7.38397	−0.434351
$290$	0	0
$291$	45.2718	2.65388
$292$	0	0
$293$	26.7553	1.56306	0.781532	−	0.623865i	$-0.214439\pi$
$293$	26.7553	1.56306	0.781532	+	0.623865i	$0.214439\pi$
$294$	0	0
$295$	−7.01210	−0.408260
$296$	0	0
$297$	−19.9899	−1.15993
$298$	0	0
$299$	18.0463	1.04364
$300$	0	0
$301$	−19.7500	−1.13837
$302$	0	0
$303$	8.93769	0.513457
$304$	0	0
$305$	0.690678	0.0395481
$306$	0	0
$307$	−7.62034	−0.434916	−0.217458	−	0.976070i	$-0.569777\pi$
$307$	−7.62034	−0.434916	−0.217458	+	0.976070i	$0.569777\pi$
$308$	0	0
$309$	−23.9524	−1.36260
$310$	0	0
$311$	−4.77251	−0.270624	−0.135312	−	0.990803i	$-0.543204\pi$
$311$	−4.77251	−0.270624	−0.135312	+	0.990803i	$0.543204\pi$
$312$	0	0
$313$	3.13058	0.176951	0.0884755	−	0.996078i	$-0.471801\pi$
$313$	3.13058	0.176951	0.0884755	+	0.996078i	$0.471801\pi$
$314$	0	0
$315$	−11.4754	−0.646563
$316$	0	0
$317$	5.34082	0.299970	0.149985	−	0.988688i	$-0.452077\pi$
$317$	5.34082	0.299970	0.149985	+	0.988688i	$0.452077\pi$
$318$	0	0
$319$	−43.1803	−2.41764
$320$	0	0
$321$	−30.3673	−1.69494
$322$	0	0
$323$	18.6763	1.03918
$324$	0	0
$325$	−3.07033	−0.170311
$326$	0	0
$327$	34.9030	1.93014
$328$	0	0
$329$	25.2517	1.39217
$330$	0	0
$331$	6.28556	0.345486	0.172743	−	0.984967i	$-0.444737\pi$
$331$	6.28556	0.345486	0.172743	+	0.984967i	$0.444737\pi$
$332$	0	0
$333$	−40.4309	−2.21560
$334$	0	0
$335$	5.87975	0.321245
$336$	0	0
$337$	−0.904843	−0.0492899	−0.0246450	−	0.999696i	$-0.507846\pi$
$337$	−0.904843	−0.0492899	−0.0246450	+	0.999696i	$0.507846\pi$
$338$	0	0
$339$	−14.7147	−0.799192
$340$	0	0
$341$	−1.18206	−0.0640124
$342$	0	0
$343$	19.4348	1.04938
$344$	0	0
$345$	16.2183	0.873164
$346$	0	0
$347$	−20.3646	−1.09323	−0.546615	−	0.837384i	$-0.684084\pi$
$347$	−20.3646	−1.09323	−0.546615	+	0.837384i	$0.684084\pi$
$348$	0	0
$349$	−8.30847	−0.444742	−0.222371	−	0.974962i	$-0.571380\pi$
$349$	−8.30847	−0.444742	−0.222371	+	0.974962i	$0.571380\pi$
$350$	0	0
$351$	13.6728	0.729800
$352$	0	0
$353$	20.7973	1.10693	0.553465	−	0.832873i	$-0.313305\pi$
$353$	20.7973	1.10693	0.553465	+	0.832873i	$0.313305\pi$
$354$	0	0
$355$	11.0447	0.586192
$356$	0	0
$357$	21.2814	1.12633
$358$	0	0
$359$	−28.1768	−1.48711	−0.743556	−	0.668673i	$-0.766862\pi$
$359$	−28.1768	−1.48711	−0.743556	+	0.668673i	$0.766862\pi$
$360$	0	0
$361$	17.2731	0.909112
$362$	0	0
$363$	−25.2483	−1.32519
$364$	0	0
$365$	0.108386	0.00567318
$366$	0	0
$367$	2.27838	0.118930	0.0594651	−	0.998230i	$-0.481060\pi$
$367$	2.27838	0.118930	0.0594651	+	0.998230i	$0.481060\pi$
$368$	0	0
$369$	29.8749	1.55522
$370$	0	0
$371$	−0.963069	−0.0500000
$372$	0	0
$373$	−22.8078	−1.18094	−0.590471	−	0.807059i	$-0.701058\pi$
$373$	−22.8078	−1.18094	−0.590471	+	0.807059i	$0.701058\pi$
$374$	0	0
$375$	−2.75932	−0.142491
$376$	0	0
$377$	29.5347	1.52111
$378$	0	0
$379$	−13.6606	−0.701700	−0.350850	−	0.936432i	$-0.614107\pi$
$379$	−13.6606	−0.701700	−0.350850	+	0.936432i	$0.614107\pi$
$380$	0	0
$381$	−11.2271	−0.575181
$382$	0	0
