LMFDB - Embedded newform 4012.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.96950 q^{3} -1.93705 q^{5} -2.26754 q^{7} +0.878948 q^{9} +O(q^{10})$	$q-1.96950 q^{3} -1.93705 q^{5} -2.26754 q^{7} +0.878948 q^{9} +4.98984 q^{11} +1.24418 q^{13} +3.81502 q^{15} +1.00000 q^{17} -2.47732 q^{19} +4.46593 q^{21} -3.49992 q^{23} -1.24786 q^{25} +4.17742 q^{27} -5.88249 q^{29} -5.18730 q^{31} -9.82751 q^{33} +4.39233 q^{35} +2.85933 q^{37} -2.45041 q^{39} +1.07365 q^{41} +5.36068 q^{43} -1.70256 q^{45} -7.66431 q^{47} -1.85826 q^{49} -1.96950 q^{51} -8.25696 q^{53} -9.66554 q^{55} +4.87910 q^{57} +1.00000 q^{59} +0.588579 q^{61} -1.99305 q^{63} -2.41003 q^{65} -7.16702 q^{67} +6.89310 q^{69} -2.36610 q^{71} +5.38279 q^{73} +2.45766 q^{75} -11.3147 q^{77} -6.34825 q^{79} -10.8643 q^{81} +6.86772 q^{83} -1.93705 q^{85} +11.5856 q^{87} +3.68832 q^{89} -2.82122 q^{91} +10.2164 q^{93} +4.79869 q^{95} +0.642587 q^{97} +4.38581 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * 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
$3$	−1.96950	−1.13709	−0.568547	−	0.822651i	$-0.692494\pi$
$3$	−1.96950	−1.13709	−0.568547	+	0.822651i	$0.692494\pi$
$4$	0	0
$5$	−1.93705	−0.866273	−0.433136	−	0.901328i	$-0.642593\pi$
$5$	−1.93705	−0.866273	−0.433136	+	0.901328i	$0.642593\pi$
$6$	0	0
$7$	−2.26754	−0.857050	−0.428525	−	0.903530i	$-0.640967\pi$
$7$	−2.26754	−0.857050	−0.428525	+	0.903530i	$0.640967\pi$
$8$	0	0
$9$	0.878948	0.292983
$10$	0	0
$11$	4.98984	1.50449	0.752246	−	0.658882i	$-0.228970\pi$
$11$	4.98984	1.50449	0.752246	+	0.658882i	$0.228970\pi$
$12$	0	0
$13$	1.24418	0.345073	0.172536	−	0.985003i	$-0.444804\pi$
$13$	1.24418	0.345073	0.172536	+	0.985003i	$0.444804\pi$
$14$	0	0
$15$	3.81502	0.985034
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.47732	−0.568337	−0.284168	−	0.958774i	$-0.591717\pi$
$19$	−2.47732	−0.568337	−0.284168	+	0.958774i	$0.591717\pi$
$20$	0	0
$21$	4.46593	0.974546
$22$	0	0
$23$	−3.49992	−0.729783	−0.364891	−	0.931050i	$-0.618894\pi$
$23$	−3.49992	−0.729783	−0.364891	+	0.931050i	$0.618894\pi$
$24$	0	0
$25$	−1.24786	−0.249571
$26$	0	0
$27$	4.17742	0.803945
$28$	0	0
$29$	−5.88249	−1.09235	−0.546176	−	0.837670i	$-0.683917\pi$
$29$	−5.88249	−1.09235	−0.546176	+	0.837670i	$0.683917\pi$
$30$	0	0
$31$	−5.18730	−0.931666	−0.465833	−	0.884873i	$-0.654245\pi$
$31$	−5.18730	−0.931666	−0.465833	+	0.884873i	$0.654245\pi$
$32$	0	0
$33$	−9.82751	−1.71075
$34$	0	0
$35$	4.39233	0.742439
$36$	0	0
$37$	2.85933	0.470071	0.235036	−	0.971987i	$-0.424479\pi$
$37$	2.85933	0.470071	0.235036	+	0.971987i	$0.424479\pi$
$38$	0	0
$39$	−2.45041	−0.392380
$40$	0	0
$41$	1.07365	0.167676	0.0838382	−	0.996479i	$-0.473282\pi$
$41$	1.07365	0.167676	0.0838382	+	0.996479i	$0.473282\pi$
$42$	0	0
$43$	5.36068	0.817496	0.408748	−	0.912647i	$-0.365965\pi$
$43$	5.36068	0.817496	0.408748	+	0.912647i	$0.365965\pi$
$44$	0	0
$45$	−1.70256	−0.253803
$46$	0	0
$47$	−7.66431	−1.11795	−0.558977	−	0.829183i	$-0.688806\pi$
$47$	−7.66431	−1.11795	−0.558977	+	0.829183i	$0.688806\pi$
$48$	0	0
$49$	−1.85826	−0.265466
$50$	0	0
$51$	−1.96950	−0.275786
$52$	0	0
$53$	−8.25696	−1.13418	−0.567090	−	0.823656i	$-0.691931\pi$
$53$	−8.25696	−1.13418	−0.567090	+	0.823656i	$0.691931\pi$
$54$	0	0
$55$	−9.66554	−1.30330
$56$	0	0
$57$	4.87910	0.646252
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	0.588579	0.0753598	0.0376799	−	0.999290i	$-0.488003\pi$
$61$	0.588579	0.0753598	0.0376799	+	0.999290i	$0.488003\pi$
$62$	0	0
$63$	−1.99305	−0.251101
$64$	0	0
$65$	−2.41003	−0.298927
$66$	0	0
$67$	−7.16702	−0.875591	−0.437795	−	0.899075i	$-0.644241\pi$
$67$	−7.16702	−0.875591	−0.437795	+	0.899075i	$0.644241\pi$
$68$	0	0
$69$	6.89310	0.829832
$70$	0	0
$71$	−2.36610	−0.280805	−0.140402	−	0.990095i	$-0.544840\pi$
$71$	−2.36610	−0.280805	−0.140402	+	0.990095i	$0.544840\pi$
$72$	0	0
$73$	5.38279	0.630008	0.315004	−	0.949090i	$-0.397994\pi$
$73$	5.38279	0.630008	0.315004	+	0.949090i	$0.397994\pi$
$74$	0	0
$75$	2.45766	0.283786
$76$	0	0
$77$	−11.3147	−1.28942
$78$	0	0
$79$	−6.34825	−0.714234	−0.357117	−	0.934060i	$-0.616240\pi$
$79$	−6.34825	−0.714234	−0.357117	+	0.934060i	$0.616240\pi$
$80$	0	0
$81$	−10.8643	−1.20714
$82$	0	0
$83$	6.86772	0.753831	0.376915	−	0.926248i	$-0.376985\pi$
$83$	6.86772	0.753831	0.376915	+	0.926248i	$0.376985\pi$
$84$	0	0
$85$	−1.93705	−0.210102
$86$	0	0
$87$	11.5856	1.24211
$88$	0	0
$89$	3.68832	0.390961	0.195480	−	0.980708i	$-0.437373\pi$
$89$	3.68832	0.390961	0.195480	+	0.980708i	$0.437373\pi$
$90$	0	0
$91$	−2.82122	−0.295745
$92$	0	0
$93$	10.2164	1.05939
$94$	0	0
$95$	4.79869	0.492335
$96$	0	0
$97$	0.642587	0.0652449	0.0326224	−	0.999468i	$-0.489614\pi$
$97$	0.642587	0.0652449	0.0326224	+	0.999468i	$0.489614\pi$
$98$	0	0
$99$	4.38581	0.440790
$100$	0	0
$101$	−17.9490	−1.78599	−0.892995	−	0.450066i	$-0.851400\pi$
$101$	−17.9490	−1.78599	−0.892995	+	0.450066i	$0.851400\pi$
$102$	0	0
$103$	14.5275	1.43143	0.715717	−	0.698390i	$-0.246100\pi$
$103$	14.5275	1.43143	0.715717	+	0.698390i	$0.246100\pi$
$104$	0	0
$105$	−8.65071	−0.844223
$106$	0	0
$107$	−1.84547	−0.178408	−0.0892040	−	0.996013i	$-0.528432\pi$
$107$	−1.84547	−0.178408	−0.0892040	+	0.996013i	$0.528432\pi$
$108$	0	0
$109$	−0.448807	−0.0429879	−0.0214939	−	0.999769i	$-0.506842\pi$
$109$	−0.448807	−0.0429879	−0.0214939	+	0.999769i	$0.506842\pi$
$110$	0	0
$111$	−5.63147	−0.534515
$112$	0	0
$113$	13.9359	1.31098	0.655492	−	0.755202i	$-0.272461\pi$
$113$	13.9359	1.31098	0.655492	+	0.755202i	$0.272461\pi$
$114$	0	0
$115$	6.77950	0.632191
$116$	0	0
$117$	1.09357	0.101100
$118$	0	0
$119$	−2.26754	−0.207865
$120$	0	0
$121$	13.8985	1.26350
$122$	0	0
$123$	−2.11457	−0.190664
$124$	0	0
$125$	12.1024	1.08247
$126$	0	0
$127$	10.7660	0.955326	0.477663	−	0.878543i	$-0.341484\pi$
$127$	10.7660	0.955326	0.477663	+	0.878543i	$0.341484\pi$
$128$	0	0
$129$	−10.5579	−0.929570
$130$	0	0
$131$	21.0970	1.84325	0.921626	−	0.388079i	$-0.126861\pi$
$131$	21.0970	1.84325	0.921626	+	0.388079i	$0.126861\pi$
$132$	0	0
$133$	5.61743	0.487093
$134$	0	0
$135$	−8.09185	−0.696436
$136$	0	0
$137$	−9.89183	−0.845116	−0.422558	−	0.906336i	$-0.638868\pi$
$137$	−9.89183	−0.845116	−0.422558	+	0.906336i	$0.638868\pi$
$138$	0	0
$139$	23.4107	1.98567	0.992833	−	0.119510i	$-0.0381324\pi$
$139$	23.4107	1.98567	0.992833	+	0.119510i	$0.0381324\pi$
$140$	0	0
$141$	15.0949	1.27122
$142$	0	0
$143$	6.20825	0.519160
$144$	0	0
$145$	11.3947	0.946275
$146$	0	0
$147$	3.65986	0.301860
$148$	0	0
$149$	1.06846	0.0875313	0.0437656	−	0.999042i	$-0.486065\pi$
$149$	1.06846	0.0875313	0.0437656	+	0.999042i	$0.486065\pi$
$150$	0	0
$151$	−1.75851	−0.143106	−0.0715528	−	0.997437i	$-0.522795\pi$
$151$	−1.75851	−0.143106	−0.0715528	+	0.997437i	$0.522795\pi$
$152$	0	0
$153$	0.878948	0.0710588
$154$	0	0
$155$	10.0480	0.807077
$156$	0	0
$157$	−18.7002	−1.49244	−0.746219	−	0.665701i	$-0.768133\pi$
$157$	−18.7002	−1.49244	−0.746219	+	0.665701i	$0.768133\pi$
$158$	0	0
$159$	16.2621	1.28967
$160$	0	0
$161$	7.93620	0.625460
$162$	0	0
$163$	9.29359	0.727930	0.363965	−	0.931413i	$-0.381423\pi$
$163$	9.29359	0.727930	0.363965	+	0.931413i	$0.381423\pi$
$164$	0	0
$165$	19.0363	1.48198
$166$	0	0
$167$	8.55273	0.661830	0.330915	−	0.943661i	$-0.392643\pi$
$167$	8.55273	0.661830	0.330915	+	0.943661i	$0.392643\pi$
$168$	0	0
$169$	−11.4520	−0.880925
$170$	0	0
$171$	−2.17744	−0.166513
$172$	0	0
$173$	6.86267	0.521759	0.260880	−	0.965371i	$-0.415987\pi$
$173$	6.86267	0.521759	0.260880	+	0.965371i	$0.415987\pi$
$174$	0	0
$175$	2.82956	0.213895
$176$	0	0
$177$	−1.96950	−0.148037
$178$	0	0
$179$	−1.13081	−0.0845209	−0.0422604	−	0.999107i	$-0.513456\pi$
$179$	−1.13081	−0.0845209	−0.0422604	+	0.999107i	$0.513456\pi$
$180$	0	0
$181$	2.28049	0.169507	0.0847537	−	0.996402i	$-0.472990\pi$
$181$	2.28049	0.169507	0.0847537	+	0.996402i	$0.472990\pi$
$182$	0	0
$183$	−1.15921	−0.0856912
$184$	0	0
$185$	−5.53866	−0.407210
$186$	0	0
$187$	4.98984	0.364893
$188$	0	0
$189$	−9.47247	−0.689021
$190$	0	0
$191$	26.3666	1.90782	0.953910	−	0.300092i	$-0.0970173\pi$
$191$	26.3666	1.90782	0.953910	+	0.300092i	$0.0970173\pi$
$192$	0	0
$193$	24.6297	1.77289	0.886443	−	0.462838i	$-0.153169\pi$
$193$	24.6297	1.77289	0.886443	+	0.462838i	$0.153169\pi$
$194$	0	0
$195$	4.74656	0.339908
$196$	0	0
$197$	−9.55211	−0.680560	−0.340280	−	0.940324i	$-0.610522\pi$
$197$	−9.55211	−0.680560	−0.340280	+	0.940324i	$0.610522\pi$
$198$	0	0
$199$	−6.96553	−0.493774	−0.246887	−	0.969044i	$-0.579408\pi$
$199$	−6.96553	−0.493774	−0.246887	+	0.969044i	$0.579408\pi$
$200$	0	0
$201$	14.1155	0.995629
$202$	0	0
$203$	13.3388	0.936200
$204$	0	0
$205$	−2.07972	−0.145254
$206$	0	0
$207$	−3.07625	−0.213814
$208$	0	0
$209$	−12.3614	−0.855059
$210$	0	0
$211$	−25.6791	−1.76782	−0.883912	−	0.467653i	$-0.845100\pi$
$211$	−25.6791	−1.76782	−0.883912	+	0.467653i	$0.845100\pi$