$383$	−5.66150	−0.289289	−0.144645	−	0.989484i	$-0.546204\pi$
$383$	−5.66150	−0.289289	−0.144645	+	0.989484i	$0.546204\pi$
$384$	0	0
$385$	−11.1645	−0.568996
$386$	0	0
$387$	36.6380	1.86242
$388$	0	0
$389$	−15.0971	−0.765454	−0.382727	−	0.923862i	$-0.625015\pi$
$389$	−15.0971	−0.765454	−0.382727	+	0.923862i	$0.625015\pi$
$390$	0	0
$391$	−18.2264	−0.921747
$392$	0	0
$393$	−23.9591	−1.20858
$394$	0	0
$395$	12.8460	0.646352
$396$	0	0
$397$	−3.32316	−0.166785	−0.0833923	−	0.996517i	$-0.526575\pi$
$397$	−3.32316	−0.166785	−0.0833923	+	0.996517i	$0.526575\pi$
$398$	0	0
$399$	41.3329	2.06923
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	0.808513	0.0402749
$404$	0	0
$405$	−1.55380	−0.0772089
$406$	0	0
$407$	−39.3357	−1.94980
$408$	0	0
$409$	−19.1552	−0.947163	−0.473581	−	0.880750i	$-0.657039\pi$
$409$	−19.1552	−0.947163	−0.473581	+	0.880750i	$0.657039\pi$
$410$	0	0
$411$	8.26997	0.407928
$412$	0	0
$413$	17.4401	0.858170
$414$	0	0
$415$	−0.0846490	−0.00415526
$416$	0	0
$417$	35.9671	1.76132
$418$	0	0
$419$	36.9539	1.80532	0.902659	−	0.430357i	$-0.141612\pi$
$419$	36.9539	1.80532	0.902659	+	0.430357i	$0.141612\pi$
$420$	0	0
$421$	−22.6312	−1.10298	−0.551489	−	0.834182i	$-0.685940\pi$
$421$	−22.6312	−1.10298	−0.551489	+	0.834182i	$0.685940\pi$
$422$	0	0
$423$	−46.8443	−2.27765
$424$	0	0
$425$	3.10097	0.150419
$426$	0	0
$427$	−1.71781	−0.0831309
$428$	0	0
$429$	38.0301	1.83611
$430$	0	0
$431$	26.2221	1.26308	0.631538	−	0.775345i	$-0.282424\pi$
$431$	26.2221	1.26308	0.631538	+	0.775345i	$0.282424\pi$
$432$	0	0
$433$	−3.08379	−0.148197	−0.0740987	−	0.997251i	$-0.523608\pi$
$433$	−3.08379	−0.148197	−0.0740987	+	0.997251i	$0.523608\pi$
$434$	0	0
$435$	26.5430	1.27264
$436$	0	0
$437$	−35.3993	−1.69338
$438$	0	0
$439$	−3.19198	−0.152345	−0.0761726	−	0.997095i	$-0.524270\pi$
$439$	−3.19198	−0.152345	−0.0761726	+	0.997095i	$0.524270\pi$
$440$	0	0
$441$	−3.75627	−0.178870
$442$	0	0
$443$	−6.33463	−0.300967	−0.150484	−	0.988612i	$-0.548083\pi$
$443$	−6.33463	−0.300967	−0.150484	+	0.988612i	$0.548083\pi$
$444$	0	0
$445$	4.79442	0.227277
$446$	0	0
$447$	−2.63973	−0.124855
$448$	0	0
$449$	11.9788	0.565314	0.282657	−	0.959221i	$-0.408784\pi$
$449$	11.9788	0.565314	0.282657	+	0.959221i	$0.408784\pi$
$450$	0	0
$451$	29.0656	1.36865
$452$	0	0
$453$	−5.65146	−0.265529
$454$	0	0
$455$	7.63635	0.357998
$456$	0	0
$457$	26.9280	1.25964	0.629820	−	0.776741i	$-0.283129\pi$
$457$	26.9280	1.25964	0.629820	+	0.776741i	$0.283129\pi$
$458$	0	0
$459$	−13.8092	−0.644560
$460$	0	0
$461$	−1.14245	−0.0532092	−0.0266046	−	0.999646i	$-0.508470\pi$
$461$	−1.14245	−0.0532092	−0.0266046	+	0.999646i	$0.508470\pi$
$462$	0	0
$463$	−31.6753	−1.47208	−0.736039	−	0.676939i	$-0.763306\pi$
$463$	−31.6753	−1.47208	−0.736039	+	0.676939i	$0.763306\pi$
$464$	0	0
$465$	0.726615	0.0336960
$466$	0	0
$467$	−7.54316	−0.349056	−0.174528	−	0.984652i	$-0.555840\pi$
$467$	−7.54316	−0.349056	−0.174528	+	0.984652i	$0.555840\pi$
$468$	0	0