$212$	0	0
$213$	4.66005	0.319301
$214$	0	0
$215$	−10.3839	−0.708175
$216$	0	0
$217$	11.7624	0.798484
$218$	0	0
$219$	−10.6014	−0.716378
$220$	0	0
$221$	1.24418	0.0836925
$222$	0	0
$223$	0.446916	0.0299277	0.0149638	−	0.999888i	$-0.495237\pi$
$223$	0.446916	0.0299277	0.0149638	+	0.999888i	$0.495237\pi$
$224$	0	0
$225$	−1.09680	−0.0731200
$226$	0	0
$227$	0.466602	0.0309694	0.0154847	−	0.999880i	$-0.495071\pi$
$227$	0.466602	0.0309694	0.0154847	+	0.999880i	$0.495071\pi$
$228$	0	0
$229$	20.5854	1.36032	0.680161	−	0.733063i	$-0.261910\pi$
$229$	20.5854	1.36032	0.680161	+	0.733063i	$0.261910\pi$
$230$	0	0
$231$	22.2843	1.46620
$232$	0	0
$233$	−1.93662	−0.126872	−0.0634360	−	0.997986i	$-0.520206\pi$
$233$	−1.93662	−0.126872	−0.0634360	+	0.997986i	$0.520206\pi$
$234$	0	0
$235$	14.8461	0.968454
$236$	0	0
$237$	12.5029	0.812151
$238$	0	0
$239$	7.69563	0.497789	0.248895	−	0.968531i	$-0.419933\pi$
$239$	7.69563	0.497789	0.248895	+	0.968531i	$0.419933\pi$
$240$	0	0
$241$	−9.88189	−0.636549	−0.318274	−	0.947999i	$-0.603103\pi$
$241$	−9.88189	−0.636549	−0.318274	+	0.947999i	$0.603103\pi$
$242$	0	0
$243$	8.86502	0.568691
$244$	0	0
$245$	3.59954	0.229966
$246$	0	0
$247$	−3.08223	−0.196118
$248$	0	0
$249$	−13.5260	−0.857176
$250$	0	0
$251$	3.70604	0.233923	0.116962	−	0.993136i	$-0.462685\pi$
$251$	3.70604	0.233923	0.116962	+	0.993136i	$0.462685\pi$
$252$	0	0
$253$	−17.4640	−1.09795
$254$	0	0
$255$	3.81502	0.238906
$256$	0	0
$257$	−22.4758	−1.40200	−0.701002	−	0.713160i	$-0.747263\pi$
$257$	−22.4758	−1.40200	−0.701002	+	0.713160i	$0.747263\pi$
$258$	0	0
$259$	−6.48365	−0.402874
$260$	0	0
$261$	−5.17041	−0.320040
$262$	0	0
$263$	11.5590	0.712761	0.356381	−	0.934341i	$-0.384011\pi$
$263$	11.5590	0.712761	0.356381	+	0.934341i	$0.384011\pi$
$264$	0	0
$265$	15.9941	0.982510
$266$	0	0
$267$	−7.26416	−0.444559
$268$	0	0
$269$	−7.48098	−0.456123	−0.228062	−	0.973647i	$-0.573239\pi$
$269$	−7.48098	−0.456123	−0.228062	+	0.973647i	$0.573239\pi$
$270$	0	0
$271$	−15.5411	−0.944052	−0.472026	−	0.881585i	$-0.656477\pi$
$271$	−15.5411	−0.944052	−0.472026	+	0.881585i	$0.656477\pi$
$272$	0	0
$273$	5.55641	0.336289
$274$	0	0
$275$	−6.22660	−0.375478
$276$	0	0
$277$	11.0713	0.665209	0.332604	−	0.943066i	$-0.392073\pi$
$277$	11.0713	0.665209	0.332604	+	0.943066i	$0.392073\pi$
$278$	0	0
$279$	−4.55937	−0.272962
$280$	0	0
$281$	14.7271	0.878543	0.439271	−	0.898354i	$-0.355237\pi$
$281$	14.7271	0.878543	0.439271	+	0.898354i	$0.355237\pi$
$282$	0	0
$283$	19.0986	1.13529	0.567646	−	0.823273i	$-0.307854\pi$
$283$	19.0986	1.13529	0.567646	+	0.823273i	$0.307854\pi$
$284$	0	0
$285$	−9.45104	−0.559831
$286$	0	0
$287$	−2.43455	−0.143707
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−1.26558	−0.0741895
$292$	0	0
$293$	22.4086	1.30912	0.654561	−	0.756009i	$-0.272853\pi$
$293$	22.4086	1.30912	0.654561	+	0.756009i	$0.272853\pi$
$294$	0	0
$295$	−1.93705	−0.112779
$296$	0	0
$297$	20.8447	1.20953
$298$	0	0
$299$	−4.35452	−0.251828
$300$	0	0
$301$	−12.1556	−0.700635
$302$	0	0
$303$	35.3506	2.03084
$304$	0	0
$305$	−1.14010	−0.0652822
$306$	0	0
$307$	2.45095	0.139883	0.0699415	−	0.997551i	$-0.477719\pi$
$307$	2.45095	0.139883	0.0699415	+	0.997551i	$0.477719\pi$
$308$	0	0
$309$	−28.6119	−1.62768
$310$	0	0
$311$	−1.35571	−0.0768754	−0.0384377	−	0.999261i	$-0.512238\pi$
$311$	−1.35571	−0.0768754	−0.0384377	+	0.999261i	$0.512238\pi$
$312$	0	0
$313$	32.8702	1.85793	0.928966	−	0.370164i	$-0.120699\pi$
$313$	32.8702	1.85793	0.928966	+	0.370164i	$0.120699\pi$
$314$	0	0
$315$	3.86063	0.217522
$316$	0	0
$317$	1.31152	0.0736623	0.0368311	−	0.999322i	$-0.488274\pi$
$317$	1.31152	0.0736623	0.0368311	+	0.999322i	$0.488274\pi$
$318$	0	0
$319$	−29.3527	−1.64344
$320$	0	0
$321$	3.63465	0.202867
$322$	0	0
$323$	−2.47732	−0.137842
$324$	0	0
$325$	−1.55255	−0.0861202
$326$	0	0
$327$	0.883927	0.0488813
$328$	0	0
$329$	17.3791	0.958143
$330$	0	0
$331$	13.9572	0.767155	0.383577	−	0.923509i	$-0.374692\pi$
$331$	13.9572	0.767155	0.383577	+	0.923509i	$0.374692\pi$
$332$	0	0
$333$	2.51321	0.137723
$334$	0	0
$335$	13.8828	0.758501
$336$	0	0
$337$	−10.6053	−0.577710	−0.288855	−	0.957373i	$-0.593275\pi$
$337$	−10.6053	−0.577710	−0.288855	+	0.957373i	$0.593275\pi$
$338$	0	0
$339$	−27.4469	−1.49071
$340$	0	0
$341$	−25.8838	−1.40168
$342$	0	0
$343$	20.0865	1.08457
$344$	0	0
$345$	−13.3522	−0.718861
$346$	0	0
$347$	2.42274	0.130060	0.0650299	−	0.997883i	$-0.479286\pi$