$469$	−14.6238	−0.675263
$470$	0	0
$471$	−26.9493	−1.24176
$472$	0	0
$473$	35.6456	1.63899
$474$	0	0
$475$	6.02272	0.276341
$476$	0	0
$477$	1.78658	0.0818019
$478$	0	0
$479$	−23.0865	−1.05485	−0.527425	−	0.849602i	$-0.676842\pi$
$479$	−23.0865	−1.05485	−0.527425	+	0.849602i	$0.676842\pi$
$480$	0	0
$481$	26.9050	1.22676
$482$	0	0
$483$	−40.3372	−1.83541
$484$	0	0
$485$	−16.4068	−0.744996
$486$	0	0
$487$	−31.4338	−1.42440	−0.712200	−	0.701977i	$-0.752301\pi$
$487$	−31.4338	−1.42440	−0.712200	+	0.701977i	$0.752301\pi$
$488$	0	0
$489$	−0.238456	−0.0107834
$490$	0	0
$491$	1.27809	0.0576793	0.0288397	−	0.999584i	$-0.490819\pi$
$491$	1.27809	0.0576793	0.0288397	+	0.999584i	$0.490819\pi$
$492$	0	0
$493$	−29.8294	−1.34345
$494$	0	0
$495$	20.7112	0.930899
$496$	0	0
$497$	−27.4697	−1.23219
$498$	0	0
$499$	5.88233	0.263329	0.131665	−	0.991294i	$-0.457968\pi$
$499$	5.88233	0.263329	0.131665	+	0.991294i	$0.457968\pi$
$500$	0	0
$501$	−9.84263	−0.439736
$502$	0	0
$503$	−1.05449	−0.0470175	−0.0235088	−	0.999724i	$-0.507484\pi$
$503$	−1.05449	−0.0470175	−0.0235088	+	0.999724i	$0.507484\pi$
$504$	0	0
$505$	−3.23908	−0.144137
$506$	0	0
$507$	9.85922	0.437863
$508$	0	0
$509$	−25.9015	−1.14807	−0.574033	−	0.818832i	$-0.694622\pi$
$509$	−25.9015	−1.14807	−0.574033	+	0.818832i	$0.694622\pi$
$510$	0	0
$511$	−0.269571	−0.0119251
$512$	0	0
$513$	−26.8204	−1.18415
$514$	0	0
$515$	8.68051	0.382509
$516$	0	0
$517$	−45.5754	−2.00440
$518$	0	0
$519$	−24.3117	−1.06716
$520$	0	0
$521$	−26.0207	−1.13999	−0.569993	−	0.821649i	$-0.693054\pi$
$521$	−26.0207	−1.13999	−0.569993	+	0.821649i	$0.693054\pi$
$522$	0	0
$523$	1.89435	0.0828344	0.0414172	−	0.999142i	$-0.486813\pi$
$523$	1.89435	0.0828344	0.0414172	+	0.999142i	$0.486813\pi$
$524$	0	0
$525$	6.86283	0.299519
$526$	0	0
$527$	−0.816582	−0.0355709
$528$	0	0
$529$	11.5466	0.502024
$530$	0	0
$531$	−32.3529	−1.40400
$532$	0	0
$533$	−19.8804	−0.861117
$534$	0	0
$535$	11.0053	0.475803
$536$	0	0
$537$	9.66577	0.417109
$538$	0	0
$539$	−3.65453	−0.157412
$540$	0	0
$541$	1.73674	0.0746683	0.0373341	−	0.999303i	$-0.488113\pi$
$541$	1.73674	0.0746683	0.0373341	+	0.999303i	$0.488113\pi$
$542$	0	0
$543$	−48.3095	−2.07316
$544$	0	0
$545$	−12.6491	−0.541828
$546$	0	0
$547$	−25.9053	−1.10763	−0.553815	−	0.832640i	$-0.686828\pi$
$547$	−25.9053	−1.10763	−0.553815	+	0.832640i	$0.686828\pi$
$548$	0	0
$549$	3.18670	0.136005
$550$	0	0
$551$	−57.9348	−2.46810
$552$	0	0
$553$	−31.9498	−1.35864
$554$	0	0
$555$	24.1797	1.02637
$556$	0	0
$557$	5.15529	0.218437	0.109218	−	0.994018i	$-0.465165\pi$
$557$	5.15529	0.218437	0.109218	+	0.994018i	$0.465165\pi$
$558$	0	0
$559$	−24.3810	−1.03121
$560$	0	0
$561$	−38.4096	−1.62166
$562$	0	0
$563$	−42.9726	−1.81108	−0.905539	−	0.424263i	$-0.860533\pi$
$563$	−42.9726	−1.81108	−0.905539	+	0.424263i	$0.860533\pi$
$564$	0	0
$565$	5.33271	0.224349
$566$	0	0
$567$	3.86452	0.162294
$568$	0	0
$569$	−17.6747	−0.740964	−0.370482	−	0.928840i	$-0.620807\pi$