$347$	2.42274	0.130060	0.0650299	+	0.997883i	$0.479286\pi$
$348$	0	0
$349$	−30.9622	−1.65737	−0.828685	−	0.559715i	$-0.810910\pi$
$349$	−30.9622	−1.65737	−0.828685	+	0.559715i	$0.810910\pi$
$350$	0	0
$351$	5.19746	0.277420
$352$	0	0
$353$	−25.3769	−1.35068	−0.675339	−	0.737508i	$-0.736003\pi$
$353$	−25.3769	−1.35068	−0.675339	+	0.737508i	$0.736003\pi$
$354$	0	0
$355$	4.58325	0.243253
$356$	0	0
$357$	4.46593	0.236362
$358$	0	0
$359$	23.3392	1.23180	0.615898	−	0.787826i	$-0.288793\pi$
$359$	23.3392	1.23180	0.615898	+	0.787826i	$0.288793\pi$
$360$	0	0
$361$	−12.8629	−0.676993
$362$	0	0
$363$	−27.3731	−1.43672
$364$	0	0
$365$	−10.4267	−0.545759
$366$	0	0
$367$	37.3250	1.94835	0.974174	−	0.225799i	$-0.0724991\pi$
$367$	37.3250	1.94835	0.974174	+	0.225799i	$0.0724991\pi$
$368$	0	0
$369$	0.943686	0.0491263
$370$	0	0
$371$	18.7230	0.972049
$372$	0	0
$373$	−13.9449	−0.722038	−0.361019	−	0.932558i	$-0.617571\pi$
$373$	−13.9449	−0.722038	−0.361019	+	0.932558i	$0.617571\pi$
$374$	0	0
$375$	−23.8357	−1.23087
$376$	0	0
$377$	−7.31887	−0.376941
$378$	0	0
$379$	−22.9156	−1.17709	−0.588546	−	0.808463i	$-0.700300\pi$
$379$	−22.9156	−1.17709	−0.588546	+	0.808463i	$0.700300\pi$
$380$	0	0
$381$	−21.2036	−1.08630
$382$	0	0
$383$	27.9889	1.43016	0.715082	−	0.699040i	$-0.246389\pi$
$383$	27.9889	1.43016	0.715082	+	0.699040i	$0.246389\pi$
$384$	0	0
$385$	21.9170	1.11699
$386$	0	0
$387$	4.71176	0.239512
$388$	0	0
$389$	−24.1302	−1.22345	−0.611724	−	0.791071i	$-0.709524\pi$
$389$	−24.1302	−1.22345	−0.611724	+	0.791071i	$0.709524\pi$
$390$	0	0
$391$	−3.49992	−0.176998
$392$	0	0
$393$	−41.5506	−2.09595
$394$	0	0
$395$	12.2968	0.618721
$396$	0	0
$397$	−20.7905	−1.04345	−0.521723	−	0.853115i	$-0.674711\pi$
$397$	−20.7905	−1.04345	−0.521723	+	0.853115i	$0.674711\pi$
$398$	0	0
$399$	−11.0636	−0.553870
$400$	0	0
$401$	7.96744	0.397875	0.198937	−	0.980012i	$-0.436251\pi$
$401$	7.96744	0.397875	0.198937	+	0.980012i	$0.436251\pi$
$402$	0	0
$403$	−6.45392	−0.321493
$404$	0	0
$405$	21.0446	1.04572
$406$	0	0
$407$	14.2676	0.707219
$408$	0	0
$409$	−3.82110	−0.188941	−0.0944707	−	0.995528i	$-0.530116\pi$
$409$	−3.82110	−0.188941	−0.0944707	+	0.995528i	$0.530116\pi$
$410$	0	0
$411$	19.4820	0.960977
$412$	0	0
$413$	−2.26754	−0.111578
$414$	0	0
$415$	−13.3031	−0.653023
$416$	0	0
$417$	−46.1074	−2.25789
$418$	0	0
$419$	−2.42594	−0.118515	−0.0592575	−	0.998243i	$-0.518873\pi$
$419$	−2.42594	−0.118515	−0.0592575	+	0.998243i	$0.518873\pi$
$420$	0	0
$421$	−18.0139	−0.877944	−0.438972	−	0.898501i	$-0.644657\pi$
$421$	−18.0139	−0.877944	−0.438972	+	0.898501i	$0.644657\pi$
$422$	0	0
$423$	−6.73653	−0.327542
$424$	0	0
$425$	−1.24786	−0.0605299
$426$	0	0
$427$	−1.33463	−0.0645871
$428$	0	0
$429$	−12.2272	−0.590333
$430$	0	0
$431$	7.96922	0.383864	0.191932	−	0.981408i	$-0.438525\pi$
$431$	7.96922	0.383864	0.191932	+	0.981408i	$0.438525\pi$
$432$	0	0
$433$	11.5574	0.555414	0.277707	−	0.960666i	$-0.410426\pi$
$433$	11.5574	0.555414	0.277707	+	0.960666i	$0.410426\pi$
$434$	0	0
$435$	−22.4418	−1.07600
$436$	0	0
$437$	8.67042	0.414763
$438$	0	0
$439$	34.7085	1.65655	0.828273	−	0.560325i	$-0.189324\pi$
$439$	34.7085	1.65655	0.828273	+	0.560325i	$0.189324\pi$
$440$	0	0
$441$	−1.63332	−0.0777770
$442$	0	0
$443$	37.5852	1.78573	0.892863	−	0.450329i	$-0.148693\pi$
$443$	37.5852	1.78573	0.892863	+	0.450329i	$0.148693\pi$
$444$	0	0
$445$	−7.14444	−0.338679
$446$	0	0
$447$	−2.10433	−0.0995313
$448$	0	0
$449$	9.86467	0.465542	0.232771	−	0.972532i	$-0.425221\pi$
$449$	9.86467	0.465542	0.232771	+	0.972532i	$0.425221\pi$
$450$	0	0
$451$	5.35736	0.252268
$452$	0	0
$453$	3.46340	0.162725
$454$	0	0
$455$	5.46484	0.256196
$456$	0	0
$457$	34.3741	1.60795	0.803977	−	0.594661i	$-0.202714\pi$
$457$	34.3741	1.60795	0.803977	+	0.594661i	$0.202714\pi$
$458$	0	0
$459$	4.17742	0.194985
$460$	0	0
$461$	40.3857	1.88095	0.940474	−	0.339866i	$-0.110382\pi$
$461$	40.3857	1.88095	0.940474	+	0.339866i	$0.110382\pi$
$462$	0	0
$463$	−26.4286	−1.22824	−0.614121	−	0.789212i	$-0.710489\pi$
$463$	−26.4286	−1.22824	−0.614121	+	0.789212i	$0.710489\pi$
$464$	0	0
$465$	−19.7896	−0.917722
$466$	0	0
$467$	8.58946	0.397473	0.198736	−	0.980053i	$-0.436316\pi$
$467$	8.58946	0.397473	0.198736	+	0.980053i	$0.436316\pi$
$468$	0	0
$469$	16.2515	0.750425
$470$	0	0
$471$	36.8301	1.69704
$472$	0	0
$473$	26.7489	1.22992
$474$	0	0
$475$	3.09134	0.141840
$476$	0	0
$477$	−7.25744	−0.332295
$478$	0	0
$479$	42.8665	1.95862	0.979309	−	0.202370i	$-0.0648643\pi$
$479$	42.8665	1.95862	0.979309	+	0.202370i	$0.0648643\pi$
$480$	0	0
$481$	3.55752	0.162209
$482$	0	0
$483$	−15.6304	−0.711207
$484$	0	0
$485$	−1.24472	−0.0565199
$486$	0	0
$487$	−14.0684	−0.637502	−0.318751	−	0.947839i	$-0.603263\pi$
$487$	−14.0684	−0.637502	−0.318751	+	0.947839i	$0.603263\pi$
$488$	0	0
$489$	−18.3038	−0.827725
$490$	0	0
$491$	−31.7641	−1.43349	−0.716747	−	0.697334i	$-0.754370\pi$
$491$	−31.7641	−1.43349	−0.716747	+	0.697334i	$0.754370\pi$
$492$	0	0
$493$	−5.88249	−0.264934
$494$	0	0
$495$	−8.49551	−0.381845
$496$	0	0
$497$	5.36523	0.240663
$498$	0	0
$499$	−7.09274	−0.317514	−0.158757	−	0.987318i	$-0.550749\pi$
$499$	−7.09274	−0.317514	−0.158757	+	0.987318i	$0.550749\pi$
$500$	0	0
$501$	−16.8446	−0.752563
$502$	0	0
$503$	11.1962	0.499213	0.249606	−	0.968347i	$-0.419699\pi$
$503$	11.1962	0.499213	0.249606	+	0.968347i	$0.419699\pi$
$504$	0	0
$505$	34.7680	1.54716
$506$	0	0
$507$	22.5548	1.00169
$508$	0	0
$509$	21.6627	0.960184	0.480092	−	0.877218i	$-0.340603\pi$
$509$	21.6627	0.960184	0.480092	+	0.877218i	$0.340603\pi$
$510$	0	0
$511$	−12.2057	−0.539948
$512$	0	0
$513$	−10.3488	−0.456912
$514$	0	0
$515$	−28.1404	−1.24001
$516$	0	0
$517$	−38.2437	−1.68195
$518$	0	0
$519$	−13.5161	−0.593290
$520$	0	0
$521$	14.4328	0.632312	0.316156	−	0.948707i	$-0.397608\pi$
$521$	14.4328	0.632312	0.316156	+	0.948707i	$0.397608\pi$
$522$	0	0
$523$	−5.83577	−0.255180	−0.127590	−	0.991827i	$-0.540724\pi$
$523$	−5.83577	−0.255180	−0.127590	+	0.991827i	$0.540724\pi$
$524$	0	0
$525$	−5.57284	−0.243219
$526$	0	0
$527$	−5.18730	−0.225962
$528$	0	0
$529$	−10.7506	−0.467417
$530$	0	0
$531$	0.878948	0.0381431
$532$	0	0
$533$	1.33582	0.0578606
$534$	0	0
$535$	3.57475	0.154550
$536$	0	0
$537$	2.22714	0.0961082
$538$	0	0
$539$	−9.27243	−0.399392
$540$	0	0
$541$	−30.3078	−1.30303	−0.651516	−	0.758635i	$-0.725867\pi$
$541$	−30.3078	−1.30303	−0.651516	+	0.758635i	$0.725867\pi$
$542$	0	0
$543$	−4.49143	−0.192746
$544$	0	0
$545$	0.869359	0.0372392
$546$	0	0
$547$	23.1105	0.988132	0.494066	−	0.869424i	$-0.335510\pi$
$547$	23.1105	0.988132	0.494066	+	0.869424i	$0.335510\pi$
$548$	0	0
$549$	0.517331	0.0220791
$550$	0	0
$551$	14.5728	0.620824
$552$	0	0
$553$	14.3949	0.612134
$554$	0	0
$555$	10.9084	0.463036
$556$	0	0
$557$	3.00248	0.127219	0.0636095	−	0.997975i	$-0.479739\pi$
$557$	3.00248	0.127219	0.0636095	+	0.997975i	$0.479739\pi$
$558$	0	0
$559$	6.66964	0.282096
$560$	0	0
$561$	−9.82751	−0.414918
$562$	0	0
$563$	−4.66274	−0.196511	−0.0982555	−	0.995161i	$-0.531326\pi$
$563$	−4.66274	−0.196511	−0.0982555	+	0.995161i	$0.531326\pi$
$564$	0	0
$565$	−26.9946	−1.13567
$566$	0	0
$567$	24.6352	1.03458
$568$	0	0
$569$	42.1133	1.76548	0.882742	−	0.469859i	$-0.155695\pi$
$569$	42.1133	1.76548	0.882742	+	0.469859i	$0.155695\pi$
$570$	0	0
$571$	16.8968	0.707109	0.353555	−	0.935414i	$-0.384973\pi$
$571$	16.8968	0.707109	0.353555	+	0.935414i	$0.384973\pi$
$572$	0	0
$573$	−51.9292	−2.16937
$574$	0	0
$575$	4.36739	0.182133
$576$	0	0
$577$	8.62473	0.359052	0.179526	−	0.983753i	$-0.442544\pi$
$577$	8.62473	0.359052	0.179526	+	0.983753i	$0.442544\pi$
$578$	0	0
$579$	−48.5084	−2.01594
$580$	0	0
$581$	−15.5728	−0.646070
$582$	0	0
$583$	−41.2009	−1.70637
$584$	0	0
$585$	−2.11829	−0.0875806
$586$	0	0
$587$	18.3612	0.757849	0.378925	−	0.925428i	$-0.376294\pi$
$587$	18.3612	0.757849	0.378925	+	0.925428i	$0.376294\pi$
$588$	0	0
$589$	12.8506	0.529500
$590$	0	0
$591$	18.8129	0.773860
$592$	0	0
$593$	15.0030	0.616101	0.308051	−	0.951370i	$-0.400323\pi$
$593$	15.0030	0.616101	0.308051	+	0.951370i	$0.400323\pi$
$594$	0	0
$595$	4.39233	0.180068
$596$	0	0
$597$	13.7187	0.561467
$598$	0	0
$599$	−44.5119	−1.81871	−0.909354	−	0.416023i	$-0.863424\pi$
$599$	−44.5119	−1.81871	−0.909354	+	0.416023i	$0.863424\pi$
$600$	0	0
$601$	23.0046	0.938379	0.469190	−	0.883097i	$-0.344546\pi$
$601$	23.0046	0.938379	0.469190	+	0.883097i	$0.344546\pi$
$602$	0	0
$603$	−6.29944	−0.256533
$604$	0	0
$605$	−26.9220	−1.09453
$606$	0	0
$607$	−24.2666	−0.984951	−0.492476	−	0.870326i	$-0.663908\pi$
$607$	−24.2666	−0.984951	−0.492476	+	0.870326i	$0.663908\pi$
$608$	0	0
$609$	−26.2708	−1.06455
$610$	0	0
$611$	−9.53577	−0.385776
$612$	0	0
$613$	−14.2874	−0.577064	−0.288532	−	0.957470i	$-0.593167\pi$
$613$	−14.2874	−0.577064	−0.288532	+	0.957470i	$0.593167\pi$
$614$	0	0