$569$	−17.6747	−0.740964	−0.370482	+	0.928840i	$0.620807\pi$
$570$	0	0
$571$	16.0366	0.671109	0.335555	−	0.942021i	$-0.391076\pi$
$571$	16.0366	0.671109	0.335555	+	0.942021i	$0.391076\pi$
$572$	0	0
$573$	65.3767	2.73115
$574$	0	0
$575$	−5.87763	−0.245114
$576$	0	0
$577$	−0.634795	−0.0264269	−0.0132134	−	0.999913i	$-0.504206\pi$
$577$	−0.634795	−0.0264269	−0.0132134	+	0.999913i	$0.504206\pi$
$578$	0	0
$579$	−39.4950	−1.64136
$580$	0	0
$581$	0.210534	0.00873443
$582$	0	0
$583$	1.73819	0.0719883
$584$	0	0
$585$	−14.1661	−0.585697
$586$	0	0
$587$	−8.24399	−0.340266	−0.170133	−	0.985421i	$-0.554420\pi$
$587$	−8.24399	−0.340266	−0.170133	+	0.985421i	$0.554420\pi$
$588$	0	0
$589$	−1.58597	−0.0653487
$590$	0	0
$591$	28.6645	1.17910
$592$	0	0
$593$	−10.7724	−0.442370	−0.221185	−	0.975232i	$-0.570992\pi$
$593$	−10.7724	−0.442370	−0.221185	+	0.975232i	$0.570992\pi$
$594$	0	0
$595$	−7.71256	−0.316184
$596$	0	0
$597$	29.2500	1.19712
$598$	0	0
$599$	7.39105	0.301990	0.150995	−	0.988535i	$-0.451752\pi$
$599$	7.39105	0.301990	0.150995	+	0.988535i	$0.451752\pi$
$600$	0	0
$601$	−37.3751	−1.52456	−0.762282	−	0.647245i	$-0.775921\pi$
$601$	−37.3751	−1.52456	−0.762282	+	0.647245i	$0.775921\pi$
$602$	0	0
$603$	27.1284	1.10475
$604$	0	0
$605$	9.15018	0.372008
$606$	0	0
$607$	−10.3841	−0.421479	−0.210740	−	0.977542i	$-0.567587\pi$
$607$	−10.3841	−0.421479	−0.210740	+	0.977542i	$0.567587\pi$
$608$	0	0
$609$	−66.0161	−2.67511
$610$	0	0
$611$	31.1728	1.26112
$612$	0	0
$613$	−12.9055	−0.521250	−0.260625	−	0.965440i	$-0.583929\pi$
$613$	−12.9055	−0.521250	−0.260625	+	0.965440i	$0.583929\pi$
$614$	0	0
$615$	−17.8666	−0.720453
$616$	0	0
$617$	−30.9708	−1.24684	−0.623420	−	0.781888i	$-0.714257\pi$
$617$	−30.9708	−1.24684	−0.623420	+	0.781888i	$0.714257\pi$
$618$	0	0
$619$	−25.8952	−1.04081	−0.520407	−	0.853918i	$-0.674220\pi$
$619$	−25.8952	−1.04081	−0.520407	+	0.853918i	$0.674220\pi$
$620$	0	0
$621$	26.1743	1.05034
$622$	0	0
$623$	−11.9244	−0.477741
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−74.5993	−2.97921
$628$	0	0
$629$	−27.1735	−1.08348
$630$	0	0
$631$	−30.8332	−1.22745	−0.613725	−	0.789520i	$-0.710330\pi$
$631$	−30.8332	−1.22745	−0.613725	+	0.789520i	$0.710330\pi$
$632$	0	0
$633$	−24.0404	−0.955520
$634$	0	0
$635$	4.06878	0.161465
$636$	0	0
$637$	2.49964	0.0990393
$638$	0	0
$639$	50.9588	2.01590
$640$	0	0
$641$	−18.8893	−0.746081	−0.373040	−	0.927815i	$-0.621685\pi$
$641$	−18.8893	−0.746081	−0.373040	+	0.927815i	$0.621685\pi$
$642$	0	0
$643$	34.6564	1.36671	0.683357	−	0.730084i	$-0.260519\pi$
$643$	34.6564	1.36671	0.683357	+	0.730084i	$0.260519\pi$
$644$	0	0
$645$	−21.9114	−0.862759
$646$	0	0
$647$	36.0994	1.41922	0.709608	−	0.704597i	$-0.248872\pi$
$647$	36.0994	1.41922	0.709608	+	0.704597i	$0.248872\pi$
$648$	0	0
$649$	−31.4766	−1.23556
$650$	0	0
$651$	−1.80720	−0.0708296
$652$	0	0
$653$	35.3342	1.38273	0.691367	−	0.722503i	$-0.257009\pi$
$653$	35.3342	1.38273	0.691367	+	0.722503i	$0.257009\pi$