$615$	4.09601	0.165167
$616$	0	0
$617$	−7.35264	−0.296006	−0.148003	−	0.988987i	$-0.547285\pi$
$617$	−7.35264	−0.296006	−0.148003	+	0.988987i	$0.547285\pi$
$618$	0	0
$619$	5.14724	0.206885	0.103443	−	0.994635i	$-0.467014\pi$
$619$	5.14724	0.206885	0.103443	+	0.994635i	$0.467014\pi$
$620$	0	0
$621$	−14.6206	−0.586705
$622$	0	0
$623$	−8.36341	−0.335073
$624$	0	0
$625$	−17.2036	−0.688143
$626$	0	0
$627$	24.3459	0.972282
$628$	0	0
$629$	2.85933	0.114009
$630$	0	0
$631$	−4.58377	−0.182477	−0.0912385	−	0.995829i	$-0.529083\pi$
$631$	−4.58377	−0.182477	−0.0912385	+	0.995829i	$0.529083\pi$
$632$	0	0
$633$	50.5752	2.01018
$634$	0	0
$635$	−20.8542	−0.827573
$636$	0	0
$637$	−2.31201	−0.0916052
$638$	0	0
$639$	−2.07968	−0.0822709
$640$	0	0
$641$	−46.5578	−1.83892	−0.919461	−	0.393181i	$-0.871375\pi$
$641$	−46.5578	−1.83892	−0.919461	+	0.393181i	$0.871375\pi$
$642$	0	0
$643$	−27.2156	−1.07328	−0.536639	−	0.843812i	$-0.680306\pi$
$643$	−27.2156	−1.07328	−0.536639	+	0.843812i	$0.680306\pi$
$644$	0	0
$645$	20.4511	0.805262
$646$	0	0
$647$	26.6272	1.04682	0.523411	−	0.852080i	$-0.324659\pi$
$647$	26.6272	1.04682	0.523411	+	0.852080i	$0.324659\pi$
$648$	0	0
$649$	4.98984	0.195868
$650$	0	0
$651$	−23.1661	−0.907951
$652$	0	0
$653$	−21.6477	−0.847139	−0.423569	−	0.905864i	$-0.639223\pi$
$653$	−21.6477	−0.847139	−0.423569	+	0.905864i	$0.639223\pi$
$654$	0	0
$655$	−40.8658	−1.59676
$656$	0	0
$657$	4.73119	0.184581
$658$	0	0
$659$	34.5895	1.34742	0.673708	−	0.738998i	$-0.264701\pi$
$659$	34.5895	1.34742	0.673708	+	0.738998i	$0.264701\pi$
$660$	0	0
$661$	16.1292	0.627353	0.313676	−	0.949530i	$-0.398439\pi$
$661$	16.1292	0.627353	0.313676	+	0.949530i	$0.398439\pi$
$662$	0	0
$663$	−2.45041	−0.0951662
$664$	0	0
$665$	−10.8812	−0.421955
$666$	0	0
$667$	20.5882	0.797180
$668$	0	0
$669$	−0.880203	−0.0340306
$670$	0	0
$671$	2.93691	0.113378
$672$	0	0
$673$	3.41665	0.131702	0.0658510	−	0.997829i	$-0.479024\pi$
$673$	3.41665	0.131702	0.0658510	+	0.997829i	$0.479024\pi$
$674$	0	0
$675$	−5.21282	−0.200641
$676$	0	0
$677$	35.6831	1.37141	0.685706	−	0.727879i	$-0.259494\pi$
$677$	35.6831	1.37141	0.685706	+	0.727879i	$0.259494\pi$
$678$	0	0
$679$	−1.45709	−0.0559181
$680$	0	0
$681$	−0.918974	−0.0352152
$682$	0	0
$683$	16.6080	0.635487	0.317743	−	0.948177i	$-0.397075\pi$
$683$	16.6080	0.635487	0.317743	+	0.948177i	$0.397075\pi$
$684$	0	0
$685$	19.1609	0.732102
$686$	0	0
$687$	−40.5431	−1.54681
$688$	0	0
$689$	−10.2731	−0.391375
$690$	0	0
$691$	−31.4500	−1.19642	−0.598208	−	0.801341i	$-0.704120\pi$
$691$	−31.4500	−1.19642	−0.598208	+	0.801341i	$0.704120\pi$
$692$	0	0
$693$	−9.94500	−0.377779
$694$	0	0
$695$	−45.3475	−1.72013
$696$	0	0
$697$	1.07365	0.0406675
$698$	0	0
$699$	3.81418	0.144265
$700$	0	0
$701$	−25.2040	−0.951940	−0.475970	−	0.879461i	$-0.657903\pi$
$701$	−25.2040	−0.951940	−0.475970	+	0.879461i	$0.657903\pi$
$702$	0	0
$703$	−7.08349	−0.267159
$704$	0	0
$705$	−29.2395	−1.10122
$706$	0	0
$707$	40.7000	1.53068
$708$	0	0
$709$	−38.0792	−1.43010	−0.715048	−	0.699075i	$-0.753595\pi$
$709$	−38.0792	−1.43010	−0.715048	+	0.699075i	$0.753595\pi$
$710$	0	0
$711$	−5.57978	−0.209258
$712$	0	0
$713$	18.1551	0.679914
$714$	0	0
$715$	−12.0257	−0.449734
$716$	0	0
$717$	−15.1566	−0.566033
$718$	0	0
$719$	−29.3734	−1.09544	−0.547721	−	0.836661i	$-0.684505\pi$
$719$	−29.3734	−1.09544	−0.547721	+	0.836661i	$0.684505\pi$
$720$	0	0
$721$	−32.9416	−1.22681
$722$	0	0
$723$	19.4624	0.723816
$724$	0	0
$725$	7.34050	0.272619
$726$	0	0
$727$	−6.61185	−0.245220	−0.122610	−	0.992455i	$-0.539126\pi$
$727$	−6.61185	−0.245220	−0.122610	+	0.992455i	$0.539126\pi$
$728$	0	0
$729$	15.1332	0.560489
$730$	0	0
$731$	5.36068	0.198272
$732$	0	0
$733$	33.4842	1.23677	0.618383	−	0.785877i	$-0.287788\pi$
$733$	33.4842	1.23677	0.618383	+	0.785877i	$0.287788\pi$
$734$	0	0
$735$	−7.08931	−0.261493
$736$	0	0
$737$	−35.7623	−1.31732
$738$	0	0
$739$	−43.0345	−1.58305	−0.791524	−	0.611138i	$-0.790712\pi$
$739$	−43.0345	−1.58305	−0.791524	+	0.611138i	$0.790712\pi$
$740$	0	0
$741$	6.07047	0.223004
$742$	0	0
$743$	4.44380	0.163027	0.0815135	−	0.996672i	$-0.474025\pi$
$743$	4.44380	0.163027	0.0815135	+	0.996672i	$0.474025\pi$
$744$	0	0
$745$	−2.06965	−0.0758260
$746$	0	0
$747$	6.03638	0.220859
$748$	0	0
$749$	4.18467	0.152904
$750$	0	0
$751$	39.4405	1.43920	0.719602	−	0.694386i	$-0.244324\pi$
$751$	39.4405	1.43920	0.719602	+	0.694386i	$0.244324\pi$