$654$	0	0
$655$	8.68296	0.339271
$656$	0	0
$657$	0.500079	0.0195100
$658$	0	0
$659$	−9.89289	−0.385372	−0.192686	−	0.981260i	$-0.561720\pi$
$659$	−9.89289	−0.385372	−0.192686	+	0.981260i	$0.561720\pi$
$660$	0	0
$661$	15.1592	0.589626	0.294813	−	0.955555i	$-0.404743\pi$
$661$	15.1592	0.589626	0.294813	+	0.955555i	$0.404743\pi$
$662$	0	0
$663$	26.2716	1.02030
$664$	0	0
$665$	−14.9794	−0.580874
$666$	0	0
$667$	56.5391	2.18920
$668$	0	0
$669$	63.9933	2.47412
$670$	0	0
$671$	3.10038	0.119689
$672$	0	0
$673$	−2.66584	−0.102761	−0.0513803	−	0.998679i	$-0.516362\pi$
$673$	−2.66584	−0.102761	−0.0513803	+	0.998679i	$0.516362\pi$
$674$	0	0
$675$	−4.45320	−0.171404
$676$	0	0
$677$	−7.51697	−0.288900	−0.144450	−	0.989512i	$-0.546141\pi$
$677$	−7.51697	−0.288900	−0.144450	+	0.989512i	$0.546141\pi$
$678$	0	0
$679$	40.8061	1.56599
$680$	0	0
$681$	51.9358	1.99018
$682$	0	0
$683$	−31.6845	−1.21238	−0.606188	−	0.795322i	$-0.707302\pi$
$683$	−31.6845	−1.21238	−0.606188	+	0.795322i	$0.707302\pi$
$684$	0	0
$685$	−2.99710	−0.114513
$686$	0	0
$687$	56.8487	2.16891
$688$	0	0
$689$	−1.18889	−0.0452932
$690$	0	0
$691$	21.2219	0.807321	0.403660	−	0.914909i	$-0.367738\pi$
$691$	21.2219	0.807321	0.403660	+	0.914909i	$0.367738\pi$
$692$	0	0
$693$	−51.5117	−1.95677
$694$	0	0
$695$	−13.0348	−0.494437
$696$	0	0
$697$	20.0788	0.760539
$698$	0	0
$699$	31.4878	1.19098
$700$	0	0
$701$	−37.6364	−1.42151	−0.710753	−	0.703442i	$-0.751646\pi$
$701$	−37.6364	−1.42151	−0.710753	+	0.703442i	$0.751646\pi$
$702$	0	0
$703$	−52.7764	−1.99050
$704$	0	0
$705$	28.0152	1.05511
$706$	0	0
$707$	8.05606	0.302979
$708$	0	0
$709$	−20.1777	−0.757790	−0.378895	−	0.925440i	$-0.623696\pi$
$709$	−20.1777	−0.757790	−0.378895	+	0.925440i	$0.623696\pi$
$710$	0	0
$711$	59.2698	2.22279
$712$	0	0
$713$	1.54776	0.0579641
$714$	0	0
$715$	−13.7824	−0.515433
$716$	0	0
$717$	−50.7125	−1.89389
$718$	0	0
$719$	−14.5142	−0.541289	−0.270644	−	0.962679i	$-0.587237\pi$
$719$	−14.5142	−0.541289	−0.270644	+	0.962679i	$0.587237\pi$
$720$	0	0
$721$	−21.5897	−0.804041
$722$	0	0
$723$	37.9972	1.41313
$724$	0	0
$725$	−9.61937	−0.357254
$726$	0	0
$727$	−16.6536	−0.617648	−0.308824	−	0.951119i	$-0.599935\pi$
$727$	−16.6536	−0.617648	−0.308824	+	0.951119i	$0.599935\pi$
$728$	0	0
$729$	−44.0325	−1.63083
$730$	0	0
$731$	24.6243	0.910763
$732$	0	0
$733$	24.3367	0.898895	0.449448	−	0.893307i	$-0.351621\pi$
$733$	24.3367	0.898895	0.449448	+	0.893307i	$0.351621\pi$
$734$	0	0
$735$	2.24644	0.0828611
$736$	0	0
$737$	26.3936	0.972220
$738$	0	0
$739$	22.5170	0.828303	0.414152	−	0.910208i	$-0.364078\pi$
$739$	22.5170	0.828303	0.414152	+	0.910208i	$0.364078\pi$
$740$	0	0
$741$	51.0247	1.87444
$742$	0	0
$743$	43.7566	1.60527	0.802636	−	0.596469i	$-0.203430\pi$
$743$	43.7566	1.60527	0.802636	+	0.596469i	$0.203430\pi$
$744$	0	0
$745$	0.956659	0.0350493
$746$	0	0
$747$	−0.390560	−0.0142898
$748$	0	0
$749$	−27.3719	−1.00015
$750$	0	0
$751$	−45.2210	−1.65014	−0.825068	−	0.565033i	$-0.808863\pi$
$751$	−45.2210	−1.65014	−0.825068	+	0.565033i	$0.808863\pi$
$752$	0	0
$753$	60.7573	2.21412
$754$	0	0
$755$	2.04813	0.0745392
$756$	0	0
$757$	−14.9461	−0.543224	−0.271612	−	0.962407i	$-0.587557\pi$
$757$	−14.9461	−0.543224	−0.271612	+	0.962407i	$0.587557\pi$
$758$	0	0
$759$	72.8022	2.64255
$760$	0	0
$761$	9.44323	0.342317	0.171158	−	0.985244i	$-0.445249\pi$
$761$	9.44323	0.342317	0.171158	+	0.985244i	$0.445249\pi$
$762$	0	0
$763$	31.4601	1.13893
$764$	0	0
$765$	14.3075	0.517288
$766$	0	0
$767$	21.5295	0.777384
$768$	0	0
$769$	−28.7635	−1.03724	−0.518620	−	0.855005i	$-0.673554\pi$
$769$	−28.7635	−1.03724	−0.518620	+	0.855005i	$0.673554\pi$
$770$	0	0
$771$	35.7120	1.28614
$772$	0	0
$773$	7.93123	0.285266	0.142633	−	0.989776i	$-0.454443\pi$
$773$	7.93123	0.285266	0.142633	+	0.989776i	$0.454443\pi$
$774$	0	0
$775$	−0.263331	−0.00945912
$776$	0	0
$777$	−60.1382	−2.15745
$778$	0	0
$779$	38.9971	1.39722
$780$	0	0
$781$	49.5785	1.77406
$782$	0	0
$783$	42.8370	1.53087
$784$	0	0
$785$	9.76661	0.348585
$786$	0	0
$787$	−39.4485	−1.40619	−0.703094	−	0.711097i	$-0.748199\pi$
$787$	−39.4485	−1.40619	−0.703094	+	0.711097i	$0.748199\pi$
$788$	0	0
$789$	−62.6514	−2.23045
$790$	0	0
$791$	−13.2632	−0.471585
$792$	0	0
$793$	−2.12061	−0.0753051
$794$	0	0
$795$	−1.06846	−0.0378945
$796$	0	0
$797$	47.4619	1.68119	0.840593	−	0.541668i	$-0.182207\pi$
$797$	47.4619	1.68119	0.840593	+	0.541668i	$0.182207\pi$
$798$	0	0
$799$	−31.4839	−1.11382
$800$	0	0
$801$	22.1208	0.781601
$802$	0	0
$803$	0.486533	0.0171694
$804$	0	0
$805$	14.6185	0.515235
$806$	0	0
$807$	−57.2939	−2.01684
$808$	0	0
$809$	−30.4292	−1.06983	−0.534917	−	0.844904i	$-0.679657\pi$
$809$	−30.4292	−1.06983	−0.534917	+	0.844904i	$0.679657\pi$
$810$	0	0
$811$	36.3006	1.27469	0.637344	−	0.770580i	$-0.280033\pi$
$811$	36.3006	1.27469	0.637344	+	0.770580i	$0.280033\pi$
$812$	0	0
$813$	−78.4119	−2.75002
$814$	0	0
$815$	0.0864182	0.00302710
$816$	0	0
$817$	47.8254	1.67320
$818$	0	0
$819$	35.2332	1.23115
$820$	0	0
$821$	8.22794	0.287157	0.143579	−	0.989639i	$-0.454139\pi$
$821$	8.22794	0.287157	0.143579	+	0.989639i	$0.454139\pi$
$822$	0	0
$823$	−30.3158	−1.05674	−0.528371	−	0.849014i	$-0.677197\pi$
$823$	−30.3158	−1.05674	−0.528371	+	0.849014i	$0.677197\pi$
$824$	0	0
$825$	−12.3863	−0.431236
$826$	0	0
$827$	−10.9900	−0.382159	−0.191080	−	0.981575i	$-0.561199\pi$
$827$	−10.9900	−0.382159	−0.191080	+	0.981575i	$0.561199\pi$
$828$	0	0
$829$	−53.6791	−1.86435	−0.932177	−	0.362003i	$-0.882093\pi$
$829$	−53.6791	−1.86435	−0.932177	+	0.362003i	$0.882093\pi$
$830$	0	0
$831$	70.5745	2.44820
$832$	0	0
$833$	−2.52458	−0.0874716
$834$	0	0
$835$	3.56704	0.123443
$836$	0	0
$837$	1.17266	0.0405332
$838$	0	0
$839$	−39.2007	−1.35336	−0.676679	−	0.736278i	$-0.736581\pi$
$839$	−39.2007	−1.35336	−0.676679	+	0.736278i	$0.736581\pi$
$840$	0	0
$841$	63.5323	2.19077
$842$	0	0
$843$	49.5881	1.70791
$844$	0	0
$845$	−3.57306	−0.122917
$846$	0	0
$847$	−22.7578	−0.781967