$752$	0	0
$753$	−7.29907	−0.265993
$754$	0	0
$755$	3.40632	0.123969
$756$	0	0
$757$	−3.96746	−0.144200	−0.0720998	−	0.997397i	$-0.522970\pi$
$757$	−3.96746	−0.144200	−0.0720998	+	0.997397i	$0.522970\pi$
$758$	0	0
$759$	34.3955	1.24848
$760$	0	0
$761$	−13.0697	−0.473775	−0.236887	−	0.971537i	$-0.576127\pi$
$761$	−13.0697	−0.473775	−0.236887	+	0.971537i	$0.576127\pi$
$762$	0	0
$763$	1.01769	0.0368427
$764$	0	0
$765$	−1.70256	−0.0615563
$766$	0	0
$767$	1.24418	0.0449247
$768$	0	0
$769$	14.6912	0.529779	0.264889	−	0.964279i	$-0.414665\pi$
$769$	14.6912	0.529779	0.264889	+	0.964279i	$0.414665\pi$
$770$	0	0
$771$	44.2662	1.59421
$772$	0	0
$773$	−13.5854	−0.488634	−0.244317	−	0.969695i	$-0.578564\pi$
$773$	−13.5854	−0.488634	−0.244317	+	0.969695i	$0.578564\pi$
$774$	0	0
$775$	6.47300	0.232517
$776$	0	0
$777$	12.7696	0.458106
$778$	0	0
$779$	−2.65979	−0.0952967
$780$	0	0
$781$	−11.8065	−0.422468
$782$	0	0
$783$	−24.5737	−0.878191
$784$	0	0
$785$	36.2231	1.29286
$786$	0	0
$787$	6.52292	0.232517	0.116258	−	0.993219i	$-0.462910\pi$
$787$	6.52292	0.232517	0.116258	+	0.993219i	$0.462910\pi$
$788$	0	0
$789$	−22.7656	−0.810476
$790$	0	0
$791$	−31.6003	−1.12358
$792$	0	0
$793$	0.732297	0.0260046
$794$	0	0
$795$	−31.5004	−1.11721
$796$	0	0
$797$	26.0758	0.923653	0.461826	−	0.886970i	$-0.347194\pi$
$797$	26.0758	0.923653	0.461826	+	0.886970i	$0.347194\pi$
$798$	0	0
$799$	−7.66431	−0.271144
$800$	0	0
$801$	3.24184	0.114545
$802$	0	0
$803$	26.8592	0.947842
$804$	0	0
$805$	−15.3728	−0.541819
$806$	0	0
$807$	14.7338	0.518655
$808$	0	0
$809$	27.0993	0.952762	0.476381	−	0.879239i	$-0.341948\pi$
$809$	27.0993	0.952762	0.476381	+	0.879239i	$0.341948\pi$
$810$	0	0
$811$	−27.1529	−0.953466	−0.476733	−	0.879048i	$-0.658179\pi$
$811$	−27.1529	−0.953466	−0.476733	+	0.879048i	$0.658179\pi$
$812$	0	0
$813$	30.6082	1.07348
$814$	0	0
$815$	−18.0021	−0.630586
$816$	0	0
$817$	−13.2801	−0.464613
$818$	0	0
$819$	−2.47971	−0.0866481
$820$	0	0
$821$	18.1944	0.634991	0.317495	−	0.948260i	$-0.397158\pi$
$821$	18.1944	0.634991	0.317495	+	0.948260i	$0.397158\pi$
$822$	0	0
$823$	3.51972	0.122690	0.0613449	−	0.998117i	$-0.480461\pi$
$823$	3.51972	0.122690	0.0613449	+	0.998117i	$0.480461\pi$
$824$	0	0
$825$	12.2633	0.426954
$826$	0	0
$827$	−5.96360	−0.207375	−0.103687	−	0.994610i	$-0.533064\pi$
$827$	−5.96360	−0.207375	−0.103687	+	0.994610i	$0.533064\pi$
$828$	0	0
$829$	−40.3290	−1.40068	−0.700342	−	0.713807i	$-0.746969\pi$
$829$	−40.3290	−1.40068	−0.700342	+	0.713807i	$0.746969\pi$
$830$	0	0
$831$	−21.8049	−0.756405
$832$	0	0
$833$	−1.85826	−0.0643850
$834$	0	0
$835$	−16.5670	−0.573325
$836$	0	0
$837$	−21.6695	−0.749008
$838$	0	0
$839$	16.0135	0.552848	0.276424	−	0.961036i	$-0.410851\pi$
$839$	16.0135	0.552848	0.276424	+	0.961036i	$0.410851\pi$
$840$	0	0
$841$	5.60375	0.193233
$842$	0	0
$843$	−29.0050	−0.998986
$844$	0	0
$845$	22.1831	0.763121
$846$	0	0
$847$	−31.5153	−1.08288
$848$	0	0
$849$	−37.6147	−1.29093
$850$	0	0
$851$	−10.0074	−0.343050
$852$	0	0
$853$	−15.1370	−0.518281	−0.259140	−	0.965840i	$-0.583439\pi$
$853$	−15.1370	−0.518281	−0.259140	+	0.965840i	$0.583439\pi$
$854$	0	0
$855$	4.21780	0.144246
$856$	0	0
$857$	−22.3174	−0.762347	−0.381174	−	0.924503i	$-0.624480\pi$
$857$	−22.3174	−0.762347	−0.381174	+	0.924503i	$0.624480\pi$
$858$	0	0
$859$	−21.4745	−0.732700	−0.366350	−	0.930477i	$-0.619393\pi$
$859$	−21.4745	−0.732700	−0.366350	+	0.930477i	$0.619393\pi$
$860$	0	0
$861$	4.79486	0.163408
$862$	0	0
$863$	2.87460	0.0978526	0.0489263	−	0.998802i	$-0.484420\pi$
$863$	2.87460	0.0978526	0.0489263	+	0.998802i	$0.484420\pi$
$864$	0	0
$865$	−13.2933	−0.451986
$866$	0	0
$867$	−1.96950	−0.0668879
$868$	0	0
$869$	−31.6767	−1.07456
$870$	0	0
$871$	−8.91705	−0.302143
$872$	0	0
$873$	0.564801	0.0191156
$874$	0	0
$875$	−27.4426	−0.927730
$876$	0	0
$877$	−9.47159	−0.319833	−0.159916	−	0.987131i	$-0.551122\pi$
$877$	−9.47159	−0.319833	−0.159916	+	0.987131i	$0.551122\pi$
$878$	0	0
$879$	−44.1338	−1.48859
$880$	0	0
$881$	3.89749	0.131310	0.0656548	−	0.997842i	$-0.479086\pi$
$881$	3.89749	0.131310	0.0656548	+	0.997842i	$0.479086\pi$
$882$	0	0
$883$	1.85619	0.0624657	0.0312329	−	0.999512i	$-0.490057\pi$
$883$	1.85619	0.0624657	0.0312329	+	0.999512i	$0.490057\pi$
$884$	0	0
$885$	3.81502	0.128240
$886$	0	0
$887$	3.19381	0.107238	0.0536188	−	0.998561i	$-0.482924\pi$
$887$	3.19381	0.107238	0.0536188	+	0.998561i	$0.482924\pi$
$888$	0	0