$848$	0	0
$849$	9.33567	0.320399
$850$	0	0
$851$	51.5050	1.76557
$852$	0	0
$853$	−17.3631	−0.594502	−0.297251	−	0.954799i	$-0.596070\pi$
$853$	−17.3631	−0.594502	−0.297251	+	0.954799i	$0.596070\pi$
$854$	0	0
$855$	27.7881	0.950331
$856$	0	0
$857$	22.1265	0.755828	0.377914	−	0.925841i	$-0.376642\pi$
$857$	22.1265	0.755828	0.377914	+	0.925841i	$0.376642\pi$
$858$	0	0
$859$	34.4829	1.17654	0.588271	−	0.808664i	$-0.299809\pi$
$859$	34.4829	1.17654	0.588271	+	0.808664i	$0.299809\pi$
$860$	0	0
$861$	44.4369	1.51440
$862$	0	0
$863$	−28.6380	−0.974848	−0.487424	−	0.873166i	$-0.662063\pi$
$863$	−28.6380	−0.974848	−0.487424	+	0.873166i	$0.662063\pi$
$864$	0	0
$865$	8.81074	0.299574
$866$	0	0
$867$	20.3748	0.691964
$868$	0	0
$869$	57.6643	1.95613
$870$	0	0
$871$	−18.0528	−0.611695
$872$	0	0
$873$	−75.6990	−2.56202
$874$	0	0
$875$	−2.48714	−0.0840807
$876$	0	0
$877$	47.2225	1.59459	0.797295	−	0.603590i	$-0.206264\pi$
$877$	47.2225	1.59459	0.797295	+	0.603590i	$0.206264\pi$
$878$	0	0
$879$	−73.8267	−2.49011
$880$	0	0
$881$	1.84295	0.0620905	0.0310452	−	0.999518i	$-0.490116\pi$
$881$	1.84295	0.0620905	0.0310452	+	0.999518i	$0.490116\pi$
$882$	0	0
$883$	−31.9932	−1.07666	−0.538328	−	0.842735i	$-0.680944\pi$
$883$	−31.9932	−1.07666	−0.538328	+	0.842735i	$0.680944\pi$
$884$	0	0
$885$	19.3487	0.650398
$886$	0	0
$887$	36.9596	1.24098	0.620491	−	0.784213i	$-0.286933\pi$
$887$	36.9596	1.24098	0.620491	+	0.784213i	$0.286933\pi$
$888$	0	0
$889$	−10.1196	−0.339402
$890$	0	0
$891$	−6.97484	−0.233666
$892$	0	0
$893$	−61.1482	−2.04624
$894$	0	0
$895$	−3.50295	−0.117091
$896$	0	0
$897$	−49.7956	−1.66263
$898$	0	0
$899$	2.53308	0.0844829
$900$	0	0
$901$	1.20076	0.0400030
$902$	0	0
$903$	54.4966	1.81353
$904$	0	0
$905$	17.5077	0.581977
$906$	0	0
$907$	11.9197	0.395786	0.197893	−	0.980224i	$-0.436590\pi$
$907$	11.9197	0.395786	0.197893	+	0.980224i	$0.436590\pi$
$908$	0	0
$909$	−14.9447	−0.495685
$910$	0	0
$911$	−46.8583	−1.55248	−0.776242	−	0.630435i	$-0.782877\pi$
$911$	−46.8583	−1.55248	−0.776242	+	0.630435i	$0.782877\pi$
$912$	0	0
$913$	−0.379981	−0.0125755
$914$	0	0
$915$	−1.90580	−0.0630040
$916$	0	0
$917$	−21.5957	−0.713154
$918$	0	0
$919$	−29.8110	−0.983375	−0.491687	−	0.870772i	$-0.663620\pi$
$919$	−29.8110	−0.983375	−0.491687	+	0.870772i	$0.663620\pi$
$920$	0	0
$921$	21.0270	0.692863
$922$	0	0
$923$	−33.9109	−1.11619
$924$	0	0
$925$	−8.76289	−0.288122
$926$	0	0
$927$	40.0508	1.31544
$928$	0	0
$929$	−12.1172	−0.397551	−0.198776	−	0.980045i	$-0.563696\pi$
$929$	−12.1172	−0.397551	−0.198776	+	0.980045i	$0.563696\pi$
$930$	0	0
$931$	−4.90325	−0.160698
$932$	0	0
$933$	13.1689	0.431131
$934$	0	0
$935$	13.9199	0.455231
$936$	0	0
$937$	−10.4329	−0.340829	−0.170415	−	0.985372i	$-0.554511\pi$
$937$	−10.4329	−0.340829	−0.170415	+	0.985372i	$0.554511\pi$
$938$	0	0
$939$	−8.63829	−0.281900
$940$	0	0
$941$	20.1513	0.656912	0.328456	−	0.944519i	$-0.393472\pi$
$941$	20.1513	0.656912	0.328456	+	0.944519i	$0.393472\pi$