$889$	−24.4123	−0.818762
$890$	0	0
$891$	−54.2111	−1.81614
$892$	0	0
$893$	18.9870	0.635375
$894$	0	0
$895$	2.19044	0.0732182
$896$	0	0
$897$	8.57624	0.286352
$898$	0	0
$899$	30.5142	1.01771
$900$	0	0
$901$	−8.25696	−0.275079
$902$	0	0
$903$	23.9404	0.796688
$904$	0	0
$905$	−4.41741	−0.146840
$906$	0	0
$907$	6.68174	0.221864	0.110932	−	0.993828i	$-0.464616\pi$
$907$	6.68174	0.221864	0.110932	+	0.993828i	$0.464616\pi$
$908$	0	0
$909$	−15.7762	−0.523264
$910$	0	0
$911$	36.7755	1.21843	0.609213	−	0.793007i	$-0.291485\pi$
$911$	36.7755	1.21843	0.609213	+	0.793007i	$0.291485\pi$
$912$	0	0
$913$	34.2688	1.13413
$914$	0	0
$915$	2.24544	0.0742320
$916$	0	0
$917$	−47.8383	−1.57976
$918$	0	0
$919$	6.23275	0.205599	0.102800	−	0.994702i	$-0.467220\pi$
$919$	6.23275	0.205599	0.102800	+	0.994702i	$0.467220\pi$
$920$	0	0
$921$	−4.82715	−0.159060
$922$	0	0
$923$	−2.94385	−0.0968981
$924$	0	0
$925$	−3.56803	−0.117316
$926$	0	0
$927$	12.7689	0.419386
$928$	0	0
$929$	39.7024	1.30259	0.651296	−	0.758824i	$-0.274226\pi$
$929$	39.7024	1.30259	0.651296	+	0.758824i	$0.274226\pi$
$930$	0	0
$931$	4.60352	0.150874
$932$	0	0
$933$	2.67008	0.0874146
$934$	0	0
$935$	−9.66554	−0.316097
$936$	0	0
$937$	48.3394	1.57918	0.789590	−	0.613635i	$-0.210293\pi$
$937$	48.3394	1.57918	0.789590	+	0.613635i	$0.210293\pi$
$938$	0	0
$939$	−64.7380	−2.11264
$940$	0	0
$941$	−34.0305	−1.10936	−0.554681	−	0.832063i	$-0.687160\pi$
$941$	−34.0305	−1.10936	−0.554681	+	0.832063i	$0.687160\pi$
$942$	0	0
$943$	−3.75770	−0.122367
$944$	0	0
$945$	18.3486	0.596880
$946$	0	0
$947$	49.0104	1.59262	0.796312	−	0.604886i	$-0.206781\pi$
$947$	49.0104	1.59262	0.796312	+	0.604886i	$0.206781\pi$
$948$	0	0
$949$	6.69715	0.217399
$950$	0	0
$951$	−2.58304	−0.0837609
$952$	0	0
$953$	−24.9340	−0.807691	−0.403846	−	0.914827i	$-0.632327\pi$
$953$	−24.9340	−0.807691	−0.403846	+	0.914827i	$0.632327\pi$
$954$	0	0
$955$	−51.0733	−1.65269
$956$	0	0
$957$	57.8103	1.86874
$958$	0	0
$959$	22.4301	0.724307
$960$	0	0
$961$	−4.09196	−0.131999
$962$	0	0
$963$	−1.62207	−0.0522705
$964$	0	0
$965$	−47.7089	−1.53580
$966$	0	0
$967$	−1.25146	−0.0402444	−0.0201222	−	0.999798i	$-0.506406\pi$
$967$	−1.25146	−0.0402444	−0.0201222	+	0.999798i	$0.506406\pi$
$968$	0	0
$969$	4.87910	0.156739
$970$	0	0
$971$	5.62603	0.180548	0.0902740	−	0.995917i	$-0.471226\pi$
$971$	5.62603	0.180548	0.0902740	+	0.995917i	$0.471226\pi$
$972$	0	0
$973$	−53.0846	−1.70181
$974$	0	0
$975$	3.05776	0.0979268
$976$	0	0
$977$	−22.2123	−0.710633	−0.355316	−	0.934746i	$-0.615627\pi$
$977$	−22.2123	−0.710633	−0.355316	+	0.934746i	$0.615627\pi$
$978$	0	0
$979$	18.4041	0.588198
$980$	0	0
$981$	−0.394478	−0.0125947
$982$	0	0
$983$	37.9591	1.21071	0.605353	−	0.795957i	$-0.293032\pi$
$983$	37.9591	1.21071	0.605353	+	0.795957i	$0.293032\pi$
$984$	0	0
$985$	18.5029	0.589550
$986$	0	0
$987$	−34.2283	−1.08950
$988$	0	0
$989$	−18.7619	−0.596595
$990$	0	0
$991$	−12.3312	−0.391713	−0.195856	−	0.980633i	$-0.562749\pi$
$991$	−12.3312	−0.391713	−0.195856	+	0.980633i	$0.562749\pi$
$992$	0	0
$993$	−27.4887	−0.872327
$994$	0	0
$995$	13.4926	0.427743
$996$	0	0
$997$	19.1592	0.606777	0.303388	−	0.952867i	$-0.401882\pi$
$997$	19.1592	0.606777	0.303388	+	0.952867i	$0.401882\pi$
$998$	0	0
$999$	11.9446	0.377911

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.4	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.4	✓	21	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-1.96950 q^{3} -1.93705 q^{5} -2.26754 q^{7} +0.878948 q^{9} +O(q^{10})\)	\(q-1.96950 q^{3} -1.93705 q^{5} -2.26754 q^{7} +0.878948 q^{9} +4.98984 q^{11} +1.24418 q^{13} +3.81502 q^{15} +1.00000 q^{17} -2.47732 q^{19} +4.46593 q^{21} -3.49992 q^{23} -1.24786 q^{25} +4.17742 q^{27} -5.88249 q^{29} -5.18730 q^{31} -9.82751 q^{33} +4.39233 q^{35} +2.85933 q^{37} -2.45041 q^{39} +1.07365 q^{41} +5.36068 q^{43} -1.70256 q^{45} -7.66431 q^{47} -1.85826 q^{49} -1.96950 q^{51} -8.25696 q^{53} -9.66554 q^{55} +4.87910 q^{57} +1.00000 q^{59} +0.588579 q^{61} -1.99305 q^{63} -2.41003 q^{65} -7.16702 q^{67} +6.89310 q^{69} -2.36610 q^{71} +5.38279 q^{73} +2.45766 q^{75} -11.3147 q^{77} -6.34825 q^{79} -10.8643 q^{81} +6.86772 q^{83} -1.93705 q^{85} +11.5856 q^{87} +3.68832 q^{89} -2.82122 q^{91} +10.2164 q^{93} +4.79869 q^{95} +0.642587 q^{97} +4.38581 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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