$942$	0	0
$943$	−38.0577	−1.23933
$944$	0	0
$945$	11.0757	0.360294
$946$	0	0
$947$	15.9102	0.517013	0.258507	−	0.966009i	$-0.416770\pi$
$947$	15.9102	0.517013	0.258507	+	0.966009i	$0.416770\pi$
$948$	0	0
$949$	−0.332781	−0.0108025
$950$	0	0
$951$	−14.7371	−0.477882
$952$	0	0
$953$	5.99579	0.194223	0.0971114	−	0.995274i	$-0.469040\pi$
$953$	5.99579	0.194223	0.0971114	+	0.995274i	$0.469040\pi$
$954$	0	0
$955$	−23.6930	−0.766688
$956$	0	0
$957$	119.149	3.85153
$958$	0	0
$959$	7.45421	0.240709
$960$	0	0
$961$	−30.9307	−0.997763
$962$	0	0
$963$	50.7773	1.63627
$964$	0	0
$965$	14.3133	0.460761
$966$	0	0
$967$	15.1716	0.487886	0.243943	−	0.969790i	$-0.421559\pi$
$967$	15.1716	0.487886	0.243943	+	0.969790i	$0.421559\pi$
$968$	0	0
$969$	−51.5339	−1.65551
$970$	0	0
$971$	−6.66716	−0.213959	−0.106980	−	0.994261i	$-0.534118\pi$
$971$	−6.66716	−0.213959	−0.106980	+	0.994261i	$0.534118\pi$
$972$	0	0
$973$	32.4193	1.03931
$974$	0	0
$975$	8.47205	0.271323
$976$	0	0
$977$	−33.4701	−1.07080	−0.535402	−	0.844597i	$-0.679840\pi$
$977$	−33.4701	−1.07080	−0.535402	+	0.844597i	$0.679840\pi$
$978$	0	0
$979$	21.5216	0.687834
$980$	0	0
$981$	−58.3614	−1.86334
$982$	0	0
$983$	48.0004	1.53098	0.765488	−	0.643450i	$-0.222498\pi$
$983$	48.0004	1.53098	0.765488	+	0.643450i	$0.222498\pi$
$984$	0	0
$985$	−10.3882	−0.330996
$986$	0	0
$987$	−69.6778	−2.21787
$988$	0	0
$989$	−46.6733	−1.48413
$990$	0	0
$991$	27.1852	0.863565	0.431783	−	0.901978i	$-0.357885\pi$
$991$	27.1852	0.863565	0.431783	+	0.901978i	$0.357885\pi$
$992$	0	0
$993$	−17.3439	−0.550392
$994$	0	0
$995$	−10.6004	−0.336056
$996$	0	0
$997$	−44.1516	−1.39829	−0.699147	−	0.714978i	$-0.746437\pi$
$997$	−44.1516	−1.39829	−0.699147	+	0.714978i	$0.746437\pi$
$998$	0	0
$999$	39.0229	1.23463

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.d.1.2	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.d.1.2	✓	29	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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75932 q^{3} +1.00000 q^{5} -2.48714 q^{7} +4.61387 q^{9} +O(q^{10})\)	\(q-2.75932 q^{3} +1.00000 q^{5} -2.48714 q^{7} +4.61387 q^{9} +4.48889 q^{11} -3.07033 q^{13} -2.75932 q^{15} +3.10097 q^{17} +6.02272 q^{19} +6.86283 q^{21} -5.87763 q^{23} +1.00000 q^{25} -4.45320 q^{27} -9.61937 q^{29} -0.263331 q^{31} -12.3863 q^{33} -2.48714 q^{35} -8.76289 q^{37} +8.47205 q^{39} +6.47501 q^{41} +7.94084 q^{43} +4.61387 q^{45} -10.1529 q^{47} -0.814126 q^{49} -8.55659 q^{51} +0.387219 q^{53} +4.48889 q^{55} -16.6186 q^{57} -7.01210 q^{59} +0.690678 q^{61} -11.4754 q^{63} -3.07033 q^{65} +5.87975 q^{67} +16.2183 q^{69} +11.0447 q^{71} +0.108386 q^{73} -2.75932 q^{75} -11.1645 q^{77} +12.8460 q^{79} -1.55380 q^{81} -0.0846490 q^{83} +3.10097 q^{85} +26.5430 q^{87} +4.79442 q^{89} +7.63635 q^{91} +0.726615 q^{93} +6.02272 q^{95} -16.4068 q^{97} +20.7112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